LMFDB - Embedded newform 3864.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.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:	$30.8541953410$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.69639$ of defining polynomial
Character	$\chi$	$=$	3864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.42586 q^{11} +4.69639 q^{13} -3.27053 q^{15} +3.96693 q^{17} -5.42586 q^{19} -1.00000 q^{21} +1.00000 q^{23} +5.69639 q^{25} +1.00000 q^{27} +3.96693 q^{29} -8.81864 q^{31} -1.42586 q^{33} +3.27053 q^{35} +4.81864 q^{37} +4.69639 q^{39} +2.57414 q^{41} -11.5481 q^{43} -3.27053 q^{45} +5.39279 q^{47} +1.00000 q^{49} +3.96693 q^{51} -12.6633 q^{53} +4.66332 q^{55} -5.42586 q^{57} +11.5150 q^{59} -13.2375 q^{61} -1.00000 q^{63} -15.3597 q^{65} +5.51504 q^{67} +1.00000 q^{69} -2.69639 q^{71} -9.35971 q^{73} +5.69639 q^{75} +1.42586 q^{77} -4.57414 q^{79} +1.00000 q^{81} -0.277576 q^{83} -12.9740 q^{85} +3.96693 q^{87} +11.4820 q^{89} -4.69639 q^{91} -8.81864 q^{93} +17.7455 q^{95} +7.96693 q^{97} -1.42586 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 5 q^{13} - 3 q^{15} - 4 q^{17} - 14 q^{19} - 3 q^{21} + 3 q^{23} + 8 q^{25} + 3 q^{27} - 4 q^{29} - 6 q^{31} - 2 q^{33} + 3 q^{35} - 6 q^{37} + 5 q^{39} + 10 q^{41} - 21 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 13 q^{53} - 11 q^{55} - 14 q^{57} + 5 q^{59} - 17 q^{61} - 3 q^{63} - 12 q^{65} - 13 q^{67} + 3 q^{69} + q^{71} + 6 q^{73} + 8 q^{75} + 2 q^{77} - 16 q^{79} + 3 q^{81} + 6 q^{83} - 23 q^{85} - 4 q^{87} - 11 q^{89} - 5 q^{91} - 6 q^{93} + q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 5 * q^13 - 3 * q^15 - 4 * q^17 - 14 * q^19 - 3 * q^21 + 3 * q^23 + 8 * q^25 + 3 * q^27 - 4 * q^29 - 6 * q^31 - 2 * q^33 + 3 * q^35 - 6 * q^37 + 5 * q^39 + 10 * q^41 - 21 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 13 * q^53 - 11 * q^55 - 14 * q^57 + 5 * q^59 - 17 * q^61 - 3 * q^63 - 12 * q^65 - 13 * q^67 + 3 * q^69 + q^71 + 6 * q^73 + 8 * q^75 + 2 * q^77 - 16 * q^79 + 3 * q^81 + 6 * q^83 - 23 * q^85 - 4 * q^87 - 11 * q^89 - 5 * q^91 - 6 * q^93 + q^95 + 8 * q^97 - 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.27053	−1.46263	−0.731314	−	0.682041i	$-0.761092\pi$
$5$	−3.27053	−1.46263	−0.731314	+	0.682041i	$0.761092\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.42586	−0.429913	−0.214956	−	0.976624i	$-0.568961\pi$
$11$	−1.42586	−0.429913	−0.214956	+	0.976624i	$0.568961\pi$
$12$	0	0
$13$	4.69639	1.30254	0.651272	−	0.758844i	$-0.274235\pi$
$13$	4.69639	1.30254	0.651272	+	0.758844i	$0.274235\pi$
$14$	0	0
$15$	−3.27053	−0.844448
$16$	0	0
$17$	3.96693	0.962121	0.481061	−	0.876687i	$-0.340252\pi$
$17$	3.96693	0.962121	0.481061	+	0.876687i	$0.340252\pi$
$18$	0	0
$19$	−5.42586	−1.24478	−0.622389	−	0.782708i	$-0.713838\pi$
$19$	−5.42586	−1.24478	−0.622389	+	0.782708i	$0.713838\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.69639	1.13928
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.96693	0.736640	0.368320	−	0.929699i	$-0.379933\pi$
$29$	3.96693	0.736640	0.368320	+	0.929699i	$0.379933\pi$
$30$	0	0
$31$	−8.81864	−1.58388	−0.791938	−	0.610602i	$-0.790928\pi$
$31$	−8.81864	−1.58388	−0.791938	+	0.610602i	$0.790928\pi$
$32$	0	0
$33$	−1.42586	−0.248210
$34$	0	0
$35$	3.27053	0.552821
$36$	0	0
$37$	4.81864	0.792180	0.396090	−	0.918212i	$-0.370367\pi$
$37$	4.81864	0.792180	0.396090	+	0.918212i	$0.370367\pi$
$38$	0	0
$39$	4.69639	0.752025
$40$	0	0
$41$	2.57414	0.402013	0.201007	−	0.979590i	$-0.435579\pi$
$41$	2.57414	0.402013	0.201007	+	0.979590i	$0.435579\pi$
$42$	0	0
$43$	−11.5481	−1.76107	−0.880535	−	0.473981i	$-0.842817\pi$
$43$	−11.5481	−1.76107	−0.880535	+	0.473981i	$0.842817\pi$
$44$	0	0
$45$	−3.27053	−0.487542
$46$	0	0
$47$	5.39279	0.786619	0.393309	−	0.919406i	$-0.371330\pi$
$47$	5.39279	0.786619	0.393309	+	0.919406i	$0.371330\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.96693	0.555481
$52$	0	0
$53$	−12.6633	−1.73944	−0.869720	−	0.493545i	$-0.835701\pi$
$53$	−12.6633	−1.73944	−0.869720	+	0.493545i	$0.835701\pi$
$54$	0	0
$55$	4.66332	0.628802
$56$	0	0
$57$	−5.42586	−0.718673
$58$	0	0
$59$	11.5150	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$59$	11.5150	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$60$	0	0
$61$	−13.2375	−1.69488	−0.847442	−	0.530889i	$-0.821858\pi$
$61$	−13.2375	−1.69488	−0.847442	+	0.530889i	$0.821858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−15.3597	−1.90514
$66$	0	0
$67$	5.51504	0.673769	0.336885	−	0.941546i	$-0.390627\pi$
$67$	5.51504	0.673769	0.336885	+	0.941546i	$0.390627\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−2.69639	−0.320003	−0.160001	−	0.987117i	$-0.551150\pi$
