LMFDB - Embedded newform 2900.2.s.c.1293.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(157,2900)] 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("2900.157"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.s (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$23.1566165862$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
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} + 36x^{14} + 468x^{12} + 2844x^{10} + 8574x^{8} + 12524x^{6} + 8404x^{4} + 2324x^{2} + 169 $ x^16 + 36x^14 + 468x^12 + 2844x^10 + 8574x^8 + 12524x^6 + 8404x^4 + 2324*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1293.5
Root			$0.722887i$ of defining polynomial
Character	$\chi$	$=$	2900.1293
Dual form			2900.2.s.c.157.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.277113 q^{3} +(3.60863 - 3.60863i) q^{7} -2.92321 q^{9} +(-2.44164 + 2.44164i) q^{11} +(2.78793 - 2.78793i) q^{13} -3.19850i q^{17} +(1.44164 + 1.44164i) q^{19} +(1.00000 - 1.00000i) q^{21} +(4.00813 + 4.00813i) q^{23} -1.64140 q^{27} +(-5.17735 - 1.48157i) q^{29} +(6.13742 - 6.13742i) q^{31} +(-0.676611 + 0.676611i) q^{33} +2.11132 q^{37} +(0.772573 - 0.772573i) q^{39} +(4.61899 + 4.61899i) q^{41} -0.0110648 q^{43} -6.65198 q^{47} -19.0445i q^{49} -0.886345i q^{51} +(-6.26354 - 6.26354i) q^{53} +(0.399497 + 0.399497i) q^{57} -9.31477i q^{59} +(0.963137 - 0.963137i) q^{61} +(-10.5488 + 10.5488i) q^{63} +(-0.532522 - 0.532522i) q^{67} +(1.11071 + 1.11071i) q^{69} -5.92321i q^{71} -6.67410i q^{73} +17.6220i q^{77} +(-0.521498 - 0.521498i) q^{79} +8.31477 q^{81} +(1.78549 + 1.78549i) q^{83} +(-1.43471 - 0.410562i) q^{87} +(-2.73571 - 2.73571i) q^{89} -20.1213i q^{91} +(1.70076 - 1.70076i) q^{93} +13.8692 q^{97} +(7.13742 - 7.13742i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9} - 4 q^{11} - 12 q^{19} + 16 q^{21} - 28 q^{29} + 28 q^{31} + 32 q^{39} - 16 q^{41} - 24 q^{61} - 72 q^{69} - 4 q^{79} + 8 q^{81} - 24 q^{89} + 44 q^{99}+O(q^{100}) $ 16 * q + 8 * q^9 - 4 * q^11 - 12 * q^19 + 16 * q^21 - 28 * q^29 + 28 * q^31 + 32 * q^39 - 16 * q^41 - 24 * q^61 - 72 * q^69 - 4 * q^79 + 8 * q^81 - 24 * q^89 + 44 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.277113			0.159991			0.0799957	−	0.996795i	$-0.474509\pi$
$3$	0.277113			0.159991			0.0799957	+	0.996795i	$0.474509\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.60863	−	3.60863i	1.36394	−	1.36394i	0.495098	−	0.868837i	$-0.335132\pi$
$7$	3.60863	−	3.60863i	1.36394	−	1.36394i	0.868837	−	0.495098i	$-0.164868\pi$
$8$		0			0
$9$	−2.92321			−0.974403
$10$		0			0
$11$	−2.44164	+	2.44164i	−0.736182	+	0.736182i	−0.971837	−	0.235655i	$-0.924277\pi$
$11$	−2.44164	+	2.44164i	−0.736182	+	0.736182i	0.235655	+	0.971837i	$0.424277\pi$
$12$		0			0
$13$	2.78793	−	2.78793i	0.773234	−	0.773234i	−0.205437	−	0.978670i	$-0.565861\pi$
$13$	2.78793	−	2.78793i	0.773234	−	0.773234i	0.978670	+	0.205437i	$0.0658615\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.19850i		−	0.775749i	−0.921712	−	0.387875i	$-0.873209\pi$
$17$		−	3.19850i		−	0.775749i	0.921712	−	0.387875i	$-0.126791\pi$
$18$		0			0
$19$	1.44164	+	1.44164i	0.330735	+	0.330735i	0.852865	−	0.522131i	$-0.174863\pi$
$19$	1.44164	+	1.44164i	0.330735	+	0.330735i	−0.522131	+	0.852865i	$0.674863\pi$
$20$		0			0
$21$	1.00000	−	1.00000i	0.218218	−	0.218218i
$22$		0			0
$23$	4.00813	+	4.00813i	0.835753	+	0.835753i	0.988297	−	0.152544i	$-0.0487465\pi$
$23$	4.00813	+	4.00813i	0.835753	+	0.835753i	−0.152544	+	0.988297i	$0.548746\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.64140			−0.315887
$28$		0			0
$29$	−5.17735	−	1.48157i	−0.961410	−	0.275120i
$29$	−5.17735	−	1.48157i	−0.961410	−	0.275120i
$30$		0			0
$31$	6.13742	−	6.13742i	1.10231	−	1.10231i	0.108182	−	0.994131i	$-0.465497\pi$
$31$	6.13742	−	6.13742i	1.10231	−	1.10231i	0.994131	−	0.108182i	$-0.0345030\pi$
$32$		0			0
$33$	−0.676611	+	0.676611i	−0.117783	+	0.117783i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.11132			0.347099			0.173550	−	0.984825i	$-0.444476\pi$
$37$	2.11132			0.347099			0.173550	+	0.984825i	$0.444476\pi$
$38$		0			0
$39$	0.772573	−	0.772573i	0.123711	−	0.123711i
$40$		0			0
$41$	4.61899	+	4.61899i	0.721365	+	0.721365i	0.968883	−	0.247518i	$-0.0796150\pi$
$41$	4.61899	+	4.61899i	0.721365	+	0.721365i	−0.247518	+	0.968883i	$0.579615\pi$
$42$		0			0
$43$	−0.0110648			−0.00168737			−0.000843687	−	1.00000i	$-0.500269\pi$
$43$	−0.0110648			−0.00168737			−0.000843687		1.00000i	$0.500269\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.65198			−0.970290			−0.485145	−	0.874434i	$-0.661233\pi$
$47$	−6.65198			−0.970290			−0.485145	+	0.874434i	$0.661233\pi$
$48$		0			0
$49$		−	19.0445i		−	2.72064i
$50$		0			0
$51$		−	0.886345i		−	0.124113i
$52$		0			0
$53$	−6.26354	−	6.26354i	−0.860364	−	0.860364i	0.131016	−	0.991380i	$-0.458176\pi$
$53$	−6.26354	−	6.26354i	−0.860364	−	0.860364i	−0.991380	+	0.131016i	$0.958176\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.399497	+	0.399497i	0.0529147	+	0.0529147i
$58$		0			0
$59$		−	9.31477i		−	1.21268i	−0.795206	−	0.606340i	$-0.792637\pi$
$59$		−	9.31477i		−	1.21268i	0.795206	−	0.606340i	$-0.207363\pi$
$60$		0			0
$61$	0.963137	−	0.963137i	0.123317	−	0.123317i	−0.642755	−	0.766072i	$-0.722209\pi$
$61$	0.963137	−	0.963137i	0.123317	−	0.123317i	0.766072	+	0.642755i	$0.222209\pi$
$62$		0			0
$63$	−10.5488	+	10.5488i	−1.32902	+	1.32902i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.532522	−	0.532522i	−0.0650579	−	0.0650579i	0.673829	−	0.738887i	$-0.264648\pi$
$67$	−0.532522	−	0.532522i	−0.0650579	−	0.0650579i	−0.738887	+	0.673829i	$0.764648\pi$
$68$		0			0
$69$	1.11071	+	1.11071i	0.133713	+	0.133713i
$70$		0			0
$71$		−	5.92321i		−	0.702955i	−0.936196	−	0.351478i	$-0.885679\pi$
$71$		−	5.92321i		−	0.702955i	0.936196	−	0.351478i	$-0.114321\pi$
$72$		0			0
