Newform orbit 900.2.br.a

• Label: 900.2.br.a
• Level: $900$
• Weight: $2$
• Character orbit: 900.br
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $1408$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.br (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(1408\)
Relative dimension:	\(176\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41412	+	0.0164544i	1.73149	+	0.0441365i	1.99946	−	0.0465370i	0.109034	+	2.23341i	−2.44925	−	0.0339235i	4.22770	+	2.44086i	−2.82670	+	0.0987088i	2.99610	+	0.152844i	−0.190936	−	3.15651i
11.2	−1.41333	+	0.0498700i	0.415230	−	1.68154i	1.99503	−	0.140966i	2.10318	+	0.759360i	−0.503000	+	2.39729i	0.0542083	+	0.0312972i	−2.81261	+	0.298724i	−2.65517	−	1.39645i	−3.01037	−	0.968344i
11.3	−1.40913	−	0.119781i	−1.60716	+	0.645795i	1.97131	+	0.337574i	−2.02406	+	0.950366i	2.34205	−	0.717504i	2.01812	+	1.16516i	−2.73739	−	0.711810i	2.16590	−	2.07579i	2.96600	−	1.09675i
11.4	−1.40910	+	0.120168i	0.557010	+	1.64004i	1.97112	−	0.338658i	0.772972	−	2.09822i	−0.981963	−	2.24405i	−3.16649	−	1.82817i	−2.73681	+	0.714069i	−2.37948	+	1.82704i	−0.837055	+	3.04948i
11.5	−1.40781	−	0.134426i	1.43423	−	0.971076i	1.96386	+	0.378493i	−1.58778	−	1.57447i	−2.14966	+	1.17429i	0.442017	+	0.255198i	−2.71386	−	0.796841i	1.11402	−	2.78549i	2.02364	+	2.43000i
11.6	−1.40757	−	0.136935i	0.442037	+	1.67469i	1.96250	+	0.385490i	−2.12734	−	0.688794i	−0.392873	−	2.41778i	2.87689	+	1.66097i	−2.70956	−	0.811338i	−2.60921	+	1.48055i	2.90005	+	1.26083i
11.7	−1.40571	−	0.154836i	1.36450	+	1.06683i	1.95205	+	0.435310i	1.94619	+	1.10107i	−1.75291	−	1.71093i	−0.768920	−	0.443936i	−2.67662	−	0.914169i	0.723739	+	2.91139i	−2.56529	−	1.84913i
11.8	−1.40329	+	0.175426i	−0.557010	−	1.64004i	1.93845	−	0.492347i	0.772972	−	2.09822i	1.06935	+	2.20374i	3.16649	+	1.82817i	−2.63384	+	1.03096i	−2.37948	+	1.82704i	−0.716623	+	3.08001i
11.9	−1.39618	−	0.225138i	0.485359	−	1.66266i	1.89863	+	0.628665i	−0.538091	+	2.17036i	−1.05197	+	2.21209i	−1.07321	−	0.619616i	−2.50928	−	1.30518i	−2.52885	−	1.61397i	1.23990	−	2.90906i
11.10	−1.39282	+	0.245068i	−0.415230	+	1.68154i	1.87988	−	0.682671i	2.10318	+	0.759360i	0.166247	−	2.44384i	−0.0542083	−	0.0312972i	−2.45103	+	1.41154i	−2.65517	−	1.39645i	−3.11544	−	0.542227i
11.11	−1.39198	−	0.249789i	−1.73158	−	0.0402466i	1.87521	+	0.695401i	1.95000	−	1.09430i	2.40027	+	0.488552i	−0.359130	−	0.207344i	−2.43655	−	1.43639i	2.99676	+	0.139381i	−2.98771	+	1.03615i
