LMFDB - Embedded newform 2340.2.y.b.1457.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(53,2340)] 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, 2, 3, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.53"); 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.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.4
Character	$\chi$	$=$	2340.1457
Dual form			2340.2.y.b.53.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.23903 + 1.86140i) q^{5} +(-0.157336 - 0.157336i) q^{7} -2.06519i q^{11} +(-0.707107 + 0.707107i) q^{13} +(-0.379176 + 0.379176i) q^{17} -5.74635i q^{19} +(1.48364 + 1.48364i) q^{23} +(-1.92963 - 4.61265i) q^{25} -1.06702 q^{29} +8.85481 q^{31} +(0.487808 - 0.0979220i) q^{35} +(-0.221577 - 0.221577i) q^{37} +1.85034i q^{41} +(3.87003 - 3.87003i) q^{43} +(-0.646634 + 0.646634i) q^{47} -6.95049i q^{49} +(5.31837 + 5.31837i) q^{53} +(3.84415 + 2.55882i) q^{55} +0.345872 q^{59} +0.645233 q^{61} +(-0.440086 - 2.19233i) q^{65} +(7.32461 + 7.32461i) q^{67} +2.07674i q^{71} +(2.76594 - 2.76594i) q^{73} +(-0.324928 + 0.324928i) q^{77} -0.527368i q^{79} +(3.39898 + 3.39898i) q^{83} +(-0.235990 - 1.17561i) q^{85} +14.3380 q^{89} +0.222506 q^{91} +(10.6963 + 7.11987i) q^{95} +(-9.21138 - 9.21138i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Character values

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

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

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.23903	+	1.86140i	−0.554109	+	0.832444i
$5$	−1.23903	+	1.86140i	−0.554109	+	0.832444i
$6$		0			0
$7$	−0.157336	−	0.157336i	−0.0594673	−	0.0594673i	0.676748	−	0.736215i	$-0.263389\pi$
$7$	−0.157336	−	0.157336i	−0.0594673	−	0.0594673i	−0.736215	+	0.676748i	$0.763389\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.06519i		−	0.622678i	−0.950299	−	0.311339i	$-0.899223\pi$
$11$		−	2.06519i		−	0.622678i	0.950299	−	0.311339i	$-0.100777\pi$
$12$		0			0
$13$	−0.707107	+	0.707107i	−0.196116	+	0.196116i
$13$	−0.707107	+	0.707107i	−0.196116	+	0.196116i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.379176	+	0.379176i	−0.0919638	+	0.0919638i	−0.751592	−	0.659628i	$-0.770714\pi$
$17$	−0.379176	+	0.379176i	−0.0919638	+	0.0919638i	0.659628	+	0.751592i	$0.270714\pi$
$18$		0			0
$19$		−	5.74635i		−	1.31830i	−0.752011	−	0.659151i	$-0.770916\pi$
$19$		−	5.74635i		−	1.31830i	0.752011	−	0.659151i	$-0.229084\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.48364	+	1.48364i	0.309361	+	0.309361i	0.844662	−	0.535300i	$-0.179802\pi$
$23$	1.48364	+	1.48364i	0.309361	+	0.309361i	−0.535300	+	0.844662i	$0.679802\pi$
$24$		0			0
$25$	−1.92963	−	4.61265i	−0.385926	−	0.922530i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.06702			−0.198141			−0.0990705	−	0.995080i	$-0.531587\pi$
$29$	−1.06702			−0.198141			−0.0990705	+	0.995080i	$0.531587\pi$
$30$		0			0
$31$	8.85481			1.59037			0.795186	−	0.606366i	$-0.207373\pi$
$31$	8.85481			1.59037			0.795186	+	0.606366i	$0.207373\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.487808	−	0.0979220i	0.0824546	−	0.0165518i
$36$		0			0
$37$	−0.221577	−	0.221577i	−0.0364270	−	0.0364270i	0.688659	−	0.725086i	$-0.258200\pi$
$37$	−0.221577	−	0.221577i	−0.0364270	−	0.0364270i	−0.725086	+	0.688659i	$0.758200\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.85034i			0.288975i	0.989507	+	0.144487i	$0.0461533\pi$
$41$			1.85034i			0.288975i	−0.989507	+	0.144487i	$0.953847\pi$
$42$		0			0
$43$	3.87003	−	3.87003i	0.590173	−	0.590173i	−0.347505	−	0.937678i	$-0.612971\pi$
$43$	3.87003	−	3.87003i	0.590173	−	0.590173i	0.937678	+	0.347505i	$0.112971\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.646634	+	0.646634i	−0.0943212	+	0.0943212i	−0.752693	−	0.658372i	$-0.771246\pi$
$47$	−0.646634	+	0.646634i	−0.0943212	+	0.0943212i	0.658372	+	0.752693i	$0.271246\pi$
$48$		0			0
$49$		−	6.95049i		−	0.992927i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.31837	+	5.31837i	0.730535	+	0.730535i	0.970726	−	0.240191i	$-0.0772101\pi$
$53$	5.31837	+	5.31837i	0.730535	+	0.730535i	−0.240191	+	0.970726i	$0.577210\pi$
$54$		0			0
$55$	3.84415	+	2.55882i	0.518345	+	0.345031i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.345872			0.0450287			0.0225144	−	0.999747i	$-0.492833\pi$
$59$	0.345872			0.0450287			0.0225144	+	0.999747i	$0.492833\pi$
$60$		0			0
$61$	0.645233			0.0826137			0.0413068	−	0.999147i	$-0.486848\pi$
$61$	0.645233			0.0826137			0.0413068	+	0.999147i	$0.486848\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.440086	−	2.19233i	−0.0545860	−	0.271925i
$66$		0			0
$67$	7.32461	+	7.32461i	0.894844	+	0.894844i	0.994974	−	0.100130i	$-0.0319260\pi$
$67$	7.32461	+	7.32461i	0.894844	+	0.894844i	−0.100130	+	0.994974i	$0.531926\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.07674i			0.246464i	0.992378	+	0.123232i	$0.0393259\pi$
$71$			2.07674i			0.246464i	−0.992378	+	0.123232i	$0.960674\pi$
$72$		0			0
$73$	2.76594	−	2.76594i	0.323728	−	0.323728i	−0.526467	−	0.850196i	$-0.676484\pi$
$73$	2.76594	−	2.76594i	0.323728	−	0.323728i	0.850196	+	0.526467i	$0.176484\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.324928	+	0.324928i	−0.0370290	+	0.0370290i
$78$		0			0
$79$		−	0.527368i		−	0.0593335i	−0.999560	−	0.0296668i	$-0.990555\pi$
$79$		−	0.527368i		−	0.0593335i	0.999560	−	0.0296668i	$-0.00944461\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.39898	+	3.39898i	0.373086	+	0.373086i	0.868600	−	0.495514i	$-0.165020\pi$
$83$	3.39898	+	3.39898i	0.373086	+	0.373086i	−0.495514	+	0.868600i	$0.665020\pi$
$84$		0			0
