Properties

 Label 363.4.a.i Level $363$ Weight $4$ Character orbit 363.a Self dual yes Analytic conductor $21.418$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 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 $$\beta = \frac{1}{2}(1 + \sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 3 q^{3} + (\beta + 16) q^{4} + ( - 2 \beta - 6) q^{5} + 3 \beta q^{6} + (4 \beta - 14) q^{7} + ( - 9 \beta - 24) q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 - 3 * q^3 + (b + 16) * q^4 + (-2*b - 6) * q^5 + 3*b * q^6 + (4*b - 14) * q^7 + (-9*b - 24) * q^8 + 9 * q^9 $$q - \beta q^{2} - 3 q^{3} + (\beta + 16) q^{4} + ( - 2 \beta - 6) q^{5} + 3 \beta q^{6} + (4 \beta - 14) q^{7} + ( - 9 \beta - 24) q^{8} + 9 q^{9} + (8 \beta + 48) q^{10} + ( - 3 \beta - 48) q^{12} + ( - 2 \beta - 14) q^{13} + (10 \beta - 96) q^{14} + (6 \beta + 18) q^{15} + (25 \beta + 88) q^{16} + (14 \beta - 60) q^{17} - 9 \beta q^{18} + (2 \beta - 26) q^{19} + ( - 40 \beta - 144) q^{20} + ( - 12 \beta + 42) q^{21} + ( - 10 \beta + 72) q^{23} + (27 \beta + 72) q^{24} + (28 \beta + 7) q^{25} + (16 \beta + 48) q^{26} - 27 q^{27} + (54 \beta - 128) q^{28} + (6 \beta + 96) q^{29} + ( - 24 \beta - 144) q^{30} + (8 \beta + 176) q^{31} + ( - 41 \beta - 408) q^{32} + (46 \beta - 336) q^{34} + ( - 4 \beta - 108) q^{35} + (9 \beta + 144) q^{36} + (52 \beta - 190) q^{37} + (24 \beta - 48) q^{38} + (6 \beta + 42) q^{39} + (120 \beta + 576) q^{40} + (14 \beta + 384) q^{41} + ( - 30 \beta + 288) q^{42} + (26 \beta - 206) q^{43} + ( - 18 \beta - 54) q^{45} + ( - 62 \beta + 240) q^{46} + (74 \beta + 96) q^{47} + ( - 75 \beta - 264) q^{48} + ( - 96 \beta + 237) q^{49} + ( - 35 \beta - 672) q^{50} + ( - 42 \beta + 180) q^{51} + ( - 48 \beta - 272) q^{52} + ( - 54 \beta - 234) q^{53} + 27 \beta q^{54} + ( - 6 \beta - 528) q^{56} + ( - 6 \beta + 78) q^{57} + ( - 102 \beta - 144) q^{58} + ( - 100 \beta - 36) q^{59} + (120 \beta + 432) q^{60} + ( - 34 \beta + 406) q^{61} + ( - 184 \beta - 192) q^{62} + (36 \beta - 126) q^{63} + (249 \beta + 280) q^{64} + (44 \beta + 180) q^{65} + ( - 96 \beta - 340) q^{67} + (178 \beta - 624) q^{68} + (30 \beta - 216) q^{69} + (112 \beta + 96) q^{70} + (54 \beta + 288) q^{71} + ( - 81 \beta - 216) q^{72} + (28 \beta - 662) q^{73} + (138 \beta - 1248) q^{74} + ( - 84 \beta - 21) q^{75} + (8 \beta - 368) q^{76} + ( - 48 \beta - 144) q^{78} + ( - 144 \beta - 254) q^{79} + ( - 376 \beta - 1728) q^{80} + 81 q^{81} + ( - 398 \beta - 336) q^{82} + ( - 156 \beta + 240) q^{83} + ( - 162 \beta + 384) q^{84} + (8 \beta - 312) q^{85} + (180 \beta - 624) q^{86} + ( - 18 \beta - 288) q^{87} + (72 \beta - 414) q^{89} + (72 \beta + 432) q^{90} + ( - 36 \beta + 4) q^{91} + ( - 98 \beta + 912) q^{92} + ( - 24 \beta - 528) q^{93} + ( - 170 \beta - 1776) q^{94} + (36 \beta + 60) q^{95} + (123 \beta + 1224) q^{96} + (192 \beta - 322) q^{97} + ( - 141 \beta + 2304) q^{98} +O(q^{100})$$ q - b * q^2 - 3 * q^3 + (b + 16) * q^4 + (-2*b - 6) * q^5 + 3*b * q^6 + (4*b - 14) * q^7 + (-9*b - 24) * q^8 + 9 * q^9 + (8*b + 48) * q^10 + (-3*b - 48) * q^12 + (-2*b - 14) * q^13 + (10*b - 96) * q^14 + (6*b + 18) * q^15 + (25*b + 88) * q^16 + (14*b - 60) * q^17 - 9*b * q^18 + (2*b - 26) * q^19 + (-40*b - 144) * q^20 + (-12*b + 42) * q^21 + (-10*b + 72) * q^23 + (27*b + 72) * q^24 + (28*b + 7) * q^25 + (16*b + 48) * q^26 - 27 * q^27 + (54*b - 128) * q^28 + (6*b + 96) * q^29 + (-24*b - 144) * q^30 + (8*b + 176) * q^31 + (-41*b - 408) * q^32 + (46*b - 336) * q^34 + (-4*b - 108) * q^35 + (9*b + 144) * q^36 + (52*b - 190) * q^37 + (24*b - 48) * q^38 + (6*b + 42) * q^39 + (120*b + 576) * q^40 + (14*b + 384) * q^41 + (-30*b + 288) * q^42 + (26*b - 206) * q^43 + (-18*b - 54) * q^45 + (-62*b + 240) * q^46 + (74*b + 96) * q^47 + (-75*b - 264) * q^48 + (-96*b + 237) * q^49 + (-35*b - 672) * q^50 + (-42*b + 180) * q^51 + (-48*b - 272) * q^52 + (-54*b - 234) * q^53 + 27*b * q^54 + (-6*b - 528) * q^56 + (-6*b + 78) * q^57 + (-102*b - 144) * q^58 + (-100*b - 36) * q^59 + (120*b + 432) * q^60 + (-34*b + 406) * q^61 + (-184*b - 192) * q^62 + (36*b - 126) * q^63 + (249*b + 280) * q^64 + (44*b + 180) * q^65 + (-96*b - 340) * q^67 + (178*b - 624) * q^68 + (30*b - 216) * q^69 + (112*b + 96) * q^70 + (54*b + 288) * q^71 + (-81*b - 216) * q^72 + (28*b - 662) * q^73 + (138*b - 1248) * q^74 + (-84*b - 21) * q^75 + (8*b - 368) * q^76 + (-48*b - 144) * q^78 + (-144*b - 254) * q^79 + (-376*b - 1728) * q^80 + 81 * q^81 + (-398*b - 336) * q^82 + (-156*b + 240) * q^83 + (-162*b + 384) * q^84 + (8*b - 312) * q^85 + (180*b - 624) * q^86 + (-18*b - 288) * q^87 + (72*b - 414) * q^89 + (72*b + 432) * q^90 + (-36*b + 4) * q^91 + (-98*b + 912) * q^92 + (-24*b - 528) * q^93 + (-170*b - 1776) * q^94 + (36*b + 60) * q^95 + (123*b + 1224) * q^96 + (192*b - 322) * q^97 + (-141*b + 2304) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 + 3 * q^6 - 24 * q^7 - 57 * q^8 + 18 * q^9 $$2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} + 104 q^{10} - 99 q^{12} - 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} - 106 q^{17} - 9 q^{18} - 50 q^{19} - 328 q^{20} + 72 q^{21} + 134 q^{23} + 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} - 202 q^{28} + 198 q^{29} - 312 q^{30} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} + 90 q^{39} + 1272 q^{40} + 782 q^{41} + 546 q^{42} - 386 q^{43} - 126 q^{45} + 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} - 1379 q^{50} + 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 1062 q^{56} + 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} - 216 q^{63} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} - 513 q^{72} - 1296 q^{73} - 2358 q^{74} - 126 q^{75} - 728 q^{76} - 336 q^{78} - 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} + 324 q^{83} + 606 q^{84} - 616 q^{85} - 1068 q^{86} - 594 q^{87} - 756 q^{89} + 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} + 156 q^{95} + 2571 q^{96} - 452 q^{97} + 4467 q^{98}+O(q^{100})$$ 2 * q - q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 + 3 * q^6 - 24 * q^7 - 57 * q^8 + 18 * q^9 + 104 * q^10 - 99 * q^12 - 30 * q^13 - 182 * q^14 + 42 * q^15 + 201 * q^16 - 106 * q^17 - 9 * q^18 - 50 * q^19 - 328 * q^20 + 72 * q^21 + 134 * q^23 + 171 * q^24 + 42 * q^25 + 112 * q^26 - 54 * q^27 - 202 * q^28 + 198 * q^29 - 312 * q^30 + 360 * q^31 - 857 * q^32 - 626 * q^34 - 220 * q^35 + 297 * q^36 - 328 * q^37 - 72 * q^38 + 90 * q^39 + 1272 * q^40 + 782 * q^41 + 546 * q^42 - 386 * q^43 - 126 * q^45 + 418 * q^46 + 266 * q^47 - 603 * q^48 + 378 * q^49 - 1379 * q^50 + 318 * q^51 - 592 * q^52 - 522 * q^53 + 27 * q^54 - 1062 * q^56 + 150 * q^57 - 390 * q^58 - 172 * q^59 + 984 * q^60 + 778 * q^61 - 568 * q^62 - 216 * q^63 + 809 * q^64 + 404 * q^65 - 776 * q^67 - 1070 * q^68 - 402 * q^69 + 304 * q^70 + 630 * q^71 - 513 * q^72 - 1296 * q^73 - 2358 * q^74 - 126 * q^75 - 728 * q^76 - 336 * q^78 - 652 * q^79 - 3832 * q^80 + 162 * q^81 - 1070 * q^82 + 324 * q^83 + 606 * q^84 - 616 * q^85 - 1068 * q^86 - 594 * q^87 - 756 * q^89 + 936 * q^90 - 28 * q^91 + 1726 * q^92 - 1080 * q^93 - 3722 * q^94 + 156 * q^95 + 2571 * q^96 - 452 * q^97 + 4467 * 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
 5.42443 −4.42443
−5.42443 −3.00000 21.4244 −16.8489 16.2733 7.69772 −72.8199 9.00000 91.3954
1.2 4.42443 −3.00000 11.5756 2.84886 −13.2733 −31.6977 15.8199 9.00000 12.6046
 $$n$$: 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.i 2
3.b odd 2 1 1089.4.a.u 2
11.b odd 2 1 33.4.a.c 2
33.d even 2 1 99.4.a.f 2
44.c even 2 1 528.4.a.p 2
55.d odd 2 1 825.4.a.l 2
55.e even 4 2 825.4.c.h 4
77.b even 2 1 1617.4.a.k 2
88.b odd 2 1 2112.4.a.bn 2
88.g even 2 1 2112.4.a.bg 2
132.d odd 2 1 1584.4.a.bj 2
165.d even 2 1 2475.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 11.b odd 2 1
99.4.a.f 2 33.d even 2 1
363.4.a.i 2 1.a even 1 1 trivial
528.4.a.p 2 44.c even 2 1
825.4.a.l 2 55.d odd 2 1
825.4.c.h 4 55.e even 4 2
1089.4.a.u 2 3.b odd 2 1
1584.4.a.bj 2 132.d odd 2 1
1617.4.a.k 2 77.b even 2 1
2112.4.a.bg 2 88.g even 2 1
2112.4.a.bn 2 88.b odd 2 1
2475.4.a.p 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(363))$$:

 $$T_{2}^{2} + T_{2} - 24$$ T2^2 + T2 - 24 $$T_{5}^{2} + 14T_{5} - 48$$ T5^2 + 14*T5 - 48 $$T_{7}^{2} + 24T_{7} - 244$$ T7^2 + 24*T7 - 244

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 24$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 14T - 48$$
$7$ $$T^{2} + 24T - 244$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 30T + 128$$
$17$ $$T^{2} + 106T - 1944$$
$19$ $$T^{2} + 50T + 528$$
$23$ $$T^{2} - 134T + 2064$$
$29$ $$T^{2} - 198T + 8928$$
$31$ $$T^{2} - 360T + 30848$$
$37$ $$T^{2} + 328T - 38676$$
$41$ $$T^{2} - 782T + 148128$$
$43$ $$T^{2} + 386T + 20856$$
$47$ $$T^{2} - 266T - 115104$$
$53$ $$T^{2} + 522T - 2592$$
$59$ $$T^{2} + 172T - 235104$$
$61$ $$T^{2} - 778T + 123288$$
$67$ $$T^{2} + 776T - 72944$$
$71$ $$T^{2} - 630T + 28512$$
$73$ $$T^{2} + 1296 T + 400892$$
$79$ $$T^{2} + 652T - 396572$$
$83$ $$T^{2} - 324T - 563904$$
$89$ $$T^{2} + 756T + 17172$$
$97$ $$T^{2} + 452T - 842876$$