LMFDB - Embedded newform 5520.2.be.c.1471.25

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:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.25
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -5.01811 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -5.01811 q^{7} -1.00000 q^{9} -2.36711 q^{11} +5.01660 q^{13} -1.00000 q^{15} -1.91641i q^{17} +4.78232 q^{19} +5.01811i q^{21} +(3.28367 + 3.49535i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +5.75850 q^{29} -3.94661i q^{31} +2.36711i q^{33} +5.01811i q^{35} +8.21675i q^{37} -5.01660i q^{39} +6.26306 q^{41} -8.50959 q^{43} +1.00000i q^{45} -8.32504i q^{47} +18.1815 q^{49} -1.91641 q^{51} +1.72989i q^{53} +2.36711i q^{55} -4.78232i q^{57} +4.37464i q^{59} -2.56392i q^{61} +5.01811 q^{63} -5.01660i q^{65} +5.18000 q^{67} +(3.49535 - 3.28367i) q^{69} +9.36201i q^{71} -6.89991 q^{73} +1.00000i q^{75} +11.8784 q^{77} +0.143570 q^{79} +1.00000 q^{81} -11.9654 q^{83} -1.91641 q^{85} -5.75850i q^{87} -11.4681i q^{89} -25.1739 q^{91} -3.94661 q^{93} -4.78232i q^{95} +14.2966i q^{97} +2.36711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * 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
$7$	−5.01811			−1.89667			−0.948335	−	0.317272i	$-0.897233\pi$
$7$	−5.01811			−1.89667			−0.948335	+	0.317272i	$0.897233\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.36711			−0.713710			−0.356855	−	0.934160i	$-0.616151\pi$
$11$	−2.36711			−0.713710			−0.356855	+	0.934160i	$0.616151\pi$
$12$		0			0
$13$	5.01660			1.39135			0.695677	−	0.718354i	$-0.255104\pi$
$13$	5.01660			1.39135			0.695677	+	0.718354i	$0.255104\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	1.91641i		−	0.464797i	−0.972621	−	0.232398i	$-0.925343\pi$
$17$		−	1.91641i		−	0.464797i	0.972621	−	0.232398i	$-0.0746573\pi$
$18$		0			0
$19$	4.78232			1.09714			0.548569	−	0.836105i	$-0.315173\pi$
$19$	4.78232			1.09714			0.548569	+	0.836105i	$0.315173\pi$
$20$		0			0
$21$			5.01811i			1.09504i
$22$		0			0
$23$	3.28367	+	3.49535i	0.684693	+	0.728831i
$23$	3.28367	+	3.49535i	0.684693	+	0.728831i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	5.75850			1.06933			0.534663	−	0.845065i	$-0.320439\pi$
$29$	5.75850			1.06933			0.534663	+	0.845065i	$0.320439\pi$
$30$		0			0
$31$		−	3.94661i		−	0.708833i	−0.935088	−	0.354416i	$-0.884680\pi$
$31$		−	3.94661i		−	0.708833i	0.935088	−	0.354416i	$-0.115320\pi$
$32$		0			0
$33$			2.36711i			0.412061i
$34$		0			0
$35$			5.01811i			0.848216i
$36$		0			0
$37$			8.21675i			1.35082i	0.737440	+	0.675412i	$0.236034\pi$
$37$			8.21675i			1.35082i	−0.737440	+	0.675412i	$0.763966\pi$
$38$		0			0
$39$		−	5.01660i		−	0.803299i
$40$		0			0
$41$	6.26306			0.978125			0.489063	−	0.872249i	$-0.337339\pi$
$41$	6.26306			0.978125			0.489063	+	0.872249i	$0.337339\pi$
$42$		0			0
$43$	−8.50959			−1.29770			−0.648850	−	0.760916i	$-0.724750\pi$
$43$	−8.50959			−1.29770			−0.648850	+	0.760916i	$0.724750\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	8.32504i		−	1.21433i	−0.794575	−	0.607166i	$-0.792306\pi$
$47$		−	8.32504i		−	1.21433i	0.794575	−	0.607166i	$-0.207694\pi$
$48$		0			0
$49$	18.1815			2.59735
$50$		0			0
$51$	−1.91641			−0.268351
$52$		0			0
$53$			1.72989i			0.237619i	0.992917	+	0.118809i	$0.0379078\pi$
$53$			1.72989i			0.237619i	−0.992917	+	0.118809i	$0.962092\pi$
$54$		0			0
$55$			2.36711i			0.319181i
$56$		0			0
$57$		−	4.78232i		−	0.633433i
$58$		0			0
$59$			4.37464i			0.569530i	0.958597	+	0.284765i	$0.0919156\pi$
$59$			4.37464i			0.569530i	−0.958597	+	0.284765i	$0.908084\pi$
$60$		0			0
$61$		−	2.56392i		−	0.328276i	−0.986437	−	0.164138i	$-0.947516\pi$
$61$		−	2.56392i		−	0.328276i	0.986437	−	0.164138i	$-0.0524843\pi$
$62$		0			0
$63$	5.01811			0.632223
$64$		0			0
$65$		−	5.01660i		−	0.622233i
$66$		0			0
$67$	5.18000			0.632838			0.316419	−	0.948620i	$-0.397519\pi$
$67$	5.18000			0.632838			0.316419	+	0.948620i	$0.397519\pi$
$68$		0			0
$69$	3.49535	−	3.28367i	0.420791	−	0.395308i
$70$		0			0
$71$			9.36201i			1.11107i	0.831494	+	0.555533i	$0.187486\pi$
$71$			9.36201i			1.11107i	−0.831494	+	0.555533i	$0.812514\pi$
$72$		0			0
$73$	−6.89991			−0.807574			−0.403787	−	0.914853i	$-0.632306\pi$
$73$	−6.89991			−0.807574			−0.403787	+	0.914853i	$0.632306\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	11.8784			1.35367
$78$		0			0
$79$	0.143570			0.0161529			0.00807644	−	0.999967i	$-0.497429\pi$
$79$	0.143570			0.0161529			0.00807644	+	0.999967i	$0.497429\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.9654			−1.31337			−0.656684	−	0.754166i	$-0.728041\pi$
$83$	−11.9654			−1.31337			−0.656684	+	0.754166i	$0.728041\pi$
$84$		0			0
$85$	−1.91641			−0.207864
$86$		0			0
$87$		−	5.75850i		−	0.617376i
$88$		0			0
$89$		−	11.4681i		−	1.21561i	−0.794085	−	0.607807i	$-0.792049\pi$
$89$		−	11.4681i		−	1.21561i	0.794085	−	0.607807i	$-0.207951\pi$
$90$		0			0
$91$	−25.1739			−2.63894
$92$		0			0
$93$	−3.94661			−0.409245
$94$		0			0
$95$		−	4.78232i		−	0.490655i
$96$		0			0
$97$			14.2966i			1.45160i	0.687908	+	0.725798i	$0.258529\pi$
