LMFDB - Embedded newform 2340.2.u.f.73.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(73,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.73"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.u (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,6,0,-8,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2340.73
Dual form			2340.2.u.f.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +(0.381966 - 0.381966i) q^{11} +(-2.00000 - 3.00000i) q^{13} +(-2.23607 - 2.23607i) q^{17} +(-0.854102 + 0.854102i) q^{19} +(-5.61803 + 5.61803i) q^{23} -5.00000 q^{25} -0.763932i q^{29} +(-0.854102 - 0.854102i) q^{31} +3.70820i q^{37} +(-8.23607 - 8.23607i) q^{41} +(2.85410 - 2.85410i) q^{43} +8.94427i q^{47} +7.00000 q^{49} +(7.47214 + 7.47214i) q^{53} +(-0.854102 - 0.854102i) q^{55} +(5.61803 + 5.61803i) q^{59} -7.70820 q^{61} +(-6.70820 + 4.47214i) q^{65} -13.7082 q^{67} +(-0.381966 - 0.381966i) q^{71} -1.70820 q^{73} -9.70820i q^{79} +1.52786i q^{83} +(-5.00000 + 5.00000i) q^{85} +(-2.23607 - 2.23607i) q^{89} +(1.90983 + 1.90983i) q^{95} -11.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{11} - 8 q^{13} + 10 q^{19} - 18 q^{23} - 20 q^{25} + 10 q^{31} - 24 q^{41} - 2 q^{43} + 28 q^{49} + 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{67} - 6 q^{71} + 20 q^{73} - 20 q^{85}+ \cdots + 8 q^{97}+O(q^{100}) $ 4 * q + 6 * q^11 - 8 * q^13 + 10 * q^19 - 18 * q^23 - 20 * q^25 + 10 * q^31 - 24 * q^41 - 2 * q^43 + 28 * q^49 + 12 * q^53 + 10 * q^55 + 18 * q^59 - 4 * q^61 - 28 * q^67 - 6 * q^71 + 20 * q^73 - 20 * q^85 + 30 * q^95 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.381966	−	0.381966i	0.115167	−	0.115167i	−0.647175	−	0.762342i	$-0.724050\pi$
$11$	0.381966	−	0.381966i	0.115167	−	0.115167i	0.762342	+	0.647175i	$0.224050\pi$
$12$		0			0
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.23607	−	2.23607i	−0.542326	−	0.542326i	0.381884	−	0.924210i	$-0.375275\pi$
$17$	−2.23607	−	2.23607i	−0.542326	−	0.542326i	−0.924210	+	0.381884i	$0.875275\pi$
$18$		0			0
$19$	−0.854102	+	0.854102i	−0.195944	+	0.195944i	−0.798259	−	0.602314i	$-0.794245\pi$
$19$	−0.854102	+	0.854102i	−0.195944	+	0.195944i	0.602314	+	0.798259i	$0.294245\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.61803	+	5.61803i	−1.17144	+	1.17144i	−0.189575	+	0.981866i	$0.560711\pi$
$23$	−5.61803	+	5.61803i	−1.17144	+	1.17144i	−0.981866	+	0.189575i	$0.939289\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	0.763932i		−	0.141859i	−0.997481	−	0.0709293i	$-0.977404\pi$
$29$		−	0.763932i		−	0.141859i	0.997481	−	0.0709293i	$-0.0225965\pi$
$30$		0			0
$31$	−0.854102	−	0.854102i	−0.153401	−	0.153401i	0.626234	−	0.779635i	$-0.284595\pi$
$31$	−0.854102	−	0.854102i	−0.153401	−	0.153401i	−0.779635	+	0.626234i	$0.784595\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			3.70820i			0.609625i	0.952412	+	0.304812i	$0.0985938\pi$
$37$			3.70820i			0.609625i	−0.952412	+	0.304812i	$0.901406\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.23607	−	8.23607i	−1.28626	−	1.28626i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−8.23607	−	8.23607i	−1.28626	−	1.28626i	−0.349215	−	0.937043i	$-0.613552\pi$
$42$		0			0
$43$	2.85410	−	2.85410i	0.435246	−	0.435246i	−0.455162	−	0.890409i	$-0.650419\pi$
$43$	2.85410	−	2.85410i	0.435246	−	0.435246i	0.890409	+	0.455162i	$0.150419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.94427i			1.30466i	0.757937	+	0.652328i	$0.226208\pi$
$47$			8.94427i			1.30466i	−0.757937	+	0.652328i	$0.773792\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.47214	+	7.47214i	1.02638	+	1.02638i	0.999643	+	0.0267342i	$0.00851078\pi$
$53$	7.47214	+	7.47214i	1.02638	+	1.02638i	0.0267342	+	0.999643i	$0.491489\pi$
$54$		0			0
$55$	−0.854102	−	0.854102i	−0.115167	−	0.115167i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.61803	+	5.61803i	0.731406	+	0.731406i	0.970898	−	0.239492i	$-0.0769810\pi$
$59$	5.61803	+	5.61803i	0.731406	+	0.731406i	−0.239492	+	0.970898i	$0.576981\pi$
$60$		0			0
$61$	−7.70820			−0.986934			−0.493467	−	0.869764i	$-0.664271\pi$
$61$	−7.70820			−0.986934			−0.493467	+	0.869764i	$0.664271\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.70820	+	4.47214i	−0.832050	+	0.554700i
$66$		0			0
$67$	−13.7082			−1.67472			−0.837362	−	0.546649i	$-0.815903\pi$
$67$	−13.7082			−1.67472			−0.837362	+	0.546649i	$0.815903\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.381966	−	0.381966i	−0.0453310	−	0.0453310i	0.684078	−	0.729409i	$-0.260205\pi$
$71$	−0.381966	−	0.381966i	−0.0453310	−	0.0453310i	−0.729409	+	0.684078i	$0.760205\pi$
$72$		0			0
$73$	−1.70820			−0.199930			−0.0999651	−	0.994991i	$-0.531873\pi$
$73$	−1.70820			−0.199930			−0.0999651	+	0.994991i	$0.531873\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	9.70820i		−	1.09226i	−0.837701	−	0.546129i	$-0.816101\pi$
$79$		−	9.70820i		−	1.09226i	0.837701	−	0.546129i	$-0.183899\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.52786i			0.167705i	0.996478	+	0.0838524i	$0.0267224\pi$
$83$			1.52786i			0.167705i	−0.996478	+	0.0838524i	$0.973278\pi$
$84$		0			0
