Newform orbit 140.2.w.b

• Label: 140.2.w.b
• Level: $140$
• Weight: $2$
• Character orbit: 140.w
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.41262	−	0.0671791i	2.38471	+	0.638980i	1.99097	+	0.189797i	−0.525600	−	2.17342i	−3.32575	−	1.06284i	2.10106	−	1.60796i	−2.79973	−	0.401862i	2.68045	+	1.54756i	0.596463	+	3.10552i
23.2	−1.40543	−	0.157385i	1.08292	+	0.290169i	1.95046	+	0.442386i	1.68711	+	1.46754i	−1.47630	−	0.578247i	−1.51730	+	2.16744i	−2.67161	−	0.928715i	−1.50955	−	0.871538i	−2.14014	−	2.32805i
23.3	−1.34754	−	0.429119i	−2.71477	−	0.727420i	1.63171	+	1.15651i	−2.01971	+	0.959579i	3.34610	+	2.14518i	2.59868	+	0.496852i	−1.70252	−	2.25864i	4.24276	+	2.44956i	3.13340	−	0.426375i
23.4	−1.24267	+	0.675113i	−0.402205	−	0.107770i	1.08845	−	1.67788i	−2.22809	−	0.188711i	0.572564	−	0.137611i	−2.30240	−	1.30345i	−0.219816	+	2.81987i	−2.44792	−	1.41331i	2.89618	−	1.26971i
23.5	−1.05431	−	0.942570i	−1.20413	−	0.322645i	0.223125	+	1.98751i	2.09885	−	0.771256i	0.965404	+	1.47514i	−0.451584	−	2.60693i	1.63813	−	2.30576i	−1.25225	−	0.722989i	−2.93979	−	1.16517i
23.6	−0.809319	+	1.15974i	−0.551941	−	0.147892i	−0.690004	−	1.87720i	1.08510	−	1.95513i	0.618214	−	0.520418i	2.54158	+	0.735093i	2.73551	+	0.719030i	−2.31531	−	1.33674i	1.38926	+	2.84077i
23.7	−0.674418	+	1.24304i	2.47915	+	0.664287i	−1.09032	−	1.67666i	1.93364	+	1.12296i	−2.49773	+	2.63369i	−1.41045	−	2.23844i	2.81950	−	0.224544i	3.10685	+	1.79374i	−2.69997	+	1.64626i
23.8	−0.667698	−	1.24667i	2.02821	+	0.543458i	−1.10836	+	1.66480i	−0.518800	+	2.17505i	−0.676724	−	2.89137i	2.64471	−	0.0742486i	2.81549	+	0.270171i	1.22023	+	0.704499i	3.05797	−	0.805507i
23.9	−0.212826	+	1.39811i	−0.807254	−	0.216303i	−1.90941	−	0.595107i	−0.780454	+	2.09545i	0.474220	−	1.08259i	−0.335905	+	2.62434i	1.23840	−	2.54291i	−1.99320	−	1.15078i	−2.76356	−	1.53712i
23.10	−0.0450897	−	1.41349i	−2.02821	−	0.543458i	−1.99593	+	0.127468i	−0.518800	+	2.17505i	−0.676724	+	2.89137i	−2.64471	+	0.0742486i	0.270171	+	2.81549i	1.22023	+	0.704499i	3.09782	+	0.635249i
23.11	0.441772	−	1.34344i	1.20413	+	0.322645i	−1.60968	−	1.18699i	2.09885	−	0.771256i	0.965404	−	1.47514i	0.451584	+	2.60693i	−2.30576	+	1.63813i	−1.25225	−	0.722989i	−0.108927	−	3.16040i
23.12	0.883367	+	1.10438i	0.807254	+	0.216303i	−0.439327	+	1.95115i	−0.780454	+	2.09545i	0.474220	+	1.08259i	0.335905	−	2.62434i	−2.54291	+	1.23840i	−1.99320	−	1.15078i	−3.00360	+	0.989126i
23.13	0.952442	−	1.04540i	2.71477	+	0.727420i	−0.185707	−	1.99136i	−2.01971	+	0.959579i	3.34610	−	2.14518i	−2.59868	−	0.496852i	−2.25864	−	1.70252i	4.24276	+	2.44956i	−0.920513	+	3.02534i
23.14	1.13844	−	0.839014i	−1.08292	−	0.290169i	0.592112	−	1.91034i	1.68711	+	1.46754i	−1.47630	+	0.578247i	1.51730	−	2.16744i	−0.928715	−	2.67161i	−1.50955	−	0.871538i	3.15196	+	0.255205i
23.15	1.18977	−	0.764487i	−2.38471	−	0.638980i	0.831118	−	1.81913i	−0.525600	−	2.17342i	−3.32575	+	1.06284i	−2.10106	+	1.60796i	−0.401862	−	2.79973i	2.68045	+	1.54756i	−2.28689	−	2.18406i
23.16	1.20559	+	0.739299i	−2.47915	−	0.664287i	0.906874	+	1.78258i	1.93364	+	1.12296i	−2.49773	−	2.63369i	1.41045	+	2.23844i	−0.224544	+	2.81950i	3.10685	+	1.79374i	1.50097	+	2.78336i
