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Properties

Label 275.3.c.f
Level $275$
Weight $3$
Character orbit 275.c
Analytic conductor $7.493$
Analytic rank $0$
Dimension $8$
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(76,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.76"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-8,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.49320726991\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_1) q^{6} - \beta_{7} q^{7} + (\beta_{6} - \beta_{5} - 3 \beta_1) q^{8} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{9}+ \cdots + (6 \beta_{7} - 4 \beta_{6} - 13 \beta_{4} + \cdots - 23) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 28 q^{4} - 4 q^{9} + 8 q^{11} + 48 q^{12} + 88 q^{16} - 80 q^{22} - 8 q^{23} - 100 q^{26} + 16 q^{27} + 36 q^{31} + 152 q^{33} + 80 q^{34} - 216 q^{36} + 88 q^{37} + 160 q^{38} - 280 q^{42}+ \cdots - 184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 25\nu^{4} - 171\nu^{2} - 275 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 29\nu^{4} + 235\nu^{2} + 415 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 29\nu^{5} + 235\nu^{3} + 415\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 29\nu^{5} + 251\nu^{3} + 591\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 27\nu^{5} - 203\nu^{3} - 345\nu ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 4\beta_{3} - 16\beta_{2} + 93 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 16\beta_{6} + 24\beta_{5} + 141\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -100\beta_{4} - 116\beta_{3} + 229\beta_{2} - 1232 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -116\beta_{7} + 229\beta_{6} - 445\beta_{5} - 1919\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
− 3.88606i
− 3.15955i
− 1.66769i
− 1.46104i
1.46104i
1.66769i
3.15955i
3.88606i
− 3.88606i −4.12151 −11.1014 0 16.0164i 3.44089i 27.5966i 7.98688 0
76.2 − 3.15955i 3.03112 −5.98274 0 − 9.57697i − 5.67155i 6.26455i 0.187686 0
76.3 − 1.66769i −2.79505 1.21881 0 4.66128i 6.72266i − 8.70336i −1.18769 0
76.4 − 1.46104i −0.114554 1.86536 0 0.167368i − 4.56066i − 8.56953i −8.98688 0
76.5 1.46104i −0.114554 1.86536 0 − 0.167368i 4.56066i 8.56953i −8.98688 0
76.6 1.66769i −2.79505 1.21881 0 − 4.66128i − 6.72266i 8.70336i −1.18769 0
76.7 3.15955i 3.03112 −5.98274 0 9.57697i 5.67155i − 6.26455i 0.187686 0
76.8 3.88606i −4.12151 −11.1014 0 − 16.0164i − 3.44089i − 27.5966i 7.98688 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.3.c.f 8
5.b even 2 1 55.3.c.a ✓ 8
5.c odd 4 2 275.3.d.c 16
11.b odd 2 1 inner 275.3.c.f 8
15.d odd 2 1 495.3.b.a 8
20.d odd 2 1 880.3.j.a 8
55.d odd 2 1 55.3.c.a ✓ 8
55.e even 4 2 275.3.d.c 16
165.d even 2 1 495.3.b.a 8
220.g even 2 1 880.3.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.c.a ✓ 8 5.b even 2 1
55.3.c.a ✓ 8 55.d odd 2 1
275.3.c.f 8 1.a even 1 1 trivial
275.3.c.f 8 11.b odd 2 1 inner
275.3.d.c 16 5.c odd 4 2
275.3.d.c 16 55.e even 4 2
495.3.b.a 8 15.d odd 2 1
495.3.b.a 8 165.d even 2 1
880.3.j.a 8 20.d odd 2 1
880.3.j.a 8 220.g even 2 1

Hecke kernels

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

\( T_{2}^{8} + 30T_{2}^{6} + 280T_{2}^{4} + 890T_{2}^{2} + 895 \) Copy content Toggle raw display
\( T_{3}^{4} + 4T_{3}^{3} - 9T_{3}^{2} - 36T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 30 T^{6} + \cdots + 895 \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{3} - 9 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 110 T^{6} + \cdots + 358000 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} + 620 T^{6} + \cdots + 27723520 \) Copy content Toggle raw display
$17$ \( T^{8} + 1270 T^{6} + \cdots + 1732720 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3931928320 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots - 27584)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 4850 T^{6} + \cdots + 27723520 \) Copy content Toggle raw display
$31$ \( (T^{4} - 18 T^{3} + \cdots + 178076)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 44 T^{3} + \cdots - 1000204)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 5420634603520 \) Copy content Toggle raw display
$43$ \( T^{8} + 4180 T^{6} + \cdots + 91648000 \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{3} + \cdots + 22336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 76 T^{3} + \cdots - 14101244)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots - 465344)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 44785710872320 \) Copy content Toggle raw display
$67$ \( (T^{4} - 44 T^{3} + \cdots + 3497456)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 138 T^{3} + \cdots - 22924)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 15327220913920 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 67909027102720 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 26187838846720 \) Copy content Toggle raw display
$89$ \( (T^{4} - 222 T^{3} + \cdots - 1268884)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 156 T^{3} + \cdots - 21090064)^{2} \) Copy content Toggle raw display
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