$85$	−5.00000	+	5.00000i	−0.542326	+	0.542326i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.23607	−	2.23607i	−0.237023	−	0.237023i	0.578593	−	0.815616i	$-0.303602\pi$
$89$	−2.23607	−	2.23607i	−0.237023	−	0.237023i	−0.815616	+	0.578593i	$0.803602\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.90983	+	1.90983i	0.195944	+	0.195944i
$96$		0			0
$97$	−11.4164			−1.15916			−0.579580	−	0.814915i	$-0.696783\pi$
$97$	−11.4164			−1.15916			−0.579580	+	0.814915i	$0.696783\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	10.4721i		−	1.04202i	−0.853552	−	0.521008i	$-0.825556\pi$
$101$		−	10.4721i		−	1.04202i	0.853552	−	0.521008i	$-0.174444\pi$
$102$		0			0
$103$	2.85410	−	2.85410i	0.281223	−	0.281223i	−0.552374	−	0.833597i	$-0.686278\pi$
$103$	2.85410	−	2.85410i	0.281223	−	0.281223i	0.833597	+	0.552374i	$0.186278\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.381966	−	0.381966i	0.0369260	−	0.0369260i	−0.688403	−	0.725329i	$-0.741688\pi$
$107$	0.381966	−	0.381966i	0.0369260	−	0.0369260i	0.725329	+	0.688403i	$0.241688\pi$
$108$		0			0
$109$	−8.70820	+	8.70820i	−0.834095	+	0.834095i	−0.988074	−	0.153979i	$-0.950791\pi$
$109$	−8.70820	+	8.70820i	−0.834095	+	0.834095i	0.153979	+	0.988074i	$0.450791\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.94427	−	5.94427i	−0.559190	−	0.559190i	0.369887	−	0.929077i	$-0.379396\pi$
$113$	−5.94427	−	5.94427i	−0.559190	−	0.559190i	−0.929077	+	0.369887i	$0.879396\pi$
$114$		0			0
$115$	12.5623	+	12.5623i	1.17144	+	1.17144i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			10.7082i			0.973473i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$	−6.85410	−	6.85410i	−0.608203	−	0.608203i	0.334273	−	0.942476i	$-0.391509\pi$
$127$	−6.85410	−	6.85410i	−0.608203	−	0.608203i	−0.942476	+	0.334273i	$0.891509\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.4164			−1.69642			−0.848210	−	0.529661i	$-0.822319\pi$
$131$	−19.4164			−1.69642			−0.848210	+	0.529661i	$0.822319\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.52786i			0.643149i	0.946884	+	0.321574i	$0.104212\pi$
$137$			7.52786i			0.643149i	−0.946884	+	0.321574i	$0.895788\pi$
$138$		0			0
$139$			17.1246i			1.45249i	0.687436	+	0.726245i	$0.258736\pi$
$139$			17.1246i			1.45249i	−0.687436	+	0.726245i	$0.741264\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.90983	−	0.381966i	−0.159708	−	0.0319416i
$144$		0			0
$145$	−1.70820			−0.141859
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.9443	−	11.9443i	0.978513	−	0.978513i	−0.0212611	−	0.999774i	$-0.506768\pi$
$149$	11.9443	−	11.9443i	0.978513	−	0.978513i	0.999774	+	0.0212611i	$0.00676812\pi$
$150$		0			0
$151$	−6.85410	+	6.85410i	−0.557779	+	0.557779i	−0.928675	−	0.370896i	$-0.879051\pi$
$151$	−6.85410	+	6.85410i	−0.557779	+	0.557779i	0.370896	+	0.928675i	$0.379051\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.90983	+	1.90983i	−0.153401	+	0.153401i
$156$		0			0
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.7082			1.38701			0.693507	−	0.720450i	$-0.256065\pi$
$163$	17.7082			1.38701			0.693507	+	0.720450i	$0.256065\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	22.4721i		−	1.73895i	−0.493980	−	0.869473i	$-0.664459\pi$
$167$		−	22.4721i		−	1.73895i	0.493980	−	0.869473i	$-0.335541\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.18034	+	5.18034i	−0.393854	+	0.393854i	−0.876059	−	0.482205i	$-0.839836\pi$
$173$	−5.18034	+	5.18034i	−0.393854	+	0.393854i	0.482205	+	0.876059i	$0.339836\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$		−	21.7082i		−	1.61356i	−0.590853	−	0.806779i	$-0.701209\pi$
$181$		−	21.7082i		−	1.61356i	0.590853	−	0.806779i	$-0.298791\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.29180			0.609625
$186$		0			0
$187$	−1.70820			−0.124916
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.4164			1.40492			0.702461	−	0.711722i	$-0.252085\pi$
$191$	19.4164			1.40492			0.702461	+	0.711722i	$0.252085\pi$
$192$		0			0
$193$	−23.4164			−1.68555			−0.842775	−	0.538266i	$-0.819080\pi$
$193$	−23.4164			−1.68555			−0.842775	+	0.538266i	$0.819080\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	15.4164			1.09284			0.546420	−	0.837511i	$-0.315990\pi$
$199$	15.4164			1.09284			0.546420	+	0.837511i	$0.315990\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−18.4164	+	18.4164i	−1.28626	+	1.28626i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.652476i			0.0451327i
$210$		0			0
$211$	−23.4164			−1.61205			−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164			−1.61205			−0.806026	+	0.591880i	$0.798386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.38197	−	6.38197i	−0.435246	−	0.435246i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.23607	+	11.1803i	−0.150414	+	0.752071i
$222$		0			0
$223$		−	19.4164i		−	1.30022i	−0.759841	−	0.650109i	$-0.774723\pi$
$223$		−	19.4164i		−	1.30022i	0.759841	−	0.650109i	$-0.225277\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.7082			−1.44082			−0.720412	−	0.693546i	$-0.756047\pi$
$227$	−21.7082			−1.44082			−0.720412	+	0.693546i	$0.756047\pi$
$228$		0			0
$229$	−18.4164	−	18.4164i	−1.21699	−	1.21699i	−0.968681	−	0.248310i	$-0.920125\pi$
