LMFDB - Embedded newform 450.6.f.e.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(107,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("450.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.f (of order $4$, degree $2$, 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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3457x^{8} + 2937456x^{4} + 12960000 $ x^12 + 3457x^8 + 2937456x^4 + 12960000
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.4
Root			$1.02615 - 1.02615i$ of defining polynomial
Character	$\chi$	$=$	450.107
Dual form			450.6.f.e.143.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-148.726 - 148.726i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})$	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-148.726 - 148.726i) q^{7} +(-45.2548 - 45.2548i) q^{8} -299.527i q^{11} +(193.262 - 193.262i) q^{13} -841.322 q^{14} -256.000 q^{16} +(1522.21 - 1522.21i) q^{17} -1748.19i q^{19} +(-847.191 - 847.191i) q^{22} +(2539.27 + 2539.27i) q^{23} -1093.25i q^{26} +(-2379.62 + 2379.62i) q^{28} -2617.43 q^{29} -8652.76 q^{31} +(-724.077 + 724.077i) q^{32} -8610.89i q^{34} +(5378.10 + 5378.10i) q^{37} +(-4944.63 - 4944.63i) q^{38} +1586.56i q^{41} +(-2005.55 + 2005.55i) q^{43} -4792.44 q^{44} +14364.3 q^{46} +(-11013.5 + 11013.5i) q^{47} +27431.9i q^{49} +(-3092.19 - 3092.19i) q^{52} +(-9879.22 - 9879.22i) q^{53} +13461.2i q^{56} +(-7403.22 + 7403.22i) q^{58} -13932.3 q^{59} -28140.7 q^{61} +(-24473.7 + 24473.7i) q^{62} +4096.00i q^{64} +(-24197.7 - 24197.7i) q^{67} +(-24355.3 - 24355.3i) q^{68} +6518.47i q^{71} +(61503.1 - 61503.1i) q^{73} +30423.1 q^{74} -27971.0 q^{76} +(-44547.6 + 44547.6i) q^{77} +42431.1i q^{79} +(4487.47 + 4487.47i) q^{82} +(-33805.9 - 33805.9i) q^{83} +11345.1i q^{86} +(-13555.1 + 13555.1i) q^{88} -63913.9 q^{89} -57486.2 q^{91} +(40628.3 - 40628.3i) q^{92} +62301.7i q^{94} +(58905.9 + 58905.9i) q^{97} +(77589.2 + 77589.2i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 144 q^{7}+O(q^{10}) $ 12 * q - 144 * q^7	$ 12 q - 144 q^{7} + 276 q^{13} - 3072 q^{16} - 384 q^{22} - 2304 q^{28} - 58512 q^{31} - 25764 q^{37} - 16080 q^{43} - 60672 q^{46} - 4416 q^{52} - 23952 q^{58} - 145200 q^{61} - 33552 q^{67} + 158988 q^{73} - 86016 q^{76} + 75024 q^{82} - 6144 q^{88} - 465024 q^{91} + 631116 q^{97}+O(q^{100}) $ 12 * q - 144 * q^7 + 276 * q^13 - 3072 * q^16 - 384 * q^22 - 2304 * q^28 - 58512 * q^31 - 25764 * q^37 - 16080 * q^43 - 60672 * q^46 - 4416 * q^52 - 23952 * q^58 - 145200 * q^61 - 33552 * q^67 + 158988 * q^73 - 86016 * q^76 + 75024 * q^82 - 6144 * q^88 - 465024 * q^91 + 631116 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$2$	2.82843	−	2.82843i	0.500000	−	0.500000i
$3$		0			0
$3$		0			0
$4$		−	16.0000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−148.726	−	148.726i	−1.14721	−	1.14721i	−0.987099	−	0.160109i	$-0.948816\pi$
$7$	−148.726	−	148.726i	−1.14721	−	1.14721i	−0.160109	−	0.987099i	$-0.551184\pi$
$8$	−45.2548	−	45.2548i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$		−	299.527i		−	0.746371i	−0.927757	−	0.373186i	$-0.878265\pi$
$11$		−	299.527i		−	0.746371i	0.927757	−	0.373186i	$-0.121735\pi$
$12$		0			0
$13$	193.262	−	193.262i	0.317167	−	0.317167i	−0.530511	−	0.847678i	$-0.678000\pi$
$13$	193.262	−	193.262i	0.317167	−	0.317167i	0.847678	+	0.530511i	$0.178000\pi$
$14$	−841.322			−1.14721
$15$		0			0
$16$	−256.000			−0.250000
$17$	1522.21	−	1522.21i	1.27747	−	1.27747i	0.335392	−	0.942079i	$-0.391131\pi$
$17$	1522.21	−	1522.21i	1.27747	−	1.27747i	0.942079	−	0.335392i	$-0.108869\pi$
$18$		0			0
$19$		−	1748.19i		−	1.11098i	−0.831525	−	0.555488i	$-0.812532\pi$
$19$		−	1748.19i		−	1.11098i	0.831525	−	0.555488i	$-0.187468\pi$
$20$		0			0
$21$		0			0
$22$	−847.191	−	847.191i	−0.373186	−	0.373186i
$23$	2539.27	+	2539.27i	1.00090	+	1.00090i	1.00000		0.000896947i	$0.000285507\pi$
$23$	2539.27	+	2539.27i	1.00090	+	1.00090i	0.000896947		1.00000i	$0.499714\pi$
$24$		0			0
$25$		0			0
$26$		−	1093.25i		−	0.317167i
$27$		0			0
$28$	−2379.62	+	2379.62i	−0.573604	+	0.573604i
$29$	−2617.43			−0.577937			−0.288969	−	0.957339i	$-0.593312\pi$
$29$	−2617.43			−0.577937			−0.288969	+	0.957339i	$0.593312\pi$
$30$		0			0
$31$	−8652.76			−1.61715			−0.808575	−	0.588393i	$-0.799761\pi$
$31$	−8652.76			−1.61715			−0.808575	+	0.588393i	$0.799761\pi$
$32$	−724.077	+	724.077i	−0.125000	+	0.125000i
$33$		0			0
$34$		−	8610.89i		−	1.27747i
$35$		0			0
$36$		0			0
$37$	5378.10	+	5378.10i	0.645840	+	0.645840i	0.951985	−	0.306145i	$-0.0990393\pi$
$37$	5378.10	+	5378.10i	0.645840	+	0.645840i	−0.306145	+	0.951985i	$0.599039\pi$
$38$	−4944.63	−	4944.63i	−0.555488	−	0.555488i
$39$		0			0
$40$		0			0
$41$			1586.56i			0.147400i	0.997280	+	0.0737000i	$0.0234807\pi$
$41$			1586.56i			0.147400i	−0.997280	+	0.0737000i	$0.976519\pi$
$42$		0			0
$43$	−2005.55	+	2005.55i	−0.165410	+	0.165410i	−0.784959	−	0.619548i	$-0.787316\pi$
$43$	−2005.55	+	2005.55i	−0.165410	+	0.165410i	0.619548	+	0.784959i	$0.287316\pi$
$44$	−4792.44			−0.373186
$45$		0			0
$46$	14364.3			1.00090
$47$	−11013.5	+	11013.5i	−0.727245	+	0.727245i	−0.970070	−	0.242825i	$-0.921926\pi$
$47$	−11013.5	+	11013.5i	−0.727245	+	0.727245i	0.242825	+	0.970070i	$0.421926\pi$
$48$		0			0
$49$			27431.9i			1.63217i
$50$		0			0
$51$		0			0
$52$	−3092.19	−	3092.19i	−0.158584	−	0.158584i
$53$	−9879.22	−	9879.22i	−0.483096	−	0.483096i	0.423023	−	0.906119i	$-0.360969\pi$
$53$	−9879.22	−	9879.22i	−0.483096	−	0.483096i	−0.906119	+	0.423023i	$0.860969\pi$
$54$		0			0
$55$		0			0
$56$			13461.2i			0.573604i
$57$		0			0
$58$	−7403.22	+	7403.22i	−0.288969	+	0.288969i
$59$	−13932.3			−0.521066			−0.260533	−	0.965465i	$-0.583898\pi$
$59$	−13932.3			−0.521066			−0.260533	+	0.965465i	$0.583898\pi$
$60$		0			0
$61$	−28140.7			−0.968301			−0.484151	−	0.874985i	$-0.660871\pi$
$61$	−28140.7			−0.968301			−0.484151	+	0.874985i	$0.660871\pi$
$62$	−24473.7	+	24473.7i	−0.808575	+	0.808575i
$63$		0			0
$64$			4096.00i			0.125000i
$65$		0			0
$66$		0			0
$67$	−24197.7	−	24197.7i	−0.658547	−	0.658547i	0.296489	−	0.955036i	$-0.404184\pi$
$67$	−24197.7	−	24197.7i	−0.658547	−	0.658547i	−0.955036	+	0.296489i	$0.904184\pi$
$68$	−24355.3	−	24355.3i	−0.638735	−	0.638735i
$69$		0			0
$70$		0			0
$71$			6518.47i			0.153462i	0.997052	+	0.0767308i	$0.0244482\pi$
$71$			6518.47i			0.153462i	−0.997052	+	0.0767308i	$0.975552\pi$
$72$		0			0
