Properties

Label 1008.6.a.bq
Level 1008
Weight 6
Character orbit 1008.a
Self dual yes
Analytic conductor 161.667
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(161.666890371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 9 + 5 \beta ) q^{5} -49 q^{7} +O(q^{10})\) \( q + ( 9 + 5 \beta ) q^{5} -49 q^{7} + ( 198 + 62 \beta ) q^{11} + ( -175 - 63 \beta ) q^{13} + ( -900 - 38 \beta ) q^{17} + ( 1633 - 9 \beta ) q^{19} + ( 1044 - 284 \beta ) q^{23} + ( -1619 + 90 \beta ) q^{25} + ( -3348 + 126 \beta ) q^{29} + ( 10 + 270 \beta ) q^{31} + ( -441 - 245 \beta ) q^{35} + ( 3116 - 270 \beta ) q^{37} + ( 3024 - 546 \beta ) q^{41} + ( 1510 - 2394 \beta ) q^{43} + ( 5850 - 1874 \beta ) q^{47} + 2401 q^{49} + ( -4734 - 104 \beta ) q^{53} + ( 19452 + 1548 \beta ) q^{55} + ( -21969 + 1025 \beta ) q^{59} + ( -32377 + 2403 \beta ) q^{61} + ( -19530 - 1442 \beta ) q^{65} + ( -12392 - 972 \beta ) q^{67} + ( 48708 + 2100 \beta ) q^{71} + ( 8726 + 2628 \beta ) q^{73} + ( -9702 - 3038 \beta ) q^{77} + ( -25628 + 7452 \beta ) q^{79} + ( 58779 - 7875 \beta ) q^{83} + ( -18930 - 4842 \beta ) q^{85} + ( -42138 - 11104 \beta ) q^{89} + ( 8575 + 3087 \beta ) q^{91} + ( 12132 + 8084 \beta ) q^{95} + ( 10388 - 4410 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 18q^{5} - 98q^{7} + O(q^{10}) \) \( 2q + 18q^{5} - 98q^{7} + 396q^{11} - 350q^{13} - 1800q^{17} + 3266q^{19} + 2088q^{23} - 3238q^{25} - 6696q^{29} + 20q^{31} - 882q^{35} + 6232q^{37} + 6048q^{41} + 3020q^{43} + 11700q^{47} + 4802q^{49} - 9468q^{53} + 38904q^{55} - 43938q^{59} - 64754q^{61} - 39060q^{65} - 24784q^{67} + 97416q^{71} + 17452q^{73} - 19404q^{77} - 51256q^{79} + 117558q^{83} - 37860q^{85} - 84276q^{89} + 17150q^{91} + 24264q^{95} + 20776q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.27492
4.27492
0 0 0 −28.7492 0 −49.0000 0 0 0
1.2 0 0 0 46.7492 0 −49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.6.a.bq 2
3.b odd 2 1 112.6.a.h 2
4.b odd 2 1 63.6.a.f 2
12.b even 2 1 7.6.a.b 2
21.c even 2 1 784.6.a.v 2
24.f even 2 1 448.6.a.w 2
24.h odd 2 1 448.6.a.u 2
28.d even 2 1 441.6.a.l 2
60.h even 2 1 175.6.a.c 2
60.l odd 4 2 175.6.b.c 4
84.h odd 2 1 49.6.a.f 2
84.j odd 6 2 49.6.c.d 4
84.n even 6 2 49.6.c.e 4
132.d odd 2 1 847.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 12.b even 2 1
49.6.a.f 2 84.h odd 2 1
49.6.c.d 4 84.j odd 6 2
49.6.c.e 4 84.n even 6 2
63.6.a.f 2 4.b odd 2 1
112.6.a.h 2 3.b odd 2 1
175.6.a.c 2 60.h even 2 1
175.6.b.c 4 60.l odd 4 2
441.6.a.l 2 28.d even 2 1
448.6.a.u 2 24.h odd 2 1
448.6.a.w 2 24.f even 2 1
784.6.a.v 2 21.c even 2 1
847.6.a.c 2 132.d odd 2 1
1008.6.a.bq 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 18 T_{5} - 1344 \)
\( T_{11}^{2} - 396 T_{11} - 179904 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 18 T + 4906 T^{2} - 56250 T^{3} + 9765625 T^{4} \)
$7$ \( ( 1 + 49 T )^{2} \)
$11$ \( 1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 + 350 T + 546978 T^{2} + 129952550 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 + 1800 T + 3567406 T^{2} + 2555742600 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 - 3266 T + 7614270 T^{2} - 8086939334 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 - 2088 T + 9365230 T^{2} - 13439084184 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 + 6696 T + 51326470 T^{2} + 137342653704 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 - 20 T + 53103102 T^{2} - 572583020 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 - 6048 T + 223864366 T^{2} - 700698303648 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 - 3020 T - 30383466 T^{2} - 443965497860 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 - 11700 T + 292735582 T^{2} - 2683336581900 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 + 9468 T + 858185230 T^{2} + 3959474927724 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 + 24784 T + 2799959190 T^{2} + 33461500651888 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 - 17452 T + 3828622374 T^{2} - 36179245441036 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 + 51256 T + 3645565854 T^{2} + 157717602787144 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 - 117558 T + 7798161502 T^{2} - 463065739909794 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 + 84276 T + 5915697430 T^{2} + 470602194123924 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 - 20776 T + 16174049358 T^{2} - 178410581179432 T^{3} + 73742412689492826049 T^{4} \)
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