LMFDB - Embedded newform 475.2.b.d.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.d.324.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} +O(q^{10})$	\(q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} -6.34017 q^{11} +4.63090i q^{12} +1.36910i q^{13} -2.34017 q^{14} -2.07838 q^{16} -3.26180i q^{17} -0.170086i q^{18} +1.00000 q^{19} -1.84324 q^{21} +13.7587i q^{22} +2.34017i q^{23} +2.63090 q^{24} +2.97107 q^{26} -5.26180i q^{27} +2.92162i q^{28} -1.41855 q^{29} +8.68035 q^{31} +7.58864i q^{32} +10.8371i q^{33} -7.07838 q^{34} -0.212347 q^{36} -5.36910i q^{37} -2.17009i q^{38} +2.34017 q^{39} -3.26180 q^{41} +4.00000i q^{42} -11.9155i q^{43} +17.1773 q^{44} +5.07838 q^{46} -1.07838i q^{47} +3.55252i q^{48} +5.83710 q^{49} -5.57531 q^{51} -3.70928i q^{52} +6.63090i q^{53} -11.4186 q^{54} +1.65983 q^{56} -1.70928i q^{57} +3.07838i q^{58} +11.4186 q^{59} +5.60197 q^{61} -18.8371i q^{62} -0.0845208i q^{63} +12.3112 q^{64} +23.5174 q^{66} -10.3896i q^{67} +8.83710i q^{68} +4.00000 q^{69} -10.8371 q^{71} +0.120638i q^{72} +5.41855i q^{73} -11.6514 q^{74} -2.70928 q^{76} +6.83710i q^{77} -5.07838i q^{78} -14.2557 q^{79} -8.75872 q^{81} +7.07838i q^{82} -14.3402i q^{83} +4.99386 q^{84} -25.8576 q^{86} +2.42469i q^{87} -9.75872i q^{88} -7.57531 q^{89} +1.47641 q^{91} -6.34017i q^{92} -14.8371i q^{93} -2.34017 q^{94} +12.9711 q^{96} +8.88655i q^{97} -12.6670i q^{98} -0.496928 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.17009i	− 1.53448i	−0.641358	−	0.767241i	$-0.721629\pi$
$2$	− 2.17009i	− 1.53448i	0.641358	−	0.767241i	$-0.278371\pi$
$3$	− 1.70928i	− 0.986851i	−0.869788	−	0.493425i	$-0.835745\pi$
$3$	− 1.70928i	− 0.986851i	0.869788	−	0.493425i	$-0.164255\pi$
$4$	−2.70928	−1.35464
$5$	0	0
$5$	0	0
$6$	−3.70928	−1.51431
$7$	− 1.07838i	− 0.407588i	−0.979014	−	0.203794i	$-0.934673\pi$
$7$	− 1.07838i	− 0.407588i	0.979014	−	0.203794i	$-0.0653274\pi$
$8$	1.53919i	0.544185i
$9$	0.0783777	0.0261259
$10$	0	0
$11$	−6.34017	−1.91163	−0.955817	−	0.293962i	$-0.905026\pi$
$11$	−6.34017	−1.91163	−0.955817	+	0.293962i	$0.905026\pi$
$12$	4.63090i	1.33682i
$13$	1.36910i	0.379721i	0.981811	+	0.189860i	$0.0608035\pi$
$13$	1.36910i	0.379721i	−0.981811	+	0.189860i	$0.939196\pi$
$14$	−2.34017	−0.625438
$15$	0	0
$16$	−2.07838	−0.519594
$17$	− 3.26180i	− 0.791102i	−0.918444	−	0.395551i	$-0.870554\pi$
$17$	− 3.26180i	− 0.791102i	0.918444	−	0.395551i	$-0.129446\pi$
$18$	− 0.170086i	− 0.0400898i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.84324	−0.402229
$22$	13.7587i	2.93337i
$23$	2.34017i	0.487960i	0.969780	+	0.243980i	$0.0784531\pi$
$23$	2.34017i	0.487960i	−0.969780	+	0.243980i	$0.921547\pi$
$24$	2.63090	0.537030
$25$	0	0
$26$	2.97107	0.582675
$27$	− 5.26180i	− 1.01263i
$28$	2.92162i	0.552135i
$29$	−1.41855	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$29$	−1.41855	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$30$	0	0
$31$	8.68035	1.55904	0.779518	−	0.626380i	$-0.215464\pi$
$31$	8.68035	1.55904	0.779518	+	0.626380i	$0.215464\pi$
$32$	7.58864i	1.34149i
$33$	10.8371i	1.88650i
$34$	−7.07838	−1.21393
$35$	0	0
$36$	−0.212347	−0.0353911
$37$	− 5.36910i	− 0.882675i	−0.897341	−	0.441337i	$-0.854504\pi$
$37$	− 5.36910i	− 0.882675i	0.897341	−	0.441337i	$-0.145496\pi$
$38$	− 2.17009i	− 0.352035i
$39$	2.34017	0.374728
$40$	0	0
$41$	−3.26180	−0.509407	−0.254703	−	0.967019i	$-0.581978\pi$
$41$	−3.26180	−0.509407	−0.254703	+	0.967019i	$0.581978\pi$
$42$	4.00000i	0.617213i
$43$	− 11.9155i	− 1.81709i	−0.417783	−	0.908547i	$-0.637193\pi$
$43$	− 11.9155i	− 1.81709i	0.417783	−	0.908547i	$-0.362807\pi$
$44$	17.1773	2.58957
$45$	0	0
$46$	5.07838	0.748766
$47$	− 1.07838i	− 0.157298i	−0.996902	−	0.0786488i	$-0.974939\pi$
$47$	− 1.07838i	− 0.157298i	0.996902	−	0.0786488i	$-0.0250606\pi$
$48$	3.55252i	0.512762i
$49$	5.83710	0.833872
$50$	0	0
$51$	−5.57531	−0.780699
$52$	− 3.70928i	− 0.514384i
$53$	6.63090i	0.910824i	0.890281	+	0.455412i	$0.150508\pi$
$53$	6.63090i	0.910824i	−0.890281	+	0.455412i	$0.849492\pi$
$54$	−11.4186	−1.55387
$55$	0	0
$56$	1.65983	0.221804
$57$	− 1.70928i	− 0.226399i
$58$	3.07838i	0.404211i
$59$	11.4186	1.48657	0.743284	−	0.668976i	$-0.233267\pi$
$59$	11.4186	1.48657	0.743284	+	0.668976i	$0.233267\pi$
$60$	0	0
$61$	5.60197	0.717259	0.358629	−	0.933480i	$-0.383244\pi$
$61$	5.60197	0.717259	0.358629	+	0.933480i	$0.383244\pi$
$62$	− 18.8371i	− 2.39231i
$63$	− 0.0845208i	− 0.0106486i
$64$	12.3112	1.53891
$65$	0	0
$66$	23.5174	2.89480
$67$	− 10.3896i	− 1.26929i	−0.772802	−	0.634647i	$-0.781145\pi$
$67$	− 10.3896i	− 1.26929i	0.772802	−	0.634647i	$-0.218855\pi$
$68$	8.83710i	1.07166i
$69$	4.00000	0.481543
$70$	0	0
$71$	−10.8371	−1.28613	−0.643064	−	0.765813i	$-0.722337\pi$
$71$	−10.8371	−1.28613	−0.643064	+	0.765813i	$0.722337\pi$
$72$	0.120638i	0.0142173i
$73$	5.41855i	0.634193i	0.948393	+	0.317097i	$0.102708\pi$
$73$	5.41855i	0.634193i	−0.948393	+	0.317097i	$0.897292\pi$
$74$	−11.6514	−1.35445
$75$	0	0
$76$	−2.70928	−0.310775
$77$	6.83710i	0.779160i
$78$	− 5.07838i	− 0.575013i
$79$	−14.2557	−1.60389	−0.801943	−	0.597400i	$-0.796200\pi$
$79$	−14.2557	−1.60389	−0.801943	+	0.597400i	$0.796200\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	7.07838i	0.781676i
$83$	− 14.3402i	− 1.57404i	−0.616928	−	0.787019i	$-0.711623\pi$
$83$	− 14.3402i	− 1.57404i	0.616928	−	0.787019i	$-0.288377\pi$
