LMFDB - Embedded newform 2900.2.s.d.1293.12

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 = [30,0,0] 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:	$30$
Relative dimension:	$15$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 580)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1293.12
Character	$\chi$	$=$	2900.1293
Dual form			2900.2.s.d.157.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61432 q^{3} +(-1.96009 + 1.96009i) q^{7} -0.393986 q^{9} +(1.97757 - 1.97757i) q^{11} +(1.79751 - 1.79751i) q^{13} -3.77934i q^{17} +(-3.84455 - 3.84455i) q^{19} +(-3.16420 + 3.16420i) q^{21} +(3.28633 + 3.28633i) q^{23} -5.47896 q^{27} +(-0.143119 - 5.38326i) q^{29} +(6.28310 - 6.28310i) q^{31} +(3.19243 - 3.19243i) q^{33} -2.84861 q^{37} +(2.90174 - 2.90174i) q^{39} +(-0.715885 - 0.715885i) q^{41} +1.15641 q^{43} +1.02010 q^{47} -0.683883i q^{49} -6.10105i q^{51} +(1.96686 + 1.96686i) q^{53} +(-6.20632 - 6.20632i) q^{57} -6.60253i q^{59} +(-2.62953 + 2.62953i) q^{61} +(0.772246 - 0.772246i) q^{63} +(5.70401 + 5.70401i) q^{67} +(5.30517 + 5.30517i) q^{69} -10.2338i q^{71} -12.4381i q^{73} +7.75244i q^{77} +(8.41306 + 8.41306i) q^{79} -7.66282 q^{81} +(-2.10823 - 2.10823i) q^{83} +(-0.231039 - 8.69028i) q^{87} +(2.72236 + 2.72236i) q^{89} +7.04653i q^{91} +(10.1429 - 10.1429i) q^{93} +1.98353 q^{97} +(-0.779136 + 0.779136i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 38 q^{9} - 4 q^{11} + 6 q^{13} - 4 q^{21} - 12 q^{27} - 4 q^{31} - 4 q^{33} + 24 q^{37} - 12 q^{39} + 10 q^{41} + 16 q^{43} + 32 q^{47} + 18 q^{53} - 24 q^{57} - 22 q^{61} - 24 q^{63} - 16 q^{67}+ \cdots + 16 q^{97}+O(q^{100}) $ 30 * q + 38 * q^9 - 4 * q^11 + 6 * q^13 - 4 * q^21 - 12 * q^27 - 4 * q^31 - 4 * q^33 + 24 * q^37 - 12 * q^39 + 10 * q^41 + 16 * q^43 + 32 * q^47 + 18 * q^53 - 24 * q^57 - 22 * q^61 - 24 * q^63 - 16 * q^67 - 8 * q^69 - 20 * q^79 + 70 * q^81 - 56 * q^83 - 48 * q^87 - 6 * q^89 - 44 * q^93 + 16 * q^97

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
$3$	1.61432			0.932025			0.466013	−	0.884778i	$-0.345690\pi$
$3$	1.61432			0.932025			0.466013	+	0.884778i	$0.345690\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.96009	+	1.96009i	−0.740843	+	0.740843i	−0.972740	−	0.231897i	$-0.925507\pi$
$7$	−1.96009	+	1.96009i	−0.740843	+	0.740843i	0.231897	+	0.972740i	$0.425507\pi$
$8$		0			0
$9$	−0.393986			−0.131329
$10$		0			0
$11$	1.97757	−	1.97757i	0.596261	−	0.596261i	−0.343054	−	0.939316i	$-0.611462\pi$
$11$	1.97757	−	1.97757i	0.596261	−	0.596261i	0.939316	+	0.343054i	$0.111462\pi$
$12$		0			0
$13$	1.79751	−	1.79751i	0.498538	−	0.498538i	−0.412444	−	0.910983i	$-0.635325\pi$
$13$	1.79751	−	1.79751i	0.498538	−	0.498538i	0.910983	+	0.412444i	$0.135325\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.77934i		−	0.916625i	−0.888791	−	0.458312i	$-0.848454\pi$
$17$		−	3.77934i		−	0.916625i	0.888791	−	0.458312i	$-0.151546\pi$
$18$		0			0
$19$	−3.84455	−	3.84455i	−0.882001	−	0.882001i	0.111737	−	0.993738i	$-0.464359\pi$
$19$	−3.84455	−	3.84455i	−0.882001	−	0.882001i	−0.993738	+	0.111737i	$0.964359\pi$
$20$		0			0
$21$	−3.16420	+	3.16420i	−0.690485	+	0.690485i
$22$		0			0
$23$	3.28633	+	3.28633i	0.685246	+	0.685246i	0.961177	−	0.275931i	$-0.0889862\pi$
$23$	3.28633	+	3.28633i	0.685246	+	0.685246i	−0.275931	+	0.961177i	$0.588986\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−5.47896			−1.05443
$28$		0			0
$29$	−0.143119	−	5.38326i	−0.0265765	−	0.999647i
$29$	−0.143119	−	5.38326i	−0.0265765	−	0.999647i
$30$		0			0
$31$	6.28310	−	6.28310i	1.12848	−	1.12848i	0.138052	−	0.990425i	$-0.455916\pi$
$31$	6.28310	−	6.28310i	1.12848	−	1.12848i	0.990425	−	0.138052i	$-0.0440842\pi$
$32$		0			0
$33$	3.19243	−	3.19243i	0.555731	−	0.555731i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.84861			−0.468309			−0.234154	−	0.972199i	$-0.575232\pi$
$37$	−2.84861			−0.468309			−0.234154	+	0.972199i	$0.575232\pi$
$38$		0			0
$39$	2.90174	−	2.90174i	0.464650	−	0.464650i
$40$		0			0
$41$	−0.715885	−	0.715885i	−0.111802	−	0.111802i	0.648992	−	0.760795i	$-0.275191\pi$
$41$	−0.715885	−	0.715885i	−0.111802	−	0.111802i	−0.760795	+	0.648992i	$0.775191\pi$
$42$		0			0
$43$	1.15641			0.176351			0.0881756	−	0.996105i	$-0.471896\pi$
$43$	1.15641			0.176351			0.0881756	+	0.996105i	$0.471896\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.02010			0.148796			0.0743982	−	0.997229i	$-0.476296\pi$
$47$	1.02010			0.148796			0.0743982	+	0.997229i	$0.476296\pi$
$48$		0			0
$49$		−	0.683883i		−	0.0976975i
$50$		0			0
$51$		−	6.10105i		−	0.854318i
$52$		0			0
$53$	1.96686	+	1.96686i	0.270168	+	0.270168i	0.829168	−	0.559000i	$-0.188815\pi$
$53$	1.96686	+	1.96686i	0.270168	+	0.270168i	−0.559000	+	0.829168i	$0.688815\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.20632	−	6.20632i	−0.822048	−	0.822048i
$58$		0			0
$59$		−	6.60253i		−	0.859576i	−0.902930	−	0.429788i	$-0.858588\pi$
$59$		−	6.60253i		−	0.859576i	0.902930	−	0.429788i	$-0.141412\pi$
$60$		0			0
$61$	−2.62953	+	2.62953i	−0.336676	+	0.336676i	−0.855115	−	0.518439i	$-0.826513\pi$
$61$	−2.62953	+	2.62953i	−0.336676	+	0.336676i	0.518439	+	0.855115i	$0.326513\pi$
$62$		0			0
$63$	0.772246	−	0.772246i	0.0972939	−	0.0972939i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.70401	+	5.70401i	0.696856	+	0.696856i	0.963731	−	0.266875i	$-0.0859910\pi$
$67$	5.70401	+	5.70401i	0.696856	+	0.696856i	−0.266875	+	0.963731i	$0.585991\pi$
$68$		0			0
$69$	5.30517	+	5.30517i	0.638667	+	0.638667i
$70$		0			0
$71$		−	10.2338i		−	1.21452i	−0.794502	−	0.607262i	$-0.792268\pi$
$71$		−	10.2338i		−	1.21452i	0.794502	−	0.607262i	$-0.207732\pi$
$72$		0			0
$73$		−	12.4381i		−	1.45577i	−0.685698	−	0.727886i	$-0.740503\pi$
$73$		−	12.4381i		−	1.45577i	0.685698	−	0.727886i	$-0.259497\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			7.75244i			0.883472i
