LMFDB - Embedded newform 5520.2.be.d.1471.10

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			1471.10
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.d.1471.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} +1.53388 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.00000i q^{5} +1.53388 q^{7} -1.00000 q^{9} -3.80458 q^{11} +2.39402 q^{13} +1.00000 q^{15} -2.51901i q^{17} +3.13707 q^{19} +1.53388i q^{21} +(3.77440 - 2.95870i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -1.76846 q^{29} -3.33803i q^{31} -3.80458i q^{33} -1.53388i q^{35} -3.84305i q^{37} +2.39402i q^{39} -12.2015 q^{41} +0.753300 q^{43} +1.00000i q^{45} +9.86269i q^{47} -4.64720 q^{49} +2.51901 q^{51} +5.02467i q^{53} +3.80458i q^{55} +3.13707i q^{57} -11.1175i q^{59} -13.1210i q^{61} -1.53388 q^{63} -2.39402i q^{65} +1.33054 q^{67} +(2.95870 + 3.77440i) q^{69} +7.54359i q^{71} +8.62861 q^{73} -1.00000i q^{75} -5.83578 q^{77} -2.23001 q^{79} +1.00000 q^{81} -16.1380 q^{83} -2.51901 q^{85} -1.76846i q^{87} -11.4152i q^{89} +3.67215 q^{91} +3.33803 q^{93} -3.13707i q^{95} -1.26542i q^{97} +3.80458 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.53388			0.579754			0.289877	−	0.957064i	$-0.406386\pi$
$7$	1.53388			0.579754			0.289877	+	0.957064i	$0.406386\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.80458			−1.14712			−0.573562	−	0.819162i	$-0.694439\pi$
$11$	−3.80458			−1.14712			−0.573562	+	0.819162i	$0.694439\pi$
$12$		0			0
$13$	2.39402			0.663982			0.331991	−	0.943283i	$-0.392280\pi$
$13$	2.39402			0.663982			0.331991	+	0.943283i	$0.392280\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$		−	2.51901i		−	0.610950i	−0.952200	−	0.305475i	$-0.901185\pi$
$17$		−	2.51901i		−	0.610950i	0.952200	−	0.305475i	$-0.0988152\pi$
$18$		0			0
$19$	3.13707			0.719694			0.359847	−	0.933011i	$-0.382829\pi$
$19$	3.13707			0.719694			0.359847	+	0.933011i	$0.382829\pi$
$20$		0			0
$21$			1.53388i			0.334721i
$22$		0			0
$23$	3.77440	−	2.95870i	0.787018	−	0.616931i
$23$	3.77440	−	2.95870i	0.787018	−	0.616931i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.76846			−0.328395			−0.164198	−	0.986427i	$-0.552504\pi$
$29$	−1.76846			−0.328395			−0.164198	+	0.986427i	$0.552504\pi$
$30$		0			0
$31$		−	3.33803i		−	0.599528i	−0.954013	−	0.299764i	$-0.903092\pi$
$31$		−	3.33803i		−	0.599528i	0.954013	−	0.299764i	$-0.0969080\pi$
$32$		0			0
$33$		−	3.80458i		−	0.662292i
$34$		0			0
$35$		−	1.53388i		−	0.259274i
$36$		0			0
$37$		−	3.84305i		−	0.631794i	−0.948794	−	0.315897i	$-0.897695\pi$
$37$		−	3.84305i		−	0.631794i	0.948794	−	0.315897i	$-0.102305\pi$
$38$		0			0
$39$			2.39402i			0.383350i
$40$		0			0
$41$	−12.2015			−1.90555			−0.952774	−	0.303680i	$-0.901785\pi$
$41$	−12.2015			−1.90555			−0.952774	+	0.303680i	$0.901785\pi$
$42$		0			0
$43$	0.753300			0.114877			0.0574386	−	0.998349i	$-0.481707\pi$
$43$	0.753300			0.114877			0.0574386	+	0.998349i	$0.481707\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$			9.86269i			1.43862i	0.694688	+	0.719311i	$0.255542\pi$
$47$			9.86269i			1.43862i	−0.694688	+	0.719311i	$0.744458\pi$
$48$		0			0
$49$	−4.64720			−0.663886
$50$		0			0
$51$	2.51901			0.352732
$52$		0			0
$53$			5.02467i			0.690191i	0.938568	+	0.345096i	$0.112154\pi$
$53$			5.02467i			0.690191i	−0.938568	+	0.345096i	$0.887846\pi$
$54$		0			0
$55$			3.80458i			0.513009i
$56$		0			0
$57$			3.13707i			0.415516i
$58$		0			0
$59$		−	11.1175i		−	1.44738i	−0.690124	−	0.723691i	$-0.742444\pi$
$59$		−	11.1175i		−	1.44738i	0.690124	−	0.723691i	$-0.257556\pi$
$60$		0			0
$61$		−	13.1210i		−	1.67997i	−0.542612	−	0.839983i	$-0.682565\pi$
$61$		−	13.1210i		−	1.67997i	0.542612	−	0.839983i	$-0.317435\pi$
$62$		0			0
$63$	−1.53388			−0.193251
$64$		0			0
$65$		−	2.39402i		−	0.296942i
$66$		0			0
$67$	1.33054			0.162551			0.0812757	−	0.996692i	$-0.474101\pi$
$67$	1.33054			0.162551			0.0812757	+	0.996692i	$0.474101\pi$
$68$		0			0
$69$	2.95870	+	3.77440i	0.356185	+	0.454385i
$70$		0			0
$71$			7.54359i			0.895259i	0.894219	+	0.447630i	$0.147732\pi$
$71$			7.54359i			0.895259i	−0.894219	+	0.447630i	$0.852268\pi$
$72$		0			0
$73$	8.62861			1.00990			0.504951	−	0.863148i	$-0.331511\pi$
$73$	8.62861			1.00990			0.504951	+	0.863148i	$0.331511\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	−5.83578			−0.665049
$78$		0			0
$79$	−2.23001			−0.250895			−0.125448	−	0.992100i	$-0.540037\pi$
$79$	−2.23001			−0.250895			−0.125448	+	0.992100i	$0.540037\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−16.1380			−1.77138			−0.885689	−	0.464279i	$-0.846314\pi$
$83$	−16.1380			−1.77138			−0.885689	+	0.464279i	$0.846314\pi$
$84$		0			0
$85$	−2.51901			−0.273225
$86$		0			0
$87$		−	1.76846i		−	0.189599i
$88$		0			0
$89$		−	11.4152i		−	1.21001i	−0.796222	−	0.605005i	$-0.793171\pi$
$89$		−	11.4152i		−	1.21001i	0.796222	−	0.605005i	$-0.206829\pi$
$90$		0			0
$91$	3.67215			0.384946
$92$		0			0
$93$	3.33803			0.346138
$94$		0			0
$95$		−	3.13707i		−	0.321857i
$96$		0			0
$97$		−	1.26542i		−	0.128484i	−0.997934	−	0.0642420i	$-0.979537\pi$
