Properties

Label 867.2.h.j
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.1963501163244660295991296.1
Defining polynomial: \(x^{16} + 1889 x^{8} + 65536\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{5} ) q^{2} + \beta_{7} q^{3} + ( 3 \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{14} - \beta_{15} ) q^{5} -\beta_{11} q^{6} + ( 5 \beta_{12} - \beta_{13} ) q^{8} + \beta_{12} q^{9} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{5} ) q^{2} + \beta_{7} q^{3} + ( 3 \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{14} - \beta_{15} ) q^{5} -\beta_{11} q^{6} + ( 5 \beta_{12} - \beta_{13} ) q^{8} + \beta_{12} q^{9} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{10} + ( -\beta_{10} + \beta_{11} ) q^{11} + ( -2 \beta_{14} - \beta_{15} ) q^{12} + ( -2 \beta_{8} + \beta_{9} ) q^{13} + ( -2 \beta_{3} - \beta_{5} ) q^{15} + 3 \beta_{4} q^{16} + \beta_{4} q^{18} + 3 \beta_{5} q^{19} + ( 4 \beta_{6} - 6 \beta_{7} ) q^{20} + 4 \beta_{14} q^{22} + ( 5 \beta_{10} - \beta_{11} ) q^{23} + ( \beta_{1} - 4 \beta_{2} ) q^{24} + ( -3 \beta_{12} + 3 \beta_{13} ) q^{25} + ( 2 \beta_{12} + 2 \beta_{13} ) q^{26} -\beta_{2} q^{27} + ( 2 \beta_{14} - 4 \beta_{15} ) q^{29} + ( -6 \beta_{8} - 2 \beta_{9} ) q^{30} + ( -2 \beta_{6} - 2 \beta_{7} ) q^{31} + ( -5 \beta_{3} - \beta_{5} ) q^{32} + ( -1 - \beta_{4} ) q^{33} + ( -3 \beta_{3} - \beta_{5} ) q^{36} -2 \beta_{6} q^{37} + 12 \beta_{8} q^{38} + ( 3 \beta_{14} - \beta_{15} ) q^{39} + ( 8 \beta_{10} + 6 \beta_{11} ) q^{40} + ( -\beta_{1} + \beta_{2} ) q^{41} -3 \beta_{13} q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( \beta_{10} + \beta_{11} ) q^{45} + ( -4 \beta_{14} + 4 \beta_{15} ) q^{46} + ( 8 \beta_{8} + 2 \beta_{9} ) q^{47} + 3 \beta_{6} q^{48} -7 \beta_{3} q^{49} + ( 12 - 3 \beta_{4} ) q^{50} + 2 q^{52} + ( -6 \beta_{3} - 4 \beta_{5} ) q^{53} + \beta_{6} q^{54} + ( 4 \beta_{8} + \beta_{9} ) q^{55} + ( 3 \beta_{10} - 3 \beta_{11} ) q^{57} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{58} + ( -4 \beta_{12} + 2 \beta_{13} ) q^{59} + ( -10 \beta_{12} + 4 \beta_{13} ) q^{60} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{61} -8 \beta_{10} q^{62} + ( -3 \beta_{8} + \beta_{9} ) q^{64} + ( -\beta_{6} - \beta_{7} ) q^{65} + 4 \beta_{3} q^{66} -4 q^{67} + ( 5 + \beta_{4} ) q^{69} + ( -4 \beta_{6} - 4 \beta_{7} ) q^{71} + ( -5 \beta_{8} - \beta_{9} ) q^{72} + ( 2 \beta_{14} + 4 \beta_{15} ) q^{73} + ( -8 \beta_{10} - 2 \beta_{11} ) q^{74} -3 \beta_{1} q^{75} + ( 12 \beta_{12} - 6 \beta_{13} ) q^{76} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{78} + ( 6 \beta_{10} - 6 \beta_{11} ) q^{79} + ( 12 \beta_{14} + 6 \beta_{15} ) q^{80} -\beta_{8} q^{81} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{82} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{83} -12 q^{86} + ( -2 \beta_{3} - 4 \beta_{5} ) q^{87} -4 \beta_{6} q^{88} + ( 2 \beta_{8} - 2 \beta_{9} ) q^{89} + ( 4 \beta_{14} + 2 \beta_{15} ) q^{90} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{92} -2 \beta_{13} q^{93} + ( 16 \beta_{12} - 8 \beta_{13} ) q^{94} + ( 3 \beta_{1} - 9 \beta_{2} ) q^{95} + ( 4 \beta_{10} + \beta_{11} ) q^{96} + ( -8 \beta_{14} + 2 \beta_{15} ) q^{97} + ( -7 \beta_{8} - 7 \beta_{9} ) q^{98} + ( -\beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 24q^{16} - 8q^{18} - 8q^{33} + 216q^{50} + 32q^{52} - 64q^{67} + 72q^{69} - 192q^{86} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 1889 x^{8} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + 1165 \nu \)\()/1764\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{10} + 2929 \nu^{2} \)\()/7056\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{8} + 724 \)\()/441\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{10} - 7589 \nu^{2} \)\()/7056\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} + 2929 \nu^{3} \)\()/7056\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{11} + 7589 \nu^{3} \)\()/28224\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{12} + 2145 \nu^{4} \)\()/12544\)
\(\beta_{9}\)\(=\)\((\)\( -29 \nu^{12} - 49661 \nu^{4} \)\()/112896\)
\(\beta_{10}\)\(=\)\((\)\( -29 \nu^{13} - 49661 \nu^{5} \)\()/451584\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{13} + 2145 \nu^{5} \)\()/12544\)
\(\beta_{12}\)\(=\)\((\)\( 65 \nu^{14} + 126881 \nu^{6} \)\()/1806336\)
\(\beta_{13}\)\(=\)\((\)\( 181 \nu^{14} + 325525 \nu^{6} \)\()/1806336\)
\(\beta_{14}\)\(=\)\((\)\( -181 \nu^{15} - 325525 \nu^{7} \)\()/7225344\)
\(\beta_{15}\)\(=\)\((\)\( 65 \nu^{15} + 126881 \nu^{7} \)\()/1806336\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 5 \beta_{3}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 5 \beta_{6}\)
\(\nu^{4}\)\(=\)\(9 \beta_{9} + 29 \beta_{8}\)
\(\nu^{5}\)\(=\)\(29 \beta_{11} + 36 \beta_{10}\)
\(\nu^{6}\)\(=\)\(-65 \beta_{13} + 181 \beta_{12}\)
\(\nu^{7}\)\(=\)\(181 \beta_{15} + 260 \beta_{14}\)
\(\nu^{8}\)\(=\)\(441 \beta_{4} - 724\)
\(\nu^{9}\)\(=\)\(1764 \beta_{2} - 1165 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-2929 \beta_{5} - 7589 \beta_{3}\)
\(\nu^{11}\)\(=\)\(11716 \beta_{7} - 7589 \beta_{6}\)
\(\nu^{12}\)\(=\)\(-19305 \beta_{9} - 49661 \beta_{8}\)
\(\nu^{13}\)\(=\)\(-49661 \beta_{11} - 77220 \beta_{10}\)
\(\nu^{14}\)\(=\)\(126881 \beta_{13} - 325525 \beta_{12}\)
\(\nu^{15}\)\(=\)\(-325525 \beta_{15} - 507524 \beta_{14}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1
−1.44269 + 0.597580i
1.44269 0.597580i
2.36657 0.980264i
−2.36657 + 0.980264i
