LMFDB - Embedded newform 375.2.b.a.124.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	375.124
Dual form			375.2.b.a.124.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966i q^{2} -1.00000i q^{3} +1.85410 q^{4} -0.381966 q^{6} -3.61803i q^{7} -1.47214i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-0.381966i q^{2} -1.00000i q^{3} +1.85410 q^{4} -0.381966 q^{6} -3.61803i q^{7} -1.47214i q^{8} -1.00000 q^{9} -1.76393 q^{11} -1.85410i q^{12} +3.00000i q^{13} -1.38197 q^{14} +3.14590 q^{16} -5.61803i q^{17} +0.381966i q^{18} +1.00000 q^{19} -3.61803 q^{21} +0.673762i q^{22} +6.70820i q^{23} -1.47214 q^{24} +1.14590 q^{26} +1.00000i q^{27} -6.70820i q^{28} -0.236068 q^{29} +8.09017 q^{31} -4.14590i q^{32} +1.76393i q^{33} -2.14590 q^{34} -1.85410 q^{36} -5.00000i q^{37} -0.381966i q^{38} +3.00000 q^{39} -10.8541 q^{41} +1.38197i q^{42} -4.61803i q^{43} -3.27051 q^{44} +2.56231 q^{46} +13.1803i q^{47} -3.14590i q^{48} -6.09017 q^{49} -5.61803 q^{51} +5.56231i q^{52} -1.38197i q^{53} +0.381966 q^{54} -5.32624 q^{56} -1.00000i q^{57} +0.0901699i q^{58} +13.7984 q^{59} +6.09017 q^{61} -3.09017i q^{62} +3.61803i q^{63} +4.70820 q^{64} +0.673762 q^{66} +8.00000i q^{67} -10.4164i q^{68} +6.70820 q^{69} +7.85410 q^{71} +1.47214i q^{72} +15.8541i q^{73} -1.90983 q^{74} +1.85410 q^{76} +6.38197i q^{77} -1.14590i q^{78} -7.23607 q^{79} +1.00000 q^{81} +4.14590i q^{82} +7.32624i q^{83} -6.70820 q^{84} -1.76393 q^{86} +0.236068i q^{87} +2.59675i q^{88} -3.47214 q^{89} +10.8541 q^{91} +12.4377i q^{92} -8.09017i q^{93} +5.03444 q^{94} -4.14590 q^{96} +10.5623i q^{97} +2.32624i q^{98} +1.76393 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9	$ 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} + 26 q^{16} + 4 q^{19} - 10 q^{21} + 12 q^{24} + 18 q^{26} + 8 q^{29} + 10 q^{31} - 22 q^{34} + 6 q^{36} + 12 q^{39} - 30 q^{41} + 54 q^{44} - 30 q^{46} - 2 q^{49} - 18 q^{51} + 6 q^{54} + 10 q^{56} + 6 q^{59} + 2 q^{61} - 8 q^{64} + 34 q^{66} + 18 q^{71} - 30 q^{74} - 6 q^{76} - 20 q^{79} + 4 q^{81} - 16 q^{86} + 4 q^{89} + 30 q^{91} - 38 q^{94} - 30 q^{96} + 16 q^{99}+O(q^{100}) $ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 + 26 * q^16 + 4 * q^19 - 10 * q^21 + 12 * q^24 + 18 * q^26 + 8 * q^29 + 10 * q^31 - 22 * q^34 + 6 * q^36 + 12 * q^39 - 30 * q^41 + 54 * q^44 - 30 * q^46 - 2 * q^49 - 18 * q^51 + 6 * q^54 + 10 * q^56 + 6 * q^59 + 2 * q^61 - 8 * q^64 + 34 * q^66 + 18 * q^71 - 30 * q^74 - 6 * q^76 - 20 * q^79 + 4 * q^81 - 16 * q^86 + 4 * q^89 + 30 * q^91 - 38 * q^94 - 30 * q^96 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 0.381966i	− 0.270091i	−0.990839	−	0.135045i	$-0.956882\pi$
$2$	− 0.381966i	− 0.270091i	0.990839	−	0.135045i	$-0.0431180\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.85410	0.927051
$5$	0	0
$5$	0	0
$6$	−0.381966	−0.155937
$7$	− 3.61803i	− 1.36749i	−0.729722	−	0.683744i	$-0.760350\pi$
$7$	− 3.61803i	− 1.36749i	0.729722	−	0.683744i	$-0.239650\pi$
$8$	− 1.47214i	− 0.520479i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.76393	−0.531846	−0.265923	−	0.963994i	$-0.585677\pi$
$11$	−1.76393	−0.531846	−0.265923	+	0.963994i	$0.585677\pi$
$12$	− 1.85410i	− 0.535233i
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−1.38197	−0.369346
$15$	0	0
$16$	3.14590	0.786475
$17$	− 5.61803i	− 1.36257i	−0.732017	−	0.681287i	$-0.761421\pi$
$17$	− 5.61803i	− 1.36257i	0.732017	−	0.681287i	$-0.238579\pi$
$18$	0.381966i	0.0900303i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−3.61803	−0.789520
$22$	0.673762i	0.143647i
$23$	6.70820i	1.39876i	0.714751	+	0.699379i	$0.246540\pi$
$23$	6.70820i	1.39876i	−0.714751	+	0.699379i	$0.753460\pi$
$24$	−1.47214	−0.300498
$25$	0	0
$26$	1.14590	0.224729
$27$	1.00000i	0.192450i
$28$	− 6.70820i	− 1.26773i
$29$	−0.236068	−0.0438367	−0.0219184	−	0.999760i	$-0.506977\pi$
$29$	−0.236068	−0.0438367	−0.0219184	+	0.999760i	$0.506977\pi$
$30$	0	0
$31$	8.09017	1.45304	0.726519	−	0.687147i	$-0.241137\pi$
$31$	8.09017	1.45304	0.726519	+	0.687147i	$0.241137\pi$
$32$	− 4.14590i	− 0.732898i
$33$	1.76393i	0.307061i
$34$	−2.14590	−0.368018
$35$	0	0
$36$	−1.85410	−0.309017
$37$	− 5.00000i	− 0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$	− 5.00000i	− 0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$	− 0.381966i	− 0.0619631i
$39$	3.00000	0.480384
$40$	0	0
$41$	−10.8541	−1.69513	−0.847563	−	0.530695i	$-0.821931\pi$
$41$	−10.8541	−1.69513	−0.847563	+	0.530695i	$0.821931\pi$
$42$	1.38197i	0.213242i
$43$	− 4.61803i	− 0.704244i	−0.935954	−	0.352122i	$-0.885460\pi$
$43$	− 4.61803i	− 0.704244i	0.935954	−	0.352122i	$-0.114540\pi$
$44$	−3.27051	−0.493048
$45$	0	0
$46$	2.56231	0.377791
$47$	13.1803i	1.92255i	0.275591	+	0.961275i	$0.411127\pi$
$47$	13.1803i	1.92255i	−0.275591	+	0.961275i	$0.588873\pi$
$48$	− 3.14590i	− 0.454071i
$49$	−6.09017	−0.870024
$50$	0	0
$51$	−5.61803	−0.786682
$52$	5.56231i	0.771353i
$53$	− 1.38197i	− 0.189828i	−0.995485	−	0.0949138i	$-0.969742\pi$
$53$	− 1.38197i	− 0.189828i	0.995485	−	0.0949138i	$-0.0302576\pi$
$54$	0.381966	0.0519790
$55$	0	0
$56$	−5.32624	−0.711748
$57$	− 1.00000i	− 0.132453i
$58$	0.0901699i	0.0118399i
$59$	13.7984	1.79640	0.898198	−	0.439592i	$-0.144877\pi$
$59$	13.7984	1.79640	0.898198	+	0.439592i	$0.144877\pi$
$60$	0	0
$61$	6.09017	0.779766	0.389883	−	0.920864i	$-0.372515\pi$
$61$	6.09017	0.779766	0.389883	+	0.920864i	$0.372515\pi$
$62$	− 3.09017i	− 0.392452i
$63$	3.61803i	0.455829i
$64$	4.70820	0.588525
$65$	0	0
$66$	0.673762	0.0829344
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	− 10.4164i	− 1.26317i
