LMFDB - Embedded newform 3822.2.a.d.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:	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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -5.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} +1.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -1.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +5.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -5.00000 q^{51} +1.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{57} +3.00000 q^{58} +14.0000 q^{59} +1.00000 q^{60} +1.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +5.00000 q^{68} +5.00000 q^{69} -2.00000 q^{71} -1.00000 q^{72} -1.00000 q^{73} +11.0000 q^{74} +4.00000 q^{75} -1.00000 q^{76} +1.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{83} -5.00000 q^{85} +1.00000 q^{86} +3.00000 q^{87} -1.00000 q^{88} -12.0000 q^{89} +1.00000 q^{90} -5.00000 q^{92} +2.00000 q^{93} -12.0000 q^{94} +1.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−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	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−1.00000	−0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−5.00000	−0.857493
$35$	0	0
$36$	1.00000	0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	1.00000	0.162221
$39$	−1.00000	−0.160128
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	5.00000	0.737210
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	−5.00000	−0.700140
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	1.00000	0.132453
$58$	3.00000	0.393919
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	1.00000	0.129099
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	5.00000	0.606339
$69$	5.00000	0.601929
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	11.0000	1.27872
$75$	4.00000	0.461880
$76$	−1.00000	−0.114708
$77$	0	0
$78$	1.00000	0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	1.00000	0.107833
$87$	3.00000	0.321634
$88$	−1.00000	−0.106600
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−5.00000	−0.521286
$93$	2.00000	0.207390
$94$	−12.0000	−1.23771
$95$	1.00000	0.102598
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−4.00000	−0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	5.00000	0.495074
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−1.00000	−0.0962250
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	1.00000	0.0953463
$111$	11.0000	1.04407
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−1.00000	−0.0936586
$115$	5.00000	0.466252
$116$	−3.00000	−0.278543
$117$	1.00000	0.0924500
$118$	−14.0000	−1.28880
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	−2.00000	−0.179605
$125$	9.00000	0.804984
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	1.00000	0.0880451
$130$	1.00000	0.0877058
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	−5.00000	−0.428746
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	−5.00000	−0.425628
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	2.00000	0.167836
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	1.00000	0.0827606
$147$	0	0
$148$	−11.0000	−0.904194
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−4.00000	−0.326599
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	1.00000	0.0811107
$153$	5.00000	0.404226
$154$	0	0
$155$	2.00000	0.160644
$156$	−1.00000	−0.0800641
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	8.00000	0.636446
$159$	−10.0000	−0.793052
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	2.00000	0.155230
$167$	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
$170$	5.00000	0.383482
$171$	−1.00000	−0.0764719
$172$	−1.00000	−0.0762493
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	1.00000	0.0753778
$177$	−14.0000	−1.05230
$178$	12.0000	0.899438
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	−1.00000	−0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	5.00000	0.368605
$185$	11.0000	0.808736
$186$	−2.00000	−0.146647
$187$	5.00000	0.365636
$188$	12.0000	0.875190
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	2.00000	0.143592
$195$	1.00000	0.0716115
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	−1.00000	−0.0710669
