LMFDB - Embedded newform 8036.2.a.o.1.7

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
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			1.7
Root			$-1.05099$ 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)

$f(q)$	$=$	$q+1.05099 q^{3} +1.02904 q^{5} -1.89542 q^{9} +O(q^{10})$	$q+1.05099 q^{3} +1.02904 q^{5} -1.89542 q^{9} -1.28502 q^{11} +2.29756 q^{13} +1.08151 q^{15} -4.44667 q^{17} +2.95049 q^{19} +4.81352 q^{23} -3.94108 q^{25} -5.14504 q^{27} +0.100729 q^{29} -0.925502 q^{31} -1.35054 q^{33} -7.07815 q^{37} +2.41471 q^{39} -1.00000 q^{41} -5.47396 q^{43} -1.95046 q^{45} +2.99968 q^{47} -4.67341 q^{51} -1.46189 q^{53} -1.32233 q^{55} +3.10094 q^{57} -4.37678 q^{59} +5.39476 q^{61} +2.36428 q^{65} -5.18998 q^{67} +5.05896 q^{69} -14.2892 q^{71} +9.89202 q^{73} -4.14203 q^{75} -7.39997 q^{79} +0.278885 q^{81} -16.1355 q^{83} -4.57580 q^{85} +0.105865 q^{87} +15.7712 q^{89} -0.972693 q^{93} +3.03617 q^{95} -3.20703 q^{97} +2.43565 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9	$ 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9} - 6 q^{13} - 4 q^{15} - 12 q^{17} - 8 q^{19} + 4 q^{23} + 16 q^{25} - 8 q^{27} + 2 q^{29} - 8 q^{31} + 6 q^{33} - 2 q^{37} - 2 q^{39} - 10 q^{41} - 2 q^{43} - 44 q^{45} + 14 q^{47} + 14 q^{51} + 8 q^{53} - 8 q^{55} - 10 q^{57} - 24 q^{59} - 14 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 10 q^{71} - 44 q^{73} + 50 q^{75} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 8 q^{85} - 20 q^{87} - 6 q^{89} + 8 q^{93} + 4 q^{95} - 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9 - 6 * q^13 - 4 * q^15 - 12 * q^17 - 8 * q^19 + 4 * q^23 + 16 * q^25 - 8 * q^27 + 2 * q^29 - 8 * q^31 + 6 * q^33 - 2 * q^37 - 2 * q^39 - 10 * q^41 - 2 * q^43 - 44 * q^45 + 14 * q^47 + 14 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^57 - 24 * q^59 - 14 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 10 * q^71 - 44 * q^73 + 50 * q^75 + 10 * q^79 - 14 * q^81 - 20 * q^83 + 8 * q^85 - 20 * q^87 - 6 * q^89 + 8 * q^93 + 4 * q^95 - 46 * 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
$3$	1.05099	0.606789	0.303395	−	0.952865i	$-0.401880\pi$
$3$	1.05099	0.606789	0.303395	+	0.952865i	$0.401880\pi$
$4$	0	0
$5$	1.02904	0.460200	0.230100	−	0.973167i	$-0.426095\pi$
$5$	1.02904	0.460200	0.230100	+	0.973167i	$0.426095\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.89542	−0.631807
$10$	0	0
$11$	−1.28502	−0.387448	−0.193724	−	0.981056i	$-0.562057\pi$
$11$	−1.28502	−0.387448	−0.193724	+	0.981056i	$0.562057\pi$
$12$	0	0
$13$	2.29756	0.637228	0.318614	−	0.947885i	$-0.396783\pi$
$13$	2.29756	0.637228	0.318614	+	0.947885i	$0.396783\pi$
$14$	0	0
$15$	1.08151	0.279244
$16$	0	0
$17$	−4.44667	−1.07848	−0.539239	−	0.842153i	$-0.681288\pi$
$17$	−4.44667	−1.07848	−0.539239	+	0.842153i	$0.681288\pi$
$18$	0	0
$19$	2.95049	0.676889	0.338445	−	0.940986i	$-0.390099\pi$
$19$	2.95049	0.676889	0.338445	+	0.940986i	$0.390099\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.81352	1.00369	0.501844	−	0.864958i	$-0.332655\pi$
$23$	4.81352	1.00369	0.501844	+	0.864958i	$0.332655\pi$
$24$	0	0
$25$	−3.94108	−0.788216
$26$	0	0
$27$	−5.14504	−0.990163
$28$	0	0
$29$	0.100729	0.0187050	0.00935248	−	0.999956i	$-0.497023\pi$
$29$	0.100729	0.0187050	0.00935248	+	0.999956i	$0.497023\pi$
$30$	0	0
$31$	−0.925502	−0.166225	−0.0831125	−	0.996540i	$-0.526486\pi$
$31$	−0.925502	−0.166225	−0.0831125	+	0.996540i	$0.526486\pi$
$32$	0	0
$33$	−1.35054	−0.235099
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.07815	−1.16364	−0.581821	−	0.813317i	$-0.697659\pi$
$37$	−7.07815	−1.16364	−0.581821	+	0.813317i	$0.697659\pi$
$38$	0	0
$39$	2.41471	0.386663
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−5.47396	−0.834770	−0.417385	−	0.908730i	$-0.637053\pi$
$43$	−5.47396	−0.834770	−0.417385	+	0.908730i	$0.637053\pi$
$44$	0	0
$45$	−1.95046	−0.290758
$46$	0	0
$47$	2.99968	0.437549	0.218774	−	0.975775i	$-0.429794\pi$
$47$	2.99968	0.437549	0.218774	+	0.975775i	$0.429794\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.67341	−0.654408
$52$	0	0
$53$	−1.46189	−0.200806	−0.100403	−	0.994947i	$-0.532013\pi$
$53$	−1.46189	−0.200806	−0.100403	+	0.994947i	$0.532013\pi$
$54$	0	0
$55$	−1.32233	−0.178304
$56$	0	0
$57$	3.10094	0.410729
$58$	0	0
$59$	−4.37678	−0.569808	−0.284904	−	0.958556i	$-0.591962\pi$
$59$	−4.37678	−0.569808	−0.284904	+	0.958556i	$0.591962\pi$
$60$	0	0
$61$	5.39476	0.690729	0.345364	−	0.938469i	$-0.387755\pi$
$61$	5.39476	0.690729	0.345364	+	0.938469i	$0.387755\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.36428	0.293252
$66$	0	0
$67$	−5.18998	−0.634056	−0.317028	−	0.948416i	$-0.602685\pi$
