LMFDB - Embedded newform 8036.2.a.k.1.5

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.287349.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 7x - 1 $ x^5 - 6x^3 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.145487$ 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)

$f(q)$	$=$	$q+1.97883 q^{3} +2.39996 q^{5} +0.915782 q^{9} +O(q^{10})$	$q+1.97883 q^{3} +2.39996 q^{5} +0.915782 q^{9} +2.34916 q^{11} +1.41222 q^{13} +4.74913 q^{15} +2.31575 q^{17} +5.93164 q^{19} -6.90130 q^{23} +0.759824 q^{25} -4.12432 q^{27} -1.33108 q^{29} -4.98191 q^{31} +4.64860 q^{33} +9.14326 q^{37} +2.79454 q^{39} +1.00000 q^{41} +9.18251 q^{43} +2.19784 q^{45} -10.6838 q^{47} +4.58247 q^{51} +12.4802 q^{53} +5.63791 q^{55} +11.7377 q^{57} -5.58221 q^{59} +12.8464 q^{61} +3.38927 q^{65} +16.3648 q^{67} -13.6565 q^{69} -13.0124 q^{71} +3.76721 q^{73} +1.50357 q^{75} +13.4804 q^{79} -10.9087 q^{81} +3.21593 q^{83} +5.55770 q^{85} -2.63398 q^{87} +7.17204 q^{89} -9.85838 q^{93} +14.2357 q^{95} +16.6599 q^{97} +2.15132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9	$ 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * 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
$3$	1.97883	1.14248	0.571240	−	0.820783i	$-0.306463\pi$
$3$	1.97883	1.14248	0.571240	+	0.820783i	$0.306463\pi$
$4$	0	0
$5$	2.39996	1.07330	0.536648	−	0.843806i	$-0.319690\pi$
$5$	2.39996	1.07330	0.536648	+	0.843806i	$0.319690\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.915782	0.305261
$10$	0	0
$11$	2.34916	0.708300	0.354150	−	0.935189i	$-0.384770\pi$
$11$	2.34916	0.708300	0.354150	+	0.935189i	$0.384770\pi$
$12$	0	0
$13$	1.41222	0.391678	0.195839	−	0.980636i	$-0.437257\pi$
$13$	1.41222	0.391678	0.195839	+	0.980636i	$0.437257\pi$
$14$	0	0
$15$	4.74913	1.22622
$16$	0	0
$17$	2.31575	0.561651	0.280825	−	0.959759i	$-0.409392\pi$
$17$	2.31575	0.561651	0.280825	+	0.959759i	$0.409392\pi$
$18$	0	0
$19$	5.93164	1.36081	0.680406	−	0.732836i	$-0.261804\pi$
$19$	5.93164	1.36081	0.680406	+	0.732836i	$0.261804\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.90130	−1.43902	−0.719510	−	0.694482i	$-0.755634\pi$
$23$	−6.90130	−1.43902	−0.719510	+	0.694482i	$0.755634\pi$
$24$	0	0
$25$	0.759824	0.151965
$26$	0	0
$27$	−4.12432	−0.793726
$28$	0	0
$29$	−1.33108	−0.247175	−0.123587	−	0.992334i	$-0.539440\pi$
$29$	−1.33108	−0.247175	−0.123587	+	0.992334i	$0.539440\pi$
$30$	0	0
$31$	−4.98191	−0.894778	−0.447389	−	0.894339i	$-0.647646\pi$
$31$	−4.98191	−0.894778	−0.447389	+	0.894339i	$0.647646\pi$
$32$	0	0
$33$	4.64860	0.809218
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.14326	1.50314	0.751571	−	0.659652i	$-0.229296\pi$
$37$	9.14326	1.50314	0.751571	+	0.659652i	$0.229296\pi$
$38$	0	0
$39$	2.79454	0.447485
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	9.18251	1.40032	0.700160	−	0.713986i	$-0.253112\pi$
$43$	9.18251	1.40032	0.700160	+	0.713986i	$0.253112\pi$
$44$	0	0
$45$	2.19784	0.327635
$46$	0	0
$47$	−10.6838	−1.55840	−0.779200	−	0.626776i	$-0.784374\pi$
$47$	−10.6838	−1.55840	−0.779200	+	0.626776i	$0.784374\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.58247	0.641675
$52$	0	0
$53$	12.4802	1.71428	0.857141	−	0.515081i	$-0.172238\pi$
$53$	12.4802	1.71428	0.857141	+	0.515081i	$0.172238\pi$
$54$	0	0
$55$	5.63791	0.760215
$56$	0	0
$57$	11.7377	1.55470
$58$	0	0
$59$	−5.58221	−0.726742	−0.363371	−	0.931645i	$-0.618374\pi$
$59$	−5.58221	−0.726742	−0.363371	+	0.931645i	$0.618374\pi$
$60$	0	0
$61$	12.8464	1.64482	0.822410	−	0.568896i	$-0.192629\pi$
$61$	12.8464	1.64482	0.822410	+	0.568896i	$0.192629\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.38927	0.420387
$66$	0	0
$67$	16.3648	1.99928	0.999642	−	0.0267719i	$-0.00852277\pi$
$67$	16.3648	1.99928	0.999642	+	0.0267719i	$0.00852277\pi$
$68$	0	0
$69$	−13.6565	−1.64405
$70$	0	0
$71$	−13.0124	−1.54428	−0.772142	−	0.635450i	$-0.780815\pi$
$71$	−13.0124	−1.54428	−0.772142	+	0.635450i	$0.780815\pi$
$72$	0	0
$73$	3.76721	0.440919	0.220460	−	0.975396i	$-0.429244\pi$
$73$	3.76721	0.440919	0.220460	+	0.975396i	$0.429244\pi$
$74$	0	0
$75$	1.50357	0.173617
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.4804	1.51667	0.758333	−	0.651867i	$-0.226014\pi$
$79$	13.4804	1.51667	0.758333	+	0.651867i	$0.226014\pi$
$80$	0	0
$81$	−10.9087	−1.21208
$82$	0	0
$83$	3.21593	0.352994	0.176497	−	0.984301i	$-0.443523\pi$
$83$	3.21593	0.352994	0.176497	+	0.984301i	$0.443523\pi$
$84$	0	0
$85$	5.55770	0.602818
$86$	0	0
$87$	−2.63398	−0.282392
$88$	0	0
$89$	7.17204	0.760235	0.380117	−	0.924938i	$-0.375884\pi$
