Properties

Label 550.2.t.a
Level $550$
Weight $2$
Character orbit 550.t
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(89,550)]
 
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([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.t (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: \(4.39177211117\)
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: 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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + ( - \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20} + 1) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{2} - \zeta_{20}) q^{5} + ( - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20} + 1) q^{6} + ( - \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{7} + \zeta_{20}^{3} q^{8} + ( - 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + ( - \zeta_{20}^{7} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20} + 1) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{2} - \zeta_{20}) q^{5} + ( - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20} + 1) q^{6} + ( - \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 1) q^{7} + \zeta_{20}^{3} q^{8} + ( - 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3}) q^{9} + (\zeta_{20}^{6} - \zeta_{20}^{5} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{10} + \zeta_{20}^{6} q^{11} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{2} + \zeta_{20}) q^{12} + ( - \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 3 \zeta_{20}^{2} + 4) q^{13} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - \zeta_{20}) q^{14} + (\zeta_{20}^{7} + 3 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - \zeta_{20}^{3} - 2 \zeta_{20} - 1) q^{15} + \zeta_{20}^{4} q^{16} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{5} + \zeta_{20}^{3} - 2 \zeta_{20} - 1) q^{17} + ( - 2 \zeta_{20}^{6} - \zeta_{20}^{5} - 2 \zeta_{20}^{4}) q^{18} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{6} + \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} - 3 \zeta_{20} + 2) q^{19} + (\zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3}) q^{20} + (2 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{3} + 4 \zeta_{20}^{2} - 2 \zeta_{20} - 4) q^{21} + \zeta_{20}^{7} q^{22} + ( - 3 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 3 \zeta_{20}^{5} - 4 \zeta_{20}^{4} + 3 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{23} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{3} + 1) q^{24} + ( - 4 \zeta_{20}^{7} + \zeta_{20}^{6} + 4 \zeta_{20}^{5} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 2) q^{25} + ( - \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 3 \zeta_{20}^{3} + 4 \zeta_{20}) q^{26} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{2} - 2) q^{27} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - \zeta_{20}^{2}) q^{28} + (2 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - \zeta_{20}^{4} + 2 \zeta_{20}^{3} + \zeta_{20}^{2} - 2 \zeta_{20} - 1) q^{29} + (3 \zeta_{20}^{7} + 3 \zeta_{20}^{6} - \zeta_{20}^{5} - 2 \zeta_{20}^{4} - \zeta_{20}^{2} - \zeta_{20} - 1) q^{30} + ( - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 3 \zeta_{20} + 3) q^{31} + \zeta_{20}^{5} q^{32} + (\zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20} + 1) q^{33} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{6} + \zeta_{20}^{4} - 2 \zeta_{20}^{2} - \zeta_{20}) q^{34} + (2 \zeta_{20}^{7} + 3 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - 3 \zeta_{20}^{4} - 3 \zeta_{20}^{3} + 3 \zeta_{20}^{2} + \cdots - 1) q^{35}+ \cdots + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 4 \zeta_{20} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{3} + 2 q^{4} + 6 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{3} + 2 q^{4} + 6 q^{6} + 2 q^{9} + 2 q^{11} + 20 q^{13} - 2 q^{14} - 2 q^{16} - 10 q^{17} + 16 q^{19} - 16 q^{21} + 10 q^{23} + 4 q^{24} - 10 q^{25} - 20 q^{27} - 10 q^{28} - 4 q^{29} + 10 q^{31} + 10 q^{33} - 10 q^{34} + 10 q^{35} - 2 q^{36} + 20 q^{37} - 10 q^{38} + 30 q^{39} + 26 q^{41} - 20 q^{42} - 2 q^{44} + 20 q^{45} + 16 q^{46} + 10 q^{47} + 8 q^{49} + 20 q^{50} - 40 q^{51} + 20 q^{52} - 