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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2816,2,Mod(1407,2816)] 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("2816.1407"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,4,0,0,0,8,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.4858732092\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 21 x^{10} - 40 x^{9} + 19 x^{8} + 98 x^{7} - 176 x^{6} + 34 x^{5} + 115 x^{4} + \cdots + 73 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 1408)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{7} q^{5} + (\beta_{4} + 1) q^{7} + (\beta_{6} + \beta_{3} + 1) q^{9} + (\beta_{11} + \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{13}+ \cdots + (\beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 8 q^{7} + 12 q^{9} - 6 q^{11} - 4 q^{13} - 8 q^{21} - 12 q^{25} + 16 q^{27} + 36 q^{29} + 4 q^{33} - 40 q^{39} + 12 q^{49} - 4 q^{55} - 20 q^{59} - 28 q^{61} + 40 q^{63} + 4 q^{67} - 52 q^{75}+ \cdots - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 21 x^{10} - 40 x^{9} + 19 x^{8} + 98 x^{7} - 176 x^{6} + 34 x^{5} + 115 x^{4} + \cdots + 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1526954786 \nu^{11} + 12088176341 \nu^{10} - 68405568646 \nu^{9} + 263317478264 \nu^{8} + \cdots + 1365130309773 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2979810375 \nu^{11} - 15559194130 \nu^{10} + 46336614578 \nu^{9} - 91101455270 \nu^{8} + \cdots + 870396159606 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5954956035 \nu^{11} + 12210516911 \nu^{10} - 15818216548 \nu^{9} - 96142186278 \nu^{8} + \cdots - 946033859905 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6707558652 \nu^{11} + 51896829036 \nu^{10} - 182663219884 \nu^{9} + 369739830397 \nu^{8} + \cdots - 1162129902116 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14652726490 \nu^{11} + 94655597176 \nu^{10} - 350610870646 \nu^{9} + 752223782575 \nu^{8} + \cdots + 1432167708553 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19614565953 \nu^{11} + 110729079950 \nu^{10} - 384724439216 \nu^{9} + 698565096878 \nu^{8} + \cdots - 576443741268 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21868125835 \nu^{11} + 110402267766 \nu^{10} - 325755376544 \nu^{9} + \cdots + 2087491582301 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2429272 \nu^{11} + 12929344 \nu^{10} - 42298230 \nu^{9} + 68672502 \nu^{8} - 4782762 \nu^{7} + \cdots + 129635798 ) / 113440793 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25907161020 \nu^{11} - 138664711514 \nu^{10} + 440856661502 \nu^{9} - 698078423367 \nu^{8} + \cdots - 1979790531563 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30440173420 \nu^{11} - 182328683312 \nu^{10} + 610863460100 \nu^{9} - 1055829605360 \nu^{8} + \cdots - 4949295090890 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43885222991 \nu^{11} - 224695503378 \nu^{10} + 738446804772 \nu^{9} - 1223614033963 \nu^{8} + \cdots - 2077590451804 ) / 1073263342573 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{11} + 2 \beta_{10} - 5 \beta_{9} + 7 \beta_{8} + 3 \beta_{7} + \beta_{6} + 10 \beta_{5} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{11} + 11 \beta_{10} - 10 \beta_{9} + 12 \beta_{8} + 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \cdots + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 52 \beta_{11} + 24 \beta_{10} + 15 \beta_{9} - 31 \beta_{8} + 7 \beta_{7} + 15 \beta_{6} - 44 \beta_{5} + \cdots + 122 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 62 \beta_{11} - 82 \beta_{10} + 85 \beta_{9} - 129 \beta_{8} + 13 \beta_{7} + 4 \beta_{6} + \cdots - 125 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 288 \beta_{11} - 492 \beta_{10} + 15 \beta_{9} - 169 \beta_{8} - 65 \beta_{7} - 59 \beta_{6} + \cdots - 1714 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 701 \beta_{11} + 267 \beta_{10} - 864 \beta_{9} + 892 \beta_{8} - 312 \beta_{7} - 70 \beta_{6} + \cdots - 393 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 766 \beta_{11} + 4746 \beta_{10} - 3289 \beta_{9} + 3643 \beta_{8} - 1077 \beta_{7} + 139 \beta_{6} + \cdots + 13192 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6949 \beta_{11} + 1165 \beta_{10} + 4473 \beta_{9} - 5169 \beta_{8} + 1245 \beta_{7} + 284 \beta_{6} + \cdots + 13632 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15112 \beta_{11} - 35648 \beta_{10} + 40585 \beta_{9} - 45751 \beta_{8} + 12193 \beta_{7} + \cdots - 66066 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1407.1
