Properties

Label 475.2.b.b
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: 6.0.1827904.1
Defining polynomial: \(x^{6} + 9 x^{4} + 14 x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{4} + \beta_{5} ) q^{2} + ( \beta_{1} + \beta_{4} ) q^{3} + ( -1 - \beta_{3} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{3} ) q^{6} + ( \beta_{1} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{8} + ( -1 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{4} + \beta_{5} ) q^{2} + ( \beta_{1} + \beta_{4} ) q^{3} + ( -1 - \beta_{3} ) q^{4} + ( -2 + \beta_{2} + 2 \beta_{3} ) q^{6} + ( \beta_{1} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{8} + ( -1 + \beta_{2} + \beta_{3} ) q^{9} + \beta_{2} q^{11} + ( -\beta_{1} - 3 \beta_{4} + \beta_{5} ) q^{12} + ( -3 \beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{13} + ( -2 - \beta_{2} ) q^{14} + ( -3 + \beta_{2} + 2 \beta_{3} ) q^{16} + ( -\beta_{1} - 5 \beta_{4} ) q^{17} + ( -\beta_{1} - 3 \beta_{4} ) q^{18} - q^{19} + ( -2 - \beta_{2} + \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{22} + ( -2 \beta_{1} + \beta_{4} + 3 \beta_{5} ) q^{23} -5 q^{24} + ( -3 + 2 \beta_{2} - 4 \beta_{3} ) q^{26} + ( 3 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{27} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{28} + ( 2 - \beta_{2} ) q^{29} + ( -2 \beta_{2} + \beta_{3} ) q^{31} + ( \beta_{1} - 2 \beta_{5} ) q^{32} + ( \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{33} + ( 2 - 5 \beta_{2} - 6 \beta_{3} ) q^{34} + ( -\beta_{2} - 2 \beta_{3} ) q^{36} + ( -5 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( \beta_{4} - \beta_{5} ) q^{38} + ( 1 + 5 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 3 - 3 \beta_{2} - 5 \beta_{3} ) q^{41} + ( -3 \beta_{1} + \beta_{5} ) q^{42} + ( 2 \beta_{1} + \beta_{4} - 6 \beta_{5} ) q^{43} + q^{44} + ( -5 + 4 \beta_{2} - \beta_{3} ) q^{46} + ( -\beta_{1} - 5 \beta_{4} + 5 \beta_{5} ) q^{47} + ( -2 \beta_{1} - \beta_{4} - 3 \beta_{5} ) q^{48} + ( 3 - 3 \beta_{2} + \beta_{3} ) q^{49} + ( 8 - 5 \beta_{2} - \beta_{3} ) q^{51} + ( 4 \beta_{1} + 11 \beta_{4} - 7 \beta_{5} ) q^{52} + ( -2 \beta_{1} + 9 \beta_{4} + 2 \beta_{5} ) q^{53} + 5 \beta_{3} q^{54} + ( -3 - 2 \beta_{2} + 2 \beta_{3} ) q^{56} + ( -\beta_{1} - \beta_{4} ) q^{57} + ( -\beta_{1} - \beta_{4} + 3 \beta_{5} ) q^{58} + ( 4 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{61} + ( -4 \beta_{1} - \beta_{4} + 4 \beta_{5} ) q^{62} -\beta_{4} q^{63} + ( -2 + 5 \beta_{3} ) q^{64} + ( 1 - 3 \beta_{2} - \beta_{3} ) q^{66} + ( \beta_{1} + 6 \beta_{4} - 4 \beta_{5} ) q^{67} + ( 5 \beta_{1} + 11 \beta_{4} - 5 \beta_{5} ) q^{68} + ( -4 + 7 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 1 + 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} ) q^{73} + ( 4 - 2 \beta_{2} - 9 \beta_{3} ) q^{74} + ( 1 + \beta_{3} ) q^{76} + ( 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{77} + ( -\beta_{1} - 15 \beta_{4} + 2 \beta_{5} ) q^{78} + ( 7 + \beta_{2} - 4 \beta_{3} ) q^{79} + ( -8 + \beta_{2} + 2 \beta_{3} ) q^{81} + ( 7 \beta_{1} + 15 \beta_{4} - 4 \beta_{5} ) q^{82} + ( 6 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} ) q^{83} + ( -1 - \beta_{2} - \beta_{3} ) q^{84} + ( 14 - 5 \beta_{2} + 3 \beta_{3} ) q^{86} + ( \beta_{1} + 4 \beta_{4} + \beta_{5} ) q^{87} + ( 2 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{88} + ( 4 + 3 \beta_{2} + 5 \beta_{3} ) q^{89} + ( 5 + 5 