LMFDB - Embedded newform 4015.2.a.i.1.11

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:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-1.50911 q^{2} -0.619433 q^{3} +0.277405 q^{4} +1.00000 q^{5} +0.934791 q^{6} +1.62740 q^{7} +2.59958 q^{8} -2.61630 q^{9} +O(q^{10})$	\(q-1.50911 q^{2} -0.619433 q^{3} +0.277405 q^{4} +1.00000 q^{5} +0.934791 q^{6} +1.62740 q^{7} +2.59958 q^{8} -2.61630 q^{9} -1.50911 q^{10} -1.00000 q^{11} -0.171834 q^{12} -6.84915 q^{13} -2.45593 q^{14} -0.619433 q^{15} -4.47786 q^{16} -4.30366 q^{17} +3.94828 q^{18} -4.24491 q^{19} +0.277405 q^{20} -1.00807 q^{21} +1.50911 q^{22} -3.05486 q^{23} -1.61027 q^{24} +1.00000 q^{25} +10.3361 q^{26} +3.47892 q^{27} +0.451450 q^{28} -8.79831 q^{29} +0.934791 q^{30} +3.50116 q^{31} +1.55841 q^{32} +0.619433 q^{33} +6.49469 q^{34} +1.62740 q^{35} -0.725776 q^{36} -8.29415 q^{37} +6.40602 q^{38} +4.24259 q^{39} +2.59958 q^{40} -5.21751 q^{41} +1.52128 q^{42} +6.43753 q^{43} -0.277405 q^{44} -2.61630 q^{45} +4.61012 q^{46} +8.33576 q^{47} +2.77373 q^{48} -4.35155 q^{49} -1.50911 q^{50} +2.66583 q^{51} -1.89999 q^{52} -2.00208 q^{53} -5.25007 q^{54} -1.00000 q^{55} +4.23057 q^{56} +2.62944 q^{57} +13.2776 q^{58} +3.57848 q^{59} -0.171834 q^{60} +10.3770 q^{61} -5.28362 q^{62} -4.25778 q^{63} +6.60391 q^{64} -6.84915 q^{65} -0.934791 q^{66} +7.09816 q^{67} -1.19386 q^{68} +1.89228 q^{69} -2.45593 q^{70} +10.5769 q^{71} -6.80129 q^{72} -1.00000 q^{73} +12.5168 q^{74} -0.619433 q^{75} -1.17756 q^{76} -1.62740 q^{77} -6.40252 q^{78} -3.17482 q^{79} -4.47786 q^{80} +5.69395 q^{81} +7.87379 q^{82} +11.4993 q^{83} -0.279643 q^{84} -4.30366 q^{85} -9.71493 q^{86} +5.44997 q^{87} -2.59958 q^{88} +14.2786 q^{89} +3.94828 q^{90} -11.1463 q^{91} -0.847435 q^{92} -2.16873 q^{93} -12.5796 q^{94} -4.24491 q^{95} -0.965328 q^{96} -6.31534 q^{97} +6.56696 q^{98} +2.61630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) $ 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * 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 $

$2$	−1.50911	−1.06710	−0.533550	−	0.845769i	$-0.679142\pi$
$2$	−1.50911	−1.06710	−0.533550	+	0.845769i	$0.679142\pi$
$3$	−0.619433	−0.357630	−0.178815	−	0.983883i	$-0.557226\pi$
$3$	−0.619433	−0.357630	−0.178815	+	0.983883i	$0.557226\pi$
$4$	0.277405	0.138703
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.934791	0.381627
$7$	1.62740	0.615101	0.307551	−	0.951532i	$-0.400491\pi$
$7$	1.62740	0.615101	0.307551	+	0.951532i	$0.400491\pi$
$8$	2.59958	0.919091
$9$	−2.61630	−0.872101
$10$	−1.50911	−0.477222
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.171834	−0.0496042
$13$	−6.84915	−1.89961	−0.949806	−	0.312839i	$-0.898720\pi$
$13$	−6.84915	−1.89961	−0.949806	+	0.312839i	$0.898720\pi$
$14$	−2.45593	−0.656374
$15$	−0.619433	−0.159937
$16$	−4.47786	−1.11946
$17$	−4.30366	−1.04379	−0.521896	−	0.853009i	$-0.674775\pi$
$17$	−4.30366	−1.04379	−0.521896	+	0.853009i	$0.674775\pi$
$18$	3.94828	0.930619
$19$	−4.24491	−0.973849	−0.486925	−	0.873444i	$-0.661881\pi$
$19$	−4.24491	−0.973849	−0.486925	+	0.873444i	$0.661881\pi$
$20$	0.277405	0.0620297
$21$	−1.00807	−0.219979
$22$	1.50911	0.321743
$23$	−3.05486	−0.636983	−0.318492	−	0.947926i	$-0.603176\pi$
$23$	−3.05486	−0.636983	−0.318492	+	0.947926i	$0.603176\pi$
$24$	−1.61027	−0.328694
$25$	1.00000	0.200000
$26$	10.3361	2.02708
$27$	3.47892	0.669519
$28$	0.451450	0.0853161
$29$	−8.79831	−1.63381	−0.816903	−	0.576776i	$-0.804311\pi$
$29$	−8.79831	−1.63381	−0.816903	+	0.576776i	$0.804311\pi$
$30$	0.934791	0.170669
$31$	3.50116	0.628827	0.314413	−	0.949286i	$-0.398192\pi$
$31$	3.50116	0.628827	0.314413	+	0.949286i	$0.398192\pi$
$32$	1.55841	0.275490
$33$	0.619433	0.107829
$34$	6.49469	1.11383
$35$	1.62740	0.275082
$36$	−0.725776	−0.120963
$37$	−8.29415	−1.36355	−0.681775	−	0.731562i	$-0.738792\pi$
$37$	−8.29415	−1.36355	−0.681775	+	0.731562i	$0.738792\pi$
$38$	6.40602	1.03919
$39$	4.24259	0.679358
$40$	2.59958	0.411030
$41$	−5.21751	−0.814839	−0.407419	−	0.913241i	$-0.633571\pi$
$41$	−5.21751	−0.814839	−0.407419	+	0.913241i	$0.633571\pi$
$42$	1.52128	0.234739
$43$	6.43753	0.981715	0.490857	−	0.871240i	$-0.336684\pi$
$43$	6.43753	0.981715	0.490857	+	0.871240i	$0.336684\pi$
$44$	−0.277405	−0.0418204
$45$	−2.61630	−0.390015
$46$	4.61012	0.679725
$47$	8.33576	1.21590	0.607948	−	0.793977i	$-0.291993\pi$
$47$	8.33576	1.21590	0.607948	+	0.793977i	$0.291993\pi$
$48$	2.77373	0.400354
$49$	−4.35155	−0.621651
$50$	−1.50911	−0.213420
$51$	2.66583	0.373291
$52$	−1.89999	−0.263481
$53$	−2.00208	−0.275007	−0.137503	−	0.990501i	$-0.543908\pi$
$53$	−2.00208	−0.275007	−0.137503	+	0.990501i	$0.543908\pi$
$54$	−5.25007	−0.714444
$55$	−1.00000	−0.134840
$56$	4.23057	0.565334
$57$	2.62944	0.348278
$58$	13.2776	1.74343
$59$	3.57848	0.465879	0.232939	−	0.972491i	$-0.425166\pi$
$59$	3.57848	0.465879	0.232939	+	0.972491i	$0.425166\pi$
$60$	−0.171834	−0.0221837
$61$	10.3770	1.32864	0.664321	−	0.747447i	$-0.268721\pi$
$61$	10.3770	1.32864	0.664321	+	0.747447i	$0.268721\pi$
$62$	−5.28362	−0.671021
$63$	−4.25778	−0.536430
$64$	6.60391	0.825489
$65$	−6.84915	−0.849532
$66$	−0.934791	−0.115065
$67$	7.09816	0.867179	0.433589	−	0.901111i	$-0.357247\pi$
$67$	7.09816	0.867179	0.433589	+	0.901111i	$0.357247\pi$
$68$	−1.19386	−0.144777
$69$	1.89228	0.227804
$70$	−2.45593	−0.293540
$71$	10.5769	1.25524	0.627621	−	0.778519i	$-0.284029\pi$
$71$	10.5769	1.25524	0.627621	+	0.778519i	$0.284029\pi$
$72$	−6.80129	−0.801540
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	12.5168	1.45504
$75$	−0.619433	−0.0715260
$76$	−1.17756	−0.135075
$77$	−1.62740	−0.185460
$78$	−6.40252	−0.724943
$79$	−3.17482	−0.357196	−0.178598	−	0.983922i	$-0.557156\pi$
$79$	−3.17482	−0.357196	−0.178598	+	0.983922i	$0.557156\pi$
$80$	−4.47786	−0.500640
$81$	5.69395	0.632661
$82$	7.87379	0.869514
