LMFDB - Embedded newform 9075.2.a.df.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:	$1$
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			1.1
Root			$1.82709$ 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)

$f(q)$	$=$	$q-1.95630 q^{2} +1.00000 q^{3} +1.82709 q^{4} -1.95630 q^{6} +1.54732 q^{7} +0.338261 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.95630 q^{2} +1.00000 q^{3} +1.82709 q^{4} -1.95630 q^{6} +1.54732 q^{7} +0.338261 q^{8} +1.00000 q^{9} +1.82709 q^{12} +4.15622 q^{13} -3.02701 q^{14} -4.31592 q^{16} +0.741591 q^{17} -1.95630 q^{18} -7.48127 q^{19} +1.54732 q^{21} +1.15622 q^{23} +0.338261 q^{24} -8.13078 q^{26} +1.00000 q^{27} +2.82709 q^{28} -6.88558 q^{29} +6.77583 q^{31} +7.76669 q^{32} -1.45077 q^{34} +1.82709 q^{36} -3.92015 q^{37} +14.6356 q^{38} +4.15622 q^{39} -0.0345680 q^{41} -3.02701 q^{42} -11.1726 q^{43} -2.26190 q^{46} +7.04179 q^{47} -4.31592 q^{48} -4.60581 q^{49} +0.741591 q^{51} +7.59378 q^{52} +1.22384 q^{53} -1.95630 q^{54} +0.523398 q^{56} -7.48127 q^{57} +13.4702 q^{58} -4.30934 q^{59} -6.55488 q^{61} -13.2555 q^{62} +1.54732 q^{63} -6.56210 q^{64} +3.55199 q^{67} +1.35495 q^{68} +1.15622 q^{69} +5.88558 q^{71} +0.338261 q^{72} +4.57880 q^{73} +7.66897 q^{74} -13.6690 q^{76} -8.13078 q^{78} -6.17758 q^{79} +1.00000 q^{81} +0.0676252 q^{82} -15.1493 q^{83} +2.82709 q^{84} +21.8569 q^{86} -6.88558 q^{87} -17.5582 q^{89} +6.43099 q^{91} +2.11251 q^{92} +6.77583 q^{93} -13.7758 q^{94} +7.76669 q^{96} +0.144318 q^{97} +9.01032 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.95630	−1.38331	−0.691655	−	0.722228i	$-0.743118\pi$
$2$	−1.95630	−1.38331	−0.691655	+	0.722228i	$0.743118\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.82709	0.913545
$5$	0	0
$5$	0	0
$6$	−1.95630	−0.798654
$7$	1.54732	0.584831	0.292416	−	0.956291i	$-0.405541\pi$
$7$	1.54732	0.584831	0.292416	+	0.956291i	$0.405541\pi$
$8$	0.338261	0.119593
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.82709	0.527436
$13$	4.15622	1.15273	0.576363	−	0.817194i	$-0.304471\pi$
$13$	4.15622	1.15273	0.576363	+	0.817194i	$0.304471\pi$
$14$	−3.02701	−0.809003
$15$	0	0
$16$	−4.31592	−1.07898
$17$	0.741591	0.179862	0.0899312	−	0.995948i	$-0.471335\pi$
$17$	0.741591	0.179862	0.0899312	+	0.995948i	$0.471335\pi$
$18$	−1.95630	−0.461103
$19$	−7.48127	−1.71632	−0.858161	−	0.513381i	$-0.828393\pi$
$19$	−7.48127	−1.71632	−0.858161	+	0.513381i	$0.828393\pi$
$20$	0	0
$21$	1.54732	0.337652
$22$	0	0
$23$	1.15622	0.241088	0.120544	−	0.992708i	$-0.461536\pi$
$23$	1.15622	0.241088	0.120544	+	0.992708i	$0.461536\pi$
$24$	0.338261	0.0690473
$25$	0	0
$26$	−8.13078	−1.59458
$27$	1.00000	0.192450
$28$	2.82709	0.534270
$29$	−6.88558	−1.27862	−0.639310	−	0.768949i	$-0.720780\pi$
$29$	−6.88558	−1.27862	−0.639310	+	0.768949i	$0.720780\pi$
$30$	0	0
$31$	6.77583	1.21697	0.608487	−	0.793564i	$-0.291777\pi$
$31$	6.77583	1.21697	0.608487	+	0.793564i	$0.291777\pi$
$32$	7.76669	1.37297
$33$	0	0
$34$	−1.45077	−0.248805
$35$	0	0
$36$	1.82709	0.304515
$37$	−3.92015	−0.644468	−0.322234	−	0.946660i	$-0.604434\pi$
$37$	−3.92015	−0.644468	−0.322234	+	0.946660i	$0.604434\pi$
$38$	14.6356	2.37420
$39$	4.15622	0.665527
$40$	0	0
$41$	−0.0345680	−0.00539862	−0.00269931	−	0.999996i	$-0.500859\pi$
$41$	−0.0345680	−0.00539862	−0.00269931	+	0.999996i	$0.500859\pi$
$42$	−3.02701	−0.467078
$43$	−11.1726	−1.70380	−0.851901	−	0.523703i	$-0.824550\pi$
$43$	−11.1726	−1.70380	−0.851901	+	0.523703i	$0.824550\pi$
$44$	0	0
$45$	0	0
$46$	−2.26190	−0.333499
$47$	7.04179	1.02715	0.513576	−	0.858044i	$-0.328321\pi$
$47$	7.04179	1.02715	0.513576	+	0.858044i	$0.328321\pi$
$48$	−4.31592	−0.622949
$49$	−4.60581	−0.657972
$50$	0	0
$51$	0.741591	0.103844
$52$	7.59378	1.05307
$53$	1.22384	0.168107	0.0840537	−	0.996461i	$-0.473213\pi$
$53$	1.22384	0.168107	0.0840537	+	0.996461i	$0.473213\pi$
$54$	−1.95630	−0.266218
$55$	0	0
$56$	0.523398	0.0699420
$57$	−7.48127	−0.990919
$58$	13.4702	1.76873
$59$	−4.30934	−0.561028	−0.280514	−	0.959850i	$-0.590505\pi$
$59$	−4.30934	−0.561028	−0.280514	+	0.959850i	$0.590505\pi$
$60$	0	0
$61$	−6.55488	−0.839266	−0.419633	−	0.907694i	$-0.637841\pi$
$61$	−6.55488	−0.839266	−0.419633	+	0.907694i	$0.637841\pi$
$62$	−13.2555	−1.68345
$63$	1.54732	0.194944
$64$	−6.56210	−0.820263
$65$	0	0
$66$	0	0
$67$	3.55199	0.433944	0.216972	−	0.976178i	$-0.430382\pi$
$67$	3.55199	0.433944	0.216972	+	0.976178i	$0.430382\pi$
$68$	1.35495	0.164312
$69$	1.15622	0.139192
$70$	0	0
$71$	5.88558	0.698490	0.349245	−	0.937031i	$-0.386438\pi$
$71$	5.88558	0.698490	0.349245	+	0.937031i	$0.386438\pi$
$72$	0.338261	0.0398645
$73$	4.57880	0.535907	0.267954	−	0.963432i	$-0.413653\pi$
$73$	4.57880	0.535907	0.267954	+	0.963432i	$0.413653\pi$
$74$	7.66897	0.891499
$75$	0	0
$76$	−13.6690	−1.56794
$77$	0	0
$78$	−8.13078	−0.920630
$79$	−6.17758	−0.695032	−0.347516	−	0.937674i	$-0.612975\pi$
$79$	−6.17758	−0.695032	−0.347516	+	0.937674i	$0.612975\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.0676252	0.00746796
$83$	−15.1493	−1.66285	−0.831424	−	0.555639i	$-0.812474\pi$
