LMFDB - Embedded newform 6036.2.a.i.1.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.23317 q^{5} -0.513502 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.23317 q^{5} -0.513502 q^{7} +1.00000 q^{9} -2.27751 q^{11} +2.85645 q^{13} -4.23317 q^{15} +6.71341 q^{17} +6.76543 q^{19} +0.513502 q^{21} +0.132693 q^{23} +12.9197 q^{25} -1.00000 q^{27} -2.95922 q^{29} -2.49313 q^{31} +2.27751 q^{33} -2.17374 q^{35} +4.59026 q^{37} -2.85645 q^{39} +2.66067 q^{41} +8.46873 q^{43} +4.23317 q^{45} -5.38139 q^{47} -6.73632 q^{49} -6.71341 q^{51} -4.52175 q^{53} -9.64109 q^{55} -6.76543 q^{57} +3.87421 q^{59} -7.60878 q^{61} -0.513502 q^{63} +12.0918 q^{65} -13.7973 q^{67} -0.132693 q^{69} -7.80452 q^{71} +7.62794 q^{73} -12.9197 q^{75} +1.16951 q^{77} +7.12986 q^{79} +1.00000 q^{81} +4.86059 q^{83} +28.4190 q^{85} +2.95922 q^{87} -3.93230 q^{89} -1.46679 q^{91} +2.49313 q^{93} +28.6392 q^{95} +5.48570 q^{97} -2.27751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.23317	1.89313	0.946565	−	0.322513i	$-0.104527\pi$
$5$	4.23317	1.89313	0.946565	+	0.322513i	$0.104527\pi$
$6$	0	0
$7$	−0.513502	−0.194085	−0.0970427	−	0.995280i	$-0.530938\pi$
$7$	−0.513502	−0.194085	−0.0970427	+	0.995280i	$0.530938\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.27751	−0.686696	−0.343348	−	0.939208i	$-0.611561\pi$
$11$	−2.27751	−0.686696	−0.343348	+	0.939208i	$0.611561\pi$
$12$	0	0
$13$	2.85645	0.792237	0.396118	−	0.918199i	$-0.370357\pi$
$13$	2.85645	0.792237	0.396118	+	0.918199i	$0.370357\pi$
$14$	0	0
$15$	−4.23317	−1.09300
$16$	0	0
$17$	6.71341	1.62824	0.814120	−	0.580697i	$-0.197220\pi$
$17$	6.71341	1.62824	0.814120	+	0.580697i	$0.197220\pi$
$18$	0	0
$19$	6.76543	1.55210	0.776048	−	0.630674i	$-0.217221\pi$
$19$	6.76543	1.55210	0.776048	+	0.630674i	$0.217221\pi$
$20$	0	0
$21$	0.513502	0.112055
$22$	0	0
$23$	0.132693	0.0276684	0.0138342	−	0.999904i	$-0.495596\pi$
$23$	0.132693	0.0276684	0.0138342	+	0.999904i	$0.495596\pi$
$24$	0	0
$25$	12.9197	2.58394
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.95922	−0.549513	−0.274756	−	0.961514i	$-0.588597\pi$
$29$	−2.95922	−0.549513	−0.274756	+	0.961514i	$0.588597\pi$
$30$	0	0
$31$	−2.49313	−0.447779	−0.223889	−	0.974615i	$-0.571875\pi$
$31$	−2.49313	−0.447779	−0.223889	+	0.974615i	$0.571875\pi$
$32$	0	0
$33$	2.27751	0.396464
$34$	0	0
$35$	−2.17374	−0.367429
$36$	0	0
$37$	4.59026	0.754634	0.377317	−	0.926084i	$-0.376847\pi$
$37$	4.59026	0.754634	0.377317	+	0.926084i	$0.376847\pi$
$38$	0	0
$39$	−2.85645	−0.457398
$40$	0	0
$41$	2.66067	0.415526	0.207763	−	0.978179i	$-0.433382\pi$
$41$	2.66067	0.415526	0.207763	+	0.978179i	$0.433382\pi$
$42$	0	0
$43$	8.46873	1.29147	0.645735	−	0.763562i	$-0.276551\pi$
$43$	8.46873	1.29147	0.645735	+	0.763562i	$0.276551\pi$
$44$	0	0
$45$	4.23317	0.631043
$46$	0	0
$47$	−5.38139	−0.784957	−0.392478	−	0.919761i	$-0.628382\pi$
$47$	−5.38139	−0.784957	−0.392478	+	0.919761i	$0.628382\pi$
$48$	0	0
$49$	−6.73632	−0.962331
$50$	0	0
$51$	−6.71341	−0.940065
$52$	0	0
$53$	−4.52175	−0.621110	−0.310555	−	0.950555i	$-0.600515\pi$
$53$	−4.52175	−0.621110	−0.310555	+	0.950555i	$0.600515\pi$
$54$	0	0
$55$	−9.64109	−1.30000
$56$	0	0
$57$	−6.76543	−0.896103
$58$	0	0
$59$	3.87421	0.504379	0.252190	−	0.967678i	$-0.418849\pi$
$59$	3.87421	0.504379	0.252190	+	0.967678i	$0.418849\pi$
$60$	0	0
$61$	−7.60878	−0.974204	−0.487102	−	0.873345i	$-0.661946\pi$
$61$	−7.60878	−0.974204	−0.487102	+	0.873345i	$0.661946\pi$
$62$	0	0
$63$	−0.513502	−0.0646952
$64$	0	0
$65$	12.0918	1.49981
$66$	0	0
$67$	−13.7973	−1.68561	−0.842804	−	0.538221i	$-0.819097\pi$
$67$	−13.7973	−1.68561	−0.842804	+	0.538221i	$0.819097\pi$
$68$	0	0
$69$	−0.132693	−0.0159744
$70$	0	0
$71$	−7.80452	−0.926226	−0.463113	−	0.886299i	$-0.653268\pi$
$71$	−7.80452	−0.926226	−0.463113	+	0.886299i	$0.653268\pi$
$72$	0	0
$73$	7.62794	0.892783	0.446392	−	0.894838i	$-0.352709\pi$
$73$	7.62794	0.892783	0.446392	+	0.894838i	$0.352709\pi$
$74$	0	0
$75$	−12.9197	−1.49184
$76$	0	0
$77$	1.16951	0.133278
$78$	0	0
$79$	7.12986	0.802172	0.401086	−	0.916040i	$-0.368633\pi$
$79$	7.12986	0.802172	0.401086	+	0.916040i	$0.368633\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.86059	0.533519	0.266760	−	0.963763i	$-0.414047\pi$
$83$	4.86059	0.533519	0.266760	+	0.963763i	$0.414047\pi$
$84$	0	0
$85$	28.4190	3.08247
$86$	0	0
$87$	2.95922	0.317261
$88$	0	0
