LMFDB - Embedded newform 2646.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(1,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, 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("2646.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 378)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2646.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{8} -3.00000 q^{10} +3.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +7.00000 q^{19} +3.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -5.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -7.00000 q^{37} -7.00000 q^{38} -3.00000 q^{40} +9.00000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} -6.00000 q^{47} -4.00000 q^{50} +4.00000 q^{52} +12.0000 q^{53} +9.00000 q^{55} +6.00000 q^{59} -8.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +9.00000 q^{71} -2.00000 q^{73} +7.00000 q^{74} +7.00000 q^{76} -10.0000 q^{79} +3.00000 q^{80} -9.00000 q^{82} +18.0000 q^{85} +10.0000 q^{86} -3.00000 q^{88} -15.0000 q^{89} -3.00000 q^{92} +6.00000 q^{94} +21.0000 q^{95} -8.00000 q^{97} +O(q^{100})$

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	−3.00000	−0.474342
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	4.00000	0.554700
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	7.00000	0.802955
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−9.00000	−0.993884
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	10.0000	1.07833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	0	0
$94$	6.00000	0.618853
$95$	21.0000	2.15455
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−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
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	−9.00000	−0.858116
$111$	0	0
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000	0.724286
$123$	0	0
$124$	−5.00000	−0.449013
$125$	−3.00000	−0.268328
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	−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
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−9.00000	−0.755263
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	−7.00000	−0.575396
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	0	0
$155$	−15.0000	−1.20483
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	−3.00000	−0.237171
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−18.0000	−1.38054
$171$	0	0
$172$	−10.0000	−0.762493
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	15.0000	1.12430
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	−21.0000	−1.54395
$186$	0	0
$187$	18.0000	1.31629
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−21.0000	−1.52350
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	27.0000	1.88576
$206$	5.00000	0.348367
$207$	0	0
$208$	4.00000	0.277350
$209$	21.0000	1.45260
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	−30.0000	−2.04598
$216$	0	0
$217$	0	0
$218$	−11.0000	−0.745014
$219$	0	0
$220$	9.00000	0.606780
$221$	24.0000	1.61441
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	0	0
$226$	18.0000	1.19734
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	9.00000	0.593442
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	28.0000	1.78160
$248$	5.00000	0.317500
$249$	0	0
$250$	3.00000	0.189737
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	6.00000	0.370681
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−12.0000	−0.724947
$275$	12.0000	0.723627
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	−12.0000	−0.709575
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	7.00000	0.406867
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	10.0000	0.575435
$303$	0	0
$304$	7.00000	0.401478
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	0	0
$309$	0	0
$310$	15.0000	0.851943
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	42.0000	2.33694
$324$	0	0
$325$	16.0000	0.887520
$326$	16.0000	0.886158
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	−18.0000	−0.984916
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	18.0000	0.976187
$341$	−15.0000	−0.812296
$342$	0	0
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	−21.0000	−1.12897
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	27.0000	1.43301
$356$	−15.0000	−0.794998
$357$	0	0
$358$	24.0000	1.26844
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	21.0000	1.09174
$371$	0	0
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	21.0000	1.07728
$381$	0	0
$382$	−15.0000	−0.767467
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−8.00000	−0.406138
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	−30.0000	−1.50946
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−7.00000	−0.350878
$399$	0	0
$400$	4.00000	0.200000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	−27.0000	−1.33343
$411$	0	0
$412$	−5.00000	−0.246332
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	−21.0000	−1.02714
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	−12.0000	−0.582772
$425$	24.0000	1.16417
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	30.0000	1.44673
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	0	0
