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Properties

Label 680.2.bd.b
Level $680$
Weight $2$
Character orbit 680.bd
Analytic conductor $5.430$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(81,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.bd (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 38 x^{18} + 597 x^{16} + 5004 x^{14} + 24072 x^{12} + 66452 x^{10} + 99328 x^{8} + 70784 x^{6} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} - \beta_{4} q^{5} - \beta_{11} q^{7} + ( - \beta_{18} - \beta_{16} + \cdots - \beta_{5}) q^{9} + (\beta_{19} + \beta_{18} + \cdots - \beta_{3}) q^{11} + ( - \beta_{17} - \beta_{16} + \cdots + 2 \beta_{3}) q^{13}+ \cdots + ( - 2 \beta_{18} - \beta_{17} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} + 4 q^{7} + 12 q^{11} + 4 q^{13} + 4 q^{17} + 16 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 16 q^{33} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} + 4 q^{47} - 4 q^{51} - 8 q^{55}+ \cdots + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 38 x^{18} + 597 x^{16} + 5004 x^{14} + 24072 x^{12} + 66452 x^{10} + 99328 x^{8} + 70784 x^{6} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7967 \nu^{18} - 299792 \nu^{16} - 4657907 \nu^{14} - 38543794 \nu^{12} - 182598508 \nu^{10} + \cdots - 3314328 ) / 4470176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 317550 \nu^{19} + 727367 \nu^{18} + 12234752 \nu^{17} + 27765188 \nu^{16} + 195676630 \nu^{15} + \cdots + 446824920 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 317550 \nu^{19} - 727367 \nu^{18} + 12234752 \nu^{17} - 27765188 \nu^{16} + 195676630 \nu^{15} + \cdots - 446824920 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 727367 \nu^{19} + 167852 \nu^{18} + 27765188 \nu^{17} + 6099280 \nu^{16} + 438515595 \nu^{15} + \cdots - 5080800 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 727367 \nu^{19} - 167852 \nu^{18} + 27765188 \nu^{17} - 6099280 \nu^{16} + 438515595 \nu^{15} + \cdots + 5080800 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 218122 \nu^{18} - 8217913 \nu^{16} - 127554672 \nu^{14} - 1050806639 \nu^{12} - 4926908906 \nu^{10} + \cdots - 4527664 ) / 18998248 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 812611 \nu^{18} - 30883886 \nu^{16} - 485157135 \nu^{14} - 4065060976 \nu^{12} + \cdots - 267545480 ) / 37996496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 719026 \nu^{19} + 1517355 \nu^{18} + 27519144 \nu^{17} + 57669124 \nu^{16} + 436970550 \nu^{15} + \cdots + 887830904 ) / 75992992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 786184 \nu^{19} + 1507459 \nu^{18} + 29978496 \nu^{17} + 57193856 \nu^{16} + 472886756 \nu^{15} + \cdots + 640790584 ) / 75992992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 786184 \nu^{19} + 1507459 \nu^{18} - 29978496 \nu^{17} + 57193856 \nu^{16} - 472886756 \nu^{15} + \cdots + 640790584 ) / 75992992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4059835 \nu^{19} - 707575 \nu^{18} - 154387652 \nu^{17} - 26814652 \nu^{16} - 2427002263 \nu^{15} + \cdots + 47255720 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4059835 \nu^{19} + 707575 \nu^{18} - 154387652 \nu^{17} + 26814652 \nu^{16} - 2427002263 \nu^{15} + \cdots - 47255720 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 414291 \nu^{19} - 15727124 \nu^{17} - 246732143 \nu^{15} - 2063796350 \nu^{13} + \cdots - 218392408 \nu ) / 8940352 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 5410853 \nu^{19} + 1965289 \nu^{18} + 205478316 \nu^{17} + 74530072 \nu^{16} + 3224946733 \nu^{15} + \cdots + 792712296 ) / 75992992 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 5410853 \nu^{19} + 1965289 \nu^{18} - 205478316 \nu^{17} + 74530072 \nu^{16} + \cdots + 792712296 ) / 75992992 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18334022 \nu^{19} - 188293 \nu^{18} + 696017112 \nu^{17} - 6907244 \nu^{16} + 10919620302 \nu^{15} + \cdots + 142981496 ) / 151985984 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 17171471 \nu^{19} + 3762726 \nu^{18} - 651590556 \nu^{17} + 142960864 \nu^{16} + \cdots + 1590505392 ) / 151985984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{11} + \beta_{6} - \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{19} - 13 \beta_{18} - \beta_{17} - 2 \beta_{16} - 9 \beta_{15} - \beta_{14} + \cdots + 41 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} + \beta_{16} + 11 \beta_{14} - 11 \beta_{13} - 12 \beta_{12} - 12 \beta_{11} - 22 \beta_{9} + \cdots - 188 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86 \beta_{19} + 129 \beta_{18} + 12 \beta_{17} + 31 \beta_{16} + 135 \beta_{15} + 13 \beta_{14} + \cdots - 301 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{17} - 12 \beta_{16} - 96 \beta_{14} + 96 \beta_{13} + 110 \beta_{12} + 110 \beta_{11} + \cdots + 1428 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 720 \beta_{19} - 1181 \beta_{18} - 108 \beta_{17} - 353 \beta_{16} - 1497 \beta_{15} + \cdots + 2305 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 114 \beta_{17} + 114 \beta_{16} + 778 \beta_{14} - 778 \beta_{13} - 928 \beta_{12} - 928 \beta_{11} + \cdots - 11164 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6002 \beta_{19} + 10463 \beta_{18} + 892 \beta_{17} + 3569 \beta_{16} + 14837 \beta_{15} + \cdots - 18113 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1022 \beta_{17} - 1022 \beta_{16} - 6124 \beta_{14} + 6124 \beta_{13} + 7576 \beta_{12} + \cdots + 88884 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 50062 \beta_{19} - 91193 \beta_{18} - 7146 \beta_{17} - 33985 \beta_{16} - 139195 \beta_{15} + \cdots + 144727 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8968 \beta_{17} + 8968 \beta_{16} + 47646 \beta_{14} - 47646 \beta_{13} - 60942 \beta_{12} + \cdots - 716272 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 418138 \beta_{19} + 787371 \beta_{18} + 56614 \beta_{17} + 312619 \beta_{16} + 1266069 \beta_{15} + \cdots - 1169605 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 77652 \beta_{17} - 77652 \beta_{16} - 369214 \beta_{14} + 369214 \beta_{13} + 486788 \beta_{12} + \cdots + 5820996 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3496898 \beta_{19} - 6757621 \beta_{18} - 446866 \beta_{17} - 2813857 \beta_{16} + \cdots + 9529195 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 664900 \beta_{17} + 664900 \beta_{16} + 2859482 \beta_{14} - 2859482 \beta_{13} - 3875618 \beta_{12} + \cdots - 47600512 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 29274934 \beta_{19} + 57760267 \beta_{18} + 3524382 \beta_{17} + 24960951 \beta_{16} + \cdots - 78110537 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/680\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(241\) \(341\) \(511\)
\(\chi(n)\) \(1\) \(-\beta_{15}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
2.89484i
− 2.25239i
− 2.14289i
1.66702i
0.748548i
0.155688i
− 0.205791i
0.914258i
− 2.69040i
2.91112i
− 2.89484i
2.25239i
2.14289i
− 1.66702i
− 0.748548i
− 0.155688i
0.205791i
− 0.914258i
2.69040i
− 2.91112i
0 −2.04696 − 2.04696i 0 0.707107 + 0.707107i 0 0.691988 − 0.691988i 0 5.38009i 0