$71$	−2.69639	−0.320003	−0.160001	+	0.987117i	$0.551150\pi$
$72$	0	0
$73$	−9.35971	−1.09547	−0.547736	−	0.836651i	$-0.684510\pi$
$73$	−9.35971	−1.09547	−0.547736	+	0.836651i	$0.684510\pi$
$74$	0	0
$75$	5.69639	0.657763
$76$	0	0
$77$	1.42586	0.162492
$78$	0	0
$79$	−4.57414	−0.514631	−0.257316	−	0.966327i	$-0.582838\pi$
$79$	−4.57414	−0.514631	−0.257316	+	0.966327i	$0.582838\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.277576	−0.0304680	−0.0152340	−	0.999884i	$-0.504849\pi$
$83$	−0.277576	−0.0304680	−0.0152340	+	0.999884i	$0.504849\pi$
$84$	0	0
$85$	−12.9740	−1.40722
$86$	0	0
$87$	3.96693	0.425299
$88$	0	0
$89$	11.4820	1.21709	0.608543	−	0.793521i	$-0.291754\pi$
$89$	11.4820	1.21709	0.608543	+	0.793521i	$0.291754\pi$
$90$	0	0
$91$	−4.69639	−0.492316
$92$	0	0
$93$	−8.81864	−0.914451
$94$	0	0
$95$	17.7455	1.82065
$96$	0	0
$97$	7.96693	0.808919	0.404459	−	0.914556i	$-0.367460\pi$
$97$	7.96693	0.808919	0.404459	+	0.914556i	$0.367460\pi$
$98$	0	0
$99$	−1.42586	−0.143304
$100$	0	0
$101$	6.97397	0.693936	0.346968	−	0.937877i	$-0.387211\pi$
$101$	6.97397	0.693936	0.346968	+	0.937877i	$0.387211\pi$
$102$	0	0
$103$	−19.8677	−1.95762	−0.978812	−	0.204763i	$-0.934358\pi$
$103$	−19.8677	−1.95762	−0.978812	+	0.204763i	$0.934358\pi$
$104$	0	0
$105$	3.27053	0.319171
$106$	0	0
$107$	−10.3998	−1.00539	−0.502695	−	0.864464i	$-0.667658\pi$
$107$	−10.3998	−1.00539	−0.502695	+	0.864464i	$0.667658\pi$
$108$	0	0
$109$	−12.3527	−1.18317	−0.591586	−	0.806242i	$-0.701498\pi$
$109$	−12.3527	−1.18317	−0.591586	+	0.806242i	$0.701498\pi$
$110$	0	0
$111$	4.81864	0.457365
$112$	0	0
$113$	−2.41882	−0.227543	−0.113772	−	0.993507i	$-0.536293\pi$
$113$	−2.41882	−0.227543	−0.113772	+	0.993507i	$0.536293\pi$
$114$	0	0
$115$	−3.27053	−0.304979
$116$	0	0
$117$	4.69639	0.434182
$118$	0	0
$119$	−3.96693	−0.363648
$120$	0	0
$121$	−8.96693	−0.815175
$122$	0	0
$123$	2.57414	0.232102
$124$	0	0
$125$	−2.27758	−0.203713
$126$	0	0
$127$	−16.9268	−1.50201	−0.751006	−	0.660296i	$-0.770431\pi$
$127$	−16.9268	−1.50201	−0.751006	+	0.660296i	$0.770431\pi$
$128$	0	0
$129$	−11.5481	−1.01675
$130$	0	0
$131$	−19.2936	−1.68569	−0.842843	−	0.538159i	$-0.819120\pi$
$131$	−19.2936	−1.68569	−0.842843	+	0.538159i	$0.819120\pi$
$132$	0	0
$133$	5.42586	0.470482
$134$	0	0
$135$	−3.27053	−0.281483
$136$	0	0
$137$	4.27758	0.365458	0.182729	−	0.983163i	$-0.441507\pi$
$137$	4.27758	0.365458	0.182729	+	0.983163i	$0.441507\pi$
$138$	0	0
$139$	−2.18840	−0.185617	−0.0928087	−	0.995684i	$-0.529585\pi$
$139$	−2.18840	−0.185617	−0.0928087	+	0.995684i	$0.529585\pi$
$140$	0	0
$141$	5.39279	0.454154
$142$	0	0
$143$	−6.69639	−0.559980
$144$	0	0
$145$	−12.9740	−1.07743
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−15.6894	−1.28532	−0.642661	−	0.766151i	$-0.722170\pi$
$149$	−15.6894	−1.28532	−0.642661	+	0.766151i	$0.722170\pi$
$150$	0	0
$151$	11.0962	0.902998	0.451499	−	0.892272i	$-0.350890\pi$
$151$	11.0962	0.902998	0.451499	+	0.892272i	$0.350890\pi$
$152$	0	0
$153$	3.96693	0.320707
$154$	0	0
$155$	28.8417	2.31662
$156$	0	0
$157$	−6.54107	−0.522034	−0.261017	−	0.965334i	$-0.584058\pi$
$157$	−6.54107	−0.522034	−0.261017	+	0.965334i	$0.584058\pi$
$158$	0	0
$159$	−12.6633	−1.00427
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.51504	0.275319	0.137659	−	0.990480i	$-0.456042\pi$
$163$	3.51504	0.275319	0.137659	+	0.990480i	$0.456042\pi$
$164$	0	0
$165$	4.66332	0.363039
$166$	0	0
$167$	−2.27758	−0.176244	−0.0881221	−	0.996110i	$-0.528087\pi$
$167$	−2.27758	−0.176244	−0.0881221	+	0.996110i	$0.528087\pi$
$168$	0	0
$169$	9.05611	0.696623
$170$	0	0
$171$	−5.42586	−0.414926
$172$	0	0
$173$	−9.35971	−0.711606	−0.355803	−	0.934561i	$-0.615793\pi$
$173$	−9.35971	−0.711606	−0.355803	+	0.934561i	$0.615793\pi$
$174$	0	0
$175$	−5.69639	−0.430607
$176$	0	0
$177$	11.5150	0.865523
$178$	0	0
$179$	−4.66332	−0.348553	−0.174276	−	0.984697i	$-0.555759\pi$
$179$	−4.66332	−0.348553	−0.174276	+	0.984697i	$0.555759\pi$
$180$	0	0
$181$	−17.3738	−1.29138	−0.645692	−	0.763598i	$-0.723431\pi$
$181$	−17.3738	−1.29138	−0.645692	+	0.763598i	$0.723431\pi$
$182$	0	0
$183$	−13.2375	−0.978541
$184$	0	0
$185$	−15.7595	−1.15866
$186$	0	0
$187$	−5.65628	−0.413628
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−11.6894	−0.845812	−0.422906	−	0.906174i	$-0.638990\pi$
$191$	−11.6894	−0.845812	−0.422906	+	0.906174i	$0.638990\pi$
$192$	0	0
$193$	4.60721	0.331635	0.165817	−	0.986156i	$-0.446974\pi$
$193$	4.60721	0.331635	0.165817	+	0.986156i	$0.446974\pi$
$194$	0	0
$195$	−15.3597	−1.09993
$196$	0	0
$197$	−24.8747	−1.77225	−0.886126	−	0.463444i	$-0.846614\pi$
