Properties

Label 12.3.d.a
Level 12
Weight 3
Character orbit 12.d
Analytic conductor 0.327
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.326976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + \beta q^{3} \) \( + ( -2 + 2 \beta ) q^{4} \) \( -2 q^{5} \) \( + ( 3 - \beta ) q^{6} \) \( -4 \beta q^{7} \) \( + 8 q^{8} \) \( -3 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta ) q^{2} \) \( + \beta q^{3} \) \( + ( -2 + 2 \beta ) q^{4} \) \( -2 q^{5} \) \( + ( 3 - \beta ) q^{6} \) \( -4 \beta q^{7} \) \( + 8 q^{8} \) \( -3 q^{9} \) \( + ( 2 + 2 \beta ) q^{10} \) \( + 4 \beta q^{11} \) \( + ( -6 - 2 \beta ) q^{12} \) \( + 2 q^{13} \) \( + ( -12 + 4 \beta ) q^{14} \) \( -2 \beta q^{15} \) \( + ( -8 - 8 \beta ) q^{16} \) \( + 10 q^{17} \) \( + ( 3 + 3 \beta ) q^{18} \) \( + 12 \beta q^{19} \) \( + ( 4 - 4 \beta ) q^{20} \) \( + 12 q^{21} \) \( + ( 12 - 4 \beta ) q^{22} \) \( -16 \beta q^{23} \) \( + 8 \beta q^{24} \) \( -21 q^{25} \) \( + ( -2 - 2 \beta ) q^{26} \) \( -3 \beta q^{27} \) \( + ( 24 + 8 \beta ) q^{28} \) \( -26 q^{29} \) \( + ( -6 + 2 \beta ) q^{30} \) \( + 4 \beta q^{31} \) \( + ( -16 + 16 \beta ) q^{32} \) \( -12 q^{33} \) \( + ( -10 - 10 \beta ) q^{34} \) \( + 8 \beta q^{35} \) \( + ( 6 - 6 \beta ) q^{36} \) \( + 26 q^{37} \) \( + ( 36 - 12 \beta ) q^{38} \) \( + 2 \beta q^{39} \) \( -16 q^{40} \) \( + 58 q^{41} \) \( + ( -12 - 12 \beta ) q^{42} \) \( -28 \beta q^{43} \) \( + ( -24 - 8 \beta ) q^{44} \) \( + 6 q^{45} \) \( + ( -48 + 16 \beta ) q^{46} \) \( + 40 \beta q^{47} \) \( + ( 24 - 8 \beta ) q^{48} \) \(+ q^{49}\) \( + ( 21 + 21 \beta ) q^{50} \) \( + 10 \beta q^{51} \) \( + ( -4 + 4 \beta ) q^{52} \) \( -74 q^{53} \) \( + ( -9 + 3 \beta ) q^{54} \) \( -8 \beta q^{55} \) \( -32 \beta q^{56} \) \( -36 q^{57} \) \( + ( 26 + 26 \beta ) q^{58} \) \( -52 \beta q^{59} \) \( + ( 12 + 4 \beta ) q^{60} \) \( + 26 q^{61} \) \( + ( 12 - 4 \beta ) q^{62} \) \( + 12 \beta q^{63} \) \( + 64 q^{64} \) \( -4 q^{65} \) \( + ( 12 + 12 \beta ) q^{66} \) \( + 4 \beta q^{67} \) \( + ( -20 + 20 \beta ) q^{68} \) \( + 48 q^{69} \) \( + ( 24 - 8 \beta ) q^{70} \) \( -24 q^{72} \) \( -46 q^{73} \) \( + ( -26 - 26 \beta ) q^{74} \) \( -21 \beta q^{75} \) \( + ( -72 - 24 \beta ) q^{76} \) \( + 48 q^{77} \) \( + ( 6 - 2 \beta ) q^{78} \) \( + 68 \beta q^{79} \) \( + ( 16 + 16 \beta ) q^{80} \) \( + 9 q^{81} \) \( + ( -58 - 58 \beta ) q^{82} \) \( + 28 \beta q^{83} \) \( + ( -24 + 24 \beta ) q^{84} \) \( -20 q^{85} \) \( + ( -84 + 28 \beta ) q^{86} \) \( -26 \beta q^{87} \) \( + 32 \beta q^{88} \) \( + 82 q^{89} \) \( + ( -6 - 6 \beta ) q^{90} \) \( -8 \beta q^{91} \) \( + ( 96 + 32 \beta ) q^{92} \) \( -12 q^{93} \) \( + ( 120 - 40 \beta ) q^{94} \) \( -24 \beta q^{95} \) \( + ( -48 - 16 \beta ) q^{96} \) \( + 