Properties

Label 20.4.c.a
Level 20
Weight 4
Character orbit 20.c
Analytic conductor 1.180
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 20.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{3} \) \( + ( 7 + \beta ) q^{5} \) \( + \beta q^{7} \) \( -49 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{3} \) \( + ( 7 + \beta ) q^{5} \) \( + \beta q^{7} \) \( -49 q^{9} \) \( + 20 q^{11} \) \( + 6 \beta q^{13} \) \( + ( 76 - 7 \beta ) q^{15} \) \( -8 \beta q^{17} \) \( -84 q^{19} \) \( + 76 q^{21} \) \( -7 \beta q^{23} \) \( + ( -27 + 14 \beta ) q^{25} \) \( + 22 \beta q^{27} \) \( + 6 q^{29} \) \( -224 q^{31} \) \( -20 \beta q^{33} \) \( + ( -76 + 7 \beta ) q^{35} \) \( -14 \beta q^{37} \) \( + 456 q^{39} \) \( + 266 q^{41} \) \( + 35 \beta q^{43} \) \( + ( -343 - 49 \beta ) q^{45} \) \( -43 \beta q^{47} \) \( + 267 q^{49} \) \( -608 q^{51} \) \( + 42 \beta q^{53} \) \( + ( 140 + 20 \beta ) q^{55} \) \( + 84 \beta q^{57} \) \( -28 q^{59} \) \( + 182 q^{61} \) \( -49 \beta q^{63} \) \( + ( -456 + 42 \beta ) q^{65} \) \( -49 \beta q^{67} \) \( -532 q^{69} \) \( + 408 q^{71} \) \( -124 \beta q^{73} \) \( + ( 1064 + 27 \beta ) q^{75} \) \( + 20 \beta q^{77} \) \( + 48 q^{79} \) \( + 349 q^{81} \) \( + 23 \beta q^{83} \) \( + ( 608 - 56 \beta ) q^{85} \) \( -6 \beta q^{87} \) \( -1526 q^{89} \) \( -456 q^{91} \) \( + 224 \beta q^{93} \) \( + ( -588 - 84 \beta ) q^{95} \) \( -64 \beta q^{97} \) \( -980 q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 98q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 98q^{9} \) \(\mathstrut +\mathstrut 40q^{11} \) \(\mathstrut +\mathstrut 152q^{15} \) \(\mathstrut -\mathstrut 168q^{19} \) \(\mathstrut +\mathstrut 152q^{21} \) \(\mathstrut -\mathstrut 54q^{25} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 448q^{31} \) \(\mathstrut -\mathstrut 152q^{35} \) \(\mathstrut +\mathstrut 912q^{39} \) \(\mathstrut +\mathstrut 532q^{41} \) \(\mathstrut -\mathstrut 686q^{45} \) \(\mathstrut +\mathstrut 534q^{49} \) \(\mathstrut -\mathstrut 1216q^{51} \) \(\mathstrut +\mathstrut 280q^{55} \) \(\mathstrut -\mathstrut 56q^{59} \) \(\mathstrut +\mathstrut 364q^{61} \) \(\mathstrut -\mathstrut 912q^{65} \) \(\mathstrut -\mathstrut 1064q^{69} \) \(\mathstrut +\mathstrut 816q^{71} \) \(\mathstrut +\mathstrut 2128q^{75} \) \(\mathstrut +\mathstrut 96q^{79} \) \(\mathstrut +\mathstrut 698q^{81} \) \(\mathstrut +\mathstrut 1216q^{85} \) \(\mathstrut -\mathstrut 3052q^{89} \) \(\mathstrut -\mathstrut 912q^{91} \) \(\mathstrut -\mathstrut 1176q^{95} \) \(\mathstrut -\mathstrut 1960q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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
0.500000 + 2.17945i
0.500000 2.17945i
0 8.71780i 0 7.00000 + 8.71780i 0 8.71780i 0 −49.0000 0
9.2 0 8.71780i 0 7.00000 8.71780i 0 8.71780i 0 −49.0000 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.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(20, [\chi])\).