LMFDB - Embedded newform 9075.2.a.cj.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:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.363328$ 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)

$f(q)$	$=$	$q+1.36333 q^{2} -1.00000 q^{3} -0.141336 q^{4} -1.36333 q^{6} +2.50466 q^{7} -2.91934 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.36333 q^{2} -1.00000 q^{3} -0.141336 q^{4} -1.36333 q^{6} +2.50466 q^{7} -2.91934 q^{8} +1.00000 q^{9} +0.141336 q^{12} +1.14134 q^{13} +3.41468 q^{14} -3.69735 q^{16} +7.64600 q^{17} +1.36333 q^{18} -1.77801 q^{19} -2.50466 q^{21} -1.41468 q^{23} +2.91934 q^{24} +1.55602 q^{26} -1.00000 q^{27} -0.354000 q^{28} +0.726656 q^{29} +2.85866 q^{31} +0.797984 q^{32} +10.4240 q^{34} -0.141336 q^{36} +8.42401 q^{37} -2.42401 q^{38} -1.14134 q^{39} -0.636672 q^{41} -3.41468 q^{42} -12.6974 q^{43} -1.92867 q^{46} -6.14134 q^{47} +3.69735 q^{48} -0.726656 q^{49} -7.64600 q^{51} -0.161312 q^{52} +12.0187 q^{53} -1.36333 q^{54} -7.31198 q^{56} +1.77801 q^{57} +0.990671 q^{58} -3.41468 q^{59} -4.59465 q^{61} +3.89730 q^{62} +2.50466 q^{63} +8.48262 q^{64} +9.32131 q^{67} -1.08066 q^{68} +1.41468 q^{69} +5.85866 q^{71} -2.91934 q^{72} +7.55602 q^{73} +11.4847 q^{74} +0.251297 q^{76} -1.55602 q^{78} -6.91934 q^{79} +1.00000 q^{81} -0.867993 q^{82} -6.17997 q^{83} +0.354000 q^{84} -17.3107 q^{86} -0.726656 q^{87} +3.45331 q^{89} +2.85866 q^{91} +0.199945 q^{92} -2.85866 q^{93} -8.37266 q^{94} -0.797984 q^{96} +19.4626 q^{97} -0.990671 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} - 3 q^{3} + 8 q^{4} - 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 - 3 * q^3 + 8 * q^4 - 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9	$ 3 q + 2 q^{2} - 3 q^{3} + 8 q^{4} - 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - 8 q^{12} - 5 q^{13} + 6 q^{14} + 10 q^{16} + 4 q^{17} + 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} - 3 q^{27} - 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} + 8 q^{36} + 18 q^{38} + 5 q^{39} - 4 q^{41} - 6 q^{42} - 17 q^{43} + 30 q^{46} - 10 q^{47} - 10 q^{48} + 2 q^{49} - 4 q^{51} - 30 q^{52} - 6 q^{53} - 2 q^{54} - 22 q^{56} - q^{57} + 24 q^{58} - 6 q^{59} + 3 q^{61} + 16 q^{62} - 3 q^{63} + 34 q^{64} + 7 q^{67} - 18 q^{68} + 26 q^{71} + 6 q^{72} + 10 q^{73} - 14 q^{74} + 24 q^{76} + 8 q^{78} - 6 q^{79} + 3 q^{81} + 10 q^{82} - 6 q^{83} + 20 q^{84} + 28 q^{86} + 2 q^{87} + 2 q^{89} + 17 q^{91} + 26 q^{92} - 17 q^{93} - 2 q^{94} - 34 q^{96} + 29 q^{97} - 24 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 - 3 * q^3 + 8 * q^4 - 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 - 8 * q^12 - 5 * q^13 + 6 * q^14 + 10 * q^16 + 4 * q^17 + 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 - 3 * q^27 - 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 + 8 * q^36 + 18 * q^38 + 5 * q^39 - 4 * q^41 - 6 * q^42 - 17 * q^43 + 30 * q^46 - 10 * q^47 - 10 * q^48 + 2 * q^49 - 4 * q^51 - 30 * q^52 - 6 * q^53 - 2 * q^54 - 22 * q^56 - q^57 + 24 * q^58 - 6 * q^59 + 3 * q^61 + 16 * q^62 - 3 * q^63 + 34 * q^64 + 7 * q^67 - 18 * q^68 + 26 * q^71 + 6 * q^72 + 10 * q^73 - 14 * q^74 + 24 * q^76 + 8 * q^78 - 6 * q^79 + 3 * q^81 + 10 * q^82 - 6 * q^83 + 20 * q^84 + 28 * q^86 + 2 * q^87 + 2 * q^89 + 17 * q^91 + 26 * q^92 - 17 * q^93 - 2 * q^94 - 34 * q^96 + 29 * q^97 - 24 * 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 $

$2$	1.36333	0.964019	0.482009	−	0.876166i	$-0.339907\pi$
$2$	1.36333	0.964019	0.482009	+	0.876166i	$0.339907\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.141336	−0.0706681
$5$	0	0
$5$	0	0
$6$	−1.36333	−0.556576
$7$	2.50466	0.946674	0.473337	−	0.880881i	$-0.343049\pi$
$7$	2.50466	0.946674	0.473337	+	0.880881i	$0.343049\pi$
$8$	−2.91934	−1.03214
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.141336	0.0408002
$13$	1.14134	0.316550	0.158275	−	0.987395i	$-0.449407\pi$
$13$	1.14134	0.316550	0.158275	+	0.987395i	$0.449407\pi$
$14$	3.41468	0.912612
$15$	0	0
$16$	−3.69735	−0.924338
$17$	7.64600	1.85443	0.927214	−	0.374533i	$-0.122197\pi$
$17$	7.64600	1.85443	0.927214	+	0.374533i	$0.122197\pi$
$18$	1.36333	0.321340
$19$	−1.77801	−0.407903	−0.203951	−	0.978981i	$-0.565378\pi$
$19$	−1.77801	−0.407903	−0.203951	+	0.978981i	$0.565378\pi$
$20$	0	0
$21$	−2.50466	−0.546563
$22$	0	0
$23$	−1.41468	−0.294981	−0.147491	−	0.989063i	$-0.547120\pi$
$23$	−1.41468	−0.294981	−0.147491	+	0.989063i	$0.547120\pi$
$24$	2.91934	0.595909
$25$	0	0
$26$	1.55602	0.305160
$27$	−1.00000	−0.192450
$28$	−0.354000	−0.0668996
$29$	0.726656	0.134937	0.0674684	−	0.997721i	$-0.478508\pi$
$29$	0.726656	0.134937	0.0674684	+	0.997721i	$0.478508\pi$
$30$	0	0
$31$	2.85866	0.513431	0.256716	−	0.966487i	$-0.417360\pi$
$31$	2.85866	0.513431	0.256716	+	0.966487i	$0.417360\pi$
$32$	0.797984	0.141065
$33$	0	0
$34$	10.4240	1.78770
$35$	0	0
$36$	−0.141336	−0.0235560
$37$	8.42401	1.38490	0.692449	−	0.721467i	$-0.256532\pi$
$37$	8.42401	1.38490	0.692449	+	0.721467i	$0.256532\pi$
$38$	−2.42401	−0.393226
$39$	−1.14134	−0.182760
$40$	0	0
$41$	−0.636672	−0.0994314	−0.0497157	−	0.998763i	$-0.515832\pi$
$41$	−0.636672	−0.0994314	−0.0497157	+	0.998763i	$0.515832\pi$
$42$	−3.41468	−0.526897
$43$	−12.6974	−1.93633	−0.968164	−	0.250317i	$-0.919465\pi$
$43$	−12.6974	−1.93633	−0.968164	+	0.250317i	$0.919465\pi$
$44$	0	0
$45$	0	0
$46$	−1.92867	−0.284367
$47$	−6.14134	−0.895806	−0.447903	−	0.894082i	$-0.647829\pi$
$47$	−6.14134	−0.895806	−0.447903	+	0.894082i	$0.647829\pi$
