LMFDB - Embedded newform 4024.2.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79909 q^{3} +0.0213618 q^{5} -3.47965 q^{7} +0.236734 q^{9} +O(q^{10})$	$q-1.79909 q^{3} +0.0213618 q^{5} -3.47965 q^{7} +0.236734 q^{9} -4.90645 q^{11} -0.505790 q^{13} -0.0384319 q^{15} +6.24486 q^{17} +5.27746 q^{19} +6.26021 q^{21} +3.59215 q^{23} -4.99954 q^{25} +4.97137 q^{27} +6.29152 q^{29} -0.796985 q^{31} +8.82715 q^{33} -0.0743317 q^{35} +4.25043 q^{37} +0.909962 q^{39} +9.96089 q^{41} -0.416794 q^{43} +0.00505707 q^{45} +0.618326 q^{47} +5.10796 q^{49} -11.2351 q^{51} +0.914981 q^{53} -0.104811 q^{55} -9.49464 q^{57} -10.6530 q^{59} -7.61687 q^{61} -0.823751 q^{63} -0.0108046 q^{65} +3.27779 q^{67} -6.46262 q^{69} -12.8188 q^{71} -12.7902 q^{73} +8.99464 q^{75} +17.0727 q^{77} -0.878524 q^{79} -9.65416 q^{81} -12.0340 q^{83} +0.133402 q^{85} -11.3190 q^{87} +12.6933 q^{89} +1.75997 q^{91} +1.43385 q^{93} +0.112736 q^{95} -5.12585 q^{97} -1.16152 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−1.79909	−1.03871	−0.519353	−	0.854560i	$-0.673827\pi$
$3$	−1.79909	−1.03871	−0.519353	+	0.854560i	$0.673827\pi$
$4$	0	0
$5$	0.0213618	0.00955330	0.00477665	−	0.999989i	$-0.498480\pi$
$5$	0.0213618	0.00955330	0.00477665	+	0.999989i	$0.498480\pi$
$6$	0	0
$7$	−3.47965	−1.31518	−0.657592	−	0.753374i	$-0.728425\pi$
$7$	−3.47965	−1.31518	−0.657592	+	0.753374i	$0.728425\pi$
$8$	0	0
$9$	0.236734	0.0789113
$10$	0	0
$11$	−4.90645	−1.47935	−0.739675	−	0.672965i	$-0.765021\pi$
$11$	−4.90645	−1.47935	−0.739675	+	0.672965i	$0.765021\pi$
$12$	0	0
$13$	−0.505790	−0.140281	−0.0701404	−	0.997537i	$-0.522345\pi$
$13$	−0.505790	−0.140281	−0.0701404	+	0.997537i	$0.522345\pi$
$14$	0	0
$15$	−0.0384319	−0.00992308
$16$	0	0
$17$	6.24486	1.51460	0.757301	−	0.653067i	$-0.226518\pi$
$17$	6.24486	1.51460	0.757301	+	0.653067i	$0.226518\pi$
$18$	0	0
$19$	5.27746	1.21073	0.605366	−	0.795947i	$-0.293027\pi$
$19$	5.27746	1.21073	0.605366	+	0.795947i	$0.293027\pi$
$20$	0	0
$21$	6.26021	1.36609
$22$	0	0
$23$	3.59215	0.749016	0.374508	−	0.927224i	$-0.377812\pi$
$23$	3.59215	0.749016	0.374508	+	0.927224i	$0.377812\pi$
$24$	0	0
$25$	−4.99954	−0.999909
$26$	0	0
$27$	4.97137	0.956741
$28$	0	0
$29$	6.29152	1.16831	0.584153	−	0.811644i	$-0.301427\pi$
$29$	6.29152	1.16831	0.584153	+	0.811644i	$0.301427\pi$
$30$	0	0
$31$	−0.796985	−0.143143	−0.0715714	−	0.997435i	$-0.522801\pi$
$31$	−0.796985	−0.143143	−0.0715714	+	0.997435i	$0.522801\pi$
$32$	0	0
$33$	8.82715	1.53661
$34$	0	0
$35$	−0.0743317	−0.0125643
$36$	0	0
$37$	4.25043	0.698766	0.349383	−	0.936980i	$-0.386391\pi$
$37$	4.25043	0.698766	0.349383	+	0.936980i	$0.386391\pi$
$38$	0	0
$39$	0.909962	0.145711
$40$	0	0
$41$	9.96089	1.55563	0.777815	−	0.628494i	$-0.216328\pi$
$41$	9.96089	1.55563	0.777815	+	0.628494i	$0.216328\pi$
$42$	0	0
$43$	−0.416794	−0.0635606	−0.0317803	−	0.999495i	$-0.510118\pi$
$43$	−0.416794	−0.0635606	−0.0317803	+	0.999495i	$0.510118\pi$
$44$	0	0
$45$	0.00505707	0.000753864 0
$46$	0	0
$47$	0.618326	0.0901921	0.0450961	−	0.998983i	$-0.485641\pi$
$47$	0.618326	0.0901921	0.0450961	+	0.998983i	$0.485641\pi$
$48$	0	0
$49$	5.10796	0.729709
$50$	0	0
$51$	−11.2351	−1.57323
$52$	0	0
$53$	0.914981	0.125682	0.0628412	−	0.998024i	$-0.479984\pi$
$53$	0.914981	0.125682	0.0628412	+	0.998024i	$0.479984\pi$
$54$	0	0
$55$	−0.104811	−0.0141327
$56$	0	0
$57$	−9.49464	−1.25760
$58$	0	0
$59$	−10.6530	−1.38691	−0.693454	−	0.720501i	$-0.743912\pi$
$59$	−10.6530	−1.38691	−0.693454	+	0.720501i	$0.743912\pi$
$60$	0	0
$61$	−7.61687	−0.975240	−0.487620	−	0.873056i	$-0.662135\pi$
$61$	−7.61687	−0.975240	−0.487620	+	0.873056i	$0.662135\pi$
$62$	0	0
$63$	−0.823751	−0.103783
$64$	0	0
$65$	−0.0108046	−0.00134014
$66$	0	0
$67$	3.27779	0.400445	0.200223	−	0.979750i	$-0.435833\pi$
$67$	3.27779	0.400445	0.200223	+	0.979750i	$0.435833\pi$
$68$	0	0
$69$	−6.46262	−0.778008
$70$	0	0
$71$	−12.8188	−1.52132	−0.760658	−	0.649152i	$-0.775124\pi$
$71$	−12.8188	−1.52132	−0.760658	+	0.649152i	$0.775124\pi$
$72$	0	0
$73$	−12.7902	−1.49698	−0.748491	−	0.663144i	$-0.769222\pi$
$73$	−12.7902	−1.49698	−0.748491	+	0.663144i	$0.769222\pi$
$74$	0	0
$75$	8.99464	1.03861
$76$	0	0
$77$	17.0727	1.94562
$78$	0	0
$79$	−0.878524	−0.0988417	−0.0494208	−	0.998778i	$-0.515738\pi$
$79$	−0.878524	−0.0988417	−0.0494208	+	0.998778i	$0.515738\pi$
$80$	0	0
$81$	−9.65416	−1.07268
$82$	0	0
$83$	−12.0340	−1.32090	−0.660451	−	0.750869i	$-0.729635\pi$
$83$	−12.0340	−1.32090	−0.660451	+	0.750869i	$0.729635\pi$
