Newform orbit 700.2.bq.a

• Label: 700.2.bq.a
• Level: $700$
• Weight: $2$
• Character orbit: 700.bq
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.bq (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(116\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 3 q^{2} - 3 q^{4} - 24 q^{5} - 30 q^{8} + 102 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.41421		0.000257337i	−0.218726	−	2.08104i	2.00000		0.000727858i	0.867744	+	2.06083i	0.309861	+	2.94298i	0.624858	−	2.57091i	−2.82843	+	0.00154402i	−1.34844	+	0.286619i	−1.22771	−	2.91423i
31.2	−1.41414	−	0.0142791i	0.0124422	+	0.118380i	1.99959	+	0.0403852i	1.35766	+	1.77673i	−0.0159047	−	0.167583i	−0.558566	+	2.58612i	−2.82713	−	0.0856628i	2.92058	−	0.620789i	−1.89455	−	2.53193i
31.3	−1.41413	+	0.0150995i	0.278739	+	2.65202i	1.99954	−	0.0427054i	2.22583	−	0.213693i	−0.434218	−	3.74610i	−0.265511	−	2.63240i	−2.82698	+	0.0905831i	−4.02108	+	0.854707i	−3.14440	+	0.335799i
31.4	−1.41211	+	0.0770899i	−0.173887	−	1.65443i	1.98811	−	0.217719i	−1.51845	−	1.64144i	0.373087	+	2.32283i	−2.52288	+	0.796932i	−2.79065	+	0.460707i	0.227557	−	0.0483687i	2.27076	+	2.20083i
31.5	−1.40315	−	0.176572i	−0.0271700	−	0.258505i	1.93765	+	0.495512i	1.72272	−	1.42557i	−0.00752116	+	0.367519i	2.60149	−	0.481927i	−2.63131	−	1.03741i	2.86836	−	0.609688i	−2.66894	+	1.69610i
31.6	−1.40167	+	0.187967i	−0.342810	−	3.26162i	1.92934	−	0.526934i	−1.78491	+	1.34688i	1.09358	+	4.50727i	1.70343	+	2.02443i	−2.60524	+	1.10124i	−7.58622	+	1.61250i	2.24868	−	2.22338i
31.7	−1.40149	−	0.189270i	0.289127	+	2.75086i	1.92835	+	0.530520i	−1.23333	−	1.86518i	0.115446	−	3.91003i	2.04777	+	1.67530i	−2.60216	−	1.10850i	−4.54921	+	0.966965i	1.37548	+	2.84747i
31.8	−1.39979	+	0.201466i	0.0582476	+	0.554189i	1.91882	−	0.564019i	−2.09353	+	0.785571i	−0.193184	−	0.764014i	2.49476	−	0.881002i	−2.57232	+	1.17608i	2.63071	−	0.559175i	2.77224	−	1.52141i
31.9	−1.39180	+	0.250783i	0.291127	+	2.76989i	1.87422	−	0.698079i	−1.29969	+	1.81957i	−1.09983	−	3.78212i	−2.28897	+	1.32688i	−2.43347	+	1.44161i	−4.65309	+	0.989045i	1.35259	−	2.85841i
31.10	−1.36556	−	0.367758i	0.131846	+	1.25443i	1.72951	+	1.00439i	0.794103	−	2.09031i	0.281283	−	1.76149i	−2.19886	+	1.47140i	−1.99238	−	2.00759i	1.37823	−	0.292952i	−1.85312	+	2.56241i
31.11	−1.36106	−	0.384073i	−0.167051	−	1.58938i	1.70498	+	1.04549i	−1.57515	+	1.58711i	−0.383072	+	2.22740i	−2.41684	−	1.07652i	−1.91903	−	2.07782i	0.436219	−	0.0927212i	2.75344	−	1.55518i
