LMFDB - Embedded newform 5520.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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([0, 0, 0, 0, 0]))

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

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

chi := DirichletCharacter("5520.1");

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.a (trivial)

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:	yes
Analytic conductor:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -4.46410 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -4.46410 q^{7} +1.00000 q^{9} +6.19615 q^{11} -6.73205 q^{13} +1.00000 q^{15} -4.26795 q^{17} +2.73205 q^{19} -4.46410 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.19615 q^{29} +1.00000 q^{31} +6.19615 q^{33} -4.46410 q^{35} +9.39230 q^{37} -6.73205 q^{39} -4.26795 q^{41} +3.46410 q^{43} +1.00000 q^{45} +2.73205 q^{47} +12.9282 q^{49} -4.26795 q^{51} -10.6603 q^{53} +6.19615 q^{55} +2.73205 q^{57} -3.19615 q^{59} -7.26795 q^{61} -4.46410 q^{63} -6.73205 q^{65} -5.00000 q^{67} +1.00000 q^{69} -10.1244 q^{71} -14.7321 q^{73} +1.00000 q^{75} -27.6603 q^{77} -4.00000 q^{79} +1.00000 q^{81} -10.6603 q^{83} -4.26795 q^{85} -3.19615 q^{87} +4.39230 q^{89} +30.0526 q^{91} +1.00000 q^{93} +2.73205 q^{95} +4.00000 q^{97} +6.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 12 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{31} + 2 q^{33} - 2 q^{35} - 2 q^{37} - 10 q^{39} - 12 q^{41} + 2 q^{45} + 2 q^{47} + 12 q^{49} - 12 q^{51} - 4 q^{53} + 2 q^{55} + 2 q^{57} + 4 q^{59} - 18 q^{61} - 2 q^{63} - 10 q^{65} - 10 q^{67} + 2 q^{69} + 4 q^{71} - 26 q^{73} + 2 q^{75} - 38 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{83} - 12 q^{85} + 4 q^{87} - 12 q^{89} + 22 q^{91} + 2 q^{93} + 2 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 10 * q^13 + 2 * q^15 - 12 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^31 + 2 * q^33 - 2 * q^35 - 2 * q^37 - 10 * q^39 - 12 * q^41 + 2 * q^45 + 2 * q^47 + 12 * q^49 - 12 * q^51 - 4 * q^53 + 2 * q^55 + 2 * q^57 + 4 * q^59 - 18 * q^61 - 2 * q^63 - 10 * q^65 - 10 * q^67 + 2 * q^69 + 4 * q^71 - 26 * q^73 + 2 * q^75 - 38 * q^77 - 8 * q^79 + 2 * q^81 - 4 * q^83 - 12 * q^85 + 4 * q^87 - 12 * q^89 + 22 * q^91 + 2 * q^93 + 2 * q^95 + 8 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.46410	−1.68727	−0.843636	−	0.536916i	$-0.819589\pi$
$7$	−4.46410	−1.68727	−0.843636	+	0.536916i	$0.819589\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.19615	1.86821	0.934105	−	0.356998i	$-0.116200\pi$
$11$	6.19615	1.86821	0.934105	+	0.356998i	$0.116200\pi$
$12$	0	0
$13$	−6.73205	−1.86713	−0.933567	−	0.358402i	$-0.883322\pi$
$13$	−6.73205	−1.86713	−0.933567	+	0.358402i	$0.883322\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.26795	−1.03513	−0.517565	−	0.855644i	$-0.673161\pi$
$17$	−4.26795	−1.03513	−0.517565	+	0.855644i	$0.673161\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	−4.46410	−0.974147
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.19615	−0.593511	−0.296755	−	0.954954i	$-0.595905\pi$
$29$	−3.19615	−0.593511	−0.296755	+	0.954954i	$0.595905\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	6.19615	1.07861
$34$	0	0
$35$	−4.46410	−0.754571
$36$	0	0
$37$	9.39230	1.54409	0.772043	−	0.635571i	$-0.219235\pi$
$37$	9.39230	1.54409	0.772043	+	0.635571i	$0.219235\pi$
$38$	0	0
$39$	−6.73205	−1.07799
$40$	0	0
$41$	−4.26795	−0.666542	−0.333271	−	0.942831i	$-0.608152\pi$
$41$	−4.26795	−0.666542	−0.333271	+	0.942831i	$0.608152\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.73205	0.398511	0.199255	−	0.979948i	$-0.436148\pi$
$47$	2.73205	0.398511	0.199255	+	0.979948i	$0.436148\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	0	0
$51$	−4.26795	−0.597632
$52$	0	0
$53$	−10.6603	−1.46430	−0.732149	−	0.681144i	$-0.761483\pi$
$53$	−10.6603	−1.46430	−0.732149	+	0.681144i	$0.761483\pi$
$54$	0	0
$55$	6.19615	0.835489
$56$	0	0
$57$	2.73205	0.361869
$58$	0	0
$59$	−3.19615	−0.416104	−0.208052	−	0.978118i	$-0.566712\pi$
$59$	−3.19615	−0.416104	−0.208052	+	0.978118i	$0.566712\pi$
$60$	0	0
$61$	−7.26795	−0.930566	−0.465283	−	0.885162i	$-0.654047\pi$
$61$	−7.26795	−0.930566	−0.465283	+	0.885162i	$0.654047\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	−6.73205	−0.835008
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−10.1244	−1.20154	−0.600770	−	0.799422i	$-0.705139\pi$
$71$	−10.1244	−1.20154	−0.600770	+	0.799422i	$0.705139\pi$
$72$	0	0
