LMFDB - Embedded newform 4730.2.a.bb.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 22 x^{9} + 21 x^{8} + 165 x^{7} - 130 x^{6} - 535 x^{5} + 323 x^{4} + 710 x^{3} + \cdots + 128 $ x^11 - x^10 - 22x^9 + 21x^8 + 165x^7 - 130x^6 - 535x^5 + 323x^4 + 710x^3 - 343x^2 - 261*x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-3.34767$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.34767 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34767 q^{6} +2.01478 q^{7} -1.00000 q^{8} +8.20687 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.34767 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34767 q^{6} +2.01478 q^{7} -1.00000 q^{8} +8.20687 q^{9} +1.00000 q^{10} -1.00000 q^{11} +3.34767 q^{12} +1.48948 q^{13} -2.01478 q^{14} -3.34767 q^{15} +1.00000 q^{16} -4.62109 q^{17} -8.20687 q^{18} +4.61164 q^{19} -1.00000 q^{20} +6.74481 q^{21} +1.00000 q^{22} -0.510515 q^{23} -3.34767 q^{24} +1.00000 q^{25} -1.48948 q^{26} +17.4309 q^{27} +2.01478 q^{28} +6.69280 q^{29} +3.34767 q^{30} -3.21861 q^{31} -1.00000 q^{32} -3.34767 q^{33} +4.62109 q^{34} -2.01478 q^{35} +8.20687 q^{36} -2.87600 q^{37} -4.61164 q^{38} +4.98630 q^{39} +1.00000 q^{40} +5.97483 q^{41} -6.74481 q^{42} +1.00000 q^{43} -1.00000 q^{44} -8.20687 q^{45} +0.510515 q^{46} +0.687284 q^{47} +3.34767 q^{48} -2.94066 q^{49} -1.00000 q^{50} -15.4699 q^{51} +1.48948 q^{52} -9.90378 q^{53} -17.4309 q^{54} +1.00000 q^{55} -2.01478 q^{56} +15.4382 q^{57} -6.69280 q^{58} +6.00237 q^{59} -3.34767 q^{60} +10.5041 q^{61} +3.21861 q^{62} +16.5350 q^{63} +1.00000 q^{64} -1.48948 q^{65} +3.34767 q^{66} +9.21417 q^{67} -4.62109 q^{68} -1.70903 q^{69} +2.01478 q^{70} -1.91041 q^{71} -8.20687 q^{72} +13.5257 q^{73} +2.87600 q^{74} +3.34767 q^{75} +4.61164 q^{76} -2.01478 q^{77} -4.98630 q^{78} +4.69668 q^{79} -1.00000 q^{80} +33.7321 q^{81} -5.97483 q^{82} -13.3863 q^{83} +6.74481 q^{84} +4.62109 q^{85} -1.00000 q^{86} +22.4053 q^{87} +1.00000 q^{88} +8.93667 q^{89} +8.20687 q^{90} +3.00098 q^{91} -0.510515 q^{92} -10.7748 q^{93} -0.687284 q^{94} -4.61164 q^{95} -3.34767 q^{96} -18.5182 q^{97} +2.94066 q^{98} -8.20687 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{2} - q^{3} + 11 q^{4} - 11 q^{5} + q^{6} + 6 q^{7} - 11 q^{8} + 12 q^{9}+O(q^{10}) $ 11 * q - 11 * q^2 - q^3 + 11 * q^4 - 11 * q^5 + q^6 + 6 * q^7 - 11 * q^8 + 12 * q^9	\( 11 q - 11 q^{2} - q^{3} + 11 q^{4} - 11 q^{5} + q^{6} + 6 q^{7} - 11 q^{8} + 12 q^{9} + 11 q^{10} - 11 q^{11} - q^{12} + 4 q^{13} - 6 q^{14} + q^{15} + 11 q^{16} - 10 q^{17} - 12 q^{18} + 14 q^{19} - 11 q^{20} - 2 q^{21} + 11 q^{22} - 18 q^{23} + q^{24} + 11 q^{25} - 4 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29} - q^{30} + 13 q^{31} - 11 q^{32} + q^{33} + 10 q^{34} - 6 q^{35} + 12 q^{36} + q^{37} - 14 q^{38} + 12 q^{39} + 11 q^{40} + 12 q^{41} + 2 q^{42} + 11 q^{43} - 11 q^{44} - 12 q^{45} + 18 q^{46} - 5 q^{47} - q^{48} + 31 q^{49} - 11 q^{50} + q^{51} + 4 q^{52} - 27 q^{53} - 2 q^{54} + 11 q^{55} - 6 q^{56} - 5 q^{57} + 6 q^{58} + 11 q^{59} + q^{60} + 36 q^{61} - 13 q^{62} + 17 q^{63} + 11 q^{64} - 4 q^{65} - q^{66} + 18 q^{67} - 10 q^{68} + 14 q^{69} + 6 q^{70} - 14 q^{71} - 12 q^{72} - 11 q^{73} - q^{74} - q^{75} + 14 q^{76} - 6 q^{77} - 12 q^{78} + 28 q^{79} - 11 q^{80} + 7 q^{81} - 12 q^{82} - 4 q^{83} - 2 q^{84} + 10 q^{85} - 11 q^{86} + 38 q^{87} + 11 q^{88} - 7 q^{89} + 12 q^{90} + 14 q^{91} - 18 q^{92} - 3 q^{93} + 5 q^{94} - 14 q^{95} + q^{96} - q^{97} - 31 q^{98} - 12 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - q^3 + 11 * q^4 - 11 * q^5 + q^6 + 6 * q^7 - 11 * q^8 + 12 * q^9 + 11 * q^10 - 11 * q^11 - q^12 + 4 * q^13 - 6 * q^14 + q^15 + 11 * q^16 - 10 * q^17 - 12 * q^18 + 14 * q^19 - 11 * q^20 - 2 * q^21 + 11 * q^22 - 18 * q^23 + q^24 + 11 * q^25 - 4 * q^26 + 2 * q^27 + 6 * q^28 - 6 * q^29 - q^30 + 13 * q^31 - 11 * q^32 + q^33 + 10 * q^34 - 6 * q^35 + 12 * q^36 + q^37 - 14 * q^38 + 12 * q^39 + 11 * q^40 + 12 * q^41 + 2 * q^42 + 11 * q^43 - 11 * q^44 - 12 * q^45 + 18 * q^46 - 5 * q^47 - q^48 + 31 * q^49 - 11 * q^50 + q^51 + 4 * q^52 - 27 * q^53 - 2 * q^54 + 11 * q^55 - 6 * q^56 - 5 * q^57 + 6 * q^58 + 11 * q^59 + q^60 + 36 * q^61 - 13 * q^62 + 17 * q^63 + 11 * q^64 - 4 * q^65 - q^66 + 18 * q^67 - 10 * q^68 + 14 * q^69 + 6 * q^70 - 14 * q^71 - 12 * q^72 - 11 * q^73 - q^74 - q^75 + 14 * q^76 - 6 * q^77 - 12 * q^78 + 28 * q^79 - 11 * q^80 + 7 * q^81 - 12 * q^82 - 4 * q^83 - 2 * q^84 + 10 * q^85 - 11 * q^86 + 38 * q^87 + 11 * q^88 - 7 * q^89 + 12 * q^90 + 14 * q^91 - 18 * q^92 - 3 * q^93 + 5 * q^94 - 14 * q^95 + q^96 - q^97 - 31 * q^98 - 12 * 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
$3$	3.34767	1.93278	0.966388	−	0.257087i	$-0.0827628\pi$
$3$	3.34767	1.93278	0.966388	+	0.257087i	$0.0827628\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.34767	−1.36668
$7$	2.01478	0.761515	0.380758	−	0.924675i	$-0.375663\pi$
$7$	2.01478	0.761515	0.380758	+	0.924675i	$0.375663\pi$
$8$	−1.00000	−0.353553
$9$	8.20687	2.73562
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.34767	0.966388
$13$	1.48948	0.413109	0.206554	−	0.978435i	$-0.433775\pi$
$13$	1.48948	0.413109	0.206554	+	0.978435i	$0.433775\pi$
$14$	−2.01478	−0.538472
$15$	−3.34767	−0.864364
$16$	1.00000	0.250000
$17$	−4.62109	−1.12078	−0.560390	−	0.828229i	$-0.689349\pi$
$17$	−4.62109	−1.12078	−0.560390	+	0.828229i	$0.689349\pi$
$18$	−8.20687	−1.93438
$19$	4.61164	1.05798	0.528991	−	0.848627i	$-0.322571\pi$
$19$	4.61164	1.05798	0.528991	+	0.848627i	$0.322571\pi$
$20$	−1.00000	−0.223607
$21$	6.74481	1.47184
$22$	1.00000	0.213201
$23$	−0.510515	−0.106450	−0.0532249	−	0.998583i	$-0.516950\pi$
$23$	−0.510515	−0.106450	−0.0532249	+	0.998583i	$0.516950\pi$
$24$	−3.34767	−0.683340
$25$	1.00000	0.200000
$26$	−1.48948	−0.292112
$27$	17.4309	3.35457
$28$	2.01478	0.380758
$29$	6.69280	1.24282	0.621411	−	0.783485i	$-0.286560\pi$
$29$	6.69280	1.24282	0.621411	+	0.783485i	$0.286560\pi$
$30$	3.34767	0.611198
$31$	−3.21861	−0.578080	−0.289040	−	0.957317i	$-0.593336\pi$
$31$	−3.21861	−0.578080	−0.289040	+	0.957317i	$0.593336\pi$
$32$	−1.00000	−0.176777
$33$	−3.34767	−0.582754
$34$	4.62109	0.792511
