LMFDB - Embedded newform 1932.2.k.a.1609.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1609,1932)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1932.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.30
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.24015 q^{5} +(0.274849 + 2.63144i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.24015 q^{5} +(0.274849 + 2.63144i) q^{7} -1.00000 q^{9} +5.61325i q^{11} -3.17811i q^{13} -2.24015i q^{15} -0.653410 q^{17} +1.73662 q^{19} +(2.63144 - 0.274849i) q^{21} +(3.98929 - 2.66187i) q^{23} +0.0182760 q^{25} +1.00000i q^{27} +0.830792 q^{29} +3.41784i q^{31} +5.61325 q^{33} +(0.615703 + 5.89481i) q^{35} +9.03579i q^{37} -3.17811 q^{39} +4.58672i q^{41} -1.46364i q^{43} -2.24015 q^{45} +10.2009i q^{47} +(-6.84892 + 1.44649i) q^{49} +0.653410i q^{51} -0.509589i q^{53} +12.5745i q^{55} -1.73662i q^{57} -1.43776i q^{59} +10.9925 q^{61} +(-0.274849 - 2.63144i) q^{63} -7.11944i q^{65} +4.99995i q^{67} +(-2.66187 - 3.98929i) q^{69} -3.32787 q^{71} -10.8962i q^{73} -0.0182760i q^{75} +(-14.7709 + 1.54279i) q^{77} +3.98034i q^{79} +1.00000 q^{81} +10.5122 q^{83} -1.46374 q^{85} -0.830792i q^{87} +5.52664 q^{89} +(8.36299 - 0.873499i) q^{91} +3.41784 q^{93} +3.89029 q^{95} -2.63063 q^{97} -5.61325i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9}+O(q^{10}) $ 32 * q - 32 * q^9	$ 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) $ 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	2.24015			1.00183			0.500913	−	0.865498i	$-0.332998\pi$
$5$	2.24015			1.00183			0.500913	+	0.865498i	$0.332998\pi$
$6$		0			0
$7$	0.274849	+	2.63144i	0.103883	+	0.994590i
$7$	0.274849	+	2.63144i	0.103883	+	0.994590i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			5.61325i			1.69246i	0.532819	+	0.846229i	$0.321133\pi$
$11$			5.61325i			1.69246i	−0.532819	+	0.846229i	$0.678867\pi$
$12$		0			0
$13$		−	3.17811i		−	0.881448i	−0.897643	−	0.440724i	$-0.854722\pi$
$13$		−	3.17811i		−	0.881448i	0.897643	−	0.440724i	$-0.145278\pi$
$14$		0			0
$15$		−	2.24015i		−	0.578404i
$16$		0			0
$17$	−0.653410			−0.158475			−0.0792376	−	0.996856i	$-0.525249\pi$
$17$	−0.653410			−0.158475			−0.0792376	+	0.996856i	$0.525249\pi$
$18$		0			0
$19$	1.73662			0.398408			0.199204	−	0.979958i	$-0.436164\pi$
$19$	1.73662			0.398408			0.199204	+	0.979958i	$0.436164\pi$
$20$		0			0
$21$	2.63144	−	0.274849i	0.574227	−	0.0599769i
$22$		0			0
$23$	3.98929	−	2.66187i	0.831824	−	0.555039i
$23$	3.98929	−	2.66187i	0.831824	−	0.555039i
$24$		0			0
$25$	0.0182760			0.00365520
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	0.830792			0.154274			0.0771371	−	0.997020i	$-0.475422\pi$
$29$	0.830792			0.154274			0.0771371	+	0.997020i	$0.475422\pi$
$30$		0			0
$31$			3.41784i			0.613862i	0.951732	+	0.306931i	$0.0993021\pi$
$31$			3.41784i			0.613862i	−0.951732	+	0.306931i	$0.900698\pi$
$32$		0			0
$33$	5.61325			0.977141
$34$		0			0
$35$	0.615703	+	5.89481i	0.104073	+	0.996406i
$36$		0			0
$37$			9.03579i			1.48547i	0.669583	+	0.742737i	$0.266473\pi$
$37$			9.03579i			1.48547i	−0.669583	+	0.742737i	$0.733527\pi$
$38$		0			0
$39$	−3.17811			−0.508904
$40$		0			0
$41$			4.58672i			0.716325i	0.933659	+	0.358162i	$0.116597\pi$
$41$			4.58672i			0.716325i	−0.933659	+	0.358162i	$0.883403\pi$
$42$		0			0
$43$		−	1.46364i		−	0.223203i	−0.993753	−	0.111601i	$-0.964402\pi$
$43$		−	1.46364i		−	0.223203i	0.993753	−	0.111601i	$-0.0355979\pi$
$44$		0			0
$45$	−2.24015			−0.333942
$46$		0			0
$47$			10.2009i			1.48796i	0.668202	+	0.743980i	$0.267064\pi$
$47$			10.2009i			1.48796i	−0.668202	+	0.743980i	$0.732936\pi$
$48$		0			0
$49$	−6.84892	+	1.44649i	−0.978417	+	0.206642i
$50$		0			0
$51$			0.653410i			0.0914957i
$52$		0			0
$53$		−	0.509589i		−	0.0699974i	−0.999387	−	0.0349987i	$-0.988857\pi$
$53$		−	0.509589i		−	0.0699974i	0.999387	−	0.0349987i	$-0.0111427\pi$
$54$		0			0
$55$			12.5745i			1.69555i
$56$		0			0
$57$		−	1.73662i		−	0.230021i
$58$		0			0
$59$		−	1.43776i		−	0.187180i	−0.995611	−	0.0935902i	$-0.970166\pi$
$59$		−	1.43776i		−	0.187180i	0.995611	−	0.0935902i	$-0.0298344\pi$
$60$		0			0
$61$	10.9925			1.40745			0.703723	−	0.710475i	$-0.251520\pi$
$61$	10.9925			1.40745			0.703723	+	0.710475i	$0.251520\pi$
$62$		0			0
$63$	−0.274849	−	2.63144i	−0.0346277	−	0.331530i
$64$		0			0
$65$		−	7.11944i		−	0.883058i
$66$		0			0
$67$			4.99995i			0.610841i	0.952218	+	0.305420i	$0.0987970\pi$
$67$			4.99995i			0.610841i	−0.952218	+	0.305420i	$0.901203\pi$
$68$		0			0
$69$	−2.66187	−	3.98929i	−0.320452	−	0.480254i
$70$		0			0
$71$	−3.32787			−0.394946			−0.197473	−	0.980308i	$-0.563273\pi$
$71$	−3.32787			−0.394946			−0.197473	+	0.980308i	$0.563273\pi$
$72$		0			0
$73$		−	10.8962i		−	1.27530i	−0.770325	−	0.637651i	$-0.779906\pi$
$73$		−	10.8962i		−	1.27530i	0.770325	−	0.637651i	$-0.220094\pi$
$74$		0			0
$75$		−	0.0182760i		−	0.00211033i
$76$		0			0
$77$	−14.7709	+	1.54279i	−1.68330	+	0.175818i
$78$		0			0
$79$			3.98034i			0.447823i	0.974609	+	0.223912i	$0.0718827\pi$
$79$			3.98034i			0.447823i	−0.974609	+	0.223912i	$0.928117\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	10.5122			1.15387			0.576933	−	0.816792i	$-0.304250\pi$
$83$	10.5122			1.15387			0.576933	+	0.816792i	$0.304250\pi$
$84$		0			0
$85$	−1.46374			−0.158765
$86$		0			0
$87$		−	0.830792i		−	0.0890702i
$88$		0			0
