Properties

 Label 363.4.a.j Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{3} + \beta q^{4} + ( - 4 \beta + 10) q^{5} - 3 \beta q^{6} + (2 \beta - 2) q^{7} + (7 \beta - 8) q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 + 3 * q^3 + b * q^4 + (-4*b + 10) * q^5 - 3*b * q^6 + (2*b - 2) * q^7 + (7*b - 8) * q^8 + 9 * q^9 $$q - \beta q^{2} + 3 q^{3} + \beta q^{4} + ( - 4 \beta + 10) q^{5} - 3 \beta q^{6} + (2 \beta - 2) q^{7} + (7 \beta - 8) q^{8} + 9 q^{9} + ( - 6 \beta + 32) q^{10} + 3 \beta q^{12} + ( - 8 \beta + 42) q^{13} - 16 q^{14} + ( - 12 \beta + 30) q^{15} + ( - 7 \beta - 56) q^{16} + ( - 30 \beta + 28) q^{17} - 9 \beta q^{18} + (18 \beta + 18) q^{19} + (6 \beta - 32) q^{20} + (6 \beta - 6) q^{21} + 112 q^{23} + (21 \beta - 24) q^{24} + ( - 64 \beta + 103) q^{25} + ( - 34 \beta + 64) q^{26} + 27 q^{27} + 16 q^{28} + ( - 46 \beta - 88) q^{29} + ( - 18 \beta + 96) q^{30} + (104 \beta - 72) q^{31} + (7 \beta + 120) q^{32} + (2 \beta + 240) q^{34} + (20 \beta - 84) q^{35} + 9 \beta q^{36} + (44 \beta - 46) q^{37} + ( - 36 \beta - 144) q^{38} + ( - 24 \beta + 126) q^{39} + (74 \beta - 304) q^{40} + ( - 2 \beta + 248) q^{41} - 48 q^{42} + (86 \beta - 10) q^{43} + ( - 36 \beta + 90) q^{45} - 112 \beta q^{46} + ( - 48 \beta - 8) q^{47} + ( - 21 \beta - 168) q^{48} + ( - 4 \beta - 307) q^{49} + ( - 39 \beta + 512) q^{50} + ( - 90 \beta + 84) q^{51} + (34 \beta - 64) q^{52} + ( - 128 \beta + 22) q^{53} - 27 \beta q^{54} + ( - 16 \beta + 128) q^{56} + (54 \beta + 54) q^{57} + (134 \beta + 368) q^{58} + 196 q^{59} + (18 \beta - 96) q^{60} + (52 \beta + 526) q^{61} + ( - 32 \beta - 832) q^{62} + (18 \beta - 18) q^{63} + ( - 71 \beta + 392) q^{64} + ( - 216 \beta + 676) q^{65} + (152 \beta + 388) q^{67} + ( - 2 \beta - 240) q^{68} + 336 q^{69} + (64 \beta - 160) q^{70} + (184 \beta + 136) q^{71} + (63 \beta - 72) q^{72} + (252 \beta + 170) q^{73} + (2 \beta - 352) q^{74} + ( - 192 \beta + 309) q^{75} + (36 \beta + 144) q^{76} + ( - 102 \beta + 192) q^{78} + (74 \beta + 78) q^{79} + (182 \beta - 336) q^{80} + 81 q^{81} + ( - 246 \beta + 16) q^{82} + (324 \beta - 336) q^{83} + 48 q^{84} + ( - 292 \beta + 1240) q^{85} + ( - 76 \beta - 688) q^{86} + ( - 138 \beta - 264) q^{87} + (8 \beta + 482) q^{89} + ( - 54 \beta + 288) q^{90} + (84 \beta - 212) q^{91} + 112 \beta q^{92} + (312 \beta - 216) q^{93} + (56 \beta + 384) q^{94} + (36 \beta - 396) q^{95} + (21 \beta + 360) q^{96} + (420 \beta - 802) q^{97} + (311 \beta + 32) q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^3 + b * q^4 + (-4*b + 10) * q^5 - 3*b * q^6 + (2*b - 2) * q^7 + (7*b - 8) * q^8 + 9 * q^9 + (-6*b + 32) * q^10 + 3*b * q^12 + (-8*b + 42) * q^13 - 16 * q^14 + (-12*b + 30) * q^15 + (-7*b - 56) * q^16 + (-30*b + 28) * q^17 - 9*b * q^18 + (18*b + 18) * q^19 + (6*b - 32) * q^20 + (6*b - 6) * q^21 + 112 * q^23 + (21*b - 24) * q^24 + (-64*b + 103) * q^25 + (-34*b + 64) * q^26 + 27 * q^27 + 16 * q^28 + (-46*b - 88) * q^29 + (-18*b + 96) * q^30 + (104*b - 72) * q^31 + (7*b + 120) * q^32 + (2*b + 240) * q^34 + (20*b - 84) * q^35 + 9*b * q^36 + (44*b - 46) * q^37 + (-36*b - 144) * q^38 + (-24*b + 126) * q^39 + (74*b - 304) * q^40 + (-2*b + 248) * q^41 - 48 * q^42 + (86*b - 10) * q^43 + (-36*b + 90) * q^45 - 112*b * q^46 + (-48*b - 8) * q^47 + (-21*b - 168) * q^48 + (-4*b - 307) * q^49 + (-39*b + 512) * q^50 + (-90*b + 84) * q^51 + (34*b - 64) * q^52 + (-128*b + 22) * q^53 - 27*b * q^54 + (-16*b + 128) * q^56 + (54*b + 54) * q^57 + (134*b + 368) * q^58 + 196 * q^59 + (18*b - 96) * q^60 + (52*b + 526) * q^61 + (-32*b - 832) * q^62 + (18*b - 18) * q^63 + (-71*b + 392) * q^64 + (-216*b + 676) * q^65 + (152*b + 388) * q^67 + (-2*b - 240) * q^68 + 336 * q^69 + (64*b - 160) * q^70 + (184*b + 136) * q^71 + (63*b - 72) * q^72 + (252*b + 170) * q^73 + (2*b - 352) * q^74 + (-192*b + 309) * q^75 + (36*b + 144) * q^76 + (-102*b + 192) * q^78 + (74*b + 78) * q^79 + (182*b - 336) * q^80 + 81 * q^81 + (-246*b + 16) * q^82 + (324*b - 336) * q^83 + 48 * q^84 + (-292*b + 1240) * q^85 + (-76*b - 688) * q^86 + (-138*b - 264) * q^87 + (8*b + 482) * q^89 + (-54*b + 288) * q^90 + (84*b - 212) * q^91 + 112*b * q^92 + (312*b - 216) * q^93 + (56*b + 384) * q^94 + (36*b - 396) * q^95 + (21*b + 360) * q^96 + (420*b - 802) * q^97 + (311*b + 32) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 $$2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20} - 6 q^{21} + 224 q^{23} - 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} + 32 q^{28} - 222 q^{29} + 174 q^{30} - 40 q^{31} + 247 q^{32} + 482 q^{34} - 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} + 228 q^{39} - 534 q^{40} + 494 q^{41} - 96 q^{42} + 66 q^{43} + 144 q^{45} - 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} + 985 q^{50} + 78 q^{51} - 94 q^{52} - 84 q^{53} - 27 q^{54} + 240 q^{56} + 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} + 1104 q^{61} - 1696 q^{62} - 18 q^{63} + 713 q^{64} + 1136 q^{65} + 928 q^{67} - 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} - 81 q^{72} + 592 q^{73} - 702 q^{74} + 426 q^{75} + 324 q^{76} + 282 q^{78} + 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} - 348 q^{83} + 96 q^{84} + 2188 q^{85} - 1452 q^{86} - 666 q^{87} + 972 q^{89} + 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} + 824 q^{94} - 756 q^{95} + 741 q^{96} - 1184 q^{97} + 375 q^{98}+O(q^{100})$$ 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 + 58 * q^10 + 3 * q^12 + 76 * q^13 - 32 * q^14 + 48 * q^15 - 119 * q^16 + 26 * q^17 - 9 * q^18 + 54 * q^19 - 58 * q^20 - 6 * q^21 + 224 * q^23 - 27 * q^24 + 142 * q^25 + 94 * q^26 + 54 * q^27 + 32 * q^28 - 222 * q^29 + 174 * q^30 - 40 * q^31 + 247 * q^32 + 482 * q^34 - 148 * q^35 + 9 * q^36 - 48 * q^37 - 324 * q^38 + 228 * q^39 - 534 * q^40 + 494 * q^41 - 96 * q^42 + 66 * q^43 + 144 * q^45 - 112 * q^46 - 64 * q^47 - 357 * q^48 - 618 * q^49 + 985 * q^50 + 78 * q^51 - 94 * q^52 - 84 * q^53 - 27 * q^54 + 240 * q^56 + 162 * q^57 + 870 * q^58 + 392 * q^59 - 174 * q^60 + 1104 * q^61 - 1696 * q^62 - 18 * q^63 + 713 * q^64 + 1136 * q^65 + 928 * q^67 - 482 * q^68 + 672 * q^69 - 256 * q^70 + 456 * q^71 - 81 * q^72 + 592 * q^73 - 702 * q^74 + 426 * q^75 + 324 * q^76 + 282 * q^78 + 230 * q^79 - 490 * q^80 + 162 * q^81 - 214 * q^82 - 348 * q^83 + 96 * q^84 + 2188 * q^85 - 1452 * q^86 - 666 * q^87 + 972 * q^89 + 522 * q^90 - 340 * q^91 + 112 * q^92 - 120 * q^93 + 824 * q^94 - 756 * q^95 + 741 * q^96 - 1184 * q^97 + 375 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
−3.37228 3.00000 3.37228 −3.48913 −10.1168 4.74456 15.6060 9.00000 11.7663
1.2 2.37228 3.00000 −2.37228 19.4891 7.11684 −6.74456 −24.6060 9.00000 46.2337
 $$n$$: 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 363.4.a.j 2
3.b odd 2 1 1089.4.a.t 2
11.b odd 2 1 33.4.a.d 2
33.d even 2 1 99.4.a.e 2
44.c even 2 1 528.4.a.o 2
55.d odd 2 1 825.4.a.k 2
55.e even 4 2 825.4.c.i 4
77.b even 2 1 1617.4.a.j 2
88.b odd 2 1 2112.4.a.ba 2
88.g even 2 1 2112.4.a.bh 2
132.d odd 2 1 1584.4.a.x 2
165.d even 2 1 2475.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 11.b odd 2 1
99.4.a.e 2 33.d even 2 1
363.4.a.j 2 1.a even 1 1 trivial
528.4.a.o 2 44.c even 2 1
825.4.a.k 2 55.d odd 2 1
825.4.c.i 4 55.e even 4 2
1089.4.a.t 2 3.b odd 2 1
1584.4.a.x 2 132.d odd 2 1
1617.4.a.j 2 77.b even 2 1
2112.4.a.ba 2 88.b odd 2 1
2112.4.a.bh 2 88.g even 2 1
2475.4.a.o 2 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(363))$$:

 $$T_{2}^{2} + T_{2} - 8$$ T2^2 + T2 - 8 $$T_{5}^{2} - 16T_{5} - 68$$ T5^2 - 16*T5 - 68 $$T_{7}^{2} + 2T_{7} - 32$$ T7^2 + 2*T7 - 32

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 8$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 16T - 68$$
$7$ $$T^{2} + 2T - 32$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 76T + 916$$
$17$ $$T^{2} - 26T - 7256$$
$19$ $$T^{2} - 54T - 1944$$
$23$ $$(T - 112)^{2}$$
$29$ $$T^{2} + 222T - 5136$$
$31$ $$T^{2} + 40T - 88832$$
$37$ $$T^{2} + 48T - 15396$$
$41$ $$T^{2} - 494T + 60976$$
$43$ $$T^{2} - 66T - 59928$$
$47$ $$T^{2} + 64T - 17984$$
$53$ $$T^{2} + 84T - 133404$$
$59$ $$(T - 196)^{2}$$
$61$ $$T^{2} - 1104 T + 282396$$
$67$ $$T^{2} - 928T + 24688$$
$71$ $$T^{2} - 456T - 227328$$
$73$ $$T^{2} - 592T - 436292$$
$79$ $$T^{2} - 230T - 31952$$
$83$ $$T^{2} + 348T - 835776$$
$89$ $$T^{2} - 972T + 235668$$
$97$ $$T^{2} + 1184 T - 1104836$$