$73$	−14.7321	−1.72426	−0.862128	−	0.506690i	$-0.830869\pi$
$73$	−14.7321	−1.72426	−0.862128	+	0.506690i	$0.830869\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−27.6603	−3.15218
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.6603	−1.17011	−0.585057	−	0.810992i	$-0.698928\pi$
$83$	−10.6603	−1.17011	−0.585057	+	0.810992i	$0.698928\pi$
$84$	0	0
$85$	−4.26795	−0.462924
$86$	0	0
$87$	−3.19615	−0.342664
$88$	0	0
$89$	4.39230	0.465583	0.232792	−	0.972527i	$-0.425214\pi$
$89$	4.39230	0.465583	0.232792	+	0.972527i	$0.425214\pi$
$90$	0	0
$91$	30.0526	3.15036
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.73205	0.280302
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	6.19615	0.622737
$100$	0	0
$101$	−13.1962	−1.31307	−0.656533	−	0.754297i	$-0.727978\pi$
$101$	−13.1962	−1.31307	−0.656533	+	0.754297i	$0.727978\pi$
$102$	0	0
$103$	−3.07180	−0.302673	−0.151337	−	0.988482i	$-0.548358\pi$
$103$	−3.07180	−0.302673	−0.151337	+	0.988482i	$0.548358\pi$
$104$	0	0
$105$	−4.46410	−0.435652
$106$	0	0
$107$	9.19615	0.889026	0.444513	−	0.895772i	$-0.353377\pi$
$107$	9.19615	0.889026	0.444513	+	0.895772i	$0.353377\pi$
$108$	0	0
$109$	−13.6603	−1.30842	−0.654208	−	0.756315i	$-0.726998\pi$
$109$	−13.6603	−1.30842	−0.654208	+	0.756315i	$0.726998\pi$
$110$	0	0
$111$	9.39230	0.891478
$112$	0	0
$113$	−4.80385	−0.451908	−0.225954	−	0.974138i	$-0.572550\pi$
$113$	−4.80385	−0.451908	−0.225954	+	0.974138i	$0.572550\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−6.73205	−0.622378
$118$	0	0
$119$	19.0526	1.74655
$120$	0	0
$121$	27.3923	2.49021
$122$	0	0
$123$	−4.26795	−0.384828
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.1244	1.16460	0.582299	−	0.812975i	$-0.302153\pi$
$127$	13.1244	1.16460	0.582299	+	0.812975i	$0.302153\pi$
$128$	0	0
$129$	3.46410	0.304997
$130$	0	0
$131$	−5.85641	−0.511677	−0.255838	−	0.966720i	$-0.582351\pi$
$131$	−5.85641	−0.511677	−0.255838	+	0.966720i	$0.582351\pi$
$132$	0	0
$133$	−12.1962	−1.05754
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	2.73205	0.230080
$142$	0	0
$143$	−41.7128	−3.48820
$144$	0	0
$145$	−3.19615	−0.265426
$146$	0	0
$147$	12.9282	1.06630
$148$	0	0
$149$	16.1962	1.32684	0.663420	−	0.748247i	$-0.269104\pi$
$149$	16.1962	1.32684	0.663420	+	0.748247i	$0.269104\pi$
$150$	0	0
$151$	−18.3923	−1.49674	−0.748372	−	0.663279i	$-0.769164\pi$
$151$	−18.3923	−1.49674	−0.748372	+	0.663279i	$0.769164\pi$
$152$	0	0
$153$	−4.26795	−0.345043
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−22.3205	−1.78137	−0.890685	−	0.454621i	$-0.849775\pi$
$157$	−22.3205	−1.78137	−0.890685	+	0.454621i	$0.849775\pi$
$158$	0	0
$159$	−10.6603	−0.845413
$160$	0	0
$161$	−4.46410	−0.351820
$162$	0	0
$163$	−23.8564	−1.86858	−0.934289	−	0.356517i	$-0.883964\pi$
$163$	−23.8564	−1.86858	−0.934289	+	0.356517i	$0.883964\pi$
$164$	0	0
$165$	6.19615	0.482370
$166$	0	0
$167$	16.1962	1.25330	0.626648	−	0.779302i	$-0.284426\pi$
$167$	16.1962	1.25330	0.626648	+	0.779302i	$0.284426\pi$
$168$	0	0
$169$	32.3205	2.48619
$170$	0	0
$171$	2.73205	0.208925
$172$	0	0
$173$	−9.85641	−0.749369	−0.374684	−	0.927152i	$-0.622249\pi$
$173$	−9.85641	−0.749369	−0.374684	+	0.927152i	$0.622249\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	0	0
$177$	−3.19615	−0.240238
$178$	0	0
$179$	−4.53590	−0.339029	−0.169514	−	0.985528i	$-0.554220\pi$
$179$	−4.53590	−0.339029	−0.169514	+	0.985528i	$0.554220\pi$
$180$	0	0
$181$	8.39230	0.623795	0.311898	−	0.950116i	$-0.399035\pi$
$181$	8.39230	0.623795	0.311898	+	0.950116i	$0.399035\pi$
$182$	0	0
$183$	−7.26795	−0.537262
$184$	0	0
$185$	9.39230	0.690536
$186$	0	0
$187$	−26.4449	−1.93384
$188$	0	0
$189$	−4.46410	−0.324716
$190$	0	0
$191$	−16.1962	−1.17191	−0.585956	−	0.810343i	$-0.699281\pi$
$191$	−16.1962	−1.17191	−0.585956	+	0.810343i	$0.699281\pi$
$192$	0	0
$193$	3.60770	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$193$	3.60770	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$194$	0	0
$195$	−6.73205	−0.482092
$196$	0	0
$197$	−0.535898	−0.0381812	−0.0190906	−	0.999818i	$-0.506077\pi$
$197$	−0.535898	−0.0381812	−0.0190906	+	0.999818i	$0.506077\pi$
$198$	0	0
$199$	11.4641	0.812669	0.406334	−	0.913724i	$-0.366807\pi$
$199$	11.4641	0.812669	0.406334	+	0.913724i	$0.366807\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	14.2679	1.00141
$204$	0	0
$205$	−4.26795	−0.298087
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	16.9282	1.17095
$210$	0	0
$211$	4.07180	0.280314	0.140157	−	0.990129i	$-0.455239\pi$
$211$	4.07180	0.280314	0.140157	+	0.990129i	$0.455239\pi$
