LMFDB - Embedded newform 5520.2.be.b.1471.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.12
Root			$-2.02116 + 2.02116i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.b.1471.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -0.279423 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -0.279423 q^{7} -1.00000 q^{9} -0.628840 q^{11} -3.03226 q^{13} -1.00000 q^{15} +2.36579i q^{17} +1.48654 q^{19} -0.279423i q^{21} +(1.08636 - 4.67117i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.72600 q^{29} +8.99698i q^{31} -0.628840i q^{33} -0.279423i q^{35} -5.99375i q^{37} -3.03226i q^{39} -7.39498 q^{41} -8.69030 q^{43} -1.00000i q^{45} +2.41437i q^{47} -6.92192 q^{49} -2.36579 q^{51} +1.22938i q^{53} -0.628840i q^{55} +1.48654i q^{57} +7.12332i q^{59} -3.34390i q^{61} +0.279423 q^{63} -3.03226i q^{65} +8.37910 q^{67} +(4.67117 + 1.08636i) q^{69} -3.78611i q^{71} +15.1169 q^{73} -1.00000i q^{75} +0.175712 q^{77} -16.5375 q^{79} +1.00000 q^{81} -7.07387 q^{83} -2.36579 q^{85} +3.72600i q^{87} -12.8162i q^{89} +0.847281 q^{91} -8.99698 q^{93} +1.48654i q^{95} -12.7589i q^{97} +0.628840 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q + 8 * q^7 - 16 * q^9	$ 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−0.279423			−0.105612			−0.0528059	−	0.998605i	$-0.516816\pi$
$7$	−0.279423			−0.105612			−0.0528059	+	0.998605i	$0.516816\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−0.628840			−0.189602			−0.0948012	−	0.995496i	$-0.530222\pi$
$11$	−0.628840			−0.189602			−0.0948012	+	0.995496i	$0.530222\pi$
$12$		0			0
$13$	−3.03226			−0.840997			−0.420499	−	0.907293i	$-0.638145\pi$
$13$	−3.03226			−0.840997			−0.420499	+	0.907293i	$0.638145\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			2.36579i			0.573787i	0.957962	+	0.286894i	$0.0926227\pi$
$17$			2.36579i			0.573787i	−0.957962	+	0.286894i	$0.907377\pi$
$18$		0			0
$19$	1.48654			0.341036			0.170518	−	0.985355i	$-0.445456\pi$
$19$	1.48654			0.341036			0.170518	+	0.985355i	$0.445456\pi$
$20$		0			0
$21$		−	0.279423i		−	0.0609750i
$22$		0			0
$23$	1.08636	−	4.67117i	0.226522	−	0.974006i
$23$	1.08636	−	4.67117i	0.226522	−	0.974006i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	3.72600			0.691900			0.345950	−	0.938253i	$-0.387557\pi$
$29$	3.72600			0.691900			0.345950	+	0.938253i	$0.387557\pi$
$30$		0			0
$31$			8.99698i			1.61591i	0.589247	+	0.807953i	$0.299425\pi$
$31$			8.99698i			1.61591i	−0.589247	+	0.807953i	$0.700575\pi$
$32$		0			0
$33$		−	0.628840i		−	0.109467i
$34$		0			0
$35$		−	0.279423i		−	0.0472310i
$36$		0			0
$37$		−	5.99375i		−	0.985366i	−0.870209	−	0.492683i	$-0.836016\pi$
$37$		−	5.99375i		−	0.985366i	0.870209	−	0.492683i	$-0.163984\pi$
$38$		0			0
$39$		−	3.03226i		−	0.485550i
$40$		0			0
$41$	−7.39498			−1.15490			−0.577451	−	0.816425i	$-0.695953\pi$
$41$	−7.39498			−1.15490			−0.577451	+	0.816425i	$0.695953\pi$
$42$		0			0
$43$	−8.69030			−1.32526			−0.662629	−	0.748948i	$-0.730559\pi$
$43$	−8.69030			−1.32526			−0.662629	+	0.748948i	$0.730559\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$			2.41437i			0.352172i	0.984375	+	0.176086i	$0.0563437\pi$
$47$			2.41437i			0.352172i	−0.984375	+	0.176086i	$0.943656\pi$
$48$		0			0
$49$	−6.92192			−0.988846
$50$		0			0
$51$	−2.36579			−0.331276
$52$		0			0
$53$			1.22938i			0.168869i	0.996429	+	0.0844343i	$0.0269083\pi$
$53$			1.22938i			0.168869i	−0.996429	+	0.0844343i	$0.973092\pi$
$54$		0			0
$55$		−	0.628840i		−	0.0847927i
$56$		0			0
$57$			1.48654i			0.196897i
$58$		0			0
$59$			7.12332i			0.927377i	0.885998	+	0.463689i	$0.153474\pi$
$59$			7.12332i			0.927377i	−0.885998	+	0.463689i	$0.846526\pi$
$60$		0			0
$61$		−	3.34390i		−	0.428142i	−0.976818	−	0.214071i	$-0.931328\pi$
$61$		−	3.34390i		−	0.428142i	0.976818	−	0.214071i	$-0.0686724\pi$
$62$		0			0
$63$	0.279423			0.0352039
$64$		0			0
$65$		−	3.03226i		−	0.376105i
$66$		0			0
$67$	8.37910			1.02367			0.511835	−	0.859084i	$-0.328966\pi$
$67$	8.37910			1.02367			0.511835	+	0.859084i	$0.328966\pi$
$68$		0			0
$69$	4.67117	+	1.08636i	0.562343	+	0.130783i
$70$		0			0
$71$		−	3.78611i		−	0.449328i	−0.974436	−	0.224664i	$-0.927872\pi$
$71$		−	3.78611i		−	0.449328i	0.974436	−	0.224664i	$-0.0721285\pi$
$72$		0			0
$73$	15.1169			1.76930			0.884651	−	0.466255i	$-0.154397\pi$
$73$	15.1169			1.76930			0.884651	+	0.466255i	$0.154397\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	0.175712			0.0200242
$78$		0			0
$79$	−16.5375			−1.86062			−0.930309	−	0.366778i	$-0.880461\pi$
$79$	−16.5375			−1.86062			−0.930309	+	0.366778i	$0.880461\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−7.07387			−0.776458			−0.388229	−	0.921563i	$-0.626913\pi$
$83$	−7.07387			−0.776458			−0.388229	+	0.921563i	$0.626913\pi$
$84$		0			0
$85$	−2.36579			−0.256606
$86$		0			0
$87$			3.72600i			0.399469i
$88$		0			0
$89$		−	12.8162i		−	1.35852i	−0.733899	−	0.679259i	$-0.762301\pi$
$89$		−	12.8162i		−	1.35852i	0.733899	−	0.679259i	$-0.237699\pi$
$90$		0			0
$91$	0.847281			0.0888192
$92$		0			0
$93$	−8.99698			−0.932944
$94$		0			0
$95$			1.48654i			0.152516i
$96$		0			0
