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Properties

Label 550.2.h.j
Level $550$
Weight $2$
Character orbit 550.h
Analytic conductor $4.392$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,2,Mod(201,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.h (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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.20164000000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{6} - 537\nu^{4} + 44034\nu^{2} - 40328 ) / 383471 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -153\nu^{6} + 4833\nu^{4} - 12835\nu^{2} - 20519 ) / 383471 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -153\nu^{7} + 4833\nu^{5} - 12835\nu^{3} - 20519\nu ) / 383471 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -585\nu^{7} - 4078\nu^{5} - 49075\nu^{3} - 461926\nu ) / 383471 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -721\nu^{6} + 218\nu^{4} - 17876\nu^{2} - 139302 ) / 383471 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -721\nu^{7} + 218\nu^{5} - 17876\nu^{3} - 139302\nu ) / 383471 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 9\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{7} - 9\beta_{5} - 8\beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{6} + 82\beta_{3} + 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{5} + 65\beta_{4} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -537\beta_{6} - 218\beta_{2} - 218 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -755\beta_{7} + 218\beta_{5} + 218\beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.839592 + 2.58400i
−0.839592 − 2.58400i
−2.50900 + 1.82290i
2.50900 − 1.82290i
0.839592 − 2.58400i
−0.839592 + 2.58400i
−2.50900 − 1.82290i
2.50900 + 1.82290i
−0.809017 + 0.587785i −0.500000 − 1.53884i 0.309017 − 0.951057i 0 1.30902 + 0.951057i −1.16751 + 3.59321i 0.309017 + 0.951057i 0.309017 − 0.224514i 0
201.2 −0.809017 + 0.587785i −0.500000 − 1.53884i 0.309017 − 0.951057i 0 1.30902 + 0.951057i 1.54947 − 4.76878i 0.309017 + 0.951057i 0.309017 − 0.224514i 0
251.1 0.309017 − 0.951057i −0.500000 + 0.363271i −0.809017 − 0.587785i 0 0.190983 + 0.587785i −0.241631 − 0.175555i −0.809017 + 0.587785i −0.809017 + 2.48990i 0
251.2 0.309017 − 0.951057i −0.500000 + 0.363271i −0.809017 − 0.587785i 0 0.190983 + 0.587785i 2.85966 + 2.07767i −0.809017 + 0.587785i −0.809017 + 2.48990i 0
301.1 −0.809017 − 0.587785i −0.500000 + 1.53884i 0.309017 + 0.951057i 0 1.30902 − 0.951057i −1.16751 − 3.59321i 0.309017 − 0.951057i 0.309017 + 0.224514i 0
301.2 −0.809017 − 0.587785i −0.500000 + 1.53884i 0.309017 + 0.951057i 0 1.30902 − 0.951057i 1.54947 + 4.76878i 0.309017 − 0.951057i 0.309017 + 0.224514i 0
401.1 0.309017 + 0.951057i −0.500000 − 0.363271i −0.809017 + 0.587785i 0 0.190983 − 0.587785i −0.241631 + 0.175555i −0.809017 − 0.587785i −0.809017 − 2.48990i 0
401.2 0.309017 + 0.951057i −0.500000 − 0.363271i −0.809017 + 0.587785i 0 0.190983 − 0.587785i 2.85966 − 2.07767i −0.809017 − 0.587785i −0.809017 − 2.48990i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.2
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 550.2.h.j 8
5.b even 2 1 550.2.h.n 8
5.c odd 4 2 110.2.j.b ✓ 16
11.c even 5 1 inner 550.2.h.j 8
11.c even 5 1 6050.2.a.di 4
11.d odd 10 1 6050.2.a.da 4
15.e even 4 2 990.2.ba.h 16
20.e even 4 2 880.2.cd.b 16
55.h odd 10 1 6050.2.a.dl 4
55.j even 10 1 550.2.h.n 8
55.j even 10 1 6050.2.a.dd 4
55.k odd 20 2 110.2.j.b ✓ 16
55.k odd 20 2 1210.2.b.k 8
55.l even 20 2 1210.2.b.l 8
165.v even 20 2 990.2.ba.h 16
220.v even 20 2 880.2.cd.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.j.b ✓ 16 5.c odd 4 2
110.2.j.b ✓ 16 55.k odd 20 2
550.2.h.j 8 1.a even 1 1 trivial
550.2.h.j 8 11.c even 5 1 inner
550.2.h.n 8 5.b even 2 1
550.2.h.n 8 55.j even 10 1
880.2.cd.b 16 20.e even 4 2
880.2.cd.b 16 220.v even 20 2
990.2.ba.h 16 15.e even 4 2
990.2.ba.h 16 165.v even 20 2
1210.2.b.k 8 55.k odd 20 2
1210.2.b.l 8 55.l even 20 2
6050.2.a.da 4 11.d odd 10 1
6050.2.a.dd 4 55.j even 10 1
6050.2.a.di 4 11.c even 5 1
6050.2.a.dl 4 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 6T_{7}^{7} + 46T_{7}^{6} - 156T_{7}^{5} + 596T_{7}^{4} - 1560T_{7}^{3} + 3640T_{7}^{2} + 2000T_{7} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{8} + 10 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{4} - 38 T^{2} + \cdots - 44)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 144400 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 3168400 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 633616 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T + 5)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} + \cdots + 3168400 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + \cdots + 144400 \) Copy content Toggle raw display
$59$ \( T^{8} + 9 T^{6} + \cdots + 57121 \) Copy content Toggle raw display
$61$ \( T^{8} - 10 T^{7} + \cdots + 1210000 \) Copy content Toggle raw display
$67$ \( (T^{4} + 10 T^{3} + \cdots - 2299)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 28 T^{7} + \cdots + 28944400 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( T^{8} + 18 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$83$ \( T^{8} - 14 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$89$ \( (T^{2} - T - 31)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} + \cdots + 104427961 \) Copy content Toggle raw display
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