$69$	6.70820	0.807573
$70$	0	0
$71$	7.85410	0.932110	0.466055	−	0.884756i	$-0.345675\pi$
$71$	7.85410	0.932110	0.466055	+	0.884756i	$0.345675\pi$
$72$	1.47214i	0.173493i
$73$	15.8541i	1.85558i	0.373100	+	0.927791i	$0.378295\pi$
$73$	15.8541i	1.85558i	−0.373100	+	0.927791i	$0.621705\pi$
$74$	−1.90983	−0.222013
$75$	0	0
$76$	1.85410	0.212680
$77$	6.38197i	0.727293i
$78$	− 1.14590i	− 0.129747i
$79$	−7.23607	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$79$	−7.23607	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.14590i	0.457838i
$83$	7.32624i	0.804159i	0.915605	+	0.402080i	$0.131712\pi$
$83$	7.32624i	0.804159i	−0.915605	+	0.402080i	$0.868288\pi$
$84$	−6.70820	−0.731925
$85$	0	0
$86$	−1.76393	−0.190210
$87$	0.236068i	0.0253091i
$88$	2.59675i	0.276814i
$89$	−3.47214	−0.368046	−0.184023	−	0.982922i	$-0.558912\pi$
$89$	−3.47214	−0.368046	−0.184023	+	0.982922i	$0.558912\pi$
$90$	0	0
$91$	10.8541	1.13782
$92$	12.4377i	1.29672i
$93$	− 8.09017i	− 0.838912i
$94$	5.03444	0.519263
$95$	0	0
$96$	−4.14590	−0.423139
$97$	10.5623i	1.07244i	0.844078	+	0.536220i	$0.180148\pi$
$97$	10.5623i	1.07244i	−0.844078	+	0.536220i	$0.819852\pi$
$98$	2.32624i	0.234986i
$99$	1.76393	0.177282
$100$	0	0
$101$	1.52786	0.152028	0.0760141	−	0.997107i	$-0.475781\pi$
$101$	1.52786	0.152028	0.0760141	+	0.997107i	$0.475781\pi$
$102$	2.14590i	0.212476i
$103$	0.909830i	0.0896482i	0.998995	+	0.0448241i	$0.0142727\pi$
$103$	0.909830i	0.0896482i	−0.998995	+	0.0448241i	$0.985727\pi$
$104$	4.41641	0.433064
$105$	0	0
$106$	−0.527864	−0.0512707
$107$	0.618034i	0.0597476i	0.999554	+	0.0298738i	$0.00951054\pi$
$107$	0.618034i	0.0597476i	−0.999554	+	0.0298738i	$0.990489\pi$
$108$	1.85410i	0.178411i
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	− 11.3820i	− 1.07549i
$113$	− 0.763932i	− 0.0718647i	−0.999354	−	0.0359323i	$-0.988560\pi$
$113$	− 0.763932i	− 0.0718647i	0.999354	−	0.0359323i	$-0.0114401\pi$
$114$	−0.381966	−0.0357744
$115$	0	0
$116$	−0.437694	−0.0406389
$117$	− 3.00000i	− 0.277350i
$118$	− 5.27051i	− 0.485190i
$119$	−20.3262	−1.86330
$120$	0	0
$121$	−7.88854	−0.717140
$122$	− 2.32624i	− 0.210608i
$123$	10.8541i	0.978681i
$124$	15.0000	1.34704
$125$	0	0
$126$	1.38197	0.123115
$127$	6.41641i	0.569364i	0.958622	+	0.284682i	$0.0918880\pi$
$127$	6.41641i	0.569364i	−0.958622	+	0.284682i	$0.908112\pi$
$128$	− 10.0902i	− 0.891853i
$129$	−4.61803	−0.406595
$130$	0	0
$131$	−10.0902	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$131$	−10.0902	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$132$	3.27051i	0.284661i
$133$	− 3.61803i	− 0.313723i
$134$	3.05573	0.263975
$135$	0	0
$136$	−8.27051	−0.709190
$137$	− 16.2361i	− 1.38714i	−0.720389	−	0.693570i	$-0.756037\pi$
$137$	− 16.2361i	− 1.38714i	0.720389	−	0.693570i	$-0.243963\pi$
$138$	− 2.56231i	− 0.218118i
$139$	1.76393	0.149615	0.0748074	−	0.997198i	$-0.476166\pi$
$139$	1.76393	0.149615	0.0748074	+	0.997198i	$0.476166\pi$
$140$	0	0
$141$	13.1803	1.10998
$142$	− 3.00000i	− 0.251754i
$143$	− 5.29180i	− 0.442522i
$144$	−3.14590	−0.262158
$145$	0	0
$146$	6.05573	0.501176
$147$	6.09017i	0.502309i
$148$	− 9.27051i	− 0.762031i
$149$	11.9443	0.978513	0.489256	−	0.872140i	$-0.337268\pi$
$149$	11.9443	0.978513	0.489256	+	0.872140i	$0.337268\pi$
$150$	0	0
$151$	−4.18034	−0.340191	−0.170096	−	0.985428i	$-0.554408\pi$
$151$	−4.18034	−0.340191	−0.170096	+	0.985428i	$0.554408\pi$
$152$	− 1.47214i	− 0.119406i
$153$	5.61803i	0.454191i
$154$	2.43769	0.196435
$155$	0	0
$156$	5.56231	0.445341
$157$	− 16.5623i	− 1.32182i	−0.750467	−	0.660908i	$-0.770171\pi$
$157$	− 16.5623i	− 1.32182i	0.750467	−	0.660908i	$-0.229829\pi$
$158$	2.76393i	0.219887i
$159$	−1.38197	−0.109597
$160$	0	0
$161$	24.2705	1.91278
$162$	− 0.381966i	− 0.0300101i
$163$	− 9.00000i	− 0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$	− 9.00000i	− 0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$	−20.1246	−1.57147
$165$	0	0
$166$	2.79837	0.217196
$167$	− 4.47214i	− 0.346064i	−0.984916	−	0.173032i	$-0.944644\pi$
$167$	− 4.47214i	− 0.346064i	0.984916	−	0.173032i	$-0.0553564\pi$
$168$	5.32624i	0.410928i
$169$	4.00000	0.307692
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 8.56231i	− 0.652870i
$173$	− 3.32624i	− 0.252889i	−0.991974	−	0.126445i	$-0.959643\pi$
$173$	− 3.32624i	− 0.252889i	0.991974	−	0.126445i	$-0.0403566\pi$
$174$	0.0901699	0.00683577
$175$	0	0
$176$	−5.54915	−0.418283
$177$	− 13.7984i	− 1.03715i
$178$	1.32624i	0.0994057i
$179$	17.7639	1.32774	0.663869	−	0.747849i	$-0.268913\pi$
$179$	17.7639	1.32774	0.663869	+	0.747849i	$0.268913\pi$
$180$	0	0
$181$	−13.4164	−0.997234	−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164	−0.997234	−0.498617	+	0.866822i	$0.666159\pi$
$182$	− 4.14590i	− 0.307314i
$183$	− 6.09017i	− 0.450198i
$184$	9.87539	0.728023
$185$	0	0
$186$	−3.09017	−0.226582
$187$	9.90983i	0.724679i
$188$	24.4377i	1.78230i
$189$	3.61803	0.263173
$190$	0	0
$191$	−17.9443	−1.29840	−0.649201	−	0.760617i	$-0.724897\pi$
$191$	−17.9443	−1.29840	−0.649201	+	0.760617i	$0.724897\pi$
$192$	− 4.70820i	− 0.339785i
$193$	8.70820i	0.626830i	0.949616	+	0.313415i	$0.101473\pi$
$193$	8.70820i	0.626830i	−0.949616	+	0.313415i	$0.898527\pi$
$194$	4.03444	0.289656
$195$	0	0
$196$	−11.2918	−0.806557
$197$	− 7.18034i	− 0.511578i	−0.966733	−	0.255789i	$-0.917665\pi$
$197$	− 7.18034i	− 0.511578i	0.966733	−	0.255789i	$-0.0823352\pi$
$198$	− 0.673762i	− 0.0478822i
$199$	−5.29180	−0.375125	−0.187563	−	0.982253i	$-0.560059\pi$
$199$	−5.29180	−0.375125	−0.187563	+	0.982253i	$0.560059\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 0.583592i	− 0.0410614i
