LMFDB - Embedded newform 5070.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} -8.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -8.00000 q^{51} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +2.00000 q^{56} -6.00000 q^{57} +4.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -6.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +6.00000 q^{76} +8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +8.00000 q^{85} +4.00000 q^{86} +4.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +6.00000 q^{92} +6.00000 q^{95} +1.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	4.00000	0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	−8.00000	−1.37199
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−8.00000	−1.12022
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	2.00000	0.267261
$57$	−6.00000	−0.794719
$58$	4.00000	0.525226
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	8.00000	0.970143
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	6.00000	0.688247
$77$	8.00000	0.911685
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	8.00000	0.867722
$86$	4.00000	0.431331
$87$	4.00000	0.428845
$88$	4.00000	0.426401
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	0	0
$95$	6.00000	0.615587
$96$	1.00000	0.102062
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	3.00000	0.303046
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	8.00000	0.792118
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	10.0000	0.971286
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	4.00000	0.381385
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	20.0000	1.88144	0.940721	−	0.339182i	$-0.110150\pi$
$113$	20.0000	1.88144	0.940721	+	0.339182i	$0.110150\pi$
$114$	6.00000	0.561951
$115$	6.00000	0.559503
$116$	−4.00000	−0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	−16.0000	−1.46672
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	4.00000	0.348155
$133$	−12.0000	−1.04053
$134$	12.0000	1.03664
$135$	−1.00000	−0.0860663
$136$	−8.00000	−0.685994
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	6.00000	0.510754
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−8.00000	−0.662085
$147$	3.00000	0.247436
$148$	2.00000	0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−6.00000	−0.486664
$153$	8.00000	0.646762
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−8.00000	−0.636446
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	2.00000	0.156174
$165$	4.00000	0.311400
$166$	12.0000	0.931381
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	−8.00000	−0.613572
$171$	6.00000	0.458831
$172$	−4.00000	−0.304997
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	−4.00000	−0.303239
$175$	−2.00000	−0.151186
$176$	−4.00000	−0.301511
$177$	4.00000	0.300658
$178$	−14.0000	−1.04934
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	1.00000	0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	−6.00000	−0.442326
$185$	2.00000	0.147043
$186$	0	0
$187$	−32.0000	−2.34007
$188$	0	0
$189$	2.00000	0.145479
$190$	−6.00000	−0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	12.0000	0.846415
$202$	16.0000	1.12576
$203$	8.00000	0.561490
$204$	−8.00000	−0.560112
$205$	2.00000	0.139686
$206$	12.0000	0.836080
$207$	6.00000	0.417029
$208$	0	0
$209$	−24.0000	−1.66011
$210$	−2.00000	−0.138013
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−10.0000	−0.686803
$213$	−8.00000	−0.548151
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	0	0
$218$	12.0000	0.812743
$219$	−8.00000	−0.540590
$220$	−4.00000	−0.269680
$221$	0	0
$222$	2.00000	0.134231
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−20.0000	−1.33038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−6.00000	−0.397360
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−6.00000	−0.395628
$231$	−8.00000	−0.526361
$232$	4.00000	0.262613
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	16.0000	1.03713
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−1.00000	−0.0645497
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	−2.00000	−0.125988
$253$	−24.0000	−1.50887
$254$	−4.00000	−0.250982
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−4.00000	−0.249029
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−10.0000	−0.617802
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	−4.00000	−0.246183
$265$	−10.0000	−0.614295
$266$	12.0000	0.735767
$267$	−14.0000	−0.856786
