Newform orbit 231.4.a.h

• Label: 231.4.a.h
• Level: $231$
• Weight: $4$
• Character orbit: 231.a
• Self dual: yes
• Analytic conductor: $13.629$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.6294412113\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^2 - 3 * q^3 + (3*b - 3) * q^4 + (-3*b + 2) * q^5 + (-3*b - 3) * q^6 + 7 * q^7 + (-5*b + 1) * q^8 + 9 * q^9 + (-4*b - 10) * q^10 - 11 * q^11 + (-9*b + 9) * q^12 + (-13*b - 6) * q^13 + (7*b + 7) * q^14 + (9*b - 6) * q^15 + (-33*b + 5) * q^16 + (-4*b - 54) * q^17 + (9*b + 9) * q^18 + (-23*b - 24) * q^19 + (6*b - 42) * q^20 - 21 * q^21 + (-11*b - 11) * q^22 + (8*b - 32) * q^23 + (15*b - 3) * q^24 + (-3*b - 85) * q^25 + (-32*b - 58) * q^26 - 27 * q^27 + (21*b - 21) * q^28 + (65*b - 38) * q^29 + (12*b + 30) * q^30 + (26*b - 168) * q^31 + (-21*b - 135) * q^32 + 33 * q^33 + (-62*b - 70) * q^34 + (-21*b + 14) * q^35 + (27*b - 27) * q^36 + (107*b - 86) * q^37 + (-70*b - 116) * q^38 + (39*b + 18) * q^39 + (2*b + 62) * q^40 + (86*b - 22) * q^41 + (-21*b - 21) * q^42 + (120*b - 76) * q^43 + (-33*b + 33) * q^44 + (-27*b + 18) * q^45 - 16*b * q^46 + (-163*b + 132) * q^47 + (99*b - 15) * q^48 + 49 * q^49 + (-91*b - 97) * q^50 + (12*b + 162) * q^51 + (-18*b - 138) * q^52 + (58*b + 54) * q^53 + (-27*b - 27) * q^54 + (33*b - 22) * q^55 + (-35*b + 7) * q^56 + (69*b + 72) * q^57 + (92*b + 222) * q^58 + (53*b - 32) * q^59 + (-18*b + 126) * q^60 + (-80*b - 178) * q^61 + (-116*b - 64) * q^62 + 63 * q^63 + (87*b - 259) * q^64 + (31*b + 144) * q^65 + (33*b + 33) * q^66 + (-95*b - 16) * q^67 + (-162*b + 114) * q^68 + (-24*b + 96) * q^69 + (-28*b - 70) * q^70 + (-56*b + 496) * q^71 + (-45*b + 9) * q^72 + (317*b - 322) * q^73 + (128*b + 342) * q^74 + (9*b + 255) * q^75 + (-72*b - 204) * q^76 - 77 * q^77 + (96*b + 174) * q^78 + (124*b - 176) * q^79 + (18*b + 406) * q^80 + 81 * q^81 + (150*b + 322) * q^82 + (-30*b - 116) * q^83 + (-63*b + 63) * q^84 + (166*b - 60) * q^85 + (164*b + 404) * q^86 + (-195*b + 114) * q^87 + (55*b - 11) * q^88 + (-216*b + 130) * q^89 + (-36*b - 90) * q^90 + (-91*b - 42) * q^91 + (-96*b + 192) * q^92 + (-78*b + 504) * q^93 + (-194*b - 520) * q^94 + (95*b + 228) * q^95 + (63*b + 405) * q^96 + (18*b - 1142) * q^97 + (49*b + 49) * q^98 - 99 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^2 - 6 * q^3 - 3 * q^4 + q^5 - 9 * q^6 + 14 * q^7 - 3 * q^8 + 18 * q^9 - 24 * q^10 - 22 * q^11 + 9 * q^12 - 25 * q^13 + 21 * q^14 - 3 * q^15 - 23 * q^16 - 112 * q^17 + 27 * q^18 - 71 * q^19 - 78 * q^20 - 42 * q^21 - 33 * q^22 - 56 * q^23 + 9 * q^24 - 173 * q^25 - 148 * q^26 - 54 * q^27 - 21 * q^28 - 11 * q^29 + 72 * q^30 - 310 * q^31 - 291 * q^32 + 66 * q^33 - 202 * q^34 + 7 * q^35 - 27 * q^36 - 65 * q^37 - 302 * q^38 + 75 * q^39 + 126 * q^40 + 42 * q^41 - 63 * q^42 - 32 * q^43 + 33 * q^44 + 9 * q^45 - 16 * q^46 + 101 * q^47 + 69 * q^48 + 98 * q^49 - 285 * q^50 + 336 * q^51 - 294 * q^52 + 166 * q^53 - 81 * q^54 - 11 * q^55 - 21 * q^56 + 213 * q^57 + 536 * q^58 - 11 * q^59 + 234 * q^60 - 436 * q^61 - 244 * q^62 + 126 * q^63 - 431 * q^64 + 319 * q^65 + 99 * q^66 - 127 * q^67 + 66 * q^68 + 168 * q^69 - 168 * q^70 + 936 * q^71 - 27 * q^72 - 327 * q^73 + 812 * q^74 + 519 * q^75 - 480 * q^76 - 154 * q^77 + 444 * q^78 - 228 * q^79 + 830 * q^80 + 162 * q^81 + 794 * q^82 - 262 * q^83 + 63 * q^84 + 46 * q^85 + 972 * q^86 + 33 * q^87 + 33 * q^88 + 44 * q^89 - 216 * q^90 - 175 * q^91 + 288 * q^92 + 930 * q^93 - 1234 * q^94 + 551 * q^95 + 873 * q^96 - 2266 * q^97 + 147 * q^98 - 198 * 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

−1.56155
2.56155

−0.561553

−3.00000

−7.68466

6.68466

1.68466

7.00000

8.80776

9.00000

−3.75379

1.2

3.56155

−3.00000

4.68466

−5.68466

−10.6847

7.00000

−11.8078

9.00000

−20.2462

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( +1 \)
\(7\)	\( -1 \)
\(11\)	\( +1 \)

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	231.4.a.h	✓	2
3.b	odd	2	1		693.4.a.g		2
7.b	odd	2	1		1617.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.4.a.h	✓	2	1.a	even	1	1	trivial
693.4.a.g		2	3.b	odd	2	1
1617.4.a.m		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(231))\):

\( T_{2}^{2} - 3T_{2} - 2 \)

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

Hecke characteristic polynomials

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