LMFDB - Embedded newform 8036.2.a.s.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.538932$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.538932 q^{3} -0.810745 q^{5} -2.70955 q^{9} +O(q^{10})$	$q-0.538932 q^{3} -0.810745 q^{5} -2.70955 q^{9} +4.92893 q^{11} -0.0654121 q^{13} +0.436937 q^{15} +1.24015 q^{17} -3.82383 q^{19} -1.71957 q^{23} -4.34269 q^{25} +3.07706 q^{27} +3.89893 q^{29} -3.19907 q^{31} -2.65636 q^{33} -0.0930044 q^{37} +0.0352527 q^{39} +1.00000 q^{41} +10.5630 q^{43} +2.19676 q^{45} +0.959640 q^{47} -0.668359 q^{51} -2.72668 q^{53} -3.99611 q^{55} +2.06078 q^{57} -11.1898 q^{59} +10.8004 q^{61} +0.0530326 q^{65} +10.3531 q^{67} +0.926733 q^{69} -5.60052 q^{71} -2.76947 q^{73} +2.34041 q^{75} -16.2539 q^{79} +6.47033 q^{81} -9.99122 q^{83} -1.00545 q^{85} -2.10126 q^{87} -11.8641 q^{89} +1.72408 q^{93} +3.10015 q^{95} -6.31445 q^{97} -13.3552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

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
$3$	−0.538932	−0.311152	−0.155576	−	0.987824i	$-0.549723\pi$
$3$	−0.538932	−0.311152	−0.155576	+	0.987824i	$0.549723\pi$
$4$	0	0
$5$	−0.810745	−0.362576	−0.181288	−	0.983430i	$-0.558027\pi$
$5$	−0.810745	−0.362576	−0.181288	+	0.983430i	$0.558027\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.70955	−0.903184
$10$	0	0
$11$	4.92893	1.48613	0.743064	−	0.669220i	$-0.233372\pi$
$11$	4.92893	1.48613	0.743064	+	0.669220i	$0.233372\pi$
$12$	0	0
$13$	−0.0654121	−0.0181421	−0.00907103	−	0.999959i	$-0.502887\pi$
$13$	−0.0654121	−0.0181421	−0.00907103	+	0.999959i	$0.502887\pi$
$14$	0	0
$15$	0.436937	0.112817
$16$	0	0
$17$	1.24015	0.300782	0.150391	−	0.988627i	$-0.451947\pi$
$17$	1.24015	0.300782	0.150391	+	0.988627i	$0.451947\pi$
$18$	0	0
$19$	−3.82383	−0.877247	−0.438623	−	0.898671i	$-0.644534\pi$
$19$	−3.82383	−0.877247	−0.438623	+	0.898671i	$0.644534\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.71957	−0.358556	−0.179278	−	0.983798i	$-0.557376\pi$
$23$	−1.71957	−0.358556	−0.179278	+	0.983798i	$0.557376\pi$
$24$	0	0
$25$	−4.34269	−0.868538
$26$	0	0
$27$	3.07706	0.592180
$28$	0	0
$29$	3.89893	0.724014	0.362007	−	0.932175i	$-0.382092\pi$
$29$	3.89893	0.724014	0.362007	+	0.932175i	$0.382092\pi$
$30$	0	0
$31$	−3.19907	−0.574570	−0.287285	−	0.957845i	$-0.592753\pi$
$31$	−3.19907	−0.574570	−0.287285	+	0.957845i	$0.592753\pi$
$32$	0	0
$33$	−2.65636	−0.462413
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0930044	−0.0152898	−0.00764492	−	0.999971i	$-0.502433\pi$
$37$	−0.0930044	−0.0152898	−0.00764492	+	0.999971i	$0.502433\pi$
$38$	0	0
$39$	0.0352527	0.00564495
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	10.5630	1.61084	0.805421	−	0.592703i	$-0.201939\pi$
$43$	10.5630	1.61084	0.805421	+	0.592703i	$0.201939\pi$
$44$	0	0
$45$	2.19676	0.327473
$46$	0	0
$47$	0.959640	0.139978	0.0699889	−	0.997548i	$-0.477704\pi$
$47$	0.959640	0.139978	0.0699889	+	0.997548i	$0.477704\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.668359	−0.0935890
$52$	0	0
$53$	−2.72668	−0.374539	−0.187269	−	0.982309i	$-0.559964\pi$
$53$	−2.72668	−0.374539	−0.187269	+	0.982309i	$0.559964\pi$
$54$	0	0
$55$	−3.99611	−0.538835
$56$	0	0
$57$	2.06078	0.272958
$58$	0	0
$59$	−11.1898	−1.45679	−0.728393	−	0.685160i	$-0.759732\pi$
$59$	−11.1898	−1.45679	−0.728393	+	0.685160i	$0.759732\pi$
$60$	0	0
$61$	10.8004	1.38285	0.691423	−	0.722450i	$-0.256984\pi$
$61$	10.8004	1.38285	0.691423	+	0.722450i	$0.256984\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.0530326	0.00657788
$66$	0	0
$67$	10.3531	1.26483	0.632414	−	0.774630i	$-0.282064\pi$
$67$	10.3531	1.26483	0.632414	+	0.774630i	$0.282064\pi$
$68$	0	0
$69$	0.926733	0.111566
$70$	0	0
$71$	−5.60052	−0.664659	−0.332330	−	0.943163i	$-0.607835\pi$
$71$	−5.60052	−0.664659	−0.332330	+	0.943163i	$0.607835\pi$
$72$	0	0
$73$	−2.76947	−0.324141	−0.162071	−	0.986779i	$-0.551817\pi$
$73$	−2.76947	−0.324141	−0.162071	+	0.986779i	$0.551817\pi$
$74$	0	0
$75$	2.34041	0.270248
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.2539	−1.82870	−0.914352	−	0.404921i	$-0.867299\pi$
$79$	−16.2539	−1.82870	−0.914352	+	0.404921i	$0.867299\pi$
$80$	0	0
$81$	6.47033	0.718926
$82$	0	0
$83$	−9.99122	−1.09668	−0.548340	−	0.836256i	$-0.684740\pi$
$83$	−9.99122	−1.09668	−0.548340	+	0.836256i	$0.684740\pi$
$84$	0	0
$85$	−1.00545	−0.109056
$86$	0	0
$87$	−2.10126	−0.225279
$88$	0	0
$89$	−11.8641	−1.25760	−0.628799	−	0.777568i	$-0.716453\pi$
