LMFDB - Embedded newform 5520.2.be.c.1471.16

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			1471.16
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +5.78590 q^{11} +0.896503 q^{13} -1.00000 q^{15} +6.83315i q^{17} +1.90832 q^{19} +3.60511i q^{21} +(4.77275 - 0.469990i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +1.39299 q^{29} +1.12061i q^{31} +5.78590i q^{33} +3.60511i q^{35} -7.48133i q^{37} +0.896503i q^{39} +3.19938 q^{41} +2.53782 q^{43} -1.00000i q^{45} -5.80647i q^{47} +5.99683 q^{49} -6.83315 q^{51} +2.67178i q^{53} +5.78590i q^{55} +1.90832i q^{57} +6.58667i q^{59} +0.597564i q^{61} -3.60511 q^{63} +0.896503i q^{65} -5.41745 q^{67} +(0.469990 + 4.77275i) q^{69} -3.72842i q^{71} +0.479048 q^{73} -1.00000i q^{75} +20.8588 q^{77} +1.03711 q^{79} +1.00000 q^{81} -11.2088 q^{83} -6.83315 q^{85} +1.39299i q^{87} -5.21573i q^{89} +3.23199 q^{91} -1.12061 q^{93} +1.90832i q^{95} -4.73433i q^{97} -5.78590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.60511			1.36260			0.681302	−	0.732002i	$-0.261414\pi$
$7$	3.60511			1.36260			0.681302	+	0.732002i	$0.261414\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.78590			1.74452			0.872258	−	0.489046i	$-0.162655\pi$
$11$	5.78590			1.74452			0.872258	+	0.489046i	$0.162655\pi$
$12$		0			0
$13$	0.896503			0.248645			0.124323	−	0.992242i	$-0.460324\pi$
$13$	0.896503			0.248645			0.124323	+	0.992242i	$0.460324\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			6.83315i			1.65728i	0.559781	+	0.828641i	$0.310885\pi$
$17$			6.83315i			1.65728i	−0.559781	+	0.828641i	$0.689115\pi$
$18$		0			0
$19$	1.90832			0.437798			0.218899	−	0.975748i	$-0.429753\pi$
$19$	1.90832			0.437798			0.218899	+	0.975748i	$0.429753\pi$
$20$		0			0
$21$			3.60511i			0.786700i
$22$		0			0
$23$	4.77275	−	0.469990i	0.995186	−	0.0979997i
$23$	4.77275	−	0.469990i	0.995186	−	0.0979997i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	1.39299			0.258672			0.129336	−	0.991601i	$-0.458715\pi$
$29$	1.39299			0.258672			0.129336	+	0.991601i	$0.458715\pi$
$30$		0			0
$31$			1.12061i			0.201268i	0.994924	+	0.100634i	$0.0320871\pi$
$31$			1.12061i			0.201268i	−0.994924	+	0.100634i	$0.967913\pi$
$32$		0			0
$33$			5.78590i			1.00720i
$34$		0			0
$35$			3.60511i			0.609375i
$36$		0			0
$37$		−	7.48133i		−	1.22992i	−0.788557	−	0.614961i	$-0.789172\pi$
$37$		−	7.48133i		−	1.22992i	0.788557	−	0.614961i	$-0.210828\pi$
$38$		0			0
$39$			0.896503i			0.143555i
$40$		0			0
$41$	3.19938			0.499660			0.249830	−	0.968290i	$-0.419625\pi$
$41$	3.19938			0.499660			0.249830	+	0.968290i	$0.419625\pi$
$42$		0			0
$43$	2.53782			0.387014			0.193507	−	0.981099i	$-0.438014\pi$
$43$	2.53782			0.387014			0.193507	+	0.981099i	$0.438014\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	5.80647i		−	0.846961i	−0.905905	−	0.423480i	$-0.860808\pi$
$47$		−	5.80647i		−	0.846961i	0.905905	−	0.423480i	$-0.139192\pi$
$48$		0			0
$49$	5.99683			0.856690
$50$		0			0
$51$	−6.83315			−0.956832
$52$		0			0
$53$			2.67178i			0.366998i	0.983020	+	0.183499i	$0.0587424\pi$
$53$			2.67178i			0.366998i	−0.983020	+	0.183499i	$0.941258\pi$
$54$		0			0
$55$			5.78590i			0.780171i
$56$		0			0
$57$			1.90832i			0.252763i
$58$		0			0
$59$			6.58667i			0.857511i	0.903420	+	0.428756i	$0.141048\pi$
$59$			6.58667i			0.857511i	−0.903420	+	0.428756i	$0.858952\pi$
$60$		0			0
$61$			0.597564i			0.0765102i	0.999268	+	0.0382551i	$0.0121800\pi$
$61$			0.597564i			0.0765102i	−0.999268	+	0.0382551i	$0.987820\pi$
$62$		0			0
$63$	−3.60511			−0.454201
$64$		0			0
$65$			0.896503i			0.111197i
$66$		0			0
$67$	−5.41745			−0.661846			−0.330923	−	0.943658i	$-0.607360\pi$
$67$	−5.41745			−0.661846			−0.330923	+	0.943658i	$0.607360\pi$
$68$		0			0
$69$	0.469990	+	4.77275i	0.0565801	+	0.574571i
$70$		0			0
$71$		−	3.72842i		−	0.442482i	−0.975219	−	0.221241i	$-0.928989\pi$
$71$		−	3.72842i		−	0.442482i	0.975219	−	0.221241i	$-0.0710108\pi$
$72$		0			0
$73$	0.479048			0.0560683			0.0280342	−	0.999607i	$-0.491075\pi$
$73$	0.479048			0.0560683			0.0280342	+	0.999607i	$0.491075\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	20.8588			2.37708
$78$		0			0
$79$	1.03711			0.116684			0.0583422	−	0.998297i	$-0.481419\pi$
$79$	1.03711			0.116684			0.0583422	+	0.998297i	$0.481419\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.2088			−1.23033			−0.615163	−	0.788400i	$-0.710910\pi$
$83$	−11.2088			−1.23033			−0.615163	+	0.788400i	$0.710910\pi$
$84$		0			0
$85$	−6.83315			−0.741159
$86$		0			0
$87$			1.39299i			0.149344i
$88$		0			0
$89$		−	5.21573i		−	0.552866i	−0.961033	−	0.276433i	$-0.910848\pi$
$89$		−	5.21573i		−	0.552866i	0.961033	−	0.276433i	$-0.0891524\pi$
$90$		0			0
$91$	3.23199			0.338805
$92$		0			0
$93$	−1.12061			−0.116202
$94$		0			0
$95$			1.90832i			0.195789i
$96$		0			0
$97$		−	4.73433i		−	0.480698i	−0.970687	−	0.240349i	$-0.922738\pi$
$97$		−	4.73433i		−	0.480698i	0.970687	−	0.240349i	$-0.0772619\pi$
$98$		0			0
$99$	−5.78590			−0.581505
$100$		0			0
