Properties

 Label 1815.2.c.a Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(364,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1815.364");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.c (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: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} + ( - 2 i - 1) q^{5} - q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9}+O(q^{10})$$ q + i * q^2 + i * q^3 + q^4 + (-2*i - 1) * q^5 - q^6 + 4*i * q^7 + 3*i * q^8 - q^9 $$q + i q^{2} + i q^{3} + q^{4} + ( - 2 i - 1) q^{5} - q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} + ( - i + 2) q^{10} + i q^{12} - 2 i q^{13} - 4 q^{14} + ( - i + 2) q^{15} - q^{16} + 2 i q^{17} - i q^{18} - 8 q^{19} + ( - 2 i - 1) q^{20} - 4 q^{21} - 4 i q^{23} - 3 q^{24} + (4 i - 3) q^{25} + 2 q^{26} - i q^{27} + 4 i q^{28} - 4 q^{29} + (2 i + 1) q^{30} - 8 q^{31} + 5 i q^{32} - 2 q^{34} + ( - 4 i + 8) q^{35} - q^{36} + 8 i q^{37} - 8 i q^{38} + 2 q^{39} + ( - 3 i + 6) q^{40} + 12 q^{41} - 4 i q^{42} + 8 i q^{43} + (2 i + 1) q^{45} + 4 q^{46} - 4 i q^{47} - i q^{48} - 9 q^{49} + ( - 3 i - 4) q^{50} - 2 q^{51} - 2 i q^{52} + 4 i q^{53} + q^{54} - 12 q^{56} - 8 i q^{57} - 4 i q^{58} + 8 q^{59} + ( - i + 2) q^{60} - 8 i q^{62} - 4 i q^{63} - 7 q^{64} + (2 i - 4) q^{65} - 4 i q^{67} + 2 i q^{68} + 4 q^{69} + (8 i + 4) q^{70} - 12 q^{71} - 3 i q^{72} - 2 i q^{73} - 8 q^{74} + ( - 3 i - 4) q^{75} - 8 q^{76} + 2 i q^{78} - 8 q^{79} + (2 i + 1) q^{80} + q^{81} + 12 i q^{82} - 4 i q^{83} - 4 q^{84} + ( - 2 i + 4) q^{85} - 8 q^{86} - 4 i q^{87} - 6 q^{89} + (i - 2) q^{90} + 8 q^{91} - 4 i q^{92} - 8 i q^{93} + 4 q^{94} + (16 i + 8) q^{95} - 5 q^{96} - 8 i q^{97} - 9 i q^{98} +O(q^{100})$$ q + i * q^2 + i * q^3 + q^4 + (-2*i - 1) * q^5 - q^6 + 4*i * q^7 + 3*i * q^8 - q^9 + (-i + 2) * q^10 + i * q^12 - 2*i * q^13 - 4 * q^14 + (-i + 2) * q^15 - q^16 + 2*i * q^17 - i * q^18 - 8 * q^19 + (-2*i - 1) * q^20 - 4 * q^21 - 4*i * q^23 - 3 * q^24 + (4*i - 3) * q^25 + 2 * q^26 - i * q^27 + 4*i * q^28 - 4 * q^29 + (2*i + 1) * q^30 - 8 * q^31 + 5*i * q^32 - 2 * q^34 + (-4*i + 8) * q^35 - q^36 + 8*i * q^37 - 8*i * q^38 + 2 * q^39 + (-3*i + 6) * q^40 + 12 * q^41 - 4*i * q^42 + 8*i * q^43 + (2*i + 1) * q^45 + 4 * q^46 - 4*i * q^47 - i * q^48 - 9 * q^49 + (-3*i - 4) * q^50 - 2 * q^51 - 2*i * q^52 + 4*i * q^53 + q^54 - 12 * q^56 - 8*i * q^57 - 4*i * q^58 + 8 * q^59 + (-i + 2) * q^60 - 8*i * q^62 - 4*i * q^63 - 7 * q^64 + (2*i - 4) * q^65 - 4*i * q^67 + 2*i * q^68 + 4 * q^69 + (8*i + 4) * q^70 - 12 * q^71 - 3*i * q^72 - 2*i * q^73 - 8 * q^74 + (-3*i - 4) * q^75 - 8 * q^76 + 2*i * q^78 - 8 * q^79 + (2*i + 1) * q^80 + q^81 + 12*i * q^82 - 4*i * q^83 - 4 * q^84 + (-2*i + 4) * q^85 - 8 * q^86 - 4*i * q^87 - 6 * q^89 + (i - 2) * q^90 + 8 * q^91 - 4*i * q^92 - 8*i * q^93 + 4 * q^94 + (16*i + 8) * q^95 - 5 * q^96 - 8*i * q^97 - 9*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{9} + 4 q^{10} - 8 q^{14} + 4 q^{15} - 2 q^{16} - 16 q^{19} - 2 q^{20} - 8 q^{21} - 6 q^{24} - 6 q^{25} + 4 q^{26} - 8 q^{29} + 2 q^{30} - 16 q^{31} - 4 q^{34} + 16 q^{35} - 2 q^{36} + 4 q^{39} + 12 q^{40} + 24 q^{41} + 2 q^{45} + 8 q^{46} - 18 q^{49} - 8 q^{50} - 4 q^{51} + 2 q^{54} - 24 q^{56} + 16 q^{59} + 4 q^{60} - 14 q^{64} - 8 q^{65} + 8 q^{69} + 8 q^{70} - 24 q^{71} - 16 q^{74} - 8 q^{75} - 16 q^{76} - 16 q^{79} + 2 q^{80} + 2 q^{81} - 8 q^{84} + 8 q^{85} - 16 q^{86} - 12 q^{89} - 4 q^{90} + 16 q^{91} + 8 q^{94} + 16 q^{95} - 10 q^{96}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^9 + 4 * q^10 - 8 * q^14 + 4 * q^15 - 2 * q^16 - 16 * q^19 - 2 * q^20 - 8 * q^21 - 6 * q^24 - 6 * q^25 + 4 * q^26 - 8 * q^29 + 2 * q^30 - 16 * q^31 - 4 * q^34 + 16 * q^35 - 2 * q^36 + 4 * q^39 + 12 * q^40 + 24 * q^41 + 2 * q^45 + 8 * q^46 - 18 * q^49 - 8 * q^50 - 4 * q^51 + 2 * q^54 - 24 * q^56 + 16 * q^59 + 4 * q^60 - 14 * q^64 - 8 * q^65 + 8 * q^69 + 8 * q^70 - 24 * q^71 - 16 * q^74 - 8 * q^75 - 16 * q^76 - 16 * q^79 + 2 * q^80 + 2 * q^81 - 8 * q^84 + 8 * q^85 - 16 * q^86 - 12 * q^89 - 4 * q^90 + 16 * q^91 + 8 * q^94 + 16 * q^95 - 10 * q^96

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 −1.00000 + 2.00000i −1.00000 4.00000i 3.00000i −1.00000 2.00000 + 1.00000i
364.2 1.00000i 1.00000i 1.00000 −1.00000 2.00000i −1.00000 4.00000i 3.00000i −1.00000 2.00000 1.00000i
 $$n$$: 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.a 2
5.b even 2 1 inner 1815.2.c.a 2
5.c odd 4 1 9075.2.a.f 1
5.c odd 4 1 9075.2.a.o 1
11.b odd 2 1 1815.2.c.b yes 2
55.d odd 2 1 1815.2.c.b yes 2
55.e even 4 1 9075.2.a.c 1
55.e even 4 1 9075.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.a 2 1.a even 1 1 trivial
1815.2.c.a 2 5.b even 2 1 inner
1815.2.c.b yes 2 11.b odd 2 1
1815.2.c.b yes 2 55.d odd 2 1
9075.2.a.c 1 55.e even 4 1
9075.2.a.f 1 5.c odd 4 1
9075.2.a.o 1 5.c odd 4 1
9075.2.a.r 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{19} + 8$$ T19 + 8

Hecke characteristic polynomials

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