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	4.00000	0.282843
$201$	−4.00000	−0.282138
$202$	6.00000	0.422159
$203$	0	0
$204$	−5.00000	−0.350070
$205$	0	0
$206$	3.00000	0.209020
$207$	−5.00000	−0.347524
$208$	1.00000	0.0693375
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	10.0000	0.686803
$213$	2.00000	0.137038
$214$	16.0000	1.09374
$215$	1.00000	0.0681994
$216$	1.00000	0.0680414
$217$	0	0
$218$	−11.0000	−0.745014
$219$	1.00000	0.0675737
$220$	−1.00000	−0.0674200
$221$	5.00000	0.336336
$222$	−11.0000	−0.738272
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	4.00000	0.266076
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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	$0.457808\pi$
$230$	−5.00000	−0.329690
$231$	0	0
$232$	3.00000	0.196960
$233$	−20.0000	−1.31024	−0.655122	−	0.755523i	$-0.727383\pi$
$233$	−20.0000	−1.31024	−0.655122	+	0.755523i	$0.727383\pi$
$234$	−1.00000	−0.0653720
$235$	−12.0000	−0.782794
$236$	14.0000	0.911322
$237$	8.00000	0.519656
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	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
$248$	2.00000	0.127000
$249$	2.00000	0.126745
$250$	−9.00000	−0.569210
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	−4.00000	−0.250982
$255$	5.00000	0.313112
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−1.00000	−0.0622573
$259$	0	0
$260$	−1.00000	−0.0620174
$261$	−3.00000	−0.185695
$262$	−13.0000	−0.803143
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	1.00000	0.0615457
$265$	−10.0000	−0.614295
$266$	0	0
$267$	12.0000	0.734388
$268$	4.00000	0.244339
$269$	28.0000	1.70719	0.853595	−	0.520937i	$-0.174417\pi$
$269$	28.0000	1.70719	0.853595	+	0.520937i	$0.174417\pi$
$270$	−1.00000	−0.0608581
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	5.00000	0.302061
$275$	−4.00000	−0.241209
$276$	5.00000	0.300965
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	8.00000	0.479808
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	12.0000	0.714590
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−2.00000	−0.118678
$285$	−1.00000	−0.0592349
$286$	−1.00000	−0.0591312
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	−3.00000	−0.176166
$291$	2.00000	0.117242
$292$	−1.00000	−0.0585206
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	11.0000	0.639362
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.00000	−0.289157
$300$	4.00000	0.230940
$301$	0	0
$302$	−17.0000	−0.978240
$303$	6.00000	0.344691
$304$	−1.00000	−0.0573539
$305$	−1.00000	−0.0572598
$306$	−5.00000	−0.285831
$307$	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
$309$	3.00000	0.170664
$310$	−2.00000	−0.113592
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	1.00000	0.0566139
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	10.0000	0.560772
$319$	−3.00000	−0.167968
$320$	−1.00000	−0.0559017
$321$	16.0000	0.893033
$322$	0	0
$323$	−5.00000	−0.278207
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	18.0000	0.996928
$327$	−11.0000	−0.608301
$328$	0	0
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−2.00000	−0.109764
$333$	−11.0000	−0.602796
$334$	−3.00000	−0.164153
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	−1.00000	−0.0543928
$339$	4.00000	0.217250
$340$	−5.00000	−0.271163
$341$	−2.00000	−0.108306
$342$	1.00000	0.0540738
$343$	0	0
$344$	1.00000	0.0539164
$345$	−5.00000	−0.269191
$346$	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	$0.660506\pi$
$348$	3.00000	0.160817
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	14.0000	0.744092
$355$	2.00000	0.106149
$356$	−12.0000	−0.635999
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	10.0000	0.525588
$363$	10.0000	0.524864
$364$	0	0
$365$	1.00000	0.0523424
$366$	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	$0.637127\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−11.0000	−0.571863
$371$	0	0
$372$	2.00000	0.103695
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	−5.00000	−0.258544
$375$	−9.00000	−0.464758
$376$	−12.0000	−0.618853
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	1.00000	0.0512989
