# Properties

 Label 21.8.a.b Level $21$ Weight $8$ Character orbit 21.a Self dual yes Analytic conductor $6.560$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.56008553517$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1065})$$ Defining polynomial: $$x^{2} - x - 266$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1065})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{2} -27 q^{3} + ( 154 + 9 \beta ) q^{4} + ( -190 + 20 \beta ) q^{5} + ( 108 + 27 \beta ) q^{6} + 343 q^{7} + ( -2498 - 71 \beta ) q^{8} + 729 q^{9} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{2} -27 q^{3} + ( 154 + 9 \beta ) q^{4} + ( -190 + 20 \beta ) q^{5} + ( 108 + 27 \beta ) q^{6} + 343 q^{7} + ( -2498 - 71 \beta ) q^{8} + 729 q^{9} + ( -4560 + 90 \beta ) q^{10} + ( -2312 - 308 \beta ) q^{11} + ( -4158 - 243 \beta ) q^{12} + ( 3710 + 288 \beta ) q^{13} + ( -1372 - 343 \beta ) q^{14} + ( 5130 - 540 \beta ) q^{15} + ( 9166 + 1701 \beta ) q^{16} + ( -14570 + 556 \beta ) q^{17} + ( -2916 - 729 \beta ) q^{18} + ( -31396 - 936 \beta ) q^{19} + ( 18620 + 1550 \beta ) q^{20} -9261 q^{21} + ( 91176 + 3852 \beta ) q^{22} + ( 41660 - 1060 \beta ) q^{23} + ( 67446 + 1917 \beta ) q^{24} + ( 64375 - 7200 \beta ) q^{25} + ( -91448 - 5150 \beta ) q^{26} -19683 q^{27} + ( 52822 + 3087 \beta ) q^{28} + ( -219042 + 2088 \beta ) q^{29} + ( 123120 - 2430 \beta ) q^{30} + ( -9544 - 10152 \beta ) q^{31} + ( -169386 - 8583 \beta ) q^{32} + ( 62424 + 8316 \beta ) q^{33} + ( -89616 + 11790 \beta ) q^{34} + ( -65170 + 6860 \beta ) q^{35} + ( 112266 + 6561 \beta ) q^{36} + ( -353266 - 3024 \beta ) q^{37} + ( 374560 + 36076 \beta ) q^{38} + ( -100170 - 7776 \beta ) q^{39} + ( 96900 - 37890 \beta ) q^{40} + ( -32298 + 39540 \beta ) q^{41} + ( 37044 + 9261 \beta ) q^{42} + ( 242132 + 11952 \beta ) q^{43} + ( -1093400 - 71012 \beta ) q^{44} + ( -138510 + 14580 \beta ) q^{45} + ( 115320 - 36360 \beta ) q^{46} + ( -795544 + 16088 \beta ) q^{47} + ( -247482 - 45927 \beta ) q^{48} + 117649 q^{49} + ( 1657700 - 28375 \beta ) q^{50} + ( 393390 - 15012 \beta ) q^{51} + ( 1260812 + 80334 \beta ) q^{52} + ( 1044286 - 31136 \beta ) q^{53} + ( 78732 + 19683 \beta ) q^{54} + ( -1199280 + 6120 \beta ) q^{55} + ( -856814 - 24353 \beta ) q^{56} + ( 847692 + 25272 \beta ) q^{57} + ( 320760 + 208602 \beta ) q^{58} + ( -523820 - 53384 \beta ) q^{59} + ( -502740 - 41850 \beta ) q^{60} + ( 33830 - 38664 \beta ) q^{61} + ( 2738608 + 60304 \beta ) q^{62} + 250047 q^{63} + ( 1787374 - 5427 \beta ) q^{64} + ( 827260 + 25240 \beta ) q^{65} + ( -2461752 - 104004 \beta ) q^{66} + ( -2258860 + 36936 \beta ) q^{67} + ( -912716 - 40502 \beta ) q^{68} + ( -1124820 + 28620 \beta ) q^{69} + ( -1564080 + 30870 \beta ) q^{70} + ( 46356 - 38172 \beta ) q^{71} + ( -1821042 - 51759 \beta ) q^{72} + ( 478706 - 290808 \beta ) q^{73} + ( 2217448 + 368386 \beta ) q^{74} + ( -1738125 + 194400 \beta ) q^{75} + ( -7075768 - 435132 \beta ) q^{76} + ( -793016 - 105644 \beta ) q^{77} + ( 2469096 + 139050 \beta ) q^{78} + ( 1134584 + 53784 \beta ) q^{79} + ( 7307780 - 105850 \beta ) q^{80} + 531441 q^{81} + ( -10388448 - 165402 \beta ) q^{82} + ( -3772188 + 159984 \beta ) q^{83} + ( -1426194 - 83349 \beta ) q^{84} + ( 5726220 - 385920 \beta ) q^{85} + ( -4147760 - 301892 \beta ) q^{86} + ( 5914134 - 56376 \beta ) q^{87} + ( 11592264 + 955404 \beta ) q^{88} + ( 649718 + 485012 \beta ) q^{89} + ( -3324240 + 65610 \beta ) q^{90} + ( 1272530 + 98784 \beta ) q^{91} + ( 3878000 + 202160 \beta ) q^{92} + ( 257688 + 274104 \beta ) q^{93} + ( -1097232 + 715104 \beta ) q^{94} + ( 985720 - 468800 \beta ) q^{95} + ( 4573422 + 231741 \beta ) q^{96} + ( 8194298 - 122184 \beta ) q^{97} + ( -470596 - 117649 \beta ) q^{98} + ( -1685448 - 224532 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{2} - 54q^{3} + 317q^{4} - 360q^{5} + 243q^{6} + 686q^{7} - 5067q^{8} + 1458q^{9} + O(q^{10})$$ $$2q - 9q^{2} - 54q^{3} + 317q^{4} - 360q^{5} + 243q^{6} + 686q^{7} - 5067q^{8} + 1458q^{9} - 9030q^{10} - 4932q^{11} - 8559q^{12} + 7708q^{13} - 3087q^{14} + 9720q^{15} + 20033q^{16} - 28584q^{17} - 6561q^{18} - 63728q^{19} + 38790q^{20} - 18522q^{21} + 186204q^{22} + 82260q^{23} + 136809q^{24} + 121550q^{25} - 188046q^{26} - 39366q^{27} + 108731q^{28} - 435996q^{29} + 243810q^{30} - 29240q^{31} - 347355q^{32} + 133164q^{33} - 167442q^{34} - 123480q^{35} + 231093q^{36} - 709556q^{37} + 785196q^{38} - 208116q^{39} + 155910q^{40} - 25056q^{41} + 83349q^{42} + 496216q^{43} - 2257812q^{44} - 262440q^{45} + 194280q^{46} - 1575000q^{47} - 540891q^{48} + 235298q^{49} + 3287025q^{50} + 771768q^{51} + 2601958q^{52} + 2057436q^{53} + 177147q^{54} - 2392440q^{55} - 1737981q^{56} + 1720656q^{57} + 850122q^{58} - 1101024q^{59} - 1047330q^{60} + 28996q^{61} + 5537520q^{62} + 500094q^{63} + 3569321q^{64} + 1679760q^{65} - 5027508q^{66} - 4480784q^{67} - 1865934q^{68} - 2221020q^{69} - 3097290q^{70} + 54540q^{71} - 3693843q^{72} + 666604q^{73} + 4803282q^{74} - 3281850q^{75} - 14586668q^{76} - 1691676q^{77} + 5077242q^{78} + 2322952q^{79} + 14509710q^{80} + 1062882q^{81} - 20942298q^{82} - 7384392q^{83} - 2935737q^{84} + 11066520q^{85} - 8597412q^{86} + 11771892q^{87} + 24139932q^{88} + 1784448q^{89} - 6582870q^{90} + 2643844q^{91} + 7958160q^{92} + 789480q^{93} - 1479360q^{94} + 1502640q^{95} + 9378585q^{96} + 16266412q^{97} - 1058841q^{98} - 3595428q^{99} + O(q^{100})$$

## 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
 16.8172 −15.8172
−20.8172 −27.0000 305.355 146.343 562.064 343.000 −3692.02 729.000 −3046.45
1.2 11.8172 −27.0000 11.6455 −506.343 −319.064 343.000 −1374.98 729.000 −5983.55
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 21.8.a.b 2
3.b odd 2 1 63.8.a.f 2
4.b odd 2 1 336.8.a.n 2
5.b even 2 1 525.8.a.e 2
7.b odd 2 1 147.8.a.c 2
7.c even 3 2 147.8.e.h 4
7.d odd 6 2 147.8.e.g 4
21.c even 2 1 441.8.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.a.b 2 1.a even 1 1 trivial
63.8.a.f 2 3.b odd 2 1
147.8.a.c 2 7.b odd 2 1
147.8.e.g 4 7.d odd 6 2
147.8.e.h 4 7.c even 3 2
336.8.a.n 2 4.b odd 2 1
441.8.a.m 2 21.c even 2 1
525.8.a.e 2 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9 T_{2} - 246$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-246 + 9 T + T^{2}$$
$3$ $$( 27 + T )^{2}$$
$5$ $$-74100 + 360 T + T^{2}$$
$7$ $$( -343 + T )^{2}$$
$11$ $$-19176384 + 4932 T + T^{2}$$
$13$ $$-7230524 - 7708 T + T^{2}$$
$17$ $$121953804 + 28584 T + T^{2}$$
$19$ $$782053936 + 63728 T + T^{2}$$
$23$ $$1392518400 - 82260 T + T^{2}$$
$29$ $$46362346164 + 435996 T + T^{2}$$
$31$ $$-27226807040 + 29240 T + T^{2}$$
$37$ $$123432685924 + 709556 T + T^{2}$$
$41$ $$-416101387716 + 25056 T + T^{2}$$
$43$ $$23523686224 - 496216 T + T^{2}$$
$47$ $$551244428160 + 1575000 T + T^{2}$$
$53$ $$800144528964 - 2057436 T + T^{2}$$
$59$ $$-455709488016 + 1101024 T + T^{2}$$
$61$ $$-397808236556 - 28996 T + T^{2}$$
$67$ $$4656119933104 + 4480784 T + T^{2}$$
$71$ $$-387209643840 - 54540 T + T^{2}$$
$73$ $$-22405484001836 - 666604 T + T^{2}$$
$79$ $$578840156416 - 2322952 T + T^{2}$$
$83$ $$6817674434256 + 7384392 T + T^{2}$$
$89$ $$-61835691772164 - 1784448 T + T^{2}$$
$97$ $$62174212264276 - 16266412 T + T^{2}$$