LMFDB - Embedded newform 6032.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} + \cdots + 68 $ x^12 - 6x^11 - 10x^10 + 98x^9 + 10x^8 - 585x^7 + 151x^6 + 1524x^5 - 445x^4 - 1567x^3 + 273x^2 + 424*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.22268$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.22268 q^{3} +2.71513 q^{5} -1.08088 q^{7} +1.94031 q^{9} +O(q^{10})$	$q-2.22268 q^{3} +2.71513 q^{5} -1.08088 q^{7} +1.94031 q^{9} -3.45501 q^{11} +1.00000 q^{13} -6.03486 q^{15} +0.0669193 q^{17} -7.68830 q^{19} +2.40246 q^{21} +6.33320 q^{23} +2.37192 q^{25} +2.35535 q^{27} +1.00000 q^{29} +6.08882 q^{31} +7.67938 q^{33} -2.93474 q^{35} +5.46042 q^{37} -2.22268 q^{39} +5.02322 q^{41} +2.71383 q^{43} +5.26820 q^{45} -1.68949 q^{47} -5.83169 q^{49} -0.148740 q^{51} -8.77658 q^{53} -9.38078 q^{55} +17.0886 q^{57} +3.74858 q^{59} -12.8369 q^{61} -2.09725 q^{63} +2.71513 q^{65} +13.7986 q^{67} -14.0767 q^{69} -3.42706 q^{71} -4.09745 q^{73} -5.27201 q^{75} +3.73446 q^{77} -13.8217 q^{79} -11.0561 q^{81} -12.4748 q^{83} +0.181694 q^{85} -2.22268 q^{87} +4.56207 q^{89} -1.08088 q^{91} -13.5335 q^{93} -20.8747 q^{95} +16.1529 q^{97} -6.70379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * 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
$3$	−2.22268	−1.28327	−0.641633	−	0.767012i	$-0.721743\pi$
$3$	−2.22268	−1.28327	−0.641633	+	0.767012i	$0.721743\pi$
$4$	0	0
$5$	2.71513	1.21424	0.607121	−	0.794609i	$-0.292324\pi$
$5$	2.71513	1.21424	0.607121	+	0.794609i	$0.292324\pi$
$6$	0	0
$7$	−1.08088	−0.408536	−0.204268	−	0.978915i	$-0.565481\pi$
$7$	−1.08088	−0.408536	−0.204268	+	0.978915i	$0.565481\pi$
$8$	0	0
$9$	1.94031	0.646771
$10$	0	0
$11$	−3.45501	−1.04172	−0.520862	−	0.853641i	$-0.674389\pi$
$11$	−3.45501	−1.04172	−0.520862	+	0.853641i	$0.674389\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−6.03486	−1.55819
$16$	0	0
$17$	0.0669193	0.0162303	0.00811516	−	0.999967i	$-0.497417\pi$
$17$	0.0669193	0.0162303	0.00811516	+	0.999967i	$0.497417\pi$
$18$	0	0
$19$	−7.68830	−1.76382	−0.881909	−	0.471420i	$-0.843742\pi$
$19$	−7.68830	−1.76382	−0.881909	+	0.471420i	$0.843742\pi$
$20$	0	0
$21$	2.40246	0.524260
$22$	0	0
$23$	6.33320	1.32056	0.660282	−	0.751018i	$-0.270437\pi$
$23$	6.33320	1.32056	0.660282	+	0.751018i	$0.270437\pi$
$24$	0	0
$25$	2.37192	0.474383
$26$	0	0
$27$	2.35535	0.453286
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.08882	1.09358	0.546792	−	0.837269i	$-0.315849\pi$
$31$	6.08882	1.09358	0.546792	+	0.837269i	$0.315849\pi$
$32$	0	0
$33$	7.67938	1.33681
$34$	0	0
$35$	−2.93474	−0.496061
$36$	0	0
$37$	5.46042	0.897687	0.448843	−	0.893610i	$-0.351836\pi$
$37$	5.46042	0.897687	0.448843	+	0.893610i	$0.351836\pi$
$38$	0	0
$39$	−2.22268	−0.355914
$40$	0	0
$41$	5.02322	0.784495	0.392248	−	0.919860i	$-0.371698\pi$
$41$	5.02322	0.784495	0.392248	+	0.919860i	$0.371698\pi$
$42$	0	0
$43$	2.71383	0.413856	0.206928	−	0.978356i	$-0.433653\pi$
$43$	2.71383	0.413856	0.206928	+	0.978356i	$0.433653\pi$
$44$	0	0
$45$	5.26820	0.785337
$46$	0	0
$47$	−1.68949	−0.246438	−0.123219	−	0.992380i	$-0.539322\pi$
$47$	−1.68949	−0.246438	−0.123219	+	0.992380i	$0.539322\pi$
$48$	0	0
$49$	−5.83169	−0.833099
$50$	0	0
$51$	−0.148740	−0.0208278
$52$	0	0
$53$	−8.77658	−1.20556	−0.602778	−	0.797909i	$-0.705940\pi$
$53$	−8.77658	−1.20556	−0.602778	+	0.797909i	$0.705940\pi$
$54$	0	0
$55$	−9.38078	−1.26490
$56$	0	0
$57$	17.0886	2.26345
$58$	0	0
$59$	3.74858	0.488024	0.244012	−	0.969772i	$-0.421536\pi$
$59$	3.74858	0.488024	0.244012	+	0.969772i	$0.421536\pi$
$60$	0	0
$61$	−12.8369	−1.64360	−0.821800	−	0.569776i	$-0.807030\pi$
$61$	−12.8369	−1.64360	−0.821800	+	0.569776i	$0.807030\pi$
$62$	0	0
$63$	−2.09725	−0.264229
$64$	0	0
$65$	2.71513	0.336770
$66$	0	0
$67$	13.7986	1.68577	0.842885	−	0.538093i	$-0.180855\pi$
$67$	13.7986	1.68577	0.842885	+	0.538093i	$0.180855\pi$
$68$	0	0
$69$	−14.0767	−1.69463
$70$	0	0
$71$	−3.42706	−0.406717	−0.203358	−	0.979104i	$-0.565186\pi$
$71$	−3.42706	−0.406717	−0.203358	+	0.979104i	$0.565186\pi$
$72$	0	0
$73$	−4.09745	−0.479571	−0.239785	−	0.970826i	$-0.577077\pi$
$73$	−4.09745	−0.479571	−0.239785	+	0.970826i	$0.577077\pi$
$74$	0	0
$75$	−5.27201	−0.608760
$76$	0	0
$77$	3.73446	0.425581
$78$	0	0
$79$	−13.8217	−1.55506	−0.777529	−	0.628847i	$-0.783527\pi$
$79$	−13.8217	−1.55506	−0.777529	+	0.628847i	$0.783527\pi$
$80$	0	0
$81$	−11.0561	−1.22846
$82$	0	0
$83$	−12.4748	−1.36929	−0.684643	−	0.728879i	$-0.740042\pi$
