Properties

Label 2394.2.b.i
Level $2394$
Weight $2$
Character orbit 2394.b
Analytic conductor $19.116$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 10 x^{6} + 30 x^{4} + 24 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} - q^{8} -\beta_{3} q^{10} -\beta_{5} q^{11} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + q^{14} + q^{16} -\beta_{5} q^{17} + ( 1 + \beta_{4} ) q^{19} + \beta_{3} q^{20} + \beta_{5} q^{22} + ( -\beta_{2} + \beta_{5} ) q^{23} + ( -3 + \beta_{4} + \beta_{7} ) q^{25} + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{26} - q^{28} + ( 2 - \beta_{4} - \beta_{7} ) q^{29} + ( \beta_{3} + \beta_{5} ) q^{31} - q^{32} + \beta_{5} q^{34} -\beta_{3} q^{35} + ( \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 - \beta_{4} ) q^{38} -\beta_{3} q^{40} + ( -5 + \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{41} + ( -5 - \beta_{1} + \beta_{4} - \beta_{6} ) q^{43} -\beta_{5} q^{44} + ( \beta_{2} - \beta_{5} ) q^{46} + ( 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + q^{49} + ( 3 - \beta_{4} - \beta_{7} ) q^{50} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( -6 + 2 \beta_{1} + \beta_{4} + \beta_{7} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{55} + q^{56} + ( -2 + \beta_{4} + \beta_{7} ) q^{58} + ( 1 - \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{61} + ( -\beta_{3} - \beta_{5} ) q^{62} + q^{64} + ( 1 - \beta_{1} - 4 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{2} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} -\beta_{5} q^{68} + \beta_{3} q^{70} + ( 2 + 2 \beta_{1} ) q^{71} + ( -1 + \beta_{1} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{73} + ( -\beta_{2} + 2 \beta_{3} ) q^{74} + ( 1 + \beta_{4} ) q^{76} + \beta_{5} q^{77} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + \beta_{3} q^{80} + ( 5 - \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -1 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{85} + ( 5 + \beta_{1} - \beta_{4} + \beta_{6} ) q^{86} + \beta_{5} q^{88} + ( 3 - 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( -\beta_{2} + \beta_{5} ) q^{92} + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{95} + ( -\beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + O(q^{10}) \) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} + 16 q^{29} - 8 q^{32} - 12 q^{38} - 24 q^{41} - 32 q^{43} + 8 q^{49} + 24 q^{50} - 48 q^{53} + 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} - 16 q^{65} + 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} + 32 q^{86} + 8 q^{89} - 8 q^{95} - 8 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 10 x^{6} + 30 x^{4} + 24 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{4} + 10 \nu^{2} + 5 \)
\(\beta_{2}\)\(=\)\( -3 \nu^{5} - 18 \nu^{3} - 21 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{7} + 8 \nu^{5} + 19 \nu^{3} + 14 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{7} + \nu^{6} + 10 \nu^{5} + 5 \nu^{4} + 29 \nu^{3} + 3 \nu^{2} + 20 \nu + 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{7} - 11 \nu^{5} - 35 \nu^{3} - 25 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{7} - \nu^{6} + 10 \nu^{5} - 7 \nu^{4} + 29 \nu^{3} - 11 \nu^{2} + 20 \nu - 1 \)
