LMFDB - Embedded newform 4620.2.h.e.1849.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4620,2,Mod(1849,4620)] 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("4620.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,0,-4,0,0,0,-14,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 12 x^{12} - 32 x^{11} + 86 x^{10} - 220 x^{9} + 585 x^{8} - 1536 x^{7} + \cdots + 78125 $ x^14 - 4x^13 + 12x^12 - 32x^11 + 86x^10 - 220x^9 + 585x^8 - 1536x^7 + 2925x^6 - 5500x^5 + 10750x^4 - 20000x^3 + 37500x^2 - 62500*x + 78125
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.10
Root			$1.56500 - 1.59711i$ of defining polynomial
Character	$\chi$	$=$	4620.1849
Dual form			4620.2.h.e.1849.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.56500 - 1.59711i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +0.122890i q^{13} +(1.59711 - 1.56500i) q^{15} -0.802598i q^{17} +7.00100 q^{19} -1.00000 q^{21} -3.44318i q^{23} +(-0.101546 + 4.99897i) q^{25} -1.00000i q^{27} -4.73924 q^{29} +1.99706 q^{31} +1.00000i q^{33} +(1.59711 - 1.56500i) q^{35} -3.45598i q^{37} -0.122890 q^{39} -11.7902 q^{41} -6.54753i q^{43} +(1.56500 + 1.59711i) q^{45} +0.536484i q^{47} -1.00000 q^{49} +0.802598 q^{51} +10.7389i q^{53} +(-1.56500 - 1.59711i) q^{55} +7.00100i q^{57} -12.8744 q^{59} +5.10051 q^{61} -1.00000i q^{63} +(0.196269 - 0.192323i) q^{65} -2.25594i q^{67} +3.44318 q^{69} +8.41853 q^{71} -5.45423i q^{73} +(-4.99897 - 0.101546i) q^{75} +1.00000i q^{77} +6.49664 q^{79} +1.00000 q^{81} +5.67053i q^{83} +(-1.28184 + 1.25607i) q^{85} -4.73924i q^{87} +13.5387 q^{89} -0.122890 q^{91} +1.99706i q^{93} +(-10.9566 - 11.1814i) q^{95} -7.67347i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) $ 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.56500	−	1.59711i	−0.699890	−	0.714251i
$5$	−1.56500	−	1.59711i	−0.699890	−	0.714251i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			0.122890i			0.0340835i	0.999855	+	0.0170418i	$0.00542483\pi$
$13$			0.122890i			0.0340835i	−0.999855	+	0.0170418i	$0.994575\pi$
$14$		0			0
$15$	1.59711	−	1.56500i	0.412373	−	0.404081i
$16$		0			0
$17$		−	0.802598i		−	0.194658i	−0.995252	−	0.0973292i	$-0.968970\pi$
$17$		−	0.802598i		−	0.194658i	0.995252	−	0.0973292i	$-0.0310300\pi$
$18$		0			0
$19$	7.00100			1.60614			0.803070	−	0.595885i	$-0.203199\pi$
$19$	7.00100			1.60614			0.803070	+	0.595885i	$0.203199\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		−	3.44318i		−	0.717953i	−0.933347	−	0.358977i	$-0.883126\pi$
$23$		−	3.44318i		−	0.717953i	0.933347	−	0.358977i	$-0.116874\pi$
$24$		0			0
$25$	−0.101546	+	4.99897i	−0.0203093	+	0.999794i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.73924			−0.880056			−0.440028	−	0.897984i	$-0.645031\pi$
$29$	−4.73924			−0.880056			−0.440028	+	0.897984i	$0.645031\pi$
$30$		0			0
$31$	1.99706			0.358683			0.179342	−	0.983787i	$-0.442603\pi$
$31$	1.99706			0.358683			0.179342	+	0.983787i	$0.442603\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	1.59711	−	1.56500i	0.269962	−	0.264533i
$36$		0			0
$37$		−	3.45598i		−	0.568160i	−0.958801	−	0.284080i	$-0.908312\pi$
$37$		−	3.45598i		−	0.568160i	0.958801	−	0.284080i	$-0.0916882\pi$
$38$		0			0
$39$	−0.122890			−0.0196781
$40$		0			0
$41$	−11.7902			−1.84133			−0.920664	−	0.390356i	$-0.872352\pi$
$41$	−11.7902			−1.84133			−0.920664	+	0.390356i	$0.872352\pi$
$42$		0			0
$43$		−	6.54753i		−	0.998489i	−0.866461	−	0.499245i	$-0.833611\pi$
$43$		−	6.54753i		−	0.998489i	0.866461	−	0.499245i	$-0.166389\pi$
$44$		0			0
$45$	1.56500	+	1.59711i	0.233297	+	0.238084i
$46$		0			0
$47$			0.536484i			0.0782543i	0.999234	+	0.0391271i	$0.0124577\pi$
$47$			0.536484i			0.0782543i	−0.999234	+	0.0391271i	$0.987542\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.802598			0.112386
$52$		0			0
$53$			10.7389i			1.47511i	0.675289	+	0.737553i	$0.264019\pi$
$53$			10.7389i			1.47511i	−0.675289	+	0.737553i	$0.735981\pi$
$54$		0			0
$55$	−1.56500	−	1.59711i	−0.211025	−	0.215355i
$56$		0			0
$57$			7.00100i			0.927305i
$58$		0			0
$59$	−12.8744			−1.67610			−0.838051	−	0.545591i	$-0.816305\pi$
$59$	−12.8744			−1.67610			−0.838051	+	0.545591i	$0.816305\pi$
$60$		0			0
$61$	5.10051			0.653053			0.326526	−	0.945188i	$-0.394122\pi$
$61$	5.10051			0.653053			0.326526	+	0.945188i	$0.394122\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	0.196269	−	0.192323i	0.0243442	−	0.0238547i
$66$		0			0
$67$		−	2.25594i		−	0.275607i	−0.990460	−	0.137803i	$-0.955996\pi$
$67$		−	2.25594i		−	0.275607i	0.990460	−	0.137803i	$-0.0440042\pi$
$68$		0			0
$69$	3.44318			0.414510
$70$		0			0
$71$	8.41853			0.999096			0.499548	−	0.866286i	$-0.333499\pi$
$71$	8.41853			0.999096			0.499548	+	0.866286i	$0.333499\pi$
$72$		0			0
$73$		−	5.45423i		−	0.638369i	−0.947693	−	0.319185i	$-0.896591\pi$
$73$		−	5.45423i		−	0.638369i	0.947693	−	0.319185i	$-0.103409\pi$
$74$		0			0
$75$	−4.99897	−	0.101546i	−0.577231	−	0.0117256i
$76$		0			0
$77$			1.00000i			0.113961i
$78$		0			0
$79$	6.49664			0.730929			0.365465	−	0.930825i	$-0.380910\pi$
$79$	6.49664			0.730929			0.365465	+	0.930825i	$0.380910\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.67053i			0.622422i	0.950341	+	0.311211i	$0.100735\pi$
$83$			5.67053i			0.622422i	−0.950341	+	0.311211i	$0.899265\pi$
$84$		0			0
$85$	−1.28184	+	1.25607i	−0.139035	+	0.136239i
$86$		0			0
