LMFDB - Embedded newform 4830.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.46410 q^{11} -1.00000 q^{12} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +1.46410 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.46410 q^{26} -1.00000 q^{27} -1.00000 q^{28} -0.535898 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -1.46410 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -8.92820 q^{37} +4.00000 q^{38} -3.46410 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -1.46410 q^{43} +1.46410 q^{44} -1.00000 q^{45} -1.00000 q^{46} -6.92820 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +3.46410 q^{52} +2.00000 q^{53} -1.00000 q^{54} -1.46410 q^{55} -1.00000 q^{56} -4.00000 q^{57} -0.535898 q^{58} +6.92820 q^{59} +1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.46410 q^{65} -1.46410 q^{66} -1.46410 q^{67} -2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{70} -9.46410 q^{71} +1.00000 q^{72} +2.00000 q^{73} -8.92820 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.46410 q^{77} -3.46410 q^{78} +10.9282 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +2.00000 q^{85} -1.46410 q^{86} +0.535898 q^{87} +1.46410 q^{88} +4.53590 q^{89} -1.00000 q^{90} -3.46410 q^{91} -1.00000 q^{92} -4.00000 q^{93} -6.92820 q^{94} -4.00000 q^{95} -1.00000 q^{96} +7.46410 q^{97} +1.00000 q^{98} +1.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 8 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} - 2 q^{28} - 8 q^{29} + 2 q^{30} + 8 q^{31} + 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 2 q^{40} + 4 q^{41} + 2 q^{42} + 4 q^{43} - 4 q^{44} - 2 q^{45} - 2 q^{46} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 4 q^{51} + 4 q^{53} - 2 q^{54} + 4 q^{55} - 2 q^{56} - 8 q^{57} - 8 q^{58} + 2 q^{60} + 20 q^{61} + 8 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} + 4 q^{67} - 4 q^{68} + 2 q^{69} + 2 q^{70} - 12 q^{71} + 2 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{75} + 8 q^{76} + 4 q^{77} + 8 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 2 q^{84} + 4 q^{85} + 4 q^{86} + 8 q^{87} - 4 q^{88} + 16 q^{89} - 2 q^{90} - 2 q^{92} - 8 q^{93} - 8 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 8 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 - 2 * q^28 - 8 * q^29 + 2 * q^30 + 8 * q^31 + 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 4 * q^37 + 8 * q^38 - 2 * q^40 + 4 * q^41 + 2 * q^42 + 4 * q^43 - 4 * q^44 - 2 * q^45 - 2 * q^46 - 2 * q^48 + 2 * q^49 + 2 * q^50 + 4 * q^51 + 4 * q^53 - 2 * q^54 + 4 * q^55 - 2 * q^56 - 8 * q^57 - 8 * q^58 + 2 * q^60 + 20 * q^61 + 8 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^66 + 4 * q^67 - 4 * q^68 + 2 * q^69 + 2 * q^70 - 12 * q^71 + 2 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^75 + 8 * q^76 + 4 * q^77 + 8 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 2 * q^84 + 4 * q^85 + 4 * q^86 + 8 * q^87 - 4 * q^88 + 16 * q^89 - 2 * q^90 - 2 * q^92 - 8 * q^93 - 8 * q^95 - 2 * q^96 + 8 * q^97 + 2 * q^98 - 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	−1.00000	−0.288675
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	1.46410	0.312148
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	3.46410	0.679366
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−0.535898	−0.0995138	−0.0497569	−	0.998761i	$-0.515845\pi$
$29$	−0.535898	−0.0995138	−0.0497569	+	0.998761i	$0.515845\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−1.46410	−0.254867
$34$	−2.00000	−0.342997
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−8.92820	−1.46779	−0.733894	−	0.679264i	$-0.762299\pi$
$37$	−8.92820	−1.46779	−0.733894	+	0.679264i	$0.762299\pi$
$38$	4.00000	0.648886
$39$	−3.46410	−0.554700
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	−1.46410	−0.223273	−0.111637	−	0.993749i	$-0.535609\pi$
$43$	−1.46410	−0.223273	−0.111637	+	0.993749i	$0.535609\pi$
$44$	1.46410	0.220722
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	3.46410	0.480384
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	−1.46410	−0.197419
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	−0.535898	−0.0703669
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	1.00000	0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−3.46410	−0.429669
$66$	−1.46410	−0.180218
$67$	−1.46410	−0.178868	−0.0894342	−	0.995993i	$-0.528506\pi$
$67$	−1.46410	−0.178868	−0.0894342	+	0.995993i	$0.528506\pi$
$68$	−2.00000	−0.242536
$69$	1.00000	0.120386
$70$	1.00000	0.119523
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.92820	−1.03788
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	−1.46410	−0.166850
$78$	−3.46410	−0.392232
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	1.00000	0.109109
$85$	2.00000	0.216930
$86$	−1.46410	−0.157878
$87$	0.535898	0.0574543
$88$	1.46410	0.156074
$89$	4.53590	0.480804	0.240402	−	0.970673i	$-0.422721\pi$
$89$	4.53590	0.480804	0.240402	+	0.970673i	$0.422721\pi$
$90$	−1.00000	−0.105409
$91$	−3.46410	−0.363137
$92$	−1.00000	−0.104257
$93$	−4.00000	−0.414781
$94$	−6.92820	−0.714590
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	7.46410	0.757865	0.378932	−	0.925424i	$-0.376291\pi$
$97$	7.46410	0.757865	0.378932	+	0.925424i	$0.376291\pi$
$98$	1.00000	0.101015
$99$	1.46410	0.147148