$97$			14.2966i			1.45160i	−0.687908	+	0.725798i	$0.741471\pi$
$98$		0			0
$99$	2.36711			0.237903
$100$		0			0
$101$	−13.9368			−1.38676			−0.693379	−	0.720573i	$-0.743879\pi$
$101$	−13.9368			−1.38676			−0.693379	+	0.720573i	$0.743879\pi$
$102$		0			0
$103$	18.5326			1.82608			0.913038	−	0.407875i	$-0.133730\pi$
$103$	18.5326			1.82608			0.913038	+	0.407875i	$0.133730\pi$
$104$		0			0
$105$	5.01811			0.489718
$106$		0			0
$107$	6.84039			0.661286			0.330643	−	0.943756i	$-0.392734\pi$
$107$	6.84039			0.661286			0.330643	+	0.943756i	$0.392734\pi$
$108$		0			0
$109$		−	16.8804i		−	1.61685i	−0.588598	−	0.808426i	$-0.700320\pi$
$109$		−	16.8804i		−	1.61685i	0.588598	−	0.808426i	$-0.299680\pi$
$110$		0			0
$111$	8.21675			0.779899
$112$		0			0
$113$		−	12.3526i		−	1.16203i	−0.813892	−	0.581016i	$-0.802655\pi$
$113$		−	12.3526i		−	1.16203i	0.813892	−	0.581016i	$-0.197345\pi$
$114$		0			0
$115$	3.49535	−	3.28367i	0.325943	−	0.306204i
$116$		0			0
$117$	−5.01660			−0.463785
$118$		0			0
$119$			9.61675i			0.881566i
$120$		0			0
$121$	−5.39679			−0.490617
$122$		0			0
$123$		−	6.26306i		−	0.564721i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	7.03827i		−	0.624545i	−0.949992	−	0.312273i	$-0.898910\pi$
$127$		−	7.03827i		−	0.624545i	0.949992	−	0.312273i	$-0.101090\pi$
$128$		0			0
$129$			8.50959i			0.749227i
$130$		0			0
$131$		−	15.8340i		−	1.38342i	−0.722174	−	0.691712i	$-0.756857\pi$
$131$		−	15.8340i		−	1.38342i	0.722174	−	0.691712i	$-0.243143\pi$
$132$		0			0
$133$	−23.9982			−2.08091
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	9.41398i		−	0.804291i	−0.915576	−	0.402145i	$-0.868265\pi$
$137$		−	9.41398i		−	0.804291i	0.915576	−	0.402145i	$-0.131735\pi$
$138$		0			0
$139$		−	5.23682i		−	0.444181i	−0.975026	−	0.222090i	$-0.928712\pi$
$139$		−	5.23682i		−	0.444181i	0.975026	−	0.222090i	$-0.0712880\pi$
$140$		0			0
$141$	−8.32504			−0.701095
$142$		0			0
$143$	−11.8748			−0.993024
$144$		0			0
$145$		−	5.75850i		−	0.478217i
$146$		0			0
$147$		−	18.1815i		−	1.49958i
$148$		0			0
$149$		−	21.1290i		−	1.73096i	−0.500947	−	0.865478i	$-0.667015\pi$
$149$		−	21.1290i		−	1.73096i	0.500947	−	0.865478i	$-0.332985\pi$
$150$		0			0
$151$		−	21.8189i		−	1.77559i	−0.460235	−	0.887797i	$-0.652235\pi$
$151$		−	21.8189i		−	1.77559i	0.460235	−	0.887797i	$-0.347765\pi$
$152$		0			0
$153$			1.91641i			0.154932i
$154$		0			0
$155$	−3.94661			−0.317000
$156$		0			0
$157$		−	5.15443i		−	0.411368i	−0.978618	−	0.205684i	$-0.934058\pi$
$157$		−	5.15443i		−	0.411368i	0.978618	−	0.205684i	$-0.0659419\pi$
$158$		0			0
$159$	1.72989			0.137189
$160$		0			0
$161$	−16.4779	−	17.5401i	−1.29864	−	1.38235i
$162$		0			0
$163$			6.43764i			0.504235i	0.967697	+	0.252117i	$0.0811269\pi$
$163$			6.43764i			0.504235i	−0.967697	+	0.252117i	$0.918873\pi$
$164$		0			0
$165$	2.36711			0.184279
$166$		0			0
$167$		−	4.86457i		−	0.376432i	−0.982128	−	0.188216i	$-0.939729\pi$
$167$		−	4.86457i		−	0.376432i	0.982128	−	0.188216i	$-0.0602705\pi$
$168$		0			0
$169$	12.1663			0.935868
$170$		0			0
$171$	−4.78232			−0.365713
$172$		0			0
$173$	20.1408			1.53128			0.765638	−	0.643272i	$-0.222424\pi$
$173$	20.1408			1.53128			0.765638	+	0.643272i	$0.222424\pi$
$174$		0			0
$175$	5.01811			0.379334
$176$		0			0
$177$	4.37464			0.328818
$178$		0			0
$179$			12.1505i			0.908175i	0.890957	+	0.454087i	$0.150035\pi$
$179$			12.1505i			0.908175i	−0.890957	+	0.454087i	$0.849965\pi$
$180$		0			0
$181$			2.58656i			0.192258i	0.995369	+	0.0961288i	$0.0306461\pi$
$181$			2.58656i			0.192258i	−0.995369	+	0.0961288i	$0.969354\pi$
$182$		0			0
$183$	−2.56392			−0.189530
$184$		0			0
$185$	8.21675			0.604107
$186$		0			0
$187$			4.53635i			0.331730i
$188$		0			0
$189$		−	5.01811i		−	0.365014i
$190$		0			0
$191$	3.97384			0.287537			0.143768	−	0.989611i	$-0.454078\pi$
$191$	3.97384			0.287537			0.143768	+	0.989611i	$0.454078\pi$
$192$		0			0
$193$	13.8995			1.00051			0.500254	−	0.865879i	$-0.333240\pi$
$193$	13.8995			1.00051			0.500254	+	0.865879i	$0.333240\pi$
$194$		0			0
$195$	−5.01660			−0.359246
$196$		0			0
$197$	21.0037			1.49646			0.748228	−	0.663442i	$-0.230905\pi$
$197$	21.0037			1.49646			0.748228	+	0.663442i	$0.230905\pi$
$198$		0			0
$199$	−24.1044			−1.70872			−0.854358	−	0.519686i	$-0.826049\pi$
$199$	−24.1044			−1.70872			−0.854358	+	0.519686i	$0.826049\pi$
$200$		0			0
$201$		−	5.18000i		−	0.365369i
$202$		0			0
$203$	−28.8968			−2.02816
$204$		0			0
$205$		−	6.26306i		−	0.437431i
$206$		0			0
$207$	−3.28367	−	3.49535i	−0.228231	−	0.242944i
$208$		0			0
$209$	−11.3203			−0.783039
$210$		0			0
$211$		−	2.03788i		−	0.140294i	−0.997537	−	0.0701469i	$-0.977653\pi$
$211$		−	2.03788i		−	0.140294i	0.997537	−	0.0701469i	$-0.0223468\pi$
$212$		0			0
$213$	9.36201			0.641475
$214$		0			0
$215$			8.50959i			0.580349i
$216$		0			0
$217$			19.8046i			1.34442i
$218$		0			0
$219$			6.89991i			0.466253i
$220$		0			0
$221$		−	9.61385i		−	0.646698i
$222$		0			0
$223$		−	17.5574i		−	1.17573i	−0.808959	−	0.587865i	$-0.799969\pi$
