LMFDB - Embedded newform 9075.2.a.ea.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} + \cdots - 5 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 18x^8 - 220x^7 + 61x^6 + 330x^5 - 145x^4 - 148x^3 + 71x^2 + 10*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.21395$ 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)

$f(q)$	$=$	$q-2.21395 q^{2} +1.00000 q^{3} +2.90157 q^{4} -2.21395 q^{6} +1.59339 q^{7} -1.99602 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.21395 q^{2} +1.00000 q^{3} +2.90157 q^{4} -2.21395 q^{6} +1.59339 q^{7} -1.99602 q^{8} +1.00000 q^{9} +2.90157 q^{12} +0.891576 q^{13} -3.52768 q^{14} -1.38404 q^{16} +4.43839 q^{17} -2.21395 q^{18} +5.84384 q^{19} +1.59339 q^{21} -3.27284 q^{23} -1.99602 q^{24} -1.97390 q^{26} +1.00000 q^{27} +4.62332 q^{28} -5.30235 q^{29} +8.91445 q^{31} +7.05624 q^{32} -9.82638 q^{34} +2.90157 q^{36} -5.53121 q^{37} -12.9380 q^{38} +0.891576 q^{39} +4.19014 q^{41} -3.52768 q^{42} +6.39321 q^{43} +7.24589 q^{46} +8.47681 q^{47} -1.38404 q^{48} -4.46111 q^{49} +4.43839 q^{51} +2.58697 q^{52} -4.38324 q^{53} -2.21395 q^{54} -3.18044 q^{56} +5.84384 q^{57} +11.7391 q^{58} -7.20613 q^{59} -2.03022 q^{61} -19.7361 q^{62} +1.59339 q^{63} -12.8541 q^{64} +13.6199 q^{67} +12.8783 q^{68} -3.27284 q^{69} +1.15336 q^{71} -1.99602 q^{72} +6.66040 q^{73} +12.2458 q^{74} +16.9563 q^{76} -1.97390 q^{78} +4.89466 q^{79} +1.00000 q^{81} -9.27676 q^{82} +13.2475 q^{83} +4.62332 q^{84} -14.1542 q^{86} -5.30235 q^{87} -8.06859 q^{89} +1.42063 q^{91} -9.49635 q^{92} +8.91445 q^{93} -18.7672 q^{94} +7.05624 q^{96} -10.7654 q^{97} +9.87667 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9	$ 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{12} + 18 q^{13} + 6 q^{14} + 24 q^{16} + 18 q^{17} + 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} + 12 q^{27} + 30 q^{28} + 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} + 18 q^{39} + 6 q^{42} + 32 q^{43} + 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} + 18 q^{51} + 48 q^{52} - 12 q^{53} + 4 q^{54} + 6 q^{56} + 16 q^{57} - 10 q^{58} + 20 q^{59} + 20 q^{61} + 20 q^{62} + 8 q^{63} - 6 q^{64} + 10 q^{67} + 26 q^{68} + 32 q^{71} + 12 q^{72} + 26 q^{73} - 68 q^{74} + 34 q^{76} + 16 q^{78} + 32 q^{79} + 12 q^{81} - 62 q^{82} + 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} - 2 q^{94} + 28 q^{96} - 22 q^{97} + 60 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9 + 12 * q^12 + 18 * q^13 + 6 * q^14 + 24 * q^16 + 18 * q^17 + 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 + 12 * q^27 + 30 * q^28 + 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 + 18 * q^39 + 6 * q^42 + 32 * q^43 + 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 + 18 * q^51 + 48 * q^52 - 12 * q^53 + 4 * q^54 + 6 * q^56 + 16 * q^57 - 10 * q^58 + 20 * q^59 + 20 * q^61 + 20 * q^62 + 8 * q^63 - 6 * q^64 + 10 * q^67 + 26 * q^68 + 32 * q^71 + 12 * q^72 + 26 * q^73 - 68 * q^74 + 34 * q^76 + 16 * q^78 + 32 * q^79 + 12 * q^81 - 62 * q^82 + 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 - 2 * q^94 + 28 * q^96 - 22 * q^97 + 60 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.21395	−1.56550	−0.782749	−	0.622338i	$-0.786183\pi$
$2$	−2.21395	−1.56550	−0.782749	+	0.622338i	$0.786183\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.90157	1.45078
$5$	0	0
$5$	0	0
$6$	−2.21395	−0.903841
$7$	1.59339	0.602244	0.301122	−	0.953586i	$-0.402639\pi$
$7$	1.59339	0.602244	0.301122	+	0.953586i	$0.402639\pi$
$8$	−1.99602	−0.705701
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.90157	0.837610
$13$	0.891576	0.247279	0.123639	−	0.992327i	$-0.460543\pi$
$13$	0.891576	0.247279	0.123639	+	0.992327i	$0.460543\pi$
$14$	−3.52768	−0.942812
$15$	0	0
$16$	−1.38404	−0.346010
$17$	4.43839	1.07647	0.538234	−	0.842795i	$-0.319092\pi$
$17$	4.43839	1.07647	0.538234	+	0.842795i	$0.319092\pi$
$18$	−2.21395	−0.521833
$19$	5.84384	1.34067	0.670335	−	0.742059i	$-0.266150\pi$
$19$	5.84384	1.34067	0.670335	+	0.742059i	$0.266150\pi$
$20$	0	0
$21$	1.59339	0.347706
$22$	0	0
$23$	−3.27284	−0.682433	−0.341217	−	0.939985i	$-0.610839\pi$
$23$	−3.27284	−0.682433	−0.341217	+	0.939985i	$0.610839\pi$
$24$	−1.99602	−0.407437
$25$	0	0
$26$	−1.97390	−0.387114
$27$	1.00000	0.192450
$28$	4.62332	0.873726
$29$	−5.30235	−0.984621	−0.492311	−	0.870420i	$-0.663848\pi$
$29$	−5.30235	−0.984621	−0.492311	+	0.870420i	$0.663848\pi$
$30$	0	0
$31$	8.91445	1.60108	0.800541	−	0.599278i	$-0.204546\pi$
$31$	8.91445	1.60108	0.800541	+	0.599278i	$0.204546\pi$
$32$	7.05624	1.24738
$33$	0	0
$34$	−9.82638	−1.68521
$35$	0	0
$36$	2.90157	0.483595
$37$	−5.53121	−0.909325	−0.454663	−	0.890664i	$-0.650240\pi$
$37$	−5.53121	−0.909325	−0.454663	+	0.890664i	$0.650240\pi$
$38$	−12.9380	−2.09882
$39$	0.891576	0.142766
$40$	0	0
$41$	4.19014	0.654391	0.327195	−	0.944957i	$-0.393896\pi$
$41$	4.19014	0.654391	0.327195	+	0.944957i	$0.393896\pi$
$42$	−3.52768	−0.544333
$43$	6.39321	0.974955	0.487477	−	0.873136i	$-0.337917\pi$
$43$	6.39321	0.974955	0.487477	+	0.873136i	$0.337917\pi$
$44$	0	0
$45$	0	0
$46$	7.24589	1.06835
$47$	8.47681	1.23647	0.618235	−	0.785993i	$-0.287848\pi$
$47$	8.47681	1.23647	0.618235	+	0.785993i	$0.287848\pi$
$48$	−1.38404	−0.199769
$49$	−4.46111	−0.637302
$50$	0	0
$51$	4.43839	0.621500
$52$	2.58697	0.358748
$53$	−4.38324	−0.602084	−0.301042	−	0.953611i	$-0.597334\pi$
$53$	−4.38324	−0.602084	−0.301042	+	0.953611i	$0.597334\pi$
$54$	−2.21395	−0.301280
$55$	0	0
