LMFDB - Embedded newform 4024.2.a.g.1.20

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:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.940347 q^{3} +2.53897 q^{5} +0.182218 q^{7} -2.11575 q^{9} +O(q^{10})$	$q+0.940347 q^{3} +2.53897 q^{5} +0.182218 q^{7} -2.11575 q^{9} +0.888982 q^{11} +0.134466 q^{13} +2.38751 q^{15} +5.15559 q^{17} -2.24525 q^{19} +0.171348 q^{21} +4.77154 q^{23} +1.44637 q^{25} -4.81058 q^{27} -3.24225 q^{29} +10.4737 q^{31} +0.835952 q^{33} +0.462645 q^{35} +6.14802 q^{37} +0.126445 q^{39} +0.268578 q^{41} -3.38909 q^{43} -5.37182 q^{45} +1.64138 q^{47} -6.96680 q^{49} +4.84804 q^{51} -0.125379 q^{53} +2.25710 q^{55} -2.11131 q^{57} +11.0523 q^{59} -3.81357 q^{61} -0.385527 q^{63} +0.341407 q^{65} +1.10648 q^{67} +4.48690 q^{69} +1.17276 q^{71} +16.4984 q^{73} +1.36009 q^{75} +0.161988 q^{77} +15.1790 q^{79} +1.82363 q^{81} -13.8440 q^{83} +13.0899 q^{85} -3.04884 q^{87} +5.14859 q^{89} +0.0245022 q^{91} +9.84891 q^{93} -5.70062 q^{95} +4.43146 q^{97} -1.88086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	0.940347	0.542909	0.271455	−	0.962451i	$-0.412495\pi$
$3$	0.940347	0.542909	0.271455	+	0.962451i	$0.412495\pi$
$4$	0	0
$5$	2.53897	1.13546	0.567731	−	0.823214i	$-0.307821\pi$
$5$	2.53897	1.13546	0.567731	+	0.823214i	$0.307821\pi$
$6$	0	0
$7$	0.182218	0.0688718	0.0344359	−	0.999407i	$-0.489037\pi$
$7$	0.182218	0.0688718	0.0344359	+	0.999407i	$0.489037\pi$
$8$	0	0
$9$	−2.11575	−0.705249
$10$	0	0
$11$	0.888982	0.268038	0.134019	−	0.990979i	$-0.457212\pi$
$11$	0.888982	0.268038	0.134019	+	0.990979i	$0.457212\pi$
$12$	0	0
$13$	0.134466	0.0372943	0.0186471	−	0.999826i	$-0.494064\pi$
$13$	0.134466	0.0372943	0.0186471	+	0.999826i	$0.494064\pi$
$14$	0	0
$15$	2.38751	0.616453
$16$	0	0
$17$	5.15559	1.25041	0.625207	−	0.780459i	$-0.285014\pi$
$17$	5.15559	1.25041	0.625207	+	0.780459i	$0.285014\pi$
$18$	0	0
$19$	−2.24525	−0.515096	−0.257548	−	0.966266i	$-0.582914\pi$
$19$	−2.24525	−0.515096	−0.257548	+	0.966266i	$0.582914\pi$
$20$	0	0
$21$	0.171348	0.0373911
$22$	0	0
$23$	4.77154	0.994935	0.497467	−	0.867483i	$-0.334263\pi$
$23$	4.77154	0.994935	0.497467	+	0.867483i	$0.334263\pi$
$24$	0	0
$25$	1.44637	0.289275
$26$	0	0
$27$	−4.81058	−0.925796
$28$	0	0
$29$	−3.24225	−0.602071	−0.301035	−	0.953613i	$-0.597332\pi$
$29$	−3.24225	−0.602071	−0.301035	+	0.953613i	$0.597332\pi$
$30$	0	0
$31$	10.4737	1.88113	0.940566	−	0.339612i	$-0.110296\pi$
$31$	10.4737	1.88113	0.940566	+	0.339612i	$0.110296\pi$
$32$	0	0
$33$	0.835952	0.145521
$34$	0	0
$35$	0.462645	0.0782013
$36$	0	0
$37$	6.14802	1.01073	0.505364	−	0.862906i	$-0.331358\pi$
$37$	6.14802	1.01073	0.505364	+	0.862906i	$0.331358\pi$
$38$	0	0
$39$	0.126445	0.0202474
$40$	0	0
$41$	0.268578	0.0419449	0.0209724	−	0.999780i	$-0.493324\pi$
$41$	0.268578	0.0419449	0.0209724	+	0.999780i	$0.493324\pi$
$42$	0	0
$43$	−3.38909	−0.516832	−0.258416	−	0.966034i	$-0.583201\pi$
$43$	−3.38909	−0.516832	−0.258416	+	0.966034i	$0.583201\pi$
$44$	0	0
$45$	−5.37182	−0.800784
$46$	0	0
$47$	1.64138	0.239420	0.119710	−	0.992809i	$-0.461803\pi$
$47$	1.64138	0.239420	0.119710	+	0.992809i	$0.461803\pi$
$48$	0	0
$49$	−6.96680	−0.995257
$50$	0	0
$51$	4.84804	0.678862
$52$	0	0
$53$	−0.125379	−0.0172221	−0.00861107	−	0.999963i	$-0.502741\pi$
$53$	−0.125379	−0.0172221	−0.00861107	+	0.999963i	$0.502741\pi$
$54$	0	0
$55$	2.25710	0.304347
$56$	0	0
$57$	−2.11131	−0.279650
$58$	0	0
$59$	11.0523	1.43889	0.719445	−	0.694550i	$-0.244396\pi$
$59$	11.0523	1.43889	0.719445	+	0.694550i	$0.244396\pi$
$60$	0	0
$61$	−3.81357	−0.488278	−0.244139	−	0.969740i	$-0.578505\pi$
$61$	−3.81357	−0.488278	−0.244139	+	0.969740i	$0.578505\pi$
$62$	0	0
$63$	−0.385527	−0.0485718
$64$	0	0
$65$	0.341407	0.0423463
$66$	0	0
$67$	1.10648	0.135177	0.0675887	−	0.997713i	$-0.478469\pi$
$67$	1.10648	0.135177	0.0675887	+	0.997713i	$0.478469\pi$
$68$	0	0
$69$	4.48690	0.540160
$70$	0	0
$71$	1.17276	0.139181	0.0695905	−	0.997576i	$-0.477831\pi$
$71$	1.17276	0.139181	0.0695905	+	0.997576i	$0.477831\pi$
$72$	0	0
$73$	16.4984	1.93099	0.965497	−	0.260414i	$-0.0838590\pi$
$73$	16.4984	1.93099	0.965497	+	0.260414i	$0.0838590\pi$
$74$	0	0
$75$	1.36009	0.157050
$76$	0	0
$77$	0.161988	0.0184603
$78$	0	0
$79$	15.1790	1.70778	0.853888	−	0.520457i	$-0.174239\pi$
$79$	15.1790	1.70778	0.853888	+	0.520457i	$0.174239\pi$
$80$	0	0
$81$	1.82363	0.202626
$82$	0	0
$83$	−13.8440	−1.51958	−0.759789	−	0.650170i	$-0.774698\pi$
