Properties

Label 475.3.c.d
Level $475$
Weight $3$
Character orbit 475.c
Analytic conductor $12.943$
Analytic rank $0$
Dimension $4$
CM discriminant -95
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.7600.1
Defining polynomial: \(x^{4} + 9 x^{2} + 19\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + ( -4 + 3 \beta_{3} ) q^{4} + ( -7 + 5 \beta_{3} ) q^{6} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -9 + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} + ( -4 + 3 \beta_{3} ) q^{4} + ( -7 + 5 \beta_{3} ) q^{6} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -9 + 2 \beta_{3} ) q^{9} + 6 \beta_{3} q^{11} + ( -13 \beta_{1} + \beta_{2} ) q^{12} + ( 5 \beta_{1} + 3 \beta_{2} ) q^{13} + ( 29 - 12 \beta_{3} ) q^{16} + ( -13 \beta_{1} + 2 \beta_{2} ) q^{18} + 19 q^{19} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{22} + ( 75 - 21 \beta_{3} ) q^{24} + ( -43 + 9 \beta_{3} ) q^{26} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{27} + 29 \beta_{1} q^{32} + ( -18 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 66 - 35 \beta_{3} ) q^{36} + ( -\beta_{1} - 15 \beta_{2} ) q^{37} + 19 \beta_{1} q^{38} + ( -2 + 34 \beta_{3} ) q^{39} + ( 90 - 24 \beta_{3} ) q^{44} + ( 65 \beta_{1} - 17 \beta_{2} ) q^{48} -49 q^{49} + ( -41 \beta_{1} + 21 \beta_{2} ) q^{52} + ( -23 \beta_{1} - 9 \beta_{2} ) q^{53} + ( 50 - 14 \beta_{3} ) q^{54} + ( 19 \beta_{1} - 19 \beta_{2} ) q^{57} + 54 \beta_{3} q^{61} + ( -116 + 39 \beta_{3} ) q^{64} + ( 150 - 42 \beta_{3} ) q^{66} + ( 19 \beta_{1} + 21 \beta_{2} ) q^{67} + ( 84 \beta_{1} - 27 \beta_{2} ) q^{72} + ( 23 + 27 \beta_{3} ) q^{74} + ( -76 + 57 \beta_{3} ) q^{76} + ( -70 \beta_{1} + 34 \beta_{2} ) q^{78} + ( -61 - 18 \beta_{3} ) q^{81} + 90 \beta_{1} q^{88} + ( -203 + 145 \beta_{3} ) q^{96} + ( 23 \beta_{1} + 33 \beta_{2} ) q^{97} -49 \beta_{1} q^{98} + ( 60 - 54 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{4} - 28q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 16q^{4} - 28q^{6} - 36q^{9} + 116q^{16} + 76q^{19} + 300q^{24} - 172q^{26} + 264q^{36} - 8q^{39} + 360q^{44} - 196q^{49} + 200q^{54} - 464q^{64} + 600q^{66} + 92q^{74} - 304q^{76} - 244q^{81} - 812q^{96} + 240q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 6 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{2} + 3 \beta_{1}\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
2.37024i
1.83901i
1.83901i
2.37024i
3.83513i 4.74048i −10.7082 0 −18.1803 0 25.7268i −13.4721 0
151.2 1.13657i 3.67802i 2.70820 0 4.18034 0 7.62436i −4.52786 0
151.3 1.13657i 3.67802i 2.70820 0 4.18034 0 7.62436i −4.52786 0
151.4 3.83513i 4.74048i −10.7082 0 −18.1803 0 25.7268i −13.4721 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
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.d 4
5.b even 2 1 inner 475.3.c.d 4
5.c odd 4 2 95.3.d.c 4
15.e even 4 2 855.3.g.d 4
19.b odd 2 1 inner 475.3.c.d 4
95.d odd 2 1 CM 475.3.c.d 4
95.g even 4 2 95.3.d.c 4
285.j odd 4 2 855.3.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.d.c 4 5.c odd 4 2
95.3.d.c 4 95.g even 4 2
475.3.c.d 4 1.a even 1 1 trivial
475.3.c.d 4 5.b even 2 1 inner
475.3.c.d 4 19.b odd 2 1 inner
475.3.c.d 4 95.d odd 2 1 CM
855.3.g.d 4 15.e even 4 2
855.3.g.d 4 285.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{4} + 16 T_{2}^{2} + 19 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 19 + 16 T^{2} + T^{4} \)
$3$ \( 304 + 36 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -180 + T^{2} )^{2} \)
$13$ \( 109744 + 676 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -19 + T )^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( 511024 + 5476 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 30935344 + 11236 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -14580 + T^{2} )^{2} \)
$67$ \( 43666864 + 17956 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 116480944 + 37636 T^{2} + T^{4} \)
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