LMFDB - Embedded newform 7225.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
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} - 20x^{10} + 135x^{8} - 400x^{6} + 515x^{4} - 222x^{2} + 25 $ x^12 - 20x^10 + 135x^8 - 400x^6 + 515x^4 - 222*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 425)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.62170$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.44718 q^{2} +1.62170 q^{3} +0.0943296 q^{4} -2.34689 q^{6} -3.81562 q^{7} +2.75785 q^{8} -0.370087 q^{9} -6.14862 q^{11} +0.152974 q^{12} -2.56919 q^{13} +5.52189 q^{14} -4.17976 q^{16} +0.535583 q^{18} +1.17142 q^{19} -6.18779 q^{21} +8.89815 q^{22} -3.57861 q^{23} +4.47240 q^{24} +3.71808 q^{26} -5.46527 q^{27} -0.359926 q^{28} +5.23551 q^{29} -0.558323 q^{31} +0.533170 q^{32} -9.97121 q^{33} -0.0349102 q^{36} -7.46770 q^{37} -1.69526 q^{38} -4.16645 q^{39} -7.57672 q^{41} +8.95485 q^{42} -0.774454 q^{43} -0.579996 q^{44} +5.17890 q^{46} +4.35087 q^{47} -6.77832 q^{48} +7.55894 q^{49} -0.242351 q^{52} +2.36008 q^{53} +7.90923 q^{54} -10.5229 q^{56} +1.89970 q^{57} -7.57672 q^{58} -12.8886 q^{59} +4.66449 q^{61} +0.807994 q^{62} +1.41211 q^{63} +7.58793 q^{64} +14.4301 q^{66} +2.97145 q^{67} -5.80344 q^{69} -8.44824 q^{71} -1.02064 q^{72} -16.7118 q^{73} +10.8071 q^{74} +0.110500 q^{76} +23.4608 q^{77} +6.02961 q^{78} -11.4516 q^{79} -7.75277 q^{81} +10.9649 q^{82} +5.07499 q^{83} -0.583692 q^{84} +1.12077 q^{86} +8.49043 q^{87} -16.9569 q^{88} -8.16031 q^{89} +9.80304 q^{91} -0.337569 q^{92} -0.905433 q^{93} -6.29650 q^{94} +0.864643 q^{96} -7.29986 q^{97} -10.9392 q^{98} +2.27553 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 12 q^{13} + 4 q^{16} + 4 q^{18} + 12 q^{19} - 8 q^{21} + 12 q^{26} + 48 q^{32} - 4 q^{33} - 20 q^{36} - 12 q^{38} + 56 q^{42} + 16 q^{43} + 36 q^{47} + 16 q^{49}+ \cdots + 76 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 12 * q^13 + 4 * q^16 + 4 * q^18 + 12 * q^19 - 8 * q^21 + 12 * q^26 + 48 * q^32 - 4 * q^33 - 20 * q^36 - 12 * q^38 + 56 * q^42 + 16 * q^43 + 36 * q^47 + 16 * q^49 + 64 * q^52 + 48 * q^53 - 32 * q^59 + 44 * q^64 + 40 * q^66 - 20 * q^67 - 72 * q^69 - 92 * q^72 + 36 * q^76 + 24 * q^77 + 8 * q^81 + 52 * q^83 + 36 * q^84 + 84 * q^86 + 48 * q^87 + 24 * q^89 + 60 * q^93 - 8 * q^94 + 76 * q^98

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 $

$2$	−1.44718	−1.02331	−0.511655	−	0.859191i	$-0.670968\pi$
$2$	−1.44718	−1.02331	−0.511655	+	0.859191i	$0.670968\pi$
$3$	1.62170	0.936289	0.468145	−	0.883652i	$-0.344923\pi$
$3$	1.62170	0.936289	0.468145	+	0.883652i	$0.344923\pi$
$4$	0.0943296	0.0471648
$5$	0	0
$5$	0	0
$6$	−2.34689	−0.958115
$7$	−3.81562	−1.44217	−0.721084	−	0.692848i	$-0.756356\pi$
$7$	−3.81562	−1.44217	−0.721084	+	0.692848i	$0.756356\pi$
$8$	2.75785	0.975046
$9$	−0.370087	−0.123362
$10$	0	0
$11$	−6.14862	−1.85388	−0.926939	−	0.375213i	$-0.877570\pi$
$11$	−6.14862	−1.85388	−0.926939	+	0.375213i	$0.877570\pi$
$12$	0.152974	0.0441599
$13$	−2.56919	−0.712565	−0.356282	−	0.934378i	$-0.615956\pi$
$13$	−2.56919	−0.712565	−0.356282	+	0.934378i	$0.615956\pi$
$14$	5.52189	1.47579
$15$	0	0
$16$	−4.17976	−1.04494
$17$	0	0
$17$	0	0
$18$	0.535583	0.126238
$19$	1.17142	0.268743	0.134371	−	0.990931i	$-0.457099\pi$
$19$	1.17142	0.268743	0.134371	+	0.990931i	$0.457099\pi$
$20$	0	0
$21$	−6.18779	−1.35029
$22$	8.89815	1.89709
$23$	−3.57861	−0.746192	−0.373096	−	0.927793i	$-0.621704\pi$
$23$	−3.57861	−0.746192	−0.373096	+	0.927793i	$0.621704\pi$
$24$	4.47240	0.912926
$25$	0	0
$26$	3.71808	0.729175
$27$	−5.46527	−1.05179
$28$	−0.359926	−0.0680196
$29$	5.23551	0.972210	0.486105	−	0.873900i	$-0.338417\pi$
$29$	5.23551	0.972210	0.486105	+	0.873900i	$0.338417\pi$
$30$	0	0
$31$	−0.558323	−0.100278	−0.0501389	−	0.998742i	$-0.515966\pi$
$31$	−0.558323	−0.100278	−0.0501389	+	0.998742i	$0.515966\pi$
$32$	0.533170	0.0942521
$33$	−9.97121	−1.73577
$34$	0	0
$35$	0	0
$36$	−0.0349102	−0.00581837
$37$	−7.46770	−1.22768	−0.613841	−	0.789430i	$-0.710376\pi$
$37$	−7.46770	−1.22768	−0.613841	+	0.789430i	$0.710376\pi$
$38$	−1.69526	−0.275007
$39$	−4.16645	−0.667167
$40$	0	0
$41$	−7.57672	−1.18329	−0.591643	−	0.806200i	$-0.701520\pi$
$41$	−7.57672	−1.18329	−0.591643	+	0.806200i	$0.701520\pi$
$42$	8.95485	1.38176
$43$	−0.774454	−0.118103	−0.0590516	−	0.998255i	$-0.518808\pi$
$43$	−0.774454	−0.118103	−0.0590516	+	0.998255i	$0.518808\pi$
$44$	−0.579996	−0.0874378
$45$	0	0
$46$	5.17890	0.763587
$47$	4.35087	0.634640	0.317320	−	0.948318i	$-0.397217\pi$
$47$	4.35087	0.634640	0.317320	+	0.948318i	$0.397217\pi$
$48$	−6.77832	−0.978366
$49$	7.55894	1.07985
$50$	0	0
$51$	0	0
$52$	−0.242351	−0.0336080
$53$	2.36008	0.324182	0.162091	−	0.986776i	$-0.448176\pi$
$53$	2.36008	0.324182	0.162091	+	0.986776i	$0.448176\pi$
$54$	7.90923	1.07631
$55$	0	0
$56$	−10.5229	−1.40618
$57$	1.89970	0.251621
$58$	−7.57672	−0.994873
$59$	−12.8886	−1.67796	−0.838978	−	0.544165i	$-0.816847\pi$
$59$	−12.8886	−1.67796	−0.838978	+	0.544165i	$0.816847\pi$
$60$	0	0
$61$	4.66449	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$61$	4.66449	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$62$	0.807994	0.102615
$63$	1.41211	0.177909
$64$	7.58793	0.948491
$65$	0	0
$66$	14.4301	1.77623