$83$	11.4993	1.26221	0.631107	−	0.775696i	$-0.282601\pi$
$83$	11.4993	1.26221	0.631107	+	0.775696i	$0.282601\pi$
$84$	−0.279643	−0.0305116
$85$	−4.30366	−0.466798
$86$	−9.71493	−1.04759
$87$	5.44997	0.584298
$88$	−2.59958	−0.277116
$89$	14.2786	1.51353	0.756764	−	0.653689i	$-0.226779\pi$
$89$	14.2786	1.51353	0.756764	+	0.653689i	$0.226779\pi$
$90$	3.94828	0.416185
$91$	−11.1463	−1.16845
$92$	−0.847435	−0.0883512
$93$	−2.16873	−0.224887
$94$	−12.5796	−1.29748
$95$	−4.24491	−0.435519
$96$	−0.965328	−0.0985234
$97$	−6.31534	−0.641226	−0.320613	−	0.947210i	$-0.603889\pi$
$97$	−6.31534	−0.641226	−0.320613	+	0.947210i	$0.603889\pi$
$98$	6.56696	0.663363
$99$	2.61630	0.262948
$100$	0.277405	0.0277405
$101$	−9.24858	−0.920268	−0.460134	−	0.887849i	$-0.652199\pi$
$101$	−9.24858	−0.920268	−0.460134	+	0.887849i	$0.652199\pi$
$102$	−4.02303	−0.398339
$103$	15.1379	1.49158	0.745791	−	0.666180i	$-0.232072\pi$
$103$	15.1379	1.49158	0.745791	+	0.666180i	$0.232072\pi$
$104$	−17.8049	−1.74592
$105$	−1.00807	−0.0983774
$106$	3.02135	0.293460
$107$	5.23545	0.506130	0.253065	−	0.967449i	$-0.418561\pi$
$107$	5.23545	0.506130	0.253065	+	0.967449i	$0.418561\pi$
$108$	0.965071	0.0928640
$109$	2.66138	0.254914	0.127457	−	0.991844i	$-0.459319\pi$
$109$	2.66138	0.254914	0.127457	+	0.991844i	$0.459319\pi$
$110$	1.50911	0.143888
$111$	5.13767	0.487646
$112$	−7.28728	−0.688584
$113$	−20.2896	−1.90869	−0.954344	−	0.298711i	$-0.903443\pi$
$113$	−20.2896	−1.90869	−0.954344	+	0.298711i	$0.903443\pi$
$114$	−3.96810	−0.371647
$115$	−3.05486	−0.284868
$116$	−2.44070	−0.226613
$117$	17.9194	1.65665
$118$	−5.40031	−0.497139
$119$	−7.00380	−0.642038
$120$	−1.61027	−0.146997
$121$	1.00000	0.0909091
$122$	−15.6601	−1.41779
$123$	3.23190	0.291411
$124$	0.971239	0.0872198
$125$	1.00000	0.0894427
$126$	6.42545	0.572425
$127$	5.58445	0.495540	0.247770	−	0.968819i	$-0.420302\pi$
$127$	5.58445	0.495540	0.247770	+	0.968819i	$0.420302\pi$
$128$	−13.0828	−1.15637
$129$	−3.98762	−0.351091
$130$	10.3361	0.906536
$131$	−3.64219	−0.318220	−0.159110	−	0.987261i	$-0.550862\pi$
$131$	−3.64219	−0.318220	−0.159110	+	0.987261i	$0.550862\pi$
$132$	0.171834	0.0149562
$133$	−6.90819	−0.599016
$134$	−10.7119	−0.925367
$135$	3.47892	0.299418
$136$	−11.1877	−0.959339
$137$	−11.4871	−0.981408	−0.490704	−	0.871326i	$-0.663260\pi$
$137$	−11.4871	−0.981408	−0.490704	+	0.871326i	$0.663260\pi$
$138$	−2.85566	−0.243090
$139$	−17.4387	−1.47913	−0.739565	−	0.673085i	$-0.764969\pi$
$139$	−17.4387	−1.47913	−0.739565	+	0.673085i	$0.764969\pi$
$140$	0.451450	0.0381545
$141$	−5.16345	−0.434841
$142$	−15.9616	−1.33947
$143$	6.84915	0.572755
$144$	11.7154	0.976286
$145$	−8.79831	−0.730660
$146$	1.50911	0.124895
$147$	2.69550	0.222321
$148$	−2.30084	−0.189128
$149$	14.8460	1.21623	0.608117	−	0.793848i	$-0.291925\pi$
$149$	14.8460	1.21623	0.608117	+	0.793848i	$0.291925\pi$
$150$	0.934791	0.0763254
$151$	8.53691	0.694724	0.347362	−	0.937731i	$-0.387078\pi$
$151$	8.53691	0.694724	0.347362	+	0.937731i	$0.387078\pi$
$152$	−11.0350	−0.895056
$153$	11.2597	0.910292
$154$	2.45593	0.197904
$155$	3.50116	0.281220
$156$	1.17692	0.0942287
$157$	−9.76299	−0.779171	−0.389586	−	0.920990i	$-0.627382\pi$
$157$	−9.76299	−0.779171	−0.389586	+	0.920990i	$0.627382\pi$
$158$	4.79115	0.381163
$159$	1.24015	0.0983506
$160$	1.55841	0.123203
$161$	−4.97150	−0.391809
$162$	−8.59278	−0.675112
$163$	−12.9845	−1.01702	−0.508510	−	0.861056i	$-0.669804\pi$
$163$	−12.9845	−1.01702	−0.508510	+	0.861056i	$0.669804\pi$
$164$	−1.44736	−0.113020
$165$	0.619433	0.0482228
$166$	−17.3537	−1.34691
$167$	−13.0140	−1.00705	−0.503527	−	0.863979i	$-0.667965\pi$
$167$	−13.0140	−1.00705	−0.503527	+	0.863979i	$0.667965\pi$
$168$	−2.62056	−0.202180
$169$	33.9108	2.60853
$170$	6.49469	0.498120
$171$	11.1060	0.849295
$172$	1.78580	0.136166
$173$	−10.4466	−0.794243	−0.397122	−	0.917766i	$-0.629991\pi$
$173$	−10.4466	−0.794243	−0.397122	+	0.917766i	$0.629991\pi$
$174$	−8.22458	−0.623504
$175$	1.62740	0.123020
$176$	4.47786	0.337531
$177$	−2.21663	−0.166612
$178$	−21.5479	−1.61509
$179$	0.254483	0.0190209	0.00951047	−	0.999955i	$-0.496973\pi$
$179$	0.254483	0.0190209	0.00951047	+	0.999955i	$0.496973\pi$
$180$	−0.725776	−0.0540961
$181$	18.4164	1.36888	0.684441	−	0.729068i	$-0.260046\pi$
$181$	18.4164	1.36888	0.684441	+	0.729068i	$0.260046\pi$
$182$	16.8210	1.24686
$183$	−6.42788	−0.475162
$184$	−7.94137	−0.585445
$185$	−8.29415	−0.609798
$186$	3.27285	0.239977
$187$	4.30366	0.314715
$188$	2.31238	0.168648
$189$	5.66162	0.411822
$190$	6.40602	0.464742
$191$	−4.83502	−0.349850	−0.174925	−	0.984582i	$-0.555968\pi$
$191$	−4.83502	−0.349850	−0.174925	+	0.984582i	$0.555968\pi$
$192$	−4.09068	−0.295220
$193$	−18.0237	−1.29737	−0.648686	−	0.761056i	$-0.724681\pi$
$193$	−18.0237	−1.29737	−0.648686	+	0.761056i	$0.724681\pi$
$194$	9.53053	0.684252
$195$	4.24259	0.303818
$196$	−1.20714	−0.0862245
$197$	22.8292	1.62651	0.813255	−	0.581907i	$-0.197693\pi$
$197$	22.8292	1.62651	0.813255	+	0.581907i	$0.197693\pi$
$198$	−3.94828	−0.280592
$199$	7.39339	0.524104	0.262052	−	0.965054i	$-0.415601\pi$
$199$	7.39339	0.524104	0.262052	+	0.965054i	$0.415601\pi$
$200$	2.59958	0.183818
$201$	−4.39684	−0.310129
$202$	13.9571	0.982018
$203$	−14.3184	−1.00496
$204$	0.739516	0.0517764
$205$	−5.21751	−0.364407
$206$	−22.8447	−1.59167
$207$	7.99245	0.555514
$208$	30.6695	2.12655
$209$	4.24491	0.293627
$210$	1.52128	0.104979
$211$	8.11729	0.558818	0.279409	−	0.960172i	$-0.409862\pi$
$211$	8.11729	0.558818	0.279409	+	0.960172i	$0.409862\pi$
$212$	−0.555387	−0.0381441
$213$	−6.55166	−0.448912
$214$	−7.90086	−0.540091
$215$	6.43753	0.439036
$216$	9.04374	0.615349
$217$	5.69780	0.386792
$218$	−4.01630	−0.272018
$219$	0.619433	0.0418574
$220$	−0.277405	−0.0187026
$221$	29.4764	1.98280
$222$	−7.75330	−0.520367
$223$	−0.0308224	−0.00206402	−0.00103201	−	0.999999i	$-0.500328\pi$
