LMFDB - Embedded newform 9075.2.a.ea.1.1

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 18x^8 - 220x^7 + 61x^6 + 330x^5 - 145x^4 - 148x^3 + 71x^2 + 10*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$-2.36377$ 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)

$f(q)$	$=$	$q-2.36377 q^{2} +1.00000 q^{3} +3.58741 q^{4} -2.36377 q^{6} +1.71423 q^{7} -3.75227 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.36377 q^{2} +1.00000 q^{3} +3.58741 q^{4} -2.36377 q^{6} +1.71423 q^{7} -3.75227 q^{8} +1.00000 q^{9} +3.58741 q^{12} +3.53338 q^{13} -4.05205 q^{14} +1.69468 q^{16} +0.373305 q^{17} -2.36377 q^{18} +5.76794 q^{19} +1.71423 q^{21} -3.80745 q^{23} -3.75227 q^{24} -8.35211 q^{26} +1.00000 q^{27} +6.14965 q^{28} +3.51670 q^{29} -10.0978 q^{31} +3.49870 q^{32} -0.882407 q^{34} +3.58741 q^{36} +11.5420 q^{37} -13.6341 q^{38} +3.53338 q^{39} +2.41898 q^{41} -4.05205 q^{42} +0.224486 q^{43} +8.99992 q^{46} -4.67637 q^{47} +1.69468 q^{48} -4.06141 q^{49} +0.373305 q^{51} +12.6757 q^{52} -2.44417 q^{53} -2.36377 q^{54} -6.43226 q^{56} +5.76794 q^{57} -8.31267 q^{58} +5.35991 q^{59} +3.40629 q^{61} +23.8688 q^{62} +1.71423 q^{63} -11.6595 q^{64} +9.08988 q^{67} +1.33920 q^{68} -3.80745 q^{69} -8.77564 q^{71} -3.75227 q^{72} +14.9622 q^{73} -27.2825 q^{74} +20.6919 q^{76} -8.35211 q^{78} -8.79204 q^{79} +1.00000 q^{81} -5.71791 q^{82} -7.44215 q^{83} +6.14965 q^{84} -0.530633 q^{86} +3.51670 q^{87} +15.5058 q^{89} +6.05704 q^{91} -13.6589 q^{92} -10.0978 q^{93} +11.0539 q^{94} +3.49870 q^{96} +14.6637 q^{97} +9.60024 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9	$ 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{12} + 18 q^{13} + 6 q^{14} + 24 q^{16} + 18 q^{17} + 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} + 12 q^{27} + 30 q^{28} + 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} + 18 q^{39} + 6 q^{42} + 32 q^{43} + 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} + 18 q^{51} + 48 q^{52} - 12 q^{53} + 4 q^{54} + 6 q^{56} + 16 q^{57} - 10 q^{58} + 20 q^{59} + 20 q^{61} + 20 q^{62} + 8 q^{63} - 6 q^{64} + 10 q^{67} + 26 q^{68} + 32 q^{71} + 12 q^{72} + 26 q^{73} - 68 q^{74} + 34 q^{76} + 16 q^{78} + 32 q^{79} + 12 q^{81} - 62 q^{82} + 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} - 2 q^{94} + 28 q^{96} - 22 q^{97} + 60 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9 + 12 * q^12 + 18 * q^13 + 6 * q^14 + 24 * q^16 + 18 * q^17 + 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 + 12 * q^27 + 30 * q^28 + 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 + 18 * q^39 + 6 * q^42 + 32 * q^43 + 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 + 18 * q^51 + 48 * q^52 - 12 * q^53 + 4 * q^54 + 6 * q^56 + 16 * q^57 - 10 * q^58 + 20 * q^59 + 20 * q^61 + 20 * q^62 + 8 * q^63 - 6 * q^64 + 10 * q^67 + 26 * q^68 + 32 * q^71 + 12 * q^72 + 26 * q^73 - 68 * q^74 + 34 * q^76 + 16 * q^78 + 32 * q^79 + 12 * q^81 - 62 * q^82 + 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 - 2 * q^94 + 28 * q^96 - 22 * q^97 + 60 * 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 $

$2$	−2.36377	−1.67144	−0.835719	−	0.549158i	$-0.814949\pi$
$2$	−2.36377	−1.67144	−0.835719	+	0.549158i	$0.814949\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.58741	1.79370
$5$	0	0
$5$	0	0
$6$	−2.36377	−0.965005
$7$	1.71423	0.647919	0.323959	−	0.946071i	$-0.394986\pi$
$7$	1.71423	0.647919	0.323959	+	0.946071i	$0.394986\pi$
$8$	−3.75227	−1.32663
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	3.58741	1.03560
$13$	3.53338	0.979984	0.489992	−	0.871727i	$-0.337000\pi$
$13$	3.53338	0.979984	0.489992	+	0.871727i	$0.337000\pi$
$14$	−4.05205	−1.08296
$15$	0	0
$16$	1.69468	0.423670
$17$	0.373305	0.0905398	0.0452699	−	0.998975i	$-0.485585\pi$
$17$	0.373305	0.0905398	0.0452699	+	0.998975i	$0.485585\pi$
$18$	−2.36377	−0.557146
$19$	5.76794	1.32326	0.661628	−	0.749832i	$-0.269866\pi$
$19$	5.76794	1.32326	0.661628	+	0.749832i	$0.269866\pi$
$20$	0	0
$21$	1.71423	0.374076
$22$	0	0
$23$	−3.80745	−0.793907	−0.396954	−	0.917839i	$-0.629933\pi$
$23$	−3.80745	−0.793907	−0.396954	+	0.917839i	$0.629933\pi$
$24$	−3.75227	−0.765929
$25$	0	0
$26$	−8.35211	−1.63798
$27$	1.00000	0.192450
$28$	6.14965	1.16217
$29$	3.51670	0.653035	0.326517	−	0.945191i	$-0.394125\pi$
$29$	3.51670	0.653035	0.326517	+	0.945191i	$0.394125\pi$
$30$	0	0
$31$	−10.0978	−1.81362	−0.906808	−	0.421545i	$-0.861488\pi$
$31$	−10.0978	−1.81362	−0.906808	+	0.421545i	$0.861488\pi$
$32$	3.49870	0.618488
$33$	0	0
$34$	−0.882407	−0.151332
$35$	0	0
$36$	3.58741	0.597901
$37$	11.5420	1.89748	0.948742	−	0.316050i	$-0.102357\pi$
$37$	11.5420	1.89748	0.948742	+	0.316050i	$0.102357\pi$
$38$	−13.6341	−2.21174
$39$	3.53338	0.565794
$40$	0	0
$41$	2.41898	0.377781	0.188890	−	0.981998i	$-0.439511\pi$
$41$	2.41898	0.377781	0.188890	+	0.981998i	$0.439511\pi$
$42$	−4.05205	−0.625245
$43$	0.224486	0.0342338	0.0171169	−	0.999853i	$-0.494551\pi$
$43$	0.224486	0.0342338	0.0171169	+	0.999853i	$0.494551\pi$
$44$	0	0
$45$	0	0
$46$	8.99992	1.32697
$47$	−4.67637	−0.682118	−0.341059	−	0.940042i	$-0.610786\pi$
$47$	−4.67637	−0.682118	−0.341059	+	0.940042i	$0.610786\pi$
$48$	1.69468	0.244606
$49$	−4.06141	−0.580201
$50$	0	0
$51$	0.373305	0.0522732
$52$	12.6757	1.75780
$53$	−2.44417	−0.335732	−0.167866	−	0.985810i	$-0.553688\pi$
$53$	−2.44417	−0.335732	−0.167866	+	0.985810i	$0.553688\pi$
$54$	−2.36377	−0.321668
$55$	0	0
$56$	−6.43226	−0.859547