$83$	−15.1493	−1.66285	−0.831424	+	0.555639i	$0.812474\pi$
$84$	2.82709	0.308461
$85$	0	0
$86$	21.8569	2.35689
$87$	−6.88558	−0.738212
$88$	0	0
$89$	−17.5582	−1.86117	−0.930585	−	0.366076i	$-0.880701\pi$
$89$	−17.5582	−1.86117	−0.930585	+	0.366076i	$0.880701\pi$
$90$	0	0
$91$	6.43099	0.674151
$92$	2.11251	0.220244
$93$	6.77583	0.702621
$94$	−13.7758	−1.42087
$95$	0	0
$96$	7.76669	0.792685
$97$	0.144318	0.0146533	0.00732665	−	0.999973i	$-0.497668\pi$
$97$	0.144318	0.0146533	0.00732665	+	0.999973i	$0.497668\pi$
$98$	9.01032	0.910179
$99$	0	0
$100$	0	0
$101$	3.50073	0.348335	0.174168	−	0.984716i	$-0.444277\pi$
$101$	3.50073	0.348335	0.174168	+	0.984716i	$0.444277\pi$
$102$	−1.45077	−0.143648
$103$	−12.1151	−1.19373	−0.596866	−	0.802341i	$-0.703588\pi$
$103$	−12.1151	−1.19373	−0.596866	+	0.802341i	$0.703588\pi$
$104$	1.40589	0.137859
$105$	0	0
$106$	−2.39419	−0.232545
$107$	−3.77307	−0.364756	−0.182378	−	0.983228i	$-0.558380\pi$
$107$	−3.77307	−0.364756	−0.182378	+	0.983228i	$0.558380\pi$
$108$	1.82709	0.175812
$109$	−18.3140	−1.75416	−0.877082	−	0.480341i	$-0.840513\pi$
$109$	−18.3140	−1.75416	−0.877082	+	0.480341i	$0.840513\pi$
$110$	0	0
$111$	−3.92015	−0.372084
$112$	−6.67810	−0.631021
$113$	14.3333	1.34836	0.674180	−	0.738567i	$-0.264497\pi$
$113$	14.3333	1.34836	0.674180	+	0.738567i	$0.264497\pi$
$114$	14.6356	1.37075
$115$	0	0
$116$	−12.5806	−1.16808
$117$	4.15622	0.384242
$118$	8.43034	0.776076
$119$	1.14748	0.105189
$120$	0	0
$121$	0	0
$122$	12.8233	1.16096
$123$	−0.0345680	−0.00311689
$124$	12.3801	1.11176
$125$	0	0
$126$	−3.02701	−0.269668
$127$	−14.4648	−1.28354	−0.641771	−	0.766896i	$-0.721800\pi$
$127$	−14.4648	−1.28354	−0.641771	+	0.766896i	$0.721800\pi$
$128$	−2.69598	−0.238293
$129$	−11.1726	−0.983691
$130$	0	0
$131$	−14.6523	−1.28017	−0.640087	−	0.768302i	$-0.721102\pi$
$131$	−14.6523	−1.28017	−0.640087	+	0.768302i	$0.721102\pi$
$132$	0	0
$133$	−11.5759	−1.00376
$134$	−6.94874	−0.600280
$135$	0	0
$136$	0.250852	0.0215103
$137$	−10.4178	−0.890051	−0.445026	−	0.895518i	$-0.646805\pi$
$137$	−10.4178	−0.890051	−0.445026	+	0.895518i	$0.646805\pi$
$138$	−2.26190	−0.192546
$139$	1.73212	0.146917	0.0734584	−	0.997298i	$-0.476596\pi$
$139$	1.73212	0.146917	0.0734584	+	0.997298i	$0.476596\pi$
$140$	0	0
$141$	7.04179	0.593026
$142$	−11.5139	−0.966228
$143$	0	0
$144$	−4.31592	−0.359660
$145$	0	0
$146$	−8.95748	−0.741326
$147$	−4.60581	−0.379881
$148$	−7.16247	−0.588751
$149$	12.4580	1.02060	0.510299	−	0.859997i	$-0.329535\pi$
$149$	12.4580	1.02060	0.510299	+	0.859997i	$0.329535\pi$
$150$	0	0
$151$	−17.6070	−1.43284	−0.716418	−	0.697671i	$-0.754220\pi$
$151$	−17.6070	−1.43284	−0.716418	+	0.697671i	$0.754220\pi$
$152$	−2.53062	−0.205261
$153$	0.741591	0.0599541
$154$	0	0
$155$	0	0
$156$	7.59378	0.607989
$157$	−15.8594	−1.26572	−0.632860	−	0.774266i	$-0.718119\pi$
$157$	−15.8594	−1.26572	−0.632860	+	0.774266i	$0.718119\pi$
$158$	12.0852	0.961444
$159$	1.22384	0.0970569
$160$	0	0
$161$	1.78903	0.140996
$162$	−1.95630	−0.153701
$163$	−8.51958	−0.667305	−0.333652	−	0.942696i	$-0.608281\pi$
$163$	−8.51958	−0.667305	−0.333652	+	0.942696i	$0.608281\pi$
$164$	−0.0631589	−0.00493188
$165$	0	0
$166$	29.6364	2.30023
$167$	14.5293	1.12431	0.562156	−	0.827031i	$-0.309972\pi$
$167$	14.5293	1.12431	0.562156	+	0.827031i	$0.309972\pi$
$168$	0.523398	0.0403810
$169$	4.27413	0.328779
$170$	0	0
$171$	−7.48127	−0.572107
$172$	−20.4133	−1.55650
$173$	−10.1366	−0.770674	−0.385337	−	0.922776i	$-0.625915\pi$
$173$	−10.1366	−0.770674	−0.385337	+	0.922776i	$0.625915\pi$
$174$	13.4702	1.02118
$175$	0	0
$176$	0	0
$177$	−4.30934	−0.323910
$178$	34.3491	2.57457
$179$	26.6506	1.99196	0.995978	−	0.0895936i	$-0.0285568\pi$
$179$	26.6506	1.99196	0.995978	+	0.0895936i	$0.0285568\pi$
$180$	0	0
$181$	22.0593	1.63966	0.819829	−	0.572609i	$-0.194069\pi$
$181$	22.0593	1.63966	0.819829	+	0.572609i	$0.194069\pi$
$182$	−12.5809	−0.932559
$183$	−6.55488	−0.484550
$184$	0.391103	0.0288325
$185$	0	0
$186$	−13.2555	−0.971942
$187$	0	0
$188$	12.8660	0.938349
$189$	1.54732	0.112551
$190$	0	0
$191$	10.6234	0.768679	0.384339	−	0.923192i	$-0.374429\pi$
$191$	10.6234	0.768679	0.384339	+	0.923192i	$0.374429\pi$
$192$	−6.56210	−0.473579
$193$	−2.68375	−0.193180	−0.0965902	−	0.995324i	$-0.530794\pi$
$193$	−2.68375	−0.193180	−0.0965902	+	0.995324i	$0.530794\pi$
$194$	−0.282329	−0.0202700
$195$	0	0
$196$	−8.41523	−0.601088
$197$	17.2838	1.23142	0.615709	−	0.787973i	$-0.288870\pi$
$197$	17.2838	1.23142	0.615709	+	0.787973i	$0.288870\pi$
$198$	0	0
$199$	−8.39386	−0.595025	−0.297513	−	0.954718i	$-0.596157\pi$
$199$	−8.39386	−0.595025	−0.297513	+	0.954718i	$0.596157\pi$
$200$	0	0
$201$	3.55199	0.250538
$202$	−6.84846	−0.481856
$203$	−10.6542	−0.747777
$204$	1.35495	0.0948658
$205$	0	0
$206$	23.7006	1.65130
$207$	1.15622	0.0803625
$208$	−17.9379	−1.24377
$209$	0	0
$210$	0	0
$211$	23.6434	1.62768	0.813840	−	0.581089i	$-0.197373\pi$
$211$	23.6434	1.62768	0.813840	+	0.581089i	$0.197373\pi$
$212$	2.23607	0.153574
$213$	5.88558	0.403273
$214$	7.38124	0.504571
$215$	0	0
$216$	0.338261	0.0230158
