Newform orbit 2205.4.a.z

• Label: 2205.4.a.z
• Level: $2205$
• Weight: $4$
• Character orbit: 2205.a
• Self dual: yes
• Analytic conductor: $130.099$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(130.099211563\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{65}) \)
Defining polynomial:	\( x^{2} - x - 16 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^2 + (b + 8) * q^4 + 5 * q^5 + (-b - 16) * q^8 - 5*b * q^10 + (2*b + 10) * q^11 + (-2*b + 12) * q^13 + (9*b - 48) * q^16 + (-16*b + 66) * q^17 + (14*b - 58) * q^19 + (5*b + 40) * q^20 + (-12*b - 32) * q^22 + (20*b - 140) * q^23 + 25 * q^25 + (-10*b + 32) * q^26 + (48*b + 74) * q^29 + (-42*b - 54) * q^31 + (47*b - 16) * q^32 + (-50*b + 256) * q^34 + (-36*b - 30) * q^37 + (44*b - 224) * q^38 + (-5*b - 80) * q^40 + (100*b - 138) * q^41 + (-32*b - 156) * q^43 + (28*b + 112) * q^44 + (120*b - 320) * q^46 + (-48*b + 304) * q^47 - 25*b * q^50 + (-6*b + 64) * q^52 + (-86*b - 120) * q^53 + (10*b + 50) * q^55 + (-122*b - 768) * q^58 + (84*b - 464) * q^59 + (-24*b + 114) * q^61 + (96*b + 672) * q^62 + (-103*b - 368) * q^64 + (-10*b + 60) * q^65 + (64*b - 84) * q^67 + (-78*b + 272) * q^68 + (-42*b - 814) * q^71 + (202*b + 92) * q^73 + (66*b + 576) * q^74 + (68*b - 240) * q^76 + (-104*b - 392) * q^79 + (45*b - 240) * q^80 + (38*b - 1600) * q^82 + (216*b + 356) * q^83 + (-80*b + 330) * q^85 + (188*b + 512) * q^86 + (-44*b - 192) * q^88 + (8*b + 290) * q^89 + (40*b - 800) * q^92 + (-256*b + 768) * q^94 + (70*b - 290) * q^95 + (-314*b - 104) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8 - 5 * q^10 + 22 * q^11 + 22 * q^13 - 87 * q^16 + 116 * q^17 - 102 * q^19 + 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 + 54 * q^26 + 196 * q^29 - 150 * q^31 + 15 * q^32 + 462 * q^34 - 96 * q^37 - 404 * q^38 - 165 * q^40 - 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 - 25 * q^50 + 122 * q^52 - 326 * q^53 + 110 * q^55 - 1658 * q^58 - 844 * q^59 + 204 * q^61 + 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 - 412 * q^76 - 888 * q^79 - 435 * q^80 - 3162 * q^82 + 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 + 588 * q^89 - 1560 * q^92 + 1280 * q^94 - 510 * q^95 - 522 * q^97

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

4.53113
−3.53113

−4.53113

0

12.5311

5.00000

0

0

−20.5311

0

−22.6556

1.2

3.53113

0

4.46887

5.00000

0

0

−12.4689

0

17.6556

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(5\)	\( -1 \)
\(7\)	\( -1 \)

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	2205.4.a.z		2
3.b	odd	2	1		735.4.a.p		2
7.b	odd	2	1		315.4.a.i		2
21.c	even	2	1		105.4.a.f	✓	2
35.c	odd	2	1		1575.4.a.w		2
84.h	odd	2	1		1680.4.a.bg		2
105.g	even	2	1		525.4.a.k		2
105.k	odd	4	2		525.4.d.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.f	✓	2	21.c	even	2	1
315.4.a.i		2	7.b	odd	2	1
525.4.a.k		2	105.g	even	2	1
525.4.d.h		4	105.k	odd	4	2
735.4.a.p		2	3.b	odd	2	1
1575.4.a.w		2	35.c	odd	2	1
1680.4.a.bg		2	84.h	odd	2	1
2205.4.a.z		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} + T_{2} - 16 \)

\( T_{11}^{2} - 22T_{11} + 56 \)

\( T_{13}^{2} - 22T_{13} + 56 \)

\( T_{17}^{2} - 116T_{17} - 796 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 16 \)
$3$	\( T^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 22T + 56 \)