LMFDB - Embedded newform 4015.2.a.e.1.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0599364115$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66032 q^{2} +3.11989 q^{3} +5.07733 q^{4} -1.00000 q^{5} +8.29991 q^{6} -0.107344 q^{7} +8.18668 q^{8} +6.73369 q^{9} +O(q^{10})$	\(q+2.66032 q^{2} +3.11989 q^{3} +5.07733 q^{4} -1.00000 q^{5} +8.29991 q^{6} -0.107344 q^{7} +8.18668 q^{8} +6.73369 q^{9} -2.66032 q^{10} +1.00000 q^{11} +15.8407 q^{12} -1.25883 q^{13} -0.285569 q^{14} -3.11989 q^{15} +11.6246 q^{16} -0.853067 q^{17} +17.9138 q^{18} +0.172455 q^{19} -5.07733 q^{20} -0.334900 q^{21} +2.66032 q^{22} -4.20870 q^{23} +25.5415 q^{24} +1.00000 q^{25} -3.34891 q^{26} +11.6487 q^{27} -0.545019 q^{28} -7.97429 q^{29} -8.29991 q^{30} -4.36092 q^{31} +14.5518 q^{32} +3.11989 q^{33} -2.26943 q^{34} +0.107344 q^{35} +34.1891 q^{36} -4.99020 q^{37} +0.458786 q^{38} -3.92742 q^{39} -8.18668 q^{40} -0.464066 q^{41} -0.890943 q^{42} +4.96142 q^{43} +5.07733 q^{44} -6.73369 q^{45} -11.1965 q^{46} -3.55141 q^{47} +36.2674 q^{48} -6.98848 q^{49} +2.66032 q^{50} -2.66147 q^{51} -6.39151 q^{52} -10.9340 q^{53} +30.9893 q^{54} -1.00000 q^{55} -0.878789 q^{56} +0.538040 q^{57} -21.2142 q^{58} +7.60293 q^{59} -15.8407 q^{60} -1.21437 q^{61} -11.6015 q^{62} -0.722820 q^{63} +15.4633 q^{64} +1.25883 q^{65} +8.29991 q^{66} +1.30636 q^{67} -4.33130 q^{68} -13.1307 q^{69} +0.285569 q^{70} +6.76002 q^{71} +55.1266 q^{72} -1.00000 q^{73} -13.2755 q^{74} +3.11989 q^{75} +0.875609 q^{76} -0.107344 q^{77} -10.4482 q^{78} +14.1526 q^{79} -11.6246 q^{80} +16.1415 q^{81} -1.23457 q^{82} +12.2986 q^{83} -1.70040 q^{84} +0.853067 q^{85} +13.1990 q^{86} -24.8789 q^{87} +8.18668 q^{88} +8.45850 q^{89} -17.9138 q^{90} +0.135128 q^{91} -21.3689 q^{92} -13.6056 q^{93} -9.44790 q^{94} -0.172455 q^{95} +45.4000 q^{96} +7.24579 q^{97} -18.5916 q^{98} +6.73369 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) $ 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.66032	1.88113	0.940567	−	0.339609i	$-0.110295\pi$
$2$	2.66032	1.88113	0.940567	+	0.339609i	$0.110295\pi$
$3$	3.11989	1.80127	0.900634	−	0.434579i	$-0.143103\pi$
$3$	3.11989	1.80127	0.900634	+	0.434579i	$0.143103\pi$
$4$	5.07733	2.53866
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	8.29991	3.38842
$7$	−0.107344	−0.0405721	−0.0202861	−	0.999794i	$-0.506458\pi$
$7$	−0.107344	−0.0405721	−0.0202861	+	0.999794i	$0.506458\pi$
$8$	8.18668	2.89443
$9$	6.73369	2.24456
$10$	−2.66032	−0.841268
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	15.8407	4.57281
$13$	−1.25883	−0.349138	−0.174569	−	0.984645i	$-0.555853\pi$
$13$	−1.25883	−0.349138	−0.174569	+	0.984645i	$0.555853\pi$
$14$	−0.285569	−0.0763216
$15$	−3.11989	−0.805551
$16$	11.6246	2.90615
$17$	−0.853067	−0.206899	−0.103450	−	0.994635i	$-0.532988\pi$
$17$	−0.853067	−0.206899	−0.103450	+	0.994635i	$0.532988\pi$
$18$	17.9138	4.22232
$19$	0.172455	0.0395639	0.0197819	−	0.999804i	$-0.493703\pi$
$19$	0.172455	0.0395639	0.0197819	+	0.999804i	$0.493703\pi$
$20$	−5.07733	−1.13532
$21$	−0.334900	−0.0730812
$22$	2.66032	0.567183
$23$	−4.20870	−0.877575	−0.438787	−	0.898591i	$-0.644592\pi$
$23$	−4.20870	−0.877575	−0.438787	+	0.898591i	$0.644592\pi$
$24$	25.5415	5.21364
$25$	1.00000	0.200000
$26$	−3.34891	−0.656775
$27$	11.6487	2.24179
$28$	−0.545019	−0.102999
$29$	−7.97429	−1.48079	−0.740394	−	0.672173i	$-0.765361\pi$
$29$	−7.97429	−1.48079	−0.740394	+	0.672173i	$0.765361\pi$
$30$	−8.29991	−1.51535
$31$	−4.36092	−0.783245	−0.391622	−	0.920126i	$-0.628086\pi$
$31$	−4.36092	−0.783245	−0.391622	+	0.920126i	$0.628086\pi$
$32$	14.5518	2.57242
$33$	3.11989	0.543103
$34$	−2.26943	−0.389205
$35$	0.107344	0.0181444
$36$	34.1891	5.69819
$37$	−4.99020	−0.820383	−0.410192	−	0.911999i	$-0.634538\pi$
$37$	−4.99020	−0.820383	−0.410192	+	0.911999i	$0.634538\pi$
$38$	0.458786	0.0744249
$39$	−3.92742	−0.628891
$40$	−8.18668	−1.29443
$41$	−0.464066	−0.0724750	−0.0362375	−	0.999343i	$-0.511537\pi$
$41$	−0.464066	−0.0724750	−0.0362375	+	0.999343i	$0.511537\pi$
$42$	−0.890943	−0.137476
$43$	4.96142	0.756609	0.378305	−	0.925681i	$-0.376507\pi$
$43$	4.96142	0.756609	0.378305	+	0.925681i	$0.376507\pi$
$44$	5.07733	0.765436
$45$	−6.73369	−1.00380
$46$	−11.1965	−1.65084
$47$	−3.55141	−0.518026	−0.259013	−	0.965874i	$-0.583397\pi$
$47$	−3.55141	−0.518026	−0.259013	+	0.965874i	$0.583397\pi$
$48$	36.2674	5.23475
$49$	−6.98848	−0.998354
$50$	2.66032	0.376227
$51$	−2.66147	−0.372680
$52$	−6.39151	−0.886343
$53$	−10.9340	−1.50190	−0.750948	−	0.660362i	$-0.770403\pi$
$53$	−10.9340	−1.50190	−0.750948	+	0.660362i	$0.770403\pi$
$54$	30.9893	4.21711
$55$	−1.00000	−0.134840
$56$	−0.878789	−0.117433
$57$	0.538040	0.0712651
$58$	−21.2142	−2.78556
$59$	7.60293	0.989817	0.494908	−	0.868945i	$-0.335202\pi$
$59$	7.60293	0.989817	0.494908	+	0.868945i	$0.335202\pi$
$60$	−15.8407	−2.04502
$61$	−1.21437	−0.155484	−0.0777418	−	0.996974i	$-0.524771\pi$
$61$	−1.21437	−0.155484	−0.0777418	+	0.996974i	$0.524771\pi$
$62$	−11.6015	−1.47339
$63$	−0.722820	−0.0910667
$64$	15.4633	1.93292
$65$	1.25883	0.156139
$66$	8.29991	1.02165
$67$	1.30636	0.159598	0.0797989	−	0.996811i	$-0.474572\pi$
$67$	1.30636	0.159598	0.0797989	+	0.996811i	$0.474572\pi$
$68$	−4.33130	−0.525247
$69$	−13.1307	−1.58075
$70$	0.285569	0.0341320
$71$	6.76002	0.802267	0.401133	−	0.916020i	$-0.368616\pi$
$71$	6.76002	0.802267	0.401133	+	0.916020i	$0.368616\pi$
$72$	55.1266	6.49673
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−13.2755	−1.54325
