LMFDB - Embedded newform 6015.2.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, 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("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.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:	$48.0300168158$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.363417 q^{2} -1.00000 q^{3} -1.86793 q^{4} -1.00000 q^{5} -0.363417 q^{6} +3.29378 q^{7} -1.40567 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.363417 q^{2} -1.00000 q^{3} -1.86793 q^{4} -1.00000 q^{5} -0.363417 q^{6} +3.29378 q^{7} -1.40567 q^{8} +1.00000 q^{9} -0.363417 q^{10} +5.11051 q^{11} +1.86793 q^{12} +3.88535 q^{13} +1.19702 q^{14} +1.00000 q^{15} +3.22501 q^{16} +1.77140 q^{17} +0.363417 q^{18} +8.58089 q^{19} +1.86793 q^{20} -3.29378 q^{21} +1.85725 q^{22} +3.04111 q^{23} +1.40567 q^{24} +1.00000 q^{25} +1.41200 q^{26} -1.00000 q^{27} -6.15254 q^{28} +0.998626 q^{29} +0.363417 q^{30} -0.261235 q^{31} +3.98337 q^{32} -5.11051 q^{33} +0.643758 q^{34} -3.29378 q^{35} -1.86793 q^{36} +1.94690 q^{37} +3.11844 q^{38} -3.88535 q^{39} +1.40567 q^{40} -0.494718 q^{41} -1.19702 q^{42} -0.551455 q^{43} -9.54607 q^{44} -1.00000 q^{45} +1.10519 q^{46} +10.5964 q^{47} -3.22501 q^{48} +3.84898 q^{49} +0.363417 q^{50} -1.77140 q^{51} -7.25755 q^{52} +7.21751 q^{53} -0.363417 q^{54} -5.11051 q^{55} -4.62997 q^{56} -8.58089 q^{57} +0.362918 q^{58} +0.336213 q^{59} -1.86793 q^{60} +3.44130 q^{61} -0.0949372 q^{62} +3.29378 q^{63} -5.00240 q^{64} -3.88535 q^{65} -1.85725 q^{66} -7.39432 q^{67} -3.30885 q^{68} -3.04111 q^{69} -1.19702 q^{70} +13.0060 q^{71} -1.40567 q^{72} -7.30978 q^{73} +0.707537 q^{74} -1.00000 q^{75} -16.0285 q^{76} +16.8329 q^{77} -1.41200 q^{78} -14.9059 q^{79} -3.22501 q^{80} +1.00000 q^{81} -0.179789 q^{82} +3.26880 q^{83} +6.15254 q^{84} -1.77140 q^{85} -0.200408 q^{86} -0.998626 q^{87} -7.18370 q^{88} -5.55115 q^{89} -0.363417 q^{90} +12.7975 q^{91} -5.68057 q^{92} +0.261235 q^{93} +3.85092 q^{94} -8.58089 q^{95} -3.98337 q^{96} -5.82593 q^{97} +1.39878 q^{98} +5.11051 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * q^98 - 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 $

$2$	0.363417	0.256975	0.128487	−	0.991711i	$-0.458988\pi$
$2$	0.363417	0.256975	0.128487	+	0.991711i	$0.458988\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.86793	−0.933964
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.363417	−0.148364
$7$	3.29378	1.24493	0.622466	−	0.782647i	$-0.286131\pi$
$7$	3.29378	1.24493	0.622466	+	0.782647i	$0.286131\pi$
$8$	−1.40567	−0.496980
$9$	1.00000	0.333333
$10$	−0.363417	−0.114923
$11$	5.11051	1.54088	0.770439	−	0.637514i	$-0.220037\pi$
$11$	5.11051	1.54088	0.770439	+	0.637514i	$0.220037\pi$
$12$	1.86793	0.539224
$13$	3.88535	1.07760	0.538801	−	0.842433i	$-0.318878\pi$
$13$	3.88535	1.07760	0.538801	+	0.842433i	$0.318878\pi$
$14$	1.19702	0.319916
$15$	1.00000	0.258199
$16$	3.22501	0.806253
$17$	1.77140	0.429628	0.214814	−	0.976655i	$-0.431085\pi$
$17$	1.77140	0.429628	0.214814	+	0.976655i	$0.431085\pi$
$18$	0.363417	0.0856582
$19$	8.58089	1.96859	0.984295	−	0.176529i	$-0.0564870\pi$
$19$	8.58089	1.96859	0.984295	+	0.176529i	$0.0564870\pi$
$20$	1.86793	0.417681
$21$	−3.29378	−0.718762
$22$	1.85725	0.395966
$23$	3.04111	0.634115	0.317058	−	0.948406i	$-0.397305\pi$
$23$	3.04111	0.634115	0.317058	+	0.948406i	$0.397305\pi$
$24$	1.40567	0.286931
$25$	1.00000	0.200000
$26$	1.41200	0.276916
$27$	−1.00000	−0.192450
$28$	−6.15254	−1.16272
$29$	0.998626	0.185440	0.0927201	−	0.995692i	$-0.470444\pi$
$29$	0.998626	0.185440	0.0927201	+	0.995692i	$0.470444\pi$
$30$	0.363417	0.0663506
$31$	−0.261235	−0.0469192	−0.0234596	−	0.999725i	$-0.507468\pi$
$31$	−0.261235	−0.0469192	−0.0234596	+	0.999725i	$0.507468\pi$
$32$	3.98337	0.704166
$33$	−5.11051	−0.889626
$34$	0.643758	0.110404
$35$	−3.29378	−0.556750
$36$	−1.86793	−0.311321
$37$	1.94690	0.320069	0.160034	−	0.987111i	$-0.448840\pi$
$37$	1.94690	0.320069	0.160034	+	0.987111i	$0.448840\pi$
$38$	3.11844	0.505878
$39$	−3.88535	−0.622154
$40$	1.40567	0.222256
$41$	−0.494718	−0.0772619	−0.0386310	−	0.999254i	$-0.512300\pi$
$41$	−0.494718	−0.0772619	−0.0386310	+	0.999254i	$0.512300\pi$
$42$	−1.19702	−0.184703
$43$	−0.551455	−0.0840962	−0.0420481	−	0.999116i	$-0.513388\pi$
$43$	−0.551455	−0.0840962	−0.0420481	+	0.999116i	$0.513388\pi$
$44$	−9.54607	−1.43912
$45$	−1.00000	−0.149071
$46$	1.10519	0.162952
$47$	10.5964	1.54565	0.772825	−	0.634620i	$-0.218843\pi$
$47$	10.5964	1.54565	0.772825	+	0.634620i	$0.218843\pi$
$48$	−3.22501	−0.465490
$49$	3.84898	0.549854
$50$	0.363417	0.0513949
$51$	−1.77140	−0.248046
$52$	−7.25755	−1.00644
$53$	7.21751	0.991401	0.495700	−	0.868494i	$-0.334911\pi$
$53$	7.21751	0.991401	0.495700	+	0.868494i	$0.334911\pi$
$54$	−0.363417	−0.0494548
$55$	−5.11051	−0.689101
$56$	−4.62997	−0.618706
$57$	−8.58089	−1.13657
$58$	0.362918	0.0476534
$59$	0.336213	0.0437712	0.0218856	−	0.999760i	$-0.493033\pi$
$59$	0.336213	0.0437712	0.0218856	+	0.999760i	$0.493033\pi$
$60$	−1.86793	−0.241148
$61$	3.44130	0.440613	0.220306	−	0.975431i	$-0.429294\pi$
$61$	3.44130	0.440613	0.220306	+	0.975431i	$0.429294\pi$
$62$	−0.0949372	−0.0120570
$63$	3.29378	0.414977
$64$	−5.00240	−0.625300
$65$	−3.88535	−0.481918
$66$	−1.85725	−0.228611
$67$	−7.39432	−0.903360	−0.451680	−	0.892180i	$-0.649175\pi$
$67$	−7.39432	−0.903360	−0.451680	+	0.892180i	$0.649175\pi$
$68$	−3.30885	−0.401257
$69$	−3.04111	−0.366107
$70$	−1.19702	−0.143071
$71$	13.0060	1.54352	0.771761	−	0.635913i	$-0.219376\pi$
$71$	13.0060	1.54352	0.771761	+	0.635913i	$0.219376\pi$