$223$		−	17.5574i		−	1.17573i	0.808959	−	0.587865i	$-0.200031\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−11.9674			−0.794307			−0.397154	−	0.917752i	$-0.630002\pi$
$227$	−11.9674			−0.794307			−0.397154	+	0.917752i	$0.630002\pi$
$228$		0			0
$229$		−	16.7251i		−	1.10523i	−0.833438	−	0.552613i	$-0.813631\pi$
$229$		−	16.7251i		−	1.10523i	0.833438	−	0.552613i	$-0.186369\pi$
$230$		0			0
$231$		−	11.8784i		−	0.781543i
$232$		0			0
$233$	−29.6780			−1.94427			−0.972134	−	0.234425i	$-0.924679\pi$
$233$	−29.6780			−1.94427			−0.972134	+	0.234425i	$0.924679\pi$
$234$		0			0
$235$	−8.32504			−0.543066
$236$		0			0
$237$		−	0.143570i		−	0.00932587i
$238$		0			0
$239$		−	15.9888i		−	1.03423i	−0.855917	−	0.517113i	$-0.827007\pi$
$239$		−	15.9888i		−	1.03423i	0.855917	−	0.517113i	$-0.172993\pi$
$240$		0			0
$241$			9.18015i			0.591345i	0.955289	+	0.295673i	$0.0955438\pi$
$241$			9.18015i			0.591345i	−0.955289	+	0.295673i	$0.904456\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	18.1815i		−	1.16157i
$246$		0			0
$247$	23.9910			1.52651
$248$		0			0
$249$			11.9654i			0.758273i
$250$		0			0
$251$	19.2154			1.21286			0.606432	−	0.795135i	$-0.292600\pi$
$251$	19.2154			1.21286			0.606432	+	0.795135i	$0.292600\pi$
$252$		0			0
$253$	−7.77282	−	8.27388i	−0.488673	−	0.520174i
$254$		0			0
$255$			1.91641i			0.120010i
$256$		0			0
$257$	−9.57519			−0.597284			−0.298642	−	0.954365i	$-0.596534\pi$
$257$	−9.57519			−0.597284			−0.298642	+	0.954365i	$0.596534\pi$
$258$		0			0
$259$		−	41.2326i		−	2.56207i
$260$		0			0
$261$	−5.75850			−0.356442
$262$		0			0
$263$	−11.3872			−0.702168			−0.351084	−	0.936344i	$-0.614187\pi$
$263$	−11.3872			−0.702168			−0.351084	+	0.936344i	$0.614187\pi$
$264$		0			0
$265$	1.72989			0.106266
$266$		0			0
$267$	−11.4681			−0.701835
$268$		0			0
$269$	20.8812			1.27315			0.636575	−	0.771215i	$-0.280351\pi$
$269$	20.8812			1.27315			0.636575	+	0.771215i	$0.280351\pi$
$270$		0			0
$271$			23.1526i			1.40642i	0.710983	+	0.703209i	$0.248250\pi$
$271$			23.1526i			1.40642i	−0.710983	+	0.703209i	$0.751750\pi$
$272$		0			0
$273$			25.1739i			1.52359i
$274$		0			0
$275$	2.36711			0.142742
$276$		0			0
$277$	5.52150			0.331755			0.165877	−	0.986146i	$-0.446954\pi$
$277$	5.52150			0.331755			0.165877	+	0.986146i	$0.446954\pi$
$278$		0			0
$279$			3.94661i			0.236278i
$280$		0			0
$281$			14.0586i			0.838665i	0.907833	+	0.419333i	$0.137736\pi$
$281$			14.0586i			0.838665i	−0.907833	+	0.419333i	$0.862264\pi$
$282$		0			0
$283$	−27.4300			−1.63054			−0.815272	−	0.579078i	$-0.803413\pi$
$283$	−27.4300			−1.63054			−0.815272	+	0.579078i	$0.803413\pi$
$284$		0			0
$285$	−4.78232			−0.283280
$286$		0			0
$287$	−31.4287			−1.85518
$288$		0			0
$289$	13.3274			0.783964
$290$		0			0
$291$	14.2966			0.838079
$292$		0			0
$293$			6.25711i			0.365545i	0.983155	+	0.182772i	$0.0585071\pi$
$293$			6.25711i			0.365545i	−0.983155	+	0.182772i	$0.941493\pi$
$294$		0			0
$295$	4.37464			0.254702
$296$		0			0
$297$		−	2.36711i		−	0.137354i
$298$		0			0
$299$	16.4729	+	17.5348i	0.952651	+	1.01406i
$300$		0			0
$301$	42.7021			2.46131
$302$		0			0
$303$			13.9368i			0.800645i
$304$		0			0
$305$	−2.56392			−0.146810
$306$		0			0
$307$			33.1297i			1.89081i	0.325899	+	0.945405i	$0.394333\pi$
$307$			33.1297i			1.89081i	−0.325899	+	0.945405i	$0.605667\pi$
$308$		0			0
$309$		−	18.5326i		−	1.05429i
$310$		0			0
$311$		−	24.1899i		−	1.37168i	−0.727752	−	0.685841i	$-0.759435\pi$
$311$		−	24.1899i		−	1.37168i	0.727752	−	0.685841i	$-0.240565\pi$
$312$		0			0
$313$			4.05898i			0.229427i	0.993399	+	0.114714i	$0.0365950\pi$
$313$			4.05898i			0.229427i	−0.993399	+	0.114714i	$0.963405\pi$
$314$		0			0
$315$		−	5.01811i		−	0.282739i
$316$		0			0
$317$	−2.25580			−0.126698			−0.0633491	−	0.997991i	$-0.520178\pi$
$317$	−2.25580			−0.126698			−0.0633491	+	0.997991i	$0.520178\pi$
$318$		0			0
$319$	−13.6310			−0.763189
$320$		0			0
$321$		−	6.84039i		−	0.381794i
$322$		0			0
$323$		−	9.16486i		−	0.509947i
$324$		0			0
$325$	−5.01660			−0.278271
$326$		0			0
$327$	−16.8804			−0.933490
$328$		0			0
$329$			41.7760i			2.30319i
$330$		0			0
$331$		−	26.5510i		−	1.45938i	−0.683781	−	0.729688i	$-0.739666\pi$
$331$		−	26.5510i		−	1.45938i	0.683781	−	0.729688i	$-0.260334\pi$
$332$		0			0
$333$		−	8.21675i		−	0.450275i
$334$		0			0
$335$		−	5.18000i		−	0.283014i
$336$		0			0
$337$			0.0158041i			0.000860902i	1.00000		0.000430451i	$0.000137017\pi$
$337$			0.0158041i			0.000860902i	−1.00000		0.000430451i	$0.999863\pi$
$338$		0			0
$339$	−12.3526			−0.670899
$340$		0			0
$341$			9.34207i			0.505901i
$342$		0			0
$343$	−56.1099			−3.02965
$344$		0			0
$345$	−3.28367	−	3.49535i	−0.176787	−	0.188183i
$346$		0			0
$347$			10.4024i			0.558431i	0.960229	+	0.279215i	$0.0900743\pi$
$347$			10.4024i			0.558431i	−0.960229	+	0.279215i	$0.909926\pi$
$348$		0			0
$349$	0.0381635			0.00204285			0.00102142	−	0.999999i	$-0.499675\pi$
$349$	0.0381635			0.00204285			0.00102142	+	0.999999i	$0.499675\pi$
$350$		0			0
$351$			5.01660i			0.267766i
$352$		0			0
