Properties

Label 96.4.d.a
Level $96$
Weight $4$
Character orbit 96.d
Analytic conductor $5.664$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 96.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.66418336055\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
Defining polynomial: \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 - 5) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_1 - 5) q^{7} - 9 q^{9} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{11} + ( - \beta_{4} + 3 \beta_{3} + 6 \beta_{2}) q^{13} + ( - \beta_{5} + 10) q^{15} + ( - 2 \beta_{5} + 2 \beta_1 + 8) q^{17} + (2 \beta_{4} + 6 \beta_{3} + 4 \beta_{2}) q^{19} + ( - 3 \beta_{4} - 5 \beta_{2}) q^{21} + ( - 2 \beta_1 - 54) q^{23} + (4 \beta_{5} + 4 \beta_1 - 19) q^{25} - 9 \beta_{2} q^{27} + (2 \beta_{4} - 5 \beta_{3} + 19 \beta_{2}) q^{29} + (4 \beta_{5} + 3 \beta_1 + 105) q^{31} + ( - 2 \beta_{5} - 6 \beta_1 + 2) q^{33} + ( - 2 \beta_{4} - 14 \beta_{3} - 12 \beta_{2}) q^{35} + (11 \beta_{4} - 3 \beta_{3} - 32 \beta_{2}) q^{37} + ( - 3 \beta_{5} - 3 \beta_1 - 51) q^{39} + ( - 2 \beta_{5} - 14 \beta_1 + 44) q^{41} + (10 \beta_{4} - 18 \beta_{3} + 28 \beta_{2}) q^{43} + ( - 9 \beta_{3} + 9 \beta_{2}) q^{45} + ( - 4 \beta_{5} - 6 \beta_1 + 70) q^{47} + (8 \beta_{5} + 109) q^{49} + ( - 6 \beta_{4} - 18 \beta_{3} + 6 \beta_{2}) q^{51} + ( - 10 \beta_{4} - 11 \beta_{3} - 123 \beta_{2}) q^{53} + (12 \beta_{5} + 4 \beta_1 - 172) q^{55} + ( - 6 \beta_{5} + 6 \beta_1 - 30) q^{57} + (8 \beta_{4} + 8 \beta_{3} - 20 \beta_{2}) q^{59} + (9 \beta_{4} + 3 \beta_{3} + 172 \beta_{2}) q^{61} + ( - 9 \beta_1 + 45) q^{63} + (6 \beta_{5} + 10 \beta_1 - 294) q^{65} + (48 \beta_{3} - 68 \beta_{2}) q^{67} + (6 \beta_{4} - 54 \beta_{2}) q^{69} + ( - 12 \beta_{5} + 6 \beta_1 + 282) q^{71} + ( - 8 \beta_{5} + 16 \beta_1 + 154) q^{73} + ( - 12 \beta_{4} + 36 \beta_{3} - 15 \beta_{2}) q^{75} + ( - 20 \beta_{4} + 20 \beta_{3} + 256 \beta_{2}) q^{77} + ( - 4 \beta_{5} + 7 \beta_1 + 5) q^{79} + 81 q^{81} + (2 \beta_{4} + 14 \beta_{3} + 128 \beta_{2}) q^{83} + ( - 28 \beta_{4} + 42 \beta_{3} - 302 \beta_{2}) q^{85} + (5 \beta_{5} + 6 \beta_1 - 176) q^{87} + ( - 4 \beta_{5} + 4 \beta_1 - 38) q^{89} + ( - 38 \beta_{4} - 18 \beta_{3} + 64 \beta_{2}) q^{91} + ( - 9 \beta_{4} + 36 \beta_{3} + 109 \beta_{2}) q^{93} + (8 \beta_{5} + 28 \beta_1 - 860) q^{95} + ( - 12 \beta_{5} - 4 \beta_1 - 406) q^{97} + (18 \beta_{4} - 18 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{7} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{7} - 54 q^{9} + 60 q^{15} + 52 q^{17} - 328 q^{23} - 106 q^{25} + 636 q^{31} - 312 q^{39} + 236 q^{41} + 408 q^{47} + 654 q^{49} - 1024 q^{55} - 168 q^{57} + 252 q^{63} - 1744 q^{65} + 1704 q^{71} + 956 q^{73} + 44 q^{79} + 486 q^{81} - 1044 q^{87} - 220 q^{89} - 5104 q^{95} - 2444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{4} + 3\nu^{2} + 12\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} - 6\nu^{4} + 9\nu^{3} + 6\nu^{2} + 24\nu - 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 6\nu^{4} - 17\nu^{3} + 90\nu^{2} - 184\nu + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{5} + 6\nu^{4} + 111\nu^{3} + 90\nu^{2} - 56\nu - 672 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 6\nu^{4} - 3\nu^{3} + 18\nu^{2} - 36\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 6\beta_{3} - 6\beta_{2} + 3\beta _1 - 1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{4} + 9\beta_{3} - 20\beta_{2} + 6\beta _1 - 18 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{5} + 12\beta_{4} - 6\beta_{3} + 6\beta_{2} - 3\beta _1 + 289 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{5} + 3\beta_{4} - 15\beta_{3} - 44\beta_{2} + 2\beta _1 - 38 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{5} + 24\beta_{4} + 42\beta_{3} - 318\beta_{2} + 15\beta _1 - 485 ) / 48 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.641412 + 1.89436i
