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Properties

Label 275.3.q.b
Level $275$
Weight $3$
Character orbit 275.q
Analytic conductor $7.493$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,3,Mod(24,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.q (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: \(7.49320726991\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5 \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{5}) q^{2} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5}) q^{3} + ( - \beta_{3} + 2 \beta_1 - 1) q^{4} + (6 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 4) q^{6} + (\beta_{7} - \beta_{6} - 5 \beta_{2}) q^{7}+ \cdots + ( - 66 \beta_{4} + 33 \beta_{3} + \cdots + 132) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 20 q^{6} + 22 q^{9} + 2 q^{11} - 40 q^{14} - 22 q^{16} + 100 q^{19} - 60 q^{24} + 130 q^{26} - 120 q^{29} - 86 q^{31} - 140 q^{34} + 22 q^{36} + 400 q^{39} + 190 q^{41} - 28 q^{44} - 40 q^{46}+ \cdots + 638 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{20}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{3} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{20}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{20}^{5} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + 2\zeta_{20} \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + 4\beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{6} + \beta_{5} + \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 4\beta_{5} - \beta_{2} ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
−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
−0.587785 + 1.80902i 2.62866 + 3.61803i 0.309017 + 0.224514i 0 −8.09017 + 2.62866i 6.74315 + 4.89919i −6.74315 + 4.89919i −3.39919 + 10.4616i 0
24.2 0.587785 − 1.80902i −2.62866 − 3.61803i 0.309017 + 0.224514i 0 −8.09017 + 2.62866i −6.74315 − 4.89919i 6.74315 − 4.89919i −3.39919 + 10.4616i 0
74.1 −0.951057 + 0.690983i −4.25325 + 1.38197i −0.809017 + 2.48990i 0 3.09017 − 4.25325i 2.40414 − 7.39919i −2.40414 − 7.39919i 8.89919 − 6.46564i 0
74.2 0.951057 − 0.690983i 4.25325 − 1.38197i −0.809017 + 2.48990i 0 3.09017 − 4.25325i −2.40414 + 7.39919i 2.40414 + 7.39919i 8.89919 − 6.46564i 0
149.1 −0.587785 − 1.80902i 2.62866 − 3.61803i 0.309017 − 0.224514i 0 −8.09017 − 2.62866i 6.74315 − 4.89919i −6.74315 − 4.89919i −3.39919 − 10.4616i 0
149.2 0.587785 + 1.80902i −2.62866 + 3.61803i 0.309017 − 0.224514i 0 −8.09017 − 2.62866i −6.74315 + 4.89919i 6.74315 + 4.89919i −3.39919 − 10.4616i 0
249.1 −0.951057 − 0.690983i −4.25325 − 1.38197i −0.809017 − 2.48990i 0 3.09017 + 4.25325i 2.40414 + 7.39919i −2.40414 + 7.39919i 8.89919 + 6.46564i 0
249.2 0.951057 + 0.690983i 4.25325 + 1.38197i −0.809017 − 2.48990i 0 3.09017 + 4.25325i −2.40414 − 7.39919i 2.40414 − 7.39919i 8.89919 + 6.46564i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.d odd 10 1 inner
55.h odd 10 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(275, [\chi])\):

\( T_{2}^{8} + 5T_{2}^{6} + 10T_{2}^{4} + 25 \) Copy content Toggle raw display
\( T_{3}^{8} - 20T_{3}^{6} + 400T_{3}^{4} - 8000T_{3}^{2} + 160000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} - 20 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 55 T^{6} + \cdots + 17682025 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 600 T^{6} + \cdots + 228765625 \) Copy content Toggle raw display
$17$ \( T^{8} + 735 T^{6} + \cdots + 144120025 \) Copy content Toggle raw display
$19$ \( (T^{4} - 50 T^{3} + \cdots + 67280)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1035 T^{2} + 42025)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 60 T^{3} + \cdots + 17405)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 43 T^{3} + \cdots + 1560001)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 99082613700625 \) Copy content Toggle raw display
$41$ \( (T^{4} - 95 T^{3} + \cdots + 3880805)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6130 T^{2} + 7925405)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1766100625 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 92352100000000 \) Copy content Toggle raw display
$59$ \( (T^{4} + 33 T^{3} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 85 T^{3} + \cdots + 8179205)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 14610 T^{2} + 51051025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 137 T^{3} + \cdots + 22005481)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 12387619356025 \) Copy content Toggle raw display
$79$ \( (T^{4} + 70 T^{3} + \cdots + 41789405)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16329721410025 \) Copy content Toggle raw display
$89$ \( (T^{2} - 159 T + 5659)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 885480536850625 \) Copy content Toggle raw display
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