$83$	−12.4748	−1.36929	−0.684643	+	0.728879i	$0.740042\pi$
$84$	0	0
$85$	0.181694	0.0197075
$86$	0	0
$87$	−2.22268	−0.238296
$88$	0	0
$89$	4.56207	0.483578	0.241789	−	0.970329i	$-0.422266\pi$
$89$	4.56207	0.483578	0.241789	+	0.970329i	$0.422266\pi$
$90$	0	0
$91$	−1.08088	−0.113307
$92$	0	0
$93$	−13.5335	−1.40336
$94$	0	0
$95$	−20.8747	−2.14170
$96$	0	0
$97$	16.1529	1.64008	0.820042	−	0.572304i	$-0.193950\pi$
$97$	16.1529	1.64008	0.820042	+	0.572304i	$0.193950\pi$
$98$	0	0
$99$	−6.70379	−0.673757
$100$	0	0
$101$	−0.343755	−0.0342049	−0.0171024	−	0.999854i	$-0.505444\pi$
$101$	−0.343755	−0.0342049	−0.0171024	+	0.999854i	$0.505444\pi$
$102$	0	0
$103$	11.7221	1.15501	0.577507	−	0.816386i	$-0.304026\pi$
$103$	11.7221	1.15501	0.577507	+	0.816386i	$0.304026\pi$
$104$	0	0
$105$	6.52299	0.636578
$106$	0	0
$107$	−5.77577	−0.558365	−0.279182	−	0.960238i	$-0.590063\pi$
$107$	−5.77577	−0.558365	−0.279182	+	0.960238i	$0.590063\pi$
$108$	0	0
$109$	5.81525	0.557000	0.278500	−	0.960436i	$-0.410163\pi$
$109$	5.81525	0.557000	0.278500	+	0.960436i	$0.410163\pi$
$110$	0	0
$111$	−12.1368	−1.15197
$112$	0	0
$113$	0.440011	0.0413927	0.0206964	−	0.999786i	$-0.493412\pi$
$113$	0.440011	0.0413927	0.0206964	+	0.999786i	$0.493412\pi$
$114$	0	0
$115$	17.1954	1.60348
$116$	0	0
$117$	1.94031	0.179382
$118$	0	0
$119$	−0.0723320	−0.00663066
$120$	0	0
$121$	0.937062	0.0851874
$122$	0	0
$123$	−11.1650	−1.00672
$124$	0	0
$125$	−7.13558	−0.638226
$126$	0	0
$127$	5.31138	0.471309	0.235654	−	0.971837i	$-0.424277\pi$
$127$	5.31138	0.471309	0.235654	+	0.971837i	$0.424277\pi$
$128$	0	0
$129$	−6.03199	−0.531087
$130$	0	0
$131$	−1.65358	−0.144474	−0.0722371	−	0.997387i	$-0.523014\pi$
$131$	−1.65358	−0.144474	−0.0722371	+	0.997387i	$0.523014\pi$
$132$	0	0
$133$	8.31016	0.720582
$134$	0	0
$135$	6.39506	0.550399
$136$	0	0
$137$	−17.5004	−1.49516	−0.747582	−	0.664170i	$-0.768785\pi$
$137$	−17.5004	−1.49516	−0.747582	+	0.664170i	$0.768785\pi$
$138$	0	0
$139$	−13.8606	−1.17564	−0.587821	−	0.808991i	$-0.700014\pi$
$139$	−13.8606	−1.17564	−0.587821	+	0.808991i	$0.700014\pi$
$140$	0	0
$141$	3.75520	0.316245
$142$	0	0
$143$	−3.45501	−0.288922
$144$	0	0
$145$	2.71513	0.225479
$146$	0	0
$147$	12.9620	1.06909
$148$	0	0
$149$	0.0906399	0.00742551	0.00371275	−	0.999993i	$-0.498818\pi$
$149$	0.0906399	0.00742551	0.00371275	+	0.999993i	$0.498818\pi$
$150$	0	0
$151$	−6.23096	−0.507069	−0.253534	−	0.967326i	$-0.581593\pi$
$151$	−6.23096	−0.507069	−0.253534	+	0.967326i	$0.581593\pi$
$152$	0	0
$153$	0.129844	0.0104973
$154$	0	0
$155$	16.5319	1.32787
$156$	0	0
$157$	−3.02217	−0.241195	−0.120598	−	0.992701i	$-0.538481\pi$
$157$	−3.02217	−0.241195	−0.120598	+	0.992701i	$0.538481\pi$
$158$	0	0
$159$	19.5075	1.54705
$160$	0	0
$161$	−6.84545	−0.539497
$162$	0	0
$163$	−11.2647	−0.882321	−0.441161	−	0.897428i	$-0.645433\pi$
$163$	−11.2647	−0.882321	−0.441161	+	0.897428i	$0.645433\pi$
$164$	0	0
$165$	20.8505	1.62321
$166$	0	0
$167$	3.50243	0.271026	0.135513	−	0.990776i	$-0.456732\pi$
$167$	3.50243	0.271026	0.135513	+	0.990776i	$0.456732\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−14.9177	−1.14079
$172$	0	0
$173$	−18.0610	−1.37315	−0.686577	−	0.727057i	$-0.740888\pi$
$173$	−18.0610	−1.37315	−0.686577	+	0.727057i	$0.740888\pi$
$174$	0	0
$175$	−2.56376	−0.193802
$176$	0	0
$177$	−8.33190	−0.626264
$178$	0	0
$179$	11.7873	0.881021	0.440511	−	0.897747i	$-0.354797\pi$
$179$	11.7873	0.881021	0.440511	+	0.897747i	$0.354797\pi$
$180$	0	0
$181$	11.5044	0.855117	0.427558	−	0.903988i	$-0.359374\pi$
$181$	11.5044	0.855117	0.427558	+	0.903988i	$0.359374\pi$
$182$	0	0
$183$	28.5324	2.10918
$184$	0	0
$185$	14.8257	1.09001
$186$	0	0
$187$	−0.231207	−0.0169075
$188$	0	0
$189$	−2.54585	−0.185184
$190$	0	0
$191$	2.88371	0.208658	0.104329	−	0.994543i	$-0.466731\pi$
$191$	2.88371	0.208658	0.104329	+	0.994543i	$0.466731\pi$
$192$	0	0
$193$	−4.57295	−0.329168	−0.164584	−	0.986363i	$-0.552628\pi$
$193$	−4.57295	−0.329168	−0.164584	+	0.986363i	$0.552628\pi$
$194$	0	0
$195$	−6.03486	−0.432166
$196$	0	0
$197$	−2.06943	−0.147441	−0.0737203	−	0.997279i	$-0.523487\pi$
$197$	−2.06943	−0.147441	−0.0737203	+	0.997279i	$0.523487\pi$
$198$	0	0
$199$	−0.0314738	−0.00223112	−0.00111556	−	0.999999i	$-0.500355\pi$
$199$	−0.0314738	−0.00223112	−0.00111556	+	0.999999i	$0.500355\pi$
$200$	0	0
$201$	−30.6700	−2.16329
$202$	0	0
$203$	−1.08088	−0.0758632
$204$	0	0
$205$	13.6387	0.952567
