LMFDB - Embedded newform 4830.2.a.cc.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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 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			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.69459 q^{11} +1.00000 q^{12} -4.36959 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.45336 q^{17} +1.00000 q^{18} +8.58172 q^{19} -1.00000 q^{20} -1.00000 q^{21} -2.69459 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.36959 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.21213 q^{29} -1.00000 q^{30} -9.06418 q^{31} +1.00000 q^{32} -2.69459 q^{33} -4.45336 q^{34} +1.00000 q^{35} +1.00000 q^{36} -0.610815 q^{37} +8.58172 q^{38} -4.36959 q^{39} -1.00000 q^{40} -9.51754 q^{41} -1.00000 q^{42} +0.822948 q^{43} -2.69459 q^{44} -1.00000 q^{45} -1.00000 q^{46} -9.06418 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.45336 q^{51} -4.36959 q^{52} +4.61081 q^{53} +1.00000 q^{54} +2.69459 q^{55} -1.00000 q^{56} +8.58172 q^{57} +8.21213 q^{58} -8.00000 q^{59} -1.00000 q^{60} -5.51754 q^{61} -9.06418 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.36959 q^{65} -2.69459 q^{66} -14.2121 q^{67} -4.45336 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.08378 q^{71} +1.00000 q^{72} +14.2567 q^{73} -0.610815 q^{74} +1.00000 q^{75} +8.58172 q^{76} +2.69459 q^{77} -4.36959 q^{78} -13.6459 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.51754 q^{82} +3.19253 q^{83} -1.00000 q^{84} +4.45336 q^{85} +0.822948 q^{86} +8.21213 q^{87} -2.69459 q^{88} -18.0155 q^{89} -1.00000 q^{90} +4.36959 q^{91} -1.00000 q^{92} -9.06418 q^{93} -9.06418 q^{94} -8.58172 q^{95} +1.00000 q^{96} -6.98040 q^{97} +1.00000 q^{98} -2.69459 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} + 3 q^{12} - 6 q^{13} - 3 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{21} - 6 q^{22} - 3 q^{23} + 3 q^{24} + 3 q^{25} - 6 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{30} - 18 q^{31} + 3 q^{32} - 6 q^{33} + 3 q^{35} + 3 q^{36} - 6 q^{37} - 6 q^{38} - 6 q^{39} - 3 q^{40} - 6 q^{41} - 3 q^{42} - 18 q^{43} - 6 q^{44} - 3 q^{45} - 3 q^{46} - 18 q^{47} + 3 q^{48} + 3 q^{49} + 3 q^{50} - 6 q^{52} + 18 q^{53} + 3 q^{54} + 6 q^{55} - 3 q^{56} - 6 q^{57} - 24 q^{59} - 3 q^{60} + 6 q^{61} - 18 q^{62} - 3 q^{63} + 3 q^{64} + 6 q^{65} - 6 q^{66} - 18 q^{67} - 3 q^{69} + 3 q^{70} - 6 q^{71} + 3 q^{72} + 6 q^{73} - 6 q^{74} + 3 q^{75} - 6 q^{76} + 6 q^{77} - 6 q^{78} - 3 q^{80} + 3 q^{81} - 6 q^{82} - 18 q^{83} - 3 q^{84} - 18 q^{86} - 6 q^{88} - 6 q^{89} - 3 q^{90} + 6 q^{91} - 3 q^{92} - 18 q^{93} - 18 q^{94} + 6 q^{95} + 3 q^{96} - 18 q^{97} + 3 q^{98} - 6 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 - 6 * q^11 + 3 * q^12 - 6 * q^13 - 3 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^18 - 6 * q^19 - 3 * q^20 - 3 * q^21 - 6 * q^22 - 3 * q^23 + 3 * q^24 + 3 * q^25 - 6 * q^26 + 3 * q^27 - 3 * q^28 - 3 * q^30 - 18 * q^31 + 3 * q^32 - 6 * q^33 + 3 * q^35 + 3 * q^36 - 6 * q^37 - 6 * q^38 - 6 * q^39 - 3 * q^40 - 6 * q^41 - 3 * q^42 - 18 * q^43 - 6 * q^44 - 3 * q^45 - 3 * q^46 - 18 * q^47 + 3 * q^48 + 3 * q^49 + 3 * q^50 - 6 * q^52 + 18 * q^53 + 3 * q^54 + 6 * q^55 - 3 * q^56 - 6 * q^57 - 24 * q^59 - 3 * q^60 + 6 * q^61 - 18 * q^62 - 3 * q^63 + 3 * q^64 + 6 * q^65 - 6 * q^66 - 18 * q^67 - 3 * q^69 + 3 * q^70 - 6 * q^71 + 3 * q^72 + 6 * q^73 - 6 * q^74 + 3 * q^75 - 6 * q^76 + 6 * q^77 - 6 * q^78 - 3 * q^80 + 3 * q^81 - 6 * q^82 - 18 * q^83 - 3 * q^84 - 18 * q^86 - 6 * q^88 - 6 * q^89 - 3 * q^90 + 6 * q^91 - 3 * q^92 - 18 * q^93 - 18 * q^94 + 6 * q^95 + 3 * q^96 - 18 * q^97 + 3 * q^98 - 6 * 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
$11$	−2.69459	−0.812450	−0.406225	−	0.913773i	$-0.633155\pi$
$11$	−2.69459	−0.812450	−0.406225	+	0.913773i	$0.633155\pi$
$12$	1.00000	0.288675
$13$	−4.36959	−1.21190	−0.605952	−	0.795501i	$-0.707208\pi$
$13$	−4.36959	−1.21190	−0.605952	+	0.795501i	$0.707208\pi$
$14$	−1.00000	−0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−4.45336	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$17$	−4.45336	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$18$	1.00000	0.235702
$19$	8.58172	1.96878	0.984391	−	0.175997i	$-0.0563150\pi$
$19$	8.58172	1.96878	0.984391	+	0.175997i	$0.0563150\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−2.69459	−0.574489
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−4.36959	−0.856946
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	8.21213	1.52495	0.762477	−	0.647015i	$-0.223983\pi$
$29$	8.21213	1.52495	0.762477	+	0.647015i	$0.223983\pi$
$30$	−1.00000	−0.182574
$31$	−9.06418	−1.62797	−0.813987	−	0.580883i	$-0.802707\pi$
$31$	−9.06418	−1.62797	−0.813987	+	0.580883i	$0.802707\pi$
$32$	1.00000	0.176777
$33$	−2.69459	−0.469068
$34$	−4.45336	−0.763745
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−0.610815	−0.100417	−0.0502086	−	0.998739i	$-0.515989\pi$
$37$	−0.610815	−0.100417	−0.0502086	+	0.998739i	$0.515989\pi$
$38$	8.58172	1.39214
$39$	−4.36959	−0.699694
$40$	−1.00000	−0.158114
$41$	−9.51754	−1.48639	−0.743195	−	0.669075i	$-0.766691\pi$
$41$	−9.51754	−1.48639	−0.743195	+	0.669075i	$0.766691\pi$
$42$	−1.00000	−0.154303
$43$	0.822948	0.125498	0.0627492	−	0.998029i	$-0.480013\pi$
$43$	0.822948	0.125498	0.0627492	+	0.998029i	$0.480013\pi$
$44$	−2.69459	−0.406225
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	−9.06418	−1.32215	−0.661073	−	0.750321i	$-0.729899\pi$
$47$	−9.06418	−1.32215	−0.661073	+	0.750321i	$0.729899\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.45336	−0.623596
$52$	−4.36959	−0.605952
$53$	4.61081	0.633344	0.316672	−	0.948535i	$-0.397435\pi$
$53$	4.61081	0.633344	0.316672	+	0.948535i	$0.397435\pi$
$54$	1.00000	0.136083
$55$	2.69459	0.363339
$56$	−1.00000	−0.133631
$57$	8.58172	1.13668
$58$	8.21213	1.07831