$67$	−5.18998	−0.634056	−0.317028	+	0.948416i	$0.602685\pi$
$68$	0	0
$69$	5.05896	0.609027
$70$	0	0
$71$	−14.2892	−1.69581	−0.847905	−	0.530148i	$-0.822137\pi$
$71$	−14.2892	−1.69581	−0.847905	+	0.530148i	$0.822137\pi$
$72$	0	0
$73$	9.89202	1.15777	0.578887	−	0.815408i	$-0.303487\pi$
$73$	9.89202	1.15777	0.578887	+	0.815408i	$0.303487\pi$
$74$	0	0
$75$	−4.14203	−0.478281
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.39997	−0.832562	−0.416281	−	0.909236i	$-0.636667\pi$
$79$	−7.39997	−0.832562	−0.416281	+	0.909236i	$0.636667\pi$
$80$	0	0
$81$	0.278885	0.0309872
$82$	0	0
$83$	−16.1355	−1.77110	−0.885548	−	0.464547i	$-0.846217\pi$
$83$	−16.1355	−1.77110	−0.885548	+	0.464547i	$0.846217\pi$
$84$	0	0
$85$	−4.57580	−0.496315
$86$	0	0
$87$	0.105865	0.0113500
$88$	0	0
$89$	15.7712	1.67175	0.835874	−	0.548922i	$-0.184961\pi$
$89$	15.7712	1.67175	0.835874	+	0.548922i	$0.184961\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.972693	−0.100864
$94$	0	0
$95$	3.03617	0.311505
$96$	0	0
$97$	−3.20703	−0.325625	−0.162812	−	0.986657i	$-0.552057\pi$
$97$	−3.20703	−0.325625	−0.162812	+	0.986657i	$0.552057\pi$
$98$	0	0
$99$	2.43565	0.244792
$100$	0	0
$101$	−12.6500	−1.25872	−0.629361	−	0.777113i	$-0.716684\pi$
$101$	−12.6500	−1.25872	−0.629361	+	0.777113i	$0.716684\pi$
$102$	0	0
$103$	−4.93576	−0.486335	−0.243168	−	0.969984i	$-0.578186\pi$
$103$	−4.93576	−0.486335	−0.243168	+	0.969984i	$0.578186\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.41487	0.233455	0.116727	−	0.993164i	$-0.462760\pi$
$107$	2.41487	0.233455	0.116727	+	0.993164i	$0.462760\pi$
$108$	0	0
$109$	16.3593	1.56694	0.783470	−	0.621430i	$-0.213448\pi$
$109$	16.3593	1.56694	0.783470	+	0.621430i	$0.213448\pi$
$110$	0	0
$111$	−7.43907	−0.706085
$112$	0	0
$113$	8.11756	0.763636	0.381818	−	0.924238i	$-0.375298\pi$
$113$	8.11756	0.763636	0.381818	+	0.924238i	$0.375298\pi$
$114$	0	0
$115$	4.95330	0.461897
$116$	0	0
$117$	−4.35484	−0.402605
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.34873	−0.849884
$122$	0	0
$123$	−1.05099	−0.0947645
$124$	0	0
$125$	−9.20072	−0.822937
$126$	0	0
$127$	−9.43718	−0.837415	−0.418707	−	0.908121i	$-0.637517\pi$
$127$	−9.43718	−0.837415	−0.418707	+	0.908121i	$0.637517\pi$
$128$	0	0
$129$	−5.75307	−0.506530
$130$	0	0
$131$	0.192974	0.0168602	0.00843009	−	0.999964i	$-0.497317\pi$
$131$	0.192974	0.0168602	0.00843009	+	0.999964i	$0.497317\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.29444	−0.455673
$136$	0	0
$137$	14.8617	1.26972	0.634861	−	0.772626i	$-0.281057\pi$
$137$	14.8617	1.26972	0.634861	+	0.772626i	$0.281057\pi$
$138$	0	0
$139$	2.89568	0.245608	0.122804	−	0.992431i	$-0.460811\pi$
$139$	2.89568	0.245608	0.122804	+	0.992431i	$0.460811\pi$
$140$	0	0
$141$	3.15263	0.265500
$142$	0	0
$143$	−2.95240	−0.246892
$144$	0	0
$145$	0.103654	0.00860802
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.5174	−1.35316	−0.676578	−	0.736371i	$-0.736538\pi$
$149$	−16.5174	−1.35316	−0.676578	+	0.736371i	$0.736538\pi$
$150$	0	0
$151$	−8.66936	−0.705503	−0.352751	−	0.935717i	$-0.614754\pi$
$151$	−8.66936	−0.705503	−0.352751	+	0.935717i	$0.614754\pi$
$152$	0	0
$153$	8.42832	0.681389
$154$	0	0
$155$	−0.952377	−0.0764968
$156$	0	0
$157$	−20.4555	−1.63253	−0.816263	−	0.577681i	$-0.803958\pi$
$157$	−20.4555	−1.63253	−0.816263	+	0.577681i	$0.803958\pi$
$158$	0	0
$159$	−1.53643	−0.121847
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.5478	1.21780	0.608898	−	0.793249i	$-0.291612\pi$
$163$	15.5478	1.21780	0.608898	+	0.793249i	$0.291612\pi$
$164$	0	0
$165$	−1.38976	−0.108193
$166$	0	0
$167$	13.6387	1.05539	0.527695	−	0.849434i	$-0.323056\pi$
$167$	13.6387	1.05539	0.527695	+	0.849434i	$0.323056\pi$
$168$	0	0
$169$	−7.72123	−0.593941
$170$	0	0
$171$	−5.59242	−0.427663
$172$	0	0
$173$	−3.02896	−0.230288	−0.115144	−	0.993349i	$-0.536733\pi$
$173$	−3.02896	−0.230288	−0.115144	+	0.993349i	$0.536733\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.59995	−0.345753
$178$	0	0
$179$	14.8778	1.11202	0.556010	−	0.831175i	$-0.312332\pi$
$179$	14.8778	1.11202	0.556010	+	0.831175i	$0.312332\pi$
$180$	0	0
$181$	−18.7524	−1.39386	−0.696928	−	0.717141i	$-0.745450\pi$
$181$	−18.7524	−1.39386	−0.696928	+	0.717141i	$0.745450\pi$
$182$	0	0
$183$	5.66984	0.419127
$184$	0	0
$185$	−7.28370	−0.535508
$186$	0	0
$187$	5.71406	0.417854
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.07373	−0.367122	−0.183561	−	0.983008i	$-0.558762\pi$
$191$	−5.07373	−0.367122	−0.183561	+	0.983008i	$0.558762\pi$
$192$	0	0
$193$	−2.47971	−0.178494	−0.0892468	−	0.996010i	$-0.528446\pi$
$193$	−2.47971	−0.178494	−0.0892468	+	0.996010i	$0.528446\pi$
$194$	0	0
$195$	2.48483	0.177942
$196$	0	0
