LMFDB - Embedded newform 108.6.i.a.13.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(13,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.8
Character	$\chi$	$=$	108.13
Dual form			108.6.i.a.25.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.96043 + 15.4647i) q^{3} +(25.7256 + 21.5863i) q^{5} +(-179.693 - 65.4027i) q^{7} +(-235.313 - 60.6348i) q^{9} +O(q^{10})$	\(q+(-1.96043 + 15.4647i) q^{3} +(25.7256 + 21.5863i) q^{5} +(-179.693 - 65.4027i) q^{7} +(-235.313 - 60.6348i) q^{9} +(94.3418 - 79.1622i) q^{11} +(170.696 - 968.062i) q^{13} +(-384.259 + 355.520i) q^{15} +(-856.170 + 1482.93i) q^{17} +(238.223 + 412.614i) q^{19} +(1363.71 - 2650.67i) q^{21} +(3031.38 - 1103.33i) q^{23} +(-346.815 - 1966.88i) q^{25} +(1399.01 - 3520.18i) q^{27} +(-1287.38 - 7301.08i) q^{29} +(-4076.01 + 1483.55i) q^{31} +(1039.27 + 1614.16i) q^{33} +(-3210.89 - 5561.42i) q^{35} +(4796.00 - 8306.91i) q^{37} +(14636.2 + 4537.57i) q^{39} +(685.513 - 3887.74i) q^{41} +(-12640.1 + 10606.3i) q^{43} +(-4744.69 - 6639.42i) q^{45} +(-23806.3 - 8664.79i) q^{47} +(15137.0 + 12701.4i) q^{49} +(-21254.6 - 16147.6i) q^{51} +20986.9 q^{53} +4135.82 q^{55} +(-6847.97 + 2875.15i) q^{57} +(-3184.72 - 2672.30i) q^{59} +(-37644.6 - 13701.5i) q^{61} +(38318.4 + 26285.8i) q^{63} +(25288.2 - 21219.3i) q^{65} +(-1195.16 + 6778.12i) q^{67} +(11119.9 + 49042.4i) q^{69} +(-10821.9 + 18744.1i) q^{71} +(5533.76 + 9584.75i) q^{73} +(31097.1 - 1507.45i) q^{75} +(-22129.9 + 8054.64i) q^{77} +(16143.9 + 91556.6i) q^{79} +(51695.8 + 28536.4i) q^{81} +(3542.69 + 20091.6i) q^{83} +(-54036.5 + 19667.7i) q^{85} +(115433. - 5595.67i) q^{87} +(-64987.0 - 112561. i) q^{89} +(-93986.7 + 162790. i) q^{91} +(-14951.9 - 65942.6i) q^{93} +(-2778.40 + 15757.1i) q^{95} +(-27001.5 + 22656.9i) q^{97} +(-26999.9 + 12907.5i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) $ 90 * q - 87 * q^5 + 330 * q^9	$ 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) $ 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.96043	+	15.4647i	−0.125761	+	0.992061i
$3$	−1.96043	+	15.4647i	−0.125761	+	0.992061i
$4$		0			0
$5$	25.7256	+	21.5863i	0.460193	+	0.386148i	0.843202	−	0.537597i	$-0.180668\pi$
$5$	25.7256	+	21.5863i	0.460193	+	0.386148i	−0.383009	+	0.923745i	$0.625112\pi$
$6$		0			0
$7$	−179.693	−	65.4027i	−1.38607	−	0.504488i	−0.462057	−	0.886850i	$-0.652888\pi$
$7$	−179.693	−	65.4027i	−1.38607	−	0.504488i	−0.924012	+	0.382362i	$0.875111\pi$
$8$		0			0
$9$	−235.313	−	60.6348i	−0.968368	−	0.249526i
$10$		0			0
$11$	94.3418	−	79.1622i	0.235084	−	0.197259i	−0.517634	−	0.855602i	$-0.673187\pi$
$11$	94.3418	−	79.1622i	0.235084	−	0.197259i	0.752718	+	0.658344i	$0.228743\pi$
$12$		0			0
$13$	170.696	−	968.062i	0.280133	−	1.58871i	−0.442038	−	0.896996i	$-0.645744\pi$
$13$	170.696	−	968.062i	0.280133	−	1.58871i	0.722171	−	0.691715i	$-0.243145\pi$
$14$		0			0
$15$	−384.259	+	355.520i	−0.440957	+	0.407977i
$16$		0			0
$17$	−856.170	+	1482.93i	−0.718518	+	1.24451i	0.243069	+	0.970009i	$0.421846\pi$
$17$	−856.170	+	1482.93i	−0.718518	+	1.24451i	−0.961587	+	0.274501i	$0.911487\pi$
$18$		0			0
$19$	238.223	+	412.614i	0.151391	+	0.262217i	0.931739	−	0.363129i	$-0.118291\pi$
$19$	238.223	+	412.614i	0.151391	+	0.262217i	−0.780348	+	0.625345i	$0.784958\pi$
$20$		0			0
$21$	1363.71	−	2650.67i	0.674797	−	1.31162i
$22$		0			0
$23$	3031.38	−	1103.33i	1.19487	−	0.434898i	0.333440	−	0.942771i	$-0.391791\pi$
$23$	3031.38	−	1103.33i	1.19487	−	0.434898i	0.861432	+	0.507874i	$0.169568\pi$
$24$		0			0
$25$	−346.815	−	1966.88i	−0.110981	−	0.629403i
$26$		0			0
$27$	1399.01	−	3520.18i	0.369328	−	0.929299i
$28$		0			0
$29$	−1287.38	−	7301.08i	−0.284257	−	1.61210i	−0.707928	−	0.706285i	$-0.750370\pi$
$29$	−1287.38	−	7301.08i	−0.284257	−	1.61210i	0.423671	−	0.905816i	$-0.360741\pi$
$30$		0			0
$31$	−4076.01	+	1483.55i	−0.761782	+	0.277266i	−0.693555	−	0.720404i	$-0.743956\pi$
$31$	−4076.01	+	1483.55i	−0.761782	+	0.277266i	−0.0682271	+	0.997670i	$0.521734\pi$
$32$		0			0
$33$	1039.27	+	1614.16i	0.166128	+	0.258025i
$34$		0			0
$35$	−3210.89	−	5561.42i	−0.443053	−	0.767390i
$36$		0			0
$37$	4796.00	−	8306.91i	0.575937	−	0.997551i	−0.420003	−	0.907523i	$-0.637971\pi$
$37$	4796.00	−	8306.91i	0.575937	−	0.997551i	0.995939	−	0.0900284i	$-0.0286958\pi$
$38$		0			0
$39$	14636.2	+	4537.57i	1.54087	+	0.477707i
$40$		0			0
$41$	685.513	−	3887.74i	0.0636877	−	0.361191i	−0.936263	−	0.351299i	$-0.885740\pi$
$41$	685.513	−	3887.74i	0.0636877	−	0.361191i	0.999951	−	0.00989213i	$-0.00314881\pi$
$42$		0			0
$43$	−12640.1	+	10606.3i	−1.04251	+	0.874766i	−0.992286	−	0.123972i	$-0.960437\pi$
$43$	−12640.1	+	10606.3i	−1.04251	+	0.874766i	−0.0502200	+	0.998738i	$0.515992\pi$
$44$		0			0
$45$	−4744.69	−	6639.42i	−0.349282	−	0.488763i
$46$		0			0
$47$	−23806.3	−	8664.79i	−1.57198	−	0.572154i	−0.598540	−	0.801093i	$-0.704252\pi$
$47$	−23806.3	−	8664.79i	−1.57198	−	0.572154i	−0.973441	+	0.228938i	$0.926475\pi$
$48$		0			0
$49$	15137.0	+	12701.4i	0.900636	+	0.755723i
$50$		0			0
$51$	−21254.6	−	16147.6i	−1.14427	−	0.869325i
$52$		0			0
$53$	20986.9			1.02626			0.513132	−	0.858310i	$-0.328485\pi$
$53$	20986.9			1.02626			0.513132	+	0.858310i	$0.328485\pi$
$54$		0			0
$55$	4135.82			0.184355
$56$		0			0
$57$	−6847.97	+	2875.15i	−0.279174	+	0.117212i
$58$		0			0
$59$	−3184.72	−	2672.30i	−0.119108	−	0.0999435i	0.581288	−	0.813698i	$-0.302549\pi$
$59$	−3184.72	−	2672.30i	−0.119108	−	0.0999435i	−0.700396	+	0.713754i	$0.746993\pi$
$60$		0			0
$61$	−37644.6	−	13701.5i	−1.29532	−	0.471459i	−0.399853	−	0.916579i	$-0.630939\pi$
$61$	−37644.6	−	13701.5i	−1.29532	−	0.471459i	−0.895471	+	0.445120i	$0.853161\pi$
$62$		0			0
$63$	38318.4	+	26285.8i	1.21634	+	0.834390i
$64$		0			0
$65$	25288.2	−	21219.3i	0.742393	−	0.622941i
$66$		0			0
$67$	−1195.16	+	6778.12i	−0.0325268	+	0.184468i	−0.996742	−	0.0806507i	$-0.974300\pi$
$67$	−1195.16	+	6778.12i	−0.0325268	+	0.184468i	0.964216	+	0.265119i	$0.0854113\pi$
$68$		0			0
$69$	11119.9	+	49042.4i	0.281176	+	1.24008i
$70$		0			0
$71$	−10821.9	+	18744.1i	−0.254775	+	0.441284i	−0.964834	−	0.262858i	$-0.915335\pi$
$71$	−10821.9	+	18744.1i	−0.254775	+	0.441284i	0.710059	+	0.704142i	$0.248668\pi$
$72$		0			0
$73$	5533.76	+	9584.75i	0.121538	+	0.210511i	0.920374	−	0.391038i	$-0.127884\pi$
$73$	5533.76	+	9584.75i	0.121538	+	0.210511i	−0.798836	+	0.601549i	$0.794551\pi$
$74$		0			0
$75$	31097.1	−	1507.45i	0.638362	−	0.0309450i
$76$		0			0
$77$	−22129.9	+	8054.64i	−0.425357	+	0.154817i
$78$		0			0
$79$	16143.9	+	91556.6i	0.291032	+	1.65052i	0.682907	+	0.730506i	$0.260716\pi$
$79$	16143.9	+	91556.6i	0.291032	+	1.65052i	−0.391875	+	0.920019i	$0.628173\pi$
$80$		0			0
$81$	51695.8	+	28536.4i	0.875474	+	0.483266i
$82$		0			0
$83$	3542.69	+	20091.6i	0.0564466	+	0.320125i	0.999936	−	0.0112769i	$-0.00358963\pi$
