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Properties

Label 1480.2.q.d
Level $1480$
Weight $2$
Character orbit 1480.q
Analytic conductor $11.818$
Analytic rank $0$
Dimension $20$
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] = [1480,2,Mod(121,1480)] 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("1480.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1480, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1480.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8178594991\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
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: \( x^{20} + 22 x^{18} - 2 x^{17} + 318 x^{16} - 28 x^{15} + 2661 x^{14} - 96 x^{13} + 16187 x^{12} + \cdots + 5476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - \beta_{3} q^{5} + \beta_{15} q^{7} + ( - \beta_{4} - \beta_{3} - 1) q^{9} + (\beta_{18} - \beta_{14} + \cdots - \beta_{9}) q^{11} - \beta_{17} q^{13} - \beta_1 q^{15} - \beta_{8} q^{17}+ \cdots + ( - 4 \beta_{12} + 3 \beta_{9} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{5} - 14 q^{9} + 8 q^{11} + 3 q^{13} + q^{17} + 9 q^{19} + 3 q^{21} + 10 q^{23} - 10 q^{25} + 6 q^{27} + 20 q^{31} - 7 q^{33} - 15 q^{37} - 19 q^{39} + 9 q^{41} + 38 q^{43} - 28 q^{45} + 10 q^{47}+ \cdots - 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 22 x^{18} - 2 x^{17} + 318 x^{16} - 28 x^{15} + 2661 x^{14} - 96 x^{13} + 16187 x^{12} + \cdots + 5476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 49\!\cdots\!37 \nu^{19} + \cdots + 14\!\cdots\!44 ) / 37\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\!\cdots\!71 \nu^{19} + \cdots + 62\!\cdots\!90 ) / 27\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39\!\cdots\!73 \nu^{19} + \cdots - 64\!\cdots\!08 ) / 13\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!79 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 47\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!45 \nu^{19} + \cdots - 15\!\cdots\!54 ) / 37\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!43 \nu^{19} + \cdots + 75\!\cdots\!64 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!31 \nu^{19} + \cdots - 48\!\cdots\!12 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!22 \nu^{19} + \cdots - 47\!\cdots\!76 ) / 30\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42\!\cdots\!20 \nu^{19} + \cdots + 57\!\cdots\!12 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 52\!\cdots\!57 \nu^{19} + \cdots - 27\!\cdots\!68 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 65\!\cdots\!67 \nu^{19} + \cdots - 58\!\cdots\!16 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35\!\cdots\!07 \nu^{19} + \cdots - 20\!\cdots\!98 ) / 30\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 77\!\cdots\!95 \nu^{19} + \cdots + 63\!\cdots\!10 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 83\!\cdots\!90 \nu^{19} + \cdots - 23\!\cdots\!40 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 21\!\cdots\!57 \nu^{19} + \cdots - 22\!\cdots\!64 ) / 12\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{19} + \cdots + 14\!\cdots\!74 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 14\!\cdots\!10 \nu^{19} + \cdots + 31\!\cdots\!84 ) / 61\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 72\!\cdots\!17 \nu^{19} + \cdots - 11\!\cdots\!08 ) / 16\!