$217$	10.4844	0.711725
$218$	35.8276	2.42655
$219$	4.57880	0.309406
$220$	0	0
$221$	3.08221	0.207332
$222$	7.66897	0.514707
$223$	23.7007	1.58712	0.793559	−	0.608494i	$-0.208226\pi$
$223$	23.7007	1.58712	0.793559	+	0.608494i	$0.208226\pi$
$224$	12.0175	0.802956
$225$	0	0
$226$	−28.0401	−1.86520
$227$	−14.4080	−0.956295	−0.478148	−	0.878279i	$-0.658692\pi$
$227$	−14.4080	−0.956295	−0.478148	+	0.878279i	$0.658692\pi$
$228$	−13.6690	−0.905249
$229$	−20.6000	−1.36129	−0.680644	−	0.732614i	$-0.738300\pi$
$229$	−20.6000	−1.36129	−0.680644	+	0.732614i	$0.738300\pi$
$230$	0	0
$231$	0	0
$232$	−2.32912	−0.152915
$233$	10.1634	0.665829	0.332915	−	0.942957i	$-0.391968\pi$
$233$	10.1634	0.665829	0.332915	+	0.942957i	$0.391968\pi$
$234$	−8.13078	−0.531526
$235$	0	0
$236$	−7.87356	−0.512525
$237$	−6.17758	−0.401277
$238$	−2.24481	−0.145509
$239$	8.52498	0.551435	0.275717	−	0.961239i	$-0.411085\pi$
$239$	8.52498	0.551435	0.275717	+	0.961239i	$0.411085\pi$
$240$	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
$244$	−11.9764	−0.766707
$245$	0	0
$246$	0.0676252	0.00431163
$247$	−31.0938	−1.97845
$248$	2.29200	0.145542
$249$	−15.1493	−0.960045
$250$	0	0
$251$	−14.0612	−0.887538	−0.443769	−	0.896141i	$-0.646359\pi$
$251$	−14.0612	−0.887538	−0.443769	+	0.896141i	$0.646359\pi$
$252$	2.82709	0.178090
$253$	0	0
$254$	28.2974	1.77554
$255$	0	0
$256$	18.3983	1.14990
$257$	−11.6340	−0.725709	−0.362854	−	0.931846i	$-0.618198\pi$
$257$	−11.6340	−0.725709	−0.362854	+	0.931846i	$0.618198\pi$
$258$	21.8569	1.36075
$259$	−6.06572	−0.376905
$260$	0	0
$261$	−6.88558	−0.426207
$262$	28.6642	1.77088
$263$	−11.0280	−0.680015	−0.340007	−	0.940423i	$-0.610430\pi$
$263$	−11.0280	−0.680015	−0.340007	+	0.940423i	$0.610430\pi$
$264$	0	0
$265$	0	0
$266$	22.6459	1.38851
$267$	−17.5582	−1.07455
$268$	6.48981	0.396428
$269$	18.9754	1.15695	0.578476	−	0.815700i	$-0.303648\pi$
$269$	18.9754	1.15695	0.578476	+	0.815700i	$0.303648\pi$
$270$	0	0
$271$	−15.5699	−0.945802	−0.472901	−	0.881116i	$-0.656793\pi$
$271$	−15.5699	−0.945802	−0.472901	+	0.881116i	$0.656793\pi$
$272$	−3.20065	−0.194068
$273$	6.43099	0.389221
$274$	20.3803	1.23122
$275$	0	0
$276$	2.11251	0.127158
$277$	5.91319	0.355289	0.177645	−	0.984095i	$-0.443152\pi$
$277$	5.91319	0.355289	0.177645	+	0.984095i	$0.443152\pi$
$278$	−3.38855	−0.203232
$279$	6.77583	0.405658
$280$	0	0
$281$	−27.7543	−1.65569	−0.827843	−	0.560961i	$-0.810432\pi$
$281$	−27.7543	−1.65569	−0.827843	+	0.560961i	$0.810432\pi$
$282$	−13.7758	−0.820339
$283$	19.7440	1.17366	0.586830	−	0.809710i	$-0.300376\pi$
$283$	19.7440	1.17366	0.586830	+	0.809710i	$0.300376\pi$
$284$	10.7535	0.638102
$285$	0	0
$286$	0	0
$287$	−0.0534877	−0.00315728
$288$	7.76669	0.457657
$289$	−16.4500	−0.967650
$290$	0	0
$291$	0.144318	0.00846008
$292$	8.36588	0.489576
$293$	19.8794	1.16137	0.580684	−	0.814129i	$-0.302785\pi$
$293$	19.8794	1.16137	0.580684	+	0.814129i	$0.302785\pi$
$294$	9.01032	0.525492
$295$	0	0
$296$	−1.32603	−0.0770741
$297$	0	0
$298$	−24.3715	−1.41180
$299$	4.80548	0.277908
$300$	0	0
$301$	−17.2875	−0.996437
$302$	34.4445	1.98206
$303$	3.50073	0.201112
$304$	32.2886	1.85188
$305$	0	0
$306$	−1.45077	−0.0829351
$307$	−10.3072	−0.588262	−0.294131	−	0.955765i	$-0.595030\pi$
$307$	−10.3072	−0.588262	−0.294131	+	0.955765i	$0.595030\pi$
$308$	0	0
$309$	−12.1151	−0.689202
$310$	0	0
$311$	8.36316	0.474231	0.237116	−	0.971481i	$-0.423798\pi$
$311$	8.36316	0.474231	0.237116	+	0.971481i	$0.423798\pi$
$312$	1.40589	0.0795926
$313$	7.12380	0.402661	0.201331	−	0.979523i	$-0.435473\pi$
$313$	7.12380	0.402661	0.201331	+	0.979523i	$0.435473\pi$
$314$	31.0257	1.75088
$315$	0	0
$316$	−11.2870	−0.634943
$317$	−27.5894	−1.54957	−0.774786	−	0.632223i	$-0.782143\pi$
$317$	−27.5894	−1.54957	−0.774786	+	0.632223i	$0.782143\pi$
$318$	−2.39419	−0.134260
$319$	0	0
$320$	0	0
$321$	−3.77307	−0.210592
$322$	−3.49988	−0.195040
$323$	−5.54805	−0.308702
$324$	1.82709	0.101505
$325$	0	0
$326$	16.6668	0.923089
$327$	−18.3140	−1.01277
$328$	−0.0116930	−0.000645639 0
$329$	10.8959	0.600710
$330$	0	0
$331$	−15.9410	−0.876195	−0.438098	−	0.898927i	$-0.644348\pi$
$331$	−15.9410	−0.876195	−0.438098	+	0.898927i	$0.644348\pi$
$332$	−27.6791	−1.51909
$333$	−3.92015	−0.214823
$334$	−28.4236	−1.55527
$335$	0	0
$336$	−6.67810	−0.364320
$337$	11.9554	0.651250	0.325625	−	0.945499i	$-0.394425\pi$
$337$	11.9554	0.651250	0.325625	+	0.945499i	$0.394425\pi$
$338$	−8.36145	−0.454803
$339$	14.3333	0.778476
$340$	0	0
$341$	0	0
$342$	14.6356	0.791401
$343$	−17.9579	−0.969634
$344$	−3.77925	−0.203764
$345$	0	0
$346$	19.8303	1.06608
$347$	−2.19367	−0.117762	−0.0588812	−	0.998265i	$-0.518753\pi$
$347$	−2.19367	−0.117762	−0.0588812	+	0.998265i	$0.518753\pi$
$348$	−12.5806	−0.674390
$349$	4.82985	0.258536	0.129268	−	0.991610i	$-0.458737\pi$
$349$	4.82985	0.258536	0.129268	+	0.991610i	$0.458737\pi$
$350$	0	0
$351$	4.15622	0.221842
$352$	0	0
$353$	11.1032	0.590962	0.295481	−	0.955349i	$-0.404520\pi$
$353$	11.1032	0.590962	0.295481	+	0.955349i	$0.404520\pi$
$354$	8.43034	0.448068
$355$	0	0
