LMFDB - Embedded newform 2900.2.s.c.157.5

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)

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")

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);

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			157.5
Root			$-0.722887i$ of defining polynomial
Character	$\chi$	$=$	2900.157
Dual form			2900.2.s.c.1293.5

$q$-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 $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

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

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	0.868837	+	0.495098i	$0.164868\pi$
$7$	3.60863	+	3.60863i	1.36394	+	1.36394i	0.495098	+	0.868837i	$0.335132\pi$
$8$		0			0
$9$	−2.92321			−0.974403
$10$		0			0
$11$	−2.44164	−	2.44164i	−0.736182	−	0.736182i	0.235655	−	0.971837i	$-0.424277\pi$
$11$	−2.44164	−	2.44164i	−0.736182	−	0.736182i	−0.971837	+	0.235655i	$0.924277\pi$
$12$		0			0
$13$	2.78793	+	2.78793i	0.773234	+	0.773234i	0.978670	−	0.205437i	$-0.0658615\pi$
$13$	2.78793	+	2.78793i	0.773234	+	0.773234i	−0.205437	+	0.978670i	$0.565861\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.19850i			0.775749i	0.921712	+	0.387875i	$0.126791\pi$
$17$			3.19850i			0.775749i	−0.921712	+	0.387875i	$0.873209\pi$
$18$		0			0
$19$	1.44164	−	1.44164i	0.330735	−	0.330735i	−0.522131	−	0.852865i	$-0.674863\pi$
$19$	1.44164	−	1.44164i	0.330735	−	0.330735i	0.852865	+	0.522131i	$0.174863\pi$
$20$		0			0
$21$	1.00000	+	1.00000i	0.218218	+	0.218218i
$22$		0			0
$23$	4.00813	−	4.00813i	0.835753	−	0.835753i	−0.152544	−	0.988297i	$-0.548746\pi$
$23$	4.00813	−	4.00813i	0.835753	−	0.835753i	0.988297	+	0.152544i	$0.0487465\pi$
$24$		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	0.994131	+	0.108182i	$0.0345030\pi$
$31$	6.13742	+	6.13742i	1.10231	+	1.10231i	0.108182	+	0.994131i	$0.465497\pi$
$32$		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	−0.247518	−	0.968883i	$-0.579615\pi$
$41$	4.61899	−	4.61899i	0.721365	−	0.721365i	0.968883	+	0.247518i	$0.0796150\pi$
$42$		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	−0.991380	−	0.131016i	$-0.958176\pi$
$53$	−6.26354	+	6.26354i	−0.860364	+	0.860364i	0.131016	+	0.991380i	$0.458176\pi$
$54$		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	$0.207363\pi$
$59$			9.31477i			1.21268i	−0.795206	+	0.606340i	$0.792637\pi$
$60$		0			0
$61$	0.963137	+	0.963137i	0.123317	+	0.123317i	0.766072	−	0.642755i	$-0.222209\pi$
$61$	0.963137	+	0.963137i	0.123317	+	0.123317i	−0.642755	+	0.766072i	$0.722209\pi$
$62$		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	−0.738887	−	0.673829i	$-0.764648\pi$
$67$	−0.532522	+	0.532522i	−0.0650579	+	0.0650579i	0.673829	+	0.738887i	$0.264648\pi$
$68$		0			0
$69$	1.11071	−	1.11071i	0.133713	−	0.133713i
$70$		0			0
$71$			5.92321i			0.702955i	0.936196	+	0.351478i	$0.114321\pi$
$71$			5.92321i			0.702955i	−0.936196	+	0.351478i	$0.885679\pi$
$72$		0			0
$73$			6.67410i			0.781145i	0.920572	+	0.390572i	$0.127723\pi$
$73$			6.67410i			0.781145i	−0.920572	+	0.390572i	$0.872277\pi$
$74$		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	−0.735834	−	0.677161i	$-0.763210\pi$
$79$	−0.521498	+	0.521498i	−0.0586731	+	0.0586731i	0.677161	+	0.735834i	$0.263210\pi$
$80$		0			0
$81$	8.31477			0.923863
$82$		0			0
$83$	1.78549	−	1.78549i	0.195983	−	0.195983i	−0.602293	−	0.798275i	$-0.705746\pi$
$83$	1.78549	−	1.78549i	0.195983	−	0.195983i	0.798275	+	0.602293i	$0.205746\pi$
