LMFDB - Embedded newform 1352.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, 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("1352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.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:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3728753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 $ x^6 - x^5 - 10x^4 + 11x^3 + 22x^2 - 18x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.83636$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.589380 q^{3} +3.45555 q^{5} +4.70032 q^{7} -2.65263 q^{9} +O(q^{10})$	$q+0.589380 q^{3} +3.45555 q^{5} +4.70032 q^{7} -2.65263 q^{9} -3.87476 q^{11} +2.03663 q^{15} +6.92820 q^{17} +1.86980 q^{19} +2.77027 q^{21} +1.64089 q^{23} +6.94085 q^{25} -3.33155 q^{27} -2.54563 q^{29} -1.55220 q^{31} -2.28371 q^{33} +16.2422 q^{35} -4.96759 q^{37} +2.76736 q^{41} +2.32469 q^{43} -9.16631 q^{45} -5.65642 q^{47} +15.0930 q^{49} +4.08334 q^{51} -2.83998 q^{53} -13.3895 q^{55} +1.10202 q^{57} -10.5071 q^{59} +4.39485 q^{61} -12.4682 q^{63} -12.4028 q^{67} +0.967109 q^{69} -5.07415 q^{71} +13.2196 q^{73} +4.09080 q^{75} -18.2126 q^{77} -4.92790 q^{79} +5.99435 q^{81} +6.02834 q^{83} +23.9408 q^{85} -1.50034 q^{87} -2.59891 q^{89} -0.914835 q^{93} +6.46118 q^{95} -7.23082 q^{97} +10.2783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.589380	0.340278	0.170139	−	0.985420i	$-0.445578\pi$
$3$	0.589380	0.340278	0.170139	+	0.985420i	$0.445578\pi$
$4$	0	0
$5$	3.45555	1.54537	0.772685	−	0.634789i	$-0.218913\pi$
$5$	3.45555	1.54537	0.772685	+	0.634789i	$0.218913\pi$
$6$	0	0
$7$	4.70032	1.77655	0.888277	−	0.459307i	$-0.151902\pi$
$7$	4.70032	1.77655	0.888277	+	0.459307i	$0.151902\pi$
$8$	0	0
$9$	−2.65263	−0.884211
$10$	0	0
$11$	−3.87476	−1.16829	−0.584143	−	0.811651i	$-0.698569\pi$
$11$	−3.87476	−1.16829	−0.584143	+	0.811651i	$0.698569\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.03663	0.525856
$16$	0	0
$17$	6.92820	1.68034	0.840168	−	0.542327i	$-0.182457\pi$
$17$	6.92820	1.68034	0.840168	+	0.542327i	$0.182457\pi$
$18$	0	0
$19$	1.86980	0.428961	0.214480	−	0.976728i	$-0.431194\pi$
$19$	1.86980	0.428961	0.214480	+	0.976728i	$0.431194\pi$
$20$	0	0
$21$	2.77027	0.604523
$22$	0	0
$23$	1.64089	0.342150	0.171075	−	0.985258i	$-0.445276\pi$
$23$	1.64089	0.342150	0.171075	+	0.985258i	$0.445276\pi$
$24$	0	0
$25$	6.94085	1.38817
$26$	0	0
$27$	−3.33155	−0.641156
$28$	0	0
$29$	−2.54563	−0.472712	−0.236356	−	0.971666i	$-0.575953\pi$
$29$	−2.54563	−0.472712	−0.236356	+	0.971666i	$0.575953\pi$
$30$	0	0
$31$	−1.55220	−0.278784	−0.139392	−	0.990237i	$-0.544515\pi$
$31$	−1.55220	−0.278784	−0.139392	+	0.990237i	$0.544515\pi$
$32$	0	0
$33$	−2.28371	−0.397542
$34$	0	0
$35$	16.2422	2.74544
$36$	0	0
$37$	−4.96759	−0.816667	−0.408333	−	0.912833i	$-0.633890\pi$
$37$	−4.96759	−0.816667	−0.408333	+	0.912833i	$0.633890\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.76736	0.432188	0.216094	−	0.976373i	$-0.430668\pi$
$41$	2.76736	0.432188	0.216094	+	0.976373i	$0.430668\pi$
$42$	0	0
$43$	2.32469	0.354511	0.177256	−	0.984165i	$-0.443278\pi$
$43$	2.32469	0.354511	0.177256	+	0.984165i	$0.443278\pi$
$44$	0	0
$45$	−9.16631	−1.36643
$46$	0	0
$47$	−5.65642	−0.825073	−0.412536	−	0.910941i	$-0.635357\pi$
$47$	−5.65642	−0.825073	−0.412536	+	0.910941i	$0.635357\pi$
$48$	0	0
$49$	15.0930	2.15615
$50$	0	0
$51$	4.08334	0.571782
$52$	0	0
$53$	−2.83998	−0.390101	−0.195051	−	0.980793i	$-0.562487\pi$
$53$	−2.83998	−0.390101	−0.195051	+	0.980793i	$0.562487\pi$
$54$	0	0
$55$	−13.3895	−1.80543
$56$	0	0
$57$	1.10202	0.145966
$58$	0	0
$59$	−10.5071	−1.36790	−0.683952	−	0.729527i	$-0.739740\pi$
$59$	−10.5071	−1.36790	−0.683952	+	0.729527i	$0.739740\pi$
$60$	0	0
$61$	4.39485	0.562703	0.281351	−	0.959605i	$-0.409217\pi$
$61$	4.39485	0.562703	0.281351	+	0.959605i	$0.409217\pi$
$62$	0	0
$63$	−12.4682	−1.57085
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.4028	−1.51524	−0.757620	−	0.652696i	$-0.773638\pi$
$67$	−12.4028	−1.51524	−0.757620	+	0.652696i	$0.773638\pi$
$68$	0	0
$69$	0.967109	0.116426
$70$	0	0
$71$	−5.07415	−0.602191	−0.301095	−	0.953594i	$-0.597352\pi$
$71$	−5.07415	−0.602191	−0.301095	+	0.953594i	$0.597352\pi$
$72$	0	0
$73$	13.2196	1.54724	0.773618	−	0.633653i	$-0.218445\pi$
$73$	13.2196	1.54724	0.773618	+	0.633653i	$0.218445\pi$
$74$	0	0
$75$	4.09080	0.472364
$76$	0	0
$77$	−18.2126	−2.07552
$78$	0	0
$79$	−4.92790	−0.554432	−0.277216	−	0.960808i	$-0.589412\pi$
$79$	−4.92790	−0.554432	−0.277216	+	0.960808i	$0.589412\pi$
$80$	0	0
$81$	5.99435	0.666039
$82$	0	0
