LMFDB - Embedded newform 5472.2.f.c.1025.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5472,2,Mod(1025,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 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("5472.1025"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,28,0,0,0,-16,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(59)] == traces)

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 38 x^{18} + 528 x^{16} + 3442 x^{14} + 11480 x^{12} + 20550 x^{10} + 20369 x^{8} + 11136 x^{6} + \cdots + 16 $ x^20 + 38x^18 + 528x^16 + 3442x^14 + 11480x^12 + 20550x^10 + 20369x^8 + 11136x^6 + 3192x^4 + 416*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.1
Root			$-1.10730i$ of defining polynomial
Character	$\chi$	$=$	5472.1025
Dual form			5472.2.f.c.1025.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.50596i q^{5} -2.78457 q^{7} +3.14374i q^{11} -2.18316i q^{13} -1.90628i q^{17} +(-0.182595 + 4.35507i) q^{19} +2.24534i q^{23} -7.29179 q^{25} +5.61691 q^{29} -2.50357i q^{31} +9.76261i q^{35} +3.48569i q^{37} -7.87921 q^{41} -1.87923 q^{43} +12.0080i q^{47} +0.753846 q^{49} +2.03716 q^{53} +11.0218 q^{55} -12.5359 q^{59} -1.36228 q^{61} -7.65407 q^{65} -2.93912i q^{67} +15.0770 q^{71} +4.42105 q^{73} -8.75398i q^{77} -13.3186i q^{79} +8.96837i q^{83} -6.68335 q^{85} +10.9667 q^{89} +6.07916i q^{91} +(15.2687 + 0.640173i) q^{95} +1.89684i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) $ 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	3.50596i		−	1.56791i	−0.620815	−	0.783957i	$-0.713198\pi$
$5$		−	3.50596i		−	1.56791i	0.620815	−	0.783957i	$-0.286802\pi$
$6$		0			0
$7$	−2.78457			−1.05247			−0.526235	−	0.850339i	$-0.676397\pi$
$7$	−2.78457			−1.05247			−0.526235	+	0.850339i	$0.676397\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.14374i			0.947874i	0.880559	+	0.473937i	$0.157168\pi$
$11$			3.14374i			0.947874i	−0.880559	+	0.473937i	$0.842832\pi$
$12$		0			0
$13$		−	2.18316i		−	0.605499i	−0.953070	−	0.302749i	$-0.902095\pi$
$13$		−	2.18316i		−	0.605499i	0.953070	−	0.302749i	$-0.0979045\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.90628i		−	0.462341i	−0.972913	−	0.231170i	$-0.925745\pi$
$17$		−	1.90628i		−	0.462341i	0.972913	−	0.231170i	$-0.0742555\pi$
$18$		0			0
$19$	−0.182595	+	4.35507i	−0.0418902	+	0.999122i
$19$	−0.182595	+	4.35507i	−0.0418902	+	0.999122i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.24534i			0.468186i	0.972214	+	0.234093i	$0.0752120\pi$
$23$			2.24534i			0.468186i	−0.972214	+	0.234093i	$0.924788\pi$
$24$		0			0
$25$	−7.29179			−1.45836
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.61691			1.04303			0.521517	−	0.853241i	$-0.325366\pi$
$29$	5.61691			1.04303			0.521517	+	0.853241i	$0.325366\pi$
$30$		0			0
$31$		−	2.50357i		−	0.449654i	−0.974399	−	0.224827i	$-0.927818\pi$
$31$		−	2.50357i		−	0.449654i	0.974399	−	0.224827i	$-0.0721817\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			9.76261i			1.65018i
$36$		0			0
$37$			3.48569i			0.573043i	0.958074	+	0.286522i	$0.0924990\pi$
$37$			3.48569i			0.573043i	−0.958074	+	0.286522i	$0.907501\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.87921			−1.23053			−0.615263	−	0.788322i	$-0.710950\pi$
$41$	−7.87921			−1.23053			−0.615263	+	0.788322i	$0.710950\pi$
$42$		0			0
$43$	−1.87923			−0.286580			−0.143290	−	0.989681i	$-0.545768\pi$
$43$	−1.87923			−0.286580			−0.143290	+	0.989681i	$0.545768\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.0080i			1.75154i	0.482729	+	0.875770i	$0.339646\pi$
$47$			12.0080i			1.75154i	−0.482729	+	0.875770i	$0.660354\pi$
$48$		0			0
$49$	0.753846			0.107692
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.03716			0.279825			0.139912	−	0.990164i	$-0.455318\pi$
$53$	2.03716			0.279825			0.139912	+	0.990164i	$0.455318\pi$
$54$		0			0
$55$	11.0218			1.48619
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.5359			−1.63203			−0.816017	−	0.578028i	$-0.803822\pi$
$59$	−12.5359			−1.63203			−0.816017	+	0.578028i	$0.803822\pi$
$60$		0			0
$61$	−1.36228			−0.174422			−0.0872112	−	0.996190i	$-0.527796\pi$
$61$	−1.36228			−0.174422			−0.0872112	+	0.996190i	$0.527796\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.65407			−0.949370
$66$		0			0
$67$		−	2.93912i		−	0.359071i	−0.983752	−	0.179535i	$-0.942541\pi$
$67$		−	2.93912i		−	0.359071i	0.983752	−	0.179535i	$-0.0574594\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.0770			1.78930			0.894652	−	0.446763i	$-0.147423\pi$
$71$	15.0770			1.78930			0.894652	+	0.446763i	$0.147423\pi$
$72$		0			0
$73$	4.42105			0.517445			0.258723	−	0.965952i	$-0.416698\pi$
$73$	4.42105			0.517445			0.258723	+	0.965952i	$0.416698\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	8.75398i		−	0.997608i
$78$		0			0
$79$		−	13.3186i		−	1.49847i	−0.662307	−	0.749233i	$-0.730423\pi$
