Properties

Label 5.8.b.a
Level 5
Weight 8
Character orbit 5.b
Analytic conductor 1.562
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.56192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-29}) \)
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 \(\beta = 2\sqrt{-29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( 75 - 25 \beta ) q^{5} -348 q^{6} -39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} +O(q^{10})\) \( q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( 75 - 25 \beta ) q^{5} -348 q^{6} -39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} + ( 2900 + 75 \beta ) q^{10} -6828 q^{11} + 36 \beta q^{12} -942 \beta q^{13} + 4524 q^{14} + ( 8700 + 225 \beta ) q^{15} -14704 q^{16} + 1456 \beta q^{17} + 1143 \beta q^{18} + 6860 q^{19} + ( 900 - 300 \beta ) q^{20} + 13572 q^{21} -6828 \beta q^{22} + 2713 \beta q^{23} -48720 q^{24} + ( -66875 - 3750 \beta ) q^{25} + 109272 q^{26} + 9990 \beta q^{27} -468 \beta q^{28} + 25590 q^{29} + ( -26100 + 8700 \beta ) q^{30} + 82112 q^{31} + 3216 \beta q^{32} -20484 \beta q^{33} -168896 q^{34} + ( -113100 - 2925 \beta ) q^{35} + 13716 q^{36} -20754 \beta q^{37} + 6860 \beta q^{38} + 327816 q^{39} + ( 406000 + 10500 \beta ) q^{40} -533118 q^{41} + 13572 \beta q^{42} + 65823 \beta q^{43} -81936 q^{44} + ( 85725 - 28575 \beta ) q^{45} -314708 q^{46} + 541 \beta q^{47} -44112 \beta q^{48} + 647107 q^{49} + ( 435000 - 66875 \beta ) q^{50} -506688 q^{51} -11304 \beta q^{52} -54722 \beta q^{53} -1158840 q^{54} + ( -512100 + 170700 \beta ) q^{55} + 633360 q^{56} + 20580 \beta q^{57} + 25590 \beta q^{58} + 1438980 q^{59} + ( 104400 + 2700 \beta ) q^{60} + 1381022 q^{61} + 82112 \beta q^{62} -44577 \beta q^{63} -2255168 q^{64} + ( -2731800 - 70650 \beta ) q^{65} + 2376144 q^{66} -252069 \beta q^{67} + 17472 \beta q^{68} -944124 q^{69} + ( 339300 - 113100 \beta ) q^{70} -481608 q^{71} + 160020 \beta q^{72} + 137988 \beta q^{73} + 2407464 q^{74} + ( 1305000 - 200625 \beta ) q^{75} + 82320 q^{76} + 266292 \beta q^{77} + 327816 \beta q^{78} -1059760 q^{79} + ( -1102800 + 367600 \beta ) q^{80} -976779 q^{81} -533118 \beta q^{82} -241757 \beta q^{83} + 162864 q^{84} + ( 4222400 + 109200 \beta ) q^{85} -7635468 q^{86} + 76770 \beta q^{87} -955920 \beta q^{88} + 5644170 q^{89} + ( 3314700 + 85725 \beta ) q^{90} -4261608 q^{91} + 32556 \beta q^{92} + 246336 \beta q^{93} -62756 q^{94} + ( 514500 - 171500 \beta ) q^{95} -1119168 q^{96} + 1115016 \beta q^{97} + 647107 \beta q^{98} -7804404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 24q^{4} + 150q^{5} - 696q^{6} + 2286q^{9} + O(q^{10}) \) \( 2q + 24q^{4} + 150q^{5} - 696q^{6} + 2286q^{9} + 5800q^{10} - 13656q^{11} + 9048q^{14} + 17400q^{15} - 29408q^{16} + 13720q^{19} + 1800q^{20} + 27144q^{21} - 97440q^{24} - 133750q^{25} + 218544q^{26} + 51180q^{29} - 52200q^{30} + 164224q^{31} - 337792q^{34} - 226200q^{35} + 27432q^{36} + 655632q^{39} + 812000q^{40} - 1066236q^{41} - 163872q^{44} + 171450q^{45} - 629416q^{46} + 