LMFDB - Embedded newform 9075.2.a.db.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.792287$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} -2.52434 q^{7} +2.67181 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} -2.52434 q^{7} +2.67181 q^{8} +1.00000 q^{9} -1.37228 q^{12} -6.78073 q^{13} +2.00000 q^{14} +0.627719 q^{16} -6.63325 q^{17} -0.792287 q^{18} -7.72049 q^{19} -2.52434 q^{21} -8.00000 q^{23} +2.67181 q^{24} +5.37228 q^{26} +1.00000 q^{27} +3.46410 q^{28} -3.16915 q^{29} -3.37228 q^{31} -5.84096 q^{32} +5.25544 q^{34} -1.37228 q^{36} -5.00000 q^{37} +6.11684 q^{38} -6.78073 q^{39} +6.92820 q^{41} +2.00000 q^{42} -9.94987 q^{43} +6.33830 q^{46} -2.74456 q^{47} +0.627719 q^{48} -0.627719 q^{49} -6.63325 q^{51} +9.30506 q^{52} +12.7446 q^{53} -0.792287 q^{54} -6.74456 q^{56} -7.72049 q^{57} +2.51087 q^{58} +4.00000 q^{59} +2.67181 q^{61} +2.67181 q^{62} -2.52434 q^{63} +3.37228 q^{64} -6.11684 q^{67} +9.10268 q^{68} -8.00000 q^{69} -0.744563 q^{71} +2.67181 q^{72} -5.19615 q^{73} +3.96143 q^{74} +10.5947 q^{76} +5.37228 q^{78} -0.147477 q^{79} +1.00000 q^{81} -5.48913 q^{82} +1.87953 q^{83} +3.46410 q^{84} +7.88316 q^{86} -3.16915 q^{87} -17.4891 q^{89} +17.1168 q^{91} +10.9783 q^{92} -3.37228 q^{93} +2.17448 q^{94} -5.84096 q^{96} +3.62772 q^{97} +0.497333 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 14 * q^16 - 32 * q^23 + 10 * q^26 + 4 * q^27 - 2 * q^31 + 44 * q^34 + 6 * q^36 - 20 * q^37 - 10 * q^38 + 8 * q^42 + 12 * q^47 + 14 * q^48 - 14 * q^49 + 28 * q^53 - 4 * q^56 + 56 * q^58 + 16 * q^59 + 2 * q^64 + 10 * q^67 - 32 * q^69 + 20 * q^71 + 10 * q^78 + 4 * q^81 + 24 * q^82 + 66 * q^86 - 24 * q^89 + 34 * q^91 - 48 * q^92 - 2 * q^93 + 26 * q^97

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 $

$2$	−0.792287	−0.560232	−0.280116	−	0.959966i	$-0.590373\pi$
$2$	−0.792287	−0.560232	−0.280116	+	0.959966i	$0.590373\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.37228	−0.686141
$5$	0	0
$5$	0	0
$6$	−0.792287	−0.323450
$7$	−2.52434	−0.954110	−0.477055	−	0.878873i	$-0.658296\pi$
$7$	−2.52434	−0.954110	−0.477055	+	0.878873i	$0.658296\pi$
$8$	2.67181	0.944629
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.37228	−0.396143
$13$	−6.78073	−1.88064	−0.940318	−	0.340298i	$-0.889472\pi$
$13$	−6.78073	−1.88064	−0.940318	+	0.340298i	$0.889472\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	0.627719	0.156930
$17$	−6.63325	−1.60880	−0.804400	−	0.594089i	$-0.797513\pi$
$17$	−6.63325	−1.60880	−0.804400	+	0.594089i	$0.797513\pi$
$18$	−0.792287	−0.186744
$19$	−7.72049	−1.77120	−0.885601	−	0.464447i	$-0.846253\pi$
$19$	−7.72049	−1.77120	−0.885601	+	0.464447i	$0.846253\pi$
$20$	0	0
$21$	−2.52434	−0.550856
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	2.67181	0.545382
$25$	0	0
$26$	5.37228	1.05359
$27$	1.00000	0.192450
$28$	3.46410	0.654654
$29$	−3.16915	−0.588496	−0.294248	−	0.955729i	$-0.595069\pi$
$29$	−3.16915	−0.588496	−0.294248	+	0.955729i	$0.595069\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	−5.84096	−1.03255
$33$	0	0
$34$	5.25544	0.901300
$35$	0	0
$36$	−1.37228	−0.228714
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	6.11684	0.992283
$39$	−6.78073	−1.08579
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	2.00000	0.308607
$43$	−9.94987	−1.51734	−0.758671	−	0.651474i	$-0.774151\pi$
$43$	−9.94987	−1.51734	−0.758671	+	0.651474i	$0.774151\pi$
$44$	0	0
$45$	0	0
$46$	6.33830	0.934531
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	0.627719	0.0906034
$49$	−0.627719	−0.0896741
$50$	0	0
$51$	−6.63325	−0.928841
$52$	9.30506	1.29038
$53$	12.7446	1.75060	0.875300	−	0.483580i	$-0.160664\pi$
$53$	12.7446	1.75060	0.875300	+	0.483580i	$0.160664\pi$
$54$	−0.792287	−0.107817
$55$	0	0
$56$	−6.74456	−0.901280
$57$	−7.72049	−1.02260
$58$	2.51087	0.329694
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.67181	0.342091	0.171045	−	0.985263i	$-0.445286\pi$
$61$	2.67181	0.342091	0.171045	+	0.985263i	$0.445286\pi$
$62$	2.67181	0.339321
$63$	−2.52434	−0.318037
$64$	3.37228	0.421535
$65$	0	0
$66$	0	0
$67$	−6.11684	−0.747291	−0.373646	−	0.927571i	$-0.621892\pi$
$67$	−6.11684	−0.747291	−0.373646	+	0.927571i	$0.621892\pi$
$68$	9.10268	1.10386
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−0.744563	−0.0883633	−0.0441817	−	0.999024i	$-0.514068\pi$
$71$	−0.744563	−0.0883633	−0.0441817	+	0.999024i	$0.514068\pi$
$72$	2.67181	0.314876
$73$	−5.19615	−0.608164	−0.304082	−	0.952646i	$-0.598350\pi$
$73$	−5.19615	−0.608164	−0.304082	+	0.952646i	$0.598350\pi$
$74$	3.96143	0.460507
$75$	0	0
$76$	10.5947	1.21529
$77$	0	0
$78$	5.37228	0.608291
$79$	−0.147477	−0.0165924	−0.00829622	−	0.999966i	$-0.502641\pi$
$79$	−0.147477	−0.0165924	−0.00829622	+	0.999966i	$0.502641\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.48913	−0.606172
$83$	1.87953	0.206305	0.103152	−	0.994666i	$-0.467107\pi$
