LMFDB - Embedded newform 8024.2.a.ba.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.477393 q^{3} +1.25877 q^{5} +1.96440 q^{7} -2.77210 q^{9} +O(q^{10})$	$q+0.477393 q^{3} +1.25877 q^{5} +1.96440 q^{7} -2.77210 q^{9} -5.00508 q^{11} -3.48240 q^{13} +0.600928 q^{15} -1.00000 q^{17} -0.386716 q^{19} +0.937789 q^{21} +4.65375 q^{23} -3.41550 q^{25} -2.75556 q^{27} +5.68991 q^{29} -5.70769 q^{31} -2.38939 q^{33} +2.47273 q^{35} +6.54964 q^{37} -1.66247 q^{39} +5.11227 q^{41} +3.18320 q^{43} -3.48943 q^{45} +3.38176 q^{47} -3.14114 q^{49} -0.477393 q^{51} +1.38359 q^{53} -6.30024 q^{55} -0.184615 q^{57} +1.00000 q^{59} -4.68212 q^{61} -5.44550 q^{63} -4.38354 q^{65} +11.7329 q^{67} +2.22167 q^{69} +4.66732 q^{71} +13.3059 q^{73} -1.63053 q^{75} -9.83196 q^{77} +4.30009 q^{79} +7.00081 q^{81} -5.58294 q^{83} -1.25877 q^{85} +2.71632 q^{87} +13.8699 q^{89} -6.84081 q^{91} -2.72481 q^{93} -0.486786 q^{95} -0.411486 q^{97} +13.8746 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	$ 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) $ 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * 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
$3$	0.477393	0.275623	0.137811	−	0.990458i	$-0.455993\pi$
$3$	0.477393	0.275623	0.137811	+	0.990458i	$0.455993\pi$
$4$	0	0
$5$	1.25877	0.562939	0.281470	−	0.959570i	$-0.409178\pi$
$5$	1.25877	0.562939	0.281470	+	0.959570i	$0.409178\pi$
$6$	0	0
$7$	1.96440	0.742473	0.371236	−	0.928538i	$-0.378934\pi$
$7$	1.96440	0.742473	0.371236	+	0.928538i	$0.378934\pi$
$8$	0	0
$9$	−2.77210	−0.924032
$10$	0	0
$11$	−5.00508	−1.50909	−0.754544	−	0.656250i	$-0.772142\pi$
$11$	−5.00508	−1.50909	−0.754544	+	0.656250i	$0.772142\pi$
$12$	0	0
$13$	−3.48240	−0.965843	−0.482922	−	0.875664i	$-0.660424\pi$
$13$	−3.48240	−0.965843	−0.482922	+	0.875664i	$0.660424\pi$
$14$	0	0
$15$	0.600928	0.155159
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.386716	−0.0887187	−0.0443594	−	0.999016i	$-0.514125\pi$
$19$	−0.386716	−0.0887187	−0.0443594	+	0.999016i	$0.514125\pi$
$20$	0	0
$21$	0.937789	0.204642
$22$	0	0
$23$	4.65375	0.970374	0.485187	−	0.874410i	$-0.338752\pi$
$23$	4.65375	0.970374	0.485187	+	0.874410i	$0.338752\pi$
$24$	0	0
$25$	−3.41550	−0.683100
$26$	0	0
$27$	−2.75556	−0.530307
$28$	0	0
$29$	5.68991	1.05659	0.528294	−	0.849061i	$-0.322832\pi$
$29$	5.68991	1.05659	0.528294	+	0.849061i	$0.322832\pi$
$30$	0	0
$31$	−5.70769	−1.02513	−0.512566	−	0.858648i	$-0.671305\pi$
$31$	−5.70769	−1.02513	−0.512566	+	0.858648i	$0.671305\pi$
$32$	0	0
$33$	−2.38939	−0.415939
$34$	0	0
$35$	2.47273	0.417967
$36$	0	0
$37$	6.54964	1.07675	0.538377	−	0.842704i	$-0.319037\pi$
$37$	6.54964	1.07675	0.538377	+	0.842704i	$0.319037\pi$
$38$	0	0
$39$	−1.66247	−0.266208
$40$	0	0
$41$	5.11227	0.798403	0.399201	−	0.916863i	$-0.369287\pi$
$41$	5.11227	0.798403	0.399201	+	0.916863i	$0.369287\pi$
$42$	0	0
$43$	3.18320	0.485433	0.242717	−	0.970097i	$-0.421961\pi$
$43$	3.18320	0.485433	0.242717	+	0.970097i	$0.421961\pi$
$44$	0	0
$45$	−3.48943	−0.520174
$46$	0	0
$47$	3.38176	0.493280	0.246640	−	0.969107i	$-0.420673\pi$
$47$	3.38176	0.493280	0.246640	+	0.969107i	$0.420673\pi$
$48$	0	0
$49$	−3.14114	−0.448734
$50$	0	0
$51$	−0.477393	−0.0668483
$52$	0	0
$53$	1.38359	0.190051	0.0950254	−	0.995475i	$-0.469707\pi$
$53$	1.38359	0.190051	0.0950254	+	0.995475i	$0.469707\pi$
$54$	0	0
$55$	−6.30024	−0.849524
$56$	0	0
$57$	−0.184615	−0.0244529
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−4.68212	−0.599484	−0.299742	−	0.954020i	$-0.596901\pi$
$61$	−4.68212	−0.599484	−0.299742	+	0.954020i	$0.596901\pi$
$62$	0	0
$63$	−5.44550	−0.686069
$64$	0	0
$65$	−4.38354	−0.543711
$66$	0	0
$67$	11.7329	1.43341	0.716703	−	0.697379i	$-0.245650\pi$
$67$	11.7329	1.43341	0.716703	+	0.697379i	$0.245650\pi$
$68$	0	0
$69$	2.22167	0.267457
$70$	0	0
$71$	4.66732	0.553909	0.276954	−	0.960883i	$-0.410675\pi$
$71$	4.66732	0.553909	0.276954	+	0.960883i	$0.410675\pi$
$72$	0	0
$73$	13.3059	1.55733	0.778667	−	0.627437i	$-0.215896\pi$
$73$	13.3059	1.55733	0.778667	+	0.627437i	$0.215896\pi$
$74$	0	0
$75$	−1.63053	−0.188278
$76$	0	0
$77$	−9.83196	−1.12046
$78$	0	0
$79$	4.30009	0.483798	0.241899	−	0.970301i	$-0.422230\pi$
$79$	4.30009	0.483798	0.241899	+	0.970301i	$0.422230\pi$
$80$	0	0
$81$	7.00081	0.777867
$82$	0	0
$83$	−5.58294	−0.612807	−0.306403	−	0.951902i	$-0.599126\pi$
$83$	−5.58294	−0.612807	−0.306403	+	0.951902i	$0.599126\pi$
$84$	0	0
$85$	−1.25877	−0.136533
$86$	0	0
$87$	2.71632	0.291220
$88$	0	0
$89$	13.8699	1.47020	0.735101	−	0.677958i	$-0.237135\pi$
$89$	13.8699	1.47020	0.735101	+	0.677958i	$0.237135\pi$