$75$	3.11989	0.360253
$76$	0.875609	0.100439
$77$	−0.107344	−0.0122330
$78$	−10.4482	−1.18303
$79$	14.1526	1.59229	0.796146	−	0.605104i	$-0.206869\pi$
$79$	14.1526	1.59229	0.796146	+	0.605104i	$0.206869\pi$
$80$	−11.6246	−1.29967
$81$	16.1415	1.79350
$82$	−1.23457	−0.136335
$83$	12.2986	1.34995	0.674973	−	0.737843i	$-0.264155\pi$
$83$	12.2986	1.34995	0.674973	+	0.737843i	$0.264155\pi$
$84$	−1.70040	−0.185529
$85$	0.853067	0.0925281
$86$	13.1990	1.42328
$87$	−24.8789	−2.66730
$88$	8.18668	0.872703
$89$	8.45850	0.896600	0.448300	−	0.893883i	$-0.352030\pi$
$89$	8.45850	0.896600	0.448300	+	0.893883i	$0.352030\pi$
$90$	−17.9138	−1.88828
$91$	0.135128	0.0141653
$92$	−21.3689	−2.22787
$93$	−13.6056	−1.41083
$94$	−9.44790	−0.974477
$95$	−0.172455	−0.0176935
$96$	45.4000	4.63361
$97$	7.24579	0.735699	0.367849	−	0.929885i	$-0.380094\pi$
$97$	7.24579	0.735699	0.367849	+	0.929885i	$0.380094\pi$
$98$	−18.5916	−1.87804
$99$	6.73369	0.676762
$100$	5.07733	0.507733
$101$	−8.00084	−0.796113	−0.398057	−	0.917361i	$-0.630315\pi$
$101$	−8.00084	−0.796113	−0.398057	+	0.917361i	$0.630315\pi$
$102$	−7.08038	−0.701062
$103$	9.18645	0.905168	0.452584	−	0.891722i	$-0.350502\pi$
$103$	9.18645	0.905168	0.452584	+	0.891722i	$0.350502\pi$
$104$	−10.3057	−1.01055
$105$	0.334900	0.0326829
$106$	−29.0879	−2.82526
$107$	8.64811	0.836045	0.418022	−	0.908437i	$-0.362723\pi$
$107$	8.64811	0.836045	0.418022	+	0.908437i	$0.362723\pi$
$108$	59.1442	5.69116
$109$	−6.00963	−0.575618	−0.287809	−	0.957688i	$-0.592927\pi$
$109$	−6.00963	−0.575618	−0.287809	+	0.957688i	$0.592927\pi$
$110$	−2.66032	−0.253652
$111$	−15.5688	−1.47773
$112$	−1.24783	−0.117909
$113$	−2.91850	−0.274550	−0.137275	−	0.990533i	$-0.543834\pi$
$113$	−2.91850	−0.274550	−0.137275	+	0.990533i	$0.543834\pi$
$114$	1.43136	0.134059
$115$	4.20870	0.392463
$116$	−40.4881	−3.75922
$117$	−8.47660	−0.783662
$118$	20.2263	1.86198
$119$	0.0915714	0.00839433
$120$	−25.5415	−2.33161
$121$	1.00000	0.0909091
$122$	−3.23061	−0.292485
$123$	−1.44783	−0.130547
$124$	−22.1418	−1.98839
$125$	−1.00000	−0.0894427
$126$	−1.92293	−0.171309
$127$	2.33139	0.206877	0.103439	−	0.994636i	$-0.467015\pi$
$127$	2.33139	0.206877	0.103439	+	0.994636i	$0.467015\pi$
$128$	12.0339	1.06365
$129$	15.4791	1.36286
$130$	3.34891	0.293719
$131$	12.5836	1.09943	0.549717	−	0.835351i	$-0.314736\pi$
$131$	12.5836	1.09943	0.549717	+	0.835351i	$0.314736\pi$
$132$	15.8407	1.37875
$133$	−0.0185119	−0.00160519
$134$	3.47535	0.300225
$135$	−11.6487	−1.00256
$136$	−6.98379	−0.598855
$137$	18.8024	1.60640	0.803201	−	0.595709i	$-0.203129\pi$
$137$	18.8024	1.60640	0.803201	+	0.595709i	$0.203129\pi$
$138$	−34.9318	−2.97360
$139$	2.43362	0.206417	0.103208	−	0.994660i	$-0.467089\pi$
$139$	2.43362	0.206417	0.103208	+	0.994660i	$0.467089\pi$
$140$	0.545019	0.0460625
$141$	−11.0800	−0.933104
$142$	17.9838	1.50917
$143$	−1.25883	−0.105269
$144$	78.2764	6.52303
$145$	7.97429	0.662229
$146$	−2.66032	−0.220170
$147$	−21.8033	−1.79830
$148$	−25.3368	−2.08268
$149$	14.4199	1.18132	0.590662	−	0.806919i	$-0.298867\pi$
$149$	14.4199	1.18132	0.590662	+	0.806919i	$0.298867\pi$
$150$	8.29991	0.677685
$151$	−12.8549	−1.04612	−0.523058	−	0.852297i	$-0.675209\pi$
$151$	−12.8549	−1.04612	−0.523058	+	0.852297i	$0.675209\pi$
$152$	1.41183	0.114515
$153$	−5.74429	−0.464398
$154$	−0.285569	−0.0230118
$155$	4.36092	0.350278
$156$	−19.9408	−1.59654
$157$	−20.3394	−1.62326	−0.811630	−	0.584172i	$-0.801419\pi$
$157$	−20.3394	−1.62326	−0.811630	+	0.584172i	$0.801419\pi$
$158$	37.6505	2.99531
$159$	−34.1127	−2.70531
$160$	−14.5518	−1.15042
$161$	0.451778	0.0356051
$162$	42.9417	3.37382
$163$	−16.2476	−1.27261	−0.636305	−	0.771438i	$-0.719538\pi$
$163$	−16.2476	−1.27261	−0.636305	+	0.771438i	$0.719538\pi$
$164$	−2.35622	−0.183989
$165$	−3.11989	−0.242883
$166$	32.7182	2.53943
$167$	0.306703	0.0237334	0.0118667	−	0.999930i	$-0.496223\pi$
$167$	0.306703	0.0237334	0.0118667	+	0.999930i	$0.496223\pi$
$168$	−2.74172	−0.211529
$169$	−11.4153	−0.878103
$170$	2.26943	0.174058
$171$	1.16126	0.0888036
$172$	25.1907	1.92078
$173$	7.57732	0.576093	0.288047	−	0.957616i	$-0.406994\pi$
$173$	7.57732	0.576093	0.288047	+	0.957616i	$0.406994\pi$
$174$	−66.1859	−5.01754
$175$	−0.107344	−0.00811442
$176$	11.6246	0.876236
$177$	23.7203	1.78292
$178$	22.5024	1.68662
$179$	1.44275	0.107836	0.0539182	−	0.998545i	$-0.482829\pi$
$179$	1.44275	0.107836	0.0539182	+	0.998545i	$0.482829\pi$
$180$	−34.1891	−2.54831
$181$	−4.54518	−0.337841	−0.168920	−	0.985630i	$-0.554028\pi$
$181$	−4.54518	−0.337841	−0.168920	+	0.985630i	$0.554028\pi$
$182$	0.359484	0.0266467
$183$	−3.78868	−0.280068
$184$	−34.4553	−2.54008
$185$	4.99020	0.366886
$186$	−36.1953	−2.65397
$187$	−0.853067	−0.0623824
$188$	−18.0317	−1.31509
$189$	−1.25041	−0.0909543
$190$	−0.458786	−0.0332838
$191$	18.7044	1.35340	0.676701	−	0.736258i	$-0.263409\pi$
$191$	18.7044	1.35340	0.676701	+	0.736258i	$0.263409\pi$
$192$	48.2438	3.48170
$193$	6.33667	0.456124	0.228062	−	0.973647i	$-0.426761\pi$
$193$	6.33667	0.456124	0.228062	+	0.973647i	$0.426761\pi$
$194$	19.2762	1.38395
$195$	3.92742	0.281248
$196$	−35.4828	−2.53448
$197$	−25.1063	−1.78875	−0.894374	−	0.447321i	$-0.852378\pi$
$197$	−25.1063	−1.78875	−0.894374	+	0.447321i	$0.852378\pi$
$198$	17.9138	1.27308
$199$	20.0683	1.42260	0.711301	−	0.702887i	$-0.248106\pi$
$199$	20.0683	1.42260	0.711301	+	0.702887i	$0.248106\pi$
$200$	8.18668	0.578886
$201$	4.07571	0.287478
$202$	−21.2848	−1.49760
$203$	0.855990	0.0600787
$204$	−13.5132	−0.946110