10 q^{53} - 8 q^{54} + 10 q^{55} + 2 q^{56} - 20 q^{58} - 16 q^{59} - 10 q^{60} - 40 q^{61} - 30 q^{63} + 2 q^{64} - 10 q^{65} + 4 q^{66} + 28 q^{69} + 10 q^{70} - 26 q^{71} - 10 q^{73} + 8 q^{74} + 10 q^{75} + 4 q^{76} + 10 q^{77} + 20 q^{78} + 32 q^{79} - 18 q^{81} + 50 q^{83} - 24 q^{84} + 10 q^{85} - 10 q^{86} - 30 q^{87} + 6 q^{89} - 10 q^{90} + 40 q^{91} - 20 q^{92} - 10 q^{94} - 30 q^{95} + 6 q^{96} - 40 q^{97} - 20 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 0.809017i 1.22123 + 0.396802i −0.309017 + 0.951057i 1.70582 + 1.44575i −0.396802 1.22123i 2.07768i 0.951057 0.309017i −1.09310 0.794181i 0.166977 2.22982i
89.2 0.587785 + 0.809017i 2.39680 + 0.778768i −0.309017 + 0.951057i 0.530249 2.17229i 0.778768 + 2.39680i 4.07768i −0.951057 + 0.309017i 2.71113 + 1.96975i 2.06909 0.847859i
309.1 −0.587785 + 0.809017i 1.22123 0.396802i −0.309017 0.951057i 1.70582 1.44575i −0.396802 + 1.22123i 2.07768i 0.951057 + 0.309017i −1.09310 + 0.794181i 0.166977 + 2.22982i
309.2 0.587785 0.809017i 2.39680 0.778768i −0.309017 0.951057i 0.530249 + 2.17229i 0.778768 2.39680i 4.07768i −0.951057 0.309017i 2.71113 1.96975i 2.06909 + 0.847859i
419.1 −0.951057 + 0.309017i −0.260074 0.357960i 0.809017 0.587785i −0.166977 + 2.22982i 0.357960 + 0.260074i 0.273457i −0.587785 + 0.809017i 0.866554 2.66698i −0.530249 2.17229i
419.2 0.951057 0.309017i 1.64204 + 2.26007i 0.809017 0.587785i −2.06909 + 0.847859i 2.26007 + 1.64204i 1.72654i 0.587785 0.809017i −1.48459 + 4.56909i −1.70582 + 1.44575i
529.1 −0.951057 0.309017i −0.260074 + 0.357960i 0.809017 + 0.587785i −0.166977 2.22982i 0.357960 0.260074i 0.273457i −0.587785 0.809017i 0.866554 + 2.66698i −0.530249 + 2.17229i
529.2 0.951057 + 0.309017i 1.64204 2.26007i 0.809017 + 0.587785i −2.06909 0.847859i 2.26007 1.64204i 1.72654i 0.587785 + 0.809017i −1.48459 4.56909i −1.70582 1.44575i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e 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.t.a 8
25.e even 10 1 inner 550.2.t.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.t.a 8 1.a even 1 1 trivial
550.2.t.a 8 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 10T_{3}^{7} + 46T_{3}^{6} - 120T_{3}^{5} + 176T_{3}^{4} - 120T_{3}^{3} + 16T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(550, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 10 T^{7} + 46 T^{6} - 120 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{6} + 10 T^{5} + 5 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 24 T^{6} + 136 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{3} + 50 T^{2} - 125 T + 125)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + 40 T^{6} + 70 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$19$ \( T^{8} - 16 T^{7} + 132 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + 46 T^{6} + \cdots + 59536 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + 32 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + 90 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{8} - 20 T^{7} + 119 T^{6} + \cdots + 606841 \) Copy content Toggle raw display
$41$ \( T^{8} - 26 T^{7} + 432 T^{6} + \cdots + 39601 \) Copy content Toggle raw display
$43$ \( T^{8} + 320 T^{6} + \cdots + 16810000 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + 19 T^{6} + \cdots + 160801 \) Copy content Toggle raw display
$59$ \( T^{8} + 16 T^{7} + 192 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$61$ \( T^{8} + 40 T^{7} + 1000 T^{6} + \cdots + 9030025 \) Copy content Toggle raw display
$67$ \( T^{8} - 176 T^{6} + \cdots + 162205696 \) Copy content Toggle raw display
$71$ \( T^{8} + 26 T^{7} + \cdots + 101687056 \) Copy content Toggle raw display
$73$ \( T^{8} + 10 T^{7} + 36 T^{6} + \cdots + 7070281 \) Copy content Toggle raw display
$79$ \( T^{8} - 32 T^{7} + 588 T^{6} + \cdots + 23078416 \) Copy content Toggle raw display
$83$ \( T^{8} - 50 T^{7} + 1270 T^{6} + \cdots + 15840400 \) Copy content Toggle raw display
$89$ \( T^{8} - 6 T^{7} + 317 T^{6} + \cdots + 2253001 \) Copy content Toggle raw display
$97$ \( T^{8} + 40 T^{7} + \cdots + 143017681 \) Copy content Toggle raw display
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