$56$	−3.18044	−0.425004
$57$	5.84384	0.774036
$58$	11.7391	1.54142
$59$	−7.20613	−0.938159	−0.469079	−	0.883156i	$-0.655414\pi$
$59$	−7.20613	−0.938159	−0.469079	+	0.883156i	$0.655414\pi$
$60$	0	0
$61$	−2.03022	−0.259942	−0.129971	−	0.991518i	$-0.541488\pi$
$61$	−2.03022	−0.259942	−0.129971	+	0.991518i	$0.541488\pi$
$62$	−19.7361	−2.50649
$63$	1.59339	0.200748
$64$	−12.8541	−1.60676
$65$	0	0
$66$	0	0
$67$	13.6199	1.66393	0.831967	−	0.554825i	$-0.187215\pi$
$67$	13.6199	1.66393	0.831967	+	0.554825i	$0.187215\pi$
$68$	12.8783	1.56172
$69$	−3.27284	−0.394003
$70$	0	0
$71$	1.15336	0.136879	0.0684394	−	0.997655i	$-0.478198\pi$
$71$	1.15336	0.136879	0.0684394	+	0.997655i	$0.478198\pi$
$72$	−1.99602	−0.235234
$73$	6.66040	0.779541	0.389770	−	0.920912i	$-0.372554\pi$
$73$	6.66040	0.779541	0.389770	+	0.920912i	$0.372554\pi$
$74$	12.2458	1.42355
$75$	0	0
$76$	16.9563	1.94502
$77$	0	0
$78$	−1.97390	−0.223501
$79$	4.89466	0.550693	0.275346	−	0.961345i	$-0.411207\pi$
$79$	4.89466	0.550693	0.275346	+	0.961345i	$0.411207\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.27676	−1.02445
$83$	13.2475	1.45410	0.727050	−	0.686585i	$-0.240891\pi$
$83$	13.2475	1.45410	0.727050	+	0.686585i	$0.240891\pi$
$84$	4.62332	0.504446
$85$	0	0
$86$	−14.1542	−1.52629
$87$	−5.30235	−0.568471
$88$	0	0
$89$	−8.06859	−0.855268	−0.427634	−	0.903952i	$-0.640653\pi$
$89$	−8.06859	−0.855268	−0.427634	+	0.903952i	$0.640653\pi$
$90$	0	0
$91$	1.42063	0.148922
$92$	−9.49635	−0.990063
$93$	8.91445	0.924385
$94$	−18.7672	−1.93569
$95$	0	0
$96$	7.05624	0.720175
$97$	−10.7654	−1.09306	−0.546528	−	0.837441i	$-0.684051\pi$
$97$	−10.7654	−1.09306	−0.546528	+	0.837441i	$0.684051\pi$
$98$	9.87667	0.997695
$99$	0	0
$100$	0	0
$101$	−13.3647	−1.32984	−0.664920	−	0.746915i	$-0.731534\pi$
$101$	−13.3647	−1.32984	−0.664920	+	0.746915i	$0.731534\pi$
$102$	−9.82638	−0.972956
$103$	−4.36350	−0.429948	−0.214974	−	0.976620i	$-0.568967\pi$
$103$	−4.36350	−0.429948	−0.214974	+	0.976620i	$0.568967\pi$
$104$	−1.77961	−0.174505
$105$	0	0
$106$	9.70426	0.942561
$107$	20.6028	1.99175	0.995873	−	0.0907584i	$-0.0289291\pi$
$107$	20.6028	1.99175	0.995873	+	0.0907584i	$0.0289291\pi$
$108$	2.90157	0.279203
$109$	4.83053	0.462681	0.231340	−	0.972873i	$-0.425689\pi$
$109$	4.83053	0.462681	0.231340	+	0.972873i	$0.425689\pi$
$110$	0	0
$111$	−5.53121	−0.524999
$112$	−2.20532	−0.208383
$113$	−15.8292	−1.48908	−0.744542	−	0.667576i	$-0.767332\pi$
$113$	−15.8292	−1.48908	−0.744542	+	0.667576i	$0.767332\pi$
$114$	−12.9380	−1.21175
$115$	0	0
$116$	−15.3851	−1.42847
$117$	0.891576	0.0824263
$118$	15.9540	1.46869
$119$	7.07209	0.648297
$120$	0	0
$121$	0	0
$122$	4.49479	0.406939
$123$	4.19014	0.377813
$124$	25.8659	2.32282
$125$	0	0
$126$	−3.52768	−0.314271
$127$	9.04884	0.802955	0.401477	−	0.915869i	$-0.368497\pi$
$127$	9.04884	0.802955	0.401477	+	0.915869i	$0.368497\pi$
$128$	14.3458	1.26800
$129$	6.39321	0.562890
$130$	0	0
$131$	−9.87408	−0.862702	−0.431351	−	0.902184i	$-0.641963\pi$
$131$	−9.87408	−0.862702	−0.431351	+	0.902184i	$0.641963\pi$
$132$	0	0
$133$	9.31151	0.807411
$134$	−30.1537	−2.60488
$135$	0	0
$136$	−8.85914	−0.759665
$137$	11.6288	0.993518	0.496759	−	0.867888i	$-0.334523\pi$
$137$	11.6288	0.993518	0.496759	+	0.867888i	$0.334523\pi$
$138$	7.24589	0.616811
$139$	1.77693	0.150717	0.0753584	−	0.997157i	$-0.475990\pi$
$139$	1.77693	0.150717	0.0753584	+	0.997157i	$0.475990\pi$
$140$	0	0
$141$	8.47681	0.713876
$142$	−2.55348	−0.214283
$143$	0	0
$144$	−1.38404	−0.115337
$145$	0	0
$146$	−14.7458	−1.22037
$147$	−4.46111	−0.367946
$148$	−16.0492	−1.31923
$149$	−21.4976	−1.76115	−0.880574	−	0.473909i	$-0.842843\pi$
$149$	−21.4976	−1.76115	−0.880574	+	0.473909i	$0.842843\pi$
$150$	0	0
$151$	8.04208	0.654455	0.327228	−	0.944946i	$-0.393886\pi$
$151$	8.04208	0.654455	0.327228	+	0.944946i	$0.393886\pi$
$152$	−11.6645	−0.946112
$153$	4.43839	0.358823
$154$	0	0
$155$	0	0
$156$	2.58697	0.207123
$157$	17.0621	1.36170	0.680852	−	0.732421i	$-0.261610\pi$
$157$	17.0621	1.36170	0.680852	+	0.732421i	$0.261610\pi$
$158$	−10.8365	−0.862108
$159$	−4.38324	−0.347613
$160$	0	0
$161$	−5.21490	−0.410992
$162$	−2.21395	−0.173944
$163$	13.4702	1.05507	0.527535	−	0.849533i	$-0.323116\pi$
$163$	13.4702	1.05507	0.527535	+	0.849533i	$0.323116\pi$
$164$	12.1580	0.949379
$165$	0	0
$166$	−29.3292	−2.27639
$167$	6.50196	0.503136	0.251568	−	0.967840i	$-0.419054\pi$
$167$	6.50196	0.503136	0.251568	+	0.967840i	$0.419054\pi$
$168$	−3.18044	−0.245376
$169$	−12.2051	−0.938853
$170$	0	0
$171$	5.84384	0.446890
$172$	18.5503	1.41445
$173$	−2.47668	−0.188298	−0.0941491	−	0.995558i	$-0.530013\pi$
$173$	−2.47668	−0.188298	−0.0941491	+	0.995558i	$0.530013\pi$
$174$	11.7391	0.889941
$175$	0	0
$176$	0	0
$177$	−7.20613	−0.541646
$178$	17.8634	1.33892
$179$	−11.6127	−0.867972	−0.433986	−	0.900920i	$-0.642893\pi$
$179$	−11.6127	−0.867972	−0.433986	+	0.900920i	$0.642893\pi$
$180$	0	0
$181$	7.06522	0.525153	0.262577	−	0.964911i	$-0.415428\pi$
$181$	7.06522	0.525153	0.262577	+	0.964911i	$0.415428\pi$
$182$	−3.14520	−0.233137
$183$	−2.03022	−0.150078
$184$	6.53266	0.481594
$185$	0	0
$186$	−19.7361	−1.44712
$187$	0	0
$188$	24.5960	1.79385
$189$	1.59339	0.115902
$190$	0	0
$191$	23.8182	1.72342	0.861712	−	0.507398i	$-0.169393\pi$
$191$	23.8182	1.72342	0.861712	+	0.507398i	$0.169393\pi$
$192$	−12.8541	−0.927663
$193$	−25.1985	−1.81382	−0.906912	−	0.421319i	$-0.861567\pi$
$193$	−25.1985	−1.81382	−0.906912	+	0.421319i	$0.861567\pi$
$194$	23.8339	1.71118
