LMFDB - Embedded newform 3822.2.a.bm.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$4.27492$ of defining polynomial
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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.27492 q^{10} -2.27492 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.27492 q^{15} +1.00000 q^{16} -0.274917 q^{17} -1.00000 q^{18} +2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} +2.27492 q^{23} -1.00000 q^{24} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{27} +8.27492 q^{29} -4.27492 q^{30} -8.00000 q^{31} -1.00000 q^{32} -2.27492 q^{33} +0.274917 q^{34} +1.00000 q^{36} -4.27492 q^{37} -2.27492 q^{38} +1.00000 q^{39} -4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} -2.27492 q^{44} +4.27492 q^{45} -2.27492 q^{46} +1.00000 q^{48} -13.2749 q^{50} -0.274917 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -9.72508 q^{55} +2.27492 q^{57} -8.27492 q^{58} -8.00000 q^{59} +4.27492 q^{60} +12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.27492 q^{65} +2.27492 q^{66} +12.5498 q^{67} -0.274917 q^{68} +2.27492 q^{69} -1.00000 q^{72} +12.8248 q^{73} +4.27492 q^{74} +13.2749 q^{75} +2.27492 q^{76} -1.00000 q^{78} +12.5498 q^{79} +4.27492 q^{80} +1.00000 q^{81} +6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} +2.27492 q^{86} +8.27492 q^{87} +2.27492 q^{88} +14.0000 q^{89} -4.27492 q^{90} +2.27492 q^{92} -8.00000 q^{93} +9.72508 q^{95} -1.00000 q^{96} -15.0997 q^{97} -2.27492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + q^{15} + 2 q^{16} + 7 q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - 3 q^{22} - 3 q^{23}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 - q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 + q^15 + 2 * q^16 + 7 * q^17 - 2 * q^18 - 3 * q^19 + q^20 - 3 * q^22 - 3 * q^23 - 2 * q^24 + 19 * q^25 - 2 * q^26 + 2 * q^27 + 9 * q^29 - q^30 - 16 * q^31 - 2 * q^32 + 3 * q^33 - 7 * q^34 + 2 * q^36 - q^37 + 3 * q^38 + 2 * q^39 - q^40 + 2 * q^41 + 3 * q^43 + 3 * q^44 + q^45 + 3 * q^46 + 2 * q^48 - 19 * q^50 + 7 * q^51 + 2 * q^52 + 20 * q^53 - 2 * q^54 - 27 * q^55 - 3 * q^57 - 9 * q^58 - 16 * q^59 + q^60 + 17 * q^61 + 16 * q^62 + 2 * q^64 + q^65 - 3 * q^66 + 10 * q^67 + 7 * q^68 - 3 * q^69 - 2 * q^72 + 3 * q^73 + q^74 + 19 * q^75 - 3 * q^76 - 2 * q^78 + 10 * q^79 + q^80 + 2 * q^81 - 2 * q^82 - 6 * q^83 - 25 * q^85 - 3 * q^86 + 9 * q^87 - 3 * q^88 + 28 * q^89 - q^90 - 3 * q^92 - 16 * q^93 + 27 * q^95 - 2 * q^96 + 3 * q^99

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
$5$	4.27492	1.91180	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	4.27492	1.91180	0.955901	+	0.293691i	$0.0948835\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−4.27492	−1.35185
$11$	−2.27492	−0.685913	−0.342957	−	0.939351i	$-0.611428\pi$
$11$	−2.27492	−0.685913	−0.342957	+	0.939351i	$0.611428\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	4.27492	1.10378
$16$	1.00000	0.250000
$17$	−0.274917	−0.0666772	−0.0333386	−	0.999444i	$-0.510614\pi$
$17$	−0.274917	−0.0666772	−0.0333386	+	0.999444i	$0.510614\pi$
$18$	−1.00000	−0.235702
$19$	2.27492	0.521902	0.260951	−	0.965352i	$-0.415964\pi$
$19$	2.27492	0.521902	0.260951	+	0.965352i	$0.415964\pi$
$20$	4.27492	0.955901
$21$	0	0
$22$	2.27492	0.485014
$23$	2.27492	0.474353	0.237177	−	0.971467i	$-0.423778\pi$
$23$	2.27492	0.474353	0.237177	+	0.971467i	$0.423778\pi$
$24$	−1.00000	−0.204124
$25$	13.2749	2.65498
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	8.27492	1.53661	0.768307	−	0.640082i	$-0.221100\pi$
$29$	8.27492	1.53661	0.768307	+	0.640082i	$0.221100\pi$
$30$	−4.27492	−0.780490
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−2.27492	−0.396012
$34$	0.274917	0.0471479
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.27492	−0.702792	−0.351396	−	0.936227i	$-0.614293\pi$
$37$	−4.27492	−0.702792	−0.351396	+	0.936227i	$0.614293\pi$
$38$	−2.27492	−0.369040
$39$	1.00000	0.160128
$40$	−4.27492	−0.675924
$41$	−6.54983	−1.02291	−0.511456	−	0.859309i	$-0.670894\pi$
$41$	−6.54983	−1.02291	−0.511456	+	0.859309i	$0.670894\pi$
$42$	0	0
$43$	−2.27492	−0.346922	−0.173461	−	0.984841i	$-0.555495\pi$
$43$	−2.27492	−0.346922	−0.173461	+	0.984841i	$0.555495\pi$
$44$	−2.27492	−0.342957
$45$	4.27492	0.637267
$46$	−2.27492	−0.335418
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−13.2749	−1.87736
$51$	−0.274917	−0.0384961
$52$	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
$55$	−9.72508	−1.31133
$56$	0	0
$57$	2.27492	0.301320
$58$	−8.27492	−1.08655
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	4.27492	0.551889
$61$	12.2749	1.57164	0.785821	−	0.618454i	$-0.212241\pi$
$61$	12.2749	1.57164	0.785821	+	0.618454i	$0.212241\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	4.27492	0.530238
$66$	2.27492	0.280023
$67$	12.5498	1.53321	0.766603	−	0.642121i	$-0.221945\pi$
$67$	12.5498	1.53321	0.766603	+	0.642121i	$0.221945\pi$
$68$	−0.274917	−0.0333386
$69$	2.27492	0.273868
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	12.8248	1.50102	0.750512	−	0.660857i	$-0.229807\pi$
$73$	12.8248	1.50102	0.750512	+	0.660857i	$0.229807\pi$
