LMFDB - Embedded newform 4830.2.a.bz.1.3

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.05086 q^{11} -1.00000 q^{12} -0.622216 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.18421 q^{17} +1.00000 q^{18} +2.42864 q^{19} -1.00000 q^{20} -1.00000 q^{21} +5.05086 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -0.622216 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.80642 q^{29} +1.00000 q^{30} -2.42864 q^{31} +1.00000 q^{32} -5.05086 q^{33} -7.18421 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.42864 q^{38} +0.622216 q^{39} -1.00000 q^{40} +9.61285 q^{41} -1.00000 q^{42} -1.05086 q^{43} +5.05086 q^{44} -1.00000 q^{45} +1.00000 q^{46} -2.42864 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.18421 q^{51} -0.622216 q^{52} -9.61285 q^{53} -1.00000 q^{54} -5.05086 q^{55} +1.00000 q^{56} -2.42864 q^{57} +9.80642 q^{58} -4.85728 q^{59} +1.00000 q^{60} -2.85728 q^{61} -2.42864 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.622216 q^{65} -5.05086 q^{66} -1.05086 q^{67} -7.18421 q^{68} -1.00000 q^{69} -1.00000 q^{70} +3.80642 q^{71} +1.00000 q^{72} +15.7146 q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.42864 q^{76} +5.05086 q^{77} +0.622216 q^{78} +8.85728 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.61285 q^{82} -7.67307 q^{83} -1.00000 q^{84} +7.18421 q^{85} -1.05086 q^{86} -9.80642 q^{87} +5.05086 q^{88} +10.7239 q^{89} -1.00000 q^{90} -0.622216 q^{91} +1.00000 q^{92} +2.42864 q^{93} -2.42864 q^{94} -2.42864 q^{95} -1.00000 q^{96} -0.622216 q^{97} +1.00000 q^{98} +5.05086 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{13} + 3 q^{14} + 3 q^{15} + 3 q^{16} - 8 q^{17} + 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{21} + 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 16 q^{29} + 3 q^{30} + 6 q^{31} + 3 q^{32} - 2 q^{33} - 8 q^{34} - 3 q^{35} + 3 q^{36} + 6 q^{37} - 6 q^{38} + 2 q^{39} - 3 q^{40} + 2 q^{41} - 3 q^{42} + 10 q^{43} + 2 q^{44} - 3 q^{45} + 3 q^{46} + 6 q^{47} - 3 q^{48} + 3 q^{49} + 3 q^{50} + 8 q^{51} - 2 q^{52} - 2 q^{53} - 3 q^{54} - 2 q^{55} + 3 q^{56} + 6 q^{57} + 16 q^{58} + 12 q^{59} + 3 q^{60} + 18 q^{61} + 6 q^{62} + 3 q^{63} + 3 q^{64} + 2 q^{65} - 2 q^{66} + 10 q^{67} - 8 q^{68} - 3 q^{69} - 3 q^{70} - 2 q^{71} + 3 q^{72} - 6 q^{73} + 6 q^{74} - 3 q^{75} - 6 q^{76} + 2 q^{77} + 2 q^{78} - 3 q^{80} + 3 q^{81} + 2 q^{82} - 10 q^{83} - 3 q^{84} + 8 q^{85} + 10 q^{86} - 16 q^{87} + 2 q^{88} + 6 q^{89} - 3 q^{90} - 2 q^{91} + 3 q^{92} - 6 q^{93} + 6 q^{94} + 6 q^{95} - 3 q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 2 * q^13 + 3 * q^14 + 3 * q^15 + 3 * q^16 - 8 * q^17 + 3 * q^18 - 6 * q^19 - 3 * q^20 - 3 * q^21 + 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 2 * q^26 - 3 * q^27 + 3 * q^28 + 16 * q^29 + 3 * q^30 + 6 * q^31 + 3 * q^32 - 2 * q^33 - 8 * q^34 - 3 * q^35 + 3 * q^36 + 6 * q^37 - 6 * q^38 + 2 * q^39 - 3 * q^40 + 2 * q^41 - 3 * q^42 + 10 * q^43 + 2 * q^44 - 3 * q^45 + 3 * q^46 + 6 * q^47 - 3 * q^48 + 3 * q^49 + 3 * q^50 + 8 * q^51 - 2 * q^52 - 2 * q^53 - 3 * q^54 - 2 * q^55 + 3 * q^56 + 6 * q^57 + 16 * q^58 + 12 * q^59 + 3 * q^60 + 18 * q^61 + 6 * q^62 + 3 * q^63 + 3 * q^64 + 2 * q^65 - 2 * q^66 + 10 * q^67 - 8 * q^68 - 3 * q^69 - 3 * q^70 - 2 * q^71 + 3 * q^72 - 6 * q^73 + 6 * q^74 - 3 * q^75 - 6 * q^76 + 2 * q^77 + 2 * q^78 - 3 * q^80 + 3 * q^81 + 2 * q^82 - 10 * q^83 - 3 * q^84 + 8 * q^85 + 10 * q^86 - 16 * q^87 + 2 * q^88 + 6 * q^89 - 3 * q^90 - 2 * q^91 + 3 * q^92 - 6 * q^93 + 6 * q^94 + 6 * q^95 - 3 * q^96 - 2 * q^97 + 3 * q^98 + 2 * 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
$11$	5.05086	1.52289	0.761445	−	0.648229i	$-0.224490\pi$
$11$	5.05086	1.52289	0.761445	+	0.648229i	$0.224490\pi$
$12$	−1.00000	−0.288675
$13$	−0.622216	−0.172572	−0.0862858	−	0.996270i	$-0.527500\pi$
$13$	−0.622216	−0.172572	−0.0862858	+	0.996270i	$0.527500\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−7.18421	−1.74243	−0.871213	−	0.490905i	$-0.836666\pi$
$17$	−7.18421	−1.74243	−0.871213	+	0.490905i	$0.836666\pi$
$18$	1.00000	0.235702
$19$	2.42864	0.557168	0.278584	−	0.960412i	$-0.410135\pi$
$19$	2.42864	0.557168	0.278584	+	0.960412i	$0.410135\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	5.05086	1.07685
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−0.622216	−0.122027
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	9.80642	1.82101	0.910504	−	0.413501i	$-0.135694\pi$
$29$	9.80642	1.82101	0.910504	+	0.413501i	$0.135694\pi$
$30$	1.00000	0.182574
$31$	−2.42864	−0.436197	−0.218098	−	0.975927i	$-0.569985\pi$
$31$	−2.42864	−0.436197	−0.218098	+	0.975927i	$0.569985\pi$
$32$	1.00000	0.176777
$33$	−5.05086	−0.879241
$34$	−7.18421	−1.23208
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.42864	0.393977
$39$	0.622216	0.0996342
$40$	−1.00000	−0.158114
$41$	9.61285	1.50127	0.750637	−	0.660715i	$-0.229747\pi$
$41$	9.61285	1.50127	0.750637	+	0.660715i	$0.229747\pi$
$42$	−1.00000	−0.154303
$43$	−1.05086	−0.160254	−0.0801270	−	0.996785i	$-0.525533\pi$
$43$	−1.05086	−0.160254	−0.0801270	+	0.996785i	$0.525533\pi$
$44$	5.05086	0.761445
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	−2.42864	−0.354253	−0.177127	−	0.984188i	$-0.556680\pi$
$47$	−2.42864	−0.354253	−0.177127	+	0.984188i	$0.556680\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	7.18421	1.00599
$52$	−0.622216	−0.0862858
$53$	−9.61285	−1.32043	−0.660213	−	0.751078i	$-0.729534\pi$
$53$	−9.61285	−1.32043	−0.660213	+	0.751078i	$0.729534\pi$
$54$	−1.00000	−0.136083
$55$	−5.05086	−0.681057
$56$	1.00000	0.133631
$57$	−2.42864	−0.321681
$58$	9.80642	1.28765
$59$	−4.85728	−0.632364	−0.316182	−	0.948699i	$-0.602401\pi$
$59$	−4.85728	−0.632364	−0.316182	+	0.948699i	$0.602401\pi$
$60$	1.00000	0.129099
$61$	−2.85728	−0.365837	−0.182919	−	0.983128i	$-0.558554\pi$
$61$	−2.85728	−0.365837	−0.182919	+	0.983128i	$0.558554\pi$
