LMFDB - Embedded newform 3822.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3822.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} -3.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +2.00000 q^{45} +1.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} +5.00000 q^{51} +1.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +1.00000 q^{57} +1.00000 q^{58} -3.00000 q^{59} -2.00000 q^{60} +5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} +5.00000 q^{67} -5.00000 q^{68} -1.00000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} +1.00000 q^{78} -6.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -16.0000 q^{83} -10.0000 q^{85} +10.0000 q^{86} +1.00000 q^{87} -3.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{93} -1.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -4.00000 q^{97} +3.00000 q^{99} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	2.00000	0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	5.00000	0.857493
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	−2.00000	−0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	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
$45$	2.00000	0.298142
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	5.00000	0.700140
$52$	1.00000	0.138675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	1.00000	0.132453
$58$	1.00000	0.131306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	−2.00000	−0.258199
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	3.00000	0.369274
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	−1.00000	−0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	1.00000	0.113228
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	10.0000	1.07833
$87$	1.00000	0.107211
$88$	−3.00000	−0.319801
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	−1.00000	−0.103142
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−5.00000	−0.495074
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	3.00000	0.291386
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−6.00000	−0.572078
$111$	2.00000	0.189832
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	1.00000	0.0924500
$118$	3.00000	0.276172
$119$	0	0
$120$	2.00000	0.182574
$121$	−2.00000	−0.181818
$122$	−5.00000	−0.452679
$123$	10.0000	0.901670
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	10.0000	0.880451
$130$	−2.00000	−0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	−5.00000	−0.431934
$135$	−2.00000	−0.172133
$136$	5.00000	0.428746
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	1.00000	0.0839181
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	12.0000	0.993127
$147$	0	0
$148$	−2.00000	−0.164399
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	−1.00000	−0.0816497
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	1.00000	0.0811107
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−1.00000	−0.0800641
$157$	−21.0000	−1.67598	−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000	−1.67598	−0.837991	+	0.545684i	$0.816270\pi$
$158$	6.00000	0.477334
$159$	3.00000	0.237915
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−10.0000	−0.780869
$165$	−6.00000	−0.467099
$166$	16.0000	1.24184
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	10.0000	0.766965
$171$	−1.00000	−0.0764719
$172$	−10.0000	−0.762493
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	3.00000	0.226134
$177$	3.00000	0.225494
$178$	−14.0000	−1.04934
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	2.00000	0.149071
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	0	0
$185$	−4.00000	−0.294086
$186$	−4.00000	−0.293294
$187$	−15.0000	−1.09691
$188$	1.00000	0.0729325
$189$	0	0
$190$	2.00000	0.145095
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	4.00000	0.287183
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	−3.00000	−0.213201
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	1.00000	0.0707107
$201$	−5.00000	−0.352673
$202$	2.00000	0.140720
$203$	0	0
$204$	5.00000	0.350070
$205$	−20.0000	−1.39686
$206$	−2.00000	−0.139347
