LMFDB - Embedded newform 9075.2.a.ck.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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} +1.53919 q^{6} +0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} +1.53919 q^{6} +0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{9} +0.369102 q^{12} +6.97107 q^{13} +0.447480 q^{14} -4.60197 q^{16} +4.78765 q^{17} +1.53919 q^{18} -7.75872 q^{19} +0.290725 q^{21} -4.00000 q^{23} -2.51026 q^{24} +10.7298 q^{26} +1.00000 q^{27} +0.107307 q^{28} +7.41855 q^{29} +6.34017 q^{31} -2.06278 q^{32} +7.36910 q^{34} +0.369102 q^{36} +3.41855 q^{37} -11.9421 q^{38} +6.97107 q^{39} +7.41855 q^{41} +0.447480 q^{42} +0.290725 q^{43} -6.15676 q^{46} -5.26180 q^{47} -4.60197 q^{48} -6.91548 q^{49} +4.78765 q^{51} +2.57304 q^{52} -5.75872 q^{53} +1.53919 q^{54} -0.729794 q^{56} -7.75872 q^{57} +11.4186 q^{58} +3.60197 q^{59} +6.68035 q^{61} +9.75872 q^{62} +0.290725 q^{63} +6.02893 q^{64} -6.15676 q^{67} +1.76713 q^{68} -4.00000 q^{69} -5.07838 q^{71} -2.51026 q^{72} -1.12783 q^{73} +5.26180 q^{74} -2.86376 q^{76} +10.7298 q^{78} +0.921622 q^{79} +1.00000 q^{81} +11.4186 q^{82} +1.70928 q^{83} +0.107307 q^{84} +0.447480 q^{86} +7.41855 q^{87} +4.34017 q^{89} +2.02666 q^{91} -1.47641 q^{92} +6.34017 q^{93} -8.09890 q^{94} -2.06278 q^{96} -4.68035 q^{97} -10.6442 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} + 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} + 2 q^{19} + 8 q^{21} - 12 q^{23} + 9 q^{24} - 8 q^{26} + 3 q^{27} + 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34} + 5 q^{36} - 4 q^{37} - 6 q^{38} + 6 q^{39} + 8 q^{41} + 2 q^{42} + 8 q^{43} - 12 q^{46} - 8 q^{47} + 5 q^{48} + 11 q^{49} + 4 q^{51} - 26 q^{52} + 8 q^{53} + 3 q^{54} + 38 q^{56} + 2 q^{57} + 20 q^{58} - 8 q^{59} - 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} - 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} + 8 q^{74} + 18 q^{76} - 8 q^{78} + 6 q^{79} + 3 q^{81} + 20 q^{82} - 2 q^{83} + 12 q^{84} + 2 q^{86} + 8 q^{87} + 2 q^{89} + 8 q^{91} - 20 q^{92} + 8 q^{93} + 12 q^{94} + 11 q^{96} + 8 q^{97} - 29 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 + 5 * q^12 + 6 * q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 + 3 * q^18 + 2 * q^19 + 8 * q^21 - 12 * q^23 + 9 * q^24 - 8 * q^26 + 3 * q^27 + 12 * q^28 + 8 * q^29 + 8 * q^31 + 11 * q^32 + 26 * q^34 + 5 * q^36 - 4 * q^37 - 6 * q^38 + 6 * q^39 + 8 * q^41 + 2 * q^42 + 8 * q^43 - 12 * q^46 - 8 * q^47 + 5 * q^48 + 11 * q^49 + 4 * q^51 - 26 * q^52 + 8 * q^53 + 3 * q^54 + 38 * q^56 + 2 * q^57 + 20 * q^58 - 8 * q^59 - 2 * q^61 + 4 * q^62 + 8 * q^63 + 33 * q^64 - 12 * q^67 + 28 * q^68 - 12 * q^69 - 12 * q^71 + 9 * q^72 + 18 * q^73 + 8 * q^74 + 18 * q^76 - 8 * q^78 + 6 * q^79 + 3 * q^81 + 20 * q^82 - 2 * q^83 + 12 * q^84 + 2 * q^86 + 8 * q^87 + 2 * q^89 + 8 * q^91 - 20 * q^92 + 8 * q^93 + 12 * q^94 + 11 * q^96 + 8 * q^97 - 29 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.369102	0.184551
$5$	0	0
$5$	0	0
$6$	1.53919	0.628371
$7$	0.290725	0.109884	0.0549418	−	0.998490i	$-0.482503\pi$
$7$	0.290725	0.109884	0.0549418	+	0.998490i	$0.482503\pi$
$8$	−2.51026	−0.887511
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.369102	0.106551
$13$	6.97107	1.93343	0.966714	−	0.255861i	$-0.0823590\pi$
$13$	6.97107	1.93343	0.966714	+	0.255861i	$0.0823590\pi$
$14$	0.447480	0.119594
$15$	0	0
$16$	−4.60197	−1.15049
$17$	4.78765	1.16118	0.580588	−	0.814197i	$-0.302823\pi$
$17$	4.78765	1.16118	0.580588	+	0.814197i	$0.302823\pi$
$18$	1.53919	0.362790
$19$	−7.75872	−1.77997	−0.889987	−	0.455987i	$-0.849286\pi$
$19$	−7.75872	−1.77997	−0.889987	+	0.455987i	$0.849286\pi$
$20$	0	0
$21$	0.290725	0.0634413
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−2.51026	−0.512405
$25$	0	0
$26$	10.7298	2.10429
$27$	1.00000	0.192450
$28$	0.107307	0.0202791
$29$	7.41855	1.37759	0.688795	−	0.724956i	$-0.258140\pi$
$29$	7.41855	1.37759	0.688795	+	0.724956i	$0.258140\pi$
$30$	0	0
$31$	6.34017	1.13873	0.569364	−	0.822085i	$-0.307189\pi$
$31$	6.34017	1.13873	0.569364	+	0.822085i	$0.307189\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	7.36910	1.26379
$35$	0	0
$36$	0.369102	0.0615171
$37$	3.41855	0.562006	0.281003	−	0.959707i	$-0.409333\pi$
$37$	3.41855	0.562006	0.281003	+	0.959707i	$0.409333\pi$
$38$	−11.9421	−1.93727
$39$	6.97107	1.11626
$40$	0	0
$41$	7.41855	1.15858	0.579291	−	0.815120i	$-0.303329\pi$
$41$	7.41855	1.15858	0.579291	+	0.815120i	$0.303329\pi$
$42$	0.447480	0.0690477
$43$	0.290725	0.0443351	0.0221675	−	0.999754i	$-0.492943\pi$
$43$	0.290725	0.0443351	0.0221675	+	0.999754i	$0.492943\pi$
$44$	0	0
$45$	0	0
$46$	−6.15676	−0.907764
$47$	−5.26180	−0.767512	−0.383756	−	0.923435i	$-0.625370\pi$
$47$	−5.26180	−0.767512	−0.383756	+	0.923435i	$0.625370\pi$
$48$	−4.60197	−0.664237
$49$	−6.91548	−0.987926
$50$	0	0
$51$	4.78765	0.670406
$52$	2.57304	0.356816
$53$	−5.75872	−0.791022	−0.395511	−	0.918461i	$-0.629432\pi$
$53$	−5.75872	−0.791022	−0.395511	+	0.918461i	$0.629432\pi$
$54$	1.53919	0.209457
$55$	0	0
$56$	−0.729794	−0.0975229
$57$	−7.75872	−1.02767
$58$	11.4186	1.49933
$59$	3.60197	0.468936	0.234468	−	0.972124i	$-0.424665\pi$
$59$	3.60197	0.468936	0.234468	+	0.972124i	$0.424665\pi$
$60$	0	0
$61$	6.68035	0.855331	0.427665	−	0.903937i	$-0.359336\pi$
$61$	6.68035	0.855331	0.427665	+	0.903937i	$0.359336\pi$
$62$	9.75872	1.23936
