# Properties

 Label 5.8.a.b Level 5 Weight 8 Character orbit 5.a Self dual yes Analytic conductor 1.562 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 5.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.56192512742$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 10 + \beta ) q^{2} + ( 10 - 8 \beta ) q^{3} + ( 48 + 20 \beta ) q^{4} -125 q^{5} + ( -508 - 70 \beta ) q^{6} + ( -50 + 56 \beta ) q^{7} + ( 720 + 120 \beta ) q^{8} + ( 2777 - 160 \beta ) q^{9} +O(q^{10})$$ $$q + ( 10 + \beta ) q^{2} + ( 10 - 8 \beta ) q^{3} + ( 48 + 20 \beta ) q^{4} -125 q^{5} + ( -508 - 70 \beta ) q^{6} + ( -50 + 56 \beta ) q^{7} + ( 720 + 120 \beta ) q^{8} + ( 2777 - 160 \beta ) q^{9} + ( -1250 - 125 \beta ) q^{10} + ( 2272 + 400 \beta ) q^{11} + ( -11680 - 184 \beta ) q^{12} + ( 1770 - 608 \beta ) q^{13} + ( 3756 + 510 \beta ) q^{14} + ( -1250 + 1000 \beta ) q^{15} + ( 10176 - 640 \beta ) q^{16} + ( -13670 - 1184 \beta ) q^{17} + ( 15610 + 1177 \beta ) q^{18} + ( 19380 - 320 \beta ) q^{19} + ( -6000 - 2500 \beta ) q^{20} + ( -34548 + 960 \beta ) q^{21} + ( 53120 + 6272 \beta ) q^{22} + ( -62070 - 408 \beta ) q^{23} + ( -65760 - 4560 \beta ) q^{24} + 15625 q^{25} + ( -28508 - 4310 \beta ) q^{26} + ( 103180 - 6320 \beta ) q^{27} + ( 82720 + 1688 \beta ) q^{28} + ( -36130 + 19520 \beta ) q^{29} + ( 63500 + 8750 \beta ) q^{30} + ( 153412 - 2800 \beta ) q^{31} + ( -39040 - 11584 \beta ) q^{32} + ( -220480 - 14176 \beta ) q^{33} + ( -226684 - 25510 \beta ) q^{34} + ( 6250 - 7000 \beta ) q^{35} + ( -109904 + 47860 \beta ) q^{36} + ( -61510 + 25536 \beta ) q^{37} + ( 169480 + 16180 \beta ) q^{38} + ( 387364 - 20240 \beta ) q^{39} + ( -90000 - 15000 \beta ) q^{40} + ( 132182 - 56800 \beta ) q^{41} + ( -272520 - 24948 \beta ) q^{42} + ( 211650 + 43192 \beta ) q^{43} + ( 717056 + 64640 \beta ) q^{44} + ( -347125 + 20000 \beta ) q^{45} + ( -651708 - 66150 \beta ) q^{46} + ( -52730 + 45496 \beta ) q^{47} + ( 490880 - 87808 \beta ) q^{48} + ( -582707 - 5600 \beta ) q^{49} + ( 156250 + 15625 \beta ) q^{50} + ( 583172 + 97520 \beta ) q^{51} + ( -839200 + 6216 \beta ) q^{52} + ( -1195790 - 53408 \beta ) q^{53} + ( 551480 + 39980 \beta ) q^{54} + ( -284000 - 50000 \beta ) q^{55} + ( 474720 + 34320 \beta ) q^{56} + ( 388360 - 158240 \beta ) q^{57} + ( 1122220 + 159070 \beta ) q^{58} + ( -560060 - 227360 \beta ) q^{59} + ( 1460000 + 23000 \beta ) q^{60} + ( 1128522 + 160000 \beta ) q^{61} + ( 1321320 + 125412 \beta ) q^{62} + ( -819810 + 163512 \beta ) q^{63} + ( -2573312 - 72960 \beta ) q^{64} + ( -221250 + 76000 \beta ) q^{65} + ( -3282176 - 362240 \beta ) q^{66} + ( 2258230 - 79384 \beta ) q^{67} + ( -2455840 - 330232 \beta ) q^{68} + ( -372636 + 492480 \beta ) q^{69} + ( -469500 - 63750 \beta ) q^{70} + ( 310892 - 70000 \beta ) q^{71} + ( 540240 + 218040 \beta ) q^{72} + ( 2284530 - 226208 \beta ) q^{73} + ( 1325636 + 193850 \beta ) q^{74} + ( 156250 - 125000 \beta ) q^{75} + ( 443840 + 372240 \beta ) q^{76} + ( 1588800 + 107232 \beta ) q^{77} + ( 2335400 + 184964 \beta ) q^{78} + ( 2166520 - 472480 \beta ) q^{79} + ( -1272000 + 80000 \beta ) q^{80} + ( -1198939 - 538720 \beta ) q^{81} + ( -2994980 - 435818 \beta ) q^{82} + ( -4896510 + 490392 \beta ) q^{83} + ( -199104 - 644880 \beta ) q^{84} + ( 1708750 + 148000 \beta ) q^{85} + ( 5399092 + 643570 \beta ) q^{86} + ( -12229460 + 484240 \beta ) q^{87} + ( 5283840 + 560640 \beta ) q^{88} + ( 3012810 + 317760 \beta ) q^{89} + ( -1951250 - 147125 \beta ) q^{90} + ( -2676148 + 129520 \beta ) q^{91} + ( -3599520 - 1260984 \beta ) q^{92} + ( 3236520 - 1255296 \beta ) q^{93} + ( 2930396 + 402230 \beta ) q^{94} + ( -2422500 + 40000 \beta ) q^{95} + ( 6652672 + 196480 \beta ) q^{96} + ( 2304770 + 561696 \beta ) q^{97} + ( -6252670 - 638707 \beta ) q^{98} + ( 1445344 + 747280 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 20q^{2} + 20q^{3} + 96q^{4} - 250q^{5} - 1016q^{6} - 100q^{7} + 1440q^{8} + 5554q^{9} + O(q^{10})$$ $$2q + 20q^{2} + 20q^{3} + 96q^{4} - 250q^{5} - 1016q^{6} - 100q^{7} + 1440q^{8} + 5554q^{9} - 2500q^{10} + 4544q^{11} - 23360q^{12} + 3540q^{13} + 7512q^{14} - 2500q^{15} + 20352q^{16} - 27340q^{17} + 31220q^{18} + 38760q^{19} - 12000q^{20} - 69096q^{21} + 106240q^{22} - 124140q^{23} - 131520q^{24} + 31250q^{25} - 57016q^{26} + 206360q^{27} + 165440q^{28} - 72260q^{29} + 127000q^{30} + 306824q^{31} - 78080q^{32} - 440960q^{33} - 453368q^{34} + 12500q^{35} - 219808q^{36} - 123020q^{37} + 338960q^{38} + 774728q^{39} - 180000q^{40} + 264364q^{41} - 545040q^{42} + 423300q^{43} + 1434112q^{44} - 694250q^{45} - 1303416q^{46} - 105460q^{47} + 981760q^{48} - 1165414q^{49} + 312500q^{50} + 1166344q^{51} - 1678400q^{52} - 2391580q^{53} + 1102960q^{54} - 568000q^{55} + 949440q^{56} + 776720q^{57} + 2244440q^{58} - 1120120q^{59} + 2920000q^{60} + 2257044q^{61} + 2642640q^{62} - 1639620q^{63} - 5146624q^{64} - 442500q^{65} - 6564352q^{66} + 4516460q^{67} - 4911680q^{68} - 745272q^{69} - 939000q^{70} + 621784q^{71} + 1080480q^{72} + 4569060q^{73} + 2651272q^{74} + 312500q^{75} + 887680q^{76} + 3177600q^{77} + 4670800q^{78} + 4333040q^{79} - 2544000q^{80} - 2397878q^{81} - 5989960q^{82} - 9793020q^{83} - 398208q^{84} + 3417500q^{85} + 10798184q^{86} - 24458920q^{87} + 10567680q^{88} + 6025620q^{89} - 3902500q^{90} - 5352296q^{91} - 7199040q^{92} + 6473040q^{93} + 5860792q^{94} - 4845000q^{95} + 13305344q^{96} + 4609540q^{97} - 12505340q^{98} + 2890688q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.35890 4.35890
1.28220 79.7424 −126.356 −125.000 102.246 −538.197 −326.136 4171.85 −160.275
1.2 18.7178 −59.7424 222.356 −125.000 −1118.25 438.197 1766.14 1382.15 −2339.72
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.8.a.b 2
3.b odd 2 1 45.8.a.h 2
4.b odd 2 1 80.8.a.g 2
5.b even 2 1 25.8.a.b 2
5.c odd 4 2 25.8.b.c 4
7.b odd 2 1 245.8.a.c 2
8.b even 2 1 320.8.a.l 2
8.d odd 2 1 320.8.a.u 2
11.b odd 2 1 605.8.a.d 2
15.d odd 2 1 225.8.a.w 2
15.e even 4 2 225.8.b.m 4