$87$		−	4.73924i		−	0.508100i
$88$		0			0
$89$	13.5387			1.43510			0.717548	−	0.696509i	$-0.245264\pi$
$89$	13.5387			1.43510			0.717548	+	0.696509i	$0.245264\pi$
$90$		0			0
$91$	−0.122890			−0.0128824
$92$		0			0
$93$			1.99706i			0.207086i
$94$		0			0
$95$	−10.9566	−	11.1814i	−1.12412	−	1.14719i
$96$		0			0
$97$		−	7.67347i		−	0.779123i	−0.921000	−	0.389561i	$-0.872627\pi$
$97$		−	7.67347i		−	0.779123i	0.921000	−	0.389561i	$-0.127373\pi$
$98$		0			0
$99$	−1.00000			−0.100504
$100$		0			0
$101$	8.09255			0.805239			0.402619	−	0.915367i	$-0.368100\pi$
$101$	8.09255			0.805239			0.402619	+	0.915367i	$0.368100\pi$
$102$		0			0
$103$		−	9.18812i		−	0.905332i	−0.891680	−	0.452666i	$-0.850473\pi$
$103$		−	9.18812i		−	0.905332i	0.891680	−	0.452666i	$-0.149527\pi$
$104$		0			0
$105$	1.56500	+	1.59711i	0.152728	+	0.155862i
$106$		0			0
$107$		−	14.0151i		−	1.35489i	−0.735573	−	0.677445i	$-0.763087\pi$
$107$		−	14.0151i		−	1.35489i	0.735573	−	0.677445i	$-0.236913\pi$
$108$		0			0
$109$	−0.988199			−0.0946523			−0.0473261	−	0.998879i	$-0.515070\pi$
$109$	−0.988199			−0.0946523			−0.0473261	+	0.998879i	$0.515070\pi$
$110$		0			0
$111$	3.45598			0.328027
$112$		0			0
$113$		−	16.1437i		−	1.51867i	−0.650701	−	0.759334i	$-0.725525\pi$
$113$		−	16.1437i		−	1.51867i	0.650701	−	0.759334i	$-0.274475\pi$
$114$		0			0
$115$	−5.49915	+	5.38858i	−0.512799	+	0.502488i
$116$		0			0
$117$		−	0.122890i		−	0.0113612i
$118$		0			0
$119$	0.802598			0.0735740
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	11.7902i		−	1.06309i
$124$		0			0
$125$	8.14284	−	7.66121i	0.728318	−	0.685239i
$126$		0			0
$127$			14.5957i			1.29516i	0.761998	+	0.647579i	$0.224218\pi$
$127$			14.5957i			1.29516i	−0.761998	+	0.647579i	$0.775782\pi$
$128$		0			0
$129$	6.54753			0.576478
$130$		0			0
$131$	−3.72330			−0.325307			−0.162653	−	0.986683i	$-0.552005\pi$
$131$	−3.72330			−0.325307			−0.162653	+	0.986683i	$0.552005\pi$
$132$		0			0
$133$			7.00100i			0.607064i
$134$		0			0
$135$	−1.59711	+	1.56500i	−0.137458	+	0.134694i
$136$		0			0
$137$		−	0.484151i		−	0.0413638i	−0.999786	−	0.0206819i	$-0.993416\pi$
$137$		−	0.484151i		−	0.0413638i	0.999786	−	0.0206819i	$-0.00658372\pi$
$138$		0			0
$139$	21.8921			1.85686			0.928431	−	0.371505i	$-0.121158\pi$
$139$	21.8921			1.85686			0.928431	+	0.371505i	$0.121158\pi$
$140$		0			0
$141$	−0.536484			−0.0451801
$142$		0			0
$143$			0.122890i			0.0102766i
$144$		0			0
$145$	7.41692	+	7.56911i	0.615942	+	0.628581i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	11.7443			0.962128			0.481064	−	0.876686i	$-0.340251\pi$
$149$	11.7443			0.962128			0.481064	+	0.876686i	$0.340251\pi$
$150$		0			0
$151$	−2.01138			−0.163684			−0.0818419	−	0.996645i	$-0.526080\pi$
$151$	−2.01138			−0.163684			−0.0818419	+	0.996645i	$0.526080\pi$
$152$		0			0
$153$			0.802598i			0.0648862i
$154$		0			0
$155$	−3.12541	−	3.18954i	−0.251039	−	0.256190i
$156$		0			0
$157$		−	23.1571i		−	1.84814i	−0.382225	−	0.924069i	$-0.624842\pi$
$157$		−	23.1571i		−	1.84814i	0.382225	−	0.924069i	$-0.375158\pi$
$158$		0			0
$159$	−10.7389			−0.851653
$160$		0			0
$161$	3.44318			0.271361
$162$		0			0
$163$		−	14.0902i		−	1.10363i	−0.833966	−	0.551816i	$-0.813935\pi$
$163$		−	14.0902i		−	1.10363i	0.833966	−	0.551816i	$-0.186065\pi$
$164$		0			0
$165$	1.59711	−	1.56500i	0.124335	−	0.121835i
$166$		0			0
$167$		−	2.58892i		−	0.200337i	−0.994971	−	0.100168i	$-0.968062\pi$
$167$		−	2.58892i		−	0.200337i	0.994971	−	0.100168i	$-0.0319381\pi$
$168$		0			0
$169$	12.9849			0.998838
$170$		0			0
$171$	−7.00100			−0.535380
$172$		0			0
$173$		−	14.3432i		−	1.09050i	−0.838275	−	0.545248i	$-0.816436\pi$
$173$		−	14.3432i		−	1.09050i	0.838275	−	0.545248i	$-0.183564\pi$
$174$		0			0
$175$	−4.99897	−	0.101546i	−0.377887	−	0.00767618i
$176$		0			0
$177$		−	12.8744i		−	0.967698i
$178$		0			0
$179$	20.2055			1.51023			0.755114	−	0.655593i	$-0.227581\pi$
$179$	20.2055			1.51023			0.755114	+	0.655593i	$0.227581\pi$
$180$		0			0
$181$	20.1269			1.49602			0.748012	−	0.663685i	$-0.231008\pi$
$181$	20.1269			1.49602			0.748012	+	0.663685i	$0.231008\pi$
$182$		0			0
$183$			5.10051i			0.377040i
$184$		0			0
$185$	−5.51960	+	5.40862i	−0.405809	+	0.397649i
$186$		0			0
$187$		−	0.802598i		−	0.0586917i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−15.3725			−1.11232			−0.556159	−	0.831076i	$-0.687725\pi$
$191$	−15.3725			−1.11232			−0.556159	+	0.831076i	$0.687725\pi$
$192$		0			0
$193$		−	2.39480i		−	0.172382i	−0.996279	−	0.0861909i	$-0.972530\pi$
$193$		−	2.39480i		−	0.172382i	0.996279	−	0.0861909i	$-0.0274695\pi$
$194$		0			0
$195$	0.192323	+	0.196269i	0.0137725	+	0.0140551i
$196$		0			0
$197$			15.6451i			1.11467i	0.830289	+	0.557334i	$0.188176\pi$
$197$			15.6451i			1.11467i	−0.830289	+	0.557334i	$0.811824\pi$
$198$		0			0
$199$	6.17533			0.437757			0.218879	−	0.975752i	$-0.429760\pi$
$199$	6.17533			0.437757			0.218879	+	0.975752i	$0.429760\pi$
$200$		0			0
$201$	2.25594			0.159122
$202$		0			0
$203$		−	4.73924i		−	0.332630i
$204$		0			0
$205$	18.4517	+	18.8304i	1.28873	+	1.31517i
$206$		0			0
$207$			3.44318i			0.239318i
$208$		0			0
$209$	7.00100			0.484269
$210$		0			0
$211$	9.01682			0.620743			0.310372	−	0.950615i	$-0.399546\pi$
$211$	9.01682			0.620743			0.310372	+	0.950615i	$0.399546\pi$
$212$		0			0
$213$			8.41853i			0.576828i
$214$		0			0
