LMFDB - Embedded newform 3822.2.a.bg.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:	$0$
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	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +3.00000 q^{20} -1.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +5.00000 q^{29} +3.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} +3.00000 q^{40} +3.00000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -1.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} +3.00000 q^{51} -1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -1.00000 q^{57} +5.00000 q^{58} +2.00000 q^{59} +3.00000 q^{60} -1.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -1.00000 q^{66} +16.0000 q^{67} +3.00000 q^{68} -1.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -5.00000 q^{73} -1.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} +12.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} +9.00000 q^{85} +3.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} +3.00000 q^{90} -1.00000 q^{92} +6.00000 q^{93} -4.00000 q^{94} -3.00000 q^{95} +1.00000 q^{96} -18.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	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	$0.536593\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	3.00000	0.547723
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	−1.00000	−0.150756
$45$	3.00000	0.447214
$46$	−1.00000	−0.147442
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	3.00000	0.420084
$52$	−1.00000	−0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−1.00000	−0.132453
$58$	5.00000	0.656532
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	3.00000	0.387298
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.00000	−0.372104
$66$	−1.00000	−0.123091
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	3.00000	0.363803
$69$	−1.00000	−0.120386
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	−1.00000	−0.116248
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	3.00000	0.323498
$87$	5.00000	0.536056
$88$	−1.00000	−0.106600
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	−4.00000	−0.412568
$95$	−3.00000	−0.307794
$96$	1.00000	0.102062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	3.00000	0.297044
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−3.00000	−0.286039
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	−1.00000	−0.0936586
$115$	−3.00000	−0.279751
$116$	5.00000	0.464238
$117$	−1.00000	−0.0924500
$118$	2.00000	0.184115
$119$	0	0
$120$	3.00000	0.273861
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	6.00000	0.538816
$125$	−3.00000	−0.268328
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	3.00000	0.264135
$130$	−3.00000	−0.263117
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	16.0000	1.38219
$135$	3.00000	0.258199
$136$	3.00000	0.257248
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−1.00000	−0.0851257
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000	0.503509
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	15.0000	1.24568
$146$	−5.00000	−0.413803
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	4.00000	0.326599
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	−1.00000	−0.0811107
$153$	3.00000	0.242536
$154$	0	0
$155$	18.0000	1.44579
$156$	−1.00000	−0.0800641
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	12.0000	0.954669
$159$	−6.00000	−0.475831
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	6.00000	0.465690
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	9.00000	0.690268
$171$	−1.00000	−0.0764719
$172$	3.00000	0.228748
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	2.00000	0.150329
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	3.00000	0.223607
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	−1.00000	−0.0737210
$185$	−3.00000	−0.220564
$186$	6.00000	0.439941
$187$	−3.00000	−0.219382
$188$	−4.00000	−0.291730
$189$	0	0
$190$	−3.00000	−0.217643
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−18.0000	−1.29232
$195$	−3.00000	−0.214834
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−1.00000	−0.0710669