$48$	3.69735	0.533667
$49$	−0.726656	−0.103808
$50$	0	0
$51$	−7.64600	−1.07065
$52$	−0.161312	−0.0223700
$53$	12.0187	1.65089	0.825445	−	0.564483i	$-0.190924\pi$
$53$	12.0187	1.65089	0.825445	+	0.564483i	$0.190924\pi$
$54$	−1.36333	−0.185525
$55$	0	0
$56$	−7.31198	−0.977104
$57$	1.77801	0.235503
$58$	0.990671	0.130082
$59$	−3.41468	−0.444553	−0.222277	−	0.974984i	$-0.571349\pi$
$59$	−3.41468	−0.444553	−0.222277	+	0.974984i	$0.571349\pi$
$60$	0	0
$61$	−4.59465	−0.588285	−0.294142	−	0.955762i	$-0.595034\pi$
$61$	−4.59465	−0.588285	−0.294142	+	0.955762i	$0.595034\pi$
$62$	3.89730	0.494957
$63$	2.50466	0.315558
$64$	8.48262	1.06033
$65$	0	0
$66$	0	0
$67$	9.32131	1.13878	0.569389	−	0.822068i	$-0.307180\pi$
$67$	9.32131	1.13878	0.569389	+	0.822068i	$0.307180\pi$
$68$	−1.08066	−0.131049
$69$	1.41468	0.170307
$70$	0	0
$71$	5.85866	0.695295	0.347648	−	0.937625i	$-0.386981\pi$
$71$	5.85866	0.695295	0.347648	+	0.937625i	$0.386981\pi$
$72$	−2.91934	−0.344048
$73$	7.55602	0.884365	0.442182	−	0.896925i	$-0.354204\pi$
$73$	7.55602	0.884365	0.442182	+	0.896925i	$0.354204\pi$
$74$	11.4847	1.33507
$75$	0	0
$76$	0.251297	0.0288257
$77$	0	0
$78$	−1.55602	−0.176184
$79$	−6.91934	−0.778487	−0.389244	−	0.921135i	$-0.627264\pi$
$79$	−6.91934	−0.778487	−0.389244	+	0.921135i	$0.627264\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.867993	−0.0958537
$83$	−6.17997	−0.678340	−0.339170	−	0.940725i	$-0.610146\pi$
$83$	−6.17997	−0.678340	−0.339170	+	0.940725i	$0.610146\pi$
$84$	0.354000	0.0386245
$85$	0	0
$86$	−17.3107	−1.86666
$87$	−0.726656	−0.0779058
$88$	0	0
$89$	3.45331	0.366050	0.183025	−	0.983108i	$-0.441411\pi$
$89$	3.45331	0.366050	0.183025	+	0.983108i	$0.441411\pi$
$90$	0	0
$91$	2.85866	0.299669
$92$	0.199945	0.0208457
$93$	−2.85866	−0.296430
$94$	−8.37266	−0.863574
$95$	0	0
$96$	−0.797984	−0.0814439
$97$	19.4626	1.97613	0.988066	−	0.154032i	$-0.0492257\pi$
$97$	19.4626	1.97613	0.988066	+	0.154032i	$0.0492257\pi$
$98$	−0.990671	−0.100073
$99$	0	0
$100$	0	0
$101$	8.37266	0.833111	0.416555	−	0.909110i	$-0.363237\pi$
$101$	8.37266	0.833111	0.416555	+	0.909110i	$0.363237\pi$
$102$	−10.4240	−1.03213
$103$	−2.54669	−0.250933	−0.125466	−	0.992098i	$-0.540043\pi$
$103$	−2.54669	−0.250933	−0.125466	+	0.992098i	$0.540043\pi$
$104$	−3.33195	−0.326725
$105$	0	0
$106$	16.3854	1.59149
$107$	−14.5653	−1.40808	−0.704042	−	0.710158i	$-0.748624\pi$
$107$	−14.5653	−1.40808	−0.704042	+	0.710158i	$0.748624\pi$
$108$	0.141336	0.0136001
$109$	−2.41468	−0.231284	−0.115642	−	0.993291i	$-0.536893\pi$
$109$	−2.41468	−0.231284	−0.115642	+	0.993291i	$0.536893\pi$
$110$	0	0
$111$	−8.42401	−0.799571
$112$	−9.26063	−0.875047
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	2.42401	0.227029
$115$	0	0
$116$	−0.102703	−0.00953572
$117$	1.14134	0.105517
$118$	−4.65533	−0.428558
$119$	19.1507	1.75554
$120$	0	0
$121$	0	0
$122$	−6.26401	−0.567117
$123$	0.636672	0.0574068
$124$	−0.404032	−0.0362832
$125$	0	0
$126$	3.41468	0.304204
$127$	−15.2020	−1.34896	−0.674480	−	0.738293i	$-0.735632\pi$
$127$	−15.2020	−1.34896	−0.674480	+	0.738293i	$0.735632\pi$
$128$	9.96862	0.881110
$129$	12.6974	1.11794
$130$	0	0
$131$	−18.7453	−1.63779	−0.818893	−	0.573946i	$-0.805412\pi$
$131$	−18.7453	−1.63779	−0.818893	+	0.573946i	$0.805412\pi$
$132$	0	0
$133$	−4.45331	−0.386151
$134$	12.7080	1.09780
$135$	0	0
$136$	−22.3213	−1.91404
$137$	−8.74531	−0.747163	−0.373581	−	0.927597i	$-0.621870\pi$
$137$	−8.74531	−0.747163	−0.373581	+	0.927597i	$0.621870\pi$
$138$	1.92867	0.164180
$139$	22.3013	1.89157	0.945787	−	0.324787i	$-0.105293\pi$
$139$	22.3013	1.89157	0.945787	+	0.324787i	$0.105293\pi$
$140$	0	0
$141$	6.14134	0.517194
$142$	7.98728	0.670278
$143$	0	0
$144$	−3.69735	−0.308113
$145$	0	0
$146$	10.3013	0.852544
$147$	0.726656	0.0599336
$148$	−1.19062	−0.0978681
$149$	15.9287	1.30493	0.652464	−	0.757820i	$-0.273735\pi$
$149$	15.9287	1.30493	0.652464	+	0.757820i	$0.273735\pi$
$150$	0	0
$151$	9.88665	0.804564	0.402282	−	0.915516i	$-0.368217\pi$
$151$	9.88665	0.804564	0.402282	+	0.915516i	$0.368217\pi$
$152$	5.19062	0.421015
$153$	7.64600	0.618143
$154$	0	0
$155$	0	0
$156$	0.161312	0.0129153
$157$	0.132007	0.0105353	0.00526767	−	0.999986i	$-0.498323\pi$
$157$	0.132007	0.0105353	0.00526767	+	0.999986i	$0.498323\pi$
$158$	−9.43334	−0.750476
$159$	−12.0187	−0.953142
$160$	0	0
$161$	−3.54330	−0.279251
$162$	1.36333	0.107113
$163$	4.31198	0.337740	0.168870	−	0.985638i	$-0.445988\pi$
$163$	4.31198	0.337740	0.168870	+	0.985638i	$0.445988\pi$
$164$	0.0899847	0.00702663
$165$	0	0
$166$	−8.42533	−0.653932
$167$	−11.2920	−0.873801	−0.436901	−	0.899510i	$-0.643924\pi$
$167$	−11.2920	−0.873801	−0.436901	+	0.899510i	$0.643924\pi$
$168$	7.31198	0.564131
$169$	−11.6974	−0.899796
$170$	0	0
$171$	−1.77801	−0.135968
$172$	1.79459	0.136837
$173$	21.6460	1.64571	0.822857	−	0.568248i	$-0.192379\pi$
$173$	21.6460	1.64571	0.822857	+	0.568248i	$0.192379\pi$
$174$	−0.990671	−0.0751026
$175$	0	0
$176$	0	0
$177$	3.41468	0.256663
$178$	4.70800	0.352879
$179$	16.1413	1.20646	0.603230	−	0.797567i	$-0.293880\pi$
$179$	16.1413	1.20646	0.603230	+	0.797567i	$0.293880\pi$
$180$	0	0
$181$	20.7546	1.54268	0.771340	−	0.636423i	$-0.219587\pi$
$181$	20.7546	1.54268	0.771340	+	0.636423i	$0.219587\pi$
$182$	3.89730	0.288887
$183$	4.59465	0.339646
$184$	4.12994	0.304463
$185$	0	0
$186$	−3.89730	−0.285764
$187$	0	0
$188$	0.867993	0.0633049
$189$	−2.50466	−0.182188
$190$	0	0