$90$	0	0
$91$	−6.84081	−0.717112
$92$	0	0
$93$	−2.72481	−0.282550
$94$	0	0
$95$	−0.486786	−0.0499432
$96$	0	0
$97$	−0.411486	−0.0417801	−0.0208901	−	0.999782i	$-0.506650\pi$
$97$	−0.411486	−0.0417801	−0.0208901	+	0.999782i	$0.506650\pi$
$98$	0	0
$99$	13.8746	1.39445
$100$	0	0
$101$	16.5113	1.64293	0.821465	−	0.570258i	$-0.193157\pi$
$101$	16.5113	1.64293	0.821465	+	0.570258i	$0.193157\pi$
$102$	0	0
$103$	−10.5457	−1.03910	−0.519548	−	0.854441i	$-0.673900\pi$
$103$	−10.5457	−1.03910	−0.519548	+	0.854441i	$0.673900\pi$
$104$	0	0
$105$	1.18046	0.115201
$106$	0	0
$107$	9.66094	0.933958	0.466979	−	0.884268i	$-0.345342\pi$
$107$	9.66094	0.933958	0.466979	+	0.884268i	$0.345342\pi$
$108$	0	0
$109$	−13.4460	−1.28789	−0.643945	−	0.765072i	$-0.722704\pi$
$109$	−13.4460	−1.28789	−0.643945	+	0.765072i	$0.722704\pi$
$110$	0	0
$111$	3.12675	0.296778
$112$	0	0
$113$	15.6175	1.46917	0.734584	−	0.678518i	$-0.237377\pi$
$113$	15.6175	1.46917	0.734584	+	0.678518i	$0.237377\pi$
$114$	0	0
$115$	5.85800	0.546262
$116$	0	0
$117$	9.65354	0.892470
$118$	0	0
$119$	−1.96440	−0.180076
$120$	0	0
$121$	14.0508	1.27734
$122$	0	0
$123$	2.44056	0.220058
$124$	0	0
$125$	−10.5932	−0.947483
$126$	0	0
$127$	20.6830	1.83532	0.917658	−	0.397371i	$-0.130077\pi$
$127$	20.6830	1.83532	0.917658	+	0.397371i	$0.130077\pi$
$128$	0	0
$129$	1.51964	0.133796
$130$	0	0
$131$	19.7876	1.72885	0.864426	−	0.502760i	$-0.167682\pi$
$131$	19.7876	1.72885	0.864426	+	0.502760i	$0.167682\pi$
$132$	0	0
$133$	−0.759664	−0.0658712
$134$	0	0
$135$	−3.46861	−0.298531
$136$	0	0
$137$	−7.27255	−0.621336	−0.310668	−	0.950518i	$-0.600553\pi$
$137$	−7.27255	−0.621336	−0.310668	+	0.950518i	$0.600553\pi$
$138$	0	0
$139$	−0.306134	−0.0259659	−0.0129830	−	0.999916i	$-0.504133\pi$
$139$	−0.306134	−0.0259659	−0.0129830	+	0.999916i	$0.504133\pi$
$140$	0	0
$141$	1.61443	0.135959
$142$	0	0
$143$	17.4297	1.45754
$144$	0	0
$145$	7.16228	0.594795
$146$	0	0
$147$	−1.49956	−0.123681
$148$	0	0
$149$	−14.5625	−1.19300	−0.596502	−	0.802611i	$-0.703443\pi$
$149$	−14.5625	−1.19300	−0.596502	+	0.802611i	$0.703443\pi$
$150$	0	0
$151$	−21.7032	−1.76618	−0.883089	−	0.469206i	$-0.844540\pi$
$151$	−21.7032	−1.76618	−0.883089	+	0.469206i	$0.844540\pi$
$152$	0	0
$153$	2.77210	0.224111
$154$	0	0
$155$	−7.18467	−0.577087
$156$	0	0
$157$	13.6215	1.08712	0.543558	−	0.839372i	$-0.317077\pi$
$157$	13.6215	1.08712	0.543558	+	0.839372i	$0.317077\pi$
$158$	0	0
$159$	0.660516	0.0523823
$160$	0	0
$161$	9.14182	0.720476
$162$	0	0
$163$	23.0601	1.80621	0.903103	−	0.429424i	$-0.141283\pi$
$163$	23.0601	1.80621	0.903103	+	0.429424i	$0.141283\pi$
$164$	0	0
$165$	−3.00769	−0.234148
$166$	0	0
$167$	−7.41534	−0.573816	−0.286908	−	0.957958i	$-0.592627\pi$
$167$	−7.41534	−0.573816	−0.286908	+	0.957958i	$0.592627\pi$
$168$	0	0
$169$	−0.872915	−0.0671473
$170$	0	0
$171$	1.07201	0.0819789
$172$	0	0
$173$	10.7404	0.816577	0.408288	−	0.912853i	$-0.366126\pi$
$173$	10.7404	0.816577	0.408288	+	0.912853i	$0.366126\pi$
$174$	0	0
$175$	−6.70940	−0.507183
$176$	0	0
$177$	0.477393	0.0358830
$178$	0	0
$179$	1.54634	0.115579	0.0577894	−	0.998329i	$-0.481595\pi$
$179$	1.54634	0.115579	0.0577894	+	0.998329i	$0.481595\pi$
$180$	0	0
$181$	13.2388	0.984033	0.492016	−	0.870586i	$-0.336260\pi$
$181$	13.2388	0.984033	0.492016	+	0.870586i	$0.336260\pi$
$182$	0	0
$183$	−2.23521	−0.165231
$184$	0	0
$185$	8.24450	0.606147
$186$	0	0
$187$	5.00508	0.366007
$188$	0	0
$189$	−5.41301	−0.393738
$190$	0	0
$191$	2.19267	0.158656	0.0793282	−	0.996849i	$-0.474722\pi$
$191$	2.19267	0.158656	0.0793282	+	0.996849i	$0.474722\pi$
$192$	0	0
$193$	−10.8521	−0.781148	−0.390574	−	0.920571i	$-0.627724\pi$
$193$	−10.8521	−0.781148	−0.390574	+	0.920571i	$0.627724\pi$
$194$	0	0
$195$	−2.09267	−0.149859
$196$	0	0
$197$	−11.6522	−0.830186	−0.415093	−	0.909779i	$-0.636251\pi$
$197$	−11.6522	−0.830186	−0.415093	+	0.909779i	$0.636251\pi$
$198$	0	0
$199$	16.2701	1.15336	0.576678	−	0.816972i	$-0.304349\pi$
$199$	16.2701	1.15336	0.576678	+	0.816972i	$0.304349\pi$
$200$	0	0
$201$	5.60121	0.395079
$202$	0	0
$203$	11.1772	0.784488
$204$	0	0
$205$	6.43518	0.449452
$206$	0	0
$207$	−12.9006	−0.896657
$208$	0	0
$209$	1.93554	0.133884
$210$	0	0
$211$	22.1313	1.52358	0.761790	−	0.647824i	$-0.224321\pi$
$211$	22.1313	1.52358	0.761790	+	0.647824i	$0.224321\pi$
$212$	0	0
$213$	2.22814	0.152670
$214$	0	0
$215$	4.00692	0.273269
$216$	0	0
$217$	−11.2122	−0.761133
$218$	0	0
$219$	6.35213	0.429237
$220$	0	0
$221$	3.48240	0.234251
$222$	0	0
$223$	11.1963	0.749762	0.374881	−	0.927073i	$-0.377684\pi$
