LMFDB - Embedded newform 3249.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3249,2,Mod(1,3249)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3249, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3249.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.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:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51414 q^{2} +4.32088 q^{4} -3.32088 q^{5} +2.32088 q^{7} -5.83502 q^{8} +O(q^{10})$	$q-2.51414 q^{2} +4.32088 q^{4} -3.32088 q^{5} +2.32088 q^{7} -5.83502 q^{8} +8.34916 q^{10} +1.70739 q^{11} -4.02827 q^{13} -5.83502 q^{14} +6.02827 q^{16} -14.3492 q^{20} -4.29261 q^{22} +2.34916 q^{23} +6.02827 q^{25} +10.1276 q^{26} +10.0283 q^{28} +6.64177 q^{29} -6.70739 q^{31} -3.48586 q^{32} -7.70739 q^{35} +1.00000 q^{37} +19.3774 q^{40} -6.64177 q^{41} +0.707389 q^{43} +7.37743 q^{44} -5.90611 q^{46} +6.00000 q^{47} -1.61350 q^{49} -15.1559 q^{50} -17.4057 q^{52} -9.96265 q^{53} -5.67004 q^{55} -13.5424 q^{56} -16.6983 q^{58} +1.70739 q^{59} +3.38650 q^{61} +16.8633 q^{62} -3.29261 q^{64} +13.3774 q^{65} +8.37743 q^{67} +19.3774 q^{70} -9.41478 q^{71} +11.6418 q^{73} -2.51414 q^{74} +3.96265 q^{77} +3.34916 q^{79} -20.0192 q^{80} +16.6983 q^{82} +10.0565 q^{83} -1.77847 q^{86} -9.96265 q^{88} -2.67912 q^{89} -9.34916 q^{91} +10.1504 q^{92} -15.0848 q^{94} -17.7266 q^{97} +4.05655 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8	$ 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} + 5 q^{16} - 22 q^{20} - 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} + 4 q^{29} - 15 q^{31} - 17 q^{32} - 18 q^{35} + 3 q^{37} + 24 q^{40} - 4 q^{41} - 3 q^{43} - 12 q^{44} - 20 q^{46} + 18 q^{47} - 2 q^{49} - 23 q^{50} - 5 q^{52} - 6 q^{53} + 12 q^{55} - 21 q^{56} - 8 q^{58} + 13 q^{61} + 23 q^{62} - 15 q^{64} + 6 q^{65} - 9 q^{67} + 24 q^{70} - 18 q^{71} + 19 q^{73} - q^{74} - 12 q^{77} - 11 q^{79} - 10 q^{80} + 8 q^{82} + 4 q^{83} - 17 q^{86} - 6 q^{88} - 16 q^{89} - 7 q^{91} + 2 q^{92} - 6 q^{94} + 2 q^{97} - 14 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 + 4 * q^10 + q^13 - 3 * q^14 + 5 * q^16 - 22 * q^20 - 18 * q^22 - 14 * q^23 + 5 * q^25 + 21 * q^26 + 17 * q^28 + 4 * q^29 - 15 * q^31 - 17 * q^32 - 18 * q^35 + 3 * q^37 + 24 * q^40 - 4 * q^41 - 3 * q^43 - 12 * q^44 - 20 * q^46 + 18 * q^47 - 2 * q^49 - 23 * q^50 - 5 * q^52 - 6 * q^53 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 13 * q^61 + 23 * q^62 - 15 * q^64 + 6 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 + 19 * q^73 - q^74 - 12 * q^77 - 11 * q^79 - 10 * q^80 + 8 * q^82 + 4 * q^83 - 17 * q^86 - 6 * q^88 - 16 * q^89 - 7 * q^91 + 2 * q^92 - 6 * q^94 + 2 * q^97 - 14 * q^98

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 $

$2$	−2.51414	−1.77776	−0.888882	−	0.458137i	$-0.848517\pi$
$2$	−2.51414	−1.77776	−0.888882	+	0.458137i	$0.848517\pi$
$3$	0	0
$3$	0	0
$4$	4.32088	2.16044
$5$	−3.32088	−1.48514	−0.742572	−	0.669766i	$-0.766394\pi$
$5$	−3.32088	−1.48514	−0.742572	+	0.669766i	$0.766394\pi$
$6$	0	0
$7$	2.32088	0.877212	0.438606	−	0.898679i	$-0.355472\pi$
$7$	2.32088	0.877212	0.438606	+	0.898679i	$0.355472\pi$
$8$	−5.83502	−2.06299
$9$	0	0
$10$	8.34916	2.64024
$11$	1.70739	0.514797	0.257399	−	0.966305i	$-0.417135\pi$
$11$	1.70739	0.514797	0.257399	+	0.966305i	$0.417135\pi$
$12$	0	0
$13$	−4.02827	−1.11724	−0.558621	−	0.829423i	$-0.688669\pi$
$13$	−4.02827	−1.11724	−0.558621	+	0.829423i	$0.688669\pi$
$14$	−5.83502	−1.55948
$15$	0	0
$16$	6.02827	1.50707
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−14.3492	−3.20857
$21$	0	0
$22$	−4.29261	−0.915188
$23$	2.34916	0.489833	0.244917	−	0.969544i	$-0.421239\pi$
$23$	2.34916	0.489833	0.244917	+	0.969544i	$0.421239\pi$
$24$	0	0
$25$	6.02827	1.20565
$26$	10.1276	1.98619
$27$	0	0
$28$	10.0283	1.89517
$29$	6.64177	1.23335	0.616673	−	0.787220i	$-0.288480\pi$
$29$	6.64177	1.23335	0.616673	+	0.787220i	$0.288480\pi$
$30$	0	0
$31$	−6.70739	−1.20468	−0.602341	−	0.798239i	$-0.705765\pi$
$31$	−6.70739	−1.20468	−0.602341	+	0.798239i	$0.705765\pi$
$32$	−3.48586	−0.616219
$33$	0	0
$34$	0	0
$35$	−7.70739	−1.30279
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	0	0
$40$	19.3774	3.06384
$41$	−6.64177	−1.03727	−0.518635	−	0.854996i	$-0.673560\pi$
$41$	−6.64177	−1.03727	−0.518635	+	0.854996i	$0.673560\pi$
$42$	0	0
$43$	0.707389	0.107876	0.0539379	−	0.998544i	$-0.482823\pi$
$43$	0.707389	0.107876	0.0539379	+	0.998544i	$0.482823\pi$
$44$	7.37743	1.11219
$45$	0	0
$46$	−5.90611	−0.870808
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−1.61350	−0.230499
$50$	−15.1559	−2.14337
$51$	0	0
$52$	−17.4057	−2.41374
$53$	−9.96265	−1.36848	−0.684238	−	0.729259i	$-0.739865\pi$
$53$	−9.96265	−1.36848	−0.684238	+	0.729259i	$0.739865\pi$
$54$	0	0
$55$	−5.67004	−0.764548
$56$	−13.5424	−1.80968
$57$	0	0
$58$	−16.6983	−2.19260
$59$	1.70739	0.222283	0.111142	−	0.993805i	$-0.464549\pi$
$59$	1.70739	0.222283	0.111142	+	0.993805i	$0.464549\pi$
$60$	0	0
$61$	3.38650	0.433598	0.216799	−	0.976216i	$-0.430438\pi$