$89$	−3.93230	−0.416823	−0.208411	−	0.978041i	$-0.566829\pi$
$89$	−3.93230	−0.416823	−0.208411	+	0.978041i	$0.566829\pi$
$90$	0	0
$91$	−1.46679	−0.153762
$92$	0	0
$93$	2.49313	0.258525
$94$	0	0
$95$	28.6392	2.93832
$96$	0	0
$97$	5.48570	0.556988	0.278494	−	0.960438i	$-0.410165\pi$
$97$	5.48570	0.556988	0.278494	+	0.960438i	$0.410165\pi$
$98$	0	0
$99$	−2.27751	−0.228899
$100$	0	0
$101$	18.8274	1.87340	0.936699	−	0.350135i	$-0.113864\pi$
$101$	18.8274	1.87340	0.936699	+	0.350135i	$0.113864\pi$
$102$	0	0
$103$	−3.35215	−0.330297	−0.165148	−	0.986269i	$-0.552810\pi$
$103$	−3.35215	−0.330297	−0.165148	+	0.986269i	$0.552810\pi$
$104$	0	0
$105$	2.17374	0.212135
$106$	0	0
$107$	−8.33922	−0.806183	−0.403092	−	0.915160i	$-0.632064\pi$
$107$	−8.33922	−0.806183	−0.403092	+	0.915160i	$0.632064\pi$
$108$	0	0
$109$	1.65856	0.158861	0.0794307	−	0.996840i	$-0.474690\pi$
$109$	1.65856	0.158861	0.0794307	+	0.996840i	$0.474690\pi$
$110$	0	0
$111$	−4.59026	−0.435688
$112$	0	0
$113$	−18.8380	−1.77213	−0.886064	−	0.463562i	$-0.846571\pi$
$113$	−18.8380	−1.77213	−0.886064	+	0.463562i	$0.846571\pi$
$114$	0	0
$115$	0.561713	0.0523800
$116$	0	0
$117$	2.85645	0.264079
$118$	0	0
$119$	−3.44735	−0.316018
$120$	0	0
$121$	−5.81294	−0.528449
$122$	0	0
$123$	−2.66067	−0.239904
$124$	0	0
$125$	33.5255	2.99861
$126$	0	0
$127$	14.8451	1.31729	0.658644	−	0.752455i	$-0.271130\pi$
$127$	14.8451	1.31729	0.658644	+	0.752455i	$0.271130\pi$
$128$	0	0
$129$	−8.46873	−0.745630
$130$	0	0
$131$	14.7216	1.28623	0.643117	−	0.765768i	$-0.277641\pi$
$131$	14.7216	1.28623	0.643117	+	0.765768i	$0.277641\pi$
$132$	0	0
$133$	−3.47406	−0.301239
$134$	0	0
$135$	−4.23317	−0.364333
$136$	0	0
$137$	12.3473	1.05490	0.527449	−	0.849587i	$-0.323149\pi$
$137$	12.3473	1.05490	0.527449	+	0.849587i	$0.323149\pi$
$138$	0	0
$139$	−16.2275	−1.37640	−0.688198	−	0.725523i	$-0.741598\pi$
$139$	−16.2275	−1.37640	−0.688198	+	0.725523i	$0.741598\pi$
$140$	0	0
$141$	5.38139	0.453195
$142$	0	0
$143$	−6.50560	−0.544025
$144$	0	0
$145$	−12.5269	−1.04030
$146$	0	0
$147$	6.73632	0.555602
$148$	0	0
$149$	3.16194	0.259037	0.129518	−	0.991577i	$-0.458657\pi$
$149$	3.16194	0.259037	0.129518	+	0.991577i	$0.458657\pi$
$150$	0	0
$151$	5.29060	0.430543	0.215271	−	0.976554i	$-0.430936\pi$
$151$	5.29060	0.430543	0.215271	+	0.976554i	$0.430936\pi$
$152$	0	0
$153$	6.71341	0.542747
$154$	0	0
$155$	−10.5538	−0.847704
$156$	0	0
$157$	−13.4518	−1.07357	−0.536785	−	0.843719i	$-0.680362\pi$
$157$	−13.4518	−1.07357	−0.536785	+	0.843719i	$0.680362\pi$
$158$	0	0
$159$	4.52175	0.358598
$160$	0	0
$161$	−0.0681382	−0.00537004
$162$	0	0
$163$	−22.0799	−1.72943	−0.864717	−	0.502259i	$-0.832502\pi$
$163$	−22.0799	−1.72943	−0.864717	+	0.502259i	$0.832502\pi$
$164$	0	0
$165$	9.64109	0.750558
$166$	0	0
$167$	12.0195	0.930098	0.465049	−	0.885285i	$-0.346037\pi$
$167$	12.0195	0.930098	0.465049	+	0.885285i	$0.346037\pi$
$168$	0	0
$169$	−4.84069	−0.372361
$170$	0	0
$171$	6.76543	0.517365
$172$	0	0
$173$	−15.2748	−1.16132	−0.580659	−	0.814147i	$-0.697205\pi$
$173$	−15.2748	−1.16132	−0.580659	+	0.814147i	$0.697205\pi$
$174$	0	0
$175$	−6.63430	−0.501506
$176$	0	0
$177$	−3.87421	−0.291204
$178$	0	0
$179$	−4.54020	−0.339350	−0.169675	−	0.985500i	$-0.554272\pi$
$179$	−4.54020	−0.339350	−0.169675	+	0.985500i	$0.554272\pi$
$180$	0	0
$181$	23.7123	1.76252	0.881262	−	0.472628i	$-0.156695\pi$
$181$	23.7123	1.76252	0.881262	+	0.472628i	$0.156695\pi$
$182$	0	0
$183$	7.60878	0.562457
$184$	0	0
$185$	19.4314	1.42862
$186$	0	0
$187$	−15.2899	−1.11811
$188$	0	0
$189$	0.513502	0.0373518
$190$	0	0
$191$	−15.1643	−1.09725	−0.548624	−	0.836069i	$-0.684848\pi$
$191$	−15.1643	−1.09725	−0.548624	+	0.836069i	$0.684848\pi$
$192$	0	0
$193$	19.0147	1.36871	0.684354	−	0.729149i	$-0.260084\pi$
$193$	19.0147	1.36871	0.684354	+	0.729149i	$0.260084\pi$
$194$	0	0
$195$	−12.0918	−0.865914
$196$	0	0
$197$	18.1286	1.29161	0.645804	−	0.763503i	$-0.276522\pi$
$197$	18.1286	1.29161	0.645804	+	0.763503i	$0.276522\pi$
$198$	0	0
$199$	−19.1556	−1.35790	−0.678952	−	0.734182i	$-0.737566\pi$
$199$	−19.1556	−1.35790	−0.678952	+	0.734182i	$0.737566\pi$
$200$	0	0
$201$	13.7973	0.973186
$202$	0	0
$203$	1.51956	0.106652
$204$	0	0
$205$	11.2630	0.786645
$206$	0	0
$207$	0.132693	0.00922282
$208$	0	0
$209$	−15.4083	−1.06582
$210$	0	0
$211$	9.90211	0.681690	0.340845	−	0.940120i	$-0.389287\pi$
$211$	9.90211	0.681690	0.340845	+	0.940120i	$0.389287\pi$