$79$		−	13.3186i		−	1.49847i	0.662307	−	0.749233i	$-0.269577\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.96837i			0.984407i	0.870480	+	0.492203i	$0.163808\pi$
$83$			8.96837i			0.984407i	−0.870480	+	0.492203i	$0.836192\pi$
$84$		0			0
$85$	−6.68335			−0.724911
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.9667			1.16246			0.581232	−	0.813738i	$-0.302571\pi$
$89$	10.9667			1.16246			0.581232	+	0.813738i	$0.302571\pi$
$90$		0			0
$91$			6.07916i			0.637269i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.2687	+	0.640173i	1.56654	+	0.0656803i
$96$		0			0
$97$			1.89684i			0.192595i	0.995353	+	0.0962975i	$0.0307000\pi$
$97$			1.89684i			0.192595i	−0.995353	+	0.0962975i	$0.969300\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			14.1865i			1.41161i	0.708406	+	0.705805i	$0.249414\pi$
$101$			14.1865i			1.41161i	−0.708406	+	0.705805i	$0.750586\pi$
$102$		0			0
$103$			14.0825i			1.38759i	0.720171	+	0.693797i	$0.244063\pi$
$103$			14.0825i			1.38759i	−0.720171	+	0.693797i	$0.755937\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.7020			−1.13128			−0.565640	−	0.824652i	$-0.691371\pi$
$107$	−11.7020			−1.13128			−0.565640	+	0.824652i	$0.691371\pi$
$108$		0			0
$109$			13.0076i			1.24591i	0.782259	+	0.622953i	$0.214067\pi$
$109$			13.0076i			1.24591i	−0.782259	+	0.622953i	$0.785933\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.70436			−0.442549			−0.221275	−	0.975212i	$-0.571022\pi$
$113$	−4.70436			−0.442549			−0.221275	+	0.975212i	$0.571022\pi$
$114$		0			0
$115$	7.87208			0.734076
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.30817i			0.486600i
$120$		0			0
$121$	1.11689			0.101535
$122$		0			0
$123$		0			0
$124$		0			0
$125$			8.03492i			0.718665i
$126$		0			0
$127$		−	12.1763i		−	1.08047i	−0.841513	−	0.540237i	$-0.818334\pi$
$127$		−	12.1763i		−	1.08047i	0.841513	−	0.540237i	$-0.181666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	0.794239i		−	0.0693930i	−0.999398	−	0.0346965i	$-0.988954\pi$
$131$		−	0.794239i		−	0.0693930i	0.999398	−	0.0346965i	$-0.0110465\pi$
$132$		0			0
$133$	0.508450	−	12.1270i	0.0440882	−	1.05155i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.8042i			0.923066i	0.887123	+	0.461533i	$0.152700\pi$
$137$			10.8042i			0.923066i	−0.887123	+	0.461533i	$0.847300\pi$
$138$		0			0
$139$	11.5620			0.980676			0.490338	−	0.871532i	$-0.336873\pi$
$139$	11.5620			0.980676			0.490338	+	0.871532i	$0.336873\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.86328			0.573936
$144$		0			0
$145$		−	19.6927i		−	1.63539i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			12.9576i			1.06153i	0.847518	+	0.530766i	$0.178096\pi$
$149$			12.9576i			1.06153i	−0.847518	+	0.530766i	$0.821904\pi$
$150$		0			0
$151$			4.40977i			0.358862i	0.983771	+	0.179431i	$0.0574257\pi$
$151$			4.40977i			0.358862i	−0.983771	+	0.179431i	$0.942574\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.77742			−0.705020
$156$		0			0
$157$	13.0038			1.03782			0.518909	−	0.854830i	$-0.326338\pi$
$157$	13.0038			1.03782			0.518909	+	0.854830i	$0.326338\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.25231i		−	0.492751i
$162$		0			0
$163$	19.0340			1.49086			0.745428	−	0.666586i	$-0.232245\pi$
$163$	19.0340			1.49086			0.745428	+	0.666586i	$0.232245\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.39760			0.108149			0.0540747	−	0.998537i	$-0.482779\pi$
$167$	1.39760			0.108149			0.0540747	+	0.998537i	$0.482779\pi$
$168$		0			0
$169$	8.23383			0.633371
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.40380			0.638929			0.319465	−	0.947598i	$-0.396497\pi$
$173$	8.40380			0.638929			0.319465	+	0.947598i	$0.396497\pi$
$174$		0			0
$175$	20.3045			1.53488
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.83876			−0.361666			−0.180833	−	0.983514i	$-0.557879\pi$
$179$	−4.83876			−0.361666			−0.180833	+	0.983514i	$0.557879\pi$
$180$		0			0
$181$		−	20.2854i		−	1.50780i	−0.656987	−	0.753902i	$-0.728169\pi$
$181$		−	20.2854i		−	1.50780i	0.656987	−	0.753902i	$-0.271831\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.2207			0.898483
$186$		0			0
$187$	5.99285			0.438241
$188$		0			0
$189$		0			0
$190$		0			0
$191$			5.41234i			0.391623i	0.980642	+	0.195812i	$0.0627341\pi$
$191$			5.41234i			0.391623i	−0.980642	+	0.195812i	$0.937266\pi$
$192$		0			0
$193$			16.0698i			1.15673i	0.815777	+	0.578366i	$0.196310\pi$
$193$			16.0698i			1.15673i	−0.815777	+	0.578366i	$0.803690\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	22.9191i		−	1.63292i	−0.577402	−	0.816460i	$-0.695933\pi$
$197$		−	22.9191i		−	1.63292i	0.577402	−	0.816460i	$-0.304067\pi$
$198$		0			0
$199$	17.5199			1.24196			0.620978	−	0.783828i	$-0.286736\pi$
$199$	17.5199			1.24196			0.620978	+	0.783828i	$0.286736\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−15.6407			−1.09776
$204$		0			0
$205$			27.6242i			1.92936i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.6912	−	0.574033i	−0.947042	−	0.0397067i