$195$	0	0
$196$	−12.9442	−0.924587
$197$	7.24398	0.516112	0.258056	−	0.966130i	$-0.416918\pi$
$197$	7.24398	0.516112	0.258056	+	0.966130i	$0.416918\pi$
$198$	0	0
$199$	−13.4111	−0.950689	−0.475345	−	0.879800i	$-0.657677\pi$
$199$	−13.4111	−0.950689	−0.475345	+	0.879800i	$0.657677\pi$
$200$	0	0
$201$	13.6199	0.960672
$202$	29.5888	2.08186
$203$	−8.44870	−0.592983
$204$	12.8783	0.901661
$205$	0	0
$206$	9.66055	0.673083
$207$	−3.27284	−0.227478
$208$	−1.23398	−0.0855610
$209$	0	0
$210$	0	0
$211$	−3.11369	−0.214356	−0.107178	−	0.994240i	$-0.534181\pi$
$211$	−3.11369	−0.214356	−0.107178	+	0.994240i	$0.534181\pi$
$212$	−12.7183	−0.873494
$213$	1.15336	0.0790270
$214$	−45.6135	−3.11807
$215$	0	0
$216$	−1.99602	−0.135812
$217$	14.2042	0.964242
$218$	−10.6945	−0.724326
$219$	6.66040	0.450068
$220$	0	0
$221$	3.95717	0.266188
$222$	12.2458	0.821885
$223$	12.6942	0.850063	0.425032	−	0.905178i	$-0.360263\pi$
$223$	12.6942	0.850063	0.425032	+	0.905178i	$0.360263\pi$
$224$	11.2433	0.751227
$225$	0	0
$226$	35.0450	2.33116
$227$	2.27677	0.151114	0.0755572	−	0.997141i	$-0.475926\pi$
$227$	2.27677	0.151114	0.0755572	+	0.997141i	$0.475926\pi$
$228$	16.9563	1.12296
$229$	20.1618	1.33233	0.666166	−	0.745803i	$-0.267934\pi$
$229$	20.1618	1.33233	0.666166	+	0.745803i	$0.267934\pi$
$230$	0	0
$231$	0	0
$232$	10.5836	0.694848
$233$	9.85555	0.645659	0.322829	−	0.946457i	$-0.395366\pi$
$233$	9.85555	0.645659	0.322829	+	0.946457i	$0.395366\pi$
$234$	−1.97390	−0.129038
$235$	0	0
$236$	−20.9091	−1.36107
$237$	4.89466	0.317942
$238$	−15.6572	−1.01491
$239$	9.55913	0.618328	0.309164	−	0.951009i	$-0.399951\pi$
$239$	9.55913	0.618328	0.309164	+	0.951009i	$0.399951\pi$
$240$	0	0
$241$	−4.58482	−0.295334	−0.147667	−	0.989037i	$-0.547176\pi$
$241$	−4.58482	−0.295334	−0.147667	+	0.989037i	$0.547176\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−5.89081	−0.377120
$245$	0	0
$246$	−9.27676	−0.591465
$247$	5.21023	0.331519
$248$	−17.7934	−1.12989
$249$	13.2475	0.839525
$250$	0	0
$251$	−13.6439	−0.861195	−0.430598	−	0.902544i	$-0.641697\pi$
$251$	−13.6439	−0.861195	−0.430598	+	0.902544i	$0.641697\pi$
$252$	4.62332	0.291242
$253$	0	0
$254$	−20.0337	−1.25702
$255$	0	0
$256$	−6.05265	−0.378291
$257$	−21.2167	−1.32346	−0.661731	−	0.749742i	$-0.730178\pi$
$257$	−21.2167	−1.32346	−0.661731	+	0.749742i	$0.730178\pi$
$258$	−14.1542	−0.881204
$259$	−8.81336	−0.547636
$260$	0	0
$261$	−5.30235	−0.328207
$262$	21.8607	1.35056
$263$	−15.4054	−0.949936	−0.474968	−	0.880003i	$-0.657540\pi$
$263$	−15.4054	−0.949936	−0.474968	+	0.880003i	$0.657540\pi$
$264$	0	0
$265$	0	0
$266$	−20.6152	−1.26400
$267$	−8.06859	−0.493789
$268$	39.5190	2.41401
$269$	7.08797	0.432161	0.216081	−	0.976376i	$-0.430673\pi$
$269$	7.08797	0.432161	0.216081	+	0.976376i	$0.430673\pi$
$270$	0	0
$271$	29.1371	1.76996	0.884978	−	0.465633i	$-0.154173\pi$
$271$	29.1371	1.76996	0.884978	+	0.465633i	$0.154173\pi$
$272$	−6.14292	−0.372469
$273$	1.42063	0.0859803
$274$	−25.7456	−1.55535
$275$	0	0
$276$	−9.49635	−0.571613
$277$	−14.1005	−0.847218	−0.423609	−	0.905845i	$-0.639237\pi$
$277$	−14.1005	−0.847218	−0.423609	+	0.905845i	$0.639237\pi$
$278$	−3.93402	−0.235947
$279$	8.91445	0.533694
$280$	0	0
$281$	−26.2478	−1.56581	−0.782907	−	0.622139i	$-0.786264\pi$
$281$	−26.2478	−1.56581	−0.782907	+	0.622139i	$0.786264\pi$
$282$	−18.7672	−1.11757
$283$	−0.834033	−0.0495781	−0.0247891	−	0.999693i	$-0.507891\pi$
$283$	−0.834033	−0.0495781	−0.0247891	+	0.999693i	$0.507891\pi$
$284$	3.34655	0.198581
$285$	0	0
$286$	0	0
$287$	6.67653	0.394103
$288$	7.05624	0.415793
$289$	2.69935	0.158785
$290$	0	0
$291$	−10.7654	−0.631077
$292$	19.3256	1.13095
$293$	−14.2041	−0.829812	−0.414906	−	0.909864i	$-0.636186\pi$
$293$	−14.2041	−0.829812	−0.414906	+	0.909864i	$0.636186\pi$
$294$	9.87667	0.576019
$295$	0	0
$296$	11.0404	0.641712
$297$	0	0
$298$	47.5945	2.75707
$299$	−2.91798	−0.168751
$300$	0	0
$301$	10.1869	0.587161
$302$	−17.8047	−1.02455
$303$	−13.3647	−0.767784
$304$	−8.08812	−0.463886
$305$	0	0
$306$	−9.82638	−0.561737
$307$	0.166296	0.00949103	0.00474552	−	0.999989i	$-0.498489\pi$
$307$	0.166296	0.00949103	0.00474552	+	0.999989i	$0.498489\pi$
$308$	0	0
$309$	−4.36350	−0.248231
$310$	0	0
$311$	6.52076	0.369758	0.184879	−	0.982761i	$-0.440811\pi$
$311$	6.52076	0.369758	0.184879	+	0.982761i	$0.440811\pi$
$312$	−1.77961	−0.100750
$313$	0.803861	0.0454369	0.0227185	−	0.999742i	$-0.492768\pi$
$313$	0.803861	0.0454369	0.0227185	+	0.999742i	$0.492768\pi$
$314$	−37.7746	−2.13175
$315$	0	0
$316$	14.2022	0.798936
$317$	14.4751	0.813000	0.406500	−	0.913651i	$-0.366749\pi$
$317$	14.4751	0.813000	0.406500	+	0.913651i	$0.366749\pi$
$318$	9.70426	0.544188
$319$	0	0
$320$	0	0
$321$	20.6028	1.14994
$322$	11.5455	0.643406
$323$	25.9373	1.44319
$324$	2.90157	0.161198
$325$	0	0
$326$	−29.8224	−1.65171
$327$	4.83053	0.267129
$328$	−8.36363	−0.461804
$329$	13.5069	0.744657
$330$	0	0
$331$	7.41532	0.407583	0.203792	−	0.979014i	$-0.434674\pi$
$331$	7.41532	0.407583	0.203792	+	0.979014i	$0.434674\pi$
$332$	38.4384	2.10958
$333$	−5.53121	−0.303108
$334$	−14.3950	−0.787659
$335$	0	0
$336$	−2.20532	−0.120310
$337$	15.3347	0.835334	0.417667	−	0.908600i	$-0.362848\pi$
$337$	15.3347	0.835334	0.417667	+	0.908600i	$0.362848\pi$
$338$	27.0214	1.46977
$339$	−15.8292	−0.859723
$340$	0	0
$341$	0	0
$342$	−12.9380	−0.699605
$343$	−18.2620	−0.986056
$344$	−12.7610	−0.688026
$345$	0	0
$346$	5.48324	0.294781
$347$	−21.6845	−1.16409	−0.582043	−	0.813158i	$-0.697746\pi$