$57$	5.76794	0.763982
$58$	−8.31267	−1.09151
$59$	5.35991	0.697800	0.348900	−	0.937160i	$-0.386555\pi$
$59$	5.35991	0.697800	0.348900	+	0.937160i	$0.386555\pi$
$60$	0	0
$61$	3.40629	0.436131	0.218066	−	0.975934i	$-0.430025\pi$
$61$	3.40629	0.436131	0.218066	+	0.975934i	$0.430025\pi$
$62$	23.8688	3.03134
$63$	1.71423	0.215973
$64$	−11.6595	−1.45744
$65$	0	0
$66$	0	0
$67$	9.08988	1.11051	0.555253	−	0.831682i	$-0.312621\pi$
$67$	9.08988	1.11051	0.555253	+	0.831682i	$0.312621\pi$
$68$	1.33920	0.162402
$69$	−3.80745	−0.458363
$70$	0	0
$71$	−8.77564	−1.04148	−0.520738	−	0.853716i	$-0.674343\pi$
$71$	−8.77564	−1.04148	−0.520738	+	0.853716i	$0.674343\pi$
$72$	−3.75227	−0.442209
$73$	14.9622	1.75120	0.875599	−	0.483040i	$-0.160467\pi$
$73$	14.9622	1.75120	0.875599	+	0.483040i	$0.160467\pi$
$74$	−27.2825	−3.17153
$75$	0	0
$76$	20.6919	2.37353
$77$	0	0
$78$	−8.35211	−0.945690
$79$	−8.79204	−0.989181	−0.494591	−	0.869126i	$-0.664682\pi$
$79$	−8.79204	−0.989181	−0.494591	+	0.869126i	$0.664682\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.71791	−0.631437
$83$	−7.44215	−0.816882	−0.408441	−	0.912785i	$-0.633927\pi$
$83$	−7.44215	−0.816882	−0.408441	+	0.912785i	$0.633927\pi$
$84$	6.14965	0.670982
$85$	0	0
$86$	−0.530633	−0.0572196
$87$	3.51670	0.377030
$88$	0	0
$89$	15.5058	1.64361	0.821806	−	0.569767i	$-0.192966\pi$
$89$	15.5058	1.64361	0.821806	+	0.569767i	$0.192966\pi$
$90$	0	0
$91$	6.05704	0.634950
$92$	−13.6589	−1.42403
$93$	−10.0978	−1.04709
$94$	11.0539	1.14012
$95$	0	0
$96$	3.49870	0.357084
$97$	14.6637	1.48887	0.744436	−	0.667694i	$-0.232719\pi$
$97$	14.6637	1.48887	0.744436	+	0.667694i	$0.232719\pi$
$98$	9.60024	0.969770
$99$	0	0
$100$	0	0
$101$	−6.13469	−0.610424	−0.305212	−	0.952284i	$-0.598727\pi$
$101$	−6.13469	−0.610424	−0.305212	+	0.952284i	$0.598727\pi$
$102$	−0.882407	−0.0873713
$103$	7.86989	0.775443	0.387722	−	0.921777i	$-0.373262\pi$
$103$	7.86989	0.775443	0.387722	+	0.921777i	$0.373262\pi$
$104$	−13.2582	−1.30007
$105$	0	0
$106$	5.77745	0.561156
$107$	4.95618	0.479132	0.239566	−	0.970880i	$-0.422995\pi$
$107$	4.95618	0.479132	0.239566	+	0.970880i	$0.422995\pi$
$108$	3.58741	0.345199
$109$	2.61667	0.250632	0.125316	−	0.992117i	$-0.460006\pi$
$109$	2.61667	0.250632	0.125316	+	0.992117i	$0.460006\pi$
$110$	0	0
$111$	11.5420	1.09551
$112$	2.90508	0.274504
$113$	−2.11658	−0.199111	−0.0995554	−	0.995032i	$-0.531742\pi$
$113$	−2.11658	−0.199111	−0.0995554	+	0.995032i	$0.531742\pi$
$114$	−13.6341	−1.27695
$115$	0	0
$116$	12.6158	1.17135
$117$	3.53338	0.326661
$118$	−12.6696	−1.16633
$119$	0.639932	0.0586624
$120$	0	0
$121$	0	0
$122$	−8.05170	−0.728966
$123$	2.41898	0.218112
$124$	−36.2249	−3.25309
$125$	0	0
$126$	−4.05205	−0.360985
$127$	11.9477	1.06018	0.530092	−	0.847940i	$-0.322157\pi$
$127$	11.9477	1.06018	0.530092	+	0.847940i	$0.322157\pi$
$128$	20.5629	1.81752
$129$	0.224486	0.0197649
$130$	0	0
$131$	−20.3359	−1.77676	−0.888380	−	0.459110i	$-0.848169\pi$
$131$	−20.3359	−1.77676	−0.888380	+	0.459110i	$0.848169\pi$
$132$	0	0
$133$	9.88758	0.857362
$134$	−21.4864	−1.85614
$135$	0	0
$136$	−1.40074	−0.120113
$137$	−5.31343	−0.453957	−0.226979	−	0.973900i	$-0.572885\pi$
$137$	−5.31343	−0.453957	−0.226979	+	0.973900i	$0.572885\pi$
$138$	8.99992	0.766124
$139$	−13.2156	−1.12093	−0.560465	−	0.828178i	$-0.689378\pi$
$139$	−13.2156	−1.12093	−0.560465	+	0.828178i	$0.689378\pi$
$140$	0	0
$141$	−4.67637	−0.393821
$142$	20.7436	1.74076
$143$	0	0
$144$	1.69468	0.141223
$145$	0	0
$146$	−35.3673	−2.92702
$147$	−4.06141	−0.334979
$148$	41.4057	3.40353
$149$	6.06286	0.496689	0.248344	−	0.968672i	$-0.420114\pi$
$149$	6.06286	0.496689	0.248344	+	0.968672i	$0.420114\pi$
$150$	0	0
$151$	7.33491	0.596907	0.298453	−	0.954424i	$-0.403529\pi$
$151$	7.33491	0.596907	0.298453	+	0.954424i	$0.403529\pi$
$152$	−21.6429	−1.75547
$153$	0.373305	0.0301799
$154$	0	0
$155$	0	0
$156$	12.6757	1.01487
$157$	−10.5558	−0.842444	−0.421222	−	0.906958i	$-0.638399\pi$
$157$	−10.5558	−0.842444	−0.421222	+	0.906958i	$0.638399\pi$
$158$	20.7824	1.65336
$159$	−2.44417	−0.193835
$160$	0	0
$161$	−6.52684	−0.514387
$162$	−2.36377	−0.185715
$163$	−10.1070	−0.791645	−0.395822	−	0.918327i	$-0.629540\pi$
$163$	−10.1070	−0.791645	−0.395822	+	0.918327i	$0.629540\pi$
$164$	8.67786	0.677627
$165$	0	0
$166$	17.5915	1.36537
$167$	13.3472	1.03284	0.516419	−	0.856336i	$-0.327265\pi$
$167$	13.3472	1.03284	0.516419	+	0.856336i	$0.327265\pi$
$168$	−6.43226	−0.496259
$169$	−0.515200	−0.0396308
$170$	0	0
$171$	5.76794	0.441085
$172$	0.805322	0.0614052
$173$	17.8603	1.35790	0.678948	−	0.734187i	$-0.262436\pi$
$173$	17.8603	1.35790	0.678948	+	0.734187i	$0.262436\pi$
$174$	−8.31267	−0.630182
$175$	0	0
$176$	0	0
$177$	5.35991	0.402875
$178$	−36.6522	−2.74720
$179$	−12.7372	−0.952024	−0.476012	−	0.879439i	$-0.657918\pi$
$179$	−12.7372	−0.952024	−0.476012	+	0.879439i	$0.657918\pi$
$180$	0	0
$181$	20.7117	1.53949	0.769746	−	0.638350i	$-0.220383\pi$
$181$	20.7117	1.53949	0.769746	+	0.638350i	$0.220383\pi$
$182$	−14.3174	−1.06128
$183$	3.40629	0.251801
$184$	14.2866	1.05322
$185$	0	0
$186$	23.8688	1.75015
$187$	0	0
$188$	−16.7760	−1.22352
$189$	1.71423	0.124692
$190$	0	0
$191$	−6.19957	−0.448585	−0.224293	−	0.974522i	$-0.572007\pi$
$191$	−6.19957	−0.448585	−0.224293	+	0.974522i	$0.572007\pi$
$192$	−11.6595	−0.841451
$193$	5.52362	0.397599	0.198800	−	0.980040i	$-0.436296\pi$
$193$	5.52362	0.397599	0.198800	+	0.980040i	$0.436296\pi$
$194$	−34.6616	−2.48856
