LMFDB - Embedded newform 4024.2.a.d.1.7

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.73728 q^{3} +2.58604 q^{5} +2.10260 q^{7} +0.0181326 q^{9} +O(q^{10})$	$q-1.73728 q^{3} +2.58604 q^{5} +2.10260 q^{7} +0.0181326 q^{9} -2.62289 q^{11} +0.715213 q^{13} -4.49267 q^{15} -5.38011 q^{17} +1.38337 q^{19} -3.65280 q^{21} +6.72492 q^{23} +1.68762 q^{25} +5.18033 q^{27} -4.19384 q^{29} -5.92164 q^{31} +4.55669 q^{33} +5.43741 q^{35} -7.49843 q^{37} -1.24252 q^{39} -2.29080 q^{41} +7.41206 q^{43} +0.0468916 q^{45} -7.28843 q^{47} -2.57908 q^{49} +9.34674 q^{51} -11.8850 q^{53} -6.78291 q^{55} -2.40330 q^{57} -7.39694 q^{59} -1.47413 q^{61} +0.0381255 q^{63} +1.84957 q^{65} -4.45942 q^{67} -11.6831 q^{69} +8.17012 q^{71} -9.67286 q^{73} -2.93186 q^{75} -5.51489 q^{77} +9.04789 q^{79} -9.05407 q^{81} +4.25798 q^{83} -13.9132 q^{85} +7.28586 q^{87} +9.05608 q^{89} +1.50381 q^{91} +10.2875 q^{93} +3.57746 q^{95} +8.08527 q^{97} -0.0475598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) $ 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * 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
$3$	−1.73728	−1.00302	−0.501509	−	0.865153i	$-0.667221\pi$
$3$	−1.73728	−1.00302	−0.501509	+	0.865153i	$0.667221\pi$
$4$	0	0
$5$	2.58604	1.15651	0.578257	−	0.815855i	$-0.303733\pi$
$5$	2.58604	1.15651	0.578257	+	0.815855i	$0.303733\pi$
$6$	0	0
$7$	2.10260	0.794708	0.397354	−	0.917665i	$-0.369929\pi$
$7$	2.10260	0.794708	0.397354	+	0.917665i	$0.369929\pi$
$8$	0	0
$9$	0.0181326	0.00604419
$10$	0	0
$11$	−2.62289	−0.790832	−0.395416	−	0.918502i	$-0.629400\pi$
$11$	−2.62289	−0.790832	−0.395416	+	0.918502i	$0.629400\pi$
$12$	0	0
$13$	0.715213	0.198364	0.0991822	−	0.995069i	$-0.468377\pi$
$13$	0.715213	0.198364	0.0991822	+	0.995069i	$0.468377\pi$
$14$	0	0
$15$	−4.49267	−1.16000
$16$	0	0
$17$	−5.38011	−1.30487	−0.652434	−	0.757846i	$-0.726252\pi$
$17$	−5.38011	−1.30487	−0.652434	+	0.757846i	$0.726252\pi$
$18$	0	0
$19$	1.38337	0.317367	0.158684	−	0.987330i	$-0.449275\pi$
$19$	1.38337	0.317367	0.158684	+	0.987330i	$0.449275\pi$
$20$	0	0
$21$	−3.65280	−0.797106
$22$	0	0
$23$	6.72492	1.40224	0.701122	−	0.713041i	$-0.252683\pi$
$23$	6.72492	1.40224	0.701122	+	0.713041i	$0.252683\pi$
$24$	0	0
$25$	1.68762	0.337523
$26$	0	0
$27$	5.18033	0.996955
$28$	0	0
$29$	−4.19384	−0.778776	−0.389388	−	0.921074i	$-0.627313\pi$
$29$	−4.19384	−0.778776	−0.389388	+	0.921074i	$0.627313\pi$
$30$	0	0
$31$	−5.92164	−1.06356	−0.531779	−	0.846883i	$-0.678476\pi$
$31$	−5.92164	−1.06356	−0.531779	+	0.846883i	$0.678476\pi$
$32$	0	0
$33$	4.55669	0.793218
$34$	0	0
$35$	5.43741	0.919090
$36$	0	0
$37$	−7.49843	−1.23273	−0.616367	−	0.787459i	$-0.711396\pi$
$37$	−7.49843	−1.23273	−0.616367	+	0.787459i	$0.711396\pi$
$38$	0	0
$39$	−1.24252	−0.198963
$40$	0	0
$41$	−2.29080	−0.357763	−0.178882	−	0.983871i	$-0.557248\pi$
$41$	−2.29080	−0.357763	−0.178882	+	0.983871i	$0.557248\pi$
$42$	0	0
$43$	7.41206	1.13033	0.565164	−	0.824979i	$-0.308813\pi$
$43$	7.41206	1.13033	0.565164	+	0.824979i	$0.308813\pi$
$44$	0	0
$45$	0.0468916	0.00699019
$46$	0	0
$47$	−7.28843	−1.06313	−0.531564	−	0.847018i	$-0.678395\pi$
$47$	−7.28843	−1.06313	−0.531564	+	0.847018i	$0.678395\pi$
$48$	0	0
$49$	−2.57908	−0.368440
$50$	0	0
$51$	9.34674	1.30881
$52$	0	0
$53$	−11.8850	−1.63253	−0.816266	−	0.577677i	$-0.803959\pi$
$53$	−11.8850	−1.63253	−0.816266	+	0.577677i	$0.803959\pi$
$54$	0	0
$55$	−6.78291	−0.914608
$56$	0	0
$57$	−2.40330	−0.318325
$58$	0	0
$59$	−7.39694	−0.963000	−0.481500	−	0.876446i	$-0.659908\pi$
$59$	−7.39694	−0.963000	−0.481500	+	0.876446i	$0.659908\pi$
$60$	0	0
$61$	−1.47413	−0.188743	−0.0943717	−	0.995537i	$-0.530084\pi$
$61$	−1.47413	−0.188743	−0.0943717	+	0.995537i	$0.530084\pi$
$62$	0	0
$63$	0.0381255	0.00480337
$64$	0	0
$65$	1.84957	0.229411
$66$	0	0
$67$	−4.45942	−0.544805	−0.272402	−	0.962183i	$-0.587818\pi$
$67$	−4.45942	−0.544805	−0.272402	+	0.962183i	$0.587818\pi$
$68$	0	0
$69$	−11.6831	−1.40648
$70$	0	0
$71$	8.17012	0.969615	0.484807	−	0.874621i	$-0.338890\pi$
$71$	8.17012	0.969615	0.484807	+	0.874621i	$0.338890\pi$
$72$	0	0
$73$	−9.67286	−1.13212	−0.566061	−	0.824363i	$-0.691533\pi$
$73$	−9.67286	−1.13212	−0.566061	+	0.824363i	$0.691533\pi$
$74$	0	0
$75$	−2.93186	−0.338542
$76$	0	0
$77$	−5.51489	−0.628480
$78$	0	0
$79$	9.04789	1.01797	0.508983	−	0.860776i	$-0.330021\pi$
$79$	9.04789	1.01797	0.508983	+	0.860776i	$0.330021\pi$
$80$	0	0
$81$	−9.05407	−1.00601
$82$	0	0
$83$	4.25798	0.467374	0.233687	−	0.972312i	$-0.424921\pi$