$89$	5.52664			0.585823			0.292911	−	0.956140i	$-0.405376\pi$
$89$	5.52664			0.585823			0.292911	+	0.956140i	$0.405376\pi$
$90$		0			0
$91$	8.36299	−	0.873499i	0.876679	−	0.0915676i
$92$		0			0
$93$	3.41784			0.354413
$94$		0			0
$95$	3.89029			0.399135
$96$		0			0
$97$	−2.63063			−0.267101			−0.133550	−	0.991042i	$-0.542638\pi$
$97$	−2.63063			−0.267101			−0.133550	+	0.991042i	$0.542638\pi$
$98$		0			0
$99$		−	5.61325i		−	0.564153i
$100$		0			0
$101$			6.81860i			0.678476i	0.940701	+	0.339238i	$0.110169\pi$
$101$			6.81860i			0.678476i	−0.940701	+	0.339238i	$0.889831\pi$
$102$		0			0
$103$	5.63264			0.555001			0.277500	−	0.960726i	$-0.410494\pi$
$103$	5.63264			0.555001			0.277500	+	0.960726i	$0.410494\pi$
$104$		0			0
$105$	5.89481	−	0.615703i	0.575275	−	0.0600864i
$106$		0			0
$107$		−	13.5267i		−	1.30767i	−0.756636	−	0.653837i	$-0.773158\pi$
$107$		−	13.5267i		−	1.30767i	0.756636	−	0.653837i	$-0.226842\pi$
$108$		0			0
$109$			10.1826i			0.975316i	0.873035	+	0.487658i	$0.162149\pi$
$109$			10.1826i			0.975316i	−0.873035	+	0.487658i	$0.837851\pi$
$110$		0			0
$111$	9.03579			0.857639
$112$		0			0
$113$		−	9.06577i		−	0.852836i	−0.904526	−	0.426418i	$-0.859775\pi$
$113$		−	9.06577i		−	0.852836i	0.904526	−	0.426418i	$-0.140225\pi$
$114$		0			0
$115$	8.93661	−	5.96300i	0.833343	−	0.556053i
$116$		0			0
$117$			3.17811i			0.293816i
$118$		0			0
$119$	−0.179589	−	1.71941i	−0.0164629	−	0.157618i
$120$		0			0
$121$	−20.5086			−1.86441
$122$		0			0
$123$	4.58672			0.413570
$124$		0			0
$125$	−11.1598			−0.998164
$126$		0			0
$127$	12.8808			1.14299			0.571493	−	0.820607i	$-0.306364\pi$
$127$	12.8808			1.14299			0.571493	+	0.820607i	$0.306364\pi$
$128$		0			0
$129$	−1.46364			−0.128866
$130$		0			0
$131$		−	16.9581i		−	1.48163i	−0.671708	−	0.740816i	$-0.734439\pi$
$131$		−	16.9581i		−	1.48163i	0.671708	−	0.740816i	$-0.265561\pi$
$132$		0			0
$133$	0.477308	+	4.56980i	0.0413878	+	0.396252i
$134$		0			0
$135$			2.24015i			0.192801i
$136$		0			0
$137$		−	0.839197i		−	0.0716974i	−0.999357	−	0.0358487i	$-0.988587\pi$
$137$		−	0.839197i		−	0.0716974i	0.999357	−	0.0358487i	$-0.0114134\pi$
$138$		0			0
$139$			19.7845i			1.67810i	0.544054	+	0.839050i	$0.316889\pi$
$139$			19.7845i			1.67810i	−0.544054	+	0.839050i	$0.683111\pi$
$140$		0			0
$141$	10.2009			0.859074
$142$		0			0
$143$	17.8395			1.49181
$144$		0			0
$145$	1.86110			0.154556
$146$		0			0
$147$	1.44649	+	6.84892i	0.119305	+	0.564889i
$148$		0			0
$149$		−	11.4198i		−	0.935543i	−0.883849	−	0.467772i	$-0.845057\pi$
$149$		−	11.4198i		−	0.935543i	0.883849	−	0.467772i	$-0.154943\pi$
$150$		0			0
$151$	−4.76752			−0.387975			−0.193988	−	0.981004i	$-0.562142\pi$
$151$	−4.76752			−0.387975			−0.193988	+	0.981004i	$0.562142\pi$
$152$		0			0
$153$	0.653410			0.0528250
$154$		0			0
$155$			7.65647i			0.614983i
$156$		0			0
$157$	−0.602641			−0.0480960			−0.0240480	−	0.999711i	$-0.507655\pi$
$157$	−0.602641			−0.0480960			−0.0240480	+	0.999711i	$0.507655\pi$
$158$		0			0
$159$	−0.509589			−0.0404130
$160$		0			0
$161$	8.10100	+	9.76595i	0.638449	+	0.769664i
$162$		0			0
$163$	5.65161			0.442668			0.221334	−	0.975198i	$-0.428959\pi$
$163$	5.65161			0.442668			0.221334	+	0.975198i	$0.428959\pi$
$164$		0			0
$165$	12.5745			0.978925
$166$		0			0
$167$			5.35531i			0.414406i	0.978298	+	0.207203i	$0.0664361\pi$
$167$			5.35531i			0.414406i	−0.978298	+	0.207203i	$0.933564\pi$
$168$		0			0
$169$	2.89963			0.223049
$170$		0			0
$171$	−1.73662			−0.132803
$172$		0			0
$173$		−	11.7452i		−	0.892971i	−0.894791	−	0.446485i	$-0.852676\pi$
$173$		−	11.7452i		−	0.892971i	0.894791	−	0.446485i	$-0.147324\pi$
$174$		0			0
$175$	0.00502314	+	0.0480922i	0.000379714	+	0.00363543i
$176$		0			0
$177$	−1.43776			−0.108069
$178$		0			0
$179$	21.3113			1.59288			0.796442	−	0.604715i	$-0.206713\pi$
$179$	21.3113			1.59288			0.796442	+	0.604715i	$0.206713\pi$
$180$		0			0
$181$	−16.4293			−1.22118			−0.610591	−	0.791946i	$-0.709068\pi$
$181$	−16.4293			−1.22118			−0.610591	+	0.791946i	$0.709068\pi$
$182$		0			0
$183$		−	10.9925i		−	0.812589i
$184$		0			0
$185$			20.2415i			1.48819i
$186$		0			0
$187$		−	3.66775i		−	0.268213i
$188$		0			0
$189$	−2.63144	+	0.274849i	−0.191409	+	0.0199923i
$190$		0			0
$191$		−	4.51368i		−	0.326599i	−0.986577	−	0.163299i	$-0.947786\pi$
$191$		−	4.51368i		−	0.326599i	0.986577	−	0.163299i	$-0.0522136\pi$
$192$		0			0
$193$	−26.5150			−1.90859			−0.954295	−	0.298868i	$-0.903391\pi$
$193$	−26.5150			−1.90859			−0.954295	+	0.298868i	$0.903391\pi$
$194$		0			0
$195$	−7.11944			−0.509834
$196$		0			0
$197$	6.43093			0.458185			0.229092	−	0.973405i	$-0.426424\pi$
$197$	6.43093			0.458185			0.229092	+	0.973405i	$0.426424\pi$
$198$		0			0
$199$	5.72596			0.405903			0.202951	−	0.979189i	$-0.434947\pi$
$199$	5.72596			0.405903			0.202951	+	0.979189i	$0.434947\pi$
$200$		0			0
$201$	4.99995			0.352669
$202$		0			0
$203$	0.228342	+	2.18618i	0.0160265	+	0.153439i
$204$		0			0
$205$			10.2749i			0.717633i
$206$		0			0
$207$	−3.98929	+	2.66187i	−0.277275	+	0.185013i
$208$		0			0
$209$			9.74808i			0.674289i
$210$		0			0
$211$	−13.1708			−0.906715			−0.453358	−	0.891329i	$-0.649774\pi$
$211$	−13.1708			−0.906715			−0.453358	+	0.891329i	$0.649774\pi$
$212$		0			0
$213$			3.32787i			0.228022i
$214$		0			0
$215$		−	3.27877i		−	0.223610i
$216$		0			0
$217$	−8.99382	+	0.939389i	−0.610541	+	0.0637699i