31.12	−1.35141	+	0.416760i	−0.0787803	−	0.749544i	1.65262	−	1.12643i	−0.501529	−	2.17910i	0.418845	+	0.980110i	0.672327	−	2.55890i	−1.76392	+	2.21102i	2.37883	−	0.505636i	1.58593	+	2.73584i
31.13	−1.33191	−	0.475406i	0.139870	+	1.33078i	1.54798	+	1.26640i	−2.18636	−	0.468853i	0.446364	−	1.83897i	−1.93252	−	1.80703i	−1.45972	−	2.42265i	1.18304	−	0.251462i	2.68915	+	1.66388i
31.14	−1.32500	−	0.494352i	−0.340812	−	3.24261i	1.51123	+	1.31003i	0.411849	−	2.19781i	−1.15142	+	4.46493i	0.0336942	−	2.64554i	−1.35476	−	2.48287i	−7.46395	+	1.58651i	−1.63219	+	2.70850i
31.15	−1.29610	+	0.565792i	−0.256213	−	2.43770i	1.35976	−	1.46665i	2.08572	−	0.806074i	1.71131	+	3.01455i	−2.46708	+	0.955785i	−0.932568	+	2.67027i	−2.94232	+	0.625408i	−2.24724	+	2.22484i
31.16	−1.29121	+	0.576863i	0.113297	+	1.07795i	1.33446	−	1.48971i	2.21461	+	0.309052i	−0.768122	−	1.32651i	−2.23840	−	1.41052i	−0.863712	+	2.69333i	1.78530	−	0.379477i	−3.03781	+	0.878474i
31.17	−1.26156	−	0.639114i	0.318849	+	3.03365i	1.18307	+	1.61256i	2.14116	+	0.644553i	1.53660	−	4.03091i	1.75842	+	1.97685i	−0.461900	−	2.79046i	−6.16690	+	1.31082i	−2.28926	−	2.18158i
31.18	−1.24395	+	0.672754i	0.314462	+	2.99191i	1.09481	−	1.67374i	−0.795595	−	2.08974i	−2.40399	−	3.51022i	0.945499	−	2.47104i	−0.235865	+	2.81858i	−5.91819	+	1.25795i	2.39556	+	2.06429i
31.19	−1.23779	−	0.684007i	−0.190106	−	1.80873i	1.06427	+	1.69332i	−1.97180	−	1.05451i	−1.00187	+	2.36887i	2.28938	+	1.32617i	−0.159106	−	2.82395i	−0.300932	+	0.0639652i	1.71940	+	2.65399i
31.20	−1.18394	−	0.773493i	0.0479679	+	0.456384i	0.803416	+	1.83154i	−0.127620	+	2.23242i	0.296219	−	0.577433i	2.60474	+	0.464056i	0.465487	−	2.78986i	2.72846	−	0.579951i	1.87786	−	2.54434i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
25.d	even	5	1	inner
28.f	even	6	1	inner
100.j	odd	10	1	inner
175.v	odd	30	1	inner
700.bq	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.bq.a	✓	928
4.b	odd	2	1	inner	700.2.bq.a	✓	928
7.d	odd	6	1	inner	700.2.bq.a	✓	928
25.d	even	5	1	inner	700.2.bq.a	✓	928
28.f	even	6	1	inner	700.2.bq.a	✓	928
100.j	odd	10	1	inner	700.2.bq.a	✓	928
175.v	odd	30	1	inner	700.2.bq.a	✓	928
700.bq	even	30	1	inner	700.2.bq.a	✓	928

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.bq.a	✓	928	1.a	even	1	1	trivial
700.2.bq.a	✓	928	4.b	odd	2	1	inner
700.2.bq.a	✓	928	7.d	odd	6	1	inner
700.2.bq.a	✓	928	25.d	even	5	1	inner
700.2.bq.a	✓	928	28.f	even	6	1	inner
700.2.bq.a	✓	928	100.j	odd	10	1	inner
700.2.bq.a	✓	928	175.v	odd	30	1	inner
700.2.bq.a	✓	928	700.bq	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).