$62$	−2.42864	−0.308438
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0.622216	0.0771764
$66$	−5.05086	−0.621717
$67$	−1.05086	−0.128382	−0.0641912	−	0.997938i	$-0.520447\pi$
$67$	−1.05086	−0.128382	−0.0641912	+	0.997938i	$0.520447\pi$
$68$	−7.18421	−0.871213
$69$	−1.00000	−0.120386
$70$	−1.00000	−0.119523
$71$	3.80642	0.451739	0.225870	−	0.974158i	$-0.427478\pi$
$71$	3.80642	0.451739	0.225870	+	0.974158i	$0.427478\pi$
$72$	1.00000	0.117851
$73$	15.7146	1.83925	0.919625	−	0.392798i	$-0.128493\pi$
$73$	15.7146	1.83925	0.919625	+	0.392798i	$0.128493\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	2.42864	0.278584
$77$	5.05086	0.575598
$78$	0.622216	0.0704520
$79$	8.85728	0.996522	0.498261	−	0.867027i	$-0.333972\pi$
$79$	8.85728	0.996522	0.498261	+	0.867027i	$0.333972\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	9.61285	1.06156
$83$	−7.67307	−0.842229	−0.421114	−	0.907008i	$-0.638361\pi$
$83$	−7.67307	−0.842229	−0.421114	+	0.907008i	$0.638361\pi$
$84$	−1.00000	−0.109109
$85$	7.18421	0.779237
$86$	−1.05086	−0.113317
$87$	−9.80642	−1.05136
$88$	5.05086	0.538423
$89$	10.7239	1.13673	0.568367	−	0.822775i	$-0.307575\pi$
$89$	10.7239	1.13673	0.568367	+	0.822775i	$0.307575\pi$
$90$	−1.00000	−0.105409
$91$	−0.622216	−0.0652259
$92$	1.00000	0.104257
$93$	2.42864	0.251838
$94$	−2.42864	−0.250495
$95$	−2.42864	−0.249173
$96$	−1.00000	−0.102062
$97$	−0.622216	−0.0631764	−0.0315882	−	0.999501i	$-0.510057\pi$
$97$	−0.622216	−0.0631764	−0.0315882	+	0.999501i	$0.510057\pi$
$98$	1.00000	0.101015
$99$	5.05086	0.507630
$100$	1.00000	0.100000
$101$	5.47949	0.545230	0.272615	−	0.962123i	$-0.412111\pi$
$101$	5.47949	0.545230	0.272615	+	0.962123i	$0.412111\pi$
$102$	7.18421	0.711343
$103$	−6.10171	−0.601219	−0.300610	−	0.953747i	$-0.597190\pi$
$103$	−6.10171	−0.601219	−0.300610	+	0.953747i	$0.597190\pi$
$104$	−0.622216	−0.0610133
$105$	1.00000	0.0975900
$106$	−9.61285	−0.933682
$107$	1.24443	0.120304	0.0601519	−	0.998189i	$-0.480842\pi$
$107$	1.24443	0.120304	0.0601519	+	0.998189i	$0.480842\pi$
$108$	−1.00000	−0.0962250
$109$	4.48886	0.429955	0.214978	−	0.976619i	$-0.431032\pi$
$109$	4.48886	0.429955	0.214978	+	0.976619i	$0.431032\pi$
$110$	−5.05086	−0.481580
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	−2.85728	−0.268790	−0.134395	−	0.990928i	$-0.542909\pi$
$113$	−2.85728	−0.268790	−0.134395	+	0.990928i	$0.542909\pi$
$114$	−2.42864	−0.227463
$115$	−1.00000	−0.0932505
$116$	9.80642	0.910504
$117$	−0.622216	−0.0575239
$118$	−4.85728	−0.447149
$119$	−7.18421	−0.658575
$120$	1.00000	0.0912871
$121$	14.5111	1.31919
$122$	−2.85728	−0.258686
$123$	−9.61285	−0.866761
$124$	−2.42864	−0.218098
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	2.94914	0.261694	0.130847	−	0.991403i	$-0.458230\pi$
$127$	2.94914	0.261694	0.130847	+	0.991403i	$0.458230\pi$
$128$	1.00000	0.0883883
$129$	1.05086	0.0925226
$130$	0.622216	0.0545719
$131$	−19.2257	−1.67976	−0.839878	−	0.542775i	$-0.817374\pi$
$131$	−19.2257	−1.67976	−0.839878	+	0.542775i	$0.817374\pi$
$132$	−5.05086	−0.439621
$133$	2.42864	0.210590
$134$	−1.05086	−0.0907801
$135$	1.00000	0.0860663
$136$	−7.18421	−0.616041
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−1.00000	−0.0851257
$139$	6.75557	0.573000	0.286500	−	0.958080i	$-0.407508\pi$
$139$	6.75557	0.573000	0.286500	+	0.958080i	$0.407508\pi$
$140$	−1.00000	−0.0845154
$141$	2.42864	0.204528
$142$	3.80642	0.319428
$143$	−3.14272	−0.262808
$144$	1.00000	0.0833333
$145$	−9.80642	−0.814379
$146$	15.7146	1.30055
$147$	−1.00000	−0.0824786
$148$	2.00000	0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−1.00000	−0.0816497
$151$	2.75557	0.224245	0.112123	−	0.993694i	$-0.464235\pi$
$151$	2.75557	0.224245	0.112123	+	0.993694i	$0.464235\pi$
$152$	2.42864	0.196989
$153$	−7.18421	−0.580809
$154$	5.05086	0.407010
$155$	2.42864	0.195073
$156$	0.622216	0.0498171
$157$	8.10171	0.646587	0.323293	−	0.946299i	$-0.395210\pi$
$157$	8.10171	0.646587	0.323293	+	0.946299i	$0.395210\pi$
$158$	8.85728	0.704647
$159$	9.61285	0.762348
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	20.8573	1.63367	0.816834	−	0.576873i	$-0.195727\pi$
$163$	20.8573	1.63367	0.816834	+	0.576873i	$0.195727\pi$
$164$	9.61285	0.750637
$165$	5.05086	0.393209
$166$	−7.67307	−0.595546
$167$	18.0415	1.39609	0.698046	−	0.716053i	$-0.254053\pi$
$167$	18.0415	1.39609	0.698046	+	0.716053i	$0.254053\pi$
$168$	−1.00000	−0.0771517
$169$	−12.6128	−0.970219
$170$	7.18421	0.551004
$171$	2.42864	0.185723
$172$	−1.05086	−0.0801270
$173$	5.28592	0.401881	0.200940	−	0.979603i	$-0.435600\pi$
$173$	5.28592	0.401881	0.200940	+	0.979603i	$0.435600\pi$
$174$	−9.80642	−0.743423
$175$	1.00000	0.0755929
$176$	5.05086	0.380723
$177$	4.85728	0.365095
$178$	10.7239	0.803792
$179$	−9.71456	−0.726100	−0.363050	−	0.931770i	$-0.618265\pi$
$179$	−9.71456	−0.726100	−0.363050	+	0.931770i	$0.618265\pi$
$180$	−1.00000	−0.0745356
$181$	−8.36842	−0.622019	−0.311010	−	0.950407i	$-0.600667\pi$
$181$	−8.36842	−0.622019	−0.311010	+	0.950407i	$0.600667\pi$
$182$	−0.622216	−0.0461217
$183$	2.85728	0.211216
$184$	1.00000	0.0737210
$185$	−2.00000	−0.147043
$186$	2.42864	0.178076
$187$	−36.2864	−2.65352
$188$	−2.42864	−0.177127
$189$	−1.00000	−0.0727393
$190$	−2.42864	−0.176192
$191$	4.85728	0.351460	0.175730	−	0.984438i	$-0.443771\pi$
$191$	4.85728	0.351460	0.175730	+	0.984438i	$0.443771\pi$
$192$	−1.00000	−0.0721688
$193$	−6.85728	−0.493598	−0.246799	−	0.969067i	$-0.579379\pi$
$193$	−6.85728	−0.493598	−0.246799	+	0.969067i	$0.579379\pi$
$194$	−0.622216	−0.0446725
$195$	−0.622216	−0.0445578
$196$	1.00000	0.0714286
$197$	1.14272	0.0814155	0.0407078	−	0.999171i	$-0.487039\pi$
$197$	1.14272	0.0814155	0.0407078	+	0.999171i	$0.487039\pi$
$198$	5.05086	0.358949
$199$	22.1017	1.56675	0.783374	−	0.621550i	$-0.213497\pi$
$199$	22.1017	1.56675	0.783374	+	0.621550i	$0.213497\pi$
