Properties

Label 1680.2.t.e
Level 1680
Weight 2
Character orbit 1680.t
Analytic conductor 13.415
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1680.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + ( 1 + 2 i ) q^{5} -i q^{7} - q^{9} -2 q^{11} + 6 i q^{13} + ( 2 - i ) q^{15} -4 i q^{17} -6 q^{19} - q^{21} + 8 i q^{23} + ( -3 + 4 i ) q^{25} + i q^{27} -6 q^{29} + 2 q^{31} + 2 i q^{33} + ( 2 - i ) q^{35} -4 i q^{37} + 6 q^{39} + 2 q^{41} -4 i q^{43} + ( -1 - 2 i ) q^{45} + 8 i q^{47} - q^{49} -4 q^{51} + 6 i q^{53} + ( -2 - 4 i ) q^{55} + 6 i q^{57} -8 q^{59} -10 q^{61} + i q^{63} + ( -12 + 6 i ) q^{65} + 8 i q^{67} + 8 q^{69} + 6 q^{71} -14 i q^{73} + ( 4 + 3 i ) q^{75} + 2 i q^{77} -12 q^{79} + q^{81} + 8 i q^{83} + ( 8 - 4 i ) q^{85} + 6 i q^{87} + 10 q^{89} + 6 q^{91} -2 i q^{93} + ( -6 - 12 i ) q^{95} + 10 i q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{9} - 4q^{11} + 4q^{15} - 12q^{19} - 2q^{21} - 6q^{25} - 12q^{29} + 4q^{31} + 4q^{35} + 12q^{39} + 4q^{41} - 2q^{45} - 2q^{49} - 8q^{51} - 4q^{55} - 16q^{59} - 20q^{61} - 24q^{65} + 16q^{69} + 12q^{71} + 8q^{75} - 24q^{79} + 2q^{81} + 16q^{85} + 20q^{89} + 12q^{91} - 12q^{95} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 1.00000i 0 −1.00000 0
1009.2 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 1680.2.t.e 2
3.b odd 2 1 5040.2.t.h 2
4.b odd 2 1 210.2.g.b 2
5.b even 2 1 inner 1680.2.t.e 2
5.c odd 4 1 8400.2.a.w 1
5.c odd 4 1 8400.2.a.bp 1
12.b even 2 1 630.2.g.c 2
15.d odd 2 1 5040.2.t.h 2
20.d odd 2 1 210.2.g.b 2
20.e even 4 1 1050.2.a.d 1
20.e even 4 1 1050.2.a.p 1
28.d even 2 1 1470.2.g.b 2
28.f even 6 2 1470.2.n.f 4
28.g odd 6 2 1470.2.n.b 4
60.h even 2 1 630.2.g.c 2
60.l odd 4 1 3150.2.a.d 1
60.l odd 4 1 3150.2.a.bk 1
140.c even 2 1 1470.2.g.b 2
140.j odd 4 1 7350.2.a.bk 1
140.j odd 4 1 7350.2.a.bz 1
140.p odd 6 2 1470.2.n.b 4
140.s even 6 2 1470.2.n.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 4.b odd 2 1
210.2.g.b 2 20.d odd 2 1
630.2.g.c 2 12.b even 2 1
630.2.g.c 2 60.h even 2 1
1050.2.a.d 1 20.e even 4 1
1050.2.a.p 1 20.e even 4 1
1470.2.g.b 2 28.d even 2 1
1470.2.g.b 2 140.c even 2 1
1470.2.n.b 4 28.g odd 6 2
1470.2.n.b 4 140.p odd 6 2
1470.2.n.f 4 28.f even 6 2
1470.2.n.f 4 140.s even 6 2
1680.2.t.e 2 1.a even 1 1 trivial
1680.2.t.e 2 5.b even 2 1 inner
3150.2.a.d 1 60.l odd 4 1
3150.2.a.bk 1 60.l odd 4 1
5040.2.t.h 2 3.b odd 2 1
5040.2.t.h 2 15.d odd 2 1
7350.2.a.bk 1 140.j odd 4 1
7350.2.a.bz 1 140.j odd 4 1
8400.2.a.w 1 5.c odd 4 1
8400.2.a.bp 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( 1 - 18 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 58 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 12 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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