$212$	0	0
$213$	7.80452	0.534757
$214$	0	0
$215$	35.8496	2.44492
$216$	0	0
$217$	1.28023	0.0869074
$218$	0	0
$219$	−7.62794	−0.515449
$220$	0	0
$221$	19.1765	1.28995
$222$	0	0
$223$	1.53797	0.102990	0.0514950	−	0.998673i	$-0.483601\pi$
$223$	1.53797	0.102990	0.0514950	+	0.998673i	$0.483601\pi$
$224$	0	0
$225$	12.9197	0.861314
$226$	0	0
$227$	20.3326	1.34952	0.674762	−	0.738035i	$-0.264246\pi$
$227$	20.3326	1.34952	0.674762	+	0.738035i	$0.264246\pi$
$228$	0	0
$229$	−4.78969	−0.316512	−0.158256	−	0.987398i	$-0.550587\pi$
$229$	−4.78969	−0.316512	−0.158256	+	0.987398i	$0.550587\pi$
$230$	0	0
$231$	−1.16951	−0.0769479
$232$	0	0
$233$	−25.6466	−1.68017	−0.840084	−	0.542457i	$-0.817494\pi$
$233$	−25.6466	−1.68017	−0.840084	+	0.542457i	$0.817494\pi$
$234$	0	0
$235$	−22.7803	−1.48603
$236$	0	0
$237$	−7.12986	−0.463134
$238$	0	0
$239$	−4.90548	−0.317309	−0.158655	−	0.987334i	$-0.550716\pi$
$239$	−4.90548	−0.317309	−0.158655	+	0.987334i	$0.550716\pi$
$240$	0	0
$241$	−7.32776	−0.472022	−0.236011	−	0.971750i	$-0.575840\pi$
$241$	−7.32776	−0.472022	−0.236011	+	0.971750i	$0.575840\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−28.5160	−1.82182
$246$	0	0
$247$	19.3251	1.22963
$248$	0	0
$249$	−4.86059	−0.308028
$250$	0	0
$251$	−17.8427	−1.12622	−0.563109	−	0.826382i	$-0.690395\pi$
$251$	−17.8427	−1.12622	−0.563109	+	0.826382i	$0.690395\pi$
$252$	0	0
$253$	−0.302210	−0.0189998
$254$	0	0
$255$	−28.4190	−1.77967
$256$	0	0
$257$	28.7648	1.79430	0.897151	−	0.441724i	$-0.145633\pi$
$257$	28.7648	1.79430	0.897151	+	0.441724i	$0.145633\pi$
$258$	0	0
$259$	−2.35711	−0.146464
$260$	0	0
$261$	−2.95922	−0.183171
$262$	0	0
$263$	29.2520	1.80376	0.901879	−	0.431989i	$-0.142188\pi$
$263$	29.2520	1.80376	0.901879	+	0.431989i	$0.142188\pi$
$264$	0	0
$265$	−19.1413	−1.17584
$266$	0	0
$267$	3.93230	0.240653
$268$	0	0
$269$	−29.7538	−1.81412	−0.907061	−	0.420999i	$-0.861680\pi$
$269$	−29.7538	−1.81412	−0.907061	+	0.420999i	$0.861680\pi$
$270$	0	0
$271$	20.5913	1.25084	0.625418	−	0.780290i	$-0.284929\pi$
$271$	20.5913	1.25084	0.625418	+	0.780290i	$0.284929\pi$
$272$	0	0
$273$	1.46679	0.0887743
$274$	0	0
$275$	−29.4248	−1.77438
$276$	0	0
$277$	0.219899	0.0132124	0.00660622	−	0.999978i	$-0.497897\pi$
$277$	0.219899	0.0132124	0.00660622	+	0.999978i	$0.497897\pi$
$278$	0	0
$279$	−2.49313	−0.149260
$280$	0	0
$281$	−24.8664	−1.48341	−0.741703	−	0.670728i	$-0.765982\pi$
$281$	−24.8664	−1.48341	−0.741703	+	0.670728i	$0.765982\pi$
$282$	0	0
$283$	−1.32266	−0.0786240	−0.0393120	−	0.999227i	$-0.512517\pi$
$283$	−1.32266	−0.0786240	−0.0393120	+	0.999227i	$0.512517\pi$
$284$	0	0
$285$	−28.6392	−1.69644
$286$	0	0
$287$	−1.36626	−0.0806476
$288$	0	0
$289$	28.0698	1.65117
$290$	0	0
$291$	−5.48570	−0.321577
$292$	0	0
$293$	−17.9775	−1.05025	−0.525127	−	0.851024i	$-0.675982\pi$
$293$	−17.9775	−1.05025	−0.525127	+	0.851024i	$0.675982\pi$
$294$	0	0
$295$	16.4002	0.954856
$296$	0	0
$297$	2.27751	0.132155
$298$	0	0
$299$	0.379032	0.0219200
$300$	0	0
$301$	−4.34871	−0.250656
$302$	0	0
$303$	−18.8274	−1.08161
$304$	0	0
$305$	−32.2092	−1.84430
$306$	0	0
$307$	6.99657	0.399315	0.199658	−	0.979866i	$-0.436017\pi$
$307$	6.99657	0.399315	0.199658	+	0.979866i	$0.436017\pi$
$308$	0	0
$309$	3.35215	0.190697
$310$	0	0
$311$	−11.1497	−0.632243	−0.316122	−	0.948719i	$-0.602381\pi$
$311$	−11.1497	−0.632243	−0.316122	+	0.948719i	$0.602381\pi$
$312$	0	0
$313$	8.43088	0.476542	0.238271	−	0.971199i	$-0.423419\pi$
$313$	8.43088	0.476542	0.238271	+	0.971199i	$0.423419\pi$
$314$	0	0
$315$	−2.17374	−0.122476
$316$	0	0
$317$	26.5152	1.48924	0.744622	−	0.667487i	$-0.232630\pi$
$317$	26.5152	1.48924	0.744622	+	0.667487i	$0.232630\pi$
$318$	0	0
$319$	6.73965	0.377348
$320$	0	0
$321$	8.33922	0.465450
$322$	0	0
$323$	45.4191	2.52718
$324$	0	0
$325$	36.9045	2.04709
$326$	0	0
$327$	−1.65856	−0.0917186
$328$	0	0
$329$	2.76336	0.152349
$330$	0	0
$331$	−4.49483	−0.247058	−0.123529	−	0.992341i	$-0.539421\pi$
$331$	−4.49483	−0.247058	−0.123529	+	0.992341i	$0.539421\pi$
$332$	0	0
$333$	4.59026	0.251545
$334$	0	0
$335$	−58.4063	−3.19107
$336$	0	0
$337$	−26.8911	−1.46485	−0.732425	−	0.680848i	$-0.761611\pi$
$337$	−26.8911	−1.46485	−0.732425	+	0.680848i	$0.761611\pi$
$338$	0	0
$339$	18.8380	1.02314
$340$	0	0
$341$	5.67812	0.307488
$342$	0	0
$343$	7.05362	0.380860
$344$	0	0
$345$	−0.561713	−0.0302416
$346$	0	0
