Newform orbit 315.2.u.a

• Label: 315.2.u.a
• Level: $315$
• Weight: $2$
• Character orbit: 315.u
• 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.u (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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 38 q^{4} - 6 q^{5} + 12 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.35415	+	2.34546i	−1.70299	+	0.315967i	−2.66745	−	4.62015i	−2.11290	+	0.731888i	1.56501	−	4.42215i	−1.60604	−	2.10253i	9.03188			2.80033	−	1.07618i	1.14457	−	5.94680i
59.2	−1.28800	+	2.23088i	1.68461	+	0.402583i	−2.31788	−	4.01469i	0.488384	+	2.18208i	−3.06790	+	3.23965i	−2.15932	+	1.52884i	6.78974			2.67585	+	1.35639i	−5.49700	−	1.72099i
59.3	−1.26586	+	2.19253i	−0.586821	−	1.62961i	−2.20480	−	3.81882i	1.14318	−	1.92175i	4.31581	+	0.776236i	−1.80040	+	1.93870i	6.10041			−2.31128	+	1.91258i	2.76640	+	4.93913i
59.4	−1.23016	+	2.13070i	0.771307	−	1.55083i	−2.02658	−	3.51014i	−2.13577	+	0.662193i	2.35553	+	3.55119i	2.39641	+	1.12126i	5.05144			−1.81017	−	2.39234i	1.21640	−	5.36528i
59.5	−1.14897	+	1.99008i	−1.60485	+	0.651500i	−1.64028	−	2.84105i	1.90726	−	1.16720i	0.547395	−	3.94234i	2.44187	+	1.01847i	2.94267			2.15110	−	2.09112i	0.131420	+	5.13669i
59.6	−1.08083	+	1.87205i	−0.102923	+	1.72899i	−1.33638	−	2.31468i	−1.85798	−	1.24416i	−3.12551	−	2.06142i	−0.599782	+	2.57687i	1.45428			−2.97881	−	0.355906i	4.33727	−	2.13351i
59.7	−1.03259	+	1.78850i	−1.10896	−	1.33049i	−1.13248	−	1.96151i	1.44626	+	1.70539i	3.52468	−	0.609527i	−0.162008	−	2.64079i	0.547180			−0.540404	+	2.95093i	−4.54346	+	0.825662i
59.8	−1.02260	+	1.77120i	1.55827	−	0.756177i	−1.09143	−	1.89041i	−0.00270873	−	2.23607i	−0.254147	+	3.53327i	−1.69883	−	2.02829i	0.373988			1.85639	−	2.35665i	3.96329	+	2.28181i
59.9	−1.00105	+	1.73387i	0.113913	+	1.72830i	−1.00421	−	1.73934i	0.439789	+	2.19239i	−3.11068	−	1.53261i	2.40459	−	1.10361i	0.0168431			−2.97405	+	0.393751i	−4.24158	−	1.43216i
59.10	−0.851246	+	1.47440i	1.72149	−	0.191018i	−0.449240	−	0.778107i	2.23006	+	0.163754i	−1.18377	+	2.70076i	2.57388	+	0.612504i	−1.87533			2.92702	−	0.657668i	−2.13977	+	3.14861i
59.11	−0.754840	+	1.30742i	−0.856257	−	1.50560i	−0.139566	−	0.241735i	−1.87872	−	1.21261i	2.61479	+	0.0169972i	1.79617	−	1.94262i	−2.59796			−1.53365	+	2.57836i	3.00352	−	1.54095i
59.12	−0.686620	+	1.18926i	0.795887	+	1.53836i	0.0571066	+	0.0989115i	2.03494	−	0.926840i	−2.37599	−	0.109754i	−2.64486	−	0.0687428i	−2.90332			−1.73313	+	2.44873i	−0.294973	+	3.05646i
59.13	−0.665500	+	1.15268i	−1.68054	−	0.419267i	0.114220	+	0.197834i	−1.38997	+	1.75157i	1.60168	−	1.65810i	1.25852	+	2.32726i	−2.96605			2.64843	+	1.40919i	−1.09398	−	2.76786i
59.14	−0.603467	+	1.04524i	−1.29781	+	1.14703i	0.271654	+	0.470519i	−0.171693	−	2.22947i	−0.415733	−	2.04872i	0.716364	−	2.54692i	−3.06961			0.368633	−	2.97727i	2.43393	+	1.16595i
59.15	−0.601450	+	1.04174i	0.375988	−	1.69075i	0.276517	+	0.478941i	1.70560	+	1.44600i	1.53519	+	1.40858i	−0.552728	+	2.58737i	−3.07104			−2.71727	−	1.27140i	−2.53219	+	0.907096i
59.16	−0.466107	+	0.807321i	−1.01831	+	1.40109i	0.565488	+	0.979455i	−1.03266	+	1.98333i	−0.656489	−	1.47516i	−2.46700	−	0.955995i	−2.91874			−0.926102	−	2.85348i	−1.11985	−	1.75814i
59.17	−0.413705	+	0.716558i	1.72082	+	0.196965i	0.657696	+	1.13916i	−2.23428	+	0.0894268i	−0.853047	+	1.15158i	−1.49622	+	2.18205i	−2.74319			2.92241	+	0.677881i	0.860253	−	1.63799i
59.18	−0.293005	+	0.507499i	1.21352	+	1.23587i	0.828297	+	1.43465i	−0.212444	−	2.22595i	−0.982768	+	0.253746i	2.62667	+	0.317199i	−2.14280			−0.0547336	+	2.99950i	1.19192	+	0.544399i
59.19	−0.252360	+	0.437100i	1.57415	−	0.722528i	0.872629	+	1.51144i	−1.06742	+	1.96485i	−0.0814357	+	0.870398i	1.93769	−	1.80149i	−1.89030			1.95591	−	2.27474i	−0.589461	−	0.962416i
59.20	−0.243540	+	0.421824i	−0.0380306	−	1.73163i	0.881376	+	1.52659i	−2.06439	−	0.859249i	0.739706	+	0.405680i	−2.62390	−	0.339358i	−1.83276			−2.99711	+	0.131710i	0.865213	−	0.661546i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
63.s	even	6	1	inner
315.u	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.u.a	✓	88
3.b	odd	2	1		945.2.u.a		88
5.b	even	2	1	inner	315.2.u.a	✓	88
7.d	odd	6	1		315.2.bq.a	yes	88
9.c	even	3	1		945.2.bq.a		88
9.d	odd	6	1		315.2.bq.a	yes	88
15.d	odd	2	1		945.2.u.a		88
21.g	even	6	1		945.2.bq.a		88
35.i	odd	6	1		315.2.bq.a	yes	88
45.h	odd	6	1		315.2.bq.a	yes	88
45.j	even	6	1		945.2.bq.a		88
63.k	odd	6	1		945.2.u.a		88
63.s	even	6	1	inner	315.2.u.a	✓	88
105.p	even	6	1		945.2.bq.a		88
315.u	even	6	1	inner	315.2.u.a	✓	88
315.bn	odd	6	1		945.2.u.a		88

    

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

Hecke kernels

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