LMFDB - Embedded newform 4015.2.a.c.1.10

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:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-0.760826 q^{2} +0.764917 q^{3} -1.42114 q^{4} +1.00000 q^{5} -0.581969 q^{6} -2.66383 q^{7} +2.60290 q^{8} -2.41490 q^{9} +O(q^{10})$	\(q-0.760826 q^{2} +0.764917 q^{3} -1.42114 q^{4} +1.00000 q^{5} -0.581969 q^{6} -2.66383 q^{7} +2.60290 q^{8} -2.41490 q^{9} -0.760826 q^{10} -1.00000 q^{11} -1.08706 q^{12} +2.62846 q^{13} +2.02671 q^{14} +0.764917 q^{15} +0.861937 q^{16} +2.00850 q^{17} +1.83732 q^{18} +1.89539 q^{19} -1.42114 q^{20} -2.03761 q^{21} +0.760826 q^{22} -4.49789 q^{23} +1.99100 q^{24} +1.00000 q^{25} -1.99980 q^{26} -4.14195 q^{27} +3.78569 q^{28} +0.374347 q^{29} -0.581969 q^{30} +6.67921 q^{31} -5.86157 q^{32} -0.764917 q^{33} -1.52812 q^{34} -2.66383 q^{35} +3.43192 q^{36} +4.71737 q^{37} -1.44206 q^{38} +2.01056 q^{39} +2.60290 q^{40} -8.97536 q^{41} +1.55027 q^{42} +7.58519 q^{43} +1.42114 q^{44} -2.41490 q^{45} +3.42211 q^{46} -3.53549 q^{47} +0.659310 q^{48} +0.0960001 q^{49} -0.760826 q^{50} +1.53633 q^{51} -3.73543 q^{52} -1.76521 q^{53} +3.15131 q^{54} -1.00000 q^{55} -6.93368 q^{56} +1.44982 q^{57} -0.284813 q^{58} +11.5159 q^{59} -1.08706 q^{60} -5.97026 q^{61} -5.08172 q^{62} +6.43289 q^{63} +2.73577 q^{64} +2.62846 q^{65} +0.581969 q^{66} +3.81211 q^{67} -2.85436 q^{68} -3.44051 q^{69} +2.02671 q^{70} -4.82216 q^{71} -6.28574 q^{72} +1.00000 q^{73} -3.58910 q^{74} +0.764917 q^{75} -2.69362 q^{76} +2.66383 q^{77} -1.52969 q^{78} -8.61316 q^{79} +0.861937 q^{80} +4.07645 q^{81} +6.82869 q^{82} -6.27943 q^{83} +2.89574 q^{84} +2.00850 q^{85} -5.77101 q^{86} +0.286344 q^{87} -2.60290 q^{88} +10.0458 q^{89} +1.83732 q^{90} -7.00179 q^{91} +6.39214 q^{92} +5.10905 q^{93} +2.68989 q^{94} +1.89539 q^{95} -4.48362 q^{96} -19.5768 q^{97} -0.0730394 q^{98} +2.41490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * q^98 - 8 * 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 $

$2$	−0.760826	−0.537985	−0.268993	−	0.963142i	$-0.586691\pi$
$2$	−0.760826	−0.537985	−0.268993	+	0.963142i	$0.586691\pi$
$3$	0.764917	0.441625	0.220813	−	0.975316i	$-0.429129\pi$
$3$	0.764917	0.441625	0.220813	+	0.975316i	$0.429129\pi$
$4$	−1.42114	−0.710572
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.581969	−0.237588
$7$	−2.66383	−1.00683	−0.503417	−	0.864044i	$-0.667924\pi$
$7$	−2.66383	−1.00683	−0.503417	+	0.864044i	$0.667924\pi$
$8$	2.60290	0.920262
$9$	−2.41490	−0.804967
$10$	−0.760826	−0.240594
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.08706	−0.313806
$13$	2.62846	0.729005	0.364502	−	0.931202i	$-0.381239\pi$
$13$	2.62846	0.729005	0.364502	+	0.931202i	$0.381239\pi$
$14$	2.02671	0.541662
$15$	0.764917	0.197501
$16$	0.861937	0.215484
$17$	2.00850	0.487132	0.243566	−	0.969884i	$-0.421683\pi$
$17$	2.00850	0.487132	0.243566	+	0.969884i	$0.421683\pi$
$18$	1.83732	0.433060
$19$	1.89539	0.434832	0.217416	−	0.976079i	$-0.430237\pi$
$19$	1.89539	0.434832	0.217416	+	0.976079i	$0.430237\pi$
$20$	−1.42114	−0.317777
$21$	−2.03761	−0.444643
$22$	0.760826	0.162209
$23$	−4.49789	−0.937874	−0.468937	−	0.883232i	$-0.655363\pi$
$23$	−4.49789	−0.937874	−0.468937	+	0.883232i	$0.655363\pi$
$24$	1.99100	0.406411
$25$	1.00000	0.200000
$26$	−1.99980	−0.392194
$27$	−4.14195	−0.797119
$28$	3.78569	0.715428
$29$	0.374347	0.0695145	0.0347572	−	0.999396i	$-0.488934\pi$
$29$	0.374347	0.0695145	0.0347572	+	0.999396i	$0.488934\pi$
$30$	−0.581969	−0.106253
$31$	6.67921	1.19962	0.599811	−	0.800142i	$-0.295242\pi$
$31$	6.67921	1.19962	0.599811	+	0.800142i	$0.295242\pi$
$32$	−5.86157	−1.03619
$33$	−0.764917	−0.133155
$34$	−1.52812	−0.262070
$35$	−2.66383	−0.450270
$36$	3.43192	0.571987
$37$	4.71737	0.775530	0.387765	−	0.921758i	$-0.373247\pi$
$37$	4.71737	0.775530	0.387765	+	0.921758i	$0.373247\pi$
$38$	−1.44206	−0.233933
$39$	2.01056	0.321947
$40$	2.60290	0.411554
$41$	−8.97536	−1.40172	−0.700858	−	0.713301i	$-0.747199\pi$
$41$	−8.97536	−1.40172	−0.700858	+	0.713301i	$0.747199\pi$
$42$	1.55027	0.239212
$43$	7.58519	1.15673	0.578365	−	0.815778i	$-0.303691\pi$
$43$	7.58519	1.15673	0.578365	+	0.815778i	$0.303691\pi$
$44$	1.42114	0.214245
$45$	−2.41490	−0.359992
$46$	3.42211	0.504562
$47$	−3.53549	−0.515704	−0.257852	−	0.966184i	$-0.583015\pi$
$47$	−3.53549	−0.515704	−0.257852	+	0.966184i	$0.583015\pi$
$48$	0.659310	0.0951633
$49$	0.0960001	0.0137143
$50$	−0.760826	−0.107597
$51$	1.53633	0.215130
$52$	−3.73543	−0.518010
$53$	−1.76521	−0.242471	−0.121235	−	0.992624i	$-0.538686\pi$
$53$	−1.76521	−0.242471	−0.121235	+	0.992624i	$0.538686\pi$
$54$	3.15131	0.428838
$55$	−1.00000	−0.134840
$56$	−6.93368	−0.926551
$57$	1.44982	0.192033
$58$	−0.284813	−0.0373978
$59$	11.5159	1.49924	0.749622	−	0.661866i	$-0.230235\pi$
$59$	11.5159	1.49924	0.749622	+	0.661866i	$0.230235\pi$
$60$	−1.08706	−0.140339
$61$	−5.97026	−0.764414	−0.382207	−	0.924077i	$-0.624836\pi$
$61$	−5.97026	−0.764414	−0.382207	+	0.924077i	$0.624836\pi$
$62$	−5.08172	−0.645379
$63$	6.43289	0.810468
$64$	2.73577	0.341971
$65$	2.62846	0.326021
$66$	0.581969	0.0716354
$67$	3.81211	0.465723	0.232861	−	0.972510i	$-0.425191\pi$
$67$	3.81211	0.465723	0.232861	+	0.972510i	$0.425191\pi$
$68$	−2.85436	−0.346142
$69$	−3.44051	−0.414189
$70$	2.02671	0.242239
$71$	−4.82216	−0.572285	−0.286142	−	0.958187i	$-0.592373\pi$
$71$	−4.82216	−0.572285	−0.286142	+	0.958187i	$0.592373\pi$
$72$	−6.28574	−0.740781
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	−3.58910	−0.417224
$75$	0.764917	0.0883251
$76$	−2.69362	−0.308979
