# Properties

 Label 363.6.a.j Level $363$ Weight $6$ Character orbit 363.a Self dual yes Analytic conductor $58.219$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ Defining polynomial: $$x^{2} - x - 44$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 - \beta ) q^{2} -9 q^{3} + ( 21 - 5 \beta ) q^{4} + ( 34 - 10 \beta ) q^{5} + ( -27 + 9 \beta ) q^{6} + ( 148 - 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} + 81 q^{9} +O(q^{10})$$ $$q + ( 3 - \beta ) q^{2} -9 q^{3} + ( 21 - 5 \beta ) q^{4} + ( 34 - 10 \beta ) q^{5} + ( -27 + 9 \beta ) q^{6} + ( 148 - 10 \beta ) q^{7} + ( 187 + \beta ) q^{8} + 81 q^{9} + ( 542 - 54 \beta ) q^{10} + ( -189 + 45 \beta ) q^{12} + ( 102 - 38 \beta ) q^{13} + ( 884 - 168 \beta ) q^{14} + ( -306 + 90 \beta ) q^{15} + ( -155 - 25 \beta ) q^{16} + ( 430 - 60 \beta ) q^{17} + ( 243 - 81 \beta ) q^{18} + ( 732 + 12 \beta ) q^{19} + ( 2914 - 330 \beta ) q^{20} + ( -1332 + 90 \beta ) q^{21} + ( -1868 + 366 \beta ) q^{23} + ( -1683 - 9 \beta ) q^{24} + ( 2431 - 580 \beta ) q^{25} + ( 1978 - 178 \beta ) q^{26} -729 q^{27} + ( 5308 - 900 \beta ) q^{28} + ( -3094 - 412 \beta ) q^{29} + ( -4878 + 486 \beta ) q^{30} + ( -3936 + 344 \beta ) q^{31} + ( -5349 + 73 \beta ) q^{32} + ( 3930 - 550 \beta ) q^{34} + ( 9432 - 1720 \beta ) q^{35} + ( 1701 - 405 \beta ) q^{36} + ( -14890 - 136 \beta ) q^{37} + ( 1668 - 708 \beta ) q^{38} + ( -918 + 342 \beta ) q^{39} + ( 5918 - 1846 \beta ) q^{40} + ( 2534 + 712 \beta ) q^{41} + ( -7956 + 1512 \beta ) q^{42} + ( 7580 + 1496 \beta ) q^{43} + ( 2754 - 810 \beta ) q^{45} + ( -21708 + 2600 \beta ) q^{46} + ( 5188 - 2526 \beta ) q^{47} + ( 1395 + 225 \beta ) q^{48} + ( 9497 - 2860 \beta ) q^{49} + ( 32813 - 3591 \beta ) q^{50} + ( -3870 + 540 \beta ) q^{51} + ( 10502 - 1118 \beta ) q^{52} + ( 5986 + 2206 \beta ) q^{53} + ( -2187 + 729 \beta ) q^{54} + ( 27236 - 1732 \beta ) q^{56} + ( -6588 - 108 \beta ) q^{57} + ( 8846 + 2270 \beta ) q^{58} + ( 9388 - 1476 \beta ) q^{59} + ( -26226 + 2970 \beta ) q^{60} + ( 2638 - 2330 \beta ) q^{61} + ( -26944 + 4624 \beta ) q^{62} + ( 11988 - 810 \beta ) q^{63} + ( -14299 + 6295 \beta ) q^{64} + ( 20188 - 1932 \beta ) q^{65} + ( 14068 + 3200 \beta ) q^{67} + ( 22230 - 3110 \beta ) q^{68} + ( 16812 - 3294 \beta ) q^{69} + ( 103976 - 12872 \beta ) q^{70} + ( -15356 - 3098 \beta ) q^{71} + ( 15147 + 81 \beta ) q^{72} + ( -26554 - 7536 \beta ) q^{73} + ( -38686 + 14618 \beta ) q^{74} + ( -21879 + 5220 \beta ) q^{75} + ( 12732 - 3468 \beta ) q^{76} + ( -17802 + 1602 \beta ) q^{78} + ( -5676 + 9482 \beta ) q^{79} + ( 5730 + 950 \beta ) q^{80} + 6561 q^{81} + ( -23726 - 1110 \beta ) q^{82} + ( 30444 - 2592 \beta ) q^{83} + ( -47772 + 8100 \beta ) q^{84} + ( 41020 - 5740 \beta ) q^{85} + ( -43084 - 4588 \beta ) q^{86} + ( 27846 + 3708 \beta ) q^{87} + ( 38666 + 15056 \beta ) q^{89} + ( 43902 - 4374 \beta ) q^{90} + ( 31816 - 6264 \beta ) q^{91} + ( -119748 + 15196 \beta ) q^{92} + ( 35424 - 3096 \beta ) q^{93} + ( 