$73$		−	6.67410i		−	0.781145i	−0.920572	−	0.390572i	$-0.872277\pi$
$73$		−	6.67410i		−	0.781145i	0.920572	−	0.390572i	$-0.127723\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			17.6220i			2.00821i
$78$		0			0
$79$	−0.521498	−	0.521498i	−0.0586731	−	0.0586731i	0.677161	−	0.735834i	$-0.263210\pi$
$79$	−0.521498	−	0.521498i	−0.0586731	−	0.0586731i	−0.735834	+	0.677161i	$0.763210\pi$
$80$		0			0
$81$	8.31477			0.923863
$82$		0			0
$83$	1.78549	+	1.78549i	0.195983	+	0.195983i	0.798275	−	0.602293i	$-0.205746\pi$
$83$	1.78549	+	1.78549i	0.195983	+	0.195983i	−0.602293	+	0.798275i	$0.705746\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.43471	−	0.410562i	−0.153817	−	0.0440169i
$88$		0			0
$89$	−2.73571	−	2.73571i	−0.289985	−	0.289985i	0.547089	−	0.837074i	$-0.315736\pi$
$89$	−2.73571	−	2.73571i	−0.289985	−	0.289985i	−0.837074	+	0.547089i	$0.815736\pi$
$90$		0			0
$91$		−	20.1213i		−	2.10928i
$92$		0			0
$93$	1.70076	−	1.70076i	0.176361	−	0.176361i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.8692			1.40821			0.704104	−	0.710097i	$-0.251349\pi$
$97$	13.8692			1.40821			0.704104	+	0.710097i	$0.251349\pi$
$98$		0			0
$99$	7.13742	−	7.13742i	0.717338	−	0.717338i
$100$		0			0
$101$	−4.57906	+	4.57906i	−0.455634	+	0.455634i	−0.897219	−	0.441586i	$-0.854416\pi$
$101$	−4.57906	+	4.57906i	−0.455634	+	0.455634i	0.441586	+	0.897219i	$0.354416\pi$
$102$		0			0
$103$	−5.84234	−	5.84234i	−0.575663	−	0.575663i	0.358042	−	0.933705i	$-0.383444\pi$
$103$	−5.84234	−	5.84234i	−0.575663	−	0.575663i	−0.933705	+	0.358042i	$0.883444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.99707	+	5.99707i	−0.579759	+	0.579759i	−0.934837	−	0.355078i	$-0.884454\pi$
$107$	−5.99707	+	5.99707i	−0.579759	+	0.579759i	0.355078	+	0.934837i	$0.384454\pi$
$108$		0			0
$109$	−16.6529			−1.59506			−0.797529	−	0.603280i	$-0.793860\pi$
$109$	−16.6529			−1.59506			−0.797529	+	0.603280i	$0.793860\pi$
$110$		0			0
$111$	0.585075			0.0555329
$112$		0			0
$113$		−	11.0432i		−	1.03886i	−0.854513	−	0.519430i	$-0.826144\pi$
$113$		−	11.0432i		−	1.03886i	0.854513	−	0.519430i	$-0.173856\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−8.14971	+	8.14971i	−0.753441	+	0.753441i
$118$		0			0
$119$	−11.5422	−	11.5422i	−1.05807	−	1.05807i
$120$		0			0
$121$		−	0.923208i		−	0.0839280i
$122$		0			0
$123$	1.27998	+	1.27998i	0.115412	+	0.115412i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			10.6086i			0.941359i	0.882304	+	0.470679i	$0.155991\pi$
$127$			10.6086i			0.941359i	−0.882304	+	0.470679i	$0.844009\pi$
$128$		0			0
$129$	−0.00306622			−0.000269965
$130$		0			0
$131$	11.0207	+	11.0207i	0.962883	+	0.962883i	0.999335	−	0.0364523i	$-0.0116057\pi$
$131$	11.0207	+	11.0207i	0.962883	+	0.962883i	−0.0364523	+	0.999335i	$0.511606\pi$
$132$		0			0
$133$	10.4047			0.902202
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	15.7035i		−	1.34164i	−0.741622	−	0.670818i	$-0.765943\pi$
$137$		−	15.7035i		−	1.34164i	0.741622	−	0.670818i	$-0.234057\pi$
$138$		0			0
$139$			3.76656i			0.319475i	0.987160	+	0.159738i	$0.0510648\pi$
$139$			3.76656i			0.319475i	−0.987160	+	0.159738i	$0.948935\pi$
$140$		0			0
$141$	−1.84335			−0.155238
$142$		0			0
$143$			13.6143i			1.13848i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	5.27747i		−	0.435279i
$148$		0			0
$149$	−1.92627			−0.157807			−0.0789033	−	0.996882i	$-0.525142\pi$
$149$	−1.92627			−0.157807			−0.0789033	+	0.996882i	$0.525142\pi$
$150$		0			0
$151$			5.33507i			0.434162i	0.976154	+	0.217081i	$0.0696535\pi$
$151$			5.33507i			0.434162i	−0.976154	+	0.217081i	$0.930346\pi$
$152$		0			0
$153$			9.34987i			0.755892i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.55991			0.284112			0.142056	−	0.989859i	$-0.454629\pi$
$157$	3.55991			0.284112			0.142056	+	0.989859i	$0.454629\pi$
$158$		0			0
$159$	−1.73571	−	1.73571i	−0.137651	−	0.137651i
$160$		0			0
$161$	28.9277			2.27983
$162$		0			0
$163$		−	24.0811i		−	1.88618i	−0.332538	−	0.943090i	$-0.607905\pi$
$163$		−	24.0811i		−	1.88618i	0.332538	−	0.943090i	$-0.392095\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.11658	−	5.11658i	−0.395933	−	0.395933i	0.480863	−	0.876796i	$-0.340324\pi$
$167$	−5.11658	−	5.11658i	−0.395933	−	0.395933i	−0.876796	+	0.480863i	$0.840324\pi$
$168$		0			0
$169$		−	2.54515i		−	0.195781i
$170$		0			0
$171$	−4.21421	−	4.21421i	−0.322269	−	0.322269i
$172$		0			0
$173$	−9.99414	+	9.99414i	−0.759840	+	0.759840i	−0.976293	−	0.216453i	$-0.930551\pi$
$173$	−9.99414	+	9.99414i	−0.759840	+	0.759840i	0.216453	+	0.976293i	$0.430551\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	2.58125i		−	0.194018i
$178$		0			0
$179$	−10.7297			−0.801975			−0.400988	−	0.916083i	$-0.631333\pi$
$179$	−10.7297			−0.801975			−0.400988	+	0.916083i	$0.631333\pi$
$180$		0			0
$181$	11.3148			0.841020			0.420510	−	0.907288i	$-0.361851\pi$
$181$	11.3148			0.841020			0.420510	+	0.907288i	$0.361851\pi$
$182$		0			0
$183$	0.266898	−	0.266898i	0.0197297	−	0.0197297i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.80957	+	7.80957i	0.571093	+	0.571093i
$188$		0			0
$189$	−5.92321	+	5.92321i	−0.430850	+	0.430850i
$190$		0			0
$191$	−5.24813	+	5.24813i	−0.379741	+	0.379741i	−0.871009	−	0.491268i	$-0.836534\pi$
$191$	−5.24813	+	5.24813i	−0.379741	+	0.379741i	0.491268	+	0.871009i	$0.336534\pi$
$192$		0			0
$193$	26.2256			1.88776			0.943882	−	0.330284i	$-0.107144\pi$
$193$	26.2256			1.88776			0.943882	+	0.330284i	$0.107144\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.89925	−	2.89925i	0.206563	−	0.206563i	−0.596242	−	0.802805i	$-0.703340\pi$
$197$	2.89925	−	2.89925i	0.206563	−	0.206563i	0.802805	+	0.596242i	$0.203340\pi$
$198$		0			0
$199$		−	18.1243i		−	1.28480i	−0.766370	−	0.642400i	$-0.777939\pi$
$199$		−	18.1243i		−	1.28480i	0.766370	−	0.642400i	$-0.222061\pi$
$200$		0			0