$436$	11.0000	0.526804
$437$	−21.0000	−1.00457
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−9.00000	−0.429058
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	−45.0000	−2.13320
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	27.0000	1.27138
$452$	−18.0000	−0.846649
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	−19.0000	−0.888783	−0.444391	−	0.895833i	$-0.646580\pi$
$457$	−19.0000	−0.888783	−0.444391	+	0.895833i	$0.646580\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	−9.00000	−0.419627
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−30.0000	−1.37940
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	26.0000	1.18427
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−24.0000	−1.08978
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	0	0
$494$	−28.0000	−1.25978
$495$	0	0
$496$	−5.00000	−0.224507
$497$	0	0
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	9.00000	0.400099
$507$	0	0
$508$	20.0000	0.887357
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.0000	−0.926270
$515$	−15.0000	−0.660979
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	0	0
$520$	−12.0000	−0.526235
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	15.0000	0.654031
$527$	−30.0000	−1.30682
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−36.0000	−1.56374
$531$	0	0
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	−36.0000	−1.55642
$536$	4.00000	0.172774
$537$	0	0
$538$	−15.0000	−0.646696
$539$	0	0
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−6.00000	−0.257248
$545$	33.0000	1.41356
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−12.0000	−0.511682
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−17.0000	−0.722261
$555$	0	0
$556$	4.00000	0.169638
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	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
$565$	−54.0000	−2.27180
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−9.00000	−0.377632
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	2.00000	0.0827606
$585$	0	0
$586$	18.0000	0.743573
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−35.0000	−1.44215
$590$	−18.0000	−0.741048
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−33.0000	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$593$	−33.0000	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	12.0000	0.490716
$599$	−33.0000	−1.34834	−0.674172	−	0.738575i	$-0.735499\pi$
$599$	−33.0000	−1.34834	−0.674172	+	0.738575i	$0.735499\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	0	0
$604$	−10.0000	−0.406894
$605$	−6.00000	−0.243935
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	24.0000	0.971732
$611$	−24.0000	−0.970936
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	29.0000	1.17034
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	−15.0000	−0.602414
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−28.0000	−1.11911
$627$	0	0
$628$	4.00000	0.159617
$629$	−42.0000	−1.67465
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	30.0000	1.19145
$635$	60.0000	2.38103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	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
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	0	0
$645$	0	0
$646$	−42.0000	−1.65247
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	−16.0000	−0.627572
$651$	0	0
$652$	−16.0000	−0.626608
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	9.00000	0.351391
$657$	0	0
$658$	0	0
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	18.0000	0.696441
$669$	0	0
$670$	12.0000	0.463600
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	3.00000	0.115385
$677$	−15.0000	−0.576497	−0.288248	−	0.957556i	$-0.593073\pi$
$677$	−15.0000	−0.576497	−0.288248	+	0.957556i	$0.593073\pi$
$678$	0	0
$679$	0	0
$680$	−18.0000	−0.690268
$681$	0	0
$682$	15.0000	0.574380
$683$	−15.0000	−0.573959	−0.286980	−	0.957937i	$-0.592651\pi$
$683$	−15.0000	−0.573959	−0.286980	+	0.957937i	$0.592651\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	0	0
$688$	−10.0000	−0.381246
$689$	48.0000	1.82865
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	−3.00000	−0.113878
$695$	12.0000	0.455186
$696$	0	0
$697$	54.0000	2.04540
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−49.0000	−1.84807
$704$	3.00000	0.113067
$705$	0	0
$706$	21.0000	0.790345
$707$	0	0
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	−27.0000	−1.01329
$711$	0	0
$712$	15.0000	0.562149
$713$	15.0000	0.561754
$714$	0	0
$715$	36.0000	1.34632
$716$	−24.0000	−0.896922
$717$	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
$721$	0	0
$722$	−30.0000	−1.11648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	6.00000	0.222070
$731$	−60.0000	−2.21918
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	3.00000	0.110581
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	13.0000	0.475964
$747$	0	0
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	0	0
$755$	−30.0000	−1.09181
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−2.00000	−0.0726433
$759$	0	0
$760$	−21.0000	−0.761750