$74$	4.27492	0.496949
$75$	13.2749	1.53286
$76$	2.27492	0.260951
$77$	0	0
$78$	−1.00000	−0.113228
$79$	12.5498	1.41197	0.705983	−	0.708228i	$-0.250505\pi$
$79$	12.5498	1.41197	0.705983	+	0.708228i	$0.250505\pi$
$80$	4.27492	0.477950
$81$	1.00000	0.111111
$82$	6.54983	0.723308
$83$	4.54983	0.499409	0.249705	−	0.968322i	$-0.419666\pi$
$83$	4.54983	0.499409	0.249705	+	0.968322i	$0.419666\pi$
$84$	0	0
$85$	−1.17525	−0.127474
$86$	2.27492	0.245311
$87$	8.27492	0.887164
$88$	2.27492	0.242507
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−4.27492	−0.450616
$91$	0	0
$92$	2.27492	0.237177
$93$	−8.00000	−0.829561
$94$	0	0
$95$	9.72508	0.997772
$96$	−1.00000	−0.102062
$97$	−15.0997	−1.53314	−0.766570	−	0.642161i	$-0.778038\pi$
$97$	−15.0997	−1.53314	−0.766570	+	0.642161i	$0.778038\pi$
$98$	0	0
$99$	−2.27492	−0.228638
$100$	13.2749	1.32749
$101$	−2.54983	−0.253718	−0.126859	−	0.991921i	$-0.540490\pi$
$101$	−2.54983	−0.253718	−0.126859	+	0.991921i	$0.540490\pi$
$102$	0.274917	0.0272209
$103$	−10.2749	−1.01242	−0.506209	−	0.862411i	$-0.668954\pi$
$103$	−10.2749	−1.01242	−0.506209	+	0.862411i	$0.668954\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	0.549834	0.0531545	0.0265773	−	0.999647i	$-0.491539\pi$
$107$	0.549834	0.0531545	0.0265773	+	0.999647i	$0.491539\pi$
$108$	1.00000	0.0962250
$109$	0.274917	0.0263323	0.0131661	−	0.999913i	$-0.495809\pi$
$109$	0.274917	0.0263323	0.0131661	+	0.999913i	$0.495809\pi$
$110$	9.72508	0.927250
$111$	−4.27492	−0.405757
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.27492	−0.213066
$115$	9.72508	0.906869
$116$	8.27492	0.768307
$117$	1.00000	0.0924500
$118$	8.00000	0.736460
$119$	0	0
$120$	−4.27492	−0.390245
$121$	−5.82475	−0.529523
$122$	−12.2749	−1.11132
$123$	−6.54983	−0.590579
$124$	−8.00000	−0.718421
$125$	35.3746	3.16400
$126$	0	0
$127$	−20.5498	−1.82350	−0.911751	−	0.410742i	$-0.865270\pi$
$127$	−20.5498	−1.82350	−0.911751	+	0.410742i	$0.865270\pi$
$128$	−1.00000	−0.0883883
$129$	−2.27492	−0.200295
$130$	−4.27492	−0.374935
$131$	−6.82475	−0.596281	−0.298141	−	0.954522i	$-0.596366\pi$
$131$	−6.82475	−0.596281	−0.298141	+	0.954522i	$0.596366\pi$
$132$	−2.27492	−0.198006
$133$	0	0
$134$	−12.5498	−1.08414
$135$	4.27492	0.367926
$136$	0.274917	0.0235740
$137$	−17.3746	−1.48441	−0.742206	−	0.670172i	$-0.766220\pi$
$137$	−17.3746	−1.48441	−0.742206	+	0.670172i	$0.766220\pi$
$138$	−2.27492	−0.193654
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.27492	−0.190238
$144$	1.00000	0.0833333
$145$	35.3746	2.93770
$146$	−12.8248	−1.06138
$147$	0	0
$148$	−4.27492	−0.351396
$149$	22.5498	1.84735	0.923677	−	0.383172i	$-0.125168\pi$
$149$	22.5498	1.84735	0.923677	+	0.383172i	$0.125168\pi$
$150$	−13.2749	−1.08389
$151$	6.27492	0.510646	0.255323	−	0.966856i	$-0.417818\pi$
$151$	6.27492	0.510646	0.255323	+	0.966856i	$0.417818\pi$
$152$	−2.27492	−0.184520
$153$	−0.274917	−0.0222257
$154$	0	0
$155$	−34.1993	−2.74696
$156$	1.00000	0.0800641
$157$	13.3746	1.06741	0.533704	−	0.845671i	$-0.320800\pi$
$157$	13.3746	1.06741	0.533704	+	0.845671i	$0.320800\pi$
$158$	−12.5498	−0.998411
$159$	10.0000	0.793052
$160$	−4.27492	−0.337962
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	12.5498	0.982979	0.491489	−	0.870884i	$-0.336453\pi$
$163$	12.5498	0.982979	0.491489	+	0.870884i	$0.336453\pi$
$164$	−6.54983	−0.511456
$165$	−9.72508	−0.757097
$166$	−4.54983	−0.353136
$167$	1.72508	0.133491	0.0667455	−	0.997770i	$-0.478738\pi$
$167$	1.72508	0.133491	0.0667455	+	0.997770i	$0.478738\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.17525	0.0901374
$171$	2.27492	0.173967
$172$	−2.27492	−0.173461
$173$	−7.09967	−0.539778	−0.269889	−	0.962891i	$-0.586987\pi$
$173$	−7.09967	−0.539778	−0.269889	+	0.962891i	$0.586987\pi$
$174$	−8.27492	−0.627320
$175$	0	0
$176$	−2.27492	−0.171478
$177$	−8.00000	−0.601317
$178$	−14.0000	−1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	4.27492	0.318634
$181$	−19.0997	−1.41967	−0.709834	−	0.704369i	$-0.751230\pi$
$181$	−19.0997	−1.41967	−0.709834	+	0.704369i	$0.751230\pi$
$182$	0	0
$183$	12.2749	0.907388
$184$	−2.27492	−0.167709
$185$	−18.2749	−1.34360
$186$	8.00000	0.586588
$187$	0.625414	0.0457348
$188$	0	0
$189$	0	0
$190$	−9.72508	−0.705532
$191$	2.27492	0.164607	0.0823036	−	0.996607i	$-0.473772\pi$
$191$	2.27492	0.164607	0.0823036	+	0.996607i	$0.473772\pi$
$192$	1.00000	0.0721688
$193$	−18.5498	−1.33525	−0.667623	−	0.744499i	$-0.732688\pi$
$193$	−18.5498	−1.33525	−0.667623	+	0.744499i	$0.732688\pi$
$194$	15.0997	1.08409
$195$	4.27492	0.306133
$196$	0	0
$197$	−2.54983	−0.181668	−0.0908341	−	0.995866i	$-0.528953\pi$
$197$	−2.54983	−0.181668	−0.0908341	+	0.995866i	$0.528953\pi$
$198$	2.27492	0.161671
$199$	−6.82475	−0.483794	−0.241897	−	0.970302i	$-0.577770\pi$
$199$	−6.82475	−0.483794	−0.241897	+	0.970302i	$0.577770\pi$
$200$	−13.2749	−0.938678
$201$	12.5498	0.885197
$202$	2.54983	0.179406
$203$	0	0
$204$	−0.274917	−0.0192481
$205$	−28.0000	−1.95560
$206$	10.2749	0.715887
$207$	2.27492	0.158118
$208$	1.00000	0.0693375
$209$	−5.17525	−0.357979
$210$	0	0
$211$	1.17525	0.0809074	0.0404537	−	0.999181i	$-0.487120\pi$