$223$	−0.0308224	−0.00206402	−0.00103201	+	0.999999i	$0.500328\pi$
$224$	2.53616	0.169454
$225$	−2.61630	−0.174420
$226$	30.6192	2.03676
$227$	−11.2333	−0.745580	−0.372790	−	0.927916i	$-0.621599\pi$
$227$	−11.2333	−0.745580	−0.372790	+	0.927916i	$0.621599\pi$
$228$	0.729420	0.0483070
$229$	−4.84599	−0.320232	−0.160116	−	0.987098i	$-0.551187\pi$
$229$	−4.84599	−0.320232	−0.160116	+	0.987098i	$0.551187\pi$
$230$	4.61012	0.303982
$231$	1.00807	0.0663260
$232$	−22.8719	−1.50161
$233$	3.67876	0.241003	0.120502	−	0.992713i	$-0.461550\pi$
$233$	3.67876	0.241003	0.120502	+	0.992713i	$0.461550\pi$
$234$	−27.0424	−1.76781
$235$	8.33576	0.543765
$236$	0.992689	0.0646185
$237$	1.96659	0.127744
$238$	10.5695	0.685118
$239$	3.09594	0.200260	0.100130	−	0.994974i	$-0.468074\pi$
$239$	3.09594	0.200260	0.100130	+	0.994974i	$0.468074\pi$
$240$	2.77373	0.179044
$241$	10.4743	0.674709	0.337354	−	0.941378i	$-0.390468\pi$
$241$	10.4743	0.674709	0.337354	+	0.941378i	$0.390468\pi$
$242$	−1.50911	−0.0970091
$243$	−13.9638	−0.895778
$244$	2.87864	0.184286
$245$	−4.35155	−0.278011
$246$	−4.87729	−0.310964
$247$	29.0740	1.84994
$248$	9.10154	0.577949
$249$	−7.12305	−0.451405
$250$	−1.50911	−0.0954443
$251$	13.4657	0.849946	0.424973	−	0.905206i	$-0.360284\pi$
$251$	13.4657	0.849946	0.424973	+	0.905206i	$0.360284\pi$
$252$	−1.18113	−0.0744042
$253$	3.05486	0.192058
$254$	−8.42754	−0.528791
$255$	2.66583	0.166941
$256$	6.53556	0.408472
$257$	30.3686	1.89434	0.947169	−	0.320734i	$-0.103930\pi$
$257$	30.3686	1.89434	0.947169	+	0.320734i	$0.103930\pi$
$258$	6.01775	0.374649
$259$	−13.4979	−0.838721
$260$	−1.89999	−0.117832
$261$	23.0190	1.42484
$262$	5.49646	0.339573
$263$	17.7974	1.09743	0.548717	−	0.836008i	$-0.315116\pi$
$263$	17.7974	1.09743	0.548717	+	0.836008i	$0.315116\pi$
$264$	1.61027	0.0991051
$265$	−2.00208	−0.122987
$266$	10.4252	0.639210
$267$	−8.84463	−0.541283
$268$	1.96907	0.120280
$269$	3.20768	0.195576	0.0977878	−	0.995207i	$-0.468823\pi$
$269$	3.20768	0.195576	0.0977878	+	0.995207i	$0.468823\pi$
$270$	−5.25007	−0.319509
$271$	−14.5863	−0.886056	−0.443028	−	0.896508i	$-0.646096\pi$
$271$	−14.5863	−0.886056	−0.443028	+	0.896508i	$0.646096\pi$
$272$	19.2712	1.16849
$273$	6.90441	0.417874
$274$	17.3352	1.04726
$275$	−1.00000	−0.0603023
$276$	0.524929	0.0315970
$277$	−16.9189	−1.01656	−0.508279	−	0.861192i	$-0.669718\pi$
$277$	−16.9189	−1.01656	−0.508279	+	0.861192i	$0.669718\pi$
$278$	26.3168	1.57838
$279$	−9.16009	−0.548400
$280$	4.23057	0.252825
$281$	30.7818	1.83629	0.918144	−	0.396247i	$-0.129688\pi$
$281$	30.7818	1.83629	0.918144	+	0.396247i	$0.129688\pi$
$282$	7.79220	0.464018
$283$	22.8537	1.35851	0.679256	−	0.733901i	$-0.262303\pi$
$283$	22.8537	1.35851	0.679256	+	0.733901i	$0.262303\pi$
$284$	2.93408	0.174105
$285$	2.62944	0.155754
$286$	−10.3361	−0.611186
$287$	−8.49100	−0.501208
$288$	−4.07726	−0.240255
$289$	1.52153	0.0895018
$290$	13.2776	0.779687
$291$	3.91193	0.229322
$292$	−0.277405	−0.0162339
$293$	−22.3629	−1.30645	−0.653227	−	0.757162i	$-0.726585\pi$
$293$	−22.3629	−1.30645	−0.653227	+	0.757162i	$0.726585\pi$
$294$	−4.06780	−0.237239
$295$	3.57848	0.208347
$296$	−21.5613	−1.25323
$297$	−3.47892	−0.201868
$298$	−22.4042	−1.29784
$299$	20.9232	1.21002
$300$	−0.171834	−0.00992084
$301$	10.4765	0.603854
$302$	−12.8831	−0.741340
$303$	5.72888	0.329116
$304$	19.0081	1.09019
$305$	10.3770	0.594187
$306$	−16.9921	−0.971373
$307$	14.9723	0.854517	0.427258	−	0.904130i	$-0.359479\pi$
$307$	14.9723	0.854517	0.427258	+	0.904130i	$0.359479\pi$
$308$	−0.451450	−0.0257238
$309$	−9.37692	−0.533434
$310$	−5.28362	−0.300090
$311$	−16.4916	−0.935152	−0.467576	−	0.883953i	$-0.654872\pi$
$311$	−16.4916	−0.935152	−0.467576	+	0.883953i	$0.654872\pi$
$312$	11.0290	0.624392
$313$	−11.6949	−0.661037	−0.330519	−	0.943799i	$-0.607224\pi$
$313$	−11.6949	−0.661037	−0.330519	+	0.943799i	$0.607224\pi$
$314$	14.7334	0.831454
$315$	−4.25778	−0.239899
$316$	−0.880712	−0.0495439
$317$	−15.5109	−0.871178	−0.435589	−	0.900146i	$-0.643460\pi$
$317$	−15.5109	−0.871178	−0.435589	+	0.900146i	$0.643460\pi$
$318$	−1.87153	−0.104950
$319$	8.79831	0.492611
$320$	6.60391	0.369170
$321$	−3.24301	−0.181007
$322$	7.50253	0.418100
$323$	18.2687	1.01650
$324$	1.57953	0.0877516
$325$	−6.84915	−0.379922
$326$	19.5949	1.08526
$327$	−1.64855	−0.0911647
$328$	−13.5633	−0.748911
$329$	13.5657	0.747899
$330$	−0.934791	−0.0514586
$331$	29.1246	1.60083	0.800416	−	0.599445i	$-0.204612\pi$
$331$	29.1246	1.60083	0.800416	+	0.599445i	$0.204612\pi$
$332$	3.18997	0.175072
$333$	21.7000	1.18915
$334$	19.6395	1.07463
$335$	7.09816	0.387814
$336$	4.51399	0.246258
$337$	33.4809	1.82382	0.911912	−	0.410387i	$-0.134606\pi$
$337$	33.4809	1.82382	0.911912	+	0.410387i	$0.134606\pi$
$338$	−51.1751	−2.78356
$339$	12.5681	0.682604
$340$	−1.19386	−0.0647461
$341$	−3.50116	−0.189598
$342$	−16.7601	−0.906282
$343$	−18.4736	−0.997479
$344$	16.7349	0.902285
$345$	1.89228	0.101877
$346$	15.7651	0.847537
$347$	−0.170878	−0.00917323	−0.00458661	−	0.999989i	$-0.501460\pi$
$347$	−0.170878	−0.00917323	−0.00458661	+	0.999989i	$0.501460\pi$
$348$	1.51185	0.0810436
$349$	−23.3852	−1.25178	−0.625892	−	0.779910i	$-0.715265\pi$
$349$	−23.3852	−1.25178	−0.625892	+	0.779910i	$0.715265\pi$
$350$	−2.45593	−0.131275
$351$	−23.8277	−1.27183
$352$	−1.55841	−0.0830633
$353$	28.5100	1.51744	0.758718	−	0.651420i	$-0.225826\pi$
$353$	28.5100	1.51744	0.758718	+	0.651420i	$0.225826\pi$
$354$	3.34513	0.177792
$355$	10.5769	0.561362
$356$	3.96095	0.209930
$357$	4.33839	0.229612
$358$	−0.384042	−0.0202972
$359$	18.9341	0.999305	0.499652	−	0.866226i	$-0.333461\pi$
$359$	18.9341	0.999305	0.499652	+	0.866226i	$0.333461\pi$
$360$	−6.80129	−0.358459
$361$	−0.980740	−0.0516179
$362$	−27.7924	−1.46073
$363$	−0.619433	−0.0325118
$364$	−3.09205	−0.162067
