Properties

Label 2475.2.c.f
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
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] = [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 55)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} + 3 i q^{8} + q^{11} + 2 i q^{13} - q^{16} + 6 i q^{17} + 4 q^{19} + i q^{22} - 4 i q^{23} - 2 q^{26} + 6 q^{29} - 8 q^{31} + 5 i q^{32} - 6 q^{34} + 2 i q^{37} + 4 i q^{38} - 2 q^{41} + 4 i q^{43} + q^{44} + 4 q^{46} - 12 i q^{47} + 7 q^{49} + 2 i q^{52} + 2 i q^{53} + 6 i q^{58} + 4 q^{59} - 10 q^{61} - 8 i q^{62} - 7 q^{64} + 16 i q^{67} + 6 i q^{68} - 8 q^{71} + 14 i q^{73} - 2 q^{74} + 4 q^{76} - 8 q^{79} - 2 i q^{82} + 4 i q^{83} - 4 q^{86} + 3 i q^{88} + 10 q^{89} - 4 i q^{92} + 12 q^{94} - 10 i q^{97} + 7 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{11} - 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 16 q^{31} - 12 q^{34} - 4 q^{41} + 2 q^{44} + 8 q^{46} + 14 q^{49} + 8 q^{59} - 20 q^{61} - 14 q^{64} - 16 q^{71} - 4 q^{74} + 8 q^{76} - 16 q^{79} - 8 q^{86} + 20 q^{89} + 24 q^{94}+O(q^{100}) \) 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 0 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 0 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.f 2
3.b odd 2 1 275.2.b.b 2
5.b even 2 1 inner 2475.2.c.f 2
5.c odd 4 1 495.2.a.a 1
5.c odd 4 1 2475.2.a.i 1
12.b even 2 1 4400.2.b.n 2
15.d odd 2 1 275.2.b.b 2
15.e even 4 1 55.2.a.a 1
15.e even 4 1 275.2.a.a 1
20.e even 4 1 7920.2.a.i 1
55.e even 4 1 5445.2.a.i 1
60.h even 2 1 4400.2.b.n 2
60.l odd 4 1 880.2.a.h 1
60.l odd 4 1 4400.2.a.p 1
105.k odd 4 1 2695.2.a.c 1
120.q odd 4 1 3520.2.a.n 1
120.w even 4 1 3520.2.a.p 1
165.l odd 4 1 605.2.a.b 1
165.l odd 4 1 3025.2.a.f 1
165.u odd 20 4 605.2.g.c 4
165.v even 20 4 605.2.g.a 4
195.s even 4 1 9295.2.a.b 1
660.q even 4 1 9680.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 15.e even 4 1
275.2.a.a 1 15.e even 4 1
275.2.b.b 2 3.b odd 2 1
275.2.b.b 2 15.d odd 2 1
495.2.a.a 1 5.c odd 4 1
605.2.a.b 1 165.l odd 4 1
605.2.g.a 4 165.v even 20 4
605.2.g.c 4 165.u odd 20 4
880.2.a.h 1 60.l odd 4 1
2475.2.a.i 1 5.c odd 4 1
2475.2.c.f 2 1.a even 1 1 trivial
2475.2.c.f 2 5.b even 2 1 inner
2695.2.a.c 1 105.k odd 4 1
3025.2.a.f 1 165.l odd 4 1
3520.2.a.n 1 120.q odd 4 1
3520.2.a.p 1 120.w even 4 1
4400.2.a.p 1 60.l odd 4 1
4400.2.b.n 2 12.b even 2 1
4400.2.b.n 2 60.h even 2 1
5445.2.a.i 1 55.e even 4 1
7920.2.a.i 1 20.e even 4 1
9295.2.a.b 1 195.s even 4 1
9680.2.a.r 1 660.q 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
\( T_{7} \) 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
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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