Properties

Label 475.3.d.b
Level $475$
Weight $3$
Character orbit 475.d
Analytic conductor $12.943$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_{1} q^{7} + 5 \beta_{3} q^{8} + 4 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + \beta_{3} q^{3} + 9 q^{4} + 13 q^{6} + \beta_{1} q^{7} + 5 \beta_{3} q^{8} + 4 q^{9} -10 q^{11} + 9 \beta_{3} q^{12} -\beta_{3} q^{13} + \beta_{2} q^{14} + 29 q^{16} -3 \beta_{1} q^{17} + 4 \beta_{3} q^{18} + ( 6 - \beta_{2} ) q^{19} + \beta_{2} q^{21} -10 \beta_{3} q^{22} + 7 \beta_{1} q^{23} + 65 q^{24} -13 q^{26} -5 \beta_{3} q^{27} + 9 \beta_{1} q^{28} -\beta_{2} q^{29} -2 \beta_{2} q^{31} + 9 \beta_{3} q^{32} -10 \beta_{3} q^{33} -3 \beta_{2} q^{34} + 36 q^{36} -6 \beta_{3} q^{37} + ( -13 \beta_{1} + 6 \beta_{3} ) q^{38} -13 q^{39} + 2 \beta_{2} q^{41} + 13 \beta_{1} q^{42} -4 \beta_{1} q^{43} -90 q^{44} + 7 \beta_{2} q^{46} -2 \beta_{1} q^{47} + 29 \beta_{3} q^{48} + 24 q^{49} -3 \beta_{2} q^{51} -9 \beta_{3} q^{52} + 21 \beta_{3} q^{53} -65 q^{54} + 5 \beta_{2} q^{56} + ( -13 \beta_{1} + 6 \beta_{3} ) q^{57} -13 \beta_{1} q^{58} -\beta_{2} q^{59} -40 q^{61} -26 \beta_{1} q^{62} + 4 \beta_{1} q^{63} + q^{64} -130 q^{66} + 11 \beta_{3} q^{67} -27 \beta_{1} q^{68} + 7 \beta_{2} q^{69} + 6 \beta_{2} q^{71} + 20 \beta_{3} q^{72} + 21 \beta_{1} q^{73} -78 q^{74} + ( 54 - 9 \beta_{2} ) q^{76} -10 \beta_{1} q^{77} -13 \beta_{3} q^{78} + 2 \beta_{2} q^{79} -101 q^{81} + 26 \beta_{1} q^{82} -8 \beta_{1} q^{83} + 9 \beta_{2} q^{84} -4 \beta_{2} q^{86} -13 \beta_{1} q^{87} -50 \beta_{3} q^{88} -\beta_{2} q^{91} + 63 \beta_{1} q^{92} -26 \beta_{1} q^{93} -2 \beta_{2} q^{94} + 117 q^{96} + 34 \beta_{3} q^{97} + 24 \beta_{3} q^{98} -40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 36q^{4} + 52q^{6} + 16q^{9} + O(q^{10}) \) \( 4q + 36q^{4} + 52q^{6} + 16q^{9} - 40q^{11} + 116q^{16} + 24q^{19} + 260q^{24} - 52q^{26} + 144q^{36} - 52q^{39} - 360q^{44} + 96q^{49} - 260q^{54} - 160q^{61} + 4q^{64} - 520q^{66} - 312q^{74} + 216q^{76} - 404q^{81} + 468q^{96} - 160q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{3} + 20 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{3} + 50 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{2} + 5 \beta_{1}\)\()/5\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
2.30278i
2.30278i
1.30278i
1.30278i
−3.60555 −3.60555 9.00000 0 13.0000 5.00000i −18.0278 4.00000 0
474.2 −3.60555 −3.60555 9.00000 0 13.0000 5.00000i −18.0278 4.00000 0
474.3 3.60555 3.60555 9.00000 0 13.0000 5.00000i 18.0278 4.00000 0
474.4 3.60555 3.60555 9.00000 0 13.0000 5.00000i 18.0278 4.00000 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.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.b 4
5.b even 2 1 inner 475.3.d.b 4
5.c odd 4 1 19.3.b.b 2
5.c odd 4 1 475.3.c.b 2
15.e even 4 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.d.b 4
20.e even 4 1 304.3.e.d 2
40.i odd 4 1 1216.3.e.g 2
40.k even 4 1 1216.3.e.h 2
60.l odd 4 1 2736.3.o.d 2
95.d odd 2 1 inner 475.3.d.b 4
95.g even 4 1 19.3.b.b 2
95.g even 4 1 475.3.c.b 2
95.l even 12 2 361.3.d.b 4
95.m odd 12 2 361.3.d.b 4
95.q odd 36 6 361.3.f.d 12
95.r even 36 6 361.3.f.d 12
285.j odd 4 1 171.3.c.b 2
380.j odd 4 1 304.3.e.d 2
760.t even 4 1 1216.3.e.g 2
760.y odd 4 1 1216.3.e.h 2
1140.w even 4 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 5.c odd 4 1
19.3.b.b 2 95.g even 4 1
171.3.c.b 2 15.e even 4 1
171.3.c.b 2 285.j odd 4 1
304.3.e.d 2 20.e even 4 1
304.3.e.d 2 380.j odd 4 1
361.3.d.b 4 95.l even 12 2
361.3.d.b 4 95.m odd 12 2
361.3.f.d 12 95.q odd 36 6
361.3.f.d 12 95.r even 36 6
475.3.c.b 2 5.c odd 4 1
475.3.c.b 2 95.g even 4 1
475.3.d.b 4 1.a even 1 1 trivial
475.3.d.b 4 5.b even 2 1 inner
475.3.d.b 4 19.b odd 2 1 inner
475.3.d.b 4 95.d odd 2 1 inner
1216.3.e.g 2 40.i odd 4 1
1216.3.e.g 2 760.t even 4 1
1216.3.e.h 2 40.k even 4 1
1216.3.e.h 2 760.y odd 4 1
2736.3.o.d 2 60.l odd 4 1
2736.3.o.d 2 1140.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 13 \) acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -13 + T^{2} )^{2} \)
$3$ \( ( -13 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 25 + T^{2} )^{2} \)
$11$ \( ( 10 + T )^{4} \)
$13$ \( ( -13 + T^{2} )^{2} \)
$17$ \( ( 225 + T^{2} )^{2} \)
$19$ \( ( 361 - 12 T + T^{2} )^{2} \)
$23$ \( ( 1225 + T^{2} )^{2} \)
$29$ \( ( 325 + T^{2} )^{2} \)
$31$ \( ( 1300 + T^{2} )^{2} \)
$37$ \( ( -468 + T^{2} )^{2} \)
$41$ \( ( 1300 + T^{2} )^{2} \)
$43$ \( ( 400 + T^{2} )^{2} \)
$47$ \( ( 100 + T^{2} )^{2} \)
$53$ \( ( -5733 + T^{2} )^{2} \)
$59$ \( ( 325 + T^{2} )^{2} \)
$61$ \( ( 40 + T )^{4} \)
$67$ \( ( -1573 + T^{2} )^{2} \)
$71$ \( ( 11700 + T^{2} )^{2} \)
$73$ \( ( 11025 + T^{2} )^{2} \)
$79$ \( ( 1300 + T^{2} )^{2} \)
$83$ \( ( 1600 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( -15028 + T^{2} )^{2} \)
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