$35$	−2.01478	−0.340560
$36$	8.20687	1.36781
$37$	−2.87600	−0.472812	−0.236406	−	0.971654i	$-0.575969\pi$
$37$	−2.87600	−0.472812	−0.236406	+	0.971654i	$0.575969\pi$
$38$	−4.61164	−0.748106
$39$	4.98630	0.798447
$40$	1.00000	0.158114
$41$	5.97483	0.933112	0.466556	−	0.884492i	$-0.345495\pi$
$41$	5.97483	0.933112	0.466556	+	0.884492i	$0.345495\pi$
$42$	−6.74481	−1.04075
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.00000	−0.150756
$45$	−8.20687	−1.22341
$46$	0.510515	0.0752714
$47$	0.687284	0.100251	0.0501253	−	0.998743i	$-0.484038\pi$
$47$	0.687284	0.100251	0.0501253	+	0.998743i	$0.484038\pi$
$48$	3.34767	0.483194
$49$	−2.94066	−0.420095
$50$	−1.00000	−0.141421
$51$	−15.4699	−2.16622
$52$	1.48948	0.206554
$53$	−9.90378	−1.36039	−0.680194	−	0.733032i	$-0.738105\pi$
$53$	−9.90378	−1.36039	−0.680194	+	0.733032i	$0.738105\pi$
$54$	−17.4309	−2.37204
$55$	1.00000	0.134840
$56$	−2.01478	−0.269236
$57$	15.4382	2.04484
$58$	−6.69280	−0.878808
$59$	6.00237	0.781442	0.390721	−	0.920509i	$-0.372226\pi$
$59$	6.00237	0.781442	0.390721	+	0.920509i	$0.372226\pi$
$60$	−3.34767	−0.432182
$61$	10.5041	1.34491	0.672457	−	0.740136i	$-0.265239\pi$
$61$	10.5041	1.34491	0.672457	+	0.740136i	$0.265239\pi$
$62$	3.21861	0.408764
$63$	16.5350	2.08322
$64$	1.00000	0.125000
$65$	−1.48948	−0.184748
$66$	3.34767	0.412069
$67$	9.21417	1.12569	0.562845	−	0.826562i	$-0.309707\pi$
$67$	9.21417	1.12569	0.562845	+	0.826562i	$0.309707\pi$
$68$	−4.62109	−0.560390
$69$	−1.70903	−0.205744
$70$	2.01478	0.240812
$71$	−1.91041	−0.226724	−0.113362	−	0.993554i	$-0.536162\pi$
$71$	−1.91041	−0.226724	−0.113362	+	0.993554i	$0.536162\pi$
$72$	−8.20687	−0.967189
$73$	13.5257	1.58307	0.791534	−	0.611125i	$-0.209283\pi$
$73$	13.5257	1.58307	0.791534	+	0.611125i	$0.209283\pi$
$74$	2.87600	0.334328
$75$	3.34767	0.386555
$76$	4.61164	0.528991
$77$	−2.01478	−0.229605
$78$	−4.98630	−0.564587
$79$	4.69668	0.528418	0.264209	−	0.964465i	$-0.414889\pi$
$79$	4.69668	0.528418	0.264209	+	0.964465i	$0.414889\pi$
$80$	−1.00000	−0.111803
$81$	33.7321	3.74802
$82$	−5.97483	−0.659810
$83$	−13.3863	−1.46934	−0.734669	−	0.678426i	$-0.762662\pi$
$83$	−13.3863	−1.46934	−0.734669	+	0.678426i	$0.762662\pi$
$84$	6.74481	0.735919
$85$	4.62109	0.501228
$86$	−1.00000	−0.107833
$87$	22.4053	2.40210
$88$	1.00000	0.106600
$89$	8.93667	0.947285	0.473642	−	0.880717i	$-0.342939\pi$
$89$	8.93667	0.947285	0.473642	+	0.880717i	$0.342939\pi$
$90$	8.20687	0.865080
$91$	3.00098	0.314589
$92$	−0.510515	−0.0532249
$93$	−10.7748	−1.11730
$94$	−0.687284	−0.0708879
$95$	−4.61164	−0.473144
$96$	−3.34767	−0.341670
$97$	−18.5182	−1.88024	−0.940121	−	0.340841i	$-0.889288\pi$
$97$	−18.5182	−1.88024	−0.940121	+	0.340841i	$0.889288\pi$
$98$	2.94066	0.297052
$99$	−8.20687	−0.824822
$100$	1.00000	0.100000
$101$	−1.14767	−0.114198	−0.0570988	−	0.998369i	$-0.518185\pi$
$101$	−1.14767	−0.114198	−0.0570988	+	0.998369i	$0.518185\pi$
$102$	15.4699	1.53175
$103$	−13.7309	−1.35294	−0.676472	−	0.736468i	$-0.736492\pi$
$103$	−13.7309	−1.35294	−0.676472	+	0.736468i	$0.736492\pi$
$104$	−1.48948	−0.146056
$105$	−6.74481	−0.658226
$106$	9.90378	0.961940
$107$	14.5724	1.40877	0.704384	−	0.709819i	$-0.251223\pi$
$107$	14.5724	1.40877	0.704384	+	0.709819i	$0.251223\pi$
$108$	17.4309	1.67729
$109$	8.75054	0.838149	0.419075	−	0.907952i	$-0.362355\pi$
$109$	8.75054	0.838149	0.419075	+	0.907952i	$0.362355\pi$
$110$	−1.00000	−0.0953463
$111$	−9.62790	−0.913839
$112$	2.01478	0.190379
$113$	−6.03707	−0.567920	−0.283960	−	0.958836i	$-0.591648\pi$
$113$	−6.03707	−0.567920	−0.283960	+	0.958836i	$0.591648\pi$
$114$	−15.4382	−1.44592
$115$	0.510515	0.0476058
$116$	6.69280	0.621411
$117$	12.2240	1.13011
$118$	−6.00237	−0.552563
$119$	−9.31048	−0.853490
$120$	3.34767	0.305599
$121$	1.00000	0.0909091
$122$	−10.5041	−0.950998
$123$	20.0017	1.80350
$124$	−3.21861	−0.289040
$125$	−1.00000	−0.0894427
$126$	−16.5350	−1.47306
$127$	−3.20196	−0.284128	−0.142064	−	0.989858i	$-0.545374\pi$
$127$	−3.20196	−0.284128	−0.142064	+	0.989858i	$0.545374\pi$
$128$	−1.00000	−0.0883883
$129$	3.34767	0.294746
$130$	1.48948	0.130636
$131$	−13.6496	−1.19258	−0.596288	−	0.802771i	$-0.703358\pi$
$131$	−13.6496	−1.19258	−0.596288	+	0.802771i	$0.703358\pi$
$132$	−3.34767	−0.291377
$133$	9.29143	0.805669
$134$	−9.21417	−0.795983
$135$	−17.4309	−1.50021
$136$	4.62109	0.396255
$137$	1.47015	0.125604	0.0628018	−	0.998026i	$-0.479996\pi$
$137$	1.47015	0.125604	0.0628018	+	0.998026i	$0.479996\pi$
$138$	1.70903	0.145483
$139$	2.94406	0.249712	0.124856	−	0.992175i	$-0.460153\pi$
$139$	2.94406	0.249712	0.124856	+	0.992175i	$0.460153\pi$
$140$	−2.01478	−0.170280
$141$	2.30080	0.193762
$142$	1.91041	0.160318
$143$	−1.48948	−0.124557
$144$	8.20687	0.683906
$145$	−6.69280	−0.555807
$146$	−13.5257	−1.11940
$147$	−9.84436	−0.811949
$148$	−2.87600	−0.236406
$149$	−17.8364	−1.46122	−0.730608	−	0.682797i	$-0.760763\pi$
$149$	−17.8364	−1.46122	−0.730608	+	0.682797i	$0.760763\pi$
$150$	−3.34767	−0.273336
$151$	5.52170	0.449349	0.224675	−	0.974434i	$-0.427868\pi$
$151$	5.52170	0.449349	0.224675	+	0.974434i	$0.427868\pi$
$152$	−4.61164	−0.374053
$153$	−37.9247	−3.06603
$154$	2.01478	0.162356
$155$	3.21861	0.258525
$156$	4.98630	0.399223
$157$	8.76485	0.699511	0.349756	−	0.936841i	$-0.386265\pi$
$157$	8.76485	0.699511	0.349756	+	0.936841i	$0.386265\pi$
$158$	−4.69668	−0.373648
$159$	−33.1546	−2.62933
$160$	1.00000	0.0790569
$161$	−1.02858	−0.0810631
$162$	−33.7321	−2.65025
$163$	9.07445	0.710765	0.355383	−	0.934721i	$-0.384351\pi$
$163$	9.07445	0.710765	0.355383	+	0.934721i	$0.384351\pi$
$164$	5.97483	0.466556
$165$	3.34767	0.260616
$166$	13.3863	1.03898
$167$	25.2448	1.95350	0.976750	−	0.214382i	$-0.0687736\pi$
$167$	25.2448	1.95350	0.976750	+	0.214382i	$0.0687736\pi$
$168$	−6.74481	−0.520373
$169$	−10.7814	−0.829341
$170$	−4.62109	−0.354421
$171$	37.8471	2.89424
$172$	1.00000	0.0762493
$173$	−11.2676	−0.856663	−0.428332	−	0.903622i	$-0.640898\pi$
$173$	−11.2676	−0.856663	−0.428332	+	0.903622i	$0.640898\pi$
$174$	−22.4053	−1.69854
$175$	2.01478	0.152303
$176$	−1.00000	−0.0753778
$177$	20.0939	1.51035
$178$	−8.93667	−0.669831
