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Properties

Label 550.2.ba.c
Level $550$
Weight $2$
Character orbit 550.ba
Analytic conductor $4.392$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,2,Mod(49,550)]
 
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([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.ba (of order \(10\), degree \(4\), not 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 \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + (\zeta_{20}^{5} + \zeta_{20}^{3} + \zeta_{20}) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} + \cdots + \zeta_{20}^{2}) q^{6}+ \cdots + (2 \zeta_{20}^{6} + 3 \zeta_{20}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + (\zeta_{20}^{5} + \zeta_{20}^{3} + \zeta_{20}) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} + \cdots + \zeta_{20}^{2}) q^{6}+ \cdots + ( - 9 \zeta_{20}^{6} + 13 \zeta_{20}^{4} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 2 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 2 q^{6} - 14 q^{9} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 10 q^{19} - 24 q^{21} - 2 q^{24} + 12 q^{26} - 20 q^{29} - 4 q^{31} - 4 q^{34} - 16 q^{36} + 12 q^{39} - 4 q^{41} - 8 q^{44} - 8 q^{46} - 6 q^{49} - 14 q^{51} - 16 q^{56} - 10 q^{59} + 16 q^{61} + 2 q^{64} + 32 q^{66} - 28 q^{69} + 16 q^{71} + 36 q^{74} - 20 q^{76} + 60 q^{79} + 28 q^{81} - 16 q^{84} + 42 q^{86} + 20 q^{89} - 24 q^{91} - 24 q^{94} - 8 q^{96} + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-\zeta_{20}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.951057 + 0.309017i
0.951057 − 0.309017i
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 − 0.809017i
−0.951057 + 0.309017i −1.53884 + 2.11803i 0.809017 − 0.587785i 0 0.809017 − 2.48990i 1.17557 + 1.61803i −0.587785 + 0.809017i −1.19098 − 3.66547i 0
49.2 0.951057 − 0.309017i 1.53884 − 2.11803i 0.809017 − 0.587785i 0 0.809017 − 2.48990i −1.17557 − 1.61803i 0.587785 − 0.809017i −1.19098 − 3.66547i 0
399.1 −0.587785 − 0.809017i 0.363271 − 0.118034i −0.309017 + 0.951057i 0 −0.309017 − 0.224514i −1.90211 − 0.618034i 0.951057 − 0.309017i −2.30902 + 1.67760i 0
399.2 0.587785 + 0.809017i −0.363271 + 0.118034i −0.309017 + 0.951057i 0 −0.309017 − 0.224514i 1.90211 + 0.618034i −0.951057 + 0.309017i −2.30902 + 1.67760i 0
449.1 −0.951057 − 0.309017i −1.53884 − 2.11803i 0.809017 + 0.587785i 0 0.809017 + 2.48990i 1.17557 − 1.61803i −0.587785 − 0.809017i −1.19098 + 3.66547i 0
449.2 0.951057 + 0.309017i 1.53884 + 2.11803i 0.809017 + 0.587785i 0 0.809017 + 2.48990i −1.17557 + 1.61803i 0.587785 + 0.809017i −1.19098 + 3.66547i 0
499.1 −0.587785 + 0.809017i 0.363271 + 0.118034i −0.309017 − 0.951057i 0 −0.309017 + 0.224514i −1.90211 + 0.618034i 0.951057 + 0.309017i −2.30902 − 1.67760i 0
499.2 0.587785 − 0.809017i −0.363271 − 0.118034i −0.309017 − 0.951057i 0 −0.309017 + 0.224514i 1.90211 − 0.618034i −0.951057 − 0.309017i −2.30902 − 1.67760i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{8} + 4T_{3}^{6} + 46T_{3}^{4} - 11T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 4T_{7}^{6} + 16T_{7}^{4} - 64T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + 21 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 36 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 207 T^{2} + 9801)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 64 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} - 176 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( (T^{4} + 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 123 T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 76 T^{6} + \cdots + 294499921 \) Copy content Toggle raw display
$79$ \( (T^{4} - 30 T^{3} + \cdots + 32400)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 11 T^{6} + \cdots + 12117361 \) Copy content Toggle raw display
$89$ \( (T^{2} - 5 T - 25)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 279 T^{6} + \cdots + 96059601 \) Copy content Toggle raw display
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