Properties

Label 5775.2.a.bq
Level $5775$
Weight $2$
Character orbit 5775.a
Self dual yes
Analytic conductor $46.114$
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] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.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: \(46.1136071673\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9} + q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} - \beta_1 q^{14} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + (\beta_{2} - 2 \beta_1 - 2) q^{17} - \beta_1 q^{18} + ( - \beta_{2} - 2 \beta_1) q^{19} - q^{21} - \beta_1 q^{22} + (\beta_{2} + 2) q^{23} + (\beta_{2} + 1) q^{24} + (4 \beta_1 - 2) q^{26} - q^{27} + (\beta_{2} + 1) q^{28} + (\beta_1 + 3) q^{29} + (\beta_{2} + \beta_1 - 1) q^{31} + (\beta_{2} + 2 \beta_1 - 3) q^{32} - q^{33} + (\beta_{2} + \beta_1 + 5) q^{34} + (\beta_{2} + 1) q^{36} + (3 \beta_{2} + \beta_1 - 3) q^{37} + (3 \beta_{2} + \beta_1 + 7) q^{38} + (\beta_{2} - \beta_1 + 3) q^{39} + ( - 3 \beta_{2} + \beta_1 + 1) q^{41} + \beta_1 q^{42} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{43} + (\beta_{2} + 1) q^{44} + ( - \beta_{2} - 3 \beta_1 - 1) q^{46} + (\beta_{2} + \beta_1 - 1) q^{47} + (\beta_{2} - 2 \beta_1 + 1) q^{48} + q^{49} + ( - \beta_{2} + 2 \beta_1 + 2) q^{51} + ( - 2 \beta_{2} - 6) q^{52} - \beta_{2} q^{53} + \beta_1 q^{54} + ( - \beta_{2} - 1) q^{56} + (\beta_{2} + 2 \beta_1) q^{57} + ( - \beta_{2} - 3 \beta_1 - 3) q^{58} + (3 \beta_1 - 5) q^{59} + (3 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - 2 \beta_{2} - 4) q^{62} + q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + \beta_1 q^{66} + 2 \beta_{2} q^{67} + ( - 4 \beta_{2} - 2 \beta_1) q^{68} + ( - \beta_{2} - 2) q^{69} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{71} + ( - \beta_{2} - 1) q^{72} + ( - 2 \beta_1 - 4) q^{73} + ( - 4 \beta_{2} - 6) q^{74} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{76} + q^{77} + ( - 4 \beta_1 + 2) q^{78} + ( - \beta_{2} - 7 \beta_1 + 3) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1) q^{82} + ( - 3 \beta_{2} + 6 \beta_1 - 4) q^{83} + ( - \beta_{2} - 1) q^{84} + ( - \beta_{2} + 3 \beta_1 - 7) q^{86} + ( - \beta_1 - 3) q^{87} + ( - \beta_{2} - 1) q^{88} + (4 \beta_{2} - \beta_1 - 3) q^{89} + ( - \beta_{2} + \beta_1 - 3) q^{91} + (2 \beta_{2} + 2 \beta_1 + 6) q^{92} + ( - \beta_{2} - \beta_1 + 1) q^{93} + ( - 2 \beta_{2} - 4) q^{94} + ( - \beta_{2} - 2 \beta_1 + 3) q^{96} + ( - 6 \beta_{2} + 5 \beta_1 - 5) q^{97} - \beta_1 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 3 q^{3} + 3 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} - 3 q^{12} - 8 q^{13} - q^{14} - q^{16} - 8 q^{17} - q^{18} - 2 q^{19} - 3 q^{21} - q^{22} + 6 q^{23} + 3 q^{24} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 10 q^{29} - 2 q^{31} - 7 q^{32} - 3 q^{33} + 16 q^{34} + 3 q^{36} - 8 q^{37} + 22 q^{38} + 8 q^{39} + 4 q^{41} + q^{42} + 3 q^{44} - 6 q^{46} - 2 q^{47} + q^{48} + 3 q^{49} + 8 q^{51} - 18 q^{52} + q^{54} - 3 q^{56} + 2 q^{57} - 12 q^{58} - 12 q^{59} + 4 q^{61} - 12 q^{62} + 3 q^{63} - 17 q^{64} + q^{66} - 2 q^{68} - 6 q^{69} - 18 q^{71} - 3 q^{72} - 14 q^{73} - 18 q^{74} - 24 q^{76} + 3 q^{77} + 2 q^{78} + 2 q^{79} + 3 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} - 18 q^{86} - 10 q^{87} - 3 q^{88} - 10 q^{89} - 8 q^{91} + 20 q^{92} + 2 q^{93} - 12 q^{94} + 7 q^{96} - 10 q^{97} - q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
0.470683
−1.81361
−2.34292 −1.00000 3.48929 0 2.34292 1.00000 −3.48929 1.00000 0
1.2 −0.470683 −1.00000 −1.77846 0 0.470683 1.00000 1.77846 1.00000 0
1.3 1.81361 −1.00000 1.28917 0 −1.81361 1.00000 −1.28917 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 5775.2.a.bq 3
5.b even 2 1 1155.2.a.t 3
15.d odd 2 1 3465.2.a.bb 3
35.c odd 2 1 8085.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.t 3 5.b even 2 1
3465.2.a.bb 3 15.d odd 2 1
5775.2.a.bq 3 1.a even 1 1 trivial
8085.2.a.bl 3 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 14T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{3} + 8T_{17}^{2} + 5T_{17} - 46 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 14 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 8 T^{2} + 5 T - 46 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 31 T + 44 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} + 5 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 29 T - 22 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 14 T - 32 \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} - 58 T - 292 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 50 T - 68 \) Copy content Toggle raw display
$43$ \( T^{3} - 43T + 44 \) Copy content Toggle raw display
$47$ \( T^{3} + 2 T^{2} - 14 T - 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 7T - 2 \) Copy content Toggle raw display
$59$ \( T^{3} + 12 T^{2} + 9 T - 76 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 51 T + 226 \) Copy content Toggle raw display
$67$ \( T^{3} - 28T + 16 \) Copy content Toggle raw display
$71$ \( T^{3} + 18 T^{2} + 42 T - 272 \) Copy content Toggle raw display
$73$ \( T^{3} + 14 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 246 T + 608 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 135 T + 292 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 67 T - 2 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} - 207 T - 1822 \) Copy content Toggle raw display
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