$381$	−4.00000	−0.204926
$382$	−15.0000	−0.767467
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	−1.00000	−0.0508329
$388$	−2.00000	−0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−1.00000	−0.0506370
$391$	−25.0000	−1.26430
$392$	0	0
$393$	−13.0000	−0.655763
$394$	14.0000	0.705310
$395$	8.00000	0.402524
$396$	1.00000	0.0502519
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	17.0000	0.852133
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	4.00000	0.199502
$403$	−2.00000	−0.0996271
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−11.0000	−0.545250
$408$	5.00000	0.247537
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	−3.00000	−0.147799
$413$	0	0
$414$	5.00000	0.245737
$415$	2.00000	0.0981761
$416$	−1.00000	−0.0490290
$417$	8.00000	0.391762
$418$	1.00000	0.0489116
$419$	−17.0000	−0.830504	−0.415252	−	0.909706i	$-0.636307\pi$
$419$	−17.0000	−0.830504	−0.415252	+	0.909706i	$0.636307\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	3.00000	0.146038
$423$	12.0000	0.583460
$424$	−10.0000	−0.485643
$425$	−20.0000	−0.970143
$426$	−2.00000	−0.0969003
$427$	0	0
$428$	−16.0000	−0.773389
$429$	−1.00000	−0.0482805
$430$	−1.00000	−0.0482243
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	−1.00000	−0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	11.0000	0.526804
$437$	5.00000	0.239182
$438$	−1.00000	−0.0477818
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−5.00000	−0.237826
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	11.0000	0.522037
$445$	12.0000	0.568855
$446$	6.00000	0.284108
$447$	0	0
$448$	0	0
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	−4.00000	−0.188144
$453$	−17.0000	−0.798730
$454$	0	0
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−4.00000	−0.186908
$459$	−5.00000	−0.233380
$460$	5.00000	0.233126
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−3.00000	−0.139272
$465$	−2.00000	−0.0927478
$466$	20.0000	0.926482
$467$	−35.0000	−1.61961	−0.809803	−	0.586701i	$-0.800426\pi$
$467$	−35.0000	−1.61961	−0.809803	+	0.586701i	$0.800426\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	12.0000	0.553519
$471$	11.0000	0.506853
$472$	−14.0000	−0.644402
$473$	−1.00000	−0.0459800
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	10.0000	0.457869
$478$	−6.00000	−0.274434
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	−1.00000	−0.0456435
$481$	−11.0000	−0.501557
$482$	22.0000	1.00207
$483$	0	0
$484$	−10.0000	−0.454545
$485$	2.00000	0.0908153
$486$	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
$489$	18.0000	0.813988
$490$	0	0
$491$	38.0000	1.71492	0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000	1.71492	0.857458	+	0.514554i	$0.172042\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	1.00000	0.0449921
$495$	−1.00000	−0.0449467
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	−2.00000	−0.0896221
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	9.00000	0.402492
$501$	−3.00000	−0.134030
$502$	15.0000	0.669483
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	5.00000	0.222277
$507$	−1.00000	−0.0444116
$508$	4.00000	0.177471
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	−5.00000	−0.221404
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	2.00000	0.0882162
$515$	3.00000	0.132196
$516$	1.00000	0.0440225
$517$	12.0000	0.527759
$518$	0	0
$519$	4.00000	0.175581
$520$	1.00000	0.0438529
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	3.00000	0.131306
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	13.0000	0.567908
$525$	0	0
$526$	12.0000	0.523225
$527$	−10.0000	−0.435607
$528$	−1.00000	−0.0435194
$529$	2.00000	0.0869565
$530$	10.0000	0.434372
$531$	14.0000	0.607548
$532$	0	0
$533$	0	0
$534$	−12.0000	−0.519291
$535$	16.0000	0.691740
$536$	−4.00000	−0.172774
$537$	−2.00000	−0.0863064
$538$	−28.0000	−1.20717
$539$	0	0
$540$	1.00000	0.0430331
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	12.0000	0.515444
$543$	10.0000	0.429141
$544$	−5.00000	−0.214373
$545$	−11.0000	−0.471188
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−5.00000	−0.213589
$549$	1.00000	0.0426790