$67$	2.97145	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$67$	2.97145	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$68$	0	0
$69$	−5.80344	−0.698652
$70$	0	0
$71$	−8.44824	−1.00262	−0.501311	−	0.865267i	$-0.667149\pi$
$71$	−8.44824	−1.00262	−0.501311	+	0.865267i	$0.667149\pi$
$72$	−1.02064	−0.120284
$73$	−16.7118	−1.95597	−0.977984	−	0.208681i	$-0.933083\pi$
$73$	−16.7118	−1.95597	−0.977984	+	0.208681i	$0.933083\pi$
$74$	10.8071	1.25630
$75$	0	0
$76$	0.110500	0.0126752
$77$	23.4608	2.67360
$78$	6.02961	0.682719
$79$	−11.4516	−1.28840	−0.644201	−	0.764857i	$-0.722810\pi$
$79$	−11.4516	−1.28840	−0.644201	+	0.764857i	$0.722810\pi$
$80$	0	0
$81$	−7.75277	−0.861419
$82$	10.9649	1.21087
$83$	5.07499	0.557052	0.278526	−	0.960429i	$-0.410154\pi$
$83$	5.07499	0.557052	0.278526	+	0.960429i	$0.410154\pi$
$84$	−0.583692	−0.0636860
$85$	0	0
$86$	1.12077	0.120856
$87$	8.49043	0.910270
$88$	−16.9569	−1.80762
$89$	−8.16031	−0.864991	−0.432496	−	0.901636i	$-0.642367\pi$
$89$	−8.16031	−0.864991	−0.432496	+	0.901636i	$0.642367\pi$
$90$	0	0
$91$	9.80304	1.02764
$92$	−0.337569	−0.0351940
$93$	−0.905433	−0.0938890
$94$	−6.29650	−0.649434
$95$	0	0
$96$	0.864643	0.0882472
$97$	−7.29986	−0.741188	−0.370594	−	0.928795i	$-0.620846\pi$
$97$	−7.29986	−0.741188	−0.370594	+	0.928795i	$0.620846\pi$
$98$	−10.9392	−1.10502
$99$	2.27553	0.228699
$100$	0	0
$101$	−9.68792	−0.963984	−0.481992	−	0.876176i	$-0.660087\pi$
$101$	−9.68792	−0.963984	−0.481992	+	0.876176i	$0.660087\pi$
$102$	0	0
$103$	11.1163	1.09532	0.547660	−	0.836701i	$-0.315519\pi$
$103$	11.1163	1.09532	0.547660	+	0.836701i	$0.315519\pi$
$104$	−7.08543	−0.694784
$105$	0	0
$106$	−3.41546	−0.331739
$107$	9.07400	0.877216	0.438608	−	0.898678i	$-0.355472\pi$
$107$	9.07400	0.877216	0.438608	+	0.898678i	$0.355472\pi$
$108$	−0.515537	−0.0496076
$109$	−9.38992	−0.899391	−0.449695	−	0.893182i	$-0.648467\pi$
$109$	−9.38992	−0.899391	−0.449695	+	0.893182i	$0.648467\pi$
$110$	0	0
$111$	−12.1104	−1.14947
$112$	15.9484	1.50698
$113$	−5.51881	−0.519166	−0.259583	−	0.965721i	$-0.583585\pi$
$113$	−5.51881	−0.519166	−0.259583	+	0.965721i	$0.583585\pi$
$114$	−2.74920	−0.257486
$115$	0	0
$116$	0.493864	0.0458541
$117$	0.950824	0.0879037
$118$	18.6522	1.71707
$119$	0	0
$120$	0	0
$121$	26.8055	2.43686
$122$	−6.75035	−0.611148
$123$	−12.2872	−1.10790
$124$	−0.0526664	−0.00472958
$125$	0	0
$126$	−2.04358	−0.182057
$127$	−8.49327	−0.753656	−0.376828	−	0.926283i	$-0.622985\pi$
$127$	−8.49327	−0.753656	−0.376828	+	0.926283i	$0.622985\pi$
$128$	−12.0474	−1.06485
$129$	−1.25593	−0.110579
$130$	0	0
$131$	4.92623	0.430407	0.215203	−	0.976569i	$-0.430959\pi$
$131$	4.92623	0.430407	0.215203	+	0.976569i	$0.430959\pi$
$132$	−0.940581	−0.0818670
$133$	−4.46970	−0.387572
$134$	−4.30023	−0.371483
$135$	0	0
$136$	0	0
$137$	15.9794	1.36521	0.682604	−	0.730788i	$-0.260847\pi$
$137$	15.9794	1.36521	0.682604	+	0.730788i	$0.260847\pi$
$138$	8.39862	0.714938
$139$	12.0611	1.02301	0.511504	−	0.859281i	$-0.329089\pi$
$139$	12.0611	1.02301	0.511504	+	0.859281i	$0.329089\pi$
$140$	0	0
$141$	7.05581	0.594207
$142$	12.2261	1.02599
$143$	15.7970	1.32101
$144$	1.54688	0.128906
$145$	0	0
$146$	24.1850	2.00156
$147$	12.2583	1.01105
$148$	−0.704425	−0.0579034
$149$	0.791691	0.0648579	0.0324289	−	0.999474i	$-0.489676\pi$
$149$	0.791691	0.0648579	0.0324289	+	0.999474i	$0.489676\pi$
$150$	0	0
$151$	−19.1605	−1.55926	−0.779630	−	0.626240i	$-0.784593\pi$
$151$	−19.1605	−1.55926	−0.779630	+	0.626240i	$0.784593\pi$
$152$	3.23060	0.262037
$153$	0	0
$154$	−33.9520	−2.73593
$155$	0	0
$156$	−0.393020	−0.0314668
$157$	18.0597	1.44132	0.720662	−	0.693287i	$-0.243838\pi$
$157$	18.0597	1.44132	0.720662	+	0.693287i	$0.243838\pi$
$158$	16.5725	1.31843
$159$	3.82735	0.303528
$160$	0	0
$161$	13.6546	1.07613
$162$	11.2197	0.881500
$163$	22.9689	1.79906	0.899532	−	0.436855i	$-0.143908\pi$
$163$	22.9689	1.79906	0.899532	+	0.436855i	$0.143908\pi$
$164$	−0.714709	−0.0558094
$165$	0	0
$166$	−7.34442	−0.570038
$167$	−2.55674	−0.197846	−0.0989231	−	0.995095i	$-0.531540\pi$
$167$	−2.55674	−0.197846	−0.0989231	+	0.995095i	$0.531540\pi$
$168$	−17.0650	−1.31659
$169$	−6.39927	−0.492252
$170$	0	0
$171$	−0.433529	−0.0331528
$172$	−0.0730539	−0.00557031
$173$	−2.58835	−0.196788	−0.0983942	−	0.995148i	$-0.531371\pi$
$173$	−2.58835	−0.196788	−0.0983942	+	0.995148i	$0.531371\pi$
$174$	−12.2872	−0.931489
$175$	0	0
$176$	25.6997	1.93719
$177$	−20.9015	−1.57105
$178$	11.8094	0.885155
$179$	20.7762	1.55289	0.776444	−	0.630186i	$-0.217021\pi$
$179$	20.7762	1.55289	0.776444	+	0.630186i	$0.217021\pi$
$180$	0	0
$181$	−7.24176	−0.538276	−0.269138	−	0.963102i	$-0.586739\pi$
$181$	−7.24176	−0.538276	−0.269138	+	0.963102i	$0.586739\pi$
$182$	−14.1868	−1.05159
$183$	7.56440	0.559177
$184$	−9.86927	−0.727572
$185$	0	0
$186$	1.31032	0.0960776
$187$	0	0
$188$	0.410416	0.0299327
$189$	20.8534	1.51686
$190$	0	0
$191$	−24.3229	−1.75994	−0.879971	−	0.475027i	$-0.842438\pi$
$191$	−24.3229	−1.75994	−0.879971	+	0.475027i	$0.842438\pi$
$192$	12.3053	0.888062
$193$	16.8034	1.20954	0.604769	−	0.796401i	$-0.293265\pi$
$193$	16.8034	1.20954	0.604769	+	0.796401i	$0.293265\pi$
$194$	10.5642	0.758466
$195$	0	0
$196$	0.713032	0.0509309
$197$	13.0123	0.927087	0.463544	−	0.886074i	$-0.346578\pi$
$197$	13.0123	0.927087	0.463544	+	0.886074i	$0.346578\pi$
