Newform orbit 680.2.h.b

• Label: 680.2.h.b
• Level: $680$
• Weight: $2$
• Character orbit: 680.h
• Analytic conductor: $5.430$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.42982733745\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 40 * q + 12 * q^4 - 56 * q^9 + 56 * q^15 - 4 * q^16 - 36 * q^26 - 28 * q^30 + 24 * q^34 + 88 * q^36 + 8 * q^49 - 16 * q^50 - 88 * q^55 - 88 * q^60 - 132 * q^64 + 208 * q^66 + 72 * q^70 - 20 * q^76 - 56 * q^81 + 144 * q^84 + 256 * q^86 - 96 * q^89 - 36 * q^94

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
509.1	−1.36282	−	0.377781i		−	0.944008i	1.71456	+	1.02970i	−0.0976904	−	2.23393i	−0.356628	+	1.28651i	2.88614			−1.94764	−	2.05102i	2.10885			−0.710803	+	3.08136i
509.2	−1.36282	−	0.377781i			0.944008i	1.71456	+	1.02970i	0.0976904	+	2.23393i	0.356628	−	1.28651i	−2.88614			−1.94764	−	2.05102i	2.10885			0.710803	−	3.08136i
509.3	−1.36282	+	0.377781i		−	0.944008i	1.71456	−	1.02970i	0.0976904	−	2.23393i	0.356628	+	1.28651i	−2.88614			−1.94764	+	2.05102i	2.10885			0.710803	+	3.08136i
509.4	−1.36282	+	0.377781i			0.944008i	1.71456	−	1.02970i	−0.0976904	+	2.23393i	−0.356628	−	1.28651i	2.88614			−1.94764	+	2.05102i	2.10885			−0.710803	−	3.08136i
509.5	−1.29647	−	0.564954i		−	2.10974i	1.36165	+	1.46489i	−2.12767	+	0.687771i	−1.19191	+	2.73521i	2.16636			−0.937746	−	2.66845i	−1.45102			3.14701	+	0.310361i
509.6	−1.29647	−	0.564954i			2.10974i	1.36165	+	1.46489i	2.12767	−	0.687771i	1.19191	−	2.73521i	−2.16636			−0.937746	−	2.66845i	−1.45102			−3.14701	−	0.310361i
509.7	−1.29647	+	0.564954i		−	2.10974i	1.36165	−	1.46489i	2.12767	+	0.687771i	1.19191	+	2.73521i	−2.16636			−0.937746	+	2.66845i	−1.45102			−3.14701	+	0.310361i
509.8	−1.29647	+	0.564954i			2.10974i	1.36165	−	1.46489i	−2.12767	−	0.687771i	−1.19191	−	2.73521i	2.16636			−0.937746	+	2.66845i	−1.45102			3.14701	−	0.310361i
509.9	−1.20224	−	0.744721i		−	1.24640i	0.890781	+	1.79067i	1.91133	−	1.16052i	−0.928222	+	1.49848i	−1.18000			0.262617	−	2.81621i	1.44648			−3.16215	−	0.0281742i
509.10	−1.20224	−	0.744721i			1.24640i	0.890781	+	1.79067i	−1.91133	+	1.16052i	0.928222	−	1.49848i	1.18000			0.262617	−	2.81621i	1.44648			3.16215	+	0.0281742i
509.11	−1.20224	+	0.744721i		−	1.24640i	0.890781	−	1.79067i	−1.91133	−	1.16052i	0.928222	+	1.49848i	1.18000			0.262617	+	2.81621i	1.44648			3.16215	−	0.0281742i
509.12	−1.20224	+	0.744721i			1.24640i	0.890781	−	1.79067i	1.91133	+	1.16052i	−0.928222	−	1.49848i	−1.18000			0.262617	+	2.81621i	1.44648			−3.16215	+	0.0281742i
509.13	−0.870262	−	1.11474i		−	3.10853i	−0.485287	+	1.94023i	−0.636925	+	2.14344i	−3.46520	+	2.70523i	1.38296			2.58518	−	1.14754i	−6.66293			2.94367	−	1.15535i
509.14	−0.870262	−	1.11474i			3.10853i	−0.485287	+	1.94023i	0.636925	−	2.14344i	3.46520	−	2.70523i	−1.38296			2.58518	−	1.14754i	−6.66293			−2.94367	+	1.15535i
509.15	−0.870262	+	1.11474i		−	3.10853i	−0.485287	−	1.94023i	0.636925	+	2.14344i	3.46520	+	2.70523i	−1.38296			2.58518	+	1.14754i	−6.66293			−2.94367	−	1.15535i
509.16	−0.870262	+	1.11474i			3.10853i	−0.485287	−	1.94023i	−0.636925	−	2.14344i	−3.46520	−	2.70523i	1.38296			2.58518	+	1.14754i	−6.66293			2.94367	+	1.15535i
509.17	−0.0956273	−	1.41098i		−	2.33268i	−1.98171	+	0.269856i	1.97601	+	1.04660i	−3.29135	+	0.223067i	4.43533			0.570266	+	2.77034i	−2.44138			1.28777	−	2.88819i
509.18	−0.0956273	−	1.41098i			2.33268i	−1.98171	+	0.269856i	−1.97601	−	1.04660i	3.29135	−	0.223067i	−4.43533			0.570266	+	2.77034i	−2.44138			−1.28777	+	2.88819i
509.19	−0.0956273	+	1.41098i		−	2.33268i	−1.98171	−	0.269856i	−1.97601	+	1.04660i	3.29135	+	0.223067i	−4.43533			0.570266	−	2.77034i	−2.44138			−1.28777	−	2.88819i
509.20	−0.0956273	+	1.41098i			2.33268i	−1.98171	−	0.269856i	1.97601	−	1.04660i	−3.29135	−	0.223067i	4.43533			0.570266	−	2.77034i	−2.44138			1.28777	+	2.88819i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
17.b	even	2	1	inner
40.f	even	2	1	inner
85.c	even	2	1	inner
136.h	even	2	1	inner
680.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	680.2.h.b	✓	40
5.b	even	2	1	inner	680.2.h.b	✓	40
8.b	even	2	1	inner	680.2.h.b	✓	40
17.b	even	2	1	inner	680.2.h.b	✓	40
40.f	even	2	1	inner	680.2.h.b	✓	40
85.c	even	2	1	inner	680.2.h.b	✓	40
136.h	even	2	1	inner	680.2.h.b	✓	40
680.h	even	2	1	inner	680.2.h.b	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
680.2.h.b	✓	40	1.a	even	1	1	trivial
680.2.h.b	✓	40	5.b	even	2	1	inner
680.2.h.b	✓	40	8.b	even	2	1	inner
680.2.h.b	✓	40	17.b	even	2	1	inner
680.2.h.b	✓	40	40.f	even	2	1	inner
680.2.h.b	✓	40	85.c	even	2	1	inner
680.2.h.b	✓	40	136.h	even	2	1	inner
680.2.h.b	✓	40	680.h	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 22T_{3}^{8} + 169T_{3}^{6} + 554T_{3}^{4} + 738T_{3}^{2} + 324 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\).