$229$	−18.4164	−	18.4164i	−1.21699	−	1.21699i	−0.248310	−	0.968681i	$-0.579875\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.47214	+	1.47214i	−0.0964428	+	0.0964428i	−0.753682	−	0.657239i	$-0.771724\pi$
$233$	−1.47214	+	1.47214i	−0.0964428	+	0.0964428i	0.657239	+	0.753682i	$0.271724\pi$
$234$		0			0
$235$	20.0000			1.30466
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.38197	−	6.38197i	0.412815	−	0.412815i	−0.469903	−	0.882718i	$-0.655711\pi$
$239$	6.38197	−	6.38197i	0.412815	−	0.412815i	0.882718	+	0.469903i	$0.155711\pi$
$240$		0			0
$241$	10.7082	−	10.7082i	0.689776	−	0.689776i	−0.272406	−	0.962182i	$-0.587819\pi$
$241$	10.7082	−	10.7082i	0.689776	−	0.689776i	0.962182	+	0.272406i	$0.0878195\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	15.6525i		−	1.00000i
$246$		0			0
$247$	4.27051	+	0.854102i	0.271726	+	0.0543452i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			3.81966i			0.241095i	0.992708	+	0.120547i	$0.0384650\pi$
$251$			3.81966i			0.241095i	−0.992708	+	0.120547i	$0.961535\pi$
$252$		0			0
$253$			4.29180i			0.269823i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.18034	+	5.18034i	0.323141	+	0.323141i	0.849971	−	0.526830i	$-0.176620\pi$
$257$	5.18034	+	5.18034i	0.323141	+	0.323141i	−0.526830	+	0.849971i	$0.676620\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.03444	+	7.03444i	0.433762	+	0.433762i	0.889906	−	0.456144i	$-0.150770\pi$
$263$	7.03444	+	7.03444i	0.433762	+	0.433762i	−0.456144	+	0.889906i	$0.650770\pi$
$264$		0			0
$265$	16.7082	−	16.7082i	1.02638	−	1.02638i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	22.4721i		−	1.37015i	−0.728472	−	0.685075i	$-0.759769\pi$
$269$		−	22.4721i		−	1.37015i	0.728472	−	0.685075i	$-0.240231\pi$
$270$		0			0
$271$	12.5623	−	12.5623i	0.763106	−	0.763106i	−0.213777	−	0.976883i	$-0.568577\pi$
$271$	12.5623	−	12.5623i	0.763106	−	0.763106i	0.976883	+	0.213777i	$0.0685765\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.90983	+	1.90983i	−0.115167	+	0.115167i
$276$		0			0
$277$	10.7082	+	10.7082i	0.643394	+	0.643394i	0.951388	−	0.307995i	$-0.0996578\pi$
$277$	10.7082	+	10.7082i	0.643394	+	0.643394i	−0.307995	+	0.951388i	$0.599658\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.23607	−	2.23607i	0.133393	−	0.133393i	−0.637258	−	0.770651i	$-0.719931\pi$
$281$	2.23607	−	2.23607i	0.133393	−	0.133393i	0.770651	+	0.637258i	$0.219931\pi$
$282$		0			0
$283$	−9.14590	+	9.14590i	−0.543667	+	0.543667i	−0.924602	−	0.380935i	$-0.875602\pi$
$283$	−9.14590	+	9.14590i	−0.543667	+	0.543667i	0.380935	+	0.924602i	$0.375602\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	7.00000i		−	0.411765i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.2918			−0.834936			−0.417468	−	0.908692i	$-0.637082\pi$
$293$	−14.2918			−0.834936			−0.417468	+	0.908692i	$0.637082\pi$
$294$		0			0
$295$	12.5623	−	12.5623i	0.731406	−	0.731406i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	28.0902	+	5.61803i	1.62450	+	0.324899i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			17.2361i			0.986934i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	20.1803i		−	1.14432i	−0.820141	−	0.572161i	$-0.806105\pi$
$311$		−	20.1803i		−	1.14432i	0.820141	−	0.572161i	$-0.193895\pi$
$312$		0			0
$313$	−1.29180	−	1.29180i	−0.0730166	−	0.0730166i	0.669655	−	0.742672i	$-0.266442\pi$
$313$	−1.29180	−	1.29180i	−0.0730166	−	0.0730166i	−0.742672	+	0.669655i	$0.766442\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.70820			−0.545267			−0.272634	−	0.962118i	$-0.587895\pi$
$317$	−9.70820			−0.545267			−0.272634	+	0.962118i	$0.587895\pi$
$318$		0			0
$319$	−0.291796	−	0.291796i	−0.0163374	−	0.0163374i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.81966			0.212532
$324$		0			0
$325$	10.0000	+	15.0000i	0.554700	+	0.832050i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−9.14590	−	9.14590i	−0.502704	−	0.502704i	0.409573	−	0.912277i	$-0.365678\pi$
$331$	−9.14590	−	9.14590i	−0.502704	−	0.502704i	−0.912277	+	0.409573i	$0.865678\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			30.6525i			1.67472i
$336$		0			0
$337$	−12.4164	−	12.4164i	−0.676365	−	0.676365i	0.282811	−	0.959176i	$-0.408733\pi$
$337$	−12.4164	−	12.4164i	−0.676365	−	0.676365i	−0.959176	+	0.282811i	$0.908733\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.652476			−0.0353335
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.09017	−	4.09017i	0.219572	−	0.219572i	−0.588746	−	0.808318i	$-0.700378\pi$
$347$	4.09017	−	4.09017i	0.219572	−	0.219572i	0.808318	+	0.588746i	$0.200378\pi$
$348$		0			0
$349$	3.29180	+	3.29180i	0.176206	+	0.176206i	0.789700	−	0.613494i	$-0.210236\pi$
$349$	3.29180	+	3.29180i	0.176206	+	0.176206i	−0.613494	+	0.789700i	$0.710236\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			17.2361i			0.917383i	0.888595	+	0.458692i	$0.151682\pi$
$353$			17.2361i			0.917383i	−0.888595	+	0.458692i	$0.848318\pi$
$354$		0			0
$355$	−0.854102	+	0.854102i	−0.0453310	+	0.0453310i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.90983	+	1.90983i	0.100797	+	0.100797i	0.755707	−	0.654910i	$-0.227293\pi$