$347$	−6.71010	−0.360217	−0.180109	−	0.983647i	$-0.557645\pi$
$347$	−6.71010	−0.360217	−0.180109	+	0.983647i	$0.557645\pi$
$348$	0	0
$349$	9.54307	0.510829	0.255415	−	0.966832i	$-0.417788\pi$
$349$	9.54307	0.510829	0.255415	+	0.966832i	$0.417788\pi$
$350$	0	0
$351$	−2.85645	−0.152466
$352$	0	0
$353$	11.2970	0.601279	0.300640	−	0.953738i	$-0.402800\pi$
$353$	11.2970	0.601279	0.300640	+	0.953738i	$0.402800\pi$
$354$	0	0
$355$	−33.0378	−1.75347
$356$	0	0
$357$	3.44735	0.182453
$358$	0	0
$359$	8.78012	0.463397	0.231699	−	0.972788i	$-0.425572\pi$
$359$	8.78012	0.463397	0.231699	+	0.972788i	$0.425572\pi$
$360$	0	0
$361$	26.7710	1.40900
$362$	0	0
$363$	5.81294	0.305100
$364$	0	0
$365$	32.2904	1.69016
$366$	0	0
$367$	−22.8956	−1.19514	−0.597571	−	0.801816i	$-0.703867\pi$
$367$	−22.8956	−1.19514	−0.597571	+	0.801816i	$0.703867\pi$
$368$	0	0
$369$	2.66067	0.138509
$370$	0	0
$371$	2.32193	0.120548
$372$	0	0
$373$	30.4149	1.57482	0.787412	−	0.616427i	$-0.211421\pi$
$373$	30.4149	1.57482	0.787412	+	0.616427i	$0.211421\pi$
$374$	0	0
$375$	−33.5255	−1.73125
$376$	0	0
$377$	−8.45285	−0.435344
$378$	0	0
$379$	32.5921	1.67414	0.837072	−	0.547092i	$-0.184265\pi$
$379$	32.5921	1.67414	0.837072	+	0.547092i	$0.184265\pi$
$380$	0	0
$381$	−14.8451	−0.760536
$382$	0	0
$383$	10.9590	0.559980	0.279990	−	0.960003i	$-0.409669\pi$
$383$	10.9590	0.559980	0.279990	+	0.960003i	$0.409669\pi$
$384$	0	0
$385$	4.95072	0.252312
$386$	0	0
$387$	8.46873	0.430490
$388$	0	0
$389$	−6.67383	−0.338376	−0.169188	−	0.985584i	$-0.554115\pi$
$389$	−6.67383	−0.338376	−0.169188	+	0.985584i	$0.554115\pi$
$390$	0	0
$391$	0.890823	0.0450509
$392$	0	0
$393$	−14.7216	−0.742607
$394$	0	0
$395$	30.1819	1.51862
$396$	0	0
$397$	15.1749	0.761604	0.380802	−	0.924657i	$-0.375648\pi$
$397$	15.1749	0.761604	0.380802	+	0.924657i	$0.375648\pi$
$398$	0	0
$399$	3.47406	0.173921
$400$	0	0
$401$	−30.9480	−1.54547	−0.772734	−	0.634730i	$-0.781111\pi$
$401$	−30.9480	−1.54547	−0.772734	+	0.634730i	$0.781111\pi$
$402$	0	0
$403$	−7.12149	−0.354747
$404$	0	0
$405$	4.23317	0.210348
$406$	0	0
$407$	−10.4544	−0.518204
$408$	0	0
$409$	−19.7223	−0.975203	−0.487601	−	0.873066i	$-0.662128\pi$
$409$	−19.7223	−0.975203	−0.487601	+	0.873066i	$0.662128\pi$
$410$	0	0
$411$	−12.3473	−0.609045
$412$	0	0
$413$	−1.98942	−0.0978927
$414$	0	0
$415$	20.5757	1.01002
$416$	0	0
$417$	16.2275	0.794662
$418$	0	0
$419$	26.1855	1.27925	0.639623	−	0.768689i	$-0.279091\pi$
$419$	26.1855	1.27925	0.639623	+	0.768689i	$0.279091\pi$
$420$	0	0
$421$	−15.8461	−0.772290	−0.386145	−	0.922438i	$-0.626194\pi$
$421$	−15.8461	−0.772290	−0.386145	+	0.922438i	$0.626194\pi$
$422$	0	0
$423$	−5.38139	−0.261652
$424$	0	0
$425$	86.7353	4.20728
$426$	0	0
$427$	3.90712	0.189079
$428$	0	0
$429$	6.50560	0.314093
$430$	0	0
$431$	−21.0142	−1.01222	−0.506108	−	0.862470i	$-0.668916\pi$
$431$	−21.0142	−1.01222	−0.506108	+	0.862470i	$0.668916\pi$
$432$	0	0
$433$	19.7843	0.950772	0.475386	−	0.879777i	$-0.342308\pi$
$433$	19.7843	0.950772	0.475386	+	0.879777i	$0.342308\pi$
$434$	0	0
$435$	12.5269	0.600617
$436$	0	0
$437$	0.897727	0.0429441
$438$	0	0
$439$	10.8080	0.515836	0.257918	−	0.966167i	$-0.416964\pi$
$439$	10.8080	0.515836	0.257918	+	0.966167i	$0.416964\pi$
$440$	0	0
$441$	−6.73632	−0.320777
$442$	0	0
$443$	−2.75360	−0.130827	−0.0654137	−	0.997858i	$-0.520837\pi$
$443$	−2.75360	−0.130827	−0.0654137	+	0.997858i	$0.520837\pi$
$444$	0	0
$445$	−16.6461	−0.789100
$446$	0	0
$447$	−3.16194	−0.149555
$448$	0	0
$449$	23.9404	1.12982	0.564908	−	0.825154i	$-0.308912\pi$
$449$	23.9404	1.12982	0.564908	+	0.825154i	$0.308912\pi$
$450$	0	0
$451$	−6.05970	−0.285340
$452$	0	0
$453$	−5.29060	−0.248574
$454$	0	0
$455$	−6.20918	−0.291091
$456$	0	0
$457$	20.2214	0.945919	0.472960	−	0.881084i	$-0.343186\pi$
$457$	20.2214	0.945919	0.472960	+	0.881084i	$0.343186\pi$
$458$	0	0
$459$	−6.71341	−0.313355
$460$	0	0
$461$	9.04164	0.421111	0.210555	−	0.977582i	$-0.432473\pi$
$461$	9.04164	0.421111	0.210555	+	0.977582i	$0.432473\pi$
$462$	0	0
$463$	−5.32845	−0.247634	−0.123817	−	0.992305i	$-0.539514\pi$
$463$	−5.32845	−0.247634	−0.123817	+	0.992305i	$0.539514\pi$
$464$	0	0
$465$	10.5538	0.489422
$466$	0	0
$467$	−22.6293	−1.04716	−0.523581	−	0.851976i	$-0.675404\pi$
$467$	−22.6293	−1.04716	−0.523581	+	0.851976i	$0.675404\pi$
$468$	0	0
$469$	7.08494	0.327152
$470$	0	0
$471$	13.4518	0.619826
$472$	0	0
$473$	−19.2876	−0.886847
$474$	0	0
