Properties

Label 475.2.b.c
Level $475$
Weight $2$
Character orbit 475.b
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{5} ) q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + ( -3 + 2 \beta_{2} + 3 \beta_{4} ) q^{4} + ( 2 - \beta_{2} ) q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( -4 \beta_{1} - \beta_{3} + 4 \beta_{5} ) q^{8} + ( 1 + \beta_{2} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{5} ) q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + ( -3 + 2 \beta_{2} + 3 \beta_{4} ) q^{4} + ( 2 - \beta_{2} ) q^{6} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( -4 \beta_{1} - \beta_{3} + 4 \beta_{5} ) q^{8} + ( 1 + \beta_{2} - \beta_{4} ) q^{9} + ( -\beta_{2} + 2 \beta_{4} ) q^{11} + ( \beta_{1} - 2 \beta_{5} ) q^{12} + ( -\beta_{1} + 2 \beta_{5} ) q^{13} + ( 4 - \beta_{2} - 4 \beta_{4} ) q^{14} + ( 9 - \beta_{2} - 8 \beta_{4} ) q^{16} + ( \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{17} + ( 3 \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{18} + q^{19} + ( -\beta_{2} + \beta_{4} ) q^{21} + ( -5 \beta_{1} + 2 \beta_{5} ) q^{22} + ( \beta_{3} + 3 \beta_{5} ) q^{23} + ( -1 + 4 \beta_{4} ) q^{24} + ( 5 - 2 \beta_{2} - 4 \beta_{4} ) q^{26} + ( \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{27} + ( 5 \beta_{1} + 2 \beta_{3} - 8 \beta_{5} ) q^{28} + ( 4 - 5 \beta_{2} ) q^{29} + ( -4 + 2 \beta_{2} + 5 \beta_{4} ) q^{31} + ( 7 \beta_{1} + 7 \beta_{3} - 9 \beta_{5} ) q^{32} + ( 5 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{33} + ( 4 - \beta_{2} - 2 \beta_{4} ) q^{34} + ( -8 + 5 \beta_{2} + 8 \beta_{4} ) q^{36} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{37} + ( \beta_{3} - \beta_{5} ) q^{38} + ( -1 + \beta_{2} + \beta_{4} ) q^{39} + ( 1 + 3 \beta_{2} - 5 \beta_{4} ) q^{41} + ( -3 \beta_{1} + \beta_{5} ) q^{42} + ( -6 \beta_{1} + 4 \beta_{3} + 5 \beta_{5} ) q^{43} + ( 9 - 4 \beta_{2} - 8 \beta_{4} ) q^{44} + ( 3 - 2 \beta_{2} - \beta_{4} ) q^{46} + ( -\beta_{1} + 2 \beta_{3} ) q^{47} + ( -6 \beta_{1} - \beta_{3} + \beta_{5} ) q^{48} + ( 1 + \beta_{2} + 3 \beta_{4} ) q^{49} + ( -2 - \beta_{2} - \beta_{4} ) q^{51} + ( 4 \beta_{1} + 5 \beta_{3} - 5 \beta_{5} ) q^{52} + ( 4 \beta_{1} - 6 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 6 - 2 \beta_{2} - 3 \beta_{4} ) q^{54} + ( -19 + 8 \beta_{2} + 14 \beta_{4} ) q^{56} + ( \beta_{1} - \beta_{3} ) q^{57} + ( -5 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} ) q^{58} + ( -4 \beta_{2} - 6 \beta_{4} ) q^{59} + ( -8 + 4 \beta_{2} + 3 \beta_{4} ) q^{61} + ( -8 \beta_{1} - 4 \beta_{3} + 9 \beta_{5} ) q^{62} + ( -4 \beta_{3} + \beta_{5} ) q^{63} + ( -28 + 14 \beta_{2} + 21 \beta_{4} ) q^{64} + ( 3 - 3 \beta_{2} + 5 \beta_{4} ) q^{66} + ( -5 \beta_{1} + 9 \beta_{3} + 5 \beta_{5} ) q^{67} + ( 5 \beta_{1} - 2 \beta_{3} - 10 \beta_{5} ) q^{68} + ( -2 + 3 \beta_{2} ) q^{69} + ( -3 - 4 \beta_{2} - 6 \beta_{4} ) q^{71} + ( -5 \beta_{1} - 6 \beta_{3} + 12 \beta_{5} ) q^{72} + ( -8 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{73} + ( 2 - \beta_{4} ) q^{74} + ( -3 + 2 \beta_{2} + 3 \beta_{4} ) q^{76} + ( -\beta_{1} + 4 \beta_{3} - 3 \beta_{5} ) q^{77} + ( -\beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{78} + ( -7 + 3 \beta_{2} ) q^{79} + ( 3 \beta_{2} - 4 \beta_{4} ) q^{81} + ( 13 \beta_{1} + \beta_{3} - 6 \beta_{5} ) q^{82} + ( 5 \beta_{3} + 6 \beta_{5} ) q^{83} + ( 5 - 3 \beta_{2} - 5 \beta_{4} ) q^{84} + ( 4 - \beta_{2} - 9 \beta_{4} ) q^{86} + ( 9 \beta_{1} - 4 \beta_{3} + 5 \beta_{5} ) q^{87} + ( 2 \beta_{1} + 9 \beta_{3} - 13 \beta_{5} ) q^{88} + ( 2 - \beta_{2} - 7 \beta_{4} ) q^{89} + ( -5 + 3 \beta_{2} + 5 \beta_{4} ) q^{91} + ( 5 \beta_{3} + 2 \beta_{5} ) q^{92} + ( 4 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{93} + ( -5 + 2 \beta_{2} + 2 \beta_{4} ) q^{94} + ( 9 - 2 \beta_{2} - 7 \beta_{4} ) q^{96} + ( \beta_{1} - 2 \beta_{5} ) q^{97} + ( -5 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{98} + ( -8 + 3 \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.24698i
1.80194i
0.445042i
0.445042i
1.80194i
1.24698i
2.80194i 0.554958i −5.85086 0 1.55496 3.04892i 10.7899i 2.69202 0
324.2 1.44504i 2.24698i −0.0881460 0 3.24698 1.35690i 2.76271i −2.04892 0
324.3 0.246980i 0.801938i 1.93900 0 0.198062 1.69202i 0.972853i 2.35690 0
324.4 0.246980i 0.801938i 1.93900 0 0.198062 1.69202i 0.972853i 2.35690 0
324.5 1.44504i 2.24698i −0.0881460 0 3.24698 1.35690i 2.76271i −2.04892 0
324.6 2.80194i 0.554958i −5.85086 0 1.55496 3.04892i 10.7899i 2.69202 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.b.c 6
5.b even 2 1 inner 475.2.b.c 6
5.c odd 4 1 475.2.a.d 3
5.c odd 4 1 475.2.a.h yes 3
15.e even 4 1 4275.2.a.z 3
15.e even 4 1 4275.2.a.bn 3
20.e even 4 1 7600.2.a.bn 3
20.e even 4 1 7600.2.a.bw 3
95.g even 4 1 9025.2.a.w 3
95.g even 4 1 9025.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.d 3 5.c odd 4 1
475.2.a.h yes 3 5.c odd 4 1
475.2.b.c 6 1.a even 1 1 trivial
475.2.b.c 6 5.b even 2 1 inner
4275.2.a.z 3 15.e even 4 1
4275.2.a.bn 3 15.e even 4 1
7600.2.a.bn 3 20.e even 4 1
7600.2.a.bw 3 20.e even 4 1
9025.2.a.w 3 95.g even 4 1
9025.2.a.be 3 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10 T_{2}^{4} + 17 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 17 T^{2} + 10 T^{4} + T^{6} \)
$3$ \( 1 + 5 T^{2} + 6 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 49 + 49 T^{2} + 14 T^{4} + T^{6} \)
$11$ \( ( -13 - 16 T - T^{2} + T^{3} )^{2} \)
$13$ \( 1 + 26 T^{2} + 13 T^{4} + T^{6} \)
$17$ \( 169 + 173 T^{2} + 34 T^{4} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( 169 + 153 T^{2} + 26 T^{4} + T^{6} \)
$29$ \( ( 91 - 42 T - 7 T^{2} + T^{3} )^{2} \)
$31$ \( ( 43 - 36 T + 5 T^{2} + T^{3} )^{2} \)
$37$ \( 1 + 54 T^{2} + 41 T^{4} + T^{6} \)
$41$ \( ( 421 - 114 T - T^{2} + T^{3} )^{2} \)
$43$ \( 85849 + 6179 T^{2} + 139 T^{4} + T^{6} \)
$47$ \( 169 + 94 T^{2} + 17 T^{4} + T^{6} \)
$53$ \( 94249 + 14691 T^{2} + 251 T^{4} + T^{6} \)
$59$ \( ( -328 - 32 T + 10 T^{2} + T^{3} )^{2} \)
$61$ \( ( -41 + 66 T + 17 T^{2} + T^{3} )^{2} \)
$67$ \( 312481 + 19046 T^{2} + 285 T^{4} + T^{6} \)
$71$ \( ( -307 + 55 T + 19 T^{2} + T^{3} )^{2} \)
$73$ \( 214369 + 29826 T^{2} + 341 T^{4} + T^{6} \)
$79$ \( ( 97 + 87 T + 18 T^{2} + T^{3} )^{2} \)
$83$ \( 19321 + 3618 T^{2} + 173 T^{4} + T^{6} \)
$89$ \( ( -281 - 99 T + 2 T^{2} + T^{3} )^{2} \)
$97$ \( 1 + 26 T^{2} + 13 T^{4} + T^{6} \)
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