$365$	−1.00000	−0.0523424
$366$	9.70036	0.507046
$367$	−33.9757	−1.77352	−0.886758	−	0.462233i	$-0.847048\pi$
$367$	−33.9757	−1.77352	−0.886758	+	0.462233i	$0.847048\pi$
$368$	13.6792	0.713080
$369$	13.6506	0.710622
$370$	12.5168	0.650715
$371$	−3.25819	−0.169157
$372$	−0.601618	−0.0311924
$373$	−10.2813	−0.532347	−0.266174	−	0.963925i	$-0.585759\pi$
$373$	−10.2813	−0.532347	−0.266174	+	0.963925i	$0.585759\pi$
$374$	−6.49469	−0.335833
$375$	−0.619433	−0.0319874
$376$	21.6695	1.11752
$377$	60.2609	3.10360
$378$	−8.54399	−0.439455
$379$	−10.1861	−0.523224	−0.261612	−	0.965173i	$-0.584254\pi$
$379$	−10.1861	−0.523224	−0.261612	+	0.965173i	$0.584254\pi$
$380$	−1.17756	−0.0604075
$381$	−3.45920	−0.177220
$382$	7.29656	0.373324
$383$	−5.07312	−0.259224	−0.129612	−	0.991565i	$-0.541373\pi$
$383$	−5.07312	−0.259224	−0.129612	+	0.991565i	$0.541373\pi$
$384$	8.10394	0.413552
$385$	−1.62740	−0.0829402
$386$	27.1996	1.38443
$387$	−16.8425	−0.856154
$388$	−1.75191	−0.0889397
$389$	−18.2803	−0.926848	−0.463424	−	0.886137i	$-0.653379\pi$
$389$	−18.2803	−0.926848	−0.463424	+	0.886137i	$0.653379\pi$
$390$	−6.40252	−0.324204
$391$	13.1471	0.664878
$392$	−11.3122	−0.571353
$393$	2.25610	0.113805
$394$	−34.4517	−1.73565
$395$	−3.17482	−0.159743
$396$	0.725776	0.0364716
$397$	−33.4301	−1.67781	−0.838905	−	0.544277i	$-0.816804\pi$
$397$	−33.4301	−1.67781	−0.838905	+	0.544277i	$0.816804\pi$
$398$	−11.1574	−0.559271
$399$	4.27916	0.214226
$400$	−4.47786	−0.223893
$401$	32.1748	1.60673	0.803366	−	0.595486i	$-0.203040\pi$
$401$	32.1748	1.60673	0.803366	+	0.595486i	$0.203040\pi$
$402$	6.63530	0.330939
$403$	−23.9800	−1.19453
$404$	−2.56560	−0.127644
$405$	5.69395	0.282934
$406$	21.6080	1.07239
$407$	8.29415	0.411126
$408$	6.93005	0.343088
$409$	−25.8826	−1.27981	−0.639906	−	0.768453i	$-0.721027\pi$
$409$	−25.8826	−1.27981	−0.639906	+	0.768453i	$0.721027\pi$
$410$	7.87379	0.388859
$411$	7.11548	0.350981
$412$	4.19933	0.206886
$413$	5.82364	0.286562
$414$	−12.0615	−0.592789
$415$	11.4993	0.564479
$416$	−10.6737	−0.523324
$417$	10.8021	0.528981
$418$	−6.40602	−0.313329
$419$	26.3918	1.28932	0.644662	−	0.764468i	$-0.276998\pi$
$419$	26.3918	1.28932	0.644662	+	0.764468i	$0.276998\pi$
$420$	−0.279643	−0.0136452
$421$	9.92001	0.483471	0.241736	−	0.970342i	$-0.422283\pi$
$421$	9.92001	0.483471	0.241736	+	0.970342i	$0.422283\pi$
$422$	−12.2499	−0.596314
$423$	−21.8089	−1.06038
$424$	−5.20457	−0.252756
$425$	−4.30366	−0.208758
$426$	9.88716	0.479034
$427$	16.8876	0.817250
$428$	1.45234	0.0702015
$429$	−4.24259	−0.204834
$430$	−9.71493	−0.468496
$431$	−1.37416	−0.0661912	−0.0330956	−	0.999452i	$-0.510537\pi$
$431$	−1.37416	−0.0661912	−0.0330956	+	0.999452i	$0.510537\pi$
$432$	−15.5781	−0.749503
$433$	−15.9514	−0.766578	−0.383289	−	0.923629i	$-0.625209\pi$
$433$	−15.9514	−0.766578	−0.383289	+	0.923629i	$0.625209\pi$
$434$	−8.59859	−0.412746
$435$	5.44997	0.261306
$436$	0.738279	0.0353572
$437$	12.9676	0.620326
$438$	−0.934791	−0.0446661
$439$	0.859343	0.0410142	0.0205071	−	0.999790i	$-0.493472\pi$
$439$	0.859343	0.0410142	0.0205071	+	0.999790i	$0.493472\pi$
$440$	−2.59958	−0.123930
$441$	11.3850	0.542142
$442$	−44.4831	−2.11585
$443$	0.285221	0.0135513	0.00677563	−	0.999977i	$-0.497843\pi$
$443$	0.285221	0.0135513	0.00677563	+	0.999977i	$0.497843\pi$
$444$	1.42522	0.0676378
$445$	14.2786	0.676870
$446$	0.0465143	0.00220251
$447$	−9.19612	−0.434961
$448$	10.7472	0.507759
$449$	36.6745	1.73078	0.865388	−	0.501103i	$-0.167072\pi$
$449$	36.6745	1.73078	0.865388	+	0.501103i	$0.167072\pi$
$450$	3.94828	0.186124
$451$	5.21751	0.245683
$452$	−5.62844	−0.264740
$453$	−5.28805	−0.248454
$454$	16.9523	0.795609
$455$	−11.1463	−0.522548
$456$	6.83544	0.320099
$457$	−27.3570	−1.27970	−0.639852	−	0.768498i	$-0.721005\pi$
$457$	−27.3570	−1.27970	−0.639852	+	0.768498i	$0.721005\pi$
$458$	7.31312	0.341720
$459$	−14.9721	−0.698839
$460$	−0.847435	−0.0395119
$461$	−0.0579708	−0.00269997	−0.00134999	−	0.999999i	$-0.500430\pi$
$461$	−0.0579708	−0.00269997	−0.00134999	+	0.999999i	$0.500430\pi$
$462$	−1.52128	−0.0707765
$463$	−36.7409	−1.70750	−0.853748	−	0.520686i	$-0.825676\pi$
$463$	−36.7409	−1.70750	−0.853748	+	0.520686i	$0.825676\pi$
$464$	39.3976	1.82899
$465$	−2.16873	−0.100573
$466$	−5.55164	−0.257175
$467$	−26.6393	−1.23272	−0.616360	−	0.787465i	$-0.711393\pi$
$467$	−26.6393	−1.23272	−0.616360	+	0.787465i	$0.711393\pi$
$468$	4.97094	0.229782
$469$	11.5516	0.533403
$470$	−12.5796	−0.580252
$471$	6.04752	0.278655
$472$	9.30255	0.428185
$473$	−6.43753	−0.295998
$474$	−2.96780	−0.136315
$475$	−4.24491	−0.194770
$476$	−1.94289	−0.0890522
$477$	5.23804	0.239834
$478$	−4.67211	−0.213697
$479$	25.3430	1.15795	0.578977	−	0.815344i	$-0.303452\pi$
$479$	25.3430	1.15795	0.578977	+	0.815344i	$0.303452\pi$
$480$	−0.965328	−0.0440610
$481$	56.8079	2.59022
$482$	−15.8068	−0.719982
$483$	3.07951	0.140123
$484$	0.277405	0.0126093
$485$	−6.31534	−0.286765
$486$	21.0729	0.955884
$487$	8.36638	0.379117	0.189558	−	0.981869i	$-0.439294\pi$
$487$	8.36638	0.379117	0.189558	+	0.981869i	$0.439294\pi$
$488$	26.9759	1.22114
$489$	8.04300	0.363717
$490$	6.56696	0.296665
$491$	−15.3434	−0.692438	−0.346219	−	0.938154i	$-0.612535\pi$
$491$	−15.3434	−0.692438	−0.346219	+	0.938154i	$0.612535\pi$
$492$	0.896546	0.0404194
$493$	37.8650	1.70535
$494$	−43.8758	−1.97407
$495$	2.61630	0.117594
$496$	−15.6777	−0.703949
$497$	17.2128	0.772101
$498$	10.7495	0.481694
$499$	14.9589	0.669652	0.334826	−	0.942280i	$-0.391322\pi$
$499$	14.9589	0.669652	0.334826	+	0.942280i	$0.391322\pi$
$500$	0.277405	0.0124059
$501$	8.06131	0.360153
$502$	−20.3212	−0.906978
$503$	26.7943	1.19470	0.597349	−	0.801981i	$-0.296221\pi$
$503$	26.7943	1.19470	0.597349	+	0.801981i	$0.296221\pi$
$504$	−11.0684	−0.493028
$505$	−9.24858	−0.411557
$506$	−4.61012	−0.204945
$507$	−21.0055	−0.932887