$84$		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	−0.837074	−	0.547089i	$-0.815736\pi$
$89$	−2.73571	+	2.73571i	−0.289985	+	0.289985i	0.547089	+	0.837074i	$0.315736\pi$
$90$		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	0.441586	−	0.897219i	$-0.354416\pi$
$101$	−4.57906	−	4.57906i	−0.455634	−	0.455634i	−0.897219	+	0.441586i	$0.854416\pi$
$102$		0			0
$103$	−5.84234	+	5.84234i	−0.575663	+	0.575663i	−0.933705	−	0.358042i	$-0.883444\pi$
$103$	−5.84234	+	5.84234i	−0.575663	+	0.575663i	0.358042	+	0.933705i	$0.383444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.99707	−	5.99707i	−0.579759	−	0.579759i	0.355078	−	0.934837i	$-0.384454\pi$
$107$	−5.99707	−	5.99707i	−0.579759	−	0.579759i	−0.934837	+	0.355078i	$0.884454\pi$
$108$		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	$0.173856\pi$
$113$			11.0432i			1.03886i	−0.854513	+	0.519430i	$0.826144\pi$
$114$		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	$-0.844009\pi$
$127$		−	10.6086i		−	0.941359i	0.882304	−	0.470679i	$-0.155991\pi$
$128$		0			0
$129$	−0.00306622			−0.000269965
$130$		0			0
$131$	11.0207	−	11.0207i	0.962883	−	0.962883i	−0.0364523	−	0.999335i	$-0.511606\pi$
$131$	11.0207	−	11.0207i	0.962883	−	0.962883i	0.999335	+	0.0364523i	$0.0116057\pi$
$132$		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	$0.234057\pi$
$137$			15.7035i			1.34164i	−0.741622	+	0.670818i	$0.765943\pi$
$138$		0			0
$139$		−	3.76656i		−	0.319475i	−0.987160	−	0.159738i	$-0.948935\pi$
$139$		−	3.76656i		−	0.319475i	0.987160	−	0.159738i	$-0.0510648\pi$
$140$		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	$-0.930346\pi$
$151$		−	5.33507i		−	0.434162i	0.976154	−	0.217081i	$-0.0696535\pi$
$152$		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	$0.392095\pi$
$163$			24.0811i			1.88618i	−0.332538	+	0.943090i	$0.607905\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.11658	+	5.11658i	−0.395933	+	0.395933i	−0.876796	−	0.480863i	$-0.840324\pi$
$167$	−5.11658	+	5.11658i	−0.395933	+	0.395933i	0.480863	+	0.876796i	$0.340324\pi$
$168$		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	0.216453	−	0.976293i	$-0.430551\pi$
$173$	−9.99414	−	9.99414i	−0.759840	−	0.759840i	−0.976293	+	0.216453i	$0.930551\pi$
$174$		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	0.491268	−	0.871009i	$-0.336534\pi$
$191$	−5.24813	−	5.24813i	−0.379741	−	0.379741i	−0.871009	+	0.491268i	$0.836534\pi$
$192$		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	0.802805	−	0.596242i	$-0.203340\pi$
$197$	2.89925	+	2.89925i	0.206563	+	0.206563i	−0.596242	+	0.802805i	$0.703340\pi$
$198$		0			0
$199$			18.1243i			1.28480i	0.766370	+	0.642400i	$0.222061\pi$
$199$			18.1243i			1.28480i	−0.766370	+	0.642400i	$0.777939\pi$
$200$		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	−0.997228	−	0.0744052i	$-0.976294\pi$
$211$	−13.4048	+	13.4048i	−0.922823	+	0.922823i	0.0744052	+	0.997228i	$0.476294\pi$
$212$		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	−0.238091	−	0.971243i	$-0.576522\pi$
$223$	10.9483	−	10.9483i	0.733151	−	0.733151i	0.971243	+	0.238091i	$0.0765218\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.95648	+	9.95648i	0.660835	+	0.660835i	0.955577	−	0.294742i	$-0.0952337\pi$
$227$	9.95648	+	9.95648i	0.660835	+	0.660835i	−0.294742	+	0.955577i	$0.595234\pi$
$228$		0			0
$229$	14.4255	+	14.4255i	0.953262	+	0.953262i	0.998956	−	0.0456931i	$-0.0145496\pi$
