Newform orbit 644.2.be.a

• Label: 644.2.be.a
• Level: $644$
• Weight: $2$
• Character orbit: 644.be
• Analytic conductor: $5.142$
• Analytic rank: $0$
• Dimension: $1840$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.be (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.14236589017\)
Analytic rank:	\(0\)
Dimension:	\(1840\)
Relative dimension:	\(92\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1840 q - 7 q^{2} - 11 q^{4} - 54 q^{5} - 40 q^{8} + 66 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.41164	+	0.0853273i	−1.20499	−	1.69218i	1.98544	−	0.240902i	0.206000	−	1.06883i	1.84540	+	2.28592i	0.242571	+	2.63461i	−2.78216	+	0.509479i	−0.430247	+	1.24312i	−0.199597	+	1.52638i
3.2	−1.41106	+	0.0943817i	0.427467	+	0.600293i	1.98218	−	0.266357i	0.0710194	−	0.368484i	−0.659838	−	0.806705i	2.42977	+	1.04700i	−2.77184	+	0.562927i	0.803580	−	2.32179i	−0.0654346	+	0.526656i
3.3	−1.40803	+	0.132147i	−0.413894	−	0.581233i	1.96507	−	0.372132i	0.315181	−	1.63532i	0.659582	+	0.763697i	0.999594	−	2.44966i	−2.71770	+	0.783650i	0.814680	−	2.35386i	−0.227682	+	2.34422i
3.4	−1.40641	+	0.148374i	1.37689	+	1.93356i	1.95597	−	0.417350i	−0.799404	+	4.14770i	−2.22335	−	2.51509i	1.96807	−	1.76825i	−2.68897	+	0.877180i	−0.861656	+	2.48959i	0.508877	−	5.95198i
3.5	−1.40075	−	0.194650i	0.955832	+	1.34228i	1.92422	+	0.545314i	0.356282	−	1.84857i	−1.07761	−	2.06625i	−1.63230	+	2.08221i	−2.58922	−	1.13840i	0.0931074	−	0.269016i	−0.858887	+	2.52004i
3.6	−1.39167	−	0.251484i	0.259455	+	0.364354i	1.87351	+	0.699967i	−0.517846	+	2.68684i	−0.269448	−	0.572310i	−2.40692	+	1.09851i	−2.43129	−	1.44528i	0.915767	−	2.64594i	1.39637	−	3.60898i
3.7	−1.37084	−	0.347569i	1.60306	+	2.25119i	1.75839	+	0.952920i	0.804530	−	4.17430i	−1.41510	−	3.64318i	2.55698	−	0.679601i	−2.07927	−	1.91746i	−1.51682	+	4.38258i	−2.55373	+	5.44265i
3.8	−1.36594	+	0.366329i	−0.721178	−	1.01275i	1.73161	−	1.00077i	−0.345451	+	1.79237i	1.35609	+	1.11917i	−1.58094	−	2.12147i	−1.99867	+	2.00133i	0.475634	−	1.37425i	−0.184730	−	2.57483i
3.9	−1.35122	−	0.417382i	−1.84195	−	2.58665i	1.65159	+	1.12795i	−0.199051	+	1.03278i	1.40925	+	4.26393i	−2.63905	−	0.188132i	−1.76087	−	2.21345i	−2.31681	+	6.69397i	0.700023	−	1.31243i
3.10	−1.33858	+	0.456307i	−1.65615	−	2.32574i	1.58357	−	1.22160i	−0.539432	+	2.79884i	3.27813	+	2.35746i	2.61862	+	0.377907i	−1.56230	+	2.35780i	−1.68502	+	4.86854i	−0.555059	−	3.99260i
3.11	−1.29403	+	0.570521i	1.46620	+	2.05899i	1.34901	−	1.47654i	−0.0982371	+	0.509703i	−3.07199	−	1.82789i	0.723046	+	2.54504i	−0.903260	+	2.68032i	−1.10849	+	3.20276i	−0.163675	−	0.715615i
