LMFDB - Embedded newform 207.6.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

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:	yes
Analytic conductor:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.93938$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.93938 q^{2} -23.3601 q^{4} -91.0649 q^{5} -103.931 q^{7} -162.724 q^{8} +O(q^{10})$	\(q+2.93938 q^{2} -23.3601 q^{4} -91.0649 q^{5} -103.931 q^{7} -162.724 q^{8} -267.674 q^{10} +213.235 q^{11} -479.651 q^{13} -305.492 q^{14} +269.215 q^{16} -838.909 q^{17} -413.545 q^{19} +2127.28 q^{20} +626.778 q^{22} +529.000 q^{23} +5167.82 q^{25} -1409.87 q^{26} +2427.83 q^{28} -853.616 q^{29} +8242.80 q^{31} +5998.49 q^{32} -2465.87 q^{34} +9464.45 q^{35} -15980.8 q^{37} -1215.56 q^{38} +14818.5 q^{40} +13358.1 q^{41} +3371.38 q^{43} -4981.18 q^{44} +1554.93 q^{46} +6899.81 q^{47} -6005.39 q^{49} +15190.2 q^{50} +11204.7 q^{52} +15760.1 q^{53} -19418.2 q^{55} +16912.0 q^{56} -2509.10 q^{58} -24689.9 q^{59} -53534.9 q^{61} +24228.7 q^{62} +9016.97 q^{64} +43679.4 q^{65} -8697.07 q^{67} +19597.0 q^{68} +27819.6 q^{70} +22263.4 q^{71} -29206.3 q^{73} -46973.7 q^{74} +9660.44 q^{76} -22161.7 q^{77} -22400.6 q^{79} -24516.0 q^{80} +39264.5 q^{82} +70660.8 q^{83} +76395.2 q^{85} +9909.75 q^{86} -34698.4 q^{88} +121399. q^{89} +49850.5 q^{91} -12357.5 q^{92} +20281.1 q^{94} +37659.5 q^{95} -25587.7 q^{97} -17652.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.93938	0.519613	0.259807	−	0.965661i	$-0.416341\pi$
$2$	2.93938	0.519613	0.259807	+	0.965661i	$0.416341\pi$
$3$	0	0
$3$	0	0
$4$	−23.3601	−0.730002
$5$	−91.0649	−1.62902	−0.814510	−	0.580150i	$-0.802994\pi$
$5$	−91.0649	−1.62902	−0.814510	+	0.580150i	$0.802994\pi$
$6$	0	0
$7$	−103.931	−0.801676	−0.400838	−	0.916149i	$-0.631281\pi$
$7$	−103.931	−0.801676	−0.400838	+	0.916149i	$0.631281\pi$
$8$	−162.724	−0.898932
$9$	0	0
$10$	−267.674	−0.846460
$11$	213.235	0.531345	0.265672	−	0.964063i	$-0.414406\pi$
$11$	213.235	0.531345	0.265672	+	0.964063i	$0.414406\pi$
$12$	0	0
$13$	−479.651	−0.787167	−0.393583	−	0.919289i	$-0.628765\pi$
$13$	−479.651	−0.787167	−0.393583	+	0.919289i	$0.628765\pi$
$14$	−305.492	−0.416562
$15$	0	0
$16$	269.215	0.262905
$17$	−838.909	−0.704033	−0.352016	−	0.935994i	$-0.614504\pi$
$17$	−838.909	−0.704033	−0.352016	+	0.935994i	$0.614504\pi$
$18$	0	0
$19$	−413.545	−0.262808	−0.131404	−	0.991329i	$-0.541949\pi$
$19$	−413.545	−0.262808	−0.131404	+	0.991329i	$0.541949\pi$
$20$	2127.28	1.18919
$21$	0	0
$22$	626.778	0.276094
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	5167.82	1.65370
$26$	−1409.87	−0.409022
$27$	0	0
$28$	2427.83	0.585225
$29$	−853.616	−0.188481	−0.0942405	−	0.995549i	$-0.530042\pi$
$29$	−853.616	−0.188481	−0.0942405	+	0.995549i	$0.530042\pi$
$30$	0	0
$31$	8242.80	1.54053	0.770266	−	0.637723i	$-0.220123\pi$
$31$	8242.80	1.54053	0.770266	+	0.637723i	$0.220123\pi$
$32$	5998.49	1.03554
$33$	0	0
$34$	−2465.87	−0.365825
$35$	9464.45	1.30595
$36$	0	0
$37$	−15980.8	−1.91909	−0.959545	−	0.281556i	$-0.909149\pi$
$37$	−15980.8	−1.91909	−0.959545	+	0.281556i	$0.909149\pi$
$38$	−1215.56	−0.136559
$39$	0	0
$40$	14818.5	1.46438
$41$	13358.1	1.24104	0.620520	−	0.784191i	$-0.286922\pi$
$41$	13358.1	1.24104	0.620520	+	0.784191i	$0.286922\pi$
$42$	0	0
$43$	3371.38	0.278059	0.139029	−	0.990288i	$-0.455602\pi$
$43$	3371.38	0.278059	0.139029	+	0.990288i	$0.455602\pi$
$44$	−4981.18	−0.387883
$45$	0	0
$46$	1554.93	0.108347
$47$	6899.81	0.455609	0.227805	−	0.973707i	$-0.426845\pi$
$47$	6899.81	0.455609	0.227805	+	0.973707i	$0.426845\pi$
$48$	0	0
$49$	−6005.39	−0.357315
$50$	15190.2	0.859287
$51$	0	0
$52$	11204.7	0.574633
$53$	15760.1	0.770670	0.385335	−	0.922777i	$-0.374086\pi$
$53$	15760.1	0.770670	0.385335	+	0.922777i	$0.374086\pi$
$54$	0	0
$55$	−19418.2	−0.865571
$56$	16912.0	0.720653
$57$	0	0
$58$	−2509.10	−0.0979372
$59$	−24689.9	−0.923399	−0.461699	−	0.887036i	$-0.652760\pi$
$59$	−24689.9	−0.923399	−0.461699	+	0.887036i	$0.652760\pi$
$60$	0	0
$61$	−53534.9	−1.84210	−0.921048	−	0.389449i	$-0.872666\pi$
$61$	−53534.9	−1.84210	−0.921048	+	0.389449i	$0.872666\pi$
$62$	24228.7	0.800481
$63$	0	0
$64$	9016.97	0.275176
$65$	43679.4	1.28231
$66$	0	0
$67$	−8697.07	−0.236693	−0.118347	−	0.992972i	$-0.537759\pi$
$67$	−8697.07	−0.236693	−0.118347	+	0.992972i	$0.537759\pi$
$68$	19597.0	0.513945
$69$	0	0
$70$	27819.6	0.678587
$71$	22263.4	0.524139	0.262069	−	0.965049i	$-0.415595\pi$
$71$	22263.4	0.524139	0.262069	+	0.965049i	$0.415595\pi$
$72$	0	0
$73$	−29206.3	−0.641461	−0.320730	−	0.947171i	$-0.603928\pi$
$73$	−29206.3	−0.641461	−0.320730	+	0.947171i	$0.603928\pi$
$74$	−46973.7	−0.997184
$75$	0	0
$76$	9660.44	0.191850
$77$	−22161.7	−0.425967
$78$	0	0
$79$	−22400.6	−0.403824	−0.201912	−	0.979404i	$-0.564715\pi$
$79$	−22400.6	−0.403824	−0.201912	+	0.979404i	$0.564715\pi$
$80$	−24516.0	−0.428277
$81$	0	0
$82$	39264.5	0.644861
$83$	70660.8	1.12586	0.562929	−	0.826505i	$-0.309675\pi$