$205$	0.464066	0.0324118
$206$	24.4389	1.70274
$207$	−28.3401	−1.96977
$208$	−14.6334	−1.01465
$209$	0.172455	0.0119290
$210$	0.890943	0.0614809
$211$	−3.54377	−0.243963	−0.121982	−	0.992532i	$-0.538925\pi$
$211$	−3.54377	−0.243963	−0.121982	+	0.992532i	$0.538925\pi$
$212$	−55.5153	−3.81281
$213$	21.0905	1.44510
$214$	23.0068	1.57271
$215$	−4.96142	−0.338366
$216$	95.3642	6.48871
$217$	0.468118	0.0317779
$218$	−15.9876	−1.08281
$219$	−3.11989	−0.210822
$220$	−5.07733	−0.342313
$221$	1.07387	0.0722363
$222$	−41.4182	−2.77981
$223$	−2.66557	−0.178500	−0.0892498	−	0.996009i	$-0.528447\pi$
$223$	−2.66557	−0.178500	−0.0892498	+	0.996009i	$0.528447\pi$
$224$	−1.56204	−0.104368
$225$	6.73369	0.448913
$226$	−7.76416	−0.516465
$227$	−1.52299	−0.101084	−0.0505422	−	0.998722i	$-0.516095\pi$
$227$	−1.52299	−0.101084	−0.0505422	+	0.998722i	$0.516095\pi$
$228$	2.73180	0.180918
$229$	−12.5214	−0.827439	−0.413719	−	0.910404i	$-0.635771\pi$
$229$	−12.5214	−0.827439	−0.413719	+	0.910404i	$0.635771\pi$
$230$	11.1965	0.738276
$231$	−0.334900	−0.0220348
$232$	−65.2830	−4.28604
$233$	14.3448	0.939761	0.469881	−	0.882730i	$-0.344297\pi$
$233$	14.3448	0.939761	0.469881	+	0.882730i	$0.344297\pi$
$234$	−22.5505	−1.47417
$235$	3.55141	0.231668
$236$	38.6025	2.51281
$237$	44.1545	2.86814
$238$	0.243609	0.0157909
$239$	−12.7950	−0.827640	−0.413820	−	0.910359i	$-0.635806\pi$
$239$	−12.7950	−0.827640	−0.413820	+	0.910359i	$0.635806\pi$
$240$	−36.2674	−2.34105
$241$	−23.2527	−1.49784	−0.748919	−	0.662661i	$-0.769427\pi$
$241$	−23.2527	−1.49784	−0.748919	+	0.662661i	$0.769427\pi$
$242$	2.66032	0.171012
$243$	15.4137	0.988787
$244$	−6.16573	−0.394721
$245$	6.98848	0.446477
$246$	−3.85171	−0.245576
$247$	−0.217092	−0.0138132
$248$	−35.7015	−2.26705
$249$	38.3702	2.43161
$250$	−2.66032	−0.168254
$251$	−18.5858	−1.17313	−0.586563	−	0.809904i	$-0.699519\pi$
$251$	−18.5858	−1.17313	−0.586563	+	0.809904i	$0.699519\pi$
$252$	−3.66999	−0.231188
$253$	−4.20870	−0.264599
$254$	6.20225	0.389164
$255$	2.66147	0.166668
$256$	1.08736	0.0679601
$257$	−6.13717	−0.382826	−0.191413	−	0.981510i	$-0.561307\pi$
$257$	−6.13717	−0.382826	−0.191413	+	0.981510i	$0.561307\pi$
$258$	41.1793	2.56371
$259$	0.535666	0.0332847
$260$	6.39151	0.396385
$261$	−53.6964	−3.32372
$262$	33.4764	2.06818
$263$	−0.0514100	−0.00317008	−0.00158504	−	0.999999i	$-0.500505\pi$
$263$	−0.0514100	−0.00317008	−0.00158504	+	0.999999i	$0.500505\pi$
$264$	25.5415	1.57197
$265$	10.9340	0.671668
$266$	−0.0492478	−0.00301958
$267$	26.3896	1.61502
$268$	6.63283	0.405165
$269$	13.0404	0.795086	0.397543	−	0.917584i	$-0.369863\pi$
$269$	13.0404	0.795086	0.397543	+	0.917584i	$0.369863\pi$
$270$	−30.9893	−1.88595
$271$	13.8175	0.839356	0.419678	−	0.907673i	$-0.362143\pi$
$271$	13.8175	0.839356	0.419678	+	0.907673i	$0.362143\pi$
$272$	−9.91654	−0.601279
$273$	0.421584	0.0255154
$274$	50.0206	3.02185
$275$	1.00000	0.0603023
$276$	−66.6687	−4.01298
$277$	20.9423	1.25830	0.629149	−	0.777285i	$-0.283404\pi$
$277$	20.9423	1.25830	0.629149	+	0.777285i	$0.283404\pi$
$278$	6.47421	0.388297
$279$	−29.3651	−1.75804
$280$	0.878789	0.0525177
$281$	−29.8715	−1.78198	−0.890991	−	0.454020i	$-0.849989\pi$
$281$	−29.8715	−1.78198	−0.890991	+	0.454020i	$0.849989\pi$
$282$	−29.4764	−1.75529
$283$	16.5107	0.981460	0.490730	−	0.871312i	$-0.336730\pi$
$283$	16.5107	0.981460	0.490730	+	0.871312i	$0.336730\pi$
$284$	34.3228	2.03668
$285$	−0.538040	−0.0318707
$286$	−3.34891	−0.198025
$287$	0.0498146	0.00294046
$288$	97.9873	5.77396
$289$	−16.2723	−0.957193
$290$	21.2142	1.24574
$291$	22.6060	1.32519
$292$	−5.07733	−0.297128
$293$	−0.758323	−0.0443017	−0.0221508	−	0.999755i	$-0.507051\pi$
$293$	−0.758323	−0.0443017	−0.0221508	+	0.999755i	$0.507051\pi$
$294$	−58.0037	−3.38285
$295$	−7.60293	−0.442659
$296$	−40.8532	−2.37454
$297$	11.6487	0.675926
$298$	38.3616	2.22223
$299$	5.29806	0.306395
$300$	15.8407	0.914562
$301$	−0.532577	−0.0306972
$302$	−34.1982	−1.96788
$303$	−24.9617	−1.43401
$304$	2.00472	0.114978
$305$	1.21437	0.0695344
$306$	−15.2817	−0.873595
$307$	0.372102	0.0212370	0.0106185	−	0.999944i	$-0.496620\pi$
$307$	0.372102	0.0212370	0.0106185	+	0.999944i	$0.496620\pi$
$308$	−0.545019	−0.0310553
$309$	28.6607	1.63045
$310$	11.6015	0.658919
$311$	−1.37217	−0.0778087	−0.0389043	−	0.999243i	$-0.512387\pi$
$311$	−1.37217	−0.0778087	−0.0389043	+	0.999243i	$0.512387\pi$
$312$	−32.1526	−1.82028
$313$	5.29907	0.299521	0.149761	−	0.988722i	$-0.452150\pi$
$313$	5.29907	0.299521	0.149761	+	0.988722i	$0.452150\pi$
$314$	−54.1094	−3.05357
$315$	0.722820	0.0407263
$316$	71.8574	4.04229
$317$	11.5995	0.651491	0.325746	−	0.945457i	$-0.394385\pi$
$317$	11.5995	0.651491	0.325746	+	0.945457i	$0.394385\pi$
$318$	−90.7509	−5.08906
$319$	−7.97429	−0.446475
$320$	−15.4633	−0.864426
$321$	26.9811	1.50594
$322$	1.20188	0.0669779
$323$	−0.147115	−0.00818572
$324$	81.9558	4.55310
$325$	−1.25883	−0.0698276
$326$	−43.2238	−2.39395
$327$	−18.7494	−1.03684
$328$	−3.79916	−0.209774
$329$	0.381222	0.0210174
$330$	−8.29991	−0.456895
$331$	−33.0798	−1.81823	−0.909115	−	0.416544i	$-0.863241\pi$
$331$	−33.0798	−1.81823	−0.909115	+	0.416544i	$0.863241\pi$
$332$	62.4439	3.42706
$333$	−33.6024	−1.84140
$334$	0.815928	0.0446456
$335$	−1.30636	−0.0713743
$336$	−3.89308	−0.212385
$337$	17.6445	0.961157	0.480579	−	0.876952i	$-0.340427\pi$
$337$	17.6445	0.961157	0.480579	+	0.876952i	$0.340427\pi$
$338$	−30.3685	−1.65183
$339$	−9.10540	−0.494537
$340$	4.33130	0.234898
$341$	−4.36092	−0.236157