$83$	−13.8440	−1.51958	−0.759789	+	0.650170i	$0.774698\pi$
$84$	0	0
$85$	13.0899	1.41980
$86$	0	0
$87$	−3.04884	−0.326870
$88$	0	0
$89$	5.14859	0.545750	0.272875	−	0.962050i	$-0.412025\pi$
$89$	5.14859	0.545750	0.272875	+	0.962050i	$0.412025\pi$
$90$	0	0
$91$	0.0245022	0.00256852
$92$	0	0
$93$	9.84891	1.02128
$94$	0	0
$95$	−5.70062	−0.584872
$96$	0	0
$97$	4.43146	0.449946	0.224973	−	0.974365i	$-0.427771\pi$
$97$	4.43146	0.449946	0.224973	+	0.974365i	$0.427771\pi$
$98$	0	0
$99$	−1.88086	−0.189034
$100$	0	0
$101$	−5.49984	−0.547255	−0.273627	−	0.961836i	$-0.588224\pi$
$101$	−5.49984	−0.547255	−0.273627	+	0.961836i	$0.588224\pi$
$102$	0	0
$103$	−0.776257	−0.0764869	−0.0382434	−	0.999268i	$-0.512176\pi$
$103$	−0.776257	−0.0764869	−0.0382434	+	0.999268i	$0.512176\pi$
$104$	0	0
$105$	0.435047	0.0424562
$106$	0	0
$107$	2.16127	0.208938	0.104469	−	0.994528i	$-0.466686\pi$
$107$	2.16127	0.208938	0.104469	+	0.994528i	$0.466686\pi$
$108$	0	0
$109$	−1.04526	−0.100118	−0.0500591	−	0.998746i	$-0.515941\pi$
$109$	−1.04526	−0.100118	−0.0500591	+	0.998746i	$0.515941\pi$
$110$	0	0
$111$	5.78127	0.548734
$112$	0	0
$113$	−13.5267	−1.27249	−0.636245	−	0.771487i	$-0.719513\pi$
$113$	−13.5267	−1.27249	−0.636245	+	0.771487i	$0.719513\pi$
$114$	0	0
$115$	12.1148	1.12971
$116$	0	0
$117$	−0.284497	−0.0263018
$118$	0	0
$119$	0.939440	0.0861183
$120$	0	0
$121$	−10.2097	−0.928155
$122$	0	0
$123$	0.252557	0.0227723
$124$	0	0
$125$	−9.02255	−0.807002
$126$	0	0
$127$	17.9355	1.59152	0.795758	−	0.605615i	$-0.207073\pi$
$127$	17.9355	1.59152	0.795758	+	0.605615i	$0.207073\pi$
$128$	0	0
$129$	−3.18692	−0.280593
$130$	0	0
$131$	2.01082	0.175686	0.0878431	−	0.996134i	$-0.472003\pi$
$131$	2.01082	0.175686	0.0878431	+	0.996134i	$0.472003\pi$
$132$	0	0
$133$	−0.409124	−0.0354755
$134$	0	0
$135$	−12.2139	−1.05121
$136$	0	0
$137$	12.7252	1.08719	0.543595	−	0.839348i	$-0.317063\pi$
$137$	12.7252	1.08719	0.543595	+	0.839348i	$0.317063\pi$
$138$	0	0
$139$	15.5490	1.31885	0.659423	−	0.751772i	$-0.270801\pi$
$139$	15.5490	1.31885	0.659423	+	0.751772i	$0.270801\pi$
$140$	0	0
$141$	1.54347	0.129984
$142$	0	0
$143$	0.119538	0.00999630
$144$	0	0
$145$	−8.23198	−0.683629
$146$	0	0
$147$	−6.55121	−0.540334
$148$	0	0
$149$	−6.62174	−0.542474	−0.271237	−	0.962513i	$-0.587433\pi$
$149$	−6.62174	−0.542474	−0.271237	+	0.962513i	$0.587433\pi$
$150$	0	0
$151$	9.45873	0.769741	0.384870	−	0.922971i	$-0.374246\pi$
$151$	9.45873	0.769741	0.384870	+	0.922971i	$0.374246\pi$
$152$	0	0
$153$	−10.9079	−0.881854
$154$	0	0
$155$	26.5924	2.13595
$156$	0	0
$157$	11.1905	0.893101	0.446550	−	0.894759i	$-0.352652\pi$
$157$	11.1905	0.893101	0.446550	+	0.894759i	$0.352652\pi$
$158$	0	0
$159$	−0.117900	−0.00935006
$160$	0	0
$161$	0.869459	0.0685229
$162$	0	0
$163$	−15.5098	−1.21482	−0.607410	−	0.794388i	$-0.707792\pi$
$163$	−15.5098	−1.21482	−0.607410	+	0.794388i	$0.707792\pi$
$164$	0	0
$165$	2.12246	0.165233
$166$	0	0
$167$	9.25286	0.716008	0.358004	−	0.933720i	$-0.383457\pi$
$167$	9.25286	0.716008	0.358004	+	0.933720i	$0.383457\pi$
$168$	0	0
$169$	−12.9819	−0.998609
$170$	0	0
$171$	4.75038	0.363271
$172$	0	0
$173$	6.87982	0.523063	0.261532	−	0.965195i	$-0.415772\pi$
$173$	6.87982	0.523063	0.261532	+	0.965195i	$0.415772\pi$
$174$	0	0
$175$	0.263555	0.0199229
$176$	0	0
$177$	10.3930	0.781187
$178$	0	0
$179$	−7.07028	−0.528458	−0.264229	−	0.964460i	$-0.585117\pi$
$179$	−7.07028	−0.528458	−0.264229	+	0.964460i	$0.585117\pi$
$180$	0	0
$181$	−10.4726	−0.778424	−0.389212	−	0.921148i	$-0.627253\pi$
$181$	−10.4726	−0.778424	−0.389212	+	0.921148i	$0.627253\pi$
$182$	0	0
$183$	−3.58608	−0.265091
$184$	0	0
$185$	15.6096	1.14764
$186$	0	0
$187$	4.58323	0.335159
$188$	0	0
$189$	−0.876572	−0.0637612
$190$	0	0
$191$	6.17867	0.447073	0.223537	−	0.974696i	$-0.428240\pi$
$191$	6.17867	0.447073	0.223537	+	0.974696i	$0.428240\pi$
$192$	0	0
$193$	−9.60019	−0.691037	−0.345518	−	0.938412i	$-0.612297\pi$
$193$	−9.60019	−0.691037	−0.345518	+	0.938412i	$0.612297\pi$
$194$	0	0
$195$	0.321041	0.0229902
$196$	0	0
$197$	0.801618	0.0571129	0.0285565	−	0.999592i	$-0.490909\pi$
$197$	0.801618	0.0571129	0.0285565	+	0.999592i	$0.490909\pi$
$198$	0	0
$199$	−8.82739	−0.625757	−0.312878	−	0.949793i	$-0.601293\pi$
$199$	−8.82739	−0.625757	−0.312878	+	0.949793i	$0.601293\pi$
$200$	0	0
$201$	1.04047	0.0733891
$202$	0	0
$203$	−0.590795	−0.0414657
$204$	0	0
$205$	0.681912	0.0476268
$206$	0	0
$207$	−10.0954	−0.701677
$208$	0	0
$209$	−1.99599	−0.138065
$210$	0	0
