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: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 2 i q^{3} -4 q^{4} + ( 5 + 10 i ) q^{5} -4 q^{6} -8 i q^{8} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{2} + 2 i q^{3} -4 q^{4} + ( 5 + 10 i ) q^{5} -4 q^{6} -8 i q^{8} + 23 q^{9} + ( -20 + 10 i ) q^{10} -28 q^{11} -8 i q^{12} + 12 i q^{13} + ( -20 + 10 i ) q^{15} + 16 q^{16} + 64 i q^{17} + 46 i q^{18} -60 q^{19} + ( -20 - 40 i ) q^{20} -56 i q^{22} + 58 i q^{23} + 16 q^{24} + ( -75 + 100 i ) q^{25} -24 q^{26} + 100 i q^{27} -90 q^{29} + ( -20 - 40 i ) q^{30} + 128 q^{31} + 32 i q^{32} -56 i q^{33} -128 q^{34} -92 q^{36} + 236 i q^{37} -120 i q^{38} -24 q^{39} + ( 80 - 40 i ) q^{40} -242 q^{41} -362 i q^{43} + 112 q^{44} + ( 115 + 230 i ) q^{45} -116 q^{46} -226 i q^{47} + 32 i q^{48} + ( -200 - 150 i ) q^{50} -128 q^{51} -48 i q^{52} + 108 i q^{53} -200 q^{54} + ( -140 - 280 i ) q^{55} -120 i q^{57} -180 i q^{58} -20 q^{59} + ( 80 - 40 i ) q^{60} -542 q^{61} + 256 i q^{62} -64 q^{64} + ( -120 + 60 i ) q^{65} + 112 q^{66} -434 i q^{67} -256 i q^{68} -116 q^{69} -1128 q^{71} -184 i q^{72} + 632 i q^{73} -472 q^{74} + ( -200 - 150 i ) q^{75} + 240 q^{76} -48 i q^{78} + 720 q^{79} + ( 80 + 160 i ) q^{80} + 421 q^{81} -484 i q^{82} -478 i q^{83} + ( -640 + 320 i ) q^{85} + 724 q^{86} -180 i q^{87} + 224 i q^{88} -490 q^{89} + ( -460 + 230 i ) q^{90} -232 i q^{92} + 256 i q^{93} + 452 q^{94} + ( -300 - 600 i ) q^{95} -64 q^{96} -1456 i q^{97} -644 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 10 q^{5} - 8 q^{6} + 46 q^{9} + O(q^{10}) \) \( 2 q - 8 q^{4} + 10 q^{5} - 8 q^{6} + 46 q^{9} - 40 q^{10} - 56 q^{11} - 40 q^{15} + 32 q^{16} - 120 q^{19} - 40 q^{20} + 32 q^{24} - 150 q^{25} - 48 q^{26} - 180 q^{29} - 40 q^{30} + 256 q^{31} - 256 q^{34} - 184 q^{36} - 48 q^{39} + 160 q^{40} - 484 q^{41} + 224 q^{44} + 230 q^{45} - 232 q^{46} - 400 q^{50} - 256 q^{51} - 400 q^{54} - 280 q^{55} - 40 q^{59} + 160 q^{60} - 1084 q^{61} - 128 q^{64} - 240 q^{65} + 224 q^{66} - 232 q^{69} - 2256 q^{71} - 944 q^{74} - 400 q^{75} + 480 q^{76} + 1440 q^{79} + 160 q^{80} + 842 q^{81} - 1280 q^{85} + 1448 q^{86} - 980 q^{89} - 920 q^{90} + 904 q^{94} - 600 q^{95} - 128 q^{96} - 1288 q^{99} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{19} + 60 \)

Hecke characteristic polynomials

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