Properties

Label 867.2.h.d
Level $867$
Weight $2$
Character orbit 867.h
Analytic conductor $6.923$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{16}^{2} q^{2} -\zeta_{16}^{3} q^{3} -\zeta_{16}^{4} q^{4} -\zeta_{16}^{5} q^{6} -4 \zeta_{16} q^{7} -3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} +O(q^{10})\) \( q + \zeta_{16}^{2} q^{2} -\zeta_{16}^{3} q^{3} -\zeta_{16}^{4} q^{4} -\zeta_{16}^{5} q^{6} -4 \zeta_{16} q^{7} -3 \zeta_{16}^{6} q^{8} + \zeta_{16}^{6} q^{9} + 4 \zeta_{16}^{5} q^{11} + \zeta_{16}^{7} q^{12} -2 \zeta_{16}^{4} q^{13} -4 \zeta_{16}^{3} q^{14} + q^{16} - q^{18} -4 \zeta_{16}^{2} q^{19} + 4 \zeta_{16}^{4} q^{21} + 4 \zeta_{16}^{7} q^{22} + 4 \zeta_{16}^{5} q^{23} -3 \zeta_{16} q^{24} + 5 \zeta_{16}^{6} q^{25} -2 \zeta_{16}^{6} q^{26} + \zeta_{16} q^{27} + 4 \zeta_{16}^{5} q^{28} -4 \zeta_{16}^{3} q^{31} -5 \zeta_{16}^{2} q^{32} + 4 q^{33} + \zeta_{16}^{2} q^{36} -8 \zeta_{16}^{3} q^{37} -4 \zeta_{16}^{4} q^{38} + 2 \zeta_{16}^{7} q^{39} -8 \zeta_{16} q^{41} + 4 \zeta_{16}^{6} q^{42} + 4 \zeta_{16}^{6} q^{43} + 4 \zeta_{16} q^{44} + 4 \zeta_{16}^{7} q^{46} + 8 \zeta_{16}^{4} q^{47} -\zeta_{16}^{3} q^{48} + 9 \zeta_{16}^{2} q^{49} -5 q^{50} -2 q^{52} + 6 \zeta_{16}^{2} q^{53} + \zeta_{16}^{3} q^{54} + 12 \zeta_{16}^{7} q^{56} + 4 \zeta_{16}^{5} q^{57} -12 \zeta_{16}^{6} q^{59} + 8 \zeta_{16} q^{61} -4 \zeta_{16}^{5} q^{62} -4 \zeta_{16}^{7} q^{63} -7 \zeta_{16}^{4} q^{64} + 4 \zeta_{16}^{2} q^{66} -12 q^{67} + 4 q^{69} -12 \zeta_{16}^{3} q^{71} + 3 \zeta_{16}^{4} q^{72} -8 \zeta_{16}^{5} q^{74} + 5 \zeta_{16} q^{75} + 4 \zeta_{16}^{6} q^{76} -16 \zeta_{16}^{6} q^{77} -2 \zeta_{16} q^{78} -4 \zeta_{16}^{5} q^{79} -\zeta_{16}^{4} q^{81} -8 \zeta_{16}^{3} q^{82} -12 \zeta_{16}^{2} q^{83} + 4 q^{84} -4 q^{86} + 12 \zeta_{16}^{3} q^{88} -10 \zeta_{16}^{4} q^{89} + 8 \zeta_{16}^{5} q^{91} + 4 \zeta_{16} q^{92} + 4 \zeta_{16}^{6} q^{93} + 8 \zeta_{16}^{6} q^{94} + 5 \zeta_{16}^{5} q^{96} -16 \zeta_{16}^{7} q^{97} + 9 \zeta_{16}^{4} q^{98} -4 \zeta_{16}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{16} - 8q^{18} + 32q^{33} - 40q^{50} - 16q^{52} - 96q^{67} + 32q^{69} + 32q^{84} - 32q^{86} + O(q^{100}) \)

