LMFDB - Embedded newform 5520.2.a.h.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{15} -3.00000 q^{17} +8.00000 q^{19} -3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.00000 q^{29} +5.00000 q^{31} -4.00000 q^{33} -3.00000 q^{35} -9.00000 q^{37} +7.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +2.00000 q^{47} +2.00000 q^{49} +3.00000 q^{51} +13.0000 q^{53} -4.00000 q^{55} -8.00000 q^{57} +3.00000 q^{59} -14.0000 q^{61} +3.00000 q^{63} -13.0000 q^{67} +1.00000 q^{69} +13.0000 q^{71} -4.00000 q^{73} -1.00000 q^{75} +12.0000 q^{77} +1.00000 q^{81} +1.00000 q^{83} +3.00000 q^{85} -9.00000 q^{87} -8.00000 q^{89} -5.00000 q^{93} -8.00000 q^{95} +10.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	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
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	13.0000	1.78569	0.892844	−	0.450367i	$-0.148707\pi$
$53$	13.0000	1.78569	0.892844	+	0.450367i	$0.148707\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−7.00000	−0.631169
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	−13.0000	−1.03097
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	27.0000	1.89503
$204$	0	0
$205$	−7.00000	−0.488901
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	0	0
$213$	−13.0000	−0.890745
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	15.0000	1.01827
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−27.0000	−1.67770
$260$	0	0
$261$	9.00000	0.557086
$262$	0	0
$263$	23.0000	1.41824	0.709120	−	0.705087i	$-0.249092\pi$
$263$	23.0000	1.41824	0.709120	+	0.705087i	$0.249092\pi$
$264$	0	0
$265$	−13.0000	−0.798584
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	21.0000	1.23959
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	−9.00000	−0.517036
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	11.0000	0.613960
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	−9.00000	−0.493197
$334$	0	0
$335$	13.0000	0.710266
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	−1.00000	−0.0543125
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−13.0000	−0.689968
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−31.0000	−1.61819	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−31.0000	−1.61819	−0.809093	+	0.587680i	$0.800041\pi$
$368$	0	0
$369$	7.00000	0.364405
$370$	0	0
$371$	39.0000	2.02478
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−18.0000	−0.922168
$382$	0	0
$383$	23.0000	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$383$	23.0000	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−36.0000	−1.78445
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	−1.00000	−0.0490881
$416$	0	0
$417$	−3.00000	−0.146911
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−42.0000	−2.03252
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	−37.0000	−1.77811	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−37.0000	−1.77811	−0.889053	+	0.457804i	$0.848636\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	0	0
$447$	14.0000	0.662177
$448$	0	0
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	0	0
$451$	28.0000	1.31847
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.0000	−1.63723	−0.818615	−	0.574342i	$-0.805258\pi$
$457$	−35.0000	−1.63723	−0.818615	+	0.574342i	$0.805258\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	0	0
$469$	−39.0000	−1.80085
$470$	0	0
$471$	3.00000	0.138233
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	13.0000	0.595229
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	−27.0000	−1.21602
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	39.0000	1.74939
$498$	0	0
$499$	−21.0000	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$499$	−21.0000	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$500$	0	0
$501$	−22.0000	−0.982888
$502$	0	0
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−15.0000	−0.653410
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	11.0000	0.475571
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.00000	0.344584
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	72.0000	3.06730
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.00000	−0.382029
$556$	0	0
$557$	21.0000	0.889799	0.444899	−	0.895581i	$-0.353239\pi$
$557$	21.0000	0.889799	0.444899	+	0.895581i	$0.353239\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−1.00000	−0.0420703
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	−2.00000	−0.0835512
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	52.0000	2.15362
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	0	0
$597$	−18.0000	−0.736691
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−41.0000	−1.67242	−0.836212	−	0.548406i	$-0.815235\pi$
$601$	−41.0000	−1.67242	−0.836212	+	0.548406i	$0.815235\pi$
$602$	0	0
$603$	−13.0000	−0.529401
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	0	0
$609$	−27.0000	−1.09410
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	7.00000	0.282267
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−32.0000	−1.27796
$628$	0	0
$629$	27.0000	1.07656
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	0	0
$633$	7.00000	0.278225
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.0000	0.514272
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−15.0000	−0.587896
$652$	0	0
$653$	16.0000	0.626128	0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000	0.626128	0.313064	+	0.949732i	$0.398644\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	−56.0000	−2.16186
