Properties

Label 475.2.p.b
Level $475$
Weight $2$
Character orbit 475.p
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -2 - 2 \zeta_{12}^{3} ) q^{7} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{9} - q^{11} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{12} + ( -2 + \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{13} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{17} + ( -5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{19} + ( -4 - 4 \zeta_{12}^{2} ) q^{21} + ( 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{23} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{28} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{29} + ( -5 + 10 \zeta_{12}^{2} ) q^{31} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + ( 6 - 6 \zeta_{12}^{2} ) q^{36} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12}^{3} q^{39} + ( -6 - 6 \zeta_{12}^{2} ) q^{41} + ( -8 + 8 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{43} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{44} + ( 5 \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{47} + ( 8 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{48} + \zeta_{12}^{3} q^{49} + ( 12 - 6 \zeta_{12}^{2} ) q^{51} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{52} + ( 2 - \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{53} + ( -7 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{57} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{59} + 7 \zeta_{12}^{2} q^{61} + ( -6 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} + ( 6 - 6 \zeta_{12}^{3} ) q^{68} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{69} + ( 5 + 5 \zeta_{12}^{2} ) q^{71} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{73} + ( -10 + 6 \zeta_{12}^{2} ) q^{76} + ( 2 + 2 \zeta_{12}^{3} ) q^{77} + ( 3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{84} + ( 3 + 3 \zeta_{12}^{3} ) q^{87} + ( -7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{89} + ( 8 - 4 \zeta_{12}^{2} ) q^{91} + ( 8 + 8 \zeta_{12} - 8 \zeta_{12}^{2} ) q^{92} + ( -15 + 15 \zeta_{12} + 15 \zeta_{12}^{2} ) q^{93} + ( -4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{97} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 8 q^{7} + O(q^{10}) \) \( 4 q + 6 q^{3} - 8 q^{7} - 4 q^{11} - 6 q^{13} + 8 q^{16} + 6 q^{17} - 24 q^{21} + 8 q^{23} - 8 q^{28} - 6 q^{33} + 12 q^{36} - 36 q^{41} - 16 q^{43} - 10 q^{47} + 24 q^{48} + 36 q^{51} + 12 q^{52} + 6 q^{53} - 12 q^{57} + 14 q^{61} - 12 q^{63} + 6 q^{67} + 24 q^{68} + 30 q^{71} + 4 q^{73} - 28 q^{76} + 8 q^{77} - 18 q^{81} + 12 q^{87} + 24 q^{91} + 16 q^{92} - 30 q^{93} - 24 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 0.633975 2.36603i −1.73205 1.00000i 0 0 −2.00000 2.00000i 0 −2.59808 1.50000i 0
293.1 0 0.633975 + 2.36603i −1.73205 + 1.00000i 0 0 −2.00000 + 2.00000i 0 −2.59808 + 1.50000i 0
407.1 0 2.36603 0.633975i 1.73205 1.00000i 0 0 −2.00000 2.00000i 0 2.59808 1.50000i 0
468.1 0 2.36603 + 0.633975i 1.73205 + 1.00000i 0 0 −2.00000 + 2.00000i 0 2.59808 + 1.50000i 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
5.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.p.b 4
5.b even 2 1 95.2.l.b 4
5.c odd 4 1 95.2.l.b 4
5.c odd 4 1 inner 475.2.p.b 4
15.d odd 2 1 855.2.cj.b 4
15.e even 4 1 855.2.cj.b 4
19.d odd 6 1 inner 475.2.p.b 4
95.h odd 6 1 95.2.l.b 4
95.l even 12 1 95.2.l.b 4
95.l even 12 1 inner 475.2.p.b 4
285.q even 6 1 855.2.cj.b 4
285.w odd 12 1 855.2.cj.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.l.b 4 5.b even 2 1
95.2.l.b 4 5.c odd 4 1
95.2.l.b 4 95.h odd 6 1
95.2.l.b 4 95.l even 12 1
475.2.p.b 4 1.a even 1 1 trivial
475.2.p.b 4 5.c odd 4 1 inner
475.2.p.b 4 19.d odd 6 1 inner
475.2.p.b 4 95.l even 12 1 inner
855.2.cj.b 4 15.d odd 2 1
855.2.cj.b 4 15.e even 4 1
855.2.cj.b 4 285.q even 6 1
855.2.cj.b 4 285.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\):

\( T_{2} \)
\( T_{3}^{4} - 6 T_{3}^{3} + 18 T_{3}^{2} - 36 T_{3} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 8 + 4 T + T^{2} )^{2} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 361 - 37 T^{2} + T^{4} \)
$23$ \( 1024 - 256 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( 9 + 3 T^{2} + T^{4} \)
$31$ \( ( 75 + T^{2} )^{2} \)
$37$ \( 576 + T^{4} \)
$41$ \( ( 108 + 18 T + T^{2} )^{2} \)
$43$ \( 16384 + 2048 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$47$ \( 2500 + 500 T + 50 T^{2} + 10 T^{3} + T^{4} \)
$53$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( ( 49 - 7 T + T^{2} )^{2} \)
$67$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 75 - 15 T + T^{2} )^{2} \)
$73$ \( 64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 729 + 27 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( 21609 + 147 T^{2} + T^{4} \)
$97$ \( 9216 + 2304 T + 288 T^{2} + 24 T^{3} + T^{4} \)
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