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 \(2\) and degree \(1\))

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: \( 2 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{8}^{\mathrm{new}}(5, [\chi])\).