$508$	1.54916	0.0687327
$509$	−13.1836	−0.584354	−0.292177	−	0.956364i	$-0.594380\pi$
$509$	−13.1836	−0.584354	−0.292177	+	0.956364i	$0.594380\pi$
$510$	−4.02303	−0.178143
$511$	−1.62740	−0.0719921
$512$	16.3028	0.720488
$513$	−14.7677	−0.652011
$514$	−45.8294	−2.02145
$515$	15.1379	0.667056
$516$	−1.10619	−0.0486972
$517$	−8.33576	−0.366606
$518$	20.3698	0.894999
$519$	6.47100	0.284045
$520$	−17.8049	−0.780797
$521$	11.9024	0.521455	0.260728	−	0.965412i	$-0.416038\pi$
$521$	11.9024	0.521455	0.260728	+	0.965412i	$0.416038\pi$
$522$	−34.7382	−1.52045
$523$	−34.4746	−1.50747	−0.753735	−	0.657179i	$-0.771750\pi$
$523$	−34.4746	−1.50747	−0.753735	+	0.657179i	$0.771750\pi$
$524$	−1.01036	−0.0441379
$525$	−1.00807	−0.0439957
$526$	−26.8582	−1.17107
$527$	−15.0678	−0.656364
$528$	−2.77373	−0.120711
$529$	−13.6678	−0.594252
$530$	3.02135	0.131239
$531$	−9.36239	−0.406293
$532$	−1.91637	−0.0830850
$533$	35.7355	1.54788
$534$	13.3475	0.577603
$535$	5.23545	0.226348
$536$	18.4523	0.797016
$537$	−0.157635	−0.00680246
$538$	−4.84073	−0.208699
$539$	4.35155	0.187435
$540$	0.965071	0.0415301
$541$	−2.79806	−0.120298	−0.0601490	−	0.998189i	$-0.519158\pi$
$541$	−2.79806	−0.120298	−0.0601490	+	0.998189i	$0.519158\pi$
$542$	22.0123	0.945510
$543$	−11.4078	−0.489553
$544$	−6.70685	−0.287554
$545$	2.66138	0.114001
$546$	−10.4195	−0.445913
$547$	16.2324	0.694048	0.347024	−	0.937856i	$-0.387192\pi$
$547$	16.2324	0.694048	0.347024	+	0.937856i	$0.387192\pi$
$548$	−3.18657	−0.136124
$549$	−27.1495	−1.15871
$550$	1.50911	0.0643486
$551$	37.3480	1.59108
$552$	4.91915	0.209373
$553$	−5.16672	−0.219711
$554$	25.5324	1.08477
$555$	5.13767	0.218082
$556$	−4.83758	−0.205159
$557$	10.1597	0.430482	0.215241	−	0.976561i	$-0.430946\pi$
$557$	10.1597	0.430482	0.215241	+	0.976561i	$0.430946\pi$
$558$	13.8236	0.585198
$559$	−44.0916	−1.86488
$560$	−7.28728	−0.307944
$561$	−2.66583	−0.112552
$562$	−46.4530	−1.95950
$563$	−40.3739	−1.70156	−0.850778	−	0.525526i	$-0.823869\pi$
$563$	−40.3739	−1.70156	−0.850778	+	0.525526i	$0.823869\pi$
$564$	−1.43237	−0.0603135
$565$	−20.2896	−0.853591
$566$	−34.4887	−1.44967
$567$	9.26635	0.389150
$568$	27.4954	1.15368
$569$	−26.0933	−1.09389	−0.546944	−	0.837169i	$-0.684209\pi$
$569$	−26.0933	−1.09389	−0.546944	+	0.837169i	$0.684209\pi$
$570$	−3.96810	−0.166206
$571$	28.4359	1.19000	0.595002	−	0.803724i	$-0.297151\pi$
$571$	28.4359	1.19000	0.595002	+	0.803724i	$0.297151\pi$
$572$	1.89999	0.0794425
$573$	2.99497	0.125117
$574$	12.8138	0.534839
$575$	−3.05486	−0.127397
$576$	−17.2778	−0.719910
$577$	−11.9113	−0.495876	−0.247938	−	0.968776i	$-0.579753\pi$
$577$	−11.9113	−0.495876	−0.247938	+	0.968776i	$0.579753\pi$
$578$	−2.29615	−0.0955073
$579$	11.1645	0.463979
$580$	−2.44070	−0.101344
$581$	18.7140	0.776389
$582$	−5.90353	−0.244709
$583$	2.00208	0.0829176
$584$	−2.59958	−0.107571
$585$	17.9194	0.740878
$586$	33.7480	1.39412
$587$	38.1437	1.57436	0.787180	−	0.616724i	$-0.211540\pi$
$587$	38.1437	1.57436	0.787180	+	0.616724i	$0.211540\pi$
$588$	0.747745	0.0308365
$589$	−14.8621	−0.612382
$590$	−5.40031	−0.222327
$591$	−14.1411	−0.581689
$592$	37.1400	1.52644
$593$	−41.5089	−1.70457	−0.852283	−	0.523082i	$-0.824782\pi$
$593$	−41.5089	−1.70457	−0.852283	+	0.523082i	$0.824782\pi$
$594$	5.25007	0.215413
$595$	−7.00380	−0.287128
$596$	4.11836	0.168695
$597$	−4.57971	−0.187435
$598$	−31.5754	−1.29121
$599$	7.82313	0.319645	0.159822	−	0.987146i	$-0.448908\pi$
$599$	7.82313	0.319645	0.159822	+	0.987146i	$0.448908\pi$
$600$	−1.61027	−0.0657389
$601$	6.09019	0.248424	0.124212	−	0.992256i	$-0.460360\pi$
$601$	6.09019	0.248424	0.124212	+	0.992256i	$0.460360\pi$
$602$	−15.8101	−0.644372
$603$	−18.5709	−0.756267
$604$	2.36818	0.0963600
$605$	1.00000	0.0406558
$606$	−8.64549	−0.351199
$607$	−7.69422	−0.312299	−0.156149	−	0.987733i	$-0.549908\pi$
$607$	−7.69422	−0.312299	−0.156149	+	0.987733i	$0.549908\pi$
$608$	−6.61529	−0.268285
$609$	8.86930	0.359402
$610$	−15.6601	−0.634057
$611$	−57.0929	−2.30973
$612$	3.12349	0.126260
$613$	−22.2080	−0.896971	−0.448486	−	0.893790i	$-0.648037\pi$
$613$	−22.2080	−0.896971	−0.448486	+	0.893790i	$0.648037\pi$
$614$	−22.5949	−0.911855
$615$	3.23190	0.130323
$616$	−4.23057	−0.170454
$617$	−23.6493	−0.952084	−0.476042	−	0.879423i	$-0.657929\pi$
$617$	−23.6493	−0.952084	−0.476042	+	0.879423i	$0.657929\pi$
$618$	14.1508	0.569228
$619$	27.3150	1.09788	0.548941	−	0.835861i	$-0.315031\pi$
$619$	27.3150	1.09788	0.548941	+	0.835861i	$0.315031\pi$
$620$	0.971239	0.0390059
$621$	−10.6276	−0.426473
$622$	24.8876	0.997900
$623$	23.2370	0.930972
$624$	−18.9977	−0.760517
$625$	1.00000	0.0400000
$626$	17.6489	0.705393
$627$	−2.62944	−0.105010
$628$	−2.70830	−0.108073
$629$	35.6952	1.42326
$630$	6.42545	0.255996
$631$	31.2664	1.24470	0.622348	−	0.782741i	$-0.286179\pi$
$631$	31.2664	1.24470	0.622348	+	0.782741i	$0.286179\pi$
$632$	−8.25321	−0.328295
$633$	−5.02812	−0.199850
$634$	23.4076	0.929634
$635$	5.58445	0.221612
$636$	0.344025	0.0136415
$637$	29.8044	1.18090
$638$	−13.2776	−0.525665
$639$	−27.6723	−1.09470
$640$	−13.0828	−0.517144
$641$	−10.1022	−0.399012	−0.199506	−	0.979897i	$-0.563934\pi$
$641$	−10.1022	−0.399012	−0.199506	+	0.979897i	$0.563934\pi$
$642$	4.89405	0.193153
$643$	−43.1651	−1.70226	−0.851132	−	0.524952i	$-0.824083\pi$
$643$	−43.1651	−1.70226	−0.851132	+	0.524952i	$0.824083\pi$
$644$	−1.37912	−0.0543449
$645$	−3.98762	−0.157012
$646$	−27.5694	−1.08470
$647$	37.6345	1.47957	0.739783	−	0.672846i	$-0.234928\pi$
$647$	37.6345	1.47957	0.739783	+	0.672846i	$0.234928\pi$
$648$	14.8019	0.581472
$649$	−3.57848	−0.140468
$650$	10.3361	0.405415
$651$	−3.52941	−0.138328
$652$	−3.60195	−0.141063
$653$	8.41111	0.329152	0.164576	−	0.986364i	$-0.447374\pi$
$653$	8.41111	0.329152	0.164576	+	0.986364i	$0.447374\pi$
$654$	2.48783	0.0972819
$655$	−3.64219	−0.142312