$97$		−	1.26542i		−	0.128484i	0.997934	−	0.0642420i	$-0.0204629\pi$
$98$		0			0
$99$	3.80458			0.382375
$100$		0			0
$101$	−9.29932			−0.925317			−0.462659	−	0.886537i	$-0.653104\pi$
$101$	−9.29932			−0.925317			−0.462659	+	0.886537i	$0.653104\pi$
$102$		0			0
$103$	15.9393			1.57055			0.785275	−	0.619148i	$-0.212522\pi$
$103$	15.9393			1.57055			0.785275	+	0.619148i	$0.212522\pi$
$104$		0			0
$105$	1.53388			0.149692
$106$		0			0
$107$	14.4419			1.39616			0.698078	−	0.716022i	$-0.254039\pi$
$107$	14.4419			1.39616			0.698078	+	0.716022i	$0.254039\pi$
$108$		0			0
$109$		−	10.4690i		−	1.00275i	−0.865231	−	0.501374i	$-0.832828\pi$
$109$		−	10.4690i		−	1.00275i	0.865231	−	0.501374i	$-0.167172\pi$
$110$		0			0
$111$	3.84305			0.364766
$112$		0			0
$113$			17.2391i			1.62171i	0.585244	+	0.810857i	$0.300999\pi$
$113$			17.2391i			1.62171i	−0.585244	+	0.810857i	$0.699001\pi$
$114$		0			0
$115$	−2.95870	−	3.77440i	−0.275900	−	0.351965i
$116$		0			0
$117$	−2.39402			−0.221327
$118$		0			0
$119$		−	3.86387i		−	0.354200i
$120$		0			0
$121$	3.47482			0.315892
$122$		0			0
$123$		−	12.2015i		−	1.10017i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	5.16262i		−	0.458109i	−0.973414	−	0.229054i	$-0.926437\pi$
$127$		−	5.16262i		−	0.458109i	0.973414	−	0.229054i	$-0.0735634\pi$
$128$		0			0
$129$			0.753300i			0.0663243i
$130$		0			0
$131$		−	14.7628i		−	1.28983i	−0.764255	−	0.644914i	$-0.776893\pi$
$131$		−	14.7628i		−	1.28983i	0.764255	−	0.644914i	$-0.223107\pi$
$132$		0			0
$133$	4.81191			0.417245
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$		−	20.7112i		−	1.76948i	−0.466088	−	0.884739i	$-0.654337\pi$
$137$		−	20.7112i		−	1.76948i	0.466088	−	0.884739i	$-0.345663\pi$
$138$		0			0
$139$		−	5.33288i		−	0.452329i	−0.974089	−	0.226165i	$-0.927381\pi$
$139$		−	5.33288i		−	0.452329i	0.974089	−	0.226165i	$-0.0726187\pi$
$140$		0			0
$141$	−9.86269			−0.830589
$142$		0			0
$143$	−9.10824			−0.761669
$144$		0			0
$145$			1.76846i			0.146863i
$146$		0			0
$147$		−	4.64720i		−	0.383295i
$148$		0			0
$149$		−	10.2771i		−	0.841936i	−0.907075	−	0.420968i	$-0.861690\pi$
$149$		−	10.2771i		−	0.841936i	0.907075	−	0.420968i	$-0.138310\pi$
$150$		0			0
$151$			11.9498i			0.972465i	0.873830	+	0.486232i	$0.161629\pi$
$151$			11.9498i			0.972465i	−0.873830	+	0.486232i	$0.838371\pi$
$152$		0			0
$153$			2.51901i			0.203650i
$154$		0			0
$155$	−3.33803			−0.268117
$156$		0			0
$157$		−	10.9153i		−	0.871135i	−0.900156	−	0.435567i	$-0.856548\pi$
$157$		−	10.9153i		−	0.871135i	0.900156	−	0.435567i	$-0.143452\pi$
$158$		0			0
$159$	−5.02467			−0.398482
$160$		0			0
$161$	5.78950	−	4.53830i	0.456276	−	0.357668i
$162$		0			0
$163$		−	20.7616i		−	1.62617i	−0.582142	−	0.813087i	$-0.697785\pi$
$163$		−	20.7616i		−	1.62617i	0.582142	−	0.813087i	$-0.302215\pi$
$164$		0			0
$165$	−3.80458			−0.296186
$166$		0			0
$167$		−	12.0149i		−	0.929738i	−0.885380	−	0.464869i	$-0.846101\pi$
$167$		−	12.0149i		−	0.929738i	0.885380	−	0.464869i	$-0.153899\pi$
$168$		0			0
$169$	−7.26867			−0.559128
$170$		0			0
$171$	−3.13707			−0.239898
$172$		0			0
$173$	−12.7396			−0.968571			−0.484285	−	0.874910i	$-0.660920\pi$
$173$	−12.7396			−0.968571			−0.484285	+	0.874910i	$0.660920\pi$
$174$		0			0
$175$	−1.53388			−0.115951
$176$		0			0
$177$	11.1175			0.835646
$178$		0			0
$179$			20.9807i			1.56817i	0.620653	+	0.784085i	$0.286868\pi$
$179$			20.9807i			1.56817i	−0.620653	+	0.784085i	$0.713132\pi$
$180$		0			0
$181$			11.2594i			0.836904i	0.908239	+	0.418452i	$0.137427\pi$
$181$			11.2594i			0.836904i	−0.908239	+	0.418452i	$0.862573\pi$
$182$		0			0
$183$	13.1210			0.969929
$184$		0			0
$185$	−3.84305			−0.282547
$186$		0			0
$187$			9.58377i			0.700835i
$188$		0			0
$189$		−	1.53388i		−	0.111574i
$190$		0			0
$191$	20.2870			1.46792			0.733959	−	0.679194i	$-0.237670\pi$
$191$	20.2870			1.46792			0.733959	+	0.679194i	$0.237670\pi$
$192$		0			0
$193$	−19.5055			−1.40404			−0.702020	−	0.712158i	$-0.747718\pi$
$193$	−19.5055			−1.40404			−0.702020	+	0.712158i	$0.747718\pi$
$194$		0			0
$195$	2.39402			0.171439
$196$		0			0
$197$	−14.6532			−1.04400			−0.522000	−	0.852946i	$-0.674814\pi$
$197$	−14.6532			−1.04400			−0.522000	+	0.852946i	$0.674814\pi$
$198$		0			0
$199$	10.0280			0.710866			0.355433	−	0.934702i	$-0.384333\pi$
$199$	10.0280			0.710866			0.355433	+	0.934702i	$0.384333\pi$
$200$		0			0
$201$			1.33054i			0.0938491i
$202$		0			0
$203$	−2.71262			−0.190388
$204$		0			0
$205$			12.2015i			0.852187i
$206$		0			0
$207$	−3.77440	+	2.95870i	−0.262339	+	0.205644i
$208$		0			0
$209$	−11.9352			−0.825578
$210$		0			0
$211$			2.41236i			0.166073i	0.996546	+	0.0830367i	$0.0264619\pi$
$211$			2.41236i			0.166073i	−0.996546	+	0.0830367i	$0.973538\pi$
$212$		0			0
$213$	−7.54359			−0.516878
$214$		0			0
$215$		−	0.753300i		−	0.0513746i
$216$		0			0
$217$		−	5.12016i		−	0.347579i
$218$		0			0
$219$			8.62861i			0.583067i
$220$		0			0
$221$		−	6.03056i		−	0.405659i
$222$		0			0
$223$		−	26.5580i		−	1.77845i	−0.457466	−	0.889227i	$-0.651243\pi$
