Newform orbit 859.2.a.a

• Label: 859.2.a.a
• Level: $859$
• Weight: $2$
• Character orbit: 859.a
• Self dual: yes
• Analytic conductor: $6.859$
• Analytic rank: $1$
• Dimension: $29$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	859.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.85914953363\)
Analytic rank:	\(1\)
Dimension:	\(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 10 q^{2} - 5 q^{3} + 20 q^{4} - 21 q^{5} - 7 q^{6} - 4 q^{7} - 27 q^{8} + 6 q^{9}+O(q^{10}) \)

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	−2.70873			1.91476			5.33722			−0.429340			−5.18657			1.12770			−9.03963			0.666311			1.16297
1.2	−2.60614			−2.47100			4.79198			2.07711			6.43979			4.76921			−7.27629			3.10586			−5.41324
1.3	−2.47666			0.394056			4.13385			−4.25168			−0.975944			3.82292			−5.28481			−2.84472			10.5300
1.4	−2.46521			1.25093			4.07728			−1.48700			−3.08382			−3.12540			−5.12093			−1.43516			3.66578
1.5	−2.32625			−2.42993			3.41144			−3.46276			5.65263			−0.283353			−3.28335			2.90457			8.05525
1.6	−2.12947			0.0307143			2.53465			3.38617			−0.0654052			−2.08550			−1.13853			−2.99906			−7.21075
1.7	−2.09435			−1.09566			2.38632			1.17212			2.29471			1.30819			−0.809088			−1.79952			−2.45484
1.8	−1.70827			2.78031			0.918179			1.18533			−4.74952			−4.21175			1.84804			4.73013			−2.02486
1.9	−1.65205			1.45271			0.729267			−1.57713			−2.39995			3.60613			2.09931			−0.889630			2.60550
1.10	−1.27576			−1.47830			−0.372430			2.27281			1.88596			−1.24359			3.02666			−0.814639			−2.89957
1.11	−1.22496			−2.11286			−0.499470			−1.75341			2.58817			−3.38969			3.06175			1.46418			2.14786
1.12	−0.863271			2.75806			−1.25476			−3.50368			−2.38095			−1.04660			2.80974			4.60689			3.02463
1.13	−0.774360			−2.36833			−1.40037			−3.40626			1.83394			2.89898			2.63311			2.60898			2.63767
1.14	−0.667252			0.862309			−1.55478			0.422005			−0.575377			0.0862506			2.37193			−2.25642			−0.281584
1.15	−0.649866			0.0403459			−1.57767			−2.70354			−0.0262194			1.88913			2.32501			−2.99837			1.75694
1.16	−0.517624			1.04008			−1.73207			0.513586			−0.538370			−0.258507			1.93181			−1.91823			−0.265844
1.17	0.0443279			−2.92540			−1.99804			0.750926			−0.129677			0.948996			−0.177225			5.55795			0.0332870
1.18	0.465620			1.32235			−1.78320			1.11564			0.615713			−4.75610			−1.76153			−1.25139			0.519465
1.19	0.577415			−1.33230			−1.66659			3.34338			−0.769287			−0.0656289			−2.11714			−1.22499			1.93052
1.20	0.821805			1.84134			−1.32464			−0.836590			1.51322			−2.53001			−2.73220			0.390539			−0.687514

Atkin-Lehner signs

\( p \)	Sign
\(859\)	\(1\)

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	859.2.a.a	✓	29
3.b	odd	2	1		7731.2.a.c		29

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
859.2.a.a	✓	29	1.a	even	1	1	trivial
7731.2.a.c		29	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^29 + 10*T2^28 + 11*T2^27 - 201*T2^26 - 639*T2^25 + 1379*T2^24 + 8131*T2^23 - 1221*T2^22 - 52280*T2^21 - 37112*T2^20 + 197613*T2^19 + 255017*T2^18 - 454866*T2^17 - 855797*T2^16 + 609080*T2^15 + 1740283*T2^14 - 356217*T2^13 - 2269187*T2^12 - 181974*T2^11 + 1919504*T2^10 + 487957*T2^9 - 1037295*T2^8 - 377746*T2^7 + 342369*T2^6 + 147606*T2^5 - 62919*T2^4 - 28904*T2^3 + 5277*T2^2 + 2219*T2 - 106