$63$	0.290725	0.0366279
$64$	6.02893	0.753616
$65$	0	0
$66$	0	0
$67$	−6.15676	−0.752167	−0.376084	−	0.926586i	$-0.622729\pi$
$67$	−6.15676	−0.752167	−0.376084	+	0.926586i	$0.622729\pi$
$68$	1.76713	0.214296
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−5.07838	−0.602693	−0.301346	−	0.953515i	$-0.597436\pi$
$71$	−5.07838	−0.602693	−0.301346	+	0.953515i	$0.597436\pi$
$72$	−2.51026	−0.295837
$73$	−1.12783	−0.132002	−0.0660010	−	0.997820i	$-0.521024\pi$
$73$	−1.12783	−0.132002	−0.0660010	+	0.997820i	$0.521024\pi$
$74$	5.26180	0.611671
$75$	0	0
$76$	−2.86376	−0.328496
$77$	0	0
$78$	10.7298	1.21491
$79$	0.921622	0.103691	0.0518453	−	0.998655i	$-0.483490\pi$
$79$	0.921622	0.103691	0.0518453	+	0.998655i	$0.483490\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.4186	1.26097
$83$	1.70928	0.187617	0.0938087	−	0.995590i	$-0.470096\pi$
$83$	1.70928	0.187617	0.0938087	+	0.995590i	$0.470096\pi$
$84$	0.107307	0.0117082
$85$	0	0
$86$	0.447480	0.0482530
$87$	7.41855	0.795352
$88$	0	0
$89$	4.34017	0.460057	0.230029	−	0.973184i	$-0.426118\pi$
$89$	4.34017	0.460057	0.230029	+	0.973184i	$0.426118\pi$
$90$	0	0
$91$	2.02666	0.212452
$92$	−1.47641	−0.153926
$93$	6.34017	0.657445
$94$	−8.09890	−0.835337
$95$	0	0
$96$	−2.06278	−0.210532
$97$	−4.68035	−0.475217	−0.237609	−	0.971361i	$-0.576364\pi$
$97$	−4.68035	−0.475217	−0.237609	+	0.971361i	$0.576364\pi$
$98$	−10.6442	−1.07523
$99$	0	0
$100$	0	0
$101$	8.58145	0.853886	0.426943	−	0.904279i	$-0.359590\pi$
$101$	8.58145	0.853886	0.426943	+	0.904279i	$0.359590\pi$
$102$	7.36910	0.729650
$103$	−6.73820	−0.663935	−0.331968	−	0.943291i	$-0.607712\pi$
$103$	−6.73820	−0.663935	−0.331968	+	0.943291i	$0.607712\pi$
$104$	−17.4992	−1.71594
$105$	0	0
$106$	−8.86376	−0.860925
$107$	12.2329	1.18260	0.591298	−	0.806453i	$-0.298616\pi$
$107$	12.2329	1.18260	0.591298	+	0.806453i	$0.298616\pi$
$108$	0.369102	0.0355169
$109$	6.31351	0.604725	0.302362	−	0.953193i	$-0.402225\pi$
$109$	6.31351	0.604725	0.302362	+	0.953193i	$0.402225\pi$
$110$	0	0
$111$	3.41855	0.324474
$112$	−1.33791	−0.126420
$113$	16.4969	1.55190	0.775950	−	0.630794i	$-0.217271\pi$
$113$	16.4969	1.55190	0.775950	+	0.630794i	$0.217271\pi$
$114$	−11.9421	−1.11848
$115$	0	0
$116$	2.73820	0.254236
$117$	6.97107	0.644476
$118$	5.54411	0.510377
$119$	1.39189	0.127594
$120$	0	0
$121$	0	0
$122$	10.2823	0.930917
$123$	7.41855	0.668908
$124$	2.34017	0.210154
$125$	0	0
$126$	0.447480	0.0398647
$127$	4.87217	0.432336	0.216168	−	0.976356i	$-0.430644\pi$
$127$	4.87217	0.432336	0.216168	+	0.976356i	$0.430644\pi$
$128$	13.4052	1.18487
$129$	0.290725	0.0255969
$130$	0	0
$131$	8.68035	0.758405	0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035	0.758405	0.379203	+	0.925314i	$0.376198\pi$
$132$	0	0
$133$	−2.25565	−0.195590
$134$	−9.47641	−0.818637
$135$	0	0
$136$	−12.0183	−1.03056
$137$	12.5958	1.07613	0.538067	−	0.842902i	$-0.319155\pi$
$137$	12.5958	1.07613	0.538067	+	0.842902i	$0.319155\pi$
$138$	−6.15676	−0.524098
$139$	9.91548	0.841020	0.420510	−	0.907288i	$-0.361851\pi$
$139$	9.91548	0.841020	0.420510	+	0.907288i	$0.361851\pi$
$140$	0	0
$141$	−5.26180	−0.443123
$142$	−7.81658	−0.655953
$143$	0	0
$144$	−4.60197	−0.383497
$145$	0	0
$146$	−1.73594	−0.143667
$147$	−6.91548	−0.570379
$148$	1.26180	0.103719
$149$	1.26180	0.103370	0.0516851	−	0.998663i	$-0.483541\pi$
$149$	1.26180	0.103370	0.0516851	+	0.998663i	$0.483541\pi$
$150$	0	0
$151$	−1.60197	−0.130366	−0.0651832	−	0.997873i	$-0.520763\pi$
$151$	−1.60197	−0.130366	−0.0651832	+	0.997873i	$0.520763\pi$
$152$	19.4764	1.57975
$153$	4.78765	0.387059
$154$	0	0
$155$	0	0
$156$	2.57304	0.206008
$157$	7.10504	0.567044	0.283522	−	0.958966i	$-0.408497\pi$
$157$	7.10504	0.567044	0.283522	+	0.958966i	$0.408497\pi$
$158$	1.41855	0.112854
$159$	−5.75872	−0.456696
$160$	0	0
$161$	−1.16290	−0.0916492
$162$	1.53919	0.120930
$163$	22.9360	1.79649	0.898243	−	0.439499i	$-0.144844\pi$
$163$	22.9360	1.79649	0.898243	+	0.439499i	$0.144844\pi$
$164$	2.73820	0.213818
$165$	0	0
$166$	2.63090	0.204197
$167$	4.81432	0.372543	0.186271	−	0.982498i	$-0.440360\pi$
$167$	4.81432	0.372543	0.186271	+	0.982498i	$0.440360\pi$
$168$	−0.729794	−0.0563049
$169$	35.5958	2.73814
$170$	0	0
$171$	−7.75872	−0.593324
$172$	0.107307	0.00818209
$173$	−12.8865	−0.979746	−0.489873	−	0.871794i	$-0.662957\pi$
$173$	−12.8865	−0.979746	−0.489873	+	0.871794i	$0.662957\pi$
$174$	11.4186	0.865638
$175$	0	0
$176$	0	0
$177$	3.60197	0.270741
$178$	6.68035	0.500713
$179$	1.84324	0.137771	0.0688853	−	0.997625i	$-0.478056\pi$
$179$	1.84324	0.137771	0.0688853	+	0.997625i	$0.478056\pi$
$180$	0	0
$181$	10.2823	0.764278	0.382139	−	0.924105i	$-0.375187\pi$
$181$	10.2823	0.764278	0.382139	+	0.924105i	$0.375187\pi$
$182$	3.11942	0.231226
$183$	6.68035	0.493825
$184$	10.0410	0.740235
$185$	0	0
$186$	9.75872	0.715544
$187$	0	0
$188$	−1.94214	−0.141645
$189$	0.290725	0.0211471
$190$	0	0
$191$	−11.5174	−0.833373	−0.416687	−	0.909050i	$-0.636809\pi$
$191$	−11.5174	−0.833373	−0.416687	+	0.909050i	$0.636809\pi$
$192$	6.02893	0.435101
$193$	3.86603	0.278283	0.139141	−	0.990273i	$-0.455566\pi$
$193$	3.86603	0.278283	0.139141	+	0.990273i	$0.455566\pi$
$194$	−7.20394	−0.517212
$195$	0	0
$196$	−2.55252	−0.182323
$197$	−8.57304	−0.610804	−0.305402	−	0.952224i	$-0.598791\pi$
$197$	−8.57304	−0.610804	−0.305402	+	0.952224i	$0.598791\pi$
$198$	0	0
