Properties

Label 2940.2.q.q
Level $2940$
Weight $2$
Character orbit 2940.q
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} + \beta_{2} q^{5} + \beta_{2} q^{9} + \beta_{1} q^{11} + ( 1 + \beta_{3} ) q^{13} - q^{15} + \beta_{1} q^{17} + 7 \beta_{2} q^{19} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} - q^{27} + ( -6 - \beta_{3} ) q^{29} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( 1 - \beta_{1} + \beta_{2} ) q^{39} -\beta_{3} q^{41} + ( -1 - \beta_{3} ) q^{43} + ( -1 - \beta_{2} ) q^{45} + 6 \beta_{2} q^{47} + ( \beta_{1} + \beta_{3} ) q^{51} -2 \beta_{1} q^{53} + \beta_{3} q^{55} -7 q^{57} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} ) q^{67} -\beta_{3} q^{69} + 3 \beta_{3} q^{71} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{73} -\beta_{2} q^{75} + 11 \beta_{2} q^{79} + ( -1 - \beta_{2} ) q^{81} + ( 6 - \beta_{3} ) q^{83} + \beta_{3} q^{85} + ( -6 + \beta_{1} - 6 \beta_{2} ) q^{87} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{89} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{93} + ( -7 - 7 \beta_{2} ) q^{95} + ( -8 - 2 \beta_{3} ) q^{97} + \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + 4 q^{13} - 4 q^{15} - 14 q^{19} - 2 q^{25} - 4 q^{27} - 24 q^{29} - 2 q^{31} + 2 q^{37} + 2 q^{39} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 28 q^{57} - 12 q^{59} - 8 q^{61} - 2 q^{65} + 2 q^{67} + 10 q^{73} + 2 q^{75} - 22 q^{79} - 2 q^{81} + 24 q^{83} - 12 q^{87} - 12 q^{89} + 2 q^{93} - 14 q^{95} - 32 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 3 \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)\(/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
361.2 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.q.q 4
7.b odd 2 1 420.2.q.d 4
7.c even 3 1 2940.2.a.p 2
7.c even 3 1 inner 2940.2.q.q 4
7.d odd 6 1 420.2.q.d 4
7.d odd 6 1 2940.2.a.r 2
21.c even 2 1 1260.2.s.e 4
21.g even 6 1 1260.2.s.e 4
21.g even 6 1 8820.2.a.bk 2
21.h odd 6 1 8820.2.a.bf 2
28.d even 2 1 1680.2.bg.t 4
28.f even 6 1 1680.2.bg.t 4
35.c odd 2 1 2100.2.q.k 4
35.f even 4 2 2100.2.bc.f 8
35.i odd 6 1 2100.2.q.k 4
35.k even 12 2 2100.2.bc.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 7.b odd 2 1
420.2.q.d 4 7.d odd 6 1
1260.2.s.e 4 21.c even 2 1
1260.2.s.e 4 21.g even 6 1
1680.2.bg.t 4 28.d even 2 1
1680.2.bg.t 4 28.f even 6 1
2100.2.q.k 4 35.c odd 2 1
2100.2.q.k 4 35.i odd 6 1
2100.2.bc.f 8 35.f even 4 2
2100.2.bc.f 8 35.k even 12 2
2940.2.a.p 2 7.c even 3 1
2940.2.a.r 2 7.d odd 6 1
2940.2.q.q 4 1.a even 1 1 trivial
2940.2.q.q 4 7.c even 3 1 inner
8820.2.a.bf 2 21.h odd 6 1
8820.2.a.bk 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11}^{4} + 18 T_{11}^{2} + 324 \)
\( T_{13}^{2} - 2 T_{13} - 17 \)
\( T_{17}^{4} + 18 T_{17}^{2} + 324 \)
\( T_{31}^{4} + 2 T_{31}^{3} + 75 T_{31}^{2} - 142 T_{31} + 5041 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 324 + 18 T^{2} + T^{4} \)
$13$ \( ( -17 - 2 T + T^{2} )^{2} \)
$17$ \( 324 + 18 T^{2} + T^{4} \)
$19$ \( ( 49 + 7 T + T^{2} )^{2} \)
$23$ \( 324 + 18 T^{2} + T^{4} \)
$29$ \( ( 18 + 12 T + T^{2} )^{2} \)
$31$ \( 5041 - 142 T + 75 T^{2} + 2 T^{3} + T^{4} \)
$37$ \( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( ( -18 + T^{2} )^{2} \)
$43$ \( ( -17 + 2 T + T^{2} )^{2} \)
$47$ \( ( 36 + 6 T + T^{2} )^{2} \)
$53$ \( 5184 + 72 T^{2} + T^{4} \)
$59$ \( 324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4} \)
$67$ \( 289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( -162 + T^{2} )^{2} \)
$73$ \( 49 - 70 T + 93 T^{2} - 10 T^{3} + T^{4} \)
$79$ \( ( 121 + 11 T + T^{2} )^{2} \)
$83$ \( ( 18 - 12 T + T^{2} )^{2} \)
$89$ \( 324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( ( -8 + 16 T + T^{2} )^{2} \)
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