LMFDB - Embedded newform 297.3.c.a.109.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [297,3,Mod(109,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("297.109"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	297.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	no
Analytic conductor:	$8.09266385150$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 212 x^{14} + 20776 x^{12} - 1001288 x^{10} + 21274804 x^{8} - 80418176 x^{6} + \cdots + 421886622784 $ x^16 - 212x^14 + 20776x^12 - 1001288x^10 + 21274804x^8 - 80418176x^6 - 1860160544x^4 + 2985017728*x^2 + 421886622784
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.11
Root			$-3.98989 - 1.29424i$ of defining polynomial
Character	$\chi$	$=$	297.109
Dual form			297.3.c.a.109.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.29424i q^{2} +2.32494 q^{4} -3.98989 q^{5} +7.86477i q^{7} +8.18599i q^{8} -5.16388i q^{10} +(-9.91473 - 4.76425i) q^{11} -0.289207i q^{13} -10.1789 q^{14} -1.29487 q^{16} -0.214945i q^{17} +25.5715i q^{19} -9.27627 q^{20} +(6.16608 - 12.8320i) q^{22} -14.1688 q^{23} -9.08076 q^{25} +0.374303 q^{26} +18.2851i q^{28} +47.4977i q^{29} -6.71234 q^{31} +31.0681i q^{32} +0.278190 q^{34} -31.3796i q^{35} +18.3694 q^{37} -33.0956 q^{38} -32.6612i q^{40} +1.67587i q^{41} -74.7886i q^{43} +(-23.0512 - 11.0766i) q^{44} -18.3378i q^{46} -51.1148 q^{47} -12.8545 q^{49} -11.7527i q^{50} -0.672389i q^{52} +56.8095 q^{53} +(39.5587 + 19.0088i) q^{55} -64.3809 q^{56} -61.4734 q^{58} +47.2789 q^{59} +53.8026i q^{61} -8.68738i q^{62} -45.3891 q^{64} +1.15390i q^{65} -12.6774 q^{67} -0.499734i q^{68} +40.6127 q^{70} +110.454 q^{71} -33.1017i q^{73} +23.7745i q^{74} +59.4522i q^{76} +(37.4697 - 77.9771i) q^{77} +73.9696i q^{79} +5.16639 q^{80} -2.16898 q^{82} +88.4291i q^{83} +0.857606i q^{85} +96.7944 q^{86} +(39.0001 - 81.1620i) q^{88} +161.349 q^{89} +2.27454 q^{91} -32.9416 q^{92} -66.1548i q^{94} -102.027i q^{95} -59.3695 q^{97} -16.6368i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 40 q^{16} + 24 q^{22} + 120 q^{25} - 88 q^{31} - 216 q^{34} + 56 q^{37} - 32 q^{49} - 280 q^{55} + 120 q^{58} + 376 q^{64} + 200 q^{67} - 192 q^{70} + 456 q^{82} - 168 q^{88} - 264 q^{91}+ \cdots + 80 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 40 * q^16 + 24 * q^22 + 120 * q^25 - 88 * q^31 - 216 * q^34 + 56 * q^37 - 32 * q^49 - 280 * q^55 + 120 * q^58 + 376 * q^64 + 200 * q^67 - 192 * q^70 + 456 * q^82 - 168 * q^88 - 264 * q^91 + 80 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.29424i			0.647120i	0.946208	+	0.323560i	$0.104880\pi$
$2$			1.29424i			0.647120i	−0.946208	+	0.323560i	$0.895120\pi$
$3$		0			0
$3$		0			0
$4$	2.32494			0.581236
$5$	−3.98989			−0.797978			−0.398989	−	0.916956i	$-0.630639\pi$
$5$	−3.98989			−0.797978			−0.398989	+	0.916956i	$0.630639\pi$
$6$		0			0
$7$			7.86477i			1.12354i	0.827294	+	0.561769i	$0.189879\pi$
$7$			7.86477i			1.12354i	−0.827294	+	0.561769i	$0.810121\pi$
$8$			8.18599i			1.02325i
$9$		0			0
$10$		−	5.16388i		−	0.516388i
$11$	−9.91473	−	4.76425i	−0.901339	−	0.433113i
$11$	−9.91473	−	4.76425i	−0.901339	−	0.433113i
$12$		0			0
$13$		−	0.289207i		−	0.0222467i	−0.999938	−	0.0111233i	$-0.996459\pi$
$13$		−	0.289207i		−	0.0222467i	0.999938	−	0.0111233i	$-0.00354074\pi$
$14$	−10.1789			−0.727064
$15$		0			0
$16$	−1.29487			−0.0809293
$17$		−	0.214945i		−	0.0126438i	−0.999980	−	0.00632190i	$-0.997988\pi$
$17$		−	0.214945i		−	0.0126438i	0.999980	−	0.00632190i	$-0.00201234\pi$
$18$		0			0
$19$			25.5715i			1.34587i	0.739703	+	0.672934i	$0.234966\pi$
$19$			25.5715i			1.34587i	−0.739703	+	0.672934i	$0.765034\pi$
$20$	−9.27627			−0.463814
$21$		0			0
$22$	6.16608	−	12.8320i	0.280276	−	0.583275i
$23$	−14.1688			−0.616034			−0.308017	−	0.951381i	$-0.599665\pi$
$23$	−14.1688			−0.616034			−0.308017	+	0.951381i	$0.599665\pi$
$24$		0			0
$25$	−9.08076			−0.363230
$26$	0.374303			0.0143963
$27$		0			0
$28$			18.2851i			0.653040i
$29$			47.4977i			1.63785i	0.573900	+	0.818926i	$0.305430\pi$
$29$			47.4977i			1.63785i	−0.573900	+	0.818926i	$0.694570\pi$
$30$		0			0
$31$	−6.71234			−0.216527			−0.108264	−	0.994122i	$-0.534529\pi$
$31$	−6.71234			−0.216527			−0.108264	+	0.994122i	$0.534529\pi$
$32$			31.0681i			0.970878i
$33$		0			0
$34$	0.278190			0.00818206
$35$		−	31.3796i		−	0.896559i
$36$		0			0
$37$	18.3694			0.496471			0.248236	−	0.968700i	$-0.420149\pi$
$37$	18.3694			0.496471			0.248236	+	0.968700i	$0.420149\pi$
$38$	−33.0956			−0.870938
$39$		0			0
$40$		−	32.6612i		−	0.816531i
$41$			1.67587i			0.0408749i	0.999791	+	0.0204374i	$0.00650589\pi$
$41$			1.67587i			0.0408749i	−0.999791	+	0.0204374i	$0.993494\pi$
$42$		0			0
$43$		−	74.7886i		−	1.73927i	−0.493696	−	0.869635i	$-0.664354\pi$
$43$		−	74.7886i		−	1.73927i	0.493696	−	0.869635i	$-0.335646\pi$
$44$	−23.0512	−	11.0766i	−0.523891	−	0.251741i
$45$		0			0
$46$		−	18.3378i		−	0.398648i
$47$	−51.1148			−1.08755			−0.543774	−	0.839232i	$-0.683005\pi$
$47$	−51.1148			−1.08755			−0.543774	+	0.839232i	$0.683005\pi$
$48$		0			0
$49$	−12.8545			−0.262337
$50$		−	11.7527i		−	0.235054i
$51$		0			0
$52$		−	0.672389i		−	0.0129306i
$53$	56.8095			1.07188			0.535938	−	0.844257i	$-0.319958\pi$
$53$	56.8095			1.07188			0.535938	+	0.844257i	$0.319958\pi$
$54$		0			0
$55$	39.5587	+	19.0088i	0.719250	+	0.345615i
$56$	−64.3809			−1.14966
$57$		0			0
$58$	−61.4734			−1.05989
$59$	47.2789			0.801337			0.400668	−	0.916223i	$-0.368778\pi$
$59$	47.2789			0.801337			0.400668	+	0.916223i	$0.368778\pi$
$60$		0			0
$61$			53.8026i			0.882009i	0.897505	+	0.441005i	$0.145378\pi$
$61$			53.8026i			0.882009i	−0.897505	+	0.441005i	$0.854622\pi$
$62$		−	8.68738i		−	0.140119i
$63$		0			0
$64$	−45.3891			−0.709204
$65$			1.15390i			0.0177524i
$66$		0			0
$67$	−12.6774			−0.189215			−0.0946075	−	0.995515i	$-0.530160\pi$
