LMFDB - Embedded newform 8036.2.a.j.1.1

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:	$5$
Coefficient field:	5.5.1935333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 $ x^5 - 2x^4 - 8x^3 + 10x^2 + 13x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$3.23378$ 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)

$f(q)$	$=$	$q-3.23378 q^{3} -1.20798 q^{5} +7.45735 q^{9} +O(q^{10})$	$q-3.23378 q^{3} -1.20798 q^{5} +7.45735 q^{9} -2.69837 q^{11} -0.474804 q^{13} +3.90635 q^{15} +0.147373 q^{17} +1.75898 q^{19} +5.23378 q^{23} -3.54078 q^{25} -14.4141 q^{27} -3.90635 q^{29} -0.792018 q^{31} +8.72594 q^{33} -4.63953 q^{37} +1.53541 q^{39} -1.00000 q^{41} +2.14737 q^{43} -9.00834 q^{45} +5.64074 q^{47} -0.476573 q^{51} -1.20798 q^{53} +3.25958 q^{55} -5.68815 q^{57} +8.72594 q^{59} -8.38319 q^{61} +0.573555 q^{65} -0.625155 q^{67} -16.9249 q^{69} -2.30284 q^{71} -2.28240 q^{73} +11.4501 q^{75} +4.02043 q^{79} +24.2400 q^{81} -3.69716 q^{83} -0.178024 q^{85} +12.6323 q^{87} +17.4171 q^{89} +2.56121 q^{93} -2.12481 q^{95} +7.70135 q^{97} -20.1227 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9	$ 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9} - 7 q^{13} + 3 q^{15} + 3 q^{17} - 10 q^{19} + 12 q^{23} + 2 q^{25} - 14 q^{27} - 3 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} + 7 q^{39} - 5 q^{41} + 13 q^{43} + 3 q^{45} - 9 q^{47} + 9 q^{51} - 3 q^{53} - 9 q^{55} - 11 q^{57} + 3 q^{59} - 16 q^{61} + 3 q^{65} + 19 q^{67} - 24 q^{69} - 12 q^{71} - 4 q^{73} - 8 q^{75} + 28 q^{79} + 5 q^{81} - 18 q^{83} - 15 q^{85} + 6 q^{87} + q^{93} + 3 q^{95} - 4 q^{97} - 33 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9 - 7 * q^13 + 3 * q^15 + 3 * q^17 - 10 * q^19 + 12 * q^23 + 2 * q^25 - 14 * q^27 - 3 * q^29 - 7 * q^31 + 3 * q^33 + q^37 + 7 * q^39 - 5 * q^41 + 13 * q^43 + 3 * q^45 - 9 * q^47 + 9 * q^51 - 3 * q^53 - 9 * q^55 - 11 * q^57 + 3 * q^59 - 16 * q^61 + 3 * q^65 + 19 * q^67 - 24 * q^69 - 12 * q^71 - 4 * q^73 - 8 * q^75 + 28 * q^79 + 5 * q^81 - 18 * q^83 - 15 * q^85 + 6 * q^87 + q^93 + 3 * q^95 - 4 * q^97 - 33 * 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
$3$	−3.23378	−1.86703	−0.933513	−	0.358545i	$-0.883273\pi$
$3$	−3.23378	−1.86703	−0.933513	+	0.358545i	$0.883273\pi$
$4$	0	0
$5$	−1.20798	−0.540226	−0.270113	−	0.962829i	$-0.587061\pi$
$5$	−1.20798	−0.540226	−0.270113	+	0.962829i	$0.587061\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.45735	2.48578
$10$	0	0
$11$	−2.69837	−0.813589	−0.406794	−	0.913520i	$-0.633353\pi$
$11$	−2.69837	−0.813589	−0.406794	+	0.913520i	$0.633353\pi$
$12$	0	0
$13$	−0.474804	−0.131687	−0.0658435	−	0.997830i	$-0.520974\pi$
$13$	−0.474804	−0.131687	−0.0658435	+	0.997830i	$0.520974\pi$
$14$	0	0
$15$	3.90635	1.00862
$16$	0	0
$17$	0.147373	0.0357433	0.0178716	−	0.999840i	$-0.494311\pi$
$17$	0.147373	0.0357433	0.0178716	+	0.999840i	$0.494311\pi$
$18$	0	0
$19$	1.75898	0.403537	0.201769	−	0.979433i	$-0.435331\pi$
$19$	1.75898	0.403537	0.201769	+	0.979433i	$0.435331\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.23378	1.09132	0.545660	−	0.838007i	$-0.316279\pi$
$23$	5.23378	1.09132	0.545660	+	0.838007i	$0.316279\pi$
$24$	0	0
$25$	−3.54078	−0.708156
$26$	0	0
$27$	−14.4141	−2.77399
$28$	0	0
$29$	−3.90635	−0.725391	−0.362696	−	0.931908i	$-0.618144\pi$
$29$	−3.90635	−0.725391	−0.362696	+	0.931908i	$0.618144\pi$
$30$	0	0
$31$	−0.792018	−0.142251	−0.0711253	−	0.997467i	$-0.522659\pi$
$31$	−0.792018	−0.142251	−0.0711253	+	0.997467i	$0.522659\pi$
$32$	0	0
$33$	8.72594	1.51899
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.63953	−0.762734	−0.381367	−	0.924424i	$-0.624547\pi$
$37$	−4.63953	−0.762734	−0.381367	+	0.924424i	$0.624547\pi$
$38$	0	0
$39$	1.53541	0.245863
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	2.14737	0.327471	0.163736	−	0.986504i	$-0.447646\pi$
$43$	2.14737	0.327471	0.163736	+	0.986504i	$0.447646\pi$
$44$	0	0
$45$	−9.00834	−1.34288
$46$	0	0
$47$	5.64074	0.822786	0.411393	−	0.911458i	$-0.365042\pi$
$47$	5.64074	0.822786	0.411393	+	0.911458i	$0.365042\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.476573	−0.0667336
$52$	0	0
$53$	−1.20798	−0.165929	−0.0829646	−	0.996552i	$-0.526439\pi$
$53$	−1.20798	−0.165929	−0.0829646	+	0.996552i	$0.526439\pi$
$54$	0	0
$55$	3.25958	0.439522
$56$	0	0
$57$	−5.68815	−0.753414
$58$	0	0
$59$	8.72594	1.13602	0.568010	−	0.823022i	$-0.307713\pi$
$59$	8.72594	1.13602	0.568010	+	0.823022i	$0.307713\pi$
$60$	0	0
$61$	−8.38319	−1.07336	−0.536679	−	0.843787i	$-0.680321\pi$
$61$	−8.38319	−1.07336	−0.536679	+	0.843787i	$0.680321\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.573555	0.0711407