$84$	4.99386	0.544874
$85$	0	0
$86$	−25.8576	−2.78830
$87$	2.42469i	0.259954i
$88$	− 9.75872i	− 1.04028i
$89$	−7.57531	−0.802981	−0.401490	−	0.915863i	$-0.631508\pi$
$89$	−7.57531	−0.802981	−0.401490	+	0.915863i	$0.631508\pi$
$90$	0	0
$91$	1.47641	0.154770
$92$	− 6.34017i	− 0.661009i
$93$	− 14.8371i	− 1.53854i
$94$	−2.34017	−0.241370
$95$	0	0
$96$	12.9711	1.32385
$97$	8.88655i	0.902292i	0.892450	+	0.451146i	$0.148985\pi$
$97$	8.88655i	0.902292i	−0.892450	+	0.451146i	$0.851015\pi$
$98$	− 12.6670i	− 1.27956i
$99$	−0.496928	−0.0499432
$100$	0	0
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	12.0989i	1.19797i
$103$	6.38962i	0.629588i	0.949160	+	0.314794i	$0.101935\pi$
$103$	6.38962i	0.629588i	−0.949160	+	0.314794i	$0.898065\pi$
$104$	−2.10731	−0.206638
$105$	0	0
$106$	14.3896	1.39764
$107$	− 2.29072i	− 0.221453i	−0.993851	−	0.110726i	$-0.964682\pi$
$107$	− 2.29072i	− 0.221453i	0.993851	−	0.110726i	$-0.0353177\pi$
$108$	14.2557i	1.37175i
$109$	12.8371	1.22957	0.614786	−	0.788694i	$-0.289243\pi$
$109$	12.8371	1.22957	0.614786	+	0.788694i	$0.289243\pi$
$110$	0	0
$111$	−9.17727	−0.871068
$112$	2.24128i	0.211781i
$113$	− 12.8865i	− 1.21226i	−0.795364	−	0.606132i	$-0.792720\pi$
$113$	− 12.8865i	− 1.21226i	0.795364	−	0.606132i	$-0.207280\pi$
$114$	−3.70928	−0.347405
$115$	0	0
$116$	3.84324	0.356836
$117$	0.107307i	0.00992055i
$118$	− 24.7792i	− 2.28111i
$119$	−3.51745	−0.322444
$120$	0	0
$121$	29.1978	2.65434
$122$	− 12.1568i	− 1.10062i
$123$	5.57531i	0.502708i
$124$	−23.5174	−2.11193
$125$	0	0
$126$	−0.183417	−0.0163401
$127$	8.23287i	0.730549i	0.930900	+	0.365274i	$0.119025\pi$
$127$	8.23287i	0.730549i	−0.930900	+	0.365274i	$0.880975\pi$
$128$	− 11.5392i	− 1.01993i
$129$	−20.3668	−1.79320
$130$	0	0
$131$	1.47641	0.128995	0.0644973	−	0.997918i	$-0.479456\pi$
$131$	1.47641	0.128995	0.0644973	+	0.997918i	$0.479456\pi$
$132$	− 29.3607i	− 2.55552i
$133$	− 1.07838i	− 0.0935072i
$134$	−22.5464	−1.94771
$135$	0	0
$136$	5.02052	0.430506
$137$	3.94214i	0.336800i	0.985719	+	0.168400i	$0.0538600\pi$
$137$	3.94214i	0.336800i	−0.985719	+	0.168400i	$0.946140\pi$
$138$	− 8.68035i	− 0.738920i
$139$	−8.86376	−0.751815	−0.375907	−	0.926657i	$-0.622669\pi$
$139$	−8.86376	−0.751815	−0.375907	+	0.926657i	$0.622669\pi$
$140$	0	0
$141$	−1.84324	−0.155229
$142$	23.5174i	1.97354i
$143$	− 8.68035i	− 0.725887i
$144$	−0.162899	−0.0135749
$145$	0	0
$146$	11.7587	0.973159
$147$	− 9.97721i	− 0.822907i
$148$	14.5464i	1.19570i
$149$	19.7587	1.61870	0.809349	−	0.587328i	$-0.199820\pi$
$149$	19.7587	1.61870	0.809349	+	0.587328i	$0.199820\pi$
$150$	0	0
$151$	3.41855	0.278198	0.139099	−	0.990279i	$-0.455579\pi$
$151$	3.41855	0.278198	0.139099	+	0.990279i	$0.455579\pi$
$152$	1.53919i	0.124845i
$153$	− 0.255652i	− 0.0206683i
$154$	14.8371	1.19561
$155$	0	0
$156$	−6.34017	−0.507620
$157$	− 9.41855i	− 0.751682i	−0.926684	−	0.375841i	$-0.877354\pi$
$157$	− 9.41855i	− 0.751682i	0.926684	−	0.375841i	$-0.122646\pi$
$158$	30.9360i	2.46114i
$159$	11.3340	0.898847
$160$	0	0
$161$	2.52359	0.198887
$162$	19.0072i	1.49335i
$163$	− 2.92162i	− 0.228839i	−0.993433	−	0.114420i	$-0.963499\pi$
$163$	− 2.92162i	− 0.228839i	0.993433	−	0.114420i	$-0.0365008\pi$
$164$	8.83710	0.690062
$165$	0	0
$166$	−31.1194	−2.41534
$167$	− 20.9132i	− 1.61831i	−0.587593	−	0.809156i	$-0.699924\pi$
$167$	− 20.9132i	− 1.61831i	0.587593	−	0.809156i	$-0.300076\pi$
$168$	− 2.83710i	− 0.218887i
$169$	11.1256	0.855812
$170$	0	0
$171$	0.0783777	0.00599370
$172$	32.2823i	2.46150i
$173$	− 1.05559i	− 0.0802551i	−0.999195	−	0.0401276i	$-0.987224\pi$
$173$	− 1.05559i	− 0.0802551i	0.999195	−	0.0401276i	$-0.0127764\pi$
$174$	5.26180	0.398896
$175$	0	0
$176$	13.1773	0.993274
$177$	− 19.5174i	− 1.46702i
$178$	16.4391i	1.23216i
$179$	−0.894960	−0.0668925	−0.0334462	−	0.999441i	$-0.510648\pi$
$179$	−0.894960	−0.0668925	−0.0334462	+	0.999441i	$0.510648\pi$
$180$	0	0
$181$	−0.837101	−0.0622213	−0.0311106	−	0.999516i	$-0.509904\pi$
$181$	−0.837101	−0.0622213	−0.0311106	+	0.999516i	$0.509904\pi$
$182$	− 3.20394i	− 0.237492i
$183$	− 9.57531i	− 0.707827i
$184$	−3.60197	−0.265541
$185$	0	0
$186$	−32.1978	−2.36086
$187$	20.6803i	1.51230i
$188$	2.92162i	0.213081i
$189$	−5.67420	−0.412738
$190$	0	0
$191$	22.0410	1.59483	0.797417	−	0.603429i	$-0.206199\pi$
$191$	22.0410	1.59483	0.797417	+	0.603429i	$0.206199\pi$
$192$	− 21.0433i	− 1.51867i
$193$	12.7877i	0.920475i	0.887796	+	0.460238i	$0.152236\pi$
$193$	12.7877i	0.920475i	−0.887796	+	0.460238i	$0.847764\pi$
$194$	19.2846	1.38455
$195$	0	0
$196$	−15.8143	−1.12959
$197$	9.20394i	0.655753i	0.944721	+	0.327877i	$0.106333\pi$
$197$	9.20394i	0.655753i	−0.944721	+	0.327877i	$0.893667\pi$
$198$	1.07838i	0.0766370i
$199$	16.1978	1.14823	0.574116	−	0.818774i	$-0.305346\pi$
$199$	16.1978	1.14823	0.574116	+	0.818774i	$0.305346\pi$
$200$	0	0
$201$	−17.7587	−1.25260
$202$	10.6803i	0.751467i
$203$	1.52973i	0.107366i
$204$	15.1050	1.05756
$205$	0	0
$206$	13.8660	0.966092
$207$	0.183417i	0.0127484i
$208$	− 2.84551i	− 0.197301i
$209$	−6.34017	−0.438559
$210$	0	0
$211$	7.78539	0.535968	0.267984	−	0.963423i	$-0.413643\pi$
$211$	7.78539	0.535968	0.267984	+	0.963423i	$0.413643\pi$
$212$	− 17.9649i	− 1.23384i
$213$	18.5236i	1.26922i
$214$	−4.97107	−0.339815
$215$	0	0
$216$	8.09890	0.551060
$217$	− 9.36069i	− 0.635445i