$77$	2.66383	0.303572
$78$	−1.52969	−0.173203
$79$	−8.61316	−0.969057	−0.484528	−	0.874776i	$-0.661009\pi$
$79$	−8.61316	−0.969057	−0.484528	+	0.874776i	$0.661009\pi$
$80$	0.861937	0.0963675
$81$	4.07645	0.452939
$82$	6.82869	0.754102
$83$	−6.27943	−0.689257	−0.344628	−	0.938739i	$-0.611995\pi$
$83$	−6.27943	−0.689257	−0.344628	+	0.938739i	$0.611995\pi$
$84$	2.89574	0.315951
$85$	2.00850	0.217852
$86$	−5.77101	−0.622304
$87$	0.286344	0.0306993
$88$	−2.60290	−0.277470
$89$	10.0458	1.06485	0.532425	−	0.846477i	$-0.321281\pi$
$89$	10.0458	1.06485	0.532425	+	0.846477i	$0.321281\pi$
$90$	1.83732	0.193671
$91$	−7.00179	−0.733987
$92$	6.39214	0.666427
$93$	5.10905	0.529783
$94$	2.68989	0.277441
$95$	1.89539	0.194463
$96$	−4.48362	−0.457608
$97$	−19.5768	−1.98772	−0.993862	−	0.110631i	$-0.964713\pi$
$97$	−19.5768	−1.98772	−0.993862	+	0.110631i	$0.964713\pi$
$98$	−0.0730394	−0.00737809
$99$	2.41490	0.242707
$100$	−1.42114	−0.142114
$101$	−0.800418	−0.0796446	−0.0398223	−	0.999207i	$-0.512679\pi$
$101$	−0.800418	−0.0796446	−0.0398223	+	0.999207i	$0.512679\pi$
$102$	−1.16888	−0.115737
$103$	0.0926071	0.00912485	0.00456242	−	0.999990i	$-0.498548\pi$
$103$	0.0926071	0.00912485	0.00456242	+	0.999990i	$0.498548\pi$
$104$	6.84162	0.670876
$105$	−2.03761	−0.198851
$106$	1.34302	0.130446
$107$	−14.1172	−1.36477	−0.682383	−	0.730995i	$-0.739056\pi$
$107$	−14.1172	−1.36477	−0.682383	+	0.730995i	$0.739056\pi$
$108$	5.88631	0.566410
$109$	4.71606	0.451717	0.225858	−	0.974160i	$-0.427481\pi$
$109$	4.71606	0.451717	0.225858	+	0.974160i	$0.427481\pi$
$110$	0.760826	0.0725419
$111$	3.60840	0.342494
$112$	−2.29605	−0.216957
$113$	−3.67765	−0.345965	−0.172982	−	0.984925i	$-0.555340\pi$
$113$	−3.67765	−0.345965	−0.172982	+	0.984925i	$0.555340\pi$
$114$	−1.10306	−0.103311
$115$	−4.49789	−0.419430
$116$	−0.532001	−0.0493950
$117$	−6.34748	−0.586825
$118$	−8.76161	−0.806572
$119$	−5.35030	−0.490461
$120$	1.99100	0.181753
$121$	1.00000	0.0909091
$122$	4.54233	0.411244
$123$	−6.86541	−0.619033
$124$	−9.49212	−0.852418
$125$	1.00000	0.0894427
$126$	−4.89431	−0.436020
$127$	−9.52739	−0.845419	−0.422710	−	0.906265i	$-0.638921\pi$
$127$	−9.52739	−0.845419	−0.422710	+	0.906265i	$0.638921\pi$
$128$	9.64171	0.852215
$129$	5.80204	0.510842
$130$	−1.99980	−0.175394
$131$	−19.2308	−1.68020	−0.840102	−	0.542428i	$-0.817505\pi$
$131$	−19.2308	−1.68020	−0.840102	+	0.542428i	$0.817505\pi$
$132$	1.08706	0.0946162
$133$	−5.04900	−0.437804
$134$	−2.90035	−0.250552
$135$	−4.14195	−0.356483
$136$	5.22791	0.448289
$137$	−9.06265	−0.774274	−0.387137	−	0.922022i	$-0.626536\pi$
$137$	−9.06265	−0.774274	−0.387137	+	0.922022i	$0.626536\pi$
$138$	2.61763	0.222828
$139$	−1.95780	−0.166058	−0.0830292	−	0.996547i	$-0.526459\pi$
$139$	−1.95780	−0.166058	−0.0830292	+	0.996547i	$0.526459\pi$
$140$	3.78569	0.319949
$141$	−2.70435	−0.227748
$142$	3.66882	0.307881
$143$	−2.62846	−0.219803
$144$	−2.08149	−0.173458
$145$	0.374347	0.0310878
$146$	−0.760826	−0.0629664
$147$	0.0734321	0.00605658
$148$	−6.70406	−0.551070
$149$	−5.90414	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$149$	−5.90414	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$150$	−0.581969	−0.0475176
$151$	7.59076	0.617727	0.308864	−	0.951106i	$-0.400051\pi$
$151$	7.59076	0.617727	0.308864	+	0.951106i	$0.400051\pi$
$152$	4.93350	0.400160
$153$	−4.85032	−0.392125
$154$	−2.02671	−0.163317
$155$	6.67921	0.536487
$156$	−2.85729	−0.228766
$157$	−19.8573	−1.58478	−0.792392	−	0.610012i	$-0.791165\pi$
$157$	−19.8573	−1.58478	−0.792392	+	0.610012i	$0.791165\pi$
$158$	6.55312	0.521338
$159$	−1.35024	−0.107081
$160$	−5.86157	−0.463398
$161$	11.9816	0.944283
$162$	−3.10147	−0.243675
$163$	10.5996	0.830226	0.415113	−	0.909770i	$-0.363742\pi$
$163$	10.5996	0.830226	0.415113	+	0.909770i	$0.363742\pi$
$164$	12.7553	0.996020
$165$	−0.764917	−0.0595487
$166$	4.77755	0.370810
$167$	5.16563	0.399729	0.199864	−	0.979824i	$-0.435950\pi$
$167$	5.16563	0.399729	0.199864	+	0.979824i	$0.435950\pi$
$168$	−5.30369	−0.409189
$169$	−6.09117	−0.468552
$170$	−1.52812	−0.117201
$171$	−4.57718	−0.350026
$172$	−10.7796	−0.821940
$173$	6.88487	0.523447	0.261724	−	0.965143i	$-0.415709\pi$
$173$	6.88487	0.523447	0.261724	+	0.965143i	$0.415709\pi$
$174$	−0.217858	−0.0165158
$175$	−2.66383	−0.201367
$176$	−0.861937	−0.0649709
$177$	8.80873	0.662104
$178$	−7.64309	−0.572874
$179$	0.600684	0.0448972	0.0224486	−	0.999748i	$-0.492854\pi$
$179$	0.600684	0.0448972	0.0224486	+	0.999748i	$0.492854\pi$
$180$	3.43192	0.255800
$181$	−19.1513	−1.42350	−0.711752	−	0.702431i	$-0.752098\pi$
$181$	−19.1513	−1.42350	−0.711752	+	0.702431i	$0.752098\pi$
$182$	5.32714	0.394874
$183$	−4.56676	−0.337585
$184$	−11.7075	−0.863090
$185$	4.71737	0.346828
$186$	−3.88710	−0.285016
$187$	−2.00850	−0.146876
$188$	5.02443	0.366444
$189$	11.0335	0.802566
$190$	−1.44206	−0.104618
$191$	−21.8640	−1.58202	−0.791011	−	0.611802i	$-0.790445\pi$
$191$	−21.8640	−1.58202	−0.791011	+	0.611802i	$0.790445\pi$
$192$	2.09263	0.151023
$193$	−15.3904	−1.10783	−0.553914	−	0.832574i	$-0.686866\pi$
$193$	−15.3904	−1.10783	−0.553914	+	0.832574i	$0.686866\pi$
$194$	14.8945	1.06937
$195$	2.01056	0.143979
$196$	−0.136430	−0.00974499
$197$	−6.27147	−0.446824	−0.223412	−	0.974724i	$-0.571720\pi$
$197$	−6.27147	−0.446824	−0.223412	+	0.974724i	$0.571720\pi$
$198$	−1.83732	−0.130573
$199$	13.3860	0.948911	0.474455	−	0.880280i	$-0.342645\pi$
$199$	13.3860	0.948911	0.474455	+	0.880280i	$0.342645\pi$
$200$	2.60290	0.184052
$201$	2.91595	0.205675
$202$	0.608979	0.0428476
$203$	−0.997197	−0.0699895
