LMFDB - Embedded newform 1638.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
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	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +4.27492 q^{10} +2.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.274917 q^{17} -2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} -2.27492 q^{23} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{28} -8.27492 q^{29} +8.00000 q^{31} +1.00000 q^{32} -0.274917 q^{34} +4.27492 q^{35} -4.27492 q^{37} -2.27492 q^{38} +4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} +2.27492 q^{44} -2.27492 q^{46} +1.00000 q^{49} +13.2749 q^{50} -1.00000 q^{52} -10.0000 q^{53} +9.72508 q^{55} +1.00000 q^{56} -8.27492 q^{58} -8.00000 q^{59} -12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.27492 q^{65} +12.5498 q^{67} -0.274917 q^{68} +4.27492 q^{70} -12.8248 q^{73} -4.27492 q^{74} -2.27492 q^{76} +2.27492 q^{77} +12.5498 q^{79} +4.27492 q^{80} -6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} -2.27492 q^{86} +2.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} -2.27492 q^{92} -9.72508 q^{95} +15.0997 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + 19 q^{25} - 2 q^{26} + 2 q^{28} - 9 q^{29} + 16 q^{31} + 2 q^{32} + 7 q^{34} + q^{35} - q^{37} + 3 q^{38} + q^{40} + 2 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 2 q^{49} + 19 q^{50} - 2 q^{52} - 20 q^{53} + 27 q^{55} + 2 q^{56} - 9 q^{58} - 16 q^{59} - 17 q^{61} + 16 q^{62} + 2 q^{64} - q^{65} + 10 q^{67} + 7 q^{68} + q^{70} - 3 q^{73} - q^{74} + 3 q^{76} - 3 q^{77} + 10 q^{79} + q^{80} + 2 q^{82} - 6 q^{83} - 25 q^{85} + 3 q^{86} - 3 q^{88} + 28 q^{89} - 2 q^{91} + 3 q^{92} - 27 q^{95} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 + q^10 - 3 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 7 * q^17 + 3 * q^19 + q^20 - 3 * q^22 + 3 * q^23 + 19 * q^25 - 2 * q^26 + 2 * q^28 - 9 * q^29 + 16 * q^31 + 2 * q^32 + 7 * q^34 + q^35 - q^37 + 3 * q^38 + q^40 + 2 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 2 * q^49 + 19 * q^50 - 2 * q^52 - 20 * q^53 + 27 * q^55 + 2 * q^56 - 9 * q^58 - 16 * q^59 - 17 * q^61 + 16 * q^62 + 2 * q^64 - q^65 + 10 * q^67 + 7 * q^68 + q^70 - 3 * q^73 - q^74 + 3 * q^76 - 3 * q^77 + 10 * q^79 + q^80 + 2 * q^82 - 6 * q^83 - 25 * q^85 + 3 * q^86 - 3 * q^88 + 28 * q^89 - 2 * q^91 + 3 * q^92 - 27 * q^95 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.27492	1.91180	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	4.27492	1.91180	0.955901	+	0.293691i	$0.0948835\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	4.27492	1.35185
$11$	2.27492	0.685913	0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492	0.685913	0.342957	+	0.939351i	$0.388572\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	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	$0.510614\pi$
$18$	0	0
$19$	−2.27492	−0.521902	−0.260951	−	0.965352i	$-0.584036\pi$
$19$	−2.27492	−0.521902	−0.260951	+	0.965352i	$0.584036\pi$
$20$	4.27492	0.955901
$21$	0	0
$22$	2.27492	0.485014
$23$	−2.27492	−0.474353	−0.237177	−	0.971467i	$-0.576222\pi$
$23$	−2.27492	−0.474353	−0.237177	+	0.971467i	$0.576222\pi$
$24$	0	0
$25$	13.2749	2.65498
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−8.27492	−1.53661	−0.768307	−	0.640082i	$-0.778900\pi$
$29$	−8.27492	−1.53661	−0.768307	+	0.640082i	$0.778900\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.274917	−0.0471479
$35$	4.27492	0.722593
$36$	0	0
$37$	−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
$39$	0	0
$40$	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
$45$	0	0
$46$	−2.27492	−0.335418
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	13.2749	1.87736
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	9.72508	1.31133
$56$	1.00000	0.133631
$57$	0	0
