# Properties

 Label 1050.6.g.l Level $1050$ Weight $6$ Character orbit 1050.g Analytic conductor $168.403$ 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] = [1050,6,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1050.799");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1050.g (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: $$168.403010804$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} - 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} - 64 i q^{8} - 81 q^{9} +O(q^{10})$$ q + 4*i * q^2 - 9*i * q^3 - 16 * q^4 + 36 * q^6 + 49*i * q^7 - 64*i * q^8 - 81 * q^9 $$q + 4 i q^{2} - 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} - 64 i q^{8} - 81 q^{9} - 470 q^{11} + 144 i q^{12} - 1158 i q^{13} - 196 q^{14} + 256 q^{16} - 1204 i q^{17} - 324 i q^{18} + 2644 q^{19} + 441 q^{21} - 1880 i q^{22} - 1190 i q^{23} - 576 q^{24} + 4632 q^{26} + 729 i q^{27} - 784 i q^{28} - 3614 q^{29} + 5616 q^{31} + 1024 i q^{32} + 4230 i q^{33} + 4816 q^{34} + 1296 q^{36} + 6478 i q^{37} + 10576 i q^{38} - 10422 q^{39} + 2856 q^{41} + 1764 i q^{42} - 13492 i q^{43} + 7520 q^{44} + 4760 q^{46} + 18372 i q^{47} - 2304 i q^{48} - 2401 q^{49} - 10836 q^{51} + 18528 i q^{52} - 4374 i q^{53} - 2916 q^{54} + 3136 q^{56} - 23796 i q^{57} - 14456 i q^{58} - 30248 q^{59} + 19542 q^{61} + 22464 i q^{62} - 3969 i q^{63} - 4096 q^{64} - 16920 q^{66} - 54328 i q^{67} + 19264 i q^{68} - 10710 q^{69} - 10730 q^{71} + 5184 i q^{72} + 35374 i q^{73} - 25912 q^{74} - 42304 q^{76} - 23030 i q^{77} - 41688 i q^{78} + 49956 q^{79} + 6561 q^{81} + 11424 i q^{82} - 26948 i q^{83} - 7056 q^{84} + 53968 q^{86} + 32526 i q^{87} + 30080 i q^{88} - 100776 q^{89} + 56742 q^{91} + 19040 i q^{92} - 50544 i q^{93} - 73488 q^{94} + 9216 q^{96} - 77134 i q^{97} - 9604 i q^{98} + 38070 q^{99} +O(q^{100})$$ q + 4*i * q^2 - 9*i * q^3 - 16 * q^4 + 36 * q^6 + 49*i * q^7 - 64*i * q^8 - 81 * q^9 - 470 * q^11 + 144*i * q^12 - 1158*i * q^13 - 196 * q^14 + 256 * q^16 - 1204*i * q^17 - 324*i * q^18 + 2644 * q^19 + 441 * q^21 - 1880*i * q^22 - 1190*i * q^23 - 576 * q^24 + 4632 * q^26 + 729*i * q^27 - 784*i * q^28 - 3614 * q^29 + 5616 * q^31 + 1024*i * q^32 + 4230*i * q^33 + 4816 * q^34 + 1296 * q^36 + 6478*i * q^37 + 10576*i * q^38 - 10422 * q^39 + 2856 * q^41 + 1764*i * q^42 - 13492*i * q^43 + 7520 * q^44 + 4760 * q^46 + 18372*i * q^47 - 2304*i * q^48 - 2401 * q^49 - 10836 * q^51 + 18528*i * q^52 - 4374*i * q^53 - 2916 * q^54 + 3136 * q^56 - 23796*i * q^57 - 14456*i * q^58 - 30248 * q^59 + 19542 * q^61 + 22464*i * q^62 - 3969*i * q^63 - 4096 * q^64 - 16920 * q^66 - 54328*i * q^67 + 19264*i * q^68 - 10710 * q^69 - 10730 * q^71 + 5184*i * q^72 + 35374*i * q^73 - 25912 * q^74 - 42304 * q^76 - 23030*i * q^77 - 41688*i * q^78 + 49956 * q^79 + 6561 * q^81 + 11424*i * q^82 - 26948*i * q^83 - 7056 * q^84 + 53968 * q^86 + 32526*i * q^87 + 30080*i * q^88 - 100776 * q^89 + 56742 * q^91 + 19040*i * q^92 - 50544*i * q^93 - 73488 * q^94 + 9216 * q^96 - 77134*i * q^97 - 9604*i * q^98 + 38070 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} + 72 q^{6} - 162 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 $$2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} - 940 q^{11} - 392 q^{14} + 512 q^{16} + 5288 q^{19} + 882 q^{21} - 1152 q^{24} + 9264 q^{26} - 7228 q^{29} + 11232 q^{31} + 9632 q^{34} + 2592 q^{36} - 20844 q^{39} + 5712 q^{41} + 15040 q^{44} + 9520 q^{46} - 4802 q^{49} - 21672 q^{51} - 5832 q^{54} + 6272 q^{56} - 60496 q^{59} + 39084 q^{61} - 8192 q^{64} - 33840 q^{66} - 21420 q^{69} - 21460 q^{71} - 51824 q^{74} - 84608 q^{76} + 99912 q^{79} + 13122 q^{81} - 14112 q^{84} + 107936 q^{86} - 201552 q^{89} + 113484 q^{91} - 146976 q^{94} + 18432 q^{96} + 76140 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 - 940 * q^11 - 392 * q^14 + 512 * q^16 + 5288 * q^19 + 882 * q^21 - 1152 * q^24 + 9264 * q^26 - 7228 * q^29 + 11232 * q^31 + 9632 * q^34 + 2592 * q^36 - 20844 * q^39 + 5712 * q^41 + 15040 * q^44 + 9520 * q^46 - 4802 * q^49 - 21672 * q^51 - 5832 * q^54 + 6272 * q^56 - 60496 * q^59 + 39084 * q^61 - 8192 * q^64 - 33840 * q^66 - 21420 * q^69 - 21460 * q^71 - 51824 * q^74 - 84608 * q^76 + 99912 * q^79 + 13122 * q^81 - 14112 * q^84 + 107936 * q^86 - 201552 * q^89 + 113484 * q^91 - 146976 * q^94 + 18432 * q^96 + 76140 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 − 1.00000i 1.00000i
4.00000i 9.00000i −16.0000 0 36.0000 49.0000i 64.0000i −81.0000 0
799.2 4.00000i 9.00000i −16.0000 0 36.0000 49.0000i 64.0000i −81.0000 0
 $$n$$: 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 1050.6.g.l 2
5.b even 2 1 inner 1050.6.g.l 2
5.c odd 4 1 42.6.a.b 1
5.c odd 4 1 1050.6.a.o 1
15.e even 4 1 126.6.a.h 1
20.e even 4 1 336.6.a.o 1
35.f even 4 1 294.6.a.f 1
35.k even 12 2 294.6.e.k 2
35.l odd 12 2 294.6.e.o 2
60.l odd 4 1 1008.6.a.g 1
105.k odd 4 1 882.6.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.b 1 5.c odd 4 1
126.6.a.h 1 15.e even 4 1
294.6.a.f 1 35.f even 4 1
294.6.e.k 2 35.k even 12 2
294.6.e.o 2 35.l odd 12 2
336.6.a.o 1 20.e even 4 1
882.6.a.v 1 105.k odd 4 1
1008.6.a.g 1 60.l odd 4 1
1050.6.a.o 1 5.c odd 4 1
1050.6.g.l 2 1.a even 1 1 trivial
1050.6.g.l 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} + 470$$ acting on $$S_{6}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 470)^{2}$$
$13$ $$T^{2} + 1340964$$
$17$ $$T^{2} + 1449616$$
$19$ $$(T - 2644)^{2}$$
$23$ $$T^{2} + 1416100$$
$29$ $$(T + 3614)^{2}$$
$31$ $$(T - 5616)^{2}$$
$37$ $$T^{2} + 41964484$$
$41$ $$(T - 2856)^{2}$$
$43$ $$T^{2} + 182034064$$
$47$ $$T^{2} + 337530384$$
$53$ $$T^{2} + 19131876$$
$59$ $$(T + 30248)^{2}$$
$61$ $$(T - 19542)^{2}$$
$67$ $$T^{2} + 2951531584$$
$71$ $$(T + 10730)^{2}$$
$73$ $$T^{2} + 1251319876$$
$79$ $$(T - 49956)^{2}$$
$83$ $$T^{2} + 726194704$$
$89$ $$(T + 100776)^{2}$$
$97$ $$T^{2} + 5949653956$$