$89$	−11.8641	−1.25760	−0.628799	+	0.777568i	$0.716453\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.72408	0.178779
$94$	0	0
$95$	3.10015	0.318069
$96$	0	0
$97$	−6.31445	−0.641136	−0.320568	−	0.947226i	$-0.603874\pi$
$97$	−6.31445	−0.641136	−0.320568	+	0.947226i	$0.603874\pi$
$98$	0	0
$99$	−13.3552	−1.34225
$100$	0	0
$101$	12.5369	1.24747	0.623735	−	0.781636i	$-0.285614\pi$
$101$	12.5369	1.24747	0.623735	+	0.781636i	$0.285614\pi$
$102$	0	0
$103$	−18.0340	−1.77695	−0.888473	−	0.458929i	$-0.848233\pi$
$103$	−18.0340	−1.77695	−0.888473	+	0.458929i	$0.848233\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.09750	0.879488	0.439744	−	0.898123i	$-0.355069\pi$
$107$	9.09750	0.879488	0.439744	+	0.898123i	$0.355069\pi$
$108$	0	0
$109$	9.88562	0.946870	0.473435	−	0.880829i	$-0.343014\pi$
$109$	9.88562	0.946870	0.473435	+	0.880829i	$0.343014\pi$
$110$	0	0
$111$	0.0501230	0.00475747
$112$	0	0
$113$	−8.45002	−0.794911	−0.397455	−	0.917622i	$-0.630107\pi$
$113$	−8.45002	−0.794911	−0.397455	+	0.917622i	$0.630107\pi$
$114$	0	0
$115$	1.39414	0.130004
$116$	0	0
$117$	0.177238	0.0163856
$118$	0	0
$119$	0	0
$120$	0	0
$121$	13.2944	1.20858
$122$	0	0
$123$	−0.538932	−0.0485938
$124$	0	0
$125$	7.57454	0.677488
$126$	0	0
$127$	6.19115	0.549376	0.274688	−	0.961533i	$-0.411425\pi$
$127$	6.19115	0.549376	0.274688	+	0.961533i	$0.411425\pi$
$128$	0	0
$129$	−5.69274	−0.501218
$130$	0	0
$131$	−0.990027	−0.0864990	−0.0432495	−	0.999064i	$-0.513771\pi$
$131$	−0.990027	−0.0864990	−0.0432495	+	0.999064i	$0.513771\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.49471	−0.214711
$136$	0	0
$137$	3.78181	0.323102	0.161551	−	0.986864i	$-0.448350\pi$
$137$	3.78181	0.323102	0.161551	+	0.986864i	$0.448350\pi$
$138$	0	0
$139$	18.3431	1.55584	0.777920	−	0.628364i	$-0.216275\pi$
$139$	18.3431	1.55584	0.777920	+	0.628364i	$0.216275\pi$
$140$	0	0
$141$	−0.517181	−0.0435545
$142$	0	0
$143$	−0.322412	−0.0269614
$144$	0	0
$145$	−3.16104	−0.262510
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.01195	−0.410595	−0.205297	−	0.978700i	$-0.565816\pi$
$149$	−5.01195	−0.410595	−0.205297	+	0.978700i	$0.565816\pi$
$150$	0	0
$151$	21.4822	1.74819	0.874097	−	0.485751i	$-0.161454\pi$
$151$	21.4822	1.74819	0.874097	+	0.485751i	$0.161454\pi$
$152$	0	0
$153$	−3.36026	−0.271661
$154$	0	0
$155$	2.59363	0.208325
$156$	0	0
$157$	10.8416	0.865255	0.432627	−	0.901573i	$-0.357587\pi$
$157$	10.8416	0.865255	0.432627	+	0.901573i	$0.357587\pi$
$158$	0	0
$159$	1.46950	0.116539
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.1435	−1.49943	−0.749717	−	0.661759i	$-0.769810\pi$
$163$	−19.1435	−1.49943	−0.749717	+	0.661759i	$0.769810\pi$
$164$	0	0
$165$	2.15363	0.167660
$166$	0	0
$167$	−21.3883	−1.65508	−0.827538	−	0.561409i	$-0.810259\pi$
$167$	−21.3883	−1.65508	−0.827538	+	0.561409i	$0.810259\pi$
$168$	0	0
$169$	−12.9957	−0.999671
$170$	0	0
$171$	10.3609	0.792316
$172$	0	0
$173$	−8.49170	−0.645612	−0.322806	−	0.946465i	$-0.604626\pi$
$173$	−8.49170	−0.645612	−0.322806	+	0.946465i	$0.604626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.03053	0.453282
$178$	0	0
$179$	4.17881	0.312339	0.156169	−	0.987730i	$-0.450085\pi$
$179$	4.17881	0.312339	0.156169	+	0.987730i	$0.450085\pi$
$180$	0	0
$181$	−7.46457	−0.554837	−0.277418	−	0.960749i	$-0.589479\pi$
$181$	−7.46457	−0.554837	−0.277418	+	0.960749i	$0.589479\pi$
$182$	0	0
$183$	−5.82066	−0.430276
$184$	0	0
$185$	0.0754029	0.00554373
$186$	0	0
$187$	6.11264	0.447000
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.0481	0.799413	0.399706	−	0.916643i	$-0.369112\pi$
$191$	11.0481	0.799413	0.399706	+	0.916643i	$0.369112\pi$
$192$	0	0
$193$	2.37310	0.170819	0.0854097	−	0.996346i	$-0.472780\pi$
$193$	2.37310	0.170819	0.0854097	+	0.996346i	$0.472780\pi$
$194$	0	0
$195$	−0.0285809	−0.00204672
$196$	0	0
$197$	−23.9369	−1.70543	−0.852717	−	0.522374i	$-0.825047\pi$
$197$	−23.9369	−1.70543	−0.852717	+	0.522374i	$0.825047\pi$
$198$	0	0
$199$	−5.28061	−0.374333	−0.187166	−	0.982328i	$-0.559930\pi$
$199$	−5.28061	−0.374333	−0.187166	+	0.982328i	$0.559930\pi$
$200$	0	0
$201$	−5.57960	−0.393555
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.810745	−0.0566249
$206$	0	0
$207$	4.65928	0.323842
$208$	0	0
$209$	−18.8474	−1.30370
$210$	0	0
$211$	−3.31065	−0.227915	−0.113957	−	0.993486i	$-0.536353\pi$
$211$	−3.31065	−0.227915	−0.113957	+	0.993486i	$0.536353\pi$
$212$	0	0
$213$	3.01830	0.206810
