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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2610,2,Mod(1061,2610)] 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("2610.1061"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2610, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2610.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-20,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.8409549276\)
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: \( x^{20} - 8 x^{19} + 52 x^{18} - 156 x^{17} + 296 x^{16} - 124 x^{15} - 456 x^{14} - 1268 x^{13} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - \beta_{9} q^{4} - q^{5} + \beta_{14} q^{7} + \beta_1 q^{8} + \beta_{10} q^{10} - \beta_{11} q^{11} + ( - \beta_{17} + \beta_{16} + \beta_{15} + \cdots + 1) q^{13} - \beta_{16} q^{14}+ \cdots + (\beta_{19} + \beta_{17} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{5} + 8 q^{7} - 4 q^{11} + 8 q^{14} - 20 q^{16} + 8 q^{17} + 8 q^{19} + 20 q^{25} + 4 q^{26} + 20 q^{29} - 8 q^{35} + 4 q^{37} - 16 q^{38} - 8 q^{43} + 4 q^{44} + 12 q^{46} + 24 q^{47} + 20 q^{49}+ \cdots - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 52 x^{18} - 156 x^{17} + 296 x^{16} - 124 x^{15} - 456 x^{14} - 1268 x^{13} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\!\cdots\!43 \nu^{19} + \cdots - 25\!\cdots\!68 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!51 \nu^{19} + \cdots - 29\!\cdots\!64 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!38 \nu^{19} + \cdots + 18\!\cdots\!88 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!99 \nu^{19} + \cdots - 13\!\cdots\!76 ) / 63\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36\!\cdots\!73 \nu^{19} + \cdots + 33\!\cdots\!64 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 31\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 98\!\cdots\!97 \nu^{19} + \cdots + 32\!\cdots\!36 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49\!\cdots\!31 \nu^{19} + \cdots - 33\!\cdots\!40 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!13 \nu^{19} + \cdots - 25\!\cdots\!92 ) / 25\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!33 \nu^{19} + \cdots + 41\!\cdots\!88 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!43 \nu^{19} + \cdots - 15\!\cdots\!80 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 90\!\cdots\!59 \nu^{19} + \cdots + 39\!\cdots\!16 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!95 \nu^{19} + \cdots - 39\!\cdots\!16 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!93 \nu^{19} + \cdots - 56\!\cdots\!64 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!01 \nu^{19} + \cdots - 77\!\cdots\!48 ) / 53\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 41\!\cdots\!73 \nu^{19} + \cdots - 82\!\cdots\!56 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 43\!\cdots\!43 \nu^{19} + \cdots - 37\!\cdots\!80 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 47\!\cdots\!83 \nu^{19} + \cdots + 49\!\cdots\!96 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 26\!\cdots\!50 \nu^{19} + \cdots - 83\!\cdots\!44 ) / 53\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - 2 \beta_{16} + 4 \beta_{14} - 3 \beta_{13} + \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{19} - 2 \beta_{18} - 4 \beta_{16} + 4 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + \cdots - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{19} + 19 \beta_{18} - 24 \beta_{17} + 12 \beta_{16} + 31 \beta_{15} - 54 \beta_{14} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16 \beta_{19} + 112 \beta_{18} - 16 \beta_{17} + 198 \beta_{16} - 22 \beta_{15} - 332 \beta_{14} + \cdots + 829 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 378 \beta_{19} + 251 \beta_{17} + 380 \beta_{16} - 1002 \beta_{15} + 545 \beta_{13} - 545 \beta_{12} + \cdots + 2665 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 474 \beta_{19} - 2894 \beta_{18} + 1272 \beta_{17} - 3612 \beta_{16} - 2894 \beta_{15} + 8818 \beta_{14} + \cdots - 9123 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4897 \beta_{19} - 11201 \beta_{18} - 24346 \beta_{16} + 13145 \beta_{15} + 34190 \beta_{14} + \cdots - 96056 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 28268 \beta_{19} + 30180 \beta_{18} - 28268 \beta_{17} + 120518 \beta_{15} - 90338 \beta_{14} + \cdots - 126035 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 380732 \beta_{18} - 104577 \beta_{17} + 587290 \beta_{16} + 139212 \beta_{15} - 1174580 \beta_{14} + \cdots + 1713437 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 636492 \beta_{19} + 722324 \beta_{18} + 260710 \beta_{17} + 2190636 \beta_{16} - 2383430 \beta_{15} + \cdots + 9230859 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2340819 \beta_{19} - 6504555 \beta_{18} + 3324416 \beta_{17} - 5805728 \beta_{16} - 12310283 \beta_{15} + \cdots - 5805728 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 6010884 \beta_{19} - 41460568 \beta_{18} + 6010884 \beta_{17} - 74152758 \beta_{16} + 10928718 \beta_{15} + \cdots - 237211213 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 76042674 \beta_{19} - 53690827 \beta_{17} - 138280820 \beta_{16} + 334083698 \beta_{15} + \cdots - 740880401 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 139585570 \beta_{19} + 985268774 \beta_{18} - 337413560 \beta_{17} + 1244614548 \beta_{16} + \cdots + 2820754651 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 1247494873 \beta_{19} + 3671577689 \beta_{18} + 7917242562 \beta_{16} - 4245664873 \beta_{15} + \cdots + 27869948992 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 7876752940 \beta_{19} - 9668027220 \beta_{18} + 7876752940 \beta_{17} - 39138343430 \beta_{15} + \cdots + 36981438571 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 122827566156 \beta_{18} + 29186944169 \beta_{17} - 187247947362 \beta_{16} - 45273108804 \beta_{15} + \cdots - 518593263797 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 184725543204 \beta_{19} - 228535140284 \beta_{18} - 76505007062 \beta_{17} - 696606943964 \beta_{16} + \cdots - 2796501450435 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2610\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(1567\)
\(\chi(n)\) \(\beta_{9}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1061.1
0.123596 − 0.0511953i
1.43337 − 0.593721i
1.26667 − 0.524671i
−2.59380 + 1.07439i
1.77016 − 0.733226i
1.85871 + 4.48732i
−0.253296 − 0.611510i
−1.02607 − 2.47715i
0.0456623 + 0.110239i
1.37499 + 3.31952i
0.123596 + 0.0511953i
1.43337 + 0.593721i
1.26667 + 0.524671i
−2.59380 − 1.07439i
1.77016 + 0.733226i
1.85871 − 4.48732i
−0.253296 + 0.611510i
−1.02607 + 2.47715i
0.0456623 − 0.110239i
1.37499 − 3.31952i
−0.707107 − 0.707107i 0 1.00000i −1.00000 0 −4.31820 0.707107 − 0.707107i 0 0.707107 + 0.707107i
1061.2 −0.707107 − 0.707107i 0 1.00000i −1.00000 0 −2.72070 0.707107 − 0.707107i 0 0.707107 + 0.707107i
1061.3 −0.707107 − 0.707107i 0 1.00000i −1.00000 0 0.472866 0.707107 − 0.707107i 0 0.707107 + 0.707107i
1061.4 −0.707107 − 0.707107i 0 1.00000i −1.00000 0 2.65003 0.707107 − 0.707107i 0 0.707107 + 0.707107i
1061.5 −0.707107 − 0.707107i 0 1.00000i −1.00000 0 3.08758 0.707107 − 0.707107i 0 0.707107 + 0.707107i
1061.6 0.707107 + 0.707107i 0 1.00000i −1.00000 0 −2.11873 −0.707107 + 0.707107i 0 −0.707107 − 0.707107i