$67$	−12.6774			−0.189215			−0.0946075	+	0.995515i	$0.530160\pi$
$68$		−	0.499734i		−	0.00734903i
$69$		0			0
$70$	40.6127			0.580181
$71$	110.454			1.55570			0.777848	−	0.628453i	$-0.216311\pi$
$71$	110.454			1.55570			0.777848	+	0.628453i	$0.216311\pi$
$72$		0			0
$73$		−	33.1017i		−	0.453448i	−0.973959	−	0.226724i	$-0.927198\pi$
$73$		−	33.1017i		−	0.453448i	0.973959	−	0.226724i	$-0.0728015\pi$
$74$			23.7745i			0.321277i
$75$		0			0
$76$			59.4522i			0.782266i
$77$	37.4697	−	77.9771i	0.486619	−	1.01269i
$78$		0			0
$79$			73.9696i			0.936324i	0.883643	+	0.468162i	$0.155084\pi$
$79$			73.9696i			0.936324i	−0.883643	+	0.468162i	$0.844916\pi$
$80$	5.16639			0.0645798
$81$		0			0
$82$	−2.16898			−0.0264510
$83$			88.4291i			1.06541i	0.846301	+	0.532705i	$0.178825\pi$
$83$			88.4291i			1.06541i	−0.846301	+	0.532705i	$0.821175\pi$
$84$		0			0
$85$			0.857606i			0.0100895i
$86$	96.7944			1.12552
$87$		0			0
$88$	39.0001	−	81.1620i	0.443183	−	0.922295i
$89$	161.349			1.81291			0.906454	−	0.422304i	$-0.138779\pi$
$89$	161.349			1.81291			0.906454	+	0.422304i	$0.138779\pi$
$90$		0			0
$91$	2.27454			0.0249950
$92$	−32.9416			−0.358061
$93$		0			0
$94$		−	66.1548i		−	0.703774i
$95$		−	102.027i		−	1.07397i
$96$		0			0
$97$	−59.3695			−0.612057			−0.306029	−	0.952022i	$-0.599000\pi$
$97$	−59.3695			−0.612057			−0.306029	+	0.952022i	$0.599000\pi$
$98$		−	16.6368i		−	0.169764i
$99$		0			0
$100$	−21.1122			−0.211122
$101$		−	139.852i		−	1.38467i	−0.721577	−	0.692334i	$-0.756582\pi$
$101$		−	139.852i		−	1.38467i	0.721577	−	0.692334i	$-0.243418\pi$
$102$		0			0
$103$	167.886			1.62997			0.814983	−	0.579485i	$-0.196746\pi$
$103$	167.886			1.62997			0.814983	+	0.579485i	$0.196746\pi$
$104$	2.36744			0.0227639
$105$		0			0
$106$			73.5251i			0.693633i
$107$		−	196.445i		−	1.83593i	−0.396660	−	0.917965i	$-0.629831\pi$
$107$		−	196.445i		−	1.83593i	0.396660	−	0.917965i	$-0.370169\pi$
$108$		0			0
$109$			68.2786i			0.626409i	0.949686	+	0.313205i	$0.101403\pi$
$109$			68.2786i			0.626409i	−0.949686	+	0.313205i	$0.898597\pi$
$110$	−24.6020	+	51.1985i	−0.223654	+	0.465441i
$111$		0			0
$112$		−	10.1838i		−	0.0909271i
$113$	60.3963			0.534480			0.267240	−	0.963630i	$-0.413888\pi$
$113$	60.3963			0.534480			0.267240	+	0.963630i	$0.413888\pi$
$114$		0			0
$115$	56.5319			0.491582
$116$			110.429i			0.951978i
$117$		0			0
$118$			61.1902i			0.518561i
$119$	1.69049			0.0142058
$120$		0			0
$121$	75.6039	+	94.4725i	0.624826	+	0.780764i
$122$	−69.6334			−0.570766
$123$		0			0
$124$	−15.6058			−0.125853
$125$	135.979			1.08783
$126$		0			0
$127$		−	100.112i		−	0.788285i	−0.919049	−	0.394142i	$-0.871042\pi$
$127$		−	100.112i		−	0.788285i	0.919049	−	0.394142i	$-0.128958\pi$
$128$			65.5281i			0.511938i
$129$		0			0
$130$	−1.49343			−0.0114879
$131$			143.403i			1.09468i	0.836911	+	0.547338i	$0.184359\pi$
$131$			143.403i			1.09468i	−0.836911	+	0.547338i	$0.815641\pi$
$132$		0			0
$133$	−201.114			−1.51213
$134$		−	16.4076i		−	0.122445i
$135$		0			0
$136$	1.75954			0.0129378
$137$	−37.8146			−0.276019			−0.138009	−	0.990431i	$-0.544070\pi$
$137$	−37.8146			−0.276019			−0.138009	+	0.990431i	$0.544070\pi$
$138$		0			0
$139$			126.595i			0.910757i	0.890298	+	0.455379i	$0.150496\pi$
$139$			126.595i			0.910757i	−0.890298	+	0.455379i	$0.849504\pi$
$140$		−	72.9557i		−	0.521112i
$141$		0			0
$142$			142.954i			1.00672i
$143$	−1.37785	+	2.86741i	−0.00963533	+	0.0200518i
$144$		0			0
$145$		−	189.511i		−	1.30697i
$146$	42.8415			0.293435
$147$		0			0
$148$	42.7079			0.288567
$149$			148.260i			0.995034i	0.867454	+	0.497517i	$0.165755\pi$
$149$			148.260i			0.995034i	−0.867454	+	0.497517i	$0.834245\pi$
$150$		0			0
$151$		−	235.934i		−	1.56248i	−0.624232	−	0.781239i	$-0.714588\pi$
$151$		−	235.934i		−	1.56248i	0.624232	−	0.781239i	$-0.285412\pi$
$152$	−209.328			−1.37716
$153$		0			0
$154$	100.921	+	48.4948i	0.655331	+	0.314901i
$155$	26.7815			0.172784
$156$		0			0
$157$	−154.184			−0.982064			−0.491032	−	0.871142i	$-0.663380\pi$
$157$	−154.184			−0.982064			−0.491032	+	0.871142i	$0.663380\pi$
$158$	−95.7344			−0.605914
$159$		0			0
$160$		−	123.958i		−	0.774740i
$161$		−	111.434i		−	0.692138i
$162$		0			0
$163$	−260.743			−1.59965			−0.799825	−	0.600233i	$-0.795075\pi$
$163$	−260.743			−1.59965			−0.799825	+	0.600233i	$0.795075\pi$
$164$			3.89630i			0.0237579i
$165$		0			0
$166$	−114.448			−0.689449
$167$			61.4058i			0.367700i	0.982954	+	0.183850i	$0.0588560\pi$
$167$			61.4058i			0.367700i	−0.982954	+	0.183850i	$0.941144\pi$
$168$		0			0
$169$	168.916			0.999505
$170$	−1.10995			−0.00652911
$171$		0			0
$172$		−	173.879i		−	1.01093i
$173$			0.445334i			0.00257419i	0.999999	+	0.00128709i	$0.000409695\pi$
$173$			0.445334i			0.00257419i	−0.999999	+	0.00128709i	$0.999590\pi$
$174$		0			0
$175$		−	71.4180i		−	0.408103i
$176$	12.8383	+	6.16907i	0.0729448	+	0.0350515i
$177$		0			0
$178$			208.824i			1.17317i
$179$	−46.4819			−0.259676			−0.129838	−	0.991535i	$-0.541446\pi$
$179$	−46.4819			−0.259676			−0.129838	+	0.991535i	$0.541446\pi$
$180$		0			0
$181$	89.3545			0.493671			0.246836	−	0.969057i	$-0.420609\pi$
$181$	89.3545			0.493671			0.246836	+	0.969057i	$0.420609\pi$
$182$			2.94380i			0.0161748i
$183$		0			0
$184$		−	115.986i		−	0.630356i
$185$	−73.2921			−0.396174
$186$		0			0
$187$	−1.02405	+	2.13112i	−0.00547620	+	0.0113964i
$188$	−118.839			−0.632122
$189$		0			0
$190$	132.048			0.694990
$191$	−93.8287			−0.491250			−0.245625	−	0.969365i	$-0.578993\pi$
$191$	−93.8287			−0.491250			−0.245625	+	0.969365i	$0.578993\pi$
$192$		0			0
$193$			154.194i			0.798935i	0.916747	+	0.399467i	$0.130805\pi$
$193$			154.194i			0.798935i	−0.916747	+	0.399467i	$0.869195\pi$
$194$		−	76.8384i		−	0.396074i
$195$		0			0
$196$	−29.8861			−0.152480
