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Properties

Label 2700.2.s.d
Level $2700$
Weight $2$
Character orbit 2700.s
Analytic conductor $21.560$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2700,2,Mod(1549,2700)] 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("2700.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,-6] 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: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.1333317747165888577536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: no (minimal twist has level 900)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{7} + ( - \beta_{3} + \beta_{2}) q^{11} + (\beta_{13} + \beta_{8} + \cdots - \beta_{6}) q^{13} + ( - 2 \beta_{13} - \beta_{12}) q^{17} + (\beta_{11} + 1) q^{19} + (\beta_{14} - \beta_{13} + \cdots + \beta_{6}) q^{23}+ \cdots + ( - \beta_{15} + 2 \beta_{13} + \cdots - 2 \beta_{6}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{12} + 7\nu^{10} + 17\nu^{8} + 19\nu^{6} + 41\nu^{4} + 112\nu^{2} + 112 ) / 48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 5\nu^{12} + 5\nu^{10} + 55\nu^{8} + 5\nu^{6} - 230\nu^{4} - 120\nu^{2} + 288 ) / 480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{14} - 25\nu^{12} - 15\nu^{10} - 45\nu^{8} - 135\nu^{6} - 180\nu^{4} + 64 ) / 960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - \nu^{12} - 7\nu^{10} - 5\nu^{8} - 79\nu^{6} - 56\nu^{4} - 224\nu^{2} - 256 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} + 5\nu^{12} - 5\nu^{10} + 65\nu^{8} + 115\nu^{6} + 260\nu^{4} + 200\nu^{2} + 1008 ) / 240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{15} - 5\nu^{13} + 45\nu^{11} + 135\nu^{9} + 405\nu^{7} - 180\nu^{5} + 880\nu^{3} + 896\nu ) / 5760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -19\nu^{15} - 5\nu^{13} - 195\nu^{11} - 345\nu^{9} + 885\nu^{7} + 240\nu^{5} - 3200\nu^{3} + 11072\nu ) / 5760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{15} - 31\nu^{13} + 39\nu^{11} - 75\nu^{9} - 33\nu^{7} + 276\nu^{5} + 224\nu^{3} - 512\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4\nu^{15} + 5\nu^{13} - 15\nu^{11} + 15\nu^{9} + 165\nu^{7} + 195\nu^{5} - 10\nu^{3} - 152\nu ) / 720 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9\nu^{14} + 7\nu^{12} + \nu^{10} + 83\nu^{8} + 121\nu^{6} + 8\nu^{4} + 304\nu^{2} + 832 ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -9\nu^{14} - 19\nu^{12} - 37\nu^{10} - 143\nu^{8} - 301\nu^{6} - 356\nu^{4} - 448\nu^{2} - 1216 ) / 192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -41\nu^{15} - 295\nu^{13} - 705\nu^{11} - 915\nu^{9} - 1785\nu^{7} - 4200\nu^{5} - 5200\nu^{3} - 3392\nu ) / 5760 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -11\nu^{15} - 10\nu^{13} - 30\nu^{11} - 150\nu^{9} - 210\nu^{7} - 285\nu^{5} - 1000\nu^{3} - 1472\nu ) / 1440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3\nu^{15} + 15\nu^{13} + 25\nu^{11} + 115\nu^{9} + 185\nu^{7} + 330\nu^{5} + 240\nu^{3} + 896\nu ) / 320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -21\nu^{15} - 15\nu^{13} - 25\nu^{11} - 75\nu^{9} - 225\nu^{7} + 260\nu^{5} - 80\nu^{3} - 512\nu ) / 640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{9} + \beta_{7} - \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2\beta_{14} - 5\beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + 3\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{5} + 2\beta_{4} + 4\beta_{3} - 3\beta_{2} - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} + 2\beta_{9} + 2\beta_{8} - 11\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{11} - \beta_{10} - 6\beta_{4} + 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{15} - \beta_{14} + 4\beta_{13} - \beta_{12} + 7\beta_{9} + \beta_{8} + 2\beta_{7} + 23\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{11} + \beta_{10} + 5\beta_{5} + 7\beta_{4} - \beta_{3} + 9\beta_{2} + 3\beta _1 - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{15} + 14\beta_{14} + 11\beta_{13} + 12\beta_{12} - 7\beta_{9} - 2\beta_{8} - 13\beta_{7} + 6\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -11\beta_{11} - 12\beta_{10} - 18\beta_{5} + 4\beta_{4} + 13\beta_{3} + 4\beta_{2} + 6\beta _1 + 46 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -12\beta_{15} + 6\beta_{14} + 31\beta_{13} - 10\beta_{12} - 29\beta_{9} + 24\beta_{8} + \beta_{7} + 13\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9\beta_{11} + 5\beta_{10} - \beta_{4} - 61\beta_{3} - \beta_{2} + 5\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 15\beta_{15} - 30\beta_{14} - 7\beta_{13} - 51\beta_{12} + 11\beta_{9} - 75\beta_{8} + 16\beta_{7} - 19\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -35\beta_{11} + 25\beta_{10} - 45\beta_{5} + 50\beta_{4} + 140\beta_{3} - 85\beta_{2} - 20\beta _1 + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 65 \beta_{15} - 5 \beta_{14} - 83 \beta_{13} + 20 \beta_{12} + 22 \beta_{9} + 50 \beta_{8} + \cdots - 393 \beta_{6} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
