Properties

 Label 363.3.h.e Level $363$ Weight $3$ Character orbit 363.h Analytic conductor $9.891$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(245,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 8]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("363.245");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.h (of order $$10$$, degree $$4$$, not minimal)

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: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.228765625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81$$ x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^3 - 18*x^2 - 27*x + 81 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + 3 \beta_{4} q^{3} - 7 \beta_{2} q^{4} + (2 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots - 9 \beta_{3} q^{9}+O(q^{10})$$ q + b7 * q^2 + 3*b4 * q^3 - 7*b2 * q^4 + (2*b7 + 2*b6 + 2*b5 + 2*b1) * q^5 + (-3*b7 - 3*b6 - 3*b5 - 3*b1) * q^6 - 8*b2 * q^7 - 3*b5 * q^8 - 9*b3 * q^9 $$q + \beta_{7} q^{2} + 3 \beta_{4} q^{3} - 7 \beta_{2} q^{4} + (2 \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots + 15 \beta_{6} q^{98}+O(q^{100})$$ q + b7 * q^2 + 3*b4 * q^3 - 7*b2 * q^4 + (2*b7 + 2*b6 + 2*b5 + 2*b1) * q^5 + (-3*b7 - 3*b6 - 3*b5 - 3*b1) * q^6 - 8*b2 * q^7 - 3*b5 * q^8 - 9*b3 * q^9 + 22 * q^10 - 21 * q^12 - 4*b3 * q^13 - 8*b5 * q^14 - 6*b1 * q^15 + (-5*b4 + 5*b3 - 5*b2 - 5) * q^16 + (-4*b7 - 4*b6 - 4*b5 - 4*b1) * q^17 + 9*b1 * q^18 - 6*b4 * q^19 + 14*b7 * q^20 - 24 * q^21 + 2*b6 * q^23 - 9*b7 * q^24 - 19*b4 * q^25 + 4*b1 * q^26 + (-27*b4 + 27*b3 - 27*b2 - 27) * q^27 + (-56*b4 + 56*b3 - 56*b2 - 56) * q^28 - 12*b1 * q^29 + 66*b4 * q^30 + 26*b3 * q^31 - 7*b6 * q^32 - 44 * q^34 + 16*b7 * q^35 - 63*b4 * q^36 + 30*b2 * q^37 + (6*b7 + 6*b6 + 6*b5 + 6*b1) * q^38 + (-12*b4 + 12*b3 - 12*b2 - 12) * q^39 - 66*b2 * q^40 + 4*b5 * q^41 - 24*b7 * q^42 + 42 * q^43 - 18*b6 * q^45 + 22*b3 * q^46 + 26*b5 * q^47 + 15*b2 * q^48 + (-15*b4 + 15*b3 - 15*b2 - 15) * q^49 + (19*b7 + 19*b6 + 19*b5 + 19*b1) * q^50 + 12*b1 * q^51 - 28*b4 * q^52 - 18*b7 * q^53 + 27*b6 * q^54 + 24*b6 * q^56 + 18*b3 * q^57 + 132*b4 * q^58 + 20*b1 * q^59 + (-42*b7 - 42*b6 - 42*b5 - 42*b1) * q^60 + (-12*b4 + 12*b3 - 12*b2 - 12) * q^61 - 26*b1 * q^62 - 72*b4 * q^63 - 97*b3 * q^64 - 8*b6 * q^65 + 2 * q^67 - 28*b7 * q^68 + 6*b5 * q^69 - 176*b2 * q^70 + (18*b7 + 18*b6 + 18*b5 + 18*b1) * q^71 + (27*b7 + 27*b6 + 27*b5 + 