Newform orbit 2178.4.a.y

• Label: 2178.4.a.y
• Level: $2178$
• Weight: $4$
• Character orbit: 2178.a
• Self dual: yes
• Analytic conductor: $128.506$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(128.506159993\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{37}) \)
Defining polynomial:	\( x^{2} - x - 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 242)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 4 * q^4 + (-3*b - 2) * q^5 + (b + 21) * q^7 - 8 * q^8 + (6*b + 4) * q^10 + (-7*b - 2) * q^13 + (-2*b - 42) * q^14 + 16 * q^16 + (-4*b - 45) * q^17 + (-11*b + 61) * q^19 + (-12*b - 8) * q^20 + (b + 73) * q^23 + (12*b + 212) * q^25 + (14*b + 4) * q^26 + (4*b + 84) * q^28 + (-17*b - 22) * q^29 + (7*b - 133) * q^31 - 32 * q^32 + (8*b + 90) * q^34 + (-65*b - 153) * q^35 + (b - 268) * q^37 + (22*b - 122) * q^38 + (24*b + 16) * q^40 + (-46*b + 97) * q^41 + (42*b + 158) * q^43 + (-2*b - 146) * q^46 + (29*b + 205) * q^47 + (42*b + 135) * q^49 + (-24*b - 424) * q^50 + (-28*b - 8) * q^52 + (-65*b + 84) * q^53 + (-8*b - 168) * q^56 + (34*b + 44) * q^58 + (6*b + 210) * q^59 + (12*b - 202) * q^61 + (-14*b + 266) * q^62 + 64 * q^64 + (20*b + 781) * q^65 + (-151*b - 103) * q^67 + (-16*b - 180) * q^68 + (130*b + 306) * q^70 + (112*b - 100) * q^71 + (-66*b + 408) * q^73 + (-2*b + 536) * q^74 + (-44*b + 244) * q^76 + (-113*b + 111) * q^79 + (-48*b - 32) * q^80 + (92*b - 194) * q^82 + (-21*b + 715) * q^83 + (143*b + 534) * q^85 + (-84*b - 316) * q^86 + (-66*b - 673) * q^89 + (-149*b - 301) * q^91 + (4*b + 292) * q^92 + (-58*b - 410) * q^94 + (-161*b + 1099) * q^95 + (72*b - 1121) * q^97 + (-84*b - 270) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 8 * q^4 - 4 * q^5 + 42 * q^7 - 16 * q^8 + 8 * q^10 - 4 * q^13 - 84 * q^14 + 32 * q^16 - 90 * q^17 + 122 * q^19 - 16 * q^20 + 146 * q^23 + 424 * q^25 + 8 * q^26 + 168 * q^28 - 44 * q^29 - 266 * q^31 - 64 * q^32 + 180 * q^34 - 306 * q^35 - 536 * q^37 - 244 * q^38 + 32 * q^40 + 194 * q^41 + 316 * q^43 - 292 * q^46 + 410 * q^47 + 270 * q^49 - 848 * q^50 - 16 * q^52 + 168 * q^53 - 336 * q^56 + 88 * q^58 + 420 * q^59 - 404 * q^61 + 532 * q^62 + 128 * q^64 + 1562 * q^65 - 206 * q^67 - 360 * q^68 + 612 * q^70 - 200 * q^71 + 816 * q^73 + 1072 * q^74 + 488 * q^76 + 222 * q^79 - 64 * q^80 - 388 * q^82 + 1430 * q^83 + 1068 * q^85 - 632 * q^86 - 1346 * q^89 - 602 * q^91 + 584 * q^92 - 820 * q^94 + 2198 * q^95 - 2242 * q^97 - 540 * 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

3.54138
−2.54138

−2.00000

0

4.00000

−20.2483

0

27.0828

−8.00000

0

40.4966

1.2

−2.00000

0

4.00000

16.2483

0

14.9172

−8.00000

0

−32.4966

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(11\)	\( -1 \)

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	2178.4.a.y		2
3.b	odd	2	1		242.4.a.m	yes	2
11.b	odd	2	1		2178.4.a.bh		2
12.b	even	2	1		1936.4.a.p		2
33.d	even	2	1		242.4.a.j	✓	2
33.f	even	10	4		242.4.c.s		8
33.h	odd	10	4		242.4.c.o		8
132.d	odd	2	1		1936.4.a.q		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
242.4.a.j	✓	2	33.d	even	2	1
242.4.a.m	yes	2	3.b	odd	2	1
242.4.c.o		8	33.h	odd	10	4
242.4.c.s		8	33.f	even	10	4
1936.4.a.p		2	12.b	even	2	1
1936.4.a.q		2	132.d	odd	2	1
2178.4.a.y		2	1.a	even	1	1	trivial
2178.4.a.bh		2	11.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):

\( T_{5}^{2} + 4T_{5} - 329 \)

\( T_{7}^{2} - 42T_{7} + 404 \)

\( T_{17}^{2} + 90T_{17} + 1433 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4T - 329 \)
$7$	\( T^{2} - 42T + 404 \)
$11$	\( T^{2} \)