\cdots\!22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{13} + \beta_{12} - \beta_{9} + 7\beta_{6} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 8 \beta_{4} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} - 14 \beta_{18} + 3 \beta_{17} - \beta_{16} + \beta_{14} + 17 \beta_{13} + \cdots + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{19} + 31 \beta_{18} - 15 \beta_{17} - 14 \beta_{16} - 19 \beta_{15} - 12 \beta_{14} + \cdots + 193 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 157 \beta_{12} + 18 \beta_{11} - 47 \beta_{10} + 207 \beta_{9} + 95 \beta_{8} + 2 \beta_{7} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 181 \beta_{19} - 376 \beta_{18} + 175 \beta_{17} + 155 \beta_{16} + 237 \beta_{15} + \cdots + 538 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 914 \beta_{19} + 1653 \beta_{18} - 562 \beta_{17} + 118 \beta_{16} - 59 \beta_{15} - 232 \beta_{14} + \cdots + 838 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4193 \beta_{12} + 1244 \beta_{11} - 1885 \beta_{10} + 4543 \beta_{9} + 2025 \beta_{8} + 2558 \beta_{7} + \cdots - 13153 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8928 \beta_{19} - 16979 \beta_{18} + 6137 \beta_{17} - 861 \beta_{16} + 1091 \beta_{15} + \cdots + 4867 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 21846 \beta_{19} + 45020 \beta_{18} - 19683 \beta_{17} - 15359 \beta_{16} - 25942 \beta_{15} + \cdots + 117086 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 172605 \beta_{12} + 29130 \beta_{11} - 64511 \beta_{10} + 236888 \beta_{9} + 88226 \beta_{8} + \cdots - 140499 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 231011 \beta_{19} - 474082 \beta_{18} + 202907 \beta_{17} + 146740 \beta_{16} + 255666 \beta_{15} + \cdots + 405014 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 879096 \beta_{19} + 1747674 \beta_{18} - 666451 \beta_{17} + 6054 \beta_{16} - 218569 \beta_{15} + \cdots + 1631054 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4938612 \beta_{12} + 1235236 \beta_{11} - 2080481 \beta_{10} + 5686061 \beta_{9} + 2414094 \beta_{8} + \cdots - 10194205 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 8810898 \beta_{19} - 17676985 \beta_{18} + 6829074 \beta_{17} + 325709 \beta_{16} + \cdots + 7365202 \beta_{2} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 25042187 \beta_{19} + 51115892 \beta_{18} - 21283204 \beta_{17} - 13128563 \beta_{16} + \cdots + 98093753 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 178841264 \beta_{12} + 33944758 \beta_{11} - 69710665 \beta_{10} + 239488305 \beta_{9} + \cdots - 198921851 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1480\mathbb{Z}\right)^\times\).

\(n\) \(297\) \(741\) \(1001\) \(1111\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−1.59636 + 2.76498i
−1.16198 + 2.01261i
−1.12611 + 1.95048i
−0.570365 + 0.987900i
−0.224535 + 0.388906i
0.222478 − 0.385344i
0.557594 − 0.965781i
1.22822 − 2.12735i
1.23220 − 2.13424i
1.43886 − 2.49217i
−1.59636 − 2.76498i
−1.16198 − 2.01261i
−1.12611 − 1.95048i
−0.570365 − 0.987900i
−0.224535 − 0.388906i
0.222478 + 0.385344i
0.557594 + 0.965781i
1.22822 + 2.12735i
1.23220 + 2.13424i
1.43886 + 2.49217i
0 −1.59636 − 2.76498i 0 0.500000 + 0.866025i 0 −0.930812 − 1.61221i 0 −3.59673 + 6.22972i 0
121.2 0 −1.16198 − 2.01261i 0 0.500000 + 0.866025i 0 0.511631 + 0.886171i 0 −1.20041 + 2.07917i 0
121.3 0 −1.12611 − 1.95048i 0 0.500000 + 0.866025i 0 1.89195 + 3.27696i 0 −1.03626 + 1.79485i 0
121.4 0 −0.570365 − 0.987900i 0 0.500000 + 0.866025i 0 −2.07678 − 3.59709i 0 0.849369 − 1.47115i 0