$353$	−3.08032			−0.163949			−0.0819743	−	0.996634i	$-0.526123\pi$
$353$	−3.08032			−0.163949			−0.0819743	+	0.996634i	$0.526123\pi$
$354$		0			0
$355$	9.36201			0.496884
$356$		0			0
$357$	9.61675			0.508972
$358$		0			0
$359$	20.2252			1.06744			0.533722	−	0.845660i	$-0.320793\pi$
$359$	20.2252			1.06744			0.533722	+	0.845660i	$0.320793\pi$
$360$		0			0
$361$	3.87054			0.203713
$362$		0			0
$363$			5.39679i			0.283258i
$364$		0			0
$365$			6.89991i			0.361158i
$366$		0			0
$367$	10.4553			0.545765			0.272882	−	0.962047i	$-0.412023\pi$
$367$	10.4553			0.545765			0.272882	+	0.962047i	$0.412023\pi$
$368$		0			0
$369$	−6.26306			−0.326042
$370$		0			0
$371$		−	8.68080i		−	0.450684i
$372$		0			0
$373$		−	6.09776i		−	0.315730i	−0.987461	−	0.157865i	$-0.949539\pi$
$373$		−	6.09776i		−	0.315730i	0.987461	−	0.157865i	$-0.0504611\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	28.8881			1.48781
$378$		0			0
$379$	2.48922			0.127863			0.0639313	−	0.997954i	$-0.479636\pi$
$379$	2.48922			0.127863			0.0639313	+	0.997954i	$0.479636\pi$
$380$		0			0
$381$	−7.03827			−0.360582
$382$		0			0
$383$	2.24159			0.114540			0.0572699	−	0.998359i	$-0.481760\pi$
$383$	2.24159			0.114540			0.0572699	+	0.998359i	$0.481760\pi$
$384$		0			0
$385$		−	11.8784i		−	0.605381i
$386$		0			0
$387$	8.50959			0.432567
$388$		0			0
$389$			30.0070i			1.52142i	0.649095	+	0.760708i	$0.275148\pi$
$389$			30.0070i			1.52142i	−0.649095	+	0.760708i	$0.724852\pi$
$390$		0			0
$391$	6.69852	−	6.29286i	0.338759	−	0.318243i
$392$		0			0
$393$	−15.8340			−0.798720
$394$		0			0
$395$		−	0.143570i		−	0.00722379i
$396$		0			0
$397$	−27.7912			−1.39480			−0.697400	−	0.716682i	$-0.745660\pi$
$397$	−27.7912			−1.39480			−0.697400	+	0.716682i	$0.745660\pi$
$398$		0			0
$399$			23.9982i			1.20141i
$400$		0			0
$401$		−	17.8010i		−	0.888939i	−0.895794	−	0.444469i	$-0.853392\pi$
$401$		−	17.8010i		−	0.888939i	0.895794	−	0.444469i	$-0.146608\pi$
$402$		0			0
$403$		−	19.7986i		−	0.986238i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$		−	19.4499i		−	0.964098i
$408$		0			0
$409$	−32.0839			−1.58644			−0.793222	−	0.608932i	$-0.791598\pi$
$409$	−32.0839			−1.58644			−0.793222	+	0.608932i	$0.791598\pi$
$410$		0			0
$411$	−9.41398			−0.464358
$412$		0			0
$413$		−	21.9525i		−	1.08021i
$414$		0			0
$415$			11.9654i			0.587356i
$416$		0			0
$417$	−5.23682			−0.256448
$418$		0			0
$419$	−0.157810			−0.00770954			−0.00385477	−	0.999993i	$-0.501227\pi$
$419$	−0.157810			−0.00770954			−0.00385477	+	0.999993i	$0.501227\pi$
$420$		0			0
$421$			16.3945i			0.799019i	0.916729	+	0.399509i	$0.130819\pi$
$421$			16.3945i			0.799019i	−0.916729	+	0.399509i	$0.869181\pi$
$422$		0			0
$423$			8.32504i			0.404777i
$424$		0			0
$425$			1.91641i			0.0929594i
$426$		0			0
$427$			12.8660i			0.622632i
$428$		0			0
$429$			11.8748i			0.573323i
$430$		0			0
$431$	28.3831			1.36717			0.683583	−	0.729873i	$-0.260421\pi$
$431$	28.3831			1.36717			0.683583	+	0.729873i	$0.260421\pi$
$432$		0			0
$433$			7.68466i			0.369301i	0.982804	+	0.184651i	$0.0591153\pi$
$433$			7.68466i			0.369301i	−0.982804	+	0.184651i	$0.940885\pi$
$434$		0			0
$435$	−5.75850			−0.276099
$436$		0			0
$437$	15.7036	+	16.7159i	0.751203	+	0.799629i
$438$		0			0
$439$			33.5914i			1.60323i	0.597839	+	0.801616i	$0.296026\pi$
$439$			33.5914i			1.60323i	−0.597839	+	0.801616i	$0.703974\pi$
$440$		0			0
$441$	−18.1815			−0.865785
$442$		0			0
$443$			16.9677i			0.806158i	0.915165	+	0.403079i	$0.132060\pi$
$443$			16.9677i			0.806158i	−0.915165	+	0.403079i	$0.867940\pi$
$444$		0			0
$445$	−11.4681			−0.543639
$446$		0			0
$447$	−21.1290			−0.999368
$448$		0			0
$449$	27.1197			1.27986			0.639928	−	0.768435i	$-0.278964\pi$
$449$	27.1197			1.27986			0.639928	+	0.768435i	$0.278964\pi$
$450$		0			0
$451$	−14.8253			−0.698098
$452$		0			0
$453$	−21.8189			−1.02514
$454$		0			0
$455$			25.1739i			1.18017i
$456$		0			0
$457$			13.9545i			0.652762i	0.945238	+	0.326381i	$0.105829\pi$
$457$			13.9545i			0.652762i	−0.945238	+	0.326381i	$0.894171\pi$
$458$		0			0
$459$	1.91641			0.0894502
$460$		0			0
$461$	32.7791			1.52668			0.763338	−	0.646000i	$-0.223559\pi$
$461$	32.7791			1.52668			0.763338	+	0.646000i	$0.223559\pi$
$462$		0			0
$463$		−	11.9485i		−	0.555295i	−0.960683	−	0.277647i	$-0.910445\pi$
$463$		−	11.9485i		−	0.555295i	0.960683	−	0.277647i	$-0.0895547\pi$
$464$		0			0
$465$			3.94661i			0.183020i
$466$		0			0
$467$	4.87190			0.225445			0.112722	−	0.993627i	$-0.464043\pi$
$467$	4.87190			0.225445			0.112722	+	0.993627i	$0.464043\pi$
$468$		0			0
$469$	−25.9938			−1.20028
$470$		0			0
$471$	−5.15443			−0.237504
$472$		0			0
$473$	20.1431			0.926182
$474$		0			0
$475$	−4.78232			−0.219428
$476$		0			0
$477$		−	1.72989i		−	0.0792063i
$478$		0			0
$479$	23.6174			1.07911			0.539553	−	0.841952i	$-0.318593\pi$
$479$	23.6174			1.07911			0.539553	+	0.841952i	$0.318593\pi$
$480$		0			0
$481$			41.2201i			1.87948i
$482$		0			0
$483$	−17.5401	+	16.4779i	−0.798101	+	0.749768i
$484$		0			0
$485$	14.2966			0.649174
$486$		0			0
