Properties

Label 2352.3.f.j
Level $2352$
Weight $3$
Character orbit 2352.f
Analytic conductor $64.087$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{5} + \beta_{4}) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{5} + \beta_{4}) q^{5} - 3 q^{9} + (\beta_{6} - \beta_1) q^{11} + (2 \beta_{7} - 4 \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{3} + 3 \beta_1) q^{15} + ( - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_{2}) q^{17} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 8 \beta_{2}) q^{19} + ( - \beta_{6} - 2 \beta_{3} + 9 \beta_1 + 2) q^{23} + ( - 2 \beta_{6} - \beta_1 + 9) q^{25} + 3 \beta_{2} q^{27} + ( - \beta_{6} + 6 \beta_{3} - 5 \beta_1 + 10) q^{29} + (3 \beta_{7} + 14 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{31} + ( - 3 \beta_{7} + \beta_{5}) q^{33} + (6 \beta_{6} + 4 \beta_{3} + 9 \beta_1 + 16) q^{37} + (2 \beta_{6} + \beta_{3} - 12 \beta_1) q^{39} + ( - 5 \beta_{7} - 13 \beta_{5} + 7 \beta_{4} - 14 \beta_{2}) q^{41} + (2 \beta_{6} - 8 \beta_{3} - 8 \beta_1 + 14) q^{43} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{45} + ( - 7 \beta_{7} - 2 \beta_{5} + 8 \beta_{4} - 22 \beta_{2}) q^{47} + ( - 3 \beta_{3} - 9 \beta_1 + 12) q^{51} + ( - 6 \beta_{6} - 4 \beta_{3} - 26 \beta_1 - 18) q^{53} + ( - \beta_{7} + 10 \beta_{5} + 6 \beta_{4}) q^{55} + ( - \beta_{6} + 2 \beta_{3} - 6 \beta_1 - 24) q^{57} + ( - 5 \beta_{7} + 22 \beta_{5} - 12 \beta_{4} - 14 \beta_{2}) q^{59} + ( - 8 \beta_{7} + 14 \beta_{5} - 5 \beta_{4} + 12 \beta_{2}) q^{61} + (5 \beta_{6} + 4 \beta_{3} - 19 \beta_1 + 30) q^{65} + (6 \beta_{6} - 4 \beta_{3} - 12 \beta_1 + 8) q^{67} + (3 \beta_{7} - 9 \beta_{5} - 6 \beta_{4} - 2 \beta_{2}) q^{69} + ( - \beta_{6} - 8 \beta_{3} - 43 \beta_1 - 28) q^{71} + (8 \beta_{7} - 6 \beta_{5} - 15 \beta_{4} + 28 \beta_{2}) q^{73} + (6 \beta_{7} + \beta_{5} - 9 \beta_{2}) q^{75} + (4 \beta_{6} - 28 \beta_1 + 54) q^{79} + 9 q^{81} + (14 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 16 \beta_{2}) q^{83} + (4 \beta_{3} - 15 \beta_1 - 12) q^{85} + (3 \beta_{7} + 5 \beta_{5} + 18 \beta_{4} - 10 \beta_{2}) q^{87} + (16 \beta_{7} + 23 \beta_{5} + 11 \beta_{4} - 8 \beta_{2}) q^{89} + (3 \beta_{6} - 4 \beta_{3} + 42 \beta_1 - 12) q^{93} + (4 \beta_{6} - 10 \beta_{3} + 36 \beta_1 + 34) q^{95} + (4 \beta_{7} - 10 \beta_{5} + 3 \beta_{4} - 28 \beta_{2}) q^{97} + ( - 3 \beta_{6} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{7} + 35\nu^{5} - 126\nu^{3} + 134\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 28\nu^{5} + 91\nu^{3} - 96\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + 3\beta_{4} + 3\beta_{3} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{2} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 8\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{2} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} + 13\beta_{4} - 13\beta_{3} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31\beta_{7} - 31\beta_{6} - 44\beta_{4} - 44\beta_{3} ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1.60021 0.923880i
0.662827 0.382683i
−1.60021 + 0.923880i
−0.662827 + 0.382683i
−0.662827 0.382683i
−1.60021 0.923880i
0.662827 + 0.382683i
1.60021 + 0.923880i
0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
97.5 0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
97.6 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.7 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.8 0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.f.j 8
4.b odd 2 1 588.3.d.c 8
7.b odd 2 1 inner 2352.3.f.j 8
12.b even 2 1 1764.3.d.h 8
28.d even 2 1 588.3.d.c 8
28.f even 6 1 588.3.m.e 8
28.f even 6 1 588.3.m.f 8
28.g odd 6 1 588.3.m.e 8
28.g odd 6 1 588.3.m.f 8
84.h odd 2 1 1764.3.d.h 8
84.j odd 6 1 1764.3.z.l 8
84.j odd 6 1 1764.3.z.m 8
84.n even 6 1 1764.3.z.l 8
84.n even 6 1 1764.3.z.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 4.b odd 2 1
588.3.d.c 8 28.d even 2 1
588.3.m.e 8 28.f even 6 1
588.3.m.e 8 28.g odd 6 1
588.3.m.f 8 28.f even 6 1
588.3.m.f 8 28.g odd 6 1
1764.3.d.h 8 12.b even 2 1
1764.3.d.h 8 84.h odd 2 1
1764.3.z.l 8 84.j odd 6 1
1764.3.z.l 8 84.n even 6 1
1764.3.z.m 8 84.j odd 6 1
1764.3.z.m 8 84.n even 6 1
2352.3.f.j 8 1.a even 1 1 trivial
2352.3.f.j 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} + 64T_{5}^{6} + 1052T_{5}^{4} + 1088T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{11}^{4} - 124T_{11}^{2} + 48T_{11} + 3292 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 64 T^{6} + 1052 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 124 T^{2} + 48 T + 3292)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 712 T^{6} + 123284 T^{4} + \cdots + 2979076 \) Copy content Toggle raw display
$17$ \( T^{8} + 768 T^{6} + \cdots + 168428484 \) Copy content Toggle raw display
$19$ \( T^{8} + 1136 T^{6} + \cdots + 285745216 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} - 612 T^{2} + 9856 T - 37988)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 40 T^{3} - 1924 T^{2} + \cdots + 468892)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 6448 T^{6} + \cdots + 2052452416 \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} - 3492 T^{2} + \cdots - 546236)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9808 T^{6} + \cdots + 26697826996036 \) Copy content Toggle raw display
$43$ \( (T^{4} - 56 T^{3} - 3784 T^{2} + \cdots + 192784)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 11488 T^{6} + \cdots + 8881401308224 \) Copy content Toggle raw display
$53$ \( (T^{4} + 72 T^{3} - 5464 T^{2} + \cdots - 8464112)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 372481662446656 \) Copy content Toggle raw display
$61$ \( T^{8} + 13192 T^{6} + \cdots + 454276 \) Copy content Toggle raw display
$67$ \( (T^{4} - 32 T^{3} - 6048 T^{2} + \cdots - 553856)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 112 T^{3} - 6460 T^{2} + \cdots - 7722596)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 22472 T^{6} + \cdots + 3712242251524 \) Copy content Toggle raw display
$79$ \( (T^{4} - 216 T^{3} + 12440 T^{2} + \cdots - 7050224)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 19712 T^{6} + \cdots + 61585579131904 \) Copy content Toggle raw display
$89$ \( T^{8} + 41600 T^{6} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + 13640 T^{6} + \cdots + 5315948141956 \) Copy content Toggle raw display
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