LMFDB - Embedded newform 1216.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -1.56155 q^{5} -3.00000 q^{7} +3.56155 q^{9} -3.56155 q^{11} -2.56155 q^{13} -4.00000 q^{15} -8.12311 q^{17} +1.00000 q^{19} -7.68466 q^{21} -1.43845 q^{23} -2.56155 q^{25} +1.43845 q^{27} +7.68466 q^{29} -0.876894 q^{31} -9.12311 q^{33} +4.68466 q^{35} +1.12311 q^{37} -6.56155 q^{39} +4.00000 q^{41} +9.56155 q^{43} -5.56155 q^{45} -8.68466 q^{47} +2.00000 q^{49} -20.8078 q^{51} +8.56155 q^{53} +5.56155 q^{55} +2.56155 q^{57} -8.56155 q^{59} +5.80776 q^{61} -10.6847 q^{63} +4.00000 q^{65} -4.56155 q^{67} -3.68466 q^{69} -12.2462 q^{71} +7.24621 q^{73} -6.56155 q^{75} +10.6847 q^{77} -10.0000 q^{79} -7.00000 q^{81} +7.36932 q^{83} +12.6847 q^{85} +19.6847 q^{87} +9.36932 q^{89} +7.68466 q^{91} -2.24621 q^{93} -1.56155 q^{95} -1.12311 q^{97} -12.6847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - 8 q^{15} - 8 q^{17} + 2 q^{19} - 3 q^{21} - 7 q^{23} - q^{25} + 7 q^{27} + 3 q^{29} - 10 q^{31} - 10 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39}+ \cdots - 13 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 - 6 * q^7 + 3 * q^9 - 3 * q^11 - q^13 - 8 * q^15 - 8 * q^17 + 2 * q^19 - 3 * q^21 - 7 * q^23 - q^25 + 7 * q^27 + 3 * q^29 - 10 * q^31 - 10 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 + 8 * q^41 + 15 * q^43 - 7 * q^45 - 5 * q^47 + 4 * q^49 - 21 * q^51 + 13 * q^53 + 7 * q^55 + q^57 - 13 * q^59 - 9 * q^61 - 9 * q^63 + 8 * q^65 - 5 * q^67 + 5 * q^69 - 8 * q^71 - 2 * q^73 - 9 * q^75 + 9 * q^77 - 20 * q^79 - 14 * q^81 - 10 * q^83 + 13 * q^85 + 27 * q^87 - 6 * q^89 + 3 * q^91 + 12 * q^93 + q^95 + 6 * q^97 - 13 * 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 $

$2$	0	0
$2$	0	0
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−3.56155	−1.07385	−0.536924	−	0.843630i	$-0.680414\pi$
$11$	−3.56155	−1.07385	−0.536924	+	0.843630i	$0.680414\pi$
$12$	0	0
$13$	−2.56155	−0.710447	−0.355223	−	0.934781i	$-0.615595\pi$
$13$	−2.56155	−0.710447	−0.355223	+	0.934781i	$0.615595\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	−8.12311	−1.97014	−0.985071	−	0.172147i	$-0.944930\pi$
$17$	−8.12311	−1.97014	−0.985071	+	0.172147i	$0.944930\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−7.68466	−1.67693
$22$	0	0
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	7.68466	1.42701	0.713503	−	0.700653i	$-0.247108\pi$
$29$	7.68466	1.42701	0.713503	+	0.700653i	$0.247108\pi$
$30$	0	0
$31$	−0.876894	−0.157495	−0.0787474	−	0.996895i	$-0.525092\pi$
$31$	−0.876894	−0.157495	−0.0787474	+	0.996895i	$0.525092\pi$
$32$	0	0
$33$	−9.12311	−1.58813
$34$	0	0
$35$	4.68466	0.791852
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	0	0
$39$	−6.56155	−1.05069
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	9.56155	1.45812	0.729062	−	0.684448i	$-0.239957\pi$
$43$	9.56155	1.45812	0.729062	+	0.684448i	$0.239957\pi$
$44$	0	0
$45$	−5.56155	−0.829067
$46$	0	0
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−20.8078	−2.91367
$52$	0	0
$53$	8.56155	1.17602	0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$54$	0	0
$55$	5.56155	0.749920
$56$	0	0
$57$	2.56155	0.339286
$58$	0	0
$59$	−8.56155	−1.11462	−0.557310	−	0.830305i	$-0.688166\pi$
$59$	−8.56155	−1.11462	−0.557310	+	0.830305i	$0.688166\pi$
$60$	0	0
$61$	5.80776	0.743608	0.371804	−	0.928311i	$-0.378739\pi$
$61$	5.80776	0.743608	0.371804	+	0.928311i	$0.378739\pi$
$62$	0	0
$63$	−10.6847	−1.34614
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−4.56155	−0.557282	−0.278641	−	0.960395i	$-0.589884\pi$
$67$	−4.56155	−0.557282	−0.278641	+	0.960395i	$0.589884\pi$
$68$	0	0
$69$	−3.68466	−0.443581
$70$	0	0
$71$	−12.2462	−1.45336	−0.726679	−	0.686977i	$-0.758937\pi$
$71$	−12.2462	−1.45336	−0.726679	+	0.686977i	$0.758937\pi$
$72$	0	0
$73$	7.24621	0.848105	0.424052	−	0.905638i	$-0.360607\pi$
$73$	7.24621	0.848105	0.424052	+	0.905638i	$0.360607\pi$
$74$	0	0
$75$	−6.56155	−0.757663
$76$	0	0
$77$	10.6847	1.21763
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	7.36932	0.808888	0.404444	−	0.914563i	$-0.367465\pi$
$83$	7.36932	0.808888	0.404444	+	0.914563i	$0.367465\pi$
$84$	0	0
$85$	12.6847	1.37584
$86$	0	0
$87$	19.6847	2.11042
$88$	0	0
$89$	9.36932	0.993146	0.496573	−	0.867995i	$-0.334592\pi$
$89$	9.36932	0.993146	0.496573	+	0.867995i	$0.334592\pi$
$90$	0	0
$91$	7.68466	0.805571
$92$	0	0
$93$	−2.24621	−0.232921
$94$	0	0
$95$	−1.56155	−0.160212
$96$	0	0
