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Properties

Label 8464.2.a.ch.1.8
Level $8464$
Weight $2$
Character 8464.1
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $15$
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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,0,-1,0,0,0,10,0,16,0,23,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.159078\) of defining polynomial
Character \(\chi\) \(=\) 8464.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.159078 q^{3} +0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} -2.40638 q^{11} -6.46974 q^{13} +0.00595930 q^{15} -1.41265 q^{17} -3.79359 q^{19} -0.505629 q^{21} -4.99860 q^{25} -0.950440 q^{27} -1.34251 q^{29} -5.26684 q^{31} -0.382801 q^{33} -0.119072 q^{35} -2.86429 q^{37} -1.02919 q^{39} +1.53077 q^{41} +3.15025 q^{43} -0.111437 q^{45} +7.00603 q^{47} +3.10291 q^{49} -0.224721 q^{51} -12.8957 q^{53} -0.0901467 q^{55} -0.603475 q^{57} +0.932170 q^{59} +4.33766 q^{61} +9.45509 q^{63} -0.242367 q^{65} +10.5734 q^{67} -12.6991 q^{71} -12.6489 q^{73} -0.795165 q^{75} +7.64869 q^{77} -7.50885 q^{79} +8.77289 q^{81} +12.5406 q^{83} -0.0529202 q^{85} -0.213563 q^{87} -7.79190 q^{89} +20.5641 q^{91} -0.837836 q^{93} -0.142114 q^{95} +2.28969 q^{97} +7.15824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) 0.159078 0.0918435 0.0459217 − 0.998945i \(-0.485378\pi\)
0.0459217 + 0.998945i \(0.485378\pi\)
\(4\) 0 0
\(5\) 0.0374616 0.0167533 0.00837667 − 0.999965i \(-0.497334\pi\)
0.00837667 + 0.999965i \(0.497334\pi\)
\(6\) 0 0
\(7\) −3.17851 −1.20136 −0.600682 − 0.799488i \(-0.705104\pi\)
−0.600682 + 0.799488i \(0.705104\pi\)
\(8\) 0 0
\(9\) −2.97469 −0.991565
\(10\) 0 0
\(11\) −2.40638 −0.725550 −0.362775 − 0.931877i \(-0.618171\pi\)
−0.362775 + 0.931877i \(0.618171\pi\)
\(12\) 0 0
\(13\) −6.46974 −1.79438 −0.897191 − 0.441642i \(-0.854396\pi\)
−0.897191 + 0.441642i \(0.854396\pi\)
\(14\) 0 0
\(15\) 0.00595930 0.00153868
\(16\) 0 0
\(17\) −1.41265 −0.342618 −0.171309 − 0.985217i \(-0.554800\pi\)
−0.171309 + 0.985217i \(0.554800\pi\)
\(18\) 0 0
\(19\) −3.79359 −0.870310 −0.435155 − 0.900356i \(-0.643306\pi\)
−0.435155 + 0.900356i \(0.643306\pi\)
\(20\) 0 0
\(21\) −0.505629 −0.110337
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.99860 −0.999719
\(26\) 0 0
\(27\) −0.950440 −0.182912
\(28\) 0 0
\(29\) −1.34251 −0.249298 −0.124649 − 0.992201i \(-0.539780\pi\)
−0.124649 + 0.992201i \(0.539780\pi\)
\(30\) 0 0
\(31\) −5.26684 −0.945952 −0.472976 − 0.881075i \(-0.656820\pi\)
−0.472976 + 0.881075i \(0.656820\pi\)
\(32\) 0 0
\(33\) −0.382801 −0.0666370
\(34\) 0 0
\(35\) −0.119072 −0.0201268
\(36\) 0 0
\(37\) −2.86429 −0.470886 −0.235443 − 0.971888i \(-0.575654\pi\)
−0.235443 + 0.971888i \(0.575654\pi\)