$191$	3.31198	0.239646	0.119823	−	0.992795i	$-0.461767\pi$
$191$	3.31198	0.239646	0.119823	+	0.992795i	$0.461767\pi$
$192$	−8.48262	−0.612180
$193$	−2.13201	−0.153465	−0.0767326	−	0.997052i	$-0.524449\pi$
$193$	−2.13201	−0.153465	−0.0767326	+	0.997052i	$0.524449\pi$
$194$	26.5340	1.90503
$195$	0	0
$196$	0.102703	0.00733591
$197$	17.9287	1.27737	0.638683	−	0.769470i	$-0.279480\pi$
$197$	17.9287	1.27737	0.638683	+	0.769470i	$0.279480\pi$
$198$	0	0
$199$	15.1600	1.07466	0.537332	−	0.843371i	$-0.319432\pi$
$199$	15.1600	1.07466	0.537332	+	0.843371i	$0.319432\pi$
$200$	0	0
$201$	−9.32131	−0.657474
$202$	11.4147	0.803134
$203$	1.82003	0.127741
$204$	1.08066	0.0756611
$205$	0	0
$206$	−3.47197	−0.241904
$207$	−1.41468	−0.0983270
$208$	−4.21992	−0.292599
$209$	0	0
$210$	0	0
$211$	13.3213	0.917076	0.458538	−	0.888675i	$-0.348373\pi$
$211$	13.3213	0.917076	0.458538	+	0.888675i	$0.348373\pi$
$212$	−1.69867	−0.116665
$213$	−5.85866	−0.401429
$214$	−19.8573	−1.35742
$215$	0	0
$216$	2.91934	0.198636
$217$	7.15999	0.486052
$218$	−3.29200	−0.222962
$219$	−7.55602	−0.510588
$220$	0	0
$221$	8.72666	0.587018
$222$	−11.4847	−0.770802
$223$	12.2534	0.820546	0.410273	−	0.911963i	$-0.365433\pi$
$223$	12.2534	0.820546	0.410273	+	0.911963i	$0.365433\pi$
$224$	1.99868	0.133543
$225$	0	0
$226$	−8.17997	−0.544123
$227$	8.74531	0.580447	0.290223	−	0.956959i	$-0.406270\pi$
$227$	8.74531	0.580447	0.290223	+	0.956959i	$0.406270\pi$
$228$	−0.251297	−0.0166425
$229$	−11.4626	−0.757473	−0.378736	−	0.925505i	$-0.623641\pi$
$229$	−11.4626	−0.757473	−0.378736	+	0.925505i	$0.623641\pi$
$230$	0	0
$231$	0	0
$232$	−2.12136	−0.139274
$233$	7.48469	0.490338	0.245169	−	0.969480i	$-0.421157\pi$
$233$	7.48469	0.490338	0.245169	+	0.969480i	$0.421157\pi$
$234$	1.55602	0.101720
$235$	0	0
$236$	0.482618	0.0314157
$237$	6.91934	0.449460
$238$	26.1086	1.69237
$239$	−12.9066	−0.834860	−0.417430	−	0.908709i	$-0.637069\pi$
$239$	−12.9066	−0.834860	−0.417430	+	0.908709i	$0.637069\pi$
$240$	0	0
$241$	16.8773	1.08716	0.543582	−	0.839356i	$-0.317068\pi$
$241$	16.8773	1.08716	0.543582	+	0.839356i	$0.317068\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0.649390	0.0415729
$245$	0	0
$246$	0.867993	0.0553412
$247$	−2.02930	−0.129122
$248$	−8.34542	−0.529935
$249$	6.17997	0.391640
$250$	0	0
$251$	2.28267	0.144081	0.0720405	−	0.997402i	$-0.477049\pi$
$251$	2.28267	0.144081	0.0720405	+	0.997402i	$0.477049\pi$
$252$	−0.354000	−0.0222999
$253$	0	0
$254$	−20.7253	−1.30042
$255$	0	0
$256$	−3.37473	−0.210920
$257$	26.8667	1.67590	0.837949	−	0.545749i	$-0.183755\pi$
$257$	26.8667	1.67590	0.837949	+	0.545749i	$0.183755\pi$
$258$	17.3107	1.07771
$259$	21.0993	1.31105
$260$	0	0
$261$	0.726656	0.0449789
$262$	−25.5560	−1.57886
$263$	12.0187	0.741102	0.370551	−	0.928812i	$-0.379169\pi$
$263$	12.0187	0.741102	0.370551	+	0.928812i	$0.379169\pi$
$264$	0	0
$265$	0	0
$266$	−6.07133	−0.372257
$267$	−3.45331	−0.211339
$268$	−1.31744	−0.0804753
$269$	−14.3854	−0.877092	−0.438546	−	0.898709i	$-0.644506\pi$
$269$	−14.3854	−0.877092	−0.438546	+	0.898709i	$0.644506\pi$
$270$	0	0
$271$	−28.6553	−1.74069	−0.870344	−	0.492445i	$-0.836103\pi$
$271$	−28.6553	−1.74069	−0.870344	+	0.492445i	$0.836103\pi$
$272$	−28.2700	−1.71412
$273$	−2.85866	−0.173014
$274$	−11.9227	−0.720279
$275$	0	0
$276$	−0.199945	−0.0120353
$277$	−29.1600	−1.75205	−0.876027	−	0.482262i	$-0.839815\pi$
$277$	−29.1600	−1.75205	−0.876027	+	0.482262i	$0.839815\pi$
$278$	30.4040	1.82351
$279$	2.85866	0.171144
$280$	0	0
$281$	5.36333	0.319949	0.159975	−	0.987121i	$-0.448859\pi$
$281$	5.36333	0.319949	0.159975	+	0.987121i	$0.448859\pi$
$282$	8.37266	0.498584
$283$	−10.0420	−0.596936	−0.298468	−	0.954420i	$-0.596476\pi$
$283$	−10.0420	−0.596936	−0.298468	+	0.954420i	$0.596476\pi$
$284$	−0.828041	−0.0491352
$285$	0	0
$286$	0	0
$287$	−1.59465	−0.0941292
$288$	0.797984	0.0470216
$289$	41.4613	2.43890
$290$	0	0
$291$	−19.4626	−1.14092
$292$	−1.06794	−0.0624963
$293$	20.3540	1.18909	0.594547	−	0.804061i	$-0.297332\pi$
$293$	20.3540	1.18909	0.594547	+	0.804061i	$0.297332\pi$
$294$	0.990671	0.0577771
$295$	0	0
$296$	−24.5926	−1.42941
$297$	0	0
$298$	21.7160	1.25797
$299$	−1.61462	−0.0933762
$300$	0	0
$301$	−31.8026	−1.83307
$302$	13.4787	0.775615
$303$	−8.37266	−0.480997
$304$	6.57392	0.377040
$305$	0	0
$306$	10.4240	0.595901
$307$	25.6226	1.46236	0.731181	−	0.682184i	$-0.238970\pi$
$307$	25.6226	1.46236	0.731181	+	0.682184i	$0.238970\pi$
$308$	0	0
$309$	2.54669	0.144876
$310$	0	0
$311$	11.9414	0.677134	0.338567	−	0.940942i	$-0.390058\pi$
$311$	11.9414	0.677134	0.338567	+	0.940942i	$0.390058\pi$
$312$	3.33195	0.188635
$313$	−17.1986	−0.972124	−0.486062	−	0.873924i	$-0.661567\pi$
$313$	−17.1986	−0.972124	−0.486062	+	0.873924i	$0.661567\pi$
$314$	0.179969	0.0101563
$315$	0	0
$316$	0.977953	0.0550142
$317$	22.7453	1.27750	0.638752	−	0.769413i	$-0.279451\pi$
$317$	22.7453	1.27750	0.638752	+	0.769413i	$0.279451\pi$
$318$	−16.3854	−0.918846
$319$	0	0
$320$	0	0
$321$	14.5653	0.812958
$322$	−4.83068	−0.269203
$323$	−13.5946	−0.756427
$324$	−0.141336	−0.00785201
$325$	0	0
$326$	5.87864	0.325588
$327$	2.41468	0.133532
$328$	1.85866	0.102628
$329$	−15.3820	−0.848036
$330$	0	0
$331$	25.2733	1.38915	0.694574	−	0.719421i	$-0.255593\pi$
$331$	25.2733	1.38915	0.694574	+	0.719421i	$0.255593\pi$
$332$	0.873453	0.0479370
$333$	8.42401	0.461633
$334$	−15.3947	−0.842361
$335$	0	0
$336$	9.26063	0.505209
$337$	−17.6880	−0.963528	−0.481764	−	0.876301i	$-0.660004\pi$
