↑

Properties

Label 64.20.a.l.1.1
Level $64$
Weight $20$
Character 64.1
Self dual yes
Analytic conductor $146.443$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,20,Mod(1,64)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("64.1"); S:= CuspForms(chi, 20); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 20, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-23732,0,-2140218] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(146.442685796\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2519x + 43659 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(20.6663\) of defining polynomial
Character \(\chi\) \(=\) 64.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-54241.2 q^{3} -7.60339e6 q^{5} +7.17920e7 q^{7} +1.77984e9 q^{9} +4.34624e9 q^{11} +7.43111e10 q^{13} +4.12417e11 q^{15} +1.24277e11 q^{17} -3.27915e11 q^{19} -3.89408e12 q^{21} +4.60752e12 q^{23} +3.87380e13 q^{25} -3.34984e13 q^{27} -5.90175e12 q^{29} -2.14979e14 q^{31} -2.35745e14 q^{33} -5.45862e14 q^{35} +1.90271e13 q^{37} -4.03072e15 q^{39} +1.20083e15 q^{41} +3.64834e15 q^{43} -1.35328e16 q^{45} +7.52366e15 q^{47} -6.24480e15 q^{49} -6.74095e15 q^{51} -2.09104e16 q^{53} -3.30462e16 q^{55} +1.77865e16 q^{57} +1.11875e17 q^{59} -8.86226e16 q^{61} +1.27779e17 q^{63} -5.65016e17 q^{65} +1.47802e17 q^{67} -2.49917e17 q^{69} +3.98293e17 q^{71} -7.71257e15 q^{73} -2.10119e18 q^{75} +3.12026e17 q^{77} -1.54385e18 q^{79} -2.51651e17 q^{81} +5.86011e17 q^{83} -9.44929e17 q^{85} +3.20118e17 q^{87} +2.54352e18 q^{89} +5.33495e18 q^{91} +1.16607e19 q^{93} +2.49327e18 q^{95} +7.18112e18 q^{97} +7.73564e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 23732 q^{3} - 2140218 q^{5} + 55851720 q^{7} + 646753951 q^{9} + 297392964 q^{11} + 14862401022 q^{13} + 292635653528 q^{15} + 803332464534 q^{17} - 3212269666884 q^{19} - 11192319829728 q^{21} + 24948509305560 q^{23}+ \cdots + 11\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −54241.2 −1.59103 −0.795513 − 0.605937i \(-0.792798\pi\)
−0.795513 + 0.605937i \(0.792798\pi\)
\(4\) 0 0
\(5\) −7.60339e6 −1.74097 −0.870486 − 0.492193i \(-0.836196\pi\)
−0.870486 + 0.492193i \(0.836196\pi\)
\(6\) 0 0
\(7\) 7.17920e7 0.672426 0.336213 − 0.941786i \(-0.390854\pi\)
0.336213 + 0.941786i \(0.390854\pi\)
\(8\) 0 0
\(9\) 1.77984e9 1.53136
\(10\) 0 0
\(11\) 4.34624e9 0.555755 0.277878 − 0.960616i \(-0.410369\pi\)
0.277878 + 0.960616i \(0.410369\pi\)
\(12\) 0 0
\(13\) 7.43111e10 1.94353 0.971767 − 0.235944i \(-0.0758183\pi\)
0.971767 + 0.235944i \(0.0758183\pi\)
\(14\) 0 0
\(15\) 4.12417e11 2.76993
\(16\) 0 0
\(17\) 1.24277e11 0.254172 0.127086 − 0.991892i \(-0.459438\pi\)
0.127086 + 0.991892i \(0.459438\pi\)
\(18\) 0 0
\(19\) −3.27915e11 −0.233132 −0.116566 − 0.993183i \(-0.537189\pi\)
−0.116566 + 0.993183i \(0.537189\pi\)
\(20\) 0 0
\(21\) −3.89408e12 −1.06985
\(22\) 0 0
\(23\) 4.60752e12 0.533399 0.266700 − 0.963780i \(-0.414067\pi\)
0.266700 + 0.963780i \(0.414067\pi\)
\(24\) 0 0
\(25\) 3.87380e13 2.03099
\(26\) 0 0