$204$	−2.18335	−0.152865
$205$	−8.97536	−0.626866
$206$	−0.0704579	−0.00490903
$207$	10.8620	0.754958
$208$	2.26557	0.157089
$209$	−1.89539	−0.131107
$210$	1.55027	0.106979
$211$	10.1923	0.701668	0.350834	−	0.936438i	$-0.385898\pi$
$211$	10.1923	0.701668	0.350834	+	0.936438i	$0.385898\pi$
$212$	2.50862	0.172293
$213$	−3.68855	−0.252736
$214$	10.7408	0.734224
$215$	7.58519	0.517306
$216$	−10.7811	−0.733559
$217$	−17.7923	−1.20782
$218$	−3.58810	−0.243017
$219$	0.764917	0.0516883
$220$	1.42114	0.0958135
$221$	5.27926	0.355122
$222$	−2.74536	−0.184257
$223$	−3.04803	−0.204111	−0.102056	−	0.994779i	$-0.532542\pi$
$223$	−3.04803	−0.204111	−0.102056	+	0.994779i	$0.532542\pi$
$224$	15.6142	1.04327
$225$	−2.41490	−0.160993
$226$	2.79806	0.186124
$227$	−0.0708750	−0.00470414	−0.00235207	−	0.999997i	$-0.500749\pi$
$227$	−0.0708750	−0.00470414	−0.00235207	+	0.999997i	$0.500749\pi$
$228$	−2.06040	−0.136453
$229$	8.02098	0.530041	0.265021	−	0.964243i	$-0.414621\pi$
$229$	8.02098	0.530041	0.265021	+	0.964243i	$0.414621\pi$
$230$	3.42211	0.225647
$231$	2.03761	0.134065
$232$	0.974386	0.0639716
$233$	13.6847	0.896517	0.448258	−	0.893904i	$-0.352044\pi$
$233$	13.6847	0.896517	0.448258	+	0.893904i	$0.352044\pi$
$234$	4.82933	0.315703
$235$	−3.53549	−0.230630
$236$	−16.3658	−1.06532
$237$	−6.58836	−0.427960
$238$	4.07065	0.263861
$239$	18.0880	1.17002	0.585009	−	0.811027i	$-0.301091\pi$
$239$	18.0880	1.17002	0.585009	+	0.811027i	$0.301091\pi$
$240$	0.659310	0.0425583
$241$	−15.2666	−0.983411	−0.491706	−	0.870761i	$-0.663626\pi$
$241$	−15.2666	−0.983411	−0.491706	+	0.870761i	$0.663626\pi$
$242$	−0.760826	−0.0489078
$243$	15.5440	0.997149
$244$	8.48460	0.543171
$245$	0.0960001	0.00613322
$246$	5.22338	0.333031
$247$	4.98196	0.316995
$248$	17.3853	1.10397
$249$	−4.80324	−0.304393
$250$	−0.760826	−0.0481189
$251$	−21.2443	−1.34093	−0.670464	−	0.741942i	$-0.733905\pi$
$251$	−21.2443	−1.34093	−0.670464	+	0.741942i	$0.733905\pi$
$252$	−9.14206	−0.575896
$253$	4.49789	0.282780
$254$	7.24869	0.454823
$255$	1.53633	0.0962090
$256$	−12.8072	−0.800450
$257$	1.56159	0.0974091	0.0487046	−	0.998813i	$-0.484491\pi$
$257$	1.56159	0.0974091	0.0487046	+	0.998813i	$0.484491\pi$
$258$	−4.41435	−0.274825
$259$	−12.5663	−0.780830
$260$	−3.73543	−0.231661
$261$	−0.904011	−0.0559569
$262$	14.6313	0.903925
$263$	23.1099	1.42502	0.712510	−	0.701662i	$-0.247558\pi$
$263$	23.1099	1.42502	0.712510	+	0.701662i	$0.247558\pi$
$264$	−1.99100	−0.122538
$265$	−1.76521	−0.108436
$266$	3.84141	0.235532
$267$	7.68419	0.470265
$268$	−5.41755	−0.330930
$269$	−27.9259	−1.70267	−0.851336	−	0.524621i	$-0.824207\pi$
$269$	−27.9259	−1.70267	−0.851336	+	0.524621i	$0.824207\pi$
$270$	3.15131	0.191782
$271$	3.04683	0.185081	0.0925407	−	0.995709i	$-0.470501\pi$
$271$	3.04683	0.185081	0.0925407	+	0.995709i	$0.470501\pi$
$272$	1.73120	0.104969
$273$	−5.35579	−0.324147
$274$	6.89510	0.416548
$275$	−1.00000	−0.0603023
$276$	4.88946	0.294311
$277$	−11.3572	−0.682385	−0.341193	−	0.939993i	$-0.610831\pi$
$277$	−11.3572	−0.682385	−0.341193	+	0.939993i	$0.610831\pi$
$278$	1.48954	0.0893369
$279$	−16.1296	−0.965656
$280$	−6.93368	−0.414366
$281$	11.4040	0.680307	0.340154	−	0.940370i	$-0.389521\pi$
$281$	11.4040	0.680307	0.340154	+	0.940370i	$0.389521\pi$
$282$	2.05754	0.122525
$283$	5.97685	0.355287	0.177643	−	0.984095i	$-0.443153\pi$
$283$	5.97685	0.355287	0.177643	+	0.984095i	$0.443153\pi$
$284$	6.85298	0.406650
$285$	1.44982	0.0858797
$286$	1.99980	0.118251
$287$	23.9088	1.41129
$288$	14.1551	0.834099
$289$	−12.9659	−0.762702
$290$	−0.284813	−0.0167248
$291$	−14.9746	−0.877829
$292$	−1.42114	−0.0831661
$293$	−7.50759	−0.438598	−0.219299	−	0.975658i	$-0.570377\pi$
$293$	−7.50759	−0.438598	−0.219299	+	0.975658i	$0.570377\pi$
$294$	−0.0558691	−0.00325835
$295$	11.5159	0.670483
$296$	12.2788	0.713691
$297$	4.14195	0.240340
$298$	4.49202	0.260216
$299$	−11.8225	−0.683715
$300$	−1.08706	−0.0627613
$301$	−20.2057	−1.16464
$302$	−5.77525	−0.332328
$303$	−0.612254	−0.0351731
$304$	1.63371	0.0936994
$305$	−5.97026	−0.341856
$306$	3.69025	0.210958
$307$	−0.706376	−0.0403150	−0.0201575	−	0.999797i	$-0.506417\pi$
$307$	−0.706376	−0.0403150	−0.0201575	+	0.999797i	$0.506417\pi$
$308$	−3.78569	−0.215710
$309$	0.0708368	0.00402976
$310$	−5.08172	−0.288622
$311$	−26.3341	−1.49327	−0.746634	−	0.665235i	$-0.768331\pi$
$311$	−26.3341	−1.49327	−0.746634	+	0.665235i	$0.768331\pi$
$312$	5.23327	0.296276
$313$	−24.4353	−1.38116	−0.690582	−	0.723254i	$-0.742646\pi$
$313$	−24.4353	−1.38116	−0.690582	+	0.723254i	$0.742646\pi$
$314$	15.1079	0.852591
$315$	6.43289	0.362452
$316$	12.2405	0.688584
$317$	11.9528	0.671334	0.335667	−	0.941981i	$-0.391038\pi$
$317$	11.9528	0.671334	0.335667	+	0.941981i	$0.391038\pi$
$318$	1.02730	0.0576081
$319$	−0.374347	−0.0209594
$320$	2.73577	0.152934
$321$	−10.7985	−0.602715
$322$	−9.11592	−0.508010
$323$	3.80688	0.211821
$324$	−5.79322	−0.321846
$325$	2.62846	0.145801
$326$	−8.06446	−0.446649
$327$	3.60740	0.199490
$328$	−23.3619	−1.28995
$329$	9.41794	0.519228
$330$	0.581969	0.0320363
$331$	−0.0560152	−0.00307887	−0.00153944	−	0.999999i	$-0.500490\pi$
$331$	−0.0560152	−0.00307887	−0.00153944	+	0.999999i	$0.500490\pi$
$332$	8.92397	0.489767
$333$	−11.3920	−0.624276
$334$	−3.93015	−0.215048
$335$	3.81211	0.208278
$336$	−1.75629	−0.0958136
$337$	−5.13374	−0.279653	−0.139826	−	0.990176i	$-0.544654\pi$
$337$	−5.13374	−0.279653	−0.139826	+	0.990176i	$0.544654\pi$
$338$	4.63432	0.252074
$339$	−2.81310	−0.152787
$340$	−2.85436	−0.154800
$341$	−6.67921	−0.361700