$197$	−8.51382	−0.606584	−0.303292	−	0.952898i	$-0.598086\pi$
$197$	−8.51382	−0.606584	−0.303292	+	0.952898i	$0.598086\pi$
$198$	0	0
$199$	−18.8343	−1.33513	−0.667563	−	0.744554i	$-0.732662\pi$
$199$	−18.8343	−1.33513	−0.667563	+	0.744554i	$0.732662\pi$
$200$	0	0
$201$	−5.45461	−0.384738
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.02904	−0.0718712
$206$	0	0
$207$	−9.12365	−0.634137
$208$	0	0
$209$	−3.79144	−0.262259
$210$	0	0
$211$	11.1392	0.766854	0.383427	−	0.923571i	$-0.374744\pi$
$211$	11.1392	0.766854	0.383427	+	0.923571i	$0.374744\pi$
$212$	0	0
$213$	−15.0178	−1.02900
$214$	0	0
$215$	−5.63291	−0.384162
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.3964	0.702525
$220$	0	0
$221$	−10.2165	−0.687235
$222$	0	0
$223$	−14.7176	−0.985565	−0.492782	−	0.870153i	$-0.664020\pi$
$223$	−14.7176	−0.985565	−0.492782	+	0.870153i	$0.664020\pi$
$224$	0	0
$225$	7.47000	0.498000
$226$	0	0
$227$	−14.2238	−0.944064	−0.472032	−	0.881581i	$-0.656479\pi$
$227$	−14.2238	−0.944064	−0.472032	+	0.881581i	$0.656479\pi$
$228$	0	0
$229$	−8.34259	−0.551294	−0.275647	−	0.961259i	$-0.588892\pi$
$229$	−8.34259	−0.551294	−0.275647	+	0.961259i	$0.588892\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.97617	0.194976	0.0974878	−	0.995237i	$-0.468919\pi$
$233$	2.97617	0.194976	0.0974878	+	0.995237i	$0.468919\pi$
$234$	0	0
$235$	3.08679	0.201360
$236$	0	0
$237$	−7.77730	−0.505190
$238$	0	0
$239$	5.63303	0.364370	0.182185	−	0.983264i	$-0.441683\pi$
$239$	5.63303	0.364370	0.182185	+	0.983264i	$0.441683\pi$
$240$	0	0
$241$	−7.29302	−0.469784	−0.234892	−	0.972021i	$-0.575474\pi$
$241$	−7.29302	−0.469784	−0.234892	+	0.972021i	$0.575474\pi$
$242$	0	0
$243$	15.7282	1.00897
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.77892	0.431332
$248$	0	0
$249$	−16.9582	−1.07468
$250$	0	0
$251$	−10.7087	−0.675926	−0.337963	−	0.941159i	$-0.609738\pi$
$251$	−10.7087	−0.675926	−0.337963	+	0.941159i	$0.609738\pi$
$252$	0	0
$253$	−6.18546	−0.388877
$254$	0	0
$255$	−4.80912	−0.301159
$256$	0	0
$257$	−6.63688	−0.413997	−0.206999	−	0.978341i	$-0.566370\pi$
$257$	−6.63688	−0.413997	−0.206999	+	0.978341i	$0.566370\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.190924	−0.0118179
$262$	0	0
$263$	−19.9365	−1.22934	−0.614669	−	0.788785i	$-0.710710\pi$
$263$	−19.9365	−1.22934	−0.614669	+	0.788785i	$0.710710\pi$
$264$	0	0
$265$	−1.50434	−0.0924111
$266$	0	0
$267$	16.5754	1.01440
$268$	0	0
$269$	−5.10946	−0.311529	−0.155765	−	0.987794i	$-0.549784\pi$
$269$	−5.10946	−0.311529	−0.155765	+	0.987794i	$0.549784\pi$
$270$	0	0
$271$	17.5747	1.06759	0.533793	−	0.845615i	$-0.320766\pi$
$271$	17.5747	1.06759	0.533793	+	0.845615i	$0.320766\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.06436	0.305393
$276$	0	0
$277$	18.5799	1.11636	0.558178	−	0.829721i	$-0.311501\pi$
$277$	18.5799	1.11636	0.558178	+	0.829721i	$0.311501\pi$
$278$	0	0
$279$	1.75422	0.105022
$280$	0	0
$281$	−11.5655	−0.689942	−0.344971	−	0.938613i	$-0.612111\pi$
$281$	−11.5655	−0.689942	−0.344971	+	0.938613i	$0.612111\pi$
$282$	0	0
$283$	12.6487	0.751887	0.375944	−	0.926643i	$-0.377319\pi$
$283$	12.6487	0.751887	0.375944	+	0.926643i	$0.377319\pi$
$284$	0	0
$285$	3.19098	0.189018
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.77292	0.163113
$290$	0	0
$291$	−3.37056	−0.197586
$292$	0	0
$293$	−22.3710	−1.30693	−0.653463	−	0.756958i	$-0.726685\pi$
$293$	−22.3710	−1.30693	−0.653463	+	0.756958i	$0.726685\pi$
$294$	0	0
$295$	−4.50388	−0.262226
$296$	0	0
$297$	6.61147	0.383636
$298$	0	0
$299$	11.0593	0.639578
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−13.2950	−0.763779
$304$	0	0
$305$	5.55142	0.317874
$306$	0	0
$307$	−8.60090	−0.490879	−0.245440	−	0.969412i	$-0.578932\pi$
$307$	−8.60090	−0.490879	−0.245440	+	0.969412i	$0.578932\pi$
$308$	0	0
$309$	−5.18743	−0.295103
$310$	0	0
$311$	16.7875	0.951933	0.475966	−	0.879464i	$-0.342098\pi$
$311$	16.7875	0.951933	0.475966	+	0.879464i	$0.342098\pi$
$312$	0	0
$313$	3.21817	0.181902	0.0909509	−	0.995855i	$-0.471009\pi$
$313$	3.21817	0.181902	0.0909509	+	0.995855i	$0.471009\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.9450	1.51338	0.756692	−	0.653772i	$-0.226814\pi$
$317$	26.9450	1.51338	0.756692	+	0.653772i	$0.226814\pi$
$318$	0	0
$319$	−0.129439	−0.00724719
$320$	0	0
$321$	2.53801	0.141658
$322$	0	0
$323$	−13.1199	−0.730010
$324$	0	0
$325$	−9.05485	−0.502273
$326$	0	0
$327$	17.1935	0.950802
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.99110	−0.439230	−0.219615	−	0.975587i	$-0.570480\pi$
$331$	−7.99110	−0.439230	−0.219615	+	0.975587i	$0.570480\pi$
$332$	0	0
$333$	13.4161	0.735197
$334$	0	0
$335$	−5.34069	−0.291793
$336$	0	0
