Properties

Label 2700.2.d.g
Level $2700$
Weight $2$
Character orbit 2700.d
Analytic conductor $21.560$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 i q^{7} +O(q^{10})\) \( q + 5 i q^{7} + 7 i q^{13} + q^{19} -4 q^{31} -i q^{37} -8 i q^{43} -18 q^{49} -13 q^{61} + 11 i q^{67} -17 i q^{73} + 13 q^{79} -35 q^{91} + 5 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{19} - 8q^{31} - 36q^{49} - 26q^{61} + 26q^{79} - 70q^{91} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
0 0 0 0 0 5.00000i 0 0 0
649.2 0 0 0 0 0 5.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 2700.2.d.g 2
3.b odd 2 1 CM 2700.2.d.g 2
5.b even 2 1 inner 2700.2.d.g 2
5.c odd 4 1 108.2.a.a 1
5.c odd 4 1 2700.2.a.b 1
15.d odd 2 1 inner 2700.2.d.g 2
15.e even 4 1 108.2.a.a 1
15.e even 4 1 2700.2.a.b 1
20.e even 4 1 432.2.a.d 1
35.f even 4 1 5292.2.a.j 1
40.i odd 4 1 1728.2.a.p 1
40.k even 4 1 1728.2.a.m 1
45.k odd 12 2 324.2.e.b 2
45.l even 12 2 324.2.e.b 2
60.l odd 4 1 432.2.a.d 1
105.k odd 4 1 5292.2.a.j 1
120.q odd 4 1 1728.2.a.m 1
120.w even 4 1 1728.2.a.p 1
180.v odd 12 2 1296.2.i.j 2
180.x even 12 2 1296.2.i.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 5.c odd 4 1
108.2.a.a 1 15.e even 4 1
324.2.e.b 2 45.k odd 12 2
324.2.e.b 2 45.l even 12 2
432.2.a.d 1 20.e even 4 1
432.2.a.d 1 60.l odd 4 1
1296.2.i.j 2 180.v odd 12 2
1296.2.i.j 2 180.x even 12 2
1728.2.a.m 1 40.k even 4 1
1728.2.a.m 1 120.q odd 4 1
1728.2.a.p 1 40.i odd 4 1
1728.2.a.p 1 120.w even 4 1
2700.2.a.b 1 5.c odd 4 1
2700.2.a.b 1 15.e even 4 1
2700.2.d.g 2 1.a even 1 1 trivial
2700.2.d.g 2 3.b odd 2 1 CM
2700.2.d.g 2 5.b even 2 1 inner
2700.2.d.g 2 15.d odd 2 1 inner
5292.2.a.j 1 35.f even 4 1
5292.2.a.j 1 105.k odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 25 \)
\( T_{11} \)
\( T_{13}^{2} + 49 \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 49 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 121 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 289 + T^{2} \)
$79$ \( ( -13 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 25 + T^{2} \)
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