Properties

Label 3.8.a.a
Level $3$
Weight $8$
Character orbit 3.a
Self dual yes
Analytic conductor $0.937$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.937155076452\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6 q^{2} - 27 q^{3} - 92 q^{4} + 390 q^{5} - 162 q^{6} - 64 q^{7} - 1320 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 q^{2} - 27 q^{3} - 92 q^{4} + 390 q^{5} - 162 q^{6} - 64 q^{7} - 1320 q^{8} + 729 q^{9} + 2340 q^{10} - 948 q^{11} + 2484 q^{12} - 5098 q^{13} - 384 q^{14} - 10530 q^{15} + 3856 q^{16} + 28386 q^{17} + 4374 q^{18} - 8620 q^{19} - 35880 q^{20} + 1728 q^{21} - 5688 q^{22} - 15288 q^{23} + 35640 q^{24} + 73975 q^{25} - 30588 q^{26} - 19683 q^{27} + 5888 q^{28} + 36510 q^{29} - 63180 q^{30} - 276808 q^{31} + 192096 q^{32} + 25596 q^{33} + 170316 q^{34} - 24960 q^{35} - 67068 q^{36} + 268526 q^{37} - 51720 q^{38} + 137646 q^{39} - 514800 q^{40} - 629718 q^{41} + 10368 q^{42} + 685772 q^{43} + 87216 q^{44} + 284310 q^{45} - 91728 q^{46} + 583296 q^{47} - 104112 q^{48} - 819447 q^{49} + 443850 q^{50} - 766422 q^{51} + 469016 q^{52} - 428058 q^{53} - 118098 q^{54} - 369720 q^{55} + 84480 q^{56} + 232740 q^{57} + 219060 q^{58} + 1306380 q^{59} + 968760 q^{60} + 300662 q^{61} - 1660848 q^{62} - 46656 q^{63} + 659008 q^{64} - 1988220 q^{65} + 153576 q^{66} - 507244 q^{67} - 2611512 q^{68} + 412776 q^{69} - 149760 q^{70} + 5560632 q^{71} - 962280 q^{72} + 1369082 q^{73} + 1611156 q^{74} - 1997325 q^{75} + 793040 q^{76} + 60672 q^{77} + 825876 q^{78} - 6913720 q^{79} + 1503840 q^{80} + 531441 q^{81} - 3778308 q^{82} - 4376748 q^{83} - 158976 q^{84} + 11070540 q^{85} + 4114632 q^{86} - 985770 q^{87} + 1251360 q^{88} - 8528310 q^{89} + 1705860 q^{90} + 326272 q^{91} + 1406496 q^{92} + 7473816 q^{93} + 3499776 q^{94} - 3361800 q^{95} - 5186592 q^{96} - 8826814 q^{97} - 4916682 q^{98} - 691092 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
6.00000 −27.0000 −92.0000 390.000 −162.000 −64.0000 −1320.00 729.000 2340.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.8.a.a 1
3.b odd 2 1 9.8.a.a 1
4.b odd 2 1 48.8.a.g 1
5.b even 2 1 75.8.a.a 1
5.c odd 4 2 75.8.b.c 2
7.b odd 2 1 147.8.a.b 1
7.c even 3 2 147.8.e.b 2
7.d odd 6 2 147.8.e.a 2
8.b even 2 1 192.8.a.i 1
8.d odd 2 1 192.8.a.a 1
9.c even 3 2 81.8.c.a 2
9.d odd 6 2 81.8.c.c 2
11.b odd 2 1 363.8.a.b 1
12.b even 2 1 144.8.a.b 1
13.b even 2 1 507.8.a.a 1
15.d odd 2 1 225.8.a.i 1
15.e even 4 2 225.8.b.f 2
21.c even 2 1 441.8.a.a 1
24.f even 2 1 576.8.a.x 1
24.h odd 2 1 576.8.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 1.a even 1 1 trivial
9.8.a.a 1 3.b odd 2 1
48.8.a.g 1 4.b odd 2 1
75.8.a.a 1 5.b even 2 1
75.8.b.c 2 5.c odd 4 2
81.8.c.a 2 9.c even 3 2
81.8.c.c 2 9.d odd 6 2
144.8.a.b 1 12.b even 2 1
147.8.a.b 1 7.b odd 2 1
147.8.e.a 2 7.d odd 6 2
147.8.e.b 2 7.c even 3 2
192.8.a.a 1 8.d odd 2 1
192.8.a.i 1 8.b even 2 1
225.8.a.i 1 15.d odd 2 1
225.8.b.f 2 15.e even 4 2
363.8.a.b 1 11.b odd 2 1
441.8.a.a 1 21.c even 2 1
507.8.a.a 1 13.b even 2 1
576.8.a.w 1 24.h odd 2 1
576.8.a.x 1 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 6 \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T - 390 \) Copy content Toggle raw display
$7$ \( T + 64 \) Copy content Toggle raw display
$11$ \( T + 948 \) Copy content Toggle raw display
$13$ \( T + 5098 \) Copy content Toggle raw display
$17$ \( T - 28386 \) Copy content Toggle raw display
$19$ \( T + 8620 \) Copy content Toggle raw display
$23$ \( T + 15288 \) Copy content Toggle raw display
$29$ \( T - 36510 \) Copy content Toggle raw display
$31$ \( T + 276808 \) Copy content Toggle raw display
$37$ \( T - 268526 \) Copy content Toggle raw display
$41$ \( T + 629718 \) Copy content Toggle raw display
$43$ \( T - 685772 \) Copy content Toggle raw display
$47$ \( T - 583296 \) Copy content Toggle raw display
$53$ \( T + 428058 \) Copy content Toggle raw display
$59$ \( T - 1306380 \) Copy content Toggle raw display
$61$ \( T - 300662 \) Copy content Toggle raw display
$67$ \( T + 507244 \) Copy content Toggle raw display
$71$ \( T - 5560632 \) Copy content Toggle raw display
$73$ \( T - 1369082 \) Copy content Toggle raw display
$79$ \( T + 6913720 \) Copy content Toggle raw display
$83$ \( T + 4376748 \) Copy content Toggle raw display
$89$ \( T + 8528310 \) Copy content Toggle raw display
$97$ \( T + 8826814 \) Copy content Toggle raw display
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