$97$	−1.12311	−0.114034	−0.0570170	−	0.998373i	$-0.518159\pi$
$97$	−1.12311	−0.114034	−0.0570170	+	0.998373i	$0.518159\pi$
$98$	0	0
$99$	−12.6847	−1.27486
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	18.4924	1.82211	0.911056	−	0.412282i	$-0.135268\pi$
$103$	18.4924	1.82211	0.911056	+	0.412282i	$0.135268\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	−11.9309	−1.15340	−0.576700	−	0.816956i	$-0.695660\pi$
$107$	−11.9309	−1.15340	−0.576700	+	0.816956i	$0.695660\pi$
$108$	0	0
$109$	−0.561553	−0.0537870	−0.0268935	−	0.999638i	$-0.508561\pi$
$109$	−0.561553	−0.0537870	−0.0268935	+	0.999638i	$0.508561\pi$
$110$	0	0
$111$	2.87689	0.273063
$112$	0	0
$113$	13.1231	1.23452	0.617259	−	0.786760i	$-0.288243\pi$
$113$	13.1231	1.23452	0.617259	+	0.786760i	$0.288243\pi$
$114$	0	0
$115$	2.24621	0.209460
$116$	0	0
$117$	−9.12311	−0.843431
$118$	0	0
$119$	24.3693	2.23393
$120$	0	0
$121$	1.68466	0.153151
$122$	0	0
$123$	10.2462	0.923870
$124$	0	0
$125$	11.8078	1.05612
$126$	0	0
$127$	−6.87689	−0.610226	−0.305113	−	0.952316i	$-0.598694\pi$
$127$	−6.87689	−0.610226	−0.305113	+	0.952316i	$0.598694\pi$
$128$	0	0
$129$	24.4924	2.15644
$130$	0	0
$131$	−17.5616	−1.53436	−0.767180	−	0.641432i	$-0.778341\pi$
$131$	−17.5616	−1.53436	−0.767180	+	0.641432i	$0.778341\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−2.24621	−0.193323
$136$	0	0
$137$	−10.3693	−0.885911	−0.442955	−	0.896544i	$-0.646070\pi$
$137$	−10.3693	−0.885911	−0.442955	+	0.896544i	$0.646070\pi$
$138$	0	0
$139$	9.56155	0.811000	0.405500	−	0.914095i	$-0.367097\pi$
$139$	9.56155	0.811000	0.405500	+	0.914095i	$0.367097\pi$
$140$	0	0
$141$	−22.2462	−1.87347
$142$	0	0
$143$	9.12311	0.762912
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	5.12311	0.422547
$148$	0	0
$149$	−6.43845	−0.527458	−0.263729	−	0.964597i	$-0.584952\pi$
$149$	−6.43845	−0.527458	−0.263729	+	0.964597i	$0.584952\pi$
$150$	0	0
$151$	−10.4924	−0.853861	−0.426931	−	0.904284i	$-0.640405\pi$
$151$	−10.4924	−0.853861	−0.426931	+	0.904284i	$0.640405\pi$
$152$	0	0
$153$	−28.9309	−2.33892
$154$	0	0
$155$	1.36932	0.109986
$156$	0	0
$157$	24.2462	1.93506	0.967529	−	0.252759i	$-0.0813381\pi$
$157$	24.2462	1.93506	0.967529	+	0.252759i	$0.0813381\pi$
$158$	0	0
$159$	21.9309	1.73923
$160$	0	0
$161$	4.31534	0.340097
$162$	0	0
$163$	−16.4924	−1.29179	−0.645893	−	0.763428i	$-0.723515\pi$
$163$	−16.4924	−1.29179	−0.645893	+	0.763428i	$0.723515\pi$
$164$	0	0
$165$	14.2462	1.10907
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−6.43845	−0.495265
$170$	0	0
$171$	3.56155	0.272359
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	7.68466	0.580906
$176$	0	0
$177$	−21.9309	−1.64843
$178$	0	0
$179$	−14.2462	−1.06481	−0.532406	−	0.846489i	$-0.678712\pi$
$179$	−14.2462	−1.06481	−0.532406	+	0.846489i	$0.678712\pi$
$180$	0	0
$181$	−16.2462	−1.20757	−0.603786	−	0.797147i	$-0.706342\pi$
$181$	−16.2462	−1.20757	−0.603786	+	0.797147i	$0.706342\pi$
$182$	0	0
$183$	14.8769	1.09973
$184$	0	0
$185$	−1.75379	−0.128941
$186$	0	0
$187$	28.9309	2.11563
$188$	0	0
$189$	−4.31534	−0.313895
$190$	0	0
$191$	1.00000	0.0723575	0.0361787	−	0.999345i	$-0.488481\pi$
$191$	1.00000	0.0723575	0.0361787	+	0.999345i	$0.488481\pi$
$192$	0	0
$193$	−14.4924	−1.04319	−0.521594	−	0.853194i	$-0.674662\pi$
$193$	−14.4924	−1.04319	−0.521594	+	0.853194i	$0.674662\pi$
$194$	0	0
$195$	10.2462	0.733746
$196$	0	0
$197$	13.3693	0.952524	0.476262	−	0.879303i	$-0.341991\pi$
$197$	13.3693	0.952524	0.476262	+	0.879303i	$0.341991\pi$
$198$	0	0
$199$	−25.7386	−1.82456	−0.912282	−	0.409563i	$-0.865681\pi$
$199$	−25.7386	−1.82456	−0.912282	+	0.409563i	$0.865681\pi$
$200$	0	0
$201$	−11.6847	−0.824172
$202$	0	0
$203$	−23.0540	−1.61807
$204$	0	0
$205$	−6.24621	−0.436254
$206$	0	0
$207$	−5.12311	−0.356080
$208$	0	0
$209$	−3.56155	−0.246358
$210$	0	0
$211$	11.4384	0.787455	0.393728	−	0.919227i	$-0.371185\pi$
$211$	11.4384	0.787455	0.393728	+	0.919227i	$0.371185\pi$
$212$	0	0
$213$	−31.3693	−2.14939
$214$	0	0
$215$	−14.9309	−1.01828
$216$	0	0
$217$	2.63068	0.178582
$218$	0	0
$219$	18.5616	1.25427
$220$	0	0
$221$	20.8078	1.39968
$222$	0	0
$223$	−23.3693	−1.56493	−0.782463	−	0.622698i	$-0.786037\pi$
$223$	−23.3693	−1.56493	−0.782463	+	0.622698i	$0.786037\pi$
$224$	0	0
$225$	−9.12311	−0.608207
$226$	0	0
$227$	4.31534	0.286419	0.143210	−	0.989692i	$-0.454258\pi$
$227$	4.31534	0.286419	0.143210	+	0.989692i	$0.454258\pi$
$228$	0	0