$212$	0	0
$213$	−10.1244	−0.693709
$214$	0	0
$215$	3.46410	0.236250
$216$	0	0
$217$	−4.46410	−0.303043
$218$	0	0
$219$	−14.7321	−0.995500
$220$	0	0
$221$	28.7321	1.93273
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	9.07180	0.602116	0.301058	−	0.953606i	$-0.402660\pi$
$227$	9.07180	0.602116	0.301058	+	0.953606i	$0.402660\pi$
$228$	0	0
$229$	10.5359	0.696232	0.348116	−	0.937452i	$-0.386822\pi$
$229$	10.5359	0.696232	0.348116	+	0.937452i	$0.386822\pi$
$230$	0	0
$231$	−27.6603	−1.81991
$232$	0	0
$233$	−20.7846	−1.36165	−0.680823	−	0.732448i	$-0.738378\pi$
$233$	−20.7846	−1.36165	−0.680823	+	0.732448i	$0.738378\pi$
$234$	0	0
$235$	2.73205	0.178219
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	22.1244	1.43111	0.715553	−	0.698559i	$-0.246175\pi$
$239$	22.1244	1.43111	0.715553	+	0.698559i	$0.246175\pi$
$240$	0	0
$241$	16.5885	1.06856	0.534278	−	0.845309i	$-0.320583\pi$
$241$	16.5885	1.06856	0.534278	+	0.845309i	$0.320583\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	12.9282	0.825953
$246$	0	0
$247$	−18.3923	−1.17027
$248$	0	0
$249$	−10.6603	−0.675566
$250$	0	0
$251$	3.85641	0.243414	0.121707	−	0.992566i	$-0.461163\pi$
$251$	3.85641	0.243414	0.121707	+	0.992566i	$0.461163\pi$
$252$	0	0
$253$	6.19615	0.389549
$254$	0	0
$255$	−4.26795	−0.267269
$256$	0	0
$257$	−15.1244	−0.943431	−0.471716	−	0.881751i	$-0.656365\pi$
$257$	−15.1244	−0.943431	−0.471716	+	0.881751i	$0.656365\pi$
$258$	0	0
$259$	−41.9282	−2.60529
$260$	0	0
$261$	−3.19615	−0.197837
$262$	0	0
$263$	28.6603	1.76727	0.883633	−	0.468179i	$-0.155090\pi$
$263$	28.6603	1.76727	0.883633	+	0.468179i	$0.155090\pi$
$264$	0	0
$265$	−10.6603	−0.654854
$266$	0	0
$267$	4.39230	0.268805
$268$	0	0
$269$	1.73205	0.105605	0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205	0.105605	0.0528025	+	0.998605i	$0.483185\pi$
$270$	0	0
$271$	−2.60770	−0.158406	−0.0792031	−	0.996858i	$-0.525238\pi$
$271$	−2.60770	−0.158406	−0.0792031	+	0.996858i	$0.525238\pi$
$272$	0	0
$273$	30.0526	1.81886
$274$	0	0
$275$	6.19615	0.373642
$276$	0	0
$277$	−31.8564	−1.91407	−0.957033	−	0.289979i	$-0.906352\pi$
$277$	−31.8564	−1.91407	−0.957033	+	0.289979i	$0.906352\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−29.5167	−1.76082	−0.880408	−	0.474217i	$-0.842731\pi$
$281$	−29.5167	−1.76082	−0.880408	+	0.474217i	$0.842731\pi$
$282$	0	0
$283$	−3.14359	−0.186867	−0.0934336	−	0.995626i	$-0.529784\pi$
$283$	−3.14359	−0.186867	−0.0934336	+	0.995626i	$0.529784\pi$
$284$	0	0
$285$	2.73205	0.161833
$286$	0	0
$287$	19.0526	1.12464
$288$	0	0
$289$	1.21539	0.0714935
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	−7.33975	−0.428793	−0.214396	−	0.976747i	$-0.568778\pi$
$293$	−7.33975	−0.428793	−0.214396	+	0.976747i	$0.568778\pi$
$294$	0	0
$295$	−3.19615	−0.186087
$296$	0	0
$297$	6.19615	0.359537
$298$	0	0
$299$	−6.73205	−0.389325
$300$	0	0
$301$	−15.4641	−0.891336
$302$	0	0
$303$	−13.1962	−0.758099
$304$	0	0
$305$	−7.26795	−0.416162
$306$	0	0
$307$	13.6603	0.779632	0.389816	−	0.920893i	$-0.372539\pi$
$307$	13.6603	0.779632	0.389816	+	0.920893i	$0.372539\pi$
$308$	0	0
$309$	−3.07180	−0.174748
$310$	0	0
$311$	−9.07180	−0.514414	−0.257207	−	0.966356i	$-0.582802\pi$
$311$	−9.07180	−0.514414	−0.257207	+	0.966356i	$0.582802\pi$
$312$	0	0
$313$	−12.3205	−0.696396	−0.348198	−	0.937421i	$-0.613206\pi$
$313$	−12.3205	−0.696396	−0.348198	+	0.937421i	$0.613206\pi$
$314$	0	0
$315$	−4.46410	−0.251524
$316$	0	0
$317$	31.5167	1.77015	0.885076	−	0.465447i	$-0.154106\pi$
$317$	31.5167	1.77015	0.885076	+	0.465447i	$0.154106\pi$
$318$	0	0
$319$	−19.8038	−1.10880
$320$	0	0
$321$	9.19615	0.513279
$322$	0	0
$323$	−11.6603	−0.648794
$324$	0	0
$325$	−6.73205	−0.373427
$326$	0	0
$327$	−13.6603	−0.755414
$328$	0	0
$329$	−12.1962	−0.672396
$330$	0	0
$331$	−21.2487	−1.16793	−0.583967	−	0.811777i	$-0.698500\pi$
$331$	−21.2487	−1.16793	−0.583967	+	0.811777i	$0.698500\pi$
$332$	0	0
$333$	9.39230	0.514695
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	28.9282	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$337$	28.9282	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$338$	0	0
$339$	−4.80385	−0.260909
$340$	0	0
$341$	6.19615	0.335540
$342$	0	0
$343$	−26.4641	−1.42893
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−28.3923	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$347$	−28.3923	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$348$	0	0
$349$	−6.46410	−0.346015	−0.173008	−	0.984920i	$-0.555349\pi$