$101$	−0.362853			−0.0361052			−0.0180526	−	0.999837i	$-0.505747\pi$
$101$	−0.362853			−0.0361052			−0.0180526	+	0.999837i	$0.505747\pi$
$102$		0			0
$103$	−11.7189			−1.15470			−0.577348	−	0.816498i	$-0.695912\pi$
$103$	−11.7189			−1.15470			−0.577348	+	0.816498i	$0.695912\pi$
$104$		0			0
$105$	−3.60511			−0.351823
$106$		0			0
$107$	12.6988			1.22764			0.613818	−	0.789447i	$-0.289633\pi$
$107$	12.6988			1.22764			0.613818	+	0.789447i	$0.289633\pi$
$108$		0			0
$109$		−	8.37783i		−	0.802451i	−0.915979	−	0.401225i	$-0.868584\pi$
$109$		−	8.37783i		−	0.802451i	0.915979	−	0.401225i	$-0.131416\pi$
$110$		0			0
$111$	7.48133			0.710096
$112$		0			0
$113$		−	7.77619i		−	0.731522i	−0.930709	−	0.365761i	$-0.880809\pi$
$113$		−	7.77619i		−	0.731522i	0.930709	−	0.365761i	$-0.119191\pi$
$114$		0			0
$115$	0.469990	+	4.77275i	0.0438268	+	0.445061i
$116$		0			0
$117$	−0.896503			−0.0828817
$118$		0			0
$119$			24.6343i			2.25822i
$120$		0			0
$121$	22.4767			2.04334
$122$		0			0
$123$			3.19938i			0.288479i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			5.02772i			0.446138i	0.974803	+	0.223069i	$0.0716075\pi$
$127$			5.02772i			0.446138i	−0.974803	+	0.223069i	$0.928392\pi$
$128$		0			0
$129$			2.53782i			0.223443i
$130$		0			0
$131$		−	16.0105i		−	1.39885i	−0.714707	−	0.699424i	$-0.753440\pi$
$131$		−	16.0105i		−	1.39885i	0.714707	−	0.699424i	$-0.246560\pi$
$132$		0			0
$133$	6.87970			0.596545
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	23.3183i		−	1.99222i	−0.0881386	−	0.996108i	$-0.528092\pi$
$137$		−	23.3183i		−	1.99222i	0.0881386	−	0.996108i	$-0.471908\pi$
$138$		0			0
$139$			2.21510i			0.187882i	0.995578	+	0.0939412i	$0.0299466\pi$
$139$			2.21510i			0.187882i	−0.995578	+	0.0939412i	$0.970053\pi$
$140$		0			0
$141$	5.80647			0.488993
$142$		0			0
$143$	5.18708			0.433765
$144$		0			0
$145$			1.39299i			0.115682i
$146$		0			0
$147$			5.99683i			0.494610i
$148$		0			0
$149$			4.16837i			0.341486i	0.985316	+	0.170743i	$0.0546168\pi$
$149$			4.16837i			0.341486i	−0.985316	+	0.170743i	$0.945383\pi$
$150$		0			0
$151$		−	5.22123i		−	0.424897i	−0.977172	−	0.212449i	$-0.931856\pi$
$151$		−	5.22123i		−	0.424897i	0.977172	−	0.212449i	$-0.0681439\pi$
$152$		0			0
$153$		−	6.83315i		−	0.552427i
$154$		0			0
$155$	−1.12061			−0.0900098
$156$		0			0
$157$		−	1.91699i		−	0.152992i	−0.997070	−	0.0764961i	$-0.975627\pi$
$157$		−	1.91699i		−	0.152992i	0.997070	−	0.0764961i	$-0.0243733\pi$
$158$		0			0
$159$	−2.67178			−0.211886
$160$		0			0
$161$	17.2063	−	1.69437i	1.35605	−	0.133535i
$162$		0			0
$163$			8.45209i			0.662019i	0.943627	+	0.331010i	$0.107389\pi$
$163$			8.45209i			0.662019i	−0.943627	+	0.331010i	$0.892611\pi$
$164$		0			0
$165$	−5.78590			−0.450432
$166$		0			0
$167$		−	19.2777i		−	1.49175i	−0.666084	−	0.745877i	$-0.732031\pi$
$167$		−	19.2777i		−	1.49175i	0.666084	−	0.745877i	$-0.267969\pi$
$168$		0			0
$169$	−12.1963			−0.938176
$170$		0			0
$171$	−1.90832			−0.145933
$172$		0			0
$173$	−1.99198			−0.151447			−0.0757236	−	0.997129i	$-0.524127\pi$
$173$	−1.99198			−0.151447			−0.0757236	+	0.997129i	$0.524127\pi$
$174$		0			0
$175$	−3.60511			−0.272521
$176$		0			0
$177$	−6.58667			−0.495084
$178$		0			0
$179$			22.5461i			1.68517i	0.538563	+	0.842586i	$0.318968\pi$
$179$			22.5461i			1.68517i	−0.538563	+	0.842586i	$0.681032\pi$
$180$		0			0
$181$		−	2.37594i		−	0.176602i	−0.996094	−	0.0883011i	$-0.971856\pi$
$181$		−	2.37594i		−	0.176602i	0.996094	−	0.0883011i	$-0.0281438\pi$
$182$		0			0
$183$	−0.597564			−0.0441732
$184$		0			0
$185$	7.48133			0.550038
$186$		0			0
$187$			39.5359i			2.89115i
$188$		0			0
$189$		−	3.60511i		−	0.262233i
$190$		0			0
$191$	2.63031			0.190323			0.0951613	−	0.995462i	$-0.469663\pi$
$191$	2.63031			0.190323			0.0951613	+	0.995462i	$0.469663\pi$
$192$		0			0
$193$	−20.1946			−1.45364			−0.726818	−	0.686830i	$-0.759002\pi$
$193$	−20.1946			−1.45364			−0.726818	+	0.686830i	$0.759002\pi$
$194$		0			0
$195$	−0.896503			−0.0641999
$196$		0			0
$197$	26.4159			1.88205			0.941027	−	0.338332i	$-0.109863\pi$
$197$	26.4159			1.88205			0.941027	+	0.338332i	$0.109863\pi$
$198$		0			0
$199$	−21.4192			−1.51837			−0.759185	−	0.650875i	$-0.774402\pi$
$199$	−21.4192			−1.51837			−0.759185	+	0.650875i	$0.774402\pi$
$200$		0			0
$201$		−	5.41745i		−	0.382117i
$202$		0			0
$203$	5.02189			0.352467
$204$		0			0
$205$			3.19938i			0.223455i
$206$		0			0
$207$	−4.77275	+	0.469990i	−0.331729	+	0.0326666i
$208$		0			0
$209$	11.0413			0.763746
$210$		0			0
$211$			0.342786i			0.0235984i	0.999930	+	0.0117992i	$0.00375588\pi$
$211$			0.342786i			0.0235984i	−0.999930	+	0.0117992i	$0.996244\pi$
$212$		0			0
$213$	3.72842			0.255467
$214$		0			0
$215$			2.53782i			0.173078i
$216$		0			0
$217$			4.03993i			0.274249i
$218$		0			0
$219$			0.479048i			0.0323711i
$220$		0			0
$221$			6.12593i			0.412075i
$222$		0			0
$223$		−	6.96417i		−	0.466355i	−0.972434	−	0.233178i	$-0.925088\pi$
$223$		−	6.96417i		−	0.466355i	0.972434	−	0.233178i	$-0.0749124\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−17.1845			−1.14058			−0.570289	−	0.821444i	$-0.693169\pi$