$66$	0	0
$67$	−0.625155	−0.0763748	−0.0381874	−	0.999271i	$-0.512158\pi$
$67$	−0.625155	−0.0763748	−0.0381874	+	0.999271i	$0.512158\pi$
$68$	0	0
$69$	−16.9249	−2.03752
$70$	0	0
$71$	−2.30284	−0.273297	−0.136648	−	0.990620i	$-0.543633\pi$
$71$	−2.30284	−0.273297	−0.136648	+	0.990620i	$0.543633\pi$
$72$	0	0
$73$	−2.28240	−0.267135	−0.133568	−	0.991040i	$-0.542643\pi$
$73$	−2.28240	−0.267135	−0.133568	+	0.991040i	$0.542643\pi$
$74$	0	0
$75$	11.4501	1.32214
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.02043	0.452334	0.226167	−	0.974089i	$-0.427380\pi$
$79$	4.02043	0.452334	0.226167	+	0.974089i	$0.427380\pi$
$80$	0	0
$81$	24.2400	2.69333
$82$	0	0
$83$	−3.69716	−0.405816	−0.202908	−	0.979198i	$-0.565039\pi$
$83$	−3.69716	−0.405816	−0.202908	+	0.979198i	$0.565039\pi$
$84$	0	0
$85$	−0.178024	−0.0193095
$86$	0	0
$87$	12.6323	1.35432
$88$	0	0
$89$	17.4171	1.84621	0.923103	−	0.384553i	$-0.125645\pi$
$89$	17.4171	1.84621	0.923103	+	0.384553i	$0.125645\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.56121	0.265585
$94$	0	0
$95$	−2.12481	−0.218001
$96$	0	0
$97$	7.70135	0.781953	0.390977	−	0.920401i	$-0.372137\pi$
$97$	7.70135	0.781953	0.390977	+	0.920401i	$0.372137\pi$
$98$	0	0
$99$	−20.1227	−2.02241
$100$	0	0
$101$	−9.17130	−0.912578	−0.456289	−	0.889832i	$-0.650822\pi$
$101$	−9.17130	−0.912578	−0.456289	+	0.889832i	$0.650822\pi$
$102$	0	0
$103$	11.7707	1.15980	0.579899	−	0.814689i	$-0.303092\pi$
$103$	11.7707	1.15980	0.579899	+	0.814689i	$0.303092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.42857	0.138105	0.0690525	−	0.997613i	$-0.478002\pi$
$107$	1.42857	0.138105	0.0690525	+	0.997613i	$0.478002\pi$
$108$	0	0
$109$	14.2803	1.36780	0.683901	−	0.729575i	$-0.260282\pi$
$109$	14.2803	1.36780	0.683901	+	0.729575i	$0.260282\pi$
$110$	0	0
$111$	15.0032	1.42404
$112$	0	0
$113$	−17.8262	−1.67694	−0.838472	−	0.544945i	$-0.816551\pi$
$113$	−17.8262	−1.67694	−0.838472	+	0.544945i	$0.816551\pi$
$114$	0	0
$115$	−6.32232	−0.589559
$116$	0	0
$117$	−3.54078	−0.327345
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.71880	−0.338073
$122$	0	0
$123$	3.23378	0.291580
$124$	0	0
$125$	10.3171	0.922790
$126$	0	0
$127$	15.2718	1.35515	0.677577	−	0.735451i	$-0.263030\pi$
$127$	15.2718	1.35515	0.677577	+	0.735451i	$0.263030\pi$
$128$	0	0
$129$	−6.94414	−0.611397
$130$	0	0
$131$	−14.4870	−1.26574	−0.632869	−	0.774259i	$-0.718123\pi$
$131$	−14.4870	−1.26574	−0.632869	+	0.774259i	$0.718123\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	17.4120	1.49858
$136$	0	0
$137$	−0.0297066	−0.00253800	−0.00126900	−	0.999999i	$-0.500404\pi$
$137$	−0.0297066	−0.00253800	−0.00126900	+	0.999999i	$0.500404\pi$
$138$	0	0
$139$	14.5422	1.23346	0.616728	−	0.787176i	$-0.288458\pi$
$139$	14.5422	1.23346	0.616728	+	0.787176i	$0.288458\pi$
$140$	0	0
$141$	−18.2409	−1.53616
$142$	0	0
$143$	1.28120	0.107139
$144$	0	0
$145$	4.71880	0.391875
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.53718	0.371700	0.185850	−	0.982578i	$-0.440496\pi$
$149$	4.53718	0.371700	0.185850	+	0.982578i	$0.440496\pi$
$150$	0	0
$151$	−0.234343	−0.0190706	−0.00953530	−	0.999955i	$-0.503035\pi$
$151$	−0.234343	−0.0190706	−0.00953530	+	0.999955i	$0.503035\pi$
$152$	0	0
$153$	1.09901	0.0888500
$154$	0	0
$155$	0.956743	0.0768475
$156$	0	0
$157$	8.51013	0.679182	0.339591	−	0.940573i	$-0.389711\pi$
$157$	8.51013	0.679182	0.339591	+	0.940573i	$0.389711\pi$
$158$	0	0
$159$	3.90635	0.309794
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.0425	1.88315	0.941576	−	0.336799i	$-0.109344\pi$
$163$	24.0425	1.88315	0.941576	+	0.336799i	$0.109344\pi$
$164$	0	0
$165$	−10.5408	−0.820599
$166$	0	0
$167$	16.9648	1.31278	0.656389	−	0.754423i	$-0.272083\pi$
$167$	16.9648	1.31278	0.656389	+	0.754423i	$0.272083\pi$
$168$	0	0
$169$	−12.7746	−0.982659
$170$	0	0
$171$	13.1173	1.00311
$172$	0	0
$173$	−21.6442	−1.64558	−0.822790	−	0.568345i	$-0.807584\pi$
$173$	−21.6442	−1.64558	−0.822790	+	0.568345i	$0.807584\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−28.2178	−2.12098
$178$	0	0
$179$	2.40722	0.179924	0.0899620	−	0.995945i	$-0.471325\pi$
$179$	2.40722	0.179924	0.0899620	+	0.995945i	$0.471325\pi$
$180$	0	0
$181$	−10.5941	−0.787457	−0.393728	−	0.919227i	$-0.628815\pi$
$181$	−10.5941	−0.787457	−0.393728	+	0.919227i	$0.628815\pi$
$182$	0	0
$183$	27.1094	2.00398
$184$	0	0
$185$	5.60447	0.412049
$186$	0	0
$187$	−0.397668	−0.0290803
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.1650	1.74852	0.874259	−	0.485459i	$-0.161348\pi$
$191$	24.1650	1.74852	0.874259	+	0.485459i	$0.161348\pi$
$192$	0	0
$193$	13.4395	0.967399	0.483699	−	0.875234i	$-0.339293\pi$
