Newform orbit 300.3.t.a

• Label: 300.3.t.a
• Level: $300$
• Weight: $3$
• Character orbit: 300.t
• Analytic conductor: $8.174$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	300.t (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.17440793081\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{5} - 30 q^{8} - 180 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.98898	−	0.209627i	−1.40126	+	1.01807i	3.91211	+	0.833889i	−1.55591	−	4.75175i	3.00050	−	1.73119i	7.92052			−7.60632	−	2.47867i	0.927051	−	2.85317i	2.09859	+	9.77732i
19.2	−1.96691	−	0.362292i	1.40126	−	1.01807i	3.73749	+	1.42519i	−0.557886	+	4.96878i	−3.12499	+	1.49480i	12.1496			−6.83498	−	4.15729i	0.927051	−	2.85317i	2.89746	−	9.57104i
19.3	−1.96407	+	0.377424i	1.40126	−	1.01807i	3.71510	−	1.48257i	3.99710	+	3.00386i	−2.36792	+	2.52843i	−8.63330			−6.73715	+	4.31403i	0.927051	−	2.85317i	−8.98430	−	4.39117i
19.4	−1.96282	+	0.383870i	−1.40126	+	1.01807i	3.70529	−	1.50693i	4.40354	−	2.36830i	2.35960	−	2.53619i	−8.67853			−6.69433	+	4.38018i	0.927051	−	2.85317i	−7.73422	+	6.33891i
19.5	−1.94080	−	0.483026i	−1.40126	+	1.01807i	3.53337	+	1.87491i	−4.91730	−	0.905607i	3.21131	−	1.29903i	−12.3316			−5.95192	−	5.34553i	0.927051	−	2.85317i	9.10605	+	4.13278i
19.6	−1.93862	+	0.491691i	1.40126	−	1.01807i	3.51648	−	1.90640i	−4.34660	+	2.47126i	−2.21593	+	2.66264i	−7.28594			−5.87975	+	5.42481i	0.927051	−	2.85317i	7.21129	−	6.92801i
19.7	−1.85738	+	0.741705i	−1.40126	+	1.01807i	2.89975	−	2.75526i	−4.34660	+	2.47126i	1.84756	−	2.93027i	7.28594			−3.34235	+	7.26833i	0.927051	−	2.85317i	6.24035	−	7.81396i
19.8	−1.85549	−	0.746441i	1.40126	−	1.01807i	2.88565	+	2.77002i	3.99388	−	3.00814i	−3.35995	+	0.843065i	−1.25731			−3.28663	−	7.29370i	0.927051	−	2.85317i	−9.65599	+	2.60036i
19.9	−1.83664	−	0.791689i	1.40126	−	1.01807i	2.74646	+	2.90809i	−4.17591	−	2.74987i	−3.37960	+	0.760470i	−1.90697			−2.74194	−	7.51543i	0.927051	−	2.85317i	5.49258	+	8.35653i
19.10	−1.81358	+	0.843156i	1.40126	−	1.01807i	2.57817	−	3.05827i	4.40354	−	2.36830i	−1.68290	+	3.02784i	8.67853			−2.09714	+	7.72023i	0.927051	−	2.85317i	−5.98935	+	8.00798i
19.11	−1.81081	+	0.849106i	−1.40126	+	1.01807i	2.55804	−	3.07513i	3.99710	+	3.00386i	1.67295	−	3.03335i	8.63330			−2.02099	+	7.74052i	0.927051	−	2.85317i	−9.78857	−	2.04544i
19.12	−1.71888	−	1.02248i	−1.40126	+	1.01807i	1.90909	+	3.51502i	4.99894	−	0.102816i	3.44955	−	0.317194i	5.17984			0.312530	−	7.99389i	0.927051	−	2.85317i	−8.69770	−	4.93457i
19.13	−1.48591	+	1.33869i	1.40126	−	1.01807i	0.415834	−	3.97833i	−1.55591	−	4.75175i	−0.719256	+	3.38861i	−7.92052			4.70784	+	6.46809i	0.927051	−	2.85317i	8.67305	+	4.97777i
19.14	−1.38659	−	1.44131i	1.40126	−	1.01807i	−0.154756	+	3.99701i	3.10974	+	3.91529i	−3.41033	−	0.608002i	−10.8238			5.97551	−	5.31914i	0.927051	−	2.85317i	1.33123	−	9.91100i
19.15	−1.37832	+	1.44922i	−1.40126	+	1.01807i	−0.200490	−	3.99497i	−0.557886	+	4.96878i	0.455962	−	3.43396i	−12.1496			6.06594	+	5.21578i	0.927051	−	2.85317i	−6.43192	−	7.65705i
19.16	−1.30758	−	1.51335i	−1.40126	+	1.01807i	−0.580459	+	3.95766i	−0.792479	+	4.93680i	3.37296	+	0.789380i	0.703844			6.74832	−	4.29653i	0.927051	−	2.85317i	8.50734	−	5.25597i
19.17	−1.30007	−	1.51981i	1.40126	−	1.01807i	−0.619642	+	3.95171i	−3.48097	+	3.58927i	−3.36901	−	0.806080i	−1.04829			6.81143	−	4.19576i	0.927051	−	2.85317i	9.98051	+	0.624108i
19.18	−1.28622	+	1.53155i	1.40126	−	1.01807i	−0.691273	−	3.93981i	−4.91730	−	0.905607i	−0.243100	+	3.45556i	12.3316			6.92314	+	4.00875i	0.927051	−	2.85317i	7.71172	−	6.36627i
19.19	−1.16714	−	1.62413i	−1.40126	+	1.01807i	−1.27558	+	3.79116i	−4.98328	−	0.408564i	3.28894	+	1.08759i	−3.47708			7.64610	−	2.35311i	0.927051	−	2.85317i	5.15262	+	8.57033i
19.20	−1.06237	+	1.69451i	−1.40126	+	1.01807i	−1.74273	−	3.60040i	3.99388	−	3.00814i	−0.236479	−	3.45602i	1.25731			7.95235	+	0.871892i	0.927051	−	2.85317i	0.854336	+	9.96344i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
25.e	even	10	1	inner
100.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	300.3.t.a	✓	240
4.b	odd	2	1	inner	300.3.t.a	✓	240
25.e	even	10	1	inner	300.3.t.a	✓	240
100.h	odd	10	1	inner	300.3.t.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
300.3.t.a	✓	240	1.a	even	1	1	trivial
300.3.t.a	✓	240	4.b	odd	2	1	inner
300.3.t.a	✓	240	25.e	even	10	1	inner
300.3.t.a	✓	240	100.h	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(300, [\chi])\).