LMFDB - Embedded newform 1215.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1215,2,Mod(1,1215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1215, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1215.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1215 = 3^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1215.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:	$9.70182384559$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{7} +3.00000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} -5.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +5.00000 q^{26} -3.00000 q^{28} +4.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} +4.00000 q^{34} +3.00000 q^{35} +5.00000 q^{37} -1.00000 q^{38} +3.00000 q^{40} -8.00000 q^{41} +7.00000 q^{43} +4.00000 q^{44} +6.00000 q^{46} -10.0000 q^{47} +2.00000 q^{49} -1.00000 q^{50} +5.00000 q^{52} +8.00000 q^{53} -4.00000 q^{55} +9.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} -14.0000 q^{61} -3.00000 q^{62} +7.00000 q^{64} -5.00000 q^{65} -12.0000 q^{67} +4.00000 q^{68} -3.00000 q^{70} -14.0000 q^{71} -2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{76} -12.0000 q^{77} +9.00000 q^{79} -1.00000 q^{80} +8.00000 q^{82} -12.0000 q^{83} -4.00000 q^{85} -7.00000 q^{86} -12.0000 q^{88} -6.00000 q^{89} -15.0000 q^{91} +6.00000 q^{92} +10.0000 q^{94} +1.00000 q^{95} -1.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	3.00000	1.06066
$9$	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
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−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
$26$	5.00000	0.980581
$27$	0	0
$28$	−3.00000	−0.566947
$29$	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
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	3.00000	0.507093
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	6.00000	0.884652
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.00000	0.693375
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	9.00000	1.20268
$57$	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
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−3.00000	−0.381000
$63$	0	0
$64$	7.00000	0.875000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−12.0000	−1.36753
$78$	0	0
$79$	9.00000	1.01258	0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000	1.01258	0.506290	+	0.862364i	$0.331017\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.00000	0.883452
$83$	−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
$85$	−4.00000	−0.433861
$86$	−7.00000	−0.754829
$87$	0	0
$88$	−12.0000	−1.27920
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−15.0000	−1.57243
$92$	6.00000	0.625543
$93$	0	0
$94$	10.0000	1.03142
$95$	1.00000	0.102598
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−15.0000	−1.47087
$105$	0	0
$106$	−8.00000	−0.777029
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−3.00000	−0.283473
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−4.00000	−0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	−12.0000	−1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	0	0
$124$	−3.00000	−0.269408
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	5.00000	0.438529
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	12.0000	1.03664
$135$	0	0
$136$	−12.0000	−1.02899
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	14.0000	1.17485
$143$	20.0000	1.67248
$144$	0	0
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	0	0
$148$	−5.00000	−0.410997
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	12.0000	0.966988
$155$	3.00000	0.240966
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−9.00000	−0.716002
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−18.0000	−1.41860
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	12.0000	0.931381
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	4.00000	0.306786
$171$	0	0
$172$	−7.00000	−0.533745
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	4.00000	0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	15.0000	1.11187
$183$	0	0
$184$	−18.0000	−1.32698
$185$	5.00000	0.367607
$186$	0	0
$187$	16.0000	1.17004
