Properties

Label 475.2.e.c
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} + 4 q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} + 4 q^{7} + 3 \zeta_{6} q^{9} - q^{11} -2 \zeta_{6} q^{13} + ( 8 - 8 \zeta_{6} ) q^{14} + ( 4 - 4 \zeta_{6} ) q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} + 6 q^{18} + ( -2 - 3 \zeta_{6} ) q^{19} + ( -2 + 2 \zeta_{6} ) q^{22} + 6 \zeta_{6} q^{23} -4 q^{26} -8 \zeta_{6} q^{28} -9 \zeta_{6} q^{29} -7 q^{31} -8 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( 6 - 6 \zeta_{6} ) q^{36} -2 q^{37} + ( -10 + 4 \zeta_{6} ) q^{38} + ( -2 + 2 \zeta_{6} ) q^{41} + ( 2 - 2 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} + 12 q^{46} + 6 \zeta_{6} q^{47} + 9 q^{49} + ( -4 + 4 \zeta_{6} ) q^{52} + 4 \zeta_{6} q^{53} -18 q^{58} + ( -9 + 9 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + ( -14 + 14 \zeta_{6} ) q^{62} + 12 \zeta_{6} q^{63} -8 q^{64} -10 \zeta_{6} q^{67} + 4 q^{68} + ( -1 + \zeta_{6} ) q^{71} + ( 10 - 10 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + ( -6 + 10 \zeta_{6} ) q^{76} -4 q^{77} + ( -1 + \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 4 \zeta_{6} q^{82} + 6 q^{83} -4 \zeta_{6} q^{86} + 11 \zeta_{6} q^{89} -8 \zeta_{6} q^{91} + ( 12 - 12 \zeta_{6} ) q^{92} + 12 q^{94} + ( -6 + 6 \zeta_{6} ) q^{97} + ( 18 - 18 \zeta_{6} ) q^{98} -3 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 3 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} + 8 q^{14} + 4 q^{16} - 2 q^{17} + 12 q^{18} - 7 q^{19} - 2 q^{22} + 6 q^{23} - 8 q^{26} - 8 q^{28} - 9 q^{29} - 14 q^{31} - 8 q^{32} + 4 q^{34} + 6 q^{36} - 4 q^{37} - 16 q^{38} - 2 q^{41} + 2 q^{43} + 2 q^{44} + 24 q^{46} + 6 q^{47} + 18 q^{49} - 4 q^{52} + 4 q^{53} - 36 q^{58} - 9 q^{59} + 7 q^{61} - 14 q^{62} + 12 q^{63} - 16 q^{64} - 10 q^{67} + 8 q^{68} - q^{71} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 8 q^{77} - q^{79} - 9 q^{81} + 4 q^{82} + 12 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} + 12 q^{92} + 24 q^{94} - 6 q^{97} + 18 q^{98} - 3 q^{99} + O(q^{100}) \)

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
1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0 0 4.00000 0 1.50000 2.59808i 0
201.1 1.00000 1.73205i 0 −1.00000 1.73205i 0 0 4.00000 0 1.50000 + 2.59808i 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.c 2
5.b even 2 1 475.2.e.a 2
5.c odd 4 2 95.2.i.a 4
15.e even 4 2 855.2.be.a 4
19.c even 3 1 inner 475.2.e.c 2
19.c even 3 1 9025.2.a.b 1
19.d odd 6 1 9025.2.a.j 1
95.h odd 6 1 9025.2.a.a 1
95.i even 6 1 475.2.e.a 2
95.i even 6 1 9025.2.a.i 1
95.l even 12 2 1805.2.b.a 2
95.m odd 12 2 95.2.i.a 4
95.m odd 12 2 1805.2.b.b 2
285.v even 12 2 855.2.be.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 5.c odd 4 2
95.2.i.a 4 95.m odd 12 2
475.2.e.a 2 5.b even 2 1
475.2.e.a 2 95.i even 6 1
475.2.e.c 2 1.a even 1 1 trivial
475.2.e.c 2 19.c even 3 1 inner
855.2.be.a 4 15.e even 4 2
855.2.be.a 4 285.v even 12 2
1805.2.b.a 2 95.l even 12 2
1805.2.b.b 2 95.m odd 12 2
9025.2.a.a 1 95.h odd 6 1
9025.2.a.b 1 19.c even 3 1
9025.2.a.i 1 95.i even 6 1
9025.2.a.j 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 19 + 7 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 81 + 9 T + T^{2} \)
$31$ \( ( 7 + T )^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( 4 + 2 T + T^{2} \)
$43$ \( 4 - 2 T + T^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 81 + 9 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( 1 + T + T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 121 - 11 T + T^{2} \)
$97$ \( 36 + 6 T + T^{2} \)
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