# Properties

 Label 490.2.c.c Level 490 Weight 2 Character orbit 490.c Analytic conductor 3.913 Analytic rank 0 Dimension 2 CM no Inner twists 2

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.91266969904$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 3 i q^{3} - q^{4} + ( 1 + 2 i ) q^{5} -3 q^{6} -i q^{8} -6 q^{9} +O(q^{10})$$ $$q + i q^{2} + 3 i q^{3} - q^{4} + ( 1 + 2 i ) q^{5} -3 q^{6} -i q^{8} -6 q^{9} + ( -2 + i ) q^{10} -3 i q^{12} + 2 i q^{13} + ( -6 + 3 i ) q^{15} + q^{16} -2 i q^{17} -6 i q^{18} + 2 q^{19} + ( -1 - 2 i ) q^{20} + i q^{23} + 3 q^{24} + ( -3 + 4 i ) q^{25} -2 q^{26} -9 i q^{27} + q^{29} + ( -3 - 6 i ) q^{30} + 10 q^{31} + i q^{32} + 2 q^{34} + 6 q^{36} -8 i q^{37} + 2 i q^{38} -6 q^{39} + ( 2 - i ) q^{40} -3 q^{41} -5 i q^{43} + ( -6 - 12 i ) q^{45} - q^{46} + 8 i q^{47} + 3 i q^{48} + ( -4 - 3 i ) q^{50} + 6 q^{51} -2 i q^{52} + 6 i q^{53} + 9 q^{54} + 6 i q^{57} + i q^{58} -2 q^{59} + ( 6 - 3 i ) q^{60} -9 q^{61} + 10 i q^{62} - q^{64} + ( -4 + 2 i ) q^{65} -7 i q^{67} + 2 i q^{68} -3 q^{69} + 6 q^{71} + 6 i q^{72} + 10 i q^{73} + 8 q^{74} + ( -12 - 9 i ) q^{75} -2 q^{76} -6 i q^{78} + 10 q^{79} + ( 1 + 2 i ) q^{80} + 9 q^{81} -3 i q^{82} + 9 i q^{83} + ( 4 - 2 i ) q^{85} + 5 q^{86} + 3 i q^{87} + 7 q^{89} + ( 12 - 6 i ) q^{90} -i q^{92} + 30 i q^{93} -8 q^{94} + ( 2 + 4 i ) q^{95} -3 q^{96} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{5} - 6q^{6} - 12q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{5} - 6q^{6} - 12q^{9} - 4q^{10} - 12q^{15} + 2q^{16} + 4q^{19} - 2q^{20} + 6q^{24} - 6q^{25} - 4q^{26} + 2q^{29} - 6q^{30} + 20q^{31} + 4q^{34} + 12q^{36} - 12q^{39} + 4q^{40} - 6q^{41} - 12q^{45} - 2q^{46} - 8q^{50} + 12q^{51} + 18q^{54} - 4q^{59} + 12q^{60} - 18q^{61} - 2q^{64} - 8q^{65} - 6q^{69} + 12q^{71} + 16q^{74} - 24q^{75} - 4q^{76} + 20q^{79} + 2q^{80} + 18q^{81} + 8q^{85} + 10q^{86} + 14q^{89} + 24q^{90} - 16q^{94} + 4q^{95} - 6q^{96} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
1.00000i 3.00000i −1.00000 1.00000 2.00000i −3.00000 0 1.00000i −6.00000 −2.00000 1.00000i
99.2 1.00000i 3.00000i −1.00000 1.00000 + 2.00000i −3.00000 0 1.00000i −6.00000 −2.00000 + 1.00000i
 $$n$$: 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 490.2.c.c 2
5.b even 2 1 inner 490.2.c.c 2
5.c odd 4 1 2450.2.a.r 1
5.c odd 4 1 2450.2.a.s 1
7.b odd 2 1 490.2.c.b 2
7.c even 3 2 70.2.i.a 4
7.d odd 6 2 490.2.i.b 4
21.h odd 6 2 630.2.u.b 4
28.g odd 6 2 560.2.bw.c 4
35.c odd 2 1 490.2.c.b 2
35.f even 4 1 2450.2.a.c 1
35.f even 4 1 2450.2.a.bh 1
35.i odd 6 2 490.2.i.b 4
35.j even 6 2 70.2.i.a 4
35.l odd 12 2 350.2.e.f 2
35.l odd 12 2 350.2.e.g 2
105.o odd 6 2 630.2.u.b 4
140.p odd 6 2 560.2.bw.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 7.c even 3 2
70.2.i.a 4 35.j even 6 2
350.2.e.f 2 35.l odd 12 2
350.2.e.g 2 35.l odd 12 2
490.2.c.b 2 7.b odd 2 1
490.2.c.b 2 35.c odd 2 1
490.2.c.c 2 1.a even 1 1 trivial
490.2.c.c 2 5.b even 2 1 inner
490.2.i.b 4 7.d odd 6 2
490.2.i.b 4 35.i odd 6 2
560.2.bw.c 4 28.g odd 6 2
560.2.bw.c 4 140.p odd 6 2
630.2.u.b 4 21.h odd 6 2
630.2.u.b 4 105.o odd 6 2
2450.2.a.c 1 35.f even 4 1
2450.2.a.r 1 5.c odd 4 1
2450.2.a.s 1 5.c odd 4 1
2450.2.a.bh 1 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 $$S_{2}^{\mathrm{new}}(490, [\chi])$$:

 $$T_{3}^{2} + 9$$ $$T_{11}$$ $$T_{19} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ 1
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 - 45 T^{2} + 529 T^{4}$$
$29$ $$( 1 - T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$1 - 61 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 2 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 9 T + 61 T^{2} )^{2}$$
$67$ $$1 - 85 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 46 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{2}$$
$83$ $$1 - 85 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 7 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$
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