$210$		0			0
$211$		−	2.93912i		−	0.202337i	−0.994869	−	0.101169i	$-0.967742\pi$
$211$		−	2.93912i		−	0.202337i	0.994869	−	0.101169i	$-0.0322582\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.58852i			0.449333i
$216$		0			0
$217$			6.97137i			0.473247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.16171			−0.279947
$222$		0			0
$223$		−	1.66938i		−	0.111790i	−0.998437	−	0.0558951i	$-0.982199\pi$
$223$		−	1.66938i		−	0.111790i	0.998437	−	0.0558951i	$-0.0178013\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.75846			0.249458			0.124729	−	0.992191i	$-0.460194\pi$
$227$	3.75846			0.249458			0.124729	+	0.992191i	$0.460194\pi$
$228$		0			0
$229$	18.7710			1.24042			0.620212	−	0.784435i	$-0.287047\pi$
$229$	18.7710			1.24042			0.620212	+	0.784435i	$0.287047\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.27259i			0.410931i	0.978664	+	0.205466i	$0.0658709\pi$
$233$			6.27259i			0.410931i	−0.978664	+	0.205466i	$0.934129\pi$
$234$		0			0
$235$	42.0995			2.74627
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.15549i		−	0.139427i	−0.997567	−	0.0697136i	$-0.977791\pi$
$239$		−	2.15549i		−	0.139427i	0.997567	−	0.0697136i	$-0.0222085\pi$
$240$		0			0
$241$		−	9.23406i		−	0.594818i	−0.954750	−	0.297409i	$-0.903878\pi$
$241$		−	9.23406i		−	0.594818i	0.954750	−	0.297409i	$-0.0961225\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	2.64296i		−	0.168852i
$246$		0			0
$247$	9.50781	+	0.398634i	0.604967	+	0.0253645i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.3023i			0.650277i	0.945666	+	0.325139i	$0.105411\pi$
$251$			10.3023i			0.650277i	−0.945666	+	0.325139i	$0.894589\pi$
$252$		0			0
$253$	−7.05877			−0.443781
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.60001			0.286941			0.143470	−	0.989655i	$-0.454174\pi$
$257$	4.60001			0.286941			0.143470	+	0.989655i	$0.454174\pi$
$258$		0			0
$259$		−	9.70614i		−	0.603110i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.5673i		−	1.14491i	−0.819936	−	0.572455i	$-0.805991\pi$
$263$		−	18.5673i		−	1.14491i	0.819936	−	0.572455i	$-0.194009\pi$
$264$		0			0
$265$		−	7.14219i		−	0.438742i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−26.6530			−1.62506			−0.812531	−	0.582917i	$-0.801911\pi$
$269$	−26.6530			−1.62506			−0.812531	+	0.582917i	$0.801911\pi$
$270$		0			0
$271$	25.0718			1.52300			0.761501	−	0.648164i	$-0.224463\pi$
$271$	25.0718			1.52300			0.761501	+	0.648164i	$0.224463\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	22.9235i		−	1.38234i
$276$		0			0
$277$	−10.9458			−0.657670			−0.328835	−	0.944387i	$-0.606656\pi$
$277$	−10.9458			−0.657670			−0.328835	+	0.944387i	$0.606656\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.9458			1.84607			0.923035	−	0.384716i	$-0.125701\pi$
$281$	30.9458			1.84607			0.923035	+	0.384716i	$0.125701\pi$
$282$		0			0
$283$	−5.25414			−0.312326			−0.156163	−	0.987731i	$-0.549913\pi$
$283$	−5.25414			−0.312326			−0.156163	+	0.987731i	$0.549913\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.9402			1.29509
$288$		0			0
$289$	13.3661			0.786241
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−27.2043			−1.58929			−0.794646	−	0.607073i	$-0.792344\pi$
$293$	−27.2043			−1.58929			−0.794646	+	0.607073i	$0.792344\pi$
$294$		0			0
$295$			43.9504i			2.55889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.90193			0.283486
$300$		0			0
$301$	5.23286			0.301617
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.77612i			0.273480i
$306$		0			0
$307$		−	9.67276i		−	0.552054i	−0.961150	−	0.276027i	$-0.910982\pi$
$307$		−	9.67276i		−	0.552054i	0.961150	−	0.276027i	$-0.0890179\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.44525i		−	0.365477i	−0.983162	−	0.182738i	$-0.941504\pi$
$311$		−	6.44525i		−	0.365477i	0.983162	−	0.182738i	$-0.0584961\pi$
$312$		0			0
$313$	−1.65031			−0.0932809			−0.0466405	−	0.998912i	$-0.514852\pi$
$313$	−1.65031			−0.0932809			−0.0466405	+	0.998912i	$0.514852\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.80495			−0.157542			−0.0787709	−	0.996893i	$-0.525100\pi$
$317$	−2.80495			−0.157542			−0.0787709	+	0.996893i	$0.525100\pi$
$318$		0			0
$319$			17.6581i			0.988665i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.30199	+	0.348078i	0.461935	+	0.0193676i
$324$		0			0
$325$			15.9191i			0.883033i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	33.4370i		−	1.84344i
$330$		0			0
$331$			4.10266i			0.225502i	0.993623	+	0.112751i	$0.0359663\pi$
$331$			4.10266i			0.225502i	−0.993623	+	0.112751i	$0.964034\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.3044			−0.562992
$336$		0			0
$337$			10.9601i			0.597033i	0.954404	+	0.298517i	$0.0964918\pi$
$337$			10.9601i			0.597033i	−0.954404	+	0.298517i	$0.903508\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.87058			0.426216
$342$		0			0
$343$	17.3929			0.939127
$344$		0			0
$345$		0			0
$346$		0			0
