# Properties

 Label 1014.4.b.d Level $1014$ Weight $4$ Character orbit 1014.b Analytic conductor $59.828$ Analytic rank $0$ 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] = [1014,4,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 6) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 3 \beta q^{5} + 3 \beta q^{6} + 8 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} +O(q^{10})$$ q - b * q^2 - 3 * q^3 - 4 * q^4 + 3*b * q^5 + 3*b * q^6 + 8*b * q^7 + 4*b * q^8 + 9 * q^9 $$q - \beta q^{2} - 3 q^{3} - 4 q^{4} + 3 \beta q^{5} + 3 \beta q^{6} + 8 \beta q^{7} + 4 \beta q^{8} + 9 q^{9} + 12 q^{10} - 6 \beta q^{11} + 12 q^{12} + 32 q^{14} - 9 \beta q^{15} + 16 q^{16} + 126 q^{17} - 9 \beta q^{18} + 10 \beta q^{19} - 12 \beta q^{20} - 24 \beta q^{21} - 24 q^{22} - 168 q^{23} - 12 \beta q^{24} + 89 q^{25} - 27 q^{27} - 32 \beta q^{28} + 30 q^{29} - 36 q^{30} - 44 \beta q^{31} - 16 \beta q^{32} + 18 \beta q^{33} - 126 \beta q^{34} - 96 q^{35} - 36 q^{36} - 127 \beta q^{37} + 40 q^{38} - 48 q^{40} + 21 \beta q^{41} - 96 q^{42} + 52 q^{43} + 24 \beta q^{44} + 27 \beta q^{45} + 168 \beta q^{46} + 48 \beta q^{47} - 48 q^{48} + 87 q^{49} - 89 \beta q^{50} - 378 q^{51} + 198 q^{53} + 27 \beta q^{54} + 72 q^{55} - 128 q^{56} - 30 \beta q^{57} - 30 \beta q^{58} + 330 \beta q^{59} + 36 \beta q^{60} - 538 q^{61} - 176 q^{62} + 72 \beta q^{63} - 64 q^{64} + 72 q^{66} + 442 \beta q^{67} - 504 q^{68} + 504 q^{69} + 96 \beta q^{70} + 396 \beta q^{71} + 36 \beta q^{72} - 109 \beta q^{73} - 508 q^{74} - 267 q^{75} - 40 \beta q^{76} + 192 q^{77} - 520 q^{79} + 48 \beta q^{80} + 81 q^{81} + 84 q^{82} - 246 \beta q^{83} + 96 \beta q^{84} + 378 \beta q^{85} - 52 \beta q^{86} - 90 q^{87} + 96 q^{88} - 405 \beta q^{89} + 108 q^{90} + 672 q^{92} + 132 \beta q^{93} + 192 q^{94} - 120 q^{95} + 48 \beta q^{96} + 577 \beta q^{97} - 87 \beta q^{98} - 54 \beta q^{99} +O(q^{100})$$ q - b * q^2 - 3 * q^3 - 4 * q^4 + 3*b * q^5 + 3*b * q^6 + 8*b * q^7 + 4*b * q^8 + 9 * q^9 + 12 * q^10 - 6*b * q^11 + 12 * q^12 + 32 * q^14 - 9*b * q^15 + 16 * q^16 + 126 * q^17 - 9*b * q^18 + 10*b * q^19 - 12*b * q^20 - 24*b * q^21 - 24 * q^22 - 168 * q^23 - 12*b * q^24 + 89 * q^25 - 27 * q^27 - 32*b * q^28 + 30 * q^29 - 36 * q^30 - 44*b * q^31 - 16*b * q^32 + 18*b * q^33 - 126*b * q^34 - 96 * q^35 - 36 * q^36 - 127*b * q^37 + 40 * q^38 - 48 * q^40 + 21*b * q^41 - 96 * q^42 + 52 * q^43 + 24*b * q^44 + 27*b * q^45 + 168*b * q^46 + 48*b * q^47 - 48 * q^48 + 87 * q^49 - 89*b * q^50 - 378 * q^51 + 198 * q^53 + 27*b * q^54 + 72 * q^55 - 128 * q^56 - 30*b * q^57 - 30*b * q^58 + 330*b * q^59 + 36*b * q^60 - 538 * q^61 - 176 * q^62 + 72*b * q^63 - 64 * q^64 + 72 * q^66 + 442*b * q^67 - 504 * q^68 + 504 * q^69 + 96*b * q^70 + 396*b * q^71 + 36*b * q^72 - 109*b * q^73 - 508 * q^74 - 267 * q^75 - 40*b * q^76 + 192 * q^77 - 520 * q^79 + 48*b * q^80 + 81 * q^81 + 84 * q^82 - 246*b * q^83 + 96*b * q^84 + 378*b * q^85 - 52*b * q^86 - 90 * q^87 + 96 * q^88 - 405*b * q^89 + 108 * q^90 + 672 * q^92 + 132*b * q^93 + 192 * q^94 - 120 * q^95 + 48*b * q^96 + 577*b * q^97 - 87*b * q^98 - 54*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 $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 24 q^{10} + 24 q^{12} + 64 q^{14} + 32 q^{16} + 252 q^{17} - 48 q^{22} - 336 q^{23} + 178 q^{25} - 54 q^{27} + 60 q^{29} - 72 q^{30} - 192 q^{35} - 72 q^{36} + 80 q^{38} - 96 q^{40} - 192 q^{42} + 104 q^{43} - 96 q^{48} + 174 q^{49} - 756 q^{51} + 396 q^{53} + 144 q^{55} - 256 q^{56} - 1076 q^{61} - 352 q^{62} - 128 q^{64} + 144 q^{66} - 1008 q^{68} + 1008 q^{69} - 1016 q^{74} - 534 q^{75} + 384 q^{77} - 1040 q^{79} + 162 q^{81} + 168 q^{82} - 180 q^{87} + 192 q^{88} + 216 q^{90} + 1344 q^{92} + 384 q^{94} - 240 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 + 24 * q^10 + 24 * q^12 + 64 * q^14 + 32 * q^16 + 252 * q^17 - 48 * q^22 - 336 * q^23 + 178 * q^25 - 54 * q^27 + 60 * q^29 - 72 * q^30 - 192 * q^35 - 72 * q^36 + 80 * q^38 - 96 * q^40 - 192 * q^42 + 104 * q^43 - 96 * q^48 + 174 * q^49 - 756 * q^51 + 396 * q^53 + 144 * q^55 - 256 * q^56 - 1076 * q^61 - 352 * q^62 - 128 * q^64 + 144 * q^66 - 1008 * q^68 + 1008 * q^69 - 1016 * q^74 - 534 * q^75 + 384 * q^77 - 1040 * q^79 + 162 * q^81 + 168 * q^82 - 180 * q^87 + 192 * q^88 + 216 * q^90 + 1344 * q^92 + 384 * q^94 - 240 * 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 6.00000i 6.00000i 16.0000i 8.00000i 9.00000 12.0000
337.2 2.00000i −3.00000 −4.00000 6.00000i 6.00000i 16.0000i 8.00000i 9.00000 12.0000
 $$n$$: 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.d 2
13.b even 2 1 inner 1014.4.b.d 2
13.d odd 4 1 6.4.a.a 1
13.d odd 4 1 1014.4.a.g 1
39.f even 4 1 18.4.a.a 1
52.f even 4 1 48.4.a.c 1
65.f even 4 1 150.4.c.d 2
65.g odd 4 1 150.4.a.i 1
65.k even 4 1 150.4.c.d 2
91.i even 4 1 294.4.a.e 1
