LMFDB - Embedded newform 9075.2.a.cm.1.4

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:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 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			1.4
Root			$2.09529$ 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)

$f(q)$	$=$	$q+1.09529 q^{2} -1.00000 q^{3} -0.800331 q^{4} -1.09529 q^{6} -0.705037 q^{7} -3.06719 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.09529 q^{2} -1.00000 q^{3} -0.800331 q^{4} -1.09529 q^{6} -0.705037 q^{7} -3.06719 q^{8} +1.00000 q^{9} +0.800331 q^{12} -4.71333 q^{13} -0.772223 q^{14} -1.75881 q^{16} -7.78051 q^{17} +1.09529 q^{18} +1.19967 q^{19} +0.705037 q^{21} +6.89318 q^{23} +3.06719 q^{24} -5.16248 q^{26} -1.00000 q^{27} +0.564263 q^{28} -1.32741 q^{29} -7.68126 q^{31} +4.20796 q^{32} -8.52195 q^{34} -0.800331 q^{36} -8.43763 q^{37} +1.31399 q^{38} +4.71333 q^{39} +0.232901 q^{41} +0.772223 q^{42} +7.32892 q^{43} +7.55006 q^{46} -8.32228 q^{47} +1.75881 q^{48} -6.50292 q^{49} +7.78051 q^{51} +3.77222 q^{52} -6.82332 q^{53} -1.09529 q^{54} +2.16248 q^{56} -1.19967 q^{57} -1.45390 q^{58} -3.54011 q^{59} +10.8719 q^{61} -8.41324 q^{62} -0.705037 q^{63} +8.12657 q^{64} +2.04036 q^{67} +6.22699 q^{68} -6.89318 q^{69} +0.670527 q^{71} -3.06719 q^{72} -5.00433 q^{73} -9.24168 q^{74} -0.960132 q^{76} +5.16248 q^{78} -2.28027 q^{79} +1.00000 q^{81} +0.255095 q^{82} -2.10999 q^{83} -0.564263 q^{84} +8.02732 q^{86} +1.32741 q^{87} -3.34722 q^{89} +3.32307 q^{91} -5.51683 q^{92} +7.68126 q^{93} -9.11534 q^{94} -4.20796 q^{96} -3.32228 q^{97} -7.12261 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 3 * q^2 - 4 * q^3 + q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 4 * q^9	$ 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9} - q^{12} - 7 q^{13} + 3 q^{14} - q^{16} - 10 q^{17} - 3 q^{18} + 9 q^{19} + 6 q^{21} + 3 q^{23} + 3 q^{24} - 4 q^{26} - 4 q^{27} + 7 q^{28} + 15 q^{29} - 13 q^{31} + 6 q^{32} - 3 q^{34} + q^{36} + 3 q^{37} - 15 q^{38} + 7 q^{39} + 22 q^{41} - 3 q^{42} + q^{46} + 2 q^{47} + q^{48} - 12 q^{49} + 10 q^{51} + 9 q^{52} - 10 q^{53} + 3 q^{54} - 8 q^{56} - 9 q^{57} - 39 q^{58} - 21 q^{59} + 11 q^{61} - 10 q^{62} - 6 q^{63} - 3 q^{64} - q^{67} - 3 q^{68} - 3 q^{69} - 13 q^{71} - 3 q^{72} - q^{73} - 11 q^{74} + 19 q^{76} + 4 q^{78} - 4 q^{79} + 4 q^{81} - 25 q^{82} - 3 q^{83} - 7 q^{84} - 15 q^{87} - 10 q^{89} + 12 q^{91} + 24 q^{92} + 13 q^{93} - 35 q^{94} - 6 q^{96} + 22 q^{97} + 11 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 - 4 * q^3 + q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 4 * q^9 - q^12 - 7 * q^13 + 3 * q^14 - q^16 - 10 * q^17 - 3 * q^18 + 9 * q^19 + 6 * q^21 + 3 * q^23 + 3 * q^24 - 4 * q^26 - 4 * q^27 + 7 * q^28 + 15 * q^29 - 13 * q^31 + 6 * q^32 - 3 * q^34 + q^36 + 3 * q^37 - 15 * q^38 + 7 * q^39 + 22 * q^41 - 3 * q^42 + q^46 + 2 * q^47 + q^48 - 12 * q^49 + 10 * q^51 + 9 * q^52 - 10 * q^53 + 3 * q^54 - 8 * q^56 - 9 * q^57 - 39 * q^58 - 21 * q^59 + 11 * q^61 - 10 * q^62 - 6 * q^63 - 3 * q^64 - q^67 - 3 * q^68 - 3 * q^69 - 13 * q^71 - 3 * q^72 - q^73 - 11 * q^74 + 19 * q^76 + 4 * q^78 - 4 * q^79 + 4 * q^81 - 25 * q^82 - 3 * q^83 - 7 * q^84 - 15 * q^87 - 10 * q^89 + 12 * q^91 + 24 * q^92 + 13 * q^93 - 35 * q^94 - 6 * q^96 + 22 * q^97 + 11 * 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 $

$2$	1.09529	0.774490	0.387245	−	0.921977i	$-0.373427\pi$
$2$	1.09529	0.774490	0.387245	+	0.921977i	$0.373427\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.800331	−0.400166
$5$	0	0
$5$	0	0
$6$	−1.09529	−0.447152
$7$	−0.705037	−0.266479	−0.133239	−	0.991084i	$-0.542538\pi$
$7$	−0.705037	−0.266479	−0.133239	+	0.991084i	$0.542538\pi$
$8$	−3.06719	−1.08441
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.800331	0.231036
$13$	−4.71333	−1.30724	−0.653621	−	0.756822i	$-0.726751\pi$
$13$	−4.71333	−1.30724	−0.653621	+	0.756822i	$0.726751\pi$
$14$	−0.772223	−0.206385
$15$	0	0
$16$	−1.75881	−0.439702
$17$	−7.78051	−1.88705	−0.943526	−	0.331299i	$-0.892513\pi$
$17$	−7.78051	−1.88705	−0.943526	+	0.331299i	$0.892513\pi$
$18$	1.09529	0.258163
$19$	1.19967	0.275223	0.137611	−	0.990486i	$-0.456057\pi$
$19$	1.19967	0.275223	0.137611	+	0.990486i	$0.456057\pi$
$20$	0	0
$21$	0.705037	0.153852
$22$	0	0
$23$	6.89318	1.43733	0.718664	−	0.695358i	$-0.244754\pi$
$23$	6.89318	1.43733	0.718664	+	0.695358i	$0.244754\pi$
$24$	3.06719	0.626087
$25$	0	0
$26$	−5.16248	−1.01245
$27$	−1.00000	−0.192450
$28$	0.564263	0.106636
$29$	−1.32741	−0.246493	−0.123246	−	0.992376i	$-0.539331\pi$
$29$	−1.32741	−0.246493	−0.123246	+	0.992376i	$0.539331\pi$
$30$	0	0
$31$	−7.68126	−1.37960	−0.689798	−	0.724002i	$-0.742301\pi$
$31$	−7.68126	−1.37960	−0.689798	+	0.724002i	$0.742301\pi$
$32$	4.20796	0.743869
$33$	0	0
$34$	−8.52195	−1.46150
$35$	0	0
$36$	−0.800331	−0.133389
$37$	−8.43763	−1.38714	−0.693569	−	0.720391i	$-0.743963\pi$
$37$	−8.43763	−1.38714	−0.693569	+	0.720391i	$0.743963\pi$
$38$	1.31399	0.213157
$39$	4.71333	0.754737
$40$	0	0
$41$	0.232901	0.0363730	0.0181865	−	0.999835i	$-0.494211\pi$
$41$	0.232901	0.0363730	0.0181865	+	0.999835i	$0.494211\pi$
$42$	0.772223	0.119157
$43$	7.32892	1.11765	0.558825	−	0.829286i	$-0.311252\pi$
$43$	7.32892	1.11765	0.558825	+	0.829286i	$0.311252\pi$
$44$	0	0
$45$	0	0
$46$	7.55006	1.11320
$47$	−8.32228	−1.21393	−0.606965	−	0.794729i	$-0.707613\pi$
$47$	−8.32228	−1.21393	−0.606965	+	0.794729i	$0.707613\pi$
$48$	1.75881	0.253862
$49$	−6.50292	−0.928989
$50$	0	0
$51$	7.78051	1.08949