$215$	−10.4572	+	10.2469i	−0.713172	+	0.698832i
$216$		0			0
$217$			1.99706i			0.135570i
$218$		0			0
$219$	5.45423			0.368563
$220$		0			0
$221$	0.0986312			0.00663465
$222$		0			0
$223$		−	15.7055i		−	1.05172i	−0.850572	−	0.525859i	$-0.823744\pi$
$223$		−	15.7055i		−	1.05172i	0.850572	−	0.525859i	$-0.176256\pi$
$224$		0			0
$225$	0.101546	−	4.99897i	0.00676975	−	0.333265i
$226$		0			0
$227$		−	25.4782i		−	1.69105i	−0.533938	−	0.845524i	$-0.679288\pi$
$227$		−	25.4782i		−	1.69105i	0.533938	−	0.845524i	$-0.320712\pi$
$228$		0			0
$229$	−16.5750			−1.09531			−0.547655	−	0.836704i	$-0.684479\pi$
$229$	−16.5750			−1.09531			−0.547655	+	0.836704i	$0.684479\pi$
$230$		0			0
$231$	−1.00000			−0.0657952
$232$		0			0
$233$		−	26.3986i		−	1.72943i	−0.502260	−	0.864716i	$-0.667498\pi$
$233$		−	26.3986i		−	1.72943i	0.502260	−	0.864716i	$-0.332502\pi$
$234$		0			0
$235$	0.856827	−	0.839598i	0.0558932	−	0.0547693i
$236$		0			0
$237$			6.49664i			0.422002i
$238$		0			0
$239$	9.75552			0.631032			0.315516	−	0.948920i	$-0.397822\pi$
$239$	9.75552			0.631032			0.315516	+	0.948920i	$0.397822\pi$
$240$		0			0
$241$	−22.4317			−1.44495			−0.722477	−	0.691395i	$-0.756996\pi$
$241$	−22.4317			−1.44495			−0.722477	+	0.691395i	$0.756996\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.56500	+	1.59711i	0.0999842	+	0.102036i
$246$		0			0
$247$			0.860353i			0.0547429i
$248$		0			0
$249$	−5.67053			−0.359355
$250$		0			0
$251$	−23.1380			−1.46045			−0.730227	−	0.683204i	$-0.760586\pi$
$251$	−23.1380			−1.46045			−0.730227	+	0.683204i	$0.760586\pi$
$252$		0			0
$253$		−	3.44318i		−	0.216471i
$254$		0			0
$255$	−1.25607	−	1.28184i	−0.0786579	−	0.0802719i
$256$		0			0
$257$		−	18.4826i		−	1.15291i	−0.817128	−	0.576456i	$-0.804435\pi$
$257$		−	18.4826i		−	1.15291i	0.817128	−	0.576456i	$-0.195565\pi$
$258$		0			0
$259$	3.45598			0.214744
$260$		0			0
$261$	4.73924			0.293352
$262$		0			0
$263$		−	3.40223i		−	0.209790i	−0.994483	−	0.104895i	$-0.966549\pi$
$263$		−	3.40223i		−	0.209790i	0.994483	−	0.104895i	$-0.0334507\pi$
$264$		0			0
$265$	17.1513	−	16.8064i	1.05360	−	1.03241i
$266$		0			0
$267$			13.5387i			0.828553i
$268$		0			0
$269$	−21.7258			−1.32465			−0.662323	−	0.749218i	$-0.730429\pi$
$269$	−21.7258			−1.32465			−0.662323	+	0.749218i	$0.730429\pi$
$270$		0			0
$271$	−18.4200			−1.11893			−0.559466	−	0.828853i	$-0.688994\pi$
$271$	−18.4200			−1.11893			−0.559466	+	0.828853i	$0.688994\pi$
$272$		0			0
$273$		−	0.122890i		−	0.00743764i
$274$		0			0
$275$	−0.101546	+	4.99897i	−0.00612347	+	0.301449i
$276$		0			0
$277$			1.10740i			0.0665373i	0.999446	+	0.0332687i	$0.0105917\pi$
$277$			1.10740i			0.0665373i	−0.999446	+	0.0332687i	$0.989408\pi$
$278$		0			0
$279$	−1.99706			−0.119561
$280$		0			0
$281$	3.87370			0.231085			0.115543	−	0.993303i	$-0.463139\pi$
$281$	3.87370			0.231085			0.115543	+	0.993303i	$0.463139\pi$
$282$		0			0
$283$		−	11.1282i		−	0.661506i	−0.943717	−	0.330753i	$-0.892697\pi$
$283$		−	11.1282i		−	0.661506i	0.943717	−	0.330753i	$-0.107303\pi$
$284$		0			0
$285$	11.1814	−	10.9566i	0.662329	−	0.649011i
$286$		0			0
$287$		−	11.7902i		−	0.695956i
$288$		0			0
$289$	16.3558			0.962108
$290$		0			0
$291$	7.67347			0.449827
$292$		0			0
$293$			27.3496i			1.59778i	0.601478	+	0.798889i	$0.294579\pi$
$293$			27.3496i			1.59778i	−0.601478	+	0.798889i	$0.705421\pi$
$294$		0			0
$295$	20.1484	+	20.5619i	1.17309	+	1.19716i
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$		0			0
$299$	0.423133			0.0244704
$300$		0			0
$301$	6.54753			0.377393
$302$		0			0
$303$			8.09255i			0.464905i
$304$		0			0
$305$	−7.98229	−	8.14609i	−0.457065	−	0.466444i
$306$		0			0
$307$		−	12.6281i		−	0.720724i	−0.932812	−	0.360362i	$-0.882653\pi$
$307$		−	12.6281i		−	0.720724i	0.932812	−	0.360362i	$-0.117347\pi$
$308$		0			0
$309$	9.18812			0.522694
$310$		0			0
$311$	−15.5412			−0.881259			−0.440630	−	0.897689i	$-0.645245\pi$
$311$	−15.5412			−0.881259			−0.440630	+	0.897689i	$0.645245\pi$
$312$		0			0
$313$			14.3797i			0.812790i	0.913698	+	0.406395i	$0.133214\pi$
$313$			14.3797i			0.812790i	−0.913698	+	0.406395i	$0.866786\pi$
$314$		0			0
$315$	−1.59711	+	1.56500i	−0.0899872	+	0.0881778i
$316$		0			0
$317$			1.38430i			0.0777501i	0.999244	+	0.0388750i	$0.0123774\pi$
$317$			1.38430i			0.0777501i	−0.999244	+	0.0388750i	$0.987623\pi$
$318$		0			0
$319$	−4.73924			−0.265347
$320$		0			0
$321$	14.0151			0.782246
$322$		0			0
$323$		−	5.61899i		−	0.312649i
$324$		0			0
$325$	−0.614323	−	0.0124790i	−0.0340765	−	0.000692212i
$326$		0			0
$327$		−	0.988199i		−	0.0546475i
$328$		0			0
$329$	−0.536484			−0.0295773
$330$		0			0
$331$	9.30072			0.511214			0.255607	−	0.966781i	$-0.417725\pi$
$331$	9.30072			0.511214			0.255607	+	0.966781i	$0.417725\pi$
$332$		0			0
$333$			3.45598i			0.189387i
$334$		0			0
$335$	−3.60299	+	3.53055i	−0.196852	+	0.192894i
$336$		0			0
$337$			13.0934i			0.713241i	0.934249	+	0.356620i	$0.116071\pi$
$337$			13.0934i			0.713241i	−0.934249	+	0.356620i	$0.883929\pi$
$338$		0			0
$339$	16.1437			0.876804
$340$		0			0
$341$	1.99706			0.108147
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−5.38858	−	5.49915i	−0.290112	−	0.296065i
$346$		0			0
$347$			12.0664i			0.647756i	0.946099	+	0.323878i	$0.104987\pi$
$347$			12.0664i			0.647756i	−0.946099	+	0.323878i	$0.895013\pi$
$348$		0			0
