Properties

Label 490.2.c.a
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -i q^{8} + 3 q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -i q^{8} + 3 q^{9} + ( 1 - 2 i ) q^{10} + 3 q^{11} + 5 i q^{13} + q^{16} -2 i q^{17} + 3 i q^{18} + 5 q^{19} + ( 2 + i ) q^{20} + 3 i q^{22} + 7 i q^{23} + ( 3 + 4 i ) q^{25} -5 q^{26} + 4 q^{29} -2 q^{31} + i q^{32} + 2 q^{34} -3 q^{36} + i q^{37} + 5 i q^{38} + ( -1 + 2 i ) q^{40} + 3 q^{41} -2 i q^{43} -3 q^{44} + ( -6 - 3 i ) q^{45} -7 q^{46} -7 i q^{47} + ( -4 + 3 i ) q^{50} -5 i q^{52} -9 i q^{53} + ( -6 - 3 i ) q^{55} + 4 i q^{58} + 4 q^{59} + 6 q^{61} -2 i q^{62} - q^{64} + ( 5 - 10 i ) q^{65} + 2 i q^{67} + 2 i q^{68} -6 q^{71} -3 i q^{72} + 16 i q^{73} - q^{74} -5 q^{76} -14 q^{79} + ( -2 - i ) q^{80} + 9 q^{81} + 3 i q^{82} + 6 i q^{83} + ( -2 + 4 i ) q^{85} + 2 q^{86} -3 i q^{88} -2 q^{89} + ( 3 - 6 i ) q^{90} -7 i q^{92} + 7 q^{94} + ( -10 - 5 i ) q^{95} -12 i q^{97} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{5} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{5} + 6q^{9} + 2q^{10} + 6q^{11} + 2q^{16} + 10q^{19} + 4q^{20} + 6q^{25} - 10q^{26} + 8q^{29} - 4q^{31} + 4q^{34} - 6q^{36} - 2q^{40} + 6q^{41} - 6q^{44} - 12q^{45} - 14q^{46} - 8q^{50} - 12q^{55} + 8q^{59} + 12q^{61} - 2q^{64} + 10q^{65} - 12q^{71} - 2q^{74} - 10q^{76} - 28q^{79} - 4q^{80} + 18q^{81} - 4q^{85} + 4q^{86} - 4q^{89} + 6q^{90} + 14q^{94} - 20q^{95} + 18q^{99} + O(q^{100}) \)

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
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 0 1.00000i 3.00000 1.00000 + 2.00000i
99.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 0 1.00000i 3.00000 1.00000 2.00000i
\(n\): 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.a 2
5.b even 2 1 inner 490.2.c.a 2
5.c odd 4 1 2450.2.a.k 1
5.c odd 4 1 2450.2.a.ba 1
7.b odd 2 1 490.2.c.d 2
7.c even 3 2 70.2.i.b 4
7.d odd 6 2 490.2.i.a 4
21.h odd 6 2 630.2.u.a 4
28.g odd 6 2 560.2.bw.d 4
35.c odd 2 1 490.2.c.d 2
35.f even 4 1 2450.2.a.j 1
35.f even 4 1 2450.2.a.bb 1
35.i odd 6 2 490.2.i.a 4
35.j even 6 2 70.2.i.b 4
35.l odd 12 2 350.2.e.c 2
35.l odd 12 2 350.2.e.j 2
105.o odd 6 2 630.2.u.a 4
140.p odd 6 2 560.2.bw.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 7.c even 3 2
70.2.i.b 4 35.j even 6 2
350.2.e.c 2 35.l odd 12 2
350.2.e.j 2 35.l odd 12 2
490.2.c.a 2 1.a even 1 1 trivial
490.2.c.a 2 5.b even 2 1 inner
490.2.c.d 2 7.b odd 2 1
490.2.c.d 2 35.c odd 2 1
490.2.i.a 4 7.d odd 6 2
490.2.i.a 4 35.i odd 6 2
560.2.bw.d 4 28.g odd 6 2
560.2.bw.d 4 140.p odd 6 2
630.2.u.a 4 21.h odd 6 2
630.2.u.a 4 105.o odd 6 2
2450.2.a.j 1 35.f even 4 1
2450.2.a.k 1 5.c odd 4 1
2450.2.a.ba 1 5.c odd 4 1
2450.2.a.bb 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 \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3} \)
\( T_{11} - 3 \)
\( T_{19} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 49 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 49 + T^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 256 + T^{2} \)
$79$ \( ( 14 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 144 + T^{2} \)
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