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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 = [16,0,4,0,-16] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 15 x^{13} + 101 x^{12} - 273 x^{11} + 1747 x^{10} - 2144 x^{9} + \cdots + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} - q^{5} - \beta_1 q^{7} - \beta_{9} q^{9} + ( - \beta_{6} + \beta_{3} + \cdots - \beta_1) q^{11} + (\beta_{10} + 2 \beta_{9} - \beta_{6} - 2) q^{13} - \beta_{10} q^{15} + ( - \beta_{12} + \beta_{9} + \beta_{6} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{11} + \beta_{8} + \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 16 q^{5} - 3 q^{7} - 4 q^{9} + 2 q^{11} - 18 q^{13} - 4 q^{15} + 20 q^{17} - q^{19} - 2 q^{21} - 12 q^{23} + 16 q^{25} + 4 q^{27} - 7 q^{29} - 3 q^{31} + 3 q^{33} + 3 q^{35} - 44 q^{37}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 15 x^{13} + 101 x^{12} - 273 x^{11} + 1747 x^{10} - 2144 x^{9} + \cdots + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 70\!\cdots\!14 \nu^{15} + \cdots + 78\!\cdots\!73 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!21 \nu^{15} + \cdots + 13\!\cdots\!61 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!82 \nu^{15} + \cdots + 11\!\cdots\!35 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!90 \nu^{15} + \cdots + 60\!\cdots\!53 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!51 \nu^{15} + \cdots + 23\!\cdots\!69 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\!\cdots\!22 \nu^{15} + \cdots - 43\!\cdots\!11 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55\!\cdots\!16 \nu^{15} + \cdots - 42\!\cdots\!12 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 63\!\cdots\!90 \nu^{15} + \cdots + 76\!\cdots\!73 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!93 \nu^{15} + \cdots + 65\!\cdots\!56 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82\!\cdots\!93 \nu^{15} + \cdots - 22\!\cdots\!79 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!63 \nu^{15} + \cdots + 37\!\cdots\!90 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!30 \nu^{15} + \cdots + 28\!\cdots\!56 ) / 65\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!91 \nu^{15} + \cdots - 48\!\cdots\!96 ) / 32\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!64 \nu^{15} + \cdots - 23\!\cdots\!57 ) / 10\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 95\!\cdots\!87 \nu^{15} + \cdots + 15\!\cdots\!52 ) / 65\!\cdots\!14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{8} - \beta_{6} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{9} - 5\beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_{8} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{15} + 11 \beta_{12} - 11 \beta_{11} + 61 \beta_{10} + 49 \beta_{9} + 5 \beta_{6} + 49 \beta_{5} + \cdots - 61 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{14} + 16 \beta_{13} - 6 \beta_{11} + 45 \beta_{10} + 10 \beta_{9} + 13 \beta_{8} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 125 \beta_{15} - 125 \beta_{14} + 122 \beta_{13} - 116 \beta_{12} - 125 \beta_{10} - 416 \beta_{9} + \cdots + 133 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 224 \beta_{15} - 18 \beta_{13} - 220 \beta_{12} + 119 \beta_{11} - 935 \beta_{10} - 491 \beta_{9} + \cdots + 224 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 100 \beta_{15} + 1480 \beta_{14} - 100 \beta_{13} - 1518 \beta_{12} + 1257 \beta_{11} - 5962 \beta_{10} + \cdots - 560 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3249 \beta_{15} + 2978 \beta_{14} - 1086 \beta_{12} + 1086 \beta_{11} + 9578 \beta_{10} + 16457 \beta_{9} + \cdots - 9578 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2130 \beta_{14} + 2130 \beta_{13} + 2878 \beta_{11} + 10377 \beta_{10} - 13940 \beta_{9} + \cdots + 68596 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 42634 \beta_{15} - 42634 \beta_{14} + 3984 \beta_{13} + 12086 \beta_{12} - 42634 \beta_{10} + \cdots + 180502 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 37575 \beta_{15} - 183608 \beta_{13} + 185669 \beta_{12} - 28625 \beta_{11} - 221980 \beta_{10} + \cdots + 37575 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 495569 \beta_{15} + 554362 \beta_{14} - 495569 \beta_{13} + 438837 \beta_{12} - 135826 \beta_{11} + \cdots - 2327765 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2748051 \beta_{15} + 599623 \beta_{14} + 1788685 \beta_{12} - 1788685 \beta_{11} + 11849418 \beta_{10} + \cdots - 11849418 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 6313233 \beta_{14} + 6313233 \beta_{13} - 3822932 \beta_{11} + 24005758 \beta_{10} + 1519044 \beta_{9} + \cdots + 10421677 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{9}\)

