Newform orbit 352.2.q.a

• Label: 352.2.q.a
• Level: $352$
• Weight: $2$
• Character orbit: 352.q
• Analytic conductor: $2.811$
• Analytic rank: $0$
• Dimension: $184$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.q (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.81073415115\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 184 * q - 8 * q^3 - 8 * q^4 - 8 * q^5 - 8 * q^9 - 4 * q^11 - 40 * q^12 + 8 * q^14 - 16 * q^15 + 32 * q^16 - 24 * q^20 - 16 * q^22 + 8 * q^23 - 8 * q^25 + 32 * q^26 - 8 * q^27 - 8 * q^33 - 64 * q^34 - 8 * q^36 - 8 * q^37 - 8 * q^38 + 32 * q^42 - 44 * q^44 - 32 * q^45 - 16 * q^47 - 8 * q^48 - 40 * q^53 - 36 * q^55 + 8 * q^56 - 8 * q^58 - 40 * q^59 - 88 * q^60 + 88 * q^64 - 12 * q^66 + 72 * q^67 - 8 * q^69 - 56 * q^70 - 40 * q^71 + 16 * q^75 + 12 * q^77 + 40 * q^78 + 16 * q^80 - 48 * q^82 + 96 * q^86 - 8 * q^89 - 104 * q^91 - 168 * q^92 + 16 * q^93 - 16 * q^97 + 48 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.41204	+	0.0783166i	1.24165	+	2.99762i	1.98773	−	0.221173i	0.124428	−	0.300396i	−1.98803	−	4.13553i	1.23021	+	1.23021i	−2.78944	+	0.467978i	−5.32269	+	5.32269i	−0.152172	+	0.433917i
43.2	−1.39860	−	0.209594i	−1.19463	−	2.88409i	1.91214	+	0.586276i	1.14836	−	2.77239i	1.06631	+	4.28407i	−2.36419	−	2.36419i	−2.55143	−	1.22074i	−4.76953	+	4.76953i	−2.18717	+	3.63677i
43.3	−1.39534	−	0.230267i	0.192477	+	0.464680i	1.89395	+	0.642602i	−0.416964	+	1.00664i	−0.161570	−	0.692708i	−1.10485	−	1.10485i	−2.49474	−	1.33276i	1.94244	−	1.94244i	0.813602	−	1.30859i
43.4	−1.39284	+	0.244949i	−0.457151	−	1.10366i	1.88000	−	0.682349i	−1.65960	+	4.00662i	0.907077	+	1.42524i	−1.50025	−	1.50025i	−2.45140	+	1.41091i	1.11224	−	1.11224i	1.33013	−	5.98709i
43.5	−1.38830	+	0.269506i	−1.02368	−	2.47139i	1.85473	−	0.748308i	−0.271219	+	0.654780i	2.08723	+	3.15513i	3.19185	+	3.19185i	−2.37325	+	1.53874i	−2.93852	+	2.93852i	0.200065	−	0.982123i
43.6	−1.33261	+	0.473459i	0.463157	+	1.11816i	1.55167	−	1.26187i	0.677436	−	1.63547i	−1.14661	−	1.27078i	1.45614	+	1.45614i	−1.47033	+	2.41622i	1.08555	−	1.08555i	−0.128425	+	2.50018i
43.7	−1.26603	−	0.630213i	−0.206112	−	0.497599i	1.20566	+	1.59574i	1.37698	−	3.32432i	−0.0526490	+	0.759870i	1.37106	+	1.37106i	−0.520752	−	2.78007i	1.91620	−	1.91620i	−3.83832	+	3.34090i
43.8	−1.23854	−	0.682656i	0.632071	+	1.52596i	1.06796	+	1.69099i	−1.18830	+	2.86881i	0.258856	−	2.32144i	3.37302	+	3.37302i	−0.168348	−	2.82341i	0.192295	−	0.192295i	3.43017	−	2.74194i
43.9	−1.18118	+	0.777690i	−0.693252	−	1.67366i	0.790395	−	1.83719i	0.0714092	−	0.172397i	2.12045	+	1.43776i	−0.502877	−	0.502877i	0.495164	+	2.78475i	−0.199211	+	0.199211i	0.0497241	+	0.259167i