$349$	27.1449			1.45303			0.726515	−	0.687150i	$-0.241139\pi$
$349$	27.1449			1.45303			0.726515	+	0.687150i	$0.241139\pi$
$350$		0			0
$351$	0.122890			0.00655938
$352$		0			0
$353$		−	20.6786i		−	1.10061i	−0.834964	−	0.550305i	$-0.814511\pi$
$353$		−	20.6786i		−	1.10061i	0.834964	−	0.550305i	$-0.185489\pi$
$354$		0			0
$355$	−13.1750	−	13.4454i	−0.699257	−	0.713605i
$356$		0			0
$357$			0.802598i			0.0424780i
$358$		0			0
$359$	18.7309			0.988577			0.494288	−	0.869298i	$-0.335429\pi$
$359$	18.7309			0.988577			0.494288	+	0.869298i	$0.335429\pi$
$360$		0			0
$361$	30.0140			1.57968
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	−8.71103	+	8.53587i	−0.455956	+	0.446788i
$366$		0			0
$367$			27.7381i			1.44792i	0.689843	+	0.723959i	$0.257680\pi$
$367$			27.7381i			1.44792i	−0.689843	+	0.723959i	$0.742320\pi$
$368$		0			0
$369$	11.7902			0.613776
$370$		0			0
$371$	−10.7389			−0.557538
$372$		0			0
$373$			6.38615i			0.330662i	0.986238	+	0.165331i	$0.0528693\pi$
$373$			6.38615i			0.330662i	−0.986238	+	0.165331i	$0.947131\pi$
$374$		0			0
$375$	7.66121	+	8.14284i	0.395623	+	0.420495i
$376$		0			0
$377$		−	0.582406i		−	0.0299954i
$378$		0			0
$379$	−0.108279			−0.00556191			−0.00278096	−	0.999996i	$-0.500885\pi$
$379$	−0.108279			−0.00556191			−0.00278096	+	0.999996i	$0.500885\pi$
$380$		0			0
$381$	−14.5957			−0.747760
$382$		0			0
$383$		−	18.8278i		−	0.962055i	−0.876705	−	0.481028i	$-0.840264\pi$
$383$		−	18.8278i		−	0.962055i	0.876705	−	0.481028i	$-0.159736\pi$
$384$		0			0
$385$	1.59711	−	1.56500i	0.0813965	−	0.0797598i
$386$		0			0
$387$			6.54753i			0.332830i
$388$		0			0
$389$	−9.94632			−0.504299			−0.252149	−	0.967688i	$-0.581137\pi$
$389$	−9.94632			−0.504299			−0.252149	+	0.967688i	$0.581137\pi$
$390$		0			0
$391$	−2.76349			−0.139756
$392$		0			0
$393$		−	3.72330i		−	0.187816i
$394$		0			0
$395$	−10.1672	−	10.3759i	−0.511570	−	0.522067i
$396$		0			0
$397$			10.7285i			0.538446i	0.963078	+	0.269223i	$0.0867669\pi$
$397$			10.7285i			0.538446i	−0.963078	+	0.269223i	$0.913233\pi$
$398$		0			0
$399$	−7.00100			−0.350488
$400$		0			0
$401$	35.0296			1.74930			0.874648	−	0.484758i	$-0.161092\pi$
$401$	35.0296			1.74930			0.874648	+	0.484758i	$0.161092\pi$
$402$		0			0
$403$			0.245419i			0.0122252i
$404$		0			0
$405$	−1.56500	−	1.59711i	−0.0777655	−	0.0793612i
$406$		0			0
$407$		−	3.45598i		−	0.171307i
$408$		0			0
$409$	−11.9422			−0.590504			−0.295252	−	0.955419i	$-0.595404\pi$
$409$	−11.9422			−0.590504			−0.295252	+	0.955419i	$0.595404\pi$
$410$		0			0
$411$	0.484151			0.0238814
$412$		0			0
$413$		−	12.8744i		−	0.633507i
$414$		0			0
$415$	9.05649	−	8.87439i	0.444566	−	0.435627i
$416$		0			0
$417$			21.8921i			1.07206i
$418$		0			0
$419$	−4.93827			−0.241250			−0.120625	−	0.992698i	$-0.538490\pi$
$419$	−4.93827			−0.241250			−0.120625	+	0.992698i	$0.538490\pi$
$420$		0			0
$421$	−37.9293			−1.84856			−0.924280	−	0.381715i	$-0.875334\pi$
$421$	−37.9293			−1.84856			−0.924280	+	0.381715i	$0.875334\pi$
$422$		0			0
$423$		−	0.536484i		−	0.0260848i
$424$		0			0
$425$	4.01216	+	0.0815008i	0.194618	+	0.00395337i
$426$		0			0
$427$			5.10051i			0.246831i
$428$		0			0
$429$	−0.122890			−0.00593318
$430$		0			0
$431$	11.4983			0.553852			0.276926	−	0.960891i	$-0.410684\pi$
$431$	11.4983			0.553852			0.276926	+	0.960891i	$0.410684\pi$
$432$		0			0
$433$		−	9.89698i		−	0.475618i	−0.971312	−	0.237809i	$-0.923571\pi$
$433$		−	9.89698i		−	0.475618i	0.971312	−	0.237809i	$-0.0764293\pi$
$434$		0			0
$435$	−7.56911	+	7.41692i	−0.362911	+	0.355614i
$436$		0			0
$437$		−	24.1057i		−	1.15313i
$438$		0			0
$439$	34.0036			1.62290			0.811452	−	0.584420i	$-0.198678\pi$
$439$	34.0036			1.62290			0.811452	+	0.584420i	$0.198678\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			14.7238i			0.699549i	0.936834	+	0.349775i	$0.113742\pi$
$443$			14.7238i			0.699549i	−0.936834	+	0.349775i	$0.886258\pi$
$444$		0			0
$445$	−21.1880	−	21.6228i	−1.00441	−	1.02502i
$446$		0			0
$447$			11.7443i			0.555485i
$448$		0			0
$449$	9.84873			0.464790			0.232395	−	0.972621i	$-0.425344\pi$
$449$	9.84873			0.464790			0.232395	+	0.972621i	$0.425344\pi$
$450$		0			0
$451$	−11.7902			−0.555181
$452$		0			0
$453$		−	2.01138i		−	0.0945029i
$454$		0			0
$455$	0.192323	+	0.196269i	0.00901623	+	0.00920125i
$456$		0			0
$457$		−	1.05919i		−	0.0495467i	−0.999693	−	0.0247733i	$-0.992114\pi$
$457$		−	1.05919i		−	0.0495467i	0.999693	−	0.0247733i	$-0.00788641\pi$
$458$		0			0
$459$	−0.802598			−0.0374620
$460$		0			0
$461$	34.6058			1.61175			0.805877	−	0.592084i	$-0.201694\pi$
$461$	34.6058			1.61175			0.805877	+	0.592084i	$0.201694\pi$
$462$		0			0
$463$			32.8173i			1.52515i	0.646901	+	0.762574i	$0.276065\pi$
$463$			32.8173i			1.52515i	−0.646901	+	0.762574i	$0.723935\pi$
$464$		0			0
$465$	3.18954	−	3.12541i	0.147911	−	0.144937i
$466$		0			0
$467$		−	24.9820i		−	1.15603i	−0.816026	−	0.578015i	$-0.803828\pi$
$467$		−	24.9820i		−	1.15603i	0.816026	−	0.578015i	$-0.196172\pi$
$468$		0			0
$469$	2.25594			0.104170
$470$		0			0
$471$	23.1571			1.06702
$472$		0			0
$473$		−	6.54753i		−	0.301056i
$474$		0			0
$475$	−0.710926	+	34.9978i	−0.0326195	+	1.60581i
$476$		0			0
$477$		−	10.7389i		−	0.491702i
$478$		0			0
$479$	−7.31361			−0.334167			−0.167084	−	0.985943i	$-0.553435\pi$
$479$	−7.31361			−0.334167			−0.167084	+	0.985943i	$0.553435\pi$
$480$		0			0
