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Properties

Label 2700.3.g.s
Level $2700$
Weight $3$
Character orbit 2700.g
Analytic conductor $73.570$
Analytic rank $0$
Dimension $8$
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] = [2700,3,Mod(701,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(2))
 
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("2700.701");
 
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.g (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.488455618816.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{7} + (\beta_{6} - \beta_{3}) q^{11} + ( - \beta_{5} + \beta_1) q^{13} - \beta_{2} q^{17} + (\beta_{7} - 1) q^{19} + (2 \beta_{4} - \beta_{2}) q^{23} + 3 \beta_{6} q^{29} + ( - 2 \beta_{7} + 3) q^{31} + (2 \beta_{5} + 2 \beta_1) q^{37} - \beta_{6} q^{41} + ( - 3 \beta_{5} - 5 \beta_1) q^{43} + (6 \beta_{4} + 4 \beta_{2}) q^{47} + (\beta_{7} - 19) q^{49} + (19 \beta_{4} + 6 \beta_{2}) q^{53} + ( - 3 \beta_{6} + 8 \beta_{3}) q^{59} + ( - \beta_{7} + 9) q^{61} + (2 \beta_{5} + 12 \beta_1) q^{67} + ( - \beta_{6} + 14 \beta_{3}) q^{71} + (4 \beta_{5} - 9 \beta_1) q^{73} + (15 \beta_{4} + \beta_{2}) q^{77} + (3 \beta_{7} + 3) q^{79} + (19 \beta_{4} - 10 \beta_{2}) q^{83} + ( - 3 \beta_{6} - 14 \beta_{3}) q^{89} + ( - 2 \beta_{7} + 20) q^{91} + (\beta_{5} - 26 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 107\nu^{7} + 1001\nu^{6} - 4543\nu^{5} - 13153\nu^{4} + 51884\nu^{3} + 72002\nu^{2} - 258088\nu - 983740 ) / 137550 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\nu^{7} + 322\nu^{6} - 756\nu^{5} - 1666\nu^{4} + 6013\nu^{3} + 70294\nu^{2} - 52076\nu - 180730 ) / 45850 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73\nu^{7} + 1120\nu^{6} - 4025\nu^{5} + 5887\nu^{4} + 4648\nu^{3} + 61978\nu^{2} - 65216\nu + 73420 ) / 68775 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -78\nu^{7} + 273\nu^{6} - 1239\nu^{5} + 2415\nu^{4} - 5607\nu^{3} + 6132\nu^{2} - 46236\nu + 22170 ) / 45850 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 107\nu^{7} - 571\nu^{6} + 173\nu^{5} + 4139\nu^{4} + 9440\nu^{3} - 38038\nu^{2} - 126040\nu + 226700 ) / 19650 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 584\nu^{7} - 2044\nu^{6} + 812\nu^{5} + 3080\nu^{4} + 70196\nu^{3} - 109396\nu^{2} + 72488\nu - 17860 ) / 68775 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -24\nu^{7} + 84\nu^{6} + 42\nu^{5} - 315\nu^{4} - 1302\nu^{3} + 2310\nu^{2} + 8628\nu - 5170 ) / 917 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{6} - 3\beta_{5} + 4\beta_{4} - 3\beta _1 + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{5} + 8\beta_{4} - 4\beta_{3} + 12\beta_{2} - 5\beta _1 + 24 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 9\beta_{6} + 3\beta_{5} + 11\beta_{4} - 3\beta_{3} + 9\beta_{2} + 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} + 5\beta_{6} + 4\beta_{5} + 8\beta_{4} + 4\beta_{3} + 8\beta_{2} - 12\beta _1 - 96 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25\beta_{7} - 13\beta_{6} + 68\beta_{5} - 209\beta_{4} + 35\beta_{3} + 45\beta_{2} - 47\beta _1 - 742 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 30\beta_{7} - 62\beta_{6} + 76\beta_{5} - 428\beta_{4} + 229\beta_{3} - 132\beta_{2} + 50\beta _1 - 924 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -217\beta_{7} - 776\beta_{6} + 13\beta_{5} - 3529\beta_{4} + 1477\beta_{3} - 1071\beta_{2} + 230\beta _1 - 4022 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
2.67945 − 1.15831i
2.67945 + 1.15831i
−1.67945 − 1.15831i
−1.67945 + 1.15831i
2.67945 − 2.15831i
2.67945 + 2.15831i
−1.67945 − 2.15831i
−1.67945 + 2.15831i
0 0 0 0 0 −7.15439 0 0 0
701.2 0 0 0 0 0 −7.15439 0 0 0
701.3 0 0 0 0 0 −2.79549 0 0 0
701.4 0 0 0 0 0 −2.79549 0 0 0
701.5 0 0 0 0 0 2.79549 0 0 0
701.6 0 0 0 0 0 2.79549 0 0 0
701.7 0 0 0 0 0 7.15439 0 0 0
701.8 0 0 0 0 0 7.15439 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.g.s 8
3.b odd 2 1 inner 2700.3.g.s 8
5.b even 2 1 inner 2700.3.g.s 8
5.c odd 4 1 540.3.b.a ✓ 4
5.c odd 4 1 540.3.b.b yes 4
15.d odd 2 1 inner 2700.3.g.s 8
15.e even 4 1 540.3.b.a ✓ 4
15.e even 4 1 540.3.b.b yes 4
20.e even 4 1 2160.3.c.h 4
20.e even 4 1 2160.3.c.l 4
45.k odd 12 2 1620.3.t.a 8
45.k odd 12 2 1620.3.t.d 8
45.l even 12 2 1620.3.t.a 8
45.l even 12 2 1620.3.t.d 8
60.l odd 4 1 2160.3.c.h 4
60.l odd 4 1 2160.3.c.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a ✓ 4 5.c odd 4 1
540.3.b.a ✓ 4 15.e even 4 1
540.3.b.b yes 4 5.c odd 4 1
540.3.b.b yes 4 15.e even 4 1
1620.3.t.a 8 45.k odd 12 2
1620.3.t.a 8 45.l even 12 2
1620.3.t.d 8 45.k odd 12 2
1620.3.t.d 8 45.l even 12 2
2160.3.c.h 4 20.e even 4 1
2160.3.c.h 4 60.l odd 4 1
2160.3.c.l 4 20.e even 4 1
2160.3.c.l 4 60.l odd 4 1
2700.3.g.s 8 1.a even 1 1 trivial
2700.3.g.s 8 3.b odd 2 1 inner
2700.3.g.s 8 5.b even 2 1 inner
2700.3.g.s 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2700, [\chi])\):

\( T_{7}^{4} - 59T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{11}^{4} + 355T_{11}^{2} + 8464 \) Copy content Toggle raw display
\( T_{13}^{4} - 540T_{13}^{2} + 5184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 59 T^{2} + 400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 355 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 540 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 109 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T - 468)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 217 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1584)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 1865)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 1216)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 176)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6620 T^{2} + 9684544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1960 T^{2} + 478864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8370 T^{2} + 178929)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 6880 T^{2} + 4129024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 19 T - 380)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 11372 T^{2} + 541696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 16956 T^{2} + 68558400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 11795 T^{2} + 6091024)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 3 T - 4230)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20818 T^{2} + 1681)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 24700 T^{2} + 19430464)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 39515 T^{2} + 266603584)^{2} \) Copy content Toggle raw display
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