$197$	−24.8747	−1.77225	−0.886126	+	0.463444i	$0.846614\pi$
$198$	0	0
$199$	7.44889	0.528038	0.264019	−	0.964517i	$-0.414952\pi$
$199$	7.44889	0.528038	0.264019	+	0.964517i	$0.414952\pi$
$200$	0	0
$201$	5.51504	0.389001
$202$	0	0
$203$	−3.96693	−0.278424
$204$	0	0
$205$	−8.41882	−0.587996
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	7.73651	0.535145
$210$	0	0
$211$	11.9810	0.824807	0.412403	−	0.911001i	$-0.364689\pi$
$211$	11.9810	0.824807	0.412403	+	0.911001i	$0.364689\pi$
$212$	0	0
$213$	−2.69639	−0.184754
$214$	0	0
$215$	37.7685	2.57579
$216$	0	0
$217$	8.81864	0.598649
$218$	0	0
$219$	−9.35971	−0.632471
$220$	0	0
$221$	18.6302	1.25321
$222$	0	0
$223$	−10.1884	−0.682266	−0.341133	−	0.940015i	$-0.610811\pi$
$223$	−10.1884	−0.682266	−0.341133	+	0.940015i	$0.610811\pi$
$224$	0	0
$225$	5.69639	0.379760
$226$	0	0
$227$	−20.0561	−1.33117	−0.665585	−	0.746322i	$-0.731818\pi$
$227$	−20.0561	−1.33117	−0.665585	+	0.746322i	$0.731818\pi$
$228$	0	0
$229$	9.28462	0.613545	0.306772	−	0.951783i	$-0.400751\pi$
$229$	9.28462	0.613545	0.306772	+	0.951783i	$0.400751\pi$
$230$	0	0
$231$	1.42586	0.0938146
$232$	0	0
$233$	5.20439	0.340951	0.170475	−	0.985362i	$-0.445470\pi$
$233$	5.20439	0.340951	0.170475	+	0.985362i	$0.445470\pi$
$234$	0	0
$235$	−17.6373	−1.15053
$236$	0	0
$237$	−4.57414	−0.297122
$238$	0	0
$239$	−23.1382	−1.49669	−0.748344	−	0.663311i	$-0.769151\pi$
$239$	−23.1382	−1.49669	−0.748344	+	0.663311i	$0.769151\pi$
$240$	0	0
$241$	1.11521	0.0718369	0.0359185	−	0.999355i	$-0.488564\pi$
$241$	1.11521	0.0718369	0.0359185	+	0.999355i	$0.488564\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.27053	−0.208947
$246$	0	0
$247$	−25.4820	−1.62138
$248$	0	0
$249$	−0.277576	−0.0175907
$250$	0	0
$251$	23.8818	1.50741	0.753703	−	0.657216i	$-0.228266\pi$
$251$	23.8818	1.50741	0.753703	+	0.657216i	$0.228266\pi$
$252$	0	0
$253$	−1.42586	−0.0896430
$254$	0	0
$255$	−12.9740	−0.812461
$256$	0	0
$257$	−22.1122	−1.37932	−0.689661	−	0.724132i	$-0.742240\pi$
$257$	−22.1122	−1.37932	−0.689661	+	0.724132i	$0.742240\pi$
$258$	0	0
$259$	−4.81864	−0.299416
$260$	0	0
$261$	3.96693	0.245547
$262$	0	0
$263$	16.1973	0.998771	0.499386	−	0.866380i	$-0.333559\pi$
$263$	16.1973	0.998771	0.499386	+	0.866380i	$0.333559\pi$
$264$	0	0
$265$	41.4158	2.54415
$266$	0	0
$267$	11.4820	0.702685
$268$	0	0
$269$	20.3196	1.23891	0.619454	−	0.785033i	$-0.287354\pi$
$269$	20.3196	1.23891	0.619454	+	0.785033i	$0.287354\pi$
$270$	0	0
$271$	−16.5080	−1.00279	−0.501395	−	0.865219i	$-0.667180\pi$
$271$	−16.5080	−1.00279	−0.501395	+	0.865219i	$0.667180\pi$
$272$	0	0
$273$	−4.69639	−0.284239
$274$	0	0
$275$	−8.12225	−0.489790
$276$	0	0
$277$	−3.30361	−0.198495	−0.0992473	−	0.995063i	$-0.531643\pi$
$277$	−3.30361	−0.198495	−0.0992473	+	0.995063i	$0.531643\pi$
$278$	0	0
$279$	−8.81864	−0.527958
$280$	0	0
$281$	32.4749	1.93729	0.968646	−	0.248446i	$-0.0799199\pi$
$281$	32.4749	1.93729	0.968646	+	0.248446i	$0.0799199\pi$
$282$	0	0
$283$	6.66332	0.396093	0.198047	−	0.980193i	$-0.436540\pi$
$283$	6.66332	0.396093	0.198047	+	0.980193i	$0.436540\pi$
$284$	0	0
$285$	17.7455	1.05115
$286$	0	0
$287$	−2.57414	−0.151947
$288$	0	0
$289$	−1.26349	−0.0743230
$290$	0	0
$291$	7.96693	0.467030
$292$	0	0
$293$	−1.08214	−0.0632191	−0.0316095	−	0.999500i	$-0.510063\pi$
$293$	−1.08214	−0.0632191	−0.0316095	+	0.999500i	$0.510063\pi$
$294$	0	0
$295$	−37.6603	−2.19267
$296$	0	0
$297$	−1.42586	−0.0827367
$298$	0	0
$299$	4.69639	0.271599
$300$	0	0
$301$	11.5481	0.665622
$302$	0	0
$303$	6.97397	0.400644
$304$	0	0
$305$	43.2936	2.47898
$306$	0	0
$307$	−8.88479	−0.507082	−0.253541	−	0.967325i	$-0.581595\pi$
$307$	−8.88479	−0.507082	−0.253541	+	0.967325i	$0.581595\pi$
$308$	0	0
$309$	−19.8677	−1.13023
$310$	0	0
$311$	15.2515	0.864836	0.432418	−	0.901673i	$-0.357661\pi$
$311$	15.2515	0.864836	0.432418	+	0.901673i	$0.357661\pi$
$312$	0	0
$313$	33.3126	1.88294	0.941468	−	0.337101i	$-0.109446\pi$
$313$	33.3126	1.88294	0.941468	+	0.337101i	$0.109446\pi$
$314$	0	0
$315$	3.27053	0.184274
$316$	0	0
$317$	−17.2044	−0.966295	−0.483147	−	0.875539i	$-0.660506\pi$
$317$	−17.2044	−0.966295	−0.483147	+	0.875539i	$0.660506\pi$
$318$	0	0
$319$	−5.65628	−0.316691
$320$	0	0
$321$	−10.3998	−0.580462
$322$	0	0
$323$	−21.5240	−1.19763
$324$	0	0
$325$	26.7525	1.48396
$326$	0	0
$327$	−12.3527	−0.683104
$328$	0	0
$329$	−5.39279	−0.297314
$330$	0	0
$331$	5.14828	0.282975	0.141488	−	0.989940i	$-0.454811\pi$
$331$	5.14828	0.282975	0.141488	+	0.989940i	$0.454811\pi$
$332$	0	0
$333$	4.81864	0.264060
$334$	0	0
$335$	−18.0371	−0.985473
$336$	0	0
$337$	−6.71048	−0.365543	−0.182771	−	0.983155i	$-0.558507\pi$
$337$	−6.71048	−0.365543	−0.182771	+	0.983155i	$0.558507\pi$