$229$	8.43845	0.557628	0.278814	−	0.960345i	$-0.410059\pi$
$229$	8.43845	0.557628	0.278814	+	0.960345i	$0.410059\pi$
$230$	0	0
$231$	27.3693	1.80077
$232$	0	0
$233$	8.93087	0.585081	0.292540	−	0.956253i	$-0.405499\pi$
$233$	8.93087	0.585081	0.292540	+	0.956253i	$0.405499\pi$
$234$	0	0
$235$	13.5616	0.884658
$236$	0	0
$237$	−25.6155	−1.66391
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	2.24621	0.144691	0.0723456	−	0.997380i	$-0.476952\pi$
$241$	2.24621	0.144691	0.0723456	+	0.997380i	$0.476952\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	−3.12311	−0.199528
$246$	0	0
$247$	−2.56155	−0.162988
$248$	0	0
$249$	18.8769	1.19627
$250$	0	0
$251$	−6.68466	−0.421932	−0.210966	−	0.977493i	$-0.567661\pi$
$251$	−6.68466	−0.421932	−0.210966	+	0.977493i	$0.567661\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	0	0
$255$	32.4924	2.03475
$256$	0	0
$257$	9.75379	0.608425	0.304212	−	0.952604i	$-0.401607\pi$
$257$	9.75379	0.608425	0.304212	+	0.952604i	$0.401607\pi$
$258$	0	0
$259$	−3.36932	−0.209359
$260$	0	0
$261$	27.3693	1.69412
$262$	0	0
$263$	26.0540	1.60656	0.803278	−	0.595604i	$-0.203087\pi$
$263$	26.0540	1.60656	0.803278	+	0.595604i	$0.203087\pi$
$264$	0	0
$265$	−13.3693	−0.821271
$266$	0	0
$267$	24.0000	1.46878
$268$	0	0
$269$	−14.4924	−0.883619	−0.441809	−	0.897109i	$-0.645663\pi$
$269$	−14.4924	−0.883619	−0.441809	+	0.897109i	$0.645663\pi$
$270$	0	0
$271$	27.0540	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$271$	27.0540	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$272$	0	0
$273$	19.6847	1.19137
$274$	0	0
$275$	9.12311	0.550144
$276$	0	0
$277$	−0.684658	−0.0411371	−0.0205686	−	0.999788i	$-0.506548\pi$
$277$	−0.684658	−0.0411371	−0.0205686	+	0.999788i	$0.506548\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	−25.3693	−1.51341	−0.756703	−	0.653758i	$-0.773191\pi$
$281$	−25.3693	−1.51341	−0.756703	+	0.653758i	$0.773191\pi$
$282$	0	0
$283$	24.4384	1.45271	0.726357	−	0.687317i	$-0.241212\pi$
$283$	24.4384	1.45271	0.726357	+	0.687317i	$0.241212\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	48.9848	2.88146
$290$	0	0
$291$	−2.87689	−0.168647
$292$	0	0
$293$	−27.6847	−1.61736	−0.808678	−	0.588252i	$-0.799816\pi$
$293$	−27.6847	−1.61736	−0.808678	+	0.588252i	$0.799816\pi$
$294$	0	0
$295$	13.3693	0.778392
$296$	0	0
$297$	−5.12311	−0.297273
$298$	0	0
$299$	3.68466	0.213089
$300$	0	0
$301$	−28.6847	−1.65336
$302$	0	0
$303$	−41.6155	−2.39075
$304$	0	0
$305$	−9.06913	−0.519297
$306$	0	0
$307$	29.3693	1.67620	0.838098	−	0.545520i	$-0.183668\pi$
$307$	29.3693	1.67620	0.838098	+	0.545520i	$0.183668\pi$
$308$	0	0
$309$	47.3693	2.69475
$310$	0	0
$311$	−2.12311	−0.120390	−0.0601951	−	0.998187i	$-0.519172\pi$
$311$	−2.12311	−0.120390	−0.0601951	+	0.998187i	$0.519172\pi$
$312$	0	0
$313$	27.9309	1.57875	0.789373	−	0.613914i	$-0.210406\pi$
$313$	27.9309	1.57875	0.789373	+	0.613914i	$0.210406\pi$
$314$	0	0
$315$	16.6847	0.940074
$316$	0	0
$317$	−20.5616	−1.15485	−0.577426	−	0.816443i	$-0.695943\pi$
$317$	−20.5616	−1.15485	−0.577426	+	0.816443i	$0.695943\pi$
$318$	0	0
$319$	−27.3693	−1.53239
$320$	0	0
$321$	−30.5616	−1.70578
$322$	0	0
$323$	−8.12311	−0.451982
$324$	0	0
$325$	6.56155	0.363969
$326$	0	0
$327$	−1.43845	−0.0795463
$328$	0	0
$329$	26.0540	1.43640
$330$	0	0
$331$	8.56155	0.470586	0.235293	−	0.971925i	$-0.424395\pi$
$331$	8.56155	0.470586	0.235293	+	0.971925i	$0.424395\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	7.12311	0.389177
$336$	0	0
$337$	12.4924	0.680506	0.340253	−	0.940334i	$-0.389487\pi$
$337$	12.4924	0.680506	0.340253	+	0.940334i	$0.389487\pi$
$338$	0	0
$339$	33.6155	1.82574
$340$	0	0
$341$	3.12311	0.169126
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	5.75379	0.309774
$346$	0	0
$347$	−0.438447	−0.0235371	−0.0117685	−	0.999931i	$-0.503746\pi$
$347$	−0.438447	−0.0235371	−0.0117685	+	0.999931i	$0.503746\pi$
$348$	0	0
$349$	−1.31534	−0.0704086	−0.0352043	−	0.999380i	$-0.511208\pi$
$349$	−1.31534	−0.0704086	−0.0352043	+	0.999380i	$0.511208\pi$
$350$	0	0
$351$	−3.68466	−0.196673
$352$	0	0
$353$	−0.0691303	−0.00367944	−0.00183972	−	0.999998i	$-0.500586\pi$
$353$	−0.0691303	−0.00367944	−0.00183972	+	0.999998i	$0.500586\pi$
$354$	0	0
$355$	19.1231	1.01495
$356$	0	0
$357$	62.4233	3.30379
$358$	0	0
$359$	−36.1231	−1.90650	−0.953252	−	0.302176i	$-0.902287\pi$
$359$	−36.1231	−1.90650	−0.953252	+	0.302176i	$0.902287\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.31534	0.226497
