# Properties

 Label 361.2.c.a Level $361$ Weight $2$ Character orbit 361.c Analytic conductor $2.883$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,2,Mod(68,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("361.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.c (of order $$3$$, degree $$2$$, 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: $$2.88259951297$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + 2 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + 2*z * q^4 + (3*z - 3) * q^5 - q^7 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + 2 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{7} - \zeta_{6} q^{9} + 3 q^{11} - 4 q^{12} - 4 \zeta_{6} q^{13} - 6 \zeta_{6} q^{15} + (4 \zeta_{6} - 4) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - 6 q^{20} + ( - 2 \zeta_{6} + 2) q^{21} - 4 \zeta_{6} q^{25} - 4 q^{27} - 2 \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} + 4 q^{31} + (6 \zeta_{6} - 6) q^{33} + ( - 3 \zeta_{6} + 3) q^{35} + ( - 2 \zeta_{6} + 2) q^{36} - 2 q^{37} + 8 q^{39} + (6 \zeta_{6} - 6) q^{41} + ( - \zeta_{6} + 1) q^{43} + 6 \zeta_{6} q^{44} + 3 q^{45} + 3 \zeta_{6} q^{47} - 8 \zeta_{6} q^{48} - 6 q^{49} + 6 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + 12 \zeta_{6} q^{53} + (9 \zeta_{6} - 9) q^{55} + (6 \zeta_{6} - 6) q^{59} + ( - 12 \zeta_{6} + 12) q^{60} + \zeta_{6} q^{61} + \zeta_{6} q^{63} - 8 q^{64} + 12 q^{65} - 4 \zeta_{6} q^{67} + 6 q^{68} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + 8 q^{75} - 3 q^{77} + ( - 8 \zeta_{6} + 8) q^{79} - 12 \zeta_{6} q^{80} + ( - 11 \zeta_{6} + 11) q^{81} + 12 q^{83} + 4 q^{84} + 9 \zeta_{6} q^{85} - 12 q^{87} + 12 \zeta_{6} q^{89} + 4 \zeta_{6} q^{91} + (8 \zeta_{6} - 8) q^{93} + ( - 8 \zeta_{6} + 8) q^{97} - 3 \zeta_{6} q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 + 2*z * q^4 + (3*z - 3) * q^5 - q^7 - z * q^9 + 3 * q^11 - 4 * q^12 - 4*z * q^13 - 6*z * q^15 + (4*z - 4) * q^16 + (-3*z + 3) * q^17 - 6 * q^20 + (-2*z + 2) * q^21 - 4*z * q^25 - 4 * q^27 - 2*z * q^28 + 6*z * q^29 + 4 * q^31 + (6*z - 6) * q^33 + (-3*z + 3) * q^35 + (-2*z + 2) * q^36 - 2 * q^37 + 8 * q^39 + (6*z - 6) * q^41 + (-z + 1) * q^43 + 6*z * q^44 + 3 * q^45 + 3*z * q^47 - 8*z * q^48 - 6 * q^49 + 6*z * q^51 + (-8*z + 8) * q^52 + 12*z * q^53 + (9*z - 9) * q^55 + (6*z - 6) * q^59 + (-12*z + 12) * q^60 + z * q^61 + z * q^63 - 8 * q^64 + 12 * q^65 - 4*z * q^67 + 6 * q^68 + (-6*z + 6) * q^71 + (-7*z + 7) * q^73 + 8 * q^75 - 3 * q^77 + (-8*z + 8) * q^79 - 12*z * q^80 + (-11*z + 11) * q^81 + 12 * q^83 + 4 * q^84 + 9*z * q^85 - 12 * q^87 + 12*z * q^89 + 4*z * q^91 + (8*z - 8) * q^93 + (-8*z + 8) * q^97 - 3*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^7 - q^9 $$2 q - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{7} - q^{9} + 6 q^{11} - 8 q^{12} - 4 q^{13} - 6 q^{15} - 4 q^{16} + 3 q^{17} - 12 q^{20} + 2 q^{21} - 4 q^{25} - 8 q^{27} - 2 q^{28} + 6 q^{29} + 8 q^{31} - 6 q^{33} + 3 q^{35} + 2 q^{36} - 4 q^{37} + 16 q^{39} - 6 q^{41} + q^{43} + 6 q^{44} + 6 