$656$	23.3633	0.912183
$657$	2.61630	0.102072
$658$	−20.4720	−0.798083
$659$	−23.3081	−0.907953	−0.453976	−	0.891014i	$-0.649995\pi$
$659$	−23.3081	−0.907953	−0.453976	+	0.891014i	$0.649995\pi$
$660$	0.171834	0.00668863
$661$	23.9986	0.933436	0.466718	−	0.884406i	$-0.345436\pi$
$661$	23.9986	0.933436	0.466718	+	0.884406i	$0.345436\pi$
$662$	−43.9521	−1.70825
$663$	−18.2587	−0.709109
$664$	29.8934	1.16009
$665$	−6.90819	−0.267888
$666$	−32.7476	−1.26894
$667$	26.8777	1.04071
$668$	−3.61015	−0.139681
$669$	0.0190924	0.000738155 0
$670$	−10.7119	−0.413837
$671$	−10.3770	−0.400601
$672$	−1.57098	−0.0606018
$673$	1.76374	0.0679871	0.0339935	−	0.999422i	$-0.489177\pi$
$673$	1.76374	0.0679871	0.0339935	+	0.999422i	$0.489177\pi$
$674$	−50.5263	−1.94620
$675$	3.47892	0.133904
$676$	9.40704	0.361809
$677$	3.57915	0.137558	0.0687789	−	0.997632i	$-0.478090\pi$
$677$	3.57915	0.137558	0.0687789	+	0.997632i	$0.478090\pi$
$678$	−18.9666	−0.728406
$679$	−10.2776	−0.394419
$680$	−11.1877	−0.429030
$681$	6.95828	0.266642
$682$	5.28362	0.202320
$683$	7.73401	0.295934	0.147967	−	0.988992i	$-0.452727\pi$
$683$	7.73401	0.295934	0.147967	+	0.988992i	$0.452727\pi$
$684$	3.08085	0.117799
$685$	−11.4871	−0.438899
$686$	27.8786	1.06441
$687$	3.00177	0.114525
$688$	−28.8264	−1.09899
$689$	13.7125	0.522406
$690$	−2.85566	−0.108713
$691$	−24.4657	−0.930719	−0.465360	−	0.885122i	$-0.654075\pi$
$691$	−24.4657	−0.930719	−0.465360	+	0.885122i	$0.654075\pi$
$692$	−2.89795	−0.110164
$693$	4.25778	0.161740
$694$	0.257874	0.00978875
$695$	−17.4387	−0.661487
$696$	14.1676	0.537022
$697$	22.4544	0.850522
$698$	35.2908	1.33578
$699$	−2.27874	−0.0861900
$700$	0.451450	0.0170632
$701$	−3.43328	−0.129673	−0.0648365	−	0.997896i	$-0.520653\pi$
$701$	−3.43328	−0.129673	−0.0648365	+	0.997896i	$0.520653\pi$
$702$	35.9585	1.35717
$703$	35.2079	1.32789
$704$	−6.60391	−0.248894
$705$	−5.16345	−0.194467
$706$	−43.0247	−1.61926
$707$	−15.0512	−0.566058
$708$	−0.614905	−0.0231095
$709$	39.5606	1.48573	0.742865	−	0.669441i	$-0.233466\pi$
$709$	39.5606	1.48573	0.742865	+	0.669441i	$0.233466\pi$
$710$	−15.9616	−0.599029
$711$	8.30630	0.311511
$712$	37.1183	1.39107
$713$	−10.6956	−0.400552
$714$	−6.54709	−0.245019
$715$	6.84915	0.256144
$716$	0.0705948	0.00263825
$717$	−1.91773	−0.0716189
$718$	−28.5736	−1.06636
$719$	41.5373	1.54908	0.774540	−	0.632525i	$-0.217982\pi$
$719$	41.5373	1.54908	0.774540	+	0.632525i	$0.217982\pi$
$720$	11.7154	0.436608
$721$	24.6355	0.917473
$722$	1.48004	0.0550815
$723$	−6.48813	−0.241296
$724$	5.10881	0.189868
$725$	−8.79831	−0.326761
$726$	0.934791	0.0346934
$727$	48.6692	1.80504	0.902521	−	0.430645i	$-0.141714\pi$
$727$	48.6692	1.80504	0.902521	+	0.430645i	$0.141714\pi$
$728$	−28.9758	−1.07391
$729$	−8.43220	−0.312304
$730$	1.50911	0.0558546
$731$	−27.7050	−1.02471
$732$	−1.78313	−0.0659062
$733$	−22.5497	−0.832893	−0.416446	−	0.909160i	$-0.636725\pi$
$733$	−22.5497	−0.832893	−0.416446	+	0.909160i	$0.636725\pi$
$734$	51.2730	1.89252
$735$	2.69550	0.0994249
$736$	−4.76072	−0.175482
$737$	−7.09816	−0.261464
$738$	−20.6002	−0.758304
$739$	16.9062	0.621906	0.310953	−	0.950425i	$-0.399352\pi$
$739$	16.9062	0.621906	0.310953	+	0.950425i	$0.399352\pi$
$740$	−2.30084	−0.0845805
$741$	−18.0094	−0.661592
$742$	4.91696	0.180507
$743$	45.7895	1.67985	0.839927	−	0.542699i	$-0.182598\pi$
$743$	45.7895	1.67985	0.839927	+	0.542699i	$0.182598\pi$
$744$	−5.63780	−0.206692
$745$	14.8460	0.543916
$746$	15.5156	0.568068
$747$	−30.0857	−1.10078
$748$	1.19386	0.0436518
$749$	8.52020	0.311321
$750$	0.934791	0.0341338
$751$	−27.2974	−0.996095	−0.498047	−	0.867150i	$-0.665949\pi$
$751$	−27.2974	−0.996095	−0.498047	+	0.867150i	$0.665949\pi$
$752$	−37.3263	−1.36115
$753$	−8.34109	−0.303966
$754$	−90.9402	−3.31185
$755$	8.53691	0.310690
$756$	1.57056	0.0571208
$757$	−18.3913	−0.668444	−0.334222	−	0.942494i	$-0.608474\pi$
$757$	−18.3913	−0.668444	−0.334222	+	0.942494i	$0.608474\pi$
$758$	15.3719	0.558333
$759$	−1.89228	−0.0686856
$760$	−11.0350	−0.400281
$761$	−5.80621	−0.210475	−0.105237	−	0.994447i	$-0.533560\pi$
$761$	−5.80621	−0.210475	−0.105237	+	0.994447i	$0.533560\pi$
$762$	5.22030	0.189111
$763$	4.33114	0.156798
$764$	−1.34126	−0.0485250
$765$	11.2597	0.407095
$766$	7.65588	0.276618
$767$	−24.5096	−0.884989
$768$	−4.04834	−0.146082
$769$	−27.7325	−1.00006	−0.500029	−	0.866008i	$-0.666677\pi$
$769$	−27.7325	−1.00006	−0.500029	+	0.866008i	$0.666677\pi$
$770$	2.45593	0.0885055
$771$	−18.8113	−0.677472
$772$	−4.99985	−0.179949
$773$	−6.82605	−0.245516	−0.122758	−	0.992437i	$-0.539174\pi$
$773$	−6.82605	−0.245516	−0.122758	+	0.992437i	$0.539174\pi$
$774$	25.4172	0.913602
$775$	3.50116	0.125765
$776$	−16.4172	−0.589345
$777$	8.36107	0.299952
$778$	27.5869	0.989040
$779$	22.1479	0.793530
$780$	1.17692	0.0421404
$781$	−10.5769	−0.378470
$782$	−19.8404	−0.709492
$783$	−30.6087	−1.09386
$784$	19.4856	0.695916
$785$	−9.76299	−0.348456
$786$	−3.40469	−0.121441
$787$	−16.2108	−0.577854	−0.288927	−	0.957351i	$-0.593298\pi$
$787$	−16.2108	−0.577854	−0.288927	+	0.957351i	$0.593298\pi$
$788$	6.33293	0.225601
$789$	−11.0243	−0.392475
$790$	4.79115	0.170461
$791$	−33.0194	−1.17404
$792$	6.80129	0.241673
$793$	−71.0738	−2.52391
$794$	50.4497	1.79039
$795$	1.24015	0.0439837
$796$	2.05096	0.0726945
$797$	23.1113	0.818643	0.409322	−	0.912390i	$-0.365765\pi$
$797$	23.1113	0.818643	0.409322	+	0.912390i	$0.365765\pi$
$798$	−6.45771	−0.228601
$799$	−35.8743	−1.26914
$800$	1.55841	0.0550979
$801$	−37.3571	−1.31995
$802$	−48.5552	−1.71454
$803$	1.00000	0.0352892
$804$	−1.21971	−0.0430157
$805$	−4.97150	−0.175222
$806$	36.1883	1.27468
$807$	−1.98694	−0.0699437
$808$	−24.0424	−0.845810
$809$	13.4602	0.473236	0.236618	−	0.971603i	$-0.423961\pi$
$809$	13.4602	0.473236	0.236618	+	0.971603i	$0.423961\pi$
$810$	−8.59278	−0.301919