$201$	−0.147569	−	0.147569i	−0.0104087	−	0.0104087i
$202$		0			0
$203$	−24.0296	+	13.3367i	−1.68655	+	0.936054i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−11.7166	−	11.7166i	−0.814360	−	0.814360i
$208$		0			0
$209$	−7.03993			−0.486962
$210$		0			0
$211$	−13.4048	−	13.4048i	−0.922823	−	0.922823i	0.0744052	−	0.997228i	$-0.476294\pi$
$211$	−13.4048	−	13.4048i	−0.922823	−	0.922823i	−0.997228	+	0.0744052i	$0.976294\pi$
$212$		0			0
$213$		−	1.64140i		−	0.112467i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	44.2954i		−	3.00697i
$218$		0			0
$219$		−	1.84948i		−	0.124976i
$220$		0			0
$221$	−8.91719	−	8.91719i	−0.599835	−	0.599835i
$222$		0			0
$223$	10.9483	+	10.9483i	0.733151	+	0.733151i	0.971243	−	0.238091i	$-0.0765218\pi$
$223$	10.9483	+	10.9483i	0.733151	+	0.733151i	−0.238091	+	0.971243i	$0.576522\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.95648	−	9.95648i	0.660835	−	0.660835i	−0.294742	−	0.955577i	$-0.595234\pi$
$227$	9.95648	−	9.95648i	0.660835	−	0.660835i	0.955577	+	0.294742i	$0.0952337\pi$
$228$		0			0
$229$	14.4255	−	14.4255i	0.953262	−	0.953262i	−0.0456931	−	0.998956i	$-0.514550\pi$
$229$	14.4255	−	14.4255i	0.953262	−	0.953262i	0.998956	+	0.0456931i	$0.0145496\pi$
$230$		0			0
$231$			4.88328i			0.321296i
$232$		0			0
$233$	18.6682	+	18.6682i	1.22300	+	1.22300i	0.966561	+	0.256436i	$0.0825483\pi$
$233$	18.6682	+	18.6682i	1.22300	+	1.22300i	0.256436	+	0.966561i	$0.417452\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.144514	−	0.144514i	−0.00938719	−	0.00938719i
$238$		0			0
$239$		−	20.9277i		−	1.35370i	−0.736119	−	0.676852i	$-0.763344\pi$
$239$		−	20.9277i		−	1.35370i	0.736119	−	0.676852i	$-0.236656\pi$
$240$		0			0
$241$			15.6130i			1.00572i	0.864368	+	0.502860i	$0.167719\pi$
$241$			15.6130i			1.00572i	−0.864368	+	0.502860i	$0.832281\pi$
$242$		0			0
$243$	7.22833			0.463698
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.03839			0.511471
$248$		0			0
$249$	0.494782	+	0.494782i	0.0313556	+	0.0313556i
$250$		0			0
$251$	18.8272	−	18.8272i	1.18836	−	1.18836i	0.210841	−	0.977520i	$-0.432380\pi$
$251$	18.8272	−	18.8272i	1.18836	−	1.18836i	0.977520	−	0.210841i	$-0.0676203\pi$
$252$		0			0
$253$	−19.5728			−1.23053
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.4538	+	10.4538i	−0.652093	+	0.652093i	−0.953497	−	0.301404i	$-0.902545\pi$
$257$	−10.4538	+	10.4538i	−0.652093	+	0.652093i	0.301404	+	0.953497i	$0.402545\pi$
$258$		0			0
$259$	7.61899	−	7.61899i	0.473421	−	0.473421i
$260$		0			0
$261$	15.1345	+	4.33093i	0.936800	+	0.268078i
$262$		0			0
$263$	27.1953			1.67694			0.838468	−	0.544951i	$-0.183452\pi$
$263$	27.1953			1.67694			0.838468	+	0.544951i	$0.183452\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.758102	−	0.758102i	−0.0463951	−	0.0463951i
$268$		0			0
$269$	5.99693	−	5.99693i	0.365640	−	0.365640i	−0.500245	−	0.865884i	$-0.666757\pi$
$269$	5.99693	−	5.99693i	0.365640	−	0.365640i	0.865884	+	0.500245i	$0.166757\pi$
$270$		0			0
$271$	2.62607	+	2.62607i	0.159522	+	0.159522i	0.782355	−	0.622833i	$-0.214018\pi$
$271$	2.62607	+	2.62607i	0.159522	+	0.159522i	−0.622833	+	0.782355i	$0.714018\pi$
$272$		0			0
$273$		−	5.57587i		−	0.337467i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.8258	+	15.8258i	−0.950882	+	0.950882i	−0.998849	−	0.0479667i	$-0.984726\pi$
$277$	−15.8258	+	15.8258i	−0.950882	+	0.950882i	0.0479667	+	0.998849i	$0.484726\pi$
$278$		0			0
$279$	−17.9410	+	17.9410i	−1.07410	+	1.07410i
$280$		0			0
$281$	−14.8004			−0.882915			−0.441458	−	0.897282i	$-0.645538\pi$
$281$	−14.8004			−0.882915			−0.441458	+	0.897282i	$0.645538\pi$
$282$		0			0
$283$	12.3338	−	12.3338i	0.733171	−	0.733171i	−0.238075	−	0.971247i	$-0.576517\pi$
$283$	12.3338	−	12.3338i	0.733171	−	0.733171i	0.971247	+	0.238075i	$0.0765165\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	33.3365			1.96779
$288$		0			0
$289$	6.76962			0.398213
$290$		0			0
$291$	3.84335			0.225301
$292$		0			0
$293$	−12.7829			−0.746786			−0.373393	−	0.927673i	$-0.621806\pi$
$293$	−12.7829			−0.746786			−0.373393	+	0.927673i	$0.621806\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.00771	−	4.00771i	0.232551	−	0.232551i
$298$		0			0
$299$	22.3488			1.29246
$300$		0			0
$301$	−0.0399290	+	0.0399290i	−0.00230147	+	0.00230147i
$302$		0			0
$303$	−1.26892	+	1.26892i	−0.0728975	+	0.0728975i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			29.7438i			1.69757i	0.528739	+	0.848785i	$0.322665\pi$
$307$			29.7438i			1.69757i	−0.528739	+	0.848785i	$0.677335\pi$
$308$		0			0
$309$	−1.61899	−	1.61899i	−0.0921011	−	0.0921011i
$310$		0			0
$311$	13.8303	−	13.8303i	0.784242	−	0.784242i	−0.196302	−	0.980544i	$-0.562893\pi$
$311$	13.8303	−	13.8303i	0.784242	−	0.784242i	0.980544	+	0.196302i	$0.0628932\pi$
$312$		0			0
$313$	2.25584	+	2.25584i	0.127507	+	0.127507i	0.767981	−	0.640473i	$-0.221262\pi$
$313$	2.25584	+	2.25584i	0.127507	+	0.127507i	−0.640473	+	0.767981i	$0.721262\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.51848			−0.534611			−0.267305	−	0.963612i	$-0.586133\pi$
$317$	−9.51848			−0.534611			−0.267305	+	0.963612i	$0.586133\pi$
$318$		0			0
$319$	16.2587	−	9.02377i	0.910311	−	0.505234i
$320$		0			0
$321$	−1.66187	+	1.66187i	−0.0927564	+	0.0927564i
$322$		0			0
$323$	4.61108	−	4.61108i	0.256567	−	0.256567i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.61474			−0.255196
$328$		0			0
$329$	−24.0045	+	24.0045i	−1.32341	+	1.32341i
$330$		0			0
$331$	−21.5660	−	21.5660i	−1.18537	−	1.18537i	−0.978332	−	0.207040i	$-0.933617\pi$
$331$	−21.5660	−	21.5660i	−1.18537	−	1.18537i	−0.207040	−	0.978332i	$-0.566383\pi$
$332$		0			0
$333$	−6.17184			−0.338215
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.82596			0.208413			0.104207	−	0.994556i	$-0.466770\pi$
$337$	3.82596			0.208413			0.104207	+	0.994556i	$0.466770\pi$
$338$		0			0