$342$	3.48244	0.188309
$343$	18.3911	0.993026
$344$	19.7435	1.06450
$345$	−3.44051	−0.185231
$346$	−5.23819	−0.281607
$347$	−12.5570	−0.674097	−0.337048	−	0.941487i	$-0.609429\pi$
$347$	−12.5570	−0.674097	−0.337048	+	0.941487i	$0.609429\pi$
$348$	−0.406937	−0.0218141
$349$	−18.2858	−0.978816	−0.489408	−	0.872055i	$-0.662787\pi$
$349$	−18.2858	−0.978816	−0.489408	+	0.872055i	$0.662787\pi$
$350$	2.02671	0.108332
$351$	−10.8870	−0.581104
$352$	5.86157	0.312423
$353$	−30.6329	−1.63042	−0.815212	−	0.579162i	$-0.803380\pi$
$353$	−30.6329	−1.63042	−0.815212	+	0.579162i	$0.803380\pi$
$354$	−6.70191	−0.356202
$355$	−4.82216	−0.255934
$356$	−14.2765	−0.756652
$357$	−4.09254	−0.216600
$358$	−0.457016	−0.0241540
$359$	18.7547	0.989837	0.494918	−	0.868939i	$-0.335198\pi$
$359$	18.7547	0.989837	0.494918	+	0.868939i	$0.335198\pi$
$360$	−6.28574	−0.331287
$361$	−15.4075	−0.810921
$362$	14.5708	0.765824
$363$	0.764917	0.0401478
$364$	9.95055	0.521550
$365$	1.00000	0.0523424
$366$	3.47451	0.181616
$367$	7.66132	0.399917	0.199959	−	0.979804i	$-0.435919\pi$
$367$	7.66132	0.399917	0.199959	+	0.979804i	$0.435919\pi$
$368$	−3.87689	−0.202097
$369$	21.6746	1.12833
$370$	−3.58910	−0.186588
$371$	4.70223	0.244128
$372$	−7.26069	−0.376449
$373$	−0.186061	−0.00963390	−0.00481695	−	0.999988i	$-0.501533\pi$
$373$	−0.186061	−0.00963390	−0.00481695	+	0.999988i	$0.501533\pi$
$374$	1.52812	0.0790170
$375$	0.764917	0.0395002
$376$	−9.20250	−0.474583
$377$	0.983957	0.0506764
$378$	−8.39455	−0.431769
$379$	−20.0002	−1.02734	−0.513671	−	0.857987i	$-0.671715\pi$
$379$	−20.0002	−1.02734	−0.513671	+	0.857987i	$0.671715\pi$
$380$	−2.69362	−0.138180
$381$	−7.28767	−0.373359
$382$	16.6347	0.851105
$383$	−6.83166	−0.349082	−0.174541	−	0.984650i	$-0.555844\pi$
$383$	−6.83166	−0.349082	−0.174541	+	0.984650i	$0.555844\pi$
$384$	7.37511	0.376360
$385$	2.66383	0.135761
$386$	11.7095	0.595995
$387$	−18.3175	−0.931130
$388$	27.8214	1.41242
$389$	−21.8249	−1.10657	−0.553284	−	0.832993i	$-0.686625\pi$
$389$	−21.8249	−1.10657	−0.553284	+	0.832993i	$0.686625\pi$
$390$	−1.52969	−0.0774586
$391$	−9.03399	−0.456869
$392$	0.249878	0.0126208
$393$	−14.7100	−0.742021
$394$	4.77150	0.240385
$395$	−8.61316	−0.433375
$396$	−3.43192	−0.172461
$397$	27.0679	1.35850	0.679250	−	0.733907i	$-0.262305\pi$
$397$	27.0679	1.35850	0.679250	+	0.733907i	$0.262305\pi$
$398$	−10.1844	−0.510500
$399$	−3.86207	−0.193345
$400$	0.861937	0.0430968
$401$	10.6156	0.530116	0.265058	−	0.964232i	$-0.414609\pi$
$401$	10.6156	0.530116	0.265058	+	0.964232i	$0.414609\pi$
$402$	−2.21853	−0.110650
$403$	17.5561	0.874530
$404$	1.13751	0.0565932
$405$	4.07645	0.202561
$406$	0.758694	0.0376533
$407$	−4.71737	−0.233831
$408$	3.99892	0.197976
$409$	19.9053	0.984253	0.492127	−	0.870524i	$-0.336220\pi$
$409$	19.9053	0.984253	0.492127	+	0.870524i	$0.336220\pi$
$410$	6.82869	0.337245
$411$	−6.93218	−0.341939
$412$	−0.131608	−0.00648386
$413$	−30.6765	−1.50949
$414$	−8.26406	−0.406156
$415$	−6.27943	−0.308245
$416$	−15.4069	−0.755387
$417$	−1.49755	−0.0733356
$418$	1.44206	0.0705335
$419$	−1.94216	−0.0948807	−0.0474404	−	0.998874i	$-0.515106\pi$
$419$	−1.94216	−0.0948807	−0.0474404	+	0.998874i	$0.515106\pi$
$420$	2.89574	0.141298
$421$	30.8299	1.50256	0.751280	−	0.659984i	$-0.229437\pi$
$421$	30.8299	1.50256	0.751280	+	0.659984i	$0.229437\pi$
$422$	−7.75458	−0.377487
$423$	8.53785	0.415124
$424$	−4.59466	−0.223137
$425$	2.00850	0.0974264
$426$	2.80635	0.135968
$427$	15.9038	0.769638
$428$	20.0626	0.969764
$429$	−2.01056	−0.0970707
$430$	−5.77101	−0.278303
$431$	−19.0322	−0.916748	−0.458374	−	0.888759i	$-0.651568\pi$
$431$	−19.0322	−0.916748	−0.458374	+	0.888759i	$0.651568\pi$
$432$	−3.57010	−0.171767
$433$	−12.4988	−0.600656	−0.300328	−	0.953836i	$-0.597096\pi$
$433$	−12.4988	−0.600656	−0.300328	+	0.953836i	$0.597096\pi$
$434$	13.5368	0.649789
$435$	0.286344	0.0137292
$436$	−6.70220	−0.320977
$437$	−8.52524	−0.407818
$438$	−0.581969	−0.0278076
$439$	−8.96056	−0.427664	−0.213832	−	0.976870i	$-0.568595\pi$
$439$	−8.96056	−0.427664	−0.213832	+	0.976870i	$0.568595\pi$
$440$	−2.60290	−0.124088
$441$	−0.231831	−0.0110396
$442$	−4.01660	−0.191050
$443$	7.90196	0.375433	0.187717	−	0.982223i	$-0.439891\pi$
$443$	7.90196	0.375433	0.187717	+	0.982223i	$0.439891\pi$
$444$	−5.12805	−0.243366
$445$	10.0458	0.476215
$446$	2.31902	0.109809
$447$	−4.51618	−0.213608
$448$	−7.28762	−0.344308
$449$	−32.9186	−1.55352	−0.776762	−	0.629794i	$-0.783139\pi$
$449$	−32.9186	−1.55352	−0.776762	+	0.629794i	$0.783139\pi$
$450$	1.83732	0.0866121
$451$	8.97536	0.422633
$452$	5.22648	0.245833
$453$	5.80630	0.272804
$454$	0.0539235	0.00253076
$455$	−7.00179	−0.328249
$456$	3.77372	0.176721
$457$	5.20533	0.243495	0.121748	−	0.992561i	$-0.461150\pi$
$457$	5.20533	0.243495	0.121748	+	0.992561i	$0.461150\pi$
$458$	−6.10257	−0.285154
$459$	−8.31910	−0.388302
$460$	6.39214	0.298035
$461$	−18.5719	−0.864982	−0.432491	−	0.901638i	$-0.642365\pi$
$461$	−18.5719	−0.864982	−0.432491	+	0.901638i	$0.642365\pi$
$462$	−1.55027	−0.0721250
$463$	−4.10861	−0.190943	−0.0954716	−	0.995432i	$-0.530436\pi$
$463$	−4.10861	−0.190943	−0.0954716	+	0.995432i	$0.530436\pi$
$464$	0.322663	0.0149793
$465$	5.10905	0.236926
$466$	−10.4117	−0.482313
$467$	13.2304	0.612230	0.306115	−	0.951995i	$-0.400971\pi$
$467$	13.2304	0.612230	0.306115	+	0.951995i	$0.400971\pi$
$468$	9.02068	0.416981
$469$	−10.1548	−0.468906
$470$	2.68989	0.124075
$471$	−15.1892	−0.699881
$472$	29.9747	1.37970
$473$	−7.58519	−0.348767
$474$	5.01260	0.230236
$475$	1.89539	0.0869664
$476$	7.60354	0.348508
