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Properties

Label 2700.3.p.c
Level $2700$
Weight $3$
Character orbit 2700.p
Analytic conductor $73.570$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2700,3,Mod(1601,2700)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2700.1601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2700.p (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: \(73.5696713773\)
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: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 3^{8} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3} + \beta_1 + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3} + \beta_1 + 2) q^{7} + (\beta_{4} + \beta_{2} + 3 \beta_1 - 2) q^{11} + (\beta_{11} - \beta_{9} + \cdots - 5 \beta_1) q^{13}+ \cdots + ( - 2 \beta_{11} - 12 \beta_{10} + \cdots - 16) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} - 48 q^{11} + 30 q^{13} + 72 q^{19} - 78 q^{23} - 150 q^{29} - 12 q^{31} + 12 q^{37} - 90 q^{41} - 114 q^{43} + 12 q^{47} + 48 q^{49} - 48 q^{59} - 78 q^{61} + 168 q^{67} + 24 q^{73} - 258 q^{77} + 120 q^{79} + 114 q^{83} + 120 q^{91} - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1264 \nu^{11} - 7514 \nu^{10} + 7570 \nu^{9} + 69045 \nu^{8} - 447600 \nu^{7} + 644742 \nu^{6} + \cdots + 160081839 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2216 \nu^{11} + 16727 \nu^{10} - 78439 \nu^{9} + 181038 \nu^{8} + 440994 \nu^{7} + \cdots - 1396567899 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 262 \nu^{11} + 2342 \nu^{10} - 7018 \nu^{9} - 7395 \nu^{8} + 133365 \nu^{7} - 477666 \nu^{6} + \cdots - 178308297 ) / 10609137 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 554 \nu^{11} + 2386 \nu^{10} - 755 \nu^{9} - 32880 \nu^{8} + 119670 \nu^{7} - 115038 \nu^{6} + \cdots - 74637936 ) / 10609137 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5945 \nu^{11} + 23266 \nu^{10} - 2231 \nu^{9} - 288570 \nu^{8} + 1334805 \nu^{7} + \cdots - 967281669 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2881 \nu^{11} + 3805 \nu^{10} - 39323 \nu^{9} + 5982 \nu^{8} + 231540 \nu^{7} - 1654632 \nu^{6} + \cdots - 476466381 ) / 31827411 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8957 \nu^{11} + 48352 \nu^{10} - 88385 \nu^{9} - 311070 \nu^{8} + 2376807 \nu^{7} + \cdots - 1484019468 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9181 \nu^{11} - 94979 \nu^{10} + 184516 \nu^{9} + 661812 \nu^{8} - 4788867 \nu^{7} + \cdots + 4166615538 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9721 \nu^{11} + 68357 \nu^{10} + 18254 \nu^{9} - 630789 \nu^{8} + 2825589 \nu^{7} + \cdots - 2612682054 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1108 \nu^{11} + 4772 \nu^{10} - 1510 \nu^{9} - 65760 \nu^{8} + 239340 \nu^{7} + \cdots - 149275872 ) / 10609137 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1417 \nu^{11} + 5975 \nu^{10} - 5086 \nu^{9} - 56283 \nu^{8} + 326775 \nu^{7} + \cdots - 189606339 ) / 10609137 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 2\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} - 5 \beta_{10} + \beta_{9} - \beta_{8} + 5 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \cdots - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 25 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 11 \beta_{6} + 8 \beta_{5} + \cdots + 84 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} + 13 \beta_{10} - 2 \beta_{9} - 52 \beta_{8} - 19 \beta_{7} - 29 \beta_{6} - 28 \beta_{5} + \cdots - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 151 \beta_{11} + 283 \beta_{10} - 11 \beta_{9} + 29 \beta_{8} - 19 \beta_{7} - 74 \beta_{6} + \cdots - 1050 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 326 \beta_{11} - 878 \beta_{10} + 