$211$	−17.2881	−1.19016	−0.595080	−	0.803667i	$-0.702880\pi$
$211$	−17.2881	−1.19016	−0.595080	+	0.803667i	$0.702880\pi$
$212$	0	0
$213$	1.10280	0.0755626
$214$	0	0
$215$	−8.60481	−0.586843
$216$	0	0
$217$	1.90849	0.129557
$218$	0	0
$219$	15.5142	1.04836
$220$	0	0
$221$	0.693254	0.0466333
$222$	0	0
$223$	−25.1275	−1.68266	−0.841331	−	0.540520i	$-0.818228\pi$
$223$	−25.1275	−1.68266	−0.841331	+	0.540520i	$0.818228\pi$
$224$	0	0
$225$	−3.06016	−0.204011
$226$	0	0
$227$	27.1487	1.80192	0.900960	−	0.433903i	$-0.142864\pi$
$227$	27.1487	1.80192	0.900960	+	0.433903i	$0.142864\pi$
$228$	0	0
$229$	−14.1587	−0.935631	−0.467815	−	0.883826i	$-0.654959\pi$
$229$	−14.1587	−0.935631	−0.467815	+	0.883826i	$0.654959\pi$
$230$	0	0
$231$	0.152325	0.0100223
$232$	0	0
$233$	0.679896	0.0445415	0.0222707	−	0.999752i	$-0.492910\pi$
$233$	0.679896	0.0445415	0.0222707	+	0.999752i	$0.492910\pi$
$234$	0	0
$235$	4.16743	0.271853
$236$	0	0
$237$	14.2736	0.927167
$238$	0	0
$239$	−22.9888	−1.48702	−0.743512	−	0.668723i	$-0.766841\pi$
$239$	−22.9888	−1.48702	−0.743512	+	0.668723i	$0.766841\pi$
$240$	0	0
$241$	−1.73150	−0.111536	−0.0557678	−	0.998444i	$-0.517761\pi$
$241$	−1.73150	−0.111536	−0.0557678	+	0.998444i	$0.517761\pi$
$242$	0	0
$243$	16.1466	1.03580
$244$	0	0
$245$	−17.6885	−1.13008
$246$	0	0
$247$	−0.301911	−0.0192101
$248$	0	0
$249$	−13.0182	−0.824994
$250$	0	0
$251$	−6.06697	−0.382944	−0.191472	−	0.981498i	$-0.561326\pi$
$251$	−6.06697	−0.382944	−0.191472	+	0.981498i	$0.561326\pi$
$252$	0	0
$253$	4.24181	0.266681
$254$	0	0
$255$	12.3090	0.770822
$256$	0	0
$257$	−27.5166	−1.71644	−0.858221	−	0.513281i	$-0.828430\pi$
$257$	−27.5166	−1.71644	−0.858221	+	0.513281i	$0.828430\pi$
$258$	0	0
$259$	1.12028	0.0696106
$260$	0	0
$261$	6.85979	0.424610
$262$	0	0
$263$	2.00972	0.123925	0.0619624	−	0.998078i	$-0.480264\pi$
$263$	2.00972	0.123925	0.0619624	+	0.998078i	$0.480264\pi$
$264$	0	0
$265$	−0.318334	−0.0195551
$266$	0	0
$267$	4.84146	0.296293
$268$	0	0
$269$	−20.2239	−1.23307	−0.616536	−	0.787326i	$-0.711465\pi$
$269$	−20.2239	−1.23307	−0.616536	+	0.787326i	$0.711465\pi$
$270$	0	0
$271$	8.29982	0.504178	0.252089	−	0.967704i	$-0.418882\pi$
$271$	8.29982	0.504178	0.252089	+	0.967704i	$0.418882\pi$
$272$	0	0
$273$	0.0230405	0.00139448
$274$	0	0
$275$	1.28580	0.0775367
$276$	0	0
$277$	22.4498	1.34888	0.674439	−	0.738331i	$-0.264386\pi$
$277$	22.4498	1.34888	0.674439	+	0.738331i	$0.264386\pi$
$278$	0	0
$279$	−22.1597	−1.32667
$280$	0	0
$281$	−15.5854	−0.929750	−0.464875	−	0.885376i	$-0.653901\pi$
$281$	−15.5854	−0.929750	−0.464875	+	0.885376i	$0.653901\pi$
$282$	0	0
$283$	11.3036	0.671930	0.335965	−	0.941874i	$-0.390938\pi$
$283$	11.3036	0.671930	0.335965	+	0.941874i	$0.390938\pi$
$284$	0	0
$285$	−5.36056	−0.317532
$286$	0	0
$287$	0.0489397	0.00288882
$288$	0	0
$289$	9.58012	0.563537
$290$	0	0
$291$	4.16711	0.244280
$292$	0	0
$293$	−14.9544	−0.873645	−0.436823	−	0.899548i	$-0.643896\pi$
$293$	−14.9544	−0.873645	−0.436823	+	0.899548i	$0.643896\pi$
$294$	0	0
$295$	28.0615	1.63380
$296$	0	0
$297$	−4.27652	−0.248149
$298$	0	0
$299$	0.641612	0.0371054
$300$	0	0
$301$	−0.617553	−0.0355951
$302$	0	0
$303$	−5.17176	−0.297110
$304$	0	0
$305$	−9.68256	−0.554421
$306$	0	0
$307$	9.30639	0.531144	0.265572	−	0.964091i	$-0.414439\pi$
$307$	9.30639	0.531144	0.265572	+	0.964091i	$0.414439\pi$
$308$	0	0
$309$	−0.729951	−0.0415255
$310$	0	0
$311$	20.4862	1.16167	0.580833	−	0.814023i	$-0.302727\pi$
$311$	20.4862	1.16167	0.580833	+	0.814023i	$0.302727\pi$
$312$	0	0
$313$	20.2950	1.14714	0.573572	−	0.819155i	$-0.305557\pi$
$313$	20.2950	1.14714	0.573572	+	0.819155i	$0.305557\pi$
$314$	0	0
$315$	−0.978841	−0.0551514
$316$	0	0
$317$	6.39147	0.358981	0.179490	−	0.983760i	$-0.442555\pi$
$317$	6.39147	0.358981	0.179490	+	0.983760i	$0.442555\pi$
$318$	0	0
$319$	−2.88230	−0.161378
$320$	0	0
$321$	2.03235	0.113435
$322$	0	0
$323$	−11.5756	−0.644083
$324$	0	0
$325$	0.194489	0.0107883
$326$	0	0
$327$	−0.982911	−0.0543551
$328$	0	0
$329$	0.299089	0.0164893
$330$	0	0
$331$	15.3663	0.844609	0.422304	−	0.906454i	$-0.361221\pi$
$331$	15.3663	0.844609	0.422304	+	0.906454i	$0.361221\pi$
$332$	0	0
$333$	−13.0077	−0.712815
$334$	0	0
$335$	2.80931	0.153489
$336$	0	0
$337$	5.45449	0.297125	0.148563	−	0.988903i	$-0.452535\pi$
$337$	5.45449	0.297125	0.148563	+	0.988903i	$0.452535\pi$
$338$	0	0
$339$	−12.7198	−0.690846
$340$	0	0
$341$	9.31093	0.504215
$342$	0	0
$343$	−2.54500	−0.137417
$344$	0	0