$337$	−14.1652	−0.771628	−0.385814	−	0.922577i	$-0.626079\pi$
$337$	−14.1652	−0.771628	−0.385814	+	0.922577i	$0.626079\pi$
$338$	0	0
$339$	8.53147	0.463366
$340$	0	0
$341$	1.18929	0.0644035
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.20586	0.280274
$346$	0	0
$347$	10.1687	0.545885	0.272942	−	0.962030i	$-0.412003\pi$
$347$	10.1687	0.545885	0.272942	+	0.962030i	$0.412003\pi$
$348$	0	0
$349$	−2.71487	−0.145324	−0.0726618	−	0.997357i	$-0.523149\pi$
$349$	−2.71487	−0.145324	−0.0726618	+	0.997357i	$0.523149\pi$
$350$	0	0
$351$	−11.8210	−0.630959
$352$	0	0
$353$	−25.2164	−1.34213	−0.671066	−	0.741398i	$-0.734163\pi$
$353$	−25.2164	−1.34213	−0.671066	+	0.741398i	$0.734163\pi$
$354$	0	0
$355$	−14.7041	−0.780413
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.9355	1.36882	0.684412	−	0.729095i	$-0.260059\pi$
$359$	25.9355	1.36882	0.684412	+	0.729095i	$0.260059\pi$
$360$	0	0
$361$	−10.2946	−0.541821
$362$	0	0
$363$	−9.82541	−0.515700
$364$	0	0
$365$	10.1793	0.532808
$366$	0	0
$367$	−11.6145	−0.606274	−0.303137	−	0.952947i	$-0.598034\pi$
$367$	−11.6145	−0.606274	−0.303137	+	0.952947i	$0.598034\pi$
$368$	0	0
$369$	1.89542	0.0986717
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.376334	−0.0194858	−0.00974291	−	0.999953i	$-0.503101\pi$
$373$	−0.376334	−0.0194858	−0.00974291	+	0.999953i	$0.503101\pi$
$374$	0	0
$375$	−9.66986	−0.499349
$376$	0	0
$377$	0.231431	0.0119193
$378$	0	0
$379$	−5.39565	−0.277156	−0.138578	−	0.990352i	$-0.544253\pi$
$379$	−5.39565	−0.277156	−0.138578	+	0.990352i	$0.544253\pi$
$380$	0	0
$381$	−9.91838	−0.508134
$382$	0	0
$383$	29.9041	1.52803	0.764014	−	0.645200i	$-0.223226\pi$
$383$	29.9041	1.52803	0.764014	+	0.645200i	$0.223226\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.3755	0.527414
$388$	0	0
$389$	10.2180	0.518071	0.259036	−	0.965868i	$-0.416595\pi$
$389$	10.2180	0.518071	0.259036	+	0.965868i	$0.416595\pi$
$390$	0	0
$391$	−21.4042	−1.08245
$392$	0	0
$393$	0.202813	0.0102306
$394$	0	0
$395$	−7.61486	−0.383145
$396$	0	0
$397$	1.21654	0.0610566	0.0305283	−	0.999534i	$-0.490281\pi$
$397$	1.21654	0.0610566	0.0305283	+	0.999534i	$0.490281\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.5983	1.27832	0.639159	−	0.769075i	$-0.279283\pi$
$401$	25.5983	1.27832	0.639159	+	0.769075i	$0.279283\pi$
$402$	0	0
$403$	−2.12639	−0.105923
$404$	0	0
$405$	0.286983	0.0142603
$406$	0	0
$407$	9.09556	0.450850
$408$	0	0
$409$	−38.4482	−1.90114	−0.950570	−	0.310510i	$-0.899500\pi$
$409$	−38.4482	−1.90114	−0.950570	+	0.310510i	$0.899500\pi$
$410$	0	0
$411$	15.6195	0.770454
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.6040	−0.815059
$416$	0	0
$417$	3.04333	0.149032
$418$	0	0
$419$	6.49022	0.317068	0.158534	−	0.987354i	$-0.449323\pi$
$419$	6.49022	0.317068	0.158534	+	0.987354i	$0.449323\pi$
$420$	0	0
$421$	11.3180	0.551605	0.275803	−	0.961214i	$-0.411056\pi$
$421$	11.3180	0.551605	0.275803	+	0.961214i	$0.411056\pi$
$422$	0	0
$423$	−5.68566	−0.276446
$424$	0	0
$425$	17.5247	0.850073
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.10295	−0.149812
$430$	0	0
$431$	−23.3926	−1.12678	−0.563391	−	0.826191i	$-0.690503\pi$
$431$	−23.3926	−1.12678	−0.563391	+	0.826191i	$0.690503\pi$
$432$	0	0
$433$	−2.82451	−0.135737	−0.0678687	−	0.997694i	$-0.521620\pi$
$433$	−2.82451	−0.135737	−0.0678687	+	0.997694i	$0.521620\pi$
$434$	0	0
$435$	0.108940	0.00522325
$436$	0	0
$437$	14.2022	0.679386
$438$	0	0
$439$	−9.09712	−0.434182	−0.217091	−	0.976151i	$-0.569657\pi$
$439$	−9.09712	−0.434182	−0.217091	+	0.976151i	$0.569657\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.12057	0.195774	0.0978870	−	0.995198i	$-0.468792\pi$
$443$	4.12057	0.195774	0.0978870	+	0.995198i	$0.468792\pi$
$444$	0	0
$445$	16.2292	0.769339
$446$	0	0
$447$	−17.3596	−0.821080
$448$	0	0
$449$	9.26797	0.437382	0.218691	−	0.975794i	$-0.429821\pi$
$449$	9.26797	0.437382	0.218691	+	0.975794i	$0.429821\pi$
$450$	0	0
$451$	1.28502	0.0605092
$452$	0	0
$453$	−9.11141	−0.428091
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.1586	−1.55109	−0.775547	−	0.631290i	$-0.782526\pi$
$457$	−33.1586	−1.55109	−0.775547	+	0.631290i	$0.782526\pi$
$458$	0	0
$459$	22.8783	1.06787
$460$	0	0
$461$	9.75466	0.454320	0.227160	−	0.973857i	$-0.427056\pi$
$461$	9.75466	0.454320	0.227160	+	0.973857i	$0.427056\pi$
$462$	0	0
$463$	32.6665	1.51814	0.759069	−	0.651010i	$-0.225654\pi$
$463$	32.6665	1.51814	0.759069	+	0.651010i	$0.225654\pi$
$464$	0	0
$465$	−1.00094	−0.0464174
$466$	0	0
$467$	34.7311	1.60716	0.803582	−	0.595194i	$-0.202925\pi$
$467$	34.7311	1.60716	0.803582	+	0.595194i	$0.202925\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−21.4985	−0.990599
$472$	0	0
$473$	7.03414	0.323430
