LMFDB - Embedded newform 9075.2.a.cz.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.18890$ 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)

$f(q)$	$=$	$q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.79129 q^{12} +0.913701 q^{13} -0.417424 q^{14} +2.79129 q^{16} -7.02355 q^{17} +0.456850 q^{18} +5.29150 q^{19} -0.913701 q^{21} +0.582576 q^{23} -1.73205 q^{24} +0.417424 q^{26} +1.00000 q^{27} +1.63670 q^{28} +4.37780 q^{29} +2.58258 q^{31} +4.73930 q^{32} -3.20871 q^{34} -1.79129 q^{36} -9.58258 q^{37} +2.41742 q^{38} +0.913701 q^{39} +6.92820 q^{41} -0.417424 q^{42} -4.37780 q^{43} +0.266150 q^{46} -6.58258 q^{47} +2.79129 q^{48} -6.16515 q^{49} -7.02355 q^{51} -1.63670 q^{52} -5.00000 q^{53} +0.456850 q^{54} +1.58258 q^{56} +5.29150 q^{57} +2.00000 q^{58} -3.58258 q^{59} +8.66025 q^{61} +1.17985 q^{62} -0.913701 q^{63} -3.41742 q^{64} +14.7477 q^{67} +12.5812 q^{68} +0.582576 q^{69} -9.16515 q^{71} -1.73205 q^{72} -15.6838 q^{73} -4.37780 q^{74} -9.47860 q^{76} +0.417424 q^{78} -2.64575 q^{79} +1.00000 q^{81} +3.16515 q^{82} +12.2197 q^{83} +1.63670 q^{84} -2.00000 q^{86} +4.37780 q^{87} +16.7477 q^{89} -0.834849 q^{91} -1.04356 q^{92} +2.58258 q^{93} -3.00725 q^{94} +4.73930 q^{96} +3.58258 q^{97} -2.81655 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 20 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} + 4 q^{27} - 8 q^{31} - 22 q^{34} + 2 q^{36} - 20 q^{37} + 28 q^{38} - 20 q^{42} - 8 q^{47} + 2 q^{48} + 12 q^{49} - 20 q^{53} - 12 q^{56} + 8 q^{58} + 4 q^{59} - 32 q^{64} + 4 q^{67} - 16 q^{69} + 20 q^{78} + 4 q^{81} - 24 q^{82} - 8 q^{86} + 12 q^{89} - 40 q^{91} - 50 q^{92} - 8 q^{93} - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^4 + 4 * q^9 + 2 * q^12 - 20 * q^14 + 2 * q^16 - 16 * q^23 + 20 * q^26 + 4 * q^27 - 8 * q^31 - 22 * q^34 + 2 * q^36 - 20 * q^37 + 28 * q^38 - 20 * q^42 - 8 * q^47 + 2 * q^48 + 12 * q^49 - 20 * q^53 - 12 * q^56 + 8 * q^58 + 4 * q^59 - 32 * q^64 + 4 * q^67 - 16 * q^69 + 20 * q^78 + 4 * q^81 - 24 * q^82 - 8 * q^86 + 12 * q^89 - 40 * q^91 - 50 * q^92 - 8 * q^93 - 4 * 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 $

$2$	0.456850	0.323042	0.161521	−	0.986869i	$-0.448360\pi$
$2$	0.456850	0.323042	0.161521	+	0.986869i	$0.448360\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.79129	−0.895644
$5$	0	0
$5$	0	0
$6$	0.456850	0.186508
$7$	−0.913701	−0.345346	−0.172673	−	0.984979i	$-0.555240\pi$
$7$	−0.913701	−0.345346	−0.172673	+	0.984979i	$0.555240\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.79129	−0.517100
$13$	0.913701	0.253415	0.126707	−	0.991940i	$-0.459559\pi$
$13$	0.913701	0.253415	0.126707	+	0.991940i	$0.459559\pi$
$14$	−0.417424	−0.111561
$15$	0	0
$16$	2.79129	0.697822
$17$	−7.02355	−1.70346	−0.851731	−	0.523979i	$-0.824447\pi$
$17$	−7.02355	−1.70346	−0.851731	+	0.523979i	$0.824447\pi$
$18$	0.456850	0.107681
$19$	5.29150	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$20$	0	0
$21$	−0.913701	−0.199386
$22$	0	0
$23$	0.582576	0.121475	0.0607377	−	0.998154i	$-0.480655\pi$
$23$	0.582576	0.121475	0.0607377	+	0.998154i	$0.480655\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	0.417424	0.0818636
$27$	1.00000	0.192450
$28$	1.63670	0.309307
$29$	4.37780	0.812937	0.406469	−	0.913665i	$-0.366760\pi$
$29$	4.37780	0.812937	0.406469	+	0.913665i	$0.366760\pi$
$30$	0	0
$31$	2.58258	0.463844	0.231922	−	0.972734i	$-0.425499\pi$
$31$	2.58258	0.463844	0.231922	+	0.972734i	$0.425499\pi$
$32$	4.73930	0.837798
$33$	0	0
$34$	−3.20871	−0.550290
$35$	0	0
$36$	−1.79129	−0.298548
$37$	−9.58258	−1.57537	−0.787683	−	0.616081i	$-0.788719\pi$
$37$	−9.58258	−1.57537	−0.787683	+	0.616081i	$0.788719\pi$
$38$	2.41742	0.392158
$39$	0.913701	0.146309
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	−0.417424	−0.0644100
$43$	−4.37780	−0.667609	−0.333804	−	0.942642i	$-0.608332\pi$
$43$	−4.37780	−0.667609	−0.333804	+	0.942642i	$0.608332\pi$
$44$	0	0
$45$	0	0
$46$	0.266150	0.0392417
$47$	−6.58258	−0.960167	−0.480084	−	0.877223i	$-0.659394\pi$
$47$	−6.58258	−0.960167	−0.480084	+	0.877223i	$0.659394\pi$
$48$	2.79129	0.402888
$49$	−6.16515	−0.880736
$50$	0	0
$51$	−7.02355	−0.983494
$52$	−1.63670	−0.226970
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0.456850	0.0621694
$55$	0	0
$56$	1.58258	0.211481
$57$	5.29150	0.700877
$58$	2.00000	0.262613
$59$	−3.58258	−0.466412	−0.233206	−	0.972427i	$-0.574922\pi$
$59$	−3.58258	−0.466412	−0.233206	+	0.972427i	$0.574922\pi$
$60$	0	0
$61$	8.66025	1.10883	0.554416	−	0.832240i	$-0.312942\pi$
$61$	8.66025	1.10883	0.554416	+	0.832240i	$0.312942\pi$
$62$	1.17985	0.149841
$63$	−0.913701	−0.115115
$64$	−3.41742	−0.427178
$65$	0	0
$66$	0	0
$67$	14.7477	1.80172	0.900861	−	0.434108i	$-0.142936\pi$
$67$	14.7477	1.80172	0.900861	+	0.434108i	$0.142936\pi$
$68$	12.5812	1.52570
$69$	0.582576	0.0701339
$70$	0	0
$71$	−9.16515	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$71$	−9.16515	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$72$	−1.73205	−0.204124