126708 - 10240 \beta ) q^{94} + ( 19608 - 7032 \beta ) q^{95} + ( 48141 - 657 \beta ) q^{96} + ( 14210 - 21300 \beta ) q^{97} + ( 154331 - 15217 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} - 45q^{6} + 286q^{7} + 375q^{8} + 162q^{9} + O(q^{10})$$ $$2q + 5q^{2} - 18q^{3} + 37q^{4} + 58q^{5} - 45q^{6} + 286q^{7} + 375q^{8} + 162q^{9} + 1030q^{10} - 333q^{12} + 166q^{13} + 1600q^{14} - 522q^{15} - 335q^{16} + 800q^{17} + 405q^{18} + 1476q^{19} + 5498q^{20} - 2574q^{21} - 3370q^{23} - 3375q^{24} + 4282q^{25} + 3778q^{26} - 1458q^{27} + 9716q^{28} - 6600q^{29} - 9270q^{30} - 7528q^{31} - 10625q^{32} + 7310q^{34} + 17144q^{35} + 2997q^{36} - 29916q^{37} + 2628q^{38} - 1494q^{39} + 9990q^{40} + 5780q^{41} - 14400q^{42} + 16656q^{43} + 4698q^{45} - 40816q^{46} + 7850q^{47} + 3015q^{48} + 16134q^{49} + 62035q^{50} - 7200q^{51} + 19886q^{52} + 14178q^{53} - 3645q^{54} + 52740q^{56} - 13284q^{57} + 19962q^{58} + 17300q^{59} - 49482q^{60} + 2946q^{61} - 49264q^{62} + 23166q^{63} - 22303q^{64} + 38444q^{65} + 31336q^{67} + 41350q^{68} + 30330q^{69} + 195080q^{70} - 33810q^{71} + 30375q^{72} - 60644q^{73} - 62754q^{74} - 38538q^{75} + 21996q^{76} - 34002q^{78} - 1870q^{79} + 12410q^{80} + 13122q^{81} - 48562q^{82} + 58296q^{83} - 87444q^{84} + 76300q^{85} - 90756q^{86} + 59400q^{87} + 92388q^{89} + 83430q^{90} + 57368q^{91} - 224300q^{92} + 67752q^{93} + 243176q^{94} + 32184q^{95} + 95625q^{96} + 7120q^{97} + 293445q^{98} + O(q^{100})$$

## 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
 7.15207 −6.15207
−4.15207 −9.00000 −14.7603 −37.5207 37.3686 76.4793 194.152 81.0000 155.788
1.2 9.15207 −9.00000 51.7603 95.5207 −82.3686 209.521 180.848 81.0000 874.212
 $$n$$: 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.j 2
3.b odd 2 1 1089.6.a.j 2
11.b odd 2 1 33.6.a.c 2
33.d even 2 1 99.6.a.f 2
44.c even 2 1 528.6.a.s 2
55.d odd 2 1 825.6.a.e 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 5 T_{2} - 38$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(363))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-38 - 5 T + T^{2}$$
$3$ $$( 9 + T )^{2}$$
$5$ $$-3584 - 58 T + T^{2}$$
$7$ $$16024 - 286 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-57008 - 166 T + T^{2}$$
$17$ $$700 - 800 T + T^{2}$$
$19$ $$538272 - 1476 T + T^{2}$$
$23$ $$-3088328 + 3370 T + T^{2}$$
$29$ $$3378828 + 6600 T + T^{2}$$
$31$ $$8931328 + 7528 T + T^{2}$$
$37$ $$222923316 + 29916 T + T^{2}$$
$41$ $$-14080172 - 5780 T + T^{2}$$
$43$ $$-29676624 - 16656 T + T^{2}$$
$47$ $$-266939288 - 7850 T + T^{2}$$
$53$ $$-165085872 - 14178 T + T^{2}$$
$59$ $$-21579488 - 17300 T + T^{2}$$
$61$ $$-238059096 - 2946 T + T^{2}$$
$67$ $$-207633776 - 31336 T + T^{2}$$
$71$ $$-138914952 + 33810 T + T^{2}$$
$73$ $$-1593591164 + 60644 T + T^{2}$$
$79$ $$-3977569112 + 1870 T + T^{2}$$
$83$ $$552313872 - 58296 T + T^{2}$$
$89$ $$-7896843132 - 92388 T + T^{2}$$
$97$ $$-20063108900 - 7120 T + T^{2}$$
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