$487$		−	31.2403i		−	1.41564i	−0.706395	−	0.707818i	$-0.749680\pi$
$487$		−	31.2403i		−	1.41564i	0.706395	−	0.707818i	$-0.250320\pi$
$488$		0			0
$489$	6.43764			0.291120
$490$		0			0
$491$		−	35.7693i		−	1.61425i	−0.590382	−	0.807124i	$-0.701023\pi$
$491$		−	35.7693i		−	1.61425i	0.590382	−	0.807124i	$-0.298977\pi$
$492$		0			0
$493$		−	11.0356i		−	0.497020i
$494$		0			0
$495$		−	2.36711i		−	0.106394i
$496$		0			0
$497$		−	46.9797i		−	2.10733i
$498$		0			0
$499$		−	19.3511i		−	0.866273i	−0.901328	−	0.433137i	$-0.857407\pi$
$499$		−	19.3511i		−	0.866273i	0.901328	−	0.433137i	$-0.142593\pi$
$500$		0			0
$501$	−4.86457			−0.217333
$502$		0			0
$503$	22.9151			1.02174			0.510868	−	0.859659i	$-0.329324\pi$
$503$	22.9151			1.02174			0.510868	+	0.859659i	$0.329324\pi$
$504$		0			0
$505$			13.9368i			0.620177i
$506$		0			0
$507$		−	12.1663i		−	0.540324i
$508$		0			0
$509$	18.5920			0.824075			0.412038	−	0.911167i	$-0.364817\pi$
$509$	18.5920			0.824075			0.412038	+	0.911167i	$0.364817\pi$
$510$		0			0
$511$	34.6245			1.53170
$512$		0			0
$513$			4.78232i			0.211144i
$514$		0			0
$515$		−	18.5326i		−	0.816646i
$516$		0			0
$517$			19.7063i			0.866681i
$518$		0			0
$519$		−	20.1408i		−	0.884082i
$520$		0			0
$521$		−	23.9529i		−	1.04940i	−0.851288	−	0.524699i	$-0.824178\pi$
$521$		−	23.9529i		−	1.04940i	0.851288	−	0.524699i	$-0.175822\pi$
$522$		0			0
$523$	−28.4800			−1.24534			−0.622671	−	0.782484i	$-0.713953\pi$
$523$	−28.4800			−1.24534			−0.622671	+	0.782484i	$0.713953\pi$
$524$		0			0
$525$		−	5.01811i		−	0.219008i
$526$		0			0
$527$	−7.56332			−0.329463
$528$		0			0
$529$	−1.43497	+	22.9552i	−0.0623900	+	0.998052i
$530$		0			0
$531$		−	4.37464i		−	0.189843i
$532$		0			0
$533$	31.4193			1.36092
$534$		0			0
$535$		−	6.84039i		−	0.295736i
$536$		0			0
$537$	12.1505			0.524335
$538$		0			0
$539$	−43.0375			−1.85376
$540$		0			0
$541$	3.40647			0.146455			0.0732277	−	0.997315i	$-0.476670\pi$
$541$	3.40647			0.146455			0.0732277	+	0.997315i	$0.476670\pi$
$542$		0			0
$543$	2.58656			0.111000
$544$		0			0
$545$	−16.8804			−0.723078
$546$		0			0
$547$			25.0754i			1.07215i	0.844171	+	0.536074i	$0.180093\pi$
$547$			25.0754i			1.07215i	−0.844171	+	0.536074i	$0.819907\pi$
$548$		0			0
$549$			2.56392i			0.109425i
$550$		0			0
$551$	27.5390			1.17320
$552$		0			0
$553$	−0.720451			−0.0306367
$554$		0			0
$555$		−	8.21675i		−	0.348781i
$556$		0			0
$557$		−	41.8190i		−	1.77193i	−0.463754	−	0.885964i	$-0.653498\pi$
$557$		−	41.8190i		−	1.77193i	0.463754	−	0.885964i	$-0.346502\pi$
$558$		0			0
$559$	−42.6892			−1.80556
$560$		0			0
$561$	4.53635			0.191525
$562$		0			0
$563$	37.1917			1.56744			0.783721	−	0.621113i	$-0.213319\pi$
$563$	37.1917			1.56744			0.783721	+	0.621113i	$0.213319\pi$
$564$		0			0
$565$	−12.3526			−0.519676
$566$		0			0
$567$	−5.01811			−0.210741
$568$		0			0
$569$		−	37.5972i		−	1.57616i	−0.615575	−	0.788078i	$-0.711076\pi$
$569$		−	37.5972i		−	1.57616i	0.615575	−	0.788078i	$-0.288924\pi$
$570$		0			0
$571$	−5.91777			−0.247651			−0.123826	−	0.992304i	$-0.539516\pi$
$571$	−5.91777			−0.247651			−0.123826	+	0.992304i	$0.539516\pi$
$572$		0			0
$573$		−	3.97384i		−	0.166009i
$574$		0			0
$575$	−3.28367	−	3.49535i	−0.136939	−	0.145766i
$576$		0			0
$577$	−41.8360			−1.74166			−0.870828	−	0.491588i	$-0.836416\pi$
$577$	−41.8360			−1.74166			−0.870828	+	0.491588i	$0.836416\pi$
$578$		0			0
$579$		−	13.8995i		−	0.577643i
$580$		0			0
$581$	60.0435			2.49102
$582$		0			0
$583$		−	4.09484i		−	0.169591i
$584$		0			0
$585$			5.01660i			0.207411i
$586$		0			0
$587$		−	28.8621i		−	1.19127i	−0.803256	−	0.595634i	$-0.796901\pi$
$587$		−	28.8621i		−	1.19127i	0.803256	−	0.595634i	$-0.203099\pi$
$588$		0			0
$589$		−	18.8740i		−	0.777688i
$590$		0			0
$591$		−	21.0037i		−	0.863979i
$592$		0			0
$593$	−25.0315			−1.02792			−0.513960	−	0.857814i	$-0.671822\pi$
$593$	−25.0315			−1.02792			−0.513960	+	0.857814i	$0.671822\pi$
$594$		0			0
$595$	9.61675			0.394248
$596$		0			0
$597$			24.1044i			0.986527i
$598$		0			0
$599$			14.7838i			0.604049i	0.953300	+	0.302025i	$0.0976625\pi$
$599$			14.7838i			0.604049i	−0.953300	+	0.302025i	$0.902338\pi$
$600$		0			0
$601$	16.5582			0.675424			0.337712	−	0.941249i	$-0.390347\pi$
$601$	16.5582			0.675424			0.337712	+	0.941249i	$0.390347\pi$
$602$		0			0
$603$	−5.18000			−0.210946
$604$		0			0
$605$			5.39679i			0.219411i
$606$		0			0
$607$			28.2229i			1.14553i	0.819719	+	0.572766i	$0.194130\pi$
$607$			28.2229i			1.14553i	−0.819719	+	0.572766i	$0.805870\pi$
$608$		0			0
$609$			28.8968i			1.17096i
$610$		0			0
$611$		−	41.7634i		−	1.68957i
$612$		0			0
$613$		−	30.7909i		−	1.24363i	−0.783164	−	0.621816i	$-0.786395\pi$
$613$		−	30.7909i		−	1.24363i	0.783164	−	0.621816i	$-0.213605\pi$
$614$		0			0
$615$	−6.26306			−0.252551
$616$		0			0
$617$			22.4530i			0.903924i	0.892037	+	0.451962i	$0.149276\pi$
$617$			22.4530i			0.903924i	−0.892037	+	0.451962i	$0.850724\pi$
$618$		0			0
$619$	−6.43745			−0.258743			−0.129371	−	0.991596i	$-0.541296\pi$
$619$	−6.43745			−0.258743			−0.129371	+	0.991596i	$0.541296\pi$