$73$	−15.6838	−1.83565	−0.917825	−	0.396984i	$-0.870057\pi$
$73$	−15.6838	−1.83565	−0.917825	+	0.396984i	$0.870057\pi$
$74$	−4.37780	−0.508909
$75$	0	0
$76$	−9.47860	−1.08727
$77$	0	0
$78$	0.417424	0.0472640
$79$	−2.64575	−0.297670	−0.148835	−	0.988862i	$-0.547552\pi$
$79$	−2.64575	−0.297670	−0.148835	+	0.988862i	$0.547552\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.16515	0.349532
$83$	12.2197	1.34129	0.670643	−	0.741780i	$-0.266018\pi$
$83$	12.2197	1.34129	0.670643	+	0.741780i	$0.266018\pi$
$84$	1.63670	0.178579
$85$	0	0
$86$	−2.00000	−0.215666
$87$	4.37780	0.469350
$88$	0	0
$89$	16.7477	1.77526	0.887628	−	0.460562i	$-0.152352\pi$
$89$	16.7477	1.77526	0.887628	+	0.460562i	$0.152352\pi$
$90$	0	0
$91$	−0.834849	−0.0875159
$92$	−1.04356	−0.108799
$93$	2.58258	0.267801
$94$	−3.00725	−0.310174
$95$	0	0
$96$	4.73930	0.483703
$97$	3.58258	0.363755	0.181878	−	0.983321i	$-0.441782\pi$
$97$	3.58258	0.363755	0.181878	+	0.983321i	$0.441782\pi$
$98$	−2.81655	−0.284515
$99$	0	0
$100$	0	0
$101$	5.29150	0.526524	0.263262	−	0.964724i	$-0.415202\pi$
$101$	5.29150	0.526524	0.263262	+	0.964724i	$0.415202\pi$
$102$	−3.20871	−0.317710
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−1.58258	−0.155184
$105$	0	0
$106$	−2.28425	−0.221866
$107$	−1.00905	−0.0975486	−0.0487743	−	0.998810i	$-0.515532\pi$
$107$	−1.00905	−0.0975486	−0.0487743	+	0.998810i	$0.515532\pi$
$108$	−1.79129	−0.172367
$109$	5.10080	0.488568	0.244284	−	0.969704i	$-0.421447\pi$
$109$	5.10080	0.488568	0.244284	+	0.969704i	$0.421447\pi$
$110$	0	0
$111$	−9.58258	−0.909538
$112$	−2.55040	−0.240990
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	2.41742	0.226413
$115$	0	0
$116$	−7.84190	−0.728102
$117$	0.913701	0.0844716
$118$	−1.63670	−0.150671
$119$	6.41742	0.588284
$120$	0	0
$121$	0	0
$122$	3.95644	0.358199
$123$	6.92820	0.624695
$124$	−4.62614	−0.415439
$125$	0	0
$126$	−0.417424	−0.0371871
$127$	−6.01450	−0.533701	−0.266850	−	0.963738i	$-0.585983\pi$
$127$	−6.01450	−0.533701	−0.266850	+	0.963738i	$0.585983\pi$
$128$	−11.0399	−0.975795
$129$	−4.37780	−0.385444
$130$	0	0
$131$	1.82740	0.159661	0.0798304	−	0.996808i	$-0.474562\pi$
$131$	1.82740	0.159661	0.0798304	+	0.996808i	$0.474562\pi$
$132$	0	0
$133$	−4.83485	−0.419235
$134$	6.73750	0.582032
$135$	0	0
$136$	12.1652	1.04315
$137$	4.16515	0.355853	0.177926	−	0.984044i	$-0.443061\pi$
$137$	4.16515	0.355853	0.177926	+	0.984044i	$0.443061\pi$
$138$	0.266150	0.0226562
$139$	−20.1570	−1.70969	−0.854846	−	0.518883i	$-0.826348\pi$
$139$	−20.1570	−1.70969	−0.854846	+	0.518883i	$0.826348\pi$
$140$	0	0
$141$	−6.58258	−0.554353
$142$	−4.18710	−0.351374
$143$	0	0
$144$	2.79129	0.232607
$145$	0	0
$146$	−7.16515	−0.592992
$147$	−6.16515	−0.508493
$148$	17.1652	1.41097
$149$	−14.7701	−1.21001	−0.605007	−	0.796220i	$-0.706830\pi$
$149$	−14.7701	−1.21001	−0.605007	+	0.796220i	$0.706830\pi$
$150$	0	0
$151$	−5.91915	−0.481694	−0.240847	−	0.970563i	$-0.577425\pi$
$151$	−5.91915	−0.481694	−0.240847	+	0.970563i	$0.577425\pi$
$152$	−9.16515	−0.743392
$153$	−7.02355	−0.567821
$154$	0	0
$155$	0	0
$156$	−1.63670	−0.131041
$157$	−12.4174	−0.991018	−0.495509	−	0.868603i	$-0.665019\pi$
$157$	−12.4174	−0.991018	−0.495509	+	0.868603i	$0.665019\pi$
$158$	−1.20871	−0.0961600
$159$	−5.00000	−0.396526
$160$	0	0
$161$	−0.532300	−0.0419511
$162$	0.456850	0.0358935
$163$	−17.5826	−1.37717	−0.688587	−	0.725154i	$-0.741769\pi$
$163$	−17.5826	−1.37717	−0.688587	+	0.725154i	$0.741769\pi$
$164$	−12.4104	−0.969090
$165$	0	0
$166$	5.58258	0.433292
$167$	18.3296	1.41838	0.709192	−	0.705015i	$-0.249060\pi$
$167$	18.3296	1.41838	0.709192	+	0.705015i	$0.249060\pi$
$168$	1.58258	0.122098
$169$	−12.1652	−0.935781
$170$	0	0
$171$	5.29150	0.404651
$172$	7.84190	0.597940
$173$	−15.6838	−1.19242	−0.596209	−	0.802829i	$-0.703327\pi$
$173$	−15.6838	−1.19242	−0.596209	+	0.802829i	$0.703327\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	−3.58258	−0.269283
$178$	7.65120	0.573482
$179$	−10.7477	−0.803323	−0.401661	−	0.915788i	$-0.631567\pi$
$179$	−10.7477	−0.803323	−0.401661	+	0.915788i	$0.631567\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−0.381401	−0.0282713
$183$	8.66025	0.640184
$184$	−1.00905	−0.0743882
$185$	0	0
$186$	1.17985	0.0865108
$187$	0	0
$188$	11.7913	0.859968
$189$	−0.913701	−0.0664619
$190$	0	0
$191$	−19.9129	−1.44085	−0.720423	−	0.693535i	$-0.756052\pi$
$191$	−19.9129	−1.44085	−0.720423	+	0.693535i	$0.756052\pi$
$192$	−3.41742	−0.246631
$193$	13.1334	0.945363	0.472682	−	0.881233i	$-0.343286\pi$
$193$	13.1334	0.945363	0.472682	+	0.881233i	$0.343286\pi$
$194$	1.63670	0.117508
$195$	0	0
$196$	11.0436	0.788826
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−17.4174	−1.23469	−0.617344	−	0.786693i	$-0.711791\pi$
$199$	−17.4174	−1.23469	−0.617344	+	0.786693i	$0.711791\pi$
$200$	0	0
$201$	14.7477	1.04022
$202$	2.41742	0.170089
$203$	−4.00000	−0.280745
$204$	12.5812	0.880861
$205$	0	0
$206$	−2.74110	−0.190982
$207$	0.582576	0.0404918
$208$	2.55040	0.176838
$209$	0	0
$210$	0	0