$218$	− 27.8576i	− 1.88676i
$219$	9.26180	0.625854
$220$	0	0
$221$	4.46573	0.300398
$222$	19.9155i	1.33664i
$223$	12.5464i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$223$	12.5464i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$224$	8.18342	0.546778
$225$	0	0
$226$	−27.9649	−1.86020
$227$	− 2.29072i	− 0.152041i	−0.997106	−	0.0760204i	$-0.975779\pi$
$227$	− 2.29072i	− 0.152041i	0.997106	−	0.0760204i	$-0.0242214\pi$
$228$	4.63090i	0.306689i
$229$	5.91548	0.390906	0.195453	−	0.980713i	$-0.437382\pi$
$229$	5.91548	0.390906	0.195453	+	0.980713i	$0.437382\pi$
$230$	0	0
$231$	11.6865	0.768915
$232$	− 2.18342i	− 0.143348i
$233$	13.5174i	0.885557i	0.896631	+	0.442779i	$0.146007\pi$
$233$	13.5174i	0.885557i	−0.896631	+	0.442779i	$0.853993\pi$
$234$	0.232866	0.0152229
$235$	0	0
$236$	−30.9360	−2.01376
$237$	24.3668i	1.58280i
$238$	7.63317i	0.494785i
$239$	−13.8432	−0.895445	−0.447723	−	0.894173i	$-0.647765\pi$
$239$	−13.8432	−0.895445	−0.447723	+	0.894173i	$0.647765\pi$
$240$	0	0
$241$	7.26180	0.467773	0.233887	−	0.972264i	$-0.424856\pi$
$241$	7.26180	0.467773	0.233887	+	0.972264i	$0.424856\pi$
$242$	− 63.3617i	− 4.07305i
$243$	− 0.814315i	− 0.0522383i
$244$	−15.1773	−0.971625
$245$	0	0
$246$	12.0989	0.771397
$247$	1.36910i	0.0871139i
$248$	13.3607i	0.848405i
$249$	−24.5113	−1.55334
$250$	0	0
$251$	10.4703	0.660877	0.330439	−	0.943827i	$-0.392803\pi$
$251$	10.4703	0.660877	0.330439	+	0.943827i	$0.392803\pi$
$252$	0.228990i	0.0144250i
$253$	− 14.8371i	− 0.932801i
$254$	17.8660	1.12101
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	23.6248i	1.47367i	0.676072	+	0.736836i	$0.263681\pi$
$257$	23.6248i	1.47367i	−0.676072	+	0.736836i	$0.736319\pi$
$258$	44.1978i	2.75163i
$259$	−5.78992	−0.359768
$260$	0	0
$261$	−0.111183	−0.00688204
$262$	− 3.20394i	− 0.197940i
$263$	5.65983i	0.349000i	0.984657	+	0.174500i	$0.0558309\pi$
$263$	5.65983i	0.349000i	−0.984657	+	0.174500i	$0.944169\pi$
$264$	−16.6803	−1.02660
$265$	0	0
$266$	−2.34017	−0.143485
$267$	12.9483i	0.792422i
$268$	28.1483i	1.71943i
$269$	2.31351	0.141057	0.0705286	−	0.997510i	$-0.477531\pi$
$269$	2.31351	0.141057	0.0705286	+	0.997510i	$0.477531\pi$
$270$	0	0
$271$	−19.7009	−1.19674	−0.598371	−	0.801219i	$-0.704185\pi$
$271$	−19.7009	−1.19674	−0.598371	+	0.801219i	$0.704185\pi$
$272$	6.77924i	0.411052i
$273$	− 2.52359i	− 0.152735i
$274$	8.55479	0.516814
$275$	0	0
$276$	−10.8371	−0.652317
$277$	25.7321i	1.54609i	0.634351	+	0.773045i	$0.281267\pi$
$277$	25.7321i	1.54609i	−0.634351	+	0.773045i	$0.718733\pi$
$278$	19.2351i	1.15365i
$279$	0.680346	0.0407312
$280$	0	0
$281$	−6.58145	−0.392616	−0.196308	−	0.980542i	$-0.562895\pi$
$281$	−6.58145	−0.392616	−0.196308	+	0.980542i	$0.562895\pi$
$282$	4.00000i	0.238197i
$283$	− 0.496928i	− 0.0295393i	−0.999891	−	0.0147697i	$-0.995298\pi$
$283$	− 0.496928i	− 0.0295393i	0.999891	−	0.0147697i	$-0.00470150\pi$
$284$	29.3607	1.74224
$285$	0	0
$286$	−18.8371	−1.11386
$287$	3.51745i	0.207628i
$288$	0.594780i	0.0350478i
$289$	6.36069	0.374158
$290$	0	0
$291$	15.1896	0.890428
$292$	− 14.6803i	− 0.859102i
$293$	6.63090i	0.387381i	0.981063	+	0.193691i	$0.0620458\pi$
$293$	6.63090i	0.387381i	−0.981063	+	0.193691i	$0.937954\pi$
$294$	−21.6514	−1.26274
$295$	0	0
$296$	8.26406	0.480339
$297$	33.3607i	1.93578i
$298$	− 42.8781i	− 2.48386i
$299$	−3.20394	−0.185288
$300$	0	0
$301$	−12.8494	−0.740626
$302$	− 7.41855i	− 0.426890i
$303$	8.41241i	0.483280i
$304$	−2.07838	−0.119203
$305$	0	0
$306$	−0.554787	−0.0317151
$307$	− 14.6042i	− 0.833508i	−0.909019	−	0.416754i	$-0.863168\pi$
$307$	− 14.6042i	− 0.833508i	0.909019	−	0.416754i	$-0.136832\pi$
$308$	− 18.5236i	− 1.05548i
$309$	10.9216	0.621309
$310$	0	0
$311$	19.3340	1.09633	0.548166	−	0.836369i	$-0.315326\pi$
$311$	19.3340	1.09633	0.548166	+	0.836369i	$0.315326\pi$
$312$	3.60197i	0.203921i
$313$	30.6803i	1.73416i	0.498173	+	0.867078i	$0.334005\pi$
$313$	30.6803i	1.73416i	−0.498173	+	0.867078i	$0.665995\pi$
$314$	−20.4391	−1.15344
$315$	0	0
$316$	38.6225	2.17268
$317$	− 18.7298i	− 1.05197i	−0.850494	−	0.525985i	$-0.823697\pi$
$317$	− 18.7298i	− 1.05197i	0.850494	−	0.525985i	$-0.176303\pi$
$318$	− 24.5958i	− 1.37927i
$319$	8.99386	0.503559
$320$	0	0
$321$	−3.91548	−0.218541
$322$	− 5.47641i	− 0.305188i
$323$	− 3.26180i	− 0.181491i
$324$	23.7298	1.31832
$325$	0	0
$326$	−6.34017	−0.351150
$327$	− 21.9421i	− 1.21340i
$328$	− 5.02052i	− 0.277212i
$329$	−1.16290	−0.0641127
$330$	0	0
$331$	−2.73820	−0.150505	−0.0752527	−	0.997164i	$-0.523976\pi$
$331$	−2.73820	−0.150505	−0.0752527	+	0.997164i	$0.523976\pi$
$332$	38.8515i	2.13225i
$333$	− 0.420818i	− 0.0230607i
$334$	−45.3835	−2.48327
$335$	0	0
$336$	3.83096	0.208996
$337$	6.04945i	0.329534i	0.986332	+	0.164767i	$0.0526873\pi$
$337$	6.04945i	0.329534i	−0.986332	+	0.164767i	$0.947313\pi$
$338$	− 24.1434i	− 1.31323i
$339$	−22.0267	−1.19632
$340$	0	0
$341$	−55.0349	−2.98031
$342$	− 0.170086i	− 0.00919722i
$343$	− 13.8432i	− 0.747465i
$344$	18.3402	0.988836
$345$	0	0
$346$	−2.29072	−0.123150
$347$	5.97334i	0.320666i	0.987063	+	0.160333i	$0.0512567\pi$
$347$	5.97334i	0.320666i	−0.987063	+	0.160333i	$0.948743\pi$
$348$	− 6.56916i	− 0.352144i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	7.20394	0.384518
$352$	− 48.1133i	− 2.56445i
$353$	− 14.0989i	− 0.750409i	−0.926942	−	0.375204i	$-0.877573\pi$