$339$		−	3.06022i		−	0.166209i
$340$		0			0
$341$			29.9707i			1.62301i
$342$		0			0
$343$	−43.4641	−	43.4641i	−2.34684	−	2.34684i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.51125	+	2.51125i	0.134811	+	0.134811i	0.771292	−	0.636481i	$-0.219611\pi$
$347$	2.51125	+	2.51125i	0.134811	+	0.134811i	−0.636481	+	0.771292i	$0.719611\pi$
$348$		0			0
$349$			29.3592i			1.57156i	0.618504	+	0.785782i	$0.287739\pi$
$349$			29.3592i			1.57156i	−0.618504	+	0.785782i	$0.712261\pi$
$350$		0			0
$351$	−4.57611	+	4.57611i	−0.244255	+	0.244255i
$352$		0			0
$353$	−21.8065	+	21.8065i	−1.16064	+	1.16064i	−0.176310	+	0.984335i	$0.556416\pi$
$353$	−21.8065	+	21.8065i	−1.16064	+	1.16064i	−0.984335	+	0.176310i	$0.943584\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.19850	−	3.19850i	−0.169282	−	0.169282i
$358$		0			0
$359$	0.291005	+	0.291005i	0.0153586	+	0.0153586i	0.714744	−	0.699386i	$-0.246543\pi$
$359$	0.291005	+	0.291005i	0.0153586	+	0.0153586i	−0.699386	+	0.714744i	$0.746543\pi$
$360$		0			0
$361$		−	14.8434i		−	0.781229i
$362$		0			0
$363$		−	0.255833i		−	0.0134278i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			21.6841i			1.13190i	0.824439	+	0.565952i	$0.191491\pi$
$367$			21.6841i			1.13190i	−0.824439	+	0.565952i	$0.808509\pi$
$368$		0			0
$369$	−13.5023	−	13.5023i	−0.702900	−	0.702900i
$370$		0			0
$371$	−45.2057			−2.34696
$372$		0			0
$373$	3.53778	+	3.53778i	0.183179	+	0.183179i	0.792740	−	0.609560i	$-0.208654\pi$
$373$	3.53778	+	3.53778i	0.183179	+	0.183179i	−0.609560	+	0.792740i	$0.708654\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.5646	+	10.3036i	−0.956127	+	0.530662i
$378$		0			0
$379$	11.4921	+	11.4921i	0.590311	+	0.590311i	0.937715	−	0.347405i	$-0.112937\pi$
$379$	11.4921	+	11.4921i	0.590311	+	0.590311i	−0.347405	+	0.937715i	$0.612937\pi$
$380$		0			0
$381$			2.93978i			0.150609i
$382$		0			0
$383$	12.1791	−	12.1791i	0.622324	−	0.622324i	−0.323801	−	0.946125i	$-0.604961\pi$
$383$	12.1791	−	12.1791i	0.622324	−	0.622324i	0.946125	+	0.323801i	$0.104961\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.0323449			0.00164418
$388$		0			0
$389$	−24.4729	+	24.4729i	−1.24082	+	1.24082i	−0.281166	+	0.959659i	$0.590721\pi$
$389$	−24.4729	+	24.4729i	−1.24082	+	1.24082i	−0.959659	+	0.281166i	$0.909279\pi$
$390$		0			0
$391$	12.8200	−	12.8200i	0.648335	−	0.648335i
$392$		0			0
$393$	3.05398	+	3.05398i	0.154053	+	0.154053i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−13.9748	+	13.9748i	−0.701376	+	0.701376i	−0.964706	−	0.263330i	$-0.915179\pi$
$397$	−13.9748	+	13.9748i	−0.701376	+	0.701376i	0.263330	+	0.964706i	$0.415179\pi$
$398$		0			0
$399$	2.88328			0.144345
$400$		0			0
$401$	−22.3930			−1.11825			−0.559127	−	0.829082i	$-0.688864\pi$
$401$	−22.3930			−1.11825			−0.559127	+	0.829082i	$0.688864\pi$
$402$		0			0
$403$		−	34.2214i		−	1.70469i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.15509	+	5.15509i	−0.255528	+	0.255528i
$408$		0			0
$409$	15.6559	+	15.6559i	0.774132	+	0.774132i	0.978826	−	0.204694i	$-0.0656199\pi$
$409$	15.6559	+	15.6559i	0.774132	+	0.774132i	−0.204694	+	0.978826i	$0.565620\pi$
$410$		0			0
$411$		−	4.35163i		−	0.214650i
$412$		0			0
$413$	−33.6136	−	33.6136i	−1.65402	−	1.65402i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			1.04376i			0.0511133i
$418$		0			0
$419$	26.4993			1.29458			0.647288	−	0.762245i	$-0.275903\pi$
$419$	26.4993			1.29458			0.647288	+	0.762245i	$0.275903\pi$
$420$		0			0
$421$	−3.05048	−	3.05048i	−0.148671	−	0.148671i	0.628853	−	0.777524i	$-0.283525\pi$
$421$	−3.05048	−	3.05048i	−0.148671	−	0.148671i	−0.777524	+	0.628853i	$0.783525\pi$
$422$		0			0
$423$	19.4451			0.945454
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	6.95122i		−	0.336393i
$428$		0			0
$429$			3.77269i			0.182147i
$430$		0			0
$431$	21.8141			1.05075			0.525374	−	0.850871i	$-0.323925\pi$
$431$	21.8141			1.05075			0.525374	+	0.850871i	$0.323925\pi$
$432$		0			0
$433$			37.4314i			1.79884i	0.437087	+	0.899419i	$0.356010\pi$
$433$			37.4314i			1.79884i	−0.437087	+	0.899419i	$0.643990\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			11.5566i			0.552825i
$438$		0			0
$439$	12.4315			0.593323			0.296661	−	0.954983i	$-0.404127\pi$
$439$	12.4315			0.593323			0.296661	+	0.954983i	$0.404127\pi$
$440$		0			0
$441$			55.6709i			2.65100i
$442$		0			0
$443$			31.2763i			1.48598i	0.669302	+	0.742990i	$0.266593\pi$
$443$			31.2763i			1.48598i	−0.669302	+	0.742990i	$0.733407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.533796			−0.0252477
$448$		0			0
$449$	14.4685	+	14.4685i	0.682809	+	0.682809i	0.960632	−	0.277823i	$-0.0896128\pi$
$449$	14.4685	+	14.4685i	0.682809	+	0.682809i	−0.277823	+	0.960632i	$0.589613\pi$
$450$		0			0
$451$	−22.5558			−1.06211
$452$		0			0
$453$			1.47842i			0.0694621i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	13.8971	+	13.8971i	0.650079	+	0.650079i	0.953012	−	0.302933i	$-0.0979657\pi$
$457$	13.8971	+	13.8971i	0.650079	+	0.650079i	−0.302933	+	0.953012i	$0.597966\pi$
$458$		0			0
$459$			5.25001i			0.245049i
$460$		0			0
$461$	25.7402	+	25.7402i	1.19884	+	1.19884i	0.974513	+	0.224330i	$0.0720192\pi$
$461$	25.7402	+	25.7402i	1.19884	+	1.19884i	0.224330	+	0.974513i	$0.427981\pi$
$462$		0			0
$463$	−27.0353	+	27.0353i	−1.25644	+	1.25644i	−0.303654	+	0.952783i	$0.598207\pi$
$463$	−27.0353	+	27.0353i	−1.25644	+	1.25644i	−0.952783	+	0.303654i	$0.901793\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	19.3698i		−	0.896327i	−0.893952	−	0.448164i	$-0.852078\pi$
$467$		−	19.3698i		−	0.896327i	0.893952	−	0.448164i	$-0.147922\pi$
$468$		0			0
$469$	−3.84335			−0.177469
$470$		0			0
$471$	0.986499			0.0454555
$472$		0			0
$473$	0.0270164	−	0.0270164i	0.00124221	−	0.00124221i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	18.3096	+	18.3096i	0.838341	+	0.838341i