$474$	0	0
$475$	−11.6281	−0.533535
$476$	0	0
$477$	2.77090	0.126871
$478$	0	0
$479$	−23.9080	−1.09239	−0.546193	−	0.837659i	$-0.683924\pi$
$479$	−23.9080	−1.09239	−0.546193	+	0.837659i	$0.683924\pi$
$480$	0	0
$481$	−16.2625	−0.741504
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.30016	−0.149853
$486$	0	0
$487$	21.4037	0.969893	0.484947	−	0.874544i	$-0.338839\pi$
$487$	21.4037	0.969893	0.484947	+	0.874544i	$0.338839\pi$
$488$	0	0
$489$	16.3405	0.738945
$490$	0	0
$491$	−15.8929	−0.717235	−0.358618	−	0.933485i	$-0.616752\pi$
$491$	−15.8929	−0.717235	−0.358618	+	0.933485i	$0.616752\pi$
$492$	0	0
$493$	−0.447910	−0.0201729
$494$	0	0
$495$	2.50638	0.112653
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.7109	1.28527	0.642637	−	0.766170i	$-0.277840\pi$
$499$	28.7109	1.28527	0.642637	+	0.766170i	$0.277840\pi$
$500$	0	0
$501$	14.3341	0.640400
$502$	0	0
$503$	3.02608	0.134926	0.0674631	−	0.997722i	$-0.478510\pi$
$503$	3.02608	0.134926	0.0674631	+	0.997722i	$0.478510\pi$
$504$	0	0
$505$	−13.0174	−0.579265
$506$	0	0
$507$	−8.11494	−0.360397
$508$	0	0
$509$	−0.818528	−0.0362806	−0.0181403	−	0.999835i	$-0.505775\pi$
$509$	−0.818528	−0.0362806	−0.0181403	+	0.999835i	$0.505775\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.1804	−0.670230
$514$	0	0
$515$	−5.07909	−0.223812
$516$	0	0
$517$	−3.85465	−0.169527
$518$	0	0
$519$	−3.18341	−0.139736
$520$	0	0
$521$	12.3960	0.543078	0.271539	−	0.962427i	$-0.412467\pi$
$521$	12.3960	0.543078	0.271539	+	0.962427i	$0.412467\pi$
$522$	0	0
$523$	−24.5318	−1.07270	−0.536351	−	0.843995i	$-0.680198\pi$
$523$	−24.5318	−1.07270	−0.536351	+	0.843995i	$0.680198\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.11541	0.179270
$528$	0	0
$529$	0.169962	0.00738967
$530$	0	0
$531$	8.29584	0.360009
$532$	0	0
$533$	−2.29756	−0.0995182
$534$	0	0
$535$	2.48500	0.107436
$536$	0	0
$537$	15.6364	0.674762
$538$	0	0
$539$	0	0
$540$	0	0
$541$	39.6480	1.70460	0.852301	−	0.523051i	$-0.175206\pi$
$541$	39.6480	1.70460	0.852301	+	0.523051i	$0.175206\pi$
$542$	0	0
$543$	−19.7086	−0.845777
$544$	0	0
$545$	16.8344	0.721106
$546$	0	0
$547$	19.1807	0.820107	0.410053	−	0.912062i	$-0.365510\pi$
$547$	19.1807	0.820107	0.410053	+	0.912062i	$0.365510\pi$
$548$	0	0
$549$	−10.2253	−0.436407
$550$	0	0
$551$	0.297201	0.0126612
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.65509	−0.324940
$556$	0	0
$557$	−4.90164	−0.207689	−0.103845	−	0.994594i	$-0.533114\pi$
$557$	−4.90164	−0.207689	−0.103845	+	0.994594i	$0.533114\pi$
$558$	0	0
$559$	−12.5767	−0.531939
$560$	0	0
$561$	6.00542	0.253549
$562$	0	0
$563$	0.494864	0.0208560	0.0104280	−	0.999946i	$-0.496681\pi$
$563$	0.494864	0.0208560	0.0104280	+	0.999946i	$0.496681\pi$
$564$	0	0
$565$	8.35329	0.351425
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.10516	−0.130175	−0.0650875	−	0.997880i	$-0.520733\pi$
$569$	−3.10516	−0.130175	−0.0650875	+	0.997880i	$0.520733\pi$
$570$	0	0
$571$	−24.7565	−1.03603	−0.518013	−	0.855373i	$-0.673328\pi$
$571$	−24.7565	−1.03603	−0.518013	+	0.855373i	$0.673328\pi$
$572$	0	0
$573$	−5.33243	−0.222766
$574$	0	0
$575$	−18.9705	−0.791123
$576$	0	0
$577$	6.90679	0.287534	0.143767	−	0.989612i	$-0.454078\pi$
$577$	6.90679	0.287534	0.143767	+	0.989612i	$0.454078\pi$
$578$	0	0
$579$	−2.60615	−0.108308
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.87856	0.0778020
$584$	0	0
$585$	−4.48130	−0.185279
$586$	0	0
$587$	−2.53709	−0.104717	−0.0523585	−	0.998628i	$-0.516674\pi$
$587$	−2.53709	−0.104717	−0.0523585	+	0.998628i	$0.516674\pi$
$588$	0	0
$589$	−2.73069	−0.112516
$590$	0	0
$591$	−8.94793	−0.368069
$592$	0	0
$593$	−36.6188	−1.50375	−0.751877	−	0.659304i	$-0.770851\pi$
$593$	−36.6188	−1.50375	−0.751877	+	0.659304i	$0.770851\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.7946	−0.810139
$598$	0	0
$599$	16.8939	0.690264	0.345132	−	0.938554i	$-0.387834\pi$
$599$	16.8939	0.690264	0.345132	+	0.938554i	$0.387834\pi$
$600$	0	0
$601$	−33.0537	−1.34829	−0.674145	−	0.738599i	$-0.735488\pi$
$601$	−33.0537	−1.34829	−0.674145	+	0.738599i	$0.735488\pi$
$602$	0	0
$603$	9.83719	0.400601
$604$	0	0
$605$	−9.62020	−0.391117
$606$	0	0
$607$	32.3474	1.31294	0.656469	−	0.754353i	$-0.272049\pi$
$607$	32.3474	1.31294	0.656469	+	0.754353i	$0.272049\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.89194	0.278818
$612$	0	0
$613$	−17.4200	−0.703587	−0.351794	−	0.936078i	$-0.614428\pi$
$613$	−17.4200	−0.703587	−0.351794	+	0.936078i	$0.614428\pi$
$614$	0	0
$615$	−1.08151	−0.0436107
$616$	0	0
$617$	30.1475	1.21369	0.606846	−	0.794819i	$-0.292434\pi$
$617$	30.1475	1.21369	0.606846	+	0.794819i	$0.292434\pi$
$618$	0	0
$619$	35.3433	1.42057	0.710283	−	0.703916i	$-0.248567\pi$