$477$	4.26281	0.195181
$478$	−13.7619	−0.629453
$479$	−22.2280	−1.01562	−0.507811	−	0.861469i	$-0.669545\pi$
$479$	−22.2280	−1.01562	−0.507811	+	0.861469i	$0.669545\pi$
$480$	−4.48362	−0.204648
$481$	12.3994	0.565365
$482$	11.6153	0.529061
$483$	9.16494	0.417019
$484$	−1.42114	−0.0645974
$485$	−19.5768	−0.888937
$486$	−11.8263	−0.536451
$487$	−9.78589	−0.443441	−0.221721	−	0.975110i	$-0.571167\pi$
$487$	−9.78589	−0.443441	−0.221721	+	0.975110i	$0.571167\pi$
$488$	−15.5400	−0.703462
$489$	8.10783	0.366649
$490$	−0.0730394	−0.00329958
$491$	18.3131	0.826457	0.413228	−	0.910627i	$-0.364401\pi$
$491$	18.3131	0.826457	0.413228	+	0.910627i	$0.364401\pi$
$492$	9.75673	0.439867
$493$	0.751875	0.0338627
$494$	−3.79041	−0.170538
$495$	2.41490	0.108542
$496$	5.75706	0.258500
$497$	12.8454	0.576196
$498$	3.65443	0.163759
$499$	−4.99544	−0.223626	−0.111813	−	0.993729i	$-0.535666\pi$
$499$	−4.99544	−0.223626	−0.111813	+	0.993729i	$0.535666\pi$
$500$	−1.42114	−0.0635555
$501$	3.95128	0.176530
$502$	16.1632	0.721399
$503$	15.7024	0.700137	0.350068	−	0.936724i	$-0.386158\pi$
$503$	15.7024	0.700137	0.350068	+	0.936724i	$0.386158\pi$
$504$	16.7441	0.745843
$505$	−0.800418	−0.0356181
$506$	−3.42211	−0.152131
$507$	−4.65925	−0.206924
$508$	13.5398	0.600731
$509$	−44.5124	−1.97298	−0.986488	−	0.163835i	$-0.947614\pi$
$509$	−44.5124	−1.97298	−0.986488	+	0.163835i	$0.947614\pi$
$510$	−1.16888	−0.0517590
$511$	−2.66383	−0.117841
$512$	−9.53937	−0.421585
$513$	−7.85061	−0.346613
$514$	−1.18810	−0.0524047
$515$	0.0926071	0.00408076
$516$	−8.24554	−0.362990
$517$	3.53549	0.155490
$518$	9.56075	0.420075
$519$	5.26636	0.231167
$520$	6.84162	0.300025
$521$	23.1104	1.01249	0.506243	−	0.862391i	$-0.331034\pi$
$521$	23.1104	1.01249	0.506243	+	0.862391i	$0.331034\pi$
$522$	0.687795	0.0301040
$523$	32.8870	1.43805	0.719024	−	0.694985i	$-0.244589\pi$
$523$	32.8870	1.43805	0.719024	+	0.694985i	$0.244589\pi$
$524$	27.3298	1.19391
$525$	−2.03761	−0.0889287
$526$	−17.5826	−0.766640
$527$	13.4152	0.584374
$528$	−0.659310	−0.0286928
$529$	−2.76902	−0.120392
$530$	1.34302	0.0583370
$531$	−27.8098	−1.20684
$532$	7.17535	0.311091
$533$	−23.5914	−1.02186
$534$	−5.84633	−0.252995
$535$	−14.1172	−0.610342
$536$	9.92251	0.428587
$537$	0.459473	0.0198277
$538$	21.2467	0.916012
$539$	−0.0960001	−0.00413502
$540$	5.88631	0.253306
$541$	18.4746	0.794285	0.397142	−	0.917757i	$-0.370002\pi$
$541$	18.4746	0.794285	0.397142	+	0.917757i	$0.370002\pi$
$542$	−2.31810	−0.0995711
$543$	−14.6492	−0.628656
$544$	−11.7730	−0.504761
$545$	4.71606	0.202014
$546$	4.07482	0.174386
$547$	−10.0218	−0.428502	−0.214251	−	0.976779i	$-0.568731\pi$
$547$	−10.0218	−0.428502	−0.214251	+	0.976779i	$0.568731\pi$
$548$	12.8793	0.550177
$549$	14.4176	0.615328
$550$	0.760826	0.0324417
$551$	0.709533	0.0302271
$552$	−8.95529	−0.381162
$553$	22.9440	0.975679
$554$	8.64082	0.367113
$555$	3.60840	0.153168
$556$	2.78231	0.117996
$557$	−22.1152	−0.937051	−0.468525	−	0.883450i	$-0.655215\pi$
$557$	−22.1152	−0.937051	−0.468525	+	0.883450i	$0.655215\pi$
$558$	12.2718	0.519509
$559$	19.9374	0.843262
$560$	−2.29605	−0.0970260
$561$	−1.53633	−0.0648641
$562$	−8.67648	−0.365995
$563$	19.1575	0.807391	0.403695	−	0.914893i	$-0.367726\pi$
$563$	19.1575	0.807391	0.403695	+	0.914893i	$0.367726\pi$
$564$	3.84328	0.161831
$565$	−3.67765	−0.154720
$566$	−4.54734	−0.191139
$567$	−10.8590	−0.456034
$568$	−12.5516	−0.526652
$569$	17.9795	0.753740	0.376870	−	0.926266i	$-0.377000\pi$
$569$	17.9795	0.753740	0.376870	+	0.926266i	$0.377000\pi$
$570$	−1.10306	−0.0462020
$571$	34.1344	1.42848	0.714241	−	0.699900i	$-0.246772\pi$
$571$	34.1344	1.42848	0.714241	+	0.699900i	$0.246772\pi$
$572$	3.73543	0.156186
$573$	−16.7241	−0.698661
$574$	−18.1905	−0.759256
$575$	−4.49789	−0.187575
$576$	−6.60660	−0.275275
$577$	44.1366	1.83743	0.918715	−	0.394922i	$-0.129228\pi$
$577$	44.1366	1.83743	0.918715	+	0.394922i	$0.129228\pi$
$578$	9.86482	0.410323
$579$	−11.7724	−0.489245
$580$	−0.532001	−0.0220901
$581$	16.7273	0.693967
$582$	11.3931	0.472259
$583$	1.76521	0.0731076
$584$	2.60290	0.107709
$585$	−6.34748	−0.262436
$586$	5.71197	0.235959
$587$	2.85259	0.117739	0.0588696	−	0.998266i	$-0.481250\pi$
$587$	2.85259	0.117739	0.0588696	+	0.998266i	$0.481250\pi$
$588$	−0.104358	−0.00430364
$589$	12.6597	0.521634
$590$	−8.76161	−0.360710
$591$	−4.79716	−0.197329
$592$	4.06607	0.167115
$593$	23.4905	0.964639	0.482320	−	0.875995i	$-0.339794\pi$
$593$	23.4905	0.964639	0.482320	+	0.875995i	$0.339794\pi$
$594$	−3.15131	−0.129300
$595$	−5.35030	−0.219341
$596$	8.39063	0.343694
$597$	10.2392	0.419063
$598$	8.99489	0.367828
$599$	21.4598	0.876824	0.438412	−	0.898774i	$-0.355541\pi$
$599$	21.4598	0.876824	0.438412	+	0.898774i	$0.355541\pi$
$600$	1.99100	0.0812822
$601$	27.2392	1.11111	0.555555	−	0.831480i	$-0.312506\pi$
$601$	27.2392	1.11111	0.555555	+	0.831480i	$0.312506\pi$
$602$	15.3730	0.626557
$603$	−9.20586	−0.374892
$604$	−10.7876	−0.438939
$605$	1.00000	0.0406558
$606$	0.465819	0.0189226
$607$	8.88731	0.360725	0.180362	−	0.983600i	$-0.442273\pi$
$607$	8.88731	0.360725	0.180362	+	0.983600i	$0.442273\pi$
$608$	−11.1100	−0.450569
$609$	−0.762773	−0.0309091
$610$	4.54233	0.183914
$611$	−9.29290	−0.375950
$612$	6.89301	0.278633
$613$	12.6798	0.512134	0.256067	−	0.966659i	$-0.417573\pi$
$613$	12.6798	0.512134	0.256067	+	0.966659i	$0.417573\pi$
$614$	0.537429	0.0216889
$615$	−6.86541	−0.276840
$616$	6.93368	0.279366
$617$	−21.9638	−0.884231	−0.442116	−	0.896958i	$-0.645772\pi$
$617$	−21.9638	−0.884231	−0.442116	+	0.896958i	$0.645772\pi$