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 2 \beta_{1} + 6 \beta_{4} - 5 \beta_{5} ) q^{92} + ( -2 \beta_{1} + 6 \beta_{4} + \beta_{5} ) q^{93} + ( -13 - 6 \beta_{3} ) q^{94} + ( 3 - 4 \beta_{2} - 3 \beta_{3} ) q^{96} + ( 3 \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{97} + ( -5 \beta_{1} - 3 \beta_{4} + 8 \beta_{5} ) q^{98} + ( 2 - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + O(q^{10}) \) \( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 14 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 5 \nu^{2} - 1 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 10 \nu^{2} + 14 \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 10 \nu^{3} + 19 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} - 25 \nu^{3} - 27 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 3 \beta_{4} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{3} + 10 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(-10 \beta_{5} - 25 \beta_{4} + 41 \beta_{1}\)

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
0.273891i
1.37720i
2.65109i
2.65109i
1.37720i
0.273891i
2.37720i 1.27389i −3.65109 0 3.02830 0.726109i 3.92498i 1.37720 0
324.2 1.65109i 2.37720i −0.726109 0 −3.92498 0.377203i 2.10331i −2.65109 0
324.3 1.27389i 1.65109i 0.377203 0 −2.10331 3.65109i 3.02830i 0.273891 0
324.4 1.27389i 1.65109i 0.377203 0 −2.10331 3.65109i 3.02830i 0.273891 0
324.5 1.65109i 2.37720i −0.726109 0 −3.92498 0.377203i 2.10331i −2.65109 0
324.6 2.37720i 1.27389i −3.65109 0 3.02830 0.726109i 3.92498i 1.37720 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.b 6
5.b even 2 1 inner 475.2.b.b 6
5.c odd 4 1 475.2.a.e 3
5.c odd 4 1 475.2.a.g yes 3
15.e even 4 1 4275.2.a.ba 3
15.e even 4 1 4275.2.a.bm 3
20.e even 4 1 7600.2.a.bh 3
20.e even 4 1 7600.2.a.cc 3
95.g even 4 1 9025.2.a.y 3
95.g even 4 1 9025.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.e 3 5.c odd 4 1
475.2.a.g yes 3 5.c odd 4 1
475.2.b.b 6 1.a even 1 1 trivial
475.2.b.b 6 5.b even 2 1 inner
4275.2.a.ba 3 15.e even 4 1
4275.2.a.bm 3 15.e even 4 1
7600.2.a.bh 3 20.e even 4 1
7600.2.a.cc 3 20.e even 4 1
9025.2.a.y 3 95.g even 4 1
9025.2.a.bc 3 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10 T_{2}^{4} + 29 T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + 29 T^{2} + 10 T^{4} + T^{6} \)
$3$ \( 25 + 29 T^{2} + 10 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 + 9 T^{2} + 14 T^{4} + T^{6} \)
$11$ \( ( -1 - 4 T - T^{2} + T^{3} )^{2} \)
$13$ \( 10609 + 1914 T^{2} + 81 T^{4} + T^{6} \)
$17$ \( 6241 + 1509 T^{2} + 74 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 15625 + 2081 T^{2} + 82 T^{4} + T^{6} \)
$29$ \( ( 5 + 4 T - 5 T^{2} + T^{3} )^{2} \)
$31$ \( ( 53 - 30 T + T^{2} + T^{3} )^{2} \)
$37$ \( 156025 + 9426 T^{2} + 173 T^{4} + T^{6} \)
$41$ \( ( 155 - 82 T - T^{2} + T^{3} )^{2} \)
$43$ \( 100489 + 15939 T^{2} + 251 T^{4} + T^{6} \)
$47$ \( 96721 + 9694 T^{2} + 209 T^{4} + T^{6} \)
$53$ \( 819025 + 35699 T^{2} + 355 T^{4} + T^{6} \)
$59$ \( ( 200 - 40 T - 6 T^{2} + T^{3} )^{2} \)
$61$ \( ( -1 - 10 T - 3 T^{2} + T^{3} )^{2} \)
$67$ \( 28561 + 4394 T^{2} + 169 T^{4} + T^{6} \)
$71$ \( ( 47 - T - 7 T^{2} + T^{3} )^{2} \)
$73$ \( 2809 + 794 T^{2} + 61 T^{4} + T^{6} \)
$79$ \( ( 395 + 17 T - 18 T^{2} + T^{3} )^{2} \)
$83$ \( 17161 + 72114 T^{2} + 549 T^{4} + T^{6} \)
$89$ \( ( 125 + 51 T - 20 T^{2} + T^{3} )^{2} \)
$97$ \( 28561 + 4394 T^{2} + 169 T^{4} + T^{6} \)
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