$195$	0	0
$196$	−14.5699	−1.04071
$197$	12.8524	0.915693	0.457846	−	0.889031i	$-0.348621\pi$
$197$	12.8524	0.915693	0.457846	+	0.889031i	$0.348621\pi$
$198$	0	0
$199$	3.42830	0.243026	0.121513	−	0.992590i	$-0.461225\pi$
$199$	3.42830	0.243026	0.121513	+	0.992590i	$0.461225\pi$
$200$	0	0
$201$	9.08988	0.641151
$202$	14.5010	1.02029
$203$	6.02844	0.423113
$204$	1.33920	0.0937626
$205$	0	0
$206$	−18.6026	−1.29610
$207$	−3.80745	−0.264636
$208$	5.98796	0.415190
$209$	0	0
$210$	0	0
$211$	−17.8292	−1.22741	−0.613705	−	0.789535i	$-0.710322\pi$
$211$	−17.8292	−1.22741	−0.613705	+	0.789535i	$0.710322\pi$
$212$	−8.76823	−0.602205
$213$	−8.77564	−0.601297
$214$	−11.7153	−0.800840
$215$	0	0
$216$	−3.75227	−0.255310
$217$	−17.3099	−1.17508
$218$	−6.18522	−0.418916
$219$	14.9622	1.01105
$220$	0	0
$221$	1.31903	0.0887276
$222$	−27.2825	−1.83108
$223$	4.31709	0.289094	0.144547	−	0.989498i	$-0.453828\pi$
$223$	4.31709	0.289094	0.144547	+	0.989498i	$0.453828\pi$
$224$	5.99758	0.400730
$225$	0	0
$226$	5.00310	0.332801
$227$	−25.4363	−1.68827	−0.844134	−	0.536133i	$-0.819885\pi$
$227$	−25.4363	−1.68827	−0.844134	+	0.536133i	$0.819885\pi$
$228$	20.6919	1.37036
$229$	14.3934	0.951139	0.475570	−	0.879678i	$-0.342242\pi$
$229$	14.3934	0.951139	0.475570	+	0.879678i	$0.342242\pi$
$230$	0	0
$231$	0	0
$232$	−13.1956	−0.866333
$233$	11.9315	0.781660	0.390830	−	0.920463i	$-0.372188\pi$
$233$	11.9315	0.781660	0.390830	+	0.920463i	$0.372188\pi$
$234$	−8.35211	−0.545994
$235$	0	0
$236$	19.2282	1.25165
$237$	−8.79204	−0.571104
$238$	−1.51265	−0.0980506
$239$	8.25748	0.534132	0.267066	−	0.963678i	$-0.413946\pi$
$239$	8.25748	0.534132	0.267066	+	0.963678i	$0.413946\pi$
$240$	0	0
$241$	11.1487	0.718154	0.359077	−	0.933308i	$-0.383092\pi$
$241$	11.1487	0.718154	0.359077	+	0.933308i	$0.383092\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	12.2198	0.782291
$245$	0	0
$246$	−5.71791	−0.364561
$247$	20.3803	1.29677
$248$	37.8896	2.40599
$249$	−7.44215	−0.471627
$250$	0	0
$251$	28.6627	1.80918	0.904588	−	0.426286i	$-0.140178\pi$
$251$	28.6627	1.80918	0.904588	+	0.426286i	$0.140178\pi$
$252$	6.14965	0.387392
$253$	0	0
$254$	−28.2415	−1.77203
$255$	0	0
$256$	−25.2871	−1.58044
$257$	2.32004	0.144720	0.0723601	−	0.997379i	$-0.476947\pi$
$257$	2.32004	0.144720	0.0723601	+	0.997379i	$0.476947\pi$
$258$	−0.530633	−0.0330358
$259$	19.7856	1.22942
$260$	0	0
$261$	3.51670	0.217678
$262$	48.0695	2.96974
$263$	2.92970	0.180653	0.0903265	−	0.995912i	$-0.471209\pi$
$263$	2.92970	0.180653	0.0903265	+	0.995912i	$0.471209\pi$
$264$	0	0
$265$	0	0
$266$	−23.3720	−1.43303
$267$	15.5058	0.948940
$268$	32.6091	1.99192
$269$	−4.53203	−0.276323	−0.138161	−	0.990410i	$-0.544119\pi$
$269$	−4.53203	−0.276323	−0.138161	+	0.990410i	$0.544119\pi$
$270$	0	0
$271$	−27.1116	−1.64692	−0.823458	−	0.567378i	$-0.807958\pi$
$271$	−27.1116	−1.64692	−0.823458	+	0.567378i	$0.807958\pi$
$272$	0.632633	0.0383590
$273$	6.05704	0.366589
$274$	12.5597	0.758761
$275$	0	0
$276$	−13.6589	−0.822167
$277$	13.2514	0.796199	0.398100	−	0.917342i	$-0.369670\pi$
$277$	13.2514	0.796199	0.398100	+	0.917342i	$0.369670\pi$
$278$	31.2386	1.87356
$279$	−10.0978	−0.604538
$280$	0	0
$281$	11.0716	0.660479	0.330239	−	0.943897i	$-0.392871\pi$
$281$	11.0716	0.660479	0.330239	+	0.943897i	$0.392871\pi$
$282$	11.0539	0.658247
$283$	10.0767	0.599000	0.299500	−	0.954096i	$-0.403180\pi$
$283$	10.0767	0.599000	0.299500	+	0.954096i	$0.403180\pi$
$284$	−31.4818	−1.86810
$285$	0	0
$286$	0	0
$287$	4.14669	0.244771
$288$	3.49870	0.206163
$289$	−16.8606	−0.991803
$290$	0	0
$291$	14.6637	0.859600
$292$	53.6756	3.14113
$293$	7.84659	0.458402	0.229201	−	0.973379i	$-0.426389\pi$
$293$	7.84659	0.458402	0.229201	+	0.973379i	$0.426389\pi$
$294$	9.60024	0.559897
$295$	0	0
$296$	−43.3085	−2.51725
$297$	0	0
$298$	−14.3312	−0.830184
$299$	−13.4532	−0.778017
$300$	0	0
$301$	0.384821	0.0221807
$302$	−17.3380	−0.997692
$303$	−6.13469	−0.352429
$304$	9.77482	0.560624
$305$	0	0
$306$	−0.882407	−0.0504439
$307$	−2.64692	−0.151068	−0.0755339	−	0.997143i	$-0.524066\pi$
$307$	−2.64692	−0.151068	−0.0755339	+	0.997143i	$0.524066\pi$
$308$	0	0
$309$	7.86989	0.447702
$310$	0	0
$311$	−4.50819	−0.255636	−0.127818	−	0.991798i	$-0.540797\pi$
$311$	−4.50819	−0.255636	−0.127818	+	0.991798i	$0.540797\pi$
$312$	−13.2582	−0.750598
$313$	−9.70923	−0.548798	−0.274399	−	0.961616i	$-0.588479\pi$
$313$	−9.70923	−0.548798	−0.274399	+	0.961616i	$0.588479\pi$
$314$	24.9515	1.40809
$315$	0	0
$316$	−31.5406	−1.77430
$317$	−17.0757	−0.959066	−0.479533	−	0.877524i	$-0.659194\pi$
$317$	−17.0757	−0.959066	−0.479533	+	0.877524i	$0.659194\pi$
$318$	5.77745	0.323984
$319$	0	0
$320$	0	0
$321$	4.95618	0.276627
$322$	15.4280	0.859766
$323$	2.15320	0.119807
$324$	3.58741	0.199300
$325$	0	0
$326$	23.8907	1.32319
$327$	2.61667	0.144702
$328$	−9.07666	−0.501174
$329$	−8.01638	−0.441957
$330$	0	0
$331$	−2.79769	−0.153775	−0.0768874	−	0.997040i	$-0.524498\pi$
$331$	−2.79769	−0.153775	−0.0768874	+	0.997040i	$0.524498\pi$
$332$	−26.6980	−1.46524
$333$	11.5420	0.632495
$334$	−31.5497	−1.72632
$335$	0	0
$336$	2.90508	0.158485
$337$	−9.28778	−0.505937	−0.252969	−	0.967474i	$-0.581407\pi$
$337$	−9.28778	−0.505937	−0.252969	+	0.967474i	$0.581407\pi$
$338$	1.21781	0.0662404
$339$	−2.11658	−0.114957
$340$	0	0
$341$	0	0
$342$	−13.6341	−0.737247
$343$	−18.9618	−1.02384
$344$	−0.842331	−0.0454154
$345$	0	0
$346$	−42.2177	−2.26964
$347$	−9.35525	−0.502216	−0.251108	−	0.967959i	$-0.580795\pi$