$83$	1.87953	0.206305	0.103152	+	0.994666i	$0.467107\pi$
$84$	3.46410	0.377964
$85$	0	0
$86$	7.88316	0.850063
$87$	−3.16915	−0.339768
$88$	0	0
$89$	−17.4891	−1.85384	−0.926922	−	0.375255i	$-0.877555\pi$
$89$	−17.4891	−1.85384	−0.926922	+	0.375255i	$0.877555\pi$
$90$	0	0
$91$	17.1168	1.79433
$92$	10.9783	1.14456
$93$	−3.37228	−0.349689
$94$	2.17448	0.224281
$95$	0	0
$96$	−5.84096	−0.596141
$97$	3.62772	0.368339	0.184170	−	0.982894i	$-0.441040\pi$
$97$	3.62772	0.368339	0.184170	+	0.982894i	$0.441040\pi$
$98$	0.497333	0.0502383
$99$	0	0
$100$	0	0
$101$	13.2665	1.32007	0.660033	−	0.751237i	$-0.270542\pi$
$101$	13.2665	1.32007	0.660033	+	0.751237i	$0.270542\pi$
$102$	5.25544	0.520366
$103$	7.74456	0.763094	0.381547	−	0.924349i	$-0.375391\pi$
$103$	7.74456	0.763094	0.381547	+	0.924349i	$0.375391\pi$
$104$	−18.1168	−1.77650
$105$	0	0
$106$	−10.0974	−0.980741
$107$	11.6819	1.12933	0.564667	−	0.825319i	$-0.309005\pi$
$107$	11.6819	1.12933	0.564667	+	0.825319i	$0.309005\pi$
$108$	−1.37228	−0.132048
$109$	−5.98844	−0.573588	−0.286794	−	0.957992i	$-0.592590\pi$
$109$	−5.98844	−0.573588	−0.286794	+	0.957992i	$0.592590\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	−1.58457	−0.149728
$113$	18.2337	1.71528	0.857641	−	0.514250i	$-0.171930\pi$
$113$	18.2337	1.71528	0.857641	+	0.514250i	$0.171930\pi$
$114$	6.11684	0.572895
$115$	0	0
$116$	4.34896	0.403791
$117$	−6.78073	−0.626878
$118$	−3.16915	−0.291744
$119$	16.7446	1.53497
$120$	0	0
$121$	0	0
$122$	−2.11684	−0.191650
$123$	6.92820	0.624695
$124$	4.62772	0.415581
$125$	0	0
$126$	2.00000	0.178174
$127$	−3.61158	−0.320476	−0.160238	−	0.987078i	$-0.551226\pi$
$127$	−3.61158	−0.320476	−0.160238	+	0.987078i	$0.551226\pi$
$128$	9.01011	0.796389
$129$	−9.94987	−0.876038
$130$	0	0
$131$	1.28962	0.112675	0.0563373	−	0.998412i	$-0.482058\pi$
$131$	1.28962	0.112675	0.0563373	+	0.998412i	$0.482058\pi$
$132$	0	0
$133$	19.4891	1.68992
$134$	4.84630	0.418656
$135$	0	0
$136$	−17.7228	−1.51972
$137$	−10.7446	−0.917970	−0.458985	−	0.888444i	$-0.651787\pi$
$137$	−10.7446	−0.917970	−0.458985	+	0.888444i	$0.651787\pi$
$138$	6.33830	0.539552
$139$	−10.3923	−0.881464	−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	0	0
$141$	−2.74456	−0.231134
$142$	0.589907	0.0495039
$143$	0	0
$144$	0.627719	0.0523099
$145$	0	0
$146$	4.11684	0.340712
$147$	−0.627719	−0.0517734
$148$	6.86141	0.564004
$149$	18.6101	1.52460	0.762301	−	0.647223i	$-0.224070\pi$
$149$	18.6101	1.52460	0.762301	+	0.647223i	$0.224070\pi$
$150$	0	0
$151$	−3.90653	−0.317909	−0.158955	−	0.987286i	$-0.550812\pi$
$151$	−3.90653	−0.317909	−0.158955	+	0.987286i	$0.550812\pi$
$152$	−20.6277	−1.67313
$153$	−6.63325	−0.536266
$154$	0	0
$155$	0	0
$156$	9.30506	0.745001
$157$	−2.62772	−0.209715	−0.104857	−	0.994487i	$-0.533439\pi$
$157$	−2.62772	−0.209715	−0.104857	+	0.994487i	$0.533439\pi$
$158$	0.116844	0.00929561
$159$	12.7446	1.01071
$160$	0	0
$161$	20.1947	1.59157
$162$	−0.792287	−0.0622479
$163$	−8.62772	−0.675775	−0.337888	−	0.941186i	$-0.609712\pi$
$163$	−8.62772	−0.675775	−0.337888	+	0.941186i	$0.609712\pi$
$164$	−9.50744	−0.742407
$165$	0	0
$166$	−1.48913	−0.115579
$167$	−1.58457	−0.122618	−0.0613090	−	0.998119i	$-0.519528\pi$
$167$	−1.58457	−0.122618	−0.0613090	+	0.998119i	$0.519528\pi$
$168$	−6.74456	−0.520354
$169$	32.9783	2.53679
$170$	0	0
$171$	−7.72049	−0.590401
$172$	13.6540	1.04111
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	2.51087	0.190349
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	13.8564	1.03858
$179$	−3.48913	−0.260789	−0.130395	−	0.991462i	$-0.541624\pi$
$179$	−3.48913	−0.260789	−0.130395	+	0.991462i	$0.541624\pi$
$180$	0	0
$181$	−14.8614	−1.10464	−0.552320	−	0.833632i	$-0.686257\pi$
$181$	−14.8614	−1.10464	−0.552320	+	0.833632i	$0.686257\pi$
$182$	−13.5615	−1.00524
$183$	2.67181	0.197506
$184$	−21.3745	−1.57575
$185$	0	0
$186$	2.67181	0.195907
$187$	0	0
$188$	3.76631	0.274687
$189$	−2.52434	−0.183619
$190$	0	0
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	3.37228	0.243373
$193$	2.08191	0.149859	0.0749295	−	0.997189i	$-0.476127\pi$
$193$	2.08191	0.149859	0.0749295	+	0.997189i	$0.476127\pi$
$194$	−2.87419	−0.206355
$195$	0	0
$196$	0.861407	0.0615290
$197$	1.87953	0.133911	0.0669554	−	0.997756i	$-0.478671\pi$
$197$	1.87953	0.133911	0.0669554	+	0.997756i	$0.478671\pi$
$198$	0	0
$199$	−11.8614	−0.840833	−0.420416	−	0.907331i	$-0.638116\pi$
$199$	−11.8614	−0.840833	−0.420416	+	0.907331i	$0.638116\pi$
$200$	0	0
$201$	−6.11684	−0.431449
$202$	−10.5109	−0.739543
$203$	8.00000	0.561490
$204$	9.10268	0.637315
$205$	0	0
$206$	−6.13592	−0.427510
$207$	−8.00000	−0.556038
$208$	−4.25639	−0.295127
$209$	0	0
$210$	0	0
$211$	14.5012	0.998305	0.499152	−	0.866514i	$-0.333645\pi$
$211$	14.5012	0.998305	0.499152	+	0.866514i	$0.333645\pi$
$212$	−17.4891	−1.20116
$213$	−0.744563	−0.0510166
$214$	−9.25544	−0.632689
$215$	0	0