$199$	8.31351	0.589329	0.294665	−	0.955601i	$-0.404792\pi$
$199$	8.31351	0.589329	0.294665	+	0.955601i	$0.404792\pi$
$200$	0	0
$201$	−6.15676	−0.434264
$202$	13.2085	0.929345
$203$	2.15676	0.151375
$204$	1.76713	0.123724
$205$	0	0
$206$	−10.3714	−0.722608
$207$	−4.00000	−0.278019
$208$	−32.0806	−2.22439
$209$	0	0
$210$	0	0
$211$	−25.9155	−1.78410	−0.892048	−	0.451941i	$-0.850732\pi$
$211$	−25.9155	−1.78410	−0.892048	+	0.451941i	$0.850732\pi$
$212$	−2.12556	−0.145984
$213$	−5.07838	−0.347965
$214$	18.8287	1.28710
$215$	0	0
$216$	−2.51026	−0.170802
$217$	1.84324	0.125128
$218$	9.71769	0.658165
$219$	−1.12783	−0.0762114
$220$	0	0
$221$	33.3751	2.24505
$222$	5.26180	0.353149
$223$	9.62863	0.644781	0.322390	−	0.946607i	$-0.395514\pi$
$223$	9.62863	0.644781	0.322390	+	0.946607i	$0.395514\pi$
$224$	−0.599701	−0.0400692
$225$	0	0
$226$	25.3919	1.68904
$227$	−18.3896	−1.22056	−0.610281	−	0.792185i	$-0.708943\pi$
$227$	−18.3896	−1.22056	−0.610281	+	0.792185i	$0.708943\pi$
$228$	−2.86376	−0.189657
$229$	17.1506	1.13334	0.566672	−	0.823943i	$-0.308231\pi$
$229$	17.1506	1.13334	0.566672	+	0.823943i	$0.308231\pi$
$230$	0	0
$231$	0	0
$232$	−18.6225	−1.22263
$233$	−4.10731	−0.269079	−0.134539	−	0.990908i	$-0.542955\pi$
$233$	−4.10731	−0.269079	−0.134539	+	0.990908i	$0.542955\pi$
$234$	10.7298	0.701429
$235$	0	0
$236$	1.32950	0.0865428
$237$	0.921622	0.0598658
$238$	2.14238	0.138870
$239$	4.36683	0.282467	0.141234	−	0.989976i	$-0.454893\pi$
$239$	4.36683	0.282467	0.141234	+	0.989976i	$0.454893\pi$
$240$	0	0
$241$	5.20394	0.335215	0.167608	−	0.985854i	$-0.446396\pi$
$241$	5.20394	0.335215	0.167608	+	0.985854i	$0.446396\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	2.46573	0.157852
$245$	0	0
$246$	11.4186	0.728020
$247$	−54.0866	−3.44145
$248$	−15.9155	−1.01063
$249$	1.70928	0.108321
$250$	0	0
$251$	−8.28231	−0.522775	−0.261388	−	0.965234i	$-0.584180\pi$
$251$	−8.28231	−0.522775	−0.261388	+	0.965234i	$0.584180\pi$
$252$	0.107307	0.00675972
$253$	0	0
$254$	7.49920	0.470541
$255$	0	0
$256$	8.57531	0.535957
$257$	−11.0205	−0.687441	−0.343721	−	0.939072i	$-0.611687\pi$
$257$	−11.0205	−0.687441	−0.343721	+	0.939072i	$0.611687\pi$
$258$	0.447480	0.0278589
$259$	0.993857	0.0617553
$260$	0	0
$261$	7.41855	0.459197
$262$	13.3607	0.825426
$263$	−2.97107	−0.183204	−0.0916020	−	0.995796i	$-0.529199\pi$
$263$	−2.97107	−0.183204	−0.0916020	+	0.995796i	$0.529199\pi$
$264$	0	0
$265$	0	0
$266$	−3.47187	−0.212874
$267$	4.34017	0.265614
$268$	−2.27247	−0.138813
$269$	−15.8432	−0.965980	−0.482990	−	0.875626i	$-0.660449\pi$
$269$	−15.8432	−0.965980	−0.482990	+	0.875626i	$0.660449\pi$
$270$	0	0
$271$	6.28231	0.381623	0.190812	−	0.981627i	$-0.438888\pi$
$271$	6.28231	0.381623	0.190812	+	0.981627i	$0.438888\pi$
$272$	−22.0326	−1.33592
$273$	2.02666	0.122659
$274$	19.3874	1.17123
$275$	0	0
$276$	−1.47641	−0.0888694
$277$	5.12783	0.308101	0.154051	−	0.988063i	$-0.450768\pi$
$277$	5.12783	0.308101	0.154051	+	0.988063i	$0.450768\pi$
$278$	15.2618	0.915342
$279$	6.34017	0.379576
$280$	0	0
$281$	−21.8888	−1.30578	−0.652889	−	0.757454i	$-0.726443\pi$
$281$	−21.8888	−1.30578	−0.652889	+	0.757454i	$0.726443\pi$
$282$	−8.09890	−0.482282
$283$	25.9649	1.54345	0.771727	−	0.635954i	$-0.219393\pi$
$283$	25.9649	1.54345	0.771727	+	0.635954i	$0.219393\pi$
$284$	−1.87444	−0.111228
$285$	0	0
$286$	0	0
$287$	2.15676	0.127309
$288$	−2.06278	−0.121550
$289$	5.92162	0.348331
$290$	0	0
$291$	−4.68035	−0.274367
$292$	−0.416283	−0.0243611
$293$	10.4163	0.608526	0.304263	−	0.952588i	$-0.401590\pi$
$293$	10.4163	0.608526	0.304263	+	0.952588i	$0.401590\pi$
$294$	−10.6442	−0.620784
$295$	0	0
$296$	−8.58145	−0.498787
$297$	0	0
$298$	1.94214	0.112505
$299$	−27.8843	−1.61259
$300$	0	0
$301$	0.0845208	0.00487170
$302$	−2.46573	−0.141887
$303$	8.58145	0.492991
$304$	35.7054	2.04785
$305$	0	0
$306$	7.36910	0.421264
$307$	−29.0700	−1.65911	−0.829555	−	0.558425i	$-0.811406\pi$
$307$	−29.0700	−1.65911	−0.829555	+	0.558425i	$0.811406\pi$
$308$	0	0
$309$	−6.73820	−0.383323
$310$	0	0
$311$	5.44521	0.308770	0.154385	−	0.988011i	$-0.450660\pi$
$311$	5.44521	0.308770	0.154385	+	0.988011i	$0.450660\pi$
$312$	−17.4992	−0.990697
$313$	−25.0928	−1.41833	−0.709163	−	0.705044i	$-0.750927\pi$
$313$	−25.0928	−1.41833	−0.709163	+	0.705044i	$0.750927\pi$
$314$	10.9360	0.617154
$315$	0	0
$316$	0.340173	0.0191362
$317$	−21.1773	−1.18943	−0.594717	−	0.803935i	$-0.702736\pi$
$317$	−21.1773	−1.18943	−0.594717	+	0.803935i	$0.702736\pi$
$318$	−8.86376	−0.497055
$319$	0	0
$320$	0	0
$321$	12.2329	0.682772
$322$	−1.78992	−0.0997484
$323$	−37.1461	−2.06686
$324$	0.369102	0.0205057
$325$	0	0
$326$	35.3028	1.95524
$327$	6.31351	0.349138
$328$	−18.6225	−1.02825
$329$	−1.52973	−0.0843369
$330$	0	0
$331$	−6.70701	−0.368650	−0.184325	−	0.982865i	$-0.559010\pi$
$331$	−6.70701	−0.368650	−0.184325	+	0.982865i	$0.559010\pi$
$332$	0.630898	0.0346250
$333$	3.41855	0.187335
$334$	7.41014	0.405465
$335$	0	0
$336$	−1.33791	−0.0729887
$337$	23.7503	1.29376	0.646881	−	0.762591i	$-0.276073\pi$
$337$	23.7503	1.29376	0.646881	+	0.762591i	$0.276073\pi$
$338$	54.7887	2.98011
$339$	16.4969	0.895990
$340$	0	0
$341$	0	0
$342$	−11.9421	−0.645757
$343$	−4.04557	−0.218440
$344$	−0.729794	−0.0393479
$345$	0	0
$346$	−19.8348	−1.06633
$347$	−31.1689	−1.67323	−0.836616	−	0.547790i	$-0.815469\pi$
$347$	−31.1689	−1.67323	−0.836616	+	0.547790i	$0.815469\pi$