$211$	1.17525	0.0809074	0.0404537	+	0.999181i	$0.487120\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	−0.549834	−0.0375859
$215$	−9.72508	−0.663245
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−0.274917	−0.0186197
$219$	12.8248	0.866616
$220$	−9.72508	−0.655665
$221$	−0.274917	−0.0184929
$222$	4.27492	0.286914
$223$	−21.6495	−1.44976	−0.724879	−	0.688876i	$-0.758104\pi$
$223$	−21.6495	−1.44976	−0.724879	+	0.688876i	$0.758104\pi$
$224$	0	0
$225$	13.2749	0.884994
$226$	6.00000	0.399114
$227$	20.5498	1.36394	0.681970	−	0.731380i	$-0.261123\pi$
$227$	20.5498	1.36394	0.681970	+	0.731380i	$0.261123\pi$
$228$	2.27492	0.150660
$229$	−19.0997	−1.26214	−0.631071	−	0.775725i	$-0.717384\pi$
$229$	−19.0997	−1.26214	−0.631071	+	0.775725i	$0.717384\pi$
$230$	−9.72508	−0.641253
$231$	0	0
$232$	−8.27492	−0.543275
$233$	5.45017	0.357052	0.178526	−	0.983935i	$-0.442867\pi$
$233$	5.45017	0.357052	0.178526	+	0.983935i	$0.442867\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−8.00000	−0.520756
$237$	12.5498	0.815199
$238$	0	0
$239$	25.0997	1.62356	0.811781	−	0.583962i	$-0.198498\pi$
$239$	25.0997	1.62356	0.811781	+	0.583962i	$0.198498\pi$
$240$	4.27492	0.275945
$241$	−15.0997	−0.972655	−0.486328	−	0.873777i	$-0.661664\pi$
$241$	−15.0997	−0.972655	−0.486328	+	0.873777i	$0.661664\pi$
$242$	5.82475	0.374429
$243$	1.00000	0.0641500
$244$	12.2749	0.785821
$245$	0	0
$246$	6.54983	0.417602
$247$	2.27492	0.144750
$248$	8.00000	0.508001
$249$	4.54983	0.288334
$250$	−35.3746	−2.23729
$251$	−23.9244	−1.51010	−0.755048	−	0.655669i	$-0.772386\pi$
$251$	−23.9244	−1.51010	−0.755048	+	0.655669i	$0.772386\pi$
$252$	0	0
$253$	−5.17525	−0.325365
$254$	20.5498	1.28941
$255$	−1.17525	−0.0735969
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	2.27492	0.141630
$259$	0	0
$260$	4.27492	0.265119
$261$	8.27492	0.512205
$262$	6.82475	0.421635
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	2.27492	0.140011
$265$	42.7492	2.62606
$266$	0	0
$267$	14.0000	0.856786
$268$	12.5498	0.766603
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−4.27492	−0.260163
$271$	−28.5498	−1.73428	−0.867139	−	0.498065i	$-0.834044\pi$
$271$	−28.5498	−1.73428	−0.867139	+	0.498065i	$0.834044\pi$
$272$	−0.274917	−0.0166693
$273$	0	0
$274$	17.3746	1.04964
$275$	−30.1993	−1.82109
$276$	2.27492	0.136934
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	15.0997	0.900771	0.450385	−	0.892834i	$-0.351287\pi$
$281$	15.0997	0.900771	0.450385	+	0.892834i	$0.351287\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	9.72508	0.576064
$286$	2.27492	0.134519
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−16.9244	−0.995554
$290$	−35.3746	−2.07727
$291$	−15.0997	−0.885158
$292$	12.8248	0.750512
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−34.1993	−1.99116
$296$	4.27492	0.248475
$297$	−2.27492	−0.132004
$298$	−22.5498	−1.30628
$299$	2.27492	0.131562
$300$	13.2749	0.766428
$301$	0	0
$302$	−6.27492	−0.361081
$303$	−2.54983	−0.146484
$304$	2.27492	0.130475
$305$	52.4743	3.00467
$306$	0.274917	0.0157160
$307$	−21.0997	−1.20422	−0.602111	−	0.798412i	$-0.705673\pi$
$307$	−21.0997	−1.20422	−0.602111	+	0.798412i	$0.705673\pi$
$308$	0	0
$309$	−10.2749	−0.584520
$310$	34.1993	1.94239
$311$	−3.45017	−0.195641	−0.0978205	−	0.995204i	$-0.531187\pi$
$311$	−3.45017	−0.195641	−0.0978205	+	0.995204i	$0.531187\pi$
$312$	−1.00000	−0.0566139
$313$	27.6495	1.56284	0.781421	−	0.624004i	$-0.214495\pi$
$313$	27.6495	1.56284	0.781421	+	0.624004i	$0.214495\pi$
$314$	−13.3746	−0.754772
$315$	0	0
$316$	12.5498	0.705983
$317$	−1.45017	−0.0814494	−0.0407247	−	0.999170i	$-0.512967\pi$
$317$	−1.45017	−0.0814494	−0.0407247	+	0.999170i	$0.512967\pi$
$318$	−10.0000	−0.560772
$319$	−18.8248	−1.05398
$320$	4.27492	0.238975
$321$	0.549834	0.0306888
$322$	0	0
$323$	−0.625414	−0.0347990
$324$	1.00000	0.0555556
$325$	13.2749	0.736360
$326$	−12.5498	−0.695071
$327$	0.274917	0.0152030
$328$	6.54983	0.361654
$329$	0	0
$330$	9.72508	0.535348
$331$	−29.6495	−1.62968	−0.814842	−	0.579683i	$-0.803176\pi$
$331$	−29.6495	−1.62968	−0.814842	+	0.579683i	$0.803176\pi$
$332$	4.54983	0.249705
$333$	−4.27492	−0.234264
$334$	−1.72508	−0.0943923
$335$	53.6495	2.93119
$336$	0	0
$337$	9.37459	0.510666	0.255333	−	0.966853i	$-0.417815\pi$
$337$	9.37459	0.510666	0.255333	+	0.966853i	$0.417815\pi$
$338$	−1.00000	−0.0543928
$339$	−6.00000	−0.325875
$340$	−1.17525	−0.0637368
$341$	18.1993	0.985549
$342$	−2.27492	−0.123013
$343$	0	0
$344$	2.27492	0.122655
$345$	9.72508	0.523581
$346$	7.09967	0.381681
$347$	5.09967	0.273765	0.136882	−	0.990587i	$-0.456292\pi$
$347$	5.09967	0.273765	0.136882	+	0.990587i	$0.456292\pi$
$348$	8.27492	0.443582
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	2.27492	0.121253
$353$	−27.0997	−1.44237	−0.721185	−	0.692743i	$-0.756402\pi$
$353$	−27.0997	−1.44237	−0.721185	+	0.692743i	$0.756402\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	−4.27492	−0.225308
$361$	−13.8248	−0.727619
$362$	19.0997	1.00386
$363$	−5.82475	−0.305720
$364$	0	0
$365$	54.8248	2.86966
$366$	−12.2749	−0.641620