$197$			65.3959i			0.331959i	0.986129	+	0.165979i	$0.0530785\pi$
$197$			65.3959i			0.331959i	−0.986129	+	0.165979i	$0.946921\pi$
$198$		0			0
$199$	−293.198			−1.47336			−0.736678	−	0.676243i	$-0.763607\pi$
$199$	−293.198			−1.47336			−0.736678	+	0.676243i	$0.763607\pi$
$200$		−	74.3350i		−	0.371675i
$201$		0			0
$202$	181.001			0.896047
$203$	−373.558			−1.84019
$204$		0			0
$205$		−	6.68654i		−	0.0326173i
$206$			217.285i			1.05478i
$207$		0			0
$208$			0.374485i			0.00180041i
$209$	121.829	−	253.534i	0.582913	−	1.21308i
$210$		0			0
$211$		−	114.256i		−	0.541499i	−0.962650	−	0.270750i	$-0.912728\pi$
$211$		−	114.256i		−	0.541499i	0.962650	−	0.270750i	$-0.0872715\pi$
$212$	132.079			0.623013
$213$		0			0
$214$	254.246			1.18807
$215$			298.398i			1.38790i
$216$		0			0
$217$		−	52.7910i		−	0.243277i
$218$	−88.3689			−0.405362
$219$		0			0
$220$	91.9718	+	44.1944i	0.418054	+	0.200884i
$221$	−0.0621635			−0.000281283
$222$		0			0
$223$	222.752			0.998888			0.499444	−	0.866346i	$-0.333538\pi$
$223$	222.752			0.998888			0.499444	+	0.866346i	$0.333538\pi$
$224$	−244.343			−1.09082
$225$		0			0
$226$			78.1673i			0.345873i
$227$			390.505i			1.72029i	0.510053	+	0.860143i	$0.329626\pi$
$227$			390.505i			1.72029i	−0.510053	+	0.860143i	$0.670374\pi$
$228$		0			0
$229$	−143.282			−0.625686			−0.312843	−	0.949805i	$-0.601281\pi$
$229$	−143.282			−0.625686			−0.312843	+	0.949805i	$0.601281\pi$
$230$			73.1659i			0.318113i
$231$		0			0
$232$	−388.816			−1.67593
$233$		−	47.9869i		−	0.205953i	−0.994684	−	0.102976i	$-0.967163\pi$
$233$		−	47.9869i		−	0.205953i	0.994684	−	0.102976i	$-0.0328366\pi$
$234$		0			0
$235$	203.942			0.867840
$236$	109.921			0.465766
$237$		0			0
$238$			2.18790i			0.00919286i
$239$		−	25.1964i		−	0.105424i	−0.998610	−	0.0527121i	$-0.983213\pi$
$239$		−	25.1964i		−	0.105424i	0.998610	−	0.0527121i	$-0.0167866\pi$
$240$		0			0
$241$		−	198.881i		−	0.825234i	−0.910905	−	0.412617i	$-0.864615\pi$
$241$		−	198.881i		−	0.825234i	0.910905	−	0.412617i	$-0.135385\pi$
$242$	−122.270	+	97.8496i	−0.505248	+	0.404337i
$243$		0			0
$244$			125.088i			0.512655i
$245$	51.2882			0.209340
$246$		0			0
$247$	7.39545			0.0299411
$248$		−	54.9472i		−	0.221561i
$249$		0			0
$250$			175.989i			0.703956i
$251$	−368.200			−1.46693			−0.733467	−	0.679725i	$-0.762099\pi$
$251$	−368.200			−1.46693			−0.733467	+	0.679725i	$0.762099\pi$
$252$		0			0
$253$	140.480	+	67.5036i	0.555256	+	0.266813i
$254$	129.569			0.510115
$255$		0			0
$256$	−266.365			−1.04049
$257$	−228.729			−0.889997			−0.444999	−	0.895531i	$-0.646796\pi$
$257$	−228.729			−0.889997			−0.444999	+	0.895531i	$0.646796\pi$
$258$		0			0
$259$			144.471i			0.557804i
$260$			2.68276i			0.0103183i
$261$		0			0
$262$	−185.597			−0.708387
$263$			37.2936i			0.141801i	0.997483	+	0.0709003i	$0.0225872\pi$
$263$			37.2936i			0.141801i	−0.997483	+	0.0709003i	$0.977413\pi$
$264$		0			0
$265$	−226.664			−0.855334
$266$		−	260.289i		−	0.978532i
$267$		0			0
$268$	−29.4742			−0.109978
$269$	135.749			0.504643			0.252322	−	0.967643i	$-0.418806\pi$
$269$	135.749			0.504643			0.252322	+	0.967643i	$0.418806\pi$
$270$		0			0
$271$			353.114i			1.30300i	0.758647	+	0.651502i	$0.225861\pi$
$271$			353.114i			1.30300i	−0.758647	+	0.651502i	$0.774139\pi$
$272$			0.278325i			0.00102325i
$273$		0			0
$274$		−	48.9411i		−	0.178617i
$275$	90.0333	+	43.2630i	0.327394	+	0.157320i
$276$		0			0
$277$		−	219.942i		−	0.794016i	−0.917815	−	0.397008i	$-0.870049\pi$
$277$		−	219.942i		−	0.794016i	0.917815	−	0.397008i	$-0.129951\pi$
$278$	−163.845			−0.589369
$279$		0			0
$280$	256.873			0.917403
$281$		−	499.010i		−	1.77583i	−0.460003	−	0.887917i	$-0.652152\pi$
$281$		−	499.010i		−	1.77583i	0.460003	−	0.887917i	$-0.347848\pi$
$282$		0			0
$283$			361.710i			1.27813i	0.769154	+	0.639064i	$0.220678\pi$
$283$			361.710i			1.27813i	−0.769154	+	0.639064i	$0.779322\pi$
$284$	256.800			0.904226
$285$		0			0
$286$	−3.71111	−	1.78327i	−0.0129759	−	0.00623521i
$287$	−13.1803			−0.0459245
$288$		0			0
$289$	288.954			0.999840
$290$	245.272			0.845766
$291$		0			0
$292$		−	76.9595i		−	0.263560i
$293$			388.428i			1.32569i	0.748756	+	0.662846i	$0.230652\pi$
$293$			388.428i			1.32569i	−0.748756	+	0.662846i	$0.769348\pi$
$294$		0			0
$295$	−188.638			−0.639450
$296$			150.372i			0.508014i
$297$		0			0
$298$	−191.884			−0.643906
$299$			4.09771i			0.0137047i
$300$		0			0
$301$	588.195			1.95413
$302$	305.356			1.01111
$303$		0			0
$304$		−	33.1117i		−	0.108920i
$305$		−	214.666i		−	0.703824i
$306$		0			0
$307$			19.1202i			0.0622806i	0.999515	+	0.0311403i	$0.00991387\pi$
$307$			19.1202i			0.0622806i	−0.999515	+	0.0311403i	$0.990086\pi$
$308$	87.1149	−	181.292i	0.282840	−	0.588611i
$309$		0			0
$310$			34.6617i			0.111812i
$311$	−200.557			−0.644877			−0.322438	−	0.946590i	$-0.604502\pi$
$311$	−200.557			−0.644877			−0.322438	+	0.946590i	$0.604502\pi$
$312$		0			0
$313$	236.316			0.755003			0.377501	−	0.926009i	$-0.376783\pi$
$313$	236.316			0.755003			0.377501	+	0.926009i	$0.376783\pi$
$314$		−	199.551i		−	0.635513i
$315$		0			0
$316$			171.975i			0.544225i
$317$	462.319			1.45842			0.729210	−	0.684290i	$-0.239888\pi$
$317$	462.319			1.45842			0.729210	+	0.684290i	$0.239888\pi$
$318$		0			0
$319$	226.291	−	470.927i	0.709375	−	1.47626i
$320$	181.097			0.565930
$321$		0			0
$322$	144.223			0.447896
$323$	5.49646			0.0170169
$324$		0			0
$325$			2.62622i			0.00808067i
$326$		−	337.464i		−	1.03517i
$327$		0			0
$328$	−13.7187			−0.0418252
$329$		−	402.006i		−	1.22190i
$330$		0			0
$331$	−579.164			−1.74974			−0.874870	−	0.484358i	$-0.839053\pi$
$331$	−579.164			−1.74974			−0.874870	+	0.484358i	$0.839053\pi$
$332$			205.593i			0.619255i
$333$		0			0
$334$	−79.4739			−0.237946
$335$	50.5815			0.150989
$336$		0			0
