LMFDB - Embedded newform 4015.2.a.h.1.15

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:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-0.446590 q^{2} +1.08482 q^{3} -1.80056 q^{4} +1.00000 q^{5} -0.484472 q^{6} +4.20380 q^{7} +1.69729 q^{8} -1.82316 q^{9} +O(q^{10})$	\(q-0.446590 q^{2} +1.08482 q^{3} -1.80056 q^{4} +1.00000 q^{5} -0.484472 q^{6} +4.20380 q^{7} +1.69729 q^{8} -1.82316 q^{9} -0.446590 q^{10} +1.00000 q^{11} -1.95329 q^{12} -2.82350 q^{13} -1.87738 q^{14} +1.08482 q^{15} +2.84312 q^{16} +4.54365 q^{17} +0.814205 q^{18} -4.76704 q^{19} -1.80056 q^{20} +4.56039 q^{21} -0.446590 q^{22} +2.76248 q^{23} +1.84126 q^{24} +1.00000 q^{25} +1.26095 q^{26} -5.23228 q^{27} -7.56919 q^{28} -2.06610 q^{29} -0.484472 q^{30} +7.10845 q^{31} -4.66429 q^{32} +1.08482 q^{33} -2.02915 q^{34} +4.20380 q^{35} +3.28270 q^{36} -5.06844 q^{37} +2.12892 q^{38} -3.06299 q^{39} +1.69729 q^{40} +9.63406 q^{41} -2.03662 q^{42} +11.0577 q^{43} -1.80056 q^{44} -1.82316 q^{45} -1.23370 q^{46} -0.673225 q^{47} +3.08428 q^{48} +10.6720 q^{49} -0.446590 q^{50} +4.92906 q^{51} +5.08386 q^{52} +12.9970 q^{53} +2.33668 q^{54} +1.00000 q^{55} +7.13508 q^{56} -5.17140 q^{57} +0.922700 q^{58} -14.0926 q^{59} -1.95329 q^{60} +1.12602 q^{61} -3.17457 q^{62} -7.66420 q^{63} -3.60321 q^{64} -2.82350 q^{65} -0.484472 q^{66} +7.73434 q^{67} -8.18111 q^{68} +2.99680 q^{69} -1.87738 q^{70} -15.4056 q^{71} -3.09443 q^{72} +1.00000 q^{73} +2.26352 q^{74} +1.08482 q^{75} +8.58334 q^{76} +4.20380 q^{77} +1.36790 q^{78} -3.88009 q^{79} +2.84312 q^{80} -0.206625 q^{81} -4.30248 q^{82} +3.76371 q^{83} -8.21123 q^{84} +4.54365 q^{85} -4.93826 q^{86} -2.24135 q^{87} +1.69729 q^{88} +3.77803 q^{89} +0.814205 q^{90} -11.8694 q^{91} -4.97400 q^{92} +7.71142 q^{93} +0.300656 q^{94} -4.76704 q^{95} -5.05994 q^{96} +14.3786 q^{97} -4.76600 q^{98} -1.82316 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9}+O(q^{10}) $ 37 * q + 5 * q^2 + 3 * q^3 + 43 * q^4 + 37 * q^5 + 9 * q^6 + 6 * q^7 + 12 * q^8 + 50 * q^9	\( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9} + 5 q^{10} + 37 q^{11} + 6 q^{12} + 11 q^{13} + 11 q^{14} + 3 q^{15} + 43 q^{16} + 38 q^{17} + 11 q^{18} + 34 q^{19} + 43 q^{20} + 39 q^{21} + 5 q^{22} + 4 q^{23} + 25 q^{24} + 37 q^{25} - 9 q^{26} + 3 q^{27} + 14 q^{28} + 58 q^{29} + 9 q^{30} + 8 q^{31} + 14 q^{32} + 3 q^{33} + 8 q^{34} + 6 q^{35} + 20 q^{36} + 2 q^{37} + 15 q^{38} + 14 q^{39} + 12 q^{40} + 62 q^{41} - 13 q^{42} + 30 q^{43} + 43 q^{44} + 50 q^{45} + 31 q^{46} + 5 q^{47} - 25 q^{48} + 59 q^{49} + 5 q^{50} + 23 q^{51} - q^{52} + 18 q^{53} + 13 q^{54} + 37 q^{55} + 22 q^{56} + 5 q^{57} - 40 q^{58} + 15 q^{59} + 6 q^{60} + 57 q^{61} + 20 q^{62} - 29 q^{63} + 10 q^{64} + 11 q^{65} + 9 q^{66} - 14 q^{67} + 53 q^{68} + 24 q^{69} + 11 q^{70} + 8 q^{71} + 15 q^{72} + 37 q^{73} + 7 q^{74} + 3 q^{75} + 59 q^{76} + 6 q^{77} + q^{78} + 42 q^{79} + 43 q^{80} + 61 q^{81} - 22 q^{82} + 44 q^{83} + 66 q^{84} + 38 q^{85} - q^{86} - 26 q^{87} + 12 q^{88} + 69 q^{89} + 11 q^{90} - 10 q^{91} - 21 q^{92} - 26 q^{93} + 29 q^{94} + 34 q^{95} - 9 q^{96} + 37 q^{97} - 15 q^{98} + 50 q^{99}+O(q^{100}) \) 37 * q + 5 * q^2 + 3 * q^3 + 43 * q^4 + 37 * q^5 + 9 * q^6 + 6 * q^7 + 12 * q^8 + 50 * q^9 + 5 * q^10 + 37 * q^11 + 6 * q^12 + 11 * q^13 + 11 * q^14 + 3 * q^15 + 43 * q^16 + 38 * q^17 + 11 * q^18 + 34 * q^19 + 43 * q^20 + 39 * q^21 + 5 * q^22 + 4 * q^23 + 25 * q^24 + 37 * q^25 - 9 * q^26 + 3 * q^27 + 14 * q^28 + 58 * q^29 + 9 * q^30 + 8 * q^31 + 14 * q^32 + 3 * q^33 + 8 * q^34 + 6 * q^35 + 20 * q^36 + 2 * q^37 + 15 * q^38 + 14 * q^39 + 12 * q^40 + 62 * q^41 - 13 * q^42 + 30 * q^43 + 43 * q^44 + 50 * q^45 + 31 * q^46 + 5 * q^47 - 25 * q^48 + 59 * q^49 + 5 * q^50 + 23 * q^51 - q^52 + 18 * q^53 + 13 * q^54 + 37 * q^55 + 22 * q^56 + 5 * q^57 - 40 * q^58 + 15 * q^59 + 6 * q^60 + 57 * q^61 + 20 * q^62 - 29 * q^63 + 10 * q^64 + 11 * q^65 + 9 * q^66 - 14 * q^67 + 53 * q^68 + 24 * q^69 + 11 * q^70 + 8 * q^71 + 15 * q^72 + 37 * q^73 + 7 * q^74 + 3 * q^75 + 59 * q^76 + 6 * q^77 + q^78 + 42 * q^79 + 43 * q^80 + 61 * q^81 - 22 * q^82 + 44 * q^83 + 66 * q^84 + 38 * q^85 - q^86 - 26 * q^87 + 12 * q^88 + 69 * q^89 + 11 * q^90 - 10 * q^91 - 21 * q^92 - 26 * q^93 + 29 * q^94 + 34 * q^95 - 9 * q^96 + 37 * q^97 - 15 * q^98 + 50 * 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 $

$2$	−0.446590	−0.315787	−0.157894	−	0.987456i	$-0.550470\pi$
$2$	−0.446590	−0.315787	−0.157894	+	0.987456i	$0.550470\pi$
$3$	1.08482	0.626323	0.313162	−	0.949700i	$-0.398612\pi$
$3$	1.08482	0.626323	0.313162	+	0.949700i	$0.398612\pi$
$4$	−1.80056	−0.900279
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.484472	−0.197785
$7$	4.20380	1.58889	0.794444	−	0.607337i	$-0.207762\pi$
$7$	4.20380	1.58889	0.794444	+	0.607337i	$0.207762\pi$
$8$	1.69729	0.600083
$9$	−1.82316	−0.607719
$10$	−0.446590	−0.141224
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.95329	−0.563865
$13$	−2.82350	−0.783097	−0.391548	−	0.920158i	$-0.628060\pi$
$13$	−2.82350	−0.783097	−0.391548	+	0.920158i	$0.628060\pi$
$14$	−1.87738	−0.501750
$15$	1.08482	0.280100
$16$	2.84312	0.710780
$17$	4.54365	1.10200	0.550999	−	0.834506i	$-0.314247\pi$
$17$	4.54365	1.10200	0.550999	+	0.834506i	$0.314247\pi$
$18$	0.814205	0.191910
$19$	−4.76704	−1.09364	−0.546818	−	0.837252i	$-0.684161\pi$
$19$	−4.76704	−1.09364	−0.546818	+	0.837252i	$0.684161\pi$
$20$	−1.80056	−0.402617
$21$	4.56039	0.995158
$22$	−0.446590	−0.0952134
$23$	2.76248	0.576017	0.288008	−	0.957628i	$-0.407007\pi$
$23$	2.76248	0.576017	0.288008	+	0.957628i	$0.407007\pi$
$24$	1.84126	0.375846
$25$	1.00000	0.200000
$26$	1.26095	0.247292
$27$	−5.23228	−1.00695
$28$	−7.56919	−1.43044
$29$	−2.06610	−0.383665	−0.191832	−	0.981428i	$-0.561443\pi$
$29$	−2.06610	−0.383665	−0.191832	+	0.981428i	$0.561443\pi$
$30$	−0.484472	−0.0884521
$31$	7.10845	1.27672	0.638358	−	0.769740i	$-0.279614\pi$
$31$	7.10845	1.27672	0.638358	+	0.769740i	$0.279614\pi$
$32$	−4.66429	−0.824539
$33$	1.08482	0.188844
$34$	−2.02915	−0.347997
$35$	4.20380	0.710572
$36$	3.28270	0.547116
$37$	−5.06844	−0.833247	−0.416624	−	0.909079i	$-0.636787\pi$