$89$	7.17204	0.760235	0.380117	+	0.924938i	$0.375884\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.85838	−1.02227
$94$	0	0
$95$	14.2357	1.46055
$96$	0	0
$97$	16.6599	1.69155	0.845776	−	0.533537i	$-0.179138\pi$
$97$	16.6599	1.69155	0.845776	+	0.533537i	$0.179138\pi$
$98$	0	0
$99$	2.15132	0.216216
$100$	0	0
$101$	−5.59473	−0.556696	−0.278348	−	0.960480i	$-0.589787\pi$
$101$	−5.59473	−0.556696	−0.278348	+	0.960480i	$0.589787\pi$
$102$	0	0
$103$	−7.97424	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$103$	−7.97424	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.69914	0.454283	0.227141	−	0.973862i	$-0.427062\pi$
$107$	4.69914	0.454283	0.227141	+	0.973862i	$0.427062\pi$
$108$	0	0
$109$	−5.20372	−0.498426	−0.249213	−	0.968449i	$-0.580172\pi$
$109$	−5.20372	−0.498426	−0.249213	+	0.968449i	$0.580172\pi$
$110$	0	0
$111$	18.0930	1.71731
$112$	0	0
$113$	0.739630	0.0695785	0.0347893	−	0.999395i	$-0.488924\pi$
$113$	0.739630	0.0695785	0.0347893	+	0.999395i	$0.488924\pi$
$114$	0	0
$115$	−16.5629	−1.54449
$116$	0	0
$117$	1.29328	0.119564
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.48143	−0.498312
$122$	0	0
$123$	1.97883	0.178425
$124$	0	0
$125$	−10.1763	−0.910193
$126$	0	0
$127$	14.2545	1.26488	0.632440	−	0.774609i	$-0.282053\pi$
$127$	14.2545	1.26488	0.632440	+	0.774609i	$0.282053\pi$
$128$	0	0
$129$	18.1707	1.59984
$130$	0	0
$131$	6.58934	0.575713	0.287857	−	0.957674i	$-0.407057\pi$
$131$	6.58934	0.575713	0.287857	+	0.957674i	$0.407057\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.89822	−0.851903
$136$	0	0
$137$	1.84252	0.157417	0.0787085	−	0.996898i	$-0.474920\pi$
$137$	1.84252	0.157417	0.0787085	+	0.996898i	$0.474920\pi$
$138$	0	0
$139$	−14.2400	−1.20782	−0.603910	−	0.797052i	$-0.706391\pi$
$139$	−14.2400	−1.20782	−0.603910	+	0.797052i	$0.706391\pi$
$140$	0	0
$141$	−21.1416	−1.78044
$142$	0	0
$143$	3.31753	0.277426
$144$	0	0
$145$	−3.19454	−0.265292
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.2625	−1.08650	−0.543251	−	0.839570i	$-0.682807\pi$
$149$	−13.2625	−1.08650	−0.543251	+	0.839570i	$0.682807\pi$
$150$	0	0
$151$	3.96632	0.322774	0.161387	−	0.986891i	$-0.448403\pi$
$151$	3.96632	0.322774	0.161387	+	0.986891i	$0.448403\pi$
$152$	0	0
$153$	2.12072	0.171450
$154$	0	0
$155$	−11.9564	−0.960362
$156$	0	0
$157$	−16.1974	−1.29269	−0.646346	−	0.763044i	$-0.723704\pi$
$157$	−16.1974	−1.29269	−0.646346	+	0.763044i	$0.723704\pi$
$158$	0	0
$159$	24.6962	1.95853
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.70238	−0.211667	−0.105833	−	0.994384i	$-0.533751\pi$
$163$	−2.70238	−0.211667	−0.105833	+	0.994384i	$0.533751\pi$
$164$	0	0
$165$	11.1565	0.868531
$166$	0	0
$167$	−3.00791	−0.232759	−0.116380	−	0.993205i	$-0.537129\pi$
$167$	−3.00791	−0.232759	−0.116380	+	0.993205i	$0.537129\pi$
$168$	0	0
$169$	−11.0056	−0.846588
$170$	0	0
$171$	5.43209	0.415402
$172$	0	0
$173$	2.32822	0.177012	0.0885058	−	0.996076i	$-0.471791\pi$
$173$	2.32822	0.177012	0.0885058	+	0.996076i	$0.471791\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.0463	−0.830288
$178$	0	0
$179$	−25.3001	−1.89102	−0.945511	−	0.325591i	$-0.894437\pi$
$179$	−25.3001	−1.89102	−0.945511	+	0.325591i	$0.894437\pi$
$180$	0	0
$181$	−18.8374	−1.40017	−0.700086	−	0.714058i	$-0.746855\pi$
$181$	−18.8374	−1.40017	−0.700086	+	0.714058i	$0.746855\pi$
$182$	0	0
$183$	25.4210	1.87917
$184$	0	0
$185$	21.9435	1.61332
$186$	0	0
$187$	5.44007	0.397817
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.9291	−0.863157	−0.431578	−	0.902075i	$-0.642043\pi$
$191$	−11.9291	−0.863157	−0.431578	+	0.902075i	$0.642043\pi$
$192$	0	0
$193$	13.0951	0.942605	0.471303	−	0.881972i	$-0.343784\pi$
$193$	13.0951	0.942605	0.471303	+	0.881972i	$0.343784\pi$
$194$	0	0
$195$	6.70679	0.480284
$196$	0	0
$197$	−0.513588	−0.0365916	−0.0182958	−	0.999833i	$-0.505824\pi$
$197$	−0.513588	−0.0365916	−0.0182958	+	0.999833i	$0.505824\pi$
$198$	0	0
$199$	−11.0967	−0.786627	−0.393313	−	0.919404i	$-0.628671\pi$
$199$	−11.0967	−0.786627	−0.393313	+	0.919404i	$0.628671\pi$
$200$	0	0
$201$	32.3833	2.28414
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.39996	0.167621
$206$	0	0
$207$	−6.32008	−0.439276
$208$	0	0
$209$	13.9344	0.963862
$210$	0	0
$211$	28.0923	1.93395	0.966977	−	0.254863i	$-0.0820302\pi$
$211$	28.0923	1.93395	0.966977	+	0.254863i	$0.0820302\pi$
$212$	0	0
$213$	−25.7493	−1.76431
$214$	0	0
$215$	22.0377	1.50296
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.45469	0.503741
$220$	0	0
$221$	3.27033	0.219986
$222$	0	0
