Newform orbit 700.2.be.e

• Label: 700.2.be.e
• Level: $700$
• Weight: $2$
• Character orbit: 700.be
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
107.1	−1.41349	+	0.0450897i	0.543458	−	2.02821i	1.99593	−	0.127468i		0		−0.676724	+	2.89137i	0.0742486	+	2.64471i	−2.81549	+	0.270171i	−1.22023	−	0.704499i		0
107.2	−1.34344	−	0.441772i	−0.322645	+	1.20413i	1.60968	+	1.18699i		0		0.965404	−	1.47514i	2.60693	−	0.451584i	−1.63813	−	2.30576i	1.25225	+	0.722989i		0
107.3	−1.24667	+	0.667698i	−0.543458	+	2.02821i	1.10836	−	1.66480i		0		−0.676724	−	2.89137i	−0.0742486	−	2.64471i	−0.270171	+	2.81549i	−1.22023	−	0.704499i		0
107.4	−1.04540	−	0.952442i	−0.727420	+	2.71477i	0.185707	+	1.99136i		0		3.34610	−	2.14518i	−0.496852	+	2.59868i	1.70252	−	2.25864i	−4.24276	−	2.44956i		0
107.5	−0.942570	+	1.05431i	0.322645	−	1.20413i	−0.223125	−	1.98751i		0		0.965404	+	1.47514i	−2.60693	+	0.451584i	2.30576	+	1.63813i	1.25225	+	0.722989i		0
107.6	−0.839014	−	1.13844i	0.290169	−	1.08292i	−0.592112	+	1.91034i		0		−1.47630	+	0.578247i	−2.16744	−	1.51730i	2.67161	−	0.928715i	1.50955	+	0.871538i		0
107.7	−0.764487	−	1.18977i	0.638980	−	2.38471i	−0.831118	+	1.81913i		0		−3.32575	+	1.06284i	1.60796	+	2.10106i	2.79973	−	0.401862i	−2.68045	−	1.54756i		0
107.8	−0.429119	+	1.34754i	0.727420	−	2.71477i	−1.63171	−	1.15651i		0		3.34610	+	2.14518i	0.496852	−	2.59868i	2.25864	−	1.70252i	−4.24276	−	2.44956i		0
107.9	−0.157385	+	1.40543i	−0.290169	+	1.08292i	−1.95046	−	0.442386i		0		−1.47630	−	0.578247i	2.16744	+	1.51730i	0.928715	−	2.67161i	1.50955	+	0.871538i		0
107.10	−0.0671791	+	1.41262i	−0.638980	+	2.38471i	−1.99097	−	0.189797i		0		−3.32575	−	1.06284i	−1.60796	−	2.10106i	0.401862	−	2.79973i	−2.68045	−	1.54756i		0
107.11	−0.0366689	−	1.41374i	−0.107770	+	0.402205i	−1.99731	+	0.103680i		0		0.572564	+	0.137611i	1.30345	−	2.30240i	0.219816	+	2.81987i	2.44792	+	1.41331i		0
107.12	0.599707	−	1.28076i	−0.147892	+	0.551941i	−1.28070	−	1.53616i		0		0.618214	+	0.520418i	−0.735093	+	2.54158i	−2.73551	+	0.719030i	2.31531	+	1.33674i		0
107.13	0.675113	+	1.24267i	0.107770	−	0.402205i	−1.08845	+	1.67788i		0		0.572564	−	0.137611i	−1.30345	+	2.30240i	−2.81987	−	0.219816i	2.44792	+	1.41331i		0
107.14	0.739299	−	1.20559i	0.664287	−	2.47915i	−0.906874	−	1.78258i		0		−2.49773	−	2.63369i	2.23844	−	1.41045i	−2.81950	−	0.224544i	−3.10685	−	1.79374i		0
107.15	1.10438	−	0.883367i	−0.216303	+	0.807254i	0.439327	−	1.95115i		0		0.474220	+	1.08259i	−2.62434	−	0.335905i	−1.23840	−	2.54291i	1.99320	+	1.15078i		0
107.16	1.15974	+	0.809319i	0.147892	−	0.551941i	0.690004	+	1.87720i		0		0.618214	−	0.520418i	0.735093	−	2.54158i	−0.719030	+	2.73551i	2.31531	+	1.33674i		0