$345$	11.3921	0.613331
$346$	0	0
$347$	35.7966	1.92166	0.960832	−	0.277131i	$-0.0893837\pi$
$347$	35.7966	1.92166	0.960832	+	0.277131i	$0.0893837\pi$
$348$	0	0
$349$	24.6612	1.32008	0.660042	−	0.751229i	$-0.270538\pi$
$349$	24.6612	1.32008	0.660042	+	0.751229i	$0.270538\pi$
$350$	0	0
$351$	−0.646861	−0.0345269
$352$	0	0
$353$	9.76819	0.519909	0.259954	−	0.965621i	$-0.416293\pi$
$353$	9.76819	0.519909	0.259954	+	0.965621i	$0.416293\pi$
$354$	0	0
$355$	2.97760	0.158035
$356$	0	0
$357$	0.883399	0.0467544
$358$	0	0
$359$	−26.1009	−1.37755	−0.688777	−	0.724973i	$-0.741852\pi$
$359$	−26.1009	−1.37755	−0.688777	+	0.724973i	$0.741852\pi$
$360$	0	0
$361$	−13.9589	−0.734677
$362$	0	0
$363$	−9.60067	−0.503904
$364$	0	0
$365$	41.8890	2.19257
$366$	0	0
$367$	−21.2517	−1.10933	−0.554665	−	0.832074i	$-0.687154\pi$
$367$	−21.2517	−1.10933	−0.554665	+	0.832074i	$0.687154\pi$
$368$	0	0
$369$	−0.568244	−0.0295816
$370$	0	0
$371$	−0.0228463	−0.00118612
$372$	0	0
$373$	0.122167	0.00632556	0.00316278	−	0.999995i	$-0.498993\pi$
$373$	0.122167	0.00632556	0.00316278	+	0.999995i	$0.498993\pi$
$374$	0	0
$375$	−8.48433	−0.438129
$376$	0	0
$377$	−0.435974	−0.0224538
$378$	0	0
$379$	4.08347	0.209754	0.104877	−	0.994485i	$-0.466555\pi$
$379$	4.08347	0.209754	0.104877	+	0.994485i	$0.466555\pi$
$380$	0	0
$381$	16.8656	0.864049
$382$	0	0
$383$	2.43875	0.124614	0.0623071	−	0.998057i	$-0.480154\pi$
$383$	2.43875	0.124614	0.0623071	+	0.998057i	$0.480154\pi$
$384$	0	0
$385$	0.411283	0.0209609
$386$	0	0
$387$	7.17047	0.364495
$388$	0	0
$389$	39.1682	1.98591	0.992954	−	0.118504i	$-0.0378099\pi$
$389$	39.1682	1.98591	0.992954	+	0.118504i	$0.0378099\pi$
$390$	0	0
$391$	24.6001	1.24408
$392$	0	0
$393$	1.89087	0.0953817
$394$	0	0
$395$	38.5391	1.93911
$396$	0	0
$397$	−33.3866	−1.67563	−0.837813	−	0.545957i	$-0.816166\pi$
$397$	−33.3866	−1.67563	−0.837813	+	0.545957i	$0.816166\pi$
$398$	0	0
$399$	−0.384718	−0.0192600
$400$	0	0
$401$	−22.9048	−1.14381	−0.571906	−	0.820319i	$-0.693796\pi$
$401$	−22.9048	−1.14381	−0.571906	+	0.820319i	$0.693796\pi$
$402$	0	0
$403$	1.40836	0.0701555
$404$	0	0
$405$	4.63015	0.230074
$406$	0	0
$407$	5.46548	0.270914
$408$	0	0
$409$	−7.76272	−0.383842	−0.191921	−	0.981410i	$-0.561472\pi$
$409$	−7.76272	−0.383842	−0.191921	+	0.981410i	$0.561472\pi$
$410$	0	0
$411$	11.9661	0.590246
$412$	0	0
$413$	2.01393	0.0990989
$414$	0	0
$415$	−35.1496	−1.72542
$416$	0	0
$417$	14.6214	0.716014
$418$	0	0
$419$	−4.09133	−0.199874	−0.0999372	−	0.994994i	$-0.531864\pi$
$419$	−4.09133	−0.199874	−0.0999372	+	0.994994i	$0.531864\pi$
$420$	0	0
$421$	−22.8486	−1.11357	−0.556787	−	0.830656i	$-0.687966\pi$
$421$	−22.8486	−1.11357	−0.556787	+	0.830656i	$0.687966\pi$
$422$	0	0
$423$	−3.47275	−0.168851
$424$	0	0
$425$	7.45692	0.361714
$426$	0	0
$427$	−0.694900	−0.0336286
$428$	0	0
$429$	0.112408	0.00542709
$430$	0	0
$431$	−34.7464	−1.67367	−0.836837	−	0.547452i	$-0.815598\pi$
$431$	−34.7464	−1.67367	−0.836837	+	0.547452i	$0.815598\pi$
$432$	0	0
$433$	20.8617	1.00255	0.501274	−	0.865289i	$-0.332865\pi$
$433$	20.8617	1.00255	0.501274	+	0.865289i	$0.332865\pi$
$434$	0	0
$435$	−7.74092	−0.371149
$436$	0	0
$437$	−10.7133	−0.512486
$438$	0	0
$439$	20.9912	1.00185	0.500927	−	0.865490i	$-0.332993\pi$
$439$	20.9912	1.00185	0.500927	+	0.865490i	$0.332993\pi$
$440$	0	0
$441$	14.7400	0.701904
$442$	0	0
$443$	11.9835	0.569355	0.284677	−	0.958623i	$-0.408114\pi$
$443$	11.9835	0.569355	0.284677	+	0.958623i	$0.408114\pi$
$444$	0	0
$445$	13.0721	0.619678
$446$	0	0
$447$	−6.22673	−0.294514
$448$	0	0
$449$	−20.0148	−0.944557	−0.472279	−	0.881449i	$-0.656568\pi$
$449$	−20.0148	−0.944557	−0.472279	+	0.881449i	$0.656568\pi$
$450$	0	0
$451$	0.238761	0.0112428
$452$	0	0
$453$	8.89449	0.417900
$454$	0	0
$455$	0.0622103	0.00291646
$456$	0	0
$457$	−19.4888	−0.911649	−0.455824	−	0.890070i	$-0.650655\pi$
$457$	−19.4888	−0.911649	−0.455824	+	0.890070i	$0.650655\pi$
$458$	0	0
$459$	−24.8014	−1.15763
$460$	0	0
$461$	−14.1966	−0.661200	−0.330600	−	0.943771i	$-0.607251\pi$
$461$	−14.1966	−0.661200	−0.330600	+	0.943771i	$0.607251\pi$
$462$	0	0
$463$	−14.1811	−0.659051	−0.329525	−	0.944147i	$-0.606889\pi$
$463$	−14.1811	−0.659051	−0.329525	+	0.944147i	$0.606889\pi$
$464$	0	0
$465$	25.0061	1.15963
$466$	0	0
$467$	−12.2806	−0.568279	−0.284140	−	0.958783i	$-0.591708\pi$
$467$	−12.2806	−0.568279	−0.284140	+	0.958783i	$0.591708\pi$
$468$	0	0
$469$	0.201619	0.00930991
$470$	0	0
$471$	10.5230	0.484873
$472$	0	0