$100$	1.00000	0.100000
$101$	−2.39230	−0.238043	−0.119022	−	0.992892i	$-0.537976\pi$
$101$	−2.39230	−0.238043	−0.119022	+	0.992892i	$0.537976\pi$
$102$	2.00000	0.198030
$103$	13.8564	1.36531	0.682656	−	0.730740i	$-0.260825\pi$
$103$	13.8564	1.36531	0.682656	+	0.730740i	$0.260825\pi$
$104$	3.46410	0.339683
$105$	−1.00000	−0.0975900
$106$	2.00000	0.194257
$107$	−2.92820	−0.283080	−0.141540	−	0.989933i	$-0.545205\pi$
$107$	−2.92820	−0.283080	−0.141540	+	0.989933i	$0.545205\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.46410	−0.139597
$111$	8.92820	0.847428
$112$	−1.00000	−0.0944911
$113$	7.85641	0.739069	0.369534	−	0.929217i	$-0.379517\pi$
$113$	7.85641	0.739069	0.369534	+	0.929217i	$0.379517\pi$
$114$	−4.00000	−0.374634
$115$	1.00000	0.0932505
$116$	−0.535898	−0.0497569
$117$	3.46410	0.320256
$118$	6.92820	0.637793
$119$	2.00000	0.183340
$120$	1.00000	0.0912871
$121$	−8.85641	−0.805128
$122$	10.0000	0.905357
$123$	−2.00000	−0.180334
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	12.3923	1.09964	0.549820	−	0.835283i	$-0.314696\pi$
$127$	12.3923	1.09964	0.549820	+	0.835283i	$0.314696\pi$
$128$	1.00000	0.0883883
$129$	1.46410	0.128907
$130$	−3.46410	−0.303822
$131$	17.8564	1.56012	0.780061	−	0.625704i	$-0.215188\pi$
$131$	17.8564	1.56012	0.780061	+	0.625704i	$0.215188\pi$
$132$	−1.46410	−0.127434
$133$	−4.00000	−0.346844
$134$	−1.46410	−0.126479
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	15.8564	1.35470	0.677352	−	0.735659i	$-0.263127\pi$
$137$	15.8564	1.35470	0.677352	+	0.735659i	$0.263127\pi$
$138$	1.00000	0.0851257
$139$	−9.85641	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$139$	−9.85641	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$140$	1.00000	0.0845154
$141$	6.92820	0.583460
$142$	−9.46410	−0.794210
$143$	5.07180	0.424125
$144$	1.00000	0.0833333
$145$	0.535898	0.0445039
$146$	2.00000	0.165521
$147$	−1.00000	−0.0824786
$148$	−8.92820	−0.733894
$149$	−8.92820	−0.731427	−0.365713	−	0.930727i	$-0.619175\pi$
$149$	−8.92820	−0.731427	−0.365713	+	0.930727i	$0.619175\pi$
$150$	−1.00000	−0.0816497
$151$	10.9282	0.889325	0.444662	−	0.895698i	$-0.353324\pi$
$151$	10.9282	0.889325	0.444662	+	0.895698i	$0.353324\pi$
$152$	4.00000	0.324443
$153$	−2.00000	−0.161690
$154$	−1.46410	−0.117981
$155$	−4.00000	−0.321288
$156$	−3.46410	−0.277350
$157$	23.8564	1.90395	0.951974	−	0.306178i	$-0.0990503\pi$
$157$	23.8564	1.90395	0.951974	+	0.306178i	$0.0990503\pi$
$158$	10.9282	0.869401
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	−14.9282	−1.16927	−0.584634	−	0.811297i	$-0.698762\pi$
$163$	−14.9282	−1.16927	−0.584634	+	0.811297i	$0.698762\pi$
$164$	2.00000	0.156174
$165$	1.46410	0.113980
$166$	4.00000	0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	1.00000	0.0771517
$169$	−1.00000	−0.0769231
$170$	2.00000	0.153393
$171$	4.00000	0.305888
$172$	−1.46410	−0.111637
$173$	−8.92820	−0.678799	−0.339399	−	0.940642i	$-0.610224\pi$
$173$	−8.92820	−0.678799	−0.339399	+	0.940642i	$0.610224\pi$
$174$	0.535898	0.0406264
$175$	−1.00000	−0.0755929
$176$	1.46410	0.110361
$177$	−6.92820	−0.520756
$178$	4.53590	0.339980
$179$	9.07180	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$179$	9.07180	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−3.46410	−0.256776
$183$	−10.0000	−0.739221
$184$	−1.00000	−0.0737210
$185$	8.92820	0.656415
$186$	−4.00000	−0.293294
$187$	−2.92820	−0.214131
$188$	−6.92820	−0.505291
$189$	1.00000	0.0727393
$190$	−4.00000	−0.290191
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	7.46410	0.535891
$195$	3.46410	0.248069
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	1.46410	0.104049
$199$	26.9282	1.90889	0.954445	−	0.298387i	$-0.0964487\pi$
$199$	26.9282	1.90889	0.954445	+	0.298387i	$0.0964487\pi$
$200$	1.00000	0.0707107
$201$	1.46410	0.103270
$202$	−2.39230	−0.168322
$203$	0.535898	0.0376127
$204$	2.00000	0.140028
$205$	−2.00000	−0.139686
$206$	13.8564	0.965422
$207$	−1.00000	−0.0695048
$208$	3.46410	0.240192
$209$	5.85641	0.405096
$210$	−1.00000	−0.0690066
$211$	−20.7846	−1.43087	−0.715436	−	0.698679i	$-0.753772\pi$
$211$	−20.7846	−1.43087	−0.715436	+	0.698679i	$0.753772\pi$
$212$	2.00000	0.137361
$213$	9.46410	0.648470
$214$	−2.92820	−0.200168
$215$	1.46410	0.0998509
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	10.0000	0.677285
$219$	−2.00000	−0.135147
$220$	−1.46410	−0.0987097
$221$	−6.92820	−0.466041
$222$	8.92820	0.599222
$223$	2.53590	0.169816	0.0849082	−	0.996389i	$-0.472940\pi$
$223$	2.53590	0.169816	0.0849082	+	0.996389i	$0.472940\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	7.85641	0.522600
$227$	1.07180	0.0711377	0.0355688	−	0.999367i	$-0.488676\pi$
$227$	1.07180	0.0711377	0.0355688	+	0.999367i	$0.488676\pi$
$228$	−4.00000	−0.264906
$229$	15.8564	1.04782	0.523910	−	0.851773i	$-0.324473\pi$
$229$	15.8564	1.04782	0.523910	+	0.851773i	$0.324473\pi$
$230$	1.00000	0.0659380
$231$	1.46410	0.0963308
$232$	−0.535898	−0.0351835
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	3.46410	0.226455
$235$	6.92820	0.451946
$236$	6.92820	0.450988
$237$	−10.9282	−0.709863
$238$	2.00000	0.129641
$239$	12.3923	0.801592	0.400796	−	0.916167i	$-0.368734\pi$
$239$	12.3923	0.801592	0.400796	+	0.916167i	$0.368734\pi$
$240$	1.00000	0.0645497