$97$		−	12.7589i		−	1.29547i	−0.761866	−	0.647734i	$-0.775717\pi$
$97$		−	12.7589i		−	1.29547i	0.761866	−	0.647734i	$-0.224283\pi$
$98$		0			0
$99$	0.628840			0.0632008
$100$		0			0
$101$	−11.6258			−1.15681			−0.578403	−	0.815751i	$-0.696324\pi$
$101$	−11.6258			−1.15681			−0.578403	+	0.815751i	$0.696324\pi$
$102$		0			0
$103$	10.8093			1.06508			0.532538	−	0.846406i	$-0.321238\pi$
$103$	10.8093			1.06508			0.532538	+	0.846406i	$0.321238\pi$
$104$		0			0
$105$	0.279423			0.0272689
$106$		0			0
$107$	−15.1779			−1.46730			−0.733651	−	0.679526i	$-0.762185\pi$
$107$	−15.1779			−1.46730			−0.733651	+	0.679526i	$0.762185\pi$
$108$		0			0
$109$		−	8.74076i		−	0.837213i	−0.908168	−	0.418607i	$-0.862519\pi$
$109$		−	8.74076i		−	0.837213i	0.908168	−	0.418607i	$-0.137481\pi$
$110$		0			0
$111$	5.99375			0.568901
$112$		0			0
$113$		−	19.5279i		−	1.83703i	−0.395382	−	0.918517i	$-0.629388\pi$
$113$		−	19.5279i		−	1.83703i	0.395382	−	0.918517i	$-0.370612\pi$
$114$		0			0
$115$	4.67117	+	1.08636i	0.435589	+	0.101304i
$116$		0			0
$117$	3.03226			0.280332
$118$		0			0
$119$		−	0.661054i		−	0.0605987i
$120$		0			0
$121$	−10.6046			−0.964051
$122$		0			0
$123$		−	7.39498i		−	0.666783i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	3.45069i		−	0.306199i	−0.988211	−	0.153100i	$-0.951074\pi$
$127$		−	3.45069i		−	0.306199i	0.988211	−	0.153100i	$-0.0489256\pi$
$128$		0			0
$129$		−	8.69030i		−	0.765138i
$130$		0			0
$131$		−	11.6225i		−	1.01546i	−0.861516	−	0.507730i	$-0.830485\pi$
$131$		−	11.6225i		−	1.01546i	0.861516	−	0.507730i	$-0.169515\pi$
$132$		0			0
$133$	−0.415374			−0.0360175
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	16.9576i		−	1.44879i	−0.689385	−	0.724395i	$-0.742119\pi$
$137$		−	16.9576i		−	1.44879i	0.689385	−	0.724395i	$-0.257881\pi$
$138$		0			0
$139$		−	5.39812i		−	0.457862i	−0.973443	−	0.228931i	$-0.926477\pi$
$139$		−	5.39812i		−	0.457862i	0.973443	−	0.228931i	$-0.0735231\pi$
$140$		0			0
$141$	−2.41437			−0.203327
$142$		0			0
$143$	1.90680			0.159455
$144$		0			0
$145$			3.72600i			0.309427i
$146$		0			0
$147$		−	6.92192i		−	0.570911i
$148$		0			0
$149$			2.66008i			0.217922i	0.994046	+	0.108961i	$0.0347523\pi$
$149$			2.66008i			0.217922i	−0.994046	+	0.108961i	$0.965248\pi$
$150$		0			0
$151$		−	8.87377i		−	0.722137i	−0.932539	−	0.361069i	$-0.882412\pi$
$151$		−	8.87377i		−	0.722137i	0.932539	−	0.361069i	$-0.117588\pi$
$152$		0			0
$153$		−	2.36579i		−	0.191262i
$154$		0			0
$155$	−8.99698			−0.722655
$156$		0			0
$157$			24.5886i			1.96239i	0.193031	+	0.981193i	$0.438168\pi$
$157$			24.5886i			1.96239i	−0.193031	+	0.981193i	$0.561832\pi$
$158$		0			0
$159$	−1.22938			−0.0974963
$160$		0			0
$161$	−0.303555	+	1.30523i	−0.0239234	+	0.102867i
$162$		0			0
$163$			19.3751i			1.51757i	0.651339	+	0.758786i	$0.274207\pi$
$163$			19.3751i			1.51757i	−0.651339	+	0.758786i	$0.725793\pi$
$164$		0			0
$165$	0.628840			0.0489551
$166$		0			0
$167$			14.5746i			1.12782i	0.825838	+	0.563908i	$0.190703\pi$
$167$			14.5746i			1.12782i	−0.825838	+	0.563908i	$0.809297\pi$
$168$		0			0
$169$	−3.80541			−0.292724
$170$		0			0
$171$	−1.48654			−0.113679
$172$		0			0
$173$	12.2951			0.934776			0.467388	−	0.884052i	$-0.345195\pi$
$173$	12.2951			0.934776			0.467388	+	0.884052i	$0.345195\pi$
$174$		0			0
$175$	0.279423			0.0211224
$176$		0			0
$177$	−7.12332			−0.535421
$178$		0			0
$179$			10.0284i			0.749557i	0.927114	+	0.374779i	$0.122281\pi$
$179$			10.0284i			0.749557i	−0.927114	+	0.374779i	$0.877719\pi$
$180$		0			0
$181$		−	12.4338i		−	0.924195i	−0.886829	−	0.462097i	$-0.847097\pi$
$181$		−	12.4338i		−	0.924195i	0.886829	−	0.462097i	$-0.152903\pi$
$182$		0			0
$183$	3.34390			0.247188
$184$		0			0
$185$	5.99375			0.440669
$186$		0			0
$187$		−	1.48770i		−	0.108791i
$188$		0			0
$189$			0.279423i			0.0203250i
$190$		0			0
$191$	0.607809			0.0439795			0.0219898	−	0.999758i	$-0.493000\pi$
$191$	0.607809			0.0439795			0.0219898	+	0.999758i	$0.493000\pi$
$192$		0			0
$193$	6.47078			0.465777			0.232888	−	0.972503i	$-0.425182\pi$
$193$	6.47078			0.465777			0.232888	+	0.972503i	$0.425182\pi$
$194$		0			0
$195$	3.03226			0.217145
$196$		0			0
$197$	−14.5268			−1.03499			−0.517495	−	0.855686i	$-0.673135\pi$
$197$	−14.5268			−1.03499			−0.517495	+	0.855686i	$0.673135\pi$
$198$		0			0
$199$	23.1375			1.64018			0.820089	−	0.572237i	$-0.193924\pi$
$199$	23.1375			1.64018			0.820089	+	0.572237i	$0.193924\pi$
$200$		0			0
$201$			8.37910i			0.591016i
$202$		0			0
$203$	−1.04113			−0.0730728
$204$		0			0
$205$		−	7.39498i		−	0.516488i
$206$		0			0
$207$	−1.08636	+	4.67117i	−0.0755075	+	0.324669i
$208$		0			0
$209$	−0.934798			−0.0646613
$210$		0			0
$211$		−	14.4669i		−	0.995943i	−0.867193	−	0.497971i	$-0.834078\pi$
$211$		−	14.4669i		−	0.995943i	0.867193	−	0.497971i	$-0.165922\pi$
$212$		0			0
$213$	3.78611			0.259420
$214$		0			0
$215$		−	8.69030i		−	0.592673i
$216$		0			0
$217$		−	2.51396i		−	0.170659i
$218$		0			0
$219$			15.1169i			1.02151i
$220$		0			0
$221$		−	7.17367i		−	0.482554i
$222$		0			0
$223$			22.1810i			1.48535i	0.669654	+	0.742673i	$0.266442\pi$