$58$	−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	$0.674350\pi$
$60$	0	0
$61$	−12.2749	−1.57164	−0.785821	−	0.618454i	$-0.787759\pi$
$61$	−12.2749	−1.57164	−0.785821	+	0.618454i	$0.787759\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.27492	−0.530238
$66$	0	0
$67$	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
$69$	0	0
$70$	4.27492	0.510950
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−12.8248	−1.50102	−0.750512	−	0.660857i	$-0.770193\pi$
$73$	−12.8248	−1.50102	−0.750512	+	0.660857i	$0.770193\pi$
$74$	−4.27492	−0.496949
$75$	0	0
$76$	−2.27492	−0.260951
$77$	2.27492	0.259251
$78$	0	0
$79$	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
$81$	0	0
$82$	−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
$87$	0	0
$88$	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	$0.233878\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−2.27492	−0.237177
$93$	0	0
$94$	0	0
$95$	−9.72508	−0.997772
$96$	0	0
$97$	15.0997	1.53314	0.766570	−	0.642161i	$-0.221962\pi$
$97$	15.0997	1.53314	0.766570	+	0.642161i	$0.221962\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	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	0
$103$	10.2749	1.01242	0.506209	−	0.862411i	$-0.331046\pi$
$103$	10.2749	1.01242	0.506209	+	0.862411i	$0.331046\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−0.549834	−0.0531545	−0.0265773	−	0.999647i	$-0.508461\pi$
$107$	−0.549834	−0.0531545	−0.0265773	+	0.999647i	$0.508461\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	1.00000	0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−9.72508	−0.906869
$116$	−8.27492	−0.768307
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−0.274917	−0.0252016
$120$	0	0
$121$	−5.82475	−0.529523
$122$	−12.2749	−1.11132
$123$	0	0
$124$	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
$129$	0	0
$130$	−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	$0.596366\pi$
$132$	0	0
$133$	−2.27492	−0.197260
$134$	12.5498	1.08414
$135$	0	0
$136$	−0.274917	−0.0235740
$137$	17.3746	1.48441	0.742206	−	0.670172i	$-0.233780\pi$
$137$	17.3746	1.48441	0.742206	+	0.670172i	$0.233780\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.27492	0.361296
$141$	0	0
$142$	0	0
$143$	−2.27492	−0.190238
$144$	0	0
$145$	−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	$-0.874832\pi$
$149$	−22.5498	−1.84735	−0.923677	+	0.383172i	$0.874832\pi$
$150$	0	0
$151$	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	0
$154$	2.27492	0.183318
$155$	34.1993	2.74696
$156$	0	0
$157$	−13.3746	−1.06741	−0.533704	−	0.845671i	$-0.679200\pi$
$157$	−13.3746	−1.06741	−0.533704	+	0.845671i	$0.679200\pi$
$158$	12.5498	0.998411
$159$	0	0
$160$	4.27492	0.337962
$161$	−2.27492	−0.179289
$162$	0	0
$163$	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
$165$	0	0
$166$	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
$171$	0	0
$172$	−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	$0.586987\pi$
$174$	0	0
$175$	13.2749	1.00349
$176$	2.27492	0.171478
$177$	0	0
$178$	14.0000	1.04934
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	19.0997	1.41967	0.709834	−	0.704369i	$-0.248770\pi$
$181$	19.0997	1.41967	0.709834	+	0.704369i	$0.248770\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	−2.27492	−0.167709
$185$	−18.2749	−1.34360
$186$	0	0
$187$	−0.625414	−0.0457348
$188$	0	0
$189$	0	0
$190$	−9.72508	−0.705532
$191$	−2.27492	−0.164607	−0.0823036	−	0.996607i	$-0.526228\pi$
$191$	−2.27492	−0.164607	−0.0823036	+	0.996607i	$0.526228\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.54983	0.181668	0.0908341	−	0.995866i	$-0.471047\pi$
$197$	2.54983	0.181668	0.0908341	+	0.995866i	$0.471047\pi$
$198$	0	0
$199$	6.82475	0.483794	0.241897	−	0.970302i	$-0.422230\pi$
$199$	6.82475	0.483794	0.241897	+	0.970302i	$0.422230\pi$
$200$	13.2749	0.938678
$201$	0	0