$72$	−1.40567	−0.165660
$73$	−7.30978	−0.855545	−0.427773	−	0.903886i	$-0.640702\pi$
$73$	−7.30978	−0.855545	−0.427773	+	0.903886i	$0.640702\pi$
$74$	0.707537	0.0822495
$75$	−1.00000	−0.115470
$76$	−16.0285	−1.83859
$77$	16.8329	1.91829
$78$	−1.41200	−0.159878
$79$	−14.9059	−1.67705	−0.838525	−	0.544863i	$-0.816582\pi$
$79$	−14.9059	−1.67705	−0.838525	+	0.544863i	$0.816582\pi$
$80$	−3.22501	−0.360567
$81$	1.00000	0.111111
$82$	−0.179789	−0.0198543
$83$	3.26880	0.358798	0.179399	−	0.983776i	$-0.442585\pi$
$83$	3.26880	0.358798	0.179399	+	0.983776i	$0.442585\pi$
$84$	6.15254	0.671297
$85$	−1.77140	−0.192136
$86$	−0.200408	−0.0216106
$87$	−0.998626	−0.107064
$88$	−7.18370	−0.765785
$89$	−5.55115	−0.588420	−0.294210	−	0.955741i	$-0.595057\pi$
$89$	−5.55115	−0.588420	−0.294210	+	0.955741i	$0.595057\pi$
$90$	−0.363417	−0.0383075
$91$	12.7975	1.34154
$92$	−5.68057	−0.592241
$93$	0.261235	0.0270888
$94$	3.85092	0.397193
$95$	−8.58089	−0.880381
$96$	−3.98337	−0.406551
$97$	−5.82593	−0.591534	−0.295767	−	0.955260i	$-0.595575\pi$
$97$	−5.82593	−0.591534	−0.295767	+	0.955260i	$0.595575\pi$
$98$	1.39878	0.141299
$99$	5.11051	0.513626
$100$	−1.86793	−0.186793
$101$	−13.8493	−1.37806	−0.689030	−	0.724732i	$-0.741963\pi$
$101$	−13.8493	−1.37806	−0.689030	+	0.724732i	$0.741963\pi$
$102$	−0.643758	−0.0637415
$103$	−17.7561	−1.74956	−0.874781	−	0.484519i	$-0.838995\pi$
$103$	−17.7561	−1.74956	−0.874781	+	0.484519i	$0.838995\pi$
$104$	−5.46152	−0.535546
$105$	3.29378	0.321440
$106$	2.62296	0.254765
$107$	10.7859	1.04271	0.521356	−	0.853339i	$-0.325426\pi$
$107$	10.7859	1.04271	0.521356	+	0.853339i	$0.325426\pi$
$108$	1.86793	0.179741
$109$	−7.49092	−0.717500	−0.358750	−	0.933434i	$-0.616797\pi$
$109$	−7.49092	−0.717500	−0.358750	+	0.933434i	$0.616797\pi$
$110$	−1.85725	−0.177082
$111$	−1.94690	−0.184792
$112$	10.6225	1.00373
$113$	−18.4060	−1.73149	−0.865747	−	0.500482i	$-0.833156\pi$
$113$	−18.4060	−1.73149	−0.865747	+	0.500482i	$0.833156\pi$
$114$	−3.11844	−0.292069
$115$	−3.04111	−0.283585
$116$	−1.86536	−0.173194
$117$	3.88535	0.359200
$118$	0.122185	0.0112481
$119$	5.83461	0.534858
$120$	−1.40567	−0.128320
$121$	15.1173	1.37430
$122$	1.25063	0.113226
$123$	0.494718	0.0446072
$124$	0.487968	0.0438208
$125$	−1.00000	−0.0894427
$126$	1.19702	0.106639
$127$	8.63413	0.766155	0.383077	−	0.923716i	$-0.374864\pi$
$127$	8.63413	0.766155	0.383077	+	0.923716i	$0.374864\pi$
$128$	−9.78469	−0.864852
$129$	0.551455	0.0485529
$130$	−1.41200	−0.123841
$131$	−13.7166	−1.19842	−0.599211	−	0.800591i	$-0.704519\pi$
$131$	−13.7166	−1.19842	−0.599211	+	0.800591i	$0.704519\pi$
$132$	9.54607	0.830878
$133$	28.2635	2.45076
$134$	−2.68722	−0.232140
$135$	1.00000	0.0860663
$136$	−2.49001	−0.213517
$137$	−0.504824	−0.0431300	−0.0215650	−	0.999767i	$-0.506865\pi$
$137$	−0.504824	−0.0431300	−0.0215650	+	0.999767i	$0.506865\pi$
$138$	−1.10519	−0.0940801
$139$	15.9202	1.35033	0.675167	−	0.737665i	$-0.264072\pi$
$139$	15.9202	1.35033	0.675167	+	0.737665i	$0.264072\pi$
$140$	6.15254	0.519985
$141$	−10.5964	−0.892381
$142$	4.72658	0.396646
$143$	19.8561	1.66045
$144$	3.22501	0.268751
$145$	−0.998626	−0.0829314
$146$	−2.65650	−0.219853
$147$	−3.84898	−0.317459
$148$	−3.63667	−0.298933
$149$	−7.80560	−0.639460	−0.319730	−	0.947509i	$-0.603592\pi$
$149$	−7.80560	−0.639460	−0.319730	+	0.947509i	$0.603592\pi$
$150$	−0.363417	−0.0296729
$151$	−1.27178	−0.103496	−0.0517480	−	0.998660i	$-0.516479\pi$
$151$	−1.27178	−0.103496	−0.0517480	+	0.998660i	$0.516479\pi$
$152$	−12.0619	−0.978350
$153$	1.77140	0.143209
$154$	6.11736	0.492951
$155$	0.261235	0.0209829
$156$	7.25755	0.581069
$157$	0.343503	0.0274145	0.0137073	−	0.999906i	$-0.495637\pi$
$157$	0.343503	0.0274145	0.0137073	+	0.999906i	$0.495637\pi$
$158$	−5.41707	−0.430959
$159$	−7.21751	−0.572385
$160$	−3.98337	−0.314913
$161$	10.0167	0.789430
$162$	0.363417	0.0285527
$163$	−11.0730	−0.867302	−0.433651	−	0.901081i	$-0.642775\pi$
$163$	−11.0730	−0.867302	−0.433651	+	0.901081i	$0.642775\pi$
$164$	0.924097	0.0721598
$165$	5.11051	0.397853
$166$	1.18794	0.0922019
$167$	17.9764	1.39106	0.695529	−	0.718498i	$-0.255170\pi$
$167$	17.9764	1.39106	0.695529	+	0.718498i	$0.255170\pi$
$168$	4.62997	0.357210
$169$	2.09592	0.161225
$170$	−0.643758	−0.0493740
$171$	8.58089	0.656197
$172$	1.03008	0.0785428
$173$	−2.41357	−0.183501	−0.0917503	−	0.995782i	$-0.529246\pi$
$173$	−2.41357	−0.183501	−0.0917503	+	0.995782i	$0.529246\pi$
$174$	−0.362918	−0.0275127
$175$	3.29378	0.248986
$176$	16.4815	1.24234
$177$	−0.336213	−0.0252713
$178$	−2.01738	−0.151209
$179$	10.7819	0.805877	0.402939	−	0.915227i	$-0.367989\pi$
$179$	10.7819	0.805877	0.402939	+	0.915227i	$0.367989\pi$
$180$	1.86793	0.139227
$181$	−2.43867	−0.181265	−0.0906325	−	0.995884i	$-0.528889\pi$
$181$	−2.43867	−0.181265	−0.0906325	+	0.995884i	$0.528889\pi$
$182$	4.65082	0.344742
$183$	−3.44130	−0.254388
$184$	−4.27480	−0.315142
$185$	−1.94690	−0.143139
$186$	0.0949372	0.00696113
$187$	9.05277	0.662004
$188$	−19.7934	−1.44358
$189$	−3.29378	−0.239587
$190$	−3.11844	−0.226235
$191$	20.8017	1.50516	0.752581	−	0.658500i	$-0.228809\pi$
$191$	20.8017	1.50516	0.752581	+	0.658500i	$0.228809\pi$
$192$	5.00240	0.361017
$193$	2.89762	0.208575	0.104288	−	0.994547i	$-0.466744\pi$
$193$	2.89762	0.208575	0.104288	+	0.994547i	$0.466744\pi$
$194$	−2.11724	−0.152009
$195$	3.88535	0.278236
$196$	−7.18962	−0.513544
$197$	20.4627	1.45790	0.728952	−	0.684564i	$-0.240008\pi$
$197$	20.4627	1.45790	0.728952	+	0.684564i	$0.240008\pi$
$198$	1.85725	0.131989
$199$	12.7862	0.906392	0.453196	−	0.891411i	$-0.350284\pi$