$223$			22.1810i			1.48535i	−0.669654	+	0.742673i	$0.733558\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	0.753365			0.0500026			0.0250013	−	0.999687i	$-0.492041\pi$
$227$	0.753365			0.0500026			0.0250013	+	0.999687i	$0.492041\pi$
$228$		0			0
$229$		−	8.34911i		−	0.551725i	−0.961197	−	0.275862i	$-0.911037\pi$
$229$		−	8.34911i		−	0.551725i	0.961197	−	0.275862i	$-0.0889634\pi$
$230$		0			0
$231$			0.175712i			0.0115610i
$232$		0			0
$233$	23.4423			1.53575			0.767877	−	0.640597i	$-0.221313\pi$
$233$	23.4423			1.53575			0.767877	+	0.640597i	$0.221313\pi$
$234$		0			0
$235$	−2.41437			−0.157496
$236$		0			0
$237$		−	16.5375i		−	1.07423i
$238$		0			0
$239$		−	13.7015i		−	0.886279i	−0.896453	−	0.443139i	$-0.853865\pi$
$239$		−	13.7015i		−	0.886279i	0.896453	−	0.443139i	$-0.146135\pi$
$240$		0			0
$241$		−	22.8899i		−	1.47447i	−0.675636	−	0.737235i	$-0.736131\pi$
$241$		−	22.8899i		−	1.47447i	0.675636	−	0.737235i	$-0.263869\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		−	6.92192i		−	0.442225i
$246$		0			0
$247$	−4.50758			−0.286811
$248$		0			0
$249$		−	7.07387i		−	0.448288i
$250$		0			0
$251$	−31.1381			−1.96542			−0.982711	−	0.185148i	$-0.940724\pi$
$251$	−31.1381			−1.96542			−0.982711	+	0.185148i	$0.940724\pi$
$252$		0			0
$253$	−0.683149	+	2.93742i	−0.0429492	+	0.184674i
$254$		0			0
$255$		−	2.36579i		−	0.148151i
$256$		0			0
$257$	−14.1422			−0.882166			−0.441083	−	0.897466i	$-0.645406\pi$
$257$	−14.1422			−0.882166			−0.441083	+	0.897466i	$0.645406\pi$
$258$		0			0
$259$			1.67479i			0.104066i
$260$		0			0
$261$	−3.72600			−0.230633
$262$		0			0
$263$	−26.4000			−1.62789			−0.813947	−	0.580939i	$-0.802686\pi$
$263$	−26.4000			−1.62789			−0.813947	+	0.580939i	$0.802686\pi$
$264$		0			0
$265$	−1.22938			−0.0755203
$266$		0			0
$267$	12.8162			0.784340
$268$		0			0
$269$	9.45718			0.576615			0.288307	−	0.957538i	$-0.406908\pi$
$269$	9.45718			0.576615			0.288307	+	0.957538i	$0.406908\pi$
$270$		0			0
$271$		−	24.3332i		−	1.47814i	−0.673629	−	0.739070i	$-0.735265\pi$
$271$		−	24.3332i		−	1.47814i	0.673629	−	0.739070i	$-0.264735\pi$
$272$		0			0
$273$			0.847281i			0.0512798i
$274$		0			0
$275$	0.628840			0.0379205
$276$		0			0
$277$	−9.92017			−0.596045			−0.298023	−	0.954559i	$-0.596327\pi$
$277$	−9.92017			−0.596045			−0.298023	+	0.954559i	$0.596327\pi$
$278$		0			0
$279$		−	8.99698i		−	0.538635i
$280$		0			0
$281$			21.7780i			1.29916i	0.760292	+	0.649582i	$0.225056\pi$
$281$			21.7780i			1.29916i	−0.760292	+	0.649582i	$0.774944\pi$
$282$		0			0
$283$	22.9722			1.36556			0.682779	−	0.730625i	$-0.260771\pi$
$283$	22.9722			1.36556			0.682779	+	0.730625i	$0.260771\pi$
$284$		0			0
$285$	−1.48654			−0.0880552
$286$		0			0
$287$	2.06633			0.121971
$288$		0			0
$289$	11.4031			0.670768
$290$		0			0
$291$	12.7589			0.747939
$292$		0			0
$293$		−	15.7683i		−	0.921193i	−0.887610	−	0.460596i	$-0.847636\pi$
$293$		−	15.7683i		−	0.921193i	0.887610	−	0.460596i	$-0.152364\pi$
$294$		0			0
$295$	−7.12332			−0.414736
$296$		0			0
$297$			0.628840i			0.0364890i
$298$		0			0
$299$	−3.29413	+	14.1642i	−0.190505	+	0.819136i
$300$		0			0
$301$	2.42827			0.139963
$302$		0			0
$303$		−	11.6258i		−	0.667882i
$304$		0			0
$305$	3.34390			0.191471
$306$		0			0
$307$			14.9802i			0.854963i	0.904024	+	0.427481i	$0.140599\pi$
$307$			14.9802i			0.854963i	−0.904024	+	0.427481i	$0.859401\pi$
$308$		0			0
$309$			10.8093i			0.614922i
$310$		0			0
$311$			6.28014i			0.356114i	0.984020	+	0.178057i	$0.0569812\pi$
$311$			6.28014i			0.356114i	−0.984020	+	0.178057i	$0.943019\pi$
$312$		0			0
$313$		−	21.0909i		−	1.19213i	−0.802937	−	0.596064i	$-0.796730\pi$
$313$		−	21.0909i		−	1.19213i	0.802937	−	0.596064i	$-0.203270\pi$
$314$		0			0
$315$			0.279423i			0.0157437i
$316$		0			0
$317$	11.4813			0.644852			0.322426	−	0.946595i	$-0.395502\pi$
$317$	11.4813			0.644852			0.322426	+	0.946595i	$0.395502\pi$
$318$		0			0
$319$	−2.34305			−0.131186
$320$		0			0
$321$		−	15.1779i		−	0.847148i
$322$		0			0
$323$			3.51684i			0.195682i
$324$		0			0
$325$	3.03226			0.168199
$326$		0			0
$327$	8.74076			0.483365
$328$		0			0
$329$		−	0.674629i		−	0.0371935i
$330$		0			0
$331$		−	5.43242i		−	0.298593i	−0.988792	−	0.149297i	$-0.952299\pi$
$331$		−	5.43242i		−	0.298593i	0.988792	−	0.149297i	$-0.0477009\pi$
$332$		0			0
$333$			5.99375i			0.328455i
$334$		0			0
$335$			8.37910i			0.457799i
$336$		0			0
$337$		−	13.9809i		−	0.761588i	−0.924660	−	0.380794i	$-0.875651\pi$
$337$		−	13.9809i		−	0.761588i	0.924660	−	0.380794i	$-0.124349\pi$
$338$		0			0
$339$	19.5279			1.06061
$340$		0			0
$341$		−	5.65766i		−	0.306380i
$342$		0			0
$343$	3.89010			0.210046
$344$		0			0
$345$	−1.08636	+	4.67117i	−0.0584878	+	0.251487i
$346$		0			0
$347$			10.9072i			0.585530i	0.956184	+	0.292765i	$0.0945754\pi$
$347$			10.9072i			0.585530i	−0.956184	+	0.292765i	$0.905425\pi$
$348$		0			0
$349$	−22.9947			−1.23088			−0.615440	−	0.788184i	$-0.711022\pi$
$349$	−22.9947			−1.23088			−0.615440	+	0.788184i	$0.711022\pi$
$350$		0			0
$351$			3.03226i			0.161850i
$352$		0			0
$353$	−18.1073			−0.963752			−0.481876	−	0.876239i	$-0.660044\pi$
$353$	−18.1073			−0.963752			−0.481876	+	0.876239i	$0.660044\pi$