$202$	−2.54983	−0.179406
$203$	−8.27492	−0.580785
$204$	0	0
$205$	−28.0000	−1.95560
$206$	10.2749	0.715887
$207$	0	0
$208$	−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
$216$	0	0
$217$	8.00000	0.543075
$218$	0.274917	0.0186197
$219$	0	0
$220$	9.72508	0.655665
$221$	0.274917	0.0184929
$222$	0	0
$223$	21.6495	1.44976	0.724879	−	0.688876i	$-0.241896\pi$
$223$	21.6495	1.44976	0.724879	+	0.688876i	$0.241896\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	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	$0.261123\pi$
$228$	0	0
$229$	19.0997	1.26214	0.631071	−	0.775725i	$-0.282616\pi$
$229$	19.0997	1.26214	0.631071	+	0.775725i	$0.282616\pi$
$230$	−9.72508	−0.641253
$231$	0	0
$232$	−8.27492	−0.543275
$233$	−5.45017	−0.357052	−0.178526	−	0.983935i	$-0.557133\pi$
$233$	−5.45017	−0.357052	−0.178526	+	0.983935i	$0.557133\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−0.274917	−0.0178202
$239$	−25.0997	−1.62356	−0.811781	−	0.583962i	$-0.801502\pi$
$239$	−25.0997	−1.62356	−0.811781	+	0.583962i	$0.801502\pi$
$240$	0	0
$241$	15.0997	0.972655	0.486328	−	0.873777i	$-0.338336\pi$
$241$	15.0997	0.972655	0.486328	+	0.873777i	$0.338336\pi$
$242$	−5.82475	−0.374429
$243$	0	0
$244$	−12.2749	−0.785821
$245$	4.27492	0.273114
$246$	0	0
$247$	2.27492	0.144750
$248$	8.00000	0.508001
$249$	0	0
$250$	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
$255$	0	0
$256$	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	$0.356166\pi$
$258$	0	0
$259$	−4.27492	−0.265630
$260$	−4.27492	−0.265119
$261$	0	0
$262$	−6.82475	−0.421635
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−42.7492	−2.62606
$266$	−2.27492	−0.139484
$267$	0	0
$268$	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	$0.558553\pi$
$270$	0	0
$271$	28.5498	1.73428	0.867139	−	0.498065i	$-0.165956\pi$
$271$	28.5498	1.73428	0.867139	+	0.498065i	$0.165956\pi$
$272$	−0.274917	−0.0166693
$273$	0	0
$274$	17.3746	1.04964
$275$	30.1993	1.82109
$276$	0	0
$277$	−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
$279$	0	0
$280$	4.27492	0.255475
$281$	−15.0997	−0.900771	−0.450385	−	0.892834i	$-0.648713\pi$
$281$	−15.0997	−0.900771	−0.450385	+	0.892834i	$0.648713\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	−2.27492	−0.134519
$287$	−6.54983	−0.386625
$288$	0	0
$289$	−16.9244	−0.995554
$290$	−35.3746	−2.07727
$291$	0	0
$292$	−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
$297$	0	0
$298$	−22.5498	−1.30628
$299$	2.27492	0.131562
$300$	0	0
$301$	−2.27492	−0.131124
$302$	6.27492	0.361081
$303$	0	0
$304$	−2.27492	−0.130475
$305$	−52.4743	−3.00467
$306$	0	0
$307$	21.0997	1.20422	0.602111	−	0.798412i	$-0.294327\pi$
$307$	21.0997	1.20422	0.602111	+	0.798412i	$0.294327\pi$
$308$	2.27492	0.129625
$309$	0	0
$310$	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	$0.531187\pi$
$312$	0	0
$313$	−27.6495	−1.56284	−0.781421	−	0.624004i	$-0.785505\pi$
$313$	−27.6495	−1.56284	−0.781421	+	0.624004i	$0.785505\pi$
$314$	−13.3746	−0.754772
$315$	0	0
$316$	12.5498	0.705983
$317$	1.45017	0.0814494	0.0407247	−	0.999170i	$-0.487033\pi$
$317$	1.45017	0.0814494	0.0407247	+	0.999170i	$0.487033\pi$
$318$	0	0
$319$	−18.8248	−1.05398
$320$	4.27492	0.238975
$321$	0	0
$322$	−2.27492	−0.126776
$323$	0.625414	0.0347990
$324$	0	0
$325$	−13.2749	−0.736360
$326$	12.5498	0.695071
$327$	0	0
$328$	−6.54983	−0.361654
$329$	0	0
$330$	0	0
$331$	−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
$333$	0	0
$334$	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
$339$	0	0
$340$	−1.17525	−0.0637368
$341$	18.1993	0.985549
$342$	0	0
$343$	1.00000	0.0539949
$344$	−2.27492	−0.122655
$345$	0	0
$346$	−7.09967	−0.381681
$347$	−5.09967	−0.273765	−0.136882	−	0.990587i	$-0.543708\pi$
$347$	−5.09967	−0.273765	−0.136882	+	0.990587i	$0.543708\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	13.2749	0.709574