$481$	0.424706			0.0193649
$482$		0			0
$483$			3.44318i			0.156670i
$484$		0			0
$485$	−12.2554	+	12.0090i	−0.556489	+	0.545300i
$486$		0			0
$487$			15.4320i			0.699289i	0.936882	+	0.349645i	$0.113698\pi$
$487$			15.4320i			0.699289i	−0.936882	+	0.349645i	$0.886302\pi$
$488$		0			0
$489$	14.0902			0.637182
$490$		0			0
$491$	−37.4234			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$491$	−37.4234			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$492$		0			0
$493$			3.80371i			0.171310i
$494$		0			0
$495$	1.56500	+	1.59711i	0.0703415	+	0.0717849i
$496$		0			0
$497$			8.41853i			0.377623i
$498$		0			0
$499$	−10.8150			−0.484146			−0.242073	−	0.970258i	$-0.577827\pi$
$499$	−10.8150			−0.484146			−0.242073	+	0.970258i	$0.577827\pi$
$500$		0			0
$501$	2.58892			0.115664
$502$		0			0
$503$		−	20.3641i		−	0.907990i	−0.891004	−	0.453995i	$-0.849998\pi$
$503$		−	20.3641i		−	0.907990i	0.891004	−	0.453995i	$-0.150002\pi$
$504$		0			0
$505$	−12.6648	−	12.9247i	−0.563578	−	0.575143i
$506$		0			0
$507$			12.9849i			0.576680i
$508$		0			0
$509$	14.4919			0.642342			0.321171	−	0.947021i	$-0.395924\pi$
$509$	14.4919			0.642342			0.321171	+	0.947021i	$0.395924\pi$
$510$		0			0
$511$	5.45423			0.241281
$512$		0			0
$513$		−	7.00100i		−	0.309102i
$514$		0			0
$515$	−14.6745	+	14.3794i	−0.646634	+	0.633632i
$516$		0			0
$517$			0.536484i			0.0235946i
$518$		0			0
$519$	14.3432			0.629598
$520$		0			0
$521$	3.21816			0.140990			0.0704951	−	0.997512i	$-0.477542\pi$
$521$	3.21816			0.140990			0.0704951	+	0.997512i	$0.477542\pi$
$522$		0			0
$523$			39.3973i			1.72272i	0.507991	+	0.861362i	$0.330388\pi$
$523$			39.3973i			1.72272i	−0.507991	+	0.861362i	$0.669612\pi$
$524$		0			0
$525$	0.101546	−	4.99897i	0.00443184	−	0.218173i
$526$		0			0
$527$		−	1.60284i		−	0.0698207i
$528$		0			0
$529$	11.1445			0.484543
$530$		0			0
$531$	12.8744			0.558701
$532$		0			0
$533$		−	1.44890i		−	0.0627590i
$534$		0			0
$535$	−22.3837	+	21.9336i	−0.967732	+	0.948274i
$536$		0			0
$537$			20.2055i			0.871931i
$538$		0			0
$539$	−1.00000			−0.0430730
$540$		0			0
$541$	−8.61874			−0.370549			−0.185274	−	0.982687i	$-0.559317\pi$
$541$	−8.61874			−0.370549			−0.185274	+	0.982687i	$0.559317\pi$
$542$		0			0
$543$			20.1269i			0.863730i
$544$		0			0
$545$	1.54653	+	1.57827i	0.0662461	+	0.0676055i
$546$		0			0
$547$		−	5.14806i		−	0.220115i	−0.993925	−	0.110058i	$-0.964896\pi$
$547$		−	5.14806i		−	0.220115i	0.993925	−	0.110058i	$-0.0351035\pi$
$548$		0			0
$549$	−5.10051			−0.217684
$550$		0			0
$551$	−33.1795			−1.41349
$552$		0			0
$553$			6.49664i			0.276265i
$554$		0			0
$555$	−5.40862	−	5.51960i	−0.229583	−	0.234294i
$556$		0			0
$557$			25.2427i			1.06957i	0.844989	+	0.534784i	$0.179607\pi$
$557$			25.2427i			1.06957i	−0.844989	+	0.534784i	$0.820393\pi$
$558$		0			0
$559$	0.804626			0.0340321
$560$		0			0
$561$	0.802598			0.0338857
$562$		0			0
$563$			12.4908i			0.526426i	0.964738	+	0.263213i	$0.0847821\pi$
$563$			12.4908i			0.526426i	−0.964738	+	0.263213i	$0.915218\pi$
$564$		0			0
$565$	−25.7833	+	25.2648i	−1.08471	+	1.06290i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−1.59550			−0.0668870			−0.0334435	−	0.999441i	$-0.510647\pi$
$569$	−1.59550			−0.0668870			−0.0334435	+	0.999441i	$0.510647\pi$
$570$		0			0
$571$	−5.82961			−0.243962			−0.121981	−	0.992532i	$-0.538925\pi$
$571$	−5.82961			−0.243962			−0.121981	+	0.992532i	$0.538925\pi$
$572$		0			0
$573$		−	15.3725i		−	0.642197i
$574$		0			0
$575$	17.2124	+	0.349642i	0.717805	+	0.0145811i
$576$		0			0
$577$		−	38.0049i		−	1.58217i	−0.611709	−	0.791083i	$-0.709518\pi$
$577$		−	38.0049i		−	1.58217i	0.611709	−	0.791083i	$-0.290482\pi$
$578$		0			0
$579$	2.39480			0.0995247
$580$		0			0
$581$	−5.67053			−0.235253
$582$		0			0
$583$			10.7389i			0.444761i
$584$		0			0
$585$	−0.196269	+	0.192323i	−0.00811474	+	0.00795157i
$586$		0			0
$587$			1.85723i			0.0766559i	0.999265	+	0.0383280i	$0.0122032\pi$
$587$			1.85723i			0.0766559i	−0.999265	+	0.0383280i	$0.987797\pi$
$588$		0			0
$589$	13.9814			0.576095
$590$		0			0
$591$	−15.6451			−0.643553
$592$		0			0
$593$		−	23.6721i		−	0.972096i	−0.873932	−	0.486048i	$-0.838438\pi$
$593$		−	23.6721i		−	0.972096i	0.873932	−	0.486048i	$-0.161562\pi$
$594$		0			0
$595$	−1.25607	−	1.28184i	−0.0514937	−	0.0525503i
$596$		0			0
$597$			6.17533i			0.252739i
$598$		0			0
$599$	17.9997			0.735450			0.367725	−	0.929935i	$-0.380137\pi$
$599$	17.9997			0.735450			0.367725	+	0.929935i	$0.380137\pi$
$600$		0			0
$601$	−18.8710			−0.769764			−0.384882	−	0.922966i	$-0.625758\pi$
$601$	−18.8710			−0.769764			−0.384882	+	0.922966i	$0.625758\pi$
$602$		0			0
$603$			2.25594i			0.0918689i
$604$		0			0
$605$	−1.56500	−	1.59711i	−0.0636263	−	0.0649319i
$606$		0			0
$607$		−	42.0340i		−	1.70611i	−0.521823	−	0.853054i	$-0.674748\pi$
$607$		−	42.0340i		−	1.70611i	0.521823	−	0.853054i	$-0.325252\pi$
$608$		0			0
$609$	4.73924			0.192044
$610$		0			0
$611$	−0.0659285			−0.00266718
$612$		0			0
$613$		−	22.3587i		−	0.903061i	−0.892256	−	0.451530i	$-0.850878\pi$
$613$		−	22.3587i		−	0.903061i	0.892256	−	0.451530i	$-0.149122\pi$
$614$		0			0
$615$	−18.8304	+	18.4517i	−0.759314	+	0.744046i
$616$		0			0
$617$			15.3348i			0.617354i	0.951167	+	0.308677i	$0.0998862\pi$
$617$			15.3348i			0.617354i	−0.951167	+	0.308677i	$0.900114\pi$
$618$		0			0
