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Properties

Label 64.8.a.i.1.2
Level $64$
Weight $8$
Character 64.1
Self dual yes
Analytic conductor $19.993$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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] = [64,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] 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: \(19.9926416310\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.87298\) 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)
 
\(f(q)\) \(=\) \(q+61.9677 q^{3} -70.0000 q^{5} -1115.42 q^{7} +1653.00 q^{9} -7250.22 q^{11} -13758.0 q^{13} -4337.74 q^{15} +16994.0 q^{17} +34020.3 q^{19} -69120.0 q^{21} +32347.2 q^{23} -73225.0 q^{25} -33090.8 q^{27} -34190.0 q^{29} +120465. q^{31} -449280. q^{33} +78079.3 q^{35} -35206.0 q^{37} -852552. q^{39} -484550. q^{41} -672040. q^{43} -115710. q^{45} +1.20688e6 q^{47} +420617. q^{49} +1.05308e6 q^{51} -851702. q^{53} +507516. q^{55} +2.10816e6 q^{57} +695464. q^{59} -71630.0 q^{61} -1.84379e6 q^{63} +963060. q^{65} -307298. q^{67} +2.00448e6 q^{69} +757370. q^{71} +3.91204e6 q^{73} -4.53759e6 q^{75} +8.08704e6 q^{77} -314548. q^{79} -5.66567e6 q^{81} -1.53649e6 q^{83} -1.18958e6 q^{85} -2.11868e6 q^{87} -2.51063e6 q^{89} +1.53459e7 q^{91} +7.46496e6 q^{93} -2.38142e6 q^{95} -50094.0 q^{97} -1.19846e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 140 q^{5} + 3306 q^{9} - 27516 q^{13} + 33988 q^{17} - 138240 q^{21} - 146450 q^{25} - 68380 q^{29} - 898560 q^{33} - 70412 q^{37} - 969100 q^{41} - 231420 q^{45} + 841234 q^{49} - 1703404 q^{53}+ \cdots - 100188 q^{97}+O(q^{100}) \) 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
\(3\) 61.9677 1.32508 0.662539 − 0.749028i \(-0.269479\pi\)
0.662539 + 0.749028i \(0.269479\pi\)
\(4\) 0 0
\(5\) −70.0000 −0.250440 −0.125220 − 0.992129i \(-0.539964\pi\)
−0.125220 + 0.992129i \(0.539964\pi\)
\(6\) 0 0
\(7\) −1115.42 −1.22912 −0.614561 − 0.788869i \(-0.710667\pi\)
−0.614561 + 0.788869i \(0.710667\pi\)
\(8\) 0 0
\(9\) 1653.00 0.755830
\(10\) 0 0
\(11\) −7250.22 −1.64239 −0.821196 − 0.570646i \(-0.806693\pi\)
−0.821196 + 0.570646i \(0.806693\pi\)
\(12\) 0 0
\(13\) −13758.0 −1.73682 −0.868408 − 0.495851i \(-0.834856\pi\)
−0.868408 + 0.495851i \(0.834856\pi\)
\(14\) 0 0
\(15\) −4337.74 −0.331852
\(16\) 0 0
\(17\) 16994.0 0.838927 0.419464 − 0.907772i \(-0.362218\pi\)
0.419464 + 0.907772i \(0.362218\pi\)
\(18\) 0 0
\(19\) 34020.3 1.13789 0.568945 − 0.822376i \(-0.307352\pi\)
0.568945 + 0.822376i \(0.307352\pi\)
\(20\) 0 0
\(21\) −69120.0 −1.62868
\(22\) 0 0
\(23\) 32347.2 0.554356 0.277178 − 0.960819i \(-0.410601\pi\)
0.277178 + 0.960819i \(0.410601\pi\)
\(24\) 0 0
\(25\) −73225.0 −0.937280
\(26\) 0 0
\(27\) −33090.8 −0.323544
\(28\) 0 0
\(29\) −34190.0 −0.260319 −0.130160 − 0.991493i \(-0.541549\pi\)
−0.130160 + 0.991493i \(0.541549\pi\)
\(30\) 0 0
\(31\) 120465. 0.726266 0.363133 − 0.931737i \(-0.381707\pi\)
0.363133 + 0.931737i \(0.381707\pi\)
\(32\) 0 0
\(33\) −449280. −2.17630
\(34\) 0 0
\(35\) 78079.3 0.307821
