Newform orbit 330.3.t.a

• Label: 330.3.t.a
• Level: $330$
• Weight: $3$
• Character orbit: 330.t
• Analytic conductor: $8.992$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	330.t (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.99184872389\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 40 q^{5} + 20 q^{7} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
61.1	−1.34500	+	0.437016i	−1.40126	−	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	2.32960	+	0.756934i	−5.16516	−	7.10924i	−1.66251	+	2.28825i	0.927051	+	2.85317i		−	3.16228i
61.2	−1.34500	+	0.437016i	−1.40126	−	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	2.32960	+	0.756934i	−0.101530	−	0.139744i	−1.66251	+	2.28825i	0.927051	+	2.85317i		−	3.16228i
61.3	−1.34500	+	0.437016i	1.40126	+	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	−2.32960	−	0.756934i	−4.46136	−	6.14053i	−1.66251	+	2.28825i	0.927051	+	2.85317i		−	3.16228i
61.4	−1.34500	+	0.437016i	1.40126	+	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	−2.32960	−	0.756934i	4.97537	+	6.84801i	−1.66251	+	2.28825i	0.927051	+	2.85317i		−	3.16228i
61.5	1.34500	−	0.437016i	−1.40126	−	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	−2.32960	−	0.756934i	−7.72406	−	10.6313i	1.66251	−	2.28825i	0.927051	+	2.85317i			3.16228i
61.6	1.34500	−	0.437016i	−1.40126	−	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	−2.32960	−	0.756934i	1.24033	+	1.70716i	1.66251	−	2.28825i	0.927051	+	2.85317i			3.16228i
61.7	1.34500	−	0.437016i	1.40126	+	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	2.32960	+	0.756934i	−0.464917	−	0.639904i	1.66251	−	2.28825i	0.927051	+	2.85317i			3.16228i
61.8	1.34500	−	0.437016i	1.40126	+	1.01807i	1.61803	−	1.17557i	−0.690983	+	2.12663i	2.32960	+	0.756934i	5.52099	+	7.59899i	1.66251	−	2.28825i	0.927051	+	2.85317i			3.16228i
151.1	−0.831254	−	1.14412i	−0.535233	−	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	−1.43977	+	1.98168i	3.04300	+	0.988730i	2.68999	−	0.874032i	−2.42705	+	1.76336i			3.16228i
151.2	−0.831254	−	1.14412i	−0.535233	−	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	−1.43977	+	1.98168i	9.00681	+	2.92649i	2.68999	−	0.874032i	−2.42705	+	1.76336i			3.16228i
151.3	−0.831254	−	1.14412i	0.535233	+	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	1.43977	−	1.98168i	−2.39043	−	0.776699i	2.68999	−	0.874032i	−2.42705	+	1.76336i			3.16228i
151.4	−0.831254	−	1.14412i	0.535233	+	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	1.43977	−	1.98168i	1.12079	+	0.364165i	2.68999	−	0.874032i	−2.42705	+	1.76336i			3.16228i
151.5	0.831254	+	1.14412i	−0.535233	−	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	1.43977	−	1.98168i	−1.95399	−	0.634888i	−2.68999	+	0.874032i	−2.42705	+	1.76336i		−	3.16228i
151.6	0.831254	+	1.14412i	−0.535233	−	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	1.43977	−	1.98168i	6.65460	+	2.16221i	−2.68999	+	0.874032i	−2.42705	+	1.76336i		−	3.16228i
151.7	0.831254	+	1.14412i	0.535233	+	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	−1.43977	+	1.98168i	−7.89814	−	2.56626i	−2.68999	+	0.874032i	−2.42705	+	1.76336i		−	3.16228i
151.8	0.831254	+	1.14412i	0.535233	+	1.64728i	−0.618034	+	1.90211i	−1.80902	−	1.31433i	−1.43977	+	1.98168i	8.59771	+	2.79356i	−2.68999	+	0.874032i	−2.42705	+	1.76336i		−	3.16228i
211.1	−1.34500	−	0.437016i	−1.40126	+	1.01807i	1.61803	+	1.17557i	−0.690983	−	2.12663i	2.32960	−	0.756934i	−5.16516	+	7.10924i	−1.66251	−	2.28825i	0.927051	−	2.85317i			3.16228i
211.2	−1.34500	−	0.437016i	−1.40126	+	1.01807i	1.61803	+	1.17557i	−0.690983	−	2.12663i	2.32960	−	0.756934i	−0.101530	+	0.139744i	−1.66251	−	2.28825i	0.927051	−	2.85317i			3.16228i
211.3	−1.34500	−	0.437016i	1.40126	−	1.01807i	1.61803	+	1.17557i	−0.690983	−	2.12663i	−2.32960	+	0.756934i	−4.46136	+	6.14053i	−1.66251	−	2.28825i	0.927051	−	2.85317i			3.16228i
211.4	−1.34500	−	0.437016i	1.40126	−	1.01807i	1.61803	+	1.17557i	−0.690983	−	2.12663i	−2.32960	+	0.756934i	4.97537	−	6.84801i	−1.66251	−	2.28825i	0.927051	−	2.85317i			3.16228i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	330.3.t.a	✓	32
11.d	odd	10	1	inner	330.3.t.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
330.3.t.a	✓	32	1.a	even	1	1	trivial
330.3.t.a	✓	32	11.d	odd	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^32 - 20*T7^31 + 94*T7^30 - 980*T7^29 + 44063*T7^28 - 516660*T7^27 + 464472*T7^26 + 18683980*T7^25 + 171515063*T7^24 - 3050808640*T7^23 - 6878146308*T7^22 + 211641809320*T7^21 + 577168512501*T7^20 - 18344917538120*T7^19 + 42970886191276*T7^18 + 316817233812560*T7^17 + 2676718193226438*T7^16 - 46836133618671460*T7^15 + 111579993012740648*T7^14 + 601438336402341100*T7^13 - 2286697467639795521*T7^12 - 2729589896248451520*T7^11 + 17850992314905611044*T7^10 - 3901650191910895140*T7^9 - 63048926271410211447*T7^8 + 74469218631826330140*T7^7 + 139032427982002907126*T7^6 - 153372024872096809800*T7^5 - 89456049635753314398*T7^4 - 8824684497048335160*T7^3 + 126883170035370551288*T7^2 + 26015120823034708860*T7 + 3787381812775541761