$206$	0	0
$207$	12.2884	0.854102
$208$	0	0
$209$	26.5631	1.83741
$210$	0	0
$211$	−1.68222	−0.115809	−0.0579044	−	0.998322i	$-0.518442\pi$
$211$	−1.68222	−0.115809	−0.0579044	+	0.998322i	$0.518442\pi$
$212$	0	0
$213$	7.61725	0.521926
$214$	0	0
$215$	7.36840	0.502521
$216$	0	0
$217$	−6.58130	−0.446768
$218$	0	0
$219$	9.10733	0.615417
$220$	0	0
$221$	0.0669193	0.00450148
$222$	0	0
$223$	−10.6297	−0.711819	−0.355910	−	0.934520i	$-0.615829\pi$
$223$	−10.6297	−0.711819	−0.355910	+	0.934520i	$0.615829\pi$
$224$	0	0
$225$	4.60226	0.306817
$226$	0	0
$227$	23.6411	1.56912	0.784558	−	0.620056i	$-0.212890\pi$
$227$	23.6411	1.56912	0.784558	+	0.620056i	$0.212890\pi$
$228$	0	0
$229$	−17.0956	−1.12971	−0.564854	−	0.825191i	$-0.691068\pi$
$229$	−17.0956	−1.12971	−0.564854	+	0.825191i	$0.691068\pi$
$230$	0	0
$231$	−8.30051	−0.546134
$232$	0	0
$233$	19.1070	1.25174	0.625871	−	0.779926i	$-0.284744\pi$
$233$	19.1070	1.25174	0.625871	+	0.779926i	$0.284744\pi$
$234$	0	0
$235$	−4.58719	−0.299235
$236$	0	0
$237$	30.7211	1.99555
$238$	0	0
$239$	−3.82089	−0.247153	−0.123576	−	0.992335i	$-0.539436\pi$
$239$	−3.82089	−0.247153	−0.123576	+	0.992335i	$0.539436\pi$
$240$	0	0
$241$	8.40987	0.541727	0.270864	−	0.962618i	$-0.412691\pi$
$241$	8.40987	0.541727	0.270864	+	0.962618i	$0.412691\pi$
$242$	0	0
$243$	17.5082	1.12315
$244$	0	0
$245$	−15.8338	−1.01158
$246$	0	0
$247$	−7.68830	−0.489195
$248$	0	0
$249$	27.7275	1.75716
$250$	0	0
$251$	−28.6882	−1.81078	−0.905391	−	0.424578i	$-0.860422\pi$
$251$	−28.6882	−1.81078	−0.905391	+	0.424578i	$0.860422\pi$
$252$	0	0
$253$	−21.8812	−1.37566
$254$	0	0
$255$	−0.403849	−0.0252900
$256$	0	0
$257$	−23.8469	−1.48753	−0.743765	−	0.668441i	$-0.766962\pi$
$257$	−23.8469	−1.48753	−0.743765	+	0.668441i	$0.766962\pi$
$258$	0	0
$259$	−5.90208	−0.366737
$260$	0	0
$261$	1.94031	0.120102
$262$	0	0
$263$	−26.7060	−1.64676	−0.823380	−	0.567491i	$-0.807914\pi$
$263$	−26.7060	−1.64676	−0.823380	+	0.567491i	$0.807914\pi$
$264$	0	0
$265$	−23.8295	−1.46384
$266$	0	0
$267$	−10.1400	−0.620559
$268$	0	0
$269$	8.49329	0.517845	0.258923	−	0.965898i	$-0.416633\pi$
$269$	8.49329	0.517845	0.258923	+	0.965898i	$0.416633\pi$
$270$	0	0
$271$	−26.7592	−1.62550	−0.812752	−	0.582610i	$-0.802031\pi$
$271$	−26.7592	−1.62550	−0.812752	+	0.582610i	$0.802031\pi$
$272$	0	0
$273$	2.40246	0.145404
$274$	0	0
$275$	−8.19498	−0.494176
$276$	0	0
$277$	−17.5195	−1.05264	−0.526322	−	0.850285i	$-0.676429\pi$
$277$	−17.5195	−1.05264	−0.526322	+	0.850285i	$0.676429\pi$
$278$	0	0
$279$	11.8142	0.707298
$280$	0	0
$281$	1.42043	0.0847360	0.0423680	−	0.999102i	$-0.486510\pi$
$281$	1.42043	0.0847360	0.0423680	+	0.999102i	$0.486510\pi$
$282$	0	0
$283$	18.0096	1.07056	0.535279	−	0.844676i	$-0.320207\pi$
$283$	18.0096	1.07056	0.535279	+	0.844676i	$0.320207\pi$
$284$	0	0
$285$	46.3979	2.74837
$286$	0	0
$287$	−5.42952	−0.320494
$288$	0	0
$289$	−16.9955	−0.999737
$290$	0	0
$291$	−35.9029	−2.10466
$292$	0	0
$293$	−24.2228	−1.41511	−0.707557	−	0.706657i	$-0.750203\pi$
$293$	−24.2228	−1.41511	−0.707557	+	0.706657i	$0.750203\pi$
$294$	0	0
$295$	10.1779	0.592579
$296$	0	0
$297$	−8.13773	−0.472199
$298$	0	0
$299$	6.33320	0.366258
$300$	0	0
$301$	−2.93334	−0.169075
$302$	0	0
$303$	0.764057	0.0438939
$304$	0	0
$305$	−34.8539	−1.99573
$306$	0	0
$307$	−19.0849	−1.08923	−0.544617	−	0.838685i	$-0.683325\pi$
$307$	−19.0849	−1.08923	−0.544617	+	0.838685i	$0.683325\pi$
$308$	0	0
$309$	−26.0545	−1.48219
$310$	0	0
$311$	13.0308	0.738910	0.369455	−	0.929249i	$-0.379544\pi$
$311$	13.0308	0.738910	0.369455	+	0.929249i	$0.379544\pi$
$312$	0	0
$313$	33.8313	1.91226	0.956130	−	0.292941i	$-0.0946341\pi$
$313$	33.8313	1.91226	0.956130	+	0.292941i	$0.0946341\pi$
$314$	0	0
$315$	−5.69431	−0.320838
$316$	0	0
$317$	−5.91436	−0.332184	−0.166092	−	0.986110i	$-0.553115\pi$
$317$	−5.91436	−0.332184	−0.166092	+	0.986110i	$0.553115\pi$
$318$	0	0
$319$	−3.45501	−0.193443
$320$	0	0
$321$	12.8377	0.716530
$322$	0	0
$323$	−0.514496	−0.0286273
$324$	0	0
$325$	2.37192	0.131570
$326$	0	0
$327$	−12.9255	−0.714779
$328$	0	0
$329$	1.82614	0.100679
$330$	0	0
$331$	4.33274	0.238149	0.119075	−	0.992885i	$-0.462007\pi$
$331$	4.33274	0.238149	0.119075	+	0.992885i	$0.462007\pi$
$332$	0	0
$333$	10.5949	0.580598
$334$	0	0
$335$	37.4650	2.04693
$336$	0	0
$337$	−30.8964	−1.68303	−0.841516	−	0.540232i	$-0.818336\pi$
$337$	−30.8964	−1.68303	−0.841516	+	0.540232i	$0.818336\pi$
$338$	0	0
$339$	−0.978003	−0.0531179