$347$	−21.6845	−1.16409	−0.582043	+	0.813158i	$0.697746\pi$
$348$	−15.3851	−0.824729
$349$	5.85266	0.313286	0.156643	−	0.987655i	$-0.449933\pi$
$349$	5.85266	0.313286	0.156643	+	0.987655i	$0.449933\pi$
$350$	0	0
$351$	0.891576	0.0475888
$352$	0	0
$353$	−8.94423	−0.476053	−0.238027	−	0.971259i	$-0.576501\pi$
$353$	−8.94423	−0.476053	−0.238027	+	0.971259i	$0.576501\pi$
$354$	15.9540	0.847946
$355$	0	0
$356$	−23.4115	−1.24081
$357$	7.07209	0.374295
$358$	25.7099	1.35881
$359$	−5.10770	−0.269574	−0.134787	−	0.990875i	$-0.543035\pi$
$359$	−5.10770	−0.269574	−0.134787	+	0.990875i	$0.543035\pi$
$360$	0	0
$361$	15.1505	0.797395
$362$	−15.6420	−0.822127
$363$	0	0
$364$	4.12205	0.216054
$365$	0	0
$366$	4.49479	0.234947
$367$	34.0987	1.77994	0.889969	−	0.456022i	$-0.150726\pi$
$367$	34.0987	1.77994	0.889969	+	0.456022i	$0.150726\pi$
$368$	4.52974	0.236129
$369$	4.19014	0.218130
$370$	0	0
$371$	−6.98420	−0.362602
$372$	25.8659	1.34108
$373$	26.0424	1.34842	0.674212	−	0.738538i	$-0.264483\pi$
$373$	26.0424	1.34842	0.674212	+	0.738538i	$0.264483\pi$
$374$	0	0
$375$	0	0
$376$	−16.9199	−0.872578
$377$	−4.72745	−0.243476
$378$	−3.52768	−0.181444
$379$	−1.46216	−0.0751060	−0.0375530	−	0.999295i	$-0.511956\pi$
$379$	−1.46216	−0.0751060	−0.0375530	+	0.999295i	$0.511956\pi$
$380$	0	0
$381$	9.04884	0.463586
$382$	−52.7322	−2.69802
$383$	3.45296	0.176438	0.0882190	−	0.996101i	$-0.471882\pi$
$383$	3.45296	0.176438	0.0882190	+	0.996101i	$0.471882\pi$
$384$	14.3458	0.732080
$385$	0	0
$386$	55.7881	2.83954
$387$	6.39321	0.324985
$388$	−31.2364	−1.58579
$389$	−4.09303	−0.207525	−0.103762	−	0.994602i	$-0.533088\pi$
$389$	−4.09303	−0.207525	−0.103762	+	0.994602i	$0.533088\pi$
$390$	0	0
$391$	−14.5261	−0.734618
$392$	8.90449	0.449745
$393$	−9.87408	−0.498082
$394$	−16.0378	−0.807972
$395$	0	0
$396$	0	0
$397$	−7.10672	−0.356676	−0.178338	−	0.983969i	$-0.557072\pi$
$397$	−7.10672	−0.356676	−0.178338	+	0.983969i	$0.557072\pi$
$398$	29.6915	1.48830
$399$	9.31151	0.466159
$400$	0	0
$401$	15.6618	0.782113	0.391056	−	0.920367i	$-0.372110\pi$
$401$	15.6618	0.782113	0.391056	+	0.920367i	$0.372110\pi$
$402$	−30.1537	−1.50393
$403$	7.94791	0.395914
$404$	−38.7787	−1.92931
$405$	0	0
$406$	18.7050	0.928313
$407$	0	0
$408$	−8.85914	−0.438593
$409$	18.5688	0.918169	0.459084	−	0.888393i	$-0.348178\pi$
$409$	18.5688	0.918169	0.459084	+	0.888393i	$0.348178\pi$
$410$	0	0
$411$	11.6288	0.573608
$412$	−12.6610	−0.623761
$413$	−11.4822	−0.565001
$414$	7.24589	0.356116
$415$	0	0
$416$	6.29118	0.308450
$417$	1.77693	0.0870164
$418$	0	0
$419$	2.39208	0.116861	0.0584303	−	0.998291i	$-0.481390\pi$
$419$	2.39208	0.116861	0.0584303	+	0.998291i	$0.481390\pi$
$420$	0	0
$421$	−16.6767	−0.812770	−0.406385	−	0.913702i	$-0.633211\pi$
$421$	−16.6767	−0.812770	−0.406385	+	0.913702i	$0.633211\pi$
$422$	6.89356	0.335573
$423$	8.47681	0.412157
$424$	8.74905	0.424891
$425$	0	0
$426$	−2.55348	−0.123717
$427$	−3.23492	−0.156549
$428$	59.7804	2.88959
$429$	0	0
$430$	0	0
$431$	20.3804	0.981690	0.490845	−	0.871247i	$-0.336688\pi$
$431$	20.3804	0.981690	0.490845	+	0.871247i	$0.336688\pi$
$432$	−1.38404	−0.0665897
$433$	13.2139	0.635018	0.317509	−	0.948255i	$-0.397154\pi$
$433$	13.2139	0.635018	0.317509	+	0.948255i	$0.397154\pi$
$434$	−31.4473	−1.50952
$435$	0	0
$436$	14.0161	0.671250
$437$	−19.1259	−0.914918
$438$	−14.7458	−0.704581
$439$	−31.6527	−1.51070	−0.755350	−	0.655321i	$-0.772533\pi$
$439$	−31.6527	−1.51070	−0.755350	+	0.655321i	$0.772533\pi$
$440$	0	0
$441$	−4.46111	−0.212434
$442$	−8.76097	−0.416717
$443$	−23.7672	−1.12921	−0.564606	−	0.825361i	$-0.690972\pi$
$443$	−23.7672	−1.12921	−0.564606	+	0.825361i	$0.690972\pi$
$444$	−16.0492	−0.761660
$445$	0	0
$446$	−28.1042	−1.33077
$447$	−21.4976	−1.01680
$448$	−20.4815	−0.967662
$449$	−14.3253	−0.676054	−0.338027	−	0.941137i	$-0.609759\pi$
$449$	−14.3253	−0.676054	−0.338027	+	0.941137i	$0.609759\pi$
$450$	0	0
$451$	0	0
$452$	−45.9294	−2.16034
$453$	8.04208	0.377850
$454$	−5.04065	−0.236569
$455$	0	0
$456$	−11.6645	−0.546238
$457$	−0.108900	−0.00509410	−0.00254705	−	0.999997i	$-0.500811\pi$
$457$	−0.108900	−0.00509410	−0.00254705	+	0.999997i	$0.500811\pi$
$458$	−44.6373	−2.08576
$459$	4.43839	0.207167
$460$	0	0
$461$	8.67446	0.404010	0.202005	−	0.979385i	$-0.435254\pi$
$461$	8.67446	0.404010	0.202005	+	0.979385i	$0.435254\pi$
$462$	0	0
$463$	−31.3509	−1.45700	−0.728501	−	0.685045i	$-0.759782\pi$
$463$	−31.3509	−1.45700	−0.728501	+	0.685045i	$0.759782\pi$
$464$	7.33867	0.340689
$465$	0	0
$466$	−21.8197	−1.01078
$467$	−23.6018	−1.09216	−0.546080	−	0.837733i	$-0.683880\pi$
$467$	−23.6018	−1.09216	−0.546080	+	0.837733i	$0.683880\pi$
$468$	2.58697	0.119583
$469$	21.7018	1.00209
$470$	0	0
$471$	17.0621	0.786180
$472$	14.3836	0.662060
$473$	0	0
$474$	−10.8365	−0.497738
$475$	0	0
$476$	20.5201	0.940539
$477$	−4.38324	−0.200695
$478$	−21.1634	−0.967992
$479$	24.7008	1.12861	0.564304	−	0.825567i	$-0.309145\pi$
$479$	24.7008	1.12861	0.564304	+	0.825567i	$0.309145\pi$
$480$	0	0
$481$	−4.93149	−0.224857
$482$	10.1506	0.462345
$483$	−5.21490	−0.237286
$484$	0	0
$485$	0	0
$486$	−2.21395	−0.100427
$487$	7.73374	0.350449	0.175225	−	0.984528i	$-0.443935\pi$
$487$	7.73374	0.350449	0.175225	+	0.984528i	$0.443935\pi$
$488$	4.05236	0.183442
$489$	13.4702	0.609145
$490$	0	0
$491$	−19.4198	−0.876403	−0.438201	−	0.898877i	$-0.644384\pi$
$491$	−19.4198	−0.876403	−0.438201	+	0.898877i	$0.644384\pi$
$492$	12.1580	0.548124
$493$	−23.5339	−1.05991