$347$			9.39498i			0.504349i	0.967682	+	0.252175i	$0.0811457\pi$
$347$			9.39498i			0.504349i	−0.967682	+	0.252175i	$0.918854\pi$
$348$		0			0
$349$	−3.92874			−0.210301			−0.105150	−	0.994456i	$-0.533532\pi$
$349$	−3.92874			−0.210301			−0.105150	+	0.994456i	$0.533532\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.5138i			0.985392i	0.870201	+	0.492696i	$0.163989\pi$
$353$			18.5138i			0.985392i	−0.870201	+	0.492696i	$0.836011\pi$
$354$		0			0
$355$		−	52.8593i		−	2.80548i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			26.0096i			1.37273i	0.727256	+	0.686367i	$0.240795\pi$
$359$			26.0096i			1.37273i	−0.727256	+	0.686367i	$0.759205\pi$
$360$		0			0
$361$	−18.9333	−	1.59043i	−0.996490	−	0.0837069i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	15.5001i		−	0.811310i
$366$		0			0
$367$	−11.1383			−0.581414			−0.290707	−	0.956812i	$-0.593890\pi$
$367$	−11.1383			−0.581414			−0.290707	+	0.956812i	$0.593890\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.67261			−0.294507
$372$		0			0
$373$		−	20.7683i		−	1.07534i	−0.843154	−	0.537672i	$-0.819304\pi$
$373$		−	20.7683i		−	1.07534i	0.843154	−	0.537672i	$-0.180696\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.2626i		−	0.631556i
$378$		0			0
$379$		−	21.3220i		−	1.09524i	−0.836727	−	0.547620i	$-0.815534\pi$
$379$		−	21.3220i		−	1.09524i	0.836727	−	0.547620i	$-0.184466\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.9157			1.01765			0.508823	−	0.860871i	$-0.330081\pi$
$383$	19.9157			1.01765			0.508823	+	0.860871i	$0.330081\pi$
$384$		0			0
$385$	−30.6911			−1.56417
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.17372i		−	0.110212i	−0.998481	−	0.0551059i	$-0.982450\pi$
$389$		−	2.17372i		−	0.110212i	0.998481	−	0.0551059i	$-0.0175496\pi$
$390$		0			0
$391$	4.28025			0.216461
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−46.6947			−2.34947
$396$		0			0
$397$	−15.7292			−0.789426			−0.394713	−	0.918805i	$-0.629156\pi$
$397$	−15.7292			−0.789426			−0.394713	+	0.918805i	$0.629156\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	27.2196			1.35928			0.679642	−	0.733544i	$-0.262135\pi$
$401$	27.2196			1.35928			0.679642	+	0.733544i	$0.262135\pi$
$402$		0			0
$403$	−5.46568			−0.272265
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.9581			−0.543173
$408$		0			0
$409$		−	6.56956i		−	0.324844i	−0.986721	−	0.162422i	$-0.948069\pi$
$409$		−	6.56956i		−	0.324844i	0.986721	−	0.162422i	$-0.0519306\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	34.9071			1.71767
$414$		0			0
$415$	31.4428			1.54347
$416$		0			0
$417$		0			0
$418$		0			0
$419$			17.1525i			0.837953i	0.907997	+	0.418976i	$0.137611\pi$
$419$			17.1525i			0.837953i	−0.907997	+	0.418976i	$0.862389\pi$
$420$		0			0
$421$		−	14.7337i		−	0.718075i	−0.933323	−	0.359037i	$-0.883105\pi$
$421$		−	14.7337i		−	0.718075i	0.933323	−	0.359037i	$-0.116895\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			13.9002i			0.674258i
$426$		0			0
$427$	3.79338			0.183574
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.9721			0.576677			0.288339	−	0.957528i	$-0.406897\pi$
$431$	11.9721			0.576677			0.288339	+	0.957528i	$0.406897\pi$
$432$		0			0
$433$			30.1154i			1.44725i	0.690192	+	0.723626i	$0.257526\pi$
$433$			30.1154i			1.44725i	−0.690192	+	0.723626i	$0.742474\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.77862	−	0.409989i	−0.467775	−	0.0196124i
$438$		0			0
$439$			24.5526i			1.17183i	0.810372	+	0.585915i	$0.199265\pi$
$439$			24.5526i			1.17183i	−0.810372	+	0.585915i	$0.800735\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			10.0911i			0.479444i	0.970842	+	0.239722i	$0.0770564\pi$
$443$			10.0911i			0.479444i	−0.970842	+	0.239722i	$0.922944\pi$
$444$		0			0
$445$		−	38.4487i		−	1.82264i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−36.9743			−1.74492			−0.872462	−	0.488681i	$-0.837478\pi$
$449$	−36.9743			−1.74492			−0.872462	+	0.488681i	$0.837478\pi$
$450$		0			0
$451$		−	24.7702i		−	1.16638i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21.3133			0.999184
$456$		0			0
$457$	−26.6132			−1.24491			−0.622457	−	0.782654i	$-0.713865\pi$
$457$	−26.6132			−1.24491			−0.622457	+	0.782654i	$0.713865\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			15.6438i			0.728605i	0.931281	+	0.364303i	$0.118693\pi$
$461$			15.6438i			0.728605i	−0.931281	+	0.364303i	$0.881307\pi$
$462$		0			0
$463$	−39.7965			−1.84950			−0.924751	−	0.380574i	$-0.875727\pi$
$463$	−39.7965			−1.84950			−0.924751	+	0.380574i	$0.875727\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			41.3231i			1.91220i	0.293033	+	0.956102i	$0.405335\pi$
$467$			41.3231i			1.91220i	−0.293033	+	0.956102i	$0.594665\pi$
$468$		0			0
$469$			8.18419i			0.377911i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.90782i		−	0.271642i
$474$		0			0