$364$	0	0
$365$	−11.3153	−0.592272
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	14.2462	0.741628
$370$	0	0
$371$	−25.6847	−1.33348
$372$	0	0
$373$	−13.9309	−0.721313	−0.360657	−	0.932699i	$-0.617447\pi$
$373$	−13.9309	−0.721313	−0.360657	+	0.932699i	$0.617447\pi$
$374$	0	0
$375$	30.2462	1.56191
$376$	0	0
$377$	−19.6847	−1.01381
$378$	0	0
$379$	−21.0540	−1.08147	−0.540735	−	0.841193i	$-0.681854\pi$
$379$	−21.0540	−1.08147	−0.540735	+	0.841193i	$0.681854\pi$
$380$	0	0
$381$	−17.6155	−0.902471
$382$	0	0
$383$	−7.75379	−0.396200	−0.198100	−	0.980182i	$-0.563477\pi$
$383$	−7.75379	−0.396200	−0.198100	+	0.980182i	$0.563477\pi$
$384$	0	0
$385$	−16.6847	−0.850329
$386$	0	0
$387$	34.0540	1.73106
$388$	0	0
$389$	−6.19224	−0.313959	−0.156979	−	0.987602i	$-0.550176\pi$
$389$	−6.19224	−0.313959	−0.156979	+	0.987602i	$0.550176\pi$
$390$	0	0
$391$	11.6847	0.590919
$392$	0	0
$393$	−44.9848	−2.26919
$394$	0	0
$395$	15.6155	0.785702
$396$	0	0
$397$	9.56155	0.479881	0.239940	−	0.970788i	$-0.422872\pi$
$397$	9.56155	0.479881	0.239940	+	0.970788i	$0.422872\pi$
$398$	0	0
$399$	−7.68466	−0.384714
$400$	0	0
$401$	−1.36932	−0.0683804	−0.0341902	−	0.999415i	$-0.510885\pi$
$401$	−1.36932	−0.0683804	−0.0341902	+	0.999415i	$0.510885\pi$
$402$	0	0
$403$	2.24621	0.111892
$404$	0	0
$405$	10.9309	0.543159
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	27.1231	1.34115	0.670576	−	0.741841i	$-0.266047\pi$
$409$	27.1231	1.34115	0.670576	+	0.741841i	$0.266047\pi$
$410$	0	0
$411$	−26.5616	−1.31018
$412$	0	0
$413$	25.6847	1.26386
$414$	0	0
$415$	−11.5076	−0.564885
$416$	0	0
$417$	24.4924	1.19940
$418$	0	0
$419$	−9.75379	−0.476504	−0.238252	−	0.971203i	$-0.576574\pi$
$419$	−9.75379	−0.476504	−0.238252	+	0.971203i	$0.576574\pi$
$420$	0	0
$421$	17.0540	0.831160	0.415580	−	0.909557i	$-0.363579\pi$
$421$	17.0540	0.831160	0.415580	+	0.909557i	$0.363579\pi$
$422$	0	0
$423$	−30.9309	−1.50391
$424$	0	0
$425$	20.8078	1.00932
$426$	0	0
$427$	−17.4233	−0.843172
$428$	0	0
$429$	23.3693	1.12828
$430$	0	0
$431$	22.8769	1.10194	0.550971	−	0.834525i	$-0.314258\pi$
$431$	22.8769	1.10194	0.550971	+	0.834525i	$0.314258\pi$
$432$	0	0
$433$	−1.61553	−0.0776373	−0.0388187	−	0.999246i	$-0.512359\pi$
$433$	−1.61553	−0.0776373	−0.0388187	+	0.999246i	$0.512359\pi$
$434$	0	0
$435$	−30.7386	−1.47380
$436$	0	0
$437$	−1.43845	−0.0688103
$438$	0	0
$439$	−18.2462	−0.870844	−0.435422	−	0.900226i	$-0.643401\pi$
$439$	−18.2462	−0.870844	−0.435422	+	0.900226i	$0.643401\pi$
$440$	0	0
$441$	7.12311	0.339196
$442$	0	0
$443$	−10.6847	−0.507643	−0.253822	−	0.967251i	$-0.581688\pi$
$443$	−10.6847	−0.507643	−0.253822	+	0.967251i	$0.581688\pi$
$444$	0	0
$445$	−14.6307	−0.693561
$446$	0	0
$447$	−16.4924	−0.780065
$448$	0	0
$449$	−6.87689	−0.324541	−0.162270	−	0.986746i	$-0.551882\pi$
$449$	−6.87689	−0.324541	−0.162270	+	0.986746i	$0.551882\pi$
$450$	0	0
$451$	−14.2462	−0.670828
$452$	0	0
$453$	−26.8769	−1.26279
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−27.8769	−1.30403	−0.652013	−	0.758208i	$-0.726075\pi$
$457$	−27.8769	−1.30403	−0.652013	+	0.758208i	$0.726075\pi$
$458$	0	0
$459$	−11.6847	−0.545393
$460$	0	0
$461$	5.06913	0.236093	0.118046	−	0.993008i	$-0.462337\pi$
$461$	5.06913	0.236093	0.118046	+	0.993008i	$0.462337\pi$
$462$	0	0
$463$	22.0540	1.02494	0.512468	−	0.858707i	$-0.328731\pi$
$463$	22.0540	1.02494	0.512468	+	0.858707i	$0.328731\pi$
$464$	0	0
$465$	3.50758	0.162660
$466$	0	0
$467$	5.31534	0.245965	0.122982	−	0.992409i	$-0.460754\pi$
$467$	5.31534	0.245965	0.122982	+	0.992409i	$0.460754\pi$
$468$	0	0
$469$	13.6847	0.631899
$470$	0	0
$471$	62.1080	2.86178
$472$	0	0
$473$	−34.0540	−1.56580
$474$	0	0
$475$	−2.56155	−0.117532
$476$	0	0
$477$	30.4924	1.39615
$478$	0	0
$479$	−2.24621	−0.102632	−0.0513160	−	0.998682i	$-0.516342\pi$
$479$	−2.24621	−0.102632	−0.0513160	+	0.998682i	$0.516342\pi$
$480$	0	0
$481$	−2.87689	−0.131175
$482$	0	0
$483$	11.0540	0.502973
$484$	0	0
$485$	1.75379	0.0796354
$486$	0	0
$487$	−26.4924	−1.20049	−0.600243	−	0.799818i	$-0.704930\pi$
$487$	−26.4924	−1.20049	−0.600243	+	0.799818i	$0.704930\pi$
$488$	0	0
$489$	−42.2462	−1.91044
$490$	0	0
$491$	0.492423	0.0222227	0.0111114	−	0.999938i	$-0.496463\pi$
$491$	0.492423	0.0222227	0.0111114	+	0.999938i	$0.496463\pi$
$492$	0	0
$493$	−62.4233	−2.81140
$494$	0	0
$495$	19.8078	0.890293
$496$	0	0
$497$	36.7386	1.64795
$498$	0	0