$550$	4.00000	0.170561
$551$	3.00000	0.127804
$552$	−5.00000	−0.212814
$553$	0	0
$554$	−12.0000	−0.509831
$555$	−11.0000	−0.466924
$556$	−8.00000	−0.339276
$557$	−8.00000	−0.338971	−0.169485	−	0.985533i	$-0.554211\pi$
$557$	−8.00000	−0.338971	−0.169485	+	0.985533i	$0.554211\pi$
$558$	2.00000	0.0846668
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−5.00000	−0.211100
$562$	14.0000	0.590554
$563$	1.00000	0.0421450	0.0210725	−	0.999778i	$-0.493292\pi$
$563$	1.00000	0.0421450	0.0210725	+	0.999778i	$0.493292\pi$
$564$	−12.0000	−0.505291
$565$	4.00000	0.168281
$566$	28.0000	1.17693
$567$	0	0
$568$	2.00000	0.0839181
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	1.00000	0.0418854
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	1.00000	0.0418121
$573$	−15.0000	−0.626634
$574$	0	0
$575$	20.0000	0.834058
$576$	1.00000	0.0416667
$577$	−46.0000	−1.91501	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−46.0000	−1.91501	−0.957503	+	0.288425i	$0.906868\pi$
$578$	−8.00000	−0.332756
$579$	18.0000	0.748054
$580$	3.00000	0.124568
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	10.0000	0.414158
$584$	1.00000	0.0413803
$585$	−1.00000	−0.0413449
$586$	2.00000	0.0826192
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	14.0000	0.576371
$591$	14.0000	0.575883
$592$	−11.0000	−0.452097
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	0	0
$597$	17.0000	0.695764
$598$	5.00000	0.204465
$599$	−11.0000	−0.449448	−0.224724	−	0.974422i	$-0.572148\pi$
$599$	−11.0000	−0.449448	−0.224724	+	0.974422i	$0.572148\pi$
$600$	−4.00000	−0.163299
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	17.0000	0.691720
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	25.0000	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$607$	25.0000	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	1.00000	0.0404888
$611$	12.0000	0.485468
$612$	5.00000	0.202113
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−3.00000	−0.120775	−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000	−0.120775	−0.0603877	+	0.998175i	$0.519234\pi$
$618$	−3.00000	−0.120678
$619$	−45.0000	−1.80870	−0.904351	−	0.426789i	$-0.859645\pi$
$619$	−45.0000	−1.80870	−0.904351	+	0.426789i	$0.859645\pi$
$620$	2.00000	0.0803219
$621$	5.00000	0.200643
$622$	−2.00000	−0.0801927
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	26.0000	1.03917
$627$	1.00000	0.0399362
$628$	−11.0000	−0.438948
$629$	−55.0000	−2.19299
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	8.00000	0.318223
$633$	3.00000	0.119239
$634$	30.0000	1.19145
$635$	−4.00000	−0.158735
$636$	−10.0000	−0.396526
$637$	0	0
$638$	3.00000	0.118771
$639$	−2.00000	−0.0791188
$640$	1.00000	0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−16.0000	−0.631470
$643$	29.0000	1.14365	0.571824	−	0.820376i	$-0.306236\pi$
$643$	29.0000	1.14365	0.571824	+	0.820376i	$0.306236\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	5.00000	0.196722
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	14.0000	0.549548
$650$	4.00000	0.156893
$651$	0	0
$652$	−18.0000	−0.704934
$653$	39.0000	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$653$	39.0000	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$654$	11.0000	0.430134
$655$	−13.0000	−0.507952
$656$	0	0
$657$	−1.00000	−0.0390137
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	1.00000	0.0389249
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−20.0000	−0.777322
$663$	−5.00000	−0.194184
$664$	2.00000	0.0776151
$665$	0	0
$666$	11.0000	0.426241
$667$	15.0000	0.580802
$668$	3.00000	0.116073
$669$	6.00000	0.231973
$670$	4.00000	0.154533
$671$	1.00000	0.0386046
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	1.00000	0.0385186
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−4.00000	−0.153619
$679$	0	0
$680$	5.00000	0.191741
$681$	0	0
$682$	2.00000	0.0765840
$683$	−29.0000	−1.10965	−0.554827	−	0.831966i	$-0.687216\pi$
$683$	−29.0000	−1.10965	−0.554827	+	0.831966i	$0.687216\pi$
$684$	−1.00000	−0.0382360
$685$	5.00000	0.191040
$686$	0	0
$687$	−4.00000	−0.152610
$688$	−1.00000	−0.0381246
$689$	10.0000	0.380970
$690$	5.00000	0.190347
