Properties

Label 20.10.c.a
Level 20
Weight 10
Character orbit 20.c
Analytic conductor 10.301
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 20.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + ( 165 + \beta_{1} - \beta_{3} ) q^{5} \) \( + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} \) \( + ( -2261 + 7 \beta_{2} + 7 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + ( 165 + \beta_{1} - \beta_{3} ) q^{5} \) \( + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} \) \( + ( -2261 + 7 \beta_{2} + 7 \beta_{3} ) q^{9} \) \( + ( -8700 + 26 \beta_{2} + 26 \beta_{3} ) q^{11} \) \( + ( -282 \beta_{1} + 57 \beta_{2} - 57 \beta_{3} ) q^{13} \) \( + ( -24940 + 739 \beta_{1} + 125 \beta_{2} + 11 \beta_{3} ) q^{15} \) \( + ( 1192 \beta_{1} + 170 \beta_{2} - 170 \beta_{3} ) q^{17} \) \( + ( -56916 + 258 \beta_{2} + 258 \beta_{3} ) q^{19} \) \( + ( -71824 + 143 \beta_{2} + 143 \beta_{3} ) q^{21} \) \( + ( -13525 \beta_{1} + 19 \beta_{2} - 19 \beta_{3} ) q^{23} \) \( + ( -50475 + 11910 \beta_{1} - 625 \beta_{2} - 35 \beta_{3} ) q^{25} \) \( + ( 9386 \beta_{1} - 798 \beta_{2} + 798 \beta_{3} ) q^{27} \) \( + ( 1566414 - 2244 \beta_{2} - 2244 \beta_{3} ) q^{29} \) \( + ( 93616 - 1860 \beta_{2} - 1860 \beta_{3} ) q^{31} \) \( + ( -38548 \beta_{1} - 2964 \beta_{2} + 2964 \beta_{3} ) q^{33} \) \( + ( -2028540 + 12649 \beta_{1} - 500 \beta_{2} - 24 \beta_{3} ) q^{35} \) \( + ( -9390 \beta_{1} - 2603 \beta_{2} + 2603 \beta_{3} ) q^{37} \) \( + ( 5846664 + 4980 \beta_{2} + 4980 \beta_{3} ) q^{39} \) \( + ( -4412034 + 851 \beta_{2} + 851 \beta_{3} ) q^{41} \) \( + ( 122901 \beta_{1} + 11078 \beta_{2} - 11078 \beta_{3} ) q^{43} \) \( + ( -13310465 - 83321 \beta_{1} + 4375 \beta_{2} + 196 \beta_{3} ) q^{45} \) \( + ( -8137 \beta_{1} + 21013 \beta_{2} - 21013 \beta_{3} ) q^{47} \) \( + ( 36224703 - 381 \beta_{2} - 381 \beta_{3} ) q^{49} \) \( + ( -27175888 + 29084 \beta_{2} + 29084 \beta_{3} ) q^{51} \) \( + ( 410106 \beta_{1} - 8721 \beta_{2} + 8721 \beta_{3} ) q^{53} \) \( + ( -49488700 - 309780 \beta_{1} + 16250 \beta_{2} + 1030 \beta_{3} ) q^{55} \) \( + ( -353100 \beta_{1} - 29412 \beta_{2} + 29412 \beta_{3} ) q^{57} \) \( + ( 109748868 + 11846 \beta_{2} + 11846 \beta_{3} ) q^{59} \) \( + ( -25761118 - 85017 \beta_{2} - 85017 \beta_{3} ) q^{61} \) \( + ( -176939 \beta_{1} + 3381 \beta_{2} - 3381 \beta_{3} ) q^{63} \) \( + ( -104328960 + 386226 \beta_{1} - 85125 \beta_{2} - 6351 \beta_{3} ) q^{65} \) \( + ( 664065 \beta_{1} - 28918 \beta_{2} + 28918 \beta_{3} ) q^{67} \) \( + ( 296678752 - 92357 \beta_{2} - 92357 \beta_{3} ) q^{69} \) \( + ( -126020472 + 42048 \beta_{2} + 42048 \beta_{3} ) q^{71} \) \( + ( -1244412 \beta_{1} - 104368 \beta_{2} + 104368 \beta_{3} ) q^{73} \) \( + ( -259585400 + 328365 \beta_{1} + 85000 \beta_{2} + 9760 \beta_{3} ) q^{75} \) \( + ( -658108 \beta_{1} + 12256 \beta_{2} - 12256 \beta_{3} ) q^{77} \) \( + ( 448738752 + 72036 \beta_{2} + 72036 \beta_{3} ) q^{79} \) \( + ( -245688031 + 106127 \beta_{2} + 106127 \beta_{3} ) q^{81} \) \( + ( -2391607 \beta_{1} + 203338 \beta_{2} - 203338 \beta_{3} ) q^{83} \) \( + ( -361860880 + 2654328 \beta_{1} + 250 \beta_{2} + 3422 \beta_{3} ) q^{85} \) \( + ( 4142526 \beta_{1} + 255816 \beta_{2} - 255816 \beta_{3} ) q^{87} \) \( + ( 595329546 + 249508 \beta_{2} + 249508 \beta_{3} ) q^{89} \) \( + ( -202811256 - 86496 \beta_{2} - 86496 \beta_{3} ) q^{91} \) \( + ( 2228896 \beta_{1} + 212040 \beta_{2} - 212040 \beta_{3} ) q^{93} \) \( + ( -486226740 - 3044556 \beta_{1} + 161250 \beta_{2} - 19194 \beta_{3} ) q^{95} \) \( + ( -6424368 \beta_{1} - 196042 \beta_{2} + 196042 \beta_{3} ) q^{97} \) \( + ( 692415500 - 119686 \beta_{2} - 119686 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 660q^{5} \) \(\mathstrut -\mathstrut 9044q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 660q^{5} \) \(\mathstrut -\mathstrut 9044q^{9} \) \(\mathstrut -\mathstrut 34800q^{11} \) \(\mathstrut -\mathstrut 99760q^{15} \) \(\mathstrut -\mathstrut 227664q^{19} \) \(\mathstrut -\mathstrut 287296q^{21} \) \(\mathstrut -\mathstrut 201900q^{25} \) \(\mathstrut +\mathstrut 6265656q^{29} \) \(\mathstrut +\mathstrut 374464q^{31} \) \(\mathstrut -\mathstrut 8114160q^{35} \) \(\mathstrut +\mathstrut 23386656q^{39} \) \(\mathstrut -\mathstrut 17648136q^{41} \) \(\mathstrut -\mathstrut 53241860q^{45} \) \(\mathstrut +\mathstrut 144898812q^{49} \) \(\mathstrut -\mathstrut 108703552q^{51} \) \(\mathstrut -\mathstrut 197954800q^{55} \) \(\mathstrut +\mathstrut 438995472q^{59} \) \(\mathstrut -\mathstrut 103044472q^{61} \) \(\mathstrut -\mathstrut 417315840q^{65} \) \(\mathstrut +\mathstrut 1186715008q^{69} \) \(\mathstrut -\mathstrut 504081888q^{71} \) \(\mathstrut -\mathstrut 1038341600q^{75} \) \(\mathstrut +\mathstrut 1794955008q^{79} \) \(\mathstrut -\mathstrut 982752124q^{81} \) \(\mathstrut -\mathstrut 1447443520q^{85} \) \(\mathstrut +\mathstrut 2381318184q^{89} \) \(\mathstrut -\mathstrut 811245024q^{91} \) \(\mathstrut -\mathstrut 1944906960q^{95} \) \(\mathstrut +\mathstrut 2769662000q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(1095\) \(x^{2}\mathstrut -\mathstrut \) \(80251\) \(x\mathstrut +\mathstrut \) \(2230844\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 82 \nu^{2} + 99 \nu + 14468 \)\()/1000\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{3} + 246 \nu^{2} + 19703 \nu - 48404 \)\()/500\)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{3} - 66 \nu^{2} - 13713 \nu + 739084 \)\()/500\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)\()/40\)
\(\nu^{2}\)\(=\)\((\)\(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(86\) \(\beta_{1}\mathstrut -\mathstrut \) \(4378\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(1640\) \(\beta_{3}\mathstrut -\mathstrut \) \(1131\) \(\beta_{2}\mathstrut -\mathstrut \) \(4146\) \(\beta_{1}\mathstrut +\mathstrut \) \(2374690\)\()/40\)

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
−23.7825 44.2040i
24.2825 + 17.1975i
24.2825 17.1975i
−23.7825 + 44.2040i
0 188.155i 0 1126.30 827.388i 0 1842.93i 0 −15719.2 0
9.2 0 92.1183i 0 −796.301 + 1148.49i 0 2204.86i 0 11197.2 0
9.3 0 92.1183i 0 −796.301 1148.49i 0 2204.86i 0 11197.2 0
9.4 0 188.155i 0 1126.30 + 827.388i 0 1842.93i 0 −15719.2 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 \(S_{10}^{\mathrm{new}}(20, [\chi])\).