Newform orbit 700.2.t.e

• Label: 700.2.t.e
• Level: $700$
• Weight: $2$
• Character orbit: 700.t
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{4} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
199.1	−1.41421	+	0.00108138i	−2.85162	−	1.64638i	2.00000	−	0.00305860i		0		4.03458	+	2.32525i	1.40586	+	2.24133i	−2.82842	+	0.00648826i	3.92116	+	6.79165i		0
199.2	−1.39417	−	0.237248i	0.623223	+	0.359818i	1.88743	+	0.661529i		0		−0.783513	−	0.649506i	0.956035	+	2.46698i	−2.47445	−	1.37007i	−1.24106	−	2.14958i		0
199.3	−1.38315	+	0.294788i	0.376945	+	0.217629i	1.82620	−	0.815471i		0		−0.585526	−	0.189895i	−2.62598	+	0.322869i	−2.28552	+	1.66626i	−1.40527	−	2.43401i		0
199.4	−1.24422	+	0.672242i	−1.55733	−	0.899123i	1.09618	−	1.67284i		0		2.54209	+	0.0718084i	−1.99604	+	1.73661i	−0.239340	+	2.81828i	0.116845	+	0.202382i		0
199.5	−1.24301	−	0.674474i	−0.461233	−	0.266293i	1.09017	+	1.67676i		0		0.393712	+	0.642096i	1.35494	−	2.27248i	−0.224164	−	2.81953i	−1.35818	−	2.35243i		0
199.6	−1.20429	+	0.741407i	1.55733	+	0.899123i	0.900630	−	1.78574i		0		−2.54209	+	0.0718084i	1.99604	−	1.73661i	0.239340	+	2.81828i	0.116845	+	0.202382i		0
199.7	−1.15273	−	0.819270i	1.52103	+	0.878165i	0.657592	+	1.88880i		0		−1.03388	−	2.25842i	−1.72616	−	2.00509i	0.789410	−	2.71603i	0.0423472	+	0.0733475i		0
199.8	−1.12270	−	0.859974i	2.33023	+	1.34536i	0.520890	+	1.93098i		0		−1.45917	−	3.51437i	0.0832596	+	2.64444i	1.07579	−	2.61585i	2.11999	+	3.67193i		0
199.9	−0.946868	+	1.05045i	−0.376945	−	0.217629i	−0.206882	−	1.98927i		0		0.585526	−	0.189895i	2.62598	−	0.322869i	2.28552	+	1.66626i	−1.40527	−	2.43401i		0
199.10	−0.920983	−	1.07322i	−2.22627	−	1.28534i	−0.303582	+	1.97683i		0		0.670911	+	3.57303i	−2.62816	−	0.304602i	2.40115	−	1.49481i	1.80417	+	3.12492i		0
199.11	−0.708043	+	1.22420i	2.85162	+	1.64638i	−0.997350	−	1.73358i		0		−4.03458	+	2.32525i	−1.40586	−	2.24133i	2.82842	+	0.00648826i	3.92116	+	6.79165i		0
199.12	−0.491623	+	1.32601i	−0.623223	−	0.359818i	−1.51661	−	1.30380i		0		0.783513	−	0.649506i	−0.956035	−	2.46698i	2.47445	−	1.37007i	−1.24106	−	2.14958i		0
199.13	−0.468940	−	1.33420i	−2.22627	−	1.28534i	−1.56019	+	1.25132i		0		−0.670911	+	3.57303i	−2.62816	−	0.304602i	2.40115	+	1.49481i	1.80417	+	3.12492i		0
199.14	−0.183411	−	1.40227i	2.33023	+	1.34536i	−1.93272	+	0.514384i		0		1.45917	−	3.51437i	0.0832596	+	2.64444i	1.07579	+	2.61585i	2.11999	+	3.67193i		0
199.15	−0.133142	−	1.40793i	1.52103	+	0.878165i	−1.96455	+	0.374909i		0		1.03388	−	2.25842i	−1.72616	−	2.00509i	0.789410	+	2.71603i	0.0423472	+	0.0733475i		0
