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Properties

Label 1860.2.z.c
Level $1860$
Weight $2$
Character orbit 1860.z
Analytic conductor $14.852$
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] = [1860,2,Mod(481,1860)] 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("1860.481"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.z (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,-5,0,-20] 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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
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: \( x^{20} - 5 x^{19} + 20 x^{18} - 62 x^{17} + 211 x^{16} - 436 x^{15} + 1144 x^{14} - 2239 x^{13} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{8} + \beta_{6} - 1) q^{3} - q^{5} + ( - \beta_{12} + \beta_{8}) q^{7} - \beta_{6} q^{9} + (\beta_{19} + \beta_{16} + \cdots + \beta_1) q^{11} + ( - \beta_{16} + \beta_{13} + \beta_{10} + \cdots - 2) q^{13}+ \cdots + ( - \beta_{19} + \beta_{18} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{3} - 20 q^{5} + 5 q^{7} - 5 q^{9} - 4 q^{11} + 5 q^{15} + 8 q^{17} + 4 q^{19} - 3 q^{23} + 20 q^{25} - 5 q^{27} + 9 q^{29} + 14 q^{31} + q^{33} - 5 q^{35} - 8 q^{37} - 5 q^{39} - 22 q^{41}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 20 x^{18} - 62 x^{17} + 211 x^{16} - 436 x^{15} + 1144 x^{14} - 2239 x^{13} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!22 \nu^{19} + \cdots + 34\!\cdots\!01 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!59 \nu^{19} + \cdots - 55\!\cdots\!00 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71\!\cdots\!38 \nu^{19} + \cdots + 11\!\cdots\!66 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 82\!\cdots\!81 \nu^{19} + \cdots + 18\!\cdots\!46 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!64 \nu^{19} + \cdots + 10\!\cdots\!38 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!56 \nu^{19} + \cdots - 18\!\cdots\!54 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!16 \nu^{19} + \cdots - 13\!\cdots\!66 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34\!\cdots\!07 \nu^{19} + \cdots - 12\!\cdots\!20 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!01 \nu^{19} + \cdots - 11\!\cdots\!67 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42\!\cdots\!12 \nu^{19} + \cdots + 71\!\cdots\!43 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 60\!\cdots\!36 \nu^{19} + \cdots + 24\!\cdots\!92 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 68\!\cdots\!53 \nu^{19} + \cdots + 32\!\cdots\!62 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 71\!\cdots\!26 \nu^{19} + \cdots - 32\!\cdots\!93 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39\!\cdots\!89 \nu^{19} + \cdots + 11\!\cdots\!46 ) / 28\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 98\!\cdots\!93 \nu^{19} + \cdots + 21\!\cdots\!73 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 99\!\cdots\!49 \nu^{19} + \cdots + 40\!\cdots\!95 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!33 \nu^{19} + \cdots - 36\!\cdots\!94 ) / 57\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13\!\cdots\!89 \nu^{19} + \cdots - 24\!\cdots\!71 ) / 57\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} + \beta_{11} + 4\beta_{10} - \beta_{9} + 4\beta_{8} + 4\beta_{6} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} - 2 \beta_{14} + 7 \beta_{11} + 8 \beta_{10} + \cdots - 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} + 3 \beta_{18} - \beta_{17} - 10 \beta_{14} - 2 \beta_{13} + 3 \beta_{11} + 4 \beta_{10} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{19} + 11 \beta_{18} - 13 \beta_{16} - 13 \beta_{14} + 26 \beta_{9} + 3 \beta_{8} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28 \beta_{19} + 12 \beta_{17} - 98 \beta_{16} + \beta_{15} - \beta_{13} + 25 \beta_{12} + \cdots - 93 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 35 \beta_{19} - 109 \beta_{18} + 109 \beta_{17} - 288 \beta_{16} + 94 \beta_{15} + 152 \beta_{14} + \cdots + 520 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 455 \beta_{18} + 315 \beta_{17} - 509 \beta_{16} + 265 \beta_{15} + 991 \beta_{14} + 39 \beta_{12} + \cdots + 1009 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 458 \beta_{19} - 1550 \beta_{18} + 458 \beta_{17} + 182 \beta_{15} + 3081 \beta_{14} + 720 \beta_{13} + \cdots + 3434 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3357 \beta_{19} - 3357 \beta_{18} + 5694 \beta_{16} + 5694 \beta_{14} + 714 \beta_{13} - 714 \beta_{12} + \cdots + 13880 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11157 \beta_{19} - 5432 \beta_{17} + 32680 \beta_{16} - 2789 \beta_{15} + 2789 \beta_{13} + \cdots + 26233 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18451 \beta_{19} + 35323 \beta_{18} - 35323 \beta_{17} + 107286 \beta_{16} - 27619 \beta_{15} + \cdots - 128657 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 177221 \beta_{18} - 115654 \beta_{17} + 204427 \beta_{16} - 89499 \beta_{15} - 346136 \beta_{14} + \cdots - 467369 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 205342 \beta_{19} + 576718 \beta_{18} - 205342 \beta_{17} - 126015 \beta_{15} - 1130818 \beta_{14} + \cdots - 1563625 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1210145 \beta_{19} + 1210145 \beta_{18} - 2202111 \beta_{16} - 2202111 \beta_{14} - 435382 \beta_{13} + \cdots - 3908922 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3913189 \beta_{19} + 2249814 \beta_{17} - 11962095 \beta_{16} + 1475051 \beta_{15} + \cdots - 8077683 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7417015 \beta_{19} - 12736858 \beta_{18} + 12736858 \beta_{17} - 38897158 \beta_{16} + \cdots + 39538327 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 65722283 \beta_{18} + 41335296 \beta_{17} - 77197618 \beta_{16} + 30694660 \beta_{15} + \cdots + 184850971 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 80080273 \beta_{19} - 214628742 \beta_{18} + 80080273 \beta_{17} + 55373983 \beta_{15} + \cdots + 606023083 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
−0.801277 + 2.46608i
0.600838 − 1.84919i
0.0649168 − 0.199793i
0.361473 − 1.11250i
−0.653001 + 2.00973i
0.748034 + 0.543479i
−1.75254 − 1.27329i
2.63761 + 1.91634i
0.0302961 + 0.0220114i
1.26365 + 0.918094i
0.748034 − 0.543479i
−1.75254 + 1.27329i
2.63761 − 1.91634i
0.0302961 − 0.0220114i
1.26365 − 0.918094i
−0.801277 − 2.46608i
0.600838 + 1.84919i
0.0649168 + 0.199793i
0.361473 + 1.11250i
−0.653001 − 2.00973i
0 −0.809017 − 0.587785i 0 −1.00000 0 −0.828611 + 2.55020i 0 0.309017 + 0.951057i 0
481.2 0 −0.809017 − 0.587785i 0 −1.00000 0 −0.487603 + 1.50069i 0 0.309017 + 0.951057i 0
481.3 0 −0.809017 − 0.587785i 0 −1.00000 0 −0.429696 + 1.32247i 0 0.309017 + 0.951057i 0
481.4 0 −0.809017 − 0.587785i 0 −1.00000 0 0.161692 − 0.497635i 0 0.309017 + 0.951057i 0
481.5 0 −0.809017 − 0.587785i 0 −1.00000 0 1.15717 − 3.56139i 0 0.309017 + 0.951057i 0
721.1 0 0.309017 + 0.951057i 0 −1.00000 0 −1.93772 − 1.40783i 0 −0.809017 + 0.587785i 0
721.2 0 0.309017 + 0.951057i 0 −1.00000 0 −1.41049 − 1.02478i 0 −0.809017 + 0.587785i 0
721.3 0 0.309017 + 0.951057i 0 −1.00000 0 0.300953 + 0.218655i 0 −0.809017 + 0.587785i 0
721.4 0 0.309017 + 0.951057i 0 −1.00000 0 2.44179 + 1.77406i 0 −0.809017 + 0.587785i 0
721.5 0 0.309017 + 0.951057i 0 −1.00000 0 3.53252 + 2.56653i 0 −0.809017 + 0.587785i 0
841.1 0 0.309017 − 0.951057i 0 −1.00000 0 −1.93772 + 1.40783i 0 −0.809017 − 0.587785i 0
841.2 0 0.309017 − 0.951057i 0 −1.00000 0 −1.41049 + 1.02478i 0 −0.809017 − 0.587785i 0
841.3 0 0.309017 − 0.951057i 0 −1.00000 0 0.300953 − 0.218655i 0 −0.809017 − 0.587785i 0
841.4 0 0.309017 − 0.951057i 0 −1.00000 0 2.44179 − 1.77406i 0 −0.809017 − 0.587785i 0
841.5 0 0.309017 − 0.951057i 0 −1.00000 0 3.53252 − 2.56653i 0 −0.809017 − 0.587785i 0
901.1 0 −0.809017 + 0.587785i 0 −1.00000 0 −0.828611 − 2.55020i 0 0.309017 − 0.951057i 0
901.2 0 −0.809017 + 0.587785i 0 −1.00000 0 −0.487603 − 1.50069i 0 0.309017 − 0.951057i 0
901.3 0 −0.809017 + 0.587785i 0 −1.00000 0 −0.429696 − 1.32247i 0 0.309017 − 0.951057i 0
901.4 0 −0.809017 + 0.587785i 0 −1.00000 0 0.161692 + 0.497635i 0 0.309017 − 0.951057i 0
901.5 0 −0.809017 + 0.587785i 0 −1.00000 0 1.15717 + 3.56139i 0 0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 481.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.z.c ✓ 20
31.d even 5 1 inner 1860.2.z.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.z.c ✓ 20 1.a even 1 1 trivial
1860.2.z.c ✓ 20 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} - 5 T_{7}^{19} + 22 T_{7}^{18} - 35 T_{7}^{17} + 181 T_{7}^{16} - 41 T_{7}^{15} + \cdots + 55696 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 5 T^{19} + \cdots + 55696 \) Copy content Toggle raw display
$11$ \( T^{20} + 4 T^{19} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{20} + 23 T^{18} + \cdots + 712336 \) Copy content Toggle raw display
$17$ \( T^{20} - 8 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{20} - 4 T^{19} + \cdots + 5517801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 4271406736 \) Copy content Toggle raw display
$29$ \( T^{20} - 9 T^{19} + \cdots + 430336 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( (T^{10} + 4 T^{9} + \cdots + 272164)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 22 T^{19} + \cdots + 66324736 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 406936823056 \) Copy content Toggle raw display
$47$ \( T^{20} + 24 T^{19} + \cdots + 79138816 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 403851788292096 \) Copy content Toggle raw display
$61$ \( (T^{10} + 16 T^{9} + \cdots - 147989349)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 23 T^{9} + \cdots - 299171556)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 33977893851136 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 68\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
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