$354$		0			0
$355$	3.78611			0.200946
$356$		0			0
$357$	0.661054			0.0349867
$358$		0			0
$359$	−10.5753			−0.558141			−0.279071	−	0.960271i	$-0.590026\pi$
$359$	−10.5753			−0.558141			−0.279071	+	0.960271i	$0.590026\pi$
$360$		0			0
$361$	−16.7902			−0.883694
$362$		0			0
$363$		−	10.6046i		−	0.556595i
$364$		0			0
$365$			15.1169i			0.791256i
$366$		0			0
$367$	29.3289			1.53096			0.765479	−	0.643461i	$-0.222502\pi$
$367$	29.3289			1.53096			0.765479	+	0.643461i	$0.222502\pi$
$368$		0			0
$369$	7.39498			0.384968
$370$		0			0
$371$		−	0.343517i		−	0.0178345i
$372$		0			0
$373$		−	28.1275i		−	1.45639i	−0.685373	−	0.728193i	$-0.740361\pi$
$373$		−	28.1275i		−	1.45639i	0.685373	−	0.728193i	$-0.259639\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−11.2982			−0.581886
$378$		0			0
$379$	−0.554605			−0.0284881			−0.0142441	−	0.999899i	$-0.504534\pi$
$379$	−0.554605			−0.0284881			−0.0142441	+	0.999899i	$0.504534\pi$
$380$		0			0
$381$	3.45069			0.176784
$382$		0			0
$383$	30.8384			1.57577			0.787883	−	0.615824i	$-0.211177\pi$
$383$	30.8384			1.57577			0.787883	+	0.615824i	$0.211177\pi$
$384$		0			0
$385$			0.175712i			0.00895512i
$386$		0			0
$387$	8.69030			0.441753
$388$		0			0
$389$			5.89847i			0.299065i	0.988757	+	0.149532i	$0.0477768\pi$
$389$			5.89847i			0.299065i	−0.988757	+	0.149532i	$0.952223\pi$
$390$		0			0
$391$	11.0510	+	2.57010i	0.558872	+	0.129976i
$392$		0			0
$393$	11.6225			0.586277
$394$		0			0
$395$		−	16.5375i		−	0.832093i
$396$		0			0
$397$	−4.49346			−0.225520			−0.112760	−	0.993622i	$-0.535969\pi$
$397$	−4.49346			−0.225520			−0.112760	+	0.993622i	$0.535969\pi$
$398$		0			0
$399$		−	0.415374i		−	0.0207947i
$400$		0			0
$401$		−	34.9973i		−	1.74768i	−0.486211	−	0.873842i	$-0.661621\pi$
$401$		−	34.9973i		−	1.74768i	0.486211	−	0.873842i	$-0.338379\pi$
$402$		0			0
$403$		−	27.2812i		−	1.35897i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			3.76911i			0.186828i
$408$		0			0
$409$	30.8491			1.52539			0.762696	−	0.646757i	$-0.223875\pi$
$409$	30.8491			1.52539			0.762696	+	0.646757i	$0.223875\pi$
$410$		0			0
$411$	16.9576			0.836459
$412$		0			0
$413$		−	1.99042i		−	0.0979420i
$414$		0			0
$415$		−	7.07387i		−	0.347242i
$416$		0			0
$417$	5.39812			0.264347
$418$		0			0
$419$	−21.6640			−1.05835			−0.529177	−	0.848511i	$-0.677499\pi$
$419$	−21.6640			−1.05835			−0.529177	+	0.848511i	$0.677499\pi$
$420$		0			0
$421$			32.5964i			1.58865i	0.607494	+	0.794325i	$0.292175\pi$
$421$			32.5964i			1.58865i	−0.607494	+	0.794325i	$0.707825\pi$
$422$		0			0
$423$		−	2.41437i		−	0.117391i
$424$		0			0
$425$		−	2.36579i		−	0.114757i
$426$		0			0
$427$			0.934360i			0.0452168i
$428$		0			0
$429$			1.90680i			0.0920614i
$430$		0			0
$431$	−3.64945			−0.175788			−0.0878938	−	0.996130i	$-0.528014\pi$
$431$	−3.64945			−0.175788			−0.0878938	+	0.996130i	$0.528014\pi$
$432$		0			0
$433$		−	23.4352i		−	1.12622i	−0.826381	−	0.563111i	$-0.809604\pi$
$433$		−	23.4352i		−	1.12622i	0.826381	−	0.563111i	$-0.190396\pi$
$434$		0			0
$435$	−3.72600			−0.178648
$436$		0			0
$437$	1.61493	−	6.94389i	0.0772524	−	0.332171i
$438$		0			0
$439$		−	7.38913i		−	0.352664i	−0.984331	−	0.176332i	$-0.943577\pi$
$439$		−	7.38913i		−	0.352664i	0.984331	−	0.176332i	$-0.0564232\pi$
$440$		0			0
$441$	6.92192			0.329615
$442$		0			0
$443$		−	25.5242i		−	1.21269i	−0.795202	−	0.606345i	$-0.792635\pi$
$443$		−	25.5242i		−	1.21269i	0.795202	−	0.606345i	$-0.207365\pi$
$444$		0			0
$445$	12.8162			0.607548
$446$		0			0
$447$	−2.66008			−0.125817
$448$		0			0
$449$	−36.1882			−1.70783			−0.853914	−	0.520414i	$-0.825778\pi$
$449$	−36.1882			−1.70783			−0.853914	+	0.520414i	$0.825778\pi$
$450$		0			0
$451$	4.65026			0.218972
$452$		0			0
$453$	8.87377			0.416926
$454$		0			0
$455$			0.847281i			0.0397212i
$456$		0			0
$457$			6.57098i			0.307378i	0.988119	+	0.153689i	$0.0491153\pi$
$457$			6.57098i			0.307378i	−0.988119	+	0.153689i	$0.950885\pi$
$458$		0			0
$459$	2.36579			0.110425
$460$		0			0
$461$	−20.5992			−0.959401			−0.479701	−	0.877432i	$-0.659255\pi$
$461$	−20.5992			−0.959401			−0.479701	+	0.877432i	$0.659255\pi$
$462$		0			0
$463$			18.3552i			0.853040i	0.904478	+	0.426520i	$0.140261\pi$
$463$			18.3552i			0.853040i	−0.904478	+	0.426520i	$0.859739\pi$
$464$		0			0
$465$		−	8.99698i		−	0.417225i
$466$		0			0
$467$	−28.2461			−1.30707			−0.653536	−	0.756895i	$-0.726715\pi$
$467$	−28.2461			−1.30707			−0.653536	+	0.756895i	$0.726715\pi$
$468$		0			0
$469$	−2.34131			−0.108112
$470$		0			0
$471$	−24.5886			−1.13298
$472$		0			0
$473$	5.46480			0.251272
$474$		0			0
$475$	−1.48654			−0.0682073
$476$		0			0
$477$		−	1.22938i		−	0.0562895i
$478$		0			0
$479$	22.7535			1.03963			0.519817	−	0.854277i	$-0.326000\pi$
$479$	22.7535			1.03963			0.519817	+	0.854277i	$0.326000\pi$
$480$		0			0
$481$			18.1746i			0.828690i
$482$		0			0
$483$	−1.30523	−	0.303555i	−0.0593900	−	0.0138122i
$484$		0			0
$485$	12.7589			0.579351
$486$		0			0
$487$			30.0261i			1.36061i	0.732928	+	0.680306i	$0.238153\pi$
$487$			30.0261i			1.36061i	−0.732928	+	0.680306i	$0.761847\pi$
$488$		0			0
