Properties

Label 55.3.i.a
Level $55$
Weight $3$
Character orbit 55.i
Analytic conductor $1.499$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,3,Mod(6,55)]
 
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, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 55.i (of order \(10\), 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: \(1.49864145398\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{2}+ \cdots - 7 \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{2}+ \cdots + ( - 21 \zeta_{10}^{3} - 21 \zeta_{10}^{2} + \cdots + 49) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} + 4 q^{3} - 9 q^{4} - 5 q^{5} - 20 q^{6} + 15 q^{7} + 15 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{2} + 4 q^{3} - 9 q^{4} - 5 q^{5} - 20 q^{6} + 15 q^{7} + 15 q^{8} - 7 q^{9} - q^{11} + 16 q^{12} + 5 q^{13} - 15 q^{14} - 20 q^{15} + 19 q^{16} - 10 q^{17} + 35 q^{18} + 50 q^{19} + 15 q^{20} + 15 q^{22} - 106 q^{23} + 20 q^{24} - 5 q^{25} - 40 q^{26} - 8 q^{27} - 50 q^{28} + 90 q^{29} + 60 q^{30} + 38 q^{31} + 64 q^{33} - 120 q^{34} - 20 q^{35} - 63 q^{36} - 34 q^{37} - 55 q^{38} + 20 q^{39} - 5 q^{40} + 85 q^{41} - 160 q^{42} - 19 q^{44} + 130 q^{46} - 24 q^{47} + 84 q^{48} + 56 q^{49} + 25 q^{50} - 280 q^{51} + 100 q^{52} + 114 q^{53} + 5 q^{55} + 70 q^{56} + 180 q^{57} - 90 q^{58} - 128 q^{59} - 60 q^{60} + 140 q^{61} - 90 q^{62} - 105 q^{63} - 149 q^{64} + 140 q^{66} + 116 q^{67} + 290 q^{68} - 116 q^{69} - 35 q^{70} + 8 q^{71} + 35 q^{72} - 20 q^{73} + 130 q^{74} + 20 q^{75} - 160 q^{77} + 280 q^{78} - 210 q^{79} + 25 q^{80} + 95 q^{81} - 270 q^{82} - 410 q^{83} + 340 q^{84} - 50 q^{85} - 170 q^{86} - 78 q^{89} - 35 q^{90} - 75 q^{91} + 241 q^{92} - 152 q^{93} + 375 q^{94} + 40 q^{95} - 460 q^{96} + 176 q^{97} + 133 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(\zeta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0.809017 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
−0.690983 + 0.224514i 3.23607 + 2.35114i −2.80902 + 2.04087i −0.690983 + 2.12663i −2.76393 0.898056i −0.163119 0.224514i 3.19098 4.39201i 2.16312 + 6.65740i 1.62460i
41.1 −1.80902 2.48990i −1.23607 3.80423i −1.69098 + 5.20431i −1.80902 1.31433i −7.23607 + 9.95959i 7.66312 + 2.48990i 4.30902 1.40008i −5.66312 + 4.11450i 6.88191i
46.1 −0.690983 0.224514i 3.23607 2.35114i −2.80902 2.04087i −0.690983 2.12663i −2.76393 + 0.898056i −0.163119 + 0.224514i 3.19098 + 4.39201i 2.16312 6.65740i 1.62460i
51.1 −1.80902 + 2.48990i −1.23607 + 3.80423i −1.69098 5.20431i −1.80902 + 1.31433i −7.23607 9.95959i 7.66312 2.48990i 4.30902 + 1.40008i −5.66312 4.11450i 6.88191i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.3.i.a 4
5.b even 2 1 275.3.x.d 4
5.c odd 4 2 275.3.q.c 8
11.c even 5 1 605.3.c.a 4
11.d odd 10 1 inner 55.3.i.a 4
11.d odd 10 1 605.3.c.a 4
55.h odd 10 1 275.3.x.d 4
55.l even 20 2 275.3.q.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.i.a 4 1.a even 1 1 trivial
55.3.i.a 4 11.d odd 10 1 inner
275.3.q.c 8 5.c odd 4 2
275.3.q.c 8 55.l even 20 2
275.3.x.d 4 5.b even 2 1
275.3.x.d 4 55.h odd 10 1
605.3.c.a 4 11.c even 5 1
605.3.c.a 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5T_{2}^{3} + 15T_{2}^{2} + 15T_{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 15 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + \cdots + 25205 \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots + 242000 \) Copy content Toggle raw display
$19$ \( T^{4} - 50 T^{3} + \cdots + 59405 \) Copy content Toggle raw display
$23$ \( (T^{2} + 53 T + 701)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 90 T^{3} + \cdots + 403280 \) Copy content Toggle raw display
$31$ \( T^{4} - 38 T^{3} + \cdots + 126736 \) Copy content Toggle raw display
$37$ \( T^{4} + 34 T^{3} + \cdots + 72361 \) Copy content Toggle raw display
$41$ \( T^{4} - 85 T^{3} + \cdots + 1017005 \) Copy content Toggle raw display
$43$ \( T^{4} + 2600 T^{2} + 1682000 \) Copy content Toggle raw display
$47$ \( T^{4} + 24 T^{3} + \cdots + 16491721 \) Copy content Toggle raw display
$53$ \( T^{4} - 114 T^{3} + \cdots + 5148361 \) Copy content Toggle raw display
$59$ \( T^{4} + 128 T^{3} + \cdots + 36348841 \) Copy content Toggle raw display
$61$ \( T^{4} - 140 T^{3} + \cdots + 3494480 \) Copy content Toggle raw display
$67$ \( (T^{2} - 58 T + 716)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + \cdots + 26896 \) Copy content Toggle raw display
$73$ \( T^{4} + 20 T^{3} + \cdots + 26083280 \) Copy content Toggle raw display
$79$ \( T^{4} + 210 T^{3} + \cdots + 20402000 \) Copy content Toggle raw display
$83$ \( T^{4} + 410 T^{3} + \cdots + 334562000 \) Copy content Toggle raw display
$89$ \( (T^{2} + 39 T - 11381)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 176 T^{3} + \cdots + 19044496 \) Copy content Toggle raw display
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