$619$	2.53454			0.101872			0.0509358	−	0.998702i	$-0.483780\pi$
$619$	2.53454			0.101872			0.0509358	+	0.998702i	$0.483780\pi$
$620$		0			0
$621$	−3.44318			−0.138170
$622$		0			0
$623$			13.5387i			0.542415i
$624$		0			0
$625$	−24.9794	−	1.01525i	−0.999175	−	0.0406101i
$626$		0			0
$627$			7.00100i			0.279593i
$628$		0			0
$629$	−2.77376			−0.110597
$630$		0			0
$631$	16.5078			0.657166			0.328583	−	0.944475i	$-0.393429\pi$
$631$	16.5078			0.657166			0.328583	+	0.944475i	$0.393429\pi$
$632$		0			0
$633$			9.01682i			0.358386i
$634$		0			0
$635$	23.3110	−	22.8423i	0.925068	−	0.906468i
$636$		0			0
$637$		−	0.122890i		−	0.00486908i
$638$		0			0
$639$	−8.41853			−0.333032
$640$		0			0
$641$	−30.3269			−1.19784			−0.598921	−	0.800808i	$-0.704404\pi$
$641$	−30.3269			−1.19784			−0.598921	+	0.800808i	$0.704404\pi$
$642$		0			0
$643$			25.2134i			0.994318i	0.867660	+	0.497159i	$0.165623\pi$
$643$			25.2134i			0.994318i	−0.867660	+	0.497159i	$0.834377\pi$
$644$		0			0
$645$	−10.2469	−	10.4572i	−0.403471	−	0.411750i
$646$		0			0
$647$			14.8775i			0.584895i	0.956282	+	0.292447i	$0.0944696\pi$
$647$			14.8775i			0.584895i	−0.956282	+	0.292447i	$0.905530\pi$
$648$		0			0
$649$	−12.8744			−0.505364
$650$		0			0
$651$	−1.99706			−0.0782711
$652$		0			0
$653$		−	1.05315i		−	0.0412130i	−0.999788	−	0.0206065i	$-0.993440\pi$
$653$		−	1.05315i		−	0.0412130i	0.999788	−	0.0206065i	$-0.00655971\pi$
$654$		0			0
$655$	5.82697	+	5.94654i	0.227679	+	0.232351i
$656$		0			0
$657$			5.45423i			0.212790i
$658$		0			0
$659$	−12.2198			−0.476015			−0.238008	−	0.971263i	$-0.576494\pi$
$659$	−12.2198			−0.476015			−0.238008	+	0.971263i	$0.576494\pi$
$660$		0			0
$661$	6.85187			0.266507			0.133253	−	0.991082i	$-0.457458\pi$
$661$	6.85187			0.266507			0.133253	+	0.991082i	$0.457458\pi$
$662$		0			0
$663$			0.0986312i			0.00383052i
$664$		0			0
$665$	11.1814	−	10.9566i	0.433596	−	0.424878i
$666$		0			0
$667$			16.3181i			0.631839i
$668$		0			0
$669$	15.7055			0.607210
$670$		0			0
$671$	5.10051			0.196903
$672$		0			0
$673$			3.15418i			0.121585i	0.998150	+	0.0607924i	$0.0193628\pi$
$673$			3.15418i			0.121585i	−0.998150	+	0.0607924i	$0.980637\pi$
$674$		0			0
$675$	4.99897	+	0.101546i	0.192410	+	0.00390852i
$676$		0			0
$677$		−	6.06029i		−	0.232916i	−0.993196	−	0.116458i	$-0.962846\pi$
$677$		−	6.06029i		−	0.232916i	0.993196	−	0.116458i	$-0.0371540\pi$
$678$		0			0
$679$	7.67347			0.294481
$680$		0			0
$681$	25.4782			0.976327
$682$		0			0
$683$		−	9.16492i		−	0.350686i	−0.984507	−	0.175343i	$-0.943897\pi$
$683$		−	9.16492i		−	0.350686i	0.984507	−	0.175343i	$-0.0561034\pi$
$684$		0			0
$685$	−0.773244	+	0.757696i	−0.0295441	+	0.0289501i
$686$		0			0
$687$		−	16.5750i		−	0.632377i
$688$		0			0
$689$	−1.31971			−0.0502768
$690$		0			0
$691$	−9.20569			−0.350201			−0.175100	−	0.984551i	$-0.556025\pi$
$691$	−9.20569			−0.350201			−0.175100	+	0.984551i	$0.556025\pi$
$692$		0			0
$693$		−	1.00000i		−	0.0379869i
$694$		0			0
$695$	−34.2611	−	34.9642i	−1.29960	−	1.32627i
$696$		0			0
$697$			9.46282i			0.358430i
$698$		0			0
$699$	26.3986			0.998489
$700$		0			0
$701$	45.6509			1.72421			0.862105	−	0.506731i	$-0.169146\pi$
$701$	45.6509			1.72421			0.862105	+	0.506731i	$0.169146\pi$
$702$		0			0
$703$		−	24.1953i		−	0.912545i
$704$		0			0
$705$	0.839598	+	0.856827i	0.0316211	+	0.0322700i
$706$		0			0
$707$			8.09255i			0.304352i
$708$		0			0
$709$	3.09014			0.116053			0.0580264	−	0.998315i	$-0.481519\pi$
$709$	3.09014			0.116053			0.0580264	+	0.998315i	$0.481519\pi$
$710$		0			0
$711$	−6.49664			−0.243643
$712$		0			0
$713$		−	6.87625i		−	0.257518i
$714$		0			0
$715$	0.196269	−	0.192323i	0.00734005	−	0.00719247i
$716$		0			0
$717$			9.75552i			0.364327i
$718$		0			0
$719$	−35.5698			−1.32653			−0.663265	−	0.748385i	$-0.730830\pi$
$719$	−35.5698			−1.32653			−0.663265	+	0.748385i	$0.730830\pi$
$720$		0			0
$721$	9.18812			0.342183
$722$		0			0
$723$		−	22.4317i		−	0.834244i
$724$		0			0
$725$	0.481253	−	23.6913i	0.0178733	−	0.879874i
$726$		0			0
$727$		−	10.9003i		−	0.404269i	−0.979358	−	0.202134i	$-0.935212\pi$
$727$		−	10.9003i		−	0.404269i	0.979358	−	0.202134i	$-0.0647878\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−5.25503			−0.194364
$732$		0			0
$733$			1.76168i			0.0650691i	0.999471	+	0.0325346i	$0.0103579\pi$
$733$			1.76168i			0.0650691i	−0.999471	+	0.0325346i	$0.989642\pi$
$734$		0			0
$735$	−1.59711	+	1.56500i	−0.0589104	+	0.0577259i
$736$		0			0
$737$		−	2.25594i		−	0.0830986i
$738$		0			0
$739$	−2.77789			−0.102186			−0.0510931	−	0.998694i	$-0.516271\pi$
$739$	−2.77789			−0.102186			−0.0510931	+	0.998694i	$0.516271\pi$
$740$		0			0
$741$	−0.860353			−0.0316058
$742$		0			0
$743$		−	23.0358i		−	0.845103i	−0.906339	−	0.422552i	$-0.861135\pi$
$743$		−	23.0358i		−	0.845103i	0.906339	−	0.422552i	$-0.138865\pi$
$744$		0			0
$745$	−18.3798	−	18.7569i	−0.673383	−	0.687201i
$746$		0			0
$747$		−	5.67053i		−	0.207474i
$748$		0			0
$749$	14.0151			0.512100
$750$		0			0
$751$	−11.9544			−0.436221			−0.218110	−	0.975924i	$-0.569989\pi$
$751$	−11.9544			−0.436221			−0.218110	+	0.975924i	$0.569989\pi$
$752$		0			0
$753$		−	23.1380i		−	0.843194i
$754$		0			0
$755$	3.14781	+	3.21240i	0.114561	+	0.116911i
$756$		0			0
$757$			49.7628i			1.80866i	0.426834	+	0.904330i	$0.359629\pi$
$757$			49.7628i			1.80866i	−0.426834	+	0.904330i	$0.640371\pi$