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−1.00000	−0.129099
$61$	−5.51754	−0.706449	−0.353224	−	0.935539i	$-0.614915\pi$
$61$	−5.51754	−0.706449	−0.353224	+	0.935539i	$0.614915\pi$
$62$	−9.06418	−1.15115
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.36959	0.541980
$66$	−2.69459	−0.331681
$67$	−14.2121	−1.73629	−0.868144	−	0.496312i	$-0.834687\pi$
$67$	−14.2121	−1.73629	−0.868144	+	0.496312i	$0.834687\pi$
$68$	−4.45336	−0.540050
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−4.08378	−0.484655	−0.242328	−	0.970194i	$-0.577911\pi$
$71$	−4.08378	−0.484655	−0.242328	+	0.970194i	$0.577911\pi$
$72$	1.00000	0.117851
$73$	14.2567	1.66862	0.834311	−	0.551294i	$-0.185866\pi$
$73$	14.2567	1.66862	0.834311	+	0.551294i	$0.185866\pi$
$74$	−0.610815	−0.0710058
$75$	1.00000	0.115470
$76$	8.58172	0.984391
$77$	2.69459	0.307077
$78$	−4.36959	−0.494758
$79$	−13.6459	−1.53528	−0.767642	−	0.640879i	$-0.778570\pi$
$79$	−13.6459	−1.53528	−0.767642	+	0.640879i	$0.778570\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−9.51754	−1.05104
$83$	3.19253	0.350426	0.175213	−	0.984531i	$-0.443939\pi$
$83$	3.19253	0.350426	0.175213	+	0.984531i	$0.443939\pi$
$84$	−1.00000	−0.109109
$85$	4.45336	0.483035
$86$	0.822948	0.0887408
$87$	8.21213	0.880433
$88$	−2.69459	−0.287245
$89$	−18.0155	−1.90964	−0.954819	−	0.297189i	$-0.903951\pi$
$89$	−18.0155	−1.90964	−0.954819	+	0.297189i	$0.903951\pi$
$90$	−1.00000	−0.105409
$91$	4.36959	0.458057
$92$	−1.00000	−0.104257
$93$	−9.06418	−0.939911
$94$	−9.06418	−0.934899
$95$	−8.58172	−0.880466
$96$	1.00000	0.102062
$97$	−6.98040	−0.708752	−0.354376	−	0.935103i	$-0.615307\pi$
$97$	−6.98040	−0.708752	−0.354376	+	0.935103i	$0.615307\pi$
$98$	1.00000	0.101015
$99$	−2.69459	−0.270817
$100$	1.00000	0.100000
$101$	−18.6655	−1.85729	−0.928643	−	0.370974i	$-0.879024\pi$
$101$	−18.6655	−1.85729	−0.928643	+	0.370974i	$0.879024\pi$
$102$	−4.45336	−0.440949
$103$	11.5175	1.13486	0.567429	−	0.823423i	$-0.307938\pi$
$103$	11.5175	1.13486	0.567429	+	0.823423i	$0.307938\pi$
$104$	−4.36959	−0.428473
$105$	1.00000	0.0975900
$106$	4.61081	0.447842
$107$	4.73917	0.458153	0.229076	−	0.973408i	$-0.426429\pi$
$107$	4.73917	0.458153	0.229076	+	0.973408i	$0.426429\pi$
$108$	1.00000	0.0962250
$109$	−9.51754	−0.911615	−0.455808	−	0.890078i	$-0.650649\pi$
$109$	−9.51754	−0.911615	−0.455808	+	0.890078i	$0.650649\pi$
$110$	2.69459	0.256919
$111$	−0.610815	−0.0579760
$112$	−1.00000	−0.0944911
$113$	−16.1284	−1.51723	−0.758614	−	0.651540i	$-0.774123\pi$
$113$	−16.1284	−1.51723	−0.758614	+	0.651540i	$0.774123\pi$
$114$	8.58172	0.803752
$115$	1.00000	0.0932505
$116$	8.21213	0.762477
$117$	−4.36959	−0.403968
$118$	−8.00000	−0.736460
$119$	4.45336	0.408239
$120$	−1.00000	−0.0912871
$121$	−3.73917	−0.339925
$122$	−5.51754	−0.499535
$123$	−9.51754	−0.858168
$124$	−9.06418	−0.813987
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	17.7297	1.57325	0.786627	−	0.617428i	$-0.211825\pi$
$127$	17.7297	1.57325	0.786627	+	0.617428i	$0.211825\pi$
$128$	1.00000	0.0883883
$129$	0.822948	0.0724566
$130$	4.36959	0.383238
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−2.69459	−0.234534
$133$	−8.58172	−0.744129
$134$	−14.2121	−1.22774
$135$	−1.00000	−0.0860663
$136$	−4.45336	−0.381873
$137$	14.2567	1.21803	0.609017	−	0.793158i	$-0.291564\pi$
$137$	14.2567	1.21803	0.609017	+	0.793158i	$0.291564\pi$
$138$	−1.00000	−0.0851257
$139$	14.2959	1.21256	0.606282	−	0.795250i	$-0.292660\pi$
$139$	14.2959	1.21256	0.606282	+	0.795250i	$0.292660\pi$
$140$	1.00000	0.0845154
$141$	−9.06418	−0.763341
$142$	−4.08378	−0.342703
$143$	11.7743	0.984612
$144$	1.00000	0.0833333
$145$	−8.21213	−0.681981
$146$	14.2567	1.17989
$147$	1.00000	0.0824786
$148$	−0.610815	−0.0502086
$149$	−20.8675	−1.70953	−0.854767	−	0.519012i	$-0.826300\pi$
$149$	−20.8675	−1.70953	−0.854767	+	0.519012i	$0.826300\pi$
$150$	1.00000	0.0816497
$151$	−19.7743	−1.60921	−0.804603	−	0.593813i	$-0.797622\pi$
$151$	−19.7743	−1.60921	−0.804603	+	0.593813i	$0.797622\pi$
$152$	8.58172	0.696069
$153$	−4.45336	−0.360033
$154$	2.69459	0.217136
$155$	9.06418	0.728052
$156$	−4.36959	−0.349847
$157$	−1.51754	−0.121113	−0.0605565	−	0.998165i	$-0.519288\pi$
$157$	−1.51754	−0.121113	−0.0605565	+	0.998165i	$0.519288\pi$
$158$	−13.6459	−1.08561
$159$	4.61081	0.365661
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	3.51754	0.275515	0.137758	−	0.990466i	$-0.456011\pi$
$163$	3.51754	0.275515	0.137758	+	0.990466i	$0.456011\pi$
$164$	−9.51754	−0.743195
$165$	2.69459	0.209774
$166$	3.19253	0.247789
$167$	18.7101	1.44783	0.723915	−	0.689890i	$-0.242341\pi$
$167$	18.7101	1.44783	0.723915	+	0.689890i	$0.242341\pi$
$168$	−1.00000	−0.0771517
$169$	6.09327	0.468713
$170$	4.45336	0.341557
$171$	8.58172	0.656260
$172$	0.822948	0.0627492
$173$	5.10338	0.388003	0.194001	−	0.981001i	$-0.437853\pi$
$173$	5.10338	0.388003	0.194001	+	0.981001i	$0.437853\pi$
$174$	8.21213	0.622560
$175$	−1.00000	−0.0755929
$176$	−2.69459	−0.203113
$177$	−8.00000	−0.601317
$178$	−18.0155	−1.35032
$179$	18.8675	1.41023	0.705113	−	0.709095i	$-0.250896\pi$
$179$	18.8675	1.41023	0.705113	+	0.709095i	$0.250896\pi$
$180$	−1.00000	−0.0745356
$181$	16.5526	1.23035	0.615173	−	0.788392i	$-0.289086\pi$
$181$	16.5526	1.23035	0.615173	+	0.788392i	$0.289086\pi$
$182$	4.36959	0.323895
$183$	−5.51754	−0.407868
$184$	−1.00000	−0.0737210
$185$	0.610815	0.0449080
$186$	−9.06418	−0.664618
$187$	12.0000	0.877527
$188$	−9.06418	−0.661073
$189$	−1.00000	−0.0727393
$190$	−8.58172	−0.622583
$191$	−7.34998	−0.531826	−0.265913	−	0.963997i	$-0.585673\pi$
$191$	−7.34998	−0.531826	−0.265913	+	0.963997i	$0.585673\pi$
$192$	1.00000	0.0721688
$193$	−4.77837	−0.343955	−0.171977	−	0.985101i	$-0.555016\pi$
$193$	−4.77837	−0.343955	−0.171977	+	0.985101i	$0.555016\pi$
$194$	−6.98040	−0.501163
$195$	4.36959	0.312912
$196$	1.00000	0.0714286
