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Properties

Label 912.2.ch.f
Level $912$
Weight $2$
Character orbit 912.ch
Analytic conductor $7.282$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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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] = [912,2,Mod(47,912)] 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("912.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 9, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,0,0,0,0,0,6,0,0,0,-30,0,27,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{9} - 30 q^{13} + 27 q^{15} + 27 q^{21} - 12 q^{25} - 27 q^{27} - 15 q^{33} + 54 q^{43} - 9 q^{45} + 36 q^{49} + 36 q^{51} - 36 q^{57} - 90 q^{61} + 36 q^{63} - 90 q^{67} - 9 q^{69} + 78 q^{73}+ \cdots - 27 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.71115 − 0.268287i 0 0.746079 − 2.04984i 0 0.320680 + 0.185145i 0 2.85604 + 0.918157i 0
47.2 0 −1.69514 + 0.355654i 0 −1.42178 + 3.90632i 0 −3.33819 − 1.92731i 0 2.74702 − 1.20577i 0
47.3 0 −1.46877 + 0.917992i 0 −0.303314 + 0.833349i 0 3.60249 + 2.07990i 0 1.31458 − 2.69664i 0
47.4 0 −1.25395 + 1.19482i 0 0.806698 − 2.21638i 0 −2.23982 − 1.29316i 0 0.144806 − 2.99650i 0
47.5 0 0.0329261 − 1.73174i 0 −0.746079 + 2.04984i 0 0.320680 + 0.185145i 0 −2.99783 − 0.114039i 0
47.6 0 0.392690 + 1.68695i 0 −0.670748 + 1.84286i 0 −3.15740 − 1.82293i 0 −2.69159 + 1.32490i 0
47.7 0 0.461602 + 1.66941i 0 0.923912 − 2.53843i 0 1.47222 + 0.849986i 0 −2.57385 + 1.54120i 0
47.8 0 0.644609 − 1.60763i 0 1.42178 − 3.90632i 0 −3.33819 − 1.92731i 0 −2.16896 − 2.07259i 0
47.9 0 1.15909 − 1.28705i 0 0.303314 − 0.833349i 0 3.60249 + 2.07990i 0 −0.312998 − 2.98363i 0
47.10 0 1.39442 − 1.02743i 0 −0.806698 + 2.21638i 0 −2.23982 − 1.29316i 0 0.888791 − 2.86532i 0
47.11 0 1.56389 + 0.744479i 0 −0.923912 + 2.53843i 0 1.47222 + 0.849986i 0 1.89150 + 2.32857i 0
47.12 0 1.59313 + 0.679660i 0 0.670748 − 1.84286i 0 −3.15740 − 1.82293i 0 2.07613 + 2.16557i 0
479.1 0 −1.71646 − 0.231868i 0 −0.795030 − 0.947480i 0 −0.821166 − 0.474101i 0 2.89247 + 0.795983i 0
479.2 0 −1.53364 + 0.804948i 0 0.795030 + 0.947480i 0 −0.821166 − 0.474101i 0 1.70412 − 2.46901i 0
479.3 0 −1.38917 − 1.03451i 0 −1.43381 − 1.70875i 0 3.96599 + 2.28977i 0 0.859580 + 2.87422i 0
479.4 0 −1.27236 − 1.17520i 0 2.08155 + 2.48069i 0 −1.70517 − 0.984482i 0 0.237806 + 2.99056i 0
479.5 0 −0.951569 + 1.44724i 0 1.43381 + 1.70875i 0 3.96599 + 2.28977i 0 −1.18903 − 2.75431i 0
479.6 0 −0.793686 + 1.53950i 0 −2.08155 − 2.48069i 0 −1.70517 − 0.984482i 0 −1.74012 − 2.44376i 0
479.7 0 0.238457 − 1.71556i 0 1.55329 + 1.85114i 0 −0.539557 − 0.311514i 0 −2.88628 − 0.818173i 0
479.8 0 0.518173 − 1.65272i 0 −0.636607 − 0.758679i 0 0.639016 + 0.368936i 0 −2.46299 − 1.71279i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 47.12
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
76.l odd 18 1 inner
228.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ch.f yes 72
3.b odd 2 1 inner 912.2.ch.f yes 72
4.b odd 2 1 912.2.ch.e ✓ 72
12.b even 2 1 912.2.ch.e ✓ 72
19.e even 9 1 912.2.ch.e ✓ 72
57.l odd 18 1 912.2.ch.e ✓ 72
76.l odd 18 1 inner 912.2.ch.f yes 72
228.v even 18 1 inner 912.2.ch.f yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ch.e ✓ 72 4.b odd 2 1
912.2.ch.e ✓ 72 12.b even 2 1
912.2.ch.e ✓ 72 19.e even 9 1
912.2.ch.e ✓ 72 57.l odd 18 1
912.2.ch.f yes 72 1.a even 1 1 trivial
912.2.ch.f yes 72 3.b odd 2 1 inner
912.2.ch.f yes 72 76.l odd 18 1 inner
912.2.ch.f yes 72 228.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{72} + 6 T_{5}^{70} + 48 T_{5}^{68} - 4399 T_{5}^{66} - 8139 T_{5}^{64} - 429600 T_{5}^{62} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
\( T_{7}^{36} - 72 T_{7}^{34} + 3135 T_{7}^{32} + 558 T_{7}^{31} - 88774 T_{7}^{30} - 32139 T_{7}^{29} + \cdots + 15513200704 \) Copy content Toggle raw display