−1.15347 + 0.818235i
−1.27069 − 0.620769i
−0.485097 + 1.32841i
0.263711 + 1.38941i
−0.263711 − 1.38941i
0.485097 − 1.32841i
1.27069 + 0.620769i
1.15347 − 0.818235i
−1.15347 − 0.818235i
−1.27069 + 0.620769i
−0.485097 − 1.32841i
0.263711 − 1.38941i
−0.263711 + 1.38941i
0.485097 + 1.32841i
1.27069 − 0.620769i
1.15347 + 0.818235i
0 0 0 0 0 −4.32643 + 2.49787i 0 0 0
1549.2 0 0 0 0 0 −2.94604 + 1.70089i 0 0 0
1549.3 0 0 0 0 0 −0.589266 + 0.340213i 0 0 0
1549.4 0 0 0 0 0 −0.0748933 + 0.0432397i 0 0 0
1549.5 0 0 0 0 0 0.0748933 − 0.0432397i 0 0 0
1549.6 0 0 0 0 0 0.589266 − 0.340213i 0 0 0
1549.7 0 0 0 0 0 2.94604 − 1.70089i 0 0 0
1549.8 0 0 0 0 0 4.32643 − 2.49787i 0 0 0
2449.1 0 0 0 0 0 −4.32643 − 2.49787i 0 0 0
2449.2 0 0 0 0 0 −2.94604 − 1.70089i 0 0 0
2449.3 0 0 0 0 0 −0.589266 − 0.340213i 0 0 0
2449.4 0 0 0 0 0 −0.0748933 − 0.0432397i 0 0 0
2449.5 0 0 0 0 0 0.0748933 + 0.0432397i 0 0 0
2449.6 0 0 0 0 0 0.589266 + 0.340213i 0 0 0
2449.7 0 0 0 0 0 2.94604 + 1.70089i 0 0 0
2449.8 0 0 0 0 0 4.32643 + 2.49787i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.2.s.d 16
3.b odd 2 1 900.2.s.d 16
5.b even 2 1 inner 2700.2.s.d 16
5.c odd 4 1 2700.2.i.d 8
5.c odd 4 1 2700.2.i.e 8
9.c even 3 1 inner 2700.2.s.d 16
9.c even 3 1 8100.2.d.s 8
9.d odd 6 1 900.2.s.d 16
9.d odd 6 1 8100.2.d.q 8
15.d odd 2 1 900.2.s.d 16
15.e even 4 1 900.2.i.d ✓ 8
15.e even 4 1 900.2.i.e yes 8
45.h odd 6 1 900.2.s.d 16
45.h odd 6 1 8100.2.d.q 8
45.j even 6 1 inner 2700.2.s.d 16
45.j even 6 1 8100.2.d.s 8
45.k odd 12 1 2700.2.i.d 8
45.k odd 12 1 2700.2.i.e 8
45.k odd 12 1 8100.2.a.y 4
45.k odd 12 1 8100.2.a.ba 4
45.l even 12 1 900.2.i.d ✓ 8
45.l even 12 1 900.2.i.e yes 8
45.l even 12 1 8100.2.a.x 4
45.l even 12 1 8100.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.i.d ✓ 8 15.e even 4 1
900.2.i.d ✓ 8 45.l even 12 1
900.2.i.e yes 8 15.e even 4 1
900.2.i.e yes 8 45.l even 12 1
900.2.s.d 16 3.b odd 2 1
900.2.s.d 16 9.d odd 6 1
900.2.s.d 16 15.d odd 2 1
900.2.s.d 16 45.h odd 6 1
2700.2.i.d 8 5.c odd 4 1
2700.2.i.d 8 45.k odd 12 1
2700.2.i.e 8 5.c odd 4 1
2700.2.i.e 8 45.k odd 12 1
2700.2.s.d 16 1.a even 1 1 trivial
2700.2.s.d 16 5.b even 2 1 inner
2700.2.s.d 16 9.c even 3 1 inner
2700.2.s.d 16 45.j even 6 1 inner
8100.2.a.x 4 45.l even 12 1
8100.2.a.y 4 45.k odd 12 1
8100.2.a.z 4 45.l even 12 1
8100.2.a.ba 4 45.k odd 12 1
8100.2.d.q 8 9.d odd 6 1
8100.2.d.q 8 45.h odd 6 1
8100.2.d.s 8 9.c even 3 1
8100.2.d.s 8 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7}^{16} - 37T_{7}^{14} + 1063T_{7}^{12} - 11050T_{7}^{10} + 88603T_{7}^{8} - 41542T_{7}^{6} + 18190T_{7}^{4} - 136T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{8} + 3T_{11}^{7} + 24T_{11}^{6} + 45T_{11}^{5} + 387T_{11}^{4} + 837T_{11}^{3} + 1620T_{11}^{2} + 1215T_{11} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 37 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 729)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 40 T^{14} + \cdots + 3418801 \) Copy content Toggle raw display
$17$ \( (T^{8} + 57 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 27 T^{2} + \cdots - 23)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 147 T^{14} + \cdots + 531441 \) Copy content Toggle raw display
$29$ \( (T^{8} + 9 T^{7} + \cdots + 1347921)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 79 T^{6} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 9 T^{7} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 172 T^{14} + \cdots + 1874161 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 996005996001 \) Copy content Toggle raw display
$53$ \( (T^{8} + 300 T^{6} + \cdots + 7733961)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 15 T^{7} + \cdots + 101425041)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - T^{7} + \cdots + 100489)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 66625573677601 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 729)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 124 T^{6} + \cdots + 265225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} + \cdots + 240033049)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 187213570125201 \) Copy content Toggle raw display
$89$ \( (T^{4} - 3 T^{3} + \cdots + 5913)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16354914662641 \) Copy content Toggle raw display
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