Newform orbit 2166.4.a.m

• Label: 2166.4.a.m
• Level: $2166$
• Weight: $4$
• Character orbit: 2166.a
• Self dual: yes
• Analytic conductor: $127.798$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(127.798137072\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{313}) \)
Defining polynomial:	\( x^{2} - x - 78 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - 2 * q^2 + 3 * q^3 + 4 * q^4 + (-b - 7) * q^5 - 6 * q^6 + (3*b + 3) * q^7 - 8 * q^8 + 9 * q^9 + (2*b + 14) * q^10 + (b - 5) * q^11 + 12 * q^12 + 24 * q^13 + (-6*b - 6) * q^14 + (-3*b - 21) * q^15 + 16 * q^16 + (-7*b + 59) * q^17 - 18 * q^18 + (-4*b - 28) * q^20 + (9*b + 9) * q^21 + (-2*b + 10) * q^22 + (-10*b - 64) * q^23 - 24 * q^24 + (15*b + 2) * q^25 - 48 * q^26 + 27 * q^27 + (12*b + 12) * q^28 + (-6*b - 192) * q^29 + (6*b + 42) * q^30 + (24*b + 84) * q^31 - 32 * q^32 + (3*b - 15) * q^33 + (14*b - 118) * q^34 + (-27*b - 255) * q^35 + 36 * q^36 - 84 * q^37 + 72 * q^39 + (8*b + 56) * q^40 + (-6*b - 96) * q^41 + (-18*b - 18) * q^42 + (-45*b - 61) * q^43 + (4*b - 20) * q^44 + (-9*b - 63) * q^45 + (20*b + 128) * q^46 + (43*b + 25) * q^47 + 48 * q^48 + (27*b + 368) * q^49 + (-30*b - 4) * q^50 + (-21*b + 177) * q^51 + 96 * q^52 + (-30*b + 96) * q^53 - 54 * q^54 + (-3*b - 43) * q^55 + (-24*b - 24) * q^56 + (12*b + 384) * q^58 + (-60*b - 24) * q^59 + (-12*b - 84) * q^60 + (-45*b + 129) * q^61 + (-48*b - 168) * q^62 + (27*b + 27) * q^63 + 64 * q^64 + (-24*b - 168) * q^65 + (-6*b + 30) * q^66 + (-36*b - 84) * q^67 + (-28*b + 236) * q^68 + (-30*b - 192) * q^69 + (54*b + 510) * q^70 + (24*b + 864) * q^71 - 72 * q^72 + (9*b - 713) * q^73 + 168 * q^74 + (45*b + 6) * q^75 + (-9*b + 219) * q^77 - 144 * q^78 + (84*b - 588) * q^79 + (-16*b - 112) * q^80 + 81 * q^81 + (12*b + 192) * q^82 + (74*b - 172) * q^83 + (36*b + 36) * q^84 + (-3*b + 133) * q^85 + (90*b + 122) * q^86 + (-18*b - 576) * q^87 + (-8*b + 40) * q^88 + (-78*b - 264) * q^89 + (18*b + 126) * q^90 + (72*b + 72) * q^91 + (-40*b - 256) * q^92 + (72*b + 252) * q^93 + (-86*b - 50) * q^94 - 96 * q^96 + (84*b + 264) * q^97 + (-54*b - 736) * q^98 + (9*b - 45) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 - 15 * q^5 - 12 * q^6 + 9 * q^7 - 16 * q^8 + 18 * q^9 + 30 * q^10 - 9 * q^11 + 24 * q^12 + 48 * q^13 - 18 * q^14 - 45 * q^15 + 32 * q^16 + 111 * q^17 - 36 * q^18 - 60 * q^20 + 27 * q^21 + 18 * q^22 - 138 * q^23 - 48 * q^24 + 19 * q^25 - 96 * q^26 + 54 * q^27 + 36 * q^28 - 390 * q^29 + 90 * q^30 + 192 * q^31 - 64 * q^32 - 27 * q^33 - 222 * q^34 - 537 * q^35 + 72 * q^36 - 168 * q^37 + 144 * q^39 + 120 * q^40 - 198 * q^41 - 54 * q^42 - 167 * q^43 - 36 * q^44 - 135 * q^45 + 276 * q^46 + 93 * q^47 + 96 * q^48 + 763 * q^49 - 38 * q^50 + 333 * q^51 + 192 * q^52 + 162 * q^53 - 108 * q^54 - 89 * q^55 - 72 * q^56 + 780 * q^58 - 108 * q^59 - 180 * q^60 + 213 * q^61 - 384 * q^62 + 81 * q^63 + 128 * q^64 - 360 * q^65 + 54 * q^66 - 204 * q^67 + 444 * q^68 - 414 * q^69 + 1074 * q^70 + 1752 * q^71 - 144 * q^72 - 1417 * q^73 + 336 * q^74 + 57 * q^75 + 429 * q^77 - 288 * q^78 - 1092 * q^79 - 240 * q^80 + 162 * q^81 + 396 * q^82 - 270 * q^83 + 108 * q^84 + 263 * q^85 + 334 * q^86 - 1170 * q^87 + 72 * q^88 - 606 * q^89 + 270 * q^90 + 216 * q^91 - 552 * q^92 + 576 * q^93 - 186 * q^94 - 192 * q^96 + 612 * q^97 - 1526 * q^98 - 81 * 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

9.34590
−8.34590

−2.00000

3.00000

4.00000

−16.3459

−6.00000

31.0377

−8.00000

9.00000

32.6918

1.2

−2.00000

3.00000

4.00000

1.34590

−6.00000

−22.0377

−8.00000

9.00000

−2.69181

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(19\)	\( +1 \)

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	2166.4.a.m	✓	2
19.b	odd	2	1		2166.4.a.o	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2166.4.a.m	✓	2	1.a	even	1	1	trivial
2166.4.a.o	yes	2	19.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):

\( T_{5}^{2} + 15T_{5} - 22 \)

\( T_{13} - 24 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( (T - 3)^{2} \)
$5$	\( T^{2} + 15T - 22 \)
$7$	\( T^{2} - 9T - 684 \)
$11$	\( T^{2} + 9T - 58 \)