$78$		0			0
$79$	8.41306	+	8.41306i	0.946544	+	0.946544i	0.998642	−	0.0520983i	$-0.0165909\pi$
$79$	8.41306	+	8.41306i	0.946544	+	0.946544i	−0.0520983	+	0.998642i	$0.516591\pi$
$80$		0			0
$81$	−7.66282			−0.851424
$82$		0			0
$83$	−2.10823	−	2.10823i	−0.231408	−	0.231408i	0.581872	−	0.813280i	$-0.302320\pi$
$83$	−2.10823	−	2.10823i	−0.231408	−	0.231408i	−0.813280	+	0.581872i	$0.802320\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.231039	−	8.69028i	−0.0247699	−	0.931696i
$88$		0			0
$89$	2.72236	+	2.72236i	0.288570	+	0.288570i	0.836515	−	0.547945i	$-0.184590\pi$
$89$	2.72236	+	2.72236i	0.288570	+	0.288570i	−0.547945	+	0.836515i	$0.684590\pi$
$90$		0			0
$91$			7.04653i			0.738677i
$92$		0			0
$93$	10.1429	−	10.1429i	1.05177	−	1.05177i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.98353			0.201397			0.100699	−	0.994917i	$-0.467892\pi$
$97$	1.98353			0.201397			0.100699	+	0.994917i	$0.467892\pi$
$98$		0			0
$99$	−0.779136	+	0.779136i	−0.0783061	+	0.0783061i
$100$		0			0
$101$	12.5084	−	12.5084i	1.24464	−	1.24464i	0.286579	−	0.958057i	$-0.407482\pi$
$101$	12.5084	−	12.5084i	1.24464	−	1.24464i	0.958057	−	0.286579i	$-0.0925182\pi$
$102$		0			0
$103$	−3.19708	−	3.19708i	−0.315017	−	0.315017i	0.531832	−	0.846850i	$-0.321504\pi$
$103$	−3.19708	−	3.19708i	−0.315017	−	0.315017i	−0.846850	+	0.531832i	$0.821504\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.37362	−	7.37362i	0.712835	−	0.712835i	−0.254293	−	0.967127i	$-0.581843\pi$
$107$	7.37362	−	7.37362i	0.712835	−	0.712835i	0.967127	+	0.254293i	$0.0818427\pi$
$108$		0			0
$109$	3.62667			0.347372			0.173686	−	0.984801i	$-0.444432\pi$
$109$	3.62667			0.347372			0.173686	+	0.984801i	$0.444432\pi$
$110$		0			0
$111$	−4.59856			−0.436476
$112$		0			0
$113$		−	12.0264i		−	1.13135i	−0.824628	−	0.565676i	$-0.808615\pi$
$113$		−	12.0264i		−	1.13135i	0.824628	−	0.565676i	$-0.191385\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.708191	+	0.708191i	−0.0654723	+	0.0654723i
$118$		0			0
$119$	7.40784	+	7.40784i	0.679075	+	0.679075i
$120$		0			0
$121$			3.17840i			0.288945i
$122$		0			0
$123$	−1.15566	−	1.15566i	−0.104203	−	0.104203i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	21.7471i		−	1.92974i	−0.262727	−	0.964870i	$-0.584622\pi$
$127$		−	21.7471i		−	1.92974i	0.262727	−	0.964870i	$-0.415378\pi$
$128$		0			0
$129$	1.86681			0.164364
$130$		0			0
$131$	9.63670	+	9.63670i	0.841962	+	0.841962i	0.989114	−	0.147151i	$-0.0470105\pi$
$131$	9.63670	+	9.63670i	0.841962	+	0.841962i	−0.147151	+	0.989114i	$0.547010\pi$
$132$		0			0
$133$	15.0713			1.30685
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	0.608748i		−	0.0520088i	−0.999662	−	0.0260044i	$-0.991722\pi$
$137$		−	0.608748i		−	0.0520088i	0.999662	−	0.0260044i	$-0.00827840\pi$
$138$		0			0
$139$			6.25065i			0.530174i	0.964225	+	0.265087i	$0.0854006\pi$
$139$			6.25065i			0.530174i	−0.964225	+	0.265087i	$0.914599\pi$
$140$		0			0
$141$	1.64676			0.138682
$142$		0			0
$143$		−	7.10940i		−	0.594518i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	1.10400i		−	0.0910566i
$148$		0			0
$149$	19.0372			1.55959			0.779795	−	0.626035i	$-0.215323\pi$
$149$	19.0372			1.55959			0.779795	+	0.626035i	$0.215323\pi$
$150$		0			0
$151$		−	17.7201i		−	1.44204i	−0.692915	−	0.721019i	$-0.743674\pi$
$151$		−	17.7201i		−	1.44204i	0.692915	−	0.721019i	$-0.256326\pi$
$152$		0			0
$153$			1.48901i			0.120379i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−22.1034			−1.76404			−0.882022	−	0.471209i	$-0.843818\pi$
$157$	−22.1034			−1.76404			−0.882022	+	0.471209i	$0.843818\pi$
$158$		0			0
$159$	3.17513	+	3.17513i	0.251804	+	0.251804i
$160$		0			0
$161$	−12.8830			−1.01532
$162$		0			0
$163$		−	0.940929i		−	0.0736992i	−0.999321	−	0.0368496i	$-0.988268\pi$
$163$		−	0.940929i		−	0.0736992i	0.999321	−	0.0368496i	$-0.0117323\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.07665	+	7.07665i	0.547607	+	0.547607i	0.925748	−	0.378141i	$-0.123436\pi$
$167$	7.07665	+	7.07665i	0.547607	+	0.547607i	−0.378141	+	0.925748i	$0.623436\pi$
$168$		0			0
$169$			6.53795i			0.502919i
$170$		0			0
$171$	1.51470	+	1.51470i	0.115832	+	0.115832i
$172$		0			0
$173$	−14.9169	+	14.9169i	−1.13411	+	1.13411i	−0.144624	+	0.989487i	$0.546197\pi$
$173$	−14.9169	+	14.9169i	−1.13411	+	1.13411i	−0.989487	+	0.144624i	$0.953803\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	10.6586i		−	0.801147i
$178$		0			0
$179$	−11.9746			−0.895022			−0.447511	−	0.894279i	$-0.647689\pi$
$179$	−11.9746			−0.895022			−0.447511	+	0.894279i	$0.647689\pi$
$180$		0			0
$181$	−26.3827			−1.96101			−0.980506	−	0.196490i	$-0.937046\pi$
$181$	−26.3827			−1.96101			−0.980506	+	0.196490i	$0.937046\pi$
$182$		0			0
$183$	−4.24488	+	4.24488i	−0.313791	+	0.313791i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.47393	−	7.47393i	−0.546548	−	0.546548i
$188$		0			0
$189$	10.7392	−	10.7392i	0.781165	−	0.781165i
$190$		0			0
$191$	−12.1388	+	12.1388i	−0.878334	+	0.878334i	−0.993362	−	0.115028i	$-0.963304\pi$
$191$	−12.1388	+	12.1388i	−0.878334	+	0.878334i	0.115028	+	0.993362i	$0.463304\pi$
$192$		0			0
$193$	−14.0307			−1.00995			−0.504975	−	0.863134i	$-0.668498\pi$
$193$	−14.0307			−1.00995			−0.504975	+	0.863134i	$0.668498\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.03169	−	6.03169i	0.429740	−	0.429740i	−0.458800	−	0.888540i	$-0.651720\pi$
$197$	6.03169	−	6.03169i	0.429740	−	0.429740i	0.888540	+	0.458800i	$0.151720\pi$
$198$		0			0
$199$			9.22916i			0.654238i	0.944983	+	0.327119i	$0.106078\pi$
$199$			9.22916i			0.654238i	−0.944983	+	0.327119i	$0.893922\pi$
$200$		0			0
$201$	9.20808	+	9.20808i	0.649488	+	0.649488i
$202$		0			0
$203$	10.8322	+	10.2711i	0.760271	+	0.720893i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.29477	−	1.29477i	−0.0899924	−	0.0899924i
