Properties

 Label 1050.6.g.i 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 $$q + 4 i q^{2} + 9 i q^{3} - 16 q^{4} - 36 q^{6} + 49 i q^{7} - 64 i q^{8} - 81 q^{9} + 664 q^{11} - 144 i q^{12} + 318 i q^{13} - 196 q^{14} + 256 q^{16} - 1582 i q^{17} - 324 i q^{18} - 236 q^{19} - 441 q^{21} + 2656 i q^{22} + 2212 i q^{23} + 576 q^{24} - 1272 q^{26} - 729 i q^{27} - 784 i q^{28} + 4954 q^{29} - 7128 q^{31} + 1024 i q^{32} + 5976 i q^{33} + 6328 q^{34} + 1296 q^{36} - 4358 i q^{37} - 944 i q^{38} - 2862 q^{39} + 10542 q^{41} - 1764 i q^{42} - 8452 i q^{43} - 10624 q^{44} - 8848 q^{46} - 5352 i q^{47} + 2304 i q^{48} - 2401 q^{49} + 14238 q^{51} - 5088 i q^{52} - 33354 i q^{53} + 2916 q^{54} + 3136 q^{56} - 2124 i q^{57} + 19816 i q^{58} + 15436 q^{59} - 36762 q^{61} - 28512 i q^{62} - 3969 i q^{63} - 4096 q^{64} - 23904 q^{66} - 40972 i q^{67} + 25312 i q^{68} - 19908 q^{69} - 9092 q^{71} + 5184 i q^{72} - 73454 i q^{73} + 17432 q^{74} + 3776 q^{76} + 32536 i q^{77} - 11448 i q^{78} - 89400 q^{79} + 6561 q^{81} + 42168 i q^{82} - 6428 i q^{83} + 7056 q^{84} + 33808 q^{86} + 44586 i q^{87} - 42496 i q^{88} + 122658 q^{89} - 15582 q^{91} - 35392 i q^{92} - 64152 i q^{93} + 21408 q^{94} - 9216 q^{96} - 21370 i q^{97} - 9604 i q^{98} - 53784 q^{99} +O(q^{100})$$ 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 + 664 * q^11 - 144*i * q^12 + 318*i * q^13 - 196 * q^14 + 256 * q^16 - 1582*i * q^17 - 324*i * q^18 - 236 * q^19 - 441 * q^21 + 2656*i * q^22 + 2212*i * q^23 + 576 * q^24 - 1272 * q^26 - 729*i * q^27 - 784*i * q^28 + 4954 * q^29 - 7128 * q^31 + 1024*i * q^32 + 5976*i * q^33 + 6328 * q^34 + 1296 * q^36 - 4358*i * q^37 - 944*i * q^38 - 2862 * q^39 + 10542 * q^41 - 1764*i * q^42 - 8452*i * q^43 - 10624 * q^44 - 8848 * q^46 - 5352*i * q^47 + 2304*i * q^48 - 2401 * q^49 + 14238 * q^51 - 5088*i * q^52 - 33354*i * q^53 + 2916 * q^54 + 3136 * q^56 - 2124*i * q^57 + 19816*i * q^58 + 15436 * q^59 - 36762 * q^61 - 28512*i * q^62 - 3969*i * q^63 - 4096 * q^64 - 23904 * q^66 - 40972*i * q^67 + 25312*i * q^68 - 19908 * q^69 - 9092 * q^71 + 5184*i * q^72 - 73454*i * q^73 + 17432 * q^74 + 3776 * q^76 + 32536*i * q^77 - 11448*i * q^78 - 89400 * q^79 + 6561 * q^81 + 42168*i * q^82 - 6428*i * q^83 + 7056 * q^84 + 33808 * q^86 + 44586*i * q^87 - 42496*i * q^88 + 122658 * q^89 - 15582 * q^91 - 35392*i * q^92 - 64152*i * q^93 + 21408 * q^94 - 9216 * q^96 - 21370*i * q^97 - 9604*i * q^98 - 53784 * 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 $$2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} + 1328 q^{11} - 392 q^{14} + 512 q^{16} - 472 q^{19} - 882 q^{21} + 1152 q^{24} - 2544 q^{26} + 9908 q^{29} - 14256 q^{31} + 12656 q^{34} + 2592 q^{36} - 5724 q^{39} + 21084 q^{41} - 21248 q^{44} - 17696 q^{46} - 4802 q^{49} + 28476 q^{51} + 5832 q^{54} + 6272 q^{56} + 30872 q^{59} - 73524 q^{61} - 8192 q^{64} - 47808 q^{66} - 39816 q^{69} - 18184 q^{71} + 34864 q^{74} + 7552 q^{76} - 178800 q^{79} + 13122 q^{81} + 14112 q^{84} + 67616 q^{86} + 245316 q^{89} - 31164 q^{91} + 42816 q^{94} - 18432 q^{96} - 107568 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 + 1328 * q^11 - 392 * q^14 + 512 * q^16 - 472 * q^19 - 882 * q^21 + 1152 * q^24 - 2544 * q^26 + 9908 * q^29 - 14256 * q^31 + 12656 * q^34 + 2592 * q^36 - 5724 * q^39 + 21084 * q^41 - 21248 * q^44 - 17696 * q^46 - 4802 * q^49 + 28476 * q^51 + 5832 * q^54 + 6272 * q^56 + 30872 * q^59 - 73524 * q^61 - 8192 * q^64 - 47808 * q^66 - 39816 * q^69 - 18184 * q^71 + 34864 * q^74 + 7552 * q^76 - 178800 * q^79 + 13122 * q^81 + 14112 * q^84 + 67616 * q^86 + 245316 * q^89 - 31164 * q^91 + 42816 * q^94 - 18432 * q^96 - 107568 * 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.i 2
5.b even 2 1 inner 1050.6.g.i 2
5.c odd 4 1 42.6.a.d 1
5.c odd 4 1 1050.6.a.k 1
15.e even 4 1 126.6.a.i 1
20.e even 4 1 336.6.a.h 1
35.f even 4 1 294.6.a.b 1
35.k even 12 2 294.6.e.p 2
35.l odd 12 2 294.6.e.i 2
60.l odd 4 1 1008.6.a.j 1
105.k odd 4 1 882.6.a.s 1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 664$$ acting on $$S_{6}^{\mathrm{new}}(1050, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 81$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 664)^{2}$$
$13$ $$T^{2} + 101124$$
$17$ $$T^{2} + 2502724$$
$19$ $$(T + 236)^{2}$$
$23$ $$T^{2} + 4892944$$
$29$ $$(T - 4954)^{2}$$
$31$ $$(T + 7128)^{2}$$
$37$ $$T^{2} + 18992164$$
$41$ $$(T - 10542)^{2}$$
$43$ $$T^{2} + 71436304$$
$47$ $$T^{2} + 28643904$$
$53$ $$T^{2} + 1112489316$$
$59$ $$(T - 15436)^{2}$$
$61$ $$(T + 36762)^{2}$$
$67$ $$T^{2} + 1678704784$$
$71$ $$(T + 9092)^{2}$$
$73$ $$T^{2} + 5395490116$$
$79$ $$(T + 89400)^{2}$$
$83$ $$T^{2} + 41319184$$
$89$ $$(T - 122658)^{2}$$
$97$ $$T^{2} + 456676900$$