$494$	−11.5352	−0.518993
$495$	0	0
$496$	−12.3380	−0.553991
$497$	1.83775	0.0824344
$498$	−29.3292	−1.31427
$499$	0.0857258	0.00383761	0.00191881	−	0.999998i	$-0.499389\pi$
$499$	0.0857258	0.00383761	0.00191881	+	0.999998i	$0.499389\pi$
$500$	0	0
$501$	6.50196	0.290486
$502$	30.2069	1.34820
$503$	13.5923	0.606053	0.303026	−	0.952982i	$-0.402003\pi$
$503$	13.5923	0.606053	0.303026	+	0.952982i	$0.402003\pi$
$504$	−3.18044	−0.141668
$505$	0	0
$506$	0	0
$507$	−12.2051	−0.542047
$508$	26.2558	1.16491
$509$	−14.7758	−0.654926	−0.327463	−	0.944864i	$-0.606194\pi$
$509$	−14.7758	−0.654926	−0.327463	+	0.944864i	$0.606194\pi$
$510$	0	0
$511$	10.6126	0.469474
$512$	−15.2913	−0.675786
$513$	5.84384	0.258012
$514$	46.9727	2.07188
$515$	0	0
$516$	18.5503	0.816632
$517$	0	0
$518$	19.5123	0.857323
$519$	−2.47668	−0.108714
$520$	0	0
$521$	−3.84342	−0.168383	−0.0841916	−	0.996450i	$-0.526831\pi$
$521$	−3.84342	−0.168383	−0.0841916	+	0.996450i	$0.526831\pi$
$522$	11.7391	0.513808
$523$	−10.4374	−0.456397	−0.228198	−	0.973615i	$-0.573284\pi$
$523$	−10.4374	−0.456397	−0.228198	+	0.973615i	$0.573284\pi$
$524$	−28.6503	−1.25159
$525$	0	0
$526$	34.1067	1.48712
$527$	39.5658	1.72351
$528$	0	0
$529$	−12.2885	−0.534285
$530$	0	0
$531$	−7.20613	−0.312720
$532$	27.0180	1.17138
$533$	3.73583	0.161817
$534$	17.8634	0.773026
$535$	0	0
$536$	−27.1856	−1.17424
$537$	−11.6127	−0.501124
$538$	−15.6924	−0.676547
$539$	0	0
$540$	0	0
$541$	24.0611	1.03447	0.517233	−	0.855845i	$-0.326962\pi$
$541$	24.0611	1.03447	0.517233	+	0.855845i	$0.326962\pi$
$542$	−64.5081	−2.77086
$543$	7.06522	0.303197
$544$	31.3184	1.34277
$545$	0	0
$546$	−3.14520	−0.134602
$547$	32.0148	1.36886	0.684428	−	0.729081i	$-0.260052\pi$
$547$	32.0148	1.36886	0.684428	+	0.729081i	$0.260052\pi$
$548$	33.7418	1.44138
$549$	−2.03022	−0.0866475
$550$	0	0
$551$	−30.9861	−1.32005
$552$	6.53266	0.278048
$553$	7.79910	0.331651
$554$	31.2178	1.32632
$555$	0	0
$556$	5.15587	0.218658
$557$	12.3241	0.522188	0.261094	−	0.965313i	$-0.415917\pi$
$557$	12.3241	0.522188	0.261094	+	0.965313i	$0.415917\pi$
$558$	−19.7361	−0.835497
$559$	5.70003	0.241086
$560$	0	0
$561$	0	0
$562$	58.1113	2.45128
$563$	−27.3735	−1.15366	−0.576829	−	0.816865i	$-0.695710\pi$
$563$	−27.3735	−1.15366	−0.576829	+	0.816865i	$0.695710\pi$
$564$	24.5960	1.03568
$565$	0	0
$566$	1.84651	0.0776144
$567$	1.59339	0.0669160
$568$	−2.30214	−0.0965955
$569$	−1.98445	−0.0831926	−0.0415963	−	0.999135i	$-0.513244\pi$
$569$	−1.98445	−0.0831926	−0.0415963	+	0.999135i	$0.513244\pi$
$570$	0	0
$571$	−27.0887	−1.13363	−0.566813	−	0.823846i	$-0.691824\pi$
$571$	−27.0887	−1.13363	−0.566813	+	0.823846i	$0.691824\pi$
$572$	0	0
$573$	23.8182	0.995019
$574$	−14.7815	−0.616967
$575$	0	0
$576$	−12.8541	−0.535587
$577$	7.14188	0.297320	0.148660	−	0.988888i	$-0.452504\pi$
$577$	7.14188	0.297320	0.148660	+	0.988888i	$0.452504\pi$
$578$	−5.97622	−0.248578
$579$	−25.1985	−1.04721
$580$	0	0
$581$	21.1084	0.875723
$582$	23.8339	0.987949
$583$	0	0
$584$	−13.2943	−0.550123
$585$	0	0
$586$	31.4471	1.29907
$587$	44.4172	1.83329	0.916646	−	0.399699i	$-0.130885\pi$
$587$	44.4172	1.83329	0.916646	+	0.399699i	$0.130885\pi$
$588$	−12.9442	−0.533811
$589$	52.0946	2.14652
$590$	0	0
$591$	7.24398	0.297977
$592$	7.65542	0.314636
$593$	15.7620	0.647268	0.323634	−	0.946182i	$-0.395095\pi$
$593$	15.7620	0.647268	0.323634	+	0.946182i	$0.395095\pi$
$594$	0	0
$595$	0	0
$596$	−62.3766	−2.55504
$597$	−13.4111	−0.548881
$598$	6.46026	0.264180
$599$	2.03104	0.0829861	0.0414930	−	0.999139i	$-0.486789\pi$
$599$	2.03104	0.0829861	0.0414930	+	0.999139i	$0.486789\pi$
$600$	0	0
$601$	−41.4508	−1.69081	−0.845406	−	0.534124i	$-0.820642\pi$
$601$	−41.4508	−1.69081	−0.845406	+	0.534124i	$0.820642\pi$
$602$	−22.5532	−0.919199
$603$	13.6199	0.554645
$604$	23.3346	0.949473
$605$	0	0
$606$	29.5888	1.20196
$607$	−6.04010	−0.245160	−0.122580	−	0.992459i	$-0.539117\pi$
$607$	−6.04010	−0.245160	−0.122580	+	0.992459i	$0.539117\pi$
$608$	41.2356	1.67232
$609$	−8.44870	−0.342359
$610$	0	0
$611$	7.55772	0.305753
$612$	12.8783	0.520574
$613$	30.0199	1.21249	0.606245	−	0.795278i	$-0.292675\pi$
$613$	30.0199	1.21249	0.606245	+	0.795278i	$0.292675\pi$
$614$	−0.368171	−0.0148582
$615$	0	0
$616$	0	0
$617$	−9.99761	−0.402488	−0.201244	−	0.979541i	$-0.564498\pi$
$617$	−9.99761	−0.402488	−0.201244	+	0.979541i	$0.564498\pi$
$618$	9.66055	0.388604
$619$	7.70735	0.309785	0.154892	−	0.987931i	$-0.450497\pi$
$619$	7.70735	0.309785	0.154892	+	0.987931i	$0.450497\pi$
$620$	0	0
$621$	−3.27284	−0.131334
$622$	−14.4366	−0.578856
$623$	−12.8564	−0.515081
$624$	−1.23398	−0.0493987
$625$	0	0
$626$	−1.77971	−0.0711314
$627$	0	0
$628$	49.5069	1.97554
$629$	−24.5497	−0.978860
$630$	0	0
$631$	−12.9428	−0.515244	−0.257622	−	0.966246i	$-0.582939\pi$
$631$	−12.9428	−0.515244	−0.257622	+	0.966246i	$0.582939\pi$
$632$	−9.76986	−0.388624
$633$	−3.11369	−0.123758
$634$	−32.0470	−1.27275
$635$	0	0
$636$	−12.7183	−0.504312
$637$	−3.97742	−0.157591
$638$	0	0
$639$	1.15336	0.0456262
$640$	0	0
$641$	−48.9112	−1.93188	−0.965938	−	0.258775i	$-0.916681\pi$
$641$	−48.9112	−1.93188	−0.965938	+	0.258775i	$0.916681\pi$
$642$	−45.6135	−1.80022
$643$	−46.9790	−1.85267	−0.926335	−	0.376700i	$-0.877059\pi$
$643$	−46.9790	−1.85267	−0.926335	+	0.376700i	$0.877059\pi$
$644$	−15.1314	−0.596260
$645$	0	0
$646$	−57.4238	−2.25931
$647$	3.87592	0.152378	0.0761891	−	0.997093i	$-0.475725\pi$
$647$	3.87592	0.152378	0.0761891	+	0.997093i	$0.475725\pi$