$83$	70660.8	1.12586	0.562929	+	0.826505i	$0.309675\pi$
$84$	0	0
$85$	76395.2	1.14688
$86$	9909.75	0.144483
$87$	0	0
$88$	−34698.4	−0.477643
$89$	121399.	1.62458	0.812290	−	0.583254i	$-0.198221\pi$
$89$	121399.	1.62458	0.812290	+	0.583254i	$0.198221\pi$
$90$	0	0
$91$	49850.5	0.631053
$92$	−12357.5	−0.152216
$93$	0	0
$94$	20281.1	0.236741
$95$	37659.5	0.428119
$96$	0	0
$97$	−25587.7	−0.276123	−0.138061	−	0.990424i	$-0.544087\pi$
$97$	−25587.7	−0.276123	−0.138061	+	0.990424i	$0.544087\pi$
$98$	−17652.1	−0.185666
$99$	0	0
$100$	−120721.	−1.20721
$101$	−108344.	−1.05682	−0.528411	−	0.848988i	$-0.677212\pi$
$101$	−108344.	−1.05682	−0.528411	+	0.848988i	$0.677212\pi$
$102$	0	0
$103$	135884.	1.26204	0.631022	−	0.775765i	$-0.282636\pi$
$103$	135884.	1.26204	0.631022	+	0.775765i	$0.282636\pi$
$104$	78050.7	0.707609
$105$	0	0
$106$	46324.8	0.400450
$107$	38399.5	0.324240	0.162120	−	0.986771i	$-0.448167\pi$
$107$	38399.5	0.324240	0.162120	+	0.986771i	$0.448167\pi$
$108$	0	0
$109$	−167628.	−1.35139	−0.675694	−	0.737183i	$-0.736156\pi$
$109$	−167628.	−1.35139	−0.675694	+	0.737183i	$0.736156\pi$
$110$	−57077.5	−0.449762
$111$	0	0
$112$	−27979.7	−0.210765
$113$	147545.	1.08699	0.543497	−	0.839411i	$-0.317100\pi$
$113$	147545.	1.08699	0.543497	+	0.839411i	$0.317100\pi$
$114$	0	0
$115$	−48173.4	−0.339674
$116$	19940.5	0.137591
$117$	0	0
$118$	−72572.9	−0.479810
$119$	87188.5	0.564406
$120$	0	0
$121$	−115582.	−0.717673
$122$	−157359.	−0.957178
$123$	0	0
$124$	−192552.	−1.12459
$125$	−186030.	−1.06490
$126$	0	0
$127$	58354.7	0.321046	0.160523	−	0.987032i	$-0.448682\pi$
$127$	58354.7	0.321046	0.160523	+	0.987032i	$0.448682\pi$
$128$	−165448.	−0.892556
$129$	0	0
$130$	128390.	0.666305
$131$	−19289.5	−0.0982071	−0.0491035	−	0.998794i	$-0.515636\pi$
$131$	−19289.5	−0.0982071	−0.0491035	+	0.998794i	$0.515636\pi$
$132$	0	0
$133$	42980.1	0.210687
$134$	−25564.0	−0.122989
$135$	0	0
$136$	136511.	0.632877
$137$	−39432.1	−0.179493	−0.0897467	−	0.995965i	$-0.528606\pi$
$137$	−39432.1	−0.179493	−0.0897467	+	0.995965i	$0.528606\pi$
$138$	0	0
$139$	335873.	1.47448	0.737239	−	0.675632i	$-0.236129\pi$
$139$	335873.	1.47448	0.737239	+	0.675632i	$0.236129\pi$
$140$	−221090.	−0.953343
$141$	0	0
$142$	65440.6	0.272349
$143$	−102278.	−0.418257
$144$	0	0
$145$	77734.5	0.307039
$146$	−85848.4	−0.333312
$147$	0	0
$148$	373313.	1.40094
$149$	−296758.	−1.09506	−0.547529	−	0.836786i	$-0.684432\pi$
$149$	−296758.	−1.09506	−0.547529	+	0.836786i	$0.684432\pi$
$150$	0	0
$151$	17568.6	0.0627041	0.0313520	−	0.999508i	$-0.490019\pi$
$151$	17568.6	0.0627041	0.0313520	+	0.999508i	$0.490019\pi$
$152$	67293.7	0.236247
$153$	0	0
$154$	−65141.5	−0.221338
$155$	−750631.	−2.50956
$156$	0	0
$157$	221932.	0.718574	0.359287	−	0.933227i	$-0.383020\pi$
$157$	221932.	0.718574	0.359287	+	0.933227i	$0.383020\pi$
$158$	−65843.8	−0.209832
$159$	0	0
$160$	−546253.	−1.68692
$161$	−54979.4	−0.167161
$162$	0	0
$163$	539439.	1.59028	0.795140	−	0.606426i	$-0.207397\pi$
$163$	539439.	1.59028	0.795140	+	0.606426i	$0.207397\pi$
$164$	−312046.	−0.905961
$165$	0	0
$166$	207699.	0.585011
$167$	254570.	0.706344	0.353172	−	0.935558i	$-0.385103\pi$
$167$	254570.	0.706344	0.353172	+	0.935558i	$0.385103\pi$
$168$	0	0
$169$	−141228.	−0.380369
$170$	224554.	0.595935
$171$	0	0
$172$	−78755.6	−0.202983
$173$	339160.	0.861567	0.430783	−	0.902455i	$-0.358237\pi$
$173$	339160.	0.861567	0.430783	+	0.902455i	$0.358237\pi$
$174$	0	0
$175$	−537096.	−1.32574
$176$	57405.9	0.139693
$177$	0	0
$178$	356838.	0.844153
$179$	−680544.	−1.58754	−0.793768	−	0.608221i	$-0.791884\pi$
$179$	−680544.	−1.58754	−0.793768	+	0.608221i	$0.791884\pi$
$180$	0	0
$181$	−603873.	−1.37009	−0.685045	−	0.728500i	$-0.740218\pi$
$181$	−603873.	−1.37009	−0.685045	+	0.728500i	$0.740218\pi$
$182$	146529.	0.327904
$183$	0	0
$184$	−86081.0	−0.187440
$185$	1.45529e6	3.12623
$186$	0	0
$187$	−178885.	−0.374084
$188$	−161180.	−0.332596
$189$	0	0
$190$	110695.	0.222457
$191$	−466620.	−0.925508	−0.462754	−	0.886487i	$-0.653139\pi$
$191$	−466620.	−0.925508	−0.462754	+	0.886487i	$0.653139\pi$
$192$	0	0
$193$	−250253.	−0.483600	−0.241800	−	0.970326i	$-0.577738\pi$
$193$	−250253.	−0.483600	−0.241800	+	0.970326i	$0.577738\pi$
$194$	−75212.0	−0.143477
$195$	0	0
$196$	140286.	0.260841
$197$	197147.	0.361930	0.180965	−	0.983490i	$-0.442078\pi$
$197$	197147.	0.361930	0.180965	+	0.983490i	$0.442078\pi$
$198$	0	0
$199$	−232494.	−0.416178	−0.208089	−	0.978110i	$-0.566724\pi$
$199$	−232494.	−0.416178	−0.208089	+	0.978110i	$0.566724\pi$
$200$	−840930.	−1.48657
$201$	0	0
$202$	−318464.	−0.549139
$203$	88717.0	0.151101
$204$	0	0
$205$	−1.21646e6	−2.02168
$206$	399414.	0.655775
$207$	0	0
$208$	−129129.	−0.206950
$209$	−88182.2	−0.139642
$210$	0	0
$211$	−201737.	−0.311946	−0.155973	−	0.987761i	$-0.549851\pi$
$211$	−201737.	−0.311946	−0.155973	+	0.987761i	$0.549851\pi$
$212$	−368156.	−0.562590
$213$	0	0