$489$	−19.3751			−0.876171
$490$		0			0
$491$			19.2126i			0.867055i	0.901140	+	0.433527i	$0.142731\pi$
$491$			19.2126i			0.867055i	−0.901140	+	0.433527i	$0.857269\pi$
$492$		0			0
$493$			8.81491i			0.397003i
$494$		0			0
$495$			0.628840i			0.0282642i
$496$		0			0
$497$			1.05792i			0.0474544i
$498$		0			0
$499$		−	9.24353i		−	0.413797i	−0.978362	−	0.206899i	$-0.933663\pi$
$499$		−	9.24353i		−	0.413797i	0.978362	−	0.206899i	$-0.0663370\pi$
$500$		0			0
$501$	−14.5746			−0.651145
$502$		0			0
$503$	−2.15661			−0.0961583			−0.0480791	−	0.998844i	$-0.515310\pi$
$503$	−2.15661			−0.0961583			−0.0480791	+	0.998844i	$0.515310\pi$
$504$		0			0
$505$		−	11.6258i		−	0.517339i
$506$		0			0
$507$		−	3.80541i		−	0.169004i
$508$		0			0
$509$	−26.3925			−1.16983			−0.584913	−	0.811096i	$-0.698871\pi$
$509$	−26.3925			−1.16983			−0.584913	+	0.811096i	$0.698871\pi$
$510$		0			0
$511$	−4.22401			−0.186859
$512$		0			0
$513$		−	1.48654i		−	0.0656325i
$514$		0			0
$515$			10.8093i			0.476317i
$516$		0			0
$517$		−	1.51825i		−	0.0667726i
$518$		0			0
$519$			12.2951i			0.539693i
$520$		0			0
$521$		−	35.5317i		−	1.55667i	−0.627848	−	0.778336i	$-0.716064\pi$
$521$		−	35.5317i		−	1.55667i	0.627848	−	0.778336i	$-0.283936\pi$
$522$		0			0
$523$	−6.64224			−0.290445			−0.145222	−	0.989399i	$-0.546390\pi$
$523$	−6.64224			−0.290445			−0.145222	+	0.989399i	$0.546390\pi$
$524$		0			0
$525$			0.279423i			0.0121950i
$526$		0			0
$527$	−21.2849			−0.927187
$528$		0			0
$529$	−20.6396	−	10.1492i	−0.897375	−	0.441268i
$530$		0			0
$531$		−	7.12332i		−	0.309126i
$532$		0			0
$533$	22.4235			0.971270
$534$		0			0
$535$		−	15.1779i		−	0.656198i
$536$		0			0
$537$	−10.0284			−0.432757
$538$		0			0
$539$	4.35278			0.187488
$540$		0			0
$541$	−15.5317			−0.667760			−0.333880	−	0.942616i	$-0.608358\pi$
$541$	−15.5317			−0.667760			−0.333880	+	0.942616i	$0.608358\pi$
$542$		0			0
$543$	12.4338			0.533584
$544$		0			0
$545$	8.74076			0.374413
$546$		0			0
$547$		−	0.526997i		−	0.0225328i	−0.999937	−	0.0112664i	$-0.996414\pi$
$547$		−	0.526997i		−	0.0225328i	0.999937	−	0.0112664i	$-0.00358628\pi$
$548$		0			0
$549$			3.34390i			0.142714i
$550$		0			0
$551$	5.53885			0.235963
$552$		0			0
$553$	4.62096			0.196503
$554$		0			0
$555$			5.99375i			0.254420i
$556$		0			0
$557$		−	5.19113i		−	0.219955i	−0.993934	−	0.109978i	$-0.964922\pi$
$557$		−	5.19113i		−	0.219955i	0.993934	−	0.109978i	$-0.0350779\pi$
$558$		0			0
$559$	26.3512			1.11454
$560$		0			0
$561$	1.48770			0.0628108
$562$		0			0
$563$	−45.5261			−1.91869			−0.959347	−	0.282228i	$-0.908926\pi$
$563$	−45.5261			−1.91869			−0.959347	+	0.282228i	$0.908926\pi$
$564$		0			0
$565$	19.5279			0.821546
$566$		0			0
$567$	−0.279423			−0.0117346
$568$		0			0
$569$			4.59772i			0.192746i	0.995345	+	0.0963732i	$0.0307242\pi$
$569$			4.59772i			0.192746i	−0.995345	+	0.0963732i	$0.969276\pi$
$570$		0			0
$571$	1.18402			0.0495498			0.0247749	−	0.999693i	$-0.492113\pi$
$571$	1.18402			0.0495498			0.0247749	+	0.999693i	$0.492113\pi$
$572$		0			0
$573$			0.607809i			0.0253916i
$574$		0			0
$575$	−1.08636	+	4.67117i	−0.0453045	+	0.194801i
$576$		0			0
$577$	−42.3261			−1.76206			−0.881029	−	0.473063i	$-0.843148\pi$
$577$	−42.3261			−1.76206			−0.881029	+	0.473063i	$0.843148\pi$
$578$		0			0
$579$			6.47078i			0.268916i
$580$		0			0
$581$	1.97660			0.0820031
$582$		0			0
$583$		−	0.773084i		−	0.0320179i
$584$		0			0
$585$			3.03226i			0.125368i
$586$		0			0
$587$			25.7260i			1.06183i	0.847426	+	0.530914i	$0.178151\pi$
$587$			25.7260i			1.06183i	−0.847426	+	0.530914i	$0.821849\pi$
$588$		0			0
$589$			13.3744i			0.551083i
$590$		0			0
$591$		−	14.5268i		−	0.597552i
$592$		0			0
$593$	9.16615			0.376409			0.188204	−	0.982130i	$-0.439733\pi$
$593$	9.16615			0.376409			0.188204	+	0.982130i	$0.439733\pi$
$594$		0			0
$595$	0.661054			0.0271006
$596$		0			0
$597$			23.1375i			0.946957i
$598$		0			0
$599$			3.47195i			0.141860i	0.997481	+	0.0709301i	$0.0225967\pi$
$599$			3.47195i			0.141860i	−0.997481	+	0.0709301i	$0.977403\pi$
$600$		0			0
$601$	−20.5833			−0.839609			−0.419805	−	0.907615i	$-0.637901\pi$
$601$	−20.5833			−0.839609			−0.419805	+	0.907615i	$0.637901\pi$
$602$		0			0
$603$	−8.37910			−0.341223
$604$		0			0
$605$		−	10.6046i		−	0.431137i
$606$		0			0
$607$		−	18.8120i		−	0.763555i	−0.924254	−	0.381778i	$-0.875312\pi$
$607$		−	18.8120i		−	0.763555i	0.924254	−	0.381778i	$-0.124688\pi$
$608$		0			0
$609$		−	1.04113i		−	0.0421886i
$610$		0			0
$611$		−	7.32099i		−	0.296176i
$612$		0			0
$613$			30.5205i			1.23271i	0.787469	+	0.616355i	$0.211391\pi$
$613$			30.5205i			1.23271i	−0.787469	+	0.616355i	$0.788609\pi$
$614$		0			0
$615$	7.39498			0.298195
$616$		0			0
$617$			6.93087i			0.279026i	0.990220	+	0.139513i	$0.0445538\pi$
$617$			6.93087i			0.279026i	−0.990220	+	0.139513i	$0.955446\pi$
$618$		0			0
$619$	3.83752			0.154243			0.0771216	−	0.997022i	$-0.475427\pi$
$619$	3.83752			0.154243			0.0771216	+	0.997022i	$0.475427\pi$
$620$		0			0
$621$	−4.67117	−	1.08636i	−0.187448	−	0.0435943i
$622$		0			0
$623$			3.58114i			0.143476i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	0.934798i		−	0.0373322i