$473$	−3.01285	−0.138531
$474$	0	0
$475$	−3.24747	−0.149004
$476$	0	0
$477$	0.265270	0.0121459
$478$	0	0
$479$	34.2723	1.56594	0.782970	−	0.622060i	$-0.213704\pi$
$479$	34.2723	1.56594	0.782970	+	0.622060i	$0.213704\pi$
$480$	0	0
$481$	0.826702	0.0376944
$482$	0	0
$483$	0.817593	0.0372018
$484$	0	0
$485$	11.2513	0.510897
$486$	0	0
$487$	23.5939	1.06914	0.534570	−	0.845124i	$-0.320474\pi$
$487$	23.5939	1.06914	0.534570	+	0.845124i	$0.320474\pi$
$488$	0	0
$489$	−14.5846	−0.659538
$490$	0	0
$491$	−18.2209	−0.822298	−0.411149	−	0.911568i	$-0.634872\pi$
$491$	−18.2209	−0.822298	−0.411149	+	0.911568i	$0.634872\pi$
$492$	0	0
$493$	−16.7157	−0.752838
$494$	0	0
$495$	−4.77546	−0.214641
$496$	0	0
$497$	0.213697	0.00958564
$498$	0	0
$499$	−6.84785	−0.306552	−0.153276	−	0.988183i	$-0.548982\pi$
$499$	−6.84785	−0.306552	−0.153276	+	0.988183i	$0.548982\pi$
$500$	0	0
$501$	8.70090	0.388728
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−13.9639	−0.621387
$506$	0	0
$507$	−12.2075	−0.542154
$508$	0	0
$509$	−28.9143	−1.28160	−0.640801	−	0.767707i	$-0.721398\pi$
$509$	−28.9143	−1.28160	−0.640801	+	0.767707i	$0.721398\pi$
$510$	0	0
$511$	3.00630	0.132991
$512$	0	0
$513$	10.8009	0.476873
$514$	0	0
$515$	−1.97089	−0.0868480
$516$	0	0
$517$	1.45916	0.0641738
$518$	0	0
$519$	6.46942	0.283976
$520$	0	0
$521$	28.1669	1.23402	0.617008	−	0.786957i	$-0.288345\pi$
$521$	28.1669	1.23402	0.617008	+	0.786957i	$0.288345\pi$
$522$	0	0
$523$	−12.0284	−0.525966	−0.262983	−	0.964800i	$-0.584706\pi$
$523$	−12.0284	−0.525966	−0.262983	+	0.964800i	$0.584706\pi$
$524$	0	0
$525$	0.247833	0.0108163
$526$	0	0
$527$	53.9981	2.35219
$528$	0	0
$529$	−0.232408	−0.0101047
$530$	0	0
$531$	−23.3839	−1.01478
$532$	0	0
$533$	0.0361148	0.00156430
$534$	0	0
$535$	5.48741	0.237241
$536$	0	0
$537$	−6.64852	−0.286905
$538$	0	0
$539$	−6.19336	−0.266767
$540$	0	0
$541$	7.07499	0.304178	0.152089	−	0.988367i	$-0.451400\pi$
$541$	7.07499	0.304178	0.152089	+	0.988367i	$0.451400\pi$
$542$	0	0
$543$	−9.84790	−0.422614
$544$	0	0
$545$	−2.65390	−0.113680
$546$	0	0
$547$	3.18218	0.136060	0.0680300	−	0.997683i	$-0.478329\pi$
$547$	3.18218	0.136060	0.0680300	+	0.997683i	$0.478329\pi$
$548$	0	0
$549$	8.06856	0.344358
$550$	0	0
$551$	7.27966	0.310124
$552$	0	0
$553$	2.76589	0.117618
$554$	0	0
$555$	14.6785	0.623066
$556$	0	0
$557$	−20.0460	−0.849375	−0.424688	−	0.905340i	$-0.639616\pi$
$557$	−20.0460	−0.849375	−0.424688	+	0.905340i	$0.639616\pi$
$558$	0	0
$559$	−0.455720	−0.0192749
$560$	0	0
$561$	4.30983	0.181961
$562$	0	0
$563$	18.5742	0.782808	0.391404	−	0.920219i	$-0.371990\pi$
$563$	18.5742	0.782808	0.391404	+	0.920219i	$0.371990\pi$
$564$	0	0
$565$	−34.3440	−1.44486
$566$	0	0
$567$	0.332298	0.0139552
$568$	0	0
$569$	21.4018	0.897211	0.448605	−	0.893730i	$-0.351921\pi$
$569$	21.4018	0.897211	0.448605	+	0.893730i	$0.351921\pi$
$570$	0	0
$571$	−9.87058	−0.413071	−0.206535	−	0.978439i	$-0.566219\pi$
$571$	−9.87058	−0.413071	−0.206535	+	0.978439i	$0.566219\pi$
$572$	0	0
$573$	5.81009	0.242720
$574$	0	0
$575$	6.90143	0.287810
$576$	0	0
$577$	1.12190	0.0467052	0.0233526	−	0.999727i	$-0.492566\pi$
$577$	1.12190	0.0467052	0.0233526	+	0.999727i	$0.492566\pi$
$578$	0	0
$579$	−9.02751	−0.375170
$580$	0	0
$581$	−2.52262	−0.104656
$582$	0	0
$583$	−0.111460	−0.00461619
$584$	0	0
$585$	−0.722330	−0.0298647
$586$	0	0
$587$	11.7439	0.484724	0.242362	−	0.970186i	$-0.422078\pi$
$587$	11.7439	0.484724	0.242362	+	0.970186i	$0.422078\pi$
$588$	0	0
$589$	−23.5161	−0.968962
$590$	0	0
$591$	0.753799	0.0310071
$592$	0	0
$593$	18.1624	0.745840	0.372920	−	0.927864i	$-0.378357\pi$
$593$	18.1624	0.745840	0.372920	+	0.927864i	$0.378357\pi$
$594$	0	0
$595$	2.38521	0.0977841
$596$	0	0
$597$	−8.30081	−0.339729
$598$	0	0
$599$	47.8526	1.95520	0.977601	−	0.210466i	$-0.0674981\pi$
$599$	47.8526	1.95520	0.977601	+	0.210466i	$0.0674981\pi$
$600$	0	0
$601$	−33.3626	−1.36089	−0.680445	−	0.732800i	$-0.738213\pi$
$601$	−33.3626	−1.36089	−0.680445	+	0.732800i	$0.738213\pi$
$602$	0	0
$603$	−2.34102	−0.0953338
$604$	0	0
$605$	−25.9222	−1.05389
$606$	0	0
$607$	26.0885	1.05890	0.529450	−	0.848341i	$-0.322398\pi$
$607$	26.0885	1.05890	0.529450	+	0.848341i	$0.322398\pi$
$608$	0	0
$609$	−0.555552	−0.0225121
$610$	0	0
$611$	0.220711	0.00892902
$612$	0	0
$613$	−30.6773	−1.23905	−0.619523	−	0.784978i	$-0.712674\pi$
$613$	−30.6773	−1.23905	−0.619523	+	0.784978i	$0.712674\pi$
$614$	0	0
$615$	0.641234	0.0258571
$616$	0	0