$337$		−	545.027i		−	1.61729i	−0.588296	−	0.808646i	$-0.700201\pi$
$337$		−	545.027i		−	1.61729i	0.588296	−	0.808646i	$-0.299799\pi$
$338$			218.618i			0.646800i
$339$		0			0
$340$			1.99389i			0.00586437i
$341$	66.5511	+	31.9793i	0.195165	+	0.0937808i
$342$		0			0
$343$			284.276i			0.828792i
$344$	612.219			1.77971
$345$		0			0
$346$	−0.576369			−0.00166581
$347$			30.4501i			0.0877524i	0.999037	+	0.0438762i	$0.0139707\pi$
$347$			30.4501i			0.0877524i	−0.999037	+	0.0438762i	$0.986029\pi$
$348$		0			0
$349$			251.999i			0.722060i	0.932554	+	0.361030i	$0.117575\pi$
$349$			251.999i			0.722060i	−0.932554	+	0.361030i	$0.882425\pi$
$350$	92.4321			0.264092
$351$		0			0
$352$	148.016	−	308.032i	0.420500	−	0.875091i
$353$	144.650			0.409773			0.204887	−	0.978786i	$-0.434317\pi$
$353$	144.650			0.409773			0.204887	+	0.978786i	$0.434317\pi$
$354$		0			0
$355$	−440.701			−1.24141
$356$	375.127			1.05373
$357$		0			0
$358$		−	60.1588i		−	0.168041i
$359$		−	248.507i		−	0.692221i	−0.938194	−	0.346111i	$-0.887502\pi$
$359$		−	248.507i		−	0.692221i	0.938194	−	0.346111i	$-0.112498\pi$
$360$		0			0
$361$	−292.901			−0.811359
$362$			115.646i			0.319464i
$363$		0			0
$364$	5.28818			0.0145280
$365$			132.072i			0.361842i
$366$		0			0
$367$	200.007			0.544977			0.272489	−	0.962159i	$-0.412153\pi$
$367$	200.007			0.544977			0.272489	+	0.962159i	$0.412153\pi$
$368$	18.3467			0.0498552
$369$		0			0
$370$		−	94.8576i		−	0.256372i
$371$			446.793i			1.20429i
$372$		0			0
$373$		−	143.432i		−	0.384535i	−0.981343	−	0.192268i	$-0.938416\pi$
$373$		−	143.432i		−	0.384535i	0.981343	−	0.192268i	$-0.0615842\pi$
$374$	−2.75818	−	1.32537i	−0.00737481	−	0.00354376i
$375$		0			0
$376$		−	418.425i		−	1.11283i
$377$	13.7367			0.0364367
$378$		0			0
$379$	97.7065			0.257801			0.128900	−	0.991658i	$-0.458855\pi$
$379$	97.7065			0.257801			0.128900	+	0.991658i	$0.458855\pi$
$380$		−	237.208i		−	0.624232i
$381$		0			0
$382$		−	121.437i		−	0.317898i
$383$	−296.262			−0.773530			−0.386765	−	0.922178i	$-0.626408\pi$
$383$	−296.262			−0.773530			−0.386765	+	0.922178i	$0.626408\pi$
$384$		0			0
$385$	−149.500	+	311.120i	−0.388312	+	0.808104i
$386$	−199.565			−0.517007
$387$		0			0
$388$	−138.031			−0.355749
$389$	397.383			1.02155			0.510775	−	0.859715i	$-0.329359\pi$
$389$	397.383			1.02155			0.510775	+	0.859715i	$0.329359\pi$
$390$		0			0
$391$			3.04551i			0.00778902i
$392$		−	105.227i		−	0.268437i
$393$		0			0
$394$	−84.6380			−0.214817
$395$		−	295.131i		−	0.747166i
$396$		0			0
$397$	−388.039			−0.977427			−0.488714	−	0.872444i	$-0.662534\pi$
$397$	−388.039			−0.977427			−0.488714	+	0.872444i	$0.662534\pi$
$398$		−	379.468i		−	0.953438i
$399$		0			0
$400$	11.7584			0.0293960
$401$	−188.010			−0.468854			−0.234427	−	0.972134i	$-0.575321\pi$
$401$	−188.010			−0.468854			−0.234427	+	0.972134i	$0.575321\pi$
$402$		0			0
$403$			1.94126i			0.00481701i
$404$		−	325.147i		−	0.804819i
$405$		0			0
$406$		−	483.474i		−	1.19082i
$407$	−182.128	−	87.5166i	−0.447489	−	0.215028i
$408$		0			0
$409$			614.023i			1.50128i	0.660711	+	0.750640i	$0.270255\pi$
$409$			614.023i			1.50128i	−0.660711	+	0.750640i	$0.729745\pi$
$410$	8.65399			0.0211073
$411$		0			0
$412$	390.326			0.947394
$413$			371.837i			0.900332i
$414$		0			0
$415$		−	352.823i		−	0.850175i
$416$	8.98511			0.0215988
$417$		0			0
$418$	328.134	+	157.676i	0.785011	+	0.377215i
$419$	−369.723			−0.882393			−0.441196	−	0.897411i	$-0.645446\pi$
$419$	−369.723			−0.882393			−0.441196	+	0.897411i	$0.645446\pi$
$420$		0			0
$421$	444.512			1.05585			0.527924	−	0.849292i	$-0.322971\pi$
$421$	444.512			1.05585			0.527924	+	0.849292i	$0.322971\pi$
$422$	147.875			0.350415
$423$		0			0
$424$			465.042i			1.09680i
$425$			1.95186i			0.00459261i
$426$		0			0
$427$	−423.144			−0.990971
$428$		−	456.723i		−	1.06711i
$429$		0			0
$430$	−386.199			−0.898137
$431$			727.577i			1.68811i	0.536254	+	0.844057i	$0.319839\pi$
$431$			727.577i			1.68811i	−0.536254	+	0.844057i	$0.680161\pi$
$432$		0			0
$433$	518.370			1.19716			0.598580	−	0.801063i	$-0.295732\pi$
$433$	518.370			1.19716			0.598580	+	0.801063i	$0.295732\pi$
$434$	68.3242			0.157429
$435$		0			0
$436$			158.744i			0.364091i
$437$		−	362.317i		−	0.829100i
$438$		0			0
$439$			422.918i			0.963366i	0.876345	+	0.481683i	$0.159974\pi$
$439$			422.918i			0.963366i	−0.876345	+	0.481683i	$0.840026\pi$
$440$	−155.606	+	323.827i	−0.353650	+	0.735972i
$441$		0			0
$442$		−	0.0804545i		−	0.000182024i
$443$	−857.361			−1.93535			−0.967676	−	0.252195i	$-0.918847\pi$
$443$	−857.361			−1.93535			−0.967676	+	0.252195i	$0.918847\pi$
$444$		0			0
$445$	−643.765			−1.44666
$446$			288.294i			0.646400i
$447$		0			0
$448$		−	356.974i		−	0.796818i
$449$	885.721			1.97265			0.986326	−	0.164806i	$-0.0526998\pi$
$449$	885.721			1.97265			0.986326	+	0.164806i	$0.0526998\pi$
$450$		0			0
$451$	7.98426	−	16.6158i	0.0177035	−	0.0368422i
$452$	140.418			0.310659
$453$		0			0
$454$	−505.407			−1.11323
$455$	−9.07518			−0.0199455
$456$		0			0
$457$		−	207.116i		−	0.453208i	−0.973987	−	0.226604i	$-0.927238\pi$
$457$		−	207.116i		−	0.453208i	0.973987	−	0.226604i	$-0.0727624\pi$
$458$		−	185.441i		−	0.404894i
$459$		0			0
$460$	131.434			0.285725
$461$			253.842i			0.550634i	0.961354	+	0.275317i	$0.0887828\pi$
$461$			253.842i			0.550634i	−0.961354	+	0.275317i	$0.911217\pi$
$462$		0			0
$463$	631.712			1.36439			0.682195	−	0.731170i	$-0.261026\pi$
$463$	631.712			1.36439			0.682195	+	0.731170i	$0.261026\pi$
$464$		−	61.5033i		−	0.132550i
$465$		0			0
$466$	62.1066			0.133276
$467$	−552.433			−1.18294			−0.591470	−	0.806327i	$-0.701452\pi$
$467$	−552.433			−1.18294			−0.591470	+	0.806327i	$0.701452\pi$
$468$		0			0
$469$		−	99.7048i		−	0.212590i
$470$			263.950i			0.561597i
$471$		0			0