$193$	13.4395	0.967399	0.483699	+	0.875234i	$0.339293\pi$
$194$	0	0
$195$	−1.85475	−0.132822
$196$	0	0
$197$	−5.67863	−0.404586	−0.202293	−	0.979325i	$-0.564839\pi$
$197$	−5.67863	−0.404586	−0.202293	+	0.979325i	$0.564839\pi$
$198$	0	0
$199$	−7.62840	−0.540763	−0.270381	−	0.962753i	$-0.587150\pi$
$199$	−7.62840	−0.540763	−0.270381	+	0.962753i	$0.587150\pi$
$200$	0	0
$201$	2.02161	0.142594
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.20798	0.0843691
$206$	0	0
$207$	39.0301	2.71278
$208$	0	0
$209$	−4.74637	−0.328313
$210$	0	0
$211$	−27.7742	−1.91205	−0.956027	−	0.293278i	$-0.905254\pi$
$211$	−27.7742	−1.91205	−0.956027	+	0.293278i	$0.905254\pi$
$212$	0	0
$213$	7.44688	0.510252
$214$	0	0
$215$	−2.59399	−0.176909
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.38080	0.498748
$220$	0	0
$221$	−0.0699735	−0.00470693
$222$	0	0
$223$	−21.8142	−1.46078	−0.730392	−	0.683028i	$-0.760663\pi$
$223$	−21.8142	−1.46078	−0.730392	+	0.683028i	$0.760663\pi$
$224$	0	0
$225$	−26.4048	−1.76032
$226$	0	0
$227$	22.3171	1.48124	0.740619	−	0.671925i	$-0.234532\pi$
$227$	22.3171	1.48124	0.740619	+	0.671925i	$0.234532\pi$
$228$	0	0
$229$	17.0617	1.12747	0.563735	−	0.825956i	$-0.309364\pi$
$229$	17.0617	1.12747	0.563735	+	0.825956i	$0.309364\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.8210	0.708910	0.354455	−	0.935073i	$-0.384666\pi$
$233$	10.8210	0.708910	0.354455	+	0.935073i	$0.384666\pi$
$234$	0	0
$235$	−6.81391	−0.444491
$236$	0	0
$237$	−13.0012	−0.844519
$238$	0	0
$239$	15.2886	0.988938	0.494469	−	0.869195i	$-0.335363\pi$
$239$	15.2886	0.988938	0.494469	+	0.869195i	$0.335363\pi$
$240$	0	0
$241$	−8.34989	−0.537863	−0.268932	−	0.963159i	$-0.586671\pi$
$241$	−8.34989	−0.537863	−0.268932	+	0.963159i	$0.586671\pi$
$242$	0	0
$243$	−35.1446	−2.25453
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.835170	−0.0531406
$248$	0	0
$249$	11.9558	0.757669
$250$	0	0
$251$	6.53151	0.412265	0.206133	−	0.978524i	$-0.433912\pi$
$251$	6.53151	0.412265	0.206133	+	0.978524i	$0.433912\pi$
$252$	0	0
$253$	−14.1227	−0.887885
$254$	0	0
$255$	0.575692	0.0360512
$256$	0	0
$257$	22.6124	1.41053	0.705263	−	0.708946i	$-0.250829\pi$
$257$	22.6124	1.41053	0.705263	+	0.708946i	$0.250829\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−29.1310	−1.80316
$262$	0	0
$263$	−27.1490	−1.67408	−0.837041	−	0.547141i	$-0.815716\pi$
$263$	−27.1490	−1.67408	−0.837041	+	0.547141i	$0.815716\pi$
$264$	0	0
$265$	1.45922	0.0896392
$266$	0	0
$267$	−56.3230	−3.44691
$268$	0	0
$269$	6.71785	0.409594	0.204797	−	0.978804i	$-0.434347\pi$
$269$	6.71785	0.409594	0.204797	+	0.978804i	$0.434347\pi$
$270$	0	0
$271$	−17.3430	−1.05351	−0.526756	−	0.850016i	$-0.676592\pi$
$271$	−17.3430	−1.05351	−0.526756	+	0.850016i	$0.676592\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.55433	0.576148
$276$	0	0
$277$	−1.30941	−0.0786750	−0.0393375	−	0.999226i	$-0.512525\pi$
$277$	−1.30941	−0.0786750	−0.0393375	+	0.999226i	$0.512525\pi$
$278$	0	0
$279$	−5.90635	−0.353604
$280$	0	0
$281$	−26.1959	−1.56272	−0.781358	−	0.624083i	$-0.785473\pi$
$281$	−26.1959	−1.56272	−0.781358	+	0.624083i	$0.785473\pi$
$282$	0	0
$283$	−25.1883	−1.49729	−0.748644	−	0.662972i	$-0.769295\pi$
$283$	−25.1883	−1.49729	−0.748644	+	0.662972i	$0.769295\pi$
$284$	0	0
$285$	6.87119	0.407014
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9783	−0.998722
$290$	0	0
$291$	−24.9045	−1.45993
$292$	0	0
$293$	14.9018	0.870574	0.435287	−	0.900292i	$-0.356647\pi$
$293$	14.9018	0.870574	0.435287	+	0.900292i	$0.356647\pi$
$294$	0	0
$295$	−10.5408	−0.613708
$296$	0	0
$297$	38.8945	2.25689
$298$	0	0
$299$	−2.48502	−0.143712
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.6580	1.70381
$304$	0	0
$305$	10.1267	0.579856
$306$	0	0
$307$	4.85884	0.277309	0.138654	−	0.990341i	$-0.455722\pi$
$307$	4.85884	0.277309	0.138654	+	0.990341i	$0.455722\pi$
$308$	0	0
$309$	−38.0637	−2.16537
$310$	0	0
$311$	−20.4260	−1.15825	−0.579125	−	0.815239i	$-0.696606\pi$
$311$	−20.4260	−1.15825	−0.579125	+	0.815239i	$0.696606\pi$
$312$	0	0
$313$	6.12936	0.346452	0.173226	−	0.984882i	$-0.444581\pi$
$313$	6.12936	0.346452	0.173226	+	0.984882i	$0.444581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.6190	−1.32658	−0.663288	−	0.748364i	$-0.730840\pi$
$317$	−23.6190	−1.32658	−0.663288	+	0.748364i	$0.730840\pi$
$318$	0	0
$319$	10.5408	0.590170
$320$	0	0
$321$	−4.61968	−0.257846
$322$	0	0
$323$	0.259227	0.0144238
$324$	0	0
$325$	1.68118	0.0932549
$326$	0	0
$327$	−46.1793	−2.55372
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.1115	−0.775639	−0.387820	−	0.921735i	$-0.626772\pi$
$331$	−14.1115	−0.775639	−0.387820	+	0.921735i	$0.626772\pi$
$332$	0	0