$73$	61503.1	−	61503.1i	1.35080	−	1.35080i	0.466025	−	0.884772i	$-0.345686\pi$
$73$	61503.1	−	61503.1i	1.35080	−	1.35080i	0.884772	−	0.466025i	$-0.154314\pi$
$74$	30423.1			0.645840
$75$		0			0
$76$	−27971.0			−0.555488
$77$	−44547.6	+	44547.6i	−0.856243	+	0.856243i
$78$		0			0
$79$			42431.1i			0.764921i	0.923972	+	0.382460i	$0.124923\pi$
$79$			42431.1i			0.764921i	−0.923972	+	0.382460i	$0.875077\pi$
$80$		0			0
$81$		0			0
$82$	4487.47	+	4487.47i	0.0737000	+	0.0737000i
$83$	−33805.9	−	33805.9i	−0.538638	−	0.538638i	0.384491	−	0.923129i	$-0.374377\pi$
$83$	−33805.9	−	33805.9i	−0.538638	−	0.538638i	−0.923129	+	0.384491i	$0.874377\pi$
$84$		0			0
$85$		0			0
$86$			11345.1i			0.165410i
$87$		0			0
$88$	−13555.1	+	13555.1i	−0.186593	+	0.186593i
$89$	−63913.9			−0.855304			−0.427652	−	0.903944i	$-0.640659\pi$
$89$	−63913.9			−0.855304			−0.427652	+	0.903944i	$0.640659\pi$
$90$		0			0
$91$	−57486.2			−0.727713
$92$	40628.3	−	40628.3i	0.500448	−	0.500448i
$93$		0			0
$94$			62301.7i			0.727245i
$95$		0			0
$96$		0			0
$97$	58905.9	+	58905.9i	0.635667	+	0.635667i	0.949484	−	0.313817i	$-0.101608\pi$
$97$	58905.9	+	58905.9i	0.635667	+	0.635667i	−0.313817	+	0.949484i	$0.601608\pi$
$98$	77589.2	+	77589.2i	0.816086	+	0.816086i
$99$		0			0
$100$		0			0
$101$		−	140291.i		−	1.36844i	−0.729276	−	0.684220i	$-0.760143\pi$
$101$		−	140291.i		−	1.36844i	0.729276	−	0.684220i	$-0.239857\pi$
$102$		0			0
$103$	−77193.5	+	77193.5i	−0.716948	+	0.716948i	−0.967979	−	0.251031i	$-0.919231\pi$
$103$	−77193.5	+	77193.5i	−0.716948	+	0.716948i	0.251031	+	0.967979i	$0.419231\pi$
$104$	−17492.1			−0.158584
$105$		0			0
$106$	−55885.3			−0.483096
$107$	−65849.8	+	65849.8i	−0.556026	+	0.556026i	−0.928174	−	0.372148i	$-0.878622\pi$
$107$	−65849.8	+	65849.8i	−0.556026	+	0.556026i	0.372148	+	0.928174i	$0.378622\pi$
$108$		0			0
$109$		−	27880.3i		−	0.224766i	−0.993665	−	0.112383i	$-0.964152\pi$
$109$		−	27880.3i		−	0.224766i	0.993665	−	0.112383i	$-0.0358483\pi$
$110$		0			0
$111$		0			0
$112$	38073.9	+	38073.9i	0.286802	+	0.286802i
$113$	−52351.3	−	52351.3i	−0.385684	−	0.385684i	0.487461	−	0.873145i	$-0.337923\pi$
$113$	−52351.3	−	52351.3i	−0.385684	−	0.385684i	−0.873145	+	0.487461i	$0.837923\pi$
$114$		0			0
$115$		0			0
$116$			41878.9i			0.288969i
$117$		0			0
$118$	−39406.5	+	39406.5i	−0.260533	+	0.260533i
$119$	−452784.			−2.93105
$120$		0			0
$121$	71334.3			0.442930
$122$	−79593.9	+	79593.9i	−0.484151	+	0.484151i
$123$		0			0
$124$			138444.i			0.808575i
$125$		0			0
$126$		0			0
$127$	185568.	+	185568.i	1.02092	+	1.02092i	0.999776	+	0.0211467i	$0.00673172\pi$
$127$	185568.	+	185568.i	1.02092	+	1.02092i	0.0211467	+	0.999776i	$0.493268\pi$
$128$	11585.2	+	11585.2i	0.0625000	+	0.0625000i
$129$		0			0
$130$		0			0
$131$			190849.i			0.971655i	0.874055	+	0.485827i	$0.161482\pi$
$131$			190849.i			0.971655i	−0.874055	+	0.485827i	$0.838518\pi$
$132$		0			0
$133$	−260001.	+	260001.i	−1.27452	+	1.27452i
$134$	−136883.			−0.658547
$135$		0			0
$136$	−137774.			−0.638735
$137$	127742.	−	127742.i	0.581475	−	0.581475i	−0.353834	−	0.935308i	$-0.615122\pi$
$137$	127742.	−	127742.i	0.581475	−	0.581475i	0.935308	+	0.353834i	$0.115122\pi$
$138$		0			0
$139$			217966.i			0.956866i	0.878124	+	0.478433i	$0.158795\pi$
$139$			217966.i			0.956866i	−0.878124	+	0.478433i	$0.841205\pi$
$140$		0			0
$141$		0			0
$142$	18437.0	+	18437.0i	0.0767308	+	0.0767308i
$143$	−57887.3	−	57887.3i	−0.236724	−	0.236724i
$144$		0			0
$145$		0			0
$146$		−	347914.i		−	1.35080i
$147$		0			0
$148$	86049.6	−	86049.6i	0.322920	−	0.322920i
$149$	461717.			1.70377			0.851883	−	0.523732i	$-0.175461\pi$
$149$	461717.			1.70377			0.851883	+	0.523732i	$0.175461\pi$
$150$		0			0
$151$	−2616.56			−0.00933874			−0.00466937	−	0.999989i	$-0.501486\pi$
$151$	−2616.56			−0.00933874			−0.00466937	+	0.999989i	$0.501486\pi$
$152$	−79114.0	+	79114.0i	−0.277744	+	0.277744i
$153$		0			0
$154$			251999.i			0.856243i
$155$		0			0
$156$		0			0
$157$	331986.	+	331986.i	1.07491	+	1.07491i	0.996957	+	0.0779479i	$0.0248368\pi$
$157$	331986.	+	331986.i	1.07491	+	1.07491i	0.0779479	+	0.996957i	$0.475163\pi$
$158$	120013.	+	120013.i	0.382460	+	0.382460i
$159$		0			0
$160$		0			0
$161$		−	755311.i		−	2.29647i
$162$		0			0
$163$	−15020.2	+	15020.2i	−0.0442800	+	0.0442800i	−0.728900	−	0.684620i	$-0.759968\pi$
$163$	−15020.2	+	15020.2i	−0.0442800	+	0.0442800i	0.684620	+	0.728900i	$0.259968\pi$
$164$	25385.0			0.0737000
$165$		0			0
$166$	−191235.			−0.538638
$167$	153251.	−	153251.i	0.425217	−	0.425217i	−0.461778	−	0.886995i	$-0.652788\pi$
$167$	153251.	−	153251.i	0.425217	−	0.425217i	0.886995	+	0.461778i	$0.152788\pi$
$168$		0			0
$169$			296593.i			0.798810i
$170$		0			0
$171$		0			0
$172$	32088.8	+	32088.8i	0.0827052	+	0.0827052i
$173$	−497806.	−	497806.i	−1.26458	−	1.26458i	−0.948851	−	0.315725i	$-0.897752\pi$
$173$	−497806.	−	497806.i	−1.26458	−	1.26458i	−0.315725	−	0.948851i	$-0.602248\pi$
$174$		0			0
$175$		0			0
$176$			76679.0i			0.186593i
$177$		0			0
$178$	−180776.	+	180776.i	−0.427652	+	0.427652i
$179$	−69083.5			−0.161154			−0.0805771	−	0.996748i	$-0.525676\pi$
$179$	−69083.5			−0.161154			−0.0805771	+	0.996748i	$0.525676\pi$
$180$		0			0
$181$	−798930.			−1.81264			−0.906321	−	0.422589i	$-0.861121\pi$
$181$	−798930.			−1.81264			−0.906321	+	0.422589i	$0.861121\pi$
$182$	−162596.	+	162596.i	−0.363857	+	0.363857i
$183$		0			0
$184$		−	229828.i		−	0.500448i
$185$		0			0
$186$		0			0
$187$	−455942.	−	455942.i	−0.953467	−	0.953467i
$188$	176216.	+	176216.i	0.363622	+	0.363622i
$189$		0			0
$190$		0			0
$191$			172241.i			0.341628i	0.985303	+	0.170814i	$0.0546397\pi$
$191$			172241.i			0.341628i	−0.985303	+	0.170814i	$0.945360\pi$
$192$		0			0
$193$	414251.	−	414251.i	0.800516	−	0.800516i	−0.182660	−	0.983176i	$-0.558471\pi$
$193$	414251.	−	414251.i	0.800516	−	0.800516i	0.983176	+	0.182660i	$0.0584708\pi$
$194$	333222.			0.635667
$195$		0			0
$196$	438911.			0.816086
$197$	−216065.	+	216065.i	−0.396660	+	0.396660i	−0.877053	−	0.480393i	$-0.840494\pi$
$197$	−216065.	+	216065.i	−0.396660	+	0.396660i	0.480393	+	0.877053i	$0.340494\pi$
$198$		0			0
$199$			684282.i			1.22491i	0.790507	+	0.612453i	$0.209817\pi$
$199$			684282.i			1.22491i	−0.790507	+	0.612453i	$0.790183\pi$
$200$		0			0
$201$		0			0
$202$	−396802.	−	396802.i	−0.684220	−	0.684220i
$203$	389281.	+	389281.i	0.663014	+	0.663014i
$204$		0			0
$205$		0			0
$206$			436673.i			0.716948i
$207$		0			0