$216$	2.67181	0.181794
$217$	8.51278	0.577885
$218$	4.74456	0.321342
$219$	−5.19615	−0.351123
$220$	0	0
$221$	44.9783	3.02556
$222$	3.96143	0.265874
$223$	−4.86141	−0.325544	−0.162772	−	0.986664i	$-0.552043\pi$
$223$	−4.86141	−0.325544	−0.162772	+	0.986664i	$0.552043\pi$
$224$	14.7446	0.985163
$225$	0	0
$226$	−14.4463	−0.960954
$227$	−17.3205	−1.14960	−0.574801	−	0.818293i	$-0.694921\pi$
$227$	−17.3205	−1.14960	−0.574801	+	0.818293i	$0.694921\pi$
$228$	10.5947	0.701650
$229$	12.4891	0.825305	0.412652	−	0.910889i	$-0.364602\pi$
$229$	12.4891	0.825305	0.412652	+	0.910889i	$0.364602\pi$
$230$	0	0
$231$	0	0
$232$	−8.46738	−0.555910
$233$	−4.75372	−0.311427	−0.155713	−	0.987802i	$-0.549768\pi$
$233$	−4.75372	−0.311427	−0.155713	+	0.987802i	$0.549768\pi$
$234$	5.37228	0.351197
$235$	0	0
$236$	−5.48913	−0.357312
$237$	−0.147477	−0.00957965
$238$	−13.2665	−0.859939
$239$	−13.2665	−0.858138	−0.429069	−	0.903272i	$-0.641158\pi$
$239$	−13.2665	−0.858138	−0.429069	+	0.903272i	$0.641158\pi$
$240$	0	0
$241$	−20.6371	−1.32935	−0.664677	−	0.747131i	$-0.731431\pi$
$241$	−20.6371	−1.32935	−0.664677	+	0.747131i	$0.731431\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−3.66648	−0.234722
$245$	0	0
$246$	−5.48913	−0.349974
$247$	52.3505	3.33098
$248$	−9.01011	−0.572143
$249$	1.87953	0.119110
$250$	0	0
$251$	−23.4891	−1.48262	−0.741310	−	0.671163i	$-0.765795\pi$
$251$	−23.4891	−1.48262	−0.741310	+	0.671163i	$0.765795\pi$
$252$	3.46410	0.218218
$253$	0	0
$254$	2.86141	0.179541
$255$	0	0
$256$	−13.8832	−0.867697
$257$	5.48913	0.342402	0.171201	−	0.985236i	$-0.445235\pi$
$257$	5.48913	0.342402	0.171201	+	0.985236i	$0.445235\pi$
$258$	7.88316	0.490784
$259$	12.6217	0.784274
$260$	0	0
$261$	−3.16915	−0.196165
$262$	−1.02175	−0.0631239
$263$	1.58457	0.0977090	0.0488545	−	0.998806i	$-0.484443\pi$
$263$	1.58457	0.0977090	0.0488545	+	0.998806i	$0.484443\pi$
$264$	0	0
$265$	0	0
$266$	−15.4410	−0.946747
$267$	−17.4891	−1.07032
$268$	8.39403	0.512747
$269$	8.74456	0.533165	0.266583	−	0.963812i	$-0.414105\pi$
$269$	8.74456	0.533165	0.266583	+	0.963812i	$0.414105\pi$
$270$	0	0
$271$	−10.3923	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$271$	−10.3923	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$272$	−4.16381	−0.252468
$273$	17.1168	1.03596
$274$	8.51278	0.514276
$275$	0	0
$276$	10.9783	0.660813
$277$	−29.2048	−1.75475	−0.877374	−	0.479808i	$-0.840706\pi$
$277$	−29.2048	−1.75475	−0.877374	+	0.479808i	$0.840706\pi$
$278$	8.23369	0.493824
$279$	−3.37228	−0.201893
$280$	0	0
$281$	−1.87953	−0.112123	−0.0560616	−	0.998427i	$-0.517854\pi$
$281$	−1.87953	−0.112123	−0.0560616	+	0.998427i	$0.517854\pi$
$282$	2.17448	0.129488
$283$	−18.4077	−1.09423	−0.547114	−	0.837058i	$-0.684273\pi$
$283$	−18.4077	−1.09423	−0.547114	+	0.837058i	$0.684273\pi$
$284$	1.02175	0.0606297
$285$	0	0
$286$	0	0
$287$	−17.4891	−1.03235
$288$	−5.84096	−0.344182
$289$	27.0000	1.58824
$290$	0	0
$291$	3.62772	0.212661
$292$	7.13058	0.417286
$293$	−26.1282	−1.52643	−0.763214	−	0.646146i	$-0.776380\pi$
$293$	−26.1282	−1.52643	−0.763214	+	0.646146i	$0.776380\pi$
$294$	0.497333	0.0290051
$295$	0	0
$296$	−13.3591	−0.776480
$297$	0	0
$298$	−14.7446	−0.854130
$299$	54.2458	3.13712
$300$	0	0
$301$	25.1168	1.44771
$302$	3.09509	0.178103
$303$	13.2665	0.762140
$304$	−4.84630	−0.277954
$305$	0	0
$306$	5.25544	0.300433
$307$	−9.89497	−0.564736	−0.282368	−	0.959306i	$-0.591120\pi$
$307$	−9.89497	−0.564736	−0.282368	+	0.959306i	$0.591120\pi$
$308$	0	0
$309$	7.74456	0.440573
$310$	0	0
$311$	24.2337	1.37417	0.687083	−	0.726579i	$-0.258891\pi$
$311$	24.2337	1.37417	0.687083	+	0.726579i	$0.258891\pi$
$312$	−18.1168	−1.02566
$313$	−19.9783	−1.12924	−0.564619	−	0.825352i	$-0.690977\pi$
$313$	−19.9783	−1.12924	−0.564619	+	0.825352i	$0.690977\pi$
$314$	2.08191	0.117489
$315$	0	0
$316$	0.202380	0.0113847
$317$	−26.9783	−1.51525	−0.757625	−	0.652690i	$-0.773640\pi$
$317$	−26.9783	−1.51525	−0.757625	+	0.652690i	$0.773640\pi$
$318$	−10.0974	−0.566231
$319$	0	0
$320$	0	0
$321$	11.6819	0.652021
$322$	−16.0000	−0.891645
$323$	51.2119	2.84951
$324$	−1.37228	−0.0762379
$325$	0	0
$326$	6.83563	0.378590
$327$	−5.98844	−0.331161
$328$	18.5109	1.02209
$329$	6.92820	0.381964
$330$	0	0
$331$	−33.2337	−1.82669	−0.913344	−	0.407188i	$-0.866509\pi$
$331$	−33.2337	−1.82669	−0.913344	+	0.407188i	$0.866509\pi$
$332$	−2.57924	−0.141554
$333$	−5.00000	−0.273998
$334$	1.25544	0.0686945
$335$	0	0
$336$	−1.58457	−0.0864456
$337$	−19.2549	−1.04888	−0.524442	−	0.851446i	$-0.675726\pi$
$337$	−19.2549	−1.04888	−0.524442	+	0.851446i	$0.675726\pi$
$338$	−26.1282	−1.42119
$339$	18.2337	0.990318
$340$	0	0
$341$	0	0
$342$	6.11684	0.330761
$343$	19.2549	1.03967
$344$	−26.5842	−1.43333
$345$	0	0
$346$	5.48913	0.295097
$347$	−22.0742	−1.18501	−0.592503	−	0.805568i	$-0.701860\pi$
$347$	−22.0742	−1.18501	−0.592503	+	0.805568i	$0.701860\pi$