$223$		−	26.5580i		−	1.77845i	0.457466	−	0.889227i	$-0.348757\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−0.381119			−0.0252957			−0.0126479	−	0.999920i	$-0.504026\pi$
$227$	−0.381119			−0.0252957			−0.0126479	+	0.999920i	$0.504026\pi$
$228$		0			0
$229$			14.2631i			0.942532i	0.881991	+	0.471266i	$0.156203\pi$
$229$			14.2631i			0.942532i	−0.881991	+	0.471266i	$0.843797\pi$
$230$		0			0
$231$		−	5.83578i		−	0.383966i
$232$		0			0
$233$	9.77344			0.640279			0.320140	−	0.947370i	$-0.396270\pi$
$233$	9.77344			0.640279			0.320140	+	0.947370i	$0.396270\pi$
$234$		0			0
$235$	9.86269			0.643371
$236$		0			0
$237$		−	2.23001i		−	0.144854i
$238$		0			0
$239$		−	21.5599i		−	1.39459i	−0.716784	−	0.697295i	$-0.754387\pi$
$239$		−	21.5599i		−	1.39459i	0.716784	−	0.697295i	$-0.245613\pi$
$240$		0			0
$241$			9.31459i			0.600005i	0.953938	+	0.300003i	$0.0969876\pi$
$241$			9.31459i			0.600005i	−0.953938	+	0.300003i	$0.903012\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$			4.64720i			0.296899i
$246$		0			0
$247$	7.51022			0.477864
$248$		0			0
$249$		−	16.1380i		−	1.02271i
$250$		0			0
$251$	22.6642			1.43055			0.715275	−	0.698843i	$-0.246301\pi$
$251$	22.6642			1.43055			0.715275	+	0.698843i	$0.246301\pi$
$252$		0			0
$253$	−14.3600	+	11.2566i	−0.902806	+	0.707696i
$254$		0			0
$255$		−	2.51901i		−	0.157746i
$256$		0			0
$257$	2.37700			0.148273			0.0741367	−	0.997248i	$-0.476380\pi$
$257$	2.37700			0.148273			0.0741367	+	0.997248i	$0.476380\pi$
$258$		0			0
$259$		−	5.89479i		−	0.366285i
$260$		0			0
$261$	1.76846			0.109465
$262$		0			0
$263$	16.9954			1.04798			0.523991	−	0.851724i	$-0.324443\pi$
$263$	16.9954			1.04798			0.523991	+	0.851724i	$0.324443\pi$
$264$		0			0
$265$	5.02467			0.308663
$266$		0			0
$267$	11.4152			0.698599
$268$		0			0
$269$	−22.4080			−1.36624			−0.683121	−	0.730305i	$-0.739378\pi$
$269$	−22.4080			−1.36624			−0.683121	+	0.730305i	$0.739378\pi$
$270$		0			0
$271$			0.440168i			0.0267383i	0.999911	+	0.0133692i	$0.00425566\pi$
$271$			0.440168i			0.0267383i	−0.999911	+	0.0133692i	$0.995744\pi$
$272$		0			0
$273$			3.67215i			0.222249i
$274$		0			0
$275$	3.80458			0.229425
$276$		0			0
$277$	24.5485			1.47498			0.737489	−	0.675360i	$-0.236012\pi$
$277$	24.5485			1.47498			0.737489	+	0.675360i	$0.236012\pi$
$278$		0			0
$279$			3.33803i			0.199843i
$280$		0			0
$281$			5.24108i			0.312656i	0.987705	+	0.156328i	$0.0499657\pi$
$281$			5.24108i			0.312656i	−0.987705	+	0.156328i	$0.950034\pi$
$282$		0			0
$283$	0.477815			0.0284032			0.0142016	−	0.999899i	$-0.495479\pi$
$283$	0.477815			0.0284032			0.0142016	+	0.999899i	$0.495479\pi$
$284$		0			0
$285$	3.13707			0.185824
$286$		0			0
$287$	−18.7156			−1.10475
$288$		0			0
$289$	10.6546			0.626741
$290$		0			0
$291$	1.26542			0.0741802
$292$		0			0
$293$		−	13.3598i		−	0.780486i	−0.920712	−	0.390243i	$-0.872391\pi$
$293$		−	13.3598i		−	0.780486i	0.920712	−	0.390243i	$-0.127609\pi$
$294$		0			0
$295$	−11.1175			−0.647289
$296$		0			0
$297$			3.80458i			0.220764i
$298$		0			0
$299$	9.03600	−	7.08317i	0.522565	−	0.409631i
$300$		0			0
$301$	1.15547			0.0666004
$302$		0			0
$303$		−	9.29932i		−	0.534232i
$304$		0			0
$305$	−13.1210			−0.751304
$306$		0			0
$307$		−	29.7216i		−	1.69630i	−0.529757	−	0.848150i	$-0.677717\pi$
$307$		−	29.7216i		−	1.69630i	0.529757	−	0.848150i	$-0.322283\pi$
$308$		0			0
$309$			15.9393i			0.906757i
$310$		0			0
$311$			28.2256i			1.60053i	0.599650	+	0.800263i	$0.295307\pi$
$311$			28.2256i			1.60053i	−0.599650	+	0.800263i	$0.704693\pi$
$312$		0			0
$313$		−	4.82571i		−	0.272765i	−0.990656	−	0.136383i	$-0.956452\pi$
$313$		−	4.82571i		−	0.272765i	0.990656	−	0.136383i	$-0.0435477\pi$
$314$		0			0
$315$			1.53388i			0.0864246i
$316$		0			0
$317$	−5.23453			−0.294001			−0.147000	−	0.989136i	$-0.546962\pi$
$317$	−5.23453			−0.294001			−0.147000	+	0.989136i	$0.546962\pi$
$318$		0			0
$319$	6.72826			0.376710
$320$		0			0
$321$			14.4419i			0.806071i
$322$		0			0
$323$		−	7.90232i		−	0.439697i
$324$		0			0
$325$	−2.39402			−0.132796
$326$		0			0
$327$	10.4690			0.578937
$328$		0			0
$329$			15.1282i			0.834046i
$330$		0			0
$331$		−	21.6135i		−	1.18799i	−0.804470	−	0.593993i	$-0.797551\pi$
$331$		−	21.6135i		−	1.18799i	0.804470	−	0.593993i	$-0.202449\pi$
$332$		0			0
$333$			3.84305i			0.210598i
$334$		0			0
$335$		−	1.33054i		−	0.0726952i
$336$		0			0
$337$			2.81143i			0.153148i	0.997064	+	0.0765742i	$0.0243982\pi$
$337$			2.81143i			0.153148i	−0.997064	+	0.0765742i	$0.975602\pi$
$338$		0			0
$339$	−17.2391			−0.936297
$340$		0			0
$341$			12.6998i			0.687733i
$342$		0			0
$343$	−17.8655			−0.964644
$344$		0			0
$345$	3.77440	−	2.95870i	0.203207	−	0.159291i
$346$		0			0
$347$		−	11.8025i		−	0.633591i	−0.948494	−	0.316796i	$-0.897393\pi$
$347$		−	11.8025i		−	0.633591i	0.948494	−	0.316796i	$-0.102607\pi$
$348$		0			0
$349$	−32.7930			−1.75537			−0.877685	−	0.479237i	$-0.840913\pi$
$349$	−32.7930			−1.75537			−0.877685	+	0.479237i	$0.840913\pi$
$350$		0			0
$351$		−	2.39402i		−	0.127783i
$352$		0			0