$268$	−12.0000	−0.733017
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	1.00000	0.0608581
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	−6.00000	−0.361158
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	2.00000	0.119523
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	−6.00000	−0.355409
$286$	0	0
$287$	−4.00000	−0.236113
$288$	−1.00000	−0.0589256
$289$	47.0000	2.76471
$290$	4.00000	0.234888
$291$	−16.0000	−0.937937
$292$	8.00000	0.468165
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−3.00000	−0.174964
$295$	−4.00000	−0.232889
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	−10.0000	−0.579284
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.00000	0.461112
$302$	0	0
$303$	16.0000	0.919176
$304$	6.00000	0.344124
$305$	−10.0000	−0.572598
$306$	−8.00000	−0.457330
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	8.00000	0.455842
$309$	12.0000	0.682656
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−22.0000	−1.24153
$315$	−2.00000	−0.112687
$316$	8.00000	0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−10.0000	−0.560772
$319$	16.0000	0.895828
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	12.0000	0.668734
$323$	48.0000	2.67079
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	12.0000	0.663602
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−12.0000	−0.658586
$333$	2.00000	0.109599
$334$	−4.00000	−0.218870
$335$	−12.0000	−0.655630
$336$	2.00000	0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	8.00000	0.433861
$341$	0	0
$342$	−6.00000	−0.324443
$343$	20.0000	1.07990
$344$	4.00000	0.215666
$345$	−6.00000	−0.323029
$346$	−22.0000	−1.18273
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	4.00000	0.214423
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−4.00000	−0.212598
$355$	8.00000	0.424596
$356$	14.0000	0.741999
$357$	16.0000	0.846810
$358$	10.0000	0.528516
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	−5.00000	−0.262432
$364$	0	0
$365$	8.00000	0.418739
$366$	−10.0000	−0.522708
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	6.00000	0.312772
$369$	2.00000	0.104116
$370$	−2.00000	−0.103975
$371$	20.0000	1.03835
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	32.0000	1.65468
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	6.00000	0.307794
$381$	−4.00000	−0.204926
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	8.00000	0.407718
$386$	−4.00000	−0.203595
$387$	−4.00000	−0.203331
$388$	16.0000	0.812277
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	3.00000	0.151523
$393$	−10.0000	−0.504433
$394$	−18.0000	−0.906827
$395$	8.00000	0.402524
$396$	−4.00000	−0.201008
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	−16.0000	−0.796030
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	−8.00000	−0.396545
$408$	8.00000	0.396059
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−2.00000	−0.0987730
$411$	−6.00000	−0.295958
$412$	−12.0000	−0.591198
$413$	8.00000	0.393654
$414$	−6.00000	−0.294884
$415$	−12.0000	−0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	24.0000	1.17388
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	2.00000	0.0975900
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	10.0000	0.485643
$425$	8.00000	0.388057
$426$	8.00000	0.387601
$427$	20.0000	0.967868
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	−12.0000	−0.574696
$437$	36.0000	1.72211
$438$	8.00000	0.382255
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	4.00000	0.190693
$441$	−3.00000	−0.142857
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	−2.00000	−0.0949158
$445$	14.0000	0.663664
$446$	−2.00000	−0.0947027
$447$	−10.0000	−0.472984
$448$	−2.00000	−0.0944911
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	−8.00000	−0.376705
$452$	20.0000	0.940721
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	6.00000	0.280976
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	4.00000	0.186908
$459$	−8.00000	−0.373408
$460$	6.00000	0.279751
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	8.00000	0.372194
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−24.0000	−1.11178
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	−22.0000	−1.01371
$472$	4.00000	0.184115
$473$	16.0000	0.735681
$474$	8.00000	0.367452
$475$	6.00000	0.275299
$476$	−16.0000	−0.733359
$477$	−10.0000	−0.457869
$478$	16.0000	0.731823
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	2.00000	0.0910975