\(27\) −3.34984e13 −0.845412
\(28\) 0 0
\(29\) −5.90175e12 −0.0755439 −0.0377720 − 0.999286i \(-0.512026\pi\)
−0.0377720 + 0.999286i \(0.512026\pi\)
\(30\) 0 0
\(31\) −2.14979e14 −1.46036 −0.730181 − 0.683254i \(-0.760564\pi\)
−0.730181 + 0.683254i \(0.760564\pi\)
\(32\) 0 0
\(33\) −2.35745e14 −0.884221
\(34\) 0 0
\(35\) −5.45862e14 −1.17068
\(36\) 0 0
\(37\) 1.90271e13 0.0240689 0.0120344 − 0.999928i \(-0.496169\pi\)
0.0120344 + 0.999928i \(0.496169\pi\)
\(38\) 0 0
\(39\) −4.03072e15 −3.09221
\(40\) 0 0
\(41\) 1.20083e15 0.572841 0.286420 − 0.958104i \(-0.407535\pi\)
0.286420 + 0.958104i \(0.407535\pi\)
\(42\) 0 0
\(43\) 3.64834e15 1.10699 0.553496 − 0.832852i \(-0.313293\pi\)
0.553496 + 0.832852i \(0.313293\pi\)
\(44\) 0 0
\(45\) −1.35328e16 −2.66606
\(46\) 0 0
\(47\) 7.52366e15 0.980617 0.490309 − 0.871549i \(-0.336884\pi\)
0.490309 + 0.871549i \(0.336884\pi\)
\(48\) 0 0
\(49\) −6.24480e15 −0.547843
\(50\) 0 0
\(51\) −6.74095e15 −0.404394
\(52\) 0 0
\(53\) −2.09104e16 −0.870444 −0.435222 − 0.900323i \(-0.643330\pi\)
−0.435222 + 0.900323i \(0.643330\pi\)
\(54\) 0 0
\(55\) −3.30462e16 −0.967555
\(56\) 0 0
\(57\) 1.77865e16 0.370919
\(58\) 0 0
\(59\) 1.11875e17 1.68128 0.840640 − 0.541595i \(-0.182179\pi\)
0.840640 + 0.541595i \(0.182179\pi\)
\(60\) 0 0
\(61\) −8.86226e16 −0.970311 −0.485155 − 0.874428i \(-0.661237\pi\)
−0.485155 + 0.874428i \(0.661237\pi\)
\(62\) 0 0
\(63\) 1.27779e17 1.02973
\(64\) 0 0
\(65\) −5.65016e17 −3.38364
\(66\) 0 0
\(67\) 1.47802e17 0.663696 0.331848 − 0.943333i \(-0.392328\pi\)
0.331848 + 0.943333i \(0.392328\pi\)
\(68\) 0 0
\(69\) −2.49917e17 −0.848652
\(70\) 0 0
\(71\) 3.98293e17 1.03097 0.515487 − 0.856897i \(-0.327611\pi\)
0.515487 + 0.856897i \(0.327611\pi\)
\(72\) 0 0
\(73\) −7.71257e15 −0.0153332 −0.00766658 − 0.999971i \(-0.502440\pi\)
−0.00766658 + 0.999971i \(0.502440\pi\)
\(74\) 0 0
\(75\) −2.10119e18 −3.23135
\(76\) 0 0
\(77\) 3.12026e17 0.373705
\(78\) 0 0
\(79\) −1.54385e18 −1.44927 −0.724633 − 0.689135i \(-0.757990\pi\)
−0.724633 + 0.689135i \(0.757990\pi\)
\(80\) 0 0
\(81\) −2.51651e17 −0.186291
\(82\) 0 0
\(83\) 5.86011e17 0.344084 0.172042 − 0.985090i \(-0.444964\pi\)
0.172042 + 0.985090i \(0.444964\pi\)
\(84\) 0 0
\(85\) −9.44929e17 −0.442506
\(86\) 0 0
\(87\) 3.20118e17 0.120192
\(88\) 0 0
\(89\) 2.54352e18 0.769538 0.384769 − 0.923013i \(-0.374281\pi\)
0.384769 + 0.923013i \(0.374281\pi\)
\(90\) 0 0
\(91\) 5.33495e18 1.30688
\(92\) 0 0
\(93\) 1.16607e19 2.32347
\(94\) 0 0
\(95\) 2.49327e18 0.405877
\(96\) 0 0
\(97\) 7.18112e18 0.959094 0.479547 − 0.877516i \(-0.340801\pi\)
0.479547 + 0.877516i \(0.340801\pi\)
\(98\) 0 0
\(99\) 7.73564e18 0.851063
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 64.20.a.l.1.1 3
4.3 odd 2 64.20.a.m.1.3 3
8.3 odd 2 16.20.a.f.1.1 3
8.5 even 2 8.20.a.b.1.3 ✓ 3
24.5 odd 2 72.20.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.20.a.b.1.3 ✓ 3 8.5 even 2
16.20.a.f.1.1 3 8.3 odd 2
64.20.a.l.1.1 3 1.1 even 1 trivial
64.20.a.m.1.3 3 4.3 odd 2
72.20.a.f.1.1 3 24.5 odd 2