$349$	−6.46410	−0.346015	−0.173008	+	0.984920i	$0.555349\pi$
$350$	0	0
$351$	−6.73205	−0.359330
$352$	0	0
$353$	32.5885	1.73451	0.867254	−	0.497865i	$-0.165883\pi$
$353$	32.5885	1.73451	0.867254	+	0.497865i	$0.165883\pi$
$354$	0	0
$355$	−10.1244	−0.537345
$356$	0	0
$357$	19.0526	1.00837
$358$	0	0
$359$	5.26795	0.278032	0.139016	−	0.990290i	$-0.455606\pi$
$359$	5.26795	0.278032	0.139016	+	0.990290i	$0.455606\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	27.3923	1.43772
$364$	0	0
$365$	−14.7321	−0.771111
$366$	0	0
$367$	23.5359	1.22856	0.614282	−	0.789087i	$-0.289446\pi$
$367$	23.5359	1.22856	0.614282	+	0.789087i	$0.289446\pi$
$368$	0	0
$369$	−4.26795	−0.222181
$370$	0	0
$371$	47.5885	2.47067
$372$	0	0
$373$	−2.53590	−0.131304	−0.0656519	−	0.997843i	$-0.520913\pi$
$373$	−2.53590	−0.131304	−0.0656519	+	0.997843i	$0.520913\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	21.5167	1.10816
$378$	0	0
$379$	17.8564	0.917222	0.458611	−	0.888637i	$-0.348347\pi$
$379$	17.8564	0.917222	0.458611	+	0.888637i	$0.348347\pi$
$380$	0	0
$381$	13.1244	0.672381
$382$	0	0
$383$	7.05256	0.360369	0.180184	−	0.983633i	$-0.442331\pi$
$383$	7.05256	0.360369	0.180184	+	0.983633i	$0.442331\pi$
$384$	0	0
$385$	−27.6603	−1.40970
$386$	0	0
$387$	3.46410	0.176090
$388$	0	0
$389$	19.4641	0.986869	0.493435	−	0.869783i	$-0.335741\pi$
$389$	19.4641	0.986869	0.493435	+	0.869783i	$0.335741\pi$
$390$	0	0
$391$	−4.26795	−0.215839
$392$	0	0
$393$	−5.85641	−0.295417
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	−12.1962	−0.610571
$400$	0	0
$401$	9.60770	0.479785	0.239893	−	0.970799i	$-0.422888\pi$
$401$	9.60770	0.479785	0.239893	+	0.970799i	$0.422888\pi$
$402$	0	0
$403$	−6.73205	−0.335347
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	58.1962	2.88468
$408$	0	0
$409$	30.1769	1.49215	0.746076	−	0.665861i	$-0.231935\pi$
$409$	30.1769	1.49215	0.746076	+	0.665861i	$0.231935\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	14.2679	0.702080
$414$	0	0
$415$	−10.6603	−0.523291
$416$	0	0
$417$	−7.00000	−0.342791
$418$	0	0
$419$	−21.1244	−1.03199	−0.515996	−	0.856591i	$-0.672578\pi$
$419$	−21.1244	−1.03199	−0.515996	+	0.856591i	$0.672578\pi$
$420$	0	0
$421$	−31.5167	−1.53603	−0.768014	−	0.640433i	$-0.778755\pi$
$421$	−31.5167	−1.53603	−0.768014	+	0.640433i	$0.778755\pi$
$422$	0	0
$423$	2.73205	0.132837
$424$	0	0
$425$	−4.26795	−0.207026
$426$	0	0
$427$	32.4449	1.57012
$428$	0	0
$429$	−41.7128	−2.01391
$430$	0	0
$431$	−24.3923	−1.17494	−0.587468	−	0.809247i	$-0.699875\pi$
$431$	−24.3923	−1.17494	−0.587468	+	0.809247i	$0.699875\pi$
$432$	0	0
$433$	−8.60770	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$433$	−8.60770	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$434$	0	0
$435$	−3.19615	−0.153244
$436$	0	0
$437$	2.73205	0.130692
$438$	0	0
$439$	22.7846	1.08745	0.543725	−	0.839263i	$-0.317013\pi$
$439$	22.7846	1.08745	0.543725	+	0.839263i	$0.317013\pi$
$440$	0	0
$441$	12.9282	0.615629
$442$	0	0
$443$	12.1962	0.579457	0.289728	−	0.957109i	$-0.406435\pi$
$443$	12.1962	0.579457	0.289728	+	0.957109i	$0.406435\pi$
$444$	0	0
$445$	4.39230	0.208215
$446$	0	0
$447$	16.1962	0.766052
$448$	0	0
$449$	7.73205	0.364898	0.182449	−	0.983215i	$-0.441598\pi$
$449$	7.73205	0.364898	0.182449	+	0.983215i	$0.441598\pi$
$450$	0	0
$451$	−26.4449	−1.24524
$452$	0	0
$453$	−18.3923	−0.864146
$454$	0	0
$455$	30.0526	1.40889
$456$	0	0
$457$	−19.5359	−0.913851	−0.456925	−	0.889505i	$-0.651049\pi$
$457$	−19.5359	−0.913851	−0.456925	+	0.889505i	$0.651049\pi$
$458$	0	0
$459$	−4.26795	−0.199211
$460$	0	0
$461$	5.32051	0.247801	0.123900	−	0.992295i	$-0.460460\pi$
$461$	5.32051	0.247801	0.123900	+	0.992295i	$0.460460\pi$
$462$	0	0
$463$	−20.5885	−0.956827	−0.478413	−	0.878135i	$-0.658788\pi$
$463$	−20.5885	−0.956827	−0.478413	+	0.878135i	$0.658788\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−29.4449	−1.36255	−0.681273	−	0.732030i	$-0.738573\pi$
$467$	−29.4449	−1.36255	−0.681273	+	0.732030i	$0.738573\pi$
$468$	0	0
$469$	22.3205	1.03067
$470$	0	0
$471$	−22.3205	−1.02847
$472$	0	0
$473$	21.4641	0.986920
$474$	0	0
$475$	2.73205	0.125355
$476$	0	0
$477$	−10.6603	−0.488100
$478$	0	0
$479$	−16.9808	−0.775871	−0.387935	−	0.921687i	$-0.626812\pi$
$479$	−16.9808	−0.775871	−0.387935	+	0.921687i	$0.626812\pi$
$480$	0	0
$481$	−63.2295	−2.88302
$482$	0	0
$483$	−4.46410	−0.203124
$484$	0	0
$485$	4.00000	0.181631
$486$	0	0
$487$	−0.732051	−0.0331724	−0.0165862	−	0.999862i	$-0.505280\pi$
$487$	−0.732051	−0.0331724	−0.0165862	+	0.999862i	$0.505280\pi$
