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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,3,Mod(24,275)] 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("275.24"); 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([5, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.q (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-22,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content 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: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{7} + \zeta_{20}^{5} + \cdots + \zeta_{20}) q^{2} + (4 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{3} + (3 \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots - 3) q^{4} + (4 \zeta_{20}^{6} - 8 \zeta_{20}^{4} + \cdots - 9) q^{6}+ \cdots + ( - 14 \zeta_{20}^{6} - 80 \zeta_{20}^{4} + \cdots - 37) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 22 q^{4} - 50 q^{6} - 26 q^{9} - 2 q^{11} - 100 q^{14} - 42 q^{16} + 90 q^{19} + 110 q^{24} - 120 q^{26} + 140 q^{29} + 196 q^{31} + 260 q^{34} + 204 q^{36} - 240 q^{39} + 80 q^{41} - 112 q^{44} + 20 q^{46}+ \cdots - 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
−0.951057 + 2.92705i 2.71441 + 3.73607i −4.42705 − 3.21644i 0 −13.5172 + 4.39201i 8.78402 + 6.38197i 3.66547 − 2.66312i −3.80902 + 11.7229i 0
24.2 0.951057 − 2.92705i −2.71441 − 3.73607i −4.42705 − 3.21644i 0 −13.5172 + 4.39201i −8.78402 − 6.38197i −3.66547 + 2.66312i −3.80902 + 11.7229i 0
74.1 −0.587785 + 0.427051i −2.26538 + 0.736068i −1.07295 + 3.30220i 0 1.01722 − 1.40008i 2.80017 − 8.61803i −1.67760 − 5.16312i −2.69098 + 1.95511i 0
74.2 0.587785 − 0.427051i 2.26538 − 0.736068i −1.07295 + 3.30220i 0 1.01722 − 1.40008i −2.80017 + 8.61803i 1.67760 + 5.16312i −2.69098 + 1.95511i 0
149.1 −0.951057 − 2.92705i 2.71441 − 3.73607i −4.42705 + 3.21644i 0 −13.5172 − 4.39201i 8.78402 − 6.38197i 3.66547 + 2.66312i −3.80902 − 11.7229i 0
149.2 0.951057 + 2.92705i −2.71441 + 3.73607i −4.42705 + 3.21644i 0 −13.5172 − 4.39201i −8.78402 + 6.38197i −3.66547 − 2.66312i −3.80902 − 11.7229i 0
249.1 −0.587785 − 0.427051i −2.26538 − 0.736068i −1.07295 − 3.30220i 0 1.01722 + 1.40008i 2.80017 + 8.61803i −1.67760 + 5.16312i −2.69098 − 1.95511i 0
249.2 0.587785 + 0.427051i 2.26538 + 0.736068i −1.07295 − 3.30220i 0 1.01722 + 1.40008i −2.80017 − 8.61803i 1.67760 − 5.16312i −2.69098 − 1.95511i 0
\(n\): e.g. 2-40 or 80-90
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.a 8
5.b even 2 1 inner 275.3.q.a 8
5.c odd 4 1 55.3.i.b ✓ 4
5.c odd 4 1 275.3.x.a 4
11.d odd 10 1 inner 275.3.q.a 8
55.h odd 10 1 inner 275.3.q.a 8
55.k odd 20 1 605.3.c.b 4
55.l even 20 1 55.3.i.b ✓ 4
55.l even 20 1 275.3.x.a 4
55.l even 20 1 605.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.i.b ✓ 4 5.c odd 4 1
55.3.i.b ✓ 4 55.l even 20 1
275.3.q.a 8 1.a even 1 1 trivial
275.3.q.a 8 5.b even 2 1 inner
275.3.q.a 8 11.d odd 10 1 inner
275.3.q.a 8 55.h odd 10 1 inner
275.3.x.a 4 5.c odd 4 1
275.3.x.a 4 55.l even 20 1
605.3.c.b 4 55.k odd 20 1
605.3.c.b 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 \(S_{3}^{\mathrm{new}}(275, [\chi])\):

\( T_{2}^{8} + 15T_{2}^{6} + 85T_{2}^{4} - 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{6} + 366T_{3}^{4} - 3751T_{3}^{2} + 14641 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 15 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 60 T^{6} + \cdots + 93702400 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 180 T^{6} + \cdots + 41990400 \) Copy content Toggle raw display
$17$ \( T^{8} + 800 T^{6} + \cdots + 346146025 \) Copy content Toggle raw display
$19$ \( (T^{4} - 45 T^{3} + \cdots + 59405)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 72 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 70 T^{3} + \cdots + 192080)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 98 T^{3} + \cdots + 2421136)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 36136489216 \) Copy content Toggle raw display
$41$ \( (T^{4} - 40 T^{3} + \cdots + 15125)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4325 T^{2} + 4560125)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 11866206109696 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 48281596662016 \) Copy content Toggle raw display
$59$ \( (T^{4} + 217 T^{3} + \cdots + 46389721)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 80 T^{3} + \cdots + 16128080)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2107 T^{2} + 259081)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 38 T^{3} + \cdots + 5216656)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( (T^{4} - 100 T^{3} + \cdots + 237774080)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 172612430622025 \) Copy content Toggle raw display
$89$ \( (T^{2} + 201 T + 10039)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 33122338550401 \) Copy content Toggle raw display
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