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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(705,1408)] 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("1408.705"); 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([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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: \( x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{11} + \beta_{4}) q^{3} - \beta_{6} q^{5} + \beta_{2} q^{7} + (\beta_{7} + \beta_{3} - 2) q^{9} - \beta_{4} q^{11} + (\beta_{10} - \beta_{9}) q^{13} + ( - \beta_{5} + \beta_{2}) q^{15} + (\beta_{7} - 2 \beta_{3}) q^{17}+ \cdots + (\beta_{11} - \beta_{8} + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{9} - 8 q^{17} + 4 q^{25} + 8 q^{33} + 24 q^{41} + 28 q^{49} - 48 q^{57} - 16 q^{65} - 24 q^{73} + 60 q^{81} - 48 q^{89} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19937 \nu^{11} - 10621 \nu^{10} + 186319 \nu^{9} - 405345 \nu^{8} + 1033899 \nu^{7} + \cdots - 3650592 ) / 1343784 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4752275 \nu^{11} + 4541619 \nu^{10} - 51358923 \nu^{9} + 124669767 \nu^{8} - 349176155 \nu^{7} + \cdots + 172138720 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2726533 \nu^{11} + 2168717 \nu^{10} - 27969673 \nu^{9} + 65297821 \nu^{8} - 178050285 \nu^{7} + \cdots + 116031512 ) / 63157848 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6765 \nu^{11} + 2010 \nu^{10} - 68487 \nu^{9} + 129388 \nu^{8} - 380069 \nu^{7} + 342022 \nu^{6} + \cdots - 625296 ) / 133104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6861369 \nu^{11} + 5745144 \nu^{10} - 71950207 \nu^{9} + 168540570 \nu^{8} - 469088145 \nu^{7} + \cdots - 69706944 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1770535 \nu^{11} - 430121 \nu^{10} + 17477065 \nu^{9} - 31961173 \nu^{8} + 92189437 \nu^{7} + \cdots + 160290560 ) / 31578924 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3885983 \nu^{11} - 3545639 \nu^{10} + 41847595 \nu^{9} - 99583247 \nu^{8} + 281560343 \nu^{7} + \cdots + 41984224 ) / 63157848 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10303349 \nu^{11} - 720543 \nu^{10} + 100063305 \nu^{9} - 168771087 \nu^{8} + 494158389 \nu^{7} + \cdots + 965517216 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 75766 \nu^{11} + 29821 \nu^{10} - 774069 \nu^{9} + 1520687 \nu^{8} - 4434385 \nu^{7} + \cdots - 6870768 ) / 535236 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9068197 \nu^{11} - 1816017 \nu^{10} + 90327829 \nu^{9} - 163061441 \nu^{8} + 479342601 \nu^{7} + \cdots + 846236960 ) / 63157848 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18928493 \nu^{11} - 6358549 \nu^{10} + 191366733 \nu^{9} - 366558165 \nu^{8} + 1068678901 \nu^{7} + \cdots + 1726100080 ) / 126315696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} + 5 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 13 \beta_{7} + 6 \beta_{6} + 20 \beta_{5} + \cdots + 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} - 10\beta_{9} + 8\beta_{8} - 13\beta_{6} + 32\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26 \beta_{11} + 33 \beta_{10} + 26 \beta_{9} - 65 \beta_{8} - 143 \beta_{7} + 78 \beta_{6} + \cdots - 244 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{11} + 12 \beta_{10} + 150 \beta_{9} - 155 \beta_{8} + 155 \beta_{7} + 209 \beta_{6} + \cdots + 405 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 794 \beta_{11} - 537 \beta_{10} - 754 \beta_{9} + 1575 \beta_{8} + 675 \beta_{7} - 2250 \beta_{6} + \cdots + 1120 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1269\beta_{7} - 1824\beta_{5} + 249\beta_{3} - 144\beta_{2} - 1056\beta _1 - 3293 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7486 \beta_{11} + 5195 \beta_{10} + 9798 \beta_{9} - 17567 \beta_{8} + 7367 \beta_{7} + 24934 \beta_{6} + \cdots + 14208 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4969 \beta_{11} - 3696 \beta_{10} - 16210 \beta_{9} + 21263 \beta_{8} + 21263 \beta_{7} - 29903 \beta_{6} + \cdots + 50493 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30478 \beta_{11} - 22507 \beta_{10} - 49918 \beta_{9} + 82313 \beta_{8} - 197843 \beta_{7} + \cdots - 414652 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