27*b1) * q^72 - 74*b2 * q^73 + 30*b5 * q^74 + 57*b3 * q^75 + 42 * q^76 + 12*b6 * q^78 + 40*b3 * q^79 - 10*b5 * q^80 + 81*b2 * q^81 + (44*b4 - 44*b3 + 44*b2 + 44) * q^82 + (12*b7 + 12*b6 + 12*b5 + 12*b1) * q^83 + 168*b2 * q^84 + 88*b4 * q^85 + 42*b7 * q^86 - 36*b6 * q^87 - 36*b6 * q^89 - 198*b3 * q^90 - 32*b4 * q^91 - 14*b1 * q^92 + (78*b4 - 78*b3 + 78*b2 + 78) * q^93 + (286*b4 - 286*b3 + 286*b2 + 286) * q^94 + 12*b1 * q^95 - 21*b5 * q^96 - 62*b3 * q^97 + 15*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{3} + 14 q^{4} + 16 q^{7} - 18 q^{9}+O(q^{10})$$ 8 * q - 6 * q^3 + 14 * q^4 + 16 * q^7 - 18 * q^9 $$8 q - 6 q^{3} + 14 q^{4} + 16 q^{7} - 18 q^{9} + 176 q^{10} - 168 q^{12} - 8 q^{13} - 10 q^{16} + 12 q^{19} - 192 q^{21} + 38 q^{25} - 54 q^{27} - 112 q^{28} - 132 q^{30} + 52 q^{31} - 352 q^{34} + 126 q^{36} - 60 q^{37} - 24 q^{39} + 132 q^{40} + 336 q^{43} + 44 q^{46} - 30 q^{48} - 30 q^{49} + 56 q^{52} + 36 q^{57} - 264 q^{58} - 24 q^{61} + 144 q^{63} - 194 q^{64} + 16 q^{67} + 352 q^{70} + 148 q^{73} + 114 q^{75} + 336 q^{76} + 80 q^{79} - 162 q^{81} + 88 q^{82} - 336 q^{84} - 176 q^{85} - 396 q^{90} + 64 q^{91} + 156 q^{93} + 572 q^{94} - 124 q^{97}+O(q^{100})$$ 8 * q - 6 * q^3 + 14 * q^4 + 16 * q^7 - 18 * q^9 + 176 * q^10 - 168 * q^12 - 8 * q^13 - 10 * q^16 + 12 * q^19 - 192 * q^21 + 38 * q^25 - 54 * q^27 - 112 * q^28 - 132 * q^30 + 52 * q^31 - 352 * q^34 + 126 * q^36 - 60 * q^37 - 24 * q^39 + 132 * q^40 + 336 * q^43 + 44 * q^46 - 30 * q^48 - 30 * q^49 + 56 * q^52 + 36 * q^57 - 264 * q^58 - 24 * q^61 + 144 * q^63 - 194 * q^64 + 16 * q^67 + 352 * q^70 + 148 * q^73 + 114 * q^75 + 336 * q^76 + 80 * q^79 - 162 * q^81 + 88 * q^82 - 336 * q^84 - 176 * q^85 - 396 * q^90 + 64 * q^91 + 156 * q^93 + 572 * q^94 - 124 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} - 10\nu ) / 3$$ (-v^6 - 10*v) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} - 16\nu ) / 3$$ (-v^6 - 16*v) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27$$ (-2*v^7 - 4*v^6 + 10*v^5 + 2*v^4 - 5*v^3 - 36*v^2 - 54*v + 162) / 27 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27$$ (-5*v^7 + 5*v^6 + 10*v^5 + 2*v^4 - 5*v^3 - 75*v^2 + 90*v + 135) / 27 $$\beta_{5}$$ $$=$$ $$( 7\nu^{7} - 7\nu^{6} - 14\nu^{5} + 8\nu^{4} + 7\nu^{3} + 105\nu^{2} - 126\nu - 189 ) / 27$$ (7*v^7 - 7*v^6 - 14*v^5 + 8*v^4 + 7*v^3 + 105*v^2 - 126*v - 189) / 27 $$\beta_{6}$$ $$=$$ $$2\nu^{5} + 31$$ 2*v^5 + 31 $$\beta_{7}$$ $$=$$ $$( 8\nu^{7} + 16\nu^{6} - 40\nu^{5} - 8\nu^{4} - 7\nu^{3} + 144\nu^{2} + 216\nu - 648 ) / 27$$ (8*v^7 + 16*v^6 - 40*v^5 - 8*v^4 - 7*v^3 + 144*v^2 + 216*v - 648) / 27