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	18.0000	0.683271
$695$	8.00000	0.303457
$696$	−3.00000	−0.113715
$697$	0	0
$698$	−8.00000	−0.302804
$699$	20.0000	0.756469
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	1.00000	0.0377426
$703$	11.0000	0.414873
$704$	1.00000	0.0376889
$705$	12.0000	0.451946
$706$	−6.00000	−0.225813
$707$	0	0
$708$	−14.0000	−0.526152
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	−2.00000	−0.0750587
$711$	−8.00000	−0.300023
$712$	12.0000	0.449719
$713$	10.0000	0.374503
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	2.00000	0.0747435
$717$	−6.00000	−0.224074
$718$	4.00000	0.149279
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	18.0000	0.669891
$723$	22.0000	0.818189
$724$	−10.0000	−0.371647
$725$	12.0000	0.445669
$726$	−10.0000	−0.371135
$727$	−43.0000	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$727$	−43.0000	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−1.00000	−0.0370117
$731$	−5.00000	−0.184932
$732$	−1.00000	−0.0369611
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	5.00000	0.184302
$737$	4.00000	0.147342
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	11.0000	0.404368
$741$	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	$0.314515\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	18.0000	0.659027
$747$	−2.00000	−0.0731762
$748$	5.00000	0.182818
$749$	0	0
$750$	9.00000	0.328634
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	12.0000	0.437595
$753$	15.0000	0.546630
$754$	3.00000	0.109254
$755$	−17.0000	−0.618693
$756$	0	0
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	14.0000	0.508503
$759$	5.00000	0.181489
$760$	−1.00000	−0.0362738
$761$	−4.00000	−0.145000	−0.0724999	−	0.997368i	$-0.523098\pi$
$761$	−4.00000	−0.145000	−0.0724999	+	0.997368i	$0.523098\pi$
$762$	4.00000	0.144905
$763$	0	0
$764$	15.0000	0.542681
$765$	−5.00000	−0.180775
$766$	1.00000	0.0361315
$767$	14.0000	0.505511
$768$	−1.00000	−0.0360844
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	−18.0000	−0.647834
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	1.00000	0.0359443
$775$	8.00000	0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	1.00000	0.0358057
$781$	−2.00000	−0.0715656
$782$	25.0000	0.893998
$783$	3.00000	0.107211
$784$	0	0
$785$	11.0000	0.392607
$786$	13.0000	0.463695
$787$	35.0000	1.24762	0.623808	−	0.781578i	$-0.285585\pi$
$787$	35.0000	1.24762	0.623808	+	0.781578i	$0.285585\pi$
$788$	−14.0000	−0.498729
$789$	12.0000	0.427211
$790$	−8.00000	−0.284627
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	1.00000	0.0355110
$794$	−34.0000	−1.20661
$795$	10.0000	0.354663
$796$	−17.0000	−0.602549
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	60.0000	2.12265
$800$	4.00000	0.141421
$801$	−12.0000	−0.423999
$802$	18.0000	0.635602
$803$	−1.00000	−0.0352892
$804$	−4.00000	−0.141069
$805$	0	0
$806$	2.00000	0.0704470
$807$	−28.0000	−0.985647
$808$	6.00000	0.211079
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	1.00000	0.0351364
$811$	−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
$813$	12.0000	0.420858
$814$	11.0000	0.385550
$815$	18.0000	0.630512
$816$	−5.00000	−0.175035
$817$	1.00000	0.0349856
$818$	−21.0000	−0.734248
$819$	0	0
$820$	0	0
$821$	−40.0000	−1.39601	−0.698005	−	0.716093i	$-0.745929\pi$
$821$	−40.0000	−1.39601	−0.698005	+	0.716093i	$0.745929\pi$
$822$	−5.00000	−0.174395
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	3.00000	0.104510
$825$	4.00000	0.139262
$826$	0	0
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	−5.00000	−0.173762
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	−2.00000	−0.0694210
$831$	−12.0000	−0.416275
$832$	1.00000	0.0346688
$833$	0	0
$834$	−8.00000	−0.277017
$835$	−3.00000	−0.103819
$836$	−1.00000	−0.0345857
$837$	2.00000	0.0691301
$838$	17.0000	0.587255
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−6.00000	−0.206774
$843$	14.0000	0.482186
$844$	−3.00000	−0.103264
$845$	−1.00000	−0.0344010
$846$	−12.0000	−0.412568
$847$	0	0
$848$	10.0000	0.343401
$849$	28.0000	0.960958
$850$	20.0000	0.685994
$851$	55.0000	1.88538
$852$	2.00000	0.0685189