$179$	−5.95539	−0.445127	−0.222564	−	0.974918i	$-0.571442\pi$
$179$	−5.95539	−0.445127	−0.222564	+	0.974918i	$0.571442\pi$
$180$	−8.20687	−0.611704
$181$	−2.55950	−0.190246	−0.0951229	−	0.995466i	$-0.530324\pi$
$181$	−2.55950	−0.190246	−0.0951229	+	0.995466i	$0.530324\pi$
$182$	−3.00098	−0.222448
$183$	35.1643	2.59942
$184$	0.510515	0.0376357
$185$	2.87600	0.211448
$186$	10.7748	0.790050
$187$	4.62109	0.337928
$188$	0.687284	0.0501253
$189$	35.1194	2.55456
$190$	4.61164	0.334563
$191$	−2.93960	−0.212702	−0.106351	−	0.994329i	$-0.533917\pi$
$191$	−2.93960	−0.212702	−0.106351	+	0.994329i	$0.533917\pi$
$192$	3.34767	0.241597
$193$	14.0810	1.01357	0.506786	−	0.862072i	$-0.330833\pi$
$193$	14.0810	1.01357	0.506786	+	0.862072i	$0.330833\pi$
$194$	18.5182	1.32953
$195$	−4.98630	−0.357076
$196$	−2.94066	−0.210047
$197$	−8.24038	−0.587103	−0.293551	−	0.955943i	$-0.594837\pi$
$197$	−8.24038	−0.587103	−0.293551	+	0.955943i	$0.594837\pi$
$198$	8.20687	0.583237
$199$	15.1524	1.07412	0.537061	−	0.843543i	$-0.319534\pi$
$199$	15.1524	1.07412	0.537061	+	0.843543i	$0.319534\pi$
$200$	−1.00000	−0.0707107
$201$	30.8460	2.17571
$202$	1.14767	0.0807499
$203$	13.4845	0.946428
$204$	−15.4699	−1.08311
$205$	−5.97483	−0.417300
$206$	13.7309	0.956677
$207$	−4.18973	−0.291207
$208$	1.48948	0.103277
$209$	−4.61164	−0.318994
$210$	6.74481	0.465436
$211$	13.1232	0.903437	0.451718	−	0.892161i	$-0.350811\pi$
$211$	13.1232	0.903437	0.451718	+	0.892161i	$0.350811\pi$
$212$	−9.90378	−0.680194
$213$	−6.39542	−0.438207
$214$	−14.5724	−0.996150
$215$	−1.00000	−0.0681994
$216$	−17.4309	−1.18602
$217$	−6.48479	−0.440217
$218$	−8.75054	−0.592661
$219$	45.2797	3.05972
$220$	1.00000	0.0674200
$221$	−6.88304	−0.463004
$222$	9.62790	0.646182
$223$	0.783875	0.0524922	0.0262461	−	0.999656i	$-0.491645\pi$
$223$	0.783875	0.0524922	0.0262461	+	0.999656i	$0.491645\pi$
$224$	−2.01478	−0.134618
$225$	8.20687	0.547125
$226$	6.03707	0.401580
$227$	−19.4564	−1.29137	−0.645685	−	0.763604i	$-0.723428\pi$
$227$	−19.4564	−1.29137	−0.645685	+	0.763604i	$0.723428\pi$
$228$	15.4382	1.02242
$229$	−11.2079	−0.740636	−0.370318	−	0.928905i	$-0.620751\pi$
$229$	−11.2079	−0.740636	−0.370318	+	0.928905i	$0.620751\pi$
$230$	−0.510515	−0.0336624
$231$	−6.74481	−0.443776
$232$	−6.69280	−0.439404
$233$	−2.07385	−0.135862	−0.0679312	−	0.997690i	$-0.521640\pi$
$233$	−2.07385	−0.135862	−0.0679312	+	0.997690i	$0.521640\pi$
$234$	−12.2240	−0.799109
$235$	−0.687284	−0.0448335
$236$	6.00237	0.390721
$237$	15.7229	1.02131
$238$	9.31048	0.603509
$239$	−12.0463	−0.779213	−0.389607	−	0.920981i	$-0.627389\pi$
$239$	−12.0463	−0.779213	−0.389607	+	0.920981i	$0.627389\pi$
$240$	−3.34767	−0.216091
$241$	7.59111	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$241$	7.59111	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$242$	−1.00000	−0.0642824
$243$	60.6313	3.88950
$244$	10.5041	0.672457
$245$	2.94066	0.187872
$246$	−20.0017	−1.27526
$247$	6.86896	0.437062
$248$	3.21861	0.204382
$249$	−44.8129	−2.83990
$250$	1.00000	0.0632456
$251$	−14.3646	−0.906685	−0.453342	−	0.891336i	$-0.649769\pi$
$251$	−14.3646	−0.906685	−0.453342	+	0.891336i	$0.649769\pi$
$252$	16.5350	1.04161
$253$	0.510515	0.0320958
$254$	3.20196	0.200909
$255$	15.4699	0.968761
$256$	1.00000	0.0625000
$257$	16.9270	1.05588	0.527939	−	0.849282i	$-0.322965\pi$
$257$	16.9270	1.05588	0.527939	+	0.849282i	$0.322965\pi$
$258$	−3.34767	−0.208417
$259$	−5.79451	−0.360053
$260$	−1.48948	−0.0923739
$261$	54.9270	3.39989
$262$	13.6496	0.843278
$263$	6.51799	0.401917	0.200958	−	0.979600i	$-0.435594\pi$
$263$	6.51799	0.401917	0.200958	+	0.979600i	$0.435594\pi$
$264$	3.34767	0.206035
$265$	9.90378	0.608384
$266$	−9.29143	−0.569694
$267$	29.9170	1.83089
$268$	9.21417	0.562845
$269$	−24.4660	−1.49172	−0.745860	−	0.666103i	$-0.767961\pi$
$269$	−24.4660	−1.49172	−0.745860	+	0.666103i	$0.767961\pi$
$270$	17.4309	1.06081
$271$	−22.0693	−1.34061	−0.670307	−	0.742084i	$-0.733837\pi$
$271$	−22.0693	−1.34061	−0.670307	+	0.742084i	$0.733837\pi$
$272$	−4.62109	−0.280195
$273$	10.0463	0.608029
$274$	−1.47015	−0.0888152
$275$	−1.00000	−0.0603023
$276$	−1.70903	−0.102872
$277$	23.2674	1.39800	0.699002	−	0.715120i	$-0.253628\pi$
$277$	23.2674	1.39800	0.699002	+	0.715120i	$0.253628\pi$
$278$	−2.94406	−0.176573
$279$	−26.4147	−1.58141
$280$	2.01478	0.120406
$281$	14.5212	0.866264	0.433132	−	0.901330i	$-0.357408\pi$
$281$	14.5212	0.866264	0.433132	+	0.901330i	$0.357408\pi$
$282$	−2.30080	−0.137011
$283$	−32.3625	−1.92375	−0.961874	−	0.273492i	$-0.911821\pi$
$283$	−32.3625	−1.92375	−0.961874	+	0.273492i	$0.911821\pi$
$284$	−1.91041	−0.113362
$285$	−15.4382	−0.914481
$286$	1.48948	0.0880751
$287$	12.0380	0.710579
$288$	−8.20687	−0.483595
$289$	4.35448	0.256146
$290$	6.69280	0.393015
$291$	−61.9929	−3.63409
$292$	13.5257	0.791534
$293$	0.869375	0.0507894	0.0253947	−	0.999678i	$-0.491916\pi$
$293$	0.869375	0.0507894	0.0253947	+	0.999678i	$0.491916\pi$
$294$	9.84436	0.574135
$295$	−6.00237	−0.349472
$296$	2.87600	0.167164
$297$	−17.4309	−1.01144
$298$	17.8364	1.03324
$299$	−0.760405	−0.0439753
$300$	3.34767	0.193278
$301$	2.01478	0.116130
$302$	−5.52170	−0.317738
$303$	−3.84202	−0.220719
$304$	4.61164	0.264496
$305$	−10.5041	−0.601464
$306$	37.9247	2.16801
$307$	12.5435	0.715896	0.357948	−	0.933742i	$-0.383477\pi$
$307$	12.5435	0.715896	0.357948	+	0.933742i	$0.383477\pi$
$308$	−2.01478	−0.114803
$309$	−45.9664	−2.61494
$310$	−3.21861	−0.182805
$311$	10.8906	0.617547	0.308773	−	0.951136i	$-0.400082\pi$
$311$	10.8906	0.617547	0.308773	+	0.951136i	$0.400082\pi$
$312$	−4.98630	−0.282294
$313$	−4.86148	−0.274787	−0.137394	−	0.990517i	$-0.543872\pi$
$313$	−4.86148	−0.274787	−0.137394	+	0.990517i	$0.543872\pi$
$314$	−8.76485	−0.494629
$315$	−16.5350	−0.931644
$316$	4.69668	0.264209
$317$	−15.3998	−0.864941	−0.432470	−	0.901648i	$-0.642358\pi$
$317$	−15.3998	−0.864941	−0.432470	+	0.901648i	$0.642358\pi$
$318$	33.1546	1.85922
$319$	−6.69280	−0.374725
$320$	−1.00000	−0.0559017
$321$	48.7836	2.72284
$322$	1.02858	0.0573203
$323$	−21.3108	−1.18576
$324$	33.7321	1.87401
$325$	1.48948	0.0826218
$326$	−9.07445	−0.502587
