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Properties

Label 64.22.a.o.1.1
Level $64$
Weight $22$
Character 64.1
Self dual yes
Analytic conductor $178.866$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-23144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(178.865500344\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5201320x^{3} - 466399708x^{2} + 4990572086304x - 1473608896916400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{2}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2112.98\) 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)
 
\(f(q)\) \(=\) \(q-161942. q^{3} -9.10999e6 q^{5} +4.58736e8 q^{7} +1.57648e10 q^{9} -1.21568e11 q^{11} -2.06102e11 q^{13} +1.47529e12 q^{15} +6.23844e11 q^{17} +1.95292e13 q^{19} -7.42887e13 q^{21} +1.83895e14 q^{23} -3.93845e14 q^{25} -8.59019e14 q^{27} -3.78214e15 q^{29} -6.71258e15 q^{31} +1.96869e16 q^{33} -4.17908e15 q^{35} -1.47214e16 q^{37} +3.33766e16 q^{39} +1.63913e17 q^{41} +9.69596e16 q^{43} -1.43617e17 q^{45} +1.37998e17 q^{47} -3.48107e17 q^{49} -1.01026e17 q^{51} +9.54452e17 q^{53} +1.10748e18 q^{55} -3.16260e18 q^{57} +5.91287e18 q^{59} +8.61362e18 q^{61} +7.23190e18 q^{63} +1.87759e18 q^{65} -1.37765e18 q^{67} -2.97803e19 q^{69} +4.70038e19 q^{71} +2.45764e19 q^{73} +6.37801e19 q^{75} -5.57676e19 q^{77} +6.92981e19 q^{79} -2.57946e19 q^{81} +7.55315e19 q^{83} -5.68321e18 q^{85} +6.12488e20 q^{87} +2.43657e20 q^{89} -9.45466e19 q^{91} +1.08705e21 q^{93} -1.77911e20 q^{95} -1.23815e21 q^{97} -1.91650e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 23144 q^{3} - 18311174 q^{5} + 63978640 q^{7} + 2839988161 q^{9} - 25629588280 q^{11} - 26739996110 q^{13} + 850898706352 q^{15} - 88104593910 q^{17} - 13998239618440 q^{19} - 11868255565952 q^{21}+ \cdots - 45\!\cdots\!40 q^{99}+O(q^{100}) \) 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
\(3\) −161942. −1.58338 −0.791692 − 0.610921i \(-0.790799\pi\)
−0.791692 + 0.610921i \(0.790799\pi\)
\(4\) 0 0
\(5\) −9.10999e6 −0.417189 −0.208594 − 0.978002i \(-0.566889\pi\)
−0.208594 + 0.978002i \(0.566889\pi\)
\(6\) 0 0
\(7\) 4.58736e8 0.613810 0.306905 − 0.951740i \(-0.400707\pi\)
0.306905 + 0.951740i \(0.400707\pi\)
\(8\) 0 0
\(9\) 1.57648e10 1.50710
\(10\) 0 0
\(11\) −1.21568e11 −1.41317 −0.706587 − 0.707627i \(-0.749766\pi\)
−0.706587 + 0.707627i \(0.749766\pi\)
\(12\) 0 0
\(13\) −2.06102e11 −0.414646 −0.207323 − 0.978273i \(-0.566475\pi\)
−0.207323 + 0.978273i \(0.566475\pi\)
\(14\) 0 0
\(15\) 1.47529e12 0.660570
\(16\) 0 0
\(17\) 6.23844e11 0.0750520 0.0375260 − 0.999296i \(-0.488052\pi\)
0.0375260 + 0.999296i \(0.488052\pi\)
\(18\) 0 0
\(19\) 1.95292e13 0.730755 0.365377 − 0.930859i \(-0.380940\pi\)
0.365377 + 0.930859i \(0.380940\pi\)
\(20\) 0 0
\(21\) −7.42887e13 −0.971896
\(22\) 0 0
\(23\) 1.83895e14 0.925608 0.462804 − 0.886461i \(-0.346843\pi\)
0.462804 + 0.886461i \(0.346843\pi\)
\(24\) 0 0
\(25\) −3.93845e14 −0.825953
\(26\) 0 0