$200$	1.00000	0.0707107
$201$	1.05086	0.0741216
$202$	5.47949	0.385536
$203$	9.80642	0.688276
$204$	7.18421	0.502995
$205$	−9.61285	−0.671390
$206$	−6.10171	−0.425126
$207$	1.00000	0.0695048
$208$	−0.622216	−0.0431429
$209$	12.2667	0.848506
$210$	1.00000	0.0690066
$211$	23.3461	1.60721	0.803607	−	0.595160i	$-0.202911\pi$
$211$	23.3461	1.60721	0.803607	+	0.595160i	$0.202911\pi$
$212$	−9.61285	−0.660213
$213$	−3.80642	−0.260812
$214$	1.24443	0.0850676
$215$	1.05086	0.0716677
$216$	−1.00000	−0.0680414
$217$	−2.42864	−0.164867
$218$	4.48886	0.304024
$219$	−15.7146	−1.06189
$220$	−5.05086	−0.340529
$221$	4.47013	0.300693
$222$	−2.00000	−0.134231
$223$	14.2351	0.953250	0.476625	−	0.879107i	$-0.341860\pi$
$223$	14.2351	0.953250	0.476625	+	0.879107i	$0.341860\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−2.85728	−0.190063
$227$	20.7971	1.38035	0.690175	−	0.723643i	$-0.257534\pi$
$227$	20.7971	1.38035	0.690175	+	0.723643i	$0.257534\pi$
$228$	−2.42864	−0.160841
$229$	25.6128	1.69254	0.846272	−	0.532751i	$-0.178842\pi$
$229$	25.6128	1.69254	0.846272	+	0.532751i	$0.178842\pi$
$230$	−1.00000	−0.0659380
$231$	−5.05086	−0.332322
$232$	9.80642	0.643823
$233$	−4.48886	−0.294075	−0.147038	−	0.989131i	$-0.546974\pi$
$233$	−4.48886	−0.294075	−0.147038	+	0.989131i	$0.546974\pi$
$234$	−0.622216	−0.0406755
$235$	2.42864	0.158427
$236$	−4.85728	−0.316182
$237$	−8.85728	−0.575342
$238$	−7.18421	−0.465683
$239$	3.80642	0.246217	0.123109	−	0.992393i	$-0.460714\pi$
$239$	3.80642	0.246217	0.123109	+	0.992393i	$0.460714\pi$
$240$	1.00000	0.0645497
$241$	−7.18421	−0.462776	−0.231388	−	0.972862i	$-0.574327\pi$
$241$	−7.18421	−0.462776	−0.231388	+	0.972862i	$0.574327\pi$
$242$	14.5111	0.932811
$243$	−1.00000	−0.0641500
$244$	−2.85728	−0.182919
$245$	−1.00000	−0.0638877
$246$	−9.61285	−0.612893
$247$	−1.51114	−0.0961514
$248$	−2.42864	−0.154219
$249$	7.67307	0.486261
$250$	−1.00000	−0.0632456
$251$	−4.99063	−0.315006	−0.157503	−	0.987519i	$-0.550344\pi$
$251$	−4.99063	−0.315006	−0.157503	+	0.987519i	$0.550344\pi$
$252$	1.00000	0.0629941
$253$	5.05086	0.317545
$254$	2.94914	0.185046
$255$	−7.18421	−0.449893
$256$	1.00000	0.0625000
$257$	25.2257	1.57354	0.786768	−	0.617249i	$-0.211753\pi$
$257$	25.2257	1.57354	0.786768	+	0.617249i	$0.211753\pi$
$258$	1.05086	0.0654234
$259$	2.00000	0.124274
$260$	0.622216	0.0385882
$261$	9.80642	0.607002
$262$	−19.2257	−1.18777
$263$	−8.59057	−0.529717	−0.264859	−	0.964287i	$-0.585325\pi$
$263$	−8.59057	−0.529717	−0.264859	+	0.964287i	$0.585325\pi$
$264$	−5.05086	−0.310859
$265$	9.61285	0.590513
$266$	2.42864	0.148909
$267$	−10.7239	−0.656294
$268$	−1.05086	−0.0641912
$269$	8.88892	0.541967	0.270984	−	0.962584i	$-0.412651\pi$
$269$	8.88892	0.541967	0.270984	+	0.962584i	$0.412651\pi$
$270$	1.00000	0.0608581
$271$	−2.42864	−0.147529	−0.0737647	−	0.997276i	$-0.523501\pi$
$271$	−2.42864	−0.147529	−0.0737647	+	0.997276i	$0.523501\pi$
$272$	−7.18421	−0.435607
$273$	0.622216	0.0376582
$274$	2.00000	0.120824
$275$	5.05086	0.304578
$276$	−1.00000	−0.0601929
$277$	−7.05086	−0.423645	−0.211822	−	0.977308i	$-0.567940\pi$
$277$	−7.05086	−0.423645	−0.211822	+	0.977308i	$0.567940\pi$
$278$	6.75557	0.405172
$279$	−2.42864	−0.145399
$280$	−1.00000	−0.0597614
$281$	−14.6637	−0.874763	−0.437382	−	0.899276i	$-0.644094\pi$
$281$	−14.6637	−0.874763	−0.437382	+	0.899276i	$0.644094\pi$
$282$	2.42864	0.144623
$283$	19.0923	1.13492	0.567461	−	0.823400i	$-0.307926\pi$
$283$	19.0923	1.13492	0.567461	+	0.823400i	$0.307926\pi$
$284$	3.80642	0.225870
$285$	2.42864	0.143860
$286$	−3.14272	−0.185833
$287$	9.61285	0.567428
$288$	1.00000	0.0589256
$289$	34.6128	2.03605
$290$	−9.80642	−0.575853
$291$	0.622216	0.0364749
$292$	15.7146	0.919625
$293$	−0.755569	−0.0441408	−0.0220704	−	0.999756i	$-0.507026\pi$
$293$	−0.755569	−0.0441408	−0.0220704	+	0.999756i	$0.507026\pi$
$294$	−1.00000	−0.0583212
$295$	4.85728	0.282802
$296$	2.00000	0.116248
$297$	−5.05086	−0.293080
$298$	10.0000	0.579284
$299$	−0.622216	−0.0359837
$300$	−1.00000	−0.0577350
$301$	−1.05086	−0.0605703
$302$	2.75557	0.158565
$303$	−5.47949	−0.314789
$304$	2.42864	0.139292
$305$	2.85728	0.163607
$306$	−7.18421	−0.410694
$307$	20.5906	1.17517	0.587583	−	0.809164i	$-0.300080\pi$
$307$	20.5906	1.17517	0.587583	+	0.809164i	$0.300080\pi$
$308$	5.05086	0.287799
$309$	6.10171	0.347114
$310$	2.42864	0.137937
$311$	−12.9906	−0.736631	−0.368316	−	0.929701i	$-0.620065\pi$
$311$	−12.9906	−0.736631	−0.368316	+	0.929701i	$0.620065\pi$
$312$	0.622216	0.0352260
$313$	11.8479	0.669684	0.334842	−	0.942274i	$-0.391317\pi$
$313$	11.8479	0.669684	0.334842	+	0.942274i	$0.391317\pi$
$314$	8.10171	0.457206
$315$	−1.00000	−0.0563436
$316$	8.85728	0.498261
$317$	−22.4701	−1.26205	−0.631024	−	0.775763i	$-0.717365\pi$
$317$	−22.4701	−1.26205	−0.631024	+	0.775763i	$0.717365\pi$
$318$	9.61285	0.539062
$319$	49.5308	2.77319
$320$	−1.00000	−0.0559017
$321$	−1.24443	−0.0694574
$322$	1.00000	0.0557278
$323$	−17.4479	−0.970824
$324$	1.00000	0.0555556
$325$	−0.622216	−0.0345143
$326$	20.8573	1.15518
$327$	−4.48886	−0.248235
$328$	9.61285	0.530781
$329$	−2.42864	−0.133895
$330$	5.05086	0.278040
$331$	−15.6128	−0.858160	−0.429080	−	0.903267i	$-0.641162\pi$
$331$	−15.6128	−0.858160	−0.429080	+	0.903267i	$0.641162\pi$
$332$	−7.67307	−0.421114
$333$	2.00000	0.109599
$334$	18.0415	0.987186
$335$	1.05086	0.0574143
$336$	−1.00000	−0.0545545
$337$	32.7654	1.78485	0.892423	−	0.451200i	$-0.149004\pi$
$337$	32.7654	1.78485	0.892423	+	0.451200i	$0.149004\pi$
$338$	−12.6128	−0.686048
$339$	2.85728	0.155186
$340$	7.18421	0.389618
$341$	−12.2667	−0.664279
$342$	2.42864	0.131326
$343$	1.00000	0.0539949
$344$	−1.05086	−0.0566583
$345$	1.00000	0.0538382
$346$	5.28592	0.284173
$347$	1.24443	0.0668046	0.0334023	−	0.999442i	$-0.489366\pi$
$347$	1.24443	0.0668046	0.0334023	+	0.999442i	$0.489366\pi$