$218$		0			0
$219$	−10.8962			−0.736297
$220$		0			0
$221$			2.07661i			0.139688i
$222$		0			0
$223$		−	12.2201i		−	0.818320i	−0.912463	−	0.409160i	$-0.865822\pi$
$223$		−	12.2201i		−	0.818320i	0.912463	−	0.409160i	$-0.134178\pi$
$224$		0			0
$225$	−0.0182760			−0.00121840
$226$		0			0
$227$	18.0199			1.19603			0.598013	−	0.801487i	$-0.295957\pi$
$227$	18.0199			1.19603			0.598013	+	0.801487i	$0.295957\pi$
$228$		0			0
$229$	−21.7204			−1.43532			−0.717662	−	0.696392i	$-0.754788\pi$
$229$	−21.7204			−1.43532			−0.717662	+	0.696392i	$0.754788\pi$
$230$		0			0
$231$	1.54279	+	14.7709i	0.101508	+	0.971854i
$232$		0			0
$233$	17.4964			1.14623			0.573113	−	0.819477i	$-0.305736\pi$
$233$	17.4964			1.14623			0.573113	+	0.819477i	$0.305736\pi$
$234$		0			0
$235$			22.8516i			1.49068i
$236$		0			0
$237$	3.98034			0.258551
$238$		0			0
$239$	8.15537			0.527527			0.263763	−	0.964587i	$-0.415036\pi$
$239$	8.15537			0.527527			0.263763	+	0.964587i	$0.415036\pi$
$240$		0			0
$241$	−2.73995			−0.176496			−0.0882478	−	0.996099i	$-0.528127\pi$
$241$	−2.73995			−0.176496			−0.0882478	+	0.996099i	$0.528127\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−15.3426	+	3.24037i	−0.980203	+	0.207019i
$246$		0			0
$247$		−	5.51916i		−	0.351176i
$248$		0			0
$249$		−	10.5122i		−	0.666184i
$250$		0			0
$251$	23.8708			1.50671			0.753356	−	0.657613i	$-0.228434\pi$
$251$	23.8708			1.50671			0.753356	+	0.657613i	$0.228434\pi$
$252$		0			0
$253$	14.9418	+	22.3929i	0.939380	+	1.40783i
$254$		0			0
$255$			1.46374i			0.0916627i
$256$		0			0
$257$		−	31.5389i		−	1.96734i	−0.179969	−	0.983672i	$-0.557600\pi$
$257$		−	31.5389i		−	1.96734i	0.179969	−	0.983672i	$-0.442400\pi$
$258$		0			0
$259$	−23.7771	+	2.48348i	−1.47744	+	0.154316i
$260$		0			0
$261$	−0.830792			−0.0514247
$262$		0			0
$263$		−	7.31871i		−	0.451291i	−0.974209	−	0.225646i	$-0.927551\pi$
$263$		−	7.31871i		−	0.451291i	0.974209	−	0.225646i	$-0.0724491\pi$
$264$		0			0
$265$		−	1.14156i		−	0.0701252i
$266$		0			0
$267$		−	5.52664i		−	0.338225i
$268$		0			0
$269$		−	1.49678i		−	0.0912604i	−0.998958	−	0.0456302i	$-0.985470\pi$
$269$		−	1.49678i		−	0.0912604i	0.998958	−	0.0456302i	$-0.0145296\pi$
$270$		0			0
$271$		−	17.3967i		−	1.05678i	−0.849003	−	0.528389i	$-0.822796\pi$
$271$		−	17.3967i		−	1.05678i	0.849003	−	0.528389i	$-0.177204\pi$
$272$		0			0
$273$	−0.873499	−	8.36299i	−0.0528666	−	0.506151i
$274$		0			0
$275$			0.102588i			0.00618628i
$276$		0			0
$277$	3.86819			0.232417			0.116208	−	0.993225i	$-0.462926\pi$
$277$	3.86819			0.232417			0.116208	+	0.993225i	$0.462926\pi$
$278$		0			0
$279$		−	3.41784i		−	0.204621i
$280$		0			0
$281$			6.08796i			0.363177i	0.983375	+	0.181589i	$0.0581239\pi$
$281$			6.08796i			0.363177i	−0.983375	+	0.181589i	$0.941876\pi$
$282$		0			0
$283$	−9.11546			−0.541858			−0.270929	−	0.962599i	$-0.587331\pi$
$283$	−9.11546			−0.541858			−0.270929	+	0.962599i	$0.587331\pi$
$284$		0			0
$285$		−	3.89029i		−	0.230441i
$286$		0			0
$287$	−12.0697	+	1.26065i	−0.712449	+	0.0744140i
$288$		0			0
$289$	−16.5731			−0.974886
$290$		0			0
$291$			2.63063i			0.154211i
$292$		0			0
$293$	−10.7778			−0.629648			−0.314824	−	0.949150i	$-0.601946\pi$
$293$	−10.7778			−0.629648			−0.314824	+	0.949150i	$0.601946\pi$
$294$		0			0
$295$		−	3.22080i		−	0.187522i
$296$		0			0
$297$	−5.61325			−0.325714
$298$		0			0
$299$	−8.45972	−	12.6784i	−0.489238	−	0.733210i
$300$		0			0
$301$	3.85147	−	0.402279i	0.221995	−	0.0231870i
$302$		0			0
$303$	6.81860			0.391718
$304$		0			0
$305$	24.6249			1.41002
$306$		0			0
$307$			27.7678i			1.58479i	0.610008	+	0.792395i	$0.291166\pi$
$307$			27.7678i			1.58479i	−0.610008	+	0.792395i	$0.708834\pi$
$308$		0			0
$309$		−	5.63264i		−	0.320430i
$310$		0			0
$311$		−	13.8390i		−	0.784739i	−0.919808	−	0.392370i	$-0.871655\pi$
$311$		−	13.8390i		−	0.784739i	0.919808	−	0.392370i	$-0.128345\pi$
$312$		0			0
$313$	14.9909			0.847336			0.423668	−	0.905818i	$-0.360742\pi$
$313$	14.9909			0.847336			0.423668	+	0.905818i	$0.360742\pi$
$314$		0			0
$315$	−0.615703	−	5.89481i	−0.0346909	−	0.332135i
$316$		0			0
$317$	−3.51805			−0.197593			−0.0987967	−	0.995108i	$-0.531499\pi$
$317$	−3.51805			−0.197593			−0.0987967	+	0.995108i	$0.531499\pi$
$318$		0			0
$319$			4.66344i			0.261102i
$320$		0			0
$321$	−13.5267			−0.754986
$322$		0			0
$323$	−1.13472			−0.0631377
$324$		0			0
$325$		−	0.0580831i		−	0.00322187i
$326$		0			0
$327$	10.1826			0.563099
$328$		0			0
$329$	−26.8431	+	2.80372i	−1.47991	+	0.154574i
$330$		0			0
$331$	−25.2005			−1.38515			−0.692574	−	0.721347i	$-0.743523\pi$
$331$	−25.2005			−1.38515			−0.692574	+	0.721347i	$0.743523\pi$
$332$		0			0
$333$		−	9.03579i		−	0.495158i
$334$		0			0
$335$			11.2006i			0.611956i
$336$		0			0
$337$		−	0.245779i		−	0.0133884i	−0.999978	−	0.00669421i	$-0.997869\pi$
$337$		−	0.245779i		−	0.0133884i	0.999978	−	0.00669421i	$-0.00213085\pi$
$338$		0			0
$339$	−9.06577			−0.492385
$340$		0			0
$341$	−19.1852			−1.03894
$342$		0			0
$343$	−5.68877	−	17.6249i	−0.307165	−	0.951656i
$344$		0			0
$345$	−5.96300	−	8.93661i	−0.321037	−	0.481131i
$346$		0			0
$347$	−27.8907			−1.49725			−0.748627	−	0.662992i	$-0.769286\pi$
$347$	−27.8907			−1.49725			−0.748627	+	0.662992i	$0.769286\pi$
$348$		0			0
$349$		−	32.0469i		−	1.71543i	−0.514126	−	0.857714i	$-0.671884\pi$
