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Properties

Label 12.18.b.a
Level $12$
Weight $18$
Character orbit 12.b
Analytic conductor $21.987$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [12,18,Mod(11,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.11");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.9866504813\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 54808 q^{4} - 4960776 q^{6} + 79874976 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 54808 q^{4} - 4960776 q^{6} + 79874976 q^{9} + 46839088 q^{10} + 2598308520 q^{12} + 221287360 q^{13} - 41721285088 q^{16} + 27568791600 q^{18} + 128365169856 q^{21} - 493958165040 q^{22} - 253565784288 q^{24} - 3063689463648 q^{25} - 2695436033040 q^{28} - 2169979068432 q^{30} - 17644793625600 q^{33} + 6944208632512 q^{34} - 12909384040056 q^{36} - 7970760699200 q^{37} + 21066454663744 q^{40} + 55835180334480 q^{42} + 154013263804416 q^{45} + 235918828815264 q^{46} + 299462725247520 q^{48} - 248591781347488 q^{49} + 544732086739120 q^{52} + 487051858173384 q^{54} - 21\!\cdots\!20 q^{57}+ \cdots + 90\!\cdots\!60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −361.348 − 22.3591i 9139.83 − 6753.05i 130072. + 16158.8i − 841783.i −3.45365e6 + 2.23584e6i − 1.94297e7i −4.66400e7 − 8.74726e6i 3.79327e7 − 1.23443e8i −1.88215e7 + 3.04176e8i
11.2 −361.348 + 22.3591i 9139.83 + 6753.05i 130072. − 16158.8i 841783.i −3.45365e6 − 2.23584e6i 1.94297e7i −4.66400e7 + 8.74726e6i 3.79327e7 + 1.23443e8i −1.88215e7 − 3.04176e8i
11.3 −347.385 − 101.959i −7870.26 + 8197.51i 110281. + 70838.2i − 134607.i 3.56982e6 − 2.04525e6i 2.80624e6i −3.10872e7 − 3.58523e7i −5.25830e6 − 1.29033e8i −1.37244e7 + 4.67604e7i
11.4 −347.385 + 101.959i −7870.26 − 8197.51i 110281. − 70838.2i 134607.i 3.56982e6 + 2.04525e6i − 2.80624e6i −3.10872e7 + 3.58523e7i −5.25830e6 + 1.29033e8i −1.37244e7 − 4.67604e7i
11.5 −296.839 − 207.264i 2157.01 − 11157.4i 45154.9 + 123048.i 1.59510e6i −2.95282e6 + 2.86488e6i 3.40503e6i 1.20998e7 − 4.58846e7i −1.19835e8 − 4.81332e7i 3.30608e8 − 4.73489e8i
11.6 −296.839 + 207.264i 2157.01 + 11157.4i 45154.9 − 123048.i − 1.59510e6i −2.95282e6 − 2.86488e6i − 3.40503e6i 1.20998e7 + 4.58846e7i −1.19835e8 + 4.81332e7i 3.30608e8 + 4.73489e8i
11.7 −267.429 − 244.037i −9292.74 − 6541.04i 11964.1 + 130525.i − 1.19755e6i 888889. + 4.01703e6i 5.99312e6i 2.86533e7 − 3.78257e7i 4.35697e7 + 1.21568e8i −2.92246e8 + 3.20259e8i
11.8 −267.429 + 244.037i −9292.74 + 6541.04i 11964.1 − 130525.i 1.19755e6i 888889. − 4.01703e6i − 5.99312e6i 2.86533e7 + 3.78257e7i 4.35697e7 − 1.21568e8i −2.92246e8 − 3.20259e8i
