Newform orbit 680.2.f.e

• Label: 680.2.f.e
• Level: $680$
• Weight: $2$
• Character orbit: 680.f
• Analytic conductor: $5.430$
• Analytic rank: $0$
• Dimension: $30$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.42982733745\)
Analytic rank:	\(0\)
Dimension:	\(30\)
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.

\(\operatorname{Tr}(f)(q) = \) 30 * q - 2 * q^4 - 8 * q^6 + 12 * q^7 + 6 * q^8 - 36 * q^9 + 2 * q^10 + 8 * q^12 - 10 * q^14 - 14 * q^15 - 2 * q^16 + 30 * q^17 + 6 * q^18 + 4 * q^20 - 6 * q^22 + 12 * q^23 - 46 * q^24 - 30 * q^25 + 4 * q^26 + 22 * q^28 + 6 * q^30 - 26 * q^31 - 10 * q^32 + 12 * q^33 + 32 * q^36 + 6 * q^38 + 2 * q^39 + 4 * q^40 - 20 * q^41 + 48 * q^42 + 22 * q^44 + 38 * q^46 - 22 * q^47 + 24 * q^48 + 30 * q^49 + 48 * q^52 - 66 * q^54 + 32 * q^55 - 6 * q^56 - 26 * q^57 - 32 * q^58 - 14 * q^60 - 46 * q^62 - 4 * q^63 - 50 * q^64 + 6 * q^65 - 44 * q^66 - 2 * q^68 - 2 * q^70 + 58 * q^71 - 14 * q^72 - 18 * q^73 + 56 * q^74 - 6 * q^76 + 12 * q^78 - 60 * q^79 - 42 * q^81 - 24 * q^82 + 72 * q^84 + 48 * q^86 + 6 * q^87 + 18 * q^88 + 6 * q^89 - 14 * q^90 + 74 * q^92 - 16 * q^94 - 30 * q^95 + 22 * q^96 + 38 * q^97 - 16 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
341.1	−1.41333	−	0.0499874i		−	0.827995i	1.99500	+	0.141297i		−	1.00000i	−0.0413893	+	1.17023i	−0.211569			−2.81253	−	0.299425i	2.31442			−0.0499874	+	1.41333i
341.2	−1.41333	+	0.0499874i			0.827995i	1.99500	−	0.141297i			1.00000i	−0.0413893	−	1.17023i	−0.211569			−2.81253	+	0.299425i	2.31442			−0.0499874	−	1.41333i
341.3	−1.25557	−	0.650807i		−	2.99039i	1.15290	+	1.63426i		−	1.00000i	−1.94617	+	3.75464i	4.36351			−0.383952	−	2.80225i	−5.94243			−0.650807	+	1.25557i
341.4	−1.25557	+	0.650807i			2.99039i	1.15290	−	1.63426i			1.00000i	−1.94617	−	3.75464i	4.36351			−0.383952	+	2.80225i	−5.94243			−0.650807	−	1.25557i
341.5	−1.12577	−	0.855946i			1.53622i	0.534711	+	1.92720i			1.00000i	1.31493	−	1.72943i	2.50805			1.04761	−	2.62726i	0.640013			0.855946	−	1.12577i
341.6	−1.12577	+	0.855946i		−	1.53622i	0.534711	−	1.92720i		−	1.00000i	1.31493	+	1.72943i	2.50805			1.04761	+	2.62726i	0.640013			0.855946	+	1.12577i
341.7	−1.08720	−	0.904426i		−	1.99420i	0.364028	+	1.96659i			1.00000i	−1.80361	+	2.16810i	0.982898			1.38286	−	2.46732i	−0.976839			0.904426	−	1.08720i
341.8	−1.08720	+	0.904426i			1.99420i	0.364028	−	1.96659i		−	1.00000i	−1.80361	−	2.16810i	0.982898			1.38286	+	2.46732i	−0.976839			0.904426	+	1.08720i
341.9	−0.901248	−	1.08984i		−	2.07328i	−0.375504	+	1.96443i		−	1.00000i	−2.25955	+	1.86854i	−2.82807			2.47934	−	1.36120i	−1.29849			−1.08984	+	0.901248i