$347$	−9.35525	−0.502216	−0.251108	+	0.967959i	$0.580795\pi$
$348$	12.6158	0.676280
$349$	1.68796	0.0903544	0.0451772	−	0.998979i	$-0.485615\pi$
$349$	1.68796	0.0903544	0.0451772	+	0.998979i	$0.485615\pi$
$350$	0	0
$351$	3.53338	0.188598
$352$	0	0
$353$	9.60028	0.510971	0.255486	−	0.966813i	$-0.417765\pi$
$353$	9.60028	0.510971	0.255486	+	0.966813i	$0.417765\pi$
$354$	−12.6696	−0.673381
$355$	0	0
$356$	55.6257	2.94816
$357$	0.639932	0.0338688
$358$	30.1078	1.59125
$359$	27.1863	1.43484	0.717420	−	0.696641i	$-0.245323\pi$
$359$	27.1863	1.43484	0.717420	+	0.696641i	$0.245323\pi$
$360$	0	0
$361$	14.2691	0.751006
$362$	−48.9578	−2.57317
$363$	0	0
$364$	21.7291	1.13891
$365$	0	0
$366$	−8.05170	−0.420869
$367$	7.71659	0.402803	0.201401	−	0.979509i	$-0.435450\pi$
$367$	7.71659	0.402803	0.201401	+	0.979509i	$0.435450\pi$
$368$	−6.45241	−0.336355
$369$	2.41898	0.125927
$370$	0	0
$371$	−4.18987	−0.217527
$372$	−36.2249	−1.87817
$373$	20.0037	1.03575	0.517876	−	0.855456i	$-0.326723\pi$
$373$	20.0037	1.03575	0.517876	+	0.855456i	$0.326723\pi$
$374$	0	0
$375$	0	0
$376$	17.5470	0.904916
$377$	12.4258	0.639964
$378$	−4.05205	−0.208415
$379$	−6.71765	−0.345063	−0.172531	−	0.985004i	$-0.555195\pi$
$379$	−6.71765	−0.345063	−0.172531	+	0.985004i	$0.555195\pi$
$380$	0	0
$381$	11.9477	0.612097
$382$	14.6544	0.749782
$383$	−11.3494	−0.579928	−0.289964	−	0.957038i	$-0.593643\pi$
$383$	−11.3494	−0.579928	−0.289964	+	0.957038i	$0.593643\pi$
$384$	20.5629	1.04935
$385$	0	0
$386$	−13.0566	−0.664562
$387$	0.224486	0.0114113
$388$	52.6046	2.67059
$389$	−5.16601	−0.261927	−0.130964	−	0.991387i	$-0.541807\pi$
$389$	−5.16601	−0.261927	−0.130964	+	0.991387i	$0.541807\pi$
$390$	0	0
$391$	−1.42134	−0.0718802
$392$	15.2395	0.769711
$393$	−20.3359	−1.02581
$394$	−30.3800	−1.53052
$395$	0	0
$396$	0	0
$397$	−7.40026	−0.371408	−0.185704	−	0.982606i	$-0.559457\pi$
$397$	−7.40026	−0.371408	−0.185704	+	0.982606i	$0.559457\pi$
$398$	−8.10371	−0.406202
$399$	9.88758	0.494998
$400$	0	0
$401$	−31.7932	−1.58768	−0.793839	−	0.608129i	$-0.791921\pi$
$401$	−31.7932	−1.58768	−0.793839	+	0.608129i	$0.791921\pi$
$402$	−21.4864	−1.07164
$403$	−35.6793	−1.77731
$404$	−22.0076	−1.09492
$405$	0	0
$406$	−14.2498	−0.707208
$407$	0	0
$408$	−1.40074	−0.0693470
$409$	−11.9000	−0.588419	−0.294210	−	0.955741i	$-0.595056\pi$
$409$	−11.9000	−0.588419	−0.294210	+	0.955741i	$0.595056\pi$
$410$	0	0
$411$	−5.31343	−0.262092
$412$	28.2325	1.39092
$413$	9.18812	0.452118
$414$	8.99992	0.442322
$415$	0	0
$416$	12.3622	0.606109
$417$	−13.2156	−0.647169
$418$	0	0
$419$	6.57012	0.320971	0.160486	−	0.987038i	$-0.448694\pi$
$419$	6.57012	0.320971	0.160486	+	0.987038i	$0.448694\pi$
$420$	0	0
$421$	31.3343	1.52714	0.763570	−	0.645725i	$-0.223445\pi$
$421$	31.3343	1.52714	0.763570	+	0.645725i	$0.223445\pi$
$422$	42.1441	2.05154
$423$	−4.67637	−0.227373
$424$	9.17118	0.445392
$425$	0	0
$426$	20.7436	1.00503
$427$	5.83918	0.282578
$428$	17.7799	0.859422
$429$	0	0
$430$	0	0
$431$	−12.5417	−0.604113	−0.302057	−	0.953290i	$-0.597673\pi$
$431$	−12.5417	−0.604113	−0.302057	+	0.953290i	$0.597673\pi$
$432$	1.69468	0.0815354
$433$	−26.0234	−1.25060	−0.625302	−	0.780383i	$-0.715024\pi$
$433$	−26.0234	−1.25060	−0.625302	+	0.780383i	$0.715024\pi$
$434$	40.9167	1.96407
$435$	0	0
$436$	9.38708	0.449560
$437$	−21.9611	−1.05054
$438$	−35.3673	−1.68991
$439$	35.0511	1.67290	0.836450	−	0.548044i	$-0.184627\pi$
$439$	35.0511	1.67290	0.836450	+	0.548044i	$0.184627\pi$
$440$	0	0
$441$	−4.06141	−0.193400
$442$	−3.11788	−0.148303
$443$	−26.0766	−1.23894	−0.619468	−	0.785022i	$-0.712652\pi$
$443$	−26.0766	−1.23894	−0.619468	+	0.785022i	$0.712652\pi$
$444$	41.4057	1.96503
$445$	0	0
$446$	−10.2046	−0.483202
$447$	6.06286	0.286763
$448$	−19.9871	−0.944300
$449$	−0.713928	−0.0336923	−0.0168462	−	0.999858i	$-0.505363\pi$
$449$	−0.713928	−0.0336923	−0.0168462	+	0.999858i	$0.505363\pi$
$450$	0	0
$451$	0	0
$452$	−7.59303	−0.357146
$453$	7.33491	0.344624
$454$	60.1256	2.82183
$455$	0	0
$456$	−21.6429	−1.01352
$457$	29.9979	1.40324	0.701621	−	0.712551i	$-0.252460\pi$
$457$	29.9979	1.40324	0.701621	+	0.712551i	$0.252460\pi$
$458$	−34.0226	−1.58977
$459$	0.373305	0.0174244
$460$	0	0
$461$	−0.0359281	−0.00167334	−0.000836670	−	1.00000i	$-0.500266\pi$
$461$	−0.0359281	−0.00167334	−0.000836670		1.00000i	$0.500266\pi$
$462$	0	0
$463$	−3.65474	−0.169850	−0.0849251	−	0.996387i	$-0.527065\pi$
$463$	−3.65474	−0.169850	−0.0849251	+	0.996387i	$0.527065\pi$
$464$	5.95969	0.276671
$465$	0	0
$466$	−28.2034	−1.30650
$467$	15.2379	0.705124	0.352562	−	0.935788i	$-0.385311\pi$
$467$	15.2379	0.705124	0.352562	+	0.935788i	$0.385311\pi$
$468$	12.6757	0.585934
$469$	15.5822	0.719517
$470$	0	0
$471$	−10.5558	−0.486385
$472$	−20.1118	−0.925721
$473$	0	0
$474$	20.7824	0.954565
$475$	0	0
$476$	2.29570	0.105223
$477$	−2.44417	−0.111911
$478$	−19.5188	−0.892768
$479$	12.5608	0.573918	0.286959	−	0.957943i	$-0.407356\pi$
$479$	12.5608	0.573918	0.286959	+	0.957943i	$0.407356\pi$
$480$	0	0
$481$	40.7821	1.85951
$482$	−26.3531	−1.20035
$483$	−6.52684	−0.296982
$484$	0	0
$485$	0	0
$486$	−2.36377	−0.107223
$487$	1.72235	0.0780470	0.0390235	−	0.999238i	$-0.487575\pi$
$487$	1.72235	0.0780470	0.0390235	+	0.999238i	$0.487575\pi$
$488$	−12.7813	−0.578584
$489$	−10.1070	−0.457056
$490$	0	0
$491$	−20.1226	−0.908122	−0.454061	−	0.890971i	$-0.650025\pi$
$491$	−20.1226	−0.908122	−0.454061	+	0.890971i	$0.650025\pi$
