LMFDB - Embedded newform 6036.2.a.i.1.3

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			1.3
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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00702 q^{5} +4.42560 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.00702 q^{5} +4.42560 q^{7} +1.00000 q^{9} -1.26137 q^{11} -7.02573 q^{13} +3.00702 q^{15} +4.46748 q^{17} -6.47908 q^{19} -4.42560 q^{21} -5.72649 q^{23} +4.04214 q^{25} -1.00000 q^{27} -5.47114 q^{29} +3.83015 q^{31} +1.26137 q^{33} -13.3078 q^{35} +1.66867 q^{37} +7.02573 q^{39} +3.44867 q^{41} +7.06939 q^{43} -3.00702 q^{45} +2.66617 q^{47} +12.5859 q^{49} -4.46748 q^{51} -11.5714 q^{53} +3.79297 q^{55} +6.47908 q^{57} +8.23464 q^{59} +12.8932 q^{61} +4.42560 q^{63} +21.1265 q^{65} +7.21874 q^{67} +5.72649 q^{69} +3.73003 q^{71} -13.1020 q^{73} -4.04214 q^{75} -5.58233 q^{77} -5.90073 q^{79} +1.00000 q^{81} +2.99590 q^{83} -13.4338 q^{85} +5.47114 q^{87} -14.3130 q^{89} -31.0931 q^{91} -3.83015 q^{93} +19.4827 q^{95} -9.56984 q^{97} -1.26137 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.00702	−1.34478	−0.672389	−	0.740198i	$-0.734732\pi$
$5$	−3.00702	−1.34478	−0.672389	+	0.740198i	$0.734732\pi$
$6$	0	0
$7$	4.42560	1.67272	0.836359	−	0.548182i	$-0.184680\pi$
$7$	4.42560	1.67272	0.836359	+	0.548182i	$0.184680\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.26137	−0.380318	−0.190159	−	0.981753i	$-0.560900\pi$
$11$	−1.26137	−0.380318	−0.190159	+	0.981753i	$0.560900\pi$
$12$	0	0
$13$	−7.02573	−1.94859	−0.974294	−	0.225281i	$-0.927670\pi$
$13$	−7.02573	−1.94859	−0.974294	+	0.225281i	$0.927670\pi$
$14$	0	0
$15$	3.00702	0.776408
$16$	0	0
$17$	4.46748	1.08352	0.541761	−	0.840533i	$-0.317758\pi$
$17$	4.46748	1.08352	0.541761	+	0.840533i	$0.317758\pi$
$18$	0	0
$19$	−6.47908	−1.48640	−0.743202	−	0.669067i	$-0.766694\pi$
$19$	−6.47908	−1.48640	−0.743202	+	0.669067i	$0.766694\pi$
$20$	0	0
$21$	−4.42560	−0.965744
$22$	0	0
$23$	−5.72649	−1.19406	−0.597028	−	0.802221i	$-0.703652\pi$
$23$	−5.72649	−1.19406	−0.597028	+	0.802221i	$0.703652\pi$
$24$	0	0
$25$	4.04214	0.808428
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.47114	−1.01597	−0.507983	−	0.861367i	$-0.669609\pi$
$29$	−5.47114	−1.01597	−0.507983	+	0.861367i	$0.669609\pi$
$30$	0	0
$31$	3.83015	0.687915	0.343958	−	0.938985i	$-0.388232\pi$
$31$	3.83015	0.687915	0.343958	+	0.938985i	$0.388232\pi$
$32$	0	0
$33$	1.26137	0.219577
$34$	0	0
$35$	−13.3078	−2.24943
$36$	0	0
$37$	1.66867	0.274328	0.137164	−	0.990548i	$-0.456201\pi$
$37$	1.66867	0.274328	0.137164	+	0.990548i	$0.456201\pi$
$38$	0	0
$39$	7.02573	1.12502
$40$	0	0
$41$	3.44867	0.538591	0.269296	−	0.963058i	$-0.413209\pi$
$41$	3.44867	0.538591	0.269296	+	0.963058i	$0.413209\pi$
$42$	0	0
$43$	7.06939	1.07807	0.539036	−	0.842283i	$-0.318789\pi$
$43$	7.06939	1.07807	0.539036	+	0.842283i	$0.318789\pi$
$44$	0	0
$45$	−3.00702	−0.448259
$46$	0	0
$47$	2.66617	0.388900	0.194450	−	0.980912i	$-0.437708\pi$
$47$	2.66617	0.388900	0.194450	+	0.980912i	$0.437708\pi$
$48$	0	0
$49$	12.5859	1.79799
$50$	0	0
$51$	−4.46748	−0.625572
$52$	0	0
$53$	−11.5714	−1.58946	−0.794729	−	0.606964i	$-0.792387\pi$
$53$	−11.5714	−1.58946	−0.794729	+	0.606964i	$0.792387\pi$
$54$	0	0
$55$	3.79297	0.511444
$56$	0	0
$57$	6.47908	0.858176
$58$	0	0
$59$	8.23464	1.07206	0.536029	−	0.844199i	$-0.319924\pi$
$59$	8.23464	1.07206	0.536029	+	0.844199i	$0.319924\pi$
$60$	0	0
$61$	12.8932	1.65081	0.825403	−	0.564544i	$-0.190948\pi$
$61$	12.8932	1.65081	0.825403	+	0.564544i	$0.190948\pi$
$62$	0	0
$63$	4.42560	0.557573
$64$	0	0
$65$	21.1265	2.62042
$66$	0	0
$67$	7.21874	0.881909	0.440954	−	0.897529i	$-0.354640\pi$
$67$	7.21874	0.881909	0.440954	+	0.897529i	$0.354640\pi$
$68$	0	0
$69$	5.72649	0.689388
$70$	0	0
$71$	3.73003	0.442673	0.221336	−	0.975198i	$-0.428958\pi$
$71$	3.73003	0.442673	0.221336	+	0.975198i	$0.428958\pi$
$72$	0	0
$73$	−13.1020	−1.53347	−0.766735	−	0.641964i	$-0.778120\pi$
$73$	−13.1020	−1.53347	−0.766735	+	0.641964i	$0.778120\pi$
$74$	0	0
$75$	−4.04214	−0.466746
$76$	0	0
$77$	−5.58233	−0.636165
$78$	0	0
$79$	−5.90073	−0.663884	−0.331942	−	0.943300i	$-0.607704\pi$
$79$	−5.90073	−0.663884	−0.331942	+	0.943300i	$0.607704\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.99590	0.328843	0.164421	−	0.986390i	$-0.447424\pi$
$83$	2.99590	0.328843	0.164421	+	0.986390i	$0.447424\pi$
$84$	0	0