$214$	112871.	0.168479
$215$	−307014.	−0.452963
$216$	0	0
$217$	−856681.	−1.23501
$218$	−492721.	−0.702199
$219$	0	0
$220$	453611.	0.631868
$221$	402383.	0.554191
$222$	0	0
$223$	1.18169e6	1.59126	0.795629	−	0.605784i	$-0.207140\pi$
$223$	1.18169e6	1.59126	0.795629	+	0.605784i	$0.207140\pi$
$224$	−623428.	−0.830169
$225$	0	0
$226$	433690.	0.564817
$227$	1.50488e6	1.93837	0.969186	−	0.246330i	$-0.0792247\pi$
$227$	1.50488e6	1.93837	0.969186	+	0.246330i	$0.0792247\pi$
$228$	0	0
$229$	866204.	1.09152	0.545760	−	0.837942i	$-0.316241\pi$
$229$	866204.	1.09152	0.545760	+	0.837942i	$0.316241\pi$
$230$	−141600.	−0.176499
$231$	0	0
$232$	138904.	0.169432
$233$	21061.5	0.0254155	0.0127078	−	0.999919i	$-0.495955\pi$
$233$	21061.5	0.0254155	0.0127078	+	0.999919i	$0.495955\pi$
$234$	0	0
$235$	−628331.	−0.742196
$236$	576758.	0.674083
$237$	0	0
$238$	256280.	0.293273
$239$	718350.	0.813470	0.406735	−	0.913546i	$-0.366667\pi$
$239$	718350.	0.813470	0.406735	+	0.913546i	$0.366667\pi$
$240$	0	0
$241$	389773.	0.432284	0.216142	−	0.976362i	$-0.430653\pi$
$241$	389773.	0.432284	0.216142	+	0.976362i	$0.430653\pi$
$242$	−339739.	−0.372912
$243$	0	0
$244$	1.25058e6	1.34473
$245$	546881.	0.582073
$246$	0	0
$247$	198357.	0.206874
$248$	−1.34130e6	−1.38483
$249$	0	0
$250$	−546812.	−0.553334
$251$	1.63365e6	1.63672	0.818359	−	0.574707i	$-0.194884\pi$
$251$	1.63365e6	1.63672	0.818359	+	0.574707i	$0.194884\pi$
$252$	0	0
$253$	112801.	0.110793
$254$	171527.	0.166820
$255$	0	0
$256$	−774856.	−0.738960
$257$	772961.	0.730003	0.365002	−	0.931007i	$-0.381068\pi$
$257$	772961.	0.730003	0.365002	+	0.931007i	$0.381068\pi$
$258$	0	0
$259$	1.66090e6	1.53849
$260$	−1.02035e6	−0.936089
$261$	0	0
$262$	−56699.1	−0.0510297
$263$	1.96961e6	1.75586	0.877932	−	0.478785i	$-0.158923\pi$
$263$	1.96961e6	1.75586	0.877932	+	0.478785i	$0.158923\pi$
$264$	0	0
$265$	−1.43519e6	−1.25544
$266$	126335.	0.109476
$267$	0	0
$268$	203164.	0.172787
$269$	1.55026e6	1.30625	0.653123	−	0.757252i	$-0.273459\pi$
$269$	1.55026e6	1.30625	0.653123	+	0.757252i	$0.273459\pi$
$270$	0	0
$271$	−93548.8	−0.0773775	−0.0386888	−	0.999251i	$-0.512318\pi$
$271$	−93548.8	−0.0773775	−0.0386888	+	0.999251i	$0.512318\pi$
$272$	−225847.	−0.185094
$273$	0	0
$274$	−115906.	−0.0932672
$275$	1.10196e6	0.878687
$276$	0	0
$277$	1.26686e6	0.992043	0.496022	−	0.868310i	$-0.334794\pi$
$277$	1.26686e6	0.992043	0.496022	+	0.868310i	$0.334794\pi$
$278$	987258.	0.766159
$279$	0	0
$280$	−1.54009e6	−1.17396
$281$	−2.00912e6	−1.51789	−0.758943	−	0.651157i	$-0.774284\pi$
$281$	−2.00912e6	−1.51789	−0.758943	+	0.651157i	$0.774284\pi$
$282$	0	0
$283$	80487.7	0.0597398	0.0298699	−	0.999554i	$-0.490491\pi$
$283$	80487.7	0.0597398	0.0298699	+	0.999554i	$0.490491\pi$
$284$	−520075.	−0.382622
$285$	0	0
$286$	−300634.	−0.217332
$287$	−1.38832e6	−0.994912
$288$	0	0
$289$	−716088.	−0.504338
$290$	228491.	0.159542
$291$	0	0
$292$	682262.	0.468268
$293$	−2.26580e6	−1.54189	−0.770945	−	0.636902i	$-0.780216\pi$
$293$	−2.26580e6	−1.54189	−0.770945	+	0.636902i	$0.780216\pi$
$294$	0	0
$295$	2.24838e6	1.50423
$296$	2.60047e6	1.72513
$297$	0	0
$298$	−872285.	−0.569007
$299$	−253735.	−0.164136
$300$	0	0
$301$	−350390.	−0.222913
$302$	51640.9	0.0325819
$303$	0	0
$304$	−111332.	−0.0690935
$305$	4.87515e6	3.00081
$306$	0	0
$307$	−387105.	−0.234413	−0.117207	−	0.993108i	$-0.537394\pi$
$307$	−387105.	−0.234413	−0.117207	+	0.993108i	$0.537394\pi$
$308$	517698.	0.310956
$309$	0	0
$310$	−2.20639e6	−1.30400
$311$	960249.	0.562967	0.281484	−	0.959566i	$-0.409174\pi$
$311$	960249.	0.562967	0.281484	+	0.959566i	$0.409174\pi$
$312$	0	0
$313$	1.66929e6	0.963099	0.481549	−	0.876419i	$-0.340074\pi$
$313$	1.66929e6	0.963099	0.481549	+	0.876419i	$0.340074\pi$
$314$	652343.	0.373381
$315$	0	0
$316$	523279.	0.294792
$317$	−1.39041e6	−0.777132	−0.388566	−	0.921421i	$-0.627029\pi$
$317$	−1.39041e6	−0.777132	−0.388566	+	0.921421i	$0.627029\pi$
$318$	0	0
$319$	−182021.	−0.100148
$320$	−821130.	−0.448267
$321$	0	0
$322$	−161605.	−0.0868591
$323$	346927.	0.185025
$324$	0	0
$325$	−2.47875e6	−1.30174
$326$	1.58562e6	0.826331
$327$	0	0
$328$	−2.17369e6	−1.11561
$329$	−717102.	−0.365251
$330$	0	0
$331$	2.98248e6	1.49626	0.748131	−	0.663551i	$-0.230951\pi$
$331$	2.98248e6	1.49626	0.748131	+	0.663551i	$0.230951\pi$
$332$	−1.65064e6	−0.821878
$333$	0	0
$334$	748277.	0.367026
$335$	791998.	0.385578
$336$	0	0
$337$	2.98554e6	1.43202	0.716009	−	0.698091i	$-0.245967\pi$
$337$	2.98554e6	1.43202	0.716009	+	0.698091i	$0.245967\pi$
$338$	−415123.	−0.197645
$339$	0	0
$340$	−1.78460e6	−0.837227
$341$	1.75765e6	0.818554
$342$	0	0
$343$	2.37091e6	1.08813
$344$	−548604.	−0.249956
$345$	0	0
$346$	996918.	0.447682
$347$	3.14102e6	1.40038	0.700192	−	0.713955i	$-0.253098\pi$
$347$	3.14102e6	1.40038	0.700192	+	0.713955i	$0.253098\pi$
$348$	0	0
$349$	1.77944e6	0.782025	0.391012	−	0.920385i	$-0.372125\pi$
$349$	1.77944e6	0.782025	0.391012	+	0.920385i	$0.372125\pi$
