# Properties

 Label 490.4.c.d Level $490$ Weight $4$ Character orbit 490.c Analytic conductor $28.911$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$490 = 2 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 490.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.9109359028$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.6654810844696576.6 Defining polynomial: $$x^{8} + 1325x^{4} + 9604$$ x^8 + 1325*x^4 + 9604 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{6} q^{3} - 4 q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{6} - 4 \beta_1 q^{8} + ( - \beta_{3} + 8) q^{9}+O(q^{10})$$ q + b1 * q^2 + b6 * q^3 - 4 * q^4 + (-2*b7 - 2*b6 - b5 - b4) * q^5 + (-b7 + b5 - b4) * q^6 - 4*b1 * q^8 + (-b3 + 8) * q^9 $$q + \beta_1 q^{2} + \beta_{6} q^{3} - 4 q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{6} - 4 \beta_1 q^{8} + ( - \beta_{3} + 8) q^{9} + (3 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4}) q^{10} + ( - 3 \beta_{3} - 3) q^{11} - 4 \beta_{6} q^{12} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5}) q^{13} + (2 \beta_{3} + \beta_{2} - 13 \beta_1 + 17) q^{15} + 16 q^{16} + ( - 13 \beta_{7} - 3 \beta_{6} - 13 \beta_{5}) q^{17} + (2 \beta_{2} + 8 \beta_1) q^{18} + ( - \beta_{7} + \beta_{5} + 16 \beta_{4}) q^{19} + (8 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{20} + (6 \beta_{2} - 3 \beta_1) q^{22} + (10 \beta_{2} - 4 \beta_1) q^{23} + (4 \beta_{7} - 4 \beta_{5} + 4 \beta_{4}) q^{24} + (\beta_{3} - 7 \beta_{2} + \beta_1 - 39) q^{25} + (\beta_{7} - \beta_{5} + 5 \beta_{4}) q^{26} + ( - 7 \beta_{7} + 15 \beta_{6} - 7 \beta_{5}) q^{27} + (\beta_{3} + 107) q^{29} + (2 \beta_{3} - 4 \beta_{2} + 17 \beta_1 + 52) q^{30} + (32 \beta_{7} - 32 \beta_{5} + 10 \beta_{4}) q^{31} + 16 \beta_1 q^{32} + ( - 21 \beta_{7} - 63 \beta_{6} - 21 \beta_{5}) q^{33} + (3 \beta_{7} - 3 \beta_{5} - 23 \beta_{4}) q^{34} + (4 \beta_{3} - 32) q^{36} + ( - 6 \beta_{2} - 96 \beta_1) q^{37} + ( - 34 \beta_{7} - 4 \beta_{6} - 34 \beta_{5}) q^{38} + (\beta_{3} + 47) q^{39} + ( - 12 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{40} + (20 \beta_{7} - 20 \beta_{5} - 26 \beta_{4}) q^{41} + ( - 8 \beta_{2} + 26 \beta_1) q^{43} + (12 \beta_{3} + 12) q^{44} + ( - 17 \beta_{7} + 3 \beta_{6} - 36 \beta_{5} - 11 \beta_{4}) q^{45} + (20 \beta_{3} + 16) q^{46} + ( - 35 \beta_{7} - 43 \beta_{6} - 35 \beta_{5}) q^{47} + 16 \beta_{6} q^{48} + ( - 14 \beta_{3} - 2 \beta_{2} - 39 \beta_1 - 