$353$	− 14.0989i	− 0.750409i	0.926942	−	0.375204i	$-0.122427\pi$
$354$	−42.3545	−2.25112
$355$	0	0
$356$	20.5236	1.08775
$357$	6.01229i	0.318204i
$358$	1.94214i	0.102645i
$359$	−6.02666	−0.318075	−0.159038	−	0.987273i	$-0.550839\pi$
$359$	−6.02666	−0.318075	−0.159038	+	0.987273i	$0.550839\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.81658i	0.0954775i
$363$	− 49.9071i	− 2.61944i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−20.7792	−1.08615
$367$	− 1.07838i	− 0.0562909i	−0.999604	−	0.0281454i	$-0.991040\pi$
$367$	− 1.07838i	− 0.0562909i	0.999604	−	0.0281454i	$-0.00896015\pi$
$368$	− 4.86376i	− 0.253541i
$369$	−0.255652	−0.0133087
$370$	0	0
$371$	7.15061	0.371241
$372$	40.1978i	2.08416i
$373$	12.3051i	0.637134i	0.947900	+	0.318567i	$0.103202\pi$
$373$	12.3051i	0.637134i	−0.947900	+	0.318567i	$0.896798\pi$
$374$	44.8781	2.32059
$375$	0	0
$376$	1.65983	0.0855990
$377$	− 1.94214i	− 0.100025i
$378$	12.3135i	0.633339i
$379$	−1.04718	−0.0537901	−0.0268950	−	0.999638i	$-0.508562\pi$
$379$	−1.04718	−0.0537901	−0.0268950	+	0.999638i	$0.508562\pi$
$380$	0	0
$381$	14.0722	0.720942
$382$	− 47.8310i	− 2.44724i
$383$	0.0806452i	0.00412078i	0.999998	+	0.00206039i	$0.000655842\pi$
$383$	0.0806452i	0.00412078i	−0.999998	+	0.00206039i	$0.999344\pi$
$384$	−19.7237	−1.00652
$385$	0	0
$386$	27.7503	1.41245
$387$	− 0.933908i	− 0.0474732i
$388$	− 24.0761i	− 1.22228i
$389$	20.5236	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$389$	20.5236	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$390$	0	0
$391$	7.63317	0.386026
$392$	8.98440i	0.453781i
$393$	− 2.52359i	− 0.127298i
$394$	19.9733	1.00624
$395$	0	0
$396$	1.34632	0.0676549
$397$	− 39.4596i	− 1.98042i	−0.139586	−	0.990210i	$-0.544577\pi$
$397$	− 39.4596i	− 1.98042i	0.139586	−	0.990210i	$-0.455423\pi$
$398$	− 35.1506i	− 1.76194i
$399$	−1.84324	−0.0922776
$400$	0	0
$401$	0.470266	0.0234840	0.0117420	−	0.999931i	$-0.496262\pi$
$401$	0.470266	0.0234840	0.0117420	+	0.999931i	$0.496262\pi$
$402$	38.5380i	1.92210i
$403$	11.8843i	0.591998i
$404$	13.3340	0.663393
$405$	0	0
$406$	3.31965	0.164752
$407$	34.0410i	1.68735i
$408$	− 8.58145i	− 0.424845i
$409$	−10.4826	−0.518329	−0.259164	−	0.965833i	$-0.583447\pi$
$409$	−10.4826	−0.518329	−0.259164	+	0.965833i	$0.583447\pi$
$410$	0	0
$411$	6.73820	0.332371
$412$	− 17.3112i	− 0.852864i
$413$	− 12.3135i	− 0.605908i
$414$	0.398032	0.0195622
$415$	0	0
$416$	−10.3896	−0.509393
$417$	15.1506i	0.741929i
$418$	13.7587i	0.672961i
$419$	−34.6681	−1.69365	−0.846823	−	0.531875i	$-0.821488\pi$
$419$	−34.6681	−1.69365	−0.846823	+	0.531875i	$0.821488\pi$
$420$	0	0
$421$	−34.6102	−1.68680	−0.843399	−	0.537288i	$-0.819449\pi$
$421$	−34.6102	−1.68680	−0.843399	+	0.537288i	$0.819449\pi$
$422$	− 16.8950i	− 0.822434i
$423$	− 0.0845208i	− 0.00410954i
$424$	−10.2062	−0.495657
$425$	0	0
$426$	40.1978	1.94759
$427$	− 6.04104i	− 0.292346i
$428$	6.20620i	0.299988i
$429$	−14.8371	−0.716342
$430$	0	0
$431$	6.73820	0.324568	0.162284	−	0.986744i	$-0.448114\pi$
$431$	6.73820	0.324568	0.162284	+	0.986744i	$0.448114\pi$
$432$	10.9360i	0.526158i
$433$	20.4741i	0.983924i	0.870617	+	0.491962i	$0.163720\pi$
$433$	20.4741i	0.983924i	−0.870617	+	0.491962i	$0.836280\pi$
$434$	−20.3135	−0.975080
$435$	0	0
$436$	−34.7792	−1.66562
$437$	2.34017i	0.111946i
$438$	− 20.0989i	− 0.960362i
$439$	21.4596	1.02421	0.512105	−	0.858923i	$-0.328866\pi$
$439$	21.4596	1.02421	0.512105	+	0.858923i	$0.328866\pi$
$440$	0	0
$441$	0.457499	0.0217857
$442$	− 9.69102i	− 0.460955i
$443$	− 21.5441i	− 1.02359i	−0.859107	−	0.511796i	$-0.828980\pi$
$443$	− 21.5441i	− 1.02359i	0.859107	−	0.511796i	$-0.171020\pi$
$444$	24.8638	1.17998
$445$	0	0
$446$	27.2267	1.28922
$447$	− 33.7731i	− 1.59741i
$448$	− 13.2762i	− 0.627240i
$449$	8.47027	0.399737	0.199868	−	0.979823i	$-0.435949\pi$
$449$	8.47027	0.399737	0.199868	+	0.979823i	$0.435949\pi$
$450$	0	0
$451$	20.6803	0.973799
$452$	34.9132i	1.64218i
$453$	− 5.84324i	− 0.274540i
$454$	−4.97107	−0.233304
$455$	0	0
$456$	2.63090	0.123203
$457$	− 11.3607i	− 0.531431i	−0.964052	−	0.265715i	$-0.914392\pi$
$457$	− 11.3607i	− 0.531431i	0.964052	−	0.265715i	$-0.0856081\pi$
$458$	− 12.8371i	− 0.599838i
$459$	−17.1629	−0.801096
$460$	0	0
$461$	3.04718	0.141921	0.0709607	−	0.997479i	$-0.477394\pi$
$461$	3.04718	0.141921	0.0709607	+	0.997479i	$0.477394\pi$
$462$	− 25.3607i	− 1.17989i
$463$	− 9.97334i	− 0.463500i	−0.972775	−	0.231750i	$-0.925555\pi$
$463$	− 9.97334i	− 0.463500i	0.972775	−	0.231750i	$-0.0744452\pi$
$464$	2.94828	0.136871
$465$	0	0
$466$	29.3340	1.35887
$467$	− 1.49079i	− 0.0689853i	−0.999405	−	0.0344927i	$-0.989018\pi$
$467$	− 1.49079i	− 0.0689853i	0.999405	−	0.0344927i	$-0.0109815\pi$
$468$	− 0.290725i	− 0.0134388i
$469$	−11.2039	−0.517350
$470$	0	0
$471$	−16.0989	−0.741798
$472$	17.5753i	0.808969i
$473$	75.5462i	3.47362i
$474$	52.8781	2.42877
$475$	0	0
$476$	9.52973	0.436795
$477$	0.519715i	0.0237961i
$478$	30.0410i	1.37405i
$479$	10.1711	0.464731	0.232365	−	0.972629i	$-0.425353\pi$
$479$	10.1711	0.464731	0.232365	+	0.972629i	$0.425353\pi$
$480$	0	0
$481$	7.35085	0.335170
$482$	− 15.7587i	− 0.717790i
$483$	− 4.31351i	− 0.196272i
$484$	−79.1049	−3.59568
$485$	0	0
$486$	−1.76713	−0.0801588
$487$	24.2784i	1.10016i	0.835112	+	0.550081i	$0.185403\pi$
$487$	24.2784i	1.10016i	−0.835112	+	0.550081i	$0.814597\pi$