$85$	−0.235990	−	1.17561i	−0.0255967	−	0.127513i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.3380			1.51983			0.759913	−	0.650025i	$-0.225241\pi$
$89$	14.3380			1.51983			0.759913	+	0.650025i	$0.225241\pi$
$90$		0			0
$91$	0.222506			0.0233250
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.6963	+	7.11987i	1.09741	+	0.730483i
$96$		0			0
$97$	−9.21138	−	9.21138i	−0.935274	−	0.935274i	0.0627548	−	0.998029i	$-0.480011\pi$
$97$	−9.21138	−	9.21138i	−0.935274	−	0.935274i	−0.998029	+	0.0627548i	$0.980011\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			9.60360i			0.955594i	0.878470	+	0.477797i	$0.158564\pi$
$101$			9.60360i			0.955594i	−0.878470	+	0.477797i	$0.841436\pi$
$102$		0			0
$103$	4.43445	−	4.43445i	0.436940	−	0.436940i	−0.454041	−	0.890981i	$-0.650018\pi$
$103$	4.43445	−	4.43445i	0.436940	−	0.436940i	0.890981	+	0.454041i	$0.150018\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.85381	−	6.85381i	0.662582	−	0.662582i	−0.293406	−	0.955988i	$-0.594789\pi$
$107$	6.85381	−	6.85381i	0.662582	−	0.662582i	0.955988	+	0.293406i	$0.0947887\pi$
$108$		0			0
$109$		−	8.33883i		−	0.798715i	−0.916795	−	0.399357i	$-0.869233\pi$
$109$		−	8.33883i		−	0.798715i	0.916795	−	0.399357i	$-0.130767\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.81192	+	4.81192i	0.452667	+	0.452667i	0.896239	−	0.443572i	$-0.146289\pi$
$113$	4.81192	+	4.81192i	0.452667	+	0.452667i	−0.443572	+	0.896239i	$0.646289\pi$
$114$		0			0
$115$	−4.59993	+	0.923385i	−0.428946	+	0.0861061i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.119316			0.0109377
$120$		0			0
$121$	6.73499			0.612272
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.9769	+	2.12337i	0.981800	+	0.189920i
$126$		0			0
$127$	8.35047	+	8.35047i	0.740984	+	0.740984i	0.972767	−	0.231783i	$-0.0744560\pi$
$127$	8.35047	+	8.35047i	0.740984	+	0.740984i	−0.231783	+	0.972767i	$0.574456\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	2.73819i		−	0.239237i	−0.992820	−	0.119619i	$-0.961833\pi$
$131$		−	2.73819i		−	0.239237i	0.992820	−	0.119619i	$-0.0381672\pi$
$132$		0			0
$133$	−0.904106	+	0.904106i	−0.0783959	+	0.0783959i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.52385	−	3.52385i	0.301063	−	0.301063i	−0.540367	−	0.841430i	$-0.681714\pi$
$137$	3.52385	−	3.52385i	0.301063	−	0.301063i	0.841430	+	0.540367i	$0.181714\pi$
$138$		0			0
$139$		−	2.93706i		−	0.249119i	−0.992212	−	0.124559i	$-0.960248\pi$
$139$		−	2.93706i		−	0.249119i	0.992212	−	0.124559i	$-0.0397517\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.46031	+	1.46031i	0.122117	+	0.122117i
$144$		0			0
$145$	1.32207	−	1.98616i	0.109792	−	0.164941i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.3578			−0.930468			−0.465234	−	0.885188i	$-0.654030\pi$
$149$	−11.3578			−0.930468			−0.465234	+	0.885188i	$0.654030\pi$
$150$		0			0
$151$	5.71085			0.464742			0.232371	−	0.972627i	$-0.425352\pi$
$151$	5.71085			0.464742			0.232371	+	0.972627i	$0.425352\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.9713	+	16.4824i	−0.881239	+	1.32390i
$156$		0			0
$157$	7.33964	+	7.33964i	0.585767	+	0.585767i	0.936482	−	0.350715i	$-0.114062\pi$
$157$	7.33964	+	7.33964i	0.585767	+	0.585767i	−0.350715	+	0.936482i	$0.614062\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.466861i		−	0.0367938i
$162$		0			0
$163$	4.98122	−	4.98122i	0.390159	−	0.390159i	−0.484585	−	0.874744i	$-0.661029\pi$
$163$	4.98122	−	4.98122i	0.390159	−	0.390159i	0.874744	+	0.484585i	$0.161029\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.59931	−	6.59931i	0.510670	−	0.510670i	−0.404062	−	0.914732i	$-0.632402\pi$
$167$	6.59931	−	6.59931i	0.510670	−	0.510670i	0.914732	+	0.404062i	$0.132402\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.45748	+	6.45748i	0.490953	+	0.490953i	0.908606	−	0.417654i	$-0.137147\pi$
$173$	6.45748	+	6.45748i	0.490953	+	0.490953i	−0.417654	+	0.908606i	$0.637147\pi$
$174$		0			0
$175$	−0.422134	+	1.02933i	−0.0319104	+	0.0778104i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−23.8338			−1.78142			−0.890711	−	0.454570i	$-0.849793\pi$
$179$	−23.8338			−1.78142			−0.890711	+	0.454570i	$0.849793\pi$
$180$		0			0
$181$	−5.15979			−0.383524			−0.191762	−	0.981441i	$-0.561420\pi$
$181$	−5.15979			−0.383524			−0.191762	+	0.981441i	$0.561420\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.686982	−	0.137904i	0.0505079	−	0.0101389i
$186$		0			0
$187$	0.783071	+	0.783071i	0.0572638	+	0.0572638i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1.62193i		−	0.117359i	−0.998277	−	0.0586793i	$-0.981311\pi$
$191$		−	1.62193i		−	0.117359i	0.998277	−	0.0586793i	$-0.0186889\pi$
$192$		0			0
$193$	10.5742	−	10.5742i	0.761148	−	0.761148i	−0.215382	−	0.976530i	$-0.569100\pi$
$193$	10.5742	−	10.5742i	0.761148	−	0.761148i	0.976530	+	0.215382i	$0.0690996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0692	−	18.0692i	1.28738	−	1.28738i	0.351004	−	0.936374i	$-0.385840\pi$
$197$	18.0692	−	18.0692i	1.28738	−	1.28738i	0.936374	−	0.351004i	$-0.114160\pi$
$198$		0			0
$199$		−	13.8118i		−	0.979090i	−0.871978	−	0.489545i	$-0.837163\pi$
$199$		−	13.8118i		−	0.979090i	0.871978	−	0.489545i	$-0.162837\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.167881	+	0.167881i	0.0117829	+	0.0117829i
$204$		0			0
$205$	−3.44423	−	2.29262i	−0.240555	−	0.160123i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.8673			−0.820878
$210$		0			0
$211$	11.7318			0.807650			0.403825	−	0.914836i	$-0.367680\pi$
$211$	11.7318			0.807650			0.403825	+	0.914836i	$0.367680\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.40861	+	11.9987i	0.164266	+	0.818307i