$83$	4.25798	0.467374	0.233687	+	0.972312i	$0.424921\pi$
$84$	0	0
$85$	−13.9132	−1.50910
$86$	0	0
$87$	7.28586	0.781126
$88$	0	0
$89$	9.05608	0.959943	0.479972	−	0.877284i	$-0.340647\pi$
$89$	9.05608	0.959943	0.479972	+	0.877284i	$0.340647\pi$
$90$	0	0
$91$	1.50381	0.157642
$92$	0	0
$93$	10.2875	1.06677
$94$	0	0
$95$	3.57746	0.367039
$96$	0	0
$97$	8.08527	0.820934	0.410467	−	0.911875i	$-0.365366\pi$
$97$	8.08527	0.820934	0.410467	+	0.911875i	$0.365366\pi$
$98$	0	0
$99$	−0.0475598	−0.00477994
$100$	0	0
$101$	18.8020	1.87087	0.935436	−	0.353497i	$-0.115008\pi$
$101$	18.8020	1.87087	0.935436	+	0.353497i	$0.115008\pi$
$102$	0	0
$103$	1.20257	0.118492	0.0592462	−	0.998243i	$-0.481130\pi$
$103$	1.20257	0.118492	0.0592462	+	0.998243i	$0.481130\pi$
$104$	0	0
$105$	−9.44629	−0.921863
$106$	0	0
$107$	−16.6032	−1.60509	−0.802544	−	0.596593i	$-0.796521\pi$
$107$	−16.6032	−1.60509	−0.802544	+	0.596593i	$0.796521\pi$
$108$	0	0
$109$	10.9284	1.04675	0.523376	−	0.852102i	$-0.324672\pi$
$109$	10.9284	1.04675	0.523376	+	0.852102i	$0.324672\pi$
$110$	0	0
$111$	13.0269	1.23645
$112$	0	0
$113$	−18.5233	−1.74252	−0.871261	−	0.490820i	$-0.836697\pi$
$113$	−18.5233	−1.74252	−0.871261	+	0.490820i	$0.836697\pi$
$114$	0	0
$115$	17.3909	1.62171
$116$	0	0
$117$	0.0129687	0.00119895
$118$	0	0
$119$	−11.3122	−1.03699
$120$	0	0
$121$	−4.12043	−0.374585
$122$	0	0
$123$	3.97976	0.358843
$124$	0	0
$125$	−8.56597	−0.766163
$126$	0	0
$127$	10.1838	0.903668	0.451834	−	0.892102i	$-0.350770\pi$
$127$	10.1838	0.903668	0.451834	+	0.892102i	$0.350770\pi$
$128$	0	0
$129$	−12.8768	−1.13374
$130$	0	0
$131$	−1.75144	−0.153024	−0.0765121	−	0.997069i	$-0.524378\pi$
$131$	−1.75144	−0.153024	−0.0765121	+	0.997069i	$0.524378\pi$
$132$	0	0
$133$	2.90867	0.252214
$134$	0	0
$135$	13.3966	1.15299
$136$	0	0
$137$	−15.0405	−1.28499	−0.642496	−	0.766289i	$-0.722101\pi$
$137$	−15.0405	−1.28499	−0.642496	+	0.766289i	$0.722101\pi$
$138$	0	0
$139$	−8.98840	−0.762386	−0.381193	−	0.924495i	$-0.624487\pi$
$139$	−8.98840	−0.762386	−0.381193	+	0.924495i	$0.624487\pi$
$140$	0	0
$141$	12.6620	1.06634
$142$	0	0
$143$	−1.87593	−0.156873
$144$	0	0
$145$	−10.8454	−0.900665
$146$	0	0
$147$	4.48057	0.369551
$148$	0	0
$149$	18.8981	1.54819	0.774097	−	0.633067i	$-0.218204\pi$
$149$	18.8981	1.54819	0.774097	+	0.633067i	$0.218204\pi$
$150$	0	0
$151$	17.6152	1.43351	0.716754	−	0.697326i	$-0.245627\pi$
$151$	17.6152	1.43351	0.716754	+	0.697326i	$0.245627\pi$
$152$	0	0
$153$	−0.0975552	−0.00788687
$154$	0	0
$155$	−15.3136	−1.23002
$156$	0	0
$157$	21.9626	1.75280	0.876401	−	0.481582i	$-0.159937\pi$
$157$	21.9626	1.75280	0.876401	+	0.481582i	$0.159937\pi$
$158$	0	0
$159$	20.6476	1.63746
$160$	0	0
$161$	14.1398	1.11437
$162$	0	0
$163$	−21.4671	−1.68143	−0.840717	−	0.541474i	$-0.817866\pi$
$163$	−21.4671	−1.68143	−0.840717	+	0.541474i	$0.817866\pi$
$164$	0	0
$165$	11.7838	0.917368
$166$	0	0
$167$	2.18943	0.169423	0.0847114	−	0.996406i	$-0.473003\pi$
$167$	2.18943	0.169423	0.0847114	+	0.996406i	$0.473003\pi$
$168$	0	0
$169$	−12.4885	−0.960652
$170$	0	0
$171$	0.0250841	0.00191823
$172$	0	0
$173$	2.58894	0.196833	0.0984167	−	0.995145i	$-0.468622\pi$
$173$	2.58894	0.196833	0.0984167	+	0.995145i	$0.468622\pi$
$174$	0	0
$175$	3.54838	0.268232
$176$	0	0
$177$	12.8505	0.965906
$178$	0	0
$179$	−14.8091	−1.10688	−0.553442	−	0.832888i	$-0.686686\pi$
$179$	−14.8091	−1.10688	−0.553442	+	0.832888i	$0.686686\pi$
$180$	0	0
$181$	−26.5615	−1.97430	−0.987152	−	0.159784i	$-0.948920\pi$
$181$	−26.5615	−1.97430	−0.987152	+	0.159784i	$0.948920\pi$
$182$	0	0
$183$	2.56098	0.189313
$184$	0	0
$185$	−19.3913	−1.42567
$186$	0	0
$187$	14.1115	1.03193
$188$	0	0
$189$	10.8922	0.792288
$190$	0	0
$191$	−7.64557	−0.553214	−0.276607	−	0.960983i	$-0.589210\pi$
$191$	−7.64557	−0.553214	−0.276607	+	0.960983i	$0.589210\pi$
$192$	0	0
$193$	18.6737	1.34417	0.672083	−	0.740476i	$-0.265400\pi$
$193$	18.6737	1.34417	0.672083	+	0.740476i	$0.265400\pi$
$194$	0	0
$195$	−3.21322	−0.230103
$196$	0	0
$197$	−23.8450	−1.69889	−0.849443	−	0.527680i	$-0.823062\pi$
$197$	−23.8450	−1.69889	−0.849443	+	0.527680i	$0.823062\pi$
$198$	0	0
$199$	−21.5661	−1.52878	−0.764390	−	0.644754i	$-0.776960\pi$
$199$	−21.5661	−1.52878	−0.764390	+	0.644754i	$0.776960\pi$
$200$	0	0
$201$	7.74724	0.546448
$202$	0	0
$203$	−8.81796	−0.618899
$204$	0	0
$205$	−5.92411	−0.413758
$206$	0	0
$207$	0.121940	0.00847543
$208$	0	0
$209$	−3.62843	−0.250984
$210$	0	0
$211$	−13.1034	−0.902078	−0.451039	−	0.892504i	$-0.648946\pi$