121.5 0 −0.224535 − 0.388906i 0 0.500000 + 0.866025i 0 1.39007 + 2.40767i 0 1.39917 − 2.42343i 0
121.6 0 0.222478 + 0.385344i 0 0.500000 + 0.866025i 0 −0.469382 − 0.812993i 0 1.40101 − 2.42661i 0
121.7 0 0.557594 + 0.965781i 0 0.500000 + 0.866025i 0 −0.894104 − 1.54863i 0 0.878178 − 1.52105i 0
121.8 0 1.22822 + 2.12735i 0 0.500000 + 0.866025i 0 0.729953 + 1.26432i 0 −1.51707 + 2.62764i 0
121.9 0 1.23220 + 2.13424i 0 0.500000 + 0.866025i 0 2.21315 + 3.83329i 0 −1.53664 + 2.66154i 0
121.10 0 1.43886 + 2.49217i 0 0.500000 + 0.866025i 0 −2.36568 − 4.09748i 0 −2.64061 + 4.57368i 0
1321.1 0 −1.59636 + 2.76498i 0 0.500000 − 0.866025i 0 −0.930812 + 1.61221i 0 −3.59673 − 6.22972i 0
1321.2 0 −1.16198 + 2.01261i 0 0.500000 − 0.866025i 0 0.511631 − 0.886171i 0 −1.20041 − 2.07917i 0
1321.3 0 −1.12611 + 1.95048i 0 0.500000 − 0.866025i 0 1.89195 − 3.27696i 0 −1.03626 − 1.79485i 0
1321.4 0 −0.570365 + 0.987900i 0 0.500000 − 0.866025i 0 −2.07678 + 3.59709i 0 0.849369 + 1.47115i 0
1321.5 0 −0.224535 + 0.388906i 0 0.500000 − 0.866025i 0 1.39007 − 2.40767i 0 1.39917 + 2.42343i 0
1321.6 0 0.222478 − 0.385344i 0 0.500000 − 0.866025i 0 −0.469382 + 0.812993i 0 1.40101 + 2.42661i 0
1321.7 0 0.557594 − 0.965781i 0 0.500000 − 0.866025i 0 −0.894104 + 1.54863i 0 0.878178 + 1.52105i 0
1321.8 0 1.22822 − 2.12735i 0 0.500000 − 0.866025i 0 0.729953 − 1.26432i 0 −1.51707 − 2.62764i 0
1321.9 0 1.23220 − 2.13424i 0 0.500000 − 0.866025i 0 2.21315 − 3.83329i 0 −1.53664 − 2.66154i 0
1321.10 0 1.43886 − 2.49217i 0 0.500000 − 0.866025i 0 −2.36568 + 4.09748i 0 −2.64061 − 4.57368i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1480.2.q.d ✓ 20
37.c even 3 1 inner 1480.2.q.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.q.d ✓ 20 1.a even 1 1 trivial
1480.2.q.d ✓ 20 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 22 T_{3}^{18} - 2 T_{3}^{17} + 318 T_{3}^{16} - 28 T_{3}^{15} + 2661 T_{3}^{14} + \cdots + 5476 \) acting on \(S_{2}^{\mathrm{new}}(1480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 22 T^{18} + \cdots + 5476 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + 46 T^{18} + \cdots + 18249984 \) Copy content Toggle raw display
$11$ \( (T^{10} - 4 T^{9} + \cdots - 2718)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} - 3 T^{19} + \cdots + 2692881 \) Copy content Toggle raw display
$17$ \( T^{20} - T^{19} + \cdots + 565504 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 116035984 \) Copy content Toggle raw display
$23$ \( (T^{10} - 5 T^{9} + \cdots + 150066)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 172 T^{8} + \cdots - 976040)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 10 T^{9} + \cdots + 502700)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{10} - 19 T^{9} + \cdots + 3255958)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 5 T^{9} + \cdots - 179158)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 375977448900 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 307570009330276 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 34825531456 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 210293154214144 \) Copy content Toggle raw display
$73$ \( (T^{10} + 8 T^{9} + \cdots + 2097424)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 134342564709376 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 54605596993600 \) Copy content Toggle raw display
$97$ \( (T^{10} - 7 T^{9} + \cdots - 12518664)^{2} \) Copy content Toggle raw display
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