$216$		0			0
$217$	−1.39318	−	1.39318i	−0.0945751	−	0.0945751i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.536237i		−	0.0360712i
$222$		0			0
$223$	−5.52112	+	5.52112i	−0.369722	+	0.369722i	−0.867376	−	0.497654i	$-0.834195\pi$
$223$	−5.52112	+	5.52112i	−0.369722	+	0.369722i	0.497654	+	0.867376i	$0.334195\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.9394	+	15.9394i	−1.05794	+	1.05794i	−0.0597213	+	0.998215i	$0.519021\pi$
$227$	−15.9394	+	15.9394i	−1.05794	+	1.05794i	−0.998215	+	0.0597213i	$0.980979\pi$
$228$		0			0
$229$		−	19.7008i		−	1.30187i	−0.759134	−	0.650934i	$-0.774377\pi$
$229$		−	19.7008i		−	1.30187i	0.759134	−	0.650934i	$-0.225623\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.9433	−	11.9433i	−0.782434	−	0.782434i	0.197807	−	0.980241i	$-0.436618\pi$
$233$	−11.9433	−	11.9433i	−0.782434	−	0.782434i	−0.980241	+	0.197807i	$0.936618\pi$
$234$		0			0
$235$	−0.402449	−	2.00484i	−0.0262529	−	0.130781i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.22997			−0.467668			−0.233834	−	0.972277i	$-0.575127\pi$
$239$	−7.22997			−0.467668			−0.233834	+	0.972277i	$0.575127\pi$
$240$		0			0
$241$	−3.21636			−0.207184			−0.103592	−	0.994620i	$-0.533034\pi$
$241$	−3.21636			−0.207184			−0.103592	+	0.994620i	$0.533034\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.9377	+	8.61184i	0.826556	+	0.550190i
$246$		0			0
$247$	4.06328	+	4.06328i	0.258540	+	0.258540i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	27.8442i		−	1.75751i	−0.477274	−	0.878755i	$-0.658375\pi$
$251$		−	27.8442i		−	1.75751i	0.477274	−	0.878755i	$-0.341625\pi$
$252$		0			0
$253$	3.06401	−	3.06401i	0.192632	−	0.192632i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.899591	+	0.899591i	−0.0561150	+	0.0561150i	−0.734607	−	0.678492i	$-0.762634\pi$
$257$	−0.899591	+	0.899591i	−0.0561150	+	0.0561150i	0.678492	+	0.734607i	$0.262634\pi$
$258$		0			0
$259$			0.0697239i			0.00433243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.2583	+	14.2583i	0.879206	+	0.879206i	0.993452	−	0.114246i	$-0.0364452\pi$
$263$	14.2583	+	14.2583i	0.879206	+	0.879206i	−0.114246	+	0.993452i	$0.536445\pi$
$264$		0			0
$265$	−16.4892	+	3.31003i	−1.01292	+	0.203333i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.2214			−0.745149			−0.372574	−	0.928002i	$-0.621525\pi$
$269$	−12.2214			−0.745149			−0.372574	+	0.928002i	$0.621525\pi$
$270$		0			0
$271$	13.1690			0.799959			0.399980	−	0.916524i	$-0.369017\pi$
$271$	13.1690			0.799959			0.399980	+	0.916524i	$0.369017\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.52599	+	3.98505i	−0.574439	+	0.240308i
$276$		0			0
$277$	−18.5257	−	18.5257i	−1.11310	−	1.11310i	−0.992729	−	0.120371i	$-0.961592\pi$
$277$	−18.5257	−	18.5257i	−1.11310	−	1.11310i	−0.120371	−	0.992729i	$-0.538408\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.0669i		−	1.19709i	−0.801089	−	0.598545i	$-0.795746\pi$
$281$		−	20.0669i		−	1.19709i	0.801089	−	0.598545i	$-0.204254\pi$
$282$		0			0
$283$	−15.3787	+	15.3787i	−0.914169	+	0.914169i	−0.996597	−	0.0824278i	$-0.973733\pi$
$283$	−15.3787	+	15.3787i	−0.914169	+	0.914169i	0.0824278	+	0.996597i	$0.473733\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.291125	−	0.291125i	0.0171846	−	0.0171846i
$288$		0			0
$289$			16.7125i			0.983085i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.3866	−	10.3866i	−0.606795	−	0.606795i	0.335312	−	0.942107i	$-0.391158\pi$
$293$	−10.3866	−	10.3866i	−0.606795	−	0.606795i	−0.942107	+	0.335312i	$0.891158\pi$
$294$		0			0
$295$	−0.428544	+	0.643807i	−0.0249508	+	0.0374839i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.09819			−0.121341
$300$		0			0
$301$	−1.21779			−0.0701921
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.799461	+	1.20104i	−0.0457770	+	0.0687713i
$306$		0			0
$307$	2.84488	+	2.84488i	0.162366	+	0.162366i	0.783614	−	0.621248i	$-0.213374\pi$
$307$	2.84488	+	2.84488i	0.162366	+	0.162366i	−0.621248	+	0.783614i	$0.713374\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.3208i			0.755356i	0.925937	+	0.377678i	$0.123277\pi$
$311$			13.3208i			0.755356i	−0.925937	+	0.377678i	$0.876723\pi$
$312$		0			0
$313$	−16.3970	+	16.3970i	−0.926815	+	0.926815i	−0.997499	−	0.0706836i	$-0.977482\pi$
$313$	−16.3970	+	16.3970i	−0.926815	+	0.926815i	0.0706836	+	0.997499i	$0.477482\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.3170	−	18.3170i	1.02879	−	1.02879i	0.0292136	−	0.999573i	$-0.490700\pi$
$317$	18.3170	−	18.3170i	1.02879	−	1.02879i	0.999573	−	0.0292136i	$-0.00930029\pi$
$318$		0			0
$319$			2.20360i			0.123378i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.17888	+	2.17888i	0.121236	+	0.121236i
$324$		0			0
$325$	4.62609	+	1.89718i	0.256609	+	0.105237i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.203477			0.0112181
$330$		0			0
$331$	8.50151			0.467285			0.233643	−	0.972323i	$-0.424935\pi$
$331$	8.50151			0.467285			0.233643	+	0.972323i	$0.424935\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.7094	+	4.55866i	−1.24075	+	0.249066i
$336$		0			0
$337$	4.61371	+	4.61371i	0.251325	+	0.251325i	0.821514	−	0.570189i	$-0.193130\pi$
$337$	4.61371	+	4.61371i	0.251325	+	0.251325i	−0.570189	+	0.821514i	$0.693130\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	18.2869i		−	0.990289i
$342$		0			0
$343$	−2.19491	+	2.19491i	−0.118514	+	0.118514i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.24911	+	8.24911i	−0.442835	+	0.442835i	−0.892964	−	0.450129i	$-0.851378\pi$
$347$	−8.24911	+	8.24911i	−0.442835	+	0.442835i	0.450129	+	0.892964i	$0.351378\pi$
