Properties

Label 475.2.e.b
Level $475$
Weight $2$
Character orbit 475.e
Analytic conductor $3.793$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} -\zeta_{6} q^{9} + 3 q^{11} -4 q^{12} + 2 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -2 - 3 \zeta_{6} ) q^{19} + ( -8 + 8 \zeta_{6} ) q^{21} -4 q^{27} + 8 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} -7 q^{31} + ( -6 + 6 \zeta_{6} ) q^{33} + ( 2 - 2 \zeta_{6} ) q^{36} -8 q^{37} -4 q^{39} + ( 6 - 6 \zeta_{6} ) q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} -8 \zeta_{6} q^{48} + 9 q^{49} + 12 \zeta_{6} q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} -6 \zeta_{6} q^{53} + ( 10 - 4 \zeta_{6} ) q^{57} + ( 15 - 15 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} -4 \zeta_{6} q^{63} -8 q^{64} + 2 \zeta_{6} q^{67} + 12 q^{68} + ( 3 - 3 \zeta_{6} ) q^{71} + ( 8 - 8 \zeta_{6} ) q^{73} + ( 6 - 10 \zeta_{6} ) q^{76} + 12 q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -12 q^{83} -16 q^{84} -6 q^{87} + 15 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( 14 - 14 \zeta_{6} ) q^{93} + ( 8 - 8 \zeta_{6} ) q^{97} -3 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9} + 6 q^{11} - 8 q^{12} + 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 8 q^{21} - 8 q^{27} + 8 q^{28} + 3 q^{29} - 14 q^{31} - 6 q^{33} + 2 q^{36} - 16 q^{37} - 8 q^{39} + 6 q^{41} - 4 q^{43} + 6 q^{44} + 6 q^{47} - 8 q^{48} + 18 q^{49} + 12 q^{51} - 4 q^{52} - 6 q^{53} + 16 q^{57} + 15 q^{59} - 5 q^{61} - 4 q^{63} - 16 q^{64} + 2 q^{67} + 24 q^{68} + 3 q^{71} + 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} + 11 q^{81} - 24 q^{83} - 32 q^{84} - 12 q^{87} + 15 q^{89} + 8 q^{91} + 14 q^{93} + 8 q^{97} - 3 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 1.00000 1.73205i 0 0 4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 1.00000 + 1.73205i 0 0 4.00000 0 −0.500000 0.866025i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.b 2
5.b even 2 1 95.2.e.a 2
5.c odd 4 2 475.2.j.a 4
15.d odd 2 1 855.2.k.b 2
19.c even 3 1 inner 475.2.e.b 2
19.c even 3 1 9025.2.a.g 1
19.d odd 6 1 9025.2.a.e 1
20.d odd 2 1 1520.2.q.c 2
95.h odd 6 1 1805.2.a.b 1
95.i even 6 1 95.2.e.a 2
95.i even 6 1 1805.2.a.a 1
95.m odd 12 2 475.2.j.a 4
285.n odd 6 1 855.2.k.b 2
380.p odd 6 1 1520.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 5.b even 2 1
95.2.e.a 2 95.i even 6 1
475.2.e.b 2 1.a even 1 1 trivial
475.2.e.b 2 19.c even 3 1 inner
475.2.j.a 4 5.c odd 4 2
475.2.j.a 4 95.m odd 12 2
855.2.k.b 2 15.d odd 2 1
855.2.k.b 2 285.n odd 6 1
1520.2.q.c 2 20.d odd 2 1
1520.2.q.c 2 380.p odd 6 1
1805.2.a.a 1 95.i even 6 1
1805.2.a.b 1 95.h odd 6 1
9025.2.a.e 1 19.d odd 6 1
9025.2.a.g 1 19.c even 3 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 19 + 7 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 9 - 3 T + T^{2} \)
$31$ \( ( 7 + T )^{2} \)
$37$ \( ( 8 + T )^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 16 + 4 T + T^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 225 - 15 T + T^{2} \)
$61$ \( 25 + 5 T + T^{2} \)
$67$ \( 4 - 2 T + T^{2} \)
$71$ \( 9 - 3 T + T^{2} \)
$73$ \( 64 - 8 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 225 - 15 T + T^{2} \)
$97$ \( 64 - 8 T + T^{2} \)
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