$351$	0	0
$352$	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	$0.756402\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−13.8248	−0.727619
$362$	19.0997	1.00386
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	−54.8248	−2.86966
$366$	0	0
$367$	−2.90033	−0.151396	−0.0756980	−	0.997131i	$-0.524119\pi$
$367$	−2.90033	−0.151396	−0.0756980	+	0.997131i	$0.524119\pi$
$368$	−2.27492	−0.118588
$369$	0	0
$370$	−18.2749	−0.950068
$371$	−10.0000	−0.519174
$372$	0	0
$373$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	−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	$0.781490\pi$
$384$	0	0
$385$	9.72508	0.495636
$386$	−18.5498	−0.944162
$387$	0	0
$388$	15.0997	0.766570
$389$	−28.1993	−1.42976	−0.714882	−	0.699246i	$-0.753519\pi$
$389$	−28.1993	−1.42976	−0.714882	+	0.699246i	$0.753519\pi$
$390$	0	0
$391$	0.625414	0.0316285
$392$	1.00000	0.0505076
$393$	0	0
$394$	2.54983	0.128459
$395$	53.6495	2.69940
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	6.82475	0.342094
$399$	0	0
$400$	13.2749	0.663746
$401$	3.09967	0.154790	0.0773950	−	0.997001i	$-0.475340\pi$
$401$	3.09967	0.154790	0.0773950	+	0.997001i	$0.475340\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−2.54983	−0.126859
$405$	0	0
$406$	−8.27492	−0.410677
$407$	−9.72508	−0.482054
$408$	0	0
$409$	−7.17525	−0.354793	−0.177397	−	0.984139i	$-0.556768\pi$
$409$	−7.17525	−0.354793	−0.177397	+	0.984139i	$0.556768\pi$
$410$	−28.0000	−1.38282
$411$	0	0
$412$	10.2749	0.506209
$413$	−8.00000	−0.393654
$414$	0	0
$415$	19.4502	0.954771
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−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
$427$	−12.2749	−0.594025
$428$	−0.549834	−0.0265773
$429$	0	0
$430$	−9.72508	−0.468985
$431$	11.4502	0.551535	0.275768	−	0.961224i	$-0.411068\pi$
$431$	11.4502	0.551535	0.275768	+	0.961224i	$0.411068\pi$
$432$	0	0
$433$	23.6495	1.13652	0.568261	−	0.822848i	$-0.307616\pi$
$433$	23.6495	1.13652	0.568261	+	0.822848i	$0.307616\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	0.274917	0.0131661
$437$	5.17525	0.247566
$438$	0	0
$439$	−14.8248	−0.707547	−0.353773	−	0.935331i	$-0.615102\pi$
$439$	−14.8248	−0.707547	−0.353773	+	0.935331i	$0.615102\pi$
$440$	9.72508	0.463625
$441$	0	0
$442$	0.274917	0.0130765
$443$	−7.45017	−0.353968	−0.176984	−	0.984214i	$-0.556634\pi$
$443$	−7.45017	−0.353968	−0.176984	+	0.984214i	$0.556634\pi$
$444$	0	0
$445$	59.8488	2.83711
$446$	21.6495	1.02513
$447$	0	0
$448$	1.00000	0.0472456
$449$	0.274917	0.0129741	0.00648707	−	0.999979i	$-0.497935\pi$
$449$	0.274917	0.0129741	0.00648707	+	0.999979i	$0.497935\pi$
$450$	0	0
$451$	−14.9003	−0.701629
$452$	6.00000	0.282216
$453$	0	0
$454$	20.5498	0.964452
$455$	−4.27492	−0.200411
$456$	0	0
$457$	−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	0
$460$	−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
$465$	0	0
$466$	−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	$0.292000\pi$
$468$	0	0
$469$	12.5498	0.579498
$470$	0	0
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−5.17525	−0.237958
$474$	0	0
$475$	−30.1993	−1.38564
$476$	−0.274917	−0.0126008
$477$	0	0
$478$	−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	$0.320686\pi$
$480$	0	0
$481$	4.27492	0.194919
$482$	15.0997	0.687771
$483$	0	0
$484$	−5.82475	−0.264761
$485$	64.5498	2.93106
$486$	0	0
$487$	−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
$489$	0	0
$490$	4.27492	0.193121
$491$	−8.54983	−0.385849	−0.192924	−	0.981214i	$-0.561797\pi$
$491$	−8.54983	−0.385849	−0.192924	+	0.981214i	$0.561797\pi$
$492$	0	0
$493$	2.27492	0.102457
$494$	2.27492	0.102353
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	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
$501$	0	0
$502$	−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