q^{45} + 3 q^{47} - 8 q^{48} - 12 q^{49} + 6 q^{51} + 8 q^{52} + 12 q^{53} - 9 q^{55} - 6 q^{59} + 12 q^{60} + q^{61} + q^{63} - 16 q^{64} + 24 q^{65} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 7 q^{73} + 16 q^{75} - 6 q^{77} + 8 q^{79} - 12 q^{80} + 11 q^{81} + 24 q^{83} + 8 q^{84} + 9 q^{85} - 24 q^{87} + 12 q^{89} + 4 q^{91} - 8 q^{93} + 8 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^7 - q^9 + 6 * q^11 - 8 * q^12 - 4 * q^13 - 6 * q^15 - 4 * q^16 + 3 * q^17 - 12 * q^20 + 2 * q^21 - 4 * q^25 - 8 * q^27 - 2 * q^28 + 6 * q^29 + 8 * q^31 - 6 * q^33 + 3 * q^35 + 2 * q^36 - 4 * q^37 + 16 * q^39 - 6 * q^41 + q^43 + 6 * q^44 + 6 * q^45 + 3 * q^47 - 8 * q^48 - 12 * q^49 + 6 * q^51 + 8 * q^52 + 12 * q^53 - 9 * q^55 - 6 * q^59 + 12 * q^60 + q^61 + q^63 - 16 * q^64 + 24 * q^65 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 7 * q^73 + 16 * q^75 - 6 * q^77 + 8 * q^79 - 12 * q^80 + 11 * q^81 + 24 * q^83 + 8 * q^84 + 9 * q^85 - 24 * q^87 + 12 * q^89 + 4 * q^91 - 8 * q^93 + 8 * q^97 - 3 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 + 1.73205i 1.00000 + 1.73205i −1.50000 + 2.59808i 0 −1.00000 0 −0.500000 0.866025i 0
292.1 0 −1.00000 1.73205i 1.00000 1.73205i −1.50000 2.59808i 0 −1.00000 0 −0.500000 + 0.866025i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.c.a 2
19.b odd 2 1 361.2.c.c 2
19.c even 3 1 361.2.a.b 1
19.c even 3 1 inner 361.2.c.a 2
19.d odd 6 1 19.2.a.a 1
19.d odd 6 1 361.2.c.c 2
19.e even 9 6 361.2.e.e 6
19.f odd 18 6 361.2.e.d 6
57.f even 6 1 171.2.a.b 1
57.h odd 6 1 3249.2.a.d 1
76.f even 6 1 304.2.a.f 1
76.g odd 6 1 5776.2.a.c 1
95.h odd 6 1 475.2.a.b 1
95.i even 6 1 9025.2.a.d 1
95.l even 12 2 475.2.b.a 2
133.i even 6 1 931.2.f.b 2
133.j odd 6 1 931.2.f.c 2
133.n odd 6 1 931.2.f.c 2
133.p even 6 1 931.2.a.a 1
133.s even 6 1 931.2.f.b 2
152.l odd 6 1 1216.2.a.o 1
152.o even 6 1 1216.2.a.b 1
209.g even 6 1 2299.2.a.b 1
228.n odd 6 1 2736.2.a.c 1
247.n odd 6 1 3211.2.a.a 1
285.q even 6 1 4275.2.a.i 1
323.i odd 6 1 5491.2.a.b 1
380.s even 6 1 7600.2.a.c 1
399.q odd 6 1 8379.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 19.d odd 6 1
171.2.a.b 1 57.f even 6 1
304.2.a.f 1 76.f even 6 1
361.2.a.b 1 19.c even 3 1
361.2.c.a 2 1.a even 1 1 trivial
361.2.c.a 2 19.c even 3 1 inner
361.2.c.c 2 19.b odd 2 1
361.2.c.c 2 19.d odd 6 1
361.2.e.d 6 19.f odd 18 6
361.2.e.e 6 19.e even 9 6
475.2.a.b 1 95.h odd 6 1
475.2.b.a 2 95.l even 12 2
931.2.a.a 1 133.p even 6 1
931.2.f.b 2 133.i even 6 1
931.2.f.b 2 133.s even 6 1
931.2.f.c 2 133.j odd 6 1
931.2.f.c 2 133.n odd 6 1
1216.2.a.b 1 152.o even 6 1
1216.2.a.o 1 152.l odd 6 1
2299.2.a.b 1 209.g even 6 1
2736.2.a.c 1 228.n odd 6 1
3211.2.a.a 1 247.n odd 6 1
3249.2.a.d 1 57.h odd 6 1
4275.2.a.i 1 285.q even 6 1
5491.2.a.b 1 323.i odd 6 1
5776.2.a.c 1 76.g odd 6 1
7600.2.a.c 1 380.s even 6 1
8379.2.a.j 1 399.q odd 6 1
9025.2.a.d 1 95.i even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(361, [\chi])$$:

 $$T_{2}$$ T2 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$T^{2} - 8T + 64$$