# Properties

 Label 1014.2.b.a Level $1014$ Weight $2$ Character orbit 1014.b Analytic conductor $8.097$ Analytic rank $1$ 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] = [1014,2,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$8.09683076496$$ Analytic rank: $$1$$ 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 78) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} - q^{4} + i q^{5} - i q^{6} - 2 i q^{7} - i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 - q^3 - q^4 + i * q^5 - i * q^6 - 2*i * q^7 - i * q^8 + q^9 $$q + i q^{2} - q^{3} - q^{4} + i q^{5} - i q^{6} - 2 i q^{7} - i q^{8} + q^{9} - q^{10} + 2 i q^{11} + q^{12} + 2 q^{14} - i q^{15} + q^{16} - 5 q^{17} + i q^{18} + 2 i q^{19} - i q^{20} + 2 i q^{21} - 2 q^{22} - 6 q^{23} + i q^{24} + 4 q^{25} - q^{27} + 2 i q^{28} - 9 q^{29} + q^{30} + 4 i q^{31} + i q^{32} - 2 i q^{33} - 5 i q^{34} + 2 q^{35} - q^{36} - 11 i q^{37} - 2 q^{38} + q^{40} - 5 i q^{41} - 2 q^{42} - 10 q^{43} - 2 i q^{44} + i q^{45} - 6 i q^{46} + 2 i q^{47} - q^{48} + 3 q^{49} + 4 i q^{50} + 5 q^{51} - q^{53} - i q^{54} - 2 q^{55} - 2 q^{56} - 2 i q^{57} - 9 i q^{58} - 8 i q^{59} + i q^{60} - 11 q^{61} - 4 q^{62} - 2 i q^{63} - q^{64} + 2 q^{66} - 2 i q^{67} + 5 q^{68} + 6 q^{69} + 2 i q^{70} + 14 i q^{71} - i q^{72} - 13 i q^{73} + 11 q^{74} - 4 q^{75} - 2 i q^{76} + 4 q^{77} - 4 q^{79} + i q^{80} + q^{81} + 5 q^{82} - 6 i q^{83} - 2 i q^{84} - 5 i q^{85} - 10 i q^{86} + 9 q^{87} + 2 q^{88} + 2 i q^{89} - q^{90} + 6 q^{92} - 4 i q^{93} - 2 q^{94} - 2 q^{95} - i q^{96} + 2 i q^{97} + 3 i q^{98} + 2 i q^{99} +O(q^{100})$$ q + i * q^2 - q^3 - q^4 + i * q^5 - i * q^6 - 2*i * q^7 - i * q^8 + q^9 - q^10 + 2*i * q^11 + q^12 + 2 * q^14 - i * q^15 + q^16 - 5 * q^17 + i * q^18 + 2*i * q^19 - i * q^20 + 2*i * q^21 - 2 * q^22 - 6 * q^23 + i * q^24 + 4 * q^25 - q^27 + 2*i * q^28 - 9 * q^29 + q^30 + 4*i * q^31 + i * q^32 - 2*i * q^33 - 5*i * q^34 + 2 * q^35 - q^36 - 11*i * q^37 - 2 * q^38 + q^40 - 5*i * q^41 - 2 * q^42 - 10 * q^43 - 2*i * q^44 + i * q^45 - 6*i * q^46 + 2*i * q^47 - q^48 + 3 * q^49 + 4*i * q^50 + 5 * q^51 - q^53 - i * q^54 - 2 * q^55 - 2 * q^56 - 2*i * q^57 - 9*i * q^58 - 8*i * q^59 + i * q^60 - 11 * q^61 - 4 * q^62 - 2*i * q^63 - q^64 + 2 * q^66 - 2*i * q^67 + 5 * q^68 + 6 * q^69 + 2*i * q^70 + 14*i * q^71 - i * q^72 - 13*i * q^73 + 11 * q^74 - 4 * q^75 - 2*i * q^76 + 4 * q^77 - 4 * q^79 + i * q^80 + q^81 + 5 * q^82 - 6*i * q^83 - 2*i * q^84 - 5*i * q^85 - 10*i * q^86 + 9 * q^87 + 2 * q^88 + 2*i * q^89 - q^90 + 6 * q^92 - 4*i * q^93 - 2 * q^94 - 2 * q^95 - i * q^96 + 2*i * q^97 + 3*i * q^98 + 2*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 4 