$353$	14.1618			0.753755			0.376877	−	0.926263i	$-0.376998\pi$
$353$	14.1618			0.753755			0.376877	+	0.926263i	$0.376998\pi$
$354$		0			0
$355$	7.54359			0.400372
$356$		0			0
$357$	3.86387			0.204498
$358$		0			0
$359$	−3.37945			−0.178360			−0.0891802	−	0.996016i	$-0.528425\pi$
$359$	−3.37945			−0.178360			−0.0891802	+	0.996016i	$0.528425\pi$
$360$		0			0
$361$	−9.15877			−0.482040
$362$		0			0
$363$			3.47482i			0.182381i
$364$		0			0
$365$		−	8.62861i		−	0.451642i
$366$		0			0
$367$	−3.36610			−0.175709			−0.0878546	−	0.996133i	$-0.528001\pi$
$367$	−3.36610			−0.175709			−0.0878546	+	0.996133i	$0.528001\pi$
$368$		0			0
$369$	12.2015			0.635183
$370$		0			0
$371$			7.70726i			0.400141i
$372$		0			0
$373$		−	28.8077i		−	1.49160i	−0.666168	−	0.745802i	$-0.732066\pi$
$373$		−	28.8077i		−	1.49160i	0.666168	−	0.745802i	$-0.267934\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	−4.23374			−0.218049
$378$		0			0
$379$	−21.3903			−1.09875			−0.549374	−	0.835577i	$-0.685134\pi$
$379$	−21.3903			−1.09875			−0.549374	+	0.835577i	$0.685134\pi$
$380$		0			0
$381$	5.16262			0.264489
$382$		0			0
$383$	−37.4579			−1.91401			−0.957005	−	0.290072i	$-0.906321\pi$
$383$	−37.4579			−1.91401			−0.957005	+	0.290072i	$0.906321\pi$
$384$		0			0
$385$			5.83578i			0.297419i
$386$		0			0
$387$	−0.753300			−0.0382924
$388$		0			0
$389$		−	33.0282i		−	1.67460i	−0.546745	−	0.837299i	$-0.684133\pi$
$389$		−	33.0282i		−	1.67460i	0.546745	−	0.837299i	$-0.315867\pi$
$390$		0			0
$391$	−7.45298	−	9.50776i	−0.376913	−	0.480828i
$392$		0			0
$393$	14.7628			0.744683
$394$		0			0
$395$			2.23001i			0.112204i
$396$		0			0
$397$	6.74610			0.338577			0.169289	−	0.985567i	$-0.445853\pi$
$397$	6.74610			0.338577			0.169289	+	0.985567i	$0.445853\pi$
$398$		0			0
$399$			4.81191i			0.240897i
$400$		0			0
$401$		−	24.2243i		−	1.20971i	−0.796337	−	0.604853i	$-0.793232\pi$
$401$		−	24.2243i		−	1.20971i	0.796337	−	0.604853i	$-0.206768\pi$
$402$		0			0
$403$		−	7.99132i		−	0.398076i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			14.6212i			0.724745i
$408$		0			0
$409$	32.7223			1.61802			0.809008	−	0.587798i	$-0.200005\pi$
$409$	32.7223			1.61802			0.809008	+	0.587798i	$0.200005\pi$
$410$		0			0
$411$	20.7112			1.02161
$412$		0			0
$413$		−	17.0530i		−	0.839125i
$414$		0			0
$415$			16.1380i			0.792185i
$416$		0			0
$417$	5.33288			0.261152
$418$		0			0
$419$	15.7062			0.767299			0.383649	−	0.923479i	$-0.374667\pi$
$419$	15.7062			0.767299			0.383649	+	0.923479i	$0.374667\pi$
$420$		0			0
$421$		−	8.90543i		−	0.434024i	−0.976169	−	0.217012i	$-0.930369\pi$
$421$		−	8.90543i		−	0.434024i	0.976169	−	0.217012i	$-0.0696311\pi$
$422$		0			0
$423$		−	9.86269i		−	0.479541i
$424$		0			0
$425$			2.51901i			0.122190i
$426$		0			0
$427$		−	20.1260i		−	0.973967i
$428$		0			0
$429$		−	9.10824i		−	0.439750i
$430$		0			0
$431$	14.1215			0.680208			0.340104	−	0.940388i	$-0.389538\pi$
$431$	14.1215			0.680208			0.340104	+	0.940388i	$0.389538\pi$
$432$		0			0
$433$			20.3867i			0.979722i	0.871801	+	0.489861i	$0.162952\pi$
$433$			20.3867i			0.979722i	−0.871801	+	0.489861i	$0.837048\pi$
$434$		0			0
$435$	−1.76846			−0.0847913
$436$		0			0
$437$	11.8406	−	9.28164i	0.566412	−	0.444001i
$438$		0			0
$439$		−	22.5795i		−	1.07766i	−0.842414	−	0.538830i	$-0.818866\pi$
$439$		−	22.5795i		−	1.07766i	0.842414	−	0.538830i	$-0.181134\pi$
$440$		0			0
$441$	4.64720			0.221295
$442$		0			0
$443$			13.2144i			0.627833i	0.949451	+	0.313916i	$0.101641\pi$
$443$			13.2144i			0.627833i	−0.949451	+	0.313916i	$0.898359\pi$
$444$		0			0
$445$	−11.4152			−0.541133
$446$		0			0
$447$	10.2771			0.486092
$448$		0			0
$449$	40.8158			1.92621			0.963107	−	0.269118i	$-0.0867320\pi$
$449$	40.8158			1.92621			0.963107	+	0.269118i	$0.0867320\pi$
$450$		0			0
$451$	46.4214			2.18590
$452$		0			0
$453$	−11.9498			−0.561453
$454$		0			0
$455$		−	3.67215i		−	0.172153i
$456$		0			0
$457$			36.8494i			1.72374i	0.507126	+	0.861872i	$0.330708\pi$
$457$			36.8494i			1.72374i	−0.507126	+	0.861872i	$0.669292\pi$
$458$		0			0
$459$	−2.51901			−0.117577
$460$		0			0
$461$	12.2462			0.570363			0.285182	−	0.958473i	$-0.407946\pi$
$461$	12.2462			0.570363			0.285182	+	0.958473i	$0.407946\pi$
$462$		0			0
$463$			0.407169i			0.0189228i	0.999955	+	0.00946138i	$0.00301170\pi$
$463$			0.407169i			0.0189228i	−0.999955	+	0.00946138i	$0.996988\pi$
$464$		0			0
$465$		−	3.33803i		−	0.154798i
$466$		0			0
$467$	12.3451			0.571262			0.285631	−	0.958340i	$-0.407797\pi$
$467$	12.3451			0.571262			0.285631	+	0.958340i	$0.407797\pi$
$468$		0			0
$469$	2.04089			0.0942398
$470$		0			0
$471$	10.9153			0.502950
$472$		0			0
$473$	−2.86599			−0.131778
$474$		0			0
$475$	−3.13707			−0.143939
$476$		0			0
$477$		−	5.02467i		−	0.230064i
$478$		0			0
$479$	−13.8805			−0.634218			−0.317109	−	0.948389i	$-0.602712\pi$
$479$	−13.8805			−0.634218			−0.317109	+	0.948389i	$0.602712\pi$
$480$		0			0
$481$		−	9.20034i		−	0.419499i
$482$		0			0
$483$	4.53830	+	5.78950i	0.206500	+	0.263431i
$484$		0			0
$485$	−1.26542			−0.0574598
$486$		0			0
