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:

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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