$208$		0			0
$209$	−15.2058			−1.05181
$210$		0			0
$211$	−1.89612	−	1.89612i	−0.130534	−	0.130534i	0.638821	−	0.769355i	$-0.279422\pi$
$211$	−1.89612	−	1.89612i	−0.130534	−	0.130534i	−0.769355	+	0.638821i	$0.779422\pi$
$212$		0			0
$213$		−	16.5205i		−	1.13197i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			24.6308i			1.67205i
$218$		0			0
$219$		−	20.0790i		−	1.35682i
$220$		0			0
$221$	−6.79338	−	6.79338i	−0.456973	−	0.456973i
$222$		0			0
$223$	−14.1110	−	14.1110i	−0.944944	−	0.944944i	0.0536178	−	0.998562i	$-0.482925\pi$
$223$	−14.1110	−	14.1110i	−0.944944	−	0.944944i	−0.998562	+	0.0536178i	$0.982925\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.1329	+	19.1329i	−1.26990	+	1.26990i	−0.323758	+	0.946140i	$0.604946\pi$
$227$	−19.1329	+	19.1329i	−1.26990	+	1.26990i	−0.946140	+	0.323758i	$0.895054\pi$
$228$		0			0
$229$	4.05720	−	4.05720i	0.268107	−	0.268107i	−0.560230	−	0.828337i	$-0.689287\pi$
$229$	4.05720	−	4.05720i	0.268107	−	0.268107i	0.828337	+	0.560230i	$0.189287\pi$
$230$		0			0
$231$			12.5149i			0.823419i
$232$		0			0
$233$	1.60471	+	1.60471i	0.105128	+	0.105128i	0.757714	−	0.652586i	$-0.226316\pi$
$233$	1.60471	+	1.60471i	0.105128	+	0.105128i	−0.652586	+	0.757714i	$0.726316\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	13.5813	+	13.5813i	0.882203	+	0.882203i
$238$		0			0
$239$			11.2091i			0.725054i	0.931973	+	0.362527i	$0.118086\pi$
$239$			11.2091i			0.725054i	−0.931973	+	0.362527i	$0.881914\pi$
$240$		0			0
$241$			19.6793i			1.26765i	0.773476	+	0.633826i	$0.218517\pi$
$241$			19.6793i			1.26765i	−0.773476	+	0.633826i	$0.781483\pi$
$242$		0			0
$243$	4.06669			0.260878
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−13.8212			−0.879423
$248$		0			0
$249$	−3.40334	−	3.40334i	−0.215678	−	0.215678i
$250$		0			0
$251$	16.4261	−	16.4261i	1.03680	−	1.03680i	0.0375077	−	0.999296i	$-0.488058\pi$
$251$	16.4261	−	16.4261i	1.03680	−	1.03680i	0.999296	−	0.0375077i	$-0.0119419\pi$
$252$		0			0
$253$	12.9979			0.817172
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.8175	+	13.8175i	−0.861909	+	0.861909i	−0.991560	−	0.129650i	$-0.958615\pi$
$257$	−13.8175	+	13.8175i	−0.861909	+	0.861909i	0.129650	+	0.991560i	$0.458615\pi$
$258$		0			0
$259$	5.58353	−	5.58353i	0.346943	−	0.346943i
$260$		0			0
$261$	0.0563867	+	2.12093i	0.00349025	+	0.131282i
$262$		0			0
$263$	−5.10568			−0.314830			−0.157415	−	0.987533i	$-0.550316\pi$
$263$	−5.10568			−0.314830			−0.157415	+	0.987533i	$0.550316\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.39475	+	4.39475i	0.268954	+	0.268954i
$268$		0			0
$269$	−0.193739	+	0.193739i	−0.0118125	+	0.0118125i	−0.712988	−	0.701176i	$-0.752659\pi$
$269$	−0.193739	+	0.193739i	−0.0118125	+	0.0118125i	0.701176	+	0.712988i	$0.252659\pi$
$270$		0			0
$271$	13.8823	+	13.8823i	0.843292	+	0.843292i	0.989285	−	0.145994i	$-0.0466380\pi$
$271$	13.8823	+	13.8823i	0.843292	+	0.843292i	−0.145994	+	0.989285i	$0.546638\pi$
$272$		0			0
$273$			11.3753i			0.688466i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.1060	−	21.1060i	1.26814	−	1.26814i	0.321087	−	0.947050i	$-0.395952\pi$
$277$	21.1060	−	21.1060i	1.26814	−	1.26814i	0.947050	−	0.321087i	$-0.104048\pi$
$278$		0			0
$279$	−2.47545	+	2.47545i	−0.148201	+	0.148201i
$280$		0			0
$281$	−6.65924			−0.397257			−0.198628	−	0.980075i	$-0.563649\pi$
$281$	−6.65924			−0.397257			−0.198628	+	0.980075i	$0.563649\pi$
$282$		0			0
$283$	−6.41389	+	6.41389i	−0.381266	+	0.381266i	−0.871558	−	0.490292i	$-0.836890\pi$
$283$	−6.41389	+	6.41389i	−0.381266	+	0.381266i	0.490292	+	0.871558i	$0.336890\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.80639			0.165656
$288$		0			0
$289$	2.71658			0.159799
$290$		0			0
$291$	3.20205			0.187707
$292$		0			0
$293$	29.6043			1.72950			0.864750	−	0.502203i	$-0.167477\pi$
$293$	29.6043			1.72950			0.864750	+	0.502203i	$0.167477\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−10.8351	+	10.8351i	−0.628714	+	0.628714i
$298$		0			0
$299$	11.8144			0.683243
$300$		0			0
$301$	−2.26667	+	2.26667i	−0.130649	+	0.130649i
$302$		0			0
$303$	20.1926	−	20.1926i	1.16003	−	1.16003i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	22.8054i		−	1.30157i	−0.759261	−	0.650786i	$-0.774440\pi$
$307$		−	22.8054i		−	1.30157i	0.759261	−	0.650786i	$-0.225560\pi$
$308$		0			0
$309$	−5.16109	−	5.16109i	−0.293604	−	0.293604i
$310$		0			0
$311$	3.18453	−	3.18453i	0.180578	−	0.180578i	−0.611030	−	0.791608i	$-0.709244\pi$
$311$	3.18453	−	3.18453i	0.180578	−	0.180578i	0.791608	+	0.611030i	$0.209244\pi$
$312$		0			0
$313$	−3.76785	−	3.76785i	−0.212972	−	0.212972i	0.592557	−	0.805529i	$-0.298119\pi$
$313$	−3.76785	−	3.76785i	−0.212972	−	0.212972i	−0.805529	+	0.592557i	$0.798119\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0726			−0.678062			−0.339031	−	0.940775i	$-0.610099\pi$
$317$	−12.0726			−0.678062			−0.339031	+	0.940775i	$0.610099\pi$
$318$		0			0
$319$	−10.9288	−	10.3628i	−0.611897	−	0.580204i
$320$		0			0
$321$	11.9033	−	11.9033i	0.664380	−	0.664380i
$322$		0			0
$323$	−14.5299	+	14.5299i	−0.808464	+	0.808464i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.85459			0.323759
$328$		0			0
$329$	−1.99948	+	1.99948i	−0.110235	+	0.110235i
$330$		0			0
$331$	4.13328	+	4.13328i	0.227186	+	0.227186i	0.811516	−	0.584330i	$-0.198643\pi$
$331$	4.13328	+	4.13328i	0.227186	+	0.227186i	−0.584330	+	0.811516i	$0.698643\pi$
$332$		0			0
$333$	1.12231			0.0615023
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.20250			−0.0655043			−0.0327522	−	0.999464i	$-0.510427\pi$
$337$	−1.20250			−0.0655043			−0.0327522	+	0.999464i	$0.510427\pi$
$338$		0			0
$339$		−	19.4145i		−	1.05445i
$340$		0			0
$341$		−	24.8506i		−	1.34573i
$342$		0			0
$343$	−12.3801	−	12.3801i	−0.668465	−	0.668465i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.2920	+	11.2920i	0.606187	+	0.606187i	0.941947	−	0.335761i	$-0.108993\pi$