$83$	6.02834	0.661696	0.330848	−	0.943684i	$-0.392665\pi$
$83$	6.02834	0.661696	0.330848	+	0.943684i	$0.392665\pi$
$84$	0	0
$85$	23.9408	2.59674
$86$	0	0
$87$	−1.50034	−0.160854
$88$	0	0
$89$	−2.59891	−0.275484	−0.137742	−	0.990468i	$-0.543984\pi$
$89$	−2.59891	−0.275484	−0.137742	+	0.990468i	$0.543984\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.914835	−0.0948640
$94$	0	0
$95$	6.46118	0.662903
$96$	0	0
$97$	−7.23082	−0.734179	−0.367089	−	0.930186i	$-0.619646\pi$
$97$	−7.23082	−0.734179	−0.367089	+	0.930186i	$0.619646\pi$
$98$	0	0
$99$	10.2783	1.03301
$100$	0	0
$101$	3.15538	0.313972	0.156986	−	0.987601i	$-0.449822\pi$
$101$	3.15538	0.313972	0.156986	+	0.987601i	$0.449822\pi$
$102$	0	0
$103$	18.0385	1.77739	0.888694	−	0.458501i	$-0.151613\pi$
$103$	18.0385	1.77739	0.888694	+	0.458501i	$0.151613\pi$
$104$	0	0
$105$	9.57283	0.934212
$106$	0	0
$107$	−4.99956	−0.483326	−0.241663	−	0.970360i	$-0.577693\pi$
$107$	−4.99956	−0.483326	−0.241663	+	0.970360i	$0.577693\pi$
$108$	0	0
$109$	14.6562	1.40381	0.701905	−	0.712270i	$-0.252333\pi$
$109$	14.6562	1.40381	0.701905	+	0.712270i	$0.252333\pi$
$110$	0	0
$111$	−2.92780	−0.277894
$112$	0	0
$113$	−9.32633	−0.877348	−0.438674	−	0.898646i	$-0.644552\pi$
$113$	−9.32633	−0.877348	−0.438674	+	0.898646i	$0.644552\pi$
$114$	0	0
$115$	5.67019	0.528748
$116$	0	0
$117$	0	0
$118$	0	0
$119$	32.5648	2.98521
$120$	0	0
$121$	4.01380	0.364891
$122$	0	0
$123$	1.63102	0.147064
$124$	0	0
$125$	6.70671	0.599867
$126$	0	0
$127$	−12.6408	−1.12169	−0.560844	−	0.827921i	$-0.689523\pi$
$127$	−12.6408	−1.12169	−0.560844	+	0.827921i	$0.689523\pi$
$128$	0	0
$129$	1.37012	0.120633
$130$	0	0
$131$	5.53014	0.483171	0.241585	−	0.970380i	$-0.422333\pi$
$131$	5.53014	0.483171	0.241585	+	0.970380i	$0.422333\pi$
$132$	0	0
$133$	8.78865	0.762072
$134$	0	0
$135$	−11.5123	−0.990824
$136$	0	0
$137$	9.75074	0.833062	0.416531	−	0.909121i	$-0.363246\pi$
$137$	9.75074	0.833062	0.416531	+	0.909121i	$0.363246\pi$
$138$	0	0
$139$	5.15245	0.437025	0.218513	−	0.975834i	$-0.429879\pi$
$139$	5.15245	0.437025	0.218513	+	0.975834i	$0.429879\pi$
$140$	0	0
$141$	−3.33378	−0.280755
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.79657	−0.730516
$146$	0	0
$147$	8.89552	0.733690
$148$	0	0
$149$	11.6568	0.954961	0.477481	−	0.878642i	$-0.341550\pi$
$149$	11.6568	0.954961	0.477481	+	0.878642i	$0.341550\pi$
$150$	0	0
$151$	−20.9836	−1.70762	−0.853810	−	0.520584i	$-0.825714\pi$
$151$	−20.9836	−1.70762	−0.853810	+	0.520584i	$0.825714\pi$
$152$	0	0
$153$	−18.3780	−1.48577
$154$	0	0
$155$	−5.36371	−0.430824
$156$	0	0
$157$	0.168337	0.0134347	0.00671737	−	0.999977i	$-0.497862\pi$
$157$	0.168337	0.0134347	0.00671737	+	0.999977i	$0.497862\pi$
$158$	0	0
$159$	−1.67383	−0.132743
$160$	0	0
$161$	7.71273	0.607848
$162$	0	0
$163$	−16.1170	−1.26238	−0.631190	−	0.775629i	$-0.717433\pi$
$163$	−16.1170	−1.26238	−0.631190	+	0.775629i	$0.717433\pi$
$164$	0	0
$165$	−7.89147	−0.614350
$166$	0	0
$167$	−11.2073	−0.867251	−0.433625	−	0.901093i	$-0.642766\pi$
$167$	−11.2073	−0.867251	−0.433625	+	0.901093i	$0.642766\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−4.95988	−0.379292
$172$	0	0
$173$	−7.23429	−0.550013	−0.275006	−	0.961442i	$-0.588680\pi$
$173$	−7.23429	−0.550013	−0.275006	+	0.961442i	$0.588680\pi$
$174$	0	0
$175$	32.6242	2.46616
$176$	0	0
$177$	−6.19265	−0.465468
$178$	0	0
$179$	−0.437475	−0.0326984	−0.0163492	−	0.999866i	$-0.505204\pi$
$179$	−0.437475	−0.0326984	−0.0163492	+	0.999866i	$0.505204\pi$
$180$	0	0
$181$	11.9980	0.891806	0.445903	−	0.895081i	$-0.352883\pi$
$181$	11.9980	0.891806	0.445903	+	0.895081i	$0.352883\pi$
$182$	0	0
$183$	2.59023	0.191476
$184$	0	0
$185$	−17.1658	−1.26205
$186$	0	0
$187$	−26.8451	−1.96311
$188$	0	0
$189$	−15.6593	−1.13905
$190$	0	0
$191$	8.05143	0.582581	0.291291	−	0.956635i	$-0.405915\pi$
$191$	8.05143	0.582581	0.291291	+	0.956635i	$0.405915\pi$
$192$	0	0
$193$	6.69436	0.481871	0.240935	−	0.970541i	$-0.422546\pi$
$193$	6.69436	0.481871	0.240935	+	0.970541i	$0.422546\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.75728	−0.338942	−0.169471	−	0.985535i	$-0.554206\pi$
$197$	−4.75728	−0.338942	−0.169471	+	0.985535i	$0.554206\pi$
$198$	0	0
$199$	−19.9465	−1.41397	−0.706984	−	0.707229i	$-0.749945\pi$
$199$	−19.9465	−1.41397	−0.706984	+	0.707229i	$0.749945\pi$
$200$	0	0
$201$	−7.30994	−0.515603
$202$	0	0
$203$	−11.9653	−0.839799
$204$	0	0
$205$	9.56274	0.667891
$206$	0	0
$207$	−4.35269	−0.302533
$208$	0	0
$209$	−7.24502	−0.501149
$210$	0	0
$211$	26.6951	1.83777	0.918884	−	0.394529i	$-0.129092\pi$