$223$	14.3955	0.963994	0.481997	−	0.876173i	$-0.339912\pi$
$223$	14.3955	0.963994	0.481997	+	0.876173i	$0.339912\pi$
$224$	0	0
$225$	0.695833	0.0463889
$226$	0	0
$227$	−0.468590	−0.0311014	−0.0155507	−	0.999879i	$-0.504950\pi$
$227$	−0.468590	−0.0311014	−0.0155507	+	0.999879i	$0.504950\pi$
$228$	0	0
$229$	0.683504	0.0451672	0.0225836	−	0.999745i	$-0.492811\pi$
$229$	0.683504	0.0451672	0.0225836	+	0.999745i	$0.492811\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.0814633	0.00533684	0.00266842	−	0.999996i	$-0.499151\pi$
$233$	0.0814633	0.00533684	0.00266842	+	0.999996i	$0.499151\pi$
$234$	0	0
$235$	−25.6408	−1.67262
$236$	0	0
$237$	26.6755	1.73276
$238$	0	0
$239$	23.2898	1.50649	0.753246	−	0.657739i	$-0.228487\pi$
$239$	23.2898	1.50649	0.753246	+	0.657739i	$0.228487\pi$
$240$	0	0
$241$	−25.0416	−1.61307	−0.806535	−	0.591186i	$-0.798660\pi$
$241$	−25.0416	−1.61307	−0.806535	+	0.591186i	$0.798660\pi$
$242$	0	0
$243$	−9.21352	−0.591047
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.37675	0.533000
$248$	0	0
$249$	6.36379	0.403289
$250$	0	0
$251$	16.6994	1.05406	0.527029	−	0.849848i	$-0.323306\pi$
$251$	16.6994	1.05406	0.527029	+	0.849848i	$0.323306\pi$
$252$	0	0
$253$	−16.2123	−1.01926
$254$	0	0
$255$	10.9978	0.688707
$256$	0	0
$257$	2.88562	0.180000	0.0900002	−	0.995942i	$-0.471313\pi$
$257$	2.88562	0.180000	0.0900002	+	0.995942i	$0.471313\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.21898	−0.0754527
$262$	0	0
$263$	−21.1140	−1.30194	−0.650971	−	0.759102i	$-0.725638\pi$
$263$	−21.1140	−1.30194	−0.650971	+	0.759102i	$0.725638\pi$
$264$	0	0
$265$	29.9519	1.83993
$266$	0	0
$267$	14.1923	0.868553
$268$	0	0
$269$	−2.95179	−0.179974	−0.0899870	−	0.995943i	$-0.528683\pi$
$269$	−2.95179	−0.179974	−0.0899870	+	0.995943i	$0.528683\pi$
$270$	0	0
$271$	10.6964	0.649757	0.324879	−	0.945756i	$-0.394677\pi$
$271$	10.6964	0.649757	0.324879	+	0.945756i	$0.394677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.78495	0.107637
$276$	0	0
$277$	−3.71111	−0.222979	−0.111489	−	0.993766i	$-0.535562\pi$
$277$	−3.71111	−0.222979	−0.111489	+	0.993766i	$0.535562\pi$
$278$	0	0
$279$	−4.56234	−0.273140
$280$	0	0
$281$	9.94492	0.593264	0.296632	−	0.954992i	$-0.404136\pi$
$281$	9.94492	0.593264	0.296632	+	0.954992i	$0.404136\pi$
$282$	0	0
$283$	1.00267	0.0596025	0.0298013	−	0.999556i	$-0.490513\pi$
$283$	1.00267	0.0596025	0.0298013	+	0.999556i	$0.490513\pi$
$284$	0	0
$285$	28.1701	1.66865
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.6373	−0.684549
$290$	0	0
$291$	32.9671	1.93257
$292$	0	0
$293$	23.4779	1.37160	0.685798	−	0.727792i	$-0.259453\pi$
$293$	23.4779	1.37160	0.685798	+	0.727792i	$0.259453\pi$
$294$	0	0
$295$	−13.3971	−0.780009
$296$	0	0
$297$	−9.68871	−0.562196
$298$	0	0
$299$	−9.74612	−0.563633
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.0710	−0.636014
$304$	0	0
$305$	30.8310	1.76538
$306$	0	0
$307$	−18.2829	−1.04346	−0.521731	−	0.853110i	$-0.674714\pi$
$307$	−18.2829	−1.04346	−0.521731	+	0.853110i	$0.674714\pi$
$308$	0	0
$309$	−15.7797	−0.897675
$310$	0	0
$311$	−27.2264	−1.54387	−0.771933	−	0.635704i	$-0.780710\pi$
$311$	−27.2264	−1.54387	−0.771933	+	0.635704i	$0.780710\pi$
$312$	0	0
$313$	15.5021	0.876230	0.438115	−	0.898919i	$-0.355646\pi$
$313$	15.5021	0.876230	0.438115	+	0.898919i	$0.355646\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.2003	1.64005	0.820027	−	0.572325i	$-0.193958\pi$
$317$	29.2003	1.64005	0.820027	+	0.572325i	$0.193958\pi$
$318$	0	0
$319$	−3.12692	−0.175074
$320$	0	0
$321$	9.29881	0.519009
$322$	0	0
$323$	13.7362	0.764301
$324$	0	0
$325$	1.07304	0.0595213
$326$	0	0
$327$	−10.2973	−0.569441
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.0206	−0.770640	−0.385320	−	0.922783i	$-0.625909\pi$
$331$	−14.0206	−0.770640	−0.385320	+	0.922783i	$0.625909\pi$
$332$	0	0
$333$	8.37323	0.458850
$334$	0	0
$335$	39.2750	2.14582
$336$	0	0
$337$	−3.72644	−0.202992	−0.101496	−	0.994836i	$-0.532363\pi$
$337$	−3.72644	−0.202992	−0.101496	+	0.994836i	$0.532363\pi$
$338$	0	0
$339$	1.46360	0.0794921
$340$	0	0
$341$	−11.7033	−0.633771
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−32.7751	−1.76455
$346$	0	0
$347$	−20.3232	−1.09101	−0.545504	−	0.838108i	$-0.683662\pi$
$347$	−20.3232	−1.09101	−0.545504	+	0.838108i	$0.683662\pi$
$348$	0	0
$349$	−20.0961	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$349$	−20.0961	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$350$	0	0
$351$	−5.82443	−0.310885
$352$	0	0
$353$	2.97043	0.158100	0.0790499	−	0.996871i	$-0.474811\pi$
