Newform orbit 88.6.a.a

• Label: 88.6.a.a
• Level: $88$
• Weight: $6$
• Character orbit: 88.a
• Self dual: yes
• Analytic conductor: $14.114$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$88 = 2^{3} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	88.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$14.1137761435$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37})$
Defining polynomial:	$x^{2} - x - 9$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{37}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-b - 7) * q^3 + 9 * q^5 + (11*b + 24) * q^7 + (14*b - 46) * q^9 + 121 * q^11 + (3*b - 354) * q^13 + (-9*b - 63) * q^15 + (-109*b - 904) * q^17 + (119*b - 798) * q^19 + (-101*b - 1796) * q^21 + (-83*b - 1227) * q^23 - 3044 * q^25 + (191*b - 49) * q^27 + (146*b - 2660) * q^29 + (-487*b - 1115) * q^31 + (-121*b - 847) * q^33 + (99*b + 216) * q^35 + (640*b - 5013) * q^37 + (333*b + 2034) * q^39 + (255*b - 3910) * q^41 + (-148*b + 5906) * q^43 + (126*b - 414) * q^45 + (-1282*b + 172) * q^47 + (528*b + 1677) * q^49 + (1667*b + 22460) * q^51 + (-1674*b - 8114) * q^53 + 1089 * q^55 + (-35*b - 12026) * q^57 + (-921*b + 9439) * q^59 + (-1786*b + 6336) * q^61 + (-170*b + 21688) * q^63 + (27*b - 3186) * q^65 + (-2345*b + 25235) * q^67 + (1808*b + 20873) * q^69 + (961*b + 33163) * q^71 + (2683*b + 7214) * q^73 + (3044*b + 21308) * q^75 + (1331*b + 2904) * q^77 + (-1632*b + 8630) * q^79 + (-4690*b - 16747) * q^81 + (6630*b + 13310) * q^83 + (-981*b - 8136) * q^85 + (1638*b - 2988) * q^87 + (-310*b - 35893) * q^89 + (-3822*b - 3612) * q^91 + (4524*b + 79881) * q^93 + (1071*b - 7182) * q^95 + (-5888*b - 51927) * q^97 + (1694*b - 5566) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 14 * q^3 + 18 * q^5 + 48 * q^7 - 92 * q^9 + 242 * q^11 - 708 * q^13 - 126 * q^15 - 1808 * q^17 - 1596 * q^19 - 3592 * q^21 - 2454 * q^23 - 6088 * q^25 - 98 * q^27 - 5320 * q^29 - 2230 * q^31 - 1694 * q^33 + 432 * q^35 - 10026 * q^37 + 4068 * q^39 - 7820 * q^41 + 11812 * q^43 - 828 * q^45 + 344 * q^47 + 3354 * q^49 + 44920 * q^51 - 16228 * q^53 + 2178 * q^55 - 24052 * q^57 + 18878 * q^59 + 12672 * q^61 + 43376 * q^63 - 6372 * q^65 + 50470 * q^67 + 41746 * q^69 + 66326 * q^71 + 14428 * q^73 + 42616 * q^75 + 5808 * q^77 + 17260 * q^79 - 33494 * q^81 + 26620 * q^83 - 16272 * q^85 - 5976 * q^87 - 71786 * q^89 - 7224 * q^91 + 159762 * q^93 - 14364 * q^95 - 103854 * q^97 - 11132 * 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

3.54138
−2.54138

0

−19.1655

0

9.00000

0

157.821

0

124.317

0

1.2

0

5.16553

0

9.00000

0

−109.821

0

−216.317

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$11$	$-1$

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	88.6.a.a	✓	2
3.b	odd	2	1		792.6.a.b		2
4.b	odd	2	1		176.6.a.h		2
8.b	even	2	1		704.6.a.o		2
8.d	odd	2	1		704.6.a.l		2
11.b	odd	2	1		968.6.a.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.6.a.a	✓	2	1.a	even	1	1	trivial
176.6.a.h		2	4.b	odd	2	1
704.6.a.l		2	8.d	odd	2	1
704.6.a.o		2	8.b	even	2	1
792.6.a.b		2	3.b	odd	2	1
968.6.a.b		2	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 14T_{3} - 99$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(88))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 14T - 99$
$5$	$(T - 9)^{2}$
$7$	$T^{2} - 48T - 17332$
$11$	$(T - 121)^{2}$