$52$	3.77222	0.523113
$53$	−6.82332	−0.937254	−0.468627	−	0.883396i	$-0.655251\pi$
$53$	−6.82332	−0.937254	−0.468627	+	0.883396i	$0.655251\pi$
$54$	−1.09529	−0.149051
$55$	0	0
$56$	2.16248	0.288974
$57$	−1.19967	−0.158900
$58$	−1.45390	−0.190906
$59$	−3.54011	−0.460883	−0.230442	−	0.973086i	$-0.574017\pi$
$59$	−3.54011	−0.460883	−0.230442	+	0.973086i	$0.574017\pi$
$60$	0	0
$61$	10.8719	1.39200	0.695999	−	0.718043i	$-0.254962\pi$
$61$	10.8719	1.39200	0.695999	+	0.718043i	$0.254962\pi$
$62$	−8.41324	−1.06848
$63$	−0.705037	−0.0888263
$64$	8.12657	1.01582
$65$	0	0
$66$	0	0
$67$	2.04036	0.249269	0.124635	−	0.992203i	$-0.460224\pi$
$67$	2.04036	0.249269	0.124635	+	0.992203i	$0.460224\pi$
$68$	6.22699	0.755133
$69$	−6.89318	−0.829841
$70$	0	0
$71$	0.670527	0.0795769	0.0397884	−	0.999208i	$-0.487332\pi$
$71$	0.670527	0.0795769	0.0397884	+	0.999208i	$0.487332\pi$
$72$	−3.06719	−0.361471
$73$	−5.00433	−0.585713	−0.292856	−	0.956156i	$-0.594606\pi$
$73$	−5.00433	−0.585713	−0.292856	+	0.956156i	$0.594606\pi$
$74$	−9.24168	−1.07432
$75$	0	0
$76$	−0.960132	−0.110135
$77$	0	0
$78$	5.16248	0.584536
$79$	−2.28027	−0.256550	−0.128275	−	0.991739i	$-0.540944\pi$
$79$	−2.28027	−0.256550	−0.128275	+	0.991739i	$0.540944\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.255095	0.0281706
$83$	−2.10999	−0.231601	−0.115801	−	0.993272i	$-0.536943\pi$
$83$	−2.10999	−0.231601	−0.115801	+	0.993272i	$0.536943\pi$
$84$	−0.564263	−0.0615662
$85$	0	0
$86$	8.02732	0.865608
$87$	1.32741	0.142313
$88$	0	0
$89$	−3.34722	−0.354805	−0.177402	−	0.984138i	$-0.556769\pi$
$89$	−3.34722	−0.354805	−0.177402	+	0.984138i	$0.556769\pi$
$90$	0	0
$91$	3.32307	0.348353
$92$	−5.51683	−0.575169
$93$	7.68126	0.796510
$94$	−9.11534	−0.940176
$95$	0	0
$96$	−4.20796	−0.429473
$97$	−3.32228	−0.337327	−0.168663	−	0.985674i	$-0.553945\pi$
$97$	−3.32228	−0.337327	−0.168663	+	0.985674i	$0.553945\pi$
$98$	−7.12261	−0.719492
$99$	0	0
$100$	0	0
$101$	18.9218	1.88279	0.941394	−	0.337310i	$-0.109517\pi$
$101$	18.9218	1.88279	0.941394	+	0.337310i	$0.109517\pi$
$102$	8.52195	0.843799
$103$	18.0964	1.78309	0.891545	−	0.452932i	$-0.149622\pi$
$103$	18.0964	1.78309	0.891545	+	0.452932i	$0.149622\pi$
$104$	14.4567	1.41759
$105$	0	0
$106$	−7.47354	−0.725894
$107$	−6.11945	−0.591589	−0.295795	−	0.955252i	$-0.595584\pi$
$107$	−6.11945	−0.591589	−0.295795	+	0.955252i	$0.595584\pi$
$108$	0.800331	0.0770119
$109$	−9.03128	−0.865039	−0.432520	−	0.901625i	$-0.642375\pi$
$109$	−9.03128	−0.865039	−0.432520	+	0.901625i	$0.642375\pi$
$110$	0	0
$111$	8.43763	0.800864
$112$	1.24002	0.117171
$113$	−3.91684	−0.368466	−0.184233	−	0.982883i	$-0.558980\pi$
$113$	−3.91684	−0.368466	−0.184233	+	0.982883i	$0.558980\pi$
$114$	−1.31399	−0.123066
$115$	0	0
$116$	1.06236	0.0986380
$117$	−4.71333	−0.435747
$118$	−3.87746	−0.356949
$119$	5.48555	0.502860
$120$	0	0
$121$	0	0
$122$	11.9079	1.07809
$123$	−0.232901	−0.0210000
$124$	6.14755	0.552067
$125$	0	0
$126$	−0.772223	−0.0687951
$127$	10.8675	0.964336	0.482168	−	0.876079i	$-0.339849\pi$
$127$	10.8675	0.964336	0.482168	+	0.876079i	$0.339849\pi$
$128$	0.485063	0.0428739
$129$	−7.32892	−0.645275
$130$	0	0
$131$	11.7094	1.02305	0.511526	−	0.859268i	$-0.329080\pi$
$131$	11.7094	1.02305	0.511526	+	0.859268i	$0.329080\pi$
$132$	0	0
$133$	−0.845811	−0.0733411
$134$	2.23479	0.193056
$135$	0	0
$136$	23.8643	2.04635
$137$	−20.4302	−1.74547	−0.872735	−	0.488194i	$-0.837656\pi$
$137$	−20.4302	−1.74547	−0.872735	+	0.488194i	$0.837656\pi$
$138$	−7.55006	−0.642704
$139$	8.17100	0.693055	0.346528	−	0.938040i	$-0.387361\pi$
$139$	8.17100	0.693055	0.346528	+	0.938040i	$0.387361\pi$
$140$	0	0
$141$	8.32228	0.700862
$142$	0.734424	0.0616315
$143$	0	0
$144$	−1.75881	−0.146567
$145$	0	0
$146$	−5.48122	−0.453629
$147$	6.50292	0.536352
$148$	6.75289	0.555084
$149$	9.60217	0.786641	0.393320	−	0.919401i	$-0.371326\pi$
$149$	9.60217	0.786641	0.393320	+	0.919401i	$0.371326\pi$
$150$	0	0
$151$	−6.90776	−0.562146	−0.281073	−	0.959686i	$-0.590690\pi$
$151$	−6.90776	−0.562146	−0.281073	+	0.959686i	$0.590690\pi$
$152$	−3.67961	−0.298456
$153$	−7.78051	−0.629017
$154$	0	0
$155$	0	0
$156$	−3.77222	−0.302020
$157$	−11.3519	−0.905980	−0.452990	−	0.891516i	$-0.649643\pi$
$157$	−11.3519	−0.905980	−0.452990	+	0.891516i	$0.649643\pi$
$158$	−2.49757	−0.198696
$159$	6.82332	0.541124
$160$	0	0
$161$	−4.85995	−0.383018
$162$	1.09529	0.0860544
$163$	1.79253	0.140402	0.0702008	−	0.997533i	$-0.477636\pi$
$163$	1.79253	0.140402	0.0702008	+	0.997533i	$0.477636\pi$
$164$	−0.186398	−0.0145552
$165$	0	0
$166$	−2.31106	−0.179373
$167$	6.02450	0.466189	0.233095	−	0.972454i	$-0.425115\pi$
$167$	6.02450	0.466189	0.233095	+	0.972454i	$0.425115\pi$
$168$	−2.16248	−0.166839
$169$	9.21546	0.708882
$170$	0	0
$171$	1.19967	0.0917410
$172$	−5.86556	−0.447245
$173$	4.18674	0.318312	0.159156	−	0.987253i	$-0.449123\pi$
$173$	4.18674	0.318312	0.159156	+	0.987253i	$0.449123\pi$
$174$	1.45390	0.110220
$175$	0	0
$176$	0	0
$177$	3.54011	0.266091
$178$	−3.66619	−0.274793
$179$	4.85166	0.362630	0.181315	−	0.983425i	$-0.441965\pi$
$179$	4.85166	0.362630	0.181315	+	0.983425i	$0.441965\pi$
$180$	0	0
$181$	−23.0877	−1.71610	−0.858049	−	0.513569i	$-0.828323\pi$
$181$	−23.0877	−1.71610	−0.858049	+	0.513569i	$0.828323\pi$
$182$	3.63974	0.269795
$183$	−10.8719	−0.803670
$184$	−21.1427	−1.55866
$185$	0	0
$186$	8.41324	0.616889
$187$	0	0
$188$	6.66058	0.485773
$189$	0.705037	0.0512839
$190$	0	0
$191$	6.22571	0.450476	0.225238	−	0.974304i	$-0.427684\pi$