$487$		−	0.727009i		−	0.0329439i	−0.999864	−	0.0164720i	$-0.994757\pi$
$487$		−	0.727009i		−	0.0329439i	0.999864	−	0.0164720i	$-0.00524343\pi$
$488$		0			0
$489$	20.7616			0.938872
$490$		0			0
$491$		−	4.39551i		−	0.198367i	−0.995069	−	0.0991833i	$-0.968377\pi$
$491$		−	4.39551i		−	0.198367i	0.995069	−	0.0991833i	$-0.0316230\pi$
$492$		0			0
$493$			4.45478i			0.200633i
$494$		0			0
$495$		−	3.80458i		−	0.171003i
$496$		0			0
$497$			11.5710i			0.519030i
$498$		0			0
$499$			17.7313i			0.793764i	0.917870	+	0.396882i	$0.129908\pi$
$499$			17.7313i			0.793764i	−0.917870	+	0.396882i	$0.870092\pi$
$500$		0			0
$501$	12.0149			0.536784
$502$		0			0
$503$	5.13524			0.228969			0.114485	−	0.993425i	$-0.463478\pi$
$503$	5.13524			0.228969			0.114485	+	0.993425i	$0.463478\pi$
$504$		0			0
$505$			9.29932i			0.413814i
$506$		0			0
$507$		−	7.26867i		−	0.322813i
$508$		0			0
$509$	−28.7319			−1.27352			−0.636760	−	0.771062i	$-0.719726\pi$
$509$	−28.7319			−1.27352			−0.636760	+	0.771062i	$0.719726\pi$
$510$		0			0
$511$	13.2353			0.585495
$512$		0			0
$513$		−	3.13707i		−	0.138505i
$514$		0			0
$515$		−	15.9393i		−	0.702371i
$516$		0			0
$517$		−	37.5234i		−	1.65028i
$518$		0			0
$519$		−	12.7396i		−	0.559205i
$520$		0			0
$521$		−	1.48894i		−	0.0652317i	−0.999468	−	0.0326159i	$-0.989616\pi$
$521$		−	1.48894i		−	0.0652317i	0.999468	−	0.0326159i	$-0.0103838\pi$
$522$		0			0
$523$	−22.5119			−0.984376			−0.492188	−	0.870489i	$-0.663803\pi$
$523$	−22.5119			−0.984376			−0.492188	+	0.870489i	$0.663803\pi$
$524$		0			0
$525$		−	1.53388i		−	0.0669442i
$526$		0			0
$527$	−8.40854			−0.366282
$528$		0			0
$529$	5.49224	−	22.3346i	0.238793	−	0.971070i
$530$		0			0
$531$			11.1175i			0.482461i
$532$		0			0
$533$	−29.2105			−1.26525
$534$		0			0
$535$		−	14.4419i		−	0.624380i
$536$		0			0
$537$	−20.9807			−0.905384
$538$		0			0
$539$	17.6806			0.761559
$540$		0			0
$541$	4.42320			0.190168			0.0950841	−	0.995469i	$-0.469688\pi$
$541$	4.42320			0.190168			0.0950841	+	0.995469i	$0.469688\pi$
$542$		0			0
$543$	−11.2594			−0.483187
$544$		0			0
$545$	−10.4690			−0.448443
$546$		0			0
$547$			19.9051i			0.851079i	0.904940	+	0.425540i	$0.139916\pi$
$547$			19.9051i			0.851079i	−0.904940	+	0.425540i	$0.860084\pi$
$548$		0			0
$549$			13.1210i			0.559989i
$550$		0			0
$551$	−5.54780			−0.236344
$552$		0			0
$553$	−3.42057			−0.145457
$554$		0			0
$555$		−	3.84305i		−	0.163128i
$556$		0			0
$557$		−	23.5795i		−	0.999094i	−0.866287	−	0.499547i	$-0.833500\pi$
$557$		−	23.5795i		−	0.999094i	0.866287	−	0.499547i	$-0.166500\pi$
$558$		0			0
$559$	1.80341			0.0762763
$560$		0			0
$561$	−9.58377			−0.404627
$562$		0			0
$563$	−19.7524			−0.832462			−0.416231	−	0.909259i	$-0.636649\pi$
$563$	−19.7524			−0.832462			−0.416231	+	0.909259i	$0.636649\pi$
$564$		0			0
$565$	17.2391			0.725253
$566$		0			0
$567$	1.53388			0.0644171
$568$		0			0
$569$		−	20.6533i		−	0.865833i	−0.901434	−	0.432916i	$-0.857484\pi$
$569$		−	20.6533i		−	0.865833i	0.901434	−	0.432916i	$-0.142516\pi$
$570$		0			0
$571$	15.4386			0.646084			0.323042	−	0.946385i	$-0.395294\pi$
$571$	15.4386			0.646084			0.323042	+	0.946385i	$0.395294\pi$
$572$		0			0
$573$			20.2870i			0.847503i
$574$		0			0
$575$	−3.77440	+	2.95870i	−0.157404	+	0.123386i
$576$		0			0
$577$	45.8416			1.90841			0.954206	−	0.299150i	$-0.0967031\pi$
$577$	45.8416			1.90841			0.954206	+	0.299150i	$0.0967031\pi$
$578$		0			0
$579$		−	19.5055i		−	0.810623i
$580$		0			0
$581$	−24.7539			−1.02696
$582$		0			0
$583$		−	19.1167i		−	0.791735i
$584$		0			0
$585$			2.39402i			0.0989805i
$586$		0			0
$587$			8.31516i			0.343203i	0.985166	+	0.171602i	$0.0548942\pi$
$587$			8.31516i			0.343203i	−0.985166	+	0.171602i	$0.945106\pi$
$588$		0			0
$589$		−	10.4717i		−	0.431477i
$590$		0			0
$591$		−	14.6532i		−	0.602753i
$592$		0			0
$593$	−43.8609			−1.80115			−0.900577	−	0.434697i	$-0.856855\pi$
$593$	−43.8609			−1.80115			−0.900577	+	0.434697i	$0.856855\pi$
$594$		0			0
$595$	−3.86387			−0.158403
$596$		0			0
$597$			10.0280i			0.410419i
$598$		0			0
$599$			30.3057i			1.23826i	0.785290	+	0.619128i	$0.212514\pi$
$599$			30.3057i			1.23826i	−0.785290	+	0.619128i	$0.787486\pi$
$600$		0			0
$601$	46.2015			1.88460			0.942300	−	0.334770i	$-0.108659\pi$
$601$	46.2015			1.88460			0.942300	+	0.334770i	$0.108659\pi$
$602$		0			0
$603$	−1.33054			−0.0541838
$604$		0			0
$605$		−	3.47482i		−	0.141271i
$606$		0			0
$607$			29.0800i			1.18032i	0.807286	+	0.590161i	$0.200936\pi$
$607$			29.0800i			1.18032i	−0.807286	+	0.590161i	$0.799064\pi$
$608$		0			0
$609$		−	2.71262i		−	0.109921i
$610$		0			0
$611$			23.6115i			0.955218i
$612$		0			0
$613$			21.3595i			0.862700i	0.902185	+	0.431350i	$0.141963\pi$
$613$			21.3595i			0.862700i	−0.902185	+	0.431350i	$0.858037\pi$
$614$		0			0
$615$	−12.2015			−0.492010
$616$		0			0
$617$			7.10553i			0.286058i	0.989719	+	0.143029i	$0.0456842\pi$
$617$			7.10553i			0.286058i	−0.989719	+	0.143029i	$0.954316\pi$
$618$		0			0
$619$	43.7859			1.75990			0.879952	−	0.475063i	$-0.157575\pi$