$84$	0	0
$85$	0.133402	0.0144694
$86$	0	0
$87$	−11.3190	−1.21353
$88$	0	0
$89$	12.6933	1.34548	0.672741	−	0.739878i	$-0.265117\pi$
$89$	12.6933	1.34548	0.672741	+	0.739878i	$0.265117\pi$
$90$	0	0
$91$	1.75997	0.184495
$92$	0	0
$93$	1.43385	0.148683
$94$	0	0
$95$	0.112736	0.0115665
$96$	0	0
$97$	−5.12585	−0.520452	−0.260226	−	0.965548i	$-0.583797\pi$
$97$	−5.12585	−0.520452	−0.260226	+	0.965548i	$0.583797\pi$
$98$	0	0
$99$	−1.16152	−0.116737
$100$	0	0
$101$	0.879553	0.0875188	0.0437594	−	0.999042i	$-0.486067\pi$
$101$	0.879553	0.0875188	0.0437594	+	0.999042i	$0.486067\pi$
$102$	0	0
$103$	−19.0962	−1.88161	−0.940803	−	0.338953i	$-0.889927\pi$
$103$	−19.0962	−1.88161	−0.940803	+	0.338953i	$0.889927\pi$
$104$	0	0
$105$	0.133730	0.0130507
$106$	0	0
$107$	1.72994	0.167240	0.0836200	−	0.996498i	$-0.473352\pi$
$107$	1.72994	0.167240	0.0836200	+	0.996498i	$0.473352\pi$
$108$	0	0
$109$	11.8583	1.13582	0.567912	−	0.823090i	$-0.307752\pi$
$109$	11.8583	1.13582	0.567912	+	0.823090i	$0.307752\pi$
$110$	0	0
$111$	−7.64692	−0.725813
$112$	0	0
$113$	−4.74880	−0.446730	−0.223365	−	0.974735i	$-0.571704\pi$
$113$	−4.74880	−0.446730	−0.223365	+	0.974735i	$0.571704\pi$
$114$	0	0
$115$	0.0767350	0.00715558
$116$	0	0
$117$	−0.119738	−0.0110697
$118$	0	0
$119$	−21.7299	−1.99198
$120$	0	0
$121$	13.0732	1.18847
$122$	0	0
$123$	−17.9206	−1.61584
$124$	0	0
$125$	−0.213609	−0.0191057
$126$	0	0
$127$	8.82038	0.782682	0.391341	−	0.920246i	$-0.372011\pi$
$127$	8.82038	0.782682	0.391341	+	0.920246i	$0.372011\pi$
$128$	0	0
$129$	0.749852	0.0660208
$130$	0	0
$131$	18.8990	1.65121	0.825606	−	0.564247i	$-0.190834\pi$
$131$	18.8990	1.65121	0.825606	+	0.564247i	$0.190834\pi$
$132$	0	0
$133$	−18.3637	−1.59234
$134$	0	0
$135$	0.106198	0.00914003
$136$	0	0
$137$	−14.8353	−1.26746	−0.633731	−	0.773553i	$-0.718477\pi$
$137$	−14.8353	−1.26746	−0.633731	+	0.773553i	$0.718477\pi$
$138$	0	0
$139$	−21.2112	−1.79911	−0.899557	−	0.436804i	$-0.856110\pi$
$139$	−21.2112	−1.79911	−0.899557	+	0.436804i	$0.856110\pi$
$140$	0	0
$141$	−1.11243	−0.0936831
$142$	0	0
$143$	2.48163	0.207524
$144$	0	0
$145$	0.134398	0.0111612
$146$	0	0
$147$	−9.18969	−0.757953
$148$	0	0
$149$	−19.4781	−1.59571	−0.797855	−	0.602850i	$-0.794032\pi$
$149$	−19.4781	−1.59571	−0.797855	+	0.602850i	$0.794032\pi$
$150$	0	0
$151$	19.8867	1.61835	0.809177	−	0.587565i	$-0.199913\pi$
$151$	19.8867	1.61835	0.809177	+	0.587565i	$0.199913\pi$
$152$	0	0
$153$	1.47837	0.119519
$154$	0	0
$155$	−0.0170251	−0.00136749
$156$	0	0
$157$	9.18761	0.733251	0.366625	−	0.930369i	$-0.380513\pi$
$157$	9.18761	0.733251	0.366625	+	0.930369i	$0.380513\pi$
$158$	0	0
$159$	−1.64614	−0.130547
$160$	0	0
$161$	−12.4994	−0.985094
$162$	0	0
$163$	−3.91496	−0.306644	−0.153322	−	0.988176i	$-0.548997\pi$
$163$	−3.91496	−0.306644	−0.153322	+	0.988176i	$0.548997\pi$
$164$	0	0
$165$	0.188564	0.0146797
$166$	0	0
$167$	−9.41233	−0.728348	−0.364174	−	0.931331i	$-0.618649\pi$
$167$	−9.41233	−0.728348	−0.364174	+	0.931331i	$0.618649\pi$
$168$	0	0
$169$	−12.7442	−0.980321
$170$	0	0
$171$	1.24935	0.0955405
$172$	0	0
$173$	3.30151	0.251009	0.125505	−	0.992093i	$-0.459945\pi$
$173$	3.30151	0.251009	0.125505	+	0.992093i	$0.459945\pi$
$174$	0	0
$175$	17.3967	1.31506
$176$	0	0
$177$	19.1658	1.44059
$178$	0	0
$179$	7.56172	0.565189	0.282595	−	0.959239i	$-0.408805\pi$
$179$	7.56172	0.565189	0.282595	+	0.959239i	$0.408805\pi$
$180$	0	0
$181$	−11.2760	−0.838137	−0.419069	−	0.907955i	$-0.637643\pi$
$181$	−11.2760	−0.838137	−0.419069	+	0.907955i	$0.637643\pi$
$182$	0	0
$183$	13.7034	1.01299
$184$	0	0
$185$	0.0907970	0.00667553
$186$	0	0
$187$	−30.6401	−2.24062
$188$	0	0
$189$	−17.2986	−1.25829
$190$	0	0
$191$	21.1526	1.53055	0.765275	−	0.643703i	$-0.222603\pi$
$191$	21.1526	1.53055	0.765275	+	0.643703i	$0.222603\pi$
$192$	0	0
$193$	3.26156	0.234772	0.117386	−	0.993086i	$-0.462549\pi$
$193$	3.26156	0.234772	0.117386	+	0.993086i	$0.462549\pi$
$194$	0	0
$195$	0.0194385	0.00139202
$196$	0	0
$197$	25.3243	1.80428	0.902142	−	0.431440i	$-0.141994\pi$
$197$	25.3243	1.80428	0.902142	+	0.431440i	$0.141994\pi$
$198$	0	0
$199$	21.6856	1.53725	0.768625	−	0.639700i	$-0.220941\pi$
$199$	21.6856	1.53725	0.768625	+	0.639700i	$0.220941\pi$
$200$	0	0
$201$	−5.89704	−0.415945
$202$	0	0
$203$	−21.8923	−1.53654
$204$	0	0
$205$	0.212783	0.0148614
$206$	0	0
$207$	0.850385	0.0591058
$208$	0	0
$209$	−25.8936	−1.79110
$210$	0	0
$211$	28.6831	1.97463	0.987313	−	0.158784i	$-0.0507573\pi$
