LMFDB - Embedded newform 6036.2.a.i.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.0581310 q^{5} -2.78315 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.0581310 q^{5} -2.78315 q^{7} +1.00000 q^{9} +3.35646 q^{11} -1.13981 q^{13} -0.0581310 q^{15} +1.84368 q^{17} -0.820062 q^{19} +2.78315 q^{21} -4.45849 q^{23} -4.99662 q^{25} -1.00000 q^{27} +2.16313 q^{29} -4.12569 q^{31} -3.35646 q^{33} -0.161787 q^{35} +2.83415 q^{37} +1.13981 q^{39} +11.8611 q^{41} -2.14201 q^{43} +0.0581310 q^{45} +2.54359 q^{47} +0.745927 q^{49} -1.84368 q^{51} +9.14941 q^{53} +0.195114 q^{55} +0.820062 q^{57} -5.43274 q^{59} -3.80269 q^{61} -2.78315 q^{63} -0.0662585 q^{65} -8.54100 q^{67} +4.45849 q^{69} +7.66410 q^{71} +6.97047 q^{73} +4.99662 q^{75} -9.34154 q^{77} +3.92009 q^{79} +1.00000 q^{81} -7.00196 q^{83} +0.107175 q^{85} -2.16313 q^{87} -0.190479 q^{89} +3.17227 q^{91} +4.12569 q^{93} -0.0476711 q^{95} -5.39203 q^{97} +3.35646 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.0581310	0.0259970	0.0129985	−	0.999916i	$-0.495862\pi$
$5$	0.0581310	0.0259970	0.0129985	+	0.999916i	$0.495862\pi$
$6$	0	0
$7$	−2.78315	−1.05193	−0.525966	−	0.850506i	$-0.676296\pi$
$7$	−2.78315	−1.05193	−0.525966	+	0.850506i	$0.676296\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.35646	1.01201	0.506006	−	0.862530i	$-0.331121\pi$
$11$	3.35646	1.01201	0.506006	+	0.862530i	$0.331121\pi$
$12$	0	0
$13$	−1.13981	−0.316127	−0.158064	−	0.987429i	$-0.550525\pi$
$13$	−1.13981	−0.316127	−0.158064	+	0.987429i	$0.550525\pi$
$14$	0	0
$15$	−0.0581310	−0.0150094
$16$	0	0
$17$	1.84368	0.447158	0.223579	−	0.974686i	$-0.428226\pi$
$17$	1.84368	0.447158	0.223579	+	0.974686i	$0.428226\pi$
$18$	0	0
$19$	−0.820062	−0.188135	−0.0940676	−	0.995566i	$-0.529987\pi$
$19$	−0.820062	−0.188135	−0.0940676	+	0.995566i	$0.529987\pi$
$20$	0	0
$21$	2.78315	0.607333
$22$	0	0
$23$	−4.45849	−0.929660	−0.464830	−	0.885400i	$-0.653884\pi$
$23$	−4.45849	−0.929660	−0.464830	+	0.885400i	$0.653884\pi$
$24$	0	0
$25$	−4.99662	−0.999324
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.16313	0.401684	0.200842	−	0.979624i	$-0.435632\pi$
$29$	2.16313	0.401684	0.200842	+	0.979624i	$0.435632\pi$
$30$	0	0
$31$	−4.12569	−0.740995	−0.370498	−	0.928833i	$-0.620813\pi$
$31$	−4.12569	−0.740995	−0.370498	+	0.928833i	$0.620813\pi$
$32$	0	0
$33$	−3.35646	−0.584285
$34$	0	0
$35$	−0.161787	−0.0273471
$36$	0	0
$37$	2.83415	0.465932	0.232966	−	0.972485i	$-0.425157\pi$
$37$	2.83415	0.465932	0.232966	+	0.972485i	$0.425157\pi$
$38$	0	0
$39$	1.13981	0.182516
$40$	0	0
$41$	11.8611	1.85239	0.926196	−	0.377042i	$-0.123059\pi$
$41$	11.8611	1.85239	0.926196	+	0.377042i	$0.123059\pi$
$42$	0	0
$43$	−2.14201	−0.326654	−0.163327	−	0.986572i	$-0.552223\pi$
$43$	−2.14201	−0.326654	−0.163327	+	0.986572i	$0.552223\pi$
$44$	0	0
$45$	0.0581310	0.00866566
$46$	0	0
$47$	2.54359	0.371021	0.185510	−	0.982642i	$-0.440606\pi$
$47$	2.54359	0.371021	0.185510	+	0.982642i	$0.440606\pi$
$48$	0	0
$49$	0.745927	0.106561
$50$	0	0
$51$	−1.84368	−0.258167
$52$	0	0
$53$	9.14941	1.25677	0.628384	−	0.777903i	$-0.283717\pi$
$53$	9.14941	1.25677	0.628384	+	0.777903i	$0.283717\pi$
$54$	0	0
$55$	0.195114	0.0263092
$56$	0	0
$57$	0.820062	0.108620
$58$	0	0
$59$	−5.43274	−0.707282	−0.353641	−	0.935381i	$-0.615057\pi$
$59$	−5.43274	−0.707282	−0.353641	+	0.935381i	$0.615057\pi$
$60$	0	0
$61$	−3.80269	−0.486885	−0.243442	−	0.969915i	$-0.578277\pi$
$61$	−3.80269	−0.486885	−0.243442	+	0.969915i	$0.578277\pi$
$62$	0	0
$63$	−2.78315	−0.350644
$64$	0	0
$65$	−0.0662585	−0.00821835
$66$	0	0
$67$	−8.54100	−1.04345	−0.521725	−	0.853114i	$-0.674711\pi$
$67$	−8.54100	−1.04345	−0.521725	+	0.853114i	$0.674711\pi$
$68$	0	0
$69$	4.45849	0.536739
$70$	0	0
$71$	7.66410	0.909562	0.454781	−	0.890603i	$-0.349718\pi$
$71$	7.66410	0.909562	0.454781	+	0.890603i	$0.349718\pi$
$72$	0	0
$73$	6.97047	0.815832	0.407916	−	0.913019i	$-0.366256\pi$
$73$	6.97047	0.815832	0.407916	+	0.913019i	$0.366256\pi$
$74$	0	0
$75$	4.99662	0.576960
$76$	0	0
$77$	−9.34154	−1.06457
$78$	0	0
$79$	3.92009	0.441044	0.220522	−	0.975382i	$-0.429224\pi$
$79$	3.92009	0.441044	0.220522	+	0.975382i	$0.429224\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.00196	−0.768565	−0.384282	−	0.923216i	$-0.625551\pi$
$83$	−7.00196	−0.768565	−0.384282	+	0.923216i	$0.625551\pi$
$84$	0	0
$85$	0.107175	0.0116248
$86$	0	0
$87$	−2.16313	−0.231912
$88$	0	0