$214$	0	0
$215$	−8.56390	−0.584053
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.49255	0.100857
$220$	0	0
$221$	−0.0811212	−0.00545680
$222$	0	0
$223$	21.6139	1.44738	0.723688	−	0.690127i	$-0.242445\pi$
$223$	21.6139	1.44738	0.723688	+	0.690127i	$0.242445\pi$
$224$	0	0
$225$	11.7668	0.784450
$226$	0	0
$227$	−8.01651	−0.532074	−0.266037	−	0.963963i	$-0.585714\pi$
$227$	−8.01651	−0.532074	−0.266037	+	0.963963i	$0.585714\pi$
$228$	0	0
$229$	−1.03296	−0.0682598	−0.0341299	−	0.999417i	$-0.510866\pi$
$229$	−1.03296	−0.0682598	−0.0341299	+	0.999417i	$0.510866\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.7626	1.36020	0.680100	−	0.733119i	$-0.261936\pi$
$233$	20.7626	1.36020	0.680100	+	0.733119i	$0.261936\pi$
$234$	0	0
$235$	−0.778024	−0.0507527
$236$	0	0
$237$	8.75973	0.569005
$238$	0	0
$239$	−7.91757	−0.512145	−0.256073	−	0.966658i	$-0.582429\pi$
$239$	−7.91757	−0.512145	−0.256073	+	0.966658i	$0.582429\pi$
$240$	0	0
$241$	−20.2523	−1.30457	−0.652283	−	0.757975i	$-0.726189\pi$
$241$	−20.2523	−1.30457	−0.652283	+	0.757975i	$0.726189\pi$
$242$	0	0
$243$	−12.7182	−0.815876
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.250125	0.0159151
$248$	0	0
$249$	5.38459	0.341234
$250$	0	0
$251$	1.28654	0.0812058	0.0406029	−	0.999175i	$-0.487072\pi$
$251$	1.28654	0.0812058	0.0406029	+	0.999175i	$0.487072\pi$
$252$	0	0
$253$	−8.47567	−0.532861
$254$	0	0
$255$	0.541869	0.0339331
$256$	0	0
$257$	−6.86057	−0.427951	−0.213975	−	0.976839i	$-0.568641\pi$
$257$	−6.86057	−0.427951	−0.213975	+	0.976839i	$0.568641\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−10.5644	−0.653918
$262$	0	0
$263$	4.28050	0.263947	0.131974	−	0.991253i	$-0.457869\pi$
$263$	4.28050	0.263947	0.131974	+	0.991253i	$0.457869\pi$
$264$	0	0
$265$	2.21065	0.135799
$266$	0	0
$267$	6.39397	0.391304
$268$	0	0
$269$	−8.24205	−0.502527	−0.251263	−	0.967919i	$-0.580846\pi$
$269$	−8.24205	−0.502527	−0.251263	+	0.967919i	$0.580846\pi$
$270$	0	0
$271$	−26.4651	−1.60764	−0.803820	−	0.594873i	$-0.797203\pi$
$271$	−26.4651	−1.60764	−0.803820	+	0.594873i	$0.797203\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.4048	−1.29076
$276$	0	0
$277$	15.1377	0.909537	0.454769	−	0.890610i	$-0.349722\pi$
$277$	15.1377	0.909537	0.454769	+	0.890610i	$0.349722\pi$
$278$	0	0
$279$	8.66805	0.518942
$280$	0	0
$281$	−4.51773	−0.269505	−0.134753	−	0.990879i	$-0.543024\pi$
$281$	−4.51773	−0.269505	−0.134753	+	0.990879i	$0.543024\pi$
$282$	0	0
$283$	−30.9377	−1.83906	−0.919529	−	0.393022i	$-0.871429\pi$
$283$	−30.9377	−1.83906	−0.919529	+	0.393022i	$0.871429\pi$
$284$	0	0
$285$	−1.67077	−0.0989679
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.4620	−0.909530
$290$	0	0
$291$	3.40306	0.199491
$292$	0	0
$293$	−12.5755	−0.734667	−0.367334	−	0.930089i	$-0.619729\pi$
$293$	−12.5755	−0.734667	−0.367334	+	0.930089i	$0.619729\pi$
$294$	0	0
$295$	9.07206	0.528196
$296$	0	0
$297$	15.1666	0.880056
$298$	0	0
$299$	0.112481	0.00650495
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.75655	−0.388154
$304$	0	0
$305$	−8.75635	−0.501387
$306$	0	0
$307$	−13.0695	−0.745919	−0.372959	−	0.927848i	$-0.621657\pi$
$307$	−13.0695	−0.745919	−0.372959	+	0.927848i	$0.621657\pi$
$308$	0	0
$309$	9.71911	0.552901
$310$	0	0
$311$	−17.9072	−1.01542	−0.507712	−	0.861527i	$-0.669508\pi$
$311$	−17.9072	−1.01542	−0.507712	+	0.861527i	$0.669508\pi$
$312$	0	0
$313$	12.1024	0.684070	0.342035	−	0.939687i	$-0.388884\pi$
$313$	12.1024	0.684070	0.342035	+	0.939687i	$0.388884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.9366	−0.895089	−0.447544	−	0.894262i	$-0.647701\pi$
$317$	−15.9366	−0.895089	−0.447544	+	0.894262i	$0.647701\pi$
$318$	0	0
$319$	19.2176	1.07598
$320$	0	0
$321$	−4.90293	−0.273655
$322$	0	0
$323$	−4.74214	−0.263860
$324$	0	0
$325$	0.284065	0.0157571
$326$	0	0
$327$	−5.32767	−0.294621
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.4334	0.573473	0.286737	−	0.958009i	$-0.407430\pi$
$331$	10.4334	0.573473	0.286737	+	0.958009i	$0.407430\pi$
$332$	0	0
$333$	0.252000	0.0138095
$334$	0	0
$335$	−8.39370	−0.458597
$336$	0	0
$337$	−7.66374	−0.417470	−0.208735	−	0.977972i	$-0.566935\pi$
$337$	−7.66374	−0.417470	−0.208735	+	0.977972i	$0.566935\pi$
$338$	0	0
$339$	4.55398	0.247338
$340$	0	0
$341$	−15.7680	−0.853885
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.751345	−0.0404510
$346$	0	0
$347$	−23.4021	−1.25629	−0.628146	−	0.778095i	$-0.716186\pi$