$211$	28.6831	1.97463	0.987313	+	0.158784i	$0.0507573\pi$
$212$	0	0
$213$	23.0623	1.58020
$214$	0	0
$215$	−0.00890349	−0.000607213 0
$216$	0	0
$217$	2.77323	0.188259
$218$	0	0
$219$	23.0108	1.55493
$220$	0	0
$221$	−3.15859	−0.212469
$222$	0	0
$223$	−28.1658	−1.88612	−0.943062	−	0.332617i	$-0.892068\pi$
$223$	−28.1658	−1.88612	−0.943062	+	0.332617i	$0.892068\pi$
$224$	0	0
$225$	−1.18356	−0.0789041
$226$	0	0
$227$	13.9556	0.926264	0.463132	−	0.886289i	$-0.346726\pi$
$227$	13.9556	0.926264	0.463132	+	0.886289i	$0.346726\pi$
$228$	0	0
$229$	0.812411	0.0536856	0.0268428	−	0.999640i	$-0.491455\pi$
$229$	0.812411	0.0536856	0.0268428	+	0.999640i	$0.491455\pi$
$230$	0	0
$231$	−30.7154	−2.02092
$232$	0	0
$233$	28.1459	1.84390	0.921950	−	0.387308i	$-0.126595\pi$
$233$	28.1459	1.84390	0.921950	+	0.387308i	$0.126595\pi$
$234$	0	0
$235$	0.0132086	0.000861632 0
$236$	0	0
$237$	1.58055	0.102668
$238$	0	0
$239$	−29.0610	−1.87980	−0.939901	−	0.341448i	$-0.889083\pi$
$239$	−29.0610	−1.87980	−0.939901	+	0.341448i	$0.889083\pi$
$240$	0	0
$241$	1.29591	0.0834769	0.0417385	−	0.999129i	$-0.486710\pi$
$241$	1.29591	0.0834769	0.0417385	+	0.999129i	$0.486710\pi$
$242$	0	0
$243$	2.45461	0.157463
$244$	0	0
$245$	0.109115	0.00697113
$246$	0	0
$247$	−2.66928	−0.169842
$248$	0	0
$249$	21.6503	1.37203
$250$	0	0
$251$	−4.21136	−0.265818	−0.132909	−	0.991128i	$-0.542432\pi$
$251$	−4.21136	−0.265818	−0.132909	+	0.991128i	$0.542432\pi$
$252$	0	0
$253$	−17.6247	−1.10806
$254$	0	0
$255$	−0.240002	−0.0150295
$256$	0	0
$257$	−8.13948	−0.507727	−0.253864	−	0.967240i	$-0.581701\pi$
$257$	−8.13948	−0.507727	−0.253864	+	0.967240i	$0.581701\pi$
$258$	0	0
$259$	−14.7900	−0.919006
$260$	0	0
$261$	1.48942	0.0921925
$262$	0	0
$263$	9.79071	0.603721	0.301861	−	0.953352i	$-0.402392\pi$
$263$	9.79071	0.603721	0.301861	+	0.953352i	$0.402392\pi$
$264$	0	0
$265$	0.0195457	0.00120068
$266$	0	0
$267$	−22.8363	−1.39756
$268$	0	0
$269$	−23.5888	−1.43823	−0.719116	−	0.694890i	$-0.755453\pi$
$269$	−23.5888	−1.43823	−0.719116	+	0.694890i	$0.755453\pi$
$270$	0	0
$271$	−12.7239	−0.772923	−0.386461	−	0.922306i	$-0.626303\pi$
$271$	−12.7239	−0.772923	−0.386461	+	0.922306i	$0.626303\pi$
$272$	0	0
$273$	−3.16635	−0.191636
$274$	0	0
$275$	24.5300	1.47921
$276$	0	0
$277$	25.1778	1.51278	0.756392	−	0.654118i	$-0.226960\pi$
$277$	25.1778	1.51278	0.756392	+	0.654118i	$0.226960\pi$
$278$	0	0
$279$	−0.188673	−0.0112956
$280$	0	0
$281$	−29.0299	−1.73178	−0.865889	−	0.500236i	$-0.833247\pi$
$281$	−29.0299	−1.73178	−0.865889	+	0.500236i	$0.833247\pi$
$282$	0	0
$283$	−2.14090	−0.127263	−0.0636317	−	0.997973i	$-0.520268\pi$
$283$	−2.14090	−0.127263	−0.0636317	+	0.997973i	$0.520268\pi$
$284$	0	0
$285$	−0.202823	−0.0120142
$286$	0	0
$287$	−34.6604	−2.04594
$288$	0	0
$289$	21.9983	1.29402
$290$	0	0
$291$	9.22189	0.540596
$292$	0	0
$293$	−2.52694	−0.147625	−0.0738127	−	0.997272i	$-0.523517\pi$
$293$	−2.52694	−0.147625	−0.0738127	+	0.997272i	$0.523517\pi$
$294$	0	0
$295$	−0.227568	−0.0132495
$296$	0	0
$297$	−24.3918	−1.41535
$298$	0	0
$299$	−1.81687	−0.105073
$300$	0	0
$301$	1.45030	0.0835938
$302$	0	0
$303$	−1.58240	−0.0909063
$304$	0	0
$305$	−0.162710	−0.00931676
$306$	0	0
$307$	−11.5740	−0.660563	−0.330281	−	0.943883i	$-0.607144\pi$
$307$	−11.5740	−0.660563	−0.330281	+	0.943883i	$0.607144\pi$
$308$	0	0
$309$	34.3559	1.95444
$310$	0	0
$311$	−25.9969	−1.47415	−0.737076	−	0.675810i	$-0.763794\pi$
$311$	−25.9969	−1.47415	−0.737076	+	0.675810i	$0.763794\pi$
$312$	0	0
$313$	−9.94757	−0.562270	−0.281135	−	0.959668i	$-0.590711\pi$
$313$	−9.94757	−0.562270	−0.281135	+	0.959668i	$0.590711\pi$
$314$	0	0
$315$	−0.0175968	−0.000991469 0
$316$	0	0
$317$	−21.2717	−1.19474	−0.597370	−	0.801966i	$-0.703788\pi$
$317$	−21.2717	−1.19474	−0.597370	+	0.801966i	$0.703788\pi$
$318$	0	0
$319$	−30.8690	−1.72833
$320$	0	0
$321$	−3.11233	−0.173713
$322$	0	0
$323$	32.9570	1.83378
$324$	0	0
$325$	2.52872	0.140268
$326$	0	0
$327$	−21.3343	−1.17979
$328$	0	0
$329$	−2.15156	−0.118619
$330$	0	0
$331$	−7.92758	−0.435739	−0.217870	−	0.975978i	$-0.569911\pi$
$331$	−7.92758	−0.435739	−0.217870	+	0.975978i	$0.569911\pi$
$332$	0	0
$333$	1.00622	0.0551406
$334$	0	0
$335$	0.0700195	0.00382558
$336$	0	0
$337$	−27.6770	−1.50766	−0.753831	−	0.657069i	$-0.771796\pi$
$337$	−27.6770	−1.50766	−0.753831	+	0.657069i	$0.771796\pi$
$338$	0	0
$339$	8.54354	0.464021
$340$	0	0
$341$	3.91036	0.211758
$342$	0	0
$343$	6.58364	0.355483
$344$	0	0
$345$	−0.138053	−0.00743254