$475$	1.33145	−	31.7563i	0.0610909	−	1.45708i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			14.3039i			0.653560i	0.945101	+	0.326780i	$0.105964\pi$
$479$			14.3039i			0.653560i	−0.945101	+	0.326780i	$0.894036\pi$
$480$		0			0
$481$	7.60980			0.346977
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.65025			0.301973
$486$		0			0
$487$			26.3112i			1.19227i	0.802883	+	0.596136i	$0.203298\pi$
$487$			26.3112i			1.19227i	−0.802883	+	0.596136i	$0.796702\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			7.81646i			0.352752i	0.984323	+	0.176376i	$0.0564375\pi$
$491$			7.81646i			0.352752i	−0.984323	+	0.176376i	$0.943563\pi$
$492$		0			0
$493$		−	10.7074i		−	0.482237i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−41.9829			−1.88319
$498$		0			0
$499$	5.88939			0.263645			0.131823	−	0.991273i	$-0.457917\pi$
$499$	5.88939			0.263645			0.131823	+	0.991273i	$0.457917\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			29.5393i			1.31709i	0.752540	+	0.658546i	$0.228828\pi$
$503$			29.5393i			1.31709i	−0.752540	+	0.658546i	$0.771172\pi$
$504$		0			0
$505$	49.7374			2.21328
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.1322			0.537749			0.268875	−	0.963175i	$-0.413348\pi$
$509$	12.1322			0.537749			0.268875	+	0.963175i	$0.413348\pi$
$510$		0			0
$511$	−12.3107			−0.544595
$512$		0			0
$513$		0			0
$514$		0			0
$515$	49.3729			2.17563
$516$		0			0
$517$	−37.7499			−1.66024
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.4327			−0.588497			−0.294248	−	0.955729i	$-0.595069\pi$
$521$	−13.4327			−0.588497			−0.294248	+	0.955729i	$0.595069\pi$
$522$		0			0
$523$			28.3638i			1.24026i	0.784498	+	0.620131i	$0.212921\pi$
$523$			28.3638i			1.24026i	−0.784498	+	0.620131i	$0.787079\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.77250			−0.207894
$528$		0			0
$529$	17.9584			0.780802
$530$		0			0
$531$		0			0
$532$		0			0
$533$			17.2015i			0.745082i
$534$		0			0
$535$			41.0270i			1.77375i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			2.36990i			0.102079i
$540$		0			0
$541$	−14.3527			−0.617073			−0.308536	−	0.951213i	$-0.599839\pi$
$541$	−14.3527			−0.617073			−0.308536	+	0.951213i	$0.599839\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	45.6043			1.95348
$546$		0			0
$547$		−	17.7618i		−	0.759440i	−0.925101	−	0.379720i	$-0.876020\pi$
$547$		−	17.7618i		−	0.759440i	0.925101	−	0.379720i	$-0.123980\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.02562	+	24.4621i	−0.0436930	+	1.04212i
$552$		0			0
$553$			37.0868i			1.57709i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			19.2505i			0.815671i	0.913055	+	0.407835i	$0.133716\pi$
$557$			19.2505i			0.815671i	−0.913055	+	0.407835i	$0.866284\pi$
$558$		0			0
$559$			4.10266i			0.173524i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−40.7756			−1.71849			−0.859244	−	0.511565i	$-0.829066\pi$
$563$	−40.7756			−1.71849			−0.859244	+	0.511565i	$0.829066\pi$
$564$		0			0
$565$			16.4933i			0.693880i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.26712			0.220809			0.110404	−	0.993887i	$-0.464785\pi$
$569$	5.26712			0.220809			0.110404	+	0.993887i	$0.464785\pi$
$570$		0			0
$571$	−35.6196			−1.49063			−0.745317	−	0.666711i	$-0.767702\pi$
$571$	−35.6196			−1.49063			−0.745317	+	0.666711i	$0.767702\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	16.3725i		−	0.682782i
$576$		0			0
$577$	−28.1508			−1.17193			−0.585966	−	0.810335i	$-0.699285\pi$
$577$	−28.1508			−1.17193			−0.585966	+	0.810335i	$0.699285\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	24.9731i		−	1.03606i
$582$		0			0
$583$			6.40429i			0.265239i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			13.1580i			0.543090i	0.962426	+	0.271545i	$0.0875345\pi$
$587$			13.1580i			0.543090i	−0.962426	+	0.271545i	$0.912465\pi$
$588$		0			0
$589$	10.9032	+	0.457140i	0.449260	+	0.0188361i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.55647i		−	0.0639164i	−0.999489	−	0.0319582i	$-0.989826\pi$
$593$		−	1.55647i		−	0.0639164i	0.999489	−	0.0319582i	$-0.0101743\pi$
$594$		0			0
$595$	18.6103			0.762947
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.4181			0.997698			0.498849	−	0.866689i	$-0.333756\pi$
$599$	24.4181			0.997698			0.498849	+	0.866689i	$0.333756\pi$
$600$		0			0
$601$			16.0698i			0.655503i	0.944764	+	0.327751i	$0.106291\pi$
$601$			16.0698i			0.655503i	−0.944764	+	0.327751i	$0.893709\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	3.91577i		−	0.159199i
$606$		0			0
$607$		−	8.58273i		−	0.348362i	−0.984714	−	0.174181i	$-0.944272\pi$
$607$		−	8.58273i		−	0.348362i	0.984714	−	0.174181i	$-0.0557278\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	26.2152			1.06056
$612$		0			0
$613$	26.5294			1.07151			0.535757	−	0.844372i	$-0.320026\pi$
$613$	26.5294			1.07151			0.535757	+	0.844372i	$0.320026\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10.1505i			0.408642i	0.978904	+	0.204321i	$0.0654987\pi$