$367$	2.90033	0.151396	0.0756980	−	0.997131i	$-0.475881\pi$
$367$	2.90033	0.151396	0.0756980	+	0.997131i	$0.475881\pi$
$368$	2.27492	0.118588
$369$	−6.54983	−0.340971
$370$	18.2749	0.950068
$371$	0	0
$372$	−8.00000	−0.414781
$373$	26.5498	1.37470	0.687349	−	0.726327i	$-0.258774\pi$
$373$	26.5498	1.37470	0.687349	+	0.726327i	$0.258774\pi$
$374$	−0.625414	−0.0323394
$375$	35.3746	1.82674
$376$	0	0
$377$	8.27492	0.426180
$378$	0	0
$379$	−33.0997	−1.70022	−0.850108	−	0.526609i	$-0.823463\pi$
$379$	−33.0997	−1.70022	−0.850108	+	0.526609i	$0.823463\pi$
$380$	9.72508	0.498886
$381$	−20.5498	−1.05280
$382$	−2.27492	−0.116395
$383$	−30.2749	−1.54698	−0.773488	−	0.633811i	$-0.781490\pi$
$383$	−30.2749	−1.54698	−0.773488	+	0.633811i	$0.781490\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	18.5498	0.944162
$387$	−2.27492	−0.115641
$388$	−15.0997	−0.766570
$389$	28.1993	1.42976	0.714882	−	0.699246i	$-0.246481\pi$
$389$	28.1993	1.42976	0.714882	+	0.699246i	$0.246481\pi$
$390$	−4.27492	−0.216469
$391$	−0.625414	−0.0316285
$392$	0	0
$393$	−6.82475	−0.344263
$394$	2.54983	0.128459
$395$	53.6495	2.69940
$396$	−2.27492	−0.114319
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	6.82475	0.342094
$399$	0	0
$400$	13.2749	0.663746
$401$	−3.09967	−0.154790	−0.0773950	−	0.997001i	$-0.524660\pi$
$401$	−3.09967	−0.154790	−0.0773950	+	0.997001i	$0.524660\pi$
$402$	−12.5498	−0.625929
$403$	−8.00000	−0.398508
$404$	−2.54983	−0.126859
$405$	4.27492	0.212422
$406$	0	0
$407$	9.72508	0.482054
$408$	0.274917	0.0136104
$409$	7.17525	0.354793	0.177397	−	0.984139i	$-0.443232\pi$
$409$	7.17525	0.354793	0.177397	+	0.984139i	$0.443232\pi$
$410$	28.0000	1.38282
$411$	−17.3746	−0.857025
$412$	−10.2749	−0.506209
$413$	0	0
$414$	−2.27492	−0.111806
$415$	19.4502	0.954771
$416$	−1.00000	−0.0490290
$417$	−4.00000	−0.195881
$418$	5.17525	0.253130
$419$	−2.27492	−0.111137	−0.0555685	−	0.998455i	$-0.517697\pi$
$419$	−2.27492	−0.111137	−0.0555685	+	0.998455i	$0.517697\pi$
$420$	0	0
$421$	3.09967	0.151069	0.0755343	−	0.997143i	$-0.475934\pi$
$421$	3.09967	0.151069	0.0755343	+	0.997143i	$0.475934\pi$
$422$	−1.17525	−0.0572102
$423$	0	0
$424$	−10.0000	−0.485643
$425$	−3.64950	−0.177027
$426$	0	0
$427$	0	0
$428$	0.549834	0.0265773
$429$	−2.27492	−0.109834
$430$	9.72508	0.468985
$431$	−11.4502	−0.551535	−0.275768	−	0.961224i	$-0.588932\pi$
$431$	−11.4502	−0.551535	−0.275768	+	0.961224i	$0.588932\pi$
$432$	1.00000	0.0481125
$433$	−23.6495	−1.13652	−0.568261	−	0.822848i	$-0.692384\pi$
$433$	−23.6495	−1.13652	−0.568261	+	0.822848i	$0.692384\pi$
$434$	0	0
$435$	35.3746	1.69608
$436$	0.274917	0.0131661
$437$	5.17525	0.247566
$438$	−12.8248	−0.612790
$439$	14.8248	0.707547	0.353773	−	0.935331i	$-0.384898\pi$
$439$	14.8248	0.707547	0.353773	+	0.935331i	$0.384898\pi$
$440$	9.72508	0.463625
$441$	0	0
$442$	0.274917	0.0130765
$443$	7.45017	0.353968	0.176984	−	0.984214i	$-0.443366\pi$
$443$	7.45017	0.353968	0.176984	+	0.984214i	$0.443366\pi$
$444$	−4.27492	−0.202879
$445$	59.8488	2.83711
$446$	21.6495	1.02513
$447$	22.5498	1.06657
$448$	0	0
$449$	−0.274917	−0.0129741	−0.00648707	−	0.999979i	$-0.502065\pi$
$449$	−0.274917	−0.0129741	−0.00648707	+	0.999979i	$0.502065\pi$
$450$	−13.2749	−0.625786
$451$	14.9003	0.701629
$452$	−6.00000	−0.282216
$453$	6.27492	0.294821
$454$	−20.5498	−0.964452
$455$	0	0
$456$	−2.27492	−0.106533
$457$	−18.5498	−0.867725	−0.433862	−	0.900979i	$-0.642850\pi$
$457$	−18.5498	−0.867725	−0.433862	+	0.900979i	$0.642850\pi$
$458$	19.0997	0.892469
$459$	−0.274917	−0.0128320
$460$	9.72508	0.453434
$461$	17.9244	0.834823	0.417412	−	0.908717i	$-0.362937\pi$
$461$	17.9244	0.834823	0.417412	+	0.908717i	$0.362937\pi$
$462$	0	0
$463$	30.2749	1.40699	0.703497	−	0.710698i	$-0.251621\pi$
$463$	30.2749	1.40699	0.703497	+	0.710698i	$0.251621\pi$
$464$	8.27492	0.384153
$465$	−34.1993	−1.58596
$466$	−5.45017	−0.252474
$467$	26.2749	1.21586	0.607929	−	0.793991i	$-0.292000\pi$
$467$	26.2749	1.21586	0.607929	+	0.793991i	$0.292000\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	13.3746	0.616268
$472$	8.00000	0.368230
$473$	5.17525	0.237958
$474$	−12.5498	−0.576433
$475$	30.1993	1.38564
$476$	0	0
$477$	10.0000	0.457869
$478$	−25.0997	−1.14803
$479$	23.3746	1.06801	0.534006	−	0.845481i	$-0.320686\pi$
$479$	23.3746	1.06801	0.534006	+	0.845481i	$0.320686\pi$
$480$	−4.27492	−0.195122
$481$	−4.27492	−0.194919
$482$	15.0997	0.687771
$483$	0	0
$484$	−5.82475	−0.264761
$485$	−64.5498	−2.93106
$486$	−1.00000	−0.0453609
$487$	−18.1993	−0.824691	−0.412345	−	0.911028i	$-0.635290\pi$
$487$	−18.1993	−0.824691	−0.412345	+	0.911028i	$0.635290\pi$
$488$	−12.2749	−0.555659
$489$	12.5498	0.567523
$490$	0	0
$491$	8.54983	0.385849	0.192924	−	0.981214i	$-0.438203\pi$
$491$	8.54983	0.385849	0.192924	+	0.981214i	$0.438203\pi$
$492$	−6.54983	−0.295289
$493$	−2.27492	−0.102457
$494$	−2.27492	−0.102353
$495$	−9.72508	−0.437110
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−4.54983	−0.203883
$499$	25.0997	1.12362	0.561808	−	0.827268i	$-0.310106\pi$
$499$	25.0997	1.12362	0.561808	+	0.827268i	$0.310106\pi$
$500$	35.3746	1.58200