$89$	−0.190479	−0.0201907	−0.0100953	−	0.999949i	$-0.503214\pi$
$89$	−0.190479	−0.0201907	−0.0100953	+	0.999949i	$0.503214\pi$
$90$	0	0
$91$	3.17227	0.332544
$92$	0	0
$93$	4.12569	0.427814
$94$	0	0
$95$	−0.0476711	−0.00489095
$96$	0	0
$97$	−5.39203	−0.547478	−0.273739	−	0.961804i	$-0.588260\pi$
$97$	−5.39203	−0.547478	−0.273739	+	0.961804i	$0.588260\pi$
$98$	0	0
$99$	3.35646	0.337337
$100$	0	0
$101$	−16.9622	−1.68780	−0.843899	−	0.536502i	$-0.819745\pi$
$101$	−16.9622	−1.68780	−0.843899	+	0.536502i	$0.819745\pi$
$102$	0	0
$103$	12.1345	1.19565	0.597826	−	0.801626i	$-0.296031\pi$
$103$	12.1345	1.19565	0.597826	+	0.801626i	$0.296031\pi$
$104$	0	0
$105$	0.161787	0.0157888
$106$	0	0
$107$	7.88121	0.761905	0.380953	−	0.924595i	$-0.375596\pi$
$107$	7.88121	0.761905	0.380953	+	0.924595i	$0.375596\pi$
$108$	0	0
$109$	13.0857	1.25338	0.626690	−	0.779268i	$-0.284409\pi$
$109$	13.0857	1.25338	0.626690	+	0.779268i	$0.284409\pi$
$110$	0	0
$111$	−2.83415	−0.269006
$112$	0	0
$113$	−19.0378	−1.79093	−0.895463	−	0.445136i	$-0.853155\pi$
$113$	−19.0378	−1.79093	−0.895463	+	0.445136i	$0.853155\pi$
$114$	0	0
$115$	−0.259177	−0.0241683
$116$	0	0
$117$	−1.13981	−0.105376
$118$	0	0
$119$	−5.13124	−0.470380
$120$	0	0
$121$	0.265834	0.0241667
$122$	0	0
$123$	−11.8611	−1.06948
$124$	0	0
$125$	−0.581114	−0.0519764
$126$	0	0
$127$	−14.0797	−1.24937	−0.624685	−	0.780877i	$-0.714772\pi$
$127$	−14.0797	−1.24937	−0.624685	+	0.780877i	$0.714772\pi$
$128$	0	0
$129$	2.14201	0.188594
$130$	0	0
$131$	13.9728	1.22081	0.610406	−	0.792089i	$-0.291006\pi$
$131$	13.9728	1.22081	0.610406	+	0.792089i	$0.291006\pi$
$132$	0	0
$133$	2.28236	0.197905
$134$	0	0
$135$	−0.0581310	−0.00500312
$136$	0	0
$137$	−4.93731	−0.421823	−0.210911	−	0.977505i	$-0.567643\pi$
$137$	−4.93731	−0.421823	−0.210911	+	0.977505i	$0.567643\pi$
$138$	0	0
$139$	14.7184	1.24840	0.624199	−	0.781266i	$-0.285426\pi$
$139$	14.7184	1.24840	0.624199	+	0.781266i	$0.285426\pi$
$140$	0	0
$141$	−2.54359	−0.214209
$142$	0	0
$143$	−3.82574	−0.319924
$144$	0	0
$145$	0.125745	0.0104426
$146$	0	0
$147$	−0.745927	−0.0615230
$148$	0	0
$149$	20.8872	1.71115	0.855575	−	0.517679i	$-0.173204\pi$
$149$	20.8872	1.71115	0.855575	+	0.517679i	$0.173204\pi$
$150$	0	0
$151$	−9.89692	−0.805400	−0.402700	−	0.915332i	$-0.631928\pi$
$151$	−9.89692	−0.805400	−0.402700	+	0.915332i	$0.631928\pi$
$152$	0	0
$153$	1.84368	0.149053
$154$	0	0
$155$	−0.239830	−0.0192636
$156$	0	0
$157$	−18.2131	−1.45357	−0.726784	−	0.686867i	$-0.758986\pi$
$157$	−18.2131	−1.45357	−0.726784	+	0.686867i	$0.758986\pi$
$158$	0	0
$159$	−9.14941	−0.725595
$160$	0	0
$161$	12.4087	0.977939
$162$	0	0
$163$	−6.70773	−0.525390	−0.262695	−	0.964879i	$-0.584611\pi$
$163$	−6.70773	−0.525390	−0.262695	+	0.964879i	$0.584611\pi$
$164$	0	0
$165$	−0.195114	−0.0151896
$166$	0	0
$167$	16.7108	1.29312	0.646560	−	0.762863i	$-0.276207\pi$
$167$	16.7108	1.29312	0.646560	+	0.762863i	$0.276207\pi$
$168$	0	0
$169$	−11.7008	−0.900064
$170$	0	0
$171$	−0.820062	−0.0627117
$172$	0	0
$173$	11.0922	0.843321	0.421661	−	0.906754i	$-0.361447\pi$
$173$	11.0922	0.843321	0.421661	+	0.906754i	$0.361447\pi$
$174$	0	0
$175$	13.9063	1.05122
$176$	0	0
$177$	5.43274	0.408349
$178$	0	0
$179$	7.06107	0.527769	0.263885	−	0.964554i	$-0.414996\pi$
$179$	7.06107	0.527769	0.263885	+	0.964554i	$0.414996\pi$
$180$	0	0
$181$	12.1676	0.904413	0.452207	−	0.891913i	$-0.350637\pi$
$181$	12.1676	0.904413	0.452207	+	0.891913i	$0.350637\pi$
$182$	0	0
$183$	3.80269	0.281103
$184$	0	0
$185$	0.164752	0.0121128
$186$	0	0
$187$	6.18824	0.452529
$188$	0	0
$189$	2.78315	0.202444
$190$	0	0
$191$	−1.17840	−0.0852661	−0.0426330	−	0.999091i	$-0.513575\pi$
$191$	−1.17840	−0.0852661	−0.0426330	+	0.999091i	$0.513575\pi$
$192$	0	0
$193$	21.4923	1.54705	0.773523	−	0.633768i	$-0.218492\pi$
$193$	21.4923	1.54705	0.773523	+	0.633768i	$0.218492\pi$
$194$	0	0
$195$	0.0662585	0.00474487
$196$	0	0
$197$	24.4163	1.73959	0.869796	−	0.493412i	$-0.164250\pi$
$197$	24.4163	1.73959	0.869796	+	0.493412i	$0.164250\pi$
$198$	0	0
$199$	−6.20550	−0.439896	−0.219948	−	0.975512i	$-0.570589\pi$
$199$	−6.20550	−0.439896	−0.219948	+	0.975512i	$0.570589\pi$
$200$	0	0
$201$	8.54100	0.602436
$202$	0	0
$203$	−6.02033	−0.422544
$204$	0	0
$205$	0.689497	0.0481566
$206$	0	0
$207$	−4.45849	−0.309887
$208$	0	0
$209$	−2.75251	−0.190395
$210$	0	0
$211$	7.54531	0.519441	0.259720	−	0.965684i	$-0.416370\pi$
$211$	7.54531	0.519441	0.259720	+	0.965684i	$0.416370\pi$