$349$		−	32.0469i		−	1.71543i	0.514126	−	0.857714i	$-0.328116\pi$
$350$		0			0
$351$	3.17811			0.169635
$352$		0			0
$353$		−	11.4404i		−	0.608911i	−0.952527	−	0.304456i	$-0.901525\pi$
$353$		−	11.4404i		−	0.608911i	0.952527	−	0.304456i	$-0.0984745\pi$
$354$		0			0
$355$	−7.45493			−0.395667
$356$		0			0
$357$	−1.71941	+	0.179589i	−0.0910006	+	0.00950485i
$358$		0			0
$359$			5.48064i			0.289257i	0.989486	+	0.144628i	$0.0461987\pi$
$359$			5.48064i			0.289257i	−0.989486	+	0.144628i	$0.953801\pi$
$360$		0			0
$361$	−15.9842			−0.841271
$362$		0			0
$363$			20.5086i			1.07642i
$364$		0			0
$365$		−	24.4091i		−	1.27763i
$366$		0			0
$367$	−13.2362			−0.690924			−0.345462	−	0.938433i	$-0.612278\pi$
$367$	−13.2362			−0.690924			−0.345462	+	0.938433i	$0.612278\pi$
$368$		0			0
$369$		−	4.58672i		−	0.238775i
$370$		0			0
$371$	1.34095	−	0.140060i	0.0696187	−	0.00727154i
$372$		0			0
$373$			4.75577i			0.246245i	0.992392	+	0.123122i	$0.0392907\pi$
$373$			4.75577i			0.246245i	−0.992392	+	0.123122i	$0.960709\pi$
$374$		0			0
$375$			11.1598i			0.576290i
$376$		0			0
$377$		−	2.64034i		−	0.135985i
$378$		0			0
$379$			19.1647i			0.984426i	0.870475	+	0.492213i	$0.163812\pi$
$379$			19.1647i			0.984426i	−0.870475	+	0.492213i	$0.836188\pi$
$380$		0			0
$381$		−	12.8808i		−	0.659903i
$382$		0			0
$383$	29.3116			1.49775			0.748877	−	0.662709i	$-0.230594\pi$
$383$	29.3116			1.49775			0.748877	+	0.662709i	$0.230594\pi$
$384$		0			0
$385$	−33.0891	+	3.45609i	−1.68637	+	0.176139i
$386$		0			0
$387$			1.46364i			0.0744008i
$388$		0			0
$389$		−	9.59069i		−	0.486267i	−0.969993	−	0.243134i	$-0.921825\pi$
$389$		−	9.59069i		−	0.486267i	0.969993	−	0.243134i	$-0.0781753\pi$
$390$		0			0
$391$	−2.60664	+	1.73929i	−0.131823	+	0.0879599i
$392$		0			0
$393$	−16.9581			−0.855421
$394$		0			0
$395$			8.91656i			0.448641i
$396$		0			0
$397$		−	21.8074i		−	1.09448i	−0.836975	−	0.547242i	$-0.815678\pi$
$397$		−	21.8074i		−	1.09448i	0.836975	−	0.547242i	$-0.184322\pi$
$398$		0			0
$399$	4.56980	−	0.477308i	0.228776	−	0.0238953i
$400$		0			0
$401$		−	12.3511i		−	0.616784i	−0.951259	−	0.308392i	$-0.900209\pi$
$401$		−	12.3511i		−	0.616784i	0.951259	−	0.308392i	$-0.0997908\pi$
$402$		0			0
$403$	10.8623			0.541087
$404$		0			0
$405$	2.24015			0.111314
$406$		0			0
$407$	−50.7201			−2.51410
$408$		0			0
$409$			7.13110i			0.352610i	0.984336	+	0.176305i	$0.0564146\pi$
$409$			7.13110i			0.352610i	−0.984336	+	0.176305i	$0.943585\pi$
$410$		0			0
$411$	−0.839197			−0.0413945
$412$		0			0
$413$	3.78338	−	0.395167i	0.186168	−	0.0194449i
$414$		0			0
$415$	23.5489			1.15597
$416$		0			0
$417$	19.7845			0.968851
$418$		0			0
$419$	−31.2928			−1.52875			−0.764376	−	0.644770i	$-0.776953\pi$
$419$	−31.2928			−1.52875			−0.764376	+	0.644770i	$0.776953\pi$
$420$		0			0
$421$			31.5390i			1.53712i	0.639780	+	0.768558i	$0.279025\pi$
$421$			31.5390i			1.53712i	−0.639780	+	0.768558i	$0.720975\pi$
$422$		0			0
$423$		−	10.2009i		−	0.495987i
$424$		0			0
$425$	−0.0119417			−0.000579259
$426$		0			0
$427$	3.02127	+	28.9261i	0.146210	+	1.39983i
$428$		0			0
$429$		−	17.8395i		−	0.861299i
$430$		0			0
$431$		−	3.76315i		−	0.181265i	−0.995884	−	0.0906323i	$-0.971111\pi$
$431$		−	3.76315i		−	0.181265i	0.995884	−	0.0906323i	$-0.0288888\pi$
$432$		0			0
$433$	12.7812			0.614226			0.307113	−	0.951673i	$-0.400637\pi$
$433$	12.7812			0.614226			0.307113	+	0.951673i	$0.400637\pi$
$434$		0			0
$435$		−	1.86110i		−	0.0892328i
$436$		0			0
$437$	6.92788	−	4.62266i	0.331405	−	0.221132i
$438$		0			0
$439$		−	22.1397i		−	1.05667i	−0.849036	−	0.528334i	$-0.822817\pi$
$439$		−	22.1397i		−	1.05667i	0.849036	−	0.528334i	$-0.177183\pi$
$440$		0			0
$441$	6.84892	−	1.44649i	0.326139	−	0.0688807i
$442$		0			0
$443$	−23.6185			−1.12215			−0.561074	−	0.827766i	$-0.689612\pi$
$443$	−23.6185			−1.12215			−0.561074	+	0.827766i	$0.689612\pi$
$444$		0			0
$445$	12.3805			0.586892
$446$		0			0
$447$	−11.4198			−0.540136
$448$		0			0
$449$	18.5276			0.874372			0.437186	−	0.899371i	$-0.355975\pi$
$449$	18.5276			0.874372			0.437186	+	0.899371i	$0.355975\pi$
$450$		0			0
$451$	−25.7464			−1.21235
$452$		0			0
$453$			4.76752i			0.223998i
$454$		0			0
$455$	18.7344	−	1.95677i	0.878280	−	0.0917348i
$456$		0			0
$457$			32.4762i			1.51917i	0.650408	+	0.759585i	$0.274598\pi$
$457$			32.4762i			1.51917i	−0.650408	+	0.759585i	$0.725402\pi$
$458$		0			0
$459$		−	0.653410i		−	0.0304986i
$460$		0			0
$461$		−	20.8513i		−	0.971144i	−0.874197	−	0.485572i	$-0.838611\pi$
$461$		−	20.8513i		−	0.971144i	0.874197	−	0.485572i	$-0.161389\pi$
$462$		0			0
$463$	7.58614			0.352558			0.176279	−	0.984340i	$-0.443594\pi$
$463$	7.58614			0.352558			0.176279	+	0.984340i	$0.443594\pi$
$464$		0			0
$465$	7.65647			0.355060
$466$		0			0
$467$	−21.0981			−0.976302			−0.488151	−	0.872759i	$-0.662328\pi$
$467$	−21.0981			−0.976302			−0.488151	+	0.872759i	$0.662328\pi$
$468$		0			0
$469$	−13.1570	+	1.37423i	−0.607536	+	0.0634560i
$470$		0			0
$471$			0.602641i			0.0277682i
$472$		0			0
$473$	8.21576			0.377761
$474$		0			0
$475$	0.0317385			0.00145626
$476$		0			0
$477$			0.509589i			0.0233325i
$478$		0			0
$479$	18.6864			0.853803			0.426902	−	0.904298i	$-0.359605\pi$
$479$	18.6864			0.853803			0.426902	+	0.904298i	$0.359605\pi$
$480$		0			0
$481$	28.7167			1.30937
$482$		0			0
$483$	9.76595	−	8.10100i	0.444366	−	0.368608i