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
1.10690 + 3.40668i
0.188157 + 0.579088i
−0.356846 − 1.09826i
−0.747224 − 2.29972i
2.73953 − 1.99039i
0.698098 − 0.507198i
0.443314 − 0.322087i
−2.57193 + 1.86861i
2.73953 + 1.99039i
0.698098 + 0.507198i
0.443314 + 0.322087i
−2.57193 − 1.86861i
1.10690 − 3.40668i
0.188157 − 0.579088i
−0.356846 + 1.09826i
−0.747224 + 2.29972i
0 0.809017 + 0.587785i 0 −1.00000 0 −1.10690 + 3.40668i 0 0.309017 + 0.951057i 0
481.2 0 0.809017 + 0.587785i 0 −1.00000 0 −0.188157 + 0.579088i 0 0.309017 + 0.951057i 0
481.3 0 0.809017 + 0.587785i 0 −1.00000 0 0.356846 − 1.09826i 0 0.309017 + 0.951057i 0
481.4 0 0.809017 + 0.587785i 0 −1.00000 0 0.747224 − 2.29972i 0 0.309017 + 0.951057i 0
721.1 0 −0.309017 − 0.951057i 0 −1.00000 0 −2.73953 − 1.99039i 0 −0.809017 + 0.587785i 0
721.2 0 −0.309017 − 0.951057i 0 −1.00000 0 −0.698098 − 0.507198i 0 −0.809017 + 0.587785i 0
721.3 0 −0.309017 − 0.951057i 0 −1.00000 0 −0.443314 − 0.322087i 0 −0.809017 + 0.587785i 0
721.4 0 −0.309017 − 0.951057i 0 −1.00000 0 2.57193 + 1.86861i 0 −0.809017 + 0.587785i 0
841.1 0 −0.309017 + 0.951057i 0 −1.00000 0 −2.73953 + 1.99039i 0 −0.809017 − 0.587785i 0
841.2 0 −0.309017 + 0.951057i 0 −1.00000 0 −0.698098 + 0.507198i 0 −0.809017 − 0.587785i 0
841.3 0 −0.309017 + 0.951057i 0 −1.00000 0 −0.443314 + 0.322087i 0 −0.809017 − 0.587785i 0
841.4 0 −0.309017 + 0.951057i 0 −1.00000 0 2.57193 − 1.86861i 0 −0.809017 − 0.587785i 0
901.1 0 0.809017 − 0.587785i 0 −1.00000 0 −1.10690 − 3.40668i 0 0.309017 − 0.951057i 0
901.2 0 0.809017 − 0.587785i 0 −1.00000 0 −0.188157 − 0.579088i 0 0.309017 − 0.951057i 0
901.3 0 0.809017 − 0.587785i 0 −1.00000 0 0.356846 + 1.09826i 0 0.309017 − 0.951057i 0
901.4 0 0.809017 − 0.587785i 0 −1.00000 0 0.747224 + 2.29972i 0 0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 481.4
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.b ✓ 16
31.d even 5 1 inner 1860.2.z.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.z.b ✓ 16 1.a even 1 1 trivial
1860.2.z.b ✓ 16 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 3 T_{7}^{15} + 14 T_{7}^{14} + 15 T_{7}^{13} + 101 T_{7}^{12} + 273 T_{7}^{11} + 1747 T_{7}^{10} + \cdots + 961 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 3 T^{15} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{16} + 18 T^{15} + \cdots + 249001 \) Copy content Toggle raw display
$17$ \( T^{16} - 20 T^{15} + \cdots + 57820816 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 175854121 \) Copy content Toggle raw display
$23$ \( T^{16} + 12 T^{15} + \cdots + 20106256 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 20541768976 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( (T^{8} + 22 T^{7} + \cdots + 1891)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 119718768016 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 22232301025 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1143565029376 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3155742976 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14098237696 \) Copy content Toggle raw display
$61$ \( (T^{8} + 29 T^{7} + \cdots - 86155)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 11 T^{7} + \cdots + 119366791)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1297594374400 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5438177536 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1346082721 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2667102736 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 6589304177296 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 46279706920561 \) Copy content Toggle raw display
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