Properties

Label 1792.2.b.h
Level 1792
Weight 2
Character orbit 1792.b
Analytic conductor 14.309
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -4 i q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -4 i q^{5} + q^{7} - q^{9} + 8 q^{15} -2 q^{17} -2 i q^{19} + 2 i q^{21} + 8 q^{23} -11 q^{25} + 4 i q^{27} -2 i q^{29} -4 q^{31} -4 i q^{35} -6 i q^{37} + 2 q^{41} -8 i q^{43} + 4 i q^{45} + 4 q^{47} + q^{49} -4 i q^{51} -10 i q^{53} + 4 q^{57} -6 i q^{59} -4 i q^{61} - q^{63} -12 i q^{67} + 16 i q^{69} + 14 q^{73} -22 i q^{75} + 8 q^{79} -11 q^{81} + 6 i q^{83} + 8 i q^{85} + 4 q^{87} -10 q^{89} -8 i q^{93} -8 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 16q^{15} - 4q^{17} + 16q^{23} - 22q^{25} - 8q^{31} + 4q^{41} + 8q^{47} + 2q^{49} + 8q^{57} - 2q^{63} + 28q^{73} + 16q^{79} - 22q^{81} + 8q^{87} - 20q^{89} - 16q^{95} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
1.00000i
1.00000i
0 2.00000i 0 4.00000i 0 1.00000 0 −1.00000 0
897.2 0 2.00000i 0 4.00000i 0 1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.b.h 2
4.b odd 2 1 1792.2.b.a 2
8.b even 2 1 inner 1792.2.b.h 2
8.d odd 2 1 1792.2.b.a 2
16.e even 4 1 112.2.a.a 1
16.e even 4 1 448.2.a.h 1
16.f odd 4 1 56.2.a.b 1
16.f odd 4 1 448.2.a.c 1
48.i odd 4 1 1008.2.a.m 1
48.i odd 4 1 4032.2.a.a 1
48.k even 4 1 504.2.a.h 1
48.k even 4 1 4032.2.a.d 1
80.i odd 4 1 2800.2.g.g 2
80.j even 4 1 1400.2.g.b 2
80.k odd 4 1 1400.2.a.a 1
80.q even 4 1 2800.2.a.bd 1
80.s even 4 1 1400.2.g.b 2
80.t odd 4 1 2800.2.g.g 2
112.j even 4 1 392.2.a.b 1
112.j even 4 1 3136.2.a.w 1
112.l odd 4 1 784.2.a.i 1
112.l odd 4 1 3136.2.a.c 1
112.u odd 12 2 392.2.i.a 2
112.v even 12 2 392.2.i.e 2
112.w even 12 2 784.2.i.j 2
112.x odd 12 2 784.2.i.b 2
176.i even 4 1 6776.2.a.h 1
208.o odd 4 1 9464.2.a.h 1
336.v odd 4 1 3528.2.a.b 1
336.y even 4 1 7056.2.a.c 1
336.br odd 12 2 3528.2.s.ba 2
336.bu even 12 2 3528.2.s.a 2
560.be even 4 1 9800.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 16.f odd 4 1
112.2.a.a 1 16.e even 4 1
392.2.a.b 1 112.j even 4 1
392.2.i.a 2 112.u odd 12 2
392.2.i.e 2 112.v even 12 2
448.2.a.c 1 16.f odd 4 1
448.2.a.h 1 16.e even 4 1
504.2.a.h 1 48.k even 4 1
784.2.a.i 1 112.l odd 4 1
784.2.i.b 2 112.x odd 12 2
784.2.i.j 2 112.w even 12 2
1008.2.a.m 1 48.i odd 4 1
1400.2.a.a 1 80.k odd 4 1
1400.2.g.b 2 80.j even 4 1
1400.2.g.b 2 80.s even 4 1
1792.2.b.a 2 4.b odd 2 1
1792.2.b.a 2 8.d odd 2 1
1792.2.b.h 2 1.a even 1 1 trivial
1792.2.b.h 2 8.b even 2 1 inner
2800.2.a.bd 1 80.q even 4 1
2800.2.g.g 2 80.i odd 4 1
2800.2.g.g 2 80.t odd 4 1
3136.2.a.c 1 112.l odd 4 1
3136.2.a.w 1 112.j even 4 1
3528.2.a.b 1 336.v odd 4 1
3528.2.s.a 2 336.bu even 12 2
3528.2.s.ba 2 336.br odd 12 2
4032.2.a.a 1 48.i odd 4 1
4032.2.a.d 1 48.k even 4 1
6776.2.a.h 1 176.i even 4 1
7056.2.a.c 1 336.y even 4 1
9464.2.a.h 1 208.o odd 4 1
9800.2.a.bj 1 560.be even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{5}^{2} + 16 \)
\( T_{11} \)
\( T_{23} - 8 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{2} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 34 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 54 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 4 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 82 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 106 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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