$223$	11.1963	0.749762	0.374881	+	0.927073i	$0.377684\pi$
$224$	0	0
$225$	9.46809	0.631206
$226$	0	0
$227$	−21.5590	−1.43092	−0.715461	−	0.698653i	$-0.753783\pi$
$227$	−21.5590	−1.43092	−0.715461	+	0.698653i	$0.753783\pi$
$228$	0	0
$229$	−16.7717	−1.10831	−0.554153	−	0.832415i	$-0.686958\pi$
$229$	−16.7717	−1.10831	−0.554153	+	0.832415i	$0.686958\pi$
$230$	0	0
$231$	−4.69371	−0.308823
$232$	0	0
$233$	8.56834	0.561331	0.280665	−	0.959806i	$-0.409445\pi$
$233$	8.56834	0.561331	0.280665	+	0.959806i	$0.409445\pi$
$234$	0	0
$235$	4.25685	0.277687
$236$	0	0
$237$	2.05283	0.133346
$238$	0	0
$239$	−3.09191	−0.199999	−0.0999996	−	0.994987i	$-0.531884\pi$
$239$	−3.09191	−0.199999	−0.0999996	+	0.994987i	$0.531884\pi$
$240$	0	0
$241$	−10.2954	−0.663183	−0.331591	−	0.943423i	$-0.607586\pi$
$241$	−10.2954	−0.663183	−0.331591	+	0.943423i	$0.607586\pi$
$242$	0	0
$243$	11.6088	0.744705
$244$	0	0
$245$	−3.95397	−0.252610
$246$	0	0
$247$	1.34670	0.0856883
$248$	0	0
$249$	−2.66525	−0.168903
$250$	0	0
$251$	−18.4658	−1.16555	−0.582776	−	0.812632i	$-0.698034\pi$
$251$	−18.4658	−1.16555	−0.582776	+	0.812632i	$0.698034\pi$
$252$	0	0
$253$	−23.2924	−1.46438
$254$	0	0
$255$	−0.600928	−0.0376315
$256$	0	0
$257$	−24.1186	−1.50448	−0.752238	−	0.658891i	$-0.771026\pi$
$257$	−24.1186	−1.50448	−0.752238	+	0.658891i	$0.771026\pi$
$258$	0	0
$259$	12.8661	0.799461
$260$	0	0
$261$	−15.7730	−0.976322
$262$	0	0
$263$	15.4708	0.953969	0.476985	−	0.878912i	$-0.341730\pi$
$263$	15.4708	0.953969	0.476985	+	0.878912i	$0.341730\pi$
$264$	0	0
$265$	1.74162	0.106987
$266$	0	0
$267$	6.62137	0.405221
$268$	0	0
$269$	−3.72046	−0.226841	−0.113420	−	0.993547i	$-0.536181\pi$
$269$	−3.72046	−0.226841	−0.113420	+	0.993547i	$0.536181\pi$
$270$	0	0
$271$	−11.0950	−0.673971	−0.336985	−	0.941510i	$-0.609407\pi$
$271$	−11.0950	−0.673971	−0.336985	+	0.941510i	$0.609407\pi$
$272$	0	0
$273$	−3.26575	−0.197652
$274$	0	0
$275$	17.0948	1.03086
$276$	0	0
$277$	1.90281	0.114329	0.0571643	−	0.998365i	$-0.481794\pi$
$277$	1.90281	0.114329	0.0571643	+	0.998365i	$0.481794\pi$
$278$	0	0
$279$	15.8223	0.947255
$280$	0	0
$281$	−9.71036	−0.579271	−0.289636	−	0.957137i	$-0.593534\pi$
$281$	−9.71036	−0.579271	−0.289636	+	0.957137i	$0.593534\pi$
$282$	0	0
$283$	27.7355	1.64870	0.824351	−	0.566079i	$-0.191540\pi$
$283$	27.7355	1.64870	0.824351	+	0.566079i	$0.191540\pi$
$284$	0	0
$285$	−0.232388	−0.0137655
$286$	0	0
$287$	10.0425	0.592792
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.196441	−0.0115155
$292$	0	0
$293$	4.03444	0.235694	0.117847	−	0.993032i	$-0.462401\pi$
$293$	4.03444	0.235694	0.117847	+	0.993032i	$0.462401\pi$
$294$	0	0
$295$	1.25877	0.0732884
$296$	0	0
$297$	13.7918	0.800279
$298$	0	0
$299$	−16.2062	−0.937229
$300$	0	0
$301$	6.25307	0.360421
$302$	0	0
$303$	7.88235	0.452829
$304$	0	0
$305$	−5.89371	−0.337473
$306$	0	0
$307$	9.28255	0.529783	0.264891	−	0.964278i	$-0.414664\pi$
$307$	9.28255	0.529783	0.264891	+	0.964278i	$0.414664\pi$
$308$	0	0
$309$	−5.03443	−0.286398
$310$	0	0
$311$	−15.0622	−0.854100	−0.427050	−	0.904228i	$-0.640447\pi$
$311$	−15.0622	−0.854100	−0.427050	+	0.904228i	$0.640447\pi$
$312$	0	0
$313$	−1.83683	−0.103823	−0.0519117	−	0.998652i	$-0.516531\pi$
$313$	−1.83683	−0.103823	−0.0519117	+	0.998652i	$0.516531\pi$
$314$	0	0
$315$	−6.85463	−0.386215
$316$	0	0
$317$	19.1301	1.07445	0.537226	−	0.843438i	$-0.319472\pi$
$317$	19.1301	1.07445	0.537226	+	0.843438i	$0.319472\pi$
$318$	0	0
$319$	−28.4784	−1.59448
$320$	0	0
$321$	4.61206	0.257420
$322$	0	0
$323$	0.386716	0.0215174
$324$	0	0
$325$	11.8941	0.659767
$326$	0	0
$327$	−6.41901	−0.354972
$328$	0	0
$329$	6.64312	0.366247
$330$	0	0
$331$	−12.9651	−0.712624	−0.356312	−	0.934367i	$-0.615966\pi$
$331$	−12.9651	−0.712624	−0.356312	+	0.934367i	$0.615966\pi$
$332$	0	0
$333$	−18.1562	−0.994956
$334$	0	0
$335$	14.7691	0.806920
$336$	0	0
$337$	−29.0159	−1.58060	−0.790299	−	0.612721i	$-0.790075\pi$
$337$	−29.0159	−1.58060	−0.790299	+	0.612721i	$0.790075\pi$
$338$	0	0
$339$	7.45567	0.404936
$340$	0	0
$341$	28.5674	1.54701
$342$	0	0
$343$	−19.9212	−1.07565
$344$	0	0
$345$	2.79657	0.150562
$346$	0	0
$347$	5.37539	0.288566	0.144283	−	0.989536i	$-0.453912\pi$
$347$	5.37539	0.288566	0.144283	+	0.989536i	$0.453912\pi$
$348$	0	0
$349$	5.51077	0.294985	0.147492	−	0.989063i	$-0.452880\pi$
$349$	5.51077	0.294985	0.147492	+	0.989063i	$0.452880\pi$
$350$	0	0
$351$	9.59594	0.512193
$352$	0	0
$353$	−10.6782	−0.568345	−0.284173	−	0.958773i	$-0.591719\pi$
$353$	−10.6782	−0.568345	−0.284173	+	0.958773i	$0.591719\pi$
$354$	0	0
$355$	5.87508	0.311817
$356$	0	0
$357$	−0.937789	−0.0496331