$348$	2.73820	0.146783
$349$	−11.0472	−0.591342	−0.295671	−	0.955290i	$-0.595543\pi$
$349$	−11.0472	−0.591342	−0.295671	+	0.955290i	$0.595543\pi$
$350$	0	0
$351$	6.97107	0.372088
$352$	0	0
$353$	27.4329	1.46011	0.730054	−	0.683390i	$-0.239495\pi$
$353$	27.4329	1.46011	0.730054	+	0.683390i	$0.239495\pi$
$354$	5.54411	0.294666
$355$	0	0
$356$	1.60197	0.0849041
$357$	1.39189	0.0736666
$358$	2.83710	0.149945
$359$	19.1506	1.01073	0.505365	−	0.862905i	$-0.331358\pi$
$359$	19.1506	1.01073	0.505365	+	0.862905i	$0.331358\pi$
$360$	0	0
$361$	41.1978	2.16830
$362$	15.8264	0.831818
$363$	0	0
$364$	0.748046	0.0392083
$365$	0	0
$366$	10.2823	0.537465
$367$	14.5692	0.760504	0.380252	−	0.924883i	$-0.375837\pi$
$367$	14.5692	0.760504	0.380252	+	0.924883i	$0.375837\pi$
$368$	18.4079	0.959577
$369$	7.41855	0.386194
$370$	0	0
$371$	−1.67420	−0.0869203
$372$	2.34017	0.121332
$373$	−8.81432	−0.456388	−0.228194	−	0.973616i	$-0.573282\pi$
$373$	−8.81432	−0.456388	−0.228194	+	0.973616i	$0.573282\pi$
$374$	0	0
$375$	0	0
$376$	13.2085	0.681175
$377$	51.7152	2.66347
$378$	0.447480	0.0230159
$379$	−23.5174	−1.20801	−0.604005	−	0.796980i	$-0.706429\pi$
$379$	−23.5174	−1.20801	−0.604005	+	0.796980i	$0.706429\pi$
$380$	0	0
$381$	4.87217	0.249609
$382$	−17.7275	−0.907019
$383$	19.3028	0.986329	0.493164	−	0.869936i	$-0.335840\pi$
$383$	19.3028	0.986329	0.493164	+	0.869936i	$0.335840\pi$
$384$	13.4052	0.684082
$385$	0	0
$386$	5.95055	0.302875
$387$	0.290725	0.0147784
$388$	−1.72753	−0.0877019
$389$	−20.0410	−1.01612	−0.508060	−	0.861321i	$-0.669637\pi$
$389$	−20.0410	−1.01612	−0.508060	+	0.861321i	$0.669637\pi$
$390$	0	0
$391$	−19.1506	−0.968488
$392$	17.3596	0.876795
$393$	8.68035	0.437866
$394$	−13.1955	−0.664781
$395$	0	0
$396$	0	0
$397$	33.3607	1.67433	0.837163	−	0.546954i	$-0.184213\pi$
$397$	33.3607	1.67433	0.837163	+	0.546954i	$0.184213\pi$
$398$	12.7961	0.641409
$399$	−2.25565	−0.112924
$400$	0	0
$401$	−37.6475	−1.88003	−0.940014	−	0.341135i	$-0.889189\pi$
$401$	−37.6475	−1.88003	−0.940014	+	0.341135i	$0.889189\pi$
$402$	−9.47641	−0.472640
$403$	44.1978	2.20165
$404$	3.16743	0.157586
$405$	0	0
$406$	3.31965	0.164752
$407$	0	0
$408$	−12.0183	−0.594992
$409$	25.2039	1.24625	0.623127	−	0.782120i	$-0.285862\pi$
$409$	25.2039	1.24625	0.623127	+	0.782120i	$0.285862\pi$
$410$	0	0
$411$	12.5958	0.621306
$412$	−2.48709	−0.122530
$413$	1.04718	0.0515284
$414$	−6.15676	−0.302588
$415$	0	0
$416$	−14.3798	−0.705027
$417$	9.91548	0.485563
$418$	0	0
$419$	17.8432	0.871700	0.435850	−	0.900019i	$-0.356448\pi$
$419$	17.8432	0.871700	0.435850	+	0.900019i	$0.356448\pi$
$420$	0	0
$421$	−11.8120	−0.575684	−0.287842	−	0.957678i	$-0.592938\pi$
$421$	−11.8120	−0.575684	−0.287842	+	0.957678i	$0.592938\pi$
$422$	−39.8888	−1.94176
$423$	−5.26180	−0.255837
$424$	14.4559	0.702040
$425$	0	0
$426$	−7.81658	−0.378715
$427$	1.94214	0.0939868
$428$	4.51518	0.218249
$429$	0	0
$430$	0	0
$431$	−16.6803	−0.803464	−0.401732	−	0.915757i	$-0.631592\pi$
$431$	−16.6803	−0.803464	−0.401732	+	0.915757i	$0.631592\pi$
$432$	−4.60197	−0.221412
$433$	28.3135	1.36066	0.680330	−	0.732906i	$-0.261836\pi$
$433$	28.3135	1.36066	0.680330	+	0.732906i	$0.261836\pi$
$434$	2.83710	0.136185
$435$	0	0
$436$	2.33033	0.111603
$437$	31.0349	1.48460
$438$	−1.73594	−0.0829463
$439$	27.4452	1.30989	0.654944	−	0.755677i	$-0.272692\pi$
$439$	27.4452	1.30989	0.654944	+	0.755677i	$0.272692\pi$
$440$	0	0
$441$	−6.91548	−0.329309
$442$	51.3705	2.44345
$443$	−0.412408	−0.0195941	−0.00979704	−	0.999952i	$-0.503119\pi$
$443$	−0.412408	−0.0195941	−0.00979704	+	0.999952i	$0.503119\pi$
$444$	1.26180	0.0598822
$445$	0	0
$446$	14.8203	0.701761
$447$	1.26180	0.0596809
$448$	1.75276	0.0828100
$449$	1.33403	0.0629568	0.0314784	−	0.999504i	$-0.489978\pi$
$449$	1.33403	0.0629568	0.0314784	+	0.999504i	$0.489978\pi$
$450$	0	0
$451$	0	0
$452$	6.08906	0.286405
$453$	−1.60197	−0.0752670
$454$	−28.3051	−1.32842
$455$	0	0
$456$	19.4764	0.912066
$457$	11.6514	0.545030	0.272515	−	0.962151i	$-0.412145\pi$
$457$	11.6514	0.545030	0.272515	+	0.962151i	$0.412145\pi$
$458$	26.3980	1.23350
$459$	4.78765	0.223469
$460$	0	0
$461$	2.05786	0.0958440	0.0479220	−	0.998851i	$-0.484740\pi$
$461$	2.05786	0.0958440	0.0479220	+	0.998851i	$0.484740\pi$
$462$	0	0
$463$	−28.7792	−1.33748	−0.668742	−	0.743494i	$-0.733167\pi$
$463$	−28.7792	−1.33748	−0.668742	+	0.743494i	$0.733167\pi$
$464$	−34.1399	−1.58491
$465$	0	0
$466$	−6.32192	−0.292857
$467$	−1.84324	−0.0852952	−0.0426476	−	0.999090i	$-0.513579\pi$
$467$	−1.84324	−0.0852952	−0.0426476	+	0.999090i	$0.513579\pi$
$468$	2.57304	0.118939
$469$	−1.78992	−0.0826509
$470$	0	0
$471$	7.10504	0.327383
$472$	−9.04187	−0.416186
$473$	0	0
$474$	1.41855	0.0651562
$475$	0	0
$476$	0.513749	0.0235477
$477$	−5.75872	−0.263674
$478$	6.72138	0.307429
$479$	−26.8371	−1.22622	−0.613109	−	0.789998i	$-0.710081\pi$
$479$	−26.8371	−1.22622	−0.613109	+	0.789998i	$0.710081\pi$
$480$	0	0
$481$	23.8310	1.08660
$482$	8.00984	0.364838
$483$	−1.16290	−0.0529137
$484$	0	0
$485$	0	0
$486$	1.53919	0.0698190
$487$	−28.5646	−1.29439	−0.647193	−	0.762326i	$-0.724057\pi$
$487$	−28.5646	−1.29439	−0.647193	+	0.762326i	$0.724057\pi$
$488$	−16.7694	−0.759115
$489$	22.9360	1.03720
$490$	0	0
$491$	−25.9877	−1.17281	−0.586405	−	0.810018i	$-0.699457\pi$
$491$	−25.9877	−1.17281	−0.586405	+	0.810018i	$0.699457\pi$
