Properties

Label 12.4.b.a
Level 12
Weight 4
Character orbit 12.b
Analytic conductor 0.708
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.708022920069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} - \beta_{3} ) q^{3} \) \( + ( -2 + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} \) \( + ( -6 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6} \) \( + ( \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{8} \) \( + ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} - \beta_{3} ) q^{3} \) \( + ( -2 + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} \) \( + ( -6 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{6} \) \( + ( \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{8} \) \( + ( -3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{9} \) \( + ( 20 - 2 \beta_{2} - 2 \beta_{3} ) q^{10} \) \( + ( 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{11} \) \( + ( 30 - 4 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{12} \) \( -10 q^{13} \) \( + ( -2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{14} \) \( + ( -10 \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{15} \) \( + ( -56 - 4 \beta_{2} - 4 \beta_{3} ) q^{16} \) \( + ( -8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{17} \) \( + ( -60 - 3 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{18} \) \( + ( -9 \beta_{2} - 9 \beta_{3} ) q^{19} \) \( + ( 24 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{20} \) \( + ( 30 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{21} \) \( + ( 60 + 10 \beta_{2} + 10 \beta_{3} ) q^{22} \) \( + ( -28 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{23} \) \( + ( 72 + 28 \beta_{1} - 8 \beta_{3} ) q^{24} \) \( + 45 q^{25} \) \( -10 \beta_{1} q^{26} \) \( + ( 33 \beta_{1} - 27 \beta_{2} + 6 \beta_{3} ) q^{27} \) \( + ( -60 - 2 \beta_{2} - 2 \beta_{3} ) q^{28} \) \( + ( 34 \beta_{1} + 17 \beta_{2} - 17 \beta_{3} ) q^{29} \) \( + ( -60 - 8 \beta_{1} + 6 \beta_{2} - 26 \beta_{3} ) q^{30} \) \( + ( 29 \beta_{2} + 29 \beta_{3} ) q^{31} \) \( + ( -48 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} ) q^{32} \) \( + ( -120 - 30 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{33} \) \( + ( 80 - 8 \beta_{2} - 8 \beta_{3} ) q^{34} \) \( + ( 20 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{35} \) \( + ( 6 - 72 \beta_{1} + 21 \beta_{2} - 27 \beta_{3} ) q^{36} \) \( -130 q^{37} \) \( + ( 18 \beta_{1} - 36 \beta_{2} + 36 \beta_{3} ) q^{38} \) \( + ( 10 \beta_{1} + 10 \beta_{3} ) q^{39} \) \( + ( 80 + 24 \beta_{2} + 24 \beta_{3} ) q^{40} \) \( + ( 28 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} ) q^{41} \) \( + ( 60 + 30 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{42} \) \( + ( -29 \beta_{2} - 29 \beta_{3} ) q^{43} \) \( + ( 40 \beta_{1} + 40 \beta_{2} - 40 \beta_{3} ) q^{44} \) \( + ( 240 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{45} \) \( + ( -168 - 28 \beta_{2} - 28 \beta_{3} ) q^{46} \) \( + ( -56 \beta_{1} + 28 \beta_{2} - 28 \beta_{3} ) q^{47} \) \( + ( -120 + 80 \beta_{1} + 12 \beta_{2} + 44 \beta_{3} ) q^{48} \) \( + 283 q^{49} \) \( + 45 \beta_{1} q^{50} \) \( + ( -40 \beta_{1} + 36 \beta_{2} - 4 \beta_{3} ) q^{51} \) \( + ( 20 - 10 \beta_{2} - 10 \beta_{3} ) q^{52} \) \( + ( -122 \beta_{1} - 61 \beta_{2} + 61 \beta_{3} ) q^{53} \) \( + ( 198 + 21 \beta_{1} - 9 \beta_{2} + 75 \beta_{3} ) q^{54} \) \( + ( -40 \beta_{2} - 40 \beta_{3} ) q^{55} \) \( + ( -56 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{56} \) \( + ( -270 + 54 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{57} \) \( + ( -340 + 34 \beta_{2} + 34 \beta_{3} ) q^{58} \) \( + ( 50 \beta_{1} - 25 \beta_{2} + 25 \beta_{3} ) q^{59} \) \( + ( -240 - 40 \beta_{1} - 48 \beta_{2} + 32 \beta_{3} ) q^{60} \) \( -442 q^{61} \) \( + ( -58 \beta_{1} + 116 \beta_{2} - 116 \beta_{3} ) q^{62} \) \( + ( -60 \beta_{1} + 27 \beta_{2} - 33 \beta_{3} ) q^{63} \) \( + ( 352 - 48 \beta_{2} - 48 \beta_{3} ) q^{64} \) \( + ( 20 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{65} \) \( + ( 300 - 120 \beta_{1} - 30 \beta_{2} - 30 \beta_{3} ) q^{66} \) \( + ( 95 \beta_{2} + 95 \beta_{3} ) q^{67} \) \( + ( 96 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{68} \) \( + ( 336 + 84 \beta_{1} + 42 \beta_{2} - 42 \beta_{3} ) q^{69} \) \( + ( 120 + 20 \beta_{2} + 20 \beta_{3} ) q^{70} \) \( + ( 300 \beta_{1} - 150 \beta_{2} + 150 \beta_{3} ) q^{71} \) \( + ( -240 + 12 \beta_{1} - 84 \beta_{2} - 60 \beta_{3} ) q^{72} \) \( + 410 q^{73} \) \( -130 \beta_{1} q^{74} \) \( + ( -45 \beta_{1} - 45 \beta_{3} ) q^{75} \) \( + ( 540 + 18 \beta_{2} + 18 \beta_{3} ) q^{76} \) \( + ( 60 \beta_{1} + 30 \beta_{2} - 30 \beta_{3} ) q^{77} \) \( + ( 60 - 10 \beta_{1} + 30 \beta_{2} - 10 \beta_{3} ) q^{78} \) \( + ( -11 \beta_{2} - 11 \beta_{3} ) q^{79} \) \( + ( 32 \beta_{1} + 96 \beta_{2} - 96 \beta_{3} ) q^{80} \) \( + ( -711 - 36 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{81} \) \( + ( -280 + 28 \beta_{2} + 28 \beta_{3} ) q^{82} \) \( + ( -362 \beta_{1} + 181 \beta_{2} - 181 \beta_{3} ) q^{83} \) \( + ( -60 + 72 \beta_{1} + 6 \beta_{2} + 54 \beta_{3} ) q^{84} \) \( -320 q^{85} \) \( + ( 58 \beta_{1} - 116 \beta_{2} + 116 \beta_{3} ) q^{86} \) \( + ( 170 \beta_{1} - 153 \beta_{2} + 17 \beta_{3} ) q^{87} \) \( + ( -720 + 40 \beta_{2} + 40 \beta_{3} ) q^{88} \) \( + ( -188 \beta_{1} - 94 \beta_{2} + 94 \beta_{3} ) q^{89} \) \( + ( -60 + 240 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{90} \) \( + ( -10 \beta_{2} - 10 \beta_{3} ) q^{91} \) \( + ( -112 \beta_{1} - 112 \beta_{2} + 112 \beta_{3} ) q^{92} \) \( + ( 870 - 174 \beta_{1} - 87 \beta_{2} + 87 \beta_{3} ) q^{93} \) \( + ( -336 - 56 \beta_{2} - 56 \beta_{3} ) q^{94} \) \( + ( -180 \beta_{1} + 90 \beta_{2} - 90 \beta_{3} ) q^{95} \) \( + ( 96 - 176 \beta_{1} + 