Newform orbit 490.4.a.s

• Label: 490.4.a.s
• Level: $490$
• Weight: $4$
• Character orbit: 490.a
• Self dual: yes
• Analytic conductor: $28.911$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(28.9109359028\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + (3*b + 1) * q^3 + 4 * q^4 - 5 * q^5 + (-6*b - 2) * q^6 - 8 * q^8 + (6*b - 8) * q^9 + 10 * q^10 + (8*b - 13) * q^11 + (12*b + 4) * q^12 + (9*b - 51) * q^13 + (-15*b - 5) * q^15 + 16 * q^16 + (-17*b + 93) * q^17 + (-12*b + 16) * q^18 + (16*b + 18) * q^19 - 20 * q^20 + (-16*b + 26) * q^22 + (101*b - 22) * q^23 + (-24*b - 8) * q^24 + 25 * q^25 + (-18*b + 102) * q^26 + (-99*b + 1) * q^27 + (72*b - 23) * q^29 + (30*b + 10) * q^30 + (113*b + 70) * q^31 - 32 * q^32 + (-31*b + 35) * q^33 + (34*b - 186) * q^34 + (24*b - 32) * q^36 + (197*b + 66) * q^37 + (-32*b - 36) * q^38 + (-144*b + 3) * q^39 + 40 * q^40 + (145*b - 66) * q^41 + (44*b + 162) * q^43 + (32*b - 52) * q^44 + (-30*b + 40) * q^45 + (-202*b + 44) * q^46 + (-59*b + 121) * q^47 + (48*b + 16) * q^48 - 50 * q^50 + (262*b - 9) * q^51 + (36*b - 204) * q^52 + (-227*b + 104) * q^53 + (198*b - 2) * q^54 + (-40*b + 65) * q^55 + (70*b + 114) * q^57 + (-144*b + 46) * q^58 + (-129*b + 390) * q^59 + (-60*b - 20) * q^60 + (-142*b + 108) * q^61 + (-226*b - 140) * q^62 + 64 * q^64 + (-45*b + 255) * q^65 + (62*b - 70) * q^66 + (-527*b - 128) * q^67 + (-68*b + 372) * q^68 + (35*b + 584) * q^69 + (-106*b - 346) * q^71 + (-48*b + 64) * q^72 + (-38*b + 416) * q^73 + (-394*b - 132) * q^74 + (75*b + 25) * q^75 + (64*b + 72) * q^76 + (288*b - 6) * q^78 + (172*b - 981) * q^79 - 80 * q^80 + (-258*b - 377) * q^81 + (-290*b + 132) * q^82 + (416*b + 856) * q^83 + (85*b - 465) * q^85 + (-88*b - 324) * q^86 + (3*b + 409) * q^87 + (-64*b + 104) * q^88 + (276*b + 980) * q^89 + (60*b - 80) * q^90 + (404*b - 88) * q^92 + (323*b + 748) * q^93 + (118*b - 242) * q^94 + (-80*b - 90) * q^95 + (-96*b - 32) * q^96 + (-829*b + 51) * q^97 + (-142*b + 200) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 - 4 * q^6 - 16 * q^8 - 16 * q^9 + 20 * q^10 - 26 * q^11 + 8 * q^12 - 102 * q^13 - 10 * q^15 + 32 * q^16 + 186 * q^17 + 32 * q^18 + 36 * q^19 - 40 * q^20 + 52 * q^22 - 44 * q^23 - 16 * q^24 + 50 * q^25 + 204 * q^26 + 2 * q^27 - 46 * q^29 + 20 * q^30 + 140 * q^31 - 64 * q^32 + 70 * q^33 - 372 * q^34 - 64 * q^36 + 132 * q^37 - 72 * q^38 + 6 * q^39 + 80 * q^40 - 132 * q^41 + 324 * q^43 - 104 * q^44 + 80 * q^45 + 88 * q^46 + 242 * q^47 + 32 * q^48 - 100 * q^50 - 18 * q^51 - 408 * q^52 + 208 * q^53 - 4 * q^54 + 130 * q^55 + 228 * q^57 + 92 * q^58 + 780 * q^59 - 40 * q^60 + 216 * q^61 - 280 * q^62 + 128 * q^64 + 510 * q^65 - 140 * q^66 - 256 * q^67 + 744 * q^68 + 1168 * q^69 - 692 * q^71 + 128 * q^72 + 832 * q^73 - 264 * q^74 + 50 * q^75 + 144 * q^76 - 12 * q^78 - 1962 * q^79 - 160 * q^80 - 754 * q^81 + 264 * q^82 + 1712 * q^83 - 930 * q^85 - 648 * q^86 + 818 * q^87 + 208 * q^88 + 1960 * q^89 - 160 * q^90 - 176 * q^92 + 1496 * q^93 - 484 * q^94 - 180 * q^95 - 64 * q^96 + 102 * q^97 + 400 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

−2.00000

−3.24264

4.00000

−5.00000

6.48528

0

−8.00000

−16.4853

10.0000

1.2

−2.00000

5.24264

4.00000

−5.00000

−10.4853

0

−8.00000

0.485281

10.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(5\)	\( +1 \)
\(7\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	490.4.a.s	yes	2
5.b	even	2	1		2450.4.a.bu		2
7.b	odd	2	1		490.4.a.q	✓	2
7.c	even	3	2		490.4.e.v		4
7.d	odd	6	2		490.4.e.w		4
35.c	odd	2	1		2450.4.a.by		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
490.4.a.q	✓	2	7.b	odd	2	1
490.4.a.s	yes	2	1.a	even	1	1	trivial
490.4.e.v		4	7.c	even	3	2
490.4.e.w		4	7.d	odd	6	2
2450.4.a.bu		2	5.b	even	2	1
2450.4.a.by		2	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(490))\):

\( T_{3}^{2} - 2T_{3} - 17 \)

\( T_{11}^{2} + 26T_{11} + 41 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} - 2T - 17 \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 26T + 41 \)