1294214q^{49} + 870000q^{50} - 1013376q^{51} - 2317680q^{54} - 1024200q^{55} + 1266720q^{56} + 2877960q^{59} + 208800q^{60} + 2762044q^{61} - 4510336q^{64} - 5463600q^{65} + 4752288q^{66} - 1888248q^{69} + 678600q^{70} - 963216q^{71} + 4814928q^{74} + 2610000q^{75} + 164640q^{76} - 2119520q^{79} - 2205600q^{80} - 1953558q^{81} + 325728q^{84} + 8444800q^{85} - 15270936q^{86} + 11288340q^{89} + 6629400q^{90} - 8523216q^{91} - 125512q^{94} + 1029000q^{95} - 2238336q^{96} - 15608808q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
5.38516i
5.38516i
10.7703i 32.3110i 12.0000 75.0000 + 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 807.775i
4.2 10.7703i 32.3110i 12.0000 75.0000 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 + 807.775i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.8.b.a 2
3.b odd 2 1 45.8.b.a 2
4.b odd 2 1 80.8.c.a 2
5.b even 2 1 inner 5.8.b.a 2
5.c odd 4 2 25.8.a.d 2
8.b even 2 1 320.8.c.d 2
8.d odd 2 1 320.8.c.c 2
15.d odd 2 1 45.8.b.a 2
15.e even 4 2 225.8.a.n 2
20.d odd 2 1 80.8.c.a 2
20.e even 4 2 400.8.a.y 2
40.e odd 2 1 320.8.c.c 2
40.f even 2 1 320.8.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 1.a even 1 1 trivial
5.8.b.a 2 5.b even 2 1 inner
25.8.a.d 2 5.c odd 4 2
45.8.b.a 2 3.b odd 2 1
45.8.b.a 2 15.d odd 2 1
80.8.c.a 2 4.b odd 2 1
80.8.c.a 2 20.d odd 2 1
225.8.a.n 2 15.e even 4 2
320.8.c.c 2 8.d odd 2 1
320.8.c.c 2 40.e odd 2 1
320.8.c.d 2 8.b even 2 1
320.8.c.d 2 40.f even 2 1
400.8.a.y 2 20.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 140 T^{2} + 16384 T^{4} \)
$3$ \( 1 - 3330 T^{2} + 4782969 T^{4} \)
$5$ \( 1 - 150 T + 78125 T^{2} \)
$7$ \( 1 - 1470650 T^{2} + 678223072849 T^{4} \)
$11$ \( ( 1 + 6828 T + 19487171 T^{2} )^{2} \)
$13$ \( 1 - 22562810 T^{2} + 3937376385699289 T^{4} \)
$17$ \( 1 - 574764770 T^{2} + 168377826559400929 T^{4} \)
$19$ \( ( 1 - 6860 T + 893871739 T^{2} )^{2} \)
$23$ \( 1 - 5955848090 T^{2} + 11592836324538749809 T^{4} \)
$29$ \( ( 1 - 25590 T + 17249876309 T^{2} )^{2} \)
$31$ \( ( 1 - 82112 T + 27512614111 T^{2} )^{2} \)
$37$ \( 1 - 139899246410 T^{2} + \)\(90\!\cdots\!89\)\( T^{4} \)
$41$ \( ( 1 + 533118 T + 194754273881 T^{2} )^{2} \)
$43$ \( 1 - 41047812050 T^{2} + \)\(73\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1013212289930 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - 2002060594730 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( ( 1 - 1438980 T + 2488651484819 T^{2} )^{2} \)
$61$ \( ( 1 - 1381022 T + 3142742836021 T^{2} )^{2} \)
$67$ \( 1 - 4750924642370 T^{2} + \)\(36\!\cdots\!29\)\( T^{4} \)
$71$ \( ( 1 + 481608 T + 9095120158391 T^{2} )^{2} \)
$73$ \( 1 - 19886077213490 T^{2} + \)\(12\!\cdots\!09\)\( T^{4} \)
$79$ \( ( 1 + 1059760 T + 19203908986159 T^{2} )^{2} \)
$83$ \( 1 - 47492314121570 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( ( 1 - 5644170 T + 44231334895529 T^{2} )^{2} \)
$97$ \( 1 - 17378330046530 T^{2} + \)\(65\!\cdots\!69\)\( T^{4} \)
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