$628$		0			0
$629$	14.1799			0.565390
$630$		0			0
$631$	7.79137			0.310169			0.155085	−	0.987901i	$-0.450435\pi$
$631$	7.79137			0.310169			0.155085	+	0.987901i	$0.450435\pi$
$632$		0			0
$633$	14.4669			0.575008
$634$		0			0
$635$	3.45069			0.136937
$636$		0			0
$637$	20.9891			0.831617
$638$		0			0
$639$			3.78611i			0.149776i
$640$		0			0
$641$			30.7791i			1.21570i	0.794052	+	0.607850i	$0.207968\pi$
$641$			30.7791i			1.21570i	−0.794052	+	0.607850i	$0.792032\pi$
$642$		0			0
$643$	30.8235			1.21556			0.607779	−	0.794106i	$-0.292061\pi$
$643$	30.8235			1.21556			0.607779	+	0.794106i	$0.292061\pi$
$644$		0			0
$645$	8.69030			0.342180
$646$		0			0
$647$		−	40.3220i		−	1.58522i	−0.609728	−	0.792611i	$-0.708721\pi$
$647$		−	40.3220i		−	1.58522i	0.609728	−	0.792611i	$-0.291279\pi$
$648$		0			0
$649$		−	4.47943i		−	0.175833i
$650$		0			0
$651$	2.51396			0.0985299
$652$		0			0
$653$	−6.20691			−0.242895			−0.121447	−	0.992598i	$-0.538754\pi$
$653$	−6.20691			−0.242895			−0.121447	+	0.992598i	$0.538754\pi$
$654$		0			0
$655$	11.6225			0.454128
$656$		0			0
$657$	−15.1169			−0.589767
$658$		0			0
$659$	−32.7113			−1.27425			−0.637126	−	0.770759i	$-0.719877\pi$
$659$	−32.7113			−1.27425			−0.637126	+	0.770759i	$0.719877\pi$
$660$		0			0
$661$			10.5475i			0.410250i	0.978736	+	0.205125i	$0.0657601\pi$
$661$			10.5475i			0.410250i	−0.978736	+	0.205125i	$0.934240\pi$
$662$		0			0
$663$	7.17367			0.278602
$664$		0			0
$665$		−	0.415374i		−	0.0161075i
$666$		0			0
$667$	4.04779	−	17.4048i	0.156731	−	0.673915i
$668$		0			0
$669$	−22.1810			−0.857565
$670$		0			0
$671$			2.10277i			0.0811767i
$672$		0			0
$673$	37.5234			1.44642			0.723210	−	0.690629i	$-0.242666\pi$
$673$	37.5234			1.44642			0.723210	+	0.690629i	$0.242666\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$		−	3.31254i		−	0.127311i	−0.997972	−	0.0636556i	$-0.979724\pi$
$677$		−	3.31254i		−	0.127311i	0.997972	−	0.0636556i	$-0.0202759\pi$
$678$		0			0
$679$			3.56512i			0.136817i
$680$		0			0
$681$			0.753365i			0.0288690i
$682$		0			0
$683$			34.1195i			1.30555i	0.757554	+	0.652773i	$0.226394\pi$
$683$			34.1195i			1.30555i	−0.757554	+	0.652773i	$0.773606\pi$
$684$		0			0
$685$	16.9576			0.647918
$686$		0			0
$687$	8.34911			0.318538
$688$		0			0
$689$		−	3.72780i		−	0.142018i
$690$		0			0
$691$			41.4908i			1.57839i	0.614145	+	0.789193i	$0.289501\pi$
$691$			41.4908i			1.57839i	−0.614145	+	0.789193i	$0.710499\pi$
$692$		0			0
$693$	−0.175712			−0.00667475
$694$		0			0
$695$	5.39812			0.204762
$696$		0			0
$697$		−	17.4950i		−	0.662669i
$698$		0			0
$699$			23.4423i			0.886668i
$700$		0			0
$701$			44.9555i			1.69795i	0.528436	+	0.848973i	$0.322779\pi$
$701$			44.9555i			1.69795i	−0.528436	+	0.848973i	$0.677221\pi$
$702$		0			0
$703$		−	8.90996i		−	0.336046i
$704$		0			0
$705$		−	2.41437i		−	0.0909304i
$706$		0			0
$707$	3.24850			0.122172
$708$		0			0
$709$			21.8350i			0.820029i	0.912079	+	0.410015i	$0.134476\pi$
$709$			21.8350i			0.820029i	−0.912079	+	0.410015i	$0.865524\pi$
$710$		0			0
$711$	16.5375			0.620206
$712$		0			0
$713$	42.0264	+	9.77400i	1.57390	+	0.366039i
$714$		0			0
$715$			1.90680i			0.0713105i
$716$		0			0
$717$	13.7015			0.511693
$718$		0			0
$719$			24.5319i			0.914887i	0.889239	+	0.457443i	$0.151235\pi$
$719$			24.5319i			0.914887i	−0.889239	+	0.457443i	$0.848765\pi$
$720$		0			0
$721$	−3.02037			−0.112485
$722$		0			0
$723$	22.8899			0.851286
$724$		0			0
$725$	−3.72600			−0.138380
$726$		0			0
$727$	−27.5981			−1.02356			−0.511778	−	0.859118i	$-0.671013\pi$
$727$	−27.5981			−1.02356			−0.511778	+	0.859118i	$0.671013\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	20.5594i		−	0.760416i
$732$		0			0
$733$			16.4949i			0.609251i	0.952472	+	0.304626i	$0.0985314\pi$
$733$			16.4949i			0.609251i	−0.952472	+	0.304626i	$0.901469\pi$
$734$		0			0
$735$	6.92192			0.255319
$736$		0			0
$737$	−5.26911			−0.194090
$738$		0			0
$739$		−	36.3941i		−	1.33878i	−0.742912	−	0.669389i	$-0.766556\pi$
$739$		−	36.3941i		−	1.33878i	0.742912	−	0.669389i	$-0.233444\pi$
$740$		0			0
$741$		−	4.50758i		−	0.165590i
$742$		0			0
$743$	−24.7192			−0.906860			−0.453430	−	0.891292i	$-0.649800\pi$
$743$	−24.7192			−0.906860			−0.453430	+	0.891292i	$0.649800\pi$
$744$		0			0
$745$	−2.66008			−0.0974576
$746$		0			0
$747$	7.07387			0.258819
$748$		0			0
$749$	4.24105			0.154964
$750$		0			0
$751$	14.5578			0.531223			0.265611	−	0.964080i	$-0.414426\pi$
$751$	14.5578			0.531223			0.265611	+	0.964080i	$0.414426\pi$
$752$		0			0
$753$		−	31.1381i		−	1.13474i
$754$		0			0
$755$	8.87377			0.322950
$756$		0			0
$757$			19.6837i			0.715416i	0.933833	+	0.357708i	$0.116442\pi$
$757$			19.6837i			0.715416i	−0.933833	+	0.357708i	$0.883558\pi$
$758$		0			0
$759$	−2.93742	−	0.683149i	−0.106621	−	0.0247967i
$760$		0			0
$761$	−23.3192			−0.845322			−0.422661	−	0.906288i	$-0.638904\pi$
$761$	−23.3192			−0.845322			−0.422661	+	0.906288i	$0.638904\pi$
$762$		0			0
$763$			2.44237i			0.0884196i
$764$		0			0
$765$	2.36579			0.0855352
$766$		0			0
$767$		−	21.5997i		−	0.779921i
$768$		0			0