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + 5\beta_{4} - 5\beta_{3} + 5\beta_{2} + \beta _1 + 5 ) / 2$$ (b7 + b6 + b5 + 5*b4 - 5*b3 + 5*b2 + b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{7} - 4\beta_{3}$$ -b7 - 4*b3 $$\nu^{4}$$ $$=$$ $$( 5\beta_{5} + 7\beta_{4} ) / 2$$ (5*b5 + 7*b4) / 2 $$\nu^{5}$$ $$=$$ $$( \beta_{6} - 31 ) / 2$$ (b6 - 31) / 2 $$\nu^{6}$$ $$=$$ $$5\beta_{2} - 8\beta_1$$ 5*b2 - 8*b1 $$\nu^{7}$$ $$=$$ $$( -13\beta_{7} - 13\beta_{6} - 13\beta_{5} - 83\beta_{4} + 83\beta_{3} - 83\beta_{2} - 13\beta _1 - 83 ) / 2$$ (-13*b7 - 13*b6 - 13*b5 - 83*b4 + 83*b3 - 83*b2 - 13*b1 - 83) / 2

Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
245.1
 1.42264 + 0.987975i −1.73166 − 0.0369185i −0.570223 + 1.63550i 1.37924 − 1.04771i −0.570223 − 1.63550i 1.37924 + 1.04771i 1.42264 − 0.987975i −1.73166 + 0.0369185i
−1.94946 2.68321i 0.927051 + 2.85317i −2.16312 + 6.65740i −3.89893 + 5.36641i 5.84839 8.04962i −2.47214 + 7.60845i 9.46289 3.07468i −7.28115 + 5.29007i 22.0000
245.2 1.94946 + 2.68321i 0.927051 + 2.85317i −2.16312 + 6.65740i 3.89893 5.36641i −5.84839 + 8.04962i −2.47214 + 7.60845i −9.46289 + 3.07468i −7.28115 + 5.29007i 22.0000
251.1 −3.15430 1.02489i −2.42705 + 1.76336i 5.66312 + 4.11450i −6.30860 + 2.04979i 9.46289 3.07468i 6.47214 + 4.70228i −5.84839 8.04962i 2.78115 8.55951i 22.0000
251.2 3.15430 + 1.02489i −2.42705 + 1.76336i 5.66312 + 4.11450i 6.30860 2.04979i −9.46289 + 3.07468i 6.47214 + 4.70228i 5.84839 + 8.04962i 2.78115 8.55951i 22.0000
269.1 −3.15430 + 1.02489i −2.42705 1.76336i 5.66312 4.11450i −6.30860 2.04979i 9.46289 + 3.07468i 6.47214 4.70228i −5.84839 + 8.04962i 2.78115 + 8.55951i 22.0000
269.2 3.15430 1.02489i −2.42705 1.76336i 5.66312 4.11450i 6.30860 + 2.04979i −9.46289 3.07468i 6.47214 4.70228i 5.84839 8.04962i 2.78115 + 8.55951i 22.0000
323.1 −1.94946 + 2.68321i 0.927051 2.85317i −2.16312 6.65740i −3.89893 5.36641i 5.84839 + 8.04962i −2.47214 7.60845i 9.46289 + 3.07468i −7.28115 5.29007i 22.0000
323.2 1.94946 2.68321i 0.927051 2.85317i −2.16312 6.65740i 3.89893 + 5.36641i −5.84839 8.04962i −2.47214 7.60845i −9.46289 3.07468i −7.28115 5.29007i 22.0000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 245.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 3 inner
