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Properties

Label 275.3.x.e
Level $275$
Weight $3$
Character orbit 275.x
Analytic conductor $7.493$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,3,Mod(51,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.51"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.x (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.49320726991\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{2} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 2) q^{3} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 6) q^{6}+ \cdots + ( - 69 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + \cdots + 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} - 9 q^{4} + 15 q^{6} - 10 q^{7} - 15 q^{8} - 11 q^{9} + q^{11} + 30 q^{12} + 20 q^{13} - 10 q^{14} + 19 q^{16} - 30 q^{18} + 25 q^{19} + 35 q^{22} + 20 q^{23} + 5 q^{24} - 10 q^{26} - 15 q^{27}+ \cdots + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−0.309017 − 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 − 0.587785i
1.80902 − 2.48990i −1.11803 + 3.44095i −1.69098 − 5.20431i 0 6.54508 + 9.00854i 0.854102 − 0.277515i −4.30902 − 1.40008i −3.30902 − 2.40414i 0
101.1 0.690983 + 0.224514i 1.11803 − 0.812299i −2.80902 − 2.04087i 0 0.954915 − 0.310271i −5.85410 + 8.05748i −3.19098 − 4.39201i −2.19098 + 6.74315i 0
151.1 1.80902 + 2.48990i −1.11803 − 3.44095i −1.69098 + 5.20431i 0 6.54508 − 9.00854i 0.854102 + 0.277515i −4.30902 + 1.40008i −3.30902 + 2.40414i 0
226.1 0.690983 − 0.224514i 1.11803 + 0.812299i −2.80902 + 2.04087i 0 0.954915 + 0.310271i −5.85410 − 8.05748i −3.19098 + 4.39201i −2.19098 − 6.74315i 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.d 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.x.e 4
5.b even 2 1 11.3.d.a ✓ 4
5.c odd 4 2 275.3.q.d 8
11.d odd 10 1 inner 275.3.x.e 4
15.d odd 2 1 99.3.k.a 4
20.d odd 2 1 176.3.n.a 4
55.d odd 2 1 121.3.d.d 4
55.h odd 10 1 11.3.d.a ✓ 4
55.h odd 10 1 121.3.b.b 4
55.h odd 10 1 121.3.d.a 4
55.h odd 10 1 121.3.d.c 4
55.j even 10 1 121.3.b.b 4
55.j even 10 1 121.3.d.a 4
55.j even 10 1 121.3.d.c 4
55.j even 10 1 121.3.d.d 4
55.l even 20 2 275.3.q.d 8
165.o odd 10 1 1089.3.c.e 4
165.r even 10 1 99.3.k.a 4
165.r even 10 1 1089.3.c.e 4
220.o even 10 1 176.3.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a ✓ 4 5.b even 2 1
11.3.d.a ✓ 4 55.h odd 10 1
99.3.k.a 4 15.d odd 2 1
99.3.k.a 4 165.r even 10 1
121.3.b.b 4 55.h odd 10 1
121.3.b.b 4 55.j even 10 1
121.3.d.a 4 55.h odd 10 1
121.3.d.a 4 55.j even 10 1
121.3.d.c 4 55.h odd 10 1
121.3.d.c 4 55.j even 10 1
121.3.d.d 4 55.d odd 2 1
121.3.d.d 4 55.j even 10 1
176.3.n.a 4 20.d odd 2 1
176.3.n.a 4 220.o even 10 1
275.3.q.d 8 5.c odd 4 2
275.3.q.d 8 55.l even 20 2
275.3.x.e 4 1.a even 1 1 trivial
275.3.x.e 4 11.d odd 10 1 inner
1089.3.c.e 4 165.o odd 10 1
1089.3.c.e 4 165.r even 10 1

Hecke kernels

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

\( T_{2}^{4} - 5T_{2}^{3} + 15T_{2}^{2} - 15T_{2} + 5 \) Copy content Toggle raw display
\( T_{3}^{4} + 10T_{3}^{2} - 25T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + \cdots + 80 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} - 20 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
$17$ \( T^{4} + 10985 T + 142805 \) Copy content Toggle raw display
$19$ \( T^{4} - 25 T^{3} + \cdots + 605 \) Copy content Toggle raw display
$23$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 40 T^{3} + \cdots + 9680 \) Copy content Toggle raw display
$31$ \( T^{4} + 58 T^{3} + \cdots + 55696 \) Copy content Toggle raw display
$37$ \( T^{4} + 90 T^{3} + \cdots + 2624400 \) Copy content Toggle raw display
$41$ \( T^{4} + 80 T^{3} + \cdots + 8405 \) Copy content Toggle raw display
$43$ \( T^{4} + 1625 T^{2} + 581405 \) Copy content Toggle raw display
$47$ \( T^{4} - 30 T^{3} + \cdots + 384400 \) Copy content Toggle raw display
$53$ \( T^{4} + 120 T^{3} + \cdots + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} - 23 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + \cdots + 403280 \) Copy content Toggle raw display
$67$ \( (T^{2} - 115 T + 2945)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 148 T^{3} + \cdots + 22619536 \) Copy content Toggle raw display
$73$ \( T^{4} + 300 T^{3} + \cdots + 93787805 \) Copy content Toggle raw display
$79$ \( T^{4} - 70 T^{3} + \cdots + 67280 \) Copy content Toggle raw display
$83$ \( T^{4} + 225 T^{3} + \cdots + 22281605 \) Copy content Toggle raw display
$89$ \( (T^{2} - 61 T - 7681)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 165 T^{3} + \cdots + 31416025 \) Copy content Toggle raw display
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