Newform orbit 342.3.bd.a

• Label: 342.3.bd.a
• Level: $342$
• Weight: $3$
• Character orbit: 342.bd
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.31882504112\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(40\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{3} + 12 q^{6} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−0.483690	+	1.32893i	−2.92828	−	0.652046i	−1.53209	−	1.28558i	−5.10811	−	1.85920i	2.28290	−	3.57608i	11.2656			2.44949	−	1.41421i	8.14967	+	3.81875i	4.94147	−	5.88902i
13.2	−0.483690	+	1.32893i	−2.81309	−	1.04236i	−1.53209	−	1.28558i	3.79650	+	1.38181i	2.74589	−	3.23421i	−6.18835			2.44949	−	1.41421i	6.82696	+	5.86452i	−3.67266	+	4.37690i
13.3	−0.483690	+	1.32893i	−2.71129	+	1.28410i	−1.53209	−	1.28558i	−5.32061	−	1.93655i	−0.395045	−	4.22421i	−4.93026			2.44949	−	1.41421i	5.70220	−	6.96311i	5.14705	−	6.13402i
13.4	−0.483690	+	1.32893i	−2.65568	+	1.39547i	−1.53209	−	1.28558i	4.62903	+	1.68483i	−0.569952	−	4.20418i	−0.0659081			2.44949	−	1.41421i	5.10532	−	7.41186i	−4.47802	+	5.33670i
13.5	−0.483690	+	1.32893i	−1.97319	−	2.25977i	−1.53209	−	1.28558i	−6.05642	−	2.20436i	3.95747	−	1.52919i	−11.2396			2.44949	−	1.41421i	−1.21308	+	8.91787i	5.85885	−	6.98231i
13.6	−0.483690	+	1.32893i	−1.60752	+	2.53296i	−1.53209	−	1.28558i	5.41184	+	1.96975i	−2.58858	−	3.36144i	−9.39397			2.44949	−	1.41421i	−3.83177	−	8.14356i	−5.23530	+	6.23919i
13.7	−0.483690	+	1.32893i	−1.56701	−	2.55822i	−1.53209	−	1.28558i	2.20033	+	0.800853i	4.15763	−	0.845055i	2.15253			2.44949	−	1.41421i	−4.08897	+	8.01750i	−2.12855	+	2.53671i
13.8	−0.483690	+	1.32893i	−1.00967	+	2.82499i	−1.53209	−	1.28558i	1.21652	+	0.442777i	−3.26583	−	2.70820i	11.1979			2.44949	−	1.41421i	−6.96112	−	5.70464i	−1.17684	+	1.40250i
13.9	−0.483690	+	1.32893i	−0.290428	+	2.98591i	−1.53209	−	1.28558i	−0.629298	−	0.229046i	−3.82757	−	1.83021i	7.75023			2.44949	−	1.41421i	−8.83130	−	1.73438i	0.608770	−	0.725504i
13.10	−0.483690	+	1.32893i	−0.273902	−	2.98747i	−1.53209	−	1.28558i	−2.17191	−	0.790511i	4.10261	+	1.08101i	4.44443			2.44949	−	1.41421i	−8.84995	+	1.63655i	2.10106	−	2.50395i
13.11	−0.483690	+	1.32893i	0.409896	+	2.97187i	−1.53209	−	1.28558i	−5.29173	−	1.92603i	−4.14765	−	0.892739i	−6.50724			2.44949	−	1.41421i	−8.66397	+	2.43631i	5.11911	−	6.10072i
13.12	−0.483690	+	1.32893i	1.03870	−	2.81444i	−1.53209	−	1.28558i	8.68273	+	3.16026i	3.23778	+	2.74168i	11.2360			2.44949	−	1.41421i	−6.84219	−	5.84674i	−8.39949	+	10.0101i
13.13	−0.483690	+	1.32893i	1.09617	−	2.79257i	−1.53209	−	1.28558i	−7.64241	−	2.78161i	3.18091	+	2.80746i	−1.27540			2.44949	−	1.41421i	−6.59684	−	6.12223i	7.39310	−	8.81076i
13.14	−0.483690	+	1.32893i	1.25025	−	2.72706i	−1.53209	−	1.28558i	5.98971	+	2.18008i	3.01933	+	2.98054i	−13.9436			2.44949	−	1.41421i	−5.87375	−	6.81902i	−5.79432	+	6.90541i
13.15	−0.483690	+	1.32893i	1.38826	+	2.65946i	−1.53209	−	1.28558i	2.94577	+	1.07217i	−4.20571	+	0.558545i	−1.54051			2.44949	−	1.41421i	−5.14546	+	7.38406i	−2.84968	+	3.39612i
13.16	−0.483690	+	1.32893i	2.34842	−	1.86680i	−1.53209	−	1.28558i	−2.33527	−	0.849970i	1.34493	+	4.02382i	4.54702			2.44949	−	1.41421i	2.03012	−	8.76805i	2.25910	−	2.69229i
13.17	−0.483690	+	1.32893i	2.50712	+	1.64753i	−1.53209	−	1.28558i	7.35959	+	2.67867i	−3.40211	+	2.53488i	4.29343			2.44949	−	1.41421i	3.57129	+	8.26111i	−7.11951	+	8.48470i
13.18	−0.483690	+	1.32893i	2.69218	+	1.32370i	−1.53209	−	1.28558i	−8.02723	−	2.92167i	−3.06127	+	2.93745i	9.73784			2.44949	−	1.41421i	5.49566	+	7.12725i	7.76537	−	9.25441i
13.19	−0.483690	+	1.32893i	2.97931	−	0.351754i	−1.53209	−	1.28558i	3.92646	+	1.42911i	−0.973605	+	4.12942i	−2.67764			2.44949	−	1.41421i	8.75254	−	2.09596i	−3.79837	+	4.52673i
13.20	−0.483690	+	1.32893i	2.98751	+	0.273509i	−1.53209	−	1.28558i	−3.57549	−	1.30137i	−1.80850	+	3.83788i	−8.51518			2.44949	−	1.41421i	8.85039	+	1.63422i	3.45886	−	4.12211i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.be	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	342.3.bd.a	yes	240
9.c	even	3	1		342.3.bc.a	✓	240
19.f	odd	18	1		342.3.bc.a	✓	240
171.be	odd	18	1	inner	342.3.bd.a	yes	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
342.3.bc.a	✓	240	9.c	even	3	1
342.3.bc.a	✓	240	19.f	odd	18	1
342.3.bd.a	yes	240	1.a	even	1	1	trivial
342.3.bd.a	yes	240	171.be	odd	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(342, [\chi])\).