Newform orbit 330.3.t.b

• Label: 330.3.t.b
• 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	−0.802695	−	1.10481i	−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	7.28643	+	10.0289i	−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	−3.84761	−	5.29578i	−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	−1.20846	−	1.66331i	−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	−2.63019	−	3.62014i	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	7.89688	+	10.8691i	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	−5.47189	−	7.53141i	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	4.95788	+	6.82393i	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	−3.28285	−	1.06666i	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	−1.41777	−	0.460660i	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	−10.7407	−	3.48986i	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	10.0411	+	3.26256i	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	−9.32509	−	3.02991i	−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	−2.72472	−	0.885315i	−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	−8.50402	−	2.76312i	−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	9.77367	+	3.17566i	−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	−0.802695	+	1.10481i	−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	7.28643	−	10.0289i	−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	−3.84761	+	5.29578i	−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	−1.20846	+	1.66331i	−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.b	✓	32
11.d	odd	10	1	inner	330.3.t.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
330.3.t.b	✓	32	1.a	even	1	1	trivial
330.3.t.b	✓	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 - 66*T7^30 - 1260*T7^29 + 63503*T7^28 + 602100*T7^27 - 8465288*T7^26 - 83000100*T7^25 + 1122658143*T7^24 + 11256513320*T7^23 - 112332891588*T7^22 - 1119018617760*T7^21 + 14788878449661*T7^20 + 211700851280840*T7^19 + 53255964531316*T7^18 - 11092784954716520*T7^17 - 26977378187045242*T7^16 + 492759061987663460*T7^15 + 2208965112071785368*T7^14 - 13790569233507683900*T7^13 + 40992470153004263359*T7^12 + 3756322089062119106080*T7^11 + 45275855490804837684404*T7^10 + 309396110670830456789700*T7^9 + 1441416769498743890755433*T7^8 + 4888859604462660540005420*T7^7 + 12414146040485651883969566*T7^6 + 23812750839607162528714120*T7^5 + 34309777934667457828462202*T7^4 + 36324661483000889893790720*T7^3 + 26971025814016036448977688*T7^2 + 12708344544389922745075220*T7 + 2901231308327088479594401