43.10	−1.15406	−	0.817398i	−0.645569	−	1.55854i	0.663720	+	1.88666i	−0.312817	+	0.755206i	−0.528922	+	2.32634i	−1.19705	−	1.19705i	0.776176	−	2.71984i	0.109031	−	0.109031i	0.978314	−	0.615859i
43.11	−1.03688	+	0.961702i	0.808539	+	1.95199i	0.150258	−	1.99435i	−0.918124	+	2.21655i	−2.71559	−	1.24641i	−1.30673	−	1.30673i	1.76217	+	2.21241i	−1.03519	+	1.03519i	−1.17967	−	3.18127i
43.12	−0.991223	−	1.00870i	0.674337	+	1.62799i	−0.0349522	+	1.99969i	0.510455	−	1.23235i	0.973739	−	2.29391i	−1.74640	−	1.74640i	2.05174	−	1.94689i	−0.0743117	+	0.0743117i	−1.74905	+	0.706636i
43.13	−0.878362	+	1.10837i	0.0685501	+	0.165495i	−0.456960	−	1.94710i	−0.311147	+	0.751176i	−0.243641	−	0.0693854i	1.54252	+	1.54252i	2.55948	+	1.20378i	2.09863	−	2.09863i	−0.559279	−	1.00467i
43.14	−0.823499	+	1.14972i	0.979896	+	2.36568i	−0.643700	−	1.89358i	1.61595	−	3.90124i	−3.52680	−	0.821529i	−2.74469	−	2.74469i	2.70717	+	0.819288i	−2.51492	+	2.51492i	3.15459	+	5.07054i
43.15	−0.672635	+	1.24401i	−0.806247	−	1.94645i	−1.09513	−	1.67353i	1.11051	−	2.68101i	2.96372	+	0.306271i	−0.200320	−	0.200320i	2.81851	−	0.236675i	−1.01732	+	1.01732i	2.58824	+	3.18483i
43.16	−0.595667	−	1.28265i	1.08655	+	2.62316i	−1.29036	+	1.52806i	0.350615	−	0.846460i	2.71737	−	2.95619i	1.08321	+	1.08321i	2.72859	+	0.744863i	−3.57906	+	3.57906i	−1.29456	+	0.0544930i
43.17	−0.541591	−	1.30640i	−0.302732	−	0.730861i	−1.41336	+	1.41507i	0.917995	−	2.21624i	−0.790839	+	0.791317i	2.14368	+	2.14368i	2.61411	+	1.08002i	1.67881	−	1.67881i	−3.39247	+	0.00102505i
43.18	−0.418196	+	1.35097i	−1.24148	−	2.99721i	−1.65022	−	1.12994i	−1.13620	+	2.74304i	4.56831	−	0.423783i	−1.23160	−	1.23160i	2.21663	−	1.75686i	−5.32066	+	5.32066i	−3.23060	−	2.68210i
43.19	−0.408652	−	1.35388i	−0.285877	−	0.690168i	−1.66601	+	1.10654i	−1.13213	+	2.73321i	−0.817583	+	0.669083i	−0.543587	−	0.543587i	2.17894	+	1.80339i	1.72671	−	1.72671i	4.16310	+	0.415844i
43.20	−0.325496	−	1.37625i	−0.893634	−	2.15742i	−1.78810	+	0.895926i	0.0418908	−	0.101133i	−2.67827	+	1.93209i	−2.88620	−	2.88620i	1.81504	+	2.16925i	−1.73457	+	1.73457i	−0.152820	−	0.0247335i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
32.h	odd	8	1	inner
352.q	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	352.2.q.a	✓	184
11.b	odd	2	1	inner	352.2.q.a	✓	184
32.h	odd	8	1	inner	352.2.q.a	✓	184
352.q	even	8	1	inner	352.2.q.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
352.2.q.a	✓	184	1.a	even	1	1	trivial
352.2.q.a	✓	184	11.b	odd	2	1	inner
352.2.q.a	✓	184	32.h	odd	8	1	inner
352.2.q.a	✓	184	352.q	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).