$229$	14.4255	+	14.4255i	0.953262	+	0.953262i	−0.0456931	+	0.998956i	$0.514550\pi$
$230$		0			0
$231$		−	4.88328i		−	0.321296i
$232$		0			0
$233$	18.6682	−	18.6682i	1.22300	−	1.22300i	0.256436	−	0.966561i	$-0.417452\pi$
$233$	18.6682	−	18.6682i	1.22300	−	1.22300i	0.966561	−	0.256436i	$-0.0825483\pi$
$234$		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	$0.236656\pi$
$239$			20.9277i			1.35370i	−0.736119	+	0.676852i	$0.763344\pi$
$240$		0			0
$241$		−	15.6130i		−	1.00572i	−0.864368	−	0.502860i	$-0.832281\pi$
$241$		−	15.6130i		−	1.00572i	0.864368	−	0.502860i	$-0.167719\pi$
$242$		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	0.977520	+	0.210841i	$0.0676203\pi$
$251$	18.8272	+	18.8272i	1.18836	+	1.18836i	0.210841	+	0.977520i	$0.432380\pi$
$252$		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	0.301404	−	0.953497i	$-0.402545\pi$
$257$	−10.4538	−	10.4538i	−0.652093	−	0.652093i	−0.953497	+	0.301404i	$0.902545\pi$
$258$		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	0.865884	−	0.500245i	$-0.166757\pi$
$269$	5.99693	+	5.99693i	0.365640	+	0.365640i	−0.500245	+	0.865884i	$0.666757\pi$
$270$		0			0
$271$	2.62607	−	2.62607i	0.159522	−	0.159522i	−0.622833	−	0.782355i	$-0.714018\pi$
$271$	2.62607	−	2.62607i	0.159522	−	0.159522i	0.782355	+	0.622833i	$0.214018\pi$
$272$		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	0.0479667	−	0.998849i	$-0.484726\pi$
$277$	−15.8258	−	15.8258i	−0.950882	−	0.950882i	−0.998849	+	0.0479667i	$0.984726\pi$
$278$		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	0.971247	−	0.238075i	$-0.0765165\pi$
$283$	12.3338	+	12.3338i	0.733171	+	0.733171i	−0.238075	+	0.971247i	$0.576517\pi$
$284$		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	$-0.677335\pi$
$307$		−	29.7438i		−	1.69757i	0.528739	−	0.848785i	$-0.322665\pi$
$308$		0			0
$309$	−1.61899	+	1.61899i	−0.0921011	+	0.0921011i
$310$		0			0
$311$	13.8303	+	13.8303i	0.784242	+	0.784242i	0.980544	−	0.196302i	$-0.0628932\pi$
$311$	13.8303	+	13.8303i	0.784242	+	0.784242i	−0.196302	+	0.980544i	$0.562893\pi$
$312$		0			0
$313$	2.25584	−	2.25584i	0.127507	−	0.127507i	−0.640473	−	0.767981i	$-0.721262\pi$
$313$	2.25584	−	2.25584i	0.127507	−	0.127507i	0.767981	+	0.640473i	$0.221262\pi$
$314$		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	−0.207040	+	0.978332i	$0.566383\pi$
$331$	−21.5660	+	21.5660i	−1.18537	+	1.18537i	−0.978332	+	0.207040i	$0.933617\pi$
$332$		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	−0.636481	−	0.771292i	$-0.719611\pi$
$347$	2.51125	−	2.51125i	0.134811	−	0.134811i	0.771292	+	0.636481i	$0.219611\pi$
$348$		0			0
$349$		−	29.3592i		−	1.57156i	−0.618504	−	0.785782i	$-0.712261\pi$
$349$		−	29.3592i		−	1.57156i	0.618504	−	0.785782i	$-0.287739\pi$
$350$		0			0
$351$	−4.57611	−	4.57611i	−0.244255	−	0.244255i
$352$		0			0
$353$	−21.8065	−	21.8065i	−1.16064	−	1.16064i	−0.984335	−	0.176310i	$-0.943584\pi$
$353$	−21.8065	−	21.8065i	−1.16064	−	1.16064i	−0.176310	−	0.984335i	$-0.556416\pi$
$354$		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	−0.699386	−	0.714744i	$-0.746543\pi$
$359$	0.291005	−	0.291005i	0.0153586	−	0.0153586i	0.714744	+	0.699386i	$0.246543\pi$
$360$		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	$-0.808509\pi$
$367$		−	21.6841i		−	1.13190i	0.824439	−	0.565952i	$-0.191491\pi$
$368$		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	−0.609560	−	0.792740i	$-0.708654\pi$
$373$	3.53778	−	3.53778i	0.183179	−	0.183179i	0.792740	+	0.609560i	$0.208654\pi$
