Newform orbit 945.2.di.a

• Label: 945.2.di.a
• Level: $945$
• Weight: $2$
• Character orbit: 945.di
• Analytic conductor: $7.546$
• Analytic rank: $0$
• Dimension: $1680$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.di (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.54586299101\)
Analytic rank:	\(0\)
Dimension:	\(1680\)
Relative dimension:	\(140\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1680 q - 24 q^{2} - 12 q^{7} - 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−2.23186	−	1.56277i	−0.188802	−	1.72173i	1.85493	+	5.09638i	−1.68115	−	1.47436i	−2.26928	+	4.13772i	1.95113	+	1.78692i	2.41415	−	9.00972i	−2.92871	+	0.650134i	1.44802	+	5.91781i
13.2	−2.23186	−	1.56277i	0.188802	+	1.72173i	1.85493	+	5.09638i	1.68115	+	1.47436i	2.26928	−	4.13772i	−0.114699	−	2.64326i	2.41415	−	9.00972i	−2.92871	+	0.650134i	−1.44802	−	5.91781i
13.3	−2.20304	−	1.54258i	−1.55032	−	0.772338i	1.78977	+	4.91734i	−1.29639	+	1.82191i	2.22402	+	4.09299i	1.40635	−	2.24102i	2.25035	−	8.39841i	1.80699	+	2.39474i	5.66645	−	2.01395i
13.4	−2.20304	−	1.54258i	1.55032	+	0.772338i	1.78977	+	4.91734i	1.29639	−	1.82191i	−2.22402	−	4.09299i	2.62071	+	0.363171i	2.25035	−	8.39841i	1.80699	+	2.39474i	−5.66645	+	2.01395i
13.5	−2.18942	−	1.53305i	−1.17801	+	1.26976i	1.75928	+	4.83359i	−1.27192	−	1.83908i	4.52576	−	0.974103i	−1.60301	−	2.10484i	2.17478	−	8.11640i	−0.224598	−	2.99158i	−0.0346433	+	5.97644i
13.6	−2.18942	−	1.53305i	1.17801	−	1.26976i	1.75928	+	4.83359i	1.27192	+	1.83908i	−4.52576	+	0.974103i	0.582007	+	2.58094i	2.17478	−	8.11640i	−0.224598	−	2.99158i	0.0346433	−	5.97644i
13.7	−2.16435	−	1.51550i	−0.918543	−	1.46843i	1.70366	+	4.68076i	2.23443	+	0.0855556i	−0.237345	+	4.57025i	−2.64242	−	0.132811i	2.03866	−	7.60839i	−1.31256	+	2.69763i	−4.70644	−	3.57145i
13.8	−2.16435	−	1.51550i	0.918543	+	1.46843i	1.70366	+	4.68076i	−2.23443	−	0.0855556i	0.237345	−	4.57025i	−1.59677	+	2.10958i	2.03866	−	7.60839i	−1.31256	+	2.69763i	4.70644	+	3.57145i
13.9	−2.06973	−	1.44924i	−1.62734	+	0.593088i	1.49944	+	4.11967i	−0.292912	+	2.21680i	4.22768	+	1.13088i	−2.50221	+	0.859606i	1.55906	−	5.81847i	2.29649	−	1.93032i	3.81892	−	4.16367i
13.10	−2.06973	−	1.44924i	1.62734	−	0.593088i	1.49944	+	4.11967i	0.292912	−	2.21680i	−4.22768	−	1.13088i	−2.26689	+	1.36426i	1.55906	−	5.81847i	2.29649	−	1.93032i	−3.81892	+	4.16367i
13.11	−2.01289	−	1.40944i	−1.37448	+	1.05394i	1.38115	+	3.79469i	−2.12880	+	0.684275i	4.25214	−	0.184224i	2.24970	+	1.39243i	1.29629	−	4.83782i	0.778401	−	2.89726i	5.24946	+	1.62304i