$241$	−4.92820	−0.317453	−0.158727	−	0.987323i	$-0.550739\pi$
$241$	−4.92820	−0.317453	−0.158727	+	0.987323i	$0.550739\pi$
$242$	−8.85641	−0.569311
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	−1.00000	−0.0638877
$246$	−2.00000	−0.127515
$247$	13.8564	0.881662
$248$	4.00000	0.254000
$249$	−4.00000	−0.253490
$250$	−1.00000	−0.0632456
$251$	−30.2487	−1.90928	−0.954641	−	0.297760i	$-0.903761\pi$
$251$	−30.2487	−1.90928	−0.954641	+	0.297760i	$0.903761\pi$
$252$	−1.00000	−0.0629941
$253$	−1.46410	−0.0920473
$254$	12.3923	0.777562
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	1.46410	0.0911510
$259$	8.92820	0.554772
$260$	−3.46410	−0.214834
$261$	−0.535898	−0.0331713
$262$	17.8564	1.10317
$263$	18.9282	1.16716	0.583582	−	0.812055i	$-0.301651\pi$
$263$	18.9282	1.16716	0.583582	+	0.812055i	$0.301651\pi$
$264$	−1.46410	−0.0901092
$265$	−2.00000	−0.122859
$266$	−4.00000	−0.245256
$267$	−4.53590	−0.277592
$268$	−1.46410	−0.0894342
$269$	−21.3205	−1.29993	−0.649967	−	0.759962i	$-0.725217\pi$
$269$	−21.3205	−1.29993	−0.649967	+	0.759962i	$0.725217\pi$
$270$	1.00000	0.0608581
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−2.00000	−0.121268
$273$	3.46410	0.209657
$274$	15.8564	0.957921
$275$	1.46410	0.0882886
$276$	1.00000	0.0601929
$277$	−0.535898	−0.0321990	−0.0160995	−	0.999870i	$-0.505125\pi$
$277$	−0.535898	−0.0321990	−0.0160995	+	0.999870i	$0.505125\pi$
$278$	−9.85641	−0.591148
$279$	4.00000	0.239474
$280$	1.00000	0.0597614
$281$	20.2487	1.20794	0.603968	−	0.797008i	$-0.293585\pi$
$281$	20.2487	1.20794	0.603968	+	0.797008i	$0.293585\pi$
$282$	6.92820	0.412568
$283$	24.3923	1.44997	0.724986	−	0.688764i	$-0.241846\pi$
$283$	24.3923	1.44997	0.724986	+	0.688764i	$0.241846\pi$
$284$	−9.46410	−0.561591
$285$	4.00000	0.236940
$286$	5.07180	0.299902
$287$	−2.00000	−0.118056
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0.535898	0.0314690
$291$	−7.46410	−0.437553
$292$	2.00000	0.117041
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	−1.00000	−0.0583212
$295$	−6.92820	−0.403376
$296$	−8.92820	−0.518941
$297$	−1.46410	−0.0849558
$298$	−8.92820	−0.517197
$299$	−3.46410	−0.200334
$300$	−1.00000	−0.0577350
$301$	1.46410	0.0843894
$302$	10.9282	0.628847
$303$	2.39230	0.137434
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	25.8564	1.47570	0.737852	−	0.674963i	$-0.235840\pi$
$307$	25.8564	1.47570	0.737852	+	0.674963i	$0.235840\pi$
$308$	−1.46410	−0.0834249
$309$	−13.8564	−0.788263
$310$	−4.00000	−0.227185
$311$	11.3205	0.641927	0.320964	−	0.947092i	$-0.395993\pi$
$311$	11.3205	0.641927	0.320964	+	0.947092i	$0.395993\pi$
$312$	−3.46410	−0.196116
$313$	−17.3205	−0.979013	−0.489506	−	0.872000i	$-0.662823\pi$
$313$	−17.3205	−0.979013	−0.489506	+	0.872000i	$0.662823\pi$
$314$	23.8564	1.34629
$315$	1.00000	0.0563436
$316$	10.9282	0.614759
$317$	−12.9282	−0.726120	−0.363060	−	0.931766i	$-0.618268\pi$
$317$	−12.9282	−0.726120	−0.363060	+	0.931766i	$0.618268\pi$
$318$	−2.00000	−0.112154
$319$	−0.784610	−0.0439297
$320$	−1.00000	−0.0559017
$321$	2.92820	0.163436
$322$	1.00000	0.0557278
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	3.46410	0.192154
$326$	−14.9282	−0.826797
$327$	−10.0000	−0.553001
$328$	2.00000	0.110432
$329$	6.92820	0.381964
$330$	1.46410	0.0805961
$331$	22.9282	1.26025	0.630124	−	0.776495i	$-0.283004\pi$
$331$	22.9282	1.26025	0.630124	+	0.776495i	$0.283004\pi$
$332$	4.00000	0.219529
$333$	−8.92820	−0.489263
$334$	12.0000	0.656611
$335$	1.46410	0.0799924
$336$	1.00000	0.0545545
$337$	−21.3205	−1.16140	−0.580701	−	0.814117i	$-0.697221\pi$
$337$	−21.3205	−1.16140	−0.580701	+	0.814117i	$0.697221\pi$
$338$	−1.00000	−0.0543928
$339$	−7.85641	−0.426701
$340$	2.00000	0.108465
$341$	5.85641	0.317142
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	−1.46410	−0.0789391
$345$	−1.00000	−0.0538382
$346$	−8.92820	−0.479983
$347$	−25.8564	−1.38804	−0.694022	−	0.719953i	$-0.744163\pi$
$347$	−25.8564	−1.38804	−0.694022	+	0.719953i	$0.744163\pi$
$348$	0.535898	0.0287272
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	−1.00000	−0.0534522
$351$	−3.46410	−0.184900
$352$	1.46410	0.0780369
$353$	−11.8564	−0.631053	−0.315526	−	0.948917i	$-0.602181\pi$
$353$	−11.8564	−0.631053	−0.315526	+	0.948917i	$0.602181\pi$
$354$	−6.92820	−0.368230
$355$	9.46410	0.502302
$356$	4.53590	0.240402
$357$	−2.00000	−0.105851
$358$	9.07180	0.479459
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	8.85641	0.464841
$364$	−3.46410	−0.181568
$365$	−2.00000	−0.104685
$366$	−10.0000	−0.522708
$367$	−19.7128	−1.02900	−0.514500	−	0.857490i	$-0.672023\pi$
$367$	−19.7128	−1.02900	−0.514500	+	0.857490i	$0.672023\pi$
$368$	−1.00000	−0.0521286
$369$	2.00000	0.104116
$370$	8.92820	0.464155
$371$	−2.00000	−0.103835
$372$	−4.00000	−0.207390
$373$	−8.92820	−0.462285	−0.231142	−	0.972920i	$-0.574246\pi$
$373$	−8.92820	−0.462285	−0.231142	+	0.972920i	$0.574246\pi$
$374$	−2.92820	−0.151414
$375$	1.00000	0.0516398
$376$	−6.92820	−0.357295
$377$	−1.85641	−0.0956098
$378$	1.00000	0.0514344
$379$	26.9282	1.38321	0.691604	−	0.722276i	$-0.256904\pi$
$379$	26.9282	1.38321	0.691604	+	0.722276i	$0.256904\pi$
$380$	−4.00000	−0.205196
$381$	−12.3923	−0.634877
$382$	8.00000	0.409316
$383$	−21.8564	−1.11681	−0.558405	−	0.829568i	$-0.688587\pi$