4) q^{50} + (3 \beta_{3} - 125) q^{51} + ( - 8 \beta_{7} + 4 \beta_{6} - 8 \beta_{5}) q^{52} + (6 \beta_{2} - 54 \beta_1) q^{53} + ( - 15 \beta_{7} + 15 \beta_{5} - 29 \beta_{4}) q^{54} + (3 \beta_{7} + 63 \beta_{6} - 81 \beta_{5} - 6 \beta_{4}) q^{55} + (2 \beta_{2} + 100 \beta_1) q^{57} + ( - 2 \beta_{2} + 107 \beta_1) q^{58} + (51 \beta_{7} - 51 \beta_{5} - 66 \beta_{4}) q^{59} + ( - 8 \beta_{3} - 4 \beta_{2} + 52 \beta_1 - 68) q^{60} + (43 \beta_{7} - 43 \beta_{5} + 104 \beta_{4}) q^{61} + (44 \beta_{7} + 128 \beta_{6} + 44 \beta_{5}) q^{62} - 64 q^{64} + ( - 8 \beta_{3} + \beta_{2} + 47 \beta_1 + 47) q^{65} + (63 \beta_{7} - 63 \beta_{5} + 21 \beta_{4}) q^{66} + (16 \beta_{2} - 148 \beta_1) q^{67} + (52 \beta_{7} + 12 \beta_{6} + 52 \beta_{5}) q^{68} + (104 \beta_{7} - 104 \beta_{5} + 34 \beta_{4}) q^{69} + ( - 18 \beta_{3} - 432) q^{71} + ( - 8 \beta_{2} - 32 \beta_1) q^{72} + ( - 8 \beta_{7} + 190 \beta_{6} - 8 \beta_{5}) q^{73} + ( - 12 \beta_{3} + 384) q^{74} + ( - 64 \beta_{7} - 19 \beta_{6} + 78 \beta_{5} - 22 \beta_{4}) q^{75} + (4 \beta_{7} - 4 \beta_{5} - 64 \beta_{4}) q^{76} + ( - 2 \beta_{2} + 47 \beta_1) q^{78} + ( - 37 \beta_{3} - 367) q^{79} + ( - 32 \beta_{7} - 32 \beta_{6} - 16 \beta_{5} - 16 \beta_{4}) q^{80} + ( - 42 \beta_{3} - 167) q^{81} + (92 \beta_{7} + 80 \beta_{6} + 92 \beta_{5}) q^{82} + ( - 125 \beta_{7} - 106 \beta_{6} - 125 \beta_{5}) q^{83} + (33 \beta_{3} - 16 \beta_{2} - 182 \beta_1 - 467) q^{85} + ( - 16 \beta_{3} - 104) q^{86} + (7 \beta_{7} + 127 \beta_{6} + 7 \beta_{5}) q^{87} + ( - 24 \beta_{2} + 12 \beta_1) q^{88} + ( - 98 \beta_{7} + 98 \beta_{5} - 46 \beta_{4}) q^{89} + (38 \beta_{7} + 38 \beta_{6} + 44 \beta_{5} - 56 \beta_{4}) q^{90} + ( - 40 \beta_{2} + 16 \beta_1) q^{92} + ( - 64 \beta_{2} + 454 \beta_1) q^{93} + (43 \beta_{7} - 43 \beta_{5} - 27 \beta_{4}) q^{94} + (19 \beta_{3} + 47 \beta_{2} + 289 \beta_1 - 526) q^{95} + ( - 16 \beta_{7} + 16 \beta_{5} - 16 \beta_{4}) q^{96} + ( - 205 \beta_{7} - 175 \beta_{6} - 205 \beta_{5}) q^{97} + ( - 18 \beta_{3} + 822) q^{99}+O(q^{100})$$ q + b1 * q^2 + b6 * q^3 - 4 * q^4 + (-2*b7 - 2*b6 - b5 - b4) * q^5 + (-b7 + b5 - b4) * q^6 - 4*b1 * q^8 + (-b3 + 8) * q^9 + (3*b7 - 2*b6 - b5 - b4) * q^10 + (-3*b3 - 3) * q^11 - 4*b6 * q^12 + (2*b7 - b6 + 2*b5) * q^13 + (2*b3 + b2 - 13*b1 + 17) * q^15 + 16 * q^16 + (-13*b7 - 3*b6 - 13*b5) * q^17 + (2*b2 + 8*b1) * q^18 + (-b7 + b5 + 16*b4) * q^19 + (8*b7 + 8*b6 + 4*b5 + 