81.2 0 −1.59268 − 1.59268i 0 −0.707107 − 0.707107i 0 2.48004 − 2.48004i 0 2.07327i 0
81.3 0 −1.51525 − 1.51525i 0 −0.707107 − 0.707107i 0 −0.860047 + 0.860047i 0 1.59199i 0
81.4 0 −1.17876 − 1.17876i 0 0.707107 + 0.707107i 0 −0.226360 + 0.226360i 0 − 0.221033i 0
81.5 0 −0.529303 − 0.529303i 0 0.707107 + 0.707107i 0 −3.59648 + 3.59648i 0 − 2.43968i 0
81.6 0 0.110088 + 0.110088i 0 −0.707107 − 0.707107i 0 −1.59814 + 1.59814i 0 − 2.97576i 0
81.7 0 0.145516 + 0.145516i 0 0.707107 + 0.707107i 0 2.63491 − 2.63491i 0 − 2.95765i 0
81.8 0 0.646478 + 0.646478i 0 −0.707107 − 0.707107i 0 −0.428593 + 0.428593i 0 − 2.16413i 0
81.9 0 1.90240 + 1.90240i 0 0.707107 + 0.707107i 0 0.0817319 − 0.0817319i 0 4.23827i 0
81.10 0 2.05847 + 2.05847i 0 −0.707107 − 0.707107i 0 2.82095 − 2.82095i 0 5.47464i 0
361.1 0 −2.04696 + 2.04696i 0 0.707107 − 0.707107i 0 0.691988 + 0.691988i 0 − 5.38009i 0
361.2 0 −1.59268 + 1.59268i 0 −0.707107 + 0.707107i 0 2.48004 + 2.48004i 0 − 2.07327i 0
361.3 0 −1.51525 + 1.51525i 0 −0.707107 + 0.707107i 0 −0.860047 − 0.860047i 0 − 1.59199i 0
361.4 0 −1.17876 + 1.17876i 0 0.707107 − 0.707107i 0 −0.226360 − 0.226360i 0 0.221033i 0
361.5 0 −0.529303 + 0.529303i 0 0.707107 − 0.707107i 0 −3.59648 − 3.59648i 0 2.43968i 0
361.6 0 0.110088 − 0.110088i 0 −0.707107 + 0.707107i 0 −1.59814 − 1.59814i 0 2.97576i 0
361.7 0 0.145516 − 0.145516i 0 0.707107 − 0.707107i 0 2.63491 + 2.63491i 0 2.95765i 0
361.8 0 0.646478 − 0.646478i 0 −0.707107 + 0.707107i 0 −0.428593 − 0.428593i 0 2.16413i 0
361.9 0 1.90240 − 1.90240i 0 0.707107 − 0.707107i 0 0.0817319 + 0.0817319i 0 − 4.23827i 0
361.10 0 2.05847 − 2.05847i 0 −0.707107 + 0.707107i 0 2.82095 + 2.82095i 0 − 5.47464i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.bd.b ✓ 20
4.b odd 2 1 1360.2.bt.f 20
17.c even 4 1 inner 680.2.bd.b ✓ 20
68.f odd 4 1 1360.2.bt.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.bd.b ✓ 20 1.a even 1 1 trivial
680.2.bd.b ✓ 20 17.c even 4 1 inner
1360.2.bt.f 20 4.b odd 2 1
1360.2.bt.f 20 68.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 4 T_{3}^{19} + 8 T_{3}^{18} + 8 T_{3}^{17} + 125 T_{3}^{16} + 496 T_{3}^{15} + 1016 T_{3}^{14} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{20} - 4 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{20} - 12 T^{19} + \cdots + 17774656 \) Copy content Toggle raw display
$13$ \( (T^{10} - 2 T^{9} + \cdots + 6896)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + 238 T^{18} + \cdots + 4194304 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 131974144 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 85488403456 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 285204544 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 416097243136 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 52535359504384 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 56280724194304 \) Copy content Toggle raw display
$47$ \( (T^{10} - 2 T^{9} + \cdots - 157412384)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 2434997960704 \) Copy content Toggle raw display
$67$ \( (T^{10} + 28 T^{9} + \cdots - 90404416)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 51717499654144 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 97\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 171469177701376 \) Copy content Toggle raw display
$89$ \( (T^{10} - 14 T^{9} + \cdots - 10883008)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 34803014750464 \) Copy content Toggle raw display
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