$338$	0	0
$339$	−2.41882	−0.131372
$340$	0	0
$341$	12.5741	0.680928
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.27053	−0.176080
$346$	0	0
$347$	−6.83273	−0.366800	−0.183400	−	0.983038i	$-0.558710\pi$
$347$	−6.83273	−0.366800	−0.183400	+	0.983038i	$0.558710\pi$
$348$	0	0
$349$	15.7455	0.842835	0.421417	−	0.906867i	$-0.361533\pi$
$349$	15.7455	0.842835	0.421417	+	0.906867i	$0.361533\pi$
$350$	0	0
$351$	4.69639	0.250675
$352$	0	0
$353$	3.78857	0.201645	0.100823	−	0.994904i	$-0.467853\pi$
$353$	3.78857	0.201645	0.100823	+	0.994904i	$0.467853\pi$
$354$	0	0
$355$	8.81864	0.468045
$356$	0	0
$357$	−3.96693	−0.209952
$358$	0	0
$359$	34.8276	1.83813	0.919065	−	0.394106i	$-0.128946\pi$
$359$	34.8276	1.83813	0.919065	+	0.394106i	$0.128946\pi$
$360$	0	0
$361$	10.4399	0.549471
$362$	0	0
$363$	−8.96693	−0.470642
$364$	0	0
$365$	30.6113	1.60227
$366$	0	0
$367$	0.926811	0.0483792	0.0241896	−	0.999707i	$-0.492299\pi$
$367$	0.926811	0.0483792	0.0241896	+	0.999707i	$0.492299\pi$
$368$	0	0
$369$	2.57414	0.134004
$370$	0	0
$371$	12.6633	0.657447
$372$	0	0
$373$	7.60422	0.393731	0.196866	−	0.980430i	$-0.436924\pi$
$373$	7.60422	0.393731	0.196866	+	0.980430i	$0.436924\pi$
$374$	0	0
$375$	−2.27758	−0.117614
$376$	0	0
$377$	18.6302	0.959507
$378$	0	0
$379$	32.7194	1.68068	0.840342	−	0.542057i	$-0.182354\pi$
$379$	32.7194	1.68068	0.840342	+	0.542057i	$0.182354\pi$
$380$	0	0
$381$	−16.9268	−0.867187
$382$	0	0
$383$	−1.78857	−0.0913917	−0.0456958	−	0.998955i	$-0.514551\pi$
$383$	−1.78857	−0.0913917	−0.0456958	+	0.998955i	$0.514551\pi$
$384$	0	0
$385$	−4.66332	−0.237665
$386$	0	0
$387$	−11.5481	−0.587023
$388$	0	0
$389$	−30.4559	−1.54418	−0.772089	−	0.635515i	$-0.780788\pi$
$389$	−30.4559	−1.54418	−0.772089	+	0.635515i	$0.780788\pi$
$390$	0	0
$391$	3.96693	0.200616
$392$	0	0
$393$	−19.2936	−0.973232
$394$	0	0
$395$	14.9599	0.752713
$396$	0	0
$397$	26.7053	1.34030	0.670151	−	0.742225i	$-0.266229\pi$
$397$	26.7053	1.34030	0.670151	+	0.742225i	$0.266229\pi$
$398$	0	0
$399$	5.42586	0.271633
$400$	0	0
$401$	−15.9669	−0.797350	−0.398675	−	0.917092i	$-0.630530\pi$
$401$	−15.9669	−0.797350	−0.398675	+	0.917092i	$0.630530\pi$
$402$	0	0
$403$	−41.4158	−2.06307
$404$	0	0
$405$	−3.27053	−0.162514
$406$	0	0
$407$	−6.87071	−0.340568
$408$	0	0
$409$	0.560057	0.0276930	0.0138465	−	0.999904i	$-0.495592\pi$
$409$	0.560057	0.0276930	0.0138465	+	0.999904i	$0.495592\pi$
$410$	0	0
$411$	4.27758	0.210997
$412$	0	0
$413$	−11.5150	−0.566618
$414$	0	0
$415$	0.907823	0.0445633
$416$	0	0
$417$	−2.18840	−0.107166
$418$	0	0
$419$	−16.8937	−0.825313	−0.412657	−	0.910887i	$-0.635399\pi$
$419$	−16.8937	−0.825313	−0.412657	+	0.910887i	$0.635399\pi$
$420$	0	0
$421$	12.0892	0.589191	0.294595	−	0.955622i	$-0.404815\pi$
$421$	12.0892	0.589191	0.294595	+	0.955622i	$0.404815\pi$
$422$	0	0
$423$	5.39279	0.262206
$424$	0	0
$425$	22.5972	1.09612
$426$	0	0
$427$	13.2375	0.640606
$428$	0	0
$429$	−6.69639	−0.323305
$430$	0	0
$431$	10.1412	0.488486	0.244243	−	0.969714i	$-0.421460\pi$
$431$	10.1412	0.488486	0.244243	+	0.969714i	$0.421460\pi$
$432$	0	0
$433$	40.4749	1.94510	0.972550	−	0.232693i	$-0.0747536\pi$
$433$	40.4749	1.94510	0.972550	+	0.232693i	$0.0747536\pi$
$434$	0	0
$435$	−12.9740	−0.622054
$436$	0	0
$437$	−5.42586	−0.259554
$438$	0	0
$439$	−6.03307	−0.287943	−0.143971	−	0.989582i	$-0.545987\pi$
$439$	−6.03307	−0.287943	−0.143971	+	0.989582i	$0.545987\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0.621299	0.0295188	0.0147594	−	0.999891i	$-0.495302\pi$
$443$	0.621299	0.0295188	0.0147594	+	0.999891i	$0.495302\pi$
$444$	0	0
$445$	−37.5522	−1.78014
$446$	0	0
$447$	−15.6894	−0.742081
$448$	0	0
$449$	31.7455	1.49816	0.749080	−	0.662479i	$-0.230496\pi$
$449$	31.7455	1.49816	0.749080	+	0.662479i	$0.230496\pi$
$450$	0	0
$451$	−3.67036	−0.172831
$452$	0	0
$453$	11.0962	0.521346
$454$	0	0
$455$	15.3597	0.720074
$456$	0	0
$457$	−41.3166	−1.93271	−0.966354	−	0.257214i	$-0.917195\pi$
$457$	−41.3166	−1.93271	−0.966354	+	0.257214i	$0.917195\pi$
$458$	0	0
$459$	3.96693	0.185160
$460$	0	0
$461$	13.7595	0.640846	0.320423	−	0.947275i	$-0.396175\pi$
$461$	13.7595	0.640846	0.320423	+	0.947275i	$0.396175\pi$
$462$	0	0
$463$	20.1312	0.935576	0.467788	−	0.883841i	$-0.345051\pi$
$463$	20.1312	0.935576	0.467788	+	0.883841i	$0.345051\pi$
$464$	0	0
$465$	28.8417	1.33750
$466$	0	0
$467$	−25.2936	−1.17045	−0.585223	−	0.810872i	$-0.698993\pi$
$467$	−25.2936	−1.17045	−0.585223	+	0.810872i	$0.698993\pi$
$468$	0	0
$469$	−5.51504	−0.254661
$470$	0	0
$471$	−6.54107	−0.301397
$472$	0	0
$473$	16.4660	0.757106
$474$	0	0
$475$	−30.9078	−1.41815
$476$	0	0
