Properties

Label 10.6.b.a
Level 10
Weight 6
Character orbit 10.b
Analytic conductor 1.604
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60383819813\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + 2 \beta q^{2} \) \( + 7 \beta q^{3} \) \( -16 q^{4} \) \( + ( 55 + 5 \beta ) q^{5} \) \( -56 q^{6} \) \( -79 \beta q^{7} \) \( -32 \beta q^{8} \) \( + 47 q^{9} \) \(+O(q^{10})\) \( q\) \( + 2 \beta q^{2} \) \( + 7 \beta q^{3} \) \( -16 q^{4} \) \( + ( 55 + 5 \beta ) q^{5} \) \( -56 q^{6} \) \( -79 \beta q^{7} \) \( -32 \beta q^{8} \) \( + 47 q^{9} \) \( + ( -40 + 110 \beta ) q^{10} \) \( -148 q^{11} \) \( -112 \beta q^{12} \) \( + 342 \beta q^{13} \) \( + 632 q^{14} \) \( + ( -140 + 385 \beta ) q^{15} \) \( + 256 q^{16} \) \( -1024 \beta q^{17} \) \( + 94 \beta q^{18} \) \( -2220 q^{19} \) \( + ( -880 - 80 \beta ) q^{20} \) \( + 2212 q^{21} \) \( -296 \beta q^{22} \) \( -623 \beta q^{23} \) \( + 896 q^{24} \) \( + ( 2925 + 550 \beta ) q^{25} \) \( -2736 q^{26} \) \( + 2030 \beta q^{27} \) \( + 1264 \beta q^{28} \) \( + 270 q^{29} \) \( + ( -3080 - 280 \beta ) q^{30} \) \( -2048 q^{31} \) \( + 512 \beta q^{32} \) \( -1036 \beta q^{33} \) \( + 8192 q^{34} \) \( + ( 1580 - 4345 \beta ) q^{35} \) \( -752 q^{36} \) \( + 2186 \beta q^{37} \) \( -4440 \beta q^{38} \) \( -9576 q^{39} \) \( + ( 640 - 1760 \beta ) q^{40} \) \( -2398 q^{41} \) \( + 4424 \beta q^{42} \) \( + 1147 \beta q^{43} \) \( + 2368 q^{44} \) \( + ( 2585 + 235 \beta ) q^{45} \) \( + 4984 q^{46} \) \( + 5341 \beta q^{47} \) \( + 1792 \beta q^{48} \) \( -8157 q^{49} \) \( + ( -4400 + 5850 \beta ) q^{50} \) \( + 28672 q^{51} \) \( -5472 \beta q^{52} \) \( + 1482 \beta q^{53} \) \( -16240 q^{54} \) \( + ( -8140 - 740 \beta ) q^{55} \) \( -10112 q^{56} \) \( -15540 \beta q^{57} \) \( + 540 \beta q^{58} \) \( + 39740 q^{59} \) \( + ( 2240 - 6160 \beta ) q^{60} \) \( -42298 q^{61} \) \( -4096 \beta q^{62} \) \( -3713 \beta q^{63} \) \( -4096 q^{64} \) \( + ( -6840 + 18810 \beta ) q^{65} \) \( + 8288 q^{66} \) \( -16049 \beta q^{67} \) \( + 16384 \beta q^{68} \) \( + 17444 q^{69} \) \( + ( 34760 + 3160 \beta ) q^{70} \) \( -4248 q^{71} \) \( -1504 \beta q^{72} \) \( + 15052 \beta q^{73} \) \( -17488 q^{74} \) \( + ( -15400 + 20475 \beta ) q^{75} \) \( + 35520 q^{76} \) \( + 11692 \beta q^{77} \) \( -19152 \beta q^{78} \) \( -35280 q^{79} \) \( + ( 14080 + 1280 \beta ) q^{80} \) \( -45419 q^{81} \) \( -4796 \beta q^{82} \) \( -13913 \beta q^{83} \) \( -35392 q^{84} \) \( + ( 20480 - 56320 \beta ) q^{85} \) \( -9176 q^{86} \) \( + 1890 \beta q^{87} \) \( + 4736 \beta q^{88} \) \( + 85210 q^{89} \) \( + ( -1880 + 5170 \beta ) q^{90} \) \( + 108072 q^{91} \) \( + 9968 \beta q^{92} \) \( -14336 \beta q^{93} \) \( -42728 q^{94} \) \( + ( -122100 - 11100 \beta ) q^{95} \) \( -14336 q^{96} \) \( + 48616 \beta