$83$	3542.69	+	20091.6i	0.0564466	+	0.320125i	−0.943490	+	0.331402i	$0.892479\pi$
$84$		0			0
$85$	−54036.5	+	19667.7i	−0.811222	+	0.295261i
$86$		0			0
$87$	115433.	−	5595.67i	1.63505	−	0.0792599i
$88$		0			0
$89$	−64987.0	−	112561.i	−0.869664	−	1.50630i	−0.862340	−	0.506330i	$-0.831002\pi$
$89$	−64987.0	−	112561.i	−0.869664	−	1.50630i	−0.00732457	−	0.999973i	$-0.502332\pi$
$90$		0			0
$91$	−93986.7	+	162790.i	−1.18977	+	2.06074i
$92$		0			0
$93$	−14951.9	−	65942.6i	−0.179262	−	0.790603i
$94$		0			0
$95$	−2778.40	+	15757.1i	−0.0315854	+	0.179130i
$96$		0			0
$97$	−27001.5	+	22656.9i	−0.291379	+	0.244496i	−0.776745	−	0.629815i	$-0.783131\pi$
$97$	−27001.5	+	22656.9i	−0.291379	+	0.244496i	0.485366	+	0.874311i	$0.338686\pi$
$98$		0			0
$99$	−26999.9	+	12907.5i	−0.276869	+	0.132359i
$100$		0			0
$101$	−110029.	−	40047.3i	−1.07326	−	0.390634i	−0.255863	−	0.966713i	$-0.582360\pi$
$101$	−110029.	−	40047.3i	−1.07326	−	0.390634i	−0.817394	+	0.576079i	$0.804582\pi$
$102$		0			0
$103$	−59987.1	−	50335.1i	−0.557140	−	0.467496i	0.320210	−	0.947347i	$-0.396247\pi$
$103$	−59987.1	−	50335.1i	−0.557140	−	0.467496i	−0.877350	+	0.479850i	$0.840691\pi$
$104$		0			0
$105$	92300.4	−	38752.7i	0.817016	−	0.343027i
$106$		0			0
$107$	−50234.6			−0.424173			−0.212087	−	0.977251i	$-0.568026\pi$
$107$	−50234.6			−0.424173			−0.212087	+	0.977251i	$0.568026\pi$
$108$		0			0
$109$	−30857.6			−0.248769			−0.124385	−	0.992234i	$-0.539696\pi$
$109$	−30857.6			−0.248769			−0.124385	+	0.992234i	$0.539696\pi$
$110$		0			0
$111$	119062.	+	90453.7i	0.917201	+	0.696817i
$112$		0			0
$113$	−60306.9	−	50603.5i	−0.444295	−	0.372807i	0.393019	−	0.919530i	$-0.371431\pi$
$113$	−60306.9	−	50603.5i	−0.444295	−	0.372807i	−0.837314	+	0.546723i	$0.815875\pi$
$114$		0			0
$115$	101801.	+	37052.5i	0.717806	+	0.261260i
$116$		0			0
$117$	−98865.2	+	217448.i	−0.667696	+	1.46856i
$118$		0			0
$119$	250835.	−	210476.i	1.62376	−	1.36249i
$120$		0			0
$121$	−25332.5	+	143668.i	−0.157295	+	0.892063i
$122$		0			0
$123$	58778.7	+	18222.9i	0.350314	+	0.108606i
$124$		0			0
$125$	86008.2	−	148971.i	0.492340	−	0.852757i
$126$		0			0
$127$	−98880.0	−	171265.i	−0.544001	−	0.942237i	−0.998669	−	0.0515759i	$-0.983576\pi$
$127$	−98880.0	−	171265.i	−0.544001	−	0.942237i	0.454669	−	0.890661i	$-0.349758\pi$
$128$		0			0
$129$	−139243.	−	216268.i	−0.736714	−	1.14424i
$130$		0			0
$131$	−25936.4	+	9440.06i	−0.132048	+	0.0480614i	−0.407199	−	0.913340i	$-0.633494\pi$
$131$	−25936.4	+	9440.06i	−0.132048	+	0.0480614i	0.275151	+	0.961401i	$0.411272\pi$
$132$		0			0
$133$	−15820.8	−	89724.2i	−0.0775531	−	0.439825i
$134$		0			0
$135$	111978.	−	60359.1i	0.528809	−	0.285042i
$136$		0			0
$137$	23978.1	+	135987.i	0.109147	+	0.619005i	0.989482	+	0.144653i	$0.0462066\pi$
$137$	23978.1	+	135987.i	0.109147	+	0.619005i	−0.880335	+	0.474352i	$0.842682\pi$
$138$		0			0
$139$	205658.	−	74853.3i	0.902835	−	0.328605i	0.151447	−	0.988465i	$-0.451607\pi$
$139$	205658.	−	74853.3i	0.902835	−	0.328605i	0.751388	+	0.659860i	$0.229385\pi$
$140$		0			0
$141$	180669.	−	351171.i	0.765306	−	1.48755i
$142$		0			0
$143$	−60530.2	−	104841.i	−0.247532	−	0.428739i
$144$		0			0
$145$	124485.	−	215614.i	0.491696	−	0.851643i
$146$		0			0
$147$	−226099.	+	209189.i	−0.862988	+	0.798444i
$148$		0			0
$149$	53839.0	−	305336.i	0.198669	−	1.12671i	−0.708426	−	0.705785i	$-0.750594\pi$
$149$	53839.0	−	305336.i	0.198669	−	1.12671i	0.907095	−	0.420925i	$-0.138294\pi$
$150$		0			0
$151$	−263535.	+	221132.i	−0.940581	+	0.789241i	−0.977686	−	0.210070i	$-0.932631\pi$
$151$	−263535.	+	221132.i	−0.940581	+	0.789241i	0.0371053	+	0.999311i	$0.488186\pi$
$152$		0			0
$153$	291386.	−	297040.i	1.00633	−	1.02585i
$154$		0			0
$155$	−136882.	−	49820.9i	−0.457633	−	0.166565i
$156$		0			0
$157$	255489.	+	214381.i	0.827223	+	0.694123i	0.954652	−	0.297725i	$-0.0962279\pi$
$157$	255489.	+	214381.i	0.827223	+	0.694123i	−0.127428	+	0.991848i	$0.540672\pi$
$158$		0			0
$159$	−41143.3	+	324556.i	−0.129064	+	1.01812i
$160$		0			0
$161$	−616878.			−1.87558
$162$		0			0
$163$	6833.03			0.0201439			0.0100720	−	0.999949i	$-0.496794\pi$
$163$	6833.03			0.0201439			0.0100720	+	0.999949i	$0.496794\pi$
$164$		0			0
$165$	−8107.97	+	63959.1i	−0.0231847	+	0.182891i
$166$		0			0
$167$	225856.	+	189515.i	0.626671	+	0.525840i	0.899893	−	0.436112i	$-0.143645\pi$
$167$	225856.	+	189515.i	0.626671	+	0.525840i	−0.273221	+	0.961951i	$0.588089\pi$
$168$		0			0
$169$	−559107.	−	203498.i	−1.50584	−	0.548080i
$170$		0			0
$171$	−31038.3	−	111538.i	−0.0811722	−	0.291698i
$172$		0			0
$173$	196232.	−	164658.i	0.498488	−	0.418281i	−0.358569	−	0.933503i	$-0.616735\pi$
$173$	196232.	−	164658.i	0.498488	−	0.418281i	0.857057	+	0.515222i	$0.172291\pi$
$174$		0			0
$175$	−66319.5	+	376117.i	−0.163699	+	0.928384i
$176$		0			0
$177$	47569.6	−	44011.9i	0.114129	−	0.105593i
$178$		0			0
$179$	−141184.	+	244538.i	−0.329347	+	0.570445i	−0.982382	−	0.186882i	$-0.940162\pi$
$179$	−141184.	+	244538.i	−0.329347	+	0.570445i	0.653036	+	0.757327i	$0.273495\pi$
$180$		0			0
$181$	−256179.	−	443715.i	−0.581229	−	1.00672i	−0.995334	−	0.0964885i	$-0.969239\pi$
$181$	−256179.	−	443715.i	−0.581229	−	1.00672i	0.414106	−	0.910229i	$-0.364094\pi$
$182$		0			0
$183$	285689.	−	555301.i	0.630618	−	1.22575i
$184$		0			0
$185$	302695.	−	110172.i	0.650244	−	0.236670i
$186$		0			0
$187$	36619.3	+	207679.i	0.0765784	+	0.434298i
$188$		0			0
$189$	−481622.	+	541051.i	−0.980735	+	1.10175i
$190$		0			0
$191$	−83373.2	−	472833.i	−0.165365	−	0.937830i	−0.948688	−	0.316215i	$-0.897588\pi$
$191$	−83373.2	−	472833.i	−0.165365	−	0.937830i	0.783323	−	0.621615i	$-0.213523\pi$
$192$		0			0
$193$	149424.	−	54386.1i	0.288754	−	0.105098i	−0.193582	−	0.981084i	$-0.562011\pi$
$193$	149424.	−	54386.1i	0.288754	−	0.105098i	0.482336	+	0.875986i	$0.339788\pi$
$194$		0			0
$195$	278574.	+	432672.i	0.524631	+	0.814840i
$196$		0			0
$197$	515806.	+	893403.i	0.946937	+	1.64014i	0.751825	+	0.659362i	$0.229174\pi$
$197$	515806.	+	893403.i	0.946937	+	1.64014i	0.195112	+	0.980781i	$0.437493\pi$
$198$		0			0
$199$	176895.	−	306391.i	0.316652	−	0.548457i	−0.663135	−	0.748499i	$-0.730775\pi$
$199$	176895.	−	306391.i	0.316652	−	0.548457i	0.979787	+	0.200042i	$0.0641080\pi$
$200$		0			0
$201$	−102478.	−	31770.9i	−0.178913	−	0.0554675i
$202$		0			0
$203$	−246179.	+	1.39615e6i	−0.419286	+	2.37789i
$204$		0			0
$205$	101557.	−	85216.5i	0.168782	−	0.141625i
$206$		0			0
$207$	−780226.	+	75821.9i	−1.26559	+	0.122990i
$208$		0			0
$209$	55137.8	+	20068.5i	0.0873140	+	0.0317797i
$210$		0			0
$211$	691154.	+	579947.i	1.06873	+	0.896772i	0.994937	−	0.100501i	$-0.0320446\pi$
$211$	691154.	+	579947.i	1.06873	+	0.896772i	0.0737946	+	0.997273i	$0.476489\pi$
$212$		0			0
$213$	−268656.	−	204104.i	−0.405739	−	0.308249i
$214$		0			0
$215$	−554124.			−0.817543
$216$		0			0
$217$	829456.			1.19576
$218$		0			0
