Newform orbit 490.4.a.p

• Label: 490.4.a.p
• Level: $490$
• Weight: $4$
• Character orbit: 490.a
• Self dual: yes
• Analytic conductor: $28.911$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{177}) \)
Defining polynomial:	\( x^{2} - x - 44 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + (-b - 2) * q^3 + 4 * q^4 - 5 * q^5 + (2*b + 4) * q^6 - 8 * q^8 + (5*b + 21) * q^9 + 10 * q^10 - 5*b * q^11 + (-4*b - 8) * q^12 + (-3*b - 46) * q^13 + (5*b + 10) * q^15 + 16 * q^16 + (b + 12) * q^17 + (-10*b - 42) * q^18 + (4*b + 58) * q^19 - 20 * q^20 + 10*b * q^22 + (20*b + 52) * q^23 + (8*b + 16) * q^24 + 25 * q^25 + (6*b + 92) * q^26 + (-9*b - 208) * q^27 + (25*b + 94) * q^29 + (-10*b - 20) * q^30 + (2*b - 36) * q^31 - 32 * q^32 + (15*b + 220) * q^33 + (-2*b - 24) * q^34 + (20*b + 84) * q^36 + (30*b - 82) * q^37 + (-8*b - 116) * q^38 + (55*b + 224) * q^39 + 40 * q^40 + (38*b - 204) * q^41 + (-10*b - 36) * q^43 - 20*b * q^44 + (-25*b - 105) * q^45 + (-40*b - 104) * q^46 + (-83*b - 16) * q^47 + (-16*b - 32) * q^48 - 50 * q^50 + (-15*b - 68) * q^51 + (-12*b - 184) * q^52 + (-20*b - 502) * q^53 + (18*b + 416) * q^54 + 25*b * q^55 + (-70*b - 292) * q^57 + (-50*b - 188) * q^58 + (36*b - 398) * q^59 + (20*b + 40) * q^60 + (66*b + 362) * q^61 + (-4*b + 72) * q^62 + 64 * q^64 + (15*b + 230) * q^65 + (-30*b - 440) * q^66 + (10*b + 420) * q^67 + (4*b + 48) * q^68 + (-112*b - 984) * q^69 + (-100*b + 236) * q^71 + (-40*b - 168) * q^72 + (106*b - 148) * q^73 + (-60*b + 164) * q^74 + (-25*b - 50) * q^75 + (16*b + 232) * q^76 + (-110*b - 448) * q^78 + (-55*b + 240) * q^79 - 80 * q^80 + (100*b + 245) * q^81 + (-76*b + 408) * q^82 + (6*b - 1058) * q^83 + (-5*b - 60) * q^85 + (20*b + 72) * q^86 + (-169*b - 1288) * q^87 + 40*b * q^88 + (160*b - 360) * q^89 + (50*b + 210) * q^90 + (80*b + 208) * q^92 + (30*b - 16) * q^93 + (166*b + 32) * q^94 + (-20*b - 290) * q^95 + (32*b + 64) * q^96 + (85*b - 380) * q^97 + (-130*b - 1100) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 - 5 * q^3 + 8 * q^4 - 10 * q^5 + 10 * q^6 - 16 * q^8 + 47 * q^9 + 20 * q^10 - 5 * q^11 - 20 * q^12 - 95 * q^13 + 25 * q^15 + 32 * q^16 + 25 * q^17 - 94 * q^18 + 120 * q^19 - 40 * q^20 + 10 * q^22 + 124 * q^23 + 40 * q^24 + 50 * q^25 + 190 * q^26 - 425 * q^27 + 213 * q^29 - 50 * q^30 - 70 * q^31 - 64 * q^32 + 455 * q^33 - 50 * q^34 + 188 * q^36 - 134 * q^37 - 240 * q^38 + 503 * q^39 + 80 * q^40 - 370 * q^41 - 82 * q^43 - 20 * q^44 - 235 * q^45 - 248 * q^46 - 115 * q^47 - 80 * q^48 - 100 * q^50 - 151 * q^51 - 380 * q^52 - 1024 * q^53 + 850 * q^54 + 25 * q^55 - 654 * q^57 - 426 * q^58 - 760 * q^59 + 100 * q^60 + 790 * q^61 + 140 * q^62 + 128 * q^64 + 475 * q^65 - 910 * q^66 + 850 * q^67 + 100 * q^68 - 2080 * q^69 + 372 * q^71 - 376 * q^72 - 190 * q^73 + 268 * q^74 - 125 * q^75 + 480 * q^76 - 1006 * q^78 + 425 * q^79 - 160 * q^80 + 590 * q^81 + 740 * q^82 - 2110 * q^83 - 125 * q^85 + 164 * q^86 - 2745 * q^87 + 40 * q^88 - 560 * q^89 + 470 * q^90 + 496 * q^92 - 2 * q^93 + 230 * q^94 - 600 * q^95 + 160 * q^96 - 675 * q^97 - 2330 * 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

7.15207
−6.15207

−2.00000

−9.15207

4.00000

−5.00000

18.3041

0

−8.00000

56.7603

10.0000

1.2

−2.00000

4.15207

4.00000

−5.00000

−8.30413

0

−8.00000

−9.76034

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.p	✓	2
5.b	even	2	1		2450.4.a.ca		2
7.b	odd	2	1		490.4.a.u	yes	2
7.c	even	3	2		490.4.e.x		4
7.d	odd	6	2		490.4.e.t		4
35.c	odd	2	1		2450.4.a.bt		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
490.4.a.p	✓	2	1.a	even	1	1	trivial
490.4.a.u	yes	2	7.b	odd	2	1
490.4.e.t		4	7.d	odd	6	2
490.4.e.x		4	7.c	even	3	2
2450.4.a.bt		2	35.c	odd	2	1
2450.4.a.ca		2	5.b	even	2	1

Hecke kernels

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

\( T_{3}^{2} + 5T_{3} - 38 \)

\( T_{11}^{2} + 5T_{11} - 1100 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} + 5T - 38 \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 5T - 1100 \)