$619$	35.3433	1.42057	0.710283	+	0.703916i	$0.248567\pi$
$620$	0	0
$621$	−24.7657	−0.993814
$622$	0	0
$623$	0	0
$624$	0	0
$625$	10.2375	0.409500
$626$	0	0
$627$	−3.98476	−0.159136
$628$	0	0
$629$	31.4743	1.25496
$630$	0	0
$631$	23.1762	0.922630	0.461315	−	0.887236i	$-0.347378\pi$
$631$	23.1762	0.922630	0.461315	+	0.887236i	$0.347378\pi$
$632$	0	0
$633$	11.7072	0.465319
$634$	0	0
$635$	−9.71123	−0.385378
$636$	0	0
$637$	0	0
$638$	0	0
$639$	27.0840	1.07143
$640$	0	0
$641$	−36.8733	−1.45641	−0.728204	−	0.685361i	$-0.759644\pi$
$641$	−36.8733	−1.45641	−0.728204	+	0.685361i	$0.759644\pi$
$642$	0	0
$643$	−48.0696	−1.89568	−0.947841	−	0.318744i	$-0.896739\pi$
$643$	−48.0696	−1.89568	−0.947841	+	0.318744i	$0.896739\pi$
$644$	0	0
$645$	−5.92013	−0.233105
$646$	0	0
$647$	−22.5246	−0.885532	−0.442766	−	0.896637i	$-0.646003\pi$
$647$	−22.5246	−0.885532	−0.442766	+	0.896637i	$0.646003\pi$
$648$	0	0
$649$	5.62425	0.220771
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29.3035	−1.14674	−0.573368	−	0.819298i	$-0.694363\pi$
$653$	−29.3035	−1.14674	−0.573368	+	0.819298i	$0.694363\pi$
$654$	0	0
$655$	0.198577	0.00775906
$656$	0	0
$657$	−18.7496	−0.731490
$658$	0	0
$659$	25.4909	0.992983	0.496491	−	0.868042i	$-0.334621\pi$
$659$	25.4909	0.992983	0.496491	+	0.868042i	$0.334621\pi$
$660$	0	0
$661$	−42.6069	−1.65722	−0.828608	−	0.559830i	$-0.810866\pi$
$661$	−42.6069	−1.65722	−0.828608	+	0.559830i	$0.810866\pi$
$662$	0	0
$663$	−10.7374	−0.417007
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.484862	0.0187739
$668$	0	0
$669$	−15.4681	−0.598030
$670$	0	0
$671$	−6.93237	−0.267621
$672$	0	0
$673$	20.2833	0.781864	0.390932	−	0.920420i	$-0.372153\pi$
$673$	20.2833	0.781864	0.390932	+	0.920420i	$0.372153\pi$
$674$	0	0
$675$	20.2770	0.780462
$676$	0	0
$677$	12.9561	0.497944	0.248972	−	0.968511i	$-0.419907\pi$
$677$	12.9561	0.497944	0.248972	+	0.968511i	$0.419907\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.9490	−0.572848
$682$	0	0
$683$	−18.5814	−0.710997	−0.355499	−	0.934677i	$-0.615689\pi$
$683$	−18.5814	−0.710997	−0.355499	+	0.934677i	$0.615689\pi$
$684$	0	0
$685$	15.2933	0.584326
$686$	0	0
$687$	−8.76798	−0.334519
$688$	0	0
$689$	−3.35878	−0.127959
$690$	0	0
$691$	16.2571	0.618449	0.309225	−	0.950989i	$-0.399931\pi$
$691$	16.2571	0.618449	0.309225	+	0.950989i	$0.399931\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.97977	0.113029
$696$	0	0
$697$	4.44667	0.168430
$698$	0	0
$699$	3.12793	0.118309
$700$	0	0
$701$	14.4286	0.544961	0.272480	−	0.962161i	$-0.412156\pi$
$701$	14.4286	0.544961	0.272480	+	0.962161i	$0.412156\pi$
$702$	0	0
$703$	−20.8840	−0.787656
$704$	0	0
$705$	3.24418	0.122183
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.2187	1.05978	0.529889	−	0.848067i	$-0.322234\pi$
$709$	28.2187	1.05978	0.529889	+	0.848067i	$0.322234\pi$
$710$	0	0
$711$	14.0261	0.526019
$712$	0	0
$713$	−4.45492	−0.166838
$714$	0	0
$715$	−3.03814	−0.113620
$716$	0	0
$717$	5.92025	0.221096
$718$	0	0
$719$	49.0294	1.82849	0.914243	−	0.405165i	$-0.132786\pi$
$719$	49.0294	1.82849	0.914243	+	0.405165i	$0.132786\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.66488	−0.285060
$724$	0	0
$725$	−0.396982	−0.0147435
$726$	0	0
$727$	−24.0066	−0.890355	−0.445178	−	0.895442i	$-0.646860\pi$
$727$	−24.0066	−0.890355	−0.445178	+	0.895442i	$0.646860\pi$
$728$	0	0
$729$	15.6935	0.581242
$730$	0	0
$731$	24.3409	0.900281
$732$	0	0
$733$	−27.9928	−1.03394	−0.516969	−	0.856004i	$-0.672940\pi$
$733$	−27.9928	−1.03394	−0.516969	+	0.856004i	$0.672940\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.66922	0.245664
$738$	0	0
$739$	18.3985	0.676801	0.338401	−	0.941002i	$-0.390114\pi$
$739$	18.3985	0.676801	0.338401	+	0.941002i	$0.390114\pi$
$740$	0	0
$741$	7.12458	0.261728
$742$	0	0
$743$	7.49198	0.274854	0.137427	−	0.990512i	$-0.456117\pi$
$743$	7.49198	0.274854	0.137427	+	0.990512i	$0.456117\pi$
$744$	0	0
$745$	−16.9970	−0.622723
$746$	0	0
$747$	30.5835	1.11899
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.2250	−0.373116	−0.186558	−	0.982444i	$-0.559733\pi$
$751$	−10.2250	−0.373116	−0.186558	+	0.982444i	$0.559733\pi$
$752$	0	0
$753$	−11.2547	−0.410144
$754$	0	0
$755$	−8.92111	−0.324672
$756$	0	0
$757$	13.8559	0.503602	0.251801	−	0.967779i	$-0.418977\pi$
$757$	13.8559	0.503602	0.251801	+	0.967779i	$0.418977\pi$
$758$	0	0
$759$	−6.50086	−0.235966
$760$	0	0
$761$	−24.0153	−0.870554	−0.435277	−	0.900297i	$-0.643350\pi$
$761$	−24.0153	−0.870554	−0.435277	+	0.900297i	$0.643350\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.67307	0.313576
$766$	0	0
$767$	−10.0559	−0.363098
$768$	0	0
$769$	−50.6700	−1.82721	−0.913604	−	0.406606i	$-0.866712\pi$