$358$	0	0
$359$	−11.7938	−0.622451	−0.311225	−	0.950336i	$-0.600739\pi$
$359$	−11.7938	−0.622451	−0.311225	+	0.950336i	$0.600739\pi$
$360$	0	0
$361$	−18.8505	−0.992129
$362$	0	0
$363$	6.70774	0.352065
$364$	0	0
$365$	16.7490	0.876685
$366$	0	0
$367$	12.8780	0.672225	0.336112	−	0.941822i	$-0.390888\pi$
$367$	12.8780	0.672225	0.336112	+	0.941822i	$0.390888\pi$
$368$	0	0
$369$	−14.1717	−0.737750
$370$	0	0
$371$	2.71792	0.141108
$372$	0	0
$373$	2.24023	0.115995	0.0579973	−	0.998317i	$-0.481529\pi$
$373$	2.24023	0.115995	0.0579973	+	0.998317i	$0.481529\pi$
$374$	0	0
$375$	−5.05710	−0.261148
$376$	0	0
$377$	−19.8145	−1.02050
$378$	0	0
$379$	11.6529	0.598568	0.299284	−	0.954164i	$-0.403252\pi$
$379$	11.6529	0.598568	0.299284	+	0.954164i	$0.403252\pi$
$380$	0	0
$381$	9.87389	0.505855
$382$	0	0
$383$	−31.7092	−1.62027	−0.810133	−	0.586246i	$-0.800605\pi$
$383$	−31.7092	−1.62027	−0.810133	+	0.586246i	$0.800605\pi$
$384$	0	0
$385$	−12.3762	−0.630749
$386$	0	0
$387$	−8.82413	−0.448556
$388$	0	0
$389$	−20.3433	−1.03144	−0.515722	−	0.856756i	$-0.672476\pi$
$389$	−20.3433	−1.03144	−0.515722	+	0.856756i	$0.672476\pi$
$390$	0	0
$391$	−4.65375	−0.235350
$392$	0	0
$393$	9.44646	0.476511
$394$	0	0
$395$	5.41282	0.272349
$396$	0	0
$397$	12.1903	0.611815	0.305908	−	0.952061i	$-0.401040\pi$
$397$	12.1903	0.611815	0.305908	+	0.952061i	$0.401040\pi$
$398$	0	0
$399$	−0.362658	−0.0181556
$400$	0	0
$401$	−14.5309	−0.725636	−0.362818	−	0.931860i	$-0.618185\pi$
$401$	−14.5309	−0.725636	−0.362818	+	0.931860i	$0.618185\pi$
$402$	0	0
$403$	19.8765	0.990117
$404$	0	0
$405$	8.81241	0.437892
$406$	0	0
$407$	−32.7815	−1.62492
$408$	0	0
$409$	6.76519	0.334517	0.167259	−	0.985913i	$-0.446509\pi$
$409$	6.76519	0.334517	0.167259	+	0.985913i	$0.446509\pi$
$410$	0	0
$411$	−3.47186	−0.171254
$412$	0	0
$413$	1.96440	0.0966617
$414$	0	0
$415$	−7.02763	−0.344973
$416$	0	0
$417$	−0.146146	−0.00715680
$418$	0	0
$419$	−14.9581	−0.730752	−0.365376	−	0.930860i	$-0.619060\pi$
$419$	−14.9581	−0.730752	−0.365376	+	0.930860i	$0.619060\pi$
$420$	0	0
$421$	−34.2639	−1.66992	−0.834960	−	0.550311i	$-0.814509\pi$
$421$	−34.2639	−1.66992	−0.834960	+	0.550311i	$0.814509\pi$
$422$	0	0
$423$	−9.37456	−0.455807
$424$	0	0
$425$	3.41550	0.165676
$426$	0	0
$427$	−9.19755	−0.445101
$428$	0	0
$429$	8.32079	0.401732
$430$	0	0
$431$	1.23109	0.0592994	0.0296497	−	0.999560i	$-0.490561\pi$
$431$	1.23109	0.0592994	0.0296497	+	0.999560i	$0.490561\pi$
$432$	0	0
$433$	11.5447	0.554803	0.277401	−	0.960754i	$-0.410527\pi$
$433$	11.5447	0.554803	0.277401	+	0.960754i	$0.410527\pi$
$434$	0	0
$435$	3.41922	0.163939
$436$	0	0
$437$	−1.79968	−0.0860903
$438$	0	0
$439$	9.51018	0.453896	0.226948	−	0.973907i	$-0.427125\pi$
$439$	9.51018	0.453896	0.226948	+	0.973907i	$0.427125\pi$
$440$	0	0
$441$	8.70754	0.414645
$442$	0	0
$443$	−9.28157	−0.440981	−0.220490	−	0.975389i	$-0.570766\pi$
$443$	−9.28157	−0.440981	−0.220490	+	0.975389i	$0.570766\pi$
$444$	0	0
$445$	17.4590	0.827634
$446$	0	0
$447$	−6.95202	−0.328819
$448$	0	0
$449$	−29.9156	−1.41180	−0.705901	−	0.708310i	$-0.749458\pi$
$449$	−29.9156	−1.41180	−0.705901	+	0.708310i	$0.749458\pi$
$450$	0	0
$451$	−25.5873	−1.20486
$452$	0	0
$453$	−10.3609	−0.486799
$454$	0	0
$455$	−8.61101	−0.403690
$456$	0	0
$457$	9.31014	0.435510	0.217755	−	0.976003i	$-0.430127\pi$
$457$	9.31014	0.435510	0.217755	+	0.976003i	$0.430127\pi$
$458$	0	0
$459$	2.75556	0.128618
$460$	0	0
$461$	−23.6300	−1.10056	−0.550279	−	0.834981i	$-0.685479\pi$
$461$	−23.6300	−1.10056	−0.550279	+	0.834981i	$0.685479\pi$
$462$	0	0
$463$	5.99631	0.278672	0.139336	−	0.990245i	$-0.455503\pi$
$463$	5.99631	0.278672	0.139336	+	0.990245i	$0.455503\pi$
$464$	0	0
$465$	−3.42991	−0.159058
$466$	0	0
$467$	6.65272	0.307851	0.153926	−	0.988082i	$-0.450808\pi$
$467$	6.65272	0.307851	0.153926	+	0.988082i	$0.450808\pi$
$468$	0	0
$469$	23.0481	1.06426
$470$	0	0
$471$	6.50281	0.299634
$472$	0	0
$473$	−15.9322	−0.732561
$474$	0	0
$475$	1.32083	0.0606037
$476$	0	0
$477$	−3.83545	−0.175613
$478$	0	0
$479$	−12.5445	−0.573172	−0.286586	−	0.958054i	$-0.592520\pi$
$479$	−12.5445	−0.573172	−0.286586	+	0.958054i	$0.592520\pi$
$480$	0	0
$481$	−22.8085	−1.03998
$482$	0	0
$483$	4.36424	0.198580
$484$	0	0
$485$	−0.517967	−0.0235197
$486$	0	0
$487$	42.2146	1.91293	0.956464	−	0.291851i	$-0.0942711\pi$
$487$	42.2146	1.91293	0.956464	+	0.291851i	$0.0942711\pi$
$488$	0	0
$489$	11.0087	0.497831
$490$	0	0
$491$	10.2515	0.462646	0.231323	−	0.972877i	$-0.425695\pi$
$491$	10.2515	0.462646	0.231323	+	0.972877i	$0.425695\pi$
$492$	0	0
$493$	−5.68991	−0.256260
$494$	0	0