$199$	12.7862	0.906392	0.453196	+	0.891411i	$0.350284\pi$
$200$	−1.40567	−0.0993959
$201$	7.39432	0.521555
$202$	−5.03309	−0.354127
$203$	3.28925	0.230860
$204$	3.30885	0.231666
$205$	0.494718	0.0345526
$206$	−6.45287	−0.449593
$207$	3.04111	0.211372
$208$	12.5303	0.868819
$209$	43.8527	3.03336
$210$	1.19702	0.0826019
$211$	−21.2491	−1.46285	−0.731423	−	0.681925i	$-0.761143\pi$
$211$	−21.2491	−1.46285	−0.731423	+	0.681925i	$0.761143\pi$
$212$	−13.4818	−0.925933
$213$	−13.0060	−0.891153
$214$	3.91978	0.267951
$215$	0.551455	0.0376089
$216$	1.40567	0.0956438
$217$	−0.860450	−0.0584111
$218$	−2.72233	−0.184379
$219$	7.30978	0.493949
$220$	9.54607	0.643596
$221$	6.88251	0.462968
$222$	−0.707537	−0.0474868
$223$	16.0003	1.07146	0.535731	−	0.844389i	$-0.320036\pi$
$223$	16.0003	1.07146	0.535731	+	0.844389i	$0.320036\pi$
$224$	13.1203	0.876639
$225$	1.00000	0.0666667
$226$	−6.68906	−0.444950
$227$	19.5658	1.29863	0.649314	−	0.760521i	$-0.275056\pi$
$227$	19.5658	1.29863	0.649314	+	0.760521i	$0.275056\pi$
$228$	16.0285	1.06151
$229$	6.56177	0.433614	0.216807	−	0.976214i	$-0.430436\pi$
$229$	6.56177	0.433614	0.216807	+	0.976214i	$0.430436\pi$
$230$	−1.10519	−0.0728741
$231$	−16.8329	−1.10752
$232$	−1.40374	−0.0921600
$233$	−3.45733	−0.226497	−0.113249	−	0.993567i	$-0.536126\pi$
$233$	−3.45733	−0.226497	−0.113249	+	0.993567i	$0.536126\pi$
$234$	1.41200	0.0923054
$235$	−10.5964	−0.691235
$236$	−0.628022	−0.0408807
$237$	14.9059	0.968245
$238$	2.12040	0.137445
$239$	−0.196749	−0.0127266	−0.00636330	−	0.999980i	$-0.502026\pi$
$239$	−0.196749	−0.0127266	−0.00636330	+	0.999980i	$0.502026\pi$
$240$	3.22501	0.208174
$241$	−16.5616	−1.06683	−0.533413	−	0.845855i	$-0.679091\pi$
$241$	−16.5616	−1.06683	−0.533413	+	0.845855i	$0.679091\pi$
$242$	5.49389	0.353161
$243$	−1.00000	−0.0641500
$244$	−6.42809	−0.411517
$245$	−3.84898	−0.245902
$246$	0.179789	0.0114629
$247$	33.3397	2.12136
$248$	0.367210	0.0233179
$249$	−3.26880	−0.207152
$250$	−0.363417	−0.0229845
$251$	15.6597	0.988433	0.494217	−	0.869339i	$-0.335455\pi$
$251$	15.6597	0.988433	0.494217	+	0.869339i	$0.335455\pi$
$252$	−6.15254	−0.387574
$253$	15.5416	0.977094
$254$	3.13779	0.196882
$255$	1.77140	0.110930
$256$	6.44888	0.403055
$257$	−5.83621	−0.364053	−0.182027	−	0.983294i	$-0.558266\pi$
$257$	−5.83621	−0.364053	−0.182027	+	0.983294i	$0.558266\pi$
$258$	0.200408	0.0124769
$259$	6.41266	0.398463
$260$	7.25755	0.450094
$261$	0.998626	0.0618134
$262$	−4.98484	−0.307964
$263$	−4.12366	−0.254276	−0.127138	−	0.991885i	$-0.540579\pi$
$263$	−4.12366	−0.254276	−0.127138	+	0.991885i	$0.540579\pi$
$264$	7.18370	0.442126
$265$	−7.21751	−0.443368
$266$	10.2715	0.629783
$267$	5.55115	0.339725
$268$	13.8121	0.843705
$269$	4.36026	0.265850	0.132925	−	0.991126i	$-0.457563\pi$
$269$	4.36026	0.265850	0.132925	+	0.991126i	$0.457563\pi$
$270$	0.363417	0.0221169
$271$	−25.5925	−1.55463	−0.777317	−	0.629109i	$-0.783420\pi$
$271$	−25.5925	−1.55463	−0.777317	+	0.629109i	$0.783420\pi$
$272$	5.71279	0.346389
$273$	−12.7975	−0.774538
$274$	−0.183462	−0.0110833
$275$	5.11051	0.308175
$276$	5.68057	0.341930
$277$	−22.0787	−1.32658	−0.663291	−	0.748362i	$-0.730841\pi$
$277$	−22.0787	−1.32658	−0.663291	+	0.748362i	$0.730841\pi$
$278$	5.78567	0.347002
$279$	−0.261235	−0.0156397
$280$	4.62997	0.276694
$281$	−16.2632	−0.970184	−0.485092	−	0.874463i	$-0.661214\pi$
$281$	−16.2632	−0.970184	−0.485092	+	0.874463i	$0.661214\pi$
$282$	−3.85092	−0.229319
$283$	−19.0670	−1.13341	−0.566706	−	0.823920i	$-0.691783\pi$
$283$	−19.0670	−1.13341	−0.566706	+	0.823920i	$0.691783\pi$
$284$	−24.2942	−1.44159
$285$	8.58089	0.508288
$286$	7.21605	0.426694
$287$	−1.62949	−0.0961858
$288$	3.98337	0.234722
$289$	−13.8621	−0.815420
$290$	−0.362918	−0.0213113
$291$	5.82593	0.341522
$292$	13.6541	0.799048
$293$	−3.11190	−0.181799	−0.0908995	−	0.995860i	$-0.528974\pi$
$293$	−3.11190	−0.181799	−0.0908995	+	0.995860i	$0.528974\pi$
$294$	−1.39878	−0.0815788
$295$	−0.336213	−0.0195751
$296$	−2.73670	−0.159068
$297$	−5.11051	−0.296542
$298$	−2.83669	−0.164325
$299$	11.8158	0.683323
$300$	1.86793	0.107845
$301$	−1.81637	−0.104694
$302$	−0.462186	−0.0265958
$303$	13.8493	0.795624
$304$	27.6735	1.58718
$305$	−3.44130	−0.197048
$306$	0.643758	0.0368012
$307$	11.5293	0.658011	0.329006	−	0.944328i	$-0.393287\pi$
$307$	11.5293	0.658011	0.329006	+	0.944328i	$0.393287\pi$
$308$	−31.4426	−1.79161
$309$	17.7561	1.01011
$310$	0.0949372	0.00539207
$311$	−27.7377	−1.57286	−0.786429	−	0.617681i	$-0.788072\pi$
$311$	−27.7377	−1.57286	−0.786429	+	0.617681i	$0.788072\pi$
$312$	5.46152	0.309198
$313$	−7.89435	−0.446215	−0.223108	−	0.974794i	$-0.571620\pi$
$313$	−7.89435	−0.446215	−0.223108	+	0.974794i	$0.571620\pi$
$314$	0.124835	0.00704484
$315$	−3.29378	−0.185583
$316$	27.8432	1.56630
$317$	−9.87380	−0.554568	−0.277284	−	0.960788i	$-0.589434\pi$
$317$	−9.87380	−0.554568	−0.277284	+	0.960788i	$0.589434\pi$
$318$	−2.62296	−0.147089
$319$	5.10349	0.285741
$320$	5.00240	0.279643
$321$	−10.7859	−0.602010
$322$	3.64025	0.202863
$323$	15.2002	0.845762
$324$	−1.86793	−0.103774
$325$	3.88535	0.215520
$326$	−4.02411	−0.222875
$327$	7.49092	0.414249
$328$	0.695410	0.0383976
$329$	34.9023	1.92423
$330$	1.85725	0.102238
$331$	18.7474	1.03045	0.515225	−	0.857055i	$-0.327708\pi$
$331$	18.7474	1.03045	0.515225	+	0.857055i	$0.327708\pi$
$332$	−6.10589	−0.335104
$333$	1.94690	0.106690
$334$	6.53294	0.357467
$335$	7.39432	0.403995
$336$	−10.6225	−0.579504
$337$	−5.39960	−0.294135	−0.147067	−	0.989126i	$-0.546983\pi$
$337$	−5.39960	−0.294135	−0.147067	+	0.989126i	$0.546983\pi$
