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Properties

Label 1680.2.di.f
Level $1680$
Weight $2$
Character orbit 1680.di
Analytic conductor $13.415$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [1680,2,Mod(289,1680)] 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("1680.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.di (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,2,0,0,0,10,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
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: \( x^{20} - 17 x^{18} + 195 x^{16} - 1242 x^{14} + 5741 x^{12} - 14387 x^{10} + 25181 x^{8} - 12094 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{17} + \beta_{12} - \beta_{7}) q^{5} + ( - \beta_{8} + \beta_{7}) q^{7} - \beta_{10} q^{9} + (\beta_{14} - \beta_{10} - \beta_{3} - 1) q^{11} + (\beta_{19} + \beta_{18} + \cdots + \beta_{8}) q^{13}+ \cdots + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{5} + 10 q^{9} - 10 q^{11} - 6 q^{19} - 2 q^{21} + 6 q^{25} + 24 q^{29} - 12 q^{31} + 2 q^{39} + 28 q^{41} - 2 q^{45} + 52 q^{49} - 8 q^{51} + 16 q^{55} - 4 q^{59} + 4 q^{61} + 8 q^{65} + 12 q^{69}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 17 x^{18} + 195 x^{16} - 1242 x^{14} + 5741 x^{12} - 14387 x^{10} + 25181 x^{8} - 12094 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 380083905971 \nu^{18} + 6346665847787 \nu^{16} - 71983253027758 \nu^{14} + \cdots + 129587132515577 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 189142039159 \nu^{18} + 3135295056139 \nu^{16} - 35560204799726 \nu^{14} + \cdots - 604149118967174 ) / 175660473484554 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1689351283610 \nu^{18} - 28596997848107 \nu^{16} + 327492769335304 \nu^{14} + \cdots - 4313473378751 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3178621143797 \nu^{18} + 52827925818323 \nu^{16} - 599169082209982 \nu^{14} + \cdots - 28\!\cdots\!63 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4313473378751 \nu^{19} - 71639696155157 \nu^{17} + 812530311008338 \nu^{15} + \cdots + 85\!\cdots\!55 \nu ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1335942273154 \nu^{19} - 22686018687151 \nu^{17} + 260094601617668 \nu^{15} + \cdots - 91274003213272 \nu ) / 87830236742277 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16199125708417 \nu^{19} + 269123980363237 \nu^{17} + \cdots - 23\!\cdots\!79 \nu ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16499125186021 \nu^{19} - 274093680131581 \nu^{17} + \cdots + 29\!\cdots\!75 \nu ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1335942273154 \nu^{18} + 22686018687151 \nu^{16} - 260094601617668 \nu^{14} + \cdots + 3443766470995 ) / 87830236742277 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9688873426107 \nu^{19} - 23830580207180 \nu^{18} - 160975953090780 \nu^{17} + \cdots + 16\!\cdots\!17 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9688873426107 \nu^{19} + 23830580207180 \nu^{18} - 160975953090780 \nu^{17} + \cdots - 16\!\cdots\!17 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42614940397829 \nu^{18} + 725202734580248 \nu^{16} + \cdots + 29\!\cdots\!74 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3113579433167 \nu^{18} + 52874764229100 \nu^{16} - 606215063005646 \nu^{14} + \cdots + 8027121780166 ) / 58553491161518 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32249672 \nu^{19} + 547746005 \nu^{17} - 6280269934 \nu^{15} + 39957879686 \nu^{13} + \cdots + 83180621 \nu ) / 253844676 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 205721526546693 \nu^{19} - 2196778903559 \nu^{18} + \cdots - 15\!\cdots\!53 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 205721526546693 \nu^{19} - 2196778903559 \nu^{18} + \cdots - 15\!