$337$	−17.6880	−0.963528	−0.481764	+	0.876301i	$0.660004\pi$
$338$	−15.9473	−0.867420
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	−2.42401	−0.131075
$343$	−19.3527	−1.04495
$344$	37.0679	1.99857
$345$	0	0
$346$	29.5106	1.58650
$347$	20.1027	1.07917	0.539585	−	0.841931i	$-0.318581\pi$
$347$	20.1027	1.07917	0.539585	+	0.841931i	$0.318581\pi$
$348$	0.102703	0.00550545
$349$	10.9907	0.588317	0.294159	−	0.955757i	$-0.404961\pi$
$349$	10.9907	0.588317	0.294159	+	0.955757i	$0.404961\pi$
$350$	0	0
$351$	−1.14134	−0.0609200
$352$	0	0
$353$	−16.7267	−0.890270	−0.445135	−	0.895463i	$-0.646844\pi$
$353$	−16.7267	−0.890270	−0.445135	+	0.895463i	$0.646844\pi$
$354$	4.65533	0.247428
$355$	0	0
$356$	−0.488078	−0.0258681
$357$	−19.1507	−1.01356
$358$	22.0059	1.16305
$359$	13.5933	0.717429	0.358714	−	0.933447i	$-0.383215\pi$
$359$	13.5933	0.717429	0.358714	+	0.933447i	$0.383215\pi$
$360$	0	0
$361$	−15.8387	−0.833615
$362$	28.2954	1.48717
$363$	0	0
$364$	−0.404032	−0.0211771
$365$	0	0
$366$	6.26401	0.327425
$367$	−28.9987	−1.51372	−0.756859	−	0.653578i	$-0.773267\pi$
$367$	−28.9987	−1.51372	−0.756859	+	0.653578i	$0.773267\pi$
$368$	5.23057	0.272662
$369$	−0.636672	−0.0331438
$370$	0	0
$371$	30.1027	1.56285
$372$	0.404032	0.0209481
$373$	−17.8867	−0.926136	−0.463068	−	0.886323i	$-0.653251\pi$
$373$	−17.8867	−0.926136	−0.463068	+	0.886323i	$0.653251\pi$
$374$	0	0
$375$	0	0
$376$	17.9287	0.924601
$377$	0.829359	0.0427142
$378$	−3.41468	−0.175632
$379$	23.7839	1.22170	0.610850	−	0.791747i	$-0.290828\pi$
$379$	23.7839	1.22170	0.610850	+	0.791747i	$0.290828\pi$
$380$	0	0
$381$	15.2020	0.778823
$382$	4.51531	0.231023
$383$	31.7360	1.62163	0.810817	−	0.585300i	$-0.199023\pi$
$383$	31.7360	1.62163	0.810817	+	0.585300i	$0.199023\pi$
$384$	−9.96862	−0.508709
$385$	0	0
$386$	−2.90663	−0.147943
$387$	−12.6974	−0.645443
$388$	−2.75077	−0.139649
$389$	25.7546	1.30581	0.652906	−	0.757439i	$-0.273550\pi$
$389$	25.7546	1.30581	0.652906	+	0.757439i	$0.273550\pi$
$390$	0	0
$391$	−10.8166	−0.547021
$392$	2.12136	0.107145
$393$	18.7453	0.945576
$394$	24.4427	1.23140
$395$	0	0
$396$	0	0
$397$	3.03863	0.152505	0.0762523	−	0.997089i	$-0.475705\pi$
$397$	3.03863	0.152505	0.0762523	+	0.997089i	$0.475705\pi$
$398$	20.6680	1.03600
$399$	4.45331	0.222945
$400$	0	0
$401$	−18.1800	−0.907864	−0.453932	−	0.891036i	$-0.649979\pi$
$401$	−18.1800	−0.907864	−0.453932	+	0.891036i	$0.649979\pi$
$402$	−12.7080	−0.633817
$403$	3.26270	0.162526
$404$	−1.18336	−0.0588743
$405$	0	0
$406$	2.48130	0.123145
$407$	0	0
$408$	22.3213	1.10507
$409$	20.5946	1.01834	0.509170	−	0.860666i	$-0.329952\pi$
$409$	20.5946	1.01834	0.509170	+	0.860666i	$0.329952\pi$
$410$	0	0
$411$	8.74531	0.431375
$412$	0.359939	0.0177329
$413$	−8.55263	−0.420847
$414$	−1.92867	−0.0947891
$415$	0	0
$416$	0.910768	0.0446541
$417$	−22.3013	−1.09210
$418$	0	0
$419$	32.7253	1.59874	0.799369	−	0.600841i	$-0.205167\pi$
$419$	32.7253	1.59874	0.799369	+	0.600841i	$0.205167\pi$
$420$	0	0
$421$	32.5640	1.58707	0.793537	−	0.608522i	$-0.208237\pi$
$421$	32.5640	1.58707	0.793537	+	0.608522i	$0.208237\pi$
$422$	18.1613	0.884079
$423$	−6.14134	−0.298602
$424$	−35.0866	−1.70396
$425$	0	0
$426$	−7.98728	−0.386985
$427$	−11.5081	−0.556914
$428$	2.05861	0.0995066
$429$	0	0
$430$	0	0
$431$	−1.82003	−0.0876678	−0.0438339	−	0.999039i	$-0.513957\pi$
$431$	−1.82003	−0.0876678	−0.0438339	+	0.999039i	$0.513957\pi$
$432$	3.69735	0.177889
$433$	29.8280	1.43344	0.716722	−	0.697359i	$-0.245642\pi$
$433$	29.8280	1.43344	0.716722	+	0.697359i	$0.245642\pi$
$434$	9.76142	0.468563
$435$	0	0
$436$	0.341281	0.0163444
$437$	2.51531	0.120324
$438$	−10.3013	−0.492217
$439$	−18.2220	−0.869688	−0.434844	−	0.900506i	$-0.643197\pi$
$439$	−18.2220	−0.869688	−0.434844	+	0.900506i	$0.643197\pi$
$440$	0	0
$441$	−0.726656	−0.0346027
$442$	11.8973	0.565897
$443$	22.2627	1.05773	0.528866	−	0.848705i	$-0.322617\pi$
$443$	22.2627	1.05773	0.528866	+	0.848705i	$0.322617\pi$
$444$	1.19062	0.0565042
$445$	0	0
$446$	16.7054	0.791022
$447$	−15.9287	−0.753400
$448$	21.2461	1.00378
$449$	25.8760	1.22116	0.610582	−	0.791953i	$-0.290936\pi$
$449$	25.8760	1.22116	0.610582	+	0.791953i	$0.290936\pi$
$450$	0	0
$451$	0	0
$452$	0.848017	0.0398874
$453$	−9.88665	−0.464515
$454$	11.9227	0.559562
$455$	0	0
$456$	−5.19062	−0.243073
$457$	−10.4626	−0.489422	−0.244711	−	0.969596i	$-0.578693\pi$
$457$	−10.4626	−0.489422	−0.244711	+	0.969596i	$0.578693\pi$
$458$	−15.6273	−0.730218
$459$	−7.64600	−0.356885
$460$	0	0
$461$	−11.8387	−0.551383	−0.275691	−	0.961246i	$-0.588907\pi$
$461$	−11.8387	−0.551383	−0.275691	+	0.961246i	$0.588907\pi$
$462$	0	0
$463$	−27.7546	−1.28987	−0.644934	−	0.764238i	$-0.723115\pi$
$463$	−27.7546	−1.28987	−0.644934	+	0.764238i	$0.723115\pi$
$464$	−2.68670	−0.124727
$465$	0	0
$466$	10.2041	0.472695
$467$	−20.8294	−0.963868	−0.481934	−	0.876208i	$-0.660065\pi$
$467$	−20.8294	−0.963868	−0.481934	+	0.876208i	$0.660065\pi$
$468$	−0.161312	−0.00745665
$469$	23.3467	1.07805
$470$	0	0
$471$	−0.132007	−0.00608258
$472$	9.96862	0.458843
$473$	0	0
$474$	9.43334	0.433288
$475$	0	0
$476$	−2.70668	−0.124061
$477$	12.0187	0.550297
$478$	−17.5960	−0.804821
$479$	−2.64939	−0.121054	−0.0605269	−	0.998167i	$-0.519278\pi$
$479$	−2.64939	−0.121054	−0.0605269	+	0.998167i	$0.519278\pi$
$480$	0	0
$481$	9.61462	0.438389
$482$	23.0093	1.04805
$483$	3.54330	0.161226
$484$	0	0
$485$	0	0
$486$	−1.36333	−0.0618418
$487$	−40.5360	−1.83686	−0.918432	−	0.395580i	$-0.870544\pi$
