Newform orbit 468.2.bm.a

• Label: 468.2.bm.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.bm
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{4} - 12 q^{5} - 6 q^{6} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
263.1	−1.41317	−	0.0542487i	−1.05857	+	1.37093i	1.99411	+	0.153326i	−0.0109731	+	0.00633532i	1.57031	−	1.87993i	2.19840	−	1.26925i	−2.80971	−	0.324854i	−0.758877	−	2.90243i	0.0158506	−	0.00835763i
263.2	−1.40866	+	0.125217i	1.69503	+	0.356177i	1.96864	−	0.352777i	−1.20784	+	0.697346i	−2.43232	−	0.289485i	0.468412	−	0.270438i	−2.72897	+	0.743451i	2.74628	+	1.20747i	1.61411	−	1.13356i
263.3	−1.40724	−	0.140261i	1.61920	−	0.614966i	1.96065	+	0.394763i	2.69163	−	1.55401i	−2.36486	+	0.638293i	−0.957376	+	0.552741i	−2.70374	−	0.830530i	2.24363	−	1.99151i	−4.00573	+	1.80934i
263.4	−1.38737	−	0.274233i	−1.64950	−	0.528362i	1.84959	+	0.760926i	2.50160	−	1.44430i	2.14357	+	1.18538i	3.98911	−	2.30311i	−2.35740	−	1.56291i	2.44167	+	1.74306i	−3.86672	+	1.31776i
263.5	−1.38729	+	0.274643i	−0.647777	−	1.60636i	1.84914	−	0.762019i	−1.32464	+	0.764782i	1.33983	+	2.05057i	1.59372	−	0.920136i	−2.35601	+	1.56500i	−2.16077	+	2.08112i	1.62762	−	1.42478i
263.6	−1.38531	−	0.284449i	−0.709205	−	1.58020i	1.83818	+	0.788102i	−0.0571275	+	0.0329826i	0.532984	+	2.39080i	−3.58033	+	2.06710i	−2.32227	−	1.61463i	−1.99406	+	2.24137i	0.0885213	−	0.0294413i
263.7	−1.36905	−	0.354544i	−1.73179	−	0.0301646i	1.74860	+	0.970776i	−3.34481	+	1.93112i	2.36021	+	0.655291i	−2.04665	+	1.18163i	−2.04974	−	1.94900i	2.99818	+	0.104477i	5.26388	−	1.45793i
263.8	−1.35834	+	0.393579i	0.850882	+	1.50864i	1.69019	−	1.06923i	3.14155	−	1.81378i	−1.74956	−	1.71436i	0.491881	−	0.283988i	−1.87503	+	2.11761i	−1.55200	+	2.56735i	−3.55344	+	3.70018i
263.9	−1.29634	−	0.565242i	0.232785	+	1.71634i	1.36100	+	1.46549i	0.363808	−	0.210044i	0.668377	−	2.35654i	−2.21953	+	1.28145i	−0.935965	−	2.66908i	−2.89162	+	0.799074i	−0.590345	+	0.0666499i
263.10	−1.28439	+	0.591889i	0.806419	−	1.53287i	1.29933	−	1.52044i	0.997107	−	0.575680i	−0.128470	+	2.44612i	−3.92843	+	2.26808i	−0.768926	+	2.72190i	−1.69938	−	2.47227i	−0.939939	+	1.32958i
263.11	−1.27752	−	0.606584i	0.968406	+	1.43603i	1.26411	+	1.54985i	−2.20444	+	1.27273i	−0.366084	−	2.42198i	3.39779	−	1.96171i	−0.674815	−	2.74675i	−1.12438	+	2.78133i	3.58823	−	0.288763i
263.12	−1.27418	+	0.613565i	−1.19334	+	1.25536i	1.24708	−	1.56359i	1.22419	−	0.706787i	0.750290	−	2.33175i	−1.63661	+	0.944897i	−0.629638	+	2.75745i	−0.151863	−	2.99615i	−1.12618	+	1.65169i
263.13	−1.26725	−	0.627762i	0.549338	−	1.64263i	1.21183	+	1.59106i	1.97731	−	1.14160i	−1.72733	+	1.73676i	1.93939	−	1.11971i	−0.536881	−	2.77701i	−2.39646	−	1.80472i	−3.22239	+	0.205409i
263.14	−1.24944	+	0.662491i	−1.72558	−	0.149530i	1.12221	−	1.65549i	−1.39708	+	0.806607i	2.25508	−	0.956355i	−0.469436	+	0.271029i	−0.305393	+	2.81189i	2.95528	+	0.516052i	1.21121	−	1.93337i
263.15	−1.21208	−	0.728606i	1.60548	−	0.649956i	0.938265	+	1.76626i	−2.65079	+	1.53044i	−2.41953	−	0.381963i	−2.32319	+	1.34129i	0.149655	−	2.82447i	2.15511	−	2.08698i	4.32805	+	0.0763773i
263.16	−1.19122	+	0.762224i	−0.0254317	+	1.73186i	0.838029	−	1.81596i	−3.62743	+	2.09430i	−1.28977	−	2.08242i	1.65943	−	0.958070i	0.385887	+	2.80198i	−2.99871	−	0.0880886i	2.72475	−	5.25969i
263.17	−1.13498	+	0.843688i	1.18305	−	1.26507i	0.576381	−	1.91515i	−3.20317	+	1.84935i	−0.275416	+	2.43396i	−1.16827	+	0.674500i	0.961602	+	2.65995i	−0.200800	−	2.99327i	2.07527	−	4.80146i
263.18	−1.09163	+	0.899080i	1.72319	−	0.175008i	0.383311	−	1.96292i	0.417266	−	0.240908i	−1.72374	+	1.74033i	4.45527	−	2.57225i	1.34639	+	2.48741i	2.93874	−	0.603144i	−0.238904	+	0.638138i
263.19	−1.03181	+	0.967144i	0.197719	−	1.72073i	0.129266	−	1.99582i	2.50186	−	1.44445i	1.46018	+	1.96669i	2.17658	−	1.25665i	1.79687	+	2.18432i	−2.92181	−	0.680441i	−1.18446	+	3.91006i
263.20	−0.978378	−	1.02116i	−1.39022	+	1.03310i	−0.0855526	+	1.99817i	−0.0309949	+	0.0178949i	2.41512	+	0.408873i	0.410610	−	0.237066i	2.12416	−	1.86760i	0.865397	−	2.87247i	0.0485983	+	0.0141429i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
117.k	odd	6	1	inner
468.bm	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.bm.a	yes	160
4.b	odd	2	1	inner	468.2.bm.a	yes	160
9.d	odd	6	1		468.2.bd.a	✓	160
13.c	even	3	1		468.2.bd.a	✓	160
36.h	even	6	1		468.2.bd.a	✓	160
52.j	odd	6	1		468.2.bd.a	✓	160
117.k	odd	6	1	inner	468.2.bm.a	yes	160
468.bm	even	6	1	inner	468.2.bm.a	yes	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.bd.a	✓	160	9.d	odd	6	1
468.2.bd.a	✓	160	13.c	even	3	1
468.2.bd.a	✓	160	36.h	even	6	1
468.2.bd.a	✓	160	52.j	odd	6	1
468.2.bm.a	yes	160	1.a	even	1	1	trivial
468.2.bm.a	yes	160	4.b	odd	2	1	inner
468.2.bm.a	yes	160	117.k	odd	6	1	inner
468.2.bm.a	yes	160	468.bm	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).