$338$	0.761695	0.0414307
$339$	18.4060	0.999678
$340$	3.30885	0.179448
$341$	−1.33504	−0.0722967
$342$	3.11844	0.168626
$343$	−10.3788	−0.560400
$344$	0.775165	0.0417941
$345$	3.04111	0.163728
$346$	−0.877133	−0.0471550
$347$	−30.8059	−1.65375	−0.826874	−	0.562387i	$-0.809883\pi$
$347$	−30.8059	−1.65375	−0.826874	+	0.562387i	$0.809883\pi$
$348$	1.86536	0.0999939
$349$	28.6051	1.53119	0.765597	−	0.643320i	$-0.222444\pi$
$349$	28.6051	1.53119	0.765597	+	0.643320i	$0.222444\pi$
$350$	1.19702	0.0639832
$351$	−3.88535	−0.207385
$352$	20.3570	1.08503
$353$	−26.9345	−1.43358	−0.716789	−	0.697290i	$-0.754389\pi$
$353$	−26.9345	−1.43358	−0.716789	+	0.697290i	$0.754389\pi$
$354$	−0.122185	−0.00649409
$355$	−13.0060	−0.690284
$356$	10.3691	0.549564
$357$	−5.83461	−0.308800
$358$	3.91833	0.207090
$359$	25.8287	1.36319	0.681593	−	0.731732i	$-0.261288\pi$
$359$	25.8287	1.36319	0.681593	+	0.731732i	$0.261288\pi$
$360$	1.40567	0.0740854
$361$	54.6316	2.87535
$362$	−0.886255	−0.0465805
$363$	−15.1173	−0.793454
$364$	−23.9048	−1.25295
$365$	7.30978	0.382611
$366$	−1.25063	−0.0653713
$367$	11.7943	0.615657	0.307828	−	0.951442i	$-0.400398\pi$
$367$	11.7943	0.615657	0.307828	+	0.951442i	$0.400398\pi$
$368$	9.80761	0.511257
$369$	−0.494718	−0.0257540
$370$	−0.707537	−0.0367831
$371$	23.7729	1.23423
$372$	−0.487968	−0.0253000
$373$	−14.7473	−0.763588	−0.381794	−	0.924247i	$-0.624694\pi$
$373$	−14.7473	−0.763588	−0.381794	+	0.924247i	$0.624694\pi$
$374$	3.28993	0.170118
$375$	1.00000	0.0516398
$376$	−14.8951	−0.768156
$377$	3.88001	0.199831
$378$	−1.19702	−0.0615678
$379$	−12.5665	−0.645497	−0.322749	−	0.946485i	$-0.604607\pi$
$379$	−12.5665	−0.645497	−0.322749	+	0.946485i	$0.604607\pi$
$380$	16.0285	0.822244
$381$	−8.63413	−0.442340
$382$	7.55971	0.386788
$383$	−25.5207	−1.30405	−0.652023	−	0.758199i	$-0.726079\pi$
$383$	−25.5207	−1.30405	−0.652023	+	0.758199i	$0.726079\pi$
$384$	9.78469	0.499323
$385$	−16.8329	−0.857884
$386$	1.05305	0.0535986
$387$	−0.551455	−0.0280321
$388$	10.8824	0.552471
$389$	−13.2971	−0.674188	−0.337094	−	0.941471i	$-0.609444\pi$
$389$	−13.2971	−0.674188	−0.337094	+	0.941471i	$0.609444\pi$
$390$	1.41200	0.0714995
$391$	5.38703	0.272434
$392$	−5.41040	−0.273266
$393$	13.7166	0.691910
$394$	7.43648	0.374645
$395$	14.9059	0.750000
$396$	−9.54607	−0.479708
$397$	8.53687	0.428453	0.214227	−	0.976784i	$-0.431277\pi$
$397$	8.53687	0.428453	0.214227	+	0.976784i	$0.431277\pi$
$398$	4.64674	0.232920
$399$	−28.2635	−1.41495
$400$	3.22501	0.161251
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	2.68722	0.134026
$403$	−1.01499	−0.0505602
$404$	25.8696	1.28706
$405$	−1.00000	−0.0496904
$406$	1.19537	0.0593253
$407$	9.94966	0.493186
$408$	2.49001	0.123274
$409$	−28.1928	−1.39405	−0.697023	−	0.717049i	$-0.745492\pi$
$409$	−28.1928	−1.39405	−0.697023	+	0.717049i	$0.745492\pi$
$410$	0.179789	0.00887913
$411$	0.504824	0.0249011
$412$	33.1671	1.63403
$413$	1.10741	0.0544921
$414$	1.10519	0.0543172
$415$	−3.26880	−0.160459
$416$	15.4768	0.758811
$417$	−15.9202	−0.779616
$418$	15.9368	0.779496
$419$	−22.9780	−1.12255	−0.561274	−	0.827630i	$-0.689689\pi$
$419$	−22.9780	−1.12255	−0.561274	+	0.827630i	$0.689689\pi$
$420$	−6.15254	−0.300213
$421$	−18.3919	−0.896368	−0.448184	−	0.893941i	$-0.647929\pi$
$421$	−18.3919	−0.896368	−0.448184	+	0.893941i	$0.647929\pi$
$422$	−7.72227	−0.375914
$423$	10.5964	0.515216
$424$	−10.1454	−0.492706
$425$	1.77140	0.0859257
$426$	−4.72658	−0.229004
$427$	11.3349	0.548533
$428$	−20.1473	−0.973856
$429$	−19.8561	−0.958662
$430$	0.200408	0.00966454
$431$	−4.67611	−0.225240	−0.112620	−	0.993638i	$-0.535924\pi$
$431$	−4.67611	−0.225240	−0.112620	+	0.993638i	$0.535924\pi$
$432$	−3.22501	−0.155163
$433$	−31.6922	−1.52303	−0.761516	−	0.648147i	$-0.775544\pi$
$433$	−31.6922	−1.52303	−0.761516	+	0.648147i	$0.775544\pi$
$434$	−0.312702	−0.0150102
$435$	0.998626	0.0478805
$436$	13.9925	0.670119
$437$	26.0954	1.24831
$438$	2.65650	0.126932
$439$	11.6761	0.557267	0.278634	−	0.960397i	$-0.410118\pi$
$439$	11.6761	0.557267	0.278634	+	0.960397i	$0.410118\pi$
$440$	7.18370	0.342469
$441$	3.84898	0.183285
$442$	2.50122	0.118971
$443$	18.9956	0.902511	0.451255	−	0.892395i	$-0.350976\pi$
$443$	18.9956	0.902511	0.451255	+	0.892395i	$0.350976\pi$
$444$	3.63667	0.172589
$445$	5.55115	0.263150
$446$	5.81479	0.275338
$447$	7.80560	0.369192
$448$	−16.4768	−0.778456
$449$	−0.103312	−0.00487559	−0.00243779	−	0.999997i	$-0.500776\pi$
$449$	−0.103312	−0.00487559	−0.00243779	+	0.999997i	$0.500776\pi$
$450$	0.363417	0.0171316
$451$	−2.52826	−0.119051
$452$	34.3811	1.61715
$453$	1.27178	0.0597534
$454$	7.11054	0.333714
$455$	−12.7975	−0.599955
$456$	12.0619	0.564850
$457$	−4.49403	−0.210222	−0.105111	−	0.994460i	$-0.533520\pi$
$457$	−4.49403	−0.210222	−0.105111	+	0.994460i	$0.533520\pi$
$458$	2.38466	0.111428
$459$	−1.77140	−0.0826820
$460$	5.68057	0.264858
$461$	7.84352	0.365309	0.182655	−	0.983177i	$-0.441531\pi$
$461$	7.84352	0.365309	0.182655	+	0.983177i	$0.441531\pi$
$462$	−6.11736	−0.284605
$463$	−18.6724	−0.867782	−0.433891	−	0.900965i	$-0.642860\pi$
$463$	−18.6724	−0.867782	−0.433891	+	0.900965i	$0.642860\pi$
$464$	3.22058	0.149512
$465$	−0.261235	−0.0121145
$466$	−1.25645	−0.0582040
$467$	−20.2087	−0.935145	−0.467572	−	0.883955i	$-0.654871\pi$
$467$	−20.2087	−0.935145	−0.467572	+	0.883955i	$0.654871\pi$
$468$	−7.25755	−0.335480
$469$	−24.3552	−1.12462
$470$	−3.85092	−0.177630
$471$	−0.343503	−0.0158278
$472$	−0.472605	−0.0217534
$473$	−2.81822	−0.129582
$474$	5.41707	0.248814
$475$	8.58089	0.393718
$476$	−10.8986	−0.499538