$188$	10.0000	0.729325
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	1.00000	0.0717958
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	6.00000	0.422159
$203$	12.0000	0.842235
$204$	0	0
$205$	−8.00000	−0.558744
$206$	−8.00000	−0.557386
$207$	0	0
$208$	5.00000	0.346688
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−8.00000	−0.549442
$213$	0	0
$214$	−12.0000	−0.820303
$215$	7.00000	0.477396
$216$	0	0
$217$	9.00000	0.610960
$218$	19.0000	1.28684
$219$	0	0
$220$	4.00000	0.269680
$221$	20.0000	1.34535
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	12.0000	0.787839
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	4.00000	0.260378
$237$	0	0
$238$	12.0000	0.777844
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	14.0000	0.896258
$245$	2.00000	0.127775
$246$	0	0
$247$	−5.00000	−0.318142
$248$	9.00000	0.571501
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−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
$259$	15.0000	0.932055
$260$	5.00000	0.310087
$261$	0	0
$262$	6.00000	0.370681
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	−3.00000	−0.183942
$267$	0	0
$268$	12.0000	0.733017
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−4.00000	−0.241209
$276$	0	0
$277$	27.0000	1.62227	0.811136	−	0.584857i	$-0.198849\pi$
$277$	27.0000	1.62227	0.811136	+	0.584857i	$0.198849\pi$
$278$	−5.00000	−0.299880
$279$	0	0
$280$	9.00000	0.537853
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	15.0000	0.891657	0.445829	−	0.895118i	$-0.352909\pi$
$283$	15.0000	0.891657	0.445829	+	0.895118i	$0.352909\pi$
$284$	14.0000	0.830747
$285$	0	0
$286$	−20.0000	−1.18262
$287$	−24.0000	−1.41668
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−4.00000	−0.234888
$291$	0	0
$292$	2.00000	0.117041
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	15.0000	0.871857
$297$	0	0
$298$	−22.0000	−1.27443
$299$	30.0000	1.73494
$300$	0	0
$301$	21.0000	1.21042
$302$	20.0000	1.15087
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	−14.0000	−0.801638
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	−3.00000	−0.170389
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−9.00000	−0.506290
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	7.00000	0.391312
$321$	0	0
$322$	18.0000	1.00310
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−5.00000	−0.277350
$326$	−11.0000	−0.609234
$327$	0	0
$328$	−24.0000	−1.32518
$329$	−30.0000	−1.65395
$330$	0	0
$331$	27.0000	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$331$	27.0000	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−6.00000	−0.328305
$335$	−12.0000	−0.655630
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	4.00000	0.216930
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−15.0000	−0.809924
$344$	21.0000	1.13224
$345$	0	0
$346$	6.00000	0.322562
$347$	14.0000	0.751559	0.375780	−	0.926709i	$-0.377375\pi$
$347$	14.0000	0.751559	0.375780	+	0.926709i	$0.377375\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	20.0000	1.06600
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	−14.0000	−0.743043
$356$	6.00000	0.317999
$357$	0	0
$358$	10.0000	0.528516
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	19.0000	0.998618
$363$	0	0
$364$	15.0000	0.786214
$365$	−2.00000	−0.104685
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−5.00000	−0.259938
$371$	24.0000	1.24602
$372$	0	0
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	−30.0000	−1.54713
$377$	−20.0000	−1.03005
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	−1.00000	−0.0512989
$381$	0	0
$382$	−10.0000	−0.511645
$383$	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
$385$	−12.0000	−0.611577
$386$	−2.00000	−0.101797
$387$	0	0
$388$	1.00000	0.0507673
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	6.00000	0.303046
$393$	0	0
$394$	12.0000	0.604551
$395$	9.00000	0.452839
$396$	0	0
$397$	25.0000	1.25471	0.627357	−	0.778732i	$-0.284137\pi$
$397$	25.0000	1.25471	0.627357	+	0.778732i	$0.284137\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	6.00000	0.298511
$405$	0	0
$406$	−12.0000	−0.595550
$407$	−20.0000	−0.991363
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	8.00000	0.395092