$348$	4.34896	0.233129
$349$	8.66025	0.463573	0.231786	−	0.972767i	$-0.425543\pi$
$349$	8.66025	0.463573	0.231786	+	0.972767i	$0.425543\pi$
$350$	0	0
$351$	−6.78073	−0.361928
$352$	0	0
$353$	−18.2337	−0.970481	−0.485241	−	0.874381i	$-0.661268\pi$
$353$	−18.2337	−0.970481	−0.485241	+	0.874381i	$0.661268\pi$
$354$	−3.16915	−0.168438
$355$	0	0
$356$	24.0000	1.27200
$357$	16.7446	0.886216
$358$	2.76439	0.146102
$359$	−6.63325	−0.350090	−0.175045	−	0.984560i	$-0.556007\pi$
$359$	−6.63325	−0.350090	−0.175045	+	0.984560i	$0.556007\pi$
$360$	0	0
$361$	40.6060	2.13716
$362$	11.7745	0.618854
$363$	0	0
$364$	−23.4891	−1.23116
$365$	0	0
$366$	−2.11684	−0.110649
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	−5.02175	−0.261777
$369$	6.92820	0.360668
$370$	0	0
$371$	−32.1716	−1.67027
$372$	4.62772	0.239936
$373$	−10.4472	−0.540936	−0.270468	−	0.962729i	$-0.587178\pi$
$373$	−10.4472	−0.540936	−0.270468	+	0.962729i	$0.587178\pi$
$374$	0	0
$375$	0	0
$376$	−7.33296	−0.378169
$377$	21.4891	1.10675
$378$	2.00000	0.102869
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−3.61158	−0.185027
$382$	−4.34896	−0.222512
$383$	−24.2337	−1.23828	−0.619142	−	0.785279i	$-0.712519\pi$
$383$	−24.2337	−1.23828	−0.619142	+	0.785279i	$0.712519\pi$
$384$	9.01011	0.459795
$385$	0	0
$386$	−1.64947	−0.0839557
$387$	−9.94987	−0.505781
$388$	−4.97825	−0.252732
$389$	21.2554	1.07769	0.538847	−	0.842404i	$-0.318860\pi$
$389$	21.2554	1.07769	0.538847	+	0.842404i	$0.318860\pi$
$390$	0	0
$391$	53.0660	2.68366
$392$	−1.67715	−0.0847088
$393$	1.28962	0.0650527
$394$	−1.48913	−0.0750210
$395$	0	0
$396$	0	0
$397$	8.37228	0.420193	0.210096	−	0.977681i	$-0.432622\pi$
$397$	8.37228	0.420193	0.210096	+	0.977681i	$0.432622\pi$
$398$	9.39764	0.471061
$399$	19.4891	0.975677
$400$	0	0
$401$	14.9783	0.747978	0.373989	−	0.927433i	$-0.377990\pi$
$401$	14.9783	0.747978	0.373989	+	0.927433i	$0.377990\pi$
$402$	4.84630	0.241711
$403$	22.8665	1.13906
$404$	−18.2054	−0.905751
$405$	0	0
$406$	−6.33830	−0.314564
$407$	0	0
$408$	−17.7228	−0.877410
$409$	8.45787	0.418215	0.209107	−	0.977893i	$-0.432944\pi$
$409$	8.45787	0.418215	0.209107	+	0.977893i	$0.432944\pi$
$410$	0	0
$411$	−10.7446	−0.529990
$412$	−10.6277	−0.523590
$413$	−10.0974	−0.496858
$414$	6.33830	0.311510
$415$	0	0
$416$	39.6060	1.94184
$417$	−10.3923	−0.508913
$418$	0	0
$419$	32.2337	1.57472	0.787359	−	0.616494i	$-0.211448\pi$
$419$	32.2337	1.57472	0.787359	+	0.616494i	$0.211448\pi$
$420$	0	0
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	−11.4891	−0.559282
$423$	−2.74456	−0.133445
$424$	34.0511	1.65367
$425$	0	0
$426$	0.589907	0.0285811
$427$	−6.74456	−0.326392
$428$	−16.0309	−0.774882
$429$	0	0
$430$	0	0
$431$	−2.87419	−0.138445	−0.0692225	−	0.997601i	$-0.522052\pi$
$431$	−2.87419	−0.138445	−0.0692225	+	0.997601i	$0.522052\pi$
$432$	0.627719	0.0302011
$433$	23.0951	1.10988	0.554940	−	0.831891i	$-0.312741\pi$
$433$	23.0951	1.10988	0.554940	+	0.831891i	$0.312741\pi$
$434$	−6.74456	−0.323749
$435$	0	0
$436$	8.21782	0.393562
$437$	61.7639	2.95457
$438$	4.11684	0.196710
$439$	−22.7190	−1.08432	−0.542160	−	0.840275i	$-0.682393\pi$
$439$	−22.7190	−1.08432	−0.542160	+	0.840275i	$0.682393\pi$
$440$	0	0
$441$	−0.627719	−0.0298914
$442$	−35.6357	−1.69502
$443$	−20.7446	−0.985604	−0.492802	−	0.870142i	$-0.664027\pi$
$443$	−20.7446	−0.985604	−0.492802	+	0.870142i	$0.664027\pi$
$444$	6.86141	0.325628
$445$	0	0
$446$	3.85163	0.182380
$447$	18.6101	0.880229
$448$	−8.51278	−0.402191
$449$	1.25544	0.0592478	0.0296239	−	0.999561i	$-0.490569\pi$
$449$	1.25544	0.0592478	0.0296239	+	0.999561i	$0.490569\pi$
$450$	0	0
$451$	0	0
$452$	−25.0217	−1.17692
$453$	−3.90653	−0.183545
$454$	13.7228	0.644044
$455$	0	0
$456$	−20.6277	−0.965981
$457$	33.4612	1.56525	0.782624	−	0.622494i	$-0.213881\pi$
$457$	33.4612	1.56525	0.782624	+	0.622494i	$0.213881\pi$
$458$	−9.89497	−0.462362
$459$	−6.63325	−0.309614
$460$	0	0
$461$	8.51278	0.396480	0.198240	−	0.980154i	$-0.436478\pi$
$461$	8.51278	0.396480	0.198240	+	0.980154i	$0.436478\pi$
$462$	0	0
$463$	18.9783	0.881994	0.440997	−	0.897509i	$-0.354625\pi$
$463$	18.9783	0.881994	0.440997	+	0.897509i	$0.354625\pi$
$464$	−1.98933	−0.0923525
$465$	0	0
$466$	3.76631	0.174471
$467$	−21.4891	−0.994398	−0.497199	−	0.867636i	$-0.665638\pi$
$467$	−21.4891	−0.994398	−0.497199	+	0.867636i	$0.665638\pi$
$468$	9.30506	0.430127
$469$	15.4410	0.712998
$470$	0	0
$471$	−2.62772	−0.121079
$472$	10.6873	0.491921
$473$	0	0
$474$	0.116844	0.00536682
$475$	0	0
$476$	−22.9783	−1.05321
$477$	12.7446	0.583533
$478$	10.5109	0.480756
$479$	−23.6588	−1.08100	−0.540499	−	0.841345i	$-0.681765\pi$
$479$	−23.6588	−1.08100	−0.540499	+	0.841345i	$0.681765\pi$
$480$	0	0
$481$	33.9036	1.54587
$482$	16.3505	0.744746
$483$	20.1947	0.918891
$484$	0	0
$485$	0	0
$486$	−0.792287	−0.0359389
$487$	−9.97825	−0.452158	−0.226079	−	0.974109i	$-0.572591\pi$
