Newform orbit 619.2.a.a

• Label: 619.2.a.a
• Level: $619$
• Weight: $2$
• Character orbit: 619.a
• Self dual: yes
• Analytic conductor: $4.943$
• Analytic rank: $1$
• Dimension: $21$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	619.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.94273988512\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 9 q^{2} - 5 q^{3} + 15 q^{4} - 21 q^{5} - 6 q^{6} - 4 q^{7} - 21 q^{8} + 6 q^{9}+O(q^{10}) \)

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	−2.73280			−1.70616			5.46819			2.03444			4.66258			3.83859			−9.47786			−0.0890335			−5.55971
1.2	−2.61324			2.79918			4.82902			−2.04890			−7.31494			−3.26475			−7.39292			4.83543			5.35427
1.3	−2.53656			−1.36544			4.43411			−3.55564			3.46351			0.675306			−6.17426			−1.13558			9.01908
1.4	−2.32447			0.941827			3.40317			−2.38281			−2.18925			2.74500			−3.26162			−2.11296			5.53878
1.5	−2.08256			0.916211			2.33704			2.62373			−1.90806			−1.16947			−0.701915			−2.16056			−5.46408
1.6	−1.67412			−2.96683			0.802676			−4.06102			4.96682			1.39415			2.00446			5.80206			6.79864
1.7	−1.42511			−0.995739			0.0309400			2.82686			1.41904			−2.89800			2.80613			−2.00850			−4.02859
1.8	−1.37173			0.756788			−0.118367			−1.92672			−1.03811			−0.911742			2.90582			−2.42727			2.64293
1.9	−0.887456			−2.66181			−1.21242			−0.744827			2.36224			−0.702395			2.85088			4.08525			0.661001
1.10	−0.788010			1.45313			−1.37904			−3.12358			−1.14508			4.52894			2.66272			−0.888427			2.46141
1.11	−0.723996			3.01745			−1.47583			−2.02994			−2.18462			−2.86695			2.51649			6.10499			1.46966
1.12	−0.691014			−0.832025			−1.52250			1.28552			0.574940			1.22773			2.43410			−2.30773			−0.888311
1.13	0.152908			1.46557			−1.97662			0.468718			0.224098			−3.63648			−0.608058			−0.852102			0.0716709
1.14	0.273610			−2.79069			−1.92514			−0.828377			−0.763560			5.25882			−1.07396			4.78794			−0.226652
1.15	0.642946			1.06499			−1.58662			−0.849273			0.684732			−0.754832			−2.30600			−1.86579			−0.546037
1.16	1.15106			−1.36322			−0.675056			1.59267			−1.56915			−0.851998			−3.07916			−1.14163			1.83326
1.17	1.32477			0.433395			−0.244976			−1.11370			0.574150			−1.01156			−2.97408			−2.81217			−1.47540
1.18	1.53518			1.64763			0.356779			−4.31073			2.52940			−5.09714			−2.52264			−0.285328			−6.61776
1.19	1.61466			−2.35583			0.607120			0.981279			−3.80386			−0.0820085			−2.24902			2.54995			1.58443
1.20	1.75261			−0.558565			1.07166			−3.69018			−0.978948			2.79513			−1.62703			−2.68801			−6.46747

Atkin-Lehner signs

\( p \)	Sign
\(619\)	\(1\)

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	619.2.a.a	✓	21
3.b	odd	2	1		5571.2.a.e		21
4.b	odd	2	1		9904.2.a.j		21

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
619.2.a.a	✓	21	1.a	even	1	1	trivial
5571.2.a.e		21	3.b	odd	2	1
9904.2.a.j		21	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^21 + 9*T2^20 + 12*T2^19 - 116*T2^18 - 371*T2^17 + 385*T2^16 + 2789*T2^15 + 957*T2^14 - 9722*T2^13 - 9809*T2^12 + 16968*T2^11 + 27508*T2^10 - 12214*T2^9 - 37037*T2^8 - 2648*T2^7 + 24373*T2^6 + 8158*T2^5 - 6740*T2^4 - 3106*T2^3 + 651*T2^2 + 272*T2 - 43