$211$	6.30055	0.433748	0.216874	−	0.976200i	$-0.430414\pi$
$211$	6.30055	0.433748	0.216874	+	0.976200i	$0.430414\pi$
$212$	8.95644	0.615131
$213$	−9.16515	−0.627986
$214$	−0.460985	−0.0315123
$215$	0	0
$216$	−1.73205	−0.117851
$217$	−2.35970	−0.160187
$218$	2.33030	0.157828
$219$	−15.6838	−1.05981
$220$	0	0
$221$	−6.41742	−0.431683
$222$	−4.37780	−0.293819
$223$	−23.1652	−1.55125	−0.775627	−	0.631192i	$-0.782566\pi$
$223$	−23.1652	−1.55125	−0.775627	+	0.631192i	$0.782566\pi$
$224$	−4.33030	−0.289331
$225$	0	0
$226$	1.37055	0.0911677
$227$	−18.3296	−1.21658	−0.608288	−	0.793717i	$-0.708143\pi$
$227$	−18.3296	−1.21658	−0.608288	+	0.793717i	$0.708143\pi$
$228$	−9.47860	−0.627736
$229$	−22.1652	−1.46471	−0.732357	−	0.680921i	$-0.761580\pi$
$229$	−22.1652	−1.46471	−0.732357	+	0.680921i	$0.761580\pi$
$230$	0	0
$231$	0	0
$232$	−7.58258	−0.497820
$233$	10.6784	0.699562	0.349781	−	0.936831i	$-0.386256\pi$
$233$	10.6784	0.699562	0.349781	+	0.936831i	$0.386256\pi$
$234$	0.417424	0.0272879
$235$	0	0
$236$	6.41742	0.417739
$237$	−2.64575	−0.171860
$238$	2.93180	0.190040
$239$	16.5975	1.07360	0.536802	−	0.843708i	$-0.319632\pi$
$239$	16.5975	1.07360	0.536802	+	0.843708i	$0.319632\pi$
$240$	0	0
$241$	−28.1896	−1.81585	−0.907925	−	0.419133i	$-0.862334\pi$
$241$	−28.1896	−1.81585	−0.907925	+	0.419133i	$0.862334\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−15.5130	−0.993119
$245$	0	0
$246$	3.16515	0.201803
$247$	4.83485	0.307634
$248$	−4.47315	−0.284045
$249$	12.2197	0.774392
$250$	0	0
$251$	−1.58258	−0.0998913	−0.0499456	−	0.998752i	$-0.515905\pi$
$251$	−1.58258	−0.0998913	−0.0499456	+	0.998752i	$0.515905\pi$
$252$	1.63670	0.103102
$253$	0	0
$254$	−2.74773	−0.172408
$255$	0	0
$256$	1.79129	0.111955
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	−2.00000	−0.124515
$259$	8.75560	0.544047
$260$	0	0
$261$	4.37780	0.270979
$262$	0.834849	0.0515771
$263$	−11.4014	−0.703038	−0.351519	−	0.936181i	$-0.614335\pi$
$263$	−11.4014	−0.703038	−0.351519	+	0.936181i	$0.614335\pi$
$264$	0	0
$265$	0	0
$266$	−2.20880	−0.135430
$267$	16.7477	1.02494
$268$	−26.4174	−1.61370
$269$	19.1652	1.16852	0.584260	−	0.811567i	$-0.301385\pi$
$269$	19.1652	1.16852	0.584260	+	0.811567i	$0.301385\pi$
$270$	0	0
$271$	23.4304	1.42329	0.711647	−	0.702538i	$-0.247950\pi$
$271$	23.4304	1.42329	0.711647	+	0.702538i	$0.247950\pi$
$272$	−19.6048	−1.18871
$273$	−0.834849	−0.0505273
$274$	1.90285	0.114955
$275$	0	0
$276$	−1.04356	−0.0628150
$277$	26.0761	1.56676	0.783381	−	0.621542i	$-0.213493\pi$
$277$	26.0761	1.56676	0.783381	+	0.621542i	$0.213493\pi$
$278$	−9.20871	−0.552302
$279$	2.58258	0.154615
$280$	0	0
$281$	−25.5438	−1.52382	−0.761908	−	0.647685i	$-0.775737\pi$
$281$	−25.5438	−1.52382	−0.761908	+	0.647685i	$0.775737\pi$
$282$	−3.00725	−0.179079
$283$	−29.5402	−1.75598	−0.877992	−	0.478676i	$-0.841117\pi$
$283$	−29.5402	−1.75598	−0.877992	+	0.478676i	$0.841117\pi$
$284$	16.4174	0.974195
$285$	0	0
$286$	0	0
$287$	−6.33030	−0.373666
$288$	4.73930	0.279266
$289$	32.3303	1.90178
$290$	0	0
$291$	3.58258	0.210014
$292$	28.0942	1.64409
$293$	27.9989	1.63571	0.817856	−	0.575424i	$-0.195163\pi$
$293$	27.9989	1.63571	0.817856	+	0.575424i	$0.195163\pi$
$294$	−2.81655	−0.164265
$295$	0	0
$296$	16.5975	0.964711
$297$	0	0
$298$	−6.74773	−0.390885
$299$	0.532300	0.0307837
$300$	0	0
$301$	4.00000	0.230556
$302$	−2.70417	−0.155607
$303$	5.29150	0.303989
$304$	14.7701	0.847124
$305$	0	0
$306$	−3.20871	−0.183430
$307$	−6.92820	−0.395413	−0.197707	−	0.980261i	$-0.563349\pi$
$307$	−6.92820	−0.395413	−0.197707	+	0.980261i	$0.563349\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−15.5826	−0.883607	−0.441803	−	0.897112i	$-0.645661\pi$
$311$	−15.5826	−0.883607	−0.441803	+	0.897112i	$0.645661\pi$
$312$	−1.58258	−0.0895957
$313$	−11.5826	−0.654686	−0.327343	−	0.944906i	$-0.606153\pi$
$313$	−11.5826	−0.654686	−0.327343	+	0.944906i	$0.606153\pi$
$314$	−5.67290	−0.320140
$315$	0	0
$316$	4.73930	0.266607
$317$	−7.83485	−0.440049	−0.220024	−	0.975494i	$-0.570614\pi$
$317$	−7.83485	−0.440049	−0.220024	+	0.975494i	$0.570614\pi$
$318$	−2.28425	−0.128094
$319$	0	0
$320$	0	0
$321$	−1.00905	−0.0563197
$322$	−0.243181	−0.0135520
$323$	−37.1652	−2.06792
$324$	−1.79129	−0.0995160
$325$	0	0
$326$	−8.03260	−0.444885
$327$	5.10080	0.282075
$328$	−12.0000	−0.662589
$329$	6.01450	0.331590
$330$	0	0
$331$	5.41742	0.297769	0.148884	−	0.988855i	$-0.452432\pi$
$331$	5.41742	0.297769	0.148884	+	0.988855i	$0.452432\pi$
$332$	−21.8890	−1.20132
$333$	−9.58258	−0.525122
$334$	8.37386	0.458197
$335$	0	0
$336$	−2.55040	−0.139136
$337$	0.190700	0.0103881	0.00519406	−	0.999987i	$-0.498347\pi$
$337$	0.190700	0.0103881	0.00519406	+	0.999987i	$0.498347\pi$
$338$	−5.55765	−0.302296
$339$	3.00000	0.162938
$340$	0	0
$341$	0	0
$342$	2.41742	0.130719
$343$	12.0290	0.649505
$344$	7.58258	0.408825
$345$	0	0
$346$	−7.16515	−0.385201
$347$	11.4014	0.612057	0.306028	−	0.952022i	$-0.401000\pi$
$347$	11.4014	0.612057	0.306028	+	0.952022i	$0.401000\pi$
$348$	−7.84190	−0.420370