$203$	0.854102i	0.0599462i
$204$	−10.4164	−0.729294
$205$	0	0
$206$	0.347524	0.0242132
$207$	− 6.70820i	− 0.466252i
$208$	9.43769i	0.654386i
$209$	−1.76393	−0.122014
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	− 2.56231i	− 0.175980i
$213$	− 7.85410i	− 0.538154i
$214$	0.236068	0.0161373
$215$	0	0
$216$	1.47214	0.100166
$217$	− 29.2705i	− 1.98701i
$218$	3.43769i	0.232830i
$219$	15.8541	1.07132
$220$	0	0
$221$	16.8541	1.13373
$222$	1.90983i	0.128179i
$223$	10.8885i	0.729151i	0.931174	+	0.364575i	$0.118786\pi$
$223$	10.8885i	0.729151i	−0.931174	+	0.364575i	$0.881214\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−0.291796	−0.0194100
$227$	− 6.18034i	− 0.410204i	−0.978741	−	0.205102i	$-0.934247\pi$
$227$	− 6.18034i	− 0.410204i	0.978741	−	0.205102i	$-0.0657525\pi$
$228$	− 1.85410i	− 0.122791i
$229$	−21.1246	−1.39595	−0.697977	−	0.716120i	$-0.745916\pi$
$229$	−21.1246	−1.39595	−0.697977	+	0.716120i	$0.745916\pi$
$230$	0	0
$231$	6.38197	0.419903
$232$	0.347524i	0.0228161i
$233$	− 12.5066i	− 0.819333i	−0.912235	−	0.409667i	$-0.865645\pi$
$233$	− 12.5066i	− 0.819333i	0.912235	−	0.409667i	$-0.134355\pi$
$234$	−1.14590	−0.0749097
$235$	0	0
$236$	25.5836	1.66535
$237$	7.23607i	0.470033i
$238$	7.76393i	0.503261i
$239$	24.0902	1.55826	0.779132	−	0.626860i	$-0.215660\pi$
$239$	24.0902	1.55826	0.779132	+	0.626860i	$0.215660\pi$
$240$	0	0
$241$	−15.7082	−1.01185	−0.505927	−	0.862576i	$-0.668850\pi$
$241$	−15.7082	−1.01185	−0.505927	+	0.862576i	$0.668850\pi$
$242$	3.01316i	0.193693i
$243$	− 1.00000i	− 0.0641500i
$244$	11.2918	0.722883
$245$	0	0
$246$	4.14590	0.264333
$247$	3.00000i	0.190885i
$248$	− 11.9098i	− 0.756275i
$249$	7.32624	0.464281
$250$	0	0
$251$	2.32624	0.146831	0.0734154	−	0.997301i	$-0.476610\pi$
$251$	2.32624	0.146831	0.0734154	+	0.997301i	$0.476610\pi$
$252$	6.70820i	0.422577i
$253$	− 11.8328i	− 0.743923i
$254$	2.45085	0.153780
$255$	0	0
$256$	5.56231	0.347644
$257$	3.18034i	0.198384i	0.995068	+	0.0991921i	$0.0316258\pi$
$257$	3.18034i	0.198384i	−0.995068	+	0.0991921i	$0.968374\pi$
$258$	1.76393i	0.109818i
$259$	−18.0902	−1.12407
$260$	0	0
$261$	0.236068	0.0146122
$262$	3.85410i	0.238107i
$263$	− 13.1459i	− 0.810611i	−0.914181	−	0.405305i	$-0.867165\pi$
$263$	− 13.1459i	− 0.810611i	0.914181	−	0.405305i	$-0.132835\pi$
$264$	2.59675	0.159819
$265$	0	0
$266$	−1.38197	−0.0847338
$267$	3.47214i	0.212491i
$268$	14.8328i	0.906058i
$269$	−20.3820	−1.24271	−0.621355	−	0.783529i	$-0.713418\pi$
$269$	−20.3820	−1.24271	−0.621355	+	0.783529i	$0.713418\pi$
$270$	0	0
$271$	1.85410	0.112629	0.0563143	−	0.998413i	$-0.482065\pi$
$271$	1.85410	0.112629	0.0563143	+	0.998413i	$0.482065\pi$
$272$	− 17.6738i	− 1.07163i
$273$	− 10.8541i	− 0.656920i
$274$	−6.20163	−0.374654
$275$	0	0
$276$	12.4377	0.748661
$277$	− 15.2705i	− 0.917516i	−0.888561	−	0.458758i	$-0.848294\pi$
$277$	− 15.2705i	− 0.917516i	0.888561	−	0.458758i	$-0.151706\pi$
$278$	− 0.673762i	− 0.0404096i
$279$	−8.09017	−0.484346
$280$	0	0
$281$	3.27051	0.195102	0.0975511	−	0.995231i	$-0.468899\pi$
$281$	3.27051	0.195102	0.0975511	+	0.995231i	$0.468899\pi$
$282$	− 5.03444i	− 0.299797i
$283$	− 21.4721i	− 1.27639i	−0.769876	−	0.638193i	$-0.779682\pi$
$283$	− 21.4721i	− 1.27639i	0.769876	−	0.638193i	$-0.220318\pi$
$284$	14.5623	0.864114
$285$	0	0
$286$	−2.02129	−0.119521
$287$	39.2705i	2.31806i
$288$	4.14590i	0.244299i
$289$	−14.5623	−0.856606
$290$	0	0
$291$	10.5623	0.619173
$292$	29.3951i	1.72022i
$293$	19.4164i	1.13432i	0.823608	+	0.567159i	$0.191958\pi$
$293$	19.4164i	1.13432i	−0.823608	+	0.567159i	$0.808042\pi$
$294$	2.32624	0.135669
$295$	0	0
$296$	−7.36068	−0.427831
$297$	− 1.76393i	− 0.102354i
$298$	− 4.56231i	− 0.264287i
$299$	−20.1246	−1.16384
$300$	0	0
$301$	−16.7082	−0.963045
$302$	1.59675i	0.0918825i
$303$	− 1.52786i	− 0.0877735i
$304$	3.14590	0.180430
$305$	0	0
$306$	2.14590	0.122673
$307$	22.1246i	1.26272i	0.775491	+	0.631359i	$0.217503\pi$
$307$	22.1246i	1.26272i	−0.775491	+	0.631359i	$0.782497\pi$
$308$	11.8328i	0.674237i
$309$	0.909830	0.0517584
$310$	0	0
$311$	19.0902	1.08250	0.541252	−	0.840860i	$-0.317950\pi$
$311$	19.0902	1.08250	0.541252	+	0.840860i	$0.317950\pi$
$312$	− 4.41641i	− 0.250030i
$313$	− 19.9787i	− 1.12926i	−0.825343	−	0.564632i	$-0.809018\pi$
$313$	− 19.9787i	− 1.12926i	0.825343	−	0.564632i	$-0.190982\pi$
$314$	−6.32624	−0.357010
$315$	0	0
$316$	−13.4164	−0.754732
$317$	− 16.4164i	− 0.922037i	−0.887390	−	0.461019i	$-0.847484\pi$
$317$	− 16.4164i	− 0.922037i	0.887390	−	0.461019i	$-0.152516\pi$
$318$	0.527864i	0.0296011i
$319$	0.416408	0.0233144
$320$	0	0
$321$	0.618034	0.0344953
$322$	− 9.27051i	− 0.516625i
$323$	− 5.61803i	− 0.312596i
$324$	1.85410	0.103006
$325$	0	0
$326$	−3.43769	−0.190396
$327$	9.00000i	0.497701i
$328$	15.9787i	0.882277i
$329$	47.6869	2.62906
$330$	0	0
$331$	6.14590	0.337809	0.168905	−	0.985632i	$-0.445977\pi$
$331$	6.14590	0.337809	0.168905	+	0.985632i	$0.445977\pi$
$332$	13.5836i	0.745496i
$333$	5.00000i	0.273998i
$334$	−1.70820	−0.0934688
$335$	0	0
$336$	−11.3820	−0.620937
$337$	18.4721i	1.00624i	0.864216	+	0.503121i	$0.167815\pi$
$337$	18.4721i	1.00624i	−0.864216	+	0.503121i	$0.832185\pi$
$338$	− 1.52786i	− 0.0831048i
$339$	−0.763932	−0.0414911
$340$	0	0
$341$	−14.2705	−0.772791
$342$	0.381966i	0.0206544i
$343$	− 3.29180i	− 0.177740i
$344$	−6.79837	−0.366544
$345$	0	0
$346$	−1.27051	−0.0683030
$347$	14.5066i	0.778754i	0.921078	+	0.389377i	$0.127310\pi$
$347$	14.5066i	0.778754i	−0.921078	+	0.389377i	$0.872690\pi$
$348$	0.437694i	0.0234629i
$349$	18.9098	1.01222	0.506110	−	0.862469i	$-0.331083\pi$