$342$	3.08932	0.167051
$343$	1.50158	0.0810775
$344$	40.6176	2.18995
$345$	13.1307	0.706931
$346$	20.1581	1.08371
$347$	23.1418	1.24232	0.621159	−	0.783685i	$-0.286662\pi$
$347$	23.1418	1.24232	0.621159	+	0.783685i	$0.286662\pi$
$348$	−126.318	−6.77137
$349$	−16.7412	−0.896134	−0.448067	−	0.894000i	$-0.647887\pi$
$349$	−16.7412	−0.896134	−0.448067	+	0.894000i	$0.647887\pi$
$350$	−0.285569	−0.0152643
$351$	−14.6638	−0.782695
$352$	14.5518	0.775613
$353$	−26.1351	−1.39103	−0.695515	−	0.718511i	$-0.744824\pi$
$353$	−26.1351	−1.39103	−0.695515	+	0.718511i	$0.744824\pi$
$354$	63.1036	3.35392
$355$	−6.76002	−0.358785
$356$	42.9466	2.27616
$357$	0.285692	0.0151204
$358$	3.83819	0.202855
$359$	12.7629	0.673601	0.336801	−	0.941576i	$-0.390655\pi$
$359$	12.7629	0.673601	0.336801	+	0.941576i	$0.390655\pi$
$360$	−55.1266	−2.90543
$361$	−18.9703	−0.998435
$362$	−12.0917	−0.635523
$363$	3.11989	0.163752
$364$	0.686089	0.0359608
$365$	1.00000	0.0523424
$366$	−10.0791	−0.526845
$367$	7.85422	0.409987	0.204993	−	0.978763i	$-0.434283\pi$
$367$	7.85422	0.409987	0.204993	+	0.978763i	$0.434283\pi$
$368$	−48.9244	−2.55036
$369$	−3.12488	−0.162675
$370$	13.2755	0.690162
$371$	1.17369	0.0609351
$372$	−69.0800	−3.58163
$373$	7.68456	0.397891	0.198946	−	0.980011i	$-0.436248\pi$
$373$	7.68456	0.397891	0.198946	+	0.980011i	$0.436248\pi$
$374$	−2.26943	−0.117350
$375$	−3.11989	−0.161110
$376$	−29.0743	−1.49939
$377$	10.0383	0.516999
$378$	−3.32651	−0.171097
$379$	−14.5700	−0.748410	−0.374205	−	0.927346i	$-0.622084\pi$
$379$	−14.5700	−0.748410	−0.374205	+	0.927346i	$0.622084\pi$
$380$	−0.875609	−0.0449178
$381$	7.27367	0.372642
$382$	49.7597	2.54593
$383$	25.4851	1.30223	0.651114	−	0.758980i	$-0.274302\pi$
$383$	25.4851	1.30223	0.651114	+	0.758980i	$0.274302\pi$
$384$	37.5443	1.91593
$385$	0.107344	0.00547074
$386$	16.8576	0.858029
$387$	33.4087	1.69826
$388$	36.7892	1.86769
$389$	−15.8704	−0.804659	−0.402329	−	0.915495i	$-0.631799\pi$
$389$	−15.8704	−0.804659	−0.402329	+	0.915495i	$0.631799\pi$
$390$	10.4482	0.529066
$391$	3.59030	0.181569
$392$	−57.2125	−2.88967
$393$	39.2594	1.98037
$394$	−66.7908	−3.36487
$395$	−14.1526	−0.712095
$396$	34.1891	1.71807
$397$	29.2504	1.46803	0.734017	−	0.679131i	$-0.237643\pi$
$397$	29.2504	1.46803	0.734017	+	0.679131i	$0.237643\pi$
$398$	53.3881	2.67610
$399$	−0.0577552	−0.00289138
$400$	11.6246	0.581229
$401$	5.19712	0.259532	0.129766	−	0.991545i	$-0.458577\pi$
$401$	5.19712	0.259532	0.129766	+	0.991545i	$0.458577\pi$
$402$	10.8427	0.540785
$403$	5.48968	0.273460
$404$	−40.6229	−2.02106
$405$	−16.1415	−0.802079
$406$	2.27721	0.113016
$407$	−4.99020	−0.247355
$408$	−21.7886	−1.07870
$409$	16.0266	0.792463	0.396231	−	0.918151i	$-0.370318\pi$
$409$	16.0266	0.792463	0.396231	+	0.918151i	$0.370318\pi$
$410$	1.23457	0.0609709
$411$	58.6615	2.89356
$412$	46.6426	2.29792
$413$	−0.816127	−0.0401590
$414$	−75.3939	−3.70541
$415$	−12.2986	−0.603714
$416$	−18.3183	−0.898129
$417$	7.59260	0.371811
$418$	0.458786	0.0224399
$419$	−23.6784	−1.15677	−0.578383	−	0.815765i	$-0.696316\pi$
$419$	−23.6784	−1.15677	−0.578383	+	0.815765i	$0.696316\pi$
$420$	1.70040	0.0829709
$421$	−17.6961	−0.862453	−0.431227	−	0.902244i	$-0.641919\pi$
$421$	−17.6961	−0.862453	−0.431227	+	0.902244i	$0.641919\pi$
$422$	−9.42757	−0.458927
$423$	−23.9141	−1.16274
$424$	−89.5129	−4.34713
$425$	−0.853067	−0.0413798
$426$	56.1076	2.71842
$427$	0.130355	0.00630830
$428$	43.9093	2.12244
$429$	−3.92742	−0.189618
$430$	−13.1990	−0.636511
$431$	19.3404	0.931596	0.465798	−	0.884891i	$-0.345767\pi$
$431$	19.3404	0.931596	0.465798	+	0.884891i	$0.345767\pi$
$432$	135.411	6.51498
$433$	6.56844	0.315659	0.157829	−	0.987466i	$-0.449550\pi$
$433$	6.56844	0.315659	0.157829	+	0.987466i	$0.449550\pi$
$434$	1.24534	0.0597785
$435$	24.8789	1.19285
$436$	−30.5128	−1.46130
$437$	−0.725811	−0.0347202
$438$	−8.29991	−0.396585
$439$	18.2410	0.870596	0.435298	−	0.900286i	$-0.356643\pi$
$439$	18.2410	0.870596	0.435298	+	0.900286i	$0.356643\pi$
$440$	−8.18668	−0.390285
$441$	−47.0583	−2.24087
$442$	2.85684	0.135886
$443$	−33.4633	−1.58989	−0.794944	−	0.606683i	$-0.792500\pi$
$443$	−33.4633	−1.58989	−0.794944	+	0.606683i	$0.792500\pi$
$444$	−79.0481	−3.75146
$445$	−8.45850	−0.400972
$446$	−7.09127	−0.335782
$447$	44.9884	2.12788
$448$	−1.65989	−0.0784225
$449$	−35.6223	−1.68112	−0.840560	−	0.541719i	$-0.817774\pi$
$449$	−35.6223	−1.68112	−0.840560	+	0.541719i	$0.817774\pi$
$450$	17.9138	0.844465
$451$	−0.464066	−0.0218520
$452$	−14.8182	−0.696989
$453$	−40.1058	−1.88433
$454$	−4.05165	−0.190153
$455$	−0.135128	−0.00633490
$456$	4.40476	0.206272
$457$	−6.72315	−0.314496	−0.157248	−	0.987559i	$-0.550262\pi$
$457$	−6.72315	−0.314496	−0.157248	+	0.987559i	$0.550262\pi$
$458$	−33.3110	−1.55652
$459$	−9.93711	−0.463825
$460$	21.3689	0.996332
$461$	3.61621	0.168424	0.0842118	−	0.996448i	$-0.473163\pi$
$461$	3.61621	0.168424	0.0842118	+	0.996448i	$0.473163\pi$
$462$	−0.890943	−0.0414504
$463$	−11.0558	−0.513808	−0.256904	−	0.966437i	$-0.582702\pi$
$463$	−11.0558	−0.513808	−0.256904	+	0.966437i	$0.582702\pi$
$464$	−92.6978	−4.30339
$465$	13.6056	0.630944
$466$	38.1619	1.76782
$467$	−20.0131	−0.926094	−0.463047	−	0.886334i	$-0.653244\pi$
$467$	−20.0131	−0.926094	−0.463047	+	0.886334i	$0.653244\pi$
$468$	−43.0385	−1.98945
$469$	−0.140230	−0.00647522
$470$	9.44790	0.435799
$471$	−63.4566	−2.92393
$472$	62.2428	2.86496
$473$	4.96142	0.228126
$474$	117.465	5.39536
$475$	0.172455	0.00791277
$476$	0.464938	0.0213104
$477$	−73.6259	−3.37110