1061.7 0.707107 + 0.707107i 0 1.00000i −1.00000 0 −0.595386 −0.707107 + 0.707107i 0 −0.707107 − 0.707107i
1061.8 0.707107 + 0.707107i 0 1.00000i −1.00000 0 −0.296182 −0.707107 + 0.707107i 0 −0.707107 − 0.707107i
1061.9 0.707107 + 0.707107i 0 1.00000i −1.00000 0 3.04827 −0.707107 + 0.707107i 0 −0.707107 − 0.707107i
1061.10 0.707107 + 0.707107i 0 1.00000i −1.00000 0 4.79045 −0.707107 + 0.707107i 0 −0.707107 − 0.707107i
1781.1 −0.707107 + 0.707107i 0 − 1.00000i −1.00000 0 −4.31820 0.707107 + 0.707107i 0 0.707107 − 0.707107i
1781.2 −0.707107 + 0.707107i 0 − 1.00000i −1.00000 0 −2.72070 0.707107 + 0.707107i 0 0.707107 − 0.707107i
1781.3 −0.707107 + 0.707107i 0 − 1.00000i −1.00000 0 0.472866 0.707107 + 0.707107i 0 0.707107 − 0.707107i
1781.4 −0.707107 + 0.707107i 0 − 1.00000i −1.00000 0 2.65003 0.707107 + 0.707107i 0 0.707107 − 0.707107i
1781.5 −0.707107 + 0.707107i 0 − 1.00000i −1.00000 0 3.08758 0.707107 + 0.707107i 0 0.707107 − 0.707107i
1781.6 0.707107 − 0.707107i 0 − 1.00000i −1.00000 0 −2.11873 −0.707107 − 0.707107i 0 −0.707107 + 0.707107i
1781.7 0.707107 − 0.707107i 0 − 1.00000i −1.00000 0 −0.595386 −0.707107 − 0.707107i 0 −0.707107 + 0.707107i
1781.8 0.707107 − 0.707107i 0 − 1.00000i −1.00000 0 −0.296182 −0.707107 − 0.707107i 0 −0.707107 + 0.707107i
1781.9 0.707107 − 0.707107i 0 − 1.00000i −1.00000 0 3.04827 −0.707107 − 0.707107i 0 −0.707107 + 0.707107i
1781.10 0.707107 − 0.707107i 0 − 1.00000i −1.00000 0 4.79045 −0.707107 − 0.707107i 0 −0.707107 + 0.707107i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1061.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2610.2.t.b ✓ 20
3.b odd 2 1 2610.2.t.d yes 20
29.c odd 4 1 2610.2.t.d yes 20
87.f even 4 1 inner 2610.2.t.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2610.2.t.b ✓ 20 1.a even 1 1 trivial
2610.2.t.b ✓ 20 87.f even 4 1 inner
2610.2.t.d yes 20 3.b odd 2 1
2610.2.t.d yes 20 29.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2610, [\chi])\):

\( T_{7}^{10} - 4 T_{7}^{9} - 32 T_{7}^{8} + 124 T_{7}^{7} + 298 T_{7}^{6} - 1096 T_{7}^{5} - 1052 T_{7}^{4} + \cdots - 248 \) Copy content Toggle raw display
\( T_{11}^{20} + 4 T_{11}^{19} + 8 T_{11}^{18} + 792 T_{11}^{16} + 3280 T_{11}^{15} + 6784 T_{11}^{14} + \cdots + 5345344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T + 1)^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} - 4 T^{9} + \cdots - 248)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 4 T^{19} + \cdots + 5345344 \) Copy content Toggle raw display
$13$ \( T^{20} + 164 T^{18} + \cdots + 67108864 \) Copy content Toggle raw display
$17$ \( T^{20} - 8 T^{19} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 238643974144 \) Copy content Toggle raw display
$23$ \( T^{20} + 196 T^{18} + \cdots + 53465344 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 420707233300201 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10382794816 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17815152640000 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 247369984 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 2620565241856 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14540860309504 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 18913174749184 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 365980162449664 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} + \cdots - 12551296)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 50925236260864 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 31017503480896 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
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