$219$	−159074.	+	66787.7i	−0.224124	+	0.0940993i
$220$		0			0
$221$	1.28942e6	+	1.08196e6i	1.77589	+	1.49015i
$222$		0			0
$223$	399482.	+	145400.i	0.537942	+	0.195795i	0.596681	−	0.802479i	$-0.296486\pi$
$223$	399482.	+	145400.i	0.537942	+	0.195795i	−0.0587392	+	0.998273i	$0.518708\pi$
$224$		0			0
$225$	−37651.4	+	483863.i	−0.0495821	+	0.637186i
$226$		0			0
$227$	435368.	−	365317.i	0.560779	−	0.470549i	−0.317793	−	0.948160i	$-0.602942\pi$
$227$	435368.	−	365317.i	0.560779	−	0.470549i	0.878571	+	0.477611i	$0.158497\pi$
$228$		0			0
$229$	230637.	−	1.30801e6i	0.290630	−	1.64824i	−0.393824	−	0.919186i	$-0.628848\pi$
$229$	230637.	−	1.30801e6i	0.290630	−	1.64824i	0.684453	−	0.729057i	$-0.260041\pi$
$230$		0			0
$231$	−81178.4	−	358023.i	−0.100095	−	0.441450i
$232$		0			0
$233$	−294463.	+	510025.i	−0.355338	+	0.615463i	−0.987176	−	0.159637i	$-0.948967\pi$
$233$	−294463.	+	510025.i	−0.355338	+	0.615463i	0.631838	+	0.775101i	$0.282301\pi$
$234$		0			0
$235$	−425390.	−	736797.i	−0.502479	−	0.870319i
$236$		0			0
$237$	−1.44754e6	+	70170.5i	−1.67402	+	0.0811491i
$238$		0			0
$239$	−256153.	+	93231.9i	−0.290071	+	0.105577i	−0.482957	−	0.875644i	$-0.660437\pi$
$239$	−256153.	+	93231.9i	−0.290071	+	0.105577i	0.192886	+	0.981221i	$0.438215\pi$
$240$		0			0
$241$	10058.2	+	57042.6i	0.0111552	+	0.0632640i	0.989877	−	0.141927i	$-0.0453297\pi$
$241$	10058.2	+	57042.6i	0.0111552	+	0.0632640i	−0.978722	+	0.205191i	$0.934219\pi$
$242$		0			0
$243$	−542652.	+	743517.i	−0.589530	+	0.807747i
$244$		0			0
$245$	115230.	+	653504.i	0.122645	+	0.695557i
$246$		0			0
$247$	440100.	−	160183.i	0.458996	−	0.167061i
$248$		0			0
$249$	−317655.	+	15398.5i	−0.324682	+	0.0157391i
$250$		0			0
$251$	−954571.	−	1.65337e6i	−0.956365	−	1.65647i	−0.731212	−	0.682150i	$-0.761045\pi$
$251$	−954571.	−	1.65337e6i	−0.956365	−	1.65647i	−0.225153	−	0.974323i	$-0.572288\pi$
$252$		0			0
$253$	198644.	−	344061.i	0.195107	−	0.337936i
$254$		0			0
$255$	−198220.	−	874215.i	−0.190896	−	0.841914i
$256$		0			0
$257$	−93924.6	+	532673.i	−0.0887047	+	0.503069i	0.907791	+	0.419423i	$0.137768\pi$
$257$	−93924.6	+	532673.i	−0.0887047	+	0.503069i	−0.996496	+	0.0836461i	$0.973343\pi$
$258$		0			0
$259$	−1.40510e6	+	1.17902e6i	−1.30154	+	1.09212i
$260$		0			0
$261$	−139762.	+	1.79610e6i	−0.126996	+	1.63204i
$262$		0			0
$263$	1.54111e6	+	560919.i	1.37387	+	0.500047i	0.920314	−	0.391182i	$-0.127934\pi$
$263$	1.54111e6	+	560919.i	1.37387	+	0.500047i	0.453554	+	0.891229i	$0.350156\pi$
$264$		0			0
$265$	539901.	+	453031.i	0.472280	+	0.396290i
$266$		0			0
$267$	1.86812e6	−	784337.i	1.60371	−	0.673325i
$268$		0			0
$269$	999215.			0.841934			0.420967	−	0.907076i	$-0.361691\pi$
$269$	999215.			0.841934			0.420967	+	0.907076i	$0.361691\pi$
$270$		0			0
$271$	650391.			0.537962			0.268981	−	0.963146i	$-0.413313\pi$
$271$	650391.			0.537962			0.268981	+	0.963146i	$0.413313\pi$
$272$		0			0
$273$	−2.33324e6	−	1.77261e6i	−1.89475	−	1.43948i
$274$		0			0
$275$	−188422.	−	158105.i	−0.150245	−	0.126070i
$276$		0			0
$277$	−649887.	−	236539.i	−0.508907	−	0.185227i	0.0747889	−	0.997199i	$-0.476172\pi$
$277$	−649887.	−	236539.i	−0.508907	−	0.185227i	−0.583696	+	0.811972i	$0.698394\pi$
$278$		0			0
$279$	1.04909e6	−	101950.i	0.806870	−	0.0784112i
$280$		0			0
$281$	−186818.	+	156759.i	−0.141141	+	0.118431i	−0.710625	−	0.703571i	$-0.751588\pi$
$281$	−186818.	+	156759.i	−0.141141	+	0.118431i	0.569484	+	0.822003i	$0.307143\pi$
$282$		0			0
$283$	60362.0	−	342330.i	0.0448020	−	0.254085i	−0.954178	−	0.299240i	$-0.903267\pi$
$283$	60362.0	−	342330.i	0.0448020	−	0.254085i	0.998980	+	0.0451548i	$0.0143781\pi$
$284$		0			0
$285$	−238232.	−	73857.8i	−0.173735	−	0.0538622i
$286$		0			0
$287$	−377450.	+	653763.i	−0.270492	+	0.468506i
$288$		0			0
$289$	−756126.	−	1.30965e6i	−0.532537	−	0.922381i
$290$		0			0
$291$	−297448.	−	461987.i	−0.205911	−	0.319814i
$292$		0			0
$293$	−2.35025e6	+	855421.i	−1.59936	+	0.582118i	−0.979295	−	0.202436i	$-0.935114\pi$
$293$	−2.35025e6	+	855421.i	−1.59936	+	0.582118i	−0.620060	+	0.784554i	$0.712892\pi$
$294$		0			0
$295$	−24243.7	−	137493.i	−0.0162197	−	0.0919866i
$296$		0			0
$297$	−146680.	−	442849.i	−0.0964892	−	0.291316i
$298$		0			0
$299$	−550652.	−	3.12290e6i	−0.356204	−	2.02014i
$300$		0			0
$301$	2.96501e6	−	1.07917e6i	1.88629	−	0.686555i
$302$		0			0
$303$	835023.	−	1.62305e6i	0.522506	−	1.01561i
$304$		0			0
$305$	−672664.	−	1.16509e6i	−0.414046	−	0.717149i
$306$		0			0
$307$	979428.	−	1.69642e6i	0.593098	−	1.02728i	−0.400714	−	0.916203i	$-0.631238\pi$
$307$	979428.	−	1.69642e6i	0.593098	−	1.02728i	0.993812	−	0.111073i	$-0.0354288\pi$
$308$		0			0
$309$	896018.	−	829003.i	0.533851	−	0.493924i
$310$		0			0
$311$	−77541.3	+	439759.i	−0.0454603	+	0.257818i	−0.999064	−	0.0432451i	$-0.986230\pi$
$311$	−77541.3	+	439759.i	−0.0454603	+	0.257818i	0.953604	+	0.301063i	$0.0973415\pi$
$312$		0			0
$313$	763237.	−	640432.i	0.440350	−	0.369498i	−0.395490	−	0.918470i	$-0.629425\pi$
$313$	763237.	−	640432.i	0.440350	−	0.369498i	0.835840	+	0.548972i	$0.184981\pi$
$314$		0			0
$315$	418350.	+	1.50337e6i	0.237554	+	0.853669i
$316$		0			0
$317$	3.18954e6	+	1.16090e6i	1.78271	+	0.648853i	0.999638	+	0.0269193i	$0.00856970\pi$
$317$	3.18954e6	+	1.16090e6i	1.78271	+	0.648853i	0.783070	+	0.621933i	$0.213653\pi$
$318$		0			0
$319$	−699423.	−	586886.i	−0.384825	−	0.322906i
$320$		0			0
$321$	98481.2	−	776862.i	0.0533446	−	0.420806i
$322$		0			0
$323$	−815838.			−0.435108
$324$		0			0
$325$	−1.96327e6			−1.03103
$326$		0			0
$327$	60494.1	−	477204.i	0.0312856	−	0.246794i
$328$		0			0
$329$	3.71111e6	+	3.11399e6i	1.89023	+	1.58609i
$330$		0			0
$331$	−1.25970e6	−	458494.i	−0.631972	−	0.230019i	0.00611757	−	0.999981i	$-0.498053\pi$
$331$	−1.25970e6	−	458494.i	−0.631972	−	0.230019i	−0.638089	+	0.769962i	$0.720275\pi$
$332$		0			0
$333$	−1.63225e6	+	1.66392e6i	−0.806633	+	0.822286i
$334$		0			0
$335$	−177061.	+	148572.i	−0.0862007	+	0.0723310i
$336$		0			0
$337$	−429819.	+	2.43763e6i	−0.206163	+	1.16921i	0.689436	+	0.724347i	$0.257859\pi$
$337$	−429819.	+	2.43763e6i	−0.206163	+	1.16921i	−0.895599	+	0.444863i	$0.853252\pi$
$338$		0			0
$339$	900795.	−	833423.i	0.425723	−	0.393882i
$340$		0			0
$341$	−267097.	+	462626.i	−0.124389	+	0.215449i
$342$		0			0
$343$	−282337.	−	489021.i	−0.129578	−	0.224436i
$344$		0			0
$345$	−772579.	+	1.50168e6i	−0.349458	+	0.679251i
$346$		0			0
$347$	2.62118e6	−	954033.i	1.16862	−	0.425343i	0.316451	−	0.948609i	$-0.397509\pi$
$347$	2.62118e6	−	954033.i	1.16862	−	0.425343i	0.852170	+	0.523265i	$0.175286\pi$
$348$		0			0
$349$	218683.	+	1.24021e6i	0.0961061	+	0.545045i	0.994403	+	0.105655i	$0.0336940\pi$
$349$	218683.	+	1.24021e6i	0.0961061	+	0.545045i	−0.898297	+	0.439389i	$0.855195\pi$
$350$		0			0
$351$	−3.16895e6	−	1.95521e6i	−1.37293	−	0.847083i
$352$		0			0
