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Properties

Label 845.2.n.h
Level $845$
Weight $2$
Character orbit 845.n
Analytic conductor $6.747$
Analytic rank $0$
Dimension $36$
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] = [845,2,Mod(484,845)] 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("845.484"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(845, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0,0,16,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 13 q^{10} - 22 q^{11} - 8 q^{14} - 8 q^{15} + 12 q^{16} + 28 q^{19} + 10 q^{20} + 52 q^{21} + 34 q^{24} - 16 q^{25} - 20 q^{29} - 31 q^{30} + 64 q^{31} - 36 q^{34}+ \cdots - 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
484.1 −2.18027 + 1.25878i 2.26868 − 1.30982i 2.16905 − 3.75690i 2.23296 − 0.117790i −3.29755 + 5.71153i −1.06580 − 0.615337i 5.88628i 1.93127 − 3.34505i −4.72019 + 3.06762i
484.2 −1.86795 + 1.07846i −1.51940 + 0.877227i 1.32615 − 2.29696i −0.176470 + 2.22909i 1.89211 − 3.27723i −3.34392 − 1.93061i 1.40696i 0.0390543 − 0.0676440i −2.07435 − 4.35414i
484.3 −1.86013 + 1.07395i 2.05625 − 1.18717i 1.30673 − 2.26333i 1.11435 + 1.93861i −2.54993 + 4.41661i 3.07334 + 1.77439i 1.31766i 1.31877 − 2.28417i −4.15482 − 2.40932i
484.4 −1.68495 + 0.972806i −0.651549 + 0.376172i 0.892702 − 1.54621i −0.809176 − 2.08452i 0.731885 − 1.26766i −1.29544 − 0.747922i − 0.417520i −1.21699 + 2.10789i 3.39126 + 2.72514i
484.5 −1.52984 + 0.883256i −0.598760 + 0.345695i 0.560281 − 0.970435i −2.04107 − 0.913245i 0.610673 − 1.05772i 3.77778 + 2.18110i − 1.55354i −1.26099 + 2.18410i 3.92915 − 0.405668i
484.6 −1.36568 + 0.788477i 2.57698 − 1.48782i 0.243391 − 0.421566i −2.01071 + 0.978285i −2.34622 + 4.06377i 2.16729 + 1.25129i − 2.38627i 2.92720 − 5.07006i 1.97464 − 2.92142i
484.7 −0.768923 + 0.443938i −1.09905 + 0.634534i −0.605838 + 1.04934i 1.33257 + 1.79562i 0.563388 − 0.975816i −1.97564 − 1.14064i − 2.85157i −0.694733 + 1.20331i −1.82179 − 0.789117i
484.8 −0.342148 + 0.197539i −2.36988 + 1.36825i −0.921957 + 1.59688i −1.15119 + 1.91697i 0.540567 − 0.936289i 0.545081 + 0.314703i − 1.51865i 2.24423 − 3.88712i 0.0151994 − 0.883291i
484.9 −0.210318 + 0.121427i 1.03358 − 0.596738i −0.970511 + 1.68097i 1.50873 − 1.65037i −0.144920 + 0.251009i −4.14000 − 2.39023i − 0.957093i −0.787808 + 1.36452i −0.116914 + 0.530303i
484.10 0.210318 − 0.121427i −1.03358 + 0.596738i −0.970511 + 1.68097i 1.50873 + 1.65037i −0.144920 + 0.251009i 4.14000 + 2.39023i 0.957093i −0.787808 + 1.36452i 0.517713 + 0.163901i
484.11 0.342148 − 0.197539i 2.36988 − 1.36825i −0.921957 + 1.59688i −1.15119 − 1.91697i 0.540567 − 0.936289i −0.545081 − 0.314703i 1.51865i 2.24423 − 3.88712i −0.772552 − 0.428482i
484.12 0.768923 − 0.443938i 1.09905 − 0.634534i −0.605838 + 1.04934i 1.33257 − 1.79562i 0.563388 − 0.975816i 1.97564 + 1.14064i 2.85157i −0.694733 + 1.20331i 0.227498 − 1.97227i
