Newform orbit 238.4.a.a

• Label: 238.4.a.a
• Level: $238$
• Weight: $4$
• Character orbit: 238.a
• Self dual: yes
• Analytic conductor: $14.042$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	238.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.0424545814\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{93}) \)
Defining polynomial:	\( x^{2} - x - 23 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{93})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + (-b - 2) * q^3 + 4 * q^4 + (-b + 5) * q^5 + (2*b + 4) * q^6 - 7 * q^7 - 8 * q^8 + 5*b * q^9 + (2*b - 10) * q^10 + (8*b - 6) * q^11 + (-4*b - 8) * q^12 + (8*b - 18) * q^13 + 14 * q^14 + (-2*b + 13) * q^15 + 16 * q^16 + 17 * q^17 - 10*b * q^18 + (4*b + 4) * q^19 + (-4*b + 20) * q^20 + (7*b + 14) * q^21 + (-16*b + 12) * q^22 + (2*b + 24) * q^23 + (8*b + 16) * q^24 + (-9*b - 77) * q^25 + (-16*b + 36) * q^26 + (12*b - 61) * q^27 - 28 * q^28 + (-28*b - 90) * q^29 + (4*b - 26) * q^30 + (27*b + 81) * q^31 - 32 * q^32 + (-18*b - 172) * q^33 - 34 * q^34 + (7*b - 35) * q^35 + 20*b * q^36 + (-32*b - 144) * q^37 + (-8*b - 8) * q^38 + (-6*b - 148) * q^39 + (8*b - 40) * q^40 + (7*b - 232) * q^41 + (-14*b - 28) * q^42 - 69*b * q^43 + (32*b - 24) * q^44 + (20*b - 115) * q^45 + (-4*b - 48) * q^46 + (-6*b - 384) * q^47 + (-16*b - 32) * q^48 + 49 * q^49 + (18*b + 154) * q^50 + (-17*b - 34) * q^51 + (32*b - 72) * q^52 + (15*b - 373) * q^53 + (-24*b + 122) * q^54 + (38*b - 214) * q^55 + 56 * q^56 + (-16*b - 100) * q^57 + (56*b + 180) * q^58 + 148 * q^59 + (-8*b + 52) * q^60 + (69*b - 204) * q^61 + (-54*b - 162) * q^62 - 35*b * q^63 + 64 * q^64 + (50*b - 274) * q^65 + (36*b + 344) * q^66 + (49*b + 209) * q^67 + 68 * q^68 + (-30*b - 94) * q^69 + (-14*b + 70) * q^70 + (-16*b + 116) * q^71 - 40*b * q^72 + (21*b + 400) * q^73 + (64*b + 288) * q^74 + (104*b + 361) * q^75 + (16*b + 16) * q^76 + (-56*b + 42) * q^77 + (12*b + 296) * q^78 + (78*b - 24) * q^79 + (-16*b + 80) * q^80 + (-110*b - 154) * q^81 + (-14*b + 464) * q^82 + (-158*b - 100) * q^83 + (28*b + 56) * q^84 + (-17*b + 85) * q^85 + 138*b * q^86 + (174*b + 824) * q^87 + (-64*b + 48) * q^88 + (-38*b - 440) * q^89 + (-40*b + 230) * q^90 + (-56*b + 126) * q^91 + (8*b + 96) * q^92 + (-162*b - 783) * q^93 + (12*b + 768) * q^94 + (12*b - 72) * q^95 + (32*b + 64) * q^96 + (-109*b - 403) * q^97 - 98 * q^98 + (10*b + 920) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 - 5 * q^3 + 8 * q^4 + 9 * q^5 + 10 * q^6 - 14 * q^7 - 16 * q^8 + 5 * q^9 - 18 * q^10 - 4 * q^11 - 20 * q^12 - 28 * q^13 + 28 * q^14 + 24 * q^15 + 32 * q^16 + 34 * q^17 - 10 * q^18 + 12 * q^19 + 36 * q^20 + 35 * q^21 + 8 * q^22 + 50 * q^23 + 40 * q^24 - 163 * q^25 + 56 * q^26 - 110 * q^27 - 56 * q^28 - 208 * q^29 - 48 * q^30 + 189 * q^31 - 64 * q^32 - 362 * q^33 - 68 * q^34 - 63 * q^35 + 20 * q^36 - 320 * q^37 - 24 * q^38 - 302 * q^39 - 72 * q^40 - 457 * q^41 - 70 * q^42 - 69 * q^43 - 16 * q^44 - 210 * q^45 - 100 * q^46 - 774 * q^47 - 80 * q^48 + 98 * q^49 + 326 * q^50 - 85 * q^51 - 112 * q^52 - 731 * q^53 + 220 * q^54 - 390 * q^55 + 112 * q^56 - 216 * q^57 + 416 * q^58 + 296 * q^59 + 96 * q^60 - 339 * q^61 - 378 * q^62 - 35 * q^63 + 128 * q^64 - 498 * q^65 + 724 * q^66 + 467 * q^67 + 136 * q^68 - 218 * q^69 + 126 * q^70 + 216 * q^71 - 40 * q^72 + 821 * q^73 + 640 * q^74 + 826 * q^75 + 48 * q^76 + 28 * q^77 + 604 * q^78 + 30 * q^79 + 144 * q^80 - 418 * q^81 + 914 * q^82 - 358 * q^83 + 140 * q^84 + 153 * q^85 + 138 * q^86 + 1822 * q^87 + 32 * q^88 - 918 * q^89 + 420 * q^90 + 196 * q^91 + 200 * q^92 - 1728 * q^93 + 1548 * q^94 - 132 * q^95 + 160 * q^96 - 915 * q^97 - 196 * q^98 + 1850 * 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

5.32183
−4.32183

−2.00000

−7.32183

4.00000

−0.321825

14.6437

−7.00000

−8.00000

26.6091

0.643651

1.2

−2.00000

2.32183

4.00000

9.32183

−4.64365

−7.00000

−8.00000

−21.6091

−18.6437

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(7\)	\( +1 \)
\(17\)	\( -1 \)

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	238.4.a.a	✓	2
3.b	odd	2	1		2142.4.a.l		2
4.b	odd	2	1		1904.4.a.b		2
7.b	odd	2	1		1666.4.a.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
238.4.a.a	✓	2	1.a	even	1	1	trivial
1666.4.a.c		2	7.b	odd	2	1
1904.4.a.b		2	4.b	odd	2	1
2142.4.a.l		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} - 17 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(238))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} + 5T - 17 \)
$5$	\( T^{2} - 9T - 3 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} + 4T - 1484 \)