$492$	8.67786	0.391228
$493$	1.31280	0.0591256
$494$	−48.1744	−2.16747
$495$	0	0
$496$	−17.1125	−0.768375
$497$	−15.0435	−0.674792
$498$	17.5915	0.788295
$499$	17.8863	0.800702	0.400351	−	0.916362i	$-0.368888\pi$
$499$	17.8863	0.800702	0.400351	+	0.916362i	$0.368888\pi$
$500$	0	0
$501$	13.3472	0.596309
$502$	−67.7521	−3.02393
$503$	−16.3835	−0.730504	−0.365252	−	0.930909i	$-0.619017\pi$
$503$	−16.3835	−0.730504	−0.365252	+	0.930909i	$0.619017\pi$
$504$	−6.43226	−0.286516
$505$	0	0
$506$	0	0
$507$	−0.515200	−0.0228808
$508$	42.8611	1.90166
$509$	0.140642	0.00623384	0.00311692	−	0.999995i	$-0.499008\pi$
$509$	0.140642	0.00623384	0.00311692	+	0.999995i	$0.499008\pi$
$510$	0	0
$511$	25.6487	1.13463
$512$	18.6470	0.824088
$513$	5.76794	0.254661
$514$	−5.48404	−0.241891
$515$	0	0
$516$	0.805322	0.0354523
$517$	0	0
$518$	−46.7686	−2.05489
$519$	17.8603	0.783981
$520$	0	0
$521$	5.91589	0.259180	0.129590	−	0.991568i	$-0.458634\pi$
$521$	5.91589	0.259180	0.129590	+	0.991568i	$0.458634\pi$
$522$	−8.31267	−0.363836
$523$	18.9759	0.829759	0.414879	−	0.909876i	$-0.363824\pi$
$523$	18.9759	0.829759	0.414879	+	0.909876i	$0.363824\pi$
$524$	−72.9533	−3.18698
$525$	0	0
$526$	−6.92513	−0.301950
$527$	−3.76955	−0.164204
$528$	0	0
$529$	−8.50336	−0.369711
$530$	0	0
$531$	5.35991	0.232600
$532$	35.4708	1.53785
$533$	8.54718	0.370219
$534$	−36.6522	−1.58609
$535$	0	0
$536$	−34.1077	−1.47323
$537$	−12.7372	−0.549651
$538$	10.7127	0.461856
$539$	0	0
$540$	0	0
$541$	−18.3241	−0.787813	−0.393907	−	0.919150i	$-0.628877\pi$
$541$	−18.3241	−0.787813	−0.393907	+	0.919150i	$0.628877\pi$
$542$	64.0857	2.75272
$543$	20.7117	0.888826
$544$	1.30608	0.0559978
$545$	0	0
$546$	−14.3174	−0.612730
$547$	−30.8579	−1.31939	−0.659694	−	0.751534i	$-0.729314\pi$
$547$	−30.8579	−1.31939	−0.659694	+	0.751534i	$0.729314\pi$
$548$	−19.0615	−0.814265
$549$	3.40629	0.145377
$550$	0	0
$551$	20.2841	0.864132
$552$	14.2866	0.608076
$553$	−15.0716	−0.640909
$554$	−31.3232	−1.33080
$555$	0	0
$556$	−47.4096	−2.01062
$557$	37.2863	1.57987	0.789936	−	0.613190i	$-0.210114\pi$
$557$	37.2863	1.57987	0.789936	+	0.613190i	$0.210114\pi$
$558$	23.8688	1.01045
$559$	0.793194	0.0335485
$560$	0	0
$561$	0	0
$562$	−26.1708	−1.10395
$563$	−18.4747	−0.778614	−0.389307	−	0.921108i	$-0.627285\pi$
$563$	−18.4747	−0.778614	−0.389307	+	0.921108i	$0.627285\pi$
$564$	−16.7760	−0.706398
$565$	0	0
$566$	−23.8191	−1.00119
$567$	1.71423	0.0719910
$568$	32.9285	1.38165
$569$	41.9737	1.75963	0.879814	−	0.475318i	$-0.157667\pi$
$569$	41.9737	1.75963	0.879814	+	0.475318i	$0.157667\pi$
$570$	0	0
$571$	9.70741	0.406243	0.203121	−	0.979154i	$-0.434891\pi$
$571$	9.70741	0.406243	0.203121	+	0.979154i	$0.434891\pi$
$572$	0	0
$573$	−6.19957	−0.258991
$574$	−9.80182	−0.409120
$575$	0	0
$576$	−11.6595	−0.485812
$577$	13.0307	0.542475	0.271237	−	0.962513i	$-0.412567\pi$
$577$	13.0307	0.542475	0.271237	+	0.962513i	$0.412567\pi$
$578$	39.8547	1.65774
$579$	5.52362	0.229554
$580$	0	0
$581$	−12.7576	−0.529273
$582$	−34.6616	−1.43677
$583$	0	0
$584$	−56.1423	−2.32319
$585$	0	0
$586$	−18.5475	−0.766191
$587$	−43.2385	−1.78464	−0.892321	−	0.451401i	$-0.850925\pi$
$587$	−43.2385	−1.78464	−0.892321	+	0.451401i	$0.850925\pi$
$588$	−14.5699	−0.600854
$589$	−58.2434	−2.39988
$590$	0	0
$591$	12.8524	0.528675
$592$	19.5599	0.803908
$593$	−34.9632	−1.43576	−0.717882	−	0.696164i	$-0.754888\pi$
$593$	−34.9632	−1.43576	−0.717882	+	0.696164i	$0.754888\pi$
$594$	0	0
$595$	0	0
$596$	21.7499	0.890912
$597$	3.42830	0.140311
$598$	31.8002	1.30041
$599$	22.5431	0.921086	0.460543	−	0.887637i	$-0.347655\pi$
$599$	22.5431	0.921086	0.460543	+	0.887637i	$0.347655\pi$
$600$	0	0
$601$	19.9404	0.813388	0.406694	−	0.913564i	$-0.366682\pi$
$601$	19.9404	0.813388	0.406694	+	0.913564i	$0.366682\pi$
$602$	−0.909628	−0.0370737
$603$	9.08988	0.370169
$604$	26.3133	1.07067
$605$	0	0
$606$	14.5010	0.589063
$607$	−16.4084	−0.665997	−0.332999	−	0.942927i	$-0.608060\pi$
$607$	−16.4084	−0.665997	−0.332999	+	0.942927i	$0.608060\pi$
$608$	20.1803	0.818418
$609$	6.02844	0.244285
$610$	0	0
$611$	−16.5234	−0.668465
$612$	1.33920	0.0541339
$613$	−37.2802	−1.50573	−0.752866	−	0.658174i	$-0.771329\pi$
$613$	−37.2802	−1.50573	−0.752866	+	0.658174i	$0.771329\pi$
$614$	6.25672	0.252501
$615$	0	0
$616$	0	0
$617$	−28.7965	−1.15930	−0.579651	−	0.814865i	$-0.696811\pi$
$617$	−28.7965	−1.15930	−0.579651	+	0.814865i	$0.696811\pi$
$618$	−18.6026	−0.748306
$619$	8.36974	0.336408	0.168204	−	0.985752i	$-0.446203\pi$
$619$	8.36974	0.336408	0.168204	+	0.985752i	$0.446203\pi$
$620$	0	0
$621$	−3.80745	−0.152788
$622$	10.6563	0.427280
$623$	26.5806	1.06493
$624$	5.98796	0.239710
$625$	0	0
$626$	22.9504	0.917282
$627$	0	0
$628$	−37.8679	−1.51109
$629$	4.30867	0.171798
$630$	0	0
$631$	−29.9307	−1.19152	−0.595762	−	0.803161i	$-0.703150\pi$
$631$	−29.9307	−1.19152	−0.595762	+	0.803161i	$0.703150\pi$
$632$	32.9901	1.31227
$633$	−17.8292	−0.708646
$634$	40.3630	1.60302
$635$	0	0
$636$	−8.76823	−0.347683
$637$	−14.3505	−0.568588
$638$	0	0
$639$	−8.77564	−0.347159
$640$	0	0
$641$	2.11788	0.0836513	0.0418257	−	0.999125i	$-0.486683\pi$
$641$	2.11788	0.0836513	0.0418257	+	0.999125i	$0.486683\pi$
$642$	−11.7153	−0.462365
$643$	19.3340	0.762457	0.381228	−	0.924481i	$-0.375501\pi$
$643$	19.3340	0.762457	0.381228	+	0.924481i	$0.375501\pi$
$644$	−23.4145	−0.922659
$645$	0	0
$646$	−5.08967	−0.200250
$647$	−2.10369	−0.0827045	−0.0413523	−	0.999145i	$-0.513167\pi$