$374$		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	−0.347405	−	0.937715i	$-0.612937\pi$
$379$	11.4921	−	11.4921i	0.590311	−	0.590311i	0.937715	+	0.347405i	$0.112937\pi$
$380$		0			0
$381$		−	2.93978i		−	0.150609i
$382$		0			0
$383$	12.1791	+	12.1791i	0.622324	+	0.622324i	0.946125	−	0.323801i	$-0.104961\pi$
$383$	12.1791	+	12.1791i	0.622324	+	0.622324i	−0.323801	+	0.946125i	$0.604961\pi$
$384$		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	−0.959659	−	0.281166i	$-0.909279\pi$
$389$	−24.4729	−	24.4729i	−1.24082	−	1.24082i	−0.281166	−	0.959659i	$-0.590721\pi$
$390$		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	0.263330	−	0.964706i	$-0.415179\pi$
$397$	−13.9748	−	13.9748i	−0.701376	−	0.701376i	−0.964706	+	0.263330i	$0.915179\pi$
$398$		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	−0.204694	−	0.978826i	$-0.565620\pi$
$409$	15.6559	−	15.6559i	0.774132	−	0.774132i	0.978826	+	0.204694i	$0.0656199\pi$
$410$		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	−0.777524	−	0.628853i	$-0.783525\pi$
$421$	−3.05048	+	3.05048i	−0.148671	+	0.148671i	0.628853	+	0.777524i	$0.283525\pi$
$422$		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	$-0.643990\pi$
$433$		−	37.4314i		−	1.79884i	0.437087	−	0.899419i	$-0.356010\pi$
$434$		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	$-0.733407\pi$
$443$		−	31.2763i		−	1.48598i	0.669302	−	0.742990i	$-0.266593\pi$
$444$		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	−0.277823	−	0.960632i	$-0.589613\pi$
$449$	14.4685	−	14.4685i	0.682809	−	0.682809i	0.960632	+	0.277823i	$0.0896128\pi$
$450$		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	−0.302933	−	0.953012i	$-0.597966\pi$
$457$	13.8971	−	13.8971i	0.650079	−	0.650079i	0.953012	+	0.302933i	$0.0979657\pi$
$458$		0			0
$459$		−	5.25001i		−	0.245049i
$460$		0			0
$461$	25.7402	−	25.7402i	1.19884	−	1.19884i	0.224330	−	0.974513i	$-0.427981\pi$
$461$	25.7402	−	25.7402i	1.19884	−	1.19884i	0.974513	−	0.224330i	$-0.0720192\pi$
$462$		0			0
$463$	−27.0353	−	27.0353i	−1.25644	−	1.25644i	−0.952783	−	0.303654i	$-0.901793\pi$
$463$	−27.0353	−	27.0353i	−1.25644	−	1.25644i	−0.303654	−	0.952783i	$-0.598207\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			19.3698i			0.896327i	0.893952	+	0.448164i	$0.147922\pi$
$467$			19.3698i			0.896327i	−0.893952	+	0.448164i	$0.852078\pi$
$468$		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	0.294306	−	0.955711i	$-0.404911\pi$
$479$	−14.4756	−	14.4756i	−0.661405	−	0.661405i	−0.955711	+	0.294306i	$0.904911\pi$
$480$		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	0.497641	−	0.867383i	$-0.334200\pi$
$487$	−8.15950	−	8.15950i	−0.369742	−	0.369742i	−0.867383	+	0.497641i	$0.834200\pi$
$488$		0			0
$489$			6.67320i			0.301773i
$490$		0			0
$491$	−3.52898	+	3.52898i	−0.159261	+	0.159261i	−0.782239	−	0.622978i	$-0.785922\pi$
$491$	−3.52898	+	3.52898i	−0.159261	+	0.159261i	0.622978	+	0.782239i	$0.285922\pi$
$492$		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	$0.253006\pi$
$503$			32.0156i			1.42751i	−0.700398	+	0.713753i	$0.746994\pi$
$504$		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	$0.0517863\pi$
$509$			7.30864i			0.323950i	−0.986795	+	0.161975i	$0.948214\pi$
$510$		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	$-0.797108\pi$
$521$		−	27.1673i		−	1.19022i	0.803644	−	0.595111i	$-0.202892\pi$
$522$		0			0
$523$	25.4884	−	25.4884i	1.11453	−	1.11453i	0.122000	−	0.992530i	$-0.461069\pi$