$350$	−1.57873e6	−0.688870
$351$	0	0
$352$	1.27909e6	0.550229
$353$	−3.00852e6	−1.28504	−0.642519	−	0.766270i	$-0.722111\pi$
$353$	−3.00852e6	−1.28504	−0.642519	+	0.766270i	$0.722111\pi$
$354$	0	0
$355$	−2.02742e6	−0.853832
$356$	−2.83589e6	−1.18595
$357$	0	0
$358$	−2.00037e6	−0.824905
$359$	−3.31576e6	−1.35783	−0.678917	−	0.734215i	$-0.737550\pi$
$359$	−3.31576e6	−1.35783	−0.678917	+	0.734215i	$0.737550\pi$
$360$	0	0
$361$	−2.30508e6	−0.930932
$362$	−1.77501e6	−0.711918
$363$	0	0
$364$	−1.16451e6	−0.460670
$365$	2.65967e6	1.04495
$366$	0	0
$367$	−3.00182e6	−1.16338	−0.581688	−	0.813412i	$-0.697608\pi$
$367$	−3.00182e6	−1.16338	−0.581688	+	0.813412i	$0.697608\pi$
$368$	142414.	0.0548194
$369$	0	0
$370$	4.27766e6	1.62443
$371$	−1.63796e6	−0.617828
$372$	0	0
$373$	2.83080e6	1.05351	0.526753	−	0.850018i	$-0.323409\pi$
$373$	2.83080e6	1.05351	0.526753	+	0.850018i	$0.323409\pi$
$374$	−525810.	−0.194379
$375$	0	0
$376$	−1.12277e6	−0.409562
$377$	409437.	0.148366
$378$	0	0
$379$	−1.95244e6	−0.698200	−0.349100	−	0.937085i	$-0.613513\pi$
$379$	−1.95244e6	−0.698200	−0.349100	+	0.937085i	$0.613513\pi$
$380$	−879727.	−0.312528
$381$	0	0
$382$	−1.37157e6	−0.480906
$383$	−4.41770e6	−1.53886	−0.769430	−	0.638731i	$-0.779460\pi$
$383$	−4.41770e6	−1.53886	−0.769430	+	0.638731i	$0.779460\pi$
$384$	0	0
$385$	2.01815e6	0.693908
$386$	−735589.	−0.251285
$387$	0	0
$388$	597731.	0.201570
$389$	−3.80978e6	−1.27651	−0.638257	−	0.769823i	$-0.720344\pi$
$389$	−3.80978e6	−1.27651	−0.638257	+	0.769823i	$0.720344\pi$
$390$	0	0
$391$	−443783.	−0.146801
$392$	977222.	0.321202
$393$	0	0
$394$	579489.	0.188063
$395$	2.03991e6	0.657837
$396$	0	0
$397$	−645315.	−0.205492	−0.102746	−	0.994708i	$-0.532763\pi$
$397$	−645315.	−0.205492	−0.102746	+	0.994708i	$0.532763\pi$
$398$	−683387.	−0.216252
$399$	0	0
$400$	1.39125e6	0.434767
$401$	19542.5	0.00606903	0.00303452	−	0.999995i	$-0.499034\pi$
$401$	19542.5	0.00606903	0.00303452	+	0.999995i	$0.499034\pi$
$402$	0	0
$403$	−3.95367e6	−1.21266
$404$	2.53093e6	0.771483
$405$	0	0
$406$	260773.	0.0785140
$407$	−3.40767e6	−1.01970
$408$	0	0
$409$	3.55172e6	1.04986	0.524929	−	0.851146i	$-0.324092\pi$
$409$	3.55172e6	1.04986	0.524929	+	0.851146i	$0.324092\pi$
$410$	−3.57562e6	−1.05049
$411$	0	0
$412$	−3.17426e6	−0.921295
$413$	2.56604e6	0.740267
$414$	0	0
$415$	−6.43472e6	−1.83404
$416$	−2.87718e6	−0.815143
$417$	0	0
$418$	−259201.	−0.0725597
$419$	−3.41260e6	−0.949621	−0.474811	−	0.880088i	$-0.657483\pi$
$419$	−3.41260e6	−0.949621	−0.474811	+	0.880088i	$0.657483\pi$
$420$	0	0
$421$	−1.64045e6	−0.451084	−0.225542	−	0.974233i	$-0.572415\pi$
$421$	−1.64045e6	−0.451084	−0.225542	+	0.974233i	$0.572415\pi$
$422$	−592982.	−0.162091
$423$	0	0
$424$	−2.56454e6	−0.692780
$425$	−4.33534e6	−1.16426
$426$	0	0
$427$	5.56392e6	1.47677
$428$	−897015.	−0.236696
$429$	0	0
$430$	−902431.	−0.235365
$431$	4.54503e6	1.17854	0.589269	−	0.807937i	$-0.299416\pi$
$431$	4.54503e6	1.17854	0.589269	+	0.807937i	$0.299416\pi$
$432$	0	0
$433$	−99857.2	−0.0255953	−0.0127976	−	0.999918i	$-0.504074\pi$
$433$	−99857.2	−0.0255953	−0.0127976	+	0.999918i	$0.504074\pi$
$434$	−2.51811e6	−0.641727
$435$	0	0
$436$	3.91580e6	0.986515
$437$	−218765.	−0.0547993
$438$	0	0
$439$	−1.19748e6	−0.296556	−0.148278	−	0.988946i	$-0.547373\pi$
$439$	−1.19748e6	−0.296556	−0.148278	+	0.988946i	$0.547373\pi$
$440$	3.15981e6	0.778090
$441$	0	0
$442$	1.18276e6	0.287965
$443$	5.09363e6	1.23316	0.616578	−	0.787294i	$-0.288518\pi$
$443$	5.09363e6	1.23316	0.616578	+	0.787294i	$0.288518\pi$
$444$	0	0
$445$	−1.10552e7	−2.64647
$446$	3.47343e6	0.826839
$447$	0	0
$448$	−937141.	−0.220602
$449$	1.07145e6	0.250816	0.125408	−	0.992105i	$-0.459976\pi$
$449$	1.07145e6	0.250816	0.125408	+	0.992105i	$0.459976\pi$
$450$	0	0
$451$	2.84842e6	0.659420
$452$	−3.44665e6	−0.793508
$453$	0	0
$454$	4.42341e6	1.00720
$455$	−4.53963e6	−1.02800
$456$	0	0
$457$	−1.27424e6	−0.285405	−0.142702	−	0.989766i	$-0.545579\pi$
$457$	−1.27424e6	−0.285405	−0.142702	+	0.989766i	$0.545579\pi$
$458$	2.54610e6	0.567168
$459$	0	0
$460$	1.12533e6	0.247963
$461$	1.61460e6	0.353846	0.176923	−	0.984225i	$-0.443386\pi$
$461$	1.61460e6	0.353846	0.176923	+	0.984225i	$0.443386\pi$
$462$	0	0
$463$	−1.65491e6	−0.358775	−0.179388	−	0.983778i	$-0.557412\pi$
$463$	−1.65491e6	−0.358775	−0.179388	+	0.983778i	$0.557412\pi$
$464$	−229806.	−0.0495525
$465$	0	0
$466$	61907.6	0.0132062
$467$	−8.07395e6	−1.71314	−0.856572	−	0.516027i	$-0.827411\pi$
$467$	−8.07395e6	−1.71314	−0.856572	+	0.516027i	$0.827411\pi$
$468$	0	0
$469$	903893.	0.189751
$470$	−1.84690e6	−0.385655
$471$	0	0
$472$	4.01764e6	0.830073
$473$	718895.	0.147745
$474$	0	0
$475$	−2.13713e6	−0.434607
$476$	−2.03673e6	−0.412018
$477$	0	0
$478$	2.11150e6	0.422690
$479$	−417825.	−0.0832061	−0.0416030	−	0.999134i	$-0.513246\pi$
$479$	−417825.	−0.0832061	−0.0416030	+	0.999134i	$0.513246\pi$
$480$	0	0
$481$	7.66522e6	1.51064
