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

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(21.9866504813\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut -\mathstrut 54808q^{4} \) \(\mathstrut -\mathstrut 4960776q^{6} \) \(\mathstrut +\mathstrut 79874976q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(32q \) \(\mathstrut -\mathstrut 54808q^{4} \) \(\mathstrut -\mathstrut 4960776q^{6} \) \(\mathstrut +\mathstrut 79874976q^{9} \) \(\mathstrut +\mathstrut 46839088q^{10} \) \(\mathstrut +\mathstrut 2598308520q^{12} \) \(\mathstrut +\mathstrut 221287360q^{13} \) \(\mathstrut -\mathstrut 41721285088q^{16} \) \(\mathstrut +\mathstrut 27568791600q^{18} \) \(\mathstrut +\mathstrut 128365169856q^{21} \) \(\mathstrut -\mathstrut 493958165040q^{22} \) \(\mathstrut -\mathstrut 253565784288q^{24} \) \(\mathstrut -\mathstrut 3063689463648q^{25} \) \(\mathstrut -\mathstrut 2695436033040q^{28} \) \(\mathstrut -\mathstrut 2169979068432q^{30} \) \(\mathstrut -\mathstrut 17644793625600q^{33} \) \(\mathstrut +\mathstrut 6944208632512q^{34} \) \(\mathstrut -\mathstrut 12909384040056q^{36} \) \(\mathstrut -\mathstrut 7970760699200q^{37} \) \(\mathstrut +\mathstrut 21066454663744q^{40} \) \(\mathstrut +\mathstrut 55835180334480q^{42} \) \(\mathstrut +\mathstrut 154013263804416q^{45} \) \(\mathstrut +\mathstrut 235918828815264q^{46} \) \(\mathstrut +\mathstrut 299462725247520q^{48} \) \(\mathstrut -\mathstrut 248591781347488q^{49} \) \(\mathstrut +\mathstrut 544732086739120q^{52} \) \(\mathstrut +\mathstrut 487051858173384q^{54} \) \(\mathstrut -\mathstrut 2151740586697920q^{57} \) \(\mathstrut +\mathstrut 230877304263760q^{58} \) \(\mathstrut -\mathstrut 345098683994304q^{60} \) \(\mathstrut -\mathstrut 4956543779318336q^{61} \) \(\mathstrut +\mathstrut 1003775054518400q^{64} \) \(\mathstrut -\mathstrut 2371521654206832q^{66} \) \(\mathstrut +\mathstrut 891375297976320q^{69} \) \(\mathstrut -\mathstrut 949041839387232q^{70} \) \(\mathstrut +\mathstrut 5817303713532480q^{72} \) \(\mathstrut -\mathstrut 4887976236943040q^{73} \) \(\mathstrut -\mathstrut 2051936325547056q^{76} \) \(\mathstrut +\mathstrut 22261766802491280q^{78} \) \(\mathstrut +\mathstrut 28021308982751520q^{81} \) \(\mathstrut +\mathstrut 7596239161270240q^{82} \) \(\mathstrut -\mathstrut 8356700998055568q^{84} \) \(\mathstrut +\mathstrut 59910783041540096q^{85} \) \(\mathstrut -\mathstrut 18469777603302720q^{88} \) \(\mathstrut -\mathstrut 68814047068261392q^{90} \) \(\mathstrut +\mathstrut 5961488029786560q^{93} \) \(\mathstrut -\mathstrut 61915224567506112q^{94} \) \(\mathstrut -\mathstrut 45410293501908864q^{96} \) \(\mathstrut +\mathstrut 9062030716970560q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −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

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

Hecke kernels

There are no other newforms in \(S_{18}^{\mathrm{new}}(12, [\chi])\).