$478$		0			0
$479$	−14.4756	+	14.4756i	−0.661405	+	0.661405i	−0.955711	−	0.294306i	$-0.904911\pi$
$479$	−14.4756	+	14.4756i	−0.661405	+	0.661405i	0.294306	+	0.955711i	$0.404911\pi$
$480$		0			0
$481$	5.88623	−	5.88623i	0.268389	−	0.268389i
$482$		0			0
$483$	8.01626			0.364753
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.15950	+	8.15950i	−0.369742	+	0.369742i	−0.867383	−	0.497641i	$-0.834200\pi$
$487$	−8.15950	+	8.15950i	−0.369742	+	0.369742i	0.497641	+	0.867383i	$0.334200\pi$
$488$		0			0
$489$		−	6.67320i		−	0.301773i
$490$		0			0
$491$	−3.52898	−	3.52898i	−0.159261	−	0.159261i	0.622978	−	0.782239i	$-0.285922\pi$
$491$	−3.52898	−	3.52898i	−0.159261	−	0.159261i	−0.782239	+	0.622978i	$0.785922\pi$
$492$		0			0
$493$	−4.73879	+	16.5597i	−0.213424	+	0.745813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−21.3747	−	21.3747i	−0.958786	−	0.958786i
$498$		0			0
$499$	−35.0662			−1.56978			−0.784890	−	0.619635i	$-0.787281\pi$
$499$	−35.0662			−1.56978			−0.784890	+	0.619635i	$0.787281\pi$
$500$		0			0
$501$	−1.41787	−	1.41787i	−0.0633459	−	0.0633459i
$502$		0			0
$503$		−	32.0156i		−	1.42751i	−0.700398	−	0.713753i	$-0.746994\pi$
$503$		−	32.0156i		−	1.42751i	0.700398	−	0.713753i	$-0.253006\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	0.705294i		−	0.0313232i
$508$		0			0
$509$		−	7.30864i		−	0.323950i	−0.986795	−	0.161975i	$-0.948214\pi$
$509$		−	7.30864i		−	0.323950i	0.986795	−	0.161975i	$-0.0517863\pi$
$510$		0			0
$511$	−24.0844	−	24.0844i	−1.06543	−	1.06543i
$512$		0			0
$513$	−2.36631	−	2.36631i	−0.104475	−	0.104475i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	16.2417	−	16.2417i	0.714310	−	0.714310i
$518$		0			0
$519$	−2.76951	+	2.76951i	−0.121568	+	0.121568i
$520$		0			0
$521$			27.1673i			1.19022i	0.803644	+	0.595111i	$0.202892\pi$
$521$			27.1673i			1.19022i	−0.803644	+	0.595111i	$0.797108\pi$
$522$		0			0
$523$	25.4884	+	25.4884i	1.11453	+	1.11453i	0.992530	+	0.122000i	$0.0389307\pi$
$523$	25.4884	+	25.4884i	1.11453	+	1.11453i	0.122000	+	0.992530i	$0.461069\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−19.6305	−	19.6305i	−0.855119	−	0.855119i
$528$		0			0
$529$			9.13022i			0.396966i
$530$		0			0
$531$			27.2290i			1.18164i
$532$		0			0
$533$	25.7549			1.11557
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.97334			−0.128309
$538$		0			0
$539$	46.4997	+	46.4997i	2.00288	+	2.00288i
$540$		0			0
$541$	−9.20112	+	9.20112i	−0.395587	+	0.395587i	−0.876673	−	0.481086i	$-0.840242\pi$
$541$	−9.20112	+	9.20112i	−0.395587	+	0.395587i	0.481086	+	0.876673i	$0.340242\pi$
$542$		0			0
$543$	3.13547			0.134556
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.30742	+	1.30742i	−0.0559014	+	0.0559014i	−0.734505	−	0.678603i	$-0.762586\pi$
$547$	−1.30742	+	1.30742i	−0.0559014	+	0.0559014i	0.678603	+	0.734505i	$0.262586\pi$
$548$		0			0
$549$	−2.81545	+	2.81545i	−0.120161	+	0.120161i
$550$		0			0
$551$	−5.32799	−	9.59976i	−0.226980	−	0.408964i
$552$		0			0
$553$	−3.76379			−0.160053
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.07074	+	8.07074i	0.341968	+	0.341968i	0.857107	−	0.515139i	$-0.172260\pi$
$557$	8.07074	+	8.07074i	0.341968	+	0.341968i	−0.515139	+	0.857107i	$0.672260\pi$
$558$		0			0
$559$	−0.0308481	+	0.0308481i	−0.00130473	+	0.00130473i
$560$		0			0
$561$	2.16414	+	2.16414i	0.0913699	+	0.0913699i
$562$		0			0
$563$		−	18.3375i		−	0.772834i	−0.922324	−	0.386417i	$-0.873713\pi$
$563$		−	18.3375i		−	0.772834i	0.922324	−	0.386417i	$-0.126287\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	30.0050	−	30.0050i	1.26009	−	1.26009i
$568$		0			0
$569$	3.84028	−	3.84028i	0.160993	−	0.160993i	−0.622013	−	0.783007i	$-0.713685\pi$
$569$	3.84028	−	3.84028i	0.160993	−	0.160993i	0.783007	+	0.622013i	$0.213685\pi$
$570$		0			0
$571$	−19.1966			−0.803352			−0.401676	−	0.915782i	$-0.631572\pi$
$571$	−19.1966			−0.803352			−0.401676	+	0.915782i	$0.631572\pi$
$572$		0			0
$573$	−1.45433	+	1.45433i	−0.0607553	+	0.0607553i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	12.0899			0.503310			0.251655	−	0.967817i	$-0.419025\pi$
$577$	12.0899			0.503310			0.251655	+	0.967817i	$0.419025\pi$
$578$		0			0
$579$	7.26747			0.302026
$580$		0			0
$581$	12.8863			0.534616
$582$		0			0
$583$	30.5866			1.26677
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.8093	−	29.8093i	1.23036	−	1.23036i	0.266539	−	0.963824i	$-0.414120\pi$
$587$	29.8093	−	29.8093i	1.23036	−	1.23036i	0.963824	−	0.266539i	$-0.0858801\pi$
$588$		0			0
$589$	17.6959			0.729147
$590$		0			0
$591$	0.803421	−	0.803421i	0.0330483	−	0.0330483i
$592$		0			0
$593$	26.4315	−	26.4315i	1.08541	−	1.08541i	0.0894166	−	0.995994i	$-0.471500\pi$
$593$	26.4315	−	26.4315i	1.08541	−	1.08541i	0.995994	−	0.0894166i	$-0.0285003\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	5.02249i		−	0.205557i
$598$		0			0
$599$	18.8303	+	18.8303i	0.769383	+	0.769383i	0.977998	−	0.208615i	$-0.0668954\pi$
$599$	18.8303	+	18.8303i	0.769383	+	0.769383i	−0.208615	+	0.977998i	$0.566895\pi$
$600$		0			0
$601$	−3.65892	+	3.65892i	−0.149250	+	0.149250i	−0.777783	−	0.628533i	$-0.783656\pi$
$601$	−3.65892	+	3.65892i	−0.149250	+	0.149250i	0.628533	+	0.777783i	$0.283656\pi$
$602$		0			0
$603$	1.55667	+	1.55667i	0.0633926	+	0.0633926i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−18.0591			−0.732998			−0.366499	−	0.930419i	$-0.619444\pi$
$607$	−18.0591			−0.732998			−0.366499	+	0.930419i	$0.619444\pi$
$608$		0			0
$609$	−6.65892	+	3.69578i	−0.269833	+	0.149761i
$610$		0			0
$611$	−18.5453	+	18.5453i	−0.750261	+	0.750261i
$612$		0			0
$613$	−23.9236	+	23.9236i	−0.966265	+	0.966265i	−0.999449	−	0.0331846i	$-0.989435\pi$
$613$	−23.9236	+	23.9236i	−0.966265	+	0.966265i	0.0331846	+	0.999449i	$0.489435\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.9182			1.12394			0.561971	−	0.827157i	$-0.310043\pi$