$495$	17.4649	0.784988
$496$	0	0
$497$	9.16847	0.411262
$498$	0	0
$499$	8.31422	0.372196	0.186098	−	0.982531i	$-0.440416\pi$
$499$	8.31422	0.372196	0.186098	+	0.982531i	$0.440416\pi$
$500$	0	0
$501$	−3.54003	−0.158157
$502$	0	0
$503$	38.6614	1.72382	0.861912	−	0.507058i	$-0.169267\pi$
$503$	38.6614	1.72382	0.861912	+	0.507058i	$0.169267\pi$
$504$	0	0
$505$	20.7839	0.924870
$506$	0	0
$507$	−0.416723	−0.0185073
$508$	0	0
$509$	42.4877	1.88323	0.941617	−	0.336687i	$-0.109306\pi$
$509$	42.4877	1.88323	0.941617	+	0.336687i	$0.109306\pi$
$510$	0	0
$511$	26.1380	1.15628
$512$	0	0
$513$	1.06562	0.0470481
$514$	0	0
$515$	−13.2746	−0.584948
$516$	0	0
$517$	−16.9260	−0.744402
$518$	0	0
$519$	5.12738	0.225067
$520$	0	0
$521$	−10.0141	−0.438724	−0.219362	−	0.975644i	$-0.570398\pi$
$521$	−10.0141	−0.438724	−0.219362	+	0.975644i	$0.570398\pi$
$522$	0	0
$523$	29.9482	1.30954	0.654771	−	0.755827i	$-0.272765\pi$
$523$	29.9482	1.30954	0.654771	+	0.755827i	$0.272765\pi$
$524$	0	0
$525$	−3.20302	−0.139791
$526$	0	0
$527$	5.70769	0.248631
$528$	0	0
$529$	−1.34261	−0.0583743
$530$	0	0
$531$	−2.77210	−0.120299
$532$	0	0
$533$	−17.8030	−0.771132
$534$	0	0
$535$	12.1609	0.525762
$536$	0	0
$537$	0.738210	0.0318561
$538$	0	0
$539$	15.7216	0.677179
$540$	0	0
$541$	3.58239	0.154019	0.0770095	−	0.997030i	$-0.475463\pi$
$541$	3.58239	0.154019	0.0770095	+	0.997030i	$0.475463\pi$
$542$	0	0
$543$	6.32011	0.271222
$544$	0	0
$545$	−16.9254	−0.725004
$546$	0	0
$547$	20.6184	0.881580	0.440790	−	0.897610i	$-0.354698\pi$
$547$	20.6184	0.881580	0.440790	+	0.897610i	$0.354698\pi$
$548$	0	0
$549$	12.9793	0.553943
$550$	0	0
$551$	−2.20038	−0.0937392
$552$	0	0
$553$	8.44709	0.359207
$554$	0	0
$555$	3.93586	0.167068
$556$	0	0
$557$	−10.2826	−0.435687	−0.217844	−	0.975984i	$-0.569902\pi$
$557$	−10.2826	−0.435687	−0.217844	+	0.975984i	$0.569902\pi$
$558$	0	0
$559$	−11.0852	−0.468852
$560$	0	0
$561$	2.38939	0.100880
$562$	0	0
$563$	−12.6466	−0.532989	−0.266495	−	0.963836i	$-0.585865\pi$
$563$	−12.6466	−0.532989	−0.266495	+	0.963836i	$0.585865\pi$
$564$	0	0
$565$	19.6588	0.827052
$566$	0	0
$567$	13.7524	0.577545
$568$	0	0
$569$	−3.45485	−0.144835	−0.0724175	−	0.997374i	$-0.523071\pi$
$569$	−3.45485	−0.144835	−0.0724175	+	0.997374i	$0.523071\pi$
$570$	0	0
$571$	−18.3565	−0.768197	−0.384098	−	0.923292i	$-0.625488\pi$
$571$	−18.3565	−0.768197	−0.384098	+	0.923292i	$0.625488\pi$
$572$	0	0
$573$	1.04677	0.0437293
$574$	0	0
$575$	−15.8949	−0.662862
$576$	0	0
$577$	11.7265	0.488179	0.244090	−	0.969753i	$-0.421511\pi$
$577$	11.7265	0.488179	0.244090	+	0.969753i	$0.421511\pi$
$578$	0	0
$579$	−5.18069	−0.215302
$580$	0	0
$581$	−10.9671	−0.454992
$582$	0	0
$583$	−6.92498	−0.286803
$584$	0	0
$585$	12.1516	0.502406
$586$	0	0
$587$	21.7934	0.899509	0.449755	−	0.893152i	$-0.351511\pi$
$587$	21.7934	0.899509	0.449755	+	0.893152i	$0.351511\pi$
$588$	0	0
$589$	2.20726	0.0909484
$590$	0	0
$591$	−5.56268	−0.228818
$592$	0	0
$593$	14.6008	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$593$	14.6008	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$594$	0	0
$595$	−2.47273	−0.101372
$596$	0	0
$597$	7.76721	0.317891
$598$	0	0
$599$	−17.5542	−0.717244	−0.358622	−	0.933483i	$-0.616753\pi$
$599$	−17.5542	−0.717244	−0.358622	+	0.933483i	$0.616753\pi$
$600$	0	0
$601$	37.7727	1.54078	0.770391	−	0.637572i	$-0.220061\pi$
$601$	37.7727	1.54078	0.770391	+	0.637572i	$0.220061\pi$
$602$	0	0
$603$	−32.5248	−1.32451
$604$	0	0
$605$	17.6867	0.719067
$606$	0	0
$607$	48.1835	1.95571	0.977853	−	0.209291i	$-0.0671155\pi$
$607$	48.1835	1.95571	0.977853	+	0.209291i	$0.0671155\pi$
$608$	0	0
$609$	5.33593	0.216223
$610$	0	0
$611$	−11.7766	−0.476431
$612$	0	0
$613$	15.4118	0.622477	0.311238	−	0.950332i	$-0.399256\pi$
$613$	15.4118	0.622477	0.311238	+	0.950332i	$0.399256\pi$
$614$	0	0
$615$	3.07211	0.123879
$616$	0	0
$617$	8.15648	0.328367	0.164184	−	0.986430i	$-0.447501\pi$
$617$	8.15648	0.328367	0.164184	+	0.986430i	$0.447501\pi$
$618$	0	0
$619$	43.9956	1.76833	0.884166	−	0.467173i	$-0.154727\pi$
$619$	43.9956	1.76833	0.884166	+	0.467173i	$0.154727\pi$
$620$	0	0
$621$	−12.8237	−0.514596
$622$	0	0
$623$	27.2459	1.09158
$624$	0	0
$625$	3.74311	0.149724
$626$	0	0
$627$	0.924014	0.0369015
$628$	0	0
$629$	−6.54964	−0.261151
$630$	0	0
$631$	10.9873	0.437396	0.218698	−	0.975793i	$-0.429819\pi$
$631$	10.9873	0.437396	0.218698	+	0.975793i	$0.429819\pi$
$632$	0	0
$633$	10.5653	0.419933
$634$	0	0
$635$	26.0351	1.03317
$636$	0	0
$637$	10.9387	0.433407
$638$	0	0
$639$	−12.9383	−0.511829
$640$	0	0
$641$	26.9068	1.06275	0.531376	−	0.847136i	$-0.321675\pi$