$340$	0	0
$341$	−21.0369	−1.13921
$342$	0	0
$343$	13.8696	0.748886
$344$	0	0
$345$	−38.2200	−2.05770
$346$	0	0
$347$	19.5059	1.04713	0.523565	−	0.851985i	$-0.324602\pi$
$347$	19.5059	1.04713	0.523565	+	0.851985i	$0.324602\pi$
$348$	0	0
$349$	−6.22426	−0.333177	−0.166589	−	0.986026i	$-0.553275\pi$
$349$	−6.22426	−0.333177	−0.166589	+	0.986026i	$0.553275\pi$
$350$	0	0
$351$	2.35535	0.125719
$352$	0	0
$353$	27.2893	1.45247	0.726233	−	0.687449i	$-0.241270\pi$
$353$	27.2893	1.45247	0.726233	+	0.687449i	$0.241270\pi$
$354$	0	0
$355$	−9.30489	−0.493852
$356$	0	0
$357$	0.160771	0.00850891
$358$	0	0
$359$	−25.5988	−1.35105	−0.675526	−	0.737336i	$-0.736083\pi$
$359$	−25.5988	−1.35105	−0.675526	+	0.737336i	$0.736083\pi$
$360$	0	0
$361$	40.1100	2.11105
$362$	0	0
$363$	−2.08279	−0.109318
$364$	0	0
$365$	−11.1251	−0.582315
$366$	0	0
$367$	−23.8682	−1.24591	−0.622954	−	0.782258i	$-0.714068\pi$
$367$	−23.8682	−1.24591	−0.622954	+	0.782258i	$0.714068\pi$
$368$	0	0
$369$	9.74662	0.507389
$370$	0	0
$371$	9.48646	0.492513
$372$	0	0
$373$	−29.3240	−1.51834	−0.759170	−	0.650893i	$-0.774395\pi$
$373$	−29.3240	−1.51834	−0.759170	+	0.650893i	$0.774395\pi$
$374$	0	0
$375$	15.8601	0.819014
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−17.9683	−0.922970	−0.461485	−	0.887148i	$-0.652683\pi$
$379$	−17.9683	−0.922970	−0.461485	+	0.887148i	$0.652683\pi$
$380$	0	0
$381$	−11.8055	−0.604815
$382$	0	0
$383$	−28.8274	−1.47301	−0.736507	−	0.676430i	$-0.763526\pi$
$383$	−28.8274	−1.47301	−0.736507	+	0.676430i	$0.763526\pi$
$384$	0	0
$385$	10.1395	0.516758
$386$	0	0
$387$	5.26569	0.267670
$388$	0	0
$389$	−10.1865	−0.516475	−0.258237	−	0.966081i	$-0.583142\pi$
$389$	−10.1865	−0.516475	−0.258237	+	0.966081i	$0.583142\pi$
$390$	0	0
$391$	0.423813	0.0214332
$392$	0	0
$393$	3.67539	0.185399
$394$	0	0
$395$	−37.5276	−1.88822
$396$	0	0
$397$	−11.0161	−0.552884	−0.276442	−	0.961031i	$-0.589155\pi$
$397$	−11.0161	−0.552884	−0.276442	+	0.961031i	$0.589155\pi$
$398$	0	0
$399$	−18.4708	−0.924699
$400$	0	0
$401$	16.1994	0.808961	0.404480	−	0.914547i	$-0.367452\pi$
$401$	16.1994	0.808961	0.404480	+	0.914547i	$0.367452\pi$
$402$	0	0
$403$	6.08882	0.303306
$404$	0	0
$405$	−30.0188	−1.49165
$406$	0	0
$407$	−18.8658	−0.935141
$408$	0	0
$409$	1.11092	0.0549314	0.0274657	−	0.999623i	$-0.491256\pi$
$409$	1.11092	0.0549314	0.0274657	+	0.999623i	$0.491256\pi$
$410$	0	0
$411$	38.8979	1.91869
$412$	0	0
$413$	−4.05178	−0.199375
$414$	0	0
$415$	−33.8706	−1.66264
$416$	0	0
$417$	30.8077	1.50866
$418$	0	0
$419$	−23.2018	−1.13348	−0.566741	−	0.823896i	$-0.691796\pi$
$419$	−23.2018	−1.13348	−0.566741	+	0.823896i	$0.691796\pi$
$420$	0	0
$421$	29.4165	1.43367	0.716836	−	0.697241i	$-0.245589\pi$
$421$	29.4165	1.43367	0.716836	+	0.697241i	$0.245589\pi$
$422$	0	0
$423$	−3.27814	−0.159389
$424$	0	0
$425$	0.158727	0.00769939
$426$	0	0
$427$	13.8752	0.671469
$428$	0	0
$429$	7.67938	0.370764
$430$	0	0
$431$	2.06829	0.0996260	0.0498130	−	0.998759i	$-0.484137\pi$
$431$	2.06829	0.0996260	0.0498130	+	0.998759i	$0.484137\pi$
$432$	0	0
$433$	−10.3663	−0.498173	−0.249086	−	0.968481i	$-0.580130\pi$
$433$	−10.3663	−0.498173	−0.249086	+	0.968481i	$0.580130\pi$
$434$	0	0
$435$	−6.03486	−0.289350
$436$	0	0
$437$	−48.6916	−2.32923
$438$	0	0
$439$	38.5820	1.84142	0.920710	−	0.390247i	$-0.127610\pi$
$439$	38.5820	1.84142	0.920710	+	0.390247i	$0.127610\pi$
$440$	0	0
$441$	−11.3153	−0.538824
$442$	0	0
$443$	−10.0153	−0.475840	−0.237920	−	0.971285i	$-0.576466\pi$
$443$	−10.0153	−0.475840	−0.237920	+	0.971285i	$0.576466\pi$
$444$	0	0
$445$	12.3866	0.587181
$446$	0	0
$447$	−0.201464	−0.00952890
$448$	0	0
$449$	−11.6569	−0.550124	−0.275062	−	0.961427i	$-0.588698\pi$
$449$	−11.6569	−0.550124	−0.275062	+	0.961427i	$0.588698\pi$
$450$	0	0
$451$	−17.3553	−0.817227
$452$	0	0
$453$	13.8494	0.650704
$454$	0	0
$455$	−2.93474	−0.137583
$456$	0	0
$457$	−9.86913	−0.461658	−0.230829	−	0.972994i	$-0.574144\pi$
$457$	−9.86913	−0.461658	−0.230829	+	0.972994i	$0.574144\pi$
$458$	0	0
$459$	0.157618	0.00735698
$460$	0	0
$461$	−39.2879	−1.82982	−0.914911	−	0.403656i	$-0.867739\pi$
$461$	−39.2879	−1.82982	−0.914911	+	0.403656i	$0.867739\pi$
$462$	0	0
$463$	23.3324	1.08435	0.542175	−	0.840266i	$-0.317601\pi$
$463$	23.3324	1.08435	0.542175	+	0.840266i	$0.317601\pi$
$464$	0	0
$465$	−36.7452	−1.70402
$466$	0	0
$467$	−41.7977	−1.93417	−0.967083	−	0.254462i	$-0.918102\pi$
$467$	−41.7977	−1.93417	−0.967083	+	0.254462i	$0.918102\pi$