$501$	1.72508	0.0770710
$502$	23.9244	1.06780
$503$	3.45017	0.153835	0.0769176	−	0.997037i	$-0.475492\pi$
$503$	3.45017	0.153835	0.0769176	+	0.997037i	$0.475492\pi$
$504$	0	0
$505$	−10.9003	−0.485058
$506$	5.17525	0.230068
$507$	1.00000	0.0444116
$508$	−20.5498	−0.911751
$509$	9.92442	0.439892	0.219946	−	0.975512i	$-0.429412\pi$
$509$	9.92442	0.439892	0.219946	+	0.975512i	$0.429412\pi$
$510$	1.17525	0.0520409
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.27492	0.100440
$514$	−14.0000	−0.617514
$515$	−43.9244	−1.93554
$516$	−2.27492	−0.100148
$517$	0	0
$518$	0	0
$519$	−7.09967	−0.311641
$520$	−4.27492	−0.187468
$521$	4.27492	0.187288	0.0936438	−	0.995606i	$-0.470149\pi$
$521$	4.27492	0.187288	0.0936438	+	0.995606i	$0.470149\pi$
$522$	−8.27492	−0.362183
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−6.82475	−0.298141
$525$	0	0
$526$	−28.0000	−1.22086
$527$	2.19934	0.0958047
$528$	−2.27492	−0.0990031
$529$	−17.8248	−0.774989
$530$	−42.7492	−1.85691
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−6.54983	−0.283705
$534$	−14.0000	−0.605839
$535$	2.35050	0.101621
$536$	−12.5498	−0.542070
$537$	12.0000	0.517838
$538$	6.00000	0.258678
$539$	0	0
$540$	4.27492	0.183963
$541$	42.4743	1.82611	0.913055	−	0.407835i	$-0.133716\pi$
$541$	42.4743	1.82611	0.913055	+	0.407835i	$0.133716\pi$
$542$	28.5498	1.22632
$543$	−19.0997	−0.819645
$544$	0.274917	0.0117870
$545$	1.17525	0.0503421
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−17.3746	−0.742206
$549$	12.2749	0.523881
$550$	30.1993	1.28770
$551$	18.8248	0.801961
$552$	−2.27492	−0.0968269
$553$	0	0
$554$	10.0000	0.424859
$555$	−18.2749	−0.775727
$556$	−4.00000	−0.169638
$557$	−44.7492	−1.89608	−0.948042	−	0.318146i	$-0.896940\pi$
$557$	−44.7492	−1.89608	−0.948042	+	0.318146i	$0.896940\pi$
$558$	8.00000	0.338667
$559$	−2.27492	−0.0962187
$560$	0	0
$561$	0.625414	0.0264050
$562$	−15.0997	−0.636941
$563$	35.3746	1.49086	0.745431	−	0.666583i	$-0.232244\pi$
$563$	35.3746	1.49086	0.745431	+	0.666583i	$0.232244\pi$
$564$	0	0
$565$	−25.6495	−1.07908
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	3.09967	0.129945	0.0649724	−	0.997887i	$-0.479304\pi$
$569$	3.09967	0.129945	0.0649724	+	0.997887i	$0.479304\pi$
$570$	−9.72508	−0.407339
$571$	−37.0997	−1.55257	−0.776286	−	0.630380i	$-0.782899\pi$
$571$	−37.0997	−1.55257	−0.776286	+	0.630380i	$0.782899\pi$
$572$	−2.27492	−0.0951191
$573$	2.27492	0.0950360
$574$	0	0
$575$	30.1993	1.25940
$576$	1.00000	0.0416667
$577$	8.90033	0.370526	0.185263	−	0.982689i	$-0.440686\pi$
$577$	8.90033	0.370526	0.185263	+	0.982689i	$0.440686\pi$
$578$	16.9244	0.703963
$579$	−18.5498	−0.770905
$580$	35.3746	1.46885
$581$	0	0
$582$	15.0997	0.625901
$583$	−22.7492	−0.942174
$584$	−12.8248	−0.530692
$585$	4.27492	0.176746
$586$	−6.00000	−0.247858
$587$	−46.7492	−1.92954	−0.964772	−	0.263086i	$-0.915260\pi$
$587$	−46.7492	−1.92954	−0.964772	+	0.263086i	$0.915260\pi$
$588$	0	0
$589$	−18.1993	−0.749891
$590$	34.1993	1.40796
$591$	−2.54983	−0.104886
$592$	−4.27492	−0.175698
$593$	7.09967	0.291548	0.145774	−	0.989318i	$-0.453433\pi$
$593$	7.09967	0.291548	0.145774	+	0.989318i	$0.453433\pi$
$594$	2.27492	0.0933410
$595$	0	0
$596$	22.5498	0.923677
$597$	−6.82475	−0.279318
$598$	−2.27492	−0.0930283
$599$	21.7251	0.887663	0.443831	−	0.896110i	$-0.353619\pi$
$599$	21.7251	0.887663	0.443831	+	0.896110i	$0.353619\pi$
$600$	−13.2749	−0.541946
$601$	−23.6495	−0.964683	−0.482342	−	0.875983i	$-0.660214\pi$
$601$	−23.6495	−0.964683	−0.482342	+	0.875983i	$0.660214\pi$
$602$	0	0
$603$	12.5498	0.511069
$604$	6.27492	0.255323
$605$	−24.9003	−1.01234
$606$	2.54983	0.103580
$607$	−9.17525	−0.372412	−0.186206	−	0.982511i	$-0.559619\pi$
$607$	−9.17525	−0.372412	−0.186206	+	0.982511i	$0.559619\pi$
$608$	−2.27492	−0.0922601
$609$	0	0
$610$	−52.4743	−2.12462
$611$	0	0
$612$	−0.274917	−0.0111129
$613$	16.2749	0.657338	0.328669	−	0.944445i	$-0.393400\pi$
$613$	16.2749	0.657338	0.328669	+	0.944445i	$0.393400\pi$
$614$	21.0997	0.851513
$615$	−28.0000	−1.12907
$616$	0	0
$617$	−20.8248	−0.838373	−0.419186	−	0.907900i	$-0.637685\pi$
$617$	−20.8248	−0.838373	−0.419186	+	0.907900i	$0.637685\pi$
$618$	10.2749	0.413318
$619$	−29.7251	−1.19475	−0.597376	−	0.801961i	$-0.703790\pi$
$619$	−29.7251	−1.19475	−0.597376	+	0.801961i	$0.703790\pi$
$620$	−34.1993	−1.37348
$621$	2.27492	0.0912893
$622$	3.45017	0.138339
$623$	0	0
$624$	1.00000	0.0400320
$625$	84.8488	3.39395
$626$	−27.6495	−1.10510
$627$	−5.17525	−0.206680
$628$	13.3746	0.533704
$629$	1.17525	0.0468602
$630$	0	0
$631$	−10.8248	−0.430927	−0.215463	−	0.976512i	$-0.569126\pi$
$631$	−10.8248	−0.430927	−0.215463	+	0.976512i	$0.569126\pi$
$632$	−12.5498	−0.499206
$633$	1.17525	0.0467119
$634$	1.45017	0.0575934
$635$	−87.8488	−3.48617
$636$	10.0000	0.396526
$637$	0	0
$638$	18.8248	0.745279
$639$	0	0
$640$	−4.27492	−0.168981
$641$	−15.0997	−0.596401	−0.298201	−	0.954503i	$-0.596386\pi$
$641$	−15.0997	−0.596401	−0.298201	+	0.954503i	$0.596386\pi$
$642$	−0.549834	−0.0217002
$643$	23.9244	0.943487	0.471744	−	0.881736i	$-0.343625\pi$
$643$	23.9244	0.943487	0.471744	+	0.881736i	$0.343625\pi$