$484$		0			0
$485$	−5.89302			−0.267588
$486$		0			0
$487$	−10.0129			−0.453728			−0.226864	−	0.973926i	$-0.572847\pi$
$487$	−10.0129			−0.453728			−0.226864	+	0.973926i	$0.572847\pi$
$488$		0			0
$489$		−	5.65161i		−	0.255575i
$490$		0			0
$491$	−25.7119			−1.16036			−0.580181	−	0.814487i	$-0.697018\pi$
$491$	−25.7119			−1.16036			−0.580181	+	0.814487i	$0.697018\pi$
$492$		0			0
$493$	−0.542847			−0.0244486
$494$		0			0
$495$		−	12.5745i		−	0.565183i
$496$		0			0
$497$	−0.914661	−	8.75708i	−0.0410282	−	0.392809i
$498$		0			0
$499$	32.6571			1.46193			0.730965	−	0.682415i	$-0.239070\pi$
$499$	32.6571			1.46193			0.730965	+	0.682415i	$0.239070\pi$
$500$		0			0
$501$	5.35531			0.239258
$502$		0			0
$503$	−39.1932			−1.74754			−0.873769	−	0.486342i	$-0.838331\pi$
$503$	−39.1932			−1.74754			−0.873769	+	0.486342i	$0.838331\pi$
$504$		0			0
$505$			15.2747i			0.679715i
$506$		0			0
$507$		−	2.89963i		−	0.128777i
$508$		0			0
$509$		−	41.6800i		−	1.84743i	−0.383078	−	0.923716i	$-0.625136\pi$
$509$		−	41.6800i		−	1.84743i	0.383078	−	0.923716i	$-0.374864\pi$
$510$		0			0
$511$	28.6726	−	2.99481i	1.26840	−	0.132482i
$512$		0			0
$513$			1.73662i			0.0766736i
$514$		0			0
$515$	12.6180			0.556014
$516$		0			0
$517$	−57.2604			−2.51831
$518$		0			0
$519$	−11.7452			−0.515557
$520$		0			0
$521$	11.6121			0.508736			0.254368	−	0.967108i	$-0.418133\pi$
$521$	11.6121			0.508736			0.254368	+	0.967108i	$0.418133\pi$
$522$		0			0
$523$	34.3129			1.50040			0.750200	−	0.661211i	$-0.229957\pi$
$523$	34.3129			1.50040			0.750200	+	0.661211i	$0.229957\pi$
$524$		0			0
$525$	0.0480922	−	0.00502314i	0.00209891	−	0.000219228i
$526$		0			0
$527$		−	2.23325i		−	0.0972818i
$528$		0			0
$529$	8.82885	−	21.2380i	0.383863	−	0.923390i
$530$		0			0
$531$			1.43776i			0.0623935i
$532$		0			0
$533$	14.5771			0.631403
$534$		0			0
$535$		−	30.3018i		−	1.31006i
$536$		0			0
$537$		−	21.3113i		−	0.919652i
$538$		0			0
$539$	−8.11953	−	38.4447i	−0.349733	−	1.65593i
$540$		0			0
$541$	18.8567			0.810712			0.405356	−	0.914159i	$-0.367148\pi$
$541$	18.8567			0.810712			0.405356	+	0.914159i	$0.367148\pi$
$542$		0			0
$543$			16.4293i			0.705050i
$544$		0			0
$545$			22.8106i			0.977097i
$546$		0			0
$547$	11.0517			0.472536			0.236268	−	0.971688i	$-0.424076\pi$
$547$	11.0517			0.472536			0.236268	+	0.971688i	$0.424076\pi$
$548$		0			0
$549$	−10.9925			−0.469148
$550$		0			0
$551$	1.44277			0.0614640
$552$		0			0
$553$	−10.4740	+	1.09399i	−0.445400	+	0.0465213i
$554$		0			0
$555$	20.2415			0.859205
$556$		0			0
$557$		−	26.1347i		−	1.10736i	−0.832728	−	0.553682i	$-0.813222\pi$
$557$		−	26.1347i		−	1.10736i	0.832728	−	0.553682i	$-0.186778\pi$
$558$		0			0
$559$	−4.65159			−0.196742
$560$		0			0
$561$	−3.66775			−0.154853
$562$		0			0
$563$	1.98118			0.0834967			0.0417484	−	0.999128i	$-0.486707\pi$
$563$	1.98118			0.0834967			0.0417484	+	0.999128i	$0.486707\pi$
$564$		0			0
$565$		−	20.3087i		−	0.854393i
$566$		0			0
$567$	0.274849	+	2.63144i	0.0115426	+	0.110510i
$568$		0			0
$569$			32.2528i			1.35211i	0.736853	+	0.676053i	$0.236311\pi$
$569$			32.2528i			1.35211i	−0.736853	+	0.676053i	$0.763689\pi$
$570$		0			0
$571$			1.82114i			0.0762125i	0.999274	+	0.0381062i	$0.0121325\pi$
$571$			1.82114i			0.0762125i	−0.999274	+	0.0381062i	$0.987867\pi$
$572$		0			0
$573$	−4.51368			−0.188562
$574$		0			0
$575$	0.0729083	−	0.0486485i	0.00304049	−	0.00202878i
$576$		0			0
$577$		−	32.5266i		−	1.35410i	−0.735937	−	0.677050i	$-0.763258\pi$
$577$		−	32.5266i		−	1.35410i	0.735937	−	0.677050i	$-0.236742\pi$
$578$		0			0
$579$			26.5150i			1.10192i
$580$		0			0
$581$	2.88927	+	27.6622i	0.119867	+	1.14762i
$582$		0			0
$583$	2.86045			0.118468
$584$		0			0
$585$			7.11944i			0.294353i
$586$		0			0
$587$			3.23188i			0.133394i	0.997773	+	0.0666970i	$0.0212461\pi$
$587$			3.23188i			0.133394i	−0.997773	+	0.0666970i	$0.978754\pi$
$588$		0			0
$589$			5.93548i			0.244567i
$590$		0			0
$591$		−	6.43093i		−	0.264533i
$592$		0			0
$593$			34.6271i			1.42197i	0.703209	+	0.710983i	$0.251750\pi$
$593$			34.6271i			1.42197i	−0.703209	+	0.710983i	$0.748250\pi$
$594$		0			0
$595$	−0.402306	−	3.85173i	−0.0164929	−	0.157906i
$596$		0			0
$597$		−	5.72596i		−	0.234348i
$598$		0			0
$599$	7.68972			0.314193			0.157097	−	0.987583i	$-0.449787\pi$
$599$	7.68972			0.314193			0.157097	+	0.987583i	$0.449787\pi$
$600$		0			0
$601$			23.9856i			0.978392i	0.872174	+	0.489196i	$0.162710\pi$
$601$			23.9856i			0.978392i	−0.872174	+	0.489196i	$0.837290\pi$
$602$		0			0
$603$		−	4.99995i		−	0.203614i
$604$		0			0
$605$	−45.9423			−1.86782
$606$		0			0
$607$			29.0807i			1.18035i	0.807276	+	0.590174i	$0.200941\pi$
$607$			29.0807i			1.18035i	−0.807276	+	0.590174i	$0.799059\pi$
$608$		0			0
$609$	2.18618	−	0.228342i	0.0885883	−	0.00925289i
$610$		0			0
$611$	32.4197			1.31156
$612$		0			0
$613$		−	28.6785i		−	1.15831i	−0.815216	−	0.579157i	$-0.803382\pi$
$613$		−	28.6785i		−	1.15831i	0.815216	−	0.579157i	$-0.196618\pi$
$614$		0			0
$615$	10.2749			0.414325
$616$		0			0
$617$		−	35.3324i		−	1.42243i	−0.702976	−	0.711214i	$-0.748146\pi$
$617$		−	35.3324i		−	1.42243i	0.702976	−	0.711214i	$-0.251854\pi$
$618$		0			0
$619$	0.724772			0.0291310			0.0145655	−	0.999894i	$-0.495363\pi$
$619$	0.724772			0.0291310			0.0145655	+	0.999894i	$0.495363\pi$
$620$		0			0
