Properties

Label 10.3.c.a
Level 10
Weight 3
Character orbit 10.c
Analytic conductor 0.272
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.27248026436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -1 - i ) q^{2} \) \( + ( -2 + 2 i ) q^{3} \) \( + 2 i q^{4} \) \( -5 i q^{5} \) \( + 4 q^{6} \) \( + ( 2 + 2 i ) q^{7} \) \( + ( 2 - 2 i ) q^{8} \) \( + i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - i ) q^{2} \) \( + ( -2 + 2 i ) q^{3} \) \( + 2 i q^{4} \) \( -5 i q^{5} \) \( + 4 q^{6} \) \( + ( 2 + 2 i ) q^{7} \) \( + ( 2 - 2 i ) q^{8} \) \( + i q^{9} \) \( + ( -5 + 5 i ) q^{10} \) \( -8 q^{11} \) \( + ( -4 - 4 i ) q^{12} \) \( + ( 3 - 3 i ) q^{13} \) \( -4 i q^{14} \) \( + ( 10 + 10 i ) q^{15} \) \( -4 q^{16} \) \( + ( 7 + 7 i ) q^{17} \) \( + ( 1 - i ) q^{18} \) \( -20 i q^{19} \) \( + 10 q^{20} \) \( -8 q^{21} \) \( + ( 8 + 8 i ) q^{22} \) \( + ( -2 + 2 i ) q^{23} \) \( + 8 i q^{24} \) \( -25 q^{25} \) \( -6 q^{26} \) \( + ( -20 - 20 i ) q^{27} \) \( + ( -4 + 4 i ) q^{28} \) \( + 40 i q^{29} \) \( -20 i q^{30} \) \( + 52 q^{31} \) \( + ( 4 + 4 i ) q^{32} \) \( + ( 16 - 16 i ) q^{33} \) \( -14 i q^{34} \) \( + ( 10 - 10 i ) q^{35} \) \( -2 q^{36} \) \( + ( -3 - 3 i ) q^{37} \) \( + ( -20 + 20 i ) q^{38} \) \( + 12 i q^{39} \) \( + ( -10 - 10 i ) q^{40} \) \( -8 q^{41} \) \( + ( 8 + 8 i ) q^{42} \) \( + ( -42 + 42 i ) q^{43} \) \( -16 i q^{44} \) \( + 5 q^{45} \) \( + 4 q^{46} \) \( + ( -18 - 18 i ) q^{47} \) \( + ( 8 - 8 i ) q^{48} \) \( -41 i q^{49} \) \( + ( 25 + 25 i ) q^{50} \) \( -28 q^{51} \) \( + ( 6 + 6 i ) q^{52} \) \( + ( 53 - 53 i ) q^{53} \) \( + 40 i q^{54} \) \( + 40 i q^{55} \) \( + 8 q^{56} \) \( + ( 40 + 40 i ) q^{57} \) \( + ( 40 - 40 i ) q^{58} \) \( -20 i q^{59} \) \( + ( -20 + 20 i ) q^{60} \) \( -48 q^{61} \) \( + ( -52 - 52 i ) q^{62} \) \( + ( -2 + 2 i ) q^{63} \) \( -8 i q^{64} \) \( + ( -15 - 15 i ) q^{65} \) \( -32 q^{66} \) \( + ( 62 + 62 i ) q^{67} \) \( + ( -14 + 14 i ) q^{68} \) \( -8 i q^{69} \) \( -20 q^{70} \) \( -28 q^{71} \) \( + ( 2 + 2 i ) q^{72} \) \( + ( -47 + 47 i ) q^{73} \) \( + 6 i q^{74} \) \( + ( 50 - 50 i ) q^{75} \) \( + 40 q^{76} \) \( + ( -16 - 16 i ) q^{77} \) \( + ( 12 - 12 i ) q^{78} \) \( + 20 i q^{80} \) \( + 71 q^{81} \) \( + ( 8 + 8 i ) q^{82} \) \( + ( 18 - 18 i ) q^{83} \) \( -16 i q^{84} \) \( + ( 35 - 35 i ) q^{85} \) \( + 84 q^{86} \) \( + ( -80 - 80 i ) q^{87} \) \( + ( -16 + 16 i ) q^{88} \) \( + 80 i q^{89} \) \( + ( -5 - 5 i ) q^{90} \) \( + 12 q^{91} \) \( + ( -4 - 4 i ) q^{92} \) \( + ( -104 + 104 i ) q^{93} \) \( + 36 i q^{94} \) \( -100 q^{95} \) \( -16 q^{96} \) \( + ( -63 - 63 i ) q^{97} \) \( + ( -41 + 41 i ) q^{98} \) \( -8 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 10q^{10} \) \(\mathstrut -\mathstrut 16q^{11} \) \(\mathstrut -\mathstrut 8q^{12} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 20q^{15} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 14q^{17} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 20q^{20} \) \(\mathstrut -\mathstrut 16q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 50q^{25} \) \(\mathstrut -\mathstrut 12q^{26} \) \(\mathstrut -\mathstrut 40q^{27} \) \(\mathstrut -\mathstrut 8q^{28} \) \(\mathstrut +\mathstrut 104q^{31} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut +\mathstrut 32q^{33} \) \(\mathstrut +\mathstrut 20q^{35} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 40q^{38} \) \(\mathstrut -\mathstrut 20q^{40} \) \(\mathstrut -\mathstrut 16q^{41} \) \(\mathstrut +\mathstrut 16q^{42} \) \(\mathstrut -\mathstrut 84q^{43} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 36q^{47} \) \(\mathstrut +\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 50q^{50} \) \(\mathstrut -\mathstrut 56q^{51} \) \(\mathstrut +\mathstrut 12q^{52} \) \(\mathstrut +\mathstrut 106q^{53} \) \(\mathstrut +\mathstrut 16q^{56} \) \(\mathstrut +\mathstrut 80q^{57} \) \(\mathstrut +\mathstrut 80q^{58} \) \(\mathstrut -\mathstrut 40q^{60} \) \(\mathstrut -\mathstrut 96q^{61} \) \(\mathstrut -\mathstrut 104q^{62} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut -\mathstrut 30q^{65} \) \(\mathstrut -\mathstrut 64q^{66} \) \(\mathstrut +\mathstrut 124q^{67} \) \(\mathstrut -\mathstrut 28q^{68} \) \(\mathstrut -\mathstrut 40q^{70} \) \(\mathstrut -\mathstrut 56q^{71} \) \(\mathstrut +\mathstrut 4q^{72} \) \(\mathstrut -\mathstrut 94q^{73} \) \(\mathstrut +\mathstrut 100q^{75} \) \(\mathstrut +\mathstrut 80q^{76} \) \(\mathstrut -\mathstrut 32q^{77} \) \(\mathstrut +\mathstrut 24q^{78} \) \(\mathstrut +\mathstrut 142q^{81} \) \(\mathstrut +\mathstrut 16q^{82} \) \(\mathstrut +\mathstrut 36q^{83} \) \(\mathstrut +\mathstrut 70q^{85} \) \(\mathstrut +\mathstrut 168q^{86} \) \(\mathstrut -\mathstrut 160q^{87} \) \(\mathstrut -\mathstrut 32q^{88} \) \(\mathstrut -\mathstrut 10q^{90} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut 8q^{92} \) \(\mathstrut -\mathstrut 208q^{93} \) \(\mathstrut -\mathstrut 200q^{95} \) \(\mathstrut -\mathstrut 32q^{96} \) \(\mathstrut -\mathstrut 126q^{97} \) \(\mathstrut -\mathstrut 82q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(7\)
\(\chi(n)\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
−1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000 2.00000 2.00000i 2.00000 + 2.00000i 1.00000i −5.00000 5.00000i
7.1 −1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000 2.00000 + 2.00000i 2.00000 2.00000i 1.00000i −5.00000 + 5.00000i
\(n\): 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.c Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(10, [\chi])\).