$348$		0			0
$349$		−	9.48532i		−	0.507737i	−0.967239	−	0.253869i	$-0.918297\pi$
$349$		−	9.48532i		−	0.507737i	0.967239	−	0.253869i	$-0.0817031\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.77661	+	3.77661i	0.201009	+	0.201009i	0.800432	−	0.599423i	$-0.204603\pi$
$353$	3.77661	+	3.77661i	0.201009	+	0.201009i	−0.599423	+	0.800432i	$0.704603\pi$
$354$		0			0
$355$	−3.86565	−	2.57314i	−0.205168	−	0.136568i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.7235			1.09374			0.546871	−	0.837217i	$-0.315819\pi$
$359$	20.7235			1.09374			0.546871	+	0.837217i	$0.315819\pi$
$360$		0			0
$361$	−14.0205			−0.737921
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.72145	+	8.57558i	0.0901050	+	0.448867i
$366$		0			0
$367$	10.2386	+	10.2386i	0.534451	+	0.534451i	0.921894	−	0.387443i	$-0.126642\pi$
$367$	10.2386	+	10.2386i	0.534451	+	0.534451i	−0.387443	+	0.921894i	$0.626642\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.67354i		−	0.0868859i
$372$		0			0
$373$	−6.20505	+	6.20505i	−0.321285	+	0.321285i	−0.849260	−	0.527975i	$-0.822952\pi$
$373$	−6.20505	+	6.20505i	−0.321285	+	0.321285i	0.527975	+	0.849260i	$0.322952\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.754499	−	0.754499i	0.0388587	−	0.0388587i
$378$		0			0
$379$			30.9627i			1.59045i	0.606317	+	0.795223i	$0.292646\pi$
$379$			30.9627i			1.59045i	−0.606317	+	0.795223i	$0.707354\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	21.6230	+	21.6230i	1.10488	+	1.10488i	0.993813	+	0.111071i	$0.0354280\pi$
$383$	21.6230	+	21.6230i	1.10488	+	1.10488i	0.111071	+	0.993813i	$0.464572\pi$
$384$		0			0
$385$	−0.202227	−	1.00742i	−0.0103065	−	0.0513427i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.6047			−0.841890			−0.420945	−	0.907086i	$-0.638302\pi$
$389$	−16.6047			−0.841890			−0.420945	+	0.907086i	$0.638302\pi$
$390$		0			0
$391$	−1.12513			−0.0569001
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.981643	+	0.653422i	0.0493918	+	0.0328772i
$396$		0			0
$397$	9.77161	+	9.77161i	0.490423	+	0.490423i	0.908439	−	0.418017i	$-0.137275\pi$
$397$	9.77161	+	9.77161i	0.490423	+	0.490423i	−0.418017	+	0.908439i	$0.637275\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	3.16171i		−	0.157888i	−0.996879	−	0.0789441i	$-0.974845\pi$
$401$		−	3.16171i		−	0.157888i	0.996879	−	0.0789441i	$-0.0251549\pi$
$402$		0			0
$403$	−6.26130	+	6.26130i	−0.311897	+	0.311897i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.457598	+	0.457598i	−0.0226823	+	0.0226823i
$408$		0			0
$409$		−	25.5853i		−	1.26511i	−0.774515	−	0.632556i	$-0.782006\pi$
$409$		−	25.5853i		−	1.26511i	0.774515	−	0.632556i	$-0.217994\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.0544181	−	0.0544181i	−0.00267774	−	0.00267774i
$414$		0			0
$415$	−10.5383	+	2.11544i	−0.517304	+	0.103843i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−29.6062			−1.44636			−0.723179	−	0.690661i	$-0.757320\pi$
$419$	−29.6062			−1.44636			−0.723179	+	0.690661i	$0.757320\pi$
$420$		0			0
$421$	−14.5564			−0.709435			−0.354717	−	0.934974i	$-0.615423\pi$
$421$	−14.5564			−0.709435			−0.354717	+	0.934974i	$0.615423\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.48068	+	1.01734i	0.120331	+	0.0493481i
$426$		0			0
$427$	−0.101518	−	0.101518i	−0.00491281	−	0.00491281i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	17.6092i		−	0.848205i	−0.905614	−	0.424102i	$-0.860590\pi$
$431$		−	17.6092i		−	0.848205i	0.905614	−	0.424102i	$-0.139410\pi$
$432$		0			0
$433$	−7.95487	+	7.95487i	−0.382287	+	0.382287i	−0.871925	−	0.489639i	$-0.837129\pi$
$433$	−7.95487	+	7.95487i	−0.382287	+	0.382287i	0.489639	+	0.871925i	$0.337129\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.52554	−	8.52554i	0.407832	−	0.407832i
$438$		0			0
$439$			12.8337i			0.612519i	0.951948	+	0.306260i	$0.0990776\pi$
$439$			12.8337i			0.612519i	−0.951948	+	0.306260i	$0.900922\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.8511	−	22.8511i	−1.08569	−	1.08569i	−0.995967	−	0.0897210i	$-0.971402\pi$
$443$	−22.8511	−	22.8511i	−1.08569	−	1.08569i	−0.0897210	−	0.995967i	$-0.528598\pi$
$444$		0			0
$445$	−17.7652	+	26.6888i	−0.842150	+	1.26517i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.71880			0.317080			0.158540	−	0.987353i	$-0.449321\pi$
$449$	6.71880			0.317080			0.158540	+	0.987353i	$0.449321\pi$
$450$		0			0
$451$	3.82130			0.179938
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.275691	+	0.414174i	−0.0129246	+	0.0194168i
$456$		0			0
$457$	21.1723	+	21.1723i	0.990400	+	0.990400i	0.999954	−	0.00955482i	$-0.00304144\pi$
$457$	21.1723	+	21.1723i	0.990400	+	0.990400i	−0.00955482	+	0.999954i	$0.503041\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.9687i		−	0.557438i	−0.960373	−	0.278719i	$-0.910090\pi$
$461$		−	11.9687i		−	0.557438i	0.960373	−	0.278719i	$-0.0899099\pi$
$462$		0			0
$463$	−13.8992	+	13.8992i	−0.645952	+	0.645952i	−0.952012	−	0.306060i	$-0.900989\pi$
$463$	−13.8992	+	13.8992i	−0.645952	+	0.645952i	0.306060	+	0.952012i	$0.400989\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.4011	−	12.4011i	0.573854	−	0.573854i	−0.359349	−	0.933203i	$-0.617001\pi$
$467$	12.4011	−	12.4011i	0.573854	−	0.573854i	0.933203	+	0.359349i	$0.117001\pi$
$468$		0			0
$469$		−	2.30485i		−	0.106428i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.99233	−	7.99233i	−0.367488	−	0.367488i
$474$		0			0
$475$	−26.5059	+	11.0883i	−1.21617	+	0.508768i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.82779			−0.266278			−0.133139	−	0.991097i	$-0.542506\pi$
$479$	−5.82779			−0.266278			−0.133139	+	0.991097i	$0.542506\pi$