$477$	7.21751	0.330467
$478$	−0.0715018	−0.00327041
$479$	9.91453	0.453007	0.226503	−	0.974010i	$-0.427271\pi$
$479$	9.91453	0.453007	0.226503	+	0.974010i	$0.427271\pi$
$480$	3.98337	0.181815
$481$	7.56439	0.344906
$482$	−6.01877	−0.274147
$483$	−10.0167	−0.455778
$484$	−28.2381	−1.28355
$485$	5.82593	0.264542
$486$	−0.363417	−0.0164849
$487$	32.8894	1.49036	0.745181	−	0.666862i	$-0.232363\pi$
$487$	32.8894	1.49036	0.745181	+	0.666862i	$0.232363\pi$
$488$	−4.83733	−0.218976
$489$	11.0730	0.500737
$490$	−1.39878	−0.0631907
$491$	−24.6593	−1.11286	−0.556430	−	0.830894i	$-0.687829\pi$
$491$	−24.6593	−1.11286	−0.556430	+	0.830894i	$0.687829\pi$
$492$	−0.924097	−0.0416615
$493$	1.76897	0.0796704
$494$	12.1162	0.545135
$495$	−5.11051	−0.229700
$496$	−0.842485	−0.0378287
$497$	42.8387	1.92158
$498$	−1.18794	−0.0532328
$499$	7.42174	0.332243	0.166121	−	0.986105i	$-0.446876\pi$
$499$	7.42174	0.332243	0.166121	+	0.986105i	$0.446876\pi$
$500$	1.86793	0.0835363
$501$	−17.9764	−0.803128
$502$	5.69101	0.254002
$503$	0.251541	0.0112157	0.00560783	−	0.999984i	$-0.498215\pi$
$503$	0.251541	0.0112157	0.00560783	+	0.999984i	$0.498215\pi$
$504$	−4.62997	−0.206235
$505$	13.8493	0.616288
$506$	5.64809	0.251088
$507$	−2.09592	−0.0930833
$508$	−16.1279	−0.715561
$509$	−34.6952	−1.53784	−0.768919	−	0.639346i	$-0.779205\pi$
$509$	−34.6952	−1.53784	−0.768919	+	0.639346i	$0.779205\pi$
$510$	0.643758	0.0285061
$511$	−24.0768	−1.06509
$512$	21.9130	0.968427
$513$	−8.58089	−0.378855
$514$	−2.12098	−0.0935524
$515$	17.7561	0.782428
$516$	−1.03008	−0.0453467
$517$	54.1532	2.38166
$518$	2.33047	0.102395
$519$	2.41357	0.105944
$520$	5.46152	0.239503
$521$	−5.74745	−0.251800	−0.125900	−	0.992043i	$-0.540182\pi$
$521$	−5.74745	−0.251800	−0.125900	+	0.992043i	$0.540182\pi$
$522$	0.362918	0.0158845
$523$	32.4893	1.42066	0.710328	−	0.703870i	$-0.248546\pi$
$523$	32.4893	1.42066	0.710328	+	0.703870i	$0.248546\pi$
$524$	25.6216	1.11928
$525$	−3.29378	−0.143752
$526$	−1.49861	−0.0653425
$527$	−0.462752	−0.0201578
$528$	−16.4815	−0.717263
$529$	−13.7517	−0.597898
$530$	−2.62296	−0.113934
$531$	0.336213	0.0145904
$532$	−52.7943	−2.28892
$533$	−1.92215	−0.0832575
$534$	2.01738	0.0873006
$535$	−10.7859	−0.466315
$536$	10.3940	0.448951
$537$	−10.7819	−0.465274
$538$	1.58459	0.0683167
$539$	19.6703	0.847258
$540$	−1.86793	−0.0803828
$541$	10.9683	0.471566	0.235783	−	0.971806i	$-0.424235\pi$
$541$	10.9683	0.471566	0.235783	+	0.971806i	$0.424235\pi$
$542$	−9.30075	−0.399502
$543$	2.43867	0.104653
$544$	7.05614	0.302530
$545$	7.49092	0.320876
$546$	−4.65082	−0.199037
$547$	31.0459	1.32743	0.663713	−	0.747987i	$-0.268980\pi$
$547$	31.0459	1.32743	0.663713	+	0.747987i	$0.268980\pi$
$548$	0.942975	0.0402819
$549$	3.44130	0.146871
$550$	1.85725	0.0791933
$551$	8.56910	0.365056
$552$	4.27480	0.181948
$553$	−49.0969	−2.08781
$554$	−8.02378	−0.340898
$555$	1.94690	0.0826413
$556$	−29.7378	−1.26116
$557$	−32.2159	−1.36503	−0.682516	−	0.730871i	$-0.739114\pi$
$557$	−32.2159	−1.36503	−0.682516	+	0.730871i	$0.739114\pi$
$558$	−0.0949372	−0.00401901
$559$	−2.14260	−0.0906221
$560$	−10.6225	−0.448882
$561$	−9.05277	−0.382208
$562$	−5.91034	−0.249313
$563$	21.5112	0.906588	0.453294	−	0.891361i	$-0.350249\pi$
$563$	21.5112	0.906588	0.453294	+	0.891361i	$0.350249\pi$
$564$	19.7934	0.833452
$565$	18.4060	0.774348
$566$	−6.92925	−0.291258
$567$	3.29378	0.138326
$568$	−18.2821	−0.767099
$569$	4.18264	0.175346	0.0876728	−	0.996149i	$-0.472057\pi$
$569$	4.18264	0.175346	0.0876728	+	0.996149i	$0.472057\pi$
$570$	3.11844	0.130617
$571$	−32.1468	−1.34530	−0.672650	−	0.739961i	$-0.734844\pi$
$571$	−32.1468	−1.34530	−0.672650	+	0.739961i	$0.734844\pi$
$572$	−37.0898	−1.55080
$573$	−20.8017	−0.869005
$574$	−0.592184	−0.0247173
$575$	3.04111	0.126823
$576$	−5.00240	−0.208433
$577$	−19.7265	−0.821227	−0.410613	−	0.911810i	$-0.634685\pi$
$577$	−19.7265	−0.821227	−0.410613	+	0.911810i	$0.634685\pi$
$578$	−5.03773	−0.209542
$579$	−2.89762	−0.120421
$580$	1.86536	0.0774549
$581$	10.7667	0.446679
$582$	2.11724	0.0877625
$583$	36.8851	1.52763
$584$	10.2751	0.425188
$585$	−3.88535	−0.160639
$586$	−1.13092	−0.0467177
$587$	0.239419	0.00988187	0.00494094	−	0.999988i	$-0.498427\pi$
$587$	0.239419	0.00988187	0.00494094	+	0.999988i	$0.498427\pi$
$588$	7.18962	0.296495
$589$	−2.24163	−0.0923646
$590$	−0.122185	−0.00503030
$591$	−20.4627	−0.841722
$592$	6.27878	0.258056
$593$	3.70770	0.152257	0.0761285	−	0.997098i	$-0.475744\pi$
$593$	3.70770	0.152257	0.0761285	+	0.997098i	$0.475744\pi$
$594$	−1.85725	−0.0762037
$595$	−5.83461	−0.239196
$596$	14.5803	0.597232
$597$	−12.7862	−0.523306
$598$	4.29405	0.175597
$599$	1.56793	0.0640641	0.0320320	−	0.999487i	$-0.489802\pi$
$599$	1.56793	0.0640641	0.0320320	+	0.999487i	$0.489802\pi$
$600$	1.40567	0.0573863
$601$	−3.67176	−0.149774	−0.0748872	−	0.997192i	$-0.523860\pi$
$601$	−3.67176	−0.149774	−0.0748872	+	0.997192i	$0.523860\pi$
$602$	−0.660100	−0.0269037
$603$	−7.39432	−0.301120
$604$	2.37559	0.0966615
$605$	−15.1173	−0.614607
$606$	5.03309	0.204455
$607$	16.1909	0.657169	0.328585	−	0.944474i	$-0.393428\pi$
$607$	16.1909	0.657169	0.328585	+	0.944474i	$0.393428\pi$
$608$	34.1808	1.38622
$609$	−3.28925	−0.133287
$610$	−1.25063	−0.0506364
$611$	41.1708	1.66559
$612$	−3.30885	−0.133752
$613$	4.81175	0.194345	0.0971724	−	0.995268i	$-0.469020\pi$
$613$	4.81175	0.194345	0.0971724	+	0.995268i	$0.469020\pi$
$614$	4.18994	0.169092
$615$	−0.494718	−0.0199489
$616$	−23.6615	−0.953349
$617$	24.2586	0.976615	0.488307	−	0.872672i	$-0.337614\pi$
$617$	24.2586	0.976615	0.488307	+	0.872672i	$0.337614\pi$