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	4.00000	0.282843
$201$	16.0000	1.12855
$202$	−14.0000	−0.985037
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	−1.00000	−0.0695048
$208$	−1.00000	−0.0693375
$209$	1.00000	0.0691714
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	−6.00000	−0.412082
$213$	6.00000	0.411113
$214$	−8.00000	−0.546869
$215$	9.00000	0.613795
$216$	1.00000	0.0680414
$217$	0	0
$218$	−7.00000	−0.474100
$219$	−5.00000	−0.337869
$220$	−3.00000	−0.202260
$221$	−3.00000	−0.201802
$222$	−1.00000	−0.0671156
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	12.0000	0.798228
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−1.00000	−0.0662266
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	5.00000	0.328266
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−1.00000	−0.0653720
$235$	−12.0000	−0.782794
$236$	2.00000	0.130189
$237$	12.0000	0.779484
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	3.00000	0.193649
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	6.00000	0.381000
$249$	6.00000	0.380235
$250$	−3.00000	−0.189737
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	16.0000	1.00393
$255$	9.00000	0.563602
$256$	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	$0.559920\pi$
$258$	3.00000	0.186772
$259$	0	0
$260$	−3.00000	−0.186052
$261$	5.00000	0.309492
$262$	−9.00000	−0.556022
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	−1.00000	−0.0615457
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	16.0000	0.977356
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	3.00000	0.182574
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−15.0000	−0.906183
$275$	−4.00000	−0.241209
$276$	−1.00000	−0.0601929
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−16.0000	−0.959616
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−4.00000	−0.238197
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	6.00000	0.356034
$285$	−3.00000	−0.177705
$286$	1.00000	0.0591312
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	15.0000	0.880830
$291$	−18.0000	−1.05518
$292$	−5.00000	−0.292603
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	−1.00000	−0.0581238
$297$	−1.00000	−0.0580259
$298$	4.00000	0.231714
$299$	1.00000	0.0578315
$300$	4.00000	0.230940
$301$	0	0
$302$	−9.00000	−0.517892
$303$	−14.0000	−0.804279
$304$	−1.00000	−0.0573539
$305$	−3.00000	−0.171780
$306$	3.00000	0.171499
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	18.0000	1.02233
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−1.00000	−0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	12.0000	0.675053
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−6.00000	−0.336463
$319$	−5.00000	−0.279946
$320$	3.00000	0.167705
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	6.00000	0.332309
$327$	−7.00000	−0.387101
$328$	0	0
$329$	0	0
$330$	−3.00000	−0.165145
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	6.00000	0.329293
$333$	−1.00000	−0.0547997
$334$	3.00000	0.164153
$335$	48.0000	2.62252
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	1.00000	0.0543928
$339$	12.0000	0.651751
$340$	9.00000	0.488094
$341$	−6.00000	−0.324918
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	3.00000	0.161749
$345$	−3.00000	−0.161515
$346$	−4.00000	−0.215041
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	5.00000	0.268028
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.00000	−0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	2.00000	0.106299
$355$	18.0000	0.955341
$356$	0	0
$357$	0	0
$358$	−18.0000	−0.951330
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	3.00000	0.158114
$361$	−18.0000	−0.947368
$362$	−14.0000	−0.735824
$363$	−10.0000	−0.524864
$364$	0	0
$365$	−15.0000	−0.785136
$366$	−1.00000	−0.0522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−3.00000	−0.155963
$371$	0	0
$372$	6.00000	0.311086
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−3.00000	−0.155126
$375$	−3.00000	−0.154919
$376$	−4.00000	−0.206284
$377$	−5.00000	−0.257513
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	−3.00000	−0.153897