$327$	29.2939	1.61996
$328$	−5.97483	−0.329905
$329$	1.38473	0.0763424
$330$	−3.34767	−0.184283
$331$	25.3548	1.39363	0.696813	−	0.717253i	$-0.254601\pi$
$331$	25.3548	1.39363	0.696813	+	0.717253i	$0.254601\pi$
$332$	−13.3863	−0.734669
$333$	−23.6030	−1.29344
$334$	−25.2448	−1.38133
$335$	−9.21417	−0.503424
$336$	6.74481	0.367960
$337$	−18.6303	−1.01486	−0.507429	−	0.861694i	$-0.669404\pi$
$337$	−18.6303	−1.01486	−0.507429	+	0.861694i	$0.669404\pi$
$338$	10.7814	0.586433
$339$	−20.2101	−1.09766
$340$	4.62109	0.250614
$341$	3.21861	0.174298
$342$	−37.8471	−2.04654
$343$	−20.0282	−1.08142
$344$	−1.00000	−0.0539164
$345$	1.70903	0.0920113
$346$	11.2676	0.605752
$347$	−21.0407	−1.12953	−0.564763	−	0.825253i	$-0.691032\pi$
$347$	−21.0407	−1.12953	−0.564763	+	0.825253i	$0.691032\pi$
$348$	22.4053	1.20105
$349$	22.2309	1.18999	0.594997	−	0.803728i	$-0.297153\pi$
$349$	22.2309	1.18999	0.594997	+	0.803728i	$0.297153\pi$
$350$	−2.01478	−0.107694
$351$	25.9630	1.38580
$352$	1.00000	0.0533002
$353$	31.5180	1.67753	0.838767	−	0.544491i	$-0.183277\pi$
$353$	31.5180	1.67753	0.838767	+	0.544491i	$0.183277\pi$
$354$	−20.0939	−1.06798
$355$	1.91041	0.101394
$356$	8.93667	0.473642
$357$	−31.1684	−1.64961
$358$	5.95539	0.314752
$359$	−17.5193	−0.924636	−0.462318	−	0.886714i	$-0.652982\pi$
$359$	−17.5193	−0.924636	−0.462318	+	0.886714i	$0.652982\pi$
$360$	8.20687	0.432540
$361$	2.26720	0.119326
$362$	2.55950	0.134524
$363$	3.34767	0.175707
$364$	3.00098	0.157294
$365$	−13.5257	−0.707970
$366$	−35.1643	−1.83807
$367$	−34.2260	−1.78658	−0.893292	−	0.449477i	$-0.851610\pi$
$367$	−34.2260	−1.78658	−0.893292	+	0.449477i	$0.851610\pi$
$368$	−0.510515	−0.0266124
$369$	49.0347	2.55264
$370$	−2.87600	−0.149516
$371$	−19.9539	−1.03596
$372$	−10.7748	−0.558650
$373$	0.0440564	0.00228115	0.00114058	−	0.999999i	$-0.499637\pi$
$373$	0.0440564	0.00228115	0.00114058	+	0.999999i	$0.499637\pi$
$374$	−4.62109	−0.238951
$375$	−3.34767	−0.172873
$376$	−0.687284	−0.0354440
$377$	9.96883	0.513421
$378$	−35.1194	−1.80635
$379$	27.8952	1.43288	0.716441	−	0.697648i	$-0.245770\pi$
$379$	27.8952	1.43288	0.716441	+	0.697648i	$0.245770\pi$
$380$	−4.61164	−0.236572
$381$	−10.7191	−0.549155
$382$	2.93960	0.150403
$383$	31.4666	1.60787	0.803933	−	0.594719i	$-0.202737\pi$
$383$	31.4666	1.60787	0.803933	+	0.594719i	$0.202737\pi$
$384$	−3.34767	−0.170835
$385$	2.01478	0.102683
$386$	−14.0810	−0.716704
$387$	8.20687	0.417179
$388$	−18.5182	−0.940121
$389$	−21.3938	−1.08471	−0.542355	−	0.840150i	$-0.682467\pi$
$389$	−21.3938	−1.08471	−0.542355	+	0.840150i	$0.682467\pi$
$390$	4.98630	0.252491
$391$	2.35914	0.119307
$392$	2.94066	0.148526
$393$	−45.6945	−2.30498
$394$	8.24038	0.415144
$395$	−4.69668	−0.236316
$396$	−8.20687	−0.412411
$397$	−25.2152	−1.26551	−0.632757	−	0.774350i	$-0.718077\pi$
$397$	−25.2152	−1.26551	−0.632757	+	0.774350i	$0.718077\pi$
$398$	−15.1524	−0.759520
$399$	31.1046	1.55718
$400$	1.00000	0.0500000
$401$	18.5312	0.925404	0.462702	−	0.886514i	$-0.346880\pi$
$401$	18.5312	0.925404	0.462702	+	0.886514i	$0.346880\pi$
$402$	−30.8460	−1.53846
$403$	−4.79407	−0.238810
$404$	−1.14767	−0.0570988
$405$	−33.7321	−1.67616
$406$	−13.4845	−0.669225
$407$	2.87600	0.142558
$408$	15.4699	0.765873
$409$	3.18044	0.157263	0.0786314	−	0.996904i	$-0.474945\pi$
$409$	3.18044	0.157263	0.0786314	+	0.996904i	$0.474945\pi$
$410$	5.97483	0.295076
$411$	4.92158	0.242764
$412$	−13.7309	−0.676472
$413$	12.0935	0.595080
$414$	4.18973	0.205914
$415$	13.3863	0.657108
$416$	−1.48948	−0.0730280
$417$	9.85575	0.482638
$418$	4.61164	0.225563
$419$	−13.4896	−0.659010	−0.329505	−	0.944154i	$-0.606882\pi$
$419$	−13.4896	−0.659010	−0.329505	+	0.944154i	$0.606882\pi$
$420$	−6.74481	−0.329113
$421$	−24.0053	−1.16995	−0.584973	−	0.811053i	$-0.698895\pi$
$421$	−24.0053	−1.16995	−0.584973	+	0.811053i	$0.698895\pi$
$422$	−13.1232	−0.638826
$423$	5.64045	0.274248
$424$	9.90378	0.480970
$425$	−4.62109	−0.224156
$426$	6.39542	0.309859
$427$	21.1635	1.02417
$428$	14.5724	0.704384
$429$	−4.98630	−0.240741
$430$	1.00000	0.0482243
$431$	−19.5818	−0.943224	−0.471612	−	0.881806i	$-0.656328\pi$
$431$	−19.5818	−0.943224	−0.471612	+	0.881806i	$0.656328\pi$
$432$	17.4309	0.838643
$433$	1.78736	0.0858952	0.0429476	−	0.999077i	$-0.486325\pi$
$433$	1.78736	0.0858952	0.0429476	+	0.999077i	$0.486325\pi$
$434$	6.48479	0.311280
$435$	−22.4053	−1.07425
$436$	8.75054	0.419075
$437$	−2.35431	−0.112622
$438$	−45.2797	−2.16355
$439$	17.1291	0.817529	0.408764	−	0.912640i	$-0.365960\pi$
$439$	17.1291	0.817529	0.408764	+	0.912640i	$0.365960\pi$
$440$	−1.00000	−0.0476731
$441$	−24.1337	−1.14922
$442$	6.88304	0.327393
$443$	−26.0378	−1.23709	−0.618547	−	0.785748i	$-0.712278\pi$
$443$	−26.0378	−1.23709	−0.618547	+	0.785748i	$0.712278\pi$
$444$	−9.62790	−0.456920
$445$	−8.93667	−0.423639
$446$	−0.783875	−0.0371176
$447$	−59.7103	−2.82420
$448$	2.01478	0.0951894
$449$	3.31664	0.156522	0.0782610	−	0.996933i	$-0.475063\pi$
$449$	3.31664	0.156522	0.0782610	+	0.996933i	$0.475063\pi$
$450$	−8.20687	−0.386876
$451$	−5.97483	−0.281344
$452$	−6.03707	−0.283960
$453$	18.4848	0.868492
$454$	19.4564	0.913136
$455$	−3.00098	−0.140688
$456$	−15.4382	−0.722961
$457$	25.1803	1.17789	0.588943	−	0.808174i	$-0.299544\pi$
$457$	25.1803	1.17789	0.588943	+	0.808174i	$0.299544\pi$
$458$	11.2079	0.523709
$459$	−80.5497	−3.75974
$460$	0.510515	0.0238029
$461$	−4.59424	−0.213975	−0.106988	−	0.994260i	$-0.534121\pi$
$461$	−4.59424	−0.213975	−0.106988	+	0.994260i	$0.534121\pi$
$462$	6.74481	0.313797
$463$	7.26300	0.337540	0.168770	−	0.985655i	$-0.446020\pi$
$463$	7.26300	0.337540	0.168770	+	0.985655i	$0.446020\pi$
$464$	6.69280	0.310706
$465$	10.7748	0.499671
$466$	2.07385	0.0960692
$467$	−24.4668	−1.13219	−0.566094	−	0.824340i	$-0.691546\pi$
$467$	−24.4668	−1.13219	−0.566094	+	0.824340i	$0.691546\pi$
$468$	12.2240	0.565055
$469$	18.5645	0.857230
$470$	0.687284	0.0317021
$471$	29.3418	1.35200
$472$	−6.00237	−0.276282
$473$	−1.00000	−0.0459800
$474$	−15.7229	−0.722177
$475$	4.61164	0.211596
$476$	−9.31048	−0.426745
$477$	−81.2791	−3.72151
$478$	12.0463	0.550987
$479$	−7.23813	−0.330718	−0.165359	−	0.986233i	$-0.552878\pi$
