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Properties

Label 475.2.b.e
Level $475$
Weight $2$
Character orbit 475.b
Analytic conductor $3.793$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [475,2,Mod(324,475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("475.324"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2058981376.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_1 q^{3} + ( - \beta_{3} - 2) q^{4} + ( - \beta_{3} - \beta_{2}) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - 2) q^{9} + ( - \beta_{4} + \beta_{3} + 1) q^{11}+ \cdots + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 120\nu^{7} - 143\nu^{6} + 138\nu^{5} + 30\nu^{4} + 2592\nu^{3} - 2311\nu^{2} + 3834\nu - 552 ) / 1631 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -240\nu^{7} + 286\nu^{6} - 276\nu^{5} - 60\nu^{4} - 5184\nu^{3} + 4622\nu^{2} - 1144\nu - 527 ) / 1631 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 388\nu^{7} - 408\nu^{6} + 120\nu^{5} + 97\nu^{4} + 7076\nu^{3} - 6548\nu^{2} + 1632\nu - 480 ) / 1631 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -582\nu^{7} + 612\nu^{6} - 180\nu^{5} - 961\nu^{4} - 10614\nu^{3} + 9822\nu^{2} - 2448\nu - 7435 ) / 1631 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1104\nu^{7} - 1968\nu^{6} + 1922\nu^{5} + 276\nu^{4} + 19932\nu^{3} - 32352\nu^{2} + 30706\nu - 4426 ) / 1631 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1104\nu^{7} + 1968\nu^{6} - 1922\nu^{5} - 276\nu^{4} - 19932\nu^{3} + 33983\nu^{2} - 30706\nu + 4426 ) / 1631 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2168\nu^{7} - 4432\nu^{6} + 3798\nu^{5} + 542\nu^{4} + 39000\nu^{3} - 76438\nu^{2} + 60134\nu - 8668 ) / 1631 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{3} - 4\beta_{2} + 8\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{4} - 3\beta_{3} - 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{7} - 12\beta_{6} + 4\beta_{5} - 12\beta_{3} - 17\beta_{2} - 34\beta _1 - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{7} - 23\beta_{6} - 17\beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{7} - 62\beta_{6} + 13\beta_{5} + 2\beta_{4} + 60\beta_{3} + 74\beta_{2} - 148\beta _1 + 50 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.52153 − 1.52153i
−1.43917 + 1.43917i
0.148421 + 0.148421i
0.769222 + 0.769222i
0.769222 − 0.769222i
0.148421 − 0.148421i
−1.43917 − 1.43917i
1.52153 + 1.52153i
− 2.63010i − 3.04306i −4.91744 0 −8.00355 − 0.574672i 7.67316i −6.26020 0
324.2 − 2.14243i 2.87834i −2.59002 0 6.16666 3.10482i 1.26409i −5.28487 0
324.3 − 1.95594i 0.296842i −1.82571 0 0.580605 3.56331i − 0.340899i 2.91188 0
324.4 − 0.816594i 1.53844i 1.33317 0 1.25628 − 5.03316i − 2.72185i 0.633188 0
324.5 0.816594i − 1.53844i 1.33317 0 1.25628 5.03316i 2.72185i 0.633188 0
324.6 1.95594i − 0.296842i −1.82571 0 0.580605 − 3.56331i 0.340899i 2.91188 0
324.7 2.14243i − 2.87834i −2.59002 0 6.16666 − 3.10482i − 1.26409i −5.28487 0
324.8 2.63010i 3.04306i −4.91744 0 −8.00355 0.574672i − 7.67316i −6.26020 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.8
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.e 8
5.b even 2 1 inner 475.2.b.e 8
5.c odd 4 1 95.2.a.b ✓ 4
5.c odd 4 1 475.2.a.i 4
15.e even 4 1 855.2.a.m 4
15.e even 4 1 4275.2.a.bo 4
20.e even 4 1 1520.2.a.t 4
20.e even 4 1 7600.2.a.cf 4
35.f even 4 1 4655.2.a.y 4
40.i odd 4 1 6080.2.a.cc 4
40.k even 4 1 6080.2.a.ch 4
95.g even 4 1 1805.2.a.p 4
95.g even 4 1 9025.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b ✓ 4 5.c odd 4 1
475.2.a.i 4 5.c odd 4 1
475.2.b.e 8 1.a even 1 1 trivial
475.2.b.e 8 5.b even 2 1 inner
855.2.a.m 4 15.e even 4 1
1520.2.a.t 4 20.e even 4 1
1805.2.a.p 4 95.g even 4 1
4275.2.a.bo 4 15.e even 4 1
4655.2.a.y 4 35.f even 4 1
6080.2.a.cc 4 40.i odd 4 1
6080.2.a.ch 4 40.k even 4 1
7600.2.a.cf 4 20.e even 4 1
9025.2.a.bf 4 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 16T_{2}^{6} + 86T_{2}^{4} + 172T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{8} + 20 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 48 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 112 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 32 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots - 640)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 84 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{4} - 16 T^{3} + \cdots - 240)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 48 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( T^{8} + 272 T^{6} + \cdots + 1115136 \) Copy content Toggle raw display
$53$ \( T^{8} + 100 T^{6} + \cdots + 121104 \) Copy content Toggle raw display
$59$ \( (T^{4} - 64 T^{2} + \cdots - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 20 T^{3} + \cdots - 2656)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 308 T^{6} + \cdots + 1157776 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots - 4224)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 272 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} + \cdots - 1856)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 144 T^{6} + \cdots + 230400 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots + 240)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 452 T^{6} + \cdots + 1926544 \) Copy content Toggle raw display
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