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.

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

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 \(S_{4}^{\mathrm{new}}(10, [\chi])\).