$472$			387.025i			0.819967i
$473$	−356.311	+	741.509i	−0.753301	+	1.56767i
$474$		0			0
$475$		−	232.208i		−	0.488860i
$476$	3.93029			0.00825692
$477$		0			0
$478$	32.6102			0.0682221
$479$		−	686.424i		−	1.43303i	−0.697569	−	0.716517i	$-0.745735\pi$
$479$		−	686.424i		−	1.43303i	0.697569	−	0.716517i	$-0.254265\pi$
$480$		0			0
$481$		−	5.31257i		−	0.0110448i
$482$	257.400			0.534025
$483$		0			0
$484$	175.775	+	219.643i	0.363171	+	0.453808i
$485$	236.878			0.488408
$486$		0			0
$487$	69.5872			0.142890			0.0714448	−	0.997445i	$-0.477239\pi$
$487$	69.5872			0.142890			0.0714448	+	0.997445i	$0.477239\pi$
$488$	−440.427			−0.902515
$489$		0			0
$490$			66.3792i			0.135468i
$491$		−	35.9475i		−	0.0732129i	−0.999330	−	0.0366064i	$-0.988345\pi$
$491$		−	35.9475i		−	0.0732129i	0.999330	−	0.0366064i	$-0.0116548\pi$
$492$		0			0
$493$	10.2094			0.0207087
$494$			9.57148i			0.0193755i
$495$		0			0
$496$	8.69160			0.0175234
$497$			868.698i			1.74788i
$498$		0			0
$499$	690.713			1.38419			0.692097	−	0.721804i	$-0.256687\pi$
$499$	690.713			1.38419			0.692097	+	0.721804i	$0.256687\pi$
$500$	316.142			0.632285
$501$		0			0
$502$		−	476.539i		−	0.949282i
$503$		−	119.045i		−	0.236669i	−0.992974	−	0.118335i	$-0.962244\pi$
$503$		−	119.045i		−	0.236669i	0.992974	−	0.118335i	$-0.0377556\pi$
$504$		0			0
$505$			557.993i			1.10494i
$506$	−87.3658	+	181.815i	−0.172660	+	0.359317i
$507$		0			0
$508$		−	232.755i		−	0.458179i
$509$	13.2666			0.0260640			0.0130320	−	0.999915i	$-0.495852\pi$
$509$	13.2666			0.0260640			0.0130320	+	0.999915i	$0.495852\pi$
$510$		0			0
$511$	260.337			0.509466
$512$		−	82.6283i		−	0.161383i
$513$		0			0
$514$		−	296.031i		−	0.575935i
$515$	−669.849			−1.30068
$516$		0			0
$517$	506.789	+	243.523i	0.980250	+	0.471031i
$518$	−186.981			−0.360966
$519$		0			0
$520$	−9.44585			−0.0181651
$521$	−223.589			−0.429153			−0.214577	−	0.976707i	$-0.568837\pi$
$521$	−223.589			−0.429153			−0.214577	+	0.976707i	$0.568837\pi$
$522$		0			0
$523$		−	846.295i		−	1.61816i	−0.587702	−	0.809078i	$-0.699967\pi$
$523$		−	846.295i		−	1.61816i	0.587702	−	0.809078i	$-0.300033\pi$
$524$			333.403i			0.636265i
$525$		0			0
$526$	−48.2668			−0.0917620
$527$			1.44278i			0.00273773i
$528$		0			0
$529$	−328.246			−0.620502
$530$		−	293.357i		−	0.553504i
$531$		0			0
$532$	−467.578			−0.878906
$533$	0.484673			0.000909330
$534$		0			0
$535$			783.793i			1.46503i
$536$		−	103.777i		−	0.193614i
$537$		0			0
$538$			175.692i			0.326565i
$539$	127.449	+	61.2422i	0.236455	+	0.113622i
$540$		0			0
$541$			285.581i			0.527877i	0.964540	+	0.263938i	$0.0850215\pi$
$541$			285.581i			0.527877i	−0.964540	+	0.263938i	$0.914978\pi$
$542$	−457.014			−0.843200
$543$		0			0
$544$	6.67793			0.0122756
$545$		−	272.424i		−	0.499861i
$546$		0			0
$547$			945.963i			1.72937i	0.502319	+	0.864683i	$0.332480\pi$
$547$			945.963i			1.72937i	−0.502319	+	0.864683i	$0.667520\pi$
$548$	−87.9167			−0.160432
$549$		0			0
$550$	−55.9927	+	116.525i	−0.101805	+	0.211863i
$551$	−1214.59			−2.20433
$552$		0			0
$553$	−581.753			−1.05200
$554$	284.658			0.513823
$555$		0			0
$556$			294.327i			0.529365i
$557$		−	969.516i		−	1.74060i	−0.492518	−	0.870302i	$-0.663924\pi$
$557$		−	969.516i		−	1.74060i	0.492518	−	0.870302i	$-0.336076\pi$
$558$		0			0
$559$	−21.6294			−0.0386930
$560$			40.6324i			0.0725579i
$561$		0			0
$562$	645.838			1.14918
$563$			728.456i			1.29388i	0.762540	+	0.646941i	$0.223952\pi$
$563$			728.456i			1.29388i	−0.762540	+	0.646941i	$0.776048\pi$
$564$		0			0
$565$	−240.975			−0.426504
$566$	−468.140			−0.827102
$567$		0			0
$568$			904.179i			1.59186i
$569$			196.453i			0.345260i	0.984987	+	0.172630i	$0.0552264\pi$
$569$			196.453i			0.345260i	−0.984987	+	0.172630i	$0.944774\pi$
$570$		0			0
$571$		−	218.092i		−	0.381947i	−0.981595	−	0.190973i	$-0.938836\pi$
$571$		−	218.092i		−	0.381947i	0.981595	−	0.190973i	$-0.0611644\pi$
$572$	−3.20343	+	6.66656i	−0.00560040	+	0.0116548i
$573$		0			0
$574$		−	17.0585i		−	0.0297187i
$575$	128.663			0.223762
$576$		0			0
$577$	−321.389			−0.557001			−0.278500	−	0.960436i	$-0.589837\pi$
$577$	−321.389			−0.557001			−0.278500	+	0.960436i	$0.589837\pi$
$578$			373.976i			0.647017i
$579$		0			0
$580$		−	440.601i		−	0.759658i
$581$	−695.474			−1.19703
$582$		0			0
$583$	−563.251	−	270.654i	−0.966125	−	0.464244i
$584$	270.970			0.463990
$585$		0			0
$586$	−502.719			−0.857882
$587$	−500.784			−0.853125			−0.426563	−	0.904458i	$-0.640276\pi$
$587$	−500.784			−0.853125			−0.426563	+	0.904458i	$0.640276\pi$
$588$		0			0
$589$		−	171.645i		−	0.291417i
$590$		−	244.142i		−	0.413801i
$591$		0			0
$592$	−23.7860			−0.0401791
$593$		−	560.696i		−	0.945525i	−0.881190	−	0.472763i	$-0.843257\pi$
$593$		−	560.696i		−	0.945525i	0.881190	−	0.472763i	$-0.156743\pi$
$594$		0			0
$595$	−6.74487			−0.0113359
$596$			344.696i			0.578349i
$597$		0			0
$598$	−5.30342			−0.00886859
$599$	862.894			1.44056			0.720279	−	0.693684i	$-0.244014\pi$
$599$	862.894			1.44056			0.720279	+	0.693684i	$0.244014\pi$
$600$		0			0
$601$		−	4.62885i		−	0.00770191i	−0.999993	−	0.00385096i	$-0.998774\pi$
$601$		−	4.62885i		−	0.00770191i	0.999993	−	0.00385096i	$-0.00122580\pi$
$602$			761.265i			1.26456i
$603$		0			0
$604$		−	548.534i		−	0.908168i
$605$	−301.651	−	376.935i	−0.498597	−	0.623033i
$606$		0			0
$607$		−	858.949i		−	1.41507i	−0.706677	−	0.707536i	$-0.749807\pi$
$607$		−	858.949i		−	1.41507i	0.706677	−	0.707536i	$-0.250193\pi$
$608$	−794.457			−1.30667
$609$		0			0
$610$	277.830			0.455459
$611$			14.7827i			0.0241943i
$612$		0			0
$613$		−	1043.02i		−	1.70150i	−0.525571	−	0.850750i	$-0.676148\pi$
$613$		−	1043.02i		−	1.70150i	0.525571	−	0.850750i	$-0.323852\pi$
$614$	−24.7461			−0.0403030
$615$		0			0
$616$	638.320	+	306.727i	1.03623	+	0.497933i