$227$	−17.1845			−1.14058			−0.570289	+	0.821444i	$0.693169\pi$
$228$		0			0
$229$			15.6155i			1.03190i	0.856619	+	0.515950i	$0.172561\pi$
$229$			15.6155i			1.03190i	−0.856619	+	0.515950i	$0.827439\pi$
$230$		0			0
$231$			20.8588i			1.37241i
$232$		0			0
$233$	12.8244			0.840153			0.420076	−	0.907489i	$-0.362003\pi$
$233$	12.8244			0.840153			0.420076	+	0.907489i	$0.362003\pi$
$234$		0			0
$235$	5.80647			0.378772
$236$		0			0
$237$			1.03711i			0.0673677i
$238$		0			0
$239$			10.9984i			0.711429i	0.934595	+	0.355714i	$0.115762\pi$
$239$			10.9984i			0.711429i	−0.934595	+	0.355714i	$0.884238\pi$
$240$		0			0
$241$			11.3738i			0.732652i	0.930487	+	0.366326i	$0.119384\pi$
$241$			11.3738i			0.732652i	−0.930487	+	0.366326i	$0.880616\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$			5.99683i			0.383123i
$246$		0			0
$247$	1.71081			0.108856
$248$		0			0
$249$		−	11.2088i		−	0.710329i
$250$		0			0
$251$	23.2561			1.46791			0.733957	−	0.679195i	$-0.237671\pi$
$251$	23.2561			1.46791			0.733957	+	0.679195i	$0.237671\pi$
$252$		0			0
$253$	27.6147	−	2.71932i	1.73612	−	0.170962i
$254$		0			0
$255$		−	6.83315i		−	0.427908i
$256$		0			0
$257$	−27.6478			−1.72462			−0.862312	−	0.506378i	$-0.830984\pi$
$257$	−27.6478			−1.72462			−0.862312	+	0.506378i	$0.830984\pi$
$258$		0			0
$259$		−	26.9710i		−	1.67590i
$260$		0			0
$261$	−1.39299			−0.0862240
$262$		0			0
$263$	−23.5536			−1.45237			−0.726187	−	0.687497i	$-0.758709\pi$
$263$	−23.5536			−1.45237			−0.726187	+	0.687497i	$0.758709\pi$
$264$		0			0
$265$	−2.67178			−0.164126
$266$		0			0
$267$	5.21573			0.319198
$268$		0			0
$269$	−10.7515			−0.655533			−0.327766	−	0.944759i	$-0.606296\pi$
$269$	−10.7515			−0.655533			−0.327766	+	0.944759i	$0.606296\pi$
$270$		0			0
$271$			20.0996i			1.22096i	0.792030	+	0.610482i	$0.209024\pi$
$271$			20.0996i			1.22096i	−0.792030	+	0.610482i	$0.790976\pi$
$272$		0			0
$273$			3.23199i			0.195609i
$274$		0			0
$275$	−5.78590			−0.348903
$276$		0			0
$277$	16.5831			0.996384			0.498192	−	0.867067i	$-0.333998\pi$
$277$	16.5831			0.996384			0.498192	+	0.867067i	$0.333998\pi$
$278$		0			0
$279$		−	1.12061i		−	0.0670893i
$280$		0			0
$281$			9.61074i			0.573329i	0.958031	+	0.286664i	$0.0925464\pi$
$281$			9.61074i			0.573329i	−0.958031	+	0.286664i	$0.907454\pi$
$282$		0			0
$283$	−28.3850			−1.68731			−0.843657	−	0.536883i	$-0.819602\pi$
$283$	−28.3850			−1.68731			−0.843657	+	0.536883i	$0.819602\pi$
$284$		0			0
$285$	−1.90832			−0.113039
$286$		0			0
$287$	11.5341			0.680838
$288$		0			0
$289$	−29.6919			−1.74658
$290$		0			0
$291$	4.73433			0.277531
$292$		0			0
$293$			24.9347i			1.45670i	0.685204	+	0.728351i	$0.259713\pi$
$293$			24.9347i			1.45670i	−0.685204	+	0.728351i	$0.740287\pi$
$294$		0			0
$295$	−6.58667			−0.383491
$296$		0			0
$297$		−	5.78590i		−	0.335732i
$298$		0			0
$299$	4.27878	−	0.421347i	0.247448	−	0.0243671i
$300$		0			0
$301$	9.14913			0.527347
$302$		0			0
$303$		−	0.362853i		−	0.0208454i
$304$		0			0
$305$	−0.597564			−0.0342164
$306$		0			0
$307$			9.09513i			0.519086i	0.965731	+	0.259543i	$0.0835720\pi$
$307$			9.09513i			0.519086i	−0.965731	+	0.259543i	$0.916428\pi$
$308$		0			0
$309$		−	11.7189i		−	0.666664i
$310$		0			0
$311$		−	1.14205i		−	0.0647596i	−0.999476	−	0.0323798i	$-0.989691\pi$
$311$		−	1.14205i		−	0.0647596i	0.999476	−	0.0323798i	$-0.0103086\pi$
$312$		0			0
$313$			28.0497i			1.58546i	0.609572	+	0.792731i	$0.291341\pi$
$313$			28.0497i			1.58546i	−0.609572	+	0.792731i	$0.708659\pi$
$314$		0			0
$315$		−	3.60511i		−	0.203125i
$316$		0			0
$317$	18.0554			1.01409			0.507047	−	0.861918i	$-0.330737\pi$
$317$	18.0554			1.01409			0.507047	+	0.861918i	$0.330737\pi$
$318$		0			0
$319$	8.05971			0.451257
$320$		0			0
$321$			12.6988i			0.708776i
$322$		0			0
$323$			13.0398i			0.725555i
$324$		0			0
$325$	−0.896503			−0.0497290
$326$		0			0
$327$	8.37783			0.463295
$328$		0			0
$329$		−	20.9330i		−	1.15407i
$330$		0			0
$331$			17.0971i			0.939742i	0.882735	+	0.469871i	$0.155700\pi$
$331$			17.0971i			0.939742i	−0.882735	+	0.469871i	$0.844300\pi$
$332$		0			0
$333$			7.48133i			0.409974i
$334$		0			0
$335$		−	5.41745i		−	0.295987i
$336$		0			0
$337$			18.9130i			1.03026i	0.857113	+	0.515129i	$0.172256\pi$
$337$			18.9130i			1.03026i	−0.857113	+	0.515129i	$0.827744\pi$
$338$		0			0
$339$	7.77619			0.422345
$340$		0			0
$341$			6.48376i			0.351115i
$342$		0			0
$343$	−3.61655			−0.195275
$344$		0			0
$345$	−4.77275	+	0.469990i	−0.256956	+	0.0253034i
$346$		0			0
$347$			26.3614i			1.41515i	0.706636	+	0.707577i	$0.250212\pi$
$347$			26.3614i			1.41515i	−0.706636	+	0.707577i	$0.749788\pi$
$348$		0			0
$349$	−7.87666			−0.421628			−0.210814	−	0.977526i	$-0.567611\pi$
$349$	−7.87666			−0.421628			−0.210814	+	0.977526i	$0.567611\pi$
$350$		0			0
$351$		−	0.896503i		−	0.0478518i
$352$		0			0
$353$	9.97831			0.531092			0.265546	−	0.964098i	$-0.414448\pi$
$353$	9.97831			0.531092			0.265546	+	0.964098i	$0.414448\pi$
