Newform orbit 345.2.i.c

• Label: 345.2.i.c
• Level: $345$
• Weight: $2$
• Character orbit: 345.i
• Analytic conductor: $2.755$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	345.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.75483886973\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} - 8 q^{6} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−1.94274	+	1.94274i	−1.12858	+	1.31389i		−	5.54846i	2.17003	+	0.539434i	−0.360021	−	4.74508i	0.211277	+	0.211277i	6.89373	+	6.89373i	−0.452630	−	2.96566i	−5.26377	+	3.16781i
47.2	−1.89873	+	1.89873i	−1.55298	−	0.766983i		−	5.21032i	−2.22534	−	0.218768i	4.40497	−	1.49239i	−2.55245	−	2.55245i	6.09552	+	6.09552i	1.82347	+	2.38221i	4.64069	−	3.80993i
47.3	−1.85094	+	1.85094i	0.596169	−	1.62622i		−	4.85199i	0.818739	−	2.08078i	1.90656	+	4.11351i	0.971365	+	0.971365i	5.27888	+	5.27888i	−2.28917	−	1.93900i	2.33598	+	5.36686i
47.4	−1.73288	+	1.73288i	1.72857	+	0.109684i		−	4.00577i	1.13830	+	1.92465i	−3.18549	+	2.80535i	2.73413	+	2.73413i	3.47577	+	3.47577i	2.97594	+	0.379193i	−5.30774	−	1.36264i
47.5	−1.66033	+	1.66033i	1.39149	+	1.03138i		−	3.51336i	1.31092	−	1.81149i	−4.02276	+	0.597904i	−3.69615	−	3.69615i	2.51268	+	2.51268i	0.872507	+	2.87032i	0.831104	+	5.18421i
47.6	−1.56421	+	1.56421i	1.73136	+	0.0489458i		−	2.89352i	−2.07072	−	0.843862i	−2.78477	+	2.63165i	0.460434	+	0.460434i	1.39765	+	1.39765i	2.99521	+	0.169485i	4.55903	−	1.91907i
47.7	−1.51670	+	1.51670i	−0.681526	−	1.59233i		−	2.60074i	1.51436	+	1.64521i	3.44875	+	1.38142i	−0.779889	−	0.779889i	0.911139	+	0.911139i	−2.07104	+	2.17043i	−4.79211	−	0.198450i
47.8	−1.47796	+	1.47796i	−0.750095	+	1.56120i		−	2.36874i	−1.69004	−	1.46416i	−1.19879	−	3.41601i	1.03816	+	1.03816i	0.544986	+	0.544986i	−1.87472	−	2.34210i	4.66179	−	0.333829i
47.9	−1.37844	+	1.37844i	−1.71528	−	0.240455i		−	1.80018i	−0.879087	+	2.05602i	2.69586	−	2.03295i	2.42822	+	2.42822i	−0.275435	−	0.275435i	2.88436	+	0.824895i	−1.62232	−	4.04586i
47.10	−1.14642	+	1.14642i	1.10668	−	1.33239i		−	0.628538i	−1.21066	+	1.87997i	0.258759	+	2.79619i	−2.15083	−	2.15083i	−1.57227	−	1.57227i	−0.550526	−	2.94905i	−0.767315	−	3.54315i
47.11	−1.09346	+	1.09346i	0.598276	+	1.62544i		−	0.391318i	1.69592	−	1.45734i	−2.43155	−	1.12317i	2.50915	+	2.50915i	−1.75903	−	1.75903i	−2.28413	+	1.94493i	−0.260875	+	3.44798i
47.12	−1.05861	+	1.05861i	−1.69276	+	0.366826i		−	0.241323i	2.18116	−	0.492476i	1.40365	−	2.18031i	−0.258418	−	0.258418i	−1.86176	−	1.86176i	2.73088	−	1.24190i	−1.78767	+	2.83035i
47.13	−0.749972	+	0.749972i	1.36099	−	1.07131i			0.875085i	−0.723244	−	2.11587i	−0.217254	+	1.82416i	−0.524629	−	0.524629i	−2.15623	−	2.15623i	0.704596	−	2.91608i	2.12926	+	1.04443i
47.14	−0.739038	+	0.739038i	1.71972	+	0.206319i			0.907645i	1.32647	+	1.80013i	−1.42342	+	1.11846i	−1.39104	−	1.39104i	−2.14886	−	2.14886i	2.91486	+	0.709621i	−2.31068	−	0.350049i
47.15	−0.557759	+	0.557759i	0.657062	−	1.60258i			1.37781i	2.18769	+	0.462593i	0.527372	+	1.26034i	3.14489	+	3.14489i	−1.88400	−	1.88400i	−2.13654	−	2.10599i	−1.47822	+	0.962191i
47.16	−0.536180	+	0.536180i	1.22678	+	1.22270i			1.42502i	−1.90983	+	1.16300i	−1.31337	+	0.00218736i	1.47708	+	1.47708i	−1.83643	−	1.83643i	0.00999275	+	2.99998i	0.400434	−	1.64759i
47.17	−0.409710	+	0.409710i	−1.68657	−	0.394298i			1.66427i	−2.13867	−	0.652742i	0.852555	−	0.529458i	−0.684032	−	0.684032i	−1.50129	−	1.50129i	2.68906	+	1.33003i	1.14367	−	0.608802i
47.18	−0.305412	+	0.305412i	0.337196	+	1.69891i			1.81345i	−1.49331	−	1.66434i	−0.621852	−	0.415884i	−2.99863	−	2.99863i	−1.16467	−	1.16467i	−2.77260	+	1.14573i	0.964384	+	0.0522344i
47.19	−0.0496600	+	0.0496600i	−0.0466959	−	1.73142i			1.99507i	−1.92669	+	1.13484i	0.0883012	+	0.0836634i	−0.331363	−	0.331363i	−0.198395	−	0.198395i	−2.99564	+	0.161700i	0.0393233	−	0.152036i
47.20	−0.0451493	+	0.0451493i	−0.881594	+	1.49090i			1.99592i	0.335629	+	2.21074i	−0.0275099	−	0.107117i	2.39272	+	2.39272i	−0.180413	−	0.180413i	−1.44558	−	2.62874i	−0.114967	−	0.0846598i

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	345.2.i.c	✓	80
3.b	odd	2	1	inner	345.2.i.c	✓	80
5.c	odd	4	1	inner	345.2.i.c	✓	80
15.e	even	4	1	inner	345.2.i.c	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
345.2.i.c	✓	80	1.a	even	1	1	trivial
345.2.i.c	✓	80	3.b	odd	2	1	inner
345.2.i.c	✓	80	5.c	odd	4	1	inner
345.2.i.c	✓	80	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^80 + 322*T2^76 + 45801*T2^72 + 3798452*T2^68 + 204239200*T2^64 + 7491412100*T2^60 + 192102575366*T2^56 + 3473172242420*T2^52 + 44120440294774*T2^48 + 388371333606420*T2^44 + 2312354583334392*T2^40 + 8987034127516076*T2^36 + 21692655382544113*T2^32 + 30541149597841322*T2^28 + 23789253363868873*T2^24 + 9613504484495712*T2^20 + 1869027432668440*T2^16 + 146522907889320*T2^12 + 3211185015680*T2^8 + 131305792*T2^4 + 1296