$644$	0	0
$645$	−9.72508	−0.382925
$646$	0.625414	0.0246066
$647$	18.1993	0.715490	0.357745	−	0.933819i	$-0.383546\pi$
$647$	18.1993	0.715490	0.357745	+	0.933819i	$0.383546\pi$
$648$	−1.00000	−0.0392837
$649$	18.1993	0.714386
$650$	−13.2749	−0.520685
$651$	0	0
$652$	12.5498	0.491489
$653$	−5.37459	−0.210324	−0.105162	−	0.994455i	$-0.533536\pi$
$653$	−5.37459	−0.210324	−0.105162	+	0.994455i	$0.533536\pi$
$654$	−0.274917	−0.0107501
$655$	−29.1752	−1.13997
$656$	−6.54983	−0.255728
$657$	12.8248	0.500341
$658$	0	0
$659$	−7.45017	−0.290217	−0.145109	−	0.989416i	$-0.546353\pi$
$659$	−7.45017	−0.290217	−0.145109	+	0.989416i	$0.546353\pi$
$660$	−9.72508	−0.378548
$661$	40.1993	1.56357	0.781787	−	0.623546i	$-0.214309\pi$
$661$	40.1993	1.56357	0.781787	+	0.623546i	$0.214309\pi$
$662$	29.6495	1.15236
$663$	−0.274917	−0.0106769
$664$	−4.54983	−0.176568
$665$	0	0
$666$	4.27492	0.165650
$667$	18.8248	0.728897
$668$	1.72508	0.0667455
$669$	−21.6495	−0.837018
$670$	−53.6495	−2.07266
$671$	−27.9244	−1.07801
$672$	0	0
$673$	16.2749	0.627352	0.313676	−	0.949530i	$-0.398439\pi$
$673$	16.2749	0.627352	0.313676	+	0.949530i	$0.398439\pi$
$674$	−9.37459	−0.361096
$675$	13.2749	0.510952
$676$	1.00000	0.0384615
$677$	−16.1993	−0.622591	−0.311296	−	0.950313i	$-0.600763\pi$
$677$	−16.1993	−0.622591	−0.311296	+	0.950313i	$0.600763\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	1.17525	0.0450687
$681$	20.5498	0.787471
$682$	−18.1993	−0.696889
$683$	3.37459	0.129125	0.0645625	−	0.997914i	$-0.479435\pi$
$683$	3.37459	0.129125	0.0645625	+	0.997914i	$0.479435\pi$
$684$	2.27492	0.0869836
$685$	−74.2749	−2.83790
$686$	0	0
$687$	−19.0997	−0.728698
$688$	−2.27492	−0.0867304
$689$	10.0000	0.380970
$690$	−9.72508	−0.370228
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−7.09967	−0.269889
$693$	0	0
$694$	−5.09967	−0.193581
$695$	−17.0997	−0.648627
$696$	−8.27492	−0.313660
$697$	1.80066	0.0682049
$698$	2.00000	0.0757011
$699$	5.45017	0.206144
$700$	0	0
$701$	−15.0997	−0.570307	−0.285153	−	0.958482i	$-0.592045\pi$
$701$	−15.0997	−0.570307	−0.285153	+	0.958482i	$0.592045\pi$
$702$	−1.00000	−0.0377426
$703$	−9.72508	−0.366788
$704$	−2.27492	−0.0857392
$705$	0	0
$706$	27.0997	1.01991
$707$	0	0
$708$	−8.00000	−0.300658
$709$	−23.0997	−0.867526	−0.433763	−	0.901027i	$-0.642815\pi$
$709$	−23.0997	−0.867526	−0.433763	+	0.901027i	$0.642815\pi$
$710$	0	0
$711$	12.5498	0.470656
$712$	−14.0000	−0.524672
$713$	−18.1993	−0.681571
$714$	0	0
$715$	−9.72508	−0.363697
$716$	12.0000	0.448461
$717$	25.0997	0.937364
$718$	32.0000	1.19423
$719$	29.6495	1.10574	0.552870	−	0.833268i	$-0.313533\pi$
$719$	29.6495	1.10574	0.552870	+	0.833268i	$0.313533\pi$
$720$	4.27492	0.159317
$721$	0	0
$722$	13.8248	0.514504
$723$	−15.0997	−0.561563
$724$	−19.0997	−0.709834
$725$	109.849	4.07968
$726$	5.82475	0.216177
$727$	−35.3746	−1.31197	−0.655985	−	0.754774i	$-0.727747\pi$
$727$	−35.3746	−1.31197	−0.655985	+	0.754774i	$0.727747\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−54.8248	−2.02916
$731$	0.625414	0.0231318
$732$	12.2749	0.453694
$733$	−14.5498	−0.537410	−0.268705	−	0.963222i	$-0.586596\pi$
$733$	−14.5498	−0.537410	−0.268705	+	0.963222i	$0.586596\pi$
$734$	−2.90033	−0.107053
$735$	0	0
$736$	−2.27492	−0.0838546
$737$	−28.5498	−1.05165
$738$	6.54983	0.241103
$739$	37.6495	1.38496	0.692480	−	0.721437i	$-0.256518\pi$
$739$	37.6495	1.38496	0.692480	+	0.721437i	$0.256518\pi$
$740$	−18.2749	−0.671799
$741$	2.27492	0.0835712
$742$	0	0
$743$	11.4502	0.420066	0.210033	−	0.977694i	$-0.432643\pi$
$743$	11.4502	0.420066	0.210033	+	0.977694i	$0.432643\pi$
$744$	8.00000	0.293294
$745$	96.3987	3.53177
$746$	−26.5498	−0.972059
$747$	4.54983	0.166470
$748$	0.625414	0.0228674
$749$	0	0
$750$	−35.3746	−1.29170
$751$	10.1993	0.372179	0.186090	−	0.982533i	$-0.440419\pi$
$751$	10.1993	0.372179	0.186090	+	0.982533i	$0.440419\pi$
$752$	0	0
$753$	−23.9244	−0.871854
$754$	−8.27492	−0.301355
$755$	26.8248	0.976253
$756$	0	0
$757$	−35.0997	−1.27572	−0.637860	−	0.770153i	$-0.720180\pi$
$757$	−35.0997	−1.27572	−0.637860	+	0.770153i	$0.720180\pi$
$758$	33.0997	1.20223
$759$	−5.17525	−0.187850
$760$	−9.72508	−0.352766
$761$	−38.5498	−1.39743	−0.698715	−	0.715400i	$-0.746245\pi$
$761$	−38.5498	−1.39743	−0.698715	+	0.715400i	$0.746245\pi$
$762$	20.5498	0.744442
$763$	0	0
$764$	2.27492	0.0823036
$765$	−1.17525	−0.0424912
$766$	30.2749	1.09388
$767$	−8.00000	−0.288863
$768$	1.00000	0.0360844
$769$	−32.8248	−1.18369	−0.591845	−	0.806051i	$-0.701600\pi$
$769$	−32.8248	−1.18369	−0.591845	+	0.806051i	$0.701600\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	−18.5498	−0.667623
$773$	9.92442	0.356957	0.178478	−	0.983944i	$-0.442883\pi$
$773$	9.92442	0.356957	0.178478	+	0.983944i	$0.442883\pi$
$774$	2.27492	0.0817702
$775$	−106.199	−3.81479
$776$	15.0997	0.542047
$777$	0	0
$778$	−28.1993	−1.01100
$779$	−14.9003	−0.533860
$780$	4.27492	0.153067
$781$	0	0
$782$	0.625414	0.0223648
$783$	8.27492	0.295721
$784$	0	0
$785$	57.1752	2.04067
$786$	6.82475	0.243431
$787$	10.2749	0.366261	0.183131	−	0.983089i	$-0.441377\pi$
