Properties

Label 6300.2.k.r
Level $6300$
Weight $2$
Character orbit 6300.k
Analytic conductor $50.306$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(50.3057532734\)
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 84)
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 \( q + i q^{7} + 6 q^{11} - 2 i q^{13} + 4 q^{19} - 6 i q^{23} + 6 q^{29} + 8 q^{31} + 2 i q^{37} - 12 q^{41} + 4 i q^{43} - 12 i q^{47} - q^{49} - 6 i q^{53} - 10 q^{61} + 8 i q^{67} - 6 q^{71} + 10 i q^{73} + 6 i q^{77} + 4 q^{79} - 12 i q^{83} + 12 q^{89} + 2 q^{91} - 10 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{11} + 8 q^{19} + 12 q^{29} + 16 q^{31} - 24 q^{41} - 2 q^{49} - 20 q^{61} - 12 q^{71} + 8 q^{79} + 24 q^{89} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
6049.2 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 6300.2.k.r 2
3.b odd 2 1 2100.2.k.a 2
5.b even 2 1 inner 6300.2.k.r 2
5.c odd 4 1 252.2.a.b 1
5.c odd 4 1 6300.2.a.p 1
15.d odd 2 1 2100.2.k.a 2
15.e even 4 1 84.2.a.b 1
15.e even 4 1 2100.2.a.a 1
20.e even 4 1 1008.2.a.g 1
35.f even 4 1 1764.2.a.g 1
35.k even 12 2 1764.2.k.d 2
35.l odd 12 2 1764.2.k.e 2
40.i odd 4 1 4032.2.a.u 1
40.k even 4 1 4032.2.a.t 1
45.k odd 12 2 2268.2.j.f 2
45.l even 12 2 2268.2.j.i 2
60.l odd 4 1 336.2.a.b 1
60.l odd 4 1 8400.2.a.ct 1
105.k odd 4 1 588.2.a.c 1
105.w odd 12 2 588.2.i.f 2
105.x even 12 2 588.2.i.c 2
120.q odd 4 1 1344.2.a.o 1
120.w even 4 1 1344.2.a.f 1
140.j odd 4 1 7056.2.a.x 1
240.z odd 4 1 5376.2.c.x 2
240.bb even 4 1 5376.2.c.i 2
240.bd odd 4 1 5376.2.c.x 2
240.bf even 4 1 5376.2.c.i 2
420.w even 4 1 2352.2.a.s 1
420.bp odd 12 2 2352.2.q.s 2
420.br even 12 2 2352.2.q.g 2
840.bm even 4 1 9408.2.a.r 1
840.bp odd 4 1 9408.2.a.co 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 15.e even 4 1
252.2.a.b 1 5.c odd 4 1
336.2.a.b 1 60.l odd 4 1
588.2.a.c 1 105.k odd 4 1
588.2.i.c 2 105.x even 12 2
588.2.i.f 2 105.w odd 12 2
1008.2.a.g 1 20.e even 4 1
1344.2.a.f 1 120.w even 4 1
1344.2.a.o 1 120.q odd 4 1
1764.2.a.g 1 35.f even 4 1
1764.2.k.d 2 35.k even 12 2
1764.2.k.e 2 35.l odd 12 2
2100.2.a.a 1 15.e even 4 1
2100.2.k.a 2 3.b odd 2 1
2100.2.k.a 2 15.d odd 2 1
2268.2.j.f 2 45.k odd 12 2
2268.2.j.i 2 45.l even 12 2
2352.2.a.s 1 420.w even 4 1
2352.2.q.g 2 420.br even 12 2
2352.2.q.s 2 420.bp odd 12 2
4032.2.a.t 1 40.k even 4 1
4032.2.a.u 1 40.i odd 4 1
5376.2.c.i 2 240.bb even 4 1
5376.2.c.i 2 240.bf even 4 1
5376.2.c.x 2 240.z odd 4 1
5376.2.c.x 2 240.bd odd 4 1
6300.2.a.p 1 5.c odd 4 1
6300.2.k.r 2 1.a even 1 1 trivial
6300.2.k.r 2 5.b even 2 1 inner
7056.2.a.x 1 140.j odd 4 1
8400.2.a.ct 1 60.l odd 4 1
9408.2.a.r 1 840.bm even 4 1
9408.2.a.co 1 840.bp odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6300, [\chi])\):

\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{41} + 12 \) 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
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) 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
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 12)^{2} \) 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
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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