$359$	1.90983	+	1.90983i	0.100797	+	0.100797i	−0.654910	+	0.755707i	$0.727293\pi$
$360$		0			0
$361$			17.5410i			0.923212i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			3.81966i			0.199930i
$366$		0			0
$367$	5.14590	−	5.14590i	0.268614	−	0.268614i	−0.559928	−	0.828541i	$-0.689171\pi$
$367$	5.14590	−	5.14590i	0.268614	−	0.268614i	0.828541	+	0.559928i	$0.189171\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	26.4164	+	26.4164i	1.36779	+	1.36779i	0.863583	+	0.504207i	$0.168215\pi$
$373$	26.4164	+	26.4164i	1.36779	+	1.36779i	0.504207	+	0.863583i	$0.331785\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.29180	+	1.52786i	−0.118034	+	0.0786890i
$378$		0			0
$379$	2.85410	−	2.85410i	0.146605	−	0.146605i	−0.629994	−	0.776600i	$-0.716943\pi$
$379$	2.85410	−	2.85410i	0.146605	−	0.146605i	0.776600	+	0.629994i	$0.216943\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	17.8885i		−	0.914062i	−0.889451	−	0.457031i	$-0.848913\pi$
$383$		−	17.8885i		−	0.914062i	0.889451	−	0.457031i	$-0.151087\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.41641			−0.0718147			−0.0359074	−	0.999355i	$-0.511432\pi$
$389$	−1.41641			−0.0718147			−0.0359074	+	0.999355i	$0.511432\pi$
$390$		0			0
$391$	25.1246			1.27061
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−21.7082			−1.09226
$396$		0			0
$397$		−	35.1246i		−	1.76285i	−0.472321	−	0.881427i	$-0.656584\pi$
$397$		−	35.1246i		−	1.76285i	0.472321	−	0.881427i	$-0.343416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.23607	−	2.23607i	0.111664	−	0.111664i	−0.649067	−	0.760731i	$-0.724841\pi$
$401$	2.23607	−	2.23607i	0.111664	−	0.111664i	0.760731	+	0.649067i	$0.224841\pi$
$402$		0			0
$403$	−0.854102	+	4.27051i	−0.0425458	+	0.212729i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.41641	+	1.41641i	0.0702087	+	0.0702087i
$408$		0			0
$409$	16.7082	−	16.7082i	0.826168	−	0.826168i	−0.160817	−	0.986984i	$-0.551413\pi$
$409$	16.7082	−	16.7082i	0.826168	−	0.826168i	0.986984	+	0.160817i	$0.0514128\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3.41641			0.167705
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	27.5967i		−	1.34819i	−0.738645	−	0.674095i	$-0.764534\pi$
$419$		−	27.5967i		−	1.34819i	0.738645	−	0.674095i	$-0.235466\pi$
$420$		0			0
$421$	16.7082	+	16.7082i	0.814308	+	0.814308i	0.985277	−	0.170968i	$-0.0546896\pi$
$421$	16.7082	+	16.7082i	0.814308	+	0.814308i	−0.170968	+	0.985277i	$0.554690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.1803	+	11.1803i	0.542326	+	0.542326i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.0902	−	28.0902i	−1.35306	−	1.35306i	−0.882222	−	0.470834i	$-0.843953\pi$
$431$	−28.0902	−	28.0902i	−1.35306	−	1.35306i	−0.470834	−	0.882222i	$-0.656047\pi$
$432$		0			0
$433$	−17.0000	+	17.0000i	−0.816968	+	0.816968i	−0.985668	−	0.168700i	$-0.946043\pi$
$433$	−17.0000	+	17.0000i	−0.816968	+	0.816968i	0.168700	+	0.985668i	$0.446043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	9.59675i		−	0.459075i
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.3262	+	15.3262i	0.728172	+	0.728172i	0.970255	−	0.242084i	$-0.0778309\pi$
$443$	15.3262	+	15.3262i	0.728172	+	0.728172i	−0.242084	+	0.970255i	$0.577831\pi$
$444$		0			0
$445$	−5.00000	+	5.00000i	−0.237023	+	0.237023i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.4721	+	19.4721i	0.918947	+	0.918947i	0.996953	−	0.0780060i	$-0.0248553\pi$
$449$	19.4721	+	19.4721i	0.918947	+	0.918947i	−0.0780060	+	0.996953i	$0.524855\pi$
$450$		0			0
$451$	−6.29180			−0.296269
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.41641			0.159813			0.0799064	−	0.996802i	$-0.474538\pi$
$457$	3.41641			0.159813			0.0799064	+	0.996802i	$0.474538\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.18034	+	5.18034i	0.241272	+	0.241272i	0.817376	−	0.576104i	$-0.195428\pi$
$461$	5.18034	+	5.18034i	0.241272	+	0.241272i	−0.576104	+	0.817376i	$0.695428\pi$
$462$		0			0
$463$	17.7082			0.822970			0.411485	−	0.911417i	$-0.365010\pi$
$463$	17.7082			0.822970			0.411485	+	0.911417i	$0.365010\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.0344	+	13.0344i	0.603162	+	0.603162i	0.941150	−	0.337988i	$-0.109746\pi$
$467$	13.0344	+	13.0344i	0.603162	+	0.603162i	−0.337988	+	0.941150i	$0.609746\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.18034i		−	0.100252i
$474$		0			0
$475$	4.27051	−	4.27051i	0.195944	−	0.195944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.7426	+	16.7426i	0.764991	+	0.764991i	0.977220	−	0.212229i	$-0.0680723\pi$
$479$	16.7426	+	16.7426i	0.764991	+	0.764991i	−0.212229	+	0.977220i	$0.568072\pi$
$480$		0			0
$481$	11.1246	−	7.41641i	0.507239	−	0.338159i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			25.5279i			1.15916i
$486$		0			0
$487$	25.1246			1.13850			0.569252	−	0.822163i	$-0.307233\pi$
$487$	25.1246			1.13850			0.569252	+	0.822163i	$0.307233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			6.65248i			0.300222i	0.988669	+	0.150111i	$0.0479631\pi$
$491$			6.65248i			0.300222i	−0.988669	+	0.150111i	$0.952037\pi$
$492$		0			0
