Properties

Label 490.3.f.b
Level $490$
Weight $3$
Character orbit 490.f
Analytic conductor $13.352$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 490.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.3515329537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{2} + ( 2 - 2 i ) q^{3} + 2 i q^{4} + 5 i q^{5} -4 q^{6} + ( 2 - 2 i ) q^{8} + i q^{9} +O(q^{10})\) \( q + ( -1 - i ) q^{2} + ( 2 - 2 i ) q^{3} + 2 i q^{4} + 5 i q^{5} -4 q^{6} + ( 2 - 2 i ) q^{8} + i q^{9} + ( 5 - 5 i ) q^{10} -8 q^{11} + ( 4 + 4 i ) q^{12} + ( -3 + 3 i ) q^{13} + ( 10 + 10 i ) q^{15} -4 q^{16} + ( -7 - 7 i ) q^{17} + ( 1 - i ) q^{18} + 20 i q^{19} -10 q^{20} + ( 8 + 8 i ) q^{22} + ( -2 + 2 i ) q^{23} -8 i q^{24} -25 q^{25} + 6 q^{26} + ( 20 + 20 i ) q^{27} + 40 i q^{29} -20 i q^{30} -52 q^{31} + ( 4 + 4 i ) q^{32} + ( -16 + 16 i ) q^{33} + 14 i q^{34} -2 q^{36} + ( -3 - 3 i ) q^{37} + ( 20 - 20 i ) q^{38} + 12 i q^{39} + ( 10 + 10 i ) q^{40} + 8 q^{41} + ( -42 + 42 i ) q^{43} -16 i q^{44} -5 q^{45} + 4 q^{46} + ( 18 + 18 i ) q^{47} + ( -8 + 8 i ) q^{48} + ( 25 + 25 i ) q^{50} -28 q^{51} + ( -6 - 6 i ) q^{52} + ( 53 - 53 i ) q^{53} -40 i q^{54} -40 i q^{55} + ( 40 + 40 i ) q^{57} + ( 40 - 40 i ) q^{58} + 20 i q^{59} + ( -20 + 20 i ) q^{60} + 48 q^{61} + ( 52 + 52 i ) q^{62} -8 i q^{64} + ( -15 - 15 i ) q^{65} + 32 q^{66} + ( 62 + 62 i ) q^{67} + ( 14 - 14 i ) q^{68} + 8 i q^{69} -28 q^{71} + ( 2 + 2 i ) q^{72} + ( 47 - 47 i ) q^{73} + 6 i q^{74} + ( -50 + 50 i ) q^{75} -40 q^{76} + ( 12 - 12 i ) q^{78} -20 i q^{80} + 71 q^{81} + ( -8 - 8 i ) q^{82} + ( -18 + 18 i ) q^{83} + ( 35 - 35 i ) q^{85} + 84 q^{86} + ( 80 + 80 i ) q^{87} + ( -16 + 16 i ) q^{88} -80 i q^{89} + ( 5 + 5 i ) q^{90} + ( -4 - 4 i ) q^{92} + ( -104 + 104 i ) q^{93} -36 i q^{94} -100 q^{95} + 16 q^{96} + ( 63 + 63 i ) q^{97} -8 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{3} - 8q^{6} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{3} - 8q^{6} + 4q^{8} + 10q^{10} - 16q^{11} + 8q^{12} - 6q^{13} + 20q^{15} - 8q^{16} - 14q^{17} + 2q^{18} - 20q^{20} + 16q^{22} - 4q^{23} - 50q^{25} + 12q^{26} + 40q^{27} - 104q^{31} + 8q^{32} - 32q^{33} - 4q^{36} - 6q^{37} + 40q^{38} + 20q^{40} + 16q^{41} - 84q^{43} - 10q^{45} + 8q^{46} + 36q^{47} - 16q^{48} + 50q^{50} - 56q^{51} - 12q^{52} + 106q^{53} + 80q^{57} + 80q^{58} - 40q^{60} + 96q^{61} + 104q^{62} - 30q^{65} + 64q^{66} + 124q^{67} + 28q^{68} - 56q^{71} + 4q^{72} + 94q^{73} - 100q^{75} - 80q^{76} + 24q^{78} + 142q^{81} - 16q^{82} - 36q^{83} + 70q^{85} + 168q^{86} + 160q^{87} - 32q^{88} + 10q^{90} - 8q^{92} - 208q^{93} - 200q^{95} + 32q^{96} + 126q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.00000i
1.00000i
−1.00000 1.00000i 2.00000 2.00000i 2.00000i 5.00000i −4.00000 0 2.00000 2.00000i 1.00000i 5.00000 5.00000i
393.1 −1.00000 + 1.00000i 2.00000 + 2.00000i 2.00000i 5.00000i −4.00000 0 2.00000 + 2.00000i 1.00000i 5.00000 + 5.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.3.f.b 2
5.c odd 4 1 inner 490.3.f.b 2
7.b odd 2 1 10.3.c.a 2
21.c even 2 1 90.3.g.b 2
28.d even 2 1 80.3.p.c 2
35.c odd 2 1 50.3.c.c 2
35.f even 4 1 10.3.c.a 2
35.f even 4 1 50.3.c.c 2
56.e even 2 1 320.3.p.a 2
56.h odd 2 1 320.3.p.h 2
84.h odd 2 1 720.3.bh.c 2
105.g even 2 1 450.3.g.b 2
105.k odd 4 1 90.3.g.b 2
105.k odd 4 1 450.3.g.b 2
140.c even 2 1 400.3.p.b 2
140.j odd 4 1 80.3.p.c 2
140.j odd 4 1 400.3.p.b 2
280.s even 4 1 320.3.p.h 2
280.y odd 4 1 320.3.p.a 2
420.w even 4 1 720.3.bh.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 7.b odd 2 1
10.3.c.a 2 35.f even 4 1
50.3.c.c 2 35.c odd 2 1
50.3.c.c 2 35.f even 4 1
80.3.p.c 2 28.d even 2 1
80.3.p.c 2 140.j odd 4 1
90.3.g.b 2 21.c even 2 1
90.3.g.b 2 105.k odd 4 1
320.3.p.a 2 56.e even 2 1
320.3.p.a 2 280.y odd 4 1
320.3.p.h 2 56.h odd 2 1
320.3.p.h 2 280.s even 4 1
400.3.p.b 2 140.c even 2 1
400.3.p.b 2 140.j odd 4 1
450.3.g.b 2 105.g even 2 1
450.3.g.b 2 105.k odd 4 1
490.3.f.b 2 1.a even 1 1 trivial
490.3.f.b 2 5.c odd 4 1 inner
720.3.bh.c 2 84.h odd 2 1
720.3.bh.c 2 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} - 4 T_{3} + 8 \)
\( T_{11} + 8 \)