$191$	6.22571	0.450476	0.225238	+	0.974304i	$0.427684\pi$
$192$	−8.12657	−0.586485
$193$	19.0868	1.37390	0.686950	−	0.726705i	$-0.258949\pi$
$193$	19.0868	1.37390	0.686950	+	0.726705i	$0.258949\pi$
$194$	−3.63887	−0.261256
$195$	0	0
$196$	5.20449	0.371749
$197$	6.80056	0.484520	0.242260	−	0.970211i	$-0.422111\pi$
$197$	6.80056	0.484520	0.242260	+	0.970211i	$0.422111\pi$
$198$	0	0
$199$	21.2972	1.50972	0.754860	−	0.655886i	$-0.227705\pi$
$199$	21.2972	1.50972	0.754860	+	0.655886i	$0.227705\pi$
$200$	0	0
$201$	−2.04036	−0.143916
$202$	20.7249	1.45820
$203$	0.935870	0.0656852
$204$	−6.22699	−0.435976
$205$	0	0
$206$	19.8209	1.38099
$207$	6.89318	0.479109
$208$	8.28984	0.574797
$209$	0	0
$210$	0	0
$211$	8.89073	0.612063	0.306032	−	0.952021i	$-0.400999\pi$
$211$	8.89073	0.612063	0.306032	+	0.952021i	$0.400999\pi$
$212$	5.46091	0.375057
$213$	−0.670527	−0.0459437
$214$	−6.70259	−0.458180
$215$	0	0
$216$	3.06719	0.208696
$217$	5.41558	0.367633
$218$	−9.89190	−0.669964
$219$	5.00433	0.338162
$220$	0	0
$221$	36.6721	2.46683
$222$	9.24168	0.620261
$223$	−5.41720	−0.362762	−0.181381	−	0.983413i	$-0.558057\pi$
$223$	−5.41720	−0.362762	−0.181381	+	0.983413i	$0.558057\pi$
$224$	−2.96677	−0.198226
$225$	0	0
$226$	−4.29009	−0.285373
$227$	−8.43842	−0.560077	−0.280039	−	0.959989i	$-0.590347\pi$
$227$	−8.43842	−0.560077	−0.280039	+	0.959989i	$0.590347\pi$
$228$	0.960132	0.0635863
$229$	−11.2053	−0.740466	−0.370233	−	0.928939i	$-0.620722\pi$
$229$	−11.2053	−0.740466	−0.370233	+	0.928939i	$0.620722\pi$
$230$	0	0
$231$	0	0
$232$	4.07140	0.267300
$233$	5.83979	0.382577	0.191289	−	0.981534i	$-0.438733\pi$
$233$	5.83979	0.382577	0.191289	+	0.981534i	$0.438733\pi$
$234$	−5.16248	−0.337482
$235$	0	0
$236$	2.83326	0.184430
$237$	2.28027	0.148119
$238$	6.00829	0.389460
$239$	16.5261	1.06899	0.534494	−	0.845173i	$-0.320502\pi$
$239$	16.5261	1.06899	0.534494	+	0.845173i	$0.320502\pi$
$240$	0	0
$241$	3.29180	0.212043	0.106022	−	0.994364i	$-0.466189\pi$
$241$	3.29180	0.212043	0.106022	+	0.994364i	$0.466189\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−8.70108	−0.557030
$245$	0	0
$246$	−0.255095	−0.0162643
$247$	−5.65443	−0.359783
$248$	23.5599	1.49605
$249$	2.10999	0.133715
$250$	0	0
$251$	−7.39934	−0.467042	−0.233521	−	0.972352i	$-0.575025\pi$
$251$	−7.39934	−0.467042	−0.233521	+	0.972352i	$0.575025\pi$
$252$	0.564263	0.0355452
$253$	0	0
$254$	11.9031	0.746869
$255$	0	0
$256$	−15.7219	−0.982616
$257$	18.6991	1.16642	0.583210	−	0.812322i	$-0.301797\pi$
$257$	18.6991	1.16642	0.583210	+	0.812322i	$0.301797\pi$
$258$	−8.02732	−0.499759
$259$	5.94884	0.369643
$260$	0	0
$261$	−1.32741	−0.0821643
$262$	12.8252	0.792344
$263$	3.99020	0.246046	0.123023	−	0.992404i	$-0.460741\pi$
$263$	3.99020	0.246046	0.123023	+	0.992404i	$0.460741\pi$
$264$	0	0
$265$	0	0
$266$	−0.926412	−0.0568020
$267$	3.34722	0.204847
$268$	−1.63296	−0.0997489
$269$	−12.7150	−0.775246	−0.387623	−	0.921818i	$-0.626704\pi$
$269$	−12.7150	−0.775246	−0.387623	+	0.921818i	$0.626704\pi$
$270$	0	0
$271$	−23.8280	−1.44745	−0.723724	−	0.690090i	$-0.757571\pi$
$271$	−23.8280	−1.44745	−0.723724	+	0.690090i	$0.757571\pi$
$272$	13.6844	0.829740
$273$	−3.32307	−0.201121
$274$	−22.3771	−1.35185
$275$	0	0
$276$	5.51683	0.332074
$277$	7.41252	0.445375	0.222688	−	0.974890i	$-0.428517\pi$
$277$	7.41252	0.445375	0.222688	+	0.974890i	$0.428517\pi$
$278$	8.94965	0.536764
$279$	−7.68126	−0.459865
$280$	0	0
$281$	29.0815	1.73485	0.867427	−	0.497564i	$-0.165772\pi$
$281$	29.0815	1.73485	0.867427	+	0.497564i	$0.165772\pi$
$282$	9.11534	0.542811
$283$	−21.6729	−1.28832	−0.644160	−	0.764891i	$-0.722793\pi$
$283$	−21.6729	−1.28832	−0.644160	+	0.764891i	$0.722793\pi$
$284$	−0.536643	−0.0318439
$285$	0	0
$286$	0	0
$287$	−0.164204	−0.00969265
$288$	4.20796	0.247956
$289$	43.5364	2.56096
$290$	0	0
$291$	3.32228	0.194756
$292$	4.00512	0.234382
$293$	−8.41220	−0.491446	−0.245723	−	0.969340i	$-0.579025\pi$
$293$	−8.41220	−0.491446	−0.245723	+	0.969340i	$0.579025\pi$
$294$	7.12261	0.415399
$295$	0	0
$296$	25.8798	1.50423
$297$	0	0
$298$	10.5172	0.609245
$299$	−32.4898	−1.87893
$300$	0	0
$301$	−5.16716	−0.297830
$302$	−7.56603	−0.435376
$303$	−18.9218	−1.08703
$304$	−2.10999	−0.121016
$305$	0	0
$306$	−8.52195	−0.487167
$307$	23.7431	1.35509	0.677545	−	0.735481i	$-0.263044\pi$
$307$	23.7431	1.35509	0.677545	+	0.735481i	$0.263044\pi$
$308$	0	0
$309$	−18.0964	−1.02947
$310$	0	0
$311$	23.0471	1.30688	0.653440	−	0.756979i	$-0.273325\pi$
$311$	23.0471	1.30688	0.653440	+	0.756979i	$0.273325\pi$
$312$	−14.4567	−0.818447
$313$	17.3638	0.981460	0.490730	−	0.871312i	$-0.336730\pi$
$313$	17.3638	0.981460	0.490730	+	0.871312i	$0.336730\pi$
$314$	−12.4337	−0.701673
$315$	0	0
$316$	1.82497	0.102663
$317$	6.15095	0.345472	0.172736	−	0.984968i	$-0.444739\pi$
$317$	6.15095	0.345472	0.172736	+	0.984968i	$0.444739\pi$
$318$	7.47354	0.419095
$319$	0	0
$320$	0	0
$321$	6.11945	0.341554
$322$	−5.32307	−0.296643
$323$	−9.33404	−0.519360
$324$	−0.800331	−0.0444628
$325$	0	0
$326$	1.96335	0.108740
$327$	9.03128	0.499431
$328$	−0.714351	−0.0394434
$329$	5.86752	0.323487
$330$	0	0
$331$	19.4191	1.06737	0.533685	−	0.845683i	$-0.320807\pi$
$331$	19.4191	1.06737	0.533685	+	0.845683i	$0.320807\pi$
$332$	1.68869	0.0926788
$333$	−8.43763	−0.462379
$334$	6.59859	0.361059
$335$	0	0
$336$	−1.24002	−0.0676489
$337$	31.6868	1.72609	0.863045	−	0.505127i	$-0.168554\pi$
$337$	31.6868	1.72609	0.863045	+	0.505127i	$0.168554\pi$
$338$	10.0936	0.549022
$339$	3.91684	0.212734
$340$	0	0
$341$	0	0
