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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3328,2,Mod(1665,3328)] 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("3328.1665"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,-10,0,-16,0,0,0,0,0,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.89857052655616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16x^{8} + 88x^{6} + 185x^{4} + 100x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1664)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + \beta_1 q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{4} - 1) q^{7} + ( - \beta_{7} + \beta_{6} - 1) q^{9} + ( - \beta_{3} + \beta_{2}) q^{11} + \beta_{2} q^{13} + ( - \beta_{4} + 2) q^{15}+ \cdots + ( - 6 \beta_{9} - 3 \beta_{8} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{7} - 16 q^{9} + 18 q^{15} + 6 q^{17} + 4 q^{23} - 20 q^{25} + 44 q^{31} - 16 q^{33} + 2 q^{39} - 20 q^{41} + 10 q^{47} + 64 q^{49} + 16 q^{55} - 12 q^{57} + 88 q^{63} + 2 q^{65} + 10 q^{71}+ \cdots + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 88x^{6} + 185x^{4} + 100x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{9} - 34\nu^{7} - 116\nu^{5} - 131\nu^{3} - 62\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 22\nu^{7} + 156\nu^{5} + 385\nu^{3} + 170\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 22\nu^{7} - 156\nu^{5} - 385\nu^{3} - 106\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{8} - 34\nu^{6} - 100\nu^{4} - 19\nu^{2} + 50 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{8} + 34\nu^{6} + 108\nu^{4} + 75\nu^{2} + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{8} - 34\nu^{6} - 100\nu^{4} - 35\nu^{2} - 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 14\nu^{6} + 64\nu^{4} + 101\nu^{2} + 26 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 50\nu^{7} + 292\nu^{5} + 675\nu^{3} + 446\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -9\nu^{9} - 134\nu^{7} - 668\nu^{5} - 1225\nu^{3} - 538\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 2\beta_{4} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 5\beta_{3} - 8\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{6} + 2\beta_{5} - 10\beta_{4} + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{9} - 6\beta_{8} + 27\beta_{3} + 51\beta_{2} - 10\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} - 45\beta_{6} - 23\beta_{5} + 48\beta_{4} - 186 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -22\beta_{9} + 34\beta_{8} - 151\beta_{3} - 313\beta_{2} + 80\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -34\beta_{7} + 283\beta_{6} + 194\beta_{5} - 234\beta_{4} + 1019 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 172\beta_{9} - 197\beta_{8} + 865\beta_{3} + 1904\beta_{2} - 585\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1665.1
2.04204i
0.208364i
− 0.857815i
− 2.49707i
2.19441i
− 2.19441i
2.49707i
0.857815i
− 0.208364i
− 2.04204i
0 − 3.21195i 0 1.62268i 0 −3.82180 0 −7.31660 0
1665.2 0 − 3.16495i 0 − 0.368078i 0 −2.90060 0 −7.01690 0
1665.3 0 − 1.40634i 0 0.422133i 0 4.72205 0 1.02221 0
1665.4 0 − 0.738305i 0 3.70891i 0 1.43136 0 2.45491 0
1665.5 0 − 0.378965i 0 4.27753i 0 −4.43100 0 2.85639 0
1665.6 0 0.378965i 0 − 4.27753i 0 −4.43100 0 2.85639 0
1665.7 0 0.738305i 0 − 3.70891i 0 1.43136 0 2.45491 0
1665.8 0 1.40634i 0 − 0.422133i 0 4.72205 0 1.02221 0
1665.9 0 3.16495i 0 0.368078i 0 −2.90060 0 −7.01690 0
1665.10 0 3.21195i 0 − 1.62268i 0 −3.82180 0 −7.31660 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1665.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.bc 10
4.b odd 2 1 3328.2.b.bd 10
8.b even 2 1 inner 3328.2.b.bc 10
8.d odd 2 1 3328.2.b.bd 10
16.e even 4 1 1664.2.a.z yes 5
16.e even 4 1 1664.2.a.ba yes 5
16.f odd 4 1 1664.2.a.y ✓ 5
16.f odd 4 1 1664.2.a.bb yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1664.2.a.y ✓ 5 16.f odd 4 1
1664.2.a.z yes 5 16.e even 4 1
1664.2.a.ba yes 5 16.e even 4 1
1664.2.a.bb yes 5 16.f odd 4 1
3328.2.b.bc 10 1.a even 1 1 trivial
3328.2.b.bc 10 8.b even 2 1 inner
3328.2.b.bd 10 4.b odd 2 1
3328.2.b.bd 10 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3328, [\chi])\):

\( T_{3}^{10} + 23T_{3}^{8} + 159T_{3}^{6} + 305T_{3}^{4} + 152T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{10} + 35T_{5}^{8} + 347T_{5}^{6} + 769T_{5}^{4} + 216T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{5} + 5T_{7}^{4} - 21T_{7}^{3} - 127T_{7}^{2} - 26T_{7} + 332 \) Copy content Toggle raw display
\( T_{11}^{10} + 68T_{11}^{8} + 1472T_{11}^{6} + 13376T_{11}^{4} + 51456T_{11}^{2} + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 23 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} + 35 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{5} + 5 T^{4} + \cdots + 332)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 68 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} + \cdots - 604)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 136 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{5} - 2 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 192 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( (T^{5} - 22 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 187 T^{8} + \cdots + 868624 \) Copy content Toggle raw display
$41$ \( (T^{5} + 10 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 151 T^{8} + \cdots + 8464 \) Copy content Toggle raw display
$47$ \( (T^{5} - 5 T^{4} + \cdots + 604)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 172 T^{8} + \cdots + 8667136 \) Copy content Toggle raw display
$59$ \( T^{10} + 68 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$61$ \( T^{10} + 364 T^{8} + \cdots + 27541504 \) Copy content Toggle raw display
$67$ \( T^{10} + 360 T^{8} + \cdots + 52765696 \) Copy content Toggle raw display
$71$ \( (T^{5} - 5 T^{4} + \cdots + 25684)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 4 T^{4} + \cdots + 9088)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 32 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 268435456 \) Copy content Toggle raw display
$89$ \( (T^{5} - 2 T^{4} + \cdots + 3296)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 160 T^{3} + \cdots - 704)^{2} \) Copy content Toggle raw display
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