$353$	−635622.	−	3.60479e6i	−0.271495	−	1.53973i	−0.749879	−	0.661575i	$-0.769888\pi$
$353$	−635622.	−	3.60479e6i	−0.271495	−	1.53973i	0.478383	−	0.878151i	$-0.341223\pi$
$354$		0			0
$355$	−683015.	+	248597.i	−0.287647	+	0.104695i
$356$		0			0
$357$	2.76320e6	+	4.29171e6i	1.14747	+	1.78221i
$358$		0			0
$359$	−205046.	−	355149.i	−0.0839681	−	0.145437i	0.820983	−	0.570953i	$-0.193426\pi$
$359$	−205046.	−	355149.i	−0.0839681	−	0.145437i	−0.904951	+	0.425516i	$0.860093\pi$
$360$		0			0
$361$	1.12455e6	−	1.94778e6i	0.454162	−	0.786631i
$362$		0			0
$363$	−2.17211e6	−	673409.i	−0.865199	−	0.268233i
$364$		0			0
$365$	−64540.4	+	366027.i	−0.0253571	+	0.143807i
$366$		0			0
$367$	−1.97829e6	+	1.65998e6i	−0.766699	+	0.643337i	−0.939861	−	0.341556i	$-0.889046\pi$
$367$	−1.97829e6	+	1.65998e6i	−0.766699	+	0.643337i	0.173162	+	0.984893i	$0.444602\pi$
$368$		0			0
$369$	−397042.	+	873271.i	−0.151800	+	0.333874i
$370$		0			0
$371$	−3.77119e6	−	1.37260e6i	−1.42247	−	0.517738i
$372$		0			0
$373$	2.58622e6	+	2.17010e6i	0.962484	+	0.807620i	0.981355	−	0.192202i	$-0.0615629\pi$
$373$	2.58622e6	+	2.17010e6i	0.962484	+	0.807620i	−0.0188714	+	0.999822i	$0.506007\pi$
$374$		0			0
$375$	2.13517e6	+	1.62214e6i	0.784070	+	0.595675i
$376$		0			0
$377$	−7.28765e6			−2.64079
$378$		0			0
$379$	−1.90592e6			−0.681565			−0.340783	−	0.940142i	$-0.610692\pi$
$379$	−1.90592e6			−0.681565			−0.340783	+	0.940142i	$0.610692\pi$
$380$		0			0
$381$	2.84241e6	−	1.19340e6i	1.00317	−	0.421184i
$382$		0			0
$383$	−895522.	−	751432.i	−0.311946	−	0.261754i	0.473350	−	0.880875i	$-0.343045\pi$
$383$	−895522.	−	751432.i	−0.311946	−	0.261754i	−0.785296	+	0.619121i	$0.787489\pi$
$384$		0			0
$385$	−743176.	−	270494.i	−0.255529	−	0.0930048i
$386$		0			0
$387$	3.61749e6	−	1.72937e6i	1.22781	−	0.586964i
$388$		0			0
$389$	2.30198e6	−	1.93159e6i	0.771307	−	0.647203i	−0.169737	−	0.985489i	$-0.554292\pi$
$389$	2.30198e6	−	1.93159e6i	0.771307	−	0.647203i	0.941043	+	0.338286i	$0.109847\pi$
$390$		0			0
$391$	−959213.	+	5.43997e6i	−0.317302	+	1.79951i
$392$		0			0
$393$	−95141.4	−	419604.i	−0.0310733	−	0.137044i
$394$		0			0
$395$	−1.56106e6	+	2.70383e6i	−0.503416	+	0.871941i
$396$		0			0
$397$	691371.	+	1.19749e6i	0.220158	+	0.381325i	0.954856	−	0.297070i	$-0.0960093\pi$
$397$	691371.	+	1.19749e6i	0.220158	+	0.381325i	−0.734698	+	0.678395i	$0.762676\pi$
$398$		0			0
$399$	1.41857e6	−	68766.0i	0.446087	−	0.0216243i
$400$		0			0
$401$	−42183.8	+	15353.6i	−0.0131004	+	0.00476816i	−0.348562	−	0.937286i	$-0.613330\pi$
$401$	−42183.8	+	15353.6i	−0.0131004	+	0.00476816i	0.335462	+	0.942054i	$0.391108\pi$
$402$		0			0
$403$	740408.	+	4.19906e6i	0.227096	+	1.28792i
$404$		0			0
$405$	713910.	+	1.85004e6i	0.216275	+	0.560458i
$406$		0			0
$407$	−205130.	−	1.16335e6i	−0.0613823	−	0.348116i
$408$		0			0
$409$	−860519.	+	313203.i	−0.254362	+	0.0925801i	−0.466054	−	0.884756i	$-0.654325\pi$
$409$	−860519.	+	313203.i	−0.254362	+	0.0925801i	0.211692	+	0.977336i	$0.432103\pi$
$410$		0			0
$411$	−2.15000e6	+	104222.i	−0.627817	+	0.0304338i
$412$		0			0
$413$	397495.	+	688481.i	0.114672	+	0.198617i
$414$		0			0
$415$	−342566.	+	593341.i	−0.0976391	+	0.169116i
$416$		0			0
$417$	754407.	+	3.32718e6i	0.212454	+	0.936993i
$418$		0			0
$419$	1.11791e6	−	6.34000e6i	0.311081	−	1.76423i	−0.282322	−	0.959320i	$-0.591105\pi$
$419$	1.11791e6	−	6.34000e6i	0.311081	−	1.76423i	0.593403	−	0.804906i	$-0.297784\pi$
$420$		0			0
$421$	−4.55878e6	+	3.82527e6i	−1.25356	+	1.05186i	−0.257218	+	0.966353i	$0.582806\pi$
$421$	−4.55878e6	+	3.82527e6i	−1.25356	+	1.05186i	−0.996338	+	0.0855051i	$0.972750\pi$
$422$		0			0
$423$	5.07656e6	+	3.48243e6i	1.37949	+	0.946306i
$424$		0			0
$425$	3.21368e6	+	1.16968e6i	0.863039	+	0.314121i
$426$		0			0
$427$	5.86834e6	+	4.92412e6i	1.55756	+	1.30695i
$428$		0			0
$429$	1.74000e6	−	730547.i	0.456465	−	0.191648i
$430$		0			0
$431$	777821.			0.201691			0.100845	−	0.994902i	$-0.467845\pi$
$431$	777821.			0.201691			0.100845	+	0.994902i	$0.467845\pi$
$432$		0			0
$433$	−436237.			−0.111816			−0.0559078	−	0.998436i	$-0.517805\pi$
$433$	−436237.			−0.111816			−0.0559078	+	0.998436i	$0.517805\pi$
$434$		0			0
$435$	3.09037e6	+	2.34782e6i	0.783045	+	0.594896i
$436$		0			0
$437$	1.17740e6	+	987953.i	0.294930	+	0.247476i
$438$		0			0
$439$	−6.85060e6	−	2.49341e6i	−1.69655	−	0.617494i	−0.701126	−	0.713037i	$-0.747319\pi$
$439$	−6.85060e6	−	2.49341e6i	−1.69655	−	0.617494i	−0.995425	+	0.0955431i	$0.969541\pi$
$440$		0			0
$441$	−2.79179e6	−	3.90665e6i	−0.683574	−	0.956550i
$442$		0			0
$443$	541304.	−	454208.i	0.131048	−	0.109963i	−0.574908	−	0.818218i	$-0.694962\pi$
$443$	541304.	−	454208.i	0.131048	−	0.109963i	0.705956	+	0.708256i	$0.250518\pi$
$444$		0			0
$445$	757946.	−	4.29852e6i	0.181442	−	1.02901i
$446$		0			0
$447$	4.61638e6	+	1.43119e6i	1.09278	+	0.338789i
$448$		0			0
$449$	3.35899e6	−	5.81793e6i	0.786307	−	1.36192i	−0.141908	−	0.989880i	$-0.545324\pi$
$449$	3.35899e6	−	5.81793e6i	0.786307	−	1.36192i	0.928215	−	0.372044i	$-0.121343\pi$
$450$		0			0
$451$	−243089.	−	421043.i	−0.0562761	−	0.0974731i
$452$		0			0
$453$	−2.90310e6	−	4.50900e6i	−0.664686	−	1.03237i
$454$		0			0
$455$	−5.93189e6	+	2.15903e6i	−1.34327	+	0.488912i
$456$		0			0
$457$	−436460.	−	2.47529e6i	−0.0977584	−	0.554416i	−0.993867	−	0.110581i	$-0.964729\pi$
$457$	−436460.	−	2.47529e6i	−0.0977584	−	0.554416i	0.896109	−	0.443835i	$-0.146382\pi$
$458$		0			0
$459$	4.02239e6	+	5.08851e6i	0.891153	+	1.12735i
$460$		0			0
$461$	926113.	+	5.25225e6i	0.202960	+	1.15105i	0.900617	+	0.434614i	$0.143115\pi$
$461$	926113.	+	5.25225e6i	0.202960	+	1.15105i	−0.697657	+	0.716432i	$0.745774\pi$
$462$		0			0
$463$	−1.87043e6	+	680781.i	−0.405498	+	0.147589i	−0.536713	−	0.843765i	$-0.680334\pi$
$463$	−1.87043e6	+	680781.i	−0.405498	+	0.147589i	0.131215	+	0.991354i	$0.458112\pi$
$464$		0			0
$465$	1.03881e6	−	2.01917e6i	0.222795	−	0.433052i
$466$		0			0
$467$	−1.90162e6	−	3.29371e6i	−0.403490	−	0.698865i	0.590655	−	0.806925i	$-0.298870\pi$
$467$	−1.90162e6	−	3.29371e6i	−0.403490	−	0.698865i	−0.994144	+	0.108060i	$0.965536\pi$
$468$		0			0
$469$	658070.	−	1.13981e6i	0.138146	−	0.239277i
$470$		0			0
$471$	−3.81620e6	+	3.53078e6i	−0.792644	+	0.733362i
$472$		0			0
$473$	−352871.	+	2.00123e6i	−0.0725209	+	0.411286i
$474$		0			0
$475$	728945.	−	611657.i	0.148238	−	0.124387i
$476$		0			0
$477$	−4.93851e6	−	1.27254e6i	−0.993801	−	0.256079i
$478$		0			0
$479$	−6.32901e6	−	2.30357e6i	−1.26037	−	0.458736i	−0.376475	−	0.926427i	$-0.622864\pi$
$479$	−6.32901e6	−	2.30357e6i	−1.26037	−	0.458736i	−0.883892	+	0.467691i	$0.845086\pi$
$480$		0			0
$481$	−7.22295e6	−	6.06078e6i	−1.42348	−	1.19444i
$482$		0			0
$483$	1.20934e6	−	9.53983e6i	0.235875	−	1.86068i
$484$		0			0
$485$	−1.18371e6			−0.228502
$486$		0			0
$487$	−1.00236e6			−0.191515			−0.0957575	−	0.995405i	$-0.530527\pi$