$356$	−32.0805	−1.70026
$357$	1.14748	0.0607310
$358$	−52.1364	−2.75549
$359$	12.5790	0.663894	0.331947	−	0.943298i	$-0.392294\pi$
$359$	12.5790	0.663894	0.331947	+	0.943298i	$0.392294\pi$
$360$	0	0
$361$	36.9694	1.94576
$362$	−43.1546	−2.26815
$363$	0	0
$364$	11.7500	0.615867
$365$	0	0
$366$	12.8233	0.670283
$367$	20.9190	1.09197	0.545983	−	0.837796i	$-0.316156\pi$
$367$	20.9190	1.09197	0.545983	+	0.837796i	$0.316156\pi$
$368$	−4.99013	−0.260129
$369$	−0.0345680	−0.00179954
$370$	0	0
$371$	1.89367	0.0983145
$372$	12.3801	0.641876
$373$	9.62019	0.498115	0.249057	−	0.968489i	$-0.419879\pi$
$373$	9.62019	0.498115	0.249057	+	0.968489i	$0.419879\pi$
$374$	0	0
$375$	0	0
$376$	2.38197	0.122841
$377$	−28.6180	−1.47390
$378$	−3.02701	−0.155693
$379$	−29.8900	−1.53534	−0.767672	−	0.640843i	$-0.778585\pi$
$379$	−29.8900	−1.53534	−0.767672	+	0.640843i	$0.778585\pi$
$380$	0	0
$381$	−14.4648	−0.741053
$382$	−20.7824	−1.06332
$383$	28.0360	1.43257	0.716287	−	0.697806i	$-0.245840\pi$
$383$	28.0360	1.43257	0.716287	+	0.697806i	$0.245840\pi$
$384$	−2.69598	−0.137578
$385$	0	0
$386$	5.25021	0.267228
$387$	−11.1726	−0.567934
$388$	0.263682	0.0133864
$389$	9.89338	0.501614	0.250807	−	0.968037i	$-0.419304\pi$
$389$	9.89338	0.501614	0.250807	+	0.968037i	$0.419304\pi$
$390$	0	0
$391$	0.857439	0.0433626
$392$	−1.55797	−0.0786892
$393$	−14.6523	−0.739109
$394$	−33.8122	−1.70343
$395$	0	0
$396$	0	0
$397$	−14.7457	−0.740067	−0.370033	−	0.929018i	$-0.620654\pi$
$397$	−14.7457	−0.740067	−0.370033	+	0.929018i	$0.620654\pi$
$398$	16.4209	0.823104
$399$	−11.5759	−0.579520
$400$	0	0
$401$	−14.3793	−0.718067	−0.359034	−	0.933325i	$-0.616894\pi$
$401$	−14.3793	−0.718067	−0.359034	+	0.933325i	$0.616894\pi$
$402$	−6.94874	−0.346572
$403$	28.1618	1.40284
$404$	6.39615	0.318220
$405$	0	0
$406$	20.8427	1.03441
$407$	0	0
$408$	0.250852	0.0124190
$409$	11.0034	0.544083	0.272042	−	0.962285i	$-0.412301\pi$
$409$	11.0034	0.544083	0.272042	+	0.962285i	$0.412301\pi$
$410$	0	0
$411$	−10.4178	−0.513871
$412$	−22.1353	−1.09053
$413$	−6.66792	−0.328107
$414$	−2.26190	−0.111166
$415$	0	0
$416$	32.2800	1.58266
$417$	1.73212	0.0848225
$418$	0	0
$419$	−2.80166	−0.136870	−0.0684350	−	0.997656i	$-0.521801\pi$
$419$	−2.80166	−0.136870	−0.0684350	+	0.997656i	$0.521801\pi$
$420$	0	0
$421$	−18.0116	−0.877830	−0.438915	−	0.898529i	$-0.644637\pi$
$421$	−18.0116	−0.877830	−0.438915	+	0.898529i	$0.644637\pi$
$422$	−46.2535	−2.25158
$423$	7.04179	0.342384
$424$	0.413978	0.0201045
$425$	0	0
$426$	−11.5139	−0.557852
$427$	−10.1425	−0.490829
$428$	−6.89374	−0.333221
$429$	0	0
$430$	0	0
$431$	13.9751	0.673159	0.336580	−	0.941655i	$-0.390730\pi$
$431$	13.9751	0.673159	0.336580	+	0.941655i	$0.390730\pi$
$432$	−4.31592	−0.207650
$433$	−0.370876	−0.0178232	−0.00891159	−	0.999960i	$-0.502837\pi$
$433$	−0.370876	−0.0178232	−0.00891159	+	0.999960i	$0.502837\pi$
$434$	−20.5105	−0.984536
$435$	0	0
$436$	−33.4614	−1.60251
$437$	−8.64996	−0.413784
$438$	−8.95748	−0.428005
$439$	11.2753	0.538141	0.269071	−	0.963120i	$-0.413284\pi$
$439$	11.2753	0.538141	0.269071	+	0.963120i	$0.413284\pi$
$440$	0	0
$441$	−4.60581	−0.219324
$442$	−6.02972	−0.286805
$443$	−29.5638	−1.40462	−0.702308	−	0.711873i	$-0.747847\pi$
$443$	−29.5638	−1.40462	−0.702308	+	0.711873i	$0.747847\pi$
$444$	−7.16247	−0.339916
$445$	0	0
$446$	−46.3656	−2.19547
$447$	12.4580	0.589243
$448$	−10.1537	−0.479715
$449$	1.13716	0.0536659	0.0268330	−	0.999640i	$-0.491458\pi$
$449$	1.13716	0.0536659	0.0268330	+	0.999640i	$0.491458\pi$
$450$	0	0
$451$	0	0
$452$	26.1882	1.23179
$453$	−17.6070	−0.827248
$454$	28.1864	1.32285
$455$	0	0
$456$	−2.53062	−0.118507
$457$	−7.03740	−0.329196	−0.164598	−	0.986361i	$-0.552633\pi$
$457$	−7.03740	−0.329196	−0.164598	+	0.986361i	$0.552633\pi$
$458$	40.2997	1.88308
$459$	0.741591	0.0346145
$460$	0	0
$461$	18.7977	0.875497	0.437748	−	0.899098i	$-0.355776\pi$
$461$	18.7977	0.875497	0.437748	+	0.899098i	$0.355776\pi$
$462$	0	0
$463$	37.0376	1.72128	0.860642	−	0.509211i	$-0.170063\pi$
$463$	37.0376	1.72128	0.860642	+	0.509211i	$0.170063\pi$
$464$	29.7176	1.37961
$465$	0	0
$466$	−19.8827	−0.921048
$467$	−0.579204	−0.0268024	−0.0134012	−	0.999910i	$-0.504266\pi$
$467$	−0.579204	−0.0268024	−0.0134012	+	0.999910i	$0.504266\pi$
$468$	7.59378	0.351023
$469$	5.49606	0.253784
$470$	0	0
$471$	−15.8594	−0.730763
$472$	−1.45768	−0.0670953
$473$	0	0
$474$	12.0852	0.555090
$475$	0	0
$476$	2.09655	0.0960950
$477$	1.22384	0.0560358
$478$	−16.6774	−0.762805
$479$	−11.9258	−0.544903	−0.272452	−	0.962169i	$-0.587835\pi$
$479$	−11.9258	−0.544903	−0.272452	+	0.962169i	$0.587835\pi$
$480$	0	0
$481$	−16.2930	−0.742896
$482$	9.42868	0.429465
$483$	1.78903	0.0814038
$484$	0	0
$485$	0	0
$486$	−1.95630	−0.0887394
$487$	−21.8411	−0.989715	−0.494857	−	0.868974i	$-0.664780\pi$
$487$	−21.8411	−0.989715	−0.494857	+	0.868974i	$0.664780\pi$
$488$	−2.21726	−0.100371
$489$	−8.51958	−0.385269
$490$	0	0
$491$	8.47542	0.382490	0.191245	−	0.981542i	$-0.438747\pi$
$491$	8.47542	0.382490	0.191245	+	0.981542i	$0.438747\pi$
$492$	−0.0631589	−0.00284742
$493$	−5.10629	−0.229976
