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Properties

Label 160.5.p.f
Level $160$
Weight $5$
Character orbit 160.p
Analytic conductor $16.539$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,5,Mod(33,160)] 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("160.33"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 160.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,24,0,48] 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: \(16.5391940934\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
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: \( x^{12} - 4 x^{11} - 10 x^{10} + 68 x^{9} - 52 x^{8} - 784 x^{7} + 960 x^{6} + 3824 x^{5} - 13583 x^{4} + \cdots + 348466 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} + ( - \beta_{6} + \beta_{2} + 2) q^{5} + (\beta_{3} + 4 \beta_{2} + 4) q^{7} + (\beta_{10} + \beta_{8} + \cdots + 2 \beta_1) q^{9} + (\beta_{9} + 2 \beta_{8} + \beta_{7} + \cdots + 27) q^{11}+ \cdots + (9 \beta_{11} + 85 \beta_{10} + \cdots + 786 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{5} + 48 q^{7} + 320 q^{11} + 260 q^{13} - 592 q^{15} - 60 q^{17} - 400 q^{21} - 176 q^{23} + 836 q^{25} + 2208 q^{27} - 480 q^{31} - 3120 q^{33} + 2208 q^{35} - 500 q^{37} + 2160 q^{41} + 3744 q^{43}+ \cdots + 36460 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 10 x^{10} + 68 x^{9} - 52 x^{8} - 784 x^{7} + 960 x^{6} + 3824 x^{5} - 13583 x^{4} + \cdots + 348466 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 332 \nu^{11} + 8319 \nu^{10} - 87146 \nu^{9} + 246235 \nu^{8} + 120796 \nu^{7} - 2635547 \nu^{6} + \cdots + 824098278 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 402 \nu^{11} + 2441 \nu^{10} - 44 \nu^{9} - 28835 \nu^{8} + 72994 \nu^{7} + 210867 \nu^{6} + \cdots - 82842358 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1206 \nu^{11} + 5943 \nu^{10} + 2428 \nu^{9} - 84105 \nu^{8} + 206982 \nu^{7} + \cdots - 178477954 ) / 1050000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7247 \nu^{11} - 21301 \nu^{10} - 197791 \nu^{9} + 1000685 \nu^{8} - 747459 \nu^{7} + \cdots + 3613226138 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 320 \nu^{11} - 2257 \nu^{10} + 1554 \nu^{9} + 24035 \nu^{8} - 84840 \nu^{7} - 96411 \nu^{6} + \cdots + 39107158 ) / 210000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10543 \nu^{11} - 45494 \nu^{10} - 106629 \nu^{9} + 991640 \nu^{8} - 1636171 \nu^{7} + \cdots + 2452132672 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4763 \nu^{11} + 30004 \nu^{10} - 28211 \nu^{9} - 275490 \nu^{8} + 1023511 \nu^{7} + \cdots - 373438052 ) / 1750000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4412 \nu^{11} + 27081 \nu^{10} - 12934 \nu^{9} - 282235 \nu^{8} + 862164 \nu^{7} + \cdots - 667141638 ) / 1050000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29669 \nu^{11} + 186552 \nu^{10} - 128493 \nu^{9} - 1765370 \nu^{8} + 5728393 \nu^{7} + \cdots - 4229312676 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49387 \nu^{11} - 264121 \nu^{10} - 55811 \nu^{9} + 3304885 \nu^{8} - 7389039 \nu^{7} + \cdots + 9224299498 ) / 5250000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56093 \nu^{11} + 357769 \nu^{10} - 223471 \nu^{9} - 3522765 \nu^{8} + 11360521 \nu^{7} + \cdots - 8597904122 ) / 5250000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 5\beta_{9} - 4\beta_{8} - 5\beta_{7} + 7\beta_{4} - 18\beta _1 + 65 ) / 200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{11} - 16\beta_{8} + 10\beta_{5} + 13\beta_{4} - 5\beta_{3} + 100\beta_{2} - 37\beta _1 + 300 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 127 \beta_{11} + 45 \beta_{10} - 150 \beta_{9} - 48 \beta_{8} - 25 \beta_{7} - 320 \beta_{6} + \cdots - 675 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 172 \beta_{11} + 15 \beta_{10} - 370 \beta_{9} + 17 \beta_{8} - 30 \beta_{7} - 315 \beta_{6} + \cdots + 1790 ) / 100 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2433 \beta_{11} - 135 \beta_{10} - 5810 \beta_{9} + 1768 \beta_{8} - 1015 \beta_{7} - 3740 \beta_{6} + \cdots + 77795 ) / 400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 617 \beta_{11} - 261 \beta_{10} - 2342 \beta_{9} + 1954 \beta_{8} - 483 \beta_{7} - 114 \beta_{6} + \cdots + 40319 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11571 \beta_{11} - 6105 \beta_{10} - 75950 \beta_{9} + 117496 \beta_{8} - 20025 \beta_{7} + \cdots + 1790125 ) / 400 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7577 \beta_{11} + 1410 \beta_{10} - 53630 \beta_{9} + 102257 \beta_{8} - 21295 \beta_{7} + \cdots + 1926435 ) / 100 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 305351 \beta_{11} + 120255 \beta_{10} - 556730 \beta_{9} + 618456 \beta_{8} - 357145 \beta_{7} + \cdots + 31477485 ) / 400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1174957 \beta_{11} + 230235 \beta_{10} - 1019630 \beta_{9} - 824338 \beta_{8} - 679395 \beta_{7} + \cdots + 56353135 ) / 200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12480179 \beta_{11} + 829515 \beta_{10} - 13822510 \beta_{9} - 9336936 \beta_{8} - 4593865 \beta_{7} + \cdots + 349962845 ) / 400 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−2.09327 + 0.657043i
2.66185 − 2.69040i
4.16222 − 0.0194614i
−1.78940 − 1.48971i
1.19719 + 3.15612i
−2.13858 + 0.386403i
−2.09327 − 0.657043i
2.66185 + 2.69040i
4.16222 + 0.0194614i
−1.78940 + 1.48971i
1.19719 − 3.15612i
−2.13858 − 0.386403i
0 −11.8869 − 11.8869i 0 24.9004 − 2.22894i 0 22.2437 − 22.2437i 0 201.595i 0
33.2 0 −5.86997 − 5.86997i 0 −23.6674 − 8.05314i 0 9.78913 − 9.78913i 0 − 12.0869i 0
33.3 0 −1.99526 − 1.99526i 0 7.58866 + 23.8204i 0 −28.3886 + 28.3886i 0 − 73.0379i 0
33.4 0 4.62067 + 4.62067i 0 −0.537417 − 24.9942i 0 −47.0851 + 47.0851i 0 − 38.2988i 0
33.5 0 4.69916 + 4.69916i 0 22.3401 − 11.2215i 0 55.9504 − 55.9504i 0 − 36.8358i 0
33.6 0 10.4323 + 10.4323i 0 −18.6243 + 16.6774i 0 11.4905 − 11.4905i 0 136.664i 0
97.1 0 −11.8869 + 11.8869i 0 24.9004 + 2.22894i 0 22.2437 + 22.2437i 0 − 201.595i 0
97.2 0 −5.86997 + 5.86997i 0 −23.6674 + 8.05314i 0 9.78913 + 9.78913i 0 12.0869i 0
97.3 0 −1.99526 + 1.99526i 0 7.58866 − 23.8204i 0 −28.3886 − 28.3886i 0 73.0379i 0
97.4 0 4.62067 − 4.62067i 0 −0.537417 + 24.9942i 0 −47.0851 − 47.0851i 0 38.2988i 0
97.5 0 4.69916 − 4.69916i 0 22.3401 + 11.2215i 0 55.9504 + 55.9504i 0 36.8358i 0
97.6 0 10.4323 − 10.4323i 0 −18.6243 − 16.6774i 0 11.4905 + 11.4905i 0 − 136.664i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.5.p.f yes 12
4.b odd 2 1 160.5.p.e ✓ 12
5.c odd 4 1 inner 160.5.p.f yes 12
8.b even 2 1 320.5.p.s 12
8.d odd 2 1 320.5.p.r 12
20.e even 4 1 160.5.p.e ✓ 12
40.i odd 4 1 320.5.p.s 12
40.k even 4 1 320.5.p.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.5.p.e ✓ 12 4.b odd 2 1
160.5.p.e ✓ 12 20.e even 4 1
160.5.p.f yes 12 1.a even 1 1 trivial
160.5.p.f yes 12 5.c odd 4 1 inner
320.5.p.r 12 8.d odd 2 1
320.5.p.r 12 40.k even 4 1
320.5.p.s 12 8.b even 2 1
320.5.p.s 12 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(160, [\chi])\):

\( T_{3}^{12} - 736 T_{3}^{9} + 67912 T_{3}^{8} - 186176 T_{3}^{7} + 270848 T_{3}^{6} - 3743616 T_{3}^{5} + \cdots + 63649234944 \) Copy content Toggle raw display
\( T_{13}^{12} - 260 T_{13}^{11} + 33800 T_{13}^{10} + 1752120 T_{13}^{9} + 4704729164 T_{13}^{8} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 63649234944 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{6} - 160 T^{5} + \cdots - 66813440000)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 69\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
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