$620$		0			0
$621$	−3.49535	+	3.28367i	−0.140264	+	0.131769i
$622$		0			0
$623$			57.5482i			2.30562i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			11.3203i			0.452088i
$628$		0			0
$629$	15.7466			0.627859
$630$		0			0
$631$	21.8844			0.871205			0.435602	−	0.900139i	$-0.356535\pi$
$631$	21.8844			0.871205			0.435602	+	0.900139i	$0.356535\pi$
$632$		0			0
$633$	−2.03788			−0.0809986
$634$		0			0
$635$	−7.03827			−0.279305
$636$		0			0
$637$	91.2092			3.61384
$638$		0			0
$639$		−	9.36201i		−	0.370355i
$640$		0			0
$641$			23.5428i			0.929886i	0.885341	+	0.464943i	$0.153925\pi$
$641$			23.5428i			0.929886i	−0.885341	+	0.464943i	$0.846075\pi$
$642$		0			0
$643$	7.72929			0.304814			0.152407	−	0.988318i	$-0.451298\pi$
$643$	7.72929			0.304814			0.152407	+	0.988318i	$0.451298\pi$
$644$		0			0
$645$	8.50959			0.335065
$646$		0			0
$647$		−	10.0794i		−	0.396260i	−0.980176	−	0.198130i	$-0.936513\pi$
$647$		−	10.0794i		−	0.396260i	0.980176	−	0.198130i	$-0.0634869\pi$
$648$		0			0
$649$		−	10.3553i		−	0.406480i
$650$		0			0
$651$	19.8046			0.776202
$652$		0			0
$653$	−22.2513			−0.870760			−0.435380	−	0.900247i	$-0.643386\pi$
$653$	−22.2513			−0.870760			−0.435380	+	0.900247i	$0.643386\pi$
$654$		0			0
$655$	−15.8340			−0.618686
$656$		0			0
$657$	6.89991			0.269191
$658$		0			0
$659$	16.4657			0.641412			0.320706	−	0.947179i	$-0.396080\pi$
$659$	16.4657			0.641412			0.320706	+	0.947179i	$0.396080\pi$
$660$		0			0
$661$		−	21.9479i		−	0.853673i	−0.904329	−	0.426837i	$-0.859628\pi$
$661$		−	21.9479i		−	0.853673i	0.904329	−	0.426837i	$-0.140372\pi$
$662$		0			0
$663$	−9.61385			−0.373371
$664$		0			0
$665$			23.9982i			0.930611i
$666$		0			0
$667$	18.9090	+	20.1280i	0.732161	+	0.779358i
$668$		0			0
$669$	−17.5574			−0.678808
$670$		0			0
$671$			6.06908i			0.234294i
$672$		0			0
$673$	−34.4405			−1.32759			−0.663793	−	0.747917i	$-0.731054\pi$
$673$	−34.4405			−1.32759			−0.663793	+	0.747917i	$0.731054\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	3.97780i		−	0.152879i	−0.997074	−	0.0764396i	$-0.975645\pi$
$677$		−	3.97780i		−	0.152879i	0.997074	−	0.0764396i	$-0.0243552\pi$
$678$		0			0
$679$		−	71.7418i		−	2.75320i
$680$		0			0
$681$			11.9674i			0.458594i
$682$		0			0
$683$		−	36.5259i		−	1.39762i	−0.715305	−	0.698812i	$-0.753712\pi$
$683$		−	36.5259i		−	1.39762i	0.715305	−	0.698812i	$-0.246288\pi$
$684$		0			0
$685$	−9.41398			−0.359690
$686$		0			0
$687$	−16.7251			−0.638103
$688$		0			0
$689$			8.67818i			0.330612i
$690$		0			0
$691$		−	19.0940i		−	0.726368i	−0.931717	−	0.363184i	$-0.881690\pi$
$691$		−	19.0940i		−	0.726368i	0.931717	−	0.363184i	$-0.118310\pi$
$692$		0			0
$693$	−11.8784			−0.451224
$694$		0			0
$695$	−5.23682			−0.198644
$696$		0			0
$697$		−	12.0026i		−	0.454630i
$698$		0			0
$699$			29.6780i			1.12252i
$700$		0			0
$701$		−	31.0819i		−	1.17395i	−0.809606	−	0.586974i	$-0.800319\pi$
$701$		−	31.0819i		−	1.17395i	0.809606	−	0.586974i	$-0.199681\pi$
$702$		0			0
$703$			39.2951i			1.48204i
$704$		0			0
$705$			8.32504i			0.313539i
$706$		0			0
$707$	69.9362			2.63022
$708$		0			0
$709$		−	13.4053i		−	0.503445i	−0.967799	−	0.251722i	$-0.919003\pi$
$709$		−	13.4053i		−	0.503445i	0.967799	−	0.251722i	$-0.0809970\pi$
$710$		0			0
$711$	−0.143570			−0.00538429
$712$		0			0
$713$	13.7948	−	12.9594i	0.516620	−	0.485333i
$714$		0			0
$715$			11.8748i			0.444094i
$716$		0			0
$717$	−15.9888			−0.597111
$718$		0			0
$719$			1.98233i			0.0739283i	0.999317	+	0.0369642i	$0.0117687\pi$
$719$			1.98233i			0.0739283i	−0.999317	+	0.0369642i	$0.988231\pi$
$720$		0			0
$721$	−92.9989			−3.46346
$722$		0			0
$723$	9.18015			0.341413
$724$		0			0
$725$	−5.75850			−0.213865
$726$		0			0
$727$	−32.7228			−1.21362			−0.606810	−	0.794847i	$-0.707551\pi$
$727$	−32.7228			−1.21362			−0.606810	+	0.794847i	$0.707551\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			16.3078i			0.603167i
$732$		0			0
$733$			42.1233i			1.55586i	0.628351	+	0.777930i	$0.283730\pi$
$733$			42.1233i			1.55586i	−0.628351	+	0.777930i	$0.716270\pi$
$734$		0			0
$735$	−18.1815			−0.670634
$736$		0			0
$737$	−12.2616			−0.451663
$738$		0			0
$739$		−	46.8040i		−	1.72171i	−0.508847	−	0.860857i	$-0.669928\pi$
$739$		−	46.8040i		−	1.72171i	0.508847	−	0.860857i	$-0.330072\pi$
$740$		0			0
$741$		−	23.9910i		−	0.881330i
$742$		0			0
$743$	28.1188			1.03158			0.515789	−	0.856716i	$-0.327499\pi$
$743$	28.1188			1.03158			0.515789	+	0.856716i	$0.327499\pi$
$744$		0			0
$745$	−21.1290			−0.774107
$746$		0			0
$747$	11.9654			0.437789
$748$		0			0
$749$	−34.3259			−1.25424
$750$		0			0
$751$	−15.6426			−0.570808			−0.285404	−	0.958407i	$-0.592128\pi$
$751$	−15.6426			−0.570808			−0.285404	+	0.958407i	$0.592128\pi$
$752$		0			0
$753$		−	19.2154i		−	0.700247i
$754$		0			0
$755$	−21.8189			−0.794070
$756$		0			0
$757$		−	1.79486i		−	0.0652352i	−0.999468	−	0.0326176i	$-0.989616\pi$
$757$		−	1.79486i		−	0.0652352i	0.999468	−	0.0326176i	$-0.0103843\pi$
$758$		0			0
$759$	−8.27388	+	7.77282i	−0.300323	+	0.282135i
