Newform orbit 671.2.ch.a

• Label: 671.2.ch.a
• Level: $671$
• Weight: $2$
• Character orbit: 671.ch
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.ch (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 2 q^{3} - 66 q^{4} - 15 q^{5} + 8 q^{6} + 2 q^{7} - 10 q^{8} - 114 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.05387	−	1.84931i	0.535672	−	0.389188i	0.589365	+	5.60743i	0.136467	−	1.29839i	−1.81993	−	0.191282i	−4.26521	−	0.448291i	5.91042	−	8.13500i	−0.791574	+	2.43622i	−2.68141	+	2.41436i
4.2	−1.99197	−	1.79357i	−2.58360	+	1.87709i	0.541963	+	5.15644i	0.357290	−	3.39938i	8.51315	+	0.894768i	3.51818	+	0.369775i	5.01781	−	6.90643i	2.22445	−	6.84614i	−6.80876	+	6.13063i
4.3	−1.93795	−	1.74494i	0.950266	−	0.690408i	0.501783	+	4.77414i	0.128067	−	1.21848i	−3.04628	−	0.320177i	4.25921	+	0.447661i	4.29253	−	5.90816i	−0.500710	+	1.54103i	−2.37435	+	2.13788i
4.4	−1.93361	−	1.74103i	−0.365227	+	0.265353i	0.498601	+	4.74387i	−0.311231	+	2.96116i	1.16819	+	0.122782i	2.11892	+	0.222708i	4.23637	−	5.83086i	−0.864072	+	2.65934i	5.75726	−	5.18386i
4.5	−1.85082	−	1.66649i	1.88314	−	1.36818i	0.439306	+	4.17972i	0.0748223	−	0.711887i	−5.76543	−	0.605972i	−0.495042	−	0.0520310i	3.22459	−	4.43826i	0.747252	−	2.29980i	−1.32484	+	1.19289i
4.6	−1.78481	−	1.60705i	2.55468	−	1.85608i	0.393882	+	3.74754i	−0.269452	+	2.56366i	−7.54244	−	0.792743i	−0.599543	−	0.0630145i	2.49612	−	3.43561i	2.15428	−	6.63020i	4.60087	−	4.14264i
4.7	−1.78450	−	1.60677i	−1.15682	+	0.840482i	0.393668	+	3.74550i	0.0866267	−	0.824198i	3.41481	+	0.358911i	−0.242759	−	0.0255150i	2.49278	−	3.43102i	−0.295218	+	0.908588i	−1.47888	+	1.33159i
4.8	−1.64784	−	1.48372i	−2.02817	+	1.47355i	0.304889	+	2.90082i	−0.166299	+	1.58223i	5.52844	+	0.581063i	−4.34742	−	0.456933i	1.19491	−	1.64466i	1.01507	−	3.12407i	2.62162	−	2.36052i
4.9	−1.54658	−	1.39255i	−0.0126779	+	0.00921102i	0.243669	+	2.31835i	0.453057	−	4.31055i	0.0324342	+	0.00340897i	−3.14356	−	0.330401i	0.405048	−	0.557501i	−0.926975	+	2.85294i	−6.70335	+	6.03572i
4.10	−1.52164	−	1.37009i	0.165607	−	0.120320i	0.229184	+	2.18054i	−0.427721	+	4.06949i	−0.416845	−	0.0438121i	0.0104816	+	0.00110166i	0.231739	−	0.318961i	−0.914102	+	2.81332i	6.22642	−	5.60629i
4.11	−1.43207	−	1.28944i	1.59403	−	1.15813i	0.179107	+	1.70409i	−0.126276	+	1.20144i	−3.77610	−	0.396884i	−2.45906	−	0.258457i	−0.324538	+	0.446688i	0.272614	−	0.839018i	1.73002	−	1.55772i
4.12	−1.42324	−	1.28149i	0.998980	−	0.725802i	0.174335	+	1.65868i	0.331723	−	3.15614i	−2.35189	−	0.247194i	3.32553	+	0.349527i	−0.373936	+	0.514679i	−0.455878	+	1.40305i	−4.51667	+	4.06683i
4.13	−1.35184	−	1.21720i	−2.49566	+	1.81320i	0.136835	+	1.30189i	−0.317111	+	3.01711i	5.58077	+	0.586563i	3.63906	+	0.382481i	−0.738768	+	1.01683i	2.01356	−	6.19709i	4.10112	−	3.69267i
4.14	−1.28371	−	1.15586i	−1.34327	+	0.975941i	0.102849	+	0.978546i	0.129911	−	1.23602i	2.85242	+	0.299802i	0.355951	+	0.0374119i	−1.03165	+	1.41995i	−0.0751443	+	0.231270i	−1.59544	+	1.43654i
4.15	−1.19556	−	1.07649i	−1.47053	+	1.06841i	0.0614828	+	0.584969i	−0.0280510	+	0.266887i	2.90824	+	0.305668i	4.55326	+	0.478567i	−1.33504	+	1.83752i	0.0939290	−	0.289084i	0.320837	−	0.288883i
4.16	−1.15996	−	1.04443i	0.323730	−	0.235204i	0.0456091	+	0.433942i	−0.140725	+	1.33891i	−0.621167	−	0.0652872i	−1.03556	−	0.108842i	−1.43460	+	1.97456i	−0.877571	+	2.70088i	1.56163	−	1.40610i
4.17	−1.14432	−	1.03035i	2.10999	−	1.53300i	0.0387876	+	0.369039i	0.284866	−	2.71031i	−3.99402	−	0.419788i	−3.59376	−	0.377719i	−1.47432	+	2.02923i	1.17493	−	3.61606i	−3.11854	+	2.80795i
4.18	−1.13991	−	1.02638i	−0.166643	+	0.121073i	0.0368848	+	0.350936i	−0.223140	+	2.12304i	0.314226	+	0.0330265i	−4.17869	−	0.439198i	−1.48507	+	2.04402i	−0.913940	+	2.81282i	2.43341	−	2.19105i
4.19	−1.12378	−	1.01186i	2.62213	−	1.90509i	0.0299721	+	0.285165i	0.247915	−	2.35876i	−4.87438	−	0.512318i	3.49030	+	0.366846i	−1.52283	+	2.09599i	2.31915	−	7.13762i	−2.66533	+	2.39987i
4.20	−1.04966	−	0.945117i	−2.73650	+	1.98818i	−0.000519424	−	0.00494199i	0.288751	−	2.74728i	4.75145	+	0.499398i	−3.81935	−	0.401430i	−1.66457	+	2.29108i	2.60850	−	8.02814i	−2.89959	+	2.61080i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
671.ch	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	671.2.ch.a	✓	480
11.c	even	5	1		671.2.ct.a	yes	480
61.k	even	30	1		671.2.ct.a	yes	480
671.ch	even	30	1	inner	671.2.ch.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
671.2.ch.a	✓	480	1.a	even	1	1	trivial
671.2.ch.a	✓	480	671.ch	even	30	1	inner
671.2.ct.a	yes	480	11.c	even	5	1
671.2.ct.a	yes	480	61.k	even	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).