$381$	16.0000	0.819705
$382$	−5.00000	−0.255822
$383$	−33.0000	−1.68622	−0.843111	−	0.537740i	$-0.819278\pi$
$383$	−33.0000	−1.68622	−0.843111	+	0.537740i	$0.819278\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	3.00000	0.152499
$388$	−18.0000	−0.913812
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	−3.00000	−0.151911
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−9.00000	−0.453990
$394$	18.0000	0.906827
$395$	36.0000	1.81136
$396$	−1.00000	−0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−3.00000	−0.150376
$399$	0	0
$400$	4.00000	0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	16.0000	0.798007
$403$	−6.00000	−0.298881
$404$	−14.0000	−0.696526
$405$	3.00000	0.149071
$406$	0	0
$407$	1.00000	0.0495682
$408$	3.00000	0.148522
$409$	−15.0000	−0.741702	−0.370851	−	0.928692i	$-0.620934\pi$
$409$	−15.0000	−0.741702	−0.370851	+	0.928692i	$0.620934\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	18.0000	0.883585
$416$	−1.00000	−0.0490290
$417$	−16.0000	−0.783523
$418$	1.00000	0.0489116
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	25.0000	1.21698
$423$	−4.00000	−0.194487
$424$	−6.00000	−0.291386
$425$	12.0000	0.582086
$426$	6.00000	0.290701
$427$	0	0
$428$	−8.00000	−0.386695
$429$	1.00000	0.0482805
$430$	9.00000	0.434019
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	1.00000	0.0481125
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	15.0000	0.719195
$436$	−7.00000	−0.335239
$437$	1.00000	0.0478365
$438$	−5.00000	−0.238909
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−3.00000	−0.142695
$443$	42.0000	1.99548	0.997740	−	0.0671913i	$-0.0214038\pi$
$443$	42.0000	1.99548	0.997740	+	0.0671913i	$0.0214038\pi$
$444$	−1.00000	−0.0474579
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	4.00000	0.189194
$448$	0	0
$449$	−39.0000	−1.84052	−0.920262	−	0.391303i	$-0.872024\pi$
$449$	−39.0000	−1.84052	−0.920262	+	0.391303i	$0.872024\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	12.0000	0.564433
$453$	−9.00000	−0.422857
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	4.00000	0.186908
$459$	3.00000	0.140028
$460$	−3.00000	−0.139876
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−21.0000	−0.975953	−0.487976	−	0.872857i	$-0.662265\pi$
$463$	−21.0000	−0.975953	−0.487976	+	0.872857i	$0.662265\pi$
$464$	5.00000	0.232119
$465$	18.0000	0.834730
$466$	−8.00000	−0.370593
$467$	−33.0000	−1.52706	−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000	−1.52706	−0.763529	+	0.645774i	$0.776535\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	−12.0000	−0.553519
$471$	−13.0000	−0.599008
$472$	2.00000	0.0920575
$473$	−3.00000	−0.137940
$474$	12.0000	0.551178
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−6.00000	−0.274434
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	3.00000	0.136931
$481$	1.00000	0.0455961
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−54.0000	−2.45201
$486$	1.00000	0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−1.00000	−0.0452679
$489$	6.00000	0.271329
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	15.0000	0.675566
$494$	1.00000	0.0449921
$495$	−3.00000	−0.134840
$496$	6.00000	0.269408
$497$	0	0
$498$	6.00000	0.268866
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	−3.00000	−0.134164
$501$	3.00000	0.134030
$502$	27.0000	1.20507
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	−42.0000	−1.86898
$506$	1.00000	0.0444554
$507$	1.00000	0.0444116
$508$	16.0000	0.709885
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	9.00000	0.398527
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−6.00000	−0.264649
$515$	−3.00000	−0.132196
$516$	3.00000	0.132068
$517$	4.00000	0.175920
$518$	0	0
$519$	−4.00000	−0.175581
$520$	−3.00000	−0.131559
$521$	17.0000	0.744784	0.372392	−	0.928076i	$-0.378538\pi$
$521$	17.0000	0.744784	0.372392	+	0.928076i	$0.378538\pi$
$522$	5.00000	0.218844
$523$	−42.0000	−1.83653	−0.918266	−	0.395964i	$-0.870410\pi$
$523$	−42.0000	−1.83653	−0.918266	+	0.395964i	$0.870410\pi$
$524$	−9.00000	−0.393167
$525$	0	0
