Properties

Label 475.2.g.a
Level $475$
Weight $2$
Character orbit 475.g
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{4} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{7} + 3 \beta_{1} q^{9} +O(q^{10})\) \( q -2 \beta_{1} q^{4} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{7} + 3 \beta_{1} q^{9} + ( \beta_{2} + \beta_{3} ) q^{11} -4 q^{16} + ( 3 - 3 \beta_{1} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{28} + 6 q^{36} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{44} + ( -7 + 7 \beta_{1} + \beta_{3} ) q^{47} + ( 3 - 10 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{49} + ( \beta_{2} + \beta_{3} ) q^{61} + ( -3 - 6 \beta_{1} + 3 \beta_{2} ) q^{63} + 8 \beta_{1} q^{64} + ( -8 - 6 \beta_{1} - 2 \beta_{2} ) q^{68} + ( 7 + 4 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 11 - 11 \beta_{1} - 3 \beta_{3} ) q^{77} -9 q^{81} + ( 7 + 9 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 4 \beta_{3} ) q^{92} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{7} + O(q^{10}) \) \( 4 q - 6 q^{7} - 16 q^{16} + 14 q^{17} - 8 q^{23} + 12 q^{28} + 24 q^{36} + 2 q^{43} - 26 q^{47} - 18 q^{63} - 28 q^{68} + 22 q^{73} + 38 q^{77} - 36 q^{81} + 32 q^{83} - 16 q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 5 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} - 5 \nu^{2} + 9 \nu + 25 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1} + 10\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−2.17945 + 0.500000i
2.17945 + 0.500000i
−2.17945 0.500000i
2.17945 0.500000i
0 0 2.00000i 0 0 −3.67945 + 3.67945i 0 3.00000i 0
18.2 0 0 2.00000i 0 0 0.679449 0.679449i 0 3.00000i 0
132.1 0 0 2.00000i 0 0 −3.67945 3.67945i 0 3.00000i 0
132.2 0 0 2.00000i 0 0 0.679449 + 0.679449i 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.c odd 4 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.g.a 4
5.b even 2 1 95.2.g.a 4
5.c odd 4 1 95.2.g.a 4
5.c odd 4 1 inner 475.2.g.a 4
15.d odd 2 1 855.2.p.a 4
15.e even 4 1 855.2.p.a 4
19.b odd 2 1 CM 475.2.g.a 4
95.d odd 2 1 95.2.g.a 4
95.g even 4 1 95.2.g.a 4
95.g even 4 1 inner 475.2.g.a 4
285.b even 2 1 855.2.p.a 4
285.j odd 4 1 855.2.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.g.a 4 5.b even 2 1
95.2.g.a 4 5.c odd 4 1
95.2.g.a 4 95.d odd 2 1
95.2.g.a 4 95.g even 4 1
475.2.g.a 4 1.a even 1 1 trivial
475.2.g.a 4 5.c odd 4 1 inner
475.2.g.a 4 19.b odd 2 1 CM
475.2.g.a 4 95.g even 4 1 inner
855.2.p.a 4 15.d odd 2 1
855.2.p.a 4 15.e even 4 1
855.2.p.a 4 285.b even 2 1
855.2.p.a 4 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 25 - 30 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( ( -19 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 225 - 210 T + 98 T^{2} - 14 T^{3} + T^{4} \)
$19$ \( ( 19 + T^{2} )^{2} \)
$23$ \( 900 - 240 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( 7225 + 170 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( 5625 + 1950 T + 338 T^{2} + 26 T^{3} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -19 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 625 + 550 T + 242 T^{2} - 22 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 8100 - 2880 T + 512 T^{2} - 32 T^{3} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
show more
show less