Properties

Label 2700.2.s.c
Level $2700$
Weight $2$
Character orbit 2700.s
Analytic conductor $21.560$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2700,2,Mod(1549,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{3}) q^{7} + (\beta_{8} + \beta_1) q^{11} + (\beta_{9} - \beta_{7} + \beta_{3} + \beta_{2}) q^{13} + \beta_{7} q^{17} + ( - \beta_1 - 2) q^{19} + ( - \beta_{10} + \beta_{9} + \beta_{3}) q^{23} + ( - \beta_{11} - \beta_{5}) q^{29} + (\beta_{8} - 2 \beta_{5} - 2) q^{31} + ( - 2 \beta_{10} + 3 \beta_{9} + \beta_{7} + 2 \beta_{6} + 2 \beta_{3}) q^{37} + (\beta_{11} + \beta_{8} + \beta_{5} - \beta_{4} + 1) q^{41} + ( - \beta_{3} - \beta_{2}) q^{43} + ( - \beta_{6} - 3 \beta_{3} - \beta_{2}) q^{47} + (\beta_{8} + 6 \beta_{5} + 6) q^{49} + (2 \beta_{10} - 2 \beta_{9} - 2 \beta_{6} - 2 \beta_{3}) q^{53} + (\beta_{11} - 2 \beta_{5} - \beta_{4} - 2) q^{59} + ( - \beta_{11} - \beta_{8} + 7 \beta_{5} - \beta_1) q^{61} + (\beta_{10} - 2 \beta_{9} + \beta_{7} - 2 \beta_{3} - \beta_{2}) q^{67} + ( - \beta_{4} - \beta_1 - 8) q^{71} - 4 \beta_{9} q^{73} + ( - 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{7} + 2 \beta_{3} - 3 \beta_{2}) q^{77} - 2 \beta_{5} q^{79} + (\beta_{6} - 3 \beta_{3}) q^{83} - 3 q^{89} + ( - 2 \beta_{4} - 3 \beta_1) q^{91} + (2 \beta_{6} + 4 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{19} + 6 q^{29} - 12 q^{31} + 6 q^{41} + 36 q^{49} - 12 q^{59} - 42 q^{61} - 96 q^{71} + 12 q^{79} - 36 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 559 \nu^{11} + 1507 \nu^{10} - 1502 \nu^{9} + 1290 \nu^{8} + 7427 \nu^{7} - 10616 \nu^{6} - 9422 \nu^{5} - 35640 \nu^{4} - 58068 \nu^{3} + 134808 \nu^{2} - 28638 \nu - 9288 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} - 11380 \nu^{5} - 57708 \nu^{4} - 98118 \nu^{3} + 132080 \nu^{2} + \cdots - 15984 ) / 51972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1969 \nu^{11} + 2148 \nu^{10} - 4654 \nu^{9} + 11338 \nu^{8} + 17186 \nu^{7} + 23628 \nu^{6} + 17534 \nu^{5} - 254444 \nu^{4} - 441804 \nu^{3} - 378072 \nu^{2} + \cdots + 442176 ) / 51972 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 5172 ) / 1704 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7540 \nu^{11} - 18514 \nu^{10} + 22979 \nu^{9} - 17400 \nu^{8} - 111056 \nu^{7} + 108746 \nu^{6} + 83576 \nu^{5} + 448092 \nu^{4} + 766926 \nu^{3} - 607272 \nu^{2} + \cdots + 125280 ) / 103944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5765 \nu^{11} + 3533 \nu^{10} - 2602 \nu^{9} + 2490 \nu^{8} + 88123 \nu^{7} + 86504 \nu^{6} - 51058 \nu^{5} - 522216 \nu^{4} - 1425948 \nu^{3} - 1357440 \nu^{2} + \cdots - 355320 ) / 51972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2887 \nu^{11} + 2220 \nu^{10} - 1402 \nu^{9} + 742 \nu^{8} + 46028 \nu^{7} + 34644 \nu^{6} - 35302 \nu^{5} - 251666 \nu^{4} - 690756 \nu^{3} - 566352 \nu^{2} + \cdots - 86112 ) / 25986 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20419 \nu^{11} + 2860 \nu^{10} + 2915 \nu^{9} - 3180 \nu^{8} + 333594 \nu^{7} + 426930 \nu^{6} - 94954 \nu^{5} - 1995204 \nu^{4} - 5955290 \nu^{3} + \cdots - 1546488 ) / 103944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17703 \nu^{11} - 13116 \nu^{10} + 5698 \nu^{9} + 458 \nu^{8} - 