$648$	−1.99602	−0.0784112
$649$	0	0
$650$	0	0
$651$	14.2042	0.556706
$652$	39.0848	1.53068
$653$	5.57951	0.218343	0.109171	−	0.994023i	$-0.465180\pi$
$653$	5.57951	0.218343	0.109171	+	0.994023i	$0.465180\pi$
$654$	−10.6945	−0.418190
$655$	0	0
$656$	−5.79933	−0.226426
$657$	6.66040	0.259847
$658$	−29.9035	−1.16576
$659$	5.45979	0.212683	0.106342	−	0.994330i	$-0.466086\pi$
$659$	5.45979	0.212683	0.106342	+	0.994330i	$0.466086\pi$
$660$	0	0
$661$	32.1316	1.24977	0.624887	−	0.780715i	$-0.285145\pi$
$661$	32.1316	1.24977	0.624887	+	0.780715i	$0.285145\pi$
$662$	−16.4171	−0.638070
$663$	3.95717	0.153684
$664$	−26.4423	−1.02616
$665$	0	0
$666$	12.2458	0.474516
$667$	17.3537	0.671938
$668$	18.8659	0.729942
$669$	12.6942	0.490784
$670$	0	0
$671$	0	0
$672$	11.2433	0.433721
$673$	−1.58054	−0.0609254	−0.0304627	−	0.999536i	$-0.509698\pi$
$673$	−1.58054	−0.0609254	−0.0304627	+	0.999536i	$0.509698\pi$
$674$	−33.9502	−1.30771
$675$	0	0
$676$	−35.4139	−1.36207
$677$	−1.81064	−0.0695887	−0.0347944	−	0.999394i	$-0.511078\pi$
$677$	−1.81064	−0.0695887	−0.0347944	+	0.999394i	$0.511078\pi$
$678$	35.0450	1.34589
$679$	−17.1534	−0.658287
$680$	0	0
$681$	2.27677	0.0872459
$682$	0	0
$683$	−11.3990	−0.436170	−0.218085	−	0.975930i	$-0.569981\pi$
$683$	−11.3990	−0.436170	−0.218085	+	0.975930i	$0.569981\pi$
$684$	16.9563	0.648341
$685$	0	0
$686$	40.4311	1.54367
$687$	20.1618	0.769222
$688$	−8.84846	−0.337344
$689$	−3.90799	−0.148883
$690$	0	0
$691$	9.31126	0.354217	0.177108	−	0.984191i	$-0.443326\pi$
$691$	9.31126	0.354217	0.177108	+	0.984191i	$0.443326\pi$
$692$	−7.18625	−0.273180
$693$	0	0
$694$	48.0084	1.82237
$695$	0	0
$696$	10.5836	0.401171
$697$	18.5975	0.704431
$698$	−12.9575	−0.490448
$699$	9.85555	0.372771
$700$	0	0
$701$	22.6245	0.854515	0.427258	−	0.904130i	$-0.359480\pi$
$701$	22.6245	0.854515	0.427258	+	0.904130i	$0.359480\pi$
$702$	−1.97390	−0.0745002
$703$	−32.3235	−1.21910
$704$	0	0
$705$	0	0
$706$	19.8021	0.745261
$707$	−21.2952	−0.800889
$708$	−20.9091	−0.785812
$709$	17.0391	0.639918	0.319959	−	0.947431i	$-0.396331\pi$
$709$	17.0391	0.639918	0.319959	+	0.947431i	$0.396331\pi$
$710$	0	0
$711$	4.89466	0.183564
$712$	16.1051	0.603564
$713$	−29.1755	−1.09263
$714$	−15.6572	−0.585957
$715$	0	0
$716$	−33.6950	−1.25924
$717$	9.55913	0.356992
$718$	11.3082	0.422018
$719$	38.3908	1.43173	0.715867	−	0.698236i	$-0.246031\pi$
$719$	38.3908	1.43173	0.715867	+	0.698236i	$0.246031\pi$
$720$	0	0
$721$	−6.95274	−0.258934
$722$	−33.5425	−1.24832
$723$	−4.58482	−0.170511
$724$	20.5002	0.761884
$725$	0	0
$726$	0	0
$727$	−35.5819	−1.31966	−0.659829	−	0.751416i	$-0.729371\pi$
$727$	−35.5819	−1.31966	−0.659829	+	0.751416i	$0.729371\pi$
$728$	−2.83561	−0.105095
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.3756	1.04951
$732$	−5.89081	−0.217730
$733$	43.3052	1.59951	0.799757	−	0.600324i	$-0.204962\pi$
$733$	43.3052	1.59951	0.799757	+	0.600324i	$0.204962\pi$
$734$	−75.4928	−2.78649
$735$	0	0
$736$	−23.0939	−0.851253
$737$	0	0
$738$	−9.27676	−0.341482
$739$	−2.05329	−0.0755314	−0.0377657	−	0.999287i	$-0.512024\pi$
$739$	−2.05329	−0.0755314	−0.0377657	+	0.999287i	$0.512024\pi$
$740$	0	0
$741$	5.21023	0.191403
$742$	15.4627	0.567652
$743$	−5.12186	−0.187903	−0.0939515	−	0.995577i	$-0.529950\pi$
$743$	−5.12186	−0.187903	−0.0939515	+	0.995577i	$0.529950\pi$
$744$	−17.7934	−0.652339
$745$	0	0
$746$	−57.6565	−2.11096
$747$	13.2475	0.484700
$748$	0	0
$749$	32.8282	1.19952
$750$	0	0
$751$	−22.7758	−0.831101	−0.415551	−	0.909570i	$-0.636411\pi$
$751$	−22.7758	−0.831101	−0.415551	+	0.909570i	$0.636411\pi$
$752$	−11.7323	−0.427831
$753$	−13.6439	−0.497211
$754$	10.4663	0.381161
$755$	0	0
$756$	4.62332	0.168149
$757$	−23.3537	−0.848806	−0.424403	−	0.905474i	$-0.639516\pi$
$757$	−23.3537	−0.848806	−0.424403	+	0.905474i	$0.639516\pi$
$758$	3.23714	0.117578
$759$	0	0
$760$	0	0
$761$	12.3032	0.445990	0.222995	−	0.974820i	$-0.428417\pi$
$761$	12.3032	0.445990	0.222995	+	0.974820i	$0.428417\pi$
$762$	−20.0337	−0.725743
$763$	7.69691	0.278647
$764$	69.1101	2.50031
$765$	0	0
$766$	−7.64467	−0.276213
$767$	−6.42482	−0.231987
$768$	−6.05265	−0.218406
$769$	−6.67277	−0.240626	−0.120313	−	0.992736i	$-0.538390\pi$
$769$	−6.67277	−0.240626	−0.120313	+	0.992736i	$0.538390\pi$
$770$	0	0
$771$	−21.2167	−0.764101
$772$	−73.1150	−2.63147
$773$	4.51340	0.162336	0.0811678	−	0.996700i	$-0.474135\pi$
$773$	4.51340	0.162336	0.0811678	+	0.996700i	$0.474135\pi$
$774$	−14.1542	−0.508763
$775$	0	0
$776$	21.4879	0.771371
$777$	−8.81336	−0.316178
$778$	9.06176	0.324880
$779$	24.4866	0.877322
$780$	0	0
$781$	0	0
$782$	32.1601	1.15004
$783$	−5.30235	−0.189490
$784$	6.17437	0.220513
$785$	0	0
$786$	21.8607	0.779746
$787$	19.7596	0.704355	0.352178	−	0.935933i	$-0.385441\pi$
$787$	19.7596	0.704355	0.352178	+	0.935933i	$0.385441\pi$
$788$	21.0189	0.748767
$789$	−15.4054	−0.548446
$790$	0	0
$791$	−25.2220	−0.896792
$792$	0	0
$793$	−1.81009	−0.0642783
$794$	15.7339	0.558376
$795$	0	0
$796$	−38.9133	−1.37924
$797$	7.73449	0.273970	0.136985	−	0.990573i	$-0.456259\pi$
$797$	7.73449	0.273970	0.136985	+	0.990573i	$0.456259\pi$
$798$	−20.6152	−0.729771
$799$	37.6234	1.33102
$800$	0	0
$801$	−8.06859	−0.285089
$802$	−34.6744	−1.22440
$803$	0	0
$804$	39.5190	1.39373
$805$	0	0
$806$	−17.5963	−0.619802
$807$	7.08797	0.249508
$808$	26.6763	0.938469
$809$	−47.2899	−1.66263	−0.831313	−	0.555805i	$-0.812410\pi$
$809$	−47.2899	−1.66263	−0.831313	+	0.555805i	$0.812410\pi$
$810$	0	0