$383$	−21.8564	−1.11681	−0.558405	+	0.829568i	$0.688587\pi$
$384$	−1.00000	−0.0510310
$385$	1.46410	0.0746175
$386$	−14.0000	−0.712581
$387$	−1.46410	−0.0744245
$388$	7.46410	0.378932
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	3.46410	0.175412
$391$	2.00000	0.101144
$392$	1.00000	0.0505076
$393$	−17.8564	−0.900737
$394$	−2.00000	−0.100759
$395$	−10.9282	−0.549858
$396$	1.46410	0.0735739
$397$	25.3205	1.27080	0.635400	−	0.772183i	$-0.280835\pi$
$397$	25.3205	1.27080	0.635400	+	0.772183i	$0.280835\pi$
$398$	26.9282	1.34979
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	−4.53590	−0.226512	−0.113256	−	0.993566i	$-0.536128\pi$
$401$	−4.53590	−0.226512	−0.113256	+	0.993566i	$0.536128\pi$
$402$	1.46410	0.0730228
$403$	13.8564	0.690237
$404$	−2.39230	−0.119022
$405$	−1.00000	−0.0496904
$406$	0.535898	0.0265962
$407$	−13.0718	−0.647945
$408$	2.00000	0.0990148
$409$	23.8564	1.17962	0.589812	−	0.807541i	$-0.299202\pi$
$409$	23.8564	1.17962	0.589812	+	0.807541i	$0.299202\pi$
$410$	−2.00000	−0.0987730
$411$	−15.8564	−0.782139
$412$	13.8564	0.682656
$413$	−6.92820	−0.340915
$414$	−1.00000	−0.0491473
$415$	−4.00000	−0.196352
$416$	3.46410	0.169842
$417$	9.85641	0.482670
$418$	5.85641	0.286446
$419$	18.5359	0.905538	0.452769	−	0.891628i	$-0.350436\pi$
$419$	18.5359	0.905538	0.452769	+	0.891628i	$0.350436\pi$
$420$	−1.00000	−0.0487950
$421$	−3.85641	−0.187950	−0.0939749	−	0.995575i	$-0.529957\pi$
$421$	−3.85641	−0.187950	−0.0939749	+	0.995575i	$0.529957\pi$
$422$	−20.7846	−1.01178
$423$	−6.92820	−0.336861
$424$	2.00000	0.0971286
$425$	−2.00000	−0.0970143
$426$	9.46410	0.458537
$427$	−10.0000	−0.483934
$428$	−2.92820	−0.141540
$429$	−5.07180	−0.244869
$430$	1.46410	0.0706052
$431$	−21.8564	−1.05279	−0.526393	−	0.850241i	$-0.676456\pi$
$431$	−21.8564	−1.05279	−0.526393	+	0.850241i	$0.676456\pi$
$432$	−1.00000	−0.0481125
$433$	−20.2487	−0.973091	−0.486545	−	0.873655i	$-0.661743\pi$
$433$	−20.2487	−0.973091	−0.486545	+	0.873655i	$0.661743\pi$
$434$	−4.00000	−0.192006
$435$	−0.535898	−0.0256944
$436$	10.0000	0.478913
$437$	−4.00000	−0.191346
$438$	−2.00000	−0.0955637
$439$	36.7846	1.75563	0.877817	−	0.478996i	$-0.158999\pi$
$439$	36.7846	1.75563	0.877817	+	0.478996i	$0.158999\pi$
$440$	−1.46410	−0.0697983
$441$	1.00000	0.0476190
$442$	−6.92820	−0.329541
$443$	6.14359	0.291891	0.145945	−	0.989293i	$-0.453378\pi$
$443$	6.14359	0.291891	0.145945	+	0.989293i	$0.453378\pi$
$444$	8.92820	0.423714
$445$	−4.53590	−0.215022
$446$	2.53590	0.120078
$447$	8.92820	0.422290
$448$	−1.00000	−0.0472456
$449$	−11.0718	−0.522510	−0.261255	−	0.965270i	$-0.584136\pi$
$449$	−11.0718	−0.522510	−0.261255	+	0.965270i	$0.584136\pi$
$450$	1.00000	0.0471405
$451$	2.92820	0.137884
$452$	7.85641	0.369534
$453$	−10.9282	−0.513452
$454$	1.07180	0.0503019
$455$	3.46410	0.162400
$456$	−4.00000	−0.187317
$457$	−4.53590	−0.212180	−0.106090	−	0.994357i	$-0.533833\pi$
$457$	−4.53590	−0.212180	−0.106090	+	0.994357i	$0.533833\pi$
$458$	15.8564	0.740921
$459$	2.00000	0.0933520
$460$	1.00000	0.0466252
$461$	−24.2487	−1.12938	−0.564688	−	0.825305i	$-0.691003\pi$
$461$	−24.2487	−1.12938	−0.564688	+	0.825305i	$0.691003\pi$
$462$	1.46410	0.0681162
$463$	−20.3923	−0.947711	−0.473855	−	0.880603i	$-0.657138\pi$
$463$	−20.3923	−0.947711	−0.473855	+	0.880603i	$0.657138\pi$
$464$	−0.535898	−0.0248785
$465$	4.00000	0.185496
$466$	7.85641	0.363941
$467$	17.8564	0.826296	0.413148	−	0.910664i	$-0.364429\pi$
$467$	17.8564	0.826296	0.413148	+	0.910664i	$0.364429\pi$
$468$	3.46410	0.160128
$469$	1.46410	0.0676059
$470$	6.92820	0.319574
$471$	−23.8564	−1.09925
$472$	6.92820	0.318896
$473$	−2.14359	−0.0985625
$474$	−10.9282	−0.501949
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	2.00000	0.0915737
$478$	12.3923	0.566811
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	1.00000	0.0456435
$481$	−30.9282	−1.41020
$482$	−4.92820	−0.224474
$483$	−1.00000	−0.0455016
$484$	−8.85641	−0.402564
$485$	−7.46410	−0.338927
$486$	−1.00000	−0.0453609
$487$	−12.3923	−0.561549	−0.280774	−	0.959774i	$-0.590591\pi$
$487$	−12.3923	−0.561549	−0.280774	+	0.959774i	$0.590591\pi$
$488$	10.0000	0.452679
$489$	14.9282	0.675077
$490$	−1.00000	−0.0451754
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−2.00000	−0.0901670
$493$	1.07180	0.0482713
$494$	13.8564	0.623429
$495$	−1.46410	−0.0658065
$496$	4.00000	0.179605
$497$	9.46410	0.424523
$498$	−4.00000	−0.179244
$499$	14.9282	0.668278	0.334139	−	0.942524i	$-0.391554\pi$
$499$	14.9282	0.668278	0.334139	+	0.942524i	$0.391554\pi$
$500$	−1.00000	−0.0447214
$501$	−12.0000	−0.536120
$502$	−30.2487	−1.35007
$503$	−10.9282	−0.487264	−0.243632	−	0.969868i	$-0.578339\pi$
$503$	−10.9282	−0.487264	−0.243632	+	0.969868i	$0.578339\pi$
$504$	−1.00000	−0.0445435
$505$	2.39230	0.106456
$506$	−1.46410	−0.0650873
$507$	1.00000	0.0444116
$508$	12.3923	0.549820
$509$	−24.2487	−1.07481	−0.537403	−	0.843326i	$-0.680594\pi$
$509$	−24.2487	−1.07481	−0.537403	+	0.843326i	$0.680594\pi$
$510$	−2.00000	−0.0885615
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−6.00000	−0.264649
$515$	−13.8564	−0.610586
$516$	1.46410	0.0644535
$517$	−10.1436	−0.446115
$518$	8.92820	0.392283
$519$	8.92820	0.391905
$520$	−3.46410	−0.151911
$521$	−19.4641	−0.852738	−0.426369	−	0.904549i	$-0.640207\pi$