$37$	−5.06844	−0.833247	−0.416624	+	0.909079i	$0.636787\pi$
$38$	2.12892	0.345356
$39$	−3.06299	−0.490472
$40$	1.69729	0.268365
$41$	9.63406	1.50459	0.752294	−	0.658828i	$-0.228947\pi$
$41$	9.63406	1.50459	0.752294	+	0.658828i	$0.228947\pi$
$42$	−2.03662	−0.314258
$43$	11.0577	1.68628	0.843141	−	0.537693i	$-0.180704\pi$
$43$	11.0577	1.68628	0.843141	+	0.537693i	$0.180704\pi$
$44$	−1.80056	−0.271444
$45$	−1.82316	−0.271780
$46$	−1.23370	−0.181899
$47$	−0.673225	−0.0981999	−0.0491000	−	0.998794i	$-0.515635\pi$
$47$	−0.673225	−0.0981999	−0.0491000	+	0.998794i	$0.515635\pi$
$48$	3.08428	0.445178
$49$	10.6720	1.52457
$50$	−0.446590	−0.0631574
$51$	4.92906	0.690207
$52$	5.08386	0.705005
$53$	12.9970	1.78528	0.892639	−	0.450772i	$-0.148851\pi$
$53$	12.9970	1.78528	0.892639	+	0.450772i	$0.148851\pi$
$54$	2.33668	0.317982
$55$	1.00000	0.134840
$56$	7.13508	0.953466
$57$	−5.17140	−0.684969
$58$	0.922700	0.121156
$59$	−14.0926	−1.83471	−0.917353	−	0.398076i	$-0.869678\pi$
$59$	−14.0926	−1.83471	−0.917353	+	0.398076i	$0.869678\pi$
$60$	−1.95329	−0.252168
$61$	1.12602	0.144172	0.0720858	−	0.997398i	$-0.477034\pi$
$61$	1.12602	0.144172	0.0720858	+	0.997398i	$0.477034\pi$
$62$	−3.17457	−0.403170
$63$	−7.66420	−0.965598
$64$	−3.60321	−0.450401
$65$	−2.82350	−0.350212
$66$	−0.484472	−0.0596344
$67$	7.73434	0.944900	0.472450	−	0.881358i	$-0.343370\pi$
$67$	7.73434	0.944900	0.472450	+	0.881358i	$0.343370\pi$
$68$	−8.18111	−0.992105
$69$	2.99680	0.360773
$70$	−1.87738	−0.224390
$71$	−15.4056	−1.82831	−0.914155	−	0.405364i	$-0.867145\pi$
$71$	−15.4056	−1.82831	−0.914155	+	0.405364i	$0.867145\pi$
$72$	−3.09443	−0.364682
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	2.26352	0.263129
$75$	1.08482	0.125265
$76$	8.58334	0.984576
$77$	4.20380	0.479068
$78$	1.36790	0.154885
$79$	−3.88009	−0.436544	−0.218272	−	0.975888i	$-0.570042\pi$
$79$	−3.88009	−0.436544	−0.218272	+	0.975888i	$0.570042\pi$
$80$	2.84312	0.317870
$81$	−0.206625	−0.0229583
$82$	−4.30248	−0.475129
$83$	3.76371	0.413121	0.206560	−	0.978434i	$-0.433773\pi$
$83$	3.76371	0.413121	0.206560	+	0.978434i	$0.433773\pi$
$84$	−8.21123	−0.895919
$85$	4.54365	0.492829
$86$	−4.93826	−0.532506
$87$	−2.24135	−0.240298
$88$	1.69729	0.180932
$89$	3.77803	0.400470	0.200235	−	0.979748i	$-0.435829\pi$
$89$	3.77803	0.400470	0.200235	+	0.979748i	$0.435829\pi$
$90$	0.814205	0.0858247
$91$	−11.8694	−1.24425
$92$	−4.97400	−0.518576
$93$	7.71142	0.799637
$94$	0.300656	0.0310103
$95$	−4.76704	−0.489088
$96$	−5.05994	−0.516428
$97$	14.3786	1.45992	0.729961	−	0.683489i	$-0.239538\pi$
$97$	14.3786	1.45992	0.729961	+	0.683489i	$0.239538\pi$
$98$	−4.76600	−0.481438
$99$	−1.82316	−0.183234
$100$	−1.80056	−0.180056
$101$	−13.3707	−1.33044	−0.665219	−	0.746649i	$-0.731662\pi$
$101$	−13.3707	−1.33044	−0.665219	+	0.746649i	$0.731662\pi$
$102$	−2.20127	−0.217958
$103$	14.1322	1.39249	0.696244	−	0.717805i	$-0.254853\pi$
$103$	14.1322	1.39249	0.696244	+	0.717805i	$0.254853\pi$
$104$	−4.79230	−0.469923
$105$	4.56039	0.445048
$106$	−5.80434	−0.563768
$107$	6.98899	0.675651	0.337826	−	0.941209i	$-0.390309\pi$
$107$	6.98899	0.675651	0.337826	+	0.941209i	$0.390309\pi$
$108$	9.42101	0.906537
$109$	−6.80765	−0.652054	−0.326027	−	0.945360i	$-0.605710\pi$
$109$	−6.80765	−0.652054	−0.326027	+	0.945360i	$0.605710\pi$
$110$	−0.446590	−0.0425807
$111$	−5.49837	−0.521882
$112$	11.9519	1.12935
$113$	18.0038	1.69366	0.846829	−	0.531865i	$-0.178509\pi$
$113$	18.0038	1.69366	0.846829	+	0.531865i	$0.178509\pi$
$114$	2.30950	0.216304
$115$	2.76248	0.257603
$116$	3.72013	0.345405
$117$	5.14768	0.475903
$118$	6.29364	0.579376
$119$	19.1006	1.75095
$120$	1.84126	0.168084
$121$	1.00000	0.0909091
$122$	−0.502868	−0.0455275
$123$	10.4513	0.942358
$124$	−12.7992	−1.14940
$125$	1.00000	0.0894427
$126$	3.42276	0.304923
$127$	6.65081	0.590164	0.295082	−	0.955472i	$-0.404653\pi$
$127$	6.65081	0.590164	0.295082	+	0.955472i	$0.404653\pi$
$128$	10.9377	0.966769
$129$	11.9956	1.05616
$130$	1.26095	0.110592
$131$	−15.2775	−1.33480	−0.667401	−	0.744698i	$-0.732593\pi$
$131$	−15.2775	−1.33480	−0.667401	+	0.744698i	$0.732593\pi$
$132$	−1.95329	−0.170012
$133$	−20.0397	−1.73766
$134$	−3.45408	−0.298387
$135$	−5.23228	−0.450323
$136$	7.71191	0.661291
$137$	−12.9074	−1.10275	−0.551375	−	0.834257i	$-0.685897\pi$
$137$	−12.9074	−1.10275	−0.551375	+	0.834257i	$0.685897\pi$
$138$	−1.33834	−0.113927
$139$	−1.39769	−0.118551	−0.0592753	−	0.998242i	$-0.518879\pi$
$139$	−1.39769	−0.118551	−0.0592753	+	0.998242i	$0.518879\pi$
$140$	−7.56919	−0.639713
$141$	−0.730330	−0.0615049
$142$	6.88000	0.577357
$143$	−2.82350	−0.236113
$144$	−5.18345	−0.431955
$145$	−2.06610	−0.171580
$146$	−0.446590	−0.0369601
$147$	11.5772	0.954871
$148$	9.12602	0.750155
$149$	−3.27780	−0.268528	−0.134264	−	0.990946i	$-0.542867\pi$
$149$	−3.27780	−0.268528	−0.134264	+	0.990946i	$0.542867\pi$
$150$	−0.484472	−0.0395570
$151$	−6.18263	−0.503135	−0.251568	−	0.967840i	$-0.580946\pi$
$151$	−6.18263	−0.503135	−0.251568	+	0.967840i	$0.580946\pi$
$152$	−8.09107	−0.656272
$153$	−8.28380	−0.669705
$154$	−1.87738	−0.151283
$155$	7.10845	0.570965
$156$	5.51510	0.441561
$157$	−8.67135	−0.692049	−0.346024	−	0.938226i	$-0.612469\pi$
$157$	−8.67135	−0.692049	−0.346024	+	0.938226i	$0.612469\pi$
$158$	1.73281	0.137855
$159$	14.0995	1.11816
$160$	−4.66429	−0.368745
$161$	11.6129	0.915227
$162$	0.0922767	0.00724994
$163$	17.7477	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$163$	17.7477	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$164$	−17.3467	−1.35455
$165$	1.08482	0.0844534
$166$	−1.68084	−0.130458
$167$	20.4600	1.58324	0.791621	−	0.611012i	$-0.209237\pi$
$167$	20.4600	1.58324	0.791621	+	0.611012i	$0.209237\pi$
$168$	7.74031	0.597178
$169$	−5.02787	−0.386759
$170$	−2.02915	−0.155629
$171$	8.69107	0.664623
$172$	−19.9100	−1.51812
$173$	−22.2156	−1.68902	−0.844510	−	0.535540i	$-0.820108\pi$
$173$	−22.2156	−1.68902	−0.844510	+	0.535540i	$0.820108\pi$
$174$	1.00097	0.0758831
$175$	4.20380	0.317778
$176$	2.84312	0.214308
$177$	−15.2880	−1.14912
$178$	−1.68723	−0.126463
$179$	5.94621	0.444440	0.222220	−	0.974997i	$-0.428670\pi$