$85$	−13.4338	−1.45710
$86$	0	0
$87$	5.47114	0.586568
$88$	0	0
$89$	−14.3130	−1.51717	−0.758586	−	0.651573i	$-0.774109\pi$
$89$	−14.3130	−1.51717	−0.758586	+	0.651573i	$0.774109\pi$
$90$	0	0
$91$	−31.0931	−3.25944
$92$	0	0
$93$	−3.83015	−0.397168
$94$	0	0
$95$	19.4827	1.99888
$96$	0	0
$97$	−9.56984	−0.971670	−0.485835	−	0.874051i	$-0.661484\pi$
$97$	−9.56984	−0.971670	−0.485835	+	0.874051i	$0.661484\pi$
$98$	0	0
$99$	−1.26137	−0.126773
$100$	0	0
$101$	−11.6943	−1.16363	−0.581814	−	0.813322i	$-0.697657\pi$
$101$	−11.6943	−1.16363	−0.581814	+	0.813322i	$0.697657\pi$
$102$	0	0
$103$	0.703986	0.0693658	0.0346829	−	0.999398i	$-0.488958\pi$
$103$	0.703986	0.0693658	0.0346829	+	0.999398i	$0.488958\pi$
$104$	0	0
$105$	13.3078	1.29871
$106$	0	0
$107$	−9.02064	−0.872058	−0.436029	−	0.899933i	$-0.643615\pi$
$107$	−9.02064	−0.872058	−0.436029	+	0.899933i	$0.643615\pi$
$108$	0	0
$109$	16.0323	1.53561	0.767806	−	0.640683i	$-0.221349\pi$
$109$	16.0323	1.53561	0.767806	+	0.640683i	$0.221349\pi$
$110$	0	0
$111$	−1.66867	−0.158383
$112$	0	0
$113$	−9.36851	−0.881315	−0.440658	−	0.897675i	$-0.645255\pi$
$113$	−9.36851	−0.881315	−0.440658	+	0.897675i	$0.645255\pi$
$114$	0	0
$115$	17.2196	1.60574
$116$	0	0
$117$	−7.02573	−0.649529
$118$	0	0
$119$	19.7712	1.81243
$120$	0	0
$121$	−9.40894	−0.855358
$122$	0	0
$123$	−3.44867	−0.310956
$124$	0	0
$125$	2.88030	0.257622
$126$	0	0
$127$	1.24996	0.110916	0.0554581	−	0.998461i	$-0.482338\pi$
$127$	1.24996	0.110916	0.0554581	+	0.998461i	$0.482338\pi$
$128$	0	0
$129$	−7.06939	−0.622425
$130$	0	0
$131$	13.7936	1.20516	0.602578	−	0.798060i	$-0.294140\pi$
$131$	13.7936	1.20516	0.602578	+	0.798060i	$0.294140\pi$
$132$	0	0
$133$	−28.6738	−2.48633
$134$	0	0
$135$	3.00702	0.258803
$136$	0	0
$137$	−17.0579	−1.45735	−0.728676	−	0.684858i	$-0.759864\pi$
$137$	−17.0579	−1.45735	−0.728676	+	0.684858i	$0.759864\pi$
$138$	0	0
$139$	−5.40359	−0.458327	−0.229163	−	0.973388i	$-0.573599\pi$
$139$	−5.40359	−0.458327	−0.229163	+	0.973388i	$0.573599\pi$
$140$	0	0
$141$	−2.66617	−0.224532
$142$	0	0
$143$	8.86207	0.741084
$144$	0	0
$145$	16.4518	1.36625
$146$	0	0
$147$	−12.5859	−1.03807
$148$	0	0
$149$	3.74723	0.306985	0.153492	−	0.988150i	$-0.450948\pi$
$149$	3.74723	0.306985	0.153492	+	0.988150i	$0.450948\pi$
$150$	0	0
$151$	23.7345	1.93149	0.965743	−	0.259500i	$-0.0835576\pi$
$151$	23.7345	1.93149	0.965743	+	0.259500i	$0.0835576\pi$
$152$	0	0
$153$	4.46748	0.361174
$154$	0	0
$155$	−11.5173	−0.925093
$156$	0	0
$157$	23.3909	1.86679	0.933397	−	0.358846i	$-0.116830\pi$
$157$	23.3909	1.86679	0.933397	+	0.358846i	$0.116830\pi$
$158$	0	0
$159$	11.5714	0.917674
$160$	0	0
$161$	−25.3431	−1.99732
$162$	0	0
$163$	13.5342	1.06008	0.530042	−	0.847971i	$-0.322176\pi$
$163$	13.5342	1.06008	0.530042	+	0.847971i	$0.322176\pi$
$164$	0	0
$165$	−3.79297	−0.295282
$166$	0	0
$167$	10.5120	0.813445	0.406723	−	0.913552i	$-0.366672\pi$
$167$	10.5120	0.813445	0.406723	+	0.913552i	$0.366672\pi$
$168$	0	0
$169$	36.3609	2.79699
$170$	0	0
$171$	−6.47908	−0.495468
$172$	0	0
$173$	11.4272	0.868796	0.434398	−	0.900721i	$-0.356961\pi$
$173$	11.4272	0.868796	0.434398	+	0.900721i	$0.356961\pi$
$174$	0	0
$175$	17.8889	1.35227
$176$	0	0
$177$	−8.23464	−0.618953
$178$	0	0
$179$	−22.8420	−1.70729	−0.853646	−	0.520854i	$-0.825614\pi$
$179$	−22.8420	−1.70729	−0.853646	+	0.520854i	$0.825614\pi$
$180$	0	0
$181$	6.70936	0.498703	0.249351	−	0.968413i	$-0.419783\pi$
$181$	6.70936	0.498703	0.249351	+	0.968413i	$0.419783\pi$
$182$	0	0
$183$	−12.8932	−0.953093
$184$	0	0
$185$	−5.01771	−0.368910
$186$	0	0
$187$	−5.63516	−0.412083
$188$	0	0
$189$	−4.42560	−0.321915
$190$	0	0
$191$	−7.17863	−0.519427	−0.259714	−	0.965686i	$-0.583628\pi$
$191$	−7.17863	−0.519427	−0.259714	+	0.965686i	$0.583628\pi$
$192$	0	0
$193$	19.2715	1.38719	0.693596	−	0.720364i	$-0.256025\pi$
$193$	19.2715	1.38719	0.693596	+	0.720364i	$0.256025\pi$
$194$	0	0
$195$	−21.1265	−1.51290
$196$	0	0
$197$	19.7295	1.40567	0.702833	−	0.711355i	$-0.251918\pi$
$197$	19.7295	1.40567	0.702833	+	0.711355i	$0.251918\pi$
$198$	0	0
$199$	10.7240	0.760204	0.380102	−	0.924945i	$-0.375889\pi$
$199$	10.7240	0.760204	0.380102	+	0.924945i	$0.375889\pi$
$200$	0	0
$201$	−7.21874	−0.509170
$202$	0	0
$203$	−24.2131	−1.69942
$204$	0	0
$205$	−10.3702	−0.724285
$206$	0	0
$207$	−5.72649	−0.398018
$208$	0	0
$209$	8.17254	0.565307
$210$	0	0
$211$	−23.1655	−1.59478	−0.797391	−	0.603463i	$-0.793787\pi$
