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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [760,2,Mod(379,760)] 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("760.379"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8,-8,0,0,8,0,16,-16,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1499238400.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} - q^{3} + 2 \beta_{5} q^{4} + ( - \beta_{6} + \beta_{5}) q^{5} + (\beta_{5} + 1) q^{6} + ( - \beta_{6} + \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{5} + 2) q^{8} - 2 q^{9}+ \cdots + ( - 2 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{6} + 16 q^{8} - 16 q^{9} + 4 q^{10} + 8 q^{11} - 32 q^{16} + 16 q^{18} + 8 q^{19} - 8 q^{20} - 8 q^{22} - 16 q^{24} - 8 q^{25} + 40 q^{27} - 4 q^{30} + 32 q^{32} - 8 q^{33}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 59x^{4} - 66x^{3} + 54x^{2} - 26x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} - 34\nu^{5} + 50\nu^{4} - 82\nu^{3} + 67\nu^{2} - 56\nu + 30 ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 3\nu^{5} + 12\nu^{4} - 19\nu^{3} + 28\nu^{2} - 19\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 13\nu^{6} + 49\nu^{5} - 105\nu^{4} + 167\nu^{3} - 162\nu^{2} + 111\nu - 20 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + 3\nu^{6} - 13\nu^{5} + 22\nu^{4} - 39\nu^{3} + 35\nu^{2} - 27\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{7} - 21\nu^{6} + 83\nu^{5} - 155\nu^{4} + 249\nu^{3} - 229\nu^{2} + 157\nu - 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{7} + 4\nu^{6} - 16\nu^{5} + 33\nu^{4} - 56\nu^{3} + 56\nu^{2} - 40\nu + 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\nu^{7} - 42\nu^{6} + 176\nu^{5} - 335\nu^{4} + 598\nu^{3} - 583\nu^{2} + 474\nu - 150 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 4\beta_{5} - 2\beta_{3} + 5\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 3\beta_{6} + 9\beta_{5} + 9\beta_{4} - 12\beta_{3} + 2\beta_{2} + 4\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} - 25\beta_{6} - 16\beta_{5} + 10\beta_{4} - \beta_{3} + 5\beta_{2} - 24\beta _1 + 56 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -17\beta_{7} - 10\beta_{6} - 71\beta_{5} - 50\beta_{4} + 66\beta_{3} - 7\beta_{2} - 44\beta _1 - 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{7} + 147\beta_{6} + 29\beta_{5} - 119\beta_{4} + 68\beta_{3} - 42\beta_{2} + 100\beta _1 - 363 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.500000 + 1.41267i
0.500000 + 0.0845405i
0.500000 − 1.08454i
0.500000 − 2.41267i
0.500000 − 1.41267i
0.500000 − 0.0845405i
0.500000 + 1.08454i
0.500000 + 2.41267i
−1.00000 − 1.00000i −1.00000 2.00000i −1.91267 − 1.15831i 1.00000 + 1.00000i −3.21974 2.00000 − 2.00000i −2.00000 0.754359 + 3.07098i
379.2 −1.00000 − 1.00000i −1.00000 2.00000i −0.584541 + 2.15831i 1.00000 + 1.00000i −4.86140 2.00000 − 2.00000i −2.00000 2.74285 − 1.57377i
379.3 −1.00000 − 1.00000i −1.00000 2.00000i 0.584541 + 2.15831i 1.00000 + 1.00000i 4.86140 2.00000 − 2.00000i −2.00000 1.57377 − 2.74285i
379.4 −1.00000 − 1.00000i −1.00000 2.00000i 1.91267 − 1.15831i 1.00000 + 1.00000i 3.21974 2.00000 − 2.00000i −2.00000 −3.07098 − 0.754359i
379.5 −1.00000 + 1.00000i −1.00000 − 2.00000i −1.91267 + 1.15831i 1.00000 − 1.00000i −3.21974 2.00000 + 2.00000i −2.00000 0.754359 − 3.07098i
379.6 −1.00000 + 1.00000i −1.00000 − 2.00000i −0.584541 − 2.15831i 1.00000 − 1.00000i −4.86140 2.00000 + 2.00000i −2.00000 2.74285 + 1.57377i
379.7 −1.00000 + 1.00000i −1.00000 − 2.00000i 0.584541 − 2.15831i 1.00000 − 1.00000i 4.86140 2.00000 + 2.00000i −2.00000 1.57377 + 2.74285i
379.8 −1.00000 + 1.00000i −1.00000 − 2.00000i 1.91267 + 1.15831i 1.00000 − 1.00000i 3.21974 2.00000 + 2.00000i −2.00000 −3.07098 + 0.754359i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 379.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
95.d odd 2 1 inner
760.p even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.p.d ✓ 8
5.b even 2 1 760.2.p.g yes 8
8.d odd 2 1 inner 760.2.p.d ✓ 8
19.b odd 2 1 760.2.p.g yes 8
40.e odd 2 1 760.2.p.g yes 8
95.d odd 2 1 inner 760.2.p.d ✓ 8
152.b even 2 1 760.2.p.g yes 8
760.p even 2 1 inner 760.2.p.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.p.d ✓ 8 1.a even 1 1 trivial
760.2.p.d ✓ 8 8.d odd 2 1 inner
760.2.p.d ✓ 8 95.d odd 2 1 inner
760.2.p.d ✓ 8 760.p even 2 1 inner
760.2.p.g yes 8 5.b even 2 1
760.2.p.g yes 8 19.b odd 2 1
760.2.p.g yes 8 40.e odd 2 1
760.2.p.g yes 8 152.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 34T_{7}^{2} + 245 \) Copy content Toggle raw display
\( T_{29}^{4} - 130T_{29}^{2} + 3125 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 34 T^{2} + 245)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 10)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14 T^{2} + 5)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 26 T^{2} + 125)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 130 T^{2} + 3125)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64 T^{2} + 980)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 80 T^{2} + 500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{2} + 2420)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 320)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 222 T^{2} + 7921)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 166 T^{2} + 6845)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 24 T^{2} + 100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T - 19)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 224 T^{2} + 7220)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 86 T^{2} + 1805)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 280 T^{2} + 2000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 136 T^{2} + 3920)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 176 T^{2} + 2420)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{8} \) Copy content Toggle raw display
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