$348$	−9.80642	−0.525679
$349$	11.5714	0.619401	0.309700	−	0.950834i	$-0.399771\pi$
$349$	11.5714	0.619401	0.309700	+	0.950834i	$0.399771\pi$
$350$	1.00000	0.0534522
$351$	0.622216	0.0332114
$352$	5.05086	0.269211
$353$	−29.6128	−1.57613	−0.788066	−	0.615590i	$-0.788918\pi$
$353$	−29.6128	−1.57613	−0.788066	+	0.615590i	$0.788918\pi$
$354$	4.85728	0.258161
$355$	−3.80642	−0.202024
$356$	10.7239	0.568367
$357$	7.18421	0.380229
$358$	−9.71456	−0.513430
$359$	−23.8796	−1.26031	−0.630157	−	0.776467i	$-0.717010\pi$
$359$	−23.8796	−1.26031	−0.630157	+	0.776467i	$0.717010\pi$
$360$	−1.00000	−0.0527046
$361$	−13.1017	−0.689564
$362$	−8.36842	−0.439834
$363$	−14.5111	−0.761637
$364$	−0.622216	−0.0326130
$365$	−15.7146	−0.822538
$366$	2.85728	0.149352
$367$	−17.1240	−0.893865	−0.446932	−	0.894568i	$-0.647484\pi$
$367$	−17.1240	−0.893865	−0.446932	+	0.894568i	$0.647484\pi$
$368$	1.00000	0.0521286
$369$	9.61285	0.500425
$370$	−2.00000	−0.103975
$371$	−9.61285	−0.499074
$372$	2.42864	0.125919
$373$	−29.2257	−1.51325	−0.756625	−	0.653850i	$-0.773153\pi$
$373$	−29.2257	−1.51325	−0.756625	+	0.653850i	$0.773153\pi$
$374$	−36.2864	−1.87632
$375$	1.00000	0.0516398
$376$	−2.42864	−0.125248
$377$	−6.10171	−0.314254
$378$	−1.00000	−0.0514344
$379$	−2.95899	−0.151993	−0.0759965	−	0.997108i	$-0.524214\pi$
$379$	−2.95899	−0.151993	−0.0759965	+	0.997108i	$0.524214\pi$
$380$	−2.42864	−0.124587
$381$	−2.94914	−0.151089
$382$	4.85728	0.248520
$383$	9.51114	0.485996	0.242998	−	0.970027i	$-0.421869\pi$
$383$	9.51114	0.485996	0.242998	+	0.970027i	$0.421869\pi$
$384$	−1.00000	−0.0510310
$385$	−5.05086	−0.257415
$386$	−6.85728	−0.349026
$387$	−1.05086	−0.0534180
$388$	−0.622216	−0.0315882
$389$	10.4701	0.530856	0.265428	−	0.964131i	$-0.414487\pi$
$389$	10.4701	0.530856	0.265428	+	0.964131i	$0.414487\pi$
$390$	−0.622216	−0.0315071
$391$	−7.18421	−0.363321
$392$	1.00000	0.0505076
$393$	19.2257	0.969808
$394$	1.14272	0.0575695
$395$	−8.85728	−0.445658
$396$	5.05086	0.253815
$397$	11.5812	0.581244	0.290622	−	0.956838i	$-0.406138\pi$
$397$	11.5812	0.581244	0.290622	+	0.956838i	$0.406138\pi$
$398$	22.1017	1.10786
$399$	−2.42864	−0.121584
$400$	1.00000	0.0500000
$401$	−9.53972	−0.476391	−0.238195	−	0.971217i	$-0.576556\pi$
$401$	−9.53972	−0.476391	−0.238195	+	0.971217i	$0.576556\pi$
$402$	1.05086	0.0524119
$403$	1.51114	0.0752751
$404$	5.47949	0.272615
$405$	−1.00000	−0.0496904
$406$	9.80642	0.486685
$407$	10.1017	0.500723
$408$	7.18421	0.355671
$409$	−20.9590	−1.03636	−0.518178	−	0.855273i	$-0.673389\pi$
$409$	−20.9590	−1.03636	−0.518178	+	0.855273i	$0.673389\pi$
$410$	−9.61285	−0.474745
$411$	−2.00000	−0.0986527
$412$	−6.10171	−0.300610
$413$	−4.85728	−0.239011
$414$	1.00000	0.0491473
$415$	7.67307	0.376656
$416$	−0.622216	−0.0305066
$417$	−6.75557	−0.330822
$418$	12.2667	0.599984
$419$	−7.86665	−0.384311	−0.192155	−	0.981365i	$-0.561548\pi$
$419$	−7.86665	−0.384311	−0.192155	+	0.981365i	$0.561548\pi$
$420$	1.00000	0.0487950
$421$	−29.4291	−1.43429	−0.717144	−	0.696925i	$-0.754551\pi$
$421$	−29.4291	−1.43429	−0.717144	+	0.696925i	$0.754551\pi$
$422$	23.3461	1.13647
$423$	−2.42864	−0.118084
$424$	−9.61285	−0.466841
$425$	−7.18421	−0.348485
$426$	−3.80642	−0.184422
$427$	−2.85728	−0.138273
$428$	1.24443	0.0601519
$429$	3.14272	0.151732
$430$	1.05086	0.0506767
$431$	−32.0830	−1.54538	−0.772691	−	0.634782i	$-0.781090\pi$
$431$	−32.0830	−1.54538	−0.772691	+	0.634782i	$0.781090\pi$
$432$	−1.00000	−0.0481125
$433$	−37.3590	−1.79536	−0.897681	−	0.440647i	$-0.854749\pi$
$433$	−37.3590	−1.79536	−0.897681	+	0.440647i	$0.854749\pi$
$434$	−2.42864	−0.116578
$435$	9.80642	0.470182
$436$	4.48886	0.214978
$437$	2.42864	0.116178
$438$	−15.7146	−0.750871
$439$	−40.6133	−1.93837	−0.969184	−	0.246338i	$-0.920773\pi$
$439$	−40.6133	−1.93837	−0.969184	+	0.246338i	$0.920773\pi$
$440$	−5.05086	−0.240790
$441$	1.00000	0.0476190
$442$	4.47013	0.212622
$443$	−6.75557	−0.320967	−0.160483	−	0.987039i	$-0.551305\pi$
$443$	−6.75557	−0.320967	−0.160483	+	0.987039i	$0.551305\pi$
$444$	−2.00000	−0.0949158
$445$	−10.7239	−0.508363
$446$	14.2351	0.674050
$447$	−10.0000	−0.472984
$448$	1.00000	0.0472456
$449$	−25.4924	−1.20306	−0.601530	−	0.798850i	$-0.705442\pi$
$449$	−25.4924	−1.20306	−0.601530	+	0.798850i	$0.705442\pi$
$450$	1.00000	0.0471405
$451$	48.5531	2.28628
$452$	−2.85728	−0.134395
$453$	−2.75557	−0.129468
$454$	20.7971	0.976054
$455$	0.622216	0.0291699
$456$	−2.42864	−0.113731
$457$	17.0321	0.796729	0.398364	−	0.917227i	$-0.369578\pi$
$457$	17.0321	0.796729	0.398364	+	0.917227i	$0.369578\pi$
$458$	25.6128	1.19681
$459$	7.18421	0.335330
$460$	−1.00000	−0.0466252
$461$	−38.7239	−1.80355	−0.901777	−	0.432203i	$-0.857737\pi$
$461$	−38.7239	−1.80355	−0.901777	+	0.432203i	$0.857737\pi$
$462$	−5.05086	−0.234987
$463$	7.41927	0.344803	0.172401	−	0.985027i	$-0.444847\pi$
$463$	7.41927	0.344803	0.172401	+	0.985027i	$0.444847\pi$
$464$	9.80642	0.455252
$465$	−2.42864	−0.112625
$466$	−4.48886	−0.207943
$467$	22.8988	1.05963	0.529814	−	0.848114i	$-0.322262\pi$
$467$	22.8988	1.05963	0.529814	+	0.848114i	$0.322262\pi$
$468$	−0.622216	−0.0287619
$469$	−1.05086	−0.0485240
$470$	2.42864	0.112025
$471$	−8.10171	−0.373307
$472$	−4.85728	−0.223574
$473$	−5.30772	−0.244049
$474$	−8.85728	−0.406828
$475$	2.42864	0.111434
$476$	−7.18421	−0.329288
$477$	−9.61285	−0.440142
$478$	3.80642	0.174102
$479$	6.63512	0.303166	0.151583	−	0.988444i	$-0.451563\pi$
$479$	6.63512	0.303166	0.151583	+	0.988444i	$0.451563\pi$
$480$	1.00000	0.0456435
$481$	−1.24443	−0.0567412
$482$	−7.18421	−0.327232
$483$	−1.00000	−0.0455016
$484$	14.5111	0.659597
$485$	0.622216	0.0282534
$486$	−1.00000	−0.0453609
$487$	34.8484	1.57913	0.789566	−	0.613666i	$-0.210306\pi$
$487$	34.8484	1.57913	0.789566	+	0.613666i	$0.210306\pi$
$488$	−2.85728	−0.129343
$489$	−20.8573	−0.943199
$490$	−1.00000	−0.0451754
