Properties

Label 2700.2.d.k
Level $2700$
Weight $2$
Character orbit 2700.d
Analytic conductor $21.560$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{7} + 6 q^{11} - i q^{13} + q^{19} + 6 i q^{23} - 6 q^{29} + 8 q^{31} + 7 i q^{37} - 6 q^{41} - 4 i q^{43} - 12 i q^{47} + 6 q^{49} - 6 i q^{53} + 11 q^{61} + 7 i q^{67} - 6 q^{71} + 11 i q^{73} + 6 i q^{77} + q^{79} + 6 i q^{83} + 12 q^{89} + q^{91} + 13 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{11} + 2 q^{19} - 12 q^{29} + 16 q^{31} - 12 q^{41} + 12 q^{49} + 22 q^{61} - 12 q^{71} + 2 q^{79} + 24 q^{89} + 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 1.00000i 0 0 0
649.2 0 0 0 0 0 1.00000i 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
5.b even 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.k 2
3.b odd 2 1 2700.2.d.a 2
5.b even 2 1 inner 2700.2.d.k 2
5.c odd 4 1 540.2.a.e yes 1
5.c odd 4 1 2700.2.a.m 1
15.d odd 2 1 2700.2.d.a 2
15.e even 4 1 540.2.a.b 1
15.e even 4 1 2700.2.a.k 1
20.e even 4 1 2160.2.a.s 1
40.i odd 4 1 8640.2.a.l 1
40.k even 4 1 8640.2.a.s 1
45.k odd 12 2 1620.2.i.d 2
45.l even 12 2 1620.2.i.j 2
60.l odd 4 1 2160.2.a.h 1
120.q odd 4 1 8640.2.a.bu 1
120.w even 4 1 8640.2.a.br 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 15.e even 4 1
540.2.a.e yes 1 5.c odd 4 1
1620.2.i.d 2 45.k odd 12 2
1620.2.i.j 2 45.l even 12 2
2160.2.a.h 1 60.l odd 4 1
2160.2.a.s 1 20.e even 4 1
2700.2.a.k 1 15.e even 4 1
2700.2.a.m 1 5.c odd 4 1
2700.2.d.a 2 3.b odd 2 1
2700.2.d.a 2 15.d odd 2 1
2700.2.d.k 2 1.a even 1 1 trivial
2700.2.d.k 2 5.b even 2 1 inner
8640.2.a.l 1 40.i odd 4 1
8640.2.a.s 1 40.k even 4 1
8640.2.a.br 1 120.w even 4 1
8640.2.a.bu 1 120.q odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 49 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 169 \) Copy content Toggle raw display
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