Properties

Label 8.4.b.a
Level $8$
Weight $4$
Character orbit 8.b
Analytic conductor $0.472$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.472015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + 2 \beta q^{3} + (2 \beta - 6) q^{4} - 4 \beta q^{5} + ( - 2 \beta + 14) q^{6} - 8 q^{7} + (4 \beta + 20) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + 2 \beta q^{3} + (2 \beta - 6) q^{4} - 4 \beta q^{5} + ( - 2 \beta + 14) q^{6} - 8 q^{7} + (4 \beta + 20) q^{8} - q^{9} + (4 \beta - 28) q^{10} - 6 \beta q^{11} + ( - 12 \beta - 28) q^{12} + 20 \beta q^{13} + (8 \beta + 8) q^{14} + 56 q^{15} + ( - 24 \beta + 8) q^{16} - 14 q^{17} + (\beta + 1) q^{18} - 14 \beta q^{19} + (24 \beta + 56) q^{20} - 16 \beta q^{21} + (6 \beta - 42) q^{22} - 152 q^{23} + (40 \beta - 56) q^{24} + 13 q^{25} + ( - 20 \beta + 140) q^{26} + 52 \beta q^{27} + ( - 16 \beta + 48) q^{28} - 60 \beta q^{29} + ( - 56 \beta - 56) q^{30} + 224 q^{31} + (16 \beta - 176) q^{32} + 84 q^{33} + (14 \beta + 14) q^{34} + 32 \beta q^{35} + ( - 2 \beta + 6) q^{36} + 92 \beta q^{37} + (14 \beta - 98) q^{38} - 280 q^{39} + ( - 80 \beta + 112) q^{40} - 70 q^{41} + (16 \beta - 112) q^{42} - 166 \beta q^{43} + (36 \beta + 84) q^{44} + 4 \beta q^{45} + (152 \beta + 152) q^{46} + 336 q^{47} + (16 \beta + 336) q^{48} - 279 q^{49} + ( - 13 \beta - 13) q^{50} - 28 \beta q^{51} + ( - 120 \beta - 280) q^{52} + 12 \beta q^{53} + ( - 52 \beta + 364) q^{54} - 168 q^{55} + ( - 32 \beta - 160) q^{56} + 196 q^{57} + (60 \beta - 420) q^{58} + 202 \beta q^{59} + (112 \beta - 336) q^{60} + 36 \beta q^{61} + ( - 224 \beta - 224) q^{62} + 8 q^{63} + (160 \beta + 288) q^{64} + 560 q^{65} + ( - 84 \beta - 84) q^{66} + 66 \beta q^{67} + ( - 28 \beta + 84) q^{68} - 304 \beta q^{69} + ( - 32 \beta + 224) q^{70} - 72 q^{71} + ( - 4 \beta - 20) q^{72} - 294 q^{73} + ( - 92 \beta + 644) q^{74} + 26 \beta q^{75} + (84 \beta + 196) q^{76} + 48 \beta q^{77} + (280 \beta + 280) q^{78} - 464 q^{79} + ( - 32 \beta - 672) q^{80} - 755 q^{81} + (70 \beta + 70) q^{82} - 206 \beta q^{83} + (96 \beta + 224) q^{84} + 56 \beta q^{85} + (166 \beta - 1162) q^{86} + 840 q^{87} + ( - 120 \beta + 168) q^{88} + 266 q^{89} + ( - 4 \beta + 28) q^{90} - 160 \beta q^{91} + ( - 304 \beta + 912) q^{92} + 448 \beta q^{93} + ( - 336 \beta - 336) q^{94} - 392 q^{95} + ( - 352 \beta - 224) q^{96} + 994 q^{97} + (279 \beta + 279) q^{98} + 6 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 12 q^{4} + 28 q^{6} - 16 q^{7} + 40 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 12 q^{4} + 28 q^{6} - 16 q^{7} + 40 q^{8} - 2 q^{9} - 56 q^{10} - 56 q^{12} + 16 q^{14} + 112 q^{15} + 16 q^{16} - 28 q^{17} + 2 q^{18} + 112 q^{20} - 84 q^{22} - 304 q^{23} - 112 q^{24} + 26 q^{25} + 280 q^{26} + 96 q^{28} - 112 q^{30} + 448 q^{31} - 352 q^{32} + 168 q^{33} + 28 q^{34} + 12 q^{36} - 196 q^{38} - 560 q^{39} + 224 q^{40} - 140 q^{41} - 224 q^{42} + 168 q^{44} + 304 q^{46} + 672 q^{47} + 672 q^{48} - 558 q^{49} - 26 q^{50} - 560 q^{52} + 728 q^{54} - 336 q^{55} - 320 q^{56} + 392 q^{57} - 840 q^{58} - 672 q^{60} - 448 q^{62} + 16 q^{63} + 576 q^{64} + 1120 q^{65} - 168 q^{66} + 168 q^{68} + 448 q^{70} - 144 q^{71} - 40 q^{72} - 588 q^{73} + 1288 q^{74} + 392 q^{76} + 560 q^{78} - 928 q^{79} - 1344 q^{80} - 1510 q^{81} + 140 q^{82} + 448 q^{84} - 2324 q^{86} + 1680 q^{87} + 336 q^{88} + 532 q^{89} + 56 q^{90} + 1824 q^{92} - 672 q^{94} - 784 q^{95} - 448 q^{96} + 1988 q^{97} + 558 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 1.32288i