$617$	27.9182			1.12394			0.561971	+	0.827157i	$0.310043\pi$
$618$		0			0
$619$	−5.79033	+	5.79033i	−0.232733	+	0.232733i	−0.813832	−	0.581100i	$-0.802623\pi$
$619$	−5.79033	+	5.79033i	−0.232733	+	0.232733i	0.581100	+	0.813832i	$0.302623\pi$
$620$		0			0
$621$	−6.57894	−	6.57894i	−0.264004	−	0.264004i
$622$		0			0
$623$	−19.7444			−0.791041
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−1.95086			−0.0779097
$628$		0			0
$629$		−	6.75306i		−	0.269262i
$630$		0			0
$631$		−	20.9105i		−	0.832435i	−0.909265	−	0.416217i	$-0.863356\pi$
$631$		−	20.9105i		−	0.832435i	0.909265	−	0.416217i	$-0.136644\pi$
$632$		0			0
$633$	−3.71464	−	3.71464i	−0.147644	−	0.147644i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−53.0947	−	53.0947i	−2.10369	−	2.10369i
$638$		0			0
$639$			17.3148i			0.684962i
$640$		0			0
$641$	27.6469	−	27.6469i	1.09199	−	1.09199i	0.0966700	−	0.995316i	$-0.469181\pi$
$641$	27.6469	−	27.6469i	1.09199	−	1.09199i	0.995316	−	0.0966700i	$-0.0308191\pi$
$642$		0			0
$643$	−21.3853	+	21.3853i	−0.843355	+	0.843355i	−0.989294	−	0.145939i	$-0.953380\pi$
$643$	−21.3853	+	21.3853i	−0.843355	+	0.843355i	0.145939	+	0.989294i	$0.453380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.24380	+	8.24380i	0.324097	+	0.324097i	0.850337	−	0.526239i	$-0.176398\pi$
$647$	8.24380	+	8.24380i	0.324097	+	0.324097i	−0.526239	+	0.850337i	$0.676398\pi$
$648$		0			0
$649$	22.7433	+	22.7433i	0.892753	+	0.892753i
$650$		0			0
$651$		−	12.2748i		−	0.481089i
$652$		0			0
$653$		−	10.6077i		−	0.415112i	−0.978223	−	0.207556i	$-0.933449\pi$
$653$		−	10.6077i		−	0.415112i	0.978223	−	0.207556i	$-0.0665510\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			19.5098i			0.761150i
$658$		0			0
$659$	24.6028	+	24.6028i	0.958390	+	0.958390i	0.999168	−	0.0407781i	$-0.0129837\pi$
$659$	24.6028	+	24.6028i	0.958390	+	0.958390i	−0.0407781	+	0.999168i	$0.512984\pi$
$660$		0			0
$661$	−33.5482			−1.30487			−0.652437	−	0.757843i	$-0.726253\pi$
$661$	−33.5482			−1.30487			−0.652437	+	0.757843i	$0.726253\pi$
$662$		0			0
$663$	−2.47107	−	2.47107i	−0.0959685	−	0.0959685i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−14.8132	−	26.6898i	−0.573568	−	1.03343i
$668$		0			0
$669$	3.03391	+	3.03391i	0.117298	+	0.117298i
$670$		0			0
$671$			4.70327i			0.181568i
$672$		0			0
$673$	25.9698	−	25.9698i	1.00106	−	1.00106i	0.00106328	−	0.999999i	$-0.499662\pi$
$673$	25.9698	−	25.9698i	1.00106	−	1.00106i	0.999999	−	0.00106328i	$-0.000338452\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.8421			1.14693			0.573463	−	0.819232i	$-0.305600\pi$
$677$	29.8421			1.14693			0.573463	+	0.819232i	$0.305600\pi$
$678$		0			0
$679$	50.0490	−	50.0490i	1.92070	−	1.92070i
$680$		0			0
$681$	2.75907	−	2.75907i	0.105728	−	0.105728i
$682$		0			0
$683$	3.23127	+	3.23127i	0.123641	+	0.123641i	0.766220	−	0.642579i	$-0.222135\pi$
$683$	3.23127	+	3.23127i	0.123641	+	0.123641i	−0.642579	+	0.766220i	$0.722135\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	3.99749	−	3.99749i	0.152514	−	0.152514i
$688$		0			0
$689$	−34.9247			−1.33052
$690$		0			0
$691$	−7.74933			−0.294798			−0.147399	−	0.989077i	$-0.547090\pi$
$691$	−7.74933			−0.294798			−0.147399	+	0.989077i	$0.547090\pi$
$692$		0			0
$693$		−	51.5127i		−	1.95680i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	14.7738	−	14.7738i	0.559598	−	0.559598i
$698$		0			0
$699$	5.17322	+	5.17322i	0.195669	+	0.195669i
$700$		0			0
$701$			51.8645i			1.95889i	0.201703	+	0.979447i	$0.435352\pi$
$701$			51.8645i			1.95889i	−0.201703	+	0.979447i	$0.564648\pi$
$702$		0			0
$703$	3.04377	+	3.04377i	0.114798	+	0.114798i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			33.0483i			1.24291i
$708$		0			0
$709$	−15.5301			−0.583243			−0.291622	−	0.956534i	$-0.594195\pi$
$709$	−15.5301			−0.583243			−0.291622	+	0.956534i	$0.594195\pi$
$710$		0			0
$711$	1.52445	+	1.52445i	0.0571712	+	0.0571712i
$712$		0			0
$713$	49.1992			1.84252
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	5.79936i		−	0.216581i
$718$		0			0
$719$		−	34.1182i		−	1.27239i	−0.771527	−	0.636197i	$-0.780507\pi$
$719$		−	34.1182i		−	1.27239i	0.771527	−	0.636197i	$-0.219493\pi$
$720$		0			0
$721$	−42.1657			−1.57033
$722$		0			0
$723$			4.32656i			0.160907i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			44.2814i			1.64231i	0.570708	+	0.821153i	$0.306669\pi$
$727$			44.2814i			1.64231i	−0.570708	+	0.821153i	$0.693331\pi$
$728$		0			0
$729$	−22.9412			−0.849676
$730$		0			0
$731$			0.0353909i			0.00130898i
$732$		0			0
$733$			10.2144i			0.377278i	0.982047	+	0.188639i	$0.0604075\pi$
$733$			10.2144i			0.377278i	−0.982047	+	0.188639i	$0.939592\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.60045			0.0957889
$738$		0			0
$739$	31.1111	+	31.1111i	1.14444	+	1.14444i	0.987628	+	0.156812i	$0.0501218\pi$
$739$	31.1111	+	31.1111i	1.14444	+	1.14444i	0.156812	+	0.987628i	$0.449878\pi$
$740$		0			0
$741$	2.22754			0.0818309
$742$		0			0
$743$		−	7.80384i		−	0.286295i	−0.989701	−	0.143148i	$-0.954278\pi$
$743$		−	7.80384i		−	0.286295i	0.989701	−	0.143148i	$-0.0457223\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.21935	−	5.21935i	−0.190966	−	0.190966i
$748$		0			0
$749$			43.2824i			1.58151i
$750$		0			0
$751$	−24.6059	−	24.6059i	−0.897882	−	0.897882i	0.0973668	−	0.995249i	$-0.468958\pi$
$751$	−24.6059	−	24.6059i	−0.897882	−	0.897882i	−0.995249	+	0.0973668i	$0.968958\pi$
$752$		0			0
$753$	5.21726	−	5.21726i	0.190128	−	0.190128i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			5.07696i			0.184525i	0.995735	+	0.0922627i	$0.0294100\pi$
$757$			5.07696i			0.184525i	−0.995735	+	0.0922627i	$0.970590\pi$
$758$		0			0
$759$	−5.42389			−0.196875
$760$		0			0
$761$	−15.5008			−0.561903			−0.280952	−	0.959722i	$-0.590650\pi$
$761$	−15.5008			−0.561903			−0.280952	+	0.959722i	$0.590650\pi$