$342$	1.31399	0.0710524
$343$	9.52006	0.514035
$344$	−22.4791	−1.21199
$345$	0	0
$346$	4.58571	0.246529
$347$	−27.9434	−1.50008	−0.750041	−	0.661392i	$-0.769966\pi$
$347$	−27.9434	−1.50008	−0.750041	+	0.661392i	$0.769966\pi$
$348$	−1.06236	−0.0569487
$349$	0.683331	0.0365779	0.0182889	−	0.999833i	$-0.494178\pi$
$349$	0.683331	0.0365779	0.0182889	+	0.999833i	$0.494178\pi$
$350$	0	0
$351$	4.71333	0.251579
$352$	0	0
$353$	−1.55900	−0.0829769	−0.0414885	−	0.999139i	$-0.513210\pi$
$353$	−1.55900	−0.0829769	−0.0414885	+	0.999139i	$0.513210\pi$
$354$	3.87746	0.206085
$355$	0	0
$356$	2.67889	0.141981
$357$	−5.48555	−0.290326
$358$	5.31399	0.280853
$359$	−15.8404	−0.836022	−0.418011	−	0.908442i	$-0.637273\pi$
$359$	−15.8404	−0.836022	−0.418011	+	0.908442i	$0.637273\pi$
$360$	0	0
$361$	−17.5608	−0.924252
$362$	−25.2878	−1.32910
$363$	0	0
$364$	−2.65956	−0.139399
$365$	0	0
$366$	−11.9079	−0.622434
$367$	−15.7361	−0.821419	−0.410710	−	0.911766i	$-0.634719\pi$
$367$	−15.7361	−0.821419	−0.410710	+	0.911766i	$0.634719\pi$
$368$	−12.1238	−0.631996
$369$	0.232901	0.0121243
$370$	0	0
$371$	4.81069	0.249759
$372$	−6.14755	−0.318736
$373$	−5.27703	−0.273234	−0.136617	−	0.990624i	$-0.543623\pi$
$373$	−5.27703	−0.273234	−0.136617	+	0.990624i	$0.543623\pi$
$374$	0	0
$375$	0	0
$376$	25.5260	1.31640
$377$	6.25650	0.322226
$378$	0.772223	0.0397189
$379$	33.5093	1.72125	0.860627	−	0.509235i	$-0.170072\pi$
$379$	33.5093	1.72125	0.860627	+	0.509235i	$0.170072\pi$
$380$	0	0
$381$	−10.8675	−0.556760
$382$	6.81898	0.348889
$383$	20.2323	1.03382	0.516910	−	0.856039i	$-0.327082\pi$
$383$	20.2323	1.03382	0.516910	+	0.856039i	$0.327082\pi$
$384$	−0.485063	−0.0247532
$385$	0	0
$386$	20.9057	1.06407
$387$	7.32892	0.372550
$388$	2.65892	0.134986
$389$	−27.5729	−1.39800	−0.699000	−	0.715122i	$-0.746371\pi$
$389$	−27.5729	−1.39800	−0.699000	+	0.715122i	$0.746371\pi$
$390$	0	0
$391$	−53.6325	−2.71231
$392$	19.9457	1.00741
$393$	−11.7094	−0.590660
$394$	7.44862	0.375256
$395$	0	0
$396$	0	0
$397$	−11.7601	−0.590222	−0.295111	−	0.955463i	$-0.595357\pi$
$397$	−11.7601	−0.590222	−0.295111	+	0.955463i	$0.595357\pi$
$398$	23.3267	1.16926
$399$	0.845811	0.0423435
$400$	0	0
$401$	−7.72406	−0.385721	−0.192861	−	0.981226i	$-0.561777\pi$
$401$	−7.72406	−0.385721	−0.192861	+	0.981226i	$0.561777\pi$
$402$	−2.23479	−0.111461
$403$	36.2043	1.80347
$404$	−15.1437	−0.753427
$405$	0	0
$406$	1.02505	0.0508725
$407$	0	0
$408$	−23.8643	−1.18146
$409$	−16.7409	−0.827783	−0.413892	−	0.910326i	$-0.635831\pi$
$409$	−16.7409	−0.827783	−0.413892	+	0.910326i	$0.635831\pi$
$410$	0	0
$411$	20.4302	1.00775
$412$	−14.4831	−0.713531
$413$	2.49591	0.122816
$414$	7.55006	0.371065
$415$	0	0
$416$	−19.8335	−0.972417
$417$	−8.17100	−0.400136
$418$	0	0
$419$	−38.0968	−1.86115	−0.930576	−	0.366100i	$-0.880693\pi$
$419$	−38.0968	−1.86115	−0.930576	+	0.366100i	$0.880693\pi$
$420$	0	0
$421$	−22.6633	−1.10454	−0.552272	−	0.833664i	$-0.686239\pi$
$421$	−22.6633	−1.10454	−0.552272	+	0.833664i	$0.686239\pi$
$422$	9.73797	0.474037
$423$	−8.32228	−0.404643
$424$	20.9284	1.01637
$425$	0	0
$426$	−0.734424	−0.0355829
$427$	−7.66506	−0.370938
$428$	4.89758	0.236734
$429$	0	0
$430$	0	0
$431$	33.9766	1.63660	0.818299	−	0.574793i	$-0.194918\pi$
$431$	33.9766	1.63660	0.818299	+	0.574793i	$0.194918\pi$
$432$	1.75881	0.0846207
$433$	36.6753	1.76250	0.881251	−	0.472649i	$-0.156702\pi$
$433$	36.6753	1.76250	0.881251	+	0.472649i	$0.156702\pi$
$434$	5.93165	0.284728
$435$	0	0
$436$	7.22801	0.346159
$437$	8.26953	0.395585
$438$	5.48122	0.261903
$439$	−1.05012	−0.0501193	−0.0250596	−	0.999686i	$-0.507978\pi$
$439$	−1.05012	−0.0501193	−0.0250596	+	0.999686i	$0.507978\pi$
$440$	0	0
$441$	−6.50292	−0.309663
$442$	40.1667	1.91054
$443$	30.5206	1.45008	0.725039	−	0.688708i	$-0.241822\pi$
$443$	30.5206	1.45008	0.725039	+	0.688708i	$0.241822\pi$
$444$	−6.75289	−0.320478
$445$	0	0
$446$	−5.93342	−0.280956
$447$	−9.60217	−0.454167
$448$	−5.72953	−0.270695
$449$	−36.5695	−1.72582	−0.862910	−	0.505357i	$-0.831361\pi$
$449$	−36.5695	−1.72582	−0.862910	+	0.505357i	$0.831361\pi$
$450$	0	0
$451$	0	0
$452$	3.13477	0.147447
$453$	6.90776	0.324555
$454$	−9.24255	−0.433774
$455$	0	0
$456$	3.67961	0.172313
$457$	−2.38409	−0.111523	−0.0557615	−	0.998444i	$-0.517759\pi$
$457$	−2.38409	−0.111523	−0.0557615	+	0.998444i	$0.517759\pi$
$458$	−12.2731	−0.573483
$459$	7.78051	0.363163
$460$	0	0
$461$	−28.5962	−1.33186	−0.665929	−	0.746015i	$-0.731965\pi$
$461$	−28.5962	−1.33186	−0.665929	+	0.746015i	$0.731965\pi$
$462$	0	0
$463$	30.5806	1.42120	0.710600	−	0.703596i	$-0.248423\pi$
$463$	30.5806	1.42120	0.710600	+	0.703596i	$0.248423\pi$
$464$	2.33465	0.108383
$465$	0	0
$466$	6.39628	0.296302
$467$	3.74219	0.173168	0.0865840	−	0.996245i	$-0.472405\pi$
$467$	3.74219	0.173168	0.0865840	+	0.996245i	$0.472405\pi$
$468$	3.77222	0.174371
$469$	−1.43853	−0.0664250
$470$	0	0
$471$	11.3519	0.523068
$472$	10.8582	0.499788
$473$	0	0
$474$	2.49757	0.114717
$475$	0	0
$476$	−4.39026	−0.201227
$477$	−6.82332	−0.312418
$478$	18.1010	0.827920
$479$	0.944951	0.0431759	0.0215880	−	0.999767i	$-0.493128\pi$
$479$	0.944951	0.0431759	0.0215880	+	0.999767i	$0.493128\pi$
$480$	0	0
$481$	39.7693	1.81332
$482$	3.60548	0.164225
$483$	4.85995	0.221135
$484$	0	0
$485$	0	0
$486$	−1.09529	−0.0496835
$487$	13.3873	0.606638	0.303319	−	0.952889i	$-0.401905\pi$
$487$	13.3873	0.606638	0.303319	+	0.952889i	$0.401905\pi$
$488$	−33.3460	−1.50950
$489$	−1.79253	−0.0810609
$490$	0	0