$475$	87.4074	4.01053
$476$	0	0
$477$	−4.52175	−0.207037
$478$	0	0
$479$	22.6369	1.03431	0.517154	−	0.855893i	$-0.326992\pi$
$479$	22.6369	1.03431	0.517154	+	0.855893i	$0.326992\pi$
$480$	0	0
$481$	13.1119	0.597849
$482$	0	0
$483$	0.0681382	0.00310040
$484$	0	0
$485$	23.2219	1.05445
$486$	0	0
$487$	−2.44980	−0.111011	−0.0555054	−	0.998458i	$-0.517677\pi$
$487$	−2.44980	−0.111011	−0.0555054	+	0.998458i	$0.517677\pi$
$488$	0	0
$489$	22.0799	0.998490
$490$	0	0
$491$	36.8319	1.66220	0.831101	−	0.556122i	$-0.187711\pi$
$491$	36.8319	1.66220	0.831101	+	0.556122i	$0.187711\pi$
$492$	0	0
$493$	−19.8664	−0.894739
$494$	0	0
$495$	−9.64109	−0.433335
$496$	0	0
$497$	4.00764	0.179767
$498$	0	0
$499$	31.5304	1.41149	0.705747	−	0.708464i	$-0.250612\pi$
$499$	31.5304	1.41149	0.705747	+	0.708464i	$0.250612\pi$
$500$	0	0
$501$	−12.0195	−0.536992
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	79.6997	3.54659
$506$	0	0
$507$	4.84069	0.214983
$508$	0	0
$509$	−18.1569	−0.804789	−0.402395	−	0.915466i	$-0.631822\pi$
$509$	−18.1569	−0.804789	−0.402395	+	0.915466i	$0.631822\pi$
$510$	0	0
$511$	−3.91696	−0.173276
$512$	0	0
$513$	−6.76543	−0.298701
$514$	0	0
$515$	−14.1902	−0.625295
$516$	0	0
$517$	12.2562	0.539026
$518$	0	0
$519$	15.2748	0.670488
$520$	0	0
$521$	−20.2404	−0.886747	−0.443374	−	0.896337i	$-0.646218\pi$
$521$	−20.2404	−0.886747	−0.443374	+	0.896337i	$0.646218\pi$
$522$	0	0
$523$	−32.4290	−1.41802	−0.709010	−	0.705198i	$-0.750858\pi$
$523$	−32.4290	−1.41802	−0.709010	+	0.705198i	$0.750858\pi$
$524$	0	0
$525$	6.63430	0.289544
$526$	0	0
$527$	−16.7374	−0.729091
$528$	0	0
$529$	−22.9824	−0.999234
$530$	0	0
$531$	3.87421	0.168126
$532$	0	0
$533$	7.60006	0.329195
$534$	0	0
$535$	−35.3013	−1.52621
$536$	0	0
$537$	4.54020	0.195924
$538$	0	0
$539$	15.3420	0.660828
$540$	0	0
$541$	26.0146	1.11846	0.559228	−	0.829014i	$-0.311098\pi$
$541$	26.0146	1.11846	0.559228	+	0.829014i	$0.311098\pi$
$542$	0	0
$543$	−23.7123	−1.01759
$544$	0	0
$545$	7.02097	0.300745
$546$	0	0
$547$	11.0583	0.472817	0.236409	−	0.971654i	$-0.424030\pi$
$547$	11.0583	0.472817	0.236409	+	0.971654i	$0.424030\pi$
$548$	0	0
$549$	−7.60878	−0.324735
$550$	0	0
$551$	−20.0204	−0.852896
$552$	0	0
$553$	−3.66120	−0.155690
$554$	0	0
$555$	−19.4314	−0.824815
$556$	0	0
$557$	0.254251	0.0107730	0.00538649	−	0.999985i	$-0.498285\pi$
$557$	0.254251	0.0107730	0.00538649	+	0.999985i	$0.498285\pi$
$558$	0	0
$559$	24.1905	1.02315
$560$	0	0
$561$	15.2899	0.645538
$562$	0	0
$563$	−7.84777	−0.330744	−0.165372	−	0.986231i	$-0.552882\pi$
$563$	−7.84777	−0.330744	−0.165372	+	0.986231i	$0.552882\pi$
$564$	0	0
$565$	−79.7444	−3.35487
$566$	0	0
$567$	−0.513502	−0.0215651
$568$	0	0
$569$	−1.88432	−0.0789950	−0.0394975	−	0.999220i	$-0.512576\pi$
$569$	−1.88432	−0.0789950	−0.0394975	+	0.999220i	$0.512576\pi$
$570$	0	0
$571$	−20.7521	−0.868449	−0.434225	−	0.900805i	$-0.642978\pi$
$571$	−20.7521	−0.868449	−0.434225	+	0.900805i	$0.642978\pi$
$572$	0	0
$573$	15.1643	0.633496
$574$	0	0
$575$	1.71436	0.0714937
$576$	0	0
$577$	−8.99037	−0.374274	−0.187137	−	0.982334i	$-0.559921\pi$
$577$	−8.99037	−0.374274	−0.187137	+	0.982334i	$0.559921\pi$
$578$	0	0
$579$	−19.0147	−0.790224
$580$	0	0
$581$	−2.49592	−0.103548
$582$	0	0
$583$	10.2983	0.426513
$584$	0	0
$585$	12.0918	0.499936
$586$	0	0
$587$	−41.2287	−1.70169	−0.850845	−	0.525417i	$-0.823909\pi$
$587$	−41.2287	−1.70169	−0.850845	+	0.525417i	$0.823909\pi$
$588$	0	0
$589$	−16.8671	−0.694996
$590$	0	0
$591$	−18.1286	−0.745710
$592$	0	0
$593$	−19.6634	−0.807478	−0.403739	−	0.914874i	$-0.632290\pi$
$593$	−19.6634	−0.807478	−0.403739	+	0.914874i	$0.632290\pi$
$594$	0	0
$595$	−14.5932	−0.598263
$596$	0	0
$597$	19.1556	0.783987
$598$	0	0
$599$	−6.97634	−0.285046	−0.142523	−	0.989792i	$-0.545521\pi$
$599$	−6.97634	−0.285046	−0.142523	+	0.989792i	$0.545521\pi$
$600$	0	0
$601$	15.7559	0.642699	0.321349	−	0.946961i	$-0.395864\pi$
$601$	15.7559	0.642699	0.321349	+	0.946961i	$0.395864\pi$
$602$	0	0
$603$	−13.7973	−0.561869
$604$	0	0
$605$	−24.6072	−1.00042
$606$	0	0
$607$	11.2950	0.458451	0.229226	−	0.973373i	$-0.426381\pi$
$607$	11.2950	0.458451	0.229226	+	0.973373i	$0.426381\pi$
$608$	0	0
$609$	−1.51956	−0.0615758
$610$	0	0
$611$	−15.3717	−0.621871
$612$	0	0
$613$	28.3623	1.14554	0.572771	−	0.819715i	$-0.305868\pi$
$613$	28.3623	1.14554	0.572771	+	0.819715i	$0.305868\pi$
$614$	0	0
$615$	−11.2630	−0.454170