$211$	26.6951	1.83777	0.918884	+	0.394529i	$0.129092\pi$
$212$	0	0
$213$	−2.99060	−0.204913
$214$	0	0
$215$	8.03308	0.547851
$216$	0	0
$217$	−7.29584	−0.495274
$218$	0	0
$219$	7.79135	0.526491
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0893	−0.943489	−0.471745	−	0.881735i	$-0.656376\pi$
$223$	−14.0893	−0.943489	−0.471745	+	0.881735i	$0.656376\pi$
$224$	0	0
$225$	−18.4115	−1.22743
$226$	0	0
$227$	−24.8778	−1.65119	−0.825597	−	0.564260i	$-0.809162\pi$
$227$	−24.8778	−1.65119	−0.825597	+	0.564260i	$0.809162\pi$
$228$	0	0
$229$	−19.4436	−1.28487	−0.642434	−	0.766341i	$-0.722075\pi$
$229$	−19.4436	−1.28487	−0.642434	+	0.766341i	$0.722075\pi$
$230$	0	0
$231$	−10.7342	−0.706256
$232$	0	0
$233$	−15.5798	−1.02066	−0.510332	−	0.859977i	$-0.670477\pi$
$233$	−15.5798	−1.02066	−0.510332	+	0.859977i	$0.670477\pi$
$234$	0	0
$235$	−19.5460	−1.27504
$236$	0	0
$237$	−2.90441	−0.188661
$238$	0	0
$239$	8.67798	0.561332	0.280666	−	0.959806i	$-0.409445\pi$
$239$	8.67798	0.561332	0.280666	+	0.959806i	$0.409445\pi$
$240$	0	0
$241$	−3.71340	−0.239201	−0.119601	−	0.992822i	$-0.538161\pi$
$241$	−3.71340	−0.239201	−0.119601	+	0.992822i	$0.538161\pi$
$242$	0	0
$243$	13.5276	0.867795
$244$	0	0
$245$	52.1548	3.33205
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.55298	0.225161
$250$	0	0
$251$	−22.6992	−1.43276	−0.716381	−	0.697709i	$-0.754203\pi$
$251$	−22.6992	−1.43276	−0.716381	+	0.697709i	$0.754203\pi$
$252$	0	0
$253$	−6.35807	−0.399729
$254$	0	0
$255$	14.1102	0.883615
$256$	0	0
$257$	−0.600786	−0.0374760	−0.0187380	−	0.999824i	$-0.505965\pi$
$257$	−0.600786	−0.0374760	−0.0187380	+	0.999824i	$0.505965\pi$
$258$	0	0
$259$	−23.3493	−1.45085
$260$	0	0
$261$	6.75263	0.417977
$262$	0	0
$263$	21.9882	1.35585	0.677925	−	0.735131i	$-0.262879\pi$
$263$	21.9882	1.35585	0.677925	+	0.735131i	$0.262879\pi$
$264$	0	0
$265$	−9.81371	−0.602851
$266$	0	0
$267$	−1.53174	−0.0937411
$268$	0	0
$269$	−25.9279	−1.58085	−0.790426	−	0.612557i	$-0.790141\pi$
$269$	−25.9279	−1.58085	−0.790426	+	0.612557i	$0.790141\pi$
$270$	0	0
$271$	18.2003	1.10559	0.552794	−	0.833318i	$-0.313562\pi$
$271$	18.2003	1.10559	0.552794	+	0.833318i	$0.313562\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−26.8942	−1.62178
$276$	0	0
$277$	25.8430	1.55276	0.776378	−	0.630268i	$-0.217055\pi$
$277$	25.8430	1.55276	0.776378	+	0.630268i	$0.217055\pi$
$278$	0	0
$279$	4.11742	0.246503
$280$	0	0
$281$	30.5562	1.82283	0.911415	−	0.411489i	$-0.134991\pi$
$281$	30.5562	1.82283	0.911415	+	0.411489i	$0.134991\pi$
$282$	0	0
$283$	22.9388	1.36357	0.681784	−	0.731553i	$-0.261204\pi$
$283$	22.9388	1.36357	0.681784	+	0.731553i	$0.261204\pi$
$284$	0	0
$285$	3.80809	0.225572
$286$	0	0
$287$	13.0075	0.767806
$288$	0	0
$289$	30.9999	1.82353
$290$	0	0
$291$	−4.26170	−0.249825
$292$	0	0
$293$	−17.1560	−1.00226	−0.501132	−	0.865371i	$-0.667083\pi$
$293$	−17.1560	−1.00226	−0.501132	+	0.865371i	$0.667083\pi$
$294$	0	0
$295$	−36.3077	−2.11392
$296$	0	0
$297$	12.9090	0.749054
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.9268	0.629809
$302$	0	0
$303$	1.85972	0.106838
$304$	0	0
$305$	15.1866	0.869585
$306$	0	0
$307$	−25.3357	−1.44599	−0.722993	−	0.690855i	$-0.757234\pi$
$307$	−25.3357	−1.44599	−0.722993	+	0.690855i	$0.757234\pi$
$308$	0	0
$309$	10.6315	0.604807
$310$	0	0
$311$	−10.7964	−0.612206	−0.306103	−	0.951998i	$-0.599025\pi$
$311$	−10.7964	−0.612206	−0.306103	+	0.951998i	$0.599025\pi$
$312$	0	0
$313$	−7.58905	−0.428958	−0.214479	−	0.976729i	$-0.568805\pi$
$313$	−7.58905	−0.428958	−0.214479	+	0.976729i	$0.568805\pi$
$314$	0	0
$315$	−43.0846	−2.42754
$316$	0	0
$317$	−30.4157	−1.70832	−0.854159	−	0.520012i	$-0.825928\pi$
$317$	−30.4157	−1.70832	−0.854159	+	0.520012i	$0.825928\pi$
$318$	0	0
$319$	9.86373	0.552263
$320$	0	0
$321$	−2.94664	−0.164465
$322$	0	0
$323$	12.9543	0.720798
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.63807	0.477687
$328$	0	0
$329$	−26.5870	−1.46579
$330$	0	0
$331$	−5.58638	−0.307055	−0.153528	−	0.988144i	$-0.549063\pi$
$331$	−5.58638	−0.307055	−0.153528	+	0.988144i	$0.549063\pi$
$332$	0	0
$333$	13.1772	0.722105
$334$	0	0
$335$	−42.8584	−2.34161
$336$	0	0
$337$	−0.479529	−0.0261216	−0.0130608	−	0.999915i	$-0.504157\pi$
$337$	−0.479529	−0.0261216	−0.0130608	+	0.999915i	$0.504157\pi$
$338$	0	0
$339$	−5.49675	−0.298542
$340$	0	0
$341$	6.01441	0.325699
$342$	0	0
$343$	38.0398	2.05396
$344$	0	0
$345$	3.34190	0.179922
$346$	0	0
$347$	−24.8595	−1.33453	−0.667265	−	0.744820i	$-0.732535\pi$
$347$	−24.8595	−1.33453	−0.667265	+	0.744820i	$0.732535\pi$
$348$	0	0
$349$	−18.9839	−1.01618	−0.508092	−	0.861303i	$-0.669649\pi$
