Properties

Label 9408.2.a.dv
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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{3} + ( -2 + \beta ) q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -2 + \beta ) q^{5} + q^{9} + ( 2 + 2 \beta ) q^{11} -3 \beta q^{13} + ( -2 + \beta ) q^{15} + ( 6 - \beta ) q^{17} + ( -4 + 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} + q^{27} + 2 \beta q^{29} + 2 \beta q^{31} + ( 2 + 2 \beta ) q^{33} + ( -4 - 4 \beta ) q^{37} -3 \beta q^{39} + ( 6 - 3 \beta ) q^{41} -8 \beta q^{43} + ( -2 + \beta ) q^{45} + ( 4 + 6 \beta ) q^{47} + ( 6 - \beta ) q^{51} + 2 q^{53} -2 \beta q^{55} + ( -4 + 2 \beta ) q^{57} + 6 \beta q^{59} + ( -4 + 5 \beta ) q^{61} + ( -6 + 6 \beta ) q^{65} + 8 \beta q^{67} + ( -2 + 2 \beta ) q^{69} + ( -2 + 6 \beta ) q^{71} + ( 12 + 3 \beta ) q^{73} + ( 1 - 4 \beta ) q^{75} + ( 8 - 4 \beta ) q^{79} + q^{81} + 4 q^{83} + ( -14 + 8 \beta ) q^{85} + 2 \beta q^{87} + ( 10 + 3 \beta ) q^{89} + 2 \beta q^{93} + ( 12 - 8 \beta ) q^{95} + ( 4 + 3 \beta ) q^{97} + ( 2 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + 4q^{11} - 4q^{15} + 12q^{17} - 8q^{19} - 4q^{23} + 2q^{25} + 2q^{27} + 4q^{33} - 8q^{37} + 12q^{41} - 4q^{45} + 8q^{47} + 12q^{51} + 4q^{53} - 8q^{57} - 8q^{61} - 12q^{65} - 4q^{69} - 4q^{71} + 24q^{73} + 2q^{75} + 16q^{79} + 2q^{81} + 8q^{83} - 28q^{85} + 20q^{89} + 24q^{95} + 8q^{97} + 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 1.00000 0 −3.41421 0 0 0 1.00000 0
1.2 0 1.00000 0 −0.585786 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.dv 2
4.b odd 2 1 9408.2.a.dh 2
7.b odd 2 1 9408.2.a.dr 2
8.b even 2 1 1176.2.a.l 2
8.d odd 2 1 2352.2.a.bg 2
24.f even 2 1 7056.2.a.ce 2
24.h odd 2 1 3528.2.a.bc 2
28.d even 2 1 9408.2.a.ed 2
56.e even 2 1 2352.2.a.z 2
56.h odd 2 1 1176.2.a.m yes 2
56.j odd 6 2 1176.2.q.m 4
56.k odd 6 2 2352.2.q.ba 4
56.m even 6 2 2352.2.q.bg 4
56.p even 6 2 1176.2.q.n 4
168.e odd 2 1 7056.2.a.cw 2
168.i even 2 1 3528.2.a.bm 2
168.s odd 6 2 3528.2.s.bl 4
168.ba even 6 2 3528.2.s.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.l 2 8.b even 2 1
1176.2.a.m yes 2 56.h odd 2 1
1176.2.q.m 4 56.j odd 6 2
1176.2.q.n 4 56.p even 6 2
2352.2.a.z 2 56.e even 2 1
2352.2.a.bg 2 8.d odd 2 1
2352.2.q.ba 4 56.k odd 6 2
2352.2.q.bg 4 56.m even 6 2
3528.2.a.bc 2 24.h odd 2 1
3528.2.a.bm 2 168.i even 2 1
3528.2.s.bc 4 168.ba even 6 2
3528.2.s.bl 4 168.s odd 6 2
7056.2.a.ce 2 24.f even 2 1
7056.2.a.cw 2 168.e odd 2 1
9408.2.a.dh 2 4.b odd 2 1
9408.2.a.dr 2 7.b odd 2 1
9408.2.a.dv 2 1.a even 1 1 trivial
9408.2.a.ed 2 28.d even 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} + 4 T_{5} + 2 \)
\( T_{11}^{2} - 4 T_{11} - 4 \)
\( T_{13}^{2} - 18 \)
\( T_{17}^{2} - 12 T_{17} + 34 \)
\( T_{19}^{2} + 8 T_{19} + 8 \)
\( T_{31}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 8 T^{2} + 169 T^{4} \)
$17$ \( 1 - 12 T + 68 T^{2} - 204 T^{3} + 289 T^{4} \)
$19$ \( 1 + 8 T + 46 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 50 T^{2} + 841 T^{4} \)
$31$ \( 1 + 54 T^{2} + 961 T^{4} \)
$37$ \( 1 + 8 T + 58 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 12 T + 100 T^{2} - 492 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 42 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 38 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 + 46 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 88 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 6 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 24 T + 272 T^{2} - 1752 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 190 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 8 T + 192 T^{2} - 776 T^{3} + 9409 T^{4} \)
show more
show less