Newform orbit 369.2.i.a

• Label: 369.2.i.a
• Level: $369$
• Weight: $2$
• Character orbit: 369.i
• Analytic conductor: $2.946$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	369.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.94647983459\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{2} - 40 q^{4} - 6 q^{5} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
40.1	−1.37186	−	2.37613i	−1.43846	+	0.964799i	−2.76400	+	4.78738i	−1.31071	+	2.27022i	4.26585	+	2.09440i	−3.43269	+	1.98187i	9.67982			1.13833	−	2.77565i	7.19244
40.2	−1.37186	−	2.37613i	1.43846	−	0.964799i	−2.76400	+	4.78738i	−1.31071	+	2.27022i	−4.26585	−	2.09440i	3.43269	−	1.98187i	9.67982			1.13833	−	2.77565i	7.19244
40.3	−1.28965	−	2.23374i	−0.334516	−	1.69944i	−2.32640	+	4.02945i	1.69944	−	2.94352i	−3.36470	+	2.93891i	0.796837	−	0.460054i	6.84239			−2.77620	+	1.13698i	−8.76675
40.4	−1.28965	−	2.23374i	0.334516	+	1.69944i	−2.32640	+	4.02945i	1.69944	−	2.94352i	3.36470	−	2.93891i	−0.796837	+	0.460054i	6.84239			−2.77620	+	1.13698i	−8.76675
40.5	−1.09219	−	1.89173i	−0.695647	−	1.58621i	−1.38576	+	2.40021i	−1.04241	+	1.80551i	−2.24091	+	3.04842i	−1.85764	+	1.07251i	1.68529			−2.03215	+	2.20689i	4.55404
40.6	−1.09219	−	1.89173i	0.695647	+	1.58621i	−1.38576	+	2.40021i	−1.04241	+	1.80551i	2.24091	−	3.04842i	1.85764	−	1.07251i	1.68529			−2.03215	+	2.20689i	4.55404
40.7	−1.02463	−	1.77471i	−1.66345	−	0.482641i	−1.09974	+	1.90481i	0.131093	−	0.227059i	0.847871	+	3.44667i	−0.206845	+	0.119422i	0.408786			2.53411	+	1.60570i	−0.537287
40.8	−1.02463	−	1.77471i	1.66345	+	0.482641i	−1.09974	+	1.90481i	0.131093	−	0.227059i	−0.847871	−	3.44667i	0.206845	−	0.119422i	0.408786			2.53411	+	1.60570i	−0.537287
40.9	−0.844220	−	1.46223i	−1.66658	+	0.471704i	−0.425416	+	0.736843i	1.76630	−	3.05932i	2.09670	+	2.03871i	−1.51492	+	0.874637i	−1.94030			2.55499	−	1.57227i	−5.96458
40.10	−0.844220	−	1.46223i	1.66658	−	0.471704i	−0.425416	+	0.736843i	1.76630	−	3.05932i	−2.09670	−	2.03871i	1.51492	−	0.874637i	−1.94030			2.55499	−	1.57227i	−5.96458
40.11	−0.754849	−	1.30744i	−0.884106	+	1.48941i	−0.139594	+	0.241783i	−0.478247	+	0.828349i	2.61468	+	0.0316297i	−0.418940	+	0.241875i	−2.59791			−1.43671	−	2.63360i	1.44402
40.12	−0.754849	−	1.30744i	0.884106	−	1.48941i	−0.139594	+	0.241783i	−0.478247	+	0.828349i	−2.61468	−	0.0316297i	0.418940	−	0.241875i	−2.59791			−1.43671	−	2.63360i	1.44402
40.13	−0.656534	−	1.13715i	−1.73005	−	0.0833104i	0.137927	−	0.238897i	−2.19233	+	3.79722i	1.04110	+	2.02202i	3.55464	−	2.05227i	−2.98835			2.98612	+	0.288262i	5.75734
40.14	−0.656534	−	1.13715i	1.73005	+	0.0833104i	0.137927	−	0.238897i	−2.19233	+	3.79722i	−1.04110	−	2.02202i	−3.55464	+	2.05227i	−2.98835			2.98612	+	0.288262i	5.75734
40.15	−0.594267	−	1.02930i	−0.217085	−	1.71839i	0.293693	−	0.508692i	0.468761	−	0.811918i	−1.63974	+	1.24463i	4.30848	−	2.48750i	−3.07520			−2.90575	+	0.746075i	−1.11428
40.16	−0.594267	−	1.02930i	0.217085	+	1.71839i	0.293693	−	0.508692i	0.468761	−	0.811918i	1.63974	−	1.24463i	−4.30848	+	2.48750i	−3.07520			−2.90575	+	0.746075i	−1.11428
40.17	−0.232257	−	0.402281i	−1.13165	−	1.31125i	0.892113	−	1.54519i	−0.820479	+	1.42111i	−0.264656	+	0.759788i	−2.48226	+	1.43313i	−1.75783			−0.438739	+	2.96774i	0.762248
40.18	−0.232257	−	0.402281i	1.13165	+	1.31125i	0.892113	−	1.54519i	−0.820479	+	1.42111i	0.264656	−	0.759788i	2.48226	−	1.43313i	−1.75783			−0.438739	+	2.96774i	0.762248
40.19	−0.231188	−	0.400430i	−1.08307	+	1.35165i	0.893104	−	1.54690i	0.589208	−	1.02054i	0.791635	+	0.121209i	2.12181	−	1.22503i	−1.75065			−0.653913	−	2.92787i	−0.544872
40.20	−0.231188	−	0.400430i	1.08307	−	1.35165i	0.893104	−	1.54690i	0.589208	−	1.02054i	−0.791635	−	0.121209i	−2.12181	+	1.22503i	−1.75065			−0.653913	−	2.92787i	−0.544872

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
41.b	even	2	1	inner
369.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	369.2.i.a	✓	80
3.b	odd	2	1		1107.2.i.a		80
9.c	even	3	1	inner	369.2.i.a	✓	80
9.d	odd	6	1		1107.2.i.a		80
41.b	even	2	1	inner	369.2.i.a	✓	80
123.b	odd	2	1		1107.2.i.a		80
369.i	even	6	1	inner	369.2.i.a	✓	80
369.k	odd	6	1		1107.2.i.a		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
369.2.i.a	✓	80	1.a	even	1	1	trivial
369.2.i.a	✓	80	9.c	even	3	1	inner
369.2.i.a	✓	80	41.b	even	2	1	inner
369.2.i.a	✓	80	369.i	even	6	1	inner
1107.2.i.a		80	3.b	odd	2	1
1107.2.i.a		80	9.d	odd	6	1
1107.2.i.a		80	123.b	odd	2	1
1107.2.i.a		80	369.k	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(369, [\chi])\).