Properties

Label 2394.2.b.h
Level $2394$
Weight $2$
Character orbit 2394.b
Analytic conductor $19.116$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2803712.1
Defining polynomial: \(x^{6} + 6 x^{4} + 8 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta_{1} q^{5} + q^{7} + q^{8} -\beta_{1} q^{10} + ( -\beta_{1} - \beta_{3} ) q^{11} + ( \beta_{3} + \beta_{4} ) q^{13} + q^{14} + q^{16} + ( \beta_{1} + 3 \beta_{3} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} ) q^{19} -\beta_{1} q^{20} + ( -\beta_{1} - \beta_{3} ) q^{22} + ( 2 \beta_{3} - \beta_{4} ) q^{23} + ( -3 + 2 \beta_{5} ) q^{25} + ( \beta_{3} + \beta_{4} ) q^{26} + q^{28} + 4 q^{29} + ( -\beta_{3} + 2 \beta_{4} ) q^{31} + q^{32} + ( \beta_{1} + 3 \beta_{3} ) q^{34} -\beta_{1} q^{35} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} ) q^{38} -\beta_{1} q^{40} + ( -3 + \beta_{2} + \beta_{5} ) q^{41} + ( -1 - \beta_{2} - \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{3} ) q^{44} + ( 2 \beta_{3} - \beta_{4} ) q^{46} + ( -\beta_{1} + \beta_{4} ) q^{47} + q^{49} + ( -3 + 2 \beta_{5} ) q^{50} + ( \beta_{3} + \beta_{4} ) q^{52} + ( -2 + 2 \beta_{2} ) q^{53} + ( -7 + \beta_{2} + \beta_{5} ) q^{55} + q^{56} + 4 q^{58} + ( 2 + 2 \beta_{5} ) q^{59} + ( -1 + \beta_{2} - \beta_{5} ) q^{61} + ( -\beta_{3} + 2 \beta_{4} ) q^{62} + q^{64} + ( -1 + \beta_{2} - \beta_{5} ) q^{65} + ( -\beta_{1} + 2 \beta_{3} ) q^{67} + ( \beta_{1} + 3 \beta_{3} ) q^{68} -\beta_{1} q^{70} + ( 2 \beta_{2} - 4 \beta_{5} ) q^{71} + ( -3 - \beta_{2} + 3 \beta_{5} ) q^{73} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} ) q^{76} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{79} -\beta_{1} q^{80} + ( -3 + \beta_{2} + \beta_{5} ) q^{82} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{83} + ( 5 - 3 \beta_{2} + \beta_{5} ) q^{85} + ( -1 - \beta_{2} - \beta_{5} ) q^{86} + ( -\beta_{1} - \beta_{3} ) q^{88} + ( 1 + \beta_{2} + \beta_{5} ) q^{89} + ( \beta_{3} + \beta_{4} ) q^{91} + ( 2 \beta_{3} - \beta_{4} ) q^{92} + ( -\beta_{1} + \beta_{4} ) q^{94} + ( -8 - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{95} + ( -4 \beta_{1} - \beta_{4} ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + O(q^{10}) \) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{14} + 6 q^{16} - 6 q^{19} - 14 q^{25} + 6 q^{28} + 24 q^{29} + 6 q^{32} - 6 q^{38} - 16 q^{41} - 8 q^{43} + 6 q^{49} - 14 q^{50} - 12 q^{53} - 40 q^{55} + 6 q^{56} + 24 q^{58} + 16 q^{59} - 8 q^{61} + 6 q^{64} - 8 q^{65} - 8 q^{71} - 12 q^{73} - 6 q^{76} - 16 q^{82} + 32 q^{85} - 8 q^{86} + 8 q^{89} - 44 q^{95} + 6 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 6 x^{4} + 8 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} - 5 \nu^{3} - 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{5} + 5 \nu^{3} + 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{5} + 7 \nu^{3} + 10 \nu \)
\(\beta_{5}\)\(=\)\( 2 \nu^{4} + 10 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{4} - 4 \beta_{3} - 3 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} - 5 \beta_{2} + 13\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{4} + 18 \beta_{3} + 11 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1709.1
1.20864i
2.05288i
0.569973i
0.569973i
2.05288i
1.20864i
1.00000 0 1.00000 3.83149i 0 1.00000 1.00000 0 3.83149i
1709.2 1.00000 0 1.00000 2.69155i 0 1.00000 1.00000 0 2.69155i
1709.3 1.00000 0 1.00000 0.274268i 0 1.00000 1.00000 0 0.274268i
1709.4 1.00000 0 1.00000 0.274268i 0 1.00000 1.00000 0 0.274268i
1709.5 1.00000 0 1.00000 2.69155i 0 1.00000 1.00000 0 2.69155i
1709.6 1.00000 0 1.00000 3.83149i 0 1.00000 1.00000 0 3.83149i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1709.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.b.h yes 6
3.b odd 2 1 2394.2.b.g 6
19.b odd 2 1 2394.2.b.g 6
57.d even 2 1 inner 2394.2.b.h yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.g 6 3.b odd 2 1
2394.2.b.g 6 19.b odd 2 1
2394.2.b.h yes 6 1.a even 1 1 trivial
2394.2.b.h yes 6 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{6} + 22 T_{5}^{4} + 108 T_{5}^{2} + 8 \)
\( T_{29} - 4 \)
\( T_{53}^{3} + 6 T_{53}^{2} - 52 T_{53} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( T^{6} \)
$5$ \( 8 + 108 T^{2} + 22 T^{4} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 128 + 128 T^{2} + 24 T^{4} + T^{6} \)
$13$ \( 128 + 192 T^{2} + 40 T^{4} + T^{6} \)
$17$ \( 128 + 768 T^{2} + 64 T^{4} + T^{6} \)
$19$ \( 6859 + 2166 T + 95 T^{2} - 20 T^{3} + 5 T^{4} + 6 T^{5} + T^{6} \)
$23$ \( 2888 + 1260 T^{2} + 70 T^{4} + T^{6} \)
$29$ \( ( -4 + T )^{6} \)
$31$ \( 150152 + 8844 T^{2} + 166 T^{4} + T^{6} \)
$37$ \( 200 + 844 T^{2} + 118 T^{4} + T^{6} \)
$41$ \( ( -160 - 16 T + 8 T^{2} + T^{3} )^{2} \)
$43$ \( ( 32 - 32 T + 4 T^{2} + T^{3} )^{2} \)
$47$ \( 2048 + 1024 T^{2} + 64 T^{4} + T^{6} \)
$53$ \( ( 8 - 52 T + 6 T^{2} + T^{3} )^{2} \)
$59$ \( ( 128 - 32 T - 8 T^{2} + T^{3} )^{2} \)
$61$ \( ( -32 - 16 T + 4 T^{2} + T^{3} )^{2} \)
$67$ \( 8 + 428 T^{2} + 54 T^{4} + T^{6} \)
$71$ \( ( -1472 - 208 T + 4 T^{2} + T^{3} )^{2} \)
$73$ \( ( 200 - 100 T + 6 T^{2} + T^{3} )^{2} \)
$79$ \( 107648 + 6848 T^{2} + 144 T^{4} + T^{6} \)
$83$ \( 7688 + 1420 T^{2} + 70 T^{4} + T^{6} \)
$89$ \( ( -32 - 32 T - 4 T^{2} + T^{3} )^{2} \)
$97$ \( 182408 + 20204 T^{2} + 374 T^{4} + T^{6} \)
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