Newform orbit 336.2.w.b

• Label: 336.2.w.b
• Level: $336$
• Weight: $2$
• Character orbit: 336.w
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
85.1	−1.40848	+	0.127230i	−0.707107	+	0.707107i	1.96762	−	0.358402i	−0.805240	−	0.805240i	0.905980	−	1.08591i		−	1.00000i	−2.72576	+	0.755143i		−	1.00000i	1.23661	+	1.03171i
85.2	−1.40007	−	0.199481i	0.707107	−	0.707107i	1.92041	+	0.558576i	1.25389	+	1.25389i	−1.13106	+	0.848947i		−	1.00000i	−2.57730	−	1.16513i		−	1.00000i	−1.50541	−	2.00566i
85.3	−1.14710	−	0.827134i	−0.707107	+	0.707107i	0.631698	+	1.89762i	2.39875	+	2.39875i	1.39600	−	0.226253i		−	1.00000i	0.844961	−	2.69927i		−	1.00000i	−0.767531	−	4.73571i
85.4	−0.884433	−	1.10353i	0.707107	−	0.707107i	−0.435558	+	1.95200i	−1.84737	−	1.84737i	−1.40570	−	0.154926i		−	1.00000i	2.53931	−	1.24576i		−	1.00000i	−0.404756	+	3.67251i
85.5	−0.819262	+	1.15274i	−0.707107	+	0.707107i	−0.657621	−	1.88879i	−0.979857	−	0.979857i	−0.235805	−	1.39442i		−	1.00000i	2.71605	+	0.789348i		−	1.00000i	1.93228	−	0.326762i
85.6	−0.394485	+	1.35808i	0.707107	−	0.707107i	−1.68876	−	1.07148i	−0.646579	−	0.646579i	0.681365	+	1.23925i		−	1.00000i	2.12135	−	1.87079i		−	1.00000i	1.13317	−	0.623041i
85.7	−0.371814	−	1.36446i	−0.707107	+	0.707107i	−1.72351	+	1.01465i	0.116928	+	0.116928i	1.22773	+	0.701908i		−	1.00000i	2.02528	+	1.97440i		−	1.00000i	0.116068	−	0.203020i
85.8	0.421626	+	1.34990i	0.707107	−	0.707107i	−1.64446	+	1.13831i	2.44528	+	2.44528i	1.25266	+	0.656389i		−	1.00000i	−2.22995	−	1.73992i		−	1.00000i	−2.26989	+	4.33188i
85.9	0.428680	+	1.34768i	−0.707107	+	0.707107i	−1.63247	+	1.15545i	−1.77230	−	1.77230i	−1.25607	−	0.649829i		−	1.00000i	−2.25697	−	1.70472i		−	1.00000i	1.62874	−	3.14824i
85.10	0.647888	−	1.25708i	0.707107	−	0.707107i	−1.16048	−	1.62889i	2.52107	+	2.52107i	−0.430761	−	1.34701i		−	1.00000i	−2.79950	+	0.403476i		−	1.00000i	4.80254	−	1.53580i
85.11	0.919506	−	1.07448i	0.707107	−	0.707107i	−0.309018	−	1.97598i	−1.77267	−	1.77267i	−0.109584	−	1.40996i		−	1.00000i	−2.40730	−	1.48489i		−	1.00000i	−3.53468	+	0.274720i
85.12	1.25104	+	0.659465i	−0.707107	+	0.707107i	1.13021	+	1.65004i	2.66347	+	2.66347i	−1.35093	+	0.418308i		−	1.00000i	0.325801	+	2.80960i		−	1.00000i	1.57565	+	5.08857i
85.13	1.35983	−	0.388411i	−0.707107	+	0.707107i	1.69827	−	1.05634i	−3.03597	−	3.03597i	−0.686897	+	1.23619i		−	1.00000i	1.89907	−	2.09608i		−	1.00000i	−5.30760	−	2.94920i
85.14	1.39708	+	0.219481i	0.707107	−	0.707107i	1.90366	+	0.613265i	−0.539395	−	0.539395i	1.14308	−	0.832687i		−	1.00000i	2.52496	+	1.27460i		−	1.00000i	−0.635190	−	0.871964i
253.1	−1.40848	−	0.127230i	−0.707107	−	0.707107i	1.96762	+	0.358402i	−0.805240	+	0.805240i	0.905980	+	1.08591i			1.00000i	−2.72576	−	0.755143i			1.00000i	1.23661	−	1.03171i
253.2	−1.40007	+	0.199481i	0.707107	+	0.707107i	1.92041	−	0.558576i	1.25389	−	1.25389i	−1.13106	−	0.848947i			1.00000i	−2.57730	+	1.16513i			1.00000i	−1.50541	+	2.00566i
253.3	−1.14710	+	0.827134i	−0.707107	−	0.707107i	0.631698	−	1.89762i	2.39875	−	2.39875i	1.39600	+	0.226253i			1.00000i	0.844961	+	2.69927i			1.00000i	−0.767531	+	4.73571i
253.4	−0.884433	+	1.10353i	0.707107	+	0.707107i	−0.435558	−	1.95200i	−1.84737	+	1.84737i	−1.40570	+	0.154926i			1.00000i	2.53931	+	1.24576i			1.00000i	−0.404756	−	3.67251i
253.5	−0.819262	−	1.15274i	−0.707107	−	0.707107i	−0.657621	+	1.88879i	−0.979857	+	0.979857i	−0.235805	+	1.39442i			1.00000i	2.71605	−	0.789348i			1.00000i	1.93228	+	0.326762i
253.6	−0.394485	−	1.35808i	0.707107	+	0.707107i	−1.68876	+	1.07148i	−0.646579	+	0.646579i	0.681365	−	1.23925i			1.00000i	2.12135	+	1.87079i			1.00000i	1.13317	+	0.623041i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	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.w.b	✓	28
4.b	odd	2	1		1344.2.w.b		28
8.b	even	2	1		2688.2.w.d		28
8.d	odd	2	1		2688.2.w.c		28
16.e	even	4	1	inner	336.2.w.b	✓	28
16.e	even	4	1		2688.2.w.d		28
16.f	odd	4	1		1344.2.w.b		28
16.f	odd	4	1		2688.2.w.c		28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.2.w.b	✓	28	1.a	even	1	1	trivial
336.2.w.b	✓	28	16.e	even	4	1	inner
1344.2.w.b		28	4.b	odd	2	1
1344.2.w.b		28	16.f	odd	4	1
2688.2.w.c		28	8.d	odd	2	1
2688.2.w.c		28	16.f	odd	4	1
2688.2.w.d		28	8.b	even	2	1
2688.2.w.d		28	16.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^28 + 24*T5^25 + 560*T5^24 + 144*T5^23 + 288*T5^22 + 13392*T5^21 + 103880*T5^20 + 103872*T5^19 + 170496*T5^18 + 1865824*T5^17 + 9658432*T5^16 + 16251712*T5^15 + 24014976*T5^14 + 99047488*T5^13 + 404937232*T5^12 + 842992384*T5^11 + 1158057984*T5^10 + 1636224512*T5^9 + 3585180160*T5^8 + 6967326720*T5^7 + 9317556224*T5^6 + 8063197184*T5^5 + 4377186304*T5^4 + 1235714048*T5^3 + 69337088*T5^2 - 42205184*T5 + 12845056