$353$	2.97043	0.158100	0.0790499	+	0.996871i	$0.474811\pi$
$354$	0	0
$355$	−31.2292	−1.65747
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.9055	1.10335	0.551674	−	0.834060i	$-0.313989\pi$
$359$	20.9055	1.10335	0.551674	+	0.834060i	$0.313989\pi$
$360$	0	0
$361$	16.1843	0.851807
$362$	0	0
$363$	−10.8468	−0.569311
$364$	0	0
$365$	9.04118	0.473237
$366$	0	0
$367$	5.20408	0.271651	0.135825	−	0.990733i	$-0.456631\pi$
$367$	5.20408	0.271651	0.135825	+	0.990733i	$0.456631\pi$
$368$	0	0
$369$	0.915782	0.0476737
$370$	0	0
$371$	0	0
$372$	0	0
$373$	28.5182	1.47662	0.738308	−	0.674463i	$-0.235625\pi$
$373$	28.5182	1.47662	0.738308	+	0.674463i	$0.235625\pi$
$374$	0	0
$375$	−20.1371	−1.03988
$376$	0	0
$377$	−1.87977	−0.0968130
$378$	0	0
$379$	28.2937	1.45335	0.726674	−	0.686982i	$-0.241065\pi$
$379$	28.2937	1.45335	0.726674	+	0.686982i	$0.241065\pi$
$380$	0	0
$381$	28.2072	1.44510
$382$	0	0
$383$	−31.9829	−1.63425	−0.817125	−	0.576460i	$-0.804434\pi$
$383$	−31.9829	−1.63425	−0.817125	+	0.576460i	$0.804434\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.40918	0.427462
$388$	0	0
$389$	−8.82599	−0.447495	−0.223748	−	0.974647i	$-0.571829\pi$
$389$	−8.82599	−0.447495	−0.223748	+	0.974647i	$0.571829\pi$
$390$	0	0
$391$	−15.9816	−0.808227
$392$	0	0
$393$	13.0392	0.657741
$394$	0	0
$395$	32.3525	1.62783
$396$	0	0
$397$	−3.85840	−0.193648	−0.0968238	−	0.995302i	$-0.530868\pi$
$397$	−3.85840	−0.193648	−0.0968238	+	0.995302i	$0.530868\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.9828	−0.848083	−0.424041	−	0.905643i	$-0.639389\pi$
$401$	−16.9828	−0.848083	−0.424041	+	0.905643i	$0.639389\pi$
$402$	0	0
$403$	−7.03554	−0.350465
$404$	0	0
$405$	−26.1805	−1.30092
$406$	0	0
$407$	21.4790	1.06468
$408$	0	0
$409$	−27.6572	−1.36756	−0.683781	−	0.729687i	$-0.739666\pi$
$409$	−27.6572	−1.36756	−0.683781	+	0.729687i	$0.739666\pi$
$410$	0	0
$411$	3.64604	0.179846
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.71811	0.378867
$416$	0	0
$417$	−28.1786	−1.37991
$418$	0	0
$419$	27.1918	1.32841	0.664204	−	0.747551i	$-0.268771\pi$
$419$	27.1918	1.32841	0.664204	+	0.747551i	$0.268771\pi$
$420$	0	0
$421$	11.7335	0.571857	0.285929	−	0.958251i	$-0.407698\pi$
$421$	11.7335	0.571857	0.285929	+	0.958251i	$0.407698\pi$
$422$	0	0
$423$	−9.78407	−0.475718
$424$	0	0
$425$	1.75956	0.0853512
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.56483	0.316953
$430$	0	0
$431$	−21.9481	−1.05720	−0.528602	−	0.848870i	$-0.677284\pi$
$431$	−21.9481	−1.05720	−0.528602	+	0.848870i	$0.677284\pi$
$432$	0	0
$433$	5.74092	0.275891	0.137946	−	0.990440i	$-0.455950\pi$
$433$	5.74092	0.275891	0.137946	+	0.990440i	$0.455950\pi$
$434$	0	0
$435$	−6.32146	−0.303091
$436$	0	0
$437$	−40.9360	−1.95823
$438$	0	0
$439$	5.25421	0.250770	0.125385	−	0.992108i	$-0.459983\pi$
$439$	5.25421	0.250770	0.125385	+	0.992108i	$0.459983\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.1523	−0.814933	−0.407466	−	0.913220i	$-0.633588\pi$
$443$	−17.1523	−0.814933	−0.407466	+	0.913220i	$0.633588\pi$
$444$	0	0
$445$	17.2126	0.815957
$446$	0	0
$447$	−26.2442	−1.24131
$448$	0	0
$449$	12.9056	0.609052	0.304526	−	0.952504i	$-0.401502\pi$
$449$	12.9056	0.609052	0.304526	+	0.952504i	$0.401502\pi$
$450$	0	0
$451$	2.34916	0.110618
$452$	0	0
$453$	7.84868	0.368763
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.69584	0.406774	0.203387	−	0.979098i	$-0.434805\pi$
$457$	8.69584	0.406774	0.203387	+	0.979098i	$0.434805\pi$
$458$	0	0
$459$	−9.55088	−0.445797
$460$	0	0
$461$	8.89235	0.414158	0.207079	−	0.978324i	$-0.433604\pi$
$461$	8.89235	0.414158	0.207079	+	0.978324i	$0.433604\pi$
$462$	0	0
$463$	−18.4573	−0.857785	−0.428893	−	0.903356i	$-0.641096\pi$
$463$	−18.4573	−0.857785	−0.428893	+	0.903356i	$0.641096\pi$
$464$	0	0
$465$	−23.6597	−1.09719
$466$	0	0
$467$	−12.8444	−0.594367	−0.297184	−	0.954820i	$-0.596047\pi$
$467$	−12.8444	−0.594367	−0.297184	+	0.954820i	$0.596047\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−32.0519	−1.47688
$472$	0	0
$473$	21.5712	0.991846
$474$	0	0
$475$	4.50700	0.206795
$476$	0	0
$477$	11.4291	0.523303
$478$	0	0
$479$	−16.1504	−0.737930	−0.368965	−	0.929443i	$-0.620288\pi$
$479$	−16.1504	−0.737930	−0.368965	+	0.929443i	$0.620288\pi$
$480$	0	0
$481$	12.9123	0.588748
$482$	0	0
$483$	0	0
$484$	0	0
$485$	39.9831	1.81554
$486$	0	0
$487$	−22.2567	−1.00855	−0.504275	−	0.863543i	$-0.668240\pi$
$487$	−22.2567	−1.00855	−0.504275	+	0.863543i	$0.668240\pi$
$488$	0	0
$489$	−5.34756	−0.241825
$490$	0	0