$349$	−18.9839	−1.01618	−0.508092	+	0.861303i	$0.669649\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.0547	−0.960955	−0.480477	−	0.877007i	$-0.659537\pi$
$353$	−18.0547	−0.960955	−0.480477	+	0.877007i	$0.659537\pi$
$354$	0	0
$355$	−17.5340	−0.930608
$356$	0	0
$357$	19.1930	1.01580
$358$	0	0
$359$	7.04655	0.371903	0.185951	−	0.982559i	$-0.440463\pi$
$359$	7.04655	0.371903	0.185951	+	0.982559i	$0.440463\pi$
$360$	0	0
$361$	−15.5039	−0.815993
$362$	0	0
$363$	2.36565	0.124165
$364$	0	0
$365$	45.6810	2.39105
$366$	0	0
$367$	25.4590	1.32895	0.664475	−	0.747310i	$-0.268655\pi$
$367$	25.4590	1.32895	0.664475	+	0.747310i	$0.268655\pi$
$368$	0	0
$369$	−7.34077	−0.382145
$370$	0	0
$371$	−13.3488	−0.693037
$372$	0	0
$373$	11.5856	0.599882	0.299941	−	0.953958i	$-0.403033\pi$
$373$	11.5856	0.599882	0.299941	+	0.953958i	$0.403033\pi$
$374$	0	0
$375$	3.95280	0.204122
$376$	0	0
$377$	0	0
$378$	0	0
$379$	21.8322	1.12145	0.560724	−	0.828003i	$-0.310523\pi$
$379$	21.8322	1.12145	0.560724	+	0.828003i	$0.310523\pi$
$380$	0	0
$381$	−7.45022	−0.381686
$382$	0	0
$383$	17.0237	0.869868	0.434934	−	0.900462i	$-0.356772\pi$
$383$	17.0237	0.869868	0.434934	+	0.900462i	$0.356772\pi$
$384$	0	0
$385$	−62.9348	−3.20745
$386$	0	0
$387$	−6.16653	−0.313463
$388$	0	0
$389$	21.2022	1.07499	0.537496	−	0.843266i	$-0.319370\pi$
$389$	21.2022	1.07499	0.537496	+	0.843266i	$0.319370\pi$
$390$	0	0
$391$	11.3684	0.574926
$392$	0	0
$393$	3.25935	0.164413
$394$	0	0
$395$	−17.0286	−0.856804
$396$	0	0
$397$	−24.4864	−1.22894	−0.614468	−	0.788942i	$-0.710629\pi$
$397$	−24.4864	−1.22894	−0.614468	+	0.788942i	$0.710629\pi$
$398$	0	0
$399$	5.17985	0.259317
$400$	0	0
$401$	−33.0189	−1.64889	−0.824443	−	0.565945i	$-0.808512\pi$
$401$	−33.0189	−1.64889	−0.824443	+	0.565945i	$0.808512\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	20.7138	1.02928
$406$	0	0
$407$	19.2482	0.954100
$408$	0	0
$409$	8.38884	0.414801	0.207401	−	0.978256i	$-0.433500\pi$
$409$	8.38884	0.414801	0.207401	+	0.978256i	$0.433500\pi$
$410$	0	0
$411$	5.74689	0.283473
$412$	0	0
$413$	−49.3866	−2.43016
$414$	0	0
$415$	20.8313	1.02257
$416$	0	0
$417$	3.03675	0.148710
$418$	0	0
$419$	−17.6948	−0.864446	−0.432223	−	0.901767i	$-0.642271\pi$
$419$	−17.6948	−0.864446	−0.432223	+	0.901767i	$0.642271\pi$
$420$	0	0
$421$	1.79649	0.0875554	0.0437777	−	0.999041i	$-0.486061\pi$
$421$	1.79649	0.0875554	0.0437777	+	0.999041i	$0.486061\pi$
$422$	0	0
$423$	15.0044	0.729538
$424$	0	0
$425$	48.0876	2.33259
$426$	0	0
$427$	20.6572	0.999673
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.51463	−0.169294	−0.0846468	−	0.996411i	$-0.526976\pi$
$431$	−3.51463	−0.169294	−0.0846468	+	0.996411i	$0.526976\pi$
$432$	0	0
$433$	14.5477	0.699120	0.349560	−	0.936914i	$-0.386331\pi$
$433$	14.5477	0.699120	0.349560	+	0.936914i	$0.386331\pi$
$434$	0	0
$435$	−5.18452	−0.248579
$436$	0	0
$437$	3.06814	0.146769
$438$	0	0
$439$	−15.8162	−0.754864	−0.377432	−	0.926037i	$-0.623193\pi$
$439$	−15.8162	−0.754864	−0.377432	+	0.926037i	$0.623193\pi$
$440$	0	0
$441$	−40.0362	−1.90649
$442$	0	0
$443$	−21.6418	−1.02823	−0.514117	−	0.857720i	$-0.671880\pi$
$443$	−21.6418	−1.02823	−0.514117	+	0.857720i	$0.671880\pi$
$444$	0	0
$445$	−8.98066	−0.425724
$446$	0	0
$447$	6.87027	0.324953
$448$	0	0
$449$	29.7964	1.40618	0.703089	−	0.711102i	$-0.251804\pi$
$449$	29.7964	1.40618	0.703089	+	0.711102i	$0.251804\pi$
$450$	0	0
$451$	−10.7229	−0.504919
$452$	0	0
$453$	−12.3673	−0.581066
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.7871	0.551378	0.275689	−	0.961247i	$-0.411094\pi$
$457$	11.7871	0.551378	0.275689	+	0.961247i	$0.411094\pi$
$458$	0	0
$459$	−23.0816	−1.07736
$460$	0	0
$461$	7.58846	0.353430	0.176715	−	0.984262i	$-0.443453\pi$
$461$	7.58846	0.353430	0.176715	+	0.984262i	$0.443453\pi$
$462$	0	0
$463$	7.92586	0.368346	0.184173	−	0.982894i	$-0.441039\pi$
$463$	7.92586	0.368346	0.184173	+	0.982894i	$0.441039\pi$
$464$	0	0
$465$	−3.16126	−0.146600
$466$	0	0
$467$	6.79727	0.314540	0.157270	−	0.987556i	$-0.449731\pi$
$467$	6.79727	0.314540	0.157270	+	0.987556i	$0.449731\pi$
$468$	0	0
$469$	−58.2970	−2.69191
$470$	0	0
$471$	0.0992142	0.00457155
$472$	0	0
$473$	−9.00761	−0.414170
$474$	0	0
$475$	12.9780	0.595470
$476$	0	0
$477$	7.53343	0.344932
$478$	0	0
$479$	−27.7981	−1.27013	−0.635063	−	0.772460i	$-0.719026\pi$
$479$	−27.7981	−1.27013	−0.635063	+	0.772460i	$0.719026\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.54572	0.206838
$484$	0	0
$485$	−24.9865	−1.13458
$486$	0	0
$487$	16.2799	0.737715	0.368857	−	0.929486i	$-0.379749\pi$