$179$	5.94621	0.444440	0.222220	+	0.974997i	$0.428670\pi$
$180$	3.28270	0.244678
$181$	4.15122	0.308558	0.154279	−	0.988027i	$-0.450695\pi$
$181$	4.15122	0.308558	0.154279	+	0.988027i	$0.450695\pi$
$182$	5.30077	0.392919
$183$	1.22153	0.0902980
$184$	4.68874	0.345658
$185$	−5.06844	−0.372639
$186$	−3.44385	−0.252515
$187$	4.54365	0.332265
$188$	1.21218	0.0884073
$189$	−21.9955	−1.59993
$190$	2.12892	0.154448
$191$	8.77712	0.635090	0.317545	−	0.948243i	$-0.397142\pi$
$191$	8.77712	0.635090	0.317545	+	0.948243i	$0.397142\pi$
$192$	−3.90885	−0.282097
$193$	−12.5672	−0.904608	−0.452304	−	0.891864i	$-0.649398\pi$
$193$	−12.5672	−0.904608	−0.452304	+	0.891864i	$0.649398\pi$
$194$	−6.42132	−0.461024
$195$	−3.06299	−0.219346
$196$	−19.2155	−1.37253
$197$	−4.61568	−0.328853	−0.164427	−	0.986389i	$-0.552577\pi$
$197$	−4.61568	−0.328853	−0.164427	+	0.986389i	$0.552577\pi$
$198$	0.814205	0.0578630
$199$	9.01902	0.639341	0.319671	−	0.947529i	$-0.396428\pi$
$199$	9.01902	0.639341	0.319671	+	0.947529i	$0.396428\pi$
$200$	1.69729	0.120017
$201$	8.39039	0.591813
$202$	5.97124	0.420135
$203$	−8.68547	−0.609601
$204$	−8.87506	−0.621379
$205$	9.63406	0.672872
$206$	−6.31131	−0.439730
$207$	−5.03644	−0.350057
$208$	−8.02754	−0.556609
$209$	−4.76704	−0.329743
$210$	−2.03662	−0.140540
$211$	14.9813	1.03135	0.515676	−	0.856783i	$-0.327541\pi$
$211$	14.9813	1.03135	0.515676	+	0.856783i	$0.327541\pi$
$212$	−23.4019	−1.60725
$213$	−16.7124	−1.14511
$214$	−3.12122	−0.213362
$215$	11.0577	0.754128
$216$	−8.88070	−0.604255
$217$	29.8825	2.02856
$218$	3.04023	0.205910
$219$	1.08482	0.0733056
$220$	−1.80056	−0.121394
$221$	−12.8290	−0.862971
$222$	2.45552	0.164804
$223$	−11.7986	−0.790094	−0.395047	−	0.918661i	$-0.629272\pi$
$223$	−11.7986	−0.790094	−0.395047	+	0.918661i	$0.629272\pi$
$224$	−19.6078	−1.31010
$225$	−1.82316	−0.121544
$226$	−8.04034	−0.534836
$227$	23.0051	1.52690	0.763451	−	0.645865i	$-0.223503\pi$
$227$	23.0051	1.52690	0.763451	+	0.645865i	$0.223503\pi$
$228$	9.31141	0.616663
$229$	18.0829	1.19495	0.597476	−	0.801887i	$-0.296170\pi$
$229$	18.0829	1.19495	0.597476	+	0.801887i	$0.296170\pi$
$230$	−1.23370	−0.0813476
$231$	4.56039	0.300051
$232$	−3.50677	−0.230231
$233$	20.0859	1.31587	0.657935	−	0.753075i	$-0.271430\pi$
$233$	20.0859	1.31587	0.657935	+	0.753075i	$0.271430\pi$
$234$	−2.29890	−0.150284
$235$	−0.673225	−0.0439163
$236$	25.3746	1.65175
$237$	−4.20921	−0.273418
$238$	−8.53016	−0.552928
$239$	16.4634	1.06493	0.532464	−	0.846453i	$-0.321266\pi$
$239$	16.4634	1.06493	0.532464	+	0.846453i	$0.321266\pi$
$240$	3.08428	0.199090
$241$	11.6755	0.752085	0.376042	−	0.926602i	$-0.377285\pi$
$241$	11.6755	0.752085	0.376042	+	0.926602i	$0.377285\pi$
$242$	−0.446590	−0.0287079
$243$	15.4727	0.992573
$244$	−2.02746	−0.129795
$245$	10.6720	0.681807
$246$	−4.66743	−0.297585
$247$	13.4597	0.856422
$248$	12.0651	0.766136
$249$	4.08296	0.258747
$250$	−0.446590	−0.0282449
$251$	30.0465	1.89652	0.948260	−	0.317496i	$-0.102842\pi$
$251$	30.0465	1.89652	0.948260	+	0.317496i	$0.102842\pi$
$252$	13.7998	0.869307
$253$	2.76248	0.173676
$254$	−2.97019	−0.186366
$255$	4.92906	0.308670
$256$	2.32173	0.145108
$257$	12.6649	0.790013	0.395007	−	0.918678i	$-0.370742\pi$
$257$	12.6649	0.790013	0.395007	+	0.918678i	$0.370742\pi$
$258$	−5.35714	−0.333521
$259$	−21.3067	−1.32394
$260$	5.08386	0.315288
$261$	3.76682	0.233161
$262$	6.82279	0.421513
$263$	23.5086	1.44960	0.724801	−	0.688958i	$-0.241932\pi$
$263$	23.5086	1.44960	0.724801	+	0.688958i	$0.241932\pi$
$264$	1.84126	0.113322
$265$	12.9970	0.798401
$266$	8.94955	0.548732
$267$	4.09850	0.250824
$268$	−13.9261	−0.850673
$269$	−8.05493	−0.491118	−0.245559	−	0.969382i	$-0.578971\pi$
$269$	−8.05493	−0.491118	−0.245559	+	0.969382i	$0.578971\pi$
$270$	2.33668	0.142206
$271$	−3.40895	−0.207079	−0.103539	−	0.994625i	$-0.533017\pi$
$271$	−3.40895	−0.207079	−0.103539	+	0.994625i	$0.533017\pi$
$272$	12.9182	0.783278
$273$	−12.8762	−0.779305
$274$	5.76430	0.348234
$275$	1.00000	0.0603023
$276$	−5.39592	−0.324796
$277$	−13.0712	−0.785374	−0.392687	−	0.919672i	$-0.628454\pi$
$277$	−13.0712	−0.785374	−0.392687	+	0.919672i	$0.628454\pi$
$278$	0.624196	0.0374368
$279$	−12.9598	−0.775885
$280$	7.13508	0.426403
$281$	−24.9860	−1.49054	−0.745270	−	0.666763i	$-0.767679\pi$
$281$	−24.9860	−1.49054	−0.745270	+	0.666763i	$0.767679\pi$
$282$	0.326158	0.0194225
$283$	−28.4917	−1.69366	−0.846829	−	0.531866i	$-0.821491\pi$
$283$	−28.4917	−1.69366	−0.846829	+	0.531866i	$0.821491\pi$
$284$	27.7387	1.64599
$285$	−5.17140	−0.306328
$286$	1.26095	0.0745613
$287$	40.4997	2.39062
$288$	8.50374	0.501088
$289$	3.64479	0.214400
$290$	0.922700	0.0541828
$291$	15.5982	0.914383
$292$	−1.80056	−0.105370
$293$	19.5783	1.14378	0.571889	−	0.820331i	$-0.306211\pi$
$293$	19.5783	1.14378	0.571889	+	0.820331i	$0.306211\pi$
$294$	−5.17027	−0.301536
$295$	−14.0926	−0.820505
$296$	−8.60263	−0.500018
$297$	−5.23228	−0.303607
$298$	1.46384	0.0847977
$299$	−7.79985	−0.451077
$300$	−1.95329	−0.112773
$301$	46.4843	2.67931
$302$	2.76110	0.158884
$303$	−14.5049	−0.833284
$304$	−13.5533	−0.777334
$305$	1.12602	0.0644755
$306$	3.69946	0.211484
$307$	3.33865	0.190547	0.0952733	−	0.995451i	$-0.469627\pi$
$307$	3.33865	0.190547	0.0952733	+	0.995451i	$0.469627\pi$
$308$	−7.56919	−0.431295
$309$	15.3310	0.872148
$310$	−3.17457	−0.180303
$311$	−25.6738	−1.45583	−0.727913	−	0.685669i	$-0.759510\pi$
$311$	−25.6738	−1.45583	−0.727913	+	0.685669i	$0.759510\pi$
$312$	−5.19880	−0.294324
$313$	−32.0967	−1.81421	−0.907107	−	0.420901i	$-0.861714\pi$
$313$	−32.0967	−1.81421	−0.907107	+	0.420901i	$0.861714\pi$
$314$	3.87254	0.218540
$315$	−7.66420	−0.431828
$316$	6.98632	0.393011
$317$	18.3511	1.03070	0.515349	−	0.856980i	$-0.327662\pi$
$317$	18.3511	1.03070	0.515349	+	0.856980i	$0.327662\pi$
$318$	−6.29669	−0.353101
$319$	−2.06610	−0.115679
$320$	−3.60321	−0.201426
$321$	7.58182	0.423176
$322$	−5.18622	−0.289017
$323$	−21.6598	−1.20518
$324$	0.372040	0.0206689
$325$	−2.82350	−0.156619
$326$	−7.92597	−0.438979
$327$	−7.38510	−0.408397
$328$	16.3518	0.902878
$329$	−2.83010	−0.156029
$330$	−0.484472	−0.0266693
