Newform orbit 735.2.bp.a

• Label: 735.2.bp.a
• Level: $735$
• Weight: $2$
• Character orbit: 735.bp
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $1296$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.bp (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(1296\)
Relative dimension:	\(108\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q + 52 q^{4} - 42 q^{6} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−2.66627	−	0.401876i	−0.0144994	+	1.73199i	5.03636	+	1.55351i	1.84933	−	1.25698i	0.734705	−	4.61213i	1.24969	+	2.33201i	−7.94527	−	3.82624i	−2.99958	−	0.0502258i	−5.43596	+	2.60825i
59.2	−2.66627	−	0.401876i	1.61756	+	0.619270i	5.03636	+	1.55351i	2.01601	+	0.967311i	−4.06399	−	2.30120i	−1.24969	−	2.33201i	−7.94527	−	3.82624i	2.23301	+	2.00341i	−4.98650	−	3.38930i
59.3	−2.57184	−	0.387643i	−1.52740	+	0.816740i	4.55297	+	1.40441i	0.503830	+	2.17857i	4.24483	−	1.50844i	1.62913	+	2.08469i	−6.47847	−	3.11987i	1.66587	−	2.49497i	−0.451266	−	5.79824i
59.4	−2.57184	−	0.387643i	1.31830	−	1.12342i	4.55297	+	1.40441i	0.173504	−	2.22933i	−3.82595	+	2.37824i	−1.62913	−	2.08469i	−6.47847	−	3.11987i	0.475838	−	2.96202i	−1.31041	+	5.66622i
59.5	−2.54468	−	0.383549i	−1.41477	−	0.999211i	4.41716	+	1.36251i	1.70688	−	1.44449i	3.21690	+	3.08531i	1.36151	−	2.26855i	−6.08052	−	2.92823i	1.00315	+	2.82731i	−4.89750	+	3.02111i
59.6	−2.54468	−	0.383549i	−0.413266	−	1.68203i	4.41716	+	1.36251i	1.90311	+	1.17396i	0.406490	+	4.43873i	−1.36151	+	2.26855i	−6.08052	−	2.92823i	−2.65842	+	1.39025i	−4.39253	−	3.71730i
59.7	−2.48912	−	0.375175i	0.276141	+	1.70990i	4.14384	+	1.27820i	−2.23070	−	0.154847i	−0.0458384	−	4.35975i	−2.50831	+	0.841639i	−5.29906	−	2.55189i	−2.84749	+	0.944345i	5.49439	+	1.22234i
59.8	−2.48912	−	0.375175i	1.49081	+	0.881748i	4.14384	+	1.27820i	−2.18271	+	0.485586i	−3.38001	−	2.75409i	2.50831	−	0.841639i	−5.29906	−	2.55189i	1.44504	+	2.62904i	5.61521	−	0.389788i
59.9	−2.45171	−	0.369535i	−1.44452	+	0.955702i	3.96316	+	1.22247i	−1.45584	−	1.69721i	3.89470	−	1.80930i	2.18095	−	1.49782i	−4.79704	−	2.31013i	1.17327	−	2.76106i	2.94211	+	4.69905i
59.10	−2.45171	−	0.369535i	1.41738	−	0.995507i	3.96316	+	1.22247i	−1.18662	+	1.89524i	−3.84287	+	1.91692i	−2.18095	+	1.49782i	−4.79704	−	2.31013i	1.01793	−	2.82202i	3.60961	−	4.20807i
59.11	−2.41080	−	0.363369i	−0.343570	+	1.69763i	3.76876	+	1.16251i	0.350890	+	2.20837i	1.44515	−	3.96781i	−0.485834	−	2.60076i	−4.27012	−	2.05638i	−2.76392	−	1.16651i	−0.0434730	−	5.45142i