$487$	16.2799	0.737715	0.368857	+	0.929486i	$0.379749\pi$
$488$	0	0
$489$	−9.49902	−0.429560
$490$	0	0
$491$	39.5178	1.78341	0.891707	−	0.452613i	$-0.149508\pi$
$491$	39.5178	1.78341	0.891707	+	0.452613i	$0.149508\pi$
$492$	0	0
$493$	−17.6367	−0.794315
$494$	0	0
$495$	35.5173	1.59638
$496$	0	0
$497$	−23.8501	−1.06982
$498$	0	0
$499$	−10.0896	−0.451671	−0.225836	−	0.974165i	$-0.572511\pi$
$499$	−10.0896	−0.451671	−0.225836	+	0.974165i	$0.572511\pi$
$500$	0	0
$501$	−6.60538	−0.295107
$502$	0	0
$503$	−5.32508	−0.237434	−0.118717	−	0.992928i	$-0.537878\pi$
$503$	−5.32508	−0.237434	−0.118717	+	0.992928i	$0.537878\pi$
$504$	0	0
$505$	10.9036	0.485204
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.54276	−0.290003	−0.145001	−	0.989431i	$-0.546319\pi$
$509$	−6.54276	−0.290003	−0.145001	+	0.989431i	$0.546319\pi$
$510$	0	0
$511$	62.1363	2.74875
$512$	0	0
$513$	−6.22931	−0.275031
$514$	0	0
$515$	62.3331	2.74672
$516$	0	0
$517$	21.9173	0.963921
$518$	0	0
$519$	−4.26374	−0.187157
$520$	0	0
$521$	−24.2101	−1.06066	−0.530331	−	0.847791i	$-0.677932\pi$
$521$	−24.2101	−1.06066	−0.530331	+	0.847791i	$0.677932\pi$
$522$	0	0
$523$	24.4025	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$523$	24.4025	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$524$	0	0
$525$	19.2281	0.839181
$526$	0	0
$527$	−10.7540	−0.468450
$528$	0	0
$529$	−20.3075	−0.882933
$530$	0	0
$531$	27.8714	1.20951
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−17.2763	−0.746918
$536$	0	0
$537$	−0.257839	−0.0111266
$538$	0	0
$539$	−58.4819	−2.51900
$540$	0	0
$541$	−8.24014	−0.354271	−0.177136	−	0.984186i	$-0.556683\pi$
$541$	−8.24014	−0.354271	−0.177136	+	0.984186i	$0.556683\pi$
$542$	0	0
$543$	7.07139	0.303462
$544$	0	0
$545$	50.6453	2.16941
$546$	0	0
$547$	24.9762	1.06791	0.533953	−	0.845514i	$-0.320706\pi$
$547$	24.9762	1.06791	0.533953	+	0.845514i	$0.320706\pi$
$548$	0	0
$549$	−11.6579	−0.497548
$550$	0	0
$551$	−4.75982	−0.202775
$552$	0	0
$553$	−23.1627	−0.984980
$554$	0	0
$555$	−10.1172	−0.429449
$556$	0	0
$557$	−22.7438	−0.963684	−0.481842	−	0.876258i	$-0.660032\pi$
$557$	−22.7438	−0.963684	−0.481842	+	0.876258i	$0.660032\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−15.8220	−0.668004
$562$	0	0
$563$	6.89896	0.290757	0.145378	−	0.989376i	$-0.453560\pi$
$563$	6.89896	0.290757	0.145378	+	0.989376i	$0.453560\pi$
$564$	0	0
$565$	−32.2276	−1.35583
$566$	0	0
$567$	28.1754	1.18325
$568$	0	0
$569$	16.0818	0.674185	0.337092	−	0.941472i	$-0.390557\pi$
$569$	16.0818	0.674185	0.337092	+	0.941472i	$0.390557\pi$
$570$	0	0
$571$	−10.5399	−0.441081	−0.220541	−	0.975378i	$-0.570782\pi$
$571$	−10.5399	−0.441081	−0.220541	+	0.975378i	$0.570782\pi$
$572$	0	0
$573$	4.74535	0.198240
$574$	0	0
$575$	11.3892	0.474962
$576$	0	0
$577$	36.4847	1.51888	0.759438	−	0.650579i	$-0.225474\pi$
$577$	36.4847	1.51888	0.759438	+	0.650579i	$0.225474\pi$
$578$	0	0
$579$	3.94552	0.163970
$580$	0	0
$581$	28.3351	1.17554
$582$	0	0
$583$	11.0043	0.455750
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.64242	−0.0677898	−0.0338949	−	0.999425i	$-0.510791\pi$
$587$	−1.64242	−0.0677898	−0.0338949	+	0.999425i	$0.510791\pi$
$588$	0	0
$589$	−2.90230	−0.119587
$590$	0	0
$591$	−2.80385	−0.115335
$592$	0	0
$593$	−3.92185	−0.161051	−0.0805255	−	0.996753i	$-0.525660\pi$
$593$	−3.92185	−0.161051	−0.0805255	+	0.996753i	$0.525660\pi$
$594$	0	0
$595$	112.529	4.61325
$596$	0	0
$597$	−11.7560	−0.481143
$598$	0	0
$599$	21.9605	0.897282	0.448641	−	0.893712i	$-0.351908\pi$
$599$	21.9605	0.897282	0.448641	+	0.893712i	$0.351908\pi$
$600$	0	0
$601$	3.66204	0.149378	0.0746889	−	0.997207i	$-0.476204\pi$
$601$	3.66204	0.149378	0.0746889	+	0.997207i	$0.476204\pi$
$602$	0	0
$603$	32.9000	1.33979
$604$	0	0
$605$	13.8699	0.563892
$606$	0	0
$607$	22.5743	0.916262	0.458131	−	0.888885i	$-0.348519\pi$
$607$	22.5743	0.916262	0.458131	+	0.888885i	$0.348519\pi$
$608$	0	0
$609$	−7.05210	−0.285766
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−31.5623	−1.27479	−0.637395	−	0.770537i	$-0.719988\pi$
$613$	−31.5623	−1.27479	−0.637395	+	0.770537i	$0.719988\pi$
$614$	0	0
$615$	5.63609	0.227269
$616$	0	0
$617$	−31.3446	−1.26189	−0.630943	−	0.775829i	$-0.717332\pi$
$617$	−31.3446	−1.26189	−0.630943	+	0.775829i	$0.717332\pi$
$618$	0	0
$619$	−8.41495	−0.338225	−0.169113	−	0.985597i	$-0.554090\pi$
$619$	−8.41495	−0.338225	−0.169113	+	0.985597i	$0.554090\pi$
$620$	0	0
$621$	−5.46671	−0.219372
$622$	0	0
$623$	−12.2157	−0.489412
$624$	0	0
$625$	−11.5288	−0.461154
$626$	0	0
$627$	−4.27007	−0.170530
$628$	0	0