$491$	−34.3051	−1.54817	−0.774084	−	0.633082i	$-0.781789\pi$
$491$	−34.3051	−1.54817	−0.774084	+	0.633082i	$0.781789\pi$
$492$	−9.61285	−0.433381
$493$	−70.4514	−3.17297
$494$	−1.51114	−0.0679893
$495$	−5.05086	−0.227019
$496$	−2.42864	−0.109049
$497$	3.80642	0.170741
$498$	7.67307	0.343839
$499$	8.65386	0.387400	0.193700	−	0.981061i	$-0.437951\pi$
$499$	8.65386	0.387400	0.193700	+	0.981061i	$0.437951\pi$
$500$	−1.00000	−0.0447214
$501$	−18.0415	−0.806034
$502$	−4.99063	−0.222743
$503$	18.5718	0.828077	0.414039	−	0.910259i	$-0.364118\pi$
$503$	18.5718	0.828077	0.414039	+	0.910259i	$0.364118\pi$
$504$	1.00000	0.0445435
$505$	−5.47949	−0.243834
$506$	5.05086	0.224538
$507$	12.6128	0.560156
$508$	2.94914	0.130847
$509$	26.9906	1.19634	0.598169	−	0.801370i	$-0.295895\pi$
$509$	26.9906	1.19634	0.598169	+	0.801370i	$0.295895\pi$
$510$	−7.18421	−0.318122
$511$	15.7146	0.695171
$512$	1.00000	0.0441942
$513$	−2.42864	−0.107227
$514$	25.2257	1.11266
$515$	6.10171	0.268873
$516$	1.05086	0.0462613
$517$	−12.2667	−0.539489
$518$	2.00000	0.0878750
$519$	−5.28592	−0.232026
$520$	0.622216	0.0272860
$521$	−2.25380	−0.0987407	−0.0493704	−	0.998781i	$-0.515721\pi$
$521$	−2.25380	−0.0987407	−0.0493704	+	0.998781i	$0.515721\pi$
$522$	9.80642	0.429216
$523$	−12.1334	−0.530554	−0.265277	−	0.964172i	$-0.585463\pi$
$523$	−12.1334	−0.530554	−0.265277	+	0.964172i	$0.585463\pi$
$524$	−19.2257	−0.839878
$525$	−1.00000	−0.0436436
$526$	−8.59057	−0.374567
$527$	17.4479	0.760040
$528$	−5.05086	−0.219810
$529$	1.00000	0.0434783
$530$	9.61285	0.417555
$531$	−4.85728	−0.210788
$532$	2.42864	0.105295
$533$	−5.98126	−0.259077
$534$	−10.7239	−0.464070
$535$	−1.24443	−0.0538015
$536$	−1.05086	−0.0453900
$537$	9.71456	0.419214
$538$	8.88892	0.383229
$539$	5.05086	0.217556
$540$	1.00000	0.0430331
$541$	0.101710	0.00437286	0.00218643	−	0.999998i	$-0.499304\pi$
$541$	0.101710	0.00437286	0.00218643	+	0.999998i	$0.499304\pi$
$542$	−2.42864	−0.104319
$543$	8.36842	0.359123
$544$	−7.18421	−0.308020
$545$	−4.48886	−0.192282
$546$	0.622216	0.0266284
$547$	45.2070	1.93291	0.966455	−	0.256836i	$-0.0826800\pi$
$547$	45.2070	1.93291	0.966455	+	0.256836i	$0.0826800\pi$
$548$	2.00000	0.0854358
$549$	−2.85728	−0.121946
$550$	5.05086	0.215369
$551$	23.8163	1.01461
$552$	−1.00000	−0.0425628
$553$	8.85728	0.376650
$554$	−7.05086	−0.299562
$555$	2.00000	0.0848953
$556$	6.75557	0.286500
$557$	−41.4291	−1.75541	−0.877704	−	0.479203i	$-0.840926\pi$
$557$	−41.4291	−1.75541	−0.877704	+	0.479203i	$0.840926\pi$
$558$	−2.42864	−0.102813
$559$	0.653858	0.0276553
$560$	−1.00000	−0.0422577
$561$	36.2864	1.53201
$562$	−14.6637	−0.618551
$563$	8.32693	0.350938	0.175469	−	0.984485i	$-0.443856\pi$
$563$	8.32693	0.350938	0.175469	+	0.984485i	$0.443856\pi$
$564$	2.42864	0.102264
$565$	2.85728	0.120207
$566$	19.0923	0.802511
$567$	1.00000	0.0419961
$568$	3.80642	0.159714
$569$	−1.53972	−0.0645483	−0.0322742	−	0.999479i	$-0.510275\pi$
$569$	−1.53972	−0.0645483	−0.0322742	+	0.999479i	$0.510275\pi$
$570$	2.42864	0.101725
$571$	−41.5308	−1.73801	−0.869005	−	0.494802i	$-0.835240\pi$
$571$	−41.5308	−1.73801	−0.869005	+	0.494802i	$0.835240\pi$
$572$	−3.14272	−0.131404
$573$	−4.85728	−0.202916
$574$	9.61285	0.401233
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−15.4479	−0.643102	−0.321551	−	0.946892i	$-0.604204\pi$
$577$	−15.4479	−0.643102	−0.321551	+	0.946892i	$0.604204\pi$
$578$	34.6128	1.43970
$579$	6.85728	0.284979
$580$	−9.80642	−0.407190
$581$	−7.67307	−0.318333
$582$	0.622216	0.0257917
$583$	−48.5531	−2.01086
$584$	15.7146	0.650273
$585$	0.622216	0.0257255
$586$	−0.755569	−0.0312123
$587$	−43.2257	−1.78412	−0.892058	−	0.451922i	$-0.850739\pi$
$587$	−43.2257	−1.78412	−0.892058	+	0.451922i	$0.850739\pi$
$588$	−1.00000	−0.0412393
$589$	−5.89829	−0.243035
$590$	4.85728	0.199971
$591$	−1.14272	−0.0470053
$592$	2.00000	0.0821995
$593$	−11.5111	−0.472706	−0.236353	−	0.971667i	$-0.575952\pi$
$593$	−11.5111	−0.472706	−0.236353	+	0.971667i	$0.575952\pi$
$594$	−5.05086	−0.207239
$595$	7.18421	0.294524
$596$	10.0000	0.409616
$597$	−22.1017	−0.904563
$598$	−0.622216	−0.0254443
$599$	14.5620	0.594987	0.297493	−	0.954724i	$-0.403849\pi$
$599$	14.5620	0.594987	0.297493	+	0.954724i	$0.403849\pi$
$600$	−1.00000	−0.0408248
$601$	−6.38715	−0.260537	−0.130269	−	0.991479i	$-0.541584\pi$
$601$	−6.38715	−0.260537	−0.130269	+	0.991479i	$0.541584\pi$
$602$	−1.05086	−0.0428297
$603$	−1.05086	−0.0427941
$604$	2.75557	0.112123
$605$	−14.5111	−0.589962
$606$	−5.47949	−0.222589
$607$	−17.6445	−0.716168	−0.358084	−	0.933689i	$-0.616570\pi$
$607$	−17.6445	−0.716168	−0.358084	+	0.933689i	$0.616570\pi$
$608$	2.42864	0.0984943
$609$	−9.80642	−0.397376
$610$	2.85728	0.115688
$611$	1.51114	0.0611341
$612$	−7.18421	−0.290404
$613$	−14.4701	−0.584443	−0.292221	−	0.956351i	$-0.594394\pi$
$613$	−14.4701	−0.584443	−0.292221	+	0.956351i	$0.594394\pi$
$614$	20.5906	0.830968
$615$	9.61285	0.387627
$616$	5.05086	0.203505
$617$	24.8385	0.999962	0.499981	−	0.866036i	$-0.333340\pi$
$617$	24.8385	0.999962	0.499981	+	0.866036i	$0.333340\pi$
$618$	6.10171	0.245447
$619$	7.28592	0.292846	0.146423	−	0.989222i	$-0.453224\pi$
$619$	7.28592	0.292846	0.146423	+	0.989222i	$0.453224\pi$
$620$	2.42864	0.0975365
$621$	−1.00000	−0.0401286
$622$	−12.9906	−0.520877
$623$	10.7239	0.429645
$624$	0.622216	0.0249086
$625$	1.00000	0.0400000
$626$	11.8479	0.473538
$627$	−12.2667	−0.489885
$628$	8.10171	0.323293
$629$	−14.3684	−0.572906
$630$	−1.00000	−0.0398410
$631$	21.0607	0.838413	0.419207	−	0.907891i	$-0.362308\pi$
$631$	21.0607	0.838413	0.419207	+	0.907891i	$0.362308\pi$
$632$	8.85728	0.352324
$633$	−23.3461	−0.927926
$634$	−22.4701	−0.892403
$635$	−2.94914	−0.117033
$636$	9.61285	0.381174
$637$	−0.622216	−0.0246531
$638$	49.5308	1.96094
$639$	3.80642	0.150580
$640$	−1.00000	−0.0395285