$483$	12.0000	0.546019
$484$	5.00000	0.227273
$485$	16.0000	0.726523
$486$	1.00000	0.0453609
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	10.0000	0.452679
$489$	−16.0000	−0.723545
$490$	3.00000	0.135526
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	−2.00000	−0.0901670
$493$	−32.0000	−1.44121
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−16.0000	−0.717698
$498$	−12.0000	−0.537733
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	1.00000	0.0447214
$501$	−4.00000	−0.178707
$502$	−6.00000	−0.267793
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	2.00000	0.0890871
$505$	−16.0000	−0.711991
$506$	24.0000	1.06693
$507$	0	0
$508$	4.00000	0.177471
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	8.00000	0.354246
$511$	−16.0000	−0.707798
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	−12.0000	−0.529297
$515$	−12.0000	−0.528783
$516$	4.00000	0.176090
$517$	0	0
$518$	4.00000	0.175750
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	4.00000	0.175075
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	10.0000	0.436852
$525$	2.00000	0.0872872
$526$	−2.00000	−0.0872041
$527$	0	0
$528$	4.00000	0.174078
$529$	13.0000	0.565217
$530$	10.0000	0.434372
$531$	−4.00000	−0.173585
$532$	−12.0000	−0.520266
$533$	0	0
$534$	14.0000	0.605839
$535$	12.0000	0.518805
$536$	12.0000	0.518321
$537$	10.0000	0.431532
$538$	24.0000	1.03471
$539$	12.0000	0.516877
$540$	−1.00000	−0.0430331
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	16.0000	0.687259
$543$	−10.0000	−0.429141
$544$	−8.00000	−0.342997
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	6.00000	0.256307
$549$	−10.0000	−0.426790
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	6.00000	0.255377
$553$	−16.0000	−0.680389
$554$	10.0000	0.424859
$555$	−2.00000	−0.0848953
$556$	−8.00000	−0.339276
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	32.0000	1.35104
$562$	−6.00000	−0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	4.00000	0.168133
$567$	−2.00000	−0.0839921
$568$	−8.00000	−0.335673
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	6.00000	0.251312
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	4.00000	0.166957
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	−47.0000	−1.95494
$579$	−4.00000	−0.166234
$580$	−4.00000	−0.166091
$581$	24.0000	0.995688
$582$	16.0000	0.663221
$583$	40.0000	1.65663
$584$	−8.00000	−0.331042
$585$	0	0
$586$	−26.0000	−1.07405
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	4.00000	0.164677
$591$	−18.0000	−0.740421
$592$	2.00000	0.0821995
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	−4.00000	−0.164122
$595$	−16.0000	−0.655936
$596$	10.0000	0.409616
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	1.00000	0.0408248
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.00000	−0.326056
$603$	−12.0000	−0.488678
$604$	0	0
$605$	5.00000	0.203279
$606$	−16.0000	−0.649956
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−6.00000	−0.243332
$609$	−8.00000	−0.324176
$610$	10.0000	0.404888
$611$	0	0
$612$	8.00000	0.323381
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	12.0000	0.484281
$615$	−2.00000	−0.0806478
$616$	−8.00000	−0.322329
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−12.0000	−0.482711
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	−24.0000	−0.962312
$623$	−28.0000	−1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	24.0000	0.958468
$628$	22.0000	0.877896
$629$	16.0000	0.637962
$630$	2.00000	0.0796819
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−8.00000	−0.318223
$633$	20.0000	0.794929
$634$	−30.0000	−1.19145
$635$	4.00000	0.158735
$636$	10.0000	0.396526
$637$	0	0
$638$	−16.0000	−0.633446
$639$	8.00000	0.316475
$640$	−1.00000	−0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	−12.0000	−0.472866
$645$	4.00000	0.157500
$646$	−48.0000	−1.88853
$647$	−30.0000	−1.17942	−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000	−1.17942	−0.589711	+	0.807614i	$0.700758\pi$
$648$	−1.00000	−0.0392837
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	16.0000	0.626608
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−12.0000	−0.469237
$655$	10.0000	0.390732
$656$	2.00000	0.0780869
$657$	8.00000	0.312110
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	4.00000	0.155700
$661$	−48.0000	−1.86698	−0.933492	−	0.358599i	$-0.883255\pi$