$621$	2.66187	+	3.98929i	0.106817	+	0.160085i
$622$		0			0
$623$	1.51899	+	14.5430i	0.0608571	+	0.582653i
$624$		0			0
$625$	−25.0910			−1.00364
$626$		0			0
$627$	9.74808			0.389301
$628$		0			0
$629$		−	5.90407i		−	0.235411i
$630$		0			0
$631$		−	9.89018i		−	0.393722i	−0.980431	−	0.196861i	$-0.936925\pi$
$631$		−	9.89018i		−	0.393722i	0.980431	−	0.196861i	$-0.0630747\pi$
$632$		0			0
$633$			13.1708i			0.523492i
$634$		0			0
$635$	28.8549			1.14507
$636$		0			0
$637$	4.59711	+	21.7666i	0.182144	+	0.862424i
$638$		0			0
$639$	3.32787			0.131649
$640$		0			0
$641$			27.6961i			1.09393i	0.837156	+	0.546964i	$0.184217\pi$
$641$			27.6961i			1.09393i	−0.837156	+	0.546964i	$0.815783\pi$
$642$		0			0
$643$	−37.3043			−1.47114			−0.735569	−	0.677450i	$-0.763085\pi$
$643$	−37.3043			−1.47114			−0.735569	+	0.677450i	$0.763085\pi$
$644$		0			0
$645$	−3.27877			−0.129101
$646$		0			0
$647$			2.93766i			0.115491i	0.998331	+	0.0577456i	$0.0183912\pi$
$647$			2.93766i			0.115491i	−0.998331	+	0.0577456i	$0.981609\pi$
$648$		0			0
$649$	8.07051			0.316795
$650$		0			0
$651$	0.939389	+	8.99382i	0.0368175	+	0.352496i
$652$		0			0
$653$	10.1339			0.396572			0.198286	−	0.980144i	$-0.436463\pi$
$653$	10.1339			0.396572			0.198286	+	0.980144i	$0.436463\pi$
$654$		0			0
$655$		−	37.9886i		−	1.48434i
$656$		0			0
$657$			10.8962i			0.425101i
$658$		0			0
$659$			38.5937i			1.50340i	0.659507	+	0.751698i	$0.270765\pi$
$659$			38.5937i			1.50340i	−0.659507	+	0.751698i	$0.729235\pi$
$660$		0			0
$661$	−41.0522			−1.59674			−0.798372	−	0.602164i	$-0.794305\pi$
$661$	−41.0522			−1.59674			−0.798372	+	0.602164i	$0.794305\pi$
$662$		0			0
$663$	2.07661			0.0806487
$664$		0			0
$665$	1.06924	+	10.2371i	0.0414634	+	0.396976i
$666$		0			0
$667$	3.31427	−	2.21146i	0.128329	−	0.0856282i
$668$		0			0
$669$	−12.2201			−0.472457
$670$		0			0
$671$			61.7036i			2.38204i
$672$		0			0
$673$	−10.5764			−0.407688			−0.203844	−	0.979003i	$-0.565344\pi$
$673$	−10.5764			−0.407688			−0.203844	+	0.979003i	$0.565344\pi$
$674$		0			0
$675$			0.0182760i			0.000703444i
$676$		0			0
$677$	−19.5142			−0.749991			−0.374995	−	0.927027i	$-0.622356\pi$
$677$	−19.5142			−0.749991			−0.374995	+	0.927027i	$0.622356\pi$
$678$		0			0
$679$	−0.723027	−	6.92235i	−0.0277472	−	0.265655i
$680$		0			0
$681$		−	18.0199i		−	0.690526i
$682$		0			0
$683$	−27.3751			−1.04748			−0.523740	−	0.851878i	$-0.675464\pi$
$683$	−27.3751			−1.04748			−0.523740	+	0.851878i	$0.675464\pi$
$684$		0			0
$685$		−	1.87993i		−	0.0718283i
$686$		0			0
$687$			21.7204i			0.828684i
$688$		0			0
$689$	−1.61953			−0.0616991
$690$		0			0
$691$		−	39.1447i		−	1.48913i	−0.667547	−	0.744567i	$-0.732656\pi$
$691$		−	39.1447i		−	1.48913i	0.667547	−	0.744567i	$-0.267344\pi$
$692$		0			0
$693$	14.7709	−	1.54279i	0.561100	−	0.0586059i
$694$		0			0
$695$			44.3203i			1.68116i
$696$		0			0
$697$		−	2.99700i		−	0.113520i
$698$		0			0
$699$		−	17.4964i		−	0.661773i
$700$		0			0
$701$		−	33.6130i		−	1.26955i	−0.772698	−	0.634773i	$-0.781093\pi$
$701$		−	33.6130i		−	1.26955i	0.772698	−	0.634773i	$-0.218907\pi$
$702$		0			0
$703$			15.6917i			0.591825i
$704$		0			0
$705$	22.8516			0.860642
$706$		0			0
$707$	−17.9427	+	1.87408i	−0.674805	+	0.0704822i
$708$		0			0
$709$			1.30941i			0.0491761i	0.999698	+	0.0245880i	$0.00782740\pi$
$709$			1.30941i			0.0491761i	−0.999698	+	0.0245880i	$0.992173\pi$
$710$		0			0
$711$		−	3.98034i		−	0.149274i
$712$		0			0
$713$	9.09785	+	13.6347i	0.340717	+	0.510625i
$714$		0			0
$715$	39.9632			1.49454
$716$		0			0
$717$		−	8.15537i		−	0.304568i
$718$		0			0
$719$		−	24.8634i		−	0.927249i	−0.886032	−	0.463625i	$-0.846549\pi$
$719$		−	24.8634i		−	0.927249i	0.886032	−	0.463625i	$-0.153451\pi$
$720$		0			0
$721$	1.54812	+	14.8219i	0.0576552	+	0.551998i
$722$		0			0
$723$			2.73995i			0.101900i
$724$		0			0
$725$	0.0151836			0.000563903
$726$		0			0
$727$	29.2261			1.08393			0.541967	−	0.840399i	$-0.317680\pi$
$727$	29.2261			1.08393			0.541967	+	0.840399i	$0.317680\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			0.956355i			0.0353721i
$732$		0			0
$733$	23.9065			0.883008			0.441504	−	0.897259i	$-0.354445\pi$
$733$	23.9065			0.883008			0.441504	+	0.897259i	$0.354445\pi$
$734$		0			0
$735$	3.24037	+	15.3426i	0.119523	+	0.565921i
$736$		0			0
$737$	−28.0659			−1.03382
$738$		0			0
$739$	−45.6322			−1.67861			−0.839304	−	0.543663i	$-0.817037\pi$
$739$	−45.6322			−1.67861			−0.839304	+	0.543663i	$0.817037\pi$
$740$		0			0
$741$	−5.51916			−0.202752
$742$		0			0
$743$		−	1.91418i		−	0.0702246i	−0.999383	−	0.0351123i	$-0.988821\pi$
$743$		−	1.91418i		−	0.0702246i	0.999383	−	0.0351123i	$-0.0111789\pi$
$744$		0			0
$745$		−	25.5820i		−	0.937252i
$746$		0			0
$747$	−10.5122			−0.384622
$748$		0			0
$749$	35.5946	−	3.71779i	1.30060	−	0.135845i
$750$		0			0
$751$		−	10.6600i		−	0.388990i	−0.980904	−	0.194495i	$-0.937693\pi$
$751$		−	10.6600i		−	0.388990i	0.980904	−	0.194495i	$-0.0623067\pi$
$752$		0			0
$753$		−	23.8708i		−	0.869901i
$754$		0			0
$755$	−10.6800			−0.388684
$756$		0			0
$757$			38.9437i			1.41543i	0.706497	+	0.707716i	$0.250274\pi$
$757$			38.9437i			1.41543i	−0.706497	+	0.707716i	$0.749726\pi$
$758$		0			0
$759$	22.3929	−	14.9418i	0.812810	−	0.542352i
$760$		0			0
$761$		−	18.1360i		−	0.657428i	−0.944430	−	0.328714i	$-0.893385\pi$
$761$		−	18.1360i		−	0.657428i	0.944430	−	0.328714i	$-0.106615\pi$