$198$	−3.29309	−0.234030
$199$	22.3245	1.58254	0.791269	−	0.611468i	$-0.209421\pi$
$199$	22.3245	1.58254	0.791269	+	0.611468i	$0.209421\pi$
$200$	0	0
$201$	4.81881	0.339892
$202$	14.0202	0.986455
$203$	−19.9767	−1.40209
$204$	0	0
$205$	0	0
$206$	−16.0873	−1.12085
$207$	1.32440	0.0920521
$208$	10.7386	0.744588
$209$	−7.20263	−0.498216
$210$	0	0
$211$	18.9114	1.30192	0.650958	−	0.759114i	$-0.274367\pi$
$211$	18.9114	1.30192	0.650958	+	0.759114i	$0.274367\pi$
$212$	0.222626	0.0152900
$213$	−13.7005	−0.938744
$214$	−13.1317	−0.897665
$215$	0	0
$216$	−15.0724	−1.02555
$217$	2.13035	0.144617
$218$	13.5889	0.920356
$219$	−27.1015	−1.83135
$220$	0	0
$221$	0	0
$222$	17.5259	1.17626
$223$	−7.58316	−0.507806	−0.253903	−	0.967230i	$-0.581714\pi$
$223$	−7.58316	−0.507806	−0.253903	+	0.967230i	$0.581714\pi$
$224$	−2.03438	−0.135927
$225$	0	0
$226$	7.98671	0.531268
$227$	−9.60428	−0.637458	−0.318729	−	0.947846i	$-0.603256\pi$
$227$	−9.60428	−0.637458	−0.318729	+	0.947846i	$0.603256\pi$
$228$	0.179198	0.0118676
$229$	4.12351	0.272489	0.136245	−	0.990675i	$-0.456497\pi$
$229$	4.12351	0.272489	0.136245	+	0.990675i	$0.456497\pi$
$230$	0	0
$231$	38.0464	2.50327
$232$	14.4387	0.947950
$233$	−2.09247	−0.137082	−0.0685412	−	0.997648i	$-0.521834\pi$
$233$	−2.09247	−0.137082	−0.0685412	+	0.997648i	$0.521834\pi$
$234$	−1.37601	−0.0899528
$235$	0	0
$236$	−1.21578	−0.0791405
$237$	−18.5710	−1.20632
$238$	0	0
$239$	−5.62948	−0.364141	−0.182070	−	0.983286i	$-0.558280\pi$
$239$	−5.62948	−0.364141	−0.182070	+	0.983286i	$0.558280\pi$
$240$	0	0
$241$	−12.7279	−0.819878	−0.409939	−	0.912113i	$-0.634450\pi$
$241$	−12.7279	−0.819878	−0.409939	+	0.912113i	$0.634450\pi$
$242$	−38.7924	−2.49367
$243$	3.82314	0.245255
$244$	0.439999	0.0281681
$245$	0	0
$246$	17.7818	1.13372
$247$	−3.00960	−0.191497
$248$	−1.53977	−0.0977755
$249$	8.23011	0.521562
$250$	0	0
$251$	11.2701	0.711361	0.355681	−	0.934608i	$-0.384249\pi$
$251$	11.2701	0.711361	0.355681	+	0.934608i	$0.384249\pi$
$252$	0.133204	0.00839106
$253$	22.0035	1.38335
$254$	12.2913	0.771224
$255$	0	0
$256$	2.25895	0.141185
$257$	−3.54606	−0.221197	−0.110598	−	0.993865i	$-0.535277\pi$
$257$	−3.54606	−0.221197	−0.110598	+	0.993865i	$0.535277\pi$
$258$	1.81756	0.113156
$259$	28.4939	1.77052
$260$	0	0
$261$	−1.93760	−0.119934
$262$	−7.12914	−0.440440
$263$	23.1294	1.42622	0.713109	−	0.701053i	$-0.247286\pi$
$263$	23.1294	1.42622	0.713109	+	0.701053i	$0.247286\pi$
$264$	−27.4991	−1.69245
$265$	0	0
$266$	6.46846	0.396607
$267$	−13.2336	−0.809882
$268$	0.280296	0.0171218
$269$	0.735073	0.0448182	0.0224091	−	0.999749i	$-0.492866\pi$
$269$	0.735073	0.0448182	0.0224091	+	0.999749i	$0.492866\pi$
$270$	0	0
$271$	9.37788	0.569666	0.284833	−	0.958577i	$-0.408062\pi$
$271$	9.37788	0.569666	0.284833	+	0.958577i	$0.408062\pi$
$272$	0	0
$273$	15.8976	0.962167
$274$	−23.1250	−1.39703
$275$	0	0
$276$	−0.547436	−0.0329518
$277$	13.5913	0.816621	0.408311	−	0.912843i	$-0.366118\pi$
$277$	13.5913	0.816621	0.408311	+	0.912843i	$0.366118\pi$
$278$	−17.4545	−1.04685
$279$	0.206628	0.0123705
$280$	0	0
$281$	−16.6410	−0.992717	−0.496358	−	0.868118i	$-0.665330\pi$
$281$	−16.6410	−0.992717	−0.496358	+	0.868118i	$0.665330\pi$
$282$	−10.2110	−0.608058
$283$	10.4897	0.623549	0.311775	−	0.950156i	$-0.399077\pi$
$283$	10.4897	0.623549	0.311775	+	0.950156i	$0.399077\pi$
$284$	−0.796919	−0.0472884
$285$	0	0
$286$	−22.8610	−1.35180
$287$	28.9099	1.70650
$288$	−0.197320	−0.0116272
$289$	0	0
$290$	0	0
$291$	−11.8382	−0.693967
$292$	−1.57642	−0.0922528
$293$	−1.47825	−0.0863603	−0.0431801	−	0.999067i	$-0.513749\pi$
$293$	−1.47825	−0.0863603	−0.0431801	+	0.999067i	$0.513749\pi$
$294$	−17.7400	−1.03462
$295$	0	0
$296$	−20.5948	−1.19705
$297$	33.6039	1.94989
$298$	−1.14572	−0.0663697
$299$	9.19413	0.531710
$300$	0	0
$301$	2.95502	0.170325
$302$	27.7287	1.59561
$303$	−15.7109	−0.902568
$304$	−4.89626	−0.280820
$305$	0	0
$306$	0	0
$307$	19.3620	1.10505	0.552524	−	0.833497i	$-0.313665\pi$
$307$	19.3620	1.10505	0.552524	+	0.833497i	$0.313665\pi$
$308$	2.21305	0.126100
$309$	18.0273	1.02554
$310$	0	0
$311$	0.840522	0.0476616	0.0238308	−	0.999716i	$-0.492414\pi$
$311$	0.840522	0.0476616	0.0238308	+	0.999716i	$0.492414\pi$
$312$	−11.4904	−0.650519
$313$	−6.93017	−0.391716	−0.195858	−	0.980632i	$-0.562749\pi$
$313$	−6.93017	−0.391716	−0.195858	+	0.980632i	$0.562749\pi$
$314$	−26.1357	−1.47492
$315$	0	0
$316$	−1.08022	−0.0607672
$317$	7.92328	0.445016	0.222508	−	0.974931i	$-0.428576\pi$
$317$	7.92328	0.445016	0.222508	+	0.974931i	$0.428576\pi$
$318$	−5.53886	−0.310604
$319$	−32.1911	−1.80236
$320$	0	0
$321$	14.7153	0.821328
$322$	−19.7607	−1.10122
$323$	0	0
$324$	−0.731316	−0.0406287
$325$	0	0
$326$	−33.2401	−1.84100
$327$	−15.2276	−0.842090
$328$	−20.8955	−1.15376
$329$	−16.6013	−0.915258
$330$	0	0
$331$	14.0945	0.774705	0.387353	−	0.921932i	$-0.373390\pi$
$331$	14.0945	0.774705	0.387353	+	0.921932i	$0.373390\pi$
$332$	0.478722	0.0262733
$333$	2.76370	0.151450
$334$	3.70006	0.202458
$335$	0	0
$336$	25.8635	1.41097
$337$	−9.72039	−0.529503	−0.264752	−	0.964317i	$-0.585290\pi$
$337$	−9.72039	−0.529503	−0.264752	+	0.964317i	$0.585290\pi$
$338$	9.26089	0.503726
$339$	−8.94986	−0.486090
$340$	0	0
$341$	3.43291	0.185903
$342$	0.627394	0.0339256
$343$	−2.13272	−0.115156
