Properties

Label 55.4.a.d
Level $55$
Weight $4$
Character orbit 55.a
Self dual yes
Analytic conductor $3.245$
Analytic rank $0$
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] = [55,4,Mod(1,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 55.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: \(3.24510505032\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1539480.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 25x^{2} + 9x + 96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 2) q^{3} + (\beta_{3} + \beta_1 + 5) q^{4} + 5 q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{7} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 10) q^{8} + ( - 3 \beta_{2} - 6 \beta_1 + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 2) q^{3} + (\beta_{3} + \beta_1 + 5) q^{4} + 5 q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{7} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 10) q^{8} + ( - 3 \beta_{2} - 6 \beta_1 + 13) q^{9} + 5 \beta_1 q^{10} + 11 q^{11} + (2 \beta_{3} - \beta_{2} - 10 \beta_1 + 22) q^{12} + (6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 22) q^{13} + ( - 2 \beta_{3} - 9 \beta_{2} + \cdots - 14) q^{14}+ \cdots + ( - 33 \beta_{2} - 66 \beta_1 + 143) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 9 q^{3} + 19 q^{4} + 20 q^{5} - 19 q^{6} + 9 q^{7} + 33 q^{8} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 9 q^{3} + 19 q^{4} + 20 q^{5} - 19 q^{6} + 9 q^{7} + 33 q^{8} + 49 q^{9} + 5 q^{10} + 44 q^{11} + 75 q^{12} + 70 q^{13} - 49 q^{14} + 45 q^{15} - 37 q^{16} + 103 q^{17} - 356 q^{18} - 205 q^{19} + 95 q^{20} - 181 q^{21} + 11 q^{22} - 56 q^{23} - 387 q^{24} + 100 q^{25} - 86 q^{26} + 405 q^{27} - 551 q^{28} - 79 q^{29} - 95 q^{30} + 49 q^{31} + 225 q^{32} + 99 q^{33} - 939 q^{34} + 45 q^{35} + 112 q^{36} + 289 q^{37} + 145 q^{38} - 26 q^{39} + 165 q^{40} + 736 q^{41} + 1141 q^{42} - 152 q^{43} + 209 q^{44} + 245 q^{45} - 334 q^{46} + 412 q^{47} - 661 q^{48} + 37 q^{49} + 25 q^{50} + 413 q^{51} + 1598 q^{52} + 1685 q^{53} - 1489 q^{54} + 220 q^{55} - 257 q^{56} - 795 q^{57} + 609 q^{58} - 842 q^{59} + 375 q^{60} - 1097 q^{61} + 1359 q^{62} - 186 q^{63} - 165 q^{64} + 350 q^{65} - 209 q^{66} - 122 q^{67} - 757 q^{68} - 58 q^{69} - 245 q^{70} - 521 q^{71} - 1620 q^{72} - 590 q^{73} + 3257 q^{74} + 225 q^{75} - 2825 q^{76} + 99 q^{77} - 3094 q^{78} - 1118 q^{79} - 185 q^{80} + 184 q^{81} - 402 q^{82} - 122 q^{83} - 605 q^{84} + 515 q^{85} + 3452 q^{86} - 4091 q^{87} + 363 q^{88} - 181 q^{89} - 1780 q^{90} - 2190 q^{91} + 430 q^{92} + 763 q^{93} + 1034 q^{94} - 1025 q^{95} + 3981 q^{96} + 1474 q^{97} - 872 q^{98} + 539 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 25x^{2} + 9x + 96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 15\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 17\beta _1 + 10 \) 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
−4.18087
−2.03085
2.30365
4.90807
−4.18087 9.33168 9.47970 5.00000 −39.0146 −14.2911 −6.18645 60.0803 −20.9044
1.2 −2.03085 −5.45953 −3.87566 5.00000 11.0875 27.2109 24.1176 2.80648 −10.1542
1.3 2.30365 6.23583 −2.69320 5.00000 14.3652 13.1506 −24.6334 11.8856 11.5182
1.4 4.90807 −1.10798 16.0892 5.00000 −5.43806 −17.0703 39.7022 −25.7724 24.5404
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 55.4.a.d 4
3.b odd 2 1 495.4.a.n 4
4.b odd 2 1 880.4.a.z 4
5.b even 2 1 275.4.a.e 4
5.c odd 4 2 275.4.b.e 8
11.b odd 2 1 605.4.a.j 4
15.d odd 2 1 2475.4.a.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.d 4 1.a even 1 1 trivial
275.4.a.e 4 5.b even 2 1
275.4.b.e 8 5.c odd 4 2
495.4.a.n 4 3.b odd 2 1
605.4.a.j 4 11.b odd 2 1
880.4.a.z 4 4.b odd 2 1
2475.4.a.bc 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} - 25T_{2}^{2} + 9T_{2} + 96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(55))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + \cdots + 96 \) Copy content Toggle raw display
$3$ \( T^{4} - 9 T^{3} + \cdots + 352 \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 9 T^{3} + \cdots + 87296 \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 70 T^{3} + \cdots - 2887232 \) Copy content Toggle raw display
$17$ \( T^{4} - 103 T^{3} + \cdots + 199152 \) Copy content Toggle raw display
$19$ \( T^{4} + 205 T^{3} + \cdots - 5224000 \) Copy content Toggle raw display
$23$ \( T^{4} + 56 T^{3} + \cdots - 221568 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 1106175480 \) Copy content Toggle raw display
$31$ \( T^{4} - 49 T^{3} + \cdots + 126259200 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 4092453032 \) Copy content Toggle raw display
$41$ \( T^{4} - 736 T^{3} + \cdots + 329735184 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 2589511936 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 19971136128 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 20698646424 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 123367943040 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 4578287464 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 316737807616 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 1139751168 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 138483587488 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 289419632640 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 340395563136 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 27515045400 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 144669247168 \) Copy content Toggle raw display
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