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−45.0000	−1.72949	−0.864745	−	0.502211i	$-0.832520\pi$
$677$	−45.0000	−1.72949	−0.864745	+	0.502211i	$0.832520\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	−3.00000	−0.113796
$696$	0	0
$697$	−21.0000	−0.795432
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−72.0000	−2.71553
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	27.0000	1.01544
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.00000	−0.187251
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.0000	0.485494
$718$	0	0
$719$	7.00000	0.261056	0.130528	−	0.991445i	$-0.458333\pi$
$719$	7.00000	0.261056	0.130528	+	0.991445i	$0.458333\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	−20.0000	−0.743808
$724$	0	0
$725$	9.00000	0.334252
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	51.0000	1.88373	0.941864	−	0.335994i	$-0.109072\pi$
$733$	51.0000	1.88373	0.941864	+	0.335994i	$0.109072\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	−52.0000	−1.91544
$738$	0	0
$739$	−27.0000	−0.993211	−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000	−0.993211	−0.496606	+	0.867976i	$0.665420\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	0	0
$747$	1.00000	0.0365881
$748$	0	0
$749$	−33.0000	−1.20579
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	45.0000	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$757$	45.0000	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−41.0000	−1.48625	−0.743124	−	0.669153i	$-0.766657\pi$
$761$	−41.0000	−1.48625	−0.743124	+	0.669153i	$0.766657\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	−46.0000	−1.65451	−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000	−1.65451	−0.827253	+	0.561830i	$0.810097\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	0	0
$777$	27.0000	0.968620
$778$	0	0
$779$	56.0000	2.00641
$780$	0	0
$781$	52.0000	1.86071
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	3.00000	0.107075
$786$	0	0
$787$	27.0000	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$787$	27.0000	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$788$	0	0
$789$	−23.0000	−0.818822
$790$	0	0
$791$	3.00000	0.106668
$792$	0	0
$793$	0	0
$794$	0	0
$795$	13.0000	0.461062
$796$	0	0
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	17.0000	0.598428
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	25.0000	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$811$	25.0000	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$812$	0	0
$813$	−23.0000	−0.806645
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−37.0000	−1.28662	−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000	−1.28662	−0.643308	+	0.765607i	$0.722439\pi$
$828$	0	0
$829$	−27.0000	−0.937749	−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000	−0.937749	−0.468874	+	0.883265i	$0.655340\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−22.0000	−0.761341
$836$	0	0
$837$	−5.00000	−0.172825
$838$	0	0
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	−24.0000	−0.826604
$844$	0	0
$845$	13.0000	0.447214
$846$	0	0
$847$	15.0000	0.515406
$848$	0	0
$849$	−5.00000	−0.171600
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	−21.0000	−0.715678
$862$	0	0
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	−19.0000	−0.640854
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	0	0
$887$	46.0000	1.54453	0.772264	−	0.635301i	$-0.219124\pi$
$887$	46.0000	1.54453	0.772264	+	0.635301i	$0.219124\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.0000	1.50083
$900$	0	0
$901$	−39.0000	−1.29928
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.0000	−1.75984	−0.879918	−	0.475125i	$-0.842403\pi$
$907$	−53.0000	−1.75984	−0.879918	+	0.475125i	$0.842403\pi$
$908$	0	0
$909$	9.00000	0.298511
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	12.0000	0.396275
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−9.00000	−0.295918
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	0	0
$931$	16.0000	0.524379
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−25.0000	−0.815844
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	−7.00000	−0.227951
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	−2.00000	−0.0647185
$956$	0	0
$957$	−36.0000	−1.16371
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−11.0000	−0.354470
$964$	0	0
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−10.0000	−0.321578	−0.160789	−	0.986989i	$-0.551404\pi$
$967$	−10.0000	−0.321578	−0.160789	+	0.986989i	$0.551404\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.0000	−0.799821	−0.399910	−	0.916554i	$-0.630959\pi$
$977$	−25.0000	−0.799821	−0.399910	+	0.916554i	$0.630959\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	−15.0000	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	−18.0000	−0.570638
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.h.1.1		1
4.3	odd	2		345.2.a.d.1.1	✓	1
12.11	even	2		1035.2.a.d.1.1		1
20.3	even	4		1725.2.b.j.1174.2		2
20.7	even	4		1725.2.b.j.1174.1		2
20.19	odd	2		1725.2.a.k.1.1		1
60.59	even	2		5175.2.a.o.1.1		1
92.91	even	2		7935.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.d.1.1	✓	1	4.3	odd	2
1035.2.a.d.1.1		1	12.11	even	2
1725.2.a.k.1.1		1	20.19	odd	2
1725.2.b.j.1174.1		2	20.7	even	4
1725.2.b.j.1174.2		2	20.3	even	4
5175.2.a.o.1.1		1	60.59	even	2
5520.2.a.h.1.1		1	1.1	even	1	trivial
7935.2.a.i.1.1		1	92.91	even	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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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