$494$	60.8286	2.73681
$495$	0	0
$496$	−29.2439	−1.31309
$497$	9.10686	0.408499
$498$	29.6364	1.32804
$499$	2.54042	0.113725	0.0568624	−	0.998382i	$-0.481890\pi$
$499$	2.54042	0.113725	0.0568624	+	0.998382i	$0.481890\pi$
$500$	0	0
$501$	14.5293	0.649122
$502$	27.5080	1.22774
$503$	32.6484	1.45572	0.727860	−	0.685725i	$-0.240515\pi$
$503$	32.6484	1.45572	0.727860	+	0.685725i	$0.240515\pi$
$504$	0.523398	0.0233140
$505$	0	0
$506$	0	0
$507$	4.27413	0.189821
$508$	−26.4285	−1.17257
$509$	−4.32103	−0.191526	−0.0957632	−	0.995404i	$-0.530529\pi$
$509$	−4.32103	−0.191526	−0.0957632	+	0.995404i	$0.530529\pi$
$510$	0	0
$511$	7.08485	0.313415
$512$	−30.6006	−1.35237
$513$	−7.48127	−0.330306
$514$	22.7595	1.00388
$515$	0	0
$516$	−20.4133	−0.898646
$517$	0	0
$518$	11.8663	0.521377
$519$	−10.1366	−0.444949
$520$	0	0
$521$	−23.3715	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$521$	−23.3715	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$522$	13.4702	0.589576
$523$	−29.1251	−1.27355	−0.636775	−	0.771050i	$-0.719732\pi$
$523$	−29.1251	−1.27355	−0.636775	+	0.771050i	$0.719732\pi$
$524$	−26.7710	−1.16950
$525$	0	0
$526$	21.5740	0.940671
$527$	5.02490	0.218888
$528$	0	0
$529$	−21.6632	−0.941877
$530$	0	0
$531$	−4.30934	−0.187009
$532$	−21.1502	−0.916979
$533$	−0.143672	−0.00622313
$534$	34.3491	1.48643
$535$	0	0
$536$	1.20150	0.0518969
$537$	26.6506	1.15006
$538$	−37.1215	−1.60042
$539$	0	0
$540$	0	0
$541$	−26.1724	−1.12524	−0.562619	−	0.826716i	$-0.690206\pi$
$541$	−26.1724	−1.12524	−0.562619	+	0.826716i	$0.690206\pi$
$542$	30.4592	1.30834
$543$	22.0593	0.946657
$544$	5.75971	0.246946
$545$	0	0
$546$	−12.5809	−0.538413
$547$	3.58440	0.153258	0.0766289	−	0.997060i	$-0.475584\pi$
$547$	3.58440	0.153258	0.0766289	+	0.997060i	$0.475584\pi$
$548$	−19.0342	−0.813102
$549$	−6.55488	−0.279755
$550$	0	0
$551$	51.5129	2.19452
$552$	0.391103	0.0166464
$553$	−9.55868	−0.406476
$554$	−11.5680	−0.491475
$555$	0	0
$556$	3.16475	0.134215
$557$	18.5029	0.783992	0.391996	−	0.919967i	$-0.371785\pi$
$557$	18.5029	0.783992	0.391996	+	0.919967i	$0.371785\pi$
$558$	−13.2555	−0.561151
$559$	−46.4356	−1.96402
$560$	0	0
$561$	0	0
$562$	54.2957	2.29033
$563$	−31.3801	−1.32251	−0.661257	−	0.750159i	$-0.729977\pi$
$563$	−31.3801	−1.32251	−0.661257	+	0.750159i	$0.729977\pi$
$564$	12.8660	0.541756
$565$	0	0
$566$	−38.6251	−1.62354
$567$	1.54732	0.0649813
$568$	1.99086	0.0835348
$569$	0.944229	0.0395841	0.0197921	−	0.999804i	$-0.493700\pi$
$569$	0.944229	0.0395841	0.0197921	+	0.999804i	$0.493700\pi$
$570$	0	0
$571$	−38.0576	−1.59266	−0.796331	−	0.604861i	$-0.793229\pi$
$571$	−38.0576	−1.59266	−0.796331	+	0.604861i	$0.793229\pi$
$572$	0	0
$573$	10.6234	0.443797
$574$	0.104638	0.00436749
$575$	0	0
$576$	−6.56210	−0.273421
$577$	46.2896	1.92706	0.963531	−	0.267596i	$-0.0862291\pi$
$577$	46.2896	1.92706	0.963531	+	0.267596i	$0.0862291\pi$
$578$	32.1811	1.33856
$579$	−2.68375	−0.111533
$580$	0	0
$581$	−23.4407	−0.972485
$582$	−0.282329	−0.0117029
$583$	0	0
$584$	1.54883	0.0640910
$585$	0	0
$586$	−38.8900	−1.60653
$587$	15.5435	0.641549	0.320774	−	0.947156i	$-0.396057\pi$
$587$	15.5435	0.641549	0.320774	+	0.947156i	$0.396057\pi$
$588$	−8.41523	−0.347038
$589$	−50.6918	−2.08872
$590$	0	0
$591$	17.2838	0.710960
$592$	16.9190	0.695368
$593$	11.8521	0.486709	0.243355	−	0.969937i	$-0.421752\pi$
$593$	11.8521	0.486709	0.243355	+	0.969937i	$0.421752\pi$
$594$	0	0
$595$	0	0
$596$	22.7619	0.932364
$597$	−8.39386	−0.343538
$598$	−9.40094	−0.384433
$599$	29.7295	1.21471	0.607357	−	0.794429i	$-0.292230\pi$
$599$	29.7295	1.21471	0.607357	+	0.794429i	$0.292230\pi$
$600$	0	0
$601$	−18.7234	−0.763743	−0.381871	−	0.924215i	$-0.624720\pi$
$601$	−18.7234	−0.763743	−0.381871	+	0.924215i	$0.624720\pi$
$602$	33.8195	1.37838
$603$	3.55199	0.144648
$604$	−32.1696	−1.30896
$605$	0	0
$606$	−6.84846	−0.278199
$607$	41.6415	1.69018	0.845088	−	0.534627i	$-0.179548\pi$
$607$	41.6415	1.69018	0.845088	+	0.534627i	$0.179548\pi$
$608$	−58.1047	−2.35646
$609$	−10.6542	−0.431729
$610$	0	0
$611$	29.2672	1.18402
$612$	1.35495	0.0547708
$613$	20.6594	0.834424	0.417212	−	0.908809i	$-0.363007\pi$
$613$	20.6594	0.834424	0.417212	+	0.908809i	$0.363007\pi$
$614$	20.1639	0.813748
$615$	0	0
$616$	0	0
$617$	−46.0784	−1.85505	−0.927524	−	0.373764i	$-0.878067\pi$
$617$	−46.0784	−1.85505	−0.927524	+	0.373764i	$0.878067\pi$
$618$	23.7006	0.953380
$619$	−20.3065	−0.816187	−0.408093	−	0.912940i	$-0.633806\pi$
$619$	−20.3065	−0.816187	−0.408093	+	0.912940i	$0.633806\pi$
$620$	0	0
$621$	1.15622	0.0463973
$622$	−16.3608	−0.656009
$623$	−27.1682	−1.08847
$624$	−17.9379	−0.718091
$625$	0	0
$626$	−13.9363	−0.557005
$627$	0	0
$628$	−28.9766	−1.15629
$629$	−2.90715	−0.115916
$630$	0	0
$631$	14.4810	0.576479	0.288240	−	0.957558i	$-0.406930\pi$
$631$	14.4810	0.576479	0.288240	+	0.957558i	$0.406930\pi$
$632$	−2.08964	−0.0831212
$633$	23.6434	0.939741
$634$	53.9729	2.14354
$635$	0	0
$636$	2.23607	0.0886659
$637$	−19.1427	−0.758462
$638$	0	0
$639$	5.88558	0.232830
$640$	0	0
$641$	2.61474	0.103276	0.0516380	−	0.998666i	$-0.483556\pi$