$526$	−4.00000	−0.174408
$527$	18.0000	0.784092
$528$	−1.00000	−0.0435194
$529$	−22.0000	−0.956522
$530$	−18.0000	−0.781870
$531$	2.00000	0.0867926
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−24.0000	−1.03761
$536$	16.0000	0.691095
$537$	−18.0000	−0.776757
$538$	−16.0000	−0.689809
$539$	0	0
$540$	3.00000	0.129099
$541$	−35.0000	−1.50477	−0.752384	−	0.658725i	$-0.771096\pi$
$541$	−35.0000	−1.50477	−0.752384	+	0.658725i	$0.771096\pi$
$542$	−8.00000	−0.343629
$543$	−14.0000	−0.600798
$544$	3.00000	0.128624
$545$	−21.0000	−0.899541
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−15.0000	−0.640768
$549$	−1.00000	−0.0426790
$550$	−4.00000	−0.170561
$551$	−5.00000	−0.213007
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−8.00000	−0.339887
$555$	−3.00000	−0.127343
$556$	−16.0000	−0.678551
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	6.00000	0.254000
$559$	−3.00000	−0.126886
$560$	0	0
$561$	−3.00000	−0.126660
$562$	−18.0000	−0.759284
$563$	19.0000	0.800755	0.400377	−	0.916350i	$-0.368879\pi$
$563$	19.0000	0.800755	0.400377	+	0.916350i	$0.368879\pi$
$564$	−4.00000	−0.168430
$565$	36.0000	1.51453
$566$	0	0
$567$	0	0
$568$	6.00000	0.251754
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−3.00000	−0.125656
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	1.00000	0.0418121
$573$	−5.00000	−0.208878
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−8.00000	−0.332756
$579$	2.00000	0.0831172
$580$	15.0000	0.622841
$581$	0	0
$582$	−18.0000	−0.746124
$583$	6.00000	0.248495
$584$	−5.00000	−0.206901
$585$	−3.00000	−0.124035
$586$	14.0000	0.578335
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	6.00000	0.247016
$591$	18.0000	0.740421
$592$	−1.00000	−0.0410997
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	4.00000	0.163846
$597$	−3.00000	−0.122782
$598$	1.00000	0.0408930
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	4.00000	0.163299
$601$	−32.0000	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	−9.00000	−0.366205
$605$	−30.0000	−1.21967
$606$	−14.0000	−0.568711
$607$	35.0000	1.42061	0.710303	−	0.703896i	$-0.248558\pi$
$607$	35.0000	1.42061	0.710303	+	0.703896i	$0.248558\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	−3.00000	−0.121466
$611$	4.00000	0.161823
$612$	3.00000	0.121268
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	−9.00000	−0.362326	−0.181163	−	0.983453i	$-0.557986\pi$
$617$	−9.00000	−0.362326	−0.181163	+	0.983453i	$0.557986\pi$
$618$	−1.00000	−0.0402259
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	18.0000	0.722897
$621$	−1.00000	−0.0401286
$622$	−6.00000	−0.240578
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	1.00000	0.0399362
$628$	−13.0000	−0.518756
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	12.0000	0.477334
$633$	25.0000	0.993661
$634$	22.0000	0.873732
$635$	48.0000	1.90482
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−5.00000	−0.197952
$639$	6.00000	0.237356
$640$	3.00000	0.118585
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	−8.00000	−0.315735
$643$	−35.0000	−1.38027	−0.690133	−	0.723683i	$-0.742448\pi$
$643$	−35.0000	−1.38027	−0.690133	+	0.723683i	$0.742448\pi$
$644$	0	0
$645$	9.00000	0.354375
$646$	−3.00000	−0.118033
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	1.00000	0.0392837
$649$	−2.00000	−0.0785069
$650$	−4.00000	−0.156893
$651$	0	0
$652$	6.00000	0.234978
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	−7.00000	−0.273722
$655$	−27.0000	−1.05498
$656$	0	0
$657$	−5.00000	−0.195069
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	−3.00000	−0.116775
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	6.00000	0.232845
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	−5.00000	−0.193601
$668$	3.00000	0.116073
$669$	−2.00000	−0.0773245
$670$	48.0000	1.85440
$671$	1.00000	0.0386046
$672$	0	0
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	31.0000	1.19408
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	12.0000	0.460857
$679$	0	0
$680$	9.00000	0.345134
$681$	−12.0000	−0.459841
$682$	−6.00000	−0.229752