$618$	6.45287	0.259573
$619$	−25.0982	−1.00878	−0.504392	−	0.863475i	$-0.668283\pi$
$619$	−25.0982	−1.00878	−0.504392	+	0.863475i	$0.668283\pi$
$620$	−0.487968	−0.0195973
$621$	−3.04111	−0.122036
$622$	−10.0803	−0.404185
$623$	−18.2843	−0.732543
$624$	−12.5303	−0.501613
$625$	1.00000	0.0400000
$626$	−2.86894	−0.114666
$627$	−43.8527	−1.75131
$628$	−0.641639	−0.0256042
$629$	3.44875	0.137510
$630$	−1.19702	−0.0476902
$631$	−25.9711	−1.03389	−0.516946	−	0.856018i	$-0.672931\pi$
$631$	−25.9711	−1.03389	−0.516946	+	0.856018i	$0.672931\pi$
$632$	20.9529	0.833460
$633$	21.2491	0.844574
$634$	−3.58831	−0.142510
$635$	−8.63413	−0.342635
$636$	13.4818	0.534587
$637$	14.9546	0.592524
$638$	1.85469	0.0734281
$639$	13.0060	0.514507
$640$	9.78469	0.386774
$641$	44.4986	1.75759	0.878795	−	0.477200i	$-0.158348\pi$
$641$	44.4986	1.75759	0.878795	+	0.477200i	$0.158348\pi$
$642$	−3.91978	−0.154701
$643$	−25.1154	−0.990454	−0.495227	−	0.868764i	$-0.664915\pi$
$643$	−25.1154	−0.990454	−0.495227	+	0.868764i	$0.664915\pi$
$644$	−18.7106	−0.737299
$645$	−0.551455	−0.0217135
$646$	5.52401	0.217339
$647$	−27.4964	−1.08099	−0.540497	−	0.841346i	$-0.681764\pi$
$647$	−27.4964	−1.08099	−0.540497	+	0.841346i	$0.681764\pi$
$648$	−1.40567	−0.0552200
$649$	1.71822	0.0674460
$650$	1.41200	0.0553832
$651$	0.860450	0.0337237
$652$	20.6835	0.810029
$653$	20.7320	0.811307	0.405654	−	0.914027i	$-0.367044\pi$
$653$	20.7320	0.811307	0.405654	+	0.914027i	$0.367044\pi$
$654$	2.72233	0.106451
$655$	13.7166	0.535951
$656$	−1.59547	−0.0622926
$657$	−7.30978	−0.285182
$658$	12.6841	0.494478
$659$	−20.1215	−0.783820	−0.391910	−	0.920003i	$-0.628186\pi$
$659$	−20.1215	−0.783820	−0.391910	+	0.920003i	$0.628186\pi$
$660$	−9.54607	−0.371580
$661$	−10.3662	−0.403199	−0.201600	−	0.979468i	$-0.564614\pi$
$661$	−10.3662	−0.403199	−0.201600	+	0.979468i	$0.564614\pi$
$662$	6.81313	0.264800
$663$	−6.88251	−0.267295
$664$	−4.59486	−0.178315
$665$	−28.2635	−1.09601
$666$	0.707537	0.0274165
$667$	3.03693	0.117590
$668$	−33.5787	−1.29920
$669$	−16.0003	−0.618609
$670$	2.68722	0.103816
$671$	17.5868	0.678930
$672$	−13.1203	−0.506128
$673$	35.5714	1.37118	0.685588	−	0.727990i	$-0.259545\pi$
$673$	35.5714	1.37118	0.685588	+	0.727990i	$0.259545\pi$
$674$	−1.96231	−0.0755852
$675$	−1.00000	−0.0384900
$676$	−3.91504	−0.150578
$677$	−30.1272	−1.15788	−0.578941	−	0.815369i	$-0.696534\pi$
$677$	−30.1272	−1.15788	−0.578941	+	0.815369i	$0.696534\pi$
$678$	6.68906	0.256892
$679$	−19.1893	−0.736419
$680$	2.49001	0.0954875
$681$	−19.5658	−0.749763
$682$	−0.485177	−0.0185784
$683$	13.3765	0.511836	0.255918	−	0.966698i	$-0.417622\pi$
$683$	13.3765	0.511836	0.255918	+	0.966698i	$0.417622\pi$
$684$	−16.0285	−0.612864
$685$	0.504824	0.0192883
$686$	−3.77182	−0.144009
$687$	−6.56177	−0.250347
$688$	−1.77845	−0.0678028
$689$	28.0425	1.06833
$690$	1.10519	0.0420739
$691$	41.7542	1.58840	0.794202	−	0.607653i	$-0.207889\pi$
$691$	41.7542	1.58840	0.794202	+	0.607653i	$0.207889\pi$
$692$	4.50838	0.171383
$693$	16.8329	0.639429
$694$	−11.1954	−0.424971
$695$	−15.9202	−0.603888
$696$	1.40374	0.0532086
$697$	−0.876344	−0.0331939
$698$	10.3956	0.393478
$699$	3.45733	0.130768
$700$	−6.15254	−0.232544
$701$	14.4486	0.545715	0.272857	−	0.962054i	$-0.412031\pi$
$701$	14.4486	0.545715	0.272857	+	0.962054i	$0.412031\pi$
$702$	−1.41200	−0.0532926
$703$	16.7061	0.630084
$704$	−25.5648	−0.963511
$705$	10.5964	0.399085
$706$	−9.78846	−0.368393
$707$	−45.6167	−1.71559
$708$	0.628022	0.0236025
$709$	−23.7016	−0.890133	−0.445067	−	0.895497i	$-0.646820\pi$
$709$	−23.7016	−0.890133	−0.445067	+	0.895497i	$0.646820\pi$
$710$	−4.72658	−0.177386
$711$	−14.9059	−0.559017
$712$	7.80309	0.292433
$713$	−0.794444	−0.0297522
$714$	−2.12040	−0.0793538
$715$	−19.8561	−0.742576
$716$	−20.1398	−0.752660
$717$	0.196749	0.00734771
$718$	9.38658	0.350304
$719$	−1.30331	−0.0486052	−0.0243026	−	0.999705i	$-0.507737\pi$
$719$	−1.30331	−0.0486052	−0.0243026	+	0.999705i	$0.507737\pi$
$720$	−3.22501	−0.120189
$721$	−58.4847	−2.17808
$722$	19.8541	0.738892
$723$	16.5616	0.615933
$724$	4.55526	0.169295
$725$	0.998626	0.0370880
$726$	−5.49389	−0.203897
$727$	−28.9989	−1.07551	−0.537755	−	0.843101i	$-0.680727\pi$
$727$	−28.9989	−1.07551	−0.537755	+	0.843101i	$0.680727\pi$
$728$	−17.9890	−0.666718
$729$	1.00000	0.0370370
$730$	2.65650	0.0983214
$731$	−0.976850	−0.0361301
$732$	6.42809	0.237589
$733$	36.8625	1.36155	0.680775	−	0.732493i	$-0.261643\pi$
$733$	36.8625	1.36155	0.680775	+	0.732493i	$0.261643\pi$
$734$	4.28624	0.158208
$735$	3.84898	0.141972
$736$	12.1139	0.446522
$737$	−37.7887	−1.39197
$738$	−0.179789	−0.00661812
$739$	−22.3227	−0.821152	−0.410576	−	0.911826i	$-0.634672\pi$
$739$	−22.3227	−0.821152	−0.410576	+	0.911826i	$0.634672\pi$
$740$	3.63667	0.133687
$741$	−33.3397	−1.22477
$742$	8.63946	0.317165
$743$	30.8212	1.13072	0.565360	−	0.824844i	$-0.308737\pi$
$743$	30.8212	1.13072	0.565360	+	0.824844i	$0.308737\pi$
$744$	−0.367210	−0.0134626
$745$	7.80560	0.285975
$746$	−5.35943	−0.196223
$747$	3.26880	0.119599
$748$	−16.9099	−0.618288
$749$	35.5264	1.29811
$750$	0.363417	0.0132701
$751$	10.5315	0.384299	0.192150	−	0.981366i	$-0.438454\pi$
$751$	10.5315	0.384299	0.192150	+	0.981366i	$0.438454\pi$
$752$	34.1736	1.24618
$753$	−15.6597	−0.570672
$754$	1.41006	0.0513514
$755$	1.27178	0.0462848
$756$	6.15254	0.223766
$757$	−12.2597	−0.445585	−0.222793	−	0.974866i	$-0.571517\pi$
$757$	−12.2597	−0.445585	−0.222793	+	0.974866i	$0.571517\pi$
$758$	−4.56688	−0.165876
$759$	−15.5416	−0.564125
$760$	12.0619	0.437531
$761$	32.7285	1.18641	0.593204	−	0.805052i	$-0.297863\pi$