$207$	0	0
$208$	1.00000	0.0693375
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−3.00000	−0.206041
$213$	1.00000	0.0685189
$214$	−12.0000	−0.820303
$215$	−20.0000	−1.36399
$216$	1.00000	0.0680414
$217$	0	0
$218$	16.0000	1.08366
$219$	12.0000	0.810885
$220$	6.00000	0.404520
$221$	−5.00000	−0.336336
$222$	−2.00000	−0.134231
$223$	−11.0000	−0.736614	−0.368307	−	0.929704i	$-0.620063\pi$
$223$	−11.0000	−0.736614	−0.368307	+	0.929704i	$0.620063\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−15.0000	−0.997785
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	1.00000	0.0662266
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	1.00000	0.0656532
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	−1.00000	−0.0653720
$235$	2.00000	0.130466
$236$	−3.00000	−0.195283
$237$	6.00000	0.389742
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	−2.00000	−0.129099
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	5.00000	0.320092
$245$	0	0
$246$	−10.0000	−0.637577
$247$	−1.00000	−0.0636285
$248$	4.00000	0.254000
$249$	16.0000	1.01396
$250$	12.0000	0.758947
$251$	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
$253$	0	0
$254$	0	0
$255$	10.0000	0.626224
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−10.0000	−0.622573
$259$	0	0
$260$	2.00000	0.124035
$261$	−1.00000	−0.0618984
$262$	−4.00000	−0.247121
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	3.00000	0.184637
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−14.0000	−0.856786
$268$	5.00000	0.305424
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	2.00000	0.121716
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−27.0000	−1.62227	−0.811136	−	0.584857i	$-0.801151\pi$
$277$	−27.0000	−1.62227	−0.811136	+	0.584857i	$0.801151\pi$
$278$	2.00000	0.119952
$279$	−4.00000	−0.239474
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	1.00000	0.0595491
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−1.00000	−0.0593391
$285$	2.00000	0.118470
$286$	−3.00000	−0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	2.00000	0.117444
$291$	4.00000	0.234484
$292$	−12.0000	−0.702247
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	2.00000	0.116248
$297$	−3.00000	−0.174078
$298$	−8.00000	−0.463428
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−19.0000	−1.09333
$303$	2.00000	0.114897
$304$	−1.00000	−0.0573539
$305$	10.0000	0.572598
$306$	5.00000	0.285831
$307$	27.0000	1.54097	0.770486	−	0.637457i	$-0.220014\pi$
$307$	27.0000	1.54097	0.770486	+	0.637457i	$0.220014\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	8.00000	0.454369
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	1.00000	0.0566139
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	21.0000	1.18510
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−3.00000	−0.168232
$319$	−3.00000	−0.167968
$320$	2.00000	0.111803
$321$	−12.0000	−0.669775
$322$	0	0
$323$	5.00000	0.278207
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−19.0000	−1.05231
$327$	16.0000	0.884802
$328$	10.0000	0.552158
$329$	0	0
$330$	6.00000	0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−16.0000	−0.878114
$333$	−2.00000	−0.109599
$334$	21.0000	1.14907
$335$	10.0000	0.546358
$336$	0	0
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	−1.00000	−0.0543928
$339$	−15.0000	−0.814688
$340$	−10.0000	−0.542326
$341$	−12.0000	−0.649836
$342$	1.00000	0.0540738
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	3.00000	0.161281
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	1.00000	0.0536056
$349$	−36.0000	−1.92704	−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000	−1.92704	−0.963518	+	0.267644i	$0.913755\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−3.00000	−0.159448
$355$	−2.00000	−0.106149
$356$	14.0000	0.741999
$357$	0	0
$358$	−2.00000	−0.105703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	−18.0000	−0.947368
$362$	3.00000	0.157676
$363$	2.00000	0.104973
$364$	0	0
$365$	−24.0000	−1.25622
$366$	5.00000	0.261354
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	4.00000	0.207950
