Properties

Label 12.16.b.a
Level 12
Weight 16
Character orbit 12.b
Analytic conductor 17.123
Analytic rank 0
Dimension 28
CM No

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 16 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(17.123220612\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 26968q^{4} + 823656q^{6} - 3812052q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 26968q^{4} + 823656q^{6} - 3812052q^{9} - 43831600q^{10} - 226414248q^{12} + 124527272q^{13} - 1459325408q^{16} - 4234777584q^{18} - 7261350648q^{21} - 2533670160q^{22} - 18487781856q^{24} - 146804950740q^{25} + 5481093840q^{28} + 8237058960q^{30} + 204574669728q^{33} + 450165745472q^{34} + 232631927160q^{36} + 386069193224q^{37} + 1945471012160q^{40} + 1938106219632q^{42} - 5007113912640q^{45} + 2270790222432q^{46} + 5846474725152q^{48} - 18480860963084q^{49} - 6113229405424q^{52} + 1626598700568q^{54} + 8085872464056q^{57} - 19395437098192q^{58} - 10924219377600q^{60} - 16392792556696q^{61} + 21633892829056q^{64} + 4928819126448q^{66} + 137029869973056q^{69} + 91772543171040q^{70} + 66924493142592q^{72} + 158451626683736q^{73} - 205265768291280q^{76} + 58364283489648q^{78} + 116126816635836q^{81} - 750029796726880q^{82} - 92605433207856q^{84} - 30099357052160q^{85} - 619691835246912q^{88} - 471443548913520q^{90} - 777661615138584q^{93} + 1301107064491200q^{94} + 525428335287168q^{96} - 824166397720648q^{97} + O(q^{100}) \)

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 −172.746 54.1000i 3105.31 + 2169.32i 26914.4 + 18691.1i 192307.i −419070. 542739.i 366926.i −3.63816e6 4.68489e6i 4.93700e6 + 1.34728e7i −1.04038e7 + 3.32203e7i
11.2 −172.746 + 54.1000i 3105.31 2169.32i 26914.4 18691.1i 192307.i −419070. + 542739.i 366926.i −3.63816e6 + 4.68489e6i 4.93700e6 1.34728e7i −1.04038e7 3.32203e7i
11.3 −171.504 57.9158i −3751.22 526.541i 26059.5 + 19865.6i 172529.i 612856. + 307559.i 3.89480e6i −3.31879e6 4.91630e6i 1.37944e7 + 3.95035e6i 9.99218e6 2.95896e7i
11.4 −171.504 + 57.9158i −3751.22 + 526.541i 26059.5 19865.6i 172529.i 612856. 307559.i 3.89480e6i −3.31879e6 + 4.91630e6i 1.37944e7 3.95035e6i 9.99218e6 + 2.95896e7i
11.5 −166.441 71.1726i −8.61090 3787.99i 22636.9 + 23692.0i 99785.5i −268168. + 631087.i 1.78364e6i −2.08148e6 5.55444e6i −1.43488e7 + 65235.9i −7.10200e6 + 1.66083e7i
11.6 −166.441 + 71.1726i −8.61090 + 3787.99i 22636.9 23692.0i 99785.5i −268168. 631087.i 1.78364e6i −2.08148e6 + 5.55444e6i −1.43488e7 65235.9i −7.10200e6 1.66083e7i
11.7 −127.986 128.014i −692.213 + 3724.21i −7.18349 + 32768.0i 230590.i 565345. 388034.i 3.60781e6i 4.19568e6 4.19292e6i −1.33906e7 5.15589e6i 2.95188e7 2.95123e7i
11.8 −127.986 + 128.014i −692.213 3724.21i −7.18349 32768.0i 230590.i 565345. + 388034.i 3.60781e6i 4.19568e6 + 4.19292e6i −1.33906e7 + 5.15589e6i 2.95188e7 + 2.95123e7i
11.9 −83.3106 160.709i 3555.47 1306.74i −18886.7 + 26777.5i 93360.9i −506212. 462530.i 1.14542e6i 5.87684e6 + 804409.i 1.09338e7 9.29212e6i 1.50039e7 7.77795e6i
11.10 −83.3106 + 160.709i 3555.47 + 1306.74i −18886.7 26777.5i 93360.9i −506212. + 462530.i 1.14542e6i 5.87684e6 804409.i 1.09338e7 + 9.29212e6i 1.50039e7 + 7.77795e6i
11.11 −80.5973 162.087i −2767.36 + 2586.62i −19776.2 + 26127.5i 332581.i 642299. + 240078.i 2.13481e6i 5.82882e6 + 1.09965e6i 967701. 1.43162e7i −5.39069e7 + 2.68051e7i
11.12 −80.5973 + 162.087i −2767.36 2586.62i −19776.2 26127.5i 332581.i 642299. 240078.i 2.13481e6i 5.82882e6 1.09965e6i 967701. + 1.43162e7i −5.39069e7 2.68051e7i
11.13 −35.8416 177.436i −2291.54 3016.25i −30198.8 + 12719.1i 33476.6i −453057. + 514708.i 694274.i 3.33920e6 + 4.90246e6i −3.84657e6 + 1.38237e7i 5.93994e6 1.19985e6i
11.14 −35.8416 + 177.436i −2291.54 + 3016.25i −30198.8 12719.1i 33476.6i −453057. 514708.i 694274.i 3.33920e6 4.90246e6i −3.84657e6 1.38237e7i 5.93994e6 + 1.19985e6i
11.15 35.8416 177.436i 2291.54 + 3016.25i −30198.8 12719.1i 33476.6i 617322. 298494.i 694274.i −3.33920e6 + 4.90246e6i −3.84657e6 + 1.38237e7i 5.93994e6 + 1.19985e6i
11.16 35.8416 + 177.436i 2291.54 3016.25i −30198.8 + 12719.1i 33476.6i 617322. + 298494.i 694274.i −3.33920e6 4.90246e6i −3.84657e6 1.38237e7i 5.93994e6 1.19985e6i
11.17 80.5973 162.087i 2767.36 2586.62i −19776.2 26127.5i 332581.i −196215. 657027.i 2.13481e6i −5.82882e6 + 1.09965e6i 967701. 1.43162e7i −5.39069e7 2.68051e7i
11.18 80.5973 + 162.087i 2767.36 + 2586.62i −19776.2 + 26127.5i 332581.i −196215. + 657027.i 2.13481e6i −5.82882e6 1.09965e6i 967701. + 1.43162e7i −5.39069e7 + 2.68051e7i
11.19 83.3106 160.709i −3555.47 + 1306.74i −18886.7 26777.5i 93360.9i −86203.8 + 680260.i 1.14542e6i −5.87684e6 + 804409.i 1.09338e7 9.29212e6i 1.50039e7 + 7.77795e6i
11.20 83.3106 + 160.709i −3555.47 1306.74i −18886.7 + 26777.5i 93360.9i −86203.8 680260.i 1.14542e6i −5.87684e6 804409.i 1.09338e7 + 9.29212e6i 1.50039e7 7.77795e6i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.28
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 \(S_{16}^{\mathrm{new}}(12, [\chi])\).