$478$	−34.0388	−1.55690
$479$	22.0393	1.00700	0.503500	−	0.863995i	$-0.332045\pi$
$479$	22.0393	1.00700	0.503500	+	0.863995i	$0.332045\pi$
$480$	−45.4000	−2.07221
$481$	6.28183	0.286427
$482$	−61.8597	−2.81763
$483$	1.40950	0.0641343
$484$	5.07733	0.230788
$485$	−7.24579	−0.329014
$486$	41.0054	1.86004
$487$	−10.7412	−0.486732	−0.243366	−	0.969935i	$-0.578252\pi$
$487$	−10.7412	−0.486732	−0.243366	+	0.969935i	$0.578252\pi$
$488$	−9.94163	−0.450037
$489$	−50.6906	−2.29231
$490$	18.5916	0.839884
$491$	17.9213	0.808776	0.404388	−	0.914588i	$-0.367485\pi$
$491$	17.9213	0.808776	0.404388	+	0.914588i	$0.367485\pi$
$492$	−7.35112	−0.331414
$493$	6.80260	0.306374
$494$	−0.577535	−0.0259845
$495$	−6.73369	−0.302657
$496$	−50.6939	−2.27622
$497$	−0.725646	−0.0325497
$498$	102.077	4.57419
$499$	22.6102	1.01217	0.506086	−	0.862483i	$-0.331092\pi$
$499$	22.6102	1.01217	0.506086	+	0.862483i	$0.331092\pi$
$500$	−5.07733	−0.227065
$501$	0.956877	0.0427501
$502$	−49.4443	−2.20681
$503$	−39.6552	−1.76814	−0.884068	−	0.467358i	$-0.845206\pi$
$503$	−39.6552	−1.76814	−0.884068	+	0.467358i	$0.845206\pi$
$504$	−5.91750	−0.263586
$505$	8.00084	0.356033
$506$	−11.1965	−0.497746
$507$	−35.6146	−1.58170
$508$	11.8372	0.525192
$509$	21.5410	0.954787	0.477394	−	0.878690i	$-0.341582\pi$
$509$	21.5410	0.954787	0.477394	+	0.878690i	$0.341582\pi$
$510$	7.08038	0.313524
$511$	0.107344	0.00474861
$512$	−21.1750	−0.935813
$513$	2.00887	0.0886940
$514$	−16.3269	−0.720147
$515$	−9.18645	−0.404803
$516$	78.5922	3.45983
$517$	−3.55141	−0.156191
$518$	1.42505	0.0626129
$519$	23.6404	1.03770
$520$	10.3057	0.451934
$521$	19.1361	0.838367	0.419183	−	0.907902i	$-0.362316\pi$
$521$	19.1361	0.838367	0.419183	+	0.907902i	$0.362316\pi$
$522$	−142.850	−6.25237
$523$	1.49191	0.0652368	0.0326184	−	0.999468i	$-0.489615\pi$
$523$	1.49191	0.0652368	0.0326184	+	0.999468i	$0.489615\pi$
$524$	63.8910	2.79109
$525$	−0.334900	−0.0146162
$526$	−0.136767	−0.00596334
$527$	3.72016	0.162053
$528$	36.2674	1.57834
$529$	−5.28684	−0.229862
$530$	29.0879	1.26350
$531$	51.1958	2.22171
$532$	−0.0939912	−0.00407503
$533$	0.584182	0.0253037
$534$	70.2048	3.03806
$535$	−8.64811	−0.373890
$536$	10.6948	0.461944
$537$	4.50123	0.194242
$538$	34.6916	1.49566
$539$	−6.98848	−0.301015
$540$	−59.1442	−2.54516
$541$	30.6970	1.31977	0.659883	−	0.751369i	$-0.270606\pi$
$541$	30.6970	1.31977	0.659883	+	0.751369i	$0.270606\pi$
$542$	36.7592	1.57894
$543$	−14.1804	−0.608541
$544$	−12.4137	−0.532231
$545$	6.00963	0.257424
$546$	1.12155	0.0479979
$547$	40.7763	1.74347	0.871733	−	0.489981i	$-0.162996\pi$
$547$	40.7763	1.74347	0.871733	+	0.489981i	$0.162996\pi$
$548$	95.4661	4.07811
$549$	−8.17717	−0.348993
$550$	2.66032	0.113437
$551$	−1.37521	−0.0585857
$552$	−107.497	−4.57536
$553$	−1.51919	−0.0646027
$554$	55.7132	2.36703
$555$	15.5688	0.660861
$556$	12.3563	0.524022
$557$	35.9547	1.52345	0.761725	−	0.647901i	$-0.224353\pi$
$557$	35.9547	1.52345	0.761725	+	0.647901i	$0.224353\pi$
$558$	−78.1207	−3.30711
$559$	−6.24560	−0.264161
$560$	1.24783	0.0527303
$561$	−2.66147	−0.112367
$562$	−79.4678	−3.35215
$563$	−27.9614	−1.17843	−0.589217	−	0.807975i	$-0.700564\pi$
$563$	−27.9614	−1.17843	−0.589217	+	0.807975i	$0.700564\pi$
$564$	−56.2568	−2.36884
$565$	2.91850	0.122782
$566$	43.9238	1.84626
$567$	−1.73269	−0.0727663
$568$	55.3421	2.32210
$569$	13.5633	0.568602	0.284301	−	0.958735i	$-0.408238\pi$
$569$	13.5633	0.568602	0.284301	+	0.958735i	$0.408238\pi$
$570$	−1.43136	−0.0599531
$571$	−28.5961	−1.19671	−0.598355	−	0.801231i	$-0.704179\pi$
$571$	−28.5961	−1.19671	−0.598355	+	0.801231i	$0.704179\pi$
$572$	−6.39151	−0.267243
$573$	58.3556	2.43784
$574$	0.132523	0.00553140
$575$	−4.20870	−0.175515
$576$	104.125	4.33855
$577$	−33.1935	−1.38186	−0.690932	−	0.722920i	$-0.742799\pi$
$577$	−33.1935	−1.38186	−0.690932	+	0.722920i	$0.742799\pi$
$578$	−43.2895	−1.80061
$579$	19.7697	0.821600
$580$	40.4881	1.68118
$581$	−1.32018	−0.0547701
$582$	60.1394	2.49286
$583$	−10.9340	−0.452838
$584$	−8.18668	−0.338767
$585$	8.47660	0.350464
$586$	−2.01738	−0.0833374
$587$	3.74745	0.154674	0.0773369	−	0.997005i	$-0.475358\pi$
$587$	3.74745	0.154674	0.0773369	+	0.997005i	$0.475358\pi$
$588$	−110.702	−4.56528
$589$	−0.752062	−0.0309882
$590$	−20.2263	−0.832702
$591$	−78.3287	−3.22201
$592$	−58.0089	−2.38415
$593$	−40.7095	−1.67174	−0.835870	−	0.548927i	$-0.815036\pi$
$593$	−40.7095	−1.67174	−0.835870	+	0.548927i	$0.815036\pi$
$594$	30.9893	1.27151
$595$	−0.0915714	−0.00375406
$596$	73.2145	2.99898
$597$	62.6107	2.56249
$598$	14.0946	0.576369
$599$	5.73039	0.234137	0.117069	−	0.993124i	$-0.462650\pi$
$599$	5.73039	0.234137	0.117069	+	0.993124i	$0.462650\pi$
$600$	25.5415	1.04273
$601$	40.0091	1.63200	0.816002	−	0.578049i	$-0.196186\pi$
$601$	40.0091	1.63200	0.816002	+	0.578049i	$0.196186\pi$
$602$	−1.41683	−0.0577456
$603$	8.79665	0.358227
$604$	−65.2684	−2.65573
$605$	−1.00000	−0.0406558
$606$	−66.4063	−2.69757
$607$	7.57581	0.307493	0.153746	−	0.988110i	$-0.450866\pi$
$607$	7.57581	0.307493	0.153746	+	0.988110i	$0.450866\pi$
$608$	2.50953	0.101775
$609$	2.67059	0.108218
$610$	3.23061	0.130803
$611$	4.47064	0.180863
$612$	−29.1656	−1.17895
$613$	23.7061	0.957479	0.478740	−	0.877957i	$-0.341094\pi$
$613$	23.7061	0.957479	0.478740	+	0.877957i	$0.341094\pi$
$614$	0.989911	0.0399496
$615$	1.44783	0.0583823
$616$	−0.878789	−0.0354074
$617$	29.4663	1.18627	0.593134	−	0.805104i	$-0.297890\pi$
$617$	29.4663	1.18627	0.593134	+	0.805104i	$0.297890\pi$
$618$	76.2467	3.06709