$523$	25.4884	−	25.4884i	1.11453	−	1.11453i	0.992530	−	0.122000i	$-0.0389307\pi$
$524$		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	0.481086	−	0.876673i	$-0.340242\pi$
$541$	−9.20112	−	9.20112i	−0.395587	−	0.395587i	−0.876673	+	0.481086i	$0.840242\pi$
$542$		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	0.678603	−	0.734505i	$-0.262586\pi$
$547$	−1.30742	−	1.30742i	−0.0559014	−	0.0559014i	−0.734505	+	0.678603i	$0.762586\pi$
$548$		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	−0.515139	−	0.857107i	$-0.672260\pi$
$557$	8.07074	−	8.07074i	0.341968	−	0.341968i	0.857107	+	0.515139i	$0.172260\pi$
$558$		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	$0.126287\pi$
$563$			18.3375i			0.772834i	−0.922324	+	0.386417i	$0.873713\pi$
$564$		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	0.783007	−	0.622013i	$-0.213685\pi$
$569$	3.84028	+	3.84028i	0.160993	+	0.160993i	−0.622013	+	0.783007i	$0.713685\pi$
$570$		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	0.963824	+	0.266539i	$0.0858801\pi$
$587$	29.8093	+	29.8093i	1.23036	+	1.23036i	0.266539	+	0.963824i	$0.414120\pi$
$588$		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	0.995994	+	0.0894166i	$0.0285003\pi$
$593$	26.4315	+	26.4315i	1.08541	+	1.08541i	0.0894166	+	0.995994i	$0.471500\pi$
$594$		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	−0.208615	−	0.977998i	$-0.566895\pi$
$599$	18.8303	−	18.8303i	0.769383	−	0.769383i	0.977998	+	0.208615i	$0.0668954\pi$
$600$		0			0
$601$	−3.65892	−	3.65892i	−0.149250	−	0.149250i	0.628533	−	0.777783i	$-0.283656\pi$
$601$	−3.65892	−	3.65892i	−0.149250	−	0.149250i	−0.777783	+	0.628533i	$0.783656\pi$
$602$		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	0.0331846	−	0.999449i	$-0.489435\pi$
$613$	−23.9236	−	23.9236i	−0.966265	−	0.966265i	−0.999449	+	0.0331846i	$0.989435\pi$
$614$		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	0.581100	−	0.813832i	$-0.302623\pi$
$619$	−5.79033	−	5.79033i	−0.232733	−	0.232733i	−0.813832	+	0.581100i	$0.802623\pi$
$620$		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	$0.136644\pi$
$631$			20.9105i			0.832435i	−0.909265	+	0.416217i	$0.863356\pi$
$632$		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	0.995316	+	0.0966700i	$0.0308191\pi$
$641$	27.6469	+	27.6469i	1.09199	+	1.09199i	0.0966700	+	0.995316i	$0.469181\pi$
$642$		0			0
$643$	−21.3853	−	21.3853i	−0.843355	−	0.843355i	0.145939	−	0.989294i	$-0.453380\pi$
$643$	−21.3853	−	21.3853i	−0.843355	−	0.843355i	−0.989294	+	0.145939i	$0.953380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.24380	−	8.24380i	0.324097	−	0.324097i	−0.526239	−	0.850337i	$-0.676398\pi$
$647$	8.24380	−	8.24380i	0.324097	−	0.324097i	0.850337	+	0.526239i	$0.176398\pi$
$648$		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	$0.0665510\pi$
$653$			10.6077i			0.415112i	−0.978223	+	0.207556i	$0.933449\pi$
$654$		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	−0.0407781	−	0.999168i	$-0.512984\pi$
$659$	24.6028	−	24.6028i	0.958390	−	0.958390i	0.999168	+	0.0407781i	$0.0129837\pi$
$660$		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	0.999999	+	0.00106328i	$0.000338452\pi$
$673$	25.9698	+	25.9698i	1.00106	+	1.00106i	0.00106328	+	0.999999i	$0.499662\pi$
$674$		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	−0.642579	−	0.766220i	$-0.722135\pi$
$683$	3.23127	−	3.23127i	0.123641	−	0.123641i	0.766220	+	0.642579i	$0.222135\pi$
