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Properties

Label 55.2.g.b
Level $55$
Weight $2$
Character orbit 55.g
Analytic conductor $0.439$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,2,Mod(16,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.g (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 1) q^{3}+ \cdots + ( - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 5 q^{3} - 2 q^{4} + 2 q^{5} - 7 q^{6} - q^{7} + 4 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 5 q^{3} - 2 q^{4} + 2 q^{5} - 7 q^{6} - q^{7} + 4 q^{8} - 5 q^{9} + 2 q^{10} + 3 q^{11} + 16 q^{12} - 2 q^{13} - 16 q^{14} + 5 q^{15} + 4 q^{16} - 13 q^{17} + 15 q^{19} - 3 q^{20} - 20 q^{21} - 7 q^{22} + 10 q^{23} + 13 q^{24} - 2 q^{25} + 10 q^{26} + 10 q^{27} - 6 q^{28} - 9 q^{29} - 8 q^{30} - 10 q^{31} + 16 q^{32} + 5 q^{33} + 4 q^{34} - 4 q^{35} - 15 q^{36} + 24 q^{37} + 21 q^{39} - 4 q^{40} + 8 q^{41} + 9 q^{42} - 38 q^{43} - 12 q^{44} + 3 q^{46} + 5 q^{48} + q^{49} - 2 q^{50} + q^{51} - 28 q^{52} + 13 q^{53} + 16 q^{54} + 7 q^{55} + 22 q^{56} - 45 q^{57} + 12 q^{58} - 27 q^{59} + 4 q^{60} + 6 q^{61} - 30 q^{62} + 25 q^{63} - 26 q^{64} + 2 q^{65} + 13 q^{66} - 38 q^{67} + 11 q^{68} - q^{69} + 16 q^{70} - 20 q^{71} - 30 q^{72} + 13 q^{73} + 20 q^{74} + 5 q^{75} + 34 q^{77} - 16 q^{78} + 37 q^{79} + q^{80} + 8 q^{81} + 28 q^{82} + 27 q^{83} + 28 q^{84} - 12 q^{85} - 3 q^{86} + 38 q^{87} - 36 q^{88} - 16 q^{89} - 10 q^{90} + 44 q^{91} + 11 q^{92} - 35 q^{93} + 17 q^{94} - 15 q^{95} - 17 q^{96} + 24 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.69513 + 1.23158i
−0.386111 − 0.280526i
0.418926 − 1.28932i
−0.227943 + 0.701538i
1.69513 − 1.23158i
−0.386111 + 0.280526i
0.418926 + 1.28932i
−0.227943 − 0.701538i
−0.647481 − 1.99274i −1.54765 − 1.12443i −1.93376 + 1.40496i −0.309017 + 0.951057i −1.23863 + 3.81211i 2.48141 − 1.80285i 0.661536 + 0.480634i 0.203814 + 0.627276i 2.09529
16.2 0.147481 + 0.453901i −0.261370 − 0.189896i 1.43376 − 1.04169i −0.309017 + 0.951057i 0.0476470 − 0.146642i −2.17239 + 1.57833i 1.45650 + 1.05821i −0.894797 − 2.75390i −0.477260
26.1 −1.09676 − 0.796845i 0.177837 − 0.547326i −0.0501062 − 0.154211i 0.809017 − 0.587785i −0.631180 + 0.458579i −1.12773 − 3.47080i −0.905781 + 2.78771i 2.15911 + 1.56869i −1.35567
26.2 0.596764 + 0.433574i −0.868820 + 2.67395i −0.449894 − 1.38463i 0.809017 − 0.587785i −1.67784 + 1.21902i 0.318714 + 0.980901i 0.787747 − 2.42443i −3.96813 − 2.88301i 0.737640
31.1 −0.647481 + 1.99274i −1.54765 + 1.12443i −1.93376 − 1.40496i −0.309017 − 0.951057i −1.23863 − 3.81211i 2.48141 + 1.80285i 0.661536 − 0.480634i 0.203814 − 0.627276i 2.09529
31.2 0.147481 − 0.453901i −0.261370 + 0.189896i 1.43376 + 1.04169i −0.309017 − 0.951057i 0.0476470 + 0.146642i −2.17239 − 1.57833i 1.45650 − 1.05821i −0.894797 + 2.75390i −0.477260
36.1 −1.09676 + 0.796845i 0.177837 + 0.547326i −0.0501062 + 0.154211i 0.809017 + 0.587785i −0.631180 − 0.458579i −1.12773 + 3.47080i −0.905781 − 2.78771i 2.15911 − 1.56869i −1.35567