$647$	−2.10369	−0.0827045	−0.0413523	+	0.999145i	$0.513167\pi$
$648$	−3.75227	−0.147403
$649$	0	0
$650$	0	0
$651$	−17.3099	−0.678430
$652$	−36.2581	−1.41998
$653$	2.62234	0.102620	0.0513100	−	0.998683i	$-0.483660\pi$
$653$	2.62234	0.102620	0.0513100	+	0.998683i	$0.483660\pi$
$654$	−6.18522	−0.241861
$655$	0	0
$656$	4.09940	0.160055
$657$	14.9622	0.583732
$658$	18.9489	0.738704
$659$	−36.2473	−1.41199	−0.705997	−	0.708215i	$-0.749501\pi$
$659$	−36.2473	−1.41199	−0.705997	+	0.708215i	$0.749501\pi$
$660$	0	0
$661$	22.6488	0.880937	0.440469	−	0.897768i	$-0.354812\pi$
$661$	22.6488	0.880937	0.440469	+	0.897768i	$0.354812\pi$
$662$	6.61309	0.257025
$663$	1.31903	0.0512269
$664$	27.9249	1.08370
$665$	0	0
$666$	−27.2825	−1.05718
$667$	−13.3896	−0.518449
$668$	47.8819	1.85261
$669$	4.31709	0.166908
$670$	0	0
$671$	0	0
$672$	5.99758	0.231362
$673$	37.0125	1.42673	0.713364	−	0.700793i	$-0.247171\pi$
$673$	37.0125	1.42673	0.713364	+	0.700793i	$0.247171\pi$
$674$	21.9542	0.845643
$675$	0	0
$676$	−1.84823	−0.0710859
$677$	29.7163	1.14209	0.571044	−	0.820919i	$-0.306538\pi$
$677$	29.7163	1.14209	0.571044	+	0.820919i	$0.306538\pi$
$678$	5.00310	0.192143
$679$	25.1370	0.964668
$680$	0	0
$681$	−25.4363	−0.974722
$682$	0	0
$683$	2.06173	0.0788899	0.0394450	−	0.999222i	$-0.487441\pi$
$683$	2.06173	0.0788899	0.0394450	+	0.999222i	$0.487441\pi$
$684$	20.6919	0.791176
$685$	0	0
$686$	44.8214	1.71129
$687$	14.3934	0.549141
$688$	0.380432	0.0145038
$689$	−8.63619	−0.329013
$690$	0	0
$691$	2.97575	0.113203	0.0566015	−	0.998397i	$-0.481974\pi$
$691$	2.97575	0.113203	0.0566015	+	0.998397i	$0.481974\pi$
$692$	64.0723	2.43566
$693$	0	0
$694$	22.1137	0.839423
$695$	0	0
$696$	−13.1956	−0.500178
$697$	0.903017	0.0342042
$698$	−3.98995	−0.151022
$699$	11.9315	0.451291
$700$	0	0
$701$	−42.6095	−1.60934	−0.804669	−	0.593724i	$-0.797657\pi$
$701$	−42.6095	−1.60934	−0.804669	+	0.593724i	$0.797657\pi$
$702$	−8.35211	−0.315230
$703$	66.5733	2.51086
$704$	0	0
$705$	0	0
$706$	−22.6928	−0.854056
$707$	−10.5163	−0.395505
$708$	19.2282	0.722639
$709$	−41.6111	−1.56274	−0.781370	−	0.624069i	$-0.785479\pi$
$709$	−41.6111	−1.56274	−0.781370	+	0.624069i	$0.785479\pi$
$710$	0	0
$711$	−8.79204	−0.329727
$712$	−58.1820	−2.18046
$713$	38.4468	1.43984
$714$	−1.51265	−0.0566095
$715$	0	0
$716$	−45.6936	−1.70765
$717$	8.25748	0.308381
$718$	−64.2622	−2.39824
$719$	4.90277	0.182843	0.0914213	−	0.995812i	$-0.470859\pi$
$719$	4.90277	0.182843	0.0914213	+	0.995812i	$0.470859\pi$
$720$	0	0
$721$	13.4908	0.502424
$722$	−33.7289	−1.25526
$723$	11.1487	0.414626
$724$	74.3015	2.76139
$725$	0	0
$726$	0	0
$727$	−9.26283	−0.343539	−0.171770	−	0.985137i	$-0.554948\pi$
$727$	−9.26283	−0.343539	−0.171770	+	0.985137i	$0.554948\pi$
$728$	−22.7276	−0.842342
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.0838017	0.00309952
$732$	12.2198	0.451656
$733$	−13.6525	−0.504268	−0.252134	−	0.967692i	$-0.581132\pi$
$733$	−13.6525	−0.504268	−0.252134	+	0.967692i	$0.581132\pi$
$734$	−18.2403	−0.673260
$735$	0	0
$736$	−13.3211	−0.491022
$737$	0	0
$738$	−5.71791	−0.210479
$739$	44.1335	1.62348	0.811739	−	0.584020i	$-0.198521\pi$
$739$	44.1335	1.62348	0.811739	+	0.584020i	$0.198521\pi$
$740$	0	0
$741$	20.3803	0.748690
$742$	9.90390	0.363583
$743$	−14.8496	−0.544780	−0.272390	−	0.962187i	$-0.587814\pi$
$743$	−14.8496	−0.544780	−0.272390	+	0.962187i	$0.587814\pi$
$744$	37.8896	1.38910
$745$	0	0
$746$	−47.2841	−1.73120
$747$	−7.44215	−0.272294
$748$	0	0
$749$	8.49605	0.310439
$750$	0	0
$751$	−3.79939	−0.138642	−0.0693208	−	0.997594i	$-0.522083\pi$
$751$	−3.79939	−0.138642	−0.0693208	+	0.997594i	$0.522083\pi$
$752$	−7.92495	−0.288993
$753$	28.6627	1.04453
$754$	−29.3718	−1.06966
$755$	0	0
$756$	6.14965	0.223661
$757$	7.11918	0.258751	0.129376	−	0.991596i	$-0.458703\pi$
$757$	7.11918	0.258751	0.129376	+	0.991596i	$0.458703\pi$
$758$	15.8790	0.576751
$759$	0	0
$760$	0	0
$761$	−5.47697	−0.198540	−0.0992699	−	0.995061i	$-0.531651\pi$
$761$	−5.47697	−0.198540	−0.0992699	+	0.995061i	$0.531651\pi$
$762$	−28.2415	−1.02308
$763$	4.48559	0.162389
$764$	−22.2404	−0.804629
$765$	0	0
$766$	26.8274	0.969313
$767$	18.9386	0.683833
$768$	−25.2871	−0.912469
$769$	−40.9023	−1.47498	−0.737488	−	0.675360i	$-0.763988\pi$
$769$	−40.9023	−1.47498	−0.737488	+	0.675360i	$0.763988\pi$
$770$	0	0
$771$	2.32004	0.0835542
$772$	19.8155	0.713175
$773$	28.2567	1.01632	0.508161	−	0.861262i	$-0.330326\pi$
$773$	28.2567	1.01632	0.508161	+	0.861262i	$0.330326\pi$
$774$	−0.530633	−0.0190732
$775$	0	0
$776$	−55.0221	−1.97518
$777$	19.7856	0.709804
$778$	12.2113	0.437795
$779$	13.9525	0.499901
$780$	0	0
$781$	0	0
$782$	3.35972	0.120143
$783$	3.51670	0.125677
$784$	−6.88280	−0.245814
$785$	0	0
$786$	48.0695	1.71458
$787$	−12.9583	−0.461915	−0.230957	−	0.972964i	$-0.574186\pi$
$787$	−12.9583	−0.461915	−0.230957	+	0.972964i	$0.574186\pi$
$788$	46.1067	1.64248
$789$	2.92970	0.104300
$790$	0	0
$791$	−3.62830	−0.129008
$792$	0	0
$793$	12.0357	0.427402
$794$	17.4925	0.620786
$795$	0	0
$796$	12.2987	0.435916
$797$	−25.2703	−0.895120	−0.447560	−	0.894254i	$-0.647707\pi$
$797$	−25.2703	−0.895120	−0.447560	+	0.894254i	$0.647707\pi$
$798$	−23.3720	−0.827359
$799$	−1.74571	−0.0617588
$800$	0	0
$801$	15.5058	0.547871
$802$	75.1518	2.65370
$803$	0	0
$804$	32.6091	1.15003
$805$	0	0
$806$	84.3377	2.97067
$807$	−4.53203	−0.159535
$808$	23.0190	0.809805
$809$	−27.4522	−0.965167	−0.482584	−	0.875850i	$-0.660302\pi$