$61$	3.38650	0.433598	0.216799	+	0.976216i	$0.430438\pi$
$62$	16.8633	2.14164
$63$	0	0
$64$	−3.29261	−0.411576
$65$	13.3774	1.65927
$66$	0	0
$67$	8.37743	1.02347	0.511733	−	0.859144i	$-0.329004\pi$
$67$	8.37743	1.02347	0.511733	+	0.859144i	$0.329004\pi$
$68$	0	0
$69$	0	0
$70$	19.3774	2.31605
$71$	−9.41478	−1.11733	−0.558664	−	0.829394i	$-0.688686\pi$
$71$	−9.41478	−1.11733	−0.558664	+	0.829394i	$0.688686\pi$
$72$	0	0
$73$	11.6418	1.36257	0.681283	−	0.732020i	$-0.261422\pi$
$73$	11.6418	1.36257	0.681283	+	0.732020i	$0.261422\pi$
$74$	−2.51414	−0.292262
$75$	0	0
$76$	0	0
$77$	3.96265	0.451586
$78$	0	0
$79$	3.34916	0.376810	0.188405	−	0.982091i	$-0.439668\pi$
$79$	3.34916	0.376810	0.188405	+	0.982091i	$0.439668\pi$
$80$	−20.0192	−2.23821
$81$	0	0
$82$	16.6983	1.84402
$83$	10.0565	1.10385	0.551925	−	0.833894i	$-0.313894\pi$
$83$	10.0565	1.10385	0.551925	+	0.833894i	$0.313894\pi$
$84$	0	0
$85$	0	0
$86$	−1.77847	−0.191778
$87$	0	0
$88$	−9.96265	−1.06202
$89$	−2.67912	−0.283986	−0.141993	−	0.989868i	$-0.545351\pi$
$89$	−2.67912	−0.283986	−0.141993	+	0.989868i	$0.545351\pi$
$90$	0	0
$91$	−9.34916	−0.980058
$92$	10.1504	1.05826
$93$	0	0
$94$	−15.0848	−1.55588
$95$	0	0
$96$	0	0
$97$	−17.7266	−1.79986	−0.899931	−	0.436032i	$-0.856384\pi$
$97$	−17.7266	−1.79986	−0.899931	+	0.436032i	$0.856384\pi$
$98$	4.05655	0.409773
$99$	0	0
$100$	26.0475	2.60475
$101$	−16.0565	−1.59769	−0.798843	−	0.601539i	$-0.794554\pi$
$101$	−16.0565	−1.59769	−0.798843	+	0.601539i	$0.794554\pi$
$102$	0	0
$103$	7.54787	0.743714	0.371857	−	0.928290i	$-0.378721\pi$
$103$	7.54787	0.743714	0.371857	+	0.928290i	$0.378721\pi$
$104$	23.5051	2.30486
$105$	0	0
$106$	25.0475	2.43283
$107$	7.28354	0.704126	0.352063	−	0.935976i	$-0.385480\pi$
$107$	7.28354	0.704126	0.352063	+	0.935976i	$0.385480\pi$
$108$	0	0
$109$	12.2553	1.17384	0.586921	−	0.809644i	$-0.300340\pi$
$109$	12.2553	1.17384	0.586921	+	0.809644i	$0.300340\pi$
$110$	14.2553	1.35919
$111$	0	0
$112$	13.9909	1.32202
$113$	7.37743	0.694010	0.347005	−	0.937863i	$-0.387199\pi$
$113$	7.37743	0.694010	0.347005	+	0.937863i	$0.387199\pi$
$114$	0	0
$115$	−7.80128	−0.727473
$116$	28.6983	2.66457
$117$	0	0
$118$	−4.29261	−0.395167
$119$	0	0
$120$	0	0
$121$	−8.08482	−0.734984
$122$	−8.51414	−0.770834
$123$	0	0
$124$	−28.9819	−2.60265
$125$	−3.41478	−0.305427
$126$	0	0
$127$	−14.6418	−1.29925	−0.649623	−	0.760256i	$-0.725073\pi$
$127$	−14.6418	−1.29925	−0.649623	+	0.760256i	$0.725073\pi$
$128$	15.2498	1.34790
$129$	0	0
$130$	−33.6327	−2.94978
$131$	−0.641769	−0.0560716	−0.0280358	−	0.999607i	$-0.508925\pi$
$131$	−0.641769	−0.0560716	−0.0280358	+	0.999607i	$0.508925\pi$
$132$	0	0
$133$	0	0
$134$	−21.0620	−1.81948
$135$	0	0
$136$	0	0
$137$	−2.58522	−0.220870	−0.110435	−	0.993883i	$-0.535224\pi$
$137$	−2.58522	−0.220870	−0.110435	+	0.993883i	$0.535224\pi$
$138$	0	0
$139$	−18.5761	−1.57561	−0.787804	−	0.615926i	$-0.788782\pi$
$139$	−18.5761	−1.57561	−0.787804	+	0.615926i	$0.788782\pi$
$140$	−33.3027	−2.81460
$141$	0	0
$142$	23.6700	1.98635
$143$	−6.87783	−0.575153
$144$	0	0
$145$	−22.0565	−1.83170
$146$	−29.2690	−2.42232
$147$	0	0
$148$	4.32088	0.355175
$149$	3.96265	0.324633	0.162317	−	0.986739i	$-0.448103\pi$
$149$	3.96265	0.324633	0.162317	+	0.986739i	$0.448103\pi$
$150$	0	0
$151$	8.69832	0.707859	0.353929	−	0.935272i	$-0.384845\pi$
$151$	8.69832	0.707859	0.353929	+	0.935272i	$0.384845\pi$
$152$	0	0
$153$	0	0
$154$	−9.96265	−0.802813
$155$	22.2745	1.78913
$156$	0	0
$157$	−5.69832	−0.454775	−0.227388	−	0.973804i	$-0.573018\pi$
$157$	−5.69832	−0.454775	−0.227388	+	0.973804i	$0.573018\pi$
$158$	−8.42024	−0.669879
$159$	0	0
$160$	11.5761	0.915175
$161$	5.45213	0.429688
$162$	0	0
$163$	18.3774	1.43943	0.719716	−	0.694269i	$-0.244272\pi$
$163$	18.3774	1.43943	0.719716	+	0.694269i	$0.244272\pi$
$164$	−28.6983	−2.24096
$165$	0	0
$166$	−25.2835	−1.96238
$167$	21.6327	1.67399	0.836994	−	0.547212i	$-0.184311\pi$
$167$	21.6327	1.67399	0.836994	+	0.547212i	$0.184311\pi$
$168$	0	0
$169$	3.22699	0.248230
$170$	0	0
$171$	0	0
$172$	3.05655	0.233060
$173$	−4.69832	−0.357206	−0.178603	−	0.983921i	$-0.557158\pi$
$173$	−4.69832	−0.357206	−0.178603	+	0.983921i	$0.557158\pi$
$174$	0	0
$175$	13.9909	1.05761
$176$	10.2926	0.775835
$177$	0	0
$178$	6.73566	0.504859
$179$	−1.06562	−0.0796482	−0.0398241	−	0.999207i	$-0.512680\pi$
$179$	−1.06562	−0.0796482	−0.0398241	+	0.999207i	$0.512680\pi$
$180$	0	0
$181$	3.67004	0.272792	0.136396	−	0.990654i	$-0.456448\pi$
$181$	3.67004	0.272792	0.136396	+	0.990654i	$0.456448\pi$
$182$	23.5051	1.74231
$183$	0	0
$184$	−13.7074	−1.01052
$185$	−3.32088	−0.244156
$186$	0	0
$187$	0	0
$188$	25.9253	1.89080
$189$	0	0
$190$	0	0
$191$	0.877832	0.0635177	0.0317588	−	0.999496i	$-0.489889\pi$
$191$	0.877832	0.0635177	0.0317588	+	0.999496i	$0.489889\pi$
$192$	0	0
$193$	8.61350	0.620013	0.310006	−	0.950734i	$-0.399669\pi$
$193$	8.61350	0.620013	0.310006	+	0.950734i	$0.399669\pi$
$194$	44.5671	3.19973
$195$	0	0
$196$	−6.97173	−0.497980