$333$	−34.5986	−1.89599
$334$	0	0
$335$	0.755176	0.0412597
$336$	0	0
$337$	−9.87695	−0.538032	−0.269016	−	0.963136i	$-0.586698\pi$
$337$	−9.87695	−0.538032	−0.269016	+	0.963136i	$0.586698\pi$
$338$	0	0
$339$	57.6459	3.13090
$340$	0	0
$341$	2.13716	0.115734
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.4450	1.10072
$346$	0	0
$347$	12.4267	0.667100	0.333550	−	0.942732i	$-0.391753\pi$
$347$	12.4267	0.667100	0.333550	+	0.942732i	$0.391753\pi$
$348$	0	0
$349$	−21.3226	−1.14137	−0.570685	−	0.821169i	$-0.693322\pi$
$349$	−21.3226	−1.14137	−0.570685	+	0.821169i	$0.693322\pi$
$350$	0	0
$351$	6.84387	0.365299
$352$	0	0
$353$	29.9471	1.59392	0.796961	−	0.604031i	$-0.206440\pi$
$353$	29.9471	1.59392	0.796961	+	0.604031i	$0.206440\pi$
$354$	0	0
$355$	2.78179	0.147642
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.8178	1.04594	0.522972	−	0.852350i	$-0.324823\pi$
$359$	19.8178	1.04594	0.522972	+	0.852350i	$0.324823\pi$
$360$	0	0
$361$	−15.9060	−0.837158
$362$	0	0
$363$	12.0258	0.631191
$364$	0	0
$365$	2.75710	0.144313
$366$	0	0
$367$	−22.0179	−1.14932	−0.574662	−	0.818391i	$-0.694866\pi$
$367$	−22.0179	−1.14932	−0.574662	+	0.818391i	$0.694866\pi$
$368$	0	0
$369$	−7.45735	−0.388214
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9.58545	−0.496316	−0.248158	−	0.968720i	$-0.579825\pi$
$373$	−9.58545	−0.496316	−0.248158	+	0.968720i	$0.579825\pi$
$374$	0	0
$375$	−33.3633	−1.72287
$376$	0	0
$377$	1.85475	0.0955246
$378$	0	0
$379$	29.2003	1.49992	0.749960	−	0.661483i	$-0.230073\pi$
$379$	29.2003	1.49992	0.749960	+	0.661483i	$0.230073\pi$
$380$	0	0
$381$	−49.3857	−2.53011
$382$	0	0
$383$	19.1820	0.980156	0.490078	−	0.871679i	$-0.336968\pi$
$383$	19.1820	0.980156	0.490078	+	0.871679i	$0.336968\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0137	0.814023
$388$	0	0
$389$	−8.19953	−0.415733	−0.207866	−	0.978157i	$-0.566652\pi$
$389$	−8.19953	−0.415733	−0.207866	+	0.978157i	$0.566652\pi$
$390$	0	0
$391$	0.771320	0.0390073
$392$	0	0
$393$	46.8479	2.36317
$394$	0	0
$395$	−4.85661	−0.244363
$396$	0	0
$397$	−20.7980	−1.04382	−0.521910	−	0.853000i	$-0.674780\pi$
$397$	−20.7980	−1.04382	−0.521910	+	0.853000i	$0.674780\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0627	−1.10176	−0.550878	−	0.834586i	$-0.685707\pi$
$401$	−22.0627	−1.10176	−0.550878	+	0.834586i	$0.685707\pi$
$402$	0	0
$403$	0.376053	0.0187325
$404$	0	0
$405$	−29.2815	−1.45501
$406$	0	0
$407$	12.5192	0.620552
$408$	0	0
$409$	−14.9955	−0.741481	−0.370741	−	0.928736i	$-0.620896\pi$
$409$	−14.9955	−0.741481	−0.370741	+	0.928736i	$0.620896\pi$
$410$	0	0
$411$	0.0960646	0.00473852
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.46611	0.219232
$416$	0	0
$417$	−47.0264	−2.30289
$418$	0	0
$419$	−11.2988	−0.551984	−0.275992	−	0.961160i	$-0.589006\pi$
$419$	−11.2988	−0.551984	−0.275992	+	0.961160i	$0.589006\pi$
$420$	0	0
$421$	17.8270	0.868834	0.434417	−	0.900712i	$-0.356955\pi$
$421$	17.8270	0.868834	0.434417	+	0.900712i	$0.356955\pi$
$422$	0	0
$423$	42.0649	2.04527
$424$	0	0
$425$	−0.521817	−0.0253118
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.14311	−0.200031
$430$	0	0
$431$	−21.3796	−1.02982	−0.514910	−	0.857244i	$-0.672175\pi$
$431$	−21.3796	−1.02982	−0.514910	+	0.857244i	$0.672175\pi$
$432$	0	0
$433$	−12.1329	−0.583070	−0.291535	−	0.956560i	$-0.594166\pi$
$433$	−12.1329	−0.583070	−0.291535	+	0.956560i	$0.594166\pi$
$434$	0	0
$435$	−15.2596	−0.731641
$436$	0	0
$437$	9.20611	0.440388
$438$	0	0
$439$	−27.7698	−1.32538	−0.662690	−	0.748894i	$-0.730585\pi$
$439$	−27.7698	−1.32538	−0.662690	+	0.748894i	$0.730585\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.57512	−0.122348	−0.0611738	−	0.998127i	$-0.519484\pi$
$443$	−2.57512	−0.122348	−0.0611738	+	0.998127i	$0.519484\pi$
$444$	0	0
$445$	−21.0395	−0.997368
$446$	0	0
$447$	−14.6723	−0.693974
$448$	0	0
$449$	11.7460	0.554330	0.277165	−	0.960822i	$-0.410605\pi$
$449$	11.7460	0.554330	0.277165	+	0.960822i	$0.410605\pi$
$450$	0	0
$451$	2.69837	0.127061
$452$	0	0
$453$	0.757816	0.0356053
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.4300	1.37668	0.688339	−	0.725389i	$-0.258340\pi$
$457$	29.4300	1.37668	0.688339	+	0.725389i	$0.258340\pi$
$458$	0	0
$459$	−2.12425	−0.0991516
$460$	0	0
$461$	25.6028	1.19244	0.596221	−	0.802820i	$-0.296668\pi$
$461$	25.6028	1.19244	0.596221	+	0.802820i	$0.296668\pi$
$462$	0	0
$463$	0.678892	0.0315508	0.0157754	−	0.999876i	$-0.494978\pi$
$463$	0.678892	0.0315508	0.0157754	+	0.999876i	$0.494978\pi$
$464$	0	0
$465$	−3.09390	−0.143476
$466$	0	0
$467$	−20.7314	−0.959336	−0.479668	−	0.877450i	$-0.659243\pi$