$762$		0			0
$763$	−60.0942	+	60.0942i	−2.17556	+	2.17556i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.9690	−	25.9690i	−0.937685	−	0.937685i
$768$		0			0
$769$	−7.35163	+	7.35163i	−0.265107	+	0.265107i	−0.827125	−	0.562018i	$-0.810025\pi$
$769$	−7.35163	+	7.35163i	−0.265107	+	0.265107i	0.562018	+	0.827125i	$0.310025\pi$
$770$		0			0
$771$	−2.89690	+	2.89690i	−0.104329	+	0.104329i
$772$		0			0
$773$	42.6208			1.53296			0.766482	−	0.642266i	$-0.222005\pi$
$773$	42.6208			1.53296			0.766482	+	0.642266i	$0.222005\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2.11132	−	2.11132i	0.0757433	−	0.0757433i
$778$		0			0
$779$			13.3178i			0.477161i
$780$		0			0
$781$	14.4623	+	14.4623i	0.517503	+	0.517503i
$782$		0			0
$783$	8.49810	+	2.43185i	0.303697	+	0.0869071i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	9.33475	+	9.33475i	0.332748	+	0.332748i	0.853629	−	0.520881i	$-0.174397\pi$
$787$	9.33475	+	9.33475i	0.332748	+	0.332748i	−0.520881	+	0.853629i	$0.674397\pi$
$788$		0			0
$789$	7.53618			0.268295
$790$		0			0
$791$	−39.8510	−	39.8510i	−1.41694	−	1.41694i
$792$		0			0
$793$		−	5.37033i		−	0.190706i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			51.1951i			1.81342i	0.421752	+	0.906711i	$0.361415\pi$
$797$			51.1951i			1.81342i	−0.421752	+	0.906711i	$0.638585\pi$
$798$		0			0
$799$			21.2763i			0.752702i
$800$		0			0
$801$	7.99705	+	7.99705i	0.282562	+	0.282562i
$802$		0			0
$803$	16.2958	+	16.2958i	0.575065	+	0.575065i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.66183	−	1.66183i	0.0584992	−	0.0584992i
$808$		0			0
$809$	−10.8570	+	10.8570i	−0.381711	+	0.381711i	−0.871718	−	0.490008i	$-0.836994\pi$
$809$	−10.8570	+	10.8570i	−0.381711	+	0.381711i	0.490008	+	0.871718i	$0.336994\pi$
$810$		0			0
$811$			1.51201i			0.0530939i	0.999648	+	0.0265470i	$0.00845115\pi$
$811$			1.51201i			0.0530939i	−0.999648	+	0.0265470i	$0.991549\pi$
$812$		0			0
$813$	0.727719	+	0.727719i	0.0255222	+	0.0255222i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.0159515	−	0.0159515i	−0.000558073	−	0.000558073i
$818$		0			0
$819$			58.8186i			2.05529i
$820$		0			0
$821$		−	37.2198i		−	1.29898i	−0.760370	−	0.649490i	$-0.774982\pi$
$821$		−	37.2198i		−	1.29898i	0.760370	−	0.649490i	$-0.225018\pi$
$822$		0			0
$823$	−10.8321			−0.377582			−0.188791	−	0.982017i	$-0.560457\pi$
$823$	−10.8321			−0.377582			−0.188791	+	0.982017i	$0.560457\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.1953			−1.08477			−0.542384	−	0.840131i	$-0.682478\pi$
$827$	−31.1953			−1.08477			−0.542384	+	0.840131i	$0.682478\pi$
$828$		0			0
$829$	5.11672	+	5.11672i	0.177711	+	0.177711i	0.790357	−	0.612646i	$-0.209895\pi$
$829$	5.11672	+	5.11672i	0.177711	+	0.177711i	−0.612646	+	0.790357i	$0.709895\pi$
$830$		0			0
$831$	−4.38555	+	4.38555i	−0.152133	+	0.152133i
$832$		0			0
$833$	−60.9136			−2.11053
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−10.0740	+	10.0740i	−0.348207	+	0.348207i
$838$		0			0
$839$	−3.91306	+	3.91306i	−0.135094	+	0.135094i	−0.771420	−	0.636326i	$-0.780453\pi$
$839$	−3.91306	+	3.91306i	−0.135094	+	0.135094i	0.636326	+	0.771420i	$0.280453\pi$
$840$		0			0
$841$	24.6099	+	15.3412i	0.848618	+	0.529007i
$842$		0			0
$843$	−4.10137			−0.141259
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.33152	−	3.33152i	−0.114472	−	0.114472i
$848$		0			0
$849$	3.41787	−	3.41787i	0.117301	−	0.117301i
$850$		0			0
$851$	8.46246	+	8.46246i	0.290089	+	0.290089i
$852$		0			0
$853$			19.3178i			0.661430i	0.943731	+	0.330715i	$0.107290\pi$
$853$			19.3178i			0.661430i	−0.943731	+	0.330715i	$0.892710\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.6114	+	13.6114i	−0.464958	+	0.464958i	−0.900277	−	0.435318i	$-0.856636\pi$
$857$	−13.6114	+	13.6114i	−0.464958	+	0.464958i	0.435318	+	0.900277i	$0.356636\pi$
$858$		0			0
$859$	15.7934	−	15.7934i	0.538864	−	0.538864i	−0.384332	−	0.923195i	$-0.625568\pi$
$859$	15.7934	−	15.7934i	0.538864	−	0.538864i	0.923195	+	0.384332i	$0.125568\pi$
$860$		0			0
$861$	9.23798			0.314830
$862$		0			0
$863$	−11.8939	+	11.8939i	−0.404872	+	0.404872i	−0.879946	−	0.475074i	$-0.842421\pi$
$863$	−11.8939	+	11.8939i	−0.404872	+	0.404872i	0.475074	+	0.879946i	$0.342421\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.87595			0.0637107
$868$		0			0
$869$	2.54662			0.0863881
$870$		0			0
$871$	−2.96927			−0.100610
$872$		0			0
$873$	−40.5427			−1.37216
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.15424	+	5.15424i	−0.174046	+	0.174046i	−0.788755	−	0.614708i	$-0.789274\pi$
$877$	−5.15424	+	5.15424i	−0.174046	+	0.174046i	0.614708	+	0.788755i	$0.289274\pi$
$878$		0			0
$879$	−3.54232			−0.119479
$880$		0			0
$881$	−16.8034	+	16.8034i	−0.566122	+	0.566122i	−0.931040	−	0.364918i	$-0.881097\pi$
$881$	−16.8034	+	16.8034i	−0.566122	+	0.566122i	0.364918	+	0.931040i	$0.381097\pi$
$882$		0			0
$883$	8.17142	−	8.17142i	0.274990	−	0.274990i	−0.556115	−	0.831105i	$-0.687709\pi$
$883$	8.17142	−	8.17142i	0.274990	−	0.274990i	0.831105	+	0.556115i	$0.187709\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			53.1622i			1.78501i	0.451035	+	0.892506i	$0.351055\pi$
$887$			53.1622i			1.78501i	−0.451035	+	0.892506i	$0.648945\pi$
$888$		0			0
$889$	38.2824	+	38.2824i	1.28395	+	1.28395i
$890$		0			0
$891$	−20.3017	+	20.3017i	−0.680132	+	0.680132i
$892$		0			0
$893$	−9.58975	−	9.58975i	−0.320909	−	0.320909i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	6.19315			0.206783
$898$		0			0
$899$	−40.8686	+	22.6826i	−1.36304	+	0.756506i
$900$		0			0
$901$	−20.0339	+	20.0339i	−0.667426	+	0.667426i
$902$		0			0
$903$	−0.0110648	+	0.0110648i	−0.000368215	+	0.000368215i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−46.2177			−1.53463			−0.767317	−	0.641268i	$-0.778409\pi$
$907$	−46.2177			−1.53463			−0.767317	+	0.641268i	$0.778409\pi$
