Newform orbit 735.4.a.k

• Label: 735.4.a.k
• Level: $735$
• Weight: $4$
• Character orbit: 735.a
• Self dual: yes
• Analytic conductor: $43.366$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(43.3664038542\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 3) * q^2 + 3 * q^3 + (7*b + 5) * q^4 + 5 * q^5 + (-3*b - 9) * q^6 + (-25*b - 19) * q^8 + 9 * q^9 + (-5*b - 15) * q^10 + (-10*b - 8) * q^11 + (21*b + 15) * q^12 + (22*b - 18) * q^13 + 15 * q^15 + (63*b + 117) * q^16 + (-28*b + 6) * q^17 + (-9*b - 27) * q^18 + (26*b - 100) * q^19 + (35*b + 25) * q^20 + (48*b + 64) * q^22 + (56*b + 64) * q^23 + (-75*b - 57) * q^24 + 25 * q^25 + (-70*b - 34) * q^26 + 27 * q^27 + (-84*b + 26) * q^29 + (-15*b - 45) * q^30 + (-18*b - 156) * q^31 + (-169*b - 451) * q^32 + (-30*b - 24) * q^33 + (106*b + 94) * q^34 + (63*b + 45) * q^36 + (24*b - 78) * q^37 + (-4*b + 196) * q^38 + (66*b - 54) * q^39 + (-125*b - 95) * q^40 + (-140*b - 30) * q^41 + (-68*b + 216) * q^43 + (-176*b - 320) * q^44 + 45 * q^45 + (-288*b - 416) * q^46 + (-108*b - 92) * q^47 + (189*b + 351) * q^48 + (-25*b - 75) * q^50 + (-84*b + 18) * q^51 + (138*b + 526) * q^52 + (214*b - 90) * q^53 + (-27*b - 81) * q^54 + (-50*b - 40) * q^55 + (78*b - 300) * q^57 + (310*b + 258) * q^58 + (-36*b - 164) * q^59 + (105*b + 75) * q^60 + (252*b - 522) * q^61 + (228*b + 540) * q^62 + (623*b + 1093) * q^64 + (110*b - 90) * q^65 + (144*b + 192) * q^66 + (28*b - 408) * q^67 + (-294*b - 754) * q^68 + (168*b + 192) * q^69 + (-330*b + 392) * q^71 + (-225*b - 171) * q^72 + (178*b - 478) * q^73 + (-18*b + 138) * q^74 + 75 * q^75 + (-388*b + 228) * q^76 + (-210*b - 102) * q^78 + (88*b + 160) * q^79 + (315*b + 585) * q^80 + 81 * q^81 + (590*b + 650) * q^82 + (264*b - 700) * q^83 + (-140*b + 30) * q^85 + (56*b - 376) * q^86 + (-252*b + 78) * q^87 + (640*b + 1152) * q^88 + (728*b - 382) * q^89 + (-45*b - 135) * q^90 + (1120*b + 1888) * q^92 + (-54*b - 468) * q^93 + (524*b + 708) * q^94 + (130*b - 500) * q^95 + (-507*b - 1353) * q^96 + (-146*b + 322) * q^97 + (-90*b - 72) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 7 * q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 - 21 * q^6 - 63 * q^8 + 18 * q^9 - 35 * q^10 - 26 * q^11 + 51 * q^12 - 14 * q^13 + 30 * q^15 + 297 * q^16 - 16 * q^17 - 63 * q^18 - 174 * q^19 + 85 * q^20 + 176 * q^22 + 184 * q^23 - 189 * q^24 + 50 * q^25 - 138 * q^26 + 54 * q^27 - 32 * q^29 - 105 * q^30 - 330 * q^31 - 1071 * q^32 - 78 * q^33 + 294 * q^34 + 153 * q^36 - 132 * q^37 + 388 * q^38 - 42 * q^39 - 315 * q^40 - 200 * q^41 + 364 * q^43 - 816 * q^44 + 90 * q^45 - 1120 * q^46 - 292 * q^47 + 891 * q^48 - 175 * q^50 - 48 * q^51 + 1190 * q^52 + 34 * q^53 - 189 * q^54 - 130 * q^55 - 522 * q^57 + 826 * q^58 - 364 * q^59 + 255 * q^60 - 792 * q^61 + 1308 * q^62 + 2809 * q^64 - 70 * q^65 + 528 * q^66 - 788 * q^67 - 1802 * q^68 + 552 * q^69 + 454 * q^71 - 567 * q^72 - 778 * q^73 + 258 * q^74 + 150 * q^75 + 68 * q^76 - 414 * q^78 + 408 * q^79 + 1485 * q^80 + 162 * q^81 + 1890 * q^82 - 1136 * q^83 - 80 * q^85 - 696 * q^86 - 96 * q^87 + 2944 * q^88 - 36 * q^89 - 315 * q^90 + 4896 * q^92 - 990 * q^93 + 1940 * q^94 - 870 * q^95 - 3213 * q^96 + 498 * q^97 - 234 * 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

2.56155
−1.56155

−5.56155

3.00000

22.9309

5.00000

−16.6847

0

−83.0388

9.00000

−27.8078

1.2

−1.43845

3.00000

−5.93087

5.00000

−4.31534

0

20.0388

9.00000

−7.19224

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	735.4.a.k		2
3.b	odd	2	1		2205.4.a.bh		2
7.b	odd	2	1		105.4.a.c	✓	2
21.c	even	2	1		315.4.a.m		2
28.d	even	2	1		1680.4.a.bk		2
35.c	odd	2	1		525.4.a.p		2
35.f	even	4	2		525.4.d.i		4
105.g	even	2	1		1575.4.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.c	✓	2	7.b	odd	2	1
315.4.a.m		2	21.c	even	2	1
525.4.a.p		2	35.c	odd	2	1
525.4.d.i		4	35.f	even	4	2
735.4.a.k		2	1.a	even	1	1	trivial
1575.4.a.m		2	105.g	even	2	1
1680.4.a.bk		2	28.d	even	2	1
2205.4.a.bh		2	3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2}^{2} + 7T_{2} + 8 \)

\( T_{11}^{2} + 26T_{11} - 256 \)

\( T_{13}^{2} + 14T_{13} - 2008 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 7T + 8 \)
$3$	\( (T - 3)^{2} \)
$5$	\( (T - 5)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 26T - 256 \)