11.12	−1.38664	+	0.277917i	−1.73149	−	0.0441365i	1.84552	−	0.770739i	0.109034	+	2.23341i	2.41321	−	0.420008i	−4.22770	−	2.44086i	−2.34487	+	1.58164i	2.99610	+	0.152844i	−0.771892	−	3.06662i
11.13	−1.38334	−	0.293885i	−1.42839	−	0.979641i	1.82726	+	0.813087i	−0.761994	−	2.10223i	1.68805	+	1.77496i	−2.58216	−	1.49081i	−2.28877	−	1.66178i	1.08061	+	2.79862i	0.436284	+	3.13204i
11.14	−1.36373	−	0.374497i	−0.602956	+	1.62371i	1.71950	+	1.02142i	1.43365	−	1.71599i	1.43034	−	1.98850i	3.67620	+	2.12245i	−1.96242	−	2.03689i	−2.27289	−	1.95806i	−2.59775	+	1.80325i
11.15	−1.35344	+	0.410138i	1.60716	−	0.645795i	1.66357	−	1.11019i	−2.02406	+	0.950366i	−1.91032	+	1.53320i	−2.01812	−	1.16516i	−1.79621	+	2.18487i	2.16590	−	2.07579i	2.34965	−	2.11640i
11.16	−1.34910	+	0.424189i	−1.43423	+	0.971076i	1.64013	−	1.14454i	−1.58778	−	1.57447i	1.52299	−	1.91846i	−0.442017	−	0.255198i	−1.72719	+	2.23983i	1.11402	−	2.78549i	2.80994	+	1.45060i
11.17	−1.34834	+	0.426592i	−0.442037	−	1.67469i	1.63604	−	1.15038i	−2.12734	−	0.688794i	1.31043	+	2.06949i	−2.87689	−	1.66097i	−1.71519	+	2.24903i	−2.60921	+	1.48055i	3.16221	+	0.0212227i
11.18	−1.34810	−	0.427357i	−1.08201	−	1.35250i	1.63473	+	1.15224i	−1.55446	+	1.60738i	0.880648	+	2.28571i	0.956559	+	0.552270i	−1.71136	−	2.25194i	−0.658525	+	2.92683i	2.78248	−	1.50259i
11.19	−1.34280	+	0.443717i	−1.36450	−	1.06683i	1.60623	−	1.19165i	1.94619	+	1.10107i	2.30563	+	0.827091i	0.768920	+	0.443936i	−1.62810	+	2.31286i	0.723739	+	2.91139i	−3.10191	−	0.614963i
11.20	−1.32919	−	0.482961i	1.70060	−	0.328557i	1.53350	+	1.28389i	1.74831	−	1.39406i	−2.41911	−	0.384609i	−1.93933	−	1.11967i	−1.41824	−	2.44716i	2.78410	−	1.11749i	−2.99712	+	1.00861i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
25.d	even	5	1	inner
36.h	even	6	1	inner
100.j	odd	10	1	inner
225.t	odd	30	1	inner
900.br	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.br.a	✓	1408
4.b	odd	2	1	inner	900.2.br.a	✓	1408
9.d	odd	6	1	inner	900.2.br.a	✓	1408
25.d	even	5	1	inner	900.2.br.a	✓	1408
36.h	even	6	1	inner	900.2.br.a	✓	1408
100.j	odd	10	1	inner	900.2.br.a	✓	1408
225.t	odd	30	1	inner	900.2.br.a	✓	1408
900.br	even	30	1	inner	900.2.br.a	✓	1408

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.br.a	✓	1408	1.a	even	1	1	trivial
900.2.br.a	✓	1408	4.b	odd	2	1	inner
900.2.br.a	✓	1408	9.d	odd	6	1	inner
900.2.br.a	✓	1408	25.d	even	5	1	inner
900.2.br.a	✓	1408	36.h	even	6	1	inner
900.2.br.a	✓	1408	100.j	odd	10	1	inner
900.2.br.a	✓	1408	225.t	odd	30	1	inner
900.2.br.a	✓	1408	900.br	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(900, [\chi])\).