0.500000 1.32288i
−1.00000 2.64575i 5.29150i −6.00000 + 5.29150i 10.5830i 14.0000 5.29150i −8.00000 20.0000 + 10.5830i −1.00000 −28.0000 + 10.5830i
5.2 −1.00000 + 2.64575i 5.29150i −6.00000 5.29150i 10.5830i 14.0000 + 5.29150i −8.00000 20.0000 10.5830i −1.00000 −28.0000 10.5830i
\(n\): e.g. 2-40 or 990-1000
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 8.4.b.a 2
3.b odd 2 1 72.4.d.b 2
4.b odd 2 1 32.4.b.a 2
5.b even 2 1 200.4.d.a 2
5.c odd 4 2 200.4.f.a 4
8.b even 2 1 inner 8.4.b.a 2
8.d odd 2 1 32.4.b.a 2
12.b even 2 1 288.4.d.a 2
16.e even 4 2 256.4.a.l 2
16.f odd 4 2 256.4.a.j 2
20.d odd 2 1 800.4.d.a 2
20.e even 4 2 800.4.f.a 4
24.f even 2 1 288.4.d.a 2
24.h odd 2 1 72.4.d.b 2
40.e odd 2 1 800.4.d.a 2
40.f even 2 1 200.4.d.a 2
40.i odd 4 2 200.4.f.a 4
40.k even 4 2 800.4.f.a 4
48.i odd 4 2 2304.4.a.bn 2
48.k even 4 2 2304.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 1.a even 1 1 trivial
8.4.b.a 2 8.b even 2 1 inner
32.4.b.a 2 4.b odd 2 1
32.4.b.a 2 8.d odd 2 1
72.4.d.b 2 3.b odd 2 1
72.4.d.b 2 24.h odd 2 1
200.4.d.a 2 5.b even 2 1
200.4.d.a 2 40.f even 2 1
200.4.f.a 4 5.c odd 4 2
200.4.f.a 4 40.i odd 4 2
256.4.a.j 2 16.f odd 4 2
256.4.a.l 2 16.e even 4 2
288.4.d.a 2 12.b even 2 1
288.4.d.a 2 24.f even 2 1
800.4.d.a 2 20.d odd 2 1
800.4.d.a 2 40.e odd 2 1
800.4.f.a 4 20.e even 4 2
800.4.f.a 4 40.k even 4 2
2304.4.a.v 2 48.k even 4 2
2304.4.a.bn 2 48.i odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} + 28 \) Copy content Toggle raw display
$5$ \( T^{2} + 112 \) Copy content Toggle raw display
$7$ \( (T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 252 \) Copy content Toggle raw display
$13$ \( T^{2} + 2800 \) Copy content Toggle raw display
$17$ \( (T + 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1372 \) Copy content Toggle raw display
$23$ \( (T + 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 25200 \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 59248 \) Copy content Toggle raw display
$41$ \( (T + 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 192892 \) Copy content Toggle raw display
$47$ \( (T - 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 1008 \) Copy content Toggle raw display
$59$ \( T^{2} + 285628 \) Copy content Toggle raw display
$61$ \( T^{2} + 9072 \) Copy content Toggle raw display
$67$ \( T^{2} + 30492 \) Copy content Toggle raw display
$71$ \( (T + 72)^{2} \) Copy content Toggle raw display
$73$ \( (T + 294)^{2} \) Copy content Toggle raw display
$79$ \( (T + 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 297052 \) Copy content Toggle raw display
$89$ \( (T - 266)^{2} \) Copy content Toggle raw display
$97$ \( (T - 994)^{2} \) Copy content Toggle raw display
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