\(36\) 0 0
\(37\) −35206.0 −0.114264 −0.0571322 − 0.998367i \(-0.518196\pi\)
−0.0571322 + 0.998367i \(0.518196\pi\)
\(38\) 0 0
\(39\) −852552. −2.30141
\(40\) 0 0
\(41\) −484550. −1.09798 −0.548991 − 0.835828i \(-0.684988\pi\)
−0.548991 + 0.835828i \(0.684988\pi\)
\(42\) 0 0
\(43\) −672040. −1.28901 −0.644504 − 0.764601i \(-0.722936\pi\)
−0.644504 + 0.764601i \(0.722936\pi\)
\(44\) 0 0
\(45\) −115710. −0.189290
\(46\) 0 0
\(47\) 1.20688e6 1.69560 0.847799 − 0.530318i \(-0.177927\pi\)
0.847799 + 0.530318i \(0.177927\pi\)
\(48\) 0 0
\(49\) 420617. 0.510741
\(50\) 0 0
\(51\) 1.05308e6 1.11164
\(52\) 0 0
\(53\) −851702. −0.785818 −0.392909 − 0.919577i \(-0.628531\pi\)
−0.392909 + 0.919577i \(0.628531\pi\)
\(54\) 0 0
\(55\) 507516. 0.411320
\(56\) 0 0
\(57\) 2.10816e6 1.50779
\(58\) 0 0
\(59\) 695464. 0.440852 0.220426 − 0.975404i \(-0.429255\pi\)
0.220426 + 0.975404i \(0.429255\pi\)
\(60\) 0 0
\(61\) −71630.0 −0.0404055 −0.0202028 − 0.999796i \(-0.506431\pi\)
−0.0202028 + 0.999796i \(0.506431\pi\)
\(62\) 0 0
\(63\) −1.84379e6 −0.929007
\(64\) 0 0
\(65\) 963060. 0.434967
\(66\) 0 0
\(67\) −307298. −0.124824 −0.0624120 − 0.998050i \(-0.519879\pi\)
−0.0624120 + 0.998050i \(0.519879\pi\)
\(68\) 0 0
\(69\) 2.00448e6 0.734564
\(70\) 0 0
\(71\) 757370. 0.251133 0.125566 − 0.992085i \(-0.459925\pi\)
0.125566 + 0.992085i \(0.459925\pi\)
\(72\) 0 0
\(73\) 3.91204e6 1.17699 0.588496 − 0.808500i \(-0.299720\pi\)
0.588496 + 0.808500i \(0.299720\pi\)
\(74\) 0 0
\(75\) −4.53759e6 −1.24197
\(76\) 0 0
\(77\) 8.08704e6 2.01870
\(78\) 0 0
\(79\) −314548. −0.0717782 −0.0358891 − 0.999356i \(-0.511426\pi\)
−0.0358891 + 0.999356i \(0.511426\pi\)
\(80\) 0 0
\(81\) −5.66567e6 −1.18455
\(82\) 0 0
\(83\) −1.53649e6 −0.294955 −0.147478 − 0.989065i \(-0.547115\pi\)
−0.147478 + 0.989065i \(0.547115\pi\)
\(84\) 0 0
\(85\) −1.18958e6 −0.210101
\(86\) 0 0
\(87\) −2.11868e6 −0.344943
\(88\) 0 0
\(89\) −2.51063e6 −0.377501 −0.188750 − 0.982025i \(-0.560444\pi\)
−0.188750 + 0.982025i \(0.560444\pi\)
\(90\) 0 0
\(91\) 1.53459e7 2.13476
\(92\) 0 0
\(93\) 7.46496e6 0.962359
\(94\) 0 0
\(95\) −2.38142e6 −0.284973
\(96\) 0 0
\(97\) −50094.0 −0.00557294 −0.00278647 − 0.999996i \(-0.500887\pi\)
−0.00278647 + 0.999996i \(0.500887\pi\)
\(98\) 0 0
\(99\) −1.19846e7 −1.24137
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 64.8.a.i.1.2 2
3.2 odd 2 576.8.a.bk.1.1 2
4.3 odd 2 inner 64.8.a.i.1.1 2
8.3 odd 2 32.8.a.c.1.2 yes 2
8.5 even 2 32.8.a.c.1.1 ✓ 2
12.11 even 2 576.8.a.bk.1.2 2
16.3 odd 4 256.8.b.i.129.2 4
16.5 even 4 256.8.b.i.129.1 4
16.11 odd 4 256.8.b.i.129.3 4
16.13 even 4 256.8.b.i.129.4 4
24.5 odd 2 288.8.a.k.1.1 2
24.11 even 2 288.8.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.c.1.1 ✓ 2 8.5 even 2
32.8.a.c.1.2 yes 2 8.3 odd 2
64.8.a.i.1.1 2 4.3 odd 2 inner
64.8.a.i.1.2 2 1.1 even 1 trivial
256.8.b.i.129.1 4 16.5 even 4
256.8.b.i.129.2 4 16.3 odd 4
256.8.b.i.129.3 4 16.11 odd 4
256.8.b.i.129.4 4 16.13 even 4
288.8.a.k.1.1 2 24.5 odd 2
288.8.a.k.1.2 2 24.11 even 2
576.8.a.bk.1.1 2 3.2 odd 2
576.8.a.bk.1.2 2 12.11 even 2