$908$		0			0
$909$	13.3855	−	13.3855i	0.443971	−	0.443971i
$910$		0			0
$911$	18.8777	+	18.8777i	0.625445	+	0.625445i	0.946919	−	0.321473i	$-0.104178\pi$
$911$	18.8777	+	18.8777i	0.625445	+	0.625445i	−0.321473	+	0.946919i	$0.604178\pi$
$912$		0			0
$913$	−8.71904			−0.288558
$914$		0			0
$915$		0			0
$916$		0			0
$917$	79.5393			2.62662
$918$		0			0
$919$		−	47.2683i		−	1.55924i	−0.626255	−	0.779618i	$-0.715413\pi$
$919$		−	47.2683i		−	1.55924i	0.626255	−	0.779618i	$-0.284587\pi$
$920$		0			0
$921$			8.24240i			0.271596i
$922$		0			0
$923$	−16.5135	−	16.5135i	−0.543549	−	0.543549i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	17.0784	+	17.0784i	0.560928	+	0.560928i
$928$		0			0
$929$		−	13.2380i		−	0.434324i	−0.976136	−	0.217162i	$-0.930320\pi$
$929$		−	13.2380i		−	0.434324i	0.976136	−	0.217162i	$-0.0696800\pi$
$930$		0			0
$931$	27.4553	−	27.4553i	0.899810	−	0.899810i
$932$		0			0
$933$	3.83255	−	3.83255i	0.125472	−	0.125472i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−25.4724	−	25.4724i	−0.832148	−	0.832148i	0.155662	−	0.987810i	$-0.450249\pi$
$937$	−25.4724	−	25.4724i	−0.832148	−	0.832148i	−0.987810	+	0.155662i	$0.950249\pi$
$938$		0			0
$939$	0.625122	+	0.625122i	0.0204001	+	0.0204001i
$940$		0			0
$941$			23.8675i			0.778059i	0.921225	+	0.389029i	$0.127190\pi$
$941$			23.8675i			0.778059i	−0.921225	+	0.389029i	$0.872810\pi$
$942$		0			0
$943$			37.0270i			1.20577i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	2.28285i		−	0.0741828i	−0.999312	−	0.0370914i	$-0.988191\pi$
$947$		−	2.28285i		−	0.0741828i	0.999312	−	0.0370914i	$-0.0118093\pi$
$948$		0			0
$949$	−18.6070	−	18.6070i	−0.604007	−	0.604007i
$950$		0			0
$951$	−2.63770			−0.0855331
$952$		0			0
$953$	−8.45384	−	8.45384i	−0.273847	−	0.273847i	0.556800	−	0.830647i	$-0.312029\pi$
$953$	−8.45384	−	8.45384i	−0.273847	−	0.273847i	−0.830647	+	0.556800i	$0.812029\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	4.50550	−	2.50061i	0.145642	−	0.0808331i
$958$		0			0
$959$	−56.6680	−	56.6680i	−1.82990	−	1.82990i
$960$		0			0
$961$		−	44.3359i		−	1.43019i
$962$		0			0
$963$	17.5307	−	17.5307i	0.564918	−	0.564918i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−27.3592			−0.879811			−0.439906	−	0.898044i	$-0.644988\pi$
$967$	−27.3592			−0.879811			−0.439906	+	0.898044i	$0.644988\pi$
$968$		0			0
$969$	1.27779	−	1.27779i	0.0410486	−	0.0410486i
$970$		0			0
$971$	−19.8439	+	19.8439i	−0.636820	+	0.636820i	−0.949770	−	0.312950i	$-0.898683\pi$
$971$	−19.8439	+	19.8439i	−0.636820	+	0.636820i	0.312950	+	0.949770i	$0.398683\pi$
$972$		0			0
$973$	13.5921	+	13.5921i	0.435744	+	0.435744i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.7220	+	37.7220i	−1.20683	+	1.20683i	−0.234787	+	0.972047i	$0.575439\pi$
$977$	−37.7220	+	37.7220i	−1.20683	+	1.20683i	−0.972047	+	0.234787i	$0.924561\pi$
$978$		0			0
$979$	13.3592			0.426963
$980$		0			0
$981$	48.6799			1.55423
$982$		0			0
$983$		−	3.04910i		−	0.0972510i	−0.998817	−	0.0486255i	$-0.984516\pi$
$983$		−	3.04910i		−	0.0972510i	0.998817	−	0.0486255i	$-0.0154841\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−6.65198	+	6.65198i	−0.211735	+	0.211735i
$988$		0			0
$989$	−0.0443494	−	0.0443494i	−0.00141023	−	0.00141023i
$990$		0			0
$991$		−	4.05650i		−	0.128859i	−0.997922	−	0.0644294i	$-0.979477\pi$
$991$		−	4.05650i		−	0.128859i	0.997922	−	0.0644294i	$-0.0205227\pi$
$992$		0			0
$993$	−5.97621	−	5.97621i	−0.189649	−	0.189649i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	42.1611i		−	1.33526i	−0.744495	−	0.667628i	$-0.767310\pi$
$997$		−	42.1611i		−	1.33526i	0.744495	−	0.667628i	$-0.232690\pi$
$998$		0			0
$999$	−3.46552			−0.109644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.s.c.1293.5	yes	16
5.2	odd	4		2900.2.j.c.1757.4	✓	16
5.3	odd	4		2900.2.j.c.1757.5	yes	16
5.4	even	2	inner	2900.2.s.c.1293.4	yes	16
29.12	odd	4		2900.2.j.c.2593.5	yes	16
145.12	even	4	inner	2900.2.s.c.157.5	yes	16
145.99	odd	4		2900.2.j.c.2593.4	yes	16
145.128	even	4	inner	2900.2.s.c.157.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.j.c.1757.4	✓	16	5.2	odd	4
2900.2.j.c.1757.5	yes	16	5.3	odd	4
2900.2.j.c.2593.4	yes	16	145.99	odd	4
2900.2.j.c.2593.5	yes	16	29.12	odd	4
2900.2.s.c.157.4	yes	16	145.128	even	4	inner
2900.2.s.c.157.5	yes	16	145.12	even	4	inner
2900.2.s.c.1293.4	yes	16	5.4	even	2	inner
2900.2.s.c.1293.5	yes	16	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+0.277113 q^{3} +(3.60863 - 3.60863i) q^{7} -2.92321 q^{9} +(-2.44164 + 2.44164i) q^{11} +(2.78793 - 2.78793i) q^{13} -3.19850i q^{17} +(1.44164 + 1.44164i) q^{19} +(1.00000 - 1.00000i) q^{21} +(4.00813 + 4.00813i) q^{23} -1.64140 q^{27} +(-5.17735 - 1.48157i) q^{29} +(6.13742 - 6.13742i) q^{31} +(-0.676611 + 0.676611i) q^{33} +2.11132 q^{37} +(0.772573 - 0.772573i) q^{39} +(4.61899 + 4.61899i) q^{41} -0.0110648 q^{43} -6.65198 q^{47} -19.0445i q^{49} -0.886345i q^{51} +(-6.26354 - 6.26354i) q^{53} +(0.399497 + 0.399497i) q^{57} -9.31477i q^{59} +(0.963137 - 0.963137i) q^{61} +(-10.5488 + 10.5488i) q^{63} +(-0.532522 - 0.532522i) q^{67} +(1.11071 + 1.11071i) q^{69} -5.92321i q^{71} -6.67410i q^{73} +17.6220i q^{77} +(-0.521498 - 0.521498i) q^{79} +8.31477 q^{81} +(1.78549 + 1.78549i) q^{83} +(-1.43471 - 0.410562i) q^{87} +(-2.73571 - 2.73571i) q^{89} -20.1213i q^{91} +(1.70076 - 1.70076i) q^{93} +13.8692 q^{97} +(7.13742 - 7.13742i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9} - 4 q^{11} - 12 q^{19} + 16 q^{21} - 28 q^{29} + 28 q^{31} + 32 q^{39} - 16 q^{41} - 24 q^{61} - 72 q^{69} - 4 q^{79} + 8 q^{81} - 24 q^{89} + 44 q^{99}+O(q^{100}) \) 16 * q + 8 * q^9 - 4 * q^11 - 12 * q^19 + 16 * q^21 - 28 * q^29 + 28 * q^31 + 32 * q^39 - 16 * q^41 - 24 * q^61 - 72 * q^69 - 4 * q^79 + 8 * q^81 - 24 * q^89 + 44 * q^99

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