$661$	−48.0000	−1.86698	−0.933492	+	0.358599i	$0.883255\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	−12.0000	−0.465340
$666$	−2.00000	−0.0774984
$667$	−24.0000	−0.929284
$668$	4.00000	0.154765
$669$	−2.00000	−0.0773245
$670$	12.0000	0.463600
$671$	40.0000	1.54418
$672$	−2.00000	−0.0771517
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−14.0000	−0.539260
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	20.0000	0.768095
$679$	−32.0000	−1.22805
$680$	−8.00000	−0.306786
$681$	−4.00000	−0.153280
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	6.00000	0.229416
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	4.00000	0.152610
$688$	−4.00000	−0.152499
$689$	0	0
$690$	6.00000	0.228416
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	22.0000	0.836315
$693$	8.00000	0.303895
$694$	24.0000	0.911028
$695$	−8.00000	−0.303457
$696$	−4.00000	−0.151620
$697$	16.0000	0.606043
$698$	−28.0000	−1.05982
$699$	−24.0000	−0.907763
$700$	−2.00000	−0.0755929
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	32.0000	1.20348
$708$	4.00000	0.150329
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	−8.00000	−0.300235
$711$	8.00000	0.300023
$712$	−14.0000	−0.524672
$713$	0	0
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−10.0000	−0.373718
$717$	16.0000	0.597531
$718$	−24.0000	−0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000	0.0372678
$721$	24.0000	0.893807
$722$	−17.0000	−0.632674
$723$	2.00000	0.0743808
$724$	10.0000	0.371647
$725$	−4.00000	−0.148556
$726$	5.00000	0.185567
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	−32.0000	−1.18356
$732$	10.0000	0.369611
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	36.0000	1.32878
$735$	3.00000	0.110657
$736$	−6.00000	−0.221163
$737$	48.0000	1.76810
$738$	−2.00000	−0.0736210
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−20.0000	−0.734223
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	2.00000	0.0732252
$747$	−12.0000	−0.439057
$748$	−32.0000	−1.17004
$749$	−24.0000	−0.876941
$750$	1.00000	0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−18.0000	−0.653789
$759$	24.0000	0.871145
$760$	−6.00000	−0.217643
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	4.00000	0.144905
$763$	24.0000	0.868858
$764$	0	0
$765$	8.00000	0.289241
$766$	−12.0000	−0.433578
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	−8.00000	−0.288300
$771$	−12.0000	−0.432169
$772$	4.00000	0.143963
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−16.0000	−0.574367
$777$	4.00000	0.143499
$778$	−8.00000	−0.286814
$779$	12.0000	0.429945
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−48.0000	−1.71648
$783$	4.00000	0.142948
$784$	−3.00000	−0.107143
$785$	22.0000	0.785214
$786$	10.0000	0.356688
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	18.0000	0.641223
$789$	−2.00000	−0.0712019
$790$	−8.00000	−0.284627
$791$	−40.0000	−1.42224
$792$	4.00000	0.142134
$793$	0	0
$794$	14.0000	0.496841
$795$	10.0000	0.354663
$796$	0	0
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−12.0000	−0.424795
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	14.0000	0.494666
$802$	30.0000	1.05934
$803$	−32.0000	−1.12926
$804$	12.0000	0.423207
$805$	−12.0000	−0.422944
$806$	0	0
$807$	24.0000	0.844840
$808$	16.0000	0.562878
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−1.00000	−0.0351364
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	8.00000	0.280745
$813$	16.0000	0.561144
$814$	8.00000	0.280400
$815$	16.0000	0.560456
$816$	−8.00000	−0.280056
$817$	−24.0000	−0.839654
$818$	−26.0000	−0.909069
$819$	0	0
$820$	2.00000	0.0698430
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	6.00000	0.209274
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	12.0000	0.418040
$825$	4.00000	0.139262
$826$	−8.00000	−0.278356
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	6.00000	0.208514
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	12.0000	0.416526
$831$	10.0000	0.346896
$832$	0	0
$833$	−24.0000	−0.831551
$834$	−8.00000	−0.277017
$835$	4.00000	0.138426
$836$	−24.0000	−0.830057
$837$	0	0
$838$	−10.0000	−0.345444
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	−2.00000	−0.0690066
$841$	−13.0000	−0.448276
$842$	−8.00000	−0.275698
$843$	−6.00000	−0.206651
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	−10.0000	−0.343604
$848$	−10.0000	−0.343401
$849$	4.00000	0.137280
$850$	−8.00000	−0.274398
$851$	12.0000	0.411355
$852$	−8.00000	−0.274075