$629$	−34.4164	−1.37227
$630$	0	0
$631$	43.7143	1.74024	0.870119	−	0.492841i	$-0.164042\pi$
$631$	43.7143	1.74024	0.870119	+	0.492841i	$0.164042\pi$
$632$	0	0
$633$	15.7336	0.625352
$634$	0	0
$635$	−43.6809	−1.73342
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.4598	0.532463
$640$	0	0
$641$	18.8081	0.742876	0.371438	−	0.928458i	$-0.378865\pi$
$641$	18.8081	0.742876	0.371438	+	0.928458i	$0.378865\pi$
$642$	0	0
$643$	−0.830165	−0.0327385	−0.0163693	−	0.999866i	$-0.505211\pi$
$643$	−0.830165	−0.0327385	−0.0163693	+	0.999866i	$0.505211\pi$
$644$	0	0
$645$	4.73453	0.186422
$646$	0	0
$647$	1.30026	0.0511185	0.0255593	−	0.999673i	$-0.491863\pi$
$647$	1.30026	0.0511185	0.0255593	+	0.999673i	$0.491863\pi$
$648$	0	0
$649$	40.7124	1.59810
$650$	0	0
$651$	−4.30002	−0.168531
$652$	0	0
$653$	−37.5774	−1.47052	−0.735258	−	0.677787i	$-0.762939\pi$
$653$	−37.5774	−1.47052	−0.735258	+	0.677787i	$0.762939\pi$
$654$	0	0
$655$	19.1097	0.746678
$656$	0	0
$657$	−35.0667	−1.36808
$658$	0	0
$659$	−44.3190	−1.72642	−0.863211	−	0.504844i	$-0.831550\pi$
$659$	−44.3190	−1.72642	−0.863211	+	0.504844i	$0.831550\pi$
$660$	0	0
$661$	−7.44580	−0.289608	−0.144804	−	0.989460i	$-0.546255\pi$
$661$	−7.44580	−0.289608	−0.144804	+	0.989460i	$0.546255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	30.3696	1.17768
$666$	0	0
$667$	−4.17711	−0.161738
$668$	0	0
$669$	−8.30394	−0.321049
$670$	0	0
$671$	−17.0290	−0.657398
$672$	0	0
$673$	19.4363	0.749213	0.374606	−	0.927184i	$-0.377778\pi$
$673$	19.4363	0.749213	0.374606	+	0.927184i	$0.377778\pi$
$674$	0	0
$675$	−23.1238	−0.890034
$676$	0	0
$677$	−23.2562	−0.893809	−0.446904	−	0.894582i	$-0.647474\pi$
$677$	−23.2562	−0.893809	−0.446904	+	0.894582i	$0.647474\pi$
$678$	0	0
$679$	−33.9872	−1.30431
$680$	0	0
$681$	−14.6624	−0.561866
$682$	0	0
$683$	−22.1834	−0.848823	−0.424412	−	0.905469i	$-0.639519\pi$
$683$	−22.1834	−0.848823	−0.424412	+	0.905469i	$0.639519\pi$
$684$	0	0
$685$	33.6942	1.28739
$686$	0	0
$687$	−11.4597	−0.437213
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−17.3521	−0.660107	−0.330053	−	0.943962i	$-0.607067\pi$
$691$	−17.3521	−0.660107	−0.330053	+	0.943962i	$0.607067\pi$
$692$	0	0
$693$	48.3114	1.83520
$694$	0	0
$695$	17.8046	0.675366
$696$	0	0
$697$	19.1728	0.726221
$698$	0	0
$699$	−9.18239	−0.347310
$700$	0	0
$701$	20.2453	0.764656	0.382328	−	0.924027i	$-0.375122\pi$
$701$	20.2453	0.764656	0.382328	+	0.924027i	$0.375122\pi$
$702$	0	0
$703$	−9.28838	−0.350318
$704$	0	0
$705$	−11.5200	−0.433870
$706$	0	0
$707$	14.8313	0.557789
$708$	0	0
$709$	32.8689	1.23442	0.617208	−	0.786800i	$-0.288264\pi$
$709$	32.8689	1.23442	0.617208	+	0.786800i	$0.288264\pi$
$710$	0	0
$711$	13.0719	0.490235
$712$	0	0
$713$	−2.54700	−0.0953857
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.11462	0.191009
$718$	0	0
$719$	−5.04224	−0.188044	−0.0940218	−	0.995570i	$-0.529972\pi$
$719$	−5.04224	−0.188044	−0.0940218	+	0.995570i	$0.529972\pi$
$720$	0	0
$721$	84.7869	3.15763
$722$	0	0
$723$	−2.18860	−0.0813950
$724$	0	0
$725$	−17.6689	−0.656205
$726$	0	0
$727$	39.9923	1.48323	0.741616	−	0.670825i	$-0.234060\pi$
$727$	39.9923	1.48323	0.741616	+	0.670825i	$0.234060\pi$
$728$	0	0
$729$	−10.0102	−0.370747
$730$	0	0
$731$	16.1059	0.595698
$732$	0	0
$733$	2.97055	0.109720	0.0548600	−	0.998494i	$-0.482529\pi$
$733$	2.97055	0.109720	0.0548600	+	0.998494i	$0.482529\pi$
$734$	0	0
$735$	30.7389	1.13382
$736$	0	0
$737$	48.0578	1.77023
$738$	0	0
$739$	−18.3042	−0.673332	−0.336666	−	0.941624i	$-0.609299\pi$
$739$	−18.3042	−0.673332	−0.336666	+	0.941624i	$0.609299\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.6150	0.462800	0.231400	−	0.972859i	$-0.425669\pi$
$743$	12.6150	0.462800	0.231400	+	0.972859i	$0.425669\pi$
$744$	0	0
$745$	40.2807	1.47577
$746$	0	0
$747$	−15.9910	−0.585079
$748$	0	0
$749$	−23.4996	−0.858655
$750$	0	0
$751$	−26.7986	−0.977893	−0.488946	−	0.872314i	$-0.662619\pi$
$751$	−26.7986	−0.977893	−0.488946	+	0.872314i	$0.662619\pi$
$752$	0	0
$753$	−13.3785	−0.487538
$754$	0	0
$755$	−72.5099	−2.63891
$756$	0	0
$757$	41.9837	1.52592	0.762962	−	0.646443i	$-0.223744\pi$
$757$	41.9837	1.52592	0.762962	+	0.646443i	$0.223744\pi$
$758$	0	0
$759$	−3.74732	−0.136019
$760$	0	0
$761$	−20.7956	−0.753840	−0.376920	−	0.926246i	$-0.623017\pi$
$761$	−20.7956	−0.753840	−0.376920	+	0.926246i	$0.623017\pi$
$762$	0	0
$763$	68.8889	2.49395
$764$	0	0
$765$	−63.5060	−2.29607
$766$	0	0
$767$	0	0
$768$	0	0
$769$	35.4352	1.27783	0.638914	−	0.769278i	$-0.279384\pi$
$769$	35.4352	1.27783	0.638914	+	0.769278i	$0.279384\pi$
$770$	0	0
