Properties

Label 2475.2.c.p
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{11} + ( - 3 \beta_{3} + 2 \beta_1) q^{13} + (\beta_{2} - 3) q^{14} + 3 \beta_{2} q^{16} + ( - \beta_{3} - \beta_1) q^{17} + ( - 6 \beta_{2} - 3) q^{19} - \beta_1 q^{22} + (4 \beta_{3} + 5 \beta_1) q^{23} + (5 \beta_{2} - 2) q^{26} + (5 \beta_{3} + 2 \beta_1) q^{28} + ( - \beta_{2} - 3) q^{29} - 3 q^{31} + (5 \beta_{3} + \beta_1) q^{32} + q^{34} + (7 \beta_{3} - 2 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_1) q^{38} + 3 q^{41} + 6 \beta_{3} q^{43} + ( - \beta_{2} - 1) q^{44} + (\beta_{2} - 5) q^{46} + (\beta_{3} + 8 \beta_1) q^{47} + ( - 3 \beta_{2} - 6) q^{49} + ( - \beta_{3} - 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 7 \beta_1) q^{53} + ( - \beta_{2} - 8) q^{56} + ( - \beta_{3} - 2 \beta_1) q^{58} + (4 \beta_{2} + 7) q^{59} + ( - 5 \beta_{2} - 8) q^{61} - 3 \beta_1 q^{62} + (2 \beta_{2} - 1) q^{64} + 8 \beta_{3} q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + (10 \beta_{2} + 8) q^{71} + ( - 12 \beta_{3} - \beta_1) q^{73} + ( - 9 \beta_{2} + 2) q^{74} + ( - 3 \beta_{2} - 9) q^{76} + ( - 2 \beta_{3} - 3 \beta_1) q^{77} + (3 \beta_{2} - 1) q^{79} + 3 \beta_1 q^{82} + ( - 15 \beta_{3} - 3 \beta_1) q^{83} - 6 \beta_{2} q^{86} + ( - \beta_{3} - 2 \beta_1) q^{88} + ( - 5 \beta_{2} - 15) q^{89} + 11 \beta_{2} q^{91} + (9 \beta_{3} + 4 \beta_1) q^{92} + (7 \beta_{2} - 8) q^{94} - \beta_1 q^{97} + ( - 3 \beta_{3} - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{11} - 14 q^{14} - 6 q^{16} - 18 q^{26} - 10 q^{29} - 12 q^{31} + 4 q^{34} + 12 q^{41} - 2 q^{44} - 22 q^{46} - 18 q^{49} - 30 q^{56} + 20 q^{59} - 22 q^{61} - 8 q^{64} + 12 q^{71} + 26 q^{74} - 30 q^{76} - 10 q^{79} + 12 q^{86} - 50 q^{89} - 22 q^{91} - 46 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 0 −0.618034 0 0 2.85410i 2.23607i 0 0
199.2 0.618034i 0 1.61803 0 0 3.85410i 2.23607i 0 0
199.3 0.618034i 0 1.61803 0 0 3.85410i 2.23607i 0 0
199.4 1.61803i 0 −0.618034 0 0 2.85410i 2.23607i 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.p 4
3.b odd 2 1 275.2.b.e 4
5.b even 2 1 inner 2475.2.c.p 4
5.c odd 4 1 2475.2.a.n 2
5.c odd 4 1 2475.2.a.s 2
12.b even 2 1 4400.2.b.x 4
15.d odd 2 1 275.2.b.e 4
15.e even 4 1 275.2.a.d 2
15.e even 4 1 275.2.a.g yes 2
60.h even 2 1 4400.2.b.x 4
60.l odd 4 1 4400.2.a.bg 2
60.l odd 4 1 4400.2.a.bv 2
165.l odd 4 1 3025.2.a.i 2
165.l odd 4 1 3025.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 15.e even 4 1
275.2.a.g yes 2 15.e even 4 1
275.2.b.e 4 3.b odd 2 1
275.2.b.e 4 15.d odd 2 1
2475.2.a.n 2 5.c odd 4 1
2475.2.a.s 2 5.c odd 4 1
2475.2.c.p 4 1.a even 1 1 trivial
2475.2.c.p 4 5.b even 2 1 inner
3025.2.a.i 2 165.l odd 4 1
3025.2.a.m 2 165.l odd 4 1
4400.2.a.bg 2 60.l odd 4 1
4400.2.a.bv 2 60.l odd 4 1
4400.2.b.x 4 12.b even 2 1
4400.2.b.x 4 60.h even 2 1

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 23T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{29}^{2} + 5T_{29} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 42T^{2} + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 67T^{2} + 841 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 138T^{2} + 3481 \) Copy content Toggle raw display
$41$ \( (T - 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 178T^{2} + 5041 \) Copy content Toggle raw display
$53$ \( T^{4} + 127T^{2} + 3481 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T - 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 116)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 267 T^{2} + 17161 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 387 T^{2} + 29241 \) Copy content Toggle raw display
$89$ \( (T^{2} + 25 T + 125)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
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