$477$	−12.6633	−0.579814
$478$	0	0
$479$	24.6864	1.12795	0.563974	−	0.825792i	$-0.309272\pi$
$479$	24.6864	1.12795	0.563974	+	0.825792i	$0.309272\pi$
$480$	0	0
$481$	22.6302	1.03185
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−26.0561	−1.18315
$486$	0	0
$487$	−37.2415	−1.68757	−0.843787	−	0.536678i	$-0.819679\pi$
$487$	−37.2415	−1.68757	−0.843787	+	0.536678i	$0.819679\pi$
$488$	0	0
$489$	3.51504	0.158955
$490$	0	0
$491$	−39.1193	−1.76543	−0.882714	−	0.469911i	$-0.844286\pi$
$491$	−39.1193	−1.76543	−0.882714	+	0.469911i	$0.844286\pi$
$492$	0	0
$493$	15.7365	0.708737
$494$	0	0
$495$	4.66332	0.209601
$496$	0	0
$497$	2.69639	0.120950
$498$	0	0
$499$	−19.3367	−0.865629	−0.432814	−	0.901483i	$-0.642479\pi$
$499$	−19.3367	−0.865629	−0.432814	+	0.901483i	$0.642479\pi$
$500$	0	0
$501$	−2.27758	−0.101755
$502$	0	0
$503$	2.95988	0.131975	0.0659874	−	0.997820i	$-0.478980\pi$
$503$	2.95988	0.131975	0.0659874	+	0.997820i	$0.478980\pi$
$504$	0	0
$505$	−22.8086	−1.01497
$506$	0	0
$507$	9.05611	0.402196
$508$	0	0
$509$	−22.2445	−0.985970	−0.492985	−	0.870038i	$-0.664094\pi$
$509$	−22.2445	−0.985970	−0.492985	+	0.870038i	$0.664094\pi$
$510$	0	0
$511$	9.35971	0.414049
$512$	0	0
$513$	−5.42586	−0.239558
$514$	0	0
$515$	64.9780	2.86327
$516$	0	0
$517$	−7.68935	−0.338177
$518$	0	0
$519$	−9.35971	−0.410846
$520$	0	0
$521$	42.4890	1.86148	0.930739	−	0.365685i	$-0.119165\pi$
$521$	42.4890	1.86148	0.930739	+	0.365685i	$0.119165\pi$
$522$	0	0
$523$	37.7354	1.65005	0.825027	−	0.565093i	$-0.191160\pi$
$523$	37.7354	1.65005	0.825027	+	0.565093i	$0.191160\pi$
$524$	0	0
$525$	−5.69639	−0.248611
$526$	0	0
$527$	−34.9829	−1.52388
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.5150	0.499710
$532$	0	0
$533$	12.0892	0.523640
$534$	0	0
$535$	34.0130	1.47051
$536$	0	0
$537$	−4.66332	−0.201237
$538$	0	0
$539$	−1.42586	−0.0614161
$540$	0	0
$541$	−23.9479	−1.02960	−0.514801	−	0.857310i	$-0.672134\pi$
$541$	−23.9479	−1.02960	−0.514801	+	0.857310i	$0.672134\pi$
$542$	0	0
$543$	−17.3738	−0.745581
$544$	0	0
$545$	40.3998	1.73054
$546$	0	0
$547$	3.44889	0.147464	0.0737320	−	0.997278i	$-0.476509\pi$
$547$	3.44889	0.147464	0.0737320	+	0.997278i	$0.476509\pi$
$548$	0	0
$549$	−13.2375	−0.564961
$550$	0	0
$551$	−21.5240	−0.916953
$552$	0	0
$553$	4.57414	0.194512
$554$	0	0
$555$	−15.7595	−0.668955
$556$	0	0
$557$	−19.0301	−0.806330	−0.403165	−	0.915127i	$-0.632090\pi$
$557$	−19.0301	−0.806330	−0.403165	+	0.915127i	$0.632090\pi$
$558$	0	0
$559$	−54.2345	−2.29387
$560$	0	0
$561$	−5.65628	−0.238808
$562$	0	0
$563$	−3.76444	−0.158652	−0.0793262	−	0.996849i	$-0.525277\pi$
$563$	−3.76444	−0.158652	−0.0793262	+	0.996849i	$0.525277\pi$
$564$	0	0
$565$	7.91082	0.332811
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−33.9339	−1.42258	−0.711291	−	0.702898i	$-0.751889\pi$
$569$	−33.9339	−1.42258	−0.711291	+	0.702898i	$0.751889\pi$
$570$	0	0
$571$	24.0791	1.00768	0.503840	−	0.863797i	$-0.331920\pi$
$571$	24.0791	1.00768	0.503840	+	0.863797i	$0.331920\pi$
$572$	0	0
$573$	−11.6894	−0.488330
$574$	0	0
$575$	5.69639	0.237556
$576$	0	0
$577$	28.1924	1.17367	0.586833	−	0.809708i	$-0.300374\pi$
$577$	28.1924	1.17367	0.586833	+	0.809708i	$0.300374\pi$
$578$	0	0
$579$	4.60721	0.191469
$580$	0	0
$581$	0.277576	0.0115158
$582$	0	0
$583$	18.0561	0.747807
$584$	0	0
$585$	−15.3597	−0.635046
$586$	0	0
$587$	24.8888	1.02727	0.513636	−	0.858008i	$-0.328298\pi$
$587$	24.8888	1.02727	0.513636	+	0.858008i	$0.328298\pi$
$588$	0	0
$589$	47.8487	1.97157
$590$	0	0
$591$	−24.8747	−1.02321
$592$	0	0
$593$	19.3928	0.796366	0.398183	−	0.917306i	$-0.369641\pi$
$593$	19.3928	0.796366	0.398183	+	0.917306i	$0.369641\pi$
$594$	0	0
$595$	12.9740	0.531881
$596$	0	0
$597$	7.44889	0.304863
$598$	0	0
$599$	4.15532	0.169782	0.0848910	−	0.996390i	$-0.472946\pi$
$599$	4.15532	0.169782	0.0848910	+	0.996390i	$0.472946\pi$
$600$	0	0
$601$	−0.300608	−0.0122621	−0.00613103	−	0.999981i	$-0.501952\pi$
$601$	−0.300608	−0.0122621	−0.00613103	+	0.999981i	$0.501952\pi$
$602$	0	0
$603$	5.51504	0.224590
$604$	0	0
$605$	29.3266	1.19230
$606$	0	0
$607$	−29.3497	−1.19127	−0.595633	−	0.803257i	$-0.703099\pi$
$607$	−29.3497	−1.19127	−0.595633	+	0.803257i	$0.703099\pi$
$608$	0	0
$609$	−3.96693	−0.160748
$610$	0	0
$611$	25.3266	1.02461
$612$	0	0
$613$	−5.72242	−0.231127	−0.115563	−	0.993300i	$-0.536867\pi$
$613$	−5.72242	−0.231127	−0.115563	+	0.993300i	$0.536867\pi$
$614$	0	0
$615$	−8.41882	−0.339479
$616$	0	0
$617$	24.6964	0.994239	0.497120	−	0.867682i	$-0.334391\pi$
$617$	24.6964	0.994239	0.497120	+	0.867682i	$0.334391\pi$
$618$	0	0
$619$	−40.7425	−1.63758	−0.818789	−	0.574095i	$-0.805354\pi$
