# Properties

 Label 1369.2.b.c Level $1369$ Weight $2$ Character orbit 1369.b Analytic conductor $10.932$ 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] = [1369,2,Mod(1368,1369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1369, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1369.1368");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1369 = 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1369.b (of order $$2$$, degree $$1$$, 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: $$10.9315200367$$ 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: $$2$$ Twist minimal: no (minimal twist has level 37) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 3 q^{3} - 2 q^{4} - \beta q^{5} + 3 \beta q^{6} - q^{7} + 6 q^{9} +O(q^{10})$$ q + b * q^2 + 3 * q^3 - 2 * q^4 - b * q^5 + 3*b * q^6 - q^7 + 6 * q^9 $$q + \beta q^{2} + 3 q^{3} - 2 q^{4} - \beta q^{5} + 3 \beta q^{6} - q^{7} + 6 q^{9} + 4 q^{10} + 5 q^{11} - 6 q^{12} - \beta q^{13} - \beta q^{14} - 3 \beta q^{15} - 4 q^{16} + 6 \beta q^{18} + 2 \beta q^{20} - 3 q^{21} + 5 \beta q^{22} + \beta q^{23} + q^{25} + 4 q^{26} + 9 q^{27} + 2 q^{28} - 3 \beta q^{29} + 12 q^{30} + 2 \beta q^{31} - 4 \beta q^{32} + 15 q^{33} + \beta q^{35} - 12 q^{36} - 3 \beta q^{39} + 9 q^{41} - 3 \beta q^{42} + \beta q^{43} - 10 q^{44} - 6 \beta q^{45} - 4 q^{46} - 9 q^{47} - 12 q^{48} - 6 q^{49} + \beta q^{50} + 2 \beta q^{52} + q^{53} + 9 \beta q^{54} - 5 \beta q^{55} + 12 q^{58} + 4 \beta q^{59} + 6 \beta q^{60} + 4 \beta q^{61} - 8 q^{62} - 6 q^{63} + 8 q^{64} - 4 q^{65} + 15 \beta q^{66} - 8 q^{67} + 3 \beta q^{69} - 4 q^{70} + 9 q^{71} + q^{73} + 3 q^{75} - 5 q^{77} + 12 q^{78} + 2 \beta q^{79} + 4 \beta q^{80} + 9 q^{81} + 9 \beta q^{82} - 15 q^{83} + 6 q^{84} - 4 q^{86} - 9 \beta q^{87} - 2 \beta q^{89} + 24 q^{90} + \beta q^{91} - 2 \beta q^{92} + 6 \beta q^{93} - 9 \beta q^{94} - 12 \beta q^{96} + 2 \beta q^{97} - 6 \beta q^{98} + 30 q^{99} +O(q^{100})$$ q + b * q^2 + 3 * q^3 - 2 * q^4 - b * q^5 + 3*b * q^6 - q^7 + 6 * q^9 + 4 * q^10 + 5 * q^11 - 6 * q^12 - b * q^13 - b * q^14 - 3*b * q^15 - 4 * q^16 + 6*b * q^18 + 2*b * q^20 - 3 * q^21 + 5*b * q^22 + b * q^23 + q^25 + 4 * q^26 + 9 * q^27 + 2 * q^28 - 3*b * q^29 + 12 * q^30 + 2*b * q^31 - 4*b * q^32 + 15 * q^33 + b * q^35 - 12 * q^36 - 3*b * q^39 + 9 * q^41 - 3*b * q^42 + b * q^43 - 10 * q^44 - 6*b * q^45 - 4 * q^46 - 9 * q^47 - 12 * q^48 - 6 * q^49 + b * q^50 + 2*b * q^52 + q^53 + 9*b * q^54 - 5*b * q^55 + 12 * q^58 + 4*b * q^59 + 6*b * q^60 + 4*b * q^61 - 8 * q^62 - 6 * q^63 + 8 * q^64 - 4 * q^65 + 15*b * q^66 - 8 * q^67 + 3*b * q^69 - 4 * q^70 + 9 * q^71 + q^73 + 3 * q^75 - 5 * q^77 + 12 * q^78 + 2*b * q^79 + 4*b * q^80 + 9 * q^81 + 9*b * q^82 - 15 * q^83 + 6 * q^84 - 4 * q^86 - 9*b * q^87 - 2*b * q^89 + 24 * q^90 + b * q^91 - 2*b * q^92 + 6*b * q^93 - 9*b * q^94 - 12*b * q^96 + 2*b * q^97 - 6*b * q^98 + 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9 $$2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9} + 8 q^{10} + 10 q^{11} - 12 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 8 q^{26} + 18 q^{27} + 4 q^{28} + 24 q^{30} + 30 q^{33} - 24 q^{36} + 18 q^{41} - 20 q^{44} - 8 q^{46} - 18 q^{47} - 24 q^{48} - 12 q^{49} + 2 q^{53} + 24 q^{58} - 16 q^{62} - 12 q^{63} + 16 q^{64} - 8 q^{65} - 16 q^{67} - 8 q^{70} + 18 q^{71} + 2 q^{73} + 6 q^{75} - 10 q^{77} + 24 q^{78} + 18 q^{81} - 30 q^{83} + 12 q^{84} - 8 q^{86} + 48 q^{90} + 60 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 4 * q^4 - 2 * q^7 + 12 * q^9 + 8 * q^10 + 10 * q^11 - 12 * q^12 - 8 * q^16 - 6 * q^21 + 2 * q^25 + 8 * q^26 + 18 * q^27 + 4 * q^28 + 24 * q^30 + 30 * q^33 - 24 * q^36 + 18 * q^41 - 20 * q^44 - 8 * q^46 - 18 * q^47 - 24 * q^48 - 12 * q^49 + 2 * q^53 + 24 * q^58 - 16 * q^62 - 12 * q^63 + 16 * q^64 - 8 * q^65 - 16 * q^67 - 8 * q^70 + 18 * q^71 + 2 * q^73 + 6 * q^75 - 10 * q^77 + 24 * q^78 + 18 * q^81 - 30 * q^83 + 12 * q^84 - 8 * q^86 + 48 * q^90 + 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1368.1
 − 1.00000i 1.00000i
2.00000i 3.00000 −2.00000 2.00000i 6.00000i −1.00000 0 6.00000 4.00000
1368.2 2.00000i 3.00000 −2.00000 2.00000i 6.00000i −1.00000 0 6.00000 4.00000
 $$n$$: 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1369.2.b.c 2
37.b even 2 1 inner 1369.2.b.c 2
37.d odd 4 1 37.2.a.a 1
37.d odd 4 1 1369.2.a.e 1
111.g even 4 1 333.2.a.d 1
148.g even 4 1 592.2.a.e 1
185.f even 4 1 925.2.b.b 2
185.j odd 4 1 925.2.a.e 1
185.k even 4 1 925.2.b.b 2
259.j even 4 1 1813.2.a.a 1
296.j even 4 1 2368.2.a.b 1
296.m odd 4 1 2368.2.a.q 1
407.f even 4 1 4477.2.a.b 1
444.j odd 4 1 5328.2.a.r 1
481.j odd 4 1 6253.2.a.c 1
555.m even 4 1 8325.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.a.a 1 37.d odd 4 1
333.2.a.d 1 111.g even 4 1
592.2.a.e 1 148.g even 4 1
925.2.a.e 1 185.j odd 4 1
925.2.b.b 2 185.f even 4 1
925.2.b.b 2 185.k even 4 1
1369.2.a.e 1 37.d odd 4 1
1369.2.b.c 2 1.a even 1 1 trivial
1369.2.b.c 2 37.b even 2 1 inner
1813.2.a.a 1 259.j even 4 1
2368.2.a.b 1 296.j even 4 1
2368.2.a.q 1 296.m odd 4 1
4477.2.a.b 1 407.f even 4 1
5328.2.a.r 1 444.j odd 4 1
6253.2.a.c 1 481.j odd 4 1
8325.2.a.e 1 555.m even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1369, [\chi])$$:

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

## Hecke characteristic polynomials

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