$617$	896.830			1.45353			0.726767	−	0.686884i	$-0.241022\pi$
$617$	896.830			1.45353			0.726767	+	0.686884i	$0.241022\pi$
$618$		0			0
$619$	897.348			1.44967			0.724837	−	0.688921i	$-0.241915\pi$
$619$	897.348			1.44967			0.724837	+	0.688921i	$0.241915\pi$
$620$	62.2655			0.100428
$621$		0			0
$622$		−	259.568i		−	0.417312i
$623$			1268.97i			2.03687i
$624$		0			0
$625$	−315.521			−0.504833
$626$			305.849i			0.488577i
$627$		0			0
$628$	−358.469			−0.570811
$629$		−	3.94842i		−	0.00627729i
$630$		0			0
$631$	498.633			0.790226			0.395113	−	0.918632i	$-0.370705\pi$
$631$	498.633			0.790226			0.395113	+	0.918632i	$0.370705\pi$
$632$	−605.515			−0.958093
$633$		0			0
$634$			598.352i			0.943772i
$635$			399.437i			0.629034i
$636$		0			0
$637$			3.71762i			0.00583613i
$638$	609.492	+	292.874i	0.955317	+	0.459051i
$639$		0			0
$640$		−	261.450i		−	0.408516i
$641$	356.073			0.555497			0.277748	−	0.960654i	$-0.410412\pi$
$641$	356.073			0.555497			0.277748	+	0.960654i	$0.410412\pi$
$642$		0			0
$643$	−972.797			−1.51290			−0.756452	−	0.654049i	$-0.773069\pi$
$643$	−972.797			−1.51290			−0.756452	+	0.654049i	$0.773069\pi$
$644$		−	259.078i		−	0.402295i
$645$		0			0
$646$			7.11373i			0.0110120i
$647$	542.229			0.838066			0.419033	−	0.907971i	$-0.362369\pi$
$647$	542.229			0.838066			0.419033	+	0.907971i	$0.362369\pi$
$648$		0			0
$649$	−468.758	−	225.248i	−0.722277	−	0.347070i
$650$	−3.39895			−0.00522916
$651$		0			0
$652$	−606.213			−0.929774
$653$	−319.159			−0.488757			−0.244379	−	0.969680i	$-0.578584\pi$
$653$	−319.159			−0.488757			−0.244379	+	0.969680i	$0.578584\pi$
$654$		0			0
$655$		−	572.161i		−	0.873529i
$656$		−	2.17003i		−	0.00330798i
$657$		0			0
$658$	520.292			0.790717
$659$		−	584.687i		−	0.887234i	−0.896216	−	0.443617i	$-0.853695\pi$
$659$		−	584.687i		−	0.887234i	0.896216	−	0.443617i	$-0.146305\pi$
$660$		0			0
$661$	1090.03			1.64906			0.824532	−	0.565816i	$-0.191439\pi$
$661$	1090.03			1.64906			0.824532	+	0.565816i	$0.191439\pi$
$662$		−	749.577i		−	1.13229i
$663$		0			0
$664$	−723.880			−1.09018
$665$	802.422			1.20665
$666$		0			0
$667$		−	672.985i		−	1.00897i
$668$			142.765i			0.213720i
$669$		0			0
$670$			65.4646i			0.0977083i
$671$	256.329	−	533.438i	0.382010	−	0.794990i
$672$		0			0
$673$			416.989i			0.619597i	0.950802	+	0.309799i	$0.100262\pi$
$673$			416.989i			0.619597i	−0.950802	+	0.309799i	$0.899738\pi$
$674$	705.396			1.04658
$675$		0			0
$676$	392.721			0.580948
$677$		−	417.681i		−	0.616959i	−0.951231	−	0.308479i	$-0.900180\pi$
$677$		−	417.681i		−	0.616959i	0.951231	−	0.308479i	$-0.0998201\pi$
$678$		0			0
$679$		−	466.927i		−	0.687669i
$680$	−7.02036			−0.0103241
$681$		0			0
$682$	−41.3888	+	86.1331i	−0.0606874	+	0.126295i
$683$	−529.179			−0.774786			−0.387393	−	0.921915i	$-0.626624\pi$
$683$	−529.179			−0.774786			−0.387393	+	0.921915i	$0.626624\pi$
$684$		0			0
$685$	150.876			0.220257
$686$	−367.921			−0.536328
$687$		0			0
$688$			96.8414i			0.140758i
$689$		−	16.4297i		−	0.0238457i
$690$		0			0
$691$	313.067			0.453064			0.226532	−	0.974004i	$-0.427261\pi$
$691$	313.067			0.453064			0.226532	+	0.974004i	$0.427261\pi$
$692$			1.03538i			0.00149621i
$693$		0			0
$694$	−39.4097			−0.0567863
$695$		−	505.101i		−	0.726765i
$696$		0			0
$697$	0.360220			0.000516814
$698$	−326.147			−0.467259
$699$		0			0
$700$		−	166.043i		−	0.237204i
$701$		−	842.113i		−	1.20130i	−0.799511	−	0.600651i	$-0.794908\pi$
$701$		−	842.113i		−	1.20130i	0.799511	−	0.600651i	$-0.205092\pi$
$702$		0			0
$703$			469.734i			0.668185i
$704$	450.020	+	216.245i	0.639234	+	0.307166i
$705$		0			0
$706$			187.212i			0.265172i
$707$	1099.90			1.55573
$708$		0			0
$709$	−306.993			−0.432995			−0.216497	−	0.976283i	$-0.569463\pi$
$709$	−306.993			−0.432995			−0.216497	+	0.976283i	$0.569463\pi$
$710$		−	570.373i		−	0.803342i
$711$		0			0
$712$			1320.80i			1.85506i
$713$	95.1058			0.133388
$714$		0			0
$715$	5.49748	−	11.4407i	0.00768879	−	0.0160009i
$716$	−108.068			−0.150933
$717$		0			0
$718$	321.628			0.447950
$719$	1103.93			1.53537			0.767686	−	0.640826i	$-0.221408\pi$
$719$	1103.93			1.53537			0.767686	+	0.640826i	$0.221408\pi$
$720$		0			0
$721$			1320.39i			1.83133i
$722$		−	379.084i		−	0.525047i
$723$		0			0
$724$	207.744			0.286939
$725$		−	431.315i		−	0.594917i
$726$		0			0
$727$	215.284			0.296126			0.148063	−	0.988978i	$-0.452696\pi$
$727$	215.284			0.296126			0.148063	+	0.988978i	$0.452696\pi$
$728$			18.6194i			0.0255761i
$729$		0			0
$730$	−170.933			−0.234155
$731$	−16.0754			−0.0219910
$732$		0			0
$733$		−	1290.93i		−	1.76115i	−0.473903	−	0.880577i	$-0.657155\pi$
$733$		−	1290.93i		−	1.76115i	0.473903	−	0.880577i	$-0.342845\pi$
$734$			258.857i			0.352666i
$735$		0			0
$736$		−	440.197i		−	0.598094i
$737$	125.693	+	60.3983i	0.170547	+	0.0819515i
$738$		0			0
$739$			406.204i			0.549667i	0.961492	+	0.274833i	$0.0886227\pi$
$739$			406.204i			0.549667i	−0.961492	+	0.274833i	$0.911377\pi$
$740$	−170.400			−0.230270
$741$		0			0
$742$	−578.257			−0.779323
$743$		−	948.076i		−	1.27601i	−0.770032	−	0.638006i	$-0.779760\pi$
$743$		−	948.076i		−	1.27601i	0.770032	−	0.638006i	$-0.220240\pi$
$744$		0			0
$745$		−	591.542i		−	0.794016i
$746$	185.635			0.248840
$747$		0			0
$748$	−2.38086	+	4.95473i	−0.00318296	+	0.00662397i
$749$	1544.99			2.06274
$750$		0			0
$751$	−807.504			−1.07524			−0.537619	−	0.843188i	$-0.680676\pi$
$751$	−807.504			−1.07524			−0.537619	+	0.843188i	$0.680676\pi$
$752$	66.1869			0.0880145
$753$		0			0
$754$			17.7785i			0.0235789i
$755$			941.352i			1.24682i
$756$		0			0
$757$	218.023			0.288010			0.144005	−	0.989577i	$-0.454002\pi$
$757$	218.023			0.288010			0.144005	+	0.989577i	$0.454002\pi$
$758$			126.456i			0.166828i
$759$		0			0
$760$	835.196			1.09894
$761$			585.248i			0.769051i	0.923114	+	0.384526i	$0.125635\pi$