107.17	1.24304	+	0.674418i	−0.664287	+	2.47915i	1.09032	+	1.67666i		0		−2.49773	+	2.63369i	−2.23844	+	1.41045i	0.224544	+	2.81950i	−3.10685	−	1.79374i		0
107.18	1.39811	+	0.212826i	0.216303	−	0.807254i	1.90941	+	0.595107i		0		0.474220	−	1.08259i	2.62434	+	0.335905i	2.54291	+	1.23840i	1.99320	+	1.15078i		0
207.1	−1.41374	−	0.0366689i	−0.402205	+	0.107770i	1.99731	+	0.103680i		0		0.572564	−	0.137611i	−2.30240	+	1.30345i	−2.81987	−	0.219816i	−2.44792	+	1.41331i		0
207.2	−1.28076	+	0.599707i	−0.551941	+	0.147892i	1.28070	−	1.53616i		0		0.618214	−	0.520418i	2.54158	−	0.735093i	−0.719030	+	2.73551i	−2.31531	+	1.33674i		0

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	700.2.be.e		72
4.b	odd	2	1	inner	700.2.be.e		72
5.b	even	2	1		140.2.w.b	✓	72
5.c	odd	4	1		140.2.w.b	✓	72
5.c	odd	4	1	inner	700.2.be.e		72
7.c	even	3	1	inner	700.2.be.e		72
20.d	odd	2	1		140.2.w.b	✓	72
20.e	even	4	1		140.2.w.b	✓	72
20.e	even	4	1	inner	700.2.be.e		72
28.g	odd	6	1	inner	700.2.be.e		72
35.c	odd	2	1		980.2.x.m		72
35.f	even	4	1		980.2.x.m		72
35.i	odd	6	1		980.2.k.j		36
35.i	odd	6	1		980.2.x.m		72
35.j	even	6	1		140.2.w.b	✓	72
35.j	even	6	1		980.2.k.k		36
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		140.2.w.b	✓	72
35.l	odd	12	1	inner	700.2.be.e		72
35.l	odd	12	1		980.2.k.k		36
140.c	even	2	1		980.2.x.m		72
140.j	odd	4	1		980.2.x.m		72
140.p	odd	6	1		140.2.w.b	✓	72
140.p	odd	6	1		980.2.k.k		36
140.s	even	6	1		980.2.k.j		36
140.s	even	6	1		980.2.x.m		72
140.w	even	12	1		140.2.w.b	✓	72
140.w	even	12	1	inner	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	5.b	even	2	1
140.2.w.b	✓	72	5.c	odd	4	1
140.2.w.b	✓	72	20.d	odd	2	1
140.2.w.b	✓	72	20.e	even	4	1
140.2.w.b	✓	72	35.j	even	6	1
140.2.w.b	✓	72	35.l	odd	12	1
140.2.w.b	✓	72	140.p	odd	6	1
140.2.w.b	✓	72	140.w	even	12	1
700.2.be.e		72	1.a	even	1	1	trivial
700.2.be.e		72	4.b	odd	2	1	inner
700.2.be.e		72	5.c	odd	4	1	inner
700.2.be.e		72	7.c	even	3	1	inner
700.2.be.e		72	20.e	even	4	1	inner
700.2.be.e		72	28.g	odd	6	1	inner
700.2.be.e		72	35.l	odd	12	1	inner
700.2.be.e		72	140.w	even	12	1	inner
980.2.k.j		36	35.i	odd	6	1
980.2.k.j		36	35.k	even	12	1
980.2.k.j		36	140.s	even	6	1
980.2.k.j		36	140.x	odd	12	1
980.2.k.k		36	35.j	even	6	1
980.2.k.k		36	35.l	odd	12	1
980.2.k.k		36	140.p	odd	6	1
980.2.k.k		36	140.w	even	12	1
980.2.x.m		72	35.c	odd	2	1
980.2.x.m		72	35.f	even	4	1
980.2.x.m		72	35.i	odd	6	1
980.2.x.m		72	35.k	even	12	1
980.2.x.m		72	140.c	even	2	1
980.2.x.m		72	140.j	odd	4	1
980.2.x.m		72	140.s	even	6	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