$619$	−40.7425	−1.63758	−0.818789	+	0.574095i	$0.805354\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−11.4820	−0.460015
$624$	0	0
$625$	−21.0331	−0.841323
$626$	0	0
$627$	7.73651	0.308966
$628$	0	0
$629$	19.1152	0.762173
$630$	0	0
$631$	−10.5221	−0.418877	−0.209439	−	0.977822i	$-0.567164\pi$
$631$	−10.5221	−0.418877	−0.209439	+	0.977822i	$0.567164\pi$
$632$	0	0
$633$	11.9810	0.476202
$634$	0	0
$635$	55.3597	2.19688
$636$	0	0
$637$	4.69639	0.186078
$638$	0	0
$639$	−2.69639	−0.106668
$640$	0	0
$641$	−2.64433	−0.104445	−0.0522224	−	0.998635i	$-0.516630\pi$
$641$	−2.64433	−0.104445	−0.0522224	+	0.998635i	$0.516630\pi$
$642$	0	0
$643$	5.18540	0.204492	0.102246	−	0.994759i	$-0.467397\pi$
$643$	5.18540	0.204492	0.102246	+	0.994759i	$0.467397\pi$
$644$	0	0
$645$	37.7685	1.48713
$646$	0	0
$647$	17.2564	0.678421	0.339211	−	0.940710i	$-0.389840\pi$
$647$	17.2564	0.678421	0.339211	+	0.940710i	$0.389840\pi$
$648$	0	0
$649$	−16.4188	−0.644495
$650$	0	0
$651$	8.81864	0.345630
$652$	0	0
$653$	−31.1523	−1.21908	−0.609542	−	0.792754i	$-0.708647\pi$
$653$	−31.1523	−1.21908	−0.609542	+	0.792754i	$0.708647\pi$
$654$	0	0
$655$	63.1003	2.46553
$656$	0	0
$657$	−9.35971	−0.365157
$658$	0	0
$659$	5.84872	0.227834	0.113917	−	0.993490i	$-0.463660\pi$
$659$	5.84872	0.227834	0.113917	+	0.993490i	$0.463660\pi$
$660$	0	0
$661$	−7.64219	−0.297247	−0.148623	−	0.988894i	$-0.547484\pi$
$661$	−7.64219	−0.297247	−0.148623	+	0.988894i	$0.547484\pi$
$662$	0	0
$663$	18.6302	0.723539
$664$	0	0
$665$	−17.7455	−0.688139
$666$	0	0
$667$	3.96693	0.153600
$668$	0	0
$669$	−10.1884	−0.393906
$670$	0	0
$671$	18.8747	0.728652
$672$	0	0
$673$	2.26349	0.0872512	0.0436256	−	0.999048i	$-0.486109\pi$
$673$	2.26349	0.0872512	0.0436256	+	0.999048i	$0.486109\pi$
$674$	0	0
$675$	5.69639	0.219254
$676$	0	0
$677$	22.8747	0.879148	0.439574	−	0.898206i	$-0.355129\pi$
$677$	22.8747	0.879148	0.439574	+	0.898206i	$0.355129\pi$
$678$	0	0
$679$	−7.96693	−0.305743
$680$	0	0
$681$	−20.0561	−0.768552
$682$	0	0
$683$	10.3437	0.395792	0.197896	−	0.980223i	$-0.436589\pi$
$683$	10.3437	0.395792	0.197896	+	0.980223i	$0.436589\pi$
$684$	0	0
$685$	−13.9900	−0.534529
$686$	0	0
$687$	9.28462	0.354230
$688$	0	0
$689$	−59.4719	−2.26570
$690$	0	0
$691$	−37.6603	−1.43267	−0.716333	−	0.697759i	$-0.754181\pi$
$691$	−37.6603	−1.43267	−0.716333	+	0.697759i	$0.754181\pi$
$692$	0	0
$693$	1.42586	0.0541639
$694$	0	0
$695$	7.15723	0.271489
$696$	0	0
$697$	10.2114	0.386785
$698$	0	0
$699$	5.20439	0.196848
$700$	0	0
$701$	−26.1223	−0.986624	−0.493312	−	0.869852i	$-0.664214\pi$
$701$	−26.1223	−0.986624	−0.493312	+	0.869852i	$0.664214\pi$
$702$	0	0
$703$	−26.1453	−0.986088
$704$	0	0
$705$	−17.6373	−0.664259
$706$	0	0
$707$	−6.97397	−0.262283
$708$	0	0
$709$	18.1412	0.681309	0.340654	−	0.940189i	$-0.389351\pi$
$709$	18.1412	0.681309	0.340654	+	0.940189i	$0.389351\pi$
$710$	0	0
$711$	−4.57414	−0.171544
$712$	0	0
$713$	−8.81864	−0.330261
$714$	0	0
$715$	21.9008	0.819043
$716$	0	0
$717$	−23.1382	−0.864113
$718$	0	0
$719$	−18.4559	−0.688290	−0.344145	−	0.938916i	$-0.611831\pi$
$719$	−18.4559	−0.688290	−0.344145	+	0.938916i	$0.611831\pi$
$720$	0	0
$721$	19.8677	0.739912
$722$	0	0
$723$	1.11521	0.0414751
$724$	0	0
$725$	22.5972	0.839238
$726$	0	0
$727$	17.6563	0.654835	0.327418	−	0.944880i	$-0.393822\pi$
$727$	17.6563	0.654835	0.327418	+	0.944880i	$0.393822\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−45.8105	−1.69436
$732$	0	0
$733$	−36.1122	−1.33383	−0.666917	−	0.745132i	$-0.732387\pi$
$733$	−36.1122	−1.33383	−0.666917	+	0.745132i	$0.732387\pi$
$734$	0	0
$735$	−3.27053	−0.120635
$736$	0	0
$737$	−7.86366	−0.289662
$738$	0	0
$739$	−19.9149	−0.732580	−0.366290	−	0.930501i	$-0.619372\pi$
$739$	−19.9149	−0.732580	−0.366290	+	0.930501i	$0.619372\pi$
$740$	0	0
$741$	−25.4820	−0.936103
$742$	0	0
$743$	−40.2865	−1.47797	−0.738985	−	0.673722i	$-0.764694\pi$
$743$	−40.2865	−1.47797	−0.738985	+	0.673722i	$0.764694\pi$
$744$	0	0
$745$	51.3126	1.87995
$746$	0	0
$747$	−0.277576	−0.0101560
$748$	0	0
$749$	10.3998	0.380001
$750$	0	0
$751$	−49.6793	−1.81282	−0.906412	−	0.422395i	$-0.861190\pi$
$751$	−49.6793	−1.81282	−0.906412	+	0.422395i	$0.861190\pi$
$752$	0	0
$753$	23.8818	0.870301
$754$	0	0
$755$	−36.2906	−1.32075
$756$	0	0
$757$	−37.7685	−1.37272	−0.686360	−	0.727262i	$-0.740792\pi$
$757$	−37.7685	−1.37272	−0.686360	+	0.727262i	$0.740792\pi$
$758$	0	0
$759$	−1.42586	−0.0517554
$760$	0	0
$761$	3.14828	0.114125	0.0570626	−	0.998371i	$-0.481827\pi$
$761$	3.14828	0.114125	0.0570626	+	0.998371i	$0.481827\pi$
$762$	0	0
$763$	12.3527	0.447197
$764$	0	0
$765$	−12.9740	−0.469075
$766$	0	0
$767$	54.0791	1.95268
$768$	0	0
