Newform orbit 605.2.w.a

• Label: 605.2.w.a
• Level: $605$
• Weight: $2$
• Character orbit: 605.w
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $5120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.w (of order \(220\), degree \(80\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(5120\)
Relative dimension:	\(64\) over \(\Q(\zeta_{220})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{220}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 5120 q - 78 q^{2} - 60 q^{3} - 86 q^{5} - 156 q^{6} - 88 q^{7} - 78 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−1.86668	−	1.97647i	−0.223065	+	1.40838i	−0.307761	+	5.38212i	−1.72207	+	1.42635i	3.20000	−	2.18810i	−1.00869	−	1.52134i	7.05246	−	5.93681i	0.919406	+	0.298733i	6.03369	+	0.741082i
2.2	−1.86517	−	1.97487i	0.517775	−	3.26910i	−0.307079	+	5.37020i	1.45529	−	1.69769i	−7.42179	+	5.07489i	−0.0523717	−	0.0789887i	7.02195	−	5.91113i	−7.56577	−	2.45827i	−6.06707	+	0.292469i
2.3	−1.82550	−	1.93287i	−0.499211	+	3.15189i	−0.289350	+	5.06016i	2.23158	−	0.141650i	7.00350	−	4.78887i	−1.28817	−	1.94286i	6.24096	−	5.25368i	−6.83205	−	2.21987i	−4.34753	−	4.05476i
2.4	−1.79170	−	1.89708i	−0.0393495	+	0.248443i	−0.274545	+	4.80124i	1.52452	+	1.63580i	0.541818	−	0.370486i	1.79950	+	2.71407i	5.60768	−	4.72058i	2.79299	+	0.907499i	0.371772	−	5.82299i
2.5	−1.73635	−	1.83847i	0.335328	−	2.11718i	−0.250899	+	4.38772i	−1.16194	+	1.91047i	−4.47462	+	3.05966i	0.600946	+	0.906367i	4.63314	−	3.90021i	−1.51682	−	0.492845i	5.52988	−	1.18104i
2.6	−1.66957	−	1.76777i	−0.214500	+	1.35430i	−0.223358	+	3.90609i	−1.67153	−	1.48525i	2.75221	−	1.88191i	−0.126444	−	0.190707i	3.55757	−	2.99479i	1.06505	+	0.346056i	0.165164	+	5.43462i
2.7	−1.63020	−	1.72608i	−0.0446050	+	0.281625i	−0.207627	+	3.63097i	1.61903	−	1.54231i	0.558823	−	0.382113i	−1.68230	−	2.53730i	2.97316	−	2.50283i	2.77585	+	0.901927i	−5.30150	−	0.280303i
2.8	−1.55775	−	1.64937i	0.240669	−	1.51952i	−0.179660	+	3.14189i	−1.15878	−	1.91239i	−2.88116	+	1.97009i	−2.21948	−	3.34750i	1.99079	−	1.67586i	0.602146	+	0.195649i	−1.34914	+	4.89029i
2.9	−1.52867	−	1.61858i	0.256577	−	1.61996i	−0.168790	+	2.95180i	1.47700	+	1.67883i	−3.01426	+	2.06110i	−1.20402	−	1.81594i	1.62933	−	1.37158i	0.294721	+	0.0957606i	0.459474	−	4.95701i
2.10	−1.49444	−	1.58234i	0.322560	−	2.03656i	−0.156261	+	2.73269i	−1.96968	−	1.05847i	−3.70458	+	2.53313i	1.84358	+	2.78055i	1.22742	−	1.03325i	−1.19037	−	0.386775i	1.26872	+	4.69852i
2.11	−1.48817	−	1.57569i	−0.399974	+	2.52534i	−0.153994	+	2.69305i	0.336311	−	2.21063i	4.57439	−	3.12789i	2.61648	+	3.94627i	1.15643	−	0.973491i	−3.36419	−	1.09309i	−3.98377	+	2.75987i
2.12	−1.34858	−	1.42789i	−0.353163	+	2.22978i	−0.106045	+	1.85451i	−2.16972	+	0.540671i	3.66016	−	2.50276i	0.961173	+	1.44967i	−0.214051	+	0.180190i	−1.99404	−	0.647904i	3.69805	+	2.36899i
2.13	−1.34797	−	1.42725i	0.193438	−	1.22132i	−0.105846	+	1.85104i	2.15715	−	0.588819i	−2.00388	+	1.37021i	1.76243	+	2.65815i	−0.219176	+	0.184504i	1.39897	+	0.454553i	−3.74816	−	2.28508i
2.14	−1.26609	−	1.34056i	−0.0572426	+	0.361416i	−0.0799287	+	1.39779i	−0.182529	+	2.22861i	0.556972	−	0.380848i	−2.44095	−	3.68153i	−0.846282	+	0.712406i	2.72583	+	0.885674i	3.21867	−	2.57693i
2.15	−1.18560	−	1.25533i	−0.529186	+	3.34115i	−0.0560322	+	0.979891i	−0.519117	+	2.17498i	4.82164	−	3.29695i	−0.359230	−	0.541803i	−1.34541	+	1.13258i	−8.03007	−	2.60913i	3.34577	−	1.92698i
2.16	−1.10482	−	1.16980i	0.0340061	−	0.214706i	−0.0336278	+	0.588084i	−2.09072	+	0.793034i	−0.288733	+	0.197431i	0.954022	+	1.43889i	−1.73683	+	1.46208i	2.80823	+	0.912448i	3.23755	+	1.56956i
2.17	−1.05807	−	1.12030i	−0.180139	+	1.13735i	−0.0213845	+	0.373973i	2.23461	−	0.0806053i	1.46477	−	1.00159i	0.333391	+	0.502832i	−1.91617	+	1.61304i	1.59205	+	0.517289i	−2.45468	−	2.41815i
2.18	−1.01696	−	1.07677i	0.243561	−	1.53779i	−0.0110539	+	0.193311i	0.540109	−	2.16986i	−1.90353	+	1.30160i	0.222191	+	0.335117i	−2.04674	+	1.72296i	0.547705	+	0.177960i	−2.88570	+	1.62508i
2.19	−0.967000	−	1.02388i	0.464579	−	2.93323i	0.000947099	−	0.0165629i	−2.23310	+	0.115139i	−3.45251	+	2.36077i	−2.25769	−	3.40513i	−2.17269	+	1.82899i	−5.53486	−	1.79839i	2.27730	+	2.17508i
2.20	−0.886874	−	0.939036i	−0.0632831	+	0.399554i	0.0189344	−	0.331126i	−0.878832	−	2.05613i	0.431319	−	0.294929i	0.577165	+	0.870500i	−2.30400	+	1.93952i	2.69753	+	0.876481i	−1.15136	+	2.64878i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
121.h	odd	110	1	inner
605.w	even	220	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	605.2.w.a	✓	5120
5.c	odd	4	1	inner	605.2.w.a	✓	5120
121.h	odd	110	1	inner	605.2.w.a	✓	5120
605.w	even	220	1	inner	605.2.w.a	✓	5120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
605.2.w.a	✓	5120	1.a	even	1	1	trivial
605.2.w.a	✓	5120	5.c	odd	4	1	inner
605.2.w.a	✓	5120	121.h	odd	110	1	inner
605.2.w.a	✓	5120	605.w	even	220	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(605, [\chi])\).