Properties

Label 840.2.t.b
Level $840$
Weight $2$
Character orbit 840.t
Analytic conductor $6.707$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.70743376979\)
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.

\(f(q)\) \(=\) \( q -i q^{3} + ( -1 + 2 i ) q^{5} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + ( -1 + 2 i ) q^{5} -i q^{7} - q^{9} + 2 q^{11} -2 i q^{13} + ( 2 + i ) q^{15} + 6 q^{19} - q^{21} + ( -3 - 4 i ) q^{25} + i q^{27} + 6 q^{29} + 10 q^{31} -2 i q^{33} + ( 2 + i ) q^{35} -2 q^{39} + 6 q^{41} -8 i q^{43} + ( 1 - 2 i ) q^{45} + 12 i q^{47} - q^{49} -6 i q^{53} + ( -2 + 4 i ) q^{55} -6 i q^{57} -6 q^{61} + i q^{63} + ( 4 + 2 i ) q^{65} -4 i q^{67} + 6 q^{71} -14 i q^{73} + ( -4 + 3 i ) q^{75} -2 i q^{77} -4 q^{79} + q^{81} -6 i q^{87} + 6 q^{89} -2 q^{91} -10 i q^{93} + ( -6 + 12 i ) q^{95} + 2 i q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{5} - 2 q^{9} + 4 q^{11} + 4 q^{15} + 12 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} + 20 q^{31} + 4 q^{35} - 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} - 4 q^{55} - 12 q^{61} + 8 q^{65} + 12 q^{71} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} - 12 q^{95} - 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(281\) \(337\) \(421\) \(631\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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 −1.00000 + 2.00000i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 −1.00000 2.00000i 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 840.2.t.b 2
3.b odd 2 1 2520.2.t.e 2
4.b odd 2 1 1680.2.t.b 2
5.b even 2 1 inner 840.2.t.b 2
5.c odd 4 1 4200.2.a.j 1
5.c odd 4 1 4200.2.a.w 1
12.b even 2 1 5040.2.t.l 2
15.d odd 2 1 2520.2.t.e 2
20.d odd 2 1 1680.2.t.b 2
20.e even 4 1 8400.2.a.v 1
20.e even 4 1 8400.2.a.bs 1
60.h even 2 1 5040.2.t.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.b 2 1.a even 1 1 trivial
840.2.t.b 2 5.b even 2 1 inner
1680.2.t.b 2 4.b odd 2 1
1680.2.t.b 2 20.d odd 2 1
2520.2.t.e 2 3.b odd 2 1
2520.2.t.e 2 15.d odd 2 1
4200.2.a.j 1 5.c odd 4 1
4200.2.a.w 1 5.c odd 4 1
5040.2.t.l 2 12.b even 2 1
5040.2.t.l 2 60.h even 2 1
8400.2.a.v 1 20.e even 4 1
8400.2.a.bs 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11} - 2 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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