$487$	−40.5360	−1.83686	−0.918432	+	0.395580i	$0.870544\pi$
$488$	13.4134	0.607194
$489$	−4.31198	−0.194994
$490$	0	0
$491$	16.1214	0.727547	0.363773	−	0.931487i	$-0.381488\pi$
$491$	16.1214	0.727547	0.363773	+	0.931487i	$0.381488\pi$
$492$	−0.0899847	−0.00405682
$493$	5.55602	0.250230
$494$	−2.76661	−0.124476
$495$	0	0
$496$	−10.5695	−0.474584
$497$	14.6740	0.658218
$498$	8.42533	0.377548
$499$	−28.0666	−1.25643	−0.628217	−	0.778038i	$-0.716215\pi$
$499$	−28.0666	−1.25643	−0.628217	+	0.778038i	$0.716215\pi$
$500$	0	0
$501$	11.2920	0.504489
$502$	3.11203	0.138897
$503$	0.906626	0.0404245	0.0202122	−	0.999796i	$-0.493566\pi$
$503$	0.906626	0.0404245	0.0202122	+	0.999796i	$0.493566\pi$
$504$	−7.31198	−0.325701
$505$	0	0
$506$	0	0
$507$	11.6974	0.519498
$508$	2.14859	0.0953284
$509$	−14.3013	−0.633895	−0.316948	−	0.948443i	$-0.602658\pi$
$509$	−14.3013	−0.633895	−0.316948	+	0.948443i	$0.602658\pi$
$510$	0	0
$511$	18.9253	0.837205
$512$	−24.5381	−1.08444
$513$	1.77801	0.0785010
$514$	36.6281	1.61560
$515$	0	0
$516$	−1.79459	−0.0790026
$517$	0	0
$518$	28.7653	1.26387
$519$	−21.6460	−0.950154
$520$	0	0
$521$	20.1214	0.881533	0.440766	−	0.897622i	$-0.354707\pi$
$521$	20.1214	0.881533	0.440766	+	0.897622i	$0.354707\pi$
$522$	0.990671	0.0433605
$523$	−12.8459	−0.561714	−0.280857	−	0.959750i	$-0.590619\pi$
$523$	−12.8459	−0.561714	−0.280857	+	0.959750i	$0.590619\pi$
$524$	2.64939	0.115739
$525$	0	0
$526$	16.3854	0.714436
$527$	21.8573	0.952121
$528$	0	0
$529$	−20.9987	−0.912986
$530$	0	0
$531$	−3.41468	−0.148184
$532$	0.629414	0.0272886
$533$	−0.726656	−0.0314750
$534$	−4.70800	−0.203735
$535$	0	0
$536$	−27.2121	−1.17538
$537$	−16.1413	−0.696550
$538$	−19.6120	−0.845533
$539$	0	0
$540$	0	0
$541$	−17.6040	−0.756854	−0.378427	−	0.925631i	$-0.623535\pi$
$541$	−17.6040	−0.756854	−0.378427	+	0.925631i	$0.623535\pi$
$542$	−39.0666	−1.67805
$543$	−20.7546	−0.890667
$544$	6.10138	0.261595
$545$	0	0
$546$	−3.89730	−0.166789
$547$	19.9873	0.854594	0.427297	−	0.904111i	$-0.359466\pi$
$547$	19.9873	0.854594	0.427297	+	0.904111i	$0.359466\pi$
$548$	1.23603	0.0528005
$549$	−4.59465	−0.196095
$550$	0	0
$551$	−1.29200	−0.0550411
$552$	−4.12994	−0.175782
$553$	−17.3306	−0.736974
$554$	−39.7546	−1.68901
$555$	0	0
$556$	−3.15198	−0.133674
$557$	−41.5933	−1.76237	−0.881183	−	0.472775i	$-0.843252\pi$
$557$	−41.5933	−1.76237	−0.881183	+	0.472775i	$0.843252\pi$
$558$	3.89730	0.164986
$559$	−14.4919	−0.612944
$560$	0	0
$561$	0	0
$562$	7.31198	0.308437
$563$	2.28267	0.0962032	0.0481016	−	0.998842i	$-0.484683\pi$
$563$	2.28267	0.0962032	0.0481016	+	0.998842i	$0.484683\pi$
$564$	−0.867993	−0.0365491
$565$	0	0
$566$	−13.6906	−0.575458
$567$	2.50466	0.105186
$568$	−17.1035	−0.717645
$569$	2.37266	0.0994670	0.0497335	−	0.998763i	$-0.484163\pi$
$569$	2.37266	0.0994670	0.0497335	+	0.998763i	$0.484163\pi$
$570$	0	0
$571$	31.9053	1.33520	0.667598	−	0.744522i	$-0.267323\pi$
$571$	31.9053	1.33520	0.667598	+	0.744522i	$0.267323\pi$
$572$	0	0
$573$	−3.31198	−0.138360
$574$	−2.17403	−0.0907423
$575$	0	0
$576$	8.48262	0.353442
$577$	−3.28267	−0.136659	−0.0683297	−	0.997663i	$-0.521767\pi$
$577$	−3.28267	−0.136659	−0.0683297	+	0.997663i	$0.521767\pi$
$578$	56.5254	2.35115
$579$	2.13201	0.0886032
$580$	0	0
$581$	−15.4787	−0.642167
$582$	−26.5340	−1.09987
$583$	0	0
$584$	−22.0586	−0.912792
$585$	0	0
$586$	27.7492	1.14631
$587$	−26.3400	−1.08717	−0.543583	−	0.839355i	$-0.682933\pi$
$587$	−26.3400	−1.08717	−0.543583	+	0.839355i	$0.682933\pi$
$588$	−0.102703	−0.00423539
$589$	−5.08273	−0.209430
$590$	0	0
$591$	−17.9287	−0.737487
$592$	−31.1465	−1.28011
$593$	−32.9694	−1.35389	−0.676945	−	0.736034i	$-0.736697\pi$
$593$	−32.9694	−1.35389	−0.676945	+	0.736034i	$0.736697\pi$
$594$	0	0
$595$	0	0
$596$	−2.25130	−0.0922167
$597$	−15.1600	−0.620457
$598$	−2.20126	−0.0900164
$599$	24.5454	1.00290	0.501448	−	0.865188i	$-0.332801\pi$
$599$	24.5454	1.00290	0.501448	+	0.865188i	$0.332801\pi$
$600$	0	0
$601$	9.70668	0.395944	0.197972	−	0.980208i	$-0.436565\pi$
$601$	9.70668	0.395944	0.197972	+	0.980208i	$0.436565\pi$
$602$	−43.3574	−1.76712
$603$	9.32131	0.379593
$604$	−1.39734	−0.0568570
$605$	0	0
$606$	−11.4147	−0.463690
$607$	1.73599	0.0704615	0.0352307	−	0.999379i	$-0.488783\pi$
$607$	1.73599	0.0704615	0.0352307	+	0.999379i	$0.488783\pi$
$608$	−1.41882	−0.0575408
$609$	−1.82003	−0.0737514
$610$	0	0
$611$	−7.00933	−0.283567
$612$	−1.08066	−0.0436829
$613$	12.0187	0.485429	0.242715	−	0.970098i	$-0.421962\pi$
$613$	12.0187	0.485429	0.242715	+	0.970098i	$0.421962\pi$
$614$	34.9321	1.40974
$615$	0	0
$616$	0	0
$617$	−42.1587	−1.69724	−0.848622	−	0.528999i	$-0.822567\pi$
$617$	−42.1587	−1.69724	−0.848622	+	0.528999i	$0.822567\pi$
$618$	3.47197	0.139663
$619$	9.03863	0.363293	0.181647	−	0.983364i	$-0.441857\pi$
$619$	9.03863	0.363293	0.181647	+	0.983364i	$0.441857\pi$
$620$	0	0
$621$	1.41468	0.0567691
$622$	16.2800	0.652770
$623$	8.64939	0.346530
$624$	4.21992	0.168932
$625$	0	0
$626$	−23.4474	−0.937146
$627$	0	0
$628$	−0.0186574	−0.000744512 0
$629$	64.4100	2.56819
$630$	0	0
$631$	10.9614	0.436365	0.218183	−	0.975908i	$-0.429987\pi$
$631$	10.9614	0.436365	0.218183	+	0.975908i	$0.429987\pi$
$632$	20.1999	0.803511
$633$	−13.3213	−0.529474
$634$	31.0093	1.23154
$635$	0	0
$636$	1.69867	0.0673567
$637$	−0.829359	−0.0328604
$638$	0	0
$639$	5.85866	0.231765
$640$	0	0
$641$	−44.1587	−1.74416	−0.872081	−	0.489361i	$-0.837230\pi$
