LMFDB - Embedded newform 9075.2.a.df.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:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.209057$ 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)

$f(q)$	$=$	$q+1.33826 q^{2} +1.00000 q^{3} -0.209057 q^{4} +1.33826 q^{6} -3.78339 q^{7} -2.95630 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.33826 q^{2} +1.00000 q^{3} -0.209057 q^{4} +1.33826 q^{6} -3.78339 q^{7} -2.95630 q^{8} +1.00000 q^{9} -0.209057 q^{12} +4.93395 q^{13} -5.06316 q^{14} -3.53818 q^{16} +3.25841 q^{17} +1.33826 q^{18} -1.37283 q^{19} -3.78339 q^{21} +1.93395 q^{23} -2.95630 q^{24} +6.60292 q^{26} +1.00000 q^{27} +0.790943 q^{28} +1.73968 q^{29} -5.92173 q^{31} +1.17758 q^{32} +4.36060 q^{34} -0.209057 q^{36} -4.69789 q^{37} -1.83720 q^{38} +4.93395 q^{39} -9.43757 q^{41} -5.06316 q^{42} +11.4086 q^{43} +2.58814 q^{46} -0.805727 q^{47} -3.53818 q^{48} +7.31401 q^{49} +3.25841 q^{51} -1.03148 q^{52} -10.6960 q^{53} +1.33826 q^{54} +11.1848 q^{56} -1.37283 q^{57} +2.32815 q^{58} +12.1634 q^{59} -8.59102 q^{61} -7.92482 q^{62} -3.78339 q^{63} +8.65227 q^{64} +2.77425 q^{67} -0.681193 q^{68} +1.93395 q^{69} -2.73968 q^{71} -2.95630 q^{72} -9.37717 q^{73} -6.28700 q^{74} +0.286999 q^{76} +6.60292 q^{78} -12.7667 q^{79} +1.00000 q^{81} -12.6299 q^{82} -3.41304 q^{83} +0.790943 q^{84} +15.2677 q^{86} +1.73968 q^{87} -7.85817 q^{89} -18.6671 q^{91} -0.404307 q^{92} -5.92173 q^{93} -1.07827 q^{94} +1.17758 q^{96} +13.6196 q^{97} +9.78806 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + q^2 + 4 * q^3 + q^4 + q^6 - 3 * q^8 + 4 * q^9	$ 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + q^{12} + 7 q^{13} - 5 q^{14} - 9 q^{16} + 8 q^{17} + q^{18} - 11 q^{19} - 5 q^{23} - 3 q^{24} - 12 q^{26} + 4 q^{27} + 5 q^{28} - 17 q^{29} - 5 q^{31} + 17 q^{34} + q^{36} - 15 q^{37} + q^{38} + 7 q^{39} - 10 q^{41} - 5 q^{42} - 4 q^{43} - 15 q^{46} + 8 q^{47} - 9 q^{48} - 8 q^{49} + 8 q^{51} - 7 q^{52} - 10 q^{53} + q^{54} + 10 q^{56} - 11 q^{57} + 7 q^{58} + 9 q^{59} - 37 q^{61} - 20 q^{62} - 7 q^{64} - 3 q^{67} + 17 q^{68} - 5 q^{69} + 13 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} - 29 q^{76} - 12 q^{78} - 20 q^{79} + 4 q^{81} - 5 q^{82} - 17 q^{83} + 5 q^{84} + 34 q^{86} - 17 q^{87} - 24 q^{89} - 20 q^{91} - 10 q^{92} - 5 q^{93} - 23 q^{94} + 32 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + q^2 + 4 * q^3 + q^4 + q^6 - 3 * q^8 + 4 * q^9 + q^12 + 7 * q^13 - 5 * q^14 - 9 * q^16 + 8 * q^17 + q^18 - 11 * q^19 - 5 * q^23 - 3 * q^24 - 12 * q^26 + 4 * q^27 + 5 * q^28 - 17 * q^29 - 5 * q^31 + 17 * q^34 + q^36 - 15 * q^37 + q^38 + 7 * q^39 - 10 * q^41 - 5 * q^42 - 4 * q^43 - 15 * q^46 + 8 * q^47 - 9 * q^48 - 8 * q^49 + 8 * q^51 - 7 * q^52 - 10 * q^53 + q^54 + 10 * q^56 - 11 * q^57 + 7 * q^58 + 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 - 3 * q^67 + 17 * q^68 - 5 * q^69 + 13 * q^71 - 3 * q^72 + 15 * q^73 + 5 * q^74 - 29 * q^76 - 12 * q^78 - 20 * q^79 + 4 * q^81 - 5 * q^82 - 17 * q^83 + 5 * q^84 + 34 * q^86 - 17 * q^87 - 24 * q^89 - 20 * q^91 - 10 * q^92 - 5 * q^93 - 23 * q^94 + 32 * q^97 + 13 * 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 $

$2$	1.33826	0.946294	0.473147	−	0.880984i	$-0.343118\pi$
$2$	1.33826	0.946294	0.473147	+	0.880984i	$0.343118\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.209057	−0.104528
$5$	0	0
$5$	0	0
$6$	1.33826	0.546343
$7$	−3.78339	−1.42999	−0.714993	−	0.699132i	$-0.753570\pi$
$7$	−3.78339	−1.42999	−0.714993	+	0.699132i	$0.753570\pi$
$8$	−2.95630	−1.04521
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.209057	−0.0603495
$13$	4.93395	1.36843	0.684216	−	0.729279i	$-0.260144\pi$
$13$	4.93395	1.36843	0.684216	+	0.729279i	$0.260144\pi$
$14$	−5.06316	−1.35319
$15$	0	0
$16$	−3.53818	−0.884545
$17$	3.25841	0.790280	0.395140	−	0.918621i	$-0.370696\pi$
$17$	3.25841	0.790280	0.395140	+	0.918621i	$0.370696\pi$
$18$	1.33826	0.315431
$19$	−1.37283	−0.314949	−0.157474	−	0.987523i	$-0.550335\pi$
$19$	−1.37283	−0.314949	−0.157474	+	0.987523i	$0.550335\pi$
$20$	0	0
$21$	−3.78339	−0.825603
$22$	0	0
$23$	1.93395	0.403257	0.201629	−	0.979462i	$-0.435377\pi$
$23$	1.93395	0.403257	0.201629	+	0.979462i	$0.435377\pi$
$24$	−2.95630	−0.603451
$25$	0	0
$26$	6.60292	1.29494
$27$	1.00000	0.192450
$28$	0.790943	0.149474
$29$	1.73968	0.323051	0.161525	−	0.986869i	$-0.448359\pi$
$29$	1.73968	0.323051	0.161525	+	0.986869i	$0.448359\pi$
$30$	0	0
$31$	−5.92173	−1.06357	−0.531787	−	0.846878i	$-0.678479\pi$
$31$	−5.92173	−1.06357	−0.531787	+	0.846878i	$0.678479\pi$
$32$	1.17758	0.208169
$33$	0	0
$34$	4.36060	0.747837
$35$	0	0
$36$	−0.209057	−0.0348428
$37$	−4.69789	−0.772328	−0.386164	−	0.922430i	$-0.626200\pi$
$37$	−4.69789	−0.772328	−0.386164	+	0.922430i	$0.626200\pi$
$38$	−1.83720	−0.298034
$39$	4.93395	0.790065
$40$	0	0
$41$	−9.43757	−1.47390	−0.736950	−	0.675947i	$-0.763735\pi$
$41$	−9.43757	−1.47390	−0.736950	+	0.675947i	$0.763735\pi$
$42$	−5.06316	−0.781262
$43$	11.4086	1.73980	0.869901	−	0.493226i	$-0.164182\pi$
$43$	11.4086	1.73980	0.869901	+	0.493226i	$0.164182\pi$
$44$	0	0
$45$	0	0
$46$	2.58814	0.381600
$47$	−0.805727	−0.117527	−0.0587637	−	0.998272i	$-0.518716\pi$
$47$	−0.805727	−0.117527	−0.0587637	+	0.998272i	$0.518716\pi$
$48$	−3.53818	−0.510692
$49$	7.31401	1.04486
$50$	0	0
$51$	3.25841	0.456268
$52$	−1.03148	−0.143040
$53$	−10.6960	−1.46921	−0.734603	−	0.678498i	$-0.762631\pi$
$53$	−10.6960	−1.46921	−0.734603	+	0.678498i	$0.762631\pi$