$344$	−2.13583	−0.115156
$345$	0	0
$346$	3.74580	0.201376
$347$	−0.113821	−0.00611026	−0.00305513	−	0.999995i	$-0.500972\pi$
$347$	−0.113821	−0.00611026	−0.00305513	+	0.999995i	$0.500972\pi$
$348$	0.800899	0.0429327
$349$	−21.3996	−1.14550	−0.572748	−	0.819732i	$-0.694122\pi$
$349$	−21.3996	−1.14550	−0.572748	+	0.819732i	$0.694122\pi$
$350$	0	0
$351$	14.0413	0.749470
$352$	−3.27826	−0.174732
$353$	−15.9897	−0.851048	−0.425524	−	0.904947i	$-0.639910\pi$
$353$	−15.9897	−0.851048	−0.425524	+	0.904947i	$0.639910\pi$
$354$	30.2482	1.60768
$355$	0	0
$356$	−0.769759	−0.0407971
$357$	0	0
$358$	−30.0669	−1.58909
$359$	7.48730	0.395164	0.197582	−	0.980286i	$-0.436691\pi$
$359$	7.48730	0.395164	0.197582	+	0.980286i	$0.436691\pi$
$360$	0	0
$361$	−17.6278	−0.927777
$362$	10.4801	0.550823
$363$	43.4705	2.28161
$364$	0.924717	0.0484683
$365$	0	0
$366$	−10.9470	−0.572211
$367$	26.2279	1.36908	0.684542	−	0.728974i	$-0.260002\pi$
$367$	26.2279	1.36908	0.684542	+	0.728974i	$0.260002\pi$
$368$	14.9577	0.779726
$369$	2.80405	0.145973
$370$	0	0
$371$	−9.00517	−0.467525
$372$	−0.0854091	−0.00442826
$373$	8.87012	0.459277	0.229639	−	0.973276i	$-0.426246\pi$
$373$	8.87012	0.459277	0.229639	+	0.973276i	$0.426246\pi$
$374$	0	0
$375$	0	0
$376$	11.9990	0.618804
$377$	−13.4510	−0.692762
$378$	−30.1786	−1.55222
$379$	3.62460	0.186183	0.0930917	−	0.995658i	$-0.470325\pi$
$379$	3.62460	0.186183	0.0930917	+	0.995658i	$0.470325\pi$
$380$	0	0
$381$	−13.7735	−0.705640
$382$	35.1996	1.80097
$383$	28.3307	1.44763	0.723815	−	0.689994i	$-0.242387\pi$
$383$	28.3307	1.44763	0.723815	+	0.689994i	$0.242387\pi$
$384$	−19.5373	−0.997011
$385$	0	0
$386$	−24.3176	−1.23773
$387$	0.286616	0.0145695
$388$	−0.688593	−0.0349580
$389$	−6.35885	−0.322407	−0.161203	−	0.986921i	$-0.551537\pi$
$389$	−6.35885	−0.322407	−0.161203	+	0.986921i	$0.551537\pi$
$390$	0	0
$391$	0	0
$392$	20.8464	1.05290
$393$	7.98887	0.402985
$394$	−18.8311	−0.948698
$395$	0	0
$396$	0.214649	0.0107865
$397$	−8.68310	−0.435792	−0.217896	−	0.975972i	$-0.569919\pi$
$397$	−8.68310	−0.435792	−0.217896	+	0.975972i	$0.569919\pi$
$398$	−32.3075	−1.61943
$399$	−7.24851	−0.362880
$400$	0	0
$401$	−27.8569	−1.39111	−0.695554	−	0.718474i	$-0.744841\pi$
$401$	−27.8569	−1.39111	−0.695554	+	0.718474i	$0.744841\pi$
$402$	−6.97368	−0.347815
$403$	1.43444	0.0714544
$404$	−0.913857	−0.0454661
$405$	0	0
$406$	28.9099	1.43477
$407$	45.9160	2.27597
$408$	0	0
$409$	17.9787	0.888989	0.444494	−	0.895782i	$-0.353383\pi$
$409$	17.9787	0.888989	0.444494	+	0.895782i	$0.353383\pi$
$410$	0	0
$411$	25.9137	1.27823
$412$	1.04859	0.0516605
$413$	49.1781	2.41990
$414$	−1.91664	−0.0941979
$415$	0	0
$416$	−1.36982	−0.0671607
$417$	19.5595	0.957831
$418$	10.4235	0.509830
$419$	−34.0826	−1.66504	−0.832522	−	0.553992i	$-0.813104\pi$
$419$	−34.0826	−1.66504	−0.832522	+	0.553992i	$0.813104\pi$
$420$	0	0
$421$	−30.0061	−1.46241	−0.731204	−	0.682159i	$-0.761041\pi$
$421$	−30.0061	−1.46241	−0.731204	+	0.682159i	$0.761041\pi$
$422$	−27.3682	−1.33226
$423$	−1.61020	−0.0782908
$424$	6.50875	0.316093
$425$	0	0
$426$	19.8271	0.960627
$427$	−17.7979	−0.861301
$428$	0.855946	0.0413737
$429$	25.6179	1.23685
$430$	0	0
$431$	−29.2342	−1.40816	−0.704082	−	0.710119i	$-0.748641\pi$
$431$	−29.2342	−1.40816	−0.704082	+	0.710119i	$0.748641\pi$
$432$	22.8435	1.09906
$433$	−9.55769	−0.459313	−0.229657	−	0.973272i	$-0.573760\pi$
$433$	−9.55769	−0.459313	−0.229657	+	0.973272i	$0.573760\pi$
$434$	−3.08300	−0.147989
$435$	0	0
$436$	−0.885747	−0.0424196
$437$	−4.19207	−0.200534
$438$	39.2208	1.87404
$439$	−30.5786	−1.45943	−0.729717	−	0.683749i	$-0.760348\pi$
$439$	−30.5786	−1.45943	−0.729717	+	0.683749i	$0.760348\pi$
$440$	0	0
$441$	−2.79747	−0.133213
$442$	0	0
$443$	4.19885	0.199493	0.0997467	−	0.995013i	$-0.468197\pi$
$443$	4.19885	0.199493	0.0997467	+	0.995013i	$0.468197\pi$
$444$	−1.14237	−0.0542143
$445$	0	0
$446$	10.9742	0.519644
$447$	1.28389	0.0607257
$448$	−28.9526	−1.36788
$449$	30.5834	1.44332	0.721660	−	0.692248i	$-0.243379\pi$
$449$	30.5834	1.44332	0.721660	+	0.692248i	$0.243379\pi$
$450$	0	0
$451$	46.5864	2.19367
$452$	−0.520587	−0.0244864
$453$	−31.0726	−1.45992
$454$	13.8991	0.652318
$455$	0	0
$456$	5.23907	0.245342
$457$	−5.94941	−0.278302	−0.139151	−	0.990271i	$-0.544437\pi$
$457$	−5.94941	−0.278302	−0.139151	+	0.990271i	$0.544437\pi$
$458$	−5.96746	−0.278841
$459$	0	0
$460$	0	0
$461$	38.0247	1.77099	0.885494	−	0.464650i	$-0.153820\pi$
$461$	38.0247	1.77099	0.885494	+	0.464650i	$0.153820\pi$
$462$	−55.0599	−2.56162
$463$	−9.32934	−0.433571	−0.216786	−	0.976219i	$-0.569557\pi$
$463$	−9.32934	−0.433571	−0.216786	+	0.976219i	$0.569557\pi$
$464$	−21.8832	−1.01590
$465$	0	0
$466$	3.02818	0.140278
$467$	16.8764	0.780949	0.390474	−	0.920614i	$-0.372311\pi$
$467$	16.8764	0.780949	0.390474	+	0.920614i	$0.372311\pi$
$468$	0.0896909	0.00414596
$469$	−11.3379	−0.523537
$470$	0	0
$471$	29.2875	1.34950
$472$	−35.5449	−1.63609
$473$	4.76182	0.218949
$474$	26.8756	1.23444
$475$	0	0
$476$	0	0
$477$	−0.873436	−0.0399919
$478$	8.14687	0.372629
$479$	−8.68906	−0.397013	−0.198507	−	0.980100i	$-0.563609\pi$
$479$	−8.68906	−0.397013	−0.198507	+	0.980100i	$0.563609\pi$
$480$	0	0
$481$	19.1859	0.874803
$482$	18.4196	0.838989
$483$	22.1437	1.00757
$484$	2.52855	0.114934
$485$	0	0
$486$	−5.53277	−0.250972