$619$	−9.87223	−0.396799	−0.198399	−	0.980121i	$-0.563574\pi$
$619$	−9.87223	−0.396799	−0.198399	+	0.980121i	$0.563574\pi$
$620$	22.1418	0.889237
$621$	−49.0259	−1.96734
$622$	−3.65042	−0.146369
$623$	−0.907967	−0.0363769
$624$	−45.6546	−1.82765
$625$	1.00000	0.0400000
$626$	14.0972	0.563439
$627$	0.538040	0.0214872
$628$	−103.270	−4.12091
$629$	4.25697	0.169736
$630$	1.92293	0.0766116
$631$	28.6302	1.13975	0.569874	−	0.821732i	$-0.306992\pi$
$631$	28.6302	1.13975	0.569874	+	0.821732i	$0.306992\pi$
$632$	115.863	4.60878
$633$	−11.0562	−0.439443
$634$	30.8584	1.22554
$635$	−2.33139	−0.0925184
$636$	−173.201	−6.86788
$637$	8.79734	0.348563
$638$	−21.2142	−0.839878
$639$	45.5199	1.80074
$640$	−12.0339	−0.475681
$641$	−15.7173	−0.620794	−0.310397	−	0.950607i	$-0.600462\pi$
$641$	−15.7173	−0.620794	−0.310397	+	0.950607i	$0.600462\pi$
$642$	71.7786	2.83287
$643$	38.3998	1.51434	0.757171	−	0.653217i	$-0.226581\pi$
$643$	38.3998	1.51434	0.757171	+	0.653217i	$0.226581\pi$
$644$	2.29382	0.0903893
$645$	−15.4791	−0.609487
$646$	−0.391375	−0.0153984
$647$	41.3362	1.62509	0.812546	−	0.582896i	$-0.198081\pi$
$647$	41.3362	1.62509	0.812546	+	0.582896i	$0.198081\pi$
$648$	132.146	5.19117
$649$	7.60293	0.298441
$650$	−3.34891	−0.131355
$651$	1.46047	0.0572405
$652$	−82.4943	−3.23073
$653$	−43.8995	−1.71792	−0.858960	−	0.512042i	$-0.828889\pi$
$653$	−43.8995	−1.71792	−0.858960	+	0.512042i	$0.828889\pi$
$654$	−49.8794	−1.95044
$655$	−12.5836	−0.491681
$656$	−5.39458	−0.210623
$657$	−6.73369	−0.262706
$658$	1.01417	0.0395366
$659$	−33.2659	−1.29586	−0.647928	−	0.761701i	$-0.724364\pi$
$659$	−33.2659	−1.29586	−0.647928	+	0.761701i	$0.724364\pi$
$660$	−15.8407	−0.616598
$661$	22.0576	0.857939	0.428970	−	0.903319i	$-0.358877\pi$
$661$	22.0576	0.857939	0.428970	+	0.903319i	$0.358877\pi$
$662$	−88.0030	−3.42033
$663$	3.35035	0.130117
$664$	100.685	3.90732
$665$	0.0185119	0.000717863 0
$666$	−89.3934	−3.46392
$667$	33.5614	1.29950
$668$	1.55723	0.0602510
$669$	−8.31627	−0.321526
$670$	−3.47535	−0.134265
$671$	−1.21437	−0.0468801
$672$	−4.87340	−0.187996
$673$	−30.6205	−1.18033	−0.590167	−	0.807281i	$-0.700938\pi$
$673$	−30.6205	−1.18033	−0.590167	+	0.807281i	$0.700938\pi$
$674$	46.9401	1.80807
$675$	11.6487	0.448359
$676$	−57.9594	−2.22921
$677$	22.4901	0.864363	0.432181	−	0.901787i	$-0.357744\pi$
$677$	22.4901	0.864363	0.432181	+	0.901787i	$0.357744\pi$
$678$	−24.2233	−0.930291
$679$	−0.777790	−0.0298489
$680$	6.98379	0.267816
$681$	−4.75155	−0.182080
$682$	−11.6015	−0.444243
$683$	−22.6394	−0.866272	−0.433136	−	0.901329i	$-0.642593\pi$
$683$	−22.6394	−0.866272	−0.433136	+	0.901329i	$0.642593\pi$
$684$	5.89608	0.225442
$685$	−18.8024	−0.718404
$686$	3.99468	0.152518
$687$	−39.0654	−1.49044
$688$	57.6744	2.19882
$689$	13.7640	0.524368
$690$	34.9318	1.32983
$691$	43.9154	1.67062	0.835311	−	0.549778i	$-0.185288\pi$
$691$	43.9154	1.67062	0.835311	+	0.549778i	$0.185288\pi$
$692$	38.4725	1.46251
$693$	−0.722820	−0.0274577
$694$	61.5647	2.33696
$695$	−2.43362	−0.0923123
$696$	−203.676	−7.72030
$697$	0.395879	0.0149950
$698$	−44.5369	−1.68575
$699$	44.7542	1.69276
$700$	−0.545019	−0.0205998
$701$	30.4599	1.15046	0.575228	−	0.817993i	$-0.304913\pi$
$701$	30.4599	1.15046	0.575228	+	0.817993i	$0.304913\pi$
$702$	−39.0104	−1.47235
$703$	−0.860583	−0.0324575
$704$	15.4633	0.582796
$705$	11.0800	0.417297
$706$	−69.5278	−2.61671
$707$	0.858840	0.0323000
$708$	120.436	4.52624
$709$	−22.5666	−0.847506	−0.423753	−	0.905778i	$-0.639288\pi$
$709$	−22.5666	−0.847506	−0.423753	+	0.905778i	$0.639288\pi$
$710$	−17.9838	−0.674922
$711$	95.2993	3.57400
$712$	69.2471	2.59514
$713$	18.3538	0.687356
$714$	0.760034	0.0284436
$715$	1.25883	0.0470777
$716$	7.32533	0.273760
$717$	−39.9189	−1.49080
$718$	33.9535	1.26713
$719$	24.7785	0.924083	0.462042	−	0.886858i	$-0.347117\pi$
$719$	24.7785	0.924083	0.462042	+	0.886858i	$0.347117\pi$
$720$	−78.2764	−2.91719
$721$	−0.986108	−0.0367246
$722$	−50.4670	−1.87819
$723$	−72.5458	−2.69801
$724$	−23.0774	−0.857663
$725$	−7.97429	−0.296158
$726$	8.29991	0.308039
$727$	9.93941	0.368632	0.184316	−	0.982867i	$-0.440993\pi$
$727$	9.93941	0.368632	0.184316	+	0.982867i	$0.440993\pi$
$728$	1.10625	0.0410004
$729$	−0.335702	−0.0124334
$730$	2.66032	0.0984630
$731$	−4.23242	−0.156542
$732$	−19.2364	−0.710997
$733$	0.894025	0.0330216	0.0165108	−	0.999864i	$-0.494744\pi$
$733$	0.894025	0.0330216	0.0165108	+	0.999864i	$0.494744\pi$
$734$	20.8948	0.771240
$735$	21.8033	0.804225
$736$	−61.2442	−2.25749
$737$	1.30636	0.0481205
$738$	−8.31319	−0.306013
$739$	4.45704	0.163955	0.0819775	−	0.996634i	$-0.473876\pi$
$739$	4.45704	0.163955	0.0819775	+	0.996634i	$0.473876\pi$
$740$	25.3368	0.931401
$741$	−0.677303	−0.0248813
$742$	3.12240	0.114627
$743$	15.2394	0.559079	0.279539	−	0.960134i	$-0.409818\pi$
$743$	15.2394	0.559079	0.279539	+	0.960134i	$0.409818\pi$
$744$	−111.385	−4.08356
$745$	−14.4199	−0.528304
$746$	20.4434	0.748486
$747$	82.8149	3.03004
$748$	−4.33130	−0.158368
$749$	−0.928321	−0.0339201
$750$	−8.29991	−0.303070
$751$	−35.4319	−1.29293	−0.646465	−	0.762944i	$-0.723753\pi$
$751$	−35.4319	−1.29293	−0.646465	+	0.762944i	$0.723753\pi$
$752$	−41.2837	−1.50546
$753$	−57.9856	−2.11311
$754$	26.7052	0.972545
$755$	12.8549	0.467837
$756$	−6.34876	−0.230902
$757$	−40.5694	−1.47452	−0.737260	−	0.675609i	$-0.763881\pi$
$757$	−40.5694	−1.47452	−0.737260	+	0.675609i	$0.763881\pi$
$758$	−38.7609	−1.40786
$759$	−13.1307	−0.476613
$760$	−1.41183	−0.0512126
