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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(1407,1408)] 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("1408.1407"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-8,0,-12,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
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: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + \beta_1 q^{5} + ( - \beta_{3} - 1) q^{7} + ( - \beta_{8} - 1) q^{9} + ( - \beta_{5} - 1) q^{11} + ( - \beta_{11} - \beta_{9}) q^{13} + (\beta_{11} + \beta_{4}) q^{15} + (\beta_{9} + \beta_{6} - \beta_{2}) q^{17}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7} - 12 q^{9} - 6 q^{11} + 4 q^{19} + 12 q^{25} + 4 q^{33} + 16 q^{35} + 40 q^{39} - 28 q^{43} + 12 q^{49} - 48 q^{51} + 16 q^{53} + 4 q^{55} + 40 q^{63} + 32 q^{69} + 24 q^{77} + 16 q^{79}+ \cdots - 42 q^{99}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( 4235588970 \nu^{11} - 187476460 \nu^{10} - 71192108680 \nu^{9} + 452378117190 \nu^{8} + \cdots - 1525720033354 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6707558652 \nu^{11} + 51896829036 \nu^{10} - 182663219884 \nu^{9} + 369739830397 \nu^{8} + \cdots - 1162129902116 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14652726490 \nu^{11} - 94655597176 \nu^{10} + 350610870646 \nu^{9} - 752223782575 \nu^{8} + \cdots - 1432167708553 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19486544494 \nu^{11} + 104700990672 \nu^{10} - 336293808425 \nu^{9} + 569933496771 \nu^{8} + \cdots - 194601194056 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2429272 \nu^{11} + 12929344 \nu^{10} - 42298230 \nu^{9} + 68672502 \nu^{8} - 4782762 \nu^{7} + \cdots + 129635798 ) / 113440793 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23049450630 \nu^{11} - 138890739065 \nu^{10} + 487243576432 \nu^{9} - 908223889445 \nu^{8} + \cdots - 3073128488976 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25569521988 \nu^{11} + 122939596861 \nu^{10} - 400542655764 \nu^{9} + \cdots - 1522477601173 ) / 1073263342573 \) 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
\(\beta_{10}\)\(=\) \( ( 49488046390 \nu^{11} - 290213463740 \nu^{10} + 1009780387641 \nu^{9} - 1903124838975 \nu^{8} + \cdots - 5483510658246 ) / 1073263342573 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 52457270576 \nu^{11} + 296621918924 \nu^{10} - 1023554888328 \nu^{9} + \cdots + 3866130617820 ) / 1073263342573 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{9} - 2\beta_{7} + 2\beta_{5} - \beta_{3} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} - \beta_{8} - 6 \beta_{7} + 4 \beta_{6} - \beta_{5} + \cdots - 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{11} + \beta_{10} - 10 \beta_{9} - \beta_{8} + 9 \beta_{7} + 5 \beta_{6} - 13 \beta_{5} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{11} - 71 \beta_{10} + 15 \beta_{9} + 2 \beta_{8} + 89 \beta_{7} - 11 \beta_{6} - 31 \beta_{5} + \cdots + 85 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9 \beta_{11} - 89 \beta_{10} + 85 \beta_{9} - 14 \beta_{8} - 2 \beta_{7} - 48 \beta_{6} + 73 \beta_{5} + \cdots - 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52 \beta_{11} + 251 \beta_{10} + 15 \beta_{9} - 148 \beta_{8} - 691 \beta_{7} - 119 \beta_{6} + \cdots - 1311 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 222 \beta_{11} + 931 \beta_{10} - 864 \beta_{9} - 62 \beta_{8} - 442 \beta_{7} + 186 \beta_{6} + \cdots - 652 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 767 \beta_{11} + 915 \beta_{10} - 3289 \beta_{9} + 1053 \beta_{8} + 4598 \beta_{7} + 886 \beta_{6} + \cdots + 9360 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 849 \beta_{11} - 6546 \beta_{10} + 4473 \beta_{9} + 1536 \beta_{8} + 6862 \beta_{7} - 1428 \beta_{6} + \cdots + 13719 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8441 \beta_{11} - 23653 \beta_{10} + 40585 \beta_{9} - 2617 \beta_{8} - 20436 \beta_{7} + \cdots - 30518 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(639\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
1.48442 + 0.678444i
0.718624 − 2.81970i
1.78529 − 1.33780i
0.614603 − 0.454341i
−1.26534 + 0.142059i
−0.337597 − 0.756748i
−0.337597 + 0.756748i
−1.26534 − 0.142059i
0.614603 + 0.454341i
1.78529 + 1.33780i
0.718624 + 2.81970i
1.48442 − 0.678444i
0 − 3.06432i 0 1.45237 0 1.31461 0 −6.39005 0
1407.2 0 − 2.55396i 0 1.64820 0 −3.17359 0 −3.52270 0
1407.3 0 − 2.27341i 0 −3.97278 0 −4.20226 0 −2.16840 0
1407.4 0 − 1.59473i 0 −1.91525 0 −0.700406 0 0.456840 0
1407.5 0 − 0.558566i 0 3.37337 0 −1.28725 0 2.68800 0
1407.6 0 − 0.252391i 0 −0.585911 0 4.04889 0 2.93630 0
1407.7 0 0.252391i 0 −0.585911 0 4.04889 0 2.93630 0
1407.8 0 0.558566i 0 3.37337 0 −1.28725 0 2.68800 0
1407.9 0 1.59473i 0 −1.91525 0 −0.700406 0 0.456840 0
1407.10 0 2.27341i 0 −3.97278 0 −4.20226 0 −2.16840 0
1407.11 0 2.55396i 0 1.64820 0 −3.17359 0 −3.52270 0
1407.12 0 3.06432i 0 1.45237 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
44.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1408, [\chi])\):

\( T_{5}^{6} - 18T_{5}^{4} + 8T_{5}^{3} + 61T_{5}^{2} - 32T_{5} - 36 \) 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
$3$ \( T^{12} + 24 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{6} - 18 T^{4} + \cdots - 36)^{2} \) 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
$13$ \( T^{12} + 92 T^{10} + \cdots + 3268864 \) Copy content Toggle raw display
$17$ \( T^{12} + 112 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} - 52 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 112 T^{10} + \cdots + 913936 \) Copy content Toggle raw display
$29$ \( T^{12} + 268 T^{10} + \cdots + 7929856 \) Copy content Toggle raw display
$31$ \( T^{12} + 176 T^{10} + \cdots + 2862864 \) Copy content Toggle raw display
$37$ \( (T^{6} - 114 T^{4} + \cdots - 22276)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 208 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( (T^{6} + 14 T^{5} + \cdots + 33792)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 268 T^{10} + \cdots + 4460544 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 2816)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 538054416 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 6217007104 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1806930064 \) 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
$83$ \( (T^{6} - 26 T^{5} + \cdots + 811008)^{2} \) 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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