$480$		0			0
$481$	0.313357			0.0142878
$482$		0			0
$483$		0			0
$484$		0			0
$485$	28.5592	−	5.73294i	1.29681	−	0.260320i
$486$		0			0
$487$	−10.9595	−	10.9595i	−0.496622	−	0.496622i	0.413763	−	0.910385i	$-0.364214\pi$
$487$	−10.9595	−	10.9595i	−0.496622	−	0.496622i	−0.910385	+	0.413763i	$0.864214\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.6730i		−	0.526796i	−0.964687	−	0.263398i	$-0.915157\pi$
$491$		−	11.6730i		−	0.526796i	0.964687	−	0.263398i	$-0.0848432\pi$
$492$		0			0
$493$	0.404590	−	0.404590i	0.0182218	−	0.0182218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.326746	−	0.326746i	0.0146566	−	0.0146566i
$498$		0			0
$499$			13.7099i			0.613738i	0.951752	+	0.306869i	$0.0992814\pi$
$499$			13.7099i			0.613738i	−0.951752	+	0.306869i	$0.900719\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.310963	−	0.310963i	−0.0138652	−	0.0138652i	0.700140	−	0.714005i	$-0.253121\pi$
$503$	−0.310963	−	0.310963i	−0.0138652	−	0.0138652i	−0.714005	+	0.700140i	$0.753121\pi$
$504$		0			0
$505$	−17.8762	−	11.8991i	−0.795478	−	0.529503i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−41.1167			−1.82247			−0.911233	−	0.411891i	$-0.864868\pi$
$509$	−41.1167			−1.82247			−0.911233	+	0.411891i	$0.864868\pi$
$510$		0			0
$511$	−0.870361			−0.0385025
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.75990	+	13.7487i	0.121616	+	0.605840i
$516$		0			0
$517$	1.33542	+	1.33542i	0.0587318	+	0.0587318i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.48076i		−	0.415360i	−0.978197	−	0.207680i	$-0.933409\pi$
$521$		−	9.48076i		−	0.415360i	0.978197	−	0.207680i	$-0.0665912\pi$
$522$		0			0
$523$	−16.0861	+	16.0861i	−0.703395	+	0.703395i	−0.965138	−	0.261743i	$-0.915703\pi$
$523$	−16.0861	+	16.0861i	−0.703395	+	0.703395i	0.261743	+	0.965138i	$0.415703\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.35754	+	3.35754i	−0.146257	+	0.146257i
$528$		0			0
$529$		−	18.5976i		−	0.808591i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.30839	−	1.30839i	−0.0566726	−	0.0566726i
$534$		0			0
$535$	4.26564	+	21.2497i	0.184420	+	0.918706i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.3541			−0.618274
$540$		0			0
$541$	4.47451			0.192374			0.0961871	−	0.995363i	$-0.469335\pi$
$541$	4.47451			0.192374			0.0961871	+	0.995363i	$0.469335\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.5219	+	10.3320i	0.664885	+	0.442575i
$546$		0			0
$547$	15.1803	+	15.1803i	0.649061	+	0.649061i	0.952766	−	0.303705i	$-0.0982237\pi$
$547$	15.1803	+	15.1803i	0.649061	+	0.649061i	−0.303705	+	0.952766i	$0.598224\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			6.13148i			0.261210i
$552$		0			0
$553$	−0.0829738	+	0.0829738i	−0.00352841	+	0.00352841i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30.9963	−	30.9963i	1.31335	−	1.31335i	0.394426	−	0.918928i	$-0.370943\pi$
$557$	30.9963	−	30.9963i	1.31335	−	1.31335i	0.918928	−	0.394426i	$-0.129057\pi$
$558$		0			0
$559$			5.47304i			0.231485i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.15424	+	8.15424i	0.343660	+	0.343660i	0.857742	−	0.514081i	$-0.171867\pi$
$563$	8.15424	+	8.15424i	0.343660	+	0.343660i	−0.514081	+	0.857742i	$0.671867\pi$
$564$		0			0
$565$	−14.9190	+	2.99482i	−0.627647	+	0.125993i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−36.8475			−1.54473			−0.772365	−	0.635179i	$-0.780926\pi$
$569$	−36.8475			−1.54473			−0.772365	+	0.635179i	$0.780926\pi$
$570$		0			0
$571$	−19.8439			−0.830442			−0.415221	−	0.909721i	$-0.636296\pi$
$571$	−19.8439			−0.830442			−0.415221	+	0.909721i	$0.636296\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.98064	−	9.70642i	0.166004	−	0.404786i
$576$		0			0
$577$	3.83030	+	3.83030i	0.159457	+	0.159457i	0.782326	−	0.622869i	$-0.214033\pi$
$577$	3.83030	+	3.83030i	0.159457	+	0.159457i	−0.622869	+	0.782326i	$0.714033\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1.06956i		−	0.0443729i
$582$		0			0
$583$	10.9834	−	10.9834i	0.454888	−	0.454888i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.4282	+	15.4282i	−0.636789	+	0.636789i	−0.949762	−	0.312973i	$-0.898675\pi$
$587$	−15.4282	+	15.4282i	−0.636789	+	0.636789i	0.312973	+	0.949762i	$0.398675\pi$
$588$		0			0
$589$		−	50.8828i		−	2.09659i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.3275	+	11.3275i	0.465166	+	0.465166i	0.900344	−	0.435178i	$-0.143315\pi$
$593$	11.3275	+	11.3275i	0.465166	+	0.465166i	−0.435178	+	0.900344i	$0.643315\pi$
$594$		0			0
$595$	−0.147836	+	0.222095i	−0.00606067	+	0.00910501i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.2523			−0.541476			−0.270738	−	0.962653i	$-0.587268\pi$
$599$	−13.2523			−0.541476			−0.270738	+	0.962653i	$0.587268\pi$
$600$		0			0
$601$	12.6411			0.515642			0.257821	−	0.966193i	$-0.416996\pi$
$601$	12.6411			0.515642			0.257821	+	0.966193i	$0.416996\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.34483	+	12.5365i	−0.339266	+	0.509682i
$606$		0			0
$607$	7.21777	+	7.21777i	0.292960	+	0.292960i	0.838249	−	0.545288i	$-0.183580\pi$
$607$	7.21777	+	7.21777i	0.292960	+	0.292960i	−0.545288	+	0.838249i	$0.683580\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.914478i		−	0.0369958i
$612$		0			0
$613$	−27.3349	+	27.3349i	−1.10405	+	1.10405i	−0.110128	+	0.993917i	$0.535126\pi$
$613$	−27.3349	+	27.3349i	−1.10405	+	1.10405i	−0.993917	+	0.110128i	$0.964874\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.5430	−	27.5430i	1.10884	−	1.10884i	0.115537	−	0.993303i	$-0.463141\pi$
$617$	27.5430	−	27.5430i	1.10884	−	1.10884i	0.993303	−	0.115537i	$-0.0368590\pi$