$769$	19.2605	0.694551	0.347276	−	0.937763i	$-0.387107\pi$
$769$	19.2605	0.694551	0.347276	+	0.937763i	$0.387107\pi$
$770$	0	0
$771$	−22.1122	−0.796352
$772$	0	0
$773$	−13.0631	−0.469849	−0.234924	−	0.972014i	$-0.575484\pi$
$773$	−13.0631	−0.469849	−0.234924	+	0.972014i	$0.575484\pi$
$774$	0	0
$775$	−50.2345	−1.80448
$776$	0	0
$777$	−4.81864	−0.172868
$778$	0	0
$779$	−13.9669	−0.500417
$780$	0	0
$781$	3.84468	0.137573
$782$	0	0
$783$	3.96693	0.141766
$784$	0	0
$785$	21.3928	0.763541
$786$	0	0
$787$	−45.7595	−1.63115	−0.815576	−	0.578650i	$-0.803580\pi$
$787$	−45.7595	−1.63115	−0.815576	+	0.578650i	$0.803580\pi$
$788$	0	0
$789$	16.1973	0.576641
$790$	0	0
$791$	2.41882	0.0860032
$792$	0	0
$793$	−62.1683	−2.20766
$794$	0	0
$795$	41.4158	1.46887
$796$	0	0
$797$	10.4749	0.371041	0.185520	−	0.982640i	$-0.440603\pi$
$797$	10.4749	0.371041	0.185520	+	0.982640i	$0.440603\pi$
$798$	0	0
$799$	21.3928	0.756822
$800$	0	0
$801$	11.4820	0.405695
$802$	0	0
$803$	13.3456	0.470957
$804$	0	0
$805$	3.27053	0.115271
$806$	0	0
$807$	20.3196	0.715284
$808$	0	0
$809$	14.0230	0.493024	0.246512	−	0.969140i	$-0.420716\pi$
$809$	14.0230	0.493024	0.246512	+	0.969140i	$0.420716\pi$
$810$	0	0
$811$	30.0791	1.05622	0.528111	−	0.849176i	$-0.322901\pi$
$811$	30.0791	1.05622	0.528111	+	0.849176i	$0.322901\pi$
$812$	0	0
$813$	−16.5080	−0.578961
$814$	0	0
$815$	−11.4960	−0.402689
$816$	0	0
$817$	62.6584	2.19214
$818$	0	0
$819$	−4.69639	−0.164105
$820$	0	0
$821$	44.4088	1.54988	0.774938	−	0.632037i	$-0.217781\pi$
$821$	44.4088	1.54988	0.774938	+	0.632037i	$0.217781\pi$
$822$	0	0
$823$	−14.8276	−0.516857	−0.258429	−	0.966030i	$-0.583205\pi$
$823$	−14.8276	−0.516857	−0.258429	+	0.966030i	$0.583205\pi$
$824$	0	0
$825$	−8.12225	−0.282781
$826$	0	0
$827$	−27.0401	−0.940277	−0.470138	−	0.882593i	$-0.655796\pi$
$827$	−27.0401	−0.940277	−0.470138	+	0.882593i	$0.655796\pi$
$828$	0	0
$829$	−11.9669	−0.415629	−0.207814	−	0.978168i	$-0.566635\pi$
$829$	−11.9669	−0.415629	−0.207814	+	0.978168i	$0.566635\pi$
$830$	0	0
$831$	−3.30361	−0.114601
$832$	0	0
$833$	3.96693	0.137446
$834$	0	0
$835$	7.44889	0.257779
$836$	0	0
$837$	−8.81864	−0.304817
$838$	0	0
$839$	−42.2535	−1.45875	−0.729376	−	0.684114i	$-0.760189\pi$
$839$	−42.2535	−1.45875	−0.729376	+	0.684114i	$0.760189\pi$
$840$	0	0
$841$	−13.2635	−0.457362
$842$	0	0
$843$	32.4749	1.11850
$844$	0	0
$845$	−29.6183	−1.01890
$846$	0	0
$847$	8.96693	0.308107
$848$	0	0
$849$	6.66332	0.228685
$850$	0	0
$851$	4.81864	0.165181
$852$	0	0
$853$	20.4749	0.701048	0.350524	−	0.936554i	$-0.386004\pi$
$853$	20.4749	0.701048	0.350524	+	0.936554i	$0.386004\pi$
$854$	0	0
$855$	17.7455	0.606882
$856$	0	0
$857$	17.6894	0.604257	0.302128	−	0.953267i	$-0.402303\pi$
$857$	17.6894	0.604257	0.302128	+	0.953267i	$0.402303\pi$
$858$	0	0
$859$	−5.01599	−0.171143	−0.0855717	−	0.996332i	$-0.527272\pi$
$859$	−5.01599	−0.171143	−0.0855717	+	0.996332i	$0.527272\pi$
$860$	0	0
$861$	−2.57414	−0.0877265
$862$	0	0
$863$	16.9699	0.577663	0.288831	−	0.957380i	$-0.406733\pi$
$863$	16.9699	0.577663	0.288831	+	0.957380i	$0.406733\pi$
$864$	0	0
$865$	30.6113	1.04081
$866$	0	0
$867$	−1.26349	−0.0429104
$868$	0	0
$869$	6.52208	0.221246
$870$	0	0
$871$	25.9008	0.877614
$872$	0	0
$873$	7.96693	0.269640
$874$	0	0
$875$	2.27758	0.0769961
$876$	0	0
$877$	0.851718	0.0287605	0.0143802	−	0.999897i	$-0.495422\pi$
$877$	0.851718	0.0287605	0.0143802	+	0.999897i	$0.495422\pi$
$878$	0	0
$879$	−1.08214	−0.0364995
$880$	0	0
$881$	−44.8316	−1.51042	−0.755208	−	0.655485i	$-0.772464\pi$
$881$	−44.8316	−1.51042	−0.755208	+	0.655485i	$0.772464\pi$
$882$	0	0
$883$	51.6462	1.73803	0.869017	−	0.494782i	$-0.164752\pi$
$883$	51.6462	1.73803	0.869017	+	0.494782i	$0.164752\pi$
$884$	0	0
$885$	−37.6603	−1.26594
$886$	0	0
$887$	54.9870	1.84628	0.923141	−	0.384462i	$-0.125613\pi$
$887$	54.9870	1.84628	0.923141	+	0.384462i	$0.125613\pi$
$888$	0	0
$889$	16.9268	0.567707
$890$	0	0
$891$	−1.42586	−0.0477681
$892$	0	0
$893$	−29.2605	−0.979165
$894$	0	0
$895$	15.2515	0.509803
$896$	0	0
$897$	4.69639	0.156808
$898$	0	0
$899$	−34.9829	−1.16675
$900$	0	0
$901$	−50.2345	−1.67355
$902$	0	0
$903$	11.5481	0.384297
$904$	0	0
$905$	56.8216	1.88881
$906$	0	0
$907$	−27.6132	−0.916880	−0.458440	−	0.888725i	$-0.651592\pi$
$907$	−27.6132	−0.916880	−0.458440	+	0.888725i	$0.651592\pi$
$908$	0	0
$909$	6.97397	0.231312
$910$	0	0
$911$	25.3928	0.841301	0.420650	−	0.907223i	$-0.361802\pi$
$911$	25.3928	0.841301	0.420650	+	0.907223i	$0.361802\pi$
$912$	0	0
$913$	0.395785	0.0130986
$914$	0	0
$915$	43.2936	1.43124
$916$	0	0
$917$	19.2936	0.637130
$918$	0	0
$919$	52.6202	1.73578	0.867890	−	0.496756i	$-0.165476\pi$