$617$	−18.7460	−0.754687	−0.377343	−	0.926073i	$-0.623162\pi$
$617$	−18.7460	−0.754687	−0.377343	+	0.926073i	$0.623162\pi$
$618$	0	0
$619$	−1.96827	−0.0791115	−0.0395558	−	0.999217i	$-0.512594\pi$
$619$	−1.96827	−0.0791115	−0.0395558	+	0.999217i	$0.512594\pi$
$620$	0	0
$621$	−22.9539	−0.921107
$622$	0	0
$623$	0.938164	0.0375868
$624$	0	0
$625$	−30.1399	−1.20559
$626$	0	0
$627$	−1.87692	−0.0749570
$628$	0	0
$629$	31.6967	1.26383
$630$	0	0
$631$	−18.3422	−0.730190	−0.365095	−	0.930970i	$-0.618964\pi$
$631$	−18.3422	−0.730190	−0.365095	+	0.930970i	$0.618964\pi$
$632$	0	0
$633$	−16.2568	−0.646149
$634$	0	0
$635$	45.5376	1.80711
$636$	0	0
$637$	−0.936801	−0.0371174
$638$	0	0
$639$	−2.48126	−0.0981572
$640$	0	0
$641$	−22.2154	−0.877455	−0.438727	−	0.898620i	$-0.644571\pi$
$641$	−22.2154	−0.877455	−0.438727	+	0.898620i	$0.644571\pi$
$642$	0	0
$643$	−41.0526	−1.61896	−0.809478	−	0.587151i	$-0.800250\pi$
$643$	−41.0526	−1.61896	−0.809478	+	0.587151i	$0.800250\pi$
$644$	0	0
$645$	−8.09151	−0.318603
$646$	0	0
$647$	−21.3454	−0.839176	−0.419588	−	0.907715i	$-0.637825\pi$
$647$	−21.3454	−0.839176	−0.419588	+	0.907715i	$0.637825\pi$
$648$	0	0
$649$	9.82532	0.385677
$650$	0	0
$651$	1.79464	0.0703377
$652$	0	0
$653$	13.9945	0.547648	0.273824	−	0.961780i	$-0.411711\pi$
$653$	13.9945	0.547648	0.273824	+	0.961780i	$0.411711\pi$
$654$	0	0
$655$	5.10542	0.199485
$656$	0	0
$657$	−34.9065	−1.36183
$658$	0	0
$659$	−7.02856	−0.273794	−0.136897	−	0.990585i	$-0.543713\pi$
$659$	−7.02856	−0.273794	−0.136897	+	0.990585i	$0.543713\pi$
$660$	0	0
$661$	−42.3996	−1.64915	−0.824577	−	0.565750i	$-0.808587\pi$
$661$	−42.3996	−1.64915	−0.824577	+	0.565750i	$0.808587\pi$
$662$	0	0
$663$	0.651899	0.0253177
$664$	0	0
$665$	−1.03875	−0.0402812
$666$	0	0
$667$	−15.4705	−0.599021
$668$	0	0
$669$	−23.6286	−0.913534
$670$	0	0
$671$	−3.39020	−0.130877
$672$	0	0
$673$	−4.18044	−0.161144	−0.0805720	−	0.996749i	$-0.525675\pi$
$673$	−4.18044	−0.161144	−0.0805720	+	0.996749i	$0.525675\pi$
$674$	0	0
$675$	−6.95790	−0.267810
$676$	0	0
$677$	−24.1684	−0.928866	−0.464433	−	0.885608i	$-0.653742\pi$
$677$	−24.1684	−0.928866	−0.464433	+	0.885608i	$0.653742\pi$
$678$	0	0
$679$	0.807490	0.0309886
$680$	0	0
$681$	25.5292	0.978279
$682$	0	0
$683$	13.5681	0.519170	0.259585	−	0.965720i	$-0.416414\pi$
$683$	13.5681	0.519170	0.259585	+	0.965720i	$0.416414\pi$
$684$	0	0
$685$	32.3090	1.23446
$686$	0	0
$687$	−13.3141	−0.507963
$688$	0	0
$689$	−0.0168593	−0.000642287 0
$690$	0	0
$691$	15.9632	0.607270	0.303635	−	0.952788i	$-0.401800\pi$
$691$	15.9632	0.607270	0.303635	+	0.952788i	$0.401800\pi$
$692$	0	0
$693$	−0.342726	−0.0130191
$694$	0	0
$695$	39.4784	1.49750
$696$	0	0
$697$	1.38468	0.0524485
$698$	0	0
$699$	0.639338	0.0241820
$700$	0	0
$701$	−47.5345	−1.79535	−0.897676	−	0.440656i	$-0.854746\pi$
$701$	−47.5345	−1.79535	−0.897676	+	0.440656i	$0.854746\pi$
$702$	0	0
$703$	−13.8038	−0.520621
$704$	0	0
$705$	3.91883	0.147592
$706$	0	0
$707$	−1.00217	−0.0376904
$708$	0	0
$709$	−15.8323	−0.594593	−0.297297	−	0.954785i	$-0.596085\pi$
$709$	−15.8323	−0.594593	−0.297297	+	0.954785i	$0.596085\pi$
$710$	0	0
$711$	−32.1150	−1.20441
$712$	0	0
$713$	49.9757	1.87160
$714$	0	0
$715$	0.303504	0.0113504
$716$	0	0
$717$	−21.6175	−0.807319
$718$	0	0
$719$	−16.5518	−0.617278	−0.308639	−	0.951179i	$-0.599874\pi$
$719$	−16.5518	−0.617278	−0.308639	+	0.951179i	$0.599874\pi$
$720$	0	0
$721$	−0.141448	−0.00526779
$722$	0	0
$723$	−1.62821	−0.0605538
$724$	0	0
$725$	−4.68951	−0.174164
$726$	0	0
$727$	8.94610	0.331792	0.165896	−	0.986143i	$-0.446948\pi$
$727$	8.94610	0.331792	0.165896	+	0.986143i	$0.446948\pi$
$728$	0	0
$729$	9.71249	0.359722
$730$	0	0
$731$	−17.4728	−0.646254
$732$	0	0
$733$	−4.93717	−0.182358	−0.0911792	−	0.995835i	$-0.529064\pi$
$733$	−4.93717	−0.182358	−0.0911792	+	0.995835i	$0.529064\pi$
$734$	0	0
$735$	−16.6333	−0.613529
$736$	0	0
$737$	0.983637	0.0362327
$738$	0	0
$739$	−40.4623	−1.48843	−0.744214	−	0.667941i	$-0.767176\pi$
$739$	−40.4623	−1.48843	−0.744214	+	0.667941i	$0.767176\pi$
$740$	0	0
$741$	−0.283901	−0.0104294
$742$	0	0
$743$	−31.5104	−1.15600	−0.578002	−	0.816035i	$-0.696168\pi$
$743$	−31.5104	−1.15600	−0.578002	+	0.816035i	$0.696168\pi$
$744$	0	0
$745$	−16.8124	−0.615959
$746$	0	0
$747$	29.2904	1.07168
$748$	0	0
$749$	0.393822	0.0143899
$750$	0	0
$751$	−5.38722	−0.196582	−0.0982912	−	0.995158i	$-0.531338\pi$
$751$	−5.38722	−0.196582	−0.0982912	+	0.995158i	$0.531338\pi$
$752$	0	0
$753$	−5.70505	−0.207904