$641$	15.7877	0.623576	0.311788	−	0.950152i	$-0.399072\pi$
$641$	15.7877	0.623576	0.311788	+	0.950152i	$0.399072\pi$
$642$	−1.24443	−0.0491138
$643$	40.7239	1.60599	0.802997	−	0.595982i	$-0.203237\pi$
$643$	40.7239	1.60599	0.802997	+	0.595982i	$0.203237\pi$
$644$	1.00000	0.0394055
$645$	−1.05086	−0.0413774
$646$	−17.4479	−0.686477
$647$	6.63206	0.260733	0.130367	−	0.991466i	$-0.458385\pi$
$647$	6.63206	0.260733	0.130367	+	0.991466i	$0.458385\pi$
$648$	1.00000	0.0392837
$649$	−24.5334	−0.963021
$650$	−0.622216	−0.0244053
$651$	2.42864	0.0951859
$652$	20.8573	0.816834
$653$	38.5531	1.50870	0.754350	−	0.656473i	$-0.227952\pi$
$653$	38.5531	1.50870	0.754350	+	0.656473i	$0.227952\pi$
$654$	−4.48886	−0.175528
$655$	19.2257	0.751210
$656$	9.61285	0.375319
$657$	15.7146	0.613083
$658$	−2.42864	−0.0946782
$659$	−20.0098	−0.779473	−0.389736	−	0.920926i	$-0.627434\pi$
$659$	−20.0098	−0.779473	−0.389736	+	0.920926i	$0.627434\pi$
$660$	5.05086	0.196604
$661$	−43.5308	−1.69315	−0.846576	−	0.532267i	$-0.821340\pi$
$661$	−43.5308	−1.69315	−0.846576	+	0.532267i	$0.821340\pi$
$662$	−15.6128	−0.606811
$663$	−4.47013	−0.173605
$664$	−7.67307	−0.297773
$665$	−2.42864	−0.0941786
$666$	2.00000	0.0774984
$667$	9.80642	0.379706
$668$	18.0415	0.698046
$669$	−14.2351	−0.550359
$670$	1.05086	0.0405981
$671$	−14.4317	−0.557130
$672$	−1.00000	−0.0385758
$673$	−38.4701	−1.48291	−0.741457	−	0.671000i	$-0.765865\pi$
$673$	−38.4701	−1.48291	−0.741457	+	0.671000i	$0.765865\pi$
$674$	32.7654	1.26208
$675$	−1.00000	−0.0384900
$676$	−12.6128	−0.485110
$677$	−48.6548	−1.86996	−0.934978	−	0.354705i	$-0.884581\pi$
$677$	−48.6548	−1.86996	−0.934978	+	0.354705i	$0.884581\pi$
$678$	2.85728	0.109733
$679$	−0.622216	−0.0238784
$680$	7.18421	0.275502
$681$	−20.7971	−0.796945
$682$	−12.2667	−0.469716
$683$	−29.4479	−1.12679	−0.563395	−	0.826187i	$-0.690505\pi$
$683$	−29.4479	−1.12679	−0.563395	+	0.826187i	$0.690505\pi$
$684$	2.42864	0.0928614
$685$	−2.00000	−0.0764161
$686$	1.00000	0.0381802
$687$	−25.6128	−0.977191
$688$	−1.05086	−0.0400635
$689$	5.98126	0.227868
$690$	1.00000	0.0380693
$691$	34.8385	1.32532	0.662660	−	0.748920i	$-0.269427\pi$
$691$	34.8385	1.32532	0.662660	+	0.748920i	$0.269427\pi$
$692$	5.28592	0.200940
$693$	5.05086	0.191866
$694$	1.24443	0.0472380
$695$	−6.75557	−0.256253
$696$	−9.80642	−0.371712
$697$	−69.0607	−2.61586
$698$	11.5714	0.437982
$699$	4.48886	0.169784
$700$	1.00000	0.0377964
$701$	25.6128	0.967384	0.483692	−	0.875238i	$-0.339296\pi$
$701$	25.6128	0.967384	0.483692	+	0.875238i	$0.339296\pi$
$702$	0.622216	0.0234840
$703$	4.85728	0.183196
$704$	5.05086	0.190361
$705$	−2.42864	−0.0914679
$706$	−29.6128	−1.11449
$707$	5.47949	0.206078
$708$	4.85728	0.182548
$709$	51.3274	1.92764	0.963821	−	0.266552i	$-0.0858843\pi$
$709$	51.3274	1.92764	0.963821	+	0.266552i	$0.0858843\pi$
$710$	−3.80642	−0.142853
$711$	8.85728	0.332174
$712$	10.7239	0.401896
$713$	−2.42864	−0.0909533
$714$	7.18421	0.268862
$715$	3.14272	0.117531
$716$	−9.71456	−0.363050
$717$	−3.80642	−0.142154
$718$	−23.8796	−0.891177
$719$	42.9086	1.60022	0.800111	−	0.599853i	$-0.204774\pi$
$719$	42.9086	1.60022	0.800111	+	0.599853i	$0.204774\pi$
$720$	−1.00000	−0.0372678
$721$	−6.10171	−0.227240
$722$	−13.1017	−0.487595
$723$	7.18421	0.267184
$724$	−8.36842	−0.311010
$725$	9.80642	0.364201
$726$	−14.5111	−0.538559
$727$	−9.44785	−0.350401	−0.175201	−	0.984533i	$-0.556057\pi$
$727$	−9.44785	−0.350401	−0.175201	+	0.984533i	$0.556057\pi$
$728$	−0.622216	−0.0230608
$729$	1.00000	0.0370370
$730$	−15.7146	−0.581622
$731$	7.54956	0.279231
$732$	2.85728	0.105608
$733$	−40.3684	−1.49104	−0.745521	−	0.666482i	$-0.767799\pi$
$733$	−40.3684	−1.49104	−0.745521	+	0.666482i	$0.767799\pi$
$734$	−17.1240	−0.632058
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	−5.30772	−0.195512
$738$	9.61285	0.353854
$739$	22.7556	0.837077	0.418539	−	0.908199i	$-0.362542\pi$
$739$	22.7556	0.837077	0.418539	+	0.908199i	$0.362542\pi$
$740$	−2.00000	−0.0735215
$741$	1.51114	0.0555130
$742$	−9.61285	−0.352899
$743$	−13.5941	−0.498720	−0.249360	−	0.968411i	$-0.580220\pi$
$743$	−13.5941	−0.498720	−0.249360	+	0.968411i	$0.580220\pi$
$744$	2.42864	0.0890382
$745$	−10.0000	−0.366372
$746$	−29.2257	−1.07003
$747$	−7.67307	−0.280743
$748$	−36.2864	−1.32676
$749$	1.24443	0.0454705
$750$	1.00000	0.0365148
$751$	37.9180	1.38365	0.691823	−	0.722067i	$-0.256808\pi$
$751$	37.9180	1.38365	0.691823	+	0.722067i	$0.256808\pi$
$752$	−2.42864	−0.0885634
$753$	4.99063	0.181869
$754$	−6.10171	−0.222211
$755$	−2.75557	−0.100285
$756$	−1.00000	−0.0363696
$757$	7.44785	0.270697	0.135348	−	0.990798i	$-0.456785\pi$
$757$	7.44785	0.270697	0.135348	+	0.990798i	$0.456785\pi$
$758$	−2.95899	−0.107475
$759$	−5.05086	−0.183334
$760$	−2.42864	−0.0880960
$761$	−13.8163	−0.500839	−0.250420	−	0.968137i	$-0.580569\pi$
$761$	−13.8163	−0.500839	−0.250420	+	0.968137i	$0.580569\pi$
$762$	−2.94914	−0.106836
$763$	4.48886	0.162508
$764$	4.85728	0.175730
$765$	7.18421	0.259746
$766$	9.51114	0.343651
$767$	3.02227	0.109128
$768$	−1.00000	−0.0360844
$769$	−51.0005	−1.83912	−0.919562	−	0.392945i	$-0.871456\pi$
$769$	−51.0005	−1.83912	−0.919562	+	0.392945i	$0.871456\pi$
$770$	−5.05086	−0.182020
$771$	−25.2257	−0.908481
$772$	−6.85728	−0.246799
$773$	−13.6128	−0.489620	−0.244810	−	0.969571i	$-0.578726\pi$
$773$	−13.6128	−0.489620	−0.244810	+	0.969571i	$0.578726\pi$
$774$	−1.05086	−0.0377722
$775$	−2.42864	−0.0872393
$776$	−0.622216	−0.0223362
$777$	−2.00000	−0.0717496
$778$	10.4701	0.375372
$779$	23.3461	0.836462
$780$	−0.622216	−0.0222789
$781$	19.2257	0.687949
$782$	−7.18421	−0.256907
$783$	−9.80642	−0.350453
$784$	1.00000	0.0357143
$785$	−8.10171	−0.289162
$786$	19.2257	0.685758
$787$	−44.4197	−1.58339	−0.791697	−	0.610915i	$-0.790802\pi$
$787$	−44.4197	−1.58339	−0.791697	+	0.610915i	$0.790802\pi$
$788$	1.14272	0.0407078
$789$	8.59057	0.305832