$371$	0	0
$372$	4.00000	0.207390
$373$	13.0000	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$374$	15.0000	0.775632
$375$	12.0000	0.619677
$376$	−1.00000	−0.0515711
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	−20.0000	−1.02329
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	−10.0000	−0.508329
$388$	−4.00000	−0.203069
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	24.0000	1.20910
$395$	−12.0000	−0.603786
$396$	3.00000	0.150756
$397$	−36.0000	−1.80679	−0.903394	−	0.428811i	$-0.858933\pi$
$397$	−36.0000	−1.80679	−0.903394	+	0.428811i	$0.858933\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	5.00000	0.249377
$403$	−4.00000	−0.199254
$404$	−2.00000	−0.0995037
$405$	2.00000	0.0993808
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−5.00000	−0.247537
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	20.0000	0.987730
$411$	−18.0000	−0.887875
$412$	2.00000	0.0985329
$413$	0	0
$414$	0	0
$415$	−32.0000	−1.57082
$416$	−1.00000	−0.0490290
$417$	2.00000	0.0979404
$418$	3.00000	0.146735
$419$	−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	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	16.0000	0.778868
$423$	1.00000	0.0486217
$424$	3.00000	0.145693
$425$	5.00000	0.242536
$426$	−1.00000	−0.0484502
$427$	0	0
$428$	12.0000	0.580042
$429$	−3.00000	−0.144841
$430$	20.0000	0.964486
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	−16.0000	−0.766261
$437$	0	0
$438$	−12.0000	−0.573382
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	5.00000	0.237826
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	2.00000	0.0949158
$445$	28.0000	1.32733
$446$	11.0000	0.520865
$447$	−8.00000	−0.378387
$448$	0	0
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	1.00000	0.0471405
$451$	−30.0000	−1.41264
$452$	15.0000	0.705541
$453$	−19.0000	−0.892698
$454$	8.00000	0.375459
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	6.00000	0.280362
$459$	5.00000	0.233380
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−1.00000	−0.0464238
$465$	8.00000	0.370991
$466$	−11.0000	−0.509565
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	−2.00000	−0.0922531
$471$	21.0000	0.967629
$472$	3.00000	0.138086
$473$	−30.0000	−1.37940
$474$	−6.00000	−0.275589
$475$	1.00000	0.0458831
$476$	0	0
$477$	−3.00000	−0.137361
$478$	−21.0000	−0.960518
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	2.00000	0.0912871
$481$	−2.00000	−0.0911922
$482$	4.00000	0.182195
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−8.00000	−0.363261
$486$	1.00000	0.0453609
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	−5.00000	−0.226339
$489$	−19.0000	−0.859210
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	10.0000	0.450835
$493$	5.00000	0.225189
$494$	1.00000	0.0449921
$495$	6.00000	0.269680
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−16.0000	−0.716977
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−12.0000	−0.536656
$501$	21.0000	0.938211
$502$	−18.0000	−0.803379
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	−10.0000	−0.442807
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	6.00000	0.264649
$515$	4.00000	0.176261
$516$	10.0000	0.440225
$517$	3.00000	0.131940
$518$	0	0
$519$	3.00000	0.131685
$520$	−2.00000	−0.0877058
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	1.00000	0.0437688
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	6.00000	0.261612
$527$	20.0000	0.871214
$528$	−3.00000	−0.130558
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	−3.00000	−0.130189
$532$	0	0
$533$	−10.0000	−0.433148
$534$	14.0000	0.605839
$535$	24.0000	1.03761
$536$	−5.00000	−0.215967
$537$	−2.00000	−0.0863064
$538$	−9.00000	−0.388018
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	11.0000	0.472490
$543$	3.00000	0.128742
$544$	5.00000	0.214373
$545$	−32.0000	−1.37073
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	18.0000	0.768922
$549$	5.00000	0.213395
$550$	3.00000	0.127920
$551$	1.00000	0.0426014
$552$	0	0
$553$	0	0
$554$	27.0000	1.14712
$555$	4.00000	0.169791