$492$	2.73820	0.123448
$493$	35.5174	1.59963
$494$	−83.2495	−3.74557
$495$	0	0
$496$	−29.1773	−1.31010
$497$	−1.47641	−0.0662260
$498$	2.63090	0.117893
$499$	−27.5174	−1.23185	−0.615925	−	0.787805i	$-0.711218\pi$
$499$	−27.5174	−1.23185	−0.615925	+	0.787805i	$0.711218\pi$
$500$	0	0
$501$	4.81432	0.215088
$502$	−12.7480	−0.568973
$503$	22.6576	1.01025	0.505125	−	0.863046i	$-0.331446\pi$
$503$	22.6576	1.01025	0.505125	+	0.863046i	$0.331446\pi$
$504$	−0.729794	−0.0325076
$505$	0	0
$506$	0	0
$507$	35.5958	1.58087
$508$	1.79833	0.0797880
$509$	−27.8432	−1.23413	−0.617065	−	0.786912i	$-0.711678\pi$
$509$	−27.8432	−1.23413	−0.617065	+	0.786912i	$0.711678\pi$
$510$	0	0
$511$	−0.327887	−0.0145049
$512$	−13.6114	−0.601546
$513$	−7.75872	−0.342556
$514$	−16.9627	−0.748191
$515$	0	0
$516$	0.107307	0.00472393
$517$	0	0
$518$	1.52973	0.0672126
$519$	−12.8865	−0.565657
$520$	0	0
$521$	29.7009	1.30122	0.650609	−	0.759413i	$-0.274514\pi$
$521$	29.7009	1.30122	0.650609	+	0.759413i	$0.274514\pi$
$522$	11.4186	0.499776
$523$	24.7565	1.08252	0.541262	−	0.840854i	$-0.317947\pi$
$523$	24.7565	1.08252	0.541262	+	0.840854i	$0.317947\pi$
$524$	3.20394	0.139965
$525$	0	0
$526$	−4.57304	−0.199394
$527$	30.3545	1.32226
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	3.60197	0.156312
$532$	−0.832567	−0.0360963
$533$	51.7152	2.24004
$534$	6.68035	0.289087
$535$	0	0
$536$	15.4551	0.667557
$537$	1.84324	0.0795419
$538$	−24.3857	−1.05134
$539$	0	0
$540$	0	0
$541$	12.5236	0.538431	0.269216	−	0.963080i	$-0.413236\pi$
$541$	12.5236	0.538431	0.269216	+	0.963080i	$0.413236\pi$
$542$	9.66967	0.415348
$543$	10.2823	0.441256
$544$	−9.87587	−0.423425
$545$	0	0
$546$	3.11942	0.133499
$547$	−14.2784	−0.610502	−0.305251	−	0.952272i	$-0.598740\pi$
$547$	−14.2784	−0.610502	−0.305251	+	0.952272i	$0.598740\pi$
$548$	4.64915	0.198602
$549$	6.68035	0.285110
$550$	0	0
$551$	−57.5585	−2.45207
$552$	10.0410	0.427375
$553$	0.267938	0.0113939
$554$	7.89269	0.335328
$555$	0	0
$556$	3.65983	0.155211
$557$	−3.57918	−0.151655	−0.0758274	−	0.997121i	$-0.524160\pi$
$557$	−3.57918	−0.151655	−0.0758274	+	0.997121i	$0.524160\pi$
$558$	9.75872	0.413120
$559$	2.02666	0.0857187
$560$	0	0
$561$	0	0
$562$	−33.6910	−1.42117
$563$	28.9588	1.22047	0.610234	−	0.792222i	$-0.291076\pi$
$563$	28.9588	1.22047	0.610234	+	0.792222i	$0.291076\pi$
$564$	−1.94214	−0.0817789
$565$	0	0
$566$	39.9649	1.67985
$567$	0.290725	0.0122093
$568$	12.7480	0.534896
$569$	25.8264	1.08270	0.541350	−	0.840797i	$-0.317913\pi$
$569$	25.8264	1.08270	0.541350	+	0.840797i	$0.317913\pi$
$570$	0	0
$571$	−17.1194	−0.716425	−0.358213	−	0.933640i	$-0.616614\pi$
$571$	−17.1194	−0.716425	−0.358213	+	0.933640i	$0.616614\pi$
$572$	0	0
$573$	−11.5174	−0.481148
$574$	3.31965	0.138560
$575$	0	0
$576$	6.02893	0.251205
$577$	−22.5692	−0.939567	−0.469783	−	0.882782i	$-0.655668\pi$
$577$	−22.5692	−0.939567	−0.469783	+	0.882782i	$0.655668\pi$
$578$	9.11450	0.379113
$579$	3.86603	0.160667
$580$	0	0
$581$	0.496928	0.0206161
$582$	−7.20394	−0.298613
$583$	0	0
$584$	2.83114	0.117153
$585$	0	0
$586$	16.0326	0.662302
$587$	−3.63317	−0.149957	−0.0749784	−	0.997185i	$-0.523889\pi$
$587$	−3.63317	−0.149957	−0.0749784	+	0.997185i	$0.523889\pi$
$588$	−2.55252	−0.105264
$589$	−49.1917	−2.02691
$590$	0	0
$591$	−8.57304	−0.352648
$592$	−15.7321	−0.646584
$593$	12.3051	0.505310	0.252655	−	0.967556i	$-0.418696\pi$
$593$	12.3051	0.505310	0.252655	+	0.967556i	$0.418696\pi$
$594$	0	0
$595$	0	0
$596$	0.465732	0.0190771
$597$	8.31351	0.340249
$598$	−42.9192	−1.75510
$599$	19.6865	0.804368	0.402184	−	0.915559i	$-0.368251\pi$
$599$	19.6865	0.804368	0.402184	+	0.915559i	$0.368251\pi$
$600$	0	0
$601$	−25.8843	−1.05584	−0.527921	−	0.849294i	$-0.677028\pi$
$601$	−25.8843	−1.05584	−0.527921	+	0.849294i	$0.677028\pi$
$602$	0.130094	0.00530222
$603$	−6.15676	−0.250722
$604$	−0.591290	−0.0240593
$605$	0	0
$606$	13.2085	0.536557
$607$	−41.5357	−1.68588	−0.842941	−	0.538006i	$-0.819178\pi$
$607$	−41.5357	−1.68588	−0.842941	+	0.538006i	$0.819178\pi$
$608$	16.0045	0.649070
$609$	2.15676	0.0873961
$610$	0	0
$611$	−36.6803	−1.48393
$612$	1.76713	0.0714322
$613$	−20.6453	−0.833855	−0.416927	−	0.908940i	$-0.636893\pi$
$613$	−20.6453	−0.833855	−0.416927	+	0.908940i	$0.636893\pi$
$614$	−44.7442	−1.80573
$615$	0	0
$616$	0	0
$617$	−8.69472	−0.350036	−0.175018	−	0.984565i	$-0.555998\pi$
$617$	−8.69472	−0.350036	−0.175018	+	0.984565i	$0.555998\pi$
$618$	−10.3714	−0.417198
$619$	2.65368	0.106661	0.0533303	−	0.998577i	$-0.483016\pi$
$619$	2.65368	0.106661	0.0533303	+	0.998577i	$0.483016\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	8.38121	0.336056
$623$	1.26180	0.0505528
$624$	−32.0806	−1.28425
$625$	0	0
$626$	−38.6225	−1.54367
$627$	0	0
$628$	2.62249	0.104649
$629$	16.3668	0.652588
$630$	0	0
$631$	−10.6393	−0.423544	−0.211772	−	0.977319i	$-0.567923\pi$
$631$	−10.6393	−0.423544	−0.211772	+	0.977319i	$0.567923\pi$
$632$	−2.31351	−0.0920265
$633$	−25.9155	−1.03005
$634$	−32.5958	−1.29455
$635$	0	0
$636$	−2.12556	−0.0842839
$637$	−48.2083	−1.91008
$638$	0	0
$639$	−5.07838	−0.200898
$640$	0	0
$641$	−35.8576	−1.41629	−0.708145	−	0.706067i	$-0.750468\pi$
$641$	−35.8576	−1.41629	−0.708145	+	0.706067i	$0.750468\pi$
$642$	18.8287	0.743109
$643$	−12.2146	−0.481697	−0.240849	−	0.970563i	$-0.577426\pi$
$643$	−12.2146	−0.481697	−0.240849	+	0.970563i	$0.577426\pi$