$487$	−1.00236e6			−0.191515			−0.0957575	+	0.995405i	$0.530527\pi$
$488$		0			0
$489$	−13395.6	+	105671.i	−0.00253333	+	0.0199840i
$490$		0			0
$491$	2.89217e6	+	2.42682e6i	0.541402	+	0.454290i	0.872017	−	0.489475i	$-0.162812\pi$
$491$	2.89217e6	+	2.42682e6i	0.541402	+	0.454290i	−0.330615	+	0.943766i	$0.607256\pi$
$492$		0			0
$493$	1.19292e7	+	4.34188e6i	2.21052	+	0.804563i
$494$		0			0
$495$	−973214.	−	250774.i	−0.178523	−	0.0460013i
$496$		0			0
$497$	3.17053e6	−	2.66039e6i	0.575759	−	0.483119i
$498$		0			0
$499$	−377863.	+	2.14297e6i	−0.0679334	+	0.385270i	0.931817	+	0.362928i	$0.118223\pi$
$499$	−377863.	+	2.14297e6i	−0.0679334	+	0.385270i	−0.999750	+	0.0223410i	$0.992888\pi$
$500$		0			0
$501$	−3.37357e6	+	3.12126e6i	−0.600476	+	0.555566i
$502$		0			0
$503$	2.24555e6	−	3.88942e6i	0.395734	−	0.685432i	−0.597460	−	0.801898i	$-0.703823\pi$
$503$	2.24555e6	−	3.88942e6i	0.395734	−	0.685432i	0.993195	+	0.116467i	$0.0371568\pi$
$504$		0			0
$505$	−1.96609e6	−	3.40536e6i	−0.343063	−	0.594203i
$506$		0			0
$507$	4.24312e6	−	8.24747e6i	0.733105	−	1.42495i
$508$		0			0
$509$	−4.46674e6	+	1.62576e6i	−0.764181	+	0.278139i	−0.694561	−	0.719434i	$-0.744401\pi$
$509$	−4.46674e6	+	1.62576e6i	−0.764181	+	0.278139i	−0.0696208	+	0.997574i	$0.522179\pi$
$510$		0			0
$511$	−367506.	−	2.08423e6i	−0.0622605	−	0.353097i
$512$		0			0
$513$	1.78575e6	−	261335.i	0.299591	−	0.0438434i
$514$		0			0
$515$	−456652.	−	2.58980e6i	−0.0758695	−	0.430277i
$516$		0			0
$517$	−2.93185e6	+	1.06711e6i	−0.482409	+	0.175583i
$518$		0			0
$519$	2.16169e6	+	3.35747e6i	0.352270	+	0.547134i
$520$		0			0
$521$	2.45845e6	+	4.25815e6i	0.396795	+	0.687269i	0.993329	−	0.115319i	$-0.0367890\pi$
$521$	2.45845e6	+	4.25815e6i	0.396795	+	0.687269i	−0.596533	+	0.802588i	$0.703456\pi$
$522$		0			0
$523$	−3.28326e6	+	5.68677e6i	−0.524869	+	0.909099i	0.474712	+	0.880141i	$0.342552\pi$
$523$	−3.28326e6	+	5.68677e6i	−0.524869	+	0.909099i	−0.999581	+	0.0289581i	$0.990781\pi$
$524$		0			0
$525$	−5.68652e6	−	1.76296e6i	−0.900426	−	0.279154i
$526$		0			0
$527$	1.28976e6	−	7.31460e6i	0.202294	−	1.14727i
$528$		0			0
$529$	3.04141e6	−	2.55205e6i	0.472537	−	0.396506i
$530$		0			0
$531$	587373.	+	821932.i	0.0904019	+	0.126503i
$532$		0			0
$533$	−3.64656e6	−	1.32724e6i	−0.555987	−	0.202363i
$534$		0			0
$535$	−1.29231e6	−	1.08438e6i	−0.195202	−	0.163794i
$536$		0			0
$537$	−3.50493e6	−	2.66277e6i	−0.524497	−	0.398472i
$538$		0			0
$539$	2.43352e6			0.360798
$540$		0			0
$541$	−4.48906e6			−0.659420			−0.329710	−	0.944082i	$-0.606951\pi$
$541$	−4.48906e6			−0.659420			−0.329710	+	0.944082i	$0.606951\pi$
$542$		0			0
$543$	7.36413e6	−	3.09186e6i	1.07182	−	0.450008i
$544$		0			0
$545$	−793831.	−	666103.i	−0.114482	−	0.0960617i
$546$		0			0
$547$	3.16346e6	+	1.15141e6i	0.452058	+	0.164536i	0.558008	−	0.829836i	$-0.311566\pi$
$547$	3.16346e6	+	1.15141e6i	0.452058	+	0.164536i	−0.105949	+	0.994372i	$0.533788\pi$
$548$		0			0
$549$	8.02749e6	+	5.50672e6i	1.13671	+	0.779763i
$550$		0			0
$551$	2.70585e6	−	2.27048e6i	0.379686	−	0.318594i
$552$		0			0
$553$	3.08711e6	−	1.75079e7i	0.429279	−	2.43456i
$554$		0			0
$555$	1.11037e6	+	4.89708e6i	0.153015	+	0.674846i
$556$		0			0
$557$	2.33948e6	−	4.05210e6i	0.319508	−	0.553404i	−0.660877	−	0.750494i	$-0.729816\pi$
$557$	2.33948e6	−	4.05210e6i	0.319508	−	0.553404i	0.980385	+	0.197090i	$0.0631490\pi$
$558$		0			0
$559$	8.10994e6	+	1.40468e7i	1.09771	+	1.90129i
$560$		0			0
$561$	−3.28347e6	+	159168.i	−0.440480	+	0.0213525i
$562$		0			0
$563$	−5.21679e6	+	1.89876e6i	−0.693637	+	0.252463i	−0.664692	−	0.747118i	$-0.731437\pi$
$563$	−5.21679e6	+	1.89876e6i	−0.693637	+	0.252463i	−0.0289455	+	0.999581i	$0.509215\pi$
$564$		0			0
$565$	−459087.	−	2.60361e6i	−0.0605025	−	0.343127i
$566$		0			0
$567$	−7.42300e6	−	8.50882e6i	−0.969665	−	1.11151i
$568$		0			0
$569$	826446.	+	4.68701e6i	0.107012	+	0.606897i	0.990397	+	0.138249i	$0.0441475\pi$
$569$	826446.	+	4.68701e6i	0.107012	+	0.606897i	−0.883385	+	0.468648i	$0.844741\pi$
$570$		0			0
$571$	4.89237e6	−	1.78068e6i	0.627956	−	0.228557i	−0.00838545	−	0.999965i	$-0.502669\pi$
$571$	4.89237e6	−	1.78068e6i	0.627956	−	0.228557i	0.636341	+	0.771408i	$0.280447\pi$
$572$		0			0
$573$	7.47566e6	−	362387.i	0.951181	−	0.0461090i
$574$		0			0
$575$	−3.22145e6	−	5.57972e6i	−0.406333	−	0.703790i
$576$		0			0
$577$	−107932.	+	186943.i	−0.0134961	+	0.0233760i	−0.872695	−	0.488266i	$-0.837629\pi$
$577$	−107932.	+	186943.i	−0.0134961	+	0.0233760i	0.859198	+	0.511642i	$0.170963\pi$
$578$		0			0
$579$	548128.	+	2.41742e6i	0.0679494	+	0.299679i
$580$		0			0
$581$	677450.	−	3.84201e6i	0.0832601	−	0.472192i
$582$		0			0
$583$	1.97994e6	−	1.66137e6i	0.241258	−	0.202439i
$584$		0			0
$585$	−7.23727e6	+	3.45984e6i	−0.874349	+	0.417990i
$586$		0			0
$587$	2.24098e6	+	815649.i	0.268437	+	0.0977030i	0.472732	−	0.881206i	$-0.343268\pi$
$587$	2.24098e6	+	815649.i	0.268437	+	0.0977030i	−0.204295	+	0.978909i	$0.565490\pi$
$588$		0			0
$589$	−1.58313e6	−	1.32840e6i	−0.188031	−	0.157776i
$590$		0			0
$591$	−1.48274e7	+	6.22534e6i	−1.74621	+	0.733152i
$592$		0			0
$593$	1.15415e7			1.34780			0.673901	−	0.738822i	$-0.264617\pi$
$593$	1.15415e7			1.34780			0.673901	+	0.738822i	$0.264617\pi$
$594$		0			0
$595$	1.09963e7			1.27337
$596$		0			0
$597$	4.39145e6	+	3.33628e6i	0.504280	+	0.383113i
$598$		0			0
$599$	8.79807e6	+	7.38246e6i	1.00189	+	0.840686i	0.987245	−	0.159206i	$-0.0508936\pi$
$599$	8.79807e6	+	7.38246e6i	1.00189	+	0.840686i	0.0146456	+	0.999893i	$0.495338\pi$
$600$		0			0
$601$	−1.01371e7	−	3.68959e6i	−1.14479	−	0.416670i	−0.301150	−	0.953577i	$-0.597370\pi$
$601$	−1.01371e7	−	3.68959e6i	−1.14479	−	0.416670i	−0.843642	+	0.536907i	$0.819593\pi$
$602$		0			0
$603$	692228.	−	1.52251e6i	0.0775275	−	0.170517i
$604$		0			0
$605$	−3.75295e6	+	3.14910e6i	−0.416854	+	0.349782i
$606$		0			0
$607$	−1.45388e6	+	8.24534e6i	−0.160161	+	0.908316i	0.793754	+	0.608239i	$0.208124\pi$
$607$	−1.45388e6	+	8.24534e6i	−0.160161	+	0.908316i	−0.953915	+	0.300077i	$0.902987\pi$
$608$		0			0
$609$	−2.11084e7	−	6.54412e6i	−2.30628	−	0.715003i
$610$		0			0
$611$	−1.24517e7	+	2.15670e7i	−1.34935	+	2.33715i
$612$		0			0
$613$	3.14294e6	+	5.44373e6i	0.337819	+	0.585120i	0.984022	−	0.178045i	$-0.0569774\pi$
$613$	3.14294e6	+	5.44373e6i	0.337819	+	0.585120i	−0.646203	+	0.763166i	$0.723644\pi$
$614$		0			0
$615$	1.11875e6	+	1.73761e6i	0.119274	+	0.185253i
$616$		0			0
$617$	−5.06838e6	+	1.84474e6i	−0.535990	+	0.195084i	−0.595811	−	0.803125i	$-0.703169\pi$
$617$	−5.06838e6	+	1.84474e6i	−0.535990	+	0.195084i	0.0598212	+	0.998209i	$0.480947\pi$
$618$		0			0
$619$	738508.	+	4.18829e6i	0.0774691	+	0.439349i	0.998729	+	0.0504016i	$0.0160501\pi$
$619$	738508.	+	4.18829e6i	0.0774691	+	0.439349i	−0.921260	+	0.388948i	$0.872839\pi$
$620$		0			0
$621$	357013.	−	1.22146e7i	0.0371496	−	1.27101i
$622$		0			0