$211$	−13.1034	−0.902078	−0.451039	+	0.892504i	$0.648946\pi$
$212$	0	0
$213$	−14.1938	−0.972541
$214$	0	0
$215$	19.1679	1.30724
$216$	0	0
$217$	−12.4508	−0.845218
$218$	0	0
$219$	16.8044	1.13554
$220$	0	0
$221$	−3.84792	−0.258839
$222$	0	0
$223$	7.00682	0.469211	0.234606	−	0.972091i	$-0.424620\pi$
$223$	7.00682	0.469211	0.234606	+	0.972091i	$0.424620\pi$
$224$	0	0
$225$	0.0306008	0.00204005
$226$	0	0
$227$	−11.6164	−0.771006	−0.385503	−	0.922707i	$-0.625972\pi$
$227$	−11.6164	−0.771006	−0.385503	+	0.922707i	$0.625972\pi$
$228$	0	0
$229$	−19.2049	−1.26909	−0.634547	−	0.772884i	$-0.718813\pi$
$229$	−19.2049	−1.26909	−0.634547	+	0.772884i	$0.718813\pi$
$230$	0	0
$231$	9.58090	0.630377
$232$	0	0
$233$	2.65846	0.174161	0.0870807	−	0.996201i	$-0.472246\pi$
$233$	2.65846	0.174161	0.0870807	+	0.996201i	$0.472246\pi$
$234$	0	0
$235$	−18.8482	−1.22952
$236$	0	0
$237$	−15.7187	−1.02104
$238$	0	0
$239$	−9.78193	−0.632740	−0.316370	−	0.948636i	$-0.602464\pi$
$239$	−9.78193	−0.632740	−0.316370	+	0.948636i	$0.602464\pi$
$240$	0	0
$241$	14.1847	0.913718	0.456859	−	0.889539i	$-0.348974\pi$
$241$	14.1847	0.913718	0.456859	+	0.889539i	$0.348974\pi$
$242$	0	0
$243$	0.188437	0.0120882
$244$	0	0
$245$	−6.66960	−0.426105
$246$	0	0
$247$	0.989405	0.0629543
$248$	0	0
$249$	−7.39730	−0.468785
$250$	0	0
$251$	7.28945	0.460106	0.230053	−	0.973178i	$-0.426110\pi$
$251$	7.28945	0.460106	0.230053	+	0.973178i	$0.426110\pi$
$252$	0	0
$253$	−17.6388	−1.10894
$254$	0	0
$255$	24.1711	1.51365
$256$	0	0
$257$	−24.3811	−1.52085	−0.760424	−	0.649427i	$-0.775009\pi$
$257$	−24.3811	−1.52085	−0.760424	+	0.649427i	$0.775009\pi$
$258$	0	0
$259$	−15.7662	−0.979664
$260$	0	0
$261$	−0.0760451	−0.00470707
$262$	0	0
$263$	10.9637	0.676052	0.338026	−	0.941137i	$-0.390241\pi$
$263$	10.9637	0.676052	0.338026	+	0.941137i	$0.390241\pi$
$264$	0	0
$265$	−30.7351	−1.88804
$266$	0	0
$267$	−15.7329	−0.962840
$268$	0	0
$269$	6.72724	0.410167	0.205084	−	0.978744i	$-0.434253\pi$
$269$	6.72724	0.410167	0.205084	+	0.978744i	$0.434253\pi$
$270$	0	0
$271$	−6.57983	−0.399696	−0.199848	−	0.979827i	$-0.564045\pi$
$271$	−6.57983	−0.399696	−0.199848	+	0.979827i	$0.564045\pi$
$272$	0	0
$273$	−2.61253	−0.158117
$274$	0	0
$275$	−4.42643	−0.266924
$276$	0	0
$277$	5.65170	0.339577	0.169789	−	0.985480i	$-0.445691\pi$
$277$	5.65170	0.339577	0.169789	+	0.985480i	$0.445691\pi$
$278$	0	0
$279$	−0.107375	−0.00642835
$280$	0	0
$281$	6.49270	0.387322	0.193661	−	0.981069i	$-0.437964\pi$
$281$	6.49270	0.387322	0.193661	+	0.981069i	$0.437964\pi$
$282$	0	0
$283$	−15.3968	−0.915246	−0.457623	−	0.889146i	$-0.651299\pi$
$283$	−15.3968	−0.915246	−0.457623	+	0.889146i	$0.651299\pi$
$284$	0	0
$285$	−6.21503	−0.368147
$286$	0	0
$287$	−4.81664	−0.284317
$288$	0	0
$289$	11.9456	0.702681
$290$	0	0
$291$	−14.0464	−0.823412
$292$	0	0
$293$	−25.3812	−1.48279	−0.741393	−	0.671071i	$-0.765835\pi$
$293$	−25.3812	−1.48279	−0.741393	+	0.671071i	$0.765835\pi$
$294$	0	0
$295$	−19.1288	−1.11372
$296$	0	0
$297$	−13.5875	−0.788424
$298$	0	0
$299$	4.80975	0.278155
$300$	0	0
$301$	15.5846	0.898281
$302$	0	0
$303$	−32.6643	−1.87652
$304$	0	0
$305$	−3.81217	−0.218284
$306$	0	0
$307$	24.9503	1.42399	0.711993	−	0.702186i	$-0.247793\pi$
$307$	24.9503	1.42399	0.711993	+	0.702186i	$0.247793\pi$
$308$	0	0
$309$	−2.08919	−0.118850
$310$	0	0
$311$	9.38125	0.531962	0.265981	−	0.963978i	$-0.414304\pi$
$311$	9.38125	0.531962	0.265981	+	0.963978i	$0.414304\pi$
$312$	0	0
$313$	−7.27785	−0.411368	−0.205684	−	0.978618i	$-0.565942\pi$
$313$	−7.27785	−0.411368	−0.205684	+	0.978618i	$0.565942\pi$
$314$	0	0
$315$	0.0985943	0.00555516
$316$	0	0
$317$	4.69633	0.263772	0.131886	−	0.991265i	$-0.457897\pi$
$317$	4.69633	0.263772	0.131886	+	0.991265i	$0.457897\pi$
$318$	0	0
$319$	11.0000	0.615881
$320$	0	0
$321$	28.8443	1.60993
$322$	0	0
$323$	−7.44268	−0.414122
$324$	0	0
$325$	1.20700	0.0669525
$326$	0	0
$327$	−18.9857	−1.04991
$328$	0	0
$329$	−15.3247	−0.844875
$330$	0	0
$331$	−19.3105	−1.06140	−0.530700	−	0.847560i	$-0.678071\pi$
$331$	−19.3105	−1.06140	−0.530700	+	0.847560i	$0.678071\pi$
$332$	0	0
$333$	−0.135966	−0.00745088
$334$	0	0
$335$	−11.5322	−0.630074
$336$	0	0
$337$	17.1916	0.936488	0.468244	−	0.883599i	$-0.344887\pi$
$337$	17.1916	0.936488	0.468244	+	0.883599i	$0.344887\pi$
$338$	0	0
$339$	32.1800	1.74778
$340$	0	0
$341$	15.5318	0.841096
$342$	0	0
$343$	−20.1410	−1.08751
$344$	0	0
$345$	−30.2129	−1.62661
$346$	0	0
$347$	−14.6384	−0.785832	−0.392916	−	0.919574i	$-0.628534\pi$
$347$	−14.6384	−0.785832	−0.392916	+	0.919574i	$0.628534\pi$
