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Properties

Label 1296.5.e.h
Level $1296$
Weight $5$
Character orbit 1296.e
Analytic conductor $133.967$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,5,Mod(161,1296)] 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("1296.161"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-96,0,0,0,0,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(133.967472157\)
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: \( x^{12} + 160x^{10} + 9733x^{8} + 278004x^{6} + 3678300x^{4} + 18632592x^{2} + 25765776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 648)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 2 \beta_1) q^{5} + (\beta_{8} - 8) q^{7} + (\beta_{9} + \beta_{6} + \cdots + 10 \beta_1) q^{11} + ( - \beta_{11} + 2 \beta_{10} + \cdots - 6) q^{13} + (2 \beta_{9} - \beta_{5} + \cdots - 4 \beta_1) q^{17}+ \cdots + (41 \beta_{11} - 11 \beta_{10} + \cdots + 6384) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 96 q^{7} - 72 q^{13} - 336 q^{19} + 84 q^{25} + 1536 q^{31} - 492 q^{37} - 10128 q^{43} - 6828 q^{49} + 13968 q^{55} + 26268 q^{61} + 20784 q^{67} - 9984 q^{73} - 44592 q^{79} - 45348 q^{85} + 11808 q^{91}+ \cdots + 76608 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 160x^{10} + 9733x^{8} + 278004x^{6} + 3678300x^{4} + 18632592x^{2} + 25765776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 116\nu^{11} + 17291\nu^{9} + 938678\nu^{7} + 22354071\nu^{5} + 218208942\nu^{3} + 588099852\nu ) / 219100464 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 179\nu^{10} + 13988\nu^{8} + 19211\nu^{6} - 17492952\nu^{4} - 323427744\nu^{2} - 1011413304 ) / 75246624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12436 \nu^{11} - 1620913 \nu^{9} - 75059506 \nu^{7} - 1478336997 \nu^{5} + \cdots - 41933179332 \nu ) / 7449415776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40898 \nu^{11} + 6598931 \nu^{9} + 403349948 \nu^{7} + 11250203703 \nu^{5} + \cdots + 388635848316 \nu ) / 7449415776 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3378 \nu^{11} - 340693 \nu^{9} - 8049592 \nu^{7} + 117084767 \nu^{5} + 5193920646 \nu^{3} + 27949722876 \nu ) / 413856432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3923 \nu^{11} - 463427 \nu^{9} - 15371885 \nu^{7} + 36608541 \nu^{5} + 8016340050 \nu^{3} + 64155434940 \nu ) / 413856432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42547 \nu^{10} + 5391676 \nu^{8} + 225752155 \nu^{6} + 3378450144 \nu^{4} + 15723372816 \nu^{2} + 123957473256 ) / 2483138592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 117209 \nu^{10} - 13818452 \nu^{8} - 576228401 \nu^{6} - 10512218976 \nu^{4} + \cdots - 251699256792 ) / 2483138592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13673 \nu^{11} + 1757413 \nu^{9} + 79876567 \nu^{7} + 1507234709 \nu^{5} + 9771458490 \nu^{3} - 9034470180 \nu ) / 413856432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 154187 \nu^{10} + 20666612 \nu^{8} + 982142915 \nu^{6} + 19673373336 \nu^{4} + \cdots + 312435972360 ) / 827712864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 467327 \nu^{10} + 58676504 \nu^{8} + 2530386431 \nu^{6} + 42153203724 \nu^{4} + \cdots + 65436512904 ) / 1241569296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - 2\beta_{5} + 3\beta_{4} - 3\beta_{3} - 9\beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - 2\beta_{8} + 8\beta_{7} - 19\beta_{2} - 480 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{9} - 4\beta_{6} + 49\beta_{5} - 60\beta_{4} - 146\beta_{3} - 28\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{11} + 77\beta_{10} + 97\beta_{8} - 380\beta_{7} + 1322\beta_{2} + 18402 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -500\beta_{9} + 354\beta_{6} - 2287\beta_{5} + 2256\beta_{4} + 9750\beta_{3} + 9630\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1301\beta_{11} - 4519\beta_{10} - 5363\beta_{8} + 16542\beta_{7} - 77548\beta_{2} - 774516 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20124\beta_{9} - 21656\beta_{6} + 107835\beta_{5} - 82494\beta_{4} - 539620\beta_{3} - 827102\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -77241\beta_{11} + 247643\beta_{10} + 309841\beta_{8} - 721456\beta_{7} + 4173956\beta_{2} + 34118214 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 879502 \beta_{9} + 1178798 \beta_{6} - 5136845 \beta_{5} + 2969238 \beta_{4} + 28691620 \beta_{3} + 56473064 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4235047 \beta_{11} - 13149085 \beta_{10} - 17771321 \beta_{8} + 31922002 \beta_{7} - 215421940 \beta_{2} - 1550284500 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40807528 \beta_{9} - 61164576 \beta_{6} + 246710921 \beta_{5} - 104536110 \beta_{4} + \cdots - 3471231846 \beta_1 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 2.58978i
1.47426i
5.87814i
6.45812i
4.92153i
− 7.11605i
7.11605i
− 4.92153i
− 6.45812i
− 5.87814i
− 1.47426i
2.58978i
0 0 0 − 35.6279i 0 −7.26473 0 0 0
161.2 0 0 0 − 33.0209i 0 −51.5500 0 0 0
161.3 0 0 0 − 27.4749i 0 2.93184 0 0 0
161.4 0 0 0 − 21.6069i 0 −69.7156 0 0 0
161.5 0 0 0 − 8.73487i 0 24.6181 0 0 0
161.6 0 0 0 − 7.08864i 0 52.9803 0 0 0
161.7 0 0 0 7.08864i 0 52.9803 0 0 0
161.8 0 0 0 8.73487i 0 24.6181 0 0 0
161.9 0 0 0 21.6069i 0 −69.7156 0 0 0
161.10 0 0 0 27.4749i 0 2.93184 0 0 0
161.11 0 0 0 33.0209i 0 −51.5500 0 0 0
161.12 0 0 0 35.6279i 0 −7.26473 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.5.e.h 12
3.b odd 2 1 inner 1296.5.e.h 12
4.b odd 2 1 648.5.e.b ✓ 12
12.b even 2 1 648.5.e.b ✓ 12
36.f odd 6 2 648.5.m.g 24
36.h even 6 2 648.5.m.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.5.e.b ✓ 12 4.b odd 2 1
648.5.e.b ✓ 12 12.b even 2 1
648.5.m.g 24 36.f odd 6 2
648.5.m.g 24 36.h even 6 2
1296.5.e.h 12 1.a even 1 1 trivial
1296.5.e.h 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 3708 T_{5}^{10} + 5076483 T_{5}^{8} + 3120866488 T_{5}^{6} + 824701196115 T_{5}^{4} + \cdots + 18\!\cdots\!81 \) acting on \(S_{5}^{\mathrm{new}}(1296, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$7$ \( (T^{6} + 48 T^{5} + \cdots - 99836064)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 20596749158509)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 317698964974432)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 67\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 23\!\cdots\!29 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 809122183875584)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 41\!\cdots\!91)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 48\!\cdots\!39)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 86\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 50\!\cdots\!59)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 20\!\cdots\!56)^{2} \) Copy content Toggle raw display
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