LMFDB - Embedded newform 9075.2.a.cq.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:	$4$
Coefficient field:	4.4.5725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 6x + 11 $ x^4 - x^3 - 8x^2 + 6x + 11
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			$-0.933531$ 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)

$f(q)$	$=$	$q-2.55157 q^{2} +1.00000 q^{3} +4.51049 q^{4} -2.55157 q^{6} +4.19499 q^{7} -6.40567 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.55157 q^{2} +1.00000 q^{3} +4.51049 q^{4} -2.55157 q^{6} +4.19499 q^{7} -6.40567 q^{8} +1.00000 q^{9} +4.51049 q^{12} +4.93353 q^{13} -10.7038 q^{14} +7.32351 q^{16} +6.10313 q^{17} -2.55157 q^{18} -2.51049 q^{19} +4.19499 q^{21} -3.08744 q^{23} -6.40567 q^{24} -12.5882 q^{26} +1.00000 q^{27} +18.9214 q^{28} -3.60066 q^{29} -1.99558 q^{31} -5.87507 q^{32} -15.5725 q^{34} +4.51049 q^{36} -7.64174 q^{37} +6.40567 q^{38} +4.93353 q^{39} -2.95892 q^{41} -10.7038 q^{42} +0.186978 q^{43} +7.87780 q^{46} -5.56399 q^{47} +7.32351 q^{48} +10.5979 q^{49} +6.10313 q^{51} +22.2526 q^{52} +6.21596 q^{53} -2.55157 q^{54} -26.8717 q^{56} -2.51049 q^{57} +9.18731 q^{58} -0.530595 q^{59} +8.14949 q^{61} +5.09186 q^{62} +4.19499 q^{63} +0.343617 q^{64} +6.64174 q^{67} +27.5281 q^{68} -3.08744 q^{69} -11.0166 q^{71} -6.40567 q^{72} -0.0446719 q^{73} +19.4984 q^{74} -11.3235 q^{76} -12.5882 q^{78} +12.5270 q^{79} +1.00000 q^{81} +7.54988 q^{82} +6.46772 q^{83} +18.9214 q^{84} -0.477086 q^{86} -3.60066 q^{87} -0.535874 q^{89} +20.6961 q^{91} -13.9259 q^{92} -1.99558 q^{93} +14.1969 q^{94} -5.87507 q^{96} +11.0621 q^{97} -27.0413 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - q^2 + 4 * q^3 + 9 * q^4 - q^6 + 8 * q^7 - 3 * q^8 + 4 * q^9	$ 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9} + 9 q^{12} + 15 q^{13} + 7 q^{14} + 7 q^{16} + 6 q^{17} - q^{18} - q^{19} + 8 q^{21} + q^{23} - 3 q^{24} - 18 q^{26} + 4 q^{27} + 31 q^{28} + 17 q^{29} + 15 q^{31} + 8 q^{32} - 35 q^{34} + 9 q^{36} + q^{37} + 3 q^{38} + 15 q^{39} - 12 q^{41} + 7 q^{42} + 14 q^{43} - 9 q^{46} - 14 q^{47} + 7 q^{48} + 20 q^{49} + 6 q^{51} + 39 q^{52} - 2 q^{53} - q^{54} - 12 q^{56} - q^{57} + 11 q^{58} - 11 q^{59} + q^{61} + 30 q^{62} + 8 q^{63} - 3 q^{64} - 5 q^{67} + 19 q^{68} + q^{69} - 3 q^{71} - 3 q^{72} + 45 q^{73} + 29 q^{74} - 23 q^{76} - 18 q^{78} + 4 q^{81} - 11 q^{82} - 15 q^{83} + 31 q^{84} - 10 q^{86} + 17 q^{87} + 2 q^{89} + 16 q^{91} - 34 q^{92} + 15 q^{93} - 29 q^{94} + 8 q^{96} + 26 q^{97} + q^{98}+O(q^{100}) $ 4 * q - q^2 + 4 * q^3 + 9 * q^4 - q^6 + 8 * q^7 - 3 * q^8 + 4 * q^9 + 9 * q^12 + 15 * q^13 + 7 * q^14 + 7 * q^16 + 6 * q^17 - q^18 - q^19 + 8 * q^21 + q^23 - 3 * q^24 - 18 * q^26 + 4 * q^27 + 31 * q^28 + 17 * q^29 + 15 * q^31 + 8 * q^32 - 35 * q^34 + 9 * q^36 + q^37 + 3 * q^38 + 15 * q^39 - 12 * q^41 + 7 * q^42 + 14 * q^43 - 9 * q^46 - 14 * q^47 + 7 * q^48 + 20 * q^49 + 6 * q^51 + 39 * q^52 - 2 * q^53 - q^54 - 12 * q^56 - q^57 + 11 * q^58 - 11 * q^59 + q^61 + 30 * q^62 + 8 * q^63 - 3 * q^64 - 5 * q^67 + 19 * q^68 + q^69 - 3 * q^71 - 3 * q^72 + 45 * q^73 + 29 * q^74 - 23 * q^76 - 18 * q^78 + 4 * q^81 - 11 * q^82 - 15 * q^83 + 31 * q^84 - 10 * q^86 + 17 * q^87 + 2 * q^89 + 16 * q^91 - 34 * q^92 + 15 * q^93 - 29 * q^94 + 8 * q^96 + 26 * q^97 + 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 $

$2$	−2.55157	−1.80423	−0.902115	−	0.431497i	$-0.857986\pi$
$2$	−2.55157	−1.80423	−0.902115	+	0.431497i	$0.857986\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.51049	2.25524
$5$	0	0
$5$	0	0
$6$	−2.55157	−1.04167
$7$	4.19499	1.58556	0.792778	−	0.609510i	$-0.208634\pi$
$7$	4.19499	1.58556	0.792778	+	0.609510i	$0.208634\pi$
$8$	−6.40567	−2.26475
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	4.51049	1.30206
$13$	4.93353	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$13$	4.93353	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$14$	−10.7038	−2.86071
$15$	0	0
$16$	7.32351	1.83088
$17$	6.10313	1.48023	0.740113	−	0.672482i	$-0.234772\pi$
$17$	6.10313	1.48023	0.740113	+	0.672482i	$0.234772\pi$
$18$	−2.55157	−0.601410
$19$	−2.51049	−0.575945	−0.287972	−	0.957639i	$-0.592981\pi$
$19$	−2.51049	−0.575945	−0.287972	+	0.957639i	$0.592981\pi$
$20$	0	0
$21$	4.19499	0.915421
$22$	0	0
$23$	−3.08744	−0.643776	−0.321888	−	0.946778i	$-0.604317\pi$
$23$	−3.08744	−0.643776	−0.321888	+	0.946778i	$0.604317\pi$
$24$	−6.40567	−1.30755
$25$	0	0
$26$	−12.5882	−2.46875
$27$	1.00000	0.192450
$28$	18.9214	3.57581
$29$	−3.60066	−0.668625	−0.334312	−	0.942462i	$-0.608504\pi$
$29$	−3.60066	−0.668625	−0.334312	+	0.942462i	$0.608504\pi$
$30$	0	0
$31$	−1.99558	−0.358417	−0.179209	−	0.983811i	$-0.557354\pi$
$31$	−1.99558	−0.358417	−0.179209	+	0.983811i	$0.557354\pi$
$32$	−5.87507	−1.03858
$33$	0	0
$34$	−15.5725	−2.67067
$35$	0	0
$36$	4.51049	0.751748
$37$	−7.64174	−1.25629	−0.628147	−	0.778095i	$-0.716186\pi$
$37$	−7.64174	−1.25629	−0.628147	+	0.778095i	$0.716186\pi$
$38$	6.40567	1.03914
$39$	4.93353	0.789997
$40$	0	0
$41$	−2.95892	−0.462106	−0.231053	−	0.972941i	$-0.574217\pi$
$41$	−2.95892	−0.462106	−0.231053	+	0.972941i	$0.574217\pi$
$42$	−10.7038	−1.65163
$43$	0.186978	0.0285139	0.0142569	−	0.999898i	$-0.495462\pi$
$43$	0.186978	0.0285139	0.0142569	+	0.999898i	$0.495462\pi$
$44$	0	0
$45$	0	0
$46$	7.87780	1.16152
$47$	−5.56399	−0.811592	−0.405796	−	0.913964i	$-0.633006\pi$
$47$	−5.56399	−0.811592	−0.405796	+	0.913964i	$0.633006\pi$
$48$	7.32351	1.05706
$49$	10.5979	1.51399