$349$	8.85095	0.473781	0.236890	−	0.971536i	$-0.423872\pi$
$349$	8.85095	0.473781	0.236890	+	0.971536i	$0.423872\pi$
$350$	0	0
$351$	0.913701	0.0487697
$352$	0	0
$353$	−15.3303	−0.815950	−0.407975	−	0.912993i	$-0.633765\pi$
$353$	−15.3303	−0.815950	−0.407975	+	0.912993i	$0.633765\pi$
$354$	−1.63670	−0.0869897
$355$	0	0
$356$	−30.0000	−1.59000
$357$	6.41742	0.339646
$358$	−4.91010	−0.259507
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	−4.56850	−0.240115
$363$	0	0
$364$	1.49545	0.0783831
$365$	0	0
$366$	3.95644	0.206806
$367$	25.9129	1.35264	0.676321	−	0.736607i	$-0.263573\pi$
$367$	25.9129	1.35264	0.676321	+	0.736607i	$0.263573\pi$
$368$	1.62614	0.0847682
$369$	6.92820	0.360668
$370$	0	0
$371$	4.56850	0.237185
$372$	−4.62614	−0.239854
$373$	14.2378	0.737206	0.368603	−	0.929587i	$-0.379836\pi$
$373$	14.2378	0.737206	0.368603	+	0.929587i	$0.379836\pi$
$374$	0	0
$375$	0	0
$376$	11.4014	0.587980
$377$	4.00000	0.206010
$378$	−0.417424	−0.0214700
$379$	6.58258	0.338124	0.169062	−	0.985605i	$-0.445926\pi$
$379$	6.58258	0.338124	0.169062	+	0.985605i	$0.445926\pi$
$380$	0	0
$381$	−6.01450	−0.308132
$382$	−9.09720	−0.465453
$383$	−33.4955	−1.71154	−0.855769	−	0.517358i	$-0.826915\pi$
$383$	−33.4955	−1.71154	−0.855769	+	0.517358i	$0.826915\pi$
$384$	−11.0399	−0.563375
$385$	0	0
$386$	6.00000	0.305392
$387$	−4.37780	−0.222536
$388$	−6.41742	−0.325795
$389$	−7.58258	−0.384452	−0.192226	−	0.981351i	$-0.561571\pi$
$389$	−7.58258	−0.384452	−0.192226	+	0.981351i	$0.561571\pi$
$390$	0	0
$391$	−4.09175	−0.206929
$392$	10.6784	0.539338
$393$	1.82740	0.0921802
$394$	−3.16515	−0.159458
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−7.95715	−0.398856
$399$	−4.83485	−0.242045
$400$	0	0
$401$	10.3303	0.515871	0.257935	−	0.966162i	$-0.416958\pi$
$401$	10.3303	0.515871	0.257935	+	0.966162i	$0.416958\pi$
$402$	6.73750	0.336036
$403$	2.35970	0.117545
$404$	−9.47860	−0.471578
$405$	0	0
$406$	−1.82740	−0.0906924
$407$	0	0
$408$	12.1652	0.602265
$409$	−3.36875	−0.166574	−0.0832870	−	0.996526i	$-0.526542\pi$
$409$	−3.36875	−0.166574	−0.0832870	+	0.996526i	$0.526542\pi$
$410$	0	0
$411$	4.16515	0.205452
$412$	10.7477	0.529503
$413$	3.27340	0.161074
$414$	0.266150	0.0130806
$415$	0	0
$416$	4.33030	0.212311
$417$	−20.1570	−0.987091
$418$	0	0
$419$	32.3303	1.57944	0.789719	−	0.613468i	$-0.210226\pi$
$419$	32.3303	1.57944	0.789719	+	0.613468i	$0.210226\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	2.87841	0.140119
$423$	−6.58258	−0.320056
$424$	8.66025	0.420579
$425$	0	0
$426$	−4.18710	−0.202866
$427$	−7.91288	−0.382931
$428$	1.80750	0.0873688
$429$	0	0
$430$	0	0
$431$	−32.4720	−1.56412	−0.782061	−	0.623202i	$-0.785831\pi$
$431$	−32.4720	−1.56412	−0.782061	+	0.623202i	$0.785831\pi$
$432$	2.79129	0.134296
$433$	−14.7477	−0.708731	−0.354365	−	0.935107i	$-0.615303\pi$
$433$	−14.7477	−0.708731	−0.354365	+	0.935107i	$0.615303\pi$
$434$	−1.07803	−0.0517471
$435$	0	0
$436$	−9.13701	−0.437583
$437$	3.08270	0.147466
$438$	−7.16515	−0.342364
$439$	−4.47315	−0.213492	−0.106746	−	0.994286i	$-0.534043\pi$
$439$	−4.47315	−0.213492	−0.106746	+	0.994286i	$0.534043\pi$
$440$	0	0
$441$	−6.16515	−0.293579
$442$	−2.93180	−0.139452
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	17.1652	0.814622
$445$	0	0
$446$	−10.5830	−0.501120
$447$	−14.7701	−0.698602
$448$	3.12250	0.147524
$449$	−2.83485	−0.133785	−0.0668924	−	0.997760i	$-0.521308\pi$
$449$	−2.83485	−0.133785	−0.0668924	+	0.997760i	$0.521308\pi$
$450$	0	0
$451$	0	0
$452$	−5.37386	−0.252765
$453$	−5.91915	−0.278106
$454$	−8.37386	−0.393005
$455$	0	0
$456$	−9.16515	−0.429198
$457$	40.5046	1.89473	0.947363	−	0.320161i	$-0.103737\pi$
$457$	40.5046	1.89473	0.947363	+	0.320161i	$0.103737\pi$
$458$	−10.1262	−0.473164
$459$	−7.02355	−0.327831
$460$	0	0
$461$	23.5257	1.09570	0.547851	−	0.836576i	$-0.315446\pi$
$461$	23.5257	1.09570	0.547851	+	0.836576i	$0.315446\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	12.2197	0.567286
$465$	0	0
$466$	4.87841	0.225988
$467$	−13.7477	−0.636169	−0.318084	−	0.948062i	$-0.603040\pi$
$467$	−13.7477	−0.636169	−0.318084	+	0.948062i	$0.603040\pi$
$468$	−1.63670	−0.0756565
$469$	−13.4750	−0.622218
$470$	0	0
$471$	−12.4174	−0.572165
$472$	6.20520	0.285618
$473$	0	0
$474$	−1.20871	−0.0555180
$475$	0	0
$476$	−11.4955	−0.526893
$477$	−5.00000	−0.228934
$478$	7.58258	0.346819
$479$	−24.9717	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$479$	−24.9717	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$480$	0	0
$481$	−8.75560	−0.399221
$482$	−12.8784	−0.586595
$483$	−0.532300	−0.0242205
$484$	0	0
$485$	0	0
$486$	0.456850	0.0207231
$487$	−21.9129	−0.992967	−0.496484	−	0.868046i	$-0.665376\pi$
$487$	−21.9129	−0.992967	−0.496484	+	0.868046i	$0.665376\pi$
$488$	−15.0000	−0.679018
$489$	−17.5826	−0.795112
$490$	0	0
$491$	8.94630	0.403741	0.201871	−	0.979412i	$-0.435298\pi$
$491$	8.94630	0.403741	0.201871	+	0.979412i	$0.435298\pi$
$492$	−12.4104	−0.559504
$493$	−30.7477	−1.38481
$494$	2.20880	0.0993787
$495$	0	0