$641$	2.61474	0.103276	0.0516380	+	0.998666i	$0.483556\pi$
$642$	7.38124	0.291314
$643$	−15.8227	−0.623985	−0.311993	−	0.950085i	$-0.600996\pi$
$643$	−15.8227	−0.623985	−0.311993	+	0.950085i	$0.600996\pi$
$644$	3.26873	0.128806
$645$	0	0
$646$	10.8536	0.427030
$647$	41.2841	1.62305	0.811523	−	0.584321i	$-0.198639\pi$
$647$	41.2841	1.62305	0.811523	+	0.584321i	$0.198639\pi$
$648$	0.338261	0.0132882
$649$	0	0
$650$	0	0
$651$	10.4844	0.410915
$652$	−15.5660	−0.609613
$653$	28.9158	1.13156	0.565781	−	0.824556i	$-0.308575\pi$
$653$	28.9158	1.13156	0.565781	+	0.824556i	$0.308575\pi$
$654$	35.8276	1.40097
$655$	0	0
$656$	0.149193	0.00582500
$657$	4.57880	0.178636
$658$	−21.3156	−0.830968
$659$	9.25143	0.360384	0.180192	−	0.983631i	$-0.442328\pi$
$659$	9.25143	0.360384	0.180192	+	0.983631i	$0.442328\pi$
$660$	0	0
$661$	18.4522	0.717706	0.358853	−	0.933394i	$-0.383168\pi$
$661$	18.4522	0.717706	0.358853	+	0.933394i	$0.383168\pi$
$662$	31.1853	1.21205
$663$	3.08221	0.119703
$664$	−5.12441	−0.198866
$665$	0	0
$666$	7.66897	0.297166
$667$	−7.96121	−0.308259
$668$	26.5464	1.02711
$669$	23.7007	0.916323
$670$	0	0
$671$	0	0
$672$	12.0175	0.463587
$673$	−12.0185	−0.463278	−0.231639	−	0.972802i	$-0.574409\pi$
$673$	−12.0185	−0.463278	−0.231639	+	0.972802i	$0.574409\pi$
$674$	−23.3882	−0.900880
$675$	0	0
$676$	7.80922	0.300354
$677$	8.09542	0.311132	0.155566	−	0.987825i	$-0.450280\pi$
$677$	8.09542	0.311132	0.155566	+	0.987825i	$0.450280\pi$
$678$	−28.0401	−1.07687
$679$	0.223306	0.00856970
$680$	0	0
$681$	−14.4080	−0.552117
$682$	0	0
$683$	17.5755	0.672507	0.336253	−	0.941772i	$-0.390840\pi$
$683$	17.5755	0.672507	0.336253	+	0.941772i	$0.390840\pi$
$684$	−13.6690	−0.522646
$685$	0	0
$686$	35.1309	1.34130
$687$	−20.6000	−0.785940
$688$	48.2200	1.83837
$689$	5.08654	0.193782
$690$	0	0
$691$	−27.1921	−1.03444	−0.517218	−	0.855854i	$-0.673032\pi$
$691$	−27.1921	−1.03444	−0.517218	+	0.855854i	$0.673032\pi$
$692$	−18.5206	−0.704046
$693$	0	0
$694$	4.29147	0.162902
$695$	0	0
$696$	−2.32912	−0.0882852
$697$	−0.0256353	−0.000971008 0
$698$	−9.44861	−0.357635
$699$	10.1634	0.384417
$700$	0	0
$701$	−0.103001	−0.00389029	−0.00194514	−	0.999998i	$-0.500619\pi$
$701$	−0.103001	−0.00389029	−0.00194514	+	0.999998i	$0.500619\pi$
$702$	−8.13078	−0.306877
$703$	29.3277	1.10611
$704$	0	0
$705$	0	0
$706$	−21.7211	−0.817484
$707$	5.41674	0.203717
$708$	−7.87356	−0.295906
$709$	−40.4949	−1.52082	−0.760408	−	0.649445i	$-0.775001\pi$
$709$	−40.4949	−1.52082	−0.760408	+	0.649445i	$0.775001\pi$
$710$	0	0
$711$	−6.17758	−0.231677
$712$	−5.93927	−0.222584
$713$	7.83432	0.293397
$714$	−2.24481	−0.0840097
$715$	0	0
$716$	48.6930	1.81974
$717$	8.52498	0.318371
$718$	−24.6082	−0.918371
$719$	48.9879	1.82694	0.913471	−	0.406905i	$-0.133392\pi$
$719$	48.9879	1.82694	0.913471	+	0.406905i	$0.133392\pi$
$720$	0	0
$721$	−18.7459	−0.698132
$722$	−72.3231	−2.69159
$723$	−4.81966	−0.179245
$724$	40.3044	1.49790
$725$	0	0
$726$	0	0
$727$	13.4370	0.498352	0.249176	−	0.968458i	$-0.419840\pi$
$727$	13.4370	0.498352	0.249176	+	0.968458i	$0.419840\pi$
$728$	2.17535	0.0806240
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.28549	−0.306450
$732$	−11.9764	−0.442659
$733$	−31.0641	−1.14738	−0.573688	−	0.819074i	$-0.694488\pi$
$733$	−31.0641	−1.14738	−0.573688	+	0.819074i	$0.694488\pi$
$734$	−40.9238	−1.51053
$735$	0	0
$736$	8.97997	0.331006
$737$	0	0
$738$	0.0676252	0.00248932
$739$	−31.4225	−1.15589	−0.577947	−	0.816074i	$-0.696146\pi$
$739$	−31.4225	−1.15589	−0.577947	+	0.816074i	$0.696146\pi$
$740$	0	0
$741$	−31.0938	−1.14226
$742$	−3.70458	−0.135999
$743$	1.74334	0.0639569	0.0319785	−	0.999489i	$-0.489819\pi$
$743$	1.74334	0.0639569	0.0319785	+	0.999489i	$0.489819\pi$
$744$	2.29200	0.0840288
$745$	0	0
$746$	−18.8199	−0.689047
$747$	−15.1493	−0.554283
$748$	0	0
$749$	−5.83814	−0.213321
$750$	0	0
$751$	21.4075	0.781172	0.390586	−	0.920566i	$-0.372272\pi$
$751$	21.4075	0.781172	0.390586	+	0.920566i	$0.372272\pi$
$752$	−30.3918	−1.10828
$753$	−14.0612	−0.512420
$754$	55.9852	2.03886
$755$	0	0
$756$	2.82709	0.102820
$757$	−0.444930	−0.0161712	−0.00808562	−	0.999967i	$-0.502574\pi$
$757$	−0.444930	−0.0161712	−0.00808562	+	0.999967i	$0.502574\pi$
$758$	58.4736	2.12386
$759$	0	0
$760$	0	0
$761$	−32.0823	−1.16298	−0.581492	−	0.813552i	$-0.697531\pi$
$761$	−32.0823	−1.16298	−0.581492	+	0.813552i	$0.697531\pi$
$762$	28.2974	1.02511
$763$	−28.3376	−1.02589
$764$	19.4098	0.702223
$765$	0	0
$766$	−54.8467	−1.98169
$767$	−17.9105	−0.646712
$768$	18.3983	0.663893
$769$	−38.9434	−1.40433	−0.702167	−	0.712012i	$-0.747784\pi$
$769$	−38.9434	−1.40433	−0.702167	+	0.712012i	$0.747784\pi$
$770$	0	0
$771$	−11.6340	−0.418988
$772$	−4.90345	−0.176479
$773$	−22.7930	−0.819809	−0.409904	−	0.912128i	$-0.634438\pi$
$773$	−22.7930	−0.819809	−0.409904	+	0.912128i	$0.634438\pi$
$774$	21.8569	0.785629
$775$	0	0
$776$	0.0488172	0.00175244
$777$	−6.06572	−0.217606
$778$	−19.3544	−0.693888
$779$	0.258613	0.00926576
$780$	0	0
$781$	0	0
$782$	−1.67740	−0.0599839
$783$	−6.88558	−0.246071
