Properties

Label 475.4.a.e
Level $475$
Weight $4$
Character orbit 475.a
Self dual yes
Analytic conductor $28.026$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.0259072527\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{2} + 5q^{3} + q^{4} + 15q^{6} - 11q^{7} - 21q^{8} - 2q^{9} + O(q^{10}) \) \( q + 3q^{2} + 5q^{3} + q^{4} + 15q^{6} - 11q^{7} - 21q^{8} - 2q^{9} - 54q^{11} + 5q^{12} - 11q^{13} - 33q^{14} - 71q^{16} + 93q^{17} - 6q^{18} + 19q^{19} - 55q^{21} - 162q^{22} - 183q^{23} - 105q^{24} - 33q^{26} - 145q^{27} - 11q^{28} - 249q^{29} + 56q^{31} - 45q^{32} - 270q^{33} + 279q^{34} - 2q^{36} + 250q^{37} + 57q^{38} - 55q^{39} + 240q^{41} - 165q^{42} + 196q^{43} - 54q^{44} - 549q^{46} + 168q^{47} - 355q^{48} - 222q^{49} + 465q^{51} - 11q^{52} - 435q^{53} - 435q^{54} + 231q^{56} + 95q^{57} - 747q^{58} + 195q^{59} - 358q^{61} + 168q^{62} + 22q^{63} + 433q^{64} - 810q^{66} + 961q^{67} + 93q^{68} - 915q^{69} - 246q^{71} + 42q^{72} - 353q^{73} + 750q^{74} + 19q^{76} + 594q^{77} - 165q^{78} - 34q^{79} - 671q^{81} + 720q^{82} - 234q^{83} - 55q^{84} + 588q^{86} - 1245q^{87} + 1134q^{88} - 168q^{89} + 121q^{91} - 183q^{92} + 280q^{93} + 504q^{94} - 225q^{96} - 758q^{97} - 666q^{98} + 108q^{99} + O(q^{100}) \)

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
0
3.00000 5.00000 1.00000 0 15.0000 −11.0000 −21.0000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 475.4.a.e 1
5.b even 2 1 19.4.a.a 1
5.c odd 4 2 475.4.b.c 2
15.d odd 2 1 171.4.a.d 1
20.d odd 2 1 304.4.a.b 1
35.c odd 2 1 931.4.a.a 1
40.e odd 2 1 1216.4.a.a 1
40.f even 2 1 1216.4.a.f 1
55.d odd 2 1 2299.4.a.b 1
95.d odd 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 5.b even 2 1
171.4.a.d 1 15.d odd 2 1
304.4.a.b 1 20.d odd 2 1
361.4.a.b 1 95.d odd 2 1
475.4.a.e 1 1.a even 1 1 trivial
475.4.b.c 2 5.c odd 4 2
931.4.a.a 1 35.c odd 2 1
1216.4.a.a 1 40.e odd 2 1
1216.4.a.f 1 40.f even 2 1
2299.4.a.b 1 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(475))\):

\( T_{2} - 3 \)
\( T_{3} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T \)
$3$ \( -5 + T \)
$5$ \( T \)
$7$ \( 11 + T \)
$11$ \( 54 + T \)
$13$ \( 11 + T \)
$17$ \( -93 + T \)
$19$ \( -19 + T \)
$23$ \( 183 + T \)
$29$ \( 249 + T \)
$31$ \( -56 + T \)
$37$ \( -250 + T \)
$41$ \( -240 + T \)
$43$ \( -196 + T \)
$47$ \( -168 + T \)
$53$ \( 435 + T \)
$59$ \( -195 + T \)
$61$ \( 358 + T \)
$67$ \( -961 + T \)
$71$ \( 246 + T \)
$73$ \( 353 + T \)
$79$ \( 34 + T \)
$83$ \( 234 + T \)
$89$ \( 168 + T \)
$97$ \( 758 + T \)
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