$507$	0	0
$508$	−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	$0.429412\pi$
$510$	0	0
$511$	−12.8248	−0.567334
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	43.9244	1.93554
$516$	0	0
$517$	0	0
$518$	−4.27492	−0.187829
$519$	0	0
$520$	−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	$0.470149\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−6.82475	−0.298141
$525$	0	0
$526$	−28.0000	−1.22086
$527$	−2.19934	−0.0958047
$528$	0	0
$529$	−17.8248	−0.774989
$530$	−42.7492	−1.85691
$531$	0	0
$532$	−2.27492	−0.0986302
$533$	6.54983	0.283705
$534$	0	0
$535$	−2.35050	−0.101621
$536$	12.5498	0.542070
$537$	0	0
$538$	−6.00000	−0.258678
$539$	2.27492	0.0979876
$540$	0	0
$541$	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
$543$	0	0
$544$	−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
$549$	0	0
$550$	30.1993	1.28770
$551$	18.8248	0.801961
$552$	0	0
$553$	12.5498	0.533673
$554$	−10.0000	−0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	44.7492	1.89608	0.948042	−	0.318146i	$-0.103060\pi$
$557$	44.7492	1.89608	0.948042	+	0.318146i	$0.103060\pi$
$558$	0	0
$559$	2.27492	0.0962187
$560$	4.27492	0.180648
$561$	0	0
$562$	−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	$-0.520696\pi$
$569$	−3.09967	−0.129945	−0.0649724	+	0.997887i	$0.520696\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	−6.54983	−0.273385
$575$	−30.1993	−1.25940
$576$	0	0
$577$	−8.90033	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$577$	−8.90033	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$578$	−16.9244	−0.703963
$579$	0	0
$580$	−35.3746	−1.46885
$581$	4.54983	0.188759
$582$	0	0
$583$	−22.7492	−0.942174
$584$	−12.8248	−0.530692
$585$	0	0
$586$	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
$591$	0	0
$592$	−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	$0.453433\pi$
$594$	0	0
$595$	−1.17525	−0.0481805
$596$	−22.5498	−0.923677
$597$	0	0
$598$	2.27492	0.0930283
$599$	−21.7251	−0.887663	−0.443831	−	0.896110i	$-0.646381\pi$
$599$	−21.7251	−0.887663	−0.443831	+	0.896110i	$0.646381\pi$
$600$	0	0
$601$	23.6495	0.964683	0.482342	−	0.875983i	$-0.339786\pi$
$601$	23.6495	0.964683	0.482342	+	0.875983i	$0.339786\pi$
$602$	−2.27492	−0.0927187
$603$	0	0
$604$	6.27492	0.255323
$605$	−24.9003	−1.01234
$606$	0	0
$607$	9.17525	0.372412	0.186206	−	0.982511i	$-0.440381\pi$
$607$	9.17525	0.372412	0.186206	+	0.982511i	$0.440381\pi$
$608$	−2.27492	−0.0922601
$609$	0	0
$610$	−52.4743	−2.12462
$611$	0	0
$612$	0	0
$613$	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
$615$	0	0
$616$	2.27492	0.0916590
$617$	20.8248	0.838373	0.419186	−	0.907900i	$-0.362315\pi$
$617$	20.8248	0.838373	0.419186	+	0.907900i	$0.362315\pi$
$618$	0	0
$619$	29.7251	1.19475	0.597376	−	0.801961i	$-0.296210\pi$
$619$	29.7251	1.19475	0.597376	+	0.801961i	$0.296210\pi$
$620$	34.1993	1.37348
$621$	0	0
$622$	−3.45017	−0.138339
$623$	14.0000	0.560898
$624$	0	0
$625$	84.8488	3.39395
$626$	−27.6495	−1.10510
$627$	0	0
$628$	−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
$633$	0	0
$634$	1.45017	0.0575934
$635$	−87.8488	−3.48617
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	−18.8248	−0.745279
$639$	0	0
$640$	4.27492	0.168981
$641$	15.0997	0.596401	0.298201	−	0.954503i	$-0.403614\pi$
$641$	15.0997	0.596401	0.298201	+	0.954503i	$0.403614\pi$
$642$	0	0
$643$	−23.9244	−0.943487	−0.471744	−	0.881736i	$-0.656375\pi$
$643$	−23.9244	−0.943487	−0.471744	+	0.881736i	$0.656375\pi$
$644$	−2.27492	−0.0896443
$645$	0	0
$646$	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	$0.383546\pi$
$648$	0	0
$649$	−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	$-0.466464\pi$
$653$	5.37459	0.210324	0.105162	+	0.994455i	$0.466464\pi$
$654$	0	0
$655$	−29.1752	−1.13997
$656$	−6.54983	−0.255728
$657$	0	0
$658$	0	0