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−12.0000	−0.589057
$416$	25.0000	1.22573
$417$	0	0
$418$	4.00000	0.195646
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	1.00000	0.0486792
$423$	0	0
$424$	24.0000	1.16554
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−42.0000	−2.03252
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−7.00000	−0.337570
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	19.0000	0.909935
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−12.0000	−0.572078
$441$	0	0
$442$	−20.0000	−0.951303
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	19.0000	0.899676
$447$	0	0
$448$	21.0000	0.992157
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	32.0000	1.50682
$452$	−14.0000	−0.658505
$453$	0	0
$454$	14.0000	0.657053
$455$	−15.0000	−0.703211
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	15.0000	0.700904
$459$	0	0
$460$	6.00000	0.279751
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−15.0000	−0.697109	−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000	−0.697109	−0.348555	+	0.937288i	$0.613327\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	18.0000	0.833834
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	10.0000	0.461266
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−28.0000	−1.28744
$474$	0	0
$475$	1.00000	0.0458831
$476$	12.0000	0.550019
$477$	0	0
$478$	−26.0000	−1.18921
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	23.0000	1.04762
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−1.00000	−0.0454077
$486$	0	0
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	−42.0000	−1.90125
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	10.0000	0.451294	0.225647	−	0.974209i	$-0.427550\pi$
$491$	10.0000	0.451294	0.225647	+	0.974209i	$0.427550\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	5.00000	0.224961
$495$	0	0
$496$	−3.00000	−0.134704
$497$	−42.0000	−1.88396
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−24.0000	−1.06693
$507$	0	0
$508$	−1.00000	−0.0443678
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	11.0000	0.486136
$513$	0	0
$514$	12.0000	0.529297
$515$	8.00000	0.352522
$516$	0	0
$517$	40.0000	1.75920
$518$	−15.0000	−0.659062
$519$	0	0
$520$	−15.0000	−0.657794
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	20.0000	0.872041
$527$	−12.0000	−0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	−8.00000	−0.347498
$531$	0	0
$532$	−3.00000	−0.130066
$533$	40.0000	1.73259
$534$	0	0
$535$	12.0000	0.518805
$536$	−36.0000	−1.55496
$537$	0	0
$538$	16.0000	0.689809
$539$	−8.00000	−0.344584
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	20.0000	0.857493
$545$	−19.0000	−0.813871
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	4.00000	0.170561
$551$	4.00000	0.170406
$552$	0	0
$553$	27.0000	1.14816
$554$	−27.0000	−1.14712
$555$	0	0
$556$	−5.00000	−0.212047
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−35.0000	−1.48034
$560$	−3.00000	−0.126773
$561$	0	0
$562$	6.00000	0.253095
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	−15.0000	−0.630497
$567$	0	0
$568$	−42.0000	−1.76228
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	−20.0000	−0.836242
$573$	0	0
$574$	24.0000	1.00174
$575$	−6.00000	−0.250217
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−4.00000	−0.166091
$581$	−36.0000	−1.49353
$582$	0	0
$583$	−32.0000	−1.32530
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	4.00000	0.164677
$591$	0	0
$592$	−5.00000	−0.205499
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	−22.0000	−0.901155
$597$	0	0
$598$	−30.0000	−1.22679
$599$	22.0000	0.898896	0.449448	−	0.893307i	$-0.351621\pi$
$599$	22.0000	0.898896	0.449448	+	0.893307i	$0.351621\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	−21.0000	−0.855896
$603$	0	0
$604$	20.0000	0.813788
$605$	5.00000	0.203279
$606$	0	0
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	14.0000	0.566843
$611$	50.0000	2.02278
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−36.0000	−1.45048
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−18.0000	−0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	−29.0000	−1.15907