$491$	−5.36579	−0.242155	−0.121077	−	0.992643i	$-0.538635\pi$
$491$	−5.36579	−0.242155	−0.121077	+	0.992643i	$0.538635\pi$
$492$	0	0
$493$	−3.08244	−0.138826
$494$	0	0
$495$	5.16309	0.232064
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.0927	−1.43667	−0.718333	−	0.695700i	$-0.755094\pi$
$499$	−32.0927	−1.43667	−0.718333	+	0.695700i	$0.755094\pi$
$500$	0	0
$501$	−5.95216	−0.265923
$502$	0	0
$503$	27.8846	1.24331	0.621657	−	0.783290i	$-0.286460\pi$
$503$	27.8846	1.24331	0.621657	+	0.783290i	$0.286460\pi$
$504$	0	0
$505$	−13.4271	−0.597500
$506$	0	0
$507$	−21.7783	−0.967210
$508$	0	0
$509$	−20.4222	−0.905198	−0.452599	−	0.891714i	$-0.649503\pi$
$509$	−20.4222	−0.905198	−0.452599	+	0.891714i	$0.649503\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−24.4640	−1.08011
$514$	0	0
$515$	−19.1379	−0.843316
$516$	0	0
$517$	−25.0981	−1.10381
$518$	0	0
$519$	4.60717	0.202232
$520$	0	0
$521$	16.3186	0.714933	0.357466	−	0.933926i	$-0.383641\pi$
$521$	16.3186	0.714933	0.357466	+	0.933926i	$0.383641\pi$
$522$	0	0
$523$	−20.6643	−0.903586	−0.451793	−	0.892123i	$-0.649215\pi$
$523$	−20.6643	−0.903586	−0.451793	+	0.892123i	$0.649215\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.5368	−0.502553
$528$	0	0
$529$	24.6279	1.07078
$530$	0	0
$531$	−5.11208	−0.221846
$532$	0	0
$533$	1.41222	0.0611699
$534$	0	0
$535$	11.2778	0.487580
$536$	0	0
$537$	−50.0648	−2.16045
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.1812	0.437723	0.218861	−	0.975756i	$-0.429766\pi$
$541$	10.1812	0.437723	0.218861	+	0.975756i	$0.429766\pi$
$542$	0	0
$543$	−37.2761	−1.59967
$544$	0	0
$545$	−12.4887	−0.534958
$546$	0	0
$547$	23.7769	1.01663	0.508314	−	0.861172i	$-0.330269\pi$
$547$	23.7769	1.01663	0.508314	+	0.861172i	$0.330269\pi$
$548$	0	0
$549$	11.7645	0.502098
$550$	0	0
$551$	−7.89547	−0.336358
$552$	0	0
$553$	0	0
$554$	0	0
$555$	43.4225	1.84318
$556$	0	0
$557$	34.4582	1.46004	0.730019	−	0.683426i	$-0.239511\pi$
$557$	34.4582	1.46004	0.730019	+	0.683426i	$0.239511\pi$
$558$	0	0
$559$	12.9677	0.548475
$560$	0	0
$561$	10.7650	0.454498
$562$	0	0
$563$	−40.9980	−1.72786	−0.863929	−	0.503614i	$-0.832004\pi$
$563$	−40.9980	−1.72786	−0.863929	+	0.503614i	$0.832004\pi$
$564$	0	0
$565$	1.77508	0.0746784
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.1797	−0.426757	−0.213379	−	0.976970i	$-0.568447\pi$
$569$	−10.1797	−0.426757	−0.213379	+	0.976970i	$0.568447\pi$
$570$	0	0
$571$	28.8634	1.20790	0.603948	−	0.797023i	$-0.293593\pi$
$571$	28.8634	1.20790	0.603948	+	0.797023i	$0.293593\pi$
$572$	0	0
$573$	−23.6056	−0.986140
$574$	0	0
$575$	−5.24377	−0.218680
$576$	0	0
$577$	16.2243	0.675428	0.337714	−	0.941249i	$-0.390346\pi$
$577$	16.2243	0.675428	0.337714	+	0.941249i	$0.390346\pi$
$578$	0	0
$579$	25.9130	1.07691
$580$	0	0
$581$	0	0
$582$	0	0
$583$	29.3180	1.21423
$584$	0	0
$585$	3.10383	0.128328
$586$	0	0
$587$	−35.9683	−1.48457	−0.742284	−	0.670085i	$-0.766258\pi$
$587$	−35.9683	−1.48457	−0.742284	+	0.670085i	$0.766258\pi$
$588$	0	0
$589$	−29.5509	−1.21762
$590$	0	0
$591$	−1.01630	−0.0418052
$592$	0	0
$593$	−8.03336	−0.329891	−0.164945	−	0.986303i	$-0.552745\pi$
$593$	−8.03336	−0.329891	−0.164945	+	0.986303i	$0.552745\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−21.9586	−0.898705
$598$	0	0
$599$	−29.0056	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$599$	−29.0056	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$600$	0	0
$601$	−2.87190	−0.117147	−0.0585737	−	0.998283i	$-0.518655\pi$
$601$	−2.87190	−0.117147	−0.0585737	+	0.998283i	$0.518655\pi$
$602$	0	0
$603$	14.9866	0.610302
$604$	0	0
$605$	−13.1552	−0.534836
$606$	0	0
$607$	8.01281	0.325230	0.162615	−	0.986690i	$-0.448007\pi$
$607$	8.01281	0.325230	0.162615	+	0.986690i	$0.448007\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.0879	−0.610391
$612$	0	0
$613$	−21.1189	−0.852985	−0.426492	−	0.904491i	$-0.640251\pi$
$613$	−21.1189	−0.852985	−0.426492	+	0.904491i	$0.640251\pi$
$614$	0	0
$615$	4.74913	0.191503
$616$	0	0
$617$	−18.3046	−0.736915	−0.368458	−	0.929645i	$-0.620114\pi$
$617$	−18.3046	−0.736915	−0.368458	+	0.929645i	$0.620114\pi$
$618$	0	0
$619$	17.5006	0.703410	0.351705	−	0.936111i	$-0.385602\pi$
$619$	17.5006	0.703410	0.351705	+	0.936111i	$0.385602\pi$
$620$	0	0
$621$	28.4632	1.14219
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.2218	−1.12887
$626$	0	0
$627$	27.5738	1.10119
$628$	0	0
$629$	21.1735	0.844241
$630$	0	0
$631$	40.2553	1.60254	0.801269	−	0.598305i	$-0.204159\pi$
$631$	40.2553	1.60254	0.801269	+	0.598305i	$0.204159\pi$
$632$	0	0
$633$	55.5900	2.20950
