# Properties

 Label 363.6.a.f Level $363$ Weight $6$ Character orbit 363.a Self dual yes Analytic conductor $58.219$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 363.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.2193265921$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 6) q^{2} + 9 q^{3} + (13 \beta + 12) q^{4} + ( - 10 \beta + 34) q^{5} + ( - 9 \beta - 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b - 6) * q^2 + 9 * q^3 + (13*b + 12) * q^4 + (-10*b + 34) * q^5 + (-9*b - 54) * q^6 + (62*b - 104) * q^7 + (-71*b + 16) * q^8 + 81 * q^9 $$q + ( - \beta - 6) q^{2} + 9 q^{3} + (13 \beta + 12) q^{4} + ( - 10 \beta + 34) q^{5} + ( - 9 \beta - 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9} + (36 \beta - 124) q^{10} + (117 \beta + 108) q^{12} + ( - 74 \beta + 102) q^{13} + ( - 330 \beta + 128) q^{14} + ( - 90 \beta + 306) q^{15} + (65 \beta + 88) q^{16} + (372 \beta + 178) q^{17} + ( - 81 \beta - 486) q^{18} + ( - 852 \beta + 840) q^{19} + (192 \beta - 632) q^{20} + (558 \beta - 936) q^{21} + (330 \beta - 284) q^{23} + ( - 639 \beta + 144) q^{24} + ( - 580 \beta - 1169) q^{25} + (416 \beta - 20) q^{26} + 729 q^{27} + (198 \beta + 5200) q^{28} + ( - 1492 \beta + 398) q^{29} + (324 \beta - 1116) q^{30} + ( - 1600 \beta - 4440) q^{31} + (1729 \beta - 1560) q^{32} + ( - 2782 \beta - 4044) q^{34} + (2528 \beta - 8496) q^{35} + (1053 \beta + 972) q^{36} + (2816 \beta - 2362) q^{37} + (5124 \beta + 1776) q^{38} + ( - 666 \beta + 918) q^{39} + ( - 1864 \beta + 6224) q^{40} + ( - 8 \beta - 18238) q^{41} + ( - 2970 \beta + 1152) q^{42} + ( - 3112 \beta - 3328) q^{43} + ( - 810 \beta + 2754) q^{45} + ( - 2026 \beta - 936) q^{46} + (390 \beta + 21676) q^{47} + (585 \beta + 792) q^{48} + ( - 9052 \beta + 24761) q^{49} + (5229 \beta + 11654) q^{50} + (3348 \beta + 1602) q^{51} + ( - 524 \beta - 6472) q^{52} + (7102 \beta - 9638) q^{53} + ( - 729 \beta - 4374) q^{54} + (3974 \beta - 36880) q^{56} + ( - 7668 \beta + 7560) q^{57} + (10046 \beta + 9548) q^{58} + ( - 1980 \beta - 404) q^{59} + (1728 \beta - 5688) q^{60} + (2026 \beta + 11638) q^{61} + (15640 \beta + 39440) q^{62} + (5022 \beta - 8424) q^{63} + ( - 12623 \beta - 7288) q^{64} + ( - 2796 \beta + 9388) q^{65} + (12704 \beta - 26612) q^{67} + (11614 \beta + 40824) q^{68} + (2970 \beta - 2556) q^{69} + ( - 9200 \beta + 30752) q^{70} + (4354 \beta + 13516) q^{71} + ( - 5751 \beta + 1296) q^{72} + (5568 \beta + 20606) q^{73} + ( - 17350 \beta - 8356) q^{74} + ( - 5220 \beta - 10521) q^{75} + ( - 10380 \beta - 78528) q^{76} + (3744 \beta - 180) q^{78} + (11426 \beta + 2712) q^{79} + (680 \beta - 2208) q^{80} + 6561 q^{81} + (18294 \beta + 109492) q^{82} + (21960 \beta - 50700) q^{83} + (1782 \beta + 46800) q^{84} + (7148 \beta - 23708) q^{85} + (25112 \beta + 44864) q^{86} + ( - 13428 \beta + 3582) q^{87} + ( - 26704 \beta - 13750) q^{89} + (2916 \beta - 10044) q^{90} + (9432 \beta - 47312) q^{91} + (4558 \beta + 30912) q^{92} + ( - 14400 \beta - 39960) q^{93} + ( - 24406 \beta - 133176) q^{94} + ( - 28848 \beta + 96720) q^{95} + (15561 \beta - 14040) q^{96} + ( - 9924 \beta - 115822) q^{97} + (38603 \beta - 76150) q^{98}+O(q^{100})$$ q + (-b - 6) * q^2 + 9 * q^3 + (13*b + 12) * q^4 + (-10*b + 34) * q^5 + (-9*b - 54) * q^6 + (62*b - 104) * q^7 + (-71*b + 16) * q^8 + 81 * q^9 + (36*b - 124) * q^10 + (117*b + 108) * q^12 + (-74*b + 102) * q^13 + (-330*b + 128) * q^14 + (-90*b + 306) * q^15 + (65*b + 88) * q^16 + (372*b + 178) * q^17 + (-81*b - 486) * q^18 + (-852*b + 840) * q^19 + (192*b - 632) * q^20 + (558*b - 936) * q^21 + (330*b - 284) * q^23 + (-639*b + 144) * q^24 + (-580*b - 1169) * q^25 + (416*b - 20) * q^26 + 729 * q^27 + (198*b + 5200) * q^28 + (-1492*b + 398) * q^29 + (324*b - 1116) * q^30 + (-1600*b - 4440) * q^31 + (1729*b - 1560) * q^32 + (-2782*b - 4044) * q^34 + (2528*b - 8496) * q^35 + (1053*b + 972) * q^36 + (2816*b - 2362) * q^37 + (5124*b + 1776) * q^38 + (-666*b + 918) * q^39 + (-1864*b + 6224) * q^40 + (-8*b - 18238) * q^41 + (-2970*b + 1152) * q^42 + (-3112*b - 3328) * q^43 + (-810*b + 2754) * q^45 + (-2026*b - 936) * q^46 + (390*b + 21676) * q^47 + (585*b + 792) * q^48 + (-9052*b + 24761) * q^49 + (5229*b + 11654) * q^50 + (3348*b + 1602) * q^51 + (-524*b - 6472) * q^52 + (7102*b - 9638) * q^53 + (-729*b - 4374) * q^54 + (3974*b - 36880) * q^56 + (-7668*b + 7560) * q^57 + (10046*b + 9548) * q^58 + (-1980*b - 404) * q^59 + (1728*b - 5688) * q^60 + (2026*b + 11638) * q^61 + (15640*b + 39440) * q^62 + (5022*b - 8424) * q^63 + (-12623*b - 7288) * q^64 + (-2796*b + 9388) * q^65 + (12704*b - 26612) * q^67 + (11614*b + 40824) * q^68 + (2970*b - 2556) * q^69 + (-9200*b + 30752) * q^70 + (4354*b + 13516) * q^71 + (-5751*b + 1296) * q^72 + (5568*b + 20606) * q^73 + (-17350*b - 8356) * q^74 + (-5220*b - 10521) * q^75 + (-10380*b - 78528) * q^76 + (3744*b - 180) * q^78 + (11426*b + 2712) * q^79 + (680*b - 2208) * q^80 + 6561 * q^81 + (18294*b + 109492) * q^82 + (21960*b - 50700) * q^83 + (1782*b + 46800) * q^84 + (7148*b - 23708) * q^85 + (25112*b + 44864) * q^86 + (-13428*b + 3582) * q^87 + (-26704*b - 13750) * q^89 + (2916*b - 10044) * q^90 + (9432*b - 47312) * q^91 + (4558*b + 30912) * q^92 + (-14400*b - 39960) * q^93 + (-24406*b - 133176) * q^94 + (-28848*b + 96720) * q^95 + (15561*b - 14040) * q^96 + (-9924*b - 115822) * q^97 + (38603*b - 76150) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} - 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 13 * q^2 + 18 * q^3 + 37 * q^4 + 58 * q^5 - 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 $$2 q - 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} - 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 212 q^{10} + 333 q^{12} + 130 q^{13} - 74 q^{14} + 522 q^{15} + 241 q^{16} + 728 q^{17} - 1053 q^{18} + 828 q^{19} - 1072 q^{20} - 1314 q^{21} - 238 q^{23} - 351 q^{24} - 2918 q^{25} + 376 q^{26} + 1458 q^{27} + 10598 q^{28} - 696 q^{29} - 1908 q^{30} - 10480 q^{31} - 1391 q^{32} - 10870 q^{34} - 14464 q^{35} + 2997 q^{36} - 1908 q^{37} + 8676 q^{38} + 1170 q^{39} + 10584 q^{40} - 36484 q^{41} - 666 q^{42} - 9768 q^{43} + 4698 q^{45} - 3898 q^{46} + 43742 q^{47} + 2169 q^{48} + 40470 q^{49} + 28537 q^{50} + 6552 q^{51} - 13468 q^{52} - 12174 q^{53} - 9477 q^{54} - 69786 q^{56} + 7452 q^{57} + 29142 q^{58} - 2788 q^{59} - 9648 q^{60} + 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} + 15980 q^{65} - 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 52304 q^{70} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} - 34062 q^{74} - 