$197$	16.1284	1.14910	0.574549	−	0.818470i	$-0.305178\pi$
$197$	16.1284	1.14910	0.574549	+	0.818470i	$0.305178\pi$
$198$	−2.69459	−0.191496
$199$	−2.61081	−0.185076	−0.0925379	−	0.995709i	$-0.529498\pi$
$199$	−2.61081	−0.185076	−0.0925379	+	0.995709i	$0.529498\pi$
$200$	1.00000	0.0707107
$201$	−14.2121	−1.00245
$202$	−18.6655	−1.31330
$203$	−8.21213	−0.576379
$204$	−4.45336	−0.311798
$205$	9.51754	0.664734
$206$	11.5175	0.802465
$207$	−1.00000	−0.0695048
$208$	−4.36959	−0.302976
$209$	−23.1242	−1.59954
$210$	1.00000	0.0690066
$211$	22.8675	1.57427	0.787133	−	0.616784i	$-0.211565\pi$
$211$	22.8675	1.57427	0.787133	+	0.616784i	$0.211565\pi$
$212$	4.61081	0.316672
$213$	−4.08378	−0.279816
$214$	4.73917	0.323963
$215$	−0.822948	−0.0561246
$216$	1.00000	0.0680414
$217$	9.06418	0.615316
$218$	−9.51754	−0.644609
$219$	14.2567	0.963379
$220$	2.69459	0.181669
$221$	19.4593	1.30898
$222$	−0.610815	−0.0409952
$223$	−9.14796	−0.612592	−0.306296	−	0.951936i	$-0.599090\pi$
$223$	−9.14796	−0.612592	−0.306296	+	0.951936i	$0.599090\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−16.1284	−1.07284
$227$	−9.54664	−0.633633	−0.316816	−	0.948487i	$-0.602614\pi$
$227$	−9.54664	−0.633633	−0.316816	+	0.948487i	$0.602614\pi$
$228$	8.58172	0.568338
$229$	18.6810	1.23447	0.617237	−	0.786777i	$-0.288252\pi$
$229$	18.6810	1.23447	0.617237	+	0.786777i	$0.288252\pi$
$230$	1.00000	0.0659380
$231$	2.69459	0.177291
$232$	8.21213	0.539153
$233$	21.5175	1.40966	0.704830	−	0.709376i	$-0.251023\pi$
$233$	21.5175	1.40966	0.704830	+	0.709376i	$0.251023\pi$
$234$	−4.36959	−0.285649
$235$	9.06418	0.591282
$236$	−8.00000	−0.520756
$237$	−13.6459	−0.886396
$238$	4.45336	0.288669
$239$	−4.08378	−0.264158	−0.132079	−	0.991239i	$-0.542165\pi$
$239$	−4.08378	−0.264158	−0.132079	+	0.991239i	$0.542165\pi$
$240$	−1.00000	−0.0645497
$241$	−4.45336	−0.286866	−0.143433	−	0.989660i	$-0.545814\pi$
$241$	−4.45336	−0.286866	−0.143433	+	0.989660i	$0.545814\pi$
$242$	−3.73917	−0.240363
$243$	1.00000	0.0641500
$244$	−5.51754	−0.353224
$245$	−1.00000	−0.0638877
$246$	−9.51754	−0.606816
$247$	−37.4986	−2.38598
$248$	−9.06418	−0.575576
$249$	3.19253	0.202319
$250$	−1.00000	−0.0632456
$251$	16.1830	1.02146	0.510732	−	0.859740i	$-0.329374\pi$
$251$	16.1830	1.02146	0.510732	+	0.859740i	$0.329374\pi$
$252$	−1.00000	−0.0629941
$253$	2.69459	0.169408
$254$	17.7297	1.11246
$255$	4.45336	0.278880
$256$	1.00000	0.0625000
$257$	27.6459	1.72450	0.862252	−	0.506480i	$-0.169054\pi$
$257$	27.6459	1.72450	0.862252	+	0.506480i	$0.169054\pi$
$258$	0.822948	0.0512345
$259$	0.610815	0.0379542
$260$	4.36959	0.270990
$261$	8.21213	0.508318
$262$	−4.00000	−0.247121
$263$	14.1284	0.871192	0.435596	−	0.900142i	$-0.356538\pi$
$263$	14.1284	0.871192	0.435596	+	0.900142i	$0.356538\pi$
$264$	−2.69459	−0.165841
$265$	−4.61081	−0.283240
$266$	−8.58172	−0.526179
$267$	−18.0155	−1.10253
$268$	−14.2121	−0.868144
$269$	1.75877	0.107234	0.0536171	−	0.998562i	$-0.482925\pi$
$269$	1.75877	0.107234	0.0536171	+	0.998562i	$0.482925\pi$
$270$	−1.00000	−0.0608581
$271$	−0.0992597	−0.00602960	−0.00301480	−	0.999995i	$-0.500960\pi$
$271$	−0.0992597	−0.00602960	−0.00301480	+	0.999995i	$0.500960\pi$
$272$	−4.45336	−0.270025
$273$	4.36959	0.264459
$274$	14.2567	0.861279
$275$	−2.69459	−0.162490
$276$	−1.00000	−0.0601929
$277$	13.1771	0.791732	0.395866	−	0.918308i	$-0.370444\pi$
$277$	13.1771	0.791732	0.395866	+	0.918308i	$0.370444\pi$
$278$	14.2959	0.857412
$279$	−9.06418	−0.542658
$280$	1.00000	0.0597614
$281$	−9.43376	−0.562771	−0.281386	−	0.959595i	$-0.590794\pi$
$281$	−9.43376	−0.562771	−0.281386	+	0.959595i	$0.590794\pi$
$282$	−9.06418	−0.539764
$283$	−12.6655	−0.752886	−0.376443	−	0.926440i	$-0.622853\pi$
$283$	−12.6655	−0.752886	−0.376443	+	0.926440i	$0.622853\pi$
$284$	−4.08378	−0.242328
$285$	−8.58172	−0.508337
$286$	11.7743	0.696226
$287$	9.51754	0.561803
$288$	1.00000	0.0589256
$289$	2.83244	0.166614
$290$	−8.21213	−0.482233
$291$	−6.98040	−0.409198
$292$	14.2567	0.834311
$293$	−18.9067	−1.10454	−0.552271	−	0.833664i	$-0.686239\pi$
$293$	−18.9067	−1.10454	−0.552271	+	0.833664i	$0.686239\pi$
$294$	1.00000	0.0583212
$295$	8.00000	0.465778
$296$	−0.610815	−0.0355029
$297$	−2.69459	−0.156356
$298$	−20.8675	−1.20882
$299$	4.36959	0.252700
$300$	1.00000	0.0577350
$301$	−0.822948	−0.0474339
$302$	−19.7743	−1.13788
$303$	−18.6655	−1.07230
$304$	8.58172	0.492195
$305$	5.51754	0.315933
$306$	−4.45336	−0.254582
$307$	0.482459	0.0275354	0.0137677	−	0.999905i	$-0.495617\pi$
$307$	0.482459	0.0275354	0.0137677	+	0.999905i	$0.495617\pi$
$308$	2.69459	0.153539
$309$	11.5175	0.655210
$310$	9.06418	0.514811
$311$	−11.0196	−0.624864	−0.312432	−	0.949940i	$-0.601144\pi$
$311$	−11.0196	−0.624864	−0.312432	+	0.949940i	$0.601144\pi$
$312$	−4.36959	−0.247379
$313$	−2.24123	−0.126682	−0.0633409	−	0.997992i	$-0.520176\pi$
$313$	−2.24123	−0.126682	−0.0633409	+	0.997992i	$0.520176\pi$
$314$	−1.51754	−0.0856398
$315$	1.00000	0.0563436
$316$	−13.6459	−0.767642
$317$	−3.96080	−0.222461	−0.111230	−	0.993795i	$-0.535479\pi$
$317$	−3.96080	−0.222461	−0.111230	+	0.993795i	$0.535479\pi$
$318$	4.61081	0.258562
$319$	−22.1284	−1.23895
$320$	−1.00000	−0.0559017
$321$	4.73917	0.264515
$322$	1.00000	0.0557278
$323$	−38.2175	−2.12648
$324$	1.00000	0.0555556
$325$	−4.36959	−0.242381
$326$	3.51754	0.194819
$327$	−9.51754	−0.526321
$328$	−9.51754	−0.525518
$329$	9.06418	0.499724
$330$	2.69459	0.148332
$331$	23.6851	1.30185	0.650925	−	0.759142i	$-0.274381\pi$
$331$	23.6851	1.30185	0.650925	+	0.759142i	$0.274381\pi$
$332$	3.19253	0.175213
$333$	−0.610815	−0.0334724
$334$	18.7101	1.02377
$335$	14.2121	0.776492
$336$	−1.00000	−0.0545545
$337$	24.9513	1.35918	0.679592	−	0.733590i	$-0.262157\pi$
$337$	24.9513	1.35918	0.679592	+	0.733590i	$0.262157\pi$
$338$	6.09327	0.331430
$339$	−16.1284	−0.875972
$340$	4.45336	0.241518