$197$	−24.7357	−1.76234	−0.881172	−	0.472797i	$-0.843244\pi$
$197$	−24.7357	−1.76234	−0.881172	+	0.472797i	$0.843244\pi$
$198$	0	0
$199$	−21.9909	−1.55890	−0.779448	−	0.626467i	$-0.784500\pi$
$199$	−21.9909	−1.55890	−0.779448	+	0.626467i	$0.784500\pi$
$200$	−35.1751	−2.48726
$201$	0	0
$202$	40.3684	2.84031
$203$	15.4148	1.08191
$204$	0	0
$205$	22.0565	1.54050
$206$	−18.9764	−1.32215
$207$	0	0
$208$	−24.2835	−1.68376
$209$	0	0
$210$	0	0
$211$	13.2179	0.909959	0.454979	−	0.890502i	$-0.349647\pi$
$211$	13.2179	0.909959	0.454979	+	0.890502i	$0.349647\pi$
$212$	−43.0475	−2.95651
$213$	0	0
$214$	−18.3118	−1.25177
$215$	−2.34916	−0.160211
$216$	0	0
$217$	−15.5671	−1.05676
$218$	−30.8114	−2.08681
$219$	0	0
$220$	−24.4996	−1.65176
$221$	0	0
$222$	0	0
$223$	5.48040	0.366995	0.183497	−	0.983020i	$-0.441258\pi$
$223$	5.48040	0.366995	0.183497	+	0.983020i	$0.441258\pi$
$224$	−8.09029	−0.540555
$225$	0	0
$226$	−18.5479	−1.23379
$227$	7.70739	0.511557	0.255779	−	0.966735i	$-0.417668\pi$
$227$	7.70739	0.511557	0.255779	+	0.966735i	$0.417668\pi$
$228$	0	0
$229$	4.02827	0.266196	0.133098	−	0.991103i	$-0.457508\pi$
$229$	4.02827	0.266196	0.133098	+	0.991103i	$0.457508\pi$
$230$	19.6135	1.29328
$231$	0	0
$232$	−38.7549	−2.54438
$233$	−26.1131	−1.71073	−0.855363	−	0.518029i	$-0.826666\pi$
$233$	−26.1131	−1.71073	−0.855363	+	0.518029i	$0.826666\pi$
$234$	0	0
$235$	−19.9253	−1.29978
$236$	7.37743	0.480230
$237$	0	0
$238$	0	0
$239$	7.70739	0.498550	0.249275	−	0.968433i	$-0.419808\pi$
$239$	7.70739	0.498550	0.249275	+	0.968433i	$0.419808\pi$
$240$	0	0
$241$	−21.6983	−1.39771	−0.698856	−	0.715263i	$-0.746307\pi$
$241$	−21.6983	−1.39771	−0.698856	+	0.715263i	$0.746307\pi$
$242$	20.3263	1.30663
$243$	0	0
$244$	14.6327	0.936762
$245$	5.35823	0.342325
$246$	0	0
$247$	0	0
$248$	39.1378	2.48525
$249$	0	0
$250$	8.58522	0.542977
$251$	−9.41478	−0.594256	−0.297128	−	0.954838i	$-0.596029\pi$
$251$	−9.41478	−0.594256	−0.297128	+	0.954838i	$0.596029\pi$
$252$	0	0
$253$	4.01093	0.252165
$254$	36.8114	2.30975
$255$	0	0
$256$	−31.7549	−1.98468
$257$	−10.1504	−0.633167	−0.316584	−	0.948565i	$-0.602536\pi$
$257$	−10.1504	−0.633167	−0.316584	+	0.948565i	$0.602536\pi$
$258$	0	0
$259$	2.32088	0.144213
$260$	57.8023	3.58475
$261$	0	0
$262$	1.61350	0.0996821
$263$	−2.58522	−0.159412	−0.0797058	−	0.996818i	$-0.525398\pi$
$263$	−2.58522	−0.159412	−0.0797058	+	0.996818i	$0.525398\pi$
$264$	0	0
$265$	33.0848	2.03238
$266$	0	0
$267$	0	0
$268$	36.1979	2.21114
$269$	0.547875	0.0334045	0.0167023	−	0.999861i	$-0.494683\pi$
$269$	0.547875	0.0334045	0.0167023	+	0.999861i	$0.494683\pi$
$270$	0	0
$271$	−20.8861	−1.26874	−0.634370	−	0.773029i	$-0.718741\pi$
$271$	−20.8861	−1.26874	−0.634370	+	0.773029i	$0.718741\pi$
$272$	0	0
$273$	0	0
$274$	6.49960	0.392655
$275$	10.2926	0.620668
$276$	0	0
$277$	−23.7831	−1.42899	−0.714495	−	0.699640i	$-0.753344\pi$
$277$	−23.7831	−1.42899	−0.714495	+	0.699640i	$0.753344\pi$
$278$	46.7030	2.80106
$279$	0	0
$280$	44.9728	2.68764
$281$	−11.9061	−0.710259	−0.355129	−	0.934817i	$-0.615563\pi$
$281$	−11.9061	−0.710259	−0.355129	+	0.934817i	$0.615563\pi$
$282$	0	0
$283$	18.0565	1.07335	0.536675	−	0.843789i	$-0.319680\pi$
$283$	18.0565	1.07335	0.536675	+	0.843789i	$0.319680\pi$
$284$	−40.6802	−2.41392
$285$	0	0
$286$	17.2918	1.02249
$287$	−15.4148	−0.909906
$288$	0	0
$289$	−17.0000	−1.00000
$290$	55.4532	3.25632
$291$	0	0
$292$	50.3027	2.94375
$293$	−27.3966	−1.60053	−0.800264	−	0.599648i	$-0.795307\pi$
$293$	−27.3966	−1.60053	−0.800264	+	0.599648i	$0.795307\pi$
$294$	0	0
$295$	−5.67004	−0.330123
$296$	−5.83502	−0.339154
$297$	0	0
$298$	−9.96265	−0.577121
$299$	−9.46305	−0.547262
$300$	0	0
$301$	1.64177	0.0946300
$302$	−21.8688	−1.25841
$303$	0	0
$304$	0	0
$305$	−11.2462	−0.643955
$306$	0	0
$307$	−3.61350	−0.206233	−0.103117	−	0.994669i	$-0.532881\pi$
$307$	−3.61350	−0.206233	−0.103117	+	0.994669i	$0.532881\pi$
$308$	17.1222	0.975626
$309$	0	0
$310$	−56.0011	−3.18065
$311$	−18.4057	−1.04369	−0.521846	−	0.853040i	$-0.674756\pi$
$311$	−18.4057	−1.04369	−0.521846	+	0.853040i	$0.674756\pi$
$312$	0	0
$313$	23.0848	1.30483	0.652416	−	0.757861i	$-0.273756\pi$
$313$	23.0848	1.30483	0.652416	+	0.757861i	$0.273756\pi$
$314$	14.3263	0.808483
$315$	0	0
$316$	14.4713	0.814076
$317$	−25.3774	−1.42534	−0.712669	−	0.701500i	$-0.752514\pi$
$317$	−25.3774	−1.42534	−0.712669	+	0.701500i	$0.752514\pi$
$318$	0	0
$319$	11.3401	0.634923
$320$	10.9344	0.611250
$321$	0	0
$322$	−13.7074	−0.763883
$323$	0	0
$324$	0	0
$325$	−24.2835	−1.34701
$326$	−46.2034	−2.55897
$327$	0	0
$328$	38.7549	2.13988
$329$	13.9253	0.767727
$330$	0	0
$331$	21.3492	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$331$	21.3492	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$332$	43.4532	2.38480
$333$	0	0
$334$	−54.3876	−2.97595
$335$	−27.8205	−1.52000
$336$	0	0
$337$	4.08482	0.222514	0.111257	−	0.993792i	$-0.464512\pi$
$337$	4.08482	0.222514	0.111257	+	0.993792i	$0.464512\pi$
$338$	−8.11310	−0.441294
$339$	0	0
$340$	0	0