$684$		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	$-0.564648\pi$
$701$		−	51.8645i		−	1.95889i	0.201703	−	0.979447i	$-0.435352\pi$
$702$		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	$0.219493\pi$
$719$			34.1182i			1.27239i	−0.771527	+	0.636197i	$0.780507\pi$
$720$		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	$-0.693331\pi$
$727$		−	44.2814i		−	1.64231i	0.570708	−	0.821153i	$-0.306669\pi$
$728$		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	$-0.939592\pi$
$733$		−	10.2144i		−	0.377278i	0.982047	−	0.188639i	$-0.0604075\pi$
$734$		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	0.156812	−	0.987628i	$-0.449878\pi$
$739$	31.1111	−	31.1111i	1.14444	−	1.14444i	0.987628	−	0.156812i	$-0.0501218\pi$
$740$		0			0
$741$	2.22754			0.0818309
$742$		0			0
$743$			7.80384i			0.286295i	0.989701	+	0.143148i	$0.0457223\pi$
$743$			7.80384i			0.286295i	−0.989701	+	0.143148i	$0.954278\pi$
$744$		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	−0.995249	−	0.0973668i	$-0.968958\pi$
$751$	−24.6059	+	24.6059i	−0.897882	+	0.897882i	0.0973668	+	0.995249i	$0.468958\pi$
$752$		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	$-0.970590\pi$
$757$		−	5.07696i		−	0.184525i	0.995735	−	0.0922627i	$-0.0294100\pi$
$758$		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	0.562018	−	0.827125i	$-0.310025\pi$
$769$	−7.35163	−	7.35163i	−0.265107	−	0.265107i	−0.827125	+	0.562018i	$0.810025\pi$
$770$		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	−0.520881	−	0.853629i	$-0.674397\pi$
$787$	9.33475	−	9.33475i	0.332748	−	0.332748i	0.853629	+	0.520881i	$0.174397\pi$
$788$		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	$-0.638585\pi$
$797$		−	51.1951i		−	1.81342i	0.421752	−	0.906711i	$-0.361415\pi$
$798$		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	0.490008	−	0.871718i	$-0.336994\pi$
$809$	−10.8570	−	10.8570i	−0.381711	−	0.381711i	−0.871718	+	0.490008i	$0.836994\pi$
$810$		0			0
$811$		−	1.51201i		−	0.0530939i	−0.999648	−	0.0265470i	$-0.991549\pi$
$811$		−	1.51201i		−	0.0530939i	0.999648	−	0.0265470i	$-0.00845115\pi$
$812$		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	$0.225018\pi$
$821$			37.2198i			1.29898i	−0.760370	+	0.649490i	$0.774982\pi$
$822$		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	−0.612646	−	0.790357i	$-0.709895\pi$
$829$	5.11672	−	5.11672i	0.177711	−	0.177711i	0.790357	+	0.612646i	$0.209895\pi$
$830$		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	0.636326	−	0.771420i	$-0.280453\pi$
$839$	−3.91306	−	3.91306i	−0.135094	−	0.135094i	−0.771420	+	0.636326i	$0.780453\pi$
$840$		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	$-0.892710\pi$
$853$		−	19.3178i		−	0.661430i	0.943731	−	0.330715i	$-0.107290\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.6114	−	13.6114i	−0.464958	−	0.464958i	0.435318	−	0.900277i	$-0.356636\pi$
$857$	−13.6114	−	13.6114i	−0.464958	−	0.464958i	−0.900277	+	0.435318i	$0.856636\pi$
$858$		0			0
$859$	15.7934	+	15.7934i	0.538864	+	0.538864i	0.923195	−	0.384332i	$-0.125568\pi$
$859$	15.7934	+	15.7934i	0.538864	+	0.538864i	−0.384332	+	0.923195i	$0.625568\pi$
$860$		0			0
$861$	9.23798			0.314830
$862$		0			0
$863$	−11.8939	−	11.8939i	−0.404872	−	0.404872i	0.475074	−	0.879946i	$-0.342421\pi$
$863$	−11.8939	−	11.8939i	−0.404872	−	0.404872i	−0.879946	+	0.475074i	$0.842421\pi$
$864$		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	0.614708	−	0.788755i	$-0.289274\pi$