$341$	24.4243	1.32265
$342$	8.58172	0.464046
$343$	−1.00000	−0.0539949
$344$	0.822948	0.0443704
$345$	1.00000	0.0538382
$346$	5.10338	0.274359
$347$	7.26083	0.389782	0.194891	−	0.980825i	$-0.437565\pi$
$347$	7.26083	0.389782	0.194891	+	0.980825i	$0.437565\pi$
$348$	8.21213	0.440217
$349$	−19.2317	−1.02945	−0.514726	−	0.857355i	$-0.672106\pi$
$349$	−19.2317	−1.02945	−0.514726	+	0.857355i	$0.672106\pi$
$350$	−1.00000	−0.0534522
$351$	−4.36959	−0.233231
$352$	−2.69459	−0.143622
$353$	−2.42427	−0.129031	−0.0645154	−	0.997917i	$-0.520550\pi$
$353$	−2.42427	−0.129031	−0.0645154	+	0.997917i	$0.520550\pi$
$354$	−8.00000	−0.425195
$355$	4.08378	0.216744
$356$	−18.0155	−0.954819
$357$	4.45336	0.235697
$358$	18.8675	0.997180
$359$	−18.1284	−0.956778	−0.478389	−	0.878148i	$-0.658779\pi$
$359$	−18.1284	−0.956778	−0.478389	+	0.878148i	$0.658779\pi$
$360$	−1.00000	−0.0527046
$361$	54.6459	2.87610
$362$	16.5526	0.869987
$363$	−3.73917	−0.196256
$364$	4.36959	0.229028
$365$	−14.2567	−0.746230
$366$	−5.51754	−0.288406
$367$	−20.9959	−1.09598	−0.547988	−	0.836486i	$-0.684606\pi$
$367$	−20.9959	−1.09598	−0.547988	+	0.836486i	$0.684606\pi$
$368$	−1.00000	−0.0521286
$369$	−9.51754	−0.495463
$370$	0.610815	0.0317547
$371$	−4.61081	−0.239382
$372$	−9.06418	−0.469956
$373$	33.2026	1.71917	0.859584	−	0.510995i	$-0.170723\pi$
$373$	33.2026	1.71917	0.859584	+	0.510995i	$0.170723\pi$
$374$	12.0000	0.620505
$375$	−1.00000	−0.0516398
$376$	−9.06418	−0.467449
$377$	−35.8836	−1.84810
$378$	−1.00000	−0.0514344
$379$	0.482459	0.0247823	0.0123911	−	0.999923i	$-0.496056\pi$
$379$	0.482459	0.0247823	0.0123911	+	0.999923i	$0.496056\pi$
$380$	−8.58172	−0.440233
$381$	17.7297	0.908319
$382$	−7.34998	−0.376058
$383$	−21.9026	−1.11917	−0.559585	−	0.828773i	$-0.689040\pi$
$383$	−21.9026	−1.11917	−0.559585	+	0.828773i	$0.689040\pi$
$384$	1.00000	0.0510310
$385$	−2.69459	−0.137329
$386$	−4.77837	−0.243213
$387$	0.822948	0.0418328
$388$	−6.98040	−0.354376
$389$	−25.0351	−1.26933	−0.634665	−	0.772788i	$-0.718862\pi$
$389$	−25.0351	−1.26933	−0.634665	+	0.772788i	$0.718862\pi$
$390$	4.36959	0.221263
$391$	4.45336	0.225216
$392$	1.00000	0.0505076
$393$	−4.00000	−0.201773
$394$	16.1284	0.812535
$395$	13.6459	0.686600
$396$	−2.69459	−0.135408
$397$	20.1438	1.01099	0.505495	−	0.862829i	$-0.331310\pi$
$397$	20.1438	1.01099	0.505495	+	0.862829i	$0.331310\pi$
$398$	−2.61081	−0.130868
$399$	−8.58172	−0.429623
$400$	1.00000	0.0500000
$401$	−24.6364	−1.23028	−0.615142	−	0.788417i	$-0.710901\pi$
$401$	−24.6364	−1.23028	−0.615142	+	0.788417i	$0.710901\pi$
$402$	−14.2121	−0.708837
$403$	39.6067	1.97295
$404$	−18.6655	−0.928643
$405$	−1.00000	−0.0496904
$406$	−8.21213	−0.407561
$407$	1.64590	0.0815841
$408$	−4.45336	−0.220474
$409$	−9.09327	−0.449633	−0.224817	−	0.974401i	$-0.572178\pi$
$409$	−9.09327	−0.449633	−0.224817	+	0.974401i	$0.572178\pi$
$410$	9.51754	0.470038
$411$	14.2567	0.703232
$412$	11.5175	0.567429
$413$	8.00000	0.393654
$414$	−1.00000	−0.0491473
$415$	−3.19253	−0.156715
$416$	−4.36959	−0.214237
$417$	14.2959	0.700074
$418$	−23.1242	−1.13104
$419$	−18.3114	−0.894570	−0.447285	−	0.894391i	$-0.647609\pi$
$419$	−18.3114	−0.894570	−0.447285	+	0.894391i	$0.647609\pi$
$420$	1.00000	0.0487950
$421$	19.8135	0.965649	0.482824	−	0.875717i	$-0.339611\pi$
$421$	19.8135	0.965649	0.482824	+	0.875717i	$0.339611\pi$
$422$	22.8675	1.11317
$423$	−9.06418	−0.440715
$424$	4.61081	0.223921
$425$	−4.45336	−0.216020
$426$	−4.08378	−0.197860
$427$	5.51754	0.267013
$428$	4.73917	0.229076
$429$	11.7743	0.568466
$430$	−0.822948	−0.0396861
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	1.00000	0.0481125
$433$	−27.5722	−1.32504	−0.662518	−	0.749046i	$-0.730512\pi$
$433$	−27.5722	−1.32504	−0.662518	+	0.749046i	$0.730512\pi$
$434$	9.06418	0.435094
$435$	−8.21213	−0.393742
$436$	−9.51754	−0.455808
$437$	−8.58172	−0.410519
$438$	14.2567	0.681212
$439$	3.67499	0.175398	0.0876989	−	0.996147i	$-0.472049\pi$
$439$	3.67499	0.175398	0.0876989	+	0.996147i	$0.472049\pi$
$440$	2.69459	0.128460
$441$	1.00000	0.0476190
$442$	19.4593	0.925587
$443$	8.73917	0.415211	0.207605	−	0.978213i	$-0.433433\pi$
$443$	8.73917	0.415211	0.207605	+	0.978213i	$0.433433\pi$
$444$	−0.610815	−0.0289880
$445$	18.0155	0.854016
$446$	−9.14796	−0.433168
$447$	−20.8675	−0.987000
$448$	−1.00000	−0.0472456
$449$	27.9026	1.31681	0.658403	−	0.752666i	$-0.271232\pi$
$449$	27.9026	1.31681	0.658403	+	0.752666i	$0.271232\pi$
$450$	1.00000	0.0471405
$451$	25.6459	1.20762
$452$	−16.1284	−0.758614
$453$	−19.7743	−0.929075
$454$	−9.54664	−0.448046
$455$	−4.36959	−0.204849
$456$	8.58172	0.401876
$457$	−33.8580	−1.58381	−0.791906	−	0.610643i	$-0.790911\pi$
$457$	−33.8580	−1.58381	−0.791906	+	0.610643i	$0.790911\pi$
$458$	18.6810	0.872905
$459$	−4.45336	−0.207865
$460$	1.00000	0.0466252
$461$	−22.4088	−1.04368	−0.521841	−	0.853043i	$-0.674754\pi$
$461$	−22.4088	−1.04368	−0.521841	+	0.853043i	$0.674754\pi$
$462$	2.69459	0.125364
$463$	−19.8580	−0.922881	−0.461440	−	0.887171i	$-0.652667\pi$
$463$	−19.8580	−0.922881	−0.461440	+	0.887171i	$0.652667\pi$
$464$	8.21213	0.381239
$465$	9.06418	0.420341
$466$	21.5175	0.996781
$467$	−13.0060	−0.601845	−0.300923	−	0.953649i	$-0.597295\pi$
$467$	−13.0060	−0.601845	−0.300923	+	0.953649i	$0.597295\pi$
$468$	−4.36959	−0.201984
$469$	14.2121	0.656255
$470$	9.06418	0.418099
$471$	−1.51754	−0.0699246
$472$	−8.00000	−0.368230
$473$	−2.21751	−0.101961
$474$	−13.6459	−0.626777
$475$	8.58172	0.393756
$476$	4.45336	0.204120
$477$	4.61081	0.211115
$478$	−4.08378	−0.186788
$479$	−4.48246	−0.204809	−0.102404	−	0.994743i	$-0.532654\pi$
$479$	−4.48246	−0.204809	−0.102404	+	0.994743i	$0.532654\pi$
$480$	−1.00000	−0.0456435
$481$	2.66901	0.121696
$482$	−4.45336	−0.202845
$483$	1.00000	0.0455016
$484$	−3.73917	−0.169962
$485$	6.98040	0.316964
$486$	1.00000	0.0453609