$211$	−23.1655	−1.59478	−0.797391	+	0.603463i	$0.793787\pi$
$212$	0	0
$213$	−3.73003	−0.255577
$214$	0	0
$215$	−21.2578	−1.44977
$216$	0	0
$217$	16.9507	1.15069
$218$	0	0
$219$	13.1020	0.885349
$220$	0	0
$221$	−31.3873	−2.11134
$222$	0	0
$223$	19.5792	1.31112	0.655561	−	0.755142i	$-0.272432\pi$
$223$	19.5792	1.31112	0.655561	+	0.755142i	$0.272432\pi$
$224$	0	0
$225$	4.04214	0.269476
$226$	0	0
$227$	13.6014	0.902757	0.451379	−	0.892333i	$-0.350932\pi$
$227$	13.6014	0.902757	0.451379	+	0.892333i	$0.350932\pi$
$228$	0	0
$229$	−12.4302	−0.821413	−0.410706	−	0.911768i	$-0.634718\pi$
$229$	−12.4302	−0.821413	−0.410706	+	0.911768i	$0.634718\pi$
$230$	0	0
$231$	5.58233	0.367290
$232$	0	0
$233$	−11.2581	−0.737544	−0.368772	−	0.929520i	$-0.620222\pi$
$233$	−11.2581	−0.737544	−0.368772	+	0.929520i	$0.620222\pi$
$234$	0	0
$235$	−8.01721	−0.522985
$236$	0	0
$237$	5.90073	0.383293
$238$	0	0
$239$	23.0467	1.49076	0.745382	−	0.666637i	$-0.232267\pi$
$239$	23.0467	1.49076	0.745382	+	0.666637i	$0.232267\pi$
$240$	0	0
$241$	25.7686	1.65990	0.829951	−	0.557836i	$-0.188368\pi$
$241$	25.7686	1.65990	0.829951	+	0.557836i	$0.188368\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−37.8460	−2.41789
$246$	0	0
$247$	45.5203	2.89639
$248$	0	0
$249$	−2.99590	−0.189857
$250$	0	0
$251$	−4.22022	−0.266378	−0.133189	−	0.991091i	$-0.542522\pi$
$251$	−4.22022	−0.266378	−0.133189	+	0.991091i	$0.542522\pi$
$252$	0	0
$253$	7.22324	0.454121
$254$	0	0
$255$	13.4338	0.841255
$256$	0	0
$257$	−15.8165	−0.986607	−0.493304	−	0.869857i	$-0.664211\pi$
$257$	−15.8165	−0.986607	−0.493304	+	0.869857i	$0.664211\pi$
$258$	0	0
$259$	7.38486	0.458873
$260$	0	0
$261$	−5.47114	−0.338655
$262$	0	0
$263$	13.1279	0.809503	0.404752	−	0.914427i	$-0.367358\pi$
$263$	13.1279	0.809503	0.404752	+	0.914427i	$0.367358\pi$
$264$	0	0
$265$	34.7955	2.13747
$266$	0	0
$267$	14.3130	0.875939
$268$	0	0
$269$	7.27608	0.443630	0.221815	−	0.975089i	$-0.428802\pi$
$269$	7.27608	0.443630	0.221815	+	0.975089i	$0.428802\pi$
$270$	0	0
$271$	5.30166	0.322053	0.161026	−	0.986950i	$-0.448520\pi$
$271$	5.30166	0.322053	0.161026	+	0.986950i	$0.448520\pi$
$272$	0	0
$273$	31.0931	1.88184
$274$	0	0
$275$	−5.09865	−0.307460
$276$	0	0
$277$	−6.17368	−0.370940	−0.185470	−	0.982650i	$-0.559381\pi$
$277$	−6.17368	−0.370940	−0.185470	+	0.982650i	$0.559381\pi$
$278$	0	0
$279$	3.83015	0.229305
$280$	0	0
$281$	22.6254	1.34972	0.674860	−	0.737946i	$-0.264204\pi$
$281$	22.6254	1.34972	0.674860	+	0.737946i	$0.264204\pi$
$282$	0	0
$283$	6.53144	0.388254	0.194127	−	0.980976i	$-0.437813\pi$
$283$	6.53144	0.388254	0.194127	+	0.980976i	$0.437813\pi$
$284$	0	0
$285$	−19.4827	−1.15406
$286$	0	0
$287$	15.2624	0.900911
$288$	0	0
$289$	2.95834	0.174020
$290$	0	0
$291$	9.56984	0.560994
$292$	0	0
$293$	31.2931	1.82816	0.914082	−	0.405530i	$-0.132913\pi$
$293$	31.2931	1.82816	0.914082	+	0.405530i	$0.132913\pi$
$294$	0	0
$295$	−24.7617	−1.44168
$296$	0	0
$297$	1.26137	0.0731923
$298$	0	0
$299$	40.2328	2.32672
$300$	0	0
$301$	31.2863	1.80331
$302$	0	0
$303$	11.6943	0.671821
$304$	0	0
$305$	−38.7701	−2.21997
$306$	0	0
$307$	22.9963	1.31247	0.656235	−	0.754557i	$-0.272148\pi$
$307$	22.9963	1.31247	0.656235	+	0.754557i	$0.272148\pi$
$308$	0	0
$309$	−0.703986	−0.0400484
$310$	0	0
$311$	−23.7421	−1.34629	−0.673147	−	0.739509i	$-0.735058\pi$
$311$	−23.7421	−1.34629	−0.673147	+	0.739509i	$0.735058\pi$
$312$	0	0
$313$	17.5847	0.993947	0.496974	−	0.867766i	$-0.334445\pi$
$313$	17.5847	0.993947	0.496974	+	0.867766i	$0.334445\pi$
$314$	0	0
$315$	−13.3078	−0.749811
$316$	0	0
$317$	−34.1952	−1.92060	−0.960298	−	0.278977i	$-0.910005\pi$
$317$	−34.1952	−1.92060	−0.960298	+	0.278977i	$0.910005\pi$
$318$	0	0
$319$	6.90115	0.386390
$320$	0	0
$321$	9.02064	0.503483
$322$	0	0
$323$	−28.9451	−1.61055
$324$	0	0
$325$	−28.3990	−1.57529
$326$	0	0
$327$	−16.0323	−0.886586
$328$	0	0
$329$	11.7994	0.650521
$330$	0	0
$331$	−12.7003	−0.698069	−0.349034	−	0.937110i	$-0.613490\pi$
$331$	−12.7003	−0.698069	−0.349034	+	0.937110i	$0.613490\pi$
$332$	0	0
$333$	1.66867	0.0914425
$334$	0	0
$335$	−21.7068	−1.18597
$336$	0	0
$337$	−17.8065	−0.969983	−0.484991	−	0.874519i	$-0.661177\pi$
$337$	−17.8065	−0.969983	−0.484991	+	0.874519i	$0.661177\pi$
$338$	0	0
$339$	9.36851	0.508828
$340$	0	0
$341$	−4.83125	−0.261627
$342$	0	0
$343$	24.7209	1.33480
$344$	0	0
$345$	−17.2196	−0.927074