$760$		0			0
$761$	9.48747			0.343921			0.171960	−	0.985104i	$-0.444990\pi$
$761$	9.48747			0.343921			0.171960	+	0.985104i	$0.444990\pi$
$762$		0			0
$763$			84.7080i			3.06663i
$764$		0			0
$765$	1.91641			0.0692878
$766$		0			0
$767$			21.9458i			0.792419i
$768$		0			0
$769$		−	11.9546i		−	0.431092i	−0.976494	−	0.215546i	$-0.930847\pi$
$769$		−	11.9546i		−	0.431092i	0.976494	−	0.215546i	$-0.0691532\pi$
$770$		0			0
$771$			9.57519i			0.344842i
$772$		0			0
$773$		−	14.5282i		−	0.522544i	−0.965265	−	0.261272i	$-0.915858\pi$
$773$		−	14.5282i		−	0.522544i	0.965265	−	0.261272i	$-0.0841420\pi$
$774$		0			0
$775$			3.94661i			0.141767i
$776$		0			0
$777$	−41.2326			−1.47921
$778$		0			0
$779$	29.9519			1.07314
$780$		0			0
$781$		−	22.1609i		−	0.792980i
$782$		0			0
$783$			5.75850i			0.205792i
$784$		0			0
$785$	−5.15443			−0.183969
$786$		0			0
$787$	−4.31988			−0.153987			−0.0769936	−	0.997032i	$-0.524532\pi$
$787$	−4.31988			−0.153987			−0.0769936	+	0.997032i	$0.524532\pi$
$788$		0			0
$789$			11.3872i			0.405397i
$790$		0			0
$791$			61.9866i			2.20399i
$792$		0			0
$793$		−	12.8622i		−	0.456749i
$794$		0			0
$795$		−	1.72989i		−	0.0613529i
$796$		0			0
$797$		−	21.6869i		−	0.768190i	−0.923294	−	0.384095i	$-0.874514\pi$
$797$		−	21.6869i		−	0.768190i	0.923294	−	0.384095i	$-0.125486\pi$
$798$		0			0
$799$	−15.9542			−0.564418
$800$		0			0
$801$			11.4681i			0.405205i
$802$		0			0
$803$	16.3328			0.576374
$804$		0			0
$805$	−17.5401	+	16.4779i	−0.618206	+	0.580768i
$806$		0			0
$807$		−	20.8812i		−	0.735053i
$808$		0			0
$809$	−1.66984			−0.0587084			−0.0293542	−	0.999569i	$-0.509345\pi$
$809$	−1.66984			−0.0587084			−0.0293542	+	0.999569i	$0.509345\pi$
$810$		0			0
$811$		−	52.4099i		−	1.84036i	−0.391494	−	0.920181i	$-0.628042\pi$
$811$		−	52.4099i		−	1.84036i	0.391494	−	0.920181i	$-0.371958\pi$
$812$		0			0
$813$	23.1526			0.811996
$814$		0			0
$815$	6.43764			0.225501
$816$		0			0
$817$	−40.6955			−1.42376
$818$		0			0
$819$	25.1739			0.879647
$820$		0			0
$821$	6.02238			0.210183			0.105091	−	0.994463i	$-0.466487\pi$
$821$	6.02238			0.210183			0.105091	+	0.994463i	$0.466487\pi$
$822$		0			0
$823$			3.44724i			0.120163i	0.998193	+	0.0600816i	$0.0191361\pi$
$823$			3.44724i			0.120163i	−0.998193	+	0.0600816i	$0.980864\pi$
$824$		0			0
$825$		−	2.36711i		−	0.0824122i
$826$		0			0
$827$	−12.8981			−0.448511			−0.224256	−	0.974530i	$-0.571995\pi$
$827$	−12.8981			−0.448511			−0.224256	+	0.974530i	$0.571995\pi$
$828$		0			0
$829$	−15.6205			−0.542522			−0.271261	−	0.962506i	$-0.587441\pi$
$829$	−15.6205			−0.542522			−0.271261	+	0.962506i	$0.587441\pi$
$830$		0			0
$831$		−	5.52150i		−	0.191539i
$832$		0			0
$833$		−	34.8431i		−	1.20724i
$834$		0			0
$835$	−4.86457			−0.168346
$836$		0			0
$837$	3.94661			0.136415
$838$		0			0
$839$	42.9432			1.48256			0.741282	−	0.671194i	$-0.234218\pi$
$839$	42.9432			1.48256			0.741282	+	0.671194i	$0.234218\pi$
$840$		0			0
$841$	4.16030			0.143458
$842$		0			0
$843$	14.0586			0.484204
$844$		0			0
$845$		−	12.1663i		−	0.418533i
$846$		0			0
$847$	27.0817			0.930539
$848$		0			0
$849$			27.4300i			0.941395i
$850$		0			0
$851$	−28.7204	+	26.9811i	−0.984523	+	0.924901i
$852$		0			0
$853$	29.2298			1.00081			0.500405	−	0.865792i	$-0.333185\pi$
$853$	29.2298			1.00081			0.500405	+	0.865792i	$0.333185\pi$
$854$		0			0
$855$			4.78232i			0.163552i
$856$		0			0
$857$	26.3368			0.899646			0.449823	−	0.893118i	$-0.351487\pi$
$857$	26.3368			0.899646			0.449823	+	0.893118i	$0.351487\pi$
$858$		0			0
$859$		−	0.244563i		−	0.00834439i	−0.999991	−	0.00417219i	$-0.998672\pi$
$859$		−	0.244563i		−	0.00834439i	0.999991	−	0.00417219i	$-0.00132805\pi$
$860$		0			0
$861$			31.4287i			1.07109i
$862$		0			0
$863$			36.3987i			1.23903i	0.784986	+	0.619513i	$0.212670\pi$
$863$			36.3987i			1.23903i	−0.784986	+	0.619513i	$0.787330\pi$
$864$		0			0
$865$		−	20.1408i		−	0.684807i
$866$		0			0
$867$		−	13.3274i		−	0.452622i
$868$		0			0
$869$	−0.339846			−0.0115285
$870$		0			0
$871$	25.9860			0.880502
$872$		0			0
$873$		−	14.2966i		−	0.483865i
$874$		0			0
$875$		−	5.01811i		−	0.169643i
$876$		0			0
$877$	2.20892			0.0745900			0.0372950	−	0.999304i	$-0.488126\pi$
$877$	2.20892			0.0745900			0.0372950	+	0.999304i	$0.488126\pi$
$878$		0			0
$879$	6.25711			0.211047
$880$		0			0
$881$		−	2.96526i		−	0.0999022i	−0.998752	−	0.0499511i	$-0.984093\pi$
$881$		−	2.96526i		−	0.0999022i	0.998752	−	0.0499511i	$-0.0159065\pi$
$882$		0			0
$883$			29.2315i			0.983717i	0.870675	+	0.491859i	$0.163682\pi$
$883$			29.2315i			0.983717i	−0.870675	+	0.491859i	$0.836318\pi$
$884$		0			0
$885$		−	4.37464i		−	0.147052i
$886$		0			0
$887$		−	2.89988i		−	0.0973684i	−0.998814	−	0.0486842i	$-0.984497\pi$
$887$		−	2.89988i		−	0.0973684i	0.998814	−	0.0486842i	$-0.0155028\pi$
$888$		0			0
$889$			35.3188i			1.18456i
$890$		0			0
$891$	−2.36711			−0.0793012
$892$		0			0
$893$		−	39.8130i		−	1.33229i
$894$		0			0
$895$	12.1505			0.406148
$896$		0			0
$897$	17.5348	−	16.4729i	0.585469	−	0.550014i
$898$		0			0