$644$	−0.429229	−0.0169140
$645$	0	0
$646$	−57.1748	−2.24951
$647$	−25.9421	−1.01989	−0.509945	−	0.860207i	$-0.670334\pi$
$647$	−25.9421	−1.01989	−0.509945	+	0.860207i	$0.670334\pi$
$648$	−2.51026	−0.0986123
$649$	0	0
$650$	0	0
$651$	1.84324	0.0722424
$652$	8.46573	0.331544
$653$	14.3402	0.561174	0.280587	−	0.959829i	$-0.409471\pi$
$653$	14.3402	0.561174	0.280587	+	0.959829i	$0.409471\pi$
$654$	9.71769	0.379992
$655$	0	0
$656$	−34.1399	−1.33294
$657$	−1.12783	−0.0440007
$658$	−2.35455	−0.0917899
$659$	−6.52359	−0.254123	−0.127062	−	0.991895i	$-0.540555\pi$
$659$	−6.52359	−0.254123	−0.127062	+	0.991895i	$0.540555\pi$
$660$	0	0
$661$	3.16290	0.123022	0.0615112	−	0.998106i	$-0.480408\pi$
$661$	3.16290	0.123022	0.0615112	+	0.998106i	$0.480408\pi$
$662$	−10.3234	−0.401228
$663$	33.3751	1.29618
$664$	−4.29072	−0.166512
$665$	0	0
$666$	5.26180	0.203890
$667$	−29.6742	−1.14899
$668$	1.77698	0.0687532
$669$	9.62863	0.372264
$670$	0	0
$671$	0	0
$672$	−0.599701	−0.0231340
$673$	−18.4885	−0.712680	−0.356340	−	0.934356i	$-0.615976\pi$
$673$	−18.4885	−0.712680	−0.356340	+	0.934356i	$0.615976\pi$
$674$	36.5562	1.40809
$675$	0	0
$676$	13.1385	0.505327
$677$	15.0966	0.580211	0.290105	−	0.956995i	$-0.406310\pi$
$677$	15.0966	0.580211	0.290105	+	0.956995i	$0.406310\pi$
$678$	25.3919	0.975170
$679$	−1.36069	−0.0522186
$680$	0	0
$681$	−18.3896	−0.704692
$682$	0	0
$683$	12.4657	0.476988	0.238494	−	0.971144i	$-0.423346\pi$
$683$	12.4657	0.476988	0.238494	+	0.971144i	$0.423346\pi$
$684$	−2.86376	−0.109499
$685$	0	0
$686$	−6.22690	−0.237744
$687$	17.1506	0.654337
$688$	−1.33791	−0.0510072
$689$	−40.1445	−1.52938
$690$	0	0
$691$	−30.7214	−1.16870	−0.584348	−	0.811503i	$-0.698650\pi$
$691$	−30.7214	−1.16870	−0.584348	+	0.811503i	$0.698650\pi$
$692$	−4.75646	−0.180813
$693$	0	0
$694$	−47.9748	−1.82110
$695$	0	0
$696$	−18.6225	−0.705884
$697$	35.5174	1.34532
$698$	−17.0037	−0.643599
$699$	−4.10731	−0.155353
$700$	0	0
$701$	−17.9955	−0.679679	−0.339840	−	0.940483i	$-0.610373\pi$
$701$	−17.9955	−0.679679	−0.339840	+	0.940483i	$0.610373\pi$
$702$	10.7298	0.404970
$703$	−26.5236	−1.00036
$704$	0	0
$705$	0	0
$706$	42.2245	1.58914
$707$	2.49484	0.0938281
$708$	1.32950	0.0499655
$709$	23.5897	0.885929	0.442965	−	0.896539i	$-0.353927\pi$
$709$	23.5897	0.885929	0.442965	+	0.896539i	$0.353927\pi$
$710$	0	0
$711$	0.921622	0.0345635
$712$	−10.8950	−0.408306
$713$	−25.3607	−0.949765
$714$	2.14238	0.0801765
$715$	0	0
$716$	0.680346	0.0254257
$717$	4.36683	0.163082
$718$	29.4764	1.10005
$719$	24.5646	0.916106	0.458053	−	0.888925i	$-0.348547\pi$
$719$	24.5646	0.916106	0.458053	+	0.888925i	$0.348547\pi$
$720$	0	0
$721$	−1.95896	−0.0729556
$722$	63.4112	2.35992
$723$	5.20394	0.193536
$724$	3.79523	0.141048
$725$	0	0
$726$	0	0
$727$	−8.51130	−0.315667	−0.157833	−	0.987466i	$-0.550451\pi$
$727$	−8.51130	−0.315667	−0.157833	+	0.987466i	$0.550451\pi$
$728$	−5.08745	−0.188553
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.39189	0.0514809
$732$	2.46573	0.0911361
$733$	−29.8615	−1.10296	−0.551480	−	0.834188i	$-0.685937\pi$
$733$	−29.8615	−1.10296	−0.551480	+	0.834188i	$0.685937\pi$
$734$	22.4247	0.827711
$735$	0	0
$736$	8.25112	0.304140
$737$	0	0
$738$	11.4186	0.420323
$739$	36.7526	1.35197	0.675983	−	0.736917i	$-0.263719\pi$
$739$	36.7526	1.35197	0.675983	+	0.736917i	$0.263719\pi$
$740$	0	0
$741$	−54.0866	−1.98692
$742$	−2.57691	−0.0946015
$743$	−2.17501	−0.0797933	−0.0398966	−	0.999204i	$-0.512703\pi$
$743$	−2.17501	−0.0797933	−0.0398966	+	0.999204i	$0.512703\pi$
$744$	−15.9155	−0.583490
$745$	0	0
$746$	−13.5669	−0.496719
$747$	1.70928	0.0625391
$748$	0	0
$749$	3.55640	0.129948
$750$	0	0
$751$	−44.4580	−1.62229	−0.811147	−	0.584842i	$-0.801157\pi$
$751$	−44.4580	−1.62229	−0.811147	+	0.584842i	$0.801157\pi$
$752$	24.2146	0.883016
$753$	−8.28231	−0.301824
$754$	79.5995	2.89884
$755$	0	0
$756$	0.107307	0.00390272
$757$	26.4247	0.960422	0.480211	−	0.877153i	$-0.340560\pi$
$757$	26.4247	0.960422	0.480211	+	0.877153i	$0.340560\pi$
$758$	−36.1978	−1.31476
$759$	0	0
$760$	0	0
$761$	27.6163	1.00109	0.500546	−	0.865710i	$-0.333133\pi$
$761$	27.6163	1.00109	0.500546	+	0.865710i	$0.333133\pi$
$762$	7.49920	0.271667
$763$	1.83549	0.0664493
$764$	−4.25112	−0.153800
$765$	0	0
$766$	29.7107	1.07349
$767$	25.1096	0.906654
$768$	8.57531	0.309435
$769$	−27.8432	−1.00405	−0.502027	−	0.864852i	$-0.667412\pi$
$769$	−27.8432	−1.00405	−0.502027	+	0.864852i	$0.667412\pi$
$770$	0	0
$771$	−11.0205	−0.396894
$772$	1.42696	0.0513575
$773$	18.0267	0.648374	0.324187	−	0.945993i	$-0.394909\pi$
$773$	18.0267	0.648374	0.324187	+	0.945993i	$0.394909\pi$
$774$	0.447480	0.0160843
$775$	0	0
$776$	11.7489	0.421760
$777$	0.993857	0.0356544
$778$	−30.8469	−1.10592
$779$	−57.5585	−2.06225
$780$	0	0
$781$	0	0
$782$	−29.4764	−1.05407
$783$	7.41855	0.265117
$784$	31.8248	1.13660
$785$	0	0
$786$	13.3607	0.476560
$787$	35.6391	1.27040	0.635199	−	0.772349i	$-0.280918\pi$
$787$	35.6391	1.27040	0.635199	+	0.772349i	$0.280918\pi$
$788$	−3.16433	−0.112725
$789$	−2.97107	−0.105773
$790$	0	0
$791$	4.79606	0.170528
$792$	0	0
$793$	46.5692	1.65372
$794$	51.3484	1.82229
$795$	0	0
$796$	3.06854	0.108761
$797$	4.97948	0.176382	0.0881911	−	0.996104i	$-0.471891\pi$
$797$	4.97948	0.176382	0.0881911	+	0.996104i	$0.471891\pi$
$798$	−3.47187	−0.122903
$799$	−25.1917	−0.891217
$800$	0	0
$801$	4.34017	0.153352