$761$	32.7285	1.18641	0.593204	+	0.805052i	$0.297863\pi$
$762$	−3.13779	−0.113670
$763$	−24.6734	−0.893238
$764$	−38.8562	−1.40577
$765$	−1.77140	−0.0640452
$766$	−9.27465	−0.335107
$767$	1.30630	0.0471679
$768$	−6.44888	−0.232704
$769$	−6.77018	−0.244139	−0.122070	−	0.992522i	$-0.538953\pi$
$769$	−6.77018	−0.244139	−0.122070	+	0.992522i	$0.538953\pi$
$770$	−6.11736	−0.220454
$771$	5.83621	0.210186
$772$	−5.41255	−0.194802
$773$	36.3172	1.30624	0.653120	−	0.757254i	$-0.273460\pi$
$773$	36.3172	1.30624	0.653120	+	0.757254i	$0.273460\pi$
$774$	−0.200408	−0.00720353
$775$	−0.261235	−0.00938383
$776$	8.18934	0.293980
$777$	−6.41266	−0.230053
$778$	−4.83238	−0.173249
$779$	−4.24512	−0.152097
$780$	−7.25755	−0.259862
$781$	66.4671	2.37838
$782$	1.95774	0.0700086
$783$	−0.998626	−0.0356880
$784$	12.4130	0.443322
$785$	−0.343503	−0.0122601
$786$	4.98484	0.177803
$787$	27.7585	0.989485	0.494743	−	0.869040i	$-0.335262\pi$
$787$	27.7585	0.989485	0.494743	+	0.869040i	$0.335262\pi$
$788$	−38.2228	−1.36163
$789$	4.12366	0.146806
$790$	5.41707	0.192731
$791$	−60.6254	−2.15559
$792$	−7.18370	−0.255262
$793$	13.3706	0.474805
$794$	3.10244	0.110102
$795$	7.21751	0.255979
$796$	−23.8838	−0.846538
$797$	23.4620	0.831068	0.415534	−	0.909578i	$-0.363595\pi$
$797$	23.4620	0.831068	0.415534	+	0.909578i	$0.363595\pi$
$798$	−10.2715	−0.363606
$799$	18.7706	0.664055
$800$	3.98337	0.140833
$801$	−5.55115	−0.196140
$802$	0.363417	0.0128327
$803$	−37.3567	−1.31829
$804$	−13.8121	−0.487114
$805$	−10.0167	−0.353044
$806$	−0.368864	−0.0129927
$807$	−4.36026	−0.153489
$808$	19.4676	0.684868
$809$	6.66931	0.234481	0.117240	−	0.993104i	$-0.462595\pi$
$809$	6.66931	0.234481	0.117240	+	0.993104i	$0.462595\pi$
$810$	−0.363417	−0.0127692
$811$	−30.6756	−1.07717	−0.538584	−	0.842572i	$-0.681040\pi$
$811$	−30.6756	−1.07717	−0.538584	+	0.842572i	$0.681040\pi$
$812$	−6.14409	−0.215615
$813$	25.5925	0.897569
$814$	3.61588	0.126736
$815$	11.0730	0.387869
$816$	−5.71279	−0.199988
$817$	−4.73198	−0.165551
$818$	−10.2458	−0.358234
$819$	12.7975	0.447180
$820$	−0.924097	−0.0322709
$821$	−37.1277	−1.29577	−0.647883	−	0.761740i	$-0.724345\pi$
$821$	−37.1277	−1.29577	−0.647883	+	0.761740i	$0.724345\pi$
$822$	0.183462	0.00639896
$823$	−6.58675	−0.229600	−0.114800	−	0.993389i	$-0.536623\pi$
$823$	−6.58675	−0.229600	−0.114800	+	0.993389i	$0.536623\pi$
$824$	24.9592	0.869497
$825$	−5.11051	−0.177925
$826$	0.402452	0.0140031
$827$	7.60936	0.264604	0.132302	−	0.991209i	$-0.457763\pi$
$827$	7.60936	0.264604	0.132302	+	0.991209i	$0.457763\pi$
$828$	−5.68057	−0.197414
$829$	−24.6174	−0.854997	−0.427498	−	0.904016i	$-0.640605\pi$
$829$	−24.6174	−0.854997	−0.427498	+	0.904016i	$0.640605\pi$
$830$	−1.18794	−0.0412339
$831$	22.0787	0.765902
$832$	−19.4361	−0.673824
$833$	6.81809	0.236233
$834$	−5.78567	−0.200341
$835$	−17.9764	−0.622100
$836$	−81.9137	−2.83305
$837$	0.261235	0.00902960
$838$	−8.35060	−0.288467
$839$	15.9596	0.550985	0.275492	−	0.961303i	$-0.411159\pi$
$839$	15.9596	0.550985	0.275492	+	0.961303i	$0.411159\pi$
$840$	−4.62997	−0.159749
$841$	−28.0027	−0.965612
$842$	−6.68394	−0.230344
$843$	16.2632	0.560136
$844$	39.6917	1.36624
$845$	−2.09592	−0.0721020
$846$	3.85092	0.132398
$847$	49.7931	1.71091
$848$	23.2765	0.799320
$849$	19.0670	0.654376
$850$	0.643758	0.0220807
$851$	5.92074	0.202960
$852$	24.2942	0.832305
$853$	−41.5807	−1.42369	−0.711847	−	0.702334i	$-0.752141\pi$
$853$	−41.5807	−1.42369	−0.711847	+	0.702334i	$0.752141\pi$
$854$	4.11928	0.140959
$855$	−8.58089	−0.293460
$856$	−15.1614	−0.518207
$857$	−26.7316	−0.913133	−0.456566	−	0.889689i	$-0.650921\pi$
$857$	−26.7316	−0.913133	−0.456566	+	0.889689i	$0.650921\pi$
$858$	−7.21605	−0.246352
$859$	−49.7408	−1.69713	−0.848567	−	0.529088i	$-0.822534\pi$
$859$	−49.7408	−1.69713	−0.848567	+	0.529088i	$0.822534\pi$
$860$	−1.03008	−0.0351254
$861$	1.62949	0.0555329
$862$	−1.69938	−0.0578810
$863$	−7.15741	−0.243641	−0.121821	−	0.992552i	$-0.538873\pi$
$863$	−7.15741	−0.243641	−0.121821	+	0.992552i	$0.538873\pi$
$864$	−3.98337	−0.135517
$865$	2.41357	0.0820639
$866$	−11.5175	−0.391380
$867$	13.8621	0.470783
$868$	1.60726	0.0545539
$869$	−76.1770	−2.58413
$870$	0.362918	0.0123041
$871$	−28.7295	−0.973462
$872$	10.5298	0.356583
$873$	−5.82593	−0.197178
$874$	9.48352	0.320785
$875$	−3.29378	−0.111350
$876$	−13.6541	−0.461331
$877$	18.4909	0.624394	0.312197	−	0.950017i	$-0.398935\pi$
$877$	18.4909	0.624394	0.312197	+	0.950017i	$0.398935\pi$
$878$	4.24328	0.143204
$879$	3.11190	0.104962
$880$	−16.4815	−0.555590
$881$	47.0419	1.58488	0.792440	−	0.609950i	$-0.208810\pi$
$881$	47.0419	1.58488	0.792440	+	0.609950i	$0.208810\pi$
$882$	1.39878	0.0470995
$883$	−32.6336	−1.09821	−0.549104	−	0.835754i	$-0.685031\pi$
$883$	−32.6336	−1.09821	−0.549104	+	0.835754i	$0.685031\pi$
$884$	−12.8560	−0.432396
$885$	0.336213	0.0113017
$886$	6.90334	0.231922
$887$	45.8744	1.54031	0.770155	−	0.637856i	$-0.220179\pi$
$887$	45.8744	1.54031	0.770155	+	0.637856i	$0.220179\pi$
$888$	2.73670	0.0918377
$889$	28.4389	0.953810
$890$	2.01738	0.0676228
$891$	5.11051	0.171209
$892$	−29.8875	−1.00071
$893$	90.9268	3.04275
$894$	2.83669	0.0948730
$895$	−10.7819	−0.360399
$896$	−32.2286	−1.07668
$897$	−11.8158	−0.394517
$898$	−0.0375453	−0.00125290
$899$	−0.260876	−0.00870070
$900$	−1.86793	−0.0622643
$901$	12.7851	0.425934
$902$	−0.918812	−0.0305931
$903$	1.81637	0.0604451
$904$	25.8728	0.860517
$905$	2.43867	0.0810642
$906$	0.462186	0.0153551
$907$	36.2847	1.20481	0.602407	−	0.798189i	$-0.294209\pi$
$907$	36.2847	1.20481	0.602407	+	0.798189i	$0.294209\pi$