$50$	0	0
$51$	6.10313	0.854609
$52$	22.2526	3.08588
$53$	6.21596	0.853828	0.426914	−	0.904292i	$-0.359601\pi$
$53$	6.21596	0.853828	0.426914	+	0.904292i	$0.359601\pi$
$54$	−2.55157	−0.347224
$55$	0	0
$56$	−26.8717	−3.59088
$57$	−2.51049	−0.332522
$58$	9.18731	1.20635
$59$	−0.530595	−0.0690775	−0.0345388	−	0.999403i	$-0.510996\pi$
$59$	−0.530595	−0.0690775	−0.0345388	+	0.999403i	$0.510996\pi$
$60$	0	0
$61$	8.14949	1.04344	0.521718	−	0.853118i	$-0.325291\pi$
$61$	8.14949	1.04344	0.521718	+	0.853118i	$0.325291\pi$
$62$	5.09186	0.646667
$63$	4.19499	0.528519
$64$	0.343617	0.0429521
$65$	0	0
$66$	0	0
$67$	6.64174	0.811417	0.405709	−	0.914003i	$-0.367025\pi$
$67$	6.64174	0.811417	0.405709	+	0.914003i	$0.367025\pi$
$68$	27.5281	3.33827
$69$	−3.08744	−0.371684
$70$	0	0
$71$	−11.0166	−1.30742	−0.653712	−	0.756743i	$-0.726789\pi$
$71$	−11.0166	−1.30742	−0.653712	+	0.756743i	$0.726789\pi$
$72$	−6.40567	−0.754915
$73$	−0.0446719	−0.00522845	−0.00261423	−	0.999997i	$-0.500832\pi$
$73$	−0.0446719	−0.00522845	−0.00261423	+	0.999997i	$0.500832\pi$
$74$	19.4984	2.26664
$75$	0	0
$76$	−11.3235	−1.29890
$77$	0	0
$78$	−12.5882	−1.42534
$79$	12.5270	1.40940	0.704701	−	0.709504i	$-0.251081\pi$
$79$	12.5270	1.40940	0.704701	+	0.709504i	$0.251081\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.54988	0.833745
$83$	6.46772	0.709924	0.354962	−	0.934881i	$-0.384494\pi$
$83$	6.46772	0.709924	0.354962	+	0.934881i	$0.384494\pi$
$84$	18.9214	2.06450
$85$	0	0
$86$	−0.477086	−0.0514456
$87$	−3.60066	−0.386031
$88$	0	0
$89$	−0.535874	−0.0568025	−0.0284013	−	0.999597i	$-0.509042\pi$
$89$	−0.535874	−0.0568025	−0.0284013	+	0.999597i	$0.509042\pi$
$90$	0	0
$91$	20.6961	2.16954
$92$	−13.9259	−1.45187
$93$	−1.99558	−0.206932
$94$	14.1969	1.46430
$95$	0	0
$96$	−5.87507	−0.599622
$97$	11.0621	1.12318	0.561591	−	0.827415i	$-0.310190\pi$
$97$	11.0621	1.12318	0.561591	+	0.827415i	$0.310190\pi$
$98$	−27.0413	−2.73158
$99$	0	0
$100$	0	0
$101$	−6.07448	−0.604433	−0.302217	−	0.953239i	$-0.597727\pi$
$101$	−6.07448	−0.604433	−0.302217	+	0.953239i	$0.597727\pi$
$102$	−15.5725	−1.54191
$103$	7.96165	0.784485	0.392242	−	0.919862i	$-0.371699\pi$
$103$	7.96165	0.784485	0.392242	+	0.919862i	$0.371699\pi$
$104$	−31.6026	−3.09889
$105$	0	0
$106$	−15.8604	−1.54050
$107$	0.0438105	0.00423532	0.00211766	−	0.999998i	$-0.499326\pi$
$107$	0.0438105	0.00423532	0.00211766	+	0.999998i	$0.499326\pi$
$108$	4.51049	0.434022
$109$	10.9879	1.05245	0.526225	−	0.850345i	$-0.323607\pi$
$109$	10.9879	1.05245	0.526225	+	0.850345i	$0.323607\pi$
$110$	0	0
$111$	−7.64174	−0.725321
$112$	30.7220	2.90296
$113$	−14.3348	−1.34850	−0.674251	−	0.738502i	$-0.735534\pi$
$113$	−14.3348	−1.34850	−0.674251	+	0.738502i	$0.735534\pi$
$114$	6.40567	0.599946
$115$	0	0
$116$	−16.2407	−1.50791
$117$	4.93353	0.456105
$118$	1.35385	0.124632
$119$	25.6026	2.34698
$120$	0	0
$121$	0	0
$122$	−20.7940	−1.88260
$123$	−2.95892	−0.266797
$124$	−9.00104	−0.808318
$125$	0	0
$126$	−10.7038	−0.953569
$127$	14.2134	1.26124	0.630618	−	0.776093i	$-0.282801\pi$
$127$	14.2134	1.26124	0.630618	+	0.776093i	$0.282801\pi$
$128$	10.8734	0.961081
$129$	0.186978	0.0164625
$130$	0	0
$131$	14.6699	1.28171	0.640856	−	0.767661i	$-0.278580\pi$
$131$	14.6699	1.28171	0.640856	+	0.767661i	$0.278580\pi$
$132$	0	0
$133$	−10.5315	−0.913193
$134$	−16.9468	−1.46398
$135$	0	0
$136$	−39.0946	−3.35234
$137$	8.61035	0.735632	0.367816	−	0.929899i	$-0.380106\pi$
$137$	8.61035	0.735632	0.367816	+	0.929899i	$0.380106\pi$
$138$	7.87780	0.670603
$139$	−3.73086	−0.316448	−0.158224	−	0.987403i	$-0.550577\pi$
$139$	−3.73086	−0.316448	−0.158224	+	0.987403i	$0.550577\pi$
$140$	0	0
$141$	−5.56399	−0.468573
$142$	28.1095	2.35889
$143$	0	0
$144$	7.32351	0.610292
$145$	0	0
$146$	0.113983	0.00943333
$147$	10.5979	0.874102
$148$	−34.4679	−2.83325
$149$	9.96251	0.816161	0.408080	−	0.912946i	$-0.366198\pi$
$149$	9.96251	0.816161	0.408080	+	0.912946i	$0.366198\pi$
$150$	0	0
$151$	−14.9220	−1.21433	−0.607166	−	0.794575i	$-0.707694\pi$
$151$	−14.9220	−1.21433	−0.607166	+	0.794575i	$0.707694\pi$
$152$	16.0813	1.30437
$153$	6.10313	0.493409
$154$	0	0
$155$	0	0
$156$	22.2526	1.78164
$157$	11.1848	0.892641	0.446320	−	0.894873i	$-0.352734\pi$
$157$	11.1848	0.892641	0.446320	+	0.894873i	$0.352734\pi$
$158$	−31.9636	−2.54288
$159$	6.21596	0.492958
$160$	0	0
$161$	−12.9518	−1.02074
$162$	−2.55157	−0.200470
$163$	−0.859052	−0.0672862	−0.0336431	−	0.999434i	$-0.510711\pi$
$163$	−0.859052	−0.0672862	−0.0336431	+	0.999434i	$0.510711\pi$
$164$	−13.3462	−1.04216
$165$	0	0
$166$	−16.5028	−1.28087
$167$	−18.8792	−1.46092	−0.730458	−	0.682958i	$-0.760693\pi$
$167$	−18.8792	−1.46092	−0.730458	+	0.682958i	$0.760693\pi$
$168$	−26.8717	−2.07320
$169$	11.3397	0.872287
$170$	0	0
$171$	−2.51049	−0.191982
$172$	0.843361	0.0643057
$173$	−9.82957	−0.747329	−0.373664	−	0.927564i	$-0.621899\pi$
$173$	−9.82957	−0.747329	−0.373664	+	0.927564i	$0.621899\pi$
$174$	9.18731	0.696488
$175$	0	0
$176$	0	0
$177$	−0.530595	−0.0398819
$178$	1.36732	0.102485
$179$	18.3008	1.36787	0.683935	−	0.729543i	$-0.260267\pi$
$179$	18.3008	1.36787	0.683935	+	0.729543i	$0.260267\pi$
$180$	0	0
$181$	−9.84422	−0.731715	−0.365858	−	0.930671i	$-0.619224\pi$
$181$	−9.84422	−0.731715	−0.365858	+	0.930671i	$0.619224\pi$
$182$	−52.8075	−3.91435
$183$	8.14949	0.602428
$184$	19.7771	1.45799
$185$	0	0
$186$	5.09186	0.373353
$187$	0	0
$188$	−25.0963	−1.83034
$189$	4.19499	0.305140
$190$	0	0
$191$	8.48869	0.614220	0.307110	−	0.951674i	$-0.400638\pi$