$618$	−0.0538945	−0.00216795
$619$	−43.2050	−1.73656	−0.868279	−	0.496077i	$-0.834773\pi$
$619$	−43.2050	−1.73656	−0.868279	+	0.496077i	$0.834773\pi$
$620$	−9.49212	−0.381213
$621$	18.6300	0.747597
$622$	20.0356	0.803356
$623$	−26.7603	−1.07213
$624$	1.73297	0.0693745
$625$	1.00000	0.0400000
$626$	18.5910	0.743046
$627$	−1.44982	−0.0579001
$628$	28.2201	1.12610
$629$	9.47482	0.377786
$630$	−4.89431	−0.194994
$631$	−44.0880	−1.75511	−0.877557	−	0.479472i	$-0.840828\pi$
$631$	−44.0880	−1.75511	−0.877557	+	0.479472i	$0.840828\pi$
$632$	−22.4192	−0.891786
$633$	7.79628	0.309874
$634$	−9.09397	−0.361168
$635$	−9.52739	−0.378083
$636$	1.91889	0.0760888
$637$	0.252333	0.00999779
$638$	0.284813	0.0112758
$639$	11.6450	0.460671
$640$	9.64171	0.381122
$641$	−34.1698	−1.34963	−0.674813	−	0.737989i	$-0.735776\pi$
$641$	−34.1698	−1.34963	−0.674813	+	0.737989i	$0.735776\pi$
$642$	8.21580	0.324252
$643$	10.9362	0.431283	0.215641	−	0.976473i	$-0.430816\pi$
$643$	10.9362	0.431283	0.215641	+	0.976473i	$0.430816\pi$
$644$	−17.0276	−0.670981
$645$	5.80204	0.228455
$646$	−2.89638	−0.113956
$647$	8.84682	0.347805	0.173902	−	0.984763i	$-0.444362\pi$
$647$	8.84682	0.347805	0.173902	+	0.984763i	$0.444362\pi$
$648$	10.6106	0.416823
$649$	−11.5159	−0.452039
$650$	−1.99980	−0.0784388
$651$	−13.6096	−0.533404
$652$	−15.0636	−0.589935
$653$	41.8891	1.63925	0.819625	−	0.572901i	$-0.194182\pi$
$653$	41.8891	1.63925	0.819625	+	0.572901i	$0.194182\pi$
$654$	−2.74460	−0.107322
$655$	−19.2308	−0.751410
$656$	−7.73619	−0.302048
$657$	−2.41490	−0.0942143
$658$	−7.16541	−0.279337
$659$	−23.6496	−0.921258	−0.460629	−	0.887593i	$-0.652376\pi$
$659$	−23.6496	−0.921258	−0.460629	+	0.887593i	$0.652376\pi$
$660$	1.08706	0.0423137
$661$	−27.8552	−1.08344	−0.541722	−	0.840558i	$-0.682227\pi$
$661$	−27.8552	−1.08344	−0.541722	+	0.840558i	$0.682227\pi$
$662$	0.0426178	0.00165639
$663$	4.03820	0.156831
$664$	−16.3447	−0.634297
$665$	−5.04900	−0.195792
$666$	8.66731	0.335851
$667$	−1.68377	−0.0651958
$668$	−7.34111	−0.284036
$669$	−2.33149	−0.0901408
$670$	−2.90035	−0.112050
$671$	5.97026	0.230480
$672$	11.9436	0.460735
$673$	−3.19450	−0.123139	−0.0615694	−	0.998103i	$-0.519611\pi$
$673$	−3.19450	−0.123139	−0.0615694	+	0.998103i	$0.519611\pi$
$674$	3.90588	0.150449
$675$	−4.14195	−0.159424
$676$	8.65643	0.332940
$677$	−5.44665	−0.209332	−0.104666	−	0.994507i	$-0.533377\pi$
$677$	−5.44665	−0.209332	−0.104666	+	0.994507i	$0.533377\pi$
$678$	2.14028	0.0821970
$679$	52.1493	2.00131
$680$	5.22791	0.200481
$681$	−0.0542135	−0.00207747
$682$	5.08172	0.194589
$683$	−35.5438	−1.36004	−0.680022	−	0.733191i	$-0.738030\pi$
$683$	−35.5438	−1.36004	−0.680022	+	0.733191i	$0.738030\pi$
$684$	6.50483	0.248718
$685$	−9.06265	−0.346266
$686$	−13.9924	−0.534233
$687$	6.13539	0.234080
$688$	6.53795	0.249257
$689$	−4.63980	−0.176762
$690$	2.61763	0.0996515
$691$	11.4939	0.437249	0.218624	−	0.975809i	$-0.429843\pi$
$691$	11.4939	0.437249	0.218624	+	0.975809i	$0.429843\pi$
$692$	−9.78439	−0.371947
$693$	−6.43289	−0.244365
$694$	9.55371	0.362654
$695$	−1.95780	−0.0742635
$696$	0.745325	0.0282515
$697$	−18.0270	−0.682821
$698$	13.9123	0.526588
$699$	10.4677	0.395925
$700$	3.78569	0.143086
$701$	−16.2767	−0.614761	−0.307380	−	0.951587i	$-0.599452\pi$
$701$	−16.2767	−0.614761	−0.307380	+	0.951587i	$0.599452\pi$
$702$	8.28309	0.312625
$703$	8.94125	0.337225
$704$	−2.73577	−0.103108
$705$	−2.70435	−0.101852
$706$	23.3063	0.877144
$707$	2.13218	0.0801889
$708$	−12.5185	−0.470473
$709$	23.8201	0.894583	0.447291	−	0.894388i	$-0.352389\pi$
$709$	23.8201	0.894583	0.447291	+	0.894388i	$0.352389\pi$
$710$	3.66882	0.137689
$711$	20.7999	0.780059
$712$	26.1481	0.979941
$713$	−30.0423	−1.12509
$714$	3.11371	0.116528
$715$	−2.62846	−0.0982990
$716$	−0.853658	−0.0319027
$717$	13.8359	0.516710
$718$	−14.2691	−0.532518
$719$	−29.5177	−1.10082	−0.550412	−	0.834893i	$-0.685529\pi$
$719$	−29.5177	−1.10082	−0.550412	+	0.834893i	$0.685529\pi$
$720$	−2.08149	−0.0775726
$721$	−0.246690	−0.00918721
$722$	11.7224	0.436264
$723$	−11.6777	−0.434299
$724$	27.2167	1.01150
$725$	0.374347	0.0139029
$726$	−0.581969	−0.0215989
$727$	39.6576	1.47082	0.735409	−	0.677623i	$-0.236990\pi$
$727$	39.6576	1.47082	0.735409	+	0.677623i	$0.236990\pi$
$728$	−18.2249	−0.675460
$729$	−0.339475	−0.0125731
$730$	−0.760826	−0.0281594
$731$	15.2348	0.563481
$732$	6.49002	0.239878
$733$	−21.5464	−0.795835	−0.397917	−	0.917421i	$-0.630267\pi$
$733$	−21.5464	−0.795835	−0.397917	+	0.917421i	$0.630267\pi$
$734$	−5.82893	−0.215150
$735$	0.0734321	0.00270859
$736$	26.3647	0.971815
$737$	−3.81211	−0.140421
$738$	−16.4906	−0.607028
$739$	−45.9394	−1.68991	−0.844953	−	0.534840i	$-0.820372\pi$
$739$	−45.9394	−1.68991	−0.844953	+	0.534840i	$0.820372\pi$
$740$	−6.70406	−0.246446
$741$	3.81079	0.139993
$742$	−3.57758	−0.131337
$743$	−22.4474	−0.823514	−0.411757	−	0.911294i	$-0.635085\pi$
$743$	−22.4474	−0.823514	−0.411757	+	0.911294i	$0.635085\pi$
$744$	13.2983	0.487540
$745$	−5.90414	−0.216311
$746$	0.141560	0.00518289
$747$	15.1642	0.554829
$748$	2.85436	0.104366
$749$	37.6060	1.37409
$750$	−0.581969	−0.0212505
$751$	−32.0938	−1.17112	−0.585560	−	0.810629i	$-0.699126\pi$
$751$	−32.0938	−1.17112	−0.585560	+	0.810629i	$0.699126\pi$
$752$	−3.04737	−0.111126
$753$	−16.2501	−0.592188
$754$	−0.748620	−0.0272632
$755$	7.59076	0.276256
$756$	−15.6801	−0.570281
$757$	16.7790	0.609841	0.304921	−	0.952378i	$-0.401370\pi$
$757$	16.7790	0.609841	0.304921	+	0.952378i	$0.401370\pi$
$758$	15.2167	0.552695
$759$	3.44051	0.124883