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	−1.00000	−0.0382360
$685$	−45.0000	−1.71936
$686$	0	0
$687$	4.00000	0.152610
$688$	3.00000	0.114374
$689$	6.00000	0.228582
$690$	−3.00000	−0.114208
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	−18.0000	−0.683271
$695$	−48.0000	−1.82074
$696$	5.00000	0.189525
$697$	0	0
$698$	4.00000	0.151402
$699$	−8.00000	−0.302588
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	−1.00000	−0.0377426
$703$	1.00000	0.0377157
$704$	−1.00000	−0.0376889
$705$	−12.0000	−0.451946
$706$	18.0000	0.677439
$707$	0	0
$708$	2.00000	0.0751646
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	18.0000	0.675528
$711$	12.0000	0.450035
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	3.00000	0.112194
$716$	−18.0000	−0.672692
$717$	−6.00000	−0.224074
$718$	8.00000	0.298557
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−18.0000	−0.669891
$723$	−14.0000	−0.520666
$724$	−14.0000	−0.520306
$725$	20.0000	0.742781
$726$	−10.0000	−0.371135
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−15.0000	−0.555175
$731$	9.00000	0.332877
$732$	−1.00000	−0.0369611
$733$	−54.0000	−1.99454	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	−54.0000	−1.99454	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−3.00000	−0.110282
$741$	1.00000	0.0367359
$742$	0	0
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	6.00000	0.219971
$745$	12.0000	0.439646
$746$	−22.0000	−0.805477
$747$	6.00000	0.219529
$748$	−3.00000	−0.109691
$749$	0	0
$750$	−3.00000	−0.109545
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	−4.00000	−0.145865
$753$	27.0000	0.983935
$754$	−5.00000	−0.182089
$755$	−27.0000	−0.982631
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	22.0000	0.799076
$759$	1.00000	0.0362977
$760$	−3.00000	−0.108821
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	−5.00000	−0.180894
$765$	9.00000	0.325396
$766$	−33.0000	−1.19234
$767$	−2.00000	−0.0722158
$768$	1.00000	0.0360844
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	2.00000	0.0719816
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	3.00000	0.107833
$775$	24.0000	0.862105
$776$	−18.0000	−0.646162
$777$	0	0
$778$	26.0000	0.932145
$779$	0	0
$780$	−3.00000	−0.107417
$781$	−6.00000	−0.214697
$782$	−3.00000	−0.107280
$783$	5.00000	0.178685
$784$	0	0
$785$	−39.0000	−1.39197
$786$	−9.00000	−0.321019
$787$	−29.0000	−1.03374	−0.516869	−	0.856064i	$-0.672903\pi$
$787$	−29.0000	−1.03374	−0.516869	+	0.856064i	$0.672903\pi$
$788$	18.0000	0.641223
$789$	−4.00000	−0.142404
$790$	36.0000	1.28082
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	1.00000	0.0355110
$794$	−2.00000	−0.0709773
$795$	−18.0000	−0.638394
$796$	−3.00000	−0.106332
$797$	−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
$799$	−12.0000	−0.424529
$800$	4.00000	0.141421
$801$	0	0
$802$	2.00000	0.0706225
$803$	5.00000	0.176446
$804$	16.0000	0.564276
$805$	0	0
$806$	−6.00000	−0.211341
$807$	−16.0000	−0.563227
$808$	−14.0000	−0.492518
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	3.00000	0.105409
$811$	−29.0000	−1.01833	−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000	−1.01833	−0.509164	+	0.860670i	$0.670045\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	1.00000	0.0350500
$815$	18.0000	0.630512
$816$	3.00000	0.105021
$817$	−3.00000	−0.104957
$818$	−15.0000	−0.524463
$819$	0	0
$820$	0	0
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	−15.0000	−0.523185
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	−1.00000	−0.0348367
$825$	−4.00000	−0.139262
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	−1.00000	−0.0347524
$829$	15.0000	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$830$	18.0000	0.624789
$831$	−8.00000	−0.277517
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−16.0000	−0.554035
$835$	9.00000	0.311458
$836$	1.00000	0.0345857
$837$	6.00000	0.207390
$838$	21.0000	0.725433
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−30.0000	−1.03387
$843$	−18.0000	−0.619953
$844$	25.0000	0.860535
$845$	3.00000	0.103203
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	12.0000	0.411597