$331$	−0.948328	−0.0521248	−0.0260624	−	0.999660i	$-0.508297\pi$
$331$	−0.948328	−0.0521248	−0.0260624	+	0.999660i	$0.508297\pi$
$332$	−6.77677	−0.371924
$333$	9.24057	0.506380
$334$	−9.13724	−0.499967
$335$	7.73434	0.422572
$336$	12.9657	0.707338
$337$	0.499871	0.0272297	0.0136149	−	0.999907i	$-0.495666\pi$
$337$	0.499871	0.0272297	0.0136149	+	0.999907i	$0.495666\pi$
$338$	2.24540	0.122134
$339$	19.5310	1.06078
$340$	−8.18111	−0.443683
$341$	7.10845	0.384944
$342$	−3.88135	−0.209879
$343$	15.4362	0.833477
$344$	18.7681	1.01191
$345$	2.99680	0.161343
$346$	9.92127	0.533371
$347$	−18.3938	−0.987431	−0.493716	−	0.869623i	$-0.664362\pi$
$347$	−18.3938	−0.987431	−0.493716	+	0.869623i	$0.664362\pi$
$348$	4.03568	0.216335
$349$	25.8683	1.38470	0.692350	−	0.721562i	$-0.256576\pi$
$349$	25.8683	1.38470	0.692350	+	0.721562i	$0.256576\pi$
$350$	−1.87738	−0.100350
$351$	14.7733	0.788541
$352$	−4.66429	−0.248608
$353$	11.5289	0.613622	0.306811	−	0.951770i	$-0.400738\pi$
$353$	11.5289	0.613622	0.306811	+	0.951770i	$0.400738\pi$
$354$	6.82749	0.362877
$355$	−15.4056	−0.817645
$356$	−6.80256	−0.360535
$357$	20.7208	1.09666
$358$	−2.65552	−0.140349
$359$	−13.3461	−0.704378	−0.352189	−	0.935929i	$-0.614563\pi$
$359$	−13.3461	−0.704378	−0.352189	+	0.935929i	$0.614563\pi$
$360$	−3.09443	−0.163091
$361$	3.72472	0.196038
$362$	−1.85389	−0.0974385
$363$	1.08482	0.0569385
$364$	21.3716	1.12017
$365$	1.00000	0.0523424
$366$	−0.545523	−0.0285149
$367$	−4.94953	−0.258364	−0.129182	−	0.991621i	$-0.541235\pi$
$367$	−4.94953	−0.258364	−0.129182	+	0.991621i	$0.541235\pi$
$368$	7.85406	0.409421
$369$	−17.5644	−0.914367
$370$	2.26352	0.117675
$371$	54.6369	2.83661
$372$	−13.8848	−0.719896
$373$	35.8998	1.85882	0.929411	−	0.369046i	$-0.120315\pi$
$373$	35.8998	1.85882	0.929411	+	0.369046i	$0.120315\pi$
$374$	−2.02915	−0.104925
$375$	1.08482	0.0560201
$376$	−1.14266	−0.0589281
$377$	5.83362	0.300447
$378$	9.82296	0.505239
$379$	−24.6918	−1.26833	−0.634165	−	0.773197i	$-0.718656\pi$
$379$	−24.6918	−1.26833	−0.634165	+	0.773197i	$0.718656\pi$
$380$	8.58334	0.440316
$381$	7.21496	0.369634
$382$	−3.91978	−0.200553
$383$	17.0405	0.870729	0.435364	−	0.900254i	$-0.356620\pi$
$383$	17.0405	0.870729	0.435364	+	0.900254i	$0.356620\pi$
$384$	11.8655	0.605510
$385$	4.20380	0.214246
$386$	5.61240	0.285664
$387$	−20.1599	−1.02479
$388$	−25.8894	−1.31434
$389$	−12.6850	−0.643155	−0.321578	−	0.946883i	$-0.604213\pi$
$389$	−12.6850	−0.643155	−0.321578	+	0.946883i	$0.604213\pi$
$390$	1.36790	0.0692665
$391$	12.5518	0.634770
$392$	18.1134	0.914867
$393$	−16.5734	−0.836018
$394$	2.06132	0.103848
$395$	−3.88009	−0.195229
$396$	3.28270	0.164962
$397$	2.35417	0.118152	0.0590762	−	0.998253i	$-0.481185\pi$
$397$	2.35417	0.118152	0.0590762	+	0.998253i	$0.481185\pi$
$398$	−4.02781	−0.201896
$399$	−21.7396	−1.08834
$400$	2.84312	0.142156
$401$	−26.1284	−1.30479	−0.652396	−	0.757878i	$-0.726236\pi$
$401$	−26.1284	−1.30479	−0.652396	+	0.757878i	$0.726236\pi$
$402$	−3.74707	−0.186887
$403$	−20.0707	−0.999792
$404$	24.0748	1.19776
$405$	−0.206625	−0.0102673
$406$	3.87885	0.192504
$407$	−5.06844	−0.251233
$408$	8.36606	0.414182
$409$	−6.17352	−0.305261	−0.152630	−	0.988283i	$-0.548774\pi$
$409$	−6.17352	−0.305261	−0.152630	+	0.988283i	$0.548774\pi$
$410$	−4.30248	−0.212484
$411$	−14.0022	−0.690678
$412$	−25.4458	−1.25363
$413$	−59.2427	−2.91514
$414$	2.24922	0.110543
$415$	3.76371	0.184753
$416$	13.1696	0.645693
$417$	−1.51625	−0.0742511
$418$	2.12892	0.104129
$419$	−12.6796	−0.619437	−0.309718	−	0.950828i	$-0.600235\pi$
$419$	−12.6796	−0.619437	−0.309718	+	0.950828i	$0.600235\pi$
$420$	−8.21123	−0.400667
$421$	−9.82317	−0.478752	−0.239376	−	0.970927i	$-0.576943\pi$
$421$	−9.82317	−0.478752	−0.239376	+	0.970927i	$0.576943\pi$
$422$	−6.69049	−0.325688
$423$	1.22739	0.0596780
$424$	22.0597	1.07132
$425$	4.54365	0.220400
$426$	7.46359	0.361612
$427$	4.73355	0.229072
$428$	−12.5841	−0.608274
$429$	−3.06299	−0.147883
$430$	−4.93826	−0.238144
$431$	27.8242	1.34024	0.670122	−	0.742251i	$-0.266242\pi$
$431$	27.8242	1.34024	0.670122	+	0.742251i	$0.266242\pi$
$432$	−14.8760	−0.715721
$433$	1.89166	0.0909074	0.0454537	−	0.998966i	$-0.485527\pi$
$433$	1.89166	0.0909074	0.0454537	+	0.998966i	$0.485527\pi$
$434$	−13.3453	−0.640593
$435$	−2.24135	−0.107465
$436$	12.2576	0.587030
$437$	−13.1689	−0.629952
$438$	−0.484472	−0.0231490
$439$	−12.7924	−0.610547	−0.305274	−	0.952265i	$-0.598748\pi$
$439$	−12.7924	−0.610547	−0.305274	+	0.952265i	$0.598748\pi$
$440$	1.69729	0.0809152
$441$	−19.4567	−0.926508
$442$	5.72930	0.272515
$443$	21.6744	1.02978	0.514891	−	0.857256i	$-0.327832\pi$
$443$	21.6744	1.02978	0.514891	+	0.857256i	$0.327832\pi$
$444$	9.90013	0.469839
$445$	3.77803	0.179096
$446$	5.26915	0.249502
$447$	−3.55584	−0.168185
$448$	−15.1472	−0.715637
$449$	21.0784	0.994750	0.497375	−	0.867536i	$-0.334297\pi$
$449$	21.0784	0.994750	0.497375	+	0.867536i	$0.334297\pi$
$450$	0.814205	0.0383820
$451$	9.63406	0.453650
$452$	−32.4169	−1.52476
$453$	−6.70706	−0.315125
$454$	−10.2739	−0.482176
$455$	−11.8694	−0.556447
$456$	−8.77738	−0.411039
$457$	24.7637	1.15839	0.579197	−	0.815187i	$-0.303366\pi$
$457$	24.7637	1.15839	0.579197	+	0.815187i	$0.303366\pi$
$458$	−8.07565	−0.377350
$459$	−23.7737	−1.10966
$460$	−4.97400	−0.231914
$461$	−10.2398	−0.476913	−0.238457	−	0.971153i	$-0.576641\pi$
$461$	−10.2398	−0.476913	−0.238457	+	0.971153i	$0.576641\pi$
$462$	−2.03662	−0.0947524
$463$	−3.21796	−0.149551	−0.0747756	−	0.997200i	$-0.523824\pi$
$463$	−3.21796	−0.149551	−0.0747756	+	0.997200i	$0.523824\pi$
$464$	−5.87417	−0.272701
$465$	7.71142	0.357608
$466$	−8.97016	−0.415535
$467$	18.9161	0.875331	0.437665	−	0.899138i	$-0.355805\pi$
$467$	18.9161	0.875331	0.437665	+	0.899138i	$0.355805\pi$
$468$	−9.26869	−0.428445
$469$	32.5136	1.50134
$470$	0.300656	0.0138682
$471$	−9.40688	−0.433446
$472$	−23.9193	−1.10098
$473$	11.0577	0.508433
$474$	1.87979	0.0863418
$475$	−4.76704	−0.218727
$476$	−34.3918	−1.57634
$477$	−23.6956	−1.08495
$478$	−7.35239	−0.336290
$479$	2.25652	0.103103	0.0515515	−	0.998670i	$-0.483583\pi$
$479$	2.25652	0.103103	0.0515515	+	0.998670i	$0.483583\pi$