4*b4) * q^20 + (6*b2 - 3*b1) * q^22 + (10*b2 - 4*b1) * q^23 + (4*b7 - 4*b5 + 4*b4) * q^24 + (b3 - 7*b2 + b1 - 39) * q^25 + (b7 - b5 + 5*b4) * q^26 + (-7*b7 + 15*b6 - 7*b5) * q^27 + (b3 + 107) * q^29 + (2*b3 - 4*b2 + 17*b1 + 52) * q^30 + (32*b7 - 32*b5 + 10*b4) * q^31 + 16*b1 * q^32 + (-21*b7 - 63*b6 - 21*b5) * q^33 + (3*b7 - 3*b5 - 23*b4) * q^34 + (4*b3 - 32) * q^36 + (-6*b2 - 96*b1) * q^37 + (-34*b7 - 4*b6 - 34*b5) * q^38 + (b3 + 47) * q^39 + (-12*b7 + 8*b6 + 4*b5 + 4*b4) * q^40 + (20*b7 - 20*b5 - 26*b4) * q^41 + (-8*b2 + 26*b1) * q^43 + (12*b3 + 12) * q^44 + (-17*b7 + 3*b6 - 36*b5 - 11*b4) * q^45 + (20*b3 + 16) * q^46 + (-35*b7 - 43*b6 - 35*b5) * q^47 + 16*b6 * q^48 + (-14*b3 - 2*b2 - 39*b1 - 4) * q^50 + (3*b3 - 125) * q^51 + (-8*b7 + 4*b6 - 8*b5) * q^52 + (6*b2 - 54*b1) * q^53 + (-15*b7 + 15*b5 - 29*b4) * q^54 + (3*b7 + 63*b6 - 81*b5 - 6*b4) * q^55 + (2*b2 + 100*b1) * q^57 + (-2*b2 + 107*b1) * q^58 + (51*b7 - 51*b5 - 66*b4) * q^59 + (-8*b3 - 4*b2 + 52*b1 - 68) * q^60 + (43*b7 - 43*b5 + 104*b4) * q^61 + (44*b7 + 128*b6 + 44*b5) * q^62 - 64 * q^64 + (-8*b3 + b2 + 47*b1 + 47) * q^65 + (63*b7 - 63*b5 + 21*b4) * q^66 + (16*b2 - 148*b1) * q^67 + (52*b7 + 12*b6 + 52*b5) * q^68 + (104*b7 - 104*b5 + 34*b4) * q^69 + (-18*b3 - 432) * q^71 + (-8*b2 - 32*b1) * q^72 + (-8*b7 + 190*b6 - 8*b5) * q^73 + (-12*b3 + 384) * q^74 + (-64*b7 - 19*b6 + 78*b5 - 22*b4) * q^75 + (4*b7 - 4*b5 - 64*b4) * q^76 + (-2*b2 + 47*b1) * q^78 + (-37*b3 - 367) * q^79 + (-32*b7 - 32*b6 - 16*b5 - 16*b4) * q^80 + (-42*b3 - 167) * q^81 + (92*b7 + 80*b6 + 92*b5) * q^82 + (-125*b7 - 106*b6 - 125*b5) * q^83 + (33*b3 - 16*b2 - 182*b1 - 467) * q^85 + (-16*b3 - 104) * q^86 + (7*b7 + 127*b6 + 7*b5) * q^87 + (-24*b2 + 12*b1) * q^88 + (-98*b7 + 98*b5 - 46*b4) * q^89 + (38*b7 + 38*b6 + 44*b5 - 56*b4) * q^90 + (-40*b2 + 16*b1) * q^92 + (-64*b2 + 454*b1) * q^93 + (43*b7 - 43*b5 - 27*b4) * q^94 + (19*b3 + 47*b2 + 289*b1 - 526) * q^95 + (-16*b7 + 16*b5 - 16*b4) * q^96 + (-205*b7 - 175*b6 - 205*b5) * q^97 + (-18*b3 + 822) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 32 q^{4} + 60 q^{9}+O(q^{10})$$ 8 * q - 32 * q^4 + 60 * q^9 $$8 q - 32 q^{4} + 60 q^{9} - 36 q^{11} + 144 q^{15} + 128 q^{16} - 308 q^{25} + 860 q^{29} + 424 q^{30} - 240 q^{36} + 380 q^{39} + 144 q^{44} + 208 q^{46} - 88 q^{50} - 988 q^{51} - 576 q^{60} - 512 q^{64} + 344 q^{65} - 3528 q^{71} + 3024 q^{74} - 3084 q^{79} - 1504 q^{81} - 3604 q^{85} - 896 q^{86} - 4132 q^{95} + 6504 