$479$	−7.23813	−0.330718	−0.165359	+	0.986233i	$0.552878\pi$
$480$	3.34767	0.152799
$481$	−4.28376	−0.195323
$482$	−7.59111	−0.345766
$483$	−3.44333	−0.156677
$484$	1.00000	0.0454545
$485$	18.5182	0.840870
$486$	−60.6313	−2.75029
$487$	−21.9913	−0.996520	−0.498260	−	0.867028i	$-0.666028\pi$
$487$	−21.9913	−0.996520	−0.498260	+	0.867028i	$0.666028\pi$
$488$	−10.5041	−0.475499
$489$	30.3782	1.37375
$490$	−2.94066	−0.132846
$491$	−10.1902	−0.459876	−0.229938	−	0.973205i	$-0.573852\pi$
$491$	−10.1902	−0.459876	−0.229938	+	0.973205i	$0.573852\pi$
$492$	20.0017	0.901748
$493$	−30.9280	−1.39293
$494$	−6.86896	−0.309049
$495$	8.20687	0.368871
$496$	−3.21861	−0.144520
$497$	−3.84906	−0.172654
$498$	44.8129	2.00811
$499$	−7.45725	−0.333832	−0.166916	−	0.985971i	$-0.553381\pi$
$499$	−7.45725	−0.333832	−0.166916	+	0.985971i	$0.553381\pi$
$500$	−1.00000	−0.0447214
$501$	84.5111	3.77568
$502$	14.3646	0.641123
$503$	3.98676	0.177761	0.0888804	−	0.996042i	$-0.471671\pi$
$503$	3.98676	0.177761	0.0888804	+	0.996042i	$0.471671\pi$
$504$	−16.5350	−0.736529
$505$	1.14767	0.0510707
$506$	−0.510515	−0.0226952
$507$	−36.0927	−1.60293
$508$	−3.20196	−0.142064
$509$	6.04174	0.267795	0.133898	−	0.990995i	$-0.457251\pi$
$509$	6.04174	0.267795	0.133898	+	0.990995i	$0.457251\pi$
$510$	−15.4699	−0.685017
$511$	27.2514	1.20553
$512$	−1.00000	−0.0441942
$513$	80.3849	3.54908
$514$	−16.9270	−0.746619
$515$	13.7309	0.605055
$516$	3.34767	0.147373
$517$	−0.687284	−0.0302267
$518$	5.79451	0.254596
$519$	−37.7203	−1.65574
$520$	1.48948	0.0653182
$521$	6.66591	0.292039	0.146019	−	0.989282i	$-0.453354\pi$
$521$	6.66591	0.292039	0.146019	+	0.989282i	$0.453354\pi$
$522$	−54.9270	−2.40409
$523$	−17.3931	−0.760547	−0.380273	−	0.924874i	$-0.624170\pi$
$523$	−17.3931	−0.760547	−0.380273	+	0.924874i	$0.624170\pi$
$524$	−13.6496	−0.596288
$525$	6.74481	0.294368
$526$	−6.51799	−0.284198
$527$	14.8735	0.647900
$528$	−3.34767	−0.145688
$529$	−22.7394	−0.988668
$530$	−9.90378	−0.430193
$531$	49.2607	2.13773
$532$	9.29143	0.402835
$533$	8.89942	0.385477
$534$	−29.9170	−1.29463
$535$	−14.5724	−0.630021
$536$	−9.21417	−0.397992
$537$	−19.9367	−0.860331
$538$	24.4660	1.05480
$539$	2.94066	0.126663
$540$	−17.4309	−0.750105
$541$	−37.9162	−1.63014	−0.815071	−	0.579360i	$-0.803302\pi$
$541$	−37.9162	−1.63014	−0.815071	+	0.579360i	$0.803302\pi$
$542$	22.0693	0.947957
$543$	−8.56834	−0.367703
$544$	4.62109	0.198128
$545$	−8.75054	−0.374832
$546$	−10.0463	−0.429942
$547$	−14.9070	−0.637379	−0.318689	−	0.947859i	$-0.603243\pi$
$547$	−14.9070	−0.637379	−0.318689	+	0.947859i	$0.603243\pi$
$548$	1.47015	0.0628018
$549$	86.2060	3.67918
$550$	1.00000	0.0426401
$551$	30.8648	1.31488
$552$	1.70903	0.0727413
$553$	9.46277	0.402398
$554$	−23.2674	−0.988538
$555$	9.62790	0.408681
$556$	2.94406	0.124856
$557$	19.1986	0.813471	0.406735	−	0.913546i	$-0.366667\pi$
$557$	19.1986	0.813471	0.406735	+	0.913546i	$0.366667\pi$
$558$	26.4147	1.11823
$559$	1.48948	0.0629985
$560$	−2.01478	−0.0851400
$561$	15.4699	0.653139
$562$	−14.5212	−0.612541
$563$	34.0338	1.43435	0.717177	−	0.696891i	$-0.245434\pi$
$563$	34.0338	1.43435	0.717177	+	0.696891i	$0.245434\pi$
$564$	2.30080	0.0968811
$565$	6.03707	0.253981
$566$	32.3625	1.36030
$567$	67.9628	2.85417
$568$	1.91041	0.0801591
$569$	20.0492	0.840506	0.420253	−	0.907407i	$-0.361941\pi$
$569$	20.0492	0.840506	0.420253	+	0.907407i	$0.361941\pi$
$570$	15.4382	0.646636
$571$	−0.789587	−0.0330432	−0.0165216	−	0.999864i	$-0.505259\pi$
$571$	−0.789587	−0.0330432	−0.0165216	+	0.999864i	$0.505259\pi$
$572$	−1.48948	−0.0622785
$573$	−9.84082	−0.411106
$574$	−12.0380	−0.502455
$575$	−0.510515	−0.0212900
$576$	8.20687	0.341953
$577$	−10.7861	−0.449032	−0.224516	−	0.974470i	$-0.572080\pi$
$577$	−10.7861	−0.449032	−0.224516	+	0.974470i	$0.572080\pi$
$578$	−4.35448	−0.181123
$579$	47.1385	1.95901
$580$	−6.69280	−0.277903
$581$	−26.9705	−1.11892
$582$	61.9929	2.56969
$583$	9.90378	0.410173
$584$	−13.5257	−0.559699
$585$	−12.2240	−0.505401
$586$	−0.869375	−0.0359135
$587$	11.5310	0.475935	0.237968	−	0.971273i	$-0.423519\pi$
$587$	11.5310	0.475935	0.237968	+	0.971273i	$0.423519\pi$
$588$	−9.84436	−0.405975
$589$	−14.8431	−0.611598
$590$	6.00237	0.247114
$591$	−27.5860	−1.13474
$592$	−2.87600	−0.118203
$593$	−16.6202	−0.682509	−0.341255	−	0.939971i	$-0.610852\pi$
$593$	−16.6202	−0.682509	−0.341255	+	0.939971i	$0.610852\pi$
$594$	17.4309	0.715197
$595$	9.31048	0.381692
$596$	−17.8364	−0.730608
$597$	50.7251	2.07604
$598$	0.760405	0.0310953
$599$	35.5350	1.45192	0.725960	−	0.687737i	$-0.241396\pi$
$599$	35.5350	1.45192	0.725960	+	0.687737i	$0.241396\pi$
$600$	−3.34767	−0.136668
$601$	28.4318	1.15976	0.579878	−	0.814704i	$-0.303100\pi$
$601$	28.4318	1.15976	0.579878	+	0.814704i	$0.303100\pi$
$602$	−2.01478	−0.0821163
$603$	75.6195	3.07947
$604$	5.52170	0.224675
$605$	−1.00000	−0.0406558
$606$	3.84202	0.156072
$607$	−17.1793	−0.697285	−0.348643	−	0.937256i	$-0.613357\pi$
$607$	−17.1793	−0.697285	−0.348643	+	0.937256i	$0.613357\pi$
$608$	−4.61164	−0.187027
$609$	45.1417	1.82923
$610$	10.5041	0.425299
$611$	1.02370	0.0414144
$612$	−37.9247	−1.53302
$613$	−5.87621	−0.237338	−0.118669	−	0.992934i	$-0.537863\pi$
$613$	−5.87621	−0.237338	−0.118669	+	0.992934i	$0.537863\pi$
$614$	−12.5435	−0.506215
$615$	−20.0017	−0.806548
$616$	2.01478	0.0811778
$617$	2.74950	0.110691	0.0553454	−	0.998467i	$-0.482374\pi$
$617$	2.74950	0.110691	0.0553454	+	0.998467i	$0.482374\pi$
$618$	45.9664	1.84904
$619$	−35.5230	−1.42779	−0.713895	−	0.700253i	$-0.753070\pi$
$619$	−35.5230	−1.42779	−0.713895	+	0.700253i	$0.753070\pi$
$620$	3.21861	0.129263
$621$	−8.89873	−0.357094
$622$	−10.8906	−0.436672
$623$	18.0054	0.721372
$624$	4.98630	0.199612
$625$	1.00000	0.0400000
$626$	4.86148	0.194304
$627$	−15.4382	−0.616543
$628$	8.76485	0.349756
$629$	13.2903	0.529918
$630$	16.5350	0.658772
$631$	−35.9310	−1.43039	−0.715196	−	0.698924i	$-0.753663\pi$
$631$	−35.9310	−1.43039	−0.715196	+	0.698924i	$0.753663\pi$
$632$	−4.69668	−0.186824
$633$	43.9320	1.74614
$634$	15.3998	0.611605
$635$	3.20196	0.127066
$636$	−33.1546	−1.31466
$637$	−4.38007	−0.173545