$54$	1.33826	0.182114
$55$	0	0
$56$	11.1848	1.49463
$57$	−1.37283	−0.181836
$58$	2.32815	0.305701
$59$	12.1634	1.58355	0.791773	−	0.610816i	$-0.209158\pi$
$59$	12.1634	1.58355	0.791773	+	0.610816i	$0.209158\pi$
$60$	0	0
$61$	−8.59102	−1.09997	−0.549984	−	0.835175i	$-0.685366\pi$
$61$	−8.59102	−1.09997	−0.549984	+	0.835175i	$0.685366\pi$
$62$	−7.92482	−1.00645
$63$	−3.78339	−0.476662
$64$	8.65227	1.08153
$65$	0	0
$66$	0	0
$67$	2.77425	0.338928	0.169464	−	0.985536i	$-0.445796\pi$
$67$	2.77425	0.338928	0.169464	+	0.985536i	$0.445796\pi$
$68$	−0.681193	−0.0826068
$69$	1.93395	0.232821
$70$	0	0
$71$	−2.73968	−0.325140	−0.162570	−	0.986697i	$-0.551978\pi$
$71$	−2.73968	−0.325140	−0.162570	+	0.986697i	$0.551978\pi$
$72$	−2.95630	−0.348403
$73$	−9.37717	−1.09751	−0.548757	−	0.835982i	$-0.684899\pi$
$73$	−9.37717	−1.09751	−0.548757	+	0.835982i	$0.684899\pi$
$74$	−6.28700	−0.730849
$75$	0	0
$76$	0.286999	0.0329211
$77$	0	0
$78$	6.60292	0.747633
$79$	−12.7667	−1.43637	−0.718183	−	0.695855i	$-0.755026\pi$
$79$	−12.7667	−1.43637	−0.718183	+	0.695855i	$0.755026\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.6299	−1.39474
$83$	−3.41304	−0.374630	−0.187315	−	0.982300i	$-0.559979\pi$
$83$	−3.41304	−0.374630	−0.187315	+	0.982300i	$0.559979\pi$
$84$	0.790943	0.0862990
$85$	0	0
$86$	15.2677	1.64636
$87$	1.73968	0.186513
$88$	0	0
$89$	−7.85817	−0.832964	−0.416482	−	0.909144i	$-0.636737\pi$
$89$	−7.85817	−0.832964	−0.416482	+	0.909144i	$0.636737\pi$
$90$	0	0
$91$	−18.6671	−1.95684
$92$	−0.404307	−0.0421519
$93$	−5.92173	−0.614055
$94$	−1.07827	−0.111215
$95$	0	0
$96$	1.17758	0.120186
$97$	13.6196	1.38286	0.691431	−	0.722442i	$-0.256981\pi$
$97$	13.6196	1.38286	0.691431	+	0.722442i	$0.256981\pi$
$98$	9.78806	0.988743
$99$	0	0
$100$	0	0
$101$	−7.93842	−0.789902	−0.394951	−	0.918702i	$-0.629239\pi$
$101$	−7.93842	−0.789902	−0.394951	+	0.918702i	$0.629239\pi$
$102$	4.36060	0.431764
$103$	12.2052	1.20262	0.601309	−	0.799017i	$-0.294646\pi$
$103$	12.2052	1.20262	0.601309	+	0.799017i	$0.294646\pi$
$104$	−14.5862	−1.43030
$105$	0	0
$106$	−14.3140	−1.39030
$107$	2.33537	0.225769	0.112885	−	0.993608i	$-0.463991\pi$
$107$	2.33537	0.225769	0.112885	+	0.993608i	$0.463991\pi$
$108$	−0.209057	−0.0201165
$109$	−6.39419	−0.612453	−0.306226	−	0.951959i	$-0.599066\pi$
$109$	−6.39419	−0.612453	−0.306226	+	0.951959i	$0.599066\pi$
$110$	0	0
$111$	−4.69789	−0.445904
$112$	13.3863	1.26489
$113$	−18.1316	−1.70568	−0.852840	−	0.522172i	$-0.825122\pi$
$113$	−18.1316	−1.70568	−0.852840	+	0.522172i	$0.825122\pi$
$114$	−1.83720	−0.172070
$115$	0	0
$116$	−0.363692	−0.0337680
$117$	4.93395	0.456144
$118$	16.2779	1.49850
$119$	−12.3278	−1.13009
$120$	0	0
$121$	0	0
$122$	−11.4970	−1.04089
$123$	−9.43757	−0.850957
$124$	1.23798	0.111174
$125$	0	0
$126$	−5.06316	−0.451062
$127$	−22.3123	−1.97990	−0.989948	−	0.141429i	$-0.954830\pi$
$127$	−22.3123	−1.97990	−0.989948	+	0.141429i	$0.954830\pi$
$128$	9.22384	0.815280
$129$	11.4086	1.00448
$130$	0	0
$131$	0.562102	0.0491111	0.0245555	−	0.999698i	$-0.492183\pi$
$131$	0.562102	0.0491111	0.0245555	+	0.999698i	$0.492183\pi$
$132$	0	0
$133$	5.19394	0.450372
$134$	3.71267	0.320726
$135$	0	0
$136$	−9.63282	−0.826007
$137$	18.2719	1.56107	0.780536	−	0.625110i	$-0.214946\pi$
$137$	18.2719	1.56107	0.780536	+	0.625110i	$0.214946\pi$
$138$	2.58814	0.220317
$139$	−14.2600	−1.20952	−0.604758	−	0.796409i	$-0.706730\pi$
$139$	−14.2600	−1.20952	−0.604758	+	0.796409i	$0.706730\pi$
$140$	0	0
$141$	−0.805727	−0.0678544
$142$	−3.66641	−0.307678
$143$	0	0
$144$	−3.53818	−0.294848
$145$	0	0
$146$	−12.5491	−1.03857
$147$	7.31401	0.603249
$148$	0.982126	0.0807302
$149$	−10.6039	−0.868705	−0.434353	−	0.900743i	$-0.643023\pi$
$149$	−10.6039	−0.868705	−0.434353	+	0.900743i	$0.643023\pi$
$150$	0	0
$151$	−12.5734	−1.02320	−0.511602	−	0.859222i	$-0.670948\pi$
$151$	−12.5734	−1.02320	−0.511602	+	0.859222i	$0.670948\pi$
$152$	4.05849	0.329187
$153$	3.25841	0.263427
$154$	0	0
$155$	0	0
$156$	−1.03148	−0.0825843
$157$	12.5332	1.00026	0.500128	−	0.865951i	$-0.333286\pi$
$157$	12.5332	1.00026	0.500128	+	0.865951i	$0.333286\pi$
$158$	−17.0852	−1.35922
$159$	−10.6960	−0.848246
$160$	0	0
$161$	−7.31690	−0.576652
$162$	1.33826	0.105144
$163$	3.10317	0.243059	0.121530	−	0.992588i	$-0.461220\pi$
$163$	3.10317	0.243059	0.121530	+	0.992588i	$0.461220\pi$
$164$	1.97299	0.154065
$165$	0	0
$166$	−4.56754	−0.354510
$167$	−8.34898	−0.646063	−0.323032	−	0.946388i	$-0.604702\pi$
$167$	−8.34898	−0.646063	−0.323032	+	0.946388i	$0.604702\pi$
$168$	11.1848	0.862927
$169$	11.3439	0.872608
$170$	0	0
$171$	−1.37283	−0.104983
$172$	−2.38506	−0.181859
$173$	−6.84208	−0.520194	−0.260097	−	0.965583i	$-0.583754\pi$
$173$	−6.84208	−0.520194	−0.260097	+	0.965583i	$0.583754\pi$
$174$	2.32815	0.176496
$175$	0	0
$176$	0	0
$177$	12.1634	0.914260
$178$	−10.5163	−0.788229
$179$	15.5085	1.15916	0.579579	−	0.814916i	$-0.303217\pi$
$179$	15.5085	1.15916	0.579579	+	0.814916i	$0.303217\pi$
$180$	0	0
$181$	−2.26096	−0.168056	−0.0840281	−	0.996463i	$-0.526779\pi$
$181$	−2.26096	−0.168056	−0.0840281	+	0.996463i	$0.526779\pi$
$182$	−24.9814	−1.85174
$183$	−8.59102	−0.635067
$184$	−5.71734	−0.421488
$185$	0	0
$186$	−7.92482	−0.581076
$187$	0	0
$188$	0.168443	0.0122850
$189$	−3.78339	−0.275201
$190$	0	0
$191$	−17.7692	−1.28574	−0.642869	−	0.765976i	$-0.722256\pi$
$191$	−17.7692	−1.28574	−0.642869	+	0.765976i	$0.722256\pi$
$192$	8.65227	0.624424
$193$	21.1559	1.52283	0.761417	−	0.648262i	$-0.224504\pi$
