Properties

Label 475.2.j.a
Level $475$
Weight $2$
Character orbit 475.j
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + 2 \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + 2 \beta_{3} q^{7} + \beta_{2} q^{9} + 3 q^{11} - 2 \beta_{3} q^{12} + ( - \beta_{3} + \beta_1) q^{13} + (4 \beta_{2} - 4) q^{16} + 3 \beta_1 q^{17} + (3 \beta_{2} + 2) q^{19} + (8 \beta_{2} - 8) q^{21} - 2 \beta_{3} q^{27} + ( - 4 \beta_{3} + 4 \beta_1) q^{28} - 3 \beta_{2} q^{29} - 7 q^{31} + 3 \beta_1 q^{33} + ( - 2 \beta_{2} + 2) q^{36} - 4 \beta_{3} q^{37} + 4 q^{39} + ( - 6 \beta_{2} + 6) q^{41} + 2 \beta_1 q^{43} - 6 \beta_{2} q^{44} + (3 \beta_{3} - 3 \beta_1) q^{47} + (4 \beta_{3} - 4 \beta_1) q^{48} - 9 q^{49} + 12 \beta_{2} q^{51} - 2 \beta_1 q^{52} + (3 \beta_{3} - 3 \beta_1) q^{53} + (3 \beta_{3} + 2 \beta_1) q^{57} + (15 \beta_{2} - 15) q^{59} - 5 \beta_{2} q^{61} + (2 \beta_{3} - 2 \beta_1) q^{63} + 8 q^{64} + (\beta_{3} - \beta_1) q^{67} - 6 \beta_{3} q^{68} + ( - 3 \beta_{2} + 3) q^{71} - 4 \beta_1 q^{73} + ( - 10 \beta_{2} + 6) q^{76} + 6 \beta_{3} q^{77} + ( - 5 \beta_{2} + 5) q^{79} + ( - 11 \beta_{2} + 11) q^{81} + 6 \beta_{3} q^{83} + 16 q^{84} - 3 \beta_{3} q^{87} - 15 \beta_{2} q^{89} + 8 \beta_{2} q^{91} - 7 \beta_1 q^{93} + 4 \beta_1 q^{97} + 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 2 q^{9} + 12 q^{11} - 8 q^{16} + 14 q^{19} - 16 q^{21} - 6 q^{29} - 28 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 12 q^{44} - 36 q^{49} + 24 q^{51} - 30 q^{59} - 10 q^{61} + 32 q^{64} + 6 q^{71} + 4 q^{76} + 10 q^{79} + 22 q^{81} + 64 q^{84} - 30 q^{89} + 16 q^{91} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −1.73205 1.00000i −1.00000 1.73205i 0 0 4.00000i 0 0.500000 + 0.866025i 0
49.2 0 1.73205 + 1.00000i −1.00000 1.73205i 0 0 4.00000i 0 0.500000 + 0.866025i 0
349.1 0 −1.73205 + 1.00000i −1.00000 + 1.73205i 0 0 4.00000i 0 0.500000 0.866025i 0
349.2 0 1.73205 1.00000i −1.00000 + 1.73205i 0 0 4.00000i 0 0.500000 0.866025i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.j.a 4
5.b even 2 1 inner 475.2.j.a 4
5.c odd 4 1 95.2.e.a 2
5.c odd 4 1 475.2.e.b 2
15.e even 4 1 855.2.k.b 2
19.c even 3 1 inner 475.2.j.a 4
20.e even 4 1 1520.2.q.c 2
95.i even 6 1 inner 475.2.j.a 4
95.l even 12 1 1805.2.a.b 1
95.l even 12 1 9025.2.a.e 1
95.m odd 12 1 95.2.e.a 2
95.m odd 12 1 475.2.e.b 2
95.m odd 12 1 1805.2.a.a 1
95.m odd 12 1 9025.2.a.g 1
285.v even 12 1 855.2.k.b 2
380.v even 12 1 1520.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 5.c odd 4 1
95.2.e.a 2 95.m odd 12 1
475.2.e.b 2 5.c odd 4 1
475.2.e.b 2 95.m odd 12 1
475.2.j.a 4 1.a even 1 1 trivial
475.2.j.a 4 5.b even 2 1 inner
475.2.j.a 4 19.c even 3 1 inner
475.2.j.a 4 95.i even 6 1 inner
855.2.k.b 2 15.e even 4 1
855.2.k.b 2 285.v even 12 1
1520.2.q.c 2 20.e even 4 1
1520.2.q.c 2 380.v even 12 1
1805.2.a.a 1 95.m odd 12 1
1805.2.a.b 1 95.l even 12 1
9025.2.a.e 1 95.l even 12 1
9025.2.a.g 1 95.m odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
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