$627$	0	0
$628$	7.00000	0.279330
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	27.0000	1.07400
$633$	0	0
$634$	−4.00000	−0.158860
$635$	1.00000	0.0396838
$636$	0	0
$637$	−10.0000	−0.396214
$638$	16.0000	0.633446
$639$	0	0
$640$	3.00000	0.118585
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	4.00000	0.157378
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	5.00000	0.196116
$651$	0	0
$652$	−11.0000	−0.430793
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	8.00000	0.312348
$657$	0	0
$658$	30.0000	1.16952
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−27.0000	−1.04938
$663$	0	0
$664$	−36.0000	−1.39707
$665$	3.00000	0.116335
$666$	0	0
$667$	−24.0000	−0.929284
$668$	−6.00000	−0.232147
$669$	0	0
$670$	12.0000	0.463600
$671$	56.0000	2.16186
$672$	0	0
$673$	9.00000	0.346925	0.173462	−	0.984841i	$-0.444505\pi$
$673$	9.00000	0.346925	0.173462	+	0.984841i	$0.444505\pi$
$674$	−7.00000	−0.269630
$675$	0	0
$676$	−12.0000	−0.461538
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	−12.0000	−0.460179
$681$	0	0
$682$	12.0000	0.459504
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	15.0000	0.572703
$687$	0	0
$688$	−7.00000	−0.266872
$689$	−40.0000	−1.52388
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−14.0000	−0.531433
$695$	5.00000	0.189661
$696$	0	0
$697$	32.0000	1.21209
$698$	−19.0000	−0.719161
$699$	0	0
$700$	−3.00000	−0.113389
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−12.0000	−0.451626
$707$	−18.0000	−0.676960
$708$	0	0
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	14.0000	0.525411
$711$	0	0
$712$	−18.0000	−0.674579
$713$	−18.0000	−0.674105
$714$	0	0
$715$	20.0000	0.747958
$716$	10.0000	0.373718
$717$	0	0
$718$	−12.0000	−0.447836
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	18.0000	0.669891
$723$	0	0
$724$	19.0000	0.706129
$725$	4.00000	0.148556
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−45.0000	−1.66781
$729$	0	0
$730$	2.00000	0.0740233
$731$	−28.0000	−1.03562
$732$	0	0
$733$	−49.0000	−1.80986	−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000	−1.80986	−0.904928	+	0.425564i	$0.860076\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	30.0000	1.10581
$737$	48.0000	1.76810
$738$	0	0
$739$	−5.00000	−0.183928	−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000	−0.183928	−0.0919640	+	0.995762i	$0.529314\pi$
$740$	−5.00000	−0.183804
$741$	0	0
$742$	−24.0000	−0.881068
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	0	0
$745$	22.0000	0.806018
$746$	−1.00000	−0.0366126
$747$	0	0
$748$	−16.0000	−0.585018
$749$	36.0000	1.31541
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	10.0000	0.364662
$753$	0	0
$754$	20.0000	0.728357
$755$	−20.0000	−0.727875
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−19.0000	−0.690111
$759$	0	0
$760$	3.00000	0.108821
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−57.0000	−2.06354
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−12.0000	−0.433578
$767$	20.0000	0.722158
$768$	0	0
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	12.0000	0.432450
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	−3.00000	−0.107694
$777$	0	0
$778$	12.0000	0.430221
$779$	−8.00000	−0.286630
$780$	0	0
$781$	56.0000	2.00384
$782$	−24.0000	−0.858238
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	−7.00000	−0.249841
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	−9.00000	−0.320206
$791$	42.0000	1.49335
$792$	0	0
$793$	70.0000	2.48577
$794$	−25.0000	−0.887217
$795$	0	0
$796$	−7.00000	−0.248108
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	−5.00000	−0.176777
$801$	0	0
$802$	12.0000	0.423735
$803$	8.00000	0.282314
$804$	0	0
$805$	−18.0000	−0.634417
$806$	15.0000	0.528352
$807$	0	0
$808$	−18.0000	−0.633238
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−15.0000	−0.526721	−0.263361	−	0.964697i	$-0.584831\pi$
$811$	−15.0000	−0.526721	−0.263361	+	0.964697i	$0.584831\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	20.0000	0.701000
$815$	11.0000	0.385313
$816$	0	0
$817$	7.00000	0.244899
$818$	−5.00000	−0.174821
$819$	0	0
$820$	8.00000	0.279372
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	12.0000	0.417533