$809$	−27.4522	−0.965167	−0.482584	+	0.875850i	$0.660302\pi$
$810$	0	0
$811$	35.8599	1.25921	0.629605	−	0.776915i	$-0.283217\pi$
$811$	35.8599	1.25921	0.629605	+	0.776915i	$0.283217\pi$
$812$	21.6265	0.758940
$813$	−27.1116	−0.950847
$814$	0	0
$815$	0	0
$816$	0.632633	0.0221466
$817$	1.29482	0.0453000
$818$	28.1290	0.983506
$819$	6.05704	0.211650
$820$	0	0
$821$	−54.2112	−1.89198	−0.945992	−	0.324190i	$-0.894908\pi$
$821$	−54.2112	−1.89198	−0.945992	+	0.324190i	$0.894908\pi$
$822$	12.5597	0.438071
$823$	−4.75059	−0.165595	−0.0827976	−	0.996566i	$-0.526385\pi$
$823$	−4.75059	−0.165595	−0.0827976	+	0.996566i	$0.526385\pi$
$824$	−29.5299	−1.02872
$825$	0	0
$826$	−21.7186	−0.755687
$827$	34.6636	1.20537	0.602685	−	0.797979i	$-0.294097\pi$
$827$	34.6636	1.20537	0.602685	+	0.797979i	$0.294097\pi$
$828$	−13.6589	−0.474678
$829$	−17.7771	−0.617424	−0.308712	−	0.951156i	$-0.599898\pi$
$829$	−17.7771	−0.617424	−0.308712	+	0.951156i	$0.599898\pi$
$830$	0	0
$831$	13.2514	0.459686
$832$	−41.1974	−1.42826
$833$	−1.51614	−0.0525313
$834$	31.2386	1.08170
$835$	0	0
$836$	0	0
$837$	−10.0978	−0.349030
$838$	−15.5302	−0.536484
$839$	41.7363	1.44090	0.720449	−	0.693508i	$-0.243936\pi$
$839$	41.7363	1.44090	0.720449	+	0.693508i	$0.243936\pi$
$840$	0	0
$841$	−16.6328	−0.573546
$842$	−74.0671	−2.55252
$843$	11.0716	0.381328
$844$	−63.9605	−2.20161
$845$	0	0
$846$	11.0539	0.380039
$847$	0	0
$848$	−4.14209	−0.142240
$849$	10.0767	0.345833
$850$	0	0
$851$	−43.9453	−1.50643
$852$	−31.4818	−1.07855
$853$	37.3697	1.27951	0.639756	−	0.768578i	$-0.279035\pi$
$853$	37.3697	1.27951	0.639756	+	0.768578i	$0.279035\pi$
$854$	−13.8025	−0.472311
$855$	0	0
$856$	−18.5969	−0.635630
$857$	14.6046	0.498884	0.249442	−	0.968390i	$-0.419753\pi$
$857$	14.6046	0.498884	0.249442	+	0.968390i	$0.419753\pi$
$858$	0	0
$859$	−2.27238	−0.0775325	−0.0387663	−	0.999248i	$-0.512343\pi$
$859$	−2.27238	−0.0775325	−0.0387663	+	0.999248i	$0.512343\pi$
$860$	0	0
$861$	4.14669	0.141319
$862$	29.6457	1.00974
$863$	49.6791	1.69110	0.845549	−	0.533898i	$-0.179273\pi$
$863$	49.6791	1.69110	0.845549	+	0.533898i	$0.179273\pi$
$864$	3.49870	0.119028
$865$	0	0
$866$	61.5133	2.09031
$867$	−16.8606	−0.572617
$868$	−62.0978	−2.10774
$869$	0	0
$870$	0	0
$871$	32.1180	1.08828
$872$	−9.81846	−0.332495
$873$	14.6637	0.496290
$874$	51.9110	1.75592
$875$	0	0
$876$	53.6756	1.81353
$877$	10.3771	0.350411	0.175205	−	0.984532i	$-0.443941\pi$
$877$	10.3771	0.350411	0.175205	+	0.984532i	$0.443941\pi$
$878$	−82.8528	−2.79615
$879$	7.84659	0.264659
$880$	0	0
$881$	38.5784	1.29974	0.649869	−	0.760046i	$-0.274824\pi$
$881$	38.5784	1.29974	0.649869	+	0.760046i	$0.274824\pi$
$882$	9.60024	0.323257
$883$	−17.8591	−0.601007	−0.300503	−	0.953781i	$-0.597155\pi$
$883$	−17.8591	−0.601007	−0.300503	+	0.953781i	$0.597155\pi$
$884$	4.73190	0.159151
$885$	0	0
$886$	61.6391	2.07081
$887$	40.6791	1.36587	0.682934	−	0.730480i	$-0.260703\pi$
$887$	40.6791	1.36587	0.682934	+	0.730480i	$0.260703\pi$
$888$	−43.3085	−1.45334
$889$	20.4811	0.686913
$890$	0	0
$891$	0	0
$892$	15.4872	0.518549
$893$	−26.9730	−0.902617
$894$	−14.3312	−0.479307
$895$	0	0
$896$	35.2496	1.17761
$897$	−13.4532	−0.449188
$898$	1.68756	0.0563146
$899$	−35.5109	−1.18435
$900$	0	0
$901$	−0.912421	−0.0303971
$902$	0	0
$903$	0.384821	0.0128060
$904$	7.94196	0.264146
$905$	0	0
$906$	−17.3380	−0.576018
$907$	−17.2956	−0.574289	−0.287145	−	0.957887i	$-0.592706\pi$
$907$	−17.2956	−0.574289	−0.287145	+	0.957887i	$0.592706\pi$
$908$	−91.2505	−3.02825
$909$	−6.13469	−0.203475
$910$	0	0
$911$	46.8317	1.55160	0.775802	−	0.630977i	$-0.217346\pi$
$911$	46.8317	1.55160	0.775802	+	0.630977i	$0.217346\pi$
$912$	9.77482	0.323677
$913$	0	0
$914$	−70.9081	−2.34543
$915$	0	0
$916$	51.6348	1.70606
$917$	−34.8605	−1.15120
$918$	−0.882407	−0.0291238
$919$	42.2482	1.39364	0.696820	−	0.717246i	$-0.254598\pi$
$919$	42.2482	1.39364	0.696820	+	0.717246i	$0.254598\pi$
$920$	0	0
$921$	−2.64692	−0.0872191
$922$	0.0849258	0.00279688
$923$	−31.0077	−1.02063
$924$	0	0
$925$	0	0
$926$	8.63897	0.283894
$927$	7.86989	0.258481
$928$	12.3039	0.403894
$929$	11.4317	0.375062	0.187531	−	0.982259i	$-0.439952\pi$
$929$	11.4317	0.375062	0.187531	+	0.982259i	$0.439952\pi$
$930$	0	0
$931$	−23.4260	−0.767755
$932$	42.8032	1.40207
$933$	−4.50819	−0.147592
$934$	−36.0188	−1.17857
$935$	0	0
$936$	−13.2582	−0.433358
$937$	1.46708	0.0479275	0.0239637	−	0.999713i	$-0.492371\pi$
$937$	1.46708	0.0479275	0.0239637	+	0.999713i	$0.492371\pi$
$938$	−36.8326	−1.20263
$939$	−9.70923	−0.316849
$940$	0	0
$941$	19.3755	0.631623	0.315811	−	0.948822i	$-0.397723\pi$
$941$	19.3755	0.631623	0.315811	+	0.948822i	$0.397723\pi$
$942$	24.9515	0.812962
$943$	−9.21013	−0.299923
$944$	9.08334	0.295637
$945$	0	0
$946$	0	0
$947$	27.4774	0.892895	0.446448	−	0.894810i	$-0.352689\pi$
$947$	27.4774	0.892895	0.446448	+	0.894810i	$0.352689\pi$
$948$	−31.5406	−1.02439
$949$	52.8673	1.71615
$950$	0	0
$951$	−17.0757	−0.553717
$952$	−2.40119	−0.0778232
$953$	12.9686	0.420095	0.210048	−	0.977691i	$-0.432638\pi$
$953$	12.9686	0.420095	0.210048	+	0.977691i	$0.432638\pi$
$954$	5.77745	0.187052
$955$	0	0
$956$	29.6230	0.958075
$957$	0	0
$958$	−29.6908	−0.959268
$959$	−9.10846	−0.294127
$960$	0	0
$961$	70.9652	2.28920
$962$	−96.3996	−3.10805
$963$	4.95618	0.159711
$964$	39.9951	1.28816
$965$	0	0
$966$	15.4280	0.496386
$967$	48.0563	1.54539	0.772694	−	0.634779i	$-0.218909\pi$
$967$	48.0563	1.54539	0.772694	+	0.634779i	$0.218909\pi$
$968$	0	0