$354$		0			0
$355$	3.72842			0.197884
$356$		0			0
$357$	−24.6343			−1.30378
$358$		0			0
$359$	−3.21862			−0.169872			−0.0849362	−	0.996386i	$-0.527069\pi$
$359$	−3.21862			−0.169872			−0.0849362	+	0.996386i	$0.527069\pi$
$360$		0			0
$361$	−15.3583			−0.808333
$362$		0			0
$363$			22.4767i			1.17972i
$364$		0			0
$365$			0.479048i			0.0250745i
$366$		0			0
$367$	−17.7708			−0.927630			−0.463815	−	0.885932i	$-0.653520\pi$
$367$	−17.7708			−0.927630			−0.463815	+	0.885932i	$0.653520\pi$
$368$		0			0
$369$	−3.19938			−0.166553
$370$		0			0
$371$			9.63208i			0.500073i
$372$		0			0
$373$		−	0.136042i		−	0.00704396i	−0.999994	−	0.00352198i	$-0.998879\pi$
$373$		−	0.136042i		−	0.00704396i	0.999994	−	0.00352198i	$-0.00112108\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	1.24882			0.0643175
$378$		0			0
$379$	36.4857			1.87415			0.937073	−	0.349135i	$-0.113524\pi$
$379$	36.4857			1.87415			0.937073	+	0.349135i	$0.113524\pi$
$380$		0			0
$381$	−5.02772			−0.257578
$382$		0			0
$383$	−9.13045			−0.466544			−0.233272	−	0.972412i	$-0.574943\pi$
$383$	−9.13045			−0.466544			−0.233272	+	0.972412i	$0.574943\pi$
$384$		0			0
$385$			20.8588i			1.06306i
$386$		0			0
$387$	−2.53782			−0.129005
$388$		0			0
$389$			16.2289i			0.822838i	0.911446	+	0.411419i	$0.134967\pi$
$389$			16.2289i			0.822838i	−0.911446	+	0.411419i	$0.865033\pi$
$390$		0			0
$391$	3.21151	+	32.6129i	0.162413	+	1.64930i
$392$		0			0
$393$	16.0105			0.807625
$394$		0			0
$395$			1.03711i			0.0521828i
$396$		0			0
$397$	−5.37479			−0.269753			−0.134877	−	0.990862i	$-0.543064\pi$
$397$	−5.37479			−0.269753			−0.134877	+	0.990862i	$0.543064\pi$
$398$		0			0
$399$			6.87970i			0.344416i
$400$		0			0
$401$			3.28928i			0.164259i	0.996622	+	0.0821294i	$0.0261721\pi$
$401$			3.28928i			0.164259i	−0.996622	+	0.0821294i	$0.973828\pi$
$402$		0			0
$403$			1.00463i			0.0500443i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	43.2862i		−	2.14562i
$408$		0			0
$409$	−20.9778			−1.03728			−0.518642	−	0.854992i	$-0.673562\pi$
$409$	−20.9778			−1.03728			−0.518642	+	0.854992i	$0.673562\pi$
$410$		0			0
$411$	23.3183			1.15021
$412$		0			0
$413$			23.7457i			1.16845i
$414$		0			0
$415$		−	11.2088i		−	0.550219i
$416$		0			0
$417$	−2.21510			−0.108474
$418$		0			0
$419$	−27.7118			−1.35381			−0.676904	−	0.736071i	$-0.736679\pi$
$419$	−27.7118			−1.35381			−0.676904	+	0.736071i	$0.736679\pi$
$420$		0			0
$421$		−	36.2909i		−	1.76871i	−0.466813	−	0.884356i	$-0.654598\pi$
$421$		−	36.2909i		−	1.76871i	0.466813	−	0.884356i	$-0.345402\pi$
$422$		0			0
$423$			5.80647i			0.282320i
$424$		0			0
$425$		−	6.83315i		−	0.331456i
$426$		0			0
$427$			2.15428i			0.104253i
$428$		0			0
$429$			5.18708i			0.250435i
$430$		0			0
$431$	20.7066			0.997401			0.498700	−	0.866774i	$-0.333811\pi$
$431$	20.7066			0.997401			0.498700	+	0.866774i	$0.333811\pi$
$432$		0			0
$433$		−	27.4933i		−	1.32125i	−0.750718	−	0.660623i	$-0.770292\pi$
$433$		−	27.4933i		−	1.32125i	0.750718	−	0.660623i	$-0.229708\pi$
$434$		0			0
$435$	−1.39299			−0.0667888
$436$		0			0
$437$	9.10792	−	0.896890i	0.435691	−	0.0429041i
$438$		0			0
$439$		−	2.92167i		−	0.139444i	−0.997566	−	0.0697219i	$-0.977789\pi$
$439$		−	2.92167i		−	0.139444i	0.997566	−	0.0697219i	$-0.0222112\pi$
$440$		0			0
$441$	−5.99683			−0.285563
$442$		0			0
$443$		−	37.6891i		−	1.79066i	−0.445401	−	0.895331i	$-0.646939\pi$
$443$		−	37.6891i		−	1.79066i	0.445401	−	0.895331i	$-0.353061\pi$
$444$		0			0
$445$	5.21573			0.247249
$446$		0			0
$447$	−4.16837			−0.197157
$448$		0			0
$449$	5.73059			0.270443			0.135222	−	0.990815i	$-0.456825\pi$
$449$	5.73059			0.270443			0.135222	+	0.990815i	$0.456825\pi$
$450$		0			0
$451$	18.5113			0.871664
$452$		0			0
$453$	5.22123			0.245315
$454$		0			0
$455$			3.23199i			0.151518i
$456$		0			0
$457$		−	27.3237i		−	1.27815i	−0.769146	−	0.639073i	$-0.779318\pi$
$457$		−	27.3237i		−	1.27815i	0.769146	−	0.639073i	$-0.220682\pi$
$458$		0			0
$459$	6.83315			0.318944
$460$		0			0
$461$	20.2426			0.942793			0.471397	−	0.881921i	$-0.343750\pi$
$461$	20.2426			0.942793			0.471397	+	0.881921i	$0.343750\pi$
$462$		0			0
$463$			15.5699i			0.723596i	0.932257	+	0.361798i	$0.117837\pi$
$463$			15.5699i			0.723596i	−0.932257	+	0.361798i	$0.882163\pi$
$464$		0			0
$465$		−	1.12061i		−	0.0519672i
$466$		0			0
$467$	20.1385			0.931898			0.465949	−	0.884812i	$-0.345713\pi$
$467$	20.1385			0.931898			0.465949	+	0.884812i	$0.345713\pi$
$468$		0			0
$469$	−19.5305			−0.901835
$470$		0			0
$471$	1.91699			0.0883301
$472$		0			0
$473$	14.6836			0.675152
$474$		0			0
$475$	−1.90832			−0.0875596
$476$		0			0
$477$		−	2.67178i		−	0.122333i
$478$		0			0
$479$	−27.6339			−1.26263			−0.631313	−	0.775528i	$-0.717484\pi$
$479$	−27.6339			−1.26263			−0.631313	+	0.775528i	$0.717484\pi$
$480$		0			0
$481$		−	6.70703i		−	0.305814i
$482$		0			0
$483$	1.69437	+	17.2063i	0.0770963	+	0.782913i
$484$		0			0
$485$	4.73433			0.214975
$486$		0			0
$487$			18.3891i			0.833289i	0.909070	+	0.416644i	$0.136794\pi$