$899$		−	22.7266i		−	0.757974i
$900$		0			0
$901$	3.31518			0.110445
$902$		0			0
$903$		−	42.7021i		−	1.42104i
$904$		0			0
$905$	2.58656			0.0859802
$906$		0			0
$907$	30.2617			1.00482			0.502412	−	0.864628i	$-0.332446\pi$
$907$	30.2617			1.00482			0.502412	+	0.864628i	$0.332446\pi$
$908$		0			0
$909$	13.9368			0.462253
$910$		0			0
$911$	28.7435			0.952315			0.476157	−	0.879360i	$-0.342029\pi$
$911$	28.7435			0.952315			0.476157	+	0.879360i	$0.342029\pi$
$912$		0			0
$913$	28.3233			0.937364
$914$		0			0
$915$			2.56392i			0.0847606i
$916$		0			0
$917$			79.4568i			2.62390i
$918$		0			0
$919$	13.8503			0.456880			0.228440	−	0.973558i	$-0.426638\pi$
$919$	13.8503			0.456880			0.228440	+	0.973558i	$0.426638\pi$
$920$		0			0
$921$	33.1297			1.09166
$922$		0			0
$923$			46.9655i			1.54589i
$924$		0			0
$925$		−	8.21675i		−	0.270165i
$926$		0			0
$927$	−18.5326			−0.608692
$928$		0			0
$929$	−5.51430			−0.180918			−0.0904591	−	0.995900i	$-0.528833\pi$
$929$	−5.51430			−0.180918			−0.0904591	+	0.995900i	$0.528833\pi$
$930$		0			0
$931$	86.9496			2.84966
$932$		0			0
$933$	−24.1899			−0.791941
$934$		0			0
$935$	4.53635			0.148354
$936$		0			0
$937$			5.62485i			0.183756i	0.995770	+	0.0918780i	$0.0292870\pi$
$937$			5.62485i			0.183756i	−0.995770	+	0.0918780i	$0.970713\pi$
$938$		0			0
$939$	4.05898			0.132460
$940$		0			0
$941$		−	15.6793i		−	0.511132i	−0.966792	−	0.255566i	$-0.917738\pi$
$941$		−	15.6793i		−	0.511132i	0.966792	−	0.255566i	$-0.0822618\pi$
$942$		0			0
$943$	20.5658	+	21.8916i	0.669716	+	0.712888i
$944$		0			0
$945$	−5.01811			−0.163239
$946$		0			0
$947$		−	4.95458i		−	0.161002i	−0.996755	−	0.0805012i	$-0.974348\pi$
$947$		−	4.95458i		−	0.161002i	0.996755	−	0.0805012i	$-0.0256521\pi$
$948$		0			0
$949$	−34.6141			−1.12362
$950$		0			0
$951$			2.25580i			0.0731492i
$952$		0			0
$953$		−	27.0234i		−	0.875374i	−0.899127	−	0.437687i	$-0.855798\pi$
$953$		−	27.0234i		−	0.875374i	0.899127	−	0.437687i	$-0.144202\pi$
$954$		0			0
$955$		−	3.97384i		−	0.128590i
$956$		0			0
$957$			13.6310i			0.440628i
$958$		0			0
$959$			47.2404i			1.52547i
$960$		0			0
$961$	15.4242			0.497556
$962$		0			0
$963$	−6.84039			−0.220429
$964$		0			0
$965$		−	13.8995i		−	0.447440i
$966$		0			0
$967$		−	42.1961i		−	1.35694i	−0.734630	−	0.678468i	$-0.762644\pi$
$967$		−	42.1961i		−	1.35694i	0.734630	−	0.678468i	$-0.237356\pi$
$968$		0			0
$969$	−9.16486			−0.294418
$970$		0			0
$971$	5.32455			0.170873			0.0854365	−	0.996344i	$-0.472772\pi$
$971$	5.32455			0.170873			0.0854365	+	0.996344i	$0.472772\pi$
$972$		0			0
$973$			26.2789i			0.842464i
$974$		0			0
$975$			5.01660i			0.160660i
$976$		0			0
$977$		−	2.22163i		−	0.0710761i	−0.999368	−	0.0355381i	$-0.988686\pi$
$977$		−	2.22163i		−	0.0710761i	0.999368	−	0.0355381i	$-0.0113145\pi$
$978$		0			0
$979$			27.1462i			0.867597i
$980$		0			0
$981$			16.8804i			0.538951i
$982$		0			0
$983$	−56.0536			−1.78783			−0.893916	−	0.448235i	$-0.852053\pi$
$983$	−56.0536			−1.78783			−0.893916	+	0.448235i	$0.852053\pi$
$984$		0			0
$985$		−	21.0037i		−	0.669235i
$986$		0			0
$987$	41.7760			1.32974
$988$		0			0
$989$	−27.9427	−	29.7440i	−0.888527	−	0.945804i
$990$		0			0
$991$			35.3205i			1.12199i	0.827818	+	0.560997i	$0.189582\pi$
$991$			35.3205i			1.12199i	−0.827818	+	0.560997i	$0.810418\pi$
$992$		0			0
$993$	−26.5510			−0.842571
$994$		0			0
$995$			24.1044i			0.764161i
$996$		0			0
$997$	−41.5485			−1.31585			−0.657927	−	0.753082i	$-0.728566\pi$
$997$	−41.5485			−1.31585			−0.657927	+	0.753082i	$0.728566\pi$
$998$		0			0
$999$	−8.21675			−0.259966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.25	✓	32
4.3	odd	2		5520.2.be.d.1471.26	yes	32
23.22	odd	2		5520.2.be.d.1471.25	yes	32
92.91	even	2	inner	5520.2.be.c.1471.26	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.25	✓	32	1.1	even	1	trivial
5520.2.be.c.1471.26	yes	32	92.91	even	2	inner
5520.2.be.d.1471.25	yes	32	23.22	odd	2
5520.2.be.d.1471.26	yes	32	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -5.01811 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -5.01811 q^{7} -1.00000 q^{9} -2.36711 q^{11} +5.01660 q^{13} -1.00000 q^{15} -1.91641i q^{17} +4.78232 q^{19} +5.01811i q^{21} +(3.28367 + 3.49535i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +5.75850 q^{29} -3.94661i q^{31} +2.36711i q^{33} +5.01811i q^{35} +8.21675i q^{37} -5.01660i q^{39} +6.26306 q^{41} -8.50959 q^{43} +1.00000i q^{45} -8.32504i q^{47} +18.1815 q^{49} -1.91641 q^{51} +1.72989i q^{53} +2.36711i q^{55} -4.78232i q^{57} +4.37464i q^{59} -2.56392i q^{61} +5.01811 q^{63} -5.01660i q^{65} +5.18000 q^{67} +(3.49535 - 3.28367i) q^{69} +9.36201i q^{71} -6.89991 q^{73} +1.00000i q^{75} +11.8784 q^{77} +0.143570 q^{79} +1.00000 q^{81} -11.9654 q^{83} -1.91641 q^{85} -5.75850i q^{87} -11.4681i q^{89} -25.1739 q^{91} -3.94661 q^{93} -4.78232i q^{95} +14.2966i q^{97} +2.36711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * 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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