$771$	−0.354091	−0.0127523
$772$	0	0
$773$	10.1102	0.363640	0.181820	−	0.983332i	$-0.441801\pi$
$773$	10.1102	0.363640	0.181820	+	0.983332i	$0.441801\pi$
$774$	0	0
$775$	−10.7736	−0.386999
$776$	0	0
$777$	−13.7616	−0.493694
$778$	0	0
$779$	5.17439	0.185392
$780$	0	0
$781$	19.6611	0.703531
$782$	0	0
$783$	8.48089	0.303082
$784$	0	0
$785$	0.581697	0.0207616
$786$	0	0
$787$	50.0757	1.78501	0.892504	−	0.451040i	$-0.148947\pi$
$787$	50.0757	1.78501	0.892504	+	0.451040i	$0.148947\pi$
$788$	0	0
$789$	12.9594	0.461367
$790$	0	0
$791$	−43.8368	−1.55866
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.78400	−0.205137
$796$	0	0
$797$	−35.4171	−1.25454	−0.627268	−	0.778803i	$-0.715827\pi$
$797$	−35.4171	−1.25454	−0.627268	+	0.778803i	$0.715827\pi$
$798$	0	0
$799$	−39.1888	−1.38640
$800$	0	0
$801$	6.89394	0.243586
$802$	0	0
$803$	−51.2228	−1.80761
$804$	0	0
$805$	26.6517	0.939350
$806$	0	0
$807$	−15.2814	−0.537930
$808$	0	0
$809$	−34.7275	−1.22095	−0.610476	−	0.792035i	$-0.709022\pi$
$809$	−34.7275	−1.22095	−0.610476	+	0.792035i	$0.709022\pi$
$810$	0	0
$811$	17.1760	0.603129	0.301565	−	0.953446i	$-0.402491\pi$
$811$	17.1760	0.603129	0.301565	+	0.953446i	$0.402491\pi$
$812$	0	0
$813$	10.7269	0.376207
$814$	0	0
$815$	−55.6931	−1.95084
$816$	0	0
$817$	4.34669	0.152071
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.3268	0.534907	0.267454	−	0.963571i	$-0.413818\pi$
$821$	15.3268	0.534907	0.267454	+	0.963571i	$0.413818\pi$
$822$	0	0
$823$	26.0982	0.909726	0.454863	−	0.890561i	$-0.349688\pi$
$823$	26.0982	0.909726	0.454863	+	0.890561i	$0.349688\pi$
$824$	0	0
$825$	−15.8509	−0.551856
$826$	0	0
$827$	9.97853	0.346987	0.173494	−	0.984835i	$-0.444494\pi$
$827$	9.97853	0.346987	0.173494	+	0.984835i	$0.444494\pi$
$828$	0	0
$829$	30.9746	1.07579	0.537896	−	0.843011i	$-0.319219\pi$
$829$	30.9746	1.07579	0.537896	+	0.843011i	$0.319219\pi$
$830$	0	0
$831$	15.2313	0.528369
$832$	0	0
$833$	104.568	3.62305
$834$	0	0
$835$	−38.7276	−1.34022
$836$	0	0
$837$	5.17123	0.178744
$838$	0	0
$839$	8.33467	0.287745	0.143872	−	0.989596i	$-0.454045\pi$
$839$	8.33467	0.287745	0.143872	+	0.989596i	$0.454045\pi$
$840$	0	0
$841$	−22.5197	−0.776543
$842$	0	0
$843$	18.0092	0.620270
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18.8662	0.648249
$848$	0	0
$849$	13.5196	0.463993
$850$	0	0
$851$	−8.15128	−0.279422
$852$	0	0
$853$	−14.5943	−0.499701	−0.249850	−	0.968284i	$-0.580381\pi$
$853$	−14.5943	−0.499701	−0.249850	+	0.968284i	$0.580381\pi$
$854$	0	0
$855$	−17.1391	−0.586146
$856$	0	0
$857$	19.6766	0.672139	0.336069	−	0.941837i	$-0.390902\pi$
$857$	19.6766	0.672139	0.336069	+	0.941837i	$0.390902\pi$
$858$	0	0
$859$	30.8957	1.05415	0.527073	−	0.849820i	$-0.323289\pi$
$859$	30.8957	1.05415	0.527073	+	0.849820i	$0.323289\pi$
$860$	0	0
$861$	7.66633	0.261268
$862$	0	0
$863$	42.1323	1.43420	0.717101	−	0.696969i	$-0.245469\pi$
$863$	42.1323	1.43420	0.717101	+	0.696969i	$0.245469\pi$
$864$	0	0
$865$	−24.9985	−0.849974
$866$	0	0
$867$	18.2707	0.620507
$868$	0	0
$869$	19.0945	0.647735
$870$	0	0
$871$	0	0
$872$	0	0
$873$	19.1807	0.649169
$874$	0	0
$875$	31.5237	1.06570
$876$	0	0
$877$	51.9262	1.75342	0.876712	−	0.481016i	$-0.159732\pi$
$877$	51.9262	1.75342	0.876712	+	0.481016i	$0.159732\pi$
$878$	0	0
$879$	−10.1114	−0.341049
$880$	0	0
$881$	42.6757	1.43778	0.718890	−	0.695124i	$-0.244650\pi$
$881$	42.6757	1.43778	0.718890	+	0.695124i	$0.244650\pi$
$882$	0	0
$883$	22.1831	0.746521	0.373260	−	0.927727i	$-0.378240\pi$
$883$	22.1831	0.746521	0.373260	+	0.927727i	$0.378240\pi$
$884$	0	0
$885$	−21.3990	−0.719321
$886$	0	0
$887$	32.3043	1.08467	0.542336	−	0.840162i	$-0.317540\pi$
$887$	32.3043	1.08467	0.542336	+	0.840162i	$0.317540\pi$
$888$	0	0
$889$	−59.4158	−1.99274
$890$	0	0
$891$	−23.2267	−0.778124
$892$	0	0
$893$	−10.5763	−0.353924
$894$	0	0
$895$	−1.51172	−0.0505312
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.95134	0.131784
$900$	0	0
$901$	−19.6760	−0.655501
$902$	0	0
$903$	6.44001	0.214310
$904$	0	0
$905$	41.4598	1.37817
$906$	0	0
$907$	7.13167	0.236803	0.118402	−	0.992966i	$-0.462223\pi$
$907$	7.13167	0.236803	0.118402	+	0.992966i	$0.462223\pi$
$908$	0	0
$909$	−8.37007	−0.277618
$910$	0	0
$911$	3.89845	0.129161	0.0645807	−	0.997912i	$-0.479429\pi$
$911$	3.89845	0.129161	0.0645807	+	0.997912i	$0.479429\pi$
$912$	0	0
$913$	−23.3584	−0.773050
$914$	0	0
$915$	8.95069	0.295901
$916$	0	0
$917$	25.9934	0.858379
$918$	0	0
$919$	−45.0782	−1.48699	−0.743496	−	0.668740i	$-0.766834\pi$
$919$	−45.0782	−1.48699	−0.743496	+	0.668740i	$0.766834\pi$
$920$	0	0
$921$	−14.9324	−0.492038
