LMFDB - Embedded newform 1170.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, 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("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.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:	$9.34249703649$
Analytic rank:	$1$
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	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{32} +8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +6.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} -1.00000 q^{65} +12.0000 q^{67} -8.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} -8.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -8.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000 q^{92} -6.00000 q^{95} -16.0000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−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	$0.706037\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	2.00000	0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	8.00000	1.37199
$35$	2.00000	0.338062
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	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	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−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	$0.583847\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−8.00000	−0.970143
$69$	0	0
$70$	−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	$0.342552\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−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
$81$	0	0
$82$	−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	$0.728845\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	4.00000	0.431331
$87$	0	0
$88$	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	$0.233878\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−6.00000	−0.625543
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	2.00000	0.188982
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	4.00000	0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	−16.0000	−1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	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
$129$	0	0
$130$	1.00000	0.0877058
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	−12.0000	−1.03664
$135$	0	0
$136$	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	$0.417494\pi$
$138$	0	0
$139$	−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
$143$	4.00000	0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	8.00000	0.662085
$147$	0	0
$148$	−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	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	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
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	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	$0.450538\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.00000	0.613572
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−14.0000	−1.04934
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	6.00000	0.442326
$185$	−2.00000	−0.147043
$186$	0	0
$187$	32.0000	2.34007
$188$	0	0
$189$	0	0
$190$	6.00000	0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	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	$0.278427\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−16.0000	−1.12576
$203$	8.00000	0.561490
$204$	0	0
$205$	2.00000	0.139686
$206$	12.0000	0.836080
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	−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
$213$	0	0
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	−12.0000	−0.812743
$219$	0	0
$220$	−4.00000	−0.269680
$221$	8.00000	0.538138
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	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	$0.457621\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	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	$0.673129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−10.0000	−0.640184
$245$	−3.00000	−0.191663
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	10.0000	0.617802
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	12.0000	0.735767
$267$	0	0
$268$	12.0000	0.733017
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−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
$285$	0	0
$286$	−4.00000	−0.236525
$287$	4.00000	0.236113
$288$	0	0
$289$	47.0000	2.76471
$290$	−4.00000	−0.234888
$291$	0	0
$292$	−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	$0.225459\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	2.00000	0.116248
$297$	0	0
$298$	−10.0000	−0.579284
$299$	6.00000	0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−10.0000	−0.572598
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	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
$315$	0	0
$316$	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	$0.181092\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	1.00000	0.0559017
$321$	0	0
$322$	12.0000	0.668734
$323$	48.0000	2.67079
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	16.0000	0.886158
$327$	0	0
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−4.00000	−0.218870
$335$	12.0000	0.655630
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−8.00000	−0.433861
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	−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	$0.378476\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	14.0000	0.741999
$357$	0	0
$358$	−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	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−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
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	4.00000	0.203595
$387$	0	0
$388$	−16.0000	−0.812277
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	3.00000	0.151523
$393$	0	0
$394$	−18.0000	−0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	0	0
$400$	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	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	16.0000	0.796030
$405$	0	0
$406$	−8.00000	−0.397033
$407$	8.00000	0.396545
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	−12.0000	−0.591198
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−12.0000	−0.589057
$416$	1.00000	0.0490290
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−10.0000	−0.485643
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−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	$0.5\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	12.0000	0.574696
$437$	36.0000	1.72211
$438$	0	0
$439$	−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
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	2.00000	0.0947027
$447$	0	0
$448$	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	$0.454782\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	−20.0000	−0.940721
$453$	0	0
$454$	−4.00000	−0.187729
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	−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	$0.455378\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	24.0000	1.11178
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	16.0000	0.735681
$474$	0	0
$475$	−6.00000	−0.275299
$476$	−16.0000	−0.733359
$477$	0	0
$478$	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	$0.760973\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	−16.0000	−0.726523
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	3.00000	0.135526
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	−6.00000	−0.269953
$495$	0	0
$496$	0	0
$497$	16.0000	0.717698
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.00000	0.267793
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	0	0
$505$	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	$0.630614\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	4.00000	0.175750
$519$	0	0
$520$	1.00000	0.0438529
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	2.00000	0.0872041
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−10.0000	−0.434372
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−24.0000	−1.03471
$539$	12.0000	0.516877
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	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