33.h odd 10 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.e 8
3.b odd 2 1 inner 363.3.h.e 8
11.b odd 2 1 363.3.h.d 8
11.c even 5 1 33.3.b.a 2
11.c even 5 3 inner 363.3.h.e 8
11.d odd 10 1 363.3.b.d 2
11.d odd 10 3 363.3.h.d 8
33.d even 2 1 363.3.h.d 8
33.f even 10 1 363.3.b.d 2
33.f even 10 3 363.3.h.d 8
33.h odd 10 1 33.3.b.a 2
33.h odd 10 3 inner 363.3.h.e 8
44.h odd 10 1 528.3.i.a 2
132.o even 10 1 528.3.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.a 2 11.c even 5 1
33.3.b.a 2 33.h odd 10 1
363.3.b.d 2 11.d odd 10 1
363.3.b.d 2 33.f even 10 1
363.3.h.d 8 11.b odd 2 1
363.3.h.d 8 11.d odd 10 3
363.3.h.d 8 33.d even 2 1
363.3.h.d 8 33.f even 10 3
363.3.h.e 8 1.a even 1 1 trivial
363.3.h.e 8 3.b odd 2 1 inner
363.3.h.e 8 11.c even 5 3 inner
363.3.h.e 8 33.h odd 10 3 inner
528.3.i.a 2 44.h odd 10 1
528.3.i.a 2 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{8} - 11T_{2}^{6} + 121T_{2}^{4} - 1331T_{2}^{2} + 14641$$ T2^8 - 11*T2^6 + 121*T2^4 - 1331*T2^2 + 14641 $$T_{5}^{8} - 44T_{5}^{6} + 1936T_{5}^{4} - 85184T_{5}^{2} + 3748096$$ T5^8 - 44*T5^6 + 1936*T5^4 - 85184*T5^2 + 3748096 $$T_{7}^{4} - 8T_{7}^{3} + 64T_{7}^{2} - 512T_{7} + 4096$$ T7^4 - 8*T7^3 + 64*T7^2 - 512*T7 + 4096

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 11 T^{6} + \cdots + 14641$$
$3$ $$(T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2}$$
$5$ $$T^{8} - 44 T^{6} + \cdots + 3748096$$
$7$ $$(T^{4} - 8 T^{3} + \cdots + 4096)^{2}$$
$11$ $$T^{8}$$
$13$ $$(T^{4} + 4 T^{3} + \cdots + 256)^{2}$$
$17$ $$T^{8} - 176 T^{6} + \cdots + 959512576$$
$19$ $$(T^{4} - 6 T^{3} + \cdots + 1296)^{2}$$
$23$ $$(T^{2} + 44)^{4}$$
$29$ $$T^{8} + \cdots + 6295362011136$$
$31$ $$(T^{4} - 26 T^{3} + \cdots + 456976)^{2}$$
$37$ $$(T^{4} + 30 T^{3} + \cdots + 810000)^{2}$$
$41$ $$T^{8} - 176 T^{6} + \cdots + 959512576$$
$43$ $$(T - 42)^{8}$$
$47$ $$T^{8} + \cdots + 30\!\cdots\!16$$
$53$ $$T^{8} + \cdots + 161343242793216$$
$59$ $$T^{8} + \cdots + 374809600000000$$
$61$ $$(T^{4} + 12 T^{3} + \cdots + 20736)^{2}$$
$67$ $$(T - 2)^{8}$$
$71$ $$T^{8} + \cdots + 161343242793216$$
$73$ $$(T^{4} - 74 T^{3} + \cdots + 29986576)^{2}$$
$79$ $$(T^{4} - 40 T^{3} + \cdots + 2560000)^{2}$$
$83$ $$T^{8} + \cdots + 6295362011136$$
$89$ $$(T^{2} + 14256)^{4}$$
$97$ $$(T^{4} + 62 T^{3} + \cdots + 14776336)^{2}$$