$618$		0			0
$619$		−	2.29212i		−	0.0921279i	−0.998938	−	0.0460640i	$-0.985332\pi$
$619$		−	2.29212i		−	0.0921279i	0.998938	−	0.0460640i	$-0.0146678\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.25588	−	2.25588i	−0.0903800	−	0.0903800i
$624$		0			0
$625$	−17.5530	+	17.8014i	−0.702122	+	0.712057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.168033			0.00669993
$630$		0			0
$631$	27.0130			1.07537			0.537686	−	0.843145i	$-0.319299\pi$
$631$	27.0130			1.07537			0.537686	+	0.843145i	$0.319299\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.8900	+	5.19713i	−1.02741	+	0.206242i
$636$		0			0
$637$	4.91474	+	4.91474i	0.194729	+	0.194729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12.0803i			0.477144i	0.971125	+	0.238572i	$0.0766793\pi$
$641$			12.0803i			0.477144i	−0.971125	+	0.238572i	$0.923321\pi$
$642$		0			0
$643$	−8.73222	+	8.73222i	−0.344365	+	0.344365i	−0.858006	−	0.513640i	$-0.828297\pi$
$643$	−8.73222	+	8.73222i	−0.344365	+	0.344365i	0.513640	+	0.858006i	$0.328297\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.04837	+	5.04837i	−0.198472	+	0.198472i	−0.799345	−	0.600873i	$-0.794820\pi$
$647$	−5.04837	+	5.04837i	−0.198472	+	0.198472i	0.600873	+	0.799345i	$0.294820\pi$
$648$		0			0
$649$		−	0.714292i		−	0.0280384i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.5504	−	24.5504i	−0.960732	−	0.960732i	0.0385260	−	0.999258i	$-0.487734\pi$
$653$	−24.5504	−	24.5504i	−0.960732	−	0.960732i	−0.999258	+	0.0385260i	$0.987734\pi$
$654$		0			0
$655$	5.09688	+	3.39269i	0.199152	+	0.132563i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−33.8861			−1.32001			−0.660007	−	0.751259i	$-0.729447\pi$
$659$	−33.8861			−1.32001			−0.660007	+	0.751259i	$0.729447\pi$
$660$		0			0
$661$	−35.7528			−1.39062			−0.695311	−	0.718709i	$-0.744734\pi$
$661$	−35.7528			−1.39062			−0.695311	+	0.718709i	$0.744734\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.562694	−	2.80311i	−0.0218203	−	0.108700i
$666$		0			0
$667$	−1.58308	−	1.58308i	−0.0612972	−	0.0612972i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.33253i		−	0.0514417i
$672$		0			0
$673$	−15.7229	+	15.7229i	−0.606072	+	0.606072i	−0.941917	−	0.335845i	$-0.890978\pi$
$673$	−15.7229	+	15.7229i	−0.606072	+	0.606072i	0.335845	+	0.941917i	$0.390978\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.4755	+	29.4755i	−1.13283	+	1.13283i	−0.143130	+	0.989704i	$0.545717\pi$
$677$	−29.4755	+	29.4755i	−1.13283	+	1.13283i	−0.989704	+	0.143130i	$0.954283\pi$
$678$		0			0
$679$			2.89856i			0.111237i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.66210	−	8.66210i	−0.331446	−	0.331446i	0.521689	−	0.853135i	$-0.325302\pi$
$683$	−8.66210	−	8.66210i	−0.331446	−	0.331446i	−0.853135	+	0.521689i	$0.825302\pi$
$684$		0			0
$685$	2.19316	+	10.9254i	0.0837964	+	0.417440i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.52131			−0.286539
$690$		0			0
$691$	4.79852			0.182544			0.0912721	−	0.995826i	$-0.470907\pi$
$691$	4.79852			0.182544			0.0912721	+	0.995826i	$0.470907\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.46706	+	3.63910i	0.207377	+	0.138039i
$696$		0			0
$697$	−0.701606	−	0.701606i	−0.0265752	−	0.0265752i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	28.9019i		−	1.09161i	−0.837913	−	0.545804i	$-0.816224\pi$
$701$		−	28.9019i		−	1.09161i	0.837913	−	0.545804i	$-0.183776\pi$
$702$		0			0
$703$	−1.27326	+	1.27326i	−0.0480218	+	0.0480218i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.51099	−	1.51099i	0.0568266	−	0.0568266i
$708$		0			0
$709$			3.45422i			0.129726i	0.997894	+	0.0648629i	$0.0206610\pi$
$709$			3.45422i			0.129726i	−0.997894	+	0.0648629i	$0.979339\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.1374	+	13.1374i	0.491999	+	0.491999i
$714$		0			0
$715$	−4.52758	+	0.908861i	−0.169322	+	0.0339895i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.8764			0.778560			0.389280	−	0.921119i	$-0.372724\pi$
$719$	20.8764			0.778560			0.389280	+	0.921119i	$0.372724\pi$
$720$		0			0
$721$	−1.39540			−0.0519673
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.05896	+	4.92180i	0.0764678	+	0.182791i
$726$		0			0
$727$	16.2764	+	16.2764i	0.603657	+	0.603657i	0.941281	−	0.337624i	$-0.109623\pi$
$727$	16.2764	+	16.2764i	0.603657	+	0.603657i	−0.337624	+	0.941281i	$0.609623\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			2.93484i			0.108549i
$732$		0			0
$733$	17.3721	−	17.3721i	0.641651	−	0.641651i	−0.309310	−	0.950961i	$-0.600098\pi$
$733$	17.3721	−	17.3721i	0.641651	−	0.641651i	0.950961	+	0.309310i	$0.100098\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.1267	−	15.1267i	0.557200	−	0.557200i
$738$		0			0
$739$		−	50.4600i		−	1.85620i	−0.372328	−	0.928101i	$-0.621440\pi$
$739$		−	50.4600i		−	1.85620i	0.372328	−	0.928101i	$-0.378560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.77471	−	5.77471i	−0.211854	−	0.211854i	0.593201	−	0.805054i	$-0.297864\pi$
$743$	−5.77471	−	5.77471i	−0.211854	−	0.211854i	−0.805054	+	0.593201i	$0.797864\pi$
$744$		0			0
$745$	14.0726	−	21.1415i	0.515581	−	0.774563i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.15670			−0.0788040
$750$		0			0
$751$	23.1099			0.843292			0.421646	−	0.906760i	$-0.361452\pi$
$751$	23.1099			0.843292			0.421646	+	0.906760i	$0.361452\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−7.07588	+	10.6302i	−0.257518	+	0.386872i
$756$		0			0
$757$	−22.3929	−	22.3929i	−0.813883	−	0.813883i	0.171331	−	0.985214i	$-0.445193\pi$
$757$	−22.3929	−	22.3929i	−0.813883	−	0.813883i	−0.985214	+	0.171331i	$0.945193\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.6165i		−	0.711097i	−0.934658	−	0.355548i	$-0.884294\pi$
