Properties

Label 1792.2.m.b
Level $1792$
Weight $2$
Character orbit 1792.m
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + \beta_{6} q^{7} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( 1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{5} + \beta_{6} q^{7} + ( -1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{9} + ( \beta_{1} - 2 \beta_{4} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{13} + \beta_{1} q^{15} + ( 1 - \beta_{2} + \beta_{3} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( 1 - \beta_{2} ) q^{21} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{23} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{33} + ( \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} + ( -1 + 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{39} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{41} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{6} - 2 \beta_{7} ) q^{43} + ( -1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{45} + ( -5 + \beta_{2} - \beta_{3} ) q^{47} - q^{49} + ( -2 + \beta_{1} - 2 \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{51} + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{6} + \beta_{7} ) q^{53} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 8 \beta_{6} + 4 \beta_{7} ) q^{55} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{57} + ( 8 - 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{59} + ( 2 - \beta_{1} - 2 \beta_{3} + 3 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{61} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( -6 - 2 \beta_{1} + 4 \beta_{3} - 6 \beta_{6} - 2 \beta_{7} ) q^{67} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{69} + ( -4 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{71} + ( 4 - 4 \beta_{2} - 4 \beta_{3} + 6 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 6 - 4 \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{75} + ( 2 - \beta_{1} - 2 \beta_{5} - \beta_{7} ) q^{77} + ( \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{79} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 2 - 3 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{83} + ( -\beta_{1} + \beta_{7} ) q^{85} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} ) q^{91} + ( 2 - 3 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{93} + ( -10 - 6 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{95} + ( 9 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{97} + ( 3 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{5} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{5} + 8q^{11} - 12q^{13} + 8q^{17} - 4q^{19} + 4q^{21} + 8q^{27} - 8q^{31} - 16q^{33} + 4q^{35} + 8q^{37} + 24q^{43} - 12q^{45} - 40q^{47} - 8q^{49} - 24q^{51} - 16q^{53} + 52q^{59} + 20q^{61} - 24q^{65} - 32q^{67} - 8q^{69} + 28q^{75} + 8q^{77} + 16q^{81} + 12q^{83} + 12q^{91} + 40q^{93} - 80q^{95} + 72q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4 \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 3 \nu^{6} + 11 \nu^{5} - 17 \nu^{4} + 26 \nu^{3} - 19 \nu^{2} + 13 \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - 4 \nu^{6} + 14 \nu^{5} - 28 \nu^{4} + 43 \nu^{3} - 43 \nu^{2} + 29 \nu - 8 \)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 17 \nu^{6} + 59 \nu^{5} - 102 \nu^{4} + 146 \nu^{3} - 120 \nu^{2} + 65 \nu - 14 \)
\(\beta_{5}\)\(=\)\( -5 \nu^{7} + 17 \nu^{6} - 59 \nu^{5} + 103 \nu^{4} - 148 \nu^{3} + 127 \nu^{2} - 71 \nu + 19 \)
\(\beta_{6}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(\beta_{7}\)\(=\)\( -10 \nu^{7} + 35 \nu^{6} - 123 \nu^{5} + 220 \nu^{4} - 325 \nu^{3} + 285 \nu^{2} - 166 \nu + 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{7} + 5 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1} - 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{7} + 11 \beta_{6} - 3 \beta_{5} - \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 5 \beta_{1} + 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{7} - 13 \beta_{6} - 5 \beta_{5} - 10 \beta_{4} - 7 \beta_{3} + 18 \beta_{2} - 10 \beta_{1} + 18\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(50 \beta_{7} - 67 \beta_{6} + 11 \beta_{5} - 9 \beta_{4} + 38 \beta_{3} + 26 \beta_{2} + 18 \beta_{1} - 48\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(8 \beta_{7} - 7 \beta_{6} + 30 \beta_{5} + 33 \beta_{4} + 72 \beta_{3} - 61 \beta_{2} + 72 \beta_{1} - 122\)\()/2\)

