# Properties

 Label 363.2.e.j Level 363 Weight 2 Character orbit 363.e Analytic conductor 2.899 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 363.e (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{5} + ( 1 + \zeta_{10}^{2} ) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{5} + ( 1 + \zeta_{10}^{2} ) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{10} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{12} + ( 5 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{13} + ( 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{14} + ( 1 - \zeta_{10} + \zeta_{10}^{3} ) q^{15} + ( 3 + 3 \zeta_{10}^{2} ) q^{16} + ( -1 + \zeta_{10} - \zeta_{10}^{2} ) q^{17} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{18} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{19} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{20} -3 q^{21} + ( -3 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{23} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{25} + ( 5 - 5 \zeta_{10} - 7 \zeta_{10}^{3} ) q^{26} -\zeta_{10} q^{27} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{28} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{29} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{30} + ( 2 + \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{32} - q^{34} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{35} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{36} + ( -2 + 2 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{37} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{38} + ( 2 + 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{39} + ( -3 + 3 \zeta_{10} - \zeta_{10}^{3} ) q^{40} + ( -8 \zeta_{10} + 7 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{41} + ( -3 + 3 \zeta_{10}^{3} ) q^{42} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{45} + ( -7 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{46} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{47} + ( -3 + 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{48} -2 \zeta_{10} q^{49} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{50} + ( 1 - \zeta_{10} ) q^{51} + ( -3 \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{52} + ( 8 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{53} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{54} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{56} + ( 1 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{57} + ( -2 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{58} + ( -7 + 7 \zeta_{10} + \zeta_{10}^{3} ) q^{59} + ( 2 - 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{60} + ( -3 + 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{61} + ( 2 - 2 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{62} -3 \zeta_{10}^{2} q^{63} + ( -3 + \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{64} + ( 3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{65} + ( 5 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{67} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} + ( -4 \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( -3 + 3 \zeta_{10} ) q^{70} + ( -9 - 9 \zeta_{10}^{2} ) q^{71} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{72} + ( -2 + 2 \zeta_{10} ) q^{73} + ( 3 \zeta_{10} + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{74} + ( 3 - 3 \zeta_{10}^{3} ) q^{75} + ( 3 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{76} + ( 7 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{78} + ( 3 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{79} + ( 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{80} -\zeta_{10}^{3} q^{81} + ( -1 - 8 \zeta_{10} - \zeta_{10}^{2} ) q^{82} + ( -6 - 3 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{83} + ( -3 + 3 \zeta_{10} ) q^{84} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{85} + ( -1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{86} + ( 2 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{87} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{89} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{90} + ( 6 \zeta_{10} + 9 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{91} + ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{92} + ( 3 - \zeta_{10} + 3 \zeta_{10}^{2} ) q^{93} -\zeta_{10} q^{94} + ( -7 + 7 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{95} + ( -\zeta_{10} + 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{96} + ( -7 - 6 \zeta_{10} + 6 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{97} + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} - q^{3} + 3q^{4} - q^{5} + 3q^{6} + 3q^{7} + 