$191$	8.48869	0.614220	0.307110	+	0.951674i	$0.400638\pi$
$192$	0.343617	0.0247984
$193$	27.2731	1.96316	0.981579	−	0.191057i	$-0.0611917\pi$
$193$	27.2731	1.96316	0.981579	+	0.191057i	$0.0611917\pi$
$194$	−28.2255	−2.02648
$195$	0	0
$196$	47.8018	3.41441
$197$	21.5632	1.53631	0.768156	−	0.640263i	$-0.221175\pi$
$197$	21.5632	1.53631	0.768156	+	0.640263i	$0.221175\pi$
$198$	0	0
$199$	23.2920	1.65113	0.825564	−	0.564309i	$-0.190857\pi$
$199$	23.2920	1.65113	0.825564	+	0.564309i	$0.190857\pi$
$200$	0	0
$201$	6.64174	0.468472
$202$	15.4994	1.09054
$203$	−15.1047	−1.06014
$204$	27.5281	1.92735
$205$	0	0
$206$	−20.3147	−1.41539
$207$	−3.08744	−0.214592
$208$	36.1308	2.50522
$209$	0	0
$210$	0	0
$211$	6.49225	0.446945	0.223472	−	0.974710i	$-0.428261\pi$
$211$	6.49225	0.446945	0.223472	+	0.974710i	$0.428261\pi$
$212$	28.0370	1.92559
$213$	−11.0166	−0.754842
$214$	−0.111785	−0.00764148
$215$	0	0
$216$	−6.40567	−0.435850
$217$	−8.37144	−0.568291
$218$	−28.0363	−1.89886
$219$	−0.0446719	−0.00301865
$220$	0	0
$221$	30.1100	2.02542
$222$	19.4984	1.30865
$223$	−6.17233	−0.413330	−0.206665	−	0.978412i	$-0.566261\pi$
$223$	−6.17233	−0.413330	−0.206665	+	0.978412i	$0.566261\pi$
$224$	−24.6459	−1.64672
$225$	0	0
$226$	36.5761	2.43301
$227$	−15.8335	−1.05090	−0.525452	−	0.850823i	$-0.676104\pi$
$227$	−15.8335	−1.05090	−0.525452	+	0.850823i	$0.676104\pi$
$228$	−11.3235	−0.749918
$229$	18.5785	1.22770	0.613852	−	0.789421i	$-0.289619\pi$
$229$	18.5785	1.22770	0.613852	+	0.789421i	$0.289619\pi$
$230$	0	0
$231$	0	0
$232$	23.0646	1.51427
$233$	−3.14368	−0.205949	−0.102975	−	0.994684i	$-0.532836\pi$
$233$	−3.14368	−0.205949	−0.102975	+	0.994684i	$0.532836\pi$
$234$	−12.5882	−0.822918
$235$	0	0
$236$	−2.39324	−0.155787
$237$	12.5270	0.813719
$238$	−65.3266	−4.23449
$239$	−1.11778	−0.0723031	−0.0361515	−	0.999346i	$-0.511510\pi$
$239$	−1.11778	−0.0723031	−0.0361515	+	0.999346i	$0.511510\pi$
$240$	0	0
$241$	−14.4423	−0.930312	−0.465156	−	0.885229i	$-0.654002\pi$
$241$	−14.4423	−0.930312	−0.465156	+	0.885229i	$0.654002\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	36.7582	2.35320
$245$	0	0
$246$	7.54988	0.481363
$247$	−12.3856	−0.788074
$248$	12.7830	0.811723
$249$	6.46772	0.409875
$250$	0	0
$251$	−0.292307	−0.0184502	−0.00922512	−	0.999957i	$-0.502936\pi$
$251$	−0.292307	−0.0184502	−0.00922512	+	0.999957i	$0.502936\pi$
$252$	18.9214	1.19194
$253$	0	0
$254$	−36.2664	−2.27556
$255$	0	0
$256$	−28.4314	−1.77696
$257$	25.1227	1.56711	0.783557	−	0.621320i	$-0.213403\pi$
$257$	25.1227	1.56711	0.783557	+	0.621320i	$0.213403\pi$
$258$	−0.477086	−0.0297021
$259$	−32.0570	−1.99192
$260$	0	0
$261$	−3.60066	−0.222875
$262$	−37.4311	−2.31250
$263$	13.8290	0.852735	0.426368	−	0.904550i	$-0.359793\pi$
$263$	13.8290	0.852735	0.426368	+	0.904550i	$0.359793\pi$
$264$	0	0
$265$	0	0
$266$	26.8717	1.64761
$267$	−0.535874	−0.0327950
$268$	29.9574	1.82994
$269$	−8.55598	−0.521668	−0.260834	−	0.965384i	$-0.583997\pi$
$269$	−8.55598	−0.521668	−0.260834	+	0.965384i	$0.583997\pi$
$270$	0	0
$271$	−9.15169	−0.555925	−0.277963	−	0.960592i	$-0.589659\pi$
$271$	−9.15169	−0.555925	−0.277963	+	0.960592i	$0.589659\pi$
$272$	44.6963	2.71011
$273$	20.6961	1.25259
$274$	−21.9699	−1.32725
$275$	0	0
$276$	−13.9259	−0.838238
$277$	19.1550	1.15091	0.575455	−	0.817834i	$-0.304825\pi$
$277$	19.1550	1.15091	0.575455	+	0.817834i	$0.304825\pi$
$278$	9.51954	0.570944
$279$	−1.99558	−0.119472
$280$	0	0
$281$	16.5350	0.986398	0.493199	−	0.869916i	$-0.335827\pi$
$281$	16.5350	0.986398	0.493199	+	0.869916i	$0.335827\pi$
$282$	14.1969	0.845413
$283$	−20.1584	−1.19829	−0.599145	−	0.800640i	$-0.704493\pi$
$283$	−20.1584	−1.19829	−0.599145	+	0.800640i	$0.704493\pi$
$284$	−49.6900	−2.94856
$285$	0	0
$286$	0	0
$287$	−12.4126	−0.732695
$288$	−5.87507	−0.346192
$289$	20.2482	1.19107
$290$	0	0
$291$	11.0621	0.648469
$292$	−0.201492	−0.0117914
$293$	32.9126	1.92277	0.961387	−	0.275199i	$-0.0887438\pi$
$293$	32.9126	1.92277	0.961387	+	0.275199i	$0.0887438\pi$
$294$	−27.0413	−1.57708
$295$	0	0
$296$	48.9504	2.84518
$297$	0	0
$298$	−25.4200	−1.47254
$299$	−15.2320	−0.880888
$300$	0	0
$301$	0.784370	0.0452104
$302$	38.0744	2.19093
$303$	−6.07448	−0.348970
$304$	−18.3856	−1.05448
$305$	0	0
$306$	−15.5725	−0.890223
$307$	−25.6235	−1.46241	−0.731206	−	0.682157i	$-0.761042\pi$
$307$	−25.6235	−1.46241	−0.731206	+	0.682157i	$0.761042\pi$
$308$	0	0
$309$	7.96165	0.452922
$310$	0	0
$311$	−0.250716	−0.0142168	−0.00710840	−	0.999975i	$-0.502263\pi$
$311$	−0.250716	−0.0142168	−0.00710840	+	0.999975i	$0.502263\pi$
$312$	−31.6026	−1.78914
$313$	12.6406	0.714488	0.357244	−	0.934011i	$-0.383716\pi$
$313$	12.6406	0.714488	0.357244	+	0.934011i	$0.383716\pi$
$314$	−28.5386	−1.61053
$315$	0	0
$316$	56.5030	3.17854
$317$	25.7048	1.44373	0.721864	−	0.692035i	$-0.243286\pi$
$317$	25.7048	1.44373	0.721864	+	0.692035i	$0.243286\pi$
$318$	−15.8604	−0.889408
$319$	0	0
$320$	0	0
$321$	0.0438105	0.00244526
$322$	33.0473	1.84165
$323$	−15.3218	−0.852529
$324$	4.51049	0.250583
$325$	0	0
$326$	2.19193	0.121400
$327$	10.9879	0.607632
$328$	18.9539	1.04655
$329$	−23.3409	−1.28682
$330$	0	0
$331$	−30.2843	−1.66458	−0.832289	−	0.554342i	$-0.812970\pi$
$331$	−30.2843	−1.66458	−0.832289	+	0.554342i	$0.812970\pi$
$332$	29.1725	1.60105
$333$	−7.64174	−0.418765
$334$	48.1715	2.63583
$335$	0	0
$336$	30.7220	1.67602
$337$	18.0350	0.982428	0.491214	−	0.871039i	$-0.336553\pi$
$337$	18.0350	0.982428	0.491214	+	0.871039i	$0.336553\pi$