Character values

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

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.500000 0.691860i
0.500000 1.44392i
0.500000 + 0.0297061i
0.500000 + 2.10607i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 0.0297061i
0.500000 2.10607i
0 −1.89897 + 1.89897i 0 0.372364 + 0.372364i 0 1.00000i 0 4.21215i 0
449.2 0 −1.23681 + 1.23681i 0 −0.571717 0.571717i 0 1.00000i 0 0.0594122i 0
449.3 0 0.236813 0.236813i 0 2.98593 + 2.98593i 0 1.00000i 0 2.88784i 0
449.4 0 0.898966 0.898966i 0 −0.786578 0.786578i 0 1.00000i 0 1.38372i 0
1345.1 0 −1.89897 1.89897i 0 0.372364 0.372364i 0 1.00000i 0 4.21215i 0
1345.2 0 −1.23681 1.23681i 0 −0.571717 + 0.571717i 0 1.00000i 0 0.0594122i 0
1345.3 0 0.236813 + 0.236813i 0 2.98593 2.98593i 0 1.00000i 0 2.88784i 0
1345.4 0 0.898966 + 0.898966i 0 −0.786578 + 0.786578i 0 1.00000i 0 1.38372i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1345.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.m.b yes 8
4.b odd 2 1 1792.2.m.d yes 8
8.b even 2 1 1792.2.m.c yes 8
8.d odd 2 1 1792.2.m.a 8
16.e even 4 1 inner 1792.2.m.b yes 8
16.e even 4 1 1792.2.m.c yes 8
16.f odd 4 1 1792.2.m.a 8
16.f odd 4 1 1792.2.m.d yes 8
32.g even 8 1 7168.2.a.s 4
32.g even 8 1 7168.2.a.w 4
32.h odd 8 1 7168.2.a.t 4
32.h odd 8 1 7168.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.a 8 8.d odd 2 1
1792.2.m.a 8 16.f odd 4 1
1792.2.m.b yes 8 1.a even 1 1 trivial
1792.2.m.b yes 8 16.e even 4 1 inner
1792.2.m.c yes 8 8.b even 2 1
1792.2.m.c yes 8 16.e even 4 1
1792.2.m.d yes 8 4.b odd 2 1
1792.2.m.d yes 8 16.f odd 4 1
7168.2.a.s 4 32.g even 8 1
7168.2.a.t 4 32.h odd 8 1
7168.2.a.w 4 32.g even 8 1
7168.2.a.x 4 32.h odd 8 1

Hecke kernels

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

\( T_{3}^{8} + 4 T_{3}^{7} + 8 T_{3}^{6} + 8 T_{3}^{3} + 32 T_{3}^{2} - 16 T_{3} + 4 \)
\( T_{5}^{8} - 4 T_{5}^{7} + 8 T_{5}^{6} + 24 T_{5}^{5} + 32 T_{5}^{4} + 8 T_{5}^{3} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 - 16 T + 32 T^{2} + 8 T^{3} + 8 T^{6} + 4 T^{7} + T^{8} \)
$5$ \( 4 + 8 T^{3} + 32 T^{4} + 24 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( 256 + 1536 T + 4608 T^{2} - 1408 T^{3} + 224 T^{4} + 32 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 3844 + 2480 T + 800 T^{2} + 184 T^{3} + 320 T^{4} + 208 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$17$ \( ( 8 + 16 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$19$ \( 31684 - 48416 T + 36992 T^{2} + 16152 T^{3} + 3488 T^{4} + 24 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( 16 + 2272 T^{2} + 792 T^{4} + 56 T^{6} + T^{8} \)
$29$ \( 16 - 512 T + 8192 T^{2} + 6656 T^{3} + 2696 T^{4} + 128 T^{5} + T^{8} \)
$31$ \( ( 8 + 32 T - 44 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$37$ \( 258064 - 268224 T + 139392 T^{2} - 35744 T^{3} + 4616 T^{4} - 48 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( 61504 + 125504 T^{2} + 8736 T^{4} + 184 T^{6} + T^{8} \)
$43$ \( 984064 - 317440 T + 51200 T^{2} - 5888 T^{3} + 5120 T^{4} - 1664 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$47$ \( ( 392 + 400 T + 140 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$53$ \( 80656 + 45440 T + 12800 T^{2} + 1344 T^{3} + 968 T^{4} + 480 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$59$ \( 21104836 - 16905920 T + 6771200 T^{2} - 1534872 T^{3} + 223136 T^{4} - 21384 T^{5} + 1352 T^{6} - 52 T^{7} + T^{8} \)
$61$ \( 40934404 - 15867040 T + 3075200 T^{2} - 301560 T^{3} + 17696 T^{4} - 1080 T^{5} + 200 T^{6} - 20 T^{7} + T^{8} \)
$67$ \( 20647936 + 5816320 T + 819200 T^{2} + 43008 T^{3} + 15488 T^{4} + 3840 T^{5} + 512 T^{6} + 32 T^{7} + T^{8} \)
$71$ \( 5837056 + 1673216 T^{2} + 44320 T^{4} + 384 T^{6} + T^{8} \)
$73$ \( 2027776 + 1834752 T^{2} + 44896 T^{4} + 368 T^{6} + T^{8} \)
$79$ \( ( 19088 + 128 T - 288 T^{2} + T^{4} )^{2} \)
$83$ \( 9604 + 76048 T + 301088 T^{2} - 50040 T^{3} + 4160 T^{4} + 16 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
$89$ \( 24760576 + 3843072 T^{2} + 96032 T^{4} + 576 T^{6} + T^{8} \)
$97$ \( ( -3064 - 960 T + 388 T^{2} - 36 T^{3} + T^{4} )^{2} \)
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