\(38\) 0 0
\(39\) −1.02919 −0.164802
\(40\) 0 0
\(41\) 1.53077 0.239066 0.119533 − 0.992830i \(-0.461860\pi\)
0.119533 + 0.992830i \(0.461860\pi\)
\(42\) 0 0
\(43\) 3.15025 0.480408 0.240204 − 0.970722i \(-0.422786\pi\)
0.240204 + 0.970722i \(0.422786\pi\)
\(44\) 0 0
\(45\) −0.111437 −0.0166120
\(46\) 0 0
\(47\) 7.00603 1.02193 0.510967 − 0.859600i \(-0.329287\pi\)
0.510967 + 0.859600i \(0.329287\pi\)
\(48\) 0 0
\(49\) 3.10291 0.443274
\(50\) 0 0
\(51\) −0.224721 −0.0314673
\(52\) 0 0
\(53\) −12.8957 −1.77136 −0.885681 − 0.464294i \(-0.846308\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(54\) 0 0
\(55\) −0.0901467 −0.0121554
\(56\) 0 0
\(57\) −0.603475 −0.0799323
\(58\) 0 0
\(59\) 0.932170 0.121358 0.0606791 − 0.998157i \(-0.480673\pi\)
0.0606791 + 0.998157i \(0.480673\pi\)
\(60\) 0 0
\(61\) 4.33766 0.555380 0.277690 − 0.960671i \(-0.410431\pi\)
0.277690 + 0.960671i \(0.410431\pi\)
\(62\) 0 0
\(63\) 9.45509 1.19123
\(64\) 0 0
\(65\) −0.242367 −0.0300619
\(66\) 0 0
\(67\) 10.5734 1.29175 0.645874 − 0.763444i \(-0.276493\pi\)
0.645874 + 0.763444i \(0.276493\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.6991 −1.50710 −0.753551 − 0.657390i \(-0.771660\pi\)
−0.753551 + 0.657390i \(0.771660\pi\)
\(72\) 0 0
\(73\) −12.6489 −1.48044 −0.740218 − 0.672367i \(-0.765278\pi\)
−0.740218 + 0.672367i \(0.765278\pi\)
\(74\) 0 0
\(75\) −0.795165 −0.0918177
\(76\) 0 0
\(77\) 7.64869 0.871649
\(78\) 0 0
\(79\) −7.50885 −0.844812 −0.422406 − 0.906407i \(-0.638814\pi\)
−0.422406 + 0.906407i \(0.638814\pi\)
\(80\) 0 0
\(81\) 8.77289 0.974765
\(82\) 0 0
\(83\) 12.5406 1.37651 0.688255 − 0.725469i \(-0.258377\pi\)
0.688255 + 0.725469i \(0.258377\pi\)
\(84\) 0 0
\(85\) −0.0529202 −0.00574000
\(86\) 0 0
\(87\) −0.213563 −0.0228964
\(88\) 0 0
\(89\) −7.79190 −0.825940 −0.412970 − 0.910745i \(-0.635509\pi\)
−0.412970 + 0.910745i \(0.635509\pi\)
\(90\) 0 0
\(91\) 20.5641 2.15571
\(92\) 0 0
\(93\) −0.837836 −0.0868795
\(94\) 0 0
\(95\) −0.142114 −0.0145806
\(96\) 0 0
\(97\) 2.28969 0.232482 0.116241 − 0.993221i \(-0.462915\pi\)
0.116241 + 0.993221i \(0.462915\pi\)
\(98\) 0 0
\(99\) 7.15824 0.719430
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 8464.2.a.ch.1.8 15
4.3 odd 2 4232.2.a.ba.1.8 15
23.5 odd 22 368.2.m.e.209.2 30
23.14 odd 22 368.2.m.e.81.2 30
23.22 odd 2 8464.2.a.cg.1.8 15
92.51 even 22 184.2.i.b.25.2 ✓ 30
92.83 even 22 184.2.i.b.81.2 yes 30
92.91 even 2 4232.2.a.bb.1.8 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.25.2 ✓ 30 92.51 even 22
184.2.i.b.81.2 yes 30 92.83 even 22
368.2.m.e.81.2 30 23.14 odd 22
368.2.m.e.209.2 30 23.5 odd 22
4232.2.a.ba.1.8 15 4.3 odd 2
4232.2.a.bb.1.8 15 92.91 even 2
8464.2.a.cg.1.8 15 23.22 odd 2
8464.2.a.ch.1.8 15 1.1 even 1 trivial