$349$	18.9098	1.01222	0.506110	+	0.862469i	$0.331083\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	7.31308i	0.389789i
$353$	33.3262i	1.77378i	0.461984	+	0.886888i	$0.347138\pi$
$353$	33.3262i	1.77378i	−0.461984	+	0.886888i	$0.652862\pi$
$354$	−5.27051	−0.280124
$355$	0	0
$356$	−6.43769	−0.341197
$357$	20.3262i	1.07578i
$358$	− 6.78522i	− 0.358610i
$359$	−11.5279	−0.608417	−0.304209	−	0.952605i	$-0.598392\pi$
$359$	−11.5279	−0.608417	−0.304209	+	0.952605i	$0.598392\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	5.12461i	0.269344i
$363$	7.88854i	0.414041i
$364$	20.1246	1.05482
$365$	0	0
$366$	−2.32624	−0.121594
$367$	20.9098i	1.09148i	0.837953	+	0.545742i	$0.183752\pi$
$367$	20.9098i	1.09148i	−0.837953	+	0.545742i	$0.816248\pi$
$368$	21.1033i	1.10009i
$369$	10.8541	0.565042
$370$	0	0
$371$	−5.00000	−0.259587
$372$	− 15.0000i	− 0.777714i
$373$	− 35.2705i	− 1.82624i	−0.407693	−	0.913119i	$-0.633667\pi$
$373$	− 35.2705i	− 1.82624i	0.407693	−	0.913119i	$-0.366333\pi$
$374$	3.78522	0.195729
$375$	0	0
$376$	19.4033	1.00065
$377$	− 0.708204i	− 0.0364744i
$378$	− 1.38197i	− 0.0710807i
$379$	2.56231	0.131617	0.0658084	−	0.997832i	$-0.479037\pi$
$379$	2.56231	0.131617	0.0658084	+	0.997832i	$0.479037\pi$
$380$	0	0
$381$	6.41641	0.328723
$382$	6.85410i	0.350686i
$383$	− 9.97871i	− 0.509888i	−0.966956	−	0.254944i	$-0.917943\pi$
$383$	− 9.97871i	− 0.509888i	0.966956	−	0.254944i	$-0.0820571\pi$
$384$	−10.0902	−0.514912
$385$	0	0
$386$	3.32624	0.169301
$387$	4.61803i	0.234748i
$388$	19.5836i	0.994206i
$389$	−18.0344	−0.914382	−0.457191	−	0.889368i	$-0.651145\pi$
$389$	−18.0344	−0.914382	−0.457191	+	0.889368i	$0.651145\pi$
$390$	0	0
$391$	37.6869	1.90591
$392$	8.96556i	0.452829i
$393$	10.0902i	0.508982i
$394$	−2.74265	−0.138172
$395$	0	0
$396$	3.27051	0.164349
$397$	− 16.9443i	− 0.850409i	−0.905097	−	0.425204i	$-0.860202\pi$
$397$	− 16.9443i	− 0.850409i	0.905097	−	0.425204i	$-0.139798\pi$
$398$	2.02129i	0.101318i
$399$	−3.61803	−0.181128
$400$	0	0
$401$	13.6525	0.681772	0.340886	−	0.940105i	$-0.389273\pi$
$401$	13.6525	0.681772	0.340886	+	0.940105i	$0.389273\pi$
$402$	− 3.05573i	− 0.152406i
$403$	24.2705i	1.20900i
$404$	2.83282	0.140938
$405$	0	0
$406$	0.326238	0.0161909
$407$	8.81966i	0.437174i
$408$	8.27051i	0.409451i
$409$	16.7639	0.828923	0.414462	−	0.910067i	$-0.363970\pi$
$409$	16.7639	0.828923	0.414462	+	0.910067i	$0.363970\pi$
$410$	0	0
$411$	−16.2361	−0.800866
$412$	1.68692i	0.0831085i
$413$	− 49.9230i	− 2.45655i
$414$	−2.56231	−0.125930
$415$	0	0
$416$	12.4377	0.609808
$417$	− 1.76393i	− 0.0863801i
$418$	0.673762i	0.0329548i
$419$	−3.27051	−0.159775	−0.0798874	−	0.996804i	$-0.525456\pi$
$419$	−3.27051	−0.159775	−0.0798874	+	0.996804i	$0.525456\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	8.94427i	0.435400i
$423$	− 13.1803i	− 0.640850i
$424$	−2.03444	−0.0988012
$425$	0	0
$426$	−3.00000	−0.145350
$427$	− 22.0344i	− 1.06632i
$428$	1.14590i	0.0553891i
$429$	−5.29180	−0.255490
$430$	0	0
$431$	−7.76393	−0.373975	−0.186988	−	0.982362i	$-0.559872\pi$
$431$	−7.76393	−0.373975	−0.186988	+	0.982362i	$0.559872\pi$
$432$	3.14590i	0.151357i
$433$	27.3050i	1.31219i	0.754677	+	0.656096i	$0.227793\pi$
$433$	27.3050i	1.31219i	−0.754677	+	0.656096i	$0.772207\pi$
$434$	−11.1803	−0.536673
$435$	0	0
$436$	−16.6869	−0.799158
$437$	6.70820i	0.320897i
$438$	− 6.05573i	− 0.289354i
$439$	−1.38197	−0.0659576	−0.0329788	−	0.999456i	$-0.510499\pi$
$439$	−1.38197	−0.0659576	−0.0329788	+	0.999456i	$0.510499\pi$
$440$	0	0
$441$	6.09017	0.290008
$442$	− 6.43769i	− 0.306210i
$443$	− 22.4508i	− 1.06667i	−0.845903	−	0.533336i	$-0.820938\pi$
$443$	− 22.4508i	− 1.06667i	0.845903	−	0.533336i	$-0.179062\pi$
$444$	−9.27051	−0.439959
$445$	0	0
$446$	4.15905	0.196937
$447$	− 11.9443i	− 0.564945i
$448$	− 17.0344i	− 0.804802i
$449$	4.50658	0.212679	0.106339	−	0.994330i	$-0.466087\pi$
$449$	4.50658	0.212679	0.106339	+	0.994330i	$0.466087\pi$
$450$	0	0
$451$	19.1459	0.901545
$452$	− 1.41641i	− 0.0666222i
$453$	4.18034i	0.196410i
$454$	−2.36068	−0.110792
$455$	0	0
$456$	−1.47214	−0.0689391
$457$	17.6869i	0.827359i	0.910423	+	0.413680i	$0.135757\pi$
$457$	17.6869i	0.827359i	−0.910423	+	0.413680i	$0.864243\pi$
$458$	8.06888i	0.377034i
$459$	5.61803	0.262227
$460$	0	0
$461$	−17.4721	−0.813758	−0.406879	−	0.913482i	$-0.633383\pi$
$461$	−17.4721	−0.813758	−0.406879	+	0.913482i	$0.633383\pi$
$462$	− 2.43769i	− 0.113412i
$463$	− 0.673762i	− 0.0313124i	−0.999877	−	0.0156562i	$-0.995016\pi$
$463$	− 0.673762i	− 0.0313124i	0.999877	−	0.0156562i	$-0.00498372\pi$
$464$	−0.742646	−0.0344765
$465$	0	0
$466$	−4.77709	−0.221294
$467$	− 4.90983i	− 0.227200i	−0.993527	−	0.113600i	$-0.963762\pi$
$467$	− 4.90983i	− 0.227200i	0.993527	−	0.113600i	$-0.0362382\pi$
$468$	− 5.56231i	− 0.257118i
$469$	28.9443	1.33652
$470$	0	0
$471$	−16.5623	−0.763151
$472$	− 20.3131i	− 0.934985i
$473$	8.14590i	0.374549i
$474$	2.76393	0.126952
$475$	0	0
$476$	−37.6869	−1.72738
$477$	1.38197i	0.0632759i
$478$	− 9.20163i	− 0.420873i
$479$	−19.5967	−0.895398	−0.447699	−	0.894184i	$-0.647756\pi$
$479$	−19.5967	−0.895398	−0.447699	+	0.894184i	$0.647756\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	6.00000i	0.273293i
$483$	− 24.2705i	− 1.10435i
$484$	−14.6262	−0.664826
$485$	0	0
$486$	−0.381966	−0.0173263
$487$	6.94427i	0.314675i	0.987545	+	0.157337i	$0.0502910\pi$
$487$	6.94427i	0.314675i	−0.987545	+	0.157337i	$0.949709\pi$
$488$	− 8.96556i	− 0.405852i
$489$	−9.00000	−0.406994
$490$	0	0
$491$	−16.7984	−0.758100	−0.379050	−	0.925376i	$-0.623749\pi$