$851$	1.00000	0.0342796
$852$	6.00000	0.205557
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	−8.00000	−0.273434
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	1.00000	0.0341394
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	9.00000	0.306897
$861$	0	0
$862$	6.00000	0.204361
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	1.00000	0.0340207
$865$	−12.0000	−0.408012
$866$	−4.00000	−0.135926
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−12.0000	−0.407072
$870$	15.0000	0.508548
$871$	−16.0000	−0.542139
$872$	−7.00000	−0.237050
$873$	−18.0000	−0.609208
$874$	1.00000	0.0338255
$875$	0	0
$876$	−5.00000	−0.168934
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	19.0000	0.641219
$879$	14.0000	0.472208
$880$	−3.00000	−0.101130
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	0	0
$883$	−25.0000	−0.841317	−0.420658	−	0.907219i	$-0.638201\pi$
$883$	−25.0000	−0.841317	−0.420658	+	0.907219i	$0.638201\pi$
$884$	−3.00000	−0.100901
$885$	6.00000	0.201688
$886$	42.0000	1.41102
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−1.00000	−0.0335578
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−2.00000	−0.0669650
$893$	4.00000	0.133855
$894$	4.00000	0.133780
$895$	−54.0000	−1.80502
$896$	0	0
$897$	1.00000	0.0333890
$898$	−39.0000	−1.30145
$899$	30.0000	1.00056
$900$	4.00000	0.133333
$901$	−18.0000	−0.599667
$902$	0	0
$903$	0	0
$904$	12.0000	0.399114
$905$	−42.0000	−1.39613
$906$	−9.00000	−0.299005
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−12.0000	−0.398234
$909$	−14.0000	−0.464351
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	−1.00000	−0.0331133
$913$	−6.00000	−0.198571
$914$	−22.0000	−0.727695
$915$	−3.00000	−0.0991769
$916$	4.00000	0.132164
$917$	0	0
$918$	3.00000	0.0990148
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	−3.00000	−0.0989071
$921$	20.0000	0.659022
$922$	33.0000	1.08680
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−21.0000	−0.690103
$927$	−1.00000	−0.0328443
$928$	5.00000	0.164133
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	18.0000	0.590243
$931$	0	0
$932$	−8.00000	−0.262049
$933$	−6.00000	−0.196431
$934$	−33.0000	−1.07979
$935$	−9.00000	−0.294331
$936$	−1.00000	−0.0326860
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	−12.0000	−0.391397
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	−13.0000	−0.423563
$943$	0	0
$944$	2.00000	0.0650945
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	12.0000	0.389742
$949$	5.00000	0.162307
$950$	−4.00000	−0.129777
$951$	22.0000	0.713399
$952$	0	0
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	−6.00000	−0.194257
$955$	−15.0000	−0.485389
$956$	−6.00000	−0.194054
$957$	−5.00000	−0.161627
$958$	5.00000	0.161543
$959$	0	0
$960$	3.00000	0.0968246
$961$	5.00000	0.161290
$962$	1.00000	0.0322413
$963$	−8.00000	−0.257796
$964$	−14.0000	−0.450910
$965$	6.00000	0.193147
$966$	0	0
$967$	41.0000	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$968$	−10.0000	−0.321412
$969$	−3.00000	−0.0963739
$970$	−54.0000	−1.73384
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−8.00000	−0.256337
$975$	−4.00000	−0.128103
$976$	−1.00000	−0.0320092
$977$	−13.0000	−0.415907	−0.207953	−	0.978139i	$-0.566680\pi$
$977$	−13.0000	−0.415907	−0.207953	+	0.978139i	$0.566680\pi$
$978$	6.00000	0.191859
$979$	0	0
$980$	0	0
$981$	−7.00000	−0.223493
$982$	18.0000	0.574403
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	15.0000	0.477697
$987$	0	0
$988$	1.00000	0.0318142
$989$	−3.00000	−0.0953945
$990$	−3.00000	−0.0953463
$991$	42.0000	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$991$	42.0000	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	0	0
$995$	−9.00000	−0.285319
$996$	6.00000	0.190117
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−6.00000	−0.189927
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.bg.1.1	yes	1
7.6	odd	2		3822.2.a.r.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.a.r.1.1	✓	1	7.6	odd	2
3822.2.a.bg.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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