$638$	6.69280	0.264971
$639$	−15.6785	−0.620232
$640$	1.00000	0.0395285
$641$	7.43849	0.293803	0.146901	−	0.989151i	$-0.453070\pi$
$641$	7.43849	0.293803	0.146901	+	0.989151i	$0.453070\pi$
$642$	−48.7836	−1.92534
$643$	−32.2934	−1.27353	−0.636764	−	0.771059i	$-0.719727\pi$
$643$	−32.2934	−1.27353	−0.636764	+	0.771059i	$0.719727\pi$
$644$	−1.02858	−0.0405315
$645$	−3.34767	−0.131814
$646$	21.3108	0.838462
$647$	−14.1027	−0.554436	−0.277218	−	0.960807i	$-0.589412\pi$
$647$	−14.1027	−0.554436	−0.277218	+	0.960807i	$0.589412\pi$
$648$	−33.7321	−1.32512
$649$	−6.00237	−0.235614
$650$	−1.48948	−0.0584224
$651$	−21.7089	−0.850840
$652$	9.07445	0.355383
$653$	34.5633	1.35257	0.676283	−	0.736642i	$-0.263590\pi$
$653$	34.5633	1.35257	0.676283	+	0.736642i	$0.263590\pi$
$654$	−29.2939	−1.14548
$655$	13.6496	0.533336
$656$	5.97483	0.233278
$657$	111.004	4.33068
$658$	−1.38473	−0.0539822
$659$	−47.7324	−1.85939	−0.929695	−	0.368331i	$-0.879929\pi$
$659$	−47.7324	−1.85939	−0.929695	+	0.368331i	$0.879929\pi$
$660$	3.34767	0.130308
$661$	−40.0085	−1.55615	−0.778075	−	0.628171i	$-0.783804\pi$
$661$	−40.0085	−1.55615	−0.778075	+	0.628171i	$0.783804\pi$
$662$	−25.3548	−0.985442
$663$	−23.0421	−0.894883
$664$	13.3863	0.519489
$665$	−9.29143	−0.360306
$666$	23.6030	0.914597
$667$	−3.41678	−0.132298
$668$	25.2448	0.976750
$669$	2.62415	0.101456
$670$	9.21417	0.355975
$671$	−10.5041	−0.405507
$672$	−6.74481	−0.260187
$673$	34.2935	1.32192	0.660959	−	0.750422i	$-0.270150\pi$
$673$	34.2935	1.32192	0.660959	+	0.750422i	$0.270150\pi$
$674$	18.6303	0.717613
$675$	17.4309	0.670915
$676$	−10.7814	−0.414671
$677$	−47.6651	−1.83192	−0.915958	−	0.401273i	$-0.868568\pi$
$677$	−47.6651	−1.83192	−0.915958	+	0.401273i	$0.868568\pi$
$678$	20.2101	0.776164
$679$	−37.3102	−1.43183
$680$	−4.62109	−0.177211
$681$	−65.1337	−2.49593
$682$	−3.21861	−0.123247
$683$	47.1483	1.80408	0.902040	−	0.431653i	$-0.142069\pi$
$683$	47.1483	1.80408	0.902040	+	0.431653i	$0.142069\pi$
$684$	37.8471	1.44712
$685$	−1.47015	−0.0561717
$686$	20.0282	0.764682
$687$	−37.5202	−1.43148
$688$	1.00000	0.0381246
$689$	−14.7515	−0.561989
$690$	−1.70903	−0.0650618
$691$	48.6227	1.84969	0.924847	−	0.380339i	$-0.124193\pi$
$691$	48.6227	1.84969	0.924847	+	0.380339i	$0.124193\pi$
$692$	−11.2676	−0.428332
$693$	−16.5350	−0.628114
$694$	21.0407	0.798696
$695$	−2.94406	−0.111675
$696$	−22.4053	−0.849270
$697$	−27.6102	−1.04581
$698$	−22.2309	−0.841453
$699$	−6.94255	−0.262592
$700$	2.01478	0.0761515
$701$	6.33047	0.239099	0.119549	−	0.992828i	$-0.461855\pi$
$701$	6.33047	0.239099	0.119549	+	0.992828i	$0.461855\pi$
$702$	−25.9630	−0.979911
$703$	−13.2631	−0.500226
$704$	−1.00000	−0.0376889
$705$	−2.30080	−0.0866531
$706$	−31.5180	−1.18619
$707$	−2.31231	−0.0869632
$708$	20.0939	0.755177
$709$	31.5740	1.18578	0.592892	−	0.805282i	$-0.297986\pi$
$709$	31.5740	1.18578	0.592892	+	0.805282i	$0.297986\pi$
$710$	−1.91041	−0.0716965
$711$	38.5450	1.44555
$712$	−8.93667	−0.334916
$713$	1.64315	0.0615365
$714$	31.1684	1.16645
$715$	1.48948	0.0557036
$716$	−5.95539	−0.222564
$717$	−40.3272	−1.50605
$718$	17.5193	0.653816
$719$	−21.1222	−0.787725	−0.393863	−	0.919169i	$-0.628861\pi$
$719$	−21.1222	−0.787725	−0.393863	+	0.919169i	$0.628861\pi$
$720$	−8.20687	−0.305852
$721$	−27.6647	−1.03029
$722$	−2.26720	−0.0843763
$723$	25.4125	0.945101
$724$	−2.55950	−0.0951229
$725$	6.69280	0.248564
$726$	−3.34767	−0.124244
$727$	10.2646	0.380693	0.190347	−	0.981717i	$-0.439039\pi$
$727$	10.2646	0.380693	0.190347	+	0.981717i	$0.439039\pi$
$728$	−3.00098	−0.111224
$729$	101.777	3.76952
$730$	13.5257	0.500610
$731$	−4.62109	−0.170917
$732$	35.1643	1.29971
$733$	20.0782	0.741604	0.370802	−	0.928712i	$-0.379083\pi$
$733$	20.0782	0.741604	0.370802	+	0.928712i	$0.379083\pi$
$734$	34.2260	1.26331
$735$	9.84436	0.363115
$736$	0.510515	0.0188178
$737$	−9.21417	−0.339408
$738$	−49.0347	−1.80499
$739$	−47.6831	−1.75405	−0.877026	−	0.480442i	$-0.840476\pi$
$739$	−47.6831	−1.75405	−0.877026	+	0.480442i	$0.840476\pi$
$740$	2.87600	0.105724
$741$	22.9950	0.844742
$742$	19.9539	0.732532
$743$	19.6593	0.721228	0.360614	−	0.932715i	$-0.382567\pi$
$743$	19.6593	0.721228	0.360614	+	0.932715i	$0.382567\pi$
$744$	10.7748	0.395025
$745$	17.8364	0.653475
$746$	−0.0440564	−0.00161302
$747$	−109.860	−4.01956
$748$	4.62109	0.168964
$749$	29.3602	1.07280
$750$	3.34767	0.122240
$751$	−5.69776	−0.207914	−0.103957	−	0.994582i	$-0.533150\pi$
$751$	−5.69776	−0.207914	−0.103957	+	0.994582i	$0.533150\pi$
$752$	0.687284	0.0250627
$753$	−48.0879	−1.75242
$754$	−9.96883	−0.363043
$755$	−5.52170	−0.200955
$756$	35.1194	1.27728
$757$	12.1647	0.442133	0.221067	−	0.975259i	$-0.429046\pi$
$757$	12.1647	0.442133	0.221067	+	0.975259i	$0.429046\pi$
$758$	−27.8952	−1.01320
$759$	1.70903	0.0620340
$760$	4.61164	0.167282
$761$	27.3010	0.989660	0.494830	−	0.868990i	$-0.335230\pi$
$761$	27.3010	0.989660	0.494830	+	0.868990i	$0.335230\pi$
$762$	10.7191	0.388311
$763$	17.6304	0.638263
$764$	−2.93960	−0.106351
$765$	37.9247	1.37117
$766$	−31.4666	−1.13693
$767$	8.94044	0.322821
$768$	3.34767	0.120799
$769$	46.3280	1.67063	0.835315	−	0.549772i	$-0.185285\pi$
$769$	46.3280	1.67063	0.835315	+	0.549772i	$0.185285\pi$
$770$	−2.01478	−0.0726076
$771$	56.6660	2.04078
$772$	14.0810	0.506786
$773$	47.4161	1.70544	0.852720	−	0.522368i	$-0.174951\pi$
$773$	47.4161	1.70544	0.852720	+	0.522368i	$0.174951\pi$
$774$	−8.20687	−0.294990
$775$	−3.21861	−0.115616
$776$	18.5182	0.664766
$777$	−19.3981	−0.695902
$778$	21.3938	0.767005
$779$	27.5537	0.987215
$780$	−4.98630	−0.178538
$781$	1.91041	0.0683599
$782$	−2.35914	−0.0843626
$783$	116.661	4.16914
$784$	−2.94066	−0.105024
$785$	−8.76485	−0.312831
$786$	45.6945	1.62987
$787$	27.1334	0.967203	0.483602	−	0.875288i	$-0.339328\pi$
$787$	27.1334	0.967203	0.483602	+	0.875288i	$0.339328\pi$
$788$	−8.24038	−0.293551
$789$	21.8201	0.776815
$790$	4.69668	0.167100
$791$	−12.1634	−0.432479
$792$	8.20687	0.291619
$793$	15.6457	0.555596
$794$	25.2152	0.894854
$795$	33.1546	1.17587
$796$	15.1524	0.537061
$797$	−46.5107	−1.64749	−0.823747	−	0.566958i	$-0.808120\pi$