$758$		0			0
$759$	3.44318			0.124980
$760$		0			0
$761$	4.32110			0.156640			0.0783198	−	0.996928i	$-0.475044\pi$
$761$	4.32110			0.156640			0.0783198	+	0.996928i	$0.475044\pi$
$762$		0			0
$763$		−	0.988199i		−	0.0357752i
$764$		0			0
$765$	1.28184	−	1.25607i	0.0463450	−	0.0454131i
$766$		0			0
$767$		−	1.58213i		−	0.0571275i
$768$		0			0
$769$	−1.35621			−0.0489062			−0.0244531	−	0.999701i	$-0.507784\pi$
$769$	−1.35621			−0.0489062			−0.0244531	+	0.999701i	$0.507784\pi$
$770$		0			0
$771$	18.4826			0.665634
$772$		0			0
$773$			30.8582i			1.10989i	0.831886	+	0.554947i	$0.187261\pi$
$773$			30.8582i			1.10989i	−0.831886	+	0.554947i	$0.812739\pi$
$774$		0			0
$775$	−0.202794	+	9.98326i	−0.00728459	+	0.358609i
$776$		0			0
$777$			3.45598i			0.123983i
$778$		0			0
$779$	−82.5435			−2.95743
$780$		0			0
$781$	8.41853			0.301239
$782$		0			0
$783$			4.73924i			0.169367i
$784$		0			0
$785$	−36.9845	+	36.2409i	−1.32003	+	1.29349i
$786$		0			0
$787$			6.69991i			0.238826i	0.992845	+	0.119413i	$0.0381012\pi$
$787$			6.69991i			0.238826i	−0.992845	+	0.119413i	$0.961899\pi$
$788$		0			0
$789$	3.40223			0.121122
$790$		0			0
$791$	16.1437			0.574003
$792$		0			0
$793$			0.626801i			0.0222584i
$794$		0			0
$795$	16.8064	+	17.1513i	0.596063	+	0.608294i
$796$		0			0
$797$			45.7012i			1.61882i	0.587245	+	0.809409i	$0.300213\pi$
$797$			45.7012i			1.61882i	−0.587245	+	0.809409i	$0.699787\pi$
$798$		0			0
$799$	0.430581			0.0152329
$800$		0			0
$801$	−13.5387			−0.478365
$802$		0			0
$803$		−	5.45423i		−	0.192476i
$804$		0			0
$805$	−5.38858	−	5.49915i	−0.189923	−	0.193820i
$806$		0			0
$807$		−	21.7258i		−	0.764785i
$808$		0			0
$809$	16.5972			0.583527			0.291763	−	0.956491i	$-0.405758\pi$
$809$	16.5972			0.583527			0.291763	+	0.956491i	$0.405758\pi$
$810$		0			0
$811$	−35.9659			−1.26294			−0.631468	−	0.775402i	$-0.717547\pi$
$811$	−35.9659			−1.26294			−0.631468	+	0.775402i	$0.717547\pi$
$812$		0			0
$813$		−	18.4200i		−	0.646016i
$814$		0			0
$815$	−22.5037	+	22.0512i	−0.788270	+	0.772420i
$816$		0			0
$817$		−	45.8393i		−	1.60371i
$818$		0			0
$819$	0.122890			0.00429412
$820$		0			0
$821$	−18.1886			−0.634786			−0.317393	−	0.948294i	$-0.602807\pi$
$821$	−18.1886			−0.634786			−0.317393	+	0.948294i	$0.602807\pi$
$822$		0			0
$823$			19.1199i			0.666479i	0.942842	+	0.333240i	$0.108142\pi$
$823$			19.1199i			0.666479i	−0.942842	+	0.333240i	$0.891858\pi$
$824$		0			0
$825$	−4.99897	−	0.101546i	−0.174042	−	0.00353539i
$826$		0			0
$827$			26.9412i			0.936838i	0.883507	+	0.468419i	$0.155176\pi$
$827$			26.9412i			0.936838i	−0.883507	+	0.468419i	$0.844824\pi$
$828$		0			0
$829$	−19.7270			−0.685146			−0.342573	−	0.939491i	$-0.611298\pi$
$829$	−19.7270			−0.685146			−0.342573	+	0.939491i	$0.611298\pi$
$830$		0			0
$831$	−1.10740			−0.0384153
$832$		0			0
$833$			0.802598i			0.0278084i
$834$		0			0
$835$	−4.13480	+	4.05166i	−0.143091	+	0.140214i
$836$		0			0
$837$		−	1.99706i		−	0.0690286i
$838$		0			0
$839$	−26.7897			−0.924885			−0.462442	−	0.886649i	$-0.653027\pi$
$839$	−26.7897			−0.924885			−0.462442	+	0.886649i	$0.653027\pi$
$840$		0			0
$841$	−6.53956			−0.225502
$842$		0			0
$843$			3.87370i			0.133417i
$844$		0			0
$845$	−20.3214	−	20.7384i	−0.699076	−	0.713421i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	11.1282			0.381920
$850$		0			0
$851$	−11.8996			−0.407912
$852$		0			0
$853$		−	26.8042i		−	0.917759i	−0.888498	−	0.458880i	$-0.848251\pi$
$853$		−	26.8042i		−	0.917759i	0.888498	−	0.458880i	$-0.151749\pi$
$854$		0			0
$855$	10.9566	+	11.1814i	0.374707	+	0.382396i
$856$		0			0
$857$		−	38.1053i		−	1.30165i	−0.759227	−	0.650826i	$-0.774423\pi$
$857$		−	38.1053i		−	1.30165i	0.759227	−	0.650826i	$-0.225577\pi$
$858$		0			0
$859$	−4.97911			−0.169885			−0.0849425	−	0.996386i	$-0.527071\pi$
$859$	−4.97911			−0.169885			−0.0849425	+	0.996386i	$0.527071\pi$
$860$		0			0
$861$	11.7902			0.401811
$862$		0			0
$863$		−	12.2098i		−	0.415627i	−0.978169	−	0.207813i	$-0.933365\pi$
$863$		−	12.2098i		−	0.415627i	0.978169	−	0.207813i	$-0.0666346\pi$
$864$		0			0
$865$	−22.9078	+	22.4472i	−0.778887	+	0.763226i
$866$		0			0
$867$			16.3558i			0.555473i
$868$		0			0
$869$	6.49664			0.220383
$870$		0			0
$871$	0.277232			0.00939366
$872$		0			0
$873$			7.67347i			0.259708i
$874$		0			0
$875$	7.66121	+	8.14284i	0.258996	+	0.275278i
$876$		0			0
$877$		−	50.7466i		−	1.71359i	−0.515656	−	0.856795i	$-0.672452\pi$
$877$		−	50.7466i		−	1.71359i	0.515656	−	0.856795i	$-0.327548\pi$
$878$		0			0
$879$	−27.3496			−0.922478
$880$		0			0
$881$	27.4523			0.924892			0.462446	−	0.886647i	$-0.346972\pi$
$881$	27.4523			0.924892			0.462446	+	0.886647i	$0.346972\pi$
$882$		0			0
$883$			46.6300i			1.56923i	0.619986	+	0.784613i	$0.287138\pi$
$883$			46.6300i			1.56923i	−0.619986	+	0.784613i	$0.712862\pi$
$884$		0			0
$885$	−20.5619	+	20.1484i	−0.691180	+	0.677282i
$886$		0			0
$887$			35.7104i			1.19904i	0.800360	+	0.599519i	$0.204641\pi$
$887$			35.7104i			1.19904i	−0.800360	+	0.599519i	$0.795359\pi$
$888$		0			0
$889$	−14.5957			−0.489524
$890$		0			0
$891$	1.00000			0.0335013
$892$		0			0
$893$			3.75593i			0.125687i
$894$		0			0
$895$	−31.6216	−	32.2705i	−1.05699	−	1.07868i
$896$		0			0
$897$			0.423133i			0.0141280i
$898$		0			0
$899$	−9.46457			−0.315661
$900$		0			0
$901$	8.61904			0.287142
$902$		0			0
