Properties

Label 5775.2.a.cb
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: 4.4.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_1 q^{6} - q^{7} + \beta_{3} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_1 q^{6} - q^{7} + \beta_{3} q^{8} + q^{9} - q^{11} + (\beta_{2} + 2) q^{12} + ( - 2 \beta_{3} - \beta_1 - 1) q^{13} - \beta_1 q^{14} + (\beta_{2} - 3) q^{16} + ( - 4 \beta_{2} - 4) q^{17} + \beta_1 q^{18} + (\beta_{3} + \beta_{2}) q^{19} - q^{21} - \beta_1 q^{22} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{23} + \beta_{3} q^{24} + ( - 7 \beta_{2} - \beta_1 - 6) q^{26} + q^{27} + ( - \beta_{2} - 2) q^{28} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{29} - \beta_{3} q^{31} + ( - \beta_{3} - 3 \beta_1) q^{32} - q^{33} + ( - 4 \beta_{3} - 4 \beta_1) q^{34} + (\beta_{2} + 2) q^{36} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{37} + (\beta_{3} + 3 \beta_{2} + 1) q^{38} + ( - 2 \beta_{3} - \beta_1 - 1) q^{39} + (6 \beta_{2} + 3) q^{41} - \beta_1 q^{42} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{43} + ( - \beta_{2} - 2) q^{44} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{46} + (4 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{47} + (\beta_{2} - 3) q^{48} + q^{49} + ( - 4 \beta_{2} - 4) q^{51} + ( - 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 2) q^{52} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{53} + \beta_1 q^{54} - \beta_{3} q^{56} + (\beta_{3} + \beta_{2}) q^{57} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{58} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{59} + (5 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{61} + ( - 3 \beta_{2} - 1) q^{62} - q^{63} + ( - 8 \beta_{2} - 7) q^{64} - \beta_1 q^{66} + (2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 8) q^{67} + ( - 8 \beta_{2} - 12) q^{68} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{69} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 3) q^{71} + \beta_{3} q^{72} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{73} + (3 \beta_{3} - 8 \beta_{2} - \beta_1 - 10) q^{74} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{76} + q^{77} + ( - 7 \beta_{2} - \beta_1 - 6) q^{78} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{79} + q^{81} + (6 \beta_{3} + 3 \beta_1) q^{82} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{83} + ( - \beta_{2} - 2) q^{84} + ( - 4 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 10) q^{86} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{87} - \beta_{3} q^{88} + (\beta_{2} - 5 \beta_1 - 4) q^{89} + (2 \beta_{3} + \beta_1 + 1) q^{91} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{92} - \beta_{3} q^{93} + ( - \beta_{3} + 13 \beta_{2} - 4 \beta_1 + 8) q^{94} + ( - \beta_{3} - 3 \beta_1) q^{96} + ( - 6 \beta_{2} + \beta_1 - 6) q^{97} + \beta_1 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 6 q^{4} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 6 q^{4} - 4 q^{7} + 4 q^{9} - 4 q^{11} + 6 q^{12} - 4 q^{13} - 14 q^{16} - 8 q^{17} - 2 q^{19} - 4 q^{21} - 2 q^{23} - 10 q^{26} + 4 q^{27} - 6 q^{28} + 4 q^{29} - 4 q^{33} + 6 q^{36} - 10 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{44} + 12 q^{46} - 14 q^{47} - 14 q^{48} + 4 q^{49} - 8 q^{51} - 6 q^{52} - 10 q^{53} - 2 q^{57} - 32 q^{58} + 16 q^{59} + 6 q^{61} + 2 q^{62} - 4 q^{63} - 12 q^{64} - 20 q^{67} - 32 q^{68} - 2 q^{69} - 6 q^{71} - 14 q^{73} - 24 q^{74} + 2 q^{76} + 4 q^{77} - 10 q^{78} - 6 q^{79} + 4 q^{81} - 30 q^{83} - 6 q^{84} - 24 q^{86} + 4 q^{87} - 18 q^{89} + 4 q^{91} - 8 q^{92} + 6 q^{94} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.14896
−1.54336
1.54336
2.14896
−2.14896 1.00000 2.61803 0 −2.14896 −1.00000 −1.32813 1.00000 0
1.2 −1.54336 1.00000 0.381966 0 −1.54336 −1.00000 2.49721 1.00000 0
1.3 1.54336 1.00000 0.381966 0 1.54336 −1.00000 −2.49721 1.00000 0
1.4 2.14896 1.00000 2.61803 0 2.14896 −1.00000 1.32813 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.cb yes 4
5.b even 2 1 5775.2.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5775.2.a.ca 4 5.b even 2 1
5775.2.a.cb yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5775))\):

\( T_{2}^{4} - 7T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} - 29T_{13}^{2} - 66T_{13} + 241 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7T^{2} + 11 \) 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
$13$ \( T^{4} + 4 T^{3} - 29 T^{2} - 66 T + 241 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} - 9 T^{2} + 5 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 14 T^{2} + 10 T + 5 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 72 T^{2} + 432 T - 496 \) Copy content Toggle raw display
$31$ \( T^{4} - 8T^{2} + 11 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} - 37 T^{2} + \cdots - 1459 \) Copy content Toggle raw display
$41$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 92 T^{2} + 400 T - 464 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} - 56 T^{2} + \cdots + 2255 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} + 18 T^{2} + \cdots - 179 \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + 21 T^{2} + 94 T + 31 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} - 186 T^{2} + \cdots + 8705 \) Copy content Toggle raw display
$67$ \( T^{4} + 20 T^{3} - 8 T^{2} + \cdots - 8384 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} - 26 T^{2} - 210 T - 295 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + 54 T^{2} + 70 T + 25 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} - 44 T^{2} - 234 T - 229 \) Copy content Toggle raw display
$83$ \( T^{4} + 30 T^{3} + 223 T^{2} + \cdots - 7499 \) Copy content Toggle raw display
$89$ \( T^{4} + 18 T^{3} - 56 T^{2} + \cdots + 2911 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 43 T^{2} + \cdots + 1019 \) Copy content Toggle raw display
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