$467$	−20.7314	−0.959336	−0.479668	+	0.877450i	$0.659243\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−27.5199	−1.26805
$472$	0	0
$473$	−5.79441	−0.266427
$474$	0	0
$475$	−6.22815	−0.285767
$476$	0	0
$477$	−9.00834	−0.412464
$478$	0	0
$479$	−9.29134	−0.424532	−0.212266	−	0.977212i	$-0.568084\pi$
$479$	−9.29134	−0.424532	−0.212266	+	0.977212i	$0.568084\pi$
$480$	0	0
$481$	2.20287	0.100442
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.30309	−0.422432
$486$	0	0
$487$	−26.8311	−1.21583	−0.607916	−	0.794002i	$-0.707994\pi$
$487$	−26.8311	−1.21583	−0.607916	+	0.794002i	$0.707994\pi$
$488$	0	0
$489$	−77.7482	−3.51589
$490$	0	0
$491$	−32.3527	−1.46006	−0.730028	−	0.683417i	$-0.760493\pi$
$491$	−32.3527	−1.46006	−0.730028	+	0.683417i	$0.760493\pi$
$492$	0	0
$493$	−0.575692	−0.0259279
$494$	0	0
$495$	24.3078	1.09256
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.37578	−0.151121	−0.0755603	−	0.997141i	$-0.524075\pi$
$499$	−3.37578	−0.151121	−0.0755603	+	0.997141i	$0.524075\pi$
$500$	0	0
$501$	−54.8605	−2.45099
$502$	0	0
$503$	19.7796	0.881927	0.440963	−	0.897525i	$-0.354637\pi$
$503$	19.7796	0.881927	0.440963	+	0.897525i	$0.354637\pi$
$504$	0	0
$505$	11.0788	0.492999
$506$	0	0
$507$	41.3101	1.83465
$508$	0	0
$509$	−16.1815	−0.717233	−0.358617	−	0.933485i	$-0.616751\pi$
$509$	−16.1815	−0.717233	−0.358617	+	0.933485i	$0.616751\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−25.3541	−1.11941
$514$	0	0
$515$	−14.2187	−0.626553
$516$	0	0
$517$	−15.2208	−0.669410
$518$	0	0
$519$	69.9927	3.07234
$520$	0	0
$521$	−40.1162	−1.75752	−0.878762	−	0.477260i	$-0.841630\pi$
$521$	−40.1162	−1.75752	−0.878762	+	0.477260i	$0.841630\pi$
$522$	0	0
$523$	−31.0687	−1.35854	−0.679269	−	0.733889i	$-0.737703\pi$
$523$	−31.0687	−1.35854	−0.679269	+	0.733889i	$0.737703\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.116722	−0.00508450
$528$	0	0
$529$	4.39248	0.190977
$530$	0	0
$531$	65.0724	2.82390
$532$	0	0
$533$	0.474804	0.0205660
$534$	0	0
$535$	−1.72569	−0.0746080
$536$	0	0
$537$	−7.78442	−0.335923
$538$	0	0
$539$	0	0
$540$	0	0
$541$	39.7647	1.70962	0.854810	−	0.518942i	$-0.173674\pi$
$541$	39.7647	1.70962	0.854810	+	0.518942i	$0.173674\pi$
$542$	0	0
$543$	34.2592	1.47020
$544$	0	0
$545$	−17.2503	−0.738922
$546$	0	0
$547$	5.42639	0.232016	0.116008	−	0.993248i	$-0.462990\pi$
$547$	5.42639	0.232016	0.116008	+	0.993248i	$0.462990\pi$
$548$	0	0
$549$	−62.5163	−2.66813
$550$	0	0
$551$	−6.87119	−0.292722
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−18.1236	−0.769305
$556$	0	0
$557$	−5.08463	−0.215443	−0.107721	−	0.994181i	$-0.534355\pi$
$557$	−5.08463	−0.215443	−0.107721	+	0.994181i	$0.534355\pi$
$558$	0	0
$559$	−1.01958	−0.0431237
$560$	0	0
$561$	1.28597	0.0542937
$562$	0	0
$563$	−21.3546	−0.899988	−0.449994	−	0.893031i	$-0.648574\pi$
$563$	−21.3546	−0.899988	−0.449994	+	0.893031i	$0.648574\pi$
$564$	0	0
$565$	21.5337	0.905929
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.55741	0.149134	0.0745672	−	0.997216i	$-0.476242\pi$
$569$	3.55741	0.149134	0.0745672	+	0.997216i	$0.476242\pi$
$570$	0	0
$571$	41.0461	1.71773	0.858863	−	0.512205i	$-0.171171\pi$
$571$	41.0461	1.71773	0.858863	+	0.512205i	$0.171171\pi$
$572$	0	0
$573$	−78.1444	−3.26453
$574$	0	0
$575$	−18.5317	−0.772824
$576$	0	0
$577$	4.88626	0.203417	0.101709	−	0.994814i	$-0.467569\pi$
$577$	4.88626	0.203417	0.101709	+	0.994814i	$0.467569\pi$
$578$	0	0
$579$	−43.4605	−1.80616
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.25958	0.134998
$584$	0	0
$585$	4.27720	0.176840
$586$	0	0
$587$	−27.4954	−1.13486	−0.567429	−	0.823422i	$-0.692062\pi$
$587$	−27.4954	−1.13486	−0.567429	+	0.823422i	$0.692062\pi$
$588$	0	0
$589$	−1.39314	−0.0574034
$590$	0	0
$591$	18.3634	0.755371
$592$	0	0
$593$	19.0415	0.781942	0.390971	−	0.920403i	$-0.372139\pi$
$593$	19.0415	0.781942	0.390971	+	0.920403i	$0.372139\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.6686	1.00962
$598$	0	0
$599$	9.16347	0.374409	0.187205	−	0.982321i	$-0.440057\pi$
$599$	9.16347	0.374409	0.187205	+	0.982321i	$0.440057\pi$
$600$	0	0
$601$	−25.0276	−1.02090	−0.510448	−	0.859908i	$-0.670521\pi$
$601$	−25.0276	−1.02090	−0.510448	+	0.859908i	$0.670521\pi$
$602$	0	0
$603$	−4.66200	−0.189851
$604$	0	0
$605$	4.49225	0.182636
$606$	0	0
$607$	−29.1581	−1.18349	−0.591746	−	0.806125i	$-0.701561\pi$
$607$	−29.1581	−1.18349	−0.591746	+	0.806125i	$0.701561\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.67825	−0.108350
$612$	0	0
$613$	−36.9437	−1.49214	−0.746070	−	0.665867i	$-0.768062\pi$
$613$	−36.9437	−1.49214	−0.746070	+	0.665867i	$0.768062\pi$
$614$	0	0
$615$	−3.90635	−0.157519
$616$	0	0
$617$	41.6304	1.67598	0.837989	−	0.545688i	$-0.183731\pi$