$468$	0	0
$469$	−14.9147	−0.688698
$470$	0	0
$471$	6.71732	0.309518
$472$	0	0
$473$	−9.37631	−0.431123
$474$	0	0
$475$	−18.2360	−0.836725
$476$	0	0
$477$	−17.0293	−0.779719
$478$	0	0
$479$	11.7437	0.536584	0.268292	−	0.963338i	$-0.413541\pi$
$479$	11.7437	0.536584	0.268292	+	0.963338i	$0.413541\pi$
$480$	0	0
$481$	5.46042	0.248974
$482$	0	0
$483$	15.2153	0.692318
$484$	0	0
$485$	43.8573	1.99146
$486$	0	0
$487$	9.59594	0.434833	0.217417	−	0.976079i	$-0.430237\pi$
$487$	9.59594	0.434833	0.217417	+	0.976079i	$0.430237\pi$
$488$	0	0
$489$	25.0379	1.13225
$490$	0	0
$491$	−6.63827	−0.299581	−0.149790	−	0.988718i	$-0.547860\pi$
$491$	−6.63827	−0.299581	−0.149790	+	0.988718i	$0.547860\pi$
$492$	0	0
$493$	0.0669193	0.00301389
$494$	0	0
$495$	−18.2017	−0.818103
$496$	0	0
$497$	3.70425	0.166158
$498$	0	0
$499$	6.53307	0.292460	0.146230	−	0.989251i	$-0.453286\pi$
$499$	6.53307	0.292460	0.146230	+	0.989251i	$0.453286\pi$
$500$	0	0
$501$	−7.78479	−0.347799
$502$	0	0
$503$	−23.5593	−1.05046	−0.525228	−	0.850961i	$-0.676020\pi$
$503$	−23.5593	−1.05046	−0.525228	+	0.850961i	$0.676020\pi$
$504$	0	0
$505$	−0.933338	−0.0415330
$506$	0	0
$507$	−2.22268	−0.0987128
$508$	0	0
$509$	22.7897	1.01013	0.505067	−	0.863080i	$-0.331468\pi$
$509$	22.7897	1.01013	0.505067	+	0.863080i	$0.331468\pi$
$510$	0	0
$511$	4.42887	0.195922
$512$	0	0
$513$	−18.1086	−0.799515
$514$	0	0
$515$	31.8270	1.40247
$516$	0	0
$517$	5.83720	0.256720
$518$	0	0
$519$	40.1439	1.76212
$520$	0	0
$521$	5.48862	0.240461	0.120230	−	0.992746i	$-0.461637\pi$
$521$	5.48862	0.240461	0.120230	+	0.992746i	$0.461637\pi$
$522$	0	0
$523$	20.2408	0.885067	0.442533	−	0.896752i	$-0.354080\pi$
$523$	20.2408	0.885067	0.442533	+	0.896752i	$0.354080\pi$
$524$	0	0
$525$	5.69843	0.248700
$526$	0	0
$527$	0.407459	0.0177492
$528$	0	0
$529$	17.1094	0.743887
$530$	0	0
$531$	7.27342	0.315640
$532$	0	0
$533$	5.02322	0.217580
$534$	0	0
$535$	−15.6819	−0.677990
$536$	0	0
$537$	−26.1993	−1.13058
$538$	0	0
$539$	20.1485	0.867858
$540$	0	0
$541$	−35.2985	−1.51760	−0.758800	−	0.651324i	$-0.774214\pi$
$541$	−35.2985	−1.51760	−0.758800	+	0.651324i	$0.774214\pi$
$542$	0	0
$543$	−25.5707	−1.09734
$544$	0	0
$545$	15.7892	0.676333
$546$	0	0
$547$	−26.3839	−1.12809	−0.564047	−	0.825743i	$-0.690756\pi$
$547$	−26.3839	−1.12809	−0.564047	+	0.825743i	$0.690756\pi$
$548$	0	0
$549$	−24.9077	−1.06303
$550$	0	0
$551$	−7.68830	−0.327533
$552$	0	0
$553$	14.9396	0.635297
$554$	0	0
$555$	−32.9529	−1.39877
$556$	0	0
$557$	1.33830	0.0567054	0.0283527	−	0.999598i	$-0.490974\pi$
$557$	1.33830	0.0567054	0.0283527	+	0.999598i	$0.490974\pi$
$558$	0	0
$559$	2.71383	0.114783
$560$	0	0
$561$	0.513899	0.0216968
$562$	0	0
$563$	−31.6606	−1.33434	−0.667168	−	0.744907i	$-0.732494\pi$
$563$	−31.6606	−1.33434	−0.667168	+	0.744907i	$0.732494\pi$
$564$	0	0
$565$	1.19468	0.0502608
$566$	0	0
$567$	11.9504	0.501869
$568$	0	0
$569$	33.6794	1.41191	0.705957	−	0.708255i	$-0.250517\pi$
$569$	33.6794	1.41191	0.705957	+	0.708255i	$0.250517\pi$
$570$	0	0
$571$	−9.08200	−0.380070	−0.190035	−	0.981777i	$-0.560860\pi$
$571$	−9.08200	−0.380070	−0.190035	+	0.981777i	$0.560860\pi$
$572$	0	0
$573$	−6.40957	−0.267764
$574$	0	0
$575$	15.0218	0.626453
$576$	0	0
$577$	23.3700	0.972906	0.486453	−	0.873707i	$-0.338290\pi$
$577$	23.3700	0.972906	0.486453	+	0.873707i	$0.338290\pi$
$578$	0	0
$579$	10.1642	0.422411
$580$	0	0
$581$	13.4838	0.559402
$582$	0	0
$583$	30.3231	1.25586
$584$	0	0
$585$	5.26820	0.217813
$586$	0	0
$587$	−7.97134	−0.329012	−0.164506	−	0.986376i	$-0.552603\pi$
$587$	−7.97134	−0.329012	−0.164506	+	0.986376i	$0.552603\pi$
$588$	0	0
$589$	−46.8127	−1.92888
$590$	0	0
$591$	4.59968	0.189206
$592$	0	0
$593$	−11.6414	−0.478057	−0.239028	−	0.971013i	$-0.576829\pi$
$593$	−11.6414	−0.478057	−0.239028	+	0.971013i	$0.576829\pi$
$594$	0	0
$595$	−0.196391	−0.00805123
$596$	0	0
$597$	0.0699563	0.00286312
$598$	0	0
$599$	21.3446	0.872118	0.436059	−	0.899918i	$-0.356374\pi$
$599$	21.3446	0.872118	0.436059	+	0.899918i	$0.356374\pi$
$600$	0	0
$601$	−39.3213	−1.60395	−0.801975	−	0.597358i	$-0.796217\pi$
$601$	−39.3213	−1.60395	−0.801975	+	0.597358i	$0.796217\pi$
$602$	0	0
$603$	26.7737	1.09031
$604$	0	0
$605$	2.54424	0.103438
$606$	0	0
$607$	40.2156	1.63230	0.816151	−	0.577839i	$-0.196104\pi$
$607$	40.2156	1.63230	0.816151	+	0.577839i	$0.196104\pi$
$608$	0	0
$609$	2.40246	0.0973526
$610$	0	0
$611$	−1.68949	−0.0683495
$612$	0	0