$623$	4.31590e6	+	2.44767e7i	0.445503	+	2.52658i
$624$		0			0
$625$	−436593.	+	158907.i	−0.0447071	+	0.0162721i
$626$		0			0
$627$	−418447.	+	813347.i	−0.0425081	+	0.0826241i
$628$		0			0
$629$	8.21238e6	+	1.42243e7i	0.827642	+	1.43352i
$630$		0			0
$631$	6.87143e6	−	1.19017e7i	0.687027	−	1.18997i	−0.285768	−	0.958299i	$-0.592249\pi$
$631$	6.87143e6	−	1.19017e7i	0.687027	−	1.18997i	0.972795	−	0.231667i	$-0.0744180\pi$
$632$		0			0
$633$	−1.03237e7	+	9.55154e6i	−1.02406	+	0.947467i
$634$		0			0
$635$	1.15324e6	−	6.54035e6i	0.113497	−	0.643675i
$636$		0			0
$637$	1.48796e7	−	1.24855e7i	1.45292	−	1.21915i
$638$		0			0
$639$	3.68308e6	−	3.75455e6i	0.356828	−	0.363752i
$640$		0			0
$641$	−9.63528e6	−	3.50696e6i	−0.926231	−	0.337121i	−0.165516	−	0.986207i	$-0.552929\pi$
$641$	−9.63528e6	−	3.50696e6i	−0.926231	−	0.337121i	−0.760715	+	0.649086i	$0.775151\pi$
$642$		0			0
$643$	−8.49607e6	−	7.12905e6i	−0.810383	−	0.679992i	0.140316	−	0.990107i	$-0.455188\pi$
$643$	−8.49607e6	−	7.12905e6i	−0.810383	−	0.679992i	−0.950699	+	0.310115i	$0.899633\pi$
$644$		0			0
$645$	1.08632e6	−	8.56935e6i	0.102815	−	0.811052i
$646$		0			0
$647$	1.70974e7			1.60572			0.802859	−	0.596169i	$-0.203311\pi$
$647$	1.70974e7			1.60572			0.802859	+	0.596169i	$0.203311\pi$
$648$		0			0
$649$	−511997.			−0.0477151
$650$		0			0
$651$	−1.62609e6	+	1.28273e7i	−0.150380	+	1.18627i
$652$		0			0
$653$	2.26081e6	+	1.89705e6i	0.207483	+	0.174099i	0.740607	−	0.671938i	$-0.234538\pi$
$653$	2.26081e6	+	1.89705e6i	0.207483	+	0.174099i	−0.533125	+	0.846037i	$0.678982\pi$
$654$		0			0
$655$	−871004.	−	317020.i	−0.0793263	−	0.0288724i
$656$		0			0
$657$	−720999.	−	2.59096e6i	−0.0651660	−	0.234179i
$658$		0			0
$659$	−1.12112e7	+	9.40733e6i	−1.00563	+	0.843826i	−0.987755	−	0.156015i	$-0.950135\pi$
$659$	−1.12112e7	+	9.40733e6i	−1.00563	+	0.843826i	−0.0178774	+	0.999840i	$0.505691\pi$
$660$		0			0
$661$	−2.69996e6	+	1.53122e7i	−0.240355	+	1.36312i	0.590683	+	0.806904i	$0.298859\pi$
$661$	−2.69996e6	+	1.53122e7i	−0.240355	+	1.36312i	−0.831038	+	0.556216i	$0.812253\pi$
$662$		0			0
$663$	−1.92599e7	+	1.78195e7i	−1.70165	+	1.57438i
$664$		0			0
$665$	1.52982e6	−	2.64972e6i	0.134148	−	0.232352i
$666$		0			0
$667$	−1.19581e7	−	2.07120e7i	−1.04075	−	1.80263i
$668$		0			0
$669$	−3.03172e6	+	5.89282e6i	−0.261893	+	0.509047i
$670$		0			0
$671$	−4.63610e6	+	1.68740e6i	−0.397509	+	0.144681i
$672$		0			0
$673$	−1.07023e6	−	6.06958e6i	−0.0910834	−	0.516560i	−0.995877	−	0.0907101i	$-0.971086\pi$
$673$	−1.07023e6	−	6.06958e6i	−0.0910834	−	0.516560i	0.904794	−	0.425850i	$-0.140025\pi$
$674$		0			0
$675$	−7.40898e6	−	1.53085e6i	−0.625891	−	0.129322i
$676$		0			0
$677$	1.64495e6	+	9.32898e6i	0.137937	+	0.782280i	0.972769	+	0.231776i	$0.0744535\pi$
$677$	1.64495e6	+	9.32898e6i	0.137937	+	0.782280i	−0.834832	+	0.550505i	$0.814435\pi$
$678$		0			0
$679$	6.33379e6	−	2.30531e6i	0.527217	−	0.191891i
$680$		0			0
$681$	4.79601e6	+	7.44901e6i	0.396289	+	0.615504i
$682$		0			0
$683$	1.94234e6	+	3.36424e6i	0.159322	+	0.275953i	0.934624	−	0.355637i	$-0.115736\pi$
$683$	1.94234e6	+	3.36424e6i	0.159322	+	0.275953i	−0.775303	+	0.631590i	$0.782403\pi$
$684$		0			0
$685$	−2.31860e6	+	4.01593e6i	−0.188799	+	0.327009i
$686$		0			0
$687$	1.97758e7	+	6.13098e6i	1.59861	+	0.495608i
$688$		0			0
$689$	3.58237e6	−	2.03167e7i	0.287490	−	1.63044i
$690$		0			0
$691$	−4.87039e6	+	4.08674e6i	−0.388033	+	0.325598i	−0.815846	−	0.578269i	$-0.803728\pi$
$691$	−4.87039e6	+	4.08674e6i	−0.388033	+	0.325598i	0.427813	+	0.903867i	$0.359284\pi$
$692$		0			0
$693$	5.69586e6	−	553521.i	0.450533	−	0.0437825i
$694$		0			0
$695$	6.90648e6	+	2.51375e6i	0.542368	+	0.197406i
$696$		0			0
$697$	5.17832e6	+	4.34513e6i	0.403745	+	0.338782i
$698$		0			0
$699$	−7.31011e6	−	5.55365e6i	−0.565889	−	0.429918i
$700$		0			0
$701$	5.25390e6			0.403819			0.201910	−	0.979404i	$-0.435285\pi$
$701$	5.25390e6			0.403819			0.201910	+	0.979404i	$0.435285\pi$
$702$		0			0
$703$	4.57007e6			0.348766
$704$		0			0
$705$	1.22283e7	−	5.13409e6i	0.926601	−	0.389037i
$706$		0			0
$707$	1.71522e7	+	1.43924e7i	1.29054	+	1.08289i
$708$		0			0
$709$	2.46642e7	+	8.97703e6i	1.84269	+	0.670683i	0.988603	+	0.150545i	$0.0481030\pi$
$709$	2.46642e7	+	8.97703e6i	1.84269	+	0.670683i	0.854083	+	0.520137i	$0.174119\pi$
$710$		0			0
$711$	1.75264e6	−	2.25234e7i	0.130022	−	1.67094i
$712$		0			0
$713$	−1.07191e7	+	8.99439e6i	−0.789649	+	0.662594i
$714$		0			0
$715$	705966.	−	4.00373e6i	0.0516438	−	0.292887i
$716$		0			0
$717$	−939635.	−	4.14409e6i	−0.0682592	−	0.301045i
$718$		0			0
$719$	−2.09714e6	+	3.63235e6i	−0.151288	+	0.262039i	−0.931701	−	0.363226i	$-0.881675\pi$
$719$	−2.09714e6	+	3.63235e6i	−0.151288	+	0.262039i	0.780413	+	0.625264i	$0.215009\pi$
$720$		0			0
$721$	7.48717e6	+	1.29682e7i	0.536389	+	0.929053i
$722$		0			0
$723$	−901865.	+	43718.4i	−0.0641646	+	0.00311042i
$724$		0			0
$725$	−1.39139e7	+	5.06424e6i	−0.983113	+	0.357824i
$726$		0			0
$727$	−4.05944e6	−	2.30222e7i	−0.284859	−	1.61552i	−0.705788	−	0.708423i	$-0.749407\pi$
$727$	−4.05944e6	−	2.30222e7i	−0.284859	−	1.61552i	0.420929	−	0.907094i	$-0.361704\pi$
$728$		0			0
$729$	−1.04344e7	−	9.84956e6i	−0.727194	−	0.686433i
$730$		0			0
$731$	−4.90632e6	−	2.78251e7i	−0.339596	−	1.92594i
$732$		0			0
$733$	−1.83398e7	+	6.67516e6i	−1.26077	+	0.458883i	−0.884027	−	0.467435i	$-0.845178\pi$
$733$	−1.83398e7	+	6.67516e6i	−1.26077	+	0.458883i	−0.376742	+	0.926318i	$0.622956\pi$
$734$		0			0
$735$	−1.03321e7	+	500856.i	−0.705459	+	0.0341975i
$736$		0			0
$737$	423816.	+	734072.i	0.0287415	+	0.0497817i
$738$		0			0
$739$	−1.40670e7	+	2.43647e7i	−0.947523	+	1.64116i	−0.196905	+	0.980422i	$0.563089\pi$
$739$	−1.40670e7	+	2.43647e7i	−0.947523	+	1.64116i	−0.750618	+	0.660736i	$0.770244\pi$
$740$		0			0
$741$	1.61440e6	+	7.12004e6i	0.108011	+	0.476362i
$742$		0			0
$743$	4.42993e6	−	2.51234e7i	0.294391	−	1.66958i	−0.375275	−	0.926913i	$-0.622452\pi$
$743$	4.42993e6	−	2.51234e7i	0.294391	−	1.66958i	0.669666	−	0.742662i	$-0.266437\pi$
$744$		0			0
$745$	7.97612e6	−	6.69276e6i	0.526503	−	0.441788i
$746$		0			0
$747$	384607.	−	4.94263e6i	0.0252183	−	0.324083i
$748$		0			0
$749$	9.02678e6	+	3.28548e6i	0.587934	+	0.213990i
$750$		0			0
$751$	−1.22911e7	−	1.03134e7i	−0.795224	−	0.667272i	0.151808	−	0.988410i	$-0.451490\pi$
$751$	−1.22911e7	−	1.03134e7i	−0.795224	−	0.667272i	−0.947033	+	0.321138i	$0.895935\pi$
$752$		0			0
$753$	2.74401e7	−	1.15208e7i	1.76360	−	0.740452i
$754$		0			0
$755$	−1.15530e7			−0.737613
$756$		0			0
$757$	2.27523e7			1.44306			0.721532	−	0.692381i	$-0.243438\pi$
$757$	2.27523e7			1.44306			0.721532	+	0.692381i	$0.243438\pi$
$758$		0			0
$759$	4.93138e6	+	3.74647e6i	0.310716	+	0.236058i
$760$		0			0
$761$	−1.92613e7	−	1.61621e7i	−1.20566	−	1.01167i	−0.999450	−	0.0331566i	$-0.989444\pi$
$761$	−1.92613e7	−	1.61621e7i	−1.20566	−	1.01167i	−0.206206	−	0.978509i	$-0.566112\pi$