\cdots\!53 ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 349450365843343 \nu^{19} + \cdots + 24\!\cdots\!76 \nu ) / 10\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 445338210032827 \nu^{19} + \cdots - 30\!\cdots\!12 \nu ) / 10\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 3\beta_{10} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - \beta_{18} - \beta_{9} - \beta_{8} + 6\beta_{7} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{14} + 8\beta_{13} + 8\beta_{12} - 8\beta_{11} - 17\beta_{10} + 8\beta_{4} - 6\beta_{3} - 8\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{19} - 10 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{12} + \cdots - 2 \beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{17} + 57\beta_{16} - 59\beta_{5} - 37\beta_{3} - 55\beta_{2} - 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24\beta_{12} + 24\beta_{11} + 55\beta_{9} + 81\beta_{8} - 14\beta_{6} - 236\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 396 \beta_{17} + 396 \beta_{16} - 236 \beta_{14} - 420 \beta_{13} - 396 \beta_{12} + \cdots - 360 \beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 360 \beta_{19} + 616 \beta_{18} + 220 \beta_{17} - 220 \beta_{16} + 52 \beta_{15} + 360 \beta_{9} + \cdots - 1533 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1533 \beta_{14} - 2949 \beta_{13} - 2729 \beta_{12} + 2729 \beta_{11} + 4679 \beta_{10} - 2305 \beta_{4} + \cdots + 4679 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2305 \beta_{19} + 4569 \beta_{18} + 1840 \beta_{17} - 1840 \beta_{16} - 76 \beta_{15} + \cdots - 76 \beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -18764\beta_{17} - 18764\beta_{16} + 20604\beta_{5} + 10050\beta_{3} + 14584\beta_{2} + 31525 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -14734\beta_{12} - 14734\beta_{11} - 14584\beta_{9} - 33498\beta_{8} - 3826\beta_{6} + 66209\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 129025 \beta_{17} - 129025 \beta_{16} + 66209 \beta_{14} + 143759 \beta_{13} + 129025 \beta_{12} + \cdots + 91551 \beta_{4} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 91551 \beta_{19} - 244049 \beta_{18} - 115024 \beta_{17} + 115024 \beta_{16} + 49606 \beta_{15} + \cdots + 437480 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 437480 \beta_{14} + 1003128 \beta_{13} + 888104 \beta_{12} - 888104 \beta_{11} - 1450937 \beta_{10} + \cdots - 1450937 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 570976 \beta_{19} - 1770880 \beta_{18} - 882776 \beta_{17} + 882776 \beta_{16} + \cdots + 500760 \beta_{6} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 6121505\beta_{17} + 6121505\beta_{16} - 7004281\beta_{5} - 2896873\beta_{3} - 3538065\beta_{2} - 9885195 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 6690848 \beta_{12} + 6690848 \beta_{11} + 3538065 \beta_{9} + 12812353 \beta_{8} + 4525688 \beta_{6} - 19217006 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\beta_{10}\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.48108 + 0.855102i
0.608642 − 0.351400i
−2.30491 + 1.33074i
2.20500 − 1.27306i
0.106328 − 0.0613887i
−2.20500 + 1.27306i
−0.106328 + 0.0613887i
−0.608642 + 0.351400i
2.30491 − 1.33074i
1.48108 − 0.855102i
−1.48108 − 0.855102i
0.608642 + 0.351400i
−2.30491 − 1.33074i
2.20500 + 1.27306i
0.106328 + 0.0613887i
−2.20500 − 1.27306i
−0.106328 − 0.0613887i
−0.608642 − 0.351400i
2.30491 + 1.33074i
1.48108 + 0.855102i
0 −0.866025 + 0.500000i 0 −2.12975 − 0.681292i 0 2.63917 + 0.186479i 0 0.500000 − 0.866025i 0
289.2 0 −0.866025 + 0.500000i 0 −1.97724 + 1.04428i 0 −2.53416 + 0.760296i 0 0.500000 − 0.866025i 0
289.3 0 −0.866025 + 0.500000i 0 0.679346 − 2.13037i 0 0.0274999 + 2.64561i 0 0.500000 − 0.866025i 0
