Properties

Label 475.2.e.d
Level $475$
Weight $2$
Character orbit 475.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3518667.1
Defining polynomial: \(x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -3 + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{4} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( -1 + \beta_{4} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -3 + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{4} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{6} + ( -1 + \beta_{4} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( -2 + \beta_{4} ) q^{11} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{12} + ( -5 - 5 \beta_{3} ) q^{13} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{14} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{16} + ( -\beta_{1} - 2 \beta_{5} ) q^{17} + ( -5 + \beta_{4} ) q^{18} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{19} + ( 2 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{21} + ( -3 \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{22} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{23} -7 \beta_{3} q^{24} -5 \beta_{2} q^{26} + ( 3 - \beta_{2} ) q^{27} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 1 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{29} -\beta_{2} q^{31} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{33} + ( 9 + 4 \beta_{1} - 3 \beta_{2} + 10 \beta_{3} + \beta_{4} - \beta_{5} ) q^{34} + ( -4 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{36} + ( 3 - 2 \beta_{2} - 5 \beta_{4} ) q^{37} + ( 7 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 3 \beta_{5} ) q^{38} -5 \beta_{4} q^{39} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{41} + ( -8 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{42} + ( -2 \beta_{1} - \beta_{5} ) q^{43} + ( 7 - 3 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{44} + ( -9 + 4 \beta_{2} - \beta_{4} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{47} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{48} + ( -2 + \beta_{2} - 2 \beta_{4} ) q^{49} + ( -1 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{51} + ( 10 \beta_{3} + 5 \beta_{5} ) q^{52} + ( 2 + 3 \beta_{1} + \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{53} + ( 3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{54} + ( 5 + \beta_{2} - \beta_{4} ) q^{56} + ( -2 + 2 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 3 - 2 \beta_{4} ) q^{58} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{59} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 4 \beta_{3} + \beta_{5} ) q^{62} + ( 4 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{63} + ( -5 + 2 \beta_{2} + 5 \beta_{4} ) q^{64} + ( -11 + 6 \beta_{1} - \beta_{2} - 6 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} ) q^{66} + ( -8 + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -14 + 7 \beta_{2} + \beta_{4} ) q^{68} + ( 1 - 5 \beta_{2} - 3 \beta_{4} ) q^{69} + ( 2 \beta_{1} - 8 \beta_{3} - 3 \beta_{5} ) q^{71} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{72} + ( \beta_{1} + 7 \beta_{3} + 2 \beta_{5} ) q^{73} + ( 8 \beta_{1} + 3 \beta_{3} - 8 \beta_{5} ) q^{74} + ( -1 - 2 