91.z odd 12 2 294.4.e.h 2
91.bb even 12 2 294.4.e.g 2
104.j odd 4 1 192.4.a.i 1
104.m even 4 1 192.4.a.c 1
117.y odd 12 2 162.4.c.f 2
117.z even 12 2 162.4.c.c 2
143.g even 4 1 726.4.a.f 1
156.l odd 4 1 144.4.a.c 1
195.j odd 4 1 450.4.c.e 2
195.n even 4 1 450.4.a.h 1
195.u odd 4 1 450.4.c.e 2
208.l even 4 1 768.4.d.c 2
208.m odd 4 1 768.4.d.n 2
208.r odd 4 1 768.4.d.n 2
208.s even 4 1 768.4.d.c 2
221.g odd 4 1 1734.4.a.d 1
247.i even 4 1 2166.4.a.i 1
260.l odd 4 1 1200.4.f.j 2
260.s odd 4 1 1200.4.f.j 2
260.u even 4 1 1200.4.a.b 1
273.o odd 4 1 882.4.a.n 1
273.cb odd 12 2 882.4.g.f 2
273.cd even 12 2 882.4.g.i 2
312.w odd 4 1 576.4.a.r 1
312.y even 4 1 576.4.a.q 1
364.p odd 4 1 2352.4.a.e 1
429.l odd 4 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 13.d odd 4 1
18.4.a.a 1 39.f even 4 1
48.4.a.c 1 52.f even 4 1
144.4.a.c 1 156.l odd 4 1
150.4.a.i 1 65.g odd 4 1
150.4.c.d 2 65.f even 4 1
150.4.c.d 2 65.k even 4 1
162.4.c.c 2 117.z even 12 2
162.4.c.f 2 117.y odd 12 2
192.4.a.c 1 104.m even 4 1
192.4.a.i 1 104.j odd 4 1
294.4.a.e 1 91.i even 4 1
294.4.e.g 2 91.bb even 12 2
294.4.e.h 2 91.z odd 12 2
450.4.a.h 1 195.n even 4 1
450.4.c.e 2 195.j odd 4 1
450.4.c.e 2 195.u odd 4 1
576.4.a.q 1 312.y even 4 1
576.4.a.r 1 312.w odd 4 1
726.4.a.f 1 143.g even 4 1
768.4.d.c 2 208.l even 4 1
768.4.d.c 2 208.s even 4 1
768.4.d.n 2 208.m odd 4 1
768.4.d.n 2 208.r odd 4 1
882.4.a.n 1 273.o odd 4 1
882.4.g.f 2 273.cb odd 12 2
882.4.g.i 2 273.cd even 12 2
1014.4.a.g 1 13.d odd 4 1
1014.4.b.d 2 1.a even 1 1 trivial
1014.4.b.d 2 13.b even 2 1 inner
1200.4.a.b 1 260.u even 4 1
1200.4.f.j 2 260.l odd 4 1
1200.4.f.j 2 260.s odd 4 1
1734.4.a.d 1 221.g odd 4 1
2166.4.a.i 1 247.i even 4 1
2178.4.a.e 1 429.l odd 4 1
2352.4.a.e 1 364.p odd 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 36$$ T5^2 + 36 $$T_{7}^{2} + 256$$ T7^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 36$$
$7$ $$T^{2} + 256$$
$11$ $$T^{2} + 144$$
$13$ $$T^{2}$$
$17$ $$(T - 126)^{2}$$
$19$ $$T^{2} + 400$$
$23$ $$(T + 168)^{2}$$
$29$ $$(T - 30)^{2}$$
$31$ $$T^{2} + 7744$$
$37$ $$T^{2} + 64516$$
$41$ $$T^{2} + 1764$$
$43$ $$(T - 52)^{2}$$
$47$ $$T^{2} + 9216$$
$53$ $$(T - 198)^{2}$$
$59$ $$T^{2} + 435600$$
$61$ $$(T + 538)^{2}$$
$67$ $$T^{2} + 781456$$
$71$ $$T^{2} + 627264$$
$73$ $$T^{2} + 47524$$
$79$ $$(T + 520)^{2}$$
$83$ $$T^{2} + 242064$$
$89$ $$T^{2} + 656100$$
$97$ $$T^{2} + 1331716$$