$341$	−11.4521	−0.620167
$342$	0	0
$343$	−19.9909	−1.07941
$344$	−4.12763	−0.222547
$345$	0	0
$346$	11.8122	0.635028
$347$	−19.7074	−1.05795	−0.528974	−	0.848638i	$-0.677423\pi$
$347$	−19.7074	−1.05795	−0.528974	+	0.848638i	$0.677423\pi$
$348$	0	0
$349$	23.3118	1.24785	0.623926	−	0.781483i	$-0.285537\pi$
$349$	23.3118	1.24785	0.623926	+	0.781483i	$0.285537\pi$
$350$	−35.1751	−1.88019
$351$	0	0
$352$	−5.95173	−0.317228
$353$	−25.1896	−1.34071	−0.670355	−	0.742041i	$-0.733858\pi$
$353$	−25.1896	−1.34071	−0.670355	+	0.742041i	$0.733858\pi$
$354$	0	0
$355$	31.2654	1.65939
$356$	−11.5761	−0.613535
$357$	0	0
$358$	2.67912	0.141596
$359$	3.22699	0.170314	0.0851570	−	0.996368i	$-0.472861\pi$
$359$	3.22699	0.170314	0.0851570	+	0.996368i	$0.472861\pi$
$360$	0	0
$361$	0	0
$362$	−9.22699	−0.484960
$363$	0	0
$364$	−40.3966	−2.11736
$365$	−38.6610	−2.02361
$366$	0	0
$367$	−29.4623	−1.53792	−0.768959	−	0.639299i	$-0.779225\pi$
$367$	−29.4623	−1.53792	−0.768959	+	0.639299i	$0.779225\pi$
$368$	14.1614	0.738212
$369$	0	0
$370$	8.34916	0.434052
$371$	−23.1222	−1.20044
$372$	0	0
$373$	−19.6700	−1.01848	−0.509238	−	0.860626i	$-0.670073\pi$
$373$	−19.6700	−1.01848	−0.509238	+	0.860626i	$0.670073\pi$
$374$	0	0
$375$	0	0
$376$	−35.0101	−1.80551
$377$	−26.7549	−1.37795
$378$	0	0
$379$	0.763937	0.0392408	0.0196204	−	0.999808i	$-0.493754\pi$
$379$	0.763937	0.0392408	0.0196204	+	0.999808i	$0.493754\pi$
$380$	0	0
$381$	0	0
$382$	−2.20699	−0.112919
$383$	3.83863	0.196145	0.0980724	−	0.995179i	$-0.468732\pi$
$383$	3.83863	0.196145	0.0980724	+	0.995179i	$0.468732\pi$
$384$	0	0
$385$	−13.1595	−0.670671
$386$	−21.6555	−1.10224
$387$	0	0
$388$	−76.5946	−3.88850
$389$	−32.2070	−1.63296	−0.816480	−	0.577374i	$-0.804077\pi$
$389$	−32.2070	−1.63296	−0.816480	+	0.577374i	$0.804077\pi$
$390$	0	0
$391$	0	0
$392$	9.41478	0.475518
$393$	0	0
$394$	62.1888	3.13303
$395$	−11.1222	−0.559617
$396$	0	0
$397$	−26.5953	−1.33478	−0.667391	−	0.744707i	$-0.732589\pi$
$397$	−26.5953	−1.33478	−0.667391	+	0.744707i	$0.732589\pi$
$398$	55.2882	2.77135
$399$	0	0
$400$	36.3401	1.81700
$401$	−9.32088	−0.465463	−0.232731	−	0.972541i	$-0.574766\pi$
$401$	−9.32088	−0.465463	−0.232731	+	0.972541i	$0.574766\pi$
$402$	0	0
$403$	27.0192	1.34592
$404$	−69.3785	−3.45171
$405$	0	0
$406$	−38.7549	−1.92337
$407$	1.70739	0.0846321
$408$	0	0
$409$	−21.8013	−1.07800	−0.539002	−	0.842304i	$-0.681198\pi$
$409$	−21.8013	−1.07800	−0.539002	+	0.842304i	$0.681198\pi$
$410$	−55.4532	−2.73864
$411$	0	0
$412$	32.6135	1.60675
$413$	3.96265	0.194989
$414$	0	0
$415$	−33.3966	−1.63938
$416$	14.0420	0.688466
$417$	0	0
$418$	0	0
$419$	−4.93438	−0.241060	−0.120530	−	0.992710i	$-0.538459\pi$
$419$	−4.93438	−0.241060	−0.120530	+	0.992710i	$0.538459\pi$
$420$	0	0
$421$	−4.25526	−0.207389	−0.103694	−	0.994609i	$-0.533066\pi$
$421$	−4.25526	−0.207389	−0.103694	+	0.994609i	$0.533066\pi$
$422$	−33.2317	−1.61769
$423$	0	0
$424$	58.1323	2.82315
$425$	0	0
$426$	0	0
$427$	7.85969	0.380357
$428$	31.4713	1.52122
$429$	0	0
$430$	5.90611	0.284818
$431$	−0.829557	−0.0399584	−0.0199792	−	0.999800i	$-0.506360\pi$
$431$	−0.829557	−0.0399584	−0.0199792	+	0.999800i	$0.506360\pi$
$432$	0	0
$433$	−16.4823	−0.792087	−0.396043	−	0.918232i	$-0.629617\pi$
$433$	−16.4823	−0.792087	−0.396043	+	0.918232i	$0.629617\pi$
$434$	39.1378	1.87867
$435$	0	0
$436$	52.9536	2.53602
$437$	0	0
$438$	0	0
$439$	−28.4340	−1.35708	−0.678540	−	0.734564i	$-0.737387\pi$
$439$	−28.4340	−1.35708	−0.678540	+	0.734564i	$0.737387\pi$
$440$	33.0848	1.57726
$441$	0	0
$442$	0	0
$443$	15.4148	0.732378	0.366189	−	0.930540i	$-0.380662\pi$
$443$	15.4148	0.732378	0.366189	+	0.930540i	$0.380662\pi$
$444$	0	0
$445$	8.89703	0.421760
$446$	−13.7785	−0.652430
$447$	0	0
$448$	−7.64177	−0.361040
$449$	−23.1523	−1.09262	−0.546312	−	0.837582i	$-0.683969\pi$
$449$	−23.1523	−1.09262	−0.546312	+	0.837582i	$0.683969\pi$
$450$	0	0
$451$	−11.3401	−0.533984
$452$	31.8770	1.49937
$453$	0	0
$454$	−19.3774	−0.909428
$455$	31.0475	1.45553
$456$	0	0
$457$	4.68819	0.219304	0.109652	−	0.993970i	$-0.465026\pi$
$457$	4.68819	0.219304	0.109652	+	0.993970i	$0.465026\pi$
$458$	−10.1276	−0.473233
$459$	0	0
$460$	−33.7084	−1.57166
$461$	−12.5479	−0.584413	−0.292206	−	0.956355i	$-0.594389\pi$
$461$	−12.5479	−0.584413	−0.292206	+	0.956355i	$0.594389\pi$
$462$	0	0
$463$	22.8880	1.06369	0.531847	−	0.846841i	$-0.321498\pi$
$463$	22.8880	1.06369	0.531847	+	0.846841i	$0.321498\pi$
$464$	40.0384	1.85874
$465$	0	0
$466$	65.6519	3.04127
$467$	11.8122	0.546604	0.273302	−	0.961928i	$-0.411884\pi$
$467$	11.8122	0.546604	0.273302	+	0.961928i	$0.411884\pi$
$468$	0	0
$469$	19.4431	0.897797
$470$	50.0950	2.31071
$471$	0	0
$472$	−9.96265	−0.458568
$473$	1.20779	0.0555342
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−19.3774	−0.886303
$479$	34.0384	1.55525	0.777627	−	0.628726i	$-0.216423\pi$
$479$	34.0384	1.55525	0.777627	+	0.628726i	$0.216423\pi$
$480$	0	0
$481$	−4.02827	−0.183673
$482$	54.5525	2.48480
$483$	0	0
$484$	−34.9336	−1.58789