285613 \nu^{7} - 212436 \nu^{6} + 218026 \nu^{5} + 1570478 \nu^{4} + 4134404 \nu^{3} + 3386076 \nu^{2} + \cdots + 765888 ) / 25986 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} - \beta_{2} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 9\beta_{3} - 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} - 8\beta_{8} + 4\beta_{7} + 12\beta_{5} + 6\beta_{3} - 8\beta_{2} - 4\beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} - 7\beta_{8} + 32\beta_{5} - 2\beta_{4} + 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{11} - 4 \beta_{10} - 38 \beta_{9} - 18 \beta_{8} - 36 \beta_{7} + 2 \beta_{6} + 40 \beta_{5} - 2 \beta_{4} - 17 \beta_{3} + 18 \beta_{2} + 18 \beta _1 + 80 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{10} - 65\beta_{9} - 40\beta_{7} + 16\beta_{6} + 16\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4 \beta_{11} - 8 \beta_{10} - 52 \beta_{9} - 43 \beta_{8} - 43 \beta_{7} + 16 \beta_{6} + 112 \beta_{5} + 4 \beta_{4} + 68 \beta_{3} - 43 \beta_{2} - 86 \beta _1 - 112 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35\beta_{11} - 214\beta_{8} + 704\beta_{5} - 214\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 48 \beta_{11} - 48 \beta_{10} - 273 \beta_{9} - 424 \beta_{8} - 212 \beta_{7} - 48 \beta_{6} + 1188 \beta_{5} - 24 \beta_{4} - 594 \beta_{3} + 424 \beta_{2} - 212 \beta _1 + 594 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -328\beta_{10} - 1580\beta_{9} - 1114\beta_{7} - 1580\beta_{3} + 1114\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 131 \beta_{11} - 524 \beta_{10} - 2818 \beta_{9} + 1062 \beta_{8} - 2124 \beta_{7} + 262 \beta_{6} - 3080 \beta_{5} + 262 \beta_{4} - 1147 \beta_{3} + 1062 \beta_{2} - 1062 \beta _1 - 6160 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1 - \beta_{5}\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
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
−0.403293 1.50511i
−0.673288 + 0.180407i
0.583700 + 2.17840i
2.17840 0.583700i
−0.180407 0.673288i
−1.50511 + 0.403293i
−0.403293 + 1.50511i
−0.673288 0.180407i
0.583700 2.17840i
2.17840 + 0.583700i
−0.180407 + 0.673288i
−1.50511 0.403293i
0 0 0 0 0 −3.55142 + 2.05042i 0 0 0
1549.2 0 0 0 0 0 −3.32123 + 1.91751i 0 0 0
1549.3 0 0 0 0 0 −2.36788 + 1.36710i 0 0 0
1549.4 0 0 0 0 0 2.36788 1.36710i 0 0 0
1549.5 0 0 0 0 0 3.32123 1.91751i 0 0 0
1549.6 0 0 0 0 0 3.55142 2.05042i 0 0 0
2449.1 0 0 0 0 0 −3.55142 2.05042i 0 0 0
2449.2 0 0 0 0 0 −3.32123 1.91751i 0 0 0
2449.3 0 0 0 0 0 −2.36788 1.36710i 0 0 0
2449.4 0 0 0 0 0 2.36788 + 1.36710i 0 0 0
2449.5 0 0 0 0 0 3.32123 + 1.91751i 0 0 0
2449.6 0 0 0 0 0 3.55142 + 2.05042i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.6
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.c 12
3.b odd 2 1 900.2.s.c 12
5.b even 2 1 inner 2700.2.s.c 12
5.c odd 4 1 540.2.i.b 6
5.c odd 4 1 2700.2.i.c 6
9.c even 3 1 inner 2700.2.s.c 12
9.c even 3 1 8100.2.d.o 6
9.d odd 6 1 900.2.s.c 12
9.d odd 6 1 8100.2.d.p 6
15.d odd 2 1 900.2.s.c 12
15.e even 4 1 180.2.i.b 6
15.e even 4 1 900.2.i.c 6
20.e even 4 1 2160.2.q.i 6
45.h odd 6 1 900.2.s.c 12
45.h odd 6 1 8100.2.d.p 6
45.j even 6 1 inner 2700.2.s.c 12
45.j even 6 1 8100.2.d.o 6
45.k odd 12 1 540.2.i.b 6
