Properties

Label 490.4.c.b
Level $490$
Weight $4$
Character orbit 490.c
Analytic conductor $28.911$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + \beta q^{3} - 4 q^{4} + (5 \beta + 5) q^{5} - 4 q^{6} - 4 \beta q^{8} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + \beta q^{3} - 4 q^{4} + (5 \beta + 5) q^{5} - 4 q^{6} - 4 \beta q^{8} + 23 q^{9} + (5 \beta - 20) q^{10} - 28 q^{11} - 4 \beta q^{12} + 6 \beta q^{13} + (5 \beta - 20) q^{15} + 16 q^{16} + 32 \beta q^{17} + 23 \beta q^{18} - 60 q^{19} + ( - 20 \beta - 20) q^{20} - 28 \beta q^{22} + 29 \beta q^{23} + 16 q^{24} + (50 \beta - 75) q^{25} - 24 q^{26} + 50 \beta q^{27} - 90 q^{29} + ( - 20 \beta - 20) q^{30} + 128 q^{31} + 16 \beta q^{32} - 28 \beta q^{33} - 128 q^{34} - 92 q^{36} + 118 \beta q^{37} - 60 \beta q^{38} - 24 q^{39} + ( - 20 \beta + 80) q^{40} - 242 q^{41} - 181 \beta q^{43} + 112 q^{44} + (115 \beta + 115) q^{45} - 116 q^{46} - 113 \beta q^{47} + 16 \beta q^{48} + ( - 75 \beta - 200) q^{50} - 128 q^{51} - 24 \beta q^{52} + 54 \beta q^{53} - 200 q^{54} + ( - 140 \beta - 140) q^{55} - 60 \beta q^{57} - 90 \beta q^{58} - 20 q^{59} + ( - 20 \beta + 80) q^{60} - 542 q^{61} + 128 \beta q^{62} - 64 q^{64} + (30 \beta - 120) q^{65} + 112 q^{66} - 217 \beta q^{67} - 128 \beta q^{68} - 116 q^{69} - 1128 q^{71} - 92 \beta q^{72} + 316 \beta q^{73} - 472 q^{74} + ( - 75 \beta - 200) q^{75} + 240 q^{76} - 24 \beta q^{78} + 720 q^{79} + (80 \beta + 80) q^{80} + 421 q^{81} - 242 \beta q^{82} - 239 \beta q^{83} + (160 \beta - 640) q^{85} + 724 q^{86} - 90 \beta q^{87} + 112 \beta q^{88} - 490 q^{89} + (115 \beta - 460) q^{90} - 116 \beta q^{92} + 128 \beta q^{93} + 452 q^{94} + ( - 300 \beta - 300) q^{95} - 64 q^{96} - 728 \beta q^{97} - 644 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 10 q^{5} - 8 q^{6} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 10 q^{5} - 8 q^{6} + 46 q^{9} - 40 q^{10} - 56 q^{11} - 40 q^{15} + 32 q^{16} - 120 q^{19} - 40 q^{20} + 32 q^{24} - 150 q^{25} - 48 q^{26} - 180 q^{29} - 40 q^{30} + 256 q^{31} - 256 q^{34} - 184 q^{36} - 48 q^{39} + 160 q^{40} - 484 q^{41} + 224 q^{44} + 230 q^{45} - 232 q^{46} - 400 q^{50} - 256 q^{51} - 400 q^{54} - 280 q^{55} - 40 q^{59} + 160 q^{60} - 1084 q^{61} - 128 q^{64} - 240 q^{65} + 224 q^{66} - 232 q^{69} - 2256 q^{71} - 944 q^{74} - 400 q^{75} + 480 q^{76} + 1440 q^{79} + 160 q^{80} + 842 q^{81} - 1280 q^{85} + 1448 q^{86} - 980 q^{89} - 920 q^{90} + 904 q^{94} - 600 q^{95} - 128 q^{96} - 1288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2.00000i 2.00000i −4.00000 5.00000 10.0000i −4.00000 0 8.00000i 23.0000 −20.0000 10.0000i
99.2 2.00000i 2.00000i −4.00000 5.00000 + 10.0000i −4.00000 0 8.00000i 23.0000 −20.0000 + 10.0000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.4.c.b 2
5.b even 2 1 inner 490.4.c.b 2
5.c odd 4 1 2450.4.a.o 1
5.c odd 4 1 2450.4.a.bb 1
7.b odd 2 1 10.4.b.a 2
21.c even 2 1 90.4.c.b 2
28.d even 2 1 80.4.c.a 2
35.c odd 2 1 10.4.b.a 2
35.f even 4 1 50.4.a.b 1
35.f even 4 1 50.4.a.d 1
56.e even 2 1 320.4.c.c 2
56.h odd 2 1 320.4.c.d 2
84.h odd 2 1 720.4.f.f 2
105.g even 2 1 90.4.c.b 2
105.k odd 4 1 450.4.a.j 1
105.k odd 4 1 450.4.a.k 1
140.c even 2 1 80.4.c.a 2
140.j odd 4 1 400.4.a.h 1
140.j odd 4 1 400.4.a.n 1
280.c odd 2 1 320.4.c.d 2
280.n even 2 1 320.4.c.c 2
280.s even 4 1 1600.4.a.u 1
280.s even 4 1 1600.4.a.bh 1
280.y odd 4 1 1600.4.a.t 1
280.y odd 4 1 1600.4.a.bg 1
420.o odd 2 1 720.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 7.b odd 2 1
10.4.b.a 2 35.c odd 2 1
50.4.a.b 1 35.f even 4 1
50.4.a.d 1 35.f even 4 1
80.4.c.a 2 28.d even 2 1
80.4.c.a 2 140.c even 2 1
90.4.c.b 2 21.c even 2 1
90.4.c.b 2 105.g even 2 1
320.4.c.c 2 56.e even 2 1
320.4.c.c 2 280.n even 2 1
320.4.c.d 2 56.h odd 2 1
320.4.c.d 2 280.c odd 2 1
400.4.a.h 1 140.j odd 4 1
400.4.a.n 1 140.j odd 4 1
450.4.a.j 1 105.k odd 4 1
450.4.a.k 1 105.k odd 4 1
490.4.c.b 2 1.a even 1 1 trivial
490.4.c.b 2 5.b even 2 1 inner
720.4.f.f 2 84.h odd 2 1
720.4.f.f 2 420.o odd 2 1
1600.4.a.t 1 280.y odd 4 1
1600.4.a.u 1 280.s even 4 1
1600.4.a.bg 1 280.y odd 4 1
1600.4.a.bh 1 280.s even 4 1
2450.4.a.o 1 5.c odd 4 1
2450.4.a.bb 1 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} + 60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 10T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 144 \) Copy content Toggle raw display
$17$ \( T^{2} + 4096 \) Copy content Toggle raw display
$19$ \( (T + 60)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3364 \) Copy content Toggle raw display
$29$ \( (T + 90)^{2} \) Copy content Toggle raw display
$31$ \( (T - 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 55696 \) Copy content Toggle raw display
$41$ \( (T + 242)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 131044 \) Copy content Toggle raw display
$47$ \( T^{2} + 51076 \) Copy content Toggle raw display
$53$ \( T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( (T + 20)^{2} \) Copy content Toggle raw display
$61$ \( (T + 542)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 188356 \) Copy content Toggle raw display
$71$ \( (T + 1128)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 399424 \) Copy content Toggle raw display
$79$ \( (T - 720)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 228484 \) Copy content Toggle raw display
$89$ \( (T + 490)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2119936 \) Copy content Toggle raw display
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