$212$	0	0
$213$	−7.66410	−0.525136
$214$	0	0
$215$	−0.124517	−0.00849201
$216$	0	0
$217$	11.4824	0.779477
$218$	0	0
$219$	−6.97047	−0.471021
$220$	0	0
$221$	−2.10145	−0.141359
$222$	0	0
$223$	12.4707	0.835098	0.417549	−	0.908654i	$-0.362889\pi$
$223$	12.4707	0.835098	0.417549	+	0.908654i	$0.362889\pi$
$224$	0	0
$225$	−4.99662	−0.333108
$226$	0	0
$227$	−0.761779	−0.0505610	−0.0252805	−	0.999680i	$-0.508048\pi$
$227$	−0.761779	−0.0505610	−0.0252805	+	0.999680i	$0.508048\pi$
$228$	0	0
$229$	26.6788	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$229$	26.6788	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$230$	0	0
$231$	9.34154	0.614628
$232$	0	0
$233$	14.9373	0.978577	0.489288	−	0.872122i	$-0.337257\pi$
$233$	14.9373	0.978577	0.489288	+	0.872122i	$0.337257\pi$
$234$	0	0
$235$	0.147861	0.00964541
$236$	0	0
$237$	−3.92009	−0.254637
$238$	0	0
$239$	23.1926	1.50020	0.750102	−	0.661322i	$-0.230004\pi$
$239$	23.1926	1.50020	0.750102	+	0.661322i	$0.230004\pi$
$240$	0	0
$241$	28.9557	1.86520	0.932599	−	0.360913i	$-0.117535\pi$
$241$	28.9557	1.86520	0.932599	+	0.360913i	$0.117535\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.0433615	0.00277026
$246$	0	0
$247$	0.934718	0.0594747
$248$	0	0
$249$	7.00196	0.443731
$250$	0	0
$251$	−18.9642	−1.19701	−0.598506	−	0.801118i	$-0.704239\pi$
$251$	−18.9642	−1.19701	−0.598506	+	0.801118i	$0.704239\pi$
$252$	0	0
$253$	−14.9648	−0.940826
$254$	0	0
$255$	−0.107175	−0.00671156
$256$	0	0
$257$	−3.05251	−0.190410	−0.0952052	−	0.995458i	$-0.530351\pi$
$257$	−3.05251	−0.190410	−0.0952052	+	0.995458i	$0.530351\pi$
$258$	0	0
$259$	−7.88788	−0.490129
$260$	0	0
$261$	2.16313	0.133895
$262$	0	0
$263$	−11.1032	−0.684655	−0.342327	−	0.939581i	$-0.611215\pi$
$263$	−11.1032	−0.684655	−0.342327	+	0.939581i	$0.611215\pi$
$264$	0	0
$265$	0.531864	0.0326722
$266$	0	0
$267$	0.190479	0.0116571
$268$	0	0
$269$	26.9690	1.64433	0.822165	−	0.569249i	$-0.192766\pi$
$269$	26.9690	1.64433	0.822165	+	0.569249i	$0.192766\pi$
$270$	0	0
$271$	18.4072	1.11816	0.559080	−	0.829114i	$-0.311154\pi$
$271$	18.4072	1.11816	0.559080	+	0.829114i	$0.311154\pi$
$272$	0	0
$273$	−3.17227	−0.191995
$274$	0	0
$275$	−16.7710	−1.01133
$276$	0	0
$277$	29.4623	1.77022	0.885109	−	0.465383i	$-0.154083\pi$
$277$	29.4623	1.77022	0.885109	+	0.465383i	$0.154083\pi$
$278$	0	0
$279$	−4.12569	−0.246998
$280$	0	0
$281$	−23.1967	−1.38380	−0.691899	−	0.721994i	$-0.743226\pi$
$281$	−23.1967	−1.38380	−0.691899	+	0.721994i	$0.743226\pi$
$282$	0	0
$283$	22.2312	1.32151	0.660753	−	0.750603i	$-0.270237\pi$
$283$	22.2312	1.32151	0.660753	+	0.750603i	$0.270237\pi$
$284$	0	0
$285$	0.0476711	0.00282379
$286$	0	0
$287$	−33.0112	−1.94859
$288$	0	0
$289$	−13.6008	−0.800050
$290$	0	0
$291$	5.39203	0.316087
$292$	0	0
$293$	12.6671	0.740019	0.370009	−	0.929028i	$-0.379355\pi$
$293$	12.6671	0.740019	0.370009	+	0.929028i	$0.379355\pi$
$294$	0	0
$295$	−0.315810	−0.0183872
$296$	0	0
$297$	−3.35646	−0.194762
$298$	0	0
$299$	5.08185	0.293891
$300$	0	0
$301$	5.96154	0.343618
$302$	0	0
$303$	16.9622	0.974451
$304$	0	0
$305$	−0.221054	−0.0126575
$306$	0	0
$307$	−24.1954	−1.38091	−0.690453	−	0.723377i	$-0.742589\pi$
$307$	−24.1954	−1.38091	−0.690453	+	0.723377i	$0.742589\pi$
$308$	0	0
$309$	−12.1345	−0.690310
$310$	0	0
$311$	−15.3821	−0.872238	−0.436119	−	0.899889i	$-0.643647\pi$
$311$	−15.3821	−0.872238	−0.436119	+	0.899889i	$0.643647\pi$
$312$	0	0
$313$	5.03233	0.284444	0.142222	−	0.989835i	$-0.454575\pi$
$313$	5.03233	0.284444	0.142222	+	0.989835i	$0.454575\pi$
$314$	0	0
$315$	−0.161787	−0.00911568
$316$	0	0
$317$	30.0336	1.68685	0.843427	−	0.537244i	$-0.180535\pi$
$317$	30.0336	1.68685	0.843427	+	0.537244i	$0.180535\pi$
$318$	0	0
$319$	7.26048	0.406509
$320$	0	0
$321$	−7.88121	−0.439886
$322$	0	0
$323$	−1.51193	−0.0841262
$324$	0	0
$325$	5.69521	0.315914
$326$	0	0
$327$	−13.0857	−0.723640
$328$	0	0
$329$	−7.07919	−0.390288
$330$	0	0
$331$	2.46324	0.135392	0.0676958	−	0.997706i	$-0.478435\pi$
$331$	2.46324	0.135392	0.0676958	+	0.997706i	$0.478435\pi$
$332$	0	0
$333$	2.83415	0.155311
$334$	0	0
$335$	−0.496497	−0.0271265
$336$	0	0
$337$	−14.3358	−0.780920	−0.390460	−	0.920620i	$-0.627684\pi$
$337$	−14.3358	−0.780920	−0.390460	+	0.920620i	$0.627684\pi$
$338$	0	0
$339$	19.0378	1.03399
$340$	0	0
$341$	−13.8477	−0.749895
$342$	0	0
$343$	17.4060	0.939837
$344$	0	0
$345$	0.259177	0.0139536
$346$	0	0