$641$	26.9068	1.06275	0.531376	+	0.847136i	$0.321675\pi$
$642$	0	0
$643$	−0.721640	−0.0284587	−0.0142294	−	0.999899i	$-0.504529\pi$
$643$	−0.721640	−0.0284587	−0.0142294	+	0.999899i	$0.504529\pi$
$644$	0	0
$645$	1.91287	0.0753193
$646$	0	0
$647$	−5.15406	−0.202627	−0.101313	−	0.994855i	$-0.532304\pi$
$647$	−5.15406	−0.202627	−0.101313	+	0.994855i	$0.532304\pi$
$648$	0	0
$649$	−5.00508	−0.196466
$650$	0	0
$651$	−5.35261	−0.209785
$652$	0	0
$653$	−0.562562	−0.0220147	−0.0110074	−	0.999939i	$-0.503504\pi$
$653$	−0.562562	−0.0220147	−0.0110074	+	0.999939i	$0.503504\pi$
$654$	0	0
$655$	24.9081	0.973238
$656$	0	0
$657$	−36.8852	−1.43903
$658$	0	0
$659$	−30.1221	−1.17339	−0.586695	−	0.809808i	$-0.699571\pi$
$659$	−30.1221	−1.17339	−0.586695	+	0.809808i	$0.699571\pi$
$660$	0	0
$661$	1.54028	0.0599098	0.0299549	−	0.999551i	$-0.490464\pi$
$661$	1.54028	0.0599098	0.0299549	+	0.999551i	$0.490464\pi$
$662$	0	0
$663$	1.66247	0.0645650
$664$	0	0
$665$	−0.956242	−0.0370815
$666$	0	0
$667$	26.4794	1.02529
$668$	0	0
$669$	5.34504	0.206651
$670$	0	0
$671$	23.4344	0.904674
$672$	0	0
$673$	3.81410	0.147023	0.0735115	−	0.997294i	$-0.476579\pi$
$673$	3.81410	0.147023	0.0735115	+	0.997294i	$0.476579\pi$
$674$	0	0
$675$	9.41159	0.362252
$676$	0	0
$677$	−17.0164	−0.653994	−0.326997	−	0.945025i	$-0.606037\pi$
$677$	−17.0164	−0.653994	−0.326997	+	0.945025i	$0.606037\pi$
$678$	0	0
$679$	−0.808323	−0.0310206
$680$	0	0
$681$	−10.2921	−0.394395
$682$	0	0
$683$	20.9857	0.802995	0.401497	−	0.915860i	$-0.368490\pi$
$683$	20.9857	0.802995	0.401497	+	0.915860i	$0.368490\pi$
$684$	0	0
$685$	−9.15447	−0.349774
$686$	0	0
$687$	−8.00669	−0.305474
$688$	0	0
$689$	−4.81821	−0.183559
$690$	0	0
$691$	−15.5043	−0.589812	−0.294906	−	0.955526i	$-0.595288\pi$
$691$	−15.5043	−0.589812	−0.294906	+	0.955526i	$0.595288\pi$
$692$	0	0
$693$	27.2551	1.03534
$694$	0	0
$695$	−0.385352	−0.0146172
$696$	0	0
$697$	−5.11227	−0.193641
$698$	0	0
$699$	4.09046	0.154716
$700$	0	0
$701$	14.8693	0.561605	0.280802	−	0.959766i	$-0.409399\pi$
$701$	14.8693	0.561605	0.280802	+	0.959766i	$0.409399\pi$
$702$	0	0
$703$	−2.53285	−0.0955283
$704$	0	0
$705$	2.03219	0.0765367
$706$	0	0
$707$	32.4347	1.21983
$708$	0	0
$709$	−35.3519	−1.32767	−0.663834	−	0.747880i	$-0.731072\pi$
$709$	−35.3519	−1.32767	−0.663834	+	0.747880i	$0.731072\pi$
$710$	0	0
$711$	−11.9203	−0.447045
$712$	0	0
$713$	−26.5622	−0.994761
$714$	0	0
$715$	21.9399	0.820507
$716$	0	0
$717$	−1.47606	−0.0551243
$718$	0	0
$719$	3.15961	0.117834	0.0589169	−	0.998263i	$-0.481235\pi$
$719$	3.15961	0.117834	0.0589169	+	0.998263i	$0.481235\pi$
$720$	0	0
$721$	−20.7159	−0.771500
$722$	0	0
$723$	−4.91493	−0.182788
$724$	0	0
$725$	−19.4339	−0.721755
$726$	0	0
$727$	36.9689	1.37110	0.685551	−	0.728025i	$-0.259561\pi$
$727$	36.9689	1.37110	0.685551	+	0.728025i	$0.259561\pi$
$728$	0	0
$729$	−15.4605	−0.572610
$730$	0	0
$731$	−3.18320	−0.117735
$732$	0	0
$733$	7.46618	0.275770	0.137885	−	0.990448i	$-0.455970\pi$
$733$	7.46618	0.275770	0.137885	+	0.990448i	$0.455970\pi$
$734$	0	0
$735$	−1.88760	−0.0696251
$736$	0	0
$737$	−58.7242	−2.16313
$738$	0	0
$739$	−9.09020	−0.334388	−0.167194	−	0.985924i	$-0.553471\pi$
$739$	−9.09020	−0.334388	−0.167194	+	0.985924i	$0.553471\pi$
$740$	0	0
$741$	0.642904	0.0236177
$742$	0	0
$743$	17.1829	0.630381	0.315191	−	0.949028i	$-0.397932\pi$
$743$	17.1829	0.630381	0.315191	+	0.949028i	$0.397932\pi$
$744$	0	0
$745$	−18.3308	−0.671589
$746$	0	0
$747$	15.4764	0.566253
$748$	0	0
$749$	18.9779	0.693439
$750$	0	0
$751$	47.5812	1.73626	0.868130	−	0.496336i	$-0.165322\pi$
$751$	47.5812	1.73626	0.868130	+	0.496336i	$0.165322\pi$
$752$	0	0
$753$	−8.81545	−0.321253
$754$	0	0
$755$	−27.3193	−0.994251
$756$	0	0
$757$	−13.2074	−0.480031	−0.240015	−	0.970769i	$-0.577153\pi$
$757$	−13.2074	−0.480031	−0.240015	+	0.970769i	$0.577153\pi$
$758$	0	0
$759$	−11.1196	−0.403616
$760$	0	0
$761$	52.9779	1.92045	0.960223	−	0.279233i	$-0.0900801\pi$
$761$	52.9779	1.92045	0.960223	+	0.279233i	$0.0900801\pi$
$762$	0	0
$763$	−26.4132	−0.956224
$764$	0	0
$765$	3.48943	0.126161
$766$	0	0
$767$	−3.48240	−0.125742
$768$	0	0
$769$	14.4797	0.522153	0.261076	−	0.965318i	$-0.415923\pi$
$769$	14.4797	0.522153	0.261076	+	0.965318i	$0.415923\pi$
$770$	0	0
$771$	−11.5140	−0.414668
$772$	0	0
$773$	−16.0208	−0.576229	−0.288115	−	0.957596i	$-0.593028\pi$
$773$	−16.0208	−0.576229	−0.288115	+	0.957596i	$0.593028\pi$
$774$	0	0
$775$	19.4946	0.700267
$776$	0	0
$777$	6.14218	0.220350
$778$	0	0
$779$	−1.97700	−0.0708333
$780$	0	0
$781$	−23.3603	−0.835896
$782$	0	0
$783$	−15.6789	−0.560316
$784$	0	0