$760$	4.93350	0.178957
$761$	20.8886	0.757211	0.378605	−	0.925558i	$-0.376404\pi$
$761$	20.8886	0.757211	0.378605	+	0.925558i	$0.376404\pi$
$762$	5.54465	0.200861
$763$	−12.5628	−0.454804
$764$	31.0719	1.12414
$765$	−4.85032	−0.175364
$766$	5.19771	0.187801
$767$	30.2692	1.09296
$768$	−9.79645	−0.353499
$769$	40.1368	1.44737	0.723685	−	0.690130i	$-0.242447\pi$
$769$	40.1368	1.44737	0.723685	+	0.690130i	$0.242447\pi$
$770$	−2.02671	−0.0730377
$771$	1.19449	0.0430183
$772$	21.8720	0.787192
$773$	9.42079	0.338842	0.169421	−	0.985544i	$-0.445810\pi$
$773$	9.42079	0.338842	0.169421	+	0.985544i	$0.445810\pi$
$774$	13.9364	0.500934
$775$	6.67921	0.239924
$776$	−50.9564	−1.82923
$777$	−9.61216	−0.344834
$778$	16.6050	0.595317
$779$	−17.0118	−0.609511
$780$	−2.85729	−0.102307
$781$	4.82216	0.172550
$782$	6.87330	0.245789
$783$	−1.55053	−0.0554113
$784$	0.0827460	0.00295521
$785$	−19.8573	−0.708737
$786$	11.1917	0.399196
$787$	−18.7596	−0.668708	−0.334354	−	0.942448i	$-0.608518\pi$
$787$	−18.7596	−0.668708	−0.334354	+	0.942448i	$0.608518\pi$
$788$	8.91266	0.317501
$789$	17.6772	0.629325
$790$	6.55312	0.233150
$791$	9.79665	0.348329
$792$	6.28574	0.223354
$793$	−15.6926	−0.557262
$794$	−20.5940	−0.730853
$795$	−1.35024	−0.0478881
$796$	−19.0235	−0.674269
$797$	−42.6720	−1.51152	−0.755760	−	0.654849i	$-0.772732\pi$
$797$	−42.6720	−1.51152	−0.755760	+	0.654849i	$0.772732\pi$
$798$	2.93836	0.104017
$799$	−7.10101	−0.251216
$800$	−5.86157	−0.207238
$801$	−24.2596	−0.857169
$802$	−8.07660	−0.285195
$803$	−1.00000	−0.0352892
$804$	−4.14398	−0.146147
$805$	11.9816	0.422296
$806$	−13.3571	−0.470484
$807$	−21.3610	−0.751943
$808$	−2.08340	−0.0732939
$809$	5.91667	0.208019	0.104010	−	0.994576i	$-0.466833\pi$
$809$	5.91667	0.208019	0.104010	+	0.994576i	$0.466833\pi$
$810$	−3.10147	−0.108975
$811$	43.1127	1.51389	0.756945	−	0.653478i	$-0.226691\pi$
$811$	43.1127	1.51389	0.756945	+	0.653478i	$0.226691\pi$
$812$	1.41716	0.0497326
$813$	2.33057	0.0817367
$814$	3.58910	0.125798
$815$	10.5996	0.371288
$816$	1.32422	0.0463571
$817$	14.3769	0.502984
$818$	−15.1445	−0.529514
$819$	16.9086	0.590835
$820$	12.7553	0.445434
$821$	41.8274	1.45979	0.729894	−	0.683560i	$-0.239569\pi$
$821$	41.8274	1.45979	0.729894	+	0.683560i	$0.239569\pi$
$822$	5.27418	0.183958
$823$	30.7551	1.07205	0.536027	−	0.844201i	$-0.319924\pi$
$823$	30.7551	1.07205	0.536027	+	0.844201i	$0.319924\pi$
$824$	0.241047	0.00839726
$825$	−0.764917	−0.0266310
$826$	23.3395	0.812084
$827$	30.5701	1.06303	0.531514	−	0.847050i	$-0.321623\pi$
$827$	30.5701	1.06303	0.531514	+	0.847050i	$0.321623\pi$
$828$	−15.4364	−0.536452
$829$	0.351452	0.0122064	0.00610321	−	0.999981i	$-0.498057\pi$
$829$	0.351452	0.0122064	0.00610321	+	0.999981i	$0.498057\pi$
$830$	4.77755	0.165831
$831$	−8.68729	−0.301359
$832$	7.19086	0.249298
$833$	0.192816	0.00668067
$834$	1.13938	0.0394534
$835$	5.16563	0.178764
$836$	2.69362	0.0931608
$837$	−27.6650	−0.956242
$838$	1.47765	0.0510444
$839$	9.01631	0.311278	0.155639	−	0.987814i	$-0.450256\pi$
$839$	9.01631	0.311278	0.155639	+	0.987814i	$0.450256\pi$
$840$	−5.30369	−0.182995
$841$	−28.8599	−0.995168
$842$	−23.4562	−0.808355
$843$	8.72314	0.300441
$844$	−14.4847	−0.498585
$845$	−6.09117	−0.209543
$846$	−6.49582	−0.223331
$847$	−2.66383	−0.0915303
$848$	−1.52150	−0.0522486
$849$	4.57179	0.156904
$850$	−1.52812	−0.0524140
$851$	−21.2182	−0.727350
$852$	5.24196	0.179587
$853$	−39.7510	−1.36105	−0.680525	−	0.732725i	$-0.738248\pi$
$853$	−39.7510	−1.36105	−0.680525	+	0.732725i	$0.738248\pi$
$854$	−12.1000	−0.414054
$855$	−4.57718	−0.156536
$856$	−36.7457	−1.25594
$857$	−51.7034	−1.76615	−0.883077	−	0.469229i	$-0.844532\pi$
$857$	−51.7034	−1.76615	−0.883077	+	0.469229i	$0.844532\pi$
$858$	1.52969	0.0522226
$859$	22.0614	0.752725	0.376362	−	0.926473i	$-0.377175\pi$
$859$	22.0614	0.752725	0.376362	+	0.926473i	$0.377175\pi$
$860$	−10.7796	−0.367583
$861$	18.2883	0.623263
$862$	14.4802	0.493197
$863$	37.7020	1.28339	0.641696	−	0.766959i	$-0.278231\pi$
$863$	37.7020	1.28339	0.641696	+	0.766959i	$0.278231\pi$
$864$	24.2784	0.825967
$865$	6.88487	0.234093
$866$	9.50944	0.323144
$867$	−9.91787	−0.336829
$868$	25.2854	0.858243
$869$	8.61316	0.292182
$870$	−0.217858	−0.00738609
$871$	10.0200	0.339514
$872$	12.2754	0.415698
$873$	47.2760	1.60005
$874$	6.48623	0.219400
$875$	−2.66383	−0.0900540
$876$	−1.08706	−0.0367283
$877$	35.0466	1.18344	0.591720	−	0.806143i	$-0.298449\pi$
$877$	35.0466	1.18344	0.591720	+	0.806143i	$0.298449\pi$
$878$	6.81743	0.230077
$879$	−5.74269	−0.193696
$880$	−0.861937	−0.0290559
$881$	−1.88625	−0.0635495	−0.0317747	−	0.999495i	$-0.510116\pi$
$881$	−1.88625	−0.0635495	−0.0317747	+	0.999495i	$0.510116\pi$
$882$	0.176383	0.00593912
$883$	25.3044	0.851562	0.425781	−	0.904826i	$-0.359999\pi$
$883$	25.3044	0.851562	0.425781	+	0.904826i	$0.359999\pi$
$884$	−7.50259	−0.252339
$885$	8.80873	0.296102
$886$	−6.01202	−0.201978
$887$	34.7772	1.16770	0.583852	−	0.811860i	$-0.301545\pi$
$887$	34.7772	1.16770	0.583852	+	0.811860i	$0.301545\pi$
$888$	9.39228	0.315184
$889$	25.3794	0.851197
$890$	−7.64309	−0.256197
$891$	−4.07645	−0.136566
$892$	4.33169	0.145036
$893$	−6.70112	−0.224244
$894$	3.43603	0.114918
$895$	0.600684	0.0200786
$896$	−25.6839	−0.858038
$897$	−9.04326	−0.301946
$898$	25.0453	0.835773
$899$	2.50034	0.0833911
$900$	3.43192	0.114397
$901$	−3.54542	−0.118115
$902$	−6.82869	−0.227370
$903$	−15.4557	−0.514333
$904$	−9.57255	−0.318378
$905$	−19.1513	−0.636611
$906$	−4.41759	−0.146764
$907$	−23.6700	−0.785951	−0.392975	−	0.919549i	$-0.628554\pi$
