Properties

Label 5.6.b.a
Level 5
Weight 6
Character orbit 5.b
Analytic conductor 0.802
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.801919099065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( - \beta q^{2} \) \( + 3 \beta q^{3} \) \( -12 q^{4} \) \( + ( -45 - 5 \beta ) q^{5} \) \( + 132 q^{6} \) \( + 9 \beta q^{7} \) \( -20 \beta q^{8} \) \( -153 q^{9} \) \(+O(q^{10})\) \( q\) \( - \beta q^{2} \) \( + 3 \beta q^{3} \) \( -12 q^{4} \) \( + ( -45 - 5 \beta ) q^{5} \) \( + 132 q^{6} \) \( + 9 \beta q^{7} \) \( -20 \beta q^{8} \) \( -153 q^{9} \) \( + ( -220 + 45 \beta ) q^{10} \) \( + 252 q^{11} \) \( -36 \beta q^{12} \) \( + 18 \beta q^{13} \) \( + 396 q^{14} \) \( + ( 660 - 135 \beta ) q^{15} \) \( -1264 q^{16} \) \( + 104 \beta q^{17} \) \( + 153 \beta q^{18} \) \( -220 q^{19} \) \( + ( 540 + 60 \beta ) q^{20} \) \( -1188 q^{21} \) \( -252 \beta q^{22} \) \( -367 \beta q^{23} \) \( + 2640 q^{24} \) \( + ( 925 + 450 \beta ) q^{25} \) \( + 792 q^{26} \) \( + 270 \beta q^{27} \) \( -108 \beta q^{28} \) \( -6930 q^{29} \) \( + ( -5940 - 660 \beta ) q^{30} \) \( + 6752 q^{31} \) \( + 624 \beta q^{32} \) \( + 756 \beta q^{33} \) \( + 4576 q^{34} \) \( + ( 1980 - 405 \beta ) q^{35} \) \( + 1836 q^{36} \) \( -2106 \beta q^{37} \) \( + 220 \beta q^{38} \) \( -2376 q^{39} \) \( + ( -4400 + 900 \beta ) q^{40} \) \( -198 q^{41} \) \( + 1188 \beta q^{42} \) \( + 63 \beta q^{43} \) \( -3024 q^{44} \) \( + ( 6885 + 765 \beta ) q^{45} \) \( -16148 q^{46} \) \( + 1589 \beta q^{47} \) \( -3792 \beta q^{48} \) \( + 13243 q^{49} \) \( + ( 19800 - 925 \beta ) q^{50} \) \( -13728 q^{51} \) \( -216 \beta q^{52} \) \( + 878 \beta q^{53} \) \( + 11880 q^{54} \) \( + ( -11340 - 1260 \beta ) q^{55} \) \( + 7920 q^{56} \) \( -660 \beta q^{57} \) \( + 6930 \beta q^{58} \) \( -24660 q^{59} \) \( + ( -7920 + 1620 \beta ) q^{60} \) \( -5698 q^{61} \) \( -6752 \beta q^{62} \) \( -1377 \beta q^{63} \) \( -12992 q^{64} \) \( + ( 3960 - 810 \beta ) q^{65} \) \( + 33264 q^{66} \) \( + 6579 \beta q^{67} \) \( -1248 \beta q^{68} \) \( + 48444 q^{69} \) \( + ( -17820 - 1980 \beta ) q^{70} \) \( + 53352 q^{71} \) \( + 3060 \beta q^{72} \) \( -10692 \beta q^{73} \) \( -92664 q^{74} \) \( + ( -59400 + 2775 \beta ) q^{75} \) \( + 2640 q^{76} \) \( + 2268 \beta q^{77} \) \( + 2376 \beta q^{78} \) \( + 51920 q^{79} \) \( + ( 56880 + 6320 \beta ) q^{80} \) \( -72819 q^{81} \) \( + 198 \beta q^{82} \) \( + 9323 \beta q^{83} \) \( + 14256 q^{84} \) \( + ( 22880 - 4680 \beta ) q^{85} \) \( + 2772 q^{86} \) \( -20790 \beta q^{87} \) \( -5040 \beta q^{88} \) \( -9990 q^{89} \) \( + ( 33660 - 6885 \beta ) q^{90} \) \( -7128 q^{91} \) \( + 4404 \beta q^{92} \) \( + 20256 \beta q^{93} \) \( + 69916 q^{94} \) \( + ( 9900 + 1100 \beta ) q^{95} \) \( -82368 q^{96} \) \( + 15264 \beta q^{97} \) \( -13243 \beta q^{98} \) \( -38556 