$769$	−50.6700	−1.82721	−0.913604	+	0.406606i	$0.866712\pi$
$770$	0	0
$771$	−6.97529	−0.251209
$772$	0	0
$773$	16.1315	0.580208	0.290104	−	0.956995i	$-0.406310\pi$
$773$	16.1315	0.580208	0.290104	+	0.956995i	$0.406310\pi$
$774$	0	0
$775$	3.64748	0.131021
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.95049	−0.105712
$780$	0	0
$781$	18.3618	0.657038
$782$	0	0
$783$	−0.518256	−0.0185209
$784$	0	0
$785$	−21.0495	−0.751289
$786$	0	0
$787$	19.9563	0.711366	0.355683	−	0.934607i	$-0.384248\pi$
$787$	19.9563	0.711366	0.355683	+	0.934607i	$0.384248\pi$
$788$	0	0
$789$	−20.9531	−0.745949
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.3948	0.440151
$794$	0	0
$795$	−1.58105	−0.0560740
$796$	0	0
$797$	−45.4738	−1.61077	−0.805383	−	0.592755i	$-0.798040\pi$
$797$	−45.4738	−1.61077	−0.805383	+	0.592755i	$0.798040\pi$
$798$	0	0
$799$	−13.3386	−0.471886
$800$	0	0
$801$	−29.8931	−1.05622
$802$	0	0
$803$	−12.7114	−0.448577
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.36999	−0.189033
$808$	0	0
$809$	18.5341	0.651623	0.325812	−	0.945435i	$-0.394363\pi$
$809$	18.5341	0.651623	0.325812	+	0.945435i	$0.394363\pi$
$810$	0	0
$811$	45.3402	1.59211	0.796055	−	0.605224i	$-0.206916\pi$
$811$	45.3402	1.59211	0.796055	+	0.605224i	$0.206916\pi$
$812$	0	0
$813$	18.4708	0.647800
$814$	0	0
$815$	15.9993	0.560430
$816$	0	0
$817$	−16.1509	−0.565047
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.18741	−0.0414410	−0.0207205	−	0.999785i	$-0.506596\pi$
$821$	−1.18741	−0.0414410	−0.0207205	+	0.999785i	$0.506596\pi$
$822$	0	0
$823$	−3.43355	−0.119686	−0.0598431	−	0.998208i	$-0.519060\pi$
$823$	−3.43355	−0.119686	−0.0598431	+	0.998208i	$0.519060\pi$
$824$	0	0
$825$	5.32259	0.185309
$826$	0	0
$827$	−9.25662	−0.321884	−0.160942	−	0.986964i	$-0.551453\pi$
$827$	−9.25662	−0.321884	−0.160942	+	0.986964i	$0.551453\pi$
$828$	0	0
$829$	3.46216	0.120246	0.0601229	−	0.998191i	$-0.480851\pi$
$829$	3.46216	0.120246	0.0601229	+	0.998191i	$0.480851\pi$
$830$	0	0
$831$	19.5272	0.677393
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14.0347	0.485691
$836$	0	0
$837$	4.76174	0.164590
$838$	0	0
$839$	29.6017	1.02196	0.510982	−	0.859591i	$-0.329282\pi$
$839$	29.6017	1.02196	0.510982	+	0.859591i	$0.329282\pi$
$840$	0	0
$841$	−28.9899	−0.999650
$842$	0	0
$843$	−12.1553	−0.418649
$844$	0	0
$845$	−7.94545	−0.273332
$846$	0	0
$847$	0	0
$848$	0	0
$849$	13.2937	0.456237
$850$	0	0
$851$	−34.0708	−1.16793
$852$	0	0
$853$	57.9438	1.98396	0.991978	−	0.126409i	$-0.0403452\pi$
$853$	57.9438	1.98396	0.991978	+	0.126409i	$0.0403452\pi$
$854$	0	0
$855$	−5.75482	−0.196811
$856$	0	0
$857$	−9.91912	−0.338831	−0.169415	−	0.985545i	$-0.554188\pi$
$857$	−9.91912	−0.338831	−0.169415	+	0.985545i	$0.554188\pi$
$858$	0	0
$859$	−47.6228	−1.62487	−0.812435	−	0.583052i	$-0.801858\pi$
$859$	−47.6228	−1.62487	−0.812435	+	0.583052i	$0.801858\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.0647289	0.00220340	0.00110170	−	0.999999i	$-0.499649\pi$
$863$	0.0647289	0.00220340	0.00110170	+	0.999999i	$0.499649\pi$
$864$	0	0
$865$	−3.11692	−0.105978
$866$	0	0
$867$	2.91431	0.0989751
$868$	0	0
$869$	9.50911	0.322574
$870$	0	0
$871$	−11.9243	−0.404038
$872$	0	0
$873$	6.07868	0.205732
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.5643	0.424266	0.212133	−	0.977241i	$-0.431959\pi$
$877$	12.5643	0.424266	0.212133	+	0.977241i	$0.431959\pi$
$878$	0	0
$879$	−23.5117	−0.793029
$880$	0	0
$881$	42.1898	1.42141	0.710705	−	0.703490i	$-0.248376\pi$
$881$	42.1898	1.42141	0.710705	+	0.703490i	$0.248376\pi$
$882$	0	0
$883$	8.61356	0.289869	0.144935	−	0.989441i	$-0.453703\pi$
$883$	8.61356	0.289869	0.144935	+	0.989441i	$0.453703\pi$
$884$	0	0
$885$	−4.73353	−0.159116
$886$	0	0
$887$	5.96189	0.200181	0.100090	−	0.994978i	$-0.468087\pi$
$887$	5.96189	0.200181	0.100090	+	0.994978i	$0.468087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.358372	−0.0120059
$892$	0	0
$893$	8.85054	0.296172
$894$	0	0
$895$	15.3099	0.511752
$896$	0	0
$897$	11.6232	0.388089
$898$	0	0
$899$	−0.0932251	−0.00310923
$900$	0	0
$901$	6.50056	0.216565
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.2970	−0.641453
$906$	0	0
$907$	−15.7086	−0.521595	−0.260797	−	0.965394i	$-0.583985\pi$
$907$	−15.7086	−0.521595	−0.260797	+	0.965394i	$0.583985\pi$
$908$	0	0
$909$	23.9771	0.795270
$910$	0	0
$911$	−45.8006	−1.51744	−0.758720	−	0.651417i	$-0.774175\pi$
$911$	−45.8006	−1.51744	−0.758720	+	0.651417i	$0.774175\pi$
$912$	0	0
$913$	20.7344	0.686208
$914$	0	0
$915$	5.83449	0.192882
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−45.2558	−1.49285	−0.746426	−	0.665469i	$-0.768232\pi$
$919$	−45.2558	−1.49285	−0.746426	+	0.665469i	$0.768232\pi$