$491$	−16.7984	−0.758100	−0.379050	+	0.925376i	$0.623749\pi$
$492$	20.1246i	0.907288i
$493$	1.32624i	0.0597308i
$494$	1.14590	0.0515564
$495$	0	0
$496$	25.4508	1.14278
$497$	− 28.4164i	− 1.27465i
$498$	− 2.79837i	− 0.125398i
$499$	−17.2918	−0.774087	−0.387044	−	0.922061i	$-0.626504\pi$
$499$	−17.2918	−0.774087	−0.387044	+	0.922061i	$0.626504\pi$
$500$	0	0
$501$	−4.47214	−0.199800
$502$	− 0.888544i	− 0.0396577i
$503$	13.8541i	0.617724i	0.951107	+	0.308862i	$0.0999481\pi$
$503$	13.8541i	0.617724i	−0.951107	+	0.308862i	$0.900052\pi$
$504$	5.32624	0.237249
$505$	0	0
$506$	−4.51973	−0.200927
$507$	− 4.00000i	− 0.177646i
$508$	11.8967i	0.527830i
$509$	34.6525	1.53594	0.767972	−	0.640483i	$-0.221266\pi$
$509$	34.6525	1.53594	0.767972	+	0.640483i	$0.221266\pi$
$510$	0	0
$511$	57.3607	2.53749
$512$	− 22.3050i	− 0.985749i
$513$	1.00000i	0.0441511i
$514$	1.21478	0.0535817
$515$	0	0
$516$	−8.56231	−0.376934
$517$	− 23.2492i	− 1.02250i
$518$	6.90983i	0.303601i
$519$	−3.32624	−0.146006
$520$	0	0
$521$	0.326238	0.0142927	0.00714637	−	0.999974i	$-0.497725\pi$
$521$	0.326238	0.0142927	0.00714637	+	0.999974i	$0.497725\pi$
$522$	− 0.0901699i	− 0.00394663i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−18.7082	−0.817272
$525$	0	0
$526$	−5.02129	−0.218938
$527$	− 45.4508i	− 1.97987i
$528$	5.54915i	0.241496i
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−13.7984	−0.598798
$532$	− 6.70820i	− 0.290838i
$533$	− 32.5623i	− 1.41043i
$534$	1.32624	0.0573919
$535$	0	0
$536$	11.7771	0.508693
$537$	− 17.7639i	− 0.766570i
$538$	7.78522i	0.335645i
$539$	10.7426	0.462719
$540$	0	0
$541$	17.0344	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$541$	17.0344	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$542$	− 0.708204i	− 0.0304200i
$543$	13.4164i	0.575753i
$544$	−23.2918	−0.998628
$545$	0	0
$546$	−4.14590	−0.177428
$547$	1.50658i	0.0644166i	0.999481	+	0.0322083i	$0.0102540\pi$
$547$	1.50658i	0.0644166i	−0.999481	+	0.0322083i	$0.989746\pi$
$548$	− 30.1033i	− 1.28595i
$549$	−6.09017	−0.259922
$550$	0	0
$551$	−0.236068	−0.0100568
$552$	− 9.87539i	− 0.420324i
$553$	26.1803i	1.11330i
$554$	−5.83282	−0.247813
$555$	0	0
$556$	3.27051	0.138701
$557$	− 20.2148i	− 0.856528i	−0.903654	−	0.428264i	$-0.859125\pi$
$557$	− 20.2148i	− 0.856528i	0.903654	−	0.428264i	$-0.140875\pi$
$558$	3.09017i	0.130817i
$559$	13.8541	0.585966
$560$	0	0
$561$	9.90983	0.418393
$562$	− 1.24922i	− 0.0526953i
$563$	30.2148i	1.27340i	0.771111	+	0.636701i	$0.219701\pi$
$563$	30.2148i	1.27340i	−0.771111	+	0.636701i	$0.780299\pi$
$564$	24.4377	1.02901
$565$	0	0
$566$	−8.20163	−0.344740
$567$	− 3.61803i	− 0.151943i
$568$	− 11.5623i	− 0.485144i
$569$	−29.6525	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$569$	−29.6525	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$570$	0	0
$571$	31.4508	1.31618	0.658089	−	0.752941i	$-0.271365\pi$
$571$	31.4508	1.31618	0.658089	+	0.752941i	$0.271365\pi$
$572$	− 9.81153i	− 0.410241i
$573$	17.9443i	0.749633i
$574$	15.0000	0.626088
$575$	0	0
$576$	−4.70820	−0.196175
$577$	9.83282i	0.409345i	0.978830	+	0.204673i	$0.0656130\pi$
$577$	9.83282i	0.409345i	−0.978830	+	0.204673i	$0.934387\pi$
$578$	5.56231i	0.231361i
$579$	8.70820	0.361901
$580$	0	0
$581$	26.5066	1.09968
$582$	− 4.03444i	− 0.167233i
$583$	2.43769i	0.100959i
$584$	23.3394	0.965791
$585$	0	0
$586$	7.41641	0.306369
$587$	25.9443i	1.07083i	0.844588	+	0.535417i	$0.179846\pi$
$587$	25.9443i	1.07083i	−0.844588	+	0.535417i	$0.820154\pi$
$588$	11.2918i	0.465666i
$589$	8.09017	0.333350
$590$	0	0
$591$	−7.18034	−0.295360
$592$	− 15.7295i	− 0.646478i
$593$	31.3607i	1.28783i	0.765098	+	0.643914i	$0.222691\pi$
$593$	31.3607i	1.28783i	−0.765098	+	0.643914i	$0.777309\pi$
$594$	−0.673762	−0.0276448
$595$	0	0
$596$	22.1459	0.907131
$597$	5.29180i	0.216579i
$598$	7.68692i	0.314341i
$599$	38.4508	1.57106	0.785530	−	0.618824i	$-0.212391\pi$
$599$	38.4508	1.57106	0.785530	+	0.618824i	$0.212391\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	6.38197i	0.260110i
$603$	− 8.00000i	− 0.325785i
$604$	−7.75078	−0.315375
$605$	0	0
$606$	−0.583592	−0.0237068
$607$	− 24.3607i	− 0.988769i	−0.869243	−	0.494385i	$-0.835393\pi$
$607$	− 24.3607i	− 0.988769i	0.869243	−	0.494385i	$-0.164607\pi$
$608$	− 4.14590i	− 0.168138i
$609$	0.854102	0.0346100
$610$	0	0
$611$	−39.5410	−1.59966
$612$	10.4164i	0.421058i
$613$	− 32.5967i	− 1.31657i	−0.752769	−	0.658285i	$-0.771282\pi$
$613$	− 32.5967i	− 1.31657i	0.752769	−	0.658285i	$-0.228718\pi$
$614$	8.45085	0.341049
$615$	0	0
$616$	9.39512	0.378540
$617$	− 6.23607i	− 0.251055i	−0.992090	−	0.125527i	$-0.959938\pi$
$617$	− 6.23607i	− 0.251055i	0.992090	−	0.125527i	$-0.0400623\pi$
$618$	− 0.347524i	− 0.0139795i
$619$	−19.4164	−0.780411	−0.390206	−	0.920728i	$-0.627596\pi$
$619$	−19.4164	−0.780411	−0.390206	+	0.920728i	$0.627596\pi$
$620$	0	0
$621$	−6.70820	−0.269191
$622$	− 7.29180i	− 0.292374i
$623$	12.5623i	0.503298i
$624$	9.43769	0.377810
$625$	0	0
$626$	−7.63119	−0.305004
$627$	1.76393i	0.0704447i
$628$	− 30.7082i	− 1.22539i
$629$	−28.0902	−1.12003
$630$	0	0
$631$	−1.29180	−0.0514256	−0.0257128	−	0.999669i	$-0.508186\pi$
$631$	−1.29180	−0.0514256	−0.0257128	+	0.999669i	$0.508186\pi$
$632$	10.6525i	0.423733i
$633$	23.4164i	0.930719i
$634$	−6.27051	−0.249034
$635$	0	0
$636$	−2.56231	−0.101602
$637$	− 18.2705i	− 0.723904i
$638$	− 0.159054i	− 0.00629699i
$639$	−7.85410	−0.310703
$640$	0	0
$641$	−9.59675	−0.379049	−0.189524	−	0.981876i	$-0.560695\pi$
$641$	−9.59675	−0.379049	−0.189524	+	0.981876i	$0.560695\pi$
$642$	− 0.236068i	− 0.00931686i
