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Properties

Label 31.6.g.a
Level $31$
Weight $6$
Character orbit 31.g
Analytic conductor $4.972$
Analytic rank $0$
Dimension $96$
Inner twists $2$

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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] = [31,6,Mod(7,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([28])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.97189841420\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6 q^{2} - 20 q^{3} - 346 q^{4} - 33 q^{5} + 283 q^{6} - 66 q^{7} - 271 q^{8} + 658 q^{9} + 896 q^{10} + 41 q^{11} - 1169 q^{12} + 149 q^{13} - 2 q^{14} + 4906 q^{15} - 2450 q^{16} - 3152 q^{17} - 3 q^{18}+ \cdots + 674326 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.97541 − 9.15736i 10.1218 − 2.15146i −49.1157 + 35.6846i −52.7703 + 91.4008i −49.8183 − 86.2879i −103.002 − 45.8594i 223.645 + 162.488i −124.169 + 55.2835i 994.003 + 211.282i
7.2 −2.83153 − 8.71456i 17.2314 − 3.66266i −42.0374 + 30.5420i 48.0550 − 83.2337i −80.7098 − 139.793i 22.3398 + 9.94630i 147.972 + 107.508i 61.5161 − 27.3887i −861.414 − 183.099i
7.3 −2.62800 − 8.08816i −19.6320 + 4.17290i −32.6234 + 23.7023i 0.998763 − 1.72991i 85.3440 + 147.820i 87.1112 + 38.7844i 57.2761 + 41.6135i 146.010 − 65.0077i −16.6165 − 3.53195i
7.4 −1.17874 − 3.62778i 16.0292 − 3.40711i 14.1172 − 10.2567i −4.26994 + 7.39576i −31.2545 − 54.1343i 158.296 + 70.4781i −152.601 − 110.871i 23.3355 − 10.3896i 31.8633 + 6.77275i
7.5 −1.17413 − 3.61362i −7.49492 + 1.59309i 14.2089 − 10.3234i 10.2511 − 17.7555i 14.5569 + 25.2133i −136.752 − 60.8857i −152.354 − 110.691i −168.356 + 74.9568i −76.1977 − 16.1963i
7.6 0.227851 + 0.701253i 26.6118 − 5.65650i 25.4487 − 18.4896i −9.46798 + 16.3990i 10.0302 + 17.3727i −184.391 − 82.0963i 37.8530 + 27.5018i 454.198 − 202.222i −13.6572 − 2.90292i
7.7 0.326144 + 1.00377i −7.10263 + 1.50971i 24.9874 − 18.1544i −45.3815 + 78.6031i −3.83188 − 6.63701i 143.301 + 63.8015i 53.6957 + 39.0122i −173.823 + 77.3912i −93.7002 − 19.9166i
7.8 0.694022 + 2.13598i −29.5682 + 6.28491i 21.8078 − 15.8443i 21.0196 − 36.4070i −33.9454 − 58.7951i 70.5928 + 31.4300i 107.121 + 77.8282i 612.785 − 272.829i 92.3528 + 19.6302i
7.9 1.40673 + 4.32947i 6.21041 − 1.32006i 9.12310 − 6.62832i 40.0688 − 69.4012i 14.4516 + 25.0308i 37.9018 + 16.8750i 159.383 + 115.798i −185.165 + 82.4407i 356.837 + 75.8480i
7.10 2.05355 + 6.32019i −11.3150 + 2.40508i −9.83914 + 7.14855i −26.3143 + 45.5776i −38.4365 − 66.5739i −171.077 − 76.1684i 106.655 + 77.4896i −99.7469 + 44.4102i −342.097 − 72.7149i
7.11 2.71446 + 8.35426i 18.8567 − 4.00812i −36.5368 + 26.5455i −18.0992 + 31.3488i 84.6707 + 146.654i 61.3399 + 27.3103i −93.5364 − 67.9581i 117.519 − 52.3229i −311.025 − 66.1105i
7.12 3.17407 + 9.76877i −14.6416 + 3.11217i −59.4656 + 43.2043i 16.0051 − 27.7216i −76.8754 − 133.152i 37.1329 + 16.5326i −344.886 − 250.575i −17.3008 + 7.70280i 321.607 + 68.3596i
9.1 −2.97541 + 9.15736i 10.1218 + 2.15146i −49.1157 − 35.6846i −52.7703 − 91.4008i −49.8183 + 86.2879i −103.002 + 45.8594i 223.645 − 162.488i −124.169 − 55.2835i 994.003 − 211.282i
9.2 −2.83153 + 8.71456i 17.2314 + 3.66266i −42.0374 − 30.5420i 48.0550 + 83.2337i −80.7098 + 139.793i 22.3398 − 9.94630i 147.972 − 107.508i 61.5161 + 27.3887i −861.414 + 183.099i
9.3 −2.62800 + 8.08816i −19.6320 − 4.17290i −32.6234 − 23.7023i 0.998763 + 1.72991i 85.3440 − 147.820i 87.1112 − 38.7844i 57.2761 − 41.6135i 146.010 + 65.0077i −16.6165 + 3.53195i
9.4 −1.17874 + 3.62778i 16.0292 + 3.40711i 14.1172 + 10.2567i −4.26994 − 7.39576i −31.2545 + 54.1343i 158.296 − 70.4781i −152.601 + 110.871i 23.3355 + 10.3896i 31.8633 − 6.77275i
9.5 −1.17413 + 3.61362i −7.49492 − 1.59309i 14.2089 + 10.3234i 10.2511 + 17.7555i 14.5569 − 25.2133i −136.752 + 60.8857i −152.354 + 110.691i −168.356 − 74.9568i −76.1977 + 16.1963i
9.6 0.227851 − 0.701253i 26.6118 + 5.65650i 25.4487 + 18.4896i −9.46798 − 16.3990i 10.0302 − 17.3727i −184.391 + 82.0963i 37.8530 − 27.5018i 454.198 + 202.222i −13.6572 + 2.90292i
9.7 0.326144 − 1.00377i −7.10263 − 1.50971i 24.9874 + 18.1544i −45.3815 − 78.6031i −3.83188 + 6.63701i 143.301 − 63.8015i 53.6957 − 39.0122i −173.823 − 77.3912i −93.7002 + 19.9166i
9.8 0.694022 − 2.13598i −29.5682 − 6.28491i 21.8078 + 15.8443i 21.0196 + 36.4070i −33.9454 + 58.7951i 70.5928 − 31.4300i 107.121 − 77.8282i 612.785 + 272.829i 92.3528 − 19.6302i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.6.g.a ✓ 96
31.g even 15 1 inner 31.6.g.a ✓ 96
31.g even 15 1 961.6.a.j 48
31.h odd 30 1 961.6.a.k 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.6.g.a ✓ 96 1.a even 1 1 trivial
31.6.g.a ✓ 96 31.g even 15 1 inner
961.6.a.j 48 31.g even 15 1
961.6.a.k 48 31.h odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(31, [\chi])\).