$802$	−57.9467	−2.04617
$803$	0	0
$804$	−2.27247	−0.0801439
$805$	0	0
$806$	68.0288	2.39621
$807$	−15.8432	−0.557709
$808$	−21.5417	−0.757833
$809$	−31.6697	−1.11345	−0.556723	−	0.830698i	$-0.687942\pi$
$809$	−31.6697	−1.11345	−0.556723	+	0.830698i	$0.687942\pi$
$810$	0	0
$811$	21.7998	0.765493	0.382747	−	0.923853i	$-0.374978\pi$
$811$	21.7998	0.765493	0.382747	+	0.923853i	$0.374978\pi$
$812$	0.796064	0.0279364
$813$	6.28231	0.220330
$814$	0	0
$815$	0	0
$816$	−22.0326	−0.771296
$817$	−2.25565	−0.0789153
$818$	38.7936	1.35639
$819$	2.02666	0.0708173
$820$	0	0
$821$	10.9360	0.381669	0.190834	−	0.981622i	$-0.438881\pi$
$821$	10.9360	0.381669	0.190834	+	0.981622i	$0.438881\pi$
$822$	19.3874	0.676212
$823$	−45.9709	−1.60244	−0.801222	−	0.598367i	$-0.795817\pi$
$823$	−45.9709	−1.60244	−0.801222	+	0.598367i	$0.795817\pi$
$824$	16.9146	0.589249
$825$	0	0
$826$	1.61181	0.0560820
$827$	−22.9711	−0.798782	−0.399391	−	0.916781i	$-0.630778\pi$
$827$	−22.9711	−0.798782	−0.399391	+	0.916781i	$0.630778\pi$
$828$	−1.47641	−0.0513088
$829$	−48.1543	−1.67247	−0.836234	−	0.548373i	$-0.815248\pi$
$829$	−48.1543	−1.67247	−0.836234	+	0.548373i	$0.815248\pi$
$830$	0	0
$831$	5.12783	0.177882
$832$	42.0281	1.45706
$833$	−33.1089	−1.14716
$834$	15.2618	0.528473
$835$	0	0
$836$	0	0
$837$	6.34017	0.219148
$838$	27.4641	0.948732
$839$	−17.9278	−0.618935	−0.309468	−	0.950910i	$-0.600151\pi$
$839$	−17.9278	−0.618935	−0.309468	+	0.950910i	$0.600151\pi$
$840$	0	0
$841$	26.0349	0.897755
$842$	−18.1810	−0.626558
$843$	−21.8888	−0.753891
$844$	−9.56547	−0.329257
$845$	0	0
$846$	−8.09890	−0.278446
$847$	0	0
$848$	26.5015	0.910064
$849$	25.9649	0.891114
$850$	0	0
$851$	−13.6742	−0.468746
$852$	−1.87444	−0.0642173
$853$	−9.54799	−0.326917	−0.163458	−	0.986550i	$-0.552265\pi$
$853$	−9.54799	−0.326917	−0.163458	+	0.986550i	$0.552265\pi$
$854$	2.98932	0.102292
$855$	0	0
$856$	−30.7077	−1.04957
$857$	10.5776	0.361323	0.180662	−	0.983545i	$-0.442176\pi$
$857$	10.5776	0.361323	0.180662	+	0.983545i	$0.442176\pi$
$858$	0	0
$859$	−31.3340	−1.06910	−0.534552	−	0.845136i	$-0.679520\pi$
$859$	−31.3340	−1.06910	−0.534552	+	0.845136i	$0.679520\pi$
$860$	0	0
$861$	2.15676	0.0735020
$862$	−25.6742	−0.874467
$863$	−56.7792	−1.93279	−0.966394	−	0.257066i	$-0.917244\pi$
$863$	−56.7792	−1.93279	−0.966394	+	0.257066i	$0.917244\pi$
$864$	−2.06278	−0.0701772
$865$	0	0
$866$	43.5798	1.48090
$867$	5.92162	0.201109
$868$	0.680346	0.0230924
$869$	0	0
$870$	0	0
$871$	−42.9192	−1.45426
$872$	−15.8486	−0.536700
$873$	−4.68035	−0.158406
$874$	47.7686	1.61580
$875$	0	0
$876$	−0.416283	−0.0140649
$877$	5.87832	0.198497	0.0992483	−	0.995063i	$-0.468356\pi$
$877$	5.87832	0.198497	0.0992483	+	0.995063i	$0.468356\pi$
$878$	42.2434	1.42564
$879$	10.4163	0.351333
$880$	0	0
$881$	8.21008	0.276605	0.138302	−	0.990390i	$-0.455835\pi$
$881$	8.21008	0.276605	0.138302	+	0.990390i	$0.455835\pi$
$882$	−10.6442	−0.358410
$883$	27.0349	0.909797	0.454898	−	0.890543i	$-0.349676\pi$
$883$	27.0349	0.909797	0.454898	+	0.890543i	$0.349676\pi$
$884$	12.3188	0.414327
$885$	0	0
$886$	−0.634773	−0.0213256
$887$	−48.7442	−1.63667	−0.818335	−	0.574742i	$-0.805102\pi$
$887$	−48.7442	−1.63667	−0.818335	+	0.574742i	$0.805102\pi$
$888$	−8.58145	−0.287975
$889$	1.41646	0.0475066
$890$	0	0
$891$	0	0
$892$	3.55395	0.118995
$893$	40.8248	1.36615
$894$	1.94214	0.0649549
$895$	0	0
$896$	3.89723	0.130197
$897$	−27.8843	−0.931029
$898$	2.05332	0.0685203
$899$	47.0349	1.56870
$900$	0	0
$901$	−27.5708	−0.918516
$902$	0	0
$903$	0.0845208	0.00281268
$904$	−41.4116	−1.37733
$905$	0	0
$906$	−2.46573	−0.0819184
$907$	32.4657	1.07801	0.539003	−	0.842304i	$-0.318801\pi$
$907$	32.4657	1.07801	0.539003	+	0.842304i	$0.318801\pi$
$908$	−6.78765	−0.225256
$909$	8.58145	0.284629
$910$	0	0
$911$	−4.84939	−0.160667	−0.0803337	−	0.996768i	$-0.525599\pi$
$911$	−4.84939	−0.160667	−0.0803337	+	0.996768i	$0.525599\pi$
$912$	35.7054	1.18232
$913$	0	0
$914$	17.9337	0.593195
$915$	0	0
$916$	6.33033	0.209160
$917$	2.52359	0.0833363
$918$	7.36910	0.243217
$919$	14.3980	0.474947	0.237474	−	0.971394i	$-0.423681\pi$
$919$	14.3980	0.474947	0.237474	+	0.971394i	$0.423681\pi$
$920$	0	0
$921$	−29.0700	−0.957888
$922$	3.16743	0.104314
$923$	−35.4017	−1.16526
$924$	0	0
$925$	0	0
$926$	−44.2967	−1.45568
$927$	−6.73820	−0.221312
$928$	−15.3028	−0.502340
$929$	−11.1629	−0.366243	−0.183121	−	0.983090i	$-0.558620\pi$
$929$	−11.1629	−0.366243	−0.183121	+	0.983090i	$0.558620\pi$
$930$	0	0
$931$	53.6553	1.75848
$932$	−1.51602	−0.0496588
$933$	5.44521	0.178268
$934$	−2.83710	−0.0928328
$935$	0	0
$936$	−17.4992	−0.571979
$937$	48.6453	1.58917	0.794586	−	0.607152i	$-0.207688\pi$
$937$	48.6453	1.58917	0.794586	+	0.607152i	$0.207688\pi$
$938$	−2.75503	−0.0899548
$939$	−25.0928	−0.818871
$940$	0	0
$941$	−16.4124	−0.535029	−0.267515	−	0.963554i	$-0.586202\pi$
$941$	−16.4124	−0.535029	−0.267515	+	0.963554i	$0.586202\pi$
$942$	10.9360	0.356314
$943$	−29.6742	−0.966325
$944$	−16.5761	−0.539508
$945$	0	0
$946$	0	0
$947$	45.7198	1.48569	0.742847	−	0.669462i	$-0.233475\pi$
$947$	45.7198	1.48569	0.742847	+	0.669462i	$0.233475\pi$
$948$	0.340173	0.0110483
$949$	−7.86216	−0.255216
$950$	0	0
$951$	−21.1773	−0.686720
$952$	−3.49400	−0.113241
$953$	−26.4619	−0.857184	−0.428592	−	0.903498i	$-0.640990\pi$
$953$	−26.4619	−0.857184	−0.428592	+	0.903498i	$0.640990\pi$
