# 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

# Learn more about

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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