Newform orbit 8325.2.a.bi

• Label: 8325.2.a.bi
• Level: $8325$
• Weight: $2$
• Character orbit: 8325.a
• Self dual: yes
• Analytic conductor: $66.475$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$8325 = 3^{2} \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13})$
Defining polynomial:	$x^{2} - x - 3$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 555)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{13})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - b * q^2 + (b + 1) * q^4 + 3 * q^7 - 3 * q^8 + (-b - 1) * q^11 + 5 * q^13 - 3*b * q^14 + (b - 2) * q^16 + (3*b - 4) * q^17 + (-3*b + 4) * q^19 + (2*b + 3) * q^22 + (2*b + 1) * q^23 - 5*b * q^26 + (3*b + 3) * q^28 + (3*b + 2) * q^29 + (4*b - 1) * q^31 + (b + 3) * q^32 + (b - 9) * q^34 - q^37 + (-b + 9) * q^38 + 11 * q^41 + (-b + 6) * q^43 + (-3*b - 4) * q^44 + (-3*b - 6) * q^46 + (-4*b + 3) * q^47 + 2 * q^49 + (5*b + 5) * q^52 - 4*b * q^53 - 9 * q^56 + (-5*b - 9) * q^58 + (5*b - 1) * q^59 + (-4*b - 1) * q^61 + (-3*b - 12) * q^62 + (-6*b + 1) * q^64 + (5*b - 2) * q^67 + (2*b + 5) * q^68 + (-5*b - 2) * q^71 + (-b + 9) * q^73 + b * q^74 + (-2*b - 5) * q^76 + (-3*b - 3) * q^77 + (-b - 5) * q^79 - 11*b * q^82 + (b + 5) * q^83 + (-5*b + 3) * q^86 + (3*b + 3) * q^88 + (4*b - 7) * q^89 + 15 * q^91 + (5*b + 7) * q^92 + (b + 12) * q^94 + (2*b + 5) * q^97 - 2*b * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q - q^2 + 3 * q^4 + 6 * q^7 - 6 * q^8 - 3 * q^11 + 10 * q^13 - 3 * q^14 - 3 * q^16 - 5 * q^17 + 5 * q^19 + 8 * q^22 + 4 * q^23 - 5 * q^26 + 9 * q^28 + 7 * q^29 + 2 * q^31 + 7 * q^32 - 17 * q^34 - 2 * q^37 + 17 * q^38 + 22 * q^41 + 11 * q^43 - 11 * q^44 - 15 * q^46 + 2 * q^47 + 4 * q^49 + 15 * q^52 - 4 * q^53 - 18 * q^56 - 23 * q^58 + 3 * q^59 - 6 * q^61 - 27 * q^62 - 4 * q^64 + q^67 + 12 * q^68 - 9 * q^71 + 17 * q^73 + q^74 - 12 * q^76 - 9 * q^77 - 11 * q^79 - 11 * q^82 + 11 * q^83 + q^86 + 9 * q^88 - 10 * q^89 + 30 * q^91 + 19 * q^92 + 25 * q^94 + 12 * q^97 - 2 * 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

2.30278
−1.30278

−2.30278

0

3.30278

0

0

3.00000

−3.00000

0

0

1.2

1.30278

0

−0.302776

0

0

3.00000

−3.00000

0

0

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$37$	$+1$

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	8325.2.a.bi		2
3.b	odd	2	1		2775.2.a.o		2
5.b	even	2	1		1665.2.a.k		2
15.d	odd	2	1		555.2.a.d	✓	2
60.h	even	2	1		8880.2.a.bt		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
555.2.a.d	✓	2	15.d	odd	2	1
1665.2.a.k		2	5.b	even	2	1
2775.2.a.o		2	3.b	odd	2	1
8325.2.a.bi		2	1.a	even	1	1	trivial
8880.2.a.bt		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8325))$:

$T_{2}^{2} + T_{2} - 3$

$T_{7} - 3$

$T_{11}^{2} + 3T_{11} - 1$

$T_{13} - 5$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + T - 3$
$3$	$T^{2}$
$5$	$T^{2}$
$7$	$(T - 3)^{2}$
$11$	$T^{2} + 3T - 1$