Properties

Label 420.2.k.b
Level $420$
Weight $2$
Character orbit 420.k
Analytic conductor $3.354$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{3} + ( 2 - i ) q^{5} + i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} + ( 2 - i ) q^{5} + i q^{7} - q^{9} + 4 q^{11} -2 i q^{13} + ( 1 + 2 i ) q^{15} + 2 i q^{17} + 2 q^{19} - q^{21} + 6 i q^{23} + ( 3 - 4 i ) q^{25} -i q^{27} -6 q^{29} + 6 q^{31} + 4 i q^{33} + ( 1 + 2 i ) q^{35} -4 i q^{37} + 2 q^{39} + 4 i q^{43} + ( -2 + i ) q^{45} + 4 i q^{47} - q^{49} -2 q^{51} + 2 i q^{53} + ( 8 - 4 i ) q^{55} + 2 i q^{57} -4 q^{59} -2 q^{61} -i q^{63} + ( -2 - 4 i ) q^{65} -12 i q^{67} -6 q^{69} -8 q^{71} -14 i q^{73} + ( 4 + 3 i ) q^{75} + 4 i q^{77} -16 q^{79} + q^{81} -16 i q^{83} + ( 2 + 4 i ) q^{85} -6 i q^{87} -16 q^{89} + 2 q^{91} + 6 i q^{93} + ( 4 - 2 i ) q^{95} -14 i q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} + 8q^{11} + 2q^{15} + 4q^{19} - 2q^{21} + 6q^{25} - 12q^{29} + 12q^{31} + 2q^{35} + 4q^{39} - 4q^{45} - 2q^{49} - 4q^{51} + 16q^{55} - 8q^{59} - 4q^{61} - 4q^{65} - 12q^{69} - 16q^{71} + 8q^{75} - 32q^{79} + 2q^{81} + 4q^{85} - 32q^{89} + 4q^{91} + 8q^{95} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
1.00000i
1.00000i
0 1.00000i 0 2.00000 + 1.00000i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 2.00000 1.00000i 0 1.00000i 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 420.2.k.b 2
3.b odd 2 1 1260.2.k.a 2
4.b odd 2 1 1680.2.t.g 2
5.b even 2 1 inner 420.2.k.b 2
5.c odd 4 1 2100.2.a.i 1
5.c odd 4 1 2100.2.a.n 1
7.b odd 2 1 2940.2.k.b 2
7.c even 3 2 2940.2.bb.a 4
7.d odd 6 2 2940.2.bb.f 4
12.b even 2 1 5040.2.t.d 2
15.d odd 2 1 1260.2.k.a 2
15.e even 4 1 6300.2.a.b 1
15.e even 4 1 6300.2.a.r 1
20.d odd 2 1 1680.2.t.g 2
20.e even 4 1 8400.2.a.o 1
20.e even 4 1 8400.2.a.bm 1
35.c odd 2 1 2940.2.k.b 2
35.i odd 6 2 2940.2.bb.f 4
35.j even 6 2 2940.2.bb.a 4
60.h even 2 1 5040.2.t.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.k.b 2 1.a even 1 1 trivial
420.2.k.b 2 5.b even 2 1 inner
1260.2.k.a 2 3.b odd 2 1
1260.2.k.a 2 15.d odd 2 1
1680.2.t.g 2 4.b odd 2 1
1680.2.t.g 2 20.d odd 2 1
2100.2.a.i 1 5.c odd 4 1
2100.2.a.n 1 5.c odd 4 1
2940.2.k.b 2 7.b odd 2 1
2940.2.k.b 2 35.c odd 2 1
2940.2.bb.a 4 7.c even 3 2
2940.2.bb.a 4 35.j even 6 2
2940.2.bb.f 4 7.d odd 6 2
2940.2.bb.f 4 35.i odd 6 2
5040.2.t.d 2 12.b even 2 1
5040.2.t.d 2 60.h even 2 1
6300.2.a.b 1 15.e even 4 1
6300.2.a.r 1 15.e even 4 1
8400.2.a.o 1 20.e even 4 1
8400.2.a.bm 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 4 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( 16 + T )^{2} \)
$97$ \( 196 + T^{2} \)
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