$613$	−20.1967	−0.815736	−0.407868	−	0.913041i	$-0.633728\pi$
$613$	−20.1967	−0.815736	−0.407868	+	0.913041i	$0.633728\pi$
$614$	0	0
$615$	−30.3144	−1.22240
$616$	0	0
$617$	−15.4396	−0.621575	−0.310788	−	0.950479i	$-0.600593\pi$
$617$	−15.4396	−0.621575	−0.310788	+	0.950479i	$0.600593\pi$
$618$	0	0
$619$	36.9373	1.48464	0.742319	−	0.670047i	$-0.233726\pi$
$619$	36.9373	1.48464	0.742319	+	0.670047i	$0.233726\pi$
$620$	0	0
$621$	14.9169	0.598593
$622$	0	0
$623$	−4.93106	−0.197559
$624$	0	0
$625$	−31.2336	−1.24934
$626$	0	0
$627$	−59.0414	−2.35789
$628$	0	0
$629$	0.365407	0.0145697
$630$	0	0
$631$	35.6659	1.41984	0.709918	−	0.704284i	$-0.248732\pi$
$631$	35.6659	1.41984	0.709918	+	0.704284i	$0.248732\pi$
$632$	0	0
$633$	3.73904	0.148613
$634$	0	0
$635$	14.4211	0.572283
$636$	0	0
$637$	−5.83169	−0.231060
$638$	0	0
$639$	−6.64956	−0.263053
$640$	0	0
$641$	4.99270	0.197200	0.0985998	−	0.995127i	$-0.468564\pi$
$641$	4.99270	0.197200	0.0985998	+	0.995127i	$0.468564\pi$
$642$	0	0
$643$	−8.13152	−0.320676	−0.160338	−	0.987062i	$-0.551258\pi$
$643$	−8.13152	−0.320676	−0.160338	+	0.987062i	$0.551258\pi$
$644$	0	0
$645$	−16.3776	−0.644868
$646$	0	0
$647$	32.2045	1.26609	0.633044	−	0.774116i	$-0.281805\pi$
$647$	32.2045	1.26609	0.633044	+	0.774116i	$0.281805\pi$
$648$	0	0
$649$	−12.9514	−0.508386
$650$	0	0
$651$	14.6281	0.573322
$652$	0	0
$653$	30.5557	1.19574	0.597869	−	0.801594i	$-0.296014\pi$
$653$	30.5557	1.19574	0.597869	+	0.801594i	$0.296014\pi$
$654$	0	0
$655$	−4.48969	−0.175427
$656$	0	0
$657$	−7.95034	−0.310172
$658$	0	0
$659$	18.9602	0.738583	0.369292	−	0.929314i	$-0.379600\pi$
$659$	18.9602	0.738583	0.369292	+	0.929314i	$0.379600\pi$
$660$	0	0
$661$	3.45977	0.134569	0.0672847	−	0.997734i	$-0.478566\pi$
$661$	3.45977	0.134569	0.0672847	+	0.997734i	$0.478566\pi$
$662$	0	0
$663$	−0.148740	−0.00577660
$664$	0	0
$665$	22.5631	0.874961
$666$	0	0
$667$	6.33320	0.245222
$668$	0	0
$669$	23.6265	0.913453
$670$	0	0
$671$	44.3516	1.71218
$672$	0	0
$673$	1.77539	0.0684363	0.0342182	−	0.999414i	$-0.489106\pi$
$673$	1.77539	0.0684363	0.0342182	+	0.999414i	$0.489106\pi$
$674$	0	0
$675$	5.58668	0.215031
$676$	0	0
$677$	−15.0176	−0.577174	−0.288587	−	0.957454i	$-0.593186\pi$
$677$	−15.0176	−0.577174	−0.288587	+	0.957454i	$0.593186\pi$
$678$	0	0
$679$	−17.4595	−0.670033
$680$	0	0
$681$	−52.5467	−2.01359
$682$	0	0
$683$	−47.1114	−1.80267	−0.901334	−	0.433124i	$-0.857411\pi$
$683$	−47.1114	−1.80267	−0.901334	+	0.433124i	$0.857411\pi$
$684$	0	0
$685$	−47.5159	−1.81549
$686$	0	0
$687$	37.9980	1.44972
$688$	0	0
$689$	−8.77658	−0.334361
$690$	0	0
$691$	−15.2077	−0.578527	−0.289264	−	0.957249i	$-0.593410\pi$
$691$	−15.2077	−0.578527	−0.289264	+	0.957249i	$0.593410\pi$
$692$	0	0
$693$	7.24602	0.275254
$694$	0	0
$695$	−37.6333	−1.42751
$696$	0	0
$697$	0.336150	0.0127326
$698$	0	0
$699$	−42.4688	−1.60632
$700$	0	0
$701$	1.72402	0.0651154	0.0325577	−	0.999470i	$-0.489635\pi$
$701$	1.72402	0.0651154	0.0325577	+	0.999470i	$0.489635\pi$
$702$	0	0
$703$	−41.9813	−1.58336
$704$	0	0
$705$	10.1959	0.383998
$706$	0	0
$707$	0.371559	0.0139739
$708$	0	0
$709$	−29.7521	−1.11736	−0.558681	−	0.829383i	$-0.688692\pi$
$709$	−29.7521	−1.11736	−0.558681	+	0.829383i	$0.688692\pi$
$710$	0	0
$711$	−26.8183	−1.00577
$712$	0	0
$713$	38.5617	1.44415
$714$	0	0
$715$	−9.38078	−0.350821
$716$	0	0
$717$	8.49262	0.317163
$718$	0	0
$719$	18.0988	0.674972	0.337486	−	0.941330i	$-0.390423\pi$
$719$	18.0988	0.674972	0.337486	+	0.941330i	$0.390423\pi$
$720$	0	0
$721$	−12.6702	−0.471865
$722$	0	0
$723$	−18.6925	−0.695180
$724$	0	0
$725$	2.37192	0.0880907
$726$	0	0
$727$	23.5027	0.871665	0.435833	−	0.900028i	$-0.356454\pi$
$727$	23.5027	0.871665	0.435833	+	0.900028i	$0.356454\pi$
$728$	0	0
$729$	−5.74680	−0.212844
$730$	0	0
$731$	0.181608	0.00671701
$732$	0	0
$733$	−6.01171	−0.222047	−0.111024	−	0.993818i	$-0.535413\pi$
$733$	−6.01171	−0.222047	−0.111024	+	0.993818i	$0.535413\pi$
$734$	0	0
$735$	35.1935	1.29813
$736$	0	0
$737$	−47.6743	−1.75611
$738$	0	0
$739$	28.0692	1.03254	0.516270	−	0.856426i	$-0.327320\pi$
$739$	28.0692	1.03254	0.516270	+	0.856426i	$0.327320\pi$
$740$	0	0
$741$	17.0886	0.627767
$742$	0	0
$743$	−9.86348	−0.361856	−0.180928	−	0.983496i	$-0.557910\pi$
$743$	−9.86348	−0.361856	−0.180928	+	0.983496i	$0.557910\pi$
$744$	0	0
$745$	0.246099	0.00901636
$746$	0	0
$747$	−24.2050	−0.885615
$748$	0	0
$749$	6.24293	0.228112
$750$	0	0
$751$	−2.83450	−0.103432	−0.0517162	−	0.998662i	$-0.516469\pi$
