Properties

Label 475.3.d.c
Level $475$
Weight $3$
Character orbit 475.d
Analytic conductor $12.943$
Analytic rank $0$
Dimension $24$
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: \(24\)
Twist minimal: no (minimal twist has level 95)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{4} - 56q^{6} + 96q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{4} - 56q^{6} + 96q^{9} + 64q^{11} - 88q^{16} - 16q^{19} - 200q^{24} + 216q^{26} - 160q^{36} - 152q^{39} + 512q^{44} - 144q^{49} + 152q^{54} - 592q^{61} - 376q^{64} + 304q^{66} - 272q^{74} + 496q^{76} - 744q^{81} - 88q^{96} + 624q^{99} + O(q^{100}) \)

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 −3.38208 4.08851 7.43849 0 −13.8277 4.56923i −11.6293 7.71590 0
474.2 −3.38208 4.08851 7.43849 0 −13.8277 4.56923i −11.6293 7.71590 0
474.3 −2.84623 2.18419 4.10101 0 −6.21669 9.15076i −0.287487 −4.22932 0
474.4 −2.84623 2.18419 4.10101 0 −6.21669 9.15076i −0.287487 −4.22932 0
474.5 −2.36559 −3.55563 1.59600 0 8.41115 11.0785i 5.68687 3.64250 0
474.6 −2.36559 −3.55563 1.59600 0 8.41115 11.0785i 5.68687 3.64250 0
474.7 −1.88109 −0.840697 −0.461500 0 1.58143 2.31342i 8.39248 −8.29323 0
474.8 −1.88109 −0.840697 −0.461500 0 1.58143 2.31342i 8.39248 −8.29323 0
474.9 −1.14149 4.13699 −2.69701 0 −4.72232 7.54444i 7.64455 8.11468 0
474.10 −1.14149 4.13699 −2.69701 0 −4.72232 7.54444i 7.64455 8.11468 0
474.11 −0.151673 −5.10387 −3.97700 0 0.774118 6.35478i 1.20989 17.0495 0
474.12 −0.151673 −5.10387 −3.97700 0 0.774118 6.35478i 1.20989 17.0495 0
474.13 0.151673 5.10387 −3.97700 0 0.774118 6.35478i −1.20989 17.0495 0
474.14 0.151673 5.10387 −3.97700 0 0.774118 6.35478i −1.20989 17.0495 0
474.15 1.14149 −4.13699 −2.69701 0 −4.72232 7.54444i −7.64455 8.11468 0
474.16 1.14149 −4.13699 −2.69701 0 −4.72232 7.54444i −7.64455 8.11468 0
474.17 1.88109 0.840697 −0.461500 0 1.58143 2.31342i −8.39248 −8.29323 0
474.18 1.88109 0.840697 −0.461500 0 1.58143 2.31342i −8.39248 −8.29323 0
474.19 2.36559 3.55563 1.59600 0 8.41115 11.0785i −5.68687 3.64250 0
474.20 2.36559 3.55563 1.59600 0 8.41115 11.0785i −5.68687 3.64250 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.24
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.c 24
5.b even 2 1 inner 475.3.d.c 24
5.c odd 4 1 95.3.c.a 12
5.c odd 4 1 475.3.c.g 12
15.e even 4 1 855.3.e.a 12
19.b odd 2 1 inner 475.3.d.c 24
20.e even 4 1 1520.3.h.a 12
95.d odd 2 1 inner 475.3.d.c 24
95.g even 4 1 95.3.c.a 12
95.g even 4 1 475.3.c.g 12
285.j odd 4 1 855.3.e.a 12
380.j odd 4 1 1520.3.h.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.c.a 12 5.c odd 4 1
95.3.c.a 12 95.g even 4 1
475.3.c.g 12 5.c odd 4 1
475.3.c.g 12 95.g even 4 1
475.3.d.c 24 1.a even 1 1 trivial
475.3.d.c 24 5.b even 2 1 inner
475.3.d.c 24 19.b odd 2 1 inner
475.3.d.c 24 95.d odd 2 1 inner
855.3.e.a 12 15.e even 4 1
855.3.e.a 12 285.j odd 4 1
1520.3.h.a 12 20.e even 4 1
1520.3.h.a 12 380.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 30 T_{2}^{10} + 329 T_{2}^{8} - 1620 T_{2}^{6} + 3479 T_{2}^{4} - 2470 T_{2}^{2} + 55 \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\).