$811$	30.8261	1.08245	0.541226	−	0.840877i	$-0.317960\pi$
$811$	30.8261	1.08245	0.541226	+	0.840877i	$0.317960\pi$
$812$	−24.5145	−0.860289
$813$	29.1371	1.02188
$814$	0	0
$815$	0	0
$816$	−6.14292	−0.215045
$817$	37.3609	1.30709
$818$	−41.1104	−1.43739
$819$	1.42063	0.0496407
$820$	0	0
$821$	−2.64720	−0.0923877	−0.0461939	−	0.998932i	$-0.514709\pi$
$821$	−2.64720	−0.0923877	−0.0461939	+	0.998932i	$0.514709\pi$
$822$	−25.7456	−0.897982
$823$	6.44694	0.224726	0.112363	−	0.993667i	$-0.464158\pi$
$823$	6.44694	0.224726	0.112363	+	0.993667i	$0.464158\pi$
$824$	8.70964	0.303415
$825$	0	0
$826$	25.4209	0.884508
$827$	26.9646	0.937652	0.468826	−	0.883291i	$-0.344677\pi$
$827$	26.9646	0.937652	0.468826	+	0.883291i	$0.344677\pi$
$828$	−9.49635	−0.330021
$829$	34.6453	1.20328	0.601640	−	0.798767i	$-0.294514\pi$
$829$	34.6453	1.20328	0.601640	+	0.798767i	$0.294514\pi$
$830$	0	0
$831$	−14.1005	−0.489142
$832$	−11.4604	−0.397318
$833$	−19.8002	−0.686036
$834$	−3.93402	−0.136224
$835$	0	0
$836$	0	0
$837$	8.91445	0.308128
$838$	−5.29594	−0.182945
$839$	15.1597	0.523372	0.261686	−	0.965153i	$-0.415722\pi$
$839$	15.1597	0.523372	0.261686	+	0.965153i	$0.415722\pi$
$840$	0	0
$841$	−0.885106	−0.0305209
$842$	36.9213	1.27239
$843$	−26.2478	−0.904023
$844$	−9.03459	−0.310984
$845$	0	0
$846$	−18.7672	−0.645230
$847$	0	0
$848$	6.06658	0.208327
$849$	−0.834033	−0.0286239
$850$	0	0
$851$	18.1027	0.620554
$852$	3.34655	0.114651
$853$	44.4195	1.52090	0.760448	−	0.649399i	$-0.224980\pi$
$853$	44.4195	1.52090	0.760448	+	0.649399i	$0.224980\pi$
$854$	7.16195	0.245077
$855$	0	0
$856$	−41.1236	−1.40558
$857$	−13.7395	−0.469333	−0.234667	−	0.972076i	$-0.575400\pi$
$857$	−13.7395	−0.469333	−0.234667	+	0.972076i	$0.575400\pi$
$858$	0	0
$859$	54.6076	1.86319	0.931594	−	0.363499i	$-0.118418\pi$
$859$	54.6076	1.86319	0.931594	+	0.363499i	$0.118418\pi$
$860$	0	0
$861$	6.67653	0.227535
$862$	−45.1212	−1.53683
$863$	42.3365	1.44115	0.720575	−	0.693377i	$-0.243878\pi$
$863$	42.3365	1.44115	0.720575	+	0.693377i	$0.243878\pi$
$864$	7.05624	0.240058
$865$	0	0
$866$	−29.2548	−0.994119
$867$	2.69935	0.0916747
$868$	41.2144	1.39891
$869$	0	0
$870$	0	0
$871$	12.1432	0.411455
$872$	−9.64185	−0.326514
$873$	−10.7654	−0.364352
$874$	42.3438	1.43230
$875$	0	0
$876$	19.3256	0.652952
$877$	2.33065	0.0787006	0.0393503	−	0.999225i	$-0.487471\pi$
$877$	2.33065	0.0787006	0.0393503	+	0.999225i	$0.487471\pi$
$878$	70.0774	2.36500
$879$	−14.2041	−0.479092
$880$	0	0
$881$	−21.4550	−0.722837	−0.361419	−	0.932404i	$-0.617707\pi$
$881$	−21.4550	−0.722837	−0.361419	+	0.932404i	$0.617707\pi$
$882$	9.87667	0.332565
$883$	−10.1555	−0.341761	−0.170880	−	0.985292i	$-0.554661\pi$
$883$	−10.1555	−0.341761	−0.170880	+	0.985292i	$0.554661\pi$
$884$	11.4820	0.386181
$885$	0	0
$886$	52.6193	1.76778
$887$	−17.2592	−0.579509	−0.289754	−	0.957101i	$-0.593574\pi$
$887$	−17.2592	−0.579509	−0.289754	+	0.957101i	$0.593574\pi$
$888$	11.0404	0.370492
$889$	14.4183	0.483575
$890$	0	0
$891$	0	0
$892$	36.8329	1.23326
$893$	49.5372	1.65770
$894$	47.5945	1.59180
$895$	0	0
$896$	22.8584	0.763645
$897$	−2.91798	−0.0974286
$898$	31.7155	1.05836
$899$	−47.2675	−1.57646
$900$	0	0
$901$	−19.4545	−0.648125
$902$	0	0
$903$	10.1869	0.338997
$904$	31.5954	1.05085
$905$	0	0
$906$	−17.8047	−0.591523
$907$	−56.3356	−1.87059	−0.935297	−	0.353865i	$-0.884867\pi$
$907$	−56.3356	−1.87059	−0.935297	+	0.353865i	$0.884867\pi$
$908$	6.60620	0.219234
$909$	−13.3647	−0.443280
$910$	0	0
$911$	55.8484	1.85034	0.925170	−	0.379554i	$-0.123923\pi$
$911$	55.8484	1.85034	0.925170	+	0.379554i	$0.123923\pi$
$912$	−8.08812	−0.267824
$913$	0	0
$914$	0.241098	0.00797481
$915$	0	0
$916$	58.5010	1.93293
$917$	−15.7332	−0.519558
$918$	−9.82638	−0.324319
$919$	−7.07031	−0.233228	−0.116614	−	0.993177i	$-0.537204\pi$
$919$	−7.07031	−0.233228	−0.116614	+	0.993177i	$0.537204\pi$
$920$	0	0
$921$	0.166296	0.00547965
$922$	−19.2048	−0.632476
$923$	1.02831	0.0338472
$924$	0	0
$925$	0	0
$926$	69.4093	2.28093
$927$	−4.36350	−0.143316
$928$	−37.4147	−1.22820
$929$	33.6177	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$929$	33.6177	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$930$	0	0
$931$	−26.0700	−0.854411
$932$	28.5965	0.936711
$933$	6.52076	0.213480
$934$	52.2531	1.70977
$935$	0	0
$936$	−1.77961	−0.0581683
$937$	−55.2419	−1.80467	−0.902337	−	0.431032i	$-0.858150\pi$
$937$	−55.2419	−1.80467	−0.902337	+	0.431032i	$0.858150\pi$
$938$	−48.0466	−1.56878
$939$	0.803861	0.0262330
$940$	0	0
$941$	47.9312	1.56251	0.781256	−	0.624211i	$-0.214579\pi$
$941$	47.9312	1.56251	0.781256	+	0.624211i	$0.214579\pi$
$942$	−37.7746	−1.23076
$943$	−13.7137	−0.446578
$944$	9.97359	0.324613
$945$	0	0
$946$	0	0
$947$	42.2493	1.37292	0.686459	−	0.727168i	$-0.259164\pi$
$947$	42.2493	1.37292	0.686459	+	0.727168i	$0.259164\pi$
$948$	14.2022	0.461266
$949$	5.93826	0.192764
$950$	0	0
$951$	14.4751	0.469386
$952$	−14.1161	−0.457504
$953$	24.5617	0.795631	0.397815	−	0.917465i	$-0.369768\pi$
$953$	24.5617	0.795631	0.397815	+	0.917465i	$0.369768\pi$
$954$	9.70426	0.314187
$955$	0	0
$956$	27.7365	0.897061
$957$	0	0
$958$	−54.6863	−1.76683
$959$	18.5292	0.598341
$960$	0	0
$961$	48.4674	1.56346
$962$	10.9181	0.352013
$963$	20.6028	0.663915
$964$	−13.3032	−0.428466
$965$	0	0
$966$	11.5455	0.371471
$967$	−6.41686	−0.206352	−0.103176	−	0.994663i	$-0.532901\pi$
$967$	−6.41686	−0.206352	−0.103176	+	0.994663i	$0.532901\pi$
$968$	0	0
$969$	25.9373	0.833226
$970$	0	0
$971$	4.82827	0.154946	0.0774732	−	0.996994i	$-0.475315\pi$
