Newform orbit 930.2.o.e

• Label: 930.2.o.e
• Level: $930$
• Weight: $2$
• Character orbit: 930.o
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{3} - 40 q^{4} - 12 q^{7} - 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
161.1		−	1.00000i	−1.67328	−	0.447368i	−1.00000			0.866025	+	0.500000i	−0.447368	+	1.67328i	−1.32225	−	2.29021i			1.00000i	2.59972	+	1.49714i	0.500000	−	0.866025i
161.2		−	1.00000i	−1.14810	+	1.29687i	−1.00000			0.866025	+	0.500000i	1.29687	+	1.14810i	0.742621	+	1.28626i			1.00000i	−0.363755	−	2.97787i	0.500000	−	0.866025i
161.3		−	1.00000i	−0.556373	+	1.64026i	−1.00000			0.866025	+	0.500000i	1.64026	+	0.556373i	−1.24695	−	2.15979i			1.00000i	−2.38090	−	1.82519i	0.500000	−	0.866025i
161.4		−	1.00000i	−0.474351	−	1.66583i	−1.00000			0.866025	+	0.500000i	−1.66583	+	0.474351i	1.75266	+	3.03570i			1.00000i	−2.54998	+	1.58038i	0.500000	−	0.866025i
161.5		−	1.00000i	−0.151443	−	1.72542i	−1.00000			0.866025	+	0.500000i	−1.72542	+	0.151443i	−1.64767	−	2.85385i			1.00000i	−2.95413	+	0.522605i	0.500000	−	0.866025i
161.6		−	1.00000i	0.433517	+	1.67692i	−1.00000			0.866025	+	0.500000i	1.67692	−	0.433517i	−1.74230	−	3.01775i			1.00000i	−2.62413	+	1.45395i	0.500000	−	0.866025i
161.7		−	1.00000i	0.500302	−	1.65822i	−1.00000			0.866025	+	0.500000i	−1.65822	−	0.500302i	−1.28701	−	2.22916i			1.00000i	−2.49940	−	1.65922i	0.500000	−	0.866025i
161.8		−	1.00000i	1.29636	+	1.14867i	−1.00000			0.866025	+	0.500000i	1.14867	−	1.29636i	1.71736	+	2.97455i			1.00000i	0.361105	+	2.97819i	0.500000	−	0.866025i
161.9		−	1.00000i	1.62650	−	0.595404i	−1.00000			0.866025	+	0.500000i	−0.595404	−	1.62650i	−0.581871	−	1.00783i			1.00000i	2.29099	−	1.93685i	0.500000	−	0.866025i
161.10		−	1.00000i	1.64686	−	0.536508i	−1.00000			0.866025	+	0.500000i	−0.536508	−	1.64686i	0.615418	+	1.06593i			1.00000i	2.42432	−	1.76711i	0.500000	−	0.866025i
161.11			1.00000i	−1.69869	−	0.338296i	−1.00000			−0.866025	−	0.500000i	0.338296	−	1.69869i	−1.24695	−	2.15979i		−	1.00000i	2.77111	+	1.14932i	0.500000	−	0.866025i
161.12			1.00000i	−1.69717	+	0.345843i	−1.00000			−0.866025	−	0.500000i	−0.345843	−	1.69717i	0.742621	+	1.28626i		−	1.00000i	2.76078	−	1.17391i	0.500000	−	0.866025i
161.13			1.00000i	−1.23550	−	1.21390i	−1.00000			−0.866025	−	0.500000i	1.21390	−	1.23550i	−1.74230	−	3.01775i		−	1.00000i	0.0529076	+	2.99953i	0.500000	−	0.866025i
161.14			1.00000i	−0.449207	+	1.67279i	−1.00000			−0.866025	−	0.500000i	−1.67279	−	0.449207i	−1.32225	−	2.29021i		−	1.00000i	−2.59643	−	1.50285i	0.500000	−	0.866025i
161.15			1.00000i	−0.346599	−	1.69702i	−1.00000			−0.866025	−	0.500000i	1.69702	−	0.346599i	1.71736	+	2.97455i		−	1.00000i	−2.75974	+	1.17637i	0.500000	−	0.866025i
161.16			1.00000i	1.20548	+	1.24372i	−1.00000			−0.866025	−	0.500000i	−1.24372	+	1.20548i	1.75266	+	3.03570i		−	1.00000i	−0.0936563	+	2.99854i	0.500000	−	0.866025i
161.17			1.00000i	1.28806	−	1.15797i	−1.00000			−0.866025	−	0.500000i	1.15797	+	1.28806i	0.615418	+	1.06593i		−	1.00000i	0.318204	−	2.98308i	0.500000	−	0.866025i
161.18			1.00000i	1.32888	−	1.11089i	−1.00000			−0.866025	−	0.500000i	1.11089	+	1.32888i	−0.581871	−	1.00783i		−	1.00000i	0.531863	−	2.95248i	0.500000	−	0.866025i
161.19			1.00000i	1.41853	+	0.993862i	−1.00000			−0.866025	−	0.500000i	−0.993862	+	1.41853i	−1.64767	−	2.85385i		−	1.00000i	1.02448	+	2.81965i	0.500000	−	0.866025i
161.20			1.00000i	1.68621	+	0.395836i	−1.00000			−0.866025	−	0.500000i	−0.395836	+	1.68621i	−1.28701	−	2.22916i		−	1.00000i	2.68663	+	1.33493i	0.500000	−	0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
31.e	odd	6	1	inner
93.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.o.e	✓	40
3.b	odd	2	1	inner	930.2.o.e	✓	40
31.e	odd	6	1	inner	930.2.o.e	✓	40
93.g	even	6	1	inner	930.2.o.e	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.o.e	✓	40	1.a	even	1	1	trivial
930.2.o.e	✓	40	3.b	odd	2	1	inner
930.2.o.e	✓	40	31.e	odd	6	1	inner
930.2.o.e	✓	40	93.g	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 \(S_{2}^{\mathrm{new}}(930, [\chi])\):

T7^20 + 6*T7^19 + 54*T7^18 + 224*T7^17 + 1356*T7^16 + 4826*T7^15 + 21966*T7^14 + 63504*T7^13 + 229137*T7^12 + 553070*T7^11 + 1635687*T7^10 + 3058704*T7^9 + 7030795*T7^8 + 9253848*T7^7 + 18688705*T7^6 + 18180020*T7^5 + 35267190*T7^4 + 19464312*T7^3 + 36838111*T7^2 + 13840596*T7 + 24930049

T11^40 + 98*T11^38 + 5805*T11^36 + 225442*T11^34 + 6505652*T11^32 + 139249788*T11^30 + 2288616925*T11^28 + 27840930644*T11^26 + 252146142291*T11^24 + 1538509889168*T11^22 + 6535310384305*T11^20 + 14864662935504*T11^18 + 23692464931121*T11^16 + 26302764236974*T11^14 + 21963995831262*T11^12 + 13301754816326*T11^10 + 5982174520306*T11^8 + 1769656705044*T11^6 + 332005777128*T11^4 + 10332663156*T11^2 + 276922881