$491$	2.78887	0.125860	0.0629300	−	0.998018i	$-0.479955\pi$
$491$	2.78887	0.125860	0.0629300	+	0.998018i	$0.479955\pi$
$492$	0.186398	0.00840347
$493$	10.3279	0.465145
$494$	−6.19327	−0.278648
$495$	0	0
$496$	13.5099	0.606611
$497$	−0.472746	−0.0212056
$498$	2.31106	0.103561
$499$	17.7790	0.795899	0.397950	−	0.917407i	$-0.369722\pi$
$499$	17.7790	0.795899	0.397950	+	0.917407i	$0.369722\pi$
$500$	0	0
$501$	−6.02450	−0.269155
$502$	−8.10445	−0.361719
$503$	25.7838	1.14964	0.574822	−	0.818279i	$-0.305071\pi$
$503$	25.7838	1.14964	0.574822	+	0.818279i	$0.305071\pi$
$504$	2.16248	0.0963245
$505$	0	0
$506$	0	0
$507$	−9.21546	−0.409273
$508$	−8.69761	−0.385894
$509$	−28.2301	−1.25128	−0.625639	−	0.780113i	$-0.715162\pi$
$509$	−28.2301	−1.25128	−0.625639	+	0.780113i	$0.715162\pi$
$510$	0	0
$511$	3.52824	0.156080
$512$	−18.1902	−0.803900
$513$	−1.19967	−0.0529667
$514$	20.4810	0.903380
$515$	0	0
$516$	5.86556	0.258217
$517$	0	0
$518$	6.51573	0.286285
$519$	−4.18674	−0.183778
$520$	0	0
$521$	11.6955	0.512388	0.256194	−	0.966625i	$-0.417531\pi$
$521$	11.6955	0.512388	0.256194	+	0.966625i	$0.417531\pi$
$522$	−1.45390	−0.0636354
$523$	−19.6871	−0.860857	−0.430429	−	0.902625i	$-0.641638\pi$
$523$	−19.6871	−0.860857	−0.430429	+	0.902625i	$0.641638\pi$
$524$	−9.37137	−0.409390
$525$	0	0
$526$	4.37044	0.190560
$527$	59.7642	2.60337
$528$	0	0
$529$	24.5159	1.06591
$530$	0	0
$531$	−3.54011	−0.153628
$532$	0.676929	0.0293486
$533$	−1.09774	−0.0475484
$534$	3.66619	0.158652
$535$	0	0
$536$	−6.25815	−0.270311
$537$	−4.85166	−0.209364
$538$	−13.9266	−0.600420
$539$	0	0
$540$	0	0
$541$	5.45092	0.234353	0.117177	−	0.993111i	$-0.462616\pi$
$541$	5.45092	0.234353	0.117177	+	0.993111i	$0.462616\pi$
$542$	−26.0987	−1.12103
$543$	23.0877	0.990789
$544$	−32.7401	−1.40372
$545$	0	0
$546$	−3.63974	−0.155766
$547$	12.7892	0.546827	0.273414	−	0.961897i	$-0.411847\pi$
$547$	12.7892	0.546827	0.273414	+	0.961897i	$0.411847\pi$
$548$	16.3509	0.698477
$549$	10.8719	0.463999
$550$	0	0
$551$	−1.59245	−0.0678405
$552$	21.1427	0.899891
$553$	1.60767	0.0683653
$554$	8.11889	0.344939
$555$	0	0
$556$	−6.53951	−0.277337
$557$	−0.354468	−0.0150193	−0.00750964	−	0.999972i	$-0.502390\pi$
$557$	−0.354468	−0.0150193	−0.00750964	+	0.999972i	$0.502390\pi$
$558$	−8.41324	−0.356161
$559$	−34.5436	−1.46104
$560$	0	0
$561$	0	0
$562$	31.8528	1.34363
$563$	35.0818	1.47852	0.739261	−	0.673419i	$-0.235175\pi$
$563$	35.0818	1.47852	0.739261	+	0.673419i	$0.235175\pi$
$564$	−6.66058	−0.280461
$565$	0	0
$566$	−23.7382	−0.997791
$567$	−0.705037	−0.0296088
$568$	−2.05663	−0.0862943
$569$	16.3179	0.684084	0.342042	−	0.939685i	$-0.388881\pi$
$569$	16.3179	0.684084	0.342042	+	0.939685i	$0.388881\pi$
$570$	0	0
$571$	14.4160	0.603291	0.301645	−	0.953420i	$-0.402464\pi$
$571$	14.4160	0.603291	0.301645	+	0.953420i	$0.402464\pi$
$572$	0	0
$573$	−6.22571	−0.260083
$574$	−0.179852	−0.00750686
$575$	0	0
$576$	8.12657	0.338607
$577$	−8.90863	−0.370871	−0.185435	−	0.982656i	$-0.559370\pi$
$577$	−8.90863	−0.370871	−0.185435	+	0.982656i	$0.559370\pi$
$578$	47.6852	1.98344
$579$	−19.0868	−0.793221
$580$	0	0
$581$	1.48762	0.0617168
$582$	3.63887	0.150836
$583$	0	0
$584$	15.3492	0.635155
$585$	0	0
$586$	−9.21383	−0.380620
$587$	13.8014	0.569644	0.284822	−	0.958580i	$-0.408066\pi$
$587$	13.8014	0.569644	0.284822	+	0.958580i	$0.408066\pi$
$588$	−5.20449	−0.214630
$589$	−9.21497	−0.379696
$590$	0	0
$591$	−6.80056	−0.279738
$592$	14.8402	0.609927
$593$	−16.4676	−0.676242	−0.338121	−	0.941103i	$-0.609791\pi$
$593$	−16.4676	−0.676242	−0.338121	+	0.941103i	$0.609791\pi$
$594$	0	0
$595$	0	0
$596$	−7.68492	−0.314787
$597$	−21.2972	−0.871638
$598$	−35.5859	−1.45522
$599$	37.8599	1.54692	0.773458	−	0.633848i	$-0.218525\pi$
$599$	37.8599	1.54692	0.773458	+	0.633848i	$0.218525\pi$
$600$	0	0
$601$	−1.56441	−0.0638135	−0.0319067	−	0.999491i	$-0.510158\pi$
$601$	−1.56441	−0.0638135	−0.0319067	+	0.999491i	$0.510158\pi$
$602$	−5.65956	−0.230666
$603$	2.04036	0.0830897
$604$	5.52850	0.224951
$605$	0	0
$606$	−20.7249	−0.841892
$607$	18.8523	0.765191	0.382595	−	0.923916i	$-0.375030\pi$
$607$	18.8523	0.765191	0.382595	+	0.923916i	$0.375030\pi$
$608$	5.04816	0.204730
$609$	−0.935870	−0.0379234
$610$	0	0
$611$	39.2256	1.58690
$612$	6.22699	0.251711
$613$	32.6700	1.31953	0.659765	−	0.751472i	$-0.270656\pi$
$613$	32.6700	1.31953	0.659765	+	0.751472i	$0.270656\pi$
$614$	26.0057	1.04950
$615$	0	0
$616$	0	0
$617$	33.6559	1.35493	0.677467	−	0.735553i	$-0.263078\pi$
$617$	33.6559	1.35493	0.677467	+	0.735553i	$0.263078\pi$
$618$	−19.8209	−0.797312
$619$	−8.94001	−0.359329	−0.179665	−	0.983728i	$-0.557501\pi$
$619$	−8.94001	−0.359329	−0.179665	+	0.983728i	$0.557501\pi$
$620$	0	0
$621$	−6.89318	−0.276614
$622$	25.2433	1.01216
$623$	2.35992	0.0945480
$624$	−8.28984	−0.331859
$625$	0	0
$626$	19.0185	0.760131
$627$	0	0
$628$	9.08528	0.362542
$629$	65.6491	2.61760
$630$	0	0
$631$	−19.2577	−0.766638	−0.383319	−	0.923616i	$-0.625219\pi$
$631$	−19.2577	−0.766638	−0.383319	+	0.923616i	$0.625219\pi$
$632$	6.99401	0.278207
$633$	−8.89073	−0.353375
$634$	6.73710	0.267565
$635$	0	0
$636$	−5.46091	−0.216539
$637$	30.6504	1.21441
$638$	0	0
$639$	0.670527	0.0265256
$640$	0	0
$641$	−14.8928	−0.588229	−0.294114	−	0.955770i	$-0.595025\pi$
$641$	−14.8928	−0.588229	−0.294114	+	0.955770i	$0.595025\pi$
$642$	6.70259	0.264530
$643$	−35.3193	−1.39286	−0.696429	−	0.717626i	$-0.745229\pi$
$643$	−35.3193	−1.39286	−0.696429	+	0.717626i	$0.745229\pi$
$644$	3.88957	0.153270
$645$	0	0
$646$	−10.2235	−0.402239