$762$		0			0
$763$	−26.7949	+	2.79868i	−0.970039	+	0.101319i
$764$		0			0
$765$	1.46374			0.0529215
$766$		0			0
$767$	−4.56936			−0.164990
$768$		0			0
$769$	12.2937			0.443321			0.221660	−	0.975124i	$-0.428852\pi$
$769$	12.2937			0.443321			0.221660	+	0.975124i	$0.428852\pi$
$770$		0			0
$771$	−31.5389			−1.13585
$772$		0			0
$773$	5.23094			0.188144			0.0940718	−	0.995565i	$-0.470012\pi$
$773$	5.23094			0.188144			0.0940718	+	0.995565i	$0.470012\pi$
$774$		0			0
$775$			0.0624645i			0.00224379i
$776$		0			0
$777$	2.48348	+	23.7771i	0.0890942	+	0.852999i
$778$		0			0
$779$			7.96538i			0.285389i
$780$		0			0
$781$		−	18.6802i		−	0.668429i
$782$		0			0
$783$			0.830792i			0.0296901i
$784$		0			0
$785$	−1.35001			−0.0481838
$786$		0			0
$787$	47.3332			1.68725			0.843623	−	0.536936i	$-0.180418\pi$
$787$	47.3332			1.68725			0.843623	+	0.536936i	$0.180418\pi$
$788$		0			0
$789$	−7.31871			−0.260553
$790$		0			0
$791$	23.8560	−	2.49172i	0.848222	−	0.0885952i
$792$		0			0
$793$		−	34.9353i		−	1.24059i
$794$		0			0
$795$	−1.14156			−0.0404868
$796$		0			0
$797$	37.1835			1.31711			0.658554	−	0.752534i	$-0.271168\pi$
$797$	37.1835			1.31711			0.658554	+	0.752534i	$0.271168\pi$
$798$		0			0
$799$		−	6.66539i		−	0.235805i
$800$		0			0
$801$	−5.52664			−0.195274
$802$		0			0
$803$	61.1630			2.15840
$804$		0			0
$805$	18.1475	+	21.8772i	0.639614	+	0.771070i
$806$		0			0
$807$	−1.49678			−0.0526892
$808$		0			0
$809$	21.0743			0.740934			0.370467	−	0.928846i	$-0.379198\pi$
$809$	21.0743			0.740934			0.370467	+	0.928846i	$0.379198\pi$
$810$		0			0
$811$			10.9326i			0.383896i	0.981405	+	0.191948i	$0.0614805\pi$
$811$			10.9326i			0.383896i	−0.981405	+	0.191948i	$0.938520\pi$
$812$		0			0
$813$	−17.3967			−0.610131
$814$		0			0
$815$	12.6605			0.443477
$816$		0			0
$817$		−	2.54178i		−	0.0889256i
$818$		0			0
$819$	−8.36299	+	0.873499i	−0.292226	+	0.0305225i
$820$		0			0
$821$	−36.1552			−1.26183			−0.630913	−	0.775853i	$-0.717320\pi$
$821$	−36.1552			−1.26183			−0.630913	+	0.775853i	$0.717320\pi$
$822$		0			0
$823$	1.08293			0.0377486			0.0188743	−	0.999822i	$-0.493992\pi$
$823$	1.08293			0.0377486			0.0188743	+	0.999822i	$0.493992\pi$
$824$		0			0
$825$	0.102588			0.00357165
$826$		0			0
$827$			37.9812i			1.32074i	0.750942	+	0.660368i	$0.229600\pi$
$827$			37.9812i			1.32074i	−0.750942	+	0.660368i	$0.770400\pi$
$828$		0			0
$829$		−	36.7573i		−	1.27664i	−0.769773	−	0.638318i	$-0.779631\pi$
$829$		−	36.7573i		−	1.27664i	0.769773	−	0.638318i	$-0.220369\pi$
$830$		0			0
$831$		−	3.86819i		−	0.134186i
$832$		0			0
$833$	4.47515	−	0.945153i	0.155055	−	0.0327476i
$834$		0			0
$835$			11.9967i			0.415163i
$836$		0			0
$837$	−3.41784			−0.118138
$838$		0			0
$839$	−39.0798			−1.34918			−0.674591	−	0.738191i	$-0.735680\pi$
$839$	−39.0798			−1.34918			−0.674591	+	0.738191i	$0.735680\pi$
$840$		0			0
$841$	−28.3098			−0.976199
$842$		0			0
$843$	6.08796			0.209681
$844$		0			0
$845$	6.49562			0.223456
$846$		0			0
$847$	−5.63675	−	53.9670i	−0.193681	−	1.85433i
$848$		0			0
$849$			9.11546i			0.312842i
$850$		0			0
$851$	24.0521	+	36.0464i	0.824497	+	1.23565i
$852$		0			0
$853$			2.72542i			0.0933166i	0.998911	+	0.0466583i	$0.0148572\pi$
$853$			2.72542i			0.0933166i	−0.998911	+	0.0466583i	$0.985143\pi$
$854$		0			0
$855$	−3.89029			−0.133045
$856$		0			0
$857$		−	8.51721i		−	0.290942i	−0.989362	−	0.145471i	$-0.953530\pi$
$857$		−	8.51721i		−	0.290942i	0.989362	−	0.145471i	$-0.0464698\pi$
$858$		0			0
$859$		−	18.2992i		−	0.624359i	−0.950023	−	0.312180i	$-0.898941\pi$
$859$		−	18.2992i		−	0.624359i	0.950023	−	0.312180i	$-0.101059\pi$
$860$		0			0
$861$	1.26065	+	12.0697i	0.0429629	+	0.411333i
$862$		0			0
$863$	1.32017			0.0449390			0.0224695	−	0.999748i	$-0.492847\pi$
$863$	1.32017			0.0449390			0.0224695	+	0.999748i	$0.492847\pi$
$864$		0			0
$865$		−	26.3110i		−	0.894601i
$866$		0			0
$867$			16.5731i			0.562850i
$868$		0			0
$869$	−22.3426			−0.757922
$870$		0			0
$871$	15.8904			0.538425
$872$		0			0
$873$	2.63063			0.0890335
$874$		0			0
$875$	−3.06726	−	29.3663i	−0.103692	−	0.992764i
$876$		0			0
$877$	26.3171			0.888666			0.444333	−	0.895862i	$-0.353441\pi$
$877$	26.3171			0.888666			0.444333	+	0.895862i	$0.353441\pi$
$878$		0			0
$879$			10.7778i			0.363528i
$880$		0			0
$881$	55.9859			1.88621			0.943107	−	0.332489i	$-0.107889\pi$
$881$	55.9859			1.88621			0.943107	+	0.332489i	$0.107889\pi$
$882$		0			0
$883$	−17.9707			−0.604764			−0.302382	−	0.953187i	$-0.597782\pi$
$883$	−17.9707			−0.604764			−0.302382	+	0.953187i	$0.597782\pi$
$884$		0			0
$885$	−3.22080			−0.108266
$886$		0			0
$887$			40.1597i			1.34843i	0.738534	+	0.674216i	$0.235518\pi$
$887$			40.1597i			1.34843i	−0.738534	+	0.674216i	$0.764482\pi$
$888$		0			0
$889$	3.54027	+	33.8950i	0.118737	+	1.13680i
$890$		0			0
$891$			5.61325i			0.188051i
$892$		0			0
$893$			17.7151i			0.592815i
$894$		0			0
$895$	47.7406			1.59579
$896$		0			0
$897$	−12.6784	+	8.45972i	−0.423319	+	0.282462i
$898$		0			0
$899$			2.83951i			0.0947030i
$900$		0			0
$901$			0.332970i			0.0110928i
$902$		0			0
$903$	−0.402279	−	3.85147i	−0.0133870	−	0.128169i
$904$		0			0
$905$	−36.8042			−1.22341
$906$		0			0
$907$		−	37.4054i		−	1.24203i	−0.783800	−	0.621013i	$-0.786722\pi$
$907$		−	37.4054i		−	1.24203i	0.783800	−	0.621013i	$-0.213278\pi$