$617$			10.1505i			0.408642i	−0.978904	+	0.204321i	$0.934501\pi$
$618$		0			0
$619$	−37.3056			−1.49944			−0.749720	−	0.661755i	$-0.769812\pi$
$619$	−37.3056			−1.49944			−0.749720	+	0.661755i	$0.769812\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.5375			−1.22346
$624$		0			0
$625$	−8.28878			−0.331551
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.64469			0.264941
$630$		0			0
$631$	−16.7181			−0.665535			−0.332768	−	0.943009i	$-0.607982\pi$
$631$	−16.7181			−0.665535			−0.332768	+	0.943009i	$0.607982\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−42.6898			−1.69409
$636$		0			0
$637$		−	1.64576i		−	0.0652075i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.70807			0.304451			0.152225	−	0.988346i	$-0.451356\pi$
$641$	7.70807			0.304451			0.152225	+	0.988346i	$0.451356\pi$
$642$		0			0
$643$	−36.0381			−1.42120			−0.710601	−	0.703595i	$-0.751577\pi$
$643$	−36.0381			−1.42120			−0.710601	+	0.703595i	$0.751577\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7.27270i		−	0.285920i	−0.989729	−	0.142960i	$-0.954338\pi$
$647$		−	7.27270i		−	0.285920i	0.989729	−	0.142960i	$-0.0456620\pi$
$648$		0			0
$649$		−	39.4096i		−	1.54696i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			13.7361i			0.537537i	0.963205	+	0.268768i	$0.0866166\pi$
$653$			13.7361i			0.537537i	−0.963205	+	0.268768i	$0.913383\pi$
$654$		0			0
$655$	−2.78457			−0.108802
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.3520			0.753845			0.376923	−	0.926245i	$-0.376982\pi$
$659$	19.3520			0.753845			0.376923	+	0.926245i	$0.376982\pi$
$660$		0			0
$661$			42.6995i			1.66082i	0.557153	+	0.830410i	$0.311894\pi$
$661$			42.6995i			1.66082i	−0.557153	+	0.830410i	$0.688106\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−42.5169	−	1.78261i	−1.64873	−	0.0691265i
$666$		0			0
$667$			12.6119i			0.488334i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.28267i		−	0.165330i
$672$		0			0
$673$		−	44.0581i		−	1.69832i	−0.528139	−	0.849158i	$-0.677110\pi$
$673$		−	44.0581i		−	1.69832i	0.528139	−	0.849158i	$-0.322890\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−33.0617			−1.27066			−0.635332	−	0.772239i	$-0.719137\pi$
$677$	−33.0617			−1.27066			−0.635332	+	0.772239i	$0.719137\pi$
$678$		0			0
$679$		−	5.28189i		−	0.202700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−40.7125			−1.55782			−0.778910	−	0.627136i	$-0.784227\pi$
$683$	−40.7125			−1.55782			−0.778910	+	0.627136i	$0.784227\pi$
$684$		0			0
$685$	37.8792			1.44729
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	4.44743i		−	0.169434i
$690$		0			0
$691$	−15.8127			−0.601544			−0.300772	−	0.953696i	$-0.597244\pi$
$691$	−15.8127			−0.601544			−0.300772	+	0.953696i	$0.597244\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	40.5360i		−	1.53762i
$696$		0			0
$697$			15.0200i			0.568922i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.35127i		−	0.0510366i	−0.999674	−	0.0255183i	$-0.991876\pi$
$701$		−	1.35127i		−	0.0510366i	0.999674	−	0.0255183i	$-0.00812362\pi$
$702$		0			0
$703$	−15.1804	−	0.636470i	−0.572540	−	0.0240049i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	39.5033i		−	1.48568i
$708$		0			0
$709$	40.8993			1.53600			0.768002	−	0.640447i	$-0.221251\pi$
$709$	40.8993			1.53600			0.768002	+	0.640447i	$0.221251\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.62137			0.210522
$714$		0			0
$715$		−	24.0624i		−	0.899883i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			41.3828i			1.54332i	0.636037	+	0.771659i	$0.280573\pi$
$719$			41.3828i			1.54332i	−0.636037	+	0.771659i	$0.719427\pi$
$720$		0			0
$721$		−	39.2139i		−	1.46040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−40.9573			−1.52112
$726$		0			0
$727$	−49.9715			−1.85334			−0.926671	−	0.375874i	$-0.877343\pi$
$727$	−49.9715			−1.85334			−0.926671	+	0.375874i	$0.877343\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			3.58234i			0.132498i
$732$		0			0
$733$	−23.1235			−0.854086			−0.427043	−	0.904231i	$-0.640445\pi$
$733$	−23.1235			−0.854086			−0.427043	+	0.904231i	$0.640445\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.23983			0.340354
$738$		0			0
$739$	49.2328			1.81106			0.905529	−	0.424283i	$-0.139474\pi$
$739$	49.2328			1.81106			0.905529	+	0.424283i	$0.139474\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.1799			0.520212			0.260106	−	0.965580i	$-0.416243\pi$
$743$	14.1799			0.520212			0.260106	+	0.965580i	$0.416243\pi$
$744$		0			0
$745$	45.4291			1.66439
$746$		0			0
$747$		0			0
$748$		0			0
$749$	32.5852			1.19064
$750$		0			0
$751$		−	16.0590i		−	0.586003i	−0.956112	−	0.293001i	$-0.905346\pi$
$751$		−	16.0590i		−	0.586003i	0.956112	−	0.293001i	$-0.0946540\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.4605			0.562666
$756$		0			0
$757$	13.5863			0.493803			0.246901	−	0.969041i	$-0.420588\pi$