$754$	0	0
$755$	24.0154	0.874012
$756$	0	0
$757$	−26.9001	−0.977701	−0.488851	−	0.872368i	$-0.662584\pi$
$757$	−26.9001	−0.977701	−0.488851	+	0.872368i	$0.662584\pi$
$758$	0	0
$759$	3.98878	0.144783
$760$	0	0
$761$	29.0912	1.05456	0.527278	−	0.849693i	$-0.323213\pi$
$761$	29.0912	1.05456	0.527278	+	0.849693i	$0.323213\pi$
$762$	0	0
$763$	−0.190466	−0.00689531
$764$	0	0
$765$	−27.6949	−1.00131
$766$	0	0
$767$	1.48617	0.0536624
$768$	0	0
$769$	32.9653	1.18876	0.594380	−	0.804185i	$-0.297398\pi$
$769$	32.9653	1.18876	0.594380	+	0.804185i	$0.297398\pi$
$770$	0	0
$771$	−25.8752	−0.931872
$772$	0	0
$773$	−25.1872	−0.905921	−0.452960	−	0.891531i	$-0.649632\pi$
$773$	−25.1872	−0.905921	−0.452960	+	0.891531i	$0.649632\pi$
$774$	0	0
$775$	15.1489	0.544164
$776$	0	0
$777$	1.05345	0.0377923
$778$	0	0
$779$	−0.603025	−0.0216056
$780$	0	0
$781$	1.04256	0.0373058
$782$	0	0
$783$	15.5971	0.557395
$784$	0	0
$785$	28.4124	1.01408
$786$	0	0
$787$	7.20074	0.256679	0.128339	−	0.991730i	$-0.459035\pi$
$787$	7.20074	0.256679	0.128339	+	0.991730i	$0.459035\pi$
$788$	0	0
$789$	1.88984	0.0672800
$790$	0	0
$791$	−2.46481	−0.0876386
$792$	0	0
$793$	−0.512798	−0.0182100
$794$	0	0
$795$	−0.299344	−0.0106166
$796$	0	0
$797$	38.5105	1.36411	0.682055	−	0.731301i	$-0.261086\pi$
$797$	38.5105	1.36411	0.682055	+	0.731301i	$0.261086\pi$
$798$	0	0
$799$	8.46230	0.299375
$800$	0	0
$801$	−10.8931	−0.384890
$802$	0	0
$803$	14.6668	0.517580
$804$	0	0
$805$	2.20753	0.0778052
$806$	0	0
$807$	−19.0175	−0.669447
$808$	0	0
$809$	49.8182	1.75152	0.875758	−	0.482751i	$-0.160362\pi$
$809$	49.8182	1.75152	0.875758	+	0.482751i	$0.160362\pi$
$810$	0	0
$811$	52.6891	1.85016	0.925081	−	0.379769i	$-0.123996\pi$
$811$	52.6891	1.85016	0.925081	+	0.379769i	$0.123996\pi$
$812$	0	0
$813$	7.80471	0.273723
$814$	0	0
$815$	−39.3789	−1.37938
$816$	0	0
$817$	7.60936	0.266218
$818$	0	0
$819$	−0.0518404	−0.00181145
$820$	0	0
$821$	−48.9512	−1.70841	−0.854204	−	0.519938i	$-0.825955\pi$
$821$	−48.9512	−1.70841	−0.854204	+	0.519938i	$0.825955\pi$
$822$	0	0
$823$	−17.1402	−0.597469	−0.298734	−	0.954336i	$-0.596564\pi$
$823$	−17.1402	−0.597469	−0.298734	+	0.954336i	$0.596564\pi$
$824$	0	0
$825$	1.20910	0.0420954
$826$	0	0
$827$	−51.2336	−1.78157	−0.890783	−	0.454429i	$-0.849843\pi$
$827$	−51.2336	−1.78157	−0.890783	+	0.454429i	$0.849843\pi$
$828$	0	0
$829$	−21.4591	−0.745304	−0.372652	−	0.927971i	$-0.621552\pi$
$829$	−21.4591	−0.745304	−0.372652	+	0.927971i	$0.621552\pi$
$830$	0	0
$831$	21.1106	0.732319
$832$	0	0
$833$	−35.9180	−1.24448
$834$	0	0
$835$	23.4927	0.813000
$836$	0	0
$837$	−50.3845	−1.74154
$838$	0	0
$839$	21.2313	0.732986	0.366493	−	0.930421i	$-0.380558\pi$
$839$	21.2313	0.732986	0.366493	+	0.930421i	$0.380558\pi$
$840$	0	0
$841$	−18.4878	−0.637511
$842$	0	0
$843$	−14.6557	−0.504770
$844$	0	0
$845$	−32.9607	−1.13388
$846$	0	0
$847$	−1.86039	−0.0639237
$848$	0	0
$849$	10.6293	0.364797
$850$	0	0
$851$	29.3355	1.00561
$852$	0	0
$853$	−28.7300	−0.983697	−0.491848	−	0.870681i	$-0.663679\pi$
$853$	−28.7300	−0.983697	−0.491848	+	0.870681i	$0.663679\pi$
$854$	0	0
$855$	12.0611	0.412480
$856$	0	0
$857$	−30.3052	−1.03521	−0.517603	−	0.855621i	$-0.673176\pi$
$857$	−30.3052	−1.03521	−0.517603	+	0.855621i	$0.673176\pi$
$858$	0	0
$859$	11.1416	0.380145	0.190072	−	0.981770i	$-0.439128\pi$
$859$	11.1416	0.380145	0.190072	+	0.981770i	$0.439128\pi$
$860$	0	0
$861$	0.0460203	0.00156837
$862$	0	0
$863$	57.0781	1.94296	0.971480	−	0.237121i	$-0.0762038\pi$
$863$	57.0781	1.94296	0.971480	+	0.237121i	$0.0762038\pi$
$864$	0	0
$865$	17.4677	0.593919
$866$	0	0
$867$	9.00864	0.305949
$868$	0	0
$869$	13.4939	0.457749
$870$	0	0
$871$	0.148784	0.00504135
$872$	0	0
$873$	−9.37585	−0.317324
$874$	0	0
$875$	−1.64407	−0.0555796
$876$	0	0
$877$	19.2340	0.649487	0.324744	−	0.945802i	$-0.394722\pi$
$877$	19.2340	0.649487	0.324744	+	0.945802i	$0.394722\pi$
$878$	0	0
$879$	−14.0623	−0.474310
$880$	0	0
$881$	43.8468	1.47724	0.738618	−	0.674125i	$-0.235479\pi$
$881$	43.8468	1.47724	0.738618	+	0.674125i	$0.235479\pi$
$882$	0	0
$883$	−32.7408	−1.10182	−0.550908	−	0.834566i	$-0.685719\pi$
$883$	−32.7408	−1.10182	−0.550908	+	0.834566i	$0.685719\pi$
$884$	0	0
$885$	26.3876	0.887008
$886$	0	0
$887$	−38.2321	−1.28371	−0.641854	−	0.766827i	$-0.721834\pi$
$887$	−38.2321	−1.28371	−0.641854	+	0.766827i	$0.721834\pi$
$888$	0	0
$889$	3.26816	0.109611
$890$	0	0
$891$	1.62118	0.0543115
$892$	0	0
$893$	−3.68532	−0.123324
$894$	0	0
