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Properties

Label 624.4.n.d
Level $624$
Weight $4$
Character orbit 624.n
Analytic conductor $36.817$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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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] = [624,4,Mod(623,624)] 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("624.623"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 64 q^{9} + 8 q^{13} + 1208 q^{25} + 4680 q^{49} + 1616 q^{61} + 480 q^{69} + 1568 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
623.1 0 −5.19318 − 0.175639i 0 −16.9102 0 −1.57591 0 26.9383 + 1.82425i 0
623.2 0 −5.19318 − 0.175639i 0 16.9102 0 1.57591 0 26.9383 + 1.82425i 0
623.3 0 −5.19318 + 0.175639i 0 −16.9102 0 −1.57591 0 26.9383 − 1.82425i 0
623.4 0 −5.19318 + 0.175639i 0 16.9102 0 1.57591 0 26.9383 − 1.82425i 0
623.5 0 −4.81359 − 1.95687i 0 −6.26074 0 −18.6667 0 19.3413 + 18.8391i 0
623.6 0 −4.81359 − 1.95687i 0 6.26074 0 18.6667 0 19.3413 + 18.8391i 0
623.7 0 −4.81359 + 1.95687i 0 −6.26074 0 −18.6667 0 19.3413 − 18.8391i 0
623.8 0 −4.81359 + 1.95687i 0 6.26074 0 18.6667 0 19.3413 − 18.8391i 0
623.9 0 −4.39217 − 2.77648i 0 −7.01259 0 33.9068 0 11.5823 + 24.3896i 0
623.10 0 −4.39217 − 2.77648i 0 7.01259 0 −33.9068 0 11.5823 + 24.3896i 0
623.11 0 −4.39217 + 2.77648i 0 −7.01259 0 33.9068 0 11.5823 − 24.3896i 0
623.12 0 −4.39217 + 2.77648i 0 7.01259 0 −33.9068 0 11.5823 − 24.3896i 0
623.13 0 −3.53056 − 3.81250i 0 −9.28291 0 −16.8768 0 −2.07031 + 26.9205i 0
623.14 0 −3.53056 − 3.81250i 0 9.28291 0 16.8768 0 −2.07031 + 26.9205i 0
623.15 0 −3.53056 + 3.81250i 0 −9.28291 0 −16.8768 0 −2.07031 − 26.9205i 0
623.16 0 −3.53056 + 3.81250i 0 9.28291 0 16.8768 0 −2.07031 − 26.9205i 0
623.17 0 −2.39946 − 4.60897i 0 −1.65893 0 4.60037 0 −15.4851 + 22.1181i 0
623.18 0 −2.39946 − 4.60897i 0 1.65893 0 −4.60037 0 −15.4851 + 22.1181i 0
623.19 0 −2.39946 + 4.60897i 0 −1.65893 0 4.60037 0 −15.4851 − 22.1181i 0
623.20 0 −2.39946 + 4.60897i 0 1.65893 0 −4.60037 0 −15.4851 − 22.1181i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 623.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.n.d ✓ 56
3.b odd 2 1 inner 624.4.n.d ✓ 56
4.b odd 2 1 inner 624.4.n.d ✓ 56
12.b even 2 1 inner 624.4.n.d ✓ 56
13.b even 2 1 inner 624.4.n.d ✓ 56
39.d odd 2 1 inner 624.4.n.d ✓ 56
52.b odd 2 1 inner 624.4.n.d ✓ 56
156.h even 2 1 inner 624.4.n.d ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.n.d ✓ 56 1.a even 1 1 trivial
624.4.n.d ✓ 56 3.b odd 2 1 inner
624.4.n.d ✓ 56 4.b odd 2 1 inner
624.4.n.d ✓ 56 12.b even 2 1 inner
624.4.n.d ✓ 56 13.b even 2 1 inner
624.4.n.d ✓ 56 39.d odd 2 1 inner
624.4.n.d ✓ 56 52.b odd 2 1 inner
624.4.n.d ✓ 56 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{14} - 1026 T_{5}^{12} + 389325 T_{5}^{10} - 68688684 T_{5}^{8} + 5900226660 T_{5}^{6} + \cdots - 8876779891200 \) Copy content Toggle raw display
\( T_{7}^{14} - 2986 T_{7}^{12} + 3067101 T_{7}^{10} - 1280089524 T_{7}^{8} + 218646373284 T_{7}^{6} + \cdots - 401059365175296 \) Copy content Toggle raw display