Properties

Label 12.9.c.b
Level 12
Weight 9
Character orbit 12.c
Analytic conductor 4.889
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.88854332073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-110}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{-110}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -51 + \beta ) q^{3} \) \( -18 \beta q^{5} \) \( -3094 q^{7} \) \( + ( -1359 - 102 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -51 + \beta ) q^{3} \) \( -18 \beta q^{5} \) \( -3094 q^{7} \) \( + ( -1359 - 102 \beta ) q^{9} \) \( -18 \beta q^{11} \) \( -7294 q^{13} \) \( + ( 71280 + 918 \beta ) q^{15} \) \( + 936 \beta q^{17} \) \( -80326 q^{19} \) \( + ( 157794 - 3094 \beta ) q^{21} \) \( -1548 \beta q^{23} \) \( -892415 q^{25} \) \( + ( 473229 + 3843 \beta ) q^{27} \) \( -13734 \beta q^{29} \) \( + 435914 q^{31} \) \( + ( 71280 + 918 \beta ) q^{33} \) \( + 55692 \beta q^{35} \) \( + 1159298 q^{37} \) \( + ( 371994 - 7294 \beta ) q^{39} \) \( -43164 \beta q^{41} \) \( + 990266 q^{43} \) \( + ( -7270560 + 24462 \beta ) q^{45} \) \( -106488 \beta q^{47} \) \( + 3808035 q^{49} \) \( + ( -3706560 - 47736 \beta ) q^{51} \) \( + 160038 \beta q^{53} \) \( -1283040 q^{55} \) \( + ( 4096626 - 80326 \beta ) q^{57} \) \( + 25398 \beta q^{59} \) \( + 19369154 q^{61} \) \( + ( 4204746 + 315588 \beta ) q^{63} \) \( + 131292 \beta q^{65} \) \( -28024294 q^{67} \) \( + ( 6130080 + 78948 \beta ) q^{69} \) \( -534852 \beta q^{71} \) \( -25230142 q^{73} \) \( + ( 45513165 - 892415 \beta ) q^{75} \) \( + 55692 \beta q^{77} \) \( -63401398 q^{79} \) \( + ( -39352959 + 277236 \beta ) q^{81} \) \( + 755514 \beta q^{83} \) \( + 66718080 q^{85} \) \( + ( 54386640 + 700434 \beta ) q^{87} \) \( -1244196 \beta q^{89} \) \( + 22567636 q^{91} \) \( + ( -22231614 + 435914 \beta ) q^{93} \) \( + 1445868 \beta q^{95} \) \( + 19550306 q^{97} \) \( + ( -7270560 + 24462 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 102q^{3} \) \(\mathstrut -\mathstrut 6188q^{7} \) \(\mathstrut -\mathstrut 2718q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 102q^{3} \) \(\mathstrut -\mathstrut 6188q^{7} \) \(\mathstrut -\mathstrut 2718q^{9} \) \(\mathstrut -\mathstrut 14588q^{13} \) \(\mathstrut +\mathstrut 142560q^{15} \) \(\mathstrut -\mathstrut 160652q^{19} \) \(\mathstrut +\mathstrut 315588q^{21} \) \(\mathstrut -\mathstrut 1784830q^{25} \) \(\mathstrut +\mathstrut 946458q^{27} \) \(\mathstrut +\mathstrut 871828q^{31} \) \(\mathstrut +\mathstrut 142560q^{33} \) \(\mathstrut +\mathstrut 2318596q^{37} \) \(\mathstrut +\mathstrut 743988q^{39} \) \(\mathstrut +\mathstrut 1980532q^{43} \) \(\mathstrut -\mathstrut 14541120q^{45} \) \(\mathstrut +\mathstrut 7616070q^{49} \) \(\mathstrut -\mathstrut 7413120q^{51} \) \(\mathstrut -\mathstrut 2566080q^{55} \) \(\mathstrut +\mathstrut 8193252q^{57} \) \(\mathstrut +\mathstrut 38738308q^{61} \) \(\mathstrut +\mathstrut 8409492q^{63} \) \(\mathstrut -\mathstrut 56048588q^{67} \) \(\mathstrut +\mathstrut 12260160q^{69} \) \(\mathstrut -\mathstrut 50460284q^{73} \) \(\mathstrut +\mathstrut 91026330q^{75} \) \(\mathstrut -\mathstrut 126802796q^{79} \) \(\mathstrut -\mathstrut 78705918q^{81} \) \(\mathstrut +\mathstrut 133436160q^{85} \) \(\mathstrut +\mathstrut 108773280q^{87} \) \(\mathstrut +\mathstrut 45135272q^{91} \) \(\mathstrut -\mathstrut 44463228q^{93} \) \(\mathstrut +\mathstrut 39100612q^{97} \) \(\mathstrut -\mathstrut 14541120q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
10.4881i
10.4881i
0 −51.0000 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 + 6418.71i 0
5.2 0 −51.0000 + 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 6418.71i 0
\(n\): 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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{2} \) \(\mathstrut +\mathstrut 1283040 \) acting on \(S_{9}^{\mathrm{new}}(12, [\chi])\).