36.2 0.596764 − 0.433574i −0.868820 − 2.67395i −0.449894 + 1.38463i 0.809017 + 0.587785i −1.67784 − 1.21902i 0.318714 − 0.980901i 0.787747 + 2.42443i −3.96813 + 2.88301i 0.737640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.g.b ✓ 8
3.b odd 2 1 495.2.n.e 8
4.b odd 2 1 880.2.bo.h 8
5.b even 2 1 275.2.h.a 8
5.c odd 4 2 275.2.z.a 16
11.b odd 2 1 605.2.g.k 8
11.c even 5 1 inner 55.2.g.b ✓ 8
11.c even 5 1 605.2.a.j 4
11.c even 5 2 605.2.g.m 8
11.d odd 10 1 605.2.a.k 4
11.d odd 10 2 605.2.g.e 8
11.d odd 10 1 605.2.g.k 8
33.f even 10 1 5445.2.a.bi 4
33.h odd 10 1 495.2.n.e 8
33.h odd 10 1 5445.2.a.bp 4
44.g even 10 1 9680.2.a.cm 4
44.h odd 10 1 880.2.bo.h 8
44.h odd 10 1 9680.2.a.cn 4
55.h odd 10 1 3025.2.a.w 4
55.j even 10 1 275.2.h.a 8
55.j even 10 1 3025.2.a.bd 4
55.k odd 20 2 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b ✓ 8 1.a even 1 1 trivial
55.2.g.b ✓ 8 11.c even 5 1 inner
275.2.h.a 8 5.b even 2 1
275.2.h.a 8 55.j even 10 1
275.2.z.a 16 5.c odd 4 2
275.2.z.a 16 55.k odd 20 2
495.2.n.e 8 3.b odd 2 1
495.2.n.e 8 33.h odd 10 1
605.2.a.j 4 11.c even 5 1
605.2.a.k 4 11.d odd 10 1
605.2.g.e 8 11.d odd 10 2
605.2.g.k 8 11.b odd 2 1
605.2.g.k 8 11.d odd 10 1
605.2.g.m 8 11.c even 5 2
880.2.bo.h 8 4.b odd 2 1
880.2.bo.h 8 44.h odd 10 1
3025.2.a.w 4 55.h odd 10 1
3025.2.a.bd 4 55.j even 10 1
5445.2.a.bi 4 33.f even 10 1
5445.2.a.bp 4 33.h odd 10 1
9680.2.a.cm 4 44.g even 10 1
9680.2.a.cn 4 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} + 2T_{2}^{5} - T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$17$ \( T^{8} + 13 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{8} - 15 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} + 4 T^{2} + \cdots - 11)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + \cdots + 203401 \) Copy content Toggle raw display
$31$ \( T^{8} + 10 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$37$ \( T^{8} - 24 T^{7} + \cdots + 1324801 \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + \cdots + 101761 \) Copy content Toggle raw display
$43$ \( (T^{4} + 19 T^{3} + \cdots + 211)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 23 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$53$ \( T^{8} - 13 T^{7} + \cdots + 885481 \) Copy content Toggle raw display
$59$ \( T^{8} + 27 T^{7} + \cdots + 687241 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$67$ \( (T^{4} + 19 T^{3} + \cdots - 4079)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 20 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$73$ \( T^{8} - 13 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{8} - 37 T^{7} + \cdots + 45954841 \) Copy content Toggle raw display
$83$ \( T^{8} - 27 T^{7} + \cdots + 2886601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} + \cdots + 1861)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + \cdots + 9066121 \) Copy content Toggle raw display
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