$493$	−1.70820	+	1.70820i	−0.0769336	+	0.0769336i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	25.9787	−	25.9787i	1.16297	−	1.16297i	0.179144	−	0.983823i	$-0.442667\pi$
$499$	25.9787	−	25.9787i	1.16297	−	1.16297i	0.983823	−	0.179144i	$-0.0573328\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.3262	+	27.3262i	1.21842	+	1.21842i	0.968186	+	0.250230i	$0.0805064\pi$
$503$	27.3262	+	27.3262i	1.21842	+	1.21842i	0.250230	+	0.968186i	$0.419494\pi$
$504$		0			0
$505$	−23.4164			−1.04202
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.2361	−	14.2361i	0.631003	−	0.631003i	−0.317317	−	0.948320i	$-0.602782\pi$
$509$	14.2361	−	14.2361i	0.631003	−	0.631003i	0.948320	+	0.317317i	$0.102782\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.38197	−	6.38197i	−0.281223	−	0.281223i
$516$		0			0
$517$	3.41641	+	3.41641i	0.150253	+	0.150253i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.875388			−0.0383515			−0.0191757	−	0.999816i	$-0.506104\pi$
$521$	−0.875388			−0.0383515			−0.0191757	+	0.999816i	$0.506104\pi$
$522$		0			0
$523$	−8.27051	−	8.27051i	−0.361644	−	0.361644i	0.502774	−	0.864418i	$-0.332313\pi$
$523$	−8.27051	−	8.27051i	−0.361644	−	0.361644i	−0.864418	+	0.502774i	$0.832313\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			3.81966i			0.166387i
$528$		0			0
$529$		−	40.1246i		−	1.74455i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.23607	+	41.1803i	−0.356744	+	1.78372i
$534$		0			0
$535$	−0.854102	−	0.854102i	−0.0369260	−	0.0369260i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.67376	−	2.67376i	0.115167	−	0.115167i
$540$		0			0
$541$	−12.4164	+	12.4164i	−0.533823	+	0.533823i	−0.921708	−	0.387885i	$-0.873206\pi$
$541$	−12.4164	+	12.4164i	−0.533823	+	0.533823i	0.387885	+	0.921708i	$0.373206\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.4721	+	19.4721i	0.834095	+	0.834095i
$546$		0			0
$547$	12.5623	−	12.5623i	0.537125	−	0.537125i	−0.385558	−	0.922684i	$-0.625991\pi$
$547$	12.5623	−	12.5623i	0.537125	−	0.537125i	0.922684	+	0.385558i	$0.125991\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.652476	+	0.652476i	0.0277964	+	0.0277964i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	30.7639i		−	1.30351i	−0.758430	−	0.651755i	$-0.774033\pi$
$557$		−	30.7639i		−	1.30351i	0.758430	−	0.651755i	$-0.225967\pi$
$558$		0			0
$559$	−14.2705	−	2.85410i	−0.603578	−	0.120716i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0902	−	10.0902i	0.425250	−	0.425250i	−0.461757	−	0.887007i	$-0.652781\pi$
$563$	10.0902	−	10.0902i	0.425250	−	0.425250i	0.887007	+	0.461757i	$0.152781\pi$
$564$		0			0
$565$	−13.2918	+	13.2918i	−0.559190	+	0.559190i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−23.1246			−0.969434			−0.484717	−	0.874671i	$-0.661077\pi$
$569$	−23.1246			−0.969434			−0.484717	+	0.874671i	$0.661077\pi$
$570$		0			0
$571$		−	21.7082i		−	0.908460i	−0.890884	−	0.454230i	$-0.849914\pi$
$571$		−	21.7082i		−	0.908460i	0.890884	−	0.454230i	$-0.150086\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	28.0902	−	28.0902i	1.17144	−	1.17144i
$576$		0			0
$577$	13.1246			0.546385			0.273192	−	0.961959i	$-0.411920\pi$
$577$	13.1246			0.546385			0.273192	+	0.961959i	$0.411920\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	5.70820			0.236410
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.1246			−0.706808			−0.353404	−	0.935471i	$-0.614976\pi$
$587$	−17.1246			−0.706808			−0.353404	+	0.935471i	$0.614976\pi$
$588$		0			0
$589$	1.45898			0.0601162
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−43.4164			−1.78290			−0.891449	−	0.453121i	$-0.850311\pi$
$593$	−43.4164			−1.78290			−0.891449	+	0.453121i	$0.850311\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	35.0132i		−	1.43060i	−0.698818	−	0.715299i	$-0.746290\pi$
$599$		−	35.0132i		−	1.43060i	0.698818	−	0.715299i	$-0.253710\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23.9443			0.973473
$606$		0			0
$607$	4.27051	−	4.27051i	0.173335	−	0.173335i	−0.615108	−	0.788443i	$-0.710888\pi$
$607$	4.27051	−	4.27051i	0.173335	−	0.173335i	0.788443	+	0.615108i	$0.210888\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	26.8328	−	17.8885i	1.08554	−	0.723693i
$612$		0			0
$613$		−	13.4164i		−	0.541884i	−0.962596	−	0.270942i	$-0.912665\pi$
$613$		−	13.4164i		−	0.541884i	0.962596	−	0.270942i	$-0.0873351\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.4164			−1.26478			−0.632388	−	0.774651i	$-0.717925\pi$
$617$	−31.4164			−1.26478			−0.632388	+	0.774651i	$0.717925\pi$
$618$		0			0
$619$	16.2705	+	16.2705i	0.653967	+	0.653967i	0.953946	−	0.299979i	$-0.0969796\pi$
$619$	16.2705	+	16.2705i	0.653967	+	0.653967i	−0.299979	+	0.953946i	$0.596980\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.29180	−	8.29180i	0.330616	−	0.330616i
$630$		0			0
$631$	4.27051	−	4.27051i	0.170006	−	0.170006i	−0.616976	−	0.786982i	$-0.711642\pi$
$631$	4.27051	−	4.27051i	0.170006	−	0.170006i	0.786982	+	0.616976i	$0.211642\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.3262	+	15.3262i	−0.608203	+	0.608203i
$636$		0			0
$637$	−14.0000	−	21.0000i	−0.554700	−	0.832050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.0557i		−	0.594666i	−0.954774	−	0.297333i	$-0.903903\pi$