$499$	8.68466	0.388779	0.194389	−	0.980924i	$-0.437727\pi$
$499$	8.68466	0.388779	0.194389	+	0.980924i	$0.437727\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.56155	0.292565	0.146283	−	0.989243i	$-0.453269\pi$
$503$	6.56155	0.292565	0.146283	+	0.989243i	$0.453269\pi$
$504$	0	0
$505$	25.3693	1.12892
$506$	0	0
$507$	−16.4924	−0.732454
$508$	0	0
$509$	−21.6155	−0.958091	−0.479046	−	0.877790i	$-0.659017\pi$
$509$	−21.6155	−0.958091	−0.479046	+	0.877790i	$0.659017\pi$
$510$	0	0
$511$	−21.7386	−0.961661
$512$	0	0
$513$	1.43845	0.0635090
$514$	0	0
$515$	−28.8769	−1.27247
$516$	0	0
$517$	30.9309	1.36034
$518$	0	0
$519$	9.61553	0.422075
$520$	0	0
$521$	−1.36932	−0.0599909	−0.0299954	−	0.999550i	$-0.509549\pi$
$521$	−1.36932	−0.0599909	−0.0299954	+	0.999550i	$0.509549\pi$
$522$	0	0
$523$	37.5464	1.64179	0.820895	−	0.571080i	$-0.193475\pi$
$523$	37.5464	1.64179	0.820895	+	0.571080i	$0.193475\pi$
$524$	0	0
$525$	19.6847	0.859109
$526$	0	0
$527$	7.12311	0.310287
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	−30.4924	−1.32326
$532$	0	0
$533$	−10.2462	−0.443813
$534$	0	0
$535$	18.6307	0.805475
$536$	0	0
$537$	−36.4924	−1.57476
$538$	0	0
$539$	−7.12311	−0.306814
$540$	0	0
$541$	39.5616	1.70088	0.850442	−	0.526069i	$-0.176335\pi$
$541$	39.5616	1.70088	0.850442	+	0.526069i	$0.176335\pi$
$542$	0	0
$543$	−41.6155	−1.78589
$544$	0	0
$545$	0.876894	0.0375620
$546$	0	0
$547$	−8.49242	−0.363110	−0.181555	−	0.983381i	$-0.558113\pi$
$547$	−8.49242	−0.363110	−0.181555	+	0.983381i	$0.558113\pi$
$548$	0	0
$549$	20.6847	0.882800
$550$	0	0
$551$	7.68466	0.327377
$552$	0	0
$553$	30.0000	1.27573
$554$	0	0
$555$	−4.49242	−0.190693
$556$	0	0
$557$	−10.9309	−0.463156	−0.231578	−	0.972816i	$-0.574389\pi$
$557$	−10.9309	−0.463156	−0.231578	+	0.972816i	$0.574389\pi$
$558$	0	0
$559$	−24.4924	−1.03592
$560$	0	0
$561$	74.1080	3.12884
$562$	0	0
$563$	−13.7538	−0.579653	−0.289827	−	0.957079i	$-0.593598\pi$
$563$	−13.7538	−0.579653	−0.289827	+	0.957079i	$0.593598\pi$
$564$	0	0
$565$	−20.4924	−0.862123
$566$	0	0
$567$	21.0000	0.881917
$568$	0	0
$569$	22.7386	0.953253	0.476627	−	0.879106i	$-0.341859\pi$
$569$	22.7386	0.953253	0.476627	+	0.879106i	$0.341859\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	2.56155	0.107010
$574$	0	0
$575$	3.68466	0.153661
$576$	0	0
$577$	−11.2462	−0.468186	−0.234093	−	0.972214i	$-0.575212\pi$
$577$	−11.2462	−0.468186	−0.234093	+	0.972214i	$0.575212\pi$
$578$	0	0
$579$	−37.1231	−1.54278
$580$	0	0
$581$	−22.1080	−0.917192
$582$	0	0
$583$	−30.4924	−1.26287
$584$	0	0
$585$	14.2462	0.589008
$586$	0	0
$587$	−5.06913	−0.209225	−0.104613	−	0.994513i	$-0.533360\pi$
$587$	−5.06913	−0.209225	−0.104613	+	0.994513i	$0.533360\pi$
$588$	0	0
$589$	−0.876894	−0.0361318
$590$	0	0
$591$	34.2462	1.40870
$592$	0	0
$593$	−16.7386	−0.687373	−0.343687	−	0.939084i	$-0.611676\pi$
$593$	−16.7386	−0.687373	−0.343687	+	0.939084i	$0.611676\pi$
$594$	0	0
$595$	−38.0540	−1.56006
$596$	0	0
$597$	−65.9309	−2.69837
$598$	0	0
$599$	13.8617	0.566375	0.283188	−	0.959065i	$-0.408608\pi$
$599$	13.8617	0.566375	0.283188	+	0.959065i	$0.408608\pi$
$600$	0	0
$601$	−29.3693	−1.19800	−0.599000	−	0.800749i	$-0.704435\pi$
$601$	−29.3693	−1.19800	−0.599000	+	0.800749i	$0.704435\pi$
$602$	0	0
$603$	−16.2462	−0.661597
$604$	0	0
$605$	−2.63068	−0.106952
$606$	0	0
$607$	21.1231	0.857360	0.428680	−	0.903456i	$-0.358979\pi$
$607$	21.1231	0.857360	0.428680	+	0.903456i	$0.358979\pi$
$608$	0	0
$609$	−59.0540	−2.39299
$610$	0	0
$611$	22.2462	0.899985
$612$	0	0
$613$	−28.5464	−1.15298	−0.576489	−	0.817105i	$-0.695578\pi$
$613$	−28.5464	−1.15298	−0.576489	+	0.817105i	$0.695578\pi$
$614$	0	0
$615$	−16.0000	−0.645182
$616$	0	0
$617$	9.31534	0.375022	0.187511	−	0.982263i	$-0.439958\pi$
$617$	9.31534	0.375022	0.187511	+	0.982263i	$0.439958\pi$
$618$	0	0
$619$	−38.8769	−1.56259	−0.781297	−	0.624159i	$-0.785442\pi$
$619$	−38.8769	−1.56259	−0.781297	+	0.624159i	$0.785442\pi$
$620$	0	0
$621$	−2.06913	−0.0830313
$622$	0	0
$623$	−28.1080	−1.12612
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	−9.12311	−0.364342
$628$	0	0
$629$	−9.12311	−0.363762
$630$	0	0
$631$	−8.30019	−0.330425	−0.165213	−	0.986258i	$-0.552831\pi$
$631$	−8.30019	−0.330425	−0.165213	+	0.986258i	$0.552831\pi$
$632$	0	0
$633$	29.3002	1.16458
$634$	0	0
$635$	10.7386	0.426150
$636$	0	0
$637$	−5.12311	−0.202985
$638$	0	0
$639$	−43.6155	−1.72540
$640$	0	0
$641$	−50.1080	−1.97915	−0.989573	−	0.144035i	$-0.953992\pi$
