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: \(x^{2} - x + 2\)
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.

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

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

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 8 T^{2} \)
$3$ \( 1 - 26 T^{2} + 729 T^{4} \)
$5$ \( 1 - 138 T^{2} + 15625 T^{4} \)
$7$ \( ( 1 + 8 T + 343 T^{2} )^{2} \)
$11$ \( 1 - 2410 T^{2} + 1771561 T^{4} \)
$13$ \( 1 - 1594 T^{2} + 4826809 T^{4} \)
$17$ \( ( 1 + 14 T + 4913 T^{2} )^{2} \)
$19$ \( 1 - 12346 T^{2} + 47045881 T^{4} \)
$23$ \( ( 1 + 152 T + 12167 T^{2} )^{2} \)
$29$ \( 1 - 23578 T^{2} + 594823321 T^{4} \)
$31$ \( ( 1 - 224 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 42058 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 + 70 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 33878 T^{2} + 6321363049 T^{4} \)
$47$ \( ( 1 - 336 T + 103823 T^{2} )^{2} \)
$53$ \( 1 - 296746 T^{2} + 22164361129 T^{4} \)
$59$ \( 1 - 125130 T^{2} + 42180533641 T^{4} \)
$61$ \( 1 - 444890 T^{2} + 51520374361 T^{4} \)
$67$ \( 1 - 571034 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 72 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 + 294 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 + 464 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 846522 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 266 T + 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 994 T + 912673 T^{2} )^{2} \)
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