$641$	−44.1587	−1.74416	−0.872081	+	0.489361i	$0.837230\pi$
$642$	19.8573	0.783707
$643$	−22.5840	−0.890626	−0.445313	−	0.895375i	$-0.646908\pi$
$643$	−22.5840	−0.890626	−0.445313	+	0.895375i	$0.646908\pi$
$644$	0.500796	0.0197341
$645$	0	0
$646$	−18.5340	−0.729209
$647$	13.3561	0.525081	0.262541	−	0.964921i	$-0.415440\pi$
$647$	13.3561	0.525081	0.262541	+	0.964921i	$0.415440\pi$
$648$	−2.91934	−0.114683
$649$	0	0
$650$	0	0
$651$	−7.15999	−0.280622
$652$	−0.609438	−0.0238674
$653$	−37.5933	−1.47114	−0.735570	−	0.677448i	$-0.763086\pi$
$653$	−37.5933	−1.47114	−0.735570	+	0.677448i	$0.763086\pi$
$654$	3.29200	0.128727
$655$	0	0
$656$	2.35400	0.0919082
$657$	7.55602	0.294788
$658$	−20.9707	−0.817523
$659$	−7.00933	−0.273045	−0.136522	−	0.990637i	$-0.543593\pi$
$659$	−7.00933	−0.273045	−0.136522	+	0.990637i	$0.543593\pi$
$660$	0	0
$661$	26.6226	1.03550	0.517750	−	0.855532i	$-0.326770\pi$
$661$	26.6226	1.03550	0.517750	+	0.855532i	$0.326770\pi$
$662$	34.4559	1.33917
$663$	−8.72666	−0.338915
$664$	18.0415	0.700144
$665$	0	0
$666$	11.4847	0.445023
$667$	−1.02799	−0.0398038
$668$	1.59597	0.0617498
$669$	−12.2534	−0.473743
$670$	0	0
$671$	0	0
$672$	−1.99868	−0.0771008
$673$	35.6960	1.37598	0.687990	−	0.725720i	$-0.258493\pi$
$673$	35.6960	1.37598	0.687990	+	0.725720i	$0.258493\pi$
$674$	−24.1146	−0.928859
$675$	0	0
$676$	1.65326	0.0635869
$677$	33.2920	1.27952	0.639758	−	0.768577i	$-0.279035\pi$
$677$	33.2920	1.27952	0.639758	+	0.768577i	$0.279035\pi$
$678$	8.17997	0.314150
$679$	48.7474	1.87075
$680$	0	0
$681$	−8.74531	−0.335121
$682$	0	0
$683$	36.1787	1.38434	0.692169	−	0.721736i	$-0.256655\pi$
$683$	36.1787	1.38434	0.692169	+	0.721736i	$0.256655\pi$
$684$	0.251297	0.00960857
$685$	0	0
$686$	−26.3841	−1.00735
$687$	11.4626	0.437327
$688$	46.9466	1.78982
$689$	13.7173	0.522589
$690$	0	0
$691$	35.3293	1.34399	0.671995	−	0.740555i	$-0.265438\pi$
$691$	35.3293	1.34399	0.671995	+	0.740555i	$0.265438\pi$
$692$	−3.05936	−0.116299
$693$	0	0
$694$	27.4066	1.04034
$695$	0	0
$696$	2.12136	0.0804100
$697$	−4.86799	−0.184388
$698$	14.9839	0.567149
$699$	−7.48469	−0.283097
$700$	0	0
$701$	49.4006	1.86584	0.932918	−	0.360088i	$-0.117253\pi$
$701$	49.4006	1.86584	0.932918	+	0.360088i	$0.117253\pi$
$702$	−1.55602	−0.0587280
$703$	−14.9780	−0.564904
$704$	0	0
$705$	0	0
$706$	−22.8039	−0.858237
$707$	20.9707	0.788684
$708$	−0.482618	−0.0181379
$709$	−11.1027	−0.416971	−0.208485	−	0.978025i	$-0.566853\pi$
$709$	−11.1027	−0.416971	−0.208485	+	0.978025i	$0.566853\pi$
$710$	0	0
$711$	−6.91934	−0.259496
$712$	−10.0814	−0.377817
$713$	−4.04409	−0.151452
$714$	−26.1086	−0.977091
$715$	0	0
$716$	−2.28135	−0.0852582
$717$	12.9066	0.482007
$718$	18.5322	0.691614
$719$	−43.9787	−1.64013	−0.820064	−	0.572271i	$-0.806062\pi$
$719$	−43.9787	−1.64013	−0.820064	+	0.572271i	$0.806062\pi$
$720$	0	0
$721$	−6.37860	−0.237551
$722$	−21.5933	−0.803621
$723$	−16.8773	−0.627674
$724$	−2.93338	−0.109018
$725$	0	0
$726$	0	0
$727$	44.1880	1.63884	0.819421	−	0.573193i	$-0.194295\pi$
$727$	44.1880	1.63884	0.819421	+	0.573193i	$0.194295\pi$
$728$	−8.34542	−0.309302
$729$	1.00000	0.0370370
$730$	0	0
$731$	−97.0840	−3.59078
$732$	−0.649390	−0.0240021
$733$	−47.3879	−1.75031	−0.875156	−	0.483840i	$-0.839242\pi$
$733$	−47.3879	−1.75031	−0.875156	+	0.483840i	$0.839242\pi$
$734$	−39.5347	−1.45925
$735$	0	0
$736$	−1.12889	−0.0416115
$737$	0	0
$738$	−0.867993	−0.0319512
$739$	13.0407	0.479710	0.239855	−	0.970809i	$-0.422900\pi$
$739$	13.0407	0.479710	0.239855	+	0.970809i	$0.422900\pi$
$740$	0	0
$741$	2.02930	0.0745484
$742$	41.0399	1.50662
$743$	−14.1800	−0.520213	−0.260106	−	0.965580i	$-0.583758\pi$
$743$	−14.1800	−0.520213	−0.260106	+	0.965580i	$0.583758\pi$
$744$	8.34542	0.305958
$745$	0	0
$746$	−24.3854	−0.892812
$747$	−6.17997	−0.226113
$748$	0	0
$749$	−36.4813	−1.33300
$750$	0	0
$751$	19.1120	0.697408	0.348704	−	0.937233i	$-0.386622\pi$
$751$	19.1120	0.697408	0.348704	+	0.937233i	$0.386622\pi$
$752$	22.7067	0.828027
$753$	−2.28267	−0.0831852
$754$	1.13069	0.0411773
$755$	0	0
$756$	0.354000	0.0128748
$757$	−24.7347	−0.898997	−0.449498	−	0.893281i	$-0.648397\pi$
$757$	−24.7347	−0.898997	−0.449498	+	0.893281i	$0.648397\pi$
$758$	32.4253	1.17774
$759$	0	0
$760$	0	0
$761$	9.82003	0.355976	0.177988	−	0.984033i	$-0.443041\pi$
$761$	9.82003	0.355976	0.177988	+	0.984033i	$0.443041\pi$
$762$	20.7253	0.750800
$763$	−6.04796	−0.218951
$764$	−0.468102	−0.0169353
$765$	0	0
$766$	43.2666	1.56328
$767$	−3.89730	−0.140723
$768$	3.37473	0.121775
$769$	15.3026	0.551828	0.275914	−	0.961182i	$-0.411020\pi$
$769$	15.3026	0.551828	0.275914	+	0.961182i	$0.411020\pi$
$770$	0	0
$771$	−26.8667	−0.967580
$772$	0.301330	0.0108451
$773$	−5.36927	−0.193119	−0.0965596	−	0.995327i	$-0.530784\pi$
$773$	−5.36927	−0.193119	−0.0965596	+	0.995327i	$0.530784\pi$
$774$	−17.3107	−0.622219
$775$	0	0
$776$	−56.8181	−2.03965
$777$	−21.0993	−0.756934
$778$	35.1120	1.25883
$779$	1.13201	0.0405584
$780$	0	0
$781$	0	0
$782$	−14.7466	−0.527338
$783$	−0.726656	−0.0259686
$784$	2.68670	0.0959537
$785$	0	0
$786$	25.5560	0.911553
$787$	15.4767	0.551684	0.275842	−	0.961203i	$-0.411043\pi$
$787$	15.4767	0.551684	0.275842	+	0.961203i	$0.411043\pi$
$788$	−2.53397	−0.0902689
$789$	−12.0187	−0.427876
$790$	0	0
$791$	−15.0280	−0.534334
$792$	0	0
$793$	−5.24404	−0.186221
$794$	4.14265	0.147017
$795$	0	0
$796$	−2.14265	−0.0759444
$797$	43.6774	1.54713	0.773566	−	0.633716i	$-0.218471\pi$
$797$	43.6774	1.54713	0.773566	+	0.633716i	$0.218471\pi$