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−35.0000	−1.21341
$833$	−8.00000	−0.277184
$834$	0	0
$835$	6.00000	0.207639
$836$	4.00000	0.138343
$837$	0	0
$838$	2.00000	0.0690889
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	35.0000	1.20618
$843$	0	0
$844$	1.00000	0.0344214
$845$	12.0000	0.412813
$846$	0	0
$847$	15.0000	0.515406
$848$	−8.00000	−0.274721
$849$	0	0
$850$	4.00000	0.137199
$851$	−30.0000	−1.02839
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	36.0000	1.23045
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	−7.00000	−0.238698
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	11.0000	0.373795
$867$	0	0
$868$	−9.00000	−0.305480
$869$	−36.0000	−1.22122
$870$	0	0
$871$	60.0000	2.03302
$872$	−57.0000	−1.93026
$873$	0	0
$874$	6.00000	0.202953
$875$	3.00000	0.101419
$876$	0	0
$877$	21.0000	0.709120	0.354560	−	0.935033i	$-0.384631\pi$
$877$	21.0000	0.709120	0.354560	+	0.935033i	$0.384631\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	4.00000	0.134840
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	−20.0000	−0.672673
$885$	0	0
$886$	−30.0000	−1.00787
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	6.00000	0.201120
$891$	0	0
$892$	19.0000	0.636167
$893$	−10.0000	−0.334637
$894$	0	0
$895$	−10.0000	−0.334263
$896$	9.00000	0.300669
$897$	0	0
$898$	−32.0000	−1.06785
$899$	12.0000	0.400222
$900$	0	0
$901$	−32.0000	−1.06607
$902$	−32.0000	−1.06548
$903$	0	0
$904$	42.0000	1.39690
$905$	−19.0000	−0.631581
$906$	0	0
$907$	−33.0000	−1.09575	−0.547874	−	0.836561i	$-0.684562\pi$
$907$	−33.0000	−1.09575	−0.547874	+	0.836561i	$0.684562\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	15.0000	0.497245
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	17.0000	0.562310
$915$	0	0
$916$	15.0000	0.495614
$917$	−18.0000	−0.594412
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	30.0000	0.987997
$923$	70.0000	2.30408
$924$	0	0
$925$	5.00000	0.164399
$926$	15.0000	0.492931
$927$	0	0
$928$	−20.0000	−0.656532
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	18.0000	0.589610
$933$	0	0
$934$	−34.0000	−1.11251
$935$	16.0000	0.523256
$936$	0	0
$937$	−3.00000	−0.0980057	−0.0490029	−	0.998799i	$-0.515604\pi$
$937$	−3.00000	−0.0980057	−0.0490029	+	0.998799i	$0.515604\pi$
$938$	36.0000	1.17544
$939$	0	0
$940$	10.0000	0.326164
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	4.00000	0.130189
$945$	0	0
$946$	28.0000	0.910359
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	−1.00000	−0.0324443
$951$	0	0
$952$	−36.0000	−1.16677
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	−26.0000	−0.840900
$957$	0	0
$958$	30.0000	0.969256
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	25.0000	0.806032
$963$	0	0
$964$	23.0000	0.740780
$965$	2.00000	0.0643823
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	1.00000	0.0321081
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	−17.0000	−0.544715
$975$	0	0
$976$	14.0000	0.448129
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	−10.0000	−0.319113
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	16.0000	0.509544
$987$	0	0
$988$	5.00000	0.159071
$989$	−42.0000	−1.33552
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−15.0000	−0.476250
$993$	0	0
$994$	42.0000	1.33216
$995$	7.00000	0.221915
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1215.2.a.c.1.1	✓	1
3.2	odd	2		1215.2.a.i.1.1	yes	1
5.4	even	2		6075.2.a.z.1.1		1
9.2	odd	6		1215.2.e.b.811.1		2
9.4	even	3		1215.2.e.h.406.1		2
9.5	odd	6		1215.2.e.b.406.1		2
9.7	even	3		1215.2.e.h.811.1		2
15.14	odd	2		6075.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1215.2.a.c.1.1	✓	1	1.1	even	1	trivial
1215.2.a.i.1.1	yes	1	3.2	odd	2
1215.2.e.b.406.1		2	9.5	odd	6
1215.2.e.b.811.1		2	9.2	odd	6
1215.2.e.h.406.1		2	9.4	even	3
1215.2.e.h.811.1		2	9.7	even	3
6075.2.a.h.1.1		1	15.14	odd	2
6075.2.a.z.1.1		1	5.4	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:	\( N \)	\(=\)	\( 1215 = 3^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1215.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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