$643$	22.9443i	0.904834i	0.891807	+	0.452417i	$0.149438\pi$
$643$	22.9443i	0.904834i	−0.891807	+	0.452417i	$0.850562\pi$
$644$	45.0000	1.77325
$645$	0	0
$646$	−2.14590	−0.0844292
$647$	− 22.8541i	− 0.898487i	−0.893409	−	0.449244i	$-0.851693\pi$
$647$	− 22.8541i	− 0.898487i	0.893409	−	0.449244i	$-0.148307\pi$
$648$	− 1.47214i	− 0.0578310i
$649$	−24.3394	−0.955405
$650$	0	0
$651$	−29.2705	−1.14720
$652$	− 16.6869i	− 0.653510i
$653$	− 37.0132i	− 1.44844i	−0.689571	−	0.724218i	$-0.742201\pi$
$653$	− 37.0132i	− 1.44844i	0.689571	−	0.724218i	$-0.257799\pi$
$654$	3.43769	0.134424
$655$	0	0
$656$	−34.1459	−1.33317
$657$	− 15.8541i	− 0.618527i
$658$	− 18.2148i	− 0.710086i
$659$	−29.0689	−1.13236	−0.566181	−	0.824281i	$-0.691580\pi$
$659$	−29.0689	−1.13236	−0.566181	+	0.824281i	$0.691580\pi$
$660$	0	0
$661$	−13.9787	−0.543709	−0.271854	−	0.962338i	$-0.587637\pi$
$661$	−13.9787	−0.543709	−0.271854	+	0.962338i	$0.587637\pi$
$662$	− 2.34752i	− 0.0912391i
$663$	− 16.8541i	− 0.654559i
$664$	10.7852	0.418548
$665$	0	0
$666$	1.90983	0.0740044
$667$	− 1.58359i	− 0.0613169i
$668$	− 8.29180i	− 0.320819i
$669$	10.8885	0.420975
$670$	0	0
$671$	−10.7426	−0.414715
$672$	15.0000i	0.578638i
$673$	− 16.5836i	− 0.639250i	−0.947544	−	0.319625i	$-0.896443\pi$
$673$	− 16.5836i	− 0.639250i	0.947544	−	0.319625i	$-0.103557\pi$
$674$	7.05573	0.271776
$675$	0	0
$676$	7.41641	0.285246
$677$	38.0689i	1.46311i	0.681785	+	0.731553i	$0.261204\pi$
$677$	38.0689i	1.46311i	−0.681785	+	0.731553i	$0.738796\pi$
$678$	0.291796i	0.0112064i
$679$	38.2148	1.46655
$680$	0	0
$681$	−6.18034	−0.236831
$682$	5.45085i	0.208724i
$683$	21.0000i	0.803543i	0.915740	+	0.401771i	$0.131605\pi$
$683$	21.0000i	0.803543i	−0.915740	+	0.401771i	$0.868395\pi$
$684$	−1.85410	−0.0708934
$685$	0	0
$686$	−1.25735	−0.0480060
$687$	21.1246i	0.805954i
$688$	− 14.5279i	− 0.553870i
$689$	4.14590	0.157946
$690$	0	0
$691$	14.8197	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$691$	14.8197	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$692$	− 6.16718i	− 0.234441i
$693$	− 6.38197i	− 0.242431i
$694$	5.54102	0.210334
$695$	0	0
$696$	0.347524	0.0131729
$697$	60.9787i	2.30973i
$698$	− 7.22291i	− 0.273391i
$699$	−12.5066	−0.473042
$700$	0	0
$701$	34.1803	1.29097	0.645487	−	0.763771i	$-0.276655\pi$
$701$	34.1803	1.29097	0.645487	+	0.763771i	$0.276655\pi$
$702$	1.14590i	0.0432491i
$703$	− 5.00000i	− 0.188579i
$704$	−8.30495	−0.313005
$705$	0	0
$706$	12.7295	0.479081
$707$	− 5.52786i	− 0.207897i
$708$	− 25.5836i	− 0.961490i
$709$	−0.708204	−0.0265972	−0.0132986	−	0.999912i	$-0.504233\pi$
$709$	−0.708204	−0.0265972	−0.0132986	+	0.999912i	$0.504233\pi$
$710$	0	0
$711$	7.23607	0.271374
$712$	5.11146i	0.191560i
$713$	54.2705i	2.03245i
$714$	7.76393	0.290558
$715$	0	0
$716$	32.9361	1.23088
$717$	− 24.0902i	− 0.899664i
$718$	4.40325i	0.164328i
$719$	−1.47214	−0.0549014	−0.0274507	−	0.999623i	$-0.508739\pi$
$719$	−1.47214	−0.0549014	−0.0274507	+	0.999623i	$0.508739\pi$
$720$	0	0
$721$	3.29180	0.122593
$722$	6.87539i	0.255875i
$723$	15.7082i	0.584194i
$724$	−24.8754	−0.924487
$725$	0	0
$726$	3.01316	0.111829
$727$	4.41641i	0.163796i	0.996641	+	0.0818978i	$0.0260981\pi$
$727$	4.41641i	0.163796i	−0.996641	+	0.0818978i	$0.973902\pi$
$728$	− 15.9787i	− 0.592211i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−25.9443	−0.959584
$732$	− 11.2918i	− 0.417357i
$733$	− 7.05573i	− 0.260609i	−0.991474	−	0.130305i	$-0.958404\pi$
$733$	− 7.05573i	− 0.260609i	0.991474	−	0.130305i	$-0.0415955\pi$
$734$	7.98684	0.294800
$735$	0	0
$736$	27.8115	1.02515
$737$	− 14.1115i	− 0.519802i
$738$	− 4.14590i	− 0.152613i
$739$	−40.2705	−1.48137	−0.740687	−	0.671850i	$-0.765500\pi$
$739$	−40.2705	−1.48137	−0.740687	+	0.671850i	$0.765500\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	1.90983i	0.0701121i
$743$	− 40.0689i	− 1.46998i	−0.678075	−	0.734992i	$-0.737186\pi$
$743$	− 40.0689i	− 1.46998i	0.678075	−	0.734992i	$-0.262814\pi$
$744$	−11.9098	−0.436636
$745$	0	0
$746$	−13.4721	−0.493250
$747$	− 7.32624i	− 0.268053i
$748$	18.3738i	0.671814i
$749$	2.23607	0.0817041
$750$	0	0
$751$	−0.785218	−0.0286530	−0.0143265	−	0.999897i	$-0.504560\pi$
$751$	−0.785218	−0.0286530	−0.0143265	+	0.999897i	$0.504560\pi$
$752$	41.4640i	1.51204i
$753$	− 2.32624i	− 0.0847728i
$754$	−0.270510	−0.00985139
$755$	0	0
$756$	6.70820	0.243975
$757$	11.6525i	0.423516i	0.977322	+	0.211758i	$0.0679189\pi$
$757$	11.6525i	0.423516i	−0.977322	+	0.211758i	$0.932081\pi$
$758$	− 0.978714i	− 0.0355485i
$759$	−11.8328	−0.429504
$760$	0	0
$761$	−35.0689	−1.27125	−0.635623	−	0.772000i	$-0.719257\pi$
$761$	−35.0689	−1.27125	−0.635623	+	0.772000i	$0.719257\pi$
$762$	− 2.45085i	− 0.0887849i
$763$	32.5623i	1.17883i
$764$	−33.2705	−1.20368
$765$	0	0
$766$	−3.81153	−0.137716
$767$	41.3951i	1.49469i
$768$	− 5.56231i	− 0.200712i
$769$	16.9443	0.611026	0.305513	−	0.952188i	$-0.401172\pi$
$769$	16.9443	0.611026	0.305513	+	0.952188i	$0.401172\pi$
$770$	0	0
$771$	3.18034	0.114537
$772$	16.1459i	0.581104i
$773$	21.0000i	0.755318i	0.925945	+	0.377659i	$0.123271\pi$
$773$	21.0000i	0.755318i	−0.925945	+	0.377659i	$0.876729\pi$
$774$	1.76393	0.0634032
$775$	0	0
$776$	15.5492	0.558182
$777$	18.0902i	0.648981i
$778$	6.88854i	0.246966i
$779$	−10.8541	−0.388889
$780$	0	0
$781$	−13.8541	−0.495739
$782$	− 14.3951i	− 0.514768i
$783$	− 0.236068i	− 0.00843638i
$784$	−19.1591	−0.684252
$785$	0	0
$786$	3.85410	0.137471
$787$	14.4377i	0.514648i	0.966325	+	0.257324i	$0.0828408\pi$
$787$	14.4377i	0.514648i	−0.966325	+	0.257324i	$0.917159\pi$
$788$	− 13.3131i	− 0.474259i
$789$	−13.1459	−0.468006