$347$	11.2920	+	11.2920i	0.606187	+	0.606187i	−0.335761	+	0.941947i	$0.608993\pi$
$348$		0			0
$349$			10.5990i			0.567353i	0.958920	+	0.283676i	$0.0915541\pi$
$349$			10.5990i			0.567353i	−0.958920	+	0.283676i	$0.908446\pi$
$350$		0			0
$351$	−9.84847	+	9.84847i	−0.525672	+	0.525672i
$352$		0			0
$353$	−25.7160	+	25.7160i	−1.36872	+	1.36872i	−0.506462	+	0.862262i	$0.669047\pi$
$353$	−25.7160	+	25.7160i	−1.36872	+	1.36872i	−0.862262	+	0.506462i	$0.830953\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	11.9586	+	11.9586i	0.632915	+	0.632915i
$358$		0			0
$359$	0.396828	+	0.396828i	0.0209438	+	0.0209438i	0.717501	−	0.696557i	$-0.245286\pi$
$359$	0.396828	+	0.396828i	0.0209438	+	0.0209438i	−0.696557	+	0.717501i	$0.745286\pi$
$360$		0			0
$361$			10.5612i			0.555852i
$362$		0			0
$363$			5.13094i			0.269304i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	2.84554i		−	0.148536i	−0.997238	−	0.0742679i	$-0.976338\pi$
$367$		−	2.84554i		−	0.148536i	0.997238	−	0.0742679i	$-0.0236620\pi$
$368$		0			0
$369$	0.282049	+	0.282049i	0.0146829	+	0.0146829i
$370$		0			0
$371$	−7.71042			−0.400305
$372$		0			0
$373$	11.3309	+	11.3309i	0.586692	+	0.586692i	0.936734	−	0.350042i	$-0.113833\pi$
$373$	11.3309	+	11.3309i	0.586692	+	0.586692i	−0.350042	+	0.936734i	$0.613833\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.93370	−	9.41919i	−0.511612	−	0.485113i
$378$		0			0
$379$	−22.5172	−	22.5172i	−1.15663	−	1.15663i	−0.985195	−	0.171437i	$-0.945159\pi$
$379$	−22.5172	−	22.5172i	−1.15663	−	1.15663i	−0.171437	−	0.985195i	$-0.554841\pi$
$380$		0			0
$381$		−	35.1066i		−	1.79857i
$382$		0			0
$383$	2.20602	−	2.20602i	0.112723	−	0.112723i	−0.648496	−	0.761218i	$-0.724602\pi$
$383$	2.20602	−	2.20602i	0.112723	−	0.112723i	0.761218	+	0.648496i	$0.224602\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.455610			−0.0231600
$388$		0			0
$389$	5.99292	−	5.99292i	0.303853	−	0.303853i	−0.538666	−	0.842519i	$-0.681072\pi$
$389$	5.99292	−	5.99292i	0.303853	−	0.303853i	0.842519	+	0.538666i	$0.181072\pi$
$390$		0			0
$391$	12.4201	−	12.4201i	0.628114	−	0.628114i
$392$		0			0
$393$	15.5567	+	15.5567i	0.784730	+	0.784730i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−6.67342	+	6.67342i	−0.334930	+	0.334930i	−0.854455	−	0.519525i	$-0.826109\pi$
$397$	−6.67342	+	6.67342i	−0.334930	+	0.334930i	0.519525	+	0.854455i	$0.326109\pi$
$398$		0			0
$399$	24.3299			1.21802
$400$		0			0
$401$	−18.0031			−0.899030			−0.449515	−	0.893273i	$-0.648403\pi$
$401$	−18.0031			−0.899030			−0.449515	+	0.893273i	$0.648403\pi$
$402$		0			0
$403$		−	22.5878i		−	1.12518i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.63334	+	5.63334i	−0.279234	+	0.279234i
$408$		0			0
$409$	−5.72639	−	5.72639i	−0.283152	−	0.283152i	0.551213	−	0.834365i	$-0.314165\pi$
$409$	−5.72639	−	5.72639i	−0.283152	−	0.283152i	−0.834365	+	0.551213i	$0.814165\pi$
$410$		0			0
$411$		−	0.982711i		−	0.0484736i
$412$		0			0
$413$	12.9415	+	12.9415i	0.636811	+	0.636811i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			10.0905i			0.494135i
$418$		0			0
$419$	−11.5150			−0.562547			−0.281273	−	0.959628i	$-0.590757\pi$
$419$	−11.5150			−0.562547			−0.281273	+	0.959628i	$0.590757\pi$
$420$		0			0
$421$	24.7177	+	24.7177i	1.20467	+	1.20467i	0.972731	+	0.231935i	$0.0745057\pi$
$421$	24.7177	+	24.7177i	1.20467	+	1.20467i	0.231935	+	0.972731i	$0.425494\pi$
$422$		0			0
$423$	−0.401904			−0.0195412
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	10.3082i		−	0.498849i
$428$		0			0
$429$		−	11.4768i		−	0.554106i
$430$		0			0
$431$	20.6075			0.992629			0.496314	−	0.868143i	$-0.334686\pi$
$431$	20.6075			0.992629			0.496314	+	0.868143i	$0.334686\pi$
$432$		0			0
$433$			15.8626i			0.762309i	0.924511	+	0.381155i	$0.124473\pi$
$433$			15.8626i			0.762309i	−0.924511	+	0.381155i	$0.875527\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	25.2689i		−	1.20878i
$438$		0			0
$439$	4.13619			0.197409			0.0987047	−	0.995117i	$-0.468530\pi$
$439$	4.13619			0.197409			0.0987047	+	0.995117i	$0.468530\pi$
$440$		0			0
$441$			0.269440i			0.0128305i
$442$		0			0
$443$			19.9797i			0.949266i	0.880184	+	0.474633i	$0.157419\pi$
$443$			19.9797i			0.949266i	−0.880184	+	0.474633i	$0.842581\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	30.7321			1.45358
$448$		0			0
$449$	13.4467	+	13.4467i	0.634591	+	0.634591i	0.949216	−	0.314625i	$-0.101879\pi$
$449$	13.4467	+	13.4467i	0.634591	+	0.634591i	−0.314625	+	0.949216i	$0.601879\pi$
$450$		0			0
$451$	−2.83143			−0.133327
$452$		0			0
$453$		−	28.6058i		−	1.34402i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.5618	+	14.5618i	0.681173	+	0.681173i	0.960265	−	0.279091i	$-0.0900332\pi$
$457$	14.5618	+	14.5618i	0.681173	+	0.681173i	−0.279091	+	0.960265i	$0.590033\pi$
$458$		0			0
$459$			20.7069i			0.966514i
$460$		0			0
$461$	−20.1277	−	20.1277i	−0.937441	−	0.937441i	0.0607141	−	0.998155i	$-0.480662\pi$
$461$	−20.1277	−	20.1277i	−0.937441	−	0.937441i	−0.998155	+	0.0607141i	$0.980662\pi$
$462$		0			0
$463$	19.1317	−	19.1317i	0.889125	−	0.889125i	−0.105314	−	0.994439i	$-0.533585\pi$
$463$	19.1317	−	19.1317i	0.889125	−	0.889125i	0.994439	+	0.105314i	$0.0335849\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			23.2734i			1.07696i	0.842637	+	0.538481i	$0.181002\pi$
$467$			23.2734i			1.07696i	−0.842637	+	0.538481i	$0.818998\pi$
$468$		0			0
$469$	−22.3607			−1.03252
$470$		0			0
$471$	−35.6819			−1.64413
$472$		0			0
$473$	2.28689	−	2.28689i	0.105151	−	0.105151i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.774913	−	0.774913i	−0.0354808	−	0.0354808i
$478$		0			0
$479$	−16.0347	+	16.0347i	−0.732643	+	0.732643i	−0.971143	−	0.238500i	$-0.923344\pi$
$479$	−16.0347	+	16.0347i	−0.732643	+	0.732643i	0.238500	+	0.971143i	$0.423344\pi$