$769$		−	35.1863i		−	1.26885i	−0.772985	−	0.634424i	$-0.781237\pi$
$769$		−	35.1863i		−	1.26885i	0.772985	−	0.634424i	$-0.218763\pi$
$770$		0			0
$771$		−	14.1422i		−	0.509319i
$772$		0			0
$773$			40.4149i			1.45362i	0.686836	+	0.726812i	$0.258999\pi$
$773$			40.4149i			1.45362i	−0.686836	+	0.726812i	$0.741001\pi$
$774$		0			0
$775$		−	8.99698i		−	0.323181i
$776$		0			0
$777$	−1.67479			−0.0600827
$778$		0			0
$779$	−10.9930			−0.393864
$780$		0			0
$781$			2.38085i			0.0851937i
$782$		0			0
$783$		−	3.72600i		−	0.133156i
$784$		0			0
$785$	−24.5886			−0.877605
$786$		0			0
$787$	6.91283			0.246416			0.123208	−	0.992381i	$-0.460682\pi$
$787$	6.91283			0.246416			0.123208	+	0.992381i	$0.460682\pi$
$788$		0			0
$789$		−	26.4000i		−	0.939865i
$790$		0			0
$791$			5.45655i			0.194012i
$792$		0			0
$793$			10.1396i			0.360066i
$794$		0			0
$795$		−	1.22938i		−	0.0436017i
$796$		0			0
$797$			30.4929i			1.08011i	0.841629	+	0.540057i	$0.181597\pi$
$797$			30.4929i			1.08011i	−0.841629	+	0.540057i	$0.818403\pi$
$798$		0			0
$799$	−5.71188			−0.202072
$800$		0			0
$801$			12.8162i			0.452839i
$802$		0			0
$803$	−9.50612			−0.335464
$804$		0			0
$805$	−1.30523	−	0.303555i	−0.0460033	−	0.0106989i
$806$		0			0
$807$			9.45718i			0.332909i
$808$		0			0
$809$	−7.49033			−0.263346			−0.131673	−	0.991293i	$-0.542035\pi$
$809$	−7.49033			−0.263346			−0.131673	+	0.991293i	$0.542035\pi$
$810$		0			0
$811$		−	28.1339i		−	0.987915i	−0.869486	−	0.493957i	$-0.835550\pi$
$811$		−	28.1339i		−	0.987915i	0.869486	−	0.493957i	$-0.164450\pi$
$812$		0			0
$813$	24.3332			0.853404
$814$		0			0
$815$	−19.3751			−0.678679
$816$		0			0
$817$	−12.9185			−0.451961
$818$		0			0
$819$	−0.847281			−0.0296064
$820$		0			0
$821$	−38.7063			−1.35086			−0.675430	−	0.737424i	$-0.736042\pi$
$821$	−38.7063			−1.35086			−0.675430	+	0.737424i	$0.736042\pi$
$822$		0			0
$823$			31.5561i			1.09998i	0.835172	+	0.549989i	$0.185368\pi$
$823$			31.5561i			1.09998i	−0.835172	+	0.549989i	$0.814632\pi$
$824$		0			0
$825$			0.628840i			0.0218934i
$826$		0			0
$827$	37.3402			1.29844			0.649222	−	0.760599i	$-0.275094\pi$
$827$	37.3402			1.29844			0.649222	+	0.760599i	$0.275094\pi$
$828$		0			0
$829$	54.0532			1.87735			0.938673	−	0.344807i	$-0.112056\pi$
$829$	54.0532			1.87735			0.938673	+	0.344807i	$0.112056\pi$
$830$		0			0
$831$		−	9.92017i		−	0.344127i
$832$		0			0
$833$		−	16.3758i		−	0.567387i
$834$		0			0
$835$	−14.5746			−0.504374
$836$		0			0
$837$	8.99698			0.310981
$838$		0			0
$839$	−53.9046			−1.86099			−0.930497	−	0.366298i	$-0.880625\pi$
$839$	−53.9046			−1.86099			−0.930497	+	0.366298i	$0.880625\pi$
$840$		0			0
$841$	−15.1170			−0.521274
$842$		0			0
$843$	−21.7780			−0.750073
$844$		0			0
$845$		−	3.80541i		−	0.130910i
$846$		0			0
$847$	2.96315			0.101815
$848$		0			0
$849$			22.9722i			0.788405i
$850$		0			0
$851$	−27.9978	−	6.51139i	−0.959752	−	0.223207i
$852$		0			0
$853$	49.0541			1.67958			0.839791	−	0.542911i	$-0.182678\pi$
$853$	49.0541			1.67958			0.839791	+	0.542911i	$0.182678\pi$
$854$		0			0
$855$		−	1.48654i		−	0.0508387i
$856$		0			0
$857$	6.48187			0.221416			0.110708	−	0.993853i	$-0.464688\pi$
$857$	6.48187			0.221416			0.110708	+	0.993853i	$0.464688\pi$
$858$		0			0
$859$		−	46.4965i		−	1.58644i	−0.608934	−	0.793221i	$-0.708403\pi$
$859$		−	46.4965i		−	1.58644i	0.608934	−	0.793221i	$-0.291597\pi$
$860$		0			0
$861$			2.06633i			0.0704202i
$862$		0			0
$863$		−	11.4359i		−	0.389281i	−0.980875	−	0.194641i	$-0.937646\pi$
$863$		−	11.4359i		−	0.389281i	0.980875	−	0.194641i	$-0.0623541\pi$
$864$		0			0
$865$			12.2951i			0.418044i
$866$		0			0
$867$			11.4031i			0.387268i
$868$		0			0
$869$	10.3995			0.352777
$870$		0			0
$871$	−25.4076			−0.860903
$872$		0			0
$873$			12.7589i			0.431823i
$874$		0			0
$875$			0.279423i			0.00944621i
$876$		0			0
$877$	−10.1834			−0.343869			−0.171935	−	0.985108i	$-0.555002\pi$
$877$	−10.1834			−0.343869			−0.171935	+	0.985108i	$0.555002\pi$
$878$		0			0
$879$	15.7683			0.531851
$880$		0			0
$881$		−	29.5646i		−	0.996057i	−0.867161	−	0.498029i	$-0.834057\pi$
$881$		−	29.5646i		−	0.996057i	0.867161	−	0.498029i	$-0.165943\pi$
$882$		0			0
$883$		−	2.80027i		−	0.0942365i	−0.998889	−	0.0471183i	$-0.984996\pi$
$883$		−	2.80027i		−	0.0942365i	0.998889	−	0.0471183i	$-0.0150038\pi$
$884$		0			0
$885$		−	7.12332i		−	0.239448i
$886$		0			0
$887$			19.1225i			0.642071i	0.947067	+	0.321035i	$0.104031\pi$
$887$			19.1225i			0.642071i	−0.947067	+	0.321035i	$0.895969\pi$
$888$		0			0
$889$			0.964201i			0.0323383i
$890$		0			0
$891$	−0.628840			−0.0210669
$892$		0			0
$893$			3.58906i			0.120103i
$894$		0			0
$895$	−10.0284			−0.335212
$896$		0			0
$897$	−14.1642	−	3.29413i	−0.472929	−	0.109988i
$898$		0			0
$899$			33.5227i			1.11805i
$900$		0			0
$901$	−2.90845			−0.0968947
$902$		0			0
$903$			2.42827i			0.0808076i
$904$		0			0
$905$	12.4338			0.413312
$906$		0			0
$907$	−47.0656			−1.56279			−0.781394	−	0.624038i	$-0.785491\pi$
$907$	−47.0656			−1.56279			−0.781394	+	0.624038i	$0.785491\pi$
$908$		0			0
$909$	11.6258			0.385602
$910$		0			0
$911$	−25.2465			−0.836453			−0.418226	−	0.908343i	$-0.637348\pi$