$811$	25.3377	0.889728	0.444864	−	0.895598i	$-0.353252\pi$
$811$	25.3377	0.889728	0.444864	+	0.895598i	$0.353252\pi$
$812$	−3.97200	−0.139390
$813$	9.03525	0.316880
$814$	−12.5168	−0.438712
$815$	−12.9845	−0.454825
$816$	−11.9372	−0.417886
$817$	−27.3268	−0.956042
$818$	39.0596	1.36569
$819$	29.1622	1.01901
$820$	−1.44736	−0.0505442
$821$	4.82769	0.168487	0.0842437	−	0.996445i	$-0.473153\pi$
$821$	4.82769	0.168487	0.0842437	+	0.996445i	$0.473153\pi$
$822$	−10.7380	−0.374532
$823$	6.19007	0.215772	0.107886	−	0.994163i	$-0.465592\pi$
$823$	6.19007	0.215772	0.107886	+	0.994163i	$0.465592\pi$
$824$	39.3522	1.37090
$825$	0.619433	0.0215659
$826$	−8.78849	−0.305791
$827$	−46.6469	−1.62207	−0.811035	−	0.584997i	$-0.801095\pi$
$827$	−46.6469	−1.62207	−0.811035	+	0.584997i	$0.801095\pi$
$828$	2.21715	0.0770512
$829$	−15.6080	−0.542088	−0.271044	−	0.962567i	$-0.587369\pi$
$829$	−15.6080	−0.542088	−0.271044	+	0.962567i	$0.587369\pi$
$830$	−17.3537	−0.602355
$831$	10.4801	0.363552
$832$	−45.2312	−1.56811
$833$	18.7276	0.648874
$834$	−16.3015	−0.564476
$835$	−13.0140	−0.450368
$836$	1.17756	0.0407267
$837$	12.1803	0.421012
$838$	−39.8280	−1.37584
$839$	56.1757	1.93940	0.969701	−	0.244296i	$-0.0785569\pi$
$839$	56.1757	1.93940	0.969701	+	0.244296i	$0.0785569\pi$
$840$	−2.62056	−0.0904177
$841$	48.4103	1.66932
$842$	−14.9704	−0.515912
$843$	−19.0673	−0.656711
$844$	2.25178	0.0775094
$845$	33.9108	1.16657
$846$	32.9119	1.13154
$847$	1.62740	0.0559183
$848$	8.96502	0.307860
$849$	−14.1564	−0.485845
$850$	6.49469	0.222766
$851$	25.3375	0.868558
$852$	−1.81746	−0.0622653
$853$	−21.1340	−0.723616	−0.361808	−	0.932253i	$-0.617840\pi$
$853$	−21.1340	−0.723616	−0.361808	+	0.932253i	$0.617840\pi$
$854$	−25.4852	−0.872087
$855$	11.1060	0.379816
$856$	13.6100	0.465179
$857$	6.88923	0.235331	0.117666	−	0.993053i	$-0.462459\pi$
$857$	6.88923	0.235331	0.117666	+	0.993053i	$0.462459\pi$
$858$	6.40252	0.218579
$859$	−24.6862	−0.842282	−0.421141	−	0.906995i	$-0.638370\pi$
$859$	−24.6862	−0.842282	−0.421141	+	0.906995i	$0.638370\pi$
$860$	1.78580	0.0608954
$861$	5.25961	0.179247
$862$	2.07376	0.0706326
$863$	48.9161	1.66512	0.832562	−	0.553932i	$-0.186873\pi$
$863$	48.9161	1.66512	0.832562	+	0.553932i	$0.186873\pi$
$864$	5.42157	0.184446
$865$	−10.4466	−0.355196
$866$	24.0725	0.818015
$867$	−0.942486	−0.0320085
$868$	1.58060	0.0536490
$869$	3.17482	0.107699
$870$	−8.22458	−0.278840
$871$	−48.6164	−1.64730
$872$	6.91846	0.234289
$873$	16.5228	0.559214
$874$	−19.5695	−0.661950
$875$	1.62740	0.0550163
$876$	0.171834	0.00580573
$877$	−30.5539	−1.03173	−0.515866	−	0.856669i	$-0.672530\pi$
$877$	−30.5539	−1.03173	−0.515866	+	0.856669i	$0.672530\pi$
$878$	−1.29684	−0.0437663
$879$	13.8523	0.467227
$880$	4.47786	0.150949
$881$	−7.56321	−0.254811	−0.127406	−	0.991851i	$-0.540665\pi$
$881$	−7.56321	−0.254811	−0.127406	+	0.991851i	$0.540665\pi$
$882$	−17.1812	−0.578520
$883$	33.7842	1.13693	0.568464	−	0.822708i	$-0.307538\pi$
$883$	33.7842	1.13693	0.568464	+	0.822708i	$0.307538\pi$
$884$	8.17691	0.275019
$885$	−2.21663	−0.0745112
$886$	−0.430429	−0.0144605
$887$	23.7014	0.795816	0.397908	−	0.917425i	$-0.369736\pi$
$887$	23.7014	0.795816	0.397908	+	0.917425i	$0.369736\pi$
$888$	13.3558	0.448191
$889$	9.08816	0.304807
$890$	−21.5479	−0.722288
$891$	−5.69395	−0.190754
$892$	−0.00855028	−0.000286285 0
$893$	−35.3845	−1.18410
$894$	13.8779	0.464147
$895$	0.254483	0.00850642
$896$	−21.2910	−0.711284
$897$	−12.9605	−0.432740
$898$	−55.3457	−1.84691
$899$	−30.8043	−1.02738
$900$	−0.725776	−0.0241925
$901$	8.61628	0.287050
$902$	−7.87379	−0.262168
$903$	−6.48948	−0.215956
$904$	−52.7445	−1.75426
$905$	18.4164	0.612183
$906$	7.98023	0.265125
$907$	−7.10415	−0.235890	−0.117945	−	0.993020i	$-0.537631\pi$
$907$	−7.10415	−0.235890	−0.117945	+	0.993020i	$0.537631\pi$
$908$	−3.11617	−0.103414
$909$	24.1971	0.802567
$910$	16.8210	0.557611
$911$	45.7718	1.51649	0.758244	−	0.651971i	$-0.226058\pi$
$911$	45.7718	1.51649	0.758244	+	0.651971i	$0.226058\pi$
$912$	−11.7742	−0.389884
$913$	−11.4993	−0.380572
$914$	41.2846	1.36557
$915$	−6.42788	−0.212499
$916$	−1.34430	−0.0444170
$917$	−5.92732	−0.195737
$918$	22.5945	0.745731
$919$	−46.4700	−1.53290	−0.766451	−	0.642302i	$-0.777979\pi$
$919$	−46.4700	−1.53290	−0.766451	+	0.642302i	$0.777979\pi$
$920$	−7.94137	−0.261819
$921$	−9.27437	−0.305601
$922$	0.0874842	0.00288114
$923$	−72.4425	−2.38447
$924$	0.279643	0.00919959
$925$	−8.29415	−0.272710
$926$	55.4460	1.82207
$927$	−39.6053	−1.30081
$928$	−13.7113	−0.450096
$929$	−30.5689	−1.00293	−0.501467	−	0.865177i	$-0.667206\pi$
$929$	−30.5689	−1.00293	−0.501467	+	0.865177i	$0.667206\pi$
$930$	3.27285	0.107321
$931$	18.4720	0.605394
$932$	1.02051	0.0334278
$933$	10.2154	0.334438
$934$	40.2016	1.31544
$935$	4.30366	0.140745
$936$	46.5830	1.52261
$937$	54.1315	1.76840	0.884200	−	0.467108i	$-0.154704\pi$
$937$	54.1315	1.76840	0.884200	+	0.467108i	$0.154704\pi$
$938$	−17.4326	−0.569194
$939$	7.24424	0.236407
$940$	2.31238	0.0754216
$941$	−14.2278	−0.463813	−0.231907	−	0.972738i	$-0.574496\pi$
$941$	−14.2278	−0.463813	−0.231907	+	0.972738i	$0.574496\pi$
$942$	−9.12635	−0.297353
$943$	15.9388	0.519039
$944$	−16.0239	−0.521534
$945$	5.66162	0.184172
$946$	9.71493	0.315860
$947$	40.4304	1.31381	0.656906	−	0.753972i	$-0.271865\pi$
$947$	40.4304	1.31381	0.656906	+	0.753972i	$0.271865\pi$
$948$	0.545543	0.0177184
$949$	6.84915	0.222333
$950$	6.40602	0.207839
$951$	9.60796	0.311559
$952$	−18.2070	−0.590091
$953$	−14.3148	−0.463703	−0.231851	−	0.972751i	$-0.574478\pi$
$953$	−14.3148	−0.463703	−0.231851	+	0.972751i	$0.574478\pi$
$954$	−7.90477	−0.255926
$955$	−4.83502	−0.156457
$956$	0.858830	0.0277765
$957$	−5.44997	−0.176172
$958$	−38.2454	−1.23565
$959$	−18.6941	−0.603665
$960$	−4.09068	−0.132026
$961$	−18.7419	−0.604577
$962$	−85.7291	−2.76402
$963$	−13.6975	−0.441397
