Properties

Label 12.11.c.b
Level 12
Weight 11
Character orbit 12.c
Analytic conductor 7.624
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(7.62428703208\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 36\sqrt{-35}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 117 + \beta ) q^{3} \) \( + 6 \beta q^{5} \) \( -10318 q^{7} \) \( + ( -31671 + 234 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 117 + \beta ) q^{3} \) \( + 6 \beta q^{5} \) \( -10318 q^{7} \) \( + ( -31671 + 234 \beta ) q^{9} \) \( + 1374 \beta q^{11} \) \( -256822 q^{13} \) \( + ( -272160 + 702 \beta ) q^{15} \) \( -2616 \beta q^{17} \) \( + 3196106 q^{19} \) \( + ( -1207206 - 10318 \beta ) q^{21} \) \( -39228 \beta q^{23} \) \( + 8132665 q^{25} \) \( + ( -14319747 - 4293 \beta ) q^{27} \) \( + 147858 \beta q^{29} \) \( + 23140994 q^{31} \) \( + ( -62324640 + 160758 \beta ) q^{33} \) \( -61908 \beta q^{35} \) \( + 29797946 q^{37} \) \( + ( -30048174 - 256822 \beta ) q^{39} \) \( + 4452 \beta q^{41} \) \( + 247522778 q^{43} \) \( + ( -63685440 - 190026 \beta ) q^{45} \) \( -1547352 \beta q^{47} \) \( -176014125 q^{49} \) \( + ( 118661760 - 306072 \beta ) q^{51} \) \( + 2588478 \beta q^{53} \) \( -373947840 q^{55} \) \( + ( 373944402 + 3196106 \beta ) q^{57} \) \( + 1686054 \beta q^{59} \) \( -1054839766 q^{61} \) \( + ( 326781378 - 2414412 \beta ) q^{63} \) \( -1540932 \beta q^{65} \) \( -361186198 q^{67} \) \( + ( 1779382080 - 4589676 \beta ) q^{69} \) \( -4373844 \beta q^{71} \) \( + 374437394 q^{73} \) \( + ( 951521805 + 8132665 \beta ) q^{75} \) \( -14176932 \beta q^{77} \) \( -113914462 q^{79} \) \( + ( -1480679919 - 14822028 \beta ) q^{81} \) \( + 23058234 \beta q^{83} \) \( + 711970560 q^{85} \) \( + ( -6706838880 + 17299386 \beta ) q^{87} \) \( + 15410172 \beta q^{89} \) \( + 2649889396 q^{91} \) \( + ( 2707496298 + 23140994 \beta ) q^{93} \) \( + 19176636 \beta q^{95} \) \( + 2809917122 q^{97} \) \( + ( -14583965760 - 43515954 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 234q^{3} \) \(\mathstrut -\mathstrut 20636q^{7} \) \(\mathstrut -\mathstrut 63342q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 234q^{3} \) \(\mathstrut -\mathstrut 20636q^{7} \) \(\mathstrut -\mathstrut 63342q^{9} \) \(\mathstrut -\mathstrut 513644q^{13} \) \(\mathstrut -\mathstrut 544320q^{15} \) \(\mathstrut +\mathstrut 6392212q^{19} \) \(\mathstrut -\mathstrut 2414412q^{21} \) \(\mathstrut +\mathstrut 16265330q^{25} \) \(\mathstrut -\mathstrut 28639494q^{27} \) \(\mathstrut +\mathstrut 46281988q^{31} \) \(\mathstrut -\mathstrut 124649280q^{33} \) \(\mathstrut +\mathstrut 59595892q^{37} \) \(\mathstrut -\mathstrut 60096348q^{39} \) \(\mathstrut +\mathstrut 495045556q^{43} \) \(\mathstrut -\mathstrut 127370880q^{45} \) \(\mathstrut -\mathstrut 352028250q^{49} \) \(\mathstrut +\mathstrut 237323520q^{51} \) \(\mathstrut -\mathstrut 747895680q^{55} \) \(\mathstrut +\mathstrut 747888804q^{57} \) \(\mathstrut -\mathstrut 2109679532q^{61} \) \(\mathstrut +\mathstrut 653562756q^{63} \) \(\mathstrut -\mathstrut 722372396q^{67} \) \(\mathstrut +\mathstrut 3558764160q^{69} \) \(\mathstrut +\mathstrut 748874788q^{73} \) \(\mathstrut +\mathstrut 1903043610q^{75} \) \(\mathstrut -\mathstrut 227828924q^{79} \) \(\mathstrut -\mathstrut 2961359838q^{81} \) \(\mathstrut +\mathstrut 1423941120q^{85} \) \(\mathstrut -\mathstrut 13413677760q^{87} \) \(\mathstrut +\mathstrut 5299778792q^{91} \) \(\mathstrut +\mathstrut 5414992596q^{93} \) \(\mathstrut +\mathstrut 5619834244q^{97} \) \(\mathstrut -\mathstrut 29167931520q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0.500000 2.95804i
0.500000 + 2.95804i
0 117.000 212.979i 0 1277.87i 0 −10318.0 0 −31671.0 49837.1i 0
5.2 0 117.000 + 212.979i 0 1277.87i 0 −10318.0 0 −31671.0 + 49837.1i 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 \(T_{5}^{2} \) \(\mathstrut +\mathstrut 1632960 \) acting on \(S_{11}^{\mathrm{new}}(12, [\chi])\).