$616$	0	0
$617$	10.1598	0.409019	0.204510	−	0.978865i	$-0.434440\pi$
$617$	10.1598	0.409019	0.204510	+	0.978865i	$0.434440\pi$
$618$	0	0
$619$	31.8530	1.28028	0.640140	−	0.768258i	$-0.278876\pi$
$619$	31.8530	1.28028	0.640140	+	0.768258i	$0.278876\pi$
$620$	0	0
$621$	−0.132693	−0.00532480
$622$	0	0
$623$	2.01924	0.0808992
$624$	0	0
$625$	77.3204	3.09282
$626$	0	0
$627$	15.4083	0.615350
$628$	0	0
$629$	30.8163	1.22873
$630$	0	0
$631$	18.9446	0.754171	0.377086	−	0.926178i	$-0.376926\pi$
$631$	18.9446	0.754171	0.377086	+	0.926178i	$0.376926\pi$
$632$	0	0
$633$	−9.90211	−0.393574
$634$	0	0
$635$	62.8417	2.49380
$636$	0	0
$637$	−19.2419	−0.762394
$638$	0	0
$639$	−7.80452	−0.308742
$640$	0	0
$641$	−6.95078	−0.274539	−0.137270	−	0.990534i	$-0.543833\pi$
$641$	−6.95078	−0.274539	−0.137270	+	0.990534i	$0.543833\pi$
$642$	0	0
$643$	23.6291	0.931840	0.465920	−	0.884827i	$-0.345724\pi$
$643$	23.6291	0.931840	0.465920	+	0.884827i	$0.345724\pi$
$644$	0	0
$645$	−35.8496	−1.41158
$646$	0	0
$647$	21.3303	0.838583	0.419291	−	0.907852i	$-0.362279\pi$
$647$	21.3303	0.838583	0.419291	+	0.907852i	$0.362279\pi$
$648$	0	0
$649$	−8.82356	−0.346355
$650$	0	0
$651$	−1.28023	−0.0501760
$652$	0	0
$653$	−29.7170	−1.16292	−0.581458	−	0.813576i	$-0.697518\pi$
$653$	−29.7170	−1.16292	−0.581458	+	0.813576i	$0.697518\pi$
$654$	0	0
$655$	62.3191	2.43501
$656$	0	0
$657$	7.62794	0.297594
$658$	0	0
$659$	−8.02020	−0.312423	−0.156211	−	0.987724i	$-0.549928\pi$
$659$	−8.02020	−0.312423	−0.156211	+	0.987724i	$0.549928\pi$
$660$	0	0
$661$	23.6408	0.919519	0.459760	−	0.888043i	$-0.347936\pi$
$661$	23.6408	0.919519	0.459760	+	0.888043i	$0.347936\pi$
$662$	0	0
$663$	−19.1765	−0.744754
$664$	0	0
$665$	−14.7063	−0.570285
$666$	0	0
$667$	−0.392668	−0.0152042
$668$	0	0
$669$	−1.53797	−0.0594613
$670$	0	0
$671$	17.3291	0.668982
$672$	0	0
$673$	2.82968	0.109076	0.0545382	−	0.998512i	$-0.482631\pi$
$673$	2.82968	0.109076	0.0545382	+	0.998512i	$0.482631\pi$
$674$	0	0
$675$	−12.9197	−0.497280
$676$	0	0
$677$	−47.5729	−1.82838	−0.914188	−	0.405289i	$-0.867171\pi$
$677$	−47.5729	−1.82838	−0.914188	+	0.405289i	$0.867171\pi$
$678$	0	0
$679$	−2.81692	−0.108103
$680$	0	0
$681$	−20.3326	−0.779148
$682$	0	0
$683$	9.82195	0.375827	0.187913	−	0.982186i	$-0.439828\pi$
$683$	9.82195	0.375827	0.187913	+	0.982186i	$0.439828\pi$
$684$	0	0
$685$	52.2680	1.99706
$686$	0	0
$687$	4.78969	0.182738
$688$	0	0
$689$	−12.9161	−0.492066
$690$	0	0
$691$	−36.8214	−1.40075	−0.700375	−	0.713775i	$-0.746984\pi$
$691$	−36.8214	−1.40075	−0.700375	+	0.713775i	$0.746984\pi$
$692$	0	0
$693$	1.16951	0.0444259
$694$	0	0
$695$	−68.6936	−2.60570
$696$	0	0
$697$	17.8621	0.676577
$698$	0	0
$699$	25.6466	0.970045
$700$	0	0
$701$	4.73554	0.178859	0.0894295	−	0.995993i	$-0.471496\pi$
$701$	4.73554	0.178859	0.0894295	+	0.995993i	$0.471496\pi$
$702$	0	0
$703$	31.0551	1.17127
$704$	0	0
$705$	22.7803	0.857957
$706$	0	0
$707$	−9.66792	−0.363599
$708$	0	0
$709$	10.7146	0.402395	0.201197	−	0.979551i	$-0.435517\pi$
$709$	10.7146	0.402395	0.201197	+	0.979551i	$0.435517\pi$
$710$	0	0
$711$	7.12986	0.267391
$712$	0	0
$713$	−0.330821	−0.0123893
$714$	0	0
$715$	−27.5393	−1.02991
$716$	0	0
$717$	4.90548	0.183199
$718$	0	0
$719$	14.3380	0.534718	0.267359	−	0.963597i	$-0.413849\pi$
$719$	14.3380	0.534718	0.267359	+	0.963597i	$0.413849\pi$
$720$	0	0
$721$	1.72133	0.0641058
$722$	0	0
$723$	7.32776	0.272522
$724$	0	0
$725$	−38.2322	−1.41991
$726$	0	0
$727$	−28.1467	−1.04390	−0.521951	−	0.852975i	$-0.674796\pi$
$727$	−28.1467	−1.04390	−0.521951	+	0.852975i	$0.674796\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	56.8541	2.10282
$732$	0	0
$733$	18.1781	0.671424	0.335712	−	0.941965i	$-0.391023\pi$
$733$	18.1781	0.671424	0.335712	+	0.941965i	$0.391023\pi$
$734$	0	0
$735$	28.5160	1.05183
$736$	0	0
$737$	31.4235	1.15750
$738$	0	0
$739$	50.7506	1.86689	0.933446	−	0.358717i	$-0.116786\pi$
$739$	50.7506	1.86689	0.933446	+	0.358717i	$0.116786\pi$
$740$	0	0
$741$	−19.3251	−0.709926
$742$	0	0
$743$	−35.3606	−1.29725	−0.648627	−	0.761106i	$-0.724657\pi$
$743$	−35.3606	−1.29725	−0.648627	+	0.761106i	$0.724657\pi$
$744$	0	0
$745$	13.3850	0.490390
$746$	0	0
$747$	4.86059	0.177840
$748$	0	0
$749$	4.28221	0.156468
$750$	0	0
$751$	27.3273	0.997186	0.498593	−	0.866836i	$-0.333850\pi$
$751$	27.3273	0.997186	0.498593	+	0.866836i	$0.333850\pi$
$752$	0	0
$753$	17.8427	0.650223
$754$	0	0