$919$	52.6202	1.73578	0.867890	+	0.496756i	$0.165476\pi$
$920$	0	0
$921$	−8.88479	−0.292764
$922$	0	0
$923$	−12.6633	−0.416818
$924$	0	0
$925$	27.4489	0.902514
$926$	0	0
$927$	−19.8677	−0.652541
$928$	0	0
$929$	−39.9900	−1.31203	−0.656014	−	0.754749i	$-0.727759\pi$
$929$	−39.9900	−1.31203	−0.656014	+	0.754749i	$0.727759\pi$
$930$	0	0
$931$	−5.42586	−0.177825
$932$	0	0
$933$	15.2515	0.499313
$934$	0	0
$935$	18.4990	0.604984
$936$	0	0
$937$	−36.6061	−1.19587	−0.597935	−	0.801545i	$-0.704012\pi$
$937$	−36.6061	−1.19587	−0.597935	+	0.801545i	$0.704012\pi$
$938$	0	0
$939$	33.3126	1.08711
$940$	0	0
$941$	31.4069	1.02383	0.511917	−	0.859035i	$-0.328935\pi$
$941$	31.4069	1.02383	0.511917	+	0.859035i	$0.328935\pi$
$942$	0	0
$943$	2.57414	0.0838256
$944$	0	0
$945$	3.27053	0.106390
$946$	0	0
$947$	−28.0661	−0.912027	−0.456014	−	0.889973i	$-0.650723\pi$
$947$	−28.0661	−0.912027	−0.456014	+	0.889973i	$0.650723\pi$
$948$	0	0
$949$	−43.9569	−1.42690
$950$	0	0
$951$	−17.2044	−0.557890
$952$	0	0
$953$	51.8105	1.67831	0.839153	−	0.543895i	$-0.183051\pi$
$953$	51.8105	1.67831	0.839153	+	0.543895i	$0.183051\pi$
$954$	0	0
$955$	38.2304	1.23711
$956$	0	0
$957$	−5.65628	−0.182841
$958$	0	0
$959$	−4.27758	−0.138130
$960$	0	0
$961$	46.7685	1.50866
$962$	0	0
$963$	−10.3998	−0.335130
$964$	0	0
$965$	−15.0681	−0.485058
$966$	0	0
$967$	25.0160	0.804460	0.402230	−	0.915539i	$-0.368235\pi$
$967$	25.0160	0.804460	0.402230	+	0.915539i	$0.368235\pi$
$968$	0	0
$969$	−21.5240	−0.691450
$970$	0	0
$971$	38.5782	1.23803	0.619016	−	0.785378i	$-0.287531\pi$
$971$	38.5782	1.23803	0.619016	+	0.785378i	$0.287531\pi$
$972$	0	0
$973$	2.18840	0.0701568
$974$	0	0
$975$	26.7525	0.856766
$976$	0	0
$977$	−10.0230	−0.320665	−0.160333	−	0.987063i	$-0.551257\pi$
$977$	−10.0230	−0.320665	−0.160333	+	0.987063i	$0.551257\pi$
$978$	0	0
$979$	−16.3717	−0.523240
$980$	0	0
$981$	−12.3527	−0.394390
$982$	0	0
$983$	10.3437	0.329914	0.164957	−	0.986301i	$-0.447252\pi$
$983$	10.3437	0.329914	0.164957	+	0.986301i	$0.447252\pi$
$984$	0	0
$985$	81.3537	2.59214
$986$	0	0
$987$	−5.39279	−0.171654
$988$	0	0
$989$	−11.5481	−0.367209
$990$	0	0
$991$	38.6302	1.22713	0.613565	−	0.789644i	$-0.289735\pi$
$991$	38.6302	1.22713	0.613565	+	0.789644i	$0.289735\pi$
$992$	0	0
$993$	5.14828	0.163376
$994$	0	0
$995$	−24.3619	−0.772323
$996$	0	0
$997$	28.0331	0.887816	0.443908	−	0.896072i	$-0.353592\pi$
$997$	28.0331	0.887816	0.443908	+	0.896072i	$0.353592\pi$
$998$	0	0
$999$	4.81864	0.152455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.a.o.1.1	✓	3
4.3	odd	2		7728.2.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.o.1.1	✓	3	1.1	even	1	trivial
7728.2.a.bq.1.1		3	4.3	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:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.42586 q^{11} +4.69639 q^{13} -3.27053 q^{15} +3.96693 q^{17} -5.42586 q^{19} -1.00000 q^{21} +1.00000 q^{23} +5.69639 q^{25} +1.00000 q^{27} +3.96693 q^{29} -8.81864 q^{31} -1.42586 q^{33} +3.27053 q^{35} +4.81864 q^{37} +4.69639 q^{39} +2.57414 q^{41} -11.5481 q^{43} -3.27053 q^{45} +5.39279 q^{47} +1.00000 q^{49} +3.96693 q^{51} -12.6633 q^{53} +4.66332 q^{55} -5.42586 q^{57} +11.5150 q^{59} -13.2375 q^{61} -1.00000 q^{63} -15.3597 q^{65} +5.51504 q^{67} +1.00000 q^{69} -2.69639 q^{71} -9.35971 q^{73} +5.69639 q^{75} +1.42586 q^{77} -4.57414 q^{79} +1.00000 q^{81} -0.277576 q^{83} -12.9740 q^{85} +3.96693 q^{87} +11.4820 q^{89} -4.69639 q^{91} -8.81864 q^{93} +17.7455 q^{95} +7.96693 q^{97} -1.42586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 5 q^{13} - 3 q^{15} - 4 q^{17} - 14 q^{19} - 3 q^{21} + 3 q^{23} + 8 q^{25} + 3 q^{27} - 4 q^{29} - 6 q^{31} - 2 q^{33} + 3 q^{35} - 6 q^{37} + 5 q^{39} + 10 q^{41} - 21 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 13 q^{53} - 11 q^{55} - 14 q^{57} + 5 q^{59} - 17 q^{61} - 3 q^{63} - 12 q^{65} - 13 q^{67} + 3 q^{69} + q^{71} + 6 q^{73} + 8 q^{75} + 2 q^{77} - 16 q^{79} + 3 q^{81} + 6 q^{83} - 23 q^{85} - 4 q^{87} - 11 q^{89} - 5 q^{91} - 6 q^{93} + q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 5 * q^13 - 3 * q^15 - 4 * q^17 - 14 * q^19 - 3 * q^21 + 3 * q^23 + 8 * q^25 + 3 * q^27 - 4 * q^29 - 6 * q^31 - 2 * q^33 + 3 * q^35 - 6 * q^37 + 5 * q^39 + 10 * q^41 - 21 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 13 * q^53 - 11 * q^55 - 14 * q^57 + 5 * q^59 - 17 * q^61 - 3 * q^63 - 12 * q^65 - 13 * q^67 + 3 * q^69 + q^71 + 6 * q^73 + 8 * q^75 + 2 * q^77 - 16 * q^79 + 3 * q^81 + 6 * q^83 - 23 * q^85 - 4 * q^87 - 11 * q^89 - 5 * q^91 - 6 * q^93 + q^95 + 8 * q^97 - 2 * q^99

Introduction

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