Newform orbit 185.2.x.a

• Label: 185.2.x.a
• Level: $185$
• Weight: $2$
• Character orbit: 185.x
• Analytic conductor: $1.477$
• Analytic rank: $0$
• Dimension: $108$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.47723243739\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 108 * q - 12 * q^4 - 9 * q^5 - 24 * q^6 - 6 * q^9 - 3 * q^10 - 30 * q^11 + 3 * q^15 - 24 * q^16 - 18 * q^19 + 36 * q^20 - 36 * q^24 - 27 * q^25 - 54 * q^26 - 6 * q^29 - 90 * q^30 + 84 * q^31 - 24 * q^34 - 18 * q^35 + 108 * q^36 - 12 * q^39 - 15 * q^40 + 30 * q^41 + 108 * q^44 + 36 * q^45 - 30 * q^46 - 42 * q^49 - 15 * q^50 - 6 * q^51 - 120 * q^54 - 6 * q^55 - 114 * q^56 + 30 * q^59 - 72 * q^60 + 36 * q^61 + 84 * q^64 - 30 * q^65 - 36 * q^66 + 54 * q^69 + 168 * q^70 - 42 * q^71 + 60 * q^74 + 96 * q^75 + 18 * q^76 - 36 * q^79 + 12 * q^80 - 114 * q^81 + 120 * q^84 - 72 * q^86 - 132 * q^89 + 60 * q^90 + 102 * q^91 - 66 * q^94 - 27 * q^95 - 156 * q^96 + 168 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−2.70414	+	0.476813i	0.681311	+	0.120134i	5.20564	−	1.89470i	1.78608	−	1.34533i	−1.89964			1.56904	−	1.86991i	−8.41740	+	4.85979i	−2.36932	−	0.862364i	−4.18833	+	4.48960i
9.2	−2.13305	+	0.376114i	0.0170209	+	0.00300124i	2.52906	−	0.920501i	−2.09924	+	0.770188i	−0.0374352			0.867316	−	1.03363i	−1.29685	+	0.748734i	−2.81880	−	1.02596i	4.18811	−	2.43240i
9.3	−2.03761	+	0.359285i	2.44602	+	0.431299i	2.14338	−	0.780126i	0.903819	+	2.04527i	−5.13899			−2.95638	+	3.52328i	−0.503391	+	0.290633i	2.97792	+	1.08387i	−2.57646	−	3.84272i
9.4	−1.84318	+	0.325002i	−1.77654	−	0.313251i	1.41229	−	0.514030i	−0.338687	−	2.21027i	3.37627			−1.05947	+	1.26263i	0.805692	−	0.465167i	0.238873	+	0.0869425i	1.34260	+	3.96384i
9.5	−1.58777	+	0.279967i	3.16824	+	0.558645i	0.563255	−	0.205008i	−1.77194	−	1.36391i	−5.18684			2.30998	−	2.75293i	1.95560	−	1.12907i	6.90655	+	2.51378i	3.19528	+	1.66949i
9.6	−1.09159	+	0.192477i	−1.78492	−	0.314730i	−0.724866	+	0.263830i	2.19145	+	0.444452i	2.00898			−1.23613	+	1.47317i	2.66033	−	1.53594i	0.267820	+	0.0974786i	−2.47771	−	0.0633562i
9.7	−0.810383	+	0.142892i	−2.83792	−	0.500402i	−1.24308	+	0.452445i	−1.67020	+	1.48675i	2.37131			1.94700	−	2.32035i	2.36800	−	1.36717i	4.98430	+	1.81414i	1.14106	−	1.44350i
9.8	−0.691512	+	0.121932i	1.39170	+	0.245395i	−1.41606	+	0.515405i	2.23160	−	0.141256i	−0.992300			1.18536	−	1.41266i	2.13259	−	1.23125i	−0.942462	−	0.343028i	−1.52596	+	0.369785i
9.9	−0.322818	+	0.0569215i	0.781488	+	0.137797i	−1.77841	+	0.647290i	−2.22706	−	0.200546i	−0.260122			−2.76435	+	3.29442i	1.10502	−	0.637985i	−2.22734	−	0.810686i	0.730349	−	0.0620274i