$482$	1.14569e6	0.224621
$483$	0	0
$484$	2.70000e6	0.523902
$485$	2.33015e6	0.449810
$486$	0	0
$487$	−9.47068e6	−1.80950	−0.904750	−	0.425943i	$-0.859942\pi$
$487$	−9.47068e6	−1.80950	−0.904750	+	0.425943i	$0.859942\pi$
$488$	8.71142e6	1.65592
$489$	0	0
$490$	1.60749e6	0.302453
$491$	8.43489e6	1.57898	0.789488	−	0.613766i	$-0.210346\pi$
$491$	8.43489e6	1.57898	0.789488	+	0.613766i	$0.210346\pi$
$492$	0	0
$493$	716106.	0.132697
$494$	583046.	0.107494
$495$	0	0
$496$	2.21908e6	0.405013
$497$	−2.31386e6	−0.420190
$498$	0	0
$499$	9.30986e6	1.67375	0.836877	−	0.547391i	$-0.184379\pi$
$499$	9.30986e6	1.67375	0.836877	+	0.547391i	$0.184379\pi$
$500$	4.34567e6	0.777376
$501$	0	0
$502$	4.80190e6	0.850461
$503$	−1.00227e6	−0.176631	−0.0883153	−	0.996093i	$-0.528148\pi$
$503$	−1.00227e6	−0.176631	−0.0883153	+	0.996093i	$0.528148\pi$
$504$	0	0
$505$	9.86636e6	1.72159
$506$	331565.	0.0575695
$507$	0	0
$508$	−1.36317e6	−0.234364
$509$	−3.65046e6	−0.624530	−0.312265	−	0.949995i	$-0.601088\pi$
$509$	−3.65046e6	−0.624530	−0.312265	+	0.949995i	$0.601088\pi$
$510$	0	0
$511$	3.03544e6	0.514244
$512$	3.01673e6	0.508582
$513$	0	0
$514$	2.27202e6	0.379320
$515$	−1.23743e7	−2.05590
$516$	0	0
$517$	1.47128e6	0.242086
$518$	4.88201e6	0.799419
$519$	0	0
$520$	−7.10769e6	−1.15271
$521$	−972884.	−0.157024	−0.0785121	−	0.996913i	$-0.525017\pi$
$521$	−972884.	−0.157024	−0.0785121	+	0.996913i	$0.525017\pi$
$522$	0	0
$523$	−3.41793e6	−0.546398	−0.273199	−	0.961958i	$-0.588082\pi$
$523$	−3.41793e6	−0.546398	−0.273199	+	0.961958i	$0.588082\pi$
$524$	450604.	0.0716913
$525$	0	0
$526$	5.78943e6	0.912370
$527$	−6.91497e6	−1.08458
$528$	0	0
$529$	279841.	0.0434783
$530$	−4.21856e6	−0.652341
$531$	0	0
$532$	−1.00402e6	−0.153802
$533$	−6.40723e6	−0.976905
$534$	0	0
$535$	−3.49685e6	−0.528193
$536$	1.41522e6	0.212771
$537$	0	0
$538$	4.55681e6	0.678743
$539$	−1.28056e6	−0.189857
$540$	0	0
$541$	891922.	0.131019	0.0655094	−	0.997852i	$-0.479133\pi$
$541$	891922.	0.131019	0.0655094	+	0.997852i	$0.479133\pi$
$542$	−274975.	−0.0402064
$543$	0	0
$544$	−5.03219e6	−0.729055
$545$	1.52650e7	2.20144
$546$	0	0
$547$	−8.43532e6	−1.20541	−0.602703	−	0.797966i	$-0.705909\pi$
$547$	−8.43532e6	−1.20541	−0.602703	+	0.797966i	$0.705909\pi$
$548$	921136.	0.131031
$549$	0	0
$550$	3.23908e6	0.456577
$551$	353009.	0.0495343
$552$	0	0
$553$	2.32811e6	0.323736
$554$	3.72379e6	0.515479
$555$	0	0
$556$	−7.84602e6	−1.07637
$557$	6.98560e6	0.954038	0.477019	−	0.878893i	$-0.341717\pi$
$557$	6.98560e6	0.954038	0.477019	+	0.878893i	$0.341717\pi$
$558$	0	0
$559$	−1.61708e6	−0.218878
$560$	2.54797e6	0.343340
$561$	0	0
$562$	−5.90555e6	−0.788714
$563$	9.20977e6	1.22455	0.612277	−	0.790644i	$-0.290254\pi$
$563$	9.20977e6	1.22455	0.612277	+	0.790644i	$0.290254\pi$
$564$	0	0
$565$	−1.34362e7	−1.77074
$566$	236584.	0.0310416
$567$	0	0
$568$	−3.62280e6	−0.471165
$569$	7.14910e6	0.925701	0.462850	−	0.886436i	$-0.346827\pi$
$569$	7.14910e6	0.925701	0.462850	+	0.886436i	$0.346827\pi$
$570$	0	0
$571$	−9.92634e6	−1.27409	−0.637043	−	0.770828i	$-0.719843\pi$
$571$	−9.92634e6	−1.27409	−0.637043	+	0.770828i	$0.719843\pi$
$572$	2.38923e6	0.305328
$573$	0	0
$574$	−4.08080e6	−0.516970
$575$	2.73378e6	0.344821
$576$	0	0
$577$	8.31660e6	1.03994	0.519968	−	0.854186i	$-0.325944\pi$
$577$	8.31660e6	1.03994	0.519968	+	0.854186i	$0.325944\pi$
$578$	−2.10485e6	−0.262061
$579$	0	0
$580$	−1.81588e6	−0.224139
$581$	−7.34383e6	−0.902573
$582$	0	0
$583$	3.36059e6	0.409491
$584$	4.75258e6	0.576630
$585$	0	0
$586$	−6.66005e6	−0.801186
$587$	−1.34237e7	−1.60797	−0.803983	−	0.594652i	$-0.797290\pi$
$587$	−1.34237e7	−1.60797	−0.803983	+	0.594652i	$0.797290\pi$
$588$	0	0
$589$	−3.40877e6	−0.404864
$590$	6.60885e6	0.781620
$591$	0	0
$592$	−4.30227e6	−0.504538
$593$	1.16418e7	1.35952	0.679758	−	0.733437i	$-0.262085\pi$
$593$	1.16418e7	1.35952	0.679758	+	0.733437i	$0.262085\pi$
$594$	0	0
$595$	−7.93982e6	−0.919429
$596$	6.93230e6	0.799395
$597$	0	0
$598$	−745824.	−0.0852871
$599$	5.13177e6	0.584386	0.292193	−	0.956359i	$-0.405615\pi$
$599$	5.13177e6	0.584386	0.292193	+	0.956359i	$0.405615\pi$
$600$	0	0
$601$	1.23161e7	1.39087	0.695434	−	0.718590i	$-0.255212\pi$
$601$	1.23161e7	1.39087	0.695434	+	0.718590i	$0.255212\pi$
$602$	−1.02993e6	−0.115829
$603$	0	0
$604$	−410405.	−0.0457741
$605$	1.05255e7	1.16910
$606$	0	0
$607$	−4.65419e6	−0.512710	−0.256355	−	0.966583i	$-0.582522\pi$
$607$	−4.65419e6	−0.512710	−0.256355	+	0.966583i	$0.582522\pi$
$608$	−2.48065e6	−0.272149
$609$	0	0
$610$	1.43299e7	1.55926
$611$	−3.30950e6	−0.358640
$612$	0	0
$613$	2.73838e6	0.294336	0.147168	−	0.989112i	$-0.452984\pi$
$613$	2.73838e6	0.294336	0.147168	+	0.989112i	$0.452984\pi$
$614$	−1.13785e6	−0.121804
$615$	0	0
$616$	3.60624e6	0.382915
$617$	8.17052e6	0.864046	0.432023	−	0.901863i	$-0.357800\pi$
$617$	8.17052e6	0.864046	0.432023	+	0.901863i	$0.357800\pi$
$618$	0	0
$619$	1.42242e7	1.49211	0.746054	−	0.665886i	$-0.231946\pi$
