Properties

Label 9408.2.a.ec
Level 9408
Weight 2
Character orbit 9408.a
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + q^{9} + \beta q^{11} + ( -3 + \beta ) q^{13} + \beta q^{15} -4 q^{17} + ( 3 - \beta ) q^{19} + 4 q^{23} + ( 9 + \beta ) q^{25} + q^{27} + ( -2 + \beta ) q^{29} - q^{31} + \beta q^{33} + ( -1 - \beta ) q^{37} + ( -3 + \beta ) q^{39} + ( 2 + 2 \beta ) q^{41} + ( 3 + \beta ) q^{43} + \beta q^{45} -6 q^{47} -4 q^{51} + ( -6 + \beta ) q^{53} + ( 14 + \beta ) q^{55} + ( 3 - \beta ) q^{57} + ( -2 - \beta ) q^{59} -10 q^{61} + ( 14 - 2 \beta ) q^{65} + ( -3 - \beta ) q^{67} + 4 q^{69} + 2 q^{71} + ( -1 + \beta ) q^{73} + ( 9 + \beta ) q^{75} + ( 5 - 2 \beta ) q^{79} + q^{81} + ( -4 + \beta ) q^{83} -4 \beta q^{85} + ( -2 + \beta ) q^{87} + ( -4 + 2 \beta ) q^{89} - q^{93} + ( -14 + 2 \beta ) q^{95} + ( 12 + \beta ) q^{97} + \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + q^{5} + 2q^{9} + q^{11} - 5q^{13} + q^{15} - 8q^{17} + 5q^{19} + 8q^{23} + 19q^{25} + 2q^{27} - 3q^{29} - 2q^{31} + q^{33} - 3q^{37} - 5q^{39} + 6q^{41} + 7q^{43} + q^{45} - 12q^{47} - 8q^{51} - 11q^{53} + 29q^{55} + 5q^{57} - 5q^{59} - 20q^{61} + 26q^{65} - 7q^{67} + 8q^{69} + 4q^{71} - q^{73} + 19q^{75} + 8q^{79} + 2q^{81} - 7q^{83} - 4q^{85} - 3q^{87} - 6q^{89} - 2q^{93} - 26q^{95} + 25q^{97} + q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.27492
4.27492
0 1.00000 0 −3.27492 0 0 0 1.00000 0
1.2 0 1.00000 0 4.27492 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.ec 2
4.b odd 2 1 9408.2.a.dp 2
7.b odd 2 1 9408.2.a.dj 2
7.c even 3 2 1344.2.q.w 4
8.b even 2 1 1176.2.a.k 2
8.d odd 2 1 2352.2.a.bf 2
24.f even 2 1 7056.2.a.cu 2
24.h odd 2 1 3528.2.a.bk 2
28.d even 2 1 9408.2.a.dw 2
28.g odd 6 2 1344.2.q.x 4
56.e even 2 1 2352.2.a.ba 2
56.h odd 2 1 1176.2.a.n 2
56.j odd 6 2 1176.2.q.l 4
56.k odd 6 2 336.2.q.g 4
56.m even 6 2 2352.2.q.bf 4
56.p even 6 2 168.2.q.c 4
168.e odd 2 1 7056.2.a.ch 2
168.i even 2 1 3528.2.a.bd 2
168.s odd 6 2 504.2.s.i 4
168.v even 6 2 1008.2.s.r 4
168.ba even 6 2 3528.2.s.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 56.p even 6 2
336.2.q.g 4 56.k odd 6 2
504.2.s.i 4 168.s odd 6 2
1008.2.s.r 4 168.v even 6 2
1176.2.a.k 2 8.b even 2 1
1176.2.a.n 2 56.h odd 2 1
1176.2.q.l 4 56.j odd 6 2
1344.2.q.w 4 7.c even 3 2
1344.2.q.x 4 28.g odd 6 2
2352.2.a.ba 2 56.e even 2 1
2352.2.a.bf 2 8.d odd 2 1
2352.2.q.bf 4 56.m even 6 2
3528.2.a.bd 2 168.i even 2 1
3528.2.a.bk 2 24.h odd 2 1
3528.2.s.bk 4 168.ba even 6 2
7056.2.a.ch 2 168.e odd 2 1
7056.2.a.cu 2 24.f even 2 1
9408.2.a.dj 2 7.b odd 2 1
9408.2.a.dp 2 4.b odd 2 1
9408.2.a.dw 2 28.d even 2 1
9408.2.a.ec 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} - T_{5} - 14 \)
\( T_{11}^{2} - T_{11} - 14 \)
\( T_{13}^{2} + 5 T_{13} - 8 \)
\( T_{17} + 4 \)
\( T_{19}^{2} - 5 T_{19} - 8 \)
\( T_{31} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - T + 8 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( 1 + 5 T + 18 T^{2} + 65 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 4 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 5 T + 30 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 3 T + 46 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + T + 31 T^{2} )^{2} \)
$37$ \( 1 + 3 T + 62 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 6 T + 34 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 7 T + 84 T^{2} - 301 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 6 T + 47 T^{2} )^{2} \)
$53$ \( 1 + 11 T + 122 T^{2} + 583 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 5 T + 110 T^{2} + 295 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 7 T + 132 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{2} \)
$73$ \( 1 + T + 132 T^{2} + 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 117 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 7 T + 164 T^{2} + 581 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 130 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 25 T + 336 T^{2} - 2425 T^{3} + 9409 T^{4} \)
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