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−34.4793	−1.13367
$926$	0	0
$927$	−47.8496	−1.57159
$928$	0	0
$929$	−43.3948	−1.42374	−0.711869	−	0.702312i	$-0.752151\pi$
$929$	−43.3948	−1.42374	−0.711869	+	0.702312i	$0.752151\pi$
$930$	0	0
$931$	28.2209	0.924902
$932$	0	0
$933$	−6.36316	−0.208320
$934$	0	0
$935$	−92.7648	−3.03373
$936$	0	0
$937$	1.62589	0.0531156	0.0265578	−	0.999647i	$-0.491545\pi$
$937$	1.62589	0.0531156	0.0265578	+	0.999647i	$0.491545\pi$
$938$	0	0
$939$	−4.47283	−0.145965
$940$	0	0
$941$	42.6032	1.38883	0.694413	−	0.719577i	$-0.255664\pi$
$941$	42.6032	1.38883	0.694413	+	0.719577i	$0.255664\pi$
$942$	0	0
$943$	4.54093	0.147873
$944$	0	0
$945$	−54.1117	−1.76025
$946$	0	0
$947$	8.63863	0.280718	0.140359	−	0.990101i	$-0.455174\pi$
$947$	8.63863	0.280718	0.140359	+	0.990101i	$0.455174\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−17.9264	−0.581304
$952$	0	0
$953$	39.6171	1.28332	0.641662	−	0.766987i	$-0.278245\pi$
$953$	39.6171	1.28332	0.641662	+	0.766987i	$0.278245\pi$
$954$	0	0
$955$	27.8222	0.900304
$956$	0	0
$957$	5.81348	0.187923
$958$	0	0
$959$	45.8316	1.47998
$960$	0	0
$961$	−28.5907	−0.922280
$962$	0	0
$963$	13.2620	0.427362
$964$	0	0
$965$	23.1327	0.744669
$966$	0	0
$967$	37.4532	1.20441	0.602207	−	0.798340i	$-0.294288\pi$
$967$	37.4532	1.20441	0.602207	+	0.798340i	$0.294288\pi$
$968$	0	0
$969$	7.63501	0.245272
$970$	0	0
$971$	38.8569	1.24698	0.623489	−	0.781832i	$-0.285715\pi$
$971$	38.8569	1.24698	0.623489	+	0.781832i	$0.285715\pi$
$972$	0	0
$973$	24.2182	0.776400
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.6461	−0.916471	−0.458236	−	0.888831i	$-0.651518\pi$
$977$	−28.6461	−0.916471	−0.458236	+	0.888831i	$0.651518\pi$
$978$	0	0
$979$	10.0702	0.321844
$980$	0	0
$981$	−38.8775	−1.24126
$982$	0	0
$983$	34.6874	1.10636	0.553178	−	0.833063i	$-0.313415\pi$
$983$	34.6874	1.10636	0.553178	+	0.833063i	$0.313415\pi$
$984$	0	0
$985$	−16.4390	−0.523792
$986$	0	0
$987$	−15.6698	−0.498776
$988$	0	0
$989$	3.81456	0.121296
$990$	0	0
$991$	−41.6043	−1.32161	−0.660803	−	0.750560i	$-0.729784\pi$
$991$	−41.6043	−1.32161	−0.660803	+	0.750560i	$0.729784\pi$
$992$	0	0
$993$	−3.29250	−0.104484
$994$	0	0
$995$	−68.9261	−2.18511
$996$	0	0
$997$	−31.8116	−1.00748	−0.503742	−	0.863854i	$-0.668044\pi$
$997$	−31.8116	−1.00748	−0.503742	+	0.863854i	$0.668044\pi$
$998$	0	0
$999$	16.5497	0.523611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.n.1.4	yes	6
4.3	odd	2		2704.2.a.bg.1.3		6
13.2	odd	12		1352.2.o.h.1161.6		24
13.3	even	3		1352.2.i.n.529.3		12
13.4	even	6		1352.2.i.m.1329.3		12
13.5	odd	4		1352.2.f.g.337.7		12
13.6	odd	12		1352.2.o.h.361.5		24
13.7	odd	12		1352.2.o.h.361.6		24
13.8	odd	4		1352.2.f.g.337.8		12
13.9	even	3		1352.2.i.n.1329.3		12
13.10	even	6		1352.2.i.m.529.3		12
13.11	odd	12		1352.2.o.h.1161.5		24
13.12	even	2		1352.2.a.m.1.4	✓	6
52.31	even	4		2704.2.f.r.337.5		12
52.47	even	4		2704.2.f.r.337.6		12
52.51	odd	2		2704.2.a.bf.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.m.1.4	✓	6	13.12	even	2
1352.2.a.n.1.4	yes	6	1.1	even	1	trivial
1352.2.f.g.337.7		12	13.5	odd	4
1352.2.f.g.337.8		12	13.8	odd	4
1352.2.i.m.529.3		12	13.10	even	6
1352.2.i.m.1329.3		12	13.4	even	6
1352.2.i.n.529.3		12	13.3	even	3
1352.2.i.n.1329.3		12	13.9	even	3
1352.2.o.h.361.5		24	13.6	odd	12
1352.2.o.h.361.6		24	13.7	odd	12
1352.2.o.h.1161.5		24	13.11	odd	12
1352.2.o.h.1161.6		24	13.2	odd	12
2704.2.a.bf.1.3		6	52.51	odd	2
2704.2.a.bg.1.3		6	4.3	odd	2
2704.2.f.r.337.5		12	52.31	even	4
2704.2.f.r.337.6		12	52.47	even	4

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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+0.589380 q^{3} +3.45555 q^{5} +4.70032 q^{7} -2.65263 q^{9} +O(q^{10})\)	\(q+0.589380 q^{3} +3.45555 q^{5} +4.70032 q^{7} -2.65263 q^{9} -3.87476 q^{11} +2.03663 q^{15} +6.92820 q^{17} +1.86980 q^{19} +2.77027 q^{21} +1.64089 q^{23} +6.94085 q^{25} -3.33155 q^{27} -2.54563 q^{29} -1.55220 q^{31} -2.28371 q^{33} +16.2422 q^{35} -4.96759 q^{37} +2.76736 q^{41} +2.32469 q^{43} -9.16631 q^{45} -5.65642 q^{47} +15.0930 q^{49} +4.08334 q^{51} -2.83998 q^{53} -13.3895 q^{55} +1.10202 q^{57} -10.5071 q^{59} +4.39485 q^{61} -12.4682 q^{63} -12.4028 q^{67} +0.967109 q^{69} -5.07415 q^{71} +13.2196 q^{73} +4.09080 q^{75} -18.2126 q^{77} -4.92790 q^{79} +5.99435 q^{81} +6.02834 q^{83} +23.9408 q^{85} -1.50034 q^{87} -2.59891 q^{89} -0.914835 q^{93} +6.46118 q^{95} -7.23082 q^{97} +10.2783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * q^99

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