$755$	22.3960	0.815074
$756$	0	0
$757$	18.0926	0.657586	0.328793	−	0.944402i	$-0.393358\pi$
$757$	18.0926	0.657586	0.328793	+	0.944402i	$0.393358\pi$
$758$	0	0
$759$	0.302210	0.0109695
$760$	0	0
$761$	−2.25721	−0.0818239	−0.0409119	−	0.999163i	$-0.513026\pi$
$761$	−2.25721	−0.0818239	−0.0409119	+	0.999163i	$0.513026\pi$
$762$	0	0
$763$	−0.851674	−0.0308327
$764$	0	0
$765$	28.4190	1.02749
$766$	0	0
$767$	11.0665	0.399588
$768$	0	0
$769$	−53.6333	−1.93407	−0.967034	−	0.254647i	$-0.918041\pi$
$769$	−53.6333	−1.93407	−0.967034	+	0.254647i	$0.918041\pi$
$770$	0	0
$771$	−28.7648	−1.03594
$772$	0	0
$773$	−43.5678	−1.56702	−0.783512	−	0.621377i	$-0.786574\pi$
$773$	−43.5678	−1.56702	−0.783512	+	0.621377i	$0.786574\pi$
$774$	0	0
$775$	−32.2105	−1.15703
$776$	0	0
$777$	2.35711	0.0845608
$778$	0	0
$779$	18.0006	0.644937
$780$	0	0
$781$	17.7749	0.636035
$782$	0	0
$783$	2.95922	0.105754
$784$	0	0
$785$	−56.9438	−2.03241
$786$	0	0
$787$	−37.0799	−1.32175	−0.660877	−	0.750494i	$-0.729816\pi$
$787$	−37.0799	−1.32175	−0.660877	+	0.750494i	$0.729816\pi$
$788$	0	0
$789$	−29.2520	−1.04140
$790$	0	0
$791$	9.67334	0.343945
$792$	0	0
$793$	−21.7341	−0.771800
$794$	0	0
$795$	19.1413	0.678873
$796$	0	0
$797$	−31.0311	−1.09918	−0.549589	−	0.835435i	$-0.685216\pi$
$797$	−31.0311	−1.09918	−0.549589	+	0.835435i	$0.685216\pi$
$798$	0	0
$799$	−36.1275	−1.27810
$800$	0	0
$801$	−3.93230	−0.138941
$802$	0	0
$803$	−17.3727	−0.613070
$804$	0	0
$805$	−0.288441	−0.0101662
$806$	0	0
$807$	29.7538	1.04738
$808$	0	0
$809$	8.05483	0.283193	0.141596	−	0.989924i	$-0.454777\pi$
$809$	8.05483	0.283193	0.141596	+	0.989924i	$0.454777\pi$
$810$	0	0
$811$	−12.7680	−0.448344	−0.224172	−	0.974550i	$-0.571968\pi$
$811$	−12.7680	−0.448344	−0.224172	+	0.974550i	$0.571968\pi$
$812$	0	0
$813$	−20.5913	−0.722170
$814$	0	0
$815$	−93.4681	−3.27405
$816$	0	0
$817$	57.2946	2.00449
$818$	0	0
$819$	−1.46679	−0.0512539
$820$	0	0
$821$	13.2596	0.462762	0.231381	−	0.972863i	$-0.425676\pi$
$821$	13.2596	0.462762	0.231381	+	0.972863i	$0.425676\pi$
$822$	0	0
$823$	−21.3373	−0.743772	−0.371886	−	0.928278i	$-0.621289\pi$
$823$	−21.3373	−0.743772	−0.371886	+	0.928278i	$0.621289\pi$
$824$	0	0
$825$	29.4248	1.02444
$826$	0	0
$827$	28.9938	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$827$	28.9938	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$828$	0	0
$829$	−1.78266	−0.0619142	−0.0309571	−	0.999521i	$-0.509856\pi$
$829$	−1.78266	−0.0619142	−0.0309571	+	0.999521i	$0.509856\pi$
$830$	0	0
$831$	−0.219899	−0.00762820
$832$	0	0
$833$	−45.2236	−1.56691
$834$	0	0
$835$	50.8806	1.76080
$836$	0	0
$837$	2.49313	0.0861751
$838$	0	0
$839$	−0.548927	−0.0189510	−0.00947552	−	0.999955i	$-0.503016\pi$
$839$	−0.548927	−0.0189510	−0.00947552	+	0.999955i	$0.503016\pi$
$840$	0	0
$841$	−20.2430	−0.698036
$842$	0	0
$843$	24.8664	0.856445
$844$	0	0
$845$	−20.4915	−0.704928
$846$	0	0
$847$	2.98496	0.102564
$848$	0	0
$849$	1.32266	0.0453936
$850$	0	0
$851$	0.609097	0.0208796
$852$	0	0
$853$	47.0915	1.61238	0.806191	−	0.591656i	$-0.201525\pi$
$853$	47.0915	1.61238	0.806191	+	0.591656i	$0.201525\pi$
$854$	0	0
$855$	28.6392	0.979440
$856$	0	0
$857$	5.42610	0.185352	0.0926761	−	0.995696i	$-0.470458\pi$
$857$	5.42610	0.185352	0.0926761	+	0.995696i	$0.470458\pi$
$858$	0	0
$859$	14.4043	0.491468	0.245734	−	0.969337i	$-0.420971\pi$
$859$	14.4043	0.491468	0.245734	+	0.969337i	$0.420971\pi$
$860$	0	0
$861$	1.36626	0.0465619
$862$	0	0
$863$	14.7519	0.502160	0.251080	−	0.967966i	$-0.419214\pi$
$863$	14.7519	0.502160	0.251080	+	0.967966i	$0.419214\pi$
$864$	0	0
$865$	−64.6607	−2.19853
$866$	0	0
$867$	−28.0698	−0.953301
$868$	0	0
$869$	−16.2383	−0.550848
$870$	0	0
$871$	−39.4113	−1.33540
$872$	0	0
$873$	5.48570	0.185663
$874$	0	0
$875$	−17.2154	−0.581987
$876$	0	0
$877$	11.4224	0.385708	0.192854	−	0.981227i	$-0.438226\pi$
$877$	11.4224	0.385708	0.192854	+	0.981227i	$0.438226\pi$
$878$	0	0
$879$	17.9775	0.606364
$880$	0	0
$881$	−13.4799	−0.454151	−0.227075	−	0.973877i	$-0.572916\pi$
$881$	−13.4799	−0.454151	−0.227075	+	0.973877i	$0.572916\pi$
$882$	0	0
$883$	−47.7047	−1.60539	−0.802695	−	0.596390i	$-0.796601\pi$
$883$	−47.7047	−1.60539	−0.802695	+	0.596390i	$0.796601\pi$
$884$	0	0
$885$	−16.4002	−0.551286
$886$	0	0
$887$	−34.2915	−1.15140	−0.575699	−	0.817662i	$-0.695270\pi$
$887$	−34.2915	−1.15140	−0.575699	+	0.817662i	$0.695270\pi$
$888$	0	0