$338$	−28.9341	−1.57381
$339$	−14.3348	−0.778558
$340$	0	0
$341$	0	0
$342$	6.40567	0.346379
$343$	15.0933	0.814959
$344$	−1.19772	−0.0645767
$345$	0	0
$346$	25.0808	1.34835
$347$	−7.32846	−0.393412	−0.196706	−	0.980462i	$-0.563024\pi$
$347$	−7.32846	−0.393412	−0.196706	+	0.980462i	$0.563024\pi$
$348$	−16.2407	−0.870593
$349$	−22.0375	−1.17964	−0.589821	−	0.807534i	$-0.700802\pi$
$349$	−22.0375	−1.17964	−0.589821	+	0.807534i	$0.700802\pi$
$350$	0	0
$351$	4.93353	0.263332
$352$	0	0
$353$	−10.7984	−0.574739	−0.287370	−	0.957820i	$-0.592781\pi$
$353$	−10.7984	−0.574739	−0.287370	+	0.957820i	$0.592781\pi$
$354$	1.35385	0.0719561
$355$	0	0
$356$	−2.41705	−0.128104
$357$	25.6026	1.35503
$358$	−46.6958	−2.46795
$359$	3.56945	0.188389	0.0941943	−	0.995554i	$-0.469973\pi$
$359$	3.56945	0.188389	0.0941943	+	0.995554i	$0.469973\pi$
$360$	0	0
$361$	−12.6975	−0.668288
$362$	25.1182	1.32018
$363$	0	0
$364$	93.3495	4.89284
$365$	0	0
$366$	−20.7940	−1.08692
$367$	25.4696	1.32950	0.664752	−	0.747064i	$-0.268537\pi$
$367$	25.4696	1.32950	0.664752	+	0.747064i	$0.268537\pi$
$368$	−22.6109	−1.17867
$369$	−2.95892	−0.154035
$370$	0	0
$371$	26.0759	1.35379
$372$	−9.00104	−0.466682
$373$	−25.4291	−1.31667	−0.658334	−	0.752726i	$-0.728738\pi$
$373$	−25.4291	−1.31667	−0.658334	+	0.752726i	$0.728738\pi$
$374$	0	0
$375$	0	0
$376$	35.6411	1.83805
$377$	−17.7639	−0.914890
$378$	−10.7038	−0.550543
$379$	−13.0436	−0.670006	−0.335003	−	0.942217i	$-0.608737\pi$
$379$	−13.0436	−0.670006	−0.335003	+	0.942217i	$0.608737\pi$
$380$	0	0
$381$	14.2134	0.728175
$382$	−21.6594	−1.10819
$383$	−23.1354	−1.18216	−0.591081	−	0.806612i	$-0.701299\pi$
$383$	−23.1354	−1.18216	−0.591081	+	0.806612i	$0.701299\pi$
$384$	10.8734	0.554880
$385$	0	0
$386$	−69.5890	−3.54199
$387$	0.186978	0.00950462
$388$	49.8952	2.53305
$389$	−4.05710	−0.205703	−0.102852	−	0.994697i	$-0.532797\pi$
$389$	−4.05710	−0.205703	−0.102852	+	0.994697i	$0.532797\pi$
$390$	0	0
$391$	−18.8430	−0.952934
$392$	−67.8868	−3.42880
$393$	14.6699	0.739996
$394$	−55.0198	−2.77186
$395$	0	0
$396$	0	0
$397$	23.2206	1.16541	0.582705	−	0.812684i	$-0.301994\pi$
$397$	23.2206	1.16541	0.582705	+	0.812684i	$0.301994\pi$
$398$	−59.4311	−2.97901
$399$	−10.5315	−0.527232
$400$	0	0
$401$	2.24630	0.112175	0.0560874	−	0.998426i	$-0.482137\pi$
$401$	2.24630	0.112175	0.0560874	+	0.998426i	$0.482137\pi$
$402$	−16.9468	−0.845231
$403$	−9.84527	−0.490428
$404$	−27.3988	−1.36314
$405$	0	0
$406$	38.5406	1.91274
$407$	0	0
$408$	−39.0946	−1.93547
$409$	−31.7870	−1.57177	−0.785883	−	0.618376i	$-0.787791\pi$
$409$	−31.7870	−1.57177	−0.785883	+	0.618376i	$0.787791\pi$
$410$	0	0
$411$	8.61035	0.424717
$412$	35.9109	1.76920
$413$	−2.22584	−0.109526
$414$	7.87780	0.387173
$415$	0	0
$416$	−28.9849	−1.42110
$417$	−3.73086	−0.182701
$418$	0	0
$419$	−7.67926	−0.375156	−0.187578	−	0.982250i	$-0.560064\pi$
$419$	−7.67926	−0.375156	−0.187578	+	0.982250i	$0.560064\pi$
$420$	0	0
$421$	−21.4238	−1.04413	−0.522066	−	0.852905i	$-0.674839\pi$
$421$	−21.4238	−1.04413	−0.522066	+	0.852905i	$0.674839\pi$
$422$	−16.5654	−0.806390
$423$	−5.56399	−0.270531
$424$	−39.8174	−1.93370
$425$	0	0
$426$	28.1095	1.36191
$427$	34.1870	1.65443
$428$	0.197606	0.00955167
$429$	0	0
$430$	0	0
$431$	6.77583	0.326380	0.163190	−	0.986595i	$-0.447822\pi$
$431$	6.77583	0.326380	0.163190	+	0.986595i	$0.447822\pi$
$432$	7.32351	0.352352
$433$	−19.9857	−0.960451	−0.480225	−	0.877145i	$-0.659445\pi$
$433$	−19.9857	−0.960451	−0.480225	+	0.877145i	$0.659445\pi$
$434$	21.3603	1.02533
$435$	0	0
$436$	49.5608	2.37353
$437$	7.75097	0.370779
$438$	0.113983	0.00544633
$439$	34.2187	1.63317	0.816585	−	0.577225i	$-0.195864\pi$
$439$	34.2187	1.63317	0.816585	+	0.577225i	$0.195864\pi$
$440$	0	0
$441$	10.5979	0.504663
$442$	−76.8276	−3.65432
$443$	26.2148	1.24550	0.622751	−	0.782420i	$-0.286015\pi$
$443$	26.2148	1.24550	0.622751	+	0.782420i	$0.286015\pi$
$444$	−34.4679	−1.63578
$445$	0	0
$446$	15.7491	0.745742
$447$	9.96251	0.471211
$448$	1.44147	0.0681029
$449$	24.8045	1.17060	0.585299	−	0.810818i	$-0.300977\pi$
$449$	24.8045	1.17060	0.585299	+	0.810818i	$0.300977\pi$
$450$	0	0
$451$	0	0
$452$	−64.6568	−3.04120
$453$	−14.9220	−0.701095
$454$	40.4001	1.89607
$455$	0	0
$456$	16.0813	0.753077
$457$	−3.84218	−0.179730	−0.0898649	−	0.995954i	$-0.528644\pi$
$457$	−3.84218	−0.179730	−0.0898649	+	0.995954i	$0.528644\pi$
$458$	−47.4043	−2.21506
$459$	6.10313	0.284870
$460$	0	0
$461$	11.3262	0.527515	0.263758	−	0.964589i	$-0.415038\pi$
$461$	11.3262	0.527515	0.263758	+	0.964589i	$0.415038\pi$
$462$	0	0
$463$	16.3682	0.760694	0.380347	−	0.924844i	$-0.375805\pi$
$463$	16.3682	0.760694	0.380347	+	0.924844i	$0.375805\pi$
$464$	−26.3694	−1.22417
$465$	0	0
$466$	8.02130	0.371580
$467$	−11.9680	−0.553812	−0.276906	−	0.960897i	$-0.589309\pi$
$467$	−11.9680	−0.553812	−0.276906	+	0.960897i	$0.589309\pi$
$468$	22.2526	1.02863
$469$	27.8620	1.28655
$470$	0	0
$471$	11.1848	0.515366
$472$	3.39881	0.156443
$473$	0	0
$474$	−31.9636	−1.46813
$475$	0	0
$476$	115.480	5.29302
$477$	6.21596	0.284609
$478$	2.85209	0.130451
$479$	24.5071	1.11976	0.559879	−	0.828574i	$-0.310848\pi$
$479$	24.5071	1.11976	0.559879	+	0.828574i	$0.310848\pi$
$480$	0	0
$481$	−37.7007	−1.71901
$482$	36.8505	1.67850
$483$	−12.9518	−0.589326
$484$	0	0
$485$	0	0
$486$	−2.55157	−0.115741
$487$	−36.7735	−1.66637	−0.833184	−	0.552997i	$-0.813484\pi$
$487$	−36.7735	−1.66637	−0.833184	+	0.552997i	$0.813484\pi$
$488$	−52.2029	−2.36312