$346$	0	0
$347$	36.7852	1.97473	0.987366	−	0.158454i	$-0.0506509\pi$
$347$	36.7852	1.97473	0.987366	+	0.158454i	$0.0506509\pi$
$348$	0	0
$349$	−2.96397	−0.158658	−0.0793288	−	0.996849i	$-0.525278\pi$
$349$	−2.96397	−0.158658	−0.0793288	+	0.996849i	$0.525278\pi$
$350$	0	0
$351$	7.02573	0.375006
$352$	0	0
$353$	19.4758	1.03659	0.518297	−	0.855201i	$-0.326566\pi$
$353$	19.4758	1.03659	0.518297	+	0.855201i	$0.326566\pi$
$354$	0	0
$355$	−11.2163	−0.595297
$356$	0	0
$357$	−19.7712	−1.04641
$358$	0	0
$359$	−27.6093	−1.45716	−0.728581	−	0.684960i	$-0.759820\pi$
$359$	−27.6093	−1.45716	−0.728581	+	0.684960i	$0.759820\pi$
$360$	0	0
$361$	22.9785	1.20940
$362$	0	0
$363$	9.40894	0.493841
$364$	0	0
$365$	39.3978	2.06218
$366$	0	0
$367$	−5.26407	−0.274782	−0.137391	−	0.990517i	$-0.543872\pi$
$367$	−5.26407	−0.274782	−0.137391	+	0.990517i	$0.543872\pi$
$368$	0	0
$369$	3.44867	0.179530
$370$	0	0
$371$	−51.2105	−2.65871
$372$	0	0
$373$	16.3581	0.846988	0.423494	−	0.905899i	$-0.360803\pi$
$373$	16.3581	0.846988	0.423494	+	0.905899i	$0.360803\pi$
$374$	0	0
$375$	−2.88030	−0.148738
$376$	0	0
$377$	38.4388	1.97970
$378$	0	0
$379$	10.1367	0.520688	0.260344	−	0.965516i	$-0.416164\pi$
$379$	10.1367	0.520688	0.260344	+	0.965516i	$0.416164\pi$
$380$	0	0
$381$	−1.24996	−0.0640375
$382$	0	0
$383$	5.44289	0.278119	0.139059	−	0.990284i	$-0.455592\pi$
$383$	5.44289	0.278119	0.139059	+	0.990284i	$0.455592\pi$
$384$	0	0
$385$	16.7861	0.855501
$386$	0	0
$387$	7.06939	0.359357
$388$	0	0
$389$	34.0346	1.72562	0.862811	−	0.505527i	$-0.168702\pi$
$389$	34.0346	1.72562	0.862811	+	0.505527i	$0.168702\pi$
$390$	0	0
$391$	−25.5829	−1.29378
$392$	0	0
$393$	−13.7936	−0.695797
$394$	0	0
$395$	17.7436	0.892776
$396$	0	0
$397$	9.82450	0.493078	0.246539	−	0.969133i	$-0.420707\pi$
$397$	9.82450	0.493078	0.246539	+	0.969133i	$0.420707\pi$
$398$	0	0
$399$	28.6738	1.43549
$400$	0	0
$401$	8.64054	0.431488	0.215744	−	0.976450i	$-0.430782\pi$
$401$	8.64054	0.431488	0.215744	+	0.976450i	$0.430782\pi$
$402$	0	0
$403$	−26.9096	−1.34046
$404$	0	0
$405$	−3.00702	−0.149420
$406$	0	0
$407$	−2.10482	−0.104332
$408$	0	0
$409$	13.5588	0.670437	0.335219	−	0.942140i	$-0.391190\pi$
$409$	13.5588	0.670437	0.335219	+	0.942140i	$0.391190\pi$
$410$	0	0
$411$	17.0579	0.841403
$412$	0	0
$413$	36.4432	1.79325
$414$	0	0
$415$	−9.00872	−0.442221
$416$	0	0
$417$	5.40359	0.264615
$418$	0	0
$419$	−16.1540	−0.789173	−0.394586	−	0.918859i	$-0.629112\pi$
$419$	−16.1540	−0.789173	−0.394586	+	0.918859i	$0.629112\pi$
$420$	0	0
$421$	−35.8761	−1.74850	−0.874248	−	0.485480i	$-0.838645\pi$
$421$	−35.8761	−1.74850	−0.874248	+	0.485480i	$0.838645\pi$
$422$	0	0
$423$	2.66617	0.129633
$424$	0	0
$425$	18.0582	0.875949
$426$	0	0
$427$	57.0601	2.76133
$428$	0	0
$429$	−8.86207	−0.427865
$430$	0	0
$431$	−18.6705	−0.899326	−0.449663	−	0.893198i	$-0.648456\pi$
$431$	−18.6705	−0.899326	−0.449663	+	0.893198i	$0.648456\pi$
$432$	0	0
$433$	−7.53304	−0.362015	−0.181007	−	0.983482i	$-0.557936\pi$
$433$	−7.53304	−0.362015	−0.181007	+	0.983482i	$0.557936\pi$
$434$	0	0
$435$	−16.4518	−0.788804
$436$	0	0
$437$	37.1024	1.77485
$438$	0	0
$439$	18.7912	0.896856	0.448428	−	0.893819i	$-0.351984\pi$
$439$	18.7912	0.896856	0.448428	+	0.893819i	$0.351984\pi$
$440$	0	0
$441$	12.5859	0.599328
$442$	0	0
$443$	−31.1465	−1.47981	−0.739907	−	0.672709i	$-0.765131\pi$
$443$	−31.1465	−1.47981	−0.739907	+	0.672709i	$0.765131\pi$
$444$	0	0
$445$	43.0393	2.04026
$446$	0	0
$447$	−3.74723	−0.177238
$448$	0	0
$449$	19.5780	0.923943	0.461971	−	0.886895i	$-0.347142\pi$
$449$	19.5780	0.923943	0.461971	+	0.886895i	$0.347142\pi$
$450$	0	0
$451$	−4.35006	−0.204836
$452$	0	0
$453$	−23.7345	−1.11514
$454$	0	0
$455$	93.4973	4.38322
$456$	0	0
$457$	−12.1175	−0.566835	−0.283417	−	0.958997i	$-0.591468\pi$
$457$	−12.1175	−0.566835	−0.283417	+	0.958997i	$0.591468\pi$
$458$	0	0
$459$	−4.46748	−0.208524
$460$	0	0
$461$	4.21556	0.196338	0.0981691	−	0.995170i	$-0.468701\pi$
$461$	4.21556	0.196338	0.0981691	+	0.995170i	$0.468701\pi$
$462$	0	0
$463$	−16.7951	−0.780535	−0.390267	−	0.920702i	$-0.627617\pi$
$463$	−16.7951	−0.780535	−0.390267	+	0.920702i	$0.627617\pi$
$464$	0	0
$465$	11.5173	0.534103
$466$	0	0
$467$	−22.3934	−1.03624	−0.518121	−	0.855307i	$-0.673368\pi$
$467$	−22.3934	−1.03624	−0.518121	+	0.855307i	$0.673368\pi$
$468$	0	0
$469$	31.9472	1.47518
$470$	0	0
$471$	−23.3909	−1.07779