$619$	1.42242e7	1.49211	0.746054	+	0.665886i	$0.231946\pi$
$620$	1.75348e7	1.83198
$621$	0	0
$622$	2.82253e6	0.292525
$623$	−1.26171e7	−1.30239
$624$	0	0
$625$	791336.	0.0810328
$626$	4.90667e6	0.500439
$627$	0	0
$628$	−5.18435e6	−0.524560
$629$	1.34065e7	1.35110
$630$	0	0
$631$	3.59410e6	0.359349	0.179674	−	0.983726i	$-0.442496\pi$
$631$	3.59410e6	0.359349	0.179674	+	0.983726i	$0.442496\pi$
$632$	3.64512e6	0.363010
$633$	0	0
$634$	−4.08694e6	−0.403808
$635$	−5.31407e6	−0.522989
$636$	0	0
$637$	2.88049e6	0.281266
$638$	−535027.	−0.0520384
$639$	0	0
$640$	1.50665e7	1.45399
$641$	−6.22791e6	−0.598683	−0.299342	−	0.954146i	$-0.596767\pi$
$641$	−6.22791e6	−0.598683	−0.299342	+	0.954146i	$0.596767\pi$
$642$	0	0
$643$	−1.35132e6	−0.128893	−0.0644465	−	0.997921i	$-0.520528\pi$
$643$	−1.35132e6	−0.128893	−0.0644465	+	0.997921i	$0.520528\pi$
$644$	1.28432e6	0.122028
$645$	0	0
$646$	1.01975e6	0.0961417
$647$	1.10461e6	0.103741	0.0518703	−	0.998654i	$-0.483482\pi$
$647$	1.10461e6	0.103741	0.0518703	+	0.998654i	$0.483482\pi$
$648$	0	0
$649$	−5.26475e6	−0.490643
$650$	−7.28598e6	−0.676402
$651$	0	0
$652$	−1.26013e7	−1.16091
$653$	−4.53680e6	−0.416358	−0.208179	−	0.978091i	$-0.566754\pi$
$653$	−4.53680e6	−0.416358	−0.208179	+	0.978091i	$0.566754\pi$
$654$	0	0
$655$	1.75660e6	0.159981
$656$	3.59620e6	0.326275
$657$	0	0
$658$	−2.10783e6	−0.189789
$659$	−1.96367e7	−1.76139	−0.880694	−	0.473685i	$-0.842924\pi$
$659$	−1.96367e7	−1.76139	−0.880694	+	0.473685i	$0.842924\pi$
$660$	0	0
$661$	3.23182e6	0.287702	0.143851	−	0.989599i	$-0.454051\pi$
$661$	3.23182e6	0.287702	0.143851	+	0.989599i	$0.454051\pi$
$662$	8.76664e6	0.777478
$663$	0	0
$664$	−1.14982e7	−1.01207
$665$	−3.91398e6	−0.343213
$666$	0	0
$667$	−451563.	−0.0393010
$668$	−5.94677e6	−0.515632
$669$	0	0
$670$	2.32798e6	0.200351
$671$	−1.14155e7	−0.978788
$672$	0	0
$673$	1.41412e7	1.20351	0.601754	−	0.798682i	$-0.294469\pi$
$673$	1.41412e7	1.20351	0.601754	+	0.798682i	$0.294469\pi$
$674$	8.77563e6	0.744095
$675$	0	0
$676$	3.29910e6	0.277670
$677$	−1.13182e7	−0.949089	−0.474545	−	0.880231i	$-0.657387\pi$
$677$	−1.13182e7	−0.949089	−0.474545	+	0.880231i	$0.657387\pi$
$678$	0	0
$679$	2.65935e6	0.221361
$680$	−1.24313e7	−1.03097
$681$	0	0
$682$	5.16641e6	0.425331
$683$	2.09612e7	1.71935	0.859675	−	0.510842i	$-0.170666\pi$
$683$	2.09612e7	1.71935	0.859675	+	0.510842i	$0.170666\pi$
$684$	0	0
$685$	3.59088e6	0.292398
$686$	6.96900e6	0.565406
$687$	0	0
$688$	907624.	0.0731029
$689$	−7.55933e6	−0.606646
$690$	0	0
$691$	−1.54753e7	−1.23295	−0.616473	−	0.787376i	$-0.711439\pi$
$691$	−1.54753e7	−1.23295	−0.616473	+	0.787376i	$0.711439\pi$
$692$	−7.92279e6	−0.628946
$693$	0	0
$694$	9.23264e6	0.727658
$695$	−3.05863e7	−2.40195
$696$	0	0
$697$	−1.12063e7	−0.873732
$698$	5.23045e6	0.406350
$699$	0	0
$700$	1.25466e7	0.967790
$701$	9.45817e6	0.726963	0.363481	−	0.931601i	$-0.381588\pi$
$701$	9.45817e6	0.726963	0.363481	+	0.931601i	$0.381588\pi$
$702$	0	0
$703$	6.60879e6	0.504352
$704$	1.92273e6	0.146213
$705$	0	0
$706$	−8.84317e6	−0.667723
$707$	1.12603e7	0.847230
$708$	0	0
$709$	−2.06815e7	−1.54514	−0.772569	−	0.634931i	$-0.781029\pi$
$709$	−2.06815e7	−1.54514	−0.772569	+	0.634931i	$0.781029\pi$
$710$	−5.95935e6	−0.443662
$711$	0	0
$712$	−1.97546e7	−1.46039
$713$	4.36044e6	0.321223
$714$	0	0
$715$	9.31396e6	0.681349
$716$	1.58975e7	1.15890
$717$	0	0
$718$	−9.74626e6	−0.705549
$719$	−2.51431e7	−1.81383	−0.906915	−	0.421314i	$-0.861569\pi$
$719$	−2.51431e7	−1.81383	−0.906915	+	0.421314i	$0.861569\pi$
$720$	0	0
$721$	−1.41225e7	−1.01175
$722$	−6.77550e6	−0.483725
$723$	0	0
$724$	1.41065e7	1.00017
$725$	−4.41134e6	−0.311692
$726$	0	0
$727$	1.11029e7	0.779110	0.389555	−	0.921003i	$-0.372629\pi$
$727$	1.11029e7	0.779110	0.389555	+	0.921003i	$0.372629\pi$
$728$	−8.11187e6	−0.567274
$729$	0	0
$730$	7.81778e6	0.542971
$731$	−2.82828e6	−0.195762
$732$	0	0
$733$	−2.43720e7	−1.67545	−0.837724	−	0.546094i	$-0.816114\pi$
$733$	−2.43720e7	−1.67545	−0.837724	+	0.546094i	$0.816114\pi$
$734$	−8.82349e6	−0.604506
$735$	0	0
$736$	3.17320e6	0.215925
$737$	−1.85452e6	−0.125766
$738$	0	0
$739$	4.98530e6	0.335799	0.167900	−	0.985804i	$-0.446302\pi$
$739$	4.98530e6	0.335799	0.167900	+	0.985804i	$0.446302\pi$
$740$	−3.39958e7	−2.28216
$741$	0	0
$742$	−4.81457e6	−0.321032
$743$	1.62876e7	1.08240	0.541198	−	0.840895i	$-0.317971\pi$
$743$	1.62876e7	1.08240	0.541198	+	0.840895i	$0.317971\pi$
$744$	0	0
$745$	2.70243e7	1.78387
$746$	8.32079e6	0.547416
$747$	0	0
$748$	4.17876e6	0.273082
$749$	−3.99089e6	−0.259935
$750$	0	0
$751$	−1.45274e7	−0.939917	−0.469958	−	0.882689i	$-0.655731\pi$
$751$	−1.45274e7	−0.939917	−0.469958	+	0.882689i	$0.655731\pi$
$752$	1.85753e6	0.119782
$753$	0	0
$754$	1.20349e6	0.0770929
$755$	−1.59989e6	−0.102146
$756$	0	0
$757$	−2.04109e7	−1.29456	−0.647280	−	0.762252i	$-0.724094\pi$
$757$	−2.04109e7	−1.29456	−0.647280	+	0.762252i	$0.724094\pi$