\(\beta_{7}\)\(=\)\( -\nu^{7} - \nu^{6} - 10 \nu^{5} - 7 \nu^{4} - 29 \nu^{3} - 11 \nu^{2} - 20 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 2 \beta_{2}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{4} + \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(15 \beta_{7} - 15 \beta_{6} - 24 \beta_{5} + 6 \beta_{3} + 4 \beta_{2}\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{4} - 4 \beta_{1} + 20\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-69 \beta_{7} + 69 \beta_{6} + 102 \beta_{5} - 36 \beta_{3} - 14 \beta_{2}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(23 \beta_{7} - \beta_{6} + 24 \beta_{4} + 17 \beta_{1} - 87\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(309 \beta_{7} - 309 \beta_{6} - 444 \beta_{5} + 186 \beta_{3} + 64 \beta_{2}\)\()/12\)

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
2.15177i
2.03530i
0.209899i
1.08784i
1.08784i
0.209899i
2.03530i
2.15177i
−1.00000 0 1.00000 3.71953i 0 −1.00000 −1.00000 0 3.71953i
1709.2 −1.00000 0 1.00000 3.02986i 0 −1.00000 −1.00000 0 3.02986i
1709.3 −1.00000 0 1.00000 2.76612i 0 −1.00000 −1.00000 0 2.76612i
1709.4 −1.00000 0 1.00000 1.15484i 0 −1.00000 −1.00000 0 1.15484i
1709.5 −1.00000 0 1.00000 1.15484i 0 −1.00000 −1.00000 0 1.15484i
1709.6 −1.00000 0 1.00000 2.76612i 0 −1.00000 −1.00000 0 2.76612i
1709.7 −1.00000 0 1.00000 3.02986i 0 −1.00000 −1.00000 0 3.02986i
1709.8 −1.00000 0 1.00000 3.71953i 0 −1.00000 −1.00000 0 3.71953i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1709.8
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.i 8
3.b odd 2 1 2394.2.b.j yes 8
19.b odd 2 1 2394.2.b.j yes 8
57.d even 2 1 inner 2394.2.b.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.i 8 1.a even 1 1 trivial
2394.2.b.i 8 57.d even 2 1 inner
2394.2.b.j yes 8 3.b odd 2 1
2394.2.b.j yes 8 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 32 T_{5}^{6} + 344 T_{5}^{4} + 1376 T_{5}^{2} + 1296 \)
\( T_{29}^{4} - 8 T_{29}^{3} - 16 T_{29}^{2} + 160 T_{29} - 192 \)
\( T_{53}^{4} + 24 T_{53}^{3} + 152 T_{53}^{2} - 128 T_{53} - 2544 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( T^{8} \)
$5$ \( 1296 + 1376 T^{2} + 344 T^{4} + 32 T^{6} + T^{8} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( 5184 + 5600 T^{2} + 992 T^{4} + 56 T^{6} + T^{8} \)
$13$ \( 14400 + 10208 T^{2} + 1536 T^{4} + 72 T^{6} + T^{8} \)
$17$ \( 5184 + 5600 T^{2} + 992 T^{4} + 56 T^{6} + T^{8} \)
$19$ \( 130321 - 82308 T + 15884 T^{2} + 1292 T^{3} - 986 T^{4} + 68 T^{5} + 44 T^{6} - 12 T^{7} + T^{8} \)
$23$ \( 144 + 992 T^{2} + 1496 T^{4} + 80 T^{6} + T^{8} \)
$29$ \( ( -192 + 160 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( 32400 + 14048 T^{2} + 1912 T^{4} + 88 T^{6} + T^{8} \)
$37$ \( 138384 + 60128 T^{2} + 6008 T^{4} + 152 T^{6} + T^{8} \)
$41$ \( ( 2592 - 944 T - 96 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$43$ \( ( -864 - 432 T + 24 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$47$ \( 3686400 + 653312 T^{2} + 24576 T^{4} + 288 T^{6} + T^{8} \)
$53$ \( ( -2544 - 128 T + 152 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( ( 2144 + 176 T - 112 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$67$ \( 10419984 + 1036640 T^{2} + 29688 T^{4} + 304 T^{6} + T^{8} \)
$71$ \( ( 768 + 256 T - 64 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$73$ \( ( 2480 - 592 T - 128 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$79$ \( 18455616 + 1894112 T^{2} + 43328 T^{4} + 360 T^{6} + T^{8} \)
$83$ \( 68757264 + 3420320 T^{2} + 59608 T^{4} + 424 T^{6} + T^{8} \)
$89$ \( ( 9312 + 208 T - 272 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$97$ \( 2322576 + 429920 T^{2} + 22328 T^{4} + 288 T^{6} + T^{8} \)
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