$617$	41.6304	1.67598	0.837989	+	0.545688i	$0.183731\pi$
$618$	0	0
$619$	−32.8517	−1.32042	−0.660210	−	0.751081i	$-0.729533\pi$
$619$	−32.8517	−1.32042	−0.660210	+	0.751081i	$0.729533\pi$
$620$	0	0
$621$	−75.4402	−3.02731
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.24101	0.209640
$626$	0	0
$627$	15.3487	0.612969
$628$	0	0
$629$	−0.683743	−0.0272626
$630$	0	0
$631$	21.7461	0.865697	0.432848	−	0.901467i	$-0.357509\pi$
$631$	21.7461	0.865697	0.432848	+	0.901467i	$0.357509\pi$
$632$	0	0
$633$	89.8157	3.56985
$634$	0	0
$635$	−18.4481	−0.732090
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17.1731	−0.679356
$640$	0	0
$641$	−9.59567	−0.379006	−0.189503	−	0.981880i	$-0.560688\pi$
$641$	−9.59567	−0.379006	−0.189503	+	0.981880i	$0.560688\pi$
$642$	0	0
$643$	−32.0494	−1.26390	−0.631952	−	0.775008i	$-0.717746\pi$
$643$	−32.0494	−1.26390	−0.631952	+	0.775008i	$0.717746\pi$
$644$	0	0
$645$	8.38840	0.330293
$646$	0	0
$647$	−4.46970	−0.175722	−0.0878611	−	0.996133i	$-0.528003\pi$
$647$	−4.46970	−0.175722	−0.0878611	+	0.996133i	$0.528003\pi$
$648$	0	0
$649$	−23.5458	−0.924254
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.31301	0.325313	0.162657	−	0.986683i	$-0.447994\pi$
$653$	8.31301	0.325313	0.162657	+	0.986683i	$0.447994\pi$
$654$	0	0
$655$	17.5001	0.683785
$656$	0	0
$657$	−17.0207	−0.664040
$658$	0	0
$659$	18.7498	0.730387	0.365193	−	0.930932i	$-0.381003\pi$
$659$	18.7498	0.730387	0.365193	+	0.930932i	$0.381003\pi$
$660$	0	0
$661$	−21.5180	−0.836951	−0.418476	−	0.908228i	$-0.637436\pi$
$661$	−21.5180	−0.836951	−0.418476	+	0.908228i	$0.637436\pi$
$662$	0	0
$663$	0.226279	0.00878795
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.4450	−0.791633
$668$	0	0
$669$	70.5423	2.72732
$670$	0	0
$671$	22.6209	0.873272
$672$	0	0
$673$	−36.4388	−1.40461	−0.702307	−	0.711874i	$-0.747847\pi$
$673$	−36.4388	−1.40461	−0.702307	+	0.711874i	$0.747847\pi$
$674$	0	0
$675$	51.0371	1.96442
$676$	0	0
$677$	−4.23741	−0.162857	−0.0814285	−	0.996679i	$-0.525948\pi$
$677$	−4.23741	−0.162857	−0.0814285	+	0.996679i	$0.525948\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−72.1687	−2.76551
$682$	0	0
$683$	−18.1298	−0.693718	−0.346859	−	0.937917i	$-0.612752\pi$
$683$	−18.1298	−0.693718	−0.346859	+	0.937917i	$0.612752\pi$
$684$	0	0
$685$	0.0358850	0.00137110
$686$	0	0
$687$	−55.1739	−2.10501
$688$	0	0
$689$	0.573555	0.0218507
$690$	0	0
$691$	−39.2798	−1.49427	−0.747136	−	0.664671i	$-0.768572\pi$
$691$	−39.2798	−1.49427	−0.747136	+	0.664671i	$0.768572\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.5668	−0.666345
$696$	0	0
$697$	−0.147373	−0.00558216
$698$	0	0
$699$	−34.9929	−1.32355
$700$	0	0
$701$	−18.2904	−0.690819	−0.345409	−	0.938452i	$-0.612260\pi$
$701$	−18.2904	−0.690819	−0.345409	+	0.938452i	$0.612260\pi$
$702$	0	0
$703$	−8.16083	−0.307792
$704$	0	0
$705$	22.0347	0.829875
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21.7329	−0.816195	−0.408097	−	0.912938i	$-0.633808\pi$
$709$	−21.7329	−0.816195	−0.408097	+	0.912938i	$0.633808\pi$
$710$	0	0
$711$	29.9818	1.12440
$712$	0	0
$713$	−4.14525	−0.155241
$714$	0	0
$715$	−1.54766	−0.0578793
$716$	0	0
$717$	−49.4400	−1.84637
$718$	0	0
$719$	−44.7184	−1.66771	−0.833857	−	0.551980i	$-0.813872\pi$
$719$	−44.7184	−1.66771	−0.833857	+	0.551980i	$0.813872\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	27.0017	1.00420
$724$	0	0
$725$	13.8315	0.513690
$726$	0	0
$727$	47.6991	1.76906	0.884531	−	0.466481i	$-0.154478\pi$
$727$	47.6991	1.76906	0.884531	+	0.466481i	$0.154478\pi$
$728$	0	0
$729$	40.9299	1.51592
$730$	0	0
$731$	0.316466	0.0117049
$732$	0	0
$733$	−38.2542	−1.41295	−0.706475	−	0.707738i	$-0.749716\pi$
$733$	−38.2542	−1.41295	−0.706475	+	0.707738i	$0.749716\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.68690	0.0621377
$738$	0	0
$739$	−13.8172	−0.508272	−0.254136	−	0.967168i	$-0.581791\pi$
$739$	−13.8172	−0.508272	−0.254136	+	0.967168i	$0.581791\pi$
$740$	0	0
$741$	2.70076	0.0992148
$742$	0	0
$743$	13.6813	0.501917	0.250959	−	0.967998i	$-0.419254\pi$
$743$	13.6813	0.501917	0.250959	+	0.967998i	$0.419254\pi$
$744$	0	0
$745$	−5.48084	−0.200802
$746$	0	0
$747$	−27.5710	−1.00877
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.4222	−1.07363	−0.536816	−	0.843699i	$-0.680373\pi$
$751$	−29.4222	−1.07363	−0.536816	+	0.843699i	$0.680373\pi$
$752$	0	0
$753$	−21.1215	−0.769709
$754$	0	0
$755$	0.283083	0.0103024
$756$	0	0
$757$	−49.0160	−1.78152	−0.890759	−	0.454476i	$-0.849827\pi$
$757$	−49.0160	−1.78152	−0.890759	+	0.454476i	$0.849827\pi$
$758$	0	0
$759$	45.6697	1.65770
$760$	0	0
$761$	−23.6872	−0.858661	−0.429330	−	0.903148i	$-0.641250\pi$