$488$	8.62249i	0.390322i
$489$	−4.99386	−0.225830
$490$	0	0
$491$	19.2039	0.866662	0.433331	−	0.901235i	$-0.357338\pi$
$491$	19.2039	0.866662	0.433331	+	0.901235i	$0.357338\pi$
$492$	− 15.1050i	− 0.680988i
$493$	4.62702i	0.208391i
$494$	2.97107	0.133675
$495$	0	0
$496$	−18.0410	−0.810067
$497$	11.6865i	0.524211i
$498$	53.1917i	2.38357i
$499$	−7.33403	−0.328316	−0.164158	−	0.986434i	$-0.552491\pi$
$499$	−7.33403	−0.328316	−0.164158	+	0.986434i	$0.552491\pi$
$500$	0	0
$501$	−35.7464	−1.59703
$502$	− 22.7214i	− 1.01410i
$503$	29.0616i	1.29579i	0.761729	+	0.647895i	$0.224351\pi$
$503$	29.0616i	1.29579i	−0.761729	+	0.647895i	$0.775649\pi$
$504$	0.130094	0.00579483
$505$	0	0
$506$	−32.1978	−1.43137
$507$	− 19.0166i	− 0.844559i
$508$	− 22.3051i	− 0.989629i
$509$	−26.0456	−1.15445	−0.577225	−	0.816585i	$-0.695864\pi$
$509$	−26.0456	−1.15445	−0.577225	+	0.816585i	$0.695864\pi$
$510$	0	0
$511$	5.84324	0.258490
$512$	− 22.1701i	− 0.979789i
$513$	− 5.26180i	− 0.232314i
$514$	51.2678	2.26132
$515$	0	0
$516$	55.1794	2.42914
$517$	6.83710i	0.300695i
$518$	12.5646i	0.552058i
$519$	−1.80430	−0.0791998
$520$	0	0
$521$	−38.8248	−1.70095	−0.850473	−	0.526019i	$-0.823684\pi$
$521$	−38.8248	−1.70095	−0.850473	+	0.526019i	$0.823684\pi$
$522$	0.241276i	0.0105604i
$523$	4.59970i	0.201131i	0.994930	+	0.100565i	$0.0320652\pi$
$523$	4.59970i	0.201131i	−0.994930	+	0.100565i	$0.967935\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	12.2823	0.535534
$527$	− 28.3135i	− 1.23336i
$528$	− 22.5236i	− 0.980213i
$529$	17.5236	0.761895
$530$	0	0
$531$	0.894960	0.0388380
$532$	2.92162i	0.126668i
$533$	− 4.46573i	− 0.193432i
$534$	28.0989	1.21596
$535$	0	0
$536$	15.9916	0.690731
$537$	1.52973i	0.0660129i
$538$	− 5.02052i	− 0.216450i
$539$	−37.0082	−1.59406
$540$	0	0
$541$	−12.1256	−0.521318	−0.260659	−	0.965431i	$-0.583940\pi$
$541$	−12.1256	−0.521318	−0.260659	+	0.965431i	$0.583940\pi$
$542$	42.7526i	1.83638i
$543$	1.43084i	0.0614031i
$544$	24.7526	1.06126
$545$	0	0
$546$	−5.47641	−0.234369
$547$	9.54023i	0.407911i	0.978980	+	0.203955i	$0.0653798\pi$
$547$	9.54023i	0.407911i	−0.978980	+	0.203955i	$0.934620\pi$
$548$	− 10.6803i	− 0.456242i
$549$	0.439070	0.0187390
$550$	0	0
$551$	−1.41855	−0.0604323
$552$	6.15676i	0.262049i
$553$	15.3730i	0.653726i
$554$	55.8408	2.37245
$555$	0	0
$556$	24.0144	1.01844
$557$	− 19.9421i	− 0.844976i	−0.906369	−	0.422488i	$-0.861157\pi$
$557$	− 19.9421i	− 0.844976i	0.906369	−	0.422488i	$-0.138843\pi$
$558$	− 1.47641i	− 0.0625014i
$559$	16.3135	0.689988
$560$	0	0
$561$	35.3484	1.49241
$562$	14.2823i	0.602463i
$563$	23.5525i	0.992620i	0.868145	+	0.496310i	$0.165312\pi$
$563$	23.5525i	0.992620i	−0.868145	+	0.496310i	$0.834688\pi$
$564$	4.99386	0.210279
$565$	0	0
$566$	−1.07838	−0.0453276
$567$	9.44521i	0.396662i
$568$	− 16.6803i	− 0.699892i
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−23.5031	−0.983573	−0.491786	−	0.870716i	$-0.663656\pi$
$571$	−23.5031	−0.983573	−0.491786	+	0.870716i	$0.663656\pi$
$572$	23.5174i	0.983314i
$573$	− 37.6742i	− 1.57386i
$574$	7.63317	0.318602
$575$	0	0
$576$	0.964928	0.0402053
$577$	− 16.4703i	− 0.685666i	−0.939396	−	0.342833i	$-0.888613\pi$
$577$	− 16.4703i	− 0.685666i	0.939396	−	0.342833i	$-0.111387\pi$
$578$	− 13.8033i	− 0.574140i
$579$	21.8576	0.908372
$580$	0	0
$581$	−15.4641	−0.641560
$582$	− 32.9627i	− 1.36635i
$583$	− 42.0410i	− 1.74116i
$584$	−8.34017	−0.345119
$585$	0	0
$586$	14.3896	0.594430
$587$	28.8104i	1.18913i	0.804046	+	0.594567i	$0.202677\pi$
$587$	28.8104i	1.18913i	−0.804046	+	0.594567i	$0.797323\pi$
$588$	27.0310i	1.11474i
$589$	8.68035	0.357667
$590$	0	0
$591$	15.7321	0.647131
$592$	11.1590i	0.458633i
$593$	− 43.2450i	− 1.77586i	−0.459980	−	0.887929i	$-0.652144\pi$
$593$	− 43.2450i	− 1.77586i	0.459980	−	0.887929i	$-0.347856\pi$
$594$	72.3956	2.97043
$595$	0	0
$596$	−53.5318	−2.19275
$597$	− 27.6865i	− 1.13313i
$598$	6.95282i	0.284322i
$599$	44.2967	1.80991	0.904957	−	0.425503i	$-0.139903\pi$
$599$	44.2967	1.80991	0.904957	+	0.425503i	$0.139903\pi$
$600$	0	0
$601$	−24.3090	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$601$	−24.3090	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$602$	27.8843i	1.13648i
$603$	− 0.814315i	− 0.0331615i
$604$	−9.26180	−0.376857
$605$	0	0
$606$	18.2557	0.741585
$607$	6.29072i	0.255333i	0.991817	+	0.127666i	$0.0407487\pi$
$607$	6.29072i	0.255333i	−0.991817	+	0.127666i	$0.959251\pi$
$608$	7.58864i	0.307760i
$609$	2.61474	0.105954
$610$	0	0
$611$	1.47641	0.0597291
$612$	0.692632i	0.0279980i
$613$	− 12.7915i	− 0.516645i	−0.966059	−	0.258322i	$-0.916830\pi$
$613$	− 12.7915i	− 0.516645i	0.966059	−	0.258322i	$-0.0831697\pi$
$614$	−31.6925	−1.27900
$615$	0	0
$616$	−10.5236	−0.424008
$617$	− 17.9299i	− 0.721829i	−0.932599	−	0.360914i	$-0.882465\pi$
$617$	− 17.9299i	− 0.721829i	0.932599	−	0.360914i	$-0.117535\pi$
$618$	− 23.7009i	− 0.953389i
$619$	−26.8515	−1.07925	−0.539626	−	0.841905i	$-0.681434\pi$
$619$	−26.8515	−1.07925	−0.539626	+	0.841905i	$0.681434\pi$
$620$	0	0
$621$	12.3135	0.494124
$622$	− 41.9565i	− 1.68230i
$623$	8.16904i	0.327286i
$624$	−4.86376	−0.194706
$625$	0	0
$626$	66.5790	2.66103
$627$	10.8371i	0.432792i
$628$	25.5174i	1.01826i
$629$	−17.5129	−0.698286
$630$	0	0
$631$	35.5318	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$631$	35.5318	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$632$	− 21.9421i	− 0.872812i