$347$	16.8417	0.904111	0.452056	−	0.891990i	$-0.350691\pi$
$347$	16.8417	0.904111	0.452056	+	0.891990i	$0.350691\pi$
$348$	0	0
$349$	16.4874	0.882552	0.441276	−	0.897372i	$-0.354526\pi$
$349$	16.4874	0.882552	0.441276	+	0.897372i	$0.354526\pi$
$350$	0	0
$351$	1.13981	0.0608387
$352$	0	0
$353$	14.7869	0.787025	0.393512	−	0.919319i	$-0.371260\pi$
$353$	14.7869	0.787025	0.393512	+	0.919319i	$0.371260\pi$
$354$	0	0
$355$	0.445522	0.0236458
$356$	0	0
$357$	5.13124	0.271574
$358$	0	0
$359$	−15.1451	−0.799327	−0.399663	−	0.916662i	$-0.630873\pi$
$359$	−15.1451	−0.799327	−0.399663	+	0.916662i	$0.630873\pi$
$360$	0	0
$361$	−18.3275	−0.964605
$362$	0	0
$363$	−0.265834	−0.0139527
$364$	0	0
$365$	0.405200	0.0212092
$366$	0	0
$367$	−2.49327	−0.130148	−0.0650739	−	0.997880i	$-0.520728\pi$
$367$	−2.49327	−0.130148	−0.0650739	+	0.997880i	$0.520728\pi$
$368$	0	0
$369$	11.8611	0.617464
$370$	0	0
$371$	−25.4642	−1.32203
$372$	0	0
$373$	1.89166	0.0979463	0.0489732	−	0.998800i	$-0.484405\pi$
$373$	1.89166	0.0979463	0.0489732	+	0.998800i	$0.484405\pi$
$374$	0	0
$375$	0.581114	0.0300086
$376$	0	0
$377$	−2.46557	−0.126983
$378$	0	0
$379$	27.2330	1.39887	0.699433	−	0.714698i	$-0.253436\pi$
$379$	27.2330	1.39887	0.699433	+	0.714698i	$0.253436\pi$
$380$	0	0
$381$	14.0797	0.721324
$382$	0	0
$383$	−19.0000	−0.970854	−0.485427	−	0.874277i	$-0.661336\pi$
$383$	−19.0000	−0.970854	−0.485427	+	0.874277i	$0.661336\pi$
$384$	0	0
$385$	−0.543033	−0.0276755
$386$	0	0
$387$	−2.14201	−0.108885
$388$	0	0
$389$	25.1295	1.27412	0.637058	−	0.770816i	$-0.280151\pi$
$389$	25.1295	1.27412	0.637058	+	0.770816i	$0.280151\pi$
$390$	0	0
$391$	−8.22003	−0.415705
$392$	0	0
$393$	−13.9728	−0.704836
$394$	0	0
$395$	0.227879	0.0114658
$396$	0	0
$397$	−19.9219	−0.999852	−0.499926	−	0.866068i	$-0.666640\pi$
$397$	−19.9219	−0.999852	−0.499926	+	0.866068i	$0.666640\pi$
$398$	0	0
$399$	−2.28236	−0.114261
$400$	0	0
$401$	−20.2457	−1.01102	−0.505511	−	0.862820i	$-0.668696\pi$
$401$	−20.2457	−1.01102	−0.505511	+	0.862820i	$0.668696\pi$
$402$	0	0
$403$	4.70251	0.234249
$404$	0	0
$405$	0.0581310	0.00288855
$406$	0	0
$407$	9.51273	0.471528
$408$	0	0
$409$	31.4494	1.55507	0.777537	−	0.628837i	$-0.216469\pi$
$409$	31.4494	1.55507	0.777537	+	0.628837i	$0.216469\pi$
$410$	0	0
$411$	4.93731	0.243539
$412$	0	0
$413$	15.1201	0.744013
$414$	0	0
$415$	−0.407031	−0.0199804
$416$	0	0
$417$	−14.7184	−0.720762
$418$	0	0
$419$	−2.65661	−0.129784	−0.0648919	−	0.997892i	$-0.520670\pi$
$419$	−2.65661	−0.129784	−0.0648919	+	0.997892i	$0.520670\pi$
$420$	0	0
$421$	16.9007	0.823688	0.411844	−	0.911254i	$-0.364885\pi$
$421$	16.9007	0.823688	0.411844	+	0.911254i	$0.364885\pi$
$422$	0	0
$423$	2.54359	0.123674
$424$	0	0
$425$	−9.21217	−0.446856
$426$	0	0
$427$	10.5835	0.512170
$428$	0	0
$429$	3.82574	0.184708
$430$	0	0
$431$	5.25025	0.252896	0.126448	−	0.991973i	$-0.459642\pi$
$431$	5.25025	0.252896	0.126448	+	0.991973i	$0.459642\pi$
$432$	0	0
$433$	−4.55545	−0.218921	−0.109461	−	0.993991i	$-0.534912\pi$
$433$	−4.55545	−0.218921	−0.109461	+	0.993991i	$0.534912\pi$
$434$	0	0
$435$	−0.125745	−0.00602902
$436$	0	0
$437$	3.65624	0.174902
$438$	0	0
$439$	24.9849	1.19247	0.596233	−	0.802812i	$-0.296663\pi$
$439$	24.9849	1.19247	0.596233	+	0.802812i	$0.296663\pi$
$440$	0	0
$441$	0.745927	0.0355203
$442$	0	0
$443$	−35.9441	−1.70776	−0.853879	−	0.520472i	$-0.825756\pi$
$443$	−35.9441	−1.70776	−0.853879	+	0.520472i	$0.825756\pi$
$444$	0	0
$445$	−0.0110727	−0.000524897 0
$446$	0	0
$447$	−20.8872	−0.987933
$448$	0	0
$449$	26.0892	1.23123	0.615613	−	0.788049i	$-0.288908\pi$
$449$	26.0892	1.23123	0.615613	+	0.788049i	$0.288908\pi$
$450$	0	0
$451$	39.8113	1.87464
$452$	0	0
$453$	9.89692	0.464998
$454$	0	0
$455$	0.184407	0.00864515
$456$	0	0
$457$	−8.81754	−0.412467	−0.206233	−	0.978503i	$-0.566121\pi$
$457$	−8.81754	−0.412467	−0.206233	+	0.978503i	$0.566121\pi$
$458$	0	0
$459$	−1.84368	−0.0860556
$460$	0	0
$461$	14.8329	0.690836	0.345418	−	0.938449i	$-0.387737\pi$
$461$	14.8329	0.690836	0.345418	+	0.938449i	$0.387737\pi$
$462$	0	0
$463$	20.0746	0.932945	0.466473	−	0.884536i	$-0.345525\pi$
$463$	20.0746	0.932945	0.466473	+	0.884536i	$0.345525\pi$
$464$	0	0
$465$	0.239830	0.0111219
$466$	0	0
$467$	−7.23995	−0.335025	−0.167512	−	0.985870i	$-0.553573\pi$
$467$	−7.23995	−0.335025	−0.167512	+	0.985870i	$0.553573\pi$
$468$	0	0
$469$	23.7709	1.09764
$470$	0	0
$471$	18.2131	0.839217
$472$	0	0