$877$	−5.15424	−	5.15424i	−0.174046	−	0.174046i	−0.788755	+	0.614708i	$0.789274\pi$
$878$		0			0
$879$	−3.54232			−0.119479
$880$		0			0
$881$	−16.8034	−	16.8034i	−0.566122	−	0.566122i	0.364918	−	0.931040i	$-0.381097\pi$
$881$	−16.8034	−	16.8034i	−0.566122	−	0.566122i	−0.931040	+	0.364918i	$0.881097\pi$
$882$		0			0
$883$	8.17142	+	8.17142i	0.274990	+	0.274990i	0.831105	−	0.556115i	$-0.187709\pi$
$883$	8.17142	+	8.17142i	0.274990	+	0.274990i	−0.556115	+	0.831105i	$0.687709\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	53.1622i		−	1.78501i	−0.451035	−	0.892506i	$-0.648945\pi$
$887$		−	53.1622i		−	1.78501i	0.451035	−	0.892506i	$-0.351055\pi$
$888$		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	−0.321473	−	0.946919i	$-0.604178\pi$
$911$	18.8777	−	18.8777i	0.625445	−	0.625445i	0.946919	+	0.321473i	$0.104178\pi$
$912$		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	$0.284587\pi$
$919$			47.2683i			1.55924i	−0.626255	+	0.779618i	$0.715413\pi$
$920$		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	$0.0696800\pi$
$929$			13.2380i			0.434324i	−0.976136	+	0.217162i	$0.930320\pi$
$930$		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	−0.987810	−	0.155662i	$-0.950249\pi$
$937$	−25.4724	+	25.4724i	−0.832148	+	0.832148i	0.155662	+	0.987810i	$0.450249\pi$
$938$		0			0
$939$	0.625122	−	0.625122i	0.0204001	−	0.0204001i
$940$		0			0
$941$		−	23.8675i		−	0.778059i	−0.921225	−	0.389029i	$-0.872810\pi$
$941$		−	23.8675i		−	0.778059i	0.921225	−	0.389029i	$-0.127190\pi$
$942$		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	$0.0118093\pi$
$947$			2.28285i			0.0741828i	−0.999312	+	0.0370914i	$0.988191\pi$
$948$		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	−0.830647	−	0.556800i	$-0.812029\pi$
$953$	−8.45384	+	8.45384i	−0.273847	+	0.273847i	0.556800	+	0.830647i	$0.312029\pi$
$954$		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	0.312950	−	0.949770i	$-0.398683\pi$
$971$	−19.8439	−	19.8439i	−0.636820	−	0.636820i	−0.949770	+	0.312950i	$0.898683\pi$
$972$		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	−0.972047	−	0.234787i	$-0.924561\pi$
$977$	−37.7220	−	37.7220i	−1.20683	−	1.20683i	−0.234787	−	0.972047i	$-0.575439\pi$
$978$		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	$0.0154841\pi$
$983$			3.04910i			0.0972510i	−0.998817	+	0.0486255i	$0.984516\pi$
$984$		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	$0.0205227\pi$
$991$			4.05650i			0.128859i	−0.997922	+	0.0644294i	$0.979477\pi$
$992$		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	$0.232690\pi$
$997$			42.1611i			1.33526i	−0.744495	+	0.667628i	$0.767310\pi$
$998$		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.157.5	yes	16
5.2	odd	4		2900.2.j.c.2593.4	yes	16
5.3	odd	4		2900.2.j.c.2593.5	yes	16
5.4	even	2	inner	2900.2.s.c.157.4	yes	16
29.17	odd	4		2900.2.j.c.1757.4	✓	16
145.17	even	4	inner	2900.2.s.c.1293.4	yes	16
145.104	odd	4		2900.2.j.c.1757.5	yes	16
145.133	even	4	inner	2900.2.s.c.1293.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2900.2.j.c.1757.4	✓	16	29.17	odd	4
2900.2.j.c.1757.5	yes	16	145.104	odd	4
2900.2.j.c.2593.4	yes	16	5.2	odd	4
2900.2.j.c.2593.5	yes	16	5.3	odd	4
2900.2.s.c.157.4	yes	16	5.4	even	2	inner
2900.2.s.c.157.5	yes	16	1.1	even	1	trivial
2900.2.s.c.1293.4	yes	16	145.17	even	4	inner
2900.2.s.c.1293.5	yes	16	145.133	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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