$634$	0	0
$635$	34.2102	1.35759
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11.9165	−0.471409
$640$	0	0
$641$	15.1018	0.596484	0.298242	−	0.954490i	$-0.403600\pi$
$641$	15.1018	0.596484	0.298242	+	0.954490i	$0.403600\pi$
$642$	0	0
$643$	−13.3216	−0.525352	−0.262676	−	0.964884i	$-0.584605\pi$
$643$	−13.3216	−0.525352	−0.262676	+	0.964884i	$0.584605\pi$
$644$	0	0
$645$	43.6089	1.71710
$646$	0	0
$647$	−5.41556	−0.212908	−0.106454	−	0.994318i	$-0.533950\pi$
$647$	−5.41556	−0.212908	−0.106454	+	0.994318i	$0.533950\pi$
$648$	0	0
$649$	−13.1135	−0.514751
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.2624	−1.81039	−0.905194	−	0.424998i	$-0.860275\pi$
$653$	−46.2624	−1.81039	−0.905194	+	0.424998i	$0.860275\pi$
$654$	0	0
$655$	15.8142	0.617911
$656$	0	0
$657$	3.44995	0.134595
$658$	0	0
$659$	41.1999	1.60492	0.802460	−	0.596706i	$-0.203524\pi$
$659$	41.1999	1.60492	0.802460	+	0.596706i	$0.203524\pi$
$660$	0	0
$661$	28.6270	1.11346	0.556731	−	0.830693i	$-0.312055\pi$
$661$	28.6270	1.11346	0.556731	+	0.830693i	$0.312055\pi$
$662$	0	0
$663$	6.47144	0.251330
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.18616	0.355690
$668$	0	0
$669$	28.4863	1.10134
$670$	0	0
$671$	30.1784	1.16502
$672$	0	0
$673$	26.5265	1.02252	0.511261	−	0.859426i	$-0.329179\pi$
$673$	26.5265	1.02252	0.511261	+	0.859426i	$0.329179\pi$
$674$	0	0
$675$	−3.13376	−0.120618
$676$	0	0
$677$	28.4442	1.09320	0.546600	−	0.837394i	$-0.315922\pi$
$677$	28.4442	1.09320	0.546600	+	0.837394i	$0.315922\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−0.927262	−0.0355328
$682$	0	0
$683$	0.104618	0.00400309	0.00200155	−	0.999998i	$-0.499363\pi$
$683$	0.104618	0.00400309	0.00200155	+	0.999998i	$0.499363\pi$
$684$	0	0
$685$	4.42198	0.168955
$686$	0	0
$687$	1.35254	0.0516027
$688$	0	0
$689$	17.6247	0.671447
$690$	0	0
$691$	−24.0620	−0.915361	−0.457681	−	0.889117i	$-0.651320\pi$
$691$	−24.0620	−0.915361	−0.457681	+	0.889117i	$0.651320\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−34.1755	−1.29635
$696$	0	0
$697$	2.31575	0.0877151
$698$	0	0
$699$	0.161202	0.00609723
$700$	0	0
$701$	21.9812	0.830220	0.415110	−	0.909771i	$-0.363743\pi$
$701$	21.9812	0.830220	0.415110	+	0.909771i	$0.363743\pi$
$702$	0	0
$703$	54.2345	2.04549
$704$	0	0
$705$	−50.7389	−1.91094
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.66268	−0.325334	−0.162667	−	0.986681i	$-0.552010\pi$
$709$	−8.66268	−0.325334	−0.162667	+	0.986681i	$0.552010\pi$
$710$	0	0
$711$	12.3451	0.462979
$712$	0	0
$713$	34.3817	1.28760
$714$	0	0
$715$	7.96195	0.297760
$716$	0	0
$717$	46.0866	1.72114
$718$	0	0
$719$	−15.3180	−0.571267	−0.285633	−	0.958339i	$-0.592204\pi$
$719$	−15.3180	−0.571267	−0.285633	+	0.958339i	$0.592204\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−49.5531	−1.84290
$724$	0	0
$725$	−1.01138	−0.0375619
$726$	0	0
$727$	−23.2144	−0.860975	−0.430487	−	0.902597i	$-0.641658\pi$
$727$	−23.2144	−0.860975	−0.430487	+	0.902597i	$0.641658\pi$
$728$	0	0
$729$	14.4941	0.536817
$730$	0	0
$731$	21.2644	0.786491
$732$	0	0
$733$	−47.5792	−1.75738	−0.878690	−	0.477394i	$-0.841582\pi$
$733$	−47.5792	−1.75738	−0.878690	+	0.477394i	$0.841582\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.4437	1.41609
$738$	0	0
$739$	−10.8040	−0.397432	−0.198716	−	0.980057i	$-0.563677\pi$
$739$	−10.8040	−0.397432	−0.198716	+	0.980057i	$0.563677\pi$
$740$	0	0
$741$	16.5762	0.608942
$742$	0	0
$743$	7.45742	0.273586	0.136793	−	0.990600i	$-0.456320\pi$
$743$	7.45742	0.273586	0.136793	+	0.990600i	$0.456320\pi$
$744$	0	0
$745$	−31.8294	−1.16614
$746$	0	0
$747$	2.94509	0.107755
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.09145	0.331752	0.165876	−	0.986147i	$-0.446955\pi$
$751$	9.09145	0.331752	0.165876	+	0.986147i	$0.446955\pi$
$752$	0	0
$753$	33.0453	1.20424
$754$	0	0
$755$	9.51901	0.346432
$756$	0	0
$757$	8.31853	0.302342	0.151171	−	0.988508i	$-0.451696\pi$
$757$	8.31853	0.302342	0.151171	+	0.988508i	$0.451696\pi$
$758$	0	0
$759$	−32.0814	−1.16448
$760$	0	0
$761$	−2.92558	−0.106052	−0.0530260	−	0.998593i	$-0.516887\pi$
$761$	−2.92558	−0.106052	−0.0530260	+	0.998593i	$0.516887\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.08964	0.184016
$766$	0	0
$767$	−7.88329	−0.284649
$768$	0	0
$769$	−25.8384	−0.931757	−0.465878	−	0.884849i	$-0.654262\pi$
$769$	−25.8384	−0.931757	−0.465878	+	0.884849i	$0.654262\pi$
$770$	0	0
$771$	5.71017	0.205647
$772$	0	0
$773$	−47.1884	−1.69725	−0.848624	−	0.528997i	$-0.822568\pi$
$773$	−47.1884	−1.69725	−0.848624	+	0.528997i	$0.822568\pi$