$472$	0	0
$473$	−8.91714	−0.410011
$474$	0	0
$475$	−26.1894	−1.20165
$476$	0	0
$477$	−11.5714	−0.529819
$478$	0	0
$479$	12.4221	0.567582	0.283791	−	0.958886i	$-0.408408\pi$
$479$	12.4221	0.567582	0.283791	+	0.958886i	$0.408408\pi$
$480$	0	0
$481$	−11.7236	−0.534551
$482$	0	0
$483$	25.3431	1.15315
$484$	0	0
$485$	28.7766	1.30668
$486$	0	0
$487$	14.5340	0.658597	0.329298	−	0.944226i	$-0.393188\pi$
$487$	14.5340	0.658597	0.329298	+	0.944226i	$0.393188\pi$
$488$	0	0
$489$	−13.5342	−0.612040
$490$	0	0
$491$	5.49906	0.248169	0.124085	−	0.992272i	$-0.460401\pi$
$491$	5.49906	0.248169	0.124085	+	0.992272i	$0.460401\pi$
$492$	0	0
$493$	−24.4422	−1.10082
$494$	0	0
$495$	3.79297	0.170481
$496$	0	0
$497$	16.5076	0.740467
$498$	0	0
$499$	39.2026	1.75495	0.877474	−	0.479623i	$-0.159227\pi$
$499$	39.2026	1.75495	0.877474	+	0.479623i	$0.159227\pi$
$500$	0	0
$501$	−10.5120	−0.469643
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	35.1650	1.56482
$506$	0	0
$507$	−36.3609	−1.61484
$508$	0	0
$509$	−18.9641	−0.840567	−0.420284	−	0.907393i	$-0.638069\pi$
$509$	−18.9641	−0.840567	−0.420284	+	0.907393i	$0.638069\pi$
$510$	0	0
$511$	−57.9840	−2.56506
$512$	0	0
$513$	6.47908	0.286059
$514$	0	0
$515$	−2.11690	−0.0932816
$516$	0	0
$517$	−3.36303	−0.147906
$518$	0	0
$519$	−11.4272	−0.501600
$520$	0	0
$521$	8.14358	0.356777	0.178388	−	0.983960i	$-0.442912\pi$
$521$	8.14358	0.356777	0.178388	+	0.983960i	$0.442912\pi$
$522$	0	0
$523$	15.6543	0.684513	0.342257	−	0.939607i	$-0.388809\pi$
$523$	15.6543	0.684513	0.342257	+	0.939607i	$0.388809\pi$
$524$	0	0
$525$	−17.8889	−0.780735
$526$	0	0
$527$	17.1111	0.745371
$528$	0	0
$529$	9.79265	0.425767
$530$	0	0
$531$	8.23464	0.357353
$532$	0	0
$533$	−24.2294	−1.04949
$534$	0	0
$535$	27.1252	1.17272
$536$	0	0
$537$	22.8420	0.985705
$538$	0	0
$539$	−15.8755	−0.683807
$540$	0	0
$541$	38.1085	1.63841	0.819206	−	0.573500i	$-0.194415\pi$
$541$	38.1085	1.63841	0.819206	+	0.573500i	$0.194415\pi$
$542$	0	0
$543$	−6.70936	−0.287926
$544$	0	0
$545$	−48.2092	−2.06506
$546$	0	0
$547$	8.73347	0.373416	0.186708	−	0.982415i	$-0.440218\pi$
$547$	8.73347	0.373416	0.186708	+	0.982415i	$0.440218\pi$
$548$	0	0
$549$	12.8932	0.550269
$550$	0	0
$551$	35.4480	1.51014
$552$	0	0
$553$	−26.1142	−1.11049
$554$	0	0
$555$	5.01771	0.212990
$556$	0	0
$557$	−0.442840	−0.0187637	−0.00938187	−	0.999956i	$-0.502986\pi$
$557$	−0.442840	−0.0187637	−0.00938187	+	0.999956i	$0.502986\pi$
$558$	0	0
$559$	−49.6676	−2.10072
$560$	0	0
$561$	5.63516	0.237916
$562$	0	0
$563$	10.1173	0.426395	0.213197	−	0.977009i	$-0.431612\pi$
$563$	10.1173	0.426395	0.213197	+	0.977009i	$0.431612\pi$
$564$	0	0
$565$	28.1713	1.18517
$566$	0	0
$567$	4.42560	0.185858
$568$	0	0
$569$	−25.1697	−1.05517	−0.527585	−	0.849502i	$-0.676902\pi$
$569$	−25.1697	−1.05517	−0.527585	+	0.849502i	$0.676902\pi$
$570$	0	0
$571$	−37.9163	−1.58675	−0.793375	−	0.608734i	$-0.791678\pi$
$571$	−37.9163	−1.58675	−0.793375	+	0.608734i	$0.791678\pi$
$572$	0	0
$573$	7.17863	0.299892
$574$	0	0
$575$	−23.1473	−0.965307
$576$	0	0
$577$	12.8925	0.536724	0.268362	−	0.963318i	$-0.413518\pi$
$577$	12.8925	0.536724	0.268362	+	0.963318i	$0.413518\pi$
$578$	0	0
$579$	−19.2715	−0.800896
$580$	0	0
$581$	13.2586	0.550061
$582$	0	0
$583$	14.5959	0.604500
$584$	0	0
$585$	21.1265	0.873473
$586$	0	0
$587$	−38.5802	−1.59238	−0.796188	−	0.605050i	$-0.793153\pi$
$587$	−38.5802	−1.59238	−0.796188	+	0.605050i	$0.793153\pi$
$588$	0	0
$589$	−24.8159	−1.02252
$590$	0	0
$591$	−19.7295	−0.811561
$592$	0	0
$593$	12.6819	0.520785	0.260393	−	0.965503i	$-0.416148\pi$
$593$	12.6819	0.520785	0.260393	+	0.965503i	$0.416148\pi$
$594$	0	0
$595$	−59.4524	−2.43731
$596$	0	0
$597$	−10.7240	−0.438904
$598$	0	0
$599$	36.6373	1.49696	0.748479	−	0.663159i	$-0.230785\pi$
$599$	36.6373	1.49696	0.748479	+	0.663159i	$0.230785\pi$
$600$	0	0
$601$	20.4538	0.834327	0.417164	−	0.908831i	$-0.363024\pi$
$601$	20.4538	0.834327	0.417164	+	0.908831i	$0.363024\pi$
$602$	0	0
$603$	7.21874	0.293970
$604$	0	0
$605$	28.2928	1.15027
$606$	0	0
$607$	−44.0882	−1.78949	−0.894743	−	0.446582i	$-0.852641\pi$
$607$	−44.0882	−1.78949	−0.894743	+	0.446582i	$0.852641\pi$
$608$	0	0
$609$	24.2131	0.981163
$610$	0	0
$611$	−18.7318	−0.757807
$612$	0	0
$613$	1.54841	0.0625396	0.0312698	−	0.999511i	$-0.490045\pi$
$613$	1.54841	0.0625396	0.0312698	+	0.999511i	$0.490045\pi$
$614$	0	0
$615$	10.3702	0.418166
