Newform orbit 1470.4.a.bf

• Label: 1470.4.a.bf
• Level: $1470$
• Weight: $4$
• Character orbit: 1470.a
• Self dual: yes
• Analytic conductor: $86.733$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(86.7328077084\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{505}) \)
Defining polynomial:	\( x^{2} - x - 126 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{505}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 5 * q^5 + 6 * q^6 - 8 * q^8 + 9 * q^9 - 10 * q^10 + (-b - 29) * q^11 - 12 * q^12 + (-2*b - 28) * q^13 - 15 * q^15 + 16 * q^16 + (-5*b + 5) * q^17 - 18 * q^18 + (-5*b + 1) * q^19 + 20 * q^20 + (2*b + 58) * q^22 + (5*b - 65) * q^23 + 24 * q^24 + 25 * q^25 + (4*b + 56) * q^26 - 27 * q^27 + (-11*b - 17) * q^29 + 30 * q^30 + (6*b - 172) * q^31 - 32 * q^32 + (3*b + 87) * q^33 + (10*b - 10) * q^34 + 36 * q^36 + (-17*b - 27) * q^37 + (10*b - 2) * q^38 + (6*b + 84) * q^39 - 40 * q^40 + (4*b - 30) * q^41 + (13*b - 103) * q^43 + (-4*b - 116) * q^44 + 45 * q^45 + (-10*b + 130) * q^46 + (3*b + 479) * q^47 - 48 * q^48 - 50 * q^50 + (15*b - 15) * q^51 + (-8*b - 112) * q^52 + (-b + 675) * q^53 + 54 * q^54 + (-5*b - 145) * q^55 + (15*b - 3) * q^57 + (22*b + 34) * q^58 + (10*b + 66) * q^59 - 60 * q^60 + (11*b - 9) * q^61 + (-12*b + 344) * q^62 + 64 * q^64 + (-10*b - 140) * q^65 + (-6*b - 174) * q^66 + (3*b + 743) * q^67 + (-20*b + 20) * q^68 + (-15*b + 195) * q^69 + (-10*b - 682) * q^71 - 72 * q^72 + (-28*b + 22) * q^73 + (34*b + 54) * q^74 - 75 * q^75 + (-20*b + 4) * q^76 + (-12*b - 168) * q^78 - 152 * q^79 + 80 * q^80 + 81 * q^81 + (-8*b + 60) * q^82 + (-6*b - 946) * q^83 + (-25*b + 25) * q^85 + (-26*b + 206) * q^86 + (33*b + 51) * q^87 + (8*b + 232) * q^88 + (16*b - 298) * q^89 - 90 * q^90 + (20*b - 260) * q^92 + (-18*b + 516) * q^93 + (-6*b - 958) * q^94 + (-25*b + 5) * q^95 + 96 * q^96 + (52*b - 254) * q^97 + (-9*b - 261) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 10 * q^5 + 12 * q^6 - 16 * q^8 + 18 * q^9 - 20 * q^10 - 58 * q^11 - 24 * q^12 - 56 * q^13 - 30 * q^15 + 32 * q^16 + 10 * q^17 - 36 * q^18 + 2 * q^19 + 40 * q^20 + 116 * q^22 - 130 * q^23 + 48 * q^24 + 50 * q^25 + 112 * q^26 - 54 * q^27 - 34 * q^29 + 60 * q^30 - 344 * q^31 - 64 * q^32 + 174 * q^33 - 20 * q^34 + 72 * q^36 - 54 * q^37 - 4 * q^38 + 168 * q^39 - 80 * q^40 - 60 * q^41 - 206 * q^43 - 232 * q^44 + 90 * q^45 + 260 * q^46 + 958 * q^47 - 96 * q^48 - 100 * q^50 - 30 * q^51 - 224 * q^52 + 1350 * q^53 + 108 * q^54 - 290 * q^55 - 6 * q^57 + 68 * q^58 + 132 * q^59 - 120 * q^60 - 18 * q^61 + 688 * q^62 + 128 * q^64 - 280 * q^65 - 348 * q^66 + 1486 * q^67 + 40 * q^68 + 390 * q^69 - 1364 * q^71 - 144 * q^72 + 44 * q^73 + 108 * q^74 - 150 * q^75 + 8 * q^76 - 336 * q^78 - 304 * q^79 + 160 * q^80 + 162 * q^81 + 120 * q^82 - 1892 * q^83 + 50 * q^85 + 412 * q^86 + 102 * q^87 + 464 * q^88 - 596 * q^89 - 180 * q^90 - 520 * q^92 + 1032 * q^93 - 1916 * q^94 + 10 * q^95 + 192 * q^96 - 508 * q^97 - 522 * 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

11.7361
−10.7361

−2.00000

−3.00000

4.00000

5.00000

6.00000

0

−8.00000

9.00000

−10.0000

1.2

−2.00000

−3.00000

4.00000

5.00000

6.00000

0

−8.00000

9.00000

−10.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( +1 \)
\(5\)	\( -1 \)
\(7\)	\( -1 \)

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	1470.4.a.bf	✓	2
7.b	odd	2	1		1470.4.a.bh	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1470.4.a.bf	✓	2	1.a	even	1	1	trivial
1470.4.a.bh	yes	2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11}^{2} + 58T_{11} + 336 \)

\( T_{13}^{2} + 56T_{13} - 1236 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( (T + 3)^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 58T + 336 \)