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\zeta_{16}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
688.1
0.923880 0.382683i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
0.707107 0.707107i −0.382683 + 0.923880i 1.00000i 0 0.382683 + 0.923880i −3.69552 + 1.53073i 2.12132 + 2.12132i −0.707107 0.707107i 0
688.2 0.707107 0.707107i 0.382683 0.923880i 1.00000i 0 −0.382683 0.923880i 3.69552 1.53073i 2.12132 + 2.12132i −0.707107 0.707107i 0
712.1 0.707107 + 0.707107i −0.382683 0.923880i 1.00000i 0 0.382683 0.923880i −3.69552 1.53073i 2.12132 2.12132i −0.707107 + 0.707107i 0
712.2 0.707107 + 0.707107i 0.382683 + 0.923880i 1.00000i 0 −0.382683 + 0.923880i 3.69552 + 1.53073i 2.12132 2.12132i −0.707107 + 0.707107i 0
733.1 −0.707107 0.707107i −0.923880 + 0.382683i 1.00000i 0 0.923880 + 0.382683i 1.53073 3.69552i −2.12132 + 2.12132i 0.707107 0.707107i 0
733.2 −0.707107 0.707107i 0.923880 0.382683i 1.00000i 0 −0.923880 0.382683i −1.53073 + 3.69552i −2.12132 + 2.12132i 0.707107 0.707107i 0
757.1 −0.707107 + 0.707107i −0.923880 0.382683i 1.00000i 0 0.923880 0.382683i 1.53073 + 3.69552i −2.12132 2.12132i 0.707107 + 0.707107i 0
757.2 −0.707107 + 0.707107i 0.923880 + 0.382683i 1.00000i 0 −0.923880 + 0.382683i −1.53073 3.69552i −2.12132 2.12132i 0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.2
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.d 8
17.b even 2 1 inner 867.2.h.d 8
17.c even 4 2 inner 867.2.h.d 8
17.d even 8 4 inner 867.2.h.d 8
17.e odd 16 2 51.2.d.b 2
17.e odd 16 1 867.2.a.a 1
17.e odd 16 1 867.2.a.b 1
17.e odd 16 4 867.2.e.d 4
51.i even 16 2 153.2.d.a 2
51.i even 16 1 2601.2.a.i 1
51.i even 16 1 2601.2.a.j 1
68.i even 16 2 816.2.c.c 2
85.o even 16 1 1275.2.d.b 2
85.o even 16 1 1275.2.d.d 2
85.p odd 16 2 1275.2.g.a 2
85.r even 16 1 1275.2.d.b 2
85.r even 16 1 1275.2.d.d 2
136.q odd 16 2 3264.2.c.e 2
136.s even 16 2 3264.2.c.d 2
204.t odd 16 2 2448.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 17.e odd 16 2
153.2.d.a 2 51.i even 16 2
816.2.c.c 2 68.i even 16 2
867.2.a.a 1 17.e odd 16 1
867.2.a.b 1 17.e odd 16 1
867.2.e.d 4 17.e odd 16 4
867.2.h.d 8 1.a even 1 1 trivial
867.2.h.d 8 17.b even 2 1 inner
867.2.h.d 8 17.c even 4 2 inner
867.2.h.d 8 17.d even 8 4 inner
1275.2.d.b 2 85.o even 16 1
1275.2.d.b 2 85.r even 16 1
1275.2.d.d 2 85.o even 16 1
1275.2.d.d 2 85.r even 16 1
1275.2.g.a 2 85.p odd 16 2
2448.2.c.j 2 204.t odd 16 2
2601.2.a.i 1 51.i even 16 1
2601.2.a.j 1 51.i even 16 1
3264.2.c.d 2 136.s even 16 2
3264.2.c.e 2 136.q odd 16 2

Hecke kernels

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

\( T_{2}^{4} + 1 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( 1 + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 65536 + T^{8} \)
$11$ \( 65536 + T^{8} \)
$13$ \( ( 4 + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( ( 256 + T^{4} )^{2} \)
$23$ \( 65536 + T^{8} \)
$29$ \( T^{8} \)
$31$ \( 65536 + T^{8} \)
$37$ \( 16777216 + T^{8} \)
$41$ \( 16777216 + T^{8} \)
$43$ \( ( 256 + T^{4} )^{2} \)
$47$ \( ( 64 + T^{2} )^{4} \)
$53$ \( ( 1296 + T^{4} )^{2} \)
$59$ \( ( 20736 + T^{4} )^{2} \)
$61$ \( 16777216 + T^{8} \)
$67$ \( ( 12 + T )^{8} \)
$71$ \( 429981696 + T^{8} \)
$73$ \( T^{8} \)
$79$ \( 65536 + T^{8} \)
$83$ \( ( 20736 + T^{4} )^{2} \)
$89$ \( ( 100 + T^{2} )^{4} \)
$97$ \( 4294967296 + T^{8} \)
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