$903$			6.54753i			0.217888i
$904$		0			0
$905$	−31.4987	−	32.1450i	−1.04705	−	1.06854i
$906$		0			0
$907$			46.5420i			1.54540i	0.634771	+	0.772700i	$0.281094\pi$
$907$			46.5420i			1.54540i	−0.634771	+	0.772700i	$0.718906\pi$
$908$		0			0
$909$	−8.09255			−0.268413
$910$		0			0
$911$	−12.3496			−0.409161			−0.204581	−	0.978850i	$-0.565583\pi$
$911$	−12.3496			−0.409161			−0.204581	+	0.978850i	$0.565583\pi$
$912$		0			0
$913$			5.67053i			0.187667i
$914$		0			0
$915$	8.14609	−	7.98229i	0.269301	−	0.263886i
$916$		0			0
$917$		−	3.72330i		−	0.122954i
$918$		0			0
$919$	12.8615			0.424262			0.212131	−	0.977241i	$-0.431960\pi$
$919$	12.8615			0.424262			0.212131	+	0.977241i	$0.431960\pi$
$920$		0			0
$921$	12.6281			0.416110
$922$		0			0
$923$			1.03455i			0.0340527i
$924$		0			0
$925$	17.2764	+	0.350942i	0.568043	+	0.0115389i
$926$		0			0
$927$			9.18812i			0.301777i
$928$		0			0
$929$	55.4391			1.81890			0.909448	−	0.415817i	$-0.136504\pi$
$929$	55.4391			1.81890			0.909448	+	0.415817i	$0.136504\pi$
$930$		0			0
$931$	−7.00100			−0.229449
$932$		0			0
$933$		−	15.5412i		−	0.508795i
$934$		0			0
$935$	−1.28184	+	1.25607i	−0.0419206	+	0.0410777i
$936$		0			0
$937$		−	57.7967i		−	1.88813i	−0.329753	−	0.944067i	$-0.606965\pi$
$937$		−	57.7967i		−	1.88813i	0.329753	−	0.944067i	$-0.393035\pi$
$938$		0			0
$939$	−14.3797			−0.469264
$940$		0			0
$941$	−45.0236			−1.46773			−0.733864	−	0.679296i	$-0.762285\pi$
$941$	−45.0236			−1.46773			−0.733864	+	0.679296i	$0.762285\pi$
$942$		0			0
$943$			40.5960i			1.32199i
$944$		0			0
$945$	−1.56500	−	1.59711i	−0.0509095	−	0.0519541i
$946$		0			0
$947$		−	22.3627i		−	0.726688i	−0.931655	−	0.363344i	$-0.881635\pi$
$947$		−	22.3627i		−	0.726688i	0.931655	−	0.363344i	$-0.118365\pi$
$948$		0			0
$949$	0.670270			0.0217579
$950$		0			0
$951$	−1.38430			−0.0448890
$952$		0			0
$953$		−	21.3671i		−	0.692147i	−0.938207	−	0.346074i	$-0.887515\pi$
$953$		−	21.3671i		−	0.692147i	0.938207	−	0.346074i	$-0.112485\pi$
$954$		0			0
$955$	24.0580	+	24.5517i	0.778500	+	0.794474i
$956$		0			0
$957$		−	4.73924i		−	0.153198i
$958$		0			0
$959$	0.484151			0.0156340
$960$		0			0
$961$	−27.0117			−0.871346
$962$		0			0
$963$			14.0151i			0.451630i
$964$		0			0
$965$	−3.82478	+	3.74787i	−0.123124	+	0.120648i
$966$		0			0
$967$		−	15.6000i		−	0.501661i	−0.968031	−	0.250831i	$-0.919296\pi$
$967$		−	15.6000i		−	0.501661i	0.968031	−	0.250831i	$-0.0807037\pi$
$968$		0			0
$969$	5.61899			0.180508
$970$		0			0
$971$	−44.8431			−1.43908			−0.719541	−	0.694450i	$-0.755648\pi$
$971$	−44.8431			−1.43908			−0.719541	+	0.694450i	$0.755648\pi$
$972$		0			0
$973$			21.8921i			0.701828i
$974$		0			0
$975$	0.0124790	−	0.614323i	0.000399649	−	0.0196741i
$976$		0			0
$977$			6.51379i			0.208395i	0.994557	+	0.104197i	$0.0332273\pi$
$977$			6.51379i			0.208395i	−0.994557	+	0.104197i	$0.966773\pi$
$978$		0			0
$979$	13.5387			0.432698
$980$		0			0
$981$	0.988199			0.0315508
$982$		0			0
$983$		−	36.7051i		−	1.17071i	−0.810777	−	0.585355i	$-0.800955\pi$
$983$		−	36.7051i		−	1.17071i	0.810777	−	0.585355i	$-0.199045\pi$
$984$		0			0
$985$	24.9870	−	24.4846i	0.796152	−	0.780144i
$986$		0			0
$987$		−	0.536484i		−	0.0170765i
$988$		0			0
$989$	−22.5443			−0.716869
$990$		0			0
$991$	28.3404			0.900262			0.450131	−	0.892962i	$-0.351377\pi$
$991$	28.3404			0.900262			0.450131	+	0.892962i	$0.351377\pi$
$992$		0			0
$993$			9.30072i			0.295149i
$994$		0			0
$995$	−9.66439	−	9.86270i	−0.306382	−	0.312669i
$996$		0			0
$997$		−	52.8494i		−	1.67376i	−0.547388	−	0.836879i	$-0.684378\pi$
$997$		−	52.8494i		−	1.67376i	0.547388	−	0.836879i	$-0.315622\pi$
$998$		0			0
$999$	−3.45598			−0.109342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.h.e.1849.10	yes	14
5.4	even	2	inner	4620.2.h.e.1849.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.h.e.1849.3	✓	14	5.4	even	2	inner
4620.2.h.e.1849.10	yes	14	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.56500 - 1.59711i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +0.122890i q^{13} +(1.59711 - 1.56500i) q^{15} -0.802598i q^{17} +7.00100 q^{19} -1.00000 q^{21} -3.44318i q^{23} +(-0.101546 + 4.99897i) q^{25} -1.00000i q^{27} -4.73924 q^{29} +1.99706 q^{31} +1.00000i q^{33} +(1.59711 - 1.56500i) q^{35} -3.45598i q^{37} -0.122890 q^{39} -11.7902 q^{41} -6.54753i q^{43} +(1.56500 + 1.59711i) q^{45} +0.536484i q^{47} -1.00000 q^{49} +0.802598 q^{51} +10.7389i q^{53} +(-1.56500 - 1.59711i) q^{55} +7.00100i q^{57} -12.8744 q^{59} +5.10051 q^{61} -1.00000i q^{63} +(0.196269 - 0.192323i) q^{65} -2.25594i q^{67} +3.44318 q^{69} +8.41853 q^{71} -5.45423i q^{73} +(-4.99897 - 0.101546i) q^{75} +1.00000i q^{77} +6.49664 q^{79} +1.00000 q^{81} +5.67053i q^{83} +(-1.28184 + 1.25607i) q^{85} -4.73924i q^{87} +13.5387 q^{89} -0.122890 q^{91} +1.99706i q^{93} +(-10.9566 - 11.1814i) q^{95} -7.67347i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69}+ \cdots - 14 q^{99}+O(q^{100}) \) 14 * q - 4 * q^5 - 14 * q^9 + 14 * q^11 + 6 * q^19 - 14 * q^21 - 8 * q^25 + 38 * q^29 + 4 * q^31 + 8 * q^39 - 12 * q^41 + 4 * q^45 - 14 * q^49 + 14 * q^51 - 4 * q^55 + 18 * q^59 - 54 * q^61 - 2 * q^65 + 26 * q^69 - 24 * q^71 + 48 * q^79 + 14 * q^81 + 12 * q^85 + 46 * q^89 + 8 * q^91 - 72 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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