$193$	21.1559	1.52283	0.761417	+	0.648262i	$0.224504\pi$
$194$	18.2266	1.30859
$195$	0	0
$196$	−1.52904	−0.109217
$197$	−24.1034	−1.71730	−0.858650	−	0.512563i	$-0.828696\pi$
$197$	−24.1034	−1.71730	−0.858650	+	0.512563i	$0.828696\pi$
$198$	0	0
$199$	4.30369	0.305081	0.152540	−	0.988297i	$-0.451255\pi$
$199$	4.30369	0.305081	0.152540	+	0.988297i	$0.451255\pi$
$200$	0	0
$201$	2.77425	0.195680
$202$	−10.6237	−0.747480
$203$	−6.58189	−0.461958
$204$	−0.681193	−0.0476930
$205$	0	0
$206$	16.3338	1.13803
$207$	1.93395	0.134419
$208$	−17.4572	−1.21044
$209$	0	0
$210$	0	0
$211$	−15.2270	−1.04827	−0.524135	−	0.851635i	$-0.675611\pi$
$211$	−15.2270	−1.04827	−0.524135	+	0.851635i	$0.675611\pi$
$212$	2.23607	0.153574
$213$	−2.73968	−0.187720
$214$	3.12534	0.213644
$215$	0	0
$216$	−2.95630	−0.201150
$217$	22.4042	1.52089
$218$	−8.55710	−0.579560
$219$	−9.37717	−0.633650
$220$	0	0
$221$	16.0768	1.08145
$222$	−6.28700	−0.421956
$223$	−28.6450	−1.91821	−0.959105	−	0.283050i	$-0.908654\pi$
$223$	−28.6450	−1.91821	−0.959105	+	0.283050i	$0.908654\pi$
$224$	−4.45524	−0.297678
$225$	0	0
$226$	−24.2649	−1.61407
$227$	3.02608	0.200848	0.100424	−	0.994945i	$-0.467980\pi$
$227$	3.02608	0.200848	0.100424	+	0.994945i	$0.467980\pi$
$228$	0.286999	0.0190070
$229$	−3.05244	−0.201711	−0.100856	−	0.994901i	$-0.532158\pi$
$229$	−3.05244	−0.201711	−0.100856	+	0.994901i	$0.532158\pi$
$230$	0	0
$231$	0	0
$232$	−5.14301	−0.337655
$233$	−6.30934	−0.413339	−0.206669	−	0.978411i	$-0.566262\pi$
$233$	−6.30934	−0.413339	−0.206669	+	0.978411i	$0.566262\pi$
$234$	6.60292	0.431646
$235$	0	0
$236$	−2.54285	−0.165526
$237$	−12.7667	−0.829286
$238$	−16.4978	−1.06940
$239$	5.71109	0.369420	0.184710	−	0.982793i	$-0.440865\pi$
$239$	5.71109	0.369420	0.184710	+	0.982793i	$0.440865\pi$
$240$	0	0
$241$	−4.81966	−0.310462	−0.155231	−	0.987878i	$-0.549612\pi$
$241$	−4.81966	−0.310462	−0.155231	+	0.987878i	$0.549612\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	1.79601	0.114978
$245$	0	0
$246$	−12.6299	−0.805255
$247$	−6.77348	−0.430986
$248$	17.5064	1.11166
$249$	−3.41304	−0.216293
$250$	0	0
$251$	−0.883023	−0.0557359	−0.0278680	−	0.999612i	$-0.508872\pi$
$251$	−0.883023	−0.0557359	−0.0278680	+	0.999612i	$0.508872\pi$
$252$	0.790943	0.0498247
$253$	0	0
$254$	−29.8597	−1.87356
$255$	0	0
$256$	−4.96064	−0.310040
$257$	−8.63651	−0.538731	−0.269365	−	0.963038i	$-0.586814\pi$
$257$	−8.63651	−0.538731	−0.269365	+	0.963038i	$0.586814\pi$
$258$	15.2677	0.950529
$259$	17.7739	1.10442
$260$	0	0
$261$	1.73968	0.107684
$262$	0.752239	0.0464735
$263$	−4.73594	−0.292031	−0.146015	−	0.989282i	$-0.546645\pi$
$263$	−4.73594	−0.292031	−0.146015	+	0.989282i	$0.546645\pi$
$264$	0	0
$265$	0	0
$266$	6.95085	0.426184
$267$	−7.85817	−0.480912
$268$	−0.579976	−0.0354277
$269$	−14.2672	−0.869887	−0.434944	−	0.900458i	$-0.643232\pi$
$269$	−14.2672	−0.869887	−0.434944	+	0.900458i	$0.643232\pi$
$270$	0	0
$271$	−22.9367	−1.39331	−0.696653	−	0.717408i	$-0.745328\pi$
$271$	−22.9367	−1.39331	−0.696653	+	0.717408i	$0.745328\pi$
$272$	−11.5288	−0.699039
$273$	−18.6671	−1.12978
$274$	24.4526	1.47723
$275$	0	0
$276$	−0.404307	−0.0243364
$277$	−5.82302	−0.349872	−0.174936	−	0.984580i	$-0.555972\pi$
$277$	−5.82302	−0.349872	−0.174936	+	0.984580i	$0.555972\pi$
$278$	−19.0836	−1.14456
$279$	−5.92173	−0.354525
$280$	0	0
$281$	0.157591	0.00940109	0.00470055	−	0.999989i	$-0.498504\pi$
$281$	0.157591	0.00940109	0.00470055	+	0.999989i	$0.498504\pi$
$282$	−1.07827	−0.0642102
$283$	−8.94565	−0.531764	−0.265882	−	0.964006i	$-0.585663\pi$
$283$	−8.94565	−0.531764	−0.265882	+	0.964006i	$0.585663\pi$
$284$	0.572749	0.0339864
$285$	0	0
$286$	0	0
$287$	35.7060	2.10766
$288$	1.17758	0.0693895
$289$	−6.38277	−0.375457
$290$	0	0
$291$	13.6196	0.798396
$292$	1.96036	0.114722
$293$	25.9879	1.51823	0.759113	−	0.650959i	$-0.225633\pi$
$293$	25.9879	1.51823	0.759113	+	0.650959i	$0.225633\pi$
$294$	9.78806	0.570851
$295$	0	0
$296$	13.8883	0.807243
$297$	0	0
$298$	−14.1908	−0.822050
$299$	9.54204	0.551831
$300$	0	0
$301$	−43.1633	−2.48789
$302$	−16.8264	−0.968252
$303$	−7.93842	−0.456050
$304$	4.85732	0.278586
$305$	0	0
$306$	4.36060	0.249279
$307$	4.72359	0.269590	0.134795	−	0.990874i	$-0.456962\pi$
$307$	4.72359	0.269590	0.134795	+	0.990874i	$0.456962\pi$
$308$	0	0
$309$	12.2052	0.694332
$310$	0	0
$311$	−19.2517	−1.09166	−0.545832	−	0.837895i	$-0.683786\pi$
$311$	−19.2517	−1.09166	−0.545832	+	0.837895i	$0.683786\pi$
$312$	−14.5862	−0.825782
$313$	−2.94346	−0.166374	−0.0831872	−	0.996534i	$-0.526510\pi$
$313$	−2.94346	−0.166374	−0.0831872	+	0.996534i	$0.526510\pi$
$314$	16.7727	0.946536
$315$	0	0
$316$	2.66897	0.150141
$317$	−1.82705	−0.102617	−0.0513086	−	0.998683i	$-0.516339\pi$
$317$	−1.82705	−0.102617	−0.0513086	+	0.998683i	$0.516339\pi$
$318$	−14.3140	−0.802690
$319$	0	0
$320$	0	0
$321$	2.33537	0.130348
$322$	−9.79192	−0.545682
$323$	−4.47324	−0.248898
$324$	−0.209057	−0.0116143
$325$	0	0
$326$	4.15285	0.230005
$327$	−6.39419	−0.353600
$328$	27.9002	1.54053
$329$	3.04838	0.168062
$330$	0	0
$331$	17.3017	0.950985	0.475493	−	0.879720i	$-0.342270\pi$
$331$	17.3017	0.950985	0.475493	+	0.879720i	$0.342270\pi$
$332$	0.713521	0.0391595
$333$	−4.69789	−0.257443
$334$	−11.1731	−0.611366
$335$	0	0
$336$	13.3863	0.730283
$337$	−10.8095	−0.588829	−0.294414	−	0.955678i	$-0.595125\pi$
$337$	−10.8095	−0.588829	−0.294414	+	0.955678i	$0.595125\pi$
$338$	15.1811	0.825744
$339$	−18.1316	−0.984775
$340$	0	0
$341$	0	0