$797$	−46.5107	−1.64749	−0.823747	+	0.566958i	$0.808120\pi$
$798$	−31.1046	−1.10109
$799$	−3.17600	−0.112359
$800$	−1.00000	−0.0353553
$801$	73.3421	2.59141
$802$	−18.5312	−0.654359
$803$	−13.5257	−0.477313
$804$	30.8460	1.08785
$805$	1.02858	0.0362525
$806$	4.79407	0.168864
$807$	−81.9041	−2.88316
$808$	1.14767	0.0403750
$809$	40.1382	1.41118	0.705592	−	0.708619i	$-0.250681\pi$
$809$	40.1382	1.41118	0.705592	+	0.708619i	$0.250681\pi$
$810$	33.7321	1.18523
$811$	30.8189	1.08220	0.541099	−	0.840959i	$-0.318008\pi$
$811$	30.8189	1.08220	0.541099	+	0.840959i	$0.318008\pi$
$812$	13.4845	0.473214
$813$	−73.8806	−2.59111
$814$	−2.87600	−0.100804
$815$	−9.07445	−0.317864
$816$	−15.4699	−0.541554
$817$	4.61164	0.161341
$818$	−3.18044	−0.111202
$819$	24.6287	0.860596
$820$	−5.97483	−0.208650
$821$	5.55019	0.193703	0.0968515	−	0.995299i	$-0.469123\pi$
$821$	5.55019	0.193703	0.0968515	+	0.995299i	$0.469123\pi$
$822$	−4.92158	−0.171660
$823$	−3.10128	−0.108104	−0.0540518	−	0.998538i	$-0.517214\pi$
$823$	−3.10128	−0.108104	−0.0540518	+	0.998538i	$0.517214\pi$
$824$	13.7309	0.478338
$825$	−3.34767	−0.116551
$826$	−12.0935	−0.420785
$827$	−45.8463	−1.59423	−0.797116	−	0.603827i	$-0.793642\pi$
$827$	−45.8463	−1.59423	−0.797116	+	0.603827i	$0.793642\pi$
$828$	−4.18973	−0.145603
$829$	29.8724	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$829$	29.8724	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$830$	−13.3863	−0.464645
$831$	77.8915	2.70203
$832$	1.48948	0.0516386
$833$	13.5891	0.470834
$834$	−9.85575	−0.341277
$835$	−25.2448	−0.873632
$836$	−4.61164	−0.159497
$837$	−56.1032	−1.93921
$838$	13.4896	0.465991
$839$	−6.98368	−0.241103	−0.120552	−	0.992707i	$-0.538466\pi$
$839$	−6.98368	−0.241103	−0.120552	+	0.992707i	$0.538466\pi$
$840$	6.74481	0.232718
$841$	15.7936	0.544607
$842$	24.0053	0.827276
$843$	48.6123	1.67430
$844$	13.1232	0.451718
$845$	10.7814	0.370893
$846$	−5.64045	−0.193923
$847$	2.01478	0.0692286
$848$	−9.90378	−0.340097
$849$	−108.339	−3.71818
$850$	4.62109	0.158502
$851$	1.46824	0.0503307
$852$	−6.39542	−0.219104
$853$	−32.6637	−1.11838	−0.559192	−	0.829038i	$-0.688889\pi$
$853$	−32.6637	−1.11838	−0.559192	+	0.829038i	$0.688889\pi$
$854$	−21.1635	−0.724200
$855$	−37.8471	−1.29434
$856$	−14.5724	−0.498075
$857$	−18.2452	−0.623244	−0.311622	−	0.950206i	$-0.600872\pi$
$857$	−18.2452	−0.623244	−0.311622	+	0.950206i	$0.600872\pi$
$858$	4.98630	0.170229
$859$	0.415306	0.0141701	0.00708503	−	0.999975i	$-0.497745\pi$
$859$	0.415306	0.0141701	0.00708503	+	0.999975i	$0.497745\pi$
$860$	−1.00000	−0.0340997
$861$	40.2991	1.37339
$862$	19.5818	0.666960
$863$	−52.6700	−1.79291	−0.896454	−	0.443137i	$-0.853866\pi$
$863$	−52.6700	−1.79291	−0.896454	+	0.443137i	$0.853866\pi$
$864$	−17.4309	−0.593010
$865$	11.2676	0.383111
$866$	−1.78736	−0.0607371
$867$	14.5774	0.495073
$868$	−6.48479	−0.220108
$869$	−4.69668	−0.159324
$870$	22.4053	0.759610
$871$	13.7244	0.465033
$872$	−8.75054	−0.296331
$873$	−151.977	−5.14363
$874$	2.35431	0.0796357
$875$	−2.01478	−0.0681120
$876$	45.2797	1.52986
$877$	−14.7608	−0.498438	−0.249219	−	0.968447i	$-0.580174\pi$
$877$	−14.7608	−0.498438	−0.249219	+	0.968447i	$0.580174\pi$
$878$	−17.1291	−0.578080
$879$	2.91038	0.0981646
$880$	1.00000	0.0337100
$881$	18.5642	0.625443	0.312722	−	0.949845i	$-0.398759\pi$
$881$	18.5642	0.625443	0.312722	+	0.949845i	$0.398759\pi$
$882$	24.1337	0.812622
$883$	−20.3161	−0.683692	−0.341846	−	0.939756i	$-0.611052\pi$
$883$	−20.3161	−0.683692	−0.341846	+	0.939756i	$0.611052\pi$
$884$	−6.88304	−0.231502
$885$	−20.0939	−0.675451
$886$	26.0378	0.874757
$887$	−20.9300	−0.702760	−0.351380	−	0.936233i	$-0.614288\pi$
$887$	−20.9300	−0.702760	−0.351380	+	0.936233i	$0.614288\pi$
$888$	9.62790	0.323091
$889$	−6.45123	−0.216367
$890$	8.93667	0.299558
$891$	−33.7321	−1.13007
$892$	0.783875	0.0262461
$893$	3.16950	0.106063
$894$	59.7103	1.99701
$895$	5.95539	0.199067
$896$	−2.01478	−0.0673091
$897$	−2.54558	−0.0849945
$898$	−3.31664	−0.110678
$899$	−21.5415	−0.718450
$900$	8.20687	0.273562
$901$	45.7663	1.52470
$902$	5.97483	0.198940
$903$	6.74481	0.224453
$904$	6.03707	0.200790
$905$	2.55950	0.0850805
$906$	−18.4848	−0.614116
$907$	−15.7810	−0.524000	−0.262000	−	0.965068i	$-0.584382\pi$
$907$	−15.7810	−0.524000	−0.262000	+	0.965068i	$0.584382\pi$
$908$	−19.4564	−0.645685
$909$	−9.41880	−0.312402
$910$	3.00098	0.0994816
$911$	5.97895	0.198092	0.0990458	−	0.995083i	$-0.468421\pi$
$911$	5.97895	0.198092	0.0990458	+	0.995083i	$0.468421\pi$
$912$	15.4382	0.511211
$913$	13.3863	0.443022
$914$	−25.1803	−0.832892
$915$	−35.1643	−1.16250
$916$	−11.2079	−0.370318
$917$	−27.5010	−0.908164
$918$	80.5497	2.65853
$919$	−1.80493	−0.0595392	−0.0297696	−	0.999557i	$-0.509477\pi$
$919$	−1.80493	−0.0595392	−0.0297696	+	0.999557i	$0.509477\pi$
$920$	−0.510515	−0.0168312
$921$	41.9915	1.38367
$922$	4.59424	0.151303
$923$	−2.84553	−0.0936618
$924$	−6.74481	−0.221888
$925$	−2.87600	−0.0945624
$926$	−7.26300	−0.238677
$927$	−112.688	−3.70115
$928$	−6.69280	−0.219702
$929$	−53.2285	−1.74637	−0.873184	−	0.487390i	$-0.837949\pi$
$929$	−53.2285	−1.74637	−0.873184	+	0.487390i	$0.837949\pi$
$930$	−10.7748	−0.353321
$931$	−13.5613	−0.444453
$932$	−2.07385	−0.0679312
$933$	36.4580	1.19358
$934$	24.4668	0.800578
$935$	−4.62109	−0.151126
$936$	−12.2240	−0.399554
$937$	29.5350	0.964865	0.482433	−	0.875933i	$-0.339753\pi$
$937$	29.5350	0.964865	0.482433	+	0.875933i	$0.339753\pi$
$938$	−18.5645	−0.606153
$939$	−16.2746	−0.531102
$940$	−0.687284	−0.0224167
$941$	14.6068	0.476168	0.238084	−	0.971245i	$-0.423481\pi$
$941$	14.6068	0.476168	0.238084	+	0.971245i	$0.423481\pi$
$942$	−29.3418	−0.956007
$943$	−3.05024	−0.0993295
$944$	6.00237	0.195361
$945$	−35.1194	−1.14243
$946$	1.00000	0.0325128
$947$	−38.7219	−1.25829	−0.629146	−	0.777287i	$-0.716595\pi$
$947$	−38.7219	−1.25829	−0.629146	+	0.777287i	$0.716595\pi$
$948$	15.7229	0.510657
$949$	20.1464	0.653980
$950$	−4.61164	−0.149621
$951$	−51.5535	−1.67174
$952$	9.31048	0.301754
$953$	−42.3834	−1.37293	−0.686466	−	0.727162i	$-0.740839\pi$
$953$	−42.3834	−1.37293	−0.686466	+	0.727162i	$0.740839\pi$
$954$	81.2791	2.63151