$348$	0	0
$349$	−9.91051	−0.530497	−0.265249	−	0.964180i	$-0.585454\pi$
$349$	−9.91051	−0.530497	−0.265249	+	0.964180i	$0.585454\pi$
$350$	0	0
$351$	3.70504	0.197760
$352$	0	0
$353$	−30.7124	−1.63466	−0.817329	−	0.576171i	$-0.804546\pi$
$353$	−30.7124	−1.63466	−0.817329	+	0.576171i	$0.804546\pi$
$354$	0	0
$355$	21.1283	1.12137
$356$	0	0
$357$	19.6524	1.04012
$358$	0	0
$359$	35.5684	1.87723	0.938614	−	0.344968i	$-0.112110\pi$
$359$	35.5684	1.87723	0.938614	+	0.344968i	$0.112110\pi$
$360$	0	0
$361$	−17.0863	−0.899278
$362$	0	0
$363$	7.15833	0.375715
$364$	0	0
$365$	−25.0144	−1.30931
$366$	0	0
$367$	32.6083	1.70214	0.851070	−	0.525052i	$-0.175954\pi$
$367$	32.6083	1.70214	0.851070	+	0.525052i	$0.175954\pi$
$368$	0	0
$369$	−0.0415382	−0.00216239
$370$	0	0
$371$	−24.9894	−1.29739
$372$	0	0
$373$	31.6305	1.63777	0.818883	−	0.573961i	$-0.194594\pi$
$373$	31.6305	1.63777	0.818883	+	0.573961i	$0.194594\pi$
$374$	0	0
$375$	14.8815	0.768475
$376$	0	0
$377$	−2.99949	−0.154481
$378$	0	0
$379$	25.6286	1.31645	0.658227	−	0.752820i	$-0.271307\pi$
$379$	25.6286	1.31645	0.658227	+	0.752820i	$0.271307\pi$
$380$	0	0
$381$	−17.6921	−0.906394
$382$	0	0
$383$	29.2140	1.49276	0.746382	−	0.665518i	$-0.231789\pi$
$383$	29.2140	1.49276	0.746382	+	0.665518i	$0.231789\pi$
$384$	0	0
$385$	−14.2617	−0.726846
$386$	0	0
$387$	0.134400	0.00683192
$388$	0	0
$389$	−18.4995	−0.937962	−0.468981	−	0.883208i	$-0.655379\pi$
$389$	−18.4995	−0.937962	−0.468981	+	0.883208i	$0.655379\pi$
$390$	0	0
$391$	−36.1808	−1.82974
$392$	0	0
$393$	3.04274	0.153486
$394$	0	0
$395$	23.3982	1.17729
$396$	0	0
$397$	−3.21806	−0.161510	−0.0807550	−	0.996734i	$-0.525733\pi$
$397$	−3.21806	−0.161510	−0.0807550	+	0.996734i	$0.525733\pi$
$398$	0	0
$399$	−5.05317	−0.252975
$400$	0	0
$401$	28.4986	1.42315	0.711576	−	0.702609i	$-0.247982\pi$
$401$	28.4986	1.42315	0.711576	+	0.702609i	$0.247982\pi$
$402$	0	0
$403$	−4.23523	−0.210972
$404$	0	0
$405$	−23.4142	−1.16346
$406$	0	0
$407$	19.6676	0.974886
$408$	0	0
$409$	9.38723	0.464168	0.232084	−	0.972696i	$-0.425446\pi$
$409$	9.38723	0.464168	0.232084	+	0.972696i	$0.425446\pi$
$410$	0	0
$411$	26.1294	1.28887
$412$	0	0
$413$	−15.5528	−0.765303
$414$	0	0
$415$	11.0113	0.540525
$416$	0	0
$417$	15.6153	0.764687
$418$	0	0
$419$	−24.9405	−1.21842	−0.609211	−	0.793008i	$-0.708514\pi$
$419$	−24.9405	−1.21842	−0.609211	+	0.793008i	$0.708514\pi$
$420$	0	0
$421$	−30.2102	−1.47236	−0.736178	−	0.676788i	$-0.763371\pi$
$421$	−30.2102	−1.47236	−0.736178	+	0.676788i	$0.763371\pi$
$422$	0	0
$423$	−0.132158	−0.00642574
$424$	0	0
$425$	−9.07955	−0.440423
$426$	0	0
$427$	−3.09951	−0.149996
$428$	0	0
$429$	3.25901	0.157346
$430$	0	0
$431$	−36.8695	−1.77594	−0.887970	−	0.459902i	$-0.847884\pi$
$431$	−36.8695	−1.77594	−0.887970	+	0.459902i	$0.847884\pi$
$432$	0	0
$433$	−18.5932	−0.893530	−0.446765	−	0.894651i	$-0.647424\pi$
$433$	−18.5932	−0.893530	−0.446765	+	0.894651i	$0.647424\pi$
$434$	0	0
$435$	18.8415	0.903383
$436$	0	0
$437$	9.30306	0.445026
$438$	0	0
$439$	−5.25442	−0.250780	−0.125390	−	0.992108i	$-0.540018\pi$
$439$	−5.25442	−0.250780	−0.125390	+	0.992108i	$0.540018\pi$
$440$	0	0
$441$	−0.0467653	−0.00222692
$442$	0	0
$443$	16.1896	0.769190	0.384595	−	0.923085i	$-0.374341\pi$
$443$	16.1896	0.769190	0.384595	+	0.923085i	$0.374341\pi$
$444$	0	0
$445$	23.4194	1.11019
$446$	0	0
$447$	−32.8313	−1.55287
$448$	0	0
$449$	−0.629326	−0.0296997	−0.0148499	−	0.999890i	$-0.504727\pi$
$449$	−0.629326	−0.0296997	−0.0148499	+	0.999890i	$0.504727\pi$
$450$	0	0
$451$	6.00853	0.282931
$452$	0	0
$453$	−30.6026	−1.43783
$454$	0	0
$455$	3.88891	0.182315
$456$	0	0
$457$	13.6403	0.638066	0.319033	−	0.947744i	$-0.396642\pi$
$457$	13.6403	0.638066	0.319033	+	0.947744i	$0.396642\pi$
$458$	0	0
$459$	−27.8707	−1.30089
$460$	0	0
$461$	13.8122	0.643298	0.321649	−	0.946859i	$-0.395763\pi$
$461$	13.8122	0.643298	0.321649	+	0.946859i	$0.395763\pi$
$462$	0	0
$463$	6.86831	0.319197	0.159599	−	0.987182i	$-0.448980\pi$
$463$	6.86831	0.319197	0.159599	+	0.987182i	$0.448980\pi$
$464$	0	0
$465$	26.6040	1.23373
$466$	0	0
$467$	1.48619	0.0687728	0.0343864	−	0.999409i	$-0.489052\pi$
$467$	1.48619	0.0687728	0.0343864	+	0.999409i	$0.489052\pi$
$468$	0	0
$469$	−9.37637	−0.432960
$470$	0	0
$471$	−38.1550	−1.75809
$472$	0	0
$473$	−19.4410	−0.893900
$474$	0	0
$475$	2.33460	0.107119
$476$	0	0
$477$	−0.215506	−0.00986733
$478$	0	0
$479$	−0.419481	−0.0191666	−0.00958329	−	0.999954i	$-0.503051\pi$
$479$	−0.419481	−0.0191666	−0.00958329	+	0.999954i	$0.503051\pi$
