Properties

Label 2475.2.c.d
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(199,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2475.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (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: \(19.7629745003\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} - q^{11} - 2 i q^{13} + 4 q^{14} - q^{16} - 2 i q^{17} - i q^{22} - 8 i q^{23} + 2 q^{26} - 4 i q^{28} - 6 q^{29} - 8 q^{31} + 5 i q^{32} + 2 q^{34} - 6 i q^{37} + 2 q^{41} - q^{44} + 8 q^{46} + 8 i q^{47} - 9 q^{49} - 2 i q^{52} - 6 i q^{53} + 12 q^{56} - 6 i q^{58} - 4 q^{59} + 6 q^{61} - 8 i q^{62} - 7 q^{64} + 4 i q^{67} - 2 i q^{68} - 14 i q^{73} + 6 q^{74} + 4 i q^{77} + 4 q^{79} + 2 i q^{82} - 12 i q^{83} - 3 i q^{88} - 6 q^{89} - 8 q^{91} - 8 i q^{92} - 8 q^{94} - 2 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{11} + 8 q^{14} - 2 q^{16} + 4 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 4 q^{41} - 2 q^{44} + 16 q^{46} - 18 q^{49} + 24 q^{56} - 8 q^{59} + 12 q^{61} - 14 q^{64} + 12 q^{74} + 8 q^{79} - 12 q^{89} - 16 q^{91} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 4.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 4.00000i 3.00000i 0 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 2475.2.c.d 2
3.b odd 2 1 825.2.c.a 2
5.b even 2 1 inner 2475.2.c.d 2
5.c odd 4 1 99.2.a.b 1
5.c odd 4 1 2475.2.a.g 1
15.d odd 2 1 825.2.c.a 2
15.e even 4 1 33.2.a.a 1
15.e even 4 1 825.2.a.a 1
20.e even 4 1 1584.2.a.o 1
35.f even 4 1 4851.2.a.b 1
40.i odd 4 1 6336.2.a.x 1
40.k even 4 1 6336.2.a.n 1
45.k odd 12 2 891.2.e.g 2
45.l even 12 2 891.2.e.e 2
55.e even 4 1 1089.2.a.j 1
60.l odd 4 1 528.2.a.g 1
105.k odd 4 1 1617.2.a.j 1
120.q odd 4 1 2112.2.a.j 1
120.w even 4 1 2112.2.a.bb 1
165.l odd 4 1 363.2.a.b 1
165.l odd 4 1 9075.2.a.q 1
165.u odd 20 4 363.2.e.g 4
165.v even 20 4 363.2.e.e 4
195.s even 4 1 5577.2.a.a 1
255.o even 4 1 9537.2.a.m 1
660.q even 4 1 5808.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 15.e even 4 1
99.2.a.b 1 5.c odd 4 1
363.2.a.b 1 165.l odd 4 1
363.2.e.e 4 165.v even 20 4
363.2.e.g 4 165.u odd 20 4
528.2.a.g 1 60.l odd 4 1
825.2.a.a 1 15.e even 4 1
825.2.c.a 2 3.b odd 2 1
825.2.c.a 2 15.d odd 2 1
891.2.e.e 2 45.l even 12 2
891.2.e.g 2 45.k odd 12 2
1089.2.a.j 1 55.e even 4 1
1584.2.a.o 1 20.e even 4 1
1617.2.a.j 1 105.k odd 4 1
2112.2.a.j 1 120.q odd 4 1
2112.2.a.bb 1 120.w even 4 1
2475.2.a.g 1 5.c odd 4 1
2475.2.c.d 2 1.a even 1 1 trivial
2475.2.c.d 2 5.b even 2 1 inner
4851.2.a.b 1 35.f even 4 1
5577.2.a.a 1 195.s even 4 1
5808.2.a.t 1 660.q even 4 1
6336.2.a.n 1 40.k even 4 1
6336.2.a.x 1 40.i odd 4 1
9075.2.a.q 1 165.l odd 4 1
9537.2.a.m 1 255.o even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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