Properties

Label 5577.2.a.r
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
Analytic rank $0$
Dimension $5$
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] = [5577,2,Mod(1,5577)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5577, 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("5577.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.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: \(44.5325692073\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1019601.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + (\beta_{4} - \beta_{3} + 1) q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + (\beta_{4} - \beta_{3} + 1) q^{7} + ( - \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + (\beta_{4} + \beta_{2} + 1) q^{10} + q^{11} + ( - \beta_{3} - \beta_{2} - 2) q^{12} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{14} - \beta_{3} q^{15} + (\beta_{4} - \beta_{3} + \beta_{2}) q^{16} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{17} + \beta_1 q^{18} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{19} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{20} + ( - \beta_{4} + \beta_{3} - 1) q^{21} + \beta_1 q^{22} + ( - \beta_{4} - 2 \beta_{2} + \beta_1 + 3) q^{23} + (\beta_{2} - \beta_1 - 1) q^{24} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{25} - q^{27} + (2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{28} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{4} - \beta_{2} - 1) q^{30} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{31} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{32} - q^{33} + (3 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 5) q^{34} + (4 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{35} + (\beta_{3} + \beta_{2} + 2) q^{36} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 3) q^{37} + (\beta_{4} - 2 \beta_{2} - 3) q^{38} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{40} - \beta_{2} q^{41} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{42} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{43} + (\beta_{3} + \beta_{2} + 2) q^{44} + \beta_{3} q^{45} + (5 \beta_{2} + 2 \beta_1 + 2) q^{46} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{47} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{48} + (2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{49} + (\beta_{4} + \beta_{3} - 3 \beta_1) q^{50} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{51} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{53} - \beta_1 q^{54} + \beta_{3} q^{55} + (2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{56} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{57} + ( - 3 \beta_{4} - \beta_{3} + 4 \beta_{2} - 1) q^{58} + ( - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{59} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{60} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + \beta_1 + 8) q^{62} + (\beta_{4} - \beta_{3} + 1) q^{63} + ( - \beta_{2} - 2 \beta_1 - 7) q^{64} - \beta_1 q^{66} + ( - 2 \beta_{4} - \beta_1 - 1) q^{67} + (2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{68} + (\beta_{4} + 2 \beta_{2} - \beta_1 - 3) q^{69} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 8) q^{70} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{71} + ( - \beta_{2} + \beta_1 + 1) q^{72} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 6) q^{73} + ( - 2 \beta_{4} + 5 \beta_1 + 3) q^{74} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{75} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{76} + (\beta_{4} - \beta_{3} + 1) q^{77} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{79} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{80} + q^{81} + (\beta_{4} + 2 \beta_{2} - \beta_1) q^{82} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{83} + ( - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{84} + (\beta_{4} + 6 \beta_{3} + 5 \beta_{2} - \beta_1 + 3) q^{85} + ( - 3 \beta_{4} + \beta_{2} + \beta_1) q^{86} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{87} + ( - \beta_{2} + \beta_1 + 1) q^{88} + ( - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{89} + (\beta_{4} + \beta_{2} + 1) q^{90} + ( - 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 2) q^{92} + (\beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{93} + (3 \beta_{4} - \beta_{2} + 4 \beta_1) q^{94} + ( - \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{95} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{96} + (2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 