$556$	−2.00000	−0.0848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	4.00000	0.169334
$559$	−10.0000	−0.422955
$560$	0	0
$561$	15.0000	0.633300
$562$	−16.0000	−0.674919
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−1.00000	−0.0421076
$565$	30.0000	1.26211
$566$	4.00000	0.168133
$567$	0	0
$568$	1.00000	0.0419591
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	−2.00000	−0.0837708
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	3.00000	0.125436
$573$	−20.0000	−0.835512
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−8.00000	−0.332756
$579$	14.0000	0.581820
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−4.00000	−0.165805
$583$	−9.00000	−0.372742
$584$	12.0000	0.496564
$585$	2.00000	0.0826898
$586$	−12.0000	−0.495715
$587$	−31.0000	−1.27951	−0.639753	−	0.768580i	$-0.720964\pi$
$587$	−31.0000	−1.27951	−0.639753	+	0.768580i	$0.720964\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	6.00000	0.247016
$591$	24.0000	0.987228
$592$	−2.00000	−0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	8.00000	0.327693
$597$	−20.0000	−0.818546
$598$	0	0
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	−1.00000	−0.0408248
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	19.0000	0.773099
$605$	−4.00000	−0.162623
$606$	−2.00000	−0.0812444
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−10.0000	−0.404888
$611$	1.00000	0.0404557
$612$	−5.00000	−0.202113
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−27.0000	−1.08963
$615$	20.0000	0.806478
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	2.00000	0.0804518
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	14.0000	0.561349
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	−19.0000	−0.760000
$626$	34.0000	1.35891
$627$	3.00000	0.119808
$628$	−21.0000	−0.837991
$629$	10.0000	0.398726
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	6.00000	0.238667
$633$	16.0000	0.635943
$634$	6.00000	0.238290
$635$	0	0
$636$	3.00000	0.118958
$637$	0	0
$638$	3.00000	0.118771
$639$	−1.00000	−0.0395594
$640$	−2.00000	−0.0790569
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000	0.473602
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	0	0
$645$	20.0000	0.787499
$646$	−5.00000	−0.196722
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	−1.00000	−0.0392837
$649$	−9.00000	−0.353281
$650$	1.00000	0.0392232
$651$	0	0
$652$	19.0000	0.744097
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−16.0000	−0.625650
$655$	8.00000	0.312586
$656$	−10.0000	−0.390434
$657$	−12.0000	−0.468165
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	−6.00000	−0.233550
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	20.0000	0.777322
$663$	5.00000	0.194184
$664$	16.0000	0.620920
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	−21.0000	−0.812514
$669$	11.0000	0.425285
$670$	−10.0000	−0.386334
$671$	15.0000	0.579069
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	17.0000	0.654816
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	15.0000	0.576072
$679$	0	0
$680$	10.0000	0.383482
$681$	8.00000	0.306561
$682$	12.0000	0.459504
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−1.00000	−0.0382360
$685$	36.0000	1.37549
$686$	0	0
$687$	6.00000	0.228914
$688$	−10.0000	−0.381246
$689$	−3.00000	−0.114291
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−3.00000	−0.114043
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	−4.00000	−0.151729
$696$	−1.00000	−0.0379049
$697$	50.0000	1.89389
$698$	36.0000	1.36262
$699$	−11.0000	−0.416058
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	1.00000	0.0377426
$703$	2.00000	0.0754314
$704$	3.00000	0.113067
$705$	−2.00000	−0.0753244
$706$	14.0000	0.526897
$707$	0	0
$708$	3.00000	0.112747
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	2.00000	0.0750587
$711$	−6.00000	−0.225018
$712$	−14.0000	−0.524672
$713$	0	0
$714$	0	0
$715$	6.00000	0.224387
$716$	2.00000	0.0747435
$717$	−21.0000	−0.784259
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	18.0000	0.669891
$723$	4.00000	0.148762
$724$	−3.00000	−0.111494