$347$	−23.4021	−1.25629	−0.628146	+	0.778095i	$0.716186\pi$
$348$	0	0
$349$	−14.4181	−0.771781	−0.385891	−	0.922545i	$-0.626106\pi$
$349$	−14.4181	−0.771781	−0.385891	+	0.922545i	$0.626106\pi$
$350$	0	0
$351$	−0.201277	−0.0107434
$352$	0	0
$353$	−34.4990	−1.83619	−0.918097	−	0.396355i	$-0.870275\pi$
$353$	−34.4990	−1.83619	−0.918097	+	0.396355i	$0.870275\pi$
$354$	0	0
$355$	4.54060	0.240990
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.75593	−0.198231	−0.0991153	−	0.995076i	$-0.531601\pi$
$359$	−3.75593	−0.198231	−0.0991153	+	0.995076i	$0.531601\pi$
$360$	0	0
$361$	−4.37832	−0.230438
$362$	0	0
$363$	−7.16476	−0.376052
$364$	0	0
$365$	2.24533	0.117526
$366$	0	0
$367$	−19.7801	−1.03251	−0.516255	−	0.856435i	$-0.672674\pi$
$367$	−19.7801	−1.03251	−0.516255	+	0.856435i	$0.672674\pi$
$368$	0	0
$369$	−2.70955	−0.141054
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−25.5124	−1.32098	−0.660490	−	0.750835i	$-0.729651\pi$
$373$	−25.5124	−1.32098	−0.660490	+	0.750835i	$0.729651\pi$
$374$	0	0
$375$	−4.08216	−0.210802
$376$	0	0
$377$	−0.255038	−0.0131351
$378$	0	0
$379$	13.1317	0.674529	0.337264	−	0.941410i	$-0.390498\pi$
$379$	13.1317	0.674529	0.337264	+	0.941410i	$0.390498\pi$
$380$	0	0
$381$	−3.33661	−0.170940
$382$	0	0
$383$	21.2494	1.08579	0.542897	−	0.839799i	$-0.317327\pi$
$383$	21.2494	1.08579	0.542897	+	0.839799i	$0.317327\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.6210	−1.45489
$388$	0	0
$389$	37.7262	1.91280	0.956398	−	0.292068i	$-0.0943432\pi$
$389$	37.7262	1.91280	0.956398	+	0.292068i	$0.0943432\pi$
$390$	0	0
$391$	−2.13254	−0.107847
$392$	0	0
$393$	0.533557	0.0269144
$394$	0	0
$395$	13.1777	0.663045
$396$	0	0
$397$	−9.20513	−0.461992	−0.230996	−	0.972955i	$-0.574198\pi$
$397$	−9.20513	−0.461992	−0.230996	+	0.972955i	$0.574198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.2942	0.813694	0.406847	−	0.913496i	$-0.366628\pi$
$401$	16.2942	0.813694	0.406847	+	0.913496i	$0.366628\pi$
$402$	0	0
$403$	0.209258	0.0104239
$404$	0	0
$405$	−5.24579	−0.260666
$406$	0	0
$407$	−0.458413	−0.0227227
$408$	0	0
$409$	0.569169	0.0281436	0.0140718	−	0.999901i	$-0.495521\pi$
$409$	0.569169	0.0281436	0.0140718	+	0.999901i	$0.495521\pi$
$410$	0	0
$411$	−2.03814	−0.100534
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.10034	0.397630
$416$	0	0
$417$	−9.88567	−0.484103
$418$	0	0
$419$	3.91313	0.191169	0.0955845	−	0.995421i	$-0.469528\pi$
$419$	3.91313	0.191169	0.0955845	+	0.995421i	$0.469528\pi$
$420$	0	0
$421$	13.6649	0.665985	0.332992	−	0.942930i	$-0.391942\pi$
$421$	13.6649	0.665985	0.332992	+	0.942930i	$0.391942\pi$
$422$	0	0
$423$	−2.60020	−0.126426
$424$	0	0
$425$	−5.38561	−0.261240
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.173758	0.00838912
$430$	0	0
$431$	6.84032	0.329487	0.164743	−	0.986336i	$-0.447320\pi$
$431$	6.84032	0.329487	0.164743	+	0.986336i	$0.447320\pi$
$432$	0	0
$433$	1.46375	0.0703434	0.0351717	−	0.999381i	$-0.488802\pi$
$433$	1.46375	0.0703434	0.0351717	+	0.999381i	$0.488802\pi$
$434$	0	0
$435$	1.70359	0.0816807
$436$	0	0
$437$	6.57536	0.314542
$438$	0	0
$439$	1.39866	0.0667542	0.0333771	−	0.999443i	$-0.489374\pi$
$439$	1.39866	0.0667542	0.0333771	+	0.999443i	$0.489374\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.1620	−0.862902	−0.431451	−	0.902136i	$-0.641998\pi$
$443$	−18.1620	−0.862902	−0.431451	+	0.902136i	$0.641998\pi$
$444$	0	0
$445$	9.61880	0.455975
$446$	0	0
$447$	2.70110	0.127758
$448$	0	0
$449$	−38.4896	−1.81644	−0.908219	−	0.418495i	$-0.862558\pi$
$449$	−38.4896	−1.81644	−0.908219	+	0.418495i	$0.862558\pi$
$450$	0	0
$451$	4.92893	0.232094
$452$	0	0
$453$	−11.5774	−0.543955
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.6702	−0.920135	−0.460067	−	0.887884i	$-0.652175\pi$
$457$	−19.6702	−0.920135	−0.460067	+	0.887884i	$0.652175\pi$
$458$	0	0
$459$	3.81603	0.178117
$460$	0	0
$461$	−21.3526	−0.994488	−0.497244	−	0.867611i	$-0.665655\pi$
$461$	−21.3526	−0.994488	−0.497244	+	0.867611i	$0.665655\pi$
$462$	0	0
$463$	−11.5941	−0.538823	−0.269412	−	0.963025i	$-0.586829\pi$
$463$	−11.5941	−0.538823	−0.269412	+	0.963025i	$0.586829\pi$
$464$	0	0
$465$	−1.39779	−0.0648210
$466$	0	0
$467$	15.4316	0.714087	0.357044	−	0.934088i	$-0.383785\pi$
$467$	15.4316	0.714087	0.357044	+	0.934088i	$0.383785\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.84289	−0.269226
$472$	0	0
$473$	52.0643	2.39392
$474$	0	0
$475$	16.6057	0.761923
$476$	0	0