$342$	−1.83720	−0.0993446
$343$	−1.18802	−0.0641472
$344$	−33.7273	−1.81846
$345$	0	0
$346$	−9.15649	−0.492256
$347$	−15.1883	−0.815350	−0.407675	−	0.913127i	$-0.633660\pi$
$347$	−15.1883	−0.815350	−0.407675	+	0.913127i	$0.633660\pi$
$348$	−0.363692	−0.0194960
$349$	−3.79541	−0.203164	−0.101582	−	0.994827i	$-0.532390\pi$
$349$	−3.79541	−0.203164	−0.101582	+	0.994827i	$0.532390\pi$
$350$	0	0
$351$	4.93395	0.263355
$352$	0	0
$353$	−0.519577	−0.0276543	−0.0138272	−	0.999904i	$-0.504401\pi$
$353$	−0.519577	−0.0276543	−0.0138272	+	0.999904i	$0.504401\pi$
$354$	16.2779	0.865159
$355$	0	0
$356$	1.64280	0.0870685
$357$	−12.3278	−0.652457
$358$	20.7544	1.09690
$359$	13.8374	0.730310	0.365155	−	0.930947i	$-0.381016\pi$
$359$	13.8374	0.730310	0.365155	+	0.930947i	$0.381016\pi$
$360$	0	0
$361$	−17.1153	−0.900807
$362$	−3.02576	−0.159030
$363$	0	0
$364$	3.90248	0.204545
$365$	0	0
$366$	−11.4970	−0.600960
$367$	20.6220	1.07646	0.538229	−	0.842799i	$-0.319093\pi$
$367$	20.6220	1.07646	0.538229	+	0.842799i	$0.319093\pi$
$368$	−6.84268	−0.356699
$369$	−9.43757	−0.491300
$370$	0	0
$371$	40.4670	2.10094
$372$	1.23798	0.0641862
$373$	8.17818	0.423450	0.211725	−	0.977329i	$-0.432092\pi$
$373$	8.17818	0.423450	0.211725	+	0.977329i	$0.432092\pi$
$374$	0	0
$375$	0	0
$376$	2.38197	0.122841
$377$	8.58351	0.442073
$378$	−5.06316	−0.260421
$379$	26.5982	1.36626	0.683128	−	0.730299i	$-0.260619\pi$
$379$	26.5982	1.36626	0.683128	+	0.730299i	$0.260619\pi$
$380$	0	0
$381$	−22.3123	−1.14309
$382$	−23.7799	−1.21669
$383$	14.5607	0.744018	0.372009	−	0.928229i	$-0.378669\pi$
$383$	14.5607	0.744018	0.372009	+	0.928229i	$0.378669\pi$
$384$	9.22384	0.470702
$385$	0	0
$386$	28.3121	1.44105
$387$	11.4086	0.579934
$388$	−2.84727	−0.144548
$389$	−3.94911	−0.200228	−0.100114	−	0.994976i	$-0.531921\pi$
$389$	−3.94911	−0.200228	−0.100114	+	0.994976i	$0.531921\pi$
$390$	0	0
$391$	6.30161	0.318686
$392$	−21.6224	−1.09209
$393$	0.562102	0.0283543
$394$	−32.2567	−1.62507
$395$	0	0
$396$	0	0
$397$	18.0162	0.904209	0.452105	−	0.891965i	$-0.350673\pi$
$397$	18.0162	0.904209	0.452105	+	0.891965i	$0.350673\pi$
$398$	5.75947	0.288696
$399$	5.19394	0.260022
$400$	0	0
$401$	−26.7798	−1.33732	−0.668659	−	0.743569i	$-0.733131\pi$
$401$	−26.7798	−1.33732	−0.668659	+	0.743569i	$0.733131\pi$
$402$	3.71267	0.185171
$403$	−29.2175	−1.45543
$404$	1.65958	0.0825673
$405$	0	0
$406$	−8.80828	−0.437148
$407$	0	0
$408$	−9.63282	−0.476896
$409$	−28.5313	−1.41078	−0.705390	−	0.708819i	$-0.749228\pi$
$409$	−28.5313	−1.41078	−0.705390	+	0.708819i	$0.749228\pi$
$410$	0	0
$411$	18.2719	0.901286
$412$	−2.55159	−0.125708
$413$	−46.0190	−2.26445
$414$	2.58814	0.127200
$415$	0	0
$416$	5.81012	0.284865
$417$	−14.2600	−0.698315
$418$	0	0
$419$	14.7459	0.720386	0.360193	−	0.932878i	$-0.382711\pi$
$419$	14.7459	0.720386	0.360193	+	0.932878i	$0.382711\pi$
$420$	0	0
$421$	19.3034	0.940788	0.470394	−	0.882456i	$-0.344112\pi$
$421$	19.3034	0.940788	0.470394	+	0.882456i	$0.344112\pi$
$422$	−20.3777	−0.991971
$423$	−0.805727	−0.0391758
$424$	31.6205	1.53563
$425$	0	0
$426$	−3.66641	−0.177638
$427$	32.5032	1.57294
$428$	−0.488226	−0.0235993
$429$	0	0
$430$	0	0
$431$	10.4970	0.505622	0.252811	−	0.967516i	$-0.418645\pi$
$431$	10.4970	0.505622	0.252811	+	0.967516i	$0.418645\pi$
$432$	−3.53818	−0.170231
$433$	−23.1357	−1.11183	−0.555915	−	0.831239i	$-0.687632\pi$
$433$	−23.1357	−1.11183	−0.555915	+	0.831239i	$0.687632\pi$
$434$	29.9826	1.43921
$435$	0	0
$436$	1.33675	0.0640187
$437$	−2.65499	−0.127005
$438$	−12.5491	−0.599619
$439$	25.2313	1.20422	0.602111	−	0.798412i	$-0.294326\pi$
$439$	25.2313	1.20422	0.602111	+	0.798412i	$0.294326\pi$
$440$	0	0
$441$	7.31401	0.348286
$442$	21.5150	1.02336
$443$	−6.68546	−0.317636	−0.158818	−	0.987308i	$-0.550768\pi$
$443$	−6.68546	−0.317636	−0.158818	+	0.987308i	$0.550768\pi$
$444$	0.982126	0.0466096
$445$	0	0
$446$	−38.3345	−1.81519
$447$	−10.6039	−0.501547
$448$	−32.7349	−1.54658
$449$	−13.1159	−0.618976	−0.309488	−	0.950903i	$-0.600158\pi$
$449$	−13.1159	−0.618976	−0.309488	+	0.950903i	$0.600158\pi$
$450$	0	0
$451$	0	0
$452$	3.79054	0.178292
$453$	−12.5734	−0.590748
$454$	4.04968	0.190061
$455$	0	0
$456$	4.05849	0.190056
$457$	35.1276	1.64320	0.821599	−	0.570066i	$-0.193082\pi$
$457$	35.1276	1.64320	0.821599	+	0.570066i	$0.193082\pi$
$458$	−4.08497	−0.190878
$459$	3.25841	0.152089
$460$	0	0
$461$	−30.4370	−1.41759	−0.708797	−	0.705413i	$-0.750762\pi$
$461$	−30.4370	−1.41759	−0.708797	+	0.705413i	$0.750762\pi$
$462$	0	0
$463$	10.0870	0.468783	0.234392	−	0.972142i	$-0.424690\pi$
$463$	10.0870	0.468783	0.234392	+	0.972142i	$0.424690\pi$
$464$	−6.15531	−0.285753
$465$	0	0
$466$	−8.44355	−0.391140
$467$	−17.0520	−0.789072	−0.394536	−	0.918881i	$-0.629095\pi$
$467$	−17.0520	−0.789072	−0.394536	+	0.918881i	$0.629095\pi$
$468$	−1.03148	−0.0476801
$469$	−10.4961	−0.484663
$470$	0	0
$471$	12.5332	0.577498
$472$	−35.9587	−1.65513
$473$	0	0
$474$	−17.0852	−0.784748
$475$	0	0
$476$	2.57722	0.118126
$477$	−10.6960	−0.489735
$478$	7.64293	0.349580
$479$	−8.92831	−0.407945	−0.203972	−	0.978977i	$-0.565385\pi$
$479$	−8.92831	−0.407945	−0.203972	+	0.978977i	$0.565385\pi$
$480$	0	0
$481$	−23.1792	−1.05688
$482$	−6.44996	−0.293788
$483$	−7.31690	−0.332930
$484$	0	0
$485$	0	0
$486$	1.33826	0.0607048
$487$	−33.4638	−1.51639	−0.758196	−	0.652027i	$-0.773919\pi$
$487$	−33.4638	−1.51639	−0.758196	+	0.652027i	$0.773919\pi$
$488$	25.3976	1.14970
$489$	3.10317	0.140330
$490$	0	0
