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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} + 1095 x^{2} - 80251 x + 2230844\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 660q^{5} - 9044q^{9} + O(q^{10}) \) \( 4q + 660q^{5} - 9044q^{9} - 34800q^{11} - 99760q^{15} - 227664q^{19} - 287296q^{21} - 201900q^{25} + 6265656q^{29} + 374464q^{31} - 8114160q^{35} + 23386656q^{39} - 17648136q^{41} - 53241860q^{45} + 144898812q^{49} - 108703552q^{51} - 197954800q^{55} + 438995472q^{59} - 103044472q^{61} - 417315840q^{65} + 1186715008q^{69} - 504081888q^{71} - 1038341600q^{75} + 1794955008q^{79} - 982752124q^{81} - 1447443520q^{85} + 2381318184q^{89} - 811245024q^{91} - 1944906960q^{95} + 2769662000q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.10.c.a 4
3.b odd 2 1 180.10.d.a 4
4.b odd 2 1 80.10.c.b 4
5.b even 2 1 inner 20.10.c.a 4
5.c odd 4 2 100.10.a.f 4
15.d odd 2 1 180.10.d.a 4
20.d odd 2 1 80.10.c.b 4
20.e even 4 2 400.10.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.c.a 4 1.a even 1 1 trivial
20.10.c.a 4 5.b even 2 1 inner
80.10.c.b 4 4.b odd 2 1
80.10.c.b 4 20.d odd 2 1
100.10.a.f 4 5.c odd 4 2
180.10.d.a 4 3.b odd 2 1
180.10.d.a 4 15.d odd 2 1
400.10.a.bb 4 20.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 34844 T^{2} + 897243462 T^{4} - 13499279518716 T^{6} + 150094635296999121 T^{8} \)
$5$ \( 1 - 660 T + 318750 T^{2} - 1289062500 T^{3} + 3814697265625 T^{4} \)
$7$ \( 1 - 153156620 T^{2} + 9120528185156598 T^{4} - \)\(24\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 17400 T + 2292818982 T^{2} + 41028289823400 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 - 14000436724 T^{2} + 83025702170976147702 T^{4} - \)\(15\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 181978394436 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 + 113832 T + 402567657014 T^{2} + 36732186013579128 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 + 820282526580 T^{2} + \)\(11\!\cdots\!38\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 3132828 T + 12854589500734 T^{2} - 45448393113289727532 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 - 187232 T + 40099942836798 T^{2} - 4950343336386752672 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 - 462603258659540 T^{2} + \)\(87\!\cdots\!58\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 + 8824068 T + 671552976228678 T^{2} + \)\(28\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 - 358707990293372 T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(90\!\cdots\!28\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 1037666802860076 T^{2} + \)\(16\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 5310844341928340 T^{2} + \)\(28\!\cdots\!78\)\( T^{4} - \)\(57\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 219497736 T + 28852098295168902 T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 + 51522236 T - 2665246277981394 T^{2} + \)\(60\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - 83417417389239260 T^{2} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 + 252040944 T + 101042798798695246 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 - 79546297979572004 T^{2} + \)\(52\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 - 897477504 T + 421888354983619742 T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 - 186261173343564060 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(65\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 1190659092 T + 825013495390166934 T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 - 899946211039106180 T^{2} + \)\(23\!\cdots\!78\)\( T^{4} - \)\(52\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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