$346$	0	0
$347$	−14.7938	−0.794171	−0.397085	−	0.917782i	$-0.629978\pi$
$347$	−14.7938	−0.794171	−0.397085	+	0.917782i	$0.629978\pi$
$348$	0	0
$349$	20.8610	1.11666	0.558332	−	0.829618i	$-0.311441\pi$
$349$	20.8610	1.11666	0.558332	+	0.829618i	$0.311441\pi$
$350$	0	0
$351$	−2.51447	−0.134212
$352$	0	0
$353$	−12.2981	−0.654564	−0.327282	−	0.944927i	$-0.606133\pi$
$353$	−12.2981	−0.654564	−0.327282	+	0.944927i	$0.606133\pi$
$354$	0	0
$355$	−0.273834	−0.0145336
$356$	0	0
$357$	39.0941	2.06908
$358$	0	0
$359$	−35.5858	−1.87815	−0.939073	−	0.343717i	$-0.888314\pi$
$359$	−35.5858	−1.87815	−0.939073	+	0.343717i	$0.888314\pi$
$360$	0	0
$361$	8.85158	0.465873
$362$	0	0
$363$	−23.5199	−1.23448
$364$	0	0
$365$	−0.273223	−0.0143011
$366$	0	0
$367$	4.45873	0.232744	0.116372	−	0.993206i	$-0.462874\pi$
$367$	4.45873	0.232744	0.116372	+	0.993206i	$0.462874\pi$
$368$	0	0
$369$	2.35808	0.122757
$370$	0	0
$371$	−3.18381	−0.165295
$372$	0	0
$373$	0.532958	0.0275955	0.0137978	−	0.999905i	$-0.495608\pi$
$373$	0.532958	0.0275955	0.0137978	+	0.999905i	$0.495608\pi$
$374$	0	0
$375$	0.384302	0.0198452
$376$	0	0
$377$	−3.18218	−0.163891
$378$	0	0
$379$	−12.9543	−0.665415	−0.332708	−	0.943030i	$-0.607962\pi$
$379$	−12.9543	−0.665415	−0.332708	+	0.943030i	$0.607962\pi$
$380$	0	0
$381$	−15.8687	−0.812977
$382$	0	0
$383$	23.6783	1.20991	0.604954	−	0.796261i	$-0.293192\pi$
$383$	23.6783	1.20991	0.604954	+	0.796261i	$0.293192\pi$
$384$	0	0
$385$	0.364704	0.0185871
$386$	0	0
$387$	−0.0986694	−0.00501565
$388$	0	0
$389$	−0.894690	−0.0453626	−0.0226813	−	0.999743i	$-0.507220\pi$
$389$	−0.894690	−0.0453626	−0.0226813	+	0.999743i	$0.507220\pi$
$390$	0	0
$391$	22.4325	1.13446
$392$	0	0
$393$	−34.0010	−1.71512
$394$	0	0
$395$	−0.0187669	−0.000944264 0
$396$	0	0
$397$	17.1080	0.858628	0.429314	−	0.903155i	$-0.358755\pi$
$397$	17.1080	0.858628	0.429314	+	0.903155i	$0.358755\pi$
$398$	0	0
$399$	33.0380	1.65397
$400$	0	0
$401$	−31.9986	−1.59793	−0.798966	−	0.601376i	$-0.794619\pi$
$401$	−31.9986	−1.59793	−0.798966	+	0.601376i	$0.794619\pi$
$402$	0	0
$403$	0.403107	0.0200802
$404$	0	0
$405$	−0.206231	−0.0102477
$406$	0	0
$407$	−20.8545	−1.03372
$408$	0	0
$409$	−0.853493	−0.0422025	−0.0211012	−	0.999777i	$-0.506717\pi$
$409$	−0.853493	−0.0422025	−0.0211012	+	0.999777i	$0.506717\pi$
$410$	0	0
$411$	26.6900	1.31652
$412$	0	0
$413$	37.0688	1.82404
$414$	0	0
$415$	−0.257068	−0.0126190
$416$	0	0
$417$	38.1610	1.86875
$418$	0	0
$419$	−31.0683	−1.51779	−0.758893	−	0.651216i	$-0.774259\pi$
$419$	−31.0683	−1.51779	−0.758893	+	0.651216i	$0.774259\pi$
$420$	0	0
$421$	−33.3633	−1.62603	−0.813015	−	0.582243i	$-0.802175\pi$
$421$	−33.3633	−1.62603	−0.813015	+	0.582243i	$0.802175\pi$
$422$	0	0
$423$	0.146379	0.00711718
$424$	0	0
$425$	−31.2215	−1.51446
$426$	0	0
$427$	26.5040	1.28262
$428$	0	0
$429$	−4.46468	−0.215557
$430$	0	0
$431$	−28.4231	−1.36909	−0.684546	−	0.728969i	$-0.740000\pi$
$431$	−28.4231	−1.36909	−0.684546	+	0.728969i	$0.740000\pi$
$432$	0	0
$433$	16.6687	0.801048	0.400524	−	0.916286i	$-0.368828\pi$
$433$	16.6687	0.801048	0.400524	+	0.916286i	$0.368828\pi$
$434$	0	0
$435$	−0.241795	−0.0115932
$436$	0	0
$437$	18.9575	0.906858
$438$	0	0
$439$	−1.11009	−0.0529817	−0.0264908	−	0.999649i	$-0.508433\pi$
$439$	−1.11009	−0.0529817	−0.0264908	+	0.999649i	$0.508433\pi$
$440$	0	0
$441$	1.20923	0.0575823
$442$	0	0
$443$	1.36909	0.0650473	0.0325237	−	0.999471i	$-0.489646\pi$
$443$	1.36909	0.0650473	0.0325237	+	0.999471i	$0.489646\pi$
$444$	0	0
$445$	0.271151	0.0128538
$446$	0	0
$447$	35.0429	1.65747
$448$	0	0
$449$	14.2075	0.670493	0.335246	−	0.942130i	$-0.391180\pi$
$449$	14.2075	0.670493	0.335246	+	0.942130i	$0.391180\pi$
$450$	0	0
$451$	−48.8725	−2.30132
$452$	0	0
$453$	−35.7780	−1.68099
$454$	0	0
$455$	0.0375962	0.00176254
$456$	0	0
$457$	17.8991	0.837282	0.418641	−	0.908152i	$-0.362506\pi$
$457$	17.8991	0.837282	0.418641	+	0.908152i	$0.362506\pi$
$458$	0	0
$459$	31.0455	1.44908
$460$	0	0
$461$	9.75644	0.454403	0.227201	−	0.973848i	$-0.427042\pi$
$461$	9.75644	0.454403	0.227201	+	0.973848i	$0.427042\pi$
$462$	0	0
$463$	36.9806	1.71863	0.859317	−	0.511443i	$-0.170889\pi$
$463$	36.9806	1.71863	0.859317	+	0.511443i	$0.170889\pi$
$464$	0	0
$465$	0.0306297	0.00142042
$466$	0	0
$467$	39.2607	1.81677	0.908385	−	0.418136i	$-0.137316\pi$
$467$	39.2607	1.81677	0.908385	+	0.418136i	$0.137316\pi$
$468$	0	0
$469$	−11.4055	−0.526659
$470$	0	0
$471$	−16.5294	−0.761632
$472$	0	0