$480$		0			0
$481$	−5.12039	+	5.12039i	−0.233470	+	0.233470i
$482$		0			0
$483$	−20.7972			−0.946304
$484$		0			0
$485$		0			0
$486$		0			0
$487$	27.3306	−	27.3306i	1.23847	−	1.23847i	0.277844	−	0.960626i	$-0.410380\pi$
$487$	27.3306	−	27.3306i	1.23847	−	1.23847i	0.960626	−	0.277844i	$-0.0896197\pi$
$488$		0			0
$489$		−	1.51896i		−	0.0686896i
$490$		0			0
$491$	−3.65833	−	3.65833i	−0.165098	−	0.165098i	0.619723	−	0.784821i	$-0.287245\pi$
$491$	−3.65833	−	3.65833i	−0.165098	−	0.165098i	−0.784821	+	0.619723i	$0.787245\pi$
$492$		0			0
$493$	−20.3452	+	0.540894i	−0.916301	+	0.0243606i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	20.0591	+	20.0591i	0.899772	+	0.899772i
$498$		0			0
$499$	22.1752			0.992699			0.496350	−	0.868123i	$-0.334673\pi$
$499$	22.1752			0.992699			0.496350	+	0.868123i	$0.334673\pi$
$500$		0			0
$501$	11.4239	+	11.4239i	0.510384	+	0.510384i
$502$		0			0
$503$		−	2.57000i		−	0.114591i	−0.998357	−	0.0572954i	$-0.981752\pi$
$503$		−	2.57000i		−	0.114591i	0.998357	−	0.0572954i	$-0.0182477\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			10.5543i			0.468734i
$508$		0			0
$509$			43.1784i			1.91385i	0.290334	+	0.956925i	$0.406234\pi$
$509$			43.1784i			1.91385i	−0.290334	+	0.956925i	$0.593766\pi$
$510$		0			0
$511$	24.3798	+	24.3798i	1.07850	+	1.07850i
$512$		0			0
$513$	21.0642	+	21.0642i	0.930006	+	0.930006i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.01732	−	2.01732i	0.0887216	−	0.0887216i
$518$		0			0
$519$	−24.0806	+	24.0806i	−1.05702	+	1.05702i
$520$		0			0
$521$			19.8679i			0.870427i	0.900327	+	0.435214i	$0.143327\pi$
$521$			19.8679i			0.870427i	−0.900327	+	0.435214i	$0.856673\pi$
$522$		0			0
$523$	10.1370	+	10.1370i	0.443258	+	0.443258i	0.893105	−	0.449847i	$-0.148522\pi$
$523$	10.1370	+	10.1370i	0.443258	+	0.443258i	−0.449847	+	0.893105i	$0.648522\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−23.7460	−	23.7460i	−1.03439	−	1.03439i
$528$		0			0
$529$		−	1.40012i		−	0.0608749i
$530$		0			0
$531$			2.60130i			0.112887i
$532$		0			0
$533$	−2.57361			−0.111476
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−19.3307			−0.834183
$538$		0			0
$539$	−1.35243	−	1.35243i	−0.0582532	−	0.0582532i
$540$		0			0
$541$	27.2063	−	27.2063i	1.16969	−	1.16969i	0.187407	−	0.982282i	$-0.439992\pi$
$541$	27.2063	−	27.2063i	1.16969	−	1.16969i	0.982282	−	0.187407i	$-0.0600082\pi$
$542$		0			0
$543$	−42.5900			−1.82771
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.0148	−	13.0148i	0.556475	−	0.556475i	−0.371827	−	0.928302i	$-0.621269\pi$
$547$	13.0148	−	13.0148i	0.556475	−	0.556475i	0.928302	+	0.371827i	$0.121269\pi$
$548$		0			0
$549$	1.03600	−	1.03600i	0.0442152	−	0.0442152i
$550$		0			0
$551$	−20.1460	+	21.2465i	−0.858249	+	0.905130i
$552$		0			0
$553$	−32.9807			−1.40248
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.92063	−	2.92063i	−0.123751	−	0.123751i	0.642519	−	0.766270i	$-0.277889\pi$
$557$	−2.92063	−	2.92063i	−0.123751	−	0.123751i	−0.766270	+	0.642519i	$0.777889\pi$
$558$		0			0
$559$	2.07866	−	2.07866i	0.0879178	−	0.0879178i
$560$		0			0
$561$	−12.0653	−	12.0653i	−0.509396	−	0.509396i
$562$		0			0
$563$		−	5.85586i		−	0.246795i	−0.992357	−	0.123397i	$-0.960621\pi$
$563$		−	5.85586i		−	0.246795i	0.992357	−	0.123397i	$-0.0393790\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.0198	−	15.0198i	0.630772	−	0.630772i
$568$		0			0
$569$	−14.9312	+	14.9312i	−0.625947	+	0.625947i	−0.947046	−	0.321099i	$-0.895948\pi$
$569$	−14.9312	+	14.9312i	−0.625947	+	0.625947i	0.321099	+	0.947046i	$0.395948\pi$
$570$		0			0
$571$	38.1793			1.59775			0.798876	−	0.601495i	$-0.205428\pi$
$571$	38.1793			1.59775			0.798876	+	0.601495i	$0.205428\pi$
$572$		0			0
$573$	−19.5959	+	19.5959i	−0.818630	+	0.818630i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−14.5914			−0.607449			−0.303724	−	0.952760i	$-0.598230\pi$
$577$	−14.5914			−0.607449			−0.303724	+	0.952760i	$0.598230\pi$
$578$		0			0
$579$	−22.6499			−0.941298
$580$		0			0
$581$	8.26462			0.342874
$582$		0			0
$583$	7.77921			0.322182
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0313	−	12.0313i	0.496587	−	0.496587i	−0.413787	−	0.910374i	$-0.635794\pi$
$587$	12.0313	−	12.0313i	0.496587	−	0.496587i	0.910374	+	0.413787i	$0.135794\pi$
$588$		0			0
$589$	−48.3114			−1.99064
$590$		0			0
$591$	9.73705	−	9.73705i	0.400529	−	0.400529i
$592$		0			0
$593$	−11.6124	+	11.6124i	−0.476865	+	0.476865i	−0.904128	−	0.427263i	$-0.859478\pi$
$593$	−11.6124	+	11.6124i	−0.476865	+	0.476865i	0.427263	+	0.904128i	$0.359478\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			14.8988i			0.609767i
$598$		0			0
$599$	−26.8052	−	26.8052i	−1.09523	−	1.09523i	−0.994960	−	0.100271i	$-0.968029\pi$
$599$	−26.8052	−	26.8052i	−1.09523	−	1.09523i	−0.100271	−	0.994960i	$-0.531971\pi$
$600$		0			0
$601$	−8.61367	+	8.61367i	−0.351359	+	0.351359i	−0.860615	−	0.509256i	$-0.829921\pi$
$601$	−8.61367	+	8.61367i	−0.351359	+	0.351359i	0.509256	+	0.860615i	$0.329921\pi$
$602$		0			0
$603$	−2.24730	−	2.24730i	−0.0915172	−	0.0915172i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	18.3784			0.745956			0.372978	−	0.927840i	$-0.378337\pi$
$607$	18.3784			0.745956			0.372978	+	0.927840i	$0.378337\pi$
$608$		0			0
$609$	17.4866	+	16.5809i	0.708592	+	0.671890i
$610$		0			0
$611$	1.83363	−	1.83363i	0.0741807	−	0.0741807i
$612$		0			0
$613$	−17.9986	+	17.9986i	−0.726959	+	0.726959i	−0.970013	−	0.243054i	$-0.921851\pi$
$613$	−17.9986	+	17.9986i	−0.726959	+	0.726959i	0.243054	+	0.970013i	$0.421851\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	28.4106			1.14377			0.571883	−	0.820335i	$-0.306213\pi$
$617$	28.4106			1.14377			0.571883	+	0.820335i	$0.306213\pi$
$618$		0			0
$619$	−9.49374	+	9.49374i	−0.381586	+	0.381586i	−0.871673	−	0.490088i	$-0.836965\pi$