$954$	−8.86376	−0.286975
$955$	0	0
$956$	1.61181	0.0521296
$957$	0	0
$958$	−41.3074	−1.33458
$959$	3.66192	0.118249
$960$	0	0
$961$	9.19779	0.296703
$962$	36.6803	1.18262
$963$	12.2329	0.394199
$964$	1.92079	0.0618643
$965$	0	0
$966$	−1.78992	−0.0575897
$967$	50.5464	1.62546	0.812731	−	0.582639i	$-0.197980\pi$
$967$	50.5464	1.62546	0.812731	+	0.582639i	$0.197980\pi$
$968$	0	0
$969$	−37.1461	−1.19330
$970$	0	0
$971$	−57.3074	−1.83908	−0.919540	−	0.392995i	$-0.871439\pi$
$971$	−57.3074	−1.83908	−0.919540	+	0.392995i	$0.871439\pi$
$972$	0.369102	0.0118390
$973$	2.88267	0.0924143
$974$	−43.9664	−1.40877
$975$	0	0
$976$	−30.7427	−0.984051
$977$	45.7875	1.46487	0.732436	−	0.680836i	$-0.238384\pi$
$977$	45.7875	1.46487	0.732436	+	0.680836i	$0.238384\pi$
$978$	35.3028	1.12886
$979$	0	0
$980$	0	0
$981$	6.31351	0.201575
$982$	−40.0000	−1.27645
$983$	−55.0805	−1.75679	−0.878397	−	0.477932i	$-0.841387\pi$
$983$	−55.0805	−1.75679	−0.878397	+	0.477932i	$0.841387\pi$
$984$	−18.6225	−0.593663
$985$	0	0
$986$	54.6681	1.74099
$987$	−1.52973	−0.0486920
$988$	−19.9635	−0.635123
$989$	−1.16290	−0.0369780
$990$	0	0
$991$	40.6947	1.29271	0.646355	−	0.763037i	$-0.276292\pi$
$991$	40.6947	1.29271	0.646355	+	0.763037i	$0.276292\pi$
$992$	−13.0784	−0.415239
$993$	−6.70701	−0.212840
$994$	−2.27247	−0.0720785
$995$	0	0
$996$	0.630898	0.0199908
$997$	−32.7610	−1.03755	−0.518775	−	0.854911i	$-0.673612\pi$
$997$	−32.7610	−1.03755	−0.518775	+	0.854911i	$0.673612\pi$
$998$	−42.3545	−1.34071
$999$	3.41855	0.108158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.ck.1.2		3
5.2	odd	4		1815.2.c.d.364.5		6
5.3	odd	4		1815.2.c.d.364.2		6
5.4	even	2		9075.2.a.cc.1.2		3
11.10	odd	2		825.2.a.h.1.2		3
33.32	even	2		2475.2.a.be.1.2		3
55.32	even	4		165.2.c.a.34.2	✓	6
55.43	even	4		165.2.c.a.34.5	yes	6
55.54	odd	2		825.2.a.n.1.2		3
165.32	odd	4		495.2.c.d.199.5		6
165.98	odd	4		495.2.c.d.199.2		6
165.164	even	2		2475.2.a.y.1.2		3
220.43	odd	4		2640.2.d.i.529.3		6
220.87	odd	4		2640.2.d.i.529.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.2	✓	6	55.32	even	4
165.2.c.a.34.5	yes	6	55.43	even	4
495.2.c.d.199.2		6	165.98	odd	4
495.2.c.d.199.5		6	165.32	odd	4
825.2.a.h.1.2		3	11.10	odd	2
825.2.a.n.1.2		3	55.54	odd	2
1815.2.c.d.364.2		6	5.3	odd	4
1815.2.c.d.364.5		6	5.2	odd	4
2475.2.a.y.1.2		3	165.164	even	2
2475.2.a.be.1.2		3	33.32	even	2
2640.2.d.i.529.3		6	220.43	odd	4
2640.2.d.i.529.6		6	220.87	odd	4
9075.2.a.cc.1.2		3	5.4	even	2
9075.2.a.ck.1.2		3	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} +1.53919 q^{6} +0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} +1.53919 q^{6} +0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{9} +0.369102 q^{12} +6.97107 q^{13} +0.447480 q^{14} -4.60197 q^{16} +4.78765 q^{17} +1.53919 q^{18} -7.75872 q^{19} +0.290725 q^{21} -4.00000 q^{23} -2.51026 q^{24} +10.7298 q^{26} +1.00000 q^{27} +0.107307 q^{28} +7.41855 q^{29} +6.34017 q^{31} -2.06278 q^{32} +7.36910 q^{34} +0.369102 q^{36} +3.41855 q^{37} -11.9421 q^{38} +6.97107 q^{39} +7.41855 q^{41} +0.447480 q^{42} +0.290725 q^{43} -6.15676 q^{46} -5.26180 q^{47} -4.60197 q^{48} -6.91548 q^{49} +4.78765 q^{51} +2.57304 q^{52} -5.75872 q^{53} +1.53919 q^{54} -0.729794 q^{56} -7.75872 q^{57} +11.4186 q^{58} +3.60197 q^{59} +6.68035 q^{61} +9.75872 q^{62} +0.290725 q^{63} +6.02893 q^{64} -6.15676 q^{67} +1.76713 q^{68} -4.00000 q^{69} -5.07838 q^{71} -2.51026 q^{72} -1.12783 q^{73} +5.26180 q^{74} -2.86376 q^{76} +10.7298 q^{78} +0.921622 q^{79} +1.00000 q^{81} +11.4186 q^{82} +1.70928 q^{83} +0.107307 q^{84} +0.447480 q^{86} +7.41855 q^{87} +4.34017 q^{89} +2.02666 q^{91} -1.47641 q^{92} +6.34017 q^{93} -8.09890 q^{94} -2.06278 q^{96} -4.68035 q^{97} -10.6442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} + 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} + 2 q^{19} + 8 q^{21} - 12 q^{23} + 9 q^{24} - 8 q^{26} + 3 q^{27} + 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34} + 5 q^{36} - 4 q^{37} - 6 q^{38} + 6 q^{39} + 8 q^{41} + 2 q^{42} + 8 q^{43} - 12 q^{46} - 8 q^{47} + 5 q^{48} + 11 q^{49} + 4 q^{51} - 26 q^{52} + 8 q^{53} + 3 q^{54} + 38 q^{56} + 2 q^{57} + 20 q^{58} - 8 q^{59} - 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} - 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} + 8 q^{74} + 18 q^{76} - 8 q^{78} + 6 q^{79} + 3 q^{81} + 20 q^{82} - 2 q^{83} + 12 q^{84} + 2 q^{86} + 8 q^{87} + 2 q^{89} + 8 q^{91} - 20 q^{92} + 8 q^{93} + 12 q^{94} + 11 q^{96} + 8 q^{97} - 29 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 + 5 * q^12 + 6 * q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 + 3 * q^18 + 2 * q^19 + 8 * q^21 - 12 * q^23 + 9 * q^24 - 8 * q^26 + 3 * q^27 + 12 * q^28 + 8 * q^29 + 8 * q^31 + 11 * q^32 + 26 * q^34 + 5 * q^36 - 4 * q^37 - 6 * q^38 + 6 * q^39 + 8 * q^41 + 2 * q^42 + 8 * q^43 - 12 * q^46 - 8 * q^47 + 5 * q^48 + 11 * q^49 + 4 * q^51 - 26 * q^52 + 8 * q^53 + 3 * q^54 + 38 * q^56 + 2 * q^57 + 20 * q^58 - 8 * q^59 - 2 * q^61 + 4 * q^62 + 8 * q^63 + 33 * q^64 - 12 * q^67 + 28 * q^68 - 12 * q^69 - 12 * q^71 + 9 * q^72 + 18 * q^73 + 8 * q^74 + 18 * q^76 - 8 * q^78 + 6 * q^79 + 3 * q^81 + 20 * q^82 - 2 * q^83 + 12 * q^84 + 2 * q^86 + 8 * q^87 + 2 * q^89 + 8 * q^91 - 20 * q^92 + 8 * q^93 + 12 * q^94 + 11 * q^96 + 8 * q^97 - 29 * q^98

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