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 90q^{5} \) \(\mathstrut +\mathstrut 264q^{6} \) \(\mathstrut -\mathstrut 306q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 24q^{4} \) \(\mathstrut -\mathstrut 90q^{5} \) \(\mathstrut +\mathstrut 264q^{6} \) \(\mathstrut -\mathstrut 306q^{9} \) \(\mathstrut -\mathstrut 440q^{10} \) \(\mathstrut +\mathstrut 504q^{11} \) \(\mathstrut +\mathstrut 792q^{14} \) \(\mathstrut +\mathstrut 1320q^{15} \) \(\mathstrut -\mathstrut 2528q^{16} \) \(\mathstrut -\mathstrut 440q^{19} \) \(\mathstrut +\mathstrut 1080q^{20} \) \(\mathstrut -\mathstrut 2376q^{21} \) \(\mathstrut +\mathstrut 5280q^{24} \) \(\mathstrut +\mathstrut 1850q^{25} \) \(\mathstrut +\mathstrut 1584q^{26} \) \(\mathstrut -\mathstrut 13860q^{29} \) \(\mathstrut -\mathstrut 11880q^{30} \) \(\mathstrut +\mathstrut 13504q^{31} \) \(\mathstrut +\mathstrut 9152q^{34} \) \(\mathstrut +\mathstrut 3960q^{35} \) \(\mathstrut +\mathstrut 3672q^{36} \) \(\mathstrut -\mathstrut 4752q^{39} \) \(\mathstrut -\mathstrut 8800q^{40} \) \(\mathstrut -\mathstrut 396q^{41} \) \(\mathstrut -\mathstrut 6048q^{44} \) \(\mathstrut +\mathstrut 13770q^{45} \) \(\mathstrut -\mathstrut 32296q^{46} \) \(\mathstrut +\mathstrut 26486q^{49} \) \(\mathstrut +\mathstrut 39600q^{50} \) \(\mathstrut -\mathstrut 27456q^{51} \) \(\mathstrut +\mathstrut 23760q^{54} \) \(\mathstrut -\mathstrut 22680q^{55} \) \(\mathstrut +\mathstrut 15840q^{56} \) \(\mathstrut -\mathstrut 49320q^{59} \) \(\mathstrut -\mathstrut 15840q^{60} \) \(\mathstrut -\mathstrut 11396q^{61} \) \(\mathstrut -\mathstrut 25984q^{64} \) \(\mathstrut +\mathstrut 7920q^{65} \) \(\mathstrut +\mathstrut 66528q^{66} \) \(\mathstrut +\mathstrut 96888q^{69} \) \(\mathstrut -\mathstrut 35640q^{70} \) \(\mathstrut +\mathstrut 106704q^{71} \) \(\mathstrut -\mathstrut 185328q^{74} \) \(\mathstrut -\mathstrut 118800q^{75} \) \(\mathstrut +\mathstrut 5280q^{76} \) \(\mathstrut +\mathstrut 103840q^{79} \) \(\mathstrut +\mathstrut 113760q^{80} \) \(\mathstrut -\mathstrut 145638q^{81} \) \(\mathstrut +\mathstrut 28512q^{84} \) \(\mathstrut +\mathstrut 45760q^{85} \) \(\mathstrut +\mathstrut 5544q^{86} \) \(\mathstrut -\mathstrut 19980q^{89} \) \(\mathstrut +\mathstrut 67320q^{90} \) \(\mathstrut -\mathstrut 14256q^{91} \) \(\mathstrut +\mathstrut 139832q^{94} \) \(\mathstrut +\mathstrut 19800q^{95} \) \(\mathstrut -\mathstrut 164736q^{96} \) \(\mathstrut -\mathstrut 77112q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0.500000 + 1.65831i
0.500000 1.65831i
6.63325i 19.8997i −12.0000 −45.0000 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 + 298.496i
4.2 6.63325i 19.8997i −12.0000 −45.0000 + 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 298.496i
\(n\): 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 \(S_{6}^{\mathrm{new}}(5, \chi)\).