0.718624 − 2.81970i
0.718624 + 2.81970i
0.614603 + 0.454341i
0.614603 − 0.454341i
−0.337597 − 0.756748i
−0.337597 + 0.756748i
−1.26534 − 0.142059i
−1.26534 + 0.142059i
1.78529 − 1.33780i
1.78529 + 1.33780i
1.48442 − 0.678444i
1.48442 + 0.678444i
0 −2.55396 0 − 1.64820i 0 3.17359 0 3.52270 0
1407.2 0 −2.55396 0 1.64820i 0 3.17359 0 3.52270 0
1407.3 0 −1.59473 0 − 1.91525i 0 0.700406 0 −0.456840 0
1407.4 0 −1.59473 0 1.91525i 0 0.700406 0 −0.456840 0
1407.5 0 0.252391 0 − 0.585911i 0 −4.04889 0 −2.93630 0
1407.6 0 0.252391 0 0.585911i 0 −4.04889 0 −2.93630 0
1407.7 0 0.558566 0 − 3.37337i 0 1.28725 0 −2.68800 0
1407.8 0 0.558566 0 3.37337i 0 1.28725 0 −2.68800 0
1407.9 0 2.27341 0 − 3.97278i 0 4.20226 0 2.16840 0
1407.10 0 2.27341 0 3.97278i 0 4.20226 0 2.16840 0
1407.11 0 3.06432 0 − 1.45237i 0 −1.31461 0 6.39005 0
1407.12 0 3.06432 0 1.45237i 0 −1.31461 0 6.39005 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1407.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2816.2.g.h 12
4.b odd 2 1 2816.2.g.e 12
8.b even 2 1 2816.2.g.f 12
8.d odd 2 1 2816.2.g.g 12
11.b odd 2 1 2816.2.g.g 12
16.e even 4 1 1408.2.e.a ✓ 12
16.e even 4 1 1408.2.e.b yes 12
16.f odd 4 1 1408.2.e.c yes 12
16.f odd 4 1 1408.2.e.d yes 12
44.c even 2 1 2816.2.g.f 12
88.b odd 2 1 2816.2.g.e 12
88.g even 2 1 inner 2816.2.g.h 12
176.i even 4 1 1408.2.e.a ✓ 12
176.i even 4 1 1408.2.e.b yes 12
176.l odd 4 1 1408.2.e.c yes 12
176.l odd 4 1 1408.2.e.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.e.a ✓ 12 16.e even 4 1
1408.2.e.a ✓ 12 176.i even 4 1
1408.2.e.b yes 12 16.e even 4 1
1408.2.e.b yes 12 176.i even 4 1
1408.2.e.c yes 12 16.f odd 4 1
1408.2.e.c yes 12 176.l odd 4 1
1408.2.e.d yes 12 16.f odd 4 1
1408.2.e.d yes 12 176.l odd 4 1
2816.2.g.e 12 4.b odd 2 1
2816.2.g.e 12 88.b odd 2 1
2816.2.g.f 12 8.b even 2 1
2816.2.g.f 12 44.c even 2 1
2816.2.g.g 12 8.d odd 2 1
2816.2.g.g 12 11.b odd 2 1
2816.2.g.h 12 1.a even 1 1 trivial
2816.2.g.h 12 88.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2816, [\chi])\):

\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 16T_{3}^{3} + 21T_{3}^{2} - 22T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 4T_{7}^{5} - 16T_{7}^{4} + 72T_{7}^{3} - 12T_{7}^{2} - 112T_{7} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} - 10 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 36 T^{10} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} - 16 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots - 1808)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 112 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( T^{12} + 108 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{12} + 112 T^{10} + \cdots + 913936 \) Copy content Toggle raw display
$29$ \( (T^{6} - 18 T^{5} + \cdots + 2816)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 176 T^{10} + \cdots + 2862864 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 496220176 \) Copy content Toggle raw display
$41$ \( T^{12} + 208 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1141899264 \) Copy content Toggle raw display
$47$ \( T^{12} + 268 T^{10} + \cdots + 4460544 \) Copy content Toggle raw display
$53$ \( T^{12} + 328 T^{10} + \cdots + 7929856 \) Copy content Toggle raw display
$59$ \( (T^{6} + 10 T^{5} + \cdots + 23196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 14 T^{5} + \cdots + 78848)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 2 T^{5} + \cdots + 42508)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 256 T^{10} + \cdots + 90000 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1270779904 \) Copy content Toggle raw display
$79$ \( (T^{6} - 8 T^{5} + \cdots + 45056)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 657733976064 \) Copy content Toggle raw display
$89$ \( (T^{6} - 94 T^{4} + \cdots + 8868)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 8 T^{5} + \cdots - 11996)^{2} \) Copy content Toggle raw display
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