$487$			18.3891i			0.833289i	−0.909070	+	0.416644i	$0.863206\pi$
$488$		0			0
$489$	−8.45209			−0.382217
$490$		0			0
$491$			20.8576i			0.941292i	0.882322	+	0.470646i	$0.155979\pi$
$491$			20.8576i			0.941292i	−0.882322	+	0.470646i	$0.844021\pi$
$492$		0			0
$493$			9.51851i			0.428692i
$494$		0			0
$495$		−	5.78590i		−	0.260057i
$496$		0			0
$497$		−	13.4414i		−	0.602928i
$498$		0			0
$499$		−	19.8572i		−	0.888930i	−0.895796	−	0.444465i	$-0.853394\pi$
$499$		−	19.8572i		−	0.888930i	0.895796	−	0.444465i	$-0.146606\pi$
$500$		0			0
$501$	19.2777			0.861264
$502$		0			0
$503$	15.9821			0.712609			0.356304	−	0.934370i	$-0.384037\pi$
$503$	15.9821			0.712609			0.356304	+	0.934370i	$0.384037\pi$
$504$		0			0
$505$		−	0.362853i		−	0.0161468i
$506$		0			0
$507$		−	12.1963i		−	0.541656i
$508$		0			0
$509$	37.8962			1.67972			0.839861	−	0.542802i	$-0.182637\pi$
$509$	37.8962			1.67972			0.839861	+	0.542802i	$0.182637\pi$
$510$		0			0
$511$	1.72702			0.0763989
$512$		0			0
$513$		−	1.90832i		−	0.0842543i
$514$		0			0
$515$		−	11.7189i		−	0.516396i
$516$		0			0
$517$		−	33.5957i		−	1.47754i
$518$		0			0
$519$		−	1.99198i		−	0.0874381i
$520$		0			0
$521$		−	5.78555i		−	0.253470i	−0.991937	−	0.126735i	$-0.959550\pi$
$521$		−	5.78555i		−	0.253470i	0.991937	−	0.126735i	$-0.0404497\pi$
$522$		0			0
$523$	−7.70649			−0.336981			−0.168491	−	0.985703i	$-0.553889\pi$
$523$	−7.70649			−0.336981			−0.168491	+	0.985703i	$0.553889\pi$
$524$		0			0
$525$		−	3.60511i		−	0.157340i
$526$		0			0
$527$	−7.65731			−0.333558
$528$		0			0
$529$	22.5582	−	4.48628i	0.980792	−	0.195056i
$530$		0			0
$531$		−	6.58667i		−	0.285837i
$532$		0			0
$533$	2.86826			0.124238
$534$		0			0
$535$			12.6988i			0.549016i
$536$		0			0
$537$	−22.5461			−0.972934
$538$		0			0
$539$	34.6971			1.49451
$540$		0			0
$541$	−17.0098			−0.731308			−0.365654	−	0.930751i	$-0.619155\pi$
$541$	−17.0098			−0.731308			−0.365654	+	0.930751i	$0.619155\pi$
$542$		0			0
$543$	2.37594			0.101961
$544$		0			0
$545$	8.37783			0.358867
$546$		0			0
$547$			42.6426i			1.82327i	0.411006	+	0.911633i	$0.365178\pi$
$547$			42.6426i			1.82327i	−0.411006	+	0.911633i	$0.634822\pi$
$548$		0			0
$549$		−	0.597564i		−	0.0255034i
$550$		0			0
$551$	2.65827			0.113246
$552$		0			0
$553$	3.73891			0.158995
$554$		0			0
$555$			7.48133i			0.317565i
$556$		0			0
$557$		−	13.2254i		−	0.560380i	−0.959945	−	0.280190i	$-0.909603\pi$
$557$		−	13.2254i		−	0.560380i	0.959945	−	0.280190i	$-0.0903974\pi$
$558$		0			0
$559$	2.27516			0.0962292
$560$		0			0
$561$	−39.5359			−1.66921
$562$		0			0
$563$	−39.8356			−1.67887			−0.839436	−	0.543459i	$-0.817114\pi$
$563$	−39.8356			−1.67887			−0.839436	+	0.543459i	$0.817114\pi$
$564$		0			0
$565$	7.77619			0.327147
$566$		0			0
$567$	3.60511			0.151400
$568$		0			0
$569$		−	32.1739i		−	1.34880i	−0.738366	−	0.674401i	$-0.764402\pi$
$569$		−	32.1739i		−	1.34880i	0.738366	−	0.674401i	$-0.235598\pi$
$570$		0			0
$571$	−7.64594			−0.319972			−0.159986	−	0.987119i	$-0.551145\pi$
$571$	−7.64594			−0.319972			−0.159986	+	0.987119i	$0.551145\pi$
$572$		0			0
$573$			2.63031i			0.109883i
$574$		0			0
$575$	−4.77275	+	0.469990i	−0.199037	+	0.0195999i
$576$		0			0
$577$	7.02954			0.292643			0.146322	−	0.989237i	$-0.453257\pi$
$577$	7.02954			0.292643			0.146322	+	0.989237i	$0.453257\pi$
$578$		0			0
$579$		−	20.1946i		−	0.839258i
$580$		0			0
$581$	−40.4090			−1.67645
$582$		0			0
$583$			15.4587i			0.640233i
$584$		0			0
$585$		−	0.896503i		−	0.0370658i
$586$		0			0
$587$			30.6939i			1.26687i	0.773794	+	0.633437i	$0.218357\pi$
$587$			30.6939i			1.26687i	−0.773794	+	0.633437i	$0.781643\pi$
$588$		0			0
$589$			2.13848i			0.0881147i
$590$		0			0
$591$			26.4159i			1.08660i
$592$		0			0
$593$	−30.5473			−1.25443			−0.627214	−	0.778847i	$-0.715805\pi$
$593$	−30.5473			−1.25443			−0.627214	+	0.778847i	$0.715805\pi$
$594$		0			0
$595$	−24.6343			−1.00991
$596$		0			0
$597$		−	21.4192i		−	0.876631i
$598$		0			0
$599$			8.82194i			0.360455i	0.983625	+	0.180227i	$0.0576834\pi$
$599$			8.82194i			0.360455i	−0.983625	+	0.180227i	$0.942317\pi$
$600$		0			0
$601$	−28.7517			−1.17280			−0.586402	−	0.810020i	$-0.699456\pi$
$601$	−28.7517			−1.17280			−0.586402	+	0.810020i	$0.699456\pi$
$602$		0			0
$603$	5.41745			0.220615
$604$		0			0
$605$			22.4767i			0.913808i
$606$		0			0
$607$		−	24.5794i		−	0.997646i	−0.866704	−	0.498823i	$-0.833766\pi$
$607$		−	24.5794i		−	0.997646i	0.866704	−	0.498823i	$-0.166234\pi$
$608$		0			0
$609$			5.02189i			0.203497i
$610$		0			0
$611$		−	5.20552i		−	0.210593i
$612$		0			0
$613$		−	28.1696i		−	1.13776i	−0.822420	−	0.568880i	$-0.807377\pi$
$613$		−	28.1696i		−	1.13776i	0.822420	−	0.568880i	$-0.192623\pi$
$614$		0			0
$615$	−3.19938			−0.129012
$616$		0			0
$617$			32.7417i			1.31813i	0.752086	+	0.659065i	$0.229048\pi$
$617$			32.7417i			1.31813i	−0.752086	+	0.659065i	$0.770952\pi$
$618$		0			0
$619$	−4.77997			−0.192123			−0.0960615	−	0.995375i	$-0.530625\pi$
$619$	−4.77997			−0.192123			−0.0960615	+	0.995375i	$0.530625\pi$