$473$	−7.18958	−0.330577
$474$	0	0
$475$	4.09754	0.188008
$476$	0	0
$477$	9.14941	0.418923
$478$	0	0
$479$	−10.6712	−0.487580	−0.243790	−	0.969828i	$-0.578391\pi$
$479$	−10.6712	−0.487580	−0.243790	+	0.969828i	$0.578391\pi$
$480$	0	0
$481$	−3.23040	−0.147294
$482$	0	0
$483$	−12.4087	−0.564613
$484$	0	0
$485$	−0.313444	−0.0142328
$486$	0	0
$487$	2.01733	0.0914139	0.0457070	−	0.998955i	$-0.485446\pi$
$487$	2.01733	0.0914139	0.0457070	+	0.998955i	$0.485446\pi$
$488$	0	0
$489$	6.70773	0.303334
$490$	0	0
$491$	−41.1887	−1.85882	−0.929411	−	0.369047i	$-0.879684\pi$
$491$	−41.1887	−1.85882	−0.929411	+	0.369047i	$0.879684\pi$
$492$	0	0
$493$	3.98813	0.179616
$494$	0	0
$495$	0.195114	0.00876974
$496$	0	0
$497$	−21.3303	−0.956797
$498$	0	0
$499$	−18.8212	−0.842554	−0.421277	−	0.906932i	$-0.638418\pi$
$499$	−18.8212	−0.842554	−0.421277	+	0.906932i	$0.638418\pi$
$500$	0	0
$501$	−16.7108	−0.746583
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−0.986027	−0.0438776
$506$	0	0
$507$	11.7008	0.519652
$508$	0	0
$509$	−11.5040	−0.509907	−0.254953	−	0.966953i	$-0.582060\pi$
$509$	−11.5040	−0.509907	−0.254953	+	0.966953i	$0.582060\pi$
$510$	0	0
$511$	−19.3999	−0.858200
$512$	0	0
$513$	0.820062	0.0362066
$514$	0	0
$515$	0.705393	0.0310833
$516$	0	0
$517$	8.53746	0.375477
$518$	0	0
$519$	−11.0922	−0.486892
$520$	0	0
$521$	29.4854	1.29178	0.645890	−	0.763430i	$-0.276486\pi$
$521$	29.4854	1.29178	0.645890	+	0.763430i	$0.276486\pi$
$522$	0	0
$523$	−32.1450	−1.40560	−0.702801	−	0.711386i	$-0.748068\pi$
$523$	−32.1450	−1.40560	−0.702801	+	0.711386i	$0.748068\pi$
$524$	0	0
$525$	−13.9063	−0.606923
$526$	0	0
$527$	−7.60644	−0.331342
$528$	0	0
$529$	−3.12186	−0.135733
$530$	0	0
$531$	−5.43274	−0.235761
$532$	0	0
$533$	−13.5194	−0.585592
$534$	0	0
$535$	0.458143	0.0198072
$536$	0	0
$537$	−7.06107	−0.304708
$538$	0	0
$539$	2.50368	0.107841
$540$	0	0
$541$	1.64479	0.0707149	0.0353575	−	0.999375i	$-0.488743\pi$
$541$	1.64479	0.0707149	0.0353575	+	0.999375i	$0.488743\pi$
$542$	0	0
$543$	−12.1676	−0.522163
$544$	0	0
$545$	0.760684	0.0325841
$546$	0	0
$547$	8.37139	0.357935	0.178967	−	0.983855i	$-0.442724\pi$
$547$	8.37139	0.357935	0.178967	+	0.983855i	$0.442724\pi$
$548$	0	0
$549$	−3.80269	−0.162295
$550$	0	0
$551$	−1.77391	−0.0755709
$552$	0	0
$553$	−10.9102	−0.463949
$554$	0	0
$555$	−0.164752	−0.00699334
$556$	0	0
$557$	−5.67421	−0.240424	−0.120212	−	0.992748i	$-0.538357\pi$
$557$	−5.67421	−0.240424	−0.120212	+	0.992748i	$0.538357\pi$
$558$	0	0
$559$	2.44149	0.103264
$560$	0	0
$561$	−6.18824	−0.261268
$562$	0	0
$563$	−19.4744	−0.820749	−0.410374	−	0.911917i	$-0.634602\pi$
$563$	−19.4744	−0.820749	−0.410374	+	0.911917i	$0.634602\pi$
$564$	0	0
$565$	−1.10669	−0.0465587
$566$	0	0
$567$	−2.78315	−0.116881
$568$	0	0
$569$	22.3961	0.938893	0.469447	−	0.882961i	$-0.344453\pi$
$569$	22.3961	0.938893	0.469447	+	0.882961i	$0.344453\pi$
$570$	0	0
$571$	44.2765	1.85292	0.926458	−	0.376398i	$-0.122838\pi$
$571$	44.2765	1.85292	0.926458	+	0.376398i	$0.122838\pi$
$572$	0	0
$573$	1.17840	0.0492284
$574$	0	0
$575$	22.2774	0.929031
$576$	0	0
$577$	46.7959	1.94814	0.974069	−	0.226253i	$-0.0726476\pi$
$577$	46.7959	1.94814	0.974069	+	0.226253i	$0.0726476\pi$
$578$	0	0
$579$	−21.4923	−0.893188
$580$	0	0
$581$	19.4875	0.808478
$582$	0	0
$583$	30.7096	1.27186
$584$	0	0
$585$	−0.0662585	−0.00273945
$586$	0	0
$587$	−3.73755	−0.154265	−0.0771327	−	0.997021i	$-0.524577\pi$
$587$	−3.73755	−0.154265	−0.0771327	+	0.997021i	$0.524577\pi$
$588$	0	0
$589$	3.38332	0.139407
$590$	0	0
$591$	−24.4163	−1.00435
$592$	0	0
$593$	21.4923	0.882584	0.441292	−	0.897364i	$-0.354520\pi$
$593$	21.4923	0.882584	0.441292	+	0.897364i	$0.354520\pi$
$594$	0	0
$595$	−0.298284	−0.0122285
$596$	0	0
$597$	6.20550	0.253974
$598$	0	0
$599$	48.3413	1.97517	0.987586	−	0.157079i	$-0.0502078\pi$
$599$	48.3413	1.97517	0.987586	+	0.157079i	$0.0502078\pi$
$600$	0	0
$601$	19.9239	0.812711	0.406356	−	0.913715i	$-0.366799\pi$
$601$	19.9239	0.812711	0.406356	+	0.913715i	$0.366799\pi$
$602$	0	0
$603$	−8.54100	−0.347816
$604$	0	0
$605$	0.0154532	0.000628262 0
$606$	0	0
$607$	−45.1116	−1.83102	−0.915511	−	0.402292i	$-0.868214\pi$
$607$	−45.1116	−1.83102	−0.915511	+	0.402292i	$0.868214\pi$
$608$	0	0
$609$	6.02033	0.243956
$610$	0	0
$611$	−2.89922	−0.117290
$612$	0	0
$613$	−42.7630	−1.72718	−0.863591	−	0.504193i	$-0.831790\pi$
$613$	−42.7630	−1.72718	−0.863591	+	0.504193i	$0.831790\pi$
$614$	0	0
