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Properties

Label 1440.5.e.b
Level $1440$
Weight $5$
Character orbit 1440.e
Analytic conductor $148.853$
Analytic rank $0$
Dimension $8$
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] = [1440,5,Mod(991,1440)] 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("1440.991"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1440.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,560,0,0,0,0,0,0,0,1000,0,0, 0,2064,0,0,0,0,0,0,0,3616] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(148.852746841\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 160)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + (\beta_{6} + 2 \beta_{2} - 5 \beta_1) q^{7} + ( - \beta_{7} - 3 \beta_{6} + \cdots - 16 \beta_1) q^{11} + ( - \beta_{4} + 8 \beta_{3} + 4) q^{13} + (3 \beta_{5} + \beta_{4} - 10 \beta_{3} + 70) q^{17}+ \cdots + ( - 31 \beta_{5} - 187 \beta_{4} + \cdots + 430) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{13} + 560 q^{17} + 1000 q^{25} + 2064 q^{29} + 3616 q^{37} + 5216 q^{41} + 4088 q^{49} - 5088 q^{53} - 5504 q^{61} + 7600 q^{65} + 17936 q^{73} + 15840 q^{77} - 9600 q^{85} + 50608 q^{89} + 3440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 56\nu^{7} + 1724\nu^{5} + 15576\nu^{3} + 42040\nu ) / 1003 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -150\nu^{7} - 4188\nu^{5} - 31978\nu^{3} - 65896\nu ) / 1003 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 50\nu^{6} + 1430\nu^{4} + 11600\nu^{2} + 28295 ) / 51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} - 124\nu^{4} - 1120\nu^{2} - 2986 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -224\nu^{6} - 6080\nu^{4} - 43808\nu^{2} - 87920 ) / 51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1702\nu^{7} + 50248\nu^{5} + 432706\nu^{3} + 1140448\nu ) / 3009 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1324\nu^{7} + 37608\nu^{5} + 299484\nu^{3} + 710240\nu ) / 1003 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + 17\beta_{2} + 5\beta_1 ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{5} + 10\beta_{4} + 36\beta_{3} - 1400 ) / 160 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -18\beta_{7} + 33\beta_{6} - 77\beta_{2} - 115\beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -50\beta_{5} - 125\beta_{4} - 394\beta_{3} + 7980 ) / 80 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{7} - 117\beta_{6} + 183\beta_{2} + 517\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 850\beta_{5} + 2415\beta_{4} + 7174\beta_{3} - 111100 ) / 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3662\beta_{7} + 9957\beta_{6} - 13133\beta_{2} - 48755\beta_1 ) / 40 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
991.1
4.10910i
− 2.49107i
2.49107i
− 4.10910i
− 2.11161i
− 2.72964i
2.72964i
2.11161i
0 0 0 −11.1803 0 − 47.9490i 0 0 0
991.2 0 0 0 −11.1803 0 − 41.7166i 0 0 0
991.3 0 0 0 −11.1803 0 41.7166i 0 0 0
991.4 0 0 0 −11.1803 0 47.9490i 0 0 0
991.5 0 0 0 11.1803 0 − 49.8485i 0 0 0
991.6 0 0 0 11.1803 0 − 32.1829i 0 0 0
991.7 0 0 0 11.1803 0 32.1829i 0 0 0
991.8 0 0 0 11.1803 0 49.8485i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.5.e.b 8
3.b odd 2 1 160.5.b.a ✓ 8
4.b odd 2 1 inner 1440.5.e.b 8
12.b even 2 1 160.5.b.a ✓ 8
15.d odd 2 1 800.5.b.g 8
15.e even 4 1 800.5.h.k 8
15.e even 4 1 800.5.h.l 8
24.f even 2 1 320.5.b.c 8
24.h odd 2 1 320.5.b.c 8
60.h even 2 1 800.5.b.g 8
60.l odd 4 1 800.5.h.k 8
60.l odd 4 1 800.5.h.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.5.b.a ✓ 8 3.b odd 2 1
160.5.b.a ✓ 8 12.b even 2 1
320.5.b.c 8 24.f even 2 1
320.5.b.c 8 24.h odd 2 1
800.5.b.g 8 15.d odd 2 1
800.5.b.g 8 60.h even 2 1
800.5.h.k 8 15.e even 4 1
800.5.h.k 8 60.l odd 4 1
800.5.h.l 8 15.e even 4 1
800.5.h.l 8 60.l odd 4 1
1440.5.e.b 8 1.a even 1 1 trivial
1440.5.e.b 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{8} + 7560T_{7}^{6} + 20795888T_{7}^{4} + 24482378880T_{7}^{2} + 10297526968576 \) Copy content Toggle raw display
\( T_{17}^{4} - 280T_{17}^{3} - 169192T_{17}^{2} + 37582240T_{17} + 3410634256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 10297526968576 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 7518651744256 \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + \cdots + 24176144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 280 T^{3} + \cdots + 3410634256)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} - 1032 T^{3} + \cdots - 131615691120)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} - 1808 T^{3} + \cdots + 482453699600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 8307294475280)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 21920933051120)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 121504764169616)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 85\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 119575623950096)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 774583434485136)^{2} \) Copy content Toggle raw display
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