Newform orbit 3211.2.a.p

• Label: 3211.2.a.p
• Level: $3211$
• Weight: $2$
• Character orbit: 3211.a
• Self dual: yes
• Analytic conductor: $25.640$
• Analytic rank: $1$
• Dimension: $21$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3211 = 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3211.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.6399640890\)
Analytic rank:	\(1\)
Dimension:	\(21\)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q + q^{2} - 5 q^{3} + 13 q^{4} - 4 q^{5} - 4 q^{6} - q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
1.1	−2.43841			0.292570			3.94587			1.07480			−0.713406			5.23955			−4.74483			−2.91440			−2.62080
1.2	−2.43254			−1.41028			3.91725			2.52995			3.43057			−0.481192			−4.66377			−1.01111			−6.15421
1.3	−2.05932			0.955151			2.24078			−1.10852			−1.96696			1.88269			−0.495845			−2.08769			2.28280
1.4	−2.04546			0.390167			2.18391			2.09456			−0.798071			−3.58087			−0.376182			−2.84777			−4.28434
1.5	−1.77331			−2.89119			1.14464			−1.31786			5.12698			0.246328			1.51682			5.35897			2.33697
1.6	−1.47172			1.59125			0.165955			−1.86156			−2.34187			3.15894			2.69920			−0.467918			2.73969
1.7	−0.915425			−1.24149			−1.16200			−3.77901			1.13649			4.09104			2.89457			−1.45869			3.45940
1.8	−0.804985			1.97139			−1.35200			0.346622			−1.58694			−0.578309			2.69831			0.886389			−0.279025
1.9	−0.524623			−0.651730			−1.72477			3.42748			0.341912			−0.428966			1.95410			−2.57525			−1.79813
1.10	0.0324255			−1.41775			−1.99895			−1.76803			−0.0459713			−4.03247			−0.129668			−0.989981			−0.0573293
1.11	0.0500860			2.23857			−1.99749			3.00481			0.112121			−4.70743			−0.200218			2.01118			0.150499
1.12	0.424427			−0.0535527			−1.81986			0.585915			−0.0227292			0.744359			−1.62125			−2.99713			0.248678
1.13	0.786123			−2.97481			−1.38201			−3.79835			−2.33857			−2.10502			−2.65868			5.84950			−2.98597
1.14	1.13611			1.45042			−0.709261			0.113386			1.64783			−2.08126			−3.07801			−0.896287			0.128819
1.15	1.35614			2.26012			−0.160874			−1.17320			3.06505			1.68827			−2.93046			2.10814			−1.59103
1.16	1.41272			−2.06299			−0.00422348			−0.901494			−2.91443			3.53459			−2.83141			1.25594			−1.27356
1.17	1.44594			−2.29423			0.0907318			2.88067			−3.31731			1.66213			−2.76068			2.26348			4.16527
1.18	2.00456			−0.706912			2.01827			3.68666			−1.41705			−3.05981			0.0366312			−2.50028			7.39015
1.19	2.01205			1.56790			2.04833			−3.92264			3.15469			−0.438032			0.0972422			−0.541682			−7.89253
1.20	2.32964			−2.94657			3.42724			−1.50497			−6.86446			−0.278735			3.32495			5.68228			−3.50605

Atkin-Lehner signs

\( p \)	Sign
\(13\)	\(-1\)
\(19\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3211.2.a.p	yes	21
13.b	even	2	1		3211.2.a.o	✓	21

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3211.2.a.o	✓	21	13.b	even	2	1
3211.2.a.p	yes	21	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^21 - T2^20 - 27*T2^19 + 28*T2^18 + 305*T2^17 - 328*T2^16 - 1871*T2^15 + 2086*T2^14 + 6767*T2^13 - 7819*T2^12 - 14660*T2^11 + 17553*T2^10 + 18490*T2^9 - 22969*T2^8 - 12598*T2^7 + 16352*T2^6 + 3976*T2^5 - 5595*T2^4 - 299*T2^3 + 668*T2^2 - 52*T2 + 1