Properties

Label 1476.2.a.g
Level $1476$
Weight $2$
Character orbit 1476.a
Self dual yes
Analytic conductor $11.786$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1476 = 2^{2} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1476.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7859193383\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25808.1
Defining polynomial: \(x^{4} - 10 x^{2} - 6 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 164)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -2 + \beta_{2} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + 2 \beta_{1} q^{13} -2 \beta_{3} q^{17} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( 2 + 2 \beta_{2} ) q^{23} + ( 3 - 2 \beta_{1} ) q^{25} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 7 - 4 \beta_{1} + \beta_{3} ) q^{35} + ( 4 - \beta_{2} - \beta_{3} ) q^{37} + q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{49} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{55} + ( -2 - 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{65} + ( 7 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( 4 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{71} + ( 4 - \beta_{2} + 3 \beta_{3} ) q^{73} + ( -2 + 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{77} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{79} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 12 - 2 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -7 + 2 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} - 4q^{11} + 4q^{17} + 6q^{19} + 12q^{23} + 12q^{25} + 4q^{29} - 8q^{31} + 26q^{35} + 16q^{37} + 4q^{41} + 4q^{43} + 6q^{47} + 16q^{49} + 16q^{53} - 2q^{55} - 12q^{59} + 24q^{61} - 4q^{65} + 28q^{67} + 2q^{71} + 8q^{73} - 8q^{77} - 18q^{79} + 12q^{83} + 32q^{85} - 4q^{89} + 36q^{91} - 14q^{95} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 10 x^{2} - 6 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 7 \nu - 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu^{2} + 7 \nu - 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 7 \beta_{1} + 3\)

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
−2.46810
0.707500
3.31526
−1.55466
0 0 0 −3.59669 0 −5.06479 0 0 0
1.2 0 0 0 −2.56613 0 −0.858626 0 0 0
1.3 0 0 0 −1.17025 0 3.14501 0 0 0
1.4 0 0 0 3.33307 0 2.77840 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(41\) \(-1\)

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 1476.2.a.g 4
3.b odd 2 1 164.2.a.a 4
4.b odd 2 1 5904.2.a.bp 4
12.b even 2 1 656.2.a.i 4
15.d odd 2 1 4100.2.a.c 4
15.e even 4 2 4100.2.d.c 8
21.c even 2 1 8036.2.a.i 4
24.f even 2 1 2624.2.a.y 4
24.h odd 2 1 2624.2.a.v 4
123.b odd 2 1 6724.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 3.b odd 2 1
656.2.a.i 4 12.b even 2 1
1476.2.a.g 4 1.a even 1 1 trivial
2624.2.a.v 4 24.h odd 2 1
2624.2.a.y 4 24.f even 2 1
4100.2.a.c 4 15.d odd 2 1
4100.2.d.c 8 15.e even 4 2
5904.2.a.bp 4 4.b odd 2 1
6724.2.a.c 4 123.b odd 2 1
8036.2.a.i 4 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4 T_{5}^{3} - 8 T_{5}^{2} - 44 T_{5} - 36 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1476))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( -36 - 44 T - 8 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 38 + 26 T - 22 T^{2} + T^{4} \)
$11$ \( 54 - 18 T - 18 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 144 - 48 T - 40 T^{2} + T^{4} \)
$17$ \( 432 + 80 T - 48 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( -186 + 134 T - 14 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( -192 + 128 T + 16 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 144 - 40 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( 64 - 32 T - 32 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( -324 + 36 T + 64 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( ( -1 + T )^{4} \)
$43$ \( -288 + 272 T - 48 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 1182 + 206 T - 62 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( -1296 + 720 T - 16 T^{3} + T^{4} \)
$59$ \( -192 - 128 T + 16 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( 288 - 432 T + 176 T^{2} - 24 T^{3} + T^{4} \)
$67$ \( 1094 - 1010 T + 270 T^{2} - 28 T^{3} + T^{4} \)
$71$ \( -426 + 694 T - 186 T^{2} - 2 T^{3} + T^{4} \)
$73$ \( -404 + 692 T - 80 T^{2} - 8 T^{3} + T^{4} \)
$79$ \( -18 - 42 T + 50 T^{2} + 18 T^{3} + T^{4} \)
$83$ \( -3456 + 1344 T - 80 T^{2} - 12 T^{3} + T^{4} \)
$89$ \( 4272 - 272 T - 128 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( 4944 + 1280 T - 120 T^{2} - 16 T^{3} + T^{4} \)
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