Newform orbit 170.4.a.d

• Label: 170.4.a.d
• 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:	\(\Q(\sqrt{145}) \)
Defining polynomial:	\( x^{2} - x - 36 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{145})\). 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 + 14 * q^7 - 8 * q^8 + (3*b + 10) * q^9 - 10 * q^10 + (6*b - 12) * q^11 + (4*b + 4) * q^12 + (-b - 1) * q^13 - 28 * q^14 + (5*b + 5) * q^15 + 16 * q^16 + 17 * q^17 + (-6*b - 20) * q^18 + (-13*b + 41) * q^19 + 20 * q^20 + (14*b + 14) * q^21 + (-12*b + 24) * q^22 + (-20*b + 42) * q^23 + (-8*b - 8) * q^24 + 25 * q^25 + (2*b + 2) * q^26 + (-11*b + 91) * q^27 + 56 * q^28 + (-b - 21) * q^29 + (-10*b - 10) * q^30 + (13*b + 113) * q^31 - 32 * q^32 + 204 * q^33 - 34 * q^34 + 70 * q^35 + (12*b + 40) * q^36 + (2*b + 188) * q^37 + (26*b - 82) * q^38 + (-3*b - 37) * q^39 - 40 * q^40 + (20*b + 54) * q^41 + (-28*b - 28) * q^42 + (2*b + 422) * q^43 + (24*b - 48) * q^44 + (15*b + 50) * q^45 + (40*b - 84) * q^46 + (-35*b - 81) * q^47 + (16*b + 16) * q^48 - 147 * q^49 - 50 * q^50 + (17*b + 17) * q^51 + (-4*b - 4) * q^52 + (-43*b + 357) * q^53 + (22*b - 182) * q^54 + (30*b - 60) * q^55 - 112 * q^56 + (15*b - 427) * q^57 + (2*b + 42) * q^58 + (-41*b - 387) * q^59 + (20*b + 20) * q^60 + (-43*b + 101) * q^61 + (-26*b - 226) * q^62 + (42*b + 140) * q^63 + 64 * q^64 + (-5*b - 5) * q^65 - 408 * q^66 + (-18*b - 250) * q^67 + 68 * q^68 + (2*b - 678) * q^69 - 140 * q^70 + (-39*b - 219) * q^71 + (-24*b - 80) * q^72 + (69*b - 151) * q^73 + (-4*b - 376) * q^74 + (25*b + 25) * q^75 + (-52*b + 164) * q^76 + (84*b - 168) * q^77 + (6*b + 74) * q^78 + (-194*b + 272) * q^79 + 80 * q^80 + (-12*b - 575) * q^81 + (-40*b - 108) * q^82 + (-40*b + 216) * q^83 + (56*b + 56) * q^84 + 85 * q^85 + (-4*b - 844) * q^86 + (-23*b - 57) * q^87 + (-48*b + 96) * q^88 + (71*b - 909) * q^89 + (-30*b - 100) * q^90 + (-14*b - 14) * q^91 + (-80*b + 168) * q^92 + (139*b + 581) * q^93 + (70*b + 162) * q^94 + (-65*b + 205) * q^95 + (-32*b - 32) * q^96 + (219*b + 107) * q^97 + 294 * q^98 + (42*b + 528) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 3 * q^3 + 8 * q^4 + 10 * q^5 - 6 * q^6 + 28 * q^7 - 16 * q^8 + 23 * q^9 - 20 * q^10 - 18 * q^11 + 12 * q^12 - 3 * q^13 - 56 * q^14 + 15 * q^15 + 32 * q^16 + 34 * q^17 - 46 * q^18 + 69 * q^19 + 40 * q^20 + 42 * q^21 + 36 * q^22 + 64 * q^23 - 24 * q^24 + 50 * q^25 + 6 * q^26 + 171 * q^27 + 112 * q^28 - 43 * q^29 - 30 * q^30 + 239 * q^31 - 64 * q^32 + 408 * q^33 - 68 * q^34 + 140 * q^35 + 92 * q^36 + 378 * q^37 - 138 * q^38 - 77 * q^39 - 80 * q^40 + 128 * q^41 - 84 * q^42 + 846 * q^43 - 72 * q^44 + 115 * q^45 - 128 * q^46 - 197 * q^47 + 48 * q^48 - 294 * q^49 - 100 * q^50 + 51 * q^51 - 12 * q^52 + 671 * q^53 - 342 * q^54 - 90 * q^55 - 224 * q^56 - 839 * q^57 + 86 * q^58 - 815 * q^59 + 60 * q^60 + 159 * q^61 - 478 * q^62 + 322 * q^63 + 128 * q^64 - 15 * q^65 - 816 * q^66 - 518 * q^67 + 136 * q^68 - 1354 * q^69 - 280 * q^70 - 477 * q^71 - 184 * q^72 - 233 * q^73 - 756 * q^74 + 75 * q^75 + 276 * q^76 - 252 * q^77 + 154 * q^78 + 350 * q^79 + 160 * q^80 - 1162 * q^81 - 256 * q^82 + 392 * q^83 + 168 * q^84 + 170 * q^85 - 1692 * q^86 - 137 * q^87 + 144 * q^88 - 1747 * q^89 - 230 * q^90 - 42 * q^91 + 256 * q^92 + 1301 * q^93 + 394 * q^94 + 345 * q^95 - 96 * q^96 + 433 * q^97 + 588 * q^98 + 1098 * 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

−5.52080
6.52080

−2.00000

−4.52080

4.00000

5.00000

9.04159

14.0000

−8.00000

−6.56239

−10.0000

1.2

−2.00000

7.52080

4.00000

5.00000

−15.0416

14.0000

−8.00000

29.5624

−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.d	✓	2
3.b	odd	2	1		1530.4.a.t		2
4.b	odd	2	1		1360.4.a.k		2
5.b	even	2	1		850.4.a.h		2
5.c	odd	4	2		850.4.c.f		4

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3T_{3} - 34 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(170))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} - 3T - 34 \)
$5$	\( (T - 5)^{2} \)
$7$	\( (T - 14)^{2} \)
$11$	\( T^{2} + 18T - 1224 \)