$798$	6.07133	0.214923
$799$	−46.9567	−1.66121
$800$	0	0
$801$	3.45331	0.122017
$802$	−24.7853	−0.875198
$803$	0	0
$804$	1.31744	0.0464624
$805$	0	0
$806$	4.44813	0.156679
$807$	14.3854	0.506389
$808$	−24.4427	−0.859890
$809$	26.9966	0.949150	0.474575	−	0.880215i	$-0.342602\pi$
$809$	26.9966	0.949150	0.474575	+	0.880215i	$0.342602\pi$
$810$	0	0
$811$	14.2220	0.499402	0.249701	−	0.968323i	$-0.419668\pi$
$811$	14.2220	0.499402	0.249701	+	0.968323i	$0.419668\pi$
$812$	−0.257236	−0.00902722
$813$	28.6553	1.00499
$814$	0	0
$815$	0	0
$816$	28.2700	0.989646
$817$	22.5760	0.789834
$818$	28.0773	0.981699
$819$	2.85866	0.0998898
$820$	0	0
$821$	−23.7801	−0.829930	−0.414965	−	0.909837i	$-0.636206\pi$
$821$	−23.7801	−0.829930	−0.414965	+	0.909837i	$0.636206\pi$
$822$	11.9227	0.415853
$823$	1.84934	0.0644638	0.0322319	−	0.999480i	$-0.489738\pi$
$823$	1.84934	0.0644638	0.0322319	+	0.999480i	$0.489738\pi$
$824$	7.43466	0.258998
$825$	0	0
$826$	−11.6600	−0.405705
$827$	15.6987	0.545896	0.272948	−	0.962029i	$-0.412001\pi$
$827$	15.6987	0.545896	0.272948	+	0.962029i	$0.412001\pi$
$828$	0.199945	0.00694858
$829$	−4.34128	−0.150779	−0.0753895	−	0.997154i	$-0.524020\pi$
$829$	−4.34128	−0.150779	−0.0753895	+	0.997154i	$0.524020\pi$
$830$	0	0
$831$	29.1600	1.01155
$832$	9.68152	0.335646
$833$	−5.55602	−0.192505
$834$	−30.4040	−1.05281
$835$	0	0
$836$	0	0
$837$	−2.85866	−0.0988099
$838$	44.6154	1.54121
$839$	28.3013	0.977070	0.488535	−	0.872544i	$-0.337531\pi$
$839$	28.3013	0.977070	0.488535	+	0.872544i	$0.337531\pi$
$840$	0	0
$841$	−28.4720	−0.981792
$842$	44.3955	1.52997
$843$	−5.36333	−0.184723
$844$	−1.88278	−0.0648080
$845$	0	0
$846$	−8.37266	−0.287858
$847$	0	0
$848$	−44.4372	−1.52598
$849$	10.0420	0.344641
$850$	0	0
$851$	−11.9173	−0.408519
$852$	0.828041	0.0283682
$853$	−35.7653	−1.22458	−0.612290	−	0.790633i	$-0.709752\pi$
$853$	−35.7653	−1.22458	−0.612290	+	0.790633i	$0.709752\pi$
$854$	−15.6893	−0.536875
$855$	0	0
$856$	42.5213	1.45335
$857$	−24.7513	−0.845487	−0.422743	−	0.906249i	$-0.638933\pi$
$857$	−24.7513	−0.845487	−0.422743	+	0.906249i	$0.638933\pi$
$858$	0	0
$859$	−10.8039	−0.368625	−0.184313	−	0.982868i	$-0.559006\pi$
$859$	−10.8039	−0.368625	−0.184313	+	0.982868i	$0.559006\pi$
$860$	0	0
$861$	1.59465	0.0543455
$862$	−2.48130	−0.0845134
$863$	16.6027	0.565161	0.282581	−	0.959244i	$-0.408810\pi$
$863$	16.6027	0.565161	0.282581	+	0.959244i	$0.408810\pi$
$864$	−0.797984	−0.0271480
$865$	0	0
$866$	40.6654	1.38187
$867$	−41.4613	−1.40810
$868$	−1.01197	−0.0343484
$869$	0	0
$870$	0	0
$871$	10.6387	0.360480
$872$	7.04928	0.238719
$873$	19.4626	0.658711
$874$	3.42920	0.115994
$875$	0	0
$876$	1.06794	0.0360823
$877$	−11.1600	−0.376846	−0.188423	−	0.982088i	$-0.560338\pi$
$877$	−11.1600	−0.376846	−0.188423	+	0.982088i	$0.560338\pi$
$878$	−24.8426	−0.838396
$879$	−20.3540	−0.686523
$880$	0	0
$881$	54.5254	1.83701	0.918504	−	0.395413i	$-0.129398\pi$
$881$	54.5254	1.83701	0.918504	+	0.395413i	$0.129398\pi$
$882$	−0.990671	−0.0333576
$883$	−12.7746	−0.429900	−0.214950	−	0.976625i	$-0.568959\pi$
$883$	−12.7746	−0.429900	−0.214950	+	0.976625i	$0.568959\pi$
$884$	−1.23339	−0.0414835
$885$	0	0
$886$	30.3514	1.01967
$887$	−47.1053	−1.58164	−0.790820	−	0.612049i	$-0.790345\pi$
$887$	−47.1053	−1.58164	−0.790820	+	0.612049i	$0.790345\pi$
$888$	24.5926	0.825273
$889$	−38.0759	−1.27703
$890$	0	0
$891$	0	0
$892$	−1.73184	−0.0579864
$893$	10.9193	0.365402
$894$	−21.7160	−0.726292
$895$	0	0
$896$	24.9681	0.834124
$897$	1.61462	0.0539108
$898$	35.2775	1.17722
$899$	2.07727	0.0692807
$900$	0	0
$901$	91.8947	3.06146
$902$	0	0
$903$	31.8026	1.05832
$904$	17.5161	0.582576
$905$	0	0
$906$	−13.4787	−0.447801
$907$	24.7826	0.822894	0.411447	−	0.911434i	$-0.365024\pi$
$907$	24.7826	0.822894	0.411447	+	0.911434i	$0.365024\pi$
$908$	−1.23603	−0.0410191
$909$	8.37266	0.277704
$910$	0	0
$911$	−6.08273	−0.201530	−0.100765	−	0.994910i	$-0.532129\pi$
$911$	−6.08273	−0.201530	−0.100765	+	0.994910i	$0.532129\pi$
$912$	−6.57392	−0.217684
$913$	0	0
$914$	−14.2640	−0.471812
$915$	0	0
$916$	1.62009	0.0535291
$917$	−46.9507	−1.55045
$918$	−10.4240	−0.344044
$919$	−17.9953	−0.593610	−0.296805	−	0.954938i	$-0.595921\pi$
$919$	−17.9953	−0.593610	−0.296805	+	0.954938i	$0.595921\pi$
$920$	0	0
$921$	−25.6226	−0.844295
$922$	−16.1400	−0.531543
$923$	6.68670	0.220096
$924$	0	0
$925$	0	0
$926$	−37.8387	−1.24346
$927$	−2.54669	−0.0836442
$928$	0.579860	0.0190348
$929$	−16.7453	−0.549396	−0.274698	−	0.961531i	$-0.588578\pi$
$929$	−16.7453	−0.549396	−0.274698	+	0.961531i	$0.588578\pi$
$930$	0	0
$931$	1.29200	0.0423436
$932$	−1.05786	−0.0346513
$933$	−11.9414	−0.390944
$934$	−28.3973	−0.929187
$935$	0	0
$936$	−3.33195	−0.108908
$937$	−31.5853	−1.03185	−0.515924	−	0.856635i	$-0.672551\pi$
$937$	−31.5853	−1.03185	−0.515924	+	0.856635i	$0.672551\pi$
$938$	31.8293	1.03926
$939$	17.1986	0.561256
$940$	0	0
$941$	14.8421	0.483838	0.241919	−	0.970296i	$-0.422223\pi$
$941$	14.8421	0.483838	0.241919	+	0.970296i	$0.422223\pi$
$942$	−0.179969	−0.00586372
$943$	0.900687	0.0293304
$944$	12.6253	0.410918
$945$	0	0
$946$	0	0
$947$	−2.68802	−0.0873490	−0.0436745	−	0.999046i	$-0.513906\pi$
$947$	−2.68802	−0.0873490	−0.0436745	+	0.999046i	$0.513906\pi$
$948$	−0.977953	−0.0317624
$949$	8.62395	0.279945
$950$	0	0
$951$	−22.7453	−0.737567
$952$	−55.9074	−1.81197
$953$	14.2700	0.462249	0.231125	−	0.972924i	$-0.425760\pi$
$953$	14.2700	0.462249	0.231125	+	0.972924i	$0.425760\pi$