$489$	−0.859052	−0.0388477
$490$	0	0
$491$	−5.84440	−0.263754	−0.131877	−	0.991266i	$-0.542100\pi$
$491$	−5.84440	−0.263754	−0.131877	+	0.991266i	$0.542100\pi$
$492$	−13.3462	−0.601692
$493$	−21.9753	−0.989716
$494$	31.6026	1.42187
$495$	0	0
$496$	−14.6147	−0.656218
$497$	−46.2143	−2.07299
$498$	−16.5028	−0.739508
$499$	0.191431	0.00856965	0.00428482	−	0.999991i	$-0.498636\pi$
$499$	0.191431	0.00856965	0.00428482	+	0.999991i	$0.498636\pi$
$500$	0	0
$501$	−18.8792	−0.843460
$502$	0.745839	0.0332884
$503$	17.0017	0.758069	0.379035	−	0.925382i	$-0.376256\pi$
$503$	17.0017	0.758069	0.379035	+	0.925382i	$0.376256\pi$
$504$	−26.8717	−1.19696
$505$	0	0
$506$	0	0
$507$	11.3397	0.503615
$508$	64.1094	2.84439
$509$	8.97547	0.397831	0.198915	−	0.980017i	$-0.436258\pi$
$509$	8.97547	0.397831	0.198915	+	0.980017i	$0.436258\pi$
$510$	0	0
$511$	−0.187398	−0.00829001
$512$	50.7978	2.24497
$513$	−2.51049	−0.110841
$514$	−64.1023	−2.82743
$515$	0	0
$516$	0.843361	0.0371269
$517$	0	0
$518$	81.7955	3.59389
$519$	−9.82957	−0.431470
$520$	0	0
$521$	−17.5152	−0.767356	−0.383678	−	0.923467i	$-0.625343\pi$
$521$	−17.5152	−0.767356	−0.383678	+	0.923467i	$0.625343\pi$
$522$	9.18731	0.402118
$523$	−30.3122	−1.32546	−0.662729	−	0.748859i	$-0.730602\pi$
$523$	−30.3122	−1.32546	−0.662729	+	0.748859i	$0.730602\pi$
$524$	66.1682	2.89057
$525$	0	0
$526$	−35.2857	−1.53853
$527$	−12.1793	−0.530539
$528$	0	0
$529$	−13.4677	−0.585553
$530$	0	0
$531$	−0.530595	−0.0230258
$532$	−47.5020	−2.05947
$533$	−14.5979	−0.632306
$534$	1.36732	0.0591696
$535$	0	0
$536$	−42.5447	−1.83765
$537$	18.3008	0.789740
$538$	21.8312	0.941208
$539$	0	0
$540$	0	0
$541$	−25.8640	−1.11198	−0.555990	−	0.831189i	$-0.687661\pi$
$541$	−25.8640	−1.11198	−0.555990	+	0.831189i	$0.687661\pi$
$542$	23.3511	1.00302
$543$	−9.84422	−0.422456
$544$	−35.8563	−1.53733
$545$	0	0
$546$	−52.8075	−2.25995
$547$	4.74056	0.202692	0.101346	−	0.994851i	$-0.467685\pi$
$547$	4.74056	0.202692	0.101346	+	0.994851i	$0.467685\pi$
$548$	38.8369	1.65903
$549$	8.14949	0.347812
$550$	0	0
$551$	9.03939	0.385091
$552$	19.7771	0.841770
$553$	52.5508	2.23469
$554$	−48.8751	−2.07650
$555$	0	0
$556$	−16.8280	−0.713666
$557$	−29.3108	−1.24194	−0.620968	−	0.783836i	$-0.713261\pi$
$557$	−29.3108	−1.24194	−0.620968	+	0.783836i	$0.713261\pi$
$558$	5.09186	0.215556
$559$	0.922462	0.0390160
$560$	0	0
$561$	0	0
$562$	−42.1903	−1.77969
$563$	36.0731	1.52030	0.760150	−	0.649747i	$-0.225125\pi$
$563$	36.0731	1.52030	0.760150	+	0.649747i	$0.225125\pi$
$564$	−25.0963	−1.05675
$565$	0	0
$566$	51.4354	2.16199
$567$	4.19499	0.176173
$568$	70.5684	2.96098
$569$	−27.7834	−1.16474	−0.582370	−	0.812924i	$-0.697875\pi$
$569$	−27.7834	−1.16474	−0.582370	+	0.812924i	$0.697875\pi$
$570$	0	0
$571$	−17.4091	−0.728547	−0.364274	−	0.931292i	$-0.618683\pi$
$571$	−17.4091	−0.728547	−0.364274	+	0.931292i	$0.618683\pi$
$572$	0	0
$573$	8.48869	0.354620
$574$	31.6716	1.32195
$575$	0	0
$576$	0.343617	0.0143174
$577$	22.0946	0.919810	0.459905	−	0.887968i	$-0.347884\pi$
$577$	22.0946	0.919810	0.459905	+	0.887968i	$0.347884\pi$
$578$	−51.6646	−2.14896
$579$	27.2731	1.13343
$580$	0	0
$581$	27.1320	1.12562
$582$	−28.2255	−1.16999
$583$	0	0
$584$	0.286154	0.0118411
$585$	0	0
$586$	−83.9786	−3.46913
$587$	−1.59538	−0.0658482	−0.0329241	−	0.999458i	$-0.510482\pi$
$587$	−1.59538	−0.0658482	−0.0329241	+	0.999458i	$0.510482\pi$
$588$	47.8018	1.97131
$589$	5.00988	0.206428
$590$	0	0
$591$	21.5632	0.886990
$592$	−55.9643	−2.30012
$593$	30.8033	1.26494	0.632470	−	0.774584i	$-0.282041\pi$
$593$	30.8033	1.26494	0.632470	+	0.774584i	$0.282041\pi$
$594$	0	0
$595$	0	0
$596$	44.9358	1.84064
$597$	23.2920	0.953279
$598$	38.8654	1.58932
$599$	9.54306	0.389919	0.194959	−	0.980811i	$-0.437542\pi$
$599$	9.54306	0.389919	0.194959	+	0.980811i	$0.437542\pi$
$600$	0	0
$601$	−6.45339	−0.263239	−0.131620	−	0.991300i	$-0.542018\pi$
$601$	−6.45339	−0.263239	−0.131620	+	0.991300i	$0.542018\pi$
$602$	−2.00137	−0.0815698
$603$	6.64174	0.270472
$604$	−67.3053	−2.73861
$605$	0	0
$606$	15.4994	0.629621
$607$	−4.09050	−0.166028	−0.0830141	−	0.996548i	$-0.526455\pi$
$607$	−4.09050	−0.166028	−0.0830141	+	0.996548i	$0.526455\pi$
$608$	14.7493	0.598162
$609$	−15.1047	−0.612074
$610$	0	0
$611$	−27.4501	−1.11051
$612$	27.5281	1.11276
$613$	42.9770	1.73583	0.867913	−	0.496716i	$-0.165461\pi$
$613$	42.9770	1.73583	0.867913	+	0.496716i	$0.165461\pi$
$614$	65.3801	2.63853
$615$	0	0
$616$	0	0
$617$	−16.2829	−0.655526	−0.327763	−	0.944760i	$-0.606295\pi$
$617$	−16.2829	−0.655526	−0.327763	+	0.944760i	$0.606295\pi$
$618$	−20.3147	−0.817176
$619$	−10.4457	−0.419848	−0.209924	−	0.977718i	$-0.567322\pi$
$619$	−10.4457	−0.419848	−0.209924	+	0.977718i	$0.567322\pi$
$620$	0	0
$621$	−3.08744	−0.123895
$622$	0.639719	0.0256504
$623$	−2.24799	−0.0900636
$624$	36.1308	1.44639
$625$	0	0
$626$	−32.2533	−1.28910
$627$	0	0
$628$	50.4487	2.01312
$629$	−46.6385	−1.85960
$630$	0	0
$631$	−8.27018	−0.329231	−0.164615	−	0.986358i	$-0.552638\pi$
$631$	−8.27018	−0.329231	−0.164615	+	0.986358i	$0.552638\pi$
$632$	−80.2440	−3.19194
$633$	6.49225	0.258044
$634$	−65.5875	−2.60481
$635$	0	0
$636$	28.0370	1.11174
$637$	52.2852	2.07161
$638$	0	0
$639$	−11.0166	−0.435808
$640$	0	0
$641$	−18.6818	−0.737887	−0.368944	−	0.929452i	$-0.620280\pi$
$641$	−18.6818	−0.737887	−0.368944	+	0.929452i	$0.620280\pi$
$642$	−0.111785	−0.00441181
$643$	−18.6302	−0.734702	−0.367351	−	0.930082i	$-0.619735\pi$
$643$	−18.6302	−0.734702	−0.367351	+	0.930082i	$0.619735\pi$
