Newform orbit 336.2.y.a

• Label: 336.2.y.a
• Level: $336$
• Weight: $2$
• Character orbit: 336.y
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(60\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 8 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
125.1	−1.41405	−	0.0212996i	−1.55654	+	0.759733i	1.99909	+	0.0602376i	−0.700186	+	0.700186i	2.21721	−	1.04115i	2.03996	−	1.68481i	−2.82554	−	0.127759i	1.84561	−	2.36510i	1.00501	−	0.975187i
125.2	−1.41405	−	0.0212996i	1.55654	−	0.759733i	1.99909	+	0.0602376i	0.700186	−	0.700186i	−2.21721	+	1.04115i	−2.03996	−	1.68481i	−2.82554	−	0.127759i	1.84561	−	2.36510i	−1.00501	+	0.975187i
125.3	−1.37838	+	0.316322i	−0.367327	−	1.69265i	1.79988	−	0.872025i	−0.461355	+	0.461355i	1.04174	+	2.21693i	−2.58125	+	0.580663i	−2.20509	+	1.77133i	−2.73014	+	1.24351i	0.489988	−	0.781861i
125.4	−1.37838	+	0.316322i	0.367327	+	1.69265i	1.79988	−	0.872025i	0.461355	−	0.461355i	−1.04174	−	2.21693i	2.58125	+	0.580663i	−2.20509	+	1.77133i	−2.73014	+	1.24351i	−0.489988	+	0.781861i
125.5	−1.37255	−	0.340756i	−1.24999	−	1.19897i	1.76777	+	0.935408i	2.22287	−	2.22287i	1.30712	+	2.07158i	1.16923	−	2.37337i	−2.10760	−	1.88627i	0.124963	+	2.99740i	−3.80846	+	2.29354i
125.6	−1.37255	−	0.340756i	1.24999	+	1.19897i	1.76777	+	0.935408i	−2.22287	+	2.22287i	−1.30712	−	2.07158i	−1.16923	−	2.37337i	−2.10760	−	1.88627i	0.124963	+	2.99740i	3.80846	−	2.29354i
125.7	−1.28455	−	0.591550i	−1.44974	+	0.947768i	1.30014	+	1.51975i	0.639017	−	0.639017i	2.42291	−	0.359865i	−1.10656	+	2.40323i	−0.771085	−	2.72129i	1.20347	−	2.74803i	−1.19886	+	0.442839i
125.8	−1.28455	−	0.591550i	1.44974	−	0.947768i	1.30014	+	1.51975i	−0.639017	+	0.639017i	−2.42291	+	0.359865i	1.10656	+	2.40323i	−0.771085	−	2.72129i	1.20347	−	2.74803i	1.19886	−	0.442839i
125.9	−1.22270	+	0.710642i	−1.68333	−	0.407913i	0.989977	−	1.73780i	2.35722	−	2.35722i	2.34809	−	0.697491i	−0.422823	+	2.61175i	0.0245110	+	2.82832i	2.66721	+	1.37331i	−1.20703	+	4.55731i
125.10	−1.22270	+	0.710642i	1.68333	+	0.407913i	0.989977	−	1.73780i	−2.35722	+	2.35722i	−2.34809	+	0.697491i	0.422823	+	2.61175i	0.0245110	+	2.82832i	2.66721	+	1.37331i	1.20703	−	4.55731i
125.11	−1.18025	+	0.779110i	−0.970819	+	1.43440i	0.785975	−	1.83909i	−1.77648	+	1.77648i	0.0282512	−	2.44933i	−2.61801	−	0.382105i	0.505204	+	2.78294i	−1.11502	−	2.78509i	0.712616	−	3.48076i
125.12	−1.18025	+	0.779110i	0.970819	−	1.43440i	0.785975	−	1.83909i	1.77648	−	1.77648i	−0.0282512	+	2.44933i	2.61801	−	0.382105i	0.505204	+	2.78294i	−1.11502	−	2.78509i	−0.712616	+	3.48076i
125.13	−1.15231	−	0.819870i	−1.57036	−	0.730740i	0.655627	+	1.88949i	−2.81393	+	2.81393i	1.21042	+	2.12953i	−2.59104	−	0.535248i	0.793648	−	2.71480i	1.93204	+	2.29504i	5.54958	−	0.935458i
125.14	−1.15231	−	0.819870i	1.57036	+	0.730740i	0.655627	+	1.88949i	2.81393	−	2.81393i	−1.21042	−	2.12953i	2.59104	−	0.535248i	0.793648	−	2.71480i	1.93204	+	2.29504i	−5.54958	+	0.935458i
125.15	−0.957005	+	1.04122i	−1.52023	−	0.830004i	−0.168282	−	1.99291i	−1.24321	+	1.24321i	2.31908	−	0.788576i	1.10620	−	2.40340i	2.23610	+	1.73200i	1.62219	+	2.52359i	−0.104698	−	2.48421i
125.16	−0.957005	+	1.04122i	1.52023	+	0.830004i	−0.168282	−	1.99291i	1.24321	−	1.24321i	−2.31908	+	0.788576i	−1.10620	−	2.40340i	2.23610	+	1.73200i	1.62219	+	2.52359i	0.104698	+	2.48421i
125.17	−0.881724	−	1.10570i	−0.285007	+	1.70844i	−0.445126	+	1.94984i	−2.11567	+	2.11567i	2.14031	−	1.19124i	2.51356	+	0.825850i	2.54840	−	1.22704i	−2.83754	−	0.973835i	4.20472	+	0.473850i
125.18	−0.881724	−	1.10570i	0.285007	−	1.70844i	−0.445126	+	1.94984i	2.11567	−	2.11567i	−2.14031	+	1.19124i	−2.51356	+	0.825850i	2.54840	−	1.22704i	−2.83754	−	0.973835i	−4.20472	−	0.473850i
125.19	−0.621008	+	1.27057i	−0.254609	−	1.71324i	−1.22870	−	1.57807i	−0.914067	+	0.914067i	2.33490	+	0.740435i	0.812405	+	2.51794i	2.76808	−	0.581151i	−2.87035	+	0.872410i	−0.593743	−	1.72903i
125.20	−0.621008	+	1.27057i	0.254609	+	1.71324i	−1.22870	−	1.57807i	0.914067	−	0.914067i	−2.33490	−	0.740435i	−0.812405	+	2.51794i	2.76808	−	0.581151i	−2.87035	+	0.872410i	0.593743	+	1.72903i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
16.e	even	4	1	inner
21.c	even	2	1	inner
48.i	odd	4	1	inner
112.l	odd	4	1	inner
336.y	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.2.y.a	✓	120
3.b	odd	2	1	inner	336.2.y.a	✓	120
7.b	odd	2	1	inner	336.2.y.a	✓	120
16.e	even	4	1	inner	336.2.y.a	✓	120
21.c	even	2	1	inner	336.2.y.a	✓	120
48.i	odd	4	1	inner	336.2.y.a	✓	120
112.l	odd	4	1	inner	336.2.y.a	✓	120
336.y	even	4	1	inner	336.2.y.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.2.y.a	✓	120	1.a	even	1	1	trivial
336.2.y.a	✓	120	3.b	odd	2	1	inner
336.2.y.a	✓	120	7.b	odd	2	1	inner
336.2.y.a	✓	120	16.e	even	4	1	inner
336.2.y.a	✓	120	21.c	even	2	1	inner
336.2.y.a	✓	120	48.i	odd	4	1	inner
336.2.y.a	✓	120	112.l	odd	4	1	inner
336.2.y.a	✓	120	336.y	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(336, [\chi])\).