160 \beta_{9} + 38 \beta_{8} + 260 \beta_{7} + 196 \beta_{6} + \cdots - 4362 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1276 \beta_{11} - 1256 \beta_{10} - 344 \beta_{9} - 412 \beta_{8} - 190 \beta_{7} - 1154 \beta_{6} + \cdots - 9591 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 686 \beta_{11} - 6440 \beta_{10} + 1906 \beta_{9} - 2644 \beta_{8} - 2746 \beta_{7} - 1856 \beta_{6} + \cdots + 2028 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7855 \beta_{11} + 23827 \beta_{10} + 79 \beta_{9} - 4201 \beta_{8} - 7543 \beta_{7} - 10928 \beta_{6} + \cdots - 38841 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7526 \beta_{11} + 526 \beta_{10} + 22444 \beta_{9} + 23834 \beta_{8} + 323 \beta_{7} + 35863 \beta_{6} + \cdots - 200643 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
0.459278 − 2.96464i
2.65605 − 1.39478i
−2.85525 − 0.920635i
−2.99781 − 0.114662i
2.89597 + 0.783177i
0.841761 + 2.87949i
0.459278 + 2.96464i
2.65605 + 1.39478i
−2.85525 + 0.920635i
−2.99781 + 0.114662i
2.89597 − 0.783177i
0.841761 − 2.87949i
0 0 0 0 0 −4.13490 + 7.16186i 0 0 0
1601.2 0 0 0 0 0 −1.41583 + 2.45229i 0 0 0
1601.3 0 0 0 0 0 −0.594587 + 1.02985i 0 0 0
1601.4 0 0 0 0 0 0.801399 − 1.38806i 0 0 0
1601.5 0 0 0 0 0 2.35650 − 4.08158i 0 0 0
1601.6 0 0 0 0 0 5.98742 − 10.3705i 0 0 0
2501.1 0 0 0 0 0 −4.13490 − 7.16186i 0 0 0
2501.2 0 0 0 0 0 −1.41583 − 2.45229i 0 0 0
2501.3 0 0 0 0 0 −0.594587 − 1.02985i 0 0 0
2501.4 0 0 0 0 0 0.801399 + 1.38806i 0 0 0
2501.5 0 0 0 0 0 2.35650 + 4.08158i 0 0 0
2501.6 0 0 0 0 0 5.98742 + 10.3705i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1601.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.p.c 12
3.b odd 2 1 900.3.p.c 12
5.b even 2 1 540.3.o.b 12
5.c odd 4 2 2700.3.u.c 24
9.c even 3 1 900.3.p.c 12
9.d odd 6 1 inner 2700.3.p.c 12
15.d odd 2 1 180.3.o.b ✓ 12
15.e even 4 2 900.3.u.c 24
20.d odd 2 1 2160.3.bs.b 12
45.h odd 6 1 540.3.o.b 12
45.h odd 6 1 1620.3.g.b 12
45.j even 6 1 180.3.o.b ✓ 12
45.j even 6 1 1620.3.g.b 12
45.k odd 12 2 900.3.u.c 24
45.l even 12 2 2700.3.u.c 24
60.h even 2 1 720.3.bs.b 12
180.n even 6 1 2160.3.bs.b 12
180.p odd 6 1 720.3.bs.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b ✓ 12 15.d odd 2 1
180.3.o.b ✓ 12 45.j even 6 1
540.3.o.b 12 5.b even 2 1
540.3.o.b 12 45.h odd 6 1
720.3.bs.b 12 60.h even 2 1
720.3.bs.b 12 180.p odd 6 1
900.3.p.c 12 3.b odd 2 1
900.3.p.c 12 9.c even 3 1
900.3.u.c 24 15.e even 4 2
900.3.u.c 24 45.k odd 12 2
1620.3.g.b 12 45.h odd 6 1
1620.3.g.b 12 45.j even 6 1
2160.3.bs.b 12 20.d odd 2 1
2160.3.bs.b 12 180.n even 6 1
2700.3.p.c 12 1.a even 1 1 trivial
2700.3.p.c 12 9.d odd 6 1 inner
2700.3.u.c 24 5.c odd 4 2
2700.3.u.c 24 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 6 T_{7}^{11} + 141 T_{7}^{10} + 50 T_{7}^{9} + 11340 T_{7}^{8} - 14346 T_{7}^{7} + \cdots + 6345361 \) acting on \(S_{3}^{\mathrm{new}}(2700, [\chi])\). 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
$7$ \( T^{12} - 6 T^{11} + \cdots + 6345361 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 416649744 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 15716943091600 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 191070131587344 \) Copy content Toggle raw display
$19$ \( (T^{6} - 36 T^{5} + \cdots - 19580204)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13626529936569 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3840796442025 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 177690926165776 \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots - 2144769884)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} - 12 T^{5} + \cdots - 2761132736)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
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