$920$	0	0
$921$	−9.03946	−0.297860
$922$	0	0
$923$	−32.8301	−1.08062
$924$	0	0
$925$	27.8956	0.917201
$926$	0	0
$927$	9.35535	0.307270
$928$	0	0
$929$	−23.9841	−0.786892	−0.393446	−	0.919348i	$-0.628717\pi$
$929$	−23.9841	−0.786892	−0.393446	+	0.919348i	$0.628717\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	17.6435	0.577622
$934$	0	0
$935$	5.87999	0.192296
$936$	0	0
$937$	18.6293	0.608594	0.304297	−	0.952577i	$-0.401578\pi$
$937$	18.6293	0.608594	0.304297	+	0.952577i	$0.401578\pi$
$938$	0	0
$939$	3.38226	0.110376
$940$	0	0
$941$	9.62697	0.313830	0.156915	−	0.987612i	$-0.449845\pi$
$941$	9.62697	0.313830	0.156915	+	0.987612i	$0.449845\pi$
$942$	0	0
$943$	−4.81352	−0.156750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.0283	−0.813310	−0.406655	−	0.913582i	$-0.633305\pi$
$947$	−25.0283	−0.813310	−0.406655	+	0.913582i	$0.633305\pi$
$948$	0	0
$949$	22.7275	0.737765
$950$	0	0
$951$	28.3189	0.918305
$952$	0	0
$953$	30.8050	0.997873	0.498936	−	0.866639i	$-0.333724\pi$
$953$	30.8050	0.997873	0.498936	+	0.866639i	$0.333724\pi$
$954$	0	0
$955$	−5.22106	−0.168950
$956$	0	0
$957$	−0.136039	−0.00439752
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.1434	−0.972369
$962$	0	0
$963$	−4.57720	−0.147498
$964$	0	0
$965$	−2.55172	−0.0821428
$966$	0	0
$967$	−26.7377	−0.859827	−0.429913	−	0.902870i	$-0.641456\pi$
$967$	−26.7377	−0.859827	−0.429913	+	0.902870i	$0.641456\pi$
$968$	0	0
$969$	−13.7889	−0.442962
$970$	0	0
$971$	−2.65019	−0.0850486	−0.0425243	−	0.999095i	$-0.513540\pi$
$971$	−2.65019	−0.0850486	−0.0425243	+	0.999095i	$0.513540\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−9.51655	−0.304774
$976$	0	0
$977$	−2.38400	−0.0762710	−0.0381355	−	0.999273i	$-0.512142\pi$
$977$	−2.38400	−0.0762710	−0.0381355	+	0.999273i	$0.512142\pi$
$978$	0	0
$979$	−20.2663	−0.647715
$980$	0	0
$981$	−31.0078	−0.990004
$982$	0	0
$983$	−27.5555	−0.878882	−0.439441	−	0.898271i	$-0.644824\pi$
$983$	−27.5555	−0.878882	−0.439441	+	0.898271i	$0.644824\pi$
$984$	0	0
$985$	−8.76105	−0.279150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.3490	−0.837849
$990$	0	0
$991$	17.8527	0.567109	0.283554	−	0.958956i	$-0.408486\pi$
$991$	17.8527	0.567109	0.283554	+	0.958956i	$0.408486\pi$
$992$	0	0
$993$	−8.39856	−0.266520
$994$	0	0
$995$	−19.3812	−0.614425
$996$	0	0
$997$	−16.4836	−0.522042	−0.261021	−	0.965333i	$-0.584059\pi$
$997$	−16.4836	−0.522042	−0.261021	+	0.965333i	$0.584059\pi$
$998$	0	0
$999$	36.4174	1.15219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.o.1.7	✓	10
7.6	odd	2		8036.2.a.p.1.4	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.7	✓	10	1.1	even	1	trivial
8036.2.a.p.1.4	yes	10	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)

\(f(q)\)	\(=\)	\(q+1.05099 q^{3} +1.02904 q^{5} -1.89542 q^{9} +O(q^{10})\)	\(q+1.05099 q^{3} +1.02904 q^{5} -1.89542 q^{9} -1.28502 q^{11} +2.29756 q^{13} +1.08151 q^{15} -4.44667 q^{17} +2.95049 q^{19} +4.81352 q^{23} -3.94108 q^{25} -5.14504 q^{27} +0.100729 q^{29} -0.925502 q^{31} -1.35054 q^{33} -7.07815 q^{37} +2.41471 q^{39} -1.00000 q^{41} -5.47396 q^{43} -1.95046 q^{45} +2.99968 q^{47} -4.67341 q^{51} -1.46189 q^{53} -1.32233 q^{55} +3.10094 q^{57} -4.37678 q^{59} +5.39476 q^{61} +2.36428 q^{65} -5.18998 q^{67} +5.05896 q^{69} -14.2892 q^{71} +9.89202 q^{73} -4.14203 q^{75} -7.39997 q^{79} +0.278885 q^{81} -16.1355 q^{83} -4.57580 q^{85} +0.105865 q^{87} +15.7712 q^{89} -0.972693 q^{93} +3.03617 q^{95} -3.20703 q^{97} +2.43565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9	\( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9} - 6 q^{13} - 4 q^{15} - 12 q^{17} - 8 q^{19} + 4 q^{23} + 16 q^{25} - 8 q^{27} + 2 q^{29} - 8 q^{31} + 6 q^{33} - 2 q^{37} - 2 q^{39} - 10 q^{41} - 2 q^{43} - 44 q^{45} + 14 q^{47} + 14 q^{51} + 8 q^{53} - 8 q^{55} - 10 q^{57} - 24 q^{59} - 14 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 10 q^{71} - 44 q^{73} + 50 q^{75} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 8 q^{85} - 20 q^{87} - 6 q^{89} + 8 q^{93} + 4 q^{95} - 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9 - 6 * q^13 - 4 * q^15 - 12 * q^17 - 8 * q^19 + 4 * q^23 + 16 * q^25 - 8 * q^27 + 2 * q^29 - 8 * q^31 + 6 * q^33 - 2 * q^37 - 2 * q^39 - 10 * q^41 - 2 * q^43 - 44 * q^45 + 14 * q^47 + 14 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^57 - 24 * q^59 - 14 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 10 * q^71 - 44 * q^73 + 50 * q^75 + 10 * q^79 - 14 * q^81 - 20 * q^83 + 8 * q^85 - 20 * q^87 - 6 * q^89 + 8 * q^93 + 4 * q^95 - 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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