$761$	26.3882	0.956572	0.478286	−	0.878204i	$-0.341258\pi$
$761$	26.3882	0.956572	0.478286	+	0.878204i	$0.341258\pi$
$762$	19.3503	0.700988
$763$	0.645096	0.0233540
$764$	94.9682	3.43583
$765$	5.74429	0.207685
$766$	67.7987	2.44967
$767$	−9.57082	−0.345582
$768$	3.39244	0.122414
$769$	3.41749	0.123238	0.0616190	−	0.998100i	$-0.480374\pi$
$769$	3.41749	0.123238	0.0616190	+	0.998100i	$0.480374\pi$
$770$	0.285569	0.0102912
$771$	−19.1473	−0.689572
$772$	32.1733	1.15794
$773$	25.3953	0.913407	0.456704	−	0.889619i	$-0.349030\pi$
$773$	25.3953	0.913407	0.456704	+	0.889619i	$0.349030\pi$
$774$	88.8779	3.19465
$775$	−4.36092	−0.156649
$776$	59.3190	2.12943
$777$	1.67122	0.0599546
$778$	−42.2203	−1.51367
$779$	−0.0800305	−0.00286739
$780$	19.9408	0.713995
$781$	6.76002	0.241893
$782$	9.55137	0.341556
$783$	−92.8901	−3.31962
$784$	−81.2381	−2.90136
$785$	20.3394	0.725944
$786$	104.443	3.72535
$787$	48.7506	1.73777	0.868886	−	0.495013i	$-0.164837\pi$
$787$	48.7506	1.73777	0.868886	+	0.495013i	$0.164837\pi$
$788$	−127.473	−4.54103
$789$	−0.160393	−0.00571016
$790$	−37.6505	−1.33955
$791$	0.313283	0.0111391
$792$	55.1266	1.95884
$793$	1.52869	0.0542852
$794$	77.8155	2.76157
$795$	34.1127	1.20985
$796$	101.893	3.61151
$797$	44.2033	1.56576	0.782880	−	0.622173i	$-0.213750\pi$
$797$	44.2033	1.56576	0.782880	+	0.622173i	$0.213750\pi$
$798$	−0.153648	−0.00543906
$799$	3.02959	0.107179
$800$	14.5518	0.514484
$801$	56.9570	2.01248
$802$	13.8260	0.488214
$803$	−1.00000	−0.0352892
$804$	20.6937	0.729810
$805$	−0.451778	−0.0159231
$806$	14.6043	0.514415
$807$	40.6845	1.43216
$808$	−65.5004	−2.30429
$809$	−21.6595	−0.761507	−0.380753	−	0.924677i	$-0.624335\pi$
$809$	−21.6595	−0.761507	−0.380753	+	0.924677i	$0.624335\pi$
$810$	−42.9417	−1.50882
$811$	−4.82924	−0.169578	−0.0847888	−	0.996399i	$-0.527022\pi$
$811$	−4.82924	−0.169578	−0.0847888	+	0.996399i	$0.527022\pi$
$812$	4.34614	0.152520
$813$	43.1092	1.51191
$814$	−13.2755	−0.465307
$815$	16.2476	0.569128
$816$	−30.9385	−1.08306
$817$	0.855621	0.0299344
$818$	42.6359	1.49073
$819$	0.909910	0.0317948
$820$	2.35622	0.0822826
$821$	−3.12948	−0.109220	−0.0546098	−	0.998508i	$-0.517391\pi$
$821$	−3.12948	−0.109220	−0.0546098	+	0.998508i	$0.517391\pi$
$822$	156.059	5.44317
$823$	−50.7794	−1.77006	−0.885030	−	0.465535i	$-0.845862\pi$
$823$	−50.7794	−1.77006	−0.885030	+	0.465535i	$0.845862\pi$
$824$	75.2066	2.61995
$825$	3.11989	0.108621
$826$	−2.17116	−0.0755444
$827$	14.8903	0.517787	0.258894	−	0.965906i	$-0.416642\pi$
$827$	14.8903	0.517787	0.258894	+	0.965906i	$0.416642\pi$
$828$	−143.892	−5.00059
$829$	18.3933	0.638827	0.319414	−	0.947615i	$-0.396514\pi$
$829$	18.3933	0.638827	0.319414	+	0.947615i	$0.396514\pi$
$830$	−32.7182	−1.13567
$831$	65.3375	2.26653
$832$	−19.4658	−0.674854
$833$	5.96164	0.206558
$834$	20.1988	0.699427
$835$	−0.306703	−0.0106139
$836$	0.875609	0.0302836
$837$	−50.7991	−1.75587
$838$	−62.9923	−2.17603
$839$	47.7511	1.64855	0.824275	−	0.566190i	$-0.191583\pi$
$839$	47.7511	1.64855	0.824275	+	0.566190i	$0.191583\pi$
$840$	2.74172	0.0945984
$841$	34.5893	1.19273
$842$	−47.0773	−1.62239
$843$	−93.1956	−3.20983
$844$	−17.9929	−0.619340
$845$	11.4153	0.392700
$846$	−63.6193	−2.18728
$847$	−0.107344	−0.00368837
$848$	−127.103	−4.36473
$849$	51.5115	1.76787
$850$	−2.26943	−0.0778409
$851$	21.0022	0.719948
$852$	107.083	3.66861
$853$	−44.6142	−1.52756	−0.763781	−	0.645475i	$-0.776659\pi$
$853$	−44.6142	−1.52756	−0.763781	+	0.645475i	$0.776659\pi$
$854$	0.346786	0.0118668
$855$	−1.16126	−0.0397142
$856$	70.7994	2.41987
$857$	41.8494	1.42955	0.714774	−	0.699356i	$-0.246530\pi$
$857$	41.8494	1.42955	0.714774	+	0.699356i	$0.246530\pi$
$858$	−10.4482	−0.356696
$859$	44.1078	1.50494	0.752470	−	0.658627i	$-0.228862\pi$
$859$	44.1078	1.50494	0.752470	+	0.658627i	$0.228862\pi$
$860$	−25.1907	−0.858997
$861$	0.155416	0.00529656
$862$	51.4519	1.75246
$863$	9.82192	0.334342	0.167171	−	0.985928i	$-0.446537\pi$
$863$	9.82192	0.334342	0.167171	+	0.985928i	$0.446537\pi$
$864$	169.509	5.76683
$865$	−7.57732	−0.257637
$866$	17.4742	0.593797
$867$	−50.7677	−1.72416
$868$	2.37679	0.0806734
$869$	14.1526	0.480094
$870$	66.1859	2.24391
$871$	−1.64450	−0.0557216
$872$	−49.1989	−1.66609
$873$	48.7909	1.65132
$874$	−1.93089	−0.0653134
$875$	0.107344	0.00362888
$876$	−15.8407	−0.535207
$877$	18.5493	0.626367	0.313183	−	0.949693i	$-0.398605\pi$
$877$	18.5493	0.626367	0.313183	+	0.949693i	$0.398605\pi$
$878$	48.5270	1.63771
$879$	−2.36588	−0.0797992
$880$	−11.6246	−0.391865
$881$	27.5988	0.929827	0.464914	−	0.885356i	$-0.346085\pi$
$881$	27.5988	0.929827	0.464914	+	0.885356i	$0.346085\pi$
$882$	−125.190	−4.21537
$883$	24.4287	0.822093	0.411046	−	0.911614i	$-0.365163\pi$
$883$	24.4287	0.822093	0.411046	+	0.911614i	$0.365163\pi$
$884$	5.45238	0.183384
$885$	−23.7203	−0.797348
$886$	−89.0231	−2.99079
$887$	18.4668	0.620053	0.310026	−	0.950728i	$-0.399662\pi$
$887$	18.4668	0.620053	0.310026	+	0.950728i	$0.399662\pi$
$888$	−127.457	−4.27718
$889$	−0.250260	−0.00839346
$890$	−22.5024	−0.754281
$891$	16.1415	0.540762
$892$	−13.5340	−0.453150
$893$	−0.612458	−0.0204951
$894$	119.684	4.00283
$895$	−1.44275	−0.0482259
$896$	−1.29176	−0.0431547
$897$	16.5293	0.551899
$898$	−94.7668	−3.16241
$899$	34.7753	1.15982
$900$	34.1891	1.13964
$901$	9.32740	0.310741
$902$	−1.23457	−0.0411066
$903$	−1.66158	−0.0552939
$904$	−23.8929	−0.794665
$905$	4.54518	0.151087
$906$	−106.694	−3.54468
$907$	34.1384	1.13355	0.566774	−	0.823873i	$-0.308191\pi$