$620$		0			0
$621$	−0.469990	−	4.77275i	−0.0188600	−	0.191524i
$622$		0			0
$623$		−	18.8033i		−	0.753338i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			11.0413i			0.440949i
$628$		0			0
$629$	51.1210			2.03833
$630$		0			0
$631$	46.4708			1.84997			0.924987	−	0.379998i	$-0.124075\pi$
$631$	46.4708			1.84997			0.924987	+	0.379998i	$0.124075\pi$
$632$		0			0
$633$	−0.342786			−0.0136245
$634$		0			0
$635$	−5.02772			−0.199519
$636$		0			0
$637$	5.37617			0.213012
$638$		0			0
$639$			3.72842i			0.147494i
$640$		0			0
$641$		−	23.9541i		−	0.946130i	−0.881027	−	0.473065i	$-0.843147\pi$
$641$		−	23.9541i		−	0.946130i	0.881027	−	0.473065i	$-0.156853\pi$
$642$		0			0
$643$	−33.7394			−1.33055			−0.665277	−	0.746597i	$-0.731686\pi$
$643$	−33.7394			−1.33055			−0.665277	+	0.746597i	$0.731686\pi$
$644$		0			0
$645$	−2.53782			−0.0999266
$646$		0			0
$647$		−	45.6913i		−	1.79631i	−0.439679	−	0.898155i	$-0.644908\pi$
$647$		−	45.6913i		−	1.79631i	0.439679	−	0.898155i	$-0.355092\pi$
$648$		0			0
$649$			38.1098i			1.49594i
$650$		0			0
$651$	−4.03993			−0.158337
$652$		0			0
$653$	45.2004			1.76883			0.884415	−	0.466701i	$-0.154558\pi$
$653$	45.2004			1.76883			0.884415	+	0.466701i	$0.154558\pi$
$654$		0			0
$655$	16.0105			0.625584
$656$		0			0
$657$	−0.479048			−0.0186894
$658$		0			0
$659$	−25.6885			−1.00068			−0.500342	−	0.865828i	$-0.666792\pi$
$659$	−25.6885			−1.00068			−0.500342	+	0.865828i	$0.666792\pi$
$660$		0			0
$661$			9.39395i			0.365382i	0.983170	+	0.182691i	$0.0584809\pi$
$661$			9.39395i			0.365382i	−0.983170	+	0.182691i	$0.941519\pi$
$662$		0			0
$663$	−6.12593			−0.237912
$664$		0			0
$665$			6.87970i			0.266783i
$666$		0			0
$667$	6.64839	−	0.654692i	0.257427	−	0.0253498i
$668$		0			0
$669$	6.96417			0.269250
$670$		0			0
$671$			3.45745i			0.133473i
$672$		0			0
$673$	−23.2138			−0.894826			−0.447413	−	0.894328i	$-0.647655\pi$
$673$	−23.2138			−0.894826			−0.447413	+	0.894328i	$0.647655\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$			46.6887i			1.79439i	0.441631	+	0.897197i	$0.354400\pi$
$677$			46.6887i			1.79439i	−0.441631	+	0.897197i	$0.645600\pi$
$678$		0			0
$679$		−	17.0678i		−	0.655001i
$680$		0			0
$681$		−	17.1845i		−	0.658513i
$682$		0			0
$683$		−	41.3680i		−	1.58290i	−0.611233	−	0.791451i	$-0.709326\pi$
$683$		−	41.3680i		−	1.58290i	0.611233	−	0.791451i	$-0.290674\pi$
$684$		0			0
$685$	23.3183			0.890946
$686$		0			0
$687$	−15.6155			−0.595768
$688$		0			0
$689$			2.39526i			0.0912522i
$690$		0			0
$691$		−	38.1418i		−	1.45098i	−0.688231	−	0.725492i	$-0.741612\pi$
$691$		−	38.1418i		−	1.45098i	0.688231	−	0.725492i	$-0.258388\pi$
$692$		0			0
$693$	−20.8588			−0.792361
$694$		0			0
$695$	−2.21510			−0.0840236
$696$		0			0
$697$			21.8619i			0.828077i
$698$		0			0
$699$			12.8244i			0.485063i
$700$		0			0
$701$			20.8658i			0.788091i	0.919091	+	0.394045i	$0.128925\pi$
$701$			20.8658i			0.788091i	−0.919091	+	0.394045i	$0.871075\pi$
$702$		0			0
$703$		−	14.2767i		−	0.538458i
$704$		0			0
$705$			5.80647i			0.218684i
$706$		0			0
$707$	−1.30813			−0.0491971
$708$		0			0
$709$		−	19.2856i		−	0.724284i	−0.932123	−	0.362142i	$-0.882045\pi$
$709$		−	19.2856i		−	0.724284i	0.932123	−	0.362142i	$-0.117955\pi$
$710$		0			0
$711$	−1.03711			−0.0388948
$712$		0			0
$713$	0.526677	+	5.34840i	0.0197242	+	0.200299i
$714$		0			0
$715$			5.18708i			0.193986i
$716$		0			0
$717$	−10.9984			−0.410744
$718$		0			0
$719$		−	6.24028i		−	0.232723i	−0.993207	−	0.116362i	$-0.962877\pi$
$719$		−	6.24028i		−	0.232723i	0.993207	−	0.116362i	$-0.0371231\pi$
$720$		0			0
$721$	−42.2479			−1.57339
$722$		0			0
$723$	−11.3738			−0.422997
$724$		0			0
$725$	−1.39299			−0.0517344
$726$		0			0
$727$	14.0658			0.521673			0.260836	−	0.965383i	$-0.416002\pi$
$727$	14.0658			0.521673			0.260836	+	0.965383i	$0.416002\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			17.3413i			0.641391i
$732$		0			0
$733$			28.8930i			1.06719i	0.845740	+	0.533594i	$0.179159\pi$
$733$			28.8930i			1.06719i	−0.845740	+	0.533594i	$0.820841\pi$
$734$		0			0
$735$	−5.99683			−0.221196
$736$		0			0
$737$	−31.3448			−1.15460
$738$		0			0
$739$			31.3784i			1.15427i	0.816648	+	0.577136i	$0.195830\pi$
$739$			31.3784i			1.15427i	−0.816648	+	0.577136i	$0.804170\pi$
$740$		0			0
$741$			1.71081i			0.0628482i
$742$		0			0
$743$	−41.8112			−1.53390			−0.766952	−	0.641705i	$-0.778227\pi$
$743$	−41.8112			−1.53390			−0.766952	+	0.641705i	$0.778227\pi$
$744$		0			0
$745$	−4.16837			−0.152717
$746$		0			0
$747$	11.2088			0.410109
$748$		0			0
$749$	45.7805			1.67278
$750$		0			0
$751$	34.4288			1.25632			0.628162	−	0.778082i	$-0.283807\pi$
$751$	34.4288			1.25632			0.628162	+	0.778082i	$0.283807\pi$
$752$		0			0
$753$			23.2561i			0.847501i
$754$		0			0
$755$	5.22123			0.190020
$756$		0			0
$757$		−	29.7038i		−	1.07960i	−0.841792	−	0.539802i	$-0.818499\pi$
$757$		−	29.7038i		−	1.07960i	0.841792	−	0.539802i	$-0.181501\pi$
$758$		0			0
$759$	2.71932	+	27.6147i	0.0987049	+	1.00235i