$480$	−5.05994	−0.230953
$481$	14.3107	0.652513
$482$	−5.21416	−0.237499
$483$	12.5980	0.573228
$484$	−1.80056	−0.0818435
$485$	14.3786	0.652897
$486$	−6.90995	−0.313442
$487$	10.7332	0.486367	0.243183	−	0.969980i	$-0.421808\pi$
$487$	10.7332	0.486367	0.243183	+	0.969980i	$0.421808\pi$
$488$	1.91118	0.0865149
$489$	19.2532	0.870658
$490$	−4.76600	−0.215306
$491$	−9.76040	−0.440481	−0.220240	−	0.975446i	$-0.570684\pi$
$491$	−9.76040	−0.440481	−0.220240	+	0.975446i	$0.570684\pi$
$492$	−18.8181	−0.848385
$493$	−9.38764	−0.422798
$494$	−6.01099	−0.270447
$495$	−1.82316	−0.0819448
$496$	20.2102	0.907464
$497$	−64.7622	−2.90498
$498$	−1.82341	−0.0817090
$499$	−8.19839	−0.367010	−0.183505	−	0.983019i	$-0.558744\pi$
$499$	−8.19839	−0.367010	−0.183505	+	0.983019i	$0.558744\pi$
$500$	−1.80056	−0.0805234
$501$	22.1955	0.991621
$502$	−13.4185	−0.598896
$503$	28.8451	1.28614	0.643071	−	0.765807i	$-0.277660\pi$
$503$	28.8451	1.28614	0.643071	+	0.765807i	$0.277660\pi$
$504$	−13.0084	−0.579439
$505$	−13.3707	−0.594990
$506$	−1.23370	−0.0548445
$507$	−5.45436	−0.242236
$508$	−11.9752	−0.531312
$509$	−12.9788	−0.575276	−0.287638	−	0.957739i	$-0.592870\pi$
$509$	−12.9788	−0.575276	−0.287638	+	0.957739i	$0.592870\pi$
$510$	−2.20127	−0.0974740
$511$	4.20380	0.185965
$512$	−22.9124	−1.01259
$513$	24.9425	1.10124
$514$	−5.65601	−0.249476
$515$	14.1322	0.622739
$516$	−21.5988	−0.950836
$517$	−0.673225	−0.0296084
$518$	9.51539	0.418082
$519$	−24.1000	−1.05787
$520$	−4.79230	−0.210156
$521$	−11.3429	−0.496941	−0.248470	−	0.968639i	$-0.579928\pi$
$521$	−11.3429	−0.496941	−0.248470	+	0.968639i	$0.579928\pi$
$522$	−1.68223	−0.0736291
$523$	−22.7127	−0.993159	−0.496579	−	0.867991i	$-0.665411\pi$
$523$	−22.7127	−0.993159	−0.496579	+	0.867991i	$0.665411\pi$
$524$	27.5080	1.20169
$525$	4.56039	0.199032
$526$	−10.4987	−0.457766
$527$	32.2984	1.40694
$528$	3.08428	0.134226
$529$	−15.3687	−0.668204
$530$	−5.80434	−0.252125
$531$	25.6931	1.11499
$532$	36.0827	1.56438
$533$	−27.2017	−1.17824
$534$	−1.83035	−0.0792069
$535$	6.98899	0.302160
$536$	13.1274	0.567019
$537$	6.45059	0.278363
$538$	3.59725	0.155089
$539$	10.6720	0.459674
$540$	9.42101	0.405416
$541$	−33.8790	−1.45657	−0.728287	−	0.685273i	$-0.759683\pi$
$541$	−33.8790	−1.45657	−0.728287	+	0.685273i	$0.759683\pi$
$542$	1.52240	0.0653929
$543$	4.50334	0.193257
$544$	−21.1929	−0.908640
$545$	−6.80765	−0.291608
$546$	5.75040	0.246094
$547$	−21.2530	−0.908713	−0.454357	−	0.890820i	$-0.650131\pi$
$547$	−21.2530	−0.908713	−0.454357	+	0.890820i	$0.650131\pi$
$548$	23.2404	0.992782
$549$	−2.05290	−0.0876158
$550$	−0.446590	−0.0190427
$551$	9.84919	0.419589
$552$	5.08645	0.216494
$553$	−16.3111	−0.693620
$554$	5.83749	0.248011
$555$	−5.49837	−0.233393
$556$	2.51662	0.106729
$557$	38.7154	1.64043	0.820213	−	0.572059i	$-0.193855\pi$
$557$	38.7154	1.64043	0.820213	+	0.572059i	$0.193855\pi$
$558$	5.78773	0.245014
$559$	−31.2213	−1.32052
$560$	11.9519	0.505061
$561$	4.92906	0.208105
$562$	11.1585	0.470693
$563$	−29.8386	−1.25755	−0.628774	−	0.777588i	$-0.716443\pi$
$563$	−29.8386	−1.25755	−0.628774	+	0.777588i	$0.716443\pi$
$564$	1.31500	0.0553715
$565$	18.0038	0.757427
$566$	12.7241	0.534835
$567$	−0.868610	−0.0364782
$568$	−26.1478	−1.09714
$569$	3.16612	0.132731	0.0663654	−	0.997795i	$-0.478860\pi$
$569$	3.16612	0.132731	0.0663654	+	0.997795i	$0.478860\pi$
$570$	2.30950	0.0967343
$571$	−6.67544	−0.279359	−0.139679	−	0.990197i	$-0.544607\pi$
$571$	−6.67544	−0.279359	−0.139679	+	0.990197i	$0.544607\pi$
$572$	5.08386	0.212567
$573$	9.52163	0.397772
$574$	−18.0868	−0.754928
$575$	2.76248	0.115203
$576$	6.56922	0.273717
$577$	24.4553	1.01809	0.509043	−	0.860741i	$-0.329999\pi$
$577$	24.4553	1.01809	0.509043	+	0.860741i	$0.329999\pi$
$578$	−1.62773	−0.0677047
$579$	−13.6332	−0.566577
$580$	3.72013	0.154470
$581$	15.8219	0.656403
$582$	−6.96601	−0.288750
$583$	12.9970	0.538282
$584$	1.69729	0.0702345
$585$	5.14768	0.212830
$586$	−8.74350	−0.361191
$587$	−18.3362	−0.756815	−0.378408	−	0.925639i	$-0.623528\pi$
$587$	−18.3362	−0.756815	−0.378408	+	0.925639i	$0.623528\pi$
$588$	−20.8454	−0.859650
$589$	−33.8863	−1.39626
$590$	6.29364	0.259105
$591$	−5.00719	−0.205968
$592$	−14.4102	−0.592255
$593$	−41.7703	−1.71530	−0.857650	−	0.514233i	$-0.828077\pi$
$593$	−41.7703	−1.71530	−0.857650	+	0.514233i	$0.828077\pi$
$594$	2.33668	0.0958753
$595$	19.1006	0.783049
$596$	5.90187	0.241750
$597$	9.78404	0.400434
$598$	3.48334	0.142444
$599$	−17.8542	−0.729501	−0.364751	−	0.931105i	$-0.618846\pi$
$599$	−17.8542	−0.729501	−0.364751	+	0.931105i	$0.618846\pi$
$600$	1.84126	0.0751692
$601$	31.6467	1.29090	0.645448	−	0.763804i	$-0.276671\pi$
$601$	31.6467	1.29090	0.645448	+	0.763804i	$0.276671\pi$
$602$	−20.7595	−0.846093
$603$	−14.1009	−0.574234
$604$	11.1322	0.452962
$605$	1.00000	0.0406558
$606$	6.47774	0.263140
$607$	39.3323	1.59645	0.798224	−	0.602361i	$-0.205773\pi$
$607$	39.3323	1.59645	0.798224	+	0.602361i	$0.205773\pi$
$608$	22.2349	0.901744
$609$	−9.42221	−0.381807
$610$	−0.502868	−0.0203605
$611$	1.90085	0.0769000
$612$	14.9154	0.602921
$613$	−20.4924	−0.827679	−0.413839	−	0.910350i	$-0.635812\pi$
$613$	−20.4924	−0.827679	−0.413839	+	0.910350i	$0.635812\pi$
$614$	−1.49101	−0.0601722
$615$	10.4513	0.421435
$616$	7.13508	0.287481
$617$	16.7332	0.673654	0.336827	−	0.941567i	$-0.390646\pi$
$617$	16.7332	0.673654	0.336827	+	0.941567i	$0.390646\pi$
$618$	−6.84666	−0.275413
$619$	28.4818	1.14478	0.572390	−	0.819982i	$-0.306016\pi$
$619$	28.4818	1.14478	0.572390	+	0.819982i	$0.306016\pi$
$620$	−12.7992	−0.514027
$621$	−14.4541	−0.580021
$622$	11.4657	0.459731
$623$	15.8821	0.636302
$624$	−8.70846	−0.348617
$625$	1.00000	0.0400000
$626$	14.3341	0.572905
$627$	−5.17140	−0.206526
$628$	15.6133	0.623037
$629$	−23.0293	−0.918237
$630$	3.42276	0.136366
$631$	−33.8003	−1.34557	−0.672784	−	0.739839i	$-0.734902\pi$
$631$	−33.8003	−1.34557	−0.672784	+	0.739839i	$0.734902\pi$
$632$	−6.58565	−0.261963
$633$	16.2520	0.645960
$634$	−8.19541	−0.325481
$635$	6.65081	0.263929
$636$	−25.3869	−1.00666
$637$	−30.1322	−1.19388
$638$	0.922700	0.0365300
$639$	28.0869	1.11110