$633$	− 13.3074i	− 0.528920i
$634$	−40.6453	−1.61423
$635$	0	0
$636$	−30.7070	−1.21761
$637$	7.99159i	0.316638i
$638$	− 19.5174i	− 0.772703i
$639$	−0.849388	−0.0336013
$640$	0	0
$641$	12.9360	0.510941	0.255471	−	0.966817i	$-0.417770\pi$
$641$	12.9360	0.510941	0.255471	+	0.966817i	$0.417770\pi$
$642$	8.49693i	0.335347i
$643$	8.49693i	0.335086i	0.985865	+	0.167543i	$0.0535833\pi$
$643$	8.49693i	0.335086i	−0.985865	+	0.167543i	$0.946417\pi$
$644$	−6.83710	−0.269420
$645$	0	0
$646$	−7.07838	−0.278495
$647$	45.4908i	1.78843i	0.447640	+	0.894214i	$0.352264\pi$
$647$	45.4908i	1.78843i	−0.447640	+	0.894214i	$0.647736\pi$
$648$	− 13.4813i	− 0.529597i
$649$	−72.3956	−2.84178
$650$	0	0
$651$	−16.0000	−0.627089
$652$	7.91548i	0.309994i
$653$	35.9421i	1.40652i	0.710930	+	0.703262i	$0.248274\pi$
$653$	35.9421i	1.40652i	−0.710930	+	0.703262i	$0.751726\pi$
$654$	−47.6163	−1.86195
$655$	0	0
$656$	6.77924	0.264685
$657$	0.424694i	0.0165689i
$658$	2.52359i	0.0983798i
$659$	−30.9360	−1.20510	−0.602548	−	0.798083i	$-0.705848\pi$
$659$	−30.9360	−1.20510	−0.602548	+	0.798083i	$0.705848\pi$
$660$	0	0
$661$	10.0989	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$661$	10.0989	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$662$	5.94214i	0.230948i
$663$	− 7.63317i	− 0.296448i
$664$	22.0722	0.856569
$665$	0	0
$666$	−0.913212	−0.0353862
$667$	− 3.31965i	− 0.128538i
$668$	56.6596i	2.19223i
$669$	21.4452	0.829120
$670$	0	0
$671$	−35.5174	−1.37114
$672$	− 13.9877i	− 0.539588i
$673$	− 8.67194i	− 0.334279i	−0.985933	−	0.167139i	$-0.946547\pi$
$673$	− 8.67194i	− 0.334279i	0.985933	−	0.167139i	$-0.0534530\pi$
$674$	13.1278	0.505665
$675$	0	0
$676$	−30.1422	−1.15932
$677$	− 41.9793i	− 1.61340i	−0.590964	−	0.806698i	$-0.701253\pi$
$677$	− 41.9793i	− 1.61340i	0.590964	−	0.806698i	$-0.298747\pi$
$678$	47.7998i	1.83574i
$679$	9.58306	0.367764
$680$	0	0
$681$	−3.91548	−0.150041
$682$	119.430i	4.57323i
$683$	46.3896i	1.77505i	0.460760	+	0.887525i	$0.347577\pi$
$683$	46.3896i	1.77505i	−0.460760	+	0.887525i	$0.652423\pi$
$684$	−0.212347	−0.00811929
$685$	0	0
$686$	−30.0410	−1.14697
$687$	− 10.1112i	− 0.385766i
$688$	24.7649i	0.944152i
$689$	−9.07838	−0.345859
$690$	0	0
$691$	−34.8515	−1.32581	−0.662906	−	0.748702i	$-0.730677\pi$
$691$	−34.8515	−1.32581	−0.662906	+	0.748702i	$0.730677\pi$
$692$	2.85989i	0.108717i
$693$	0.535877i	0.0203563i
$694$	12.9627	0.492056
$695$	0	0
$696$	−3.73206	−0.141463
$697$	10.6393i	0.402993i
$698$	− 21.7009i	− 0.821390i
$699$	23.1050	0.873913
$700$	0	0
$701$	35.6430	1.34622	0.673109	−	0.739543i	$-0.264959\pi$
$701$	35.6430	1.34622	0.673109	+	0.739543i	$0.264959\pi$
$702$	− 15.6332i	− 0.590036i
$703$	− 5.36910i	− 0.202500i
$704$	−78.0554	−2.94182
$705$	0	0
$706$	−30.5958	−1.15149
$707$	5.30737i	0.199604i
$708$	52.8781i	1.98728i
$709$	−16.7214	−0.627985	−0.313992	−	0.949426i	$-0.601667\pi$
$709$	−16.7214	−0.627985	−0.313992	+	0.949426i	$0.601667\pi$
$710$	0	0
$711$	−1.11733	−0.0419030
$712$	− 11.6598i	− 0.436970i
$713$	20.3135i	0.760747i
$714$	13.0472	0.488278
$715$	0	0
$716$	2.42469	0.0906151
$717$	23.6619i	0.883670i
$718$	13.0784i	0.488081i
$719$	6.85148	0.255517	0.127758	−	0.991805i	$-0.459222\pi$
$719$	6.85148	0.255517	0.127758	+	0.991805i	$0.459222\pi$
$720$	0	0
$721$	6.89043	0.256613
$722$	− 2.17009i	− 0.0807623i
$723$	− 12.4124i	− 0.461622i
$724$	2.26794	0.0842873
$725$	0	0
$726$	−108.303	−4.01949
$727$	− 34.4391i	− 1.27727i	−0.769508	−	0.638637i	$-0.779498\pi$
$727$	− 34.4391i	− 1.27727i	0.769508	−	0.638637i	$-0.220502\pi$
$728$	2.27247i	0.0842235i
$729$	−27.6681	−1.02474
$730$	0	0
$731$	−38.8659	−1.43751
$732$	25.9421i	0.958849i
$733$	− 19.3607i	− 0.715103i	−0.933893	−	0.357552i	$-0.883612\pi$
$733$	− 19.3607i	− 0.715103i	0.933893	−	0.357552i	$-0.116388\pi$
$734$	−2.34017	−0.0863774
$735$	0	0
$736$	−17.7587	−0.654595
$737$	65.8720i	2.42643i
$738$	0.554787i	0.0204220i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	2.34017	0.0859684
$742$	− 15.5174i	− 0.569663i
$743$	− 12.7154i	− 0.466483i	−0.972419	−	0.233242i	$-0.925067\pi$
$743$	− 12.7154i	− 0.466483i	0.972419	−	0.233242i	$-0.0749333\pi$
$744$	22.8371	0.837249
$745$	0	0
$746$	26.7031	0.977671
$747$	− 1.12395i	− 0.0411232i
$748$	− 56.0288i	− 2.04861i
$749$	−2.47027	−0.0902616
$750$	0	0
$751$	3.15836	0.115250	0.0576252	−	0.998338i	$-0.481647\pi$
$751$	3.15836	0.115250	0.0576252	+	0.998338i	$0.481647\pi$
$752$	2.24128i	0.0817309i
$753$	− 17.8966i	− 0.652187i
$754$	−4.21461	−0.153487
$755$	0	0
$756$	15.3730	0.559110
$757$	− 28.0410i	− 1.01917i	−0.860421	−	0.509584i	$-0.829799\pi$
$757$	− 28.0410i	− 1.01917i	0.860421	−	0.509584i	$-0.170201\pi$
$758$	2.27247i	0.0825399i
$759$	−25.3607	−0.920535
$760$	0	0
$761$	−16.4924	−0.597849	−0.298924	−	0.954277i	$-0.596628\pi$
$761$	−16.4924	−0.597849	−0.298924	+	0.954277i	$0.596628\pi$
$762$	− 30.5380i	− 1.10627i
$763$	− 13.8432i	− 0.501159i
$764$	−59.7152	−2.16042
$765$	0	0
$766$	0.175007	0.00632326
$767$	15.6332i	0.564481i
$768$	0.715418i	0.0258154i
$769$	−26.9627	−0.972298	−0.486149	−	0.873876i	$-0.661599\pi$
$769$	−26.9627	−0.972298	−0.486149	+	0.873876i	$0.661599\pi$
$770$	0	0
$771$	40.3812	1.45429
$772$	− 34.6453i	− 1.24691i
$773$	42.1939i	1.51761i	0.651318	+	0.758805i	$0.274216\pi$
