# Properties

 Label 363.4.d.b Level $363$ Weight $4$ Character orbit 363.d Analytic conductor $21.418$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(362,363)]

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([1, 1]))

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

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

chi := DirichletCharacter("363.362");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.d (of order $$2$$, degree $$1$$, 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: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - x^{2} + 2x + 27$$ x^4 - 2*x^3 - x^2 + 2*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_1 + 1) q^{3} + 5 q^{4} + 4 \beta_1 q^{5} + ( - \beta_{3} + 13 \beta_{2}) q^{6} - 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} + ( - 2 \beta_1 - 25) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-b1 + 1) * q^3 + 5 * q^4 + 4*b1 * q^5 + (-b3 + 13*b2) * q^6 - 7*b2 * q^7 + 3*b3 * q^8 + (-2*b1 - 25) * q^9 $$q - \beta_{3} q^{2} + ( - \beta_1 + 1) q^{3} + 5 q^{4} + 4 \beta_1 q^{5} + ( - \beta_{3} + 13 \beta_{2}) q^{6} - 7 \beta_{2} q^{7} + 3 \beta_{3} q^{8} + ( - 2 \beta_1 - 25) q^{9} - 52 \beta_{2} q^{10} + ( - 5 \beta_1 + 5) q^{12} - 15 \beta_{2} q^{13} + 7 \beta_1 q^{14} + (4 \beta_1 + 104) q^{15} - 79 q^{16} + 30 \beta_{3} q^{17} + (25 \beta_{3} + 26 \beta_{2}) q^{18} - 39 \beta_{2} q^{19} + 20 \beta_1 q^{20} + ( - 14 \beta_{3} - 7 \beta_{2}) q^{21} + 18 \beta_1 q^{23} + (3 \beta_{3} - 39 \beta_{2}) q^{24} - 291 q^{25} + 15 \beta_1 q^{26} + (23 \beta_1 - 77) q^{27} - 35 \beta_{2} q^{28} - 70 \beta_{3} q^{29} + ( - 104 \beta_{3} - 52 \beta_{2}) q^{30} + 82 q^{31} + 55 \beta_{3} q^{32} - 390 q^{34} + 56 \beta_{3} q^{35} + ( - 10 \beta_1 - 125) q^{36} + 188 q^{37} + 39 \beta_1 q^{38} + ( - 30 \beta_{3} - 15 \beta_{2}) q^{39} + 156 \beta_{2} q^{40} - 6 \beta_{3} q^{41} + (7 \beta_1 + 182) q^{42} + 213 \beta_{2} q^{43} + ( - 100 \beta_1 + 208) q^{45} - 234 \beta_{2} q^{46} + 52 \beta_1 q^{47} + (79 \beta_1 - 79) q^{48} + 245 q^{49} + 291 \beta_{3} q^{50} + (30 \beta_{3} - 390 \beta_{2}) q^{51} - 75 \beta_{2} q^{52} - 34 \beta_1 q^{53} + (77 \beta_{3} - 299 \beta_{2}) q^{54} - 21 \beta_1 q^{56} + ( - 78 \beta_{3} - 39 \beta_{2}) q^{57} + 910 q^{58} + 56 \beta_1 q^{59} + (20 \beta_1 + 520) q^{60} + 131 \beta_{2} q^{61} - 82 \beta_{3} q^{62} + ( - 28 \beta_{3} + 175 \beta_{2}) q^{63} - 83 q^{64} + 120 \beta_{3} q^{65} + 602 q^{67} + 150 \beta_{3} q^{68} + (18 \beta_1 + 468) q^{69} - 728 q^{70} + 136 \beta_1 q^{71} + ( - 75 \beta_{3} - 78 \beta_{2}) q^{72} + 215 \beta_{2} q^{73} - 188 \beta_{3} q^{74} + (291 \beta_1 - 291) q^{75} - 195 \beta_{2} q^{76} + (15 \beta_1 + 390) q^{78} + 129 \beta_{2} q^{79} - 316 \beta_1 q^{80} + (100 \beta_1 + 521) q^{81} + 78 q^{82} + 124 \beta_{3} q^{83} + ( - 70 \beta_{3} - 35 \beta_{2}) q^{84} + 1560 \beta_{2} q^{85} - 213 \beta_1 q^{86} + ( - 70 \beta_{3} + 910 \beta_{2}) q^{87} + 180 \beta_1 q^{89} + ( - 208 \beta_{3} + 1300 \beta_{2}) q^{90} - 210 q^{91} + 90 \beta_1 q^{92} + ( - 82 \beta_1 + 82) q^{93} - 676 \beta_{2} q^{94} + 312 \beta_{3} q^{95} + (55 \beta_{3} - 715 \beta_{2}) q^{96} + 238 q^{97} - 245 \beta_{3} q^{98}+O(q^{100})$$ q - b3 * q^2 + (-b1 + 1) * q^3 + 5 * q^4 + 4*b1 * q^5 + (-b3 + 13*b2) * q^6 - 7*b2 * q^7 + 3*b3 * q^8 + (-2*b1 - 25) * q^9 - 52*b2 * q^10 + (-5*b1 + 5) * q^12 - 15*b2 * q^13 + 7*b1 * q^14 + (4*b1 + 104) * q^15 - 79 * q^16 + 30*b3 * q^17 + (25*b3 + 26*b2) * q^18 - 39*b2 * q^19 + 20*b1 * q^20 + (-14*b3 - 7*b2) * q^21 + 18*b1 * q^23 + (3*b3 - 39*b2) * q^24 - 291 * q^25 + 15*b1 * q^26 + (23*b1 - 77) * q^27 - 35*b2 * q^28 - 70*b3 * q^29 + (-104*b3 - 52*b2) * q^30 + 82 * q^31 + 55*b3 * q^32 - 390 * q^34 + 56*b3 * q^35 + (-10*b1 - 125) * q^36 + 188 * q^37 + 39*b1 * q^38 + (-30*b3 - 15*b2) * q^39 + 156*b2 * q^40 - 6*b3 * q^41 + (7*b1 + 182) * q^42 + 213*b2 * q^43 + (-100*b1 + 208) * q^45 - 234*b2 * q^46 + 52*b1 * q^47 + (79*b1 - 79) * q^48 + 245 * q^49 + 291*b3 * q^50 + (30*b3 - 390*b2) * q^51 - 75*b2 * q^52 - 34*b1 * q^53 + (77*b3 - 299*b2) * q^54 - 21*b1 * q^56 + (-78*b3 - 39*b2) * q^57 + 910 * q^58 + 56*b1 * q^59 + (20*b1 + 520) * q^60 + 131*b2 * q^61 - 82*b3 * q^62 + (-28*b3 + 175*b2) * q^63 - 83 * q^64 + 120*b3 * q^65 + 602 * q^67 + 150*b3 * q^68 + (18*b1 + 468) * q^69 - 728 * q^70 + 136*b1 * q^71 + (-75*b3 - 78*b2) * q^72 + 215*b2 * q^73 - 188*b3 * q^74 + (291*b1 - 291) * q^75 - 195*b2 * q^76 + (15*b1 + 390) * q^78 + 129*b2 * q^79 - 316*b1 * q^80 + (100*b1 + 521) * q^81 + 78 * q^82 + 124*b3 * q^83 + (-70*b3 - 35*b2) * q^84 + 1560*b2 * q^85 - 213*b1 * q^86 + (-70*b3 + 910*b2) * q^87 + 180*b1 * q^89 + (-208*b3 + 1300*b2) * q^90 - 210 * q^91 + 90*b1 * q^92 + (-82*b1 + 82) * q^93 - 676*b2 * q^94 + 312*b3 * q^95 + (55*b3 - 715*b2) * q^96 + 238 * q^97 - 245*b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 20 q^{4} - 100 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 20 * q^4 - 100 * q^9 $$4 q + 4 q^{3} + 20 q^{4} - 100 q^{9} + 20 q^{12} + 416 q^{15} - 316 q^{16} - 1164 q^{25} - 308 q^{27} + 328 q^{31} - 1560 q^{34} - 500 q^{36} + 752 q^{37} + 728 q^{42} + 832 q^{45} - 316 q^{48} + 980 q^{49} + 3640 q^{58} + 2080 q^{60} - 332 q^{64} + 2408 q^{67} + 1872 q^{69} - 2912 q^{70} - 1164 q^{75} + 1560 q^{78} + 2084 q^{81} + 312 q^{82} - 840 q^{91} + 328 q^{93} + 952 q^{97}+O(q^{100})$$ 4 * q + 4 * q^3 + 20 * q^4 - 100 * q^9 + 20 * q^12 + 416 * q^15 - 316 * q^16 - 1164 * q^25 - 308 * q^27 + 328 * q^31 - 1560 * q^34 - 500 * q^36 + 752 * q^37 + 728 * q^42 + 832 * q^45 - 316 * q^48 + 980 * q^49 + 3640 * q^58 + 2080 * q^60 - 332 * q^64 + 2408 * q^67 + 1872 * q^69 - 2912 * q^70 - 1164 * q^75 + 1560 * q^78 + 2084 * q^81 + 312 * q^82 - 840 * q^91 + 328 * q^93 + 952 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - x^{2} + 2x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 1$$ v^2 - v - 1 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} + 7\nu - 3 ) / 21$$ (2*v^3 - 3*v^2 + 7*v - 3) / 21 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} + 28\nu - 15 ) / 21$$ (-4*v^3 + 6*v^2 + 28*v - 15) / 21
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 1 ) / 2$$ (b3 + 2*b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 2\beta _1 + 3 ) / 2$$ (b3 + 2*b2 + 2*b1 + 3) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 17\beta_{2} + 3\beta _1 + 4 ) / 2$$ (-2*b3 + 17*b2 + 3*b1 + 4) / 2

