↑

Properties

Label 475.3.c.b
Level $475$
Weight $3$
Character orbit 475.c
Analytic conductor $12.943$
Analytic rank $0$
Dimension $2$
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] = [475,3,Mod(151,475)] 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("475.151"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-18,0,26,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - \beta q^{3} - 9 q^{4} + 13 q^{6} + 5 q^{7} - 5 \beta q^{8} - 4 q^{9} - 10 q^{11} + 9 \beta q^{12} + \beta q^{13} + 5 \beta q^{14} + 29 q^{16} - 15 q^{17} - 4 \beta q^{18} + ( - 5 \beta - 6) q^{19} + \cdots + 40 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{4} + 26 q^{6} + 10 q^{7} - 8 q^{9} - 20 q^{11} + 58 q^{16} - 30 q^{17} - 12 q^{19} - 70 q^{23} - 130 q^{24} - 26 q^{26} - 90 q^{28} + 72 q^{36} + 130 q^{38} + 26 q^{39} + 130 q^{42} + 40 q^{43}+ \cdots + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
− 3.60555i
3.60555i
− 3.60555i 3.60555i −9.00000 0 13.0000 5.00000 18.0278i −4.00000 0
151.2 3.60555i − 3.60555i −9.00000 0 13.0000 5.00000 − 18.0278i −4.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.b 2
5.b even 2 1 19.3.b.b ✓ 2
5.c odd 4 2 475.3.d.b 4
15.d odd 2 1 171.3.c.b 2
19.b odd 2 1 inner 475.3.c.b 2
20.d odd 2 1 304.3.e.d 2
40.e odd 2 1 1216.3.e.h 2
40.f even 2 1 1216.3.e.g 2
60.h even 2 1 2736.3.o.d 2
95.d odd 2 1 19.3.b.b ✓ 2
95.g even 4 2 475.3.d.b 4
95.h odd 6 2 361.3.d.b 4
95.i even 6 2 361.3.d.b 4
95.o odd 18 6 361.3.f.d 12
95.p even 18 6 361.3.f.d 12
285.b even 2 1 171.3.c.b 2
380.d even 2 1 304.3.e.d 2
760.b odd 2 1 1216.3.e.g 2
760.p even 2 1 1216.3.e.h 2
1140.p odd 2 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b ✓ 2 5.b even 2 1
19.3.b.b ✓ 2 95.d odd 2 1
171.3.c.b 2 15.d odd 2 1
171.3.c.b 2 285.b even 2 1
304.3.e.d 2 20.d odd 2 1
304.3.e.d 2 380.d even 2 1
361.3.d.b 4 95.h odd 6 2
361.3.d.b 4 95.i even 6 2
361.3.f.d 12 95.o odd 18 6
361.3.f.d 12 95.p even 18 6
475.3.c.b 2 1.a even 1 1 trivial
475.3.c.b 2 19.b odd 2 1 inner
475.3.d.b 4 5.c odd 4 2
475.3.d.b 4 95.g even 4 2
1216.3.e.g 2 40.f even 2 1
1216.3.e.g 2 760.b odd 2 1
1216.3.e.h 2 40.e odd 2 1
1216.3.e.h 2 760.p even 2 1
2736.3.o.d 2 60.h even 2 1
2736.3.o.d 2 1140.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{2} + 13 \) Copy content Toggle raw display
\( T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 13 \) Copy content Toggle raw display
$3$ \( T^{2} + 13 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 5)^{2} \) Copy content Toggle raw display
$11$ \( (T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 13 \) Copy content Toggle raw display
$17$ \( (T + 15)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12T + 361 \) Copy content Toggle raw display
$23$ \( (T + 35)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 325 \) Copy content Toggle raw display
$31$ \( T^{2} + 1300 \) Copy content Toggle raw display
$37$ \( T^{2} + 468 \) Copy content Toggle raw display
$41$ \( T^{2} + 1300 \) Copy content Toggle raw display
$43$ \( (T - 20)^{2} \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 5733 \) Copy content Toggle raw display
$59$ \( T^{2} + 325 \) Copy content Toggle raw display
$61$ \( (T + 40)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1573 \) Copy content Toggle raw display
$71$ \( T^{2} + 11700 \) Copy content Toggle raw display
$73$ \( (T + 105)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1300 \) Copy content Toggle raw display
$83$ \( (T - 40)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 15028 \) Copy content Toggle raw display
show more
show less