5q^{8} - q^{9} + O(q^{10})$$ $$4q + 3q^{2} - q^{3} + 3q^{4} - q^{5} + 3q^{6} + 3q^{7} + 5q^{8} - q^{9} - 2q^{10} - 2q^{12} + 9q^{13} + 6q^{14} + 4q^{15} + 9q^{16} - 2q^{17} - 2q^{18} + 10q^{19} + 3q^{20} - 12q^{21} - 4q^{23} - 5q^{24} - 6q^{25} + 8q^{26} - q^{27} + 6q^{28} + 10q^{29} + 3q^{30} + 8q^{31} + 18q^{32} - 4q^{34} + 3q^{35} + 3q^{36} - 3q^{37} + 9q^{39} - 10q^{40} - 23q^{41} - 9q^{42} - 16q^{43} - 6q^{45} - 13q^{46} - 3q^{47} - 6q^{48} - 2q^{49} - 12q^{50} + 3q^{51} - 7q^{52} + 6q^{53} - 2q^{54} - 5q^{57} - 20q^{59} + 3q^{60} - 3q^{61} + q^{62} + 3q^{63} - 7q^{64} + 14q^{65} + 2q^{67} + q^{68} - 9q^{69} - 9q^{70} - 27q^{71} - 5q^{72} - 6q^{73} + 4q^{74} + 9q^{75} + 20q^{76} + 18q^{78} - 5q^{79} + 9q^{80} - q^{81} - 11q^{82} - 21q^{83} - 9q^{84} - 7q^{85} - 7q^{86} + 20q^{89} - 2q^{90} + 3q^{91} + 7q^{92} + 8q^{93} - q^{94} - 25q^{95} - 7q^{96} - 33q^{97} - 4q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
1.30902 0.951057i 0.309017 + 0.951057i 0.190983 0.587785i 0.309017 + 0.224514i 1.30902 + 0.951057i −0.927051 + 2.85317i 0.690983 + 2.12663i −0.809017 + 0.587785i 0.618034
130.1 0.190983 0.587785i −0.809017 + 0.587785i 1.30902 + 0.951057i −0.809017 2.48990i 0.190983 + 0.587785i 2.42705 + 1.76336i 1.80902 1.31433i 0.309017 0.951057i −1.61803
148.1 0.190983 + 0.587785i −0.809017 0.587785i 1.30902 0.951057i −0.809017 + 2.48990i 0.190983 0.587785i 2.42705 1.76336i 1.80902 + 1.31433i 0.309017 + 0.951057i −1.61803
202.1 1.30902 + 0.951057i 0.309017 0.951057i 0.190983 + 0.587785i 0.309017 0.224514i 1.30902 0.951057i −0.927051 2.85317i 0.690983 2.12663i −0.809017 0.587785i 0.618034
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.e.j 4
11.b odd 2 1 33.2.e.a 4
11.c even 5 1 363.2.a.e 2
11.c even 5 2 363.2.e.c 4
11.c even 5 1 inner 363.2.e.j 4
11.d odd 10 1 33.2.e.a 4
11.d odd 10 1 363.2.a.h 2
11.d odd 10 2 363.2.e.h 4
33.d even 2 1 99.2.f.b 4
33.f even 10 1 99.2.f.b 4
33.f even 10 1 1089.2.a.m 2
33.h odd 10 1 1089.2.a.s 2
44.c even 2 1 528.2.y.f 4
44.g even 10 1 528.2.y.f 4
44.g even 10 1 5808.2.a.bl 2
44.h odd 10 1 5808.2.a.bm 2
55.d odd 2 1 825.2.n.f 4
55.e even 4 2 825.2.bx.b 8
55.h odd 10 1 825.2.n.f 4
55.h odd 10 1 9075.2.a.x 2
55.j even 10 1 9075.2.a.bv 2
55.l even 20 2 825.2.bx.b 8
99.g even 6 2 891.2.n.a 8
99.h odd 6 2 891.2.n.d 8
99.o odd 30 2 891.2.n.d 8
99.p even 30 2 891.2.n.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 11.b odd 2 1
33.2.e.a 4 11.d odd 10 1
99.2.f.b 4 33.d even 2 1
99.2.f.b 4 33.f even 10 1
363.2.a.e 2 11.c even 5 1
363.2.a.h 2 11.d odd 10 1
363.2.e.c 4 11.c even 5 2
363.2.e.h 4 11.d odd 10 2
363.2.e.j 4 1.a even 1 1 trivial
363.2.e.j 4 11.c even 5 1 inner
528.2.y.f 4 44.c even 2 1
528.2.y.f 4 44.g even 10 1
825.2.n.f 4 55.d odd 2 1
825.2.n.f 4 55.h odd 10 1
825.2.bx.b 8 55.e even 4 2
825.2.bx.b 8 55.l even 20 2
891.2.n.a 8 99.g even 6 2
891.2.n.a 8 99.p even 30 2
891.2.n.d 8 99.h odd 6 2
891.2.n.d 8 99.o odd 30 2
1089.2.a.m 2 33.f even 10 1
1089.2.a.s 2 33.h odd 10 1
5808.2.a.bl 2 44.g even 10 1
5808.2.a.bm 2 44.h odd 10 1
9075.2.a.x 2 55.h odd 10 1
9075.2.a.bv 2 55.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 2 T^{2} + T^{4} + 8 T^{6} - 24 T^{7} + 16 T^{8}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8}$$
$7$ $$1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 105 T^{5} + 98 T^{6} - 1029 T^{7} + 2401 T^{8}$$
$11$ 1
$13$ $$1 - 9 T + 18 T^{2} + 115 T^{3} - 789 T^{4} + 1495 T^{5} + 3042 T^{6} - 19773 T^{7} + 28561 T^{8}$$
$17$ $$1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 340 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 1330 T^{5} + 7581 T^{6} - 68590 T^{7} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8}$$
$31$ $$1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 1426 T^{5} + 2883 T^{6} - 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 23 T + 208 T^{2} + 961 T^{3} + 3975 T^{4} + 39401 T^{5} + 349648 T^{6} + 1585183 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 2115 T^{5} - 94987 T^{6} + 311469 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8}$$
$61$ $$1 + 3 T - 7 T^{2} + 441 T^{3} + 4900 T^{4} + 26901 T^{5} - 26047 T^{6} + 680943 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 58149 T^{5} + 1275373 T^{6} + 9663597 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 6 T - 57 T^{2} - 130 T^{3} + 4761 T^{4} - 9490 T^{5} - 303753 T^{6} + 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 5 T + 6 T^{2} + 715 T^{3} + 9821 T^{4} + 56485 T^{5} + 37446 T^{6} + 2465195 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 75945 T^{5} + 606232 T^{6} + 12007527 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8}$$