$758$	−5.73896e6	−0.362794
$759$	0	0
$760$	−6.12810e6	−0.384850
$761$	−1.63305e7	−1.02220	−0.511102	−	0.859520i	$-0.670762\pi$
$761$	−1.63305e7	−1.02220	−0.511102	+	0.859520i	$0.670762\pi$
$762$	0	0
$763$	1.74217e7	1.08338
$764$	1.09003e7	0.675623
$765$	0	0
$766$	−1.29853e7	−0.799612
$767$	1.18425e7	0.726869
$768$	0	0
$769$	5.66474e6	0.345433	0.172717	−	0.984972i	$-0.444746\pi$
$769$	5.66474e6	0.345433	0.172717	+	0.984972i	$0.444746\pi$
$770$	5.93211e6	0.360564
$771$	0	0
$772$	5.84593e6	0.353029
$773$	9.07134e6	0.546038	0.273019	−	0.962009i	$-0.411978\pi$
$773$	9.07134e6	0.546038	0.273019	+	0.962009i	$0.411978\pi$
$774$	0	0
$775$	4.25974e7	2.54758
$776$	4.16374e6	0.248216
$777$	0	0
$778$	−1.11984e7	−0.663294
$779$	−5.52418e6	−0.326155
$780$	0	0
$781$	4.74734e6	0.278498
$782$	−1.30445e6	−0.0762797
$783$	0	0
$784$	−1.61674e6	−0.0939398
$785$	−2.02103e7	−1.17057
$786$	0	0
$787$	−1.04842e7	−0.603388	−0.301694	−	0.953405i	$-0.597552\pi$
$787$	−1.04842e7	−0.603388	−0.301694	+	0.953405i	$0.597552\pi$
$788$	−4.60536e6	−0.264209
$789$	0	0
$790$	5.99606e6	0.341821
$791$	−1.53344e7	−0.871418
$792$	0	0
$793$	2.56781e7	1.45004
$794$	−1.89682e6	−0.106776
$795$	0	0
$796$	5.43107e6	0.303811
$797$	3.00338e6	0.167480	0.0837402	−	0.996488i	$-0.473313\pi$
$797$	3.00338e6	0.167480	0.0837402	+	0.996488i	$0.473313\pi$
$798$	0	0
$799$	−5.78831e6	−0.320764
$800$	3.09992e7	1.71248
$801$	0	0
$802$	57442.8	0.00315355
$803$	−6.22781e6	−0.340837
$804$	0	0
$805$	5.00670e6	0.272309
$806$	−1.16213e7	−0.630112
$807$	0	0
$808$	1.76302e7	0.950012
$809$	−3.59510e7	−1.93126	−0.965628	−	0.259928i	$-0.916301\pi$
$809$	−3.59510e7	−1.93126	−0.965628	+	0.259928i	$0.916301\pi$
$810$	0	0
$811$	−8.94130e6	−0.477363	−0.238681	−	0.971098i	$-0.576715\pi$
$811$	−8.94130e6	−0.477363	−0.238681	+	0.971098i	$0.576715\pi$
$812$	−2.07243e6	−0.110304
$813$	0	0
$814$	−1.00164e7	−0.529849
$815$	−4.91240e7	−2.59060
$816$	0	0
$817$	−1.39422e6	−0.0730760
$818$	1.04398e7	0.545520
$819$	0	0
$820$	2.84165e7	1.47583
$821$	3.40960e7	1.76541	0.882705	−	0.469927i	$-0.155720\pi$
$821$	3.40960e7	1.76541	0.882705	+	0.469927i	$0.155720\pi$
$822$	0	0
$823$	−1.89504e7	−0.975254	−0.487627	−	0.873052i	$-0.662137\pi$
$823$	−1.89504e7	−0.975254	−0.487627	+	0.873052i	$0.662137\pi$
$824$	−2.21116e7	−1.13449
$825$	0	0
$826$	7.54256e6	0.384653
$827$	−1.03460e7	−0.526026	−0.263013	−	0.964792i	$-0.584716\pi$
$827$	−1.03460e7	−0.526026	−0.263013	+	0.964792i	$0.584716\pi$
$828$	0	0
$829$	2.56154e7	1.29454	0.647269	−	0.762262i	$-0.275911\pi$
$829$	2.56154e7	1.29454	0.647269	+	0.762262i	$0.275911\pi$
$830$	−1.89141e7	−0.952993
$831$	0	0
$832$	−4.32500e6	−0.216609
$833$	5.03798e6	0.251561
$834$	0	0
$835$	−2.31824e7	−1.15065
$836$	2.05994e6	0.101939
$837$	0	0
$838$	−1.00309e7	−0.493436
$839$	1.99793e7	0.979886	0.489943	−	0.871754i	$-0.337018\pi$
$839$	1.99793e7	0.979886	0.489943	+	0.871754i	$0.337018\pi$
$840$	0	0
$841$	−1.97825e7	−0.964475
$842$	−4.82189e6	−0.234389
$843$	0	0
$844$	4.71259e6	0.227721
$845$	1.28609e7	0.619628
$846$	0	0
$847$	1.20125e7	0.575341
$848$	4.24284e6	0.202613
$849$	0	0
$850$	−1.27432e7	−0.604966
$851$	−8.45386e6	−0.400158
$852$	0	0
$853$	−1.62847e7	−0.766313	−0.383156	−	0.923683i	$-0.625163\pi$
$853$	−1.62847e7	−0.766313	−0.383156	+	0.923683i	$0.625163\pi$
$854$	1.63545e7	0.767347
$855$	0	0
$856$	−6.24852e6	−0.291469
$857$	3.53094e7	1.64224	0.821122	−	0.570752i	$-0.193348\pi$
$857$	3.53094e7	1.64224	0.821122	+	0.570752i	$0.193348\pi$
$858$	0	0
$859$	3.23051e7	1.49379	0.746893	−	0.664944i	$-0.231545\pi$
$859$	3.23051e7	1.49379	0.746893	+	0.664944i	$0.231545\pi$
$860$	7.17187e6	0.330664
$861$	0	0
$862$	1.33596e7	0.612385
$863$	3.40923e6	0.155822	0.0779112	−	0.996960i	$-0.475175\pi$
$863$	3.40923e6	0.155822	0.0779112	+	0.996960i	$0.475175\pi$
$864$	0	0
$865$	−3.08856e7	−1.40351
$866$	−293518.	−0.0132996
$867$	0	0
$868$	2.00121e7	0.901559
$869$	−4.77659e6	−0.214570
$870$	0	0
$871$	4.17156e6	0.186317
$872$	2.72771e7	1.21481
$873$	0	0
$874$	−643034.	−0.0284744
$875$	1.93342e7	0.853702
$876$	0	0
$877$	−4.50571e7	−1.97817	−0.989086	−	0.147338i	$-0.952929\pi$
$877$	−4.50571e7	−1.97817	−0.989086	+	0.147338i	$0.952929\pi$
$878$	−3.51985e6	−0.154095
$879$	0	0
$880$	−5.22767e6	−0.227563
$881$	1.16604e7	0.506143	0.253072	−	0.967448i	$-0.418559\pi$
$881$	1.16604e7	0.506143	0.253072	+	0.967448i	$0.418559\pi$
$882$	0	0
$883$	1.45999e7	0.630158	0.315079	−	0.949066i	$-0.397969\pi$
$883$	1.45999e7	0.630158	0.315079	+	0.949066i	$0.397969\pi$
$884$	−9.39970e6	−0.404560
$885$	0	0
$886$	1.49721e7	0.640764
$887$	−1.46833e7	−0.626635	−0.313318	−	0.949648i	$-0.601440\pi$
$887$	−1.46833e7	−0.626635	−0.313318	+	0.949648i	$0.601440\pi$
$888$	0	0
$889$	−6.06485e6	−0.257375
$890$	−3.24954e7	−1.37514
$891$	0	0
$892$	−2.76043e7	−1.16162
$893$	−2.85338e6	−0.119738
$894$	0	0
$895$	6.19737e7	2.58613
$896$	1.71951e7	0.715541
$897$	0	0
$898$	3.14939e6	0.130327
$899$	−7.03619e6	−0.290361