$761$	−23.6872	−0.858661	−0.429330	+	0.903148i	$0.641250\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.32759	−0.0479991
$766$	0	0
$767$	−4.14311	−0.149599
$768$	0	0
$769$	18.2561	0.658331	0.329166	−	0.944272i	$-0.393233\pi$
$769$	18.2561	0.658331	0.329166	+	0.944272i	$0.393233\pi$
$770$	0	0
$771$	−73.1237	−2.63349
$772$	0	0
$773$	−25.5675	−0.919598	−0.459799	−	0.888023i	$-0.652079\pi$
$773$	−25.5675	−0.919598	−0.459799	+	0.888023i	$0.652079\pi$
$774$	0	0
$775$	2.80436	0.100736
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.75898	−0.0630219
$780$	0	0
$781$	6.21391	0.222351
$782$	0	0
$783$	56.3065	2.01223
$784$	0	0
$785$	−10.2801	−0.366912
$786$	0	0
$787$	2.51622	0.0896936	0.0448468	−	0.998994i	$-0.485720\pi$
$787$	2.51622	0.0896936	0.0448468	+	0.998994i	$0.485720\pi$
$788$	0	0
$789$	87.7941	3.12555
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.98037	0.141347
$794$	0	0
$795$	−4.71880	−0.167359
$796$	0	0
$797$	41.1615	1.45801	0.729007	−	0.684506i	$-0.239982\pi$
$797$	41.1615	1.45801	0.729007	+	0.684506i	$0.239982\pi$
$798$	0	0
$799$	0.831295	0.0294091
$800$	0	0
$801$	129.885	4.58927
$802$	0	0
$803$	6.15877	0.217338
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.7241	−0.764723
$808$	0	0
$809$	32.5513	1.14444	0.572221	−	0.820100i	$-0.306082\pi$
$809$	32.5513	1.14444	0.572221	+	0.820100i	$0.306082\pi$
$810$	0	0
$811$	48.0992	1.68899	0.844496	−	0.535562i	$-0.179900\pi$
$811$	48.0992	1.68899	0.844496	+	0.535562i	$0.179900\pi$
$812$	0	0
$813$	56.0835	1.96693
$814$	0	0
$815$	−29.0429	−1.01733
$816$	0	0
$817$	3.77718	0.132147
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.9418	1.67318	0.836591	−	0.547828i	$-0.184545\pi$
$821$	47.9418	1.67318	0.836591	+	0.547828i	$0.184545\pi$
$822$	0	0
$823$	11.2491	0.392118	0.196059	−	0.980592i	$-0.437186\pi$
$823$	11.2491	0.392118	0.196059	+	0.980592i	$0.437186\pi$
$824$	0	0
$825$	−30.8966	−1.07568
$826$	0	0
$827$	−6.02903	−0.209650	−0.104825	−	0.994491i	$-0.533428\pi$
$827$	−6.02903	−0.209650	−0.104825	+	0.994491i	$0.533428\pi$
$828$	0	0
$829$	−56.1382	−1.94976	−0.974880	−	0.222731i	$-0.928503\pi$
$829$	−56.1382	−1.94976	−0.974880	+	0.222731i	$0.928503\pi$
$830$	0	0
$831$	4.23436	0.146888
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−20.4932	−0.709197
$836$	0	0
$837$	11.4162	0.394602
$838$	0	0
$839$	34.4399	1.18900	0.594498	−	0.804097i	$-0.297351\pi$
$839$	34.4399	1.18900	0.594498	+	0.804097i	$0.297351\pi$
$840$	0	0
$841$	−13.7404	−0.473808
$842$	0	0
$843$	84.7118	2.91763
$844$	0	0
$845$	15.4314	0.530858
$846$	0	0
$847$	0	0
$848$	0	0
$849$	81.4535	2.79548
$850$	0	0
$851$	−24.2823	−0.832386
$852$	0	0
$853$	28.8984	0.989462	0.494731	−	0.869046i	$-0.335267\pi$
$853$	28.8984	0.989462	0.494731	+	0.869046i	$0.335267\pi$
$854$	0	0
$855$	−15.8455	−0.541904
$856$	0	0
$857$	2.46633	0.0842482	0.0421241	−	0.999112i	$-0.486588\pi$
$857$	2.46633	0.0842482	0.0421241	+	0.999112i	$0.486588\pi$
$858$	0	0
$859$	−19.5051	−0.665507	−0.332753	−	0.943014i	$-0.607978\pi$
$859$	−19.5051	−0.665507	−0.332753	+	0.943014i	$0.607978\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.0532	0.342215	0.171108	−	0.985252i	$-0.445265\pi$
$863$	10.0532	0.342215	0.171108	+	0.985252i	$0.445265\pi$
$864$	0	0
$865$	26.1458	0.888985
$866$	0	0
$867$	54.9041	1.86464
$868$	0	0
$869$	−10.8486	−0.368014
$870$	0	0
$871$	0.296826	0.0100576
$872$	0	0
$873$	57.4316	1.94377
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.5198	0.355229	0.177614	−	0.984100i	$-0.443162\pi$
$877$	10.5198	0.355229	0.177614	+	0.984100i	$0.443162\pi$
$878$	0	0
$879$	−48.1893	−1.62538
$880$	0	0
$881$	−10.4063	−0.350596	−0.175298	−	0.984515i	$-0.556089\pi$
$881$	−10.4063	−0.350596	−0.175298	+	0.984515i	$0.556089\pi$
$882$	0	0
$883$	45.5590	1.53318	0.766590	−	0.642136i	$-0.221952\pi$
$883$	45.5590	1.53318	0.766590	+	0.642136i	$0.221952\pi$
$884$	0	0
$885$	34.0866	1.14581
$886$	0	0
$887$	16.6990	0.560697	0.280348	−	0.959898i	$-0.409550\pi$
$887$	16.6990	0.560697	0.280348	+	0.959898i	$0.409550\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−65.4084	−2.19127
$892$	0	0
$893$	9.92193	0.332025
$894$	0	0
$895$	−2.90788	−0.0971996
$896$	0	0
$897$	8.03602	0.268315
$898$	0	0
$899$	3.09390	0.103187
$900$	0	0
$901$	−0.178024	−0.00593085
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.7975	0.425405
$906$	0	0
$907$	9.46860	0.314400	0.157200	−	0.987567i	$-0.449753\pi$
$907$	9.46860	0.314400	0.157200	+	0.987567i	$0.449753\pi$
$908$	0	0
$909$	−68.3936	−2.26847
$910$	0	0
$911$	−43.0113	−1.42503	−0.712514	−	0.701658i	$-0.752443\pi$
$911$	−43.0113	−1.42503	−0.712514	+	0.701658i	$0.752443\pi$
$912$	0	0
$913$	9.97631	0.330168