$641$	−50.1080	−1.97915	−0.989573	+	0.144035i	$0.953992\pi$
$642$	0	0
$643$	−26.3002	−1.03718	−0.518589	−	0.855024i	$-0.673543\pi$
$643$	−26.3002	−1.03718	−0.518589	+	0.855024i	$0.673543\pi$
$644$	0	0
$645$	−38.2462	−1.50594
$646$	0	0
$647$	−47.0000	−1.84776	−0.923880	−	0.382682i	$-0.875001\pi$
$647$	−47.0000	−1.84776	−0.923880	+	0.382682i	$0.875001\pi$
$648$	0	0
$649$	30.4924	1.19693
$650$	0	0
$651$	6.73863	0.264108
$652$	0	0
$653$	11.8078	0.462074	0.231037	−	0.972945i	$-0.425788\pi$
$653$	11.8078	0.462074	0.231037	+	0.972945i	$0.425788\pi$
$654$	0	0
$655$	27.4233	1.07152
$656$	0	0
$657$	25.8078	1.00686
$658$	0	0
$659$	37.5464	1.46260	0.731300	−	0.682056i	$-0.238914\pi$
$659$	37.5464	1.46260	0.731300	+	0.682056i	$0.238914\pi$
$660$	0	0
$661$	−24.1771	−0.940379	−0.470190	−	0.882565i	$-0.655815\pi$
$661$	−24.1771	−0.940379	−0.470190	+	0.882565i	$0.655815\pi$
$662$	0	0
$663$	53.3002	2.07001
$664$	0	0
$665$	4.68466	0.181663
$666$	0	0
$667$	−11.0540	−0.428012
$668$	0	0
$669$	−59.8617	−2.31439
$670$	0	0
$671$	−20.6847	−0.798522
$672$	0	0
$673$	−12.8769	−0.496368	−0.248184	−	0.968713i	$-0.579834\pi$
$673$	−12.8769	−0.496368	−0.248184	+	0.968713i	$0.579834\pi$
$674$	0	0
$675$	−3.68466	−0.141823
$676$	0	0
$677$	6.56155	0.252181	0.126090	−	0.992019i	$-0.459757\pi$
$677$	6.56155	0.252181	0.126090	+	0.992019i	$0.459757\pi$
$678$	0	0
$679$	3.36932	0.129303
$680$	0	0
$681$	11.0540	0.423589
$682$	0	0
$683$	−34.2462	−1.31039	−0.655197	−	0.755458i	$-0.727415\pi$
$683$	−34.2462	−1.31039	−0.655197	+	0.755458i	$0.727415\pi$
$684$	0	0
$685$	16.1922	0.618674
$686$	0	0
$687$	21.6155	0.824684
$688$	0	0
$689$	−21.9309	−0.835500
$690$	0	0
$691$	−7.94602	−0.302281	−0.151141	−	0.988512i	$-0.548295\pi$
$691$	−7.94602	−0.302281	−0.151141	+	0.988512i	$0.548295\pi$
$692$	0	0
$693$	38.0540	1.44555
$694$	0	0
$695$	−14.9309	−0.566360
$696$	0	0
$697$	−32.4924	−1.23074
$698$	0	0
$699$	22.8769	0.865284
$700$	0	0
$701$	27.6155	1.04302	0.521512	−	0.853244i	$-0.325368\pi$
$701$	27.6155	1.04302	0.521512	+	0.853244i	$0.325368\pi$
$702$	0	0
$703$	1.12311	0.0423587
$704$	0	0
$705$	34.7386	1.30833
$706$	0	0
$707$	48.7386	1.83300
$708$	0	0
$709$	−17.5076	−0.657511	−0.328755	−	0.944415i	$-0.606629\pi$
$709$	−17.5076	−0.657511	−0.328755	+	0.944415i	$0.606629\pi$
$710$	0	0
$711$	−35.6155	−1.33569
$712$	0	0
$713$	1.26137	0.0472385
$714$	0	0
$715$	−14.2462	−0.532778
$716$	0	0
$717$	−38.4233	−1.43494
$718$	0	0
$719$	9.49242	0.354008	0.177004	−	0.984210i	$-0.443360\pi$
$719$	9.49242	0.354008	0.177004	+	0.984210i	$0.443360\pi$
$720$	0	0
$721$	−55.4773	−2.06608
$722$	0	0
$723$	5.75379	0.213986
$724$	0	0
$725$	−19.6847	−0.731070
$726$	0	0
$727$	2.36932	0.0878731	0.0439365	−	0.999034i	$-0.486010\pi$
$727$	2.36932	0.0878731	0.0439365	+	0.999034i	$0.486010\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−77.6695	−2.87271
$732$	0	0
$733$	21.3693	0.789294	0.394647	−	0.918833i	$-0.370867\pi$
$733$	21.3693	0.789294	0.394647	+	0.918833i	$0.370867\pi$
$734$	0	0
$735$	−8.00000	−0.295084
$736$	0	0
$737$	16.2462	0.598437
$738$	0	0
$739$	5.56155	0.204585	0.102293	−	0.994754i	$-0.467382\pi$
$739$	5.56155	0.204585	0.102293	+	0.994754i	$0.467382\pi$
$740$	0	0
$741$	−6.56155	−0.241045
$742$	0	0
$743$	−31.1231	−1.14180	−0.570898	−	0.821021i	$-0.693405\pi$
$743$	−31.1231	−1.14180	−0.570898	+	0.821021i	$0.693405\pi$
$744$	0	0
$745$	10.0540	0.368349
$746$	0	0
$747$	26.2462	0.960299
$748$	0	0
$749$	35.7926	1.30783
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	−17.1231	−0.624001
$754$	0	0
$755$	16.3845	0.596292
$756$	0	0
$757$	16.0540	0.583492	0.291746	−	0.956496i	$-0.405764\pi$
$757$	16.0540	0.583492	0.291746	+	0.956496i	$0.405764\pi$
$758$	0	0
$759$	13.1231	0.476339
$760$	0	0
$761$	16.8617	0.611238	0.305619	−	0.952154i	$-0.401137\pi$
$761$	16.8617	0.611238	0.305619	+	0.952154i	$0.401137\pi$
$762$	0	0
$763$	1.68466	0.0609887
$764$	0	0
$765$	45.1771	1.63338
$766$	0	0
$767$	21.9309	0.791878
$768$	0	0
$769$	−12.3693	−0.446049	−0.223024	−	0.974813i	$-0.571593\pi$
$769$	−12.3693	−0.446049	−0.223024	+	0.974813i	$0.571593\pi$
$770$	0	0
$771$	24.9848	0.899807
$772$	0	0
$773$	−5.05398	−0.181779	−0.0908894	−	0.995861i	$-0.528971\pi$
$773$	−5.05398	−0.181779	−0.0908894	+	0.995861i	$0.528971\pi$
$774$	0	0
$775$	2.24621	0.0806863
$776$	0	0
$777$	−8.63068	−0.309624
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	43.6155	1.56069