$473$	2.04498	0.0940283
$474$	0	0
$475$	−26.3849	−1.21062
$476$	0	0
$477$	0.216607	0.00991776
$478$	0	0
$479$	−9.98350	−0.456158	−0.228079	−	0.973643i	$-0.573244\pi$
$479$	−9.98350	−0.456158	−0.228079	+	0.973643i	$0.573244\pi$
$480$	0	0
$481$	−2.14982	−0.0980235
$482$	0	0
$483$	22.4876	1.02322
$484$	0	0
$485$	−0.109498	−0.00497203
$486$	0	0
$487$	−18.6292	−0.844171	−0.422085	−	0.906556i	$-0.638702\pi$
$487$	−18.6292	−0.844171	−0.422085	+	0.906556i	$0.638702\pi$
$488$	0	0
$489$	7.04338	0.318513
$490$	0	0
$491$	−16.2059	−0.731361	−0.365680	−	0.930741i	$-0.619164\pi$
$491$	−16.2059	−0.731361	−0.365680	+	0.930741i	$0.619164\pi$
$492$	0	0
$493$	39.2896	1.76952
$494$	0	0
$495$	−0.0248122	−0.00111523
$496$	0	0
$497$	44.6051	2.00081
$498$	0	0
$499$	−31.1303	−1.39358	−0.696792	−	0.717273i	$-0.745390\pi$
$499$	−31.1303	−1.39358	−0.696792	+	0.717273i	$0.745390\pi$
$500$	0	0
$501$	16.9337	0.756540
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0.0187889	0.000836093 0
$506$	0	0
$507$	22.9280	1.01827
$508$	0	0
$509$	21.7263	0.963000	0.481500	−	0.876446i	$-0.340092\pi$
$509$	21.7263	0.963000	0.481500	+	0.876446i	$0.340092\pi$
$510$	0	0
$511$	44.5055	1.96881
$512$	0	0
$513$	26.2362	1.15836
$514$	0	0
$515$	−0.407930	−0.0179756
$516$	0	0
$517$	−3.03378	−0.133426
$518$	0	0
$519$	−5.93972	−0.260725
$520$	0	0
$521$	−21.3540	−0.935534	−0.467767	−	0.883852i	$-0.654941\pi$
$521$	−21.3540	−0.935534	−0.467767	+	0.883852i	$0.654941\pi$
$522$	0	0
$523$	29.9469	1.30949	0.654743	−	0.755852i	$-0.272777\pi$
$523$	29.9469	1.30949	0.654743	+	0.755852i	$0.272777\pi$
$524$	0	0
$525$	−31.2982	−1.36597
$526$	0	0
$527$	−4.97706	−0.216804
$528$	0	0
$529$	−10.0964	−0.438975
$530$	0	0
$531$	−2.52194	−0.109443
$532$	0	0
$533$	−5.03811	−0.218225
$534$	0	0
$535$	0.0369548	0.00159769
$536$	0	0
$537$	−13.6042	−0.587066
$538$	0	0
$539$	−25.0619	−1.07949
$540$	0	0
$541$	−16.2106	−0.696947	−0.348473	−	0.937319i	$-0.613300\pi$
$541$	−16.2106	−0.696947	−0.348473	+	0.937319i	$0.613300\pi$
$542$	0	0
$543$	20.2865	0.870579
$544$	0	0
$545$	0.253316	0.0108509
$546$	0	0
$547$	−18.5702	−0.794005	−0.397002	−	0.917818i	$-0.629949\pi$
$547$	−18.5702	−0.794005	−0.397002	+	0.917818i	$0.629949\pi$
$548$	0	0
$549$	−1.80317	−0.0769574
$550$	0	0
$551$	33.2032	1.41451
$552$	0	0
$553$	3.05696	0.129995
$554$	0	0
$555$	−0.163352	−0.00693391
$556$	0	0
$557$	13.1049	0.555272	0.277636	−	0.960686i	$-0.410449\pi$
$557$	13.1049	0.555272	0.277636	+	0.960686i	$0.410449\pi$
$558$	0	0
$559$	0.210810	0.00891633
$560$	0	0
$561$	55.1243	2.32735
$562$	0	0
$563$	22.0067	0.927473	0.463736	−	0.885973i	$-0.346509\pi$
$563$	22.0067	0.927473	0.463736	+	0.885973i	$0.346509\pi$
$564$	0	0
$565$	−0.101443	−0.00426775
$566$	0	0
$567$	33.5931	1.41078
$568$	0	0
$569$	−8.07487	−0.338516	−0.169258	−	0.985572i	$-0.554137\pi$
$569$	−8.07487	−0.338516	−0.169258	+	0.985572i	$0.554137\pi$
$570$	0	0
$571$	22.5224	0.942533	0.471266	−	0.881991i	$-0.343797\pi$
$571$	22.5224	0.942533	0.471266	+	0.881991i	$0.343797\pi$
$572$	0	0
$573$	−38.0555	−1.58979
$574$	0	0
$575$	−17.9591	−0.748948
$576$	0	0
$577$	−36.9964	−1.54018	−0.770089	−	0.637936i	$-0.779788\pi$
$577$	−36.9964	−1.54018	−0.770089	+	0.637936i	$0.779788\pi$
$578$	0	0
$579$	−5.86784	−0.243859
$580$	0	0
$581$	41.8741	1.73723
$582$	0	0
$583$	−4.48931	−0.185928
$584$	0	0
$585$	−0.00255781	−0.000105753 0
$586$	0	0
$587$	25.7776	1.06395	0.531977	−	0.846759i	$-0.321449\pi$
$587$	25.7776	1.06395	0.531977	+	0.846759i	$0.321449\pi$
$588$	0	0
$589$	−4.20606	−0.173308
$590$	0	0
$591$	−45.5608	−1.87412
$592$	0	0
$593$	−4.51796	−0.185531	−0.0927653	−	0.995688i	$-0.529571\pi$
$593$	−4.51796	−0.185531	−0.0927653	+	0.995688i	$0.529571\pi$
$594$	0	0
$595$	−0.464191	−0.0190300
$596$	0	0
$597$	−39.0144	−1.59675
$598$	0	0
$599$	−6.96732	−0.284677	−0.142338	−	0.989818i	$-0.545462\pi$
$599$	−6.96732	−0.284677	−0.142338	+	0.989818i	$0.545462\pi$
$600$	0	0
$601$	−25.0153	−1.02040	−0.510198	−	0.860057i	$-0.670428\pi$
$601$	−25.0153	−1.02040	−0.510198	+	0.860057i	$0.670428\pi$
$602$	0	0
$603$	0.775963	0.0315997
$604$	0	0
$605$	0.279268	0.0113538
$606$	0	0
$607$	−3.76693	−0.152895	−0.0764476	−	0.997074i	$-0.524358\pi$
$607$	−3.76693	−0.152895	−0.0764476	+	0.997074i	$0.524358\pi$
$608$	0	0
$609$	39.3862	1.59601
$610$	0	0
$611$	−0.312743	−0.0126522
$612$	0	0
$613$	0.608437	0.0245745	0.0122873	−	0.999925i	$-0.496089\pi$
$613$	0.608437	0.0245745	0.0122873	+	0.999925i	$0.496089\pi$
$614$	0	0
$615$	−0.382816	−0.0154366