59.12	−2.41080	−	0.363369i	1.70580	+	0.300395i	3.76876	+	1.16251i	0.0178312	−	2.23600i	−4.00319	−	1.34403i	0.485834	+	2.60076i	−4.27012	−	2.05638i	2.81953	+	1.02483i	−0.855480	+	5.38405i
59.13	−2.17656	−	0.328063i	−1.62099	−	0.610233i	2.71863	+	0.838585i	−0.271483	−	2.21953i	3.32799	+	1.85999i	−1.56197	+	2.13548i	−1.67581	−	0.807029i	2.25523	+	1.97837i	−0.137246	+	4.91999i
59.14	−2.17656	−	0.328063i	0.0241653	−	1.73188i	2.71863	+	0.838585i	0.0623522	+	2.23520i	−0.620764	+	3.76161i	1.56197	−	2.13548i	−1.67581	−	0.807029i	−2.99883	−	0.0837031i	0.597573	−	4.88549i
59.15	−2.10339	−	0.317035i	−1.71479	+	0.243903i	2.41260	+	0.744189i	2.08402	+	0.810458i	3.68421	+	0.0306270i	−2.60331	−	0.471967i	−1.00572	−	0.484328i	2.88102	−	0.836484i	−4.12658	−	2.36542i
59.16	−2.10339	−	0.317035i	0.853526	−	1.50715i	2.41260	+	0.744189i	1.93996	−	1.11201i	−2.27312	+	2.89952i	2.60331	+	0.471967i	−1.00572	−	0.484328i	−1.54299	−	2.57278i	−4.43303	+	1.72397i
59.17	−2.07484	−	0.312732i	−1.41112	−	1.00436i	2.29601	+	0.708226i	−1.90245	+	1.17502i	2.61375	+	2.52519i	2.04420	+	1.67966i	−0.761408	−	0.366675i	0.982514	+	2.83455i	4.31475	−	1.84302i
59.18	−2.07484	−	0.312732i	−0.419395	−	1.68051i	2.29601	+	0.708226i	−2.05633	−	0.878348i	0.344628	+	3.61794i	−2.04420	−	1.67966i	−0.761408	−	0.366675i	−2.64822	+	1.40959i	3.99187	+	2.46551i
59.19	−1.95993	−	0.295411i	−0.497986	+	1.65892i	1.84290	+	0.568459i	0.582179	−	2.15895i	1.46608	−	3.10425i	−1.89069	−	1.85076i	0.127542	+	0.0614208i	−2.50402	−	1.65224i	−1.77881	+	4.05940i
59.20	−1.95993	−	0.295411i	1.72618	+	0.142509i	1.84290	+	0.568459i	0.897452	+	2.04807i	−3.34108	−	0.789240i	1.89069	+	1.85076i	0.127542	+	0.0614208i	2.95938	+	0.491992i	−1.15392	−	4.27918i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
49.h	odd	42	1	inner
147.o	even	42	1	inner
245.u	odd	42	1	inner
735.bp	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.bp.a	✓	1296
3.b	odd	2	1	inner	735.2.bp.a	✓	1296
5.b	even	2	1	inner	735.2.bp.a	✓	1296
15.d	odd	2	1	inner	735.2.bp.a	✓	1296
49.h	odd	42	1	inner	735.2.bp.a	✓	1296
147.o	even	42	1	inner	735.2.bp.a	✓	1296
245.u	odd	42	1	inner	735.2.bp.a	✓	1296
735.bp	even	42	1	inner	735.2.bp.a	✓	1296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.bp.a	✓	1296	1.a	even	1	1	trivial
735.2.bp.a	✓	1296	3.b	odd	2	1	inner
735.2.bp.a	✓	1296	5.b	even	2	1	inner
735.2.bp.a	✓	1296	15.d	odd	2	1	inner
735.2.bp.a	✓	1296	49.h	odd	42	1	inner
735.2.bp.a	✓	1296	147.o	even	42	1	inner
735.2.bp.a	✓	1296	245.u	odd	42	1	inner
735.2.bp.a	✓	1296	735.bp	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(735, [\chi])\).