$914$	0	0
$915$	−32.7477	−1.08260
$916$	0	0
$917$	0	0
$918$	0	0
$919$	13.3859	0.441559	0.220779	−	0.975324i	$-0.429140\pi$
$919$	13.3859	0.441559	0.220779	+	0.975324i	$0.429140\pi$
$920$	0	0
$921$	−15.7124	−0.517743
$922$	0	0
$923$	1.09340	0.0359896
$924$	0	0
$925$	16.4275	0.540134
$926$	0	0
$927$	87.7779	2.88300
$928$	0	0
$929$	−21.8532	−0.716982	−0.358491	−	0.933533i	$-0.616709\pi$
$929$	−21.8532	−0.716982	−0.358491	+	0.933533i	$0.616709\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	66.0531	2.16248
$934$	0	0
$935$	0.480376	0.0157100
$936$	0	0
$937$	−2.86181	−0.0934913	−0.0467456	−	0.998907i	$-0.514885\pi$
$937$	−2.86181	−0.0934913	−0.0467456	+	0.998907i	$0.514885\pi$
$938$	0	0
$939$	−19.8210	−0.646834
$940$	0	0
$941$	12.0554	0.392995	0.196498	−	0.980504i	$-0.437043\pi$
$941$	12.0554	0.392995	0.196498	+	0.980504i	$0.437043\pi$
$942$	0	0
$943$	−5.23378	−0.170435
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.9301	1.62251	0.811255	−	0.584693i	$-0.198785\pi$
$947$	49.9301	1.62251	0.811255	+	0.584693i	$0.198785\pi$
$948$	0	0
$949$	1.08370	0.0351782
$950$	0	0
$951$	76.3788	2.47675
$952$	0	0
$953$	30.6051	0.991396	0.495698	−	0.868495i	$-0.334912\pi$
$953$	30.6051	0.991396	0.495698	+	0.868495i	$0.334912\pi$
$954$	0	0
$955$	−29.1909	−0.944595
$956$	0	0
$957$	−34.0866	−1.10186
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.3727	−0.979765
$962$	0	0
$963$	10.6533	0.343299
$964$	0	0
$965$	−16.2347	−0.522614
$966$	0	0
$967$	3.19204	0.102649	0.0513246	−	0.998682i	$-0.483656\pi$
$967$	3.19204	0.102649	0.0513246	+	0.998682i	$0.483656\pi$
$968$	0	0
$969$	−0.838282	−0.0269295
$970$	0	0
$971$	29.7007	0.953142	0.476571	−	0.879136i	$-0.341879\pi$
$971$	29.7007	0.953142	0.476571	+	0.879136i	$0.341879\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.43656	−0.174109
$976$	0	0
$977$	31.0377	0.992982	0.496491	−	0.868042i	$-0.334621\pi$
$977$	31.0377	0.992982	0.496491	+	0.868042i	$0.334621\pi$
$978$	0	0
$979$	−46.9977	−1.50205
$980$	0	0
$981$	106.493	3.40006
$982$	0	0
$983$	4.26197	0.135936	0.0679679	−	0.997688i	$-0.478348\pi$
$983$	4.26197	0.135936	0.0679679	+	0.997688i	$0.478348\pi$
$984$	0	0
$985$	6.85968	0.218568
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.2389	0.357376
$990$	0	0
$991$	51.6861	1.64186	0.820932	−	0.571026i	$-0.193454\pi$
$991$	51.6861	1.64186	0.820932	+	0.571026i	$0.193454\pi$
$992$	0	0
$993$	45.6336	1.44814
$994$	0	0
$995$	9.21497	0.292134
$996$	0	0
$997$	−24.3253	−0.770391	−0.385195	−	0.922835i	$-0.625866\pi$
$997$	−24.3253	−0.770391	−0.385195	+	0.922835i	$0.625866\pi$
$998$	0	0
$999$	66.8746	2.11582

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.e.1.5	✓	5	7.6	odd	2
4592.2.a.bd.1.1		5	28.27	even	2
8036.2.a.j.1.1		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)

\(f(q)\)	\(=\)	\(q-3.23378 q^{3} -1.20798 q^{5} +7.45735 q^{9} +O(q^{10})\)	\(q-3.23378 q^{3} -1.20798 q^{5} +7.45735 q^{9} -2.69837 q^{11} -0.474804 q^{13} +3.90635 q^{15} +0.147373 q^{17} +1.75898 q^{19} +5.23378 q^{23} -3.54078 q^{25} -14.4141 q^{27} -3.90635 q^{29} -0.792018 q^{31} +8.72594 q^{33} -4.63953 q^{37} +1.53541 q^{39} -1.00000 q^{41} +2.14737 q^{43} -9.00834 q^{45} +5.64074 q^{47} -0.476573 q^{51} -1.20798 q^{53} +3.25958 q^{55} -5.68815 q^{57} +8.72594 q^{59} -8.38319 q^{61} +0.573555 q^{65} -0.625155 q^{67} -16.9249 q^{69} -2.30284 q^{71} -2.28240 q^{73} +11.4501 q^{75} +4.02043 q^{79} +24.2400 q^{81} -3.69716 q^{83} -0.178024 q^{85} +12.6323 q^{87} +17.4171 q^{89} +2.56121 q^{93} -2.12481 q^{95} +7.70135 q^{97} -20.1227 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9	\( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9} - 7 q^{13} + 3 q^{15} + 3 q^{17} - 10 q^{19} + 12 q^{23} + 2 q^{25} - 14 q^{27} - 3 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} + 7 q^{39} - 5 q^{41} + 13 q^{43} + 3 q^{45} - 9 q^{47} + 9 q^{51} - 3 q^{53} - 9 q^{55} - 11 q^{57} + 3 q^{59} - 16 q^{61} + 3 q^{65} + 19 q^{67} - 24 q^{69} - 12 q^{71} - 4 q^{73} - 8 q^{75} + 28 q^{79} + 5 q^{81} - 18 q^{83} - 15 q^{85} + 6 q^{87} + q^{93} + 3 q^{95} - 4 q^{97} - 33 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 3 * q^5 + 5 * q^9 - 7 * q^13 + 3 * q^15 + 3 * q^17 - 10 * q^19 + 12 * q^23 + 2 * q^25 - 14 * q^27 - 3 * q^29 - 7 * q^31 + 3 * q^33 + q^37 + 7 * q^39 - 5 * q^41 + 13 * q^43 + 3 * q^45 - 9 * q^47 + 9 * q^51 - 3 * q^53 - 9 * q^55 - 11 * q^57 + 3 * q^59 - 16 * q^61 + 3 * q^65 + 19 * q^67 - 24 * q^69 - 12 * q^71 - 4 * q^73 - 8 * q^75 + 28 * q^79 + 5 * q^81 - 18 * q^83 - 15 * q^85 + 6 * q^87 + q^93 + 3 * q^95 - 4 * q^97 - 33 * q^99

Introduction

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