Newform orbit 315.3.bd.a

• Label: 315.3.bd.a
• Level: $315$
• Weight: $3$
• Character orbit: 315.bd
• Analytic conductor: $8.583$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.58312832735\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 128 q^{4} + 8 q^{6} + 2 q^{7} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
191.1	−3.40422	+	1.96543i	−2.76795	+	1.15690i	5.72580	−	9.91738i		−	2.23607i	7.14891	−	9.37856i	−5.48428	+	4.35002i			29.2911i	6.32315	−	6.40451i	4.39483	+	7.61206i
191.2	−3.37734	+	1.94991i	0.862317	−	2.87340i	5.60430	−	9.70693i			2.23607i	2.69052	+	11.3859i	−5.81707	−	3.89380i			28.1122i	−7.51282	−	4.95556i	−4.36013	−	7.55197i
191.3	−3.31611	+	1.91456i	−0.472550	+	2.96255i	5.33107	−	9.23369i			2.23607i	−4.10495	−	10.7289i	6.99168	+	0.341145i			25.5102i	−8.55339	−	2.79990i	−4.28108	−	7.41506i
191.4	−3.13136	+	1.80789i	−2.12443	−	2.11820i	4.53696	−	7.85824i		−	2.23607i	10.4819	+	2.79212i	6.87314	−	1.32662i			18.3462i	0.0264256	+	8.99996i	4.04257	+	7.00194i
191.5	−2.97211	+	1.71595i	1.59841	+	2.53872i	3.88897	−	6.73590i		−	2.23607i	−9.10697	−	4.80256i	0.867017	+	6.94610i			12.9655i	−3.89017	+	8.11582i	3.83698	+	6.64585i
191.6	−2.90242	+	1.67571i	−2.87794	−	0.847025i	3.61601	−	6.26311i			2.23607i	9.77235	−	2.36418i	1.78745	−	6.76794i			10.8319i	7.56510	+	4.87538i	−3.74700	−	6.49000i
191.7	−2.87401	+	1.65931i	2.99999	+	0.00928441i	3.50663	−	6.07367i			2.23607i	−8.63740	+	4.95123i	−0.00730779	+	7.00000i			9.99990i	8.99983	+	0.0557062i	−3.71033	−	6.42649i
191.8	−2.75421	+	1.59014i	2.63214	−	1.43939i	3.05711	−	5.29507i		−	2.23607i	−4.96063	+	8.14985i	−6.29506	+	3.06140i			6.72381i	4.85633	−	7.57734i	3.55567	+	6.15860i
191.9	−2.68460	+	1.54995i	−2.96155	+	0.478752i	2.80470	−	4.85789i			2.23607i	7.20853	−	5.87552i	−3.08652	+	6.28279i			4.98901i	8.54159	−	2.83570i	−3.46580	−	6.00294i
191.10	−2.64609	+	1.52772i	0.856453	+	2.87515i	2.66786	−	4.62086i		−	2.23607i	−6.65867	−	6.29948i	−4.66718	−	5.21704i			4.08119i	−7.53298	+	4.92486i	3.41609	+	5.91683i
191.11	−2.62606	+	1.51616i	−1.33415	−	2.68701i	2.59748	−	4.49896i		−	2.23607i	7.57751	+	5.03348i	−6.79864	+	1.66689i			3.62348i	−5.44008	+	7.16977i	3.39023	+	5.87206i
191.12	−2.39536	+	1.38296i	1.51876	−	2.58716i	1.82516	−	3.16127i		−	2.23607i	−0.0600361	+	8.29754i	3.42753	−	6.10344i		−	0.967195i	−4.38675	−	7.85853i	3.09239	+	5.35618i
191.13	−2.34079	+	1.35145i	1.92287	−	2.30274i	1.65286	−	2.86283i			2.23607i	−1.38898	+	7.98888i	6.85313	+	1.42641i		−	1.87659i	−1.60518	−	8.85570i	−3.02194	−	5.23416i
191.14	−2.24728	+	1.29747i	−1.45768	+	2.62205i	1.36684	−	2.36744i		−	2.23607i	−0.126206	−	7.78378i	6.94463	+	0.878685i		−	3.28600i	−4.75032	−	7.64424i	2.90123	+	5.02507i
191.15	−2.05535	+	1.18666i	2.83379	+	0.984707i	0.816307	−	1.41388i			2.23607i	−6.99293	+	1.33882i	−5.07212	−	4.82427i		−	5.61855i	7.06071	+	5.58090i	−2.65344	−	4.59590i
191.16	−2.02974	+	1.17187i	1.90305	+	2.31914i	0.746558	−	1.29308i			2.23607i	−6.58042	−	2.47711i	6.01727	−	3.57665i		−	5.87548i	−1.75679	+	8.82687i	−2.62038	−	4.53863i
191.17	−1.90090	+	1.09748i	−0.361145	+	2.97818i	0.408938	−	0.708302i			2.23607i	−2.58201	−	6.05757i	−2.67852	+	6.46726i		−	6.98465i	−8.73915	−	2.15111i	−2.45405	−	4.25053i
191.18	−1.79899	+	1.03865i	−1.92533	−	2.30067i	0.157588	−	0.272950i			2.23607i	5.85326	+	2.13915i	−6.91080	+	1.11393i		−	7.65449i	−1.58620	+	8.85912i	−2.32249	−	4.02267i
191.19	−1.77004	+	1.02193i	−2.97367	−	0.396585i	0.0886935	−	0.153622i		−	2.23607i	5.66880	−	2.33692i	−2.88346	−	6.37853i		−	7.81291i	8.68544	+	2.35862i	2.28511	+	3.95793i
191.20	−1.57000	+	0.906438i	2.26447	−	1.96778i	−0.356739	+	0.617891i			2.23607i	−1.77154	+	5.14202i	−5.05153	−	4.84583i		−	8.54495i	1.25567	−	8.91197i	−2.02686	−	3.51062i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.n	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.3.bd.a	yes	128
7.c	even	3	1		315.3.s.a	✓	128
9.d	odd	6	1		315.3.s.a	✓	128
63.n	odd	6	1	inner	315.3.bd.a	yes	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.3.s.a	✓	128	7.c	even	3	1
315.3.s.a	✓	128	9.d	odd	6	1
315.3.bd.a	yes	128	1.a	even	1	1	trivial
315.3.bd.a	yes	128	63.n	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(315, [\chi])\).