$757$	13.5863			0.493803			0.246901	+	0.969041i	$0.420588\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			39.3183i			1.42529i	0.701526	+	0.712644i	$0.252502\pi$
$761$			39.3183i			1.42529i	−0.701526	+	0.712644i	$0.747498\pi$
$762$		0			0
$763$		−	36.2207i		−	1.31128i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			27.3678i			0.988194i
$768$		0			0
$769$	37.8829			1.36609			0.683047	−	0.730375i	$-0.260654\pi$
$769$	37.8829			1.36609			0.683047	+	0.730375i	$0.260654\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	25.0425			0.900716			0.450358	−	0.892848i	$-0.351296\pi$
$773$	25.0425			0.900716			0.450358	+	0.892848i	$0.351296\pi$
$774$		0			0
$775$			18.2555i			0.655757i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.43871	−	34.3145i	0.0515470	−	1.22945i
$780$		0			0
$781$			47.3980i			1.69604i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	45.5909i		−	1.62721i
$786$		0			0
$787$			51.0994i			1.82150i	0.412963	+	0.910748i	$0.364494\pi$
$787$			51.0994i			1.82150i	−0.412963	+	0.910748i	$0.635506\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.0996			0.465770
$792$		0			0
$793$			2.97408i			0.105613i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	46.7671			1.65658			0.828288	−	0.560303i	$-0.189315\pi$
$797$	46.7671			1.65658			0.828288	+	0.560303i	$0.189315\pi$
$798$		0			0
$799$	22.8905			0.809808
$800$		0			0
$801$		0			0
$802$		0			0
$803$			13.8987i			0.490473i
$804$		0			0
$805$	−21.9204			−0.772592
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	21.2107i		−	0.745730i	−0.927886	−	0.372865i	$-0.878376\pi$
$809$		−	21.2107i		−	0.745730i	0.927886	−	0.372865i	$-0.121624\pi$
$810$		0			0
$811$			1.21261i			0.0425806i	0.999773	+	0.0212903i	$0.00677743\pi$
$811$			1.21261i			0.0425806i	−0.999773	+	0.0212903i	$0.993223\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	66.7325i		−	2.33754i
$816$		0			0
$817$	0.343139	−	8.18419i	0.0120049	−	0.286329i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	8.75664i		−	0.305609i	−0.988256	−	0.152804i	$-0.951170\pi$
$821$		−	8.75664i		−	0.305609i	0.988256	−	0.152804i	$-0.0488304\pi$
$822$		0			0
$823$	9.81448			0.342111			0.171056	−	0.985261i	$-0.445282\pi$
$823$	9.81448			0.342111			0.171056	+	0.985261i	$0.445282\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.2379			−0.842836			−0.421418	−	0.906867i	$-0.638467\pi$
$827$	−24.2379			−0.842836			−0.421418	+	0.906867i	$0.638467\pi$
$828$		0			0
$829$			5.39609i			0.187414i	0.995600	+	0.0937069i	$0.0298717\pi$
$829$			5.39609i			0.187414i	−0.995600	+	0.0937069i	$0.970128\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1.43704i		−	0.0497905i
$834$		0			0
$835$		−	4.89993i		−	0.169569i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.4745			0.568765			0.284382	−	0.958711i	$-0.408212\pi$
$839$	16.4745			0.568765			0.284382	+	0.958711i	$0.408212\pi$
$840$		0			0
$841$	2.54972			0.0879214
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	28.8675i		−	0.993072i
$846$		0			0
$847$	−3.11005			−0.106863
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.82655			−0.268291
$852$		0			0
$853$	25.3081			0.866534			0.433267	−	0.901266i	$-0.357361\pi$
$853$	25.3081			0.866534			0.433267	+	0.901266i	$0.357361\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.7158			1.08339			0.541696	−	0.840575i	$-0.317783\pi$
$857$	31.7158			1.08339			0.541696	+	0.840575i	$0.317783\pi$
$858$		0			0
$859$	−2.26803			−0.0773841			−0.0386920	−	0.999251i	$-0.512319\pi$
$859$	−2.26803			−0.0773841			−0.0386920	+	0.999251i	$0.512319\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.8581			0.573857			0.286928	−	0.957952i	$-0.407366\pi$
$863$	16.8581			0.573857			0.286928	+	0.957952i	$0.407366\pi$
$864$		0			0
$865$		−	29.4634i		−	1.00179i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	41.8704			1.42036
$870$		0			0
$871$	−6.41656			−0.217417
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	22.3738i		−	0.756373i
$876$		0			0
$877$		−	37.8920i		−	1.27952i	−0.768575	−	0.639760i	$-0.779034\pi$
$877$		−	37.8920i		−	1.27952i	0.768575	−	0.639760i	$-0.220966\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			33.6386i			1.13331i	0.823954	+	0.566657i	$0.191764\pi$
$881$			33.6386i			1.13331i	−0.823954	+	0.566657i	$0.808236\pi$
$882$		0			0
$883$	−39.3255			−1.32341			−0.661704	−	0.749765i	$-0.730167\pi$
$883$	−39.3255			−1.32341			−0.661704	+	0.749765i	$0.730167\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	41.6126			1.39721			0.698607	−	0.715506i	$-0.253804\pi$
$887$	41.6126			1.39721			0.698607	+	0.715506i	$0.253804\pi$
$888$		0			0
$889$			33.9059i			1.13717i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−52.2955	−	2.19260i	−1.75000	−	0.0733724i
$894$		0			0
$895$			16.9645i			0.567062i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	14.0623i		−	0.469005i
$900$		0			0
$901$		−	3.88339i		−	0.129374i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−71.1200			−2.36411