$491$	−9.07217	−0.409421	−0.204711	−	0.978823i	$-0.565625\pi$
$491$	−9.07217	−0.409421	−0.204711	+	0.978823i	$0.565625\pi$
$492$	1.97299	0.0889492
$493$	5.66859	0.255301
$494$	−9.06468	−0.407839
$495$	0	0
$496$	20.9521	0.940779
$497$	10.3653	0.464946
$498$	−4.56754	−0.204677
$499$	39.0776	1.74935	0.874677	−	0.484706i	$-0.161073\pi$
$499$	39.0776	1.74935	0.874677	+	0.484706i	$0.161073\pi$
$500$	0	0
$501$	−8.34898	−0.373005
$502$	−1.18172	−0.0527425
$503$	22.9483	1.02322	0.511608	−	0.859219i	$-0.329050\pi$
$503$	22.9483	1.02322	0.511608	+	0.859219i	$0.329050\pi$
$504$	11.1848	0.498211
$505$	0	0
$506$	0	0
$507$	11.3439	0.503801
$508$	4.66454	0.206956
$509$	40.0637	1.77579	0.887896	−	0.460045i	$-0.152167\pi$
$509$	40.0637	1.77579	0.887896	+	0.460045i	$0.152167\pi$
$510$	0	0
$511$	35.4775	1.56943
$512$	−25.0863	−1.10867
$513$	−1.37283	−0.0606119
$514$	−11.5579	−0.509797
$515$	0	0
$516$	−2.38506	−0.104996
$517$	0	0
$518$	23.7861	1.04510
$519$	−6.84208	−0.300334
$520$	0	0
$521$	−30.3711	−1.33058	−0.665292	−	0.746583i	$-0.731693\pi$
$521$	−30.3711	−1.33058	−0.665292	+	0.746583i	$0.731693\pi$
$522$	2.32815	0.101900
$523$	19.0349	0.832337	0.416169	−	0.909287i	$-0.363373\pi$
$523$	19.0349	0.832337	0.416169	+	0.909287i	$0.363373\pi$
$524$	−0.117511	−0.00513350
$525$	0	0
$526$	−6.33793	−0.276347
$527$	−19.2954	−0.840521
$528$	0	0
$529$	−19.2598	−0.837383
$530$	0	0
$531$	12.1634	0.527848
$532$	−1.08583	−0.0470767
$533$	−46.5645	−2.01693
$534$	−10.5163	−0.455084
$535$	0	0
$536$	−8.20150	−0.354251
$537$	15.5085	0.669241
$538$	−19.0933	−0.823169
$539$	0	0
$540$	0	0
$541$	11.6232	0.499721	0.249861	−	0.968282i	$-0.419615\pi$
$541$	11.6232	0.499721	0.249861	+	0.968282i	$0.419615\pi$
$542$	−30.6953	−1.31848
$543$	−2.26096	−0.0970273
$544$	3.83704	0.164512
$545$	0	0
$546$	−24.9814	−1.06911
$547$	13.6517	0.583703	0.291852	−	0.956464i	$-0.405729\pi$
$547$	13.6517	0.583703	0.291852	+	0.956464i	$0.405729\pi$
$548$	−3.81986	−0.163177
$549$	−8.59102	−0.366656
$550$	0	0
$551$	−2.38829	−0.101744
$552$	−5.71734	−0.243346
$553$	48.3013	2.05398
$554$	−7.79273	−0.331081
$555$	0	0
$556$	2.98115	0.126429
$557$	5.62173	0.238200	0.119100	−	0.992882i	$-0.461999\pi$
$557$	5.62173	0.238200	0.119100	+	0.992882i	$0.461999\pi$
$558$	−7.92482	−0.335484
$559$	56.2897	2.38080
$560$	0	0
$561$	0	0
$562$	0.210898	0.00889619
$563$	−26.0494	−1.09785	−0.548926	−	0.835871i	$-0.684963\pi$
$563$	−26.0494	−1.09785	−0.548926	+	0.835871i	$0.684963\pi$
$564$	0.168443	0.00709272
$565$	0	0
$566$	−11.9716	−0.503204
$567$	−3.78339	−0.158887
$568$	8.09931	0.339839
$569$	28.7427	1.20496	0.602478	−	0.798135i	$-0.294180\pi$
$569$	28.7427	1.20496	0.602478	+	0.798135i	$0.294180\pi$
$570$	0	0
$571$	−17.8096	−0.745310	−0.372655	−	0.927970i	$-0.621552\pi$
$571$	−17.8096	−0.745310	−0.372655	+	0.927970i	$0.621552\pi$
$572$	0	0
$573$	−17.7692	−0.742321
$574$	47.7839	1.99446
$575$	0	0
$576$	8.65227	0.360511
$577$	16.3416	0.680308	0.340154	−	0.940370i	$-0.389521\pi$
$577$	16.3416	0.680308	0.340154	+	0.940370i	$0.389521\pi$
$578$	−8.54182	−0.355293
$579$	21.1559	0.879209
$580$	0	0
$581$	12.9129	0.535716
$582$	18.2266	0.755517
$583$	0	0
$584$	27.7217	1.14713
$585$	0	0
$586$	34.7785	1.43669
$587$	−12.0714	−0.498238	−0.249119	−	0.968473i	$-0.580141\pi$
$587$	−12.0714	−0.498238	−0.249119	+	0.968473i	$0.580141\pi$
$588$	−1.52904	−0.0630567
$589$	8.12952	0.334971
$590$	0	0
$591$	−24.1034	−0.991483
$592$	16.6220	0.683159
$593$	28.5085	1.17071	0.585353	−	0.810779i	$-0.300956\pi$
$593$	28.5085	1.17071	0.585353	+	0.810779i	$0.300956\pi$
$594$	0	0
$595$	0	0
$596$	2.21682	0.0908044
$597$	4.30369	0.176138
$598$	12.7697	0.522194
$599$	−1.66060	−0.0678503	−0.0339252	−	0.999424i	$-0.510801\pi$
$599$	−1.66060	−0.0678503	−0.0339252	+	0.999424i	$0.510801\pi$
$600$	0	0
$601$	11.5218	0.469982	0.234991	−	0.971997i	$-0.424494\pi$
$601$	11.5218	0.469982	0.234991	+	0.971997i	$0.424494\pi$
$602$	−57.7638	−2.35428
$603$	2.77425	0.112976
$604$	2.62855	0.106954
$605$	0	0
$606$	−10.6237	−0.431558
$607$	−8.37099	−0.339768	−0.169884	−	0.985464i	$-0.554339\pi$
$607$	−8.37099	−0.339768	−0.169884	+	0.985464i	$0.554339\pi$
$608$	−1.61662	−0.0655624
$609$	−6.58189	−0.266711
$610$	0	0
$611$	−3.97542	−0.160828
$612$	−0.681193	−0.0275356
$613$	−21.2085	−0.856604	−0.428302	−	0.903636i	$-0.640888\pi$
$613$	−21.2085	−0.856604	−0.428302	+	0.903636i	$0.640888\pi$
$614$	6.32140	0.255111
$615$	0	0
$616$	0	0
$617$	−19.6085	−0.789408	−0.394704	−	0.918808i	$-0.629153\pi$
$617$	−19.6085	−0.789408	−0.394704	+	0.918808i	$0.629153\pi$
$618$	16.3338	0.657042
$619$	26.7786	1.07632	0.538162	−	0.842841i	$-0.319119\pi$
$619$	26.7786	1.07632	0.538162	+	0.842841i	$0.319119\pi$
$620$	0	0
$621$	1.93395	0.0776069
$622$	−25.7638	−1.03303
$623$	29.7305	1.19113
$624$	−17.4572	−0.698848
$625$	0	0
$626$	−3.93912	−0.157439
$627$	0	0
$628$	−2.62015	−0.104555
$629$	−15.3076	−0.610355
$630$	0	0
$631$	38.1370	1.51821	0.759106	−	0.650967i	$-0.225637\pi$
$631$	38.1370	1.51821	0.759106	+	0.650967i	$0.225637\pi$
$632$	37.7421	1.50130
$633$	−15.2270	−0.605219
$634$	−2.44507	−0.0971060
$635$	0	0
$636$	2.23607	0.0886659
$637$	36.0870	1.42982
$638$	0	0
$639$	−2.73968	−0.108380
$640$	0	0
$641$	−30.6279	−1.20973	−0.604865	−	0.796328i	$-0.706773\pi$
$641$	−30.6279	−1.20973	−0.604865	+	0.796328i	$0.706773\pi$
$642$	3.12534	0.123347
$643$	3.35053	0.132132	0.0660660	−	0.997815i	$-0.478955\pi$
$643$	3.35053	0.132132	0.0660660	+	0.997815i	$0.478955\pi$
$644$	1.52965	0.0602766