11.9 −232.258 − 277.720i 5373.62 + 10013.2i −23184.2 + 129005.i 293794.i 1.53280e6 − 3.81801e6i − 2.56096e7i 4.12120e7 − 2.35238e7i −7.13885e7 + 1.07614e8i 8.15922e7 − 6.82360e7i
11.10 −232.258 + 277.720i 5373.62 − 10013.2i −23184.2 − 129005.i − 293794.i 1.53280e6 + 3.81801e6i 2.56096e7i 4.12120e7 + 2.35238e7i −7.13885e7 − 1.07614e8i 8.15922e7 + 6.82360e7i
11.11 −207.324 − 296.798i 11323.4 − 959.175i −45105.9 + 123066.i − 804702.i −2.63230e6 − 3.16191e6i 2.45214e7i 4.58773e7 − 1.21272e7i 1.27300e8 − 2.17223e7i −2.38834e8 + 1.66834e8i
11.12 −207.324 + 296.798i 11323.4 + 959.175i −45105.9 − 123066.i 804702.i −2.63230e6 + 3.16191e6i − 2.45214e7i 4.58773e7 + 1.21272e7i 1.27300e8 + 2.17223e7i −2.38834e8 − 1.66834e8i
11.13 −97.3155 − 348.714i −11352.9 + 502.937i −112131. + 67870.6i 1.05410e6i 1.28019e6 + 3.90996e6i 199128.i 3.45796e7 + 3.24970e7i 1.28634e8 − 1.14195e7i 3.67578e8 − 1.02580e8i
11.14 −97.3155 + 348.714i −11352.9 − 502.937i −112131. − 67870.6i − 1.05410e6i 1.28019e6 − 3.90996e6i − 199128.i 3.45796e7 − 3.24970e7i 1.28634e8 + 1.14195e7i 3.67578e8 + 1.02580e8i
11.15 −12.6455 − 361.818i 2019.11 − 11183.2i −130752. + 9150.75i − 565044.i −4.07180e6 − 589135.i − 1.52677e7i 4.96433e6 + 4.71927e7i −1.20987e8 − 4.51602e7i −2.04443e8 + 7.14527e6i
11.16 −12.6455 + 361.818i 2019.11 + 11183.2i −130752. − 9150.75i 565044.i −4.07180e6 + 589135.i 1.52677e7i 4.96433e6 − 4.71927e7i −1.20987e8 + 4.51602e7i −2.04443e8 − 7.14527e6i
11.17 12.6455 − 361.818i −2019.11 + 11183.2i −130752. − 9150.75i − 565044.i 4.02074e6 + 871969.i 1.52677e7i −4.96433e6 + 4.71927e7i −1.20987e8 − 4.51602e7i −2.04443e8 − 7.14527e6i
11.18 12.6455 + 361.818i −2019.11 − 11183.2i −130752. + 9150.75i 565044.i 4.02074e6 − 871969.i − 1.52677e7i −4.96433e6 − 4.71927e7i −1.20987e8 + 4.51602e7i −2.04443e8 + 7.14527e6i
11.19 97.3155 − 348.714i 11352.9 − 502.937i −112131. − 67870.6i 1.05410e6i 929427. − 4.00785e6i − 199128.i −3.45796e7 + 3.24970e7i 1.28634e8 − 1.14195e7i 3.67578e8 + 1.02580e8i
11.20 97.3155 + 348.714i 11352.9 + 502.937i −112131. + 67870.6i − 1.05410e6i 929427. + 4.00785e6i 199128.i −3.45796e7 − 3.24970e7i 1.28634e8 + 1.14195e7i 3.67578e8 − 1.02580e8i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.32
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 12.18.b.a ✓ 32
3.b odd 2 1 inner 12.18.b.a ✓ 32
4.b odd 2 1 inner 12.18.b.a ✓ 32
12.b even 2 1 inner 12.18.b.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.18.b.a ✓ 32 1.a even 1 1 trivial
12.18.b.a ✓ 32 3.b odd 2 1 inner
12.18.b.a ✓ 32 4.b odd 2 1 inner
12.18.b.a ✓ 32 12.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(12, [\chi])\).