Newform orbit 175.2.q.a

• Label: 175.2.q.a
• Level: $175$
• Weight: $2$
• Character orbit: 175.q
• 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.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(18\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 3 q^{2} - 3 q^{3} + 13 q^{4} - 3 q^{5} - 12 q^{6} - 22 q^{7} - 2 q^{8} + 11 q^{9}+O(q^{10}) \)

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	−2.26854	−	1.01002i	−1.03919	−	1.15413i	2.78788	+	3.09625i	0.598087	−	2.15460i	1.19174	+	3.66780i	1.63141	−	2.08291i	−1.66242	−	5.11641i	0.0614698	−	0.584846i	−3.53297	+	4.28371i
11.2	−2.25839	−	1.00550i	0.712831	+	0.791679i	2.75102	+	3.05531i	−2.23601	−	0.0162832i	−0.813815	−	2.50467i	−2.33148	+	1.25068i	−1.61290	−	4.96398i	0.194958	−	1.85490i	5.03340	+	2.28508i
11.3	−2.16145	−	0.962341i	2.09781	+	2.32985i	2.40752	+	2.67382i	2.08640	+	0.804329i	−2.29220	−	7.05468i	−0.0714197	−	2.64479i	−1.16834	−	3.59577i	−0.713827	+	6.79161i	−3.73561	−	3.74634i
11.4	−1.94331	−	0.865215i	−1.55527	−	1.72730i	1.68958	+	1.87647i	0.566195	+	2.16320i	1.52788	+	4.70232i	−2.64421	+	0.0903750i	−0.345128	−	1.06219i	−0.251123	+	2.38928i	0.771342	−	4.69363i
11.5	−1.40506	−	0.625574i	0.893605	+	0.992448i	0.244597	+	0.271652i	−1.97083	+	1.05633i	−0.634720	−	1.95347i	2.64452	+	0.0807455i	0.776821	+	2.39081i	0.127161	−	1.20985i	3.42995	−	0.251317i
11.6	−1.26136	−	0.561596i	−0.564524	−	0.626968i	−0.0626100	−	0.0695354i	2.11758	+	0.718244i	0.359969	+	1.10787i	2.36555	+	1.18498i	0.893265	+	2.74919i	0.239184	−	2.27569i	−2.26767	−	2.09519i
11.7	−0.853279	−	0.379904i	−1.30628	−	1.45077i	−0.754503	−	0.837961i	−1.24736	−	1.85583i	0.563466	+	1.73417i	−0.508729	+	2.59638i	0.902719	+	2.77828i	−0.0847812	+	0.806639i	0.359309	+	2.05742i
11.8	−0.677204	−	0.301511i	0.992974	+	1.10281i	−0.970565	−	1.07792i	−0.0581969	−	2.23531i	−0.339937	−	1.04622i	−1.08929	−	2.41111i	0.790409	+	2.43263i	0.0833944	−	0.793444i	−0.634559	+	1.53131i
11.9	−0.428333	−	0.190706i	1.73082	+	1.92227i	−1.19116	−	1.32292i	0.335942	+	2.21069i	−0.374778	−	1.15345i	−1.07014	+	2.41967i	0.547701	+	1.68565i	−0.385798	+	3.67063i	0.277696	−	1.01098i
11.10	0.189312	+	0.0842871i	−0.609162	−	0.676543i	−1.30953	−	1.45438i	−1.65641	+	1.50211i	−0.0582978	−	0.179422i	−2.20881	−	1.45642i	−0.253398	−	0.779878i	0.226953	−	2.15932i	−0.440185	+	0.144753i
11.11	0.529445	+	0.235724i	−0.229691	−	0.255097i	−1.11352	−	1.23668i	1.94658	+	1.10038i	−0.0614760	−	0.189204i	1.74489	−	1.98881i	−0.656210	−	2.01961i	0.301269	−	2.86638i	0.771220	+	1.04145i
11.12	0.669019	+	0.297867i	1.50997	+	1.67699i	−0.979399	−	1.08773i	1.33545	−	1.79348i	0.510679	+	1.57171i	1.21370	+	2.35094i	−0.783844	−	2.41242i	−0.218706	+	2.08085i	1.42766	−	0.802086i
11.13	0.956389	+	0.425812i	−1.22328	−	1.35859i	−0.604896	−	0.671805i	1.72038	−	1.42838i	−0.591430	−	1.82023i	−2.07580	+	1.64044i	−0.939473	−	2.89140i	−0.0357700	+	0.340329i	2.25358	−	0.633531i
11.14	1.37473	+	0.612069i	2.04628	+	2.27262i	0.176990	+	0.196568i	−2.23021	+	0.161703i	1.42208	+	4.37671i	0.781664	−	2.52765i	−0.807034	−	2.48380i	−0.663974	+	6.31729i	−3.16491	−	1.14275i
11.15	1.53521	+	0.683522i	−1.65355	−	1.83645i	0.551421	+	0.612415i	−1.53440	−	1.62653i	−1.28330	−	3.94959i	2.14197	−	1.55304i	−0.610656	−	1.87941i	−0.324749	+	3.08978i	−1.24387	−	3.54587i
11.16	1.86868	+	0.831989i	−0.100275	−	0.111367i	1.46149	+	1.62315i	−0.993637	+	2.00317i	−0.0947260	−	0.291537i	1.42653	+	2.22823i	0.116408	+	0.358267i	0.311238	−	2.96123i	−3.52340	+	2.91658i
11.17	2.21264	+	0.985130i	0.483361	+	0.536826i	2.58702	+	2.87318i	−1.04758	−	1.97550i	0.540658	+	1.66398i	−2.59815	+	0.499637i	1.39679	+	4.29888i	0.259040	−	2.46460i	−0.371785	−	5.40306i
11.18	2.44336	+	1.08785i	−1.77288	−	1.96899i	3.44830	+	3.82973i	1.66349	+	1.49426i	−2.18982	−	6.73956i	−1.54320	−	2.14908i	2.60627	+	8.02126i	−0.420206	+	3.99799i	2.43896	+	5.46063i
16.1	−2.26854	+	1.01002i	−1.03919	+	1.15413i	2.78788	−	3.09625i	0.598087	+	2.15460i	1.19174	−	3.66780i	1.63141	+	2.08291i	−1.66242	+	5.11641i	0.0614698	+	0.584846i	−3.53297	−	4.28371i
16.2	−2.25839	+	1.00550i	0.712831	−	0.791679i	2.75102	−	3.05531i	−2.23601	+	0.0162832i	−0.813815	+	2.50467i	−2.33148	−	1.25068i	−1.61290	+	4.96398i	0.194958	+	1.85490i	5.03340	−	2.28508i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
25.d	even	5	1	inner
175.q	even	15	1	inner

Twists

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

    

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

Hecke kernels

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