$889$	−7.62297	−0.255666
$890$	0	0
$891$	−2.27751	−0.0762995
$892$	0	0
$893$	−36.4074	−1.21833
$894$	0	0
$895$	−19.2194	−0.642435
$896$	0	0
$897$	−0.379032	−0.0126555
$898$	0	0
$899$	7.37770	0.246060
$900$	0	0
$901$	−30.3563	−1.01132
$902$	0	0
$903$	4.34871	0.144716
$904$	0	0
$905$	100.378	3.33669
$906$	0	0
$907$	−32.5922	−1.08221	−0.541103	−	0.840956i	$-0.681993\pi$
$907$	−32.5922	−1.08221	−0.541103	+	0.840956i	$0.681993\pi$
$908$	0	0
$909$	18.8274	0.624466
$910$	0	0
$911$	25.8652	0.856951	0.428476	−	0.903553i	$-0.359051\pi$
$911$	25.8652	0.856951	0.428476	+	0.903553i	$0.359051\pi$
$912$	0	0
$913$	−11.0701	−0.366365
$914$	0	0
$915$	32.2092	1.06480
$916$	0	0
$917$	−7.55958	−0.249639
$918$	0	0
$919$	17.4233	0.574742	0.287371	−	0.957819i	$-0.407219\pi$
$919$	17.4233	0.574742	0.287371	+	0.957819i	$0.407219\pi$
$920$	0	0
$921$	−6.99657	−0.230545
$922$	0	0
$923$	−22.2932	−0.733790
$924$	0	0
$925$	59.3049	1.94993
$926$	0	0
$927$	−3.35215	−0.110099
$928$	0	0
$929$	−16.5349	−0.542494	−0.271247	−	0.962510i	$-0.587436\pi$
$929$	−16.5349	−0.542494	−0.271247	+	0.962510i	$0.587436\pi$
$930$	0	0
$931$	−45.5741	−1.49363
$932$	0	0
$933$	11.1497	0.365026
$934$	0	0
$935$	−64.7245	−2.11672
$936$	0	0
$937$	−39.5059	−1.29060	−0.645301	−	0.763929i	$-0.723268\pi$
$937$	−39.5059	−1.29060	−0.645301	+	0.763929i	$0.723268\pi$
$938$	0	0
$939$	−8.43088	−0.275131
$940$	0	0
$941$	−14.3452	−0.467641	−0.233820	−	0.972280i	$-0.575123\pi$
$941$	−14.3452	−0.467641	−0.233820	+	0.972280i	$0.575123\pi$
$942$	0	0
$943$	0.353052	0.0114970
$944$	0	0
$945$	2.17374	0.0707118
$946$	0	0
$947$	−0.178723	−0.00580772	−0.00290386	−	0.999996i	$-0.500924\pi$
$947$	−0.178723	−0.00580772	−0.00290386	+	0.999996i	$0.500924\pi$
$948$	0	0
$949$	21.7888	0.707296
$950$	0	0
$951$	−26.5152	−0.859815
$952$	0	0
$953$	−21.6960	−0.702804	−0.351402	−	0.936225i	$-0.614295\pi$
$953$	−21.6960	−0.702804	−0.351402	+	0.936225i	$0.614295\pi$
$954$	0	0
$955$	−64.1929	−2.07723
$956$	0	0
$957$	−6.73965	−0.217862
$958$	0	0
$959$	−6.34034	−0.204740
$960$	0	0
$961$	−24.7843	−0.799494
$962$	0	0
$963$	−8.33922	−0.268728
$964$	0	0
$965$	80.4925	2.59114
$966$	0	0
$967$	−13.4825	−0.433568	−0.216784	−	0.976220i	$-0.569557\pi$
$967$	−13.4825	−0.433568	−0.216784	+	0.976220i	$0.569557\pi$
$968$	0	0
$969$	−45.4191	−1.45907
$970$	0	0
$971$	−34.2465	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$971$	−34.2465	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$972$	0	0
$973$	8.33283	0.267138
$974$	0	0
$975$	−36.9045	−1.18189
$976$	0	0
$977$	20.9575	0.670490	0.335245	−	0.942131i	$-0.391181\pi$
$977$	20.9575	0.670490	0.335245	+	0.942131i	$0.391181\pi$
$978$	0	0
$979$	8.95585	0.286230
$980$	0	0
$981$	1.65856	0.0529538
$982$	0	0
$983$	−21.7622	−0.694107	−0.347054	−	0.937845i	$-0.612818\pi$
$983$	−21.7622	−0.694107	−0.347054	+	0.937845i	$0.612818\pi$
$984$	0	0
$985$	76.7413	2.44518
$986$	0	0
$987$	−2.76336	−0.0879586
$988$	0	0
$989$	1.12374	0.0357330
$990$	0	0
$991$	29.3159	0.931251	0.465626	−	0.884982i	$-0.345829\pi$
$991$	29.3159	0.931251	0.465626	+	0.884982i	$0.345829\pi$
$992$	0	0
$993$	4.49483	0.142639
$994$	0	0
$995$	−81.0889	−2.57069
$996$	0	0
$997$	−11.3722	−0.360160	−0.180080	−	0.983652i	$-0.557636\pi$
$997$	−11.3722	−0.360160	−0.180080	+	0.983652i	$0.557636\pi$
$998$	0	0
$999$	−4.59026	−0.145229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.26	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.26	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.23317 q^{5} -0.513502 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.23317 q^{5} -0.513502 q^{7} +1.00000 q^{9} -2.27751 q^{11} +2.85645 q^{13} -4.23317 q^{15} +6.71341 q^{17} +6.76543 q^{19} +0.513502 q^{21} +0.132693 q^{23} +12.9197 q^{25} -1.00000 q^{27} -2.95922 q^{29} -2.49313 q^{31} +2.27751 q^{33} -2.17374 q^{35} +4.59026 q^{37} -2.85645 q^{39} +2.66067 q^{41} +8.46873 q^{43} +4.23317 q^{45} -5.38139 q^{47} -6.73632 q^{49} -6.71341 q^{51} -4.52175 q^{53} -9.64109 q^{55} -6.76543 q^{57} +3.87421 q^{59} -7.60878 q^{61} -0.513502 q^{63} +12.0918 q^{65} -13.7973 q^{67} -0.132693 q^{69} -7.80452 q^{71} +7.62794 q^{73} -12.9197 q^{75} +1.16951 q^{77} +7.12986 q^{79} +1.00000 q^{81} +4.86059 q^{83} +28.4190 q^{85} +2.95922 q^{87} -3.93230 q^{89} -1.46679 q^{91} +2.49313 q^{93} +28.6392 q^{95} +5.48570 q^{97} -2.27751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more