\(27\) −8.59019e14 −0.802940
\(28\) 0 0
\(29\) −3.78214e15 −1.66940 −0.834698 − 0.550709i \(-0.814358\pi\)
−0.834698 + 0.550709i \(0.814358\pi\)
\(30\) 0 0
\(31\) −6.71258e15 −1.47093 −0.735465 − 0.677563i \(-0.763036\pi\)
−0.735465 + 0.677563i \(0.763036\pi\)
\(32\) 0 0
\(33\) 1.96869e16 2.23760
\(34\) 0 0
\(35\) −4.17908e15 −0.256075
\(36\) 0 0
\(37\) −1.47214e16 −0.503304 −0.251652 − 0.967818i \(-0.580974\pi\)
−0.251652 + 0.967818i \(0.580974\pi\)
\(38\) 0 0
\(39\) 3.33766e16 0.656544
\(40\) 0 0
\(41\) 1.63913e17 1.90715 0.953573 − 0.301162i \(-0.0973745\pi\)
0.953573 + 0.301162i \(0.0973745\pi\)
\(42\) 0 0
\(43\) 9.69596e16 0.684182 0.342091 − 0.939667i \(-0.388865\pi\)
0.342091 + 0.939667i \(0.388865\pi\)
\(44\) 0 0
\(45\) −1.43617e17 −0.628747
\(46\) 0 0
\(47\) 1.37998e17 0.382689 0.191344 − 0.981523i \(-0.438715\pi\)
0.191344 + 0.981523i \(0.438715\pi\)
\(48\) 0 0
\(49\) −3.48107e17 −0.623238
\(50\) 0 0
\(51\) −1.01026e17 −0.118836
\(52\) 0 0
\(53\) 9.54452e17 0.749648 0.374824 − 0.927096i \(-0.377703\pi\)
0.374824 + 0.927096i \(0.377703\pi\)
\(54\) 0 0
\(55\) 1.10748e18 0.589560
\(56\) 0 0
\(57\) −3.16260e18 −1.15707
\(58\) 0 0
\(59\) 5.91287e18 1.50609 0.753047 − 0.657966i \(-0.228583\pi\)
0.753047 + 0.657966i \(0.228583\pi\)
\(60\) 0 0
\(61\) 8.61362e18 1.54605 0.773023 − 0.634378i \(-0.218744\pi\)
0.773023 + 0.634378i \(0.218744\pi\)
\(62\) 0 0
\(63\) 7.23190e18 0.925075
\(64\) 0 0
\(65\) 1.87759e18 0.172986
\(66\) 0 0
\(67\) −1.37765e18 −0.0923323 −0.0461662 − 0.998934i \(-0.514700\pi\)
−0.0461662 + 0.998934i \(0.514700\pi\)
\(68\) 0 0
\(69\) −2.97803e19 −1.46559
\(70\) 0 0
\(71\) 4.70038e19 1.71364 0.856822 − 0.515613i \(-0.172436\pi\)
0.856822 + 0.515613i \(0.172436\pi\)
\(72\) 0 0
\(73\) 2.45764e19 0.669312 0.334656 − 0.942340i \(-0.391380\pi\)
0.334656 + 0.942340i \(0.391380\pi\)
\(74\) 0 0
\(75\) 6.37801e19 1.30780
\(76\) 0 0
\(77\) −5.57676e19 −0.867419
\(78\) 0 0
\(79\) 6.92981e19 0.823449 0.411725 − 0.911308i \(-0.364927\pi\)
0.411725 + 0.911308i \(0.364927\pi\)
\(80\) 0 0
\(81\) −2.57946e19 −0.235742
\(82\) 0 0
\(83\) 7.55315e19 0.534329 0.267164 − 0.963651i \(-0.413913\pi\)
0.267164 + 0.963651i \(0.413913\pi\)
\(84\) 0 0
\(85\) −5.68321e18 −0.0313108
\(86\) 0 0
\(87\) 6.12488e20 2.64329
\(88\) 0 0
\(89\) 2.43657e20 0.828292 0.414146 − 0.910211i \(-0.364080\pi\)
0.414146 + 0.910211i \(0.364080\pi\)
\(90\) 0 0
\(91\) −9.45466e19 −0.254514
\(92\) 0 0
\(93\) 1.08705e21 2.32905
\(94\) 0 0
\(95\) −1.77911e20 −0.304863
\(96\) 0 0
\(97\) −1.23815e21 −1.70479 −0.852397 − 0.522896i \(-0.824852\pi\)
−0.852397 + 0.522896i \(0.824852\pi\)
\(98\) 0 0
\(99\) −1.91650e21 −2.12980
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 64.22.a.o.1.1 5
4.3 odd 2 64.22.a.p.1.5 5
8.3 odd 2 32.22.a.c.1.1 ✓ 5
8.5 even 2 32.22.a.d.1.5 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.22.a.c.1.1 ✓ 5 8.3 odd 2
32.22.a.d.1.5 yes 5 8.5 even 2
64.22.a.o.1.1 5 1.1 even 1 trivial
64.22.a.p.1.5 5 4.3 odd 2