20.d odd 2 1 400.8.a.bb 2
20.e even 4 2 400.8.c.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.b 2 1.a even 1 1 trivial
25.8.a.b 2 5.b even 2 1
25.8.b.c 4 5.c odd 4 2
45.8.a.h 2 3.b odd 2 1
80.8.a.g 2 4.b odd 2 1
225.8.a.w 2 15.d odd 2 1
225.8.b.m 4 15.e even 4 2
245.8.a.c 2 7.b odd 2 1
320.8.a.l 2 8.b even 2 1
320.8.a.u 2 8.d odd 2 1
400.8.a.bb 2 20.d odd 2 1
400.8.c.m 4 20.e even 4 2
605.8.a.d 2 11.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 20 T_{2} + 24$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 20 T + 280 T^{2} - 2560 T^{3} + 16384 T^{4}$$
$3$ $$1 - 20 T - 390 T^{2} - 43740 T^{3} + 4782969 T^{4}$$
$5$ $$( 1 + 125 T )^{2}$$
$7$ $$1 + 100 T + 1411250 T^{2} + 82354300 T^{3} + 678223072849 T^{4}$$
$11$ $$1 - 4544 T + 31976326 T^{2} - 88549705024 T^{3} + 379749833583241 T^{4}$$
$13$ $$1 - 3540 T + 100535470 T^{2} - 222129750180 T^{3} + 3937376385699289 T^{4}$$
$17$ $$1 + 27340 T + 901005190 T^{2} + 11218659319820 T^{3} + 168377826559400929 T^{4}$$
$19$ $$1 - 38760 T + 2155545478 T^{2} - 34646468603640 T^{3} + 799006685782884121 T^{4}$$
$23$ $$1 + 124140 T + 10649684530 T^{2} + 422675030990580 T^{3} + 11592836324538749809 T^{4}$$
$29$ $$1 + 72260 T + 6846819118 T^{2} + 1246476062088340 T^{3} +$$$$29\!\cdots\!81$$$$T^{4}$$
$31$ $$1 - 306824 T + 77964629966 T^{2} - 8441530311993464 T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$37$ $$1 + 123020 T + 144088599870 T^{2} + 11678519524901660 T^{3} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$1 - 264364 T + 161786388886 T^{2} - 51486018860276684 T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$
$43$ $$1 - 423300 T + 446651231050 T^{2} - 115060818081593100 T^{3} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 105460 T + 858715356610 T^{2} + 53428474284027980 T^{3} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 + 2391580 T + 3562552504510 T^{2} + 2809415667811372460 T^{3} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$1 + 1120120 T + 1362334883638 T^{2} + 2787588301175458280 T^{3} +$$$$61\!\cdots\!61$$$$T^{4}$$
$61$ $$1 - 2257044 T + 5613447576526 T^{2} - 7093308861584181924 T^{3} +$$$$98\!\cdots\!41$$$$T^{4}$$
$67$ $$1 - 4516460 T + 16742087664890 T^{2} - 27372961536977116580 T^{3} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$1 - 621784 T + 17914494152446 T^{2} - 5655200192564989544 T^{3} +$$$$82\!\cdots\!81$$$$T^{4}$$
$73$ $$1 - 4569060 T + 23424949855030 T^{2} - 50476226677665338820 T^{3} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$1 - 4333040 T + 26135588252318 T^{2} - 83211305793386393360 T^{3} +$$$$36\!\cdots\!81$$$$T^{4}$$
$83$ $$1 + 9793020 T + 59971104320890 T^{2} +$$$$26\!\cdots\!40$$$$T^{3} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$1 - 6025620 T + 89865866149558 T^{2} -$$$$26\!\cdots\!80$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4}$$
$97$ $$1 - 4609540 T + 142930351581510 T^{2} -$$$$37\!\cdots\!20$$$$T^{3} +$$$$65\!\cdots\!69$$$$T^{4}$$