$619$	−9.49374	+	9.49374i	−0.381586	+	0.381586i	0.490088	+	0.871673i	$0.336965\pi$
$620$		0			0
$621$	−18.0057	−	18.0057i	−0.722542	−	0.722542i
$622$		0			0
$623$	−10.6721			−0.427570
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−24.5469			−0.980310
$628$		0			0
$629$			10.7659i			0.429264i
$630$		0			0
$631$		−	38.3228i		−	1.52561i	−0.646630	−	0.762804i	$-0.723822\pi$
$631$		−	38.3228i		−	1.52561i	0.646630	−	0.762804i	$-0.276178\pi$
$632$		0			0
$633$	−3.06094	−	3.06094i	−0.121661	−	0.121661i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.22928	−	1.22928i	−0.0487059	−	0.0487059i
$638$		0			0
$639$			4.03196i			0.159502i
$640$		0			0
$641$	−14.9702	+	14.9702i	−0.591286	+	0.591286i	−0.937979	−	0.346693i	$-0.887305\pi$
$641$	−14.9702	+	14.9702i	−0.591286	+	0.591286i	0.346693	+	0.937979i	$0.387305\pi$
$642$		0			0
$643$	1.96517	−	1.96517i	0.0774987	−	0.0774987i	−0.667295	−	0.744794i	$-0.732548\pi$
$643$	1.96517	−	1.96517i	0.0774987	−	0.0774987i	0.744794	+	0.667295i	$0.232548\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.5149	−	22.5149i	−0.885152	−	0.885152i	0.108901	−	0.994053i	$-0.465267\pi$
$647$	−22.5149	−	22.5149i	−0.885152	−	0.885152i	−0.994053	+	0.108901i	$0.965267\pi$
$648$		0			0
$649$	−13.0570	−	13.0570i	−0.512532	−	0.512532i
$650$		0			0
$651$			39.7619i			1.55839i
$652$		0			0
$653$		−	26.8513i		−	1.05077i	−0.850864	−	0.525386i	$-0.823921\pi$
$653$		−	26.8513i		−	1.05077i	0.850864	−	0.525386i	$-0.176079\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			4.90044i			0.191184i
$658$		0			0
$659$	−21.5709	−	21.5709i	−0.840282	−	0.840282i	0.148613	−	0.988895i	$-0.452519\pi$
$659$	−21.5709	−	21.5709i	−0.840282	−	0.840282i	−0.988895	+	0.148613i	$0.952519\pi$
$660$		0			0
$661$	44.9398			1.74795			0.873977	−	0.485967i	$-0.161532\pi$
$661$	44.9398			1.74795			0.873977	+	0.485967i	$0.161532\pi$
$662$		0			0
$663$	−10.9667	−	10.9667i	−0.425910	−	0.425910i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.2208	−	18.1615i	0.666793	−	0.703216i
$668$		0			0
$669$	−22.7796	−	22.7796i	−0.880712	−	0.880712i
$670$		0			0
$671$			10.4002i			0.401494i
$672$		0			0
$673$	−24.6538	+	24.6538i	−0.950335	+	0.950335i	−0.998824	−	0.0484883i	$-0.984560\pi$
$673$	−24.6538	+	24.6538i	−0.950335	+	0.950335i	0.0484883	+	0.998824i	$0.484560\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.78008			0.106847			0.0534235	−	0.998572i	$-0.482987\pi$
$677$	2.78008			0.106847			0.0534235	+	0.998572i	$0.482987\pi$
$678$		0			0
$679$	−3.88790	+	3.88790i	−0.149204	+	0.149204i
$680$		0			0
$681$	−30.8866	+	30.8866i	−1.18358	+	1.18358i
$682$		0			0
$683$	24.9472	+	24.9472i	0.954577	+	0.954577i	0.999012	−	0.0444355i	$-0.0141489\pi$
$683$	24.9472	+	24.9472i	0.954577	+	0.954577i	−0.0444355	+	0.999012i	$0.514149\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6.54960	−	6.54960i	0.249883	−	0.249883i
$688$		0			0
$689$	7.07087			0.269379
$690$		0			0
$691$	17.0310			0.647891			0.323945	−	0.946076i	$-0.394991\pi$
$691$	17.0310			0.647891			0.323945	+	0.946076i	$0.394991\pi$
$692$		0			0
$693$		−	3.05435i		−	0.116025i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.70557	+	2.70557i	−0.102481	+	0.102481i
$698$		0			0
$699$	2.59050	+	2.59050i	0.0979819	+	0.0979819i
$700$		0			0
$701$			28.6390i			1.08168i	0.841125	+	0.540841i	$0.181894\pi$
$701$			28.6390i			1.08168i	−0.841125	+	0.540841i	$0.818106\pi$
$702$		0			0
$703$	10.9516	+	10.9516i	0.413049	+	0.413049i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			49.0352i			1.84416i
$708$		0			0
$709$	−0.203100			−0.00762757			−0.00381378	−	0.999993i	$-0.501214\pi$
$709$	−0.203100			−0.00762757			−0.00381378	+	0.999993i	$0.501214\pi$
$710$		0			0
$711$	−3.31463	−	3.31463i	−0.124308	−	0.124308i
$712$		0			0
$713$	41.2966			1.54657
$714$		0			0
$715$		0			0
$716$		0			0
$717$			18.0950i			0.675769i
$718$		0			0
$719$		−	34.6022i		−	1.29044i	−0.763995	−	0.645222i	$-0.776765\pi$
$719$		−	34.6022i		−	1.29044i	0.763995	−	0.645222i	$-0.223235\pi$
$720$		0			0
$721$	12.5331			0.466757
$722$		0			0
$723$			31.7685i			1.18148i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	16.1265i		−	0.598100i	−0.954237	−	0.299050i	$-0.903330\pi$
$727$		−	16.1265i		−	0.598100i	0.954237	−	0.299050i	$-0.0966698\pi$
$728$		0			0
$729$	29.5534			1.09457
$730$		0			0
$731$		−	4.37048i		−	0.161648i
$732$		0			0
$733$			26.3707i			0.974025i	0.873395	+	0.487012i	$0.161913\pi$
$733$			26.3707i			0.974025i	−0.873395	+	0.487012i	$0.838087\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	22.5602			0.831017
$738$		0			0
$739$	18.9424	+	18.9424i	0.696806	+	0.696806i	0.963720	−	0.266914i	$-0.0860041\pi$
$739$	18.9424	+	18.9424i	0.696806	+	0.696806i	−0.266914	+	0.963720i	$0.586004\pi$
$740$		0			0
$741$	−22.3118			−0.819644
$742$		0			0
$743$		−	3.57379i		−	0.131110i	−0.997849	−	0.0655548i	$-0.979118\pi$
$743$		−	3.57379i		−	0.131110i	0.997849	−	0.0655548i	$-0.0208817\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.830612	+	0.830612i	0.0303905	+	0.0303905i
$748$		0			0
$749$			28.9059i			1.05620i
$750$		0			0
$751$	−11.8193	−	11.8193i	−0.431294	−	0.431294i	0.457775	−	0.889068i	$-0.348647\pi$
$751$	−11.8193	−	11.8193i	−0.431294	−	0.431294i	−0.889068	+	0.457775i	$0.848647\pi$
$752$		0			0
$753$	26.5168	−	26.5168i	0.966328	−	0.966328i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			28.9459i			1.05206i	0.850467	+	0.526029i	$0.176320\pi$
$757$			28.9459i			1.05206i	−0.850467	+	0.526029i	$0.823680\pi$
$758$		0			0
$759$	20.9827			0.761625
$760$		0			0
$761$	−4.54292			−0.164681			−0.0823404	−	0.996604i	$-0.526239\pi$
$761$	−4.54292			−0.164681			−0.0823404	+	0.996604i	$0.526239\pi$
$762$		0			0
$763$	−7.10859	+	7.10859i	−0.257348	+	0.257348i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.8681	−	11.8681i	−0.428532	−	0.428532i
