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Properties

Label 575.4.b.g
Level $575$
Weight $4$
Character orbit 575.b
Analytic conductor $33.926$
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] = [575,4,Mod(24,575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("575.24"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-40,0,-34] 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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 36x^{6} + 244x^{4} + 153x^{2} + 16 \) 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 23)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3}) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{3} + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots - 5) q^{4} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 5) q^{6} + (4 \beta_{7} - 4 \beta_{6} + \cdots - 4 \beta_1) q^{7}+ \cdots + ( - 48 \beta_{5} - 46 \beta_{4} + \cdots + 386) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{4} - 34 q^{6} + 66 q^{9} + 16 q^{11} + 288 q^{14} + 128 q^{16} - 192 q^{19} + 360 q^{21} + 376 q^{24} - 458 q^{26} - 42 q^{29} - 386 q^{31} - 1332 q^{34} - 1258 q^{36} + 582 q^{39} - 250 q^{41}+ \cdots + 2996 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 36x^{6} + 244x^{4} + 153x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} + 77\nu^{4} + 580\nu^{2} - 53 ) / 201 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 72\nu^{5} + 1496\nu^{3} + 7377\nu ) / 804 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{6} + 467\nu^{4} + 3167\nu^{2} + 1699 ) / 201 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -22\nu^{6} - 780\nu^{4} - 4973\nu^{2} - 1494 ) / 201 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -37\nu^{7} - 1324\nu^{5} - 8720\nu^{3} - 3341\nu ) / 804 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -63\nu^{7} - 2258\nu^{5} - 15054\nu^{3} - 8347\nu ) / 402 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} + 17\beta_{6} - \beta_{3} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -18\beta_{5} - 42\beta_{4} + 75\beta_{2} + 241 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 174\beta_{7} - 591\beta_{6} + 57\beta_{3} + 634\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 403\beta_{5} + 1037\beta_{4} - 2207\beta_{2} - 6352 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5048\beta_{7} + 17120\beta_{6} - 1804\beta_{3} - 17121\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 0.743529i
− 0.362907i
2.83969i
5.22031i
− 5.22031i
− 2.83969i
0.362907i
0.743529i
− 5.07751i − 1.55870i −17.7811 0 −7.91434 24.3381i 49.6639i 24.5704 0
24.2 − 4.24143i − 4.41777i −9.98977 0 −18.7377 27.0572i 8.43948i 7.48328 0
24.3 − 2.86845i 3.43737i −0.228032 0 9.85995 − 32.7301i − 22.2935i 15.1845 0
24.4 − 0.0323756i − 6.42170i 7.99895 0 −0.207906 − 14.0109i − 0.517976i −14.2382 0
24.5 0.0323756i 6.42170i 7.99895 0 −0.207906 14.0109i 0.517976i −14.2382 0
24.6 2.86845i − 3.43737i −0.228032 0 9.85995 32.7301i 22.2935i 15.1845 0
24.7 4.24143i 4.41777i −9.98977 0 −18.7377 − 27.0572i − 8.43948i 7.48328 0
24.8 5.07751i 1.55870i −17.7811 0 −7.91434 − 24.3381i − 49.6639i 24.5704 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.8
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 575.4.b.g 8
5.b even 2 1 inner 575.4.b.g 8
5.c odd 4 1 23.4.a.b ✓ 4
5.c odd 4 1 575.4.a.i 4
15.e even 4 1 207.4.a.e 4
20.e even 4 1 368.4.a.l 4
35.f even 4 1 1127.4.a.c 4
40.i odd 4 1 1472.4.a.y 4
40.k even 4 1 1472.4.a.bf 4
115.e even 4 1 529.4.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.b ✓ 4 5.c odd 4 1
207.4.a.e 4 15.e even 4 1
368.4.a.l 4 20.e even 4 1
529.4.a.g 4 115.e even 4 1
575.4.a.i 4 5.c odd 4 1
575.4.b.g 8 1.a even 1 1 trivial
575.4.b.g 8 5.b even 2 1 inner
1127.4.a.c 4 35.f even 4 1
1472.4.a.y 4 40.i odd 4 1
1472.4.a.bf 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{8} + 52T_{2}^{6} + 824T_{2}^{4} + 3817T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{8} + 75T_{3}^{6} + 1699T_{3}^{4} + 13209T_{3}^{2} + 23104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 52 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} + 75 T^{6} + \cdots + 23104 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 91194336256 \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{3} + \cdots - 81440)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 1749424184964 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 731503878400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 96 T^{3} + \cdots + 66996944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 21 T^{3} + \cdots + 325399050)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 193 T^{3} + \cdots - 58104720)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + 125 T^{3} + \cdots + 29467114)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} - 1140 T^{3} + \cdots + 1146071296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 754 T^{3} + \cdots - 621762112)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + 401 T^{3} + \cdots - 5581505296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} - 838 T^{3} + \cdots - 61908677856)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2342 T^{3} + \cdots - 213195182848)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
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