$496$	7.20871	0.323681
$497$	8.37420	0.375634
$498$	5.58258	0.250161
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	18.3296	0.818904
$502$	−0.723000	−0.0322691
$503$	−35.6501	−1.58956	−0.794779	−	0.606899i	$-0.792413\pi$
$503$	−35.6501	−1.58956	−0.794779	+	0.606899i	$0.792413\pi$
$504$	1.58258	0.0704935
$505$	0	0
$506$	0	0
$507$	−12.1652	−0.540273
$508$	10.7737	0.478006
$509$	3.91288	0.173435	0.0867176	−	0.996233i	$-0.472362\pi$
$509$	3.91288	0.173435	0.0867176	+	0.996233i	$0.472362\pi$
$510$	0	0
$511$	14.3303	0.633935
$512$	22.8981	1.01196
$513$	5.29150	0.233626
$514$	−5.93905	−0.261960
$515$	0	0
$516$	7.84190	0.345221
$517$	0	0
$518$	4.00000	0.175750
$519$	−15.6838	−0.688443
$520$	0	0
$521$	26.4174	1.15737	0.578684	−	0.815552i	$-0.303566\pi$
$521$	26.4174	1.15737	0.578684	+	0.815552i	$0.303566\pi$
$522$	2.00000	0.0875376
$523$	26.9898	1.18018	0.590091	−	0.807337i	$-0.299092\pi$
$523$	26.9898	1.18018	0.590091	+	0.807337i	$0.299092\pi$
$524$	−3.27340	−0.142999
$525$	0	0
$526$	−5.20871	−0.227111
$527$	−18.1389	−0.790141
$528$	0	0
$529$	−22.6606	−0.985244
$530$	0	0
$531$	−3.58258	−0.155471
$532$	8.66061	0.375485
$533$	6.33030	0.274196
$534$	7.65120	0.331100
$535$	0	0
$536$	−25.5438	−1.10332
$537$	−10.7477	−0.463799
$538$	8.75560	0.377481
$539$	0	0
$540$	0	0
$541$	−38.2958	−1.64647	−0.823233	−	0.567704i	$-0.807832\pi$
$541$	−38.2958	−1.64647	−0.823233	+	0.567704i	$0.807832\pi$
$542$	10.7042	0.459783
$543$	−10.0000	−0.429141
$544$	−33.2867	−1.42716
$545$	0	0
$546$	−0.381401	−0.0163224
$547$	12.9427	0.553390	0.276695	−	0.960958i	$-0.410761\pi$
$547$	12.9427	0.553390	0.276695	+	0.960958i	$0.410761\pi$
$548$	−7.46099	−0.318717
$549$	8.66025	0.369611
$550$	0	0
$551$	23.1652	0.986869
$552$	−1.00905	−0.0429481
$553$	2.41742	0.102799
$554$	11.9129	0.506130
$555$	0	0
$556$	36.1069	1.53127
$557$	−13.9518	−0.591155	−0.295577	−	0.955319i	$-0.595512\pi$
$557$	−13.9518	−0.591155	−0.295577	+	0.955319i	$0.595512\pi$
$558$	1.17985	0.0499470
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	−11.6697	−0.492256
$563$	36.6591	1.54500	0.772499	−	0.635016i	$-0.219007\pi$
$563$	36.6591	1.54500	0.772499	+	0.635016i	$0.219007\pi$
$564$	11.7913	0.496503
$565$	0	0
$566$	−13.4955	−0.567256
$567$	−0.913701	−0.0383718
$568$	15.8745	0.666080
$569$	2.55040	0.106918	0.0534592	−	0.998570i	$-0.482975\pi$
$569$	2.55040	0.106918	0.0534592	+	0.998570i	$0.482975\pi$
$570$	0	0
$571$	34.0134	1.42342	0.711708	−	0.702476i	$-0.247922\pi$
$571$	34.0134	1.42342	0.711708	+	0.702476i	$0.247922\pi$
$572$	0	0
$573$	−19.9129	−0.831872
$574$	−2.89200	−0.120710
$575$	0	0
$576$	−3.41742	−0.142393
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	14.7701	0.614355
$579$	13.1334	0.545806
$580$	0	0
$581$	−11.1652	−0.463209
$582$	1.63670	0.0678434
$583$	0	0
$584$	27.1652	1.12410
$585$	0	0
$586$	12.7913	0.528403
$587$	4.25227	0.175510	0.0877550	−	0.996142i	$-0.472031\pi$
$587$	4.25227	0.175510	0.0877550	+	0.996142i	$0.472031\pi$
$588$	11.0436	0.455429
$589$	13.6657	0.563086
$590$	0	0
$591$	−6.92820	−0.284988
$592$	−26.7477	−1.09932
$593$	−17.5112	−0.719099	−0.359550	−	0.933126i	$-0.617070\pi$
$593$	−17.5112	−0.719099	−0.359550	+	0.933126i	$0.617070\pi$
$594$	0	0
$595$	0	0
$596$	26.4575	1.08374
$597$	−17.4174	−0.712848
$598$	0.243181	0.00994442
$599$	31.9129	1.30392	0.651962	−	0.758251i	$-0.273946\pi$
$599$	31.9129	1.30392	0.651962	+	0.758251i	$0.273946\pi$
$600$	0	0
$601$	17.5112	0.714297	0.357149	−	0.934048i	$-0.383749\pi$
$601$	17.5112	0.714297	0.357149	+	0.934048i	$0.383749\pi$
$602$	1.82740	0.0744793
$603$	14.7477	0.600574
$604$	10.6029	0.431426
$605$	0	0
$606$	2.41742	0.0982011
$607$	−13.5148	−0.548549	−0.274275	−	0.961651i	$-0.588438\pi$
$607$	−13.5148	−0.548549	−0.274275	+	0.961651i	$0.588438\pi$
$608$	25.0780	1.01705
$609$	−4.00000	−0.162088
$610$	0	0
$611$	−6.01450	−0.243321
$612$	12.5812	0.508565
$613$	1.44600	0.0584034	0.0292017	−	0.999574i	$-0.490703\pi$
$613$	1.44600	0.0584034	0.0292017	+	0.999574i	$0.490703\pi$
$614$	−3.16515	−0.127735
$615$	0	0
$616$	0	0
$617$	−35.4955	−1.42899	−0.714497	−	0.699639i	$-0.753344\pi$
$617$	−35.4955	−1.42899	−0.714497	+	0.699639i	$0.753344\pi$
$618$	−2.74110	−0.110263
$619$	−9.49545	−0.381655	−0.190827	−	0.981624i	$-0.561117\pi$
$619$	−9.49545	−0.381655	−0.190827	+	0.981624i	$0.561117\pi$
$620$	0	0
$621$	0.582576	0.0233780
$622$	−7.11890	−0.285442
$623$	−15.3024	−0.613078
$624$	2.55040	0.102098
$625$	0	0
$626$	−5.29150	−0.211491
$627$	0	0
$628$	22.2432	0.887600
$629$	67.3037	2.68358
$630$	0	0
$631$	−20.5826	−0.819379	−0.409690	−	0.912225i	$-0.634363\pi$
$631$	−20.5826	−0.819379	−0.409690	+	0.912225i	$0.634363\pi$
$632$	4.58258	0.182285
$633$	6.30055	0.250425
$634$	−3.57935	−0.142154
$635$	0	0
$636$	8.95644	0.355146
$637$	−5.63310	−0.223192
$638$	0	0
$639$	−9.16515	−0.362568
$640$	0	0
$641$	23.5826	0.931456	0.465728	−	0.884928i	$-0.345793\pi$
$641$	23.5826	0.931456	0.465728	+	0.884928i	$0.345793\pi$
$642$	−0.460985	−0.0181936
$643$	−18.3303	−0.722877	−0.361438	−	0.932396i	$-0.617714\pi$
