Properties

Label 1792.2.b.g
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 14)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + q^{7} - q^{9} -4 i q^{13} + 6 q^{17} -2 i q^{19} + 2 i q^{21} + 5 q^{25} + 4 i q^{27} -6 i q^{29} + 4 q^{31} -2 i q^{37} + 8 q^{39} -6 q^{41} + 8 i q^{43} + 12 q^{47} + q^{49} + 12 i q^{51} -6 i q^{53} + 4 q^{57} -6 i q^{59} + 8 i q^{61} - q^{63} + 4 i q^{67} -2 q^{73} + 10 i q^{75} -8 q^{79} -11 q^{81} + 6 i q^{83} + 12 q^{87} + 6 q^{89} -4 i q^{91} + 8 i q^{93} -10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} + 12q^{17} + 10q^{25} + 8q^{31} + 16q^{39} - 12q^{41} + 24q^{47} + 2q^{49} + 8q^{57} - 2q^{63} - 4q^{73} - 16q^{79} - 22q^{81} + 24q^{87} + 12q^{89} - 20q^{97} + O(q^{100}) \)

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 0 0 1.00000 0 −1.00000 0
897.2 0 2.00000i 0 0 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.g 2
4.b odd 2 1 1792.2.b.c 2
8.b even 2 1 inner 1792.2.b.g 2
8.d odd 2 1 1792.2.b.c 2
16.e even 4 1 112.2.a.c 1
16.e even 4 1 448.2.a.a 1
16.f odd 4 1 14.2.a.a 1
16.f odd 4 1 448.2.a.g 1
48.i odd 4 1 1008.2.a.h 1
48.i odd 4 1 4032.2.a.r 1
48.k even 4 1 126.2.a.b 1
48.k even 4 1 4032.2.a.w 1
80.i odd 4 1 2800.2.g.h 2
80.j even 4 1 350.2.c.d 2
80.k odd 4 1 350.2.a.f 1
80.q even 4 1 2800.2.a.g 1
80.s even 4 1 350.2.c.d 2
80.t odd 4 1 2800.2.g.h 2
112.j even 4 1 98.2.a.a 1
112.j even 4 1 3136.2.a.e 1
112.l odd 4 1 784.2.a.b 1
112.l odd 4 1 3136.2.a.z 1
112.u odd 12 2 98.2.c.b 2
112.v even 12 2 98.2.c.a 2
112.w even 12 2 784.2.i.c 2
112.x odd 12 2 784.2.i.i 2
144.u even 12 2 1134.2.f.f 2
144.v odd 12 2 1134.2.f.l 2
176.i even 4 1 1694.2.a.e 1
208.l even 4 1 2366.2.d.b 2
208.o odd 4 1 2366.2.a.j 1
208.s even 4 1 2366.2.d.b 2
240.t even 4 1 3150.2.a.i 1
240.z odd 4 1 3150.2.g.j 2
240.bd odd 4 1 3150.2.g.j 2
272.k odd 4 1 4046.2.a.f 1
304.m even 4 1 5054.2.a.c 1
336.v odd 4 1 882.2.a.i 1
336.y even 4 1 7056.2.a.bd 1
336.br odd 12 2 882.2.g.d 2
336.bu even 12 2 882.2.g.c 2
368.i even 4 1 7406.2.a.a 1
560.u odd 4 1 2450.2.c.c 2
560.be even 4 1 2450.2.a.t 1
560.bm odd 4 1 2450.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 16.f odd 4 1
98.2.a.a 1 112.j even 4 1
98.2.c.a 2 112.v even 12 2
98.2.c.b 2 112.u odd 12 2
112.2.a.c 1 16.e even 4 1
126.2.a.b 1 48.k even 4 1
350.2.a.f 1 80.k odd 4 1
350.2.c.d 2 80.j even 4 1
350.2.c.d 2 80.s even 4 1
448.2.a.a 1 16.e even 4 1
448.2.a.g 1 16.f odd 4 1
784.2.a.b 1 112.l odd 4 1
784.2.i.c 2 112.w even 12 2
784.2.i.i 2 112.x odd 12 2
882.2.a.i 1 336.v odd 4 1
882.2.g.c 2 336.bu even 12 2
882.2.g.d 2 336.br odd 12 2
1008.2.a.h 1 48.i odd 4 1
1134.2.f.f 2 144.u even 12 2
1134.2.f.l 2 144.v odd 12 2
1694.2.a.e 1 176.i even 4 1
1792.2.b.c 2 4.b odd 2 1
1792.2.b.c 2 8.d odd 2 1
1792.2.b.g 2 1.a even 1 1 trivial
1792.2.b.g 2 8.b even 2 1 inner
2366.2.a.j 1 208.o odd 4 1
2366.2.d.b 2 208.l even 4 1
2366.2.d.b 2 208.s even 4 1
2450.2.a.t 1 560.be even 4 1
2450.2.c.c 2 560.u odd 4 1
2450.2.c.c 2 560.bm odd 4 1
2800.2.a.g 1 80.q even 4 1
2800.2.g.h 2 80.i odd 4 1
2800.2.g.h 2 80.t odd 4 1
3136.2.a.e 1 112.j even 4 1
3136.2.a.z 1 112.l odd 4 1
3150.2.a.i 1 240.t even 4 1
3150.2.g.j 2 240.z odd 4 1
3150.2.g.j 2 240.bd odd 4 1
4032.2.a.r 1 48.i odd 4 1
4032.2.a.w 1 48.k even 4 1
4046.2.a.f 1 272.k odd 4 1
5054.2.a.c 1 304.m even 4 1
7056.2.a.bd 1 336.y even 4 1
7406.2.a.a 1 368.i even 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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