$659$	7.45017	0.290217	0.145109	−	0.989416i	$-0.453647\pi$
$659$	7.45017	0.290217	0.145109	+	0.989416i	$0.453647\pi$
$660$	0	0
$661$	−40.1993	−1.56357	−0.781787	−	0.623546i	$-0.785691\pi$
$661$	−40.1993	−1.56357	−0.781787	+	0.623546i	$0.785691\pi$
$662$	−29.6495	−1.15236
$663$	0	0
$664$	4.54983	0.176568
$665$	−9.72508	−0.377123
$666$	0	0
$667$	18.8248	0.728897
$668$	1.72508	0.0667455
$669$	0	0
$670$	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
$675$	0	0
$676$	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	$0.600763\pi$
$678$	0	0
$679$	15.0997	0.579472
$680$	−1.17525	−0.0450687
$681$	0	0
$682$	18.1993	0.696889
$683$	−3.37459	−0.129125	−0.0645625	−	0.997914i	$-0.520565\pi$
$683$	−3.37459	−0.129125	−0.0645625	+	0.997914i	$0.520565\pi$
$684$	0	0
$685$	74.2749	2.83790
$686$	1.00000	0.0381802
$687$	0	0
$688$	−2.27492	−0.0867304
$689$	10.0000	0.380970
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−7.09967	−0.269889
$693$	0	0
$694$	−5.09967	−0.193581
$695$	17.0997	0.648627
$696$	0	0
$697$	1.80066	0.0682049
$698$	2.00000	0.0757011
$699$	0	0
$700$	13.2749	0.501745
$701$	15.0997	0.570307	0.285153	−	0.958482i	$-0.407955\pi$
$701$	15.0997	0.570307	0.285153	+	0.958482i	$0.407955\pi$
$702$	0	0
$703$	9.72508	0.366788
$704$	2.27492	0.0857392
$705$	0	0
$706$	−27.0997	−1.01991
$707$	−2.54983	−0.0958964
$708$	0	0
$709$	−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
$711$	0	0
$712$	14.0000	0.524672
$713$	−18.1993	−0.681571
$714$	0	0
$715$	−9.72508	−0.363697
$716$	−12.0000	−0.448461
$717$	0	0
$718$	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	$0.313533\pi$
$720$	0	0
$721$	10.2749	0.382658
$722$	−13.8248	−0.514504
$723$	0	0
$724$	19.0997	0.709834
$725$	−109.849	−4.07968
$726$	0	0
$727$	35.3746	1.31197	0.655985	−	0.754774i	$-0.272253\pi$
$727$	35.3746	1.31197	0.655985	+	0.754774i	$0.272253\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−54.8248	−2.02916
$731$	0.625414	0.0231318
$732$	0	0
$733$	14.5498	0.537410	0.268705	−	0.963222i	$-0.413404\pi$
$733$	14.5498	0.537410	0.268705	+	0.963222i	$0.413404\pi$
$734$	−2.90033	−0.107053
$735$	0	0
$736$	−2.27492	−0.0838546
$737$	28.5498	1.05165
$738$	0	0
$739$	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
$741$	0	0
$742$	−10.0000	−0.367112
$743$	−11.4502	−0.420066	−0.210033	−	0.977694i	$-0.567357\pi$
$743$	−11.4502	−0.420066	−0.210033	+	0.977694i	$0.567357\pi$
$744$	0	0
$745$	−96.3987	−3.53177
$746$	26.5498	0.972059
$747$	0	0
$748$	−0.625414	−0.0228674
$749$	−0.549834	−0.0200905
$750$	0	0
$751$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	−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	$0.746245\pi$
$762$	0	0
$763$	0.274917	0.00995267
$764$	−2.27492	−0.0823036
$765$	0	0
$766$	−30.2749	−1.09388
$767$	8.00000	0.288863
$768$	0	0
$769$	32.8248	1.18369	0.591845	−	0.806051i	$-0.298400\pi$
$769$	32.8248	1.18369	0.591845	+	0.806051i	$0.298400\pi$
$770$	9.72508	0.350468
$771$	0	0
$772$	−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	$0.442883\pi$
$774$	0	0
$775$	106.199	3.81479
$776$	15.0997	0.542047
$777$	0	0
$778$	−28.1993	−1.01100
$779$	14.9003	0.533860
$780$	0	0
$781$	0	0
$782$	0.625414	0.0223648
$783$	0	0
$784$	1.00000	0.0357143
$785$	−57.1752	−2.04067
$786$	0	0
$787$	−10.2749	−0.366261	−0.183131	−	0.983089i	$-0.558623\pi$
$787$	−10.2749	−0.366261	−0.183131	+	0.983089i	$0.558623\pi$
$788$	2.54983	0.0908341
$789$	0	0
$790$	53.6495	1.90876
$791$	6.00000	0.213335
$792$	0	0
$793$	12.2749	0.435895
$794$	−14.0000	−0.496841
$795$	0	0
$796$	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
$801$	0	0
$802$	3.09967	0.109453
$803$	−29.1752	−1.02957
$804$	0	0
$805$	−9.72508	−0.342764
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−2.54983	−0.0897029
$809$	56.1993	1.97586	0.987932	−	0.154890i	$-0.0495023\pi$
$809$	56.1993	1.97586	0.987932	+	0.154890i	$0.0495023\pi$