$480$	0	0
$481$	−5.36297	−0.244531
$482$	0	0
$483$	−24.5648	−1.11774
$484$	0	0
$485$	20.9088	0.949422
$486$	0	0
$487$	−25.9633	−1.17651	−0.588255	−	0.808676i	$-0.700185\pi$
$487$	−25.9633	−1.17651	−0.588255	+	0.808676i	$0.700185\pi$
$488$	0	0
$489$	37.2943	1.68651
$490$	0	0
$491$	−36.8094	−1.66119	−0.830594	−	0.556879i	$-0.811999\pi$
$491$	−36.8094	−1.66119	−0.830594	+	0.556879i	$0.811999\pi$
$492$	0	0
$493$	22.5633	1.01620
$494$	0	0
$495$	−0.122992	−0.00552807
$496$	0	0
$497$	17.1785	0.770561
$498$	0	0
$499$	2.13357	0.0955116	0.0477558	−	0.998859i	$-0.484793\pi$
$499$	2.13357	0.0955116	0.0477558	+	0.998859i	$0.484793\pi$
$500$	0	0
$501$	−3.80364	−0.169934
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	48.6228	2.16369
$506$	0	0
$507$	21.6959	0.963550
$508$	0	0
$509$	−30.2938	−1.34275	−0.671375	−	0.741118i	$-0.734296\pi$
$509$	−30.2938	−1.34275	−0.671375	+	0.741118i	$0.734296\pi$
$510$	0	0
$511$	−20.3381	−0.899706
$512$	0	0
$513$	7.16632	0.316401
$514$	0	0
$515$	3.10989	0.137038
$516$	0	0
$517$	19.1168	0.840755
$518$	0	0
$519$	−4.49771	−0.197427
$520$	0	0
$521$	−27.1463	−1.18930	−0.594650	−	0.803985i	$-0.702709\pi$
$521$	−27.1463	−1.18930	−0.594650	+	0.803985i	$0.702709\pi$
$522$	0	0
$523$	21.3731	0.934580	0.467290	−	0.884104i	$-0.345230\pi$
$523$	21.3731	0.934580	0.467290	+	0.884104i	$0.345230\pi$
$524$	0	0
$525$	−6.16452	−0.269042
$526$	0	0
$527$	31.8591	1.38780
$528$	0	0
$529$	22.2246	0.966287
$530$	0	0
$531$	−0.134126	−0.00582056
$532$	0	0
$533$	−1.63841	−0.0709675
$534$	0	0
$535$	−42.9365	−1.85631
$536$	0	0
$537$	25.7275	1.11022
$538$	0	0
$539$	6.76464	0.291374
$540$	0	0
$541$	29.9286	1.28673	0.643366	−	0.765559i	$-0.277537\pi$
$541$	29.9286	1.28673	0.643366	+	0.765559i	$0.277537\pi$
$542$	0	0
$543$	46.1448	1.98026
$544$	0	0
$545$	28.2613	1.21058
$546$	0	0
$547$	−29.9207	−1.27931	−0.639657	−	0.768660i	$-0.720924\pi$
$547$	−29.9207	−1.27931	−0.639657	+	0.768660i	$0.720924\pi$
$548$	0	0
$549$	−0.0267298	−0.00114080
$550$	0	0
$551$	−5.80163	−0.247158
$552$	0	0
$553$	19.0241	0.808986
$554$	0	0
$555$	33.6880	1.42998
$556$	0	0
$557$	−13.2317	−0.560644	−0.280322	−	0.959906i	$-0.590441\pi$
$557$	−13.2317	−0.560644	−0.280322	+	0.959906i	$0.590441\pi$
$558$	0	0
$559$	5.30120	0.224217
$560$	0	0
$561$	−24.5155	−1.03505
$562$	0	0
$563$	15.4871	0.652701	0.326351	−	0.945249i	$-0.394181\pi$
$563$	15.4871	0.652701	0.326351	+	0.945249i	$0.394181\pi$
$564$	0	0
$565$	−47.9019	−2.01525
$566$	0	0
$567$	−19.0371	−0.799482
$568$	0	0
$569$	16.7971	0.704170	0.352085	−	0.935968i	$-0.385473\pi$
$569$	16.7971	0.704170	0.352085	+	0.935968i	$0.385473\pi$
$570$	0	0
$571$	−30.2291	−1.26505	−0.632525	−	0.774540i	$-0.717982\pi$
$571$	−30.2291	−1.26505	−0.632525	+	0.774540i	$0.717982\pi$
$572$	0	0
$573$	13.2825	0.554883
$574$	0	0
$575$	11.3491	0.473290
$576$	0	0
$577$	1.03748	0.0431910	0.0215955	−	0.999767i	$-0.493125\pi$
$577$	1.03748	0.0431910	0.0215955	+	0.999767i	$0.493125\pi$
$578$	0	0
$579$	−32.4415	−1.34822
$580$	0	0
$581$	8.95283	0.371426
$582$	0	0
$583$	31.1731	1.29106
$584$	0	0
$585$	0.0335375	0.00138660
$586$	0	0
$587$	−26.9328	−1.11164	−0.555818	−	0.831304i	$-0.687595\pi$
$587$	−26.9328	−1.11164	−0.555818	+	0.831304i	$0.687595\pi$
$588$	0	0
$589$	−8.19183	−0.337538
$590$	0	0
$591$	41.4254	1.70401
$592$	0	0
$593$	8.67227	0.356127	0.178064	−	0.984019i	$-0.443017\pi$
$593$	8.67227	0.356127	0.178064	+	0.984019i	$0.443017\pi$
$594$	0	0
$595$	−29.2539	−1.19929
$596$	0	0
$597$	37.4663	1.53339
$598$	0	0
$599$	7.03128	0.287290	0.143645	−	0.989629i	$-0.454118\pi$
$599$	7.03128	0.287290	0.143645	+	0.989629i	$0.454118\pi$
$600$	0	0
$601$	16.9913	0.693088	0.346544	−	0.938034i	$-0.387355\pi$
$601$	16.9913	0.693088	0.346544	+	0.938034i	$0.387355\pi$
$602$	0	0
$603$	−0.0808607	−0.00329290
$604$	0	0
$605$	−10.6556	−0.433212
$606$	0	0
$607$	33.6055	1.36400	0.682002	−	0.731350i	$-0.261110\pi$
$607$	33.6055	1.36400	0.682002	+	0.731350i	$0.261110\pi$
$608$	0	0
$609$	15.3192	0.620767
$610$	0	0
$611$	−5.21278	−0.210887
$612$	0	0
$613$	−41.5746	−1.67918	−0.839591	−	0.543219i	$-0.817205\pi$
$613$	−41.5746	−1.67918	−0.839591	+	0.543219i	$0.817205\pi$
$614$	0	0
$615$	10.2918	0.415007
$616$	0	0
$617$	10.0211	0.403434	0.201717	−	0.979444i	$-0.435348\pi$
$617$	10.0211	0.403434	0.201717	+	0.979444i	$0.435348\pi$
$618$	0	0
$619$	−10.8542	−0.436267	−0.218133	−	0.975919i	$-0.569997\pi$
$619$	−10.8542	−0.436267	−0.218133	+	0.975919i	$0.569997\pi$
$620$	0	0
$621$	34.8373	1.39797
$622$	0	0
$623$	19.0413	0.762874