$645$	0	0
$646$	−5.98636	−0.235530
$647$	24.5143	0.963755	0.481878	−	0.876239i	$-0.339955\pi$
$647$	24.5143	0.963755	0.481878	+	0.876239i	$0.339955\pi$
$648$	−2.95630	−0.116134
$649$	0	0
$650$	0	0
$651$	22.4042	0.878089
$652$	−0.648739	−0.0254066
$653$	46.1663	1.80663	0.903313	−	0.428982i	$-0.141128\pi$
$653$	46.1663	1.80663	0.903313	+	0.428982i	$0.141128\pi$
$654$	−8.55710	−0.334609
$655$	0	0
$656$	33.3918	1.30373
$657$	−9.37717	−0.365838
$658$	4.07952	0.159036
$659$	11.4010	0.444122	0.222061	−	0.975033i	$-0.428722\pi$
$659$	11.4010	0.444122	0.222061	+	0.975033i	$0.428722\pi$
$660$	0	0
$661$	34.7413	1.35128	0.675641	−	0.737231i	$-0.263867\pi$
$661$	34.7413	1.35128	0.675641	+	0.737231i	$0.263867\pi$
$662$	23.1541	0.899911
$663$	16.0768	0.624373
$664$	10.0900	0.391567
$665$	0	0
$666$	−6.28700	−0.243616
$667$	3.36446	0.130273
$668$	1.74541	0.0675320
$669$	−28.6450	−1.10748
$670$	0	0
$671$	0	0
$672$	−4.45524	−0.171865
$673$	−15.0160	−0.578823	−0.289411	−	0.957205i	$-0.593460\pi$
$673$	−15.0160	−0.578823	−0.289411	+	0.957205i	$0.593460\pi$
$674$	−14.4659	−0.557205
$675$	0	0
$676$	−2.37152	−0.0912124
$677$	24.6817	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$677$	24.6817	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$678$	−24.2649	−0.931886
$679$	−51.5283	−1.97747
$680$	0	0
$681$	3.02608	0.115960
$682$	0	0
$683$	48.9656	1.87361	0.936807	−	0.349846i	$-0.113766\pi$
$683$	48.9656	1.87361	0.936807	+	0.349846i	$0.113766\pi$
$684$	0.286999	0.0109737
$685$	0	0
$686$	−1.58989	−0.0607021
$687$	−3.05244	−0.116458
$688$	−40.3659	−1.53893
$689$	−52.7735	−2.01051
$690$	0	0
$691$	45.6987	1.73846	0.869230	−	0.494408i	$-0.164615\pi$
$691$	45.6987	1.73846	0.869230	+	0.494408i	$0.164615\pi$
$692$	1.43038	0.0543750
$693$	0	0
$694$	−20.3259	−0.771561
$695$	0	0
$696$	−5.14301	−0.194945
$697$	−30.7515	−1.16479
$698$	−5.07925	−0.192252
$699$	−6.30934	−0.238641
$700$	0	0
$701$	−6.87571	−0.259692	−0.129846	−	0.991534i	$-0.541448\pi$
$701$	−6.87571	−0.259692	−0.129846	+	0.991534i	$0.541448\pi$
$702$	6.60292	0.249211
$703$	6.44940	0.243244
$704$	0	0
$705$	0	0
$706$	−0.695330	−0.0261691
$707$	30.0341	1.12955
$708$	−2.54285	−0.0955662
$709$	13.7735	0.517275	0.258638	−	0.965974i	$-0.416726\pi$
$709$	13.7735	0.517275	0.258638	+	0.965974i	$0.416726\pi$
$710$	0	0
$711$	−12.7667	−0.478788
$712$	23.2311	0.870621
$713$	−11.4524	−0.428894
$714$	−16.4978	−0.617416
$715$	0	0
$716$	−3.24216	−0.121165
$717$	5.71109	0.213285
$718$	18.5181	0.691088
$719$	−2.09938	−0.0782935	−0.0391468	−	0.999233i	$-0.512464\pi$
$719$	−2.09938	−0.0782935	−0.0391468	+	0.999233i	$0.512464\pi$
$720$	0	0
$721$	−46.1771	−1.71973
$722$	−22.9048	−0.852428
$723$	−4.81966	−0.179245
$724$	0.472670	0.0175667
$725$	0	0
$726$	0	0
$727$	14.9925	0.556042	0.278021	−	0.960575i	$-0.410322\pi$
$727$	14.9925	0.556042	0.278021	+	0.960575i	$0.410322\pi$
$728$	55.1853	2.04530
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.1740	1.37493
$732$	1.79601	0.0663825
$733$	16.5017	0.609506	0.304753	−	0.952431i	$-0.401426\pi$
$733$	16.5017	0.609506	0.304753	+	0.952431i	$0.401426\pi$
$734$	27.5976	1.01865
$735$	0	0
$736$	2.27739	0.0839455
$737$	0	0
$738$	−12.6299	−0.464914
$739$	−7.28575	−0.268011	−0.134005	−	0.990981i	$-0.542784\pi$
$739$	−7.28575	−0.268011	−0.134005	+	0.990981i	$0.542784\pi$
$740$	0	0
$741$	−6.77348	−0.248830
$742$	54.1554	1.98811
$743$	−27.6106	−1.01293	−0.506467	−	0.862259i	$-0.669049\pi$
$743$	−27.6106	−1.01293	−0.506467	+	0.862259i	$0.669049\pi$
$744$	17.5064	0.641815
$745$	0	0
$746$	10.9445	0.400708
$747$	−3.41304	−0.124877
$748$	0	0
$749$	−8.83562	−0.322847
$750$	0	0
$751$	−2.24849	−0.0820486	−0.0410243	−	0.999158i	$-0.513062\pi$
$751$	−2.24849	−0.0820486	−0.0410243	+	0.999158i	$0.513062\pi$
$752$	2.85081	0.103958
$753$	−0.883023	−0.0321792
$754$	11.4870	0.418331
$755$	0	0
$756$	0.790943	0.0287663
$757$	−33.9846	−1.23519	−0.617596	−	0.786495i	$-0.711893\pi$
$757$	−33.9846	−1.23519	−0.617596	+	0.786495i	$0.711893\pi$
$758$	35.5953	1.29288
$759$	0	0
$760$	0	0
$761$	27.7004	1.00414	0.502069	−	0.864828i	$-0.332572\pi$
$761$	27.7004	1.00414	0.502069	+	0.864828i	$0.332572\pi$
$762$	−29.8597	−1.08170
$763$	24.1917	0.875798
$764$	3.71478	0.134396
$765$	0	0
$766$	19.4861	0.704060
$767$	60.0139	2.16698
$768$	−4.96064	−0.179002
$769$	−2.29270	−0.0826770	−0.0413385	−	0.999145i	$-0.513162\pi$
$769$	−2.29270	−0.0826770	−0.0413385	+	0.999145i	$0.513162\pi$
$770$	0	0
$771$	−8.63651	−0.311036
$772$	−4.42278	−0.159179
$773$	30.9947	1.11480	0.557400	−	0.830244i	$-0.311799\pi$
$773$	30.9947	1.11480	0.557400	+	0.830244i	$0.311799\pi$
$774$	15.2677	0.548788
$775$	0	0
$776$	−40.2636	−1.44538
$777$	17.7739	0.637636
$778$	−5.28494	−0.189474
$779$	12.9562	0.464203
$780$	0	0
$781$	0	0
$782$	8.43321	0.301571
$783$	1.73968	0.0621711
$784$	−25.8783	−0.924225
$785$	0	0
$786$	0.752239	0.0268315
$787$	−49.1385	−1.75160	−0.875798	−	0.482677i	$-0.839665\pi$
$787$	−49.1385	−1.75160	−0.875798	+	0.482677i	$0.839665\pi$
$788$	5.03899	0.179507
$789$	−4.73594	−0.168604
$790$	0	0
$791$	68.5990	2.43910
$792$	0	0
$793$	−42.3877	−1.50523
$794$	24.1104	0.855647
$795$	0	0
$796$	−0.899717	−0.0318896
$797$	−2.77965	−0.0984602	−0.0492301	−	0.998787i	$-0.515677\pi$
$797$	−2.77965	−0.0984602	−0.0492301	+	0.998787i	$0.515677\pi$
$798$	6.95085	0.246058
$799$	−2.62539	−0.0928795
$800$	0	0
$801$	−7.85817	−0.277655
$802$	−35.8383	−1.26550
$803$	0	0
$804$	−0.579976	−0.0204542
$805$	0	0
$806$	−39.1007	−1.37726