$853$	−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
$855$	1.00000	0.0341993
$856$	16.0000	0.546869
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	1.00000	0.0341394
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	30.0000	1.02180
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	28.0000	0.951479
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−8.00000	−0.271381
$870$	3.00000	0.101710
$871$	4.00000	0.135535
$872$	−11.0000	−0.372507
$873$	−2.00000	−0.0676897
$874$	−5.00000	−0.169128
$875$	0	0
$876$	1.00000	0.0337869
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−25.0000	−0.843709
$879$	2.00000	0.0674583
$880$	−1.00000	−0.0337100
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	5.00000	0.168168
$885$	14.0000	0.470605
$886$	−18.0000	−0.604722
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	−11.0000	−0.369136
$889$	0	0
$890$	−12.0000	−0.402241
$891$	1.00000	0.0335013
$892$	−6.00000	−0.200895
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−2.00000	−0.0668526
$896$	0	0
$897$	5.00000	0.166945
$898$	5.00000	0.166852
$899$	6.00000	0.200111
$900$	−4.00000	−0.133333
$901$	50.0000	1.66574
$902$	0	0
$903$	0	0
$904$	4.00000	0.133038
$905$	10.0000	0.332411
$906$	17.0000	0.564787
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−3.00000	−0.0993944	−0.0496972	−	0.998764i	$-0.515826\pi$
$911$	−3.00000	−0.0993944	−0.0496972	+	0.998764i	$0.515826\pi$
$912$	1.00000	0.0331133
$913$	−2.00000	−0.0661903
$914$	2.00000	0.0661541
$915$	1.00000	0.0330590
$916$	4.00000	0.132164
$917$	0	0
$918$	5.00000	0.165025
$919$	42.0000	1.38545	0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000	1.38545	0.692726	+	0.721201i	$0.256409\pi$
$920$	−5.00000	−0.164845
$921$	−20.0000	−0.659022
$922$	−37.0000	−1.21853
$923$	−2.00000	−0.0658308
$924$	0	0
$925$	44.0000	1.44671
$926$	−13.0000	−0.427207
$927$	−3.00000	−0.0985329
$928$	3.00000	0.0984798
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	−20.0000	−0.655122
$933$	−2.00000	−0.0654771
$934$	35.0000	1.14523
$935$	−5.00000	−0.163517
$936$	−1.00000	−0.0326860
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	−12.0000	−0.391397
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−11.0000	−0.358399
$943$	0	0
$944$	14.0000	0.455661
$945$	0	0
$946$	1.00000	0.0325128
$947$	17.0000	0.552426	0.276213	−	0.961096i	$-0.410921\pi$
$947$	17.0000	0.552426	0.276213	+	0.961096i	$0.410921\pi$
$948$	8.00000	0.259828
$949$	−1.00000	−0.0324614
$950$	−4.00000	−0.129777
$951$	30.0000	0.972817
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	−10.0000	−0.323762
$955$	−15.0000	−0.485389
$956$	6.00000	0.194054
$957$	3.00000	0.0969762
$958$	−21.0000	−0.678479
$959$	0	0
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	11.0000	0.354654
$963$	−16.0000	−0.515593
$964$	−22.0000	−0.708572
$965$	18.0000	0.579441
$966$	0	0
$967$	47.0000	1.51142	0.755709	−	0.654907i	$-0.227292\pi$
$967$	47.0000	1.51142	0.755709	+	0.654907i	$0.227292\pi$
$968$	10.0000	0.321412
$969$	5.00000	0.160623
$970$	−2.00000	−0.0642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	4.00000	0.128103
$976$	1.00000	0.0320092
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	−18.0000	−0.575577
$979$	−12.0000	−0.383522
$980$	0	0
$981$	11.0000	0.351203
$982$	−38.0000	−1.21263
$983$	−23.0000	−0.733586	−0.366793	−	0.930303i	$-0.619544\pi$
$983$	−23.0000	−0.733586	−0.366793	+	0.930303i	$0.619544\pi$
$984$	0	0
$985$	14.0000	0.446077
$986$	15.0000	0.477697
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	5.00000	0.158991
$990$	1.00000	0.0317821
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	2.00000	0.0635001
$993$	−20.0000	−0.634681
$994$	0	0
$995$	17.0000	0.538936
$996$	2.00000	0.0633724
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	22.0000	0.696398
$999$	11.0000	0.348025

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.a.d.1.1	✓	1	1.1	even	1	trivial
3822.2.a.o.1.1	yes	1	7.6	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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
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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