$624$	0	0
$625$	−30.5900	−1.22360
$626$	0	0
$627$	6.30360	0.251741
$628$	0	0
$629$	40.3424	1.60856
$630$	0	0
$631$	18.4761	0.735524	0.367762	−	0.929920i	$-0.380124\pi$
$631$	18.4761	0.735524	0.367762	+	0.929920i	$0.380124\pi$
$632$	0	0
$633$	22.7643	0.904800
$634$	0	0
$635$	26.3358	1.04510
$636$	0	0
$637$	−1.84459	−0.0730853
$638$	0	0
$639$	0.148145	0.00586054
$640$	0	0
$641$	0.947579	0.0374271	0.0187136	−	0.999825i	$-0.494043\pi$
$641$	0.947579	0.0374271	0.0187136	+	0.999825i	$0.494043\pi$
$642$	0	0
$643$	−22.9716	−0.905913	−0.452956	−	0.891533i	$-0.649631\pi$
$643$	−22.9716	−0.905913	−0.452956	+	0.891533i	$0.649631\pi$
$644$	0	0
$645$	−33.2999	−1.31118
$646$	0	0
$647$	29.7014	1.16768	0.583842	−	0.811868i	$-0.301549\pi$
$647$	29.7014	1.16768	0.583842	+	0.811868i	$0.301549\pi$
$648$	0	0
$649$	19.4014	0.761571
$650$	0	0
$651$	21.6306	0.847768
$652$	0	0
$653$	42.3223	1.65620	0.828099	−	0.560581i	$-0.189422\pi$
$653$	42.3223	1.65620	0.828099	+	0.560581i	$0.189422\pi$
$654$	0	0
$655$	−4.52930	−0.176974
$656$	0	0
$657$	−0.175394	−0.00684276
$658$	0	0
$659$	15.9106	0.619789	0.309895	−	0.950771i	$-0.399706\pi$
$659$	15.9106	0.619789	0.309895	+	0.950771i	$0.399706\pi$
$660$	0	0
$661$	0.347986	0.0135351	0.00676755	−	0.999977i	$-0.497846\pi$
$661$	0.347986	0.0135351	0.00676755	+	0.999977i	$0.497846\pi$
$662$	0	0
$663$	6.68491	0.259620
$664$	0	0
$665$	7.52195	0.291689
$666$	0	0
$667$	−28.2032	−1.09203
$668$	0	0
$669$	−12.1728	−0.470627
$670$	0	0
$671$	3.86649	0.149264
$672$	0	0
$673$	18.8705	0.727406	0.363703	−	0.931515i	$-0.381512\pi$
$673$	18.8705	0.727406	0.363703	+	0.931515i	$0.381512\pi$
$674$	0	0
$675$	8.74241	0.336495
$676$	0	0
$677$	15.1564	0.582508	0.291254	−	0.956646i	$-0.405928\pi$
$677$	15.1564	0.582508	0.291254	+	0.956646i	$0.405928\pi$
$678$	0	0
$679$	17.0001	0.652403
$680$	0	0
$681$	20.1809	0.773332
$682$	0	0
$683$	39.5457	1.51318	0.756588	−	0.653892i	$-0.226865\pi$
$683$	39.5457	1.51318	0.756588	+	0.653892i	$0.226865\pi$
$684$	0	0
$685$	−38.8953	−1.48611
$686$	0	0
$687$	33.3642	1.27292
$688$	0	0
$689$	−8.50031	−0.323836
$690$	0	0
$691$	−33.2527	−1.26499	−0.632495	−	0.774564i	$-0.717969\pi$
$691$	−33.2527	−1.26499	−0.632495	+	0.774564i	$0.717969\pi$
$692$	0	0
$693$	−0.0999992	−0.00379866
$694$	0	0
$695$	−23.2444	−0.881710
$696$	0	0
$697$	12.3248	0.466834
$698$	0	0
$699$	−4.61848	−0.174687
$700$	0	0
$701$	38.7601	1.46395	0.731975	−	0.681331i	$-0.238599\pi$
$701$	38.7601	1.46395	0.731975	+	0.681331i	$0.238599\pi$
$702$	0	0
$703$	−10.3731	−0.391229
$704$	0	0
$705$	32.7445	1.23323
$706$	0	0
$707$	39.5331	1.48680
$708$	0	0
$709$	30.6093	1.14956	0.574778	−	0.818310i	$-0.305089\pi$
$709$	30.6093	1.14956	0.574778	+	0.818310i	$0.305089\pi$
$710$	0	0
$711$	0.164061	0.00615279
$712$	0	0
$713$	−39.8226	−1.49137
$714$	0	0
$715$	−4.85123	−0.181426
$716$	0	0
$717$	16.9939	0.634650
$718$	0	0
$719$	51.1073	1.90598	0.952990	−	0.303002i	$-0.0979888\pi$
$719$	51.1073	1.90598	0.952990	+	0.303002i	$0.0979888\pi$
$720$	0	0
$721$	2.52851	0.0941668
$722$	0	0
$723$	−24.6428	−0.916475
$724$	0	0
$725$	−7.07758	−0.262855
$726$	0	0
$727$	2.23928	0.0830505	0.0415252	−	0.999137i	$-0.486778\pi$
$727$	2.23928	0.0830505	0.0415252	+	0.999137i	$0.486778\pi$
$728$	0	0
$729$	26.8348	0.993883
$730$	0	0
$731$	−39.8777	−1.47493
$732$	0	0
$733$	2.66853	0.0985643	0.0492821	−	0.998785i	$-0.484307\pi$
$733$	2.66853	0.0985643	0.0492821	+	0.998785i	$0.484307\pi$
$734$	0	0
$735$	11.5869	0.427391
$736$	0	0
$737$	11.6966	0.430849
$738$	0	0
$739$	23.9717	0.881815	0.440907	−	0.897553i	$-0.354657\pi$
$739$	23.9717	0.881815	0.440907	+	0.897553i	$0.354657\pi$
$740$	0	0
$741$	−1.71887	−0.0631443
$742$	0	0
$743$	−6.79980	−0.249460	−0.124730	−	0.992191i	$-0.539807\pi$
$743$	−6.79980	−0.249460	−0.124730	+	0.992191i	$0.539807\pi$
$744$	0	0
$745$	48.8713	1.79051
$746$	0	0
$747$	0.0772082	0.00282490
$748$	0	0
$749$	−34.9098	−1.27558
$750$	0	0
$751$	−28.4576	−1.03843	−0.519217	−	0.854642i	$-0.673776\pi$
$751$	−28.4576	−1.03843	−0.519217	+	0.854642i	$0.673776\pi$
$752$	0	0
$753$	−12.6638	−0.461495
$754$	0	0
$755$	45.5538	1.65787
$756$	0	0
$757$	−25.9573	−0.943434	−0.471717	−	0.881750i	$-0.656366\pi$
$757$	−25.9573	−0.943434	−0.471717	+	0.881750i	$0.656366\pi$
$758$	0	0
$759$	30.6434	1.11229
$760$	0	0
$761$	−38.7281	−1.40389	−0.701946	−	0.712230i	$-0.747685\pi$
$761$	−38.7281	−1.40389	−0.701946	+	0.712230i	$0.747685\pi$
$762$	0	0
$763$	22.9781	0.831862
$764$	0	0
$765$	−0.252282	−0.00912127
$766$	0	0
$767$	−5.29039	−0.191025
$768$	0	0