$615$	−0.689497	−0.0278032
$616$	0	0
$617$	−21.5529	−0.867688	−0.433844	−	0.900988i	$-0.642843\pi$
$617$	−21.5529	−0.867688	−0.433844	+	0.900988i	$0.642843\pi$
$618$	0	0
$619$	−30.7810	−1.23719	−0.618597	−	0.785709i	$-0.712299\pi$
$619$	−30.7810	−1.23719	−0.618597	+	0.785709i	$0.712299\pi$
$620$	0	0
$621$	4.45849	0.178913
$622$	0	0
$623$	0.530131	0.0212392
$624$	0	0
$625$	24.9493	0.997973
$626$	0	0
$627$	2.75251	0.109925
$628$	0	0
$629$	5.22527	0.208345
$630$	0	0
$631$	43.8485	1.74558	0.872790	−	0.488096i	$-0.162308\pi$
$631$	43.8485	1.74558	0.872790	+	0.488096i	$0.162308\pi$
$632$	0	0
$633$	−7.54531	−0.299899
$634$	0	0
$635$	−0.818466	−0.0324798
$636$	0	0
$637$	−0.850217	−0.0336868
$638$	0	0
$639$	7.66410	0.303187
$640$	0	0
$641$	13.4722	0.532119	0.266059	−	0.963957i	$-0.414278\pi$
$641$	13.4722	0.532119	0.266059	+	0.963957i	$0.414278\pi$
$642$	0	0
$643$	10.9000	0.429855	0.214928	−	0.976630i	$-0.431048\pi$
$643$	10.9000	0.429855	0.214928	+	0.976630i	$0.431048\pi$
$644$	0	0
$645$	0.124517	0.00490287
$646$	0	0
$647$	−30.0981	−1.18328	−0.591639	−	0.806203i	$-0.701519\pi$
$647$	−30.0981	−1.18328	−0.591639	+	0.806203i	$0.701519\pi$
$648$	0	0
$649$	−18.2348	−0.715777
$650$	0	0
$651$	−11.4824	−0.450031
$652$	0	0
$653$	−31.5775	−1.23572	−0.617862	−	0.786287i	$-0.712001\pi$
$653$	−31.5775	−1.23572	−0.617862	+	0.786287i	$0.712001\pi$
$654$	0	0
$655$	0.812254	0.0317374
$656$	0	0
$657$	6.97047	0.271944
$658$	0	0
$659$	−36.5330	−1.42312	−0.711562	−	0.702624i	$-0.752012\pi$
$659$	−36.5330	−1.42312	−0.711562	+	0.702624i	$0.752012\pi$
$660$	0	0
$661$	−10.1541	−0.394948	−0.197474	−	0.980308i	$-0.563274\pi$
$661$	−10.1541	−0.394948	−0.197474	+	0.980308i	$0.563274\pi$
$662$	0	0
$663$	2.10145	0.0816136
$664$	0	0
$665$	0.132676	0.00514494
$666$	0	0
$667$	−9.64432	−0.373429
$668$	0	0
$669$	−12.4707	−0.482144
$670$	0	0
$671$	−12.7636	−0.492733
$672$	0	0
$673$	29.0219	1.11871	0.559355	−	0.828928i	$-0.311049\pi$
$673$	29.0219	1.11871	0.559355	+	0.828928i	$0.311049\pi$
$674$	0	0
$675$	4.99662	0.192320
$676$	0	0
$677$	13.8316	0.531591	0.265795	−	0.964029i	$-0.414365\pi$
$677$	13.8316	0.531591	0.265795	+	0.964029i	$0.414365\pi$
$678$	0	0
$679$	15.0068	0.575910
$680$	0	0
$681$	0.761779	0.0291914
$682$	0	0
$683$	−23.7315	−0.908061	−0.454030	−	0.890986i	$-0.650014\pi$
$683$	−23.7315	−0.908061	−0.454030	+	0.890986i	$0.650014\pi$
$684$	0	0
$685$	−0.287011	−0.0109661
$686$	0	0
$687$	−26.6788	−1.01786
$688$	0	0
$689$	−10.4286	−0.397299
$690$	0	0
$691$	−2.48459	−0.0945184	−0.0472592	−	0.998883i	$-0.515049\pi$
$691$	−2.48459	−0.0945184	−0.0472592	+	0.998883i	$0.515049\pi$
$692$	0	0
$693$	−9.34154	−0.354856
$694$	0	0
$695$	0.855595	0.0324546
$696$	0	0
$697$	21.8681	0.828312
$698$	0	0
$699$	−14.9373	−0.564981
$700$	0	0
$701$	−24.7033	−0.933030	−0.466515	−	0.884513i	$-0.654491\pi$
$701$	−24.7033	−0.933030	−0.466515	+	0.884513i	$0.654491\pi$
$702$	0	0
$703$	−2.32418	−0.0876582
$704$	0	0
$705$	−0.147861	−0.00556878
$706$	0	0
$707$	47.2082	1.77545
$708$	0	0
$709$	16.7783	0.630124	0.315062	−	0.949071i	$-0.397975\pi$
$709$	16.7783	0.630124	0.315062	+	0.949071i	$0.397975\pi$
$710$	0	0
$711$	3.92009	0.147015
$712$	0	0
$713$	18.3943	0.688873
$714$	0	0
$715$	−0.222394	−0.00831706
$716$	0	0
$717$	−23.1926	−0.866143
$718$	0	0
$719$	45.2929	1.68914	0.844569	−	0.535447i	$-0.179857\pi$
$719$	45.2929	1.68914	0.844569	+	0.535447i	$0.179857\pi$
$720$	0	0
$721$	−33.7723	−1.25774
$722$	0	0
$723$	−28.9557	−1.07687
$724$	0	0
$725$	−10.8084	−0.401413
$726$	0	0
$727$	41.3389	1.53317	0.766587	−	0.642141i	$-0.221954\pi$
$727$	41.3389	1.53317	0.766587	+	0.642141i	$0.221954\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.94918	−0.146066
$732$	0	0
$733$	36.1694	1.33595	0.667974	−	0.744185i	$-0.267162\pi$
$733$	36.1694	1.33595	0.667974	+	0.744185i	$0.267162\pi$
$734$	0	0
$735$	−0.0433615	−0.00159941
$736$	0	0
$737$	−28.6675	−1.05598
$738$	0	0
$739$	−29.4739	−1.08421	−0.542107	−	0.840309i	$-0.682373\pi$
$739$	−29.4739	−1.08421	−0.542107	+	0.840309i	$0.682373\pi$
$740$	0	0
$741$	−0.934718	−0.0343377
$742$	0	0
$743$	−32.7791	−1.20255	−0.601274	−	0.799043i	$-0.705340\pi$
$743$	−32.7791	−1.20255	−0.601274	+	0.799043i	$0.705340\pi$
$744$	0	0
$745$	1.21420	0.0444847
$746$	0	0
$747$	−7.00196	−0.256188
$748$	0	0
$749$	−21.9346	−0.801473
$750$	0	0
$751$	35.4109	1.29216	0.646082	−	0.763268i	$-0.276407\pi$
$751$	35.4109	1.29216	0.646082	+	0.763268i	$0.276407\pi$