$643$	−18.3303	−0.722877	−0.361438	+	0.932396i	$0.617714\pi$
$644$	0.953502	0.0375732
$645$	0	0
$646$	−16.9789	−0.668026
$647$	32.5826	1.28095	0.640477	−	0.767978i	$-0.278737\pi$
$647$	32.5826	1.28095	0.640477	+	0.767978i	$0.278737\pi$
$648$	−1.73205	−0.0680414
$649$	0	0
$650$	0	0
$651$	−2.35970	−0.0924840
$652$	31.4955	1.23346
$653$	8.33030	0.325990	0.162995	−	0.986627i	$-0.447885\pi$
$653$	8.33030	0.325990	0.162995	+	0.986627i	$0.447885\pi$
$654$	2.33030	0.0911220
$655$	0	0
$656$	19.3386	0.755046
$657$	−15.6838	−0.611884
$658$	2.74773	0.107118
$659$	35.9361	1.39987	0.699936	−	0.714205i	$-0.253212\pi$
$659$	35.9361	1.39987	0.699936	+	0.714205i	$0.253212\pi$
$660$	0	0
$661$	−21.1652	−0.823229	−0.411614	−	0.911358i	$-0.635035\pi$
$661$	−21.1652	−0.823229	−0.411614	+	0.911358i	$0.635035\pi$
$662$	2.47495	0.0961917
$663$	−6.41742	−0.249232
$664$	−21.1652	−0.821367
$665$	0	0
$666$	−4.37780	−0.169636
$667$	2.55040	0.0987519
$668$	−32.8335	−1.27037
$669$	−23.1652	−0.895616
$670$	0	0
$671$	0	0
$672$	−4.33030	−0.167045
$673$	7.46050	0.287581	0.143791	−	0.989608i	$-0.454071\pi$
$673$	7.46050	0.287581	0.143791	+	0.989608i	$0.454071\pi$
$674$	0.0871215	0.00335580
$675$	0	0
$676$	21.7913	0.838126
$677$	−5.10080	−0.196040	−0.0980199	−	0.995184i	$-0.531251\pi$
$677$	−5.10080	−0.196040	−0.0980199	+	0.995184i	$0.531251\pi$
$678$	1.37055	0.0526357
$679$	−3.27340	−0.125622
$680$	0	0
$681$	−18.3296	−0.702390
$682$	0	0
$683$	−10.3303	−0.395278	−0.197639	−	0.980275i	$-0.563327\pi$
$683$	−10.3303	−0.395278	−0.197639	+	0.980275i	$0.563327\pi$
$684$	−9.47860	−0.362423
$685$	0	0
$686$	5.49545	0.209817
$687$	−22.1652	−0.845653
$688$	−12.2197	−0.465872
$689$	−4.56850	−0.174046
$690$	0	0
$691$	−21.4174	−0.814757	−0.407379	−	0.913259i	$-0.633557\pi$
$691$	−21.4174	−0.814757	−0.407379	+	0.913259i	$0.633557\pi$
$692$	28.0942	1.06798
$693$	0	0
$694$	5.20871	0.197720
$695$	0	0
$696$	−7.58258	−0.287417
$697$	−48.6606	−1.84315
$698$	4.04356	0.153051
$699$	10.6784	0.403892
$700$	0	0
$701$	46.3284	1.74980	0.874900	−	0.484303i	$-0.160927\pi$
$701$	46.3284	1.74980	0.874900	+	0.484303i	$0.160927\pi$
$702$	0.417424	0.0157547
$703$	−50.7062	−1.91242
$704$	0	0
$705$	0	0
$706$	−7.00365	−0.263586
$707$	−4.83485	−0.181833
$708$	6.41742	0.241182
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	0	0
$711$	−2.64575	−0.0992234
$712$	−29.0079	−1.08712
$713$	1.50455	0.0563457
$714$	2.93180	0.109720
$715$	0	0
$716$	19.2523	0.719491
$717$	16.5975	0.619845
$718$	−11.0780	−0.413428
$719$	−45.4955	−1.69669	−0.848347	−	0.529441i	$-0.822402\pi$
$719$	−45.4955	−1.69669	−0.848347	+	0.529441i	$0.822402\pi$
$720$	0	0
$721$	5.48220	0.204168
$722$	4.11165	0.153020
$723$	−28.1896	−1.04838
$724$	17.9129	0.665727
$725$	0	0
$726$	0	0
$727$	23.4955	0.871398	0.435699	−	0.900092i	$-0.356501\pi$
$727$	23.4955	0.871398	0.435699	+	0.900092i	$0.356501\pi$
$728$	1.44600	0.0535923
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.7477	1.13725
$732$	−15.5130	−0.573377
$733$	−17.7019	−0.653835	−0.326917	−	0.945053i	$-0.606010\pi$
$733$	−17.7019	−0.653835	−0.326917	+	0.945053i	$0.606010\pi$
$734$	11.8383	0.436960
$735$	0	0
$736$	2.76100	0.101772
$737$	0	0
$738$	3.16515	0.116511
$739$	−18.1389	−0.667249	−0.333624	−	0.942706i	$-0.608272\pi$
$739$	−18.1389	−0.667249	−0.333624	+	0.942706i	$0.608272\pi$
$740$	0	0
$741$	4.83485	0.177613
$742$	2.08712	0.0766206
$743$	0.436950	0.0160301	0.00801506	−	0.999968i	$-0.497449\pi$
$743$	0.436950	0.0160301	0.00801506	+	0.999968i	$0.497449\pi$
$744$	−4.47315	−0.163994
$745$	0	0
$746$	6.50455	0.238148
$747$	12.2197	0.447096
$748$	0	0
$749$	0.921970	0.0336881
$750$	0	0
$751$	−11.7477	−0.428681	−0.214340	−	0.976759i	$-0.568760\pi$
$751$	−11.7477	−0.428681	−0.214340	+	0.976759i	$0.568760\pi$
$752$	−18.3739	−0.670026
$753$	−1.58258	−0.0576723
$754$	1.82740	0.0665500
$755$	0	0
$756$	1.63670	0.0595262
$757$	−33.4955	−1.21741	−0.608706	−	0.793395i	$-0.708311\pi$
$757$	−33.4955	−1.21741	−0.608706	+	0.793395i	$0.708311\pi$
$758$	3.00725	0.109228
$759$	0	0
$760$	0	0
$761$	48.3465	1.75256	0.876280	−	0.481802i	$-0.160018\pi$
$761$	48.3465	1.75256	0.876280	+	0.481802i	$0.160018\pi$
$762$	−2.74773	−0.0995396
$763$	−4.66061	−0.168725
$764$	35.6697	1.29048
$765$	0	0
$766$	−15.3024	−0.552898
$767$	−3.27340	−0.118196
$768$	1.79129	0.0646375
$769$	−22.8981	−0.825725	−0.412863	−	0.910793i	$-0.635471\pi$
$769$	−22.8981	−0.825725	−0.412863	+	0.910793i	$0.635471\pi$
$770$	0	0
$771$	−13.0000	−0.468184
$772$	−23.5257	−0.846709
$773$	−6.16515	−0.221745	−0.110873	−	0.993835i	$-0.535365\pi$
$773$	−6.16515	−0.221745	−0.110873	+	0.993835i	$0.535365\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−6.20520	−0.222754
$777$	8.75560	0.314106
$778$	−3.46410	−0.124194
$779$	36.6606	1.31350
$780$	0	0
$781$	0	0
$782$	−1.86932	−0.0668467
$783$	4.37780	0.156450
$784$	−17.2087	−0.614597
$785$	0	0
$786$	0.834849	0.0297781
$787$	−46.8607	−1.67040	−0.835202	−	0.549943i	$-0.814649\pi$
$787$	−46.8607	−1.67040	−0.835202	+	0.549943i	$0.814649\pi$