$640$	10.9377	0.432352
$641$	−24.2574	−0.958109	−0.479054	−	0.877785i	$-0.659020\pi$
$641$	−24.2574	−0.958109	−0.479054	+	0.877785i	$0.659020\pi$
$642$	−3.38597	−0.133634
$643$	−20.4656	−0.807086	−0.403543	−	0.914961i	$-0.632221\pi$
$643$	−20.4656	−0.807086	−0.403543	+	0.914961i	$0.632221\pi$
$644$	−20.9097	−0.823959
$645$	11.9956	0.472328
$646$	9.67306	0.380581
$647$	−26.6310	−1.04697	−0.523486	−	0.852034i	$-0.675369\pi$
$647$	−26.6310	−1.04697	−0.523486	+	0.852034i	$0.675369\pi$
$648$	−0.350703	−0.0137769
$649$	−14.0926	−0.553184
$650$	1.26095	0.0494584
$651$	32.4173	1.27053
$652$	−31.9558	−1.25149
$653$	4.56814	0.178765	0.0893825	−	0.995997i	$-0.471511\pi$
$653$	4.56814	0.178765	0.0893825	+	0.995997i	$0.471511\pi$
$654$	3.29811	0.128966
$655$	−15.2775	−0.596942
$656$	27.3908	1.06943
$657$	−1.82316	−0.0711281
$658$	1.26390	0.0492718
$659$	1.77338	0.0690811	0.0345405	−	0.999403i	$-0.489003\pi$
$659$	1.77338	0.0690811	0.0345405	+	0.999403i	$0.489003\pi$
$660$	−1.95329	−0.0760316
$661$	−25.9729	−1.01023	−0.505114	−	0.863053i	$-0.668550\pi$
$661$	−25.9729	−1.01023	−0.505114	+	0.863053i	$0.668550\pi$
$662$	0.423514	0.0164603
$663$	−13.9172	−0.540499
$664$	6.38811	0.247907
$665$	−20.0397	−0.777107
$666$	−4.12675	−0.159908
$667$	−5.70756	−0.220998
$668$	−36.8394	−1.42536
$669$	−12.7994	−0.494854
$670$	−3.45408	−0.133443
$671$	1.12602	0.0434693
$672$	−21.2710	−0.820546
$673$	−19.3947	−0.747610	−0.373805	−	0.927507i	$-0.621947\pi$
$673$	−19.3947	−0.747610	−0.373805	+	0.927507i	$0.621947\pi$
$674$	−0.223238	−0.00859879
$675$	−5.23228	−0.201390
$676$	9.05297	0.348191
$677$	−38.8712	−1.49394	−0.746970	−	0.664858i	$-0.768492\pi$
$677$	−38.8712	−1.49394	−0.746970	+	0.664858i	$0.768492\pi$
$678$	−8.72235	−0.334980
$679$	60.4446	2.31965
$680$	7.71191	0.295738
$681$	24.9565	0.956335
$682$	−3.17457	−0.121560
$683$	−35.1512	−1.34502	−0.672512	−	0.740086i	$-0.734785\pi$
$683$	−35.1512	−1.34502	−0.672512	+	0.740086i	$0.734785\pi$
$684$	−15.6488	−0.598346
$685$	−12.9074	−0.493165
$686$	−6.89366	−0.263201
$687$	19.6168	0.748426
$688$	31.4383	1.19858
$689$	−36.6970	−1.39805
$690$	−1.33834	−0.0509499
$691$	−32.5538	−1.23841	−0.619203	−	0.785231i	$-0.712544\pi$
$691$	−32.5538	−1.23841	−0.619203	+	0.785231i	$0.712544\pi$
$692$	40.0004	1.52059
$693$	−7.66420	−0.291139
$694$	8.21450	0.311818
$695$	−1.39769	−0.0530175
$696$	−3.80423	−0.144199
$697$	43.7738	1.65805
$698$	−11.5525	−0.437270
$699$	21.7896	0.824160
$700$	−7.56919	−0.286088
$701$	19.3571	0.731109	0.365554	−	0.930790i	$-0.380879\pi$
$701$	19.3571	0.731109	0.365554	+	0.930790i	$0.380879\pi$
$702$	−6.59762	−0.249011
$703$	24.1615	0.911268
$704$	−3.60321	−0.135801
$705$	−0.730330	−0.0275058
$706$	−5.14870	−0.193774
$707$	−56.2079	−2.11392
$708$	27.5270	1.03453
$709$	−28.2575	−1.06123	−0.530616	−	0.847612i	$-0.678039\pi$
$709$	−28.2575	−1.06123	−0.530616	+	0.847612i	$0.678039\pi$
$710$	6.88000	0.258202
$711$	7.07401	0.265296
$712$	6.41242	0.240316
$713$	19.6370	0.735410
$714$	−9.25372	−0.346312
$715$	−2.82350	−0.105593
$716$	−10.7065	−0.400120
$717$	17.8599	0.666989
$718$	5.96022	0.222433
$719$	13.7927	0.514382	0.257191	−	0.966361i	$-0.417203\pi$
$719$	13.7927	0.514382	0.257191	+	0.966361i	$0.417203\pi$
$720$	−5.18345	−0.193176
$721$	59.4090	2.21251
$722$	−1.66342	−0.0619062
$723$	12.6659	0.471048
$724$	−7.47450	−0.277788
$725$	−2.06610	−0.0767330
$726$	−0.484472	−0.0179804
$727$	2.82936	0.104935	0.0524676	−	0.998623i	$-0.483291\pi$
$727$	2.82936	0.104935	0.0524676	+	0.998623i	$0.483291\pi$
$728$	−20.1459	−0.746656
$729$	17.4050	0.644630
$730$	−0.446590	−0.0165291
$731$	50.2423	1.85828
$732$	−2.19943	−0.0812933
$733$	−47.5696	−1.75702	−0.878511	−	0.477722i	$-0.841463\pi$
$733$	−47.5696	−1.75702	−0.878511	+	0.477722i	$0.841463\pi$
$734$	2.21041	0.0815879
$735$	11.5772	0.427031
$736$	−12.8850	−0.474948
$737$	7.73434	0.284898
$738$	7.84410	0.288745
$739$	17.2405	0.634203	0.317101	−	0.948392i	$-0.397290\pi$
$739$	17.2405	0.634203	0.317101	+	0.948392i	$0.397290\pi$
$740$	9.12602	0.335479
$741$	14.6014	0.536397
$742$	−24.4003	−0.895764
$743$	−31.7628	−1.16527	−0.582633	−	0.812736i	$-0.697977\pi$
$743$	−31.7628	−1.16527	−0.582633	+	0.812736i	$0.697977\pi$
$744$	13.0885	0.479849
$745$	−3.27780	−0.120089
$746$	−16.0325	−0.586992
$747$	−6.86183	−0.251061
$748$	−8.18111	−0.299131
$749$	29.3803	1.07353
$750$	−0.484472	−0.0176904
$751$	−40.7925	−1.48854	−0.744271	−	0.667878i	$-0.767203\pi$
$751$	−40.7925	−1.48854	−0.744271	+	0.667878i	$0.767203\pi$
$752$	−1.91406	−0.0697985
$753$	32.5952	1.18783
$754$	−2.60524	−0.0948772
$755$	−6.18263	−0.225009
$756$	39.6041	1.44039
$757$	−39.6054	−1.43948	−0.719742	−	0.694242i	$-0.755740\pi$
$757$	−39.6054	−1.43948	−0.719742	+	0.694242i	$0.755740\pi$
$758$	11.0271	0.400523
$759$	2.99680	0.108777
$760$	−8.09107	−0.293494
$761$	1.53536	0.0556566	0.0278283	−	0.999613i	$-0.491141\pi$
$761$	1.53536	0.0556566	0.0278283	+	0.999613i	$0.491141\pi$
$762$	−3.22213	−0.116726
$763$	−28.6180	−1.03604
$764$	−15.8037	−0.571758
$765$	−8.28380	−0.299501
$766$	−7.61012	−0.274965
$767$	39.7905	1.43675
$768$	2.51866	0.0908845
$769$	26.4808	0.954923	0.477461	−	0.878653i	$-0.341557\pi$
$769$	26.4808	0.954923	0.477461	+	0.878653i	$0.341557\pi$
$770$	−1.87738	−0.0676560
$771$	13.7392	0.494804
$772$	22.6280	0.814399
$773$	14.5416	0.523024	0.261512	−	0.965200i	$-0.415779\pi$
$773$	14.5416	0.523024	0.261512	+	0.965200i	$0.415779\pi$
$774$	9.00322	0.323614
$775$	7.10845	0.255343
$776$	24.4046	0.876074
$777$	−23.1141	−0.829212
$778$	5.66500	0.203100
$779$	−45.9260	−1.64547
$780$	5.51510	0.197472
$781$	−15.4056	−0.551256
$782$	−5.60549	−0.200452
$783$	10.8104	0.386332
$784$	30.3417	1.08363
$785$	−8.67135	−0.309494
$786$	7.40153	0.264004
$787$	24.9365	0.888889	0.444444	−	0.895806i	$-0.353401\pi$
$787$	24.9365	0.888889	0.444444	+	0.895806i	$0.353401\pi$
$788$	8.31079	0.296060
$789$	25.5027	0.907919
$790$	1.73281	0.0616506
$791$	75.6846	2.69103
$792$	−3.09443	−0.109956
$793$	−3.17930	−0.112900
$794$	−1.05135	−0.0373110
$795$	14.0995	0.500057
$796$	−16.2393	−0.575585
$797$	−18.9986	−0.672964	−0.336482	−	0.941690i	$-0.609237\pi$