341.10	−0.901248	+	1.08984i			2.07328i	−0.375504	−	1.96443i			1.00000i	−2.25955	−	1.86854i	−2.82807			2.47934	+	1.36120i	−1.29849			−1.08984	−	0.901248i
341.11	−0.276941	−	1.38683i			2.54394i	−1.84661	+	0.768141i		−	1.00000i	3.52802	−	0.704520i	3.47689			1.57668	+	2.34821i	−3.47163			−1.38683	+	0.276941i
341.12	−0.276941	+	1.38683i		−	2.54394i	−1.84661	−	0.768141i			1.00000i	3.52802	+	0.704520i	3.47689			1.57668	−	2.34821i	−3.47163			−1.38683	−	0.276941i
341.13	−0.246245	−	1.39261i		−	1.67155i	−1.87873	+	0.685848i			1.00000i	−2.32781	+	0.411610i	2.42659			1.41775	+	2.44745i	0.205935			1.39261	−	0.246245i
341.14	−0.246245	+	1.39261i			1.67155i	−1.87873	−	0.685848i		−	1.00000i	−2.32781	−	0.411610i	2.42659			1.41775	−	2.44745i	0.205935			1.39261	+	0.246245i
341.15	−0.162515	−	1.40484i		−	1.05749i	−1.94718	+	0.456616i		−	1.00000i	−1.48560	+	0.171857i	−4.12826			0.957920	+	2.66128i	1.88172			−1.40484	+	0.162515i
341.16	−0.162515	+	1.40484i			1.05749i	−1.94718	−	0.456616i			1.00000i	−1.48560	−	0.171857i	−4.12826			0.957920	−	2.66128i	1.88172			−1.40484	−	0.162515i
341.17	−0.0347735	−	1.41379i			3.25046i	−1.99758	+	0.0983247i			1.00000i	4.59545	−	0.113030i	−1.97707			0.208473	+	2.82073i	−7.56546			1.41379	−	0.0347735i
341.18	−0.0347735	+	1.41379i		−	3.25046i	−1.99758	−	0.0983247i		−	1.00000i	4.59545	+	0.113030i	−1.97707			0.208473	−	2.82073i	−7.56546			1.41379	+	0.0347735i
341.19	0.615350	−	1.27332i		−	2.36983i	−1.24269	−	1.56708i			1.00000i	−3.01755	−	1.45828i	−4.23147			−2.76008	+	0.618040i	−2.61609			1.27332	+	0.615350i
341.20	0.615350	+	1.27332i			2.36983i	−1.24269	+	1.56708i		−	1.00000i	−3.01755	+	1.45828i	−4.23147			−2.76008	−	0.618040i	−2.61609			1.27332	−	0.615350i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	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.f.e	✓	30
4.b	odd	2	1		2720.2.f.e		30
8.b	even	2	1	inner	680.2.f.e	✓	30
8.d	odd	2	1		2720.2.f.e		30

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
680.2.f.e	✓	30	1.a	even	1	1	trivial
680.2.f.e	✓	30	8.b	even	2	1	inner
2720.2.f.e		30	4.b	odd	2	1
2720.2.f.e		30	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\):

T3^30 + 63*T3^28 + 1779*T3^26 + 29781*T3^24 + 329324*T3^22 + 2536540*T3^20 + 13975408*T3^18 + 55654464*T3^16 + 159829696*T3^14 + 326182336*T3^12 + 459219008*T3^10 + 423668032*T3^8 + 234516224*T3^6 + 66162432*T3^4 + 6620160*T3^2 + 173056

T7^15 - 6*T7^14 - 42*T7^13 + 300*T7^12 + 508*T7^11 - 5408*T7^10 - 280*T7^9 + 42928*T7^8 - 26040*T7^7 - 151952*T7^6 + 125200*T7^5 + 211936*T7^4 - 160032*T7^3 - 55936*T7^2 + 16384*T7 + 4096