$488$	0	0
$489$	−23.8564	−1.07882
$490$	0	0
$491$	19.0526	0.859830	0.429915	−	0.902869i	$-0.358544\pi$
$491$	19.0526	0.859830	0.429915	+	0.902869i	$0.358544\pi$
$492$	0	0
$493$	13.6410	0.614360
$494$	0	0
$495$	6.19615	0.278496
$496$	0	0
$497$	45.1962	2.02732
$498$	0	0
$499$	21.5359	0.964079	0.482040	−	0.876149i	$-0.339896\pi$
$499$	21.5359	0.964079	0.482040	+	0.876149i	$0.339896\pi$
$500$	0	0
$501$	16.1962	0.723591
$502$	0	0
$503$	−6.80385	−0.303369	−0.151684	−	0.988429i	$-0.548470\pi$
$503$	−6.80385	−0.303369	−0.151684	+	0.988429i	$0.548470\pi$
$504$	0	0
$505$	−13.1962	−0.587221
$506$	0	0
$507$	32.3205	1.43540
$508$	0	0
$509$	8.78461	0.389371	0.194685	−	0.980866i	$-0.437631\pi$
$509$	8.78461	0.389371	0.194685	+	0.980866i	$0.437631\pi$
$510$	0	0
$511$	65.7654	2.90929
$512$	0	0
$513$	2.73205	0.120623
$514$	0	0
$515$	−3.07180	−0.135360
$516$	0	0
$517$	16.9282	0.744502
$518$	0	0
$519$	−9.85641	−0.432648
$520$	0	0
$521$	−16.0526	−0.703275	−0.351638	−	0.936136i	$-0.614375\pi$
$521$	−16.0526	−0.703275	−0.351638	+	0.936136i	$0.614375\pi$
$522$	0	0
$523$	4.78461	0.209216	0.104608	−	0.994514i	$-0.466641\pi$
$523$	4.78461	0.209216	0.104608	+	0.994514i	$0.466641\pi$
$524$	0	0
$525$	−4.46410	−0.194829
$526$	0	0
$527$	−4.26795	−0.185915
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.19615	−0.138701
$532$	0	0
$533$	28.7321	1.24452
$534$	0	0
$535$	9.19615	0.397584
$536$	0	0
$537$	−4.53590	−0.195738
$538$	0	0
$539$	80.1051	3.45037
$540$	0	0
$541$	−1.07180	−0.0460801	−0.0230401	−	0.999735i	$-0.507335\pi$
$541$	−1.07180	−0.0460801	−0.0230401	+	0.999735i	$0.507335\pi$
$542$	0	0
$543$	8.39230	0.360148
$544$	0	0
$545$	−13.6603	−0.585141
$546$	0	0
$547$	−10.9282	−0.467256	−0.233628	−	0.972326i	$-0.575060\pi$
$547$	−10.9282	−0.467256	−0.233628	+	0.972326i	$0.575060\pi$
$548$	0	0
$549$	−7.26795	−0.310189
$550$	0	0
$551$	−8.73205	−0.371998
$552$	0	0
$553$	17.8564	0.759332
$554$	0	0
$555$	9.39230	0.398681
$556$	0	0
$557$	35.9808	1.52455	0.762277	−	0.647251i	$-0.224081\pi$
$557$	35.9808	1.52455	0.762277	+	0.647251i	$0.224081\pi$
$558$	0	0
$559$	−23.3205	−0.986352
$560$	0	0
$561$	−26.4449	−1.11650
$562$	0	0
$563$	15.5885	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$563$	15.5885	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$564$	0	0
$565$	−4.80385	−0.202099
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	7.46410	0.312911	0.156456	−	0.987685i	$-0.449993\pi$
$569$	7.46410	0.312911	0.156456	+	0.987685i	$0.449993\pi$
$570$	0	0
$571$	−6.58846	−0.275718	−0.137859	−	0.990452i	$-0.544022\pi$
$571$	−6.58846	−0.275718	−0.137859	+	0.990452i	$0.544022\pi$
$572$	0	0
$573$	−16.1962	−0.676604
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−24.7846	−1.03180	−0.515898	−	0.856650i	$-0.672542\pi$
$577$	−24.7846	−1.03180	−0.515898	+	0.856650i	$0.672542\pi$
$578$	0	0
$579$	3.60770	0.149931
$580$	0	0
$581$	47.5885	1.97430
$582$	0	0
$583$	−66.0526	−2.73562
$584$	0	0
$585$	−6.73205	−0.278336
$586$	0	0
$587$	11.0718	0.456982	0.228491	−	0.973546i	$-0.426621\pi$
$587$	11.0718	0.456982	0.228491	+	0.973546i	$0.426621\pi$
$588$	0	0
$589$	2.73205	0.112572
$590$	0	0
$591$	−0.535898	−0.0220439
$592$	0	0
$593$	−34.7321	−1.42627	−0.713137	−	0.701024i	$-0.752726\pi$
$593$	−34.7321	−1.42627	−0.713137	+	0.701024i	$0.752726\pi$
$594$	0	0
$595$	19.0526	0.781079
$596$	0	0
$597$	11.4641	0.469194
$598$	0	0
$599$	−24.3923	−0.996643	−0.498321	−	0.866992i	$-0.666050\pi$
$599$	−24.3923	−0.996643	−0.498321	+	0.866992i	$0.666050\pi$
$600$	0	0
$601$	−27.3923	−1.11736	−0.558678	−	0.829385i	$-0.688691\pi$
$601$	−27.3923	−1.11736	−0.558678	+	0.829385i	$0.688691\pi$
$602$	0	0
$603$	−5.00000	−0.203616
$604$	0	0
$605$	27.3923	1.11366
$606$	0	0
$607$	−22.9808	−0.932760	−0.466380	−	0.884584i	$-0.654442\pi$
$607$	−22.9808	−0.932760	−0.466380	+	0.884584i	$0.654442\pi$
$608$	0	0
$609$	14.2679	0.578166
$610$	0	0
$611$	−18.3923	−0.744073
$612$	0	0
$613$	13.7128	0.553855	0.276928	−	0.960891i	$-0.410684\pi$
$613$	13.7128	0.553855	0.276928	+	0.960891i	$0.410684\pi$
$614$	0	0
$615$	−4.26795	−0.172100
$616$	0	0
$617$	−36.1244	−1.45431	−0.727156	−	0.686472i	$-0.759158\pi$
$617$	−36.1244	−1.45431	−0.727156	+	0.686472i	$0.759158\pi$
$618$	0	0
$619$	2.92820	0.117694	0.0588472	−	0.998267i	$-0.481258\pi$
$619$	2.92820	0.117694	0.0588472	+	0.998267i	$0.481258\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−19.6077	−0.785566
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.9282	0.676047
$628$	0	0
$629$	−40.0859	−1.59833