192 \beta_{2} - 32 \beta_{3} ) q^{96} \) \( + 770 q^{97} \) \( + 283 \beta_{1} q^{98} \) \( + ( -30 \beta_{1} + 135 \beta_{2} + 105 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut 80q^{10} \) \(\mathstrut +\mathstrut 120q^{12} \) \(\mathstrut -\mathstrut 40q^{13} \) \(\mathstrut -\mathstrut 224q^{16} \) \(\mathstrut -\mathstrut 240q^{18} \) \(\mathstrut +\mathstrut 120q^{21} \) \(\mathstrut +\mathstrut 240q^{22} \) \(\mathstrut +\mathstrut 288q^{24} \) \(\mathstrut +\mathstrut 180q^{25} \) \(\mathstrut -\mathstrut 240q^{28} \) \(\mathstrut -\mathstrut 240q^{30} \) \(\mathstrut -\mathstrut 480q^{33} \) \(\mathstrut +\mathstrut 320q^{34} \) \(\mathstrut +\mathstrut 24q^{36} \) \(\mathstrut -\mathstrut 520q^{37} \) \(\mathstrut +\mathstrut 320q^{40} \) \(\mathstrut +\mathstrut 240q^{42} \) \(\mathstrut +\mathstrut 960q^{45} \) \(\mathstrut -\mathstrut 672q^{46} \) \(\mathstrut -\mathstrut 480q^{48} \) \(\mathstrut +\mathstrut 1132q^{49} \) \(\mathstrut +\mathstrut 80q^{52} \) \(\mathstrut +\mathstrut 792q^{54} \) \(\mathstrut -\mathstrut 1080q^{57} \) \(\mathstrut -\mathstrut 1360q^{58} \) \(\mathstrut -\mathstrut 960q^{60} \) \(\mathstrut -\mathstrut 1768q^{61} \) \(\mathstrut +\mathstrut 1408q^{64} \) \(\mathstrut +\mathstrut 1200q^{66} \) \(\mathstrut +\mathstrut 1344q^{69} \) \(\mathstrut +\mathstrut 480q^{70} \) \(\mathstrut -\mathstrut 960q^{72} \) \(\mathstrut +\mathstrut 1640q^{73} \) \(\mathstrut +\mathstrut 2160q^{76} \) \(\mathstrut +\mathstrut 240q^{78} \) \(\mathstrut -\mathstrut 2844q^{81} \) \(\mathstrut -\mathstrut 1120q^{82} \) \(\mathstrut -\mathstrut 240q^{84} \) \(\mathstrut -\mathstrut 1280q^{85} \) \(\mathstrut -\mathstrut 2880q^{88} \) \(\mathstrut -\mathstrut 240q^{90} \) \(\mathstrut +\mathstrut 3480q^{93} \) \(\mathstrut -\mathstrut 1344q^{94} \) \(\mathstrut +\mathstrut 384q^{96} \) \(\mathstrut +\mathstrut 3080q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 2 \nu^{2} + \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 2 \nu^{2} - \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 1.11803i
−0.866025 + 1.11803i
0.866025 1.11803i
0.866025 + 1.11803i
−1.73205 2.23607i 3.46410 3.87298i −2.00000 + 7.74597i 8.94427i −14.6603 1.03776i 7.74597i 20.7846 8.94427i −3.00000 26.8328i 20.0000 15.4919i
11.2 −1.73205 + 2.23607i 3.46410 + 3.87298i −2.00000 7.74597i 8.94427i −14.6603 + 1.03776i 7.74597i 20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 + 15.4919i
11.3 1.73205 2.23607i −3.46410 + 3.87298i −2.00000 7.74597i 8.94427i 2.66025 + 14.4542i 7.74597i −20.7846 8.94427i −3.00000 26.8328i 20.0000 + 15.4919i
11.4 1.73205 + 2.23607i −3.46410 3.87298i −2.00000 + 7.74597i 8.94427i 2.66025 14.4542i 7.74597i −20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 15.4919i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
4.b Odd 1 yes
12.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(12, [\chi])\).