$208$	−49475.1	+	49475.1i	−0.0792918	+	0.0792918i
$209$	−523631.			−0.829200
$210$		0			0
$211$	764382.			1.18196			0.590982	−	0.806685i	$-0.298740\pi$
$211$	764382.			1.18196			0.590982	+	0.806685i	$0.298740\pi$
$212$	−158068.	+	158068.i	−0.241548	+	0.241548i
$213$		0			0
$214$			372503.i			0.556026i
$215$		0			0
$216$		0			0
$217$	1.28689e6	+	1.28689e6i	1.85521	+	1.85521i
$218$	−78857.3	−	78857.3i	−0.112383	−	0.112383i
$219$		0			0
$220$		0			0
$221$		−	588369.i		−	0.810343i
$222$		0			0
$223$	74975.5	−	74975.5i	0.100962	−	0.100962i	−0.654822	−	0.755783i	$-0.727256\pi$
$223$	74975.5	−	74975.5i	0.100962	−	0.100962i	0.755783	+	0.654822i	$0.227256\pi$
$224$	215378.			0.286802
$225$		0			0
$226$	−296144.			−0.385684
$227$	691711.	−	691711.i	0.890963	−	0.890963i	−0.103650	−	0.994614i	$-0.533052\pi$
$227$	691711.	−	691711.i	0.890963	−	0.890963i	0.994614	+	0.103650i	$0.0330523\pi$
$228$		0			0
$229$		−	465990.i		−	0.587202i	−0.955928	−	0.293601i	$-0.905146\pi$
$229$		−	465990.i		−	0.587202i	0.955928	−	0.293601i	$-0.0948537\pi$
$230$		0			0
$231$		0			0
$232$	118451.	+	118451.i	0.144484	+	0.144484i
$233$	−118173.	−	118173.i	−0.142603	−	0.142603i	0.632201	−	0.774804i	$-0.282152\pi$
$233$	−118173.	−	118173.i	−0.142603	−	0.142603i	−0.774804	+	0.632201i	$0.782152\pi$
$234$		0			0
$235$		0			0
$236$			222917.i			0.260533i
$237$		0			0
$238$	−1.28067e6	+	1.28067e6i	−1.46552	+	1.46552i
$239$	−1.50955e6			−1.70943			−0.854715	−	0.519097i	$-0.826268\pi$
$239$	−1.50955e6			−1.70943			−0.854715	+	0.519097i	$0.826268\pi$
$240$		0			0
$241$	1.19777e6			1.32840			0.664200	−	0.747555i	$-0.268772\pi$
$241$	1.19777e6			1.32840			0.664200	+	0.747555i	$0.268772\pi$
$242$	201764.	−	201764.i	0.221465	−	0.221465i
$243$		0			0
$244$			450251.i			0.484151i
$245$		0			0
$246$		0			0
$247$	−337859.	−	337859.i	−0.352365	−	0.352365i
$248$	391579.	+	391579.i	0.404287	+	0.404287i
$249$		0			0
$250$		0			0
$251$			314772.i			0.315364i	0.987490	+	0.157682i	$0.0504021\pi$
$251$			314772.i			0.315364i	−0.987490	+	0.157682i	$0.949598\pi$
$252$		0			0
$253$	760581.	−	760581.i	0.747040	−	0.747040i
$254$	1.04973e6			1.02092
$255$		0			0
$256$	65536.0			0.0625000
$257$	−605352.	+	605352.i	−0.571709	+	0.571709i	−0.932606	−	0.360897i	$-0.882471\pi$
$257$	−605352.	+	605352.i	−0.571709	+	0.571709i	0.360897	+	0.932606i	$0.382471\pi$
$258$		0			0
$259$		−	1.59973e6i		−	1.48183i
$260$		0			0
$261$		0			0
$262$	539803.	+	539803.i	0.485827	+	0.485827i
$263$	−463557.	−	463557.i	−0.413251	−	0.413251i	0.469618	−	0.882870i	$-0.344392\pi$
$263$	−463557.	−	463557.i	−0.413251	−	0.413251i	−0.882870	+	0.469618i	$0.844392\pi$
$264$		0			0
$265$		0			0
$266$			1.47079e6i			1.27452i
$267$		0			0
$268$	−387163.	+	387163.i	−0.329273	+	0.329273i
$269$	−1.07602e6			−0.906646			−0.453323	−	0.891346i	$-0.649762\pi$
$269$	−1.07602e6			−0.906646			−0.453323	+	0.891346i	$0.649762\pi$
$270$		0			0
$271$	−1.74821e6			−1.44600			−0.723002	−	0.690846i	$-0.757238\pi$
$271$	−1.74821e6			−1.44600			−0.723002	+	0.690846i	$0.757238\pi$
$272$	−389685.	+	389685.i	−0.319368	+	0.319368i
$273$		0			0
$274$		−	722616.i		−	0.581475i
$275$		0			0
$276$		0			0
$277$	−720546.	−	720546.i	−0.564238	−	0.564238i	0.366271	−	0.930508i	$-0.380634\pi$
$277$	−720546.	−	720546.i	−0.564238	−	0.564238i	−0.930508	+	0.366271i	$0.880634\pi$
$278$	616500.	+	616500.i	0.478433	+	0.478433i
$279$		0			0
$280$		0			0
$281$			273453.i			0.206593i	0.994651	+	0.103297i	$0.0329391\pi$
$281$			273453.i			0.206593i	−0.994651	+	0.103297i	$0.967061\pi$
$282$		0			0
$283$	1.10987e6	−	1.10987e6i	0.823769	−	0.823769i	−0.162877	−	0.986646i	$-0.552077\pi$
$283$	1.10987e6	−	1.10987e6i	0.823769	−	0.823769i	0.986646	+	0.162877i	$0.0520774\pi$
$284$	104296.			0.0767308
$285$		0			0
$286$	−327460.			−0.236724
$287$	235963.	−	235963.i	0.169098	−	0.169098i
$288$		0			0
$289$		−	3.21436e6i		−	2.26386i
$290$		0			0
$291$		0			0
$292$	−984050.	−	984050.i	−0.675398	−	0.675398i
$293$	−182132.	−	182132.i	−0.123941	−	0.123941i	0.642415	−	0.766357i	$-0.277932\pi$
$293$	−182132.	−	182132.i	−0.123941	−	0.123941i	−0.766357	+	0.642415i	$0.777932\pi$
$294$		0			0
$295$		0			0
$296$		−	486770.i		−	0.322920i
$297$		0			0
$298$	1.30593e6	−	1.30593e6i	0.851883	−	0.851883i
$299$	981489.			0.634903
$300$		0			0
$301$	596556.			0.379520
$302$	−7400.75	+	7400.75i	−0.00466937	+	0.00466937i
$303$		0			0
$304$			447537.i			0.277744i
$305$		0			0
$306$		0			0
$307$	−2.01256e6	−	2.01256e6i	−1.21872	−	1.21872i	−0.968081	−	0.250638i	$-0.919360\pi$
$307$	−2.01256e6	−	2.01256e6i	−1.21872	−	1.21872i	−0.250638	−	0.968081i	$-0.580640\pi$
$308$	712761.	+	712761.i	0.428122	+	0.428122i
$309$		0			0
$310$		0			0
$311$		−	1.05508e6i		−	0.618563i	−0.950970	−	0.309282i	$-0.899911\pi$
$311$		−	1.05508e6i		−	0.618563i	0.950970	−	0.309282i	$-0.100089\pi$
$312$		0			0
$313$	−228345.	+	228345.i	−0.131744	+	0.131744i	−0.769904	−	0.638160i	$-0.779696\pi$
$313$	−228345.	+	228345.i	−0.131744	+	0.131744i	0.638160	+	0.769904i	$0.279696\pi$
$314$	1.87799e6			1.07491
$315$		0			0
$316$	678897.			0.382460
$317$	1.58626e6	−	1.58626e6i	0.886599	−	0.886599i	−0.107596	−	0.994195i	$-0.534315\pi$
$317$	1.58626e6	−	1.58626e6i	0.886599	−	0.886599i	0.994195	+	0.107596i	$0.0343153\pi$
$318$		0			0
$319$			783993.i			0.431356i
$320$		0			0
$321$		0			0
$322$	−2.13634e6	−	2.13634e6i	−1.14824	−	1.14824i
$323$	−2.66110e6	−	2.66110e6i	−1.41924	−	1.41924i
$324$		0			0
$325$		0			0
$326$			84967.3i			0.0442800i
$327$		0			0
$328$	71799.6	−	71799.6i	0.0368500	−	0.0368500i
$329$	3.27599e6			1.66860
$330$		0			0
$331$	2.34167e6			1.17478			0.587390	−	0.809304i	$-0.300155\pi$
$331$	2.34167e6			1.17478			0.587390	+	0.809304i	$0.300155\pi$
$332$	−540894.	+	540894.i	−0.269319	+	0.269319i
$333$		0			0
$334$		−	866916.i		−	0.425217i
$335$		0			0
$336$		0			0
$337$	−554612.	−	554612.i	−0.266020	−	0.266020i	0.561474	−	0.827494i	$-0.310234\pi$
$337$	−554612.	−	554612.i	−0.266020	−	0.266020i	−0.827494	+	0.561474i	$0.810234\pi$
$338$	838891.	+	838891.i	0.399405	+	0.399405i
$339$		0			0
$340$		0			0
$341$			2.59174e6i			1.20699i
$342$		0			0
$343$	1.58020e6	−	1.58020e6i	0.725234	−	0.725234i
$344$	181522.			0.0827052
$345$		0			0
$346$	−2.81602e6			−1.26458
$347$	468716.	−	468716.i	0.208971	−	0.208971i	−0.594859	−	0.803830i	$-0.702792\pi$
$347$	468716.	−	468716.i	0.208971	−	0.208971i	0.803830	+	0.594859i	$0.202792\pi$
$348$		0			0