$487$	−9.97825	−0.452158	−0.226079	+	0.974109i	$0.572591\pi$
$488$	7.13859	0.323149
$489$	−8.62772	−0.390159
$490$	0	0
$491$	6.63325	0.299354	0.149677	−	0.988735i	$-0.452177\pi$
$491$	6.63325	0.299354	0.149677	+	0.988735i	$0.452177\pi$
$492$	−9.50744	−0.428629
$493$	21.0217	0.946772
$494$	−41.4766	−1.86612
$495$	0	0
$496$	−2.11684	−0.0950491
$497$	1.87953	0.0843083
$498$	−1.48913	−0.0667293
$499$	3.11684	0.139529	0.0697645	−	0.997563i	$-0.477775\pi$
$499$	3.11684	0.139529	0.0697645	+	0.997563i	$0.477775\pi$
$500$	0	0
$501$	−1.58457	−0.0707935
$502$	18.6101	0.830611
$503$	10.0974	0.450219	0.225109	−	0.974334i	$-0.427726\pi$
$503$	10.0974	0.450219	0.225109	+	0.974334i	$0.427726\pi$
$504$	−6.74456	−0.300427
$505$	0	0
$506$	0	0
$507$	32.9783	1.46462
$508$	4.95610	0.219891
$509$	14.9783	0.663899	0.331950	−	0.943297i	$-0.392294\pi$
$509$	14.9783	0.663899	0.331950	+	0.943297i	$0.392294\pi$
$510$	0	0
$511$	13.1168	0.580255
$512$	−7.02078	−0.310277
$513$	−7.72049	−0.340868
$514$	−4.34896	−0.191825
$515$	0	0
$516$	13.6540	0.601085
$517$	0	0
$518$	−10.0000	−0.439375
$519$	−6.92820	−0.304114
$520$	0	0
$521$	−25.2554	−1.10646	−0.553230	−	0.833028i	$-0.686605\pi$
$521$	−25.2554	−1.10646	−0.553230	+	0.833028i	$0.686605\pi$
$522$	2.51087	0.109898
$523$	37.0179	1.61868	0.809339	−	0.587341i	$-0.199825\pi$
$523$	37.0179	1.61868	0.809339	+	0.587341i	$0.199825\pi$
$524$	−1.76972	−0.0773107
$525$	0	0
$526$	−1.25544	−0.0547397
$527$	22.3692	0.974417
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	4.00000	0.173585
$532$	−26.7446	−1.15952
$533$	−46.9783	−2.03485
$534$	13.8564	0.599625
$535$	0	0
$536$	−16.3431	−0.705913
$537$	−3.48913	−0.150567
$538$	−6.92820	−0.298696
$539$	0	0
$540$	0	0
$541$	−9.65492	−0.415097	−0.207549	−	0.978225i	$-0.566549\pi$
$541$	−9.65492	−0.415097	−0.207549	+	0.978225i	$0.566549\pi$
$542$	8.23369	0.353667
$543$	−14.8614	−0.637764
$544$	38.7446	1.66116
$545$	0	0
$546$	−13.5615	−0.580377
$547$	40.6844	1.73954	0.869769	−	0.493459i	$-0.164268\pi$
$547$	40.6844	1.73954	0.869769	+	0.493459i	$0.164268\pi$
$548$	14.7446	0.629857
$549$	2.67181	0.114030
$550$	0	0
$551$	24.4674	1.04235
$552$	−21.3745	−0.909760
$553$	0.372281	0.0158310
$554$	23.1386	0.983065
$555$	0	0
$556$	14.2612	0.604808
$557$	20.1947	0.855677	0.427839	−	0.903855i	$-0.359275\pi$
$557$	20.1947	0.855677	0.427839	+	0.903855i	$0.359275\pi$
$558$	2.67181	0.113107
$559$	67.4674	2.85357
$560$	0	0
$561$	0	0
$562$	1.48913	0.0628150
$563$	9.21249	0.388260	0.194130	−	0.980976i	$-0.437812\pi$
$563$	9.21249	0.388260	0.194130	+	0.980976i	$0.437812\pi$
$564$	3.76631	0.158590
$565$	0	0
$566$	14.5842	0.613020
$567$	−2.52434	−0.106012
$568$	−1.98933	−0.0834706
$569$	−29.9971	−1.25754	−0.628772	−	0.777590i	$-0.716442\pi$
$569$	−29.9971	−1.25754	−0.628772	+	0.777590i	$0.716442\pi$
$570$	0	0
$571$	39.7446	1.66326	0.831630	−	0.555330i	$-0.187408\pi$
$571$	39.7446	1.66326	0.831630	+	0.555330i	$0.187408\pi$
$572$	0	0
$573$	5.48913	0.229311
$574$	13.8564	0.578355
$575$	0	0
$576$	3.37228	0.140512
$577$	−39.0951	−1.62755	−0.813775	−	0.581180i	$-0.802591\pi$
$577$	−39.0951	−1.62755	−0.813775	+	0.581180i	$0.802591\pi$
$578$	−21.3917	−0.889779
$579$	2.08191	0.0865211
$580$	0	0
$581$	−4.74456	−0.196838
$582$	−2.87419	−0.119139
$583$	0	0
$584$	−13.8832	−0.574489
$585$	0	0
$586$	20.7011	0.855153
$587$	−23.2554	−0.959855	−0.479927	−	0.877308i	$-0.659337\pi$
$587$	−23.2554	−0.959855	−0.479927	+	0.877308i	$0.659337\pi$
$588$	0.861407	0.0355238
$589$	26.0357	1.07278
$590$	0	0
$591$	1.87953	0.0773134
$592$	−3.13859	−0.128995
$593$	37.8102	1.55268	0.776339	−	0.630316i	$-0.217075\pi$
$593$	37.8102	1.55268	0.776339	+	0.630316i	$0.217075\pi$
$594$	0	0
$595$	0	0
$596$	−25.5383	−1.04609
$597$	−11.8614	−0.485455
$598$	−42.9783	−1.75751
$599$	−44.7446	−1.82821	−0.914107	−	0.405474i	$-0.867107\pi$
$599$	−44.7446	−1.82821	−0.914107	+	0.405474i	$0.867107\pi$
$600$	0	0
$601$	−1.08724	−0.0443495	−0.0221747	−	0.999754i	$-0.507059\pi$
$601$	−1.08724	−0.0443495	−0.0221747	+	0.999754i	$0.507059\pi$
$602$	−19.8997	−0.811053
$603$	−6.11684	−0.249097
$604$	5.36086	0.218130
$605$	0	0
$606$	−10.5109	−0.426975
$607$	6.04334	0.245292	0.122646	−	0.992451i	$-0.460862\pi$
$607$	6.04334	0.245292	0.122646	+	0.992451i	$0.460862\pi$
$608$	45.0951	1.82885
$609$	8.00000	0.324176
$610$	0	0
$611$	18.6101	0.752885
$612$	9.10268	0.367954
$613$	−11.5344	−0.465872	−0.232936	−	0.972492i	$-0.574833\pi$
$613$	−11.5344	−0.465872	−0.232936	+	0.972492i	$0.574833\pi$
$614$	7.83966	0.316383
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−6.13592	−0.246823
$619$	−25.5109	−1.02537	−0.512684	−	0.858577i	$-0.671349\pi$
$619$	−25.5109	−1.02537	−0.512684	+	0.858577i	$0.671349\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	−19.2000	−0.769851
$623$	44.1485	1.76877
$624$	−4.25639	−0.170392
$625$	0	0
$626$	15.8285	0.632634
$627$	0	0
$628$	3.60597	0.143894