$644$	−58.4188	−2.30202
$645$	0	0
$646$	39.0946	1.53816
$647$	−16.6330	−0.653911	−0.326956	−	0.945040i	$-0.606023\pi$
$647$	−16.6330	−0.653911	−0.326956	+	0.945040i	$0.606023\pi$
$648$	−6.40567	−0.251638
$649$	0	0
$650$	0	0
$651$	−8.37144	−0.328103
$652$	−3.87474	−0.151747
$653$	−33.7510	−1.32078	−0.660389	−	0.750924i	$-0.729609\pi$
$653$	−33.7510	−1.32078	−0.660389	+	0.750924i	$0.729609\pi$
$654$	−28.0363	−1.09631
$655$	0	0
$656$	−21.6697	−0.846059
$657$	−0.0446719	−0.00174282
$658$	59.5558	2.32173
$659$	41.7854	1.62773	0.813864	−	0.581055i	$-0.197360\pi$
$659$	41.7854	1.62773	0.813864	+	0.581055i	$0.197360\pi$
$660$	0	0
$661$	−37.7152	−1.46695	−0.733476	−	0.679715i	$-0.762104\pi$
$661$	−37.7152	−1.46695	−0.733476	+	0.679715i	$0.762104\pi$
$662$	77.2724	3.00328
$663$	30.1100	1.16937
$664$	−41.4300	−1.60780
$665$	0	0
$666$	19.4984	0.755547
$667$	11.1168	0.430444
$668$	−85.1543	−3.29472
$669$	−6.17233	−0.238636
$670$	0	0
$671$	0	0
$672$	−24.6459	−0.950735
$673$	3.94406	0.152032	0.0760161	−	0.997107i	$-0.475780\pi$
$673$	3.94406	0.152032	0.0760161	+	0.997107i	$0.475780\pi$
$674$	−46.0174	−1.77252
$675$	0	0
$676$	51.1477	1.96722
$677$	−24.7342	−0.950611	−0.475306	−	0.879821i	$-0.657662\pi$
$677$	−24.7342	−0.950611	−0.475306	+	0.879821i	$0.657662\pi$
$678$	36.5761	1.40470
$679$	46.4052	1.78087
$680$	0	0
$681$	−15.8335	−0.606740
$682$	0	0
$683$	−16.4724	−0.630298	−0.315149	−	0.949042i	$-0.602054\pi$
$683$	−16.4724	−0.630298	−0.315149	+	0.949042i	$0.602054\pi$
$684$	−11.3235	−0.432965
$685$	0	0
$686$	−38.5114	−1.47037
$687$	18.5785	0.708815
$688$	1.36933	0.0522054
$689$	30.6666	1.16831
$690$	0	0
$691$	−19.7812	−0.752512	−0.376256	−	0.926516i	$-0.622789\pi$
$691$	−19.7812	−0.752512	−0.376256	+	0.926516i	$0.622789\pi$
$692$	−44.3362	−1.68541
$693$	0	0
$694$	18.6990	0.709806
$695$	0	0
$696$	23.0646	0.874261
$697$	−18.0587	−0.684021
$698$	56.2302	2.12834
$699$	−3.14368	−0.118905
$700$	0	0
$701$	1.29453	0.0488936	0.0244468	−	0.999701i	$-0.492218\pi$
$701$	1.29453	0.0488936	0.0244468	+	0.999701i	$0.492218\pi$
$702$	−12.5882	−0.475112
$703$	19.1845	0.723556
$704$	0	0
$705$	0	0
$706$	27.5528	1.03696
$707$	−25.4824	−0.958363
$708$	−2.39324	−0.0899434
$709$	7.43927	0.279388	0.139694	−	0.990195i	$-0.455388\pi$
$709$	7.43927	0.279388	0.139694	+	0.990195i	$0.455388\pi$
$710$	0	0
$711$	12.5270	0.469801
$712$	3.43263	0.128643
$713$	6.16124	0.230740
$714$	−65.3266	−2.44479
$715$	0	0
$716$	82.5457	3.08488
$717$	−1.11778	−0.0417442
$718$	−9.10770	−0.339896
$719$	−38.3198	−1.42909	−0.714544	−	0.699590i	$-0.753366\pi$
$719$	−38.3198	−1.42909	−0.714544	+	0.699590i	$0.753366\pi$
$720$	0	0
$721$	33.3990	1.24384
$722$	32.3984	1.20574
$723$	−14.4423	−0.537116
$724$	−44.4022	−1.65020
$725$	0	0
$726$	0	0
$727$	−40.7149	−1.51003	−0.755016	−	0.655707i	$-0.772371\pi$
$727$	−40.7149	−1.51003	−0.755016	+	0.655707i	$0.772371\pi$
$728$	−132.572	−4.91346
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.14115	0.0422070
$732$	36.7582	1.35862
$733$	4.27733	0.157987	0.0789934	−	0.996875i	$-0.474829\pi$
$733$	4.27733	0.157987	0.0789934	+	0.996875i	$0.474829\pi$
$734$	−64.9874	−2.39873
$735$	0	0
$736$	18.1389	0.668610
$737$	0	0
$738$	7.54988	0.277915
$739$	−17.0467	−0.627073	−0.313536	−	0.949576i	$-0.601514\pi$
$739$	−17.0467	−0.627073	−0.313536	+	0.949576i	$0.601514\pi$
$740$	0	0
$741$	−12.3856	−0.454995
$742$	−66.5343	−2.44255
$743$	−27.7262	−1.01718	−0.508589	−	0.861010i	$-0.669833\pi$
$743$	−27.7262	−1.01718	−0.508589	+	0.861010i	$0.669833\pi$
$744$	12.7830	0.468649
$745$	0	0
$746$	64.8839	2.37557
$747$	6.46772	0.236641
$748$	0	0
$749$	0.183784	0.00671533
$750$	0	0
$751$	−21.2301	−0.774698	−0.387349	−	0.921933i	$-0.626609\pi$
$751$	−21.2301	−0.774698	−0.387349	+	0.921933i	$0.626609\pi$
$752$	−40.7479	−1.48592
$753$	−0.292307	−0.0106522
$754$	45.3259	1.65067
$755$	0	0
$756$	18.9214	0.688166
$757$	−1.21898	−0.0443047	−0.0221523	−	0.999755i	$-0.507052\pi$
$757$	−1.21898	−0.0443047	−0.0221523	+	0.999755i	$0.507052\pi$
$758$	33.2817	1.20884
$759$	0	0
$760$	0	0
$761$	−30.3177	−1.09902	−0.549508	−	0.835489i	$-0.685185\pi$
$761$	−30.3177	−1.09902	−0.549508	+	0.835489i	$0.685185\pi$
$762$	−36.2664	−1.31379
$763$	46.0941	1.66872
$764$	38.2881	1.38522
$765$	0	0
$766$	59.0314	2.13289
$767$	−2.61770	−0.0945198
$768$	−28.4314	−1.02593
$769$	7.60170	0.274124	0.137062	−	0.990562i	$-0.456234\pi$
$769$	7.60170	0.274124	0.137062	+	0.990562i	$0.456234\pi$
$770$	0	0
$771$	25.1227	0.904773
$772$	123.015	4.42740
$773$	−40.8530	−1.46938	−0.734690	−	0.678403i	$-0.762672\pi$
$773$	−40.8530	−1.46938	−0.734690	+	0.678403i	$0.762672\pi$
$774$	−0.477086	−0.0171485
$775$	0	0
$776$	−70.8598	−2.54372
$777$	−32.0570	−1.15004
$778$	10.3520	0.371136
$779$	7.42833	0.266147
$780$	0	0
$781$	0	0
$782$	48.0793	1.71931
$783$	−3.60066	−0.128677
$784$	77.6140	2.77193
$785$	0	0
$786$	−37.4311	−1.33512
$787$	23.7941	0.848168	0.424084	−	0.905623i	$-0.360596\pi$
$787$	23.7941	0.848168	0.424084	+	0.905623i	$0.360596\pi$
$788$	97.2604	3.46476
$789$	13.8290	0.492327
$790$	0	0
$791$	−60.1342	−2.13813
$792$	0	0
$793$	40.2058	1.42775
$794$	−59.2489	−2.10267
$795$	0	0
$796$	105.058	3.72369
$797$	9.71675	0.344185	0.172092	−	0.985081i	$-0.444947\pi$
$797$	9.71675	0.344185	0.172092	+	0.985081i	$0.444947\pi$
$798$	26.8717	0.951248
$799$	−33.9578	−1.20134
$800$	0	0
$801$	−0.535874	−0.0189342
$802$	−5.73158	−0.202389
$803$	0	0
$804$	29.9574	1.05652
$805$	0	0
$806$	25.1208	0.884844