$521$	−19.4641	−0.852738	−0.426369	+	0.904549i	$0.640207\pi$
$522$	−0.535898	−0.0234556
$523$	−5.46410	−0.238928	−0.119464	−	0.992839i	$-0.538118\pi$
$523$	−5.46410	−0.238928	−0.119464	+	0.992839i	$0.538118\pi$
$524$	17.8564	0.780061
$525$	1.00000	0.0436436
$526$	18.9282	0.825309
$527$	−8.00000	−0.348485
$528$	−1.46410	−0.0637168
$529$	1.00000	0.0434783
$530$	−2.00000	−0.0868744
$531$	6.92820	0.300658
$532$	−4.00000	−0.173422
$533$	6.92820	0.300094
$534$	−4.53590	−0.196288
$535$	2.92820	0.126597
$536$	−1.46410	−0.0632396
$537$	−9.07180	−0.391477
$538$	−21.3205	−0.919192
$539$	1.46410	0.0630633
$540$	1.00000	0.0430331
$541$	3.07180	0.132067	0.0660334	−	0.997817i	$-0.478966\pi$
$541$	3.07180	0.132067	0.0660334	+	0.997817i	$0.478966\pi$
$542$	−4.00000	−0.171815
$543$	−2.00000	−0.0858282
$544$	−2.00000	−0.0857493
$545$	−10.0000	−0.428353
$546$	3.46410	0.148250
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	15.8564	0.677352
$549$	10.0000	0.426790
$550$	1.46410	0.0624295
$551$	−2.14359	−0.0913202
$552$	1.00000	0.0425628
$553$	−10.9282	−0.464714
$554$	−0.535898	−0.0227681
$555$	−8.92820	−0.378981
$556$	−9.85641	−0.418005
$557$	21.7128	0.920001	0.460001	−	0.887919i	$-0.347849\pi$
$557$	21.7128	0.920001	0.460001	+	0.887919i	$0.347849\pi$
$558$	4.00000	0.169334
$559$	−5.07180	−0.214514
$560$	1.00000	0.0422577
$561$	2.92820	0.123629
$562$	20.2487	0.854140
$563$	−14.1436	−0.596081	−0.298041	−	0.954553i	$-0.596333\pi$
$563$	−14.1436	−0.596081	−0.298041	+	0.954553i	$0.596333\pi$
$564$	6.92820	0.291730
$565$	−7.85641	−0.330522
$566$	24.3923	1.02529
$567$	−1.00000	−0.0419961
$568$	−9.46410	−0.397105
$569$	13.6077	0.570464	0.285232	−	0.958458i	$-0.407929\pi$
$569$	13.6077	0.570464	0.285232	+	0.958458i	$0.407929\pi$
$570$	4.00000	0.167542
$571$	−18.9282	−0.792121	−0.396060	−	0.918224i	$-0.629623\pi$
$571$	−18.9282	−0.792121	−0.396060	+	0.918224i	$0.629623\pi$
$572$	5.07180	0.212062
$573$	−8.00000	−0.334205
$574$	−2.00000	−0.0834784
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	15.8564	0.660111	0.330055	−	0.943962i	$-0.392933\pi$
$577$	15.8564	0.660111	0.330055	+	0.943962i	$0.392933\pi$
$578$	−13.0000	−0.540729
$579$	14.0000	0.581820
$580$	0.535898	0.0222520
$581$	−4.00000	−0.165948
$582$	−7.46410	−0.309397
$583$	2.92820	0.121274
$584$	2.00000	0.0827606
$585$	−3.46410	−0.143223
$586$	2.00000	0.0826192
$587$	−6.92820	−0.285958	−0.142979	−	0.989726i	$-0.545668\pi$
$587$	−6.92820	−0.285958	−0.142979	+	0.989726i	$0.545668\pi$
$588$	−1.00000	−0.0412393
$589$	16.0000	0.659269
$590$	−6.92820	−0.285230
$591$	2.00000	0.0822690
$592$	−8.92820	−0.366947
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	−1.46410	−0.0600728
$595$	−2.00000	−0.0819920
$596$	−8.92820	−0.365713
$597$	−26.9282	−1.10210
$598$	−3.46410	−0.141658
$599$	25.4641	1.04043	0.520217	−	0.854034i	$-0.325851\pi$
$599$	25.4641	1.04043	0.520217	+	0.854034i	$0.325851\pi$
$600$	−1.00000	−0.0408248
$601$	−0.143594	−0.00585730	−0.00292865	−	0.999996i	$-0.500932\pi$
$601$	−0.143594	−0.00585730	−0.00292865	+	0.999996i	$0.500932\pi$
$602$	1.46410	0.0596723
$603$	−1.46410	−0.0596228
$604$	10.9282	0.444662
$605$	8.85641	0.360064
$606$	2.39230	0.0971807
$607$	0.392305	0.0159232	0.00796158	−	0.999968i	$-0.497466\pi$
$607$	0.392305	0.0159232	0.00796158	+	0.999968i	$0.497466\pi$
$608$	4.00000	0.162221
$609$	−0.535898	−0.0217157
$610$	−10.0000	−0.404888
$611$	−24.0000	−0.970936
$612$	−2.00000	−0.0808452
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	25.8564	1.04348
$615$	2.00000	0.0806478
$616$	−1.46410	−0.0589903
$617$	−11.0718	−0.445734	−0.222867	−	0.974849i	$-0.571542\pi$
$617$	−11.0718	−0.445734	−0.222867	+	0.974849i	$0.571542\pi$
$618$	−13.8564	−0.557386
$619$	6.14359	0.246932	0.123466	−	0.992349i	$-0.460599\pi$
$619$	6.14359	0.246932	0.123466	+	0.992349i	$0.460599\pi$
$620$	−4.00000	−0.160644
$621$	1.00000	0.0401286
$622$	11.3205	0.453911
$623$	−4.53590	−0.181727
$624$	−3.46410	−0.138675
$625$	1.00000	0.0400000
$626$	−17.3205	−0.692267
$627$	−5.85641	−0.233882
$628$	23.8564	0.951974
$629$	17.8564	0.711982
$630$	1.00000	0.0398410
$631$	42.9282	1.70894	0.854472	−	0.519497i	$-0.173881\pi$
$631$	42.9282	1.70894	0.854472	+	0.519497i	$0.173881\pi$
$632$	10.9282	0.434701
$633$	20.7846	0.826114
$634$	−12.9282	−0.513445
$635$	−12.3923	−0.491774
$636$	−2.00000	−0.0793052
$637$	3.46410	0.137253
$638$	−0.784610	−0.0310630
$639$	−9.46410	−0.374394
$640$	−1.00000	−0.0395285
$641$	−23.4641	−0.926776	−0.463388	−	0.886155i	$-0.653366\pi$
$641$	−23.4641	−0.926776	−0.463388	+	0.886155i	$0.653366\pi$
$642$	2.92820	0.115567
$643$	14.2487	0.561914	0.280957	−	0.959720i	$-0.409348\pi$
$643$	14.2487	0.561914	0.280957	+	0.959720i	$0.409348\pi$
$644$	1.00000	0.0394055
$645$	−1.46410	−0.0576489
$646$	−8.00000	−0.314756
$647$	−25.0718	−0.985674	−0.492837	−	0.870122i	$-0.664040\pi$
$647$	−25.0718	−0.985674	−0.492837	+	0.870122i	$0.664040\pi$
$648$	1.00000	0.0392837
$649$	10.1436	0.398171
$650$	3.46410	0.135873
$651$	4.00000	0.156772
$652$	−14.9282	−0.584634
$653$	8.92820	0.349388	0.174694	−	0.984623i	$-0.444106\pi$
$653$	8.92820	0.349388	0.174694	+	0.984623i	$0.444106\pi$
$654$	−10.0000	−0.391031
$655$	−17.8564	−0.697708
$656$	2.00000	0.0780869