$487$	−18.2608	−0.827476	−0.413738	−	0.910396i	$-0.635777\pi$
$487$	−18.2608	−0.827476	−0.413738	+	0.910396i	$0.635777\pi$
$488$	12.8639	0.582323
$489$	37.2487	1.68444
$490$	0	0
$491$	8.09277	0.365222	0.182611	−	0.983185i	$-0.441545\pi$
$491$	8.09277	0.365222	0.182611	+	0.983185i	$0.441545\pi$
$492$	−1.15904	−0.0522538
$493$	0	0
$494$	4.35544	0.195960
$495$	0	0
$496$	2.33366	0.104784
$497$	32.2353	1.44595
$498$	−11.9105	−0.533720
$499$	−8.98098	−0.402044	−0.201022	−	0.979587i	$-0.564426\pi$
$499$	−8.98098	−0.402044	−0.201022	+	0.979587i	$0.564426\pi$
$500$	0	0
$501$	−4.14626	−0.185241
$502$	−16.3098	−0.727944
$503$	−15.5919	−0.695207	−0.347604	−	0.937642i	$-0.613005\pi$
$503$	−15.5919	−0.695207	−0.347604	+	0.937642i	$0.613005\pi$
$504$	3.89439	0.173470
$505$	0	0
$506$	−31.8430	−1.41560
$507$	−10.3777	−0.460890
$508$	−0.801167	−0.0355460
$509$	35.3211	1.56558	0.782790	−	0.622286i	$-0.213796\pi$
$509$	35.3211	1.56558	0.782790	+	0.622286i	$0.213796\pi$
$510$	0	0
$511$	63.7658	2.82083
$512$	20.8258	0.920377
$513$	−6.40214	−0.282661
$514$	5.13178	0.226353
$515$	0	0
$516$	−0.118472	−0.00521542
$517$	−26.7519	−1.17655
$518$	−41.2358	−1.81180
$519$	−4.19752	−0.184251
$520$	0	0
$521$	−24.8263	−1.08766	−0.543830	−	0.839196i	$-0.683026\pi$
$521$	−24.8263	−1.08766	−0.543830	+	0.839196i	$0.683026\pi$
$522$	2.80405	0.122730
$523$	−11.7480	−0.513704	−0.256852	−	0.966451i	$-0.582685\pi$
$523$	−11.7480	−0.513704	−0.256852	+	0.966451i	$0.582685\pi$
$524$	0.464689	0.0203000
$525$	0	0
$526$	−33.4724	−1.45946
$527$	0	0
$528$	41.6773	1.81377
$529$	−10.1935	−0.443197
$530$	0	0
$531$	4.76992	0.206997
$532$	−0.421625	−0.0182798
$533$	19.4660	0.843168
$534$	19.1514	0.828761
$535$	0	0
$536$	8.19481	0.353962
$537$	33.6928	1.45395
$538$	−1.06378	−0.0458630
$539$	−46.4771	−2.00191
$540$	0	0
$541$	20.9757	0.901815	0.450908	−	0.892571i	$-0.351100\pi$
$541$	20.9757	0.901815	0.450908	+	0.892571i	$0.351100\pi$
$542$	−13.5715	−0.582945
$543$	−11.7440	−0.503982
$544$	0	0
$545$	0	0
$546$	−23.0067	−0.984595
$547$	−5.51818	−0.235940	−0.117970	−	0.993017i	$-0.537639\pi$
$547$	−5.51818	−0.235940	−0.117970	+	0.993017i	$0.537639\pi$
$548$	1.50733	0.0643898
$549$	−1.72627	−0.0736753
$550$	0	0
$551$	6.13299	0.261274
$552$	−16.0050	−0.681218
$553$	43.6948	1.85809
$554$	−19.6690	−0.835657
$555$	0	0
$556$	1.13772	0.0482499
$557$	−25.1974	−1.06765	−0.533823	−	0.845596i	$-0.679245\pi$
$557$	−25.1974	−1.06765	−0.533823	+	0.845596i	$0.679245\pi$
$558$	−0.299028	−0.0126589
$559$	1.98972	0.0841561
$560$	0	0
$561$	0	0
$562$	24.0825	1.01586
$563$	−11.1335	−0.469219	−0.234610	−	0.972090i	$-0.575381\pi$
$563$	−11.1335	−0.469219	−0.234610	+	0.972090i	$0.575381\pi$
$564$	0.665572	0.0280256
$565$	0	0
$566$	−15.1805	−0.638085
$567$	29.5816	1.24231
$568$	−23.2990	−0.977603
$569$	8.18349	0.343070	0.171535	−	0.985178i	$-0.445127\pi$
$569$	8.18349	0.343070	0.171535	+	0.985178i	$0.445127\pi$
$570$	0	0
$571$	−11.3240	−0.473895	−0.236948	−	0.971522i	$-0.576147\pi$
$571$	−11.3240	−0.473895	−0.236948	+	0.971522i	$0.576147\pi$
$572$	1.49012	0.0623051
$573$	−39.4444	−1.64782
$574$	−41.8378	−1.74628
$575$	0	0
$576$	−2.80820	−0.117008
$577$	−12.5131	−0.520925	−0.260463	−	0.965484i	$-0.583875\pi$
$577$	−12.5131	−0.520925	−0.260463	+	0.965484i	$0.583875\pi$
$578$	0	0
$579$	27.2501	1.13248
$580$	0	0
$581$	−19.3642	−0.803363
$582$	17.1320	0.710143
$583$	−14.5112	−0.600994
$584$	−46.0886	−1.90716
$585$	0	0
$586$	2.13929	0.0883734
$587$	−23.6400	−0.975726	−0.487863	−	0.872920i	$-0.662223\pi$
$587$	−23.6400	−0.975726	−0.487863	+	0.872920i	$0.662223\pi$
$588$	1.15632	0.0476860
$589$	−0.654032	−0.0269489
$590$	0	0
$591$	21.1020	0.868022
$592$	31.2132	1.28285
$593$	−21.6317	−0.888306	−0.444153	−	0.895951i	$-0.646495\pi$
$593$	−21.6317	−0.888306	−0.444153	+	0.895951i	$0.646495\pi$
$594$	−48.6308	−1.99535
$595$	0	0
$596$	0.0746799	0.00305901
$597$	36.2036	1.48171
$598$	−13.3056	−0.544105
$599$	−8.41511	−0.343832	−0.171916	−	0.985112i	$-0.554996\pi$
$599$	−8.41511	−0.343832	−0.171916	+	0.985112i	$0.554996\pi$
$600$	0	0
$601$	−45.0998	−1.83966	−0.919829	−	0.392319i	$-0.871673\pi$
$601$	−45.0998	−1.83966	−0.919829	+	0.392319i	$0.871673\pi$
$602$	−4.27645	−0.174295
$603$	−1.09970	−0.0447831
$604$	−1.80740	−0.0735422
$605$	0	0
$606$	22.7365	0.923607
$607$	−16.9707	−0.688818	−0.344409	−	0.938820i	$-0.611921\pi$
$607$	−16.9707	−0.688818	−0.344409	+	0.938820i	$0.611921\pi$
$608$	0.624568	0.0253296
$609$	−32.3962	−1.31276
$610$	0	0
$611$	−11.1782	−0.452222
$612$	0	0
$613$	−40.8575	−1.65022	−0.825109	−	0.564974i	$-0.808886\pi$
$613$	−40.8575	−1.65022	−0.825109	+	0.564974i	$0.808886\pi$
$614$	−28.0203	−1.13081
$615$	0	0
$616$	64.7012	2.60689
$617$	4.36716	0.175815	0.0879076	−	0.996129i	$-0.471982\pi$
$617$	4.36716	0.175815	0.0879076	+	0.996129i	$0.471982\pi$
$618$	−26.0887	−1.04944
$619$	2.19834	0.0883586	0.0441793	−	0.999024i	$-0.485933\pi$
$619$	2.19834	0.0883586	0.0441793	+	0.999024i	$0.485933\pi$
$620$	0	0
$621$	19.5581	0.784839
$622$	−1.21639	−0.0487727
$623$	31.1366	1.24746
$624$	17.4148	0.697149
$625$	0	0
$626$	10.0292	0.400847
$627$	−11.6805	−0.466474
$628$	1.70357	0.0679797
$629$	0	0
$630$	0	0
$631$	−30.4636	−1.21274	−0.606368	−	0.795184i	$-0.707374\pi$
$631$	−30.4636	−1.21274	−0.606368	+	0.795184i	$0.707374\pi$
