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Properties

Label 8464.2.a.cj.1.6
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 = [15,0,1,0,10,0,-10,0,16,0,9,0,0] 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: \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) 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 1.6
Root \(-0.407461\) 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)
 
\(f(q)\) \(=\) \(q-0.407461 q^{3} -2.35109 q^{5} -3.33026 q^{7} -2.83398 q^{9} +4.73053 q^{11} +4.54627 q^{13} +0.957979 q^{15} +8.21061 q^{17} -0.775701 q^{19} +1.35695 q^{21} +0.527630 q^{25} +2.37712 q^{27} -3.80043 q^{29} +3.89696 q^{31} -1.92751 q^{33} +7.82974 q^{35} +3.79517 q^{37} -1.85243 q^{39} -6.95843 q^{41} -11.9919 q^{43} +6.66293 q^{45} +0.399131 q^{47} +4.09060 q^{49} -3.34550 q^{51} -5.42339 q^{53} -11.1219 q^{55} +0.316068 q^{57} -10.0256 q^{59} -0.371568 q^{61} +9.43786 q^{63} -10.6887 q^{65} +6.40822 q^{67} -2.59838 q^{71} +1.21855 q^{73} -0.214989 q^{75} -15.7539 q^{77} +5.73941 q^{79} +7.53334 q^{81} -3.62209 q^{83} -19.3039 q^{85} +1.54853 q^{87} +1.67496 q^{89} -15.1402 q^{91} -1.58786 q^{93} +1.82374 q^{95} +15.7692 q^{97} -13.4062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} + 20 q^{17} + 3 q^{19} + 21 q^{21} + 11 q^{25} + 7 q^{27} - 4 q^{29} + 14 q^{31} + 34 q^{33} - 18 q^{35} + 26 q^{37} - 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) 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
\(3\) −0.407461 −0.235248 −0.117624 − 0.993058i \(-0.537528\pi\)
−0.117624 + 0.993058i \(0.537528\pi\)
\(4\) 0 0
\(5\) −2.35109 −1.05144 −0.525720 − 0.850658i \(-0.676204\pi\)
−0.525720 + 0.850658i \(0.676204\pi\)
\(6\) 0 0
\(7\) −3.33026 −1.25872 −0.629359 − 0.777115i \(-0.716683\pi\)
−0.629359 + 0.777115i \(0.716683\pi\)
\(8\) 0 0
\(9\) −2.83398 −0.944658
\(10\) 0 0
\(11\) 4.73053 1.42631 0.713155 − 0.701006i \(-0.247266\pi\)
0.713155 + 0.701006i \(0.247266\pi\)
\(12\) 0 0
\(13\) 4.54627 1.26091 0.630454 − 0.776226i \(-0.282869\pi\)
0.630454 + 0.776226i \(0.282869\pi\)
\(14\) 0 0
\(15\) 0.957979 0.247349
\(16\) 0 0
\(17\) 8.21061 1.99136 0.995682 − 0.0928269i \(-0.0295903\pi\)
0.995682 + 0.0928269i \(0.0295903\pi\)
\(18\) 0 0
\(19\) −0.775701 −0.177958 −0.0889790 − 0.996033i \(-0.528360\pi\)
−0.0889790 + 0.996033i \(0.528360\pi\)
\(20\) 0 0
\(21\) 1.35695 0.296111
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 0.527630 0.105526
\(26\) 0 0
\(27\) 2.37712 0.457477
\(28\) 0 0
\(29\) −3.80043 −0.705722 −0.352861 − 0.935676i \(-0.614791\pi\)
−0.352861 + 0.935676i \(0.614791\pi\)
\(30\) 0 0
\(31\) 3.89696 0.699914 0.349957 − 0.936766i \(-0.386196\pi\)
0.349957 + 0.936766i \(0.386196\pi\)
\(32\) 0 0
\(33\) −1.92751 −0.335536
\(34\) 0 0
\(35\) 7.82974 1.32347
\(36\) 0 0
