Newform orbit 401.2.a.b

• Label: 401.2.a.b
• Level: $401$
• Weight: $2$
• Character orbit: 401.a
• Self dual: yes
• Analytic conductor: $3.202$
• Analytic rank: $0$
• Dimension: $21$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	401.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.20200112105\)
Analytic rank:	\(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 3 q^{3} + 28 q^{4} + 3 q^{5} + 9 q^{6} + 24 q^{7} + 20 q^{9}+O(q^{10}) \)

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	−2.65486			−3.06708			5.04826			0.873516			8.14265			5.21071			−8.09269			6.40698			−2.31906
1.2	−2.63618			2.29386			4.94946			−0.321335			−6.04702			1.15338			−7.77532			2.26177			0.847098
1.3	−2.53392			−1.39170			4.42076			−2.76066			3.52647			−2.64239			−6.13403			−1.06316			6.99531
1.4	−2.01811			0.367197			2.07278			3.63025			−0.741045			4.97025			−0.146883			−2.86517			−7.32625
1.5	−1.84210			0.198794			1.39332			−4.07083			−0.366198			3.95639			1.11757			−2.96048			7.49886
1.6	−1.82046			−1.70632			1.31407			3.24938			3.10629			−1.73574			1.24872			−0.0884609			−5.91536
1.7	−1.47194			3.10691			0.166608			0.927390			−4.57319			2.05052			2.69864			6.65291			−1.36506
1.8	−0.904202			0.329206			−1.18242			−1.52249			−0.297669			−2.57929			2.87755			−2.89162			1.37664
1.9	−0.735155			−2.45199			−1.45955			−2.62157			1.80259			−2.65010			2.54330			3.01223			1.92726
1.10	−0.408416			2.03855			−1.83320			2.83987			−0.832575			0.0804913			1.56554			1.15567			−1.15985
1.11	0.0613701			−1.09364			−1.99623			1.73745			−0.0671166			2.50533			−0.245249			−1.80396			0.106627
1.12	0.0949446			2.62130			−1.99099			−2.91857			0.248878			4.78946			−0.378923			3.87121			−0.277102
1.13	0.810218			−2.74982			−1.34355			−3.18967			−2.22795			3.44937			−2.70900			4.56151			−2.58433
1.14	1.18685			2.58203			−0.591390			1.62486			3.06448			−1.13871			−3.07559			3.66690			1.92846
1.15	1.44326			1.51643			0.0829945			3.53957			2.18860			3.53188			−2.76673			−0.700443			5.10852
1.16	1.49056			−1.86572			0.221757			3.32711			−2.78096			2.03070			−2.65057			0.480908			4.95924
1.17	2.09378			0.864353			2.38392			−1.16166			1.80977			3.77012			0.803848			−2.25289			−2.43227
1.18	2.29447			1.59521			3.26458			1.06578			3.66016			−3.84486			2.90154			−0.455297			2.44539
1.19	2.38038			−2.05906			3.66620			1.26495			−4.90133			0.729050			3.96618			1.23972			3.01106
1.20	2.42563			2.67074			3.88369			−3.36559			6.47824			−0.400214			4.56914			4.13287			−8.16369

Atkin-Lehner signs

\( p \)	Sign
\(401\)	\(-1\)

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	401.2.a.b	✓	21
3.b	odd	2	1		3609.2.a.g		21
4.b	odd	2	1		6416.2.a.m		21

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
401.2.a.b	✓	21	1.a	even	1	1	trivial
3609.2.a.g		21	3.b	odd	2	1
6416.2.a.m		21	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^21 - 35*T2^19 + 521*T2^17 + 2*T2^16 - 4305*T2^15 - 51*T2^14 + 21617*T2^13 + 519*T2^12 - 67876*T2^11 - 2749*T2^10 + 132085*T2^9 + 8292*T2^8 - 152221*T2^7 - 14353*T2^6 + 93934*T2^5 + 12831*T2^4 - 24699*T2^3 - 4111*T2^2 + 1058*T2 - 44