$774$	0	0
$775$	−3.78538	−0.135975
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.93164	0.212523
$780$	0	0
$781$	−30.5682	−1.09382
$782$	0	0
$783$	5.48979	0.196189
$784$	0	0
$785$	−38.8732	−1.38744
$786$	0	0
$787$	−53.1847	−1.89583	−0.947915	−	0.318524i	$-0.896813\pi$
$787$	−53.1847	−1.89583	−0.947915	+	0.318524i	$0.896813\pi$
$788$	0	0
$789$	−41.7810	−1.48744
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.1420	0.644240
$794$	0	0
$795$	59.2699	2.10209
$796$	0	0
$797$	12.5167	0.443364	0.221682	−	0.975119i	$-0.428845\pi$
$797$	12.5167	0.443364	0.221682	+	0.975119i	$0.428845\pi$
$798$	0	0
$799$	−24.7411	−0.875276
$800$	0	0
$801$	6.56802	0.232070
$802$	0	0
$803$	8.84981	0.312303
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.84111	−0.205617
$808$	0	0
$809$	24.3264	0.855272	0.427636	−	0.903951i	$-0.359347\pi$
$809$	24.3264	0.855272	0.427636	+	0.903951i	$0.359347\pi$
$810$	0	0
$811$	−6.12666	−0.215136	−0.107568	−	0.994198i	$-0.534306\pi$
$811$	−6.12666	−0.215136	−0.107568	+	0.994198i	$0.534306\pi$
$812$	0	0
$813$	21.1663	0.742335
$814$	0	0
$815$	−6.48561	−0.227181
$816$	0	0
$817$	54.4673	1.90557
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.00991	−0.105047	−0.0525233	−	0.998620i	$-0.516726\pi$
$821$	−3.00991	−0.105047	−0.0525233	+	0.998620i	$0.516726\pi$
$822$	0	0
$823$	−20.0821	−0.700017	−0.350008	−	0.936747i	$-0.613821\pi$
$823$	−20.0821	−0.700017	−0.350008	+	0.936747i	$0.613821\pi$
$824$	0	0
$825$	3.53212	0.122973
$826$	0	0
$827$	−42.3948	−1.47421	−0.737105	−	0.675778i	$-0.763808\pi$
$827$	−42.3948	−1.47421	−0.737105	+	0.675778i	$0.763808\pi$
$828$	0	0
$829$	−24.2774	−0.843189	−0.421595	−	0.906784i	$-0.638529\pi$
$829$	−24.2774	−0.843189	−0.421595	+	0.906784i	$0.638529\pi$
$830$	0	0
$831$	−7.34366	−0.254749
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7.21889	−0.249820
$836$	0	0
$837$	20.5470	0.710208
$838$	0	0
$839$	−50.5278	−1.74441	−0.872206	−	0.489138i	$-0.837311\pi$
$839$	−50.5278	−1.74441	−0.872206	+	0.489138i	$0.837311\pi$
$840$	0	0
$841$	−27.2282	−0.938905
$842$	0	0
$843$	19.6793	0.677793
$844$	0	0
$845$	−26.4131	−0.908640
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.98412	0.0680947
$850$	0	0
$851$	−63.1003	−2.16305
$852$	0	0
$853$	3.31868	0.113629	0.0568147	−	0.998385i	$-0.481906\pi$
$853$	3.31868	0.113629	0.0568147	+	0.998385i	$0.481906\pi$
$854$	0	0
$855$	13.0368	0.445849
$856$	0	0
$857$	21.4783	0.733684	0.366842	−	0.930283i	$-0.380439\pi$
$857$	21.4783	0.733684	0.366842	+	0.930283i	$0.380439\pi$
$858$	0	0
$859$	50.4406	1.72101	0.860505	−	0.509442i	$-0.170148\pi$
$859$	50.4406	1.72101	0.860505	+	0.509442i	$0.170148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.6212	1.62104	0.810522	−	0.585708i	$-0.199183\pi$
$863$	47.6212	1.62104	0.810522	+	0.585708i	$0.199183\pi$
$864$	0	0
$865$	5.58765	0.189986
$866$	0	0
$867$	−23.0283	−0.782083
$868$	0	0
$869$	31.6677	1.07425
$870$	0	0
$871$	23.1107	0.783076
$872$	0	0
$873$	15.2568	0.516364
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.3997	1.53304	0.766520	−	0.642221i	$-0.221987\pi$
$877$	45.3997	1.53304	0.766520	+	0.642221i	$0.221987\pi$
$878$	0	0
$879$	46.4589	1.56702
$880$	0	0
$881$	14.2445	0.479909	0.239954	−	0.970784i	$-0.422868\pi$
$881$	14.2445	0.479909	0.239954	+	0.970784i	$0.422868\pi$
$882$	0	0
$883$	2.93736	0.0988501	0.0494251	−	0.998778i	$-0.484261\pi$
$883$	2.93736	0.0988501	0.0494251	+	0.998778i	$0.484261\pi$
$884$	0	0
$885$	−26.5106	−0.891145
$886$	0	0
$887$	−48.0250	−1.61252	−0.806261	−	0.591559i	$-0.798513\pi$
$887$	−48.0250	−1.61252	−0.806261	+	0.591559i	$0.798513\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−25.6263	−0.858513
$892$	0	0
$893$	−63.3727	−2.12069
$894$	0	0
$895$	−60.7194	−2.02963
$896$	0	0
$897$	−19.2860	−0.643939
$898$	0	0
$899$	6.63131	0.221167
$900$	0	0
$901$	28.9009	0.962828
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45.2090	−1.50280
$906$	0	0
$907$	−58.5690	−1.94475	−0.972375	−	0.233424i	$-0.925007\pi$
$907$	−58.5690	−1.94475	−0.972375	+	0.233424i	$0.925007\pi$
$908$	0	0
$909$	−5.12355	−0.169937
$910$	0	0
$911$	7.14793	0.236821	0.118411	−	0.992965i	$-0.462220\pi$
$911$	7.14793	0.236821	0.118411	+	0.992965i	$0.462220\pi$
$912$	0	0
$913$	7.55475	0.250026
$914$	0	0
$915$	61.0094	2.01691
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.0249	0.891468	0.445734	−	0.895165i	$-0.352943\pi$
$919$	27.0249	0.891468	0.445734	+	0.895165i	$0.352943\pi$
$920$	0	0
$921$	−36.1789	−1.19214
$922$	0	0
$923$	−18.3763	−0.604862
$924$	0	0
$925$	6.94727	0.228425