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 6 + \beta_{2} - 3 \beta_{4} ) q^{77} + ( 5 \beta_{1} - 5 \beta_{3} - 10 \beta_{5} ) q^{78} + ( 2 \beta_{1} - 8 \beta_{3} + 2 \beta_{5} ) q^{79} + ( \beta_{1} - 7 \beta_{3} + \beta_{5} ) q^{81} + ( 13 - 4 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{82} + ( 2 - 3 \beta_{2} ) q^{83} + ( -11 - \beta_{2} + 6 \beta_{4} ) q^{84} + ( 12 + 2 \beta_{1} - 3 \beta_{2} + 11 \beta_{3} - \beta_{4} + \beta_{5} ) q^{86} + ( -4 - \beta_{2} + \beta_{4} ) q^{87} + ( 3 + 2 \beta_{2} - 2 \beta_{4} ) q^{88} + ( 2 + 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{89} + ( 5 - 5 \beta_{1} - 5 \beta_{4} + 5 \beta_{5} ) q^{91} + ( -6 \beta_{1} - 15 \beta_{3} - 2 \beta_{5} ) q^{92} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{93} + 7 q^{94} + ( 5 + 3 \beta_{2} - \beta_{4} ) q^{96} + ( \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{97} + ( -6 \beta_{3} - 5 \beta_{5} ) q^{98} + ( 5 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} + O(q^{10}) \) \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} - 10 q^{11} + 8 q^{12} - 15 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} - 28 q^{18} + 12 q^{21} - 8 q^{22} + 4 q^{23} + 21 q^{24} - 10 q^{26} + 16 q^{27} + 11 q^{28} + 2 q^{29} - 2 q^{31} - 6 q^{32} + 11 q^{33} + 25 q^{34} - 3 q^{36} + 4 q^{37} + 19 q^{38} - 10 q^{39} + 2 q^{41} - 23 q^{42} - q^{43} + 18 q^{44} - 48 q^{46} + 6 q^{47} + q^{48} - 14 q^{49} + 6 q^{51} - 35 q^{52} + 11 q^{53} - 10 q^{54} + 30 q^{56} + 19 q^{57} + 14 q^{58} - 6 q^{59} + 9 q^{61} - 13 q^{62} + 9 q^{63} - 16 q^{64} - 29 q^{66} - 20 q^{67} - 68 q^{68} - 10 q^{69} + 29 q^{71} + 11 q^{72} - 22 q^{73} + 7 q^{74} - 19 q^{76} + 32 q^{77} + 30 q^{78} + 24 q^{79} + 21 q^{81} + 31 q^{82} + 6 q^{83} - 56 q^{84} + 32 q^{86} - 24 q^{87} + 18 q^{88} + 14 q^{89} + 10 q^{91} + 41 q^{92} + 6 q^{93} + 42 q^{94} + 34 q^{96} + 7 q^{97} + 23 q^{98} + 13 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 7 x^{4} - 8 x^{3} + 43 x^{2} - 42 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 7 \nu^{4} - 49 \nu^{3} + 43 \nu^{2} - 42 \nu + 294 \)\()/259\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{5} + 5 \nu^{4} - 35 \nu^{3} - \nu^{2} - 215 \nu - 49 \)\()/259\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{5} + 35 \nu^{4} + 14 \nu^{3} + 215 \nu^{2} - 210 \nu + 952 \)\()/259\)
\(\beta_{5}\)\(=\)\((\)\( 18 \nu^{5} + 22 \nu^{4} + 105 \nu^{3} + 3 \nu^{2} + 608 \nu + 147 \)\()/259\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_{2} - 5\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 21 \beta_{3} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(6 \beta_{5} - 6 \beta_{4} - 25 \beta_{3} + 29 \beta_{2} - 35 \beta_{1} - 19\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\)

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
−1.25351 2.17114i
0.610938 + 1.05818i
1.14257 + 1.97899i
−1.25351 + 2.17114i
0.610938 1.05818i
1.14257 1.97899i
−1.25351 2.17114i 0.610938 + 1.05818i −2.14257 + 3.71104i 0 1.53163 2.65287i 0.221876 5.72889 0.753509 1.30512i 0
26.2 0.610938 + 1.05818i 1.14257 + 1.97899i 0.253509 0.439091i 0 −1.39608 + 2.41808i 1.28514 3.06327 −1.11094 + 1.92420i 0