23.17	1.28076	+	0.599707i	0.551941	+	0.147892i	1.28070	+	1.53616i	1.08510	−	1.95513i	0.618214	+	0.520418i	−2.54158	−	0.735093i	0.719030	+	2.73551i	−2.31531	−	1.33674i	2.56227	−	1.85332i
23.18	1.41374	−	0.0366689i	0.402205	+	0.107770i	1.99731	−	0.103680i	−2.22809	−	0.188711i	0.572564	+	0.137611i	2.30240	+	1.30345i	2.81987	−	0.219816i	−2.44792	−	1.41331i	−3.15686	−	0.185086i
67.1	−1.41262	+	0.0671791i	2.38471	−	0.638980i	1.99097	−	0.189797i	−0.525600	+	2.17342i	−3.32575	+	1.06284i	2.10106	+	1.60796i	−2.79973	+	0.401862i	2.68045	−	1.54756i	0.596463	−	3.10552i
67.2	−1.40543	+	0.157385i	1.08292	−	0.290169i	1.95046	−	0.442386i	1.68711	−	1.46754i	−1.47630	+	0.578247i	−1.51730	−	2.16744i	−2.67161	+	0.928715i	−1.50955	+	0.871538i	−2.14014	+	2.32805i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
7.c	even	3	1	inner
20.e	even	4	1	inner
28.g	odd	6	1	inner
35.l	odd	12	1	inner
140.w	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	140.2.w.b	✓	72
4.b	odd	2	1	inner	140.2.w.b	✓	72
5.b	even	2	1		700.2.be.e		72
5.c	odd	4	1	inner	140.2.w.b	✓	72
5.c	odd	4	1		700.2.be.e		72
7.b	odd	2	1		980.2.x.m		72
7.c	even	3	1	inner	140.2.w.b	✓	72
7.c	even	3	1		980.2.k.k		36
7.d	odd	6	1		980.2.k.j		36
7.d	odd	6	1		980.2.x.m		72
20.d	odd	2	1		700.2.be.e		72
20.e	even	4	1	inner	140.2.w.b	✓	72
20.e	even	4	1		700.2.be.e		72
28.d	even	2	1		980.2.x.m		72
28.f	even	6	1		980.2.k.j		36
28.f	even	6	1		980.2.x.m		72
28.g	odd	6	1	inner	140.2.w.b	✓	72
28.g	odd	6	1		980.2.k.k		36
35.f	even	4	1		980.2.x.m		72
35.j	even	6	1		700.2.be.e		72
35.k	even	12	1		980.2.k.j		36
35.k	even	12	1		980.2.x.m		72
35.l	odd	12	1	inner	140.2.w.b	✓	72
35.l	odd	12	1		700.2.be.e		72
35.l	odd	12	1		980.2.k.k		36
140.j	odd	4	1		980.2.x.m		72
140.p	odd	6	1		700.2.be.e		72
140.w	even	12	1	inner	140.2.w.b	✓	72
140.w	even	12	1		700.2.be.e		72
140.w	even	12	1		980.2.k.k		36
140.x	odd	12	1		980.2.k.j		36
140.x	odd	12	1		980.2.x.m		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.w.b	✓	72	1.a	even	1	1	trivial
140.2.w.b	✓	72	4.b	odd	2	1	inner
140.2.w.b	✓	72	5.c	odd	4	1	inner
140.2.w.b	✓	72	7.c	even	3	1	inner
140.2.w.b	✓	72	20.e	even	4	1	inner
140.2.w.b	✓	72	28.g	odd	6	1	inner
140.2.w.b	✓	72	35.l	odd	12	1	inner
140.2.w.b	✓	72	140.w	even	12	1	inner
700.2.be.e		72	5.b	even	2	1
700.2.be.e		72	5.c	odd	4	1
700.2.be.e		72	20.d	odd	2	1
700.2.be.e		72	20.e	even	4	1
700.2.be.e		72	35.j	even	6	1
700.2.be.e		72	35.l	odd	12	1
700.2.be.e		72	140.p	odd	6	1
700.2.be.e		72	140.w	even	12	1
980.2.k.j		36	7.d	odd	6	1
980.2.k.j		36	28.f	even	6	1
980.2.k.j		36	35.k	even	12	1
980.2.k.j		36	140.x	odd	12	1
980.2.k.k		36	7.c	even	3	1
980.2.k.k		36	28.g	odd	6	1
980.2.k.k		36	35.l	odd	12	1
980.2.k.k		36	140.w	even	12	1
980.2.x.m		72	7.b	odd	2	1
980.2.x.m		72	7.d	odd	6	1
980.2.x.m		72	28.d	even	2	1
980.2.x.m		72	28.f	even	6	1
980.2.x.m		72	35.f	even	4	1
980.2.x.m		72	35.k	even	12	1
980.2.x.m		72	140.j	odd	4	1
980.2.x.m		72	140.x	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^72 - 167*T3^68 + 17716*T3^64 - 1150565*T3^60 + 54628057*T3^56 - 1771981650*T3^52 + 42135437541*T3^48 - 631596499705*T3^44 + 6449394079565*T3^40 - 25251533070706*T3^36 + 69024525652545*T3^32 - 105455688189725*T3^28 + 114237438624740*T3^24 - 57770759256399*T3^20 + 20656199378249*T3^16 - 2513981899288*T3^12 + 224644412848*T3^8 - 6239120256*T3^4 + 136048896