$616$	0	0
$617$	17.2014	0.692503	0.346251	−	0.938142i	$-0.387454\pi$
$617$	17.2014	0.692503	0.346251	+	0.938142i	$0.387454\pi$
$618$	0	0
$619$	−37.5963	−1.51112	−0.755561	−	0.655078i	$-0.772636\pi$
$619$	−37.5963	−1.51112	−0.755561	+	0.655078i	$0.772636\pi$
$620$	0	0
$621$	17.8579	0.716614
$622$	0	0
$623$	−44.1681	−1.76956
$624$	0	0
$625$	24.9932	0.999726
$626$	0	0
$627$	46.5849	1.86042
$628$	0	0
$629$	26.5433	1.05835
$630$	0	0
$631$	24.1057	0.959634	0.479817	−	0.877369i	$-0.340703\pi$
$631$	24.1057	0.959634	0.479817	+	0.877369i	$0.340703\pi$
$632$	0	0
$633$	−51.6036	−2.05106
$634$	0	0
$635$	0.188420	0.00747720
$636$	0	0
$637$	−2.58355	−0.102364
$638$	0	0
$639$	−3.03466	−0.120049
$640$	0	0
$641$	−5.91643	−0.233685	−0.116843	−	0.993150i	$-0.537277\pi$
$641$	−5.91643	−0.233685	−0.116843	+	0.993150i	$0.537277\pi$
$642$	0	0
$643$	−6.84116	−0.269789	−0.134894	−	0.990860i	$-0.543070\pi$
$643$	−6.84116	−0.269789	−0.134894	+	0.990860i	$0.543070\pi$
$644$	0	0
$645$	0.0160182	0.000630716 0
$646$	0	0
$647$	2.29029	0.0900404	0.0450202	−	0.998986i	$-0.485665\pi$
$647$	2.29029	0.0900404	0.0450202	+	0.998986i	$0.485665\pi$
$648$	0	0
$649$	52.2686	2.05172
$650$	0	0
$651$	−4.98929	−0.195546
$652$	0	0
$653$	−36.2536	−1.41871	−0.709357	−	0.704849i	$-0.751015\pi$
$653$	−36.2536	−1.41871	−0.709357	+	0.704849i	$0.751015\pi$
$654$	0	0
$655$	0.403717	0.0157745
$656$	0	0
$657$	−3.02788	−0.118129
$658$	0	0
$659$	29.6369	1.15449	0.577244	−	0.816571i	$-0.304128\pi$
$659$	29.6369	1.15449	0.577244	+	0.816571i	$0.304128\pi$
$660$	0	0
$661$	−18.9241	−0.736063	−0.368031	−	0.929813i	$-0.619968\pi$
$661$	−18.9241	−0.736063	−0.368031	+	0.929813i	$0.619968\pi$
$662$	0	0
$663$	5.68259	0.220693
$664$	0	0
$665$	−0.392282	−0.0152121
$666$	0	0
$667$	22.6001	0.875080
$668$	0	0
$669$	50.6730	1.95913
$670$	0	0
$671$	37.3717	1.44272
$672$	0	0
$673$	−43.2686	−1.66788	−0.833941	−	0.551854i	$-0.813921\pi$
$673$	−43.2686	−1.66788	−0.833941	+	0.551854i	$0.813921\pi$
$674$	0	0
$675$	−24.8546	−0.956654
$676$	0	0
$677$	−2.53157	−0.0972962	−0.0486481	−	0.998816i	$-0.515491\pi$
$677$	−2.53157	−0.0972962	−0.0486481	+	0.998816i	$0.515491\pi$
$678$	0	0
$679$	17.8362	0.684490
$680$	0	0
$681$	−25.1074	−0.962116
$682$	0	0
$683$	8.40412	0.321575	0.160787	−	0.986989i	$-0.448597\pi$
$683$	8.40412	0.321575	0.160787	+	0.986989i	$0.448597\pi$
$684$	0	0
$685$	−0.316908	−0.0121084
$686$	0	0
$687$	−1.46160	−0.0557636
$688$	0	0
$689$	−0.462788	−0.0176308
$690$	0	0
$691$	−50.9900	−1.93975	−0.969875	−	0.243602i	$-0.921671\pi$
$691$	−50.9900	−1.93975	−0.969875	+	0.243602i	$0.921671\pi$
$692$	0	0
$693$	4.04169	0.153531
$694$	0	0
$695$	−0.453111	−0.0171875
$696$	0	0
$697$	62.2043	2.35616
$698$	0	0
$699$	−50.6371	−1.91527
$700$	0	0
$701$	−23.0464	−0.870449	−0.435224	−	0.900322i	$-0.643331\pi$
$701$	−23.0464	−0.870449	−0.435224	+	0.900322i	$0.643331\pi$
$702$	0	0
$703$	22.4315	0.846019
$704$	0	0
$705$	−0.0237634	−0.000894983 0
$706$	0	0
$707$	−3.06053	−0.115103
$708$	0	0
$709$	7.89735	0.296591	0.148296	−	0.988943i	$-0.452621\pi$
$709$	7.89735	0.296591	0.148296	+	0.988943i	$0.452621\pi$
$710$	0	0
$711$	−0.207976	−0.00779973
$712$	0	0
$713$	−2.86289	−0.107216
$714$	0	0
$715$	0.0530121	0.00198254
$716$	0	0
$717$	52.2835	1.95256
$718$	0	0
$719$	45.6895	1.70393	0.851965	−	0.523599i	$-0.175411\pi$
$719$	45.6895	1.70393	0.851965	+	0.523599i	$0.175411\pi$
$720$	0	0
$721$	66.4482	2.47466
$722$	0	0
$723$	−2.33146	−0.0867080
$724$	0	0
$725$	−31.4547	−1.16820
$726$	0	0
$727$	−42.3768	−1.57167	−0.785834	−	0.618437i	$-0.787766\pi$
$727$	−42.3768	−1.57167	−0.785834	+	0.618437i	$0.787766\pi$
$728$	0	0
$729$	24.5464	0.909126
$730$	0	0
$731$	−2.60282	−0.0962689
$732$	0	0
$733$	17.0603	0.630137	0.315069	−	0.949069i	$-0.397972\pi$
$733$	17.0603	0.630137	0.315069	+	0.949069i	$0.397972\pi$
$734$	0	0
$735$	−0.196309	−0.00724095
$736$	0	0
$737$	−16.0823	−0.592398
$738$	0	0
$739$	−30.4336	−1.11952	−0.559758	−	0.828656i	$-0.689106\pi$
$739$	−30.4336	−1.11952	−0.559758	+	0.828656i	$0.689106\pi$
$740$	0	0
$741$	4.80229	0.176417
$742$	0	0
$743$	−8.02867	−0.294543	−0.147272	−	0.989096i	$-0.547049\pi$
$743$	−8.02867	−0.294543	−0.147272	+	0.989096i	$0.547049\pi$
$744$	0	0
$745$	−0.416088	−0.0152443
$746$	0	0
$747$	−2.84886	−0.104234
$748$	0	0
$749$	−6.01960	−0.219951
$750$	0	0
$751$	−39.9968	−1.45951	−0.729753	−	0.683711i	$-0.760365\pi$
$751$	−39.9968	−1.45951	−0.729753	+	0.683711i	$0.760365\pi$
$752$	0	0
$753$	7.57662	0.276107
$754$	0	0