26.3 1.14257 + 1.97899i −1.25351 2.17114i −1.61094 + 2.79023i 0 2.86445 4.96137i −3.50702 −2.79216 −1.64257 + 2.84502i 0
201.1 −1.25351 + 2.17114i 0.610938 1.05818i −2.14257 3.71104i 0 1.53163 + 2.65287i 0.221876 5.72889 0.753509 + 1.30512i 0
201.2 0.610938 1.05818i 1.14257 1.97899i 0.253509 + 0.439091i 0 −1.39608 2.41808i 1.28514 3.06327 −1.11094 1.92420i 0
201.3 1.14257 1.97899i −1.25351 + 2.17114i −1.61094 2.79023i 0 2.86445 + 4.96137i −3.50702 −2.79216 −1.64257 2.84502i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.3
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.d 6
5.b even 2 1 95.2.e.b 6
5.c odd 4 2 475.2.j.b 12
15.d odd 2 1 855.2.k.g 6
19.c even 3 1 inner 475.2.e.d 6
19.c even 3 1 9025.2.a.z 3
19.d odd 6 1 9025.2.a.ba 3
20.d odd 2 1 1520.2.q.j 6
95.h odd 6 1 1805.2.a.g 3
95.i even 6 1 95.2.e.b 6
95.i even 6 1 1805.2.a.h 3
95.m odd 12 2 475.2.j.b 12
285.n odd 6 1 855.2.k.g 6
380.p odd 6 1 1520.2.q.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 5.b even 2 1
95.2.e.b 6 95.i even 6 1
475.2.e.d 6 1.a even 1 1 trivial
475.2.e.d 6 19.c even 3 1 inner
475.2.j.b 12 5.c odd 4 2
475.2.j.b 12 95.m odd 12 2
855.2.k.g 6 15.d odd 2 1
855.2.k.g 6 285.n odd 6 1
1520.2.q.j 6 20.d odd 2 1
1520.2.q.j 6 380.p odd 6 1
1805.2.a.g 3 95.h odd 6 1
1805.2.a.h 3 95.i even 6 1
9025.2.a.z 3 19.c even 3 1
9025.2.a.ba 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 7 T_{2}^{4} - 8 T_{2}^{3} + 43 T_{2}^{2} - 42 T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 - 42 T + 43 T^{2} - 8 T^{3} + 7 T^{4} - T^{5} + T^{6} \)
$3$ \( 49 - 42 T + 43 T^{2} - 8 T^{3} + 7 T^{4} - T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( ( 1 - 5 T + 2 T^{2} + T^{3} )^{2} \)
$11$ \( ( -1 + 2 T + 5 T^{2} + T^{3} )^{2} \)
$13$ \( ( 25 + 5 T + T^{2} )^{3} \)
$17$ \( 49 - 308 T + 1943 T^{2} + 30 T^{3} + 45 T^{4} - T^{5} + T^{6} \)
$19$ \( 6859 + 133 T^{3} + T^{6} \)
$23$ \( 2401 - 1911 T + 1717 T^{2} + 58 T^{3} + 55 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( 1 + 5 T + 23 T^{2} + 12 T^{3} + 9 T^{4} - 2 T^{5} + T^{6} \)
$31$ \( ( -7 - 6 T + T^{2} + T^{3} )^{2} \)
$37$ \( ( 227 - 119 T - 2 T^{2} + T^{3} )^{2} \)
$41$ \( 1369 - 1591 T + 1923 T^{2} + 12 T^{3} + 47 T^{4} - 2 T^{5} + T^{6} \)
$43$ \( 14641 + 5324 T + 2057 T^{2} + 198 T^{3} + 45 T^{4} + T^{5} + T^{6} \)
$47$ \( 2401 - 343 T + 343 T^{2} - 56 T^{3} + 43 T^{4} - 6 T^{5} + T^{6} \)
$53$ \( 96721 - 13062 T + 5185 T^{2} - 160 T^{3} + 163 T^{4} - 11 T^{5} + T^{6} \)
$59$ \( 2401 + 343 T + 343 T^{2} + 56 T^{3} + 43 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( 2401 - 2401 T + 2842 T^{2} + 343 T^{3} + 130 T^{4} - 9 T^{5} + T^{6} \)
$67$ \( 7744 + 9504 T + 9904 T^{2} + 1984 T^{3} + 292 T^{4} + 20 T^{5} + T^{6} \)
$71$ \( 218089 - 110212 T + 42153 T^{2} - 5910 T^{3} + 605 T^{4} - 29 T^{5} + T^{6} \)
$73$ \( 5929 + 9009 T + 11995 T^{2} + 2420 T^{3} + 367 T^{4} + 22 T^{5} + T^{6} \)
$79$ \( 61504 + 28768 T + 19408 T^{2} - 3280 T^{3} + 460 T^{4} - 24 T^{5} + T^{6} \)
$83$ \( ( -77 - 54 T - 3 T^{2} + T^{3} )^{2} \)
$89$ \( 3136 - 2016 T + 2080 T^{2} + 392 T^{3} + 232 T^{4} - 14 T^{5} + T^{6} \)
$97$ \( 14641 - 7986 T + 5203 T^{2} + 220 T^{3} + 115 T^{4} - 7 T^{5} + T^{6} \)
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