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−20.0000	−0.684386
$855$	6.00000	0.205196
$856$	−12.0000	−0.410152
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	−4.00000	−0.136399
$861$	4.00000	0.136320
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	22.0000	0.748022
$866$	2.00000	0.0679628
$867$	−47.0000	−1.59620
$868$	0	0
$869$	−32.0000	−1.08553
$870$	−4.00000	−0.135613
$871$	0	0
$872$	12.0000	0.406371
$873$	16.0000	0.541518
$874$	−36.0000	−1.21772
$875$	−2.00000	−0.0676123
$876$	−8.00000	−0.270295
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	32.0000	1.07995
$879$	−26.0000	−0.876958
$880$	−4.00000	−0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000	0.101015
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	−16.0000	−0.537531
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	2.00000	0.0671156
$889$	−8.00000	−0.268311
$890$	−14.0000	−0.469281
$891$	−4.00000	−0.134005
$892$	2.00000	0.0669650
$893$	0	0
$894$	10.0000	0.334450
$895$	−10.0000	−0.334263
$896$	2.00000	0.0668153
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	1.00000	0.0333333
$901$	−80.0000	−2.66519
$902$	8.00000	0.266371
$903$	−8.00000	−0.266223
$904$	−20.0000	−0.665190
$905$	10.0000	0.332411
$906$	0	0
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	4.00000	0.132745
$909$	−16.0000	−0.530687
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−6.00000	−0.198680
$913$	48.0000	1.58857
$914$	8.00000	0.264616
$915$	10.0000	0.330590
$916$	−4.00000	−0.132164
$917$	−20.0000	−0.660458
$918$	8.00000	0.264039
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−6.00000	−0.197814
$921$	12.0000	0.395413
$922$	−6.00000	−0.197599
$923$	0	0
$924$	−8.00000	−0.263181
$925$	2.00000	0.0657596
$926$	−26.0000	−0.854413
$927$	−12.0000	−0.394132
$928$	4.00000	0.131306
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	24.0000	0.786146
$933$	−24.0000	−0.785725
$934$	−28.0000	−0.916188
$935$	−32.0000	−1.04651
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	−24.0000	−0.783628
$939$	−6.00000	−0.195803
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	22.0000	0.716799
$943$	12.0000	0.390774
$944$	−4.00000	−0.130189
$945$	2.00000	0.0650600
$946$	−16.0000	−0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	−6.00000	−0.194666
$951$	−30.0000	−0.972817
$952$	16.0000	0.518563
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	−16.0000	−0.517477
$957$	−16.0000	−0.517207
$958$	32.0000	1.03387
$959$	−12.0000	−0.387500
$960$	−1.00000	−0.0322749
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.0000	0.386695
$964$	−2.00000	−0.0644157
$965$	4.00000	0.128765
$966$	−12.0000	−0.386094
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	−5.00000	−0.160706
$969$	−48.0000	−1.54198
$970$	−16.0000	−0.513729
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	26.0000	0.833094
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	16.0000	0.511624
$979$	−56.0000	−1.78977
$980$	−3.00000	−0.0958315
$981$	−12.0000	−0.383131
$982$	−42.0000	−1.34027
$983$	52.0000	1.65854	0.829271	−	0.558846i	$-0.188756\pi$
$983$	52.0000	1.65854	0.829271	+	0.558846i	$0.188756\pi$
$984$	2.00000	0.0637577
$985$	18.0000	0.573528
$986$	32.0000	1.01909
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	4.00000	0.127128
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	16.0000	0.507489
$995$	0	0
$996$	12.0000	0.380235
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	38.0000	1.20287
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.e.1.1		1
13.5	odd	4		5070.2.b.e.1351.2		2
13.8	odd	4		5070.2.b.e.1351.1		2
13.12	even	2		390.2.a.e.1.1	✓	1
39.38	odd	2		1170.2.a.e.1.1		1
52.51	odd	2		3120.2.a.o.1.1		1
65.12	odd	4		1950.2.e.f.1249.2		2
65.38	odd	4		1950.2.e.f.1249.1		2
65.64	even	2		1950.2.a.h.1.1		1
156.155	even	2		9360.2.a.bh.1.1		1
195.38	even	4		5850.2.e.i.5149.2		2
195.77	even	4		5850.2.e.i.5149.1		2
195.194	odd	2		5850.2.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.e.1.1	✓	1	13.12	even	2
1170.2.a.e.1.1		1	39.38	odd	2
1950.2.a.h.1.1		1	65.64	even	2
1950.2.e.f.1249.1		2	65.38	odd	4
1950.2.e.f.1249.2		2	65.12	odd	4
3120.2.a.o.1.1		1	52.51	odd	2
5070.2.a.e.1.1		1	1.1	even	1	trivial
5070.2.b.e.1351.1		2	13.8	odd	4
5070.2.b.e.1351.2		2	13.5	odd	4
5850.2.a.bi.1.1		1	195.194	odd	2
5850.2.e.i.5149.1		2	195.77	even	4
5850.2.e.i.5149.2		2	195.38	even	4
9360.2.a.bh.1.1		1	156.155	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more