q^{99}+O(q^{100})$$ 8 * q - 32 * q^4 + 60 * q^9 - 36 * q^11 + 144 * q^15 + 128 * q^16 - 308 * q^25 + 860 * q^29 + 424 * q^30 - 240 * q^36 + 380 * q^39 + 144 * q^44 + 208 * q^46 - 88 * q^50 - 988 * q^51 - 576 * q^60 - 512 * q^64 + 344 * q^65 - 3528 * q^71 + 3024 * q^74 - 3084 * q^79 - 1504 * q^81 - 3604 * q^85 - 896 * q^86 - 4132 * q^95 + 6504 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 1325x^{4} + 9604$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 1423\nu^{2} ) / 1911$$ (v^6 + 1423*v^2) / 1911 $$\beta_{2}$$ $$=$$ $$( -10\nu^{6} - 12319\nu^{2} ) / 1911$$ (-10*v^6 - 12319*v^2) / 1911 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 682 ) / 39$$ (v^4 + 682) / 39 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - 1423\nu^{3} + 3822\nu ) / 3822$$ (-v^7 - 1423*v^3 + 3822*v) / 3822 $$\beta_{5}$$ $$=$$ $$( 25\nu^{7} - 98\nu^{5} + 31753\nu^{3} - 139454\nu ) / 53508$$ (25*v^7 - 98*v^5 + 31753*v^3 - 139454*v) / 53508 $$\beta_{6}$$ $$=$$ $$( 39\nu^{7} + 98\nu^{5} + 51675\nu^{3} + 139454\nu ) / 53508$$ (39*v^7 + 98*v^5 + 51675*v^3 + 139454*v) / 53508 $$\beta_{7}$$ $$=$$ $$( -39\nu^{7} + 98\nu^{5} - 51675\nu^{3} + 85946\nu ) / 53508$$ (-39*v^7 + 98*v^5 - 51675*v^3 + 85946*v) / 53508
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2$$ (-b7 - b5 + b4) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 10\beta_1$$ b2 + 10*b1 $$\nu^{3}$$ $$=$$ $$-16\beta_{7} - 7\beta_{6} - 23\beta_{5} - 16\beta_{4}$$ -16*b7 - 7*b6 - 23*b5 - 16*b4 $$\nu^{4}$$ $$=$$ $$39\beta_{3} - 682$$ 39*b3 - 682 $$\nu^{5}$$ $$=$$ $$848\beta_{7} + 273\beta_{6} + 575\beta_{5} - 575\beta_{4}$$ 848*b7 + 273*b6 + 575*b5 - 575*b4 $$\nu^{6}$$ $$=$$ $$-1423\beta_{2} - 12319\beta_1$$ -1423*b2 - 12319*b1 $$\nu^{7}$$ $$=$$ $$20857\beta_{7} + 9961\beta_{6} + 30818\beta_{5} + 20857\beta_{4}$$ 20857*b7 + 9961*b6 + 30818*b5 + 20857*b4

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.16183 − 1.16183i 4.26030 − 4.26030i −4.26030 + 4.26030i −1.16183 + 1.16183i −1.16183 − 1.16183i −4.26030 − 4.26030i 4.26030 + 4.26030i 1.16183 + 1.16183i
2.00000i 6.02497i −4.00000 −7.18680 + 8.56445i −12.0499 0 8.00000i −9.30030 17.1289 + 14.3736i
99.2 2.00000i 1.64308i −4.00000 −5.90338 9.49474i −3.28615 0 8.00000i 24.3003 −18.9895 + 11.8068i
99.3 2.00000i 1.64308i −4.00000 5.90338 + 9.49474i 3.28615 0 8.00000i 24.3003 18.9895 11.8068i
99.4 2.00000i 6.02497i −4.00000 7.18680 8.56445i 12.0499 0 8.00000i −9.30030 −17.1289 14.3736i
