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: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{6} + (\beta_{4} - 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{6} + (\beta_{4} - 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9} + (\beta_{4} - 2) q^{11} + ( - \beta_{4} + 2 \beta_{2} + 1) q^{12} + ( - 5 \beta_{3} - 5) q^{13} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{14} + (\beta_{5} + \beta_{3} + \beta_1) q^{16} + ( - 2 \beta_{5} - \beta_1) q^{17} + (\beta_{4} - 5) q^{18} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{21} + (2 \beta_{5} + \beta_{3} - 3 \beta_1) q^{22} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{23} - 7 \beta_{3} q^{24} - 5 \beta_{2} q^{26} + ( - \beta_{2} + 3) q^{27} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{28} + (\beta_{5} - \beta_{4} - \beta_1 + 1) q^{29} - \beta_{2} q^{31} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{32} + ( - 2 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{33} + ( - \beta_{5} + \beta_{4} + 10 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 9) q^{34} + (2 \beta_{5} - \beta_{3} - 4 \beta_1) q^{36} + ( - 5 \beta_{4} - 2 \beta_{2} + 3) q^{37} + (3 \beta_{5} + 5 \beta_{3} - 2 \beta_{2} - \beta_1 + 7) q^{38} - 5 \beta_{4} q^{39} + (\beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{41} + ( - 3 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 8) q^{42} + ( - \beta_{5} - 2 \beta_1) q^{43} + (3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 3 \beta_1 + 7) q^{44} + ( - \beta_{4} + 4 \beta_{2} - 9) q^{46} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{47} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + \beta_1 + 2) q^{48} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{49} + ( - 4 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - \beta_1 - 1) q^{51} + (5 \beta_{5} + 10 \beta_{3}) q^{52} + ( - 4 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{53} + (\beta_{5} + 4 \beta_{3} + 3 \beta_1) q^{54} + ( - \beta_{4} + \beta_{2} + 5) q^{56} + ( - 2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{57} + ( - 2 \beta_{4} + 3) q^{58} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{59} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 5) q^{61} + (\beta_{5} + 4 \beta_{3}) q^{62} + (3 \beta_{5} - 3 \beta_{4} + \beta_{3} - 3 \beta_1 + 4) q^{63} + (5 \beta_{4} + 2 \beta_{2} - 5) q^{64} + ( - 5 \beta_{5} + 5 \beta_{4} - 6 \beta_{3} - \beta_{2} + 6 \beta_1 - 11) q^{66} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 8) q^{67} + (\beta_{4} + 7 \beta_{2} - 14) q^{68} + ( - 3 \beta_{4} - 5 \beta_{2} + 1) q^{69} + ( - 3 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{71} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{72} + (2 \beta_{5} + 7 \beta_{3} + \beta_1) q^{73} + ( - 8 \beta_{5} + 3 \beta_{3} + 8 \beta_1) q^{74} + (2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 1) q^{76} + ( - 3 \beta_{4} + \beta_{2} + 6) q^{77} + ( - 10 \beta_{5} - 5 \beta_{3} + 5 \beta_1) q^{78} + (2 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{79} + (\beta_{5} - 7 \beta_{3} + \beta_1) q^{81} + (4 \beta_{5} - 4 \beta_{4} + 9 \beta_{3} - 4 \beta_{2} + 13) q^{82} + ( - 3 \beta_{2} + 2) q^{83} + (6 \beta_{4} - \beta_{2} - 11) q^{84} + (\beta_{5} - \beta_{4} + 11 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 12) q^{86} + (\beta_{4} - \beta_{2} - 4) q^{87} + ( - 2 \beta_{4} + 2 \beta_{2} + 3) q^{88} + ( - 4 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 2) q^{89} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_1 + 5) q^{91} + ( - 2 \beta_{5} - 15 \beta_{3} - 6 \beta_1) q^{92} + ( - 2 \beta_{5} - \beta_{3} + \beta_1) q^{93} + 7 q^{94} + ( - \beta_{4} + 3 \beta_{2} + 5) q^{96} + (3 \beta_{5} - 3 \beta_{3} + \beta_1) q^{97} + ( - 5 \beta_{5} - 6 \beta_{3}) q^{98} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} - 10 q^{11} + 8 q^{12} - 15 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} - 28 q^{18} + 12 q^{21} - 8 q^{22} + 4 q^{23} + 21 q^{24} - 10 q^{26} + 16 q^{27} + 11 q^{28} + 2 q^{29} - 2 q^{31} - 6 q^{32} + 11 q^{33} + 25 q^{34} - 3 q^{36} + 4 q^{37} + 19 q^{38} - 10 q^{39} + 2 q^{41} - 23 q^{42} - q^{43} + 18 q^{44} - 48 q^{46} + 6 q^{47} + q^{48} - 14 q^{49} + 6 q^{51} - 35 q^{52} + 11 q^{53} - 10 q^{54} + 30 q^{56} + 19 q^{57} + 14 q^{58} - 6 q^{59} + 9 q^{61} - 13 q^{62} + 9 q^{63} - 16 q^{64} - 29 q^{66} - 20 q^{67} - 68 q^{68} - 10 q^{69} + 29 q^{71} + 11 q^{72} - 22 q^{73} + 7 q^{74} - 19 q^{76} + 32 q^{77} + 30 q^{78} + 24 q^{79} + 21 q^{81} + 31 q^{82} + 6 q^{83} - 56 q^{84} + 32 q^{86} - 24 q^{87} + 18 q^{88} + 14 q^{89} + 10 q^{91} + 41 q^{92} + 6 q^{93} + 42 q^{94} + 34 q^{96} + 7 q^{97} + 23 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 21\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \) Copy content Toggle raw display

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 \( T_{2}^{6} - T_{2}^{5} + 7T_{2}^{4} - 8T_{2}^{3} + 43T_{2}^{2} - 42T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 7 T^{4} - 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + 7 T^{4} - 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 5 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} + 2 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 25)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{5} + 45 T^{4} + 30 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} + 133T^{3} + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + 55 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + 9 T^{4} + 12 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 6 T - 7)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} - 119 T + 227)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + 47 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{5} + 45 T^{4} + 198 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + 43 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( T^{6} - 11 T^{5} + 163 T^{4} + \cdots + 96721 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 43 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} + 130 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} + 292 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$71$ \( T^{6} - 29 T^{5} + 605 T^{4} + \cdots + 218089 \) Copy content Toggle raw display
$73$ \( T^{6} + 22 T^{5} + 367 T^{4} + \cdots + 5929 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 460 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
$83$ \( (T^{3} - 3 T^{2} - 54 T - 77)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + 232 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} + 115 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
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