$647$	−31.1428	−1.22435	−0.612175	−	0.790722i	$-0.709705\pi$
$647$	−31.1428	−1.22435	−0.612175	+	0.790722i	$0.709705\pi$
$648$	−3.06719	−0.120490
$649$	0	0
$650$	0	0
$651$	−5.41558	−0.212253
$652$	−1.43462	−0.0561839
$653$	−5.05494	−0.197815	−0.0989075	−	0.995097i	$-0.531535\pi$
$653$	−5.05494	−0.197815	−0.0989075	+	0.995097i	$0.531535\pi$
$654$	9.89190	0.386804
$655$	0	0
$656$	−0.409628	−0.0159933
$657$	−5.00433	−0.195238
$658$	6.42666	0.250537
$659$	14.4486	0.562837	0.281419	−	0.959585i	$-0.409195\pi$
$659$	14.4486	0.562837	0.281419	+	0.959585i	$0.409195\pi$
$660$	0	0
$661$	14.8696	0.578361	0.289181	−	0.957275i	$-0.406617\pi$
$661$	14.8696	0.578361	0.289181	+	0.957275i	$0.406617\pi$
$662$	21.2696	0.826667
$663$	−36.6721	−1.42423
$664$	6.47172	0.251152
$665$	0	0
$666$	−9.24168	−0.358108
$667$	−9.15004	−0.354291
$668$	−4.82159	−0.186553
$669$	5.41720	0.209441
$670$	0	0
$671$	0	0
$672$	2.96677	0.114446
$673$	−41.6153	−1.60415	−0.802075	−	0.597223i	$-0.796271\pi$
$673$	−41.6153	−1.60415	−0.802075	+	0.597223i	$0.796271\pi$
$674$	34.7064	1.33684
$675$	0	0
$676$	−7.37542	−0.283670
$677$	44.2366	1.70015	0.850076	−	0.526660i	$-0.176556\pi$
$677$	44.2366	1.70015	0.850076	+	0.526660i	$0.176556\pi$
$678$	4.29009	0.164760
$679$	2.34233	0.0898904
$680$	0	0
$681$	8.43842	0.323361
$682$	0	0
$683$	−24.5318	−0.938683	−0.469342	−	0.883017i	$-0.655509\pi$
$683$	−24.5318	−0.938683	−0.469342	+	0.883017i	$0.655509\pi$
$684$	−0.960132	−0.0367116
$685$	0	0
$686$	10.4273	0.398115
$687$	11.2053	0.427508
$688$	−12.8902	−0.491433
$689$	32.1605	1.22522
$690$	0	0
$691$	23.7670	0.904139	0.452069	−	0.891983i	$-0.350686\pi$
$691$	23.7670	0.904139	0.452069	+	0.891983i	$0.350686\pi$
$692$	−3.35078	−0.127378
$693$	0	0
$694$	−30.6063	−1.16180
$695$	0	0
$696$	−4.07140	−0.154326
$697$	−1.81209	−0.0686378
$698$	0.748449	0.0283292
$699$	−5.83979	−0.220881
$700$	0	0
$701$	32.6939	1.23483	0.617416	−	0.786637i	$-0.288180\pi$
$701$	32.6939	1.23483	0.617416	+	0.786637i	$0.288180\pi$
$702$	5.16248	0.194845
$703$	−10.1224	−0.381772
$704$	0	0
$705$	0	0
$706$	−1.70756	−0.0642648
$707$	−13.3406	−0.501723
$708$	−2.83326	−0.106480
$709$	23.8264	0.894821	0.447410	−	0.894329i	$-0.352346\pi$
$709$	23.8264	0.894821	0.447410	+	0.894329i	$0.352346\pi$
$710$	0	0
$711$	−2.28027	−0.0855168
$712$	10.2666	0.384755
$713$	−52.9483	−1.98293
$714$	−6.00829	−0.224855
$715$	0	0
$716$	−3.88293	−0.145112
$717$	−16.5261	−0.617180
$718$	−17.3499	−0.647491
$719$	−2.84680	−0.106168	−0.0530838	−	0.998590i	$-0.516905\pi$
$719$	−2.84680	−0.106168	−0.0530838	+	0.998590i	$0.516905\pi$
$720$	0	0
$721$	−12.7586	−0.475156
$722$	−19.2342	−0.715824
$723$	−3.29180	−0.122423
$724$	18.4778	0.686723
$725$	0	0
$726$	0	0
$727$	13.5192	0.501399	0.250700	−	0.968065i	$-0.419339\pi$
$727$	13.5192	0.501399	0.250700	+	0.968065i	$0.419339\pi$
$728$	−10.1925	−0.377758
$729$	1.00000	0.0370370
$730$	0	0
$731$	−57.0227	−2.10906
$732$	8.70108	0.321601
$733$	32.8352	1.21280	0.606399	−	0.795161i	$-0.292614\pi$
$733$	32.8352	1.21280	0.606399	+	0.795161i	$0.292614\pi$
$734$	−17.2357	−0.636181
$735$	0	0
$736$	29.0062	1.06918
$737$	0	0
$738$	0.255095	0.00939018
$739$	−50.8927	−1.87212	−0.936058	−	0.351844i	$-0.885555\pi$
$739$	−50.8927	−1.87212	−0.936058	+	0.351844i	$0.885555\pi$
$740$	0	0
$741$	5.65443	0.207721
$742$	5.26912	0.193435
$743$	−15.4855	−0.568107	−0.284054	−	0.958808i	$-0.591679\pi$
$743$	−15.4855	−0.568107	−0.284054	+	0.958808i	$0.591679\pi$
$744$	−23.5599	−0.863746
$745$	0	0
$746$	−5.77990	−0.211617
$747$	−2.10999	−0.0772004
$748$	0	0
$749$	4.31444	0.157646
$750$	0	0
$751$	11.0575	0.403494	0.201747	−	0.979438i	$-0.435338\pi$
$751$	11.0575	0.403494	0.201747	+	0.979438i	$0.435338\pi$
$752$	14.6373	0.533767
$753$	7.39934	0.269647
$754$	6.85270	0.249561
$755$	0	0
$756$	−0.564263	−0.0205221
$757$	−9.00282	−0.327213	−0.163607	−	0.986526i	$-0.552313\pi$
$757$	−9.00282	−0.327213	−0.163607	+	0.986526i	$0.552313\pi$
$758$	36.7025	1.33309
$759$	0	0
$760$	0	0
$761$	−8.29644	−0.300746	−0.150373	−	0.988629i	$-0.548047\pi$
$761$	−8.29644	−0.300746	−0.150373	+	0.988629i	$0.548047\pi$
$762$	−11.9031	−0.431205
$763$	6.36738	0.230515
$764$	−4.98263	−0.180265
$765$	0	0
$766$	22.1603	0.800684
$767$	16.6857	0.602486
$768$	15.7219	0.567313
$769$	−2.89088	−0.104248	−0.0521239	−	0.998641i	$-0.516599\pi$
$769$	−2.89088	−0.104248	−0.0521239	+	0.998641i	$0.516599\pi$
$770$	0	0
$771$	−18.6991	−0.673432
$772$	−15.2758	−0.549787
$773$	27.8477	1.00161	0.500807	−	0.865559i	$-0.333037\pi$
$773$	27.8477	1.00161	0.500807	+	0.865559i	$0.333037\pi$
$774$	8.02732	0.288536
$775$	0	0
$776$	10.1901	0.365802
$777$	−5.94884	−0.213413
$778$	−30.2004	−1.08274
$779$	0.279404	0.0100107
$780$	0	0
$781$	0	0
$782$	−58.7433	−2.10066
$783$	1.32741	0.0474376
$784$	11.4374	0.408478
$785$	0	0
$786$	−12.8252	−0.457460
$787$	−24.8356	−0.885292	−0.442646	−	0.896696i	$-0.645960\pi$
$787$	−24.8356	−0.885292	−0.442646	+	0.896696i	$0.645960\pi$
$788$	−5.44270	−0.193888
$789$	−3.99020	−0.142055
$790$	0	0
$791$	2.76152	0.0981883
$792$	0	0
$793$	−51.2426	−1.81968
$794$	−12.8808	−0.457121
$795$	0	0
$796$	−17.0448	−0.604138
$797$	2.81107	0.0995731	0.0497866	−	0.998760i	$-0.484146\pi$
$797$	2.81107	0.0995731	0.0497866	+	0.998760i	$0.484146\pi$
$798$	0.926412	0.0327946
$799$	64.7516	2.29075
$800$	0	0
$801$	−3.34722	−0.118268
$802$	−8.46012	−0.298737
$803$	0	0
$804$	1.63296	0.0575901
$805$	0	0
$806$	39.6544	1.39677
$807$	12.7150	0.447589
$808$	−58.0366	−2.04172
$809$	−11.6766	−0.410526	−0.205263	−	0.978707i	$-0.565805\pi$