$349$			445258.i			0.195681i	0.995202	+	0.0978404i	$0.0311935\pi$
$349$			445258.i			0.195681i	−0.995202	+	0.0978404i	$0.968807\pi$
$350$		0			0
$351$		0			0
$352$	216881.	+	216881.i	0.0932964	+	0.0932964i
$353$	597724.	+	597724.i	0.255308	+	0.255308i	0.823142	−	0.567835i	$-0.192219\pi$
$353$	597724.	+	597724.i	0.255308	+	0.255308i	−0.567835	+	0.823142i	$0.692219\pi$
$354$		0			0
$355$		0			0
$356$			1.02262e6i			0.427652i
$357$		0			0
$358$	−195398.	+	195398.i	−0.0805771	+	0.0805771i
$359$	2.56995e6			1.05242			0.526208	−	0.850356i	$-0.323613\pi$
$359$	2.56995e6			1.05242			0.526208	+	0.850356i	$0.323613\pi$
$360$		0			0
$361$	−580068.			−0.234267
$362$	−2.25971e6	+	2.25971e6i	−0.906321	+	0.906321i
$363$		0			0
$364$			919780.i			0.363857i
$365$		0			0
$366$		0			0
$367$	834880.	+	834880.i	0.323563	+	0.323563i	0.850132	−	0.526569i	$-0.176522\pi$
$367$	834880.	+	834880.i	0.323563	+	0.323563i	−0.526569	+	0.850132i	$0.676522\pi$
$368$	−650053.	−	650053.i	−0.250224	−	0.250224i
$369$		0			0
$370$		0			0
$371$			2.93860e6i			1.10842i
$372$		0			0
$373$	−680958.	+	680958.i	−0.253424	+	0.253424i	−0.822373	−	0.568949i	$-0.807350\pi$
$373$	−680958.	+	680958.i	−0.253424	+	0.253424i	0.568949	+	0.822373i	$0.307350\pi$
$374$	−2.57920e6			−0.953467
$375$		0			0
$376$	996828.			0.363622
$377$	−505850.	+	505850.i	−0.183303	+	0.183303i
$378$		0			0
$379$		−	4.63614e6i		−	1.65790i	−0.559322	−	0.828951i	$-0.688938\pi$
$379$		−	4.63614e6i		−	1.65790i	0.559322	−	0.828951i	$-0.311062\pi$
$380$		0			0
$381$		0			0
$382$	487171.	+	487171.i	0.170814	+	0.170814i
$383$	−492138.	−	492138.i	−0.171431	−	0.171431i	0.616177	−	0.787608i	$-0.288681\pi$
$383$	−492138.	−	492138.i	−0.171431	−	0.171431i	−0.787608	+	0.616177i	$0.788681\pi$
$384$		0			0
$385$		0			0
$386$		−	2.34336e6i		−	0.800516i
$387$		0			0
$388$	942495.	−	942495.i	0.317833	−	0.317833i
$389$	1.09867e6			0.368123			0.184061	−	0.982915i	$-0.441075\pi$
$389$	1.09867e6			0.368123			0.184061	+	0.982915i	$0.441075\pi$
$390$		0			0
$391$	7.73058e6			2.55723
$392$	1.24143e6	−	1.24143e6i	0.408043	−	0.408043i
$393$		0			0
$394$			1.22225e6i			0.396660i
$395$		0			0
$396$		0			0
$397$	1.07385e6	+	1.07385e6i	0.341955	+	0.341955i	0.857102	−	0.515147i	$-0.172263\pi$
$397$	1.07385e6	+	1.07385e6i	0.341955	+	0.341955i	−0.515147	+	0.857102i	$0.672263\pi$
$398$	1.93544e6	+	1.93544e6i	0.612453	+	0.612453i
$399$		0			0
$400$		0			0
$401$			1.01424e6i			0.314978i	0.987521	+	0.157489i	$0.0503398\pi$
$401$			1.01424e6i			0.314978i	−0.987521	+	0.157489i	$0.949660\pi$
$402$		0			0
$403$	−1.67225e6	+	1.67225e6i	−0.512907	+	0.512907i
$404$	−2.24465e6			−0.684220
$405$		0			0
$406$	2.20210e6			0.663014
$407$	1.61089e6	−	1.61089e6i	0.482036	−	0.482036i
$408$		0			0
$409$		−	4.68373e6i		−	1.38447i	−0.721673	−	0.692235i	$-0.756626\pi$
$409$		−	4.68373e6i		−	1.38447i	0.721673	−	0.692235i	$-0.243374\pi$
$410$		0			0
$411$		0			0
$412$	1.23510e6	+	1.23510e6i	0.358474	+	0.358474i
$413$	2.07210e6	+	2.07210e6i	0.597771	+	0.597771i
$414$		0			0
$415$		0			0
$416$			279873.i			0.0792918i
$417$		0			0
$418$	−1.48105e6	+	1.48105e6i	−0.414600	+	0.414600i
$419$	−3.35986e6			−0.934946			−0.467473	−	0.884007i	$-0.654836\pi$
$419$	−3.35986e6			−0.934946			−0.467473	+	0.884007i	$0.654836\pi$
$420$		0			0
$421$	−3.24290e6			−0.891719			−0.445859	−	0.895103i	$-0.647102\pi$
$421$	−3.24290e6			−0.891719			−0.445859	+	0.895103i	$0.647102\pi$
$422$	2.16200e6	−	2.16200e6i	0.590982	−	0.590982i
$423$		0			0
$424$			894165.i			0.241548i
$425$		0			0
$426$		0			0
$427$	4.18526e6	+	4.18526e6i	1.11084	+	1.11084i
$428$	1.05360e6	+	1.05360e6i	0.278013	+	0.278013i
$429$		0			0
$430$		0			0
$431$		−	6.54531e6i		−	1.69722i	−0.529022	−	0.848608i	$-0.677441\pi$
$431$		−	6.54531e6i		−	1.69722i	0.529022	−	0.848608i	$-0.322559\pi$
$432$		0			0
$433$	229102.	−	229102.i	0.0587232	−	0.0587232i	−0.677135	−	0.735859i	$-0.736779\pi$
$433$	229102.	−	229102.i	0.0587232	−	0.0587232i	0.735859	+	0.677135i	$0.236779\pi$
$434$	7.27975e6			1.85521
$435$		0			0
$436$	−446084.			−0.112383
$437$	4.43912e6	−	4.43912e6i	1.11197	−	1.11197i
$438$		0			0
$439$			1.97352e6i			0.488744i	0.969682	+	0.244372i	$0.0785817\pi$
$439$			1.97352e6i			0.488744i	−0.969682	+	0.244372i	$0.921418\pi$
$440$		0			0
$441$		0			0
$442$	−1.66416e6	−	1.66416e6i	−0.405172	−	0.405172i
$443$	2.63389e6	+	2.63389e6i	0.637660	+	0.637660i	0.949978	−	0.312318i	$-0.101105\pi$
$443$	2.63389e6	+	2.63389e6i	0.637660	+	0.637660i	−0.312318	+	0.949978i	$0.601105\pi$
$444$		0			0
$445$		0			0
$446$		−	424126.i		−	0.100962i
$447$		0			0
$448$	609182.	−	609182.i	0.143401	−	0.143401i
$449$	−7.82998e6			−1.83293			−0.916463	−	0.400120i	$-0.868968\pi$
$449$	−7.82998e6			−1.83293			−0.916463	+	0.400120i	$0.868968\pi$
$450$		0			0
$451$	475219.			0.110015
$452$	−837621.	+	837621.i	−0.192842	+	0.192842i
$453$		0			0
$454$		−	3.91291e6i		−	0.890963i
$455$		0			0
$456$		0			0
$457$	−5.32966e6	−	5.32966e6i	−1.19374	−	1.19374i	−0.976009	−	0.217730i	$-0.930135\pi$
$457$	−5.32966e6	−	5.32966e6i	−1.19374	−	1.19374i	−0.217730	−	0.976009i	$-0.569865\pi$
$458$	−1.31802e6	−	1.31802e6i	−0.293601	−	0.293601i
$459$		0			0
$460$		0			0
$461$		−	1.29678e6i		−	0.284192i	−0.989853	−	0.142096i	$-0.954616\pi$
$461$		−	1.29678e6i		−	0.284192i	0.989853	−	0.142096i	$-0.0453842\pi$
$462$		0			0
$463$	5.30186e6	−	5.30186e6i	1.14941	−	1.14941i	0.162745	−	0.986668i	$-0.447965\pi$
$463$	5.30186e6	−	5.30186e6i	1.14941	−	1.14941i	0.986668	−	0.162745i	$-0.0520347\pi$
$464$	670063.			0.144484
$465$		0			0
$466$	−668488.			−0.142603
$467$	−885971.	+	885971.i	−0.187987	+	0.187987i	−0.794825	−	0.606838i	$-0.792438\pi$
$467$	−885971.	+	885971.i	−0.187987	+	0.187987i	0.606838	+	0.794825i	$0.292438\pi$
$468$		0			0
$469$			7.19765e6i			1.51098i
$470$		0			0
$471$		0			0
$472$	630504.	+	630504.i	0.130266	+	0.130266i
$473$	600718.	+	600718.i	0.123458	+	0.123458i
$474$		0			0
$475$		0			0
$476$			7.24454e6i			1.46552i
$477$		0			0
$478$	−4.26964e6	+	4.26964e6i	−0.854715	+	0.854715i
$479$	1.43334e6			0.285438			0.142719	−	0.989763i	$-0.454416\pi$
$479$	1.43334e6			0.285438			0.142719	+	0.989763i	$0.454416\pi$
$480$		0			0
$481$	2.07877e6			0.409678
$482$	3.38779e6	−	3.38779e6i	0.664200	−	0.664200i
$483$		0			0
$484$		−	1.14135e6i		−	0.221465i
$485$		0			0
$486$		0			0
$487$	−994929.	−	994929.i	−0.190094	−	0.190094i	0.605642	−	0.795737i	$-0.292916\pi$
$487$	−994929.	−	994929.i	−0.190094	−	0.190094i	−0.795737	+	0.605642i	$0.792916\pi$