$807$	−8.55598	−0.301185
$808$	38.9111	1.36889
$809$	−4.86398	−0.171008	−0.0855042	−	0.996338i	$-0.527250\pi$
$809$	−4.86398	−0.171008	−0.0855042	+	0.996338i	$0.527250\pi$
$810$	0	0
$811$	−33.1229	−1.16310	−0.581552	−	0.813509i	$-0.697554\pi$
$811$	−33.1229	−1.16310	−0.581552	+	0.813509i	$0.697554\pi$
$812$	−68.1296	−2.39088
$813$	−9.15169	−0.320964
$814$	0	0
$815$	0	0
$816$	44.6963	1.56468
$817$	−0.469405	−0.0164224
$818$	81.1066	2.83582
$819$	20.6961	0.723180
$820$	0	0
$821$	19.0770	0.665793	0.332896	−	0.942963i	$-0.391974\pi$
$821$	19.0770	0.665793	0.332896	+	0.942963i	$0.391974\pi$
$822$	−21.9699	−0.766287
$823$	−40.3403	−1.40617	−0.703087	−	0.711104i	$-0.748195\pi$
$823$	−40.3403	−1.40617	−0.703087	+	0.711104i	$0.748195\pi$
$824$	−50.9997	−1.77666
$825$	0	0
$826$	5.67937	0.197611
$827$	−29.9059	−1.03993	−0.519965	−	0.854187i	$-0.674055\pi$
$827$	−29.9059	−1.03993	−0.519965	+	0.854187i	$0.674055\pi$
$828$	−13.9259	−0.483957
$829$	51.6807	1.79494	0.897472	−	0.441072i	$-0.145402\pi$
$829$	51.6807	1.79494	0.897472	+	0.441072i	$0.145402\pi$
$830$	0	0
$831$	19.1550	0.664478
$832$	1.69524	0.0587720
$833$	64.6805	2.24105
$834$	9.51954	0.329635
$835$	0	0
$836$	0	0
$837$	−1.99558	−0.0689774
$838$	19.5941	0.676868
$839$	9.07332	0.313246	0.156623	−	0.987658i	$-0.449939\pi$
$839$	9.07332	0.313246	0.156623	+	0.987658i	$0.449939\pi$
$840$	0	0
$841$	−16.0353	−0.552941
$842$	54.6642	1.88385
$843$	16.5350	0.569497
$844$	29.2832	1.00797
$845$	0	0
$846$	14.1969	0.488099
$847$	0	0
$848$	45.5226	1.56325
$849$	−20.1584	−0.691833
$850$	0	0
$851$	23.5934	0.808771
$852$	−49.6900	−1.70235
$853$	−15.6305	−0.535178	−0.267589	−	0.963533i	$-0.586227\pi$
$853$	−15.6305	−0.535178	−0.267589	+	0.963533i	$0.586227\pi$
$854$	−87.2304	−2.98496
$855$	0	0
$856$	−0.280635	−0.00959191
$857$	44.3092	1.51357	0.756787	−	0.653662i	$-0.226768\pi$
$857$	44.3092	1.51357	0.756787	+	0.653662i	$0.226768\pi$
$858$	0	0
$859$	−12.3194	−0.420332	−0.210166	−	0.977666i	$-0.567400\pi$
$859$	−12.3194	−0.420332	−0.210166	+	0.977666i	$0.567400\pi$
$860$	0	0
$861$	−12.4126	−0.423021
$862$	−17.2890	−0.588865
$863$	32.9021	1.12000	0.560000	−	0.828493i	$-0.310801\pi$
$863$	32.9021	1.12000	0.560000	+	0.828493i	$0.310801\pi$
$864$	−5.87507	−0.199874
$865$	0	0
$866$	50.9948	1.73287
$867$	20.2482	0.687665
$868$	−37.7593	−1.28163
$869$	0	0
$870$	0	0
$871$	32.7672	1.11027
$872$	−70.3848	−2.38353
$873$	11.0621	0.374394
$874$	−19.7771	−0.668971
$875$	0	0
$876$	−0.201492	−0.00680779
$877$	−36.9714	−1.24843	−0.624217	−	0.781251i	$-0.714582\pi$
$877$	−36.9714	−1.24843	−0.624217	+	0.781251i	$0.714582\pi$
$878$	−87.3113	−2.94661
$879$	32.9126	1.11011
$880$	0	0
$881$	30.9007	1.04107	0.520536	−	0.853840i	$-0.325732\pi$
$881$	30.9007	1.04107	0.520536	+	0.853840i	$0.325732\pi$
$882$	−27.0413	−0.910528
$883$	16.9777	0.571345	0.285672	−	0.958327i	$-0.407783\pi$
$883$	16.9777	0.571345	0.285672	+	0.958327i	$0.407783\pi$
$884$	135.811	4.56781
$885$	0	0
$886$	−66.8887	−2.24717
$887$	8.67618	0.291318	0.145659	−	0.989335i	$-0.453470\pi$
$887$	8.67618	0.291318	0.145659	+	0.989335i	$0.453470\pi$
$888$	48.9504	1.64267
$889$	59.6251	1.99976
$890$	0	0
$891$	0	0
$892$	−27.8402	−0.932159
$893$	13.9683	0.467432
$894$	−25.4200	−0.850172
$895$	0	0
$896$	45.6137	1.52385
$897$	−15.2320	−0.508581
$898$	−63.2903	−2.11203
$899$	7.18540	0.239647
$900$	0	0
$901$	37.9368	1.26386
$902$	0	0
$903$	0.784370	0.0261022
$904$	91.8238	3.05402
$905$	0	0
$906$	38.0744	1.26494
$907$	49.4081	1.64057	0.820284	−	0.571956i	$-0.193815\pi$
$907$	49.4081	1.64057	0.820284	+	0.571956i	$0.193815\pi$
$908$	−71.4166	−2.37004
$909$	−6.07448	−0.201478
$910$	0	0
$911$	−57.2632	−1.89722	−0.948608	−	0.316453i	$-0.897508\pi$
$911$	−57.2632	−1.89722	−0.948608	+	0.316453i	$0.897508\pi$
$912$	−18.3856	−0.608807
$913$	0	0
$914$	9.80358	0.324274
$915$	0	0
$916$	83.7982	2.76877
$917$	61.5399	2.03223
$918$	−15.5725	−0.513970
$919$	9.03446	0.298019	0.149010	−	0.988836i	$-0.452391\pi$
$919$	9.03446	0.298019	0.149010	+	0.988836i	$0.452391\pi$
$920$	0	0
$921$	−25.6235	−0.844324
$922$	−28.8996	−0.951759
$923$	−54.3505	−1.78897
$924$	0	0
$925$	0	0
$926$	−41.7645	−1.37247
$927$	7.96165	0.261495
$928$	21.1541	0.694418
$929$	18.8513	0.618490	0.309245	−	0.950982i	$-0.399924\pi$
$929$	18.8513	0.618490	0.309245	+	0.950982i	$0.399924\pi$
$930$	0	0
$931$	−26.6059	−0.871974
$932$	−14.1795	−0.464465
$933$	−0.250716	−0.00820808
$934$	30.5371	0.999203
$935$	0	0
$936$	−31.6026	−1.03296
$937$	54.0534	1.76585	0.882924	−	0.469515i	$-0.155571\pi$
$937$	54.0534	1.76585	0.882924	+	0.469515i	$0.155571\pi$
$938$	−71.0917	−2.32123
$939$	12.6406	0.412510
$940$	0	0
$941$	1.37058	0.0446797	0.0223398	−	0.999750i	$-0.492888\pi$
$941$	1.37058	0.0446797	0.0223398	+	0.999750i	$0.492888\pi$
$942$	−28.5386	−0.929839
$943$	9.13549	0.297492
$944$	−3.88581	−0.126472
$945$	0	0
$946$	0	0
$947$	56.2685	1.82848	0.914240	−	0.405172i	$-0.132788\pi$
$947$	56.2685	1.82848	0.914240	+	0.405172i	$0.132788\pi$
$948$	56.5030	1.83513
$949$	−0.220390	−0.00715417
$950$	0	0
$951$	25.7048	0.833536
$952$	−164.001	−5.31532
$953$	−15.3372	−0.496821	−0.248411	−	0.968655i	$-0.579908\pi$
$953$	−15.3372	−0.496821	−0.248411	+	0.968655i	$0.579908\pi$
$954$	−15.8604	−0.513500
$955$	0	0
$956$	−5.04172	−0.163061
$957$	0	0
$958$	−62.5315	−2.02030
$959$	36.1203	1.16639
$960$	0	0
$961$	−27.0177	−0.871537
$962$	96.1959	3.10148
$963$	0.0438105	0.00141177
$964$	−65.1419	−2.09808
$965$	0	0
$966$	33.0473	1.06328