$485$	58.8680	2.67306
$486$	0	0
$487$	7.41478	0.335996	0.167998	−	0.985787i	$-0.446270\pi$
$487$	7.41478	0.335996	0.167998	+	0.985787i	$0.446270\pi$
$488$	−19.7603	−0.894508
$489$	0	0
$490$	−13.4713	−0.608572
$491$	−3.86876	−0.174595	−0.0872973	−	0.996182i	$-0.527823\pi$
$491$	−3.86876	−0.174595	−0.0872973	+	0.996182i	$0.527823\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−40.4340	−1.81554
$497$	−21.8506	−0.980134
$498$	0	0
$499$	−35.5863	−1.59306	−0.796530	−	0.604599i	$-0.793334\pi$
$499$	−35.5863	−1.59306	−0.796530	+	0.604599i	$0.793334\pi$
$500$	−14.7549	−0.659858
$501$	0	0
$502$	23.6700	1.05645
$503$	−8.58522	−0.382796	−0.191398	−	0.981512i	$-0.561302\pi$
$503$	−8.58522	−0.382796	−0.191398	+	0.981512i	$0.561302\pi$
$504$	0	0
$505$	53.3219	2.37280
$506$	−10.0840	−0.448289
$507$	0	0
$508$	−63.2654	−2.80695
$509$	−7.94345	−0.352087	−0.176044	−	0.984382i	$-0.556330\pi$
$509$	−7.94345	−0.352087	−0.176044	+	0.984382i	$0.556330\pi$
$510$	0	0
$511$	27.0192	1.19526
$512$	49.3365	2.18038
$513$	0	0
$514$	25.5196	1.12562
$515$	−25.0656	−1.10452
$516$	0	0
$517$	10.2443	0.450545
$518$	−5.83502	−0.256376
$519$	0	0
$520$	−78.0576	−3.42305
$521$	−0.0757489	−0.00331862	−0.00165931	−	0.999999i	$-0.500528\pi$
$521$	−0.0757489	−0.00331862	−0.00165931	+	0.999999i	$0.500528\pi$
$522$	0	0
$523$	−20.3027	−0.887777	−0.443888	−	0.896082i	$-0.646401\pi$
$523$	−20.3027	−0.887777	−0.443888	+	0.896082i	$0.646401\pi$
$524$	−2.77301	−0.121139
$525$	0	0
$526$	6.49960	0.283396
$527$	0	0
$528$	0	0
$529$	−17.4815	−0.760063
$530$	−83.1798	−3.61310
$531$	0	0
$532$	0	0
$533$	26.7549	1.15888
$534$	0	0
$535$	−24.1878	−1.04573
$536$	−48.8825	−2.11140
$537$	0	0
$538$	−1.37743	−0.0593853
$539$	−2.75486	−0.118660
$540$	0	0
$541$	−33.0950	−1.42286	−0.711432	−	0.702755i	$-0.751953\pi$
$541$	−33.0950	−1.42286	−0.711432	+	0.702755i	$0.751953\pi$
$542$	52.5105	2.25552
$543$	0	0
$544$	0	0
$545$	−40.6983	−1.74332
$546$	0	0
$547$	35.9162	1.53567	0.767834	−	0.640649i	$-0.221335\pi$
$547$	35.9162	1.53567	0.767834	+	0.640649i	$0.221335\pi$
$548$	−11.1704	−0.477178
$549$	0	0
$550$	−25.8770	−1.10340
$551$	0	0
$552$	0	0
$553$	7.77301	0.330542
$554$	59.7941	2.54041
$555$	0	0
$556$	−80.2654	−3.40401
$557$	1.92531	0.0815779	0.0407889	−	0.999168i	$-0.487013\pi$
$557$	1.92531	0.0815779	0.0407889	+	0.999168i	$0.487013\pi$
$558$	0	0
$559$	−2.84956	−0.120523
$560$	−46.4623	−1.96339
$561$	0	0
$562$	29.9336	1.26267
$563$	10.8861	0.458795	0.229397	−	0.973333i	$-0.426324\pi$
$563$	10.8861	0.458795	0.229397	+	0.973333i	$0.426324\pi$
$564$	0	0
$565$	−24.4996	−1.03071
$566$	−45.3966	−1.90816
$567$	0	0
$568$	54.9354	2.30504
$569$	−36.4358	−1.52747	−0.763735	−	0.645530i	$-0.776636\pi$
$569$	−36.4358	−1.52747	−0.763735	+	0.645530i	$0.776636\pi$
$570$	0	0
$571$	−7.54787	−0.315869	−0.157934	−	0.987450i	$-0.550483\pi$
$571$	−7.54787	−0.315869	−0.157934	+	0.987450i	$0.550483\pi$
$572$	−29.7183	−1.24259
$573$	0	0
$574$	38.7549	1.61760
$575$	14.1614	0.590570
$576$	0	0
$577$	−15.4823	−0.644535	−0.322267	−	0.946649i	$-0.604445\pi$
$577$	−15.4823	−0.644535	−0.322267	+	0.946649i	$0.604445\pi$
$578$	42.7403	1.77776
$579$	0	0
$580$	−95.3038	−3.95727
$581$	23.3401	0.968310
$582$	0	0
$583$	−17.0101	−0.704488
$584$	−67.9300	−2.81096
$585$	0	0
$586$	68.8789	2.84536
$587$	29.7458	1.22774	0.613870	−	0.789407i	$-0.289612\pi$
$587$	29.7458	1.22774	0.613870	+	0.789407i	$0.289612\pi$
$588$	0	0
$589$	0	0
$590$	14.2553	0.586880
$591$	0	0
$592$	6.02827	0.247761
$593$	0.547875	0.0224985	0.0112493	−	0.999937i	$-0.496419\pi$
$593$	0.547875	0.0224985	0.0112493	+	0.999937i	$0.496419\pi$
$594$	0	0
$595$	0	0
$596$	17.1222	0.701351
$597$	0	0
$598$	23.7914	0.972903
$599$	−45.6327	−1.86450	−0.932251	−	0.361813i	$-0.882158\pi$
$599$	−45.6327	−1.86450	−0.932251	+	0.361813i	$0.882158\pi$
$600$	0	0
$601$	6.85783	0.279737	0.139868	−	0.990170i	$-0.455332\pi$
$601$	6.85783	0.279737	0.139868	+	0.990170i	$0.455332\pi$
$602$	−4.12763	−0.168230
$603$	0	0
$604$	37.5844	1.52929
$605$	26.8488	1.09156
$606$	0	0
$607$	6.76394	0.274540	0.137270	−	0.990534i	$-0.456167\pi$
$607$	6.76394	0.274540	0.137270	+	0.990534i	$0.456167\pi$
$608$	0	0
$609$	0	0
$610$	28.2745	1.14480
$611$	−24.1696	−0.977799
$612$	0	0
$613$	4.44305	0.179453	0.0897266	−	0.995966i	$-0.471401\pi$
$613$	4.44305	0.179453	0.0897266	+	0.995966i	$0.471401\pi$
$614$	9.08482	0.366634
$615$	0	0
$616$	−23.1222	−0.931619
$617$	32.0011	1.28831	0.644157	−	0.764893i	$-0.277208\pi$
$617$	32.0011	1.28831	0.644157	+	0.764893i	$0.277208\pi$
$618$	0	0
$619$	−14.3774	−0.577878	−0.288939	−	0.957348i	$-0.593302\pi$
$619$	−14.3774	−0.577878	−0.288939	+	0.957348i	$0.593302\pi$
$620$	96.2454	3.86531
$621$	0	0
$622$	46.2745	1.85544
$623$	−6.21792	−0.249116
$624$	0	0
$625$	−18.8013	−0.752051
$626$	−58.0384	−2.31968
$627$	0	0
$628$	−24.6218	−0.982516
$629$	0	0
$630$	0	0
$631$	11.0939	0.441641	0.220820	−	0.975314i	$-0.429127\pi$
$631$	11.0939	0.441641	0.220820	+	0.975314i	$0.429127\pi$
$632$	−19.5424	−0.777355
$633$	0	0