$790$	−8.85728	−0.315128
$791$	−2.85728	−0.101593
$792$	5.05086	0.179474
$793$	1.77784	0.0631331
$794$	11.5812	0.411002
$795$	−9.61285	−0.340933
$796$	22.1017	0.783374
$797$	11.5941	0.410685	0.205342	−	0.978690i	$-0.434169\pi$
$797$	11.5941	0.410685	0.205342	+	0.978690i	$0.434169\pi$
$798$	−2.42864	−0.0859729
$799$	17.4479	0.617261
$800$	1.00000	0.0353553
$801$	10.7239	0.378911
$802$	−9.53972	−0.336859
$803$	79.3720	2.80098
$804$	1.05086	0.0370608
$805$	−1.00000	−0.0352454
$806$	1.51114	0.0532275
$807$	−8.88892	−0.312905
$808$	5.47949	0.192768
$809$	−14.0830	−0.495131	−0.247566	−	0.968871i	$-0.579631\pi$
$809$	−14.0830	−0.495131	−0.247566	+	0.968871i	$0.579631\pi$
$810$	−1.00000	−0.0351364
$811$	−2.95899	−0.103904	−0.0519521	−	0.998650i	$-0.516544\pi$
$811$	−2.95899	−0.103904	−0.0519521	+	0.998650i	$0.516544\pi$
$812$	9.80642	0.344138
$813$	2.42864	0.0851761
$814$	10.1017	0.354065
$815$	−20.8573	−0.730599
$816$	7.18421	0.251498
$817$	−2.55215	−0.0892884
$818$	−20.9590	−0.732814
$819$	−0.622216	−0.0217420
$820$	−9.61285	−0.335695
$821$	−42.6637	−1.48897	−0.744487	−	0.667637i	$-0.767306\pi$
$821$	−42.6637	−1.48897	−0.744487	+	0.667637i	$0.767306\pi$
$822$	−2.00000	−0.0697580
$823$	−21.0509	−0.733787	−0.366893	−	0.930263i	$-0.619579\pi$
$823$	−21.0509	−0.733787	−0.366893	+	0.930263i	$0.619579\pi$
$824$	−6.10171	−0.212563
$825$	−5.05086	−0.175848
$826$	−4.85728	−0.169006
$827$	17.2444	0.599648	0.299824	−	0.953995i	$-0.403072\pi$
$827$	17.2444	0.599648	0.299824	+	0.953995i	$0.403072\pi$
$828$	1.00000	0.0347524
$829$	−6.79706	−0.236072	−0.118036	−	0.993009i	$-0.537660\pi$
$829$	−6.79706	−0.236072	−0.118036	+	0.993009i	$0.537660\pi$
$830$	7.67307	0.266336
$831$	7.05086	0.244591
$832$	−0.622216	−0.0215714
$833$	−7.18421	−0.248918
$834$	−6.75557	−0.233926
$835$	−18.0415	−0.624351
$836$	12.2667	0.424253
$837$	2.42864	0.0839461
$838$	−7.86665	−0.271749
$839$	15.1427	0.522785	0.261392	−	0.965233i	$-0.415818\pi$
$839$	15.1427	0.522785	0.261392	+	0.965233i	$0.415818\pi$
$840$	1.00000	0.0345033
$841$	67.1659	2.31607
$842$	−29.4291	−1.01419
$843$	14.6637	0.505045
$844$	23.3461	0.803607
$845$	12.6128	0.433895
$846$	−2.42864	−0.0834983
$847$	14.5111	0.498609
$848$	−9.61285	−0.330107
$849$	−19.0923	−0.655247
$850$	−7.18421	−0.246416
$851$	2.00000	0.0685591
$852$	−3.80642	−0.130406
$853$	50.8069	1.73960	0.869798	−	0.493409i	$-0.164249\pi$
$853$	50.8069	1.73960	0.869798	+	0.493409i	$0.164249\pi$
$854$	−2.85728	−0.0977741
$855$	−2.42864	−0.0830577
$856$	1.24443	0.0425338
$857$	−45.6128	−1.55811	−0.779053	−	0.626959i	$-0.784300\pi$
$857$	−45.6128	−1.55811	−0.779053	+	0.626959i	$0.784300\pi$
$858$	3.14272	0.107291
$859$	34.8385	1.18868	0.594338	−	0.804215i	$-0.297414\pi$
$859$	34.8385	1.18868	0.594338	+	0.804215i	$0.297414\pi$
$860$	1.05086	0.0358339
$861$	−9.61285	−0.327605
$862$	−32.0830	−1.09275
$863$	−8.26671	−0.281402	−0.140701	−	0.990052i	$-0.544936\pi$
$863$	−8.26671	−0.281402	−0.140701	+	0.990052i	$0.544936\pi$
$864$	−1.00000	−0.0340207
$865$	−5.28592	−0.179727
$866$	−37.3590	−1.26951
$867$	−34.6128	−1.17551
$868$	−2.42864	−0.0824334
$869$	44.7368	1.51759
$870$	9.80642	0.332469
$871$	0.653858	0.0221551
$872$	4.48886	0.152012
$873$	−0.622216	−0.0210588
$874$	2.42864	0.0821500
$875$	−1.00000	−0.0338062
$876$	−15.7146	−0.530946
$877$	40.0544	1.35254	0.676270	−	0.736653i	$-0.263595\pi$
$877$	40.0544	1.35254	0.676270	+	0.736653i	$0.263595\pi$
$878$	−40.6133	−1.37063
$879$	0.755569	0.0254847
$880$	−5.05086	−0.170264
$881$	20.3180	0.684532	0.342266	−	0.939603i	$-0.388806\pi$
$881$	20.3180	0.684532	0.342266	+	0.939603i	$0.388806\pi$
$882$	1.00000	0.0336718
$883$	48.2864	1.62497	0.812483	−	0.582984i	$-0.198115\pi$
$883$	48.2864	1.62497	0.812483	+	0.582984i	$0.198115\pi$
$884$	4.47013	0.150347
$885$	−4.85728	−0.163276
$886$	−6.75557	−0.226958
$887$	−33.5339	−1.12596	−0.562979	−	0.826471i	$-0.690345\pi$
$887$	−33.5339	−1.12596	−0.562979	+	0.826471i	$0.690345\pi$
$888$	−2.00000	−0.0671156
$889$	2.94914	0.0989111
$890$	−10.7239	−0.359467
$891$	5.05086	0.169210
$892$	14.2351	0.476625
$893$	−5.89829	−0.197379
$894$	−10.0000	−0.334450
$895$	9.71456	0.324722
$896$	1.00000	0.0334077
$897$	0.622216	0.0207752
$898$	−25.4924	−0.850692
$899$	−23.8163	−0.794317
$900$	1.00000	0.0333333
$901$	69.0607	2.30075
$902$	48.5531	1.61664
$903$	1.05086	0.0349703
$904$	−2.85728	−0.0950317
$905$	8.36842	0.278176
$906$	−2.75557	−0.0915476
$907$	−45.9081	−1.52435	−0.762177	−	0.647368i	$-0.775870\pi$
$907$	−45.9081	−1.52435	−0.762177	+	0.647368i	$0.775870\pi$
$908$	20.7971	0.690175
$909$	5.47949	0.181743
$910$	0.622216	0.0206262
$911$	17.9813	0.595746	0.297873	−	0.954606i	$-0.403723\pi$
$911$	17.9813	0.595746	0.297873	+	0.954606i	$0.403723\pi$
$912$	−2.42864	−0.0804203
$913$	−38.7556	−1.28262
$914$	17.0321	0.563372
$915$	−2.85728	−0.0944587
$916$	25.6128	0.846272
$917$	−19.2257	−0.634888
$918$	7.18421	0.237114
$919$	46.9590	1.54903	0.774517	−	0.632553i	$-0.217993\pi$
$919$	46.9590	1.54903	0.774517	+	0.632553i	$0.217993\pi$
$920$	−1.00000	−0.0329690
$921$	−20.5906	−0.678482
$922$	−38.7239	−1.27530
$923$	−2.36842	−0.0779574
$924$	−5.05086	−0.166161
$925$	2.00000	0.0657596
$926$	7.41927	0.243812
$927$	−6.10171	−0.200406
$928$	9.80642	0.321912
$929$	−19.5941	−0.642862	−0.321431	−	0.946933i	$-0.604164\pi$
$929$	−19.5941	−0.642862	−0.321431	+	0.946933i	$0.604164\pi$
$930$	−2.42864	−0.0796382
$931$	2.42864	0.0795954
$932$	−4.48886	−0.147038
$933$	12.9906	0.425294
$934$	22.8988	0.749271
$935$	36.2864	1.18669
$936$	−0.622216	−0.0203378
$937$	−6.25380	−0.204303	−0.102151	−	0.994769i	$-0.532573\pi$
$937$	−6.25380	−0.204303	−0.102151	+	0.994769i	$0.532573\pi$
$938$	−1.05086	−0.0343116
$939$	−11.8479	−0.386642
$940$	2.42864	0.0792135
$941$	−3.51114	−0.114460	−0.0572299	−	0.998361i	$-0.518227\pi$
$941$	−3.51114	−0.114460	−0.0572299	+	0.998361i	$0.518227\pi$
