Newform orbit 380.3.q.a

• Label: 380.3.q.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.q
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9}+O(q^{10}) \)

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	−1.99987	−	0.0230852i	0.798339	+	0.460921i	3.99893	+	0.0923347i	−1.11803	+	1.93649i	−1.58593	−	0.940211i		−	13.2021i	−7.99520	−	0.276973i	−4.07510	−	7.05829i	2.28062	−	3.84692i
11.2	−1.99382	+	0.157147i	3.38036	+	1.95165i	3.95061	−	0.626645i	1.11803	−	1.93649i	−7.04652	−	3.36002i		−	4.79106i	−7.77832	+	1.87024i	3.11789	+	5.40035i	−1.92484	+	4.03670i
11.3	−1.99167	−	0.182322i	2.87776	+	1.66148i	3.93352	+	0.726252i	1.11803	−	1.93649i	−5.42864	−	3.83380i			8.40212i	−7.70187	−	2.16362i	1.02102	+	1.76846i	−2.57982	+	3.65301i
11.4	−1.98625	+	0.234140i	−4.22803	−	2.44105i	3.89036	−	0.930120i	−1.11803	+	1.93649i	8.96946	+	3.85858i		−	5.28526i	−7.50943	+	2.75834i	7.41748	+	12.8474i	1.76728	−	4.10813i
11.5	−1.98163	−	0.270469i	−0.144974	−	0.0837008i	3.85369	+	1.07194i	1.11803	−	1.93649i	0.264646	+	0.205075i			9.41366i	−7.34666	−	3.16648i	−4.48599	−	7.76996i	−2.73929	+	3.53501i
11.6	−1.95337	+	0.429344i	0.161149	+	0.0930393i	3.63133	−	1.67734i	−1.11803	+	1.93649i	−0.354730	−	0.112552i			4.44882i	−6.37318	+	4.83555i	−4.48269	−	7.76424i	1.35252	−	4.26271i
11.7	−1.92805	−	0.531634i	−1.89857	−	1.09614i	3.43473	+	2.05003i	−1.11803	+	1.93649i	3.07778	+	3.12275i			2.39245i	−5.53245	−	5.77858i	−2.09696	−	3.63204i	3.18513	−	3.13926i
11.8	−1.92602	−	0.538947i	−4.13473	−	2.38719i	3.41907	+	2.07604i	1.11803	−	1.93649i	6.67698	+	6.82616i		−	2.68655i	−5.46631	−	5.84119i	6.89731	+	11.9465i	−3.19702	+	3.12715i
11.9	−1.89765	+	0.631592i	−2.04261	−	1.17930i	3.20218	−	2.39709i	1.11803	−	1.93649i	4.62101	+	0.947812i		−	9.35011i	−4.56266	+	6.57131i	−1.71849	−	2.97651i	−0.898570	+	4.38093i
11.10	−1.87590	+	0.693529i	−0.511454	−	0.295288i	3.03804	−	2.60199i	1.11803	−	1.93649i	1.16423	+	0.199224i		−	0.545604i	−3.89451	+	6.98805i	−4.32561	−	7.49218i	−0.754312	+	4.40806i
11.11	−1.86315	+	0.727102i	4.59307	+	2.65181i	2.94265	−	2.70940i	−1.11803	+	1.93649i	−10.4857	−	1.60109i			10.5847i	−3.51258	+	7.18761i	9.56418	+	16.5656i	0.675037	−	4.42090i
11.12	−1.72531	−	1.01159i	2.17479	+	1.25562i	1.95337	+	3.49061i	−1.11803	+	1.93649i	−2.48202	−	4.36632i			9.80901i	0.160888	−	7.99838i	−1.34685	−	2.33282i	3.88789	−	2.21005i
11.13	−1.72336	−	1.01491i	4.96001	+	2.86366i	1.93992	+	3.49810i	1.11803	−	1.93649i	−5.64152	−	9.96907i		−	1.47840i	0.207068	−	7.99732i	11.9011	+	20.6134i	−3.89213	+	2.20257i
11.14	−1.61822	+	1.17532i	−2.81695	−	1.62637i	1.23726	−	3.80384i	1.11803	−	1.93649i	6.46995	−	0.678995i			13.4914i	2.46856	+	7.60961i	0.790156	+	1.36859i	0.466769	+	4.44771i
11.15	−1.60232	−	1.19690i	−2.99135	−	1.72706i	1.13487	+	3.83563i	−1.11803	+	1.93649i	2.72600	+	6.34765i		−	8.07705i	2.77243	−	7.50424i	1.46546	+	2.53825i	4.10923	−	1.76471i
11.16	−1.58052	−	1.22554i	2.60636	+	1.50478i	0.996097	+	3.87399i	−1.11803	+	1.93649i	−2.27523	−	5.57254i		−	3.75845i	3.17338	−	7.34368i	0.0287371	+	0.0497742i	4.14033	−	1.69047i
11.17	−1.57411	+	1.23377i	3.64654	+	2.10533i	0.955631	−	3.88417i	−1.11803	+	1.93649i	−8.33754	+	1.18497i		−	11.0873i	3.28790	+	7.29313i	4.36484	+	7.56012i	−0.629276	−	4.42764i
11.18	−1.55939	−	1.25232i	1.80145	+	1.04007i	0.863372	+	3.90571i	1.11803	−	1.93649i	−1.50665	−	3.87786i		−	10.0188i	3.54489	−	7.17174i	−2.33652	−	4.04698i	−4.16856	+	1.61960i
11.19	−1.53570	+	1.28126i	0.685837	+	0.395968i	0.716728	−	3.93526i	−1.11803	+	1.93649i	−1.56058	+	0.270651i			4.43960i	3.94143	+	6.96169i	−4.18642	−	7.25109i	−0.764195	−	4.40636i
11.20	−1.52236	−	1.29708i	−2.74905	−	1.58716i	0.635162	+	3.94925i	1.11803	−	1.93649i	2.12636	+	5.98197i			8.52208i	4.15555	−	6.83604i	0.538166	+	0.932130i	−4.21384	+	1.49786i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.c	even	3	1	inner
76.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.q.a	✓	160
4.b	odd	2	1	inner	380.3.q.a	✓	160
19.c	even	3	1	inner	380.3.q.a	✓	160
76.g	odd	6	1	inner	380.3.q.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.q.a	✓	160	1.a	even	1	1	trivial
380.3.q.a	✓	160	4.b	odd	2	1	inner
380.3.q.a	✓	160	19.c	even	3	1	inner
380.3.q.a	✓	160	76.g	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).