$773$	42.1939i	1.51761i	−0.651318	+	0.758805i	$0.725784\pi$
$774$	−2.02666	−0.0728469
$775$	0	0
$776$	−13.6781	−0.491014
$777$	9.89657i	0.355037i
$778$	− 44.5380i	− 1.59676i
$779$	−3.26180	−0.116866
$780$	0	0
$781$	68.7091	2.45860
$782$	− 16.5646i	− 0.592350i
$783$	7.46412i	0.266746i
$784$	−12.1317	−0.433275
$785$	0	0
$786$	−5.47641	−0.195337
$787$	35.4368i	1.26319i	0.775300	+	0.631593i	$0.217599\pi$
$787$	35.4368i	1.26319i	−0.775300	+	0.631593i	$0.782401\pi$
$788$	− 24.9360i	− 0.888308i
$789$	9.67420	0.344411
$790$	0	0
$791$	−13.8966	−0.494105
$792$	− 0.764867i	− 0.0271784i
$793$	7.66967i	0.272358i
$794$	−85.6307	−3.03892
$795$	0	0
$796$	−43.8843	−1.55544
$797$	− 15.2579i	− 0.540463i	−0.962795	−	0.270232i	$-0.912900\pi$
$797$	− 15.2579i	− 0.540463i	0.962795	−	0.270232i	$-0.0871003\pi$
$798$	4.00000i	0.141598i
$799$	−3.51745	−0.124438
$800$	0	0
$801$	−0.593735	−0.0209786
$802$	− 1.02052i	− 0.0360358i
$803$	− 34.3545i	− 1.21235i
$804$	48.1133	1.69682
$805$	0	0
$806$	25.7899	0.908411
$807$	− 3.95443i	− 0.139202i
$808$	− 7.57531i	− 0.266498i
$809$	7.16290	0.251834	0.125917	−	0.992041i	$-0.459813\pi$
$809$	7.16290	0.251834	0.125917	+	0.992041i	$0.459813\pi$
$810$	0	0
$811$	−50.3545	−1.76819	−0.884094	−	0.467310i	$-0.845223\pi$
$811$	−50.3545	−1.76819	−0.884094	+	0.467310i	$0.845223\pi$
$812$	− 4.14447i	− 0.145442i
$813$	33.6742i	1.18101i
$814$	73.8720	2.58921
$815$	0	0
$816$	11.5876	0.405647
$817$	− 11.9155i	− 0.416870i
$818$	22.7480i	0.795367i
$819$	0.115718	0.00404350
$820$	0	0
$821$	0.952819	0.0332536	0.0166268	−	0.999862i	$-0.494707\pi$
$821$	0.952819	0.0332536	0.0166268	+	0.999862i	$0.494707\pi$
$822$	− 14.6225i	− 0.510018i
$823$	− 44.2290i	− 1.54173i	−0.637001	−	0.770863i	$-0.719825\pi$
$823$	− 44.2290i	− 1.54173i	0.637001	−	0.770863i	$-0.280175\pi$
$824$	−9.83483	−0.342613
$825$	0	0
$826$	−26.7214	−0.929756
$827$	37.8615i	1.31657i	0.752767	+	0.658287i	$0.228719\pi$
$827$	37.8615i	1.31657i	−0.752767	+	0.658287i	$0.771281\pi$
$828$	− 0.496928i	− 0.0172695i
$829$	56.7214	1.97002	0.985008	−	0.172511i	$-0.0551881\pi$
$829$	56.7214	1.97002	0.985008	+	0.172511i	$0.0551881\pi$
$830$	0	0
$831$	43.9832	1.52576
$832$	16.8554i	0.584354i
$833$	− 19.0394i	− 0.659677i
$834$	32.8781	1.13848
$835$	0	0
$836$	17.1773	0.594088
$837$	− 45.6742i	− 1.57873i
$838$	75.2327i	2.59887i
$839$	28.3591	0.979064	0.489532	−	0.871985i	$-0.337168\pi$
$839$	28.3591	0.979064	0.489532	+	0.871985i	$0.337168\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	75.1071i	2.58836i
$843$	11.2495i	0.387454i
$844$	−21.0928	−0.726043
$845$	0	0
$846$	−0.183417	−0.00630602
$847$	− 31.4863i	− 1.08188i
$848$	− 13.7815i	− 0.473259i
$849$	−0.849388	−0.0291509
$850$	0	0
$851$	12.5646	0.430710
$852$	− 50.1855i	− 1.71933i
$853$	− 17.0061i	− 0.582279i	−0.956681	−	0.291140i	$-0.905966\pi$
$853$	− 17.0061i	− 0.582279i	0.956681	−	0.291140i	$-0.0940344\pi$
$854$	−13.1096	−0.448600
$855$	0	0
$856$	3.52586	0.120511
$857$	− 15.4101i	− 0.526400i	−0.964741	−	0.263200i	$-0.915222\pi$
$857$	− 15.4101i	− 0.526400i	0.964741	−	0.263200i	$-0.0847780\pi$
$858$	32.1978i	1.09921i
$859$	37.7275	1.28725	0.643623	−	0.765342i	$-0.277430\pi$
$859$	37.7275	1.28725	0.643623	+	0.765342i	$0.277430\pi$
$860$	0	0
$861$	6.01229	0.204898
$862$	− 14.6225i	− 0.498044i
$863$	32.1340i	1.09385i	0.837181	+	0.546927i	$0.184202\pi$
$863$	32.1340i	1.09385i	−0.837181	+	0.546927i	$0.815798\pi$
$864$	39.9299	1.35844
$865$	0	0
$866$	44.4307	1.50982
$867$	− 10.8722i	− 0.369238i
$868$	25.3607i	0.860798i
$869$	90.3833	3.06604
$870$	0	0
$871$	14.2245	0.481977
$872$	19.7587i	0.669115i
$873$	0.696508i	0.0235732i
$874$	5.07838	0.171779
$875$	0	0
$876$	−25.0928	−0.847806
$877$	19.8927i	0.671729i	0.941910	+	0.335864i	$0.109028\pi$
$877$	19.8927i	0.671729i	−0.941910	+	0.335864i	$0.890972\pi$
$878$	− 46.5692i	− 1.57163i
$879$	11.3340	0.382287
$880$	0	0
$881$	24.0722	0.811014	0.405507	−	0.914092i	$-0.367095\pi$
$881$	24.0722	0.811014	0.405507	+	0.914092i	$0.367095\pi$
$882$	− 0.992812i	− 0.0334297i
$883$	− 31.1727i	− 1.04905i	−0.851396	−	0.524523i	$-0.824244\pi$
$883$	− 31.1727i	− 1.04905i	0.851396	−	0.524523i	$-0.175756\pi$
$884$	−12.0989	−0.406930
$885$	0	0
$886$	−46.7526	−1.57068
$887$	− 4.86764i	− 0.163439i	−0.996655	−	0.0817197i	$-0.973959\pi$
$887$	− 4.86764i	− 0.163439i	0.996655	−	0.0817197i	$-0.0260412\pi$
$888$	− 14.1256i	− 0.474023i
$889$	8.87814	0.297763
$890$	0	0
$891$	55.5318	1.86039
$892$	− 33.9916i	− 1.13812i
$893$	− 1.07838i	− 0.0360865i
$894$	−73.2905	−2.45120
$895$	0	0
$896$	−12.4436	−0.415712
$897$	5.47641i	0.182852i
$898$	− 18.3812i	− 0.613389i
$899$	−12.3135	−0.410679
$900$	0	0
$901$	21.6286	0.720554
$902$	− 44.8781i	− 1.49428i
$903$	21.9631i	0.730888i
$904$	19.8348	0.659697
$905$	0	0
$906$	−12.6803	−0.421276
$907$	45.4778i	1.51007i	0.655686	+	0.755033i	$0.272379\pi$
$907$	45.4778i	1.51007i	−0.655686	+	0.755033i	$0.727621\pi$
$908$	6.20620i	0.205960i
$909$	−0.385746	−0.0127944
$910$	0	0
$911$	−20.9483	−0.694048	−0.347024	−	0.937856i	$-0.612808\pi$
$911$	−20.9483	−0.694048	−0.347024	+	0.937856i	$0.612808\pi$
$912$	3.55252i	0.117636i
$913$	90.9192i	3.00899i
$914$	−24.6537	−0.815471
$915$	0	0
$916$	−16.0267	−0.529536
$917$	− 1.59213i	− 0.0525767i
$918$	37.2450i	1.22927i
$919$	59.5174	1.96330	0.981650	−	0.190693i	$-0.0610735\pi$
$919$	59.5174	1.96330	0.981650	+	0.190693i	$0.0610735\pi$
