Properties

Label 968.2.i.i
Level $968$
Weight $2$
Character orbit 968.i
Analytic conductor $7.730$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.i (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{9} - 9 \zeta_{10}^{3} q^{15} - 6 \zeta_{10} q^{17} - 4 \zeta_{10}^{2} q^{19} - 6 q^{21} + q^{23} + 4 \zeta_{10}^{2} q^{25} + 9 \zeta_{10} q^{27} - 8 \zeta_{10}^{3} q^{29} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 7) q^{31} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{35} + \zeta_{10}^{3} q^{37} - 4 \zeta_{10}^{2} q^{41} - 6 q^{43} - 18 q^{45} - 8 \zeta_{10}^{2} q^{47} + 3 \zeta_{10} q^{49} + 18 \zeta_{10}^{3} q^{51} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{53} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 12) q^{57} + \zeta_{10}^{3} q^{59} + 4 \zeta_{10} q^{61} + 12 \zeta_{10}^{2} q^{63} - 5 q^{67} - 3 \zeta_{10}^{2} q^{69} - 3 \zeta_{10} q^{71} + 16 \zeta_{10}^{3} q^{73} + ( - 12 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} + 12) q^{75} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{79} - 9 \zeta_{10}^{3} q^{81} - 2 \zeta_{10} q^{83} - 18 \zeta_{10}^{2} q^{85} - 24 q^{87} + 15 q^{89} - 21 \zeta_{10} q^{93} - 12 \zeta_{10}^{3} q^{95} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 7) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9} - 9 q^{15} - 6 q^{17} + 4 q^{19} - 24 q^{21} + 4 q^{23} - 4 q^{25} + 9 q^{27} - 8 q^{29} + 7 q^{31} + 6 q^{35} + q^{37} + 4 q^{41} - 24 q^{43} - 72 q^{45} + 8 q^{47} + 3 q^{49} + 18 q^{51} - 2 q^{53} - 12 q^{57} + q^{59} + 4 q^{61} - 12 q^{63} - 20 q^{67} + 3 q^{69} - 3 q^{71} + 16 q^{73} + 12 q^{75} + 2 q^{79} - 9 q^{81} - 2 q^{83} + 18 q^{85} - 96 q^{87} + 60 q^{89} - 21 q^{93} - 12 q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 2.42705 1.76336i 0 −0.927051 2.85317i 0 −1.61803 1.17557i 0 1.85410 5.70634i 0
81.1 0 −0.927051 + 2.85317i 0 2.42705 1.76336i 0 0.618034 + 1.90211i 0 −4.85410 3.52671i 0
729.1 0 −0.927051 2.85317i 0 2.42705 + 1.76336i 0 0.618034 1.90211i 0 −4.85410 + 3.52671i 0
753.1 0 2.42705 + 1.76336i 0 −0.927051 + 2.85317i 0 −1.61803 + 1.17557i 0 1.85410 + 5.70634i 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
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.i.i 4
11.b odd 2 1 968.2.i.j 4
11.c even 5 1 968.2.a.a 1
11.c even 5 3 inner 968.2.i.i 4
11.d odd 10 1 88.2.a.a 1
11.d odd 10 3 968.2.i.j 4
33.f even 10 1 792.2.a.g 1
33.h odd 10 1 8712.2.a.x 1
44.g even 10 1 176.2.a.c 1
44.h odd 10 1 1936.2.a.l 1
55.h odd 10 1 2200.2.a.k 1
55.l even 20 2 2200.2.b.a 2
77.l even 10 1 4312.2.a.l 1
88.k even 10 1 704.2.a.b 1
88.l odd 10 1 7744.2.a.b 1
88.o even 10 1 7744.2.a.bk 1
88.p odd 10 1 704.2.a.l 1
132.n odd 10 1 1584.2.a.q 1
176.u odd 20 2 2816.2.c.i 2
176.x even 20 2 2816.2.c.d 2
220.o even 10 1 4400.2.a.a 1
220.w odd 20 2 4400.2.b.b 2
264.r odd 10 1 6336.2.a.k 1
264.u even 10 1 6336.2.a.h 1
308.s odd 10 1 8624.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 11.d odd 10 1
176.2.a.c 1 44.g even 10 1
704.2.a.b 1 88.k even 10 1
704.2.a.l 1 88.p odd 10 1
792.2.a.g 1 33.f even 10 1
968.2.a.a 1 11.c even 5 1
968.2.i.i 4 1.a even 1 1 trivial
968.2.i.i 4 11.c even 5 3 inner
968.2.i.j 4 11.b odd 2 1
968.2.i.j 4 11.d odd 10 3
1584.2.a.q 1 132.n odd 10 1
1936.2.a.l 1 44.h odd 10 1
2200.2.a.k 1 55.h odd 10 1
2200.2.b.a 2 55.l even 20 2
2816.2.c.d 2 176.x even 20 2
2816.2.c.i 2 176.u odd 20 2
4312.2.a.l 1 77.l even 10 1
4400.2.a.a 1 220.o even 10 1
4400.2.b.b 2 220.w odd 20 2
6336.2.a.h 1 264.u even 10 1
6336.2.a.k 1 264.r odd 10 1
7744.2.a.b 1 88.l odd 10 1
7744.2.a.bk 1 88.o even 10 1
8624.2.a.c 1 308.s odd 10 1
8712.2.a.x 1 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{4} - 3T_{3}^{3} + 9T_{3}^{2} - 27T_{3} + 81 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$43$ \( (T + 6)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$67$ \( (T + 5)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} + 256 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$89$ \( (T - 15)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
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