9.10	0.322818	−	0.0569215i	−0.781488	−	0.137797i	−1.77841	+	0.647290i	−0.584224	−	2.15840i	−0.260122			2.76435	−	3.29442i	−1.10502	+	0.637985i	−2.22734	−	0.810686i	−0.311457	−	0.663515i
9.11	0.691512	−	0.121932i	−1.39170	−	0.245395i	−1.41606	+	0.515405i	0.248403	+	2.22223i	−0.992300			−1.18536	+	1.41266i	−2.13259	+	1.23125i	−0.942462	−	0.343028i	0.442735	+	1.50641i
9.12	0.810383	−	0.142892i	2.83792	+	0.500402i	−1.24308	+	0.452445i	1.17413	−	1.90300i	2.37131			−1.94700	+	2.32035i	−2.36800	+	1.36717i	4.98430	+	1.81414i	0.679575	−	1.70993i
9.13	1.09159	−	0.192477i	1.78492	+	0.314730i	−0.724866	+	0.263830i	0.818242	+	2.08098i	2.00898			1.23613	−	1.47317i	−2.66033	+	1.53594i	0.267820	+	0.0974786i	1.29372	+	2.11408i
9.14	1.58777	−	0.279967i	−3.16824	−	0.558645i	0.563255	−	0.205008i	−1.65088	−	1.50818i	−5.18684			−2.30998	+	2.75293i	−1.95560	+	1.12907i	6.90655	+	2.51378i	−3.04346	−	1.93245i
9.15	1.84318	−	0.325002i	1.77654	+	0.313251i	1.41229	−	0.514030i	−2.23550	+	0.0502676i	3.37627			1.05947	−	1.26263i	−0.805692	+	0.465167i	0.238873	+	0.0869425i	−4.10409	+	0.819194i
9.16	2.03761	−	0.359285i	−2.44602	−	0.431299i	2.14338	−	0.780126i	2.17114	+	0.534931i	−5.13899			2.95638	−	3.52328i	0.503391	−	0.290633i	2.97792	+	1.08387i	4.61613	+	0.309922i
9.17	2.13305	−	0.376114i	−0.0170209	−	0.00300124i	2.52906	−	0.920501i	0.393958	−	2.20109i	−0.0374352			−0.867316	+	1.03363i	1.29685	−	0.748734i	−2.81880	−	1.02596i	0.0124705	−	4.84321i
9.18	2.70414	−	0.476813i	−0.681311	−	0.120134i	5.20564	−	1.89470i	−1.01475	+	1.99256i	−1.89964			−1.56904	+	1.86991i	8.41740	−	4.85979i	−2.36932	−	0.862364i	−1.79394	+	5.87200i
34.1	−1.65342	+	1.97047i	−2.05477	−	2.44877i	−0.801655	−	4.54641i	−2.03941	−	0.916960i	8.22263			−0.255705	+	0.702544i	5.82875	+	3.36523i	−1.25349	+	7.10889i	5.17884	−	2.50247i
34.2	−1.63794	+	1.95202i	−0.119452	−	0.142357i	−0.780243	−	4.42498i	1.73311	+	1.41292i	0.473538			1.61021	−	4.42403i	5.50206	+	3.17662i	0.514948	−	2.92041i	−5.59677	+	1.06879i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
37.f	even	9	1	inner
185.x	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	185.2.x.a	✓	108
5.b	even	2	1	inner	185.2.x.a	✓	108
5.c	odd	4	2		925.2.p.g		108
37.f	even	9	1	inner	185.2.x.a	✓	108
185.x	even	18	1	inner	185.2.x.a	✓	108
185.bd	odd	36	2		925.2.p.g		108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
185.2.x.a	✓	108	1.a	even	1	1	trivial
185.2.x.a	✓	108	5.b	even	2	1	inner
185.2.x.a	✓	108	37.f	even	9	1	inner
185.2.x.a	✓	108	185.x	even	18	1	inner
925.2.p.g		108	5.c	odd	4	2
925.2.p.g		108	185.bd	odd	36	2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).