$630$	0	0
$631$	44.8372	1.78494	0.892470	−	0.451107i	$-0.148971\pi$
$631$	44.8372	1.78494	0.892470	+	0.451107i	$0.148971\pi$
$632$	0	0
$633$	4.07180	0.161839
$634$	0	0
$635$	13.1244	0.520824
$636$	0	0
$637$	−87.0333	−3.44839
$638$	0	0
$639$	−10.1244	−0.400513
$640$	0	0
$641$	−28.0526	−1.10801	−0.554005	−	0.832514i	$-0.686901\pi$
$641$	−28.0526	−1.10801	−0.554005	+	0.832514i	$0.686901\pi$
$642$	0	0
$643$	8.46410	0.333792	0.166896	−	0.985975i	$-0.446626\pi$
$643$	8.46410	0.333792	0.166896	+	0.985975i	$0.446626\pi$
$644$	0	0
$645$	3.46410	0.136399
$646$	0	0
$647$	23.2679	0.914757	0.457379	−	0.889272i	$-0.348788\pi$
$647$	23.2679	0.914757	0.457379	+	0.889272i	$0.348788\pi$
$648$	0	0
$649$	−19.8038	−0.777369
$650$	0	0
$651$	−4.46410	−0.174962
$652$	0	0
$653$	39.3731	1.54079	0.770394	−	0.637569i	$-0.220060\pi$
$653$	39.3731	1.54079	0.770394	+	0.637569i	$0.220060\pi$
$654$	0	0
$655$	−5.85641	−0.228829
$656$	0	0
$657$	−14.7321	−0.574752
$658$	0	0
$659$	−8.44486	−0.328965	−0.164483	−	0.986380i	$-0.552595\pi$
$659$	−8.44486	−0.328965	−0.164483	+	0.986380i	$0.552595\pi$
$660$	0	0
$661$	35.4641	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$661$	35.4641	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$662$	0	0
$663$	28.7321	1.11586
$664$	0	0
$665$	−12.1962	−0.472947
$666$	0	0
$667$	−3.19615	−0.123756
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−45.0333	−1.73849
$672$	0	0
$673$	37.9090	1.46128	0.730642	−	0.682761i	$-0.239221\pi$
$673$	37.9090	1.46128	0.730642	+	0.682761i	$0.239221\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−31.1962	−1.19897	−0.599483	−	0.800388i	$-0.704627\pi$
$677$	−31.1962	−1.19897	−0.599483	+	0.800388i	$0.704627\pi$
$678$	0	0
$679$	−17.8564	−0.685266
$680$	0	0
$681$	9.07180	0.347632
$682$	0	0
$683$	10.7321	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$683$	10.7321	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	10.5359	0.401970
$688$	0	0
$689$	71.7654	2.73404
$690$	0	0
$691$	7.46410	0.283948	0.141974	−	0.989870i	$-0.454655\pi$
$691$	7.46410	0.283948	0.141974	+	0.989870i	$0.454655\pi$
$692$	0	0
$693$	−27.6603	−1.05073
$694$	0	0
$695$	−7.00000	−0.265525
$696$	0	0
$697$	18.2154	0.689957
$698$	0	0
$699$	−20.7846	−0.786146
$700$	0	0
$701$	12.5885	0.475459	0.237730	−	0.971331i	$-0.423597\pi$
$701$	12.5885	0.475459	0.237730	+	0.971331i	$0.423597\pi$
$702$	0	0
$703$	25.6603	0.967795
$704$	0	0
$705$	2.73205	0.102895
$706$	0	0
$707$	58.9090	2.21550
$708$	0	0
$709$	6.87564	0.258220	0.129110	−	0.991630i	$-0.458788\pi$
$709$	6.87564	0.258220	0.129110	+	0.991630i	$0.458788\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	−41.7128	−1.55997
$716$	0	0
$717$	22.1244	0.826249
$718$	0	0
$719$	−17.7321	−0.661294	−0.330647	−	0.943755i	$-0.607267\pi$
$719$	−17.7321	−0.661294	−0.330647	+	0.943755i	$0.607267\pi$
$720$	0	0
$721$	13.7128	0.510692
$722$	0	0
$723$	16.5885	0.616931
$724$	0	0
$725$	−3.19615	−0.118702
$726$	0	0
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.7846	−0.546829
$732$	0	0
$733$	−53.6410	−1.98128	−0.990638	−	0.136515i	$-0.956410\pi$
$733$	−53.6410	−1.98128	−0.990638	+	0.136515i	$0.956410\pi$
$734$	0	0
$735$	12.9282	0.476864
$736$	0	0
$737$	−30.9808	−1.14119
$738$	0	0
$739$	36.3205	1.33607	0.668036	−	0.744129i	$-0.267135\pi$
$739$	36.3205	1.33607	0.668036	+	0.744129i	$0.267135\pi$
$740$	0	0
$741$	−18.3923	−0.675658
$742$	0	0
$743$	−9.85641	−0.361596	−0.180798	−	0.983520i	$-0.557868\pi$
$743$	−9.85641	−0.361596	−0.180798	+	0.983520i	$0.557868\pi$
$744$	0	0
$745$	16.1962	0.593381
$746$	0	0
$747$	−10.6603	−0.390038
$748$	0	0
$749$	−41.0526	−1.50003
$750$	0	0
$751$	24.8756	0.907725	0.453863	−	0.891072i	$-0.350046\pi$
$751$	24.8756	0.907725	0.453863	+	0.891072i	$0.350046\pi$
$752$	0	0
$753$	3.85641	0.140535
$754$	0	0
$755$	−18.3923	−0.669365
$756$	0	0
$757$	50.8564	1.84841	0.924204	−	0.381900i	$-0.124730\pi$
$757$	50.8564	1.84841	0.924204	+	0.381900i	$0.124730\pi$
$758$	0	0
$759$	6.19615	0.224906
$760$	0	0
$761$	−5.19615	−0.188360	−0.0941802	−	0.995555i	$-0.530023\pi$
$761$	−5.19615	−0.188360	−0.0941802	+	0.995555i	$0.530023\pi$
$762$	0	0
$763$	60.9808	2.20765
$764$	0	0
$765$	−4.26795	−0.154308
$766$	0	0
$767$	21.5167	0.776922
$768$	0	0
$769$	7.66025	0.276236	0.138118	−	0.990416i	$-0.455895\pi$
$769$	7.66025	0.276236	0.138118	+	0.990416i	$0.455895\pi$
$770$	0	0
$771$	−15.1244	−0.544690
$772$	0	0
$773$	−45.0333	−1.61974	−0.809868	−	0.586612i	$-0.800461\pi$