$797$	−18.9986	−0.672964	−0.336482	+	0.941690i	$0.609237\pi$
$798$	9.70868	0.343684
$799$	−3.05890	−0.108216
$800$	−4.66429	−0.164908
$801$	−6.88794	−0.243373
$802$	11.6687	0.412037
$803$	1.00000	0.0352892
$804$	−15.1074	−0.532796
$805$	11.6129	0.409302
$806$	8.96337	0.315721
$807$	−8.73818	−0.307598
$808$	−22.6940	−0.798373
$809$	−14.7798	−0.519629	−0.259815	−	0.965659i	$-0.583661\pi$
$809$	−14.7798	−0.519629	−0.259815	+	0.965659i	$0.583661\pi$
$810$	0.0922767	0.00324227
$811$	31.0921	1.09179	0.545896	−	0.837853i	$-0.316189\pi$
$811$	31.0921	1.09179	0.545896	+	0.837853i	$0.316189\pi$
$812$	15.6387	0.548810
$813$	−3.69811	−0.129698
$814$	2.26352	0.0793363
$815$	17.7477	0.621676
$816$	14.0139	0.490585
$817$	−52.7125	−1.84418
$818$	2.75703	0.0963974
$819$	21.6398	0.756157
$820$	−17.3467	−0.605772
$821$	−11.3198	−0.395064	−0.197532	−	0.980296i	$-0.563293\pi$
$821$	−11.3198	−0.395064	−0.197532	+	0.980296i	$0.563293\pi$
$822$	6.25325	0.218107
$823$	−40.2698	−1.40372	−0.701859	−	0.712316i	$-0.747646\pi$
$823$	−40.2698	−1.40372	−0.701859	+	0.712316i	$0.747646\pi$
$824$	23.9865	0.835609
$825$	1.08482	0.0377687
$826$	26.4572	0.920564
$827$	−7.73425	−0.268946	−0.134473	−	0.990917i	$-0.542934\pi$
$827$	−7.73425	−0.268946	−0.134473	+	0.990917i	$0.542934\pi$
$828$	9.06839	0.315148
$829$	−3.54130	−0.122994	−0.0614972	−	0.998107i	$-0.519588\pi$
$829$	−3.54130	−0.122994	−0.0614972	+	0.998107i	$0.519588\pi$
$830$	−1.68084	−0.0583427
$831$	−14.1800	−0.491898
$832$	10.1736	0.352708
$833$	48.4897	1.68007
$834$	0.677142	0.0234475
$835$	20.4600	0.708047
$836$	8.58334	0.296861
$837$	−37.1934	−1.28559
$838$	5.66257	0.195610
$839$	31.3132	1.08105	0.540526	−	0.841327i	$-0.318225\pi$
$839$	31.3132	1.08105	0.540526	+	0.841327i	$0.318225\pi$
$840$	7.74031	0.267066
$841$	−24.7312	−0.852801
$842$	4.38693	0.151184
$843$	−27.1054	−0.933560
$844$	−26.9746	−0.928505
$845$	−5.02787	−0.172964
$846$	−0.548143	−0.0188455
$847$	4.20380	0.144444
$848$	36.9521	1.26894
$849$	−30.9085	−1.06078
$850$	−2.02915	−0.0695994
$851$	−14.0015	−0.479965
$852$	30.0916	1.03092
$853$	17.8102	0.609809	0.304905	−	0.952383i	$-0.401375\pi$
$853$	17.8102	0.609809	0.304905	+	0.952383i	$0.401375\pi$
$854$	−2.11396	−0.0723381
$855$	8.69107	0.297228
$856$	11.8624	0.405447
$857$	−4.76167	−0.162656	−0.0813278	−	0.996687i	$-0.525916\pi$
$857$	−4.76167	−0.162656	−0.0813278	+	0.996687i	$0.525916\pi$
$858$	1.36790	0.0466995
$859$	−54.5172	−1.86010	−0.930051	−	0.367431i	$-0.880237\pi$
$859$	−54.5172	−1.86010	−0.930051	+	0.367431i	$0.880237\pi$
$860$	−19.9100	−0.678925
$861$	43.9350	1.49730
$862$	−12.4260	−0.423232
$863$	−50.5845	−1.72192	−0.860959	−	0.508675i	$-0.830136\pi$
$863$	−50.5845	−1.72192	−0.860959	+	0.508675i	$0.830136\pi$
$864$	24.4049	0.830271
$865$	−22.2156	−0.755353
$866$	−0.844797	−0.0287074
$867$	3.95396	0.134284
$868$	−53.8052	−1.82627
$869$	−3.88009	−0.131623
$870$	1.00097	0.0339360
$871$	−21.8379	−0.739948
$872$	−11.5546	−0.391287
$873$	−26.2144	−0.887222
$874$	5.88109	0.198931
$875$	4.20380	0.142114
$876$	−1.95329	−0.0659955
$877$	−0.191186	−0.00645590	−0.00322795	−	0.999995i	$-0.501027\pi$
$877$	−0.191186	−0.00645590	−0.00322795	+	0.999995i	$0.501027\pi$
$878$	5.71296	0.192803
$879$	21.2390	0.716375
$880$	2.84312	0.0958415
$881$	32.7952	1.10490	0.552450	−	0.833546i	$-0.313693\pi$
$881$	32.7952	1.10490	0.552450	+	0.833546i	$0.313693\pi$
$882$	8.68916	0.292579
$883$	0.495202	0.0166649	0.00833244	−	0.999965i	$-0.497348\pi$
$883$	0.495202	0.0166649	0.00833244	+	0.999965i	$0.497348\pi$
$884$	23.0993	0.776914
$885$	−15.2880	−0.513901
$886$	−9.67958	−0.325192
$887$	−30.3178	−1.01797	−0.508986	−	0.860775i	$-0.669979\pi$
$887$	−30.3178	−1.01797	−0.508986	+	0.860775i	$0.669979\pi$
$888$	−9.33234	−0.313173
$889$	27.9587	0.937705
$890$	−1.68723	−0.0565561
$891$	−0.206625	−0.00692219
$892$	21.2441	0.711305
$893$	3.20929	0.107395
$894$	1.58800	0.0531108
$895$	5.94621	0.198760
$896$	45.9801	1.53609
$897$	−8.46146	−0.282520
$898$	−9.41340	−0.314129
$899$	−14.6868	−0.489831
$900$	3.28270	0.109423
$901$	59.0540	1.96737
$902$	−4.30248	−0.143257
$903$	50.4273	1.67812
$904$	30.5578	1.01634
$905$	4.15122	0.137991
$906$	2.99531	0.0995125
$907$	−29.1269	−0.967142	−0.483571	−	0.875305i	$-0.660661\pi$
$907$	−29.1269	−0.967142	−0.483571	+	0.875305i	$0.660661\pi$
$908$	−41.4220	−1.37464
$909$	24.3769	0.808532
$910$	5.30077	0.175719
$911$	−25.3571	−0.840119	−0.420060	−	0.907497i	$-0.637991\pi$
$911$	−25.3571	−0.840119	−0.420060	+	0.907497i	$0.637991\pi$
$912$	−14.7029	−0.486862
$913$	3.76371	0.124561
$914$	−11.0592	−0.365806
$915$	1.22153	0.0403825
$916$	−32.5593	−1.07579
$917$	−64.2237	−2.12085
$918$	10.6171	0.350416
$919$	−14.1662	−0.467299	−0.233649	−	0.972321i	$-0.575067\pi$
$919$	−14.1662	−0.467299	−0.233649	+	0.972321i	$0.575067\pi$
$920$	4.68874	0.154583
$921$	3.62184	0.119344
$922$	4.57298	0.150603
$923$	43.4977	1.43174
$924$	−8.21123	−0.270130
$925$	−5.06844	−0.166649
$926$	1.43711	0.0472263
$927$	−25.7652	−0.846241
$928$	9.63689	0.316347
$929$	47.3845	1.55464	0.777318	−	0.629108i	$-0.216580\pi$
$929$	47.3845	1.55464	0.777318	+	0.629108i	$0.216580\pi$
$930$	−3.44385	−0.112928
$931$	−50.8737	−1.66732
$932$	−36.1658	−1.18465
$933$	−27.8515	−0.911818
$934$	−8.44773	−0.276418
$935$	4.54365	0.148593
$936$	8.73711	0.285581
$937$	28.4129	0.928209	0.464104	−	0.885781i	$-0.346376\pi$
$937$	28.4129	0.928209	0.464104	+	0.885781i	$0.346376\pi$
$938$	−14.5203	−0.474104
$939$	−34.8193	−1.13628
$940$	1.21218	0.0395369
$941$	−2.83238	−0.0923328	−0.0461664	−	0.998934i	$-0.514700\pi$
$941$	−2.83238	−0.0923328	−0.0461664	+	0.998934i	$0.514700\pi$
$942$	4.20102	0.136877
$943$	26.6139	0.866668
$944$	−40.0671	−1.30407
$945$	−21.9955	−0.715512
$946$	−4.93826	−0.160557
$947$	38.7106	1.25792	0.628962	−	0.777436i	$-0.283480\pi$
$947$	38.7106	1.25792	0.628962	+	0.777436i	$0.283480\pi$
$948$	7.57893	0.246152
$949$	−2.82350	−0.0916545
$950$	2.12892	0.0690712
$951$	19.9077	0.645550
$952$	32.4194	1.05072
$953$	30.4692	0.986994	0.493497	−	0.869747i	$-0.335718\pi$
$953$	30.4692	0.986994	0.493497	+	0.869747i	$0.335718\pi$
$954$	10.5822	0.342612