$755$	0.424816	0.0154606
$756$	0	0
$757$	36.6151	1.33080	0.665399	−	0.746488i	$-0.268262\pi$
$757$	36.6151	1.33080	0.665399	+	0.746488i	$0.268262\pi$
$758$	0	0
$759$	31.7085	1.15095
$760$	0	0
$761$	12.6486	0.458510	0.229255	−	0.973366i	$-0.426371\pi$
$761$	12.6486	0.458510	0.229255	+	0.973366i	$0.426371\pi$
$762$	0	0
$763$	−41.2629	−1.49382
$764$	0	0
$765$	0.0315807	0.00114180
$766$	0	0
$767$	5.38820	0.194556
$768$	0	0
$769$	−19.2986	−0.695927	−0.347963	−	0.937508i	$-0.613127\pi$
$769$	−19.2986	−0.695927	−0.347963	+	0.937508i	$0.613127\pi$
$770$	0	0
$771$	14.6437	0.527379
$772$	0	0
$773$	−26.2734	−0.944990	−0.472495	−	0.881333i	$-0.656646\pi$
$773$	−26.2734	−0.944990	−0.472495	+	0.881333i	$0.656646\pi$
$774$	0	0
$775$	3.98456	0.143130
$776$	0	0
$777$	26.6086	0.954578
$778$	0	0
$779$	52.5682	1.88345
$780$	0	0
$781$	62.8950	2.25056
$782$	0	0
$783$	31.2775	1.11777
$784$	0	0
$785$	0.196264	0.00700497
$786$	0	0
$787$	35.6584	1.27108	0.635542	−	0.772066i	$-0.280777\pi$
$787$	35.6584	1.27108	0.635542	+	0.772066i	$0.280777\pi$
$788$	0	0
$789$	−17.6144	−0.627089
$790$	0	0
$791$	16.5242	0.587532
$792$	0	0
$793$	3.85253	0.136807
$794$	0	0
$795$	−0.0351645	−0.00124716
$796$	0	0
$797$	2.70502	0.0958167	0.0479083	−	0.998852i	$-0.484744\pi$
$797$	2.70502	0.0958167	0.0479083	+	0.998852i	$0.484744\pi$
$798$	0	0
$799$	3.86136	0.136605
$800$	0	0
$801$	3.00492	0.106174
$802$	0	0
$803$	62.7546	2.21456
$804$	0	0
$805$	−0.267011	−0.00941090
$806$	0	0
$807$	42.4384	1.49390
$808$	0	0
$809$	−26.6344	−0.936416	−0.468208	−	0.883618i	$-0.655100\pi$
$809$	−26.6344	−0.936416	−0.468208	+	0.883618i	$0.655100\pi$
$810$	0	0
$811$	47.3204	1.66164	0.830822	−	0.556538i	$-0.187871\pi$
$811$	47.3204	1.66164	0.830822	+	0.556538i	$0.187871\pi$
$812$	0	0
$813$	22.8915	0.802840
$814$	0	0
$815$	−0.0836308	−0.00292946
$816$	0	0
$817$	−2.19962	−0.0769548
$818$	0	0
$819$	0.416645	0.0145587
$820$	0	0
$821$	−43.6293	−1.52267	−0.761337	−	0.648357i	$-0.775457\pi$
$821$	−43.6293	−1.52267	−0.761337	+	0.648357i	$0.775457\pi$
$822$	0	0
$823$	−10.4701	−0.364963	−0.182482	−	0.983209i	$-0.558413\pi$
$823$	−10.4701	−0.364963	−0.182482	+	0.983209i	$0.558413\pi$
$824$	0	0
$825$	−44.1317	−1.53647
$826$	0	0
$827$	−4.35556	−0.151458	−0.0757289	−	0.997128i	$-0.524128\pi$
$827$	−4.35556	−0.151458	−0.0757289	+	0.997128i	$0.524128\pi$
$828$	0	0
$829$	−28.2476	−0.981080	−0.490540	−	0.871419i	$-0.663200\pi$
$829$	−28.2476	−0.981080	−0.490540	+	0.871419i	$0.663200\pi$
$830$	0	0
$831$	−45.2971	−1.57134
$832$	0	0
$833$	31.8985	1.10522
$834$	0	0
$835$	−0.201065	−0.00695813
$836$	0	0
$837$	−3.96211	−0.136951
$838$	0	0
$839$	7.22547	0.249451	0.124725	−	0.992191i	$-0.460195\pi$
$839$	7.22547	0.249451	0.124725	+	0.992191i	$0.460195\pi$
$840$	0	0
$841$	10.5832	0.364937
$842$	0	0
$843$	52.2275	1.79881
$844$	0	0
$845$	−0.272239	−0.00936531
$846$	0	0
$847$	−45.4902	−1.56306
$848$	0	0
$849$	3.85168	0.132189
$850$	0	0
$851$	15.2682	0.523387
$852$	0	0
$853$	26.7825	0.917014	0.458507	−	0.888691i	$-0.348384\pi$
$853$	26.7825	0.917014	0.458507	+	0.888691i	$0.348384\pi$
$854$	0	0
$855$	0.0266885	0.000912727 0
$856$	0	0
$857$	19.5478	0.667739	0.333870	−	0.942619i	$-0.391645\pi$
$857$	19.5478	0.667739	0.333870	+	0.942619i	$0.391645\pi$
$858$	0	0
$859$	30.9854	1.05721	0.528604	−	0.848868i	$-0.322716\pi$
$859$	30.9854	1.05721	0.528604	+	0.848868i	$0.322716\pi$
$860$	0	0
$861$	62.3573	2.12513
$862$	0	0
$863$	27.9105	0.950084	0.475042	−	0.879963i	$-0.342433\pi$
$863$	27.9105	0.950084	0.475042	+	0.879963i	$0.342433\pi$
$864$	0	0
$865$	0.0705263	0.00239797
$866$	0	0
$867$	−39.5769	−1.34410
$868$	0	0
$869$	4.31043	0.146221
$870$	0	0
$871$	−1.65787	−0.0561748
$872$	0	0
$873$	−1.21346	−0.0410695
$874$	0	0
$875$	0.743283	0.0251276
$876$	0	0
$877$	0.676009	0.0228272	0.0114136	−	0.999935i	$-0.496367\pi$
$877$	0.676009	0.0228272	0.0114136	+	0.999935i	$0.496367\pi$
$878$	0	0
$879$	4.54620	0.153339
$880$	0	0
$881$	−58.7113	−1.97803	−0.989017	−	0.147803i	$-0.952780\pi$
$881$	−58.7113	−1.97803	−0.989017	+	0.147803i	$0.952780\pi$
$882$	0	0
$883$	35.1814	1.18395	0.591974	−	0.805957i	$-0.298349\pi$
$883$	35.1814	1.18395	0.591974	+	0.805957i	$0.298349\pi$
$884$	0	0
$885$	0.409417	0.0137624
$886$	0	0
$887$	−20.1096	−0.675213	−0.337606	−	0.941287i	$-0.609617\pi$
$887$	−20.1096	−0.675213	−0.337606	+	0.941287i	$0.609617\pi$
$888$	0	0
$889$	−30.6918	−1.02937
$890$	0	0
$891$	47.3676	1.58687
$892$	0	0
$893$	3.26319	0.109199
$894$	0	0