45.k odd 12 1 1620.2.a.j 3
45.k odd 12 1 2700.2.i.c 6
45.k odd 12 1 8100.2.a.u 3
45.l even 12 1 180.2.i.b 6
45.l even 12 1 900.2.i.c 6
45.l even 12 1 1620.2.a.i 3
45.l even 12 1 8100.2.a.v 3
60.l odd 4 1 720.2.q.k 6
180.v odd 12 1 720.2.q.k 6
180.v odd 12 1 6480.2.a.bt 3
180.x even 12 1 2160.2.q.i 6
180.x even 12 1 6480.2.a.bw 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 15.e even 4 1
180.2.i.b 6 45.l even 12 1
540.2.i.b 6 5.c odd 4 1
540.2.i.b 6 45.k odd 12 1
720.2.q.k 6 60.l odd 4 1
720.2.q.k 6 180.v odd 12 1
900.2.i.c 6 15.e even 4 1
900.2.i.c 6 45.l even 12 1
900.2.s.c 12 3.b odd 2 1
900.2.s.c 12 9.d odd 6 1
900.2.s.c 12 15.d odd 2 1
900.2.s.c 12 45.h odd 6 1
1620.2.a.i 3 45.l even 12 1
1620.2.a.j 3 45.k odd 12 1
2160.2.q.i 6 20.e even 4 1
2160.2.q.i 6 180.x even 12 1
2700.2.i.c 6 5.c odd 4 1
2700.2.i.c 6 45.k odd 12 1
2700.2.s.c 12 1.a even 1 1 trivial
2700.2.s.c 12 5.b even 2 1 inner
2700.2.s.c 12 9.c even 3 1 inner
2700.2.s.c 12 45.j even 6 1 inner
6480.2.a.bt 3 180.v odd 12 1
6480.2.a.bw 3 180.x even 12 1
8100.2.a.u 3 45.k odd 12 1
8100.2.a.v 3 45.l even 12 1
8100.2.d.o 6 9.c even 3 1
8100.2.d.o 6 45.j even 6 1
8100.2.d.p 6 9.d odd 6 1
8100.2.d.p 6 45.h odd 6 1

Hecke kernels

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

\( T_{7}^{12} - 39T_{7}^{10} + 1038T_{7}^{8} - 15139T_{7}^{6} + 161178T_{7}^{4} - 893067T_{7}^{2} + 3418801 \) Copy content Toggle raw display
\( T_{11}^{6} + 24T_{11}^{4} + 72T_{11}^{3} + 576T_{11}^{2} + 864T_{11} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 39 T^{10} + 1038 T^{8} + \cdots + 3418801 \) Copy content Toggle raw display
$11$ \( (T^{6} + 24 T^{4} + 72 T^{3} + 576 T^{2} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 60 T^{10} + 2544 T^{8} + \cdots + 33362176 \) Copy content Toggle raw display
$17$ \( (T^{6} + 48 T^{4} + 576 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 12 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} + 78 T^{4} - 351 T^{3} + \cdots + 77841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 6 T^{5} + 48 T^{4} - 64 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 216 T^{4} + 11760 T^{2} + \cdots + 190096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + 90 T^{4} + 405 T^{3} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 60 T^{10} + 2544 T^{8} + \cdots + 33362176 \) Copy content Toggle raw display
$47$ \( T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$53$ \( (T^{6} + 156 T^{4} + 2736 T^{2} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 6 T^{5} + 96 T^{4} - 504 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 21 T^{5} + 378 T^{4} + \cdots + 167281)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 123 T^{10} + \cdots + 519885601 \) Copy content Toggle raw display
$71$ \( (T^{3} + 24 T^{2} + 108 T - 324)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{6} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 4)^{6} \) Copy content Toggle raw display
$83$ \( T^{12} - 183 T^{10} + \cdots + 3486784401 \) Copy content Toggle raw display
$89$ \( (T + 3)^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - 252 T^{10} + \cdots + 32319410176 \) Copy content Toggle raw display
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