Properties

Label 48.2.j.a
Level $48$
Weight $2$
Character orbit 48.j
Analytic conductor $0.383$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{7} + \beta_{3}) q^{6} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{8} - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4}) q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{5} + (\beta_{7} + \beta_{3}) q^{6} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{8} - \beta_{7} q^{9} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{10} + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1 - 2) q^{11} + (\beta_{6} + \beta_{2} - \beta_1) q^{12} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{13} + (2 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{14} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{15} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{16} + (2 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{6} - \beta_{2} + 1) q^{18} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{19} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2) q^{20} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{21} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{22} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{23} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \beta_1 + 1) q^{24} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{25}+ \cdots + (2 \beta_{7} + 2 \beta_{3} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{4} - 12 q^{8} - 8 q^{10} - 8 q^{11} + 8 q^{12} + 12 q^{14} - 8 q^{15} + 4 q^{18} - 8 q^{19} + 16 q^{20} + 4 q^{24} + 20 q^{26} + 8 q^{28} - 16 q^{29} - 8 q^{30} + 24 q^{31} + 24 q^{35} - 4 q^{36} - 16 q^{37} - 8 q^{38} + 16 q^{40} - 20 q^{42} - 8 q^{43} - 40 q^{44} - 8 q^{46} - 16 q^{48} - 8 q^{49} - 36 q^{50} + 8 q^{51} - 16 q^{52} + 16 q^{53} + 4 q^{54} - 16 q^{58} + 32 q^{59} + 24 q^{60} + 16 q^{61} - 12 q^{62} + 8 q^{63} + 8 q^{64} - 16 q^{65} + 24 q^{66} - 16 q^{67} + 32 q^{68} + 16 q^{69} + 32 q^{70} - 4 q^{72} + 52 q^{74} + 16 q^{75} + 8 q^{76} + 16 q^{77} - 12 q^{78} - 24 q^{79} + 8 q^{80} - 8 q^{81} + 40 q^{82} - 40 q^{83} - 24 q^{84} - 16 q^{85} - 16 q^{86} + 32 q^{88} - 8 q^{90} - 8 q^{91} - 16 q^{92} + 8 q^{94} - 48 q^{95} - 40 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 + 2.10607i
0.500000 1.44392i
0.500000 0.691860i
0.500000 + 0.0297061i
0.500000 2.10607i
0.500000 + 1.44392i
0.500000 + 0.691860i
0.500000 0.0297061i
−1.34277 0.443806i 0.707107 + 0.707107i 1.60607 + 1.19186i 1.27133 1.27133i −0.635665 1.26330i 0.158942i −1.62764 2.31318i 1.00000i −2.27133 + 1.14288i
13.2 −0.167452 1.40426i −0.707107 0.707107i −1.94392 + 0.470294i 1.74912 1.74912i −0.874559 + 1.11137i 2.55765i 0.985930 + 2.65103i 1.00000i −2.74912 2.16333i
13.3 0.635665 1.26330i 0.707107 + 0.707107i −1.19186 1.60607i −2.68554 + 2.68554i 1.34277 0.443806i 2.15894i −2.78658 + 0.484753i 1.00000i 1.68554 + 5.09976i
13.4 0.874559 + 1.11137i −0.707107 0.707107i −0.470294 + 1.94392i −0.334904 + 0.334904i 0.167452 1.40426i 4.55765i −2.57172 + 1.17740i 1.00000i −0.665096 0.0793096i
37.1 −1.34277 + 0.443806i 0.707107 0.707107i 1.60607 1.19186i 1.27133 + 1.27133i −0.635665 + 1.26330i 0.158942i −1.62764 + 2.31318i 1.00000i −2.27133 1.14288i
37.2 −0.167452 + 1.40426i −0.707107 + 0.707107i −1.94392 0.470294i 1.74912 + 1.74912i −0.874559 1.11137i 2.55765i 0.985930 2.65103i 1.00000i −2.74912 + 2.16333i
37.3 0.635665 + 1.26330i 0.707107 0.707107i −1.19186 + 1.60607i −2.68554 2.68554i 1.34277 + 0.443806i 2.15894i −2.78658 0.484753i 1.00000i 1.68554 5.09976i
37.4 0.874559 1.11137i −0.707107 + 0.707107i −0.470294 1.94392i −0.334904 0.334904i 0.167452 + 1.40426i 4.55765i −2.57172 1.17740i 1.00000i −0.665096 + 0.0793096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.j.a 8
3.b odd 2 1 144.2.k.b 8
4.b odd 2 1 192.2.j.a 8
8.b even 2 1 384.2.j.b 8
8.d odd 2 1 384.2.j.a 8
12.b even 2 1 576.2.k.b 8
16.e even 4 1 inner 48.2.j.a 8
16.e even 4 1 384.2.j.b 8
16.f odd 4 1 192.2.j.a 8
16.f odd 4 1 384.2.j.a 8
24.f even 2 1 1152.2.k.f 8
24.h odd 2 1 1152.2.k.c 8
32.g even 8 1 3072.2.a.i 4
32.g even 8 1 3072.2.a.t 4
32.g even 8 2 3072.2.d.f 8
32.h odd 8 1 3072.2.a.n 4
32.h odd 8 1 3072.2.a.o 4
32.h odd 8 2 3072.2.d.i 8
48.i odd 4 1 144.2.k.b 8
48.i odd 4 1 1152.2.k.c 8
48.k even 4 1 576.2.k.b 8
48.k even 4 1 1152.2.k.f 8
96.o even 8 1 9216.2.a.x 4
96.o even 8 1 9216.2.a.bn 4
96.p odd 8 1 9216.2.a.y 4
96.p odd 8 1 9216.2.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 1.a even 1 1 trivial
48.2.j.a 8 16.e even 4 1 inner
144.2.k.b 8 3.b odd 2 1
144.2.k.b 8 48.i odd 4 1
192.2.j.a 8 4.b odd 2 1
192.2.j.a 8 16.f odd 4 1
384.2.j.a 8 8.d odd 2 1
384.2.j.a 8 16.f odd 4 1
384.2.j.b 8 8.b even 2 1
384.2.j.b 8 16.e even 4 1
576.2.k.b 8 12.b even 2 1
576.2.k.b 8 48.k even 4 1
1152.2.k.c 8 24.h odd 2 1
1152.2.k.c 8 48.i odd 4 1
1152.2.k.f 8 24.f even 2 1
1152.2.k.f 8 48.k even 4 1
3072.2.a.i 4 32.g even 8 1
3072.2.a.n 4 32.h odd 8 1
3072.2.a.o 4 32.h odd 8 1
3072.2.a.t 4 32.g even 8 1
3072.2.d.f 8 32.g even 8 2
3072.2.d.i 8 32.h odd 8 2
9216.2.a.x 4 96.o even 8 1
9216.2.a.y 4 96.p odd 8 1
9216.2.a.bn 4 96.o even 8 1
9216.2.a.bo 4 96.p odd 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{6} + 4 T^{5} + 2 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{5} + 128 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} - 64 T^{5} + 776 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} + 64 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 32 T^{6} - 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + 40 T^{2} - 24 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$41$ \( T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 32)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{7} + 800 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216 \) Copy content Toggle raw display
$97$ \( (T^{4} - 224 T^{2} + 768 T + 512)^{2} \) Copy content Toggle raw display
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