3.12	−1.29158	−	0.576046i	−0.478288	−	0.671662i	1.33634	+	1.48801i	0.600321	−	3.11476i	0.230838	+	1.14302i	−1.72374	−	2.00717i	−0.868826	−	2.69168i	0.758834	−	2.19251i	−2.56960	+	3.67714i
3.13	−1.25358	−	0.654629i	−1.00100	−	1.40570i	1.14292	+	1.64126i	−0.309262	+	1.60460i	0.334615	+	2.41744i	2.25999	−	1.37567i	−0.358328	−	2.80564i	0.00719605	−	0.0207916i	1.43811	−	1.80905i
3.14	−1.25061	−	0.660288i	0.867270	+	1.21791i	1.12804	+	1.65152i	−0.330914	+	1.71694i	−0.280443	−	2.09578i	−0.473219	−	2.60309i	−0.320254	−	2.81024i	0.250055	−	0.722488i	1.54752	−	1.92872i
3.15	−1.24384	−	0.672947i	−0.949023	−	1.33272i	1.09428	+	1.67408i	0.651971	−	3.38275i	0.283586	+	2.29633i	0.689441	+	2.55434i	−0.234547	−	2.81869i	0.105716	−	0.305446i	−3.08736	+	3.76886i
3.16	−1.23972	+	0.680517i	0.264333	+	0.371203i	1.07379	−	1.68730i	−0.477021	+	2.47502i	−0.580308	−	0.280304i	−2.61029	+	0.431699i	−0.182967	+	2.82250i	0.913284	−	2.63876i	−1.09292	−	3.39294i
3.17	−1.21427	+	0.724939i	0.527049	+	0.740137i	0.948927	−	1.76055i	0.631760	−	3.27788i	−1.17654	−	0.516651i	−2.62757	−	0.309604i	0.124034	+	2.82571i	0.711182	−	2.05482i	1.60913	+	4.43824i
3.18	−1.18864	−	0.766247i	1.85049	+	2.59865i	0.825731	+	1.82158i	−0.416423	+	2.16061i	−0.208360	−	4.50680i	−0.600669	+	2.57666i	0.414286	−	2.79792i	−2.34747	+	6.78257i	2.15054	−	2.24910i
3.19	−1.14767	+	0.826349i	1.31132	+	1.84149i	0.634296	−	1.89675i	0.215878	−	1.12008i	−3.02667	−	1.02982i	1.19752	−	2.35922i	0.839415	+	2.70100i	−0.690318	+	1.99454i	0.677820	+	1.46387i
3.20	−1.08471	−	0.907418i	1.34904	+	1.89446i	0.353184	+	1.96857i	0.0520097	−	0.269852i	0.255754	−	3.27907i	−1.15982	−	2.37798i	1.40321	−	2.45581i	−0.787864	+	2.27638i	−0.301284	+	0.245516i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
23.c	even	11	1	inner
28.f	even	6	1	inner
92.g	odd	22	1	inner
161.n	odd	66	1	inner
644.be	even	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.2.be.a	✓	1840
4.b	odd	2	1	inner	644.2.be.a	✓	1840
7.d	odd	6	1	inner	644.2.be.a	✓	1840
23.c	even	11	1	inner	644.2.be.a	✓	1840
28.f	even	6	1	inner	644.2.be.a	✓	1840
92.g	odd	22	1	inner	644.2.be.a	✓	1840
161.n	odd	66	1	inner	644.2.be.a	✓	1840
644.be	even	66	1	inner	644.2.be.a	✓	1840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.2.be.a	✓	1840	1.a	even	1	1	trivial
644.2.be.a	✓	1840	4.b	odd	2	1	inner
644.2.be.a	✓	1840	7.d	odd	6	1	inner
644.2.be.a	✓	1840	23.c	even	11	1	inner
644.2.be.a	✓	1840	28.f	even	6	1	inner
644.2.be.a	✓	1840	92.g	odd	22	1	inner
644.2.be.a	✓	1840	161.n	odd	66	1	inner
644.2.be.a	✓	1840	644.be	even	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).