Newform orbit 336.4.bj.f

• Label: 336.4.bj.f
• Level: $336$
• Weight: $4$
• Character orbit: 336.bj
• Analytic conductor: $19.825$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bj (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 28 * q + 38 * q^7 - 70 * q^9 + 124 * q^13 - 462 * q^19 + 500 * q^21 + 566 * q^25 + 1266 * q^31 + 64 * q^33 + 338 * q^37 + 1254 * q^39 - 488 * q^45 - 206 * q^49 + 522 * q^51 + 2324 * q^57 - 340 * q^61 - 840 * q^63 + 2934 * q^67 - 776 * q^69 + 2050 * q^73 - 1458 * q^75 + 2070 * q^79 + 910 * q^81 + 2400 * q^85 + 3132 * q^87 - 5170 * q^91 + 714 * q^93 - 3344 * 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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
95.1		0		−5.18844	+	0.283083i		0		−8.90809	+	5.14309i		0		9.06303	+	16.1512i		0		26.8397	−	2.93751i		0
95.2		0		−5.04533	+	1.24284i		0		11.5701	−	6.68000i		0		−16.4792	+	8.45200i		0		23.9107	−	12.5411i		0
95.3		0		−4.15916	+	3.11470i		0		−17.2240	+	9.94429i		0		−5.66612	−	17.6322i		0		7.59723	−	25.9091i		0
95.4		0		−2.63380	+	4.47918i		0		10.1094	−	5.83669i		0		17.9159	−	4.69245i		0		−13.1262	−	23.5946i		0
95.5		0		−2.34906	−	4.63486i		0		8.90809	−	5.14309i		0		9.06303	+	16.1512i		0		−15.9638	+	21.7751i		0
95.6		0		−1.44633	−	4.99080i		0		−11.5701	+	6.68000i		0		−16.4792	+	8.45200i		0		−22.8163	+	14.4367i		0
95.7		0		−0.692259	+	5.14983i		0		4.30532	−	2.48568i		0		−14.4573	−	11.5752i		0		−26.0416	−	7.13004i		0
95.8		0		0.538038	+	5.16822i		0		−7.60125	+	4.38858i		0		3.39378	+	18.2067i		0		−26.4210	+	5.56140i		0
95.9		0		0.617833	−	5.15929i		0		17.2240	−	9.94429i		0		−5.66612	−	17.6322i		0		−26.2366	−	6.37516i		0
95.10		0		2.56219	−	4.52053i		0		−10.1094	+	5.83669i		0		17.9159	−	4.69245i		0		−13.8704	−	23.1649i		0
95.11		0		4.11376	−	3.17443i		0		−4.30532	+	2.48568i		0		−14.4573	−	11.5752i		0		6.84598	−	26.1177i		0
95.12		0		4.16378	+	3.10853i		0		13.4200	−	7.74804i		0		15.7299	−	9.77601i		0		7.67408	+	25.8865i		0
95.13		0		4.74483	−	2.11816i		0		7.60125	−	4.38858i		0		3.39378	+	18.2067i		0		18.0268	−	20.1006i		0
95.14		0		4.77395	+	2.05167i		0		−13.4200	+	7.74804i		0		15.7299	−	9.77601i		0		18.5813	+	19.5892i		0
191.1		0		−5.18844	−	0.283083i		0		−8.90809	−	5.14309i		0		9.06303	−	16.1512i		0		26.8397	+	2.93751i		0
191.2		0		−5.04533	−	1.24284i		0		11.5701	+	6.68000i		0		−16.4792	−	8.45200i		0		23.9107	+	12.5411i		0
191.3		0		−4.15916	−	3.11470i		0		−17.2240	−	9.94429i		0		−5.66612	+	17.6322i		0		7.59723	+	25.9091i		0
191.4		0		−2.63380	−	4.47918i		0		10.1094	+	5.83669i		0		17.9159	+	4.69245i		0		−13.1262	+	23.5946i		0
191.5		0		−2.34906	+	4.63486i		0		8.90809	+	5.14309i		0		9.06303	−	16.1512i		0		−15.9638	−	21.7751i		0
191.6		0		−1.44633	+	4.99080i		0		−11.5701	−	6.68000i		0		−16.4792	−	8.45200i		0		−22.8163	−	14.4367i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
28.g	odd	6	1	inner
84.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.4.bj.f	yes	28
3.b	odd	2	1	inner	336.4.bj.f	yes	28
4.b	odd	2	1		336.4.bj.e	✓	28
7.c	even	3	1		336.4.bj.e	✓	28
12.b	even	2	1		336.4.bj.e	✓	28
21.h	odd	6	1		336.4.bj.e	✓	28
28.g	odd	6	1	inner	336.4.bj.f	yes	28
84.n	even	6	1	inner	336.4.bj.f	yes	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.bj.e	✓	28	4.b	odd	2	1
336.4.bj.e	✓	28	7.c	even	3	1
336.4.bj.e	✓	28	12.b	even	2	1
336.4.bj.e	✓	28	21.h	odd	6	1
336.4.bj.f	yes	28	1.a	even	1	1	trivial
336.4.bj.f	yes	28	3.b	odd	2	1	inner
336.4.bj.f	yes	28	28.g	odd	6	1	inner
336.4.bj.f	yes	28	84.n	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(336, [\chi])$:

T5^28 - 1158*T5^26 + 811629*T5^24 - 365180866*T5^22 + 120762762282*T5^20 - 29730148967502*T5^18 + 5681673112348005*T5^16 - 838493921558959758*T5^14 + 96980557607708372298*T5^12 - 8584404034950529467682*T5^10 + 580367740977433706877069*T5^8 - 28102878716418293861451798*T5^6 + 954619034782622058128117713*T5^4 - 17875031833233881864331390288*T5^2 + 216594421992348078302697556224

T13^7 - 31*T13^6 - 7296*T13^5 + 191240*T13^4 + 8458096*T13^3 - 165766128*T13^2 - 1671949568*T13 + 5757261056

T19^14 + 231*T19^13 + 1995*T19^12 - 3647952*T19^11 - 13805967*T19^10 + 37127164455*T19^9 + 1406994917675*T19^8 - 125506127417712*T19^7 - 5733729396587295*T19^6 + 327632920689790359*T19^5 + 24575870297264341083*T19^4 + 333892326800271503520*T19^3 - 1776787352814933169520*T19^2 - 45964584628253295628800*T19 + 456822063402445977772800