$477$	7.38809	0.338278
$478$	0	0
$479$	−20.3619	−0.930361	−0.465180	−	0.885216i	$-0.654011\pi$
$479$	−20.3619	−0.930361	−0.465180	+	0.885216i	$0.654011\pi$
$480$	0	0
$481$	0.00608362	0.000277389 0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.11942	0.232461
$486$	0	0
$487$	−9.90723	−0.448939	−0.224470	−	0.974481i	$-0.572065\pi$
$487$	−9.90723	−0.448939	−0.224470	+	0.974481i	$0.572065\pi$
$488$	0	0
$489$	10.3170	0.466552
$490$	0	0
$491$	−9.11988	−0.411574	−0.205787	−	0.978597i	$-0.565975\pi$
$491$	−9.11988	−0.411574	−0.205787	+	0.978597i	$0.565975\pi$
$492$	0	0
$493$	4.83528	0.217770
$494$	0	0
$495$	10.8277	0.486667
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.1763	−0.589852	−0.294926	−	0.955520i	$-0.595295\pi$
$499$	−13.1763	−0.589852	−0.294926	+	0.955520i	$0.595295\pi$
$500$	0	0
$501$	11.5268	0.514981
$502$	0	0
$503$	33.6192	1.49901	0.749504	−	0.662000i	$-0.230292\pi$
$503$	33.6192	1.49901	0.749504	+	0.662000i	$0.230292\pi$
$504$	0	0
$505$	−10.1643	−0.452303
$506$	0	0
$507$	7.00381	0.311050
$508$	0	0
$509$	−29.7381	−1.31812	−0.659059	−	0.752091i	$-0.729045\pi$
$509$	−29.7381	−1.31812	−0.659059	+	0.752091i	$0.729045\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−11.7662	−0.519488
$514$	0	0
$515$	14.6210	0.644279
$516$	0	0
$517$	4.73000	0.208025
$518$	0	0
$519$	4.57645	0.200884
$520$	0	0
$521$	0.710103	0.0311102	0.0155551	−	0.999879i	$-0.495048\pi$
$521$	0.710103	0.0311102	0.0155551	+	0.999879i	$0.495048\pi$
$522$	0	0
$523$	−3.85319	−0.168488	−0.0842441	−	0.996445i	$-0.526848\pi$
$523$	−3.85319	−0.168488	−0.0842441	+	0.996445i	$0.526848\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.96734	−0.172820
$528$	0	0
$529$	−20.0431	−0.871438
$530$	0	0
$531$	30.3193	1.31575
$532$	0	0
$533$	−0.0654121	−0.00283331
$534$	0	0
$535$	−7.37575	−0.318882
$536$	0	0
$537$	−2.25209	−0.0971849
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15.9199	−0.684448	−0.342224	−	0.939618i	$-0.611180\pi$
$541$	−15.9199	−0.684448	−0.342224	+	0.939618i	$0.611180\pi$
$542$	0	0
$543$	4.02289	0.172639
$544$	0	0
$545$	−8.01472	−0.343313
$546$	0	0
$547$	4.91366	0.210093	0.105047	−	0.994467i	$-0.466501\pi$
$547$	4.91366	0.210093	0.105047	+	0.994467i	$0.466501\pi$
$548$	0	0
$549$	−29.2642	−1.24896
$550$	0	0
$551$	−14.9089	−0.635139
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.0406370	−0.00172495
$556$	0	0
$557$	−26.2303	−1.11141	−0.555706	−	0.831379i	$-0.687552\pi$
$557$	−26.2303	−1.11141	−0.555706	+	0.831379i	$0.687552\pi$
$558$	0	0
$559$	−0.690948	−0.0292240
$560$	0	0
$561$	−3.29430	−0.139085
$562$	0	0
$563$	−4.73653	−0.199621	−0.0998105	−	0.995006i	$-0.531824\pi$
$563$	−4.73653	−0.199621	−0.0998105	+	0.995006i	$0.531824\pi$
$564$	0	0
$565$	6.85081	0.288216
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.8284	0.705482	0.352741	−	0.935721i	$-0.385250\pi$
$569$	16.8284	0.705482	0.352741	+	0.935721i	$0.385250\pi$
$570$	0	0
$571$	−20.4408	−0.855420	−0.427710	−	0.903916i	$-0.640680\pi$
$571$	−20.4408	−0.855420	−0.427710	+	0.903916i	$0.640680\pi$
$572$	0	0
$573$	−5.95417	−0.248739
$574$	0	0
$575$	7.46758	0.311420
$576$	0	0
$577$	15.1297	0.629859	0.314930	−	0.949115i	$-0.398019\pi$
$577$	15.1297	0.629859	0.314930	+	0.949115i	$0.398019\pi$
$578$	0	0
$579$	−1.27894	−0.0531509
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.4396	−0.556613
$584$	0	0
$585$	−0.143695	−0.00594104
$586$	0	0
$587$	44.9665	1.85597	0.927984	−	0.372620i	$-0.121541\pi$
$587$	44.9665	1.85597	0.927984	+	0.372620i	$0.121541\pi$
$588$	0	0
$589$	12.2327	0.504040
$590$	0	0
$591$	12.9004	0.530650
$592$	0	0
$593$	−21.8232	−0.896171	−0.448085	−	0.893991i	$-0.647894\pi$
$593$	−21.8232	−0.896171	−0.448085	+	0.893991i	$0.647894\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.84589	0.116474
$598$	0	0
$599$	22.1256	0.904028	0.452014	−	0.892011i	$-0.350706\pi$
$599$	22.1256	0.904028	0.452014	+	0.892011i	$0.350706\pi$
$600$	0	0
$601$	−41.2542	−1.68279	−0.841396	−	0.540418i	$-0.818266\pi$
$601$	−41.2542	−1.68279	−0.841396	+	0.540418i	$0.818266\pi$
$602$	0	0
$603$	−28.0522	−1.14237
$604$	0	0
$605$	−10.7784	−0.438202
$606$	0	0
$607$	2.98200	0.121036	0.0605178	−	0.998167i	$-0.480725\pi$
$607$	2.98200	0.121036	0.0605178	+	0.998167i	$0.480725\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.0627721	−0.00253949
$612$	0	0
$613$	−22.1475	−0.894527	−0.447264	−	0.894402i	$-0.647601\pi$
$613$	−22.1475	−0.894527	−0.447264	+	0.894402i	$0.647601\pi$