$487$	20.9905	0.951171	0.475585	−	0.879670i	$-0.342236\pi$
$487$	20.9905	0.951171	0.475585	+	0.879670i	$0.342236\pi$
$488$	−5.51754	−0.249767
$489$	3.51754	0.159069
$490$	−1.00000	−0.0451754
$491$	−30.3851	−1.37126	−0.685629	−	0.727951i	$-0.740473\pi$
$491$	−30.3851	−1.37126	−0.685629	+	0.727951i	$0.740473\pi$
$492$	−9.51754	−0.429084
$493$	−36.5716	−1.64710
$494$	−37.4986	−1.68714
$495$	2.69459	0.121113
$496$	−9.06418	−0.406994
$497$	4.08378	0.183182
$498$	3.19253	0.143061
$499$	33.6459	1.50620	0.753099	−	0.657908i	$-0.228558\pi$
$499$	33.6459	1.50620	0.753099	+	0.657908i	$0.228558\pi$
$500$	−1.00000	−0.0447214
$501$	18.7101	0.835904
$502$	16.1830	0.722284
$503$	7.34998	0.327720	0.163860	−	0.986484i	$-0.447606\pi$
$503$	7.34998	0.327720	0.163860	+	0.986484i	$0.447606\pi$
$504$	−1.00000	−0.0445435
$505$	18.6655	0.830604
$506$	2.69459	0.119789
$507$	6.09327	0.270612
$508$	17.7297	0.786627
$509$	20.8830	0.925623	0.462812	−	0.886457i	$-0.346841\pi$
$509$	20.8830	0.925623	0.462812	+	0.886457i	$0.346841\pi$
$510$	4.45336	0.197198
$511$	−14.2567	−0.630680
$512$	1.00000	0.0441942
$513$	8.58172	0.378892
$514$	27.6459	1.21941
$515$	−11.5175	−0.507523
$516$	0.822948	0.0362283
$517$	24.4243	1.07418
$518$	0.610815	0.0268377
$519$	5.10338	0.224013
$520$	4.36959	0.191619
$521$	−38.6073	−1.69142	−0.845708	−	0.533645i	$-0.820822\pi$
$521$	−38.6073	−1.69142	−0.845708	+	0.533645i	$0.820822\pi$
$522$	8.21213	0.359435
$523$	39.4748	1.72611	0.863057	−	0.505107i	$-0.168547\pi$
$523$	39.4748	1.72611	0.863057	+	0.505107i	$0.168547\pi$
$524$	−4.00000	−0.174741
$525$	−1.00000	−0.0436436
$526$	14.1284	0.616026
$527$	40.3661	1.75837
$528$	−2.69459	−0.117267
$529$	1.00000	0.0434783
$530$	−4.61081	−0.200281
$531$	−8.00000	−0.347170
$532$	−8.58172	−0.372065
$533$	41.5877	1.80136
$534$	−18.0155	−0.779606
$535$	−4.73917	−0.204892
$536$	−14.2121	−0.613871
$537$	18.8675	0.814194
$538$	1.75877	0.0758260
$539$	−2.69459	−0.116064
$540$	−1.00000	−0.0430331
$541$	−21.4593	−0.922609	−0.461305	−	0.887242i	$-0.652619\pi$
$541$	−21.4593	−0.922609	−0.461305	+	0.887242i	$0.652619\pi$
$542$	−0.0992597	−0.00426357
$543$	16.5526	0.710341
$544$	−4.45336	−0.190936
$545$	9.51754	0.407687
$546$	4.36959	0.187001
$547$	−11.2918	−0.482802	−0.241401	−	0.970425i	$-0.577607\pi$
$547$	−11.2918	−0.482802	−0.241401	+	0.970425i	$0.577607\pi$
$548$	14.2567	0.609017
$549$	−5.51754	−0.235483
$550$	−2.69459	−0.114898
$551$	70.4742	3.00230
$552$	−1.00000	−0.0425628
$553$	13.6459	0.580283
$554$	13.1771	0.559839
$555$	0.610815	0.0259276
$556$	14.2959	0.606282
$557$	20.5526	0.870843	0.435421	−	0.900227i	$-0.356599\pi$
$557$	20.5526	0.870843	0.435421	+	0.900227i	$0.356599\pi$
$558$	−9.06418	−0.383717
$559$	−3.59594	−0.152092
$560$	1.00000	0.0422577
$561$	12.0000	0.506640
$562$	−9.43376	−0.397939
$563$	24.7493	1.04306	0.521529	−	0.853234i	$-0.325362\pi$
$563$	24.7493	1.04306	0.521529	+	0.853234i	$0.325362\pi$
$564$	−9.06418	−0.381671
$565$	16.1284	0.678525
$566$	−12.6655	−0.532371
$567$	−1.00000	−0.0419961
$568$	−4.08378	−0.171352
$569$	28.4688	1.19348	0.596738	−	0.802436i	$-0.296463\pi$
$569$	28.4688	1.19348	0.596738	+	0.802436i	$0.296463\pi$
$570$	−8.58172	−0.359449
$571$	33.6459	1.40804	0.704018	−	0.710182i	$-0.251387\pi$
$571$	33.6459	1.40804	0.704018	+	0.710182i	$0.251387\pi$
$572$	11.7743	0.492306
$573$	−7.34998	−0.307050
$574$	9.51754	0.397254
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	36.8675	1.53482	0.767408	−	0.641160i	$-0.221546\pi$
$577$	36.8675	1.53482	0.767408	+	0.641160i	$0.221546\pi$
$578$	2.83244	0.117814
$579$	−4.77837	−0.198582
$580$	−8.21213	−0.340990
$581$	−3.19253	−0.132449
$582$	−6.98040	−0.289347
$583$	−12.4243	−0.514561
$584$	14.2567	0.589947
$585$	4.36959	0.180660
$586$	−18.9067	−0.781030
$587$	−0.424267	−0.0175114	−0.00875569	−	0.999962i	$-0.502787\pi$
$587$	−0.424267	−0.0175114	−0.00875569	+	0.999962i	$0.502787\pi$
$588$	1.00000	0.0412393
$589$	−77.7862	−3.20513
$590$	8.00000	0.329355
$591$	16.1284	0.663432
$592$	−0.610815	−0.0251043
$593$	15.9026	0.653042	0.326521	−	0.945190i	$-0.394124\pi$
$593$	15.9026	0.653042	0.326521	+	0.945190i	$0.394124\pi$
$594$	−2.69459	−0.110560
$595$	−4.45336	−0.182570
$596$	−20.8675	−0.854767
$597$	−2.61081	−0.106854
$598$	4.36959	0.178686
$599$	−14.0446	−0.573846	−0.286923	−	0.957954i	$-0.592632\pi$
$599$	−14.0446	−0.573846	−0.286923	+	0.957954i	$0.592632\pi$
$600$	1.00000	0.0408248
$601$	−6.96492	−0.284105	−0.142052	−	0.989859i	$-0.545370\pi$
$601$	−6.96492	−0.284105	−0.142052	+	0.989859i	$0.545370\pi$
$602$	−0.822948	−0.0335409
$603$	−14.2121	−0.578763
$604$	−19.7743	−0.804603
$605$	3.73917	0.152019
$606$	−18.6655	−0.758234
$607$	−0.497941	−0.0202108	−0.0101054	−	0.999949i	$-0.503217\pi$
$607$	−0.497941	−0.0202108	−0.0101054	+	0.999949i	$0.503217\pi$
$608$	8.58172	0.348035
$609$	−8.21213	−0.332772
$610$	5.51754	0.223399
$611$	39.6067	1.60232
$612$	−4.45336	−0.180017
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0.482459	0.0194705
$615$	9.51754	0.383784
$616$	2.69459	0.108568
$617$	5.57573	0.224471	0.112235	−	0.993682i	$-0.464199\pi$
$617$	5.57573	0.224471	0.112235	+	0.993682i	$0.464199\pi$
$618$	11.5175	0.463303
$619$	−15.2817	−0.614223	−0.307111	−	0.951674i	$-0.599362\pi$
$619$	−15.2817	−0.614223	−0.307111	+	0.951674i	$0.599362\pi$
$620$	9.06418	0.364026
$621$	−1.00000	−0.0401286
$622$	−11.0196	−0.441846
$623$	18.0155	0.721775
$624$	−4.36959	−0.174923
$625$	1.00000	0.0400000
$626$	−2.24123	−0.0895775
$627$	−23.1242	−0.923493
$628$	−1.51754	−0.0605565
$629$	2.72018	0.108461
$630$	1.00000	0.0398410
$631$	−15.1242	−0.602086	−0.301043	−	0.953611i	$-0.597335\pi$
$631$	−15.1242	−0.602086	−0.301043	+	0.953611i	$0.597335\pi$
$632$	−13.6459	−0.542805
$633$	22.8675	0.908903
$634$	−3.96080	−0.157303
$635$	−17.7297	−0.703581
$636$	4.61081	0.182831
$637$	−4.36959	−0.173129
$638$	−22.1284	−0.876070
