↑

Properties

Label 475.4.b.f
Level $475$
Weight $4$
Character orbit 475.b
Analytic conductor $28.026$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [475,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-42,0,-130] 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: \(28.0259072527\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.158155776.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 19)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{2} + ( - \beta_{5} + 2 \beta_{4}) q^{3} + (\beta_{3} + 2 \beta_1 - 8) q^{4} + (3 \beta_{3} + 4 \beta_1 - 24) q^{6} + ( - 4 \beta_{5} - 13 \beta_{2}) q^{7}+ \cdots + ( - 21 \beta_{3} + 110 \beta_1 - 113) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 42 q^{4} - 130 q^{6} - 96 q^{9} + 32 q^{11} - 74 q^{14} + 66 q^{16} + 114 q^{19} - 50 q^{21} + 978 q^{24} + 598 q^{26} - 754 q^{29} - 280 q^{31} - 658 q^{34} + 2148 q^{36} + 742 q^{39} + 1912 q^{41}+ \cdots - 500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{5} - 34\nu^{4} + 121\nu^{3} + 132\nu^{2} + 66\nu + 1203 ) / 681 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -29\nu^{5} + 69\nu^{4} - 92\nu^{3} - 575\nu^{2} - 2217\nu - 819 ) / 681 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\nu^{5} - 167\nu^{4} + 374\nu^{3} + 408\nu^{2} + 204\nu - 2163 ) / 681 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\nu^{5} - 103\nu^{4} + 213\nu^{3} + 707\nu^{2} + 3645\nu + 1341 ) / 681 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -123\nu^{5} + 277\nu^{4} - 218\nu^{3} - 3292\nu^{2} - 8229\nu - 3051 ) / 681 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 13\beta_{4} - 2\beta_{3} + 29\beta_{2} + 13\beta _1 - 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{3} + 34\beta _1 - 95 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 46\beta_{5} - 197\beta_{4} - 46\beta_{3} - 493\beta_{2} + 197\beta _1 - 493 ) / 2 \) 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
2.79911 − 2.79911i
−0.376763 − 0.376763i
−1.42234 + 1.42234i
−1.42234 − 1.42234i
−0.376763 + 0.376763i
2.79911 + 2.79911i
− 5.07177i − 8.66998i −17.7229 0 −43.9722 15.1058i 49.3121i −48.1686 0
324.2 − 3.96257i − 6.71610i −7.70200 0 −26.6130 − 25.8362i − 1.18085i −18.1060 0
324.3 − 1.89080i 2.95388i 4.42486 0 5.58521 − 5.94196i − 23.4930i 18.2746 0
324.4 1.89080i − 2.95388i 4.42486 0 5.58521 5.94196i 23.4930i 18.2746 0
324.5 3.96257i 6.71610i −7.70200 0 −26.6130 25.8362i 1.18085i −18.1060 0
324.6 5.07177i 8.66998i −17.7229 0 −43.9722 − 15.1058i − 49.3121i −48.1686 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 475.4.b.f 6
5.b even 2 1 inner 475.4.b.f 6
5.c odd 4 1 19.4.a.b ✓ 3
5.c odd 4 1 475.4.a.f 3
15.e even 4 1 171.4.a.f 3
20.e even 4 1 304.4.a.i 3
35.f even 4 1 931.4.a.c 3
40.i odd 4 1 1216.4.a.s 3
40.k even 4 1 1216.4.a.u 3
55.e even 4 1 2299.4.a.h 3
95.g even 4 1 361.4.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.b ✓ 3 5.c odd 4 1
171.4.a.f 3 15.e even 4 1
304.4.a.i 3 20.e even 4 1
361.4.a.i 3 95.g even 4 1
475.4.a.f 3 5.c odd 4 1
475.4.b.f 6 1.a even 1 1 trivial
475.4.b.f 6 5.b even 2 1 inner
931.4.a.c 3 35.f even 4 1
1216.4.a.s 3 40.i odd 4 1
1216.4.a.u 3 40.k even 4 1
2299.4.a.h 3 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{6} + 45T_{2}^{4} + 552T_{2}^{2} + 1444 \) Copy content Toggle raw display
\( T_{3}^{6} + 129T_{3}^{4} + 4440T_{3}^{2} + 29584 \) Copy content Toggle raw display
\( T_{7}^{6} + 931T_{7}^{4} + 183939T_{7}^{2} + 5377761 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 45 T^{4} + \cdots + 1444 \) Copy content Toggle raw display
$3$ \( T^{6} + 129 T^{4} + \cdots + 29584 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 931 T^{4} + \cdots + 5377761 \) Copy content Toggle raw display
$11$ \( (T^{3} - 16 T^{2} + \cdots + 1182)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 2737 T^{4} + \cdots + 23503104 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 47794267161 \) Copy content Toggle raw display
$19$ \( (T - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 143017086976 \) Copy content Toggle raw display
$29$ \( (T^{3} + 377 T^{2} + \cdots - 4544396)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 140 T^{2} + \cdots - 2444352)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 100028962096704 \) Copy content Toggle raw display
$41$ \( (T^{3} - 956 T^{2} + \cdots - 31578144)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 8647269509376 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 283074433433856 \) Copy content Toggle raw display
$59$ \( (T^{3} + 265 T^{2} + \cdots - 31557612)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 988 T^{2} + \cdots + 76875874)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 56478952501504 \) Copy content Toggle raw display
$71$ \( (T^{3} - 846 T^{2} + \cdots + 1727928)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( (T^{3} + 382 T^{2} + \cdots + 56023488)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 61\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} - 172 T^{2} + \cdots + 76923456)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 38\!\cdots\!44 \) Copy content Toggle raw display
show more
show less