$760$		0			0
$761$	20.5094			0.743464			0.371732	−	0.928340i	$-0.378764\pi$
$761$	20.5094			0.743464			0.371732	+	0.928340i	$0.378764\pi$
$762$		0			0
$763$		−	30.2030i		−	1.09342i
$764$		0			0
$765$	6.83315			0.247053
$766$		0			0
$767$			5.90497i			0.213216i
$768$		0			0
$769$		−	15.5406i		−	0.560407i	−0.959941	−	0.280203i	$-0.909598\pi$
$769$		−	15.5406i		−	0.560407i	0.959941	−	0.280203i	$-0.0904019\pi$
$770$		0			0
$771$		−	27.6478i		−	0.995712i
$772$		0			0
$773$			4.79100i			0.172320i	0.996281	+	0.0861601i	$0.0274597\pi$
$773$			4.79100i			0.172320i	−0.996281	+	0.0861601i	$0.972540\pi$
$774$		0			0
$775$		−	1.12061i		−	0.0402536i
$776$		0			0
$777$	26.9710			0.967580
$778$		0			0
$779$	6.10544			0.218750
$780$		0			0
$781$		−	21.5723i		−	0.771917i
$782$		0			0
$783$		−	1.39299i		−	0.0497814i
$784$		0			0
$785$	1.91699			0.0684202
$786$		0			0
$787$	−10.9512			−0.390369			−0.195185	−	0.980767i	$-0.562531\pi$
$787$	−10.9512			−0.390369			−0.195185	+	0.980767i	$0.562531\pi$
$788$		0			0
$789$		−	23.5536i		−	0.838529i
$790$		0			0
$791$		−	28.0340i		−	0.996775i
$792$		0			0
$793$			0.535718i			0.0190239i
$794$		0			0
$795$		−	2.67178i		−	0.0947584i
$796$		0			0
$797$		−	25.2060i		−	0.892842i	−0.894823	−	0.446421i	$-0.852698\pi$
$797$		−	25.2060i		−	0.892842i	0.894823	−	0.446421i	$-0.147302\pi$
$798$		0			0
$799$	39.6765			1.40365
$800$		0			0
$801$			5.21573i			0.184289i
$802$		0			0
$803$	2.77173			0.0978121
$804$		0			0
$805$	1.69437	+	17.2063i	0.0597185	+	0.606442i
$806$		0			0
$807$		−	10.7515i		−	0.378472i
$808$		0			0
$809$	25.1100			0.882822			0.441411	−	0.897305i	$-0.354478\pi$
$809$	25.1100			0.882822			0.441411	+	0.897305i	$0.354478\pi$
$810$		0			0
$811$		−	23.2579i		−	0.816694i	−0.912827	−	0.408347i	$-0.866105\pi$
$811$		−	23.2579i		−	0.816694i	0.912827	−	0.408347i	$-0.133895\pi$
$812$		0			0
$813$	−20.0996			−0.704924
$814$		0			0
$815$	−8.45209			−0.296064
$816$		0			0
$817$	4.84297			0.169434
$818$		0			0
$819$	−3.23199			−0.112935
$820$		0			0
$821$	−7.86939			−0.274643			−0.137322	−	0.990526i	$-0.543849\pi$
$821$	−7.86939			−0.274643			−0.137322	+	0.990526i	$0.543849\pi$
$822$		0			0
$823$			7.03369i			0.245179i	0.992457	+	0.122590i	$0.0391199\pi$
$823$			7.03369i			0.245179i	−0.992457	+	0.122590i	$0.960880\pi$
$824$		0			0
$825$		−	5.78590i		−	0.201439i
$826$		0			0
$827$	−40.1544			−1.39631			−0.698153	−	0.715948i	$-0.745995\pi$
$827$	−40.1544			−1.39631			−0.698153	+	0.715948i	$0.745995\pi$
$828$		0			0
$829$	12.2379			0.425039			0.212520	−	0.977157i	$-0.431833\pi$
$829$	12.2379			0.425039			0.212520	+	0.977157i	$0.431833\pi$
$830$		0			0
$831$			16.5831i			0.575262i
$832$		0			0
$833$			40.9772i			1.41978i
$834$		0			0
$835$	19.2777			0.667132
$836$		0			0
$837$	1.12061			0.0387340
$838$		0			0
$839$	34.4547			1.18951			0.594755	−	0.803907i	$-0.297249\pi$
$839$	34.4547			1.18951			0.594755	+	0.803907i	$0.297249\pi$
$840$		0			0
$841$	−27.0596			−0.933089
$842$		0			0
$843$	−9.61074			−0.331011
$844$		0			0
$845$		−	12.1963i		−	0.419565i
$846$		0			0
$847$	81.0310			2.78426
$848$		0			0
$849$		−	28.3850i		−	0.974171i
$850$		0			0
$851$	−3.51615	−	35.7065i	−0.120532	−	1.22400i
$852$		0			0
$853$	−0.870115			−0.0297922			−0.0148961	−	0.999889i	$-0.504742\pi$
$853$	−0.870115			−0.0297922			−0.0148961	+	0.999889i	$0.504742\pi$
$854$		0			0
$855$		−	1.90832i		−	0.0652631i
$856$		0			0
$857$	−22.2563			−0.760262			−0.380131	−	0.924933i	$-0.624121\pi$
$857$	−22.2563			−0.760262			−0.380131	+	0.924933i	$0.624121\pi$
$858$		0			0
$859$		−	2.78660i		−	0.0950776i	−0.998869	−	0.0475388i	$-0.984862\pi$
$859$		−	2.78660i		−	0.0950776i	0.998869	−	0.0475388i	$-0.0151378\pi$
$860$		0			0
$861$			11.5341i			0.393082i
$862$		0			0
$863$		−	37.0567i		−	1.26142i	−0.776017	−	0.630712i	$-0.782763\pi$
$863$		−	37.0567i		−	1.26142i	0.776017	−	0.630712i	$-0.217237\pi$
$864$		0			0
$865$		−	1.99198i		−	0.0677293i
$866$		0			0
$867$		−	29.6919i		−	1.00839i
$868$		0			0
$869$	6.00064			0.203558
$870$		0			0
$871$	−4.85676			−0.164565
$872$		0			0
$873$			4.73433i			0.160233i
$874$		0			0
$875$		−	3.60511i		−	0.121875i
$876$		0			0
$877$	−20.3614			−0.687556			−0.343778	−	0.939051i	$-0.611707\pi$
$877$	−20.3614			−0.687556			−0.343778	+	0.939051i	$0.611707\pi$
$878$		0			0
$879$	−24.9347			−0.841027
$880$		0			0
$881$		−	3.41977i		−	0.115215i	−0.998339	−	0.0576075i	$-0.981653\pi$
$881$		−	3.41977i		−	0.115215i	0.998339	−	0.0576075i	$-0.0183472\pi$
$882$		0			0
$883$			4.38232i			0.147477i	0.997278	+	0.0737383i	$0.0234930\pi$
$883$			4.38232i			0.147477i	−0.997278	+	0.0737383i	$0.976507\pi$
$884$		0			0
$885$		−	6.58667i		−	0.221408i
$886$		0			0
$887$		−	22.1798i		−	0.744725i	−0.928087	−	0.372363i	$-0.878548\pi$
$887$		−	22.1798i		−	0.744725i	0.928087	−	0.372363i	$-0.121452\pi$
$888$		0			0
$889$			18.1255i			0.607910i
$890$		0			0
$891$	5.78590			0.193835
$892$		0			0
$893$		−	11.0806i		−	0.370798i
$894$		0			0
$895$	−22.5461			−0.753631
$896$		0			0