$488$	1.27350e6	+	1.27350e6i	0.242075	+	0.242075i
$489$		0			0
$490$		0			0
$491$		−	709316.i		−	0.132781i	−0.997794	−	0.0663905i	$-0.978852\pi$
$491$		−	709316.i		−	0.132781i	0.997794	−	0.0663905i	$-0.0211483\pi$
$492$		0			0
$493$	−3.98427e6	+	3.98427e6i	−0.738298	+	0.738298i
$494$	−1.91122e6			−0.352365
$495$		0			0
$496$	2.21511e6			0.404287
$497$	969467.	−	969467.i	0.176052	−	0.176052i
$498$		0			0
$499$			2.02748e6i			0.364507i	0.983252	+	0.182253i	$0.0583391\pi$
$499$			2.02748e6i			0.364507i	−0.983252	+	0.182253i	$0.941661\pi$
$500$		0			0
$501$		0			0
$502$	890311.	+	890311.i	0.157682	+	0.157682i
$503$	−696310.	−	696310.i	−0.122711	−	0.122711i	0.643084	−	0.765795i	$-0.277654\pi$
$503$	−696310.	−	696310.i	−0.122711	−	0.122711i	−0.765795	+	0.643084i	$0.777654\pi$
$504$		0			0
$505$		0			0
$506$		−	4.30249e6i		−	0.747040i
$507$		0			0
$508$	2.96908e6	−	2.96908e6i	0.510462	−	0.510462i
$509$	4.63077e6			0.792244			0.396122	−	0.918198i	$-0.370356\pi$
$509$	4.63077e6			0.792244			0.396122	+	0.918198i	$0.370356\pi$
$510$		0			0
$511$	−1.82942e7			−3.09929
$512$	185364.	−	185364.i	0.0312500	−	0.0312500i
$513$		0			0
$514$			3.42439e6i			0.571709i
$515$		0			0
$516$		0			0
$517$	3.29884e6	+	3.29884e6i	0.542794	+	0.542794i
$518$	−4.52472e6	−	4.52472e6i	−0.740913	−	0.740913i
$519$		0			0
$520$		0			0
$521$		−	1.05159e7i		−	1.69728i	−0.528971	−	0.848640i	$-0.677422\pi$
$521$		−	1.05159e7i		−	1.69728i	0.528971	−	0.848640i	$-0.322578\pi$
$522$		0			0
$523$	−4.51262e6	+	4.51262e6i	−0.721397	+	0.721397i	−0.968890	−	0.247493i	$-0.920393\pi$
$523$	−4.51262e6	+	4.51262e6i	−0.721397	+	0.721397i	0.247493	+	0.968890i	$0.420393\pi$
$524$	3.05359e6			0.485827
$525$		0			0
$526$	−2.62228e6			−0.413251
$527$	−1.31713e7	+	1.31713e7i	−2.06586	+	2.06586i
$528$		0			0
$529$			6.45944e6i			1.00359i
$530$		0			0
$531$		0			0
$532$	4.16002e6	+	4.16002e6i	0.637260	+	0.637260i
$533$	306622.	+	306622.i	0.0467504	+	0.0467504i
$534$		0			0
$535$		0			0
$536$			2.19012e6i			0.329273i
$537$		0			0
$538$	−3.04343e6	+	3.04343e6i	−0.453323	+	0.453323i
$539$	8.21661e6			1.21821
$540$		0			0
$541$	1.01225e7			1.48695			0.743475	−	0.668764i	$-0.233176\pi$
$541$	1.01225e7			1.48695			0.743475	+	0.668764i	$0.233176\pi$
$542$	−4.94467e6	+	4.94467e6i	−0.723002	+	0.723002i
$543$		0			0
$544$			2.20439e6i			0.319368i
$545$		0			0
$546$		0			0
$547$	4.06823e6	+	4.06823e6i	0.581349	+	0.581349i	0.935274	−	0.353925i	$-0.115153\pi$
$547$	4.06823e6	+	4.06823e6i	0.581349	+	0.581349i	−0.353925	+	0.935274i	$0.615153\pi$
$548$	−2.04387e6	−	2.04387e6i	−0.290737	−	0.290737i
$549$		0			0
$550$		0			0
$551$			4.57577e6i			0.642074i
$552$		0			0
$553$	6.31061e6	−	6.31061e6i	0.877523	−	0.877523i
$554$	−4.07602e6			−0.564238
$555$		0			0
$556$	3.48745e6			0.478433
$557$	910288.	−	910288.i	0.124320	−	0.124320i	−0.642209	−	0.766529i	$-0.721982\pi$
$557$	910288.	−	910288.i	0.124320	−	0.124320i	0.766529	+	0.642209i	$0.221982\pi$
$558$		0			0
$559$			775194.i			0.104925i
$560$		0			0
$561$		0			0
$562$	773441.	+	773441.i	0.103297	+	0.103297i
$563$	−8.14509e6	−	8.14509e6i	−1.08299	−	1.08299i	−0.996229	−	0.0867627i	$-0.972348\pi$
$563$	−8.14509e6	−	8.14509e6i	−1.08299	−	1.08299i	−0.0867627	−	0.996229i	$-0.527652\pi$
$564$		0			0
$565$		0			0
$566$		−	6.27837e6i		−	0.823769i
$567$		0			0
$568$	294992.	−	294992.i	0.0383654	−	0.0383654i
$569$	−1.33483e7			−1.72840			−0.864200	−	0.503149i	$-0.832175\pi$
$569$	−1.33483e7			−1.72840			−0.864200	+	0.503149i	$0.832175\pi$
$570$		0			0
$571$	−1.21487e7			−1.55934			−0.779669	−	0.626192i	$-0.784612\pi$
$571$	−1.21487e7			−1.55934			−0.779669	+	0.626192i	$0.784612\pi$
$572$	−926196.	+	926196.i	−0.118362	+	0.118362i
$573$		0			0
$574$		−	1.33481e6i		−	0.169098i
$575$		0			0
$576$		0			0
$577$	−248053.	−	248053.i	−0.0310174	−	0.0310174i	0.691428	−	0.722445i	$-0.256982\pi$
$577$	−248053.	−	248053.i	−0.0310174	−	0.0310174i	−0.722445	+	0.691428i	$0.756982\pi$
$578$	−9.09159e6	−	9.09159e6i	−1.13193	−	1.13193i
$579$		0			0
$580$		0			0
$581$			1.00556e7i			1.23586i
$582$		0			0
$583$	−2.95910e6	+	2.95910e6i	−0.360569	+	0.360569i
$584$	−5.56663e6			−0.675398
$585$		0			0
$586$	−1.03029e6			−0.123941
$587$	−5.30886e6	+	5.30886e6i	−0.635925	+	0.635925i	−0.949548	−	0.313622i	$-0.898457\pi$
$587$	−5.30886e6	+	5.30886e6i	−0.635925	+	0.635925i	0.313622	+	0.949548i	$0.398457\pi$
$588$		0			0
$589$			1.51267e7i			1.79661i
$590$		0			0
$591$		0			0
$592$	−1.37679e6	−	1.37679e6i	−0.161460	−	0.161460i
$593$	−493935.	−	493935.i	−0.0576811	−	0.0576811i	0.677678	−	0.735359i	$-0.262986\pi$
$593$	−493935.	−	493935.i	−0.0576811	−	0.0576811i	−0.735359	+	0.677678i	$0.762986\pi$
$594$		0			0
$595$		0			0
$596$		−	7.38747e6i		−	0.851883i
$597$		0			0
$598$	2.77607e6	−	2.77607e6i	0.317451	−	0.317451i
$599$	−9.20782e6			−1.04855			−0.524276	−	0.851549i	$-0.675664\pi$
$599$	−9.20782e6			−1.04855			−0.524276	+	0.851549i	$0.675664\pi$
$600$		0			0
$601$	2.89317e6			0.326729			0.163364	−	0.986566i	$-0.447765\pi$
$601$	2.89317e6			0.326729			0.163364	+	0.986566i	$0.447765\pi$
$602$	1.68732e6	−	1.68732e6i	0.189760	−	0.189760i
$603$		0			0
$604$			41865.0i			0.00466937i
$605$		0			0
$606$		0			0
$607$	−1.30019e6	−	1.30019e6i	−0.143230	−	0.143230i	0.631856	−	0.775086i	$-0.282293\pi$
$607$	−1.30019e6	−	1.30019e6i	−0.143230	−	0.143230i	−0.775086	+	0.631856i	$0.782293\pi$
$608$	1.26582e6	+	1.26582e6i	0.138872	+	0.138872i
$609$		0			0
$610$		0			0
$611$			4.25698e6i			0.461316i
$612$		0			0
$613$	−8.21096e6	+	8.21096e6i	−0.882558	+	0.882558i	−0.993794	−	0.111236i	$-0.964519\pi$
$613$	−8.21096e6	+	8.21096e6i	−0.882558	+	0.882558i	0.111236	+	0.993794i	$0.464519\pi$
$614$	−1.13848e7			−1.21872
$615$		0			0
$616$	4.03198e6			0.428122
$617$	−4.68212e6	+	4.68212e6i	−0.495142	+	0.495142i	−0.909922	−	0.414780i	$-0.863859\pi$
$617$	−4.68212e6	+	4.68212e6i	−0.495142	+	0.495142i	0.414780	+	0.909922i	$0.363859\pi$
$618$		0			0
$619$			2.83268e6i			0.297147i	0.988901	+	0.148573i	$0.0474681\pi$
$619$			2.83268e6i			0.297147i	−0.988901	+	0.148573i	$0.952532\pi$
$620$		0			0
$621$		0			0
$622$	−2.98422e6	−	2.98422e6i	−0.309282	−	0.309282i
$623$	9.50567e6	+	9.50567e6i	0.981211	+	0.981211i
$624$		0			0
$625$		0			0
$626$			1.29171e6i			0.131744i
$627$		0			0
$628$	5.31177e6	−	5.31177e6i	0.537453	−	0.537453i
$629$	1.63732e7			1.65008
$630$		0			0
$631$	−8.82438e6			−0.882289			−0.441144	−	0.897436i	$-0.645427\pi$