99.5 2.00000i 6.02497i −4.00000 7.18680 + 8.56445i 12.0499 0 8.00000i −9.30030 −17.1289 + 14.3736i
99.6 2.00000i 1.64308i −4.00000 5.90338 9.49474i 3.28615 0 8.00000i 24.3003 18.9895 + 11.8068i
99.7 2.00000i 1.64308i −4.00000 −5.90338 + 9.49474i −3.28615 0 8.00000i 24.3003 −18.9895 11.8068i
99.8 2.00000i 6.02497i −4.00000 −7.18680 8.56445i −12.0499 0 8.00000i −9.30030 17.1289 14.3736i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.8 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.4.c.d 8
5.b even 2 1 inner 490.4.c.d 8
5.c odd 4 1 2450.4.a.cm 4
5.c odd 4 1 2450.4.a.cs 4
7.b odd 2 1 inner 490.4.c.d 8
35.c odd 2 1 inner 490.4.c.d 8
35.f even 4 1 2450.4.a.cm 4
35.f even 4 1 2450.4.a.cs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.4.c.d 8 1.a even 1 1 trivial
490.4.c.d 8 5.b even 2 1 inner
490.4.c.d 8 7.b odd 2 1 inner
490.4.c.d 8 35.c odd 2 1 inner
2450.4.a.cm 4 5.c odd 4 1
2450.4.a.cm 4 35.f even 4 1
2450.4.a.cs 4 5.c odd 4 1
2450.4.a.cs 4 35.f even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 39T_{3}^{2} + 98$$ T3^4 + 39*T3^2 + 98 $$T_{19}^{4} - 20794T_{19}^{2} + 15103008$$ T19^4 - 20794*T19^2 + 15103008

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{4}$$
$3$ $$(T^{4} + 39 T^{2} + 98)^{2}$$
$5$ $$T^{8} + 154 T^{6} + \cdots + 244140625$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 9 T - 2520)^{4}$$
$13$ $$(T^{4} + 463 T^{2} + 39762)^{2}$$
$17$ $$(T^{4} + 11349 T^{2} + 1648928)^{2}$$
$19$ $$(T^{4} - 20794 T^{2} + 15103008)^{2}$$
$23$ $$(T^{4} + 56788 T^{2} + \cdots + 787139136)^{2}$$
$29$ $$(T^{2} - 215 T + 11274)^{4}$$
$31$ $$(T^{4} - 118648 T^{2} + \cdots + 757694592)^{2}$$
$37$ $$(T^{4} + 91764 T^{2} + \cdots + 653313600)^{2}$$
$41$ $$(T^{4} - 124408 T^{2} + \cdots + 1914072192)^{2}$$
$43$ $$(T^{4} + 42400 T^{2} + \cdots + 222845184)^{2}$$
$47$ $$(T^{4} + 83381 T^{2} + \cdots + 1636148808)^{2}$$
$53$ $$(T^{4} + 44964 T^{2} + 4665600)^{2}$$
$59$ $$(T^{4} - 805194 T^{2} + \cdots + 80763412608)^{2}$$
$61$ $$(T^{4} - 872458 T^{2} + \cdots + 190295143200)^{2}$$
$67$ $$(T^{4} + 329344 T^{2} + \cdots + 406425600)^{2}$$
$71$ $$(T^{2} + 882 T + 103032)^{4}$$
$73$ $$(T^{4} + 1498012 T^{2} + \cdots + 195782782752)^{2}$$
$79$ $$(T^{2} + 771 T - 237790)^{4}$$
$83$ $$(T^{4} + 914954 T^{2} + \cdots + 96227090208)^{2}$$
$89$ $$(T^{4} - 1138384 T^{2} + \cdots + 16460236800)^{2}$$
$97$ $$(T^{4} + 2463325 T^{2} + \cdots + 711385920000)^{2}$$
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