Properties

Label 20.3.f.a
Level 20
Weight 3
Character orbit 20.f
Analytic conductor 0.545
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 20.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( 1 - i ) q^{3} \) \( + ( -3 + 4 i ) q^{5} \) \( + ( -7 - 7 i ) q^{7} \) \( + 7 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - i ) q^{3} \) \( + ( -3 + 4 i ) q^{5} \) \( + ( -7 - 7 i ) q^{7} \) \( + 7 i q^{9} \) \( + 10 q^{11} \) \( + ( 9 - 9 i ) q^{13} \) \( + ( 1 + 7 i ) q^{15} \) \( + ( 1 + i ) q^{17} \) \( -8 i q^{19} \) \( -14 q^{21} \) \( + ( -23 + 23 i ) q^{23} \) \( + ( -7 - 24 i ) q^{25} \) \( + ( 16 + 16 i ) q^{27} \) \( -8 i q^{29} \) \( -14 q^{31} \) \( + ( 10 - 10 i ) q^{33} \) \( + ( 49 - 7 i ) q^{35} \) \( + ( 33 + 33 i ) q^{37} \) \( -18 i q^{39} \) \( -14 q^{41} \) \( + ( -15 + 15 i ) q^{43} \) \( + ( -28 - 21 i ) q^{45} \) \( + ( -39 - 39 i ) q^{47} \) \( + 49 i q^{49} \) \( + 2 q^{51} \) \( + ( -7 + 7 i ) q^{53} \) \( + ( -30 + 40 i ) q^{55} \) \( + ( -8 - 8 i ) q^{57} \) \( -56 i q^{59} \) \( + 42 q^{61} \) \( + ( 49 - 49 i ) q^{63} \) \( + ( 9 + 63 i ) q^{65} \) \( + ( -7 - 7 i ) q^{67} \) \( + 46 i q^{69} \) \( + 98 q^{71} \) \( + ( 49 - 49 i ) q^{73} \) \( + ( -31 - 17 i ) q^{75} \) \( + ( -70 - 70 i ) q^{77} \) \( + 96 i q^{79} \) \( -31 q^{81} \) \( + ( -63 + 63 i ) q^{83} \) \( + ( -7 + i ) q^{85} \) \( + ( -8 - 8 i ) q^{87} \) \( -112 i q^{89} \) \( -126 q^{91} \) \( + ( -14 + 14 i ) q^{93} \) \( + ( 32 + 24 i ) q^{95} \) \( + ( 33 + 33 i ) q^{97} \) \( + 70 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 20q^{11} \) \(\mathstrut +\mathstrut 18q^{13} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 28q^{21} \) \(\mathstrut -\mathstrut 46q^{23} \) \(\mathstrut -\mathstrut 14q^{25} \) \(\mathstrut +\mathstrut 32q^{27} \) \(\mathstrut -\mathstrut 28q^{31} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut +\mathstrut 98q^{35} \) \(\mathstrut +\mathstrut 66q^{37} \) \(\mathstrut -\mathstrut 28q^{41} \) \(\mathstrut -\mathstrut 30q^{43} \) \(\mathstrut -\mathstrut 56q^{45} \) \(\mathstrut -\mathstrut 78q^{47} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 14q^{53} \) \(\mathstrut -\mathstrut 60q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 84q^{61} \) \(\mathstrut +\mathstrut 98q^{63} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut -\mathstrut 14q^{67} \) \(\mathstrut +\mathstrut 196q^{71} \) \(\mathstrut +\mathstrut 98q^{73} \) \(\mathstrut -\mathstrut 62q^{75} \) \(\mathstrut -\mathstrut 140q^{77} \) \(\mathstrut -\mathstrut 62q^{81} \) \(\mathstrut -\mathstrut 126q^{83} \) \(\mathstrut -\mathstrut 14q^{85} \) \(\mathstrut -\mathstrut 16q^{87} \) \(\mathstrut -\mathstrut 252q^{91} \) \(\mathstrut -\mathstrut 28q^{93} \) \(\mathstrut +\mathstrut 64q^{95} \) \(\mathstrut +\mathstrut 66q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
0 1.00000 + 1.00000i 0 −3.00000 4.00000i 0 −7.00000 + 7.00000i 0 7.00000i 0
17.1 0 1.00000 1.00000i 0 −3.00000 + 4.00000i 0 −7.00000 7.00000i 0 7.00000i 0
\(n\): 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 \(S_{3}^{\mathrm{new}}(20, [\chi])\).