Properties

Label 96.2.c.a
Level 96
Weight 2
Character orbit 96.c
Analytic conductor 0.767
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 96.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.766563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + 2 \zeta_{8}^{2} q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} -2 q^{13} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} -6 \zeta_{8}^{2} q^{19} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} -3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{31} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{35} + 6 q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + 2 \zeta_{8}^{2} q^{43} + ( 8 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{45} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{47} + 3 q^{49} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{53} + 8 \zeta_{8}^{2} q^{55} + ( 6 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{57} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} + 2 \zeta_{8}^{2} q^{67} + ( -8 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} -6 q^{73} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{77} -14 \zeta_{8}^{2} q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{87} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{89} -4 \zeta_{8}^{2} q^{91} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{93} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{95} + 10 q^{97} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 8q^{13} - 8q^{21} - 12q^{25} - 16q^{33} + 24q^{37} + 32q^{45} + 12q^{49} + 24q^{57} - 8q^{61} - 32q^{69} - 24q^{73} - 28q^{81} - 8q^{93} + 40q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 0
95.2 0 −1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
95.3 0 1.41421 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 2.82843i 0
95.4 0 1.41421 + 1.00000i 0 2.82843i 0 2.00000i 0 1.00000 + 2.82843i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.2.c.a 4
3.b odd 2 1 inner 96.2.c.a 4
4.b odd 2 1 inner 96.2.c.a 4
5.b even 2 1 2400.2.h.c 4
5.c odd 4 1 2400.2.o.a 4
5.c odd 4 1 2400.2.o.h 4
8.b even 2 1 192.2.c.b 4
8.d odd 2 1 192.2.c.b 4
9.c even 3 2 2592.2.s.e 8
9.d odd 6 2 2592.2.s.e 8
12.b even 2 1 inner 96.2.c.a 4
15.d odd 2 1 2400.2.h.c 4
15.e even 4 1 2400.2.o.a 4
15.e even 4 1 2400.2.o.h 4
16.e even 4 1 768.2.f.a 4
16.e even 4 1 768.2.f.g 4
16.f odd 4 1 768.2.f.a 4
16.f odd 4 1 768.2.f.g 4
20.d odd 2 1 2400.2.h.c 4
20.e even 4 1 2400.2.o.a 4
20.e even 4 1 2400.2.o.h 4
24.f even 2 1 192.2.c.b 4
24.h odd 2 1 192.2.c.b 4
36.f odd 6 2 2592.2.s.e 8
36.h even 6 2 2592.2.s.e 8
48.i odd 4 1 768.2.f.a 4
48.i odd 4 1 768.2.f.g 4
48.k even 4 1 768.2.f.a 4
48.k even 4 1 768.2.f.g 4
60.h even 2 1 2400.2.h.c 4
60.l odd 4 1 2400.2.o.a 4
60.l odd 4 1 2400.2.o.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 1.a even 1 1 trivial
96.2.c.a 4 3.b odd 2 1 inner
96.2.c.a 4 4.b odd 2 1 inner
96.2.c.a 4 12.b even 2 1 inner
192.2.c.b 4 8.b even 2 1
192.2.c.b 4 8.d odd 2 1
192.2.c.b 4 24.f even 2 1
192.2.c.b 4 24.h odd 2 1
768.2.f.a 4 16.e even 4 1
768.2.f.a 4 16.f odd 4 1
768.2.f.a 4 48.i odd 4 1
768.2.f.a 4 48.k even 4 1
768.2.f.g 4 16.e even 4 1
768.2.f.g 4 16.f odd 4 1
768.2.f.g 4 48.i odd 4 1
768.2.f.g 4 48.k even 4 1
2400.2.h.c 4 5.b even 2 1
2400.2.h.c 4 15.d odd 2 1
2400.2.h.c 4 20.d odd 2 1
2400.2.h.c 4 60.h even 2 1
2400.2.o.a 4 5.c odd 4 1
2400.2.o.a 4 15.e even 4 1
2400.2.o.a 4 20.e even 4 1
2400.2.o.a 4 60.l odd 4 1
2400.2.o.h 4 5.c odd 4 1
2400.2.o.h 4 15.e even 4 1
2400.2.o.h 4 20.e even 4 1
2400.2.o.h 4 60.l odd 4 1
2592.2.s.e 8 9.c even 3 2
2592.2.s.e 8 9.d odd 6 2
2592.2.s.e 8 36.f odd 6 2
2592.2.s.e 8 36.h even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(96, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 58 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 34 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 34 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 130 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 + 38 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 110 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{4} \)
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