$768$		0			0
$769$	4.63210	−	4.63210i	0.167038	−	0.167038i	−0.618638	−	0.785676i	$-0.712315\pi$
$769$	4.63210	−	4.63210i	0.167038	−	0.167038i	0.785676	+	0.618638i	$0.212315\pi$
$770$		0			0
$771$	−22.3057	+	22.3057i	−0.803322	+	0.803322i
$772$		0			0
$773$	−23.2773			−0.837226			−0.418613	−	0.908165i	$-0.637484\pi$
$773$	−23.2773			−0.837226			−0.418613	+	0.908165i	$0.637484\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.01357	−	9.01357i	0.323360	−	0.323360i
$778$		0			0
$779$			5.50452i			0.197220i
$780$		0			0
$781$	−20.2380	−	20.2380i	−0.724174	−	0.724174i
$782$		0			0
$783$	0.784142	+	29.4947i	0.0280229	+	1.05405i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	27.3371	+	27.3371i	0.974463	+	0.974463i	0.999682	−	0.0252189i	$-0.00802827\pi$
$787$	27.3371	+	27.3371i	0.974463	+	0.974463i	−0.0252189	+	0.999682i	$0.508028\pi$
$788$		0			0
$789$	−8.24218			−0.293429
$790$		0			0
$791$	23.5729	+	23.5729i	0.838154	+	0.838154i
$792$		0			0
$793$			9.45317i			0.335692i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			37.4688i			1.32721i	0.748082	+	0.663606i	$0.230975\pi$
$797$			37.4688i			1.32721i	−0.748082	+	0.663606i	$0.769025\pi$
$798$		0			0
$799$		−	3.85530i		−	0.136391i
$800$		0			0
$801$	−1.07257	−	1.07257i	−0.0378975	−	0.0378975i
$802$		0			0
$803$	−24.5973	−	24.5973i	−0.868020	−	0.868020i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.312756	+	0.312756i	−0.0110095	+	0.0110095i
$808$		0			0
$809$	−2.68633	+	2.68633i	−0.0944463	+	0.0944463i	−0.752751	−	0.658305i	$-0.771274\pi$
$809$	−2.68633	+	2.68633i	−0.0944463	+	0.0944463i	0.658305	+	0.752751i	$0.271274\pi$
$810$		0			0
$811$			7.94175i			0.278872i	0.990231	+	0.139436i	$0.0445290\pi$
$811$			7.94175i			0.278872i	−0.990231	+	0.139436i	$0.955471\pi$
$812$		0			0
$813$	22.4105	+	22.4105i	0.785969	+	0.785969i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.44589	−	4.44589i	−0.155542	−	0.155542i
$818$		0			0
$819$		−	2.77623i		−	0.0970095i
$820$		0			0
$821$			39.1659i			1.36690i	0.729998	+	0.683449i	$0.239521\pi$
$821$			39.1659i			1.36690i	−0.729998	+	0.683449i	$0.760479\pi$
$822$		0			0
$823$	−17.0068			−0.592821			−0.296411	−	0.955061i	$-0.595790\pi$
$823$	−17.0068			−0.592821			−0.296411	+	0.955061i	$0.595790\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.6233			−0.786691			−0.393345	−	0.919391i	$-0.628682\pi$
$827$	−22.6233			−0.786691			−0.393345	+	0.919391i	$0.628682\pi$
$828$		0			0
$829$	−34.8056	−	34.8056i	−1.20885	−	1.20885i	−0.971400	−	0.237450i	$-0.923688\pi$
$829$	−34.8056	−	34.8056i	−1.20885	−	1.20885i	−0.237450	−	0.971400i	$-0.576312\pi$
$830$		0			0
$831$	34.0717	−	34.0717i	1.18194	−	1.18194i
$832$		0			0
$833$	−2.58463			−0.0895520
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−34.4249	+	34.4249i	−1.18990	+	1.18990i
$838$		0			0
$839$	24.3908	−	24.3908i	0.842064	−	0.842064i	−0.147063	−	0.989127i	$-0.546982\pi$
$839$	24.3908	−	24.3908i	0.842064	−	0.842064i	0.989127	+	0.147063i	$0.0469820\pi$
$840$		0			0
$841$	−28.9590	+	1.54089i	−0.998587	+	0.0531341i
$842$		0			0
$843$	−10.7501			−0.370253
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−6.22994	−	6.22994i	−0.214063	−	0.214063i
$848$		0			0
$849$	−10.3540	+	10.3540i	−0.355350	+	0.355350i
$850$		0			0
$851$	−9.36147	−	9.36147i	−0.320907	−	0.320907i
$852$		0			0
$853$			36.0783i			1.23530i	0.786455	+	0.617648i	$0.211914\pi$
$853$			36.0783i			1.23530i	−0.786455	+	0.617648i	$0.788086\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.5320	+	26.5320i	−0.906316	+	0.906316i	−0.995973	−	0.0896568i	$-0.971423\pi$
$857$	−26.5320	+	26.5320i	−0.906316	+	0.906316i	0.0896568	+	0.995973i	$0.471423\pi$
$858$		0			0
$859$	9.99547	−	9.99547i	0.341041	−	0.341041i	−0.515718	−	0.856759i	$-0.672475\pi$
$859$	9.99547	−	9.99547i	0.341041	−	0.341041i	0.856759	+	0.515718i	$0.172475\pi$
$860$		0			0
$861$	4.53041			0.154396
$862$		0			0
$863$	4.24981	−	4.24981i	0.144665	−	0.144665i	−0.631065	−	0.775730i	$-0.717382\pi$
$863$	4.24981	−	4.24981i	0.144665	−	0.144665i	0.775730	+	0.631065i	$0.217382\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4.38542			0.148937
$868$		0			0
$869$	33.2749			1.12877
$870$		0			0
$871$	20.5060			0.694819
$872$		0			0
$873$	−0.781483			−0.0264492
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.45331	−	6.45331i	0.217913	−	0.217913i	−0.589706	−	0.807618i	$-0.700756\pi$
$877$	6.45331	−	6.45331i	0.217913	−	0.217913i	0.807618	+	0.589706i	$0.200756\pi$
$878$		0			0
$879$	47.7906			1.61194
$880$		0			0
$881$	21.5282	−	21.5282i	0.725303	−	0.725303i	−0.244377	−	0.969680i	$-0.578584\pi$
$881$	21.5282	−	21.5282i	0.725303	−	0.725303i	0.969680	+	0.244377i	$0.0785836\pi$
$882$		0			0
$883$	20.4272	−	20.4272i	0.687429	−	0.687429i	−0.274234	−	0.961663i	$-0.588424\pi$
$883$	20.4272	−	20.4272i	0.687429	−	0.687429i	0.961663	+	0.274234i	$0.0884242\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			39.3078i			1.31983i	0.751342	+	0.659913i	$0.229407\pi$
$887$			39.3078i			1.31983i	−0.751342	+	0.659913i	$0.770593\pi$
$888$		0			0
$889$	42.6262	+	42.6262i	1.42964	+	1.42964i
$890$		0			0
$891$	−15.1538	+	15.1538i	−0.507671	+	0.507671i
$892$		0			0
$893$	−3.92182	−	3.92182i	−0.131239	−	0.131239i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	19.0721			0.636800
$898$		0			0
$899$	−34.7228	−	32.9243i	−1.15807	−	1.09809i
$900$		0			0
$901$	7.43342	−	7.43342i	0.247643	−	0.247643i
$902$		0			0
$903$	−3.65912	+	3.65912i	−0.121768	+	0.121768i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	57.9918			1.92559			0.962794	−	0.270237i	$-0.0871021\pi$
$907$	57.9918			1.92559			0.962794	+	0.270237i	$0.0871021\pi$
$908$		0			0
$909$	−4.92815	+	4.92815i	−0.163456	+	0.163456i
$910$		0			0
$911$	−18.3105	−	18.3105i	−0.606653	−	0.606653i	0.335417	−	0.942070i	$-0.391123\pi$
