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Properties

Label 550.2.h.h
Level $550$
Weight $2$
Character orbit 550.h
Analytic conductor $4.392$
Analytic rank $0$
Dimension $4$
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: \(4\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - 4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 4 q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 4 q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} + 7 q^{9} - q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - q^{16} - 2 q^{17} + 8 q^{18} - 5 q^{19} - 12 q^{21} + 11 q^{22} + 4 q^{23} + q^{24} + 6 q^{26} - 5 q^{27} - 2 q^{28} + 10 q^{29} - 2 q^{31} - 4 q^{32} - 11 q^{33} + 2 q^{34} - 8 q^{36} + 18 q^{37} - 10 q^{38} - 6 q^{39} - 2 q^{41} + 2 q^{42} - 6 q^{43} + 4 q^{44} - 4 q^{46} + 8 q^{47} + 4 q^{48} + 3 q^{49} - 7 q^{51} - 6 q^{52} + 4 q^{53} - 8 q^{56} - 5 q^{57} - 10 q^{58} + 5 q^{59} + 8 q^{61} + 2 q^{62} - 16 q^{63} - q^{64} + 16 q^{66} - 22 q^{67} - 2 q^{68} + 14 q^{69} + 8 q^{71} - 7 q^{72} + 14 q^{73} - 18 q^{74} - 10 q^{76} + 8 q^{77} + 16 q^{78} - 30 q^{79} + 14 q^{81} + 7 q^{82} + 19 q^{83} + 8 q^{84} + 21 q^{86} + 20 q^{87} + q^{88} - 10 q^{89} - 12 q^{91} - 6 q^{92} - 2 q^{93} + 12 q^{94} - 4 q^{96} + 3 q^{97} + 12 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\)

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
0.809017 + 0.587785i
−0.309017 − 0.951057i
0.809017 − 0.587785i
−0.309017 + 0.951057i
0.809017 − 0.587785i −0.118034 − 0.363271i 0.309017 − 0.951057i 0 −0.309017 − 0.224514i 0.618034 − 1.90211i −0.309017 − 0.951057i 2.30902 − 1.67760i 0
251.1 −0.309017 + 0.951057i 2.11803 − 1.53884i −0.809017 − 0.587785i 0 0.809017 + 2.48990i −1.61803 − 1.17557i 0.809017 − 0.587785i 1.19098 − 3.66547i 0
301.1 0.809017 + 0.587785i −0.118034 + 0.363271i 0.309017 + 0.951057i 0 −0.309017 + 0.224514i 0.618034 + 1.90211i −0.309017 + 0.951057i 2.30902 + 1.67760i 0
401.1 −0.309017 − 0.951057i 2.11803 + 1.53884i −0.809017 + 0.587785i 0 0.809017 − 2.48990i −1.61803 + 1.17557i 0.809017 + 0.587785i 1.19098 + 3.66547i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.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.h 4
5.b even 2 1 22.2.c.a ✓ 4
5.c odd 4 2 550.2.ba.c 8
11.c even 5 1 inner 550.2.h.h 4
11.c even 5 1 6050.2.a.bs 2
11.d odd 10 1 6050.2.a.ci 2
15.d odd 2 1 198.2.f.e 4
20.d odd 2 1 176.2.m.c 4
40.e odd 2 1 704.2.m.a 4
40.f even 2 1 704.2.m.h 4
55.d odd 2 1 242.2.c.c 4
55.h odd 10 1 242.2.a.d 2
55.h odd 10 1 242.2.c.c 4
55.h odd 10 2 242.2.c.d 4
55.j even 10 1 22.2.c.a ✓ 4
55.j even 10 1 242.2.a.f 2
55.j even 10 2 242.2.c.a 4
55.k odd 20 2 550.2.ba.c 8
165.o odd 10 1 198.2.f.e 4
165.o odd 10 1 2178.2.a.p 2
165.r even 10 1 2178.2.a.x 2
220.n odd 10 1 176.2.m.c 4
220.n odd 10 1 1936.2.a.o 2
220.o even 10 1 1936.2.a.n 2
440.ba odd 10 1 7744.2.a.bn 2
440.bd even 10 1 704.2.m.h 4
440.bd even 10 1 7744.2.a.bm 2
440.bh odd 10 1 704.2.m.a 4
440.bh odd 10 1 7744.2.a.cz 2
440.bm even 10 1 7744.2.a.cy 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.2.c.a ✓ 4 5.b even 2 1
22.2.c.a ✓ 4 55.j even 10 1
176.2.m.c 4 20.d odd 2 1
176.2.m.c 4 220.n odd 10 1
198.2.f.e 4 15.d odd 2 1
198.2.f.e 4 165.o odd 10 1
242.2.a.d 2 55.h odd 10 1
242.2.a.f 2 55.j even 10 1
242.2.c.a 4 55.j even 10 2
242.2.c.c 4 55.d odd 2 1
242.2.c.c 4 55.h odd 10 1
242.2.c.d 4 55.h odd 10 2
550.2.h.h 4 1.a even 1 1 trivial
550.2.h.h 4 11.c even 5 1 inner
550.2.ba.c 8 5.c odd 4 2
550.2.ba.c 8 55.k odd 20 2
704.2.m.a 4 40.e odd 2 1
704.2.m.a 4 440.bh odd 10 1
704.2.m.h 4 40.f even 2 1
704.2.m.h 4 440.bd even 10 1
1936.2.a.n 2 220.o even 10 1
1936.2.a.o 2 220.n odd 10 1
2178.2.a.p 2 165.o odd 10 1
2178.2.a.x 2 165.r even 10 1
6050.2.a.bs 2 11.c even 5 1
6050.2.a.ci 2 11.d odd 10 1
7744.2.a.bm 2 440.bd even 10 1
7744.2.a.bn 2 440.ba odd 10 1
7744.2.a.cy 2 440.bm even 10 1
7744.2.a.cz 2 440.bh odd 10 1

Hecke kernels

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

\( T_{3}^{4} - 4T_{3}^{3} + 6T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 18 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$43$ \( (T^{2} + 3 T - 99)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} + 11 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + \cdots + 17161 \) Copy content Toggle raw display
$79$ \( T^{4} + 30 T^{3} + \cdots + 32400 \) Copy content Toggle raw display
$83$ \( T^{4} - 19 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$89$ \( (T^{2} + 5 T - 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
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