$639$	−4.08378	−0.161552
$640$	−1.00000	−0.0395285
$641$	−43.7606	−1.72844	−0.864221	−	0.503113i	$-0.832188\pi$
$641$	−43.7606	−1.72844	−0.864221	+	0.503113i	$0.832188\pi$
$642$	4.73917	0.187040
$643$	−21.3155	−0.840602	−0.420301	−	0.907385i	$-0.638075\pi$
$643$	−21.3155	−0.840602	−0.420301	+	0.907385i	$0.638075\pi$
$644$	1.00000	0.0394055
$645$	−0.822948	−0.0324036
$646$	−38.2175	−1.50365
$647$	−43.7060	−1.71826	−0.859129	−	0.511759i	$-0.828994\pi$
$647$	−43.7060	−1.71826	−0.859129	+	0.511759i	$0.828994\pi$
$648$	1.00000	0.0392837
$649$	21.5567	0.846176
$650$	−4.36959	−0.171389
$651$	9.06418	0.355253
$652$	3.51754	0.137758
$653$	33.4593	1.30937	0.654683	−	0.755904i	$-0.272802\pi$
$653$	33.4593	1.30937	0.654683	+	0.755904i	$0.272802\pi$
$654$	−9.51754	−0.372165
$655$	4.00000	0.156293
$656$	−9.51754	−0.371598
$657$	14.2567	0.556207
$658$	9.06418	0.353358
$659$	6.46884	0.251990	0.125995	−	0.992031i	$-0.459788\pi$
$659$	6.46884	0.251990	0.125995	+	0.992031i	$0.459788\pi$
$660$	2.69459	0.104887
$661$	−22.1676	−0.862218	−0.431109	−	0.902300i	$-0.641878\pi$
$661$	−22.1676	−0.862218	−0.431109	+	0.902300i	$0.641878\pi$
$662$	23.6851	0.920547
$663$	19.4593	0.755739
$664$	3.19253	0.123894
$665$	8.58172	0.332785
$666$	−0.610815	−0.0236686
$667$	−8.21213	−0.317975
$668$	18.7101	0.723915
$669$	−9.14796	−0.353680
$670$	14.2121	0.549063
$671$	14.8675	0.573954
$672$	−1.00000	−0.0385758
$673$	−23.3892	−0.901587	−0.450793	−	0.892628i	$-0.648859\pi$
$673$	−23.3892	−0.901587	−0.450793	+	0.892628i	$0.648859\pi$
$674$	24.9513	0.961088
$675$	1.00000	0.0384900
$676$	6.09327	0.234357
$677$	−15.8135	−0.607760	−0.303880	−	0.952710i	$-0.598282\pi$
$677$	−15.8135	−0.607760	−0.303880	+	0.952710i	$0.598282\pi$
$678$	−16.1284	−0.619406
$679$	6.98040	0.267883
$680$	4.45336	0.170779
$681$	−9.54664	−0.365828
$682$	24.4243	0.935254
$683$	−19.5175	−0.746818	−0.373409	−	0.927667i	$-0.621811\pi$
$683$	−19.5175	−0.746818	−0.373409	+	0.927667i	$0.621811\pi$
$684$	8.58172	0.328130
$685$	−14.2567	−0.544721
$686$	−1.00000	−0.0381802
$687$	18.6810	0.712724
$688$	0.822948	0.0313746
$689$	−20.1473	−0.767553
$690$	1.00000	0.0380693
$691$	27.0351	1.02846	0.514231	−	0.857651i	$-0.328077\pi$
$691$	27.0351	1.02846	0.514231	+	0.857651i	$0.328077\pi$
$692$	5.10338	0.194001
$693$	2.69459	0.102359
$694$	7.26083	0.275617
$695$	−14.2959	−0.542275
$696$	8.21213	0.311280
$697$	42.3851	1.60545
$698$	−19.2317	−0.727932
$699$	21.5175	0.813868
$700$	−1.00000	−0.0377964
$701$	−29.0351	−1.09664	−0.548320	−	0.836269i	$-0.684732\pi$
$701$	−29.0351	−1.09664	−0.548320	+	0.836269i	$0.684732\pi$
$702$	−4.36959	−0.164919
$703$	−5.24184	−0.197700
$704$	−2.69459	−0.101556
$705$	9.06418	0.341377
$706$	−2.42427	−0.0912385
$707$	18.6655	0.701988
$708$	−8.00000	−0.300658
$709$	9.42839	0.354090	0.177045	−	0.984203i	$-0.443346\pi$
$709$	9.42839	0.354090	0.177045	+	0.984203i	$0.443346\pi$
$710$	4.08378	0.153261
$711$	−13.6459	−0.511761
$712$	−18.0155	−0.675159
$713$	9.06418	0.339456
$714$	4.45336	0.166663
$715$	−11.7743	−0.440332
$716$	18.8675	0.705113
$717$	−4.08378	−0.152511
$718$	−18.1284	−0.676544
$719$	−21.3155	−0.794934	−0.397467	−	0.917616i	$-0.630111\pi$
$719$	−21.3155	−0.794934	−0.397467	+	0.917616i	$0.630111\pi$
$720$	−1.00000	−0.0372678
$721$	−11.5175	−0.428936
$722$	54.6459	2.03371
$723$	−4.45336	−0.165622
$724$	16.5526	0.615173
$725$	8.21213	0.304991
$726$	−3.73917	−0.138774
$727$	−30.2175	−1.12071	−0.560353	−	0.828254i	$-0.689334\pi$
$727$	−30.2175	−1.12071	−0.560353	+	0.828254i	$0.689334\pi$
$728$	4.36959	0.161948
$729$	1.00000	0.0370370
$730$	−14.2567	−0.527665
$731$	−3.66489	−0.135551
$732$	−5.51754	−0.203934
$733$	−24.1284	−0.891201	−0.445601	−	0.895232i	$-0.647010\pi$
$733$	−24.1284	−0.891201	−0.445601	+	0.895232i	$0.647010\pi$
$734$	−20.9959	−0.774972
$735$	−1.00000	−0.0368856
$736$	−1.00000	−0.0368605
$737$	38.2959	1.41065
$738$	−9.51754	−0.350346
$739$	−10.2175	−0.375857	−0.187929	−	0.982183i	$-0.560177\pi$
$739$	−10.2175	−0.375857	−0.187929	+	0.982183i	$0.560177\pi$
$740$	0.610815	0.0224540
$741$	−37.4986	−1.37754
$742$	−4.61081	−0.169268
$743$	−23.1242	−0.848346	−0.424173	−	0.905581i	$-0.639435\pi$
$743$	−23.1242	−0.848346	−0.424173	+	0.905581i	$0.639435\pi$
$744$	−9.06418	−0.332309
$745$	20.8675	0.764527
$746$	33.2026	1.21563
$747$	3.19253	0.116809
$748$	12.0000	0.438763
$749$	−4.73917	−0.173166
$750$	−1.00000	−0.0365148
$751$	9.81345	0.358098	0.179049	−	0.983840i	$-0.442698\pi$
$751$	9.81345	0.358098	0.179049	+	0.983840i	$0.442698\pi$
$752$	−9.06418	−0.330537
$753$	16.1830	0.589743
$754$	−35.8836	−1.30680
$755$	19.7743	0.719659
$756$	−1.00000	−0.0363696
$757$	4.18655	0.152163	0.0760813	−	0.997102i	$-0.475759\pi$
$757$	4.18655	0.152163	0.0760813	+	0.997102i	$0.475759\pi$
$758$	0.482459	0.0175237
$759$	2.69459	0.0978075
$760$	−8.58172	−0.311292
$761$	44.4944	1.61292	0.806461	−	0.591287i	$-0.201380\pi$
$761$	44.4944	1.61292	0.806461	+	0.591287i	$0.201380\pi$
$762$	17.7297	0.642278
$763$	9.51754	0.344558
$764$	−7.34998	−0.265913
$765$	4.45336	0.161012
$766$	−21.9026	−0.791373
$767$	34.9567	1.26221
$768$	1.00000	0.0360844
$769$	29.5276	1.06479	0.532397	−	0.846495i	$-0.321291\pi$
$769$	29.5276	1.06479	0.532397	+	0.846495i	$0.321291\pi$
$770$	−2.69459	−0.0971064
$771$	27.6459	0.995643
$772$	−4.77837	−0.171977
$773$	−2.25671	−0.0811683	−0.0405841	−	0.999176i	$-0.512922\pi$
$773$	−2.25671	−0.0811683	−0.0405841	+	0.999176i	$0.512922\pi$
$774$	0.822948	0.0295803
$775$	−9.06418	−0.325595
$776$	−6.98040	−0.250582
$777$	0.610815	0.0219129
$778$	−25.0351	−0.897551
$779$	−81.6769	−2.92638
$780$	4.36959	0.156456
$781$	11.0041	0.393758
$782$	4.45336	0.159252
$783$	8.21213	0.293478
$784$	1.00000	0.0357143
$785$	1.51754	0.0541634
$786$	−4.00000	−0.142675
$787$	−11.5021	−0.410004	−0.205002	−	0.978762i	$-0.565720\pi$
$787$	−11.5021	−0.410004	−0.205002	+	0.978762i	$0.565720\pi$