$634$	63.8023	2.53391
$635$	48.6236	1.92957
$636$	0	0
$637$	6.49960	0.257524
$638$	−28.5105	−1.12874
$639$	0	0
$640$	−50.6428	−2.00183
$641$	43.4713	1.71701	0.858507	−	0.512802i	$-0.171392\pi$
$641$	43.4713	1.71701	0.858507	+	0.512802i	$0.171392\pi$
$642$	0	0
$643$	−23.6226	−0.931583	−0.465792	−	0.884894i	$-0.654230\pi$
$643$	−23.6226	−0.931583	−0.465792	+	0.884894i	$0.654230\pi$
$644$	23.5580	0.928315
$645$	0	0
$646$	0	0
$647$	19.9253	0.783345	0.391672	−	0.920105i	$-0.371897\pi$
$647$	19.9253	0.783345	0.391672	+	0.920105i	$0.371897\pi$
$648$	0	0
$649$	2.91518	0.114431
$650$	61.0521	2.39466
$651$	0	0
$652$	79.4068	3.10981
$653$	2.11310	0.0826918	0.0413459	−	0.999145i	$-0.486835\pi$
$653$	2.11310	0.0826918	0.0413459	+	0.999145i	$0.486835\pi$
$654$	0	0
$655$	2.13124	0.0832745
$656$	−40.0384	−1.56324
$657$	0	0
$658$	−35.0101	−1.36484
$659$	43.5015	1.69458	0.847288	−	0.531134i	$-0.178234\pi$
$659$	43.5015	1.69458	0.847288	+	0.531134i	$0.178234\pi$
$660$	0	0
$661$	−22.1131	−0.860100	−0.430050	−	0.902805i	$-0.641504\pi$
$661$	−22.1131	−0.860100	−0.430050	+	0.902805i	$0.641504\pi$
$662$	−53.6747	−2.08613
$663$	0	0
$664$	−58.6802	−2.27723
$665$	0	0
$666$	0	0
$667$	15.6026	0.604134
$668$	93.4724	3.61656
$669$	0	0
$670$	69.9445	2.70219
$671$	5.78208	0.223215
$672$	0	0
$673$	−12.5953	−0.485515	−0.242758	−	0.970087i	$-0.578052\pi$
$673$	−12.5953	−0.485515	−0.242758	+	0.970087i	$0.578052\pi$
$674$	−10.2698	−0.395578
$675$	0	0
$676$	13.9435	0.536287
$677$	−32.9427	−1.26609	−0.633045	−	0.774115i	$-0.718195\pi$
$677$	−32.9427	−1.26609	−0.633045	+	0.774115i	$0.718195\pi$
$678$	0	0
$679$	−41.1414	−1.57886
$680$	0	0
$681$	0	0
$682$	28.7922	1.10251
$683$	−18.3876	−0.703580	−0.351790	−	0.936079i	$-0.614427\pi$
$683$	−18.3876	−0.703580	−0.351790	+	0.936079i	$0.614427\pi$
$684$	0	0
$685$	8.58522	0.328024
$686$	50.2599	1.91893
$687$	0	0
$688$	4.26434	0.162576
$689$	40.1323	1.52892
$690$	0	0
$691$	−16.6599	−0.633773	−0.316887	−	0.948463i	$-0.602637\pi$
$691$	−16.6599	−0.633773	−0.316887	+	0.948463i	$0.602637\pi$
$692$	−20.3009	−0.771724
$693$	0	0
$694$	49.5471	1.88078
$695$	61.6892	2.34001
$696$	0	0
$697$	0	0
$698$	−58.6091	−2.21839
$699$	0	0
$700$	60.4532	2.28492
$701$	48.7175	1.84004	0.920018	−	0.391877i	$-0.128174\pi$
$701$	48.7175	1.84004	0.920018	+	0.391877i	$0.128174\pi$
$702$	0	0
$703$	0	0
$704$	−5.62177	−0.211878
$705$	0	0
$706$	63.3302	2.38346
$707$	−37.2654	−1.40151
$708$	0	0
$709$	29.4996	1.10788	0.553940	−	0.832556i	$-0.313124\pi$
$709$	29.4996	1.10788	0.553940	+	0.832556i	$0.313124\pi$
$710$	−78.6055	−2.95001
$711$	0	0
$712$	15.6327	0.585860
$713$	−15.7567	−0.590094
$714$	0	0
$715$	22.8405	0.854186
$716$	−4.60442	−0.172075
$717$	0	0
$718$	−8.11310	−0.302778
$719$	−22.6500	−0.844704	−0.422352	−	0.906432i	$-0.638795\pi$
$719$	−22.6500	−0.844704	−0.422352	+	0.906432i	$0.638795\pi$
$720$	0	0
$721$	17.5177	0.652395
$722$	0	0
$723$	0	0
$724$	15.8578	0.589352
$725$	40.0384	1.48699
$726$	0	0
$727$	52.7276	1.95556	0.977780	−	0.209633i	$-0.0672270\pi$
$727$	52.7276	1.95556	0.977780	+	0.209633i	$0.0672270\pi$
$728$	54.5525	2.02185
$729$	0	0
$730$	97.1990	3.59750
$731$	0	0
$732$	0	0
$733$	5.72659	0.211516	0.105758	−	0.994392i	$-0.466273\pi$
$733$	5.72659	0.211516	0.105758	+	0.994392i	$0.466273\pi$
$734$	74.0721	2.73405
$735$	0	0
$736$	−8.18884	−0.301845
$737$	14.3035	0.526878
$738$	0	0
$739$	33.4166	1.22925	0.614625	−	0.788819i	$-0.289307\pi$
$739$	33.4166	1.22925	0.614625	+	0.788819i	$0.289307\pi$
$740$	−14.3492	−0.527486
$741$	0	0
$742$	58.1323	2.13410
$743$	−35.9517	−1.31894	−0.659470	−	0.751730i	$-0.729219\pi$
$743$	−35.9517	−1.31894	−0.659470	+	0.751730i	$0.729219\pi$
$744$	0	0
$745$	−13.1595	−0.482127
$746$	49.4532	1.81061
$747$	0	0
$748$	0	0
$749$	16.9043	0.617668
$750$	0	0
$751$	23.1504	0.844772	0.422386	−	0.906416i	$-0.361193\pi$
$751$	23.1504	0.844772	0.422386	+	0.906416i	$0.361193\pi$
$752$	36.1696	1.31897
$753$	0	0
$754$	67.2654	2.44966
$755$	−28.8861	−1.05127
$756$	0	0
$757$	12.9891	0.472096	0.236048	−	0.971741i	$-0.424148\pi$
$757$	12.9891	0.472096	0.236048	+	0.971741i	$0.424148\pi$
$758$	−1.92064	−0.0697609
$759$	0	0
$760$	0	0
$761$	−13.3593	−0.484274	−0.242137	−	0.970242i	$-0.577848\pi$
$761$	−13.3593	−0.484274	−0.242137	+	0.970242i	$0.577848\pi$
$762$	0	0
$763$	28.4431	1.02971
$764$	3.79301	0.137226
$765$	0	0
$766$	−9.65084	−0.348699
$767$	−6.87783	−0.248344
$768$	0	0
$769$	11.6236	0.419159	0.209579	−	0.977792i	$-0.432791\pi$
$769$	11.6236	0.419159	0.209579	+	0.977792i	$0.432791\pi$
$770$	33.0848	1.19229
$771$	0	0
$772$	37.2179	1.33950
$773$	−7.75566	−0.278952	−0.139476	−	0.990225i	$-0.544542\pi$
$773$	−7.75566	−0.278952	−0.139476	+	0.990225i	$0.544542\pi$
$774$	0	0
$775$	−40.4340	−1.45243
$776$	103.435	3.71310
$777$	0	0
$778$	80.9728	2.90301
$779$	0	0
$780$	0	0
$781$	−16.0747	−0.575198
$782$	0	0
$783$	0	0
$784$	−9.72659	−0.347378
$785$	18.9235	0.675407
$786$	0	0
$787$	18.8880	0.673283	0.336642	−	0.941633i	$-0.390709\pi$
$787$	18.8880	0.673283	0.336642	+	0.941633i	$0.390709\pi$