$897$	0.421347	+	4.27878i	0.0140684	+	0.142864i
$898$		0			0
$899$			1.56100i			0.0520624i
$900$		0			0
$901$	−18.2567			−0.608219
$902$		0			0
$903$			9.14913i			0.304464i
$904$		0			0
$905$	2.37594			0.0789789
$906$		0			0
$907$	−43.2997			−1.43774			−0.718872	−	0.695143i	$-0.755341\pi$
$907$	−43.2997			−1.43774			−0.718872	+	0.695143i	$0.755341\pi$
$908$		0			0
$909$	0.362853			0.0120351
$910$		0			0
$911$	29.4352			0.975231			0.487616	−	0.873058i	$-0.337867\pi$
$911$	29.4352			0.975231			0.487616	+	0.873058i	$0.337867\pi$
$912$		0			0
$913$	−64.8531			−2.14632
$914$		0			0
$915$		−	0.597564i		−	0.0197549i
$916$		0			0
$917$		−	57.7198i		−	1.90608i
$918$		0			0
$919$	34.5277			1.13896			0.569482	−	0.822004i	$-0.307144\pi$
$919$	34.5277			1.13896			0.569482	+	0.822004i	$0.307144\pi$
$920$		0			0
$921$	−9.09513			−0.299695
$922$		0			0
$923$		−	3.34254i		−	0.110021i
$924$		0			0
$925$			7.48133i			0.245984i
$926$		0			0
$927$	11.7189			0.384899
$928$		0			0
$929$	15.6789			0.514408			0.257204	−	0.966357i	$-0.417199\pi$
$929$	15.6789			0.514408			0.257204	+	0.966357i	$0.417199\pi$
$930$		0			0
$931$	11.4439			0.375057
$932$		0			0
$933$	1.14205			0.0373890
$934$		0			0
$935$	−39.5359			−1.29296
$936$		0			0
$937$			20.0438i			0.654802i	0.944885	+	0.327401i	$0.106173\pi$
$937$			20.0438i			0.654802i	−0.944885	+	0.327401i	$0.893827\pi$
$938$		0			0
$939$	−28.0497			−0.915367
$940$		0			0
$941$		−	55.6519i		−	1.81420i	−0.420914	−	0.907100i	$-0.638291\pi$
$941$		−	55.6519i		−	1.81420i	0.420914	−	0.907100i	$-0.361709\pi$
$942$		0			0
$943$	15.2698	−	1.50368i	0.497255	−	0.0489665i
$944$		0			0
$945$	3.60511			0.117274
$946$		0			0
$947$			14.0016i			0.454992i	0.973779	+	0.227496i	$0.0730539\pi$
$947$			14.0016i			0.454992i	−0.973779	+	0.227496i	$0.926946\pi$
$948$		0			0
$949$	0.429468			0.0139411
$950$		0			0
$951$			18.0554i			0.585487i
$952$		0			0
$953$			5.92621i			0.191969i	0.995383	+	0.0959843i	$0.0305999\pi$
$953$			5.92621i			0.191969i	−0.995383	+	0.0959843i	$0.969400\pi$
$954$		0			0
$955$			2.63031i			0.0851148i
$956$		0			0
$957$			8.05971i			0.260534i
$958$		0			0
$959$		−	84.0651i		−	2.71460i
$960$		0			0
$961$	29.7442			0.959491
$962$		0			0
$963$	−12.6988			−0.409212
$964$		0			0
$965$		−	20.1946i		−	0.650086i
$966$		0			0
$967$		−	2.92295i		−	0.0939957i	−0.998895	−	0.0469979i	$-0.985035\pi$
$967$		−	2.92295i		−	0.0939957i	0.998895	−	0.0469979i	$-0.0149654\pi$
$968$		0			0
$969$	−13.0398			−0.418899
$970$		0			0
$971$	31.1539			0.999777			0.499888	−	0.866090i	$-0.333374\pi$
$971$	31.1539			0.999777			0.499888	+	0.866090i	$0.333374\pi$
$972$		0			0
$973$			7.98569i			0.256009i
$974$		0			0
$975$		−	0.896503i		−	0.0287111i
$976$		0			0
$977$		−	23.0997i		−	0.739025i	−0.929226	−	0.369513i	$-0.879525\pi$
$977$		−	23.0997i		−	0.739025i	0.929226	−	0.369513i	$-0.120475\pi$
$978$		0			0
$979$		−	30.1777i		−	0.964484i
$980$		0			0
$981$			8.37783i			0.267484i
$982$		0			0
$983$	16.8305			0.536809			0.268404	−	0.963306i	$-0.413504\pi$
$983$	16.8305			0.536809			0.268404	+	0.963306i	$0.413504\pi$
$984$		0			0
$985$			26.4159i			0.841680i
$986$		0			0
$987$	20.9330			0.666304
$988$		0			0
$989$	12.1124	−	1.19275i	0.385151	−	0.0379272i
$990$		0			0
$991$			26.9784i			0.856998i	0.903542	+	0.428499i	$0.140958\pi$
$991$			26.9784i			0.856998i	−0.903542	+	0.428499i	$0.859042\pi$
$992$		0			0
$993$	−17.0971			−0.542560
$994$		0			0
$995$		−	21.4192i		−	0.679035i
$996$		0			0
$997$	39.7616			1.25926			0.629632	−	0.776894i	$-0.283206\pi$
$997$	39.7616			1.25926			0.629632	+	0.776894i	$0.283206\pi$
$998$		0			0
$999$	−7.48133			−0.236699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.16	yes	32
4.3	odd	2		5520.2.be.d.1471.15	yes	32
23.22	odd	2		5520.2.be.d.1471.16	yes	32
92.91	even	2	inner	5520.2.be.c.1471.15	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.15	✓	32	92.91	even	2	inner
5520.2.be.c.1471.16	yes	32	1.1	even	1	trivial
5520.2.be.d.1471.15	yes	32	4.3	odd	2
5520.2.be.d.1471.16	yes	32	23.22	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.60511 q^{7} -1.00000 q^{9} +5.78590 q^{11} +0.896503 q^{13} -1.00000 q^{15} +6.83315i q^{17} +1.90832 q^{19} +3.60511i q^{21} +(4.77275 - 0.469990i) q^{23} -1.00000 q^{25} -1.00000i q^{27} +1.39299 q^{29} +1.12061i q^{31} +5.78590i q^{33} +3.60511i q^{35} -7.48133i q^{37} +0.896503i q^{39} +3.19938 q^{41} +2.53782 q^{43} -1.00000i q^{45} -5.80647i q^{47} +5.99683 q^{49} -6.83315 q^{51} +2.67178i q^{53} +5.78590i q^{55} +1.90832i q^{57} +6.58667i q^{59} +0.597564i q^{61} -3.60511 q^{63} +0.896503i q^{65} -5.41745 q^{67} +(0.469990 + 4.77275i) q^{69} -3.72842i q^{71} +0.479048 q^{73} -1.00000i q^{75} +20.8588 q^{77} +1.03711 q^{79} +1.00000 q^{81} -11.2088 q^{83} -6.83315 q^{85} +1.39299i q^{87} -5.21573i q^{89} +3.23199 q^{91} -1.12061 q^{93} +1.90832i q^{95} -4.73433i q^{97} -5.78590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 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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