# Properties

 Label 1014.4.b.e Level $1014$ Weight $4$ Character orbit 1014.b Analytic conductor $59.828$ 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] = [1014,4,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, 4, names="a")

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$59.8279367458$$ 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 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 10 \beta q^{5} + 3 \beta q^{6} - 16 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 - 3 * q^3 - 4 * q^4 + 10*b * q^5 + 3*b * q^6 - 16*b * q^7 + 4*b * q^8 + 9 * q^9 $$q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 10 \beta q^{5} + 3 \beta q^{6} - 16 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} + 40 q^{10} + 25 \beta q^{11} + 12 q^{12} - 64 q^{14} - 30 \beta q^{15} + 16 q^{16} + 30 q^{17} - 9 \beta q^{18} + 60 \beta q^{19} - 40 \beta q^{20} + 48 \beta q^{21} + 100 q^{22} + 20 q^{23} - 12 \beta q^{24} - 275 q^{25} - 27 q^{27} + 64 \beta q^{28} + 82 q^{29} - 120 q^{30} + 22 \beta q^{31} - 16 \beta q^{32} - 75 \beta q^{33} - 30 \beta q^{34} + 640 q^{35} - 36 q^{36} - 153 \beta q^{37} + 240 q^{38} - 160 q^{40} - 54 \beta q^{41} + 192 q^{42} + 356 q^{43} - 100 \beta q^{44} + 90 \beta q^{45} - 20 \beta q^{46} - 89 \beta q^{47} - 48 q^{48} - 681 q^{49} + 275 \beta q^{50} - 90 q^{51} + 198 q^{53} + 27 \beta q^{54} - 1000 q^{55} + 256 q^{56} - 180 \beta q^{57} - 82 \beta q^{58} + 47 \beta q^{59} + 120 \beta q^{60} - 62 q^{61} + 88 q^{62} - 144 \beta q^{63} - 64 q^{64} - 300 q^{66} + 70 \beta q^{67} - 120 q^{68} - 60 q^{69} - 640 \beta q^{70} + 389 \beta q^{71} + 36 \beta q^{72} + 31 \beta q^{73} - 612 q^{74} + 825 q^{75} - 240 \beta q^{76} + 1600 q^{77} - 1096 q^{79} + 160 \beta q^{80} + 81 q^{81} - 216 q^{82} + 231 \beta q^{83} - 192 \beta q^{84} + 300 \beta q^{85} - 356 \beta q^{86} - 246 q^{87} - 400 q^{88} + 612 \beta q^{89} + 360 q^{90} - 80 q^{92} - 66 \beta q^{93} - 356 q^{94} - 2400 q^{95} + 48 \beta q^{96} - 307 \beta q^{97} + 681 \beta q^{98} + 225 \beta q^{99} +O(q^{100})$$ q - b * q^2 - 3 * q^3 - 4 * q^4 + 10*b * q^5 + 3*b * q^6 - 16*b * q^7 + 4*b * q^8 + 9 * q^9 + 40 * q^10 + 25*b * q^11 + 12 * q^12 - 64 * q^14 - 30*b * q^15 + 16 * q^16 + 30 * q^17 - 9*b * q^18 + 60*b * q^19 - 40*b * q^20 + 48*b * q^21 + 100 * q^22 + 20 * q^23 - 12*b * q^24 - 275 * q^25 - 27 * q^27 + 64*b * q^28 + 82 * q^29 - 120 * q^30 + 22*b * q^31 - 16*b * q^32 - 75*b * q^33 - 30*b * q^34 + 640 * q^35 - 36 * q^36 - 153*b * q^37 + 240 * q^38 - 160 * q^40 - 54*b * q^41 + 192 * q^42 + 356 * q^43 - 100*b * q^44 + 90*b * q^45 - 20*b * q^46 - 89*b * q^47 - 48 * q^48 - 681 * q^49 + 275*b * q^50 - 90 * q^51 + 198 * q^53 + 27*b * q^54 - 1000 * q^55 + 256 * q^56 - 180*b * q^57 - 82*b * q^58 + 47*b * q^59 + 120*b * q^60 - 62 * q^61 + 88 * q^62 - 144*b * q^63 - 64 * q^64 - 300 * q^66 + 70*b * q^67 - 120 * q^68 - 60 * q^69 - 640*b * q^70 + 389*b * q^71 + 36*b * q^72 + 31*b * q^73 - 612 * q^74 + 825 * q^75 - 240*b * q^76 + 1600 * q^77 - 1096 * q^79 + 160*b * q^80 + 81 * q^81 - 216 * q^82 + 231*b * q^83 - 192*b * q^84 + 300*b * q^85 - 356*b * q^86 - 246 * q^87 - 400 * q^88 + 612*b * q^89 + 360 * q^90 - 80 * q^92 - 66*b * q^93 - 356 * q^94 - 2400 * q^95 + 48*b * q^96 - 307*b * q^97 + 681*b * q^98 + 225*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 80 q^{10} + 24 q^{12} - 128 q^{14} + 32 q^{16} + 60 q^{17} + 200 q^{22} + 40 q^{23} - 550 q^{25} - 54 q^{27} + 164 q^{29} - 240 q^{30} + 1280 q^{35} - 72 q^{36} + 480 q^{38} - 320 q^{40} + 384 q^{42} + 712 q^{43} - 96 q^{48} - 1362 q^{49} - 180 q^{51} + 396 q^{53} - 2000 q^{55} + 512 q^{56} - 124 q^{61} + 176 q^{62} - 128 q^{64} - 600 q^{66} - 240 q^{68} - 120 q^{69} - 1224 q^{74} + 1650 q^{75} + 3200 q^{77} - 2192 q^{79} + 162 q^{81} - 432 q^{82} - 492 q^{87} - 800 q^{88} + 720 q^{90} - 160 q^{92} - 712 q^{94} - 4800 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 + 80 * q^10 + 24 * q^12 - 128 * q^14 + 32 * q^16 + 60 * q^17 + 200 * q^22 + 40 * q^23 - 550 * q^25 - 54 * q^27 + 164 * q^29 - 240 * q^30 + 1280 * q^35 - 72 * q^36 + 480 * q^38 - 320 * q^40 + 384 * q^42 + 712 * q^43 - 96 * q^48 - 1362 * q^49 - 180 * q^51 + 396 * q^53 - 2000 * q^55 + 512 * q^56 - 124 * q^61 + 176 * q^62 - 128 * q^64 - 600 * q^66 - 240 * q^68 - 120 * q^69 - 1224 * q^74 + 1650 * q^75 + 3200 * q^77 - 2192 * q^79 + 162 * q^81 - 432 * q^82 - 492 * q^87 - 800 * q^88 + 720 * q^90 - 160 * q^92 - 712 * q^94 - 4800 * 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
2.00000i −3.00000 −4.00000 20.0000i 6.00000i 32.0000i 8.00000i 9.00000 40.0000
337.2 2.00000i −3.00000 −4.00000 20.0000i 6.00000i 32.0000i 8.00000i 9.00000 40.0000
 $$n$$: 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 1014.4.b.e 2
13.b even 2 1 inner 1014.4.b.e 2
13.d odd 4 1 78.4.a.d 1
13.d odd 4 1 1014.4.a.d 1
39.f even 4 1 234.4.a.f 1
52.f even 4 1 624.4.a.e 1
65.g odd 4 1 1950.4.a.h 1
104.j odd 4 1 2496.4.a.r 1
104.m even 4 1 2496.4.a.i 1
156.l odd 4 1 1872.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.d 1 13.d odd 4 1
234.4.a.f 1 39.f even 4 1
624.4.a.e 1 52.f even 4 1
1014.4.a.d 1 13.d odd 4 1
1014.4.b.e 2 1.a even 1 1 trivial
1014.4.b.e 2 13.b even 2 1 inner
1872.4.a.r 1 156.l odd 4 1
1950.4.a.h 1 65.g odd 4 1
2496.4.a.i 1 104.m even 4 1
2496.4.a.r 1 104.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{2} + 400$$ T5^2 + 400 $$T_{7}^{2} + 1024$$ T7^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 400$$
$7$ $$T^{2} + 1024$$
$11$ $$T^{2} + 2500$$
$13$ $$T^{2}$$
$17$ $$(T - 30)^{2}$$
$19$ $$T^{2} + 14400$$
$23$ $$(T - 20)^{2}$$
$29$ $$(T - 82)^{2}$$
$31$ $$T^{2} + 1936$$
$37$ $$T^{2} + 93636$$
$41$ $$T^{2} + 11664$$
$43$ $$(T - 356)^{2}$$
$47$ $$T^{2} + 31684$$
$53$ $$(T - 198)^{2}$$
$59$ $$T^{2} + 8836$$
$61$ $$(T + 62)^{2}$$
$67$ $$T^{2} + 19600$$
$71$ $$T^{2} + 605284$$
$73$ $$T^{2} + 3844$$
$79$ $$(T + 1096)^{2}$$
$83$ $$T^{2} + 213444$$
$89$ $$T^{2} + 1498176$$
$97$ $$T^{2} + 376996$$