Newform orbit 700.2.p.c

• Label: 700.2.p.c
• Level: $700$
• Weight: $2$
• Character orbit: 700.p
• 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.p (of order \(6\), degree \(2\), 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 2 q^{4} + 4 q^{8} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
451.1	−1.38207	+	0.299797i	1.36859	−	2.37047i	1.82024	−	0.828682i		0		−1.18083	+	3.68646i	−1.02440	−	2.43939i	−2.26727	+	1.69100i	−2.24609	−	3.89033i		0
451.2	−1.37405	−	0.334637i	0.556469	−	0.963833i	1.77604	+	0.919616i		0		−1.08715	+	1.13814i	2.32410	+	1.26433i	−2.13263	−	1.85793i	0.880685	+	1.52539i		0
451.3	−1.17265	+	0.790498i	−0.331177	+	0.573616i	0.750225	−	1.85396i		0		−0.0650866	−	0.934447i	1.68748	+	2.03775i	0.585797	+	2.76710i	1.28064	+	2.21814i		0
451.4	−1.05472	−	0.942109i	−0.450639	+	0.780530i	0.224860	+	1.98732i		0		1.21064	−	0.398687i	−2.29962	+	1.30833i	1.63511	−	2.30790i	1.09385	+	1.89460i		0
451.5	−0.906567	+	1.08542i	−0.406021	+	0.703249i	−0.356272	−	1.96801i		0		−0.395235	−	1.07825i	0.336411	−	2.62428i	2.45910	+	1.39743i	1.17029	+	2.02701i		0
451.6	−0.836177	−	1.14053i	−1.51353	+	2.62152i	−0.601615	+	1.90737i		0		4.25550	−	0.465823i	2.57616	−	0.602834i	2.67847	−	0.908739i	−3.08156	−	5.33743i		0
451.7	−0.569639	−	1.29442i	1.51353	−	2.62152i	−1.35102	+	1.47470i		0		−4.25550	−	0.465823i	−2.57616	+	0.602834i	2.67847	+	0.908739i	−3.08156	−	5.33743i		0
451.8	−0.288532	−	1.38447i	0.450639	−	0.780530i	−1.83350	+	0.798926i		0		−1.21064	−	0.398687i	2.29962	−	1.30833i	1.63511	+	2.30790i	1.09385	+	1.89460i		0
451.9	−0.0915727	+	1.41125i	−1.49907	+	2.59647i	−1.98323	−	0.258463i		0		−3.52698	−	2.35332i	−2.06101	−	1.65899i	0.546365	−	2.77516i	−2.99443	−	5.18651i		0
451.10	0.397222	−	1.35728i	−0.556469	+	0.963833i	−1.68443	−	1.07828i		0		1.08715	+	1.13814i	−2.32410	−	1.26433i	−2.13263	+	1.85793i	0.880685	+	1.52539i		0
451.11	0.501653	+	1.32225i	0.895374	−	1.55083i	−1.49669	+	1.32662i		0		2.49976	+	0.405928i	0.644798	−	2.56598i	−2.50494	−	1.31349i	−0.103389	−	0.179074i		0
451.12	0.894275	+	1.09557i	−0.895374	+	1.55083i	−0.400544	+	1.95948i		0		−2.49976	+	0.405928i	−0.644798	+	2.56598i	−2.50494	+	1.31349i	−0.103389	−	0.179074i		0
451.13	0.950668	−	1.04701i	−1.36859	+	2.37047i	−0.192463	−	1.99072i		0		1.18083	+	3.68646i	1.02440	+	2.43939i	−2.26727	−	1.69100i	−2.24609	−	3.89033i		0
451.14	1.26796	+	0.626319i	1.49907	−	2.59647i	1.21545	+	1.58830i		0		3.52698	−	2.35332i	2.06101	+	1.65899i	0.546365	+	2.77516i	−2.99443	−	5.18651i		0
451.15	1.27092	−	0.620297i	0.331177	−	0.573616i	1.23046	−	1.57669i		0		0.0650866	−	0.934447i	−1.68748	−	2.03775i	0.585797	−	2.76710i	1.28064	+	2.21814i		0
451.16	1.39328	−	0.242400i	0.406021	−	0.703249i	1.88248	−	0.675465i		0		0.395235	−	1.07825i	−0.336411	+	2.62428i	2.45910	−	1.39743i	1.17029	+	2.02701i		0
551.1	−1.38207	−	0.299797i	1.36859	+	2.37047i	1.82024	+	0.828682i		0		−1.18083	−	3.68646i	−1.02440	+	2.43939i	−2.26727	−	1.69100i	−2.24609	+	3.89033i		0
551.2	−1.37405	+	0.334637i	0.556469	+	0.963833i	1.77604	−	0.919616i		0		−1.08715	−	1.13814i	2.32410	−	1.26433i	−2.13263	+	1.85793i	0.880685	−	1.52539i		0
551.3	−1.17265	−	0.790498i	−0.331177	−	0.573616i	0.750225	+	1.85396i		0		−0.0650866	+	0.934447i	1.68748	−	2.03775i	0.585797	−	2.76710i	1.28064	−	2.21814i		0
551.4	−1.05472	+	0.942109i	−0.450639	−	0.780530i	0.224860	−	1.98732i		0		1.21064	+	0.398687i	−2.29962	−	1.30833i	1.63511	+	2.30790i	1.09385	−	1.89460i		0

Inner twists

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

    

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

T17^16 - 64*T17^14 + 2988*T17^12 - 672*T17^11 - 61696*T17^10 + 39360*T17^9 + 922000*T17^8 - 1774464*T17^7 - 3374592*T17^6 + 10285056*T17^5 + 10186752*T17^4 - 58392576*T17^3 + 76677120*T17^2 - 42467328*T17 + 9437184