Newform orbit 592.2.o.a

• Label: 592.2.o.a
• Level: $592$
• Weight: $2$
• Character orbit: 592.o
• Analytic conductor: $4.727$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.72714379966\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(72\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{6} - 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
149.1	−1.41339	+	0.0482083i	2.43090	−	2.43090i	1.99535	−	0.136274i	−2.13872	−	2.13872i	−3.31863	+	3.55301i		−	3.32818i	−2.81364	+	0.288801i		−	8.81858i	3.12595	+	2.91975i
149.2	−1.41271	−	0.0652795i	−0.836073	+	0.836073i	1.99148	+	0.184441i	0.655181	+	0.655181i	1.23570	−	1.12655i		−	5.19811i	−2.80133	−	0.390564i			1.60196i	−0.882809	−	0.968348i
149.3	−1.41125	−	0.0915211i	−0.779067	+	0.779067i	1.98325	+	0.258318i	−2.16749	−	2.16749i	1.17076	−	1.02816i			0.954430i	−2.77522	−	0.546060i			1.78611i	2.86050	+	3.25724i
149.4	−1.40214	−	0.184393i	1.90182	−	1.90182i	1.93200	+	0.517089i	1.52178	+	1.52178i	−3.01730	+	2.31594i			4.52500i	−2.61359	−	1.08128i		−	4.23383i	−1.85314	−	2.41435i
149.5	−1.38400	+	0.290787i	1.16669	−	1.16669i	1.83089	−	0.804897i	−0.696435	−	0.696435i	−1.27544	+	1.95395i		−	0.336874i	−2.29988	+	1.64637i			0.277662i	1.16638	+	0.761349i
149.6	−1.37230	+	0.341743i	−1.67587	+	1.67587i	1.76642	−	0.937950i	1.91387	+	1.91387i	1.72708	−	2.87251i		−	1.74540i	−2.10353	+	1.89081i		−	2.61706i	−3.28046	−	1.97235i
149.7	−1.36926	+	0.353745i	0.732177	−	0.732177i	1.74973	−	0.968736i	1.04669	+	1.04669i	−0.743535	+	1.26154i			0.582507i	−2.05314	+	1.94541i			1.92783i	−1.80345	−	1.06292i
149.8	−1.35207	−	0.414627i	−2.15739	+	2.15739i	1.65617	+	1.12121i	0.718026	+	0.718026i	3.81144	−	2.02242i			2.21369i	−1.77437	−	2.20264i		−	6.30863i	−0.673107	−	1.26853i
149.9	−1.33749	−	0.459490i	−0.797589	+	0.797589i	1.57774	+	1.22912i	1.78233	+	1.78233i	1.43325	−	0.700280i			2.38057i	−1.54543	−	2.36889i			1.72770i	−1.56487	−	3.20280i
149.10	−1.30801	+	0.537686i	−1.87352	+	1.87352i	1.42179	−	1.40660i	−1.02360	−	1.02360i	1.44322	−	3.45796i			2.08295i	−1.10340	+	2.60432i		−	4.02018i	1.88925	+	0.788501i
149.11	−1.30766	−	0.538538i	0.499166	−	0.499166i	1.41995	+	1.40845i	−0.840334	−	0.840334i	−0.921560	+	0.383920i		−	1.46432i	−1.09832	−	2.60647i			2.50167i	0.646321	+	1.55142i
149.12	−1.20792	−	0.735473i	1.06020	−	1.06020i	0.918160	+	1.77679i	−2.42490	−	2.42490i	−2.06039	+	0.500893i			3.23667i	0.197714	−	2.82151i			0.751951i	1.14564	+	4.71253i
149.13	−1.17968	+	0.779965i	−0.425641	+	0.425641i	0.783310	−	1.84022i	1.33130	+	1.33130i	0.170137	−	0.834107i			3.79214i	0.511251	+	2.78184i			2.63766i	−2.60889	−	0.532149i
149.14	−1.17854	+	0.781691i	1.99902	−	1.99902i	0.777919	−	1.84251i	3.02093	+	3.02093i	−0.793311	+	3.91854i		−	3.16661i	0.523463	+	2.77957i		−	4.99214i	−5.92173	−	1.19886i
149.15	−1.16214	−	0.805866i	1.69401	−	1.69401i	0.701161	+	1.87307i	1.66918	+	1.66918i	−3.33384	+	0.603542i		−	3.85810i	0.694588	−	2.74181i		−	2.73937i	−0.594695	−	3.28497i
149.16	−1.12378	+	0.858550i	−0.618144	+	0.618144i	0.525782	−	1.92965i	−2.89025	−	2.89025i	0.163953	−	1.22537i		−	3.33195i	1.06584	+	2.61992i			2.23580i	5.72944	+	0.766592i
149.17	−1.08797	−	0.903505i	−0.0335636	+	0.0335636i	0.367358	+	1.96597i	2.53952	+	2.53952i	0.0668411	−	0.00619133i			0.568937i	1.37659	−	2.47083i			2.99775i	−0.468453	−	5.05738i
149.18	−1.07639	+	0.917266i	0.359338	−	0.359338i	0.317247	−	1.97468i	−0.0542447	−	0.0542447i	−0.0571808	+	0.716398i		−	3.31452i	1.46982	+	2.41653i			2.74175i	0.108145	+	0.00863185i
149.19	−1.01695	+	0.982757i	1.38850	−	1.38850i	0.0683769	−	1.99883i	−2.03754	−	2.03754i	−0.0474777	+	2.77660i			4.55238i	1.89483	+	2.09991i		−	0.855883i	4.07448	+	0.0696704i
149.20	−0.855943	−	1.12577i	−1.67210	+	1.67210i	−0.534725	+	1.92719i	−1.36908	−	1.36908i	3.31362	+	0.451181i			4.93129i	2.62727	−	1.04759i		−	2.59183i	−0.369419	+	2.71313i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	592.2.o.a	✓	144
16.e	even	4	1	inner	592.2.o.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
592.2.o.a	✓	144	1.a	even	1	1	trivial
592.2.o.a	✓	144	16.e	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(592, [\chi])\).