$784$	19.8783	0.709939
$785$	0	0
$786$	28.6642	1.02242
$787$	−19.7845	−0.705242	−0.352621	−	0.935766i	$-0.614710\pi$
$787$	−19.7845	−0.705242	−0.352621	+	0.935766i	$0.614710\pi$
$788$	31.5790	1.12496
$789$	−11.0280	−0.392607
$790$	0	0
$791$	22.1781	0.788563
$792$	0	0
$793$	−27.2435	−0.967444
$794$	28.8470	1.02374
$795$	0	0
$796$	−15.3364	−0.543583
$797$	−12.3662	−0.438035	−0.219018	−	0.975721i	$-0.570285\pi$
$797$	−12.3662	−0.438035	−0.219018	+	0.975721i	$0.570285\pi$
$798$	22.6459	0.801656
$799$	5.22213	0.184746
$800$	0	0
$801$	−17.5582	−0.620390
$802$	28.1301	0.993309
$803$	0	0
$804$	6.48981	0.228878
$805$	0	0
$806$	−55.0928	−1.94056
$807$	18.9754	0.667966
$808$	1.18416	0.0416586
$809$	−20.4155	−0.717772	−0.358886	−	0.933381i	$-0.616843\pi$
$809$	−20.4155	−0.717772	−0.358886	+	0.933381i	$0.616843\pi$
$810$	0	0
$811$	10.0367	0.352437	0.176219	−	0.984351i	$-0.443613\pi$
$811$	10.0367	0.352437	0.176219	+	0.984351i	$0.443613\pi$
$812$	−19.4662	−0.683128
$813$	−15.5699	−0.546059
$814$	0	0
$815$	0	0
$816$	−3.20065	−0.112045
$817$	83.5851	2.92427
$818$	−21.5259	−0.752636
$819$	6.43099	0.224717
$820$	0	0
$821$	10.3699	0.361913	0.180957	−	0.983491i	$-0.442081\pi$
$821$	10.3699	0.361913	0.180957	+	0.983491i	$0.442081\pi$
$822$	20.3803	0.710843
$823$	−48.6023	−1.69417	−0.847085	−	0.531457i	$-0.821645\pi$
$823$	−48.6023	−1.69417	−0.847085	+	0.531457i	$0.821645\pi$
$824$	−4.09806	−0.142763
$825$	0	0
$826$	13.0444	0.453873
$827$	10.5295	0.366147	0.183074	−	0.983099i	$-0.441395\pi$
$827$	10.5295	0.366147	0.183074	+	0.983099i	$0.441395\pi$
$828$	2.11251	0.0734148
$829$	−29.2410	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$829$	−29.2410	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$830$	0	0
$831$	5.91319	0.205126
$832$	−27.2735	−0.945539
$833$	−3.41563	−0.118344
$834$	−3.38855	−0.117336
$835$	0	0
$836$	0	0
$837$	6.77583	0.234207
$838$	5.48087	0.189334
$839$	−22.0625	−0.761681	−0.380840	−	0.924641i	$-0.624365\pi$
$839$	−22.0625	−0.761681	−0.380840	+	0.924641i	$0.624365\pi$
$840$	0	0
$841$	18.4112	0.634869
$842$	35.2359	1.21431
$843$	−27.7543	−0.955910
$844$	43.1987	1.48696
$845$	0	0
$846$	−13.7758	−0.473623
$847$	0	0
$848$	−5.28200	−0.181385
$849$	19.7440	0.677613
$850$	0	0
$851$	−4.53253	−0.155373
$852$	10.7535	0.368408
$853$	−1.43827	−0.0492455	−0.0246227	−	0.999697i	$-0.507838\pi$
$853$	−1.43827	−0.0492455	−0.0246227	+	0.999697i	$0.507838\pi$
$854$	19.8417	0.678968
$855$	0	0
$856$	−1.27628	−0.0436224
$857$	34.3766	1.17428	0.587141	−	0.809485i	$-0.300254\pi$
$857$	34.3766	1.17428	0.587141	+	0.809485i	$0.300254\pi$
$858$	0	0
$859$	−14.4635	−0.493489	−0.246744	−	0.969081i	$-0.579361\pi$
$859$	−14.4635	−0.493489	−0.246744	+	0.969081i	$0.579361\pi$
$860$	0	0
$861$	−0.0534877	−0.00182286
$862$	−27.3395	−0.931187
$863$	−21.9433	−0.746958	−0.373479	−	0.927639i	$-0.621835\pi$
$863$	−21.9433	−0.746958	−0.373479	+	0.927639i	$0.621835\pi$
$864$	7.76669	0.264228
$865$	0	0
$866$	0.725544	0.0246550
$867$	−16.4500	−0.558673
$868$	19.1559	0.650193
$869$	0	0
$870$	0	0
$871$	14.7628	0.500219
$872$	−6.19492	−0.209786
$873$	0.144318	0.00488443
$874$	16.9219	0.572391
$875$	0	0
$876$	8.36588	0.282657
$877$	−12.6435	−0.426940	−0.213470	−	0.976950i	$-0.568477\pi$
$877$	−12.6435	−0.426940	−0.213470	+	0.976950i	$0.568477\pi$
$878$	−22.0578	−0.744416
$879$	19.8794	0.670516
$880$	0	0
$881$	−6.38769	−0.215207	−0.107603	−	0.994194i	$-0.534318\pi$
$881$	−6.38769	−0.215207	−0.107603	+	0.994194i	$0.534318\pi$
$882$	9.01032	0.303393
$883$	40.8573	1.37496	0.687478	−	0.726205i	$-0.258718\pi$
$883$	40.8573	1.37496	0.687478	+	0.726205i	$0.258718\pi$
$884$	5.63148	0.189407
$885$	0	0
$886$	57.8354	1.94302
$887$	2.81682	0.0945794	0.0472897	−	0.998881i	$-0.484942\pi$
$887$	2.81682	0.0945794	0.0472897	+	0.998881i	$0.484942\pi$
$888$	−1.32603	−0.0444988
$889$	−22.3816	−0.750655
$890$	0	0
$891$	0	0
$892$	43.3034	1.44990
$893$	−52.6816	−1.76292
$894$	−24.3715	−0.815106
$895$	0	0
$896$	−4.17153	−0.139361
$897$	4.80548	0.160450
$898$	−2.22462	−0.0742366
$899$	−46.6555	−1.55605
$900$	0	0
$901$	0.907590	0.0302362
$902$	0	0
$903$	−17.2875	−0.575293
$904$	4.84839	0.161255
$905$	0	0
$906$	34.4445	1.14434
$907$	13.0949	0.434808	0.217404	−	0.976082i	$-0.430241\pi$
$907$	13.0949	0.434808	0.217404	+	0.976082i	$0.430241\pi$
$908$	−26.3248	−0.873619
$909$	3.50073	0.116112
$910$	0	0
$911$	11.5351	0.382175	0.191087	−	0.981573i	$-0.438799\pi$
$911$	11.5351	0.382175	0.191087	+	0.981573i	$0.438799\pi$
$912$	32.2886	1.06918
$913$	0	0
$914$	13.7672	0.455379
$915$	0	0
$916$	−37.6381	−1.24360
$917$	−22.6717	−0.748686
$918$	−1.45077	−0.0478826
$919$	−7.82709	−0.258192	−0.129096	−	0.991632i	$-0.541208\pi$
$919$	−7.82709	−0.258192	−0.129096	+	0.991632i	$0.541208\pi$
$920$	0	0
$921$	−10.3072	−0.339633
$922$	−36.7739	−1.21108
$923$	24.4617	0.805168
$924$	0	0
$925$	0	0
$926$	−72.4565	−2.38107
$927$	−12.1151	−0.397911
$928$	−53.4782	−1.75551
$929$	−28.5710	−0.937386	−0.468693	−	0.883361i	$-0.655275\pi$
$929$	−28.5710	−0.937386	−0.468693	+	0.883361i	$0.655275\pi$
$930$	0	0
$931$	34.4573	1.12929