484.13 1.36568 − 0.788477i −2.57698 + 1.48782i 0.243391 − 0.421566i −2.01071 − 0.978285i −2.34622 + 4.06377i −2.16729 − 1.25129i 2.38627i 2.92720 − 5.07006i −3.51735 + 0.249373i
484.14 1.52984 − 0.883256i 0.598760 − 0.345695i 0.560281 − 0.970435i −2.04107 + 0.913245i 0.610673 − 1.05772i −3.77778 − 2.18110i 1.55354i −1.26099 + 2.18410i −2.31590 + 3.19991i
484.15 1.68495 − 0.972806i 0.651549 − 0.376172i 0.892702 − 1.54621i −0.809176 + 2.08452i 0.731885 − 1.26766i 1.29544 + 0.747922i 0.417520i −1.21699 + 2.10789i 0.664415 + 4.29949i
484.16 1.86013 − 1.07395i −2.05625 + 1.18717i 1.30673 − 2.26333i 1.11435 − 1.93861i −2.54993 + 4.41661i −3.07334 − 1.77439i − 1.31766i 1.31877 − 2.28417i −0.00912812 − 4.80284i
484.17 1.86795 − 1.07846i 1.51940 − 0.877227i 1.32615 − 2.29696i −0.176470 − 2.22909i 1.89211 − 3.27723i 3.34392 + 1.93061i − 1.40696i 0.0390543 − 0.0676440i −2.73362 − 3.97351i
484.18 2.18027 − 1.25878i −2.26868 + 1.30982i 2.16905 − 3.75690i 2.23296 + 0.117790i −3.29755 + 5.71153i 1.06580 + 0.615337i − 5.88628i 1.93127 − 3.34505i 5.01673 − 2.55399i
529.1 −2.18027 − 1.25878i 2.26868 + 1.30982i 2.16905 + 3.75690i 2.23296 + 0.117790i −3.29755 − 5.71153i −1.06580 + 0.615337i − 5.88628i 1.93127 + 3.34505i −4.72019 − 3.06762i
529.2 −1.86795 − 1.07846i −1.51940 − 0.877227i 1.32615 + 2.29696i −0.176470 − 2.22909i 1.89211 + 3.27723i −3.34392 + 1.93061i − 1.40696i 0.0390543 + 0.0676440i −2.07435 + 4.35414i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 484.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.n.h 36
5.b even 2 1 inner 845.2.n.h 36
13.b even 2 1 845.2.n.i 36
13.c even 3 1 845.2.b.h yes 18
13.c even 3 1 inner 845.2.n.h 36
13.d odd 4 2 845.2.l.g 72
13.e even 6 1 845.2.b.g ✓ 18
13.e even 6 1 845.2.n.i 36
13.f odd 12 2 845.2.d.e 36
13.f odd 12 2 845.2.l.g 72
65.d even 2 1 845.2.n.i 36
65.g odd 4 2 845.2.l.g 72
65.l even 6 1 845.2.b.g ✓ 18
65.l even 6 1 845.2.n.i 36
65.n even 6 1 845.2.b.h yes 18
65.n even 6 1 inner 845.2.n.h 36
65.q odd 12 2 4225.2.a.cb 18
65.r odd 12 2 4225.2.a.ca 18
65.s odd 12 2 845.2.d.e 36
65.s odd 12 2 845.2.l.g 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.b.g ✓ 18 13.e even 6 1
845.2.b.g ✓ 18 65.l even 6 1
845.2.b.h yes 18 13.c even 3 1
845.2.b.h yes 18 65.n even 6 1
845.2.d.e 36 13.f odd 12 2
845.2.d.e 36 65.s odd 12 2
845.2.l.g 72 13.d odd 4 2
845.2.l.g 72 13.f odd 12 2
845.2.l.g 72 65.g odd 4 2
845.2.l.g 72 65.s odd 12 2
845.2.n.h 36 1.a even 1 1 trivial
845.2.n.h 36 5.b even 2 1 inner
845.2.n.h 36 13.c even 3 1 inner
845.2.n.h 36 65.n even 6 1 inner
845.2.n.i 36 13.b even 2 1
845.2.n.i 36 13.e even 6 1
845.2.n.i 36 65.d even 2 1
845.2.n.i 36 65.l even 6 1
4225.2.a.ca 18 65.r odd 12 2
4225.2.a.cb 18 65.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{36} - 26 T_{2}^{34} + 395 T_{2}^{32} - 4042 T_{2}^{30} + 31047 T_{2}^{28} - 184148 T_{2}^{26} + \cdots + 841 \) Copy content Toggle raw display
\( T_{11}^{18} + 11 T_{11}^{17} + 122 T_{11}^{16} + 659 T_{11}^{15} + 4263 T_{11}^{14} + 16144 T_{11}^{13} + \cdots + 235192896 \) Copy content Toggle raw display