$810$	0	0
$811$	−11.3746	−0.399416	−0.199708	−	0.979855i	$-0.563999\pi$
$811$	−11.3746	−0.399416	−0.199708	+	0.979855i	$0.563999\pi$
$812$	−8.27492	−0.290393
$813$	0	0
$814$	−9.72508	−0.340864
$815$	53.6495	1.87926
$816$	0	0
$817$	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	$-0.686245\pi$
$821$	−31.6495	−1.10458	−0.552288	+	0.833654i	$0.686245\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	−8.00000	−0.278356
$827$	26.2749	0.913668	0.456834	−	0.889552i	$-0.348983\pi$
$827$	26.2749	0.913668	0.456834	+	0.889552i	$0.348983\pi$
$828$	0	0
$829$	11.7251	0.407229	0.203614	−	0.979051i	$-0.434731\pi$
$829$	11.7251	0.407229	0.203614	+	0.979051i	$0.434731\pi$
$830$	19.4502	0.675125
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−0.274917	−0.00952532
$834$	0	0
$835$	7.37459	0.255208
$836$	−5.17525	−0.178990
$837$	0	0
$838$	−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
$843$	0	0
$844$	1.17525	0.0404537
$845$	4.27492	0.147062
$846$	0	0
$847$	−5.82475	−0.200141
$848$	−10.0000	−0.343401
$849$	0	0
$850$	−3.64950	−0.125177
$851$	9.72508	0.333372
$852$	0	0
$853$	−27.6495	−0.946701	−0.473350	−	0.880874i	$-0.656956\pi$
$853$	−27.6495	−0.946701	−0.473350	+	0.880874i	$0.656956\pi$
$854$	−12.2749	−0.420039
$855$	0	0
$856$	−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	$0.605774\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−9.72508	−0.331623
$861$	0	0
$862$	11.4502	0.389994
$863$	−29.6495	−1.00928	−0.504640	−	0.863330i	$-0.668375\pi$
$863$	−29.6495	−1.00928	−0.504640	+	0.863330i	$0.668375\pi$
$864$	0	0
$865$	−30.3505	−1.03195
$866$	23.6495	0.803643
$867$	0	0
$868$	8.00000	0.271538
$869$	28.5498	0.968487
$870$	0	0
$871$	−12.5498	−0.425235
$872$	0.274917	0.00930987
$873$	0	0
$874$	5.17525	0.175055
$875$	35.3746	1.19588
$876$	0	0
$877$	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
$879$	0	0
$880$	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
$885$	0	0
$886$	−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	$0.312455\pi$
$888$	0	0
$889$	−20.5498	−0.689219
$890$	59.8488	2.00614
$891$	0	0
$892$	21.6495	0.724879
$893$	0	0
$894$	0	0
$895$	−51.2990	−1.71474
$896$	1.00000	0.0334077
$897$	0	0
$898$	0.274917	0.00917411
$899$	−66.1993	−2.20787
$900$	0	0
$901$	2.74917	0.0915882
$902$	−14.9003	−0.496127
$903$	0	0
$904$	6.00000	0.199557
$905$	81.6495	2.71412
$906$	0	0
$907$	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
$909$	0	0
$910$	−4.27492	−0.141712
$911$	52.4743	1.73855	0.869275	−	0.494329i	$-0.164586\pi$
$911$	52.4743	1.73855	0.869275	+	0.494329i	$0.164586\pi$
$912$	0	0
$913$	10.3505	0.342551
$914$	−18.5498	−0.613574
$915$	0	0
$916$	19.0997	0.631071
$917$	−6.82475	−0.225373
$918$	0	0
$919$	−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
$921$	0	0
$922$	17.9244	0.590309
$923$	0	0
$924$	0	0
$925$	−56.7492	−1.86590
$926$	30.2749	0.994896
$927$	0	0
$928$	−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	$0.576715\pi$
$930$	0	0
$931$	−2.27492	−0.0745574
$932$	−5.45017	−0.178526
$933$	0	0
$934$	26.2749	0.859742
$935$	−2.67359	−0.0874358
$936$	0	0
$937$	16.9003	0.552110	0.276055	−	0.961142i	$-0.410973\pi$
$937$	16.9003	0.552110	0.276055	+	0.961142i	$0.410973\pi$
$938$	12.5498	0.409767
$939$	0	0
$940$	0	0
$941$	−51.0997	−1.66580	−0.832901	−	0.553422i	$-0.813322\pi$
$941$	−51.0997	−1.66580	−0.832901	+	0.553422i	$0.813322\pi$
$942$	0	0
$943$	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	$-0.409977\pi$
$947$	17.1752	0.558121	0.279060	+	0.960274i	$0.409977\pi$
$948$	0	0
$949$	12.8248	0.416309
$950$	−30.1993	−0.979796
$951$	0	0
$952$	−0.274917	−0.00891012
$953$	−31.6495	−1.02523	−0.512614	−	0.858619i	$-0.671323\pi$
$953$	−31.6495	−1.02523	−0.512614	+	0.858619i	$0.671323\pi$
$954$	0	0
$955$	−9.72508	−0.314696
$956$	−25.0997	−0.811781
$957$	0	0