$641$		−	15.0557i		−	0.594666i	0.954774	−	0.297333i	$-0.0960971\pi$
$642$		0			0
$643$			16.5836i			0.653993i	0.945026	+	0.326997i	$0.106037\pi$
$643$			16.5836i			0.653993i	−0.945026	+	0.326997i	$0.893963\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.0902	−	10.0902i	−0.396686	−	0.396686i	0.480377	−	0.877062i	$-0.340500\pi$
$647$	−10.0902	−	10.0902i	−0.396686	−	0.396686i	−0.877062	+	0.480377i	$0.840500\pi$
$648$		0			0
$649$	4.29180			0.168468
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−29.0689	−	29.0689i	−1.13755	−	1.13755i	−0.988887	−	0.148666i	$-0.952502\pi$
$653$	−29.0689	−	29.0689i	−1.13755	−	1.13755i	−0.148666	−	0.988887i	$-0.547498\pi$
$654$		0			0
$655$			43.4164i			1.69642i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.0689i			1.01550i	0.861505	+	0.507750i	$0.169523\pi$
$659$			26.0689i			1.01550i	−0.861505	+	0.507750i	$0.830477\pi$
$660$		0			0
$661$	−2.70820	+	2.70820i	−0.105337	+	0.105337i	−0.757811	−	0.652474i	$-0.773731\pi$
$661$	−2.70820	+	2.70820i	−0.105337	+	0.105337i	0.652474	+	0.757811i	$0.273731\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.29180	+	4.29180i	0.166179	+	0.166179i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.94427	+	2.94427i	−0.113662	+	0.113662i
$672$		0			0
$673$	−10.1246	+	10.1246i	−0.390275	+	0.390275i	−0.874786	−	0.484510i	$-0.838998\pi$
$673$	−10.1246	+	10.1246i	−0.390275	+	0.390275i	0.484510	+	0.874786i	$0.338998\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.9443	−	17.9443i	0.689654	−	0.689654i	−0.272501	−	0.962155i	$-0.587851\pi$
$677$	17.9443	−	17.9443i	0.689654	−	0.689654i	0.962155	+	0.272501i	$0.0878508\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.12461			0.196088			0.0980439	−	0.995182i	$-0.468741\pi$
$683$	5.12461			0.196088			0.0980439	+	0.995182i	$0.468741\pi$
$684$		0			0
$685$	16.8328			0.643149
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.47214	−	37.3607i	0.284666	−	1.42333i
$690$		0			0
$691$	−11.9787	−	11.9787i	−0.455692	−	0.455692i	0.441547	−	0.897238i	$-0.354430\pi$
$691$	−11.9787	−	11.9787i	−0.455692	−	0.455692i	−0.897238	+	0.441547i	$0.854430\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	38.2918			1.45249
$696$		0			0
$697$			36.8328i			1.39514i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			16.3607i			0.617934i	0.951073	+	0.308967i	$0.0999833\pi$
$701$			16.3607i			0.617934i	−0.951073	+	0.308967i	$0.900017\pi$
$702$		0			0
$703$	−3.16718	−	3.16718i	−0.119453	−	0.119453i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	8.41641	+	8.41641i	0.316085	+	0.316085i	0.847261	−	0.531176i	$-0.178250\pi$
$709$	8.41641	+	8.41641i	0.316085	+	0.316085i	−0.531176	+	0.847261i	$0.678250\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.59675			0.359401
$714$		0			0
$715$	−0.854102	+	4.27051i	−0.0319416	+	0.159708i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.58359			−0.170939			−0.0854696	−	0.996341i	$-0.527239\pi$
$719$	−4.58359			−0.170939			−0.0854696	+	0.996341i	$0.527239\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$			3.81966i			0.141859i
$726$		0			0
$727$	−15.1459	−	15.1459i	−0.561730	−	0.561730i	0.368068	−	0.929799i	$-0.380019\pi$
$727$	−15.1459	−	15.1459i	−0.561730	−	0.561730i	−0.929799	+	0.368068i	$0.880019\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.7639			−0.472091
$732$		0			0
$733$		−	30.0000i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$		−	30.0000i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.23607	+	5.23607i	−0.192873	+	0.192873i
$738$		0			0
$739$	12.5623	+	12.5623i	0.462112	+	0.462112i	0.899347	−	0.437235i	$-0.144042\pi$
$739$	12.5623	+	12.5623i	0.462112	+	0.462112i	−0.437235	+	0.899347i	$0.644042\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.3050i		−	0.928349i	−0.885744	−	0.464174i	$-0.846351\pi$
$743$		−	25.3050i		−	0.928349i	0.885744	−	0.464174i	$-0.153649\pi$
$744$		0			0
$745$	−26.7082	−	26.7082i	−0.978513	−	0.978513i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			29.1246i			1.06277i	0.847130	+	0.531386i	$0.178329\pi$
$751$			29.1246i			1.06277i	−0.847130	+	0.531386i	$0.821671\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.3262	+	15.3262i	0.557779	+	0.557779i
$756$		0			0
$757$	−26.7082	+	26.7082i	−0.970726	+	0.970726i	−0.999584	−	0.0288574i	$-0.990813\pi$
$757$	−26.7082	+	26.7082i	−0.970726	+	0.970726i	0.0288574	+	0.999584i	$0.490813\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.5279	−	16.5279i	0.599135	−	0.599135i	−0.340948	−	0.940082i	$-0.610748\pi$
$761$	16.5279	−	16.5279i	0.599135	−	0.599135i	0.940082	+	0.340948i	$0.110748\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.61803	−	28.0902i	0.202855	−	1.01428i
$768$		0			0
$769$	−5.00000	+	5.00000i	−0.180305	+	0.180305i	−0.791489	−	0.611184i	$-0.790694\pi$
$769$	−5.00000	+	5.00000i	−0.180305	+	0.180305i	0.611184	+	0.791489i	$0.290694\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			0.652476i			0.0234679i	0.999931	+	0.0117340i	$0.00373512\pi$
$773$			0.652476i			0.0234679i	−0.999931	+	0.0117340i	$0.996265\pi$
$774$		0			0