26262 q^{75} - 167436 q^{76} + 3384 q^{78} + 16850 q^{79} - 3736 q^{80} + 13122 q^{81} + 237278 q^{82} - 79440 q^{83} + 95382 q^{84} - 40268 q^{85} + 114840 q^{86} - 6264 q^{87} - 54204 q^{89} - 17172 q^{90} - 85192 q^{91} + 66382 q^{92} - 94320 q^{93} - 290758 q^{94} + 164592 q^{95} - 12519 q^{96} - 241568 q^{97} - 113697 q^{98}+O(q^{100})$$ 2 * q - 13 * q^2 + 18 * q^3 + 37 * q^4 + 58 * q^5 - 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 212 * q^10 + 333 * q^12 + 130 * q^13 - 74 * q^14 + 522 * q^15 + 241 * q^16 + 728 * q^17 - 1053 * q^18 + 828 * q^19 - 1072 * q^20 - 1314 * q^21 - 238 * q^23 - 351 * q^24 - 2918 * q^25 + 376 * q^26 + 1458 * q^27 + 10598 * q^28 - 696 * q^29 - 1908 * q^30 - 10480 * q^31 - 1391 * q^32 - 10870 * q^34 - 14464 * q^35 + 2997 * q^36 - 1908 * q^37 + 8676 * q^38 + 1170 * q^39 + 10584 * q^40 - 36484 * q^41 - 666 * q^42 - 9768 * q^43 + 4698 * q^45 - 3898 * q^46 + 43742 * q^47 + 2169 * q^48 + 40470 * q^49 + 28537 * q^50 + 6552 * q^51 - 13468 * q^52 - 12174 * q^53 - 9477 * q^54 - 69786 * q^56 + 7452 * q^57 + 29142 * q^58 - 2788 * q^59 - 9648 * q^60 + 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 + 15980 * q^65 - 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 52304 * q^70 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 - 34062 * q^74 - 26262 * q^75 - 167436 * q^76 + 3384 * q^78 + 16850 * q^79 - 3736 * q^80 + 13122 * q^81 + 237278 * q^82 - 79440 * q^83 + 95382 * q^84 - 40268 * q^85 + 114840 * q^86 - 6264 * q^87 - 54204 * q^89 - 17172 * q^90 - 85192 * q^91 + 66382 * q^92 - 94320 * q^93 - 290758 * q^94 + 164592 * q^95 - 12519 * q^96 - 241568 * q^97 - 113697 * q^98

## 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
 3.37228 −2.37228
−9.37228 9.00000 55.8397 0.277187 −84.3505 105.081 −223.432 81.0000 −2.59787
1.2 −3.62772 9.00000 −18.8397 57.7228 −32.6495 −251.081 184.432 81.0000 −209.402
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 363.6.a.f 2
3.b odd 2 1 1089.6.a.p 2
11.b odd 2 1 33.6.a.e 2
33.d even 2 1 99.6.a.d 2
44.c even 2 1 528.6.a.o 2
55.d odd 2 1 825.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 11.b odd 2 1
99.6.a.d 2 33.d even 2 1
363.6.a.f 2 1.a even 1 1 trivial
528.6.a.o 2 44.c even 2 1
825.6.a.c 2 55.d odd 2 1
1089.6.a.p 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 13T_{2} + 34$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(363))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 13T + 34$$
$3$ $$(T - 9)^{2}$$
$5$ $$T^{2} - 58T + 16$$
$7$ $$T^{2} + 146T - 26384$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 130T - 40952$$
$17$ $$T^{2} - 728 T - 1009172$$
$19$ $$T^{2} - 828 T - 5817312$$
$23$ $$T^{2} + 238T - 884264$$
$29$ $$T^{2} + 696 T - 18243924$$
$31$ $$T^{2} + 10480 T + 6337600$$
$37$ $$T^{2} + 1908 T - 64511196$$
$41$ $$T^{2} + 36484 T + 332770036$$
$43$ $$T^{2} + 9768 T - 56044032$$
$47$ $$T^{2} - 43742 T + 477085816$$
$53$ $$T^{2} + 12174 T - 379065264$$
$59$ $$T^{2} + 2788 T - 30400064$$
$61$ $$T^{2} - 25302 T + 126184224$$
$67$ $$T^{2} + 40520 T - 921013232$$
$71$ $$T^{2} - 31386 T + 89872392$$
$73$ $$T^{2} - 46780 T + 291320452$$
$79$ $$T^{2} - 16850 T - 1006085552$$
$83$ $$T^{2} + 79440 T - 2400814800$$
$89$ $$T^{2} + 54204 T - 5148586428$$
$97$ $$T^{2} + 241568 T + 13776267004$$