$614$	0	0
$615$	0.436937	0.0176190
$616$	0	0
$617$	−4.94230	−0.198970	−0.0994848	−	0.995039i	$-0.531719\pi$
$617$	−4.94230	−0.198970	−0.0994848	+	0.995039i	$0.531719\pi$
$618$	0	0
$619$	35.4913	1.42651	0.713257	−	0.700902i	$-0.247219\pi$
$619$	35.4913	1.42651	0.713257	+	0.700902i	$0.247219\pi$
$620$	0	0
$621$	−5.29123	−0.212330
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15.5724	0.622897
$626$	0	0
$627$	10.1575	0.405650
$628$	0	0
$629$	−0.115340	−0.00459890
$630$	0	0
$631$	−40.6549	−1.61844	−0.809222	−	0.587503i	$-0.800111\pi$
$631$	−40.6549	−1.61844	−0.809222	+	0.587503i	$0.800111\pi$
$632$	0	0
$633$	1.78422	0.0709163
$634$	0	0
$635$	−5.01945	−0.199191
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.1749	0.600310
$640$	0	0
$641$	41.3461	1.63307	0.816536	−	0.577295i	$-0.195892\pi$
$641$	41.3461	1.63307	0.816536	+	0.577295i	$0.195892\pi$
$642$	0	0
$643$	−24.9995	−0.985883	−0.492942	−	0.870062i	$-0.664078\pi$
$643$	−24.9995	−0.985883	−0.492942	+	0.870062i	$0.664078\pi$
$644$	0	0
$645$	4.61536	0.181730
$646$	0	0
$647$	25.1819	0.990003	0.495002	−	0.868892i	$-0.335168\pi$
$647$	25.1819	0.990003	0.495002	+	0.868892i	$0.335168\pi$
$648$	0	0
$649$	−55.1537	−2.16497
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.3606	1.50117	0.750583	−	0.660776i	$-0.229773\pi$
$653$	38.3606	1.50117	0.750583	+	0.660776i	$0.229773\pi$
$654$	0	0
$655$	0.802660	0.0313625
$656$	0	0
$657$	7.50401	0.292759
$658$	0	0
$659$	38.7750	1.51046	0.755230	−	0.655459i	$-0.227525\pi$
$659$	38.7750	1.51046	0.755230	+	0.655459i	$0.227525\pi$
$660$	0	0
$661$	−37.8665	−1.47284	−0.736419	−	0.676526i	$-0.763485\pi$
$661$	−37.8665	−1.47284	−0.736419	+	0.676526i	$0.763485\pi$
$662$	0	0
$663$	0.0437188	0.00169790
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.70451	−0.259600
$668$	0	0
$669$	−11.6484	−0.450355
$670$	0	0
$671$	53.2343	2.05509
$672$	0	0
$673$	−5.05883	−0.195004	−0.0975018	−	0.995235i	$-0.531085\pi$
$673$	−5.05883	−0.195004	−0.0975018	+	0.995235i	$0.531085\pi$
$674$	0	0
$675$	−13.3627	−0.514331
$676$	0	0
$677$	44.1411	1.69648	0.848241	−	0.529611i	$-0.177662\pi$
$677$	44.1411	1.69648	0.848241	+	0.529611i	$0.177662\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.32035	0.165556
$682$	0	0
$683$	10.5732	0.404572	0.202286	−	0.979326i	$-0.435163\pi$
$683$	10.5732	0.404572	0.202286	+	0.979326i	$0.435163\pi$
$684$	0	0
$685$	−3.06609	−0.117149
$686$	0	0
$687$	0.556694	0.0212392
$688$	0	0
$689$	0.178358	0.00679491
$690$	0	0
$691$	28.1100	1.06936	0.534678	−	0.845056i	$-0.320433\pi$
$691$	28.1100	1.06936	0.534678	+	0.845056i	$0.320433\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.8716	−0.564111
$696$	0	0
$697$	1.24015	0.0469742
$698$	0	0
$699$	−11.1896	−0.423230
$700$	0	0
$701$	−3.99389	−0.150847	−0.0754236	−	0.997152i	$-0.524031\pi$
$701$	−3.99389	−0.150847	−0.0754236	+	0.997152i	$0.524031\pi$
$702$	0	0
$703$	0.355633	0.0134130
$704$	0	0
$705$	0.419302	0.0157918
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.43794	0.0540028	0.0270014	−	0.999635i	$-0.491404\pi$
$709$	1.43794	0.0540028	0.0270014	+	0.999635i	$0.491404\pi$
$710$	0	0
$711$	44.0407	1.65166
$712$	0	0
$713$	5.50104	0.206016
$714$	0	0
$715$	0.261394	0.00977558
$716$	0	0
$717$	4.26703	0.159355
$718$	0	0
$719$	−4.38732	−0.163619	−0.0818097	−	0.996648i	$-0.526070\pi$
$719$	−4.38732	−0.163619	−0.0818097	+	0.996648i	$0.526070\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.9146	0.405919
$724$	0	0
$725$	−16.9319	−0.628834
$726$	0	0
$727$	5.95216	0.220753	0.110377	−	0.993890i	$-0.464794\pi$
$727$	5.95216	0.220753	0.110377	+	0.993890i	$0.464794\pi$
$728$	0	0
$729$	−12.5567	−0.465064
$730$	0	0
$731$	13.0998	0.484512
$732$	0	0
$733$	−31.7441	−1.17249	−0.586247	−	0.810132i	$-0.699395\pi$
$733$	−31.7441	−1.17249	−0.586247	+	0.810132i	$0.699395\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	51.0296	1.87970
$738$	0	0
$739$	1.49001	0.0548109	0.0274055	−	0.999624i	$-0.491275\pi$
$739$	1.49001	0.0548109	0.0274055	+	0.999624i	$0.491275\pi$
$740$	0	0
$741$	−0.134800	−0.00495201
$742$	0	0
$743$	−2.63999	−0.0968519	−0.0484259	−	0.998827i	$-0.515420\pi$
$743$	−2.63999	−0.0968519	−0.0484259	+	0.998827i	$0.515420\pi$
$744$	0	0
$745$	4.06341	0.148872
$746$	0	0
$747$	27.0717	0.990503
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.58728	−0.0579206	−0.0289603	−	0.999581i	$-0.509220\pi$
$751$	−1.58728	−0.0579206	−0.0289603	+	0.999581i	$0.509220\pi$
