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Properties

Label 12.22.b.a
Level $12$
Weight $22$
Character orbit 12.b
Analytic conductor $33.537$
Analytic rank $0$
Dimension $40$
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] = [12,22,Mod(11,12)] 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("12.11"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), 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: \(33.5372813144\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 1212040 q^{4} - 73477608 q^{6} + 4862757768 q^{9} + 70993722608 q^{10} + 230311800264 q^{12} + 532697931824 q^{13} - 3296861950688 q^{16} - 38138435545488 q^{18} + 6345498814320 q^{21} + 345245332418448 q^{22}+ \cdots + 19\!\cdots\!68 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1448.15 − 1.26410i 80355.2 + 63272.4i 2.09715e6 + 3661.23i − 1.80238e7i −1.16287e8 − 9.17298e7i − 1.08064e9i −3.03699e9 − 7.95303e6i 2.45355e9 + 1.01685e10i −2.27840e7 + 2.61013e10i
11.2 −1448.15 + 1.26410i 80355.2 − 63272.4i 2.09715e6 − 3661.23i 1.80238e7i −1.16287e8 + 9.17298e7i 1.08064e9i −3.03699e9 + 7.95303e6i 2.45355e9 − 1.01685e10i −2.27840e7 − 2.61013e10i
11.3 −1422.90 − 269.282i −24738.8 − 99238.8i 1.95213e6 + 766321.i 8.02021e6i 8.47766e6 + 1.47868e8i − 9.13804e8i −2.57132e9 − 1.61607e9i −9.23633e9 + 4.91011e9i 2.15970e9 − 1.14119e10i
11.4 −1422.90 + 269.282i −24738.8 + 99238.8i 1.95213e6 − 766321.i − 8.02021e6i 8.47766e6 − 1.47868e8i 9.13804e8i −2.57132e9 + 1.61607e9i −9.23633e9 − 4.91011e9i 2.15970e9 + 1.14119e10i
11.5 −1309.28 − 618.823i −101361. + 13651.6i 1.33127e6 + 1.62042e6i − 2.42790e7i 1.41157e8 + 4.48505e7i 2.63476e8i −7.40247e8 − 2.94540e9i 1.00876e10 − 2.76748e9i −1.50244e10 + 3.17879e10i
11.6 −1309.28 + 618.823i −101361. − 13651.6i 1.33127e6 − 1.62042e6i 2.42790e7i 1.41157e8 − 4.48505e7i − 2.63476e8i −7.40247e8 + 2.94540e9i 1.00876e10 + 2.76748e9i −1.50244e10 − 3.17879e10i
11.7 −1218.33 − 782.825i −54958.0 + 86255.2i 871523. + 1.90748e6i 3.43031e7i 1.34480e8 − 6.20651e7i − 6.03889e8i 4.31420e8 − 3.00620e9i −4.41958e9 − 9.48084e9i 2.68533e10 − 4.17926e10i
11.8 −1218.33 + 782.825i −54958.0 − 86255.2i 871523. − 1.90748e6i − 3.43031e7i 1.34480e8 + 6.20651e7i 6.03889e8i 4.31420e8 + 3.00620e9i −4.41958e9 + 9.48084e9i 2.68533e10 + 4.17926e10i
11.9 −1118.98 − 919.257i 89974.0 + 48631.5i 407085. + 2.05726e6i 1.11352e7i −5.59744e7 − 1.37127e8i 4.14938e8i 1.43563e9 − 2.67625e9i 5.73030e9 + 8.75115e9i 1.02361e10 − 1.24600e10i
11.10 −1118.98 + 919.257i 89974.0 − 48631.5i 407085. − 2.05726e6i − 1.11352e7i −5.59744e7 + 1.37127e8i − 4.14938e8i 1.43563e9 + 2.67625e9i 5.73030e9 − 8.75115e9i 1.02361e10 + 1.24600e10i
11.11 −1006.34 − 1041.36i 68874.9 − 75608.2i −71700.6 + 2.09593e6i − 3.89323e7i −1.48047e8 + 4.36439e6i − 1.54048e8i 2.25476e9 − 2.03455e9i −9.72855e8 − 1.04150e10i −4.05424e10 + 3.91792e10i
11.12 −1006.34 + 1041.36i 68874.9 + 75608.2i −71700.6 − 2.09593e6i 3.89323e7i −1.48047e8 − 4.36439e6i 1.54048e8i 2.25476e9 + 2.03455e9i −9.72855e8 + 1.04150e10i −4.05424e10 − 3.91792e10i
11.13 −795.891 − 1209.84i −53927.7 − 86903.1i −830267. + 1.92580e6i 1.55115e7i −6.22182e7 + 1.34409e8i 1.05551e9i 2.99071e9 − 5.28237e8i −4.64396e9 + 9.37297e9i 1.87665e10 − 1.23455e10i
11.14 −795.891 + 1209.84i −53927.7 + 86903.1i −830267. − 1.92580e6i − 1.55115e7i −6.22182e7 − 1.34409e8i − 1.05551e9i 2.99071e9 + 5.28237e8i −4.64396e9 − 9.37297e9i 1.87665e10 + 1.23455e10i
11.15 −471.986 − 1369.08i 2769.76 + 102238.i −1.65161e6 + 1.29237e6i − 1.59334e7i 1.38665e8 − 5.20471e7i − 2.91508e7i 2.54890e9 + 1.65120e9i −1.04450e10 + 5.66352e8i −2.18140e10 + 7.52033e9i
11.16 −471.986 + 1369.08i 2769.76 − 102238.i −1.65161e6 − 1.29237e6i 1.59334e7i 1.38665e8 + 5.20471e7i 2.91508e7i 2.54890e9 − 1.65120e9i −1.04450e10 − 5.66352e8i −2.18140e10 − 7.52033e9i
11.17 −384.131 − 1396.28i 80301.8 − 63340.1i −1.80204e6 + 1.07271e6i 3.38573e7i −1.19287e8 − 8.77928e7i − 1.23992e9i 2.19002e9 + 2.10409e9i 2.43641e9 − 1.01727e10i 4.72743e10 − 1.30057e10i
11.18 −384.131 + 1396.28i 80301.8 + 63340.1i −1.80204e6 − 1.07271e6i − 3.38573e7i −1.19287e8 + 8.77928e7i 1.23992e9i 2.19002e9 − 2.10409e9i 2.43641e9 + 1.01727e10i 4.72743e10 + 1.30057e10i
11.19 −219.802 − 1431.38i −101700. − 10835.4i −2.00053e6 + 629239.i − 7.08257e6i 6.84438e6 + 1.47953e8i − 8.25046e8i 1.34040e9 + 2.72520e9i 1.02255e10 + 2.20393e9i −1.01378e10 + 1.55676e9i
11.20 −219.802 + 1431.38i −101700. + 10835.4i −2.00053e6 − 629239.i 7.08257e6i 6.84438e6 − 1.47953e8i 8.25046e8i 1.34040e9 − 2.72520e9i 1.02255e10 − 2.20393e9i −1.01378e10 − 1.55676e9i
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.40
Significant digits:
Format:

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.22.b.a ✓ 40
3.b odd 2 1 inner 12.22.b.a ✓ 40
4.b odd 2 1 inner 12.22.b.a ✓ 40
12.b even 2 1 inner 12.22.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.22.b.a ✓ 40 1.a even 1 1 trivial
12.22.b.a ✓ 40 3.b odd 2 1 inner
12.22.b.a ✓ 40 4.b odd 2 1 inner
12.22.b.a ✓ 40 12.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(12, [\chi])\).