$785$	17.1464	0.611980
$786$	0	0
$787$	6.67349	0.237884	0.118942	−	0.992901i	$-0.462050\pi$
$787$	6.67349	0.237884	0.118942	+	0.992901i	$0.462050\pi$
$788$	0	0
$789$	7.38563	0.262936
$790$	0	0
$791$	30.6789	1.09082
$792$	0	0
$793$	16.3050	0.579007
$794$	0	0
$795$	0.831438	0.0294881
$796$	0	0
$797$	4.94310	0.175094	0.0875468	−	0.996160i	$-0.472097\pi$
$797$	4.94310	0.175094	0.0875468	+	0.996160i	$0.472097\pi$
$798$	0	0
$799$	−3.38176	−0.119638
$800$	0	0
$801$	−38.4486	−1.35851
$802$	0	0
$803$	−66.5969	−2.35015
$804$	0	0
$805$	11.5074	0.405584
$806$	0	0
$807$	−1.77612	−0.0625224
$808$	0	0
$809$	−22.4583	−0.789593	−0.394797	−	0.918769i	$-0.629185\pi$
$809$	−22.4583	−0.789593	−0.394797	+	0.918769i	$0.629185\pi$
$810$	0	0
$811$	42.1794	1.48112	0.740560	−	0.671990i	$-0.234560\pi$
$811$	42.1794	1.48112	0.740560	+	0.671990i	$0.234560\pi$
$812$	0	0
$813$	−5.29665	−0.185762
$814$	0	0
$815$	29.0274	1.01678
$816$	0	0
$817$	−1.23099	−0.0430670
$818$	0	0
$819$	18.9634	0.662635
$820$	0	0
$821$	48.8537	1.70501	0.852503	−	0.522723i	$-0.175084\pi$
$821$	48.8537	1.70501	0.852503	+	0.522723i	$0.175084\pi$
$822$	0	0
$823$	−37.3452	−1.30177	−0.650886	−	0.759176i	$-0.725602\pi$
$823$	−37.3452	−1.30177	−0.650886	+	0.759176i	$0.725602\pi$
$824$	0	0
$825$	8.16094	0.284128
$826$	0	0
$827$	−36.6719	−1.27521	−0.637603	−	0.770365i	$-0.720074\pi$
$827$	−36.6719	−1.27521	−0.637603	+	0.770365i	$0.720074\pi$
$828$	0	0
$829$	−11.5294	−0.400434	−0.200217	−	0.979752i	$-0.564165\pi$
$829$	−11.5294	−0.400434	−0.200217	+	0.979752i	$0.564165\pi$
$830$	0	0
$831$	0.908386	0.0315116
$832$	0	0
$833$	3.14114	0.108834
$834$	0	0
$835$	−9.33421	−0.323024
$836$	0	0
$837$	15.7279	0.543635
$838$	0	0
$839$	−5.65315	−0.195168	−0.0975842	−	0.995227i	$-0.531112\pi$
$839$	−5.65315	−0.195168	−0.0975842	+	0.995227i	$0.531112\pi$
$840$	0	0
$841$	3.37503	0.116380
$842$	0	0
$843$	−4.63565	−0.159660
$844$	0	0
$845$	−1.09880	−0.0377998
$846$	0	0
$847$	27.6013	0.948393
$848$	0	0
$849$	13.2407	0.454420
$850$	0	0
$851$	30.4804	1.04485
$852$	0	0
$853$	−7.13396	−0.244262	−0.122131	−	0.992514i	$-0.538973\pi$
$853$	−7.13396	−0.244262	−0.122131	+	0.992514i	$0.538973\pi$
$854$	0	0
$855$	1.34942	0.0461492
$856$	0	0
$857$	−12.1533	−0.415149	−0.207575	−	0.978219i	$-0.566557\pi$
$857$	−12.1533	−0.415149	−0.207575	+	0.978219i	$0.566557\pi$
$858$	0	0
$859$	17.0384	0.581342	0.290671	−	0.956823i	$-0.406121\pi$
$859$	17.0384	0.581342	0.290671	+	0.956823i	$0.406121\pi$
$860$	0	0
$861$	4.79423	0.163387
$862$	0	0
$863$	37.4354	1.27432	0.637158	−	0.770733i	$-0.280110\pi$
$863$	37.4354	1.27432	0.637158	+	0.770733i	$0.280110\pi$
$864$	0	0
$865$	13.5197	0.459683
$866$	0	0
$867$	0.477393	0.0162131
$868$	0	0
$869$	−21.5223	−0.730093
$870$	0	0
$871$	−40.8587	−1.38444
$872$	0	0
$873$	1.14068	0.0386062
$874$	0	0
$875$	−20.8092	−0.703480
$876$	0	0
$877$	−16.2115	−0.547425	−0.273712	−	0.961812i	$-0.588252\pi$
$877$	−16.2115	−0.547425	−0.273712	+	0.961812i	$0.588252\pi$
$878$	0	0
$879$	1.92601	0.0649627
$880$	0	0
$881$	−7.20827	−0.242853	−0.121426	−	0.992600i	$-0.538747\pi$
$881$	−7.20827	−0.242853	−0.121426	+	0.992600i	$0.538747\pi$
$882$	0	0
$883$	30.1247	1.01378	0.506888	−	0.862012i	$-0.330796\pi$
$883$	30.1247	1.01378	0.506888	+	0.862012i	$0.330796\pi$
$884$	0	0
$885$	0.600928	0.0202000
$886$	0	0
$887$	2.81202	0.0944183	0.0472092	−	0.998885i	$-0.484967\pi$
$887$	2.81202	0.0944183	0.0472092	+	0.998885i	$0.484967\pi$
$888$	0	0
$889$	40.6296	1.36267
$890$	0	0
$891$	−35.0396	−1.17387
$892$	0	0
$893$	−1.30778	−0.0437632
$894$	0	0
$895$	1.94648	0.0650638
$896$	0	0
$897$	−7.73672	−0.258322
$898$	0	0
$899$	−32.4762	−1.08314
$900$	0	0
$901$	−1.38359	−0.0460941
$902$	0	0
$903$	2.98517	0.0993402
$904$	0	0
$905$	16.6646	0.553951
$906$	0	0
$907$	53.5727	1.77885	0.889426	−	0.457078i	$-0.151104\pi$
$907$	53.5727	1.77885	0.889426	+	0.457078i	$0.151104\pi$
$908$	0	0
$909$	−45.7708	−1.51812
$910$	0	0
$911$	−11.2100	−0.371405	−0.185703	−	0.982606i	$-0.559456\pi$
$911$	−11.2100	−0.371405	−0.185703	+	0.982606i	$0.559456\pi$
$912$	0	0
$913$	27.9430	0.924779
$914$	0	0
$915$	−2.81361	−0.0930152
$916$	0	0
$917$	38.8708	1.28363
$918$	0	0
$919$	−9.12202	−0.300908	−0.150454	−	0.988617i	$-0.548073\pi$
$919$	−9.12202	−0.300908	−0.150454	+	0.988617i	$0.548073\pi$
$920$	0	0
$921$	4.43142	0.146020
$922$	0	0
$923$	−16.2534	−0.534989
$924$	0	0
$925$	−22.3703	−0.735531
$926$	0	0
$927$	29.2336	0.960158
$928$	0	0
$929$	7.33093	0.240520	0.120260	−	0.992742i	$-0.461627\pi$
$929$	7.33093	0.240520	0.120260	+	0.992742i	$0.461627\pi$
$930$	0	0
$931$	1.21473	0.0398111
$932$	0	0