$752$	0	0
$753$	−0.693358	−0.0252674
$754$	0	0
$755$	−17.4166	−0.633854
$756$	0	0
$757$	−1.41445	−0.0514090	−0.0257045	−	0.999670i	$-0.508183\pi$
$757$	−1.41445	−0.0514090	−0.0257045	+	0.999670i	$0.508183\pi$
$758$	0	0
$759$	4.56781	0.165801
$760$	0	0
$761$	−34.5791	−1.25349	−0.626746	−	0.779224i	$-0.715613\pi$
$761$	−34.5791	−1.25349	−0.626746	+	0.779224i	$0.715613\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.72432	0.0984980
$766$	0	0
$767$	0.731947	0.0264291
$768$	0	0
$769$	−45.3841	−1.63659	−0.818297	−	0.574796i	$-0.805081\pi$
$769$	−45.3841	−1.63659	−0.818297	+	0.574796i	$0.805081\pi$
$770$	0	0
$771$	3.69738	0.133158
$772$	0	0
$773$	9.04090	0.325179	0.162589	−	0.986694i	$-0.448015\pi$
$773$	9.04090	0.325179	0.162589	+	0.986694i	$0.448015\pi$
$774$	0	0
$775$	13.8926	0.499036
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.82383	−0.137003
$780$	0	0
$781$	−27.6046	−0.987770
$782$	0	0
$783$	11.9973	0.428747
$784$	0	0
$785$	−8.78979	−0.313721
$786$	0	0
$787$	−14.1214	−0.503372	−0.251686	−	0.967809i	$-0.580985\pi$
$787$	−14.1214	−0.503372	−0.251686	+	0.967809i	$0.580985\pi$
$788$	0	0
$789$	−2.30690	−0.0821278
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.706475	−0.0250877
$794$	0	0
$795$	−1.19139	−0.0422542
$796$	0	0
$797$	52.9069	1.87406	0.937029	−	0.349252i	$-0.113564\pi$
$797$	52.9069	1.87406	0.937029	+	0.349252i	$0.113564\pi$
$798$	0	0
$799$	1.19010	0.0421028
$800$	0	0
$801$	32.1465	1.13584
$802$	0	0
$803$	−13.6505	−0.481716
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.44190	0.156362
$808$	0	0
$809$	−38.4156	−1.35062	−0.675311	−	0.737533i	$-0.735991\pi$
$809$	−38.4156	−1.35062	−0.675311	+	0.737533i	$0.735991\pi$
$810$	0	0
$811$	−10.9375	−0.384068	−0.192034	−	0.981388i	$-0.561508\pi$
$811$	−10.9375	−0.384068	−0.192034	+	0.981388i	$0.561508\pi$
$812$	0	0
$813$	14.2629	0.500221
$814$	0	0
$815$	15.5205	0.543659
$816$	0	0
$817$	−40.3911	−1.41311
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−52.1638	−1.82053	−0.910265	−	0.414027i	$-0.864122\pi$
$821$	−52.1638	−1.82053	−0.910265	+	0.414027i	$0.864122\pi$
$822$	0	0
$823$	23.4953	0.818995	0.409498	−	0.912311i	$-0.365704\pi$
$823$	23.4953	0.818995	0.409498	+	0.912311i	$0.365704\pi$
$824$	0	0
$825$	11.5357	0.401623
$826$	0	0
$827$	2.66381	0.0926298	0.0463149	−	0.998927i	$-0.485252\pi$
$827$	2.66381	0.0926298	0.0463149	+	0.998927i	$0.485252\pi$
$828$	0	0
$829$	−39.4035	−1.36854	−0.684270	−	0.729229i	$-0.739879\pi$
$829$	−39.4035	−1.36854	−0.684270	+	0.729229i	$0.739879\pi$
$830$	0	0
$831$	−8.15820	−0.283005
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.3405	0.600092
$836$	0	0
$837$	−9.84373	−0.340249
$838$	0	0
$839$	−23.2979	−0.804334	−0.402167	−	0.915566i	$-0.631743\pi$
$839$	−23.2979	−0.804334	−0.402167	+	0.915566i	$0.631743\pi$
$840$	0	0
$841$	−13.7983	−0.475804
$842$	0	0
$843$	2.43475	0.0838572
$844$	0	0
$845$	10.5362	0.362457
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.6733	0.572227
$850$	0	0
$851$	0.159928	0.00548226
$852$	0	0
$853$	34.1155	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$853$	34.1155	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$854$	0	0
$855$	−8.40003	−0.287275
$856$	0	0
$857$	17.7004	0.604633	0.302317	−	0.953208i	$-0.402240\pi$
$857$	17.7004	0.604633	0.302317	+	0.953208i	$0.402240\pi$
$858$	0	0
$859$	−20.5431	−0.700922	−0.350461	−	0.936577i	$-0.613975\pi$
$859$	−20.5431	−0.700922	−0.350461	+	0.936577i	$0.613975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.0834	0.479404	0.239702	−	0.970847i	$-0.422950\pi$
$863$	14.0834	0.479404	0.239702	+	0.970847i	$0.422950\pi$
$864$	0	0
$865$	6.88460	0.234084
$866$	0	0
$867$	8.33297	0.283003
$868$	0	0
$869$	−80.1142	−2.71769
$870$	0	0
$871$	−0.677216	−0.0229466
$872$	0	0
$873$	17.1093	0.579064
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.75794	−0.228199	−0.114100	−	0.993469i	$-0.536398\pi$
$877$	−6.75794	−0.228199	−0.114100	+	0.993469i	$0.536398\pi$
$878$	0	0
$879$	6.77733	0.228594
$880$	0	0
$881$	−30.5704	−1.02994	−0.514972	−	0.857207i	$-0.672198\pi$
$881$	−30.5704	−1.02994	−0.514972	+	0.857207i	$0.672198\pi$
$882$	0	0
$883$	55.0224	1.85165	0.925825	−	0.377952i	$-0.123372\pi$
$883$	55.0224	1.85165	0.925825	+	0.377952i	$0.123372\pi$
$884$	0	0
$885$	−4.88922	−0.164349
$886$	0	0
$887$	28.5153	0.957450	0.478725	−	0.877965i	$-0.341099\pi$