$769$	−31.1088	−1.12181	−0.560906	−	0.827879i	$-0.689547\pi$
$769$	−31.1088	−1.12181	−0.560906	+	0.827879i	$0.689547\pi$
$770$	0	0
$771$	42.3567	1.52544
$772$	0	0
$773$	38.9856	1.40222	0.701108	−	0.713055i	$-0.252689\pi$
$773$	38.9856	1.40222	0.701108	+	0.713055i	$0.252689\pi$
$774$	0	0
$775$	−9.99345	−0.358975
$776$	0	0
$777$	27.3903	0.982620
$778$	0	0
$779$	−3.16903	−0.113542
$780$	0	0
$781$	−21.4294	−0.766803
$782$	0	0
$783$	−21.7255	−0.776405
$784$	0	0
$785$	56.7961	2.02714
$786$	0	0
$787$	−9.31965	−0.332210	−0.166105	−	0.986108i	$-0.553119\pi$
$787$	−9.31965	−0.332210	−0.166105	+	0.986108i	$0.553119\pi$
$788$	0	0
$789$	−19.0470	−0.678092
$790$	0	0
$791$	−38.9470	−1.38480
$792$	0	0
$793$	−1.05432	−0.0374399
$794$	0	0
$795$	53.3954	1.89374
$796$	0	0
$797$	−6.06555	−0.214853	−0.107426	−	0.994213i	$-0.534261\pi$
$797$	−6.06555	−0.214853	−0.107426	+	0.994213i	$0.534261\pi$
$798$	0	0
$799$	39.2126	1.38724
$800$	0	0
$801$	0.164210	0.00580208
$802$	0	0
$803$	25.3709	0.895319
$804$	0	0
$805$	36.5662	1.28879
$806$	0	0
$807$	−11.6871	−0.411405
$808$	0	0
$809$	−10.5655	−0.371463	−0.185732	−	0.982601i	$-0.559466\pi$
$809$	−10.5655	−0.371463	−0.185732	+	0.982601i	$0.559466\pi$
$810$	0	0
$811$	29.8315	1.04752	0.523762	−	0.851865i	$-0.324528\pi$
$811$	29.8315	1.04752	0.523762	+	0.851865i	$0.324528\pi$
$812$	0	0
$813$	11.4310	0.400902
$814$	0	0
$815$	−55.5149	−1.94460
$816$	0	0
$817$	10.2536	0.358729
$818$	0	0
$819$	0.0272679	0.000952817 0
$820$	0	0
$821$	−23.0775	−0.805409	−0.402705	−	0.915330i	$-0.631930\pi$
$821$	−23.0775	−0.805409	−0.402705	+	0.915330i	$0.631930\pi$
$822$	0	0
$823$	5.76982	0.201123	0.100562	−	0.994931i	$-0.467936\pi$
$823$	5.76982	0.201123	0.100562	+	0.994931i	$0.467936\pi$
$824$	0	0
$825$	7.68994	0.267730
$826$	0	0
$827$	−6.15648	−0.214082	−0.107041	−	0.994255i	$-0.534138\pi$
$827$	−6.15648	−0.214082	−0.107041	+	0.994255i	$0.534138\pi$
$828$	0	0
$829$	−20.9851	−0.728842	−0.364421	−	0.931234i	$-0.618733\pi$
$829$	−20.9851	−0.728842	−0.364421	+	0.931234i	$0.618733\pi$
$830$	0	0
$831$	−9.81856	−0.340602
$832$	0	0
$833$	13.8757	0.480765
$834$	0	0
$835$	5.66195	0.195940
$836$	0	0
$837$	−30.6761	−1.06032
$838$	0	0
$839$	32.9640	1.13804	0.569022	−	0.822323i	$-0.307322\pi$
$839$	32.9640	1.13804	0.569022	+	0.822323i	$0.307322\pi$
$840$	0	0
$841$	−11.4117	−0.393508
$842$	0	0
$843$	−11.2796	−0.388491
$844$	0	0
$845$	−32.2957	−1.11101
$846$	0	0
$847$	−8.66361	−0.297685
$848$	0	0
$849$	26.7486	0.918008
$850$	0	0
$851$	−50.4264	−1.72859
$852$	0	0
$853$	−46.8181	−1.60302	−0.801511	−	0.597980i	$-0.795970\pi$
$853$	−46.8181	−1.60302	−0.801511	+	0.597980i	$0.795970\pi$
$854$	0	0
$855$	0.0648685	0.00221846
$856$	0	0
$857$	−21.3371	−0.728861	−0.364430	−	0.931231i	$-0.618736\pi$
$857$	−21.3371	−0.728861	−0.364430	+	0.931231i	$0.618736\pi$
$858$	0	0
$859$	−3.48916	−0.119049	−0.0595243	−	0.998227i	$-0.518958\pi$
$859$	−3.48916	−0.119049	−0.0595243	+	0.998227i	$0.518958\pi$
$860$	0	0
$861$	8.36784	0.285175
$862$	0	0
$863$	37.4202	1.27380	0.636900	−	0.770947i	$-0.280217\pi$
$863$	37.4202	1.27380	0.636900	+	0.770947i	$0.280217\pi$
$864$	0	0
$865$	6.69511	0.227641
$866$	0	0
$867$	−20.7528	−0.704801
$868$	0	0
$869$	−23.7316	−0.805041
$870$	0	0
$871$	−3.18943	−0.108070
$872$	0	0
$873$	0.146607	0.00496189
$874$	0	0
$875$	−18.0108	−0.608876
$876$	0	0
$877$	27.9015	0.942166	0.471083	−	0.882089i	$-0.343863\pi$
$877$	27.9015	0.942166	0.471083	+	0.882089i	$0.343863\pi$
$878$	0	0
$879$	44.0942	1.48726
$880$	0	0
$881$	−40.3310	−1.35879	−0.679393	−	0.733775i	$-0.737757\pi$
$881$	−40.3310	−1.35879	−0.679393	+	0.733775i	$0.737757\pi$
$882$	0	0
$883$	52.3288	1.76100	0.880502	−	0.474043i	$-0.157206\pi$
$883$	52.3288	1.76100	0.880502	+	0.474043i	$0.157206\pi$
$884$	0	0
$885$	33.2320	1.11708
$886$	0	0
$887$	28.4277	0.954510	0.477255	−	0.878765i	$-0.341632\pi$
$887$	28.4277	0.954510	0.477255	+	0.878765i	$0.341632\pi$
$888$	0	0
$889$	21.4125	0.718152
$890$	0	0
$891$	23.7479	0.795583
$892$	0	0
$893$	−10.0826	−0.337401
$894$	0	0
$895$	−38.2970	−1.28013
$896$	0	0
$897$	−8.35587	−0.278995
$898$	0	0
$899$	24.8344	0.828274
$900$	0	0
$901$	63.9426	2.13024
$902$	0	0
$903$	−27.0747	−0.900991
$904$	0	0
$905$	−68.6893	−2.28331
$906$	0	0
$907$	48.2852	1.60328	0.801641	−	0.597805i	$-0.203960\pi$
$907$	48.2852	1.60328	0.801641	+	0.597805i	$0.203960\pi$
$908$	0	0
$909$	0.340929	0.0113079
$910$	0	0
$911$	−19.7120	−0.653087	−0.326543	−	0.945182i	$-0.605884\pi$
$911$	−19.7120	−0.653087	−0.326543	+	0.945182i	$0.605884\pi$