$773$	−45.0333	−1.61974	−0.809868	+	0.586612i	$0.800461\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−41.9282	−1.50417
$778$	0	0
$779$	−11.6603	−0.417772
$780$	0	0
$781$	−62.7321	−2.24473
$782$	0	0
$783$	−3.19615	−0.114221
$784$	0	0
$785$	−22.3205	−0.796653
$786$	0	0
$787$	−24.6077	−0.877170	−0.438585	−	0.898690i	$-0.644520\pi$
$787$	−24.6077	−0.877170	−0.438585	+	0.898690i	$0.644520\pi$
$788$	0	0
$789$	28.6603	1.02033
$790$	0	0
$791$	21.4449	0.762492
$792$	0	0
$793$	48.9282	1.73749
$794$	0	0
$795$	−10.6603	−0.378080
$796$	0	0
$797$	7.73205	0.273883	0.136942	−	0.990579i	$-0.456273\pi$
$797$	7.73205	0.273883	0.136942	+	0.990579i	$0.456273\pi$
$798$	0	0
$799$	−11.6603	−0.412510
$800$	0	0
$801$	4.39230	0.155194
$802$	0	0
$803$	−91.2820	−3.22127
$804$	0	0
$805$	−4.46410	−0.157339
$806$	0	0
$807$	1.73205	0.0609711
$808$	0	0
$809$	−12.8038	−0.450159	−0.225080	−	0.974340i	$-0.572264\pi$
$809$	−12.8038	−0.450159	−0.225080	+	0.974340i	$0.572264\pi$
$810$	0	0
$811$	20.0718	0.704816	0.352408	−	0.935846i	$-0.385363\pi$
$811$	20.0718	0.704816	0.352408	+	0.935846i	$0.385363\pi$
$812$	0	0
$813$	−2.60770	−0.0914559
$814$	0	0
$815$	−23.8564	−0.835653
$816$	0	0
$817$	9.46410	0.331107
$818$	0	0
$819$	30.0526	1.05012
$820$	0	0
$821$	25.8564	0.902395	0.451197	−	0.892424i	$-0.350997\pi$
$821$	25.8564	0.902395	0.451197	+	0.892424i	$0.350997\pi$
$822$	0	0
$823$	20.7846	0.724506	0.362253	−	0.932080i	$-0.382008\pi$
$823$	20.7846	0.724506	0.362253	+	0.932080i	$0.382008\pi$
$824$	0	0
$825$	6.19615	0.215722
$826$	0	0
$827$	−4.51666	−0.157060	−0.0785298	−	0.996912i	$-0.525023\pi$
$827$	−4.51666	−0.157060	−0.0785298	+	0.996912i	$0.525023\pi$
$828$	0	0
$829$	−41.4974	−1.44127	−0.720633	−	0.693317i	$-0.756148\pi$
$829$	−41.4974	−1.44127	−0.720633	+	0.693317i	$0.756148\pi$
$830$	0	0
$831$	−31.8564	−1.10509
$832$	0	0
$833$	−55.1769	−1.91177
$834$	0	0
$835$	16.1962	0.560491
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	0	0
$843$	−29.5167	−1.01661
$844$	0	0
$845$	32.3205	1.11186
$846$	0	0
$847$	−122.282	−4.20166
$848$	0	0
$849$	−3.14359	−0.107888
$850$	0	0
$851$	9.39230	0.321964
$852$	0	0
$853$	−24.5359	−0.840093	−0.420047	−	0.907503i	$-0.637986\pi$
$853$	−24.5359	−0.840093	−0.420047	+	0.907503i	$0.637986\pi$
$854$	0	0
$855$	2.73205	0.0934342
$856$	0	0
$857$	−21.7128	−0.741696	−0.370848	−	0.928694i	$-0.620933\pi$
$857$	−21.7128	−0.741696	−0.370848	+	0.928694i	$0.620933\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	19.0526	0.649309
$862$	0	0
$863$	51.9615	1.76879	0.884395	−	0.466738i	$-0.154571\pi$
$863$	51.9615	1.76879	0.884395	+	0.466738i	$0.154571\pi$
$864$	0	0
$865$	−9.85641	−0.335128
$866$	0	0
$867$	1.21539	0.0412768
$868$	0	0
$869$	−24.7846	−0.840760
$870$	0	0
$871$	33.6603	1.14053
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−4.46410	−0.150914
$876$	0	0
$877$	−0.535898	−0.0180960	−0.00904800	−	0.999959i	$-0.502880\pi$
$877$	−0.535898	−0.0180960	−0.00904800	+	0.999959i	$0.502880\pi$
$878$	0	0
$879$	−7.33975	−0.247563
$880$	0	0
$881$	46.0526	1.55155	0.775775	−	0.631010i	$-0.217359\pi$
$881$	46.0526	1.55155	0.775775	+	0.631010i	$0.217359\pi$
$882$	0	0
$883$	28.3397	0.953708	0.476854	−	0.878982i	$-0.341777\pi$
$883$	28.3397	0.953708	0.476854	+	0.878982i	$0.341777\pi$
$884$	0	0
$885$	−3.19615	−0.107437
$886$	0	0
$887$	−17.3205	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$887$	−17.3205	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$888$	0	0
$889$	−58.5885	−1.96499
$890$	0	0
$891$	6.19615	0.207579
$892$	0	0
$893$	7.46410	0.249777
$894$	0	0
$895$	−4.53590	−0.151618
$896$	0	0
$897$	−6.73205	−0.224777
$898$	0	0
$899$	−3.19615	−0.106598
$900$	0	0
$901$	45.4974	1.51574
$902$	0	0
$903$	−15.4641	−0.514613
$904$	0	0
$905$	8.39230	0.278970
$906$	0	0
$907$	−6.07180	−0.201611	−0.100805	−	0.994906i	$-0.532142\pi$
$907$	−6.07180	−0.201611	−0.100805	+	0.994906i	$0.532142\pi$
$908$	0	0
$909$	−13.1962	−0.437689
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−66.0526	−2.18602
$914$	0	0
$915$	−7.26795	−0.240271
$916$	0	0
$917$	26.1436	0.863338
$918$	0	0
$919$	9.07180	0.299251	0.149625	−	0.988743i	$-0.452193\pi$
$919$	9.07180	0.299251	0.149625	+	0.988743i	$0.452193\pi$
$920$	0	0
$921$	13.6603	0.450121
$922$	0	0
$923$	68.1577	2.24344
$924$	0	0
$925$	9.39230	0.308817
$926$	0	0
$927$	−3.07180	−0.100891
$928$	0	0
$929$	−34.5167	−1.13245	−0.566227	−	0.824249i	$-0.691598\pi$
$929$	−34.5167	−1.13245	−0.566227	+	0.824249i	$0.691598\pi$
$930$	0	0
$931$	35.3205	1.15758
$932$	0	0
$933$	−9.07180	−0.296997
$934$	0	0
$935$	−26.4449	−0.864840
