Newform orbit 170.4.a.f

• Label: 170.4.a.f
• Level: $170$
• Weight: $4$
• Character orbit: 170.a
• Self dual: yes
• Analytic conductor: $10.030$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	170.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.0303247010\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{19}) \)
Defining polynomial:	\( x^{2} - 19 \)
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 \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + 5 * q^5 + (2*b + 2) * q^6 + (3*b + 23) * q^7 + 8 * q^8 + (2*b - 7) * q^9 + 10 * q^10 + (-9*b - 7) * q^11 + (4*b + 4) * q^12 + (-16*b - 16) * q^13 + (6*b + 46) * q^14 + (5*b + 5) * q^15 + 16 * q^16 - 17 * q^17 + (4*b - 14) * q^18 + (-2*b + 130) * q^19 + 20 * q^20 + (26*b + 80) * q^21 + (-18*b - 14) * q^22 + (13*b + 1) * q^23 + (8*b + 8) * q^24 + 25 * q^25 + (-32*b - 32) * q^26 + (-32*b + 4) * q^27 + (12*b + 92) * q^28 + (-38*b + 64) * q^29 + (10*b + 10) * q^30 + (41*b - 37) * q^31 + 32 * q^32 + (-16*b - 178) * q^33 - 34 * q^34 + (15*b + 115) * q^35 + (8*b - 28) * q^36 + (-74*b + 44) * q^37 + (-4*b + 260) * q^38 + (-32*b - 320) * q^39 + 40 * q^40 + (10*b - 224) * q^41 + (52*b + 160) * q^42 + (84*b - 18) * q^43 + (-36*b - 28) * q^44 + (10*b - 35) * q^45 + (26*b + 2) * q^46 + (6*b - 88) * q^47 + (16*b + 16) * q^48 + (138*b + 357) * q^49 + 50 * q^50 + (-17*b - 17) * q^51 + (-64*b - 64) * q^52 + (-64*b - 106) * q^53 + (-64*b + 8) * q^54 + (-45*b - 35) * q^55 + (24*b + 184) * q^56 + (128*b + 92) * q^57 + (-76*b + 128) * q^58 + (18*b - 534) * q^59 + (20*b + 20) * q^60 + (20*b - 654) * q^61 + (82*b - 74) * q^62 + (25*b - 47) * q^63 + 64 * q^64 + (-80*b - 80) * q^65 + (-32*b - 356) * q^66 + (-18*b - 380) * q^67 - 68 * q^68 + (14*b + 248) * q^69 + (30*b + 230) * q^70 + (211*b + 37) * q^71 + (16*b - 56) * q^72 + (-142*b + 572) * q^73 + (-148*b + 88) * q^74 + (25*b + 25) * q^75 + (-8*b + 520) * q^76 + (-228*b - 674) * q^77 + (-64*b - 640) * q^78 + (-45*b + 117) * q^79 + 80 * q^80 + (-82*b - 415) * q^81 + (20*b - 448) * q^82 + (76*b + 322) * q^83 + (104*b + 320) * q^84 - 85 * q^85 + (168*b - 36) * q^86 + (26*b - 658) * q^87 + (-72*b - 56) * q^88 + (-34*b + 938) * q^89 + (20*b - 70) * q^90 + (-416*b - 1280) * q^91 + (52*b + 4) * q^92 + (4*b + 742) * q^93 + (12*b - 176) * q^94 + (-10*b + 650) * q^95 + (32*b + 32) * q^96 + (-68*b + 122) * q^97 + (276*b + 714) * q^98 + (49*b - 293) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 10 * q^5 + 4 * q^6 + 46 * q^7 + 16 * q^8 - 14 * q^9 + 20 * q^10 - 14 * q^11 + 8 * q^12 - 32 * q^13 + 92 * q^14 + 10 * q^15 + 32 * q^16 - 34 * q^17 - 28 * q^18 + 260 * q^19 + 40 * q^20 + 160 * q^21 - 28 * q^22 + 2 * q^23 + 16 * q^24 + 50 * q^25 - 64 * q^26 + 8 * q^27 + 184 * q^28 + 128 * q^29 + 20 * q^30 - 74 * q^31 + 64 * q^32 - 356 * q^33 - 68 * q^34 + 230 * q^35 - 56 * q^36 + 88 * q^37 + 520 * q^38 - 640 * q^39 + 80 * q^40 - 448 * q^41 + 320 * q^42 - 36 * q^43 - 56 * q^44 - 70 * q^45 + 4 * q^46 - 176 * q^47 + 32 * q^48 + 714 * q^49 + 100 * q^50 - 34 * q^51 - 128 * q^52 - 212 * q^53 + 16 * q^54 - 70 * q^55 + 368 * q^56 + 184 * q^57 + 256 * q^58 - 1068 * q^59 + 40 * q^60 - 1308 * q^61 - 148 * q^62 - 94 * q^63 + 128 * q^64 - 160 * q^65 - 712 * q^66 - 760 * q^67 - 136 * q^68 + 496 * q^69 + 460 * q^70 + 74 * q^71 - 112 * q^72 + 1144 * q^73 + 176 * q^74 + 50 * q^75 + 1040 * q^76 - 1348 * q^77 - 1280 * q^78 + 234 * q^79 + 160 * q^80 - 830 * q^81 - 896 * q^82 + 644 * q^83 + 640 * q^84 - 170 * q^85 - 72 * q^86 - 1316 * q^87 - 112 * q^88 + 1876 * q^89 - 140 * q^90 - 2560 * q^91 + 8 * q^92 + 1484 * q^93 - 352 * q^94 + 1300 * q^95 + 64 * q^96 + 244 * q^97 + 1428 * q^98 - 586 * 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

−4.35890
4.35890

2.00000

−3.35890

4.00000

5.00000

−6.71780

9.92330

8.00000

−15.7178

10.0000

1.2

2.00000

5.35890

4.00000

5.00000

10.7178

36.0767

8.00000

1.71780

10.0000

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(5\)	\( -1 \)
\(17\)	\( +1 \)

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	170.4.a.f	✓	2
3.b	odd	2	1		1530.4.a.n		2
4.b	odd	2	1		1360.4.a.l		2
5.b	even	2	1		850.4.a.f		2
5.c	odd	4	2		850.4.c.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
170.4.a.f	✓	2	1.a	even	1	1	trivial
850.4.a.f		2	5.b	even	2	1
850.4.c.g		4	5.c	odd	4	2
1360.4.a.l		2	4.b	odd	2	1
1530.4.a.n		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(170))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} - 2T - 18 \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} - 46T + 358 \)
$11$	\( T^{2} + 14T - 1490 \)