$761$			585.248i			0.769051i	−0.923114	+	0.384526i	$0.874365\pi$
$762$		0			0
$763$	−536.995			−0.703794
$764$	−218.146			−0.285532
$765$		0			0
$766$		−	383.434i		−	0.500567i
$767$		−	13.6734i		−	0.0178271i
$768$		0			0
$769$		−	86.3137i		−	0.112241i	−0.998424	−	0.0561207i	$-0.982127\pi$
$769$		−	86.3137i		−	0.112241i	0.998424	−	0.0561207i	$-0.0178732\pi$
$770$	−402.664	−	193.489i	−0.522940	−	0.251284i
$771$		0			0
$772$			358.493i			0.464370i
$773$	1098.73			1.42139			0.710695	−	0.703500i	$-0.248381\pi$
$773$	1098.73			1.42139			0.710695	+	0.703500i	$0.248381\pi$
$774$		0			0
$775$	60.9532			0.0786493
$776$		−	485.999i		−	0.626287i
$777$		0			0
$778$			514.309i			0.661065i
$779$	−42.8545			−0.0550122
$780$		0			0
$781$	−1095.13	−	526.232i	−1.40221	−	0.673792i
$782$	−3.94162			−0.00504043
$783$		0			0
$784$	16.6449			0.0212308
$785$	615.178			0.783666
$786$		0			0
$787$			405.193i			0.514858i	0.966297	+	0.257429i	$0.0828754\pi$
$787$			405.193i			0.514858i	−0.966297	+	0.257429i	$0.917125\pi$
$788$			152.042i			0.192946i
$789$		0			0
$790$	381.970			0.483506
$791$			475.002i			0.600509i
$792$		0			0
$793$	15.5601			0.0196218
$794$		−	502.215i		−	0.632513i
$795$		0			0
$796$	−681.668			−0.856367
$797$	−71.4746			−0.0896796			−0.0448398	−	0.998994i	$-0.514278\pi$
$797$	−71.4746			−0.0896796			−0.0448398	+	0.998994i	$0.514278\pi$
$798$		0			0
$799$			10.9868i			0.0137507i
$800$		−	282.122i		−	0.352652i
$801$		0			0
$802$		−	243.330i		−	0.303404i
$803$	−157.705	+	328.194i	−0.196394	+	0.408710i
$804$		0			0
$805$			444.610i			0.552311i
$806$	−2.51245			−0.00311718
$807$		0			0
$808$	1144.82			1.41686
$809$		−	118.353i		−	0.146296i	−0.997321	−	0.0731480i	$-0.976695\pi$
$809$		−	118.353i		−	0.146296i	0.997321	−	0.0731480i	$-0.0233045\pi$
$810$		0			0
$811$			1418.76i			1.74939i	0.484670	+	0.874697i	$0.338940\pi$
$811$			1418.76i			1.74939i	−0.484670	+	0.874697i	$0.661060\pi$
$812$	−868.501			−1.06958
$813$		0			0
$814$	113.267	−	235.718i	0.139149	−	0.289579i
$815$	1040.34			1.27649
$816$		0			0
$817$	1912.45			2.34083
$818$	−794.694			−0.971508
$819$		0			0
$820$		−	15.5458i		−	0.0189583i
$821$			236.466i			0.288022i	0.989576	+	0.144011i	$0.0460001\pi$
$821$			236.466i			0.288022i	−0.989576	+	0.144011i	$0.954000\pi$
$822$		0			0
$823$	−492.472			−0.598386			−0.299193	−	0.954193i	$-0.596717\pi$
$823$	−492.472			−0.598386			−0.299193	+	0.954193i	$0.596717\pi$
$824$			1374.32i			1.66786i
$825$		0			0
$826$	−481.247			−0.582623
$827$		−	441.151i		−	0.533435i	−0.963775	−	0.266717i	$-0.914061\pi$
$827$		−	441.151i		−	0.533435i	0.963775	−	0.266717i	$-0.0859390\pi$
$828$		0			0
$829$	−127.128			−0.153351			−0.0766755	−	0.997056i	$-0.524431\pi$
$829$	−127.128			−0.153351			−0.0766755	+	0.997056i	$0.524431\pi$
$830$	456.637			0.550165
$831$		0			0
$832$			13.1268i			0.0157774i
$833$			2.76301i			0.00331694i
$834$		0			0
$835$		−	245.003i		−	0.293416i
$836$	283.245	−	589.453i	0.338810	−	0.705087i
$837$		0			0
$838$		−	478.510i		−	0.571014i
$839$	756.938			0.902190			0.451095	−	0.892476i	$-0.351034\pi$
$839$	756.938			0.902190			0.451095	+	0.892476i	$0.351034\pi$
$840$		0			0
$841$	−1415.03			−1.68256
$842$			575.305i			0.683260i
$843$		0			0
$844$		−	265.639i		−	0.314739i
$845$	−673.958			−0.797584
$846$		0			0
$847$	−743.004	+	594.607i	−0.877218	+	0.702015i
$848$	−73.5608			−0.0867462
$849$		0			0
$850$	−2.52618			−0.00297197
$851$	−260.273			−0.305843
$852$		0			0
$853$			1444.56i			1.69351i	0.531984	+	0.846754i	$0.321447\pi$
$853$			1444.56i			1.69351i	−0.531984	+	0.846754i	$0.678553\pi$
$854$		−	547.650i		−	0.641277i
$855$		0			0
$856$	1608.09			1.87861
$857$		−	535.734i		−	0.625127i	−0.949897	−	0.312564i	$-0.898812\pi$
$857$		−	535.734i		−	0.625127i	0.949897	−	0.312564i	$-0.101188\pi$
$858$		0			0
$859$	1487.77			1.73197			0.865987	−	0.500066i	$-0.166691\pi$
$859$	1487.77			1.73197			0.865987	+	0.500066i	$0.166691\pi$
$860$			693.759i			0.806697i
$861$		0			0
$862$	−941.659			−1.09241
$863$	−434.830			−0.503858			−0.251929	−	0.967746i	$-0.581065\pi$
$863$	−434.830			−0.503858			−0.251929	+	0.967746i	$0.581065\pi$
$864$		0			0
$865$		−	1.77684i		−	0.00205415i
$866$			670.895i			0.774706i
$867$		0			0
$868$		−	122.736i		−	0.141401i
$869$	352.409	−	733.389i	0.405534	−	0.843946i
$870$		0			0
$871$			3.66639i			0.00420940i
$872$	−558.928			−0.640973
$873$		0			0
$874$	468.925			0.536527
$875$			1069.44i			1.22222i
$876$		0			0
$877$			594.675i			0.678079i	0.940772	+	0.339039i	$0.110102\pi$
$877$			594.675i			0.678079i	−0.940772	+	0.339039i	$0.889898\pi$
$878$	−547.357			−0.623414
$879$		0			0
$880$	−51.2233	−	24.6139i	−0.0582083	−	0.0279704i
$881$	−896.033			−1.01706			−0.508532	−	0.861043i	$-0.669812\pi$
$881$	−896.033			−1.01706			−0.508532	+	0.861043i	$0.669812\pi$
$882$		0			0
$883$	406.258			0.460088			0.230044	−	0.973180i	$-0.426113\pi$
$883$	406.258			0.460088			0.230044	+	0.973180i	$0.426113\pi$
$884$	−0.144527			−0.000163492
$885$		0			0
$886$		−	1109.63i		−	1.25241i
$887$		−	1293.40i		−	1.45817i	−0.684421	−	0.729087i	$-0.739945\pi$
$887$		−	1293.40i		−	1.45817i	0.684421	−	0.729087i	$-0.260055\pi$
$888$		0			0
$889$	787.358			0.885668
$890$		−	833.186i		−	0.936164i
$891$		0			0
$892$	517.886			0.580589
$893$		−	1307.08i		−	1.46370i
$894$		0			0
$895$	185.458			0.207216
$896$	−515.363			−0.575182
$897$		0			0
$898$			1146.34i			1.27654i
$899$		−	318.821i		−	0.354639i
$900$		0			0
$901$		−	12.2109i		−	0.0135526i
$902$	21.5048	+	10.3335i	0.0238413	+	0.0114563i
$903$		0			0
$904$			494.403i			0.546907i
$905$	−356.515			−0.393939
$906$		0			0
$907$	71.9478			0.0793250			0.0396625	−	0.999213i	$-0.487372\pi$
$907$	71.9478			0.0793250			0.0396625	+	0.999213i	$0.487372\pi$
$908$			907.901i			0.999891i
$909$		0			0