$549$	0	0
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	0	0
$553$	16.0000	0.680389
$554$	10.0000	0.424859
$555$	0	0
$556$	−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
$559$	4.00000	0.169182
$560$	2.00000	0.0845154
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	4.00000	0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−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
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−4.00000	−0.166957
$575$	−6.00000	−0.250217
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	−47.0000	−1.95494
$579$	0	0
$580$	4.00000	0.166091
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−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	$0.766569\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	0	0
$592$	−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	$0.649188\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	10.0000	0.409616
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−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
$609$	0	0
$610$	10.0000	0.404888
$611$	0	0
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	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	$0.461462\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	28.0000	1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	22.0000	0.877896
$629$	16.0000	0.637962
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−30.0000	−1.19145
$635$	4.00000	0.158735
$636$	0	0
$637$	3.00000	0.118864
$638$	16.0000	0.633446
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−48.0000	−1.88853
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	1.00000	0.0392232
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	−2.00000	−0.0779089	−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000	−0.0779089	−0.0389545	+	0.999241i	$0.512403\pi$
$660$	0	0
$661$	48.0000	1.86698	0.933492	−	0.358599i	$-0.116745\pi$
$661$	48.0000	1.86698	0.933492	+	0.358599i	$0.116745\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−24.0000	−0.929284
$668$	4.00000	0.154765
$669$	0	0
$670$	−12.0000	−0.463600
$671$	40.0000	1.54418
$672$	0	0
$673$	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
$675$	0	0
$676$	1.00000	0.0384615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	8.00000	0.306786
$681$	0	0
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−22.0000	−0.836315
$693$	0	0
$694$	−24.0000	−0.911028
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−16.0000	−0.606043
$698$	28.0000	1.05982
$699$	0	0
$700$	2.00000	0.0755929
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	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
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	−8.00000	−0.300235
$711$	0	0
$712$	−14.0000	−0.524672
$713$	0	0
$714$	0	0
$715$	4.00000	0.149592
$716$	10.0000	0.373718
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	−17.0000	−0.632674
$723$	0	0
$724$	10.0000	0.371647
$725$	4.00000	0.148556
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	8.00000	0.296093
$731$	32.0000	1.18356
$732$	0	0
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	6.00000	0.221163
$737$	−48.0000	−1.76810
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	−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
$747$	0	0
$748$	32.0000	1.17004
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−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
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	−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
$759$	0	0
$760$	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	$0.630556\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	4.00000	0.144432
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	−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	$0.739492\pi$
$774$	0	0
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	8.00000	0.286814
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−48.0000	−1.71648
$783$	0	0
$784$	−3.00000	−0.107143
$785$	22.0000	0.785214
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−40.0000	−1.42224
$792$	0	0
$793$	10.0000	0.355110
$794$	−14.0000	−0.496841
$795$	0	0
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	30.0000	1.05934
$803$	32.0000	1.12926
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	−16.0000	−0.562878
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	8.00000	0.280745
$813$	0	0
$814$	−8.00000	−0.280400
$815$	−16.0000	−0.560456
$816$	0	0
$817$	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	$0.444169\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	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	$0.433096\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	24.0000	0.831551
$834$	0	0
$835$	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	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	8.00000	0.275698
$843$	0	0
$844$	−20.0000	−0.688428
$845$	1.00000	0.0344010
$846$	0	0
$847$	10.0000	0.343604
$848$	10.0000	0.343401
$849$	0	0
$850$	8.00000	0.274398
$851$	12.0000	0.411355
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	12.0000	0.410152
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	2.00000	0.0679628
$867$	0	0
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−12.0000	−0.406604
$872$	−12.0000	−0.406371
$873$	0	0
$874$	−36.0000	−1.21772
$875$	2.00000	0.0676123
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	32.0000	1.07995
$879$	0	0
$880$	−4.00000	−0.134840
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	16.0000	0.537531
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−14.0000	−0.469281
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	0	0
$894$	0	0
$895$	10.0000	0.334263
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−6.00000	−0.200223
$899$	0	0
$900$	0	0
$901$	−80.0000	−2.66519
$902$	8.00000	0.266371
$903$	0	0
$904$	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
$909$	0	0
$910$	2.00000	0.0662994
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	−8.00000	−0.264616
$915$	0	0
$916$	4.00000	0.132164
$917$	−20.0000	−0.660458
$918$	0	0
$919$	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
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	26.0000	0.854413
$927$	0	0
$928$	−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
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−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	$0.520702\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	6.00000	0.194666
$951$	0	0
$952$	16.0000	0.518563
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	32.0000	1.03387
$959$	12.0000	0.387500
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	2.00000	0.0644157
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	16.0000	0.513729
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	−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	$0.659324\pi$
$978$	0	0
$979$	−56.0000	−1.78977
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	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	$0.188756\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	32.0000	1.01909
$987$	0	0
$988$	6.00000	0.190885
$989$	24.0000	0.763156
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.e.1.1		1
3.2	odd	2		390.2.a.e.1.1	✓	1
4.3	odd	2		9360.2.a.bh.1.1		1
5.2	odd	4		5850.2.e.i.5149.1		2
5.3	odd	4		5850.2.e.i.5149.2		2
5.4	even	2		5850.2.a.bi.1.1		1
12.11	even	2		3120.2.a.o.1.1		1
15.2	even	4		1950.2.e.f.1249.2		2
15.8	even	4		1950.2.e.f.1249.1		2
15.14	odd	2		1950.2.a.h.1.1		1
39.5	even	4		5070.2.b.e.1351.1		2
39.8	even	4		5070.2.b.e.1351.2		2
39.38	odd	2		5070.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.e.1.1	✓	1	3.2	odd	2
1170.2.a.e.1.1		1	1.1	even	1	trivial
1950.2.a.h.1.1		1	15.14	odd	2
1950.2.e.f.1249.1		2	15.8	even	4
1950.2.e.f.1249.2		2	15.2	even	4
3120.2.a.o.1.1		1	12.11	even	2
5070.2.a.e.1.1		1	39.38	odd	2
5070.2.b.e.1351.1		2	39.5	even	4
5070.2.b.e.1351.2		2	39.8	even	4
5850.2.a.bi.1.1		1	5.4	even	2
5850.2.e.i.5149.1		2	5.2	odd	4
5850.2.e.i.5149.2		2	5.3	odd	4
9360.2.a.bh.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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