$752$	0	0
$753$	18.9642	0.691096
$754$	0	0
$755$	−0.575318	−0.0209380
$756$	0	0
$757$	−29.8838	−1.08614	−0.543072	−	0.839686i	$-0.682739\pi$
$757$	−29.8838	−1.08614	−0.543072	+	0.839686i	$0.682739\pi$
$758$	0	0
$759$	14.9648	0.543186
$760$	0	0
$761$	39.6385	1.43689	0.718447	−	0.695582i	$-0.244853\pi$
$761$	39.6385	1.43689	0.718447	+	0.695582i	$0.244853\pi$
$762$	0	0
$763$	−36.4194	−1.31847
$764$	0	0
$765$	0.107175	0.00387492
$766$	0	0
$767$	6.19230	0.223591
$768$	0	0
$769$	−49.8197	−1.79654	−0.898272	−	0.439439i	$-0.855177\pi$
$769$	−49.8197	−1.79654	−0.898272	+	0.439439i	$0.855177\pi$
$770$	0	0
$771$	3.05251	0.109933
$772$	0	0
$773$	−4.77408	−0.171712	−0.0858559	−	0.996308i	$-0.527362\pi$
$773$	−4.77408	−0.171712	−0.0858559	+	0.996308i	$0.527362\pi$
$774$	0	0
$775$	20.6145	0.740494
$776$	0	0
$777$	7.88788	0.282976
$778$	0	0
$779$	−9.72684	−0.348500
$780$	0	0
$781$	25.7243	0.920486
$782$	0	0
$783$	−2.16313	−0.0773041
$784$	0	0
$785$	−1.05875	−0.0377883
$786$	0	0
$787$	11.6565	0.415508	0.207754	−	0.978181i	$-0.433385\pi$
$787$	11.6565	0.415508	0.207754	+	0.978181i	$0.433385\pi$
$788$	0	0
$789$	11.1032	0.395286
$790$	0	0
$791$	52.9851	1.88393
$792$	0	0
$793$	4.33436	0.153918
$794$	0	0
$795$	−0.531864	−0.0188633
$796$	0	0
$797$	−8.06073	−0.285526	−0.142763	−	0.989757i	$-0.545599\pi$
$797$	−8.06073	−0.285526	−0.142763	+	0.989757i	$0.545599\pi$
$798$	0	0
$799$	4.68956	0.165905
$800$	0	0
$801$	−0.190479	−0.00673023
$802$	0	0
$803$	23.3961	0.825631
$804$	0	0
$805$	0.721327	0.0254234
$806$	0	0
$807$	−26.9690	−0.949355
$808$	0	0
$809$	13.9913	0.491908	0.245954	−	0.969282i	$-0.420899\pi$
$809$	13.9913	0.491908	0.245954	+	0.969282i	$0.420899\pi$
$810$	0	0
$811$	−15.1025	−0.530319	−0.265160	−	0.964205i	$-0.585425\pi$
$811$	−15.1025	−0.530319	−0.265160	+	0.964205i	$0.585425\pi$
$812$	0	0
$813$	−18.4072	−0.645570
$814$	0	0
$815$	−0.389927	−0.0136585
$816$	0	0
$817$	1.75658	0.0614551
$818$	0	0
$819$	3.17227	0.110848
$820$	0	0
$821$	26.3031	0.917983	0.458992	−	0.888441i	$-0.348211\pi$
$821$	26.3031	0.917983	0.458992	+	0.888441i	$0.348211\pi$
$822$	0	0
$823$	−3.70649	−0.129200	−0.0646000	−	0.997911i	$-0.520577\pi$
$823$	−3.70649	−0.129200	−0.0646000	+	0.997911i	$0.520577\pi$
$824$	0	0
$825$	16.7710	0.583890
$826$	0	0
$827$	−29.6775	−1.03199	−0.515993	−	0.856593i	$-0.672577\pi$
$827$	−29.6775	−1.03199	−0.515993	+	0.856593i	$0.672577\pi$
$828$	0	0
$829$	44.6254	1.54990	0.774952	−	0.632020i	$-0.217774\pi$
$829$	44.6254	1.54990	0.774952	+	0.632020i	$0.217774\pi$
$830$	0	0
$831$	−29.4623	−1.02204
$832$	0	0
$833$	1.37525	0.0476496
$834$	0	0
$835$	0.971415	0.0336172
$836$	0	0
$837$	4.12569	0.142605
$838$	0	0
$839$	−20.9652	−0.723798	−0.361899	−	0.932217i	$-0.617872\pi$
$839$	−20.9652	−0.723798	−0.361899	+	0.932217i	$0.617872\pi$
$840$	0	0
$841$	−24.3208	−0.838650
$842$	0	0
$843$	23.1967	0.798937
$844$	0	0
$845$	−0.680181	−0.0233989
$846$	0	0
$847$	−0.739856	−0.0254218
$848$	0	0
$849$	−22.2312	−0.762972
$850$	0	0
$851$	−12.6360	−0.433158
$852$	0	0
$853$	−29.6837	−1.01635	−0.508176	−	0.861253i	$-0.669680\pi$
$853$	−29.6837	−1.01635	−0.508176	+	0.861253i	$0.669680\pi$
$854$	0	0
$855$	−0.0476711	−0.00163032
$856$	0	0
$857$	9.66189	0.330044	0.165022	−	0.986290i	$-0.447230\pi$
$857$	9.66189	0.330044	0.165022	+	0.986290i	$0.447230\pi$
$858$	0	0
$859$	3.90188	0.133131	0.0665653	−	0.997782i	$-0.478796\pi$
$859$	3.90188	0.133131	0.0665653	+	0.997782i	$0.478796\pi$
$860$	0	0
$861$	33.0112	1.12502
$862$	0	0
$863$	45.5026	1.54893	0.774463	−	0.632619i	$-0.218020\pi$
$863$	45.5026	1.54893	0.774463	+	0.632619i	$0.218020\pi$
$864$	0	0
$865$	0.644798	0.0219238
$866$	0	0
$867$	13.6008	0.461909
$868$	0	0
$869$	13.1576	0.446342
$870$	0	0
$871$	9.73514	0.329863
$872$	0	0
$873$	−5.39203	−0.182493
$874$	0	0
$875$	1.61733	0.0546756
$876$	0	0
$877$	3.98385	0.134525	0.0672626	−	0.997735i	$-0.478573\pi$
$877$	3.98385	0.134525	0.0672626	+	0.997735i	$0.478573\pi$
$878$	0	0
$879$	−12.6671	−0.427250
$880$	0	0
$881$	4.40623	0.148450	0.0742249	−	0.997242i	$-0.476352\pi$
$881$	4.40623	0.148450	0.0742249	+	0.997242i	$0.476352\pi$
$882$	0	0
$883$	41.9372	1.41130	0.705650	−	0.708561i	$-0.250655\pi$
$883$	41.9372	1.41130	0.705650	+	0.708561i	$0.250655\pi$
$884$	0	0
$885$	0.315810	0.0106159
$886$	0	0
$887$	−39.7860	−1.33588	−0.667942	−	0.744213i	$-0.732825\pi$
$887$	−39.7860	−1.33588	−0.667942	+	0.744213i	$0.732825\pi$
$888$	0	0