$631$	−8.82438e6			−0.882289			−0.441144	+	0.897436i	$0.645427\pi$
$632$	1.92021e6	−	1.92021e6i	0.191230	−	0.191230i
$633$		0			0
$634$		−	8.97326e6i		−	0.886599i
$635$		0			0
$636$		0			0
$637$	5.30155e6	+	5.30155e6i	0.517671	+	0.517671i
$638$	2.21747e6	+	2.21747e6i	0.215678	+	0.215678i
$639$		0			0
$640$		0			0
$641$		−	2.02407e6i		−	0.194572i	−0.995256	−	0.0972859i	$-0.968984\pi$
$641$		−	2.02407e6i		−	0.194572i	0.995256	−	0.0972859i	$-0.0310161\pi$
$642$		0			0
$643$	−1.17738e7	+	1.17738e7i	−1.12303	+	1.12303i	−0.131744	+	0.991284i	$0.542058\pi$
$643$	−1.17738e7	+	1.17738e7i	−1.12303	+	1.12303i	−0.991284	+	0.131744i	$0.957942\pi$
$644$	−1.20850e7			−1.14824
$645$		0			0
$646$	−1.50535e7			−1.41924
$647$	3.10170e6	−	3.10170e6i	0.291299	−	0.291299i	−0.546294	−	0.837593i	$-0.683962\pi$
$647$	3.10170e6	−	3.10170e6i	0.291299	−	0.291299i	0.837593	+	0.546294i	$0.183962\pi$
$648$		0			0
$649$			4.17310e6i			0.388909i
$650$		0			0
$651$		0			0
$652$	240324.	+	240324.i	0.0221400	+	0.0221400i
$653$	1.37285e7	+	1.37285e7i	1.25991	+	1.25991i	0.951136	+	0.308773i	$0.0999184\pi$
$653$	1.37285e7	+	1.37285e7i	1.25991	+	1.25991i	0.308773	+	0.951136i	$0.400082\pi$
$654$		0			0
$655$		0			0
$656$		−	406160.i		−	0.0368500i
$657$		0			0
$658$	9.26590e6	−	9.26590e6i	0.834301	−	0.834301i
$659$	−8.37025e6			−0.750801			−0.375401	−	0.926863i	$-0.622495\pi$
$659$	−8.37025e6			−0.750801			−0.375401	+	0.926863i	$0.622495\pi$
$660$		0			0
$661$	−9.92386e6			−0.883440			−0.441720	−	0.897153i	$-0.645631\pi$
$661$	−9.92386e6			−0.883440			−0.441720	+	0.897153i	$0.645631\pi$
$662$	6.62326e6	−	6.62326e6i	0.587390	−	0.587390i
$663$		0			0
$664$			3.05976e6i			0.269319i
$665$		0			0
$666$		0			0
$667$	−6.64637e6	−	6.64637e6i	−0.578455	−	0.578455i
$668$	−2.45201e6	−	2.45201e6i	−0.212609	−	0.212609i
$669$		0			0
$670$		0			0
$671$			8.42891e6i			0.722712i
$672$		0			0
$673$	−3.57695e6	+	3.57695e6i	−0.304422	+	0.304422i	−0.842741	−	0.538319i	$-0.819059\pi$
$673$	−3.57695e6	+	3.57695e6i	−0.304422	+	0.304422i	0.538319	+	0.842741i	$0.319059\pi$
$674$	−3.13736e6			−0.266020
$675$		0			0
$676$	4.74548e6			0.399405
$677$	1.16921e7	−	1.16921e7i	0.980441	−	0.980441i	−0.0193711	−	0.999812i	$-0.506166\pi$
$677$	1.16921e7	−	1.16921e7i	0.980441	−	0.980441i	0.999812	+	0.0193711i	$0.00616640\pi$
$678$		0			0
$679$		−	1.75217e7i		−	1.45848i
$680$		0			0
$681$		0			0
$682$	7.33054e6	+	7.33054e6i	0.603497	+	0.603497i
$683$	−2.80028e6	−	2.80028e6i	−0.229694	−	0.229694i	0.582871	−	0.812565i	$-0.301929\pi$
$683$	−2.80028e6	−	2.80028e6i	−0.229694	−	0.229694i	−0.812565	+	0.582871i	$0.801929\pi$
$684$		0			0
$685$		0			0
$686$		−	8.93899e6i		−	0.725234i
$687$		0			0
$688$	513421.	−	513421.i	0.0413526	−	0.0413526i
$689$	−3.81856e6			−0.306444
$690$		0			0
$691$	1.64582e6			0.131125			0.0655626	−	0.997848i	$-0.479116\pi$
$691$	1.64582e6			0.131125			0.0655626	+	0.997848i	$0.479116\pi$
$692$	−7.96489e6	+	7.96489e6i	−0.632288	+	0.632288i
$693$		0			0
$694$		−	2.65146e6i		−	0.208971i
$695$		0			0
$696$		0			0
$697$	2.41507e6	+	2.41507e6i	0.188299	+	0.188299i
$698$	1.25938e6	+	1.25938e6i	0.0978404	+	0.0978404i
$699$		0			0
$700$		0			0
$701$		−	6.83500e6i		−	0.525344i	−0.964885	−	0.262672i	$-0.915396\pi$
$701$		−	6.83500e6i		−	0.525344i	0.964885	−	0.262672i	$-0.0846037\pi$
$702$		0			0
$703$	9.40194e6	−	9.40194e6i	0.717512	−	0.717512i
$704$	1.22686e6			0.0932964
$705$		0			0
$706$	3.38124e6			0.255308
$707$	−2.08649e7	+	2.08649e7i	−1.56989	+	1.56989i
$708$		0			0
$709$			1.15425e7i			0.862352i	0.902268	+	0.431176i	$0.141901\pi$
$709$			1.15425e7i			0.862352i	−0.902268	+	0.431176i	$0.858099\pi$
$710$		0			0
$711$		0			0
$712$	2.89241e6	+	2.89241e6i	0.213826	+	0.213826i
$713$	−2.19717e7	−	2.19717e7i	−1.61860	−	1.61860i
$714$		0			0
$715$		0			0
$716$			1.10534e6i			0.0805771i
$717$		0			0
$718$	7.26890e6	−	7.26890e6i	0.526208	−	0.526208i
$719$	−1.27402e6			−0.0919084			−0.0459542	−	0.998944i	$-0.514633\pi$
$719$	−1.27402e6			−0.0919084			−0.0459542	+	0.998944i	$0.514633\pi$
$720$		0			0
$721$	2.29614e7			1.64498
$722$	−1.64068e6	+	1.64068e6i	−0.117133	+	0.117133i
$723$		0			0
$724$			1.27829e7i			0.906321i
$725$		0			0
$726$		0			0
$727$	−5.16629e6	−	5.16629e6i	−0.362529	−	0.362529i	0.502214	−	0.864743i	$-0.332519\pi$
$727$	−5.16629e6	−	5.16629e6i	−0.362529	−	0.362529i	−0.864743	+	0.502214i	$0.832519\pi$
$728$	2.60153e6	+	2.60153e6i	0.181928	+	0.181928i
$729$		0			0
$730$		0			0
$731$			6.10572e6i			0.422614i
$732$		0			0
$733$	−5.10747e6	+	5.10747e6i	−0.351112	+	0.351112i	−0.860523	−	0.509411i	$-0.829863\pi$
$733$	−5.10747e6	+	5.10747e6i	−0.351112	+	0.351112i	0.509411	+	0.860523i	$0.329863\pi$
$734$	4.72280e6			0.323563
$735$		0			0
$736$	−3.67725e6			−0.250224
$737$	−7.24787e6	+	7.24787e6i	−0.491520	+	0.491520i
$738$		0			0
$739$		−	1.03581e7i		−	0.697698i	−0.937179	−	0.348849i	$-0.886573\pi$
$739$		−	1.03581e7i		−	0.697698i	0.937179	−	0.348849i	$-0.113427\pi$
$740$		0			0
$741$		0			0
$742$	8.31161e6	+	8.31161e6i	0.554211	+	0.554211i
$743$	1.53246e7	+	1.53246e7i	1.01840	+	1.01840i	0.999828	+	0.0185675i	$0.00591055\pi$
$743$	1.53246e7	+	1.53246e7i	1.01840	+	1.01840i	0.0185675	+	0.999828i	$0.494089\pi$
$744$		0			0
$745$		0			0
$746$			3.85208e6i			0.253424i
$747$		0			0
$748$	−7.29508e6	+	7.29508e6i	−0.476734	+	0.476734i
$749$	1.95872e7			1.27575
$750$		0			0
$751$	−5.50584e6			−0.356224			−0.178112	−	0.984010i	$-0.556999\pi$
$751$	−5.50584e6			−0.356224			−0.178112	+	0.984010i	$0.556999\pi$
$752$	2.81945e6	−	2.81945e6i	0.181811	−	0.181811i
$753$		0			0
$754$			2.86152e6i			0.183303i
$755$		0			0
$756$		0			0
$757$	−72342.5	−	72342.5i	−0.00458832	−	0.00458832i	0.704809	−	0.709397i	$-0.251033\pi$
$757$	−72342.5	−	72342.5i	−0.00458832	−	0.00458832i	−0.709397	+	0.704809i	$0.751033\pi$
$758$	−1.31130e7	−	1.31130e7i	−0.828951	−	0.828951i
$759$		0			0
$760$		0			0
$761$		−	2.39854e6i		−	0.150136i	−0.997178	−	0.0750680i	$-0.976083\pi$
$761$		−	2.39854e6i		−	0.150136i	0.997178	−	0.0750680i	$-0.0239174\pi$
$762$		0			0
$763$	−4.14652e6	+	4.14652e6i	−0.257853	+	0.257853i
$764$	2.75585e6			0.170814
$765$		0			0
$766$	−2.78396e6			−0.171431
$767$	−2.69258e6	+	2.69258e6i	−0.165265	+	0.165265i
$768$		0			0
$769$		−	3.62066e6i		−	0.220786i	−0.993888	−	0.110393i	$-0.964789\pi$
$769$		−	3.62066e6i		−	0.220786i	0.993888	−	0.110393i	$-0.0352110\pi$
$770$		0			0
$771$		0			0