13.12	−2.01289	−	1.40944i	1.37448	−	1.05394i	1.38115	+	3.79469i	2.12880	−	0.684275i	−4.25214	+	0.184224i	0.379418	−	2.61840i	1.29629	−	4.83782i	0.778401	−	2.89726i	−5.24946	−	1.62304i
13.13	−1.97645	−	1.38392i	−1.68367	−	0.406506i	1.30706	+	3.59111i	−0.176115	−	2.22912i	2.76511	+	3.13351i	−0.353046	+	2.62209i	1.13753	−	4.24534i	2.66951	+	1.36885i	−2.73685	+	4.64947i
13.14	−1.97645	−	1.38392i	1.68367	+	0.406506i	1.30706	+	3.59111i	0.176115	+	2.22912i	−2.76511	−	3.13351i	−2.23557	−	1.41500i	1.13753	−	4.24534i	2.66951	+	1.36885i	2.73685	−	4.64947i
13.15	−1.86385	−	1.30508i	−0.269430	+	1.71097i	1.08665	+	2.98555i	0.200151	−	2.22709i	2.73512	−	2.83735i	2.64495	+	0.0652897i	0.693225	−	2.58715i	−2.85481	−	0.921972i	−3.27958	+	3.88974i
13.16	−1.86385	−	1.30508i	0.269430	−	1.71097i	1.08665	+	2.98555i	−0.200151	+	2.22709i	−2.73512	+	2.83735i	1.65012	−	2.06811i	0.693225	−	2.58715i	−2.85481	−	0.921972i	3.27958	−	3.88974i
13.17	−1.85841	−	1.30127i	−1.69310	−	0.365267i	1.07633	+	2.95721i	1.71668	+	1.43284i	2.67116	+	2.88200i	1.95976	+	1.77746i	0.673494	−	2.51351i	2.73316	+	1.23686i	−1.32578	−	4.89666i
13.18	−1.85841	−	1.30127i	1.69310	+	0.365267i	1.07633	+	2.95721i	−1.71668	−	1.43284i	−2.67116	−	2.88200i	−0.101903	−	2.64379i	0.673494	−	2.51351i	2.73316	+	1.23686i	1.32578	+	4.89666i
13.19	−1.77405	−	1.24221i	−0.0211915	−	1.73192i	0.920150	+	2.52809i	−1.98532	+	1.02884i	−2.11381	+	3.09884i	−2.60042	+	0.487680i	0.386955	−	1.44414i	−2.99910	+	0.0734040i	4.80009	+	0.640953i
13.20	−1.77405	−	1.24221i	0.0211915	+	1.73192i	0.920150	+	2.52809i	1.98532	−	1.02884i	2.11381	−	3.09884i	−2.04510	+	1.67856i	0.386955	−	1.44414i	−2.99910	+	0.0734040i	−4.80009	−	0.640953i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.b	odd	2	1	inner
27.e	even	9	1	inner
35.f	even	4	1	inner
135.r	odd	36	1	inner
189.y	odd	18	1	inner
945.di	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	945.2.di.a	✓	1680
5.c	odd	4	1	inner	945.2.di.a	✓	1680
7.b	odd	2	1	inner	945.2.di.a	✓	1680
27.e	even	9	1	inner	945.2.di.a	✓	1680
35.f	even	4	1	inner	945.2.di.a	✓	1680
135.r	odd	36	1	inner	945.2.di.a	✓	1680
189.y	odd	18	1	inner	945.2.di.a	✓	1680
945.di	even	36	1	inner	945.2.di.a	✓	1680

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
945.2.di.a	✓	1680	1.a	even	1	1	trivial
945.2.di.a	✓	1680	5.c	odd	4	1	inner
945.2.di.a	✓	1680	7.b	odd	2	1	inner
945.2.di.a	✓	1680	27.e	even	9	1	inner
945.2.di.a	✓	1680	35.f	even	4	1	inner
945.2.di.a	✓	1680	135.r	odd	36	1	inner
945.2.di.a	✓	1680	189.y	odd	18	1	inner
945.2.di.a	✓	1680	945.di	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).