$657$	2.00000	0.0780274
$658$	6.92820	0.270089
$659$	−18.2487	−0.710869	−0.355434	−	0.934701i	$-0.615667\pi$
$659$	−18.2487	−0.710869	−0.355434	+	0.934701i	$0.615667\pi$
$660$	1.46410	0.0569901
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	22.9282	0.891130
$663$	6.92820	0.269069
$664$	4.00000	0.155230
$665$	4.00000	0.155113
$666$	−8.92820	−0.345961
$667$	0.535898	0.0207501
$668$	12.0000	0.464294
$669$	−2.53590	−0.0980435
$670$	1.46410	0.0565632
$671$	14.6410	0.565210
$672$	1.00000	0.0385758
$673$	−27.0718	−1.04354	−0.521771	−	0.853086i	$-0.674728\pi$
$673$	−27.0718	−1.04354	−0.521771	+	0.853086i	$0.674728\pi$
$674$	−21.3205	−0.821235
$675$	−1.00000	−0.0384900
$676$	−1.00000	−0.0384615
$677$	37.7128	1.44942	0.724711	−	0.689053i	$-0.241973\pi$
$677$	37.7128	1.44942	0.724711	+	0.689053i	$0.241973\pi$
$678$	−7.85641	−0.301723
$679$	−7.46410	−0.286446
$680$	2.00000	0.0766965
$681$	−1.07180	−0.0410713
$682$	5.85641	0.224253
$683$	15.7128	0.601234	0.300617	−	0.953745i	$-0.402807\pi$
$683$	15.7128	0.601234	0.300617	+	0.953745i	$0.402807\pi$
$684$	4.00000	0.152944
$685$	−15.8564	−0.605842
$686$	−1.00000	−0.0381802
$687$	−15.8564	−0.604960
$688$	−1.46410	−0.0558184
$689$	6.92820	0.263944
$690$	−1.00000	−0.0380693
$691$	42.6410	1.62214	0.811070	−	0.584949i	$-0.198885\pi$
$691$	42.6410	1.62214	0.811070	+	0.584949i	$0.198885\pi$
$692$	−8.92820	−0.339399
$693$	−1.46410	−0.0556166
$694$	−25.8564	−0.981496
$695$	9.85641	0.373875
$696$	0.535898	0.0203132
$697$	−4.00000	−0.151511
$698$	−6.00000	−0.227103
$699$	−7.85641	−0.297157
$700$	−1.00000	−0.0377964
$701$	−27.8564	−1.05212	−0.526061	−	0.850447i	$-0.676332\pi$
$701$	−27.8564	−1.05212	−0.526061	+	0.850447i	$0.676332\pi$
$702$	−3.46410	−0.130744
$703$	−35.7128	−1.34693
$704$	1.46410	0.0551804
$705$	−6.92820	−0.260931
$706$	−11.8564	−0.446222
$707$	2.39230	0.0899719
$708$	−6.92820	−0.260378
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	9.46410	0.355181
$711$	10.9282	0.409840
$712$	4.53590	0.169990
$713$	−4.00000	−0.149801
$714$	−2.00000	−0.0748481
$715$	−5.07180	−0.189674
$716$	9.07180	0.339029
$717$	−12.3923	−0.462799
$718$	0	0
$719$	−3.32051	−0.123834	−0.0619170	−	0.998081i	$-0.519721\pi$
$719$	−3.32051	−0.123834	−0.0619170	+	0.998081i	$0.519721\pi$
$720$	−1.00000	−0.0372678
$721$	−13.8564	−0.516040
$722$	−3.00000	−0.111648
$723$	4.92820	0.183282
$724$	2.00000	0.0743294
$725$	−0.535898	−0.0199028
$726$	8.85641	0.328692
$727$	−30.6410	−1.13641	−0.568206	−	0.822886i	$-0.692362\pi$
$727$	−30.6410	−1.13641	−0.568206	+	0.822886i	$0.692362\pi$
$728$	−3.46410	−0.128388
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	2.92820	0.108304
$732$	−10.0000	−0.369611
$733$	−41.7128	−1.54070	−0.770349	−	0.637623i	$-0.779918\pi$
$733$	−41.7128	−1.54070	−0.770349	+	0.637623i	$0.779918\pi$
$734$	−19.7128	−0.727613
$735$	1.00000	0.0368856
$736$	−1.00000	−0.0368605
$737$	−2.14359	−0.0789603
$738$	2.00000	0.0736210
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	8.92820	0.328207
$741$	−13.8564	−0.509028
$742$	−2.00000	−0.0734223
$743$	−32.7846	−1.20275	−0.601375	−	0.798967i	$-0.705380\pi$
$743$	−32.7846	−1.20275	−0.601375	+	0.798967i	$0.705380\pi$
$744$	−4.00000	−0.146647
$745$	8.92820	0.327104
$746$	−8.92820	−0.326885
$747$	4.00000	0.146352
$748$	−2.92820	−0.107066
$749$	2.92820	0.106994
$750$	1.00000	0.0365148
$751$	21.8564	0.797552	0.398776	−	0.917048i	$-0.369435\pi$
$751$	21.8564	0.797552	0.398776	+	0.917048i	$0.369435\pi$
$752$	−6.92820	−0.252646
$753$	30.2487	1.10232
$754$	−1.85641	−0.0676063
$755$	−10.9282	−0.397718
$756$	1.00000	0.0363696
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	26.9282	0.978076
$759$	1.46410	0.0531435
$760$	−4.00000	−0.145095
$761$	−52.6410	−1.90824	−0.954118	−	0.299432i	$-0.903203\pi$
$761$	−52.6410	−1.90824	−0.954118	+	0.299432i	$0.903203\pi$
$762$	−12.3923	−0.448926
$763$	−10.0000	−0.362024
$764$	8.00000	0.289430
$765$	2.00000	0.0723102
$766$	−21.8564	−0.789704
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	25.7128	0.927228	0.463614	−	0.886037i	$-0.346552\pi$
$769$	25.7128	0.927228	0.463614	+	0.886037i	$0.346552\pi$
$770$	1.46410	0.0527626
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	−1.46410	−0.0526260
$775$	4.00000	0.143684
$776$	7.46410	0.267946
$777$	−8.92820	−0.320298
$778$	18.0000	0.645331
$779$	8.00000	0.286630
$780$	3.46410	0.124035
$781$	−13.8564	−0.495821
$782$	2.00000	0.0715199
$783$	0.535898	0.0191514
$784$	1.00000	0.0357143
$785$	−23.8564	−0.851472
$786$	−17.8564	−0.636917
$787$	−35.3205	−1.25904	−0.629520	−	0.776984i	$-0.716748\pi$
$787$	−35.3205	−1.25904	−0.629520	+	0.776984i	$0.716748\pi$
$788$	−2.00000	−0.0712470
$789$	−18.9282	−0.673862
$790$	−10.9282	−0.388808
$791$	−7.85641	−0.279342
$792$	1.46410	0.0520246
$793$	34.6410	1.23014
$794$	25.3205	0.898591
$795$	2.00000	0.0709327
$796$	26.9282	0.954445
$797$	35.5692	1.25993	0.629963	−	0.776625i	$-0.283070\pi$
$797$	35.5692	1.25993	0.629963	+	0.776625i	$0.283070\pi$
$798$	4.00000	0.141598
$799$	13.8564	0.490204
$800$	1.00000	0.0353553
$801$	4.53590	0.160268
$802$	−4.53590	−0.160168
$803$	2.92820	0.103334
$804$	1.46410	0.0516349
$805$	−1.00000	−0.0352454
$806$	13.8564	0.488071
$807$	21.3205	0.750517
$808$	−2.39230	−0.0841610
$809$	15.8564	0.557482	0.278741	−	0.960366i	$-0.410083\pi$