$782$	0	0
$783$	11.0540	0.395037
$784$	0	0
$785$	−37.8617	−1.35134
$786$	0	0
$787$	26.5616	0.946817	0.473409	−	0.880843i	$-0.343023\pi$
$787$	26.5616	0.946817	0.473409	+	0.880843i	$0.343023\pi$
$788$	0	0
$789$	66.7386	2.37596
$790$	0	0
$791$	−39.3693	−1.39981
$792$	0	0
$793$	−14.8769	−0.528294
$794$	0	0
$795$	−34.2462	−1.21459
$796$	0	0
$797$	−9.68466	−0.343048	−0.171524	−	0.985180i	$-0.554869\pi$
$797$	−9.68466	−0.343048	−0.171524	+	0.985180i	$0.554869\pi$
$798$	0	0
$799$	70.5464	2.49575
$800$	0	0
$801$	33.3693	1.17905
$802$	0	0
$803$	−25.8078	−0.910736
$804$	0	0
$805$	−6.73863	−0.237506
$806$	0	0
$807$	−37.1231	−1.30680
$808$	0	0
$809$	−2.12311	−0.0746444	−0.0373222	−	0.999303i	$-0.511883\pi$
$809$	−2.12311	−0.0746444	−0.0373222	+	0.999303i	$0.511883\pi$
$810$	0	0
$811$	50.4233	1.77060	0.885301	−	0.465019i	$-0.153953\pi$
$811$	50.4233	1.77060	0.885301	+	0.465019i	$0.153953\pi$
$812$	0	0
$813$	69.3002	2.43046
$814$	0	0
$815$	25.7538	0.902116
$816$	0	0
$817$	9.56155	0.334516
$818$	0	0
$819$	27.3693	0.956361
$820$	0	0
$821$	−16.6847	−0.582299	−0.291149	−	0.956678i	$-0.594038\pi$
$821$	−16.6847	−0.582299	−0.291149	+	0.956678i	$0.594038\pi$
$822$	0	0
$823$	4.36932	0.152305	0.0761524	−	0.997096i	$-0.475736\pi$
$823$	4.36932	0.152305	0.0761524	+	0.997096i	$0.475736\pi$
$824$	0	0
$825$	23.3693	0.813615
$826$	0	0
$827$	15.4384	0.536847	0.268424	−	0.963301i	$-0.413497\pi$
$827$	15.4384	0.536847	0.268424	+	0.963301i	$0.413497\pi$
$828$	0	0
$829$	51.5464	1.79028	0.895140	−	0.445785i	$-0.147075\pi$
$829$	51.5464	1.79028	0.895140	+	0.445785i	$0.147075\pi$
$830$	0	0
$831$	−1.75379	−0.0608383
$832$	0	0
$833$	−16.2462	−0.562898
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.26137	−0.0435992
$838$	0	0
$839$	11.6155	0.401013	0.200506	−	0.979692i	$-0.435741\pi$
$839$	11.6155	0.401013	0.200506	+	0.979692i	$0.435741\pi$
$840$	0	0
$841$	30.0540	1.03634
$842$	0	0
$843$	−64.9848	−2.23820
$844$	0	0
$845$	10.0540	0.345867
$846$	0	0
$847$	−5.05398	−0.173657
$848$	0	0
$849$	62.6004	2.14844
$850$	0	0
$851$	−1.61553	−0.0553796
$852$	0	0
$853$	−15.7538	−0.539399	−0.269700	−	0.962944i	$-0.586924\pi$
$853$	−15.7538	−0.539399	−0.269700	+	0.962944i	$0.586924\pi$
$854$	0	0
$855$	−5.56155	−0.190201
$856$	0	0
$857$	37.3693	1.27651	0.638256	−	0.769824i	$-0.279656\pi$
$857$	37.3693	1.27651	0.638256	+	0.769824i	$0.279656\pi$
$858$	0	0
$859$	−7.06913	−0.241196	−0.120598	−	0.992701i	$-0.538481\pi$
$859$	−7.06913	−0.241196	−0.120598	+	0.992701i	$0.538481\pi$
$860$	0	0
$861$	−30.7386	−1.04757
$862$	0	0
$863$	−47.3693	−1.61247	−0.806235	−	0.591595i	$-0.798498\pi$
$863$	−47.3693	−1.61247	−0.806235	+	0.591595i	$0.798498\pi$
$864$	0	0
$865$	−5.86174	−0.199305
$866$	0	0
$867$	125.477	4.26143
$868$	0	0
$869$	35.6155	1.20817
$870$	0	0
$871$	11.6847	0.395920
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	−35.4233	−1.19753
$876$	0	0
$877$	9.68466	0.327028	0.163514	−	0.986541i	$-0.447717\pi$
$877$	9.68466	0.327028	0.163514	+	0.986541i	$0.447717\pi$
$878$	0	0
$879$	−70.9157	−2.39193
$880$	0	0
$881$	37.3153	1.25719	0.628593	−	0.777735i	$-0.283631\pi$
$881$	37.3153	1.25719	0.628593	+	0.777735i	$0.283631\pi$
$882$	0	0
$883$	28.9309	0.973601	0.486801	−	0.873513i	$-0.338164\pi$
$883$	28.9309	0.973601	0.486801	+	0.873513i	$0.338164\pi$
$884$	0	0
$885$	34.2462	1.15117
$886$	0	0
$887$	−45.2311	−1.51871	−0.759355	−	0.650676i	$-0.774485\pi$
$887$	−45.2311	−1.51871	−0.759355	+	0.650676i	$0.774485\pi$
$888$	0	0
$889$	20.6307	0.691931
$890$	0	0
$891$	24.9309	0.835216
$892$	0	0
$893$	−8.68466	−0.290621
$894$	0	0
$895$	22.2462	0.743609
$896$	0	0
$897$	9.43845	0.315141
$898$	0	0
$899$	−6.73863	−0.224746
$900$	0	0
$901$	−69.5464	−2.31693
$902$	0	0
$903$	−73.4773	−2.44517
$904$	0	0
$905$	25.3693	0.843305
$906$	0	0
$907$	−11.4384	−0.379807	−0.189904	−	0.981803i	$-0.560818\pi$
$907$	−11.4384	−0.379807	−0.189904	+	0.981803i	$0.560818\pi$
$908$	0	0
$909$	−57.8617	−1.91915
$910$	0	0
$911$	0.876894	0.0290528	0.0145264	−	0.999894i	$-0.495376\pi$
$911$	0.876894	0.0290528	0.0145264	+	0.999894i	$0.495376\pi$
$912$	0	0
$913$	−26.2462	−0.868623
$914$	0	0
$915$	−23.2311	−0.767995
$916$	0	0
$917$	52.6847	1.73980
$918$	0	0
$919$	−29.3002	−0.966524	−0.483262	−	0.875476i	$-0.660548\pi$
$919$	−29.3002	−0.966524	−0.483262	+	0.875476i	$0.660548\pi$
$920$	0	0
$921$	75.2311	2.47895
$922$	0	0
$923$	31.3693	1.03253
$924$	0	0
$925$	−2.87689	−0.0945917
$926$	0	0