$906$		0			0
$907$		−	40.4688i		−	1.34375i	−0.740667	−	0.671873i	$-0.765490\pi$
$907$		−	40.4688i		−	1.34375i	0.740667	−	0.671873i	$-0.234510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28.1766			−0.933532			−0.466766	−	0.884381i	$-0.654581\pi$
$911$	−28.1766			−0.933532			−0.466766	+	0.884381i	$0.654581\pi$
$912$		0			0
$913$	−28.1943			−0.933094
$914$		0			0
$915$		0			0
$916$		0			0
$917$			2.21162i			0.0730340i
$918$		0			0
$919$	11.1939			0.369251			0.184626	−	0.982809i	$-0.440893\pi$
$919$	11.1939			0.369251			0.184626	+	0.982809i	$0.440893\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	32.9153i		−	1.08342i
$924$		0			0
$925$		−	25.4169i		−	0.835702i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			38.4148i			1.26035i	0.776454	+	0.630174i	$0.217016\pi$
$929$			38.4148i			1.26035i	−0.776454	+	0.630174i	$0.782984\pi$
$930$		0			0
$931$	−0.137649	+	3.28305i	−0.00451125	+	0.107598i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	21.0107i		−	0.687124i
$936$		0			0
$937$	−43.0028			−1.40484			−0.702421	−	0.711762i	$-0.747897\pi$
$937$	−43.0028			−1.40484			−0.702421	+	0.711762i	$0.747897\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−47.4419			−1.54656			−0.773282	−	0.634063i	$-0.781386\pi$
$941$	−47.4419			−1.54656			−0.773282	+	0.634063i	$0.781386\pi$
$942$		0			0
$943$		−	17.6915i		−	0.576115i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.23073i		−	0.104985i	−0.998621	−	0.0524923i	$-0.983284\pi$
$947$		−	3.23073i		−	0.104985i	0.998621	−	0.0524923i	$-0.0167165\pi$
$948$		0			0
$949$		−	9.65185i		−	0.313312i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.95190			−0.257587			−0.128794	−	0.991671i	$-0.541110\pi$
$953$	−7.95190			−0.257587			−0.128794	+	0.991671i	$0.541110\pi$
$954$		0			0
$955$	18.9755			0.614032
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	30.0851i		−	0.971499i
$960$		0			0
$961$	24.7321			0.797811
$962$		0			0
$963$		0			0
$964$		0			0
$965$	56.3403			1.81366
$966$		0			0
$967$	−43.3014			−1.39248			−0.696240	−	0.717809i	$-0.745145\pi$
$967$	−43.3014			−1.39248			−0.696240	+	0.717809i	$0.745145\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−58.1503			−1.86613			−0.933066	−	0.359705i	$-0.882877\pi$
$971$	−58.1503			−1.86613			−0.933066	+	0.359705i	$0.882877\pi$
$972$		0			0
$973$	−32.1952			−1.03213
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10.5957			−0.338985			−0.169492	−	0.985531i	$-0.554213\pi$
$977$	−10.5957			−0.338985			−0.169492	+	0.985531i	$0.554213\pi$
$978$		0			0
$979$			34.4763i			1.10187i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	56.6895			1.80812			0.904058	−	0.427410i	$-0.140574\pi$
$983$	56.6895			1.80812			0.904058	+	0.427410i	$0.140574\pi$
$984$		0			0
$985$	−80.3536			−2.56028
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	4.21952i		−	0.134173i
$990$		0			0
$991$		−	48.3971i		−	1.53738i	−0.639619	−	0.768692i	$-0.720908\pi$
$991$		−	48.3971i		−	1.53738i	0.639619	−	0.768692i	$-0.279092\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	61.4243i		−	1.94728i
$996$		0			0
$997$	−14.1796			−0.449073			−0.224537	−	0.974466i	$-0.572087\pi$
$997$	−14.1796			−0.449073			−0.224537	+	0.974466i	$0.572087\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.f.c.1025.1	✓	20
3.2	odd	2		5472.2.f.d.1025.19	yes	20
4.3	odd	2	inner	5472.2.f.c.1025.2	yes	20
12.11	even	2		5472.2.f.d.1025.20	yes	20
19.18	odd	2		5472.2.f.d.1025.1	yes	20
57.56	even	2	inner	5472.2.f.c.1025.19	yes	20
76.75	even	2		5472.2.f.d.1025.2	yes	20
228.227	odd	2	inner	5472.2.f.c.1025.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5472.2.f.c.1025.1	✓	20	1.1	even	1	trivial
5472.2.f.c.1025.2	yes	20	4.3	odd	2	inner
5472.2.f.c.1025.19	yes	20	57.56	even	2	inner
5472.2.f.c.1025.20	yes	20	228.227	odd	2	inner
5472.2.f.d.1025.1	yes	20	19.18	odd	2
5472.2.f.d.1025.2	yes	20	76.75	even	2
5472.2.f.d.1025.19	yes	20	3.2	odd	2
5472.2.f.d.1025.20	yes	20	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.50596i q^{5} -2.78457 q^{7} +3.14374i q^{11} -2.18316i q^{13} -1.90628i q^{17} +(-0.182595 + 4.35507i) q^{19} +2.24534i q^{23} -7.29179 q^{25} +5.61691 q^{29} -2.50357i q^{31} +9.76261i q^{35} +3.48569i q^{37} -7.87921 q^{41} -1.87923 q^{43} +12.0080i q^{47} +0.753846 q^{49} +2.03716 q^{53} +11.0218 q^{55} -12.5359 q^{59} -1.36228 q^{61} -7.65407 q^{65} -2.93912i q^{67} +15.0770 q^{71} +4.42105 q^{73} -8.75398i q^{77} -13.3186i q^{79} +8.96837i q^{83} -6.68335 q^{85} +10.9667 q^{89} +6.07916i q^{91} +(15.2687 + 0.640173i) q^{95} +1.89684i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{25} - 16 q^{41} + 28 q^{49} - 16 q^{53} + 16 q^{65} + 16 q^{73} - 32 q^{85} + 48 q^{89}+O(q^{100}) \) 20 * q - 4 * q^25 - 16 * q^41 + 28 * q^49 - 16 * q^53 + 16 * q^65 + 16 * q^73 - 32 * q^85 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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