$807$	−14.2672	−0.502230
$808$	23.4683	0.825612
$809$	44.0336	1.54814	0.774069	−	0.633101i	$-0.218218\pi$
$809$	44.0336	1.54814	0.774069	+	0.633101i	$0.218218\pi$
$810$	0	0
$811$	17.9977	0.631985	0.315993	−	0.948762i	$-0.397662\pi$
$811$	17.9977	0.631985	0.315993	+	0.948762i	$0.397662\pi$
$812$	1.37599	0.0482877
$813$	−22.9367	−0.804426
$814$	0	0
$815$	0	0
$816$	−11.5288	−0.403590
$817$	−15.6621	−0.547948
$818$	−38.1823	−1.33501
$819$	−18.6671	−0.652280
$820$	0	0
$821$	13.6645	0.476894	0.238447	−	0.971155i	$-0.423362\pi$
$821$	13.6645	0.476894	0.238447	+	0.971155i	$0.423362\pi$
$822$	24.4526	0.852881
$823$	9.55474	0.333057	0.166529	−	0.986037i	$-0.446744\pi$
$823$	9.55474	0.333057	0.166529	+	0.986037i	$0.446744\pi$
$824$	−36.0823	−1.25699
$825$	0	0
$826$	−61.5854	−2.14283
$827$	2.86560	0.0996467	0.0498233	−	0.998758i	$-0.484134\pi$
$827$	2.86560	0.0996467	0.0498233	+	0.998758i	$0.484134\pi$
$828$	−0.404307	−0.0140506
$829$	−21.2099	−0.736649	−0.368325	−	0.929697i	$-0.620068\pi$
$829$	−21.2099	−0.736649	−0.368325	+	0.929697i	$0.620068\pi$
$830$	0	0
$831$	−5.82302	−0.201998
$832$	42.6899	1.48001
$833$	23.8320	0.825731
$834$	−19.0836	−0.660811
$835$	0	0
$836$	0	0
$837$	−5.92173	−0.204685
$838$	19.7339	0.681697
$839$	1.40998	0.0486779	0.0243390	−	0.999704i	$-0.492252\pi$
$839$	1.40998	0.0486779	0.0243390	+	0.999704i	$0.492252\pi$
$840$	0	0
$841$	−25.9735	−0.895638
$842$	25.8329	0.890262
$843$	0.157591	0.00542772
$844$	3.18331	0.109574
$845$	0	0
$846$	−1.07827	−0.0370718
$847$	0	0
$848$	37.8443	1.29958
$849$	−8.94565	−0.307014
$850$	0	0
$851$	−9.08550	−0.311447
$852$	0.572749	0.0196221
$853$	−13.4716	−0.461257	−0.230629	−	0.973042i	$-0.574078\pi$
$853$	−13.4716	−0.461257	−0.230629	+	0.973042i	$0.574078\pi$
$854$	43.4977	1.48846
$855$	0	0
$856$	−6.90406	−0.235976
$857$	25.5677	0.873376	0.436688	−	0.899613i	$-0.356151\pi$
$857$	25.5677	0.873376	0.436688	+	0.899613i	$0.356151\pi$
$858$	0	0
$859$	21.7766	0.743008	0.371504	−	0.928431i	$-0.378842\pi$
$859$	21.7766	0.743008	0.371504	+	0.928431i	$0.378842\pi$
$860$	0	0
$861$	35.7060	1.21686
$862$	14.0477	0.478467
$863$	51.9088	1.76700	0.883499	−	0.468434i	$-0.155181\pi$
$863$	51.9088	1.76700	0.883499	+	0.468434i	$0.155181\pi$
$864$	1.17758	0.0400621
$865$	0	0
$866$	−30.9616	−1.05212
$867$	−6.38277	−0.216770
$868$	−4.68375	−0.158977
$869$	0	0
$870$	0	0
$871$	13.6880	0.463801
$872$	18.9031	0.640141
$873$	13.6196	0.460954
$874$	−3.55307	−0.120184
$875$	0	0
$876$	1.96036	0.0662345
$877$	−10.9745	−0.370584	−0.185292	−	0.982684i	$-0.559323\pi$
$877$	−10.9745	−0.370584	−0.185292	+	0.982684i	$0.559323\pi$
$878$	33.7660	1.13955
$879$	25.9879	0.876548
$880$	0	0
$881$	−8.42384	−0.283806	−0.141903	−	0.989881i	$-0.545322\pi$
$881$	−8.42384	−0.283806	−0.141903	+	0.989881i	$0.545322\pi$
$882$	9.78806	0.329581
$883$	13.9067	0.467997	0.233998	−	0.972237i	$-0.424819\pi$
$883$	13.9067	0.467997	0.233998	+	0.972237i	$0.424819\pi$
$884$	−3.36097	−0.113042
$885$	0	0
$886$	−8.94690	−0.300577
$887$	−27.7955	−0.933283	−0.466641	−	0.884447i	$-0.654536\pi$
$887$	−27.7955	−0.933283	−0.466641	+	0.884447i	$0.654536\pi$
$888$	13.8883	0.466062
$889$	84.4161	2.83122
$890$	0	0
$891$	0	0
$892$	5.98843	0.200508
$893$	1.10613	0.0370151
$894$	−14.1908	−0.474611
$895$	0	0
$896$	−34.8974	−1.16584
$897$	9.54204	0.318600
$898$	−17.5525	−0.585733
$899$	−10.3019	−0.343588
$900$	0	0
$901$	−34.8519	−1.16108
$902$	0	0
$903$	−43.1633	−1.43639
$904$	53.6025	1.78279
$905$	0	0
$906$	−16.8264	−0.559021
$907$	−10.1506	−0.337046	−0.168523	−	0.985698i	$-0.553900\pi$
$907$	−10.1506	−0.337046	−0.168523	+	0.985698i	$0.553900\pi$
$908$	−0.632622	−0.0209943
$909$	−7.93842	−0.263301
$910$	0	0
$911$	45.0748	1.49339	0.746697	−	0.665164i	$-0.231638\pi$
$911$	45.0748	1.49339	0.746697	+	0.665164i	$0.231638\pi$
$912$	4.85732	0.160842
$913$	0	0
$914$	47.0099	1.55495
$915$	0	0
$916$	0.638134	0.0210845
$917$	−2.12665	−0.0702281
$918$	4.36060	0.143921
$919$	−5.79094	−0.191026	−0.0955128	−	0.995428i	$-0.530449\pi$
$919$	−5.79094	−0.191026	−0.0955128	+	0.995428i	$0.530449\pi$
$920$	0	0
$921$	4.72359	0.155648
$922$	−40.7327	−1.34146
$923$	−13.5175	−0.444933
$924$	0	0
$925$	0	0
$926$	13.4991	0.443606
$927$	12.2052	0.400873
$928$	2.04861	0.0672490
$929$	−42.8241	−1.40501	−0.702506	−	0.711678i	$-0.747936\pi$
$929$	−42.8241	−1.40501	−0.702506	+	0.711678i	$0.747936\pi$
$930$	0	0
$931$	−10.0409	−0.329077
$932$	1.31901	0.0432056
$933$	−19.2517	−0.630272
$934$	−22.8200	−0.746694
$935$	0	0
$936$	−14.5862	−0.476766
$937$	−11.8861	−0.388302	−0.194151	−	0.980972i	$-0.562195\pi$
$937$	−11.8861	−0.388302	−0.194151	+	0.980972i	$0.562195\pi$
$938$	−14.0465	−0.458633
$939$	−2.94346	−0.0960563
$940$	0	0
$941$	−40.3081	−1.31401	−0.657003	−	0.753888i	$-0.728176\pi$
$941$	−40.3081	−1.31401	−0.657003	+	0.753888i	$0.728176\pi$
$942$	16.7727	0.546483
$943$	−18.2518	−0.594361
$944$	−43.0365	−1.40072
$945$	0	0
$946$	0	0
$947$	−53.3167	−1.73256	−0.866280	−	0.499560i	$-0.833495\pi$
$947$	−53.3167	−1.73256	−0.866280	+	0.499560i	$0.833495\pi$
$948$	2.66897	0.0866840
$949$	−46.2665	−1.50188
$950$	0	0
$951$	−1.82705	−0.0592461
$952$	36.4447	1.18118
$953$	23.1339	0.749379	0.374690	−	0.927150i	$-0.377749\pi$
$953$	23.1339	0.749379	0.374690	+	0.927150i	$0.377749\pi$
$954$	−14.3140	−0.463433
$955$	0	0
$956$	−1.19394	−0.0386149
$957$	0	0
$958$	−11.9484	−0.386036
$959$	−69.1296	−2.23231
$960$	0	0
$961$	4.06685	0.131189
$962$	−31.0198	−1.00012
$963$	2.33537	0.0752564