$911$	−18.3105	−	18.3105i	−0.606653	−	0.606653i	−0.942070	+	0.335417i	$0.891123\pi$
$912$		0			0
$913$	−8.33836			−0.275959
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−37.7775			−1.24752
$918$		0			0
$919$		−	24.1742i		−	0.797432i	−0.917075	−	0.398716i	$-0.869456\pi$
$919$		−	24.1742i		−	0.797432i	0.917075	−	0.398716i	$-0.130544\pi$
$920$		0			0
$921$		−	36.8151i		−	1.21310i
$922$		0			0
$923$	−18.3952	−	18.3952i	−0.605487	−	0.605487i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.25960	+	1.25960i	0.0413708	+	0.0413708i
$928$		0			0
$929$			33.7664i			1.10784i	0.832570	+	0.553920i	$0.186869\pi$
$929$			33.7664i			1.10784i	−0.832570	+	0.553920i	$0.813131\pi$
$930$		0			0
$931$	−2.62922	+	2.62922i	−0.0861693	+	0.0861693i
$932$		0			0
$933$	5.14083	−	5.14083i	0.168303	−	0.168303i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.0508	−	20.0508i	−0.655031	−	0.655031i	0.299169	−	0.954200i	$-0.403291\pi$
$937$	−20.0508	−	20.0508i	−0.655031	−	0.655031i	−0.954200	+	0.299169i	$0.903291\pi$
$938$		0			0
$939$	−6.08250	−	6.08250i	−0.198495	−	0.198495i
$940$		0			0
$941$			36.0818i			1.17623i	0.808776	+	0.588117i	$0.200131\pi$
$941$			36.0818i			1.17623i	−0.808776	+	0.588117i	$0.799869\pi$
$942$		0			0
$943$		−	4.70526i		−	0.153224i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			4.44219i			0.144352i	0.997392	+	0.0721759i	$0.0229943\pi$
$947$			4.44219i			0.144352i	−0.997392	+	0.0721759i	$0.977006\pi$
$948$		0			0
$949$	−22.3576	−	22.3576i	−0.725758	−	0.725758i
$950$		0			0
$951$	−19.4889			−0.631971
$952$		0			0
$953$	26.7670	+	26.7670i	0.867067	+	0.867067i	0.992147	−	0.125080i	$-0.0399187\pi$
$953$	26.7670	+	26.7670i	0.867067	+	0.867067i	−0.125080	+	0.992147i	$0.539919\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−17.6426	−	16.7288i	−0.570304	−	0.540765i
$958$		0			0
$959$	1.19320	+	1.19320i	0.0385304	+	0.0385304i
$960$		0			0
$961$		−	47.9546i		−	1.54692i
$962$		0			0
$963$	−2.90510	+	2.90510i	−0.0936156	+	0.0936156i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	23.1984			0.746010			0.373005	−	0.927829i	$-0.378327\pi$
$967$	23.1984			0.746010			0.373005	+	0.927829i	$0.378327\pi$
$968$		0			0
$969$	−23.4558	+	23.4558i	−0.753509	+	0.753509i
$970$		0			0
$971$	−7.51735	+	7.51735i	−0.241243	+	0.241243i	−0.817364	−	0.576121i	$-0.804566\pi$
$971$	−7.51735	+	7.51735i	−0.241243	+	0.241243i	0.576121	+	0.817364i	$0.304566\pi$
$972$		0			0
$973$	−12.2518	−	12.2518i	−0.392775	−	0.392775i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.14969	+	4.14969i	−0.132760	+	0.132760i	−0.770364	−	0.637604i	$-0.779926\pi$
$977$	−4.14969	+	4.14969i	−0.132760	+	0.132760i	0.637604	+	0.770364i	$0.279926\pi$
$978$		0			0
$979$	10.7673			0.344126
$980$		0			0
$981$	−1.42886			−0.0456199
$982$		0			0
$983$		−	15.4286i		−	0.492095i	−0.969258	−	0.246047i	$-0.920868\pi$
$983$		−	15.4286i		−	0.492095i	0.969258	−	0.246047i	$-0.0791319\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.22779	+	3.22779i	−0.102742	+	0.102742i
$988$		0			0
$989$	3.80035	+	3.80035i	0.120844	+	0.120844i
$990$		0			0
$991$			22.2159i			0.705711i	0.935678	+	0.352856i	$0.114789\pi$
$991$			22.2159i			0.705711i	−0.935678	+	0.352856i	$0.885211\pi$
$992$		0			0
$993$	6.67241	+	6.67241i	0.211743	+	0.211743i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	32.3920i		−	1.02586i	−0.858429	−	0.512932i	$-0.828559\pi$
$997$		−	32.3920i		−	1.02586i	0.858429	−	0.512932i	$-0.171441\pi$
$998$		0			0
$999$	15.6074			0.493798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.s.d.1293.12		30
5.2	odd	4		2900.2.j.d.1757.4		30
5.3	odd	4		580.2.j.a.17.12	✓	30
5.4	even	2		580.2.s.a.133.4	yes	30
29.12	odd	4		2900.2.j.d.2593.12		30
145.12	even	4	inner	2900.2.s.d.157.12		30
145.99	odd	4		580.2.j.a.273.4	yes	30
145.128	even	4		580.2.s.a.157.4	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.j.a.17.12	✓	30	5.3	odd	4
580.2.j.a.273.4	yes	30	145.99	odd	4
580.2.s.a.133.4	yes	30	5.4	even	2
580.2.s.a.157.4	yes	30	145.128	even	4
2900.2.j.d.1757.4		30	5.2	odd	4
2900.2.j.d.2593.12		30	29.12	odd	4
2900.2.s.d.157.12		30	145.12	even	4	inner
2900.2.s.d.1293.12		30	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)

\(f(q)\)	\(=\)	\(q+1.61432 q^{3} +(-1.96009 + 1.96009i) q^{7} -0.393986 q^{9} +(1.97757 - 1.97757i) q^{11} +(1.79751 - 1.79751i) q^{13} -3.77934i q^{17} +(-3.84455 - 3.84455i) q^{19} +(-3.16420 + 3.16420i) q^{21} +(3.28633 + 3.28633i) q^{23} -5.47896 q^{27} +(-0.143119 - 5.38326i) q^{29} +(6.28310 - 6.28310i) q^{31} +(3.19243 - 3.19243i) q^{33} -2.84861 q^{37} +(2.90174 - 2.90174i) q^{39} +(-0.715885 - 0.715885i) q^{41} +1.15641 q^{43} +1.02010 q^{47} -0.683883i q^{49} -6.10105i q^{51} +(1.96686 + 1.96686i) q^{53} +(-6.20632 - 6.20632i) q^{57} -6.60253i q^{59} +(-2.62953 + 2.62953i) q^{61} +(0.772246 - 0.772246i) q^{63} +(5.70401 + 5.70401i) q^{67} +(5.30517 + 5.30517i) q^{69} -10.2338i q^{71} -12.4381i q^{73} +7.75244i q^{77} +(8.41306 + 8.41306i) q^{79} -7.66282 q^{81} +(-2.10823 - 2.10823i) q^{83} +(-0.231039 - 8.69028i) q^{87} +(2.72236 + 2.72236i) q^{89} +7.04653i q^{91} +(10.1429 - 10.1429i) q^{93} +1.98353 q^{97} +(-0.779136 + 0.779136i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 38 q^{9} - 4 q^{11} + 6 q^{13} - 4 q^{21} - 12 q^{27} - 4 q^{31} - 4 q^{33} + 24 q^{37} - 12 q^{39} + 10 q^{41} + 16 q^{43} + 32 q^{47} + 18 q^{53} - 24 q^{57} - 22 q^{61} - 24 q^{63} - 16 q^{67}+ \cdots + 16 q^{97}+O(q^{100}) \) 30 * q + 38 * q^9 - 4 * q^11 + 6 * q^13 - 4 * q^21 - 12 * q^27 - 4 * q^31 - 4 * q^33 + 24 * q^37 - 12 * q^39 + 10 * q^41 + 16 * q^43 + 32 * q^47 + 18 * q^53 - 24 * q^57 - 22 * q^61 - 24 * q^63 - 16 * q^67 - 8 * q^69 - 20 * q^79 + 70 * q^81 - 56 * q^83 - 48 * q^87 - 6 * q^89 - 44 * q^93 + 16 * q^97

Introduction

L-functions

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