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1
 2.30278 + 1.41421i 2.30278 − 1.41421i −1.30278 − 1.41421i −1.30278 + 1.41421i
−3.60555 1.00000 5.09902i 5.00000 20.3961i −3.60555 + 18.3848i 9.89949i 10.8167 −25.0000 10.1980i 73.5391i
362.2 −3.60555 1.00000 + 5.09902i 5.00000 20.3961i −3.60555 18.3848i 9.89949i 10.8167 −25.0000 + 10.1980i 73.5391i
362.3 3.60555 1.00000 5.09902i 5.00000 20.3961i 3.60555 18.3848i 9.89949i −10.8167 −25.0000 10.1980i 73.5391i
362.4 3.60555 1.00000 + 5.09902i 5.00000 20.3961i 3.60555 + 18.3848i 9.89949i −10.8167 −25.0000 + 10.1980i 73.5391i
 $$n$$: 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
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.d.b 4
3.b odd 2 1 inner 363.4.d.b 4
11.b odd 2 1 inner 363.4.d.b 4
33.d even 2 1 inner 363.4.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.d.b 4 1.a even 1 1 trivial
363.4.d.b 4 3.b odd 2 1 inner
363.4.d.b 4 11.b odd 2 1 inner
363.4.d.b 4 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 13$$ acting on $$S_{4}^{\mathrm{new}}(363, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 13)^{2}$$
$3$ $$(T^{2} - 2 T + 27)^{2}$$
$5$ $$(T^{2} + 416)^{2}$$
$7$ $$(T^{2} + 98)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 450)^{2}$$
$17$ $$(T^{2} - 11700)^{2}$$
$19$ $$(T^{2} + 3042)^{2}$$
$23$ $$(T^{2} + 8424)^{2}$$
$29$ $$(T^{2} - 63700)^{2}$$
$31$ $$(T - 82)^{4}$$
$37$ $$(T - 188)^{4}$$
$41$ $$(T^{2} - 468)^{2}$$
$43$ $$(T^{2} + 90738)^{2}$$
$47$ $$(T^{2} + 70304)^{2}$$
$53$ $$(T^{2} + 30056)^{2}$$
$59$ $$(T^{2} + 81536)^{2}$$
$61$ $$(T^{2} + 34322)^{2}$$
$67$ $$(T - 602)^{4}$$
$71$ $$(T^{2} + 480896)^{2}$$
$73$ $$(T^{2} + 92450)^{2}$$
$79$ $$(T^{2} + 33282)^{2}$$
$83$ $$(T^{2} - 199888)^{2}$$
$89$ $$(T^{2} + 842400)^{2}$$
$97$ $$(T - 238)^{4}$$