Properties

Label 475.3.d.a
Level $475$
Weight $3$
Character orbit 475.d
Analytic conductor $12.943$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{4} -5 i q^{7} -9 q^{9} +O(q^{10})\) \( q -4 q^{4} -5 i q^{7} -9 q^{9} + 3 q^{11} + 16 q^{16} + 15 i q^{17} + 19 q^{19} + 30 i q^{23} + 20 i q^{28} + 36 q^{36} + 85 i q^{43} -12 q^{44} + 75 i q^{47} + 24 q^{49} + 103 q^{61} + 45 i q^{63} -64 q^{64} -60 i q^{68} + 25 i q^{73} -76 q^{76} -15 i q^{77} + 81 q^{81} -90 i q^{83} -120 i q^{92} -27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{4} - 18q^{9} + O(q^{10}) \) \( 2q - 8q^{4} - 18q^{9} + 6q^{11} + 32q^{16} + 38q^{19} + 72q^{36} - 24q^{44} + 48q^{49} + 206q^{61} - 128q^{64} - 152q^{76} + 162q^{81} - 54q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
1.00000i
1.00000i
0 0 −4.00000 0 0 5.00000i 0 −9.00000 0
474.2 0 0 −4.00000 0 0 5.00000i 0 −9.00000 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.b even 2 1 inner
95.d 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.d.a 2
5.b even 2 1 inner 475.3.d.a 2
5.c odd 4 1 19.3.b.a 1
5.c odd 4 1 475.3.c.a 1
15.e even 4 1 171.3.c.a 1
19.b odd 2 1 CM 475.3.d.a 2
20.e even 4 1 304.3.e.a 1
40.i odd 4 1 1216.3.e.a 1
40.k even 4 1 1216.3.e.b 1
60.l odd 4 1 2736.3.o.a 1
95.d odd 2 1 inner 475.3.d.a 2
95.g even 4 1 19.3.b.a 1
95.g even 4 1 475.3.c.a 1
95.l even 12 2 361.3.d.a 2
95.m odd 12 2 361.3.d.a 2
95.q odd 36 6 361.3.f.a 6
95.r even 36 6 361.3.f.a 6
285.j odd 4 1 171.3.c.a 1
380.j odd 4 1 304.3.e.a 1
760.t even 4 1 1216.3.e.a 1
760.y odd 4 1 1216.3.e.b 1
1140.w even 4 1 2736.3.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 5.c odd 4 1
19.3.b.a 1 95.g even 4 1
171.3.c.a 1 15.e even 4 1
171.3.c.a 1 285.j odd 4 1
304.3.e.a 1 20.e even 4 1
304.3.e.a 1 380.j odd 4 1
361.3.d.a 2 95.l even 12 2
361.3.d.a 2 95.m odd 12 2
361.3.f.a 6 95.q odd 36 6
361.3.f.a 6 95.r even 36 6
475.3.c.a 1 5.c odd 4 1
475.3.c.a 1 95.g even 4 1
475.3.d.a 2 1.a even 1 1 trivial
475.3.d.a 2 5.b even 2 1 inner
475.3.d.a 2 19.b odd 2 1 CM
475.3.d.a 2 95.d odd 2 1 inner
1216.3.e.a 1 40.i odd 4 1
1216.3.e.a 1 760.t even 4 1
1216.3.e.b 1 40.k even 4 1
1216.3.e.b 1 760.y odd 4 1
2736.3.o.a 1 60.l odd 4 1
2736.3.o.a 1 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( 225 + T^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( 900 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 7225 + T^{2} \)
$47$ \( 5625 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -103 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 625 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 8100 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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