q^{97} \) \( -16314 \beta q^{98} \) \( -6956 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 110q^{5} \) \(\mathstrut -\mathstrut 112q^{6} \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 32q^{4} \) \(\mathstrut +\mathstrut 110q^{5} \) \(\mathstrut -\mathstrut 112q^{6} \) \(\mathstrut +\mathstrut 94q^{9} \) \(\mathstrut -\mathstrut 80q^{10} \) \(\mathstrut -\mathstrut 296q^{11} \) \(\mathstrut +\mathstrut 1264q^{14} \) \(\mathstrut -\mathstrut 280q^{15} \) \(\mathstrut +\mathstrut 512q^{16} \) \(\mathstrut -\mathstrut 4440q^{19} \) \(\mathstrut -\mathstrut 1760q^{20} \) \(\mathstrut +\mathstrut 4424q^{21} \) \(\mathstrut +\mathstrut 1792q^{24} \) \(\mathstrut +\mathstrut 5850q^{25} \) \(\mathstrut -\mathstrut 5472q^{26} \) \(\mathstrut +\mathstrut 540q^{29} \) \(\mathstrut -\mathstrut 6160q^{30} \) \(\mathstrut -\mathstrut 4096q^{31} \) \(\mathstrut +\mathstrut 16384q^{34} \) \(\mathstrut +\mathstrut 3160q^{35} \) \(\mathstrut -\mathstrut 1504q^{36} \) \(\mathstrut -\mathstrut 19152q^{39} \) \(\mathstrut +\mathstrut 1280q^{40} \) \(\mathstrut -\mathstrut 4796q^{41} \) \(\mathstrut +\mathstrut 4736q^{44} \) \(\mathstrut +\mathstrut 5170q^{45} \) \(\mathstrut +\mathstrut 9968q^{46} \) \(\mathstrut -\mathstrut 16314q^{49} \) \(\mathstrut -\mathstrut 8800q^{50} \) \(\mathstrut +\mathstrut 57344q^{51} \) \(\mathstrut -\mathstrut 32480q^{54} \) \(\mathstrut -\mathstrut 16280q^{55} \) \(\mathstrut -\mathstrut 20224q^{56} \) \(\mathstrut +\mathstrut 79480q^{59} \) \(\mathstrut +\mathstrut 4480q^{60} \) \(\mathstrut -\mathstrut 84596q^{61} \) \(\mathstrut -\mathstrut 8192q^{64} \) \(\mathstrut -\mathstrut 13680q^{65} \) \(\mathstrut +\mathstrut 16576q^{66} \) \(\mathstrut +\mathstrut 34888q^{69} \) \(\mathstrut +\mathstrut 69520q^{70} \) \(\mathstrut -\mathstrut 8496q^{71} \) \(\mathstrut -\mathstrut 34976q^{74} \) \(\mathstrut -\mathstrut 30800q^{75} \) \(\mathstrut +\mathstrut 71040q^{76} \) \(\mathstrut -\mathstrut 70560q^{79} \) \(\mathstrut +\mathstrut 28160q^{80} \) \(\mathstrut -\mathstrut 90838q^{81} \) \(\mathstrut -\mathstrut 70784q^{84} \) \(\mathstrut +\mathstrut 40960q^{85} \) \(\mathstrut -\mathstrut 18352q^{86} \) \(\mathstrut +\mathstrut 170420q^{89} \) \(\mathstrut -\mathstrut 3760q^{90} \) \(\mathstrut +\mathstrut 216144q^{91} \) \(\mathstrut -\mathstrut 85456q^{94} \) \(\mathstrut -\mathstrut 244200q^{95} \) \(\mathstrut -\mathstrut 28672q^{96} \) \(\mathstrut -\mathstrut 13912q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
1.00000i
1.00000i
4.00000i 14.0000i −16.0000 55.0000 10.0000i −56.0000 158.000i 64.0000i 47.0000 −40.0000 220.000i
9.2 4.00000i 14.0000i −16.0000 55.0000 + 10.0000i −56.0000 158.000i 64.0000i 47.0000 −40.0000 + 220.000i
\(n\): 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 \(S_{6}^{\mathrm{new}}(10, [\chi])\).