$761$		−	19.6165i		−	0.711097i	0.934658	−	0.355548i	$-0.115706\pi$
$762$		0			0
$763$	−1.31200	+	1.31200i	−0.0474974	+	0.0474974i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.244569	+	0.244569i	−0.00883086	+	0.00883086i
$768$		0			0
$769$			2.31455i			0.0834647i	0.999129	+	0.0417324i	$0.0132877\pi$
$769$			2.31455i			0.0834647i	−0.999129	+	0.0417324i	$0.986712\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.2277	−	10.2277i	−0.367864	−	0.367864i	0.498834	−	0.866698i	$-0.333762\pi$
$773$	−10.2277	−	10.2277i	−0.367864	−	0.367864i	−0.866698	+	0.498834i	$0.833762\pi$
$774$		0			0
$775$	−17.0865	−	40.8441i	−0.613766	−	1.46716i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.6327			0.380956
$780$		0			0
$781$	4.28887			0.153468
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−22.7560	+	4.56801i	−0.812197	+	0.163039i
$786$		0			0
$787$	−8.90292	−	8.90292i	−0.317355	−	0.317355i	0.530396	−	0.847750i	$-0.322043\pi$
$787$	−8.90292	−	8.90292i	−0.317355	−	0.317355i	−0.847750	+	0.530396i	$0.822043\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.51417i		−	0.0538378i
$792$		0			0
$793$	−0.456249	+	0.456249i	−0.0162019	+	0.0162019i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.3510	−	12.3510i	0.437495	−	0.437495i	−0.453673	−	0.891168i	$-0.649887\pi$
$797$	12.3510	−	12.3510i	0.437495	−	0.437495i	0.891168	+	0.453673i	$0.149887\pi$
$798$		0			0
$799$		−	0.490377i		−	0.0173483i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.71218	−	5.71218i	−0.201579	−	0.201579i
$804$		0			0
$805$	0.869015	+	0.578452i	0.0306288	+	0.0203878i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.98770			0.175358			0.0876790	−	0.996149i	$-0.472055\pi$
$809$	4.98770			0.175358			0.0876790	+	0.996149i	$0.472055\pi$
$810$		0			0
$811$	38.7073			1.35920			0.679599	−	0.733584i	$-0.262154\pi$
$811$	38.7073			1.35920			0.679599	+	0.733584i	$0.262154\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.10019	+	15.4439i	0.108595	+	0.540977i
$816$		0			0
$817$	−22.2385	−	22.2385i	−0.778027	−	0.778027i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			7.81673i			0.272806i	0.990653	+	0.136403i	$0.0435542\pi$
$821$			7.81673i			0.272806i	−0.990653	+	0.136403i	$0.956446\pi$
$822$		0			0
$823$	28.8818	−	28.8818i	1.00676	−	1.00676i	0.00677868	−	0.999977i	$-0.497842\pi$
$823$	28.8818	−	28.8818i	1.00676	−	1.00676i	0.999977	−	0.00677868i	$-0.00215774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.9633	−	29.9633i	1.04193	−	1.04193i	0.0428449	−	0.999082i	$-0.486358\pi$
$827$	29.9633	−	29.9633i	1.04193	−	1.04193i	0.999082	−	0.0428449i	$-0.0136421\pi$
$828$		0			0
$829$			43.9719i			1.52721i	0.645685	+	0.763604i	$0.276572\pi$
$829$			43.9719i			1.52721i	−0.645685	+	0.763604i	$0.723428\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.63546	+	2.63546i	0.0913134	+	0.0913134i
$834$		0			0
$835$	4.10725	+	20.4607i	0.142137	+	0.708071i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.34527			0.115491			0.0577457	−	0.998331i	$-0.481609\pi$
$839$	3.34527			0.115491			0.0577457	+	0.998331i	$0.481609\pi$
$840$		0			0
$841$	−27.8615			−0.960740
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.86140	+	1.23903i	0.0640342	+	0.0426238i
$846$		0			0
$847$	−1.05966	−	1.05966i	−0.0364102	−	0.0364102i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	0.657482i		−	0.0225382i
$852$		0			0
$853$	32.7803	−	32.7803i	1.12238	−	1.12238i	0.130994	−	0.991383i	$-0.458183\pi$
$853$	32.7803	−	32.7803i	1.12238	−	1.12238i	0.991383	−	0.130994i	$-0.0418169\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.7227	+	17.7227i	−0.605397	+	0.605397i	−0.941740	−	0.336343i	$-0.890810\pi$
$857$	−17.7227	+	17.7227i	−0.605397	+	0.605397i	0.336343	+	0.941740i	$0.390810\pi$
$858$		0			0
$859$			33.6637i			1.14859i	0.818648	+	0.574295i	$0.194724\pi$
$859$			33.6637i			1.14859i	−0.818648	+	0.574295i	$0.805276\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	21.2827	+	21.2827i	0.724473	+	0.724473i	0.969513	−	0.245040i	$-0.0788012\pi$
$863$	21.2827	+	21.2827i	0.724473	+	0.724473i	−0.245040	+	0.969513i	$0.578801\pi$
$864$		0			0
$865$	−20.0209	+	4.01898i	−0.680732	+	0.136649i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.08911			−0.0369457
$870$		0			0
$871$	−10.3586			−0.350987
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.39297	−	2.06113i	−0.0470910	−	0.0696790i
$876$		0			0
$877$	9.89063	+	9.89063i	0.333983	+	0.333983i	0.854097	−	0.520114i	$-0.174111\pi$
$877$	9.89063	+	9.89063i	0.333983	+	0.333983i	−0.520114	+	0.854097i	$0.674111\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			10.0955i			0.340127i	0.985433	+	0.170064i	$0.0543974\pi$
$881$			10.0955i			0.340127i	−0.985433	+	0.170064i	$0.945603\pi$
$882$		0			0
$883$	−35.5629	+	35.5629i	−1.19679	+	1.19679i	−0.221664	+	0.975123i	$0.571149\pi$
$883$	−35.5629	+	35.5629i	−1.19679	+	1.19679i	−0.975123	+	0.221664i	$0.928851\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.5912	+	33.5912i	−1.12788	+	1.12788i	−0.137362	+	0.990521i	$0.543862\pi$
$887$	−33.5912	+	33.5912i	−1.12788	+	1.12788i	−0.990521	+	0.137362i	$0.956138\pi$
$888$		0			0
$889$		−	2.62765i		−	0.0881287i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.71578	+	3.71578i	0.124344	+	0.124344i
$894$		0			0
$895$	29.5307	−	44.3643i	0.987102	−	1.48293i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.44828			−0.315118
$900$		0			0
$901$	−4.03320			−0.134365
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.39312	−	9.60445i	0.212514	−	0.319263i