$920$	0	0
$921$	−24.9627	−0.822548
$922$	− 6.61265i	− 0.217776i
$923$	− 14.8371i	− 0.488369i
$924$	−31.6619	−1.04160
$925$	0	0
$926$	−21.6430	−0.711233
$927$	0.500804i	0.0164486i
$928$	− 10.7649i	− 0.353374i
$929$	16.7214	0.548611	0.274305	−	0.961643i	$-0.411552\pi$
$929$	16.7214	0.548611	0.274305	+	0.961643i	$0.411552\pi$
$930$	0	0
$931$	5.83710	0.191303
$932$	− 36.6225i	− 1.19961i
$933$	− 33.0472i	− 1.08192i
$934$	−3.23513	−0.105857
$935$	0	0
$936$	−0.165166	−0.00539862
$937$	32.4534i	1.06021i	0.847933	+	0.530104i	$0.177847\pi$
$937$	32.4534i	1.06021i	−0.847933	+	0.530104i	$0.822153\pi$
$938$	24.3135i	0.793864i
$939$	52.4412	1.71135
$940$	0	0
$941$	−23.6742	−0.771757	−0.385878	−	0.922550i	$-0.626102\pi$
$941$	−23.6742	−0.771757	−0.385878	+	0.922550i	$0.626102\pi$
$942$	34.9360i	1.13828i
$943$	− 7.63317i	− 0.248570i
$944$	−23.7321	−0.772413
$945$	0	0
$946$	163.942	5.33021
$947$	− 21.9733i	− 0.714038i	−0.934097	−	0.357019i	$-0.883793\pi$
$947$	− 21.9733i	− 0.714038i	0.934097	−	0.357019i	$-0.116207\pi$
$948$	− 66.0165i	− 2.14412i
$949$	−7.41855	−0.240816
$950$	0	0
$951$	−32.0144	−1.03814
$952$	− 5.41402i	− 0.175469i
$953$	− 53.2990i	− 1.72652i	−0.504757	−	0.863261i	$-0.668418\pi$
$953$	− 53.2990i	− 1.72652i	0.504757	−	0.863261i	$-0.331582\pi$
$954$	1.12783	0.0365147
$955$	0	0
$956$	37.5052	1.21300
$957$	− 15.3730i	− 0.496938i
$958$	− 22.0722i	− 0.713122i
$959$	4.25112	0.137276
$960$	0	0
$961$	44.3484	1.43059
$962$	− 15.9520i	− 0.514313i
$963$	− 0.179542i	− 0.00578565i
$964$	−19.6742	−0.633663
$965$	0	0
$966$	−9.36069	−0.301175
$967$	− 15.8166i	− 0.508627i	−0.967122	−	0.254314i	$-0.918150\pi$
$967$	− 15.8166i	− 0.508627i	0.967122	−	0.254314i	$-0.0818495\pi$
$968$	44.9409i	1.44446i
$969$	−5.57531	−0.179105
$970$	0	0
$971$	−43.1506	−1.38477	−0.692385	−	0.721529i	$-0.743440\pi$
$971$	−43.1506	−1.38477	−0.692385	+	0.721529i	$0.743440\pi$
$972$	2.20620i	0.0707640i
$973$	9.55849i	0.306431i
$974$	52.6863	1.68818
$975$	0	0
$976$	−11.6430	−0.372684
$977$	8.47414i	0.271112i	0.990770	+	0.135556i	$0.0432820\pi$
$977$	8.47414i	0.271112i	−0.990770	+	0.135556i	$0.956718\pi$
$978$	10.8371i	0.346532i
$979$	48.0288	1.53501
$980$	0	0
$981$	1.00614	0.0321237
$982$	− 41.6742i	− 1.32988i
$983$	− 5.59356i	− 0.178407i	−0.996013	−	0.0892034i	$-0.971568\pi$
$983$	− 5.59356i	− 0.178407i	0.996013	−	0.0892034i	$-0.0284321\pi$
$984$	−8.58145	−0.273567
$985$	0	0
$986$	10.0410	0.319772
$987$	1.98771i	0.0632696i
$988$	− 3.70928i	− 0.118008i
$989$	27.8843	0.886669
$990$	0	0
$991$	32.8950	1.04494	0.522471	−	0.852657i	$-0.325010\pi$
$991$	32.8950	1.04494	0.522471	+	0.852657i	$0.325010\pi$
$992$	65.8720i	2.09144i
$993$	4.68035i	0.148526i
$994$	25.3607	0.804392
$995$	0	0
$996$	66.4079	2.10421
$997$	− 23.2618i	− 0.736708i	−0.929686	−	0.368354i	$-0.879921\pi$
$997$	− 23.2618i	− 0.736708i	0.929686	−	0.368354i	$-0.120079\pi$
$998$	15.9155i	0.503796i
$999$	−28.2511	−0.893826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.d.324.1		6
5.2	odd	4		95.2.a.a.1.3	✓	3
5.3	odd	4		475.2.a.f.1.1		3
5.4	even	2	inner	475.2.b.d.324.6		6
15.2	even	4		855.2.a.i.1.1		3
15.8	even	4		4275.2.a.bk.1.3		3
20.3	even	4		7600.2.a.bx.1.1		3
20.7	even	4		1520.2.a.p.1.3		3
35.27	even	4		4655.2.a.u.1.3		3
40.27	even	4		6080.2.a.by.1.1		3
40.37	odd	4		6080.2.a.bo.1.3		3
95.18	even	4		9025.2.a.bb.1.3		3
95.37	even	4		1805.2.a.f.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.3	✓	3	5.2	odd	4
475.2.a.f.1.1		3	5.3	odd	4
475.2.b.d.324.1		6	1.1	even	1	trivial
475.2.b.d.324.6		6	5.4	even	2	inner
855.2.a.i.1.1		3	15.2	even	4
1520.2.a.p.1.3		3	20.7	even	4
1805.2.a.f.1.1		3	95.37	even	4
4275.2.a.bk.1.3		3	15.8	even	4
4655.2.a.u.1.3		3	35.27	even	4
6080.2.a.bo.1.3		3	40.37	odd	4
6080.2.a.by.1.1		3	40.27	even	4
7600.2.a.bx.1.1		3	20.3	even	4
9025.2.a.bb.1.3		3	95.18	even	4

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} +O(q^{10})\)	\(q-2.17009i q^{2} -1.70928i q^{3} -2.70928 q^{4} -3.70928 q^{6} -1.07838i q^{7} +1.53919i q^{8} +0.0783777 q^{9} -6.34017 q^{11} +4.63090i q^{12} +1.36910i q^{13} -2.34017 q^{14} -2.07838 q^{16} -3.26180i q^{17} -0.170086i q^{18} +1.00000 q^{19} -1.84324 q^{21} +13.7587i q^{22} +2.34017i q^{23} +2.63090 q^{24} +2.97107 q^{26} -5.26180i q^{27} +2.92162i q^{28} -1.41855 q^{29} +8.68035 q^{31} +7.58864i q^{32} +10.8371i q^{33} -7.07838 q^{34} -0.212347 q^{36} -5.36910i q^{37} -2.17009i q^{38} +2.34017 q^{39} -3.26180 q^{41} +4.00000i q^{42} -11.9155i q^{43} +17.1773 q^{44} +5.07838 q^{46} -1.07838i q^{47} +3.55252i q^{48} +5.83710 q^{49} -5.57531 q^{51} -3.70928i q^{52} +6.63090i q^{53} -11.4186 q^{54} +1.65983 q^{56} -1.70928i q^{57} +3.07838i q^{58} +11.4186 q^{59} +5.60197 q^{61} -18.8371i q^{62} -0.0845208i q^{63} +12.3112 q^{64} +23.5174 q^{66} -10.3896i q^{67} +8.83710i q^{68} +4.00000 q^{69} -10.8371 q^{71} +0.120638i q^{72} +5.41855i q^{73} -11.6514 q^{74} -2.70928 q^{76} +6.83710i q^{77} -5.07838i q^{78} -14.2557 q^{79} -8.75872 q^{81} +7.07838i q^{82} -14.3402i q^{83} +4.99386 q^{84} -25.8576 q^{86} +2.42469i q^{87} -9.75872i q^{88} -7.57531 q^{89} +1.47641 q^{91} -6.34017i q^{92} -14.8371i q^{93} -2.34017 q^{94} +12.9711 q^{96} +8.88655i q^{97} -12.6670i q^{98} -0.496928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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