Properties

Label 304.2.a.g
Level $304$
Weight $2$
Character orbit 304.a
Self dual yes
Analytic conductor $2.427$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
Defining polynomial: \( x^{3} - x^{2} - 10x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1 - 2) q^{7} + (2 \beta_{2} - \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1 - 2) q^{7} + (2 \beta_{2} - \beta_1 + 5) q^{9} + (\beta_{2} + 2) q^{11} + ( - \beta_1 + 2) q^{13} + (2 \beta_{2} + 4) q^{15} + ( - \beta_{2} + \beta_1) q^{17} + q^{19} + (3 \beta_1 - 4) q^{21} + ( - 2 \beta_{2} + 3 \beta_1) q^{23} + ( - \beta_{2} + 2 \beta_1 + 1) q^{25} + ( - 2 \beta_{2} - 3 \beta_1) q^{27} + (2 \beta_{2} - \beta_1 - 2) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{33} + ( - \beta_{2} + 2 \beta_1 + 2) q^{35} - 2 q^{37} + (2 \beta_{2} - 3 \beta_1 + 8) q^{39} + (2 \beta_1 + 2) q^{41} + (\beta_{2} + 2 \beta_1 - 6) q^{43} + ( - \beta_{2} - 4 \beta_1 - 8) q^{45} + (3 \beta_{2} - 2 \beta_1 + 2) q^{47} + (\beta_{2} - 3 \beta_1 + 3) q^{49} + (\beta_1 - 4) q^{51} + (4 \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{2} - 2 \beta_1 - 6) q^{55} - \beta_1 q^{57} + ( - \beta_1 + 8) q^{59} + ( - \beta_{2} + 2 \beta_1) q^{61} + ( - 3 \beta_{2} + 4 \beta_1 - 18) q^{63} + 4 q^{65} + (2 \beta_{2} - \beta_1 - 4) q^{67} + ( - 2 \beta_{2} + 3 \beta_1 - 16) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{2} + 3 \beta_1) q^{73} + ( - 2 \beta_{2} + \beta_1 - 12) q^{75} + ( - \beta_{2} - 6) q^{77} + ( - 2 \beta_1 - 8) q^{79} + (4 \beta_{2} + 17) q^{81} + (4 \beta_{2} - 2 \beta_1 + 4) q^{83} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{85} + ( - 2 \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + ( - 2 \beta_{2} + 5 \beta_1 - 8) q^{91} - \beta_{2} q^{95} + (2 \beta_{2} - 2) q^{97} + (5 \beta_{2} + 2 \beta_1 + 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + q^{5} - 4 q^{7} + 12 q^{9} + 5 q^{11} + 5 q^{13} + 10 q^{15} + 2 q^{17} + 3 q^{19} - 9 q^{21} + 5 q^{23} + 6 q^{25} - q^{27} - 9 q^{29} - 12 q^{33} + 9 q^{35} - 6 q^{37} + 19 q^{39} + 8 q^{41} - 17 q^{43} - 27 q^{45} + q^{47} + 5 q^{49} - 11 q^{51} + q^{53} - 19 q^{55} - q^{57} + 23 q^{59} + 3 q^{61} - 47 q^{63} + 12 q^{65} - 15 q^{67} - 43 q^{69} + 12 q^{71} + 4 q^{73} - 33 q^{75} - 17 q^{77} - 26 q^{79} + 47 q^{81} + 6 q^{83} + 11 q^{85} + 3 q^{87} + 18 q^{89} - 17 q^{91} + q^{95} - 8 q^{97} + 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 10x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} - \beta _1 + 8 \) Copy content Toggle raw display

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
3.29707
0.786802
−3.08387
0 −3.29707 0 −3.08387 0 −1.78680 0 7.87067 0
1.2 0 −0.786802 0 3.29707 0 2.08387 0 −2.38094 0
1.3 0 3.08387 0 0.786802 0 −4.29707 0 6.51027 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)

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 304.2.a.g 3
3.b odd 2 1 2736.2.a.bd 3
4.b odd 2 1 152.2.a.c 3
5.b even 2 1 7600.2.a.bv 3
8.b even 2 1 1216.2.a.v 3
8.d odd 2 1 1216.2.a.u 3
12.b even 2 1 1368.2.a.n 3
19.b odd 2 1 5776.2.a.bp 3
20.d odd 2 1 3800.2.a.r 3
20.e even 4 2 3800.2.d.j 6
28.d even 2 1 7448.2.a.bf 3
76.d even 2 1 2888.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 4.b odd 2 1
304.2.a.g 3 1.a even 1 1 trivial
1216.2.a.u 3 8.d odd 2 1
1216.2.a.v 3 8.b even 2 1
1368.2.a.n 3 12.b even 2 1
2736.2.a.bd 3 3.b odd 2 1
2888.2.a.o 3 76.d even 2 1
3800.2.a.r 3 20.d odd 2 1
3800.2.d.j 6 20.e even 4 2
5776.2.a.bp 3 19.b odd 2 1
7448.2.a.bf 3 28.d even 2 1
7600.2.a.bv 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(304))\):

\( T_{3}^{3} + T_{3}^{2} - 10T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{3} - T_{5}^{2} - 10T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 5T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 10T - 8 \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 10T + 8 \) Copy content Toggle raw display
$7$ \( T^{3} + 4 T^{2} - 5 T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 5 T^{2} - 2 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 5 T^{2} - 2 T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 9 T + 2 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 5 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$29$ \( T^{3} + 9 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( (T + 2)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 8 T^{2} - 20 T + 128 \) Copy content Toggle raw display
$43$ \( T^{3} + 17 T^{2} + 24 T - 368 \) Copy content Toggle raw display
$47$ \( T^{3} - T^{2} - 72 T + 256 \) Copy content Toggle raw display
$53$ \( T^{3} - T^{2} - 134 T + 256 \) Copy content Toggle raw display
$59$ \( T^{3} - 23 T^{2} + 166 T - 376 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} - 28 T + 92 \) Copy content Toggle raw display
$67$ \( T^{3} + 15 T^{2} + 44 T + 32 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 76 T + 928 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} - 67 T + 326 \) Copy content Toggle raw display
$79$ \( T^{3} + 26 T^{2} + 184 T + 256 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 112 T + 736 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} - 16 T + 1024 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 20 T - 128 \) Copy content Toggle raw display
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