$788$	16.1284	0.574549
$789$	14.1284	0.502983
$790$	13.6459	0.485499
$791$	16.1284	0.573458
$792$	−2.69459	−0.0957482
$793$	24.1094	0.856149
$794$	20.1438	0.714878
$795$	−4.61081	−0.163529
$796$	−2.61081	−0.0925379
$797$	−13.6851	−0.484751	−0.242376	−	0.970183i	$-0.577927\pi$
$797$	−13.6851	−0.484751	−0.242376	+	0.970183i	$0.577927\pi$
$798$	−8.58172	−0.303790
$799$	40.3661	1.42805
$800$	1.00000	0.0353553
$801$	−18.0155	−0.636546
$802$	−24.6364	−0.869942
$803$	−38.4160	−1.35567
$804$	−14.2121	−0.501223
$805$	−1.00000	−0.0352454
$806$	39.6067	1.39509
$807$	1.75877	0.0619117
$808$	−18.6655	−0.656650
$809$	22.5134	0.791530	0.395765	−	0.918352i	$-0.370480\pi$
$809$	22.5134	0.791530	0.395765	+	0.918352i	$0.370480\pi$
$810$	−1.00000	−0.0351364
$811$	−26.3851	−0.926505	−0.463253	−	0.886226i	$-0.653318\pi$
$811$	−26.3851	−0.926505	−0.463253	+	0.886226i	$0.653318\pi$
$812$	−8.21213	−0.288189
$813$	−0.0992597	−0.00348119
$814$	1.64590	0.0576886
$815$	−3.51754	−0.123214
$816$	−4.45336	−0.155899
$817$	7.06231	0.247079
$818$	−9.09327	−0.317939
$819$	4.36959	0.152686
$820$	9.51754	0.332367
$821$	4.95130	0.172802	0.0864008	−	0.996260i	$-0.472463\pi$
$821$	4.95130	0.172802	0.0864008	+	0.996260i	$0.472463\pi$
$822$	14.2567	0.497260
$823$	−9.56212	−0.333314	−0.166657	−	0.986015i	$-0.553297\pi$
$823$	−9.56212	−0.333314	−0.166657	+	0.986015i	$0.553297\pi$
$824$	11.5175	0.401233
$825$	−2.69459	−0.0938137
$826$	8.00000	0.278356
$827$	−40.9959	−1.42557	−0.712783	−	0.701384i	$-0.752566\pi$
$827$	−40.9959	−1.42557	−0.712783	+	0.701384i	$0.752566\pi$
$828$	−1.00000	−0.0347524
$829$	−42.5817	−1.47892	−0.739462	−	0.673198i	$-0.764920\pi$
$829$	−42.5817	−1.47892	−0.739462	+	0.673198i	$0.764920\pi$
$830$	−3.19253	−0.110814
$831$	13.1771	0.457107
$832$	−4.36959	−0.151488
$833$	−4.45336	−0.154300
$834$	14.2959	0.495027
$835$	−18.7101	−0.647489
$836$	−23.1242	−0.799768
$837$	−9.06418	−0.313304
$838$	−18.3114	−0.632557
$839$	9.55674	0.329935	0.164968	−	0.986299i	$-0.447248\pi$
$839$	9.55674	0.329935	0.164968	+	0.986299i	$0.447248\pi$
$840$	1.00000	0.0345033
$841$	38.4391	1.32549
$842$	19.8135	0.682817
$843$	−9.43376	−0.324916
$844$	22.8675	0.787133
$845$	−6.09327	−0.209615
$846$	−9.06418	−0.311633
$847$	3.73917	0.128479
$848$	4.61081	0.158336
$849$	−12.6655	−0.434679
$850$	−4.45336	−0.152749
$851$	0.610815	0.0209385
$852$	−4.08378	−0.139908
$853$	12.1438	0.415797	0.207899	−	0.978150i	$-0.433338\pi$
$853$	12.1438	0.415797	0.207899	+	0.978150i	$0.433338\pi$
$854$	5.51754	0.188806
$855$	−8.58172	−0.293489
$856$	4.73917	0.161982
$857$	13.1242	0.448315	0.224158	−	0.974553i	$-0.428037\pi$
$857$	13.1242	0.448315	0.224158	+	0.974553i	$0.428037\pi$
$858$	11.7743	0.401966
$859$	−48.2567	−1.64650	−0.823249	−	0.567681i	$-0.807841\pi$
$859$	−48.2567	−1.64650	−0.823249	+	0.567681i	$0.807841\pi$
$860$	−0.822948	−0.0280623
$861$	9.51754	0.324357
$862$	−4.00000	−0.136241
$863$	−46.1011	−1.56930	−0.784650	−	0.619939i	$-0.787157\pi$
$863$	−46.1011	−1.56930	−0.784650	+	0.619939i	$0.787157\pi$
$864$	1.00000	0.0340207
$865$	−5.10338	−0.173520
$866$	−27.5722	−0.936942
$867$	2.83244	0.0961948
$868$	9.06418	0.307658
$869$	36.7701	1.24734
$870$	−8.21213	−0.278417
$871$	62.1011	2.10422
$872$	−9.51754	−0.322305
$873$	−6.98040	−0.236251
$874$	−8.58172	−0.290281
$875$	1.00000	0.0338062
$876$	14.2567	0.481690
$877$	51.1106	1.72588	0.862942	−	0.505304i	$-0.168620\pi$
$877$	51.1106	1.72588	0.862942	+	0.505304i	$0.168620\pi$
$878$	3.67499	0.124025
$879$	−18.9067	−0.637708
$880$	2.69459	0.0908347
$881$	27.7398	0.934577	0.467288	−	0.884105i	$-0.345231\pi$
$881$	27.7398	0.934577	0.467288	+	0.884105i	$0.345231\pi$
$882$	1.00000	0.0336718
$883$	−15.2608	−0.513568	−0.256784	−	0.966469i	$-0.582663\pi$
$883$	−15.2608	−0.513568	−0.256784	+	0.966469i	$0.582663\pi$
$884$	19.4593	0.654489
$885$	8.00000	0.268917
$886$	8.73917	0.293598
$887$	−9.37908	−0.314919	−0.157459	−	0.987525i	$-0.550330\pi$
$887$	−9.37908	−0.314919	−0.157459	+	0.987525i	$0.550330\pi$
$888$	−0.610815	−0.0204976
$889$	−17.7297	−0.594634
$890$	18.0155	0.603880
$891$	−2.69459	−0.0902723
$892$	−9.14796	−0.306296
$893$	−77.7862	−2.60302
$894$	−20.8675	−0.697914
$895$	−18.8675	−0.630672
$896$	−1.00000	−0.0334077
$897$	4.36959	0.145896
$898$	27.9026	0.931122
$899$	−74.4362	−2.48259
$900$	1.00000	0.0333333
$901$	−20.5336	−0.684074
$902$	25.6459	0.853915
$903$	−0.822948	−0.0273860
$904$	−16.1284	−0.536421
$905$	−16.5526	−0.550228
$906$	−19.7743	−0.656956
$907$	16.1729	0.537013	0.268507	−	0.963278i	$-0.413470\pi$
$907$	16.1729	0.537013	0.268507	+	0.963278i	$0.413470\pi$
$908$	−9.54664	−0.316816
$909$	−18.6655	−0.619095
$910$	−4.36959	−0.144850
$911$	36.7392	1.21722	0.608612	−	0.793468i	$-0.291727\pi$
$911$	36.7392	1.21722	0.608612	+	0.793468i	$0.291727\pi$
$912$	8.58172	0.284169
$913$	−8.60258	−0.284704
$914$	−33.8580	−1.11992
$915$	5.51754	0.182404
$916$	18.6810	0.617237
$917$	4.00000	0.132092
$918$	−4.45336	−0.146983
$919$	49.5093	1.63316	0.816581	−	0.577231i	$-0.195867\pi$
$919$	49.5093	1.63316	0.816581	+	0.577231i	$0.195867\pi$
$920$	1.00000	0.0329690
$921$	0.482459	0.0158976
$922$	−22.4088	−0.737994
$923$	17.8444	0.587356
$924$	2.69459	0.0886456
$925$	−0.610815	−0.0200835
$926$	−19.8580	−0.652575
$927$	11.5175	0.378286
$928$	8.21213	0.269576
$929$	28.9769	0.950701	0.475350	−	0.879797i	$-0.342321\pi$
$929$	28.9769	0.950701	0.475350	+	0.879797i	$0.342321\pi$
$930$	9.06418	0.297226
$931$	8.58172	0.281254
$932$	21.5175	0.704830
$933$	−11.0196	−0.360766
$934$	−13.0060	−0.425569
$935$	−12.0000	−0.392442
$936$	−4.36959	−0.142824
$937$	41.2181	1.34654	0.673269	−	0.739398i	$-0.264890\pi$
$937$	41.2181	1.34654	0.673269	+	0.739398i	$0.264890\pi$
$938$	14.2121	0.464043
$939$	−2.24123	−0.0731398
$940$	9.06418	0.295641
$941$	22.2567	0.725548	0.362774	−	0.931877i	$-0.381830\pi$