$887$	28.5153	0.957450	0.478725	+	0.877965i	$0.341099\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	31.8918	1.06842
$892$	0	0
$893$	−3.66950	−0.122795
$894$	0	0
$895$	−3.38795	−0.113247
$896$	0	0
$897$	−0.0606196	−0.00202403
$898$	0	0
$899$	−12.4730	−0.415997
$900$	0	0
$901$	−3.38151	−0.112654
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.05186	0.201171
$906$	0	0
$907$	−30.7081	−1.01965	−0.509823	−	0.860279i	$-0.670289\pi$
$907$	−30.7081	−1.01965	−0.509823	+	0.860279i	$0.670289\pi$
$908$	0	0
$909$	−33.9695	−1.12670
$910$	0	0
$911$	−24.7191	−0.818981	−0.409491	−	0.912314i	$-0.634294\pi$
$911$	−24.7191	−0.818981	−0.409491	+	0.912314i	$0.634294\pi$
$912$	0	0
$913$	−49.2461	−1.62981
$914$	0	0
$915$	4.71907	0.156008
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36.6530	−1.20907	−0.604535	−	0.796579i	$-0.706641\pi$
$919$	−36.6530	−1.20907	−0.604535	+	0.796579i	$0.706641\pi$
$920$	0	0
$921$	7.04360	0.232094
$922$	0	0
$923$	0.366342	0.0120583
$924$	0	0
$925$	0.403890	0.0132798
$926$	0	0
$927$	48.8642	1.60491
$928$	0	0
$929$	−31.8880	−1.04621	−0.523105	−	0.852268i	$-0.675226\pi$
$929$	−31.8880	−1.04621	−0.523105	+	0.852268i	$0.675226\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.65075	0.315951
$934$	0	0
$935$	−4.95579	−0.162072
$936$	0	0
$937$	−49.5949	−1.62019	−0.810097	−	0.586295i	$-0.800586\pi$
$937$	−49.5949	−1.62019	−0.810097	+	0.586295i	$0.800586\pi$
$938$	0	0
$939$	−6.52238	−0.212850
$940$	0	0
$941$	1.24569	0.0406084	0.0203042	−	0.999794i	$-0.493537\pi$
$941$	1.24569	0.0406084	0.0203042	+	0.999794i	$0.493537\pi$
$942$	0	0
$943$	−1.71957	−0.0559971
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.8351	−0.482076	−0.241038	−	0.970516i	$-0.577488\pi$
$947$	−14.8351	−0.482076	−0.241038	+	0.970516i	$0.577488\pi$
$948$	0	0
$949$	0.181157	0.00588059
$950$	0	0
$951$	8.58874	0.278509
$952$	0	0
$953$	31.5263	1.02124	0.510618	−	0.859808i	$-0.329417\pi$
$953$	31.5263	1.02124	0.510618	+	0.859808i	$0.329417\pi$
$954$	0	0
$955$	−8.95720	−0.289848
$956$	0	0
$957$	−10.3570	−0.334793
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.7660	−0.669869
$962$	0	0
$963$	−24.6501	−0.794340
$964$	0	0
$965$	−1.92398	−0.0619351
$966$	0	0
$967$	7.43038	0.238945	0.119472	−	0.992838i	$-0.461880\pi$
$967$	7.43038	0.238945	0.119472	+	0.992838i	$0.461880\pi$
$968$	0	0
$969$	2.55569	0.0821006
$970$	0	0
$971$	20.3316	0.652472	0.326236	−	0.945288i	$-0.394220\pi$
$971$	20.3316	0.652472	0.326236	+	0.945288i	$0.394220\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.153092	−0.00490285
$976$	0	0
$977$	−9.88193	−0.316151	−0.158076	−	0.987427i	$-0.550529\pi$
$977$	−9.88193	−0.316151	−0.158076	+	0.987427i	$0.550529\pi$
$978$	0	0
$979$	−58.4776	−1.86895
$980$	0	0
$981$	−26.7856	−0.855198
$982$	0	0
$983$	−30.4507	−0.971225	−0.485612	−	0.874174i	$-0.661403\pi$
$983$	−30.4507	−0.971225	−0.485612	+	0.874174i	$0.661403\pi$
$984$	0	0
$985$	19.4067	0.618350
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.1639	−0.577577
$990$	0	0
$991$	37.0806	1.17791	0.588953	−	0.808167i	$-0.299540\pi$
$991$	37.0806	1.17791	0.588953	+	0.808167i	$0.299540\pi$
$992$	0	0
$993$	−5.62291	−0.178438
$994$	0	0
$995$	4.28123	0.135724
$996$	0	0
$997$	1.49925	0.0474818	0.0237409	−	0.999718i	$-0.492442\pi$
$997$	1.49925	0.0474818	0.0237409	+	0.999718i	$0.492442\pi$
$998$	0	0
$999$	−0.286180	−0.00905434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.9	✓	20
7.6	odd	2		8036.2.a.t.1.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.9	✓	20	1.1	even	1	trivial
8036.2.a.t.1.12	yes	20	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-0.538932 q^{3} -0.810745 q^{5} -2.70955 q^{9} +O(q^{10})\)	\(q-0.538932 q^{3} -0.810745 q^{5} -2.70955 q^{9} +4.92893 q^{11} -0.0654121 q^{13} +0.436937 q^{15} +1.24015 q^{17} -3.82383 q^{19} -1.71957 q^{23} -4.34269 q^{25} +3.07706 q^{27} +3.89893 q^{29} -3.19907 q^{31} -2.65636 q^{33} -0.0930044 q^{37} +0.0352527 q^{39} +1.00000 q^{41} +10.5630 q^{43} +2.19676 q^{45} +0.959640 q^{47} -0.668359 q^{51} -2.72668 q^{53} -3.99611 q^{55} +2.06078 q^{57} -11.1898 q^{59} +10.8004 q^{61} +0.0530326 q^{65} +10.3531 q^{67} +0.926733 q^{69} -5.60052 q^{71} -2.76947 q^{73} +2.34041 q^{75} -16.2539 q^{79} +6.47033 q^{81} -9.99122 q^{83} -1.00545 q^{85} -2.10126 q^{87} -11.8641 q^{89} +1.72408 q^{93} +3.10015 q^{95} -6.31445 q^{97} -13.3552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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