$969$	2.15320	0.0691708
$970$	0	0
$971$	51.7067	1.65935	0.829674	−	0.558249i	$-0.188527\pi$
$971$	51.7067	1.65935	0.829674	+	0.558249i	$0.188527\pi$
$972$	3.58741	0.115066
$973$	−22.6545	−0.726271
$974$	−4.07123	−0.130451
$975$	0	0
$976$	5.77259	0.184776
$977$	−12.7980	−0.409444	−0.204722	−	0.978820i	$-0.565629\pi$
$977$	−12.7980	−0.409444	−0.204722	+	0.978820i	$0.565629\pi$
$978$	23.8907	0.763941
$979$	0	0
$980$	0	0
$981$	2.61667	0.0835440
$982$	47.5653	1.51787
$983$	−13.6790	−0.436293	−0.218147	−	0.975916i	$-0.570001\pi$
$983$	−13.6790	−0.436293	−0.218147	+	0.975916i	$0.570001\pi$
$984$	−9.07666	−0.289353
$985$	0	0
$986$	−3.10316	−0.0988248
$987$	−8.01638	−0.255164
$988$	73.1126	2.32602
$989$	−0.854717	−0.0271784
$990$	0	0
$991$	34.7067	1.10250	0.551248	−	0.834342i	$-0.314152\pi$
$991$	34.7067	1.10250	0.551248	+	0.834342i	$0.314152\pi$
$992$	−35.3291	−1.12170
$993$	−2.79769	−0.0887819
$994$	35.5593	1.12787
$995$	0	0
$996$	−26.6980	−0.845959
$997$	30.9145	0.979072	0.489536	−	0.871983i	$-0.337166\pi$
$997$	30.9145	0.979072	0.489536	+	0.871983i	$0.337166\pi$
$998$	−42.2791	−1.33832
$999$	11.5420	0.365171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.ea.1.1		12
5.2	odd	4		1815.2.c.k.364.3		24
5.3	odd	4		1815.2.c.k.364.22		24
5.4	even	2		9075.2.a.dx.1.12		12
11.2	odd	10		825.2.n.p.301.1		24
11.6	odd	10		825.2.n.p.751.1		24
11.10	odd	2		9075.2.a.dy.1.12		12
55.2	even	20		165.2.s.a.4.11	yes	48
55.13	even	20		165.2.s.a.4.2	✓	48
55.17	even	20		165.2.s.a.124.2	yes	48
55.24	odd	10		825.2.n.o.301.6		24
55.28	even	20		165.2.s.a.124.11	yes	48
55.32	even	4		1815.2.c.j.364.22		24
55.39	odd	10		825.2.n.o.751.6		24
55.43	even	4		1815.2.c.j.364.3		24
55.54	odd	2		9075.2.a.dz.1.1		12
165.2	odd	20		495.2.ba.c.334.2		48
165.17	odd	20		495.2.ba.c.289.11		48
165.68	odd	20		495.2.ba.c.334.11		48
165.83	odd	20		495.2.ba.c.289.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.4.2	✓	48	55.13	even	20
165.2.s.a.4.11	yes	48	55.2	even	20
165.2.s.a.124.2	yes	48	55.17	even	20
165.2.s.a.124.11	yes	48	55.28	even	20
495.2.ba.c.289.2		48	165.83	odd	20
495.2.ba.c.289.11		48	165.17	odd	20
495.2.ba.c.334.2		48	165.2	odd	20
495.2.ba.c.334.11		48	165.68	odd	20
825.2.n.o.301.6		24	55.24	odd	10
825.2.n.o.751.6		24	55.39	odd	10
825.2.n.p.301.1		24	11.2	odd	10
825.2.n.p.751.1		24	11.6	odd	10
1815.2.c.j.364.3		24	55.43	even	4
1815.2.c.j.364.22		24	55.32	even	4
1815.2.c.k.364.3		24	5.2	odd	4
1815.2.c.k.364.22		24	5.3	odd	4
9075.2.a.dx.1.12		12	5.4	even	2
9075.2.a.dy.1.12		12	11.10	odd	2
9075.2.a.dz.1.1		12	55.54	odd	2
9075.2.a.ea.1.1		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.36377 q^{2} +1.00000 q^{3} +3.58741 q^{4} -2.36377 q^{6} +1.71423 q^{7} -3.75227 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.36377 q^{2} +1.00000 q^{3} +3.58741 q^{4} -2.36377 q^{6} +1.71423 q^{7} -3.75227 q^{8} +1.00000 q^{9} +3.58741 q^{12} +3.53338 q^{13} -4.05205 q^{14} +1.69468 q^{16} +0.373305 q^{17} -2.36377 q^{18} +5.76794 q^{19} +1.71423 q^{21} -3.80745 q^{23} -3.75227 q^{24} -8.35211 q^{26} +1.00000 q^{27} +6.14965 q^{28} +3.51670 q^{29} -10.0978 q^{31} +3.49870 q^{32} -0.882407 q^{34} +3.58741 q^{36} +11.5420 q^{37} -13.6341 q^{38} +3.53338 q^{39} +2.41898 q^{41} -4.05205 q^{42} +0.224486 q^{43} +8.99992 q^{46} -4.67637 q^{47} +1.69468 q^{48} -4.06141 q^{49} +0.373305 q^{51} +12.6757 q^{52} -2.44417 q^{53} -2.36377 q^{54} -6.43226 q^{56} +5.76794 q^{57} -8.31267 q^{58} +5.35991 q^{59} +3.40629 q^{61} +23.8688 q^{62} +1.71423 q^{63} -11.6595 q^{64} +9.08988 q^{67} +1.33920 q^{68} -3.80745 q^{69} -8.77564 q^{71} -3.75227 q^{72} +14.9622 q^{73} -27.2825 q^{74} +20.6919 q^{76} -8.35211 q^{78} -8.79204 q^{79} +1.00000 q^{81} -5.71791 q^{82} -7.44215 q^{83} +6.14965 q^{84} -0.530633 q^{86} +3.51670 q^{87} +15.5058 q^{89} +6.05704 q^{91} -13.6589 q^{92} -10.0978 q^{93} +11.0539 q^{94} +3.49870 q^{96} +14.6637 q^{97} +9.60024 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9	\( 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{12} + 18 q^{13} + 6 q^{14} + 24 q^{16} + 18 q^{17} + 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} + 12 q^{27} + 30 q^{28} + 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} + 18 q^{39} + 6 q^{42} + 32 q^{43} + 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} + 18 q^{51} + 48 q^{52} - 12 q^{53} + 4 q^{54} + 6 q^{56} + 16 q^{57} - 10 q^{58} + 20 q^{59} + 20 q^{61} + 20 q^{62} + 8 q^{63} - 6 q^{64} + 10 q^{67} + 26 q^{68} + 32 q^{71} + 12 q^{72} + 26 q^{73} - 68 q^{74} + 34 q^{76} + 16 q^{78} + 32 q^{79} + 12 q^{81} - 62 q^{82} + 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} - 2 q^{94} + 28 q^{96} - 22 q^{97} + 60 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9 + 12 * q^12 + 18 * q^13 + 6 * q^14 + 24 * q^16 + 18 * q^17 + 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 + 12 * q^27 + 30 * q^28 + 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 + 18 * q^39 + 6 * q^42 + 32 * q^43 + 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 + 18 * q^51 + 48 * q^52 - 12 * q^53 + 4 * q^54 + 6 * q^56 + 16 * q^57 - 10 * q^58 + 20 * q^59 + 20 * q^61 + 20 * q^62 + 8 * q^63 - 6 * q^64 + 10 * q^67 + 26 * q^68 + 32 * q^71 + 12 * q^72 + 26 * q^73 - 68 * q^74 + 34 * q^76 + 16 * q^78 + 32 * q^79 + 12 * q^81 - 62 * q^82 + 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 - 2 * q^94 + 28 * q^96 - 22 * q^97 + 60 * q^98

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