$619$	43.7859			1.75990			0.879952	+	0.475063i	$0.157575\pi$
$620$		0			0
$621$	−2.95870	−	3.77440i	−0.118728	−	0.151462i
$622$		0			0
$623$		−	17.5096i		−	0.701508i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	11.9352i		−	0.476648i
$628$		0			0
$629$	−9.68068			−0.385994
$630$		0			0
$631$	23.3043			0.927728			0.463864	−	0.885906i	$-0.346463\pi$
$631$	23.3043			0.927728			0.463864	+	0.885906i	$0.346463\pi$
$632$		0			0
$633$	−2.41236			−0.0958826
$634$		0			0
$635$	−5.16262			−0.204872
$636$		0			0
$637$	−11.1255			−0.440808
$638$		0			0
$639$		−	7.54359i		−	0.298420i
$640$		0			0
$641$			16.9408i			0.669123i	0.942374	+	0.334562i	$0.108588\pi$
$641$			16.9408i			0.669123i	−0.942374	+	0.334562i	$0.891412\pi$
$642$		0			0
$643$	−12.8234			−0.505705			−0.252853	−	0.967505i	$-0.581369\pi$
$643$	−12.8234			−0.505705			−0.252853	+	0.967505i	$0.581369\pi$
$644$		0			0
$645$	0.753300			0.0296611
$646$		0			0
$647$		−	3.68104i		−	0.144717i	−0.997379	−	0.0723583i	$-0.976948\pi$
$647$		−	3.68104i		−	0.144717i	0.997379	−	0.0723583i	$-0.0230525\pi$
$648$		0			0
$649$			42.2976i			1.66033i
$650$		0			0
$651$	5.12016			0.200675
$652$		0			0
$653$	−16.5223			−0.646567			−0.323284	−	0.946302i	$-0.604787\pi$
$653$	−16.5223			−0.646567			−0.323284	+	0.946302i	$0.604787\pi$
$654$		0			0
$655$	−14.7628			−0.576829
$656$		0			0
$657$	−8.62861			−0.336634
$658$		0			0
$659$	25.6540			0.999338			0.499669	−	0.866216i	$-0.333455\pi$
$659$	25.6540			0.999338			0.499669	+	0.866216i	$0.333455\pi$
$660$		0			0
$661$			1.84288i			0.0716795i	0.999358	+	0.0358398i	$0.0114106\pi$
$661$			1.84288i			0.0716795i	−0.999358	+	0.0358398i	$0.988589\pi$
$662$		0			0
$663$	6.03056			0.234207
$664$		0			0
$665$		−	4.81191i		−	0.186598i
$666$		0			0
$667$	−6.67490	+	5.23235i	−0.258453	+	0.202597i
$668$		0			0
$669$	26.5580			1.02679
$670$		0			0
$671$			49.9197i			1.92713i
$672$		0			0
$673$	−14.6701			−0.565489			−0.282745	−	0.959195i	$-0.591245\pi$
$673$	−14.6701			−0.565489			−0.282745	+	0.959195i	$0.591245\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$		−	21.5348i		−	0.827649i	−0.910357	−	0.413825i	$-0.864193\pi$
$677$		−	21.5348i		−	0.827649i	0.910357	−	0.413825i	$-0.135807\pi$
$678$		0			0
$679$		−	1.94101i		−	0.0744890i
$680$		0			0
$681$		−	0.381119i		−	0.0146045i
$682$		0			0
$683$		−	2.94492i		−	0.112684i	−0.998412	−	0.0563421i	$-0.982056\pi$
$683$		−	2.94492i		−	0.112684i	0.998412	−	0.0563421i	$-0.0179437\pi$
$684$		0			0
$685$	−20.7112			−0.791334
$686$		0			0
$687$	−14.2631			−0.544171
$688$		0			0
$689$			12.0292i			0.458274i
$690$		0			0
$691$			39.4261i			1.49984i	0.661529	+	0.749920i	$0.269908\pi$
$691$			39.4261i			1.49984i	−0.661529	+	0.749920i	$0.730092\pi$
$692$		0			0
$693$	5.83578			0.221683
$694$		0			0
$695$	−5.33288			−0.202288
$696$		0			0
$697$			30.7356i			1.16419i
$698$		0			0
$699$			9.77344i			0.369665i
$700$		0			0
$701$		−	31.2217i		−	1.17923i	−0.807686	−	0.589613i	$-0.799280\pi$
$701$		−	31.2217i		−	1.17923i	0.807686	−	0.589613i	$-0.200720\pi$
$702$		0			0
$703$		−	12.0559i		−	0.454698i
$704$		0			0
$705$			9.86269i			0.371451i
$706$		0			0
$707$	−14.2641			−0.536456
$708$		0			0
$709$			32.9346i			1.23689i	0.785830	+	0.618443i	$0.212236\pi$
$709$			32.9346i			1.23689i	−0.785830	+	0.618443i	$0.787764\pi$
$710$		0			0
$711$	2.23001			0.0836317
$712$		0			0
$713$	−9.87622	−	12.5991i	−0.369867	−	0.471839i
$714$		0			0
$715$			9.10824i			0.340629i
$716$		0			0
$717$	21.5599			0.805167
$718$		0			0
$719$		−	2.50314i		−	0.0933514i	−0.998910	−	0.0466757i	$-0.985137\pi$
$719$		−	2.50314i		−	0.0933514i	0.998910	−	0.0466757i	$-0.0148627\pi$
$720$		0			0
$721$	24.4491			0.910532
$722$		0			0
$723$	−9.31459			−0.346413
$724$		0			0
$725$	1.76846			0.0656791
$726$		0			0
$727$	38.9275			1.44374			0.721871	−	0.692028i	$-0.243282\pi$
$727$	38.9275			1.44374			0.721871	+	0.692028i	$0.243282\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	1.89757i		−	0.0701841i
$732$		0			0
$733$			39.0232i			1.44135i	0.693271	+	0.720677i	$0.256169\pi$
$733$			39.0232i			1.44135i	−0.693271	+	0.720677i	$0.743831\pi$
$734$		0			0
$735$	−4.64720			−0.171415
$736$		0			0
$737$	−5.06215			−0.186467
$738$		0			0
$739$			7.48023i			0.275165i	0.990490	+	0.137582i	$0.0439332\pi$
$739$			7.48023i			0.275165i	−0.990490	+	0.137582i	$0.956067\pi$
$740$		0			0
$741$			7.51022i			0.275895i
$742$		0			0
$743$	−23.9955			−0.880310			−0.440155	−	0.897922i	$-0.645076\pi$
$743$	−23.9955			−0.880310			−0.440155	+	0.897922i	$0.645076\pi$
$744$		0			0
$745$	−10.2771			−0.376525
$746$		0			0
$747$	16.1380			0.590459
$748$		0			0
$749$	22.1523			0.809426
$750$		0			0
$751$	−0.141614			−0.00516756			−0.00258378	−	0.999997i	$-0.500822\pi$
$751$	−0.141614			−0.00516756			−0.00258378	+	0.999997i	$0.500822\pi$
$752$		0			0
$753$			22.6642i			0.825929i
$754$		0			0
$755$	11.9498			0.434899
$756$		0			0
$757$			6.84512i			0.248790i	0.992233	+	0.124395i	$0.0396990\pi$
$757$			6.84512i			0.248790i	−0.992233	+	0.124395i	$0.960301\pi$
$758$		0			0
$759$	−11.2566	−	14.3600i	−0.408588	−	0.521235i