$958$	23.3746	0.755199
$959$	17.3746	0.561055
$960$	0	0
$961$	33.0000	1.06452
$962$	4.27492	0.137829
$963$	0	0
$964$	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	0
$970$	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	$0.345361\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	−18.1993	−0.583144
$975$	0	0
$976$	−12.2749	−0.392911
$977$	13.9244	0.445482	0.222741	−	0.974878i	$-0.428500\pi$
$977$	13.9244	0.445482	0.222741	+	0.974878i	$0.428500\pi$
$978$	0	0
$979$	31.8488	1.01789
$980$	4.27492	0.136557
$981$	0	0
$982$	−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	$0.926117\pi$
$984$	0	0
$985$	10.9003	0.347313
$986$	2.27492	0.0724481
$987$	0	0
$988$	2.27492	0.0723748
$989$	5.17525	0.164563
$990$	0	0
$991$	−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
$993$	0	0
$994$	0	0
$995$	29.1752	0.924918
$996$	0	0
$997$	−12.9003	−0.408558	−0.204279	−	0.978913i	$-0.565485\pi$
$997$	−12.9003	−0.408558	−0.204279	+	0.978913i	$0.565485\pi$
$998$	25.0997	0.794516
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.y.1.2		2
3.2	odd	2		546.2.a.h.1.1	✓	2
12.11	even	2		4368.2.a.bh.1.1		2
21.20	even	2		3822.2.a.bm.1.2		2
39.38	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	3.2	odd	2
1638.2.a.y.1.2		2	1.1	even	1	trivial
3822.2.a.bm.1.2		2	21.20	even	2
4368.2.a.bh.1.1		2	12.11	even	2
7098.2.a.bu.1.2		2	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.27492 q^{5} +1.00000 q^{7} +1.00000 q^{8} +4.27492 q^{10} +2.27492 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -0.274917 q^{17} -2.27492 q^{19} +4.27492 q^{20} +2.27492 q^{22} -2.27492 q^{23} +13.2749 q^{25} -1.00000 q^{26} +1.00000 q^{28} -8.27492 q^{29} +8.00000 q^{31} +1.00000 q^{32} -0.274917 q^{34} +4.27492 q^{35} -4.27492 q^{37} -2.27492 q^{38} +4.27492 q^{40} -6.54983 q^{41} -2.27492 q^{43} +2.27492 q^{44} -2.27492 q^{46} +1.00000 q^{49} +13.2749 q^{50} -1.00000 q^{52} -10.0000 q^{53} +9.72508 q^{55} +1.00000 q^{56} -8.27492 q^{58} -8.00000 q^{59} -12.2749 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.27492 q^{65} +12.5498 q^{67} -0.274917 q^{68} +4.27492 q^{70} -12.8248 q^{73} -4.27492 q^{74} -2.27492 q^{76} +2.27492 q^{77} +12.5498 q^{79} +4.27492 q^{80} -6.54983 q^{82} +4.54983 q^{83} -1.17525 q^{85} -2.27492 q^{86} +2.27492 q^{88} +14.0000 q^{89} -1.00000 q^{91} -2.27492 q^{92} -9.72508 q^{95} +15.0997 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} - 3 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{19} + q^{20} - 3 q^{22} + 3 q^{23} + 19 q^{25} - 2 q^{26} + 2 q^{28} - 9 q^{29} + 16 q^{31} + 2 q^{32} + 7 q^{34} + q^{35} - q^{37} + 3 q^{38} + q^{40} + 2 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 2 q^{49} + 19 q^{50} - 2 q^{52} - 20 q^{53} + 27 q^{55} + 2 q^{56} - 9 q^{58} - 16 q^{59} - 17 q^{61} + 16 q^{62} + 2 q^{64} - q^{65} + 10 q^{67} + 7 q^{68} + q^{70} - 3 q^{73} - q^{74} + 3 q^{76} - 3 q^{77} + 10 q^{79} + q^{80} + 2 q^{82} - 6 q^{83} - 25 q^{85} + 3 q^{86} - 3 q^{88} + 28 q^{89} - 2 q^{91} + 3 q^{92} - 27 q^{95} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 + 2 * q^7 + 2 * q^8 + q^10 - 3 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 7 * q^17 + 3 * q^19 + q^20 - 3 * q^22 + 3 * q^23 + 19 * q^25 - 2 * q^26 + 2 * q^28 - 9 * q^29 + 16 * q^31 + 2 * q^32 + 7 * q^34 + q^35 - q^37 + 3 * q^38 + q^40 + 2 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 2 * q^49 + 19 * q^50 - 2 * q^52 - 20 * q^53 + 27 * q^55 + 2 * q^56 - 9 * q^58 - 16 * q^59 - 17 * q^61 + 16 * q^62 + 2 * q^64 - q^65 + 10 * q^67 + 7 * q^68 + q^70 - 3 * q^73 - q^74 + 3 * q^76 - 3 * q^77 + 10 * q^79 + q^80 + 2 * q^82 - 6 * q^83 - 25 * q^85 + 3 * q^86 - 3 * q^88 + 28 * q^89 - 2 * q^91 + 3 * q^92 - 27 * q^95 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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