$912$	0	0
$913$	−11.1682	−0.369615
$914$	0	0
$915$	6.62279	0.218943
$916$	0	0
$917$	−3.68258	−0.121609
$918$	0	0
$919$	28.4127	0.937248	0.468624	−	0.883398i	$-0.344750\pi$
$919$	28.4127	0.937248	0.468624	+	0.883398i	$0.344750\pi$
$920$	0	0
$921$	−43.3455	−1.42828
$922$	0	0
$923$	5.84338	0.192337
$924$	0	0
$925$	−12.6545	−0.416076
$926$	0	0
$927$	0.0218056	0.000716190 0
$928$	0	0
$929$	−53.6116	−1.75894	−0.879470	−	0.475954i	$-0.842103\pi$
$929$	−53.6116	−1.75894	−0.879470	+	0.475954i	$0.842103\pi$
$930$	0	0
$931$	−3.56782	−0.116931
$932$	0	0
$933$	−16.2978	−0.533567
$934$	0	0
$935$	36.4928	1.19344
$936$	0	0
$937$	37.8601	1.23684	0.618418	−	0.785849i	$-0.287774\pi$
$937$	37.8601	1.23684	0.618418	+	0.785849i	$0.287774\pi$
$938$	0	0
$939$	12.6436	0.412610
$940$	0	0
$941$	−56.8829	−1.85433	−0.927165	−	0.374652i	$-0.877762\pi$
$941$	−56.8829	−1.85433	−0.927165	+	0.374652i	$0.877762\pi$
$942$	0	0
$943$	−15.4055	−0.501671
$944$	0	0
$945$	28.1676	0.916292
$946$	0	0
$947$	0.601397	0.0195428	0.00977140	−	0.999952i	$-0.496890\pi$
$947$	0.601397	0.0195428	0.00977140	+	0.999952i	$0.496890\pi$
$948$	0	0
$949$	−6.91815	−0.224573
$950$	0	0
$951$	−8.15883	−0.264568
$952$	0	0
$953$	−11.9862	−0.388270	−0.194135	−	0.980975i	$-0.562190\pi$
$953$	−11.9862	−0.388270	−0.194135	+	0.980975i	$0.562190\pi$
$954$	0	0
$955$	−19.7718	−0.639799
$956$	0	0
$957$	−19.1100	−0.617740
$958$	0	0
$959$	−31.6240	−1.02119
$960$	0	0
$961$	4.06584	0.131156
$962$	0	0
$963$	−0.301058	−0.00970146
$964$	0	0
$965$	48.2911	1.55455
$966$	0	0
$967$	40.9121	1.31564	0.657822	−	0.753173i	$-0.271478\pi$
$967$	40.9121	1.31564	0.657822	+	0.753173i	$0.271478\pi$
$968$	0	0
$969$	12.9300	0.415372
$970$	0	0
$971$	−55.2770	−1.77392	−0.886962	−	0.461842i	$-0.847189\pi$
$971$	−55.2770	−1.77392	−0.886962	+	0.461842i	$0.847189\pi$
$972$	0	0
$973$	−18.8990	−0.605874
$974$	0	0
$975$	−2.09690	−0.0671546
$976$	0	0
$977$	13.0269	0.416767	0.208384	−	0.978047i	$-0.433180\pi$
$977$	13.0269	0.416767	0.208384	+	0.978047i	$0.433180\pi$
$978$	0	0
$979$	−23.7531	−0.759154
$980$	0	0
$981$	0.198160	0.00632677
$982$	0	0
$983$	45.8814	1.46339	0.731695	−	0.681632i	$-0.238730\pi$
$983$	45.8814	1.46339	0.731695	+	0.681632i	$0.238730\pi$
$984$	0	0
$985$	−61.6642	−1.96478
$986$	0	0
$987$	26.6232	0.847425
$988$	0	0
$989$	49.8455	1.58500
$990$	0	0
$991$	50.5955	1.60722	0.803609	−	0.595158i	$-0.202910\pi$
$991$	50.5955	1.60722	0.803609	+	0.595158i	$0.202910\pi$
$992$	0	0
$993$	33.5477	1.06460
$994$	0	0
$995$	−55.7708	−1.76805
$996$	0	0
$997$	42.6441	1.35055	0.675276	−	0.737565i	$-0.264025\pi$
$997$	42.6441	1.35055	0.675276	+	0.737565i	$0.264025\pi$
$998$	0	0
$999$	−38.8444	−1.22898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.d.1.7	✓	28
4.3	odd	2		8048.2.a.v.1.22		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.7	✓	28	1.1	even	1	trivial
8048.2.a.v.1.22		28	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)

\(f(q)\)	\(=\)	\(q-1.73728 q^{3} +2.58604 q^{5} +2.10260 q^{7} +0.0181326 q^{9} +O(q^{10})\)	\(q-1.73728 q^{3} +2.58604 q^{5} +2.10260 q^{7} +0.0181326 q^{9} -2.62289 q^{11} +0.715213 q^{13} -4.49267 q^{15} -5.38011 q^{17} +1.38337 q^{19} -3.65280 q^{21} +6.72492 q^{23} +1.68762 q^{25} +5.18033 q^{27} -4.19384 q^{29} -5.92164 q^{31} +4.55669 q^{33} +5.43741 q^{35} -7.49843 q^{37} -1.24252 q^{39} -2.29080 q^{41} +7.41206 q^{43} +0.0468916 q^{45} -7.28843 q^{47} -2.57908 q^{49} +9.34674 q^{51} -11.8850 q^{53} -6.78291 q^{55} -2.40330 q^{57} -7.39694 q^{59} -1.47413 q^{61} +0.0381255 q^{63} +1.84957 q^{65} -4.45942 q^{67} -11.6831 q^{69} +8.17012 q^{71} -9.67286 q^{73} -2.93186 q^{75} -5.51489 q^{77} +9.04789 q^{79} -9.05407 q^{81} +4.25798 q^{83} -13.9132 q^{85} +7.28586 q^{87} +9.05608 q^{89} +1.50381 q^{91} +10.2875 q^{93} +3.57746 q^{95} +8.08527 q^{97} -0.0475598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) \) 28 * q + 2 * q^3 - 12 * q^5 + 18 * q^9 - 14 * q^11 - 31 * q^13 - 2 * q^15 - 9 * q^17 - 8 * q^19 - 28 * q^21 - 4 * q^23 + 22 * q^25 - 4 * q^27 - 47 * q^29 - 5 * q^31 - 26 * q^33 - 13 * q^35 - 67 * q^37 - 9 * q^39 - 28 * q^41 + 15 * q^43 - 57 * q^45 - 10 * q^47 + 20 * q^49 - 11 * q^51 - 58 * q^53 + 15 * q^55 - 31 * q^57 - 32 * q^59 - 55 * q^61 - 16 * q^63 - 44 * q^65 + 22 * q^67 - 44 * q^69 - 47 * q^71 - 5 * q^73 - 25 * q^75 - 50 * q^77 - 14 * q^79 - 28 * q^81 - 16 * q^83 - 78 * q^85 - 11 * q^87 - 20 * q^89 - 15 * q^91 - 83 * q^93 - 27 * q^95 - 8 * q^97 - 70 * q^99

Introduction

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