$790$	0	0
$791$	−2.76393	−0.0982741
$792$	− 2.59675i	− 0.0922714i
$793$	18.2705i	0.648805i
$794$	−6.47214	−0.229688
$795$	0	0
$796$	−9.81153	−0.347760
$797$	32.3262i	1.14505i	0.819886	+	0.572527i	$0.194037\pi$
$797$	32.3262i	1.14505i	−0.819886	+	0.572527i	$0.805963\pi$
$798$	1.38197i	0.0489211i
$799$	74.0476	2.61962
$800$	0	0
$801$	3.47214	0.122682
$802$	− 5.21478i	− 0.184140i
$803$	− 27.9656i	− 0.986883i
$804$	14.8328	0.523113
$805$	0	0
$806$	9.27051	0.326540
$807$	20.3820i	0.717479i
$808$	− 2.24922i	− 0.0791274i
$809$	−19.7984	−0.696074	−0.348037	−	0.937481i	$-0.613152\pi$
$809$	−19.7984	−0.696074	−0.348037	+	0.937481i	$0.613152\pi$
$810$	0	0
$811$	−0.0344419	−0.00120942	−0.000604709	−	1.00000i	$-0.500192\pi$
$811$	−0.0344419	−0.00120942	−0.000604709		1.00000i	$0.500192\pi$
$812$	1.58359i	0.0555732i
$813$	− 1.85410i	− 0.0650262i
$814$	3.36881	0.118077
$815$	0	0
$816$	−17.6738	−0.618705
$817$	− 4.61803i	− 0.161565i
$818$	− 6.40325i	− 0.223884i
$819$	−10.8541	−0.379273
$820$	0	0
$821$	−8.23607	−0.287441	−0.143720	−	0.989618i	$-0.545907\pi$
$821$	−8.23607	−0.287441	−0.143720	+	0.989618i	$0.545907\pi$
$822$	6.20163i	0.216307i
$823$	54.3607i	1.89489i	0.319913	+	0.947447i	$0.396346\pi$
$823$	54.3607i	1.89489i	−0.319913	+	0.947447i	$0.603654\pi$
$824$	1.33939	0.0466600
$825$	0	0
$826$	−19.0689	−0.663491
$827$	− 17.3262i	− 0.602492i	−0.953546	−	0.301246i	$-0.902597\pi$
$827$	− 17.3262i	− 0.602492i	0.953546	−	0.301246i	$-0.0974026\pi$
$828$	− 12.4377i	− 0.432240i
$829$	45.1246	1.56724	0.783621	−	0.621239i	$-0.213370\pi$
$829$	45.1246	1.56724	0.783621	+	0.621239i	$0.213370\pi$
$830$	0	0
$831$	−15.2705	−0.529728
$832$	14.1246i	0.489683i
$833$	34.2148i	1.18547i
$834$	−0.673762	−0.0233305
$835$	0	0
$836$	−3.27051	−0.113113
$837$	8.09017i	0.279637i
$838$	1.24922i	0.0431537i
$839$	−0.527864	−0.0182239	−0.00911195	−	0.999958i	$-0.502900\pi$
$839$	−0.527864	−0.0182239	−0.00911195	+	0.999958i	$0.502900\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	9.93112i	0.342249i
$843$	− 3.27051i	− 0.112642i
$844$	−43.4164	−1.49445
$845$	0	0
$846$	−5.03444	−0.173088
$847$	28.5410i	0.980681i
$848$	− 4.34752i	− 0.149295i
$849$	−21.4721	−0.736922
$850$	0	0
$851$	33.5410	1.14977
$852$	− 14.5623i	− 0.498896i
$853$	9.29180i	0.318145i	0.987267	+	0.159073i	$0.0508504\pi$
$853$	9.29180i	0.318145i	−0.987267	+	0.159073i	$0.949150\pi$
$854$	−8.41641	−0.288004
$855$	0	0
$856$	0.909830	0.0310974
$857$	10.2148i	0.348930i	0.984663	+	0.174465i	$0.0558196\pi$
$857$	10.2148i	0.348930i	−0.984663	+	0.174465i	$0.944180\pi$
$858$	2.02129i	0.0690056i
$859$	8.81966	0.300923	0.150461	−	0.988616i	$-0.451924\pi$
$859$	8.81966	0.300923	0.150461	+	0.988616i	$0.451924\pi$
$860$	0	0
$861$	39.2705	1.33834
$862$	2.96556i	0.101007i
$863$	32.2148i	1.09660i	0.836280	+	0.548302i	$0.184726\pi$
$863$	32.2148i	1.09660i	−0.836280	+	0.548302i	$0.815274\pi$
$864$	4.14590	0.141046
$865$	0	0
$866$	10.4296	0.354411
$867$	14.5623i	0.494562i
$868$	− 54.2705i	− 1.84206i
$869$	12.7639	0.432987
$870$	0	0
$871$	−24.0000	−0.813209
$872$	13.2492i	0.448675i
$873$	− 10.5623i	− 0.357480i
$874$	2.56231	0.0866713
$875$	0	0
$876$	29.3951	0.993169
$877$	33.9443i	1.14622i	0.819480	+	0.573108i	$0.194263\pi$
$877$	33.9443i	1.14622i	−0.819480	+	0.573108i	$0.805737\pi$
$878$	0.527864i	0.0178145i
$879$	19.4164	0.654899
$880$	0	0
$881$	46.7639	1.57552	0.787758	−	0.615984i	$-0.211242\pi$
$881$	46.7639	1.57552	0.787758	+	0.615984i	$0.211242\pi$
$882$	− 2.32624i	− 0.0783285i
$883$	7.94427i	0.267346i	0.991025	+	0.133673i	$0.0426772\pi$
$883$	7.94427i	0.267346i	−0.991025	+	0.133673i	$0.957323\pi$
$884$	31.2492	1.05103
$885$	0	0
$886$	−8.57546	−0.288098
$887$	− 54.3050i	− 1.82338i	−0.410877	−	0.911691i	$-0.634777\pi$
$887$	− 54.3050i	− 1.82338i	0.410877	−	0.911691i	$-0.365223\pi$
$888$	7.36068i	0.247008i
$889$	23.2148	0.778599
$890$	0	0
$891$	−1.76393	−0.0590939
$892$	20.1885i	0.675960i
$893$	13.1803i	0.441063i
$894$	−4.56231	−0.152586
$895$	0	0
$896$	−36.5066	−1.21960
$897$	20.1246i	0.671941i
$898$	− 1.72136i	− 0.0574425i
$899$	−1.90983	−0.0636964
$900$	0	0
$901$	−7.76393	−0.258654
$902$	− 7.31308i	− 0.243499i
$903$	16.7082i	0.556014i
$904$	−1.12461	−0.0374040
$905$	0	0
$906$	1.59675	0.0530484
$907$	− 51.5410i	− 1.71139i	−0.517479	−	0.855696i	$-0.673130\pi$
$907$	− 51.5410i	− 1.71139i	0.517479	−	0.855696i	$-0.326870\pi$
$908$	− 11.4590i	− 0.380280i
$909$	−1.52786	−0.0506761
$910$	0	0
$911$	−12.2705	−0.406540	−0.203270	−	0.979123i	$-0.565157\pi$
$911$	−12.2705	−0.406540	−0.203270	+	0.979123i	$0.565157\pi$
$912$	− 3.14590i	− 0.104171i
$913$	− 12.9230i	− 0.427688i
$914$	6.75580	0.223462
$915$	0	0
$916$	−39.1672	−1.29412
$917$	36.5066i	1.20555i
$918$	− 2.14590i	− 0.0708252i
$919$	39.2705	1.29541	0.647707	−	0.761889i	$-0.275728\pi$
$919$	39.2705	1.29541	0.647707	+	0.761889i	$0.275728\pi$
$920$	0	0
$921$	22.1246	0.729031
$922$	6.67376i	0.219789i
$923$	23.5623i	0.775563i
$924$	11.8328	0.389271
$925$	0	0
$926$	−0.257354	−0.00845718
$927$	− 0.909830i	− 0.0298827i
$928$	0.978714i	0.0321279i
$929$	−28.9230	−0.948932	−0.474466	−	0.880274i	$-0.657359\pi$
$929$	−28.9230	−0.948932	−0.474466	+	0.880274i	$0.657359\pi$
$930$	0	0
$931$	−6.09017	−0.199597
$932$	− 23.1885i	− 0.759564i
$933$	− 19.0902i	− 0.624984i
$934$	−1.87539	−0.0613646
$935$	0	0
$936$	−4.41641	−0.144355
$937$	− 29.8197i	− 0.974166i	−0.873356	−	0.487083i	$-0.838061\pi$
$937$	− 29.8197i	− 0.974166i	0.873356	−	0.487083i	$-0.161939\pi$
$938$	− 11.0557i	− 0.360982i
$939$	−19.9787	−0.651981
$940$	0	0
$941$	32.5623	1.06150	0.530750	−	0.847528i	$-0.321910\pi$