$632$	−31.5817	−1.25625
$633$	30.6687	1.21897
$634$	−11.4664	−0.455390
$635$	0	0
$636$	0.361032	0.0143158
$637$	−19.4204	−0.769463
$638$	46.5864	1.84437
$639$	3.12659	0.123686
$640$	0	0
$641$	−27.0255	−1.06744	−0.533721	−	0.845661i	$-0.679207\pi$
$641$	−27.0255	−1.06744	−0.533721	+	0.845661i	$0.679207\pi$
$642$	−21.2957	−0.840474
$643$	32.3036	1.27393	0.636965	−	0.770892i	$-0.280189\pi$
$643$	32.3036	1.27393	0.636965	+	0.770892i	$0.280189\pi$
$644$	1.28803	0.0507557
$645$	0	0
$646$	0	0
$647$	−3.21308	−0.126319	−0.0631595	−	0.998003i	$-0.520118\pi$
$647$	−3.21308	−0.126319	−0.0631595	+	0.998003i	$0.520118\pi$
$648$	−21.3810	−0.839924
$649$	79.2473	3.11073
$650$	0	0
$651$	3.45479	0.135404
$652$	2.16665	0.0848525
$653$	−14.0028	−0.547972	−0.273986	−	0.961734i	$-0.588342\pi$
$653$	−14.0028	−0.547972	−0.273986	+	0.961734i	$0.588342\pi$
$654$	22.0371	0.861720
$655$	0	0
$656$	31.6689	1.23646
$657$	6.18482	0.241293
$658$	24.0250	0.936593
$659$	48.2924	1.88120	0.940602	−	0.339512i	$-0.110262\pi$
$659$	48.2924	1.88120	0.940602	+	0.339512i	$0.110262\pi$
$660$	0	0
$661$	−31.8807	−1.24001	−0.620007	−	0.784596i	$-0.712870\pi$
$661$	−31.8807	−1.24001	−0.620007	+	0.784596i	$0.712870\pi$
$662$	−20.3973	−0.792764
$663$	0	0
$664$	13.9960	0.543152
$665$	0	0
$666$	−3.99957	−0.154980
$667$	−18.7359	−0.725455
$668$	−0.241176	−0.00933138
$669$	−12.2976	−0.475453
$670$	0	0
$671$	−28.6801	−1.10718
$672$	−3.29915	−0.127267
$673$	−9.44276	−0.363992	−0.181996	−	0.983299i	$-0.558256\pi$
$673$	−9.44276	−0.363992	−0.181996	+	0.983299i	$0.558256\pi$
$674$	14.0672	0.541846
$675$	0	0
$676$	−0.603640	−0.0232169
$677$	29.9668	1.15172	0.575858	−	0.817549i	$-0.304668\pi$
$677$	29.9668	1.15172	0.575858	+	0.817549i	$0.304668\pi$
$678$	12.9521	0.497421
$679$	27.8535	1.06892
$680$	0	0
$681$	−15.5753	−0.596845
$682$	−4.96804	−0.190236
$683$	40.3774	1.54500	0.772499	−	0.635015i	$-0.219006\pi$
$683$	40.3774	1.54500	0.772499	+	0.635015i	$0.219006\pi$
$684$	−0.0408946	−0.00156364
$685$	0	0
$686$	3.08643	0.117840
$687$	6.68710	0.255129
$688$	3.23703	0.123411
$689$	−6.06349	−0.231001
$690$	0	0
$691$	27.9371	1.06278	0.531388	−	0.847128i	$-0.321671\pi$
$691$	27.9371	1.06278	0.531388	+	0.847128i	$0.321671\pi$
$692$	−0.244158	−0.00928148
$693$	−8.68254	−0.329822
$694$	0.164720	0.00625269
$695$	0	0
$696$	23.4153	0.887555
$697$	0	0
$698$	30.9691	1.17220
$699$	−3.39336	−0.128349
$700$	0	0
$701$	14.7138	0.555732	0.277866	−	0.960620i	$-0.410373\pi$
$701$	14.7138	0.555732	0.277866	+	0.960620i	$0.410373\pi$
$702$	−20.3203	−0.766941
$703$	−8.74783	−0.329931
$704$	−46.6553	−1.75839
$705$	0	0
$706$	23.1400	0.870886
$707$	36.9654	1.39023
$708$	−1.97163	−0.0740984
$709$	−13.3825	−0.502588	−0.251294	−	0.967911i	$-0.580856\pi$
$709$	−13.3825	−0.502588	−0.251294	+	0.967911i	$0.580856\pi$
$710$	0	0
$711$	4.23808	0.158940
$712$	−22.5049	−0.843407
$713$	1.99802	0.0748265
$714$	0	0
$715$	0	0
$716$	1.95981	0.0732417
$717$	−9.12933	−0.340941
$718$	−10.8355	−0.404376
$719$	23.7717	0.886535	0.443268	−	0.896389i	$-0.353819\pi$
$719$	23.7717	0.886535	0.443268	+	0.896389i	$0.353819\pi$
$720$	0	0
$721$	−42.4155	−1.57964
$722$	25.5106	0.949405
$723$	−20.6409	−0.767643
$724$	−0.683112	−0.0253877
$725$	0	0
$726$	−62.9096	−2.33479
$727$	4.40311	0.163302	0.0816512	−	0.996661i	$-0.473981\pi$
$727$	4.40311	0.163302	0.0816512	+	0.996661i	$0.473981\pi$
$728$	27.0353	1.00200
$729$	29.4583	1.09105
$730$	0	0
$731$	0	0
$732$	0.713547	0.0263735
$733$	42.2976	1.56230	0.781149	−	0.624344i	$-0.214634\pi$
$733$	42.2976	1.56230	0.781149	+	0.624344i	$0.214634\pi$
$734$	−37.9564	−1.40100
$735$	0	0
$736$	−1.90801	−0.0703302
$737$	−18.2703	−0.672996
$738$	−4.05796	−0.149376
$739$	−23.1126	−0.850212	−0.425106	−	0.905144i	$-0.639763\pi$
$739$	−23.1126	−0.850212	−0.425106	+	0.905144i	$0.639763\pi$
$740$	0	0
$741$	−4.88068	−0.179296
$742$	13.0321	0.478423
$743$	0.752277	0.0275984	0.0137992	−	0.999905i	$-0.495607\pi$
$743$	0.752277	0.0275984	0.0137992	+	0.999905i	$0.495607\pi$
$744$	−2.49705	−0.0915461
$745$	0	0
$746$	−12.8367	−0.469984
$747$	−1.87819	−0.0687194
$748$	0	0
$749$	−34.6229	−1.26509
$750$	0	0
$751$	−13.3283	−0.486357	−0.243178	−	0.969982i	$-0.578190\pi$
$751$	−13.3283	−0.486357	−0.243178	+	0.969982i	$0.578190\pi$
$752$	−18.1856	−0.663161
$753$	18.2767	0.666040
$754$	19.4660	0.708911
$755$	0	0
$756$	1.96709	0.0715425
$757$	−36.9030	−1.34126	−0.670630	−	0.741792i	$-0.733976\pi$
$757$	−36.9030	−1.34126	−0.670630	+	0.741792i	$0.733976\pi$
$758$	−5.24545	−0.190523
$759$	35.6831	1.29521
$760$	0	0
$761$	35.4100	1.28361	0.641805	−	0.766868i	$-0.278186\pi$
$761$	35.4100	1.28361	0.641805	+	0.766868i	$0.278186\pi$
$762$	19.9328	0.722089
$763$	35.8283	1.29707
$764$	−2.29437	−0.0830073
$765$	0	0
$766$	−40.9996	−1.48138
$767$	33.1133	1.19565
$768$	3.66335	0.132190
$769$	−2.67233	−0.0963666	−0.0481833	−	0.998839i	$-0.515343\pi$
$769$	−2.67233	−0.0963666	−0.0481833	+	0.998839i	$0.515343\pi$
$770$	0	0
$771$	−5.75064	−0.207104
$772$	1.58506	0.0570476
$773$	47.9406	1.72430	0.862152	−	0.506649i	$-0.169116\pi$
$773$	47.9406	1.72430	0.862152	+	0.506649i	$0.169116\pi$
$774$	−0.414784	−0.0149091
$775$	0	0
$776$	−20.1319	−0.722693
$777$	46.2086	1.65772
$778$	9.20240	0.329922
$779$	−8.87554	−0.317999
$780$	0	0
$781$	51.9450	1.85874
$782$	0	0
