Newform orbit 304.3.bh.a

• Label: 304.3.bh.a
• Level: $304$
• Weight: $3$
• Character orbit: 304.bh
• Analytic conductor: $8.283$
• Analytic rank: $0$
• Dimension: $936$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.bh (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.28340003655\)
Analytic rank:	\(0\)
Dimension:	\(936\)
Relative dimension:	\(78\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 936 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 6 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
35.1	−1.99877	−	0.0700000i	0.596738	−	1.27971i	3.99020	+	0.279829i	−3.76565	−	5.37791i	−1.28233	+	2.51608i	3.86036	+	6.68634i	−7.95592	−	0.838628i	4.50353	+	5.36710i	7.15023	+	11.0128i
35.2	−1.99631	+	0.121453i	1.47708	−	3.16762i	3.97050	−	0.484914i	3.39524	+	4.84890i	−2.56400	+	6.50294i	−2.93641	−	5.08602i	−7.86745	+	1.45027i	−2.06693	−	2.46327i	−7.36685	−	9.26754i
35.3	−1.98494	+	0.244993i	−0.522169	+	1.11979i	3.87996	−	0.972592i	2.79648	+	3.99379i	0.762130	−	2.35065i	−4.81927	−	8.34722i	−7.46319	+	2.88110i	4.80381	+	5.72496i	−6.52930	−	7.24231i
35.4	−1.98044	+	0.279002i	−0.300039	+	0.643436i	3.84432	−	1.10510i	−2.03092	−	2.90045i	0.414690	−	1.35800i	−0.166344	−	0.288116i	−7.30513	+	3.26116i	5.46110	+	6.50829i	4.83135	+	5.17755i
35.5	−1.97420	+	0.320191i	−1.20274	+	2.57929i	3.79496	−	1.26424i	2.60065	+	3.71411i	1.54859	−	5.47715i	4.12750	+	7.14904i	−7.08721	+	3.71099i	0.578942	+	0.689956i	−6.32343	−	6.49970i
35.6	−1.97151	+	0.336378i	2.29492	−	4.92147i	3.77370	−	1.32634i	1.89449	+	2.70561i	−2.86898	+	10.4747i	0.338061	+	0.585538i	−6.99373	+	3.88429i	−13.1691	−	15.6943i	−4.64512	−	4.69688i
35.7	−1.96388	−	0.378360i	−2.37005	+	5.08260i	3.71369	+	1.48611i	2.75097	+	3.92880i	6.57757	−	9.08490i	−1.56067	−	2.70317i	−6.73097	−	4.32367i	−14.4306	−	17.1977i	−3.91610	−	8.75657i
35.8	−1.89946	−	0.626140i	−1.78717	+	3.83259i	3.21590	+	2.37865i	−4.08731	−	5.83729i	5.79438	−	6.16083i	0.529967	+	0.917929i	−4.61910	−	6.53176i	−5.70968	−	6.80454i	4.10873	+	13.6469i
35.9	−1.84134	+	0.780686i	1.84039	−	3.94673i	2.78106	−	2.87501i	−4.91941	−	7.02565i	−0.307627	+	8.70403i	−1.92890	−	3.34096i	−2.87639	+	7.46501i	−6.40454	−	7.63264i	14.5431	+	9.09609i
35.10	−1.83561	−	0.794061i	1.09322	−	2.34442i	2.73893	+	2.91518i	−2.73318	−	3.90338i	−3.86834	+	3.43535i	−5.72380	−	9.91392i	−2.71279	−	7.52601i	1.48392	+	1.76847i	1.91753	+	9.33540i
35.11	−1.81771	−	0.834226i	1.58428	−	3.39749i	2.60814	+	3.03276i	1.73815	+	2.48234i	−5.71403	+	4.85401i	1.92678	+	3.33728i	−2.21083	−	7.68845i	−3.24792	−	3.87072i	−1.08863	−	5.96219i
35.12	−1.78533	−	0.901448i	−0.895589	+	1.92060i	2.37478	+	3.21876i	−0.640658	−	0.914954i	3.33024	−	2.62157i	−2.29508	−	3.97519i	−1.33822	−	7.88728i	2.89847	+	3.45427i	0.319000	+	2.21101i
35.13	−1.78135	+	0.909273i	−2.11405	+	4.53360i	2.34645	−	3.23947i	−2.11586	−	3.02176i	−0.356403	−	9.99820i	5.92888	+	10.2691i	−1.23429	+	7.90421i	−10.2992	−	12.2742i	6.51671	+	3.45894i
35.14	−1.76372	−	0.943016i	−0.220217	+	0.472257i	2.22144	+	3.32644i	4.20530	+	6.00578i	0.833747	−	0.625262i	3.54145	+	6.13398i	−0.781124	−	7.96177i	5.61056	+	6.68640i	−1.75343	−	14.5582i
35.15	−1.70238	+	1.04972i	−1.67073	+	3.58288i	1.79617	−	3.57404i	−1.98287	−	2.83183i	−0.916824	−	7.85321i	−5.15841	−	8.93462i	0.693988	+	7.96984i	−4.26064	−	5.07763i	6.34822	+	2.73938i
35.16	−1.67184	+	1.09770i	0.473368	−	1.01514i	1.59009	−	3.67037i	3.28117	+	4.68599i	0.322930	+	2.21677i	2.87505	+	4.97974i	1.37061	+	7.88172i	4.97865	+	5.93333i	−10.6294	−	4.23247i
35.17	−1.53296	+	1.28453i	1.68589	−	3.61541i	0.699959	−	3.93828i	−0.528798	−	0.755202i	2.05969	+	7.70788i	5.38321	+	9.32399i	3.98583	+	6.93636i	−4.44388	−	5.29601i	1.78071	+	0.478440i
35.18	−1.49224	+	1.33162i	−0.0107702	+	0.0230967i	0.453586	−	3.97420i	−2.03156	−	2.90137i	−0.0146843	−	0.0488077i	−0.822434	−	1.42450i	4.61526	+	6.53448i	5.78467	+	6.89390i	6.89511	+	1.62429i
35.19	−1.49100	−	1.33301i	2.46315	−	5.28223i	0.446174	+	3.97504i	−3.80096	−	5.42833i	−10.7138	+	4.59242i	2.76585	+	4.79060i	4.63352	−	6.52154i	−16.0498	−	19.1274i	−1.56878	+	13.1604i
35.20	−1.42450	−	1.40386i	−1.03577	+	2.22121i	0.0583763	+	3.99957i	−3.15235	−	4.50202i	4.59371	−	1.71004i	5.53973	+	9.59509i	5.53167	−	5.77933i	1.92412	+	2.29308i	−1.82968	+	10.8386i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner
19.e	even	9	1	inner
304.bh	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	304.3.bh.a	✓	936
16.f	odd	4	1	inner	304.3.bh.a	✓	936
19.e	even	9	1	inner	304.3.bh.a	✓	936
304.bh	odd	36	1	inner	304.3.bh.a	✓	936

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
304.3.bh.a	✓	936	1.a	even	1	1	trivial
304.3.bh.a	✓	936	16.f	odd	4	1	inner
304.3.bh.a	✓	936	19.e	even	9	1	inner
304.3.bh.a	✓	936	304.bh	odd	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(304, [\chi])\).