$955$	8.77712	0.284021
$956$	−29.6432	−0.958731
$957$	−2.24135	−0.0724527
$958$	−1.00774	−0.0325586
$959$	−54.2600	−1.75215
$960$	−3.90885	−0.126158
$961$	19.5301	0.630003
$962$	−6.39104	−0.206055
$963$	−12.7420	−0.410606
$964$	−21.0224	−0.677086
$965$	−12.5672	−0.404553
$966$	−5.62614	−0.181018
$967$	21.5064	0.691600	0.345800	−	0.938308i	$-0.387608\pi$
$967$	21.5064	0.691600	0.345800	+	0.938308i	$0.387608\pi$
$968$	1.69729	0.0545530
$969$	−23.4971	−0.754835
$970$	−6.42132	−0.206176
$971$	−1.68790	−0.0541673	−0.0270837	−	0.999633i	$-0.508622\pi$
$971$	−1.68790	−0.0541673	−0.0270837	+	0.999633i	$0.508622\pi$
$972$	−27.8594	−0.893592
$973$	−5.87562	−0.188364
$974$	−4.79334	−0.153588
$975$	−3.06299	−0.0980943
$976$	3.20140	0.102474
$977$	8.68726	0.277930	0.138965	−	0.990297i	$-0.455622\pi$
$977$	8.68726	0.277930	0.138965	+	0.990297i	$0.455622\pi$
$978$	−8.59828	−0.274943
$979$	3.77803	0.120746
$980$	−19.2155	−0.613816
$981$	12.4114	0.396266
$982$	4.35890	0.139098
$983$	47.5368	1.51619	0.758095	−	0.652144i	$-0.226130\pi$
$983$	47.5368	1.51619	0.758095	+	0.652144i	$0.226130\pi$
$984$	17.7388	0.565494
$985$	−4.61568	−0.147068
$986$	4.19243	0.133514
$987$	−3.07016	−0.0977244
$988$	−24.2350	−0.771018
$989$	30.5466	0.971327
$990$	0.814205	0.0258771
$991$	−19.6531	−0.624301	−0.312150	−	0.950033i	$-0.601049\pi$
$991$	−19.6531	−0.624301	−0.312150	+	0.950033i	$0.601049\pi$
$992$	−33.1559	−1.05270
$993$	−1.02877	−0.0326470
$994$	28.9222	0.917356
$995$	9.01902	0.285922
$996$	−7.35160	−0.232944
$997$	−6.91185	−0.218900	−0.109450	−	0.993992i	$-0.534909\pi$
$997$	−6.91185	−0.218900	−0.109450	+	0.993992i	$0.534909\pi$
$998$	3.66132	0.115897
$999$	26.5195	0.839040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.h.1.15	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.h.1.15	✓	37	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)

\(f(q)\)	\(=\)	\(q-0.446590 q^{2} +1.08482 q^{3} -1.80056 q^{4} +1.00000 q^{5} -0.484472 q^{6} +4.20380 q^{7} +1.69729 q^{8} -1.82316 q^{9} +O(q^{10})\)	\(q-0.446590 q^{2} +1.08482 q^{3} -1.80056 q^{4} +1.00000 q^{5} -0.484472 q^{6} +4.20380 q^{7} +1.69729 q^{8} -1.82316 q^{9} -0.446590 q^{10} +1.00000 q^{11} -1.95329 q^{12} -2.82350 q^{13} -1.87738 q^{14} +1.08482 q^{15} +2.84312 q^{16} +4.54365 q^{17} +0.814205 q^{18} -4.76704 q^{19} -1.80056 q^{20} +4.56039 q^{21} -0.446590 q^{22} +2.76248 q^{23} +1.84126 q^{24} +1.00000 q^{25} +1.26095 q^{26} -5.23228 q^{27} -7.56919 q^{28} -2.06610 q^{29} -0.484472 q^{30} +7.10845 q^{31} -4.66429 q^{32} +1.08482 q^{33} -2.02915 q^{34} +4.20380 q^{35} +3.28270 q^{36} -5.06844 q^{37} +2.12892 q^{38} -3.06299 q^{39} +1.69729 q^{40} +9.63406 q^{41} -2.03662 q^{42} +11.0577 q^{43} -1.80056 q^{44} -1.82316 q^{45} -1.23370 q^{46} -0.673225 q^{47} +3.08428 q^{48} +10.6720 q^{49} -0.446590 q^{50} +4.92906 q^{51} +5.08386 q^{52} +12.9970 q^{53} +2.33668 q^{54} +1.00000 q^{55} +7.13508 q^{56} -5.17140 q^{57} +0.922700 q^{58} -14.0926 q^{59} -1.95329 q^{60} +1.12602 q^{61} -3.17457 q^{62} -7.66420 q^{63} -3.60321 q^{64} -2.82350 q^{65} -0.484472 q^{66} +7.73434 q^{67} -8.18111 q^{68} +2.99680 q^{69} -1.87738 q^{70} -15.4056 q^{71} -3.09443 q^{72} +1.00000 q^{73} +2.26352 q^{74} +1.08482 q^{75} +8.58334 q^{76} +4.20380 q^{77} +1.36790 q^{78} -3.88009 q^{79} +2.84312 q^{80} -0.206625 q^{81} -4.30248 q^{82} +3.76371 q^{83} -8.21123 q^{84} +4.54365 q^{85} -4.93826 q^{86} -2.24135 q^{87} +1.69729 q^{88} +3.77803 q^{89} +0.814205 q^{90} -11.8694 q^{91} -4.97400 q^{92} +7.71142 q^{93} +0.300656 q^{94} -4.76704 q^{95} -5.05994 q^{96} +14.3786 q^{97} -4.76600 q^{98} -1.82316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9}+O(q^{10}) \) 37 * q + 5 * q^2 + 3 * q^3 + 43 * q^4 + 37 * q^5 + 9 * q^6 + 6 * q^7 + 12 * q^8 + 50 * q^9	\( 37 q + 5 q^{2} + 3 q^{3} + 43 q^{4} + 37 q^{5} + 9 q^{6} + 6 q^{7} + 12 q^{8} + 50 q^{9} + 5 q^{10} + 37 q^{11} + 6 q^{12} + 11 q^{13} + 11 q^{14} + 3 q^{15} + 43 q^{16} + 38 q^{17} + 11 q^{18} + 34 q^{19} + 43 q^{20} + 39 q^{21} + 5 q^{22} + 4 q^{23} + 25 q^{24} + 37 q^{25} - 9 q^{26} + 3 q^{27} + 14 q^{28} + 58 q^{29} + 9 q^{30} + 8 q^{31} + 14 q^{32} + 3 q^{33} + 8 q^{34} + 6 q^{35} + 20 q^{36} + 2 q^{37} + 15 q^{38} + 14 q^{39} + 12 q^{40} + 62 q^{41} - 13 q^{42} + 30 q^{43} + 43 q^{44} + 50 q^{45} + 31 q^{46} + 5 q^{47} - 25 q^{48} + 59 q^{49} + 5 q^{50} + 23 q^{51} - q^{52} + 18 q^{53} + 13 q^{54} + 37 q^{55} + 22 q^{56} + 5 q^{57} - 40 q^{58} + 15 q^{59} + 6 q^{60} + 57 q^{61} + 20 q^{62} - 29 q^{63} + 10 q^{64} + 11 q^{65} + 9 q^{66} - 14 q^{67} + 53 q^{68} + 24 q^{69} + 11 q^{70} + 8 q^{71} + 15 q^{72} + 37 q^{73} + 7 q^{74} + 3 q^{75} + 59 q^{76} + 6 q^{77} + q^{78} + 42 q^{79} + 43 q^{80} + 61 q^{81} - 22 q^{82} + 44 q^{83} + 66 q^{84} + 38 q^{85} - q^{86} - 26 q^{87} + 12 q^{88} + 69 q^{89} + 11 q^{90} - 10 q^{91} - 21 q^{92} - 26 q^{93} + 29 q^{94} + 34 q^{95} - 9 q^{96} + 37 q^{97} - 15 q^{98} + 50 q^{99}+O(q^{100}) \) 37 * q + 5 * q^2 + 3 * q^3 + 43 * q^4 + 37 * q^5 + 9 * q^6 + 6 * q^7 + 12 * q^8 + 50 * q^9 + 5 * q^10 + 37 * q^11 + 6 * q^12 + 11 * q^13 + 11 * q^14 + 3 * q^15 + 43 * q^16 + 38 * q^17 + 11 * q^18 + 34 * q^19 + 43 * q^20 + 39 * q^21 + 5 * q^22 + 4 * q^23 + 25 * q^24 + 37 * q^25 - 9 * q^26 + 3 * q^27 + 14 * q^28 + 58 * q^29 + 9 * q^30 + 8 * q^31 + 14 * q^32 + 3 * q^33 + 8 * q^34 + 6 * q^35 + 20 * q^36 + 2 * q^37 + 15 * q^38 + 14 * q^39 + 12 * q^40 + 62 * q^41 - 13 * q^42 + 30 * q^43 + 43 * q^44 + 50 * q^45 + 31 * q^46 + 5 * q^47 - 25 * q^48 + 59 * q^49 + 5 * q^50 + 23 * q^51 - q^52 + 18 * q^53 + 13 * q^54 + 37 * q^55 + 22 * q^56 + 5 * q^57 - 40 * q^58 + 15 * q^59 + 6 * q^60 + 57 * q^61 + 20 * q^62 - 29 * q^63 + 10 * q^64 + 11 * q^65 + 9 * q^66 - 14 * q^67 + 53 * q^68 + 24 * q^69 + 11 * q^70 + 8 * q^71 + 15 * q^72 + 37 * q^73 + 7 * q^74 + 3 * q^75 + 59 * q^76 + 6 * q^77 + q^78 + 42 * q^79 + 43 * q^80 + 61 * q^81 - 22 * q^82 + 44 * q^83 + 66 * q^84 + 38 * q^85 - q^86 - 26 * q^87 + 12 * q^88 + 69 * q^89 + 11 * q^90 - 10 * q^91 - 21 * q^92 - 26 * q^93 + 29 * q^94 + 34 * q^95 - 9 * q^96 + 37 * q^97 - 15 * q^98 + 50 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more