$783$	−28.6135	−1.02256
$784$	−31.5946	−1.12838
$785$	0	0
$786$	−11.5613	−0.412379
$787$	−15.8202	−0.563929	−0.281965	−	0.959425i	$-0.590986\pi$
$787$	−15.8202	−0.563929	−0.281965	+	0.959425i	$0.590986\pi$
$788$	1.22744	0.0437259
$789$	37.5089	1.33535
$790$	0	0
$791$	21.0577	0.748725
$792$	6.27555	0.222992
$793$	−11.9839	−0.425562
$794$	12.5660	0.445951
$795$	0	0
$796$	2.10586	0.0746401
$797$	5.01840	0.177761	0.0888804	−	0.996042i	$-0.471671\pi$
$797$	5.01840	0.177761	0.0888804	+	0.996042i	$0.471671\pi$
$798$	10.4899	0.371339
$799$	0	0
$800$	0	0
$801$	3.02003	0.106707
$802$	40.3140	1.42354
$803$	102.754	3.62612
$804$	0.454556	0.0160310
$805$	0	0
$806$	−2.07589	−0.0731201
$807$	1.19207	0.0419628
$808$	−26.7178	−0.939929
$809$	−7.06907	−0.248535	−0.124268	−	0.992249i	$-0.539658\pi$
$809$	−7.06907	−0.248535	−0.124268	+	0.992249i	$0.539658\pi$
$810$	0	0
$811$	10.7641	0.377977	0.188989	−	0.981979i	$-0.439479\pi$
$811$	10.7641	0.377977	0.188989	+	0.981979i	$0.439479\pi$
$812$	−1.88439	−0.0661293
$813$	15.2081	0.533372
$814$	−66.4487	−2.32903
$815$	0	0
$816$	0	0
$817$	−0.907213	−0.0317393
$818$	−26.0184	−0.909712
$819$	−3.62798	−0.126772
$820$	0	0
$821$	29.5285	1.03055	0.515276	−	0.857024i	$-0.327689\pi$
$821$	29.5285	1.03055	0.515276	+	0.857024i	$0.327689\pi$
$822$	−37.5018	−1.30803
$823$	25.6974	0.895755	0.447877	−	0.894095i	$-0.352180\pi$
$823$	25.6974	0.895755	0.447877	+	0.894095i	$0.352180\pi$
$824$	30.6570	1.06799
$825$	0	0
$826$	−71.1696	−2.47631
$827$	−12.2393	−0.425604	−0.212802	−	0.977095i	$-0.568259\pi$
$827$	−12.2393	−0.425604	−0.212802	+	0.977095i	$0.568259\pi$
$828$	0.124930	0.00434162
$829$	−50.2516	−1.74531	−0.872656	−	0.488336i	$-0.837604\pi$
$829$	−50.2516	−1.74531	−0.872656	+	0.488336i	$0.837604\pi$
$830$	0	0
$831$	22.0410	0.764594
$832$	−19.4948	−0.675861
$833$	0	0
$834$	−28.3060	−0.980158
$835$	0	0
$836$	−0.679421	−0.0234983
$837$	3.05139	0.105471
$838$	49.3237	1.70386
$839$	4.63496	0.160017	0.0800083	−	0.996794i	$-0.474505\pi$
$839$	4.63496	0.160017	0.0800083	+	0.996794i	$0.474505\pi$
$840$	0	0
$841$	−1.58944	−0.0548082
$842$	43.4242	1.49650
$843$	−26.9867	−0.929470
$844$	1.78391	0.0614046
$845$	0	0
$846$	2.33025	0.0801158
$847$	−102.280	−3.51437
$848$	−9.86458	−0.338751
$849$	17.0112	0.583822
$850$	0	0
$851$	26.7240	0.916087
$852$	−1.29236	−0.0442757
$853$	−32.8889	−1.12609	−0.563047	−	0.826425i	$-0.690371\pi$
$853$	−32.8889	−1.12609	−0.563047	+	0.826425i	$0.690371\pi$
$854$	25.7568	0.881378
$855$	0	0
$856$	25.0247	0.855327
$857$	4.67903	0.159833	0.0799164	−	0.996802i	$-0.474535\pi$
$857$	4.67903	0.159833	0.0799164	+	0.996802i	$0.474535\pi$
$858$	−37.0738	−1.26568
$859$	−0.831102	−0.0283568	−0.0141784	−	0.999899i	$-0.504513\pi$
$859$	−0.831102	−0.0283568	−0.0141784	+	0.999899i	$0.504513\pi$
$860$	0	0
$861$	46.8832	1.59777
$862$	42.3072	1.44099
$863$	23.4587	0.798545	0.399272	−	0.916832i	$-0.369263\pi$
$863$	23.4587	0.798545	0.399272	+	0.916832i	$0.369263\pi$
$864$	−2.91392	−0.0991336
$865$	0	0
$866$	13.8317	0.470020
$867$	0	0
$868$	0.200955	0.00682085
$869$	70.4113	2.38854
$870$	0	0
$871$	−7.63422	−0.258676
$872$	−25.8960	−0.876948
$873$	2.70159	0.0914348
$874$	6.06667	0.205208
$875$	0	0
$876$	−2.55648	−0.0863753
$877$	−30.5812	−1.03265	−0.516327	−	0.856391i	$-0.672701\pi$
$877$	−30.5812	−1.03265	−0.516327	+	0.856391i	$0.672701\pi$
$878$	44.2527	1.49346
$879$	−2.39728	−0.0808582
$880$	0	0
$881$	0.185968	0.00626543	0.00313271	−	0.999995i	$-0.499003\pi$
$881$	0.185968	0.00626543	0.00313271	+	0.999995i	$0.499003\pi$
$882$	4.04844	0.136318
$883$	−19.3509	−0.651211	−0.325605	−	0.945506i	$-0.605568\pi$
$883$	−19.3509	−0.651211	−0.325605	+	0.945506i	$0.605568\pi$
$884$	0	0
$885$	0	0
$886$	−6.07649	−0.204144
$887$	28.5188	0.957566	0.478783	−	0.877933i	$-0.341078\pi$
$887$	28.5188	0.957566	0.478783	+	0.877933i	$0.341078\pi$
$888$	−33.3986	−1.12078
$889$	32.4071	1.08690
$890$	0	0
$891$	47.6688	1.59697
$892$	−0.715317	−0.0239506
$893$	5.09671	0.170555
$894$	−1.85801	−0.0621413
$895$	0	0
$896$	45.9684	1.53570
$897$	14.9101	0.497835
$898$	−44.2597	−1.47696
$899$	−2.92311	−0.0974910
$900$	0	0
$901$	0	0
$902$	−67.4189	−2.24480
$903$	4.79216	0.159473
$904$	−15.2200	−0.506211
$905$	0	0
$906$	44.9677	1.49395
$907$	17.1889	0.570749	0.285375	−	0.958416i	$-0.407882\pi$
$907$	17.1889	0.570749	0.285375	+	0.958416i	$0.407882\pi$
$908$	−0.905967	−0.0300656
$909$	3.58538	0.118919
$910$	0	0
$911$	−31.2141	−1.03417	−0.517085	−	0.855934i	$-0.672983\pi$
$911$	−31.2141	−1.03417	−0.517085	+	0.855934i	$0.672983\pi$
$912$	−7.94027	−0.262929
$913$	−31.2042	−1.03271
$914$	8.60986	0.284789
$915$	0	0
$916$	0.388969	0.0128519
$917$	−18.7966	−0.620719
$918$	0	0
$919$	24.5006	0.808202	0.404101	−	0.914714i	$-0.367585\pi$
$919$	24.5006	0.808202	0.404101	+	0.914714i	$0.367585\pi$
$920$	0	0
$921$	31.3993	1.03464
$922$	−55.0286	−1.81227
$923$	21.7051	0.714433
$924$	3.58890	0.118066
$925$	0	0
$926$	13.5012	0.443678
$927$	−4.11400	−0.135121
$928$	2.79142	0.0916328
$929$	44.3523	1.45515	0.727576	−	0.686027i	$-0.240647\pi$
$929$	44.3523	1.45515	0.727576	+	0.686027i	$0.240647\pi$
$930$	0	0
$931$	8.85472	0.290202
$932$	−0.197382	−0.00646546
$933$	1.36308	0.0446251
$934$	−24.4233	−0.799153
$935$	0	0
$936$	2.62223	0.0857102
$937$	−32.0828	−1.04810	−0.524050	−	0.851688i	$-0.675579\pi$
