Properties

Label 475.3.d.d
Level $475$
Weight $3$
Character orbit 475.d
Analytic conductor $12.943$
Analytic rank $0$
Dimension $28$
CM no
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: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 56q^{4} - 8q^{6} + 72q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 56q^{4} - 8q^{6} + 72q^{9} - 8q^{11} + 72q^{16} - 78q^{19} + 88q^{24} + 60q^{26} + 8q^{36} + 64q^{39} + 104q^{44} - 468q^{49} - 196q^{54} + 444q^{61} + 436q^{64} + 184q^{66} + 184q^{74} - 702q^{76} + 804q^{81} + 380q^{96} + 360q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1 −3.82982 0.102062 10.6675 0 −0.390878 9.96826i −25.5353 −8.98958 0
474.2 −3.82982 0.102062 10.6675 0 −0.390878 9.96826i −25.5353 −8.98958 0
474.3 −3.01757 −4.83172 5.10575 0 14.5801 6.15539i −3.33669 14.3455 0
474.4 −3.01757 −4.83172 5.10575 0 14.5801 6.15539i −3.33669 14.3455 0
474.5 −2.92865 2.02259 4.57698 0 −5.92346 8.32815i −1.68976 −4.90912 0
474.6 −2.92865 2.02259 4.57698 0 −5.92346 8.32815i −1.68976 −4.90912 0
474.7 −2.34822 5.73703 1.51415 0 −13.4718 9.56678i 5.83734 23.9135 0
474.8 −2.34822 5.73703 1.51415 0 −13.4718 9.56678i 5.83734 23.9135 0
474.9 −1.41833 1.48141 −1.98835 0 −2.10112 2.50491i 8.49344 −6.80543 0
474.10 −1.41833 1.48141 −1.98835 0 −2.10112 2.50491i 8.49344 −6.80543 0
474.11 −1.40632 −2.91656 −2.02225 0 4.10162 11.7208i 8.46924 −0.493705 0
474.12 −1.40632 −2.91656 −2.02225 0 4.10162 11.7208i 8.46924 −0.493705 0
474.13 −0.382406 −3.15259 −3.85377 0 1.20557 4.26742i 3.00333 0.938823 0
474.14 −0.382406 −3.15259 −3.85377 0 1.20557 4.26742i 3.00333 0.938823 0
474.15 0.382406 3.15259 −3.85377 0 1.20557 4.26742i −3.00333 0.938823 0
474.16 0.382406 3.15259 −3.85377 0 1.20557 4.26742i −3.00333 0.938823 0
474.17 1.40632 2.91656 −2.02225 0 4.10162 11.7208i −8.46924 −0.493705 0
474.18 1.40632 2.91656 −2.02225 0 4.10162 11.7208i −8.46924 −0.493705 0
474.19 1.41833 −1.48141 −1.98835 0 −2.10112 2.50491i −8.49344 −6.80543 0
474.20 1.41833 −1.48141 −1.98835 0 −2.10112 2.50491i −8.49344 −6.80543 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.b odd 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.d 28
5.b even 2 1 inner 475.3.d.d 28
5.c odd 4 1 475.3.c.h 14
5.c odd 4 1 475.3.c.i yes 14
19.b odd 2 1 inner 475.3.d.d 28
95.d odd 2 1 inner 475.3.d.d 28
95.g even 4 1 475.3.c.h 14
95.g even 4 1 475.3.c.i yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.3.c.h 14 5.c odd 4 1
475.3.c.h 14 95.g even 4 1
475.3.c.i yes 14 5.c odd 4 1
475.3.c.i yes 14 95.g even 4 1
475.3.d.d 28 1.a even 1 1 trivial
475.3.d.d 28 5.b even 2 1 inner
475.3.d.d 28 19.b odd 2 1 inner
475.3.d.d 28 95.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 42 T_{2}^{12} + 677 T_{2}^{10} - 5313 T_{2}^{8} + 21125 T_{2}^{6} - 40138 T_{2}^{4} + 30565 T_{2}^{2} - 3675 \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\).