$906$		0			0
$907$	34.5195	+	34.5195i	1.14620	+	1.14620i	0.987293	+	0.158910i	$0.0507978\pi$
$907$	34.5195	+	34.5195i	1.14620	+	1.14620i	0.158910	+	0.987293i	$0.449202\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	29.4643i		−	0.976197i	−0.872788	−	0.488099i	$-0.837691\pi$
$911$		−	29.4643i		−	0.976197i	0.872788	−	0.488099i	$-0.162309\pi$
$912$		0			0
$913$	7.01953	−	7.01953i	0.232313	−	0.232313i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.430816	+	0.430816i	−0.0142268	+	0.0142268i
$918$		0			0
$919$			2.35784i			0.0777780i	0.999244	+	0.0388890i	$0.0123819\pi$
$919$			2.35784i			0.0777780i	−0.999244	+	0.0388890i	$0.987618\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.46848	−	1.46848i	−0.0483356	−	0.0483356i
$924$		0			0
$925$	−0.594494	+	1.44962i	−0.0195468	+	0.0476631i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.92173			0.227095			0.113547	−	0.993533i	$-0.463779\pi$
$929$	6.92173			0.227095			0.113547	+	0.993533i	$0.463779\pi$
$930$		0			0
$931$	−39.9399			−1.30898
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.42785	+	0.487365i	−0.0793993	+	0.0159385i
$936$		0			0
$937$	5.33755	+	5.33755i	0.174370	+	0.174370i	0.788896	−	0.614526i	$-0.210653\pi$
$937$	5.33755	+	5.33755i	0.174370	+	0.174370i	−0.614526	+	0.788896i	$0.710653\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			35.9395i			1.17160i	0.810457	+	0.585798i	$0.199219\pi$
$941$			35.9395i			1.17160i	−0.810457	+	0.585798i	$0.800781\pi$
$942$		0			0
$943$	−2.74525	+	2.74525i	−0.0893976	+	0.0893976i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.5481	−	33.5481i	1.09017	−	1.09017i	0.0946575	−	0.995510i	$-0.469824\pi$
$947$	33.5481	−	33.5481i	1.09017	−	1.09017i	0.995510	−	0.0946575i	$-0.0301756\pi$
$948$		0			0
$949$			3.91163i			0.126977i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.2332	+	30.2332i	0.979348	+	0.979348i	0.999791	−	0.0204427i	$-0.00650757\pi$
$953$	30.2332	+	30.2332i	0.979348	+	0.979348i	−0.0204427	+	0.999791i	$0.506508\pi$
$954$		0			0
$955$	3.01906	+	2.00961i	0.0976945	+	0.0650295i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.10886			−0.0358068
$960$		0			0
$961$	47.4077			1.52928
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.58113	+	32.7846i	0.211854	+	1.05537i
$966$		0			0
$967$	−8.39257	−	8.39257i	−0.269887	−	0.269887i	0.559168	−	0.829055i	$-0.311121\pi$
$967$	−8.39257	−	8.39257i	−0.269887	−	0.269887i	−0.829055	+	0.559168i	$0.811121\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	41.8984i		−	1.34458i	−0.740286	−	0.672292i	$-0.765310\pi$
$971$		−	41.8984i		−	1.34458i	0.740286	−	0.672292i	$-0.234690\pi$
$972$		0			0
$973$	−0.462105	+	0.462105i	−0.0148144	+	0.0148144i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.2785	+	19.2785i	−0.616775	+	0.616775i	−0.944703	−	0.327928i	$-0.893650\pi$
$977$	−19.2785	+	19.2785i	−0.616775	+	0.616775i	0.327928	+	0.944703i	$0.393650\pi$
$978$		0			0
$979$		−	29.6107i		−	0.946362i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.6437	+	24.6437i	0.786013	+	0.786013i	0.980838	−	0.194825i	$-0.0624139\pi$
$983$	24.6437	+	24.6437i	0.786013	+	0.786013i	−0.194825	+	0.980838i	$0.562414\pi$
$984$		0			0
$985$	11.2458	+	56.0223i	0.358322	+	1.78502i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.4835			0.365154
$990$		0			0
$991$	33.8263			1.07453			0.537263	−	0.843415i	$-0.319458\pi$
$991$	33.8263			1.07453			0.537263	+	0.843415i	$0.319458\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25.7092	+	17.1131i	0.815038	+	0.542523i
$996$		0			0
$997$	9.98800	+	9.98800i	0.316323	+	0.316323i	0.847353	−	0.531030i	$-0.178195\pi$
$997$	9.98800	+	9.98800i	0.316323	+	0.316323i	−0.531030	+	0.847353i	$0.678195\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.y.b.1457.4	yes	24
3.2	odd	2		2340.2.y.a.1457.9	yes	24
5.3	odd	4		2340.2.y.a.53.9	✓	24
15.8	even	4	inner	2340.2.y.b.53.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.9	✓	24	5.3	odd	4
2340.2.y.a.1457.9	yes	24	3.2	odd	2
2340.2.y.b.53.4	yes	24	15.8	even	4	inner
2340.2.y.b.1457.4	yes	24	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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.23903 + 1.86140i) q^{5} +(-0.157336 - 0.157336i) q^{7} -2.06519i q^{11} +(-0.707107 + 0.707107i) q^{13} +(-0.379176 + 0.379176i) q^{17} -5.74635i q^{19} +(1.48364 + 1.48364i) q^{23} +(-1.92963 - 4.61265i) q^{25} -1.06702 q^{29} +8.85481 q^{31} +(0.487808 - 0.0979220i) q^{35} +(-0.221577 - 0.221577i) q^{37} +1.85034i q^{41} +(3.87003 - 3.87003i) q^{43} +(-0.646634 + 0.646634i) q^{47} -6.95049i q^{49} +(5.31837 + 5.31837i) q^{53} +(3.84415 + 2.55882i) q^{55} +0.345872 q^{59} +0.645233 q^{61} +(-0.440086 - 2.19233i) q^{65} +(7.32461 + 7.32461i) q^{67} +2.07674i q^{71} +(2.76594 - 2.76594i) q^{73} +(-0.324928 + 0.324928i) q^{77} -0.527368i q^{79} +(3.39898 + 3.39898i) q^{83} +(-0.235990 - 1.17561i) q^{85} +14.3380 q^{89} +0.222506 q^{91} +(10.6963 + 7.11987i) q^{95} +(-9.21138 - 9.21138i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{5} - 8 q^{7} + 8 q^{17} + 8 q^{23} + 16 q^{25} - 32 q^{29} - 8 q^{35} + 16 q^{37} - 8 q^{43} + 40 q^{47} + 8 q^{53} + 8 q^{55} - 56 q^{59} + 8 q^{61} + 4 q^{65} - 16 q^{67} + 72 q^{77} + 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^5 - 8 * q^7 + 8 * q^17 + 8 * q^23 + 16 * q^25 - 32 * q^29 - 8 * q^35 + 16 * q^37 - 8 * q^43 + 40 * q^47 + 8 * q^53 + 8 * q^55 - 56 * q^59 + 8 * q^61 + 4 * q^65 - 16 * q^67 + 72 * q^77 + 32 * q^83 - 32 * q^85 - 128 * q^89 - 8 * q^91 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more