$629$	33.1662	1.32242
$630$	0	0
$631$	−13.5109	−0.537859	−0.268930	−	0.963160i	$-0.586670\pi$
$631$	−13.5109	−0.537859	−0.268930	+	0.963160i	$0.586670\pi$
$632$	−0.394031	−0.0156737
$633$	14.5012	0.576372
$634$	21.3745	0.848891
$635$	0	0
$636$	−17.4891	−0.693489
$637$	4.25639	0.168644
$638$	0	0
$639$	−0.744563	−0.0294544
$640$	0	0
$641$	−16.9783	−0.670601	−0.335300	−	0.942111i	$-0.608838\pi$
$641$	−16.9783	−0.670601	−0.335300	+	0.942111i	$0.608838\pi$
$642$	−9.25544	−0.365283
$643$	−25.2337	−0.995120	−0.497560	−	0.867430i	$-0.665770\pi$
$643$	−25.2337	−0.995120	−0.497560	+	0.867430i	$0.665770\pi$
$644$	−27.7128	−1.09204
$645$	0	0
$646$	−40.5746	−1.59638
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	2.67181	0.104959
$649$	0	0
$650$	0	0
$651$	8.51278	0.333642
$652$	11.8397	0.463677
$653$	−22.5109	−0.880919	−0.440459	−	0.897773i	$-0.645184\pi$
$653$	−22.5109	−0.880919	−0.440459	+	0.897773i	$0.645184\pi$
$654$	4.74456	0.185527
$655$	0	0
$656$	4.34896	0.169798
$657$	−5.19615	−0.202721
$658$	−5.48913	−0.213988
$659$	−18.9051	−0.736437	−0.368219	−	0.929739i	$-0.620032\pi$
$659$	−18.9051	−0.736437	−0.368219	+	0.929739i	$0.620032\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	26.3306	1.02337
$663$	44.9783	1.74681
$664$	5.02175	0.194882
$665$	0	0
$666$	3.96143	0.153502
$667$	25.3532	0.981679
$668$	2.17448	0.0841332
$669$	−4.86141	−0.187953
$670$	0	0
$671$	0	0
$672$	14.7446	0.568784
$673$	2.32196	0.0895049	0.0447525	−	0.998998i	$-0.485750\pi$
$673$	2.32196	0.0895049	0.0447525	+	0.998998i	$0.485750\pi$
$674$	15.2554	0.587617
$675$	0	0
$676$	−45.2554	−1.74059
$677$	−37.5152	−1.44183	−0.720913	−	0.693025i	$-0.756277\pi$
$677$	−37.5152	−1.44183	−0.720913	+	0.693025i	$0.756277\pi$
$678$	−14.4463	−0.554807
$679$	−9.15759	−0.351436
$680$	0	0
$681$	−17.3205	−0.663723
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	10.5947	0.405098
$685$	0	0
$686$	−15.2554	−0.582455
$687$	12.4891	0.476490
$688$	−6.24572	−0.238116
$689$	−86.4174	−3.29224
$690$	0	0
$691$	−13.7446	−0.522868	−0.261434	−	0.965221i	$-0.584195\pi$
$691$	−13.7446	−0.522868	−0.261434	+	0.965221i	$0.584195\pi$
$692$	9.50744	0.361419
$693$	0	0
$694$	17.4891	0.663878
$695$	0	0
$696$	−8.46738	−0.320955
$697$	−45.9565	−1.74073
$698$	−6.86141	−0.259708
$699$	−4.75372	−0.179802
$700$	0	0
$701$	−30.5870	−1.15526	−0.577628	−	0.816300i	$-0.696021\pi$
$701$	−30.5870	−1.15526	−0.577628	+	0.816300i	$0.696021\pi$
$702$	5.37228	0.202764
$703$	38.6025	1.45592
$704$	0	0
$705$	0	0
$706$	14.4463	0.543694
$707$	−33.4891	−1.25949
$708$	−5.48913	−0.206294
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	−0.147477	−0.00553081
$712$	−46.7277	−1.75119
$713$	26.9783	1.01034
$714$	−13.2665	−0.496486
$715$	0	0
$716$	4.78806	0.178938
$717$	−13.2665	−0.495446
$718$	5.25544	0.196131
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	−19.5499	−0.728076
$722$	−32.1716	−1.19730
$723$	−20.6371	−0.767503
$724$	20.3940	0.757938
$725$	0	0
$726$	0	0
$727$	47.4674	1.76047	0.880234	−	0.474540i	$-0.157386\pi$
$727$	47.4674	1.76047	0.880234	+	0.474540i	$0.157386\pi$
$728$	45.7330	1.69498
$729$	1.00000	0.0370370
$730$	0	0
$731$	66.0000	2.44110
$732$	−3.66648	−0.135517
$733$	−13.8564	−0.511798	−0.255899	−	0.966704i	$-0.582371\pi$
$733$	−13.8564	−0.511798	−0.255899	+	0.966704i	$0.582371\pi$
$734$	−11.8843	−0.438658
$735$	0	0
$736$	46.7277	1.72241
$737$	0	0
$738$	−5.48913	−0.202057
$739$	−12.8241	−0.471741	−0.235870	−	0.971784i	$-0.575794\pi$
$739$	−12.8241	−0.471741	−0.235870	+	0.971784i	$0.575794\pi$
$740$	0	0
$741$	52.3505	1.92314
$742$	25.4891	0.935735
$743$	17.7253	0.650277	0.325138	−	0.945666i	$-0.394589\pi$
$743$	17.7253	0.650277	0.325138	+	0.945666i	$0.394589\pi$
$744$	−9.01011	−0.330327
$745$	0	0
$746$	8.27719	0.303049
$747$	1.87953	0.0687683
$748$	0	0
$749$	−29.4891	−1.07751
$750$	0	0
$751$	−5.23369	−0.190980	−0.0954900	−	0.995430i	$-0.530442\pi$
$751$	−5.23369	−0.190980	−0.0954900	+	0.995430i	$0.530442\pi$
$752$	−1.72281	−0.0628245
$753$	−23.4891	−0.855991
$754$	−17.0256	−0.620034
$755$	0	0
$756$	3.46410	0.125988
$757$	19.3505	0.703307	0.351654	−	0.936130i	$-0.385620\pi$
$757$	19.3505	0.703307	0.351654	+	0.936130i	$0.385620\pi$
$758$	3.96143	0.143886
$759$	0	0
$760$	0	0
$761$	−1.28962	−0.0467487	−0.0233744	−	0.999727i	$-0.507441\pi$
$761$	−1.28962	−0.0467487	−0.0233744	+	0.999727i	$0.507441\pi$
$762$	2.86141	0.103658
$763$	15.1168	0.547266
$764$	−7.53262	−0.272521
$765$	0	0
$766$	19.2000	0.693725
$767$	−27.1229	−0.979351
$768$	−13.8832	−0.500965
$769$	−13.7638	−0.496336	−0.248168	−	0.968717i	$-0.579829\pi$
$769$	−13.7638	−0.496336	−0.248168	+	0.968717i	$0.579829\pi$
$770$	0	0
$771$	5.48913	0.197686
$772$	−2.85696	−0.102824
$773$	−23.4891	−0.844845	−0.422423	−	0.906399i	$-0.638820\pi$
$773$	−23.4891	−0.844845	−0.422423	+	0.906399i	$0.638820\pi$
$774$	7.88316	0.283354
$775$	0	0
$776$	9.69259	0.347944