$616$	0	0
$617$	33.6995	1.35669	0.678346	−	0.734743i	$-0.262697\pi$
$617$	33.6995	1.35669	0.678346	+	0.734743i	$0.262697\pi$
$618$	0	0
$619$	−0.599738	−0.0241055	−0.0120528	−	0.999927i	$-0.503837\pi$
$619$	−0.599738	−0.0241055	−0.0120528	+	0.999927i	$0.503837\pi$
$620$	0	0
$621$	5.72649	0.229796
$622$	0	0
$623$	−63.3434	−2.53780
$624$	0	0
$625$	−28.8718	−1.15487
$626$	0	0
$627$	−8.17254	−0.326380
$628$	0	0
$629$	7.45474	0.297240
$630$	0	0
$631$	2.82660	0.112525	0.0562627	−	0.998416i	$-0.482082\pi$
$631$	2.82660	0.112525	0.0562627	+	0.998416i	$0.482082\pi$
$632$	0	0
$633$	23.1655	0.920747
$634$	0	0
$635$	−3.75865	−0.149158
$636$	0	0
$637$	−88.4251	−3.50353
$638$	0	0
$639$	3.73003	0.147558
$640$	0	0
$641$	−10.0004	−0.394993	−0.197496	−	0.980304i	$-0.563281\pi$
$641$	−10.0004	−0.394993	−0.197496	+	0.980304i	$0.563281\pi$
$642$	0	0
$643$	−32.5066	−1.28194	−0.640968	−	0.767567i	$-0.721467\pi$
$643$	−32.5066	−1.28194	−0.640968	+	0.767567i	$0.721467\pi$
$644$	0	0
$645$	21.2578	0.837024
$646$	0	0
$647$	38.0913	1.49752	0.748761	−	0.662840i	$-0.230649\pi$
$647$	38.0913	1.49752	0.748761	+	0.662840i	$0.230649\pi$
$648$	0	0
$649$	−10.3870	−0.407724
$650$	0	0
$651$	−16.9507	−0.664350
$652$	0	0
$653$	28.0839	1.09901	0.549504	−	0.835491i	$-0.314817\pi$
$653$	28.0839	1.09901	0.549504	+	0.835491i	$0.314817\pi$
$654$	0	0
$655$	−41.4777	−1.62067
$656$	0	0
$657$	−13.1020	−0.511157
$658$	0	0
$659$	20.8790	0.813329	0.406664	−	0.913578i	$-0.366692\pi$
$659$	20.8790	0.813329	0.406664	+	0.913578i	$0.366692\pi$
$660$	0	0
$661$	4.73847	0.184305	0.0921525	−	0.995745i	$-0.470625\pi$
$661$	4.73847	0.184305	0.0921525	+	0.995745i	$0.470625\pi$
$662$	0	0
$663$	31.3873	1.21898
$664$	0	0
$665$	86.2226	3.34357
$666$	0	0
$667$	31.3304	1.21312
$668$	0	0
$669$	−19.5792	−0.756977
$670$	0	0
$671$	−16.2631	−0.627832
$672$	0	0
$673$	−32.5041	−1.25294	−0.626470	−	0.779446i	$-0.715501\pi$
$673$	−32.5041	−1.25294	−0.626470	+	0.779446i	$0.715501\pi$
$674$	0	0
$675$	−4.04214	−0.155582
$676$	0	0
$677$	22.9804	0.883207	0.441604	−	0.897210i	$-0.354410\pi$
$677$	22.9804	0.883207	0.441604	+	0.897210i	$0.354410\pi$
$678$	0	0
$679$	−42.3522	−1.62533
$680$	0	0
$681$	−13.6014	−0.521207
$682$	0	0
$683$	−14.9887	−0.573528	−0.286764	−	0.958001i	$-0.592580\pi$
$683$	−14.9887	−0.573528	−0.286764	+	0.958001i	$0.592580\pi$
$684$	0	0
$685$	51.2933	1.95982
$686$	0	0
$687$	12.4302	0.474243
$688$	0	0
$689$	81.2978	3.09720
$690$	0	0
$691$	−23.7088	−0.901925	−0.450963	−	0.892543i	$-0.648919\pi$
$691$	−23.7088	−0.901925	−0.450963	+	0.892543i	$0.648919\pi$
$692$	0	0
$693$	−5.58233	−0.212055
$694$	0	0
$695$	16.2487	0.616348
$696$	0	0
$697$	15.4068	0.583575
$698$	0	0
$699$	11.2581	0.425821
$700$	0	0
$701$	12.0436	0.454879	0.227440	−	0.973792i	$-0.426965\pi$
$701$	12.0436	0.454879	0.227440	+	0.973792i	$0.426965\pi$
$702$	0	0
$703$	−10.8114	−0.407762
$704$	0	0
$705$	8.01721	0.301945
$706$	0	0
$707$	−51.7543	−1.94642
$708$	0	0
$709$	−23.2067	−0.871545	−0.435773	−	0.900057i	$-0.643525\pi$
$709$	−23.2067	−0.871545	−0.435773	+	0.900057i	$0.643525\pi$
$710$	0	0
$711$	−5.90073	−0.221295
$712$	0	0
$713$	−21.9333	−0.821409
$714$	0	0
$715$	−26.6484	−0.996593
$716$	0	0
$717$	−23.0467	−0.860693
$718$	0	0
$719$	2.51137	0.0936582	0.0468291	−	0.998903i	$-0.485088\pi$
$719$	2.51137	0.0936582	0.0468291	+	0.998903i	$0.485088\pi$
$720$	0	0
$721$	3.11556	0.116029
$722$	0	0
$723$	−25.7686	−0.958345
$724$	0	0
$725$	−22.1151	−0.821335
$726$	0	0
$727$	−7.34411	−0.272378	−0.136189	−	0.990683i	$-0.543485\pi$
$727$	−7.34411	−0.272378	−0.136189	+	0.990683i	$0.543485\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.5823	1.16811
$732$	0	0
$733$	−4.56543	−0.168628	−0.0843140	−	0.996439i	$-0.526870\pi$
$733$	−4.56543	−0.168628	−0.0843140	+	0.996439i	$0.526870\pi$
$734$	0	0
$735$	37.8460	1.39597
$736$	0	0
$737$	−9.10552	−0.335406
$738$	0	0
$739$	0.437296	0.0160862	0.00804310	−	0.999968i	$-0.497440\pi$
$739$	0.437296	0.0160862	0.00804310	+	0.999968i	$0.497440\pi$
$740$	0	0
$741$	−45.5203	−1.67223
$742$	0	0
$743$	46.7944	1.71672	0.858361	−	0.513046i	$-0.171483\pi$
$743$	46.7944	1.71672	0.858361	+	0.513046i	$0.171483\pi$
$744$	0	0
$745$	−11.2680	−0.412826
$746$	0	0
$747$	2.99590	0.109614
$748$	0	0
$749$	−39.9217	−1.45871
$750$	0	0
$751$	5.60629	0.204576	0.102288	−	0.994755i	$-0.467384\pi$
$751$	5.60629	0.204576	0.102288	+	0.994755i	$0.467384\pi$
