Newform orbit 175.2.t.a

• Label: 175.2.t.a
• Level: $175$
• Weight: $2$
• Character orbit: 175.t
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.t (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 5 q^{2} - 5 q^{3} - 19 q^{4} - 3 q^{5} - 12 q^{6} - 50 q^{8} - 17 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−0.561630	−	2.64226i	−0.981775	−	2.20510i	−4.83903	+	2.15447i	2.14180	−	0.642423i	−5.27506	+	3.83256i	−2.42948	−	1.04767i	5.23487	+	7.20518i	−1.89120	+	2.10039i	−2.90035	−	5.29838i
4.2	−0.500096	−	2.35277i	0.932348	+	2.09409i	−3.45832	+	1.53974i	0.169836	+	2.22961i	4.46063	−	3.24084i	−0.802852	+	2.52100i	2.52452	+	3.47470i	−1.50854	+	1.67540i	5.16081	−	1.51460i
4.3	−0.482008	−	2.26767i	0.709985	+	1.59465i	−3.08291	+	1.37260i	−1.11212	−	1.93989i	3.27393	−	2.37865i	1.21596	−	2.34978i	1.87323	+	2.57828i	−0.0314466	+	0.0349249i	−3.86299	+	3.45697i
4.4	−0.369436	−	1.73806i	−0.0553691	−	0.124361i	−1.05727	+	0.470728i	2.20183	−	0.389777i	−0.195691	+	0.142178i	1.92211	+	1.81810i	−0.880110	−	1.21137i	1.99499	−	2.21566i	−1.49089	−	3.68292i
4.5	−0.335784	−	1.57974i	−1.11543	−	2.50529i	−0.555728	+	0.247426i	−1.59383	−	1.56835i	−3.58316	+	2.60332i	2.23049	+	1.42300i	−1.32111	−	1.81835i	−3.02490	+	3.35950i	−1.94240	+	3.04445i
4.6	−0.330614	−	1.55542i	−0.597202	−	1.34134i	−0.482928	+	0.215013i	−1.77722	+	1.35701i	−1.88890	+	1.37236i	−2.59765	−	0.502201i	−1.37525	−	1.89288i	0.564855	−	0.627335i	2.69830	+	2.31568i
4.7	−0.175806	−	0.827100i	0.253333	+	0.568996i	1.17390	−	0.522655i	−0.0289171	−	2.23588i	0.426079	−	0.309565i	−2.63834	+	0.197839i	−1.63270	−	2.24722i	1.74781	−	1.94114i	−1.84421	+	0.416998i
4.8	−0.147876	−	0.695704i	0.986806	+	2.21640i	1.36495	−	0.607717i	−1.89164	+	1.19235i	1.39603	−	1.01428i	2.10313	−	1.60526i	−1.46076	−	2.01056i	−1.93126	+	2.14488i	1.10925	+	1.13970i
4.9	−0.0662594	−	0.311726i	1.14522	+	2.57220i	1.73431	−	0.772164i	2.09734	+	0.775343i	0.725939	−	0.527426i	−2.42570	−	1.05639i	−0.730260	−	1.00512i	−3.29729	+	3.66201i	0.102726	−	0.705170i
4.10	−0.0586647	−	0.275996i	−1.26264	−	2.83593i	1.75436	−	0.781091i	1.62491	+	1.53612i	−0.708633	+	0.514852i	1.20265	−	2.35661i	−0.650198	−	0.894920i	−4.44086	+	4.93208i	0.328639	−	0.538584i
4.11	0.0169101	+	0.0795555i	−0.239789	−	0.538575i	1.82105	−	0.810783i	−0.445317	+	2.19128i	0.0387918	−	0.0281839i	0.791023	+	2.52473i	0.190909	+	0.262763i	1.77483	−	1.97115i	−0.181859	+	0.00162719i
4.12	0.172534	+	0.811707i	−0.515214	−	1.15719i	1.19799	−	0.533380i	−1.94638	−	1.10073i	0.850407	−	0.617857i	−0.234193	−	2.63537i	1.61518	+	2.22310i	0.933750	−	1.03703i	0.557658	−	1.76980i
