Properties

Label 1680.2.t.f
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: \(\Q(\sqrt{-1}) \)
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 105)
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.

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

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.f 2
3.b odd 2 1 5040.2.t.e 2
4.b odd 2 1 105.2.d.a 2
5.b even 2 1 inner 1680.2.t.f 2
5.c odd 4 1 8400.2.a.bj 1
5.c odd 4 1 8400.2.a.ch 1
12.b even 2 1 315.2.d.c 2
15.d odd 2 1 5040.2.t.e 2
20.d odd 2 1 105.2.d.a 2
20.e even 4 1 525.2.a.b 1
20.e even 4 1 525.2.a.c 1
28.d even 2 1 735.2.d.a 2
28.f even 6 2 735.2.q.b 4
28.g odd 6 2 735.2.q.a 4
60.h even 2 1 315.2.d.c 2
60.l odd 4 1 1575.2.a.e 1
60.l odd 4 1 1575.2.a.i 1
84.h odd 2 1 2205.2.d.f 2
140.c even 2 1 735.2.d.a 2
140.j odd 4 1 3675.2.a.d 1
140.j odd 4 1 3675.2.a.l 1
140.p odd 6 2 735.2.q.a 4
140.s even 6 2 735.2.q.b 4
420.o odd 2 1 2205.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 4.b odd 2 1
105.2.d.a 2 20.d odd 2 1
315.2.d.c 2 12.b even 2 1
315.2.d.c 2 60.h even 2 1
525.2.a.b 1 20.e even 4 1
525.2.a.c 1 20.e even 4 1
735.2.d.a 2 28.d even 2 1
735.2.d.a 2 140.c even 2 1
735.2.q.a 4 28.g odd 6 2
735.2.q.a 4 140.p odd 6 2
735.2.q.b 4 28.f even 6 2
735.2.q.b 4 140.s even 6 2
1575.2.a.e 1 60.l odd 4 1
1575.2.a.i 1 60.l odd 4 1
1680.2.t.f 2 1.a even 1 1 trivial
1680.2.t.f 2 5.b even 2 1 inner
2205.2.d.f 2 84.h odd 2 1
2205.2.d.f 2 420.o odd 2 1
3675.2.a.d 1 140.j odd 4 1
3675.2.a.l 1 140.j odd 4 1
5040.2.t.e 2 3.b odd 2 1
5040.2.t.e 2 15.d odd 2 1
8400.2.a.bj 1 5.c odd 4 1
8400.2.a.ch 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 \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( -10 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 256 + T^{2} \)
$71$ \( ( 10 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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