Properties

Label 7605.2.a.ce
Level $7605$
Weight $2$
Character orbit 7605.a
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2535)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + ( - \beta_1 + 4) q^{4} - q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 3 \beta_1 + 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{2} + ( - \beta_1 + 4) q^{4} - q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 3 \beta_1 + 5) q^{8} + (\beta_{2} + \beta_1 - 1) q^{10} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{11} + ( - \beta_{2} - 5) q^{14} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + (2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (4 \beta_{2} - 3 \beta_1 + 3) q^{19} + (\beta_1 - 4) q^{20} + ( - 3 \beta_{2} + 7 \beta_1 - 5) q^{22} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{23} + q^{25} + (2 \beta_{2} + 4 \beta_1 - 3) q^{28} + (2 \beta_{2} - \beta_1 - 4) q^{29} + ( - 2 \beta_{2} + 6 \beta_1 + 1) q^{31} + (7 \beta_{2} - 7 \beta_1 + 12) q^{32} + (2 \beta_1 - 12) q^{34} + ( - \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{2} - \beta_1 - 6) q^{37} + (7 \beta_{2} - 10 \beta_1 + 4) q^{38} + (3 \beta_1 - 5) q^{40} + ( - 3 \beta_{2} + 5 \beta_1 - 6) q^{41} + ( - 3 \beta_{2} + \beta_1 - 10) q^{43} + ( - 8 \beta_{2} + 11 \beta_1 - 14) q^{44} + ( - 9 \beta_{2} + 13 \beta_1 - 6) q^{46} + (\beta_{2} - \beta_1 + 4) q^{47} + (2 \beta_{2} + \beta_1 - 2) q^{49} + ( - \beta_{2} - \beta_1 + 1) q^{50} + (5 \beta_{2} - 9 \beta_1 + 1) q^{53} + (2 \beta_{2} - 2 \beta_1 + 3) q^{55} + ( - \beta_{2} + 5 \beta_1 - 9) q^{56} + (8 \beta_{2} + \beta_1 - 5) q^{58} + (7 \beta_{2} - 7 \beta_1 + 5) q^{59} + ( - \beta_{2} + 7) q^{61} + ( - 15 \beta_{2} + 7 \beta_1 - 13) q^{62} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + (7 \beta_{2} - 4 \beta_1 + 9) q^{67} + (4 \beta_{2} + 10 \beta_1 - 14) q^{68} + (\beta_{2} + 5) q^{70} + (4 \beta_{2} - 3 \beta_1 - 7) q^{71} + ( - 6 \beta_{2} - \beta_1 - 3) q^{73} + (6 \beta_{2} + 7 \beta_1 + 1) q^{74} + (15 \beta_{2} - 15 \beta_1 + 14) q^{76} + (\beta_{2} - 5 \beta_1 + 2) q^{77} + ( - 7 \beta_{2} + 5 \beta_1 - 2) q^{79} + ( - \beta_{2} + 6 \beta_1 - 6) q^{80} + ( - 7 \beta_{2} + 14 \beta_1 - 15) q^{82} + (11 \beta_1 - 2) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{85} + (5 \beta_{2} + 14 \beta_1 - 7) q^{86} + ( - 10 \beta_{2} + 19 \beta_1 - 21) q^{88} + (2 \beta_{2} - 3 \beta_1 + 1) q^{89} + ( - 19 \beta_{2} + 20 \beta_1 - 19) q^{92} + ( - \beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{95} + ( - 4 \beta_{2} - 2 \beta_1 + 1) q^{97} + (2 \beta_{2} + \beta_1 - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 11 q^{4} - 3 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 11 q^{4} - 3 q^{5} + 12 q^{8} - 3 q^{10} - 5 q^{11} - 14 q^{14} + 11 q^{16} - 6 q^{17} + 2 q^{19} - 11 q^{20} - 5 q^{22} - 3 q^{23} + 3 q^{25} - 7 q^{28} - 15 q^{29} + 11 q^{31} + 22 q^{32} - 34 q^{34} - 17 q^{37} - 5 q^{38} - 12 q^{40} - 10 q^{41} - 26 q^{43} - 23 q^{44} + 4 q^{46} + 10 q^{47} - 7 q^{49} + 3 q^{50} - 11 q^{53} + 5 q^{55} - 21 q^{56} - 22 q^{58} + q^{59} + 22 q^{61} - 17 q^{62} + 16 q^{67} - 36 q^{68} + 14 q^{70} - 28 q^{71} - 4 q^{73} + 4 q^{74} + 12 q^{76} + 6 q^{79} - 11 q^{80} - 24 q^{82} + 5 q^{83} + 6 q^{85} - 12 q^{86} - 34 q^{88} - 2 q^{89} - 18 q^{92} + 10 q^{94} - 2 q^{95} + 5 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−2.04892 0 2.19806 −1.00000 0 3.04892 −0.405813 0 2.04892
1.2 2.35690 0 3.55496 −1.00000 0 −1.35690 3.66487 0 −2.35690
1.3 2.69202 0 5.24698 −1.00000 0 −1.69202 8.74094 0 −2.69202
\(n\): 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.ce 3
3.b odd 2 1 2535.2.a.t 3
13.b even 2 1 7605.2.a.bm 3
39.d odd 2 1 2535.2.a.bh yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.t 3 3.b odd 2 1
2535.2.a.bh yes 3 39.d odd 2 1
7605.2.a.bm 3 13.b even 2 1
7605.2.a.ce 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7605))\):

\( T_{2}^{3} - 3T_{2}^{2} - 4T_{2} + 13 \) Copy content Toggle raw display
\( T_{7}^{3} - 7T_{7} - 7 \) Copy content Toggle raw display
\( T_{11}^{3} + 5T_{11}^{2} - T_{11} - 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$11$ \( T^{3} + 5T^{2} - T - 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 6 T^{2} - 16 T - 104 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 29 T + 71 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 46 T - 139 \) Copy content Toggle raw display
$29$ \( T^{3} + 15 T^{2} + 68 T + 97 \) Copy content Toggle raw display
$31$ \( T^{3} - 11 T^{2} - 25 T + 379 \) Copy content Toggle raw display
$37$ \( T^{3} + 17 T^{2} + 80 T + 113 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} - 11 T - 13 \) Copy content Toggle raw display
$43$ \( T^{3} + 26 T^{2} + 209 T + 491 \) Copy content Toggle raw display
$47$ \( T^{3} - 10 T^{2} + 31 T - 29 \) Copy content Toggle raw display
$53$ \( T^{3} + 11 T^{2} - 102 T - 1079 \) Copy content Toggle raw display
$59$ \( T^{3} - T^{2} - 114 T + 127 \) Copy content Toggle raw display
$61$ \( T^{3} - 22 T^{2} + 159 T - 377 \) Copy content Toggle raw display
$67$ \( T^{3} - 16 T^{2} - T + 617 \) Copy content Toggle raw display
$71$ \( T^{3} + 28 T^{2} + 231 T + 581 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 95 T + 83 \) Copy content Toggle raw display
$79$ \( T^{3} - 6 T^{2} - 79 T - 113 \) Copy content Toggle raw display
$83$ \( T^{3} - 5 T^{2} - 274 T + 811 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 15 T - 29 \) Copy content Toggle raw display
$97$ \( T^{3} - 5 T^{2} - 57 T + 293 \) Copy content Toggle raw display
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