Newform orbit 345.3.u.a

• Label: 345.3.u.a
• Level: $345$
• Weight: $3$
• Character orbit: 345.u
• Analytic conductor: $9.401$
• Analytic rank: $0$
• Dimension: $1840$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	345.u (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.40056912043\)
Analytic rank:	\(0\)
Dimension:	\(1840\)
Relative dimension:	\(92\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1840 q - 14 q^{3} - 52 q^{6} - 44 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1	−1.38559	−	3.71490i	−2.99994	−	0.0182131i	−8.85763	+	7.67518i	−2.54663	−	4.30287i	4.08902	+	11.1697i	1.84769	+	2.46823i	26.8659	+	14.6699i	8.99934	+	0.109277i	−12.4562	+	15.4225i
17.2	−1.31975	−	3.53838i	2.22170	−	2.01595i	−7.75541	+	6.72010i	4.72833	−	1.62570i	−10.0653	−	5.20067i	7.82099	+	10.4476i	20.7553	+	11.3332i	0.871879	−	8.95767i	−11.9926	−	14.5851i
17.3	−1.30232	−	3.49166i	−1.01765	+	2.82212i	−7.47266	+	6.47509i	4.05854	+	2.92032i	11.1792	−	0.122019i	0.0842355	+	0.112525i	19.2575	+	10.5154i	−6.92877	−	5.74388i	4.91124	−	17.9742i
17.4	−1.29725	−	3.47806i	2.97234	−	0.406451i	−7.39103	+	6.40436i	−0.121964	+	4.99851i	−5.26952	−	9.81070i	−5.40921	−	7.22586i	18.8306	+	10.2823i	8.66959	−	2.41622i	17.5433	−	6.06011i
17.5	−1.28990	−	3.45836i	2.24482	+	1.99018i	−7.27340	+	6.30244i	0.586915	−	4.96543i	3.98715	−	10.3305i	−2.76283	−	3.69071i	18.2197	+	9.94871i	1.07841	+	8.93516i	−17.9293	+	4.37515i
17.6	−1.25908	−	3.37572i	−0.550411	−	2.94908i	−6.78722	+	5.88116i	4.99449	−	0.234666i	−9.26225	+	5.57115i	−8.00685	−	10.6959i	15.7501	+	8.60020i	−8.39410	+	3.24641i	−7.08062	−	16.5645i
17.7	−1.24503	−	3.33807i	−2.81157	−	1.04647i	−6.56957	+	5.69257i	0.552929	+	4.96933i	0.00730675	+	10.6881i	0.440379	+	0.588277i	14.6739	+	8.01256i	6.80980	+	5.88444i	15.8995	−	8.03270i
17.8	−1.23257	−	3.30466i	−0.208898	−	2.99272i	−6.37851	+	5.52701i	−4.59590	+	1.96919i	−9.63242	+	4.37908i	2.29408	+	3.06454i	13.7444	+	7.50502i	−8.91272	+	1.25035i	12.1723	+	12.7607i
17.9	−1.18790	−	3.18489i	0.645163	+	2.92981i	−5.70944	+	4.94726i	−4.61825	+	1.91619i	8.56473	−	5.53511i	−3.22491	−	4.30798i	10.6051	+	5.79081i	−8.16753	+	3.78041i	11.5889	+	12.4324i
17.10	−1.15410	−	3.09427i	2.83924	−	0.968879i	−5.21958	+	4.52279i	−4.74090	−	1.58867i	−6.27476	−	7.66720i	0.412880	+	0.551544i	8.42457	+	4.60017i	7.12255	−	5.50176i	0.555716	+	16.5031i
17.11	−1.09206	−	2.92793i	−1.10434	−	2.78934i	−4.35720	+	3.77554i	0.578882	−	4.96638i	−6.96101	+	6.27957i	0.692853	+	0.925544i	4.84200	+	2.64393i	−6.56087	+	6.16076i	−15.1734	+	3.72867i