$787$	10.2749	0.366261	0.183131	+	0.983089i	$0.441377\pi$
$788$	−2.54983	−0.0908341
$789$	28.0000	0.996826
$790$	−53.6495	−1.90876
$791$	0	0
$792$	2.27492	0.0808357
$793$	12.2749	0.435895
$794$	−14.0000	−0.496841
$795$	42.7492	1.51616
$796$	−6.82475	−0.241897
$797$	15.6495	0.554334	0.277167	−	0.960822i	$-0.410604\pi$
$797$	15.6495	0.554334	0.277167	+	0.960822i	$0.410604\pi$
$798$	0	0
$799$	0	0
$800$	−13.2749	−0.469339
$801$	14.0000	0.494666
$802$	3.09967	0.109453
$803$	−29.1752	−1.02957
$804$	12.5498	0.442599
$805$	0	0
$806$	8.00000	0.281788
$807$	−6.00000	−0.211210
$808$	2.54983	0.0897029
$809$	−56.1993	−1.97586	−0.987932	−	0.154890i	$-0.950498\pi$
$809$	−56.1993	−1.97586	−0.987932	+	0.154890i	$0.950498\pi$
$810$	−4.27492	−0.150205
$811$	11.3746	0.399416	0.199708	−	0.979855i	$-0.436001\pi$
$811$	11.3746	0.399416	0.199708	+	0.979855i	$0.436001\pi$
$812$	0	0
$813$	−28.5498	−1.00129
$814$	−9.72508	−0.340864
$815$	53.6495	1.87926
$816$	−0.274917	−0.00962403
$817$	−5.17525	−0.181059
$818$	−7.17525	−0.250877
$819$	0	0
$820$	−28.0000	−0.977802
$821$	31.6495	1.10458	0.552288	−	0.833654i	$-0.313755\pi$
$821$	31.6495	1.10458	0.552288	+	0.833654i	$0.313755\pi$
$822$	17.3746	0.606008
$823$	−25.0997	−0.874919	−0.437460	−	0.899238i	$-0.644122\pi$
$823$	−25.0997	−0.874919	−0.437460	+	0.899238i	$0.644122\pi$
$824$	10.2749	0.357944
$825$	−30.1993	−1.05141
$826$	0	0
$827$	−26.2749	−0.913668	−0.456834	−	0.889552i	$-0.651017\pi$
$827$	−26.2749	−0.913668	−0.456834	+	0.889552i	$0.651017\pi$
$828$	2.27492	0.0790588
$829$	−11.7251	−0.407229	−0.203614	−	0.979051i	$-0.565269\pi$
$829$	−11.7251	−0.407229	−0.203614	+	0.979051i	$0.565269\pi$
$830$	−19.4502	−0.675125
$831$	−10.0000	−0.346896
$832$	1.00000	0.0346688
$833$	0	0
$834$	4.00000	0.138509
$835$	7.37459	0.255208
$836$	−5.17525	−0.178990
$837$	−8.00000	−0.276520
$838$	2.27492	0.0785857
$839$	−22.9003	−0.790607	−0.395304	−	0.918551i	$-0.629361\pi$
$839$	−22.9003	−0.790607	−0.395304	+	0.918551i	$0.629361\pi$
$840$	0	0
$841$	39.4743	1.36118
$842$	−3.09967	−0.106822
$843$	15.0997	0.520060
$844$	1.17525	0.0404537
$845$	4.27492	0.147062
$846$	0	0
$847$	0	0
$848$	10.0000	0.343401
$849$	4.00000	0.137280
$850$	3.64950	0.125177
$851$	−9.72508	−0.333372
$852$	0	0
$853$	27.6495	0.946701	0.473350	−	0.880874i	$-0.343044\pi$
$853$	27.6495	0.946701	0.473350	+	0.880874i	$0.343044\pi$
$854$	0	0
$855$	9.72508	0.332591
$856$	−0.549834	−0.0187930
$857$	−19.0997	−0.652432	−0.326216	−	0.945295i	$-0.605774\pi$
$857$	−19.0997	−0.652432	−0.326216	+	0.945295i	$0.605774\pi$
$858$	2.27492	0.0776644
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−9.72508	−0.331623
$861$	0	0
$862$	11.4502	0.389994
$863$	29.6495	1.00928	0.504640	−	0.863330i	$-0.331625\pi$
$863$	29.6495	1.00928	0.504640	+	0.863330i	$0.331625\pi$
$864$	−1.00000	−0.0340207
$865$	−30.3505	−1.03195
$866$	23.6495	0.803643
$867$	−16.9244	−0.574783
$868$	0	0
$869$	−28.5498	−0.968487
$870$	−35.3746	−1.19931
$871$	12.5498	0.425235
$872$	−0.274917	−0.00930987
$873$	−15.0997	−0.511046
$874$	−5.17525	−0.175055
$875$	0	0
$876$	12.8248	0.433308
$877$	51.0997	1.72551	0.862757	−	0.505619i	$-0.168736\pi$
$877$	51.0997	1.72551	0.862757	+	0.505619i	$0.168736\pi$
$878$	−14.8248	−0.500311
$879$	6.00000	0.202375
$880$	−9.72508	−0.327832
$881$	31.7251	1.06885	0.534423	−	0.845217i	$-0.320529\pi$
$881$	31.7251	1.06885	0.534423	+	0.845217i	$0.320529\pi$
$882$	0	0
$883$	31.9244	1.07434	0.537171	−	0.843473i	$-0.319493\pi$
$883$	31.9244	1.07434	0.537171	+	0.843473i	$0.319493\pi$
$884$	−0.274917	−0.00924647
$885$	−34.1993	−1.14960
$886$	−7.45017	−0.250293
$887$	33.0997	1.11138	0.555689	−	0.831390i	$-0.312455\pi$
$887$	33.0997	1.11138	0.555689	+	0.831390i	$0.312455\pi$
$888$	4.27492	0.143457
$889$	0	0
$890$	−59.8488	−2.00614
$891$	−2.27492	−0.0762126
$892$	−21.6495	−0.724879
$893$	0	0
$894$	−22.5498	−0.754179
$895$	51.2990	1.71474
$896$	0	0
$897$	2.27492	0.0759573
$898$	0.274917	0.00917411
$899$	−66.1993	−2.20787
$900$	13.2749	0.442497
$901$	−2.74917	−0.0915882
$902$	−14.9003	−0.496127
$903$	0	0
$904$	6.00000	0.199557
$905$	−81.6495	−2.71412
$906$	−6.27492	−0.208470
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	20.5498	0.681970
$909$	−2.54983	−0.0845727
$910$	0	0
$911$	−52.4743	−1.73855	−0.869275	−	0.494329i	$-0.835414\pi$
$911$	−52.4743	−1.73855	−0.869275	+	0.494329i	$0.835414\pi$
$912$	2.27492	0.0753300
$913$	−10.3505	−0.342551
$914$	18.5498	0.613574
$915$	52.4743	1.73475
$916$	−19.0997	−0.631071
$917$	0	0
$918$	0.274917	0.00907362
$919$	−12.5498	−0.413981	−0.206990	−	0.978343i	$-0.566367\pi$
$919$	−12.5498	−0.413981	−0.206990	+	0.978343i	$0.566367\pi$
$920$	−9.72508	−0.320626
$921$	−21.0997	−0.695258
$922$	−17.9244	−0.590309
$923$	0	0
$924$	0	0
$925$	−56.7492	−1.86590
$926$	−30.2749	−0.994896
$927$	−10.2749	−0.337473
$928$	−8.27492	−0.271637
$929$	−14.5498	−0.477365	−0.238682	−	0.971098i	$-0.576715\pi$
$929$	−14.5498	−0.477365	−0.238682	+	0.971098i	$0.576715\pi$
$930$	34.1993	1.12144
$931$	0	0
$932$	5.45017	0.178526
$933$	−3.45017	−0.112953
$934$	−26.2749	−0.859742
$935$	2.67359	0.0874358
$936$	−1.00000	−0.0326860