$908$	−36.5475	−1.21287
$909$	−13.8493	−0.459354
$910$	−4.65082	−0.154173
$911$	−4.82560	−0.159879	−0.0799396	−	0.996800i	$-0.525473\pi$
$911$	−4.82560	−0.159879	−0.0799396	+	0.996800i	$0.525473\pi$
$912$	−27.6735	−0.916360
$913$	16.7053	0.552863
$914$	−1.63321	−0.0540217
$915$	3.44130	0.113766
$916$	−12.2569	−0.404980
$917$	−45.1794	−1.49195
$918$	−0.643758	−0.0212472
$919$	−16.9362	−0.558675	−0.279337	−	0.960193i	$-0.590115\pi$
$919$	−16.9362	−0.558675	−0.279337	+	0.960193i	$0.590115\pi$
$920$	4.27480	0.140936
$921$	−11.5293	−0.379903
$922$	2.85047	0.0938752
$923$	50.5326	1.66330
$924$	31.4426	1.03439
$925$	1.94690	0.0640137
$926$	−6.78588	−0.222998
$927$	−17.7561	−0.583187
$928$	3.97789	0.130581
$929$	15.0219	0.492853	0.246426	−	0.969162i	$-0.420744\pi$
$929$	15.0219	0.492853	0.246426	+	0.969162i	$0.420744\pi$
$930$	−0.0949372	−0.00311311
$931$	33.0277	1.08244
$932$	6.45804	0.211540
$933$	27.7377	0.908090
$934$	−7.34417	−0.240309
$935$	−9.05277	−0.296057
$936$	−5.46152	−0.178515
$937$	8.76107	0.286212	0.143106	−	0.989707i	$-0.454291\pi$
$937$	8.76107	0.286212	0.143106	+	0.989707i	$0.454291\pi$
$938$	−8.85111	−0.288999
$939$	7.89435	0.257622
$940$	19.7934	0.645589
$941$	−47.5972	−1.55162	−0.775812	−	0.630964i	$-0.782660\pi$
$941$	−47.5972	−1.55162	−0.775812	+	0.630964i	$0.782660\pi$
$942$	−0.124835	−0.00406734
$943$	−1.50449	−0.0489929
$944$	1.08429	0.0352907
$945$	3.29378	0.107147
$946$	−1.02419	−0.0332992
$947$	−51.4873	−1.67311	−0.836556	−	0.547881i	$-0.815435\pi$
$947$	−51.4873	−1.67311	−0.836556	+	0.547881i	$0.815435\pi$
$948$	−27.8432	−0.904306
$949$	−28.4010	−0.921937
$950$	3.11844	0.101176
$951$	9.87380	0.320180
$952$	−8.20154	−0.265813
$953$	57.4911	1.86232	0.931160	−	0.364610i	$-0.118798\pi$
$953$	57.4911	1.86232	0.931160	+	0.364610i	$0.118798\pi$
$954$	2.62296	0.0849216
$955$	−20.8017	−0.673128
$956$	0.367512	0.0118862
$957$	−5.10349	−0.164972
$958$	3.60311	0.116411
$959$	−1.66278	−0.0536939
$960$	−5.00240	−0.161452
$961$	−30.9318	−0.997799
$962$	2.74903	0.0886322
$963$	10.7859	0.347571
$964$	30.9359	0.996378
$965$	−2.89762	−0.0932778
$966$	−3.64025	−0.117123
$967$	−5.55760	−0.178720	−0.0893602	−	0.995999i	$-0.528482\pi$
$967$	−5.55760	−0.178720	−0.0893602	+	0.995999i	$0.528482\pi$
$968$	−21.2500	−0.683000
$969$	−15.2002	−0.488301
$970$	2.11724	0.0679805
$971$	18.6237	0.597663	0.298831	−	0.954306i	$-0.403403\pi$
$971$	18.6237	0.597663	0.298831	+	0.954306i	$0.403403\pi$
$972$	1.86793	0.0599138
$973$	52.4376	1.68107
$974$	11.9526	0.382985
$975$	−3.88535	−0.124431
$976$	11.0982	0.355245
$977$	15.2720	0.488594	0.244297	−	0.969700i	$-0.421443\pi$
$977$	15.2720	0.488594	0.244297	+	0.969700i	$0.421443\pi$
$978$	4.02411	0.128677
$979$	−28.3692	−0.906684
$980$	7.18962	0.229664
$981$	−7.49092	−0.239167
$982$	−8.96162	−0.285977
$983$	9.69738	0.309298	0.154649	−	0.987969i	$-0.450575\pi$
$983$	9.69738	0.309298	0.154649	+	0.987969i	$0.450575\pi$
$984$	−0.695410	−0.0221689
$985$	−20.4627	−0.651995
$986$	0.642873	0.0204733
$987$	−34.9023	−1.11095
$988$	−62.2762	−1.98127
$989$	−1.67704	−0.0533267
$990$	−1.85725	−0.0590272
$991$	−14.6621	−0.465757	−0.232878	−	0.972506i	$-0.574814\pi$
$991$	−14.6621	−0.465757	−0.232878	+	0.972506i	$0.574814\pi$
$992$	−1.04059	−0.0330389
$993$	−18.7474	−0.594931
$994$	15.5683	0.493797
$995$	−12.7862	−0.405351
$996$	6.10589	0.193472
$997$	−8.42553	−0.266839	−0.133420	−	0.991060i	$-0.542596\pi$
$997$	−8.42553	−0.266839	−0.133420	+	0.991060i	$0.542596\pi$
$998$	2.69719	0.0853780
$999$	−1.94690	−0.0615972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.e.1.17	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.e.1.17	✓	31	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:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.363417 q^{2} -1.00000 q^{3} -1.86793 q^{4} -1.00000 q^{5} -0.363417 q^{6} +3.29378 q^{7} -1.40567 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.363417 q^{2} -1.00000 q^{3} -1.86793 q^{4} -1.00000 q^{5} -0.363417 q^{6} +3.29378 q^{7} -1.40567 q^{8} +1.00000 q^{9} -0.363417 q^{10} +5.11051 q^{11} +1.86793 q^{12} +3.88535 q^{13} +1.19702 q^{14} +1.00000 q^{15} +3.22501 q^{16} +1.77140 q^{17} +0.363417 q^{18} +8.58089 q^{19} +1.86793 q^{20} -3.29378 q^{21} +1.85725 q^{22} +3.04111 q^{23} +1.40567 q^{24} +1.00000 q^{25} +1.41200 q^{26} -1.00000 q^{27} -6.15254 q^{28} +0.998626 q^{29} +0.363417 q^{30} -0.261235 q^{31} +3.98337 q^{32} -5.11051 q^{33} +0.643758 q^{34} -3.29378 q^{35} -1.86793 q^{36} +1.94690 q^{37} +3.11844 q^{38} -3.88535 q^{39} +1.40567 q^{40} -0.494718 q^{41} -1.19702 q^{42} -0.551455 q^{43} -9.54607 q^{44} -1.00000 q^{45} +1.10519 q^{46} +10.5964 q^{47} -3.22501 q^{48} +3.84898 q^{49} +0.363417 q^{50} -1.77140 q^{51} -7.25755 q^{52} +7.21751 q^{53} -0.363417 q^{54} -5.11051 q^{55} -4.62997 q^{56} -8.58089 q^{57} +0.362918 q^{58} +0.336213 q^{59} -1.86793 q^{60} +3.44130 q^{61} -0.0949372 q^{62} +3.29378 q^{63} -5.00240 q^{64} -3.88535 q^{65} -1.85725 q^{66} -7.39432 q^{67} -3.30885 q^{68} -3.04111 q^{69} -1.19702 q^{70} +13.0060 q^{71} -1.40567 q^{72} -7.30978 q^{73} +0.707537 q^{74} -1.00000 q^{75} -16.0285 q^{76} +16.8329 q^{77} -1.41200 q^{78} -14.9059 q^{79} -3.22501 q^{80} +1.00000 q^{81} -0.179789 q^{82} +3.26880 q^{83} +6.15254 q^{84} -1.77140 q^{85} -0.200408 q^{86} -0.998626 q^{87} -7.18370 q^{88} -5.55115 q^{89} -0.363417 q^{90} +12.7975 q^{91} -5.68057 q^{92} +0.261235 q^{93} +3.85092 q^{94} -8.58089 q^{95} -3.98337 q^{96} -5.82593 q^{97} +1.39878 q^{98} +5.11051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more