$900$	0	0
$901$	−1.32213e7	−0.542577
$902$	8.37257e6	0.342643
$903$	0	0
$904$	−2.40091e7	−0.977135
$905$	5.49917e7	2.23190
$906$	0	0
$907$	−3.96001e7	−1.59837	−0.799186	−	0.601084i	$-0.794736\pi$
$907$	−3.96001e7	−1.59837	−0.799186	+	0.601084i	$0.794736\pi$
$908$	−3.51541e7	−1.41502
$909$	0	0
$910$	−1.33437e7	−0.534161
$911$	−1.46325e7	−0.584146	−0.292073	−	0.956396i	$-0.594345\pi$
$911$	−1.46325e7	−0.584146	−0.292073	+	0.956396i	$0.594345\pi$
$912$	0	0
$913$	1.50673e7	0.598218
$914$	−3.74547e6	−0.148300
$915$	0	0
$916$	−2.02346e7	−0.796811
$917$	2.00477e6	0.0787303
$918$	0	0
$919$	−3.84096e7	−1.50021	−0.750104	−	0.661320i	$-0.769996\pi$
$919$	−3.84096e7	−1.50021	−0.750104	+	0.661320i	$0.769996\pi$
$920$	7.83897e6	0.305344
$921$	0	0
$922$	4.74593e6	0.183863
$923$	−1.06787e7	−0.412584
$924$	0	0
$925$	−8.25862e7	−3.17361
$926$	−4.86441e6	−0.186425
$927$	0	0
$928$	−5.12041e6	−0.195180
$929$	−827018.	−0.0314395	−0.0157198	−	0.999876i	$-0.505004\pi$
$929$	−827018.	−0.0314395	−0.0157198	+	0.999876i	$0.505004\pi$
$930$	0	0
$931$	2.48350e6	0.0939052
$932$	−491997.	−0.0185534
$933$	0	0
$934$	−2.37324e7	−0.890173
$935$	1.62901e7	0.609390
$936$	0	0
$937$	4.93119e7	1.83486	0.917430	−	0.397896i	$-0.130260\pi$
$937$	4.93119e7	1.83486	0.917430	+	0.397896i	$0.130260\pi$
$938$	2.65688e6	0.0985974
$939$	0	0
$940$	1.46778e7	0.541805
$941$	−2.09444e7	−0.771071	−0.385536	−	0.922693i	$-0.625983\pi$
$941$	−2.09444e7	−0.771071	−0.385536	+	0.922693i	$0.625983\pi$
$942$	0	0
$943$	7.06644e6	0.258775
$944$	−6.64688e6	−0.242766
$945$	0	0
$946$	2.11310e6	0.0767702
$947$	9.11296e6	0.330206	0.165103	−	0.986276i	$-0.447204\pi$
$947$	9.11296e6	0.330206	0.165103	+	0.986276i	$0.447204\pi$
$948$	0	0
$949$	1.40088e7	0.504936
$950$	−6.28182e6	−0.225827
$951$	0	0
$952$	−1.41877e7	−0.507363
$953$	1.18036e7	0.421000	0.210500	−	0.977594i	$-0.432491\pi$
$953$	1.18036e7	0.421000	0.210500	+	0.977594i	$0.432491\pi$
$954$	0	0
$955$	4.24927e7	1.50767
$956$	−1.67807e7	−0.593835
$957$	0	0
$958$	−1.22814e6	−0.0432350
$959$	4.09821e6	0.143896
$960$	0	0
$961$	3.93147e7	1.37324
$962$	2.25310e7	0.784950
$963$	0	0
$964$	−9.10512e6	−0.315568
$965$	2.27893e7	0.787794
$966$	0	0
$967$	3.22752e7	1.10995	0.554975	−	0.831867i	$-0.312728\pi$
$967$	3.22752e7	1.10995	0.554975	+	0.831867i	$0.312728\pi$
$968$	1.88080e7	0.645139
$969$	0	0
$970$	6.84918e6	0.233727
$971$	−2.56069e7	−0.871585	−0.435792	−	0.900047i	$-0.643532\pi$
$971$	−2.56069e7	−0.871585	−0.435792	+	0.900047i	$0.643532\pi$
$972$	0	0
$973$	−3.49076e7	−1.18205
$974$	−2.78379e7	−0.940240
$975$	0	0
$976$	−1.44124e7	−0.484296
$977$	4.18582e7	1.40296	0.701478	−	0.712691i	$-0.252524\pi$
$977$	4.18582e7	1.40296	0.701478	+	0.712691i	$0.252524\pi$
$978$	0	0
$979$	2.58865e7	0.863212
$980$	−1.27752e7	−0.424914
$981$	0	0
$982$	2.47933e7	0.820457
$983$	2.90753e7	0.959711	0.479856	−	0.877347i	$-0.340689\pi$
$983$	2.90753e7	0.959711	0.479856	+	0.877347i	$0.340689\pi$
$984$	0	0
$985$	−1.79532e7	−0.589590
$986$	2.10491e6	0.0689510
$987$	0	0
$988$	−4.63364e6	−0.151018
$989$	1.78346e6	0.0579792
$990$	0	0
$991$	−1.24529e7	−0.402796	−0.201398	−	0.979509i	$-0.564548\pi$
$991$	−1.24529e7	−0.402796	−0.201398	+	0.979509i	$0.564548\pi$
$992$	4.94444e7	1.59528
$993$	0	0
$994$	−6.80129e6	−0.218336
$995$	2.11720e7	0.677962
$996$	0	0
$997$	−2.78009e7	−0.885771	−0.442886	−	0.896578i	$-0.646045\pi$
$997$	−2.78009e7	−0.885771	−0.442886	+	0.896578i	$0.646045\pi$
$998$	2.73652e7	0.869705
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.i.1.7	yes	10
3.2	odd	2		207.6.a.h.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.4	✓	10	3.2	odd	2
207.6.a.i.1.7	yes	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+2.93938 q^{2} -23.3601 q^{4} -91.0649 q^{5} -103.931 q^{7} -162.724 q^{8} +O(q^{10})\)	\(q+2.93938 q^{2} -23.3601 q^{4} -91.0649 q^{5} -103.931 q^{7} -162.724 q^{8} -267.674 q^{10} +213.235 q^{11} -479.651 q^{13} -305.492 q^{14} +269.215 q^{16} -838.909 q^{17} -413.545 q^{19} +2127.28 q^{20} +626.778 q^{22} +529.000 q^{23} +5167.82 q^{25} -1409.87 q^{26} +2427.83 q^{28} -853.616 q^{29} +8242.80 q^{31} +5998.49 q^{32} -2465.87 q^{34} +9464.45 q^{35} -15980.8 q^{37} -1215.56 q^{38} +14818.5 q^{40} +13358.1 q^{41} +3371.38 q^{43} -4981.18 q^{44} +1554.93 q^{46} +6899.81 q^{47} -6005.39 q^{49} +15190.2 q^{50} +11204.7 q^{52} +15760.1 q^{53} -19418.2 q^{55} +16912.0 q^{56} -2509.10 q^{58} -24689.9 q^{59} -53534.9 q^{61} +24228.7 q^{62} +9016.97 q^{64} +43679.4 q^{65} -8697.07 q^{67} +19597.0 q^{68} +27819.6 q^{70} +22263.4 q^{71} -29206.3 q^{73} -46973.7 q^{74} +9660.44 q^{76} -22161.7 q^{77} -22400.6 q^{79} -24516.0 q^{80} +39264.5 q^{82} +70660.8 q^{83} +76395.2 q^{85} +9909.75 q^{86} -34698.4 q^{88} +121399. q^{89} +49850.5 q^{91} -12357.5 q^{92} +20281.1 q^{94} +37659.5 q^{95} -25587.7 q^{97} -17652.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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