$937$	−32.0828	−1.04810	−0.524050	+	0.851688i	$0.675579\pi$
$938$	16.4080	0.535741
$939$	−11.2387	−0.366760
$940$	0	0
$941$	−15.1855	−0.495032	−0.247516	−	0.968884i	$-0.579614\pi$
$941$	−15.1855	−0.495032	−0.247516	+	0.968884i	$0.579614\pi$
$942$	−42.3842	−1.38095
$943$	27.1142	0.882959
$944$	53.8714	1.75336
$945$	0	0
$946$	−6.89121	−0.224053
$947$	−36.8958	−1.19895	−0.599477	−	0.800392i	$-0.704625\pi$
$947$	−36.8958	−1.19895	−0.599477	+	0.800392i	$0.704625\pi$
$948$	−1.75179	−0.0568957
$949$	42.9357	1.39375
$950$	0	0
$951$	12.8492	0.416664
$952$	0	0
$953$	22.8613	0.740551	0.370276	−	0.928922i	$-0.379263\pi$
$953$	22.8613	0.740551	0.370276	+	0.928922i	$0.379263\pi$
$954$	1.26402	0.0409241
$955$	0	0
$956$	−0.531026	−0.0171746
$957$	−52.2044	−1.68753
$958$	12.5746	0.406268
$959$	−60.9711	−1.96886
$960$	0	0
$961$	−30.6883	−0.989944
$962$	−27.7655	−0.895195
$963$	−3.35817	−0.108216
$964$	−1.20062	−0.0386694
$965$	0	0
$966$	−32.0459	−1.03106
$967$	31.3390	1.00779	0.503897	−	0.863764i	$-0.331899\pi$
$967$	31.3390	1.00779	0.503897	+	0.863764i	$0.331899\pi$
$968$	73.9254	2.37605
$969$	0	0
$970$	0	0
$971$	0.826892	0.0265362	0.0132681	−	0.999912i	$-0.495777\pi$
$971$	0.826892	0.0265362	0.0132681	+	0.999912i	$0.495777\pi$
$972$	0.360635	0.0115674
$973$	−46.0205	−1.47535
$974$	26.4267	0.846765
$975$	0	0
$976$	−19.4964	−0.624066
$977$	−21.5432	−0.689227	−0.344614	−	0.938745i	$-0.611990\pi$
$977$	−21.5432	−0.689227	−0.344614	+	0.938745i	$0.611990\pi$
$978$	−53.9056	−1.72371
$979$	50.1746	1.60359
$980$	0	0
$981$	3.47509	0.110951
$982$	−11.7117	−0.373735
$983$	−26.0044	−0.829410	−0.414705	−	0.909956i	$-0.636115\pi$
$983$	−26.0044	−0.829410	−0.414705	+	0.909956i	$0.636115\pi$
$984$	−33.8862	−1.08025
$985$	0	0
$986$	0	0
$987$	−26.9223	−0.856946
$988$	−0.283895	−0.00903190
$989$	2.77147	0.0881276
$990$	0	0
$991$	54.8263	1.74162	0.870808	−	0.491623i	$-0.163596\pi$
$991$	54.8263	1.74162	0.870808	+	0.491623i	$0.163596\pi$
$992$	−0.297681	−0.00945139
$993$	22.8571	0.725348
$994$	−46.6502	−1.47966
$995$	0	0
$996$	0.776343	0.0245994
$997$	−38.6480	−1.22400	−0.611998	−	0.790859i	$-0.709634\pi$
$997$	−38.6480	−1.22400	−0.611998	+	0.790859i	$0.709634\pi$
$998$	12.9971	0.411416
$999$	40.8130	1.29127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.br.1.4		12
5.4	even	2		7225.2.a.bm.1.9		12
17.2	even	8		425.2.e.e.276.2	yes	12
17.9	even	8		425.2.e.e.251.5	yes	12
17.16	even	2	inner	7225.2.a.br.1.3		12
85.2	odd	8		425.2.j.a.174.5		12
85.9	even	8		425.2.e.c.251.2	✓	12
85.19	even	8		425.2.e.c.276.5	yes	12
85.43	odd	8		425.2.j.a.149.5		12
85.53	odd	8		425.2.j.d.174.2		12
85.77	odd	8		425.2.j.d.149.2		12
85.84	even	2		7225.2.a.bm.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.e.c.251.2	✓	12	85.9	even	8
425.2.e.c.276.5	yes	12	85.19	even	8
425.2.e.e.251.5	yes	12	17.9	even	8
425.2.e.e.276.2	yes	12	17.2	even	8
425.2.j.a.149.5		12	85.43	odd	8
425.2.j.a.174.5		12	85.2	odd	8
425.2.j.d.149.2		12	85.77	odd	8
425.2.j.d.174.2		12	85.53	odd	8
7225.2.a.bm.1.9		12	5.4	even	2
7225.2.a.bm.1.10		12	85.84	even	2
7225.2.a.br.1.3		12	17.16	even	2	inner
7225.2.a.br.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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44718 q^{2} +1.62170 q^{3} +0.0943296 q^{4} -2.34689 q^{6} -3.81562 q^{7} +2.75785 q^{8} -0.370087 q^{9} -6.14862 q^{11} +0.152974 q^{12} -2.56919 q^{13} +5.52189 q^{14} -4.17976 q^{16} +0.535583 q^{18} +1.17142 q^{19} -6.18779 q^{21} +8.89815 q^{22} -3.57861 q^{23} +4.47240 q^{24} +3.71808 q^{26} -5.46527 q^{27} -0.359926 q^{28} +5.23551 q^{29} -0.558323 q^{31} +0.533170 q^{32} -9.97121 q^{33} -0.0349102 q^{36} -7.46770 q^{37} -1.69526 q^{38} -4.16645 q^{39} -7.57672 q^{41} +8.95485 q^{42} -0.774454 q^{43} -0.579996 q^{44} +5.17890 q^{46} +4.35087 q^{47} -6.77832 q^{48} +7.55894 q^{49} -0.242351 q^{52} +2.36008 q^{53} +7.90923 q^{54} -10.5229 q^{56} +1.89970 q^{57} -7.57672 q^{58} -12.8886 q^{59} +4.66449 q^{61} +0.807994 q^{62} +1.41211 q^{63} +7.58793 q^{64} +14.4301 q^{66} +2.97145 q^{67} -5.80344 q^{69} -8.44824 q^{71} -1.02064 q^{72} -16.7118 q^{73} +10.8071 q^{74} +0.110500 q^{76} +23.4608 q^{77} +6.02961 q^{78} -11.4516 q^{79} -7.75277 q^{81} +10.9649 q^{82} +5.07499 q^{83} -0.583692 q^{84} +1.12077 q^{86} +8.49043 q^{87} -16.9569 q^{88} -8.16031 q^{89} +9.80304 q^{91} -0.337569 q^{92} -0.905433 q^{93} -6.29650 q^{94} +0.864643 q^{96} -7.29986 q^{97} -10.9392 q^{98} +2.27553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 12 q^{13} + 4 q^{16} + 4 q^{18} + 12 q^{19} - 8 q^{21} + 12 q^{26} + 48 q^{32} - 4 q^{33} - 20 q^{36} - 12 q^{38} + 56 q^{42} + 16 q^{43} + 36 q^{47} + 16 q^{49}+ \cdots + 76 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 12 * q^13 + 4 * q^16 + 4 * q^18 + 12 * q^19 - 8 * q^21 + 12 * q^26 + 48 * q^32 - 4 * q^33 - 20 * q^36 - 12 * q^38 + 56 * q^42 + 16 * q^43 + 36 * q^47 + 16 * q^49 + 64 * q^52 + 48 * q^53 - 32 * q^59 + 44 * q^64 + 40 * q^66 - 20 * q^67 - 72 * q^69 - 92 * q^72 + 36 * q^76 + 24 * q^77 + 8 * q^81 + 52 * q^83 + 36 * q^84 + 84 * q^86 + 48 * q^87 + 24 * q^89 + 60 * q^93 - 8 * q^94 + 76 * q^98

Introduction

L-functions

Modular forms

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Database

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