Properties

Label 7605.2.a.ci
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

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: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13824.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{2} + \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{2} + \beta_1) q^{7} + \beta_{3} q^{8} + \beta_1 q^{10} + \beta_{3} q^{11} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{14} - 2 q^{16} + (\beta_{3} + \beta_1) q^{17} + (2 \beta_{3} + \beta_{2} + 3) q^{19} + (\beta_{2} + 1) q^{20} + 2 \beta_{2} q^{22} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{23} + q^{25} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{28} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{29} + ( - 2 \beta_{3} + 3) q^{31} + ( - 2 \beta_{3} - 2 \beta_1) q^{32} + (3 \beta_{2} + 3) q^{34} + (\beta_{2} + \beta_1) q^{35} + ( - 2 \beta_1 + 6) q^{37} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{38} + \beta_{3} q^{40} + ( - 3 \beta_{2} - \beta_1 - 3) q^{41} + (2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{43} + 2 \beta_1 q^{44} + ( - 2 \beta_{3} - 2 \beta_1 - 6) q^{46} + (\beta_{3} - \beta_1 - 6) q^{47} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{49} + \beta_1 q^{50} + ( - 2 \beta_{3} + 4 \beta_1) q^{53} + \beta_{3} q^{55} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{56} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{58} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{59} + ( - \beta_{2} - 8) q^{61} + ( - 4 \beta_{2} + 3 \beta_1) q^{62} + ( - 6 \beta_{2} - 2) q^{64} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 3) q^{67} + (\beta_{3} + 4 \beta_1) q^{68} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{70} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{2} - 3 \beta_1 + 6) q^{73} + ( - 2 \beta_{2} + 6 \beta_1 - 6) q^{74} + (4 \beta_{2} + 4 \beta_1 + 6) q^{76} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{77} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{79} - 2 q^{80} + ( - 3 \beta_{3} - \beta_{2} - 6 \beta_1 - 3) q^{82} + ( - 3 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{3} + \beta_1) q^{85} + (\beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{86} + ( - 2 \beta_{2} + 6) q^{88} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{89} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{92} + (\beta_{2} - 6 \beta_1 - 3) q^{94} + (2 \beta_{3} + \beta_{2} + 3) q^{95} + ( - \beta_{3} + \beta_1 + 9) q^{97} + (\beta_{3} + 6 \beta_{2} + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{5} + 12 q^{14} - 8 q^{16} + 12 q^{19} + 4 q^{20} + 4 q^{25} + 12 q^{28} + 12 q^{29} + 12 q^{31} + 12 q^{34} + 24 q^{37} - 12 q^{41} + 16 q^{43} - 24 q^{46} - 24 q^{47} - 4 q^{49} + 12 q^{58} + 12 q^{59} - 32 q^{61} - 8 q^{64} - 12 q^{67} + 12 q^{70} + 12 q^{71} + 24 q^{73} - 24 q^{74} + 24 q^{76} - 8 q^{79} - 8 q^{80} - 12 q^{82} - 12 q^{86} + 24 q^{88} - 12 q^{89} - 24 q^{92} - 12 q^{94} + 12 q^{95} + 36 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.17533
−1.12603
1.12603
2.17533
−2.17533 0 2.73205 1.00000 0 −0.443277 −1.59245 0 −2.17533
1.2 −1.12603 0 −0.732051 1.00000 0 −2.85808 3.07638 0 −1.12603
1.3 1.12603 0 −0.732051 1.00000 0 −0.606018 −3.07638 0 1.12603
1.4 2.17533 0 2.73205 1.00000 0 3.90738 1.59245 0 2.17533
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7605.2.a.ci 4
3.b odd 2 1 2535.2.a.bj 4
13.b even 2 1 7605.2.a.ch 4
13.f odd 12 2 585.2.bu.d 8
39.d odd 2 1 2535.2.a.bk 4
39.k even 12 2 195.2.bb.b 8
195.bc odd 12 2 975.2.w.i 8
195.bh even 12 2 975.2.bc.j 8
195.bn odd 12 2 975.2.w.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.b 8 39.k even 12 2
585.2.bu.d 8 13.f odd 12 2
975.2.w.h 8 195.bn odd 12 2
975.2.w.i 8 195.bc odd 12 2
975.2.bc.j 8 195.bh even 12 2
2535.2.a.bj 4 3.b odd 2 1
2535.2.a.bk 4 39.d odd 2 1
7605.2.a.ch 4 13.b even 2 1
7605.2.a.ci 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{4} - 6T_{2}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{4} - 12T_{7}^{2} - 12T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 12T_{11}^{2} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6T^{2} + 6 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 12 T^{2} - 12 T - 3 \) Copy content Toggle raw display
$11$ \( T^{4} - 12T^{2} + 24 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 18T^{2} + 54 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 312 T - 444 \) Copy content Toggle raw display
$23$ \( T^{4} - 60 T^{2} - 144 T - 72 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 30 T^{2} + \cdots - 234 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 6 T^{2} + 180 T + 33 \) Copy content Toggle raw display
$37$ \( T^{4} - 24 T^{3} + 192 T^{2} + \cdots + 528 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} - 6 T^{2} - 216 T + 222 \) Copy content Toggle raw display
$43$ \( T^{4} - 16 T^{3} + 36 T^{2} + \cdots - 1523 \) Copy content Toggle raw display
$47$ \( T^{4} + 24 T^{3} + 198 T^{2} + \cdots + 654 \) Copy content Toggle raw display
$53$ \( T^{4} - 144T^{2} + 3456 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} - 54 T^{2} + \cdots - 2106 \) Copy content Toggle raw display
$61$ \( (T^{2} + 16 T + 61)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 60 T^{2} + \cdots + 981 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} - 114 T^{2} + \cdots - 1146 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + 156 T^{2} + \cdots - 1179 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 102 T^{2} + \cdots - 1703 \) Copy content Toggle raw display
$83$ \( T^{4} - 132T^{2} + 4056 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} - 114 T^{2} + \cdots - 66 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + 468 T^{2} + \cdots + 5109 \) Copy content Toggle raw display
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