$751$	−2.83450	−0.103432	−0.0517162	+	0.998662i	$0.516469\pi$
$752$	0	0
$753$	63.7647	2.32372
$754$	0	0
$755$	−16.9179	−0.615704
$756$	0	0
$757$	−30.1452	−1.09564	−0.547822	−	0.836595i	$-0.684543\pi$
$757$	−30.1452	−1.09564	−0.547822	+	0.836595i	$0.684543\pi$
$758$	0	0
$759$	48.6350	1.76534
$760$	0	0
$761$	−19.7757	−0.716868	−0.358434	−	0.933555i	$-0.616689\pi$
$761$	−19.7757	−0.716868	−0.358434	+	0.933555i	$0.616689\pi$
$762$	0	0
$763$	−6.28561	−0.227554
$764$	0	0
$765$	0.352544	0.0127463
$766$	0	0
$767$	3.74858	0.135353
$768$	0	0
$769$	−9.00566	−0.324753	−0.162376	−	0.986729i	$-0.551916\pi$
$769$	−9.00566	−0.324753	−0.162376	+	0.986729i	$0.551916\pi$
$770$	0	0
$771$	53.0041	1.90890
$772$	0	0
$773$	−51.4107	−1.84911	−0.924557	−	0.381044i	$-0.875564\pi$
$773$	−51.4107	−1.84911	−0.924557	+	0.381044i	$0.875564\pi$
$774$	0	0
$775$	14.4422	0.518778
$776$	0	0
$777$	13.1184	0.470621
$778$	0	0
$779$	−38.6200	−1.38371
$780$	0	0
$781$	11.8405	0.423686
$782$	0	0
$783$	2.35535	0.0841732
$784$	0	0
$785$	−8.20557	−0.292869
$786$	0	0
$787$	6.53638	0.232997	0.116498	−	0.993191i	$-0.462833\pi$
$787$	6.53638	0.232997	0.116498	+	0.993191i	$0.462833\pi$
$788$	0	0
$789$	59.3588	2.11323
$790$	0	0
$791$	−0.475600	−0.0169104
$792$	0	0
$793$	−12.8369	−0.455853
$794$	0	0
$795$	52.9655	1.87849
$796$	0	0
$797$	7.77047	0.275244	0.137622	−	0.990485i	$-0.456054\pi$
$797$	7.77047	0.275244	0.137622	+	0.990485i	$0.456054\pi$
$798$	0	0
$799$	−0.113060	−0.00399976
$800$	0	0
$801$	8.85184	0.312764
$802$	0	0
$803$	14.1567	0.499580
$804$	0	0
$805$	−18.5863	−0.655080
$806$	0	0
$807$	−18.8779	−0.664533
$808$	0	0
$809$	−23.9942	−0.843593	−0.421796	−	0.906691i	$-0.638600\pi$
$809$	−23.9942	−0.843593	−0.421796	+	0.906691i	$0.638600\pi$
$810$	0	0
$811$	−35.9008	−1.26065	−0.630324	−	0.776332i	$-0.717078\pi$
$811$	−35.9008	−1.26065	−0.630324	+	0.776332i	$0.717078\pi$
$812$	0	0
$813$	59.4771	2.08595
$814$	0	0
$815$	−30.5852	−1.07135
$816$	0	0
$817$	−20.8648	−0.729966
$818$	0	0
$819$	−2.09725	−0.0732840
$820$	0	0
$821$	33.8997	1.18311	0.591554	−	0.806265i	$-0.298515\pi$
$821$	33.8997	1.18311	0.591554	+	0.806265i	$0.298515\pi$
$822$	0	0
$823$	33.9774	1.18438	0.592189	−	0.805799i	$-0.298264\pi$
$823$	33.9774	1.18438	0.592189	+	0.805799i	$0.298264\pi$
$824$	0	0
$825$	18.2148	0.634159
$826$	0	0
$827$	43.0295	1.49628	0.748142	−	0.663539i	$-0.230946\pi$
$827$	43.0295	1.49628	0.748142	+	0.663539i	$0.230946\pi$
$828$	0	0
$829$	−53.1127	−1.84468	−0.922341	−	0.386377i	$-0.873726\pi$
$829$	−53.1127	−1.84468	−0.922341	+	0.386377i	$0.873726\pi$
$830$	0	0
$831$	38.9403	1.35082
$832$	0	0
$833$	−0.390253	−0.0135215
$834$	0	0
$835$	9.50954	0.329091
$836$	0	0
$837$	14.3413	0.495707
$838$	0	0
$839$	−14.5039	−0.500730	−0.250365	−	0.968151i	$-0.580551\pi$
$839$	−14.5039	−0.500730	−0.250365	+	0.968151i	$0.580551\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.15717	−0.108739
$844$	0	0
$845$	2.71513	0.0934032
$846$	0	0
$847$	−1.01285	−0.0348021
$848$	0	0
$849$	−40.0295	−1.37381
$850$	0	0
$851$	34.5819	1.18545
$852$	0	0
$853$	−5.64199	−0.193178	−0.0965890	−	0.995324i	$-0.530793\pi$
$853$	−5.64199	−0.193178	−0.0965890	+	0.995324i	$0.530793\pi$
$854$	0	0
$855$	−40.5035	−1.38519
$856$	0	0
$857$	45.4490	1.55251	0.776255	−	0.630419i	$-0.217117\pi$
$857$	45.4490	1.55251	0.776255	+	0.630419i	$0.217117\pi$
$858$	0	0
$859$	−31.5302	−1.07580	−0.537899	−	0.843009i	$-0.680782\pi$
$859$	−31.5302	−1.07580	−0.537899	+	0.843009i	$0.680782\pi$
$860$	0	0
$861$	12.0681	0.411279
$862$	0	0
$863$	−51.2123	−1.74329	−0.871643	−	0.490141i	$-0.836945\pi$
$863$	−51.2123	−1.74329	−0.871643	+	0.490141i	$0.836945\pi$
$864$	0	0
$865$	−49.0380	−1.66734
$866$	0	0
$867$	37.7756	1.28293
$868$	0	0
$869$	47.7539	1.61994
$870$	0	0
$871$	13.7986	0.467549
$872$	0	0
$873$	31.3418	1.06076
$874$	0	0
$875$	7.71274	0.260738
$876$	0	0
$877$	−30.2884	−1.02277	−0.511383	−	0.859353i	$-0.670867\pi$
$877$	−30.2884	−1.02277	−0.511383	+	0.859353i	$0.670867\pi$
$878$	0	0
$879$	53.8397	1.81597
$880$	0	0
$881$	42.8750	1.44450	0.722248	−	0.691635i	$-0.243109\pi$
$881$	42.8750	1.44450	0.722248	+	0.691635i	$0.243109\pi$
$882$	0	0
$883$	−46.6460	−1.56976	−0.784882	−	0.619645i	$-0.787277\pi$
$883$	−46.6460	−1.56976	−0.784882	+	0.619645i	$0.787277\pi$
$884$	0	0
$885$	−22.6222	−0.760436
$886$	0	0
$887$	−12.2700	−0.411986	−0.205993	−	0.978553i	$-0.566042\pi$
$887$	−12.2700	−0.411986	−0.205993	+	0.978553i	$0.566042\pi$