$895$	0.161532	0.00539943
$896$	0	0
$897$	3.26873	0.109140
$898$	0	0
$899$	−5.01424	−0.167234
$900$	0	0
$901$	5.71393	0.190359
$902$	0	0
$903$	−2.60922	−0.0868295
$904$	0	0
$905$	−0.240876	−0.00800698
$906$	0	0
$907$	43.9291	1.45864	0.729321	−	0.684172i	$-0.239836\pi$
$907$	43.9291	1.45864	0.729321	+	0.684172i	$0.239836\pi$
$908$	0	0
$909$	0.208220	0.00690622
$910$	0	0
$911$	−43.6633	−1.44663	−0.723314	−	0.690519i	$-0.757382\pi$
$911$	−43.6633	−1.44663	−0.723314	+	0.690519i	$0.757382\pi$
$912$	0	0
$913$	59.0441	1.95408
$914$	0	0
$915$	0.292731	0.00967738
$916$	0	0
$917$	−65.7618	−2.17165
$918$	0	0
$919$	20.8443	0.687591	0.343795	−	0.939045i	$-0.388287\pi$
$919$	20.8443	0.687591	0.343795	+	0.939045i	$0.388287\pi$
$920$	0	0
$921$	20.8227	0.686131
$922$	0	0
$923$	6.48364	0.213412
$924$	0	0
$925$	−21.2502	−0.698703
$926$	0	0
$927$	−4.52072	−0.148480
$928$	0	0
$929$	−14.4995	−0.475714	−0.237857	−	0.971300i	$-0.576445\pi$
$929$	−14.4995	−0.475714	−0.237857	+	0.971300i	$0.576445\pi$
$930$	0	0
$931$	26.9571	0.883482
$932$	0	0
$933$	46.7709	1.53121
$934$	0	0
$935$	−0.654528	−0.0214054
$936$	0	0
$937$	−16.0038	−0.522821	−0.261410	−	0.965228i	$-0.584188\pi$
$937$	−16.0038	−0.522821	−0.261410	+	0.965228i	$0.584188\pi$
$938$	0	0
$939$	17.8966	0.584034
$940$	0	0
$941$	30.8028	1.00414	0.502072	−	0.864826i	$-0.332571\pi$
$941$	30.8028	1.00414	0.502072	+	0.864826i	$0.332571\pi$
$942$	0	0
$943$	35.7810	1.16519
$944$	0	0
$945$	−0.369530	−0.0120208
$946$	0	0
$947$	−13.1416	−0.427044	−0.213522	−	0.976938i	$-0.568494\pi$
$947$	−13.1416	−0.427044	−0.213522	+	0.976938i	$0.568494\pi$
$948$	0	0
$949$	6.46916	0.209998
$950$	0	0
$951$	38.2698	1.24098
$952$	0	0
$953$	−28.8980	−0.936097	−0.468049	−	0.883703i	$-0.655043\pi$
$953$	−28.8980	−0.936097	−0.468049	+	0.883703i	$0.655043\pi$
$954$	0	0
$955$	0.451859	0.0146218
$956$	0	0
$957$	55.5362	1.79523
$958$	0	0
$959$	51.6215	1.66695
$960$	0	0
$961$	−30.3648	−0.979510
$962$	0	0
$963$	0.409536	0.0131971
$964$	0	0
$965$	0.0696729	0.00224285
$966$	0	0
$967$	−36.0778	−1.16019	−0.580093	−	0.814551i	$-0.696984\pi$
$967$	−36.0778	−1.16019	−0.580093	+	0.814551i	$0.696984\pi$
$968$	0	0
$969$	−59.2927	−1.90476
$970$	0	0
$971$	5.71857	0.183518	0.0917588	−	0.995781i	$-0.470751\pi$
$971$	5.71857	0.183518	0.0917588	+	0.995781i	$0.470751\pi$
$972$	0	0
$973$	73.8077	2.36617
$974$	0	0
$975$	−4.54940	−0.145697
$976$	0	0
$977$	−46.8628	−1.49927	−0.749637	−	0.661849i	$-0.769772\pi$
$977$	−46.8628	−1.49927	−0.749637	+	0.661849i	$0.769772\pi$
$978$	0	0
$979$	−62.2788	−1.99044
$980$	0	0
$981$	2.80727	0.0896293
$982$	0	0
$983$	−52.8601	−1.68598	−0.842988	−	0.537933i	$-0.819205\pi$
$983$	−52.8601	−1.68598	−0.842988	+	0.537933i	$0.819205\pi$
$984$	0	0
$985$	0.540974	0.0172369
$986$	0	0
$987$	3.87085	0.123211
$988$	0	0
$989$	−1.49719	−0.0476079
$990$	0	0
$991$	−19.2804	−0.612462	−0.306231	−	0.951957i	$-0.599068\pi$
$991$	−19.2804	−0.612462	−0.306231	+	0.951957i	$0.599068\pi$
$992$	0	0
$993$	14.2624	0.452605
$994$	0	0
$995$	0.463244	0.0146858
$996$	0	0
$997$	15.3625	0.486534	0.243267	−	0.969959i	$-0.421781\pi$
$997$	15.3625	0.486534	0.243267	+	0.969959i	$0.421781\pi$
$998$	0	0
$999$	21.1305	0.668538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.9	✓	29
4.3	odd	2		8048.2.a.w.1.21		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.9	✓	29	1.1	even	1	trivial
8048.2.a.w.1.21		29	4.3	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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79909 q^{3} +0.0213618 q^{5} -3.47965 q^{7} +0.236734 q^{9} +O(q^{10})\)	\(q-1.79909 q^{3} +0.0213618 q^{5} -3.47965 q^{7} +0.236734 q^{9} -4.90645 q^{11} -0.505790 q^{13} -0.0384319 q^{15} +6.24486 q^{17} +5.27746 q^{19} +6.26021 q^{21} +3.59215 q^{23} -4.99954 q^{25} +4.97137 q^{27} +6.29152 q^{29} -0.796985 q^{31} +8.82715 q^{33} -0.0743317 q^{35} +4.25043 q^{37} +0.909962 q^{39} +9.96089 q^{41} -0.416794 q^{43} +0.00505707 q^{45} +0.618326 q^{47} +5.10796 q^{49} -11.2351 q^{51} +0.914981 q^{53} -0.104811 q^{55} -9.49464 q^{57} -10.6530 q^{59} -7.61687 q^{61} -0.823751 q^{63} -0.0108046 q^{65} +3.27779 q^{67} -6.46262 q^{69} -12.8188 q^{71} -12.7902 q^{73} +8.99464 q^{75} +17.0727 q^{77} -0.878524 q^{79} -9.65416 q^{81} -12.0340 q^{83} +0.133402 q^{85} -11.3190 q^{87} +12.6933 q^{89} +1.75997 q^{91} +1.43385 q^{93} +0.112736 q^{95} -5.12585 q^{97} -1.16152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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