Properties

Label 10.4.b.a
Level 10
Weight 4
Character orbit 10.b
Analytic conductor 0.590
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.590019100057\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -\beta q^{3} \) \( -4 q^{4} \) \( + ( -5 - 5 \beta ) q^{5} \) \( + 4 q^{6} \) \( + 13 \beta q^{7} \) \( -4 \beta q^{8} \) \( + 23 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -\beta q^{3} \) \( -4 q^{4} \) \( + ( -5 - 5 \beta ) q^{5} \) \( + 4 q^{6} \) \( + 13 \beta q^{7} \) \( -4 \beta q^{8} \) \( + 23 q^{9} \) \( + ( 20 - 5 \beta ) q^{10} \) \( -28 q^{11} \) \( + 4 \beta q^{12} \) \( -6 \beta q^{13} \) \( -52 q^{14} \) \( + ( -20 + 5 \beta ) q^{15} \) \( + 16 q^{16} \) \( -32 \beta q^{17} \) \( + 23 \beta q^{18} \) \( + 60 q^{19} \) \( + ( 20 + 20 \beta ) q^{20} \) \( + 52 q^{21} \) \( -28 \beta q^{22} \) \( + 29 \beta q^{23} \) \( -16 q^{24} \) \( + ( -75 + 50 \beta ) q^{25} \) \( + 24 q^{26} \) \( -50 \beta q^{27} \) \( -52 \beta q^{28} \) \( -90 q^{29} \) \( + ( -20 - 20 \beta ) q^{30} \) \( -128 q^{31} \) \( + 16 \beta q^{32} \) \( + 28 \beta q^{33} \) \( + 128 q^{34} \) \( + ( 260 - 65 \beta ) q^{35} \) \( -92 q^{36} \) \( + 118 \beta q^{37} \) \( + 60 \beta q^{38} \) \( -24 q^{39} \) \( + ( -80 + 20 \beta ) q^{40} \) \( + 242 q^{41} \) \( + 52 \beta q^{42} \) \( -181 \beta q^{43} \) \( + 112 q^{44} \) \( + ( -115 - 115 \beta ) q^{45} \) \( -116 q^{46} \) \( + 113 \beta q^{47} \) \( -16 \beta q^{48} \) \( -333 q^{49} \) \( + ( -200 - 75 \beta ) q^{50} \) \( -128 q^{51} \) \( + 24 \beta q^{52} \) \( + 54 \beta q^{53} \) \( + 200 q^{54} \) \( + ( 140 + 140 \beta ) q^{55} \) \( + 208 q^{56} \) \( -60 \beta q^{57} \) \( -90 \beta q^{58} \) \( + 20 q^{59} \) \( + ( 80 - 20 \beta ) q^{60} \) \( + 542 q^{61} \) \( -128 \beta q^{62} \) \( + 299 \beta q^{63} \) \( -64 q^{64} \) \( + ( -120 + 30 \beta ) q^{65} \) \( -112 q^{66} \) \( -217 \beta q^{67} \) \( + 128 \beta q^{68} \) \( + 116 q^{69} \) \( + ( 260 + 260 \beta ) q^{70} \) \( -1128 q^{71} \) \( -92 \beta q^{72} \) \( -316 \beta q^{73} \) \( -472 q^{74} \) \( + ( 200 + 75 \beta ) q^{75} \) \( -240 q^{76} \) \( -364 \beta q^{77} \) \( -24 \beta q^{78} \) \( + 720 q^{79} \) \( + ( -80 - 80 \beta ) q^{80} \) \( + 421 q^{81} \) \( + 242 \beta q^{82} \) \( + 239 \beta q^{83} \) \( -208 q^{84} \) \( + ( -640 + 160 \beta ) q^{85} \) \( + 724 q^{86} \) \( + 90 \beta q^{87} \) \( + 112 \beta q^{88} \) \( + 490 q^{89} \) \( + ( 460 - 115 \beta ) q^{90} \) \( + 312 q^{91} \) \( -116 \beta q^{92} \) \( + 128 \beta q^{93} \) \( -452 q^{94} \) \( + ( -300 - 300 \beta ) q^{95} \) \( + 64 q^{96} \) \( + 728 \beta q^{97} \) \( -333 \beta q^{98} \) \( -644 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 46q^{9} \) \(\mathstrut +\mathstrut 40q^{10} \) \(\mathstrut -\mathstrut 56q^{11} \) \(\mathstrut -\mathstrut 104q^{14} \) \(\mathstrut -\mathstrut 40q^{15} \) \(\mathstrut +\mathstrut 32q^{16} \) \(\mathstrut +\mathstrut 120q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut +\mathstrut 104q^{21} \) \(\mathstrut -\mathstrut 32q^{24} \) \(\mathstrut -\mathstrut 150q^{25} \) \(\mathstrut +\mathstrut 48q^{26} \) \(\mathstrut -\mathstrut 180q^{29} \) \(\mathstrut -\mathstrut 40q^{30} \) \(\mathstrut -\mathstrut 256q^{31} \) \(\mathstrut +\mathstrut 256q^{34} \) \(\mathstrut +\mathstrut 520q^{35} \) \(\mathstrut -\mathstrut 184q^{36} \) \(\mathstrut -\mathstrut 48q^{39} \) \(\mathstrut -\mathstrut 160q^{40} \) \(\mathstrut +\mathstrut 484q^{41} \) \(\mathstrut +\mathstrut 224q^{44} \) \(\mathstrut -\mathstrut 230q^{45} \) \(\mathstrut -\mathstrut 232q^{46} \) \(\mathstrut -\mathstrut 666q^{49} \) \(\mathstrut -\mathstrut 400q^{50} \) \(\mathstrut -\mathstrut 256q^{51} \) \(\mathstrut +\mathstrut 400q^{54} \) \(\mathstrut +\mathstrut 280q^{55} \) \(\mathstrut +\mathstrut 416q^{56} \) \(\mathstrut +\mathstrut 40q^{59} \) \(\mathstrut +\mathstrut 160q^{60} \) \(\mathstrut +\mathstrut 1084q^{61} \) \(\mathstrut -\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 240q^{65} \) \(\mathstrut -\mathstrut 224q^{66} \) \(\mathstrut +\mathstrut 232q^{69} \) \(\mathstrut +\mathstrut 520q^{70} \) \(\mathstrut -\mathstrut 2256q^{71} \) \(\mathstrut -\mathstrut 944q^{74} \) \(\mathstrut +\mathstrut 400q^{75} \) \(\mathstrut -\mathstrut 480q^{76} \) \(\mathstrut +\mathstrut 1440q^{79} \) \(\mathstrut -\mathstrut 160q^{80} \) \(\mathstrut +\mathstrut 842q^{81} \) \(\mathstrut -\mathstrut 416q^{84} \) \(\mathstrut -\mathstrut 1280q^{85} \) \(\mathstrut +\mathstrut 1448q^{86} \) \(\mathstrut +\mathstrut 980q^{89} \) \(\mathstrut +\mathstrut 920q^{90} \) \(\mathstrut +\mathstrut 624q^{91} \) \(\mathstrut -\mathstrut 904q^{94} \) \(\mathstrut -\mathstrut 600q^{95} \) \(\mathstrut +\mathstrut 128q^{96} \) \(\mathstrut -\mathstrut 1288q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
2.00000i 2.00000i −4.00000 −5.00000 + 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 + 10.0000i
9.2 2.00000i 2.00000i −4.00000 −5.00000 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 10.0000i
\(n\): 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 \(S_{4}^{\mathrm{new}}(10, [\chi])\).