$762$		0			0
$763$	5.54489e6	+	2.01817e6i	0.344811	+	0.125501i
$764$		0			0
$765$	1.39081e7	−	1.35158e6i	0.859237	−	0.0835002i
$766$		0			0
$767$	−3.13057e6	+	2.62686e6i	−0.192147	+	0.161231i
$768$		0			0
$769$	5.44413e6	−	3.08752e7i	0.331981	−	1.88276i	−0.123239	−	0.992377i	$-0.539328\pi$
$769$	5.44413e6	−	3.08752e7i	0.331981	−	1.88276i	0.455219	−	0.890379i	$-0.349561\pi$
$770$		0			0
$771$	−8.05349e6	−	2.49678e6i	−0.487920	−	0.151267i
$772$		0			0
$773$	1.00446e7	−	1.73977e7i	0.604620	−	1.04723i	−0.387491	−	0.921873i	$-0.626658\pi$
$773$	1.00446e7	−	1.73977e7i	0.604620	−	1.04723i	0.992111	−	0.125359i	$-0.0400084\pi$
$774$		0			0
$775$	4.33158e6	+	7.50251e6i	0.259055	+	0.448696i
$776$		0			0
$777$	−1.54786e7	−	2.40408e7i	−0.919768	−	1.42855i
$778$		0			0
$779$	1.76744e6	−	643296.i	0.104352	−	0.0379810i
$780$		0			0
$781$	462864.	+	2.62503e6i	0.0271535	+	0.153995i
$782$		0			0
$783$	−2.75022e7	−	5.68251e6i	−1.60311	−	0.331234i
$784$		0			0
$785$	1.94491e6	+	1.10301e7i	0.112648	+	0.638861i
$786$		0			0
$787$	192854.	−	70193.0i	0.0110992	−	0.00403978i	−0.336465	−	0.941696i	$-0.609231\pi$
$787$	192854.	−	70193.0i	0.0110992	−	0.00403978i	0.347564	+	0.937656i	$0.387009\pi$
$788$		0			0
$789$	−1.16957e7	+	2.27332e7i	−0.668857	+	1.30007i
$790$		0			0
$791$	7.52709e6	+	1.30373e7i	0.427746	+	0.740878i
$792$		0			0
$793$	−1.96897e7	+	3.41035e7i	−1.11187	+	1.92582i
$794$		0			0
$795$	−8.06441e6	+	7.46127e6i	−0.452538	+	0.418692i
$796$		0			0
$797$	2.37030e6	−	1.34427e7i	0.132178	−	0.749617i	−0.844606	−	0.535389i	$-0.820165\pi$
$797$	2.37030e6	−	1.34427e7i	0.132178	−	0.749617i	0.976784	−	0.214228i	$-0.0687237\pi$
$798$		0			0
$799$	3.32315e7	−	2.78846e7i	1.84155	−	1.54524i
$800$		0			0
$801$	8.46722e6	+	3.04276e7i	0.466294	+	1.67566i
$802$		0			0
$803$	1.28081e6	+	466178.i	0.0700967	+	0.0255131i
$804$		0			0
$805$	−1.58695e7	−	1.33161e7i	−0.863127	−	0.724249i
$806$		0			0
$807$	−1.95889e6	+	1.54526e7i	−0.105883	+	0.835250i
$808$		0			0
$809$	1.73943e6			0.0934408			0.0467204	−	0.998908i	$-0.485123\pi$
$809$	1.73943e6			0.0934408			0.0467204	+	0.998908i	$0.485123\pi$
$810$		0			0
$811$	−3.05982e7			−1.63359			−0.816797	−	0.576925i	$-0.804252\pi$
$811$	−3.05982e7			−1.63359			−0.816797	+	0.576925i	$0.804252\pi$
$812$		0			0
$813$	−1.27504e6	+	1.00581e7i	−0.0676548	+	0.533691i
$814$		0			0
$815$	175784.	+	147500.i	0.00927010	+	0.00777854i
$816$		0			0
$817$	−7.38746e6	−	2.68882e6i	−0.387204	−	0.140931i
$818$		0			0
$819$	3.19870e7	−	3.26077e7i	1.66634	−	1.69868i
$820$		0			0
$821$	−5.59491e6	+	4.69469e6i	−0.289691	+	0.243080i	−0.776038	−	0.630686i	$-0.782774\pi$
$821$	−5.59491e6	+	4.69469e6i	−0.289691	+	0.243080i	0.486347	+	0.873766i	$0.338329\pi$
$822$		0			0
$823$	1.52770e6	−	8.66404e6i	0.0786212	−	0.445883i	−0.919931	−	0.392081i	$-0.871755\pi$
$823$	1.52770e6	−	8.66404e6i	0.0786212	−	0.445883i	0.998552	−	0.0538013i	$-0.0171338\pi$
$824$		0			0
$825$	2.81443e6	−	2.60393e6i	0.143964	−	0.133197i
$826$		0			0
$827$	1.48974e7	−	2.58031e7i	0.757439	−	1.31192i	−0.186714	−	0.982414i	$-0.559784\pi$
$827$	1.48974e7	−	2.58031e7i	0.757439	−	1.31192i	0.944153	−	0.329508i	$-0.106883\pi$
$828$		0			0
$829$	−6.05308e6	−	1.04842e7i	−0.305908	−	0.529847i	0.671555	−	0.740954i	$-0.265627\pi$
$829$	−6.05308e6	−	1.04842e7i	−0.305908	−	0.529847i	−0.977463	+	0.211107i	$0.932293\pi$
$830$		0			0
$831$	4.93207e6	−	9.58658e6i	0.247757	−	0.481572i
$832$		0			0
$833$	−3.17952e7	+	1.15725e7i	−1.58763	+	0.577849i
$834$		0			0
$835$	1.71933e6	+	9.75079e6i	0.0853380	+	0.483976i
$836$		0			0
$837$	−480040.	+	1.64238e7i	−0.0236845	+	0.810325i
$838$		0			0
$839$	13674.7	+	77553.1i	0.000670676	+	0.00380359i	0.985141	−	0.171746i	$-0.0549409\pi$
$839$	13674.7	+	77553.1i	0.000670676	+	0.00380359i	−0.984471	+	0.175550i	$0.943830\pi$
$840$		0			0
$841$	−3.23743e7	+	1.17833e7i	−1.57837	+	0.574481i
$842$		0			0
$843$	−2.05798e6	−	3.19640e6i	−0.0997409	−	0.154914i
$844$		0			0
$845$	−9.99057e6	−	1.73042e7i	−0.481336	−	0.833698i
$846$		0			0
$847$	1.39483e7	−	2.41592e7i	0.668057	−	1.15711i
$848$		0			0
$849$	5.17569e6	+	1.60459e6i	0.246433	+	0.0764004i
$850$		0			0
$851$	5.37321e6	−	3.04730e7i	0.254337	−	1.44242i
$852$		0			0
$853$	−7.73079e6	+	6.48690e6i	−0.363790	+	0.305256i	−0.806299	−	0.591508i	$-0.798533\pi$
$853$	−7.73079e6	+	6.48690e6i	−0.363790	+	0.305256i	0.442509	+	0.896764i	$0.354088\pi$
$854$		0			0
$855$	1.60922e6	−	3.53939e6i	0.0752837	−	0.165582i
$856$		0			0
$857$	2.77346e7	+	1.00946e7i	1.28994	+	0.469501i	0.893708	−	0.448649i	$-0.148095\pi$
$857$	2.77346e7	+	1.00946e7i	1.28994	+	0.469501i	0.396234	+	0.918149i	$0.370317\pi$
$858$		0			0
$859$	1.14351e7	+	9.59519e6i	0.528758	+	0.443681i	0.867672	−	0.497136i	$-0.165615\pi$
$859$	1.14351e7	+	9.59519e6i	0.528758	+	0.443681i	−0.338914	+	0.940817i	$0.610060\pi$
$860$		0			0
$861$	−9.37028e6	−	7.11880e6i	−0.430769	−	0.327265i
$862$		0			0
$863$	8.75294e6			0.400062			0.200031	−	0.979790i	$-0.435896\pi$
$863$	8.75294e6			0.400062			0.200031	+	0.979790i	$0.435896\pi$
$864$		0			0
$865$	8.60255e6			0.390919
$866$		0			0
$867$	2.17356e7	−	9.12579e6i	0.982030	−	0.412309i
$868$		0			0
$869$	8.77086e6	+	7.35963e6i	0.393997	+	0.330603i
$870$		0			0
$871$	6.35763e6	+	2.31399e6i	0.283955	+	0.103351i
$872$		0			0
$873$	7.72762e6	−	3.69425e6i	0.343170	−	0.164056i
$874$		0			0
$875$	−2.51981e7	+	2.11437e7i	−1.11262	+	0.933601i
$876$		0			0
$877$	−2.99514e6	+	1.69863e7i	−0.131497	+	0.745759i	0.845737	+	0.533599i	$0.179161\pi$
$877$	−2.99514e6	+	1.69863e7i	−0.131497	+	0.745759i	−0.977235	+	0.212160i	$0.931950\pi$
$878$		0			0
$879$	−8.62133e6	−	3.80229e7i	−0.376359	−	1.65987i
$880$		0			0
$881$	1.56234e7	−	2.70605e7i	0.678166	−	1.17462i	−0.297367	−	0.954763i	$-0.596108\pi$
$881$	1.56234e7	−	2.70605e7i	0.678166	−	1.17462i	0.975533	−	0.219855i	$-0.0705583\pi$
$882$		0			0
$883$	1.16017e7	+	2.00947e7i	0.500748	+	0.867321i	1.00000		0.000863784i	$0.000274951\pi$
$883$	1.16017e7	+	2.00947e7i	0.500748	+	0.867321i	−0.499252	+	0.866457i	$0.666392\pi$
$884$		0			0
$885$	2.17381e6	−	105377.i	0.0932961	−	0.00452258i
$886$		0			0
$887$	−3.53036e7	+	1.28494e7i	−1.50664	+	0.548372i	−0.957770	−	0.287536i	$-0.907164\pi$
$887$	−3.53036e7	+	1.28494e7i	−1.50664	+	0.548372i	−0.548870	+	0.835908i	$0.684942\pi$
$888$		0			0
$889$	6.56679e6	+	3.72421e7i	0.278675	+	1.58045i
$890$		0			0
$891$	7.13608e6	−	1.40018e6i	0.301138	−	0.0590868i
$892$		0			0
$893$	−2.09599e6	−	1.18870e7i	−0.0879552	−	0.498819i
$894$		0			0
$895$	−8.91072e6	+	3.24324e6i	−0.371839	+	0.135338i
$896$		0			0
$897$	4.93742e7	−	2.39344e6i	2.04889	−	0.0993212i
$898$		0			0
$899$	1.60788e7	+	2.78494e7i	0.663522	+	1.14925i
$900$		0			0
$901$	−1.79684e7	+	3.11221e7i	−0.737389	+	1.27720i
$902$		0			0
$903$	1.08764e7	+	4.79686e7i	0.443881	+	1.95766i
$904$		0			0