$911$	−25.2465			−0.836453			−0.418226	+	0.908343i	$0.637348\pi$
$912$		0			0
$913$	4.44833			0.147218
$914$		0			0
$915$			3.34390i			0.110546i
$916$		0			0
$917$			3.24758i			0.107245i
$918$		0			0
$919$	−14.3700			−0.474022			−0.237011	−	0.971507i	$-0.576168\pi$
$919$	−14.3700			−0.474022			−0.237011	+	0.971507i	$0.576168\pi$
$920$		0			0
$921$	−14.9802			−0.493613
$922$		0			0
$923$			11.4805i			0.377884i
$924$		0			0
$925$			5.99375i			0.197073i
$926$		0			0
$927$	−10.8093			−0.355025
$928$		0			0
$929$	46.8359			1.53664			0.768319	−	0.640067i	$-0.221094\pi$
$929$	46.8359			1.53664			0.768319	+	0.640067i	$0.221094\pi$
$930$		0			0
$931$	−10.2897			−0.337233
$932$		0			0
$933$	−6.28014			−0.205603
$934$		0			0
$935$	1.48770			0.0486530
$936$		0			0
$937$			38.5818i			1.26041i	0.776427	+	0.630207i	$0.217030\pi$
$937$			38.5818i			1.26041i	−0.776427	+	0.630207i	$0.782970\pi$
$938$		0			0
$939$	21.0909			0.688276
$940$		0			0
$941$			56.3991i			1.83856i	0.393606	+	0.919279i	$0.371228\pi$
$941$			56.3991i			1.83856i	−0.393606	+	0.919279i	$0.628772\pi$
$942$		0			0
$943$	−8.03364	+	34.5432i	−0.261611	+	1.12488i
$944$		0			0
$945$	−0.279423			−0.00908962
$946$		0			0
$947$			11.0677i			0.359653i	0.983698	+	0.179826i	$0.0575536\pi$
$947$			11.0677i			0.359653i	−0.983698	+	0.179826i	$0.942446\pi$
$948$		0			0
$949$	−45.8384			−1.48798
$950$		0			0
$951$			11.4813i			0.372306i
$952$		0			0
$953$		−	24.6481i		−	0.798429i	−0.916858	−	0.399215i	$-0.869283\pi$
$953$		−	24.6481i		−	0.798429i	0.916858	−	0.399215i	$-0.130717\pi$
$954$		0			0
$955$			0.607809i			0.0196682i
$956$		0			0
$957$		−	2.34305i		−	0.0757402i
$958$		0			0
$959$			4.73835i			0.153009i
$960$		0			0
$961$	−49.9457			−1.61115
$962$		0			0
$963$	15.1779			0.489101
$964$		0			0
$965$			6.47078i			0.208302i
$966$		0			0
$967$			24.6252i			0.791893i	0.918274	+	0.395946i	$0.129583\pi$
$967$			24.6252i			0.791893i	−0.918274	+	0.395946i	$0.870417\pi$
$968$		0			0
$969$	−3.51684			−0.112977
$970$		0			0
$971$	−27.9286			−0.896271			−0.448135	−	0.893966i	$-0.647912\pi$
$971$	−27.9286			−0.896271			−0.448135	+	0.893966i	$0.647912\pi$
$972$		0			0
$973$			1.50836i			0.0483557i
$974$		0			0
$975$			3.03226i			0.0971100i
$976$		0			0
$977$			6.99122i			0.223669i	0.993727	+	0.111834i	$0.0356726\pi$
$977$			6.99122i			0.223669i	−0.993727	+	0.111834i	$0.964327\pi$
$978$		0			0
$979$			8.05936i			0.257578i
$980$		0			0
$981$			8.74076i			0.279071i
$982$		0			0
$983$	17.3211			0.552456			0.276228	−	0.961092i	$-0.410915\pi$
$983$	17.3211			0.552456			0.276228	+	0.961092i	$0.410915\pi$
$984$		0			0
$985$		−	14.5268i		−	0.462862i
$986$		0			0
$987$	0.674629			0.0214737
$988$		0			0
$989$	−9.44082	+	40.5938i	−0.300201	+	1.29081i
$990$		0			0
$991$		−	2.79069i		−	0.0886491i	−0.999017	−	0.0443246i	$-0.985886\pi$
$991$		−	2.79069i		−	0.0886491i	0.999017	−	0.0443246i	$-0.0141136\pi$
$992$		0			0
$993$	5.43242			0.172393
$994$		0			0
$995$			23.1375i			0.733509i
$996$		0			0
$997$	19.5149			0.618044			0.309022	−	0.951055i	$-0.399998\pi$
$997$	19.5149			0.618044			0.309022	+	0.951055i	$0.399998\pi$
$998$		0			0
$999$	−5.99375			−0.189634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.b.1471.12	yes	16
4.3	odd	2		5520.2.be.a.1471.5	✓	16
23.22	odd	2		5520.2.be.a.1471.13	yes	16
92.91	even	2	inner	5520.2.be.b.1471.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.a.1471.5	✓	16	4.3	odd	2
5520.2.be.a.1471.13	yes	16	23.22	odd	2
5520.2.be.b.1471.4	yes	16	92.91	even	2	inner
5520.2.be.b.1471.12	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -0.279423 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -0.279423 q^{7} -1.00000 q^{9} -0.628840 q^{11} -3.03226 q^{13} -1.00000 q^{15} +2.36579i q^{17} +1.48654 q^{19} -0.279423i q^{21} +(1.08636 - 4.67117i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.72600 q^{29} +8.99698i q^{31} -0.628840i q^{33} -0.279423i q^{35} -5.99375i q^{37} -3.03226i q^{39} -7.39498 q^{41} -8.69030 q^{43} -1.00000i q^{45} +2.41437i q^{47} -6.92192 q^{49} -2.36579 q^{51} +1.22938i q^{53} -0.628840i q^{55} +1.48654i q^{57} +7.12332i q^{59} -3.34390i q^{61} +0.279423 q^{63} -3.03226i q^{65} +8.37910 q^{67} +(4.67117 + 1.08636i) q^{69} -3.78611i q^{71} +15.1169 q^{73} -1.00000i q^{75} +0.175712 q^{77} -16.5375 q^{79} +1.00000 q^{81} -7.07387 q^{83} -2.36579 q^{85} +3.72600i q^{87} -12.8162i q^{89} +0.847281 q^{91} -8.99698 q^{93} +1.48654i q^{95} -12.7589i q^{97} +0.628840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q + 8 * q^7 - 16 * q^9	\( 16 q + 8 q^{7} - 16 q^{9} - 8 q^{11} + 8 q^{13} - 16 q^{15} - 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} + 4 q^{51} - 8 q^{63} + 16 q^{67} + 40 q^{73} + 24 q^{77} - 32 q^{79} + 16 q^{81} + 4 q^{85} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^7 - 16 * q^9 - 8 * q^11 + 8 * q^13 - 16 * q^15 - 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 + 4 * q^51 - 8 * q^63 + 16 * q^67 + 40 * q^73 + 24 * q^77 - 32 * q^79 + 16 * q^81 + 4 * q^85 + 48 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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