$936$	0	0
$937$	18.9282	0.618357	0.309179	−	0.951004i	$-0.399946\pi$
$937$	18.9282	0.618357	0.309179	+	0.951004i	$0.399946\pi$
$938$	0	0
$939$	−12.3205	−0.402065
$940$	0	0
$941$	11.8038	0.384794	0.192397	−	0.981317i	$-0.438374\pi$
$941$	11.8038	0.384794	0.192397	+	0.981317i	$0.438374\pi$
$942$	0	0
$943$	−4.26795	−0.138984
$944$	0	0
$945$	−4.46410	−0.145217
$946$	0	0
$947$	43.3205	1.40773	0.703864	−	0.710335i	$-0.251457\pi$
$947$	43.3205	1.40773	0.703864	+	0.710335i	$0.251457\pi$
$948$	0	0
$949$	99.1769	3.21942
$950$	0	0
$951$	31.5167	1.02200
$952$	0	0
$953$	29.7128	0.962492	0.481246	−	0.876585i	$-0.340184\pi$
$953$	29.7128	0.962492	0.481246	+	0.876585i	$0.340184\pi$
$954$	0	0
$955$	−16.1962	−0.524095
$956$	0	0
$957$	−19.8038	−0.640167
$958$	0	0
$959$	53.5692	1.72984
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	9.19615	0.296342
$964$	0	0
$965$	3.60770	0.116136
$966$	0	0
$967$	41.1244	1.32247	0.661235	−	0.750179i	$-0.270033\pi$
$967$	41.1244	1.32247	0.661235	+	0.750179i	$0.270033\pi$
$968$	0	0
$969$	−11.6603	−0.374581
$970$	0	0
$971$	12.6795	0.406904	0.203452	−	0.979085i	$-0.434784\pi$
$971$	12.6795	0.406904	0.203452	+	0.979085i	$0.434784\pi$
$972$	0	0
$973$	31.2487	1.00179
$974$	0	0
$975$	−6.73205	−0.215598
$976$	0	0
$977$	34.1244	1.09173	0.545867	−	0.837872i	$-0.316200\pi$
$977$	34.1244	1.09173	0.545867	+	0.837872i	$0.316200\pi$
$978$	0	0
$979$	27.2154	0.869808
$980$	0	0
$981$	−13.6603	−0.436138
$982$	0	0
$983$	−0.803848	−0.0256388	−0.0128194	−	0.999918i	$-0.504081\pi$
$983$	−0.803848	−0.0256388	−0.0128194	+	0.999918i	$0.504081\pi$
$984$	0	0
$985$	−0.535898	−0.0170751
$986$	0	0
$987$	−12.1962	−0.388208
$988$	0	0
$989$	3.46410	0.110152
$990$	0	0
$991$	26.7128	0.848560	0.424280	−	0.905531i	$-0.360527\pi$
$991$	26.7128	0.848560	0.424280	+	0.905531i	$0.360527\pi$
$992$	0	0
$993$	−21.2487	−0.674307
$994$	0	0
$995$	11.4641	0.363436
$996$	0	0
$997$	−46.5359	−1.47381	−0.736903	−	0.675998i	$-0.763713\pi$
$997$	−46.5359	−1.47381	−0.736903	+	0.675998i	$0.763713\pi$
$998$	0	0
$999$	9.39230	0.297159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bp.1.1		2
4.3	odd	2		1380.2.a.h.1.2	✓	2
12.11	even	2		4140.2.a.o.1.2		2
20.3	even	4		6900.2.f.i.6349.1		4
20.7	even	4		6900.2.f.i.6349.4		4
20.19	odd	2		6900.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.h.1.2	✓	2	4.3	odd	2
4140.2.a.o.1.2		2	12.11	even	2
5520.2.a.bp.1.1		2	1.1	even	1	trivial
6900.2.a.s.1.1		2	20.19	odd	2
6900.2.f.i.6349.1		4	20.3	even	4
6900.2.f.i.6349.4		4	20.7	even	4

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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.46410 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.46410 q^{7} +1.00000 q^{9} +6.19615 q^{11} -6.73205 q^{13} +1.00000 q^{15} -4.26795 q^{17} +2.73205 q^{19} -4.46410 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.19615 q^{29} +1.00000 q^{31} +6.19615 q^{33} -4.46410 q^{35} +9.39230 q^{37} -6.73205 q^{39} -4.26795 q^{41} +3.46410 q^{43} +1.00000 q^{45} +2.73205 q^{47} +12.9282 q^{49} -4.26795 q^{51} -10.6603 q^{53} +6.19615 q^{55} +2.73205 q^{57} -3.19615 q^{59} -7.26795 q^{61} -4.46410 q^{63} -6.73205 q^{65} -5.00000 q^{67} +1.00000 q^{69} -10.1244 q^{71} -14.7321 q^{73} +1.00000 q^{75} -27.6603 q^{77} -4.00000 q^{79} +1.00000 q^{81} -10.6603 q^{83} -4.26795 q^{85} -3.19615 q^{87} +4.39230 q^{89} +30.0526 q^{91} +1.00000 q^{93} +2.73205 q^{95} +4.00000 q^{97} +6.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 12 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{31} + 2 q^{33} - 2 q^{35} - 2 q^{37} - 10 q^{39} - 12 q^{41} + 2 q^{45} + 2 q^{47} + 12 q^{49} - 12 q^{51} - 4 q^{53} + 2 q^{55} + 2 q^{57} + 4 q^{59} - 18 q^{61} - 2 q^{63} - 10 q^{65} - 10 q^{67} + 2 q^{69} + 4 q^{71} - 26 q^{73} + 2 q^{75} - 38 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{83} - 12 q^{85} + 4 q^{87} - 12 q^{89} + 22 q^{91} + 2 q^{93} + 2 q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 10 * q^13 + 2 * q^15 - 12 * q^17 + 2 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^31 + 2 * q^33 - 2 * q^35 - 2 * q^37 - 10 * q^39 - 12 * q^41 + 2 * q^45 + 2 * q^47 + 12 * q^49 - 12 * q^51 - 4 * q^53 + 2 * q^55 + 2 * q^57 + 4 * q^59 - 18 * q^61 - 2 * q^63 - 10 * q^65 - 10 * q^67 + 2 * q^69 + 4 * q^71 - 26 * q^73 + 2 * q^75 - 38 * q^77 - 8 * q^79 + 2 * q^81 - 4 * q^83 - 12 * q^85 + 4 * q^87 - 12 * q^89 + 22 * q^91 + 2 * q^93 + 2 * q^95 + 8 * q^97 + 2 * q^99

Introduction

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