$964$	1.00758	0.0324521
$965$	0	0
$966$	−9.79192	−0.315050
$967$	32.8910	1.05770	0.528851	−	0.848715i	$-0.322623\pi$
$967$	32.8910	1.05770	0.528851	+	0.848715i	$0.322623\pi$
$968$	0	0
$969$	−4.47324	−0.143701
$970$	0	0
$971$	22.5974	0.725185	0.362592	−	0.931948i	$-0.381892\pi$
$971$	22.5974	0.725185	0.362592	+	0.931948i	$0.381892\pi$
$972$	−0.209057	−0.00670550
$973$	53.9510	1.72959
$974$	−44.7834	−1.43495
$975$	0	0
$976$	30.3966	0.972971
$977$	−30.7844	−0.984882	−0.492441	−	0.870346i	$-0.663895\pi$
$977$	−30.7844	−0.984882	−0.492441	+	0.870346i	$0.663895\pi$
$978$	4.15285	0.132794
$979$	0	0
$980$	0	0
$981$	−6.39419	−0.204151
$982$	−12.1409	−0.387433
$983$	−34.8963	−1.11302	−0.556509	−	0.830841i	$-0.687860\pi$
$983$	−34.8963	−1.11302	−0.556509	+	0.830841i	$0.687860\pi$
$984$	27.9002	0.889427
$985$	0	0
$986$	7.58606	0.241589
$987$	3.04838	0.0970309
$988$	1.41604	0.0450503
$989$	22.0638	0.701588
$990$	0	0
$991$	−41.7439	−1.32604	−0.663019	−	0.748602i	$-0.730725\pi$
$991$	−41.7439	−1.32604	−0.663019	+	0.748602i	$0.730725\pi$
$992$	−6.97330	−0.221403
$993$	17.3017	0.549051
$994$	13.8714	0.439975
$995$	0	0
$996$	0.713521	0.0226088
$997$	−33.7340	−1.06836	−0.534182	−	0.845369i	$-0.679380\pi$
$997$	−33.7340	−1.06836	−0.534182	+	0.845369i	$0.679380\pi$
$998$	52.2961	1.65540
$999$	−4.69789	−0.148635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.df.1.3		4
5.4	even	2		1815.2.a.q.1.2		4
11.2	odd	10		825.2.n.j.301.2		8
11.6	odd	10		825.2.n.j.751.2		8
11.10	odd	2		9075.2.a.co.1.2		4
15.14	odd	2		5445.2.a.bq.1.3		4
55.2	even	20		825.2.bx.e.499.2		16
55.13	even	20		825.2.bx.e.499.3		16
55.17	even	20		825.2.bx.e.124.3		16
55.24	odd	10		165.2.m.c.136.1	yes	8
55.28	even	20		825.2.bx.e.124.2		16
55.39	odd	10		165.2.m.c.91.1	✓	8
55.54	odd	2		1815.2.a.u.1.3		4
165.134	even	10		495.2.n.c.136.2		8
165.149	even	10		495.2.n.c.91.2		8
165.164	even	2		5445.2.a.bj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.c.91.1	✓	8	55.39	odd	10
165.2.m.c.136.1	yes	8	55.24	odd	10
495.2.n.c.91.2		8	165.149	even	10
495.2.n.c.136.2		8	165.134	even	10
825.2.n.j.301.2		8	11.2	odd	10
825.2.n.j.751.2		8	11.6	odd	10
825.2.bx.e.124.2		16	55.28	even	20
825.2.bx.e.124.3		16	55.17	even	20
825.2.bx.e.499.2		16	55.2	even	20
825.2.bx.e.499.3		16	55.13	even	20
1815.2.a.q.1.2		4	5.4	even	2
1815.2.a.u.1.3		4	55.54	odd	2
5445.2.a.bj.1.2		4	165.164	even	2
5445.2.a.bq.1.3		4	15.14	odd	2
9075.2.a.co.1.2		4	11.10	odd	2
9075.2.a.df.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)

\(f(q)\)	\(=\)	\(q+1.33826 q^{2} +1.00000 q^{3} -0.209057 q^{4} +1.33826 q^{6} -3.78339 q^{7} -2.95630 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.33826 q^{2} +1.00000 q^{3} -0.209057 q^{4} +1.33826 q^{6} -3.78339 q^{7} -2.95630 q^{8} +1.00000 q^{9} -0.209057 q^{12} +4.93395 q^{13} -5.06316 q^{14} -3.53818 q^{16} +3.25841 q^{17} +1.33826 q^{18} -1.37283 q^{19} -3.78339 q^{21} +1.93395 q^{23} -2.95630 q^{24} +6.60292 q^{26} +1.00000 q^{27} +0.790943 q^{28} +1.73968 q^{29} -5.92173 q^{31} +1.17758 q^{32} +4.36060 q^{34} -0.209057 q^{36} -4.69789 q^{37} -1.83720 q^{38} +4.93395 q^{39} -9.43757 q^{41} -5.06316 q^{42} +11.4086 q^{43} +2.58814 q^{46} -0.805727 q^{47} -3.53818 q^{48} +7.31401 q^{49} +3.25841 q^{51} -1.03148 q^{52} -10.6960 q^{53} +1.33826 q^{54} +11.1848 q^{56} -1.37283 q^{57} +2.32815 q^{58} +12.1634 q^{59} -8.59102 q^{61} -7.92482 q^{62} -3.78339 q^{63} +8.65227 q^{64} +2.77425 q^{67} -0.681193 q^{68} +1.93395 q^{69} -2.73968 q^{71} -2.95630 q^{72} -9.37717 q^{73} -6.28700 q^{74} +0.286999 q^{76} +6.60292 q^{78} -12.7667 q^{79} +1.00000 q^{81} -12.6299 q^{82} -3.41304 q^{83} +0.790943 q^{84} +15.2677 q^{86} +1.73968 q^{87} -7.85817 q^{89} -18.6671 q^{91} -0.404307 q^{92} -5.92173 q^{93} -1.07827 q^{94} +1.17758 q^{96} +13.6196 q^{97} +9.78806 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + q^2 + 4 * q^3 + q^4 + q^6 - 3 * q^8 + 4 * q^9	\( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + q^{12} + 7 q^{13} - 5 q^{14} - 9 q^{16} + 8 q^{17} + q^{18} - 11 q^{19} - 5 q^{23} - 3 q^{24} - 12 q^{26} + 4 q^{27} + 5 q^{28} - 17 q^{29} - 5 q^{31} + 17 q^{34} + q^{36} - 15 q^{37} + q^{38} + 7 q^{39} - 10 q^{41} - 5 q^{42} - 4 q^{43} - 15 q^{46} + 8 q^{47} - 9 q^{48} - 8 q^{49} + 8 q^{51} - 7 q^{52} - 10 q^{53} + q^{54} + 10 q^{56} - 11 q^{57} + 7 q^{58} + 9 q^{59} - 37 q^{61} - 20 q^{62} - 7 q^{64} - 3 q^{67} + 17 q^{68} - 5 q^{69} + 13 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} - 29 q^{76} - 12 q^{78} - 20 q^{79} + 4 q^{81} - 5 q^{82} - 17 q^{83} + 5 q^{84} + 34 q^{86} - 17 q^{87} - 24 q^{89} - 20 q^{91} - 10 q^{92} - 5 q^{93} - 23 q^{94} + 32 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + q^2 + 4 * q^3 + q^4 + q^6 - 3 * q^8 + 4 * q^9 + q^12 + 7 * q^13 - 5 * q^14 - 9 * q^16 + 8 * q^17 + q^18 - 11 * q^19 - 5 * q^23 - 3 * q^24 - 12 * q^26 + 4 * q^27 + 5 * q^28 - 17 * q^29 - 5 * q^31 + 17 * q^34 + q^36 - 15 * q^37 + q^38 + 7 * q^39 - 10 * q^41 - 5 * q^42 - 4 * q^43 - 15 * q^46 + 8 * q^47 - 9 * q^48 - 8 * q^49 + 8 * q^51 - 7 * q^52 - 10 * q^53 + q^54 + 10 * q^56 - 11 * q^57 + 7 * q^58 + 9 * q^59 - 37 * q^61 - 20 * q^62 - 7 * q^64 - 3 * q^67 + 17 * q^68 - 5 * q^69 + 13 * q^71 - 3 * q^72 + 15 * q^73 + 5 * q^74 - 29 * q^76 - 12 * q^78 - 20 * q^79 + 4 * q^81 - 5 * q^82 - 17 * q^83 + 5 * q^84 + 34 * q^86 - 17 * q^87 - 24 * q^89 - 20 * q^91 - 10 * q^92 - 5 * q^93 - 23 * q^94 + 32 * q^97 + 13 * q^98

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