$927$	65.8617	2.16318
$928$	0	0
$929$	−9.82292	−0.322280	−0.161140	−	0.986932i	$-0.551517\pi$
$929$	−9.82292	−0.322280	−0.161140	+	0.986932i	$0.551517\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	−5.43845	−0.178047
$934$	0	0
$935$	−45.1771	−1.47745
$936$	0	0
$937$	−29.2462	−0.955432	−0.477716	−	0.878514i	$-0.658535\pi$
$937$	−29.2462	−0.955432	−0.477716	+	0.878514i	$0.658535\pi$
$938$	0	0
$939$	71.5464	2.33483
$940$	0	0
$941$	35.3002	1.15075	0.575377	−	0.817889i	$-0.304856\pi$
$941$	35.3002	1.15075	0.575377	+	0.817889i	$0.304856\pi$
$942$	0	0
$943$	−5.75379	−0.187369
$944$	0	0
$945$	6.73863	0.219208
$946$	0	0
$947$	−28.9848	−0.941881	−0.470940	−	0.882165i	$-0.656085\pi$
$947$	−28.9848	−0.941881	−0.470940	+	0.882165i	$0.656085\pi$
$948$	0	0
$949$	−18.5616	−0.602534
$950$	0	0
$951$	−52.6695	−1.70793
$952$	0	0
$953$	8.24621	0.267121	0.133560	−	0.991041i	$-0.457359\pi$
$953$	8.24621	0.267121	0.133560	+	0.991041i	$0.457359\pi$
$954$	0	0
$955$	−1.56155	−0.0505307
$956$	0	0
$957$	−70.1080	−2.26627
$958$	0	0
$959$	31.1080	1.00453
$960$	0	0
$961$	−30.2311	−0.975195
$962$	0	0
$963$	−42.4924	−1.36930
$964$	0	0
$965$	22.6307	0.728507
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	0	0
$969$	−20.8078	−0.668442
$970$	0	0
$971$	−56.4924	−1.81293	−0.906464	−	0.422283i	$-0.861229\pi$
$971$	−56.4924	−1.81293	−0.906464	+	0.422283i	$0.861229\pi$
$972$	0	0
$973$	−28.6847	−0.919588
$974$	0	0
$975$	16.8078	0.538279
$976$	0	0
$977$	−12.3845	−0.396214	−0.198107	−	0.980180i	$-0.563479\pi$
$977$	−12.3845	−0.396214	−0.198107	+	0.980180i	$0.563479\pi$
$978$	0	0
$979$	−33.3693	−1.06649
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	29.6155	0.944589	0.472294	−	0.881441i	$-0.343426\pi$
$983$	29.6155	0.944589	0.472294	+	0.881441i	$0.343426\pi$
$984$	0	0
$985$	−20.8769	−0.665193
$986$	0	0
$987$	66.7386	2.12431
$988$	0	0
$989$	−13.7538	−0.437345
$990$	0	0
$991$	1.75379	0.0557109	0.0278555	−	0.999612i	$-0.491132\pi$
$991$	1.75379	0.0557109	0.0278555	+	0.999612i	$0.491132\pi$
$992$	0	0
$993$	21.9309	0.695955
$994$	0	0
$995$	40.1922	1.27418
$996$	0	0
$997$	−47.9157	−1.51751	−0.758753	−	0.651379i	$-0.774191\pi$
$997$	−47.9157	−1.51751	−0.758753	+	0.651379i	$0.774191\pi$
$998$	0	0
$999$	1.61553	0.0511130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.t.1.2		2
4.3	odd	2		1216.2.a.s.1.1		2
8.3	odd	2		608.2.a.h.1.2	yes	2
8.5	even	2		608.2.a.g.1.1	✓	2
24.5	odd	2		5472.2.a.bc.1.1		2
24.11	even	2		5472.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.g.1.1	✓	2	8.5	even	2
608.2.a.h.1.2	yes	2	8.3	odd	2
1216.2.a.s.1.1		2	4.3	odd	2
1216.2.a.t.1.2		2	1.1	even	1	trivial
5472.2.a.bc.1.1		2	24.5	odd	2
5472.2.a.bf.1.1		2	24.11	even	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -1.56155 q^{5} -3.00000 q^{7} +3.56155 q^{9} -3.56155 q^{11} -2.56155 q^{13} -4.00000 q^{15} -8.12311 q^{17} +1.00000 q^{19} -7.68466 q^{21} -1.43845 q^{23} -2.56155 q^{25} +1.43845 q^{27} +7.68466 q^{29} -0.876894 q^{31} -9.12311 q^{33} +4.68466 q^{35} +1.12311 q^{37} -6.56155 q^{39} +4.00000 q^{41} +9.56155 q^{43} -5.56155 q^{45} -8.68466 q^{47} +2.00000 q^{49} -20.8078 q^{51} +8.56155 q^{53} +5.56155 q^{55} +2.56155 q^{57} -8.56155 q^{59} +5.80776 q^{61} -10.6847 q^{63} +4.00000 q^{65} -4.56155 q^{67} -3.68466 q^{69} -12.2462 q^{71} +7.24621 q^{73} -6.56155 q^{75} +10.6847 q^{77} -10.0000 q^{79} -7.00000 q^{81} +7.36932 q^{83} +12.6847 q^{85} +19.6847 q^{87} +9.36932 q^{89} +7.68466 q^{91} -2.24621 q^{93} -1.56155 q^{95} -1.12311 q^{97} -12.6847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} - q^{13} - 8 q^{15} - 8 q^{17} + 2 q^{19} - 3 q^{21} - 7 q^{23} - q^{25} + 7 q^{27} + 3 q^{29} - 10 q^{31} - 10 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39}+ \cdots - 13 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 - 6 * q^7 + 3 * q^9 - 3 * q^11 - q^13 - 8 * q^15 - 8 * q^17 + 2 * q^19 - 3 * q^21 - 7 * q^23 - q^25 + 7 * q^27 + 3 * q^29 - 10 * q^31 - 10 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 + 8 * q^41 + 15 * q^43 - 7 * q^45 - 5 * q^47 + 4 * q^49 - 21 * q^51 + 13 * q^53 + 7 * q^55 + q^57 - 13 * q^59 - 9 * q^61 - 9 * q^63 + 8 * q^65 - 5 * q^67 + 5 * q^69 - 8 * q^71 - 2 * q^73 - 9 * q^75 + 9 * q^77 - 20 * q^79 - 14 * q^81 - 10 * q^83 + 13 * q^85 + 27 * q^87 - 6 * q^89 + 3 * q^91 + 12 * q^93 + q^95 + 6 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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