$772$	−6.62801e6	−	6.62801e6i	−0.400258	−	0.400258i
$773$	1.86029e6	+	1.86029e6i	0.111978	+	0.111978i	0.760876	−	0.648898i	$-0.224770\pi$
$773$	1.86029e6	+	1.86029e6i	0.111978	+	0.111978i	−0.648898	+	0.760876i	$0.724770\pi$
$774$		0			0
$775$		0			0
$776$		−	5.33156e6i		−	0.317833i
$777$		0			0
$778$	3.10750e6	−	3.10750e6i	0.184061	−	0.184061i
$779$	2.77361e6			0.163758
$780$		0			0
$781$	1.95246e6			0.114539
$782$	2.18654e7	−	2.18654e7i	1.27862	−	1.27862i
$783$		0			0
$784$		−	7.02257e6i		−	0.408043i
$785$		0			0
$786$		0			0
$787$	−1.24400e7	−	1.24400e7i	−0.715951	−	0.715951i	0.251822	−	0.967774i	$-0.418970\pi$
$787$	−1.24400e7	−	1.24400e7i	−0.715951	−	0.715951i	−0.967774	+	0.251822i	$0.918970\pi$
$788$	3.45704e6	+	3.45704e6i	0.198330	+	0.198330i
$789$		0			0
$790$		0			0
$791$			1.55720e7i			0.884920i
$792$		0			0
$793$	−5.43853e6	+	5.43853e6i	−0.307113	+	0.307113i
$794$	6.07463e6			0.341955
$795$		0			0
$796$	1.09485e7			0.612453
$797$	9.06451e6	−	9.06451e6i	0.505473	−	0.505473i	−0.407660	−	0.913134i	$-0.633655\pi$
$797$	9.06451e6	−	9.06451e6i	0.505473	−	0.505473i	0.913134	+	0.407660i	$0.133655\pi$
$798$		0			0
$799$			3.35296e7i			1.85807i
$800$		0			0
$801$		0			0
$802$	2.86870e6	+	2.86870e6i	0.157489	+	0.157489i
$803$	−1.84219e7	−	1.84219e7i	−1.00820	−	1.00820i
$804$		0			0
$805$		0			0
$806$			9.45967e6i			0.512907i
$807$		0			0
$808$	−6.34884e6	+	6.34884e6i	−0.342110	+	0.342110i
$809$	2.71042e7			1.45601			0.728006	−	0.685571i	$-0.240447\pi$
$809$	2.71042e7			1.45601			0.728006	+	0.685571i	$0.240447\pi$
$810$		0			0
$811$	1.95248e7			1.04240			0.521199	−	0.853435i	$-0.325485\pi$
$811$	1.95248e7			1.04240			0.521199	+	0.853435i	$0.325485\pi$
$812$	6.22849e6	−	6.22849e6i	0.331507	−	0.331507i
$813$		0			0
$814$		−	9.11257e6i		−	0.482036i
$815$		0			0
$816$		0			0
$817$	3.50608e6	+	3.50608e6i	0.183767	+	0.183767i
$818$	−1.32476e7	−	1.32476e7i	−0.692235	−	0.692235i
$819$		0			0
$820$		0			0
$821$		−	3.24232e7i		−	1.67880i	−0.543517	−	0.839398i	$-0.682908\pi$
$821$		−	3.24232e7i		−	1.67880i	0.543517	−	0.839398i	$-0.317092\pi$
$822$		0			0
$823$	2.33214e7	−	2.33214e7i	1.20021	−	1.20021i	0.226101	−	0.974104i	$-0.427402\pi$
$823$	2.33214e7	−	2.33214e7i	1.20021	−	1.20021i	0.974104	−	0.226101i	$-0.0725981\pi$
$824$	6.98676e6			0.358474
$825$		0			0
$826$	1.17215e7			0.597771
$827$	2.67706e6	−	2.67706e6i	0.136111	−	0.136111i	−0.635769	−	0.771880i	$-0.719317\pi$
$827$	2.67706e6	−	2.67706e6i	0.136111	−	0.136111i	0.771880	+	0.635769i	$0.219317\pi$
$828$		0			0
$829$		−	2.75723e6i		−	0.139343i	−0.997570	−	0.0696716i	$-0.977805\pi$
$829$		−	2.75723e6i		−	0.139343i	0.997570	−	0.0696716i	$-0.0221952\pi$
$830$		0			0
$831$		0			0
$832$	791601.	+	791601.i	0.0396459	+	0.0396459i
$833$	4.17570e7	+	4.17570e7i	2.08505	+	2.08505i
$834$		0			0
$835$		0			0
$836$			8.37809e6i			0.414600i
$837$		0			0
$838$	−9.50313e6	+	9.50313e6i	−0.467473	+	0.467473i
$839$	−1.04487e7			−0.512458			−0.256229	−	0.966616i	$-0.582480\pi$
$839$	−1.04487e7			−0.512458			−0.256229	+	0.966616i	$0.582480\pi$
$840$		0			0
$841$	−1.36602e7			−0.665989
$842$	−9.17230e6	+	9.17230e6i	−0.445859	+	0.445859i
$843$		0			0
$844$		−	1.22301e7i		−	0.590982i
$845$		0			0
$846$		0			0
$847$	−1.06093e7	−	1.06093e7i	−0.508133	−	0.508133i
$848$	2.52908e6	+	2.52908e6i	0.120774	+	0.120774i
$849$		0			0
$850$		0			0
$851$			2.73129e7i			1.29284i
$852$		0			0
$853$	3.15656e6	−	3.15656e6i	0.148539	−	0.148539i	−0.628926	−	0.777465i	$-0.716505\pi$
$853$	3.15656e6	−	3.15656e6i	0.148539	−	0.148539i	0.777465	+	0.628926i	$0.216505\pi$
$854$	2.36754e7			1.11084
$855$		0			0
$856$	5.96004e6			0.278013
$857$	1.85837e7	−	1.85837e7i	0.864331	−	0.864331i	−0.127507	−	0.991838i	$-0.540697\pi$
$857$	1.85837e7	−	1.85837e7i	0.864331	−	0.864331i	0.991838	+	0.127507i	$0.0406975\pi$
$858$		0			0
$859$		−	2.32905e7i		−	1.07695i	−0.842641	−	0.538475i	$-0.819001\pi$
$859$		−	2.32905e7i		−	1.07695i	0.842641	−	0.538475i	$-0.180999\pi$
$860$		0			0
$861$		0			0
$862$	−1.85129e7	−	1.85129e7i	−0.848608	−	0.848608i
$863$	5.36715e6	+	5.36715e6i	0.245311	+	0.245311i	0.819043	−	0.573732i	$-0.194505\pi$
$863$	5.36715e6	+	5.36715e6i	0.245311	+	0.245311i	−0.573732	+	0.819043i	$0.694505\pi$
$864$		0			0
$865$		0			0
$866$		−	1.29600e6i		−	0.0587232i
$867$		0			0
$868$	2.05903e7	−	2.05903e7i	0.927604	−	0.927604i
$869$	1.27093e7			0.570915
$870$		0			0
$871$	−9.35298e6			−0.417739
$872$	−1.26172e6	+	1.26172e6i	−0.0561915	+	0.0561915i
$873$		0			0
$874$		−	2.51115e7i		−	1.11197i
$875$		0			0
$876$		0			0
$877$	1.49483e7	+	1.49483e7i	0.656285	+	0.656285i	0.954499	−	0.298214i	$-0.0963909\pi$
$877$	1.49483e7	+	1.49483e7i	0.656285	+	0.656285i	−0.298214	+	0.954499i	$0.596391\pi$
$878$	5.58197e6	+	5.58197e6i	0.244372	+	0.244372i
$879$		0			0
$880$		0			0
$881$		−	3.98161e7i		−	1.72830i	−0.503234	−	0.864150i	$-0.667857\pi$
$881$		−	3.98161e7i		−	1.72830i	0.503234	−	0.864150i	$-0.332143\pi$
$882$		0			0
$883$	8.44942e6	−	8.44942e6i	0.364691	−	0.364691i	−0.500846	−	0.865537i	$-0.666978\pi$
$883$	8.44942e6	−	8.44942e6i	0.364691	−	0.364691i	0.865537	+	0.500846i	$0.166978\pi$
$884$	−9.41390e6			−0.405172
$885$		0			0
$886$	1.48996e7			0.637660
$887$	1.01948e7	−	1.01948e7i	0.435081	−	0.435081i	−0.455272	−	0.890353i	$-0.650458\pi$
$887$	1.01948e7	−	1.01948e7i	0.435081	−	0.435081i	0.890353	+	0.455272i	$0.150458\pi$
$888$		0			0
$889$		−	5.51975e7i		−	2.34242i
$890$		0			0
$891$		0			0
$892$	−1.19961e6	−	1.19961e6i	−0.0504809	−	0.0504809i
$893$	1.92537e7	+	1.92537e7i	0.807951	+	0.807951i
$894$		0			0
$895$		0			0
$896$		−	3.44606e6i		−	0.143401i
$897$		0			0
$898$	−2.21465e7	+	2.21465e7i	−0.916463	+	0.916463i
$899$	2.26480e7			0.934611
$900$		0			0
$901$	−3.00764e7			−1.23428
$902$	1.34412e6	−	1.34412e6i	0.0550075	−	0.0550075i
$903$		0			0
$904$			4.73830e6i			0.192842i
$905$		0			0
$906$		0			0
$907$	−2.30348e7	−	2.30348e7i	−0.929749	−	0.929749i	0.0679406	−	0.997689i	$-0.478357\pi$
$907$	−2.30348e7	−	2.30348e7i	−0.929749	−	0.929749i	−0.997689	+	0.0679406i	$0.978357\pi$
$908$	−1.10674e7	−	1.10674e7i	−0.445482	−	0.445482i
$909$		0			0
$910$		0			0
$911$		−	7.74315e6i		−	0.309116i	−0.987984	−	0.154558i	$-0.950605\pi$
$911$		−	7.74315e6i		−	0.309116i	0.987984	−	0.154558i	$-0.0493954\pi$
$912$		0			0
$913$	−1.01258e7	+	1.01258e7i	−0.402024	+	0.402024i
$914$	−3.01491e7			−1.19374
$915$		0			0
$916$	−7.45584e6			−0.293601
$917$	2.83843e7	−	2.83843e7i	1.11469	−	1.11469i