289.4 0 −0.866025 + 0.500000i 0 1.74693 + 1.39580i 0 −2.36879 − 1.17849i 0 0.500000 − 0.866025i 0
289.5 0 −0.866025 + 0.500000i 0 2.18072 − 0.494442i 0 2.23627 − 1.41389i 0 0.500000 − 0.866025i 0
289.6 0 0.866025 − 0.500000i 0 −2.08226 − 0.814984i 0 2.36879 + 1.17849i 0 0.500000 − 0.866025i 0
289.7 0 0.866025 − 0.500000i 0 −0.662159 − 2.13578i 0 −2.23627 + 1.41389i 0 0.500000 − 0.866025i 0
289.8 0 0.866025 − 0.500000i 0 0.0842423 + 2.23448i 0 2.53416 − 0.760296i 0 0.500000 − 0.866025i 0
289.9 0 0.866025 − 0.500000i 0 1.50528 − 1.65352i 0 −0.0274999 − 2.64561i 0 0.500000 − 0.866025i 0
289.10 0 0.866025 − 0.500000i 0 1.65489 + 1.50377i 0 −2.63917 − 0.186479i 0 0.500000 − 0.866025i 0
529.1 0 −0.866025 − 0.500000i 0 −2.12975 + 0.681292i 0 2.63917 − 0.186479i 0 0.500000 + 0.866025i 0
529.2 0 −0.866025 − 0.500000i 0 −1.97724 − 1.04428i 0 −2.53416 − 0.760296i 0 0.500000 + 0.866025i 0
529.3 0 −0.866025 − 0.500000i 0 0.679346 + 2.13037i 0 0.0274999 − 2.64561i 0 0.500000 + 0.866025i 0
529.4 0 −0.866025 − 0.500000i 0 1.74693 − 1.39580i 0 −2.36879 + 1.17849i 0 0.500000 + 0.866025i 0
529.5 0 −0.866025 − 0.500000i 0 2.18072 + 0.494442i 0 2.23627 + 1.41389i 0 0.500000 + 0.866025i 0
529.6 0 0.866025 + 0.500000i 0 −2.08226 + 0.814984i 0 2.36879 − 1.17849i 0 0.500000 + 0.866025i 0
529.7 0 0.866025 + 0.500000i 0 −0.662159 + 2.13578i 0 −2.23627 − 1.41389i 0 0.500000 + 0.866025i 0
529.8 0 0.866025 + 0.500000i 0 0.0842423 − 2.23448i 0 2.53416 + 0.760296i 0 0.500000 + 0.866025i 0
529.9 0 0.866025 + 0.500000i 0 1.50528 + 1.65352i 0 −0.0274999 + 2.64561i 0 0.500000 + 0.866025i 0
529.10 0 0.866025 + 0.500000i 0 1.65489 − 1.50377i 0 −2.63917 + 0.186479i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.di.f 20
4.b odd 2 1 840.2.cc.b ✓ 20
5.b even 2 1 inner 1680.2.di.f 20
7.c even 3 1 inner 1680.2.di.f 20
20.d odd 2 1 840.2.cc.b ✓ 20
28.g odd 6 1 840.2.cc.b ✓ 20
35.j even 6 1 inner 1680.2.di.f 20
140.p odd 6 1 840.2.cc.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cc.b ✓ 20 4.b odd 2 1
840.2.cc.b ✓ 20 20.d odd 2 1
840.2.cc.b ✓ 20 28.g odd 6 1
840.2.cc.b ✓ 20 140.p odd 6 1
1680.2.di.f 20 1.a even 1 1 trivial
1680.2.di.f 20 5.b even 2 1 inner
1680.2.di.f 20 7.c even 3 1 inner
1680.2.di.f 20 35.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{10} + 5 T_{11}^{9} + 31 T_{11}^{8} + 34 T_{11}^{7} + 196 T_{11}^{6} + 206 T_{11}^{5} + \cdots + 196 \) acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( (T^{10} + 5 T^{9} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 53 T^{8} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} - 56 T^{18} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{10} + 3 T^{9} + \cdots + 13225)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 303120718096 \) Copy content Toggle raw display
$29$ \( (T^{5} - 6 T^{4} + \cdots - 112)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + 6 T^{9} + \cdots + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 200510386108561 \) Copy content Toggle raw display
$41$ \( (T^{5} - 7 T^{4} + \cdots + 1126)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + 164 T^{8} + \cdots + 11236)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + 2 T^{9} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 2 T^{9} + \cdots + 10705984)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} - 20 T^{4} + \cdots - 14956)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{10} - 2 T^{9} + \cdots + 5152900)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 648 T^{8} + \cdots + 4345973776)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 18 T^{9} + \cdots + 427827856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 644 T^{8} + \cdots + 1063281664)^{2} \) Copy content Toggle raw display
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