Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 2 q^{4} - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} + 2 q^{4} - i q^{7} - q^{9} + 3 q^{11} + 4 i q^{12} + 4 i q^{13} + 4 q^{16} - 3 i q^{17} - q^{19} + 2 q^{21} + 4 i q^{27} - 2 i q^{28} - 6 q^{29} - 4 q^{31} + 6 i q^{33} - 2 q^{36} + 2 i q^{37} - 8 q^{39} - 6 q^{41} + i q^{43} + 6 q^{44} - 3 i q^{47} + 8 i q^{48} + 6 q^{49} + 6 q^{51} + 8 i q^{52} - 12 i q^{53} - 2 i q^{57} + 6 q^{59} - q^{61} + i q^{63} + 8 q^{64} - 4 i q^{67} - 6 i q^{68} + 6 q^{71} + 7 i q^{73} - 2 q^{76} - 3 i q^{77} - 8 q^{79} - 11 q^{81} - 12 i q^{83} + 4 q^{84} - 12 i q^{87} - 12 q^{89} + 4 q^{91} - 8 i q^{93} + 8 i q^{97} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 2 q^{19} + 4 q^{21} - 12 q^{29} - 8 q^{31} - 4 q^{36} - 16 q^{39} - 12 q^{41} + 12 q^{44} + 12 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} + 12 q^{71} - 4 q^{76} - 16 q^{79} - 22 q^{81} + 8 q^{84} - 24 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
1.00000i
1.00000i
0 2.00000i 2.00000 0 0 1.00000i 0 −1.00000 0
324.2 0 2.00000i 2.00000 0 0 1.00000i 0 −1.00000 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
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.a 2
5.b even 2 1 inner 475.2.b.a 2
5.c odd 4 1 19.2.a.a 1
5.c odd 4 1 475.2.a.b 1
15.e even 4 1 171.2.a.b 1
15.e even 4 1 4275.2.a.i 1
20.e even 4 1 304.2.a.f 1
20.e even 4 1 7600.2.a.c 1
35.f even 4 1 931.2.a.a 1
35.k even 12 2 931.2.f.b 2
35.l odd 12 2 931.2.f.c 2
40.i odd 4 1 1216.2.a.o 1
40.k even 4 1 1216.2.a.b 1
55.e even 4 1 2299.2.a.b 1
60.l odd 4 1 2736.2.a.c 1
65.h odd 4 1 3211.2.a.a 1
85.g odd 4 1 5491.2.a.b 1
95.g even 4 1 361.2.a.b 1
95.g even 4 1 9025.2.a.d 1
95.l even 12 2 361.2.c.a 2
95.m odd 12 2 361.2.c.c 2
95.q odd 36 6 361.2.e.d 6
95.r even 36 6 361.2.e.e 6
105.k odd 4 1 8379.2.a.j 1
285.j odd 4 1 3249.2.a.d 1
380.j odd 4 1 5776.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 5.c odd 4 1
171.2.a.b 1 15.e even 4 1
304.2.a.f 1 20.e even 4 1
361.2.a.b 1 95.g even 4 1
361.2.c.a 2 95.l even 12 2
361.2.c.c 2 95.m odd 12 2
361.2.e.d 6 95.q odd 36 6
361.2.e.e 6 95.r even 36 6
475.2.a.b 1 5.c odd 4 1
475.2.b.a 2 1.a even 1 1 trivial
475.2.b.a 2 5.b even 2 1 inner
931.2.a.a 1 35.f even 4 1
931.2.f.b 2 35.k even 12 2
931.2.f.c 2 35.l odd 12 2
1216.2.a.b 1 40.k even 4 1
1216.2.a.o 1 40.i odd 4 1
2299.2.a.b 1 55.e even 4 1
2736.2.a.c 1 60.l odd 4 1
3211.2.a.a 1 65.h odd 4 1
3249.2.a.d 1 285.j odd 4 1
4275.2.a.i 1 15.e even 4 1
5491.2.a.b 1 85.g odd 4 1
5776.2.a.c 1 380.j odd 4 1
7600.2.a.c 1 20.e even 4 1
8379.2.a.j 1 105.k odd 4 1
9025.2.a.d 1 95.g even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 49 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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