Properties

Label 867.2.e.d
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} -\zeta_{8}^{3} q^{3} + q^{4} -\zeta_{8} q^{6} -4 \zeta_{8} q^{7} -3 \zeta_{8}^{2} q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} -\zeta_{8}^{3} q^{3} + q^{4} -\zeta_{8} q^{6} -4 \zeta_{8} q^{7} -3 \zeta_{8}^{2} q^{8} -\zeta_{8}^{2} q^{9} -4 \zeta_{8} q^{11} -\zeta_{8}^{3} q^{12} -2 q^{13} + 4 \zeta_{8}^{3} q^{14} - q^{16} - q^{18} + 4 \zeta_{8}^{2} q^{19} -4 q^{21} + 4 \zeta_{8}^{3} q^{22} + 4 \zeta_{8} q^{23} -3 \zeta_{8} q^{24} + 5 \zeta_{8}^{2} q^{25} + 2 \zeta_{8}^{2} q^{26} -\zeta_{8} q^{27} -4 \zeta_{8} q^{28} + 4 \zeta_{8}^{3} q^{31} -5 \zeta_{8}^{2} q^{32} -4 q^{33} -\zeta_{8}^{2} q^{36} -8 \zeta_{8}^{3} q^{37} + 4 q^{38} + 2 \zeta_{8}^{3} q^{39} -8 \zeta_{8} q^{41} + 4 \zeta_{8}^{2} q^{42} -4 \zeta_{8}^{2} q^{43} -4 \zeta_{8} q^{44} -4 \zeta_{8}^{3} q^{46} + 8 q^{47} + \zeta_{8}^{3} q^{48} + 9 \zeta_{8}^{2} q^{49} + 5 q^{50} -2 q^{52} -6 \zeta_{8}^{2} q^{53} + \zeta_{8}^{3} q^{54} + 12 \zeta_{8}^{3} q^{56} + 4 \zeta_{8} q^{57} -12 \zeta_{8}^{2} q^{59} -8 \zeta_{8} q^{61} + 4 \zeta_{8} q^{62} + 4 \zeta_{8}^{3} q^{63} -7 q^{64} + 4 \zeta_{8}^{2} q^{66} + 12 q^{67} + 4 q^{69} -12 \zeta_{8}^{3} q^{71} -3 q^{72} -8 \zeta_{8} q^{74} + 5 \zeta_{8} q^{75} + 4 \zeta_{8}^{2} q^{76} + 16 \zeta_{8}^{2} q^{77} + 2 \zeta_{8} q^{78} + 4 \zeta_{8} q^{79} - q^{81} + 8 \zeta_{8}^{3} q^{82} -12 \zeta_{8}^{2} q^{83} -4 q^{84} -4 q^{86} + 12 \zeta_{8}^{3} q^{88} + 10 q^{89} + 8 \zeta_{8} q^{91} + 4 \zeta_{8} q^{92} + 4 \zeta_{8}^{2} q^{93} -8 \zeta_{8}^{2} q^{94} -5 \zeta_{8} q^{96} + 16 \zeta_{8}^{3} q^{97} + 9 q^{98} + 4 \zeta_{8}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 8q^{13} - 4q^{16} - 4q^{18} - 16q^{21} - 16q^{33} + 16q^{38} + 32q^{47} + 20q^{50} - 8q^{52} - 28q^{64} + 48q^{67} + 16q^{69} - 12q^{72} - 4q^{81} - 16q^{84} - 16q^{86} + 40q^{89} + 36q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
616.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000i −0.707107 + 0.707107i 1.00000 0 0.707107 + 0.707107i 2.82843 + 2.82843i 3.00000i 1.00000i 0
616.2 1.00000i 0.707107 0.707107i 1.00000 0 −0.707107 0.707107i −2.82843 2.82843i 3.00000i 1.00000i 0
829.1 1.00000i −0.707107 0.707107i 1.00000 0 0.707107 0.707107i 2.82843 2.82843i 3.00000i 1.00000i 0
829.2 1.00000i 0.707107 + 0.707107i 1.00000 0 −0.707107 + 0.707107i −2.82843 + 2.82843i 3.00000i 1.00000i 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
17.b even 2 1 inner
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.e.d 4
17.b even 2 1 inner 867.2.e.d 4
17.c even 4 2 inner 867.2.e.d 4
17.d even 8 2 51.2.d.b 2
17.d even 8 1 867.2.a.a 1
17.d even 8 1 867.2.a.b 1
17.e odd 16 8 867.2.h.d 8
51.g odd 8 2 153.2.d.a 2
51.g odd 8 1 2601.2.a.i 1
51.g odd 8 1 2601.2.a.j 1
68.g odd 8 2 816.2.c.c 2
85.k odd 8 1 1275.2.d.b 2
85.k odd 8 1 1275.2.d.d 2
85.m even 8 2 1275.2.g.a 2
85.n odd 8 1 1275.2.d.b 2
85.n odd 8 1 1275.2.d.d 2
136.o even 8 2 3264.2.c.e 2
136.p odd 8 2 3264.2.c.d 2
204.p even 8 2 2448.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 17.d even 8 2
153.2.d.a 2 51.g odd 8 2
816.2.c.c 2 68.g odd 8 2
867.2.a.a 1 17.d even 8 1
867.2.a.b 1 17.d even 8 1
867.2.e.d 4 1.a even 1 1 trivial
867.2.e.d 4 17.b even 2 1 inner
867.2.e.d 4 17.c even 4 2 inner
867.2.h.d 8 17.e odd 16 8
1275.2.d.b 2 85.k odd 8 1
1275.2.d.b 2 85.n odd 8 1
1275.2.d.d 2 85.k odd 8 1
1275.2.d.d 2 85.n odd 8 1
1275.2.g.a 2 85.m even 8 2
2448.2.c.j 2 204.p even 8 2
2601.2.a.i 1 51.g odd 8 1
2601.2.a.j 1 51.g odd 8 1
3264.2.c.d 2 136.p odd 8 2
3264.2.c.e 2 136.o even 8 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 256 + T^{4} \)
$11$ \( 256 + T^{4} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( T^{4} \)
$31$ \( 256 + T^{4} \)
$37$ \( 4096 + T^{4} \)
$41$ \( 4096 + T^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( ( -8 + T )^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 144 + T^{2} )^{2} \)
$61$ \( 4096 + T^{4} \)
$67$ \( ( -12 + T )^{4} \)
$71$ \( 20736 + T^{4} \)
$73$ \( T^{4} \)
$79$ \( 256 + T^{4} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( -10 + T )^{4} \)
$97$ \( 65536 + T^{4} \)
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