$942$	−8.10171	−0.263968
$943$	9.61285	0.313037
$944$	−4.85728	−0.158091
$945$	1.00000	0.0325300
$946$	−5.30772	−0.172569
$947$	−53.2641	−1.73085	−0.865426	−	0.501037i	$-0.832952\pi$
$947$	−53.2641	−1.73085	−0.865426	+	0.501037i	$0.832952\pi$
$948$	−8.85728	−0.287671
$949$	−9.77784	−0.317402
$950$	2.42864	0.0787955
$951$	22.4701	0.728644
$952$	−7.18421	−0.232842
$953$	1.87955	0.0608847	0.0304424	−	0.999537i	$-0.490308\pi$
$953$	1.87955	0.0608847	0.0304424	+	0.999537i	$0.490308\pi$
$954$	−9.61285	−0.311227
$955$	−4.85728	−0.157178
$956$	3.80642	0.123109
$957$	−49.5308	−1.60110
$958$	6.63512	0.214371
$959$	2.00000	0.0645834
$960$	1.00000	0.0322749
$961$	−25.1017	−0.809733
$962$	−1.24443	−0.0401221
$963$	1.24443	0.0401012
$964$	−7.18421	−0.231388
$965$	6.85728	0.220744
$966$	−1.00000	−0.0321745
$967$	−7.03212	−0.226138	−0.113069	−	0.993587i	$-0.536068\pi$
$967$	−7.03212	−0.226138	−0.113069	+	0.993587i	$0.536068\pi$
$968$	14.5111	0.466406
$969$	17.4479	0.560506
$970$	0.622216	0.0199781
$971$	−46.7052	−1.49884	−0.749420	−	0.662094i	$-0.769668\pi$
$971$	−46.7052	−1.49884	−0.749420	+	0.662094i	$0.769668\pi$
$972$	−1.00000	−0.0320750
$973$	6.75557	0.216574
$974$	34.8484	1.11661
$975$	0.622216	0.0199268
$976$	−2.85728	−0.0914593
$977$	22.3497	0.715030	0.357515	−	0.933907i	$-0.383624\pi$
$977$	22.3497	0.715030	0.357515	+	0.933907i	$0.383624\pi$
$978$	−20.8573	−0.666942
$979$	54.1650	1.73112
$980$	−1.00000	−0.0319438
$981$	4.48886	0.143318
$982$	−34.3051	−1.09472
$983$	−21.7146	−0.692587	−0.346293	−	0.938126i	$-0.612560\pi$
$983$	−21.7146	−0.692587	−0.346293	+	0.938126i	$0.612560\pi$
$984$	−9.61285	−0.306446
$985$	−1.14272	−0.0364101
$986$	−70.4514	−2.24363
$987$	2.42864	0.0773044
$988$	−1.51114	−0.0480757
$989$	−1.05086	−0.0334152
$990$	−5.05086	−0.160527
$991$	−15.4924	−0.492132	−0.246066	−	0.969253i	$-0.579138\pi$
$991$	−15.4924	−0.492132	−0.246066	+	0.969253i	$0.579138\pi$
$992$	−2.42864	−0.0771094
$993$	15.6128	0.495459
$994$	3.80642	0.120732
$995$	−22.1017	−0.700671
$996$	7.67307	0.243131
$997$	−19.6445	−0.622147	−0.311074	−	0.950386i	$-0.600689\pi$
$997$	−19.6445	−0.622147	−0.311074	+	0.950386i	$0.600689\pi$
$998$	8.65386	0.273933
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.bz.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.bz.1.3	✓	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.05086 q^{11} -1.00000 q^{12} -0.622216 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.18421 q^{17} +1.00000 q^{18} +2.42864 q^{19} -1.00000 q^{20} -1.00000 q^{21} +5.05086 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -0.622216 q^{26} -1.00000 q^{27} +1.00000 q^{28} +9.80642 q^{29} +1.00000 q^{30} -2.42864 q^{31} +1.00000 q^{32} -5.05086 q^{33} -7.18421 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.42864 q^{38} +0.622216 q^{39} -1.00000 q^{40} +9.61285 q^{41} -1.00000 q^{42} -1.05086 q^{43} +5.05086 q^{44} -1.00000 q^{45} +1.00000 q^{46} -2.42864 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.18421 q^{51} -0.622216 q^{52} -9.61285 q^{53} -1.00000 q^{54} -5.05086 q^{55} +1.00000 q^{56} -2.42864 q^{57} +9.80642 q^{58} -4.85728 q^{59} +1.00000 q^{60} -2.85728 q^{61} -2.42864 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.622216 q^{65} -5.05086 q^{66} -1.05086 q^{67} -7.18421 q^{68} -1.00000 q^{69} -1.00000 q^{70} +3.80642 q^{71} +1.00000 q^{72} +15.7146 q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.42864 q^{76} +5.05086 q^{77} +0.622216 q^{78} +8.85728 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.61285 q^{82} -7.67307 q^{83} -1.00000 q^{84} +7.18421 q^{85} -1.05086 q^{86} -9.80642 q^{87} +5.05086 q^{88} +10.7239 q^{89} -1.00000 q^{90} -0.622216 q^{91} +1.00000 q^{92} +2.42864 q^{93} -2.42864 q^{94} -2.42864 q^{95} -1.00000 q^{96} -0.622216 q^{97} +1.00000 q^{98} +5.05086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{13} + 3 q^{14} + 3 q^{15} + 3 q^{16} - 8 q^{17} + 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{21} + 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 16 q^{29} + 3 q^{30} + 6 q^{31} + 3 q^{32} - 2 q^{33} - 8 q^{34} - 3 q^{35} + 3 q^{36} + 6 q^{37} - 6 q^{38} + 2 q^{39} - 3 q^{40} + 2 q^{41} - 3 q^{42} + 10 q^{43} + 2 q^{44} - 3 q^{45} + 3 q^{46} + 6 q^{47} - 3 q^{48} + 3 q^{49} + 3 q^{50} + 8 q^{51} - 2 q^{52} - 2 q^{53} - 3 q^{54} - 2 q^{55} + 3 q^{56} + 6 q^{57} + 16 q^{58} + 12 q^{59} + 3 q^{60} + 18 q^{61} + 6 q^{62} + 3 q^{63} + 3 q^{64} + 2 q^{65} - 2 q^{66} + 10 q^{67} - 8 q^{68} - 3 q^{69} - 3 q^{70} - 2 q^{71} + 3 q^{72} - 6 q^{73} + 6 q^{74} - 3 q^{75} - 6 q^{76} + 2 q^{77} + 2 q^{78} - 3 q^{80} + 3 q^{81} + 2 q^{82} - 10 q^{83} - 3 q^{84} + 8 q^{85} + 10 q^{86} - 16 q^{87} + 2 q^{88} + 6 q^{89} - 3 q^{90} - 2 q^{91} + 3 q^{92} - 6 q^{93} + 6 q^{94} + 6 q^{95} - 3 q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 2 * q^13 + 3 * q^14 + 3 * q^15 + 3 * q^16 - 8 * q^17 + 3 * q^18 - 6 * q^19 - 3 * q^20 - 3 * q^21 + 2 * q^22 + 3 * q^23 - 3 * q^24 + 3 * q^25 - 2 * q^26 - 3 * q^27 + 3 * q^28 + 16 * q^29 + 3 * q^30 + 6 * q^31 + 3 * q^32 - 2 * q^33 - 8 * q^34 - 3 * q^35 + 3 * q^36 + 6 * q^37 - 6 * q^38 + 2 * q^39 - 3 * q^40 + 2 * q^41 - 3 * q^42 + 10 * q^43 + 2 * q^44 - 3 * q^45 + 3 * q^46 + 6 * q^47 - 3 * q^48 + 3 * q^49 + 3 * q^50 + 8 * q^51 - 2 * q^52 - 2 * q^53 - 3 * q^54 - 2 * q^55 + 3 * q^56 + 6 * q^57 + 16 * q^58 + 12 * q^59 + 3 * q^60 + 18 * q^61 + 6 * q^62 + 3 * q^63 + 3 * q^64 + 2 * q^65 - 2 * q^66 + 10 * q^67 - 8 * q^68 - 3 * q^69 - 3 * q^70 - 2 * q^71 + 3 * q^72 - 6 * q^73 + 6 * q^74 - 3 * q^75 - 6 * q^76 + 2 * q^77 + 2 * q^78 - 3 * q^80 + 3 * q^81 + 2 * q^82 - 10 * q^83 - 3 * q^84 + 8 * q^85 + 10 * q^86 - 16 * q^87 + 2 * q^88 + 6 * q^89 - 3 * q^90 - 2 * q^91 + 3 * q^92 - 6 * q^93 + 6 * q^94 + 6 * q^95 - 3 * q^96 - 2 * q^97 + 3 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more