705.1
1.85301 − 0.767544i
−0.393753 − 0.950605i
0.981031 − 2.36842i
−0.331199 − 0.137187i
−1.29438 − 3.12492i
−0.814709 + 0.337464i
−0.814709 − 0.337464i
−1.29438 + 3.12492i
−0.331199 + 0.137187i
0.981031 + 2.36842i
−0.393753 + 0.950605i
1.85301 + 0.767544i
0 − 3.17741i 0 − 2.06371i 0 −4.49354 0 −7.09593 0
705.2 0 − 3.17741i 0 2.06371i 0 4.49354 0 −7.09593 0
705.3 0 − 1.85577i 0 − 0.919001i 0 −2.62446 0 −0.443892 0
705.4 0 − 1.85577i 0 0.919001i 0 2.62446 0 −0.443892 0
705.5 0 − 0.678363i 0 − 2.98271i 0 0.959350 0 2.53982 0
705.6 0 − 0.678363i 0 2.98271i 0 −0.959350 0 2.53982 0
705.7 0 0.678363i 0 − 2.98271i 0 −0.959350 0 2.53982 0
705.8 0 0.678363i 0 2.98271i 0 0.959350 0 2.53982 0
705.9 0 1.85577i 0 − 0.919001i 0 2.62446 0 −0.443892 0
705.10 0 1.85577i 0 0.919001i 0 −2.62446 0 −0.443892 0
705.11 0 3.17741i 0 − 2.06371i 0 4.49354 0 −7.09593 0
705.12 0 3.17741i 0 2.06371i 0 −4.49354 0 −7.09593 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 705.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1408.2.c.f ✓ 12
4.b odd 2 1 inner 1408.2.c.f ✓ 12
8.b even 2 1 inner 1408.2.c.f ✓ 12
8.d odd 2 1 inner 1408.2.c.f ✓ 12
16.e even 4 1 2816.2.a.s 6
16.e even 4 1 2816.2.a.v 6
16.f odd 4 1 2816.2.a.s 6
16.f odd 4 1 2816.2.a.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.c.f ✓ 12 1.a even 1 1 trivial
1408.2.c.f ✓ 12 4.b odd 2 1 inner
1408.2.c.f ✓ 12 8.b even 2 1 inner
1408.2.c.f ✓ 12 8.d odd 2 1 inner
2816.2.a.s 6 16.e even 4 1
2816.2.a.s 6 16.f odd 4 1
2816.2.a.v 6 16.e even 4 1
2816.2.a.v 6 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{6} + 14T_{3}^{4} + 41T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} - 28T_{7}^{4} + 164T_{7}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 14 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 28 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} + 36 T^{4} + \cdots + 288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 30 T + 36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 76 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 138 T^{4} + \cdots - 69192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} - 90 T^{4} + \cdots - 72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 110 T^{4} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} - 2 T + 36)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 216 T^{4} + \cdots + 215296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 192 T^{4} + \cdots - 194688)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 136 T^{4} + \cdots + 73728)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 190 T^{4} + \cdots + 99856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 232 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 110 T^{4} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 74 T^{4} + \cdots - 648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} - 2 T - 4)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 200 T^{4} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 216 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} - 11 T + 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 4 T^{2} + \cdots - 774)^{4} \) Copy content Toggle raw display
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