$788$	12.4104	0.442102
$789$	−11.4014	−0.405899
$790$	0	0
$791$	−2.74110	−0.0974623
$792$	0	0
$793$	7.91288	0.280995
$794$	−0.913701	−0.0324260
$795$	0	0
$796$	31.1996	1.10584
$797$	9.16515	0.324646	0.162323	−	0.986738i	$-0.448101\pi$
$797$	9.16515	0.324646	0.162323	+	0.986738i	$0.448101\pi$
$798$	−2.20880	−0.0781907
$799$	46.2331	1.63561
$800$	0	0
$801$	16.7477	0.591752
$802$	4.71940	0.166648
$803$	0	0
$804$	−26.4174	−0.931671
$805$	0	0
$806$	1.07803	0.0379720
$807$	19.1652	0.674645
$808$	−9.16515	−0.322429
$809$	−26.6482	−0.936901	−0.468451	−	0.883490i	$-0.655188\pi$
$809$	−26.6482	−0.936901	−0.468451	+	0.883490i	$0.655188\pi$
$810$	0	0
$811$	−9.19255	−0.322794	−0.161397	−	0.986890i	$-0.551600\pi$
$811$	−9.19255	−0.322794	−0.161397	+	0.986890i	$0.551600\pi$
$812$	7.16515	0.251448
$813$	23.4304	0.821739
$814$	0	0
$815$	0	0
$816$	−19.6048	−0.686304
$817$	−23.1652	−0.810446
$818$	−1.53901	−0.0538104
$819$	−0.834849	−0.0291720
$820$	0	0
$821$	−12.0290	−0.419815	−0.209908	−	0.977721i	$-0.567316\pi$
$821$	−12.0290	−0.419815	−0.209908	+	0.977721i	$0.567316\pi$
$822$	1.90285	0.0663695
$823$	−4.83485	−0.168532	−0.0842661	−	0.996443i	$-0.526855\pi$
$823$	−4.83485	−0.168532	−0.0842661	+	0.996443i	$0.526855\pi$
$824$	10.3923	0.362033
$825$	0	0
$826$	1.49545	0.0520335
$827$	10.7737	0.374638	0.187319	−	0.982299i	$-0.440020\pi$
$827$	10.7737	0.374638	0.187319	+	0.982299i	$0.440020\pi$
$828$	−1.04356	−0.0362662
$829$	10.1652	0.353050	0.176525	−	0.984296i	$-0.443514\pi$
$829$	10.1652	0.353050	0.176525	+	0.984296i	$0.443514\pi$
$830$	0	0
$831$	26.0761	0.904570
$832$	−3.12250	−0.108253
$833$	43.3013	1.50030
$834$	−9.20871	−0.318872
$835$	0	0
$836$	0	0
$837$	2.58258	0.0892669
$838$	14.7701	0.510225
$839$	−1.16515	−0.0402255	−0.0201127	−	0.999798i	$-0.506403\pi$
$839$	−1.16515	−0.0402255	−0.0201127	+	0.999798i	$0.506403\pi$
$840$	0	0
$841$	−9.83485	−0.339133
$842$	−0.456850	−0.0157441
$843$	−25.5438	−0.879776
$844$	−11.2861	−0.388484
$845$	0	0
$846$	−3.00725	−0.103391
$847$	0	0
$848$	−13.9564	−0.479266
$849$	−29.5402	−1.01382
$850$	0	0
$851$	−5.58258	−0.191368
$852$	16.4174	0.562452
$853$	24.8208	0.849848	0.424924	−	0.905229i	$-0.360301\pi$
$853$	24.8208	0.849848	0.424924	+	0.905229i	$0.360301\pi$
$854$	−3.61500	−0.123703
$855$	0	0
$856$	1.74773	0.0597361
$857$	−5.38685	−0.184011	−0.0920057	−	0.995758i	$-0.529328\pi$
$857$	−5.38685	−0.184011	−0.0920057	+	0.995758i	$0.529328\pi$
$858$	0	0
$859$	−26.3303	−0.898378	−0.449189	−	0.893437i	$-0.648287\pi$
$859$	−26.3303	−0.898378	−0.449189	+	0.893437i	$0.648287\pi$
$860$	0	0
$861$	−6.33030	−0.215736
$862$	−14.8348	−0.505277
$863$	18.3303	0.623971	0.311985	−	0.950087i	$-0.399006\pi$
$863$	18.3303	0.623971	0.311985	+	0.950087i	$0.399006\pi$
$864$	4.73930	0.161234
$865$	0	0
$866$	−6.73750	−0.228950
$867$	32.3303	1.09799
$868$	4.22690	0.143470
$869$	0	0
$870$	0	0
$871$	13.4750	0.456583
$872$	−8.83485	−0.299186
$873$	3.58258	0.121252
$874$	1.40833	0.0476376
$875$	0	0
$876$	28.0942	0.949216
$877$	24.6301	0.831700	0.415850	−	0.909433i	$-0.363484\pi$
$877$	24.6301	0.831700	0.415850	+	0.909433i	$0.363484\pi$
$878$	−2.04356	−0.0689668
$879$	27.9989	0.944378
$880$	0	0
$881$	−3.49545	−0.117765	−0.0588824	−	0.998265i	$-0.518754\pi$
$881$	−3.49545	−0.117765	−0.0588824	+	0.998265i	$0.518754\pi$
$882$	−2.81655	−0.0948382
$883$	32.7477	1.10205	0.551024	−	0.834489i	$-0.314237\pi$
$883$	32.7477	1.10205	0.551024	+	0.834489i	$0.314237\pi$
$884$	11.4955	0.386634
$885$	0	0
$886$	−16.4466	−0.552535
$887$	47.2421	1.58624	0.793118	−	0.609069i	$-0.208457\pi$
$887$	47.2421	1.58624	0.793118	+	0.609069i	$0.208457\pi$
$888$	16.5975	0.556976
$889$	5.49545	0.184312
$890$	0	0
$891$	0	0
$892$	41.4955	1.38937
$893$	−34.8317	−1.16560
$894$	−6.74773	−0.225678
$895$	0	0
$896$	10.0871	0.336987
$897$	0.532300	0.0177730
$898$	−1.29510	−0.0432181
$899$	11.3060	0.377076
$900$	0	0
$901$	35.1178	1.16994
$902$	0	0
$903$	4.00000	0.133112
$904$	−5.19615	−0.172821
$905$	0	0
$906$	−2.70417	−0.0898399
$907$	−10.8348	−0.359765	−0.179883	−	0.983688i	$-0.557572\pi$
$907$	−10.8348	−0.359765	−0.179883	+	0.983688i	$0.557572\pi$
$908$	32.8335	1.08962
$909$	5.29150	0.175508
$910$	0	0
$911$	25.5826	0.847589	0.423794	−	0.905758i	$-0.360698\pi$
$911$	25.5826	0.847589	0.423794	+	0.905758i	$0.360698\pi$
$912$	14.7701	0.489087
$913$	0	0
$914$	18.5045	0.612076
$915$	0	0
$916$	39.7042	1.31186
$917$	−1.66970	−0.0551383
$918$	−3.20871	−0.105903
$919$	42.1413	1.39011	0.695057	−	0.718955i	$-0.255379\pi$
$919$	42.1413	1.39011	0.695057	+	0.718955i	$0.255379\pi$
$920$	0	0
$921$	−6.92820	−0.228292
$922$	10.7477	0.353958
$923$	−8.37420	−0.275640
$924$	0	0
$925$	0	0
$926$	−1.82740	−0.0600521
$927$	−6.00000	−0.197066
$928$	20.7477	0.681078
$929$	6.33030	0.207690	0.103845	−	0.994593i	$-0.466885\pi$
$929$	6.33030	0.207690	0.103845	+	0.994593i	$0.466885\pi$
$930$	0	0
$931$	−32.6229	−1.06917
$932$	−19.1280	−0.626559
$933$	−15.5826	−0.510151
$934$	−6.28065	−0.205509
$935$	0	0
$936$	−1.58258	−0.0517281