$907$	34.1384	1.13355	0.566774	+	0.823873i	$0.308191\pi$
$908$	−7.73271	−0.256619
$909$	−53.8752	−1.78693
$910$	−0.359484	−0.0119168
$911$	33.4125	1.10701	0.553503	−	0.832847i	$-0.313291\pi$
$911$	33.4125	1.10701	0.553503	+	0.832847i	$0.313291\pi$
$912$	6.25449	0.207107
$913$	12.2986	0.407024
$914$	−17.8858	−0.591609
$915$	3.78868	0.125250
$916$	−63.5753	−2.10059
$917$	−1.35077	−0.0446063
$918$	−26.4359	−0.872516
$919$	−9.31586	−0.307302	−0.153651	−	0.988125i	$-0.549103\pi$
$919$	−9.31586	−0.307302	−0.153651	+	0.988125i	$0.549103\pi$
$920$	34.4553	1.13596
$921$	1.16092	0.0382535
$922$	9.62029	0.316827
$923$	−8.50974	−0.280102
$924$	−1.70040	−0.0559390
$925$	−4.99020	−0.164077
$926$	−29.4121	−0.966541
$927$	61.8587	2.03171
$928$	−116.040	−3.80921
$929$	46.7213	1.53287	0.766437	−	0.642319i	$-0.222028\pi$
$929$	46.7213	1.53287	0.766437	+	0.642319i	$0.222028\pi$
$930$	36.1953	1.18689
$931$	−1.20520	−0.0394987
$932$	72.8334	2.38574
$933$	−4.28102	−0.140154
$934$	−53.2412	−1.74211
$935$	0.853067	0.0278983
$936$	−69.3953	−2.26826
$937$	−29.8313	−0.974546	−0.487273	−	0.873250i	$-0.662008\pi$
$937$	−29.8313	−0.974546	−0.487273	+	0.873250i	$0.662008\pi$
$938$	−0.373057	−0.0121807
$939$	16.5325	0.539517
$940$	18.0317	0.588128
$941$	1.78184	0.0580863	0.0290432	−	0.999578i	$-0.490754\pi$
$941$	1.78184	0.0580863	0.0290432	+	0.999578i	$0.490754\pi$
$942$	−168.815	−5.50029
$943$	1.95312	0.0636022
$944$	88.3809	2.87655
$945$	1.25041	0.0406760
$946$	13.1990	0.429136
$947$	−1.51038	−0.0490807	−0.0245404	−	0.999699i	$-0.507812\pi$
$947$	−1.51038	−0.0490807	−0.0245404	+	0.999699i	$0.507812\pi$
$948$	224.187	7.28125
$949$	1.25883	0.0408635
$950$	0.458786	0.0148850
$951$	36.1890	1.17351
$952$	0.749666	0.0242968
$953$	−41.6352	−1.34870	−0.674349	−	0.738413i	$-0.735576\pi$
$953$	−41.6352	−1.34870	−0.674349	+	0.738413i	$0.735576\pi$
$954$	−195.869	−6.34149
$955$	−18.7044	−0.605260
$956$	−64.9644	−2.10110
$957$	−24.8789	−0.804220
$958$	58.6316	1.89430
$959$	−2.01832	−0.0651751
$960$	−48.2438	−1.55706
$961$	−11.9824	−0.386528
$962$	16.7117	0.538807
$963$	58.2337	1.87656
$964$	−118.062	−3.80251
$965$	−6.33667	−0.203985
$966$	3.74971	0.120645
$967$	−17.2758	−0.555552	−0.277776	−	0.960646i	$-0.589597\pi$
$967$	−17.2758	−0.555552	−0.277776	+	0.960646i	$0.589597\pi$
$968$	8.18668	0.263130
$969$	−0.458984	−0.0147447
$970$	−19.2762	−0.618920
$971$	−0.609864	−0.0195715	−0.00978574	−	0.999952i	$-0.503115\pi$
$971$	−0.609864	−0.0195715	−0.00978574	+	0.999952i	$0.503115\pi$
$972$	78.2602	2.51020
$973$	−0.261233	−0.00837476
$974$	−28.5752	−0.915608
$975$	−3.92742	−0.125778
$976$	−14.1165	−0.451858
$977$	31.7991	1.01734	0.508671	−	0.860961i	$-0.330137\pi$
$977$	31.7991	1.01734	0.508671	+	0.860961i	$0.330137\pi$
$978$	−134.854	−4.31214
$979$	8.45850	0.270335
$980$	35.4828	1.13346
$981$	−40.4670	−1.29201
$982$	47.6764	1.52141
$983$	−4.45863	−0.142208	−0.0711041	−	0.997469i	$-0.522652\pi$
$983$	−4.45863	−0.142208	−0.0711041	+	0.997469i	$0.522652\pi$
$984$	−11.8530	−0.377859
$985$	25.1063	0.799952
$986$	18.0971	0.576330
$987$	1.18937	0.0378580
$988$	−1.10225	−0.0350672
$989$	−20.8811	−0.663981
$990$	−17.9138	−0.569338
$991$	−39.3198	−1.24903	−0.624517	−	0.781011i	$-0.714704\pi$
$991$	−39.3198	−1.24903	−0.624517	+	0.781011i	$0.714704\pi$
$992$	−63.4592	−2.01483
$993$	−103.205	−3.27512
$994$	−1.93045	−0.0612303
$995$	−20.0683	−0.636207
$996$	194.818	6.17304
$997$	18.4506	0.584336	0.292168	−	0.956367i	$-0.405623\pi$
$997$	18.4506	0.584336	0.292168	+	0.956367i	$0.405623\pi$
$998$	60.1505	1.90403
$999$	−58.1293	−1.83913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.e.1.26	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.e.1.26	✓	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66032 q^{2} +3.11989 q^{3} +5.07733 q^{4} -1.00000 q^{5} +8.29991 q^{6} -0.107344 q^{7} +8.18668 q^{8} +6.73369 q^{9} +O(q^{10})\)	\(q+2.66032 q^{2} +3.11989 q^{3} +5.07733 q^{4} -1.00000 q^{5} +8.29991 q^{6} -0.107344 q^{7} +8.18668 q^{8} +6.73369 q^{9} -2.66032 q^{10} +1.00000 q^{11} +15.8407 q^{12} -1.25883 q^{13} -0.285569 q^{14} -3.11989 q^{15} +11.6246 q^{16} -0.853067 q^{17} +17.9138 q^{18} +0.172455 q^{19} -5.07733 q^{20} -0.334900 q^{21} +2.66032 q^{22} -4.20870 q^{23} +25.5415 q^{24} +1.00000 q^{25} -3.34891 q^{26} +11.6487 q^{27} -0.545019 q^{28} -7.97429 q^{29} -8.29991 q^{30} -4.36092 q^{31} +14.5518 q^{32} +3.11989 q^{33} -2.26943 q^{34} +0.107344 q^{35} +34.1891 q^{36} -4.99020 q^{37} +0.458786 q^{38} -3.92742 q^{39} -8.18668 q^{40} -0.464066 q^{41} -0.890943 q^{42} +4.96142 q^{43} +5.07733 q^{44} -6.73369 q^{45} -11.1965 q^{46} -3.55141 q^{47} +36.2674 q^{48} -6.98848 q^{49} +2.66032 q^{50} -2.66147 q^{51} -6.39151 q^{52} -10.9340 q^{53} +30.9893 q^{54} -1.00000 q^{55} -0.878789 q^{56} +0.538040 q^{57} -21.2142 q^{58} +7.60293 q^{59} -15.8407 q^{60} -1.21437 q^{61} -11.6015 q^{62} -0.722820 q^{63} +15.4633 q^{64} +1.25883 q^{65} +8.29991 q^{66} +1.30636 q^{67} -4.33130 q^{68} -13.1307 q^{69} +0.285569 q^{70} +6.76002 q^{71} +55.1266 q^{72} -1.00000 q^{73} -13.2755 q^{74} +3.11989 q^{75} +0.875609 q^{76} -0.107344 q^{77} -10.4482 q^{78} +14.1526 q^{79} -11.6246 q^{80} +16.1415 q^{81} -1.23457 q^{82} +12.2986 q^{83} -1.70040 q^{84} +0.853067 q^{85} +13.1990 q^{86} -24.8789 q^{87} +8.18668 q^{88} +8.45850 q^{89} -17.9138 q^{90} +0.135128 q^{91} -21.3689 q^{92} -13.6056 q^{93} -9.44790 q^{94} -0.172455 q^{95} +45.4000 q^{96} +7.24579 q^{97} -18.5916 q^{98} +6.73369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more