$809$	−11.6766	−0.410526	−0.205263	+	0.978707i	$0.565805\pi$
$810$	0	0
$811$	−24.0690	−0.845175	−0.422588	−	0.906322i	$-0.638878\pi$
$811$	−24.0690	−0.845175	−0.422588	+	0.906322i	$0.638878\pi$
$812$	−0.749006	−0.0262849
$813$	23.8280	0.835684
$814$	0	0
$815$	0	0
$816$	−13.6844	−0.479051
$817$	8.79227	0.307603
$818$	−18.3362	−0.641110
$819$	3.32307	0.116118
$820$	0	0
$821$	−4.85561	−0.169462	−0.0847310	−	0.996404i	$-0.527003\pi$
$821$	−4.85561	−0.169462	−0.0847310	+	0.996404i	$0.527003\pi$
$822$	22.3771	0.780490
$823$	−26.0601	−0.908397	−0.454198	−	0.890901i	$-0.650074\pi$
$823$	−26.0601	−0.908397	−0.454198	+	0.890901i	$0.650074\pi$
$824$	−55.5050	−1.93361
$825$	0	0
$826$	2.73376	0.0951195
$827$	−16.2843	−0.566260	−0.283130	−	0.959081i	$-0.591373\pi$
$827$	−16.2843	−0.566260	−0.283130	+	0.959081i	$0.591373\pi$
$828$	−5.51683	−0.191723
$829$	−10.0141	−0.347805	−0.173903	−	0.984763i	$-0.555638\pi$
$829$	−10.0141	−0.347805	−0.173903	+	0.984763i	$0.555638\pi$
$830$	0	0
$831$	−7.41252	−0.257137
$832$	−38.3032	−1.32792
$833$	50.5961	1.75305
$834$	−8.94965	−0.309901
$835$	0	0
$836$	0	0
$837$	7.68126	0.265503
$838$	−41.7272	−1.44144
$839$	11.7532	0.405766	0.202883	−	0.979203i	$-0.434969\pi$
$839$	11.7532	0.405766	0.202883	+	0.979203i	$0.434969\pi$
$840$	0	0
$841$	−27.2380	−0.939241
$842$	−24.8230	−0.855458
$843$	−29.0815	−1.00162
$844$	−7.11553	−0.244927
$845$	0	0
$846$	−9.11534	−0.313392
$847$	0	0
$848$	12.0009	0.412113
$849$	21.6729	0.743812
$850$	0	0
$851$	−58.1621	−1.99377
$852$	0.536643	0.0183851
$853$	−45.6353	−1.56252	−0.781262	−	0.624203i	$-0.785424\pi$
$853$	−45.6353	−1.56252	−0.781262	+	0.624203i	$0.785424\pi$
$854$	−8.39549	−0.287288
$855$	0	0
$856$	18.7695	0.641527
$857$	−8.41558	−0.287471	−0.143735	−	0.989616i	$-0.545911\pi$
$857$	−8.41558	−0.287471	−0.143735	+	0.989616i	$0.545911\pi$
$858$	0	0
$859$	−45.3009	−1.54565	−0.772823	−	0.634622i	$-0.781156\pi$
$859$	−45.3009	−1.54565	−0.772823	+	0.634622i	$0.781156\pi$
$860$	0	0
$861$	0.164204	0.00559606
$862$	37.2144	1.26753
$863$	12.0590	0.410493	0.205247	−	0.978710i	$-0.434200\pi$
$863$	12.0590	0.410493	0.205247	+	0.978710i	$0.434200\pi$
$864$	−4.20796	−0.143158
$865$	0	0
$866$	40.1702	1.36504
$867$	−43.5364	−1.47857
$868$	−4.33425	−0.147114
$869$	0	0
$870$	0	0
$871$	−9.61687	−0.325855
$872$	27.7006	0.938061
$873$	−3.32228	−0.112442
$874$	9.05757	0.306377
$875$	0	0
$876$	−4.00512	−0.135321
$877$	−21.1439	−0.713979	−0.356989	−	0.934108i	$-0.616197\pi$
$877$	−21.1439	−0.713979	−0.356989	+	0.934108i	$0.616197\pi$
$878$	−1.15019	−0.0388169
$879$	8.41220	0.283736
$880$	0	0
$881$	−13.1669	−0.443605	−0.221803	−	0.975092i	$-0.571194\pi$
$881$	−13.1669	−0.443605	−0.221803	+	0.975092i	$0.571194\pi$
$882$	−7.12261	−0.239831
$883$	34.7697	1.17009	0.585047	−	0.810999i	$-0.301076\pi$
$883$	34.7697	1.17009	0.585047	+	0.810999i	$0.301076\pi$
$884$	−29.3498	−0.987142
$885$	0	0
$886$	33.4290	1.12307
$887$	−20.9565	−0.703651	−0.351826	−	0.936066i	$-0.614439\pi$
$887$	−20.9565	−0.703651	−0.351826	+	0.936066i	$0.614439\pi$
$888$	−25.8798	−0.868468
$889$	−7.66200	−0.256975
$890$	0	0
$891$	0	0
$892$	4.33555	0.145165
$893$	−9.98398	−0.334101
$894$	−10.5172	−0.351748
$895$	0	0
$896$	−0.341987	−0.0114250
$897$	32.4898	1.08480
$898$	−40.0543	−1.33663
$899$	10.1961	0.340061
$900$	0	0
$901$	53.0889	1.76865
$902$	0	0
$903$	5.16716	0.171952
$904$	12.0137	0.399569
$905$	0	0
$906$	7.56603	0.251365
$907$	−26.2408	−0.871312	−0.435656	−	0.900113i	$-0.643484\pi$
$907$	−26.2408	−0.871312	−0.435656	+	0.900113i	$0.643484\pi$
$908$	6.75353	0.224124
$909$	18.9218	0.627596
$910$	0	0
$911$	−39.6599	−1.31399	−0.656995	−	0.753895i	$-0.728173\pi$
$911$	−39.6599	−1.31399	−0.656995	+	0.753895i	$0.728173\pi$
$912$	2.10999	0.0698687
$913$	0	0
$914$	−2.61128	−0.0863734
$915$	0	0
$916$	8.96793	0.296309
$917$	−8.25554	−0.272622
$918$	8.52195	0.281266
$919$	−14.9758	−0.494005	−0.247003	−	0.969015i	$-0.579446\pi$
$919$	−14.9758	−0.494005	−0.247003	+	0.969015i	$0.579446\pi$
$920$	0	0
$921$	−23.7431	−0.782361
$922$	−31.3213	−1.03151
$923$	−3.16041	−0.104026
$924$	0	0
$925$	0	0
$926$	33.4947	1.10071
$927$	18.0964	0.594364
$928$	−5.58567	−0.183359
$929$	−49.6461	−1.62884	−0.814418	−	0.580279i	$-0.802943\pi$
$929$	−49.6461	−1.62884	−0.814418	+	0.580279i	$0.802943\pi$
$930$	0	0
$931$	−7.80135	−0.255679
$932$	−4.67376	−0.153094
$933$	−23.0471	−0.754527
$934$	4.09880	0.134117
$935$	0	0
$936$	14.4567	0.472530
$937$	−54.2914	−1.77362	−0.886811	−	0.462132i	$-0.847085\pi$
$937$	−54.2914	−1.77362	−0.886811	+	0.462132i	$0.847085\pi$
$938$	−1.57561	−0.0514455
$939$	−17.3638	−0.566646
$940$	0	0
$941$	−6.66066	−0.217131	−0.108566	−	0.994089i	$-0.534626\pi$
$941$	−6.66066	−0.217131	−0.108566	+	0.994089i	$0.534626\pi$
$942$	12.4337	0.405111
$943$	1.60543	0.0522800
$944$	6.22638	0.202651
$945$	0	0
$946$	0	0
$947$	42.6250	1.38513	0.692563	−	0.721358i	$-0.256482\pi$
$947$	42.6250	1.38513	0.692563	+	0.721358i	$0.256482\pi$
$948$	−1.82497	−0.0592723
$949$	23.5871	0.765669
$950$	0	0
$951$	−6.15095	−0.199458
$952$	−16.8252	−0.545308
$953$	−16.7539	−0.542713	−0.271357	−	0.962479i	$-0.587472\pi$
$953$	−16.7539	−0.542713	−0.271357	+	0.962479i	$0.587472\pi$
$954$	−7.47354	−0.241965
$955$	0	0
$956$	−13.2264	−0.427772
$957$	0	0
$958$	1.03500	0.0334393
$959$	14.4040	0.465131
$960$	0	0
$961$	28.0018	0.903284
$962$	43.5591	1.40440
$963$	−6.11945	−0.197196
$964$	−2.63453	−0.0848524
$965$	0	0
$966$	5.32307	0.171267
$967$	−50.1233	−1.61186	−0.805928	−	0.592014i	$-0.798333\pi$