$971$	4.82827	0.154946	0.0774732	+	0.996994i	$0.475315\pi$
$972$	2.90157	0.0930678
$973$	2.83133	0.0907683
$974$	−17.1221	−0.548628
$975$	0	0
$976$	2.80990	0.0899428
$977$	−42.7689	−1.36830	−0.684149	−	0.729343i	$-0.739826\pi$
$977$	−42.7689	−1.36830	−0.684149	+	0.729343i	$0.739826\pi$
$978$	−29.8224	−0.953616
$979$	0	0
$980$	0	0
$981$	4.83053	0.154227
$982$	42.9944	1.37201
$983$	−4.63530	−0.147843	−0.0739215	−	0.997264i	$-0.523551\pi$
$983$	−4.63530	−0.147843	−0.0739215	+	0.997264i	$0.523551\pi$
$984$	−8.36363	−0.266623
$985$	0	0
$986$	52.1029	1.65929
$987$	13.5069	0.429928
$988$	15.1178	0.480963
$989$	−20.9239	−0.665342
$990$	0	0
$991$	17.0726	0.542330	0.271165	−	0.962533i	$-0.412591\pi$
$991$	17.0726	0.542330	0.271165	+	0.962533i	$0.412591\pi$
$992$	62.9025	1.99716
$993$	7.41532	0.235318
$994$	−4.06869	−0.129051
$995$	0	0
$996$	38.4384	1.21797
$997$	−6.59685	−0.208924	−0.104462	−	0.994529i	$-0.533312\pi$
$997$	−6.59685	−0.208924	−0.104462	+	0.994529i	$0.533312\pi$
$998$	−0.189792	−0.00600777
$999$	−5.53121	−0.175000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.ea.1.2		12
5.2	odd	4		1815.2.c.k.364.4		24
5.3	odd	4		1815.2.c.k.364.21		24
5.4	even	2		9075.2.a.dx.1.11		12
11.7	odd	10		825.2.n.p.676.6		24
11.8	odd	10		825.2.n.p.526.6		24
11.10	odd	2		9075.2.a.dy.1.11		12
55.7	even	20		165.2.s.a.49.2	✓	48
55.8	even	20		165.2.s.a.64.2	yes	48
55.18	even	20		165.2.s.a.49.11	yes	48
55.19	odd	10		825.2.n.o.526.1		24
55.29	odd	10		825.2.n.o.676.1		24
55.32	even	4		1815.2.c.j.364.21		24
55.43	even	4		1815.2.c.j.364.4		24
55.52	even	20		165.2.s.a.64.11	yes	48
55.54	odd	2		9075.2.a.dz.1.2		12
165.8	odd	20		495.2.ba.c.64.11		48
165.62	odd	20		495.2.ba.c.379.11		48
165.107	odd	20		495.2.ba.c.64.2		48
165.128	odd	20		495.2.ba.c.379.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.s.a.49.2	✓	48	55.7	even	20
165.2.s.a.49.11	yes	48	55.18	even	20
165.2.s.a.64.2	yes	48	55.8	even	20
165.2.s.a.64.11	yes	48	55.52	even	20
495.2.ba.c.64.2		48	165.107	odd	20
495.2.ba.c.64.11		48	165.8	odd	20
495.2.ba.c.379.2		48	165.128	odd	20
495.2.ba.c.379.11		48	165.62	odd	20
825.2.n.o.526.1		24	55.19	odd	10
825.2.n.o.676.1		24	55.29	odd	10
825.2.n.p.526.6		24	11.8	odd	10
825.2.n.p.676.6		24	11.7	odd	10
1815.2.c.j.364.4		24	55.43	even	4
1815.2.c.j.364.21		24	55.32	even	4
1815.2.c.k.364.4		24	5.2	odd	4
1815.2.c.k.364.21		24	5.3	odd	4
9075.2.a.dx.1.11		12	5.4	even	2
9075.2.a.dy.1.11		12	11.10	odd	2
9075.2.a.dz.1.2		12	55.54	odd	2
9075.2.a.ea.1.2		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.21395 q^{2} +1.00000 q^{3} +2.90157 q^{4} -2.21395 q^{6} +1.59339 q^{7} -1.99602 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.21395 q^{2} +1.00000 q^{3} +2.90157 q^{4} -2.21395 q^{6} +1.59339 q^{7} -1.99602 q^{8} +1.00000 q^{9} +2.90157 q^{12} +0.891576 q^{13} -3.52768 q^{14} -1.38404 q^{16} +4.43839 q^{17} -2.21395 q^{18} +5.84384 q^{19} +1.59339 q^{21} -3.27284 q^{23} -1.99602 q^{24} -1.97390 q^{26} +1.00000 q^{27} +4.62332 q^{28} -5.30235 q^{29} +8.91445 q^{31} +7.05624 q^{32} -9.82638 q^{34} +2.90157 q^{36} -5.53121 q^{37} -12.9380 q^{38} +0.891576 q^{39} +4.19014 q^{41} -3.52768 q^{42} +6.39321 q^{43} +7.24589 q^{46} +8.47681 q^{47} -1.38404 q^{48} -4.46111 q^{49} +4.43839 q^{51} +2.58697 q^{52} -4.38324 q^{53} -2.21395 q^{54} -3.18044 q^{56} +5.84384 q^{57} +11.7391 q^{58} -7.20613 q^{59} -2.03022 q^{61} -19.7361 q^{62} +1.59339 q^{63} -12.8541 q^{64} +13.6199 q^{67} +12.8783 q^{68} -3.27284 q^{69} +1.15336 q^{71} -1.99602 q^{72} +6.66040 q^{73} +12.2458 q^{74} +16.9563 q^{76} -1.97390 q^{78} +4.89466 q^{79} +1.00000 q^{81} -9.27676 q^{82} +13.2475 q^{83} +4.62332 q^{84} -14.1542 q^{86} -5.30235 q^{87} -8.06859 q^{89} +1.42063 q^{91} -9.49635 q^{92} +8.91445 q^{93} -18.7672 q^{94} +7.05624 q^{96} -10.7654 q^{97} +9.87667 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9	\( 12 q + 4 q^{2} + 12 q^{3} + 12 q^{4} + 4 q^{6} + 8 q^{7} + 12 q^{8} + 12 q^{9} + 12 q^{12} + 18 q^{13} + 6 q^{14} + 24 q^{16} + 18 q^{17} + 4 q^{18} + 16 q^{19} + 8 q^{21} + 12 q^{24} + 16 q^{26} + 12 q^{27} + 30 q^{28} + 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} + 18 q^{39} + 6 q^{42} + 32 q^{43} + 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} + 18 q^{51} + 48 q^{52} - 12 q^{53} + 4 q^{54} + 6 q^{56} + 16 q^{57} - 10 q^{58} + 20 q^{59} + 20 q^{61} + 20 q^{62} + 8 q^{63} - 6 q^{64} + 10 q^{67} + 26 q^{68} + 32 q^{71} + 12 q^{72} + 26 q^{73} - 68 q^{74} + 34 q^{76} + 16 q^{78} + 32 q^{79} + 12 q^{81} - 62 q^{82} + 26 q^{83} + 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} - 2 q^{94} + 28 q^{96} - 22 q^{97} + 60 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^3 + 12 * q^4 + 4 * q^6 + 8 * q^7 + 12 * q^8 + 12 * q^9 + 12 * q^12 + 18 * q^13 + 6 * q^14 + 24 * q^16 + 18 * q^17 + 4 * q^18 + 16 * q^19 + 8 * q^21 + 12 * q^24 + 16 * q^26 + 12 * q^27 + 30 * q^28 + 28 * q^32 + 6 * q^34 + 12 * q^36 - 28 * q^38 + 18 * q^39 + 6 * q^42 + 32 * q^43 + 28 * q^46 - 4 * q^47 + 24 * q^48 + 12 * q^49 + 18 * q^51 + 48 * q^52 - 12 * q^53 + 4 * q^54 + 6 * q^56 + 16 * q^57 - 10 * q^58 + 20 * q^59 + 20 * q^61 + 20 * q^62 + 8 * q^63 - 6 * q^64 + 10 * q^67 + 26 * q^68 + 32 * q^71 + 12 * q^72 + 26 * q^73 - 68 * q^74 + 34 * q^76 + 16 * q^78 + 32 * q^79 + 12 * q^81 - 62 * q^82 + 26 * q^83 + 30 * q^84 - 36 * q^86 - 10 * q^89 + 8 * q^92 - 2 * q^94 + 28 * q^96 - 22 * q^97 + 60 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more