17.12	−1.09143	−	2.92624i	−2.21528	+	2.02300i	−4.34867	+	3.76815i	−4.72667	−	1.63052i	8.33762	+	4.27448i	−7.32431	−	9.78413i	4.80826	+	2.62551i	0.814938	−	8.96303i	0.387541	+	15.6110i
17.13	−1.04634	−	2.80534i	−0.581837	+	2.94304i	−3.75209	+	3.25121i	0.762185	−	4.94157i	8.86500	−	1.44716i	5.38525	+	7.19385i	2.53519	+	1.38432i	−8.32293	−	3.42474i	−14.6603	+	3.03236i
17.14	−1.03980	−	2.78781i	−2.31990	+	1.90213i	−3.66769	+	3.17807i	−3.79813	+	3.25180i	7.71499	+	4.48959i	6.46749	+	8.63956i	2.22770	+	1.21642i	1.76383	−	8.82547i	13.0147	+	7.20725i
17.15	−1.02569	−	2.74999i	−2.99684	+	0.137660i	−3.48739	+	3.02184i	4.77612	−	1.47943i	3.45240	+	8.10008i	−2.39283	−	3.19645i	1.58291	+	0.864332i	8.96210	−	0.825090i	−8.96725	−	11.6168i
17.16	−1.01166	−	2.71236i	2.30496	+	1.92020i	−3.31045	+	2.86852i	4.65088	+	1.83557i	2.87645	−	8.19446i	3.39165	+	4.53071i	0.966397	+	0.527693i	1.62565	+	8.85196i	0.273629	−	14.4718i
17.17	−0.965490	−	2.58858i	2.83907	+	0.969381i	−2.74557	+	2.37905i	−4.52109	−	2.13535i	−0.231770	−	8.28508i	4.81348	+	6.43005i	−0.890128	−	0.486046i	7.12060	+	5.50428i	−1.16245	+	13.7649i
17.18	−0.964140	−	2.58496i	1.71177	−	2.46371i	−2.72945	+	2.36508i	1.06579	+	4.88509i	−8.01896	−	2.04950i	0.506939	+	0.677191i	−0.940539	−	0.513573i	−3.13969	−	8.43459i	11.6002	−	7.46494i
17.19	−0.893679	−	2.39605i	−2.81272	−	1.04337i	−1.91938	+	1.66315i	4.51674	−	2.14455i	0.0136999	+	7.67184i	6.14131	+	8.20383i	−3.27761	−	1.78971i	6.82275	+	5.86942i	−9.17495	−	8.90577i
17.20	−0.875543	−	2.34742i	2.10038	−	2.14206i	−1.72081	+	1.49109i	0.529254	−	4.97191i	−6.86728	−	3.05502i	−1.93010	−	2.57831i	−3.78884	−	2.06886i	−0.176800	−	8.99826i	−12.1345	+	3.11074i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
23.d	odd	22	1	inner
69.g	even	22	1	inner
115.l	even	44	1	inner
345.u	odd	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	345.3.u.a	✓	1840
3.b	odd	2	1	inner	345.3.u.a	✓	1840
5.c	odd	4	1	inner	345.3.u.a	✓	1840
15.e	even	4	1	inner	345.3.u.a	✓	1840
23.d	odd	22	1	inner	345.3.u.a	✓	1840
69.g	even	22	1	inner	345.3.u.a	✓	1840
115.l	even	44	1	inner	345.3.u.a	✓	1840
345.u	odd	44	1	inner	345.3.u.a	✓	1840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.3.u.a	✓	1840	1.a	even	1	1	trivial
345.3.u.a	✓	1840	3.b	odd	2	1	inner
345.3.u.a	✓	1840	5.c	odd	4	1	inner
345.3.u.a	✓	1840	15.e	even	4	1	inner
345.3.u.a	✓	1840	23.d	odd	22	1	inner
345.3.u.a	✓	1840	69.g	even	22	1	inner
345.3.u.a	✓	1840	115.l	even	44	1	inner
345.3.u.a	✓	1840	345.u	odd	44	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(345, [\chi])\).