Properties

Label 12.21.c.a
Level 12
Weight 21
Character orbit 12.c
Self dual Yes
Analytic conductor 30.422
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(30.4216518123\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 59049q^{3} \) \(\mathstrut -\mathstrut 77335774q^{7} \) \(\mathstrut +\mathstrut 3486784401q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 59049q^{3} \) \(\mathstrut -\mathstrut 77335774q^{7} \) \(\mathstrut +\mathstrut 3486784401q^{9} \) \(\mathstrut +\mathstrut 217393135826q^{13} \) \(\mathstrut -\mathstrut 3843000838126q^{19} \) \(\mathstrut -\mathstrut 4566600118926q^{21} \) \(\mathstrut +\mathstrut 95367431640625q^{25} \) \(\mathstrut +\mathstrut 205891132094649q^{27} \) \(\mathstrut +\mathstrut 793864193940674q^{31} \) \(\mathstrut +\mathstrut 8665815522315698q^{37} \) \(\mathstrut +\mathstrut 12836847277389474q^{39} \) \(\mathstrut +\mathstrut 36742119154315826q^{43} \) \(\mathstrut -\mathstrut 73811444357432925q^{49} \) \(\mathstrut -\mathstrut 226925356490502174q^{57} \) \(\mathstrut -\mathstrut 1387788230779990126q^{61} \) \(\mathstrut -\mathstrut 269653170422461374q^{63} \) \(\mathstrut -\mathstrut 1579433466595168174q^{67} \) \(\mathstrut +\mathstrut 8577821547816235298q^{73} \) \(\mathstrut +\mathstrut 5631351470947265625q^{75} \) \(\mathstrut -\mathstrut 3262829215661625598q^{79} \) \(\mathstrut +\mathstrut 12157665459056928801q^{81} \) \(\mathstrut -\mathstrut 16812266421390839324q^{91} \) \(\mathstrut +\mathstrut 46876886788002859026q^{93} \) \(\mathstrut -\mathstrut 146701748198870395774q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0 59049.0 0 0 0 −7.73358e7 0 3.48678e9 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 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5} \) acting on \(S_{21}^{\mathrm{new}}(12, [\chi])\).