$910$		−	11.7455i		−	0.0129071i
$911$	906.523			0.995086			0.497543	−	0.867439i	$-0.334236\pi$
$911$	906.523			0.995086			0.497543	+	0.867439i	$0.334236\pi$
$912$		0			0
$913$	421.298	−	876.751i	0.461444	−	0.960297i
$914$	268.058			0.293280
$915$		0			0
$916$	−333.123			−0.363671
$917$	−1127.83			−1.22991
$918$		0			0
$919$		−	835.506i		−	0.909146i	−0.890709	−	0.454573i	$-0.849792\pi$
$919$		−	835.506i		−	0.909146i	0.890709	−	0.454573i	$-0.150208\pi$
$920$			462.770i			0.503011i
$921$		0			0
$922$	−328.533			−0.356326
$923$		−	31.9442i		−	0.0346091i
$924$		0			0
$925$	−166.808			−0.180333
$926$			817.587i			0.882924i
$927$		0			0
$928$	−1475.66			−1.59015
$929$	509.067			0.547974			0.273987	−	0.961733i	$-0.411658\pi$
$929$	509.067			0.547974			0.273987	+	0.961733i	$0.411658\pi$
$930$		0			0
$931$		−	328.709i		−	0.353071i
$932$		−	111.567i		−	0.119707i
$933$		0			0
$934$		−	714.981i		−	0.765504i
$935$	4.08585	−	8.50294i	0.00436989	−	0.00909405i
$936$		0			0
$937$		−	1128.46i		−	1.20433i	−0.798371	−	0.602165i	$-0.794305\pi$
$937$		−	1128.46i		−	1.20433i	0.798371	−	0.602165i	$-0.205695\pi$
$938$	129.042			0.137571
$939$		0			0
$940$	474.154			0.504420
$941$			870.345i			0.924915i	0.886641	+	0.462457i	$0.153032\pi$
$941$			870.345i			0.924915i	−0.886641	+	0.462457i	$0.846968\pi$
$942$		0			0
$943$		−	23.7451i		−	0.0251803i
$944$	−61.2199			−0.0648516
$945$		0			0
$946$	−959.690	−	461.152i	−1.01447	−	0.487476i
$947$	−1249.30			−1.31922			−0.659609	−	0.751609i	$-0.729278\pi$
$947$	−1249.30			−1.31922			−0.659609	+	0.751609i	$0.729278\pi$
$948$		0			0
$949$	−9.57323			−0.0100877
$950$	300.533			0.316351
$951$		0			0
$952$			13.8383i			0.0145361i
$953$			908.782i			0.953601i	0.879011	+	0.476801i	$0.158204\pi$
$953$			908.782i			0.953601i	−0.879011	+	0.476801i	$0.841796\pi$
$954$		0			0
$955$	374.366			0.392007
$956$		−	58.5802i		−	0.0612763i
$957$		0			0
$958$	888.397			0.927345
$959$		−	297.403i		−	0.310118i
$960$		0			0
$961$	−915.944			−0.953116
$962$	6.87574			0.00714734
$963$		0			0
$964$		−	462.388i		−	0.479656i
$965$		−	615.219i		−	0.637533i
$966$		0			0
$967$			647.176i			0.669262i	0.942349	+	0.334631i	$0.108612\pi$
$967$			647.176i			0.669262i	−0.942349	+	0.334631i	$0.891388\pi$
$968$	−773.351	+	618.893i	−0.798916	+	0.639352i
$969$		0			0
$970$			306.577i			0.316059i
$971$	−1516.83			−1.56214			−0.781068	−	0.624446i	$-0.785325\pi$
$971$	−1516.83			−1.56214			−0.781068	+	0.624446i	$0.785325\pi$
$972$		0			0
$973$	−995.642			−1.02327
$974$			90.0625i			0.0924667i
$975$		0			0
$976$		−	69.6672i		−	0.0713804i
$977$	631.357			0.646220			0.323110	−	0.946361i	$-0.395272\pi$
$977$	631.357			0.646220			0.323110	+	0.946361i	$0.395272\pi$
$978$		0			0
$979$	−1599.73	−	768.706i	−1.63405	−	0.785195i
$980$	119.242			0.121676
$981$		0			0
$982$	46.5247			0.0473775
$983$	722.669			0.735167			0.367583	−	0.929991i	$-0.380185\pi$
$983$	722.669			0.735167			0.367583	+	0.929991i	$0.380185\pi$
$984$		0			0
$985$		−	260.923i		−	0.264896i
$986$			13.2134i			0.0134010i
$987$		0			0
$988$	17.1940			0.0174028
$989$			1059.66i			1.07145i
$990$		0			0
$991$	180.088			0.181724			0.0908618	−	0.995864i	$-0.471038\pi$
$991$	180.088			0.181724			0.0908618	+	0.995864i	$0.471038\pi$
$992$		−	208.540i		−	0.210222i
$993$		0			0
$994$	−1124.30			−1.13109
$995$	1169.83			1.17571
$996$		0			0
$997$			844.273i			0.846813i	0.905940	+	0.423406i	$0.139166\pi$
$997$			844.273i			0.846813i	−0.905940	+	0.423406i	$0.860834\pi$
$998$			893.948i			0.895740i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.3.c.a.109.11	yes	16
3.2	odd	2	inner	297.3.c.a.109.6	yes	16
11.10	odd	2	inner	297.3.c.a.109.5	✓	16
33.32	even	2	inner	297.3.c.a.109.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.3.c.a.109.5	✓	16	11.10	odd	2	inner
297.3.c.a.109.6	yes	16	3.2	odd	2	inner
297.3.c.a.109.11	yes	16	1.1	even	1	trivial
297.3.c.a.109.12	yes	16	33.32	even	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	297.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.29424i q^{2} +2.32494 q^{4} -3.98989 q^{5} +7.86477i q^{7} +8.18599i q^{8} -5.16388i q^{10} +(-9.91473 - 4.76425i) q^{11} -0.289207i q^{13} -10.1789 q^{14} -1.29487 q^{16} -0.214945i q^{17} +25.5715i q^{19} -9.27627 q^{20} +(6.16608 - 12.8320i) q^{22} -14.1688 q^{23} -9.08076 q^{25} +0.374303 q^{26} +18.2851i q^{28} +47.4977i q^{29} -6.71234 q^{31} +31.0681i q^{32} +0.278190 q^{34} -31.3796i q^{35} +18.3694 q^{37} -33.0956 q^{38} -32.6612i q^{40} +1.67587i q^{41} -74.7886i q^{43} +(-23.0512 - 11.0766i) q^{44} -18.3378i q^{46} -51.1148 q^{47} -12.8545 q^{49} -11.7527i q^{50} -0.672389i q^{52} +56.8095 q^{53} +(39.5587 + 19.0088i) q^{55} -64.3809 q^{56} -61.4734 q^{58} +47.2789 q^{59} +53.8026i q^{61} -8.68738i q^{62} -45.3891 q^{64} +1.15390i q^{65} -12.6774 q^{67} -0.499734i q^{68} +40.6127 q^{70} +110.454 q^{71} -33.1017i q^{73} +23.7745i q^{74} +59.4522i q^{76} +(37.4697 - 77.9771i) q^{77} +73.9696i q^{79} +5.16639 q^{80} -2.16898 q^{82} +88.4291i q^{83} +0.857606i q^{85} +96.7944 q^{86} +(39.0001 - 81.1620i) q^{88} +161.349 q^{89} +2.27454 q^{91} -32.9416 q^{92} -66.1548i q^{94} -102.027i q^{95} -59.3695 q^{97} -16.6368i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 40 q^{16} + 24 q^{22} + 120 q^{25} - 88 q^{31} - 216 q^{34} + 56 q^{37} - 32 q^{49} - 280 q^{55} + 120 q^{58} + 376 q^{64} + 200 q^{67} - 192 q^{70} + 456 q^{82} - 168 q^{88} - 264 q^{91}+ \cdots + 80 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 40 * q^16 + 24 * q^22 + 120 * q^25 - 88 * q^31 - 216 * q^34 + 56 * q^37 - 32 * q^49 - 280 * q^55 + 120 * q^58 + 376 * q^64 + 200 * q^67 - 192 * q^70 + 456 * q^82 - 168 * q^88 - 264 * q^91 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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