Newform orbit 847.2.u.a

• Label: 847.2.u.a
• Level: $847$
• Weight: $2$
• Character orbit: 847.u
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $1720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.u (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(1720\)
Relative dimension:	\(86\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
23.1	−1.58299	−	2.22300i	−0.529306	−	0.916785i	−1.78173	+	5.14797i	−2.73161	+	2.60458i	−1.20013	+	2.62791i	0.944173	−	2.47155i	9.02743	−	2.65069i	0.939671	−	1.62756i	10.1141	+	1.94933i
23.2	−1.54654	−	2.17181i	0.114913	+	0.199036i	−1.67084	+	4.82758i	0.314136	−	0.299528i	0.254550	−	0.557387i	2.59071	+	0.536841i	7.95224	−	2.33499i	1.47359	−	2.55233i	−1.13634	−	0.219012i
23.3	−1.54389	−	2.16809i	−1.53393	−	2.65685i	−1.66287	+	4.80455i	1.34609	−	1.28349i	−3.39205	+	7.42756i	−1.07048	−	2.41952i	7.87637	−	2.31271i	−3.20589	+	5.55276i	−4.86092	−	0.936866i
23.4	−1.54361	−	2.16770i	1.30997	+	2.26894i	−1.66206	+	4.80220i	0.519449	−	0.495293i	2.89630	−	6.34201i	2.54979	−	0.706084i	7.86862	−	2.31044i	−1.93206	+	3.34643i	−1.87548	−	0.361469i
23.5	−1.53915	−	2.16144i	−0.737739	−	1.27780i	−1.64868	+	4.76356i	1.21935	−	1.16265i	−1.62639	+	3.56131i	−2.34511	+	1.22493i	7.74177	−	2.27319i	0.411483	−	0.712709i	−4.38976	−	0.846058i
23.6	−1.41974	−	1.99374i	−1.04317	−	1.80682i	−1.30522	+	3.77118i	−1.72874	+	1.64835i	−2.12130	+	4.64501i	−0.295823	+	2.62916i	4.67495	−	1.37269i	−0.676390	+	1.17154i	5.74074	+	1.10644i
23.7	−1.41639	−	1.98904i	0.754247	+	1.30639i	−1.29599	+	3.74452i	2.88425	−	2.75013i	1.53017	−	3.35060i	−0.521292	−	2.59389i	4.59784	−	1.35005i	0.362222	−	0.627387i	−9.55536	−	1.84164i
23.8	−1.39839	−	1.96377i	0.752732	+	1.30377i	−1.24675	+	3.60224i	−1.42488	+	1.35862i	1.50769	−	3.30138i	−2.64184	−	0.143811i	4.19115	−	1.23063i	0.366789	−	0.635297i	4.66055	+	0.898247i
23.9	−1.30730	−	1.83585i	1.04438	+	1.80891i	−1.00717	+	2.91001i	−0.634596	+	0.605086i	1.95557	−	4.28211i	−0.553085	+	2.58730i	2.33411	−	0.685355i	−0.681441	+	1.18029i	1.94046	+	0.373992i
23.10	−1.28860	−	1.80958i	0.333684	+	0.577958i	−0.959972	+	2.77366i	−1.44180	+	1.37475i	0.615878	−	1.34859i	−0.464073	−	2.60473i	1.99315	−	0.585241i	1.27731	−	2.21237i	4.34563	+	0.837552i
23.11	−1.28779	−	1.80845i	−1.05900	−	1.83424i	−0.957945	+	2.76780i	2.85294	−	2.72028i	−1.95336	+	4.27725i	2.60448	−	0.465469i	1.97870	−	0.580997i	−0.742952	+	1.28683i	−8.59347	−	1.65626i
23.12	−1.26662	−	1.77872i	−0.0320353	−	0.0554867i	−0.905377	+	2.61591i	0.962361	−	0.917609i	−0.0581187	+	0.127262i	1.16758	+	2.37418i	1.60942	−	0.472568i	1.49795	−	2.59452i	−2.85111	−	0.549507i
23.13	−1.26589	−	1.77769i	1.32996	+	2.30355i	−0.903582	+	2.61073i	−3.13876	+	2.99281i	2.41143	−	5.28030i	1.96223	+	1.77473i	1.59701	−	0.468923i	−2.03757	+	3.52918i	9.29362	+	1.79120i
23.14	−1.24481	−	1.74809i	−0.628335	−	1.08831i	−0.852131	+	2.46207i	−0.293339	+	0.279699i	−1.12030	+	2.45312i	−2.19146	−	1.48239i	1.24649	−	0.366003i	0.710390	−	1.23043i	0.854089	+	0.164612i
23.15	−1.23063	−	1.72817i	−1.49574	−	2.59069i	−0.818005	+	2.36347i	−1.74478	+	1.66364i	−2.63647	+	5.77306i	2.63832	−	0.198173i	1.01990	−	0.299469i	−2.97445	+	5.15190i	5.02223	+	0.967955i
23.16	−1.16442	−	1.63519i	1.21406	+	2.10281i	−0.663855	+	1.91808i	2.54551	−	2.42714i	2.02483	−	4.43376i	−2.62336	+	0.343509i	0.0572360	−	0.0168060i	−1.44787	+	2.50779i	−6.93286	−	1.33620i
23.17	−1.08714	−	1.52668i	1.52438	+	2.64031i	−0.494736	+	1.42945i	0.554663	−	0.528870i	2.37369	−	5.19765i	0.743974	−	2.53900i	−0.876412	+	0.257338i	−3.14749	+	5.45162i	−1.41042	−	0.271835i
23.18	−1.06888	−	1.50103i	−1.66808	−	2.88919i	−0.456452	+	1.31883i	1.17953	−	1.12468i	−2.55379	+	5.59203i	−0.367949	+	2.62004i	−1.06864	+	0.313782i	−4.06496	+	7.04072i	−2.94896	−	0.568365i
23.19	−1.05144	−	1.47654i	−0.136956	−	0.237214i	−0.420511	+	1.21499i	1.03151	−	0.983547i	−0.206256	+	0.451637i	1.54090	−	2.15073i	−1.24233	+	0.364781i	1.46249	−	2.53310i	−2.53682	−	0.488933i
23.20	−1.02678	−	1.44191i	0.607575	+	1.05235i	−0.370686	+	1.07103i	2.00114	−	1.90808i	0.893547	−	1.95660i	0.986575	+	2.45493i	−1.47193	+	0.432196i	0.761704	−	1.31931i	−4.80599	−	0.926279i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
121.e	even	11	1	inner
847.u	even	33	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.u.a	✓	1720
7.c	even	3	1	inner	847.2.u.a	✓	1720
121.e	even	11	1	inner	847.2.u.a	✓	1720
847.u	even	33	1	inner	847.2.u.a	✓	1720

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.u.a	✓	1720	1.a	even	1	1	trivial
847.2.u.a	✓	1720	7.c	even	3	1	inner
847.2.u.a	✓	1720	121.e	even	11	1	inner
847.2.u.a	✓	1720	847.u	even	33	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).