Newform orbit 225.4.w.a

• Label: 225.4.w.a
• Level: $225$
• Weight: $4$
• Character orbit: 225.w
• Analytic conductor: $13.275$
• Analytic rank: $0$
• Dimension: $1408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2754297513\)
Analytic rank:	\(0\)
Dimension:	\(1408\)
Relative dimension:	\(88\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1408 q - 24 q^{2} - 20 q^{3} - 10 q^{4} - 24 q^{5} - 12 q^{6} - 8 q^{7} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−4.34305	−	3.51693i	5.17306	+	0.489355i	4.82996	+	22.7232i	−1.32572	−	11.1015i	−20.7458	−	20.3186i	16.6107	−	4.45083i	38.6423	−	75.8397i	26.5211	+	5.06292i	−33.2854	+	52.8766i
2.2	−4.30462	−	3.48581i	−5.14615	+	0.719092i	4.71556	+	22.1850i	−10.7396	+	3.10814i	24.6588	+	14.8431i	−21.5496	+	5.77420i	36.9167	−	72.4531i	25.9658	−	7.40112i	57.0644	+	24.0569i
2.3	−4.21318	−	3.41176i	3.23134	−	4.06920i	4.44744	+	20.9235i	4.30303	+	10.3191i	−27.4974	+	6.11970i	−10.0061	+	2.68112i	32.9584	−	64.6846i	−6.11685	−	26.2980i	17.0769	−	58.1571i
2.4	−4.06587	−	3.29247i	−0.761351	+	5.14007i	4.02760	+	18.9483i	10.6190	−	3.49824i	20.0191	−	18.3921i	−24.4627	+	6.55475i	27.0098	−	53.0097i	−25.8407	−	7.82680i	−54.6932	−	20.7393i
2.5	−4.05443	−	3.28321i	−2.87084	−	4.33108i	3.99563	+	18.7980i	4.48143	−	10.2429i	−2.58024	+	26.9857i	10.4257	−	2.79356i	26.5697	−	52.1459i	−10.5166	+	24.8677i	−51.7992	+	26.8156i
2.6	−3.93158	−	3.18373i	−5.18779	−	0.294593i	3.65788	+	17.2090i	10.7908	+	2.92566i	19.4583	+	17.6748i	18.0371	−	4.83302i	22.0336	−	43.2434i	26.8264	+	3.05658i	−33.1102	−	45.8573i
2.7	−3.83852	−	3.10837i	−1.62193	+	4.93653i	3.40897	+	16.0379i	−2.11395	+	10.9787i	21.5704	−	13.9074i	30.7108	−	8.22892i	18.8275	−	36.9510i	−21.7387	−	16.0134i	42.2402	−	35.5709i
2.8	−3.69121	−	2.98908i	4.05500	+	3.24915i	3.02711	+	14.2414i	−2.68391	+	10.8534i	−5.25589	−	24.1140i	−13.5034	+	3.61821i	14.1446	−	27.7603i	5.88609	+	26.3506i	42.3486	−	32.0398i
2.9	−3.67205	−	2.97357i	−1.35099	−	5.01745i	2.97855	+	14.0130i	−11.1798	−	0.110934i	−9.95884	+	22.4416i	1.73352	−	0.464495i	13.5701	−	26.6329i	−23.3497	+	13.5570i	40.7229	+	33.6512i
2.10	−3.56922	−	2.89030i	0.113008	+	5.19492i	2.72222	+	12.8071i	−8.02307	−	7.78655i	14.6115	−	18.8685i	2.79210	−	0.748141i	10.6195	−	20.8420i	−26.9745	+	1.17414i	6.13063	+	50.9810i
2.11	−3.45931	−	2.80130i	4.09157	+	3.20297i	2.45629	+	11.5559i	11.0346	+	1.79948i	−5.18157	−	22.5418i	17.8438	−	4.78123i	7.70773	−	15.1273i	6.48197	+	26.2104i	−33.1312	−	37.1361i
2.12	−3.43604	−	2.78245i	−4.40205	+	2.76079i	2.40104	+	11.2960i	−6.72318	−	8.93302i	22.8074	+	2.76228i	13.4014	−	3.59090i	7.12244	−	13.9786i	11.7561	−	24.3063i	−1.75456	+	49.4011i
2.13	−3.39360	−	2.74808i	4.48081	−	2.63104i	2.30126	+	10.8266i	−10.5716	+	3.63888i	−22.4364	−	3.38492i	16.4577	−	4.40984i	6.08311	−	11.9388i	13.1552	−	23.5784i	45.8757	+	16.7027i
2.14	−3.37705	−	2.73468i	2.42726	−	4.59439i	2.26269	+	10.6451i	−0.386586	−	11.1737i	−20.7612	+	8.87771i	−26.6297	+	7.13540i	5.68746	−	11.1623i	−15.2168	−	22.3035i	−29.2509	+	38.7912i
2.15	−3.26919	−	2.64734i	−3.05579	−	4.20263i	2.01591	+	9.48411i	8.15595	+	7.64725i	−1.13582	+	21.8289i	−26.6828	+	7.14965i	3.23897	−	6.35685i	−8.32424	+	25.6848i	−6.41849	−	46.5919i
2.16	−3.17679	−	2.57251i	1.91656	−	4.82978i	1.81087	+	8.51949i	11.0680	−	1.58080i	−18.5132	+	10.4128i	21.5972	−	5.78695i	1.31729	−	2.58533i	−19.6536	−	18.5131i	−39.2274	−	23.4508i
2.17	−3.13886	−	2.54180i	5.02715	+	1.31446i	1.72840	+	8.13149i	−9.77243	−	5.43136i	−12.4384	−	16.9039i	−29.7876	+	7.98155i	0.574202	−	1.12693i	23.5444	+	13.2159i	16.8688	+	41.8878i
2.18	−2.92432	−	2.36807i	−4.88673	−	1.76633i	1.28061	+	6.02479i	−3.89595	+	10.4796i	10.1076	+	16.7374i	7.55424	−	2.02415i	−3.14435	+	6.17113i	20.7602	+	17.2631i	36.2094	−	21.4198i
2.19	−2.86259	−	2.31808i	−4.98594	+	1.46300i	1.15763	+	5.44620i	6.04904	−	9.40261i	17.6640	+	7.36982i	−11.6332	+	3.11711i	−4.06714	+	7.98221i	22.7192	−	14.5889i	−39.1119	+	12.8937i
2.20	−2.75509	−	2.23103i	5.07482	−	1.11632i	0.949743	+	4.46819i	9.57836	−	5.76672i	−16.4721	−	8.24653i	−6.27542	+	1.68149i	−5.52365	+	10.8408i	24.5077	−	11.3302i	−39.2550	−	5.48176i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
25.f	odd	20	1	inner
225.w	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.4.w.a	✓	1408
9.d	odd	6	1	inner	225.4.w.a	✓	1408
25.f	odd	20	1	inner	225.4.w.a	✓	1408
225.w	even	60	1	inner	225.4.w.a	✓	1408

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.4.w.a	✓	1408	1.a	even	1	1	trivial
225.4.w.a	✓	1408	9.d	odd	6	1	inner
225.4.w.a	✓	1408	25.f	odd	20	1	inner
225.4.w.a	✓	1408	225.w	even	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(225, [\chi])\).