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	0	0
$764$	15.0000	0.542681
$765$	0	0
$766$	−30.0000	−1.08394
$767$	24.0000	0.866590
$768$	0	0
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	0	0
$772$	14.0000	0.503871
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	8.00000	0.287183
$777$	0	0
$778$	−12.0000	−0.430221
$779$	63.0000	2.25721
$780$	0	0
$781$	27.0000	0.966136
$782$	18.0000	0.643679
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	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
$790$	30.0000	1.06735
$791$	0	0
$792$	0	0
$793$	−32.0000	−1.13635
$794$	2.00000	0.0709773
$795$	0	0
$796$	7.00000	0.248108
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−24.0000	−0.847469
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	25.0000	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$811$	25.0000	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$812$	0	0
$813$	0	0
$814$	21.0000	0.736050
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−70.0000	−2.44899
$818$	−22.0000	−0.769212
$819$	0	0
$820$	27.0000	0.942881
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	5.00000	0.174183
$825$	0	0
$826$	0	0
$827$	9.00000	0.312961	0.156480	−	0.987681i	$-0.449985\pi$
$827$	9.00000	0.312961	0.156480	+	0.987681i	$0.449985\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	54.0000	1.86875
$836$	21.0000	0.726300
$837$	0	0
$838$	18.0000	0.621800
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−17.0000	−0.585859
$843$	0	0
$844$	14.0000	0.481900
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	12.0000	0.412082
$849$	0	0
$850$	−24.0000	−0.823193
$851$	21.0000	0.719871
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	−30.0000	−1.02299
$861$	0	0
$862$	−3.00000	−0.102180
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	63.0000	2.14206
$866$	−16.0000	−0.543702
$867$	0	0
$868$	0	0
$869$	−30.0000	−1.01768
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−11.0000	−0.372507
$873$	0	0
$874$	21.0000	0.710336
$875$	0	0
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	9.00000	0.303390
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	3.00000	0.100787
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	0	0
$890$	45.0000	1.50840
$891$	0	0
$892$	1.00000	0.0334825
$893$	−42.0000	−1.40548
$894$	0	0
$895$	−72.0000	−2.40669
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	0	0
$900$	0	0
$901$	72.0000	2.39867
$902$	−27.0000	−0.899002
$903$	0	0
$904$	18.0000	0.598671
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	19.0000	0.628464
$915$	0	0
$916$	4.00000	0.132164
$917$	0	0
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	27.0000	0.889198
$923$	36.0000	1.18495
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−14.0000	−0.460069
$927$	0	0
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	6.00000	0.196326
$935$	54.0000	1.76599
$936$	0	0
$937$	−20.0000	−0.653372	−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000	−0.653372	−0.326686	+	0.945133i	$0.605932\pi$
$938$	0	0
$939$	0	0
$940$	−18.0000	−0.587095
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	6.00000	0.195283
$945$	0	0
$946$	30.0000	0.975384
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	−28.0000	−0.908440
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	45.0000	1.45617
$956$	0	0
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	28.0000	0.902756
$963$	0	0
$964$	−26.0000	−0.837404
$965$	42.0000	1.35203
$966$	0	0
$967$	−58.0000	−1.86515	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−58.0000	−1.86515	−0.932577	+	0.360971i	$0.882445\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	24.0000	0.770594
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	0	0
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−8.00000	−0.256074
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	−45.0000	−1.43821
$980$	0	0
$981$	0	0
$982$	9.00000	0.287202
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−54.0000	−1.72058
$986$	0	0
$987$	0	0
$988$	28.0000	0.890799
$989$	30.0000	0.953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	5.00000	0.158750
$993$	0	0
$994$	0	0
$995$	21.0000	0.665745
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	22.0000	0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.n.1.1		1
3.2	odd	2		2646.2.a.q.1.1		1
7.6	odd	2		378.2.a.b.1.1	✓	1
21.20	even	2		378.2.a.g.1.1	yes	1
28.27	even	2		3024.2.a.c.1.1		1
35.34	odd	2		9450.2.a.cu.1.1		1
63.13	odd	6		1134.2.f.o.379.1		2
63.20	even	6		1134.2.f.b.757.1		2
63.34	odd	6		1134.2.f.o.757.1		2
63.41	even	6		1134.2.f.b.379.1		2
84.83	odd	2		3024.2.a.bb.1.1		1
105.104	even	2		9450.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.a.b.1.1	✓	1	7.6	odd	2
378.2.a.g.1.1	yes	1	21.20	even	2
1134.2.f.b.379.1		2	63.41	even	6
1134.2.f.b.757.1		2	63.20	even	6
1134.2.f.o.379.1		2	63.13	odd	6
1134.2.f.o.757.1		2	63.34	odd	6
2646.2.a.n.1.1		1	1.1	even	1	trivial
2646.2.a.q.1.1		1	3.2	odd	2
3024.2.a.c.1.1		1	28.27	even	2
3024.2.a.bb.1.1		1	84.83	odd	2
9450.2.a.h.1.1		1	105.104	even	2
9450.2.a.cu.1.1		1	35.34	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more