\(37\) 3.79517 0.623923 0.311961 − 0.950095i \(-0.399014\pi\)
0.311961 + 0.950095i \(0.399014\pi\)
\(38\) 0 0
\(39\) −1.85243 −0.296626
\(40\) 0 0
\(41\) −6.95843 −1.08672 −0.543362 − 0.839498i \(-0.682849\pi\)
−0.543362 + 0.839498i \(0.682849\pi\)
\(42\) 0 0
\(43\) −11.9919 −1.82875 −0.914375 − 0.404868i \(-0.867317\pi\)
−0.914375 + 0.404868i \(0.867317\pi\)
\(44\) 0 0
\(45\) 6.66293 0.993252
\(46\) 0 0
\(47\) 0.399131 0.0582192 0.0291096 − 0.999576i \(-0.490733\pi\)
0.0291096 + 0.999576i \(0.490733\pi\)
\(48\) 0 0
\(49\) 4.09060 0.584372
\(50\) 0 0
\(51\) −3.34550 −0.468464
\(52\) 0 0
\(53\) −5.42339 −0.744960 −0.372480 − 0.928040i \(-0.621493\pi\)
−0.372480 + 0.928040i \(0.621493\pi\)
\(54\) 0 0
\(55\) −11.1219 −1.49968
\(56\) 0 0
\(57\) 0.316068 0.0418643
\(58\) 0 0
\(59\) −10.0256 −1.30522 −0.652610 − 0.757694i \(-0.726326\pi\)
−0.652610 + 0.757694i \(0.726326\pi\)
\(60\) 0 0
\(61\) −0.371568 −0.0475744 −0.0237872 − 0.999717i \(-0.507572\pi\)
−0.0237872 + 0.999717i \(0.507572\pi\)
\(62\) 0 0
\(63\) 9.43786 1.18906
\(64\) 0 0
\(65\) −10.6887 −1.32577
\(66\) 0 0
\(67\) 6.40822 0.782888 0.391444 − 0.920202i \(-0.371976\pi\)
0.391444 + 0.920202i \(0.371976\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.59838 −0.308371 −0.154185 − 0.988042i \(-0.549275\pi\)
−0.154185 + 0.988042i \(0.549275\pi\)
\(72\) 0 0
\(73\) 1.21855 0.142621 0.0713104 − 0.997454i \(-0.477282\pi\)
0.0713104 + 0.997454i \(0.477282\pi\)
\(74\) 0 0
\(75\) −0.214989 −0.0248248
\(76\) 0 0
\(77\) −15.7539 −1.79532
\(78\) 0 0
\(79\) 5.73941 0.645735 0.322867 − 0.946444i \(-0.395353\pi\)
0.322867 + 0.946444i \(0.395353\pi\)
\(80\) 0 0
\(81\) 7.53334 0.837038
\(82\) 0 0
\(83\) −3.62209 −0.397577 −0.198788 − 0.980042i \(-0.563701\pi\)
−0.198788 + 0.980042i \(0.563701\pi\)
\(84\) 0 0
\(85\) −19.3039 −2.09380
\(86\) 0 0
\(87\) 1.54853 0.166020
\(88\) 0 0
\(89\) 1.67496 0.177545 0.0887726 − 0.996052i \(-0.471706\pi\)
0.0887726 + 0.996052i \(0.471706\pi\)
\(90\) 0 0
\(91\) −15.1402 −1.58713
\(92\) 0 0
\(93\) −1.58786 −0.164653
\(94\) 0 0
\(95\) 1.82374 0.187112
\(96\) 0 0
\(97\) 15.7692 1.60112 0.800561 − 0.599251i \(-0.204535\pi\)
0.800561 + 0.599251i \(0.204535\pi\)
\(98\) 0 0
\(99\) −13.4062 −1.34738
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.cj.1.6 15
4.3 odd 2 4232.2.a.z.1.10 15
23.5 odd 22 368.2.m.f.209.2 30
23.14 odd 22 368.2.m.f.81.2 30
23.22 odd 2 8464.2.a.ci.1.6 15
92.51 even 22 184.2.i.a.25.2 ✓ 30
92.83 even 22 184.2.i.a.81.2 yes 30
92.91 even 2 4232.2.a.y.1.10 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.25.2 ✓ 30 92.51 even 22
184.2.i.a.81.2 yes 30 92.83 even 22
368.2.m.f.81.2 30 23.14 odd 22
368.2.m.f.209.2 30 23.5 odd 22
4232.2.a.y.1.10 15 92.91 even 2
4232.2.a.z.1.10 15 4.3 odd 2
8464.2.a.ci.1.6 15 23.22 odd 2
8464.2.a.cj.1.6 15 1.1 even 1 trivial