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Properties

Label 357.2.i.d
Level $357$
Weight $2$
Character orbit 357.i
Analytic conductor $2.851$
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] = [357,2,Mod(205,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.205"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 357 = 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 357.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.85065935216\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{3}) q^{2} - \beta_{6} q^{3} + ( - \beta_{7} - 2 \beta_{6} + \cdots + \beta_{4}) q^{4} + (2 \beta_{6} - \beta_{4} + \beta_1 - 2) q^{5} - \beta_{3} q^{6} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} - 5 q^{4} - 4 q^{5} - 2 q^{6} + 7 q^{7} + 6 q^{8} - 4 q^{9} + 4 q^{10} - q^{11} - 5 q^{12} + 20 q^{13} - 4 q^{14} + 8 q^{15} + q^{16} + 4 q^{17} + q^{18} - 13 q^{19} + 22 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\nu^{7} - 55\nu^{6} + 338\nu^{5} - 909\nu^{4} + 2308\nu^{3} - 5301\nu^{2} + 11263\nu - 6620 ) / 8102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -655\nu^{7} + 938\nu^{6} - 314\nu^{5} + 6885\nu^{4} - 22495\nu^{3} + 14321\nu^{2} - 38221\nu + 14204 ) / 8102 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -367\nu^{7} + 674\nu^{6} - 312\nu^{5} + 3332\nu^{4} - 13037\nu^{3} + 12372\nu^{2} - 18187\nu + 6734 ) / 4051 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} + \cdots + 88979 ) / 8102 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{6} - 4\beta_{4} + 2\beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 5\beta_{6} - 10\beta_{5} + 2\beta_{2} + 11\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{6} - 22\beta_{5} - 20\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{7} - 42\beta_{6} + \beta_{5} - 2\beta_{4} + 4\beta_{3} + 62\beta_{2} - 11\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + 5\beta_{6} - 115\beta_{5} + 88\beta_{4} - 62\beta_{3} + 68\beta_{2} + 30\beta _1 - 118 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/357\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(190\) \(239\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
0.373419 − 0.0835272i
2.11692 − 0.978886i
0.0512865 + 1.21608i
−1.54162 − 1.88572i
0.373419 + 0.0835272i
2.11692 + 0.978886i
0.0512865 − 1.21608i
−1.54162 + 1.88572i
−1.18584 + 2.05393i −0.500000 − 0.866025i −1.81242 − 3.13920i −0.740954 + 1.28337i 2.37167 2.18584 − 1.49068i 3.85358 −0.500000 + 0.866025i −1.75730 − 3.04373i
205.2 −0.186423 + 0.322894i −0.500000 − 0.866025i 0.930493 + 1.61166i 0.906198 − 1.56958i 0.372845 1.18642 + 2.36483i −1.43955 −0.500000 + 0.866025i 0.337872 + 0.585211i
205.3 0.768262 − 1.33067i −0.500000 − 0.866025i −0.180452 − 0.312552i −2.02752 + 3.51176i −1.53652 0.231738 + 2.63558i 2.51851 −0.500000 + 0.866025i 3.11533 + 5.39590i
205.4 1.10400 − 1.91218i −0.500000 − 0.866025i −1.43762 − 2.49004i −0.137728 + 0.238552i −2.20800 −0.103998 − 2.64371i −1.93254 −0.500000 + 0.866025i 0.304103 + 0.526722i
256.1 −1.18584 − 2.05393i −0.500000 + 0.866025i −1.81242 + 3.13920i −0.740954 − 1.28337i 2.37167 2.18584 + 1.49068i 3.85358 −0.500000 − 0.866025i −1.75730 + 3.04373i
256.2 −0.186423 − 0.322894i −0.500000 + 0.866025i 0.930493 − 1.61166i 0.906198 + 1.56958i 0.372845 1.18642 − 2.36483i −1.43955 −0.500000 − 0.866025i 0.337872 − 0.585211i
256.3 0.768262 + 1.33067i −0.500000 + 0.866025i −0.180452 + 0.312552i −2.02752 − 3.51176i −1.53652 0.231738 − 2.63558i 2.51851 −0.500000 − 0.866025i 3.11533 − 5.39590i
256.4 1.10400 + 1.91218i −0.500000 + 0.866025i −1.43762 + 2.49004i −0.137728 − 0.238552i −2.20800 −0.103998 + 2.64371i −1.93254 −0.500000 − 0.866025i 0.304103 − 0.526722i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 357.2.i.d ✓ 8
3.b odd 2 1 1071.2.i.f 8
7.c even 3 1 inner 357.2.i.d ✓ 8
7.c even 3 1 2499.2.a.w 4
7.d odd 6 1 2499.2.a.v 4
21.g even 6 1 7497.2.a.bn 4
21.h odd 6 1 1071.2.i.f 8
21.h odd 6 1 7497.2.a.bm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.i.d ✓ 8 1.a even 1 1 trivial
357.2.i.d ✓ 8 7.c even 3 1 inner
1071.2.i.f 8 3.b odd 2 1
1071.2.i.f 8 21.h odd 6 1
2499.2.a.v 4 7.d odd 6 1
2499.2.a.w 4 7.c even 3 1
7497.2.a.bm 4 21.h odd 6 1
7497.2.a.bn 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} - 6T_{2}^{5} + 39T_{2}^{4} - 30T_{2}^{3} + 54T_{2}^{2} + 18T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(357, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} - 7 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + 7 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{3} + \cdots - 103)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 13 T^{7} + \cdots + 134689 \) Copy content Toggle raw display
$23$ \( T^{8} + 3 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( (T^{4} - 7 T^{3} + \cdots + 147)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 3969 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$41$ \( (T^{4} - 26 T^{3} + \cdots - 4527)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots - 929)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( T^{8} + 15 T^{7} + \cdots + 3337929 \) Copy content Toggle raw display
$59$ \( T^{8} + 15 T^{7} + \cdots + 6036849 \) Copy content Toggle raw display
$61$ \( T^{8} - 11 T^{7} + \cdots + 2181529 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 267289801 \) Copy content Toggle raw display
$71$ \( (T^{4} - 15 T^{3} + \cdots - 531)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 3 T^{7} + \cdots + 48316401 \) Copy content Toggle raw display
$79$ \( T^{8} + 5 T^{7} + \cdots + 6538249 \) Copy content Toggle raw display
$83$ \( (T^{4} + 12 T^{3} + \cdots - 783)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + \cdots + 157176369 \) Copy content Toggle raw display
$97$ \( (T^{4} - 3 T^{3} + \cdots + 15567)^{2} \) Copy content Toggle raw display
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