$941$	32.5623	1.06150	0.530750	+	0.847528i	$0.321910\pi$
$942$	6.32624i	0.206120i
$943$	− 72.8115i	− 2.37107i
$944$	43.4083	1.41282
$945$	0	0
$946$	3.11146	0.101162
$947$	38.8885i	1.26371i	0.775087	+	0.631854i	$0.217706\pi$
$947$	38.8885i	1.26371i	−0.775087	+	0.631854i	$0.782294\pi$
$948$	13.4164i	0.435745i
$949$	−47.5623	−1.54394
$950$	0	0
$951$	−16.4164	−0.532338
$952$	29.9230i	0.969810i
$953$	7.92299i	0.256651i	0.991732	+	0.128325i	$0.0409602\pi$
$953$	7.92299i	0.256651i	−0.991732	+	0.128325i	$0.959040\pi$
$954$	0.527864	0.0170902
$955$	0	0
$956$	44.6656	1.44459
$957$	− 0.416408i	− 0.0134606i
$958$	7.48529i	0.241839i
$959$	−58.7426	−1.89690
$960$	0	0
$961$	34.4508	1.11132
$962$	− 5.72949i	− 0.184726i
$963$	− 0.618034i	− 0.0199159i
$964$	−29.1246	−0.938041
$965$	0	0
$966$	−9.27051	−0.298274
$967$	− 20.4164i	− 0.656547i	−0.944583	−	0.328274i	$-0.893533\pi$
$967$	− 20.4164i	− 0.656547i	0.944583	−	0.328274i	$-0.106467\pi$
$968$	11.6130i	0.373256i
$969$	−5.61803	−0.180477
$970$	0	0
$971$	39.5967	1.27072	0.635360	−	0.772216i	$-0.280852\pi$
$971$	39.5967	1.27072	0.635360	+	0.772216i	$0.280852\pi$
$972$	− 1.85410i	− 0.0594703i
$973$	− 6.38197i	− 0.204596i
$974$	2.65248	0.0849908
$975$	0	0
$976$	19.1591	0.613266
$977$	43.8541i	1.40302i	0.712661	+	0.701509i	$0.247490\pi$
$977$	43.8541i	1.40302i	−0.712661	+	0.701509i	$0.752510\pi$
$978$	3.43769i	0.109925i
$979$	6.12461	0.195743
$980$	0	0
$981$	9.00000	0.287348
$982$	6.41641i	0.204756i
$983$	− 2.03444i	− 0.0648886i	−0.999474	−	0.0324443i	$-0.989671\pi$
$983$	− 2.03444i	− 0.0648886i	0.999474	−	0.0324443i	$-0.0103292\pi$
$984$	15.9787	0.509383
$985$	0	0
$986$	0.506578	0.0161327
$987$	− 47.6869i	− 1.51789i
$988$	5.56231i	0.176961i
$989$	30.9787	0.985066
$990$	0	0
$991$	−32.1459	−1.02115	−0.510574	−	0.859834i	$-0.670567\pi$
$991$	−32.1459	−1.02115	−0.510574	+	0.859834i	$0.670567\pi$
$992$	− 33.5410i	− 1.06493i
$993$	− 6.14590i	− 0.195034i
$994$	−10.8541	−0.344271
$995$	0	0
$996$	13.5836	0.430413
$997$	− 43.7214i	− 1.38467i	−0.721577	−	0.692335i	$-0.756582\pi$
$997$	− 43.7214i	− 1.38467i	0.721577	−	0.692335i	$-0.243418\pi$
$998$	6.60488i	0.209074i
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.b.a.124.2		4
3.2	odd	2		1125.2.b.b.874.3		4
4.3	odd	2		6000.2.f.n.1249.4		4
5.2	odd	4		375.2.a.d.1.1	yes	2
5.3	odd	4		375.2.a.a.1.2	✓	2
5.4	even	2	inner	375.2.b.a.124.3		4
15.2	even	4		1125.2.a.a.1.2		2
15.8	even	4		1125.2.a.f.1.1		2
15.14	odd	2		1125.2.b.b.874.2		4
20.3	even	4		6000.2.a.m.1.2		2
20.7	even	4		6000.2.a.q.1.1		2
20.19	odd	2		6000.2.f.n.1249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.a.1.2	✓	2	5.3	odd	4
375.2.a.d.1.1	yes	2	5.2	odd	4
375.2.b.a.124.2		4	1.1	even	1	trivial
375.2.b.a.124.3		4	5.4	even	2	inner
1125.2.a.a.1.2		2	15.2	even	4
1125.2.a.f.1.1		2	15.8	even	4
1125.2.b.b.874.2		4	15.14	odd	2
1125.2.b.b.874.3		4	3.2	odd	2
6000.2.a.m.1.2		2	20.3	even	4
6000.2.a.q.1.1		2	20.7	even	4
6000.2.f.n.1249.1		4	20.19	odd	2
6000.2.f.n.1249.4		4	4.3	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:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.381966i q^{2} -1.00000i q^{3} +1.85410 q^{4} -0.381966 q^{6} -3.61803i q^{7} -1.47214i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-0.381966i q^{2} -1.00000i q^{3} +1.85410 q^{4} -0.381966 q^{6} -3.61803i q^{7} -1.47214i q^{8} -1.00000 q^{9} -1.76393 q^{11} -1.85410i q^{12} +3.00000i q^{13} -1.38197 q^{14} +3.14590 q^{16} -5.61803i q^{17} +0.381966i q^{18} +1.00000 q^{19} -3.61803 q^{21} +0.673762i q^{22} +6.70820i q^{23} -1.47214 q^{24} +1.14590 q^{26} +1.00000i q^{27} -6.70820i q^{28} -0.236068 q^{29} +8.09017 q^{31} -4.14590i q^{32} +1.76393i q^{33} -2.14590 q^{34} -1.85410 q^{36} -5.00000i q^{37} -0.381966i q^{38} +3.00000 q^{39} -10.8541 q^{41} +1.38197i q^{42} -4.61803i q^{43} -3.27051 q^{44} +2.56231 q^{46} +13.1803i q^{47} -3.14590i q^{48} -6.09017 q^{49} -5.61803 q^{51} +5.56231i q^{52} -1.38197i q^{53} +0.381966 q^{54} -5.32624 q^{56} -1.00000i q^{57} +0.0901699i q^{58} +13.7984 q^{59} +6.09017 q^{61} -3.09017i q^{62} +3.61803i q^{63} +4.70820 q^{64} +0.673762 q^{66} +8.00000i q^{67} -10.4164i q^{68} +6.70820 q^{69} +7.85410 q^{71} +1.47214i q^{72} +15.8541i q^{73} -1.90983 q^{74} +1.85410 q^{76} +6.38197i q^{77} -1.14590i q^{78} -7.23607 q^{79} +1.00000 q^{81} +4.14590i q^{82} +7.32624i q^{83} -6.70820 q^{84} -1.76393 q^{86} +0.236068i q^{87} +2.59675i q^{88} -3.47214 q^{89} +10.8541 q^{91} +12.4377i q^{92} -8.09017i q^{93} +5.03444 q^{94} -4.14590 q^{96} +10.5623i q^{97} +2.32624i q^{98} +1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9	\( 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} + 26 q^{16} + 4 q^{19} - 10 q^{21} + 12 q^{24} + 18 q^{26} + 8 q^{29} + 10 q^{31} - 22 q^{34} + 6 q^{36} + 12 q^{39} - 30 q^{41} + 54 q^{44} - 30 q^{46} - 2 q^{49} - 18 q^{51} + 6 q^{54} + 10 q^{56} + 6 q^{59} + 2 q^{61} - 8 q^{64} + 34 q^{66} + 18 q^{71} - 30 q^{74} - 6 q^{76} - 20 q^{79} + 4 q^{81} - 16 q^{86} + 4 q^{89} + 30 q^{91} - 38 q^{94} - 30 q^{96} + 16 q^{99}+O(q^{100}) \) 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 + 26 * q^16 + 4 * q^19 - 10 * q^21 + 12 * q^24 + 18 * q^26 + 8 * q^29 + 10 * q^31 - 22 * q^34 + 6 * q^36 + 12 * q^39 - 30 * q^41 + 54 * q^44 - 30 * q^46 - 2 * q^49 - 18 * q^51 + 6 * q^54 + 10 * q^56 + 6 * q^59 + 2 * q^61 - 8 * q^64 + 34 * q^66 + 18 * q^71 - 30 * q^74 - 6 * q^76 - 20 * q^79 + 4 * q^81 - 16 * q^86 + 4 * q^89 + 30 * q^91 - 38 * q^94 - 30 * q^96 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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