$809$	15.8564	0.557482	0.278741	+	0.960366i	$0.410083\pi$
$810$	−1.00000	−0.0351364
$811$	26.6410	0.935493	0.467746	−	0.883863i	$-0.345066\pi$
$811$	26.6410	0.935493	0.467746	+	0.883863i	$0.345066\pi$
$812$	0.535898	0.0188063
$813$	4.00000	0.140286
$814$	−13.0718	−0.458166
$815$	14.9282	0.522912
$816$	2.00000	0.0700140
$817$	−5.85641	−0.204890
$818$	23.8564	0.834120
$819$	−3.46410	−0.121046
$820$	−2.00000	−0.0698430
$821$	38.1051	1.32988	0.664939	−	0.746898i	$-0.268458\pi$
$821$	38.1051	1.32988	0.664939	+	0.746898i	$0.268458\pi$
$822$	−15.8564	−0.553056
$823$	40.1051	1.39798	0.698988	−	0.715133i	$-0.253634\pi$
$823$	40.1051	1.39798	0.698988	+	0.715133i	$0.253634\pi$
$824$	13.8564	0.482711
$825$	−1.46410	−0.0509735
$826$	−6.92820	−0.241063
$827$	−2.92820	−0.101824	−0.0509118	−	0.998703i	$-0.516213\pi$
$827$	−2.92820	−0.101824	−0.0509118	+	0.998703i	$0.516213\pi$
$828$	−1.00000	−0.0347524
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	−4.00000	−0.138842
$831$	0.535898	0.0185901
$832$	3.46410	0.120096
$833$	−2.00000	−0.0692959
$834$	9.85641	0.341299
$835$	−12.0000	−0.415277
$836$	5.85641	0.202548
$837$	−4.00000	−0.138260
$838$	18.5359	0.640312
$839$	−42.9282	−1.48205	−0.741023	−	0.671480i	$-0.765659\pi$
$839$	−42.9282	−1.48205	−0.741023	+	0.671480i	$0.765659\pi$
$840$	−1.00000	−0.0345033
$841$	−28.7128	−0.990097
$842$	−3.85641	−0.132901
$843$	−20.2487	−0.697403
$844$	−20.7846	−0.715436
$845$	1.00000	0.0344010
$846$	−6.92820	−0.238197
$847$	8.85641	0.304310
$848$	2.00000	0.0686803
$849$	−24.3923	−0.837142
$850$	−2.00000	−0.0685994
$851$	8.92820	0.306055
$852$	9.46410	0.324235
$853$	−26.3923	−0.903655	−0.451828	−	0.892105i	$-0.649228\pi$
$853$	−26.3923	−0.903655	−0.451828	+	0.892105i	$0.649228\pi$
$854$	−10.0000	−0.342193
$855$	−4.00000	−0.136797
$856$	−2.92820	−0.100084
$857$	−31.5692	−1.07838	−0.539192	−	0.842183i	$-0.681270\pi$
$857$	−31.5692	−1.07838	−0.539192	+	0.842183i	$0.681270\pi$
$858$	−5.07180	−0.173148
$859$	−6.92820	−0.236387	−0.118194	−	0.992991i	$-0.537710\pi$
$859$	−6.92820	−0.236387	−0.118194	+	0.992991i	$0.537710\pi$
$860$	1.46410	0.0499255
$861$	2.00000	0.0681598
$862$	−21.8564	−0.744432
$863$	−0.784610	−0.0267084	−0.0133542	−	0.999911i	$-0.504251\pi$
$863$	−0.784610	−0.0267084	−0.0133542	+	0.999911i	$0.504251\pi$
$864$	−1.00000	−0.0340207
$865$	8.92820	0.303568
$866$	−20.2487	−0.688079
$867$	13.0000	0.441503
$868$	−4.00000	−0.135769
$869$	16.0000	0.542763
$870$	−0.535898	−0.0181687
$871$	−5.07180	−0.171851
$872$	10.0000	0.338643
$873$	7.46410	0.252622
$874$	−4.00000	−0.135302
$875$	1.00000	0.0338062
$876$	−2.00000	−0.0675737
$877$	−16.5359	−0.558378	−0.279189	−	0.960236i	$-0.590066\pi$
$877$	−16.5359	−0.558378	−0.279189	+	0.960236i	$0.590066\pi$
$878$	36.7846	1.24142
$879$	−2.00000	−0.0674583
$880$	−1.46410	−0.0493549
$881$	10.3923	0.350126	0.175063	−	0.984557i	$-0.443987\pi$
$881$	10.3923	0.350126	0.175063	+	0.984557i	$0.443987\pi$
$882$	1.00000	0.0336718
$883$	20.7846	0.699458	0.349729	−	0.936851i	$-0.386274\pi$
$883$	20.7846	0.699458	0.349729	+	0.936851i	$0.386274\pi$
$884$	−6.92820	−0.233021
$885$	6.92820	0.232889
$886$	6.14359	0.206398
$887$	−38.9282	−1.30708	−0.653541	−	0.756891i	$-0.726717\pi$
$887$	−38.9282	−1.30708	−0.653541	+	0.756891i	$0.726717\pi$
$888$	8.92820	0.299611
$889$	−12.3923	−0.415625
$890$	−4.53590	−0.152044
$891$	1.46410	0.0490492
$892$	2.53590	0.0849082
$893$	−27.7128	−0.927374
$894$	8.92820	0.298604
$895$	−9.07180	−0.303237
$896$	−1.00000	−0.0334077
$897$	3.46410	0.115663
$898$	−11.0718	−0.369471
$899$	−2.14359	−0.0714928
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	2.92820	0.0974985
$903$	−1.46410	−0.0487223
$904$	7.85641	0.261300
$905$	−2.00000	−0.0664822
$906$	−10.9282	−0.363065
$907$	44.3923	1.47402	0.737011	−	0.675881i	$-0.236237\pi$
$907$	44.3923	1.47402	0.737011	+	0.675881i	$0.236237\pi$
$908$	1.07180	0.0355688
$909$	−2.39230	−0.0793477
$910$	3.46410	0.114834
$911$	−48.7846	−1.61631	−0.808153	−	0.588972i	$-0.799533\pi$
$911$	−48.7846	−1.61631	−0.808153	+	0.588972i	$0.799533\pi$
$912$	−4.00000	−0.132453
$913$	5.85641	0.193819
$914$	−4.53590	−0.150034
$915$	10.0000	0.330590
$916$	15.8564	0.523910
$917$	−17.8564	−0.589670
$918$	2.00000	0.0660098
$919$	−41.5692	−1.37124	−0.685621	−	0.727959i	$-0.740469\pi$
$919$	−41.5692	−1.37124	−0.685621	+	0.727959i	$0.740469\pi$
$920$	1.00000	0.0329690
$921$	−25.8564	−0.851998
$922$	−24.2487	−0.798589
$923$	−32.7846	−1.07912
$924$	1.46410	0.0481654
$925$	−8.92820	−0.293558
$926$	−20.3923	−0.670133
$927$	13.8564	0.455104
$928$	−0.535898	−0.0175917
$929$	−11.8564	−0.388996	−0.194498	−	0.980903i	$-0.562308\pi$
$929$	−11.8564	−0.388996	−0.194498	+	0.980903i	$0.562308\pi$
$930$	4.00000	0.131165
$931$	4.00000	0.131095
$932$	7.85641	0.257345
$933$	−11.3205	−0.370617
$934$	17.8564	0.584279
$935$	2.92820	0.0957625
$936$	3.46410	0.113228
$937$	3.75129	0.122549	0.0612746	−	0.998121i	$-0.480483\pi$
$937$	3.75129	0.122549	0.0612746	+	0.998121i	$0.480483\pi$
$938$	1.46410	0.0478046
$939$	17.3205	0.565233
$940$	6.92820	0.225973
$941$	13.7128	0.447025	0.223512	−	0.974701i	$-0.428248\pi$
$941$	13.7128	0.447025	0.223512	+	0.974701i	$0.428248\pi$
$942$	−23.8564	−0.777284
$943$	−2.00000	−0.0651290
$944$	6.92820	0.225494