$907$	−23.6700	−0.785951	−0.392975	+	0.919549i	$0.628554\pi$
$908$	0.100724	0.00334263
$909$	1.93293	0.0641113
$910$	5.32714	0.176593
$911$	−28.3475	−0.939194	−0.469597	−	0.882881i	$-0.655601\pi$
$911$	−28.3475	−0.939194	−0.469597	+	0.882881i	$0.655601\pi$
$912$	1.24965	0.0413800
$913$	6.27943	0.207819
$914$	−3.96035	−0.130997
$915$	−4.56676	−0.150972
$916$	−11.3990	−0.376632
$917$	51.2277	1.69169
$918$	6.32939	0.208901
$919$	−5.59339	−0.184509	−0.0922544	−	0.995735i	$-0.529407\pi$
$919$	−5.59339	−0.184509	−0.0922544	+	0.995735i	$0.529407\pi$
$920$	−11.7075	−0.385986
$921$	−0.540319	−0.0178041
$922$	14.1300	0.465347
$923$	−12.6749	−0.417199
$924$	−2.89574	−0.0952628
$925$	4.71737	0.155106
$926$	3.12594	0.102725
$927$	−0.223637	−0.00734520
$928$	−2.19426	−0.0720302
$929$	41.6696	1.36714	0.683568	−	0.729886i	$-0.260427\pi$
$929$	41.6696	1.36714	0.683568	+	0.729886i	$0.260427\pi$
$930$	−3.88710	−0.127463
$931$	0.181958	0.00596342
$932$	−19.4480	−0.637040
$933$	−20.1434	−0.659465
$934$	−10.0660	−0.329371
$935$	−2.00850	−0.0656849
$936$	−16.5218	−0.540033
$937$	52.0729	1.70115	0.850574	−	0.525856i	$-0.176255\pi$
$937$	52.0729	1.70115	0.850574	+	0.525856i	$0.176255\pi$
$938$	7.72605	0.252264
$939$	−18.6910	−0.609957
$940$	5.02443	0.163879
$941$	−51.7661	−1.68753	−0.843763	−	0.536716i	$-0.819665\pi$
$941$	−51.7661	−1.68753	−0.843763	+	0.536716i	$0.819665\pi$
$942$	11.5563	0.376526
$943$	40.3701	1.31463
$944$	9.92599	0.323064
$945$	11.0335	0.358919
$946$	5.77101	0.187632
$947$	25.5129	0.829057	0.414528	−	0.910036i	$-0.363947\pi$
$947$	25.5129	0.829057	0.414528	+	0.910036i	$0.363947\pi$
$948$	9.36300	0.304096
$949$	2.62846	0.0853236
$950$	−1.44206	−0.0467867
$951$	9.14287	0.296478
$952$	−13.9263	−0.451353
$953$	5.43875	0.176178	0.0880892	−	0.996113i	$-0.471924\pi$
$953$	5.43875	0.176178	0.0880892	+	0.996113i	$0.471924\pi$
$954$	−3.24326	−0.105004
$955$	−21.8640	−0.707502
$956$	−25.7057	−0.831382
$957$	−0.286344	−0.00925620
$958$	16.9116	0.546389
$959$	24.1414	0.779565
$960$	2.09263	0.0675395
$961$	13.6119	0.439093
$962$	−9.43381	−0.304158
$963$	34.0917	1.09859
$964$	21.6961	0.698784
$965$	−15.3904	−0.495436
$966$	−6.97293	−0.224350
$967$	18.2870	0.588069	0.294035	−	0.955795i	$-0.405002\pi$
$967$	18.2870	0.588069	0.294035	+	0.955795i	$0.405002\pi$
$968$	2.60290	0.0836602
$969$	2.91195	0.0935454
$970$	14.8945	0.478235
$971$	13.8968	0.445970	0.222985	−	0.974822i	$-0.428420\pi$
$971$	13.8968	0.445970	0.222985	+	0.974822i	$0.428420\pi$
$972$	−22.0903	−0.708546
$973$	5.21525	0.167193
$974$	7.44536	0.238565
$975$	2.01056	0.0643894
$976$	−5.14599	−0.164719
$977$	51.4250	1.64523	0.822616	−	0.568597i	$-0.192514\pi$
$977$	51.4250	1.64523	0.822616	+	0.568597i	$0.192514\pi$
$978$	−6.16865	−0.197252
$979$	−10.0458	−0.321064
$980$	−0.136430	−0.00435809
$981$	−11.3888	−0.363617
$982$	−13.9330	−0.444621
$983$	43.4514	1.38588	0.692942	−	0.720993i	$-0.256314\pi$
$983$	43.4514	1.38588	0.692942	+	0.720993i	$0.256314\pi$
$984$	−17.8699	−0.569673
$985$	−6.27147	−0.199826
$986$	−0.572046	−0.0182177
$987$	7.20395	0.229304
$988$	−7.08009	−0.225248
$989$	−34.1173	−1.08487
$990$	−1.83732	−0.0583939
$991$	−29.0281	−0.922108	−0.461054	−	0.887372i	$-0.652529\pi$
$991$	−29.0281	−0.922108	−0.461054	+	0.887372i	$0.652529\pi$
$992$	−39.1507	−1.24304
$993$	−0.0428470	−0.00135971
$994$	−9.77313	−0.309985
$995$	13.3860	0.424366
$996$	6.82610	0.216293
$997$	27.4986	0.870890	0.435445	−	0.900215i	$-0.356591\pi$
$997$	27.4986	0.870890	0.435445	+	0.900215i	$0.356591\pi$
$998$	3.80066	0.120308
$999$	−19.5391	−0.618190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.c.1.10	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.c.1.10	✓	23	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)

\(f(q)\)	\(=\)	\(q-0.760826 q^{2} +0.764917 q^{3} -1.42114 q^{4} +1.00000 q^{5} -0.581969 q^{6} -2.66383 q^{7} +2.60290 q^{8} -2.41490 q^{9} +O(q^{10})\)	\(q-0.760826 q^{2} +0.764917 q^{3} -1.42114 q^{4} +1.00000 q^{5} -0.581969 q^{6} -2.66383 q^{7} +2.60290 q^{8} -2.41490 q^{9} -0.760826 q^{10} -1.00000 q^{11} -1.08706 q^{12} +2.62846 q^{13} +2.02671 q^{14} +0.764917 q^{15} +0.861937 q^{16} +2.00850 q^{17} +1.83732 q^{18} +1.89539 q^{19} -1.42114 q^{20} -2.03761 q^{21} +0.760826 q^{22} -4.49789 q^{23} +1.99100 q^{24} +1.00000 q^{25} -1.99980 q^{26} -4.14195 q^{27} +3.78569 q^{28} +0.374347 q^{29} -0.581969 q^{30} +6.67921 q^{31} -5.86157 q^{32} -0.764917 q^{33} -1.52812 q^{34} -2.66383 q^{35} +3.43192 q^{36} +4.71737 q^{37} -1.44206 q^{38} +2.01056 q^{39} +2.60290 q^{40} -8.97536 q^{41} +1.55027 q^{42} +7.58519 q^{43} +1.42114 q^{44} -2.41490 q^{45} +3.42211 q^{46} -3.53549 q^{47} +0.659310 q^{48} +0.0960001 q^{49} -0.760826 q^{50} +1.53633 q^{51} -3.73543 q^{52} -1.76521 q^{53} +3.15131 q^{54} -1.00000 q^{55} -6.93368 q^{56} +1.44982 q^{57} -0.284813 q^{58} +11.5159 q^{59} -1.08706 q^{60} -5.97026 q^{61} -5.08172 q^{62} +6.43289 q^{63} +2.73577 q^{64} +2.62846 q^{65} +0.581969 q^{66} +3.81211 q^{67} -2.85436 q^{68} -3.44051 q^{69} +2.02671 q^{70} -4.82216 q^{71} -6.28574 q^{72} +1.00000 q^{73} -3.58910 q^{74} +0.764917 q^{75} -2.69362 q^{76} +2.66383 q^{77} -1.52969 q^{78} -8.61316 q^{79} +0.861937 q^{80} +4.07645 q^{81} +6.82869 q^{82} -6.27943 q^{83} +2.89574 q^{84} +2.00850 q^{85} -5.77101 q^{86} +0.286344 q^{87} -2.60290 q^{88} +10.0458 q^{89} +1.83732 q^{90} -7.00179 q^{91} +6.39214 q^{92} +5.10905 q^{93} +2.68989 q^{94} +1.89539 q^{95} -4.48362 q^{96} -19.5768 q^{97} -0.0730394 q^{98} +2.41490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more