199.16	−0.0373956	+	1.41372i	0.461233	+	0.266293i	−1.99720	−	0.105734i		0		−0.393712	+	0.642096i	−1.35494	+	2.27248i	0.224164	−	2.81953i	−1.35818	−	2.35243i		0
199.17	0.0373956	−	1.41372i	−0.461233	−	0.266293i	−1.99720	−	0.105734i		0		−0.393712	+	0.642096i	1.35494	−	2.27248i	−0.224164	+	2.81953i	−1.35818	−	2.35243i		0
199.18	0.133142	+	1.40793i	−1.52103	−	0.878165i	−1.96455	+	0.374909i		0		1.03388	−	2.25842i	1.72616	+	2.00509i	−0.789410	−	2.71603i	0.0423472	+	0.0733475i		0
199.19	0.183411	+	1.40227i	−2.33023	−	1.34536i	−1.93272	+	0.514384i		0		1.45917	−	3.51437i	−0.0832596	−	2.64444i	−1.07579	−	2.61585i	2.11999	+	3.67193i		0
199.20	0.468940	+	1.33420i	2.22627	+	1.28534i	−1.56019	+	1.25132i		0		−0.670911	+	3.57303i	2.62816	+	0.304602i	−2.40115	−	1.49481i	1.80417	+	3.12492i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
7.d	odd	6	1	inner
20.d	odd	2	1	inner
28.f	even	6	1	inner
35.i	odd	6	1	inner
140.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.t.e		64
4.b	odd	2	1	inner	700.2.t.e		64
5.b	even	2	1	inner	700.2.t.e		64
5.c	odd	4	1		700.2.p.d	✓	32
5.c	odd	4	1		700.2.p.f	yes	32
7.d	odd	6	1	inner	700.2.t.e		64
20.d	odd	2	1	inner	700.2.t.e		64
20.e	even	4	1		700.2.p.d	✓	32
20.e	even	4	1		700.2.p.f	yes	32
28.f	even	6	1	inner	700.2.t.e		64
35.i	odd	6	1	inner	700.2.t.e		64
35.k	even	12	1		700.2.p.d	✓	32
35.k	even	12	1		700.2.p.f	yes	32
140.s	even	6	1	inner	700.2.t.e		64
140.x	odd	12	1		700.2.p.d	✓	32
140.x	odd	12	1		700.2.p.f	yes	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.p.d	✓	32	5.c	odd	4	1
700.2.p.d	✓	32	20.e	even	4	1
700.2.p.d	✓	32	35.k	even	12	1
700.2.p.d	✓	32	140.x	odd	12	1
700.2.p.f	yes	32	5.c	odd	4	1
700.2.p.f	yes	32	20.e	even	4	1
700.2.p.f	yes	32	35.k	even	12	1
700.2.p.f	yes	32	140.x	odd	12	1
700.2.t.e		64	1.a	even	1	1	trivial
700.2.t.e		64	4.b	odd	2	1	inner
700.2.t.e		64	5.b	even	2	1	inner
700.2.t.e		64	7.d	odd	6	1	inner
700.2.t.e		64	20.d	odd	2	1	inner
700.2.t.e		64	28.f	even	6	1	inner
700.2.t.e		64	35.i	odd	6	1	inner
700.2.t.e		64	140.s	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

T3^32 - 32*T3^30 + 629*T3^28 - 7868*T3^26 + 72313*T3^24 - 482422*T3^22 + 2447558*T3^20 - 9095068*T3^18 + 25383397*T3^16 - 49348752*T3^14 + 67198632*T3^12 - 49666320*T3^10 + 25944705*T3^8 - 8260380*T3^6 + 1890432*T3^4 - 243648*T3^2 + 20736

T13^16 - 100*T13^14 + 3966*T13^12 - 81358*T13^10 + 943513*T13^8 - 6278490*T13^6 + 22890801*T13^4 - 40073508*T13^2 + 24800400