# Properties

 Label 1050.6.g.o 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 $$q - 4 i q^{2} + 9 i q^{3} - 16 q^{4} + 36 q^{6} + 49 i q^{7} + 64 i q^{8} - 81 q^{9} + 216 q^{11} - 144 i q^{12} - 998 i q^{13} + 196 q^{14} + 256 q^{16} + 1302 i q^{17} + 324 i q^{18} - 884 q^{19} - 441 q^{21} - 864 i q^{22} + 2268 i q^{23} - 576 q^{24} - 3992 q^{26} - 729 i q^{27} - 784 i q^{28} + 1482 q^{29} + 8360 q^{31} - 1024 i q^{32} + 1944 i q^{33} + 5208 q^{34} + 1296 q^{36} - 4714 i q^{37} + 3536 i q^{38} + 8982 q^{39} - 9786 q^{41} + 1764 i q^{42} - 19436 i q^{43} - 3456 q^{44} + 9072 q^{46} + 22200 i q^{47} + 2304 i q^{48} - 2401 q^{49} - 11718 q^{51} + 15968 i q^{52} - 26790 i q^{53} - 2916 q^{54} - 3136 q^{56} - 7956 i q^{57} - 5928 i q^{58} - 28092 q^{59} - 38866 q^{61} - 33440 i q^{62} - 3969 i q^{63} - 4096 q^{64} + 7776 q^{66} + 23948 i q^{67} - 20832 i q^{68} - 20412 q^{69} - 20628 q^{71} - 5184 i q^{72} - 290 i q^{73} - 18856 q^{74} + 14144 q^{76} + 10584 i q^{77} - 35928 i q^{78} + 99544 q^{79} + 6561 q^{81} + 39144 i q^{82} - 19308 i q^{83} + 7056 q^{84} - 77744 q^{86} + 13338 i q^{87} + 13824 i q^{88} - 36390 q^{89} + 48902 q^{91} - 36288 i q^{92} + 75240 i q^{93} + 88800 q^{94} + 9216 q^{96} - 79078 i q^{97} + 9604 i q^{98} - 17496 q^{99} +O(q^{100})$$ 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 + 216 * q^11 - 144*i * q^12 - 998*i * q^13 + 196 * q^14 + 256 * q^16 + 1302*i * q^17 + 324*i * q^18 - 884 * q^19 - 441 * q^21 - 864*i * q^22 + 2268*i * q^23 - 576 * q^24 - 3992 * q^26 - 729*i * q^27 - 784*i * q^28 + 1482 * q^29 + 8360 * q^31 - 1024*i * q^32 + 1944*i * q^33 + 5208 * q^34 + 1296 * q^36 - 4714*i * q^37 + 3536*i * q^38 + 8982 * q^39 - 9786 * q^41 + 1764*i * q^42 - 19436*i * q^43 - 3456 * q^44 + 9072 * q^46 + 22200*i * q^47 + 2304*i * q^48 - 2401 * q^49 - 11718 * q^51 + 15968*i * q^52 - 26790*i * q^53 - 2916 * q^54 - 3136 * q^56 - 7956*i * q^57 - 5928*i * q^58 - 28092 * q^59 - 38866 * q^61 - 33440*i * q^62 - 3969*i * q^63 - 4096 * q^64 + 7776 * q^66 + 23948*i * q^67 - 20832*i * q^68 - 20412 * q^69 - 20628 * q^71 - 5184*i * q^72 - 290*i * q^73 - 18856 * q^74 + 14144 * q^76 + 10584*i * q^77 - 35928*i * q^78 + 99544 * q^79 + 6561 * q^81 + 39144*i * q^82 - 19308*i * q^83 + 7056 * q^84 - 77744 * q^86 + 13338*i * q^87 + 13824*i * q^88 - 36390 * q^89 + 48902 * q^91 - 36288*i * q^92 + 75240*i * q^93 + 88800 * q^94 + 9216 * q^96 - 79078*i * q^97 + 9604*i * q^98 - 17496 * 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 $$2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 432 q^{11} + 392 q^{14} + 512 q^{16} - 1768 q^{19} - 882 q^{21} - 1152 q^{24} - 7984 q^{26} + 2964 q^{29} + 16720 q^{31} + 10416 q^{34} + 2592 q^{36} + 17964 q^{39} - 19572 q^{41} - 6912 q^{44} + 18144 q^{46} - 4802 q^{49} - 23436 q^{51} - 5832 q^{54} - 6272 q^{56} - 56184 q^{59} - 77732 q^{61} - 8192 q^{64} + 15552 q^{66} - 40824 q^{69} - 41256 q^{71} - 37712 q^{74} + 28288 q^{76} + 199088 q^{79} + 13122 q^{81} + 14112 q^{84} - 155488 q^{86} - 72780 q^{89} + 97804 q^{91} + 177600 q^{94} + 18432 q^{96} - 34992 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 + 432 * q^11 + 392 * q^14 + 512 * q^16 - 1768 * q^19 - 882 * q^21 - 1152 * q^24 - 7984 * q^26 + 2964 * q^29 + 16720 * q^31 + 10416 * q^34 + 2592 * q^36 + 17964 * q^39 - 19572 * q^41 - 6912 * q^44 + 18144 * q^46 - 4802 * q^49 - 23436 * q^51 - 5832 * q^54 - 6272 * q^56 - 56184 * q^59 - 77732 * q^61 - 8192 * q^64 + 15552 * q^66 - 40824 * q^69 - 41256 * q^71 - 37712 * q^74 + 28288 * q^76 + 199088 * q^79 + 13122 * q^81 + 14112 * q^84 - 155488 * q^86 - 72780 * q^89 + 97804 * q^91 + 177600 * q^94 + 18432 * q^96 - 34992 * 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.o 2
5.b even 2 1 inner 1050.6.g.o 2
5.c odd 4 1 42.6.a.a 1
5.c odd 4 1 1050.6.a.n 1
15.e even 4 1 126.6.a.k 1
20.e even 4 1 336.6.a.j 1
35.f even 4 1 294.6.a.h 1
35.k even 12 2 294.6.e.h 2
35.l odd 12 2 294.6.e.r 2
60.l odd 4 1 1008.6.a.x 1
105.k odd 4 1 882.6.a.o 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 216$$ acting on $$S_{6}^{\mathrm{new}}(1050, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 216)^{2}$$
$13$ $$T^{2} + 996004$$
$17$ $$T^{2} + 1695204$$
$19$ $$(T + 884)^{2}$$
$23$ $$T^{2} + 5143824$$
$29$ $$(T - 1482)^{2}$$
$31$ $$(T - 8360)^{2}$$
$37$ $$T^{2} + 22221796$$
$41$ $$(T + 9786)^{2}$$
$43$ $$T^{2} + 377758096$$
$47$ $$T^{2} + 492840000$$
$53$ $$T^{2} + 717704100$$
$59$ $$(T + 28092)^{2}$$
$61$ $$(T + 38866)^{2}$$
$67$ $$T^{2} + 573506704$$
$71$ $$(T + 20628)^{2}$$
$73$ $$T^{2} + 84100$$
$79$ $$(T - 99544)^{2}$$
$83$ $$T^{2} + 372798864$$
$89$ $$(T + 36390)^{2}$$
$97$ $$T^{2} + 6253330084$$