$777$	12.6217	0.452801
$778$	−16.8404	−0.603758
$779$	−53.4891	−1.91645
$780$	0	0
$781$	0	0
$782$	−42.0435	−1.50347
$783$	−3.16915	−0.113256
$784$	−0.394031	−0.0140725
$785$	0	0
$786$	−1.02175	−0.0364446
$787$	6.48577	0.231193	0.115596	−	0.993296i	$-0.463122\pi$
$787$	6.48577	0.231193	0.115596	+	0.993296i	$0.463122\pi$
$788$	−2.57924	−0.0918816
$789$	1.58457	0.0564123
$790$	0	0
$791$	−46.0280	−1.63657
$792$	0	0
$793$	−18.1168	−0.643348
$794$	−6.63325	−0.235405
$795$	0	0
$796$	16.2772	0.576930
$797$	−27.7228	−0.981992	−0.490996	−	0.871162i	$-0.663367\pi$
$797$	−27.7228	−0.981992	−0.490996	+	0.871162i	$0.663367\pi$
$798$	−15.4410	−0.546605
$799$	18.2054	0.644060
$800$	0	0
$801$	−17.4891	−0.617948
$802$	−11.8671	−0.419041
$803$	0	0
$804$	8.39403	0.296035
$805$	0	0
$806$	−18.1168	−0.638139
$807$	8.74456	0.307823
$808$	35.4456	1.24697
$809$	−7.22316	−0.253953	−0.126976	−	0.991906i	$-0.540527\pi$
$809$	−7.22316	−0.253953	−0.126976	+	0.991906i	$0.540527\pi$
$810$	0	0
$811$	44.7933	1.57290	0.786452	−	0.617651i	$-0.211916\pi$
$811$	44.7933	1.57290	0.786452	+	0.617651i	$0.211916\pi$
$812$	−10.9783	−0.385261
$813$	−10.3923	−0.364474
$814$	0	0
$815$	0	0
$816$	−4.16381	−0.145763
$817$	76.8179	2.68752
$818$	−6.70106	−0.234297
$819$	17.1168	0.598111
$820$	0	0
$821$	−12.8617	−0.448878	−0.224439	−	0.974488i	$-0.572055\pi$
$821$	−12.8617	−0.448878	−0.224439	+	0.974488i	$0.572055\pi$
$822$	8.51278	0.296917
$823$	49.3505	1.72025	0.860126	−	0.510082i	$-0.170385\pi$
$823$	49.3505	1.72025	0.860126	+	0.510082i	$0.170385\pi$
$824$	20.6920	0.720841
$825$	0	0
$826$	8.00000	0.278356
$827$	36.9253	1.28402	0.642009	−	0.766697i	$-0.278101\pi$
$827$	36.9253	1.28402	0.642009	+	0.766697i	$0.278101\pi$
$828$	10.9783	0.381521
$829$	−27.4674	−0.953981	−0.476991	−	0.878908i	$-0.658272\pi$
$829$	−27.4674	−0.953981	−0.476991	+	0.878908i	$0.658272\pi$
$830$	0	0
$831$	−29.2048	−1.01310
$832$	−22.8665	−0.792754
$833$	4.16381	0.144268
$834$	8.23369	0.285109
$835$	0	0
$836$	0	0
$837$	−3.37228	−0.116563
$838$	−25.5383	−0.882207
$839$	34.7446	1.19952	0.599758	−	0.800182i	$-0.295264\pi$
$839$	34.7446	1.19952	0.599758	+	0.800182i	$0.295264\pi$
$840$	0	0
$841$	−18.9565	−0.653672
$842$	−14.2612	−0.491472
$843$	−1.87953	−0.0647344
$844$	−19.8997	−0.684978
$845$	0	0
$846$	2.17448	0.0747602
$847$	0	0
$848$	8.00000	0.274721
$849$	−18.4077	−0.631752
$850$	0	0
$851$	40.0000	1.37118
$852$	1.02175	0.0350046
$853$	−26.0357	−0.891444	−0.445722	−	0.895171i	$-0.647053\pi$
$853$	−26.0357	−0.891444	−0.445722	+	0.895171i	$0.647053\pi$
$854$	5.34363	0.182855
$855$	0	0
$856$	31.2119	1.06680
$857$	−9.39764	−0.321017	−0.160509	−	0.987034i	$-0.551313\pi$
$857$	−9.39764	−0.321017	−0.160509	+	0.987034i	$0.551313\pi$
$858$	0	0
$859$	−21.7446	−0.741915	−0.370957	−	0.928650i	$-0.620970\pi$
$859$	−21.7446	−0.741915	−0.370957	+	0.928650i	$0.620970\pi$
$860$	0	0
$861$	−17.4891	−0.596028
$862$	2.27719	0.0775613
$863$	−14.2337	−0.484520	−0.242260	−	0.970211i	$-0.577889\pi$
$863$	−14.2337	−0.484520	−0.242260	+	0.970211i	$0.577889\pi$
$864$	−5.84096	−0.198714
$865$	0	0
$866$	−18.2979	−0.621789
$867$	27.0000	0.916968
$868$	−11.6819	−0.396510
$869$	0	0
$870$	0	0
$871$	41.4766	1.40538
$872$	−16.0000	−0.541828
$873$	3.62772	0.122780
$874$	−48.9348	−1.65524
$875$	0	0
$876$	7.13058	0.240920
$877$	10.0048	0.337837	0.168919	−	0.985630i	$-0.445972\pi$
$877$	10.0048	0.337837	0.168919	+	0.985630i	$0.445972\pi$
$878$	18.0000	0.607471
$879$	−26.1282	−0.881284
$880$	0	0
$881$	−22.4674	−0.756945	−0.378473	−	0.925613i	$-0.623551\pi$
$881$	−22.4674	−0.756945	−0.378473	+	0.925613i	$0.623551\pi$
$882$	0.497333	0.0167461
$883$	−39.8614	−1.34144	−0.670722	−	0.741709i	$-0.734015\pi$
$883$	−39.8614	−1.34144	−0.670722	+	0.741709i	$0.734015\pi$
$884$	−61.7228	−2.07596
$885$	0	0
$886$	16.4356	0.552166
$887$	−24.9484	−0.837686	−0.418843	−	0.908059i	$-0.637564\pi$
$887$	−24.9484	−0.837686	−0.418843	+	0.908059i	$0.637564\pi$
$888$	−13.3591	−0.448301
$889$	9.11684	0.305769
$890$	0	0
$891$	0	0
$892$	6.67122	0.223369
$893$	21.1894	0.709075
$894$	−14.7446	−0.493132
$895$	0	0
$896$	−22.7446	−0.759843
$897$	54.2458	1.81121
$898$	−0.994667	−0.0331925
$899$	10.6873	0.356440
$900$	0	0
$901$	−84.5379	−2.81636
$902$	0	0
$903$	25.1168	0.835836
$904$	48.7170	1.62030
$905$	0	0
$906$	3.09509	0.102828
$907$	−15.2337	−0.505826	−0.252913	−	0.967489i	$-0.581389\pi$
$907$	−15.2337	−0.505826	−0.252913	+	0.967489i	$0.581389\pi$
$908$	23.7686	0.788789
$909$	13.2665	0.440022
$910$	0	0
$911$	27.4891	0.910755	0.455378	−	0.890298i	$-0.349504\pi$
$911$	27.4891	0.910755	0.455378	+	0.890298i	$0.349504\pi$
$912$	−4.84630	−0.160477
$913$	0	0
$914$	−26.5109	−0.876902
$915$	0	0
$916$	−17.1386	−0.566275
$917$	−3.25544	−0.107504
$918$	5.25544	0.173455
$919$	−5.69349	−0.187811	−0.0939054	−	0.995581i	$-0.529935\pi$
$919$	−5.69349	−0.187811	−0.0939054	+	0.995581i	$0.529935\pi$
