Properties

Label 5775.2.a.bx
Level $5775$
Weight $2$
Character orbit 5775.a
Self dual yes
Analytic conductor $46.114$
Analytic rank $1$
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] = [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: \(4\)
Coefficient field: 4.4.2624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 - 1) q^{2} + q^{3} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{4} + ( - \beta_{3} + \beta_1 - 1) q^{6} + q^{7} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 - 1) q^{2} + q^{3} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{4} + ( - \beta_{3} + \beta_1 - 1) q^{6} + q^{7} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{8} + q^{9} + q^{11} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{12} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} + \beta_1 - 1) q^{14} + (6 \beta_{3} - \beta_{2} - 4 \beta_1 + 7) q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{17} + ( - \beta_{3} + \beta_1 - 1) q^{18} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{19} + q^{21} + ( - \beta_{3} + \beta_1 - 1) q^{22} + (2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{23} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{24} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{26} + q^{27} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{28} + (2 \beta_{3} - 2 \beta_1 + 4) q^{29} + (3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{31} + ( - 8 \beta_{3} + 2 \beta_{2} + 9 \beta_1 - 12) q^{32} + q^{33} + (8 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{34} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{36} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{37} + (5 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{38} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{39} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{41} + ( - \beta_{3} + \beta_1 - 1) q^{42} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{43} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{44} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{46} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{47} + (6 \beta_{3} - \beta_{2} - 4 \beta_1 + 7) q^{48} + q^{49} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{51} + ( - \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 1) q^{52} + (\beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{3} + \beta_1 - 1) q^{54} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{56} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{57} + ( - 6 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 10) q^{58} + (2 \beta_{2} - 3 \beta_1 - 5) q^{59} + ( - 3 \beta_{2} + \beta_1 - 3) q^{61} + (2 \beta_{3} + 3 \beta_{2} - 3) q^{62} + q^{63} + (10 \beta_{3} - 6 \beta_{2} - 12 \beta_1 + 21) q^{64} + ( - \beta_{3} + \beta_1 - 1) q^{66} + (2 \beta_{3} + 2) q^{67} + ( - 12 \beta_{3} + 4 \beta_{2} + 14 \beta_1 - 18) q^{68} + (2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{69} + (4 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{71} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{72} + (4 \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{73} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{74} + ( - 10 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 9) q^{76} + q^{77} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{78} + (\beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{79} + q^{81} + (3 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 15) q^{82} + ( - 8 \beta_{3} - 3 \beta_{2} - 1) q^{83} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{84} + ( - 12 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 10) q^{86} + (2 \beta_{3} - 2 \beta_1 + 4) q^{87} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{88} + (7 \beta_{3} + \beta_{2} - 9 \beta_1 + 3) q^{89} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{91} + ( - 3 \beta_{3} - \beta_1) q^{92} + (3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{93} + (\beta_{3} + \beta_{2} - 5) q^{94} + ( - 8 \beta_{3} + 2 \beta_{2} + 9 \beta_1 - 12) q^{96} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 - 7) q^{97} + ( - \beta_{3} + \beta_1 - 1) q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{14} + 18 q^{16} - 12 q^{17} - 2 q^{18} - 14 q^{19} + 4 q^{21} - 2 q^{22} - 8 q^{23} - 12 q^{24} + 8 q^{26} + 4 q^{27} + 2 q^{28} + 12 q^{29} - 4 q^{31} - 26 q^{32} + 4 q^{33} + 8 q^{34} + 2 q^{36} + 2 q^{37} + 10 q^{38} - 6 q^{39} - 2 q^{42} + 4 q^{43} + 2 q^{44} - 2 q^{46} - 8 q^{47} + 18 q^{48} + 4 q^{49} - 12 q^{51} + 10 q^{52} - 8 q^{53} - 2 q^{54} - 12 q^{56} - 14 q^{57} - 24 q^{58} - 22 q^{59} - 16 q^{61} - 6 q^{62} + 4 q^{63} + 48 q^{64} - 2 q^{66} + 8 q^{67} - 36 q^{68} - 8 q^{69} - 12 q^{72} - 24 q^{73} - 6 q^{74} - 20 q^{76} + 4 q^{77} + 8 q^{78} - 8 q^{79} + 4 q^{81} + 42 q^{82} - 10 q^{83} + 2 q^{84} - 16 q^{86} + 12 q^{87} - 12 q^{88} - 4 q^{89} - 6 q^{91} - 2 q^{92} - 4 q^{93} - 18 q^{94} - 26 q^{96} - 22 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \) 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
−0.360409
−1.22833
2.77462
0.814115
−2.77462 1.00000 5.69853 0 −2.77462 1.00000 −10.2620 1.00000 0
1.2 −0.814115 1.00000 −1.33722 0 −0.814115 1.00000 2.71688 1.00000 0
1.3 0.360409 1.00000 −1.87011 0 0.360409 1.00000 −1.39482 1.00000 0
1.4 1.22833 1.00000 −0.491210 0 1.22833 1.00000 −3.06002 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.bx 4
5.b even 2 1 5775.2.a.cc yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5775.2.a.bx 4 1.a even 1 1 trivial
5775.2.a.cc yes 4 5.b even 2 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} - 3T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} - 19T_{13}^{2} - 114T_{13} - 73 \) Copy content Toggle raw display
\( T_{17}^{4} + 12T_{17}^{3} + 12T_{17}^{2} - 272T_{17} - 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} - 3 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 19 T^{2} - 114 T - 73 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + 12 T^{2} + \cdots - 784 \) Copy content Toggle raw display
$19$ \( T^{4} + 14 T^{3} + 43 T^{2} + \cdots - 553 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 8 T^{2} - 116 T - 73 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 36 T^{2} - 16 T - 16 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 86 T^{2} - 308 T + 73 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} - 61 T^{2} - 58 T + 17 \) Copy content Toggle raw display
$41$ \( T^{4} - 122 T^{2} - 320 T + 1225 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 124 T^{2} + \cdots + 272 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 24 T^{2} - 180 T + 263 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + 16 T^{2} - 4 T - 17 \) Copy content Toggle raw display
$59$ \( T^{4} + 22 T^{3} + 97 T^{2} + \cdots - 2441 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + 24 T^{2} + \cdots - 521 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 56 T^{2} - 28 T + 439 \) Copy content Toggle raw display
$73$ \( T^{4} + 24 T^{3} + 128 T^{2} + \cdots - 3361 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 64 T^{2} - 44 T + 383 \) Copy content Toggle raw display
$83$ \( T^{4} + 10 T^{3} - 277 T^{2} + \cdots + 4297 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} - 336 T^{2} + \cdots + 20143 \) Copy content Toggle raw display
$97$ \( T^{4} + 22 T^{3} + 109 T^{2} + \cdots - 743 \) Copy content Toggle raw display
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