$918$		0			0
$919$			2.22571e6i			0.0869319i	0.999055	+	0.0434660i	$0.0138400\pi$
$919$			2.22571e6i			0.0869319i	−0.999055	+	0.0434660i	$0.986160\pi$
$920$		0			0
$921$		0			0
$922$	−3.66783e6	−	3.66783e6i	−0.142096	−	0.142096i
$923$	1.25977e6	+	1.25977e6i	0.0486730	+	0.0486730i
$924$		0			0
$925$		0			0
$926$		−	2.99919e7i		−	1.14941i
$927$		0			0
$928$	1.89522e6	−	1.89522e6i	0.0722421	−	0.0722421i
$929$	6.15139e6			0.233848			0.116924	−	0.993141i	$-0.462697\pi$
$929$	6.15139e6			0.233848			0.116924	+	0.993141i	$0.462697\pi$
$930$		0			0
$931$	4.79562e7			1.81330
$932$	−1.89077e6	+	1.89077e6i	−0.0713015	+	0.0713015i
$933$		0			0
$934$			5.01181e6i			0.187987i
$935$		0			0
$936$		0			0
$937$	1.13103e7	+	1.13103e7i	0.420846	+	0.420846i	0.885495	−	0.464649i	$-0.153819\pi$
$937$	1.13103e7	+	1.13103e7i	0.420846	+	0.420846i	−0.464649	+	0.885495i	$0.653819\pi$
$938$	2.03580e7	+	2.03580e7i	0.755490	+	0.755490i
$939$		0			0
$940$		0			0
$941$			4.63930e7i			1.70796i	0.520304	+	0.853981i	$0.325819\pi$
$941$			4.63930e7i			1.70796i	−0.520304	+	0.853981i	$0.674181\pi$
$942$		0			0
$943$	−4.02871e6	+	4.02871e6i	−0.147532	+	0.147532i
$944$	3.56667e6			0.130266
$945$		0			0
$946$	3.39817e6			0.123458
$947$	9.83274e6	−	9.83274e6i	0.356287	−	0.356287i	−0.506155	−	0.862442i	$-0.668934\pi$
$947$	9.83274e6	−	9.83274e6i	0.356287	−	0.356287i	0.862442	+	0.506155i	$0.168934\pi$
$948$		0			0
$949$		−	2.37724e7i		−	0.856856i
$950$		0			0
$951$		0			0
$952$	2.04906e7	+	2.04906e7i	0.732762	+	0.732762i
$953$	7.65696e6	+	7.65696e6i	0.273102	+	0.273102i	0.830348	−	0.557246i	$-0.188142\pi$
$953$	7.65696e6	+	7.65696e6i	0.273102	+	0.273102i	−0.557246	+	0.830348i	$0.688142\pi$
$954$		0			0
$955$		0			0
$956$			2.41527e7i			0.854715i
$957$		0			0
$958$	4.05411e6	−	4.05411e6i	0.142719	−	0.142719i
$959$	−3.79970e7			−1.33415
$960$		0			0
$961$	4.62410e7			1.61517
$962$	5.87964e6	−	5.87964e6i	0.204839	−	0.204839i
$963$		0			0
$964$		−	1.91642e7i		−	0.664200i
$965$		0			0
$966$		0			0
$967$	−7.72851e6	−	7.72851e6i	−0.265784	−	0.265784i	0.561615	−	0.827399i	$-0.310180\pi$
$967$	−7.72851e6	−	7.72851e6i	−0.265784	−	0.265784i	−0.827399	+	0.561615i	$0.810180\pi$
$968$	−3.22822e6	−	3.22822e6i	−0.110733	−	0.110733i
$969$		0			0
$970$		0			0
$971$			3.78060e7i			1.28680i	0.765529	+	0.643402i	$0.222477\pi$
$971$			3.78060e7i			1.28680i	−0.765529	+	0.643402i	$0.777523\pi$
$972$		0			0
$973$	3.24172e7	−	3.24172e7i	1.09772	−	1.09772i
$974$	−5.62817e6			−0.190094
$975$		0			0
$976$	7.20402e6			0.242075
$977$	−4.81434e6	+	4.81434e6i	−0.161362	+	0.161362i	−0.783170	−	0.621808i	$-0.786398\pi$
$977$	−4.81434e6	+	4.81434e6i	−0.161362	+	0.161362i	0.621808	+	0.783170i	$0.286398\pi$
$978$		0			0
$979$			1.91440e7i			0.638374i
$980$		0			0
$981$		0			0
$982$	−2.00625e6	−	2.00625e6i	−0.0663905	−	0.0663905i
$983$	−1.43163e6	−	1.43163e6i	−0.0472549	−	0.0472549i	0.683084	−	0.730339i	$-0.260638\pi$
$983$	−1.43163e6	−	1.43163e6i	−0.0472549	−	0.0472549i	−0.730339	+	0.683084i	$0.760638\pi$
$984$		0			0
$985$		0			0
$986$			2.25384e7i			0.738298i
$987$		0			0
$988$	−5.40574e6	+	5.40574e6i	−0.176182	+	0.176182i
$989$	−1.01853e7			−0.331117
$990$		0			0
$991$	−2.93087e7			−0.948009			−0.474004	−	0.880522i	$-0.657192\pi$
$991$	−2.93087e7			−0.948009			−0.474004	+	0.880522i	$0.657192\pi$
$992$	6.26526e6	−	6.26526e6i	0.202144	−	0.202144i
$993$		0			0
$994$		−	5.48413e6i		−	0.176052i
$995$		0			0
$996$		0			0
$997$	4.28898e6	+	4.28898e6i	0.136652	+	0.136652i	0.772124	−	0.635472i	$-0.219194\pi$
$997$	4.28898e6	+	4.28898e6i	0.136652	+	0.136652i	−0.635472	+	0.772124i	$0.719194\pi$
$998$	5.73458e6	+	5.73458e6i	0.182253	+	0.182253i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.f.e.107.4		12
3.2	odd	2	inner	450.6.f.e.107.1		12
5.2	odd	4		90.6.f.c.53.6	yes	12
5.3	odd	4	inner	450.6.f.e.143.1		12
5.4	even	2		90.6.f.c.17.1	✓	12
15.2	even	4		90.6.f.c.53.1	yes	12
15.8	even	4	inner	450.6.f.e.143.4		12
15.14	odd	2		90.6.f.c.17.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.f.c.17.1	✓	12	5.4	even	2
90.6.f.c.17.6	yes	12	15.14	odd	2
90.6.f.c.53.1	yes	12	15.2	even	4
90.6.f.c.53.6	yes	12	5.2	odd	4
450.6.f.e.107.1		12	3.2	odd	2	inner
450.6.f.e.107.4		12	1.1	even	1	trivial
450.6.f.e.143.1		12	5.3	odd	4	inner
450.6.f.e.143.4		12	15.8	even	4	inner

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-148.726 - 148.726i) q^{7} +(-45.2548 - 45.2548i) q^{8} +O(q^{10})\)	\(q+(2.82843 - 2.82843i) q^{2} -16.0000i q^{4} +(-148.726 - 148.726i) q^{7} +(-45.2548 - 45.2548i) q^{8} -299.527i q^{11} +(193.262 - 193.262i) q^{13} -841.322 q^{14} -256.000 q^{16} +(1522.21 - 1522.21i) q^{17} -1748.19i q^{19} +(-847.191 - 847.191i) q^{22} +(2539.27 + 2539.27i) q^{23} -1093.25i q^{26} +(-2379.62 + 2379.62i) q^{28} -2617.43 q^{29} -8652.76 q^{31} +(-724.077 + 724.077i) q^{32} -8610.89i q^{34} +(5378.10 + 5378.10i) q^{37} +(-4944.63 - 4944.63i) q^{38} +1586.56i q^{41} +(-2005.55 + 2005.55i) q^{43} -4792.44 q^{44} +14364.3 q^{46} +(-11013.5 + 11013.5i) q^{47} +27431.9i q^{49} +(-3092.19 - 3092.19i) q^{52} +(-9879.22 - 9879.22i) q^{53} +13461.2i q^{56} +(-7403.22 + 7403.22i) q^{58} -13932.3 q^{59} -28140.7 q^{61} +(-24473.7 + 24473.7i) q^{62} +4096.00i q^{64} +(-24197.7 - 24197.7i) q^{67} +(-24355.3 - 24355.3i) q^{68} +6518.47i q^{71} +(61503.1 - 61503.1i) q^{73} +30423.1 q^{74} -27971.0 q^{76} +(-44547.6 + 44547.6i) q^{77} +42431.1i q^{79} +(4487.47 + 4487.47i) q^{82} +(-33805.9 - 33805.9i) q^{83} +11345.1i q^{86} +(-13555.1 + 13555.1i) q^{88} -63913.9 q^{89} -57486.2 q^{91} +(40628.3 - 40628.3i) q^{92} +62301.7i q^{94} +(58905.9 + 58905.9i) q^{97} +(77589.2 + 77589.2i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 144 q^{7}+O(q^{10}) \) 12 * q - 144 * q^7	\( 12 q - 144 q^{7} + 276 q^{13} - 3072 q^{16} - 384 q^{22} - 2304 q^{28} - 58512 q^{31} - 25764 q^{37} - 16080 q^{43} - 60672 q^{46} - 4416 q^{52} - 23952 q^{58} - 145200 q^{61} - 33552 q^{67} + 158988 q^{73} - 86016 q^{76} + 75024 q^{82} - 6144 q^{88} - 465024 q^{91} + 631116 q^{97}+O(q^{100}) \) 12 * q - 144 * q^7 + 276 * q^13 - 3072 * q^16 - 384 * q^22 - 2304 * q^28 - 58512 * q^31 - 25764 * q^37 - 16080 * q^43 - 60672 * q^46 - 4416 * q^52 - 23952 * q^58 - 145200 * q^61 - 33552 * q^67 + 158988 * q^73 - 86016 * q^76 + 75024 * q^82 - 6144 * q^88 - 465024 * q^91 + 631116 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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