Newform orbit 239.4.a.a

• Label: 239.4.a.a
• Level: $239$
• Weight: $4$
• Character orbit: 239.a
• Self dual: yes
• Analytic conductor: $14.101$
• Analytic rank: $1$
• Dimension: $22$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	239.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.1014564914\)
Analytic rank:	\(1\)
Dimension:	\(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{2} - 13 q^{3} + 50 q^{4} - 37 q^{5} - 42 q^{6} - 52 q^{7} - 69 q^{8} + 69 q^{9}+O(q^{10}) \)

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	−5.49818			−2.46853			22.2300			9.50920			13.5724			4.48068			−78.2390			−20.9064			−52.2833
1.2	−4.47711			2.26082			12.0445			13.5922			−10.1220			−3.12004			−18.1077			−21.8887			−60.8535
1.3	−4.43192			6.11724			11.6419			−11.7413			−27.1111			6.42321			−16.1407			10.4206			52.0364
1.4	−4.40109			−5.79161			11.3696			−3.94300			25.4894			−14.1222			−14.8300			6.54271			17.3535
1.5	−3.42794			9.47202			3.75080			−7.18364			−32.4695			−16.4073			14.5660			62.7191			24.6251
1.6	−3.24958			−6.78099			2.55975			−21.6437			22.0353			3.86879			17.6785			18.9818			70.3329
1.7	−2.38671			1.00725			−2.30360			−5.10344			−2.40401			12.4633			24.5917			−25.9854			12.1804
1.8	−1.51884			−9.16488			−5.69312			−1.30911			13.9200			−3.41938			20.7977			56.9950			1.98833
1.9	−1.49092			−3.24670			−5.77714			16.6313			4.84059			−17.5795			20.5407			−16.4589			−24.7960
1.10	−0.976149			−7.86917			−7.04713			−2.51110			7.68148			23.2375			14.6882			34.9238			2.45121
1.11	−0.850702			4.89751			−7.27631			12.0519			−4.16632			−14.5326			12.9956			−3.01442			−10.2525
1.12	−0.594431			6.68117			−7.64665			−5.61897			−3.97150			−8.56684			9.30086			17.6380			3.34009
1.13	0.530015			1.65632			−7.71908			−0.726105			0.877873			27.7214			−8.33135			−24.2566			−0.384847
1.14	1.83829			−7.48019			−4.62067			9.40555			−13.7508			19.7665			−23.2005			28.9532			17.2902
1.15	2.21535			1.44058			−3.09221			10.2488			3.19139			−27.8405			−24.5732			−24.9247			22.7048
1.16	2.22859			4.77446			−3.03338			−7.56659			10.6403			−1.62323			−24.5889			−4.20457			−16.8628
1.17	2.32829			6.80655			−2.57908			−14.1221			15.8476			−19.8005			−24.6311			19.3291			−32.8804
1.18	3.32790			−3.19896			3.07490			7.52905			−10.6458			−10.0131			−16.3902			−16.7667			25.0559
1.19	3.64923			−0.563189			5.31692			−17.2106			−2.05521			24.0056			−9.79121			−26.6828			−62.8056
1.20	3.67099			−2.66709			5.47619			0.939206			−9.79088			−4.16512			−9.26490			−19.8866			3.44782

Atkin-Lehner signs

\( p \)	Sign
\(239\)	\(-1\)

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	239.4.a.a	✓	22
3.b	odd	2	1		2151.4.a.a		22

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
239.4.a.a	✓	22	1.a	even	1	1	trivial
2151.4.a.a		22	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^22 + 4*T2^21 - 105*T2^20 - 397*T2^19 + 4675*T2^18 + 16564*T2^17 - 115426*T2^16 - 380134*T2^15 + 1732419*T2^14 + 5279488*T2^13 - 16254921*T2^12 - 46071157*T2^11 + 94010673*T2^10 + 254270000*T2^9 - 313902372*T2^8 - 869060736*T2^7 + 498323776*T2^6 + 1716476928*T2^5 - 64604352*T2^4 - 1612901760*T2^3 - 581139200*T2^2 + 331715584*T2 + 161051648