# Properties

 Label 490.4.e.a Level $490$ Weight $4$ Character orbit 490.e Analytic conductor $28.911$ 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] = [490,4,Mod(361,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("490.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 490.e (of order $$3$$, degree $$2$$, 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: $$28.9109359028$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 16 q^{6} + 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^2 + (8*z - 8) * q^3 + (4*z - 4) * q^4 + 5*z * q^5 + 16 * q^6 + 8 * q^8 - 37*z * q^9 $$q - 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 16 q^{6} + 8 q^{8} - 37 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + (12 \zeta_{6} - 12) q^{11} - 32 \zeta_{6} q^{12} + 58 q^{13} - 40 q^{15} - 16 \zeta_{6} q^{16} + ( - 66 \zeta_{6} + 66) q^{17} + (74 \zeta_{6} - 74) q^{18} - 100 \zeta_{6} q^{19} - 20 q^{20} + 24 q^{22} - 132 \zeta_{6} q^{23} + (64 \zeta_{6} - 64) q^{24} + (25 \zeta_{6} - 25) q^{25} - 116 \zeta_{6} q^{26} + 80 q^{27} - 90 q^{29} + 80 \zeta_{6} q^{30} + ( - 152 \zeta_{6} + 152) q^{31} + (32 \zeta_{6} - 32) q^{32} - 96 \zeta_{6} q^{33} - 132 q^{34} + 148 q^{36} + 34 \zeta_{6} q^{37} + (200 \zeta_{6} - 200) q^{38} + (464 \zeta_{6} - 464) q^{39} + 40 \zeta_{6} q^{40} + 438 q^{41} + 32 q^{43} - 48 \zeta_{6} q^{44} + ( - 185 \zeta_{6} + 185) q^{45} + (264 \zeta_{6} - 264) q^{46} - 204 \zeta_{6} q^{47} + 128 q^{48} + 50 q^{50} + 528 \zeta_{6} q^{51} + (232 \zeta_{6} - 232) q^{52} + (222 \zeta_{6} - 222) q^{53} - 160 \zeta_{6} q^{54} - 60 q^{55} + 800 q^{57} + 180 \zeta_{6} q^{58} + ( - 420 \zeta_{6} + 420) q^{59} + ( - 160 \zeta_{6} + 160) q^{60} + 902 \zeta_{6} q^{61} - 304 q^{62} + 64 q^{64} + 290 \zeta_{6} q^{65} + (192 \zeta_{6} - 192) q^{66} + ( - 1024 \zeta_{6} + 1024) q^{67} + 264 \zeta_{6} q^{68} + 1056 q^{69} + 432 q^{71} - 296 \zeta_{6} q^{72} + ( - 362 \zeta_{6} + 362) q^{73} + ( - 68 \zeta_{6} + 68) q^{74} - 200 \zeta_{6} q^{75} + 400 q^{76} + 928 q^{78} + 160 \zeta_{6} q^{79} + ( - 80 \zeta_{6} + 80) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} - 876 \zeta_{6} q^{82} - 72 q^{83} + 330 q^{85} - 64 \zeta_{6} q^{86} + ( - 720 \zeta_{6} + 720) q^{87} + (96 \zeta_{6} - 96) q^{88} + 810 \zeta_{6} q^{89} - 370 q^{90} + 528 q^{92} + 1216 \zeta_{6} q^{93} + (408 \zeta_{6} - 408) q^{94} + ( - 500 \zeta_{6} + 500) q^{95} - 256 \zeta_{6} q^{96} - 1106 q^{97} + 444 q^{99} +O(q^{100})$$ q - 2*z * q^2 + (8*z - 8) * q^3 + (4*z - 4) * q^4 + 5*z * q^5 + 16 * q^6 + 8 * q^8 - 37*z * q^9 + (-10*z + 10) * q^10 + (12*z - 12) * q^11 - 32*z * q^12 + 58 * q^13 - 40 * q^15 - 16*z * q^16 + (-66*z + 66) * q^17 + (74*z - 74) * q^18 - 100*z * q^19 - 20 * q^20 + 24 * q^22 - 132*z * q^23 + (64*z - 64) * q^24 + (25*z - 25) * q^25 - 116*z * q^26 + 80 * q^27 - 90 * q^29 + 80*z * q^30 + (-152*z + 152) * q^31 + (32*z - 32) * q^32 - 96*z * q^33 - 132 * q^34 + 148 * q^36 + 34*z * q^37 + (200*z - 200) * q^38 + (464*z - 464) * q^39 + 40*z * q^40 + 438 * q^41 + 32 * q^43 - 48*z * q^44 + (-185*z + 185) * q^45 + (264*z - 264) * q^46 - 204*z * q^47 + 128 * q^48 + 50 * q^50 + 528*z * q^51 + (232*z - 232) * q^52 + (222*z - 222) * q^53 - 160*z * q^54 - 60 * q^55 + 800 * q^57 + 180*z * q^58 + (-420*z + 420) * q^59 + (-160*z + 160) * q^60 + 902*z * q^61 - 304 * q^62 + 64 * q^64 + 290*z * q^65 + (192*z - 192) * q^66 + (-1024*z + 1024) * q^67 + 264*z * q^68 + 1056 * q^69 + 432 * q^71 - 296*z * q^72 + (-362*z + 362) * q^73 + (-68*z + 68) * q^74 - 200*z * q^75 + 400 * q^76 + 928 * q^78 + 160*z * q^79 + (-80*z + 80) * q^80 + (-359*z + 359) * q^81 - 876*z * q^82 - 72 * q^83 + 330 * q^85 - 64*z * q^86 + (-720*z + 720) * q^87 + (96*z - 96) * q^88 + 810*z * q^89 - 370 * q^90 + 528 * q^92 + 1216*z * q^93 + (408*z - 408) * q^94 + (-500*z + 500) * q^95 - 256*z * q^96 - 1106 * q^97 + 444 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 8 q^{3} - 4 q^{4} + 5 q^{5} + 32 q^{6} + 16 q^{8} - 37 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 8 * q^3 - 4 * q^4 + 5 * q^5 + 32 * q^6 + 16 * q^8 - 37 * q^9 $$2 q - 2 q^{2} - 8 q^{3} - 4 q^{4} + 5 q^{5} + 32 q^{6} + 16 q^{8} - 37 q^{9} + 10 q^{10} - 12 q^{11} - 32 q^{12} + 116 q^{13} - 80 q^{15} - 16 q^{16} + 66 q^{17} - 74 q^{18} - 100 q^{19} - 40 q^{20} + 48 q^{22} - 132 q^{23} - 64 q^{24} - 25 q^{25} - 116 q^{26} + 160 q^{27} - 180 q^{29} + 80 q^{30} + 152 q^{31} - 32 q^{32} - 96 q^{33} - 264 q^{34} + 296 q^{36} + 34 q^{37} - 200 q^{38} - 464 q^{39} + 40 q^{40} + 876 q^{41} + 64 q^{43} - 48 q^{44} + 185 q^{45} - 264 q^{46} - 204 q^{47} + 256 q^{48} + 100 q^{50} + 528 q^{51} - 232 q^{52} - 222 q^{53} - 160 q^{54} - 120 q^{55} + 1600 q^{57} + 180 q^{58} + 420 q^{59} + 160 q^{60} + 902 q^{61} - 608 q^{62} + 128 q^{64} + 290 q^{65} - 192 q^{66} + 1024 q^{67} + 264 q^{68} + 2112 q^{69} + 864 q^{71} - 296 q^{72} + 362 q^{73} + 68 q^{74} - 200 q^{75} + 800 q^{76} + 1856 q^{78} + 160 q^{79} + 80 q^{80} + 359 q^{81} - 876 q^{82} - 144 q^{83} + 660 q^{85} - 64 q^{86} + 720 q^{87} - 96 q^{88} + 810 q^{89} - 740 q^{90} + 1056 q^{92} + 1216 q^{93} - 408 q^{94} + 500 q^{95} - 256 q^{96} - 2212 q^{97} + 888 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 8 * q^3 - 4 * q^4 + 5 * q^5 + 32 * q^6 + 16 * q^8 - 37 * q^9 + 10 * q^10 - 12 * q^11 - 32 * q^12 + 116 * q^13 - 80 * q^15 - 16 * q^16 + 66 * q^17 - 74 * q^18 - 100 * q^19 - 40 * q^20 + 48 * q^22 - 132 * q^23 - 64 * q^24 - 25 * q^25 - 116 * q^26 + 160 * q^27 - 180 * q^29 + 80 * q^30 + 152 * q^31 - 32 * q^32 - 96 * q^33 - 264 * q^34 + 296 * q^36 + 34 * q^37 - 200 * q^38 - 464 * q^39 + 40 * q^40 + 876 * q^41 + 64 * q^43 - 48 * q^44 + 185 * q^45 - 264 * q^46 - 204 * q^47 + 256 * q^48 + 100 * q^50 + 528 * q^51 - 232 * q^52 - 222 * q^53 - 160 * q^54 - 120 * q^55 + 1600 * q^57 + 180 * q^58 + 420 * q^59 + 160 * q^60 + 902 * q^61 - 608 * q^62 + 128 * q^64 + 290 * q^65 - 192 * q^66 + 1024 * q^67 + 264 * q^68 + 2112 * q^69 + 864 * q^71 - 296 * q^72 + 362 * q^73 + 68 * q^74 - 200 * q^75 + 800 * q^76 + 1856 * q^78 + 160 * q^79 + 80 * q^80 + 359 * q^81 - 876 * q^82 - 144 * q^83 + 660 * q^85 - 64 * q^86 + 720 * q^87 - 96 * q^88 + 810 * q^89 - 740 * q^90 + 1056 * q^92 + 1216 * q^93 - 408 * q^94 + 500 * q^95 - 256 * q^96 - 2212 * q^97 + 888 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i −4.00000 + 6.92820i −2.00000 + 3.46410i 2.50000 + 4.33013i 16.0000 0 8.00000 −18.5000 32.0429i 5.00000 8.66025i
471.1 −1.00000 + 1.73205i −4.00000 6.92820i −2.00000 3.46410i 2.50000 4.33013i 16.0000 0 8.00000 −18.5000 + 32.0429i 5.00000 + 8.66025i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.4.e.a 2
7.b odd 2 1 490.4.e.i 2
7.c even 3 1 490.4.a.o 1
7.c even 3 1 inner 490.4.e.a 2
7.d odd 6 1 10.4.a.a 1
7.d odd 6 1 490.4.e.i 2
21.g even 6 1 90.4.a.a 1
28.f even 6 1 80.4.a.f 1
35.i odd 6 1 50.4.a.c 1
35.j even 6 1 2450.4.a.b 1
35.k even 12 2 50.4.b.a 2
56.j odd 6 1 320.4.a.m 1
56.m even 6 1 320.4.a.b 1
63.i even 6 1 810.4.e.w 2
63.k odd 6 1 810.4.e.c 2
63.s even 6 1 810.4.e.w 2
63.t odd 6 1 810.4.e.c 2
77.i even 6 1 1210.4.a.b 1
84.j odd 6 1 720.4.a.j 1
91.s odd 6 1 1690.4.a.a 1
105.p even 6 1 450.4.a.q 1
105.w odd 12 2 450.4.c.d 2
112.v even 12 2 1280.4.d.g 2
112.x odd 12 2 1280.4.d.j 2
140.s even 6 1 400.4.a.b 1
140.x odd 12 2 400.4.c.c 2
280.ba even 6 1 1600.4.a.bx 1
280.bk odd 6 1 1600.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 7.d odd 6 1
50.4.a.c 1 35.i odd 6 1
50.4.b.a 2 35.k even 12 2
80.4.a.f 1 28.f even 6 1
90.4.a.a 1 21.g even 6 1
320.4.a.b 1 56.m even 6 1
320.4.a.m 1 56.j odd 6 1
400.4.a.b 1 140.s even 6 1
400.4.c.c 2 140.x odd 12 2
450.4.a.q 1 105.p even 6 1
450.4.c.d 2 105.w odd 12 2
490.4.a.o 1 7.c even 3 1
490.4.e.a 2 1.a even 1 1 trivial
490.4.e.a 2 7.c even 3 1 inner
490.4.e.i 2 7.b odd 2 1
490.4.e.i 2 7.d odd 6 1
720.4.a.j 1 84.j odd 6 1
810.4.e.c 2 63.k odd 6 1
810.4.e.c 2 63.t odd 6 1
810.4.e.w 2 63.i even 6 1
810.4.e.w 2 63.s even 6 1
1210.4.a.b 1 77.i even 6 1
1280.4.d.g 2 112.v even 12 2
1280.4.d.j 2 112.x odd 12 2
1600.4.a.d 1 280.bk odd 6 1
1600.4.a.bx 1 280.ba even 6 1
1690.4.a.a 1 91.s odd 6 1
2450.4.a.b 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{2} + 8T_{3} + 64$$ T3^2 + 8*T3 + 64 $$T_{11}^{2} + 12T_{11} + 144$$ T11^2 + 12*T11 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + 8T + 64$$
$5$ $$T^{2} - 5T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 12T + 144$$
$13$ $$(T - 58)^{2}$$
$17$ $$T^{2} - 66T + 4356$$
$19$ $$T^{2} + 100T + 10000$$
$23$ $$T^{2} + 132T + 17424$$
$29$ $$(T + 90)^{2}$$
$31$ $$T^{2} - 152T + 23104$$
$37$ $$T^{2} - 34T + 1156$$
$41$ $$(T - 438)^{2}$$
$43$ $$(T - 32)^{2}$$
$47$ $$T^{2} + 204T + 41616$$
$53$ $$T^{2} + 222T + 49284$$
$59$ $$T^{2} - 420T + 176400$$
$61$ $$T^{2} - 902T + 813604$$
$67$ $$T^{2} - 1024 T + 1048576$$
$71$ $$(T - 432)^{2}$$
$73$ $$T^{2} - 362T + 131044$$
$79$ $$T^{2} - 160T + 25600$$
$83$ $$(T + 72)^{2}$$
$89$ $$T^{2} - 810T + 656100$$
$97$ $$(T + 1106)^{2}$$