$908$		0			0
$909$		−	6.81860i		−	0.226159i
$910$		0			0
$911$			0.712490i			0.0236059i	0.999930	+	0.0118029i	$0.00375708\pi$
$911$			0.712490i			0.0236059i	−0.999930	+	0.0118029i	$0.996243\pi$
$912$		0			0
$913$			59.0076i			1.95287i
$914$		0			0
$915$		−	24.6249i		−	0.814073i
$916$		0			0
$917$	44.6241	−	4.66090i	1.47362	−	0.153917i
$918$		0			0
$919$		−	35.7956i		−	1.18079i	−0.807115	−	0.590394i	$-0.798972\pi$
$919$		−	35.7956i		−	1.18079i	0.807115	−	0.590394i	$-0.201028\pi$
$920$		0			0
$921$	27.7678			0.914979
$922$		0			0
$923$			10.5763i			0.348124i
$924$		0			0
$925$			0.165138i			0.00542971i
$926$		0			0
$927$	−5.63264			−0.185000
$928$		0			0
$929$		−	53.2590i		−	1.74737i	−0.486492	−	0.873685i	$-0.661724\pi$
$929$		−	53.2590i		−	1.74737i	0.486492	−	0.873685i	$-0.338276\pi$
$930$		0			0
$931$	−11.8940	+	2.51201i	−0.389809	+	0.0823278i
$932$		0			0
$933$	−13.8390			−0.453069
$934$		0			0
$935$		−	8.21632i		−	0.268702i
$936$		0			0
$937$	52.2739			1.70771			0.853857	−	0.520508i	$-0.174258\pi$
$937$	52.2739			1.70771			0.853857	+	0.520508i	$0.174258\pi$
$938$		0			0
$939$		−	14.9909i		−	0.489210i
$940$		0			0
$941$	6.17579			0.201325			0.100662	−	0.994921i	$-0.467904\pi$
$941$	6.17579			0.201325			0.100662	+	0.994921i	$0.467904\pi$
$942$		0			0
$943$	12.2093	+	18.2977i	0.397588	+	0.595856i
$944$		0			0
$945$	−5.89481	+	0.615703i	−0.191758	+	0.0200288i
$946$		0			0
$947$	3.23532			0.105134			0.0525670	−	0.998617i	$-0.483260\pi$
$947$	3.23532			0.105134			0.0525670	+	0.998617i	$0.483260\pi$
$948$		0			0
$949$	−34.6293			−1.12411
$950$		0			0
$951$			3.51805i			0.114081i
$952$		0			0
$953$		−	28.3571i		−	0.918577i	−0.888287	−	0.459289i	$-0.848104\pi$
$953$		−	28.3571i		−	0.918577i	0.888287	−	0.459289i	$-0.151896\pi$
$954$		0			0
$955$		−	10.1113i		−	0.327195i
$956$		0			0
$957$	4.66344			0.150748
$958$		0			0
$959$	2.20829	−	0.230652i	0.0713095	−	0.00744815i
$960$		0			0
$961$	19.3184			0.623174
$962$		0			0
$963$			13.5267i			0.435891i
$964$		0			0
$965$	−59.3975			−1.91207
$966$		0			0
$967$	44.6691			1.43646			0.718231	−	0.695805i	$-0.244952\pi$
$967$	44.6691			1.43646			0.718231	+	0.695805i	$0.244952\pi$
$968$		0			0
$969$			1.13472i			0.0364526i
$970$		0			0
$971$	25.5905			0.821239			0.410619	−	0.911807i	$-0.365313\pi$
$971$	25.5905			0.821239			0.410619	+	0.911807i	$0.365313\pi$
$972$		0			0
$973$	−52.0617	+	5.43775i	−1.66902	+	0.174326i
$974$		0			0
$975$	−0.0580831			−0.00186015
$976$		0			0
$977$			14.6184i			0.467683i	0.972275	+	0.233842i	$0.0751298\pi$
$977$			14.6184i			0.467683i	−0.972275	+	0.233842i	$0.924870\pi$
$978$		0			0
$979$			31.0224i			0.991481i
$980$		0			0
$981$		−	10.1826i		−	0.325105i
$982$		0			0
$983$	−0.237973			−0.00759016			−0.00379508	−	0.999993i	$-0.501208\pi$
$983$	−0.237973			−0.00759016			−0.00379508	+	0.999993i	$0.501208\pi$
$984$		0			0
$985$	14.4062			0.459021
$986$		0			0
$987$	2.80372	+	26.8431i	0.0892432	+	0.854426i
$988$		0			0
$989$	−3.89602	−	5.83887i	−0.123886	−	0.185665i
$990$		0			0
$991$	−39.8227			−1.26501			−0.632505	−	0.774556i	$-0.717973\pi$
$991$	−39.8227			−1.26501			−0.632505	+	0.774556i	$0.717973\pi$
$992$		0			0
$993$			25.2005i			0.799715i
$994$		0			0
$995$	12.8270			0.406644
$996$		0			0
$997$			60.7737i			1.92472i	0.271774	+	0.962361i	$0.412390\pi$
$997$			60.7737i			1.92472i	−0.271774	+	0.962361i	$0.587610\pi$
$998$		0			0
$999$	−9.03579			−0.285880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.30	yes	32
3.2	odd	2		5796.2.k.d.5473.7		32
7.6	odd	2	inner	1932.2.k.a.1609.31	yes	32
21.20	even	2		5796.2.k.d.5473.25		32
23.22	odd	2	inner	1932.2.k.a.1609.29	✓	32
69.68	even	2		5796.2.k.d.5473.26		32
161.160	even	2	inner	1932.2.k.a.1609.32	yes	32
483.482	odd	2		5796.2.k.d.5473.8		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.29	✓	32	23.22	odd	2	inner
1932.2.k.a.1609.30	yes	32	1.1	even	1	trivial
1932.2.k.a.1609.31	yes	32	7.6	odd	2	inner
1932.2.k.a.1609.32	yes	32	161.160	even	2	inner
5796.2.k.d.5473.7		32	3.2	odd	2
5796.2.k.d.5473.8		32	483.482	odd	2
5796.2.k.d.5473.25		32	21.20	even	2
5796.2.k.d.5473.26		32	69.68	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.24015 q^{5} +(0.274849 + 2.63144i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.24015 q^{5} +(0.274849 + 2.63144i) q^{7} -1.00000 q^{9} +5.61325i q^{11} -3.17811i q^{13} -2.24015i q^{15} -0.653410 q^{17} +1.73662 q^{19} +(2.63144 - 0.274849i) q^{21} +(3.98929 - 2.66187i) q^{23} +0.0182760 q^{25} +1.00000i q^{27} +0.830792 q^{29} +3.41784i q^{31} +5.61325 q^{33} +(0.615703 + 5.89481i) q^{35} +9.03579i q^{37} -3.17811 q^{39} +4.58672i q^{41} -1.46364i q^{43} -2.24015 q^{45} +10.2009i q^{47} +(-6.84892 + 1.44649i) q^{49} +0.653410i q^{51} -0.509589i q^{53} +12.5745i q^{55} -1.73662i q^{57} -1.43776i q^{59} +10.9925 q^{61} +(-0.274849 - 2.63144i) q^{63} -7.11944i q^{65} +4.99995i q^{67} +(-2.66187 - 3.98929i) q^{69} -3.32787 q^{71} -10.8962i q^{73} -0.0182760i q^{75} +(-14.7709 + 1.54279i) q^{77} +3.98034i q^{79} +1.00000 q^{81} +10.5122 q^{83} -1.46374 q^{85} -0.830792i q^{87} +5.52664 q^{89} +(8.36299 - 0.873499i) q^{91} +3.41784 q^{93} +3.89029 q^{95} -2.63063 q^{97} -5.61325i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 32 * q - 32 * q^9	\( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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