−1.44269 0.597580i
1.44269 + 0.597580i
2.36657 + 0.980264i
−2.36657 0.980264i
−0.980264 + 2.36657i
0.980264 2.36657i
0.597580 1.44269i
−0.597580 + 1.44269i
−0.980264 2.36657i
0.980264 + 2.36657i
0.597580 + 1.44269i
−0.597580 1.44269i
−1.10418 + 1.10418i −0.382683 + 0.923880i 0.438447i −0.518807 0.214897i −0.597580 1.44269i 0 −1.72424 1.72424i −0.707107 0.707107i 0.810145 0.335573i
688.2 −1.10418 + 1.10418i 0.382683 0.923880i 0.438447i 0.518807 + 0.214897i 0.597580 + 1.44269i 0 −1.72424 1.72424i −0.707107 0.707107i −0.810145 + 0.335573i
688.3 1.81129 1.81129i −0.382683 + 0.923880i 4.56155i 3.29045 + 1.36295i 0.980264 + 2.36657i 0 −4.63972 4.63972i −0.707107 0.707107i 8.42865 3.49126i
688.4 1.81129 1.81129i 0.382683 0.923880i 4.56155i −3.29045 1.36295i −0.980264 2.36657i 0 −4.63972 4.63972i −0.707107 0.707107i −8.42865 + 3.49126i
712.1 −1.10418 1.10418i −0.382683 0.923880i 0.438447i −0.518807 + 0.214897i −0.597580 + 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i 0.810145 + 0.335573i
712.2 −1.10418 1.10418i 0.382683 + 0.923880i 0.438447i 0.518807 0.214897i 0.597580 1.44269i 0 −1.72424 + 1.72424i −0.707107 + 0.707107i −0.810145 0.335573i
712.3 1.81129 + 1.81129i −0.382683 0.923880i 4.56155i 3.29045 1.36295i 0.980264 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i 8.42865 + 3.49126i
712.4 1.81129 + 1.81129i 0.382683 + 0.923880i 4.56155i −3.29045 + 1.36295i −0.980264 + 2.36657i 0 −4.63972 + 4.63972i −0.707107 + 0.707107i −8.42865 3.49126i
733.1 −1.81129 1.81129i −0.923880 + 0.382683i 4.56155i −1.36295 3.29045i 2.36657 + 0.980264i 0 4.63972 4.63972i 0.707107 0.707107i −3.49126 + 8.42865i
733.2 −1.81129 1.81129i 0.923880 0.382683i 4.56155i 1.36295 + 3.29045i −2.36657 0.980264i 0 4.63972 4.63972i 0.707107 0.707107i 3.49126 8.42865i
733.3 1.10418 + 1.10418i −0.923880 + 0.382683i 0.438447i 0.214897 + 0.518807i −1.44269 0.597580i 0 1.72424 1.72424i 0.707107 0.707107i −0.335573 + 0.810145i
733.4 1.10418 + 1.10418i 0.923880 0.382683i 0.438447i −0.214897 0.518807i 1.44269 + 0.597580i 0 1.72424 1.72424i 0.707107 0.707107i 0.335573 0.810145i
757.1 −1.81129 + 1.81129i −0.923880 0.382683i 4.56155i −1.36295 + 3.29045i 2.36657 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i −3.49126 8.42865i
757.2 −1.81129 + 1.81129i 0.923880 + 0.382683i 4.56155i 1.36295 3.29045i −2.36657 + 0.980264i 0 4.63972 + 4.63972i 0.707107 + 0.707107i 3.49126 + 8.42865i
757.3 1.10418 1.10418i −0.923880 0.382683i 0.438447i 0.214897 0.518807i −1.44269 + 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i −0.335573 0.810145i
757.4 1.10418 1.10418i 0.923880 + 0.382683i 0.438447i −0.214897 + 0.518807i 1.44269 0.597580i 0 1.72424 + 1.72424i 0.707107 + 0.707107i 0.335573 + 0.810145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.h.j 16