$760$		0			0
$761$	−30.8852			−1.11959			−0.559794	−	0.828632i	$-0.689120\pi$
$761$	−30.8852			−1.11959			−0.559794	+	0.828632i	$0.689120\pi$
$762$		0			0
$763$		−	16.0582i		−	0.581347i
$764$		0			0
$765$	2.51901			0.0910750
$766$		0			0
$767$		−	26.6156i		−	0.961035i
$768$		0			0
$769$			12.3453i			0.445183i	0.974912	+	0.222592i	$0.0714516\pi$
$769$			12.3453i			0.445183i	−0.974912	+	0.222592i	$0.928548\pi$
$770$		0			0
$771$			2.37700i			0.0856057i
$772$		0			0
$773$		−	19.8084i		−	0.712457i	−0.934399	−	0.356228i	$-0.884062\pi$
$773$		−	19.8084i		−	0.712457i	0.934399	−	0.356228i	$-0.115938\pi$
$774$		0			0
$775$			3.33803i			0.119906i
$776$		0			0
$777$	5.89479			0.211475
$778$		0			0
$779$	−38.2769			−1.37141
$780$		0			0
$781$		−	28.7002i		−	1.02697i
$782$		0			0
$783$			1.76846i			0.0631997i
$784$		0			0
$785$	−10.9153			−0.389583
$786$		0			0
$787$	−15.0357			−0.535965			−0.267983	−	0.963424i	$-0.586357\pi$
$787$	−15.0357			−0.535965			−0.267983	+	0.963424i	$0.586357\pi$
$788$		0			0
$789$			16.9954i			0.605052i
$790$		0			0
$791$			26.4427i			0.940195i
$792$		0			0
$793$		−	31.4118i		−	1.11547i
$794$		0			0
$795$			5.02467i			0.178207i
$796$		0			0
$797$		−	28.5067i		−	1.00976i	−0.863190	−	0.504880i	$-0.831537\pi$
$797$		−	28.5067i		−	1.00976i	0.863190	−	0.504880i	$-0.168463\pi$
$798$		0			0
$799$	24.8442			0.878925
$800$		0			0
$801$			11.4152i			0.403337i
$802$		0			0
$803$	−32.8282			−1.15848
$804$		0			0
$805$	−4.53830	−	5.78950i	−0.159954	−	0.204053i
$806$		0			0
$807$		−	22.4080i		−	0.788800i
$808$		0			0
$809$	−18.7619			−0.659633			−0.329817	−	0.944045i	$-0.606987\pi$
$809$	−18.7619			−0.659633			−0.329817	+	0.944045i	$0.606987\pi$
$810$		0			0
$811$			14.3089i			0.502453i	0.967928	+	0.251226i	$0.0808339\pi$
$811$			14.3089i			0.502453i	−0.967928	+	0.251226i	$0.919166\pi$
$812$		0			0
$813$	−0.440168			−0.0154374
$814$		0			0
$815$	−20.7616			−0.727247
$816$		0			0
$817$	2.36316			0.0826764
$818$		0			0
$819$	−3.67215			−0.128315
$820$		0			0
$821$	24.3378			0.849396			0.424698	−	0.905335i	$-0.360380\pi$
$821$	24.3378			0.849396			0.424698	+	0.905335i	$0.360380\pi$
$822$		0			0
$823$		−	6.39279i		−	0.222839i	−0.993773	−	0.111419i	$-0.964460\pi$
$823$		−	6.39279i		−	0.222839i	0.993773	−	0.111419i	$-0.0355397\pi$
$824$		0			0
$825$			3.80458i			0.132458i
$826$		0			0
$827$	42.7033			1.48494			0.742470	−	0.669879i	$-0.233654\pi$
$827$	42.7033			1.48494			0.742470	+	0.669879i	$0.233654\pi$
$828$		0			0
$829$	−38.6182			−1.34127			−0.670633	−	0.741789i	$-0.733977\pi$
$829$	−38.6182			−1.34127			−0.670633	+	0.741789i	$0.733977\pi$
$830$		0			0
$831$			24.5485i			0.851578i
$832$		0			0
$833$			11.7063i			0.405601i
$834$		0			0
$835$	−12.0149			−0.415791
$836$		0			0
$837$	−3.33803			−0.115379
$838$		0			0
$839$	−29.6475			−1.02355			−0.511773	−	0.859121i	$-0.671011\pi$
$839$	−29.6475			−1.02355			−0.511773	+	0.859121i	$0.671011\pi$
$840$		0			0
$841$	−25.8725			−0.892156
$842$		0			0
$843$	−5.24108			−0.180512
$844$		0			0
$845$			7.26867i			0.250050i
$846$		0			0
$847$	5.32997			0.183140
$848$		0			0
$849$			0.477815i			0.0163986i
$850$		0			0
$851$	−11.3704	−	14.5052i	−0.389773	−	0.497233i
$852$		0			0
$853$	−18.4911			−0.633123			−0.316562	−	0.948572i	$-0.602528\pi$
$853$	−18.4911			−0.633123			−0.316562	+	0.948572i	$0.602528\pi$
$854$		0			0
$855$			3.13707i			0.107286i
$856$		0			0
$857$	−22.5100			−0.768926			−0.384463	−	0.923140i	$-0.625613\pi$
$857$	−22.5100			−0.768926			−0.384463	+	0.923140i	$0.625613\pi$
$858$		0			0
$859$			30.0079i			1.02386i	0.859028	+	0.511928i	$0.171069\pi$
$859$			30.0079i			1.02386i	−0.859028	+	0.511928i	$0.828931\pi$
$860$		0			0
$861$		−	18.7156i		−	0.637827i
$862$		0			0
$863$			56.5548i			1.92515i	0.271020	+	0.962574i	$0.412639\pi$
$863$			56.5548i			1.92515i	−0.271020	+	0.962574i	$0.587361\pi$
$864$		0			0
$865$			12.7396i			0.433158i
$866$		0			0
$867$			10.6546i			0.361849i
$868$		0			0
$869$	8.48423			0.287808
$870$		0			0
$871$	3.18534			0.107931
$872$		0			0
$873$			1.26542i			0.0428280i
$874$		0			0
$875$			1.53388i			0.0518547i
$876$		0			0
$877$	−10.3790			−0.350475			−0.175238	−	0.984526i	$-0.556069\pi$
$877$	−10.3790			−0.350475			−0.175238	+	0.984526i	$0.556069\pi$
$878$		0			0
$879$	13.3598			0.450614
$880$		0			0
$881$			26.2502i			0.884391i	0.896919	+	0.442195i	$0.145800\pi$
$881$			26.2502i			0.884391i	−0.896919	+	0.442195i	$0.854200\pi$
$882$		0			0
$883$		−	4.30518i		−	0.144881i	−0.997373	−	0.0724404i	$-0.976921\pi$
$883$		−	4.30518i		−	0.144881i	0.997373	−	0.0724404i	$-0.0230787\pi$
$884$		0			0
$885$		−	11.1175i		−	0.373712i
$886$		0			0
$887$			39.4933i			1.32606i	0.748594	+	0.663028i	$0.230729\pi$
$887$			39.4933i			1.32606i	−0.748594	+	0.663028i	$0.769271\pi$
$888$		0			0
$889$		−	7.91887i		−	0.265590i
$890$		0			0
$891$	−3.80458			−0.127458
$892$		0			0
$893$			30.9400i			1.03537i
$894$		0			0
$895$	20.9807			0.701307
$896$		0			0
$897$	7.08317	+	9.03600i	0.236500	+	0.301703i
$898$		0			0
$899$			5.90319i			0.196882i