$752$	0	0
$753$	4.22022	0.153793
$754$	0	0
$755$	−71.3700	−2.59742
$756$	0	0
$757$	29.2598	1.06347	0.531734	−	0.846912i	$-0.321541\pi$
$757$	29.2598	1.06347	0.531734	+	0.846912i	$0.321541\pi$
$758$	0	0
$759$	−7.22324	−0.262187
$760$	0	0
$761$	35.1392	1.27379	0.636897	−	0.770949i	$-0.280218\pi$
$761$	35.1392	1.27379	0.636897	+	0.770949i	$0.280218\pi$
$762$	0	0
$763$	70.9523	2.56864
$764$	0	0
$765$	−13.4338	−0.485699
$766$	0	0
$767$	−57.8544	−2.08900
$768$	0	0
$769$	−17.1973	−0.620152	−0.310076	−	0.950712i	$-0.600355\pi$
$769$	−17.1973	−0.620152	−0.310076	+	0.950712i	$0.600355\pi$
$770$	0	0
$771$	15.8165	0.569618
$772$	0	0
$773$	−21.6372	−0.778237	−0.389119	−	0.921188i	$-0.627220\pi$
$773$	−21.6372	−0.778237	−0.389119	+	0.921188i	$0.627220\pi$
$774$	0	0
$775$	15.4820	0.556130
$776$	0	0
$777$	−7.38486	−0.264930
$778$	0	0
$779$	−22.3442	−0.800564
$780$	0	0
$781$	−4.70496	−0.168357
$782$	0	0
$783$	5.47114	0.195523
$784$	0	0
$785$	−70.3367	−2.51042
$786$	0	0
$787$	7.86308	0.280289	0.140144	−	0.990131i	$-0.455243\pi$
$787$	7.86308	0.280289	0.140144	+	0.990131i	$0.455243\pi$
$788$	0	0
$789$	−13.1279	−0.467367
$790$	0	0
$791$	−41.4612	−1.47419
$792$	0	0
$793$	−90.5842	−3.21674
$794$	0	0
$795$	−34.7955	−1.23407
$796$	0	0
$797$	6.02181	0.213304	0.106652	−	0.994296i	$-0.465987\pi$
$797$	6.02181	0.213304	0.106652	+	0.994296i	$0.465987\pi$
$798$	0	0
$799$	11.9110	0.421382
$800$	0	0
$801$	−14.3130	−0.505724
$802$	0	0
$803$	16.5265	0.583207
$804$	0	0
$805$	76.2071	2.68595
$806$	0	0
$807$	−7.27608	−0.256130
$808$	0	0
$809$	−8.89558	−0.312752	−0.156376	−	0.987698i	$-0.549981\pi$
$809$	−8.89558	−0.312752	−0.156376	+	0.987698i	$0.549981\pi$
$810$	0	0
$811$	−33.0037	−1.15892	−0.579459	−	0.815002i	$-0.696736\pi$
$811$	−33.0037	−1.15892	−0.579459	+	0.815002i	$0.696736\pi$
$812$	0	0
$813$	−5.30166	−0.185937
$814$	0	0
$815$	−40.6977	−1.42558
$816$	0	0
$817$	−45.8032	−1.60245
$818$	0	0
$819$	−31.0931	−1.08648
$820$	0	0
$821$	−27.6415	−0.964695	−0.482347	−	0.875980i	$-0.660216\pi$
$821$	−27.6415	−0.964695	−0.482347	+	0.875980i	$0.660216\pi$
$822$	0	0
$823$	−14.2051	−0.495159	−0.247579	−	0.968868i	$-0.579635\pi$
$823$	−14.2051	−0.495159	−0.247579	+	0.968868i	$0.579635\pi$
$824$	0	0
$825$	5.09865	0.177512
$826$	0	0
$827$	3.86280	0.134323	0.0671614	−	0.997742i	$-0.478606\pi$
$827$	3.86280	0.134323	0.0671614	+	0.997742i	$0.478606\pi$
$828$	0	0
$829$	50.8147	1.76487	0.882433	−	0.470438i	$-0.155904\pi$
$829$	50.8147	1.76487	0.882433	+	0.470438i	$0.155904\pi$
$830$	0	0
$831$	6.17368	0.214163
$832$	0	0
$833$	56.2272	1.94816
$834$	0	0
$835$	−31.6098	−1.09390
$836$	0	0
$837$	−3.83015	−0.132389
$838$	0	0
$839$	−27.4864	−0.948935	−0.474468	−	0.880273i	$-0.657359\pi$
$839$	−27.4864	−0.948935	−0.474468	+	0.880273i	$0.657359\pi$
$840$	0	0
$841$	0.933398	0.0321861
$842$	0	0
$843$	−22.6254	−0.779261
$844$	0	0
$845$	−109.338	−3.76134
$846$	0	0
$847$	−41.6402	−1.43077
$848$	0	0
$849$	−6.53144	−0.224159
$850$	0	0
$851$	−9.55561	−0.327562
$852$	0	0
$853$	−3.12491	−0.106995	−0.0534974	−	0.998568i	$-0.517037\pi$
$853$	−3.12491	−0.106995	−0.0534974	+	0.998568i	$0.517037\pi$
$854$	0	0
$855$	19.4827	0.666294
$856$	0	0
$857$	−7.14433	−0.244046	−0.122023	−	0.992527i	$-0.538938\pi$
$857$	−7.14433	−0.244046	−0.122023	+	0.992527i	$0.538938\pi$
$858$	0	0
$859$	43.1328	1.47167	0.735836	−	0.677159i	$-0.236789\pi$
$859$	43.1328	1.47167	0.735836	+	0.677159i	$0.236789\pi$
$860$	0	0
$861$	−15.2624	−0.520141
$862$	0	0
$863$	−51.4061	−1.74988	−0.874941	−	0.484229i	$-0.839100\pi$
$863$	−51.4061	−1.74988	−0.874941	+	0.484229i	$0.839100\pi$
$864$	0	0
$865$	−34.3619	−1.16834
$866$	0	0
$867$	−2.95834	−0.100470
$868$	0	0
$869$	7.44302	0.252487
$870$	0	0
$871$	−50.7169	−1.71848
$872$	0	0
$873$	−9.56984	−0.323890
$874$	0	0
$875$	12.7470	0.430929
$876$	0	0
$877$	15.9358	0.538113	0.269056	−	0.963124i	$-0.413288\pi$
$877$	15.9358	0.538113	0.269056	+	0.963124i	$0.413288\pi$
$878$	0	0
$879$	−31.2931	−1.05549
$880$	0	0
$881$	7.57491	0.255205	0.127603	−	0.991825i	$-0.459272\pi$
$881$	7.57491	0.255205	0.127603	+	0.991825i	$0.459272\pi$
$882$	0	0
$883$	22.9900	0.773673	0.386837	−	0.922148i	$-0.373568\pi$
$883$	22.9900	0.773673	0.386837	+	0.922148i	$0.373568\pi$
$884$	0	0
$885$	24.7617	0.832355
$886$	0	0
$887$	−19.1168	−0.641881	−0.320940	−	0.947099i	$-0.603999\pi$
$887$	−19.1168	−0.641881	−0.320940	+	0.947099i	$0.603999\pi$
$888$	0	0