$775$	4.27051	+	4.27051i	0.153401	+	0.153401i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.0689			0.504070
$780$		0			0
$781$	−0.291796			−0.0104413
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.1803	+	11.1803i	0.399043	+	0.399043i
$786$		0			0
$787$		−	43.4164i		−	1.54763i	−0.633413	−	0.773814i	$-0.718347\pi$
$787$		−	43.4164i		−	1.54763i	0.633413	−	0.773814i	$-0.281653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	15.4164	+	23.1246i	0.547453	+	0.821179i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.88854	+	8.88854i	0.314848	+	0.314848i	0.846784	−	0.531936i	$-0.178535\pi$
$797$	8.88854	+	8.88854i	0.314848	+	0.314848i	−0.531936	+	0.846784i	$0.678535\pi$
$798$		0			0
$799$	20.0000	−	20.0000i	0.707549	−	0.707549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.652476	+	0.652476i	−0.0230254	+	0.0230254i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.81966i			0.134292i	0.997743	+	0.0671460i	$0.0213893\pi$
$809$			3.81966i			0.134292i	−0.997743	+	0.0671460i	$0.978611\pi$
$810$		0			0
$811$	−16.5623	−	16.5623i	−0.581581	−	0.581581i	0.353756	−	0.935338i	$-0.384904\pi$
$811$	−16.5623	−	16.5623i	−0.581581	−	0.581581i	−0.935338	+	0.353756i	$0.884904\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	39.5967i		−	1.38701i
$816$		0			0
$817$			4.87539i			0.170568i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.36068	−	7.36068i	−0.256889	−	0.256889i	0.566898	−	0.823788i	$-0.308143\pi$
$821$	−7.36068	−	7.36068i	−0.256889	−	0.256889i	−0.823788	+	0.566898i	$0.808143\pi$
$822$		0			0
$823$	−23.9787	+	23.9787i	−0.835845	+	0.835845i	−0.988309	−	0.152464i	$-0.951279\pi$
$823$	−23.9787	+	23.9787i	−0.835845	+	0.835845i	0.152464	+	0.988309i	$0.451279\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			13.5279i			0.470410i	0.971946	+	0.235205i	$0.0755761\pi$
$827$			13.5279i			0.470410i	−0.971946	+	0.235205i	$0.924424\pi$
$828$		0			0
$829$	−0.291796			−0.0101345			−0.00506725	−	0.999987i	$-0.501613\pi$
$829$	−0.291796			−0.0101345			−0.00506725	+	0.999987i	$0.501613\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−15.6525	−	15.6525i	−0.542326	−	0.542326i
$834$		0			0
$835$	−50.2492			−1.73895
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.79837	−	1.79837i	−0.0620868	−	0.0620868i	0.675382	−	0.737468i	$-0.263979\pi$
$839$	−1.79837	−	1.79837i	−0.0620868	−	0.0620868i	−0.737468	+	0.675382i	$0.763979\pi$
$840$		0			0
$841$	28.4164			0.979876
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.8328	+	11.1803i	0.923077	+	0.384615i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20.8328	−	20.8328i	−0.714140	−	0.714140i
$852$		0			0
$853$	−45.1246			−1.54504			−0.772519	−	0.634992i	$-0.781003\pi$
$853$	−45.1246			−1.54504			−0.772519	+	0.634992i	$0.781003\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.6525	−	21.6525i	−0.739634	−	0.739634i	0.232873	−	0.972507i	$-0.425187\pi$
$857$	−21.6525	−	21.6525i	−0.739634	−	0.739634i	−0.972507	+	0.232873i	$0.925187\pi$
$858$		0			0
$859$			38.2918i			1.30650i	0.757143	+	0.653250i	$0.226595\pi$
$859$			38.2918i			1.30650i	−0.757143	+	0.653250i	$0.773405\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			25.5279i			0.868979i	0.900677	+	0.434489i	$0.143071\pi$
$863$			25.5279i			0.868979i	−0.900677	+	0.434489i	$0.856929\pi$
$864$		0			0
$865$	11.5836	+	11.5836i	0.393854	+	0.393854i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.70820	−	3.70820i	−0.125792	−	0.125792i
$870$		0			0
$871$	27.4164	+	41.1246i	0.928970	+	1.39345i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−6.83282			−0.230728			−0.115364	−	0.993323i	$-0.536803\pi$
$877$	−6.83282			−0.230728			−0.115364	+	0.993323i	$0.536803\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			38.0689i			1.28257i	0.767301	+	0.641287i	$0.221599\pi$
$881$			38.0689i			1.28257i	−0.767301	+	0.641287i	$0.778401\pi$
$882$		0			0
$883$	−21.1459	+	21.1459i	−0.711616	+	0.711616i	−0.966873	−	0.255257i	$-0.917840\pi$
$883$	−21.1459	+	21.1459i	−0.711616	+	0.711616i	0.255257	+	0.966873i	$0.417840\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−23.6180	+	23.6180i	−0.793016	+	0.793016i	−0.981983	−	0.188967i	$-0.939486\pi$
$887$	−23.6180	+	23.6180i	−0.793016	+	0.793016i	0.188967	+	0.981983i	$0.439486\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.63932	−	7.63932i	−0.255640	−	0.255640i
$894$		0			0
$895$		−	26.8328i		−	0.896922i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.652476	+	0.652476i	−0.0217613	+	0.0217613i
$900$		0			0
$901$		−	33.4164i		−	1.11326i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−48.5410			−1.61356
$906$		0			0
$907$	28.2705	+	28.2705i	0.938707	+	0.938707i	0.998227	−	0.0595202i	$-0.0189571\pi$
$907$	28.2705	+	28.2705i	0.938707	+	0.938707i	−0.0595202	+	0.998227i	$0.518957\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.83282			0.0938554			0.0469277	−	0.998898i	$-0.485057\pi$
$911$	2.83282			0.0938554			0.0469277	+	0.998898i	$0.485057\pi$
$912$		0			0
$913$	0.583592	+	0.583592i	0.0193141	+	0.0193141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	5.12461i		−	0.169045i	−0.996422	−	0.0845227i	$-0.973063\pi$