$926$	0	0
$927$	−7.30266	−0.239851
$928$	0	0
$929$	39.8861	1.30862	0.654310	−	0.756226i	$-0.272959\pi$
$929$	39.8861	1.30862	0.654310	+	0.756226i	$0.272959\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−53.8764	−1.76384
$934$	0	0
$935$	13.0560	0.426976
$936$	0	0
$937$	−12.4056	−0.405274	−0.202637	−	0.979254i	$-0.564951\pi$
$937$	−12.4056	−0.405274	−0.202637	+	0.979254i	$0.564951\pi$
$938$	0	0
$939$	30.6760	1.00107
$940$	0	0
$941$	7.18199	0.234126	0.117063	−	0.993124i	$-0.462652\pi$
$941$	7.18199	0.234126	0.117063	+	0.993124i	$0.462652\pi$
$942$	0	0
$943$	−6.90130	−0.224737
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.8341	1.13196	0.565978	−	0.824421i	$-0.308499\pi$
$947$	34.8341	1.13196	0.565978	+	0.824421i	$0.308499\pi$
$948$	0	0
$949$	5.32012	0.172698
$950$	0	0
$951$	57.7826	1.87373
$952$	0	0
$953$	20.2941	0.657391	0.328695	−	0.944436i	$-0.393391\pi$
$953$	20.2941	0.657391	0.328695	+	0.944436i	$0.393391\pi$
$954$	0	0
$955$	−28.6293	−0.926423
$956$	0	0
$957$	−6.18765	−0.200018
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.18054	−0.199372
$962$	0	0
$963$	4.30339	0.138675
$964$	0	0
$965$	31.4277	1.01169
$966$	0	0
$967$	−6.28348	−0.202063	−0.101032	−	0.994883i	$-0.532214\pi$
$967$	−6.28348	−0.202063	−0.101032	+	0.994883i	$0.532214\pi$
$968$	0	0
$969$	27.1816	0.873198
$970$	0	0
$971$	−2.04260	−0.0655501	−0.0327750	−	0.999463i	$-0.510434\pi$
$971$	−2.04260	−0.0655501	−0.0327750	+	0.999463i	$0.510434\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.12336	0.0680019
$976$	0	0
$977$	27.4777	0.879089	0.439544	−	0.898221i	$-0.355140\pi$
$977$	27.4777	0.879089	0.439544	+	0.898221i	$0.355140\pi$
$978$	0	0
$979$	16.8483	0.538474
$980$	0	0
$981$	−4.76547	−0.152150
$982$	0	0
$983$	19.2231	0.613120	0.306560	−	0.951851i	$-0.400822\pi$
$983$	19.2231	0.613120	0.306560	+	0.951851i	$0.400822\pi$
$984$	0	0
$985$	−1.23259	−0.0392736
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63.3712	−2.01509
$990$	0	0
$991$	−27.9908	−0.889158	−0.444579	−	0.895740i	$-0.646647\pi$
$991$	−27.9908	−0.889158	−0.444579	+	0.895740i	$0.646647\pi$
$992$	0	0
$993$	−27.7444	−0.880441
$994$	0	0
$995$	−26.6318	−0.844284
$996$	0	0
$997$	−34.0980	−1.07989	−0.539947	−	0.841699i	$-0.681556\pi$
$997$	−34.0980	−1.07989	−0.539947	+	0.841699i	$0.681556\pi$
$998$	0	0
$999$	−37.7097	−1.19308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.k.1.5		5
7.6	odd	2		1148.2.a.d.1.1	✓	5
28.27	even	2		4592.2.a.bc.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.d.1.1	✓	5	7.6	odd	2
4592.2.a.bc.1.5		5	28.27	even	2
8036.2.a.k.1.5		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.97883 q^{3} +2.39996 q^{5} +0.915782 q^{9} +O(q^{10})\)	\(q+1.97883 q^{3} +2.39996 q^{5} +0.915782 q^{9} +2.34916 q^{11} +1.41222 q^{13} +4.74913 q^{15} +2.31575 q^{17} +5.93164 q^{19} -6.90130 q^{23} +0.759824 q^{25} -4.12432 q^{27} -1.33108 q^{29} -4.98191 q^{31} +4.64860 q^{33} +9.14326 q^{37} +2.79454 q^{39} +1.00000 q^{41} +9.18251 q^{43} +2.19784 q^{45} -10.6838 q^{47} +4.58247 q^{51} +12.4802 q^{53} +5.63791 q^{55} +11.7377 q^{57} -5.58221 q^{59} +12.8464 q^{61} +3.38927 q^{65} +16.3648 q^{67} -13.6565 q^{69} -13.0124 q^{71} +3.76721 q^{73} +1.50357 q^{75} +13.4804 q^{79} -10.9087 q^{81} +3.21593 q^{83} +5.55770 q^{85} -2.63398 q^{87} +7.17204 q^{89} -9.85838 q^{93} +14.2357 q^{95} +16.6599 q^{97} +2.15132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9	\( 5 q - 2 q^{3} + 3 q^{5} + q^{9} + 6 q^{11} - 7 q^{13} + 9 q^{15} - q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 8 q^{27} + 11 q^{29} - 13 q^{31} - 11 q^{33} + 5 q^{37} + 23 q^{39} + 5 q^{41} + 29 q^{43} - 11 q^{45} - 7 q^{47} - 3 q^{51} + 21 q^{53} - 19 q^{55} + 9 q^{57} - 3 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} - 10 q^{69} + 22 q^{71} + 16 q^{73} + 18 q^{75} + 4 q^{79} - 15 q^{81} + 6 q^{83} + 13 q^{85} - 6 q^{87} + 20 q^{89} - 5 q^{93} - 7 q^{95} + 24 q^{97} + 27 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 3 * q^5 + q^9 + 6 * q^11 - 7 * q^13 + 9 * q^15 - q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 - 8 * q^27 + 11 * q^29 - 13 * q^31 - 11 * q^33 + 5 * q^37 + 23 * q^39 + 5 * q^41 + 29 * q^43 - 11 * q^45 - 7 * q^47 - 3 * q^51 + 21 * q^53 - 19 * q^55 + 9 * q^57 - 3 * q^59 + 8 * q^61 - 5 * q^65 + 3 * q^67 - 10 * q^69 + 22 * q^71 + 16 * q^73 + 18 * q^75 + 4 * q^79 - 15 * q^81 + 6 * q^83 + 13 * q^85 - 6 * q^87 + 20 * q^89 - 5 * q^93 - 7 * q^95 + 24 * q^97 + 27 * q^99

Introduction

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