$889$	5.53182	0.185531
$890$	0	0
$891$	−1.26137	−0.0422576
$892$	0	0
$893$	−17.2743	−0.578063
$894$	0	0
$895$	68.6863	2.29593
$896$	0	0
$897$	−40.2328	−1.34333
$898$	0	0
$899$	−20.9553	−0.698898
$900$	0	0
$901$	−51.6951	−1.72221
$902$	0	0
$903$	−31.2863	−1.04114
$904$	0	0
$905$	−20.1752	−0.670645
$906$	0	0
$907$	−7.37065	−0.244739	−0.122369	−	0.992485i	$-0.539049\pi$
$907$	−7.37065	−0.244739	−0.122369	+	0.992485i	$0.539049\pi$
$908$	0	0
$909$	−11.6943	−0.387876
$910$	0	0
$911$	−0.0486751	−0.00161268	−0.000806339	−	1.00000i	$-0.500257\pi$
$911$	−0.0486751	−0.00161268	−0.000806339		1.00000i	$0.500257\pi$
$912$	0	0
$913$	−3.77895	−0.125065
$914$	0	0
$915$	38.7701	1.28170
$916$	0	0
$917$	61.0451	2.01589
$918$	0	0
$919$	27.6991	0.913709	0.456855	−	0.889541i	$-0.348976\pi$
$919$	27.6991	0.913709	0.456855	+	0.889541i	$0.348976\pi$
$920$	0	0
$921$	−22.9963	−0.757755
$922$	0	0
$923$	−26.2062	−0.862587
$924$	0	0
$925$	6.74500	0.221774
$926$	0	0
$927$	0.703986	0.0231219
$928$	0	0
$929$	30.4208	0.998073	0.499036	−	0.866581i	$-0.333687\pi$
$929$	30.4208	0.998073	0.499036	+	0.866581i	$0.333687\pi$
$930$	0	0
$931$	−81.5451	−2.67253
$932$	0	0
$933$	23.7421	0.777283
$934$	0	0
$935$	16.9450	0.554161
$936$	0	0
$937$	34.6412	1.13168	0.565839	−	0.824516i	$-0.308552\pi$
$937$	34.6412	1.13168	0.565839	+	0.824516i	$0.308552\pi$
$938$	0	0
$939$	−17.5847	−0.573856
$940$	0	0
$941$	51.4359	1.67676	0.838382	−	0.545083i	$-0.183502\pi$
$941$	51.4359	1.67676	0.838382	+	0.545083i	$0.183502\pi$
$942$	0	0
$943$	−19.7487	−0.643107
$944$	0	0
$945$	13.3078	0.432904
$946$	0	0
$947$	23.8989	0.776610	0.388305	−	0.921531i	$-0.373061\pi$
$947$	23.8989	0.776610	0.388305	+	0.921531i	$0.373061\pi$
$948$	0	0
$949$	92.0510	2.98810
$950$	0	0
$951$	34.1952	1.10886
$952$	0	0
$953$	9.72615	0.315061	0.157531	−	0.987514i	$-0.449647\pi$
$953$	9.72615	0.315061	0.157531	+	0.987514i	$0.449647\pi$
$954$	0	0
$955$	21.5863	0.698515
$956$	0	0
$957$	−6.90115	−0.223083
$958$	0	0
$959$	−75.4913	−2.43774
$960$	0	0
$961$	−16.3300	−0.526773
$962$	0	0
$963$	−9.02064	−0.290686
$964$	0	0
$965$	−57.9496	−1.86547
$966$	0	0
$967$	18.0681	0.581031	0.290516	−	0.956870i	$-0.406173\pi$
$967$	18.0681	0.581031	0.290516	+	0.956870i	$0.406173\pi$
$968$	0	0
$969$	28.9451	0.929852
$970$	0	0
$971$	−26.1902	−0.840484	−0.420242	−	0.907412i	$-0.638055\pi$
$971$	−26.1902	−0.840484	−0.420242	+	0.907412i	$0.638055\pi$
$972$	0	0
$973$	−23.9141	−0.766651
$974$	0	0
$975$	28.3990	0.909496
$976$	0	0
$977$	−23.0168	−0.736374	−0.368187	−	0.929752i	$-0.620021\pi$
$977$	−23.0168	−0.736374	−0.368187	+	0.929752i	$0.620021\pi$
$978$	0	0
$979$	18.0540	0.577008
$980$	0	0
$981$	16.0323	0.511870
$982$	0	0
$983$	5.01061	0.159814	0.0799069	−	0.996802i	$-0.474538\pi$
$983$	5.01061	0.159814	0.0799069	+	0.996802i	$0.474538\pi$
$984$	0	0
$985$	−59.3268	−1.89031
$986$	0	0
$987$	−11.7994	−0.375578
$988$	0	0
$989$	−40.4828	−1.28728
$990$	0	0
$991$	4.98445	0.158336	0.0791681	−	0.996861i	$-0.474774\pi$
$991$	4.98445	0.158336	0.0791681	+	0.996861i	$0.474774\pi$
$992$	0	0
$993$	12.7003	0.403030
$994$	0	0
$995$	−32.2472	−1.02231
$996$	0	0
$997$	30.0604	0.952022	0.476011	−	0.879439i	$-0.342082\pi$
$997$	30.0604	0.952022	0.476011	+	0.879439i	$0.342082\pi$
$998$	0	0
$999$	−1.66867	−0.0527944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.3	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00702 q^{5} +4.42560 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.00702 q^{5} +4.42560 q^{7} +1.00000 q^{9} -1.26137 q^{11} -7.02573 q^{13} +3.00702 q^{15} +4.46748 q^{17} -6.47908 q^{19} -4.42560 q^{21} -5.72649 q^{23} +4.04214 q^{25} -1.00000 q^{27} -5.47114 q^{29} +3.83015 q^{31} +1.26137 q^{33} -13.3078 q^{35} +1.66867 q^{37} +7.02573 q^{39} +3.44867 q^{41} +7.06939 q^{43} -3.00702 q^{45} +2.66617 q^{47} +12.5859 q^{49} -4.46748 q^{51} -11.5714 q^{53} +3.79297 q^{55} +6.47908 q^{57} +8.23464 q^{59} +12.8932 q^{61} +4.42560 q^{63} +21.1265 q^{65} +7.21874 q^{67} +5.72649 q^{69} +3.73003 q^{71} -13.1020 q^{73} -4.04214 q^{75} -5.58233 q^{77} -5.90073 q^{79} +1.00000 q^{81} +2.99590 q^{83} -13.4338 q^{85} +5.47114 q^{87} -14.3130 q^{89} -31.0931 q^{91} -3.83015 q^{93} +19.4827 q^{95} -9.56984 q^{97} -1.26137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 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

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