$788$	−106.880	−3.80744
$789$	0	0
$790$	27.9627	0.994867
$791$	17.1222	0.608794
$792$	0	0
$793$	−13.6418	−0.484433
$794$	66.8644	2.37293
$795$	0	0
$796$	−95.0203	−3.36790
$797$	34.5105	1.22243	0.611213	−	0.791466i	$-0.290682\pi$
$797$	34.5105	1.22243	0.611213	+	0.791466i	$0.290682\pi$
$798$	0	0
$799$	0	0
$800$	−21.0137	−0.742948
$801$	0	0
$802$	23.4340	0.827483
$803$	19.8770	0.701445
$804$	0	0
$805$	−18.1059	−0.638148
$806$	−67.9300	−2.39273
$807$	0	0
$808$	93.6903	3.29601
$809$	−40.6044	−1.42758	−0.713788	−	0.700362i	$-0.753022\pi$
$809$	−40.6044	−1.42758	−0.713788	+	0.700362i	$0.753022\pi$
$810$	0	0
$811$	17.0957	0.600313	0.300156	−	0.953890i	$-0.402961\pi$
$811$	17.0957	0.600313	0.300156	+	0.953890i	$0.402961\pi$
$812$	66.6055	2.33739
$813$	0	0
$814$	−4.29261	−0.150456
$815$	−61.0293	−2.13776
$816$	0	0
$817$	0	0
$818$	54.8114	1.91644
$819$	0	0
$820$	95.3038	3.32815
$821$	38.0950	1.32952	0.664761	−	0.747056i	$-0.268533\pi$
$821$	38.0950	1.32952	0.664761	+	0.747056i	$0.268533\pi$
$822$	0	0
$823$	−13.2088	−0.460431	−0.230216	−	0.973140i	$-0.573943\pi$
$823$	−13.2088	−0.460431	−0.230216	+	0.973140i	$0.573943\pi$
$824$	−44.0420	−1.53428
$825$	0	0
$826$	−9.96265	−0.346645
$827$	−33.8688	−1.17773	−0.588866	−	0.808231i	$-0.700425\pi$
$827$	−33.8688	−1.17773	−0.588866	+	0.808231i	$0.700425\pi$
$828$	0	0
$829$	−16.8397	−0.584866	−0.292433	−	0.956286i	$-0.594465\pi$
$829$	−16.8397	−0.584866	−0.292433	+	0.956286i	$0.594465\pi$
$830$	83.9637	2.91442
$831$	0	0
$832$	13.2635	0.459830
$833$	0	0
$834$	0	0
$835$	−71.8397	−2.48611
$836$	0	0
$837$	0	0
$838$	12.4057	0.428548
$839$	−2.53695	−0.0875851	−0.0437926	−	0.999041i	$-0.513944\pi$
$839$	−2.53695	−0.0875851	−0.0437926	+	0.999041i	$0.513944\pi$
$840$	0	0
$841$	15.1131	0.521141
$842$	10.6983	0.368688
$843$	0	0
$844$	57.1131	1.96591
$845$	−10.7165	−0.368658
$846$	0	0
$847$	−18.7639	−0.644737
$848$	−60.0576	−2.06239
$849$	0	0
$850$	0	0
$851$	2.34916	0.0805281
$852$	0	0
$853$	−0.624423	−0.0213798	−0.0106899	−	0.999943i	$-0.503403\pi$
$853$	−0.624423	−0.0213798	−0.0106899	+	0.999943i	$0.503403\pi$
$854$	−19.7603	−0.676185
$855$	0	0
$856$	−42.4996	−1.45261
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−27.2070	−0.928291	−0.464145	−	0.885759i	$-0.653638\pi$
$859$	−27.2070	−0.928291	−0.464145	+	0.885759i	$0.653638\pi$
$860$	−10.1504	−0.346127
$861$	0	0
$862$	2.08562	0.0710365
$863$	−26.1131	−0.888900	−0.444450	−	0.895804i	$-0.646601\pi$
$863$	−26.1131	−0.888900	−0.444450	+	0.895804i	$0.646601\pi$
$864$	0	0
$865$	15.6026	0.530503
$866$	41.4386	1.40814
$867$	0	0
$868$	−67.2635	−2.28307
$869$	5.71832	0.193981
$870$	0	0
$871$	−33.7466	−1.14346
$872$	−71.5097	−2.42162
$873$	0	0
$874$	0	0
$875$	−7.92531	−0.267924
$876$	0	0
$877$	26.6591	0.900214	0.450107	−	0.892975i	$-0.351386\pi$
$877$	26.6591	0.900214	0.450107	+	0.892975i	$0.351386\pi$
$878$	71.4869	2.41257
$879$	0	0
$880$	−34.1806	−1.15223
$881$	−6.73566	−0.226930	−0.113465	−	0.993542i	$-0.536195\pi$
$881$	−6.73566	−0.226930	−0.113465	+	0.993542i	$0.536195\pi$
$882$	0	0
$883$	32.1606	1.08229	0.541145	−	0.840930i	$-0.317991\pi$
$883$	32.1606	1.08229	0.541145	+	0.840930i	$0.317991\pi$
$884$	0	0
$885$	0	0
$886$	−38.7549	−1.30200
$887$	3.83863	0.128889	0.0644443	−	0.997921i	$-0.479473\pi$
$887$	3.83863	0.128889	0.0644443	+	0.997921i	$0.479473\pi$
$888$	0	0
$889$	−33.9819	−1.13971
$890$	−22.3684	−0.749789
$891$	0	0
$892$	23.6802	0.792871
$893$	0	0
$894$	0	0
$895$	3.53880	0.118289
$896$	35.3930	1.18240
$897$	0	0
$898$	58.2080	1.94243
$899$	−44.5489	−1.48579
$900$	0	0
$901$	0	0
$902$	28.5105	0.949297
$903$	0	0
$904$	−43.0475	−1.43174
$905$	−12.1878	−0.405136
$906$	0	0
$907$	39.9637	1.32697	0.663487	−	0.748188i	$-0.269076\pi$
$907$	39.9637	1.32697	0.663487	+	0.748188i	$0.269076\pi$
$908$	33.3027	1.10519
$909$	0	0
$910$	−78.0576	−2.58758
$911$	29.3219	0.971479	0.485740	−	0.874104i	$-0.338550\pi$
$911$	29.3219	0.971479	0.485740	+	0.874104i	$0.338550\pi$
$912$	0	0
$913$	17.1704	0.568259
$914$	−11.7867	−0.389871
$915$	0	0
$916$	17.4057	0.575101
$917$	−1.48947	−0.0491867
$918$	0	0
$919$	−30.6218	−1.01012	−0.505059	−	0.863085i	$-0.668529\pi$
$919$	−30.6218	−1.01012	−0.505059	+	0.863085i	$0.668529\pi$
$920$	45.5207	1.50077
$921$	0	0
$922$	31.5471	1.03895
$923$	37.9253	1.24833
$924$	0	0
$925$	6.02827	0.198208
$926$	−57.5435	−1.89100
$927$	0	0
$928$	−23.1523	−0.760011
$929$	5.92425	0.194368	0.0971842	−	0.995266i	$-0.469016\pi$
$929$	5.92425	0.194368	0.0971842	+	0.995266i	$0.469016\pi$
$930$	0	0
$931$	0	0
$932$	−112.832	−3.69592
$933$	0	0
$934$	−29.6975	−0.971732
$935$	0	0
$936$	0	0
$937$	−5.01093	−0.163700	−0.0818499	−	0.996645i	$-0.526083\pi$
$937$	−5.01093	−0.163700	−0.0818499	+	0.996645i	$0.526083\pi$
$938$	−48.8825	−1.59607
$939$	0	0
$940$	−86.0950	−2.80811
$941$	8.39743	0.273748	0.136874	−	0.990588i	$-0.456294\pi$
$941$	8.39743	0.273748	0.136874	+	0.990588i	$0.456294\pi$
$942$	0	0
$943$	−15.6026	−0.508089
$944$	10.2926	0.334996
$945$	0	0
$946$	−3.03655	−0.0987267