$920$	0	0
$921$	−9.89497	−0.326050
$922$	−6.74456	−0.222120
$923$	5.04868	0.166179
$924$	0	0
$925$	0	0
$926$	−15.0362	−0.494121
$927$	7.74456	0.254365
$928$	18.5109	0.607649
$929$	24.7446	0.811843	0.405921	−	0.913908i	$-0.366951\pi$
$929$	24.7446	0.811843	0.405921	+	0.913908i	$0.366951\pi$
$930$	0	0
$931$	4.84630	0.158831
$932$	6.52344	0.213683
$933$	24.2337	0.793375
$934$	17.0256	0.557093
$935$	0	0
$936$	−18.1168	−0.592168
$937$	−24.8935	−0.813236	−0.406618	−	0.913598i	$-0.633292\pi$
$937$	−24.8935	−0.813236	−0.406618	+	0.913598i	$0.633292\pi$
$938$	−12.2337	−0.399444
$939$	−19.9783	−0.651966
$940$	0	0
$941$	17.0256	0.555017	0.277509	−	0.960723i	$-0.410491\pi$
$941$	17.0256	0.555017	0.277509	+	0.960723i	$0.410491\pi$
$942$	2.08191	0.0678322
$943$	−55.4256	−1.80491
$944$	2.51087	0.0817220
$945$	0	0
$946$	0	0
$947$	1.48913	0.0483901	0.0241950	−	0.999707i	$-0.492298\pi$
$947$	1.48913	0.0483901	0.0241950	+	0.999707i	$0.492298\pi$
$948$	0.202380	0.00657299
$949$	35.2337	1.14373
$950$	0	0
$951$	−26.9783	−0.874830
$952$	44.7384	1.44998
$953$	7.92287	0.256647	0.128323	−	0.991732i	$-0.459040\pi$
$953$	7.92287	0.256647	0.128323	+	0.991732i	$0.459040\pi$
$954$	−10.0974	−0.326914
$955$	0	0
$956$	18.2054	0.588804
$957$	0	0
$958$	18.7446	0.605609
$959$	27.1229	0.875844
$960$	0	0
$961$	−19.6277	−0.633152
$962$	−26.8614	−0.866047
$963$	11.6819	0.376445
$964$	28.3200	0.912124
$965$	0	0
$966$	−16.0000	−0.514792
$967$	54.3933	1.74917	0.874585	−	0.484872i	$-0.161134\pi$
$967$	54.3933	1.74917	0.874585	+	0.484872i	$0.161134\pi$
$968$	0	0
$969$	51.2119	1.64516
$970$	0	0
$971$	26.5109	0.850774	0.425387	−	0.905011i	$-0.360138\pi$
$971$	26.5109	0.850774	0.425387	+	0.905011i	$0.360138\pi$
$972$	−1.37228	−0.0440159
$973$	26.2337	0.841013
$974$	7.90564	0.253313
$975$	0	0
$976$	1.67715	0.0536842
$977$	−27.2554	−0.871979	−0.435989	−	0.899952i	$-0.643601\pi$
$977$	−27.2554	−0.871979	−0.435989	+	0.899952i	$0.643601\pi$
$978$	6.83563	0.218579
$979$	0	0
$980$	0	0
$981$	−5.98844	−0.191196
$982$	−5.25544	−0.167708
$983$	14.5109	0.462825	0.231413	−	0.972856i	$-0.425665\pi$
$983$	14.5109	0.462825	0.231413	+	0.972856i	$0.425665\pi$
$984$	18.5109	0.590105
$985$	0	0
$986$	−16.6553	−0.530411
$987$	6.92820	0.220527
$988$	−71.8397	−2.28552
$989$	79.5990	2.53110
$990$	0	0
$991$	−44.9565	−1.42809	−0.714045	−	0.700100i	$-0.753139\pi$
$991$	−44.9565	−1.42809	−0.714045	+	0.700100i	$0.753139\pi$
$992$	19.6974	0.625392
$993$	−33.2337	−1.05464
$994$	−1.48913	−0.0472322
$995$	0	0
$996$	−2.57924	−0.0817264
$997$	52.5138	1.66313	0.831564	−	0.555429i	$-0.187446\pi$
$997$	52.5138	1.66313	0.831564	+	0.555429i	$0.187446\pi$
$998$	−2.46943	−0.0781686
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.db.1.2	yes	4
5.4	even	2		9075.2.a.cw.1.3	yes	4
11.10	odd	2	inner	9075.2.a.db.1.3	yes	4
55.54	odd	2		9075.2.a.cw.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.cw.1.2	✓	4	55.54	odd	2
9075.2.a.cw.1.3	yes	4	5.4	even	2
9075.2.a.db.1.2	yes	4	1.1	even	1	trivial
9075.2.a.db.1.3	yes	4	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} -2.52434 q^{7} +2.67181 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} -0.792287 q^{6} -2.52434 q^{7} +2.67181 q^{8} +1.00000 q^{9} -1.37228 q^{12} -6.78073 q^{13} +2.00000 q^{14} +0.627719 q^{16} -6.63325 q^{17} -0.792287 q^{18} -7.72049 q^{19} -2.52434 q^{21} -8.00000 q^{23} +2.67181 q^{24} +5.37228 q^{26} +1.00000 q^{27} +3.46410 q^{28} -3.16915 q^{29} -3.37228 q^{31} -5.84096 q^{32} +5.25544 q^{34} -1.37228 q^{36} -5.00000 q^{37} +6.11684 q^{38} -6.78073 q^{39} +6.92820 q^{41} +2.00000 q^{42} -9.94987 q^{43} +6.33830 q^{46} -2.74456 q^{47} +0.627719 q^{48} -0.627719 q^{49} -6.63325 q^{51} +9.30506 q^{52} +12.7446 q^{53} -0.792287 q^{54} -6.74456 q^{56} -7.72049 q^{57} +2.51087 q^{58} +4.00000 q^{59} +2.67181 q^{61} +2.67181 q^{62} -2.52434 q^{63} +3.37228 q^{64} -6.11684 q^{67} +9.10268 q^{68} -8.00000 q^{69} -0.744563 q^{71} +2.67181 q^{72} -5.19615 q^{73} +3.96143 q^{74} +10.5947 q^{76} +5.37228 q^{78} -0.147477 q^{79} +1.00000 q^{81} -5.48913 q^{82} +1.87953 q^{83} +3.46410 q^{84} +7.88316 q^{86} -3.16915 q^{87} -17.4891 q^{89} +17.1168 q^{91} +10.9783 q^{92} -3.37228 q^{93} +2.17448 q^{94} -5.84096 q^{96} +3.62772 q^{97} +0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 6 * q^4 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 14 * q^16 - 32 * q^23 + 10 * q^26 + 4 * q^27 - 2 * q^31 + 44 * q^34 + 6 * q^36 - 20 * q^37 - 10 * q^38 + 8 * q^42 + 12 * q^47 + 14 * q^48 - 14 * q^49 + 28 * q^53 - 4 * q^56 + 56 * q^58 + 16 * q^59 + 2 * q^64 + 10 * q^67 - 32 * q^69 + 20 * q^71 + 10 * q^78 + 4 * q^81 + 24 * q^82 + 66 * q^86 - 24 * q^89 + 34 * q^91 - 48 * q^92 - 2 * q^93 + 26 * q^97

Introduction

L-functions

Modular forms

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Database

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