$955$	2.93960	0.0951234
$956$	−12.0463	−0.389607
$957$	−22.4053	−0.724260
$958$	7.23813	0.233853
$959$	2.96203	0.0956491
$960$	−3.34767	−0.108045
$961$	−20.6405	−0.665824
$962$	4.28376	0.138114
$963$	119.594	3.85386
$964$	7.59111	0.244493
$965$	−14.0810	−0.453283
$966$	3.44333	0.110787
$967$	13.1072	0.421500	0.210750	−	0.977540i	$-0.432409\pi$
$967$	13.1072	0.421500	0.210750	+	0.977540i	$0.432409\pi$
$968$	−1.00000	−0.0321412
$969$	−71.3414	−2.29182
$970$	−18.5182	−0.594585
$971$	43.1723	1.38546	0.692732	−	0.721195i	$-0.256407\pi$
$971$	43.1723	1.38546	0.692732	+	0.721195i	$0.256407\pi$
$972$	60.6313	1.94475
$973$	5.93164	0.190160
$974$	21.9913	0.704646
$975$	4.98630	0.159689
$976$	10.5041	0.336229
$977$	−0.939950	−0.0300717	−0.0150358	−	0.999887i	$-0.504786\pi$
$977$	−0.939950	−0.0300717	−0.0150358	+	0.999887i	$0.504786\pi$
$978$	−30.3782	−0.971388
$979$	−8.93667	−0.285617
$980$	2.94066	0.0939361
$981$	71.8145	2.29286
$982$	10.1902	0.325182
$983$	−34.7224	−1.10747	−0.553736	−	0.832692i	$-0.686798\pi$
$983$	−34.7224	−1.10747	−0.553736	+	0.832692i	$0.686798\pi$
$984$	−20.0017	−0.637632
$985$	8.24038	0.262560
$986$	30.9280	0.984950
$987$	4.63560	0.147553
$988$	6.86896	0.218531
$989$	−0.510515	−0.0162334
$990$	−8.20687	−0.260832
$991$	−24.8675	−0.789943	−0.394971	−	0.918693i	$-0.629245\pi$
$991$	−24.8675	−0.789943	−0.394971	+	0.918693i	$0.629245\pi$
$992$	3.21861	0.102191
$993$	84.8794	2.69357
$994$	3.84906	0.122085
$995$	−15.1524	−0.480362
$996$	−44.8129	−1.41995
$997$	7.44447	0.235769	0.117884	−	0.993027i	$-0.462389\pi$
$997$	7.44447	0.235769	0.117884	+	0.993027i	$0.462389\pi$
$998$	7.45725	0.236055
$999$	−50.1312	−1.58608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.bb.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.bb.1.11	✓	11	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:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.34767 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34767 q^{6} +2.01478 q^{7} -1.00000 q^{8} +8.20687 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.34767 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.34767 q^{6} +2.01478 q^{7} -1.00000 q^{8} +8.20687 q^{9} +1.00000 q^{10} -1.00000 q^{11} +3.34767 q^{12} +1.48948 q^{13} -2.01478 q^{14} -3.34767 q^{15} +1.00000 q^{16} -4.62109 q^{17} -8.20687 q^{18} +4.61164 q^{19} -1.00000 q^{20} +6.74481 q^{21} +1.00000 q^{22} -0.510515 q^{23} -3.34767 q^{24} +1.00000 q^{25} -1.48948 q^{26} +17.4309 q^{27} +2.01478 q^{28} +6.69280 q^{29} +3.34767 q^{30} -3.21861 q^{31} -1.00000 q^{32} -3.34767 q^{33} +4.62109 q^{34} -2.01478 q^{35} +8.20687 q^{36} -2.87600 q^{37} -4.61164 q^{38} +4.98630 q^{39} +1.00000 q^{40} +5.97483 q^{41} -6.74481 q^{42} +1.00000 q^{43} -1.00000 q^{44} -8.20687 q^{45} +0.510515 q^{46} +0.687284 q^{47} +3.34767 q^{48} -2.94066 q^{49} -1.00000 q^{50} -15.4699 q^{51} +1.48948 q^{52} -9.90378 q^{53} -17.4309 q^{54} +1.00000 q^{55} -2.01478 q^{56} +15.4382 q^{57} -6.69280 q^{58} +6.00237 q^{59} -3.34767 q^{60} +10.5041 q^{61} +3.21861 q^{62} +16.5350 q^{63} +1.00000 q^{64} -1.48948 q^{65} +3.34767 q^{66} +9.21417 q^{67} -4.62109 q^{68} -1.70903 q^{69} +2.01478 q^{70} -1.91041 q^{71} -8.20687 q^{72} +13.5257 q^{73} +2.87600 q^{74} +3.34767 q^{75} +4.61164 q^{76} -2.01478 q^{77} -4.98630 q^{78} +4.69668 q^{79} -1.00000 q^{80} +33.7321 q^{81} -5.97483 q^{82} -13.3863 q^{83} +6.74481 q^{84} +4.62109 q^{85} -1.00000 q^{86} +22.4053 q^{87} +1.00000 q^{88} +8.93667 q^{89} +8.20687 q^{90} +3.00098 q^{91} -0.510515 q^{92} -10.7748 q^{93} -0.687284 q^{94} -4.61164 q^{95} -3.34767 q^{96} -18.5182 q^{97} +2.94066 q^{98} -8.20687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{2} - q^{3} + 11 q^{4} - 11 q^{5} + q^{6} + 6 q^{7} - 11 q^{8} + 12 q^{9}+O(q^{10}) \) 11 * q - 11 * q^2 - q^3 + 11 * q^4 - 11 * q^5 + q^6 + 6 * q^7 - 11 * q^8 + 12 * q^9	\( 11 q - 11 q^{2} - q^{3} + 11 q^{4} - 11 q^{5} + q^{6} + 6 q^{7} - 11 q^{8} + 12 q^{9} + 11 q^{10} - 11 q^{11} - q^{12} + 4 q^{13} - 6 q^{14} + q^{15} + 11 q^{16} - 10 q^{17} - 12 q^{18} + 14 q^{19} - 11 q^{20} - 2 q^{21} + 11 q^{22} - 18 q^{23} + q^{24} + 11 q^{25} - 4 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29} - q^{30} + 13 q^{31} - 11 q^{32} + q^{33} + 10 q^{34} - 6 q^{35} + 12 q^{36} + q^{37} - 14 q^{38} + 12 q^{39} + 11 q^{40} + 12 q^{41} + 2 q^{42} + 11 q^{43} - 11 q^{44} - 12 q^{45} + 18 q^{46} - 5 q^{47} - q^{48} + 31 q^{49} - 11 q^{50} + q^{51} + 4 q^{52} - 27 q^{53} - 2 q^{54} + 11 q^{55} - 6 q^{56} - 5 q^{57} + 6 q^{58} + 11 q^{59} + q^{60} + 36 q^{61} - 13 q^{62} + 17 q^{63} + 11 q^{64} - 4 q^{65} - q^{66} + 18 q^{67} - 10 q^{68} + 14 q^{69} + 6 q^{70} - 14 q^{71} - 12 q^{72} - 11 q^{73} - q^{74} - q^{75} + 14 q^{76} - 6 q^{77} - 12 q^{78} + 28 q^{79} - 11 q^{80} + 7 q^{81} - 12 q^{82} - 4 q^{83} - 2 q^{84} + 10 q^{85} - 11 q^{86} + 38 q^{87} + 11 q^{88} - 7 q^{89} + 12 q^{90} + 14 q^{91} - 18 q^{92} - 3 q^{93} + 5 q^{94} - 14 q^{95} + q^{96} - q^{97} - 31 q^{98} - 12 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 - q^3 + 11 * q^4 - 11 * q^5 + q^6 + 6 * q^7 - 11 * q^8 + 12 * q^9 + 11 * q^10 - 11 * q^11 - q^12 + 4 * q^13 - 6 * q^14 + q^15 + 11 * q^16 - 10 * q^17 - 12 * q^18 + 14 * q^19 - 11 * q^20 - 2 * q^21 + 11 * q^22 - 18 * q^23 + q^24 + 11 * q^25 - 4 * q^26 + 2 * q^27 + 6 * q^28 - 6 * q^29 - q^30 + 13 * q^31 - 11 * q^32 + q^33 + 10 * q^34 - 6 * q^35 + 12 * q^36 + q^37 - 14 * q^38 + 12 * q^39 + 11 * q^40 + 12 * q^41 + 2 * q^42 + 11 * q^43 - 11 * q^44 - 12 * q^45 + 18 * q^46 - 5 * q^47 - q^48 + 31 * q^49 - 11 * q^50 + q^51 + 4 * q^52 - 27 * q^53 - 2 * q^54 + 11 * q^55 - 6 * q^56 - 5 * q^57 + 6 * q^58 + 11 * q^59 + q^60 + 36 * q^61 - 13 * q^62 + 17 * q^63 + 11 * q^64 - 4 * q^65 - q^66 + 18 * q^67 - 10 * q^68 + 14 * q^69 + 6 * q^70 - 14 * q^71 - 12 * q^72 - 11 * q^73 - q^74 - q^75 + 14 * q^76 - 6 * q^77 - 12 * q^78 + 28 * q^79 - 11 * q^80 + 7 * q^81 - 12 * q^82 - 4 * q^83 - 2 * q^84 + 10 * q^85 - 11 * q^86 + 38 * q^87 + 11 * q^88 - 7 * q^89 + 12 * q^90 + 14 * q^91 - 18 * q^92 - 3 * q^93 + 5 * q^94 - 14 * q^95 + q^96 - q^97 - 31 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more