$937$	2.93180	0.0957778	0.0478889	−	0.998853i	$-0.484751\pi$
$937$	2.93180	0.0957778	0.0478889	+	0.998853i	$0.484751\pi$
$938$	−6.15606	−0.201002
$939$	−11.5826	−0.377983
$940$	0	0
$941$	23.9071	0.779350	0.389675	−	0.920953i	$-0.372587\pi$
$941$	23.9071	0.779350	0.389675	+	0.920953i	$0.372587\pi$
$942$	−5.67290	−0.184833
$943$	4.03620	0.131437
$944$	−10.0000	−0.325472
$945$	0	0
$946$	0	0
$947$	36.9129	1.19951	0.599754	−	0.800185i	$-0.295265\pi$
$947$	36.9129	1.19951	0.599754	+	0.800185i	$0.295265\pi$
$948$	4.73930	0.153925
$949$	−14.3303	−0.465181
$950$	0	0
$951$	−7.83485	−0.254062
$952$	−11.1153	−0.360249
$953$	−45.6054	−1.47730	−0.738652	−	0.674087i	$-0.764537\pi$
$953$	−45.6054	−1.47730	−0.738652	+	0.674087i	$0.764537\pi$
$954$	−2.28425	−0.0739554
$955$	0	0
$956$	−29.7309	−0.961566
$957$	0	0
$958$	−11.4083	−0.368586
$959$	−3.80570	−0.122892
$960$	0	0
$961$	−24.3303	−0.784848
$962$	−4.00000	−0.128965
$963$	−1.00905	−0.0325162
$964$	50.4956	1.62635
$965$	0	0
$966$	−0.243181	−0.00782423
$967$	−16.0652	−0.516622	−0.258311	−	0.966062i	$-0.583166\pi$
$967$	−16.0652	−0.516622	−0.258311	+	0.966062i	$0.583166\pi$
$968$	0	0
$969$	−37.1652	−1.19392
$970$	0	0
$971$	53.5826	1.71955	0.859773	−	0.510676i	$-0.170605\pi$
$971$	53.5826	1.71955	0.859773	+	0.510676i	$0.170605\pi$
$972$	−1.79129	−0.0574556
$973$	18.4174	0.590436
$974$	−10.0109	−0.320770
$975$	0	0
$976$	24.1733	0.773767
$977$	−17.8348	−0.570587	−0.285294	−	0.958440i	$-0.592091\pi$
$977$	−17.8348	−0.570587	−0.285294	+	0.958440i	$0.592091\pi$
$978$	−8.03260	−0.256854
$979$	0	0
$980$	0	0
$981$	5.10080	0.162856
$982$	4.08712	0.130425
$983$	24.0780	0.767970	0.383985	−	0.923339i	$-0.374551\pi$
$983$	24.0780	0.767970	0.383985	+	0.923339i	$0.374551\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	−14.0471	−0.447351
$987$	6.01450	0.191444
$988$	−8.66061	−0.275531
$989$	−2.55040	−0.0810980
$990$	0	0
$991$	−19.7477	−0.627307	−0.313654	−	0.949537i	$-0.601553\pi$
$991$	−19.7477	−0.627307	−0.313654	+	0.949537i	$0.601553\pi$
$992$	12.2396	0.388608
$993$	5.41742	0.171917
$994$	3.82576	0.121346
$995$	0	0
$996$	−21.8890	−0.693580
$997$	−6.20520	−0.196521	−0.0982604	−	0.995161i	$-0.531328\pi$
$997$	−6.20520	−0.196521	−0.0982604	+	0.995161i	$0.531328\pi$
$998$	10.9644	0.347072
$999$	−9.58258	−0.303179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cz.1.3		4
5.2	odd	4		1815.2.c.f.364.5	yes	8
5.3	odd	4		1815.2.c.f.364.4	yes	8
5.4	even	2		9075.2.a.cs.1.2		4
11.10	odd	2	inner	9075.2.a.cz.1.2		4
55.32	even	4		1815.2.c.f.364.3	✓	8
55.43	even	4		1815.2.c.f.364.6	yes	8
55.54	odd	2		9075.2.a.cs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.c.f.364.3	✓	8	55.32	even	4
1815.2.c.f.364.4	yes	8	5.3	odd	4
1815.2.c.f.364.5	yes	8	5.2	odd	4
1815.2.c.f.364.6	yes	8	55.43	even	4
9075.2.a.cs.1.2		4	5.4	even	2
9075.2.a.cs.1.3		4	55.54	odd	2
9075.2.a.cz.1.2		4	11.10	odd	2	inner
9075.2.a.cz.1.3		4	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.456850 q^{2} +1.00000 q^{3} -1.79129 q^{4} +0.456850 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} -1.79129 q^{12} +0.913701 q^{13} -0.417424 q^{14} +2.79129 q^{16} -7.02355 q^{17} +0.456850 q^{18} +5.29150 q^{19} -0.913701 q^{21} +0.582576 q^{23} -1.73205 q^{24} +0.417424 q^{26} +1.00000 q^{27} +1.63670 q^{28} +4.37780 q^{29} +2.58258 q^{31} +4.73930 q^{32} -3.20871 q^{34} -1.79129 q^{36} -9.58258 q^{37} +2.41742 q^{38} +0.913701 q^{39} +6.92820 q^{41} -0.417424 q^{42} -4.37780 q^{43} +0.266150 q^{46} -6.58258 q^{47} +2.79129 q^{48} -6.16515 q^{49} -7.02355 q^{51} -1.63670 q^{52} -5.00000 q^{53} +0.456850 q^{54} +1.58258 q^{56} +5.29150 q^{57} +2.00000 q^{58} -3.58258 q^{59} +8.66025 q^{61} +1.17985 q^{62} -0.913701 q^{63} -3.41742 q^{64} +14.7477 q^{67} +12.5812 q^{68} +0.582576 q^{69} -9.16515 q^{71} -1.73205 q^{72} -15.6838 q^{73} -4.37780 q^{74} -9.47860 q^{76} +0.417424 q^{78} -2.64575 q^{79} +1.00000 q^{81} +3.16515 q^{82} +12.2197 q^{83} +1.63670 q^{84} -2.00000 q^{86} +4.37780 q^{87} +16.7477 q^{89} -0.834849 q^{91} -1.04356 q^{92} +2.58258 q^{93} -3.00725 q^{94} +4.73930 q^{96} +3.58258 q^{97} -2.81655 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{4} + 4 q^{9} + 2 q^{12} - 20 q^{14} + 2 q^{16} - 16 q^{23} + 20 q^{26} + 4 q^{27} - 8 q^{31} - 22 q^{34} + 2 q^{36} - 20 q^{37} + 28 q^{38} - 20 q^{42} - 8 q^{47} + 2 q^{48} + 12 q^{49} - 20 q^{53} - 12 q^{56} + 8 q^{58} + 4 q^{59} - 32 q^{64} + 4 q^{67} - 16 q^{69} + 20 q^{78} + 4 q^{81} - 24 q^{82} - 8 q^{86} + 12 q^{89} - 40 q^{91} - 50 q^{92} - 8 q^{93} - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^4 + 4 * q^9 + 2 * q^12 - 20 * q^14 + 2 * q^16 - 16 * q^23 + 20 * q^26 + 4 * q^27 - 8 * q^31 - 22 * q^34 + 2 * q^36 - 20 * q^37 + 28 * q^38 - 20 * q^42 - 8 * q^47 + 2 * q^48 + 12 * q^49 - 20 * q^53 - 12 * q^56 + 8 * q^58 + 4 * q^59 - 32 * q^64 + 4 * q^67 - 16 * q^69 + 20 * q^78 + 4 * q^81 - 24 * q^82 - 8 * q^86 + 12 * q^89 - 40 * q^91 - 50 * q^92 - 8 * q^93 - 4 * q^97

Introduction

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