$905$	2.98782e6	−	1.69448e7i	0.121264	−	0.687725i
$906$		0			0
$907$	−3.20451e7	+	2.68890e7i	−1.29343	+	1.08532i	−0.302192	+	0.953247i	$0.597718\pi$
$907$	−3.20451e7	+	2.68890e7i	−1.29343	+	1.08532i	−0.991240	+	0.132071i	$0.957837\pi$
$908$		0			0
$909$	2.34630e7	+	1.60952e7i	0.941835	+	0.646082i
$910$		0			0
$911$	−3.01786e7	−	1.09841e7i	−1.20477	−	0.438499i	−0.339880	−	0.940469i	$-0.610387\pi$
$911$	−3.01786e7	−	1.09841e7i	−1.20477	−	0.438499i	−0.864885	+	0.501970i	$0.832609\pi$
$912$		0			0
$913$	1.92472e6	+	1.61503e6i	0.0764170	+	0.0641215i
$914$		0			0
$915$	1.93364e7	−	8.11847e6i	0.763526	−	0.320569i
$916$		0			0
$917$	5.27798e6			0.207274
$918$		0			0
$919$	917925.			0.0358524			0.0179262	−	0.999839i	$-0.494294\pi$
$919$	917925.			0.0358524			0.0179262	+	0.999839i	$0.494294\pi$
$920$		0			0
$921$	2.43145e7	+	1.84723e7i	0.944532	+	0.717581i
$922$		0			0
$923$	1.62982e7	+	1.36758e7i	0.629702	+	0.528382i
$924$		0			0
$925$	−1.80020e7	−	6.55221e6i	−0.691779	−	0.251787i
$926$		0			0
$927$	1.10637e7	+	1.54818e7i	0.422865	+	0.591729i
$928$		0			0
$929$	−2.50009e7	+	2.09782e7i	−0.950422	+	0.797499i	−0.979369	−	0.202082i	$-0.935229\pi$
$929$	−2.50009e7	+	2.09782e7i	−0.950422	+	0.797499i	0.0289467	+	0.999581i	$0.490785\pi$
$930$		0			0
$931$	−1.63482e6	+	9.27151e6i	−0.0618152	+	0.350571i
$932$		0			0
$933$	−6.64872e6	−	2.06127e6i	−0.250054	−	0.0775229i
$934$		0			0
$935$	−3.54096e6	+	6.13313e6i	−0.132462	+	0.229431i
$936$		0			0
$937$	1.83457e7	+	3.17756e7i	0.682629	+	1.18235i	0.974176	+	0.225791i	$0.0724967\pi$
$937$	1.83457e7	+	3.17756e7i	0.682629	+	1.18235i	−0.291547	+	0.956557i	$0.594170\pi$
$938$		0			0
$939$	8.40781e6	+	1.30587e7i	0.311185	+	0.483323i
$940$		0			0
$941$	−2.12388e7	+	7.73031e6i	−0.781910	+	0.284592i	−0.701969	−	0.712208i	$-0.747695\pi$
$941$	−2.12388e7	+	7.73031e6i	−0.781910	+	0.284592i	−0.0799411	+	0.996800i	$0.525473\pi$
$942$		0			0
$943$	−2.21142e6	−	1.25416e7i	−0.0809825	−	0.459275i
$944$		0			0
$945$	−2.40693e7	+	3.52241e6i	−0.876766	+	0.128310i
$946$		0			0
$947$	−9.47873e6	−	5.37566e7i	−0.343459	−	1.94785i	−0.317715	−	0.948186i	$-0.602916\pi$
$947$	−9.47873e6	−	5.37566e7i	−0.343459	−	1.94785i	−0.0257443	−	0.999669i	$-0.508196\pi$
$948$		0			0
$949$	1.02232e7	−	3.72095e6i	0.368487	−	0.134118i
$950$		0			0
$951$	−2.42058e7	+	4.70494e7i	−0.867897	+	1.68695i
$952$		0			0
$953$	−1.57388e7	−	2.72604e7i	−0.561358	−	0.972301i	−0.997378	−	0.0723639i	$-0.976946\pi$
$953$	−1.57388e7	−	2.72604e7i	−0.561358	−	0.972301i	0.436020	−	0.899937i	$-0.356388\pi$
$954$		0			0
$955$	8.06190e6	−	1.39636e7i	0.286041	−	0.495438i
$956$		0			0
$957$	1.04472e7	−	9.66582e6i	0.368739	−	0.341160i
$958$		0			0
$959$	4.58521e6	−	2.60040e7i	0.160995	−	0.913048i
$960$		0			0
$961$	−7.51828e6	+	6.30858e6i	−0.262609	+	0.220355i
$962$		0			0
$963$	1.18209e7	+	3.04596e6i	0.410756	+	0.105842i
$964$		0			0
$965$	5.01803e6	+	1.82641e6i	0.173466	+	0.0631365i
$966$		0			0
$967$	−1.25504e7	−	1.05311e7i	−0.431611	−	0.362164i	0.400948	−	0.916101i	$-0.368681\pi$
$967$	−1.25504e7	−	1.05311e7i	−0.431611	−	0.362164i	−0.832559	+	0.553936i	$0.813125\pi$
$968$		0			0
$969$	1.59939e6	−	1.26167e7i	0.0547198	−	0.431654i
$970$		0			0
$971$	−5.52426e6			−0.188029			−0.0940147	−	0.995571i	$-0.529970\pi$
$971$	−5.52426e6			−0.188029			−0.0940147	+	0.995571i	$0.529970\pi$
$972$		0			0
$973$	−4.18508e7			−1.41717
$974$		0			0
$975$	3.84884e6	−	3.03613e7i	0.129664	−	1.02284i
$976$		0			0
$977$	1.59343e7	+	1.33705e7i	0.534068	+	0.448136i	0.869503	−	0.493927i	$-0.164439\pi$
$977$	1.59343e7	+	1.33705e7i	0.534068	+	0.448136i	−0.335436	+	0.942063i	$0.608883\pi$
$978$		0			0
$979$	−1.50416e7	−	5.47468e6i	−0.501575	−	0.182558i
$980$		0			0
$981$	7.26122e6	+	1.87105e6i	0.240900	+	0.0620743i
$982$		0			0
$983$	3.47957e7	−	2.91971e7i	1.14853	−	0.963730i	0.148845	−	0.988861i	$-0.452445\pi$
$983$	3.47957e7	−	2.91971e7i	1.14853	−	0.963730i	0.999684	+	0.0251303i	$0.00800006\pi$
$984$		0			0
$985$	−6.01587e6	+	3.41177e7i	−0.197564	+	1.12044i
$986$		0			0
$987$	−5.54323e7	+	5.12865e7i	−1.81122	+	1.67575i
$988$		0			0
$989$	−2.66146e7	+	4.60979e7i	−0.865227	+	1.49862i
$990$		0			0
$991$	−1.10278e7	−	1.91006e7i	−0.356700	−	0.617822i	0.630707	−	0.776021i	$-0.282765\pi$
$991$	−1.10278e7	−	1.91006e7i	−0.356700	−	0.617822i	−0.987407	+	0.158198i	$0.949431\pi$
$992$		0			0
$993$	9.56002e6	−	1.85821e7i	0.307670	−	0.598027i
$994$		0			0
$995$	1.11646e7	−	4.06357e6i	0.357507	−	0.130122i
$996$		0			0
$997$	−3.00854e6	−	1.70623e7i	−0.0958556	−	0.543624i	−0.994482	−	0.104909i	$-0.966545\pi$
$997$	−3.00854e6	−	1.70623e7i	−0.0958556	−	0.543624i	0.898626	−	0.438715i	$-0.144566\pi$
$998$		0			0
$999$	−2.25322e7	−	2.85043e7i	−0.714314	−	0.903641i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.13.8	✓	90
3.2	odd	2		324.6.i.a.253.5		90
27.2	odd	18		324.6.i.a.73.5		90
27.25	even	9	inner	108.6.i.a.25.8	yes	90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.8	✓	90	1.1	even	1	trivial
108.6.i.a.25.8	yes	90	27.25	even	9	inner
324.6.i.a.73.5		90	27.2	odd	18
324.6.i.a.253.5		90	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.96043 + 15.4647i) q^{3} +(25.7256 + 21.5863i) q^{5} +(-179.693 - 65.4027i) q^{7} +(-235.313 - 60.6348i) q^{9} +O(q^{10})\)	\(q+(-1.96043 + 15.4647i) q^{3} +(25.7256 + 21.5863i) q^{5} +(-179.693 - 65.4027i) q^{7} +(-235.313 - 60.6348i) q^{9} +(94.3418 - 79.1622i) q^{11} +(170.696 - 968.062i) q^{13} +(-384.259 + 355.520i) q^{15} +(-856.170 + 1482.93i) q^{17} +(238.223 + 412.614i) q^{19} +(1363.71 - 2650.67i) q^{21} +(3031.38 - 1103.33i) q^{23} +(-346.815 - 1966.88i) q^{25} +(1399.01 - 3520.18i) q^{27} +(-1287.38 - 7301.08i) q^{29} +(-4076.01 + 1483.55i) q^{31} +(1039.27 + 1614.16i) q^{33} +(-3210.89 - 5561.42i) q^{35} +(4796.00 - 8306.91i) q^{37} +(14636.2 + 4537.57i) q^{39} +(685.513 - 3887.74i) q^{41} +(-12640.1 + 10606.3i) q^{43} +(-4744.69 - 6639.42i) q^{45} +(-23806.3 - 8664.79i) q^{47} +(15137.0 + 12701.4i) q^{49} +(-21254.6 - 16147.6i) q^{51} +20986.9 q^{53} +4135.82 q^{55} +(-6847.97 + 2875.15i) q^{57} +(-3184.72 - 2672.30i) q^{59} +(-37644.6 - 13701.5i) q^{61} +(38318.4 + 26285.8i) q^{63} +(25288.2 - 21219.3i) q^{65} +(-1195.16 + 6778.12i) q^{67} +(11119.9 + 49042.4i) q^{69} +(-10821.9 + 18744.1i) q^{71} +(5533.76 + 9584.75i) q^{73} +(31097.1 - 1507.45i) q^{75} +(-22129.9 + 8054.64i) q^{77} +(16143.9 + 91556.6i) q^{79} +(51695.8 + 28536.4i) q^{81} +(3542.69 + 20091.6i) q^{83} +(-54036.5 + 19667.7i) q^{85} +(115433. - 5595.67i) q^{87} +(-64987.0 - 112561. i) q^{89} +(-93986.7 + 162790. i) q^{91} +(-14951.9 - 65942.6i) q^{93} +(-2778.40 + 15757.1i) q^{95} +(-27001.5 + 22656.9i) q^{97} +(-26999.9 + 12907.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 90 * q - 87 * q^5 + 330 * q^9	\( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

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