2 q^{97} \) \( + ( -1 - \beta ) q^{98} \) \( -12 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 6q^{6} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 6q^{6} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 12q^{12} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 24q^{14} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 20q^{17} \) \(\mathstrut +\mathstrut 6q^{18} \) \(\mathstrut +\mathstrut 8q^{20} \) \(\mathstrut +\mathstrut 24q^{21} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut -\mathstrut 42q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 48q^{28} \) \(\mathstrut -\mathstrut 52q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut -\mathstrut 32q^{32} \) \(\mathstrut -\mathstrut 24q^{33} \) \(\mathstrut -\mathstrut 20q^{34} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 52q^{37} \) \(\mathstrut +\mathstrut 72q^{38} \) \(\mathstrut -\mathstrut 32q^{40} \) \(\mathstrut +\mathstrut 116q^{41} \) \(\mathstrut -\mathstrut 24q^{42} \) \(\mathstrut -\mathstrut 48q^{44} \) \(\mathstrut +\mathstrut 12q^{45} \) \(\mathstrut -\mathstrut 96q^{46} \) \(\mathstrut +\mathstrut 48q^{48} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 42q^{50} \) \(\mathstrut -\mathstrut 8q^{52} \) \(\mathstrut -\mathstrut 148q^{53} \) \(\mathstrut -\mathstrut 18q^{54} \) \(\mathstrut -\mathstrut 72q^{57} \) \(\mathstrut +\mathstrut 52q^{58} \) \(\mathstrut +\mathstrut 24q^{60} \) \(\mathstrut +\mathstrut 52q^{61} \) \(\mathstrut +\mathstrut 24q^{62} \) \(\mathstrut +\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 24q^{66} \) \(\mathstrut -\mathstrut 40q^{68} \) \(\mathstrut +\mathstrut 96q^{69} \) \(\mathstrut +\mathstrut 48q^{70} \) \(\mathstrut -\mathstrut 48q^{72} \) \(\mathstrut -\mathstrut 92q^{73} \) \(\mathstrut -\mathstrut 52q^{74} \) \(\mathstrut -\mathstrut 144q^{76} \) \(\mathstrut +\mathstrut 96q^{77} \) \(\mathstrut +\mathstrut 12q^{78} \) \(\mathstrut +\mathstrut 32q^{80} \) \(\mathstrut +\mathstrut 18q^{81} \) \(\mathstrut -\mathstrut 116q^{82} \) \(\mathstrut -\mathstrut 48q^{84} \) \(\mathstrut -\mathstrut 40q^{85} \) \(\mathstrut -\mathstrut 168q^{86} \) \(\mathstrut +\mathstrut 164q^{89} \) \(\mathstrut -\mathstrut 12q^{90} \) \(\mathstrut +\mathstrut 192q^{92} \) \(\mathstrut -\mathstrut 24q^{93} \) \(\mathstrut +\mathstrut 240q^{94} \) \(\mathstrut -\mathstrut 96q^{96} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 1.73205i −2.00000 + 3.46410i −2.00000 3.00000 1.73205i 6.92820i 8.00000 −3.00000 2.00000 + 3.46410i
7.2 −1.00000 + 1.73205i 1.73205i −2.00000 3.46410i −2.00000 3.00000 + 1.73205i 6.92820i 8.00000 −3.00000 2.00000 3.46410i
\(n\): 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
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(12, [\chi])\).