# Properties

 Label 1210.2.b.a Level $1210$ Weight $2$ Character orbit 1210.b Analytic conductor $9.662$ 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] = [1210,2,Mod(969,1210)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1210, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1210.969");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1210 = 2 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1210.b (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: $$9.66189864457$$ 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: no (minimal twist has level 110) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} - q^{4} + (i - 2) q^{5} + q^{6} - 3 i q^{7} + i q^{8} + 2 q^{9} +O(q^{10})$$ q - i * q^2 + i * q^3 - q^4 + (i - 2) * q^5 + q^6 - 3*i * q^7 + i * q^8 + 2 * q^9 $$q - i q^{2} + i q^{3} - q^{4} + (i - 2) q^{5} + q^{6} - 3 i q^{7} + i q^{8} + 2 q^{9} + (2 i + 1) q^{10} - i q^{12} + 4 i q^{13} - 3 q^{14} + ( - 2 i - 1) q^{15} + q^{16} - 3 i q^{17} - 2 i q^{18} - 5 q^{19} + ( - i + 2) q^{20} + 3 q^{21} - 4 i q^{23} - q^{24} + ( - 4 i + 3) q^{25} + 4 q^{26} + 5 i q^{27} + 3 i q^{28} + 5 q^{29} + (i - 2) q^{30} + 7 q^{31} - i q^{32} - 3 q^{34} + (6 i + 3) q^{35} - 2 q^{36} - 7 i q^{37} + 5 i q^{38} - 4 q^{39} + ( - 2 i - 1) q^{40} + 8 q^{41} - 3 i q^{42} - 6 i q^{43} + (2 i - 4) q^{45} - 4 q^{46} + 8 i q^{47} + i q^{48} - 2 q^{49} + ( - 3 i - 4) q^{50} + 3 q^{51} - 4 i q^{52} - 9 i q^{53} + 5 q^{54} + 3 q^{56} - 5 i q^{57} - 5 i q^{58} + (2 i + 1) q^{60} + 13 q^{61} - 7 i q^{62} - 6 i q^{63} - q^{64} + ( - 8 i - 4) q^{65} - 12 i q^{67} + 3 i q^{68} + 4 q^{69} + ( - 3 i + 6) q^{70} - 3 q^{71} + 2 i q^{72} - 6 i q^{73} - 7 q^{74} + (3 i + 4) q^{75} + 5 q^{76} + 4 i q^{78} + (i - 2) q^{80} + q^{81} - 8 i q^{82} + 4 i q^{83} - 3 q^{84} + (6 i + 3) q^{85} - 6 q^{86} + 5 i q^{87} + 15 q^{89} + (4 i + 2) q^{90} + 12 q^{91} + 4 i q^{92} + 7 i q^{93} + 8 q^{94} + ( - 5 i + 10) q^{95} + q^{96} - 12 i q^{97} + 2 i q^{98} +O(q^{100})$$ q - i * q^2 + i * q^3 - q^4 + (i - 2) * q^5 + q^6 - 3*i * q^7 + i * q^8 + 2 * q^9 + (2*i + 1) * q^10 - i * q^12 + 4*i * q^13 - 3 * q^14 + (-2*i - 1) * q^15 + q^16 - 3*i * q^17 - 2*i * q^18 - 5 * q^19 + (-i + 2) * q^20 + 3 * q^21 - 4*i * q^23 - q^24 + (-4*i + 3) * q^25 + 4 * q^26 + 5*i * q^27 + 3*i * q^28 + 5 * q^29 + (i - 2) * q^30 + 7 * q^31 - i * q^32 - 3 * q^34 + (6*i + 3) * q^35 - 2 * q^36 - 7*i * q^37 + 5*i * q^38 - 4 * q^39 + (-2*i - 1) * q^40 + 8 * q^41 - 3*i * q^42 - 6*i * q^43 + (2*i - 4) * q^45 - 4 * q^46 + 8*i * q^47 + i * q^48 - 2 * q^49 + (-3*i - 4) * q^50 + 3 * q^51 - 4*i * q^52 - 9*i * q^53 + 5 * q^54 + 3 * q^56 - 5*i * q^57 - 5*i * q^58 + (2*i + 1) * q^60 + 13 * q^61 - 7*i * q^62 - 6*i * q^63 - q^64 + (-8*i - 4) * q^65 - 12*i * q^67 + 3*i * q^68 + 4 * q^69 + (-3*i + 6) * q^70 - 3 * q^71 + 2*i * q^72 - 6*i * q^73 - 7 * q^74 + (3*i + 4) * q^75 + 5 * q^76 + 4*i * q^78 + (i - 2) * q^80 + q^81 - 8*i * q^82 + 4*i * q^83 - 3 * q^84 + (6*i + 3) * q^85 - 6 * q^86 + 5*i * q^87 + 15 * q^89 + (4*i + 2) * q^90 + 12 * q^91 + 4*i * q^92 + 7*i * q^93 + 8 * q^94 + (-5*i + 10) * q^95 + q^96 - 12*i * q^97 + 2*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{9} + 2 q^{10} - 6 q^{14} - 2 q^{15} + 2 q^{16} - 10 q^{19} + 4 q^{20} + 6 q^{21} - 2 q^{24} + 6 q^{25} + 8 q^{26} + 10 q^{29} - 4 q^{30} + 14 q^{31} - 6 q^{34} + 6 q^{35} - 4 q^{36} - 8 q^{39} - 2 q^{40} + 16 q^{41} - 8 q^{45} - 8 q^{46} - 4 q^{49} - 8 q^{50} + 6 q^{51} + 10 q^{54} + 6 q^{56} + 2 q^{60} + 26 q^{61} - 2 q^{64} - 8 q^{65} + 8 q^{69} + 12 q^{70} - 6 q^{71} - 14 q^{74} + 8 q^{75} + 10 q^{76} - 4 q^{80} + 2 q^{81} - 6 q^{84} + 6 q^{85} - 12 q^{86} + 30 q^{89} + 4 q^{90} + 24 q^{91} + 16 q^{94} + 20 q^{95} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^9 + 2 * q^10 - 6 * q^14 - 2 * q^15 + 2 * q^16 - 10 * q^19 + 4 * q^20 + 6 * q^21 - 2 * q^24 + 6 * q^25 + 8 * q^26 + 10 * q^29 - 4 * q^30 + 14 * q^31 - 6 * q^34 + 6 * q^35 - 4 * q^36 - 8 * q^39 - 2 * q^40 + 16 * q^41 - 8 * q^45 - 8 * q^46 - 4 * q^49 - 8 * q^50 + 6 * q^51 + 10 * q^54 + 6 * q^56 + 2 * q^60 + 26 * q^61 - 2 * q^64 - 8 * q^65 + 8 * q^69 + 12 * q^70 - 6 * q^71 - 14 * q^74 + 8 * q^75 + 10 * q^76 - 4 * q^80 + 2 * q^81 - 6 * q^84 + 6 * q^85 - 12 * q^86 + 30 * q^89 + 4 * q^90 + 24 * q^91 + 16 * q^94 + 20 * q^95 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1091$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
969.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 3.00000i 1.00000i 2.00000 1.00000 + 2.00000i
969.2 1.00000i 1.00000i −1.00000 −2.00000 1.00000i 1.00000 3.00000i 1.00000i 2.00000 1.00000 2.00000i
 $$n$$: 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 1210.2.b.a 2
5.b even 2 1 inner 1210.2.b.a 2
5.c odd 4 1 6050.2.a.f 1
5.c odd 4 1 6050.2.a.bk 1
11.b odd 2 1 110.2.b.a 2
33.d even 2 1 990.2.c.d 2
44.c even 2 1 880.2.b.a 2
55.d odd 2 1 110.2.b.a 2
55.e even 4 1 550.2.a.e 1
55.e even 4 1 550.2.a.j 1
165.d even 2 1 990.2.c.d 2
165.l odd 4 1 4950.2.a.q 1
165.l odd 4 1 4950.2.a.ba 1
220.g even 2 1 880.2.b.a 2
220.i odd 4 1 4400.2.a.k 1
220.i odd 4 1 4400.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 11.b odd 2 1
110.2.b.a 2 55.d odd 2 1
550.2.a.e 1 55.e even 4 1
550.2.a.j 1 55.e even 4 1
880.2.b.a 2 44.c even 2 1
880.2.b.a 2 220.g even 2 1
990.2.c.d 2 33.d even 2 1
990.2.c.d 2 165.d even 2 1
1210.2.b.a 2 1.a even 1 1 trivial
1210.2.b.a 2 5.b even 2 1 inner
4400.2.a.k 1 220.i odd 4 1
4400.2.a.s 1 220.i odd 4 1
4950.2.a.q 1 165.l odd 4 1
4950.2.a.ba 1 165.l odd 4 1
6050.2.a.f 1 5.c odd 4 1
6050.2.a.bk 1 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1210, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{19} + 5$$ T19 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 9$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$T^{2}$$
$61$ $$(T - 13)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 144$$