1.88322 + 0.673417i
−1.24181 1.56777i
−1.24181 + 1.56777i
1.88322 0.673417i
−0.641412 1.89436i
0 3.00000i 0 9.15486i 0 −27.4175 0 −9.00000 0
49.2 0 3.00000i 0 0.612661i 0 22.7441 0 −9.00000 0
49.3 0 3.00000i 0 18.5422i 0 −9.32669 0 −9.00000 0
49.4 0 3.00000i 0 18.5422i 0 −9.32669 0 −9.00000 0
49.5 0 3.00000i 0 0.612661i 0 22.7441 0 −9.00000 0
49.6 0 3.00000i 0 9.15486i 0 −27.4175 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.4.d.a 6
3.b odd 2 1 288.4.d.d 6
4.b odd 2 1 24.4.d.a 6
8.b even 2 1 inner 96.4.d.a 6
8.d odd 2 1 24.4.d.a 6
12.b even 2 1 72.4.d.d 6
16.e even 4 1 768.4.a.q 3
16.e even 4 1 768.4.a.t 3
16.f odd 4 1 768.4.a.r 3
16.f odd 4 1 768.4.a.s 3
24.f even 2 1 72.4.d.d 6
24.h odd 2 1 288.4.d.d 6
48.i odd 4 1 2304.4.a.bu 3
48.i odd 4 1 2304.4.a.bw 3
48.k even 4 1 2304.4.a.bt 3
48.k even 4 1 2304.4.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 4.b odd 2 1
24.4.d.a 6 8.d odd 2 1
72.4.d.d 6 12.b even 2 1
72.4.d.d 6 24.f even 2 1
96.4.d.a 6 1.a even 1 1 trivial
96.4.d.a 6 8.b even 2 1 inner
288.4.d.d 6 3.b odd 2 1
288.4.d.d 6 24.h odd 2 1
768.4.a.q 3 16.e even 4 1
768.4.a.r 3 16.f odd 4 1
768.4.a.s 3 16.f odd 4 1
768.4.a.t 3 16.e even 4 1
2304.4.a.bt 3 48.k even 4 1
2304.4.a.bu 3 48.i odd 4 1
2304.4.a.bv 3 48.k even 4 1
2304.4.a.bw 3 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 428 T^{4} + 28976 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$7$ \( (T^{3} + 14 T^{2} - 580 T - 5816)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 5632 T^{4} + \cdots + 2415919104 \) Copy content Toggle raw display
$13$ \( T^{6} + 4912 T^{4} + \cdots + 3121680384 \) Copy content Toggle raw display
$17$ \( (T^{3} - 26 T^{2} - 11124 T + 477576)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 22960 T^{4} + \cdots + 75488661504 \) Copy content Toggle raw display
$23$ \( (T^{3} + 164 T^{2} + 6384 T + 45504)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 22348 T^{4} + \cdots + 3766031424 \) Copy content Toggle raw display
$31$ \( (T^{3} - 318 T^{2} + 4476 T + 3749624)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 179776 T^{4} + \cdots + 6879707136 \) Copy content Toggle raw display
$41$ \( (T^{3} - 118 T^{2} - 117300 T + 19985976)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 229552 T^{4} + \cdots + 73984219582464 \) Copy content Toggle raw display
$47$ \( (T^{3} - 204 T^{2} - 27792 T + 1964736)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 427051482970176 \) Copy content Toggle raw display
$59$ \( T^{6} + 138416 T^{4} + \cdots + 72651484205056 \) Copy content Toggle raw display
$61$ \( T^{6} + 902016 T^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{6} + 1054512 T^{4} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{3} - 852 T^{2} - 66960 T + 85084992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 478 T^{2} - 255956 T + 120833304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 22 T^{2} - 71524 T + 7902616)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 520448 T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} + 110 T^{2} - 41364 T + 1423656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 1222 T^{2} + 251660 T - 74802424)^{2} \) Copy content Toggle raw display
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