$947$	19.4713	0.632733	0.316367	−	0.948637i	$-0.397537\pi$
$947$	19.4713	0.632733	0.316367	+	0.948637i	$0.397537\pi$
$948$	0	0
$949$	−46.8962	−1.52232
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.5087	0.696734	0.348367	−	0.937358i	$-0.386736\pi$
$953$	21.5087	0.696734	0.348367	+	0.937358i	$0.386736\pi$
$954$	0	0
$955$	−2.91518	−0.0943330
$956$	33.3027	1.07709
$957$	0	0
$958$	−85.5772	−2.76487
$959$	−6.00000	−0.193750
$960$	0	0
$961$	13.9891	0.451260
$962$	10.1276	0.326528
$963$	0	0
$964$	−93.7559	−3.01967
$965$	−28.6044	−0.920809
$966$	0	0
$967$	−10.8205	−0.347963	−0.173982	−	0.984749i	$-0.555663\pi$
$967$	−10.8205	−0.347963	−0.173982	+	0.984749i	$0.555663\pi$
$968$	47.1751	1.51627
$969$	0	0
$970$	−148.002	−4.75206
$971$	30.9992	0.994812	0.497406	−	0.867518i	$-0.334286\pi$
$971$	30.9992	0.994812	0.497406	+	0.867518i	$0.334286\pi$
$972$	0	0
$973$	−43.1131	−1.38214
$974$	−18.6418	−0.597321
$975$	0	0
$976$	20.4148	0.653461
$977$	56.0950	1.79464	0.897318	−	0.441384i	$-0.145512\pi$
$977$	56.0950	1.79464	0.897318	+	0.441384i	$0.145512\pi$
$978$	0	0
$979$	−4.57429	−0.146195
$980$	23.1523	0.739573
$981$	0	0
$982$	9.72659	0.310388
$983$	−33.4449	−1.06673	−0.533363	−	0.845886i	$-0.679072\pi$
$983$	−33.4449	−1.06673	−0.533363	+	0.845886i	$0.679072\pi$
$984$	0	0
$985$	82.1443	2.61733
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.66177	0.0528412
$990$	0	0
$991$	−8.44491	−0.268261	−0.134131	−	0.990964i	$-0.542824\pi$
$991$	−8.44491	−0.268261	−0.134131	+	0.990964i	$0.542824\pi$
$992$	23.3810	0.742349
$993$	0	0
$994$	54.9354	1.74245
$995$	73.0293	2.31519
$996$	0	0
$997$	50.4148	1.59665	0.798326	−	0.602225i	$-0.205719\pi$
$997$	50.4148	1.59665	0.798326	+	0.602225i	$0.205719\pi$
$998$	89.4688	2.83208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.t.1.1		3
3.2	odd	2		1083.2.a.o.1.3		3
19.8	odd	6		171.2.f.b.64.1		6
19.12	odd	6		171.2.f.b.163.1		6
19.18	odd	2		3249.2.a.y.1.3		3
57.8	even	6		57.2.e.b.7.3	✓	6
57.50	even	6		57.2.e.b.49.3	yes	6
57.56	even	2		1083.2.a.l.1.1		3
76.27	even	6		2736.2.s.z.577.3		6
76.31	even	6		2736.2.s.z.1873.3		6
228.107	odd	6		912.2.q.l.49.1		6
228.179	odd	6		912.2.q.l.577.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.3	✓	6	57.8	even	6
57.2.e.b.49.3	yes	6	57.50	even	6
171.2.f.b.64.1		6	19.8	odd	6
171.2.f.b.163.1		6	19.12	odd	6
912.2.q.l.49.1		6	228.107	odd	6
912.2.q.l.577.1		6	228.179	odd	6
1083.2.a.l.1.1		3	57.56	even	2
1083.2.a.o.1.3		3	3.2	odd	2
2736.2.s.z.577.3		6	76.27	even	6
2736.2.s.z.1873.3		6	76.31	even	6
3249.2.a.t.1.1		3	1.1	even	1	trivial
3249.2.a.y.1.3		3	19.18	odd	2

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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51414 q^{2} +4.32088 q^{4} -3.32088 q^{5} +2.32088 q^{7} -5.83502 q^{8} +O(q^{10})\)	\(q-2.51414 q^{2} +4.32088 q^{4} -3.32088 q^{5} +2.32088 q^{7} -5.83502 q^{8} +8.34916 q^{10} +1.70739 q^{11} -4.02827 q^{13} -5.83502 q^{14} +6.02827 q^{16} -14.3492 q^{20} -4.29261 q^{22} +2.34916 q^{23} +6.02827 q^{25} +10.1276 q^{26} +10.0283 q^{28} +6.64177 q^{29} -6.70739 q^{31} -3.48586 q^{32} -7.70739 q^{35} +1.00000 q^{37} +19.3774 q^{40} -6.64177 q^{41} +0.707389 q^{43} +7.37743 q^{44} -5.90611 q^{46} +6.00000 q^{47} -1.61350 q^{49} -15.1559 q^{50} -17.4057 q^{52} -9.96265 q^{53} -5.67004 q^{55} -13.5424 q^{56} -16.6983 q^{58} +1.70739 q^{59} +3.38650 q^{61} +16.8633 q^{62} -3.29261 q^{64} +13.3774 q^{65} +8.37743 q^{67} +19.3774 q^{70} -9.41478 q^{71} +11.6418 q^{73} -2.51414 q^{74} +3.96265 q^{77} +3.34916 q^{79} -20.0192 q^{80} +16.6983 q^{82} +10.0565 q^{83} -1.77847 q^{86} -9.96265 q^{88} -2.67912 q^{89} -9.34916 q^{91} +10.1504 q^{92} -15.0848 q^{94} -17.7266 q^{97} +4.05655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} + 5 q^{16} - 22 q^{20} - 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} + 4 q^{29} - 15 q^{31} - 17 q^{32} - 18 q^{35} + 3 q^{37} + 24 q^{40} - 4 q^{41} - 3 q^{43} - 12 q^{44} - 20 q^{46} + 18 q^{47} - 2 q^{49} - 23 q^{50} - 5 q^{52} - 6 q^{53} + 12 q^{55} - 21 q^{56} - 8 q^{58} + 13 q^{61} + 23 q^{62} - 15 q^{64} + 6 q^{65} - 9 q^{67} + 24 q^{70} - 18 q^{71} + 19 q^{73} - q^{74} - 12 q^{77} - 11 q^{79} - 10 q^{80} + 8 q^{82} + 4 q^{83} - 17 q^{86} - 6 q^{88} - 16 q^{89} - 7 q^{91} + 2 q^{92} - 6 q^{94} + 2 q^{97} - 14 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 + 4 * q^10 + q^13 - 3 * q^14 + 5 * q^16 - 22 * q^20 - 18 * q^22 - 14 * q^23 + 5 * q^25 + 21 * q^26 + 17 * q^28 + 4 * q^29 - 15 * q^31 - 17 * q^32 - 18 * q^35 + 3 * q^37 + 24 * q^40 - 4 * q^41 - 3 * q^43 - 12 * q^44 - 20 * q^46 + 18 * q^47 - 2 * q^49 - 23 * q^50 - 5 * q^52 - 6 * q^53 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 13 * q^61 + 23 * q^62 - 15 * q^64 + 6 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 + 19 * q^73 - q^74 - 12 * q^77 - 11 * q^79 - 10 * q^80 + 8 * q^82 + 4 * q^83 - 17 * q^86 - 6 * q^88 - 16 * q^89 - 7 * q^91 + 2 * q^92 - 6 * q^94 + 2 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more