$937$	−16.9003	−0.552110	−0.276055	−	0.961142i	$-0.589027\pi$
$937$	−16.9003	−0.552110	−0.276055	+	0.961142i	$0.589027\pi$
$938$	0	0
$939$	27.6495	0.902307
$940$	0	0
$941$	−51.0997	−1.66580	−0.832901	−	0.553422i	$-0.813322\pi$
$941$	−51.0997	−1.66580	−0.832901	+	0.553422i	$0.813322\pi$
$942$	−13.3746	−0.435768
$943$	−14.9003	−0.485222
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−5.17525	−0.168262
$947$	−17.1752	−0.558121	−0.279060	−	0.960274i	$-0.590023\pi$
$947$	−17.1752	−0.558121	−0.279060	+	0.960274i	$0.590023\pi$
$948$	12.5498	0.407600
$949$	12.8248	0.416309
$950$	−30.1993	−0.979796
$951$	−1.45017	−0.0470248
$952$	0	0
$953$	31.6495	1.02523	0.512614	−	0.858619i	$-0.328677\pi$
$953$	31.6495	1.02523	0.512614	+	0.858619i	$0.328677\pi$
$954$	−10.0000	−0.323762
$955$	9.72508	0.314696
$956$	25.0997	0.811781
$957$	−18.8248	−0.608518
$958$	−23.3746	−0.755199
$959$	0	0
$960$	4.27492	0.137972
$961$	33.0000	1.06452
$962$	4.27492	0.137829
$963$	0.549834	0.0177182
$964$	−15.0997	−0.486328
$965$	−79.2990	−2.55273
$966$	0	0
$967$	42.8248	1.37715	0.688576	−	0.725165i	$-0.258236\pi$
$967$	42.8248	1.37715	0.688576	+	0.725165i	$0.258236\pi$
$968$	5.82475	0.187215
$969$	−0.625414	−0.0200912
$970$	64.5498	2.07257
$971$	29.0997	0.933853	0.466926	−	0.884296i	$-0.345361\pi$
$971$	29.0997	0.933853	0.466926	+	0.884296i	$0.345361\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	18.1993	0.583144
$975$	13.2749	0.425138
$976$	12.2749	0.392911
$977$	−13.9244	−0.445482	−0.222741	−	0.974878i	$-0.571500\pi$
$977$	−13.9244	−0.445482	−0.222741	+	0.974878i	$0.571500\pi$
$978$	−12.5498	−0.401299
$979$	−31.8488	−1.01789
$980$	0	0
$981$	0.274917	0.00877743
$982$	−8.54983	−0.272836
$983$	−61.0241	−1.94637	−0.973183	−	0.230032i	$-0.926117\pi$
$983$	−61.0241	−1.94637	−0.973183	+	0.230032i	$0.926117\pi$
$984$	6.54983	0.208801
$985$	−10.9003	−0.347313
$986$	2.27492	0.0724481
$987$	0	0
$988$	2.27492	0.0723748
$989$	−5.17525	−0.164563
$990$	9.72508	0.309083
$991$	−46.7492	−1.48504	−0.742518	−	0.669826i	$-0.766369\pi$
$991$	−46.7492	−1.48504	−0.742518	+	0.669826i	$0.766369\pi$
$992$	8.00000	0.254000
$993$	−29.6495	−0.940899
$994$	0	0
$995$	−29.1752	−0.924918
$996$	4.54983	0.144167
$997$	12.9003	0.408558	0.204279	−	0.978913i	$-0.434515\pi$
$997$	12.9003	0.408558	0.204279	+	0.978913i	$0.434515\pi$
$998$	−25.0997	−0.794516
$999$	−4.27492	−0.135252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.bm.1.2		2
7.6	odd	2		546.2.a.h.1.1	✓	2
21.20	even	2		1638.2.a.y.1.2		2
28.27	even	2		4368.2.a.bh.1.1		2
91.90	odd	2		7098.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.h.1.1	✓	2	7.6	odd	2
1638.2.a.y.1.2		2	21.20	even	2
3822.2.a.bm.1.2		2	1.1	even	1	trivial
4368.2.a.bh.1.1		2	28.27	even	2
7098.2.a.bu.1.2		2	91.90	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.27492 q^{10} -2.27492 q^{11} +1.00000 q^{12} +1.00000 q^{13} +4.27492 q^{15} +1.00000 q^{16} -0.274917 q^{17} -1.00000 q^{18} +2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} +2.27492 q^{23} -1.00000 q^{24} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{27} +8.27492 q^{29} -4.27492 q^{30} -8.00000 q^{31} -1.00000 q^{32} -2.27492 q^{33} +0.274917 q^{34} +1.00000 q^{36} -4.27492 q^{37} -2.27492 q^{38} +1.00000 q^{39} -4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} -2.27492 q^{44} +4.27492 q^{45} -2.27492 q^{46} +1.00000 q^{48} -13.2749 q^{50} -0.274917 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} -9.72508 q^{55} +2.27492 q^{57} -8.27492 q^{58} -8.00000 q^{59} +4.27492 q^{60} +12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.27492 q^{65} +2.27492 q^{66} +12.5498 q^{67} -0.274917 q^{68} +2.27492 q^{69} -1.00000 q^{72} +12.8248 q^{73} +4.27492 q^{74} +13.2749 q^{75} +2.27492 q^{76} -1.00000 q^{78} +12.5498 q^{79} +4.27492 q^{80} +1.00000 q^{81} +6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} +2.27492 q^{86} +8.27492 q^{87} +2.27492 q^{88} +14.0000 q^{89} -4.27492 q^{90} +2.27492 q^{92} -8.00000 q^{93} +9.72508 q^{95} -1.00000 q^{96} -15.0997 q^{97} -2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + q^{15} + 2 q^{16} + 7 q^{17} - 2 q^{18} - 3 q^{19} + q^{20} - 3 q^{22} - 3 q^{23}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 - 2 * q^8 + 2 * q^9 - q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 + q^15 + 2 * q^16 + 7 * q^17 - 2 * q^18 - 3 * q^19 + q^20 - 3 * q^22 - 3 * q^23 - 2 * q^24 + 19 * q^25 - 2 * q^26 + 2 * q^27 + 9 * q^29 - q^30 - 16 * q^31 - 2 * q^32 + 3 * q^33 - 7 * q^34 + 2 * q^36 - q^37 + 3 * q^38 + 2 * q^39 - q^40 + 2 * q^41 + 3 * q^43 + 3 * q^44 + q^45 + 3 * q^46 + 2 * q^48 - 19 * q^50 + 7 * q^51 + 2 * q^52 + 20 * q^53 - 2 * q^54 - 27 * q^55 - 3 * q^57 - 9 * q^58 - 16 * q^59 + q^60 + 17 * q^61 + 16 * q^62 + 2 * q^64 + q^65 - 3 * q^66 + 10 * q^67 + 7 * q^68 - 3 * q^69 - 2 * q^72 + 3 * q^73 + q^74 + 19 * q^75 - 3 * q^76 - 2 * q^78 + 10 * q^79 + q^80 + 2 * q^81 - 2 * q^82 - 6 * q^83 - 25 * q^85 - 3 * q^86 + 9 * q^87 - 3 * q^88 + 28 * q^89 - q^90 - 3 * q^92 - 16 * q^93 + 27 * q^95 - 2 * q^96 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more