Newform orbit 234.4.a.m

• Label: 234.4.a.m
• Level: $234$
• Weight: $4$
• Character orbit: 234.a
• Self dual: yes
• Analytic conductor: $13.806$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.8064469413\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{22}) \)
Defining polynomial:	\( x^{2} - 22 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = 2\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 2 * q^2 + 4 * q^4 + (b + 4) * q^5 + (b + 6) * q^7 + 8 * q^8 + (2*b + 8) * q^10 + (-5*b + 22) * q^11 + 13 * q^13 + (2*b + 12) * q^14 + 16 * q^16 + (4*b + 28) * q^17 + (-11*b + 30) * q^19 + (4*b + 16) * q^20 + (-10*b + 44) * q^22 + (-12*b + 84) * q^23 + (8*b - 21) * q^25 + 26 * q^26 + (4*b + 24) * q^28 + (12*b + 108) * q^29 + (21*b - 58) * q^31 + 32 * q^32 + (8*b + 56) * q^34 + (10*b + 112) * q^35 + (-32*b - 54) * q^37 + (-22*b + 60) * q^38 + (8*b + 32) * q^40 + (7*b + 232) * q^41 + (22*b - 216) * q^43 + (-20*b + 88) * q^44 + (-24*b + 168) * q^46 + (7*b + 154) * q^47 + (12*b - 219) * q^49 + (16*b - 42) * q^50 + 52 * q^52 + (-2*b + 208) * q^53 + (2*b - 352) * q^55 + (8*b + 48) * q^56 + (24*b + 216) * q^58 + (-43*b + 74) * q^59 + (-44*b - 426) * q^61 + (42*b - 116) * q^62 + 64 * q^64 + (13*b + 52) * q^65 + (-21*b - 190) * q^67 + (16*b + 112) * q^68 + (20*b + 224) * q^70 + (-25*b - 430) * q^71 + (48*b - 202) * q^73 + (-64*b - 108) * q^74 + (-44*b + 120) * q^76 + (-8*b - 308) * q^77 + (-96*b - 364) * q^79 + (16*b + 64) * q^80 + (14*b + 464) * q^82 + (75*b - 546) * q^83 + (44*b + 464) * q^85 + (44*b - 432) * q^86 + (-40*b + 176) * q^88 + (13*b - 896) * q^89 + (13*b + 78) * q^91 + (-48*b + 336) * q^92 + (14*b + 308) * q^94 + (-14*b - 848) * q^95 + (76*b + 342) * q^97 + (24*b - 438) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^2 + 8 * q^4 + 8 * q^5 + 12 * q^7 + 16 * q^8 + 16 * q^10 + 44 * q^11 + 26 * q^13 + 24 * q^14 + 32 * q^16 + 56 * q^17 + 60 * q^19 + 32 * q^20 + 88 * q^22 + 168 * q^23 - 42 * q^25 + 52 * q^26 + 48 * q^28 + 216 * q^29 - 116 * q^31 + 64 * q^32 + 112 * q^34 + 224 * q^35 - 108 * q^37 + 120 * q^38 + 64 * q^40 + 464 * q^41 - 432 * q^43 + 176 * q^44 + 336 * q^46 + 308 * q^47 - 438 * q^49 - 84 * q^50 + 104 * q^52 + 416 * q^53 - 704 * q^55 + 96 * q^56 + 432 * q^58 + 148 * q^59 - 852 * q^61 - 232 * q^62 + 128 * q^64 + 104 * q^65 - 380 * q^67 + 224 * q^68 + 448 * q^70 - 860 * q^71 - 404 * q^73 - 216 * q^74 + 240 * q^76 - 616 * q^77 - 728 * q^79 + 128 * q^80 + 928 * q^82 - 1092 * q^83 + 928 * q^85 - 864 * q^86 + 352 * q^88 - 1792 * q^89 + 156 * q^91 + 672 * q^92 + 616 * q^94 - 1696 * q^95 + 684 * q^97 - 876 * q^98

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

−4.69042
4.69042

2.00000

0

4.00000

−5.38083

0

−3.38083

8.00000

0

−10.7617

1.2

2.00000

0

4.00000

13.3808

0

15.3808

8.00000

0

26.7617

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( +1 \)
\(13\)	\( -1 \)

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	234.4.a.m	yes	2
3.b	odd	2	1		234.4.a.l	✓	2
4.b	odd	2	1		1872.4.a.bg		2
12.b	even	2	1		1872.4.a.w		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
234.4.a.l	✓	2	3.b	odd	2	1
234.4.a.m	yes	2	1.a	even	1	1	trivial
1872.4.a.w		2	12.b	even	2	1
1872.4.a.bg		2	4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(234))\):

\( T_{5}^{2} - 8T_{5} - 72 \)

\( T_{7}^{2} - 12T_{7} - 52 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 8T - 72 \)
$7$	\( T^{2} - 12T - 52 \)
$11$	\( T^{2} - 44T - 1716 \)