$954$	16.3854	0.530496
$955$	0	0
$956$	1.82417	0.0589980
$957$	0	0
$958$	−3.61199	−0.116698
$959$	−21.9041	−0.707320
$960$	0	0
$961$	−22.8280	−0.736388
$962$	13.1079	0.422615
$963$	−14.5653	−0.469362
$964$	−2.38538	−0.0768278
$965$	0	0
$966$	4.83068	0.155425
$967$	10.0586	0.323463	0.161732	−	0.986835i	$-0.448292\pi$
$967$	10.0586	0.323463	0.161732	+	0.986835i	$0.448292\pi$
$968$	0	0
$969$	13.5946	0.436723
$970$	0	0
$971$	−13.4520	−0.431695	−0.215848	−	0.976427i	$-0.569251\pi$
$971$	−13.4520	−0.431695	−0.215848	+	0.976427i	$0.569251\pi$
$972$	0.141336	0.00453336
$973$	55.8573	1.79070
$974$	−55.2639	−1.77077
$975$	0	0
$976$	16.9880	0.543774
$977$	32.8294	1.05030	0.525152	−	0.851008i	$-0.324008\pi$
$977$	32.8294	1.05030	0.525152	+	0.851008i	$0.324008\pi$
$978$	−5.87864	−0.187978
$979$	0	0
$980$	0	0
$981$	−2.41468	−0.0770948
$982$	21.9787	0.701369
$983$	−43.7746	−1.39619	−0.698097	−	0.716003i	$-0.745969\pi$
$983$	−43.7746	−1.39619	−0.698097	+	0.716003i	$0.745969\pi$
$984$	−1.85866	−0.0592520
$985$	0	0
$986$	7.57467	0.241227
$987$	15.3820	0.489614
$988$	0.286814	0.00912477
$989$	17.9627	0.571180
$990$	0	0
$991$	42.1507	1.33896	0.669480	−	0.742830i	$-0.266517\pi$
$991$	42.1507	1.33896	0.669480	+	0.742830i	$0.266517\pi$
$992$	2.28117	0.0724271
$993$	−25.2733	−0.802025
$994$	20.0055	0.634535
$995$	0	0
$996$	−0.873453	−0.0276764
$997$	−17.4347	−0.552161	−0.276081	−	0.961135i	$-0.589036\pi$
$997$	−17.4347	−0.552161	−0.276081	+	0.961135i	$0.589036\pi$
$998$	−38.2640	−1.21123
$999$	−8.42401	−0.266524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cj.1.2		3
5.4	even	2		9075.2.a.cd.1.2		3
11.10	odd	2		825.2.a.i.1.2	✓	3
33.32	even	2		2475.2.a.bd.1.2		3
55.32	even	4		825.2.c.f.199.3		6
55.43	even	4		825.2.c.f.199.4		6
55.54	odd	2		825.2.a.m.1.2	yes	3
165.32	odd	4		2475.2.c.q.199.4		6
165.98	odd	4		2475.2.c.q.199.3		6
165.164	even	2		2475.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.i.1.2	✓	3	11.10	odd	2
825.2.a.m.1.2	yes	3	55.54	odd	2
825.2.c.f.199.3		6	55.32	even	4
825.2.c.f.199.4		6	55.43	even	4
2475.2.a.z.1.2		3	165.164	even	2
2475.2.a.bd.1.2		3	33.32	even	2
2475.2.c.q.199.3		6	165.98	odd	4
2475.2.c.q.199.4		6	165.32	odd	4
9075.2.a.cd.1.2		3	5.4	even	2
9075.2.a.cj.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.36333 q^{2} -1.00000 q^{3} -0.141336 q^{4} -1.36333 q^{6} +2.50466 q^{7} -2.91934 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.36333 q^{2} -1.00000 q^{3} -0.141336 q^{4} -1.36333 q^{6} +2.50466 q^{7} -2.91934 q^{8} +1.00000 q^{9} +0.141336 q^{12} +1.14134 q^{13} +3.41468 q^{14} -3.69735 q^{16} +7.64600 q^{17} +1.36333 q^{18} -1.77801 q^{19} -2.50466 q^{21} -1.41468 q^{23} +2.91934 q^{24} +1.55602 q^{26} -1.00000 q^{27} -0.354000 q^{28} +0.726656 q^{29} +2.85866 q^{31} +0.797984 q^{32} +10.4240 q^{34} -0.141336 q^{36} +8.42401 q^{37} -2.42401 q^{38} -1.14134 q^{39} -0.636672 q^{41} -3.41468 q^{42} -12.6974 q^{43} -1.92867 q^{46} -6.14134 q^{47} +3.69735 q^{48} -0.726656 q^{49} -7.64600 q^{51} -0.161312 q^{52} +12.0187 q^{53} -1.36333 q^{54} -7.31198 q^{56} +1.77801 q^{57} +0.990671 q^{58} -3.41468 q^{59} -4.59465 q^{61} +3.89730 q^{62} +2.50466 q^{63} +8.48262 q^{64} +9.32131 q^{67} -1.08066 q^{68} +1.41468 q^{69} +5.85866 q^{71} -2.91934 q^{72} +7.55602 q^{73} +11.4847 q^{74} +0.251297 q^{76} -1.55602 q^{78} -6.91934 q^{79} +1.00000 q^{81} -0.867993 q^{82} -6.17997 q^{83} +0.354000 q^{84} -17.3107 q^{86} -0.726656 q^{87} +3.45331 q^{89} +2.85866 q^{91} +0.199945 q^{92} -2.85866 q^{93} -8.37266 q^{94} -0.797984 q^{96} +19.4626 q^{97} -0.990671 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} - 3 q^{3} + 8 q^{4} - 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 - 3 * q^3 + 8 * q^4 - 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9	\( 3 q + 2 q^{2} - 3 q^{3} + 8 q^{4} - 2 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - 8 q^{12} - 5 q^{13} + 6 q^{14} + 10 q^{16} + 4 q^{17} + 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} - 3 q^{27} - 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} + 8 q^{36} + 18 q^{38} + 5 q^{39} - 4 q^{41} - 6 q^{42} - 17 q^{43} + 30 q^{46} - 10 q^{47} - 10 q^{48} + 2 q^{49} - 4 q^{51} - 30 q^{52} - 6 q^{53} - 2 q^{54} - 22 q^{56} - q^{57} + 24 q^{58} - 6 q^{59} + 3 q^{61} + 16 q^{62} - 3 q^{63} + 34 q^{64} + 7 q^{67} - 18 q^{68} + 26 q^{71} + 6 q^{72} + 10 q^{73} - 14 q^{74} + 24 q^{76} + 8 q^{78} - 6 q^{79} + 3 q^{81} + 10 q^{82} - 6 q^{83} + 20 q^{84} + 28 q^{86} + 2 q^{87} + 2 q^{89} + 17 q^{91} + 26 q^{92} - 17 q^{93} - 2 q^{94} - 34 q^{96} + 29 q^{97} - 24 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 - 3 * q^3 + 8 * q^4 - 2 * q^6 - 3 * q^7 + 6 * q^8 + 3 * q^9 - 8 * q^12 - 5 * q^13 + 6 * q^14 + 10 * q^16 + 4 * q^17 + 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 - 3 * q^27 - 20 * q^28 - 2 * q^29 + 17 * q^31 + 34 * q^32 + 6 * q^34 + 8 * q^36 + 18 * q^38 + 5 * q^39 - 4 * q^41 - 6 * q^42 - 17 * q^43 + 30 * q^46 - 10 * q^47 - 10 * q^48 + 2 * q^49 - 4 * q^51 - 30 * q^52 - 6 * q^53 - 2 * q^54 - 22 * q^56 - q^57 + 24 * q^58 - 6 * q^59 + 3 * q^61 + 16 * q^62 - 3 * q^63 + 34 * q^64 + 7 * q^67 - 18 * q^68 + 26 * q^71 + 6 * q^72 + 10 * q^73 - 14 * q^74 + 24 * q^76 + 8 * q^78 - 6 * q^79 + 3 * q^81 + 10 * q^82 - 6 * q^83 + 20 * q^84 + 28 * q^86 + 2 * q^87 + 2 * q^89 + 17 * q^91 + 26 * q^92 - 17 * q^93 - 2 * q^94 - 34 * q^96 + 29 * q^97 - 24 * q^98

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