Newform orbit 775.2.cz.a

• Label: 775.2.cz.a
• Level: $775$
• Weight: $2$
• Character orbit: 775.cz
• Analytic conductor: $6.188$
• Analytic rank: $0$
• Dimension: $1248$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.cz (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(0\)
Dimension:	\(1248\)
Relative dimension:	\(78\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1248 q - 12 q^{2} - 14 q^{3} - 20 q^{4} - 8 q^{5} - 18 q^{6} - 14 q^{7} + 60 q^{8} - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.47449	−	1.26081i	−0.0918643	−	1.75288i	3.35787	+	4.62171i	0.172466	−	2.22941i	−1.98273	+	4.45330i	−2.21834	+	3.41595i	−1.61299	−	10.1840i	−0.0805694	+	0.00846818i	−3.23763	+	5.29919i
3.2	−2.41505	−	1.23053i	0.0302866	+	0.577903i	3.14268	+	4.32552i	−1.38552	+	1.75509i	0.637981	−	1.43293i	1.14179	−	1.75820i	−1.41901	−	8.95929i	2.65051	−	0.278580i	5.50578	−	2.53370i
3.3	−2.38191	−	1.21364i	−0.0156580	−	0.298772i	3.02500	+	4.16355i	1.74650	−	1.39633i	−0.325307	+	0.730652i	2.13596	−	3.28908i	−1.31582	−	8.30774i	2.89455	−	0.304229i	−5.85465	+	1.20632i
3.4	−2.34630	−	1.19550i	−0.142908	−	2.72684i	2.90032	+	3.99195i	−0.848942	+	2.06865i	−2.92463	+	6.56883i	−0.0209862	+	0.0323159i	−1.20876	−	7.63184i	−4.43169	+	0.465789i	4.46493	−	3.83875i
3.5	−2.34535	−	1.19502i	0.119218	+	2.27481i	2.89705	+	3.98744i	−1.38783	−	1.75326i	2.43883	−	5.47769i	0.626066	−	0.964056i	−1.20599	−	7.61430i	−2.17697	+	0.228808i	1.15978	+	5.77050i
3.6	−2.24295	−	1.14284i	0.173096	+	3.30287i	2.54915	+	3.50861i	2.19469	+	0.428159i	3.38640	−	7.60598i	0.808947	−	1.24567i	−0.920252	−	5.81024i	−7.89543	+	0.829843i	−4.43326	−	3.46851i
3.7	−2.21871	−	1.13049i	0.118579	+	2.26262i	2.46909	+	3.39841i	−0.343069	+	2.20959i	2.29478	−	5.15415i	−1.63502	+	2.51771i	−0.857248	−	5.41245i	−2.12184	+	0.223014i	3.25909	−	4.51461i
3.8	−2.12776	−	1.08415i	−0.0849775	−	1.62147i	2.17641	+	2.99557i	1.55053	+	1.61117i	−1.57710	+	3.54222i	−1.25082	+	1.92609i	−0.636089	−	4.01611i	0.361630	−	0.0380089i	−1.55240	−	5.10917i
3.9	−2.11437	−	1.07733i	−0.153045	−	2.92028i	2.13436	+	2.93770i	−1.84957	−	1.25661i	−2.82250	+	6.33943i	2.37571	−	3.65827i	−0.605534	−	3.82319i	−5.52103	+	0.580283i	2.55690	+	4.64954i
3.10	−1.98802	−	1.01295i	0.0198464	+	0.378691i	1.75060	+	2.40950i	1.89980	+	1.17931i	0.344140	−	0.772951i	0.537458	−	0.827612i	−0.341465	−	2.15593i	2.84055	−	0.298554i	−2.58227	−	4.26889i
3.11	−1.98161	−	1.00968i	0.0406530	+	0.775706i	1.73174	+	2.38353i	−2.21850	+	0.279717i	0.702655	−	1.57819i	−2.84224	+	4.37666i	−0.329198	−	2.07847i	2.38350	−	0.250516i	4.67862	+	1.68569i
3.12	−1.90288	−	0.969567i	0.0507767	+	0.968877i	1.50533	+	2.07191i	1.82650	−	1.28992i	0.842770	−	1.89289i	−0.811762	+	1.25000i	−0.187433	−	1.18341i	2.04742	−	0.215193i	−4.72628	+	0.683639i
3.13	−1.88412	−	0.960006i	−0.0194581	−	0.371283i	1.45272	+	1.99950i	−1.44845	−	1.70353i	−0.319773	+	0.718221i	−0.179204	+	0.275950i	−0.155975	−	0.984789i	2.84609	−	0.299136i	1.09365	+	4.60016i
3.14	−1.78753	−	0.910792i	−0.111430	−	2.12620i	1.19015	+	1.63810i	−0.111868	−	2.23327i	−1.73735	+	3.90214i	−0.0266914	+	0.0411011i	−0.00778579	−	0.0491575i	−1.52476	+	0.160258i	−1.83407	+	4.09392i
3.15	−1.72648	−	0.879686i	−0.0193240	−	0.368724i	1.03132	+	1.41949i	−0.979467	+	2.01014i	−0.290999	+	0.653593i	1.79719	−	2.76744i	0.0743904	+	0.469683i	2.84798	−	0.299335i	3.45932	−	2.60884i
3.16	−1.65561	−	0.843574i	−0.133153	−	2.54071i	0.853846	+	1.17522i	−2.02733	+	0.943354i	−1.92283	+	4.31874i	−1.76714	+	2.72116i	0.159101	+	1.00453i	−3.45390	+	0.363019i	4.15226	+	0.148383i
3.17	−1.62051	−	0.825693i	0.140122	+	2.67368i	0.768725	+	1.05806i	−2.22359	+	0.235865i	1.98057	−	4.44844i	1.11523	−	1.71730i	0.196932	+	1.24338i	−4.14538	+	0.435697i	3.79812	+	1.45378i
3.18	−1.50392	−	0.766287i	−0.0975982	−	1.86228i	0.499019	+	0.686840i	1.83120	−	1.28324i	−1.28026	+	2.87552i	−0.135159	+	0.208126i	0.303920	+	1.91888i	−0.475013	+	0.0499259i	−3.73731	+	0.526675i
3.19	−1.49860	−	0.763573i	0.0896573	+	1.71076i	0.487179	+	0.670544i	0.220252	−	2.22519i	1.17193	−	2.63221i	2.82976	−	4.35745i	0.308144	+	1.94554i	0.0648931	−	0.00682054i	−2.02917	+	3.16649i
3.20	−1.30696	−	0.665931i	−0.123742	−	2.36114i	0.0891170	+	0.122659i	0.734437	+	2.11201i	−1.41063	+	3.16833i	2.10687	−	3.24429i	0.424138	+	2.67790i	−2.57611	+	0.270760i	0.446573	−	3.24941i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
775.cz	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.2.cz.a	✓	1248
25.f	odd	20	1		775.2.da.a	yes	1248
31.h	odd	30	1		775.2.da.a	yes	1248
775.cz	even	60	1	inner	775.2.cz.a	✓	1248

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
775.2.cz.a	✓	1248	1.a	even	1	1	trivial
775.2.cz.a	✓	1248	775.cz	even	60	1	inner
775.2.da.a	yes	1248	25.f	odd	20	1
775.2.da.a	yes	1248	31.h	odd	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).