Newform orbit 315.2.bq.a

• Label: 315.2.bq.a
• Level: $315$
• Weight: $2$
• Character orbit: 315.bq
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bq (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
164.1	−2.70830			0.577858	+	1.63281i	5.33489			−0.422616	+	2.19577i	−1.56501	−	4.42215i	−1.01783	+	2.44214i	−9.03188			−2.33216	+	1.88707i	1.14457	−	5.94680i
164.2	−2.57600			−1.19095	−	1.25763i	4.63577			2.13393	+	0.668088i	3.06790	+	3.23965i	2.40367	+	1.10561i	−6.78974			−0.163255	+	2.99555i	−5.49700	−	1.72099i
164.3	−2.53172			1.70470	−	0.306605i	4.40959			−1.09270	−	1.95090i	−4.31581	+	0.776236i	2.57916	+	0.589845i	−6.10041			2.81199	−	1.04534i	2.76640	+	4.93913i
164.4	−2.46032			0.957408	−	1.44339i	4.05317			−0.494408	+	2.18072i	−2.35553	+	3.55119i	−0.227160	−	2.63598i	−5.05144			−1.16674	−	2.76382i	1.21640	−	5.36528i
164.5	−2.29795			0.238210	+	1.71559i	3.28056			−0.0571900	−	2.23534i	−0.547395	−	3.94234i	−0.338914	−	2.62395i	−2.94267			−2.88651	+	0.817344i	0.131420	+	5.13669i
164.6	−2.16166			−1.44589	+	0.953629i	2.67276			−2.00646	+	0.986977i	3.12551	−	2.06142i	2.53153	−	0.769008i	−1.45428			1.18118	−	2.75768i	4.33727	−	2.13351i
164.7	−2.06518			1.70672	+	0.295145i	2.26496			2.20004	−	0.399802i	−3.52468	−	0.609527i	−2.20598	+	1.46070i	−0.547180			2.82578	+	1.00746i	−4.54346	+	0.825662i
164.8	−2.04520			−0.124265	−	1.72759i	2.18286			−1.93784	−	1.11569i	0.254147	+	3.53327i	−0.907139	+	2.48538i	−0.373988			−2.96912	+	0.429357i	3.96329	+	2.28181i
164.9	−2.00210			−1.55371	+	0.765499i	2.00841			2.11856	+	0.715328i	3.11068	−	1.53261i	−2.15805	−	1.53063i	−0.0168431			1.82802	−	2.37873i	−4.24158	−	1.43216i
164.10	−1.70249			−0.695317	−	1.58636i	0.898480			1.25685	−	1.84941i	1.18377	+	2.70076i	−0.756494	−	2.53529i	1.87533			−2.03307	+	2.20604i	−2.13977	+	3.14861i
164.11	−1.50968			1.73201	−	0.0112588i	0.279131			−1.98951	+	1.02071i	−2.61479	+	0.0169972i	−2.58044	−	0.584218i	2.59796			2.99975	−	0.0390010i	3.00352	−	1.54095i
164.12	−1.37324			−1.73021	+	0.0799234i	−0.114213			0.214801	−	2.22573i	2.37599	−	0.109754i	1.26290	+	2.32489i	2.90332			2.98722	−	0.276568i	−0.294973	+	3.05646i
164.13	−1.33100			1.20337	+	1.24576i	−0.228439			0.821920	+	2.07953i	−1.60168	−	1.65810i	1.38620	−	2.25354i	2.96605			−0.103821	+	2.99820i	−1.09398	−	2.76786i
164.14	−1.20693			−0.344453	+	1.69745i	−0.543308			−2.01662	−	0.966042i	0.415733	−	2.04872i	−2.56388	+	0.653073i	3.06961			−2.76270	−	1.16939i	2.43393	+	1.16595i
164.15	−1.20290			1.27624	−	1.17099i	−0.553033			2.10508	−	0.754091i	−1.53519	+	1.40858i	2.51709	−	0.815010i	3.07104			0.257567	−	2.98892i	−2.53219	+	0.907096i
164.16	−0.932214			−0.704225	+	1.58242i	−1.13098			1.20128	+	1.88598i	0.656489	−	1.47516i	0.405582	+	2.61448i	2.91874			−2.00813	−	2.22877i	−1.11985	−	1.75814i
164.17	−0.827410			−1.03098	−	1.39179i	−1.31539			−1.03969	+	1.97966i	0.853047	+	1.15158i	2.63782	+	0.204740i	2.74319			−0.874143	+	2.86982i	0.860253	−	1.63799i
164.18	−0.586009			−1.67705	−	0.433007i	−1.65659			−2.03395	−	0.928995i	0.982768	+	0.253746i	−1.03863	−	2.43336i	2.14280			2.62501	+	1.45235i	1.19192	+	0.544399i
164.19	−0.504719			−0.161348	−	1.72452i	−1.74526			1.16790	+	1.90683i	0.0814357	+	0.870398i	−2.52898	−	0.777347i	1.89030			−2.94793	+	0.556497i	−0.589461	−	0.962416i
164.20	−0.487080			1.51865	−	0.832881i	−1.76275			−1.77632	+	1.35819i	−0.739706	+	0.405680i	1.01806	+	2.44204i	1.83276			1.61262	−	2.52972i	0.865213	−	0.661546i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
63.i	even	6	1	inner
315.bq	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.bq.a	yes	88
3.b	odd	2	1		945.2.bq.a		88
5.b	even	2	1	inner	315.2.bq.a	yes	88
7.d	odd	6	1		315.2.u.a	✓	88
9.c	even	3	1		945.2.u.a		88
9.d	odd	6	1		315.2.u.a	✓	88
15.d	odd	2	1		945.2.bq.a		88
21.g	even	6	1		945.2.u.a		88
35.i	odd	6	1		315.2.u.a	✓	88
45.h	odd	6	1		315.2.u.a	✓	88
45.j	even	6	1		945.2.u.a		88
63.i	even	6	1	inner	315.2.bq.a	yes	88
63.t	odd	6	1		945.2.bq.a		88
105.p	even	6	1		945.2.u.a		88
315.q	odd	6	1		945.2.bq.a		88
315.bq	even	6	1	inner	315.2.bq.a	yes	88

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.u.a	✓	88	7.d	odd	6	1
315.2.u.a	✓	88	9.d	odd	6	1
315.2.u.a	✓	88	35.i	odd	6	1
315.2.u.a	✓	88	45.h	odd	6	1
315.2.bq.a	yes	88	1.a	even	1	1	trivial
315.2.bq.a	yes	88	5.b	even	2	1	inner
315.2.bq.a	yes	88	63.i	even	6	1	inner
315.2.bq.a	yes	88	315.bq	even	6	1	inner
945.2.u.a		88	9.c	even	3	1
945.2.u.a		88	21.g	even	6	1
945.2.u.a		88	45.j	even	6	1
945.2.u.a		88	105.p	even	6	1
945.2.bq.a		88	3.b	odd	2	1
945.2.bq.a		88	15.d	odd	2	1
945.2.bq.a		88	63.t	odd	6	1
945.2.bq.a		88	315.q	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).