q^{14} + 2 q^{16} - 10 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 18 q^{29} + 2 q^{30} + 4 q^{35} - 2 q^{36} - 4 q^{38} + 2 q^{40} - 4 q^{42} - 20 q^{43} - 2 q^{48} + 6 q^{49} + 10 q^{51} - 2 q^{53} - 4 q^{55} - 4 q^{56} - 22 q^{61} - 8 q^{62} - 2 q^{64} + 4 q^{66} + 10 q^{68} + 12 q^{69} + 22 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{82} + 18 q^{87} + 4 q^{88} - 2 q^{90} + 12 q^{92} - 4 q^{94} - 4 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 4 * q^14 + 2 * q^16 - 10 * q^17 - 4 * q^22 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 18 * q^29 + 2 * q^30 + 4 * q^35 - 2 * q^36 - 4 * q^38 + 2 * q^40 - 4 * q^42 - 20 * q^43 - 2 * q^48 + 6 * q^49 + 10 * q^51 - 2 * q^53 - 4 * q^55 - 4 * q^56 - 22 * q^61 - 8 * q^62 - 2 * q^64 + 4 * q^66 + 10 * q^68 + 12 * q^69 + 22 * q^74 - 8 * q^75 + 8 * q^77 - 8 * q^79 + 2 * q^81 + 10 * q^82 + 18 * q^87 + 4 * q^88 - 2 * q^90 + 12 * q^92 - 4 * q^94 - 4 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
337.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000i 1.00000i 1.00000 −1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.b.a 2
3.b odd 2 1 3042.2.b.d 2
13.b even 2 1 inner 1014.2.b.a 2
13.c even 3 2 1014.2.i.e 4
13.d odd 4 1 1014.2.a.a 1
13.d odd 4 1 1014.2.a.e 1
13.e even 6 2 1014.2.i.e 4
13.f odd 12 2 78.2.e.b 2
13.f odd 12 2 1014.2.e.d 2
39.d odd 2 1 3042.2.b.d 2
39.f even 4 1 3042.2.a.d 1
39.f even 4 1 3042.2.a.m 1
39.k even 12 2 234.2.h.b 2
52.f even 4 1 8112.2.a.x 1
52.f even 4 1 8112.2.a.bb 1
52.l even 12 2 624.2.q.b 2
65.o even 12 2 1950.2.z.b 4
65.s odd 12 2 1950.2.i.b 2
65.t even 12 2 1950.2.z.b 4
156.v odd 12 2 1872.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 13.f odd 12 2
234.2.h.b 2 39.k even 12 2
624.2.q.b 2 52.l even 12 2
1014.2.a.a 1 13.d odd 4 1
1014.2.a.e 1 13.d odd 4 1
1014.2.b.a 2 1.a even 1 1 trivial
1014.2.b.a 2 13.b even 2 1 inner
1014.2.e.d 2 13.f odd 12 2
1014.2.i.e 4 13.c even 3 2
1014.2.i.e 4 13.e even 6 2
1872.2.t.i 2 156.v odd 12 2
1950.2.i.b 2 65.s odd 12 2
1950.2.z.b 4 65.o even 12 2
1950.2.z.b 4 65.t even 12 2
3042.2.a.d 1 39.f even 4 1
3042.2.a.m 1 39.f even 4 1
3042.2.b.d 2 3.b odd 2 1
3042.2.b.d 2 39.d odd 2 1
8112.2.a.x 1 52.f even 4 1
8112.2.a.bb 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T + 5)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T + 6)^{2}$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 121$$
$41$ $$T^{2} + 25$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} + 4$$
$53$ $$(T + 1)^{2}$$
$59$ $$T^{2} + 64$$
$61$ $$(T + 11)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 196$$
$73$ $$T^{2} + 169$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 4$$