$932$	18.5695	0.608265
$933$	8.36316	0.273798
$934$	1.13309	0.0370760
$935$	0	0
$936$	1.40589	0.0459528
$937$	39.0877	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$937$	39.0877	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$938$	−10.7519	−0.351062
$939$	7.12380	0.232477
$940$	0	0
$941$	−13.3575	−0.435443	−0.217721	−	0.976011i	$-0.569862\pi$
$941$	−13.3575	−0.435443	−0.217721	+	0.976011i	$0.569862\pi$
$942$	31.0257	1.01087
$943$	−0.0399681	−0.00130154
$944$	18.5988	0.605338
$945$	0	0
$946$	0	0
$947$	−8.22435	−0.267255	−0.133628	−	0.991032i	$-0.542663\pi$
$947$	−8.22435	−0.267255	−0.133628	+	0.991032i	$0.542663\pi$
$948$	−11.2870	−0.366585
$949$	19.0305	0.617755
$950$	0	0
$951$	−27.5894	−0.894646
$952$	0.388147	0.0125799
$953$	−55.7519	−1.80598	−0.902991	−	0.429661i	$-0.858633\pi$
$953$	−55.7519	−1.80598	−0.902991	+	0.429661i	$0.858633\pi$
$954$	−2.39419	−0.0775149
$955$	0	0
$956$	15.5759	0.503761
$957$	0	0
$958$	23.3304	0.753770
$959$	−16.1196	−0.520530
$960$	0	0
$961$	14.9119	0.481028
$962$	31.8739	1.02765
$963$	−3.77307	−0.121585
$964$	−8.80596	−0.283621
$965$	0	0
$966$	−3.49988	−0.112607
$967$	1.68449	0.0541695	0.0270847	−	0.999633i	$-0.491378\pi$
$967$	1.68449	0.0541695	0.0270847	+	0.999633i	$0.491378\pi$
$968$	0	0
$969$	−5.54805	−0.178229
$970$	0	0
$971$	−61.1384	−1.96202	−0.981012	−	0.193945i	$-0.937872\pi$
$971$	−61.1384	−1.96202	−0.981012	+	0.193945i	$0.937872\pi$
$972$	1.82709	0.0586040
$973$	2.68015	0.0859216
$974$	42.7276	1.36908
$975$	0	0
$976$	28.2903	0.905551
$977$	−22.9369	−0.733817	−0.366909	−	0.930257i	$-0.619584\pi$
$977$	−22.9369	−0.733817	−0.366909	+	0.930257i	$0.619584\pi$
$978$	16.6668	0.532946
$979$	0	0
$980$	0	0
$981$	−18.3140	−0.584721
$982$	−16.5804	−0.529103
$983$	−6.68730	−0.213292	−0.106646	−	0.994297i	$-0.534011\pi$
$983$	−6.68730	−0.213292	−0.106646	+	0.994297i	$0.534011\pi$
$984$	−0.0116930	−0.000372760 0
$985$	0	0
$986$	9.98940	0.318127
$987$	10.8959	0.346820
$988$	−56.8112	−1.80740
$989$	−12.9179	−0.410766
$990$	0	0
$991$	−3.65123	−0.115985	−0.0579925	−	0.998317i	$-0.518470\pi$
$991$	−3.65123	−0.115985	−0.0579925	+	0.998317i	$0.518470\pi$
$992$	52.6258	1.67087
$993$	−15.9410	−0.505872
$994$	−17.8157	−0.565080
$995$	0	0
$996$	−27.6791	−0.877045
$997$	8.31754	0.263419	0.131710	−	0.991288i	$-0.457953\pi$
$997$	8.31754	0.263419	0.131710	+	0.991288i	$0.457953\pi$
$998$	−4.96981	−0.157317
$999$	−3.92015	−0.124028

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.df.1.1		4
5.4	even	2		1815.2.a.q.1.4		4
11.2	odd	10		825.2.n.j.301.1		8
11.6	odd	10		825.2.n.j.751.1		8
11.10	odd	2		9075.2.a.co.1.4		4
15.14	odd	2		5445.2.a.bq.1.1		4
55.2	even	20		825.2.bx.e.499.4		16
55.13	even	20		825.2.bx.e.499.1		16
55.17	even	20		825.2.bx.e.124.1		16
55.24	odd	10		165.2.m.c.136.2	yes	8
55.28	even	20		825.2.bx.e.124.4		16
55.39	odd	10		165.2.m.c.91.2	✓	8
55.54	odd	2		1815.2.a.u.1.1		4
165.134	even	10		495.2.n.c.136.1		8
165.149	even	10		495.2.n.c.91.1		8
165.164	even	2		5445.2.a.bj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.c.91.2	✓	8	55.39	odd	10
165.2.m.c.136.2	yes	8	55.24	odd	10
495.2.n.c.91.1		8	165.149	even	10
495.2.n.c.136.1		8	165.134	even	10
825.2.n.j.301.1		8	11.2	odd	10
825.2.n.j.751.1		8	11.6	odd	10
825.2.bx.e.124.1		16	55.17	even	20
825.2.bx.e.124.4		16	55.28	even	20
825.2.bx.e.499.1		16	55.13	even	20
825.2.bx.e.499.4		16	55.2	even	20
1815.2.a.q.1.4		4	5.4	even	2
1815.2.a.u.1.1		4	55.54	odd	2
5445.2.a.bj.1.4		4	165.164	even	2
5445.2.a.bq.1.1		4	15.14	odd	2
9075.2.a.co.1.4		4	11.10	odd	2
9075.2.a.df.1.1		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)

\(f(q)\)	\(=\)	\(q-1.95630 q^{2} +1.00000 q^{3} +1.82709 q^{4} -1.95630 q^{6} +1.54732 q^{7} +0.338261 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.95630 q^{2} +1.00000 q^{3} +1.82709 q^{4} -1.95630 q^{6} +1.54732 q^{7} +0.338261 q^{8} +1.00000 q^{9} +1.82709 q^{12} +4.15622 q^{13} -3.02701 q^{14} -4.31592 q^{16} +0.741591 q^{17} -1.95630 q^{18} -7.48127 q^{19} +1.54732 q^{21} +1.15622 q^{23} +0.338261 q^{24} -8.13078 q^{26} +1.00000 q^{27} +2.82709 q^{28} -6.88558 q^{29} +6.77583 q^{31} +7.76669 q^{32} -1.45077 q^{34} +1.82709 q^{36} -3.92015 q^{37} +14.6356 q^{38} +4.15622 q^{39} -0.0345680 q^{41} -3.02701 q^{42} -11.1726 q^{43} -2.26190 q^{46} +7.04179 q^{47} -4.31592 q^{48} -4.60581 q^{49} +0.741591 q^{51} +7.59378 q^{52} +1.22384 q^{53} -1.95630 q^{54} +0.523398 q^{56} -7.48127 q^{57} +13.4702 q^{58} -4.30934 q^{59} -6.55488 q^{61} -13.2555 q^{62} +1.54732 q^{63} -6.56210 q^{64} +3.55199 q^{67} +1.35495 q^{68} +1.15622 q^{69} +5.88558 q^{71} +0.338261 q^{72} +4.57880 q^{73} +7.66897 q^{74} -13.6690 q^{76} -8.13078 q^{78} -6.17758 q^{79} +1.00000 q^{81} +0.0676252 q^{82} -15.1493 q^{83} +2.82709 q^{84} +21.8569 q^{86} -6.88558 q^{87} -17.5582 q^{89} +6.43099 q^{91} +2.11251 q^{92} +6.77583 q^{93} -13.7758 q^{94} +7.76669 q^{96} +0.144318 q^{97} +9.01032 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 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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