$941$	22.2567	0.725548	0.362774	+	0.931877i	$0.381830\pi$
$942$	−1.51754	−0.0494442
$943$	9.51754	0.309934
$944$	−8.00000	−0.260378
$945$	1.00000	0.0325300
$946$	−2.21751	−0.0720975
$947$	−45.9026	−1.49163	−0.745817	−	0.666151i	$-0.767941\pi$
$947$	−45.9026	−1.49163	−0.745817	+	0.666151i	$0.767941\pi$
$948$	−13.6459	−0.443198
$949$	−62.2959	−2.02221
$950$	8.58172	0.278428
$951$	−3.96080	−0.128438
$952$	4.45336	0.144334
$953$	−4.20676	−0.136270	−0.0681351	−	0.997676i	$-0.521705\pi$
$953$	−4.20676	−0.136270	−0.0681351	+	0.997676i	$0.521705\pi$
$954$	4.61081	0.149281
$955$	7.34998	0.237840
$956$	−4.08378	−0.132079
$957$	−22.1284	−0.715308
$958$	−4.48246	−0.144822
$959$	−14.2567	−0.460373
$960$	−1.00000	−0.0322749
$961$	51.1593	1.65030
$962$	2.66901	0.0860522
$963$	4.73917	0.152718
$964$	−4.45336	−0.143433
$965$	4.77837	0.153821
$966$	1.00000	0.0321745
$967$	22.1337	0.711773	0.355886	−	0.934529i	$-0.384179\pi$
$967$	22.1337	0.711773	0.355886	+	0.934529i	$0.384179\pi$
$968$	−3.73917	−0.120181
$969$	−38.2175	−1.22772
$970$	6.98040	0.224127
$971$	−31.3655	−1.00657	−0.503283	−	0.864122i	$-0.667875\pi$
$971$	−31.3655	−1.00657	−0.503283	+	0.864122i	$0.667875\pi$
$972$	1.00000	0.0320750
$973$	−14.2959	−0.458306
$974$	20.9905	0.672579
$975$	−4.36959	−0.139939
$976$	−5.51754	−0.176612
$977$	−50.9377	−1.62964	−0.814821	−	0.579713i	$-0.803165\pi$
$977$	−50.9377	−1.62964	−0.814821	+	0.579713i	$0.803165\pi$
$978$	3.51754	0.112479
$979$	48.5444	1.55149
$980$	−1.00000	−0.0319438
$981$	−9.51754	−0.303872
$982$	−30.3851	−0.969626
$983$	1.64590	0.0524959	0.0262480	−	0.999655i	$-0.491644\pi$
$983$	1.64590	0.0524959	0.0262480	+	0.999655i	$0.491644\pi$
$984$	−9.51754	−0.303408
$985$	−16.1284	−0.513892
$986$	−36.5716	−1.16468
$987$	9.06418	0.288516
$988$	−37.4986	−1.19299
$989$	−0.822948	−0.0261682
$990$	2.69459	0.0856398
$991$	−17.8444	−0.566847	−0.283423	−	0.958995i	$-0.591470\pi$
$991$	−17.8444	−0.566847	−0.283423	+	0.958995i	$0.591470\pi$
$992$	−9.06418	−0.287788
$993$	23.6851	0.751624
$994$	4.08378	0.129530
$995$	2.61081	0.0827684
$996$	3.19253	0.101159
$997$	36.4789	1.15530	0.577650	−	0.816285i	$-0.303970\pi$
$997$	36.4789	1.15530	0.577650	+	0.816285i	$0.303970\pi$
$998$	33.6459	1.06504
$999$	−0.610815	−0.0193253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.cc.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.cc.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.69459 q^{11} +1.00000 q^{12} -4.36959 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.45336 q^{17} +1.00000 q^{18} +8.58172 q^{19} -1.00000 q^{20} -1.00000 q^{21} -2.69459 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.36959 q^{26} +1.00000 q^{27} -1.00000 q^{28} +8.21213 q^{29} -1.00000 q^{30} -9.06418 q^{31} +1.00000 q^{32} -2.69459 q^{33} -4.45336 q^{34} +1.00000 q^{35} +1.00000 q^{36} -0.610815 q^{37} +8.58172 q^{38} -4.36959 q^{39} -1.00000 q^{40} -9.51754 q^{41} -1.00000 q^{42} +0.822948 q^{43} -2.69459 q^{44} -1.00000 q^{45} -1.00000 q^{46} -9.06418 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.45336 q^{51} -4.36959 q^{52} +4.61081 q^{53} +1.00000 q^{54} +2.69459 q^{55} -1.00000 q^{56} +8.58172 q^{57} +8.21213 q^{58} -8.00000 q^{59} -1.00000 q^{60} -5.51754 q^{61} -9.06418 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.36959 q^{65} -2.69459 q^{66} -14.2121 q^{67} -4.45336 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.08378 q^{71} +1.00000 q^{72} +14.2567 q^{73} -0.610815 q^{74} +1.00000 q^{75} +8.58172 q^{76} +2.69459 q^{77} -4.36959 q^{78} -13.6459 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.51754 q^{82} +3.19253 q^{83} -1.00000 q^{84} +4.45336 q^{85} +0.822948 q^{86} +8.21213 q^{87} -2.69459 q^{88} -18.0155 q^{89} -1.00000 q^{90} +4.36959 q^{91} -1.00000 q^{92} -9.06418 q^{93} -9.06418 q^{94} -8.58172 q^{95} +1.00000 q^{96} -6.98040 q^{97} +1.00000 q^{98} -2.69459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 6 q^{11} + 3 q^{12} - 6 q^{13} - 3 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{21} - 6 q^{22} - 3 q^{23} + 3 q^{24} + 3 q^{25} - 6 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{30} - 18 q^{31} + 3 q^{32} - 6 q^{33} + 3 q^{35} + 3 q^{36} - 6 q^{37} - 6 q^{38} - 6 q^{39} - 3 q^{40} - 6 q^{41} - 3 q^{42} - 18 q^{43} - 6 q^{44} - 3 q^{45} - 3 q^{46} - 18 q^{47} + 3 q^{48} + 3 q^{49} + 3 q^{50} - 6 q^{52} + 18 q^{53} + 3 q^{54} + 6 q^{55} - 3 q^{56} - 6 q^{57} - 24 q^{59} - 3 q^{60} + 6 q^{61} - 18 q^{62} - 3 q^{63} + 3 q^{64} + 6 q^{65} - 6 q^{66} - 18 q^{67} - 3 q^{69} + 3 q^{70} - 6 q^{71} + 3 q^{72} + 6 q^{73} - 6 q^{74} + 3 q^{75} - 6 q^{76} + 6 q^{77} - 6 q^{78} - 3 q^{80} + 3 q^{81} - 6 q^{82} - 18 q^{83} - 3 q^{84} - 18 q^{86} - 6 q^{88} - 6 q^{89} - 3 q^{90} + 6 q^{91} - 3 q^{92} - 18 q^{93} - 18 q^{94} + 6 q^{95} + 3 q^{96} - 18 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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 - 6 * q^11 + 3 * q^12 - 6 * q^13 - 3 * q^14 - 3 * q^15 + 3 * q^16 + 3 * q^18 - 6 * q^19 - 3 * q^20 - 3 * q^21 - 6 * q^22 - 3 * q^23 + 3 * q^24 + 3 * q^25 - 6 * q^26 + 3 * q^27 - 3 * q^28 - 3 * q^30 - 18 * q^31 + 3 * q^32 - 6 * q^33 + 3 * q^35 + 3 * q^36 - 6 * q^37 - 6 * q^38 - 6 * q^39 - 3 * q^40 - 6 * q^41 - 3 * q^42 - 18 * q^43 - 6 * q^44 - 3 * q^45 - 3 * q^46 - 18 * q^47 + 3 * q^48 + 3 * q^49 + 3 * q^50 - 6 * q^52 + 18 * q^53 + 3 * q^54 + 6 * q^55 - 3 * q^56 - 6 * q^57 - 24 * q^59 - 3 * q^60 + 6 * q^61 - 18 * q^62 - 3 * q^63 + 3 * q^64 + 6 * q^65 - 6 * q^66 - 18 * q^67 - 3 * q^69 + 3 * q^70 - 6 * q^71 + 3 * q^72 + 6 * q^73 - 6 * q^74 + 3 * q^75 - 6 * q^76 + 6 * q^77 - 6 * q^78 - 3 * q^80 + 3 * q^81 - 6 * q^82 - 18 * q^83 - 3 * q^84 - 18 * q^86 - 6 * q^88 - 6 * q^89 - 3 * q^90 + 6 * q^91 - 3 * q^92 - 18 * q^93 - 18 * q^94 + 6 * q^95 + 3 * q^96 - 18 * q^97 + 3 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more