$964$	2.90562	0.0935838
$965$	−18.0237	−0.580202
$966$	−4.64732	−0.149525
$967$	−32.1016	−1.03232	−0.516158	−	0.856493i	$-0.672638\pi$
$967$	−32.1016	−1.03232	−0.516158	+	0.856493i	$0.672638\pi$
$968$	2.59958	0.0835537
$969$	−11.3162	−0.363529
$970$	9.53053	0.306007
$971$	48.9298	1.57023	0.785116	−	0.619349i	$-0.212604\pi$
$971$	48.9298	1.57023	0.785116	+	0.619349i	$0.212604\pi$
$972$	−3.87363	−0.124247
$973$	−28.3798	−0.909814
$974$	−12.6258	−0.404556
$975$	4.24259	0.135872
$976$	−46.4669	−1.48737
$977$	29.6043	0.947127	0.473563	−	0.880760i	$-0.342967\pi$
$977$	29.6043	0.947127	0.473563	+	0.880760i	$0.342967\pi$
$978$	−12.1378	−0.388122
$979$	−14.2786	−0.456346
$980$	−1.20714	−0.0385608
$981$	−6.96297	−0.222310
$982$	23.1548	0.738900
$983$	−11.0533	−0.352545	−0.176272	−	0.984341i	$-0.556404\pi$
$983$	−11.0533	−0.352545	−0.176272	+	0.984341i	$0.556404\pi$
$984$	8.40159	0.267833
$985$	22.8292	0.727398
$986$	−57.1423	−1.81978
$987$	−8.40302	−0.267471
$988$	8.06528	0.256591
$989$	−19.6658	−0.625336
$990$	−3.94828	−0.125485
$991$	49.2478	1.56441	0.782204	−	0.623023i	$-0.214096\pi$
$991$	49.2478	1.56441	0.782204	+	0.623023i	$0.214096\pi$
$992$	5.45622	0.173235
$993$	−18.0407	−0.572506
$994$	−25.9760	−0.823909
$995$	7.39339	0.234386
$996$	−1.97597	−0.0626110
$997$	47.3005	1.49802	0.749010	−	0.662558i	$-0.230529\pi$
$997$	47.3005	1.49802	0.749010	+	0.662558i	$0.230529\pi$
$998$	−22.5746	−0.714586
$999$	−28.8547	−0.912923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.i.1.11	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.i.1.11	✓	38	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)

\(f(q)\)	\(=\)	\(q-1.50911 q^{2} -0.619433 q^{3} +0.277405 q^{4} +1.00000 q^{5} +0.934791 q^{6} +1.62740 q^{7} +2.59958 q^{8} -2.61630 q^{9} +O(q^{10})\)	\(q-1.50911 q^{2} -0.619433 q^{3} +0.277405 q^{4} +1.00000 q^{5} +0.934791 q^{6} +1.62740 q^{7} +2.59958 q^{8} -2.61630 q^{9} -1.50911 q^{10} -1.00000 q^{11} -0.171834 q^{12} -6.84915 q^{13} -2.45593 q^{14} -0.619433 q^{15} -4.47786 q^{16} -4.30366 q^{17} +3.94828 q^{18} -4.24491 q^{19} +0.277405 q^{20} -1.00807 q^{21} +1.50911 q^{22} -3.05486 q^{23} -1.61027 q^{24} +1.00000 q^{25} +10.3361 q^{26} +3.47892 q^{27} +0.451450 q^{28} -8.79831 q^{29} +0.934791 q^{30} +3.50116 q^{31} +1.55841 q^{32} +0.619433 q^{33} +6.49469 q^{34} +1.62740 q^{35} -0.725776 q^{36} -8.29415 q^{37} +6.40602 q^{38} +4.24259 q^{39} +2.59958 q^{40} -5.21751 q^{41} +1.52128 q^{42} +6.43753 q^{43} -0.277405 q^{44} -2.61630 q^{45} +4.61012 q^{46} +8.33576 q^{47} +2.77373 q^{48} -4.35155 q^{49} -1.50911 q^{50} +2.66583 q^{51} -1.89999 q^{52} -2.00208 q^{53} -5.25007 q^{54} -1.00000 q^{55} +4.23057 q^{56} +2.62944 q^{57} +13.2776 q^{58} +3.57848 q^{59} -0.171834 q^{60} +10.3770 q^{61} -5.28362 q^{62} -4.25778 q^{63} +6.60391 q^{64} -6.84915 q^{65} -0.934791 q^{66} +7.09816 q^{67} -1.19386 q^{68} +1.89228 q^{69} -2.45593 q^{70} +10.5769 q^{71} -6.80129 q^{72} -1.00000 q^{73} +12.5168 q^{74} -0.619433 q^{75} -1.17756 q^{76} -1.62740 q^{77} -6.40252 q^{78} -3.17482 q^{79} -4.47786 q^{80} +5.69395 q^{81} +7.87379 q^{82} +11.4993 q^{83} -0.279643 q^{84} -4.30366 q^{85} -9.71493 q^{86} +5.44997 q^{87} -2.59958 q^{88} +14.2786 q^{89} +3.94828 q^{90} -11.1463 q^{91} -0.847435 q^{92} -2.16873 q^{93} -12.5796 q^{94} -4.24491 q^{95} -0.965328 q^{96} -6.31534 q^{97} +6.56696 q^{98} +2.61630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9}+O(q^{10}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9	\( 38 q + 4 q^{2} + 5 q^{3} + 50 q^{4} + 38 q^{5} + 11 q^{6} + 15 q^{8} + 63 q^{9} + 4 q^{10} - 38 q^{11} + 12 q^{12} - q^{13} + 23 q^{14} + 5 q^{15} + 74 q^{16} + 26 q^{17} + 16 q^{18} - 10 q^{19} + 50 q^{20} + 21 q^{21} - 4 q^{22} + 10 q^{23} + 41 q^{24} + 38 q^{25} + 25 q^{26} + 5 q^{27} + 2 q^{28} + 28 q^{29} + 11 q^{30} + 24 q^{31} + 39 q^{32} - 5 q^{33} + 38 q^{34} + 111 q^{36} + 12 q^{37} + 19 q^{38} - 18 q^{39} + 15 q^{40} + 62 q^{41} - 17 q^{42} - 32 q^{43} - 50 q^{44} + 63 q^{45} - 9 q^{46} + 31 q^{47} + 53 q^{48} + 88 q^{49} + 4 q^{50} - 3 q^{51} - 21 q^{52} + 30 q^{53} + 49 q^{54} - 38 q^{55} + 32 q^{56} + 49 q^{57} + 12 q^{58} + 31 q^{59} + 12 q^{60} + 25 q^{61} + 12 q^{62} + 15 q^{63} + 137 q^{64} - q^{65} - 11 q^{66} + 20 q^{67} + 75 q^{68} + 92 q^{69} + 23 q^{70} + 32 q^{71} + 6 q^{72} - 38 q^{73} + 55 q^{74} + 5 q^{75} - 57 q^{76} - 17 q^{78} - 2 q^{79} + 74 q^{80} + 118 q^{81} + 14 q^{82} + 4 q^{83} + 22 q^{84} + 26 q^{85} + 5 q^{86} + 24 q^{87} - 15 q^{88} + 143 q^{89} + 16 q^{90} + 66 q^{91} + 29 q^{92} - 8 q^{93} - 7 q^{94} - 10 q^{95} + 59 q^{96} + 41 q^{97} - 10 q^{98} - 63 q^{99}+O(q^{100}) \) 38 * q + 4 * q^2 + 5 * q^3 + 50 * q^4 + 38 * q^5 + 11 * q^6 + 15 * q^8 + 63 * q^9 + 4 * q^10 - 38 * q^11 + 12 * q^12 - q^13 + 23 * q^14 + 5 * q^15 + 74 * q^16 + 26 * q^17 + 16 * q^18 - 10 * q^19 + 50 * q^20 + 21 * q^21 - 4 * q^22 + 10 * q^23 + 41 * q^24 + 38 * q^25 + 25 * q^26 + 5 * q^27 + 2 * q^28 + 28 * q^29 + 11 * q^30 + 24 * q^31 + 39 * q^32 - 5 * q^33 + 38 * q^34 + 111 * q^36 + 12 * q^37 + 19 * q^38 - 18 * q^39 + 15 * q^40 + 62 * q^41 - 17 * q^42 - 32 * q^43 - 50 * q^44 + 63 * q^45 - 9 * q^46 + 31 * q^47 + 53 * q^48 + 88 * q^49 + 4 * q^50 - 3 * q^51 - 21 * q^52 + 30 * q^53 + 49 * q^54 - 38 * q^55 + 32 * q^56 + 49 * q^57 + 12 * q^58 + 31 * q^59 + 12 * q^60 + 25 * q^61 + 12 * q^62 + 15 * q^63 + 137 * q^64 - q^65 - 11 * q^66 + 20 * q^67 + 75 * q^68 + 92 * q^69 + 23 * q^70 + 32 * q^71 + 6 * q^72 - 38 * q^73 + 55 * q^74 + 5 * q^75 - 57 * q^76 - 17 * q^78 - 2 * q^79 + 74 * q^80 + 118 * q^81 + 14 * q^82 + 4 * q^83 + 22 * q^84 + 26 * q^85 + 5 * q^86 + 24 * q^87 - 15 * q^88 + 143 * q^89 + 16 * q^90 + 66 * q^91 + 29 * q^92 - 8 * q^93 - 7 * q^94 - 10 * q^95 + 59 * q^96 + 41 * q^97 - 10 * q^98 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more