$725$	1.00000	0.0371391
$726$	−2.00000	−0.0742270
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	24.0000	0.888280
$731$	50.0000	1.84932
$732$	−5.00000	−0.184805
$733$	40.0000	1.47743	0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000	1.47743	0.738717	+	0.674016i	$0.235432\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	0	0
$737$	15.0000	0.552532
$738$	10.0000	0.368105
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−4.00000	−0.147043
$741$	1.00000	0.0367359
$742$	0	0
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	−4.00000	−0.146647
$745$	16.0000	0.586195
$746$	−13.0000	−0.475964
$747$	−16.0000	−0.585409
$748$	−15.0000	−0.548454
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	1.00000	0.0364662
$753$	−18.0000	−0.655956
$754$	1.00000	0.0364179
$755$	38.0000	1.38296
$756$	0	0
$757$	−37.0000	−1.34479	−0.672394	−	0.740193i	$-0.734734\pi$
$757$	−37.0000	−1.34479	−0.672394	+	0.740193i	$0.734734\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	2.00000	0.0725476
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	20.0000	0.723575
$765$	−10.0000	−0.361551
$766$	−32.0000	−1.15621
$767$	−3.00000	−0.108324
$768$	−1.00000	−0.0360844
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	10.0000	0.359443
$775$	4.00000	0.143684
$776$	4.00000	0.143592
$777$	0	0
$778$	−9.00000	−0.322666
$779$	10.0000	0.358287
$780$	−2.00000	−0.0716115
$781$	−3.00000	−0.107348
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−42.0000	−1.49904
$786$	4.00000	0.142675
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	−24.0000	−0.854965
$789$	6.00000	0.213606
$790$	12.0000	0.426941
$791$	0	0
$792$	−3.00000	−0.106600
$793$	5.00000	0.177555
$794$	36.0000	1.27759
$795$	6.00000	0.212798
$796$	20.0000	0.708881
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−5.00000	−0.176887
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	36.0000	1.27120
$803$	−36.0000	−1.27041
$804$	−5.00000	−0.176336
$805$	0	0
$806$	4.00000	0.140894
$807$	−9.00000	−0.316815
$808$	2.00000	0.0703598
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	−2.00000	−0.0702728
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	11.0000	0.385787
$814$	6.00000	0.210300
$815$	38.0000	1.33108
$816$	5.00000	0.175035
$817$	10.0000	0.349856
$818$	−30.0000	−1.04893
$819$	0	0
$820$	−20.0000	−0.698430
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	18.0000	0.627822
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	−2.00000	−0.0696733
$825$	3.00000	0.104447
$826$	0	0
$827$	−35.0000	−1.21707	−0.608535	−	0.793527i	$-0.708242\pi$
$827$	−35.0000	−1.21707	−0.608535	+	0.793527i	$0.708242\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	32.0000	1.11074
$831$	27.0000	0.936620
$832$	1.00000	0.0346688
$833$	0	0
$834$	−2.00000	−0.0692543
$835$	−42.0000	−1.45347
$836$	−3.00000	−0.103757
$837$	4.00000	0.138260
$838$	10.0000	0.345444
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−8.00000	−0.275698
$843$	−16.0000	−0.551069
$844$	−16.0000	−0.550743
$845$	2.00000	0.0688021
$846$	−1.00000	−0.0343807
$847$	0	0
$848$	−3.00000	−0.103020
$849$	4.00000	0.137280
$850$	−5.00000	−0.171499
$851$	0	0
$852$	1.00000	0.0342594
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	−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	$0.516317\pi$
$858$	3.00000	0.102418
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−20.0000	−0.681994
$861$	0	0
$862$	8.00000	0.272481
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	−19.0000	−0.645646
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−18.0000	−0.610608
$870$	−2.00000	−0.0678064
$871$	5.00000	0.169419
$872$	16.0000	0.541828
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	40.0000	1.34993
$879$	−12.0000	−0.404750
$880$	6.00000	0.202260
$881$	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
$883$	−38.0000	−1.27880	−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000	−1.27880	−0.639401	+	0.768874i	$0.720818\pi$
$884$	−5.00000	−0.168168
$885$	6.00000	0.201688
$886$	18.0000	0.604722