$933$	−7.19059	−0.235409
$934$	0	0
$935$	6.30024	0.206040
$936$	0	0
$937$	−49.9502	−1.63180	−0.815901	−	0.578191i	$-0.803759\pi$
$937$	−49.9502	−1.63180	−0.815901	+	0.578191i	$0.803759\pi$
$938$	0	0
$939$	−0.876887	−0.0286161
$940$	0	0
$941$	−37.2463	−1.21419	−0.607097	−	0.794628i	$-0.707666\pi$
$941$	−37.2463	−1.21419	−0.607097	+	0.794628i	$0.707666\pi$
$942$	0	0
$943$	23.7912	0.774749
$944$	0	0
$945$	−6.81373	−0.221651
$946$	0	0
$947$	9.27783	0.301489	0.150745	−	0.988573i	$-0.451833\pi$
$947$	9.27783	0.301489	0.150745	+	0.988573i	$0.451833\pi$
$948$	0	0
$949$	−46.3363	−1.50414
$950$	0	0
$951$	9.13255	0.296143
$952$	0	0
$953$	−22.7888	−0.738202	−0.369101	−	0.929389i	$-0.620334\pi$
$953$	−22.7888	−0.738202	−0.369101	+	0.929389i	$0.620334\pi$
$954$	0	0
$955$	2.76007	0.0893139
$956$	0	0
$957$	−13.5954	−0.439476
$958$	0	0
$959$	−14.2862	−0.461325
$960$	0	0
$961$	1.57776	0.0508956
$962$	0	0
$963$	−26.7811	−0.863008
$964$	0	0
$965$	−13.6603	−0.439739
$966$	0	0
$967$	−1.05956	−0.0340732	−0.0170366	−	0.999855i	$-0.505423\pi$
$967$	−1.05956	−0.0340732	−0.0170366	+	0.999855i	$0.505423\pi$
$968$	0	0
$969$	0.184615	0.00593070
$970$	0	0
$971$	30.6178	0.982571	0.491285	−	0.870999i	$-0.336527\pi$
$971$	30.6178	0.982571	0.491285	+	0.870999i	$0.336527\pi$
$972$	0	0
$973$	−0.601369	−0.0192790
$974$	0	0
$975$	5.67816	0.181847
$976$	0	0
$977$	44.4028	1.42057	0.710285	−	0.703914i	$-0.248566\pi$
$977$	44.4028	1.42057	0.710285	+	0.703914i	$0.248566\pi$
$978$	0	0
$979$	−69.4197	−2.21866
$980$	0	0
$981$	37.2735	1.19005
$982$	0	0
$983$	2.66012	0.0848448	0.0424224	−	0.999100i	$-0.486492\pi$
$983$	2.66012	0.0848448	0.0424224	+	0.999100i	$0.486492\pi$
$984$	0	0
$985$	−14.6675	−0.467344
$986$	0	0
$987$	3.17137	0.100946
$988$	0	0
$989$	14.8138	0.471052
$990$	0	0
$991$	−59.1190	−1.87798	−0.938988	−	0.343950i	$-0.888235\pi$
$991$	−59.1190	−1.87798	−0.938988	+	0.343950i	$0.888235\pi$
$992$	0	0
$993$	−6.18942	−0.196415
$994$	0	0
$995$	20.4803	0.649269
$996$	0	0
$997$	0.444818	0.0140875	0.00704377	−	0.999975i	$-0.497758\pi$
$997$	0.444818	0.0140875	0.00704377	+	0.999975i	$0.497758\pi$
$998$	0	0
$999$	−18.0479	−0.571011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.ba.1.16	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.ba.1.16	✓	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+0.477393 q^{3} +1.25877 q^{5} +1.96440 q^{7} -2.77210 q^{9} +O(q^{10})\)	\(q+0.477393 q^{3} +1.25877 q^{5} +1.96440 q^{7} -2.77210 q^{9} -5.00508 q^{11} -3.48240 q^{13} +0.600928 q^{15} -1.00000 q^{17} -0.386716 q^{19} +0.937789 q^{21} +4.65375 q^{23} -3.41550 q^{25} -2.75556 q^{27} +5.68991 q^{29} -5.70769 q^{31} -2.38939 q^{33} +2.47273 q^{35} +6.54964 q^{37} -1.66247 q^{39} +5.11227 q^{41} +3.18320 q^{43} -3.48943 q^{45} +3.38176 q^{47} -3.14114 q^{49} -0.477393 q^{51} +1.38359 q^{53} -6.30024 q^{55} -0.184615 q^{57} +1.00000 q^{59} -4.68212 q^{61} -5.44550 q^{63} -4.38354 q^{65} +11.7329 q^{67} +2.22167 q^{69} +4.66732 q^{71} +13.3059 q^{73} -1.63053 q^{75} -9.83196 q^{77} +4.30009 q^{79} +7.00081 q^{81} -5.58294 q^{83} -1.25877 q^{85} +2.71632 q^{87} +13.8699 q^{89} -6.84081 q^{91} -2.72481 q^{93} -0.486786 q^{95} -0.411486 q^{97} +13.8746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9}+O(q^{10}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9	\( 30 q + 4 q^{3} + 2 q^{5} + 3 q^{7} + 34 q^{9} + 3 q^{11} + 9 q^{13} + 14 q^{15} - 30 q^{17} + 24 q^{19} + 7 q^{21} + 9 q^{23} + 40 q^{25} + 19 q^{27} + 9 q^{29} + 11 q^{31} - 14 q^{33} + 30 q^{35} - 13 q^{37} + 16 q^{39} - 13 q^{41} + 23 q^{43} + 12 q^{45} + 43 q^{47} + 35 q^{49} - 4 q^{51} - 4 q^{53} + 43 q^{55} + 3 q^{57} + 30 q^{59} + 43 q^{61} + 38 q^{63} + 3 q^{65} + 50 q^{67} + 34 q^{69} + 3 q^{71} - 16 q^{73} + 21 q^{75} + 18 q^{77} + 45 q^{79} + 6 q^{81} + 63 q^{83} - 2 q^{85} + 42 q^{87} + 6 q^{89} + 22 q^{91} - 2 q^{93} + 19 q^{95} - 28 q^{97} + 51 q^{99}+O(q^{100}) \) 30 * q + 4 * q^3 + 2 * q^5 + 3 * q^7 + 34 * q^9 + 3 * q^11 + 9 * q^13 + 14 * q^15 - 30 * q^17 + 24 * q^19 + 7 * q^21 + 9 * q^23 + 40 * q^25 + 19 * q^27 + 9 * q^29 + 11 * q^31 - 14 * q^33 + 30 * q^35 - 13 * q^37 + 16 * q^39 - 13 * q^41 + 23 * q^43 + 12 * q^45 + 43 * q^47 + 35 * q^49 - 4 * q^51 - 4 * q^53 + 43 * q^55 + 3 * q^57 + 30 * q^59 + 43 * q^61 + 38 * q^63 + 3 * q^65 + 50 * q^67 + 34 * q^69 + 3 * q^71 - 16 * q^73 + 21 * q^75 + 18 * q^77 + 45 * q^79 + 6 * q^81 + 63 * q^83 - 2 * q^85 + 42 * q^87 + 6 * q^89 + 22 * q^91 - 2 * q^93 + 19 * q^95 - 28 * q^97 + 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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