$967$	42.0912	1.35356	0.676782	−	0.736184i	$-0.263374\pi$
$967$	42.0912	1.35356	0.676782	+	0.736184i	$0.263374\pi$
$968$	0	0
$969$	−15.3218	−0.492208
$970$	0	0
$971$	17.1979	0.551908	0.275954	−	0.961171i	$-0.411006\pi$
$971$	17.1979	0.551908	0.275954	+	0.961171i	$0.411006\pi$
$972$	4.51049	0.144674
$973$	−15.6509	−0.501746
$974$	93.8300	3.00651
$975$	0	0
$976$	59.6829	1.91040
$977$	−0.606761	−0.0194120	−0.00970600	−	0.999953i	$-0.503090\pi$
$977$	−0.606761	−0.0194120	−0.00970600	+	0.999953i	$0.503090\pi$
$978$	2.19193	0.0700901
$979$	0	0
$980$	0	0
$981$	10.9879	0.350817
$982$	14.9124	0.475873
$983$	32.5867	1.03935	0.519677	−	0.854363i	$-0.326052\pi$
$983$	32.5867	1.03935	0.519677	+	0.854363i	$0.326052\pi$
$984$	18.9539	0.604227
$985$	0	0
$986$	56.0713	1.78568
$987$	−23.3409	−0.742949
$988$	−55.8649	−1.77730
$989$	−0.577283	−0.0183565
$990$	0	0
$991$	−19.8486	−0.630512	−0.315256	−	0.949007i	$-0.602090\pi$
$991$	−19.8486	−0.630512	−0.315256	+	0.949007i	$0.602090\pi$
$992$	11.7242	0.372243
$993$	−30.2843	−0.961044
$994$	117.919	3.74016
$995$	0	0
$996$	29.1725	0.924368
$997$	−26.1241	−0.827359	−0.413679	−	0.910423i	$-0.635757\pi$
$997$	−26.1241	−0.827359	−0.413679	+	0.910423i	$0.635757\pi$
$998$	−0.488450	−0.0154616
$999$	−7.64174	−0.241774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cq.1.1		4
5.4	even	2		1815.2.a.v.1.4		4
11.3	even	5		825.2.n.i.526.1		8
11.4	even	5		825.2.n.i.676.1		8
11.10	odd	2		9075.2.a.dg.1.4		4
15.14	odd	2		5445.2.a.bk.1.1		4
55.3	odd	20		825.2.bx.g.724.4		16
55.4	even	10		165.2.m.b.16.2	✓	8
55.14	even	10		165.2.m.b.31.2	yes	8
55.37	odd	20		825.2.bx.g.49.4		16
55.47	odd	20		825.2.bx.g.724.1		16
55.48	odd	20		825.2.bx.g.49.1		16
55.54	odd	2		1815.2.a.r.1.1		4
165.14	odd	10		495.2.n.b.361.1		8
165.59	odd	10		495.2.n.b.181.1		8
165.164	even	2		5445.2.a.br.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.b.16.2	✓	8	55.4	even	10
165.2.m.b.31.2	yes	8	55.14	even	10
495.2.n.b.181.1		8	165.59	odd	10
495.2.n.b.361.1		8	165.14	odd	10
825.2.n.i.526.1		8	11.3	even	5
825.2.n.i.676.1		8	11.4	even	5
825.2.bx.g.49.1		16	55.48	odd	20
825.2.bx.g.49.4		16	55.37	odd	20
825.2.bx.g.724.1		16	55.47	odd	20
825.2.bx.g.724.4		16	55.3	odd	20
1815.2.a.r.1.1		4	55.54	odd	2
1815.2.a.v.1.4		4	5.4	even	2
5445.2.a.bk.1.1		4	15.14	odd	2
5445.2.a.br.1.4		4	165.164	even	2
9075.2.a.cq.1.1		4	1.1	even	1	trivial
9075.2.a.dg.1.4		4	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.55157 q^{2} +1.00000 q^{3} +4.51049 q^{4} -2.55157 q^{6} +4.19499 q^{7} -6.40567 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.55157 q^{2} +1.00000 q^{3} +4.51049 q^{4} -2.55157 q^{6} +4.19499 q^{7} -6.40567 q^{8} +1.00000 q^{9} +4.51049 q^{12} +4.93353 q^{13} -10.7038 q^{14} +7.32351 q^{16} +6.10313 q^{17} -2.55157 q^{18} -2.51049 q^{19} +4.19499 q^{21} -3.08744 q^{23} -6.40567 q^{24} -12.5882 q^{26} +1.00000 q^{27} +18.9214 q^{28} -3.60066 q^{29} -1.99558 q^{31} -5.87507 q^{32} -15.5725 q^{34} +4.51049 q^{36} -7.64174 q^{37} +6.40567 q^{38} +4.93353 q^{39} -2.95892 q^{41} -10.7038 q^{42} +0.186978 q^{43} +7.87780 q^{46} -5.56399 q^{47} +7.32351 q^{48} +10.5979 q^{49} +6.10313 q^{51} +22.2526 q^{52} +6.21596 q^{53} -2.55157 q^{54} -26.8717 q^{56} -2.51049 q^{57} +9.18731 q^{58} -0.530595 q^{59} +8.14949 q^{61} +5.09186 q^{62} +4.19499 q^{63} +0.343617 q^{64} +6.64174 q^{67} +27.5281 q^{68} -3.08744 q^{69} -11.0166 q^{71} -6.40567 q^{72} -0.0446719 q^{73} +19.4984 q^{74} -11.3235 q^{76} -12.5882 q^{78} +12.5270 q^{79} +1.00000 q^{81} +7.54988 q^{82} +6.46772 q^{83} +18.9214 q^{84} -0.477086 q^{86} -3.60066 q^{87} -0.535874 q^{89} +20.6961 q^{91} -13.9259 q^{92} -1.99558 q^{93} +14.1969 q^{94} -5.87507 q^{96} +11.0621 q^{97} -27.0413 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - q^2 + 4 * q^3 + 9 * q^4 - q^6 + 8 * q^7 - 3 * q^8 + 4 * q^9	\( 4 q - q^{2} + 4 q^{3} + 9 q^{4} - q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9} + 9 q^{12} + 15 q^{13} + 7 q^{14} + 7 q^{16} + 6 q^{17} - q^{18} - q^{19} + 8 q^{21} + q^{23} - 3 q^{24} - 18 q^{26} + 4 q^{27} + 31 q^{28} + 17 q^{29} + 15 q^{31} + 8 q^{32} - 35 q^{34} + 9 q^{36} + q^{37} + 3 q^{38} + 15 q^{39} - 12 q^{41} + 7 q^{42} + 14 q^{43} - 9 q^{46} - 14 q^{47} + 7 q^{48} + 20 q^{49} + 6 q^{51} + 39 q^{52} - 2 q^{53} - q^{54} - 12 q^{56} - q^{57} + 11 q^{58} - 11 q^{59} + q^{61} + 30 q^{62} + 8 q^{63} - 3 q^{64} - 5 q^{67} + 19 q^{68} + q^{69} - 3 q^{71} - 3 q^{72} + 45 q^{73} + 29 q^{74} - 23 q^{76} - 18 q^{78} + 4 q^{81} - 11 q^{82} - 15 q^{83} + 31 q^{84} - 10 q^{86} + 17 q^{87} + 2 q^{89} + 16 q^{91} - 34 q^{92} + 15 q^{93} - 29 q^{94} + 8 q^{96} + 26 q^{97} + q^{98}+O(q^{100}) \) 4 * q - q^2 + 4 * q^3 + 9 * q^4 - q^6 + 8 * q^7 - 3 * q^8 + 4 * q^9 + 9 * q^12 + 15 * q^13 + 7 * q^14 + 7 * q^16 + 6 * q^17 - q^18 - q^19 + 8 * q^21 + q^23 - 3 * q^24 - 18 * q^26 + 4 * q^27 + 31 * q^28 + 17 * q^29 + 15 * q^31 + 8 * q^32 - 35 * q^34 + 9 * q^36 + q^37 + 3 * q^38 + 15 * q^39 - 12 * q^41 + 7 * q^42 + 14 * q^43 - 9 * q^46 - 14 * q^47 + 7 * q^48 + 20 * q^49 + 6 * q^51 + 39 * q^52 - 2 * q^53 - q^54 - 12 * q^56 - q^57 + 11 * q^58 - 11 * q^59 + q^61 + 30 * q^62 + 8 * q^63 - 3 * q^64 - 5 * q^67 + 19 * q^68 + q^69 - 3 * q^71 - 3 * q^72 + 45 * q^73 + 29 * q^74 - 23 * q^76 - 18 * q^78 + 4 * q^81 - 11 * q^82 - 15 * q^83 + 31 * q^84 - 10 * q^86 + 17 * q^87 + 2 * q^89 + 16 * q^91 - 34 * q^92 + 15 * q^93 - 29 * q^94 + 8 * q^96 + 26 * q^97 + q^98

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