$895$	−17.9512	−0.600044
$896$	0	0
$897$	0.603338	0.0201449
$898$	0	0
$899$	−33.9584	−1.13257
$900$	0	0
$901$	−0.646403	−0.0215348
$902$	0	0
$903$	−0.580714	−0.0193249
$904$	0	0
$905$	−26.5897	−0.883871
$906$	0	0
$907$	−15.2419	−0.506098	−0.253049	−	0.967453i	$-0.581433\pi$
$907$	−15.2419	−0.506098	−0.253049	+	0.967453i	$0.581433\pi$
$908$	0	0
$909$	11.6363	0.385951
$910$	0	0
$911$	−4.81932	−0.159671	−0.0798356	−	0.996808i	$-0.525440\pi$
$911$	−4.81932	−0.159671	−0.0798356	+	0.996808i	$0.525440\pi$
$912$	0	0
$913$	−12.3071	−0.407305
$914$	0	0
$915$	−9.10496	−0.301001
$916$	0	0
$917$	0.366407	0.0120998
$918$	0	0
$919$	−40.5164	−1.33651	−0.668256	−	0.743931i	$-0.732959\pi$
$919$	−40.5164	−1.33651	−0.668256	+	0.743931i	$0.732959\pi$
$920$	0	0
$921$	8.75123	0.288363
$922$	0	0
$923$	0.157697	0.00519065
$924$	0	0
$925$	8.89233	0.292378
$926$	0	0
$927$	1.64236	0.0539423
$928$	0	0
$929$	−54.9994	−1.80447	−0.902236	−	0.431242i	$-0.858076\pi$
$929$	−54.9994	−1.80447	−0.902236	+	0.431242i	$0.858076\pi$
$930$	0	0
$931$	15.6422	0.512652
$932$	0	0
$933$	19.2641	0.630679
$934$	0	0
$935$	11.6367	0.380560
$936$	0	0
$937$	−36.0859	−1.17887	−0.589437	−	0.807814i	$-0.700650\pi$
$937$	−36.0859	−1.17887	−0.589437	+	0.807814i	$0.700650\pi$
$938$	0	0
$939$	19.0844	0.622795
$940$	0	0
$941$	−1.50153	−0.0489486	−0.0244743	−	0.999700i	$-0.507791\pi$
$941$	−1.50153	−0.0489486	−0.0244743	+	0.999700i	$0.507791\pi$
$942$	0	0
$943$	1.28153	0.0417324
$944$	0	0
$945$	−2.22559	−0.0723985
$946$	0	0
$947$	19.3758	0.629628	0.314814	−	0.949153i	$-0.398058\pi$
$947$	19.3758	0.629628	0.314814	+	0.949153i	$0.398058\pi$
$948$	0	0
$949$	2.21848	0.0720151
$950$	0	0
$951$	6.01020	0.194894
$952$	0	0
$953$	−21.5020	−0.696518	−0.348259	−	0.937398i	$-0.613227\pi$
$953$	−21.5020	−0.696518	−0.348259	+	0.937398i	$0.613227\pi$
$954$	0	0
$955$	15.6875	0.507635
$956$	0	0
$957$	−2.71037	−0.0876137
$958$	0	0
$959$	2.31876	0.0748767
$960$	0	0
$961$	78.6983	2.53866
$962$	0	0
$963$	−4.57271	−0.147354
$964$	0	0
$965$	−24.3746	−0.784646
$966$	0	0
$967$	41.8438	1.34561	0.672803	−	0.739822i	$-0.265090\pi$
$967$	41.8438	1.34561	0.672803	+	0.739822i	$0.265090\pi$
$968$	0	0
$969$	−10.8851	−0.349679
$970$	0	0
$971$	−15.2751	−0.490201	−0.245100	−	0.969498i	$-0.578821\pi$
$971$	−15.2751	−0.490201	−0.245100	+	0.969498i	$0.578821\pi$
$972$	0	0
$973$	2.83330	0.0908313
$974$	0	0
$975$	0.182887	0.00585707
$976$	0	0
$977$	47.6296	1.52380	0.761902	−	0.647692i	$-0.224266\pi$
$977$	47.6296	1.52380	0.761902	+	0.647692i	$0.224266\pi$
$978$	0	0
$979$	4.57701	0.146282
$980$	0	0
$981$	2.21152	0.0706082
$982$	0	0
$983$	−23.6697	−0.754947	−0.377473	−	0.926020i	$-0.623207\pi$
$983$	−23.6697	−0.754947	−0.377473	+	0.926020i	$0.623207\pi$
$984$	0	0
$985$	2.03528	0.0648496
$986$	0	0
$987$	0.281247	0.00895220
$988$	0	0
$989$	−16.1712	−0.514214
$990$	0	0
$991$	−32.7467	−1.04023	−0.520117	−	0.854095i	$-0.674112\pi$
$991$	−32.7467	−1.04023	−0.520117	+	0.854095i	$0.674112\pi$
$992$	0	0
$993$	14.4497	0.458546
$994$	0	0
$995$	−22.4125	−0.710523
$996$	0	0
$997$	23.8367	0.754917	0.377458	−	0.926027i	$-0.376798\pi$
$997$	23.8367	0.754917	0.377458	+	0.926027i	$0.376798\pi$
$998$	0	0
$999$	−29.5755	−0.935728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.20	✓	33
4.3	odd	2		8048.2.a.x.1.14		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.20	✓	33	1.1	even	1	trivial
8048.2.a.x.1.14		33	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)

\(f(q)\)	\(=\)	\(q+0.940347 q^{3} +2.53897 q^{5} +0.182218 q^{7} -2.11575 q^{9} +O(q^{10})\)	\(q+0.940347 q^{3} +2.53897 q^{5} +0.182218 q^{7} -2.11575 q^{9} +0.888982 q^{11} +0.134466 q^{13} +2.38751 q^{15} +5.15559 q^{17} -2.24525 q^{19} +0.171348 q^{21} +4.77154 q^{23} +1.44637 q^{25} -4.81058 q^{27} -3.24225 q^{29} +10.4737 q^{31} +0.835952 q^{33} +0.462645 q^{35} +6.14802 q^{37} +0.126445 q^{39} +0.268578 q^{41} -3.38909 q^{43} -5.37182 q^{45} +1.64138 q^{47} -6.96680 q^{49} +4.84804 q^{51} -0.125379 q^{53} +2.25710 q^{55} -2.11131 q^{57} +11.0523 q^{59} -3.81357 q^{61} -0.385527 q^{63} +0.341407 q^{65} +1.10648 q^{67} +4.48690 q^{69} +1.17276 q^{71} +16.4984 q^{73} +1.36009 q^{75} +0.161988 q^{77} +15.1790 q^{79} +1.82363 q^{81} -13.8440 q^{83} +13.0899 q^{85} -3.04884 q^{87} +5.14859 q^{89} +0.0245022 q^{91} +9.84891 q^{93} -5.70062 q^{95} +4.43146 q^{97} -1.88086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

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