Newform orbit 700.2.t.d

• Label: 700.2.t.d
• Level: $700$
• Weight: $2$
• Character orbit: 700.t
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

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:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{4} + 16 q^{9}+O(q^{10}) \)

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	−1.41125	−	0.0915727i	2.59647	+	1.49907i	1.98323	+	0.258463i		0		−3.52698	−	2.35332i	1.65899	−	2.06101i	−2.77516	−	0.546365i	2.99443	+	5.18651i		0
199.2	−1.35728	−	0.397222i	−0.963833	−	0.556469i	1.68443	+	1.07828i		0		1.08715	+	1.13814i	−1.26433	+	2.32410i	−1.85793	−	2.13263i	−0.880685	−	1.52539i		0
199.3	−1.29442	+	0.569639i	2.62152	+	1.51353i	1.35102	−	1.47470i		0		−4.25550	−	0.465823i	0.602834	+	2.57616i	−0.908739	+	2.67847i	3.08156	+	5.33743i		0
199.4	−1.14053	+	0.836177i	−2.62152	−	1.51353i	0.601615	−	1.90737i		0		4.25550	−	0.465823i	−0.602834	−	2.57616i	0.908739	+	2.67847i	3.08156	+	5.33743i		0
199.5	−0.790498	−	1.17265i	0.573616	+	0.331177i	−0.750225	+	1.85396i		0		−0.0650866	−	0.934447i	−2.03775	+	1.68748i	2.76710	−	0.585797i	−1.28064	−	2.21814i		0
199.6	−0.626319	+	1.26796i	−2.59647	−	1.49907i	−1.21545	−	1.58830i		0		3.52698	−	2.35332i	−1.65899	+	2.06101i	2.77516	−	0.546365i	2.99443	+	5.18651i		0
199.7	−0.334637	+	1.37405i	0.963833	+	0.556469i	−1.77604	−	0.919616i		0		−1.08715	+	1.13814i	1.26433	−	2.32410i	1.85793	−	2.13263i	−0.880685	−	1.52539i		0
199.8	−0.299797	−	1.38207i	−2.37047	−	1.36859i	−1.82024	+	0.828682i		0		−1.18083	+	3.68646i	2.43939	−	1.02440i	1.69100	+	2.26727i	2.24609	+	3.89033i		0
199.9	−0.242400	−	1.39328i	0.703249	+	0.406021i	−1.88248	+	0.675465i		0		0.395235	−	1.07825i	2.62428	+	0.336411i	1.39743	+	2.45910i	−1.17029	−	2.02701i		0
199.10	0.620297	+	1.27092i	−0.573616	−	0.331177i	−1.23046	+	1.57669i		0		0.0650866	−	0.934447i	2.03775	−	1.68748i	−2.76710	−	0.585797i	−1.28064	−	2.21814i		0
199.11	0.942109	−	1.05472i	0.780530	+	0.450639i	−0.224860	−	1.98732i		0		1.21064	−	0.398687i	−1.30833	−	2.29962i	−2.30790	−	1.63511i	−1.09385	−	1.89460i		0
199.12	1.04701	+	0.950668i	2.37047	+	1.36859i	0.192463	+	1.99072i		0		1.18083	+	3.68646i	−2.43939	+	1.02440i	−1.69100	+	2.26727i	2.24609	+	3.89033i		0
199.13	1.08542	+	0.906567i	−0.703249	−	0.406021i	0.356272	+	1.96801i		0		−0.395235	−	1.07825i	−2.62428	−	0.336411i	−1.39743	+	2.45910i	−1.17029	−	2.02701i		0
199.14	1.09557	−	0.894275i	−1.55083	−	0.895374i	0.400544	−	1.95948i		0		−2.49976	+	0.405928i	2.56598	+	0.644798i	−1.31349	−	2.50494i	0.103389	+	0.179074i		0
199.15	1.32225	−	0.501653i	1.55083	+	0.895374i	1.49669	−	1.32662i		0		2.49976	+	0.405928i	−2.56598	−	0.644798i	1.31349	−	2.50494i	0.103389	+	0.179074i		0
199.16	1.38447	−	0.288532i	−0.780530	−	0.450639i	1.83350	−	0.798926i		0		−1.21064	−	0.398687i	1.30833	+	2.29962i	2.30790	−	1.63511i	−1.09385	−	1.89460i		0
299.1	−1.41125	+	0.0915727i	2.59647	−	1.49907i	1.98323	−	0.258463i		0		−3.52698	+	2.35332i	1.65899	+	2.06101i	−2.77516	+	0.546365i	2.99443	−	5.18651i		0
299.2	−1.35728	+	0.397222i	−0.963833	+	0.556469i	1.68443	−	1.07828i		0		1.08715	−	1.13814i	−1.26433	−	2.32410i	−1.85793	+	2.13263i	−0.880685	+	1.52539i		0
299.3	−1.29442	−	0.569639i	2.62152	−	1.51353i	1.35102	+	1.47470i		0		−4.25550	+	0.465823i	0.602834	−	2.57616i	−0.908739	−	2.67847i	3.08156	−	5.33743i		0
299.4	−1.14053	−	0.836177i	−2.62152	+	1.51353i	0.601615	+	1.90737i		0		4.25550	+	0.465823i	−0.602834	+	2.57616i	0.908739	−	2.67847i	3.08156	−	5.33743i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	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.d		32
4.b	odd	2	1	inner	700.2.t.d		32
5.b	even	2	1		700.2.t.c		32
5.c	odd	4	1		140.2.o.a	✓	32
5.c	odd	4	1		700.2.p.c		32
7.d	odd	6	1		700.2.t.c		32
20.d	odd	2	1		700.2.t.c		32
20.e	even	4	1		140.2.o.a	✓	32
20.e	even	4	1		700.2.p.c		32
28.f	even	6	1		700.2.t.c		32
35.f	even	4	1		980.2.o.f		32
35.i	odd	6	1	inner	700.2.t.d		32
35.k	even	12	1		140.2.o.a	✓	32
35.k	even	12	1		700.2.p.c		32
35.k	even	12	1		980.2.g.a		32
35.l	odd	12	1		980.2.g.a		32
35.l	odd	12	1		980.2.o.f		32
140.j	odd	4	1		980.2.o.f		32
140.s	even	6	1	inner	700.2.t.d		32
140.w	even	12	1		980.2.g.a		32
140.w	even	12	1		980.2.o.f		32
140.x	odd	12	1		140.2.o.a	✓	32
140.x	odd	12	1		700.2.p.c		32
140.x	odd	12	1		980.2.g.a		32

    

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

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 - 7904*T3^26 + 73006*T3^24 - 483232*T3^22 + 2399561*T3^20 - 8429248*T3^18 + 21956782*T3^16 - 39983424*T3^14 + 54411189*T3^12 - 53885664*T3^10 + 40385601*T3^8 - 21827232*T3^6 + 8634816*T3^4 - 2156544*T3^2 + 331776

\( T_{13}^{8} - 2T_{13}^{7} - 51T_{13}^{6} + 112T_{13}^{5} + 724T_{13}^{4} - 1392T_{13}^{3} - 3708T_{13}^{2} + 4752T_{13} + 5856 \)