$919$		−	5.12461i		−	0.169045i	0.996422	−	0.0845227i	$-0.0269365\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.381966	+	1.90983i	−0.0125726	+	0.0628628i
$924$		0			0
$925$		−	18.5410i		−	0.609625i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−32.8885	+	32.8885i	−1.07904	+	1.07904i	−0.0824423	+	0.996596i	$0.526272\pi$
$929$	−32.8885	+	32.8885i	−1.07904	+	1.07904i	−0.996596	+	0.0824423i	$0.973728\pi$
$930$		0			0
$931$	−5.97871	+	5.97871i	−0.195944	+	0.195944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			3.81966i			0.124916i
$936$		0			0
$937$	5.58359	−	5.58359i	0.182408	−	0.182408i	−0.609996	−	0.792404i	$-0.708829\pi$
$937$	5.58359	−	5.58359i	0.182408	−	0.182408i	0.792404	+	0.609996i	$0.208829\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31.3607	−	31.3607i	−1.02233	−	1.02233i	−0.999745	−	0.0225840i	$-0.992811\pi$
$941$	−31.3607	−	31.3607i	−1.02233	−	1.02233i	−0.0225840	−	0.999745i	$-0.507189\pi$
$942$		0			0
$943$	92.5410			3.01355
$944$		0			0
$945$		0			0
$946$		0			0
$947$			20.9443i			0.680597i	0.940317	+	0.340299i	$0.110528\pi$
$947$			20.9443i			0.680597i	−0.940317	+	0.340299i	$0.889472\pi$
$948$		0			0
$949$	3.41641	+	5.12461i	0.110901	+	0.166352i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.2361	−	32.2361i	1.04423	−	1.04423i	0.0452531	−	0.998976i	$-0.485591\pi$
$953$	32.2361	−	32.2361i	1.04423	−	1.04423i	0.998976	−	0.0452531i	$-0.0144094\pi$
$954$		0			0
$955$		−	43.4164i		−	1.40492i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	29.5410i		−	0.952936i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			52.3607i			1.68555i
$966$		0			0
$967$	−33.1246			−1.06522			−0.532608	−	0.846362i	$-0.678788\pi$
$967$	−33.1246			−1.06522			−0.532608	+	0.846362i	$0.678788\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.70820			0.310593			0.155296	−	0.987868i	$-0.450367\pi$
$977$	9.70820			0.310593			0.155296	+	0.987868i	$0.450367\pi$
$978$		0			0
$979$	−1.70820			−0.0545944
$980$		0			0
$981$		0			0
$982$		0			0
$983$	45.7082			1.45787			0.728933	−	0.684585i	$-0.240017\pi$
$983$	45.7082			1.45787			0.728933	+	0.684585i	$0.240017\pi$
$984$		0			0
$985$			26.8328i			0.854965i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			32.0689i			1.01973i
$990$		0			0
$991$	−47.4164			−1.50623			−0.753116	−	0.657888i	$-0.771450\pi$
$991$	−47.4164			−1.50623			−0.753116	+	0.657888i	$0.771450\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	34.4721i		−	1.09284i
$996$		0			0
$997$	−29.0000	+	29.0000i	−0.918439	+	0.918439i	−0.996916	−	0.0784767i	$-0.974994\pi$
$997$	−29.0000	+	29.0000i	−0.918439	+	0.918439i	0.0784767	+	0.996916i	$0.474994\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.u.f.73.1		4
3.2	odd	2		260.2.m.b.73.2	yes	4
5.2	odd	4		2340.2.bp.e.1477.1		4
12.11	even	2		1040.2.bg.j.593.1		4
13.5	odd	4		2340.2.bp.e.1513.1		4
15.2	even	4		260.2.r.b.177.2	yes	4
15.8	even	4		1300.2.r.b.957.1		4
15.14	odd	2		1300.2.m.b.593.1		4
39.5	even	4		260.2.r.b.213.2	yes	4
60.47	odd	4		1040.2.cd.j.177.1		4
65.57	even	4	inner	2340.2.u.f.577.2		4
156.83	odd	4		1040.2.cd.j.993.1		4
195.44	even	4		1300.2.r.b.993.1		4
195.83	odd	4		1300.2.m.b.57.1		4
195.122	odd	4		260.2.m.b.57.2	✓	4
780.707	even	4		1040.2.bg.j.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.m.b.57.2	✓	4	195.122	odd	4
260.2.m.b.73.2	yes	4	3.2	odd	2
260.2.r.b.177.2	yes	4	15.2	even	4
260.2.r.b.213.2	yes	4	39.5	even	4
1040.2.bg.j.577.1		4	780.707	even	4
1040.2.bg.j.593.1		4	12.11	even	2
1040.2.cd.j.177.1		4	60.47	odd	4
1040.2.cd.j.993.1		4	156.83	odd	4
1300.2.m.b.57.1		4	195.83	odd	4
1300.2.m.b.593.1		4	15.14	odd	2
1300.2.r.b.957.1		4	15.8	even	4
1300.2.r.b.993.1		4	195.44	even	4
2340.2.u.f.73.1		4	1.1	even	1	trivial
2340.2.u.f.577.2		4	65.57	even	4	inner
2340.2.bp.e.1477.1		4	5.2	odd	4
2340.2.bp.e.1513.1		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +(0.381966 - 0.381966i) q^{11} +(-2.00000 - 3.00000i) q^{13} +(-2.23607 - 2.23607i) q^{17} +(-0.854102 + 0.854102i) q^{19} +(-5.61803 + 5.61803i) q^{23} -5.00000 q^{25} -0.763932i q^{29} +(-0.854102 - 0.854102i) q^{31} +3.70820i q^{37} +(-8.23607 - 8.23607i) q^{41} +(2.85410 - 2.85410i) q^{43} +8.94427i q^{47} +7.00000 q^{49} +(7.47214 + 7.47214i) q^{53} +(-0.854102 - 0.854102i) q^{55} +(5.61803 + 5.61803i) q^{59} -7.70820 q^{61} +(-6.70820 + 4.47214i) q^{65} -13.7082 q^{67} +(-0.381966 - 0.381966i) q^{71} -1.70820 q^{73} -9.70820i q^{79} +1.52786i q^{83} +(-5.00000 + 5.00000i) q^{85} +(-2.23607 - 2.23607i) q^{89} +(1.90983 + 1.90983i) q^{95} -11.4164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{11} - 8 q^{13} + 10 q^{19} - 18 q^{23} - 20 q^{25} + 10 q^{31} - 24 q^{41} - 2 q^{43} + 28 q^{49} + 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{67} - 6 q^{71} + 20 q^{73} - 20 q^{85}+ \cdots + 8 q^{97}+O(q^{100}) \) 4 * q + 6 * q^11 - 8 * q^13 + 10 * q^19 - 18 * q^23 - 20 * q^25 + 10 * q^31 - 24 * q^41 - 2 * q^43 + 28 * q^49 + 12 * q^53 + 10 * q^55 + 18 * q^59 - 4 * q^61 - 28 * q^67 - 6 * q^71 + 20 * q^73 - 20 * q^85 + 30 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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