Properties

Label 2940.2.k.d
Level $2940$
Weight $2$
Character orbit 2940.k
Analytic conductor $23.476$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q + i q^{3} + ( 2 - i ) q^{5} - q^{9} +O(q^{10})\) \( q + i q^{3} + ( 2 - i ) q^{5} - q^{9} -4 q^{11} + 6 i q^{13} + ( 1 + 2 i ) q^{15} + 2 i q^{17} + 6 q^{19} + 2 i q^{23} + ( 3 - 4 i ) q^{25} -i q^{27} -6 q^{29} + 2 q^{31} -4 i q^{33} + 4 i q^{37} -6 q^{39} -8 q^{41} -4 i q^{43} + ( -2 + i ) q^{45} + 4 i q^{47} -2 q^{51} + 6 i q^{53} + ( -8 + 4 i ) q^{55} + 6 i q^{57} + 4 q^{59} -14 q^{61} + ( 6 + 12 i ) q^{65} -4 i q^{67} -2 q^{69} + 10 i q^{73} + ( 4 + 3 i ) q^{75} + q^{81} + 16 i q^{83} + ( 2 + 4 i ) q^{85} -6 i q^{87} + 8 q^{89} + 2 i q^{93} + ( 12 - 6 i ) q^{95} + 10 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} - 8q^{11} + 2q^{15} + 12q^{19} + 6q^{25} - 12q^{29} + 4q^{31} - 12q^{39} - 16q^{41} - 4q^{45} - 4q^{51} - 16q^{55} + 8q^{59} - 28q^{61} + 12q^{65} - 4q^{69} + 8q^{75} + 2q^{81} + 4q^{85} + 16q^{89} + 24q^{95} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
1.00000i
1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 0 0 −1.00000 0
589.2 0 1.00000i 0 2.00000 1.00000i 0 0 0 −1.00000 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 2940.2.k.d 2
5.b even 2 1 inner 2940.2.k.d 2
7.b odd 2 1 420.2.k.a 2
7.c even 3 2 2940.2.bb.c 4
7.d odd 6 2 2940.2.bb.h 4
21.c even 2 1 1260.2.k.d 2
28.d even 2 1 1680.2.t.a 2
35.c odd 2 1 420.2.k.a 2
35.f even 4 1 2100.2.a.e 1
35.f even 4 1 2100.2.a.j 1
35.i odd 6 2 2940.2.bb.h 4
35.j even 6 2 2940.2.bb.c 4
84.h odd 2 1 5040.2.t.o 2
105.g even 2 1 1260.2.k.d 2
105.k odd 4 1 6300.2.a.n 1
105.k odd 4 1 6300.2.a.bc 1
140.c even 2 1 1680.2.t.a 2
140.j odd 4 1 8400.2.a.bh 1
140.j odd 4 1 8400.2.a.cd 1
420.o odd 2 1 5040.2.t.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.a 2 7.b odd 2 1
420.2.k.a 2 35.c odd 2 1
1260.2.k.d 2 21.c even 2 1
1260.2.k.d 2 105.g even 2 1
1680.2.t.a 2 28.d even 2 1
1680.2.t.a 2 140.c even 2 1
2100.2.a.e 1 35.f even 4 1
2100.2.a.j 1 35.f even 4 1
2940.2.k.d 2 1.a even 1 1 trivial
2940.2.k.d 2 5.b even 2 1 inner
2940.2.bb.c 4 7.c even 3 2
2940.2.bb.c 4 35.j even 6 2
2940.2.bb.h 4 7.d odd 6 2
2940.2.bb.h 4 35.i odd 6 2
5040.2.t.o 2 84.h odd 2 1
5040.2.t.o 2 420.o odd 2 1
6300.2.a.n 1 105.k odd 4 1
6300.2.a.bc 1 105.k odd 4 1
8400.2.a.bh 1 140.j odd 4 1
8400.2.a.cd 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11} + 4 \)
\( T_{13}^{2} + 36 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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