3) q^{97} + (2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 10 q^{4} - 2 q^{5} + 7 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 10 q^{4} - 2 q^{5} + 7 q^{7} + 3 q^{8} + 5 q^{9} + 7 q^{10} + 5 q^{11} - 10 q^{12} + q^{14} + 2 q^{15} + 4 q^{16} + 3 q^{17} + 7 q^{19} + 8 q^{20} - 7 q^{21} + 11 q^{23} - 3 q^{24} - 3 q^{25} - 5 q^{27} + 5 q^{28} - 2 q^{29} - 7 q^{30} + 10 q^{31} - 9 q^{32} - 5 q^{33} + 29 q^{34} - 14 q^{35} + 10 q^{36} + 15 q^{37} - 19 q^{38} + 15 q^{40} - 2 q^{41} - q^{42} + 7 q^{43} + 10 q^{44} - 2 q^{45} + 20 q^{46} + 18 q^{47} - 4 q^{48} + 14 q^{49} - 2 q^{50} - 3 q^{51} + 15 q^{53} - 2 q^{55} + 3 q^{56} - 7 q^{57} + 5 q^{58} - 4 q^{59} - 8 q^{60} - 14 q^{61} + 46 q^{62} + 7 q^{63} - 37 q^{64} - 5 q^{67} - 24 q^{68} - 11 q^{69} + 40 q^{70} - 13 q^{71} + 3 q^{72} + 28 q^{73} + 15 q^{74} + 3 q^{75} + 2 q^{76} + 7 q^{77} + 16 q^{79} - 22 q^{80} + 5 q^{81} + 4 q^{82} + 12 q^{83} - 5 q^{84} + 13 q^{85} + 2 q^{86} + 2 q^{87} + 3 q^{88} - 6 q^{89} + 7 q^{90} - 2 q^{92} - 10 q^{93} - 2 q^{94} - 21 q^{95} + 9 q^{96} + 9 q^{97} + 16 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 5\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 2\nu^{3} - 5\nu^{2} - 10\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{3} + 7\beta_{2} + 20 \) 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
−2.48141
−1.80092
−0.298978
2.20194
2.37936
−2.48141 −1.00000 4.15740 −1.71458 2.48141 2.09718 −5.35339 1.00000 4.25457
1.2 −1.80092 −1.00000 1.24330 1.40697 1.80092 −1.77714 1.36275 1.00000 −2.53384
1.3 −0.298978 −1.00000 −1.91061 −3.44245 0.298978 4.93983 1.16919 1.00000 1.02922
1.4 2.20194 −1.00000 2.84856 −0.484911 −2.20194 −1.91630 1.86848 1.00000 −1.06775
1.5 2.37936 −1.00000 3.66136 2.23497 −2.37936 3.65643 3.95297 1.00000 5.31779
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

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 5577.2.a.r 5
13.b even 2 1 5577.2.a.s 5
13.c even 3 2 429.2.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.d 10 13.c even 3 2
5577.2.a.r 5 1.a even 1 1 trivial
5577.2.a.s 5 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2}^{5} - 10T_{2}^{3} - T_{2}^{2} + 24T_{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{5} + 2T_{5}^{4} - 9T_{5}^{3} - 10T_{5}^{2} + 16T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{5} - 7T_{7}^{4} + 59T_{7}^{2} - 17T_{7} - 129 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 10 T^{3} - T^{2} + 24 T + 7 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 2 T^{4} - 9 T^{3} - 10 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{5} - 7 T^{4} + 59 T^{2} - 17 T - 129 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 3 T^{4} - 73 T^{3} + 230 T^{2} + \cdots - 581 \) Copy content Toggle raw display
$19$ \( T^{5} - 7 T^{4} - 9 T^{3} + 120 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$23$ \( T^{5} - 11 T^{4} - 9 T^{3} + \cdots + 1389 \) Copy content Toggle raw display
$29$ \( T^{5} + 2 T^{4} - 96 T^{3} + \cdots - 5019 \) Copy content Toggle raw display
$31$ \( T^{5} - 10 T^{4} - 81 T^{3} + \cdots - 381 \) Copy content Toggle raw display
$37$ \( T^{5} - 15 T^{4} + 57 T^{3} + \cdots + 207 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} - 9 T^{3} - 4 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$43$ \( T^{5} - 7 T^{4} - 39 T^{3} + 216 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$47$ \( T^{5} - 18 T^{4} + 71 T^{3} + \cdots - 1057 \) Copy content Toggle raw display
$53$ \( T^{5} - 15 T^{4} - 87 T^{3} + \cdots - 19467 \) Copy content Toggle raw display
$59$ \( T^{5} + 4 T^{4} - 249 T^{3} + \cdots + 68649 \) Copy content Toggle raw display
$61$ \( T^{5} + 14 T^{4} - 15 T^{3} + \cdots + 147 \) Copy content Toggle raw display
$67$ \( T^{5} + 5 T^{4} - 92 T^{3} + 223 T^{2} + \cdots - 773 \) Copy content Toggle raw display
$71$ \( T^{5} + 13 T^{4} - 51 T^{3} + \cdots - 14241 \) Copy content Toggle raw display
$73$ \( T^{5} - 28 T^{4} + 228 T^{3} + \cdots + 7071 \) Copy content Toggle raw display
$79$ \( T^{5} - 16 T^{4} + 34 T^{3} + \cdots + 2239 \) Copy content Toggle raw display
$83$ \( T^{5} - 12 T^{4} - 63 T^{3} + \cdots - 1701 \) Copy content Toggle raw display
$89$ \( T^{5} + 6 T^{4} - 273 T^{3} - 795 T^{2} + \cdots + 63 \) Copy content Toggle raw display
$97$ \( T^{5} - 9 T^{4} - 223 T^{3} + \cdots - 93247 \) Copy content Toggle raw display
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