$887$	14.0000	0.470074	0.235037	−	0.971986i	$-0.424479\pi$
$887$	14.0000	0.470074	0.235037	+	0.971986i	$0.424479\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	−28.0000	−0.938562
$891$	3.00000	0.100504
$892$	−11.0000	−0.368307
$893$	−1.00000	−0.0334637
$894$	8.00000	0.267560
$895$	4.00000	0.133705
$896$	0	0
$897$	0	0
$898$	8.00000	0.266963
$899$	4.00000	0.133407
$900$	−1.00000	−0.0333333
$901$	15.0000	0.499722
$902$	30.0000	0.998891
$903$	0	0
$904$	−15.0000	−0.498893
$905$	−6.00000	−0.199447
$906$	19.0000	0.631233
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−8.00000	−0.265489
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	1.00000	0.0331133
$913$	−48.0000	−1.58857
$914$	16.0000	0.529233
$915$	−10.0000	−0.330590
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−5.00000	−0.165025
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	−27.0000	−0.889680
$922$	20.0000	0.658665
$923$	−1.00000	−0.0329154
$924$	0	0
$925$	2.00000	0.0657596
$926$	−8.00000	−0.262896
$927$	2.00000	0.0656886
$928$	1.00000	0.0328266
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	11.0000	0.360317
$933$	14.0000	0.458339
$934$	−24.0000	−0.785304
$935$	−30.0000	−0.981105
$936$	−1.00000	−0.0326860
$937$	5.00000	0.163343	0.0816714	−	0.996659i	$-0.473974\pi$
$937$	5.00000	0.163343	0.0816714	+	0.996659i	$0.473974\pi$
$938$	0	0
$939$	34.0000	1.10955
$940$	2.00000	0.0652328
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	−21.0000	−0.684217
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	30.0000	0.975384
$947$	−25.0000	−0.812391	−0.406195	−	0.913786i	$-0.633145\pi$
$947$	−25.0000	−0.812391	−0.406195	+	0.913786i	$0.633145\pi$
$948$	6.00000	0.194871
$949$	−12.0000	−0.389536
$950$	−1.00000	−0.0324443
$951$	6.00000	0.194563
$952$	0	0
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	3.00000	0.0971286
$955$	40.0000	1.29437
$956$	21.0000	0.679189
$957$	3.00000	0.0969762
$958$	−33.0000	−1.06618
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	12.0000	0.386695
$964$	−4.00000	−0.128831
$965$	−28.0000	−0.901352
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	2.00000	0.0642824
$969$	−5.00000	−0.160623
$970$	8.00000	0.256865
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−11.0000	−0.352463
$975$	1.00000	0.0320256
$976$	5.00000	0.160046
$977$	62.0000	1.98356	0.991778	−	0.127971i	$-0.0408466\pi$
$977$	62.0000	1.98356	0.991778	+	0.127971i	$0.0408466\pi$
$978$	19.0000	0.607553
$979$	42.0000	1.34233
$980$	0	0
$981$	−16.0000	−0.510841
$982$	−22.0000	−0.702048
$983$	59.0000	1.88181	0.940904	−	0.338674i	$-0.109978\pi$
$983$	59.0000	1.88181	0.940904	+	0.338674i	$0.109978\pi$
$984$	−10.0000	−0.318788
$985$	−48.0000	−1.52941
$986$	−5.00000	−0.159232
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	0	0
$990$	−6.00000	−0.190693
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	4.00000	0.127000
$993$	20.0000	0.634681
$994$	0	0
$995$	40.0000	1.26809
$996$	16.0000	0.506979
$997$	−19.0000	−0.601736	−0.300868	−	0.953666i	$-0.597276\pi$
$997$	−19.0000	−0.601736	−0.300868	+	0.953666i	$0.597276\pi$
$998$	28.0000	0.886325
$999$	2.00000	0.0632772

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.i.1.1		1
7.3	odd	6		546.2.i.e.79.1	✓	2
7.5	odd	6		546.2.i.e.235.1	yes	2
7.6	odd	2		3822.2.a.k.1.1		1
21.5	even	6		1638.2.j.a.235.1		2
21.17	even	6		1638.2.j.a.1171.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.e.79.1	✓	2	7.3	odd	6
546.2.i.e.235.1	yes	2	7.5	odd	6
1638.2.j.a.235.1		2	21.5	even	6
1638.2.j.a.1171.1		2	21.17	even	6
3822.2.a.i.1.1		1	1.1	even	1	trivial
3822.2.a.k.1.1		1	7.6	odd	2

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Hilbert	Bianchi

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

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Modular forms

Varieties

Fields

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Groups

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3822.2.a.i.1.1

Level	$3822$
Weight	$2$
Character	3822.1
Self dual	yes
Analytic conductor	$30.519$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$