$889$	39.1859	1.31425
$890$	0	0
$891$	3.35646	0.112446
$892$	0	0
$893$	−2.08590	−0.0698020
$894$	0	0
$895$	0.410467	0.0137204
$896$	0	0
$897$	−5.08185	−0.169678
$898$	0	0
$899$	−8.92441	−0.297646
$900$	0	0
$901$	16.8686	0.561974
$902$	0	0
$903$	−5.96154	−0.198388
$904$	0	0
$905$	0.707317	0.0235120
$906$	0	0
$907$	−47.0209	−1.56130	−0.780652	−	0.624966i	$-0.785113\pi$
$907$	−47.0209	−1.56130	−0.780652	+	0.624966i	$0.785113\pi$
$908$	0	0
$909$	−16.9622	−0.562599
$910$	0	0
$911$	−58.4812	−1.93757	−0.968784	−	0.247906i	$-0.920258\pi$
$911$	−58.4812	−1.93757	−0.968784	+	0.247906i	$0.920258\pi$
$912$	0	0
$913$	−23.5018	−0.777796
$914$	0	0
$915$	0.221054	0.00730783
$916$	0	0
$917$	−38.8885	−1.28421
$918$	0	0
$919$	53.1235	1.75238	0.876192	−	0.481962i	$-0.160076\pi$
$919$	53.1235	1.75238	0.876192	+	0.481962i	$0.160076\pi$
$920$	0	0
$921$	24.1954	0.797267
$922$	0	0
$923$	−8.73564	−0.287537
$924$	0	0
$925$	−14.1612	−0.465617
$926$	0	0
$927$	12.1345	0.398551
$928$	0	0
$929$	−31.2556	−1.02546	−0.512732	−	0.858549i	$-0.671366\pi$
$929$	−31.2556	−1.02546	−0.512732	+	0.858549i	$0.671366\pi$
$930$	0	0
$931$	−0.611707	−0.0200479
$932$	0	0
$933$	15.3821	0.503587
$934$	0	0
$935$	0.359729	0.0117644
$936$	0	0
$937$	−8.11164	−0.264996	−0.132498	−	0.991183i	$-0.542300\pi$
$937$	−8.11164	−0.264996	−0.132498	+	0.991183i	$0.542300\pi$
$938$	0	0
$939$	−5.03233	−0.164224
$940$	0	0
$941$	3.77000	0.122898	0.0614492	−	0.998110i	$-0.480428\pi$
$941$	3.77000	0.122898	0.0614492	+	0.998110i	$0.480428\pi$
$942$	0	0
$943$	−52.8826	−1.72209
$944$	0	0
$945$	0.161787	0.00526294
$946$	0	0
$947$	−41.2792	−1.34139	−0.670696	−	0.741732i	$-0.734005\pi$
$947$	−41.2792	−1.34139	−0.670696	+	0.741732i	$0.734005\pi$
$948$	0	0
$949$	−7.94503	−0.257907
$950$	0	0
$951$	−30.0336	−0.973906
$952$	0	0
$953$	31.0552	1.00598	0.502989	−	0.864293i	$-0.332234\pi$
$953$	31.0552	1.00598	0.502989	+	0.864293i	$0.332234\pi$
$954$	0	0
$955$	−0.0685016	−0.00221666
$956$	0	0
$957$	−7.26048	−0.234698
$958$	0	0
$959$	13.7413	0.443729
$960$	0	0
$961$	−13.9787	−0.450926
$962$	0	0
$963$	7.88121	0.253968
$964$	0	0
$965$	1.24937	0.0402185
$966$	0	0
$967$	46.1758	1.48491	0.742457	−	0.669894i	$-0.233660\pi$
$967$	46.1758	1.48491	0.742457	+	0.669894i	$0.233660\pi$
$968$	0	0
$969$	1.51193	0.0485703
$970$	0	0
$971$	40.9441	1.31396	0.656980	−	0.753908i	$-0.271834\pi$
$971$	40.9441	1.31396	0.656980	+	0.753908i	$0.271834\pi$
$972$	0	0
$973$	−40.9635	−1.31323
$974$	0	0
$975$	−5.69521	−0.182393
$976$	0	0
$977$	41.1402	1.31619	0.658095	−	0.752935i	$-0.271362\pi$
$977$	41.1402	1.31619	0.658095	+	0.752935i	$0.271362\pi$
$978$	0	0
$979$	−0.639334	−0.0204332
$980$	0	0
$981$	13.0857	0.417794
$982$	0	0
$983$	17.1244	0.546184	0.273092	−	0.961988i	$-0.411954\pi$
$983$	17.1244	0.546184	0.273092	+	0.961988i	$0.411954\pi$
$984$	0	0
$985$	1.41935	0.0452241
$986$	0	0
$987$	7.07919	0.225333
$988$	0	0
$989$	9.55014	0.303677
$990$	0	0
$991$	−8.63117	−0.274178	−0.137089	−	0.990559i	$-0.543775\pi$
$991$	−8.63117	−0.274178	−0.137089	+	0.990559i	$0.543775\pi$
$992$	0	0
$993$	−2.46324	−0.0781684
$994$	0	0
$995$	−0.360732	−0.0114360
$996$	0	0
$997$	−39.3075	−1.24488	−0.622441	−	0.782667i	$-0.713859\pi$
$997$	−39.3075	−1.24488	−0.622441	+	0.782667i	$0.713859\pi$
$998$	0	0
$999$	−2.83415	−0.0896686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.13	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.0581310 q^{5} -2.78315 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.0581310 q^{5} -2.78315 q^{7} +1.00000 q^{9} +3.35646 q^{11} -1.13981 q^{13} -0.0581310 q^{15} +1.84368 q^{17} -0.820062 q^{19} +2.78315 q^{21} -4.45849 q^{23} -4.99662 q^{25} -1.00000 q^{27} +2.16313 q^{29} -4.12569 q^{31} -3.35646 q^{33} -0.161787 q^{35} +2.83415 q^{37} +1.13981 q^{39} +11.8611 q^{41} -2.14201 q^{43} +0.0581310 q^{45} +2.54359 q^{47} +0.745927 q^{49} -1.84368 q^{51} +9.14941 q^{53} +0.195114 q^{55} +0.820062 q^{57} -5.43274 q^{59} -3.80269 q^{61} -2.78315 q^{63} -0.0662585 q^{65} -8.54100 q^{67} +4.45849 q^{69} +7.66410 q^{71} +6.97047 q^{73} +4.99662 q^{75} -9.34154 q^{77} +3.92009 q^{79} +1.00000 q^{81} -7.00196 q^{83} +0.107175 q^{85} -2.16313 q^{87} -0.190479 q^{89} +3.17227 q^{91} +4.12569 q^{93} -0.0476711 q^{95} -5.39203 q^{97} +3.35646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 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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