17.b even 2 1 inner 867.2.h.j 16
17.c even 4 2 inner 867.2.h.j 16
17.d even 8 4 inner 867.2.h.j 16
17.e odd 16 1 51.2.a.b 2
17.e odd 16 1 867.2.a.f 2
17.e odd 16 2 867.2.d.c 4
17.e odd 16 4 867.2.e.f 8
51.i even 16 1 153.2.a.e 2
51.i even 16 1 2601.2.a.t 2
68.i even 16 1 816.2.a.m 2
85.o even 16 1 1275.2.b.d 4
85.p odd 16 1 1275.2.a.n 2
85.r even 16 1 1275.2.b.d 4
119.p even 16 1 2499.2.a.o 2
136.q odd 16 1 3264.2.a.bl 2
136.s even 16 1 3264.2.a.bg 2
187.m even 16 1 6171.2.a.p 2
204.t odd 16 1 2448.2.a.v 2
221.y odd 16 1 8619.2.a.q 2
255.be even 16 1 3825.2.a.s 2
357.be odd 16 1 7497.2.a.v 2
408.bg odd 16 1 9792.2.a.cz 2
408.bm even 16 1 9792.2.a.cy 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b 2 17.e odd 16 1
153.2.a.e 2 51.i even 16 1
816.2.a.m 2 68.i even 16 1
867.2.a.f 2 17.e odd 16 1
867.2.d.c 4 17.e odd 16 2
867.2.e.f 8 17.e odd 16 4
867.2.h.j 16 1.a even 1 1 trivial
867.2.h.j 16 17.b even 2 1 inner
867.2.h.j 16 17.c even 4 2 inner
867.2.h.j 16 17.d even 8 4 inner
1275.2.a.n 2 85.p odd 16 1
1275.2.b.d 4 85.o even 16 1
1275.2.b.d 4 85.r even 16 1
2448.2.a.v 2 204.t odd 16 1
2499.2.a.o 2 119.p even 16 1
2601.2.a.t 2 51.i even 16 1
3264.2.a.bg 2 136.s even 16 1
3264.2.a.bl 2 136.q odd 16 1
3825.2.a.s 2 255.be even 16 1
6171.2.a.p 2 187.m even 16 1
7497.2.a.v 2 357.be odd 16 1
8619.2.a.q 2 221.y odd 16 1
9792.2.a.cy 2 408.bm even 16 1
9792.2.a.cz 2 408.bg odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{8} + 49 T_{2}^{4} + 256 \)
\( T_{5}^{16} + 25889 T_{5}^{8} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 + 49 T^{4} + T^{8} )^{2} \)
$3$ \( ( 1 + T^{8} )^{2} \)
$5$ \( 256 + 25889 T^{8} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( 65536 + 1889 T^{8} + T^{16} \)
$13$ \( ( 4 + 21 T^{2} + T^{4} )^{4} \)
$17$ \( T^{16} \)
$19$ \( ( 1679616 + 3969 T^{4} + T^{8} )^{2} \)
$23$ \( 4294967296 + 3437249 T^{8} + T^{16} \)
$29$ \( ( 21381376 + T^{8} )^{2} \)
$31$ \( 4294967296 + 483584 T^{8} + T^{16} \)
$37$ \( 4294967296 + 483584 T^{8} + T^{16} \)
$41$ \( 256 + 25889 T^{8} + T^{16} \)
$43$ \( ( 1679616 + 3969 T^{4} + T^{8} )^{2} \)
$47$ \( ( 1024 + 132 T^{2} + T^{4} )^{4} \)
$53$ \( ( 7311616 + 22816 T^{4} + T^{8} )^{2} \)
$59$ \( ( 4096 + 2576 T^{4} + T^{8} )^{2} \)
$61$ \( 16777216 + 47988992 T^{8} + T^{16} \)
$67$ \( ( 4 + T )^{16} \)
$71$ \( 281474976710656 + 123797504 T^{8} + T^{16} \)
$73$ \( 53459728531456 + 505946624 T^{8} + T^{16} \)
$79$ \( 184884258895036416 + 3172794624 T^{8} + T^{16} \)
$83$ \( ( 4096 + 6928 T^{4} + T^{8} )^{2} \)
$89$ \( ( 64 + 52 T^{2} + T^{4} )^{4} \)
$97$ \( 1099511627776 + 234324224 T^{8} + T^{16} \)
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