$900$		0			0
$901$	12.6572			0.421672
$902$		0			0
$903$			1.15547i			0.0384518i
$904$		0			0
$905$	11.2594			0.374275
$906$		0			0
$907$	−47.4505			−1.57557			−0.787784	−	0.615952i	$-0.788772\pi$
$907$	−47.4505			−1.57557			−0.787784	+	0.615952i	$0.788772\pi$
$908$		0			0
$909$	9.29932			0.308439
$910$		0			0
$911$	−30.1923			−1.00031			−0.500157	−	0.865935i	$-0.666724\pi$
$911$	−30.1923			−1.00031			−0.500157	+	0.865935i	$0.666724\pi$
$912$		0			0
$913$	61.3984			2.03199
$914$		0			0
$915$		−	13.1210i		−	0.433765i
$916$		0			0
$917$		−	22.6444i		−	0.747783i
$918$		0			0
$919$	46.1559			1.52254			0.761272	−	0.648433i	$-0.224575\pi$
$919$	46.1559			1.52254			0.761272	+	0.648433i	$0.224575\pi$
$920$		0			0
$921$	29.7216			0.979359
$922$		0			0
$923$			18.0595i			0.594436i
$924$		0			0
$925$			3.84305i			0.126359i
$926$		0			0
$927$	−15.9393			−0.523516
$928$		0			0
$929$	23.6158			0.774809			0.387405	−	0.921910i	$-0.373372\pi$
$929$	23.6158			0.774809			0.387405	+	0.921910i	$0.373372\pi$
$930$		0			0
$931$	−14.5786			−0.477795
$932$		0			0
$933$	−28.2256			−0.924064
$934$		0			0
$935$	9.58377			0.313423
$936$		0			0
$937$		−	25.1461i		−	0.821486i	−0.911751	−	0.410743i	$-0.865269\pi$
$937$		−	25.1461i		−	0.821486i	0.911751	−	0.410743i	$-0.134731\pi$
$938$		0			0
$939$	4.82571			0.157481
$940$		0			0
$941$		−	4.37908i		−	0.142754i	−0.997449	−	0.0713770i	$-0.977261\pi$
$941$		−	4.37908i		−	0.142754i	0.997449	−	0.0713770i	$-0.0227393\pi$
$942$		0			0
$943$	−46.0532	+	36.1004i	−1.49970	+	1.17559i
$944$		0			0
$945$	−1.53388			−0.0498973
$946$		0			0
$947$			44.6220i			1.45002i	0.688738	+	0.725011i	$0.258165\pi$
$947$			44.6220i			1.45002i	−0.688738	+	0.725011i	$0.741835\pi$
$948$		0			0
$949$	20.6571			0.670556
$950$		0			0
$951$		−	5.23453i		−	0.169741i
$952$		0			0
$953$			16.5579i			0.536364i	0.963368	+	0.268182i	$0.0864228\pi$
$953$			16.5579i			0.536364i	−0.963368	+	0.268182i	$0.913577\pi$
$954$		0			0
$955$		−	20.2870i		−	0.656473i
$956$		0			0
$957$			6.72826i			0.217494i
$958$		0			0
$959$		−	31.7686i		−	1.02586i
$960$		0			0
$961$	19.8575			0.640566
$962$		0			0
$963$	−14.4419			−0.465385
$964$		0			0
$965$			19.5055i			0.627906i
$966$		0			0
$967$			56.4688i			1.81592i	0.419062	+	0.907958i	$0.362359\pi$
$967$			56.4688i			1.81592i	−0.419062	+	0.907958i	$0.637641\pi$
$968$		0			0
$969$	7.90232			0.253859
$970$		0			0
$971$	22.1815			0.711839			0.355919	−	0.934517i	$-0.384168\pi$
$971$	22.1815			0.711839			0.355919	+	0.934517i	$0.384168\pi$
$972$		0			0
$973$		−	8.18002i		−	0.262240i
$974$		0			0
$975$		−	2.39402i		−	0.0766700i
$976$		0			0
$977$			8.59078i			0.274843i	0.990513	+	0.137422i	$0.0438815\pi$
$977$			8.59078i			0.274843i	−0.990513	+	0.137422i	$0.956118\pi$
$978$		0			0
$979$			43.4301i			1.38803i
$980$		0			0
$981$			10.4690i			0.334249i
$982$		0			0
$983$	−4.30444			−0.137290			−0.0686451	−	0.997641i	$-0.521868\pi$
$983$	−4.30444			−0.137290			−0.0686451	+	0.997641i	$0.521868\pi$
$984$		0			0
$985$			14.6532i			0.466891i
$986$		0			0
$987$	−15.1282			−0.481537
$988$		0			0
$989$	2.84326	−	2.22878i	0.0904103	−	0.0708712i
$990$		0			0
$991$		−	16.3295i		−	0.518724i	−0.965780	−	0.259362i	$-0.916488\pi$
$991$		−	16.3295i		−	0.518724i	0.965780	−	0.259362i	$-0.0835123\pi$
$992$		0			0
$993$	21.6135			0.685884
$994$		0			0
$995$		−	10.0280i		−	0.317909i
$996$		0			0
$997$	−10.6093			−0.335999			−0.168000	−	0.985787i	$-0.553731\pi$
$997$	−10.6093			−0.335999			−0.168000	+	0.985787i	$0.553731\pi$
$998$		0			0
$999$	−3.84305			−0.121589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.d.1471.10	yes	32
4.3	odd	2		5520.2.be.c.1471.9	✓	32
23.22	odd	2		5520.2.be.c.1471.10	yes	32
92.91	even	2	inner	5520.2.be.d.1471.9	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.9	✓	32	4.3	odd	2
5520.2.be.c.1471.10	yes	32	23.22	odd	2
5520.2.be.d.1471.9	yes	32	92.91	even	2	inner
5520.2.be.d.1471.10	yes	32	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} +1.53388 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.00000i q^{5} +1.53388 q^{7} -1.00000 q^{9} -3.80458 q^{11} +2.39402 q^{13} +1.00000 q^{15} -2.51901i q^{17} +3.13707 q^{19} +1.53388i q^{21} +(3.77440 - 2.95870i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -1.76846 q^{29} -3.33803i q^{31} -3.80458i q^{33} -1.53388i q^{35} -3.84305i q^{37} +2.39402i q^{39} -12.2015 q^{41} +0.753300 q^{43} +1.00000i q^{45} +9.86269i q^{47} -4.64720 q^{49} +2.51901 q^{51} +5.02467i q^{53} +3.80458i q^{55} +3.13707i q^{57} -11.1175i q^{59} -13.1210i q^{61} -1.53388 q^{63} -2.39402i q^{65} +1.33054 q^{67} +(2.95870 + 3.77440i) q^{69} +7.54359i q^{71} +8.62861 q^{73} -1.00000i q^{75} -5.83578 q^{77} -2.23001 q^{79} +1.00000 q^{81} -16.1380 q^{83} -2.51901 q^{85} -1.76846i q^{87} -11.4152i q^{89} +3.67215 q^{91} +3.33803 q^{93} -3.13707i q^{95} -1.26542i q^{97} +3.80458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{7} - 32 q^{9}+O(q^{10}) \) 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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