$945$	−1.00000	−0.0325300
$946$	−2.14359	−0.0696942
$947$	−44.7846	−1.45530	−0.727652	−	0.685946i	$-0.759388\pi$
$947$	−44.7846	−1.45530	−0.727652	+	0.685946i	$0.759388\pi$
$948$	−10.9282	−0.354932
$949$	6.92820	0.224899
$950$	4.00000	0.129777
$951$	12.9282	0.419226
$952$	2.00000	0.0648204
$953$	7.85641	0.254494	0.127247	−	0.991871i	$-0.459386\pi$
$953$	7.85641	0.254494	0.127247	+	0.991871i	$0.459386\pi$
$954$	2.00000	0.0647524
$955$	−8.00000	−0.258874
$956$	12.3923	0.400796
$957$	0.784610	0.0253628
$958$	8.00000	0.258468
$959$	−15.8564	−0.512030
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−30.9282	−0.997165
$963$	−2.92820	−0.0943600
$964$	−4.92820	−0.158727
$965$	14.0000	0.450676
$966$	−1.00000	−0.0321745
$967$	−32.1051	−1.03243	−0.516215	−	0.856459i	$-0.672660\pi$
$967$	−32.1051	−1.03243	−0.516215	+	0.856459i	$0.672660\pi$
$968$	−8.85641	−0.284656
$969$	8.00000	0.256997
$970$	−7.46410	−0.239658
$971$	−16.3923	−0.526054	−0.263027	−	0.964788i	$-0.584721\pi$
$971$	−16.3923	−0.526054	−0.263027	+	0.964788i	$0.584721\pi$
$972$	−1.00000	−0.0320750
$973$	9.85641	0.315982
$974$	−12.3923	−0.397075
$975$	−3.46410	−0.110940
$976$	10.0000	0.320092
$977$	−54.7846	−1.75271	−0.876357	−	0.481661i	$-0.840034\pi$
$977$	−54.7846	−1.75271	−0.876357	+	0.481661i	$0.840034\pi$
$978$	14.9282	0.477351
$979$	6.64102	0.212248
$980$	−1.00000	−0.0319438
$981$	10.0000	0.319275
$982$	−20.0000	−0.638226
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−2.00000	−0.0637577
$985$	2.00000	0.0637253
$986$	1.07180	0.0341330
$987$	−6.92820	−0.220527
$988$	13.8564	0.440831
$989$	1.46410	0.0465557
$990$	−1.46410	−0.0465322
$991$	24.7846	0.787309	0.393655	−	0.919258i	$-0.371211\pi$
$991$	24.7846	0.787309	0.393655	+	0.919258i	$0.371211\pi$
$992$	4.00000	0.127000
$993$	−22.9282	−0.727605
$994$	9.46410	0.300183
$995$	−26.9282	−0.853681
$996$	−4.00000	−0.126745
$997$	9.32051	0.295183	0.147592	−	0.989048i	$-0.452848\pi$
$997$	9.32051	0.295183	0.147592	+	0.989048i	$0.452848\pi$
$998$	14.9282	0.472544
$999$	8.92820	0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.bt.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.bt.1.2	✓	2	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:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.46410 q^{11} -1.00000 q^{12} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +1.46410 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.46410 q^{26} -1.00000 q^{27} -1.00000 q^{28} -0.535898 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -1.46410 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -8.92820 q^{37} +4.00000 q^{38} -3.46410 q^{39} -1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -1.46410 q^{43} +1.46410 q^{44} -1.00000 q^{45} -1.00000 q^{46} -6.92820 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +3.46410 q^{52} +2.00000 q^{53} -1.00000 q^{54} -1.46410 q^{55} -1.00000 q^{56} -4.00000 q^{57} -0.535898 q^{58} +6.92820 q^{59} +1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.46410 q^{65} -1.46410 q^{66} -1.46410 q^{67} -2.00000 q^{68} +1.00000 q^{69} +1.00000 q^{70} -9.46410 q^{71} +1.00000 q^{72} +2.00000 q^{73} -8.92820 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.46410 q^{77} -3.46410 q^{78} +10.9282 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +2.00000 q^{85} -1.46410 q^{86} +0.535898 q^{87} +1.46410 q^{88} +4.53590 q^{89} -1.00000 q^{90} -3.46410 q^{91} -1.00000 q^{92} -4.00000 q^{93} -6.92820 q^{94} -4.00000 q^{95} -1.00000 q^{96} +7.46410 q^{97} +1.00000 q^{98} +1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 8 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} - 2 q^{28} - 8 q^{29} + 2 q^{30} + 8 q^{31} + 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 2 q^{40} + 4 q^{41} + 2 q^{42} + 4 q^{43} - 4 q^{44} - 2 q^{45} - 2 q^{46} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 4 q^{51} + 4 q^{53} - 2 q^{54} + 4 q^{55} - 2 q^{56} - 8 q^{57} - 8 q^{58} + 2 q^{60} + 20 q^{61} + 8 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} + 4 q^{67} - 4 q^{68} + 2 q^{69} + 2 q^{70} - 12 q^{71} + 2 q^{72} + 4 q^{73} - 4 q^{74} - 2 q^{75} + 8 q^{76} + 4 q^{77} + 8 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 2 q^{84} + 4 q^{85} + 4 q^{86} + 8 q^{87} - 4 q^{88} + 16 q^{89} - 2 q^{90} - 2 q^{92} - 8 q^{93} - 8 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 8 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 - 2 * q^28 - 8 * q^29 + 2 * q^30 + 8 * q^31 + 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 4 * q^37 + 8 * q^38 - 2 * q^40 + 4 * q^41 + 2 * q^42 + 4 * q^43 - 4 * q^44 - 2 * q^45 - 2 * q^46 - 2 * q^48 + 2 * q^49 + 2 * q^50 + 4 * q^51 + 4 * q^53 - 2 * q^54 + 4 * q^55 - 2 * q^56 - 8 * q^57 - 8 * q^58 + 2 * q^60 + 20 * q^61 + 8 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^66 + 4 * q^67 - 4 * q^68 + 2 * q^69 + 2 * q^70 - 12 * q^71 + 2 * q^72 + 4 * q^73 - 4 * q^74 - 2 * q^75 + 8 * q^76 + 4 * q^77 + 8 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 2 * q^84 + 4 * q^85 + 4 * q^86 + 8 * q^87 - 4 * q^88 + 16 * q^89 - 2 * q^90 - 2 * q^92 - 8 * q^93 - 8 * q^95 - 2 * q^96 + 8 * q^97 + 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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