$888$	0	0
$889$	−5.74099	−0.192547
$890$	0	0
$891$	38.1990	1.27971
$892$	0	0
$893$	12.9893	0.434671
$894$	0	0
$895$	32.0039	1.06977
$896$	0	0
$897$	−14.0767	−0.470007
$898$	0	0
$899$	6.08882	0.203073
$900$	0	0
$901$	−0.587323	−0.0195666
$902$	0	0
$903$	6.51988	0.216968
$904$	0	0
$905$	31.2360	1.03832
$906$	0	0
$907$	7.62541	0.253197	0.126599	−	0.991954i	$-0.459594\pi$
$907$	7.62541	0.253197	0.126599	+	0.991954i	$0.459594\pi$
$908$	0	0
$909$	−0.666992	−0.0221227
$910$	0	0
$911$	−31.4528	−1.04208	−0.521039	−	0.853533i	$-0.674455\pi$
$911$	−31.4528	−1.04208	−0.521039	+	0.853533i	$0.674455\pi$
$912$	0	0
$913$	43.1005	1.42642
$914$	0	0
$915$	77.4691	2.56105
$916$	0	0
$917$	1.78733	0.0590229
$918$	0	0
$919$	−16.6728	−0.549985	−0.274992	−	0.961446i	$-0.588675\pi$
$919$	−16.6728	−0.549985	−0.274992	+	0.961446i	$0.588675\pi$
$920$	0	0
$921$	42.4197	1.39778
$922$	0	0
$923$	−3.42706	−0.112803
$924$	0	0
$925$	12.9516	0.425847
$926$	0	0
$927$	22.7446	0.747030
$928$	0	0
$929$	−22.3582	−0.733549	−0.366774	−	0.930310i	$-0.619538\pi$
$929$	−22.3582	−0.733549	−0.366774	+	0.930310i	$0.619538\pi$
$930$	0	0
$931$	44.8358	1.46943
$932$	0	0
$933$	−28.9633	−0.948217
$934$	0	0
$935$	−0.627755	−0.0205298
$936$	0	0
$937$	4.46764	0.145952	0.0729758	−	0.997334i	$-0.476750\pi$
$937$	4.46764	0.145952	0.0729758	+	0.997334i	$0.476750\pi$
$938$	0	0
$939$	−75.1963	−2.45394
$940$	0	0
$941$	15.8186	0.515671	0.257835	−	0.966189i	$-0.416991\pi$
$941$	15.8186	0.515671	0.257835	+	0.966189i	$0.416991\pi$
$942$	0	0
$943$	31.8131	1.03598
$944$	0	0
$945$	−6.91232	−0.224858
$946$	0	0
$947$	6.07359	0.197365	0.0986826	−	0.995119i	$-0.468537\pi$
$947$	6.07359	0.197365	0.0986826	+	0.995119i	$0.468537\pi$
$948$	0	0
$949$	−4.09745	−0.133009
$950$	0	0
$951$	13.1457	0.426280
$952$	0	0
$953$	−29.5428	−0.956986	−0.478493	−	0.878091i	$-0.658817\pi$
$953$	−29.5428	−0.956986	−0.478493	+	0.878091i	$0.658817\pi$
$954$	0	0
$955$	7.82964	0.253361
$956$	0	0
$957$	7.67938	0.248239
$958$	0	0
$959$	18.9159	0.610828
$960$	0	0
$961$	6.07368	0.195925
$962$	0	0
$963$	−11.2068	−0.361134
$964$	0	0
$965$	−12.4162	−0.399690
$966$	0	0
$967$	−37.2366	−1.19745	−0.598725	−	0.800955i	$-0.704326\pi$
$967$	−37.2366	−1.19745	−0.598725	+	0.800955i	$0.704326\pi$
$968$	0	0
$969$	1.14356	0.0367365
$970$	0	0
$971$	−13.0335	−0.418266	−0.209133	−	0.977887i	$-0.567064\pi$
$971$	−13.0335	−0.418266	−0.209133	+	0.977887i	$0.567064\pi$
$972$	0	0
$973$	14.9817	0.480291
$974$	0	0
$975$	−5.27201	−0.168840
$976$	0	0
$977$	39.3299	1.25827	0.629137	−	0.777295i	$-0.283409\pi$
$977$	39.3299	1.25827	0.629137	+	0.777295i	$0.283409\pi$
$978$	0	0
$979$	−15.7620	−0.503755
$980$	0	0
$981$	11.2834	0.360252
$982$	0	0
$983$	−16.3232	−0.520629	−0.260314	−	0.965524i	$-0.583826\pi$
$983$	−16.3232	−0.520629	−0.260314	+	0.965524i	$0.583826\pi$
$984$	0	0
$985$	−5.61876	−0.179029
$986$	0	0
$987$	−4.05894	−0.129197
$988$	0	0
$989$	17.1872	0.546523
$990$	0	0
$991$	18.2213	0.578820	0.289410	−	0.957205i	$-0.406541\pi$
$991$	18.2213	0.578820	0.289410	+	0.957205i	$0.406541\pi$
$992$	0	0
$993$	−9.63031	−0.305609
$994$	0	0
$995$	−0.0854554	−0.00270912
$996$	0	0
$997$	45.1081	1.42859	0.714294	−	0.699846i	$-0.246748\pi$
$997$	45.1081	1.42859	0.714294	+	0.699846i	$0.246748\pi$
$998$	0	0
$999$	12.8612	0.406909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bd.1.4		12
4.3	odd	2		3016.2.a.j.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.j.1.9	✓	12	4.3	odd	2
6032.2.a.bd.1.4		12	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.22268 q^{3} +2.71513 q^{5} -1.08088 q^{7} +1.94031 q^{9} +O(q^{10})\)	\(q-2.22268 q^{3} +2.71513 q^{5} -1.08088 q^{7} +1.94031 q^{9} -3.45501 q^{11} +1.00000 q^{13} -6.03486 q^{15} +0.0669193 q^{17} -7.68830 q^{19} +2.40246 q^{21} +6.33320 q^{23} +2.37192 q^{25} +2.35535 q^{27} +1.00000 q^{29} +6.08882 q^{31} +7.67938 q^{33} -2.93474 q^{35} +5.46042 q^{37} -2.22268 q^{39} +5.02322 q^{41} +2.71383 q^{43} +5.26820 q^{45} -1.68949 q^{47} -5.83169 q^{49} -0.148740 q^{51} -8.77658 q^{53} -9.38078 q^{55} +17.0886 q^{57} +3.74858 q^{59} -12.8369 q^{61} -2.09725 q^{63} +2.71513 q^{65} +13.7986 q^{67} -14.0767 q^{69} -3.42706 q^{71} -4.09745 q^{73} -5.27201 q^{75} +3.73446 q^{77} -13.8217 q^{79} -11.0561 q^{81} -12.4748 q^{83} +0.181694 q^{85} -2.22268 q^{87} +4.56207 q^{89} -1.08088 q^{91} -13.5335 q^{93} -20.8747 q^{95} +16.1529 q^{97} -6.70379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more