Properties

Label 12.20.b.a
Level 12
Weight 20
Character orbit 12.b
Analytic conductor 27.458
Analytic rank 0
Dimension 36
CM No

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(27.4580035868\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 47880q^{4} + 21771144q^{6} - 809473644q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 47880q^{4} + 21771144q^{6} - 809473644q^{9} - 4162950000q^{10} - 2268767880q^{12} - 25913431656q^{13} + 791689296672q^{16} + 2852271258192q^{18} - 1650825908424q^{21} + 7247870602416q^{22} + 23719435803936q^{24} - 118879690462380q^{25} - 7629081673968q^{28} + 37456727138640q^{30} + 529254701828640q^{33} - 248241675948480q^{34} - 1177548745309416q^{36} + 1847838600168120q^{37} - 40433256181440q^{40} - 2709671988768336q^{42} - 4365410114088000q^{45} + 17206125364795104q^{46} + 1561106052682272q^{48} - 49399737041580084q^{49} + 16558671470242896q^{52} + 40087412948587896q^{54} + 44803667812766472q^{57} - 34645536255764880q^{58} + 50260381053157440q^{60} + 174742350239997144q^{61} - 139766490424918656q^{64} - 87772097712685968q^{66} - 957190369821780672q^{69} + 244663962371376480q^{70} + 54134736422921280q^{72} - 1445122541399481432q^{73} + 2744130151937256048q^{76} + 1125465917961999024q^{78} + 1033298241353042436q^{81} + 1064985418859680800q^{82} + 1920709484261873424q^{84} - 1802899595088057600q^{85} - 9209792381828477760q^{88} - 5194944179528261040q^{90} + 6719944190337314136q^{93} - 3835186345170303552q^{94} - 11947184361208227456q^{96} + 12765226159947004488q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −723.867 17.4378i 2865.95 33971.3i 523680. + 25245.3i 5.49376e6i −2.66695e6 + 2.45407e7i 1.05257e8i −3.78635e8 2.74060e7i −1.14583e9 1.94720e8i −9.57989e7 + 3.97675e9i
11.2 −723.867 + 17.4378i 2865.95 + 33971.3i 523680. 25245.3i 5.49376e6i −2.66695e6 2.45407e7i 1.05257e8i −3.78635e8 + 2.74060e7i −1.14583e9 + 1.94720e8i −9.57989e7 3.97675e9i
11.3 −706.401 159.015i −32549.6 10138.3i 473717. + 224656.i 5.54418e6i 2.13809e7 + 1.23376e7i 1.49965e8i −2.98910e8 2.34025e8i 9.56692e8 + 6.59995e8i 8.81605e8 3.91641e9i
11.4 −706.401 + 159.015i −32549.6 + 10138.3i 473717. 224656.i 5.54418e6i 2.13809e7 1.23376e7i 1.49965e8i −2.98910e8 + 2.34025e8i 9.56692e8 6.59995e8i 8.81605e8 + 3.91641e9i
11.5 −641.248 336.287i 33097.3 8174.89i 298110. + 431287.i 2.18420e6i −2.39727e7 5.88806e6i 4.59862e7i −4.61264e7 3.76812e8i 1.02860e9 5.41134e8i 7.34518e8 1.40061e9i
11.6 −641.248 + 336.287i 33097.3 + 8174.89i 298110. 431287.i 2.18420e6i −2.39727e7 + 5.88806e6i 4.59862e7i −4.61264e7 + 3.76812e8i 1.02860e9 + 5.41134e8i 7.34518e8 + 1.40061e9i
11.7 −627.004 362.153i 8684.36 + 32967.3i 261979. + 454142.i 6.03352e6i 6.49408e6 2.38157e7i 9.28886e7i 207167. 3.79625e8i −1.01143e9 + 5.72600e8i −2.18506e9 + 3.78304e9i
11.8 −627.004 + 362.153i 8684.36 32967.3i 261979. 454142.i 6.03352e6i 6.49408e6 + 2.38157e7i 9.28886e7i 207167. + 3.79625e8i −1.01143e9 5.72600e8i −2.18506e9 3.78304e9i
11.9 −519.517 504.371i −19784.5 27763.9i 15507.7 + 524059.i 1.34492e6i −3.72492e6 + 2.44026e7i 1.72563e8i 2.56263e8 2.80079e8i −3.79408e8 + 1.09859e9i −6.78338e8 + 6.98708e8i
11.10 −519.517 + 504.371i −19784.5 + 27763.9i 15507.7 524059.i 1.34492e6i −3.72492e6 2.44026e7i 1.72563e8i 2.56263e8 + 2.80079e8i −3.79408e8 1.09859e9i −6.78338e8 6.98708e8i
11.11 −395.354 606.616i −27131.3 + 20643.4i −211678. + 479656.i 730760.i 2.32491e7 + 8.29684e6i 7.02914e6i 3.74655e8 6.12270e7i 3.09959e8 1.12017e9i 4.43291e8 2.88909e8i
11.12 −395.354 + 606.616i −27131.3 20643.4i −211678. 479656.i 730760.i 2.32491e7 8.29684e6i 7.02914e6i 3.74655e8 + 6.12270e7i 3.09959e8 + 1.12017e9i 4.43291e8 + 2.88909e8i
11.13 −246.303 680.898i 12845.9 31579.2i −402957. + 335415.i 225554.i −2.46662e7 968699.i 9.84546e7i 3.27633e8 + 1.91759e8i −8.32227e8 8.11326e8i 1.53579e8 5.55547e7i
11.14 −246.303 + 680.898i 12845.9 + 31579.2i −402957. 335415.i 225554.i −2.46662e7 + 968699.i 9.84546e7i 3.27633e8 1.91759e8i −8.32227e8 + 8.11326e8i 1.53579e8 + 5.55547e7i
11.15 −183.227 700.511i 22981.7 + 25181.4i −457144. + 256705.i 6.84881e6i 1.34289e7 2.07129e7i 8.10387e7i 2.63586e8 + 2.73199e8i −1.05940e8 + 1.15742e9i 4.79767e9 1.25489e9i
11.16 −183.227 + 700.511i 22981.7 25181.4i −457144. 256705.i 6.84881e6i 1.34289e7 + 2.07129e7i 8.10387e7i 2.63586e8 2.73199e8i −1.05940e8 1.15742e9i 4.79767e9 + 1.25489e9i
11.17 −74.5111 720.233i 32706.8 + 9618.97i −513184. + 107331.i 7.07022e6i 4.49088e6 2.42733e7i 1.56654e8i 1.15541e8 + 3.61615e8i 9.77212e8 + 6.29212e8i −5.09221e9 + 5.26810e8i
11.18 −74.5111 + 720.233i 32706.8 9618.97i −513184. 107331.i 7.07022e6i 4.49088e6 + 2.42733e7i 1.56654e8i 1.15541e8 3.61615e8i 9.77212e8 6.29212e8i −5.09221e9 5.26810e8i
11.19 74.5111 720.233i −32706.8 9618.97i −513184. 107331.i 7.07022e6i −9.36493e6 + 2.28398e7i 1.56654e8i −1.15541e8 + 3.61615e8i 9.77212e8 + 6.29212e8i −5.09221e9 5.26810e8i
11.20 74.5111 + 720.233i −32706.8 + 9618.97i −513184. + 107331.i 7.07022e6i −9.36493e6 2.28398e7i 1.56654e8i −1.15541e8 3.61615e8i 9.77212e8 6.29212e8i −5.09221e9 + 5.26810e8i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.36
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{20}^{\mathrm{new}}(12, [\chi])\).