4.13	0.208393	+	0.980411i	0.782030	+	1.75647i	0.909313	−	0.404852i	−0.701900	−	2.12305i	−1.55909	+	1.13275i	1.25436	+	2.32950i	1.76471	+	2.42891i	−0.466216	+	0.517785i	1.93519	−	1.13058i
4.14	0.327402	+	1.54030i	−0.263147	−	0.591039i	−0.438251	+	0.195122i	2.12141	+	0.706829i	0.824224	−	0.598834i	−2.63114	+	0.277644i	1.40716	+	1.93679i	1.72731	−	1.91837i	−0.394177	+	3.49904i
4.15	0.417054	+	1.96208i	0.502581	+	1.12882i	−1.84875	+	0.823117i	−0.364398	+	2.20618i	−2.00523	+	1.45688i	1.38727	−	2.25288i	−0.0279546	−	0.0384762i	0.985753	−	1.09479i	−4.48068	+	0.205115i
4.16	0.436418	+	2.05318i	−0.881466	−	1.97981i	−2.19801	+	0.978618i	1.11577	−	1.93779i	3.68022	−	2.67383i	2.59389	+	0.521300i	−0.500948	−	0.689496i	−1.13526	+	1.26083i	4.46559	+	1.44520i
4.17	0.516009	+	2.42763i	1.10824	+	2.48915i	−3.80004	+	1.69189i	0.782280	−	2.09476i	−5.47088	+	3.97483i	−0.792837	−	2.52417i	−3.15053	−	4.33633i	−2.96028	+	3.28772i	5.48898	+	0.818170i
4.18	0.537983	+	2.53101i	−0.0303653	−	0.0682015i	−4.28951	+	1.90981i	−2.22433	+	0.228782i	0.156283	−	0.113546i	−0.537537	+	2.59057i	−4.09959	−	5.64260i	2.00366	−	2.22529i	−1.77571	−	5.50673i
9.1	−1.02647	−	2.30548i	0.763033	+	0.687038i	−2.92336	+	3.24672i	1.22529	+	1.87047i	0.800727	−	2.46438i	2.48641	−	0.904292i	5.68568	+	1.84739i	−0.203387	−	1.93510i	3.05462	−	4.74486i
9.2	−0.885632	−	1.98916i	−1.52446	−	1.37263i	−1.83416	+	2.03704i	−0.250174	+	2.22203i	−1.38027	+	4.24804i	−1.40656	+	2.24089i	1.53472	+	0.498662i	0.126279	+	1.20147i	4.64154	−	1.47026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
25.e	even	10	1	inner
175.t	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.2.t.a	✓	144
5.b	even	2	1		875.2.u.a		144
5.c	odd	4	2		875.2.q.b		288
7.c	even	3	1	inner	175.2.t.a	✓	144
25.d	even	5	1		875.2.u.a		144
25.e	even	10	1	inner	175.2.t.a	✓	144
25.f	odd	20	2		875.2.q.b		288
35.j	even	6	1		875.2.u.a		144
35.l	odd	12	2		875.2.q.b		288
175.q	even	15	1		875.2.u.a		144
175.t	even	30	1	inner	175.2.t.a	✓	144
175.w	odd	60	2		875.2.q.b		288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
175.2.t.a	✓	144	1.a	even	1	1	trivial
175.2.t.a	✓	144	7.c	even	3	1	inner
175.2.t.a	✓	144	25.e	even	10	1	inner
175.2.t.a	✓	144	175.t	even	30	1	inner
875.2.q.b		288	5.c	odd	4	2
875.2.q.b		288	25.f	odd	20	2
875.2.q.b		288	35.l	odd	12	2
875.2.q.b		288	175.w	odd	60	2
875.2.u.a		144	5.b	even	2	1
875.2.u.a		144	25.d	even	5	1
875.2.u.a		144	35.j	even	6	1
875.2.u.a		144	175.q	even	15	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).