$967$	−50.1233	−1.61186	−0.805928	+	0.592014i	$0.798333\pi$
$968$	0	0
$969$	9.33404	0.299853
$970$	0	0
$971$	40.1230	1.28761	0.643804	−	0.765190i	$-0.277355\pi$
$971$	40.1230	1.28761	0.643804	+	0.765190i	$0.277355\pi$
$972$	0.800331	0.0256706
$973$	−5.76086	−0.184685
$974$	14.6631	0.469835
$975$	0	0
$976$	−19.1215	−0.612064
$977$	24.1682	0.773209	0.386604	−	0.922246i	$-0.373648\pi$
$977$	24.1682	0.773209	0.386604	+	0.922246i	$0.373648\pi$
$978$	−1.96335	−0.0627809
$979$	0	0
$980$	0	0
$981$	−9.03128	−0.288346
$982$	3.05464	0.0974774
$983$	−23.3021	−0.743220	−0.371610	−	0.928389i	$-0.621194\pi$
$983$	−23.3021	−0.743220	−0.371610	+	0.928389i	$0.621194\pi$
$984$	0.714351	0.0227727
$985$	0	0
$986$	11.3121	0.360250
$987$	−5.86752	−0.186765
$988$	4.52542	0.143973
$989$	50.5195	1.60643
$990$	0	0
$991$	−54.0689	−1.71756	−0.858778	−	0.512348i	$-0.828776\pi$
$991$	−54.0689	−1.71756	−0.858778	+	0.512348i	$0.828776\pi$
$992$	−32.3224	−1.02624
$993$	−19.4191	−0.616246
$994$	−0.517796	−0.0164235
$995$	0	0
$996$	−1.68869	−0.0535081
$997$	50.7109	1.60603	0.803015	−	0.595959i	$-0.203228\pi$
$997$	50.7109	1.60603	0.803015	+	0.595959i	$0.203228\pi$
$998$	19.4733	0.616416
$999$	8.43763	0.266955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cm.1.4		4
5.4	even	2		1815.2.a.w.1.1		4
11.7	odd	10		825.2.n.g.676.1		8
11.8	odd	10		825.2.n.g.526.1		8
11.10	odd	2		9075.2.a.di.1.1		4
15.14	odd	2		5445.2.a.bf.1.4		4
55.7	even	20		825.2.bx.f.49.3		16
55.8	even	20		825.2.bx.f.724.3		16
55.18	even	20		825.2.bx.f.49.2		16
55.19	odd	10		165.2.m.d.31.2	yes	8
55.29	odd	10		165.2.m.d.16.2	✓	8
55.52	even	20		825.2.bx.f.724.2		16
55.54	odd	2		1815.2.a.p.1.4		4
165.29	even	10		495.2.n.a.181.1		8
165.74	even	10		495.2.n.a.361.1		8
165.164	even	2		5445.2.a.bt.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.d.16.2	✓	8	55.29	odd	10
165.2.m.d.31.2	yes	8	55.19	odd	10
495.2.n.a.181.1		8	165.29	even	10
495.2.n.a.361.1		8	165.74	even	10
825.2.n.g.526.1		8	11.8	odd	10
825.2.n.g.676.1		8	11.7	odd	10
825.2.bx.f.49.2		16	55.18	even	20
825.2.bx.f.49.3		16	55.7	even	20
825.2.bx.f.724.2		16	55.52	even	20
825.2.bx.f.724.3		16	55.8	even	20
1815.2.a.p.1.4		4	55.54	odd	2
1815.2.a.w.1.1		4	5.4	even	2
5445.2.a.bf.1.4		4	15.14	odd	2
5445.2.a.bt.1.1		4	165.164	even	2
9075.2.a.cm.1.4		4	1.1	even	1	trivial
9075.2.a.di.1.1		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)

\(f(q)\)	\(=\)	\(q+1.09529 q^{2} -1.00000 q^{3} -0.800331 q^{4} -1.09529 q^{6} -0.705037 q^{7} -3.06719 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.09529 q^{2} -1.00000 q^{3} -0.800331 q^{4} -1.09529 q^{6} -0.705037 q^{7} -3.06719 q^{8} +1.00000 q^{9} +0.800331 q^{12} -4.71333 q^{13} -0.772223 q^{14} -1.75881 q^{16} -7.78051 q^{17} +1.09529 q^{18} +1.19967 q^{19} +0.705037 q^{21} +6.89318 q^{23} +3.06719 q^{24} -5.16248 q^{26} -1.00000 q^{27} +0.564263 q^{28} -1.32741 q^{29} -7.68126 q^{31} +4.20796 q^{32} -8.52195 q^{34} -0.800331 q^{36} -8.43763 q^{37} +1.31399 q^{38} +4.71333 q^{39} +0.232901 q^{41} +0.772223 q^{42} +7.32892 q^{43} +7.55006 q^{46} -8.32228 q^{47} +1.75881 q^{48} -6.50292 q^{49} +7.78051 q^{51} +3.77222 q^{52} -6.82332 q^{53} -1.09529 q^{54} +2.16248 q^{56} -1.19967 q^{57} -1.45390 q^{58} -3.54011 q^{59} +10.8719 q^{61} -8.41324 q^{62} -0.705037 q^{63} +8.12657 q^{64} +2.04036 q^{67} +6.22699 q^{68} -6.89318 q^{69} +0.670527 q^{71} -3.06719 q^{72} -5.00433 q^{73} -9.24168 q^{74} -0.960132 q^{76} +5.16248 q^{78} -2.28027 q^{79} +1.00000 q^{81} +0.255095 q^{82} -2.10999 q^{83} -0.564263 q^{84} +8.02732 q^{86} +1.32741 q^{87} -3.34722 q^{89} +3.32307 q^{91} -5.51683 q^{92} +7.68126 q^{93} -9.11534 q^{94} -4.20796 q^{96} -3.32228 q^{97} -7.12261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 3 * q^2 - 4 * q^3 + q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 4 * q^9	\( 4 q - 3 q^{2} - 4 q^{3} + q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 4 q^{9} - q^{12} - 7 q^{13} + 3 q^{14} - q^{16} - 10 q^{17} - 3 q^{18} + 9 q^{19} + 6 q^{21} + 3 q^{23} + 3 q^{24} - 4 q^{26} - 4 q^{27} + 7 q^{28} + 15 q^{29} - 13 q^{31} + 6 q^{32} - 3 q^{34} + q^{36} + 3 q^{37} - 15 q^{38} + 7 q^{39} + 22 q^{41} - 3 q^{42} + q^{46} + 2 q^{47} + q^{48} - 12 q^{49} + 10 q^{51} + 9 q^{52} - 10 q^{53} + 3 q^{54} - 8 q^{56} - 9 q^{57} - 39 q^{58} - 21 q^{59} + 11 q^{61} - 10 q^{62} - 6 q^{63} - 3 q^{64} - q^{67} - 3 q^{68} - 3 q^{69} - 13 q^{71} - 3 q^{72} - q^{73} - 11 q^{74} + 19 q^{76} + 4 q^{78} - 4 q^{79} + 4 q^{81} - 25 q^{82} - 3 q^{83} - 7 q^{84} - 15 q^{87} - 10 q^{89} + 12 q^{91} + 24 q^{92} + 13 q^{93} - 35 q^{94} - 6 q^{96} + 22 q^{97} + 11 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 - 4 * q^3 + q^4 + 3 * q^6 - 6 * q^7 - 3 * q^8 + 4 * q^9 - q^12 - 7 * q^13 + 3 * q^14 - q^16 - 10 * q^17 - 3 * q^18 + 9 * q^19 + 6 * q^21 + 3 * q^23 + 3 * q^24 - 4 * q^26 - 4 * q^27 + 7 * q^28 + 15 * q^29 - 13 * q^31 + 6 * q^32 - 3 * q^34 + q^36 + 3 * q^37 - 15 * q^38 + 7 * q^39 + 22 * q^41 - 3 * q^42 + q^46 + 2 * q^47 + q^48 - 12 * q^49 + 10 * q^51 + 9 * q^52 - 10 * q^53 + 3 * q^54 - 8 * q^56 - 9 * q^57 - 39 * q^58 - 21 * q^59 + 11 * q^61 - 10 * q^62 - 6 * q^63 - 3 * q^64 - q^67 - 3 * q^68 - 3 * q^69 - 13 * q^71 - 3 * q^72 - q^73 - 11 * q^74 + 19 * q^76 + 4 * q^78 - 4 * q^79 + 4 * q^81 - 25 * q^82 - 3 * q^83 - 7 * q^84 - 15 * q^87 - 10 * q^89 + 12 * q^91 + 24 * q^92 + 13 * q^93 - 35 * q^94 - 6 * q^96 + 22 * q^97 + 11 * q^98

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