Newform orbit 8649.2.a.bu

• Label: 8649.2.a.bu
• Level: $8649$
• Weight: $2$
• Character orbit: 8649.a
• Self dual: yes
• Analytic conductor: $69.063$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$8649 = 3^{2} \cdot 31^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8649.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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	−2.64601				0		5.00135			0.929477				0		−4.25556			−7.94161				0		−2.45940
1.2	−2.46988				0		4.10029			0.240205				0		−1.25177			−5.18746				0		−0.593276
1.3	−2.39335				0		3.72812			3.81642				0		3.08197			−4.13599				0		−9.13404
1.4	−2.22192				0		2.93691			−1.02426				0		0.246895			−2.08174				0		2.27581
1.5	−1.86657				0		1.48408			3.06179				0		3.20321			0.963008				0		−5.71504
1.6	−1.73783				0		1.02004			−3.85240				0		−2.44804			1.70300				0		6.69480
1.7	−1.62315				0		0.634610			−1.02432				0		1.99793			2.21623				0		1.66262
1.8	−1.21162				0		−0.531985			−1.58010				0		4.11574			3.06780				0		1.91448
1.9	−1.14647				0		−0.685607			−2.53076				0		−1.30842			3.07897				0		2.90144
1.10	−0.959557				0		−1.07925			4.21749				0		−1.12111			2.95472				0		−4.04692
1.11	−0.583104				0		−1.65999			−2.31932				0		−3.54758			2.13416				0		1.35240
1.12	−0.294138				0		−1.91348			1.97505				0		−3.94391			1.15111				0		−0.580938
1.13	−0.286378				0		−1.91799			3.36659				0		5.00540			1.12203				0		−0.964118
1.14	0.256648				0		−1.93413			−2.52738				0		4.03776			−1.00969				0		−0.648646
1.15	0.544587				0		−1.70342			−0.493609				0		−0.226241			−2.01684				0		−0.268813
1.16	0.849623				0		−1.27814			−0.881731				0		3.72139			−2.78518				0		−0.749139
1.17	1.14050				0		−0.699260			−2.66755				0		0.198186			−3.07851				0		−3.04235
1.18	1.75069				0		1.06491			1.92100				0		4.93164			−1.63706				0		3.36307
1.19	2.00987				0		2.03956			−3.19190				0		0.292061			0.0795053				0		−6.41528
1.20	2.33041				0		3.43080			4.15417				0		1.65165			3.33434				0		9.68090

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$31$	$+1$

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	8649.2.a.bu		24
3.b	odd	2	1		2883.2.a.v	yes	24
31.b	odd	2	1		8649.2.a.bv		24
93.c	even	2	1		2883.2.a.u	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2883.2.a.u	✓	24	93.c	even	2	1
2883.2.a.v	yes	24	3.b	odd	2	1
8649.2.a.bu		24	1.a	even	1	1	trivial
8649.2.a.bv		24	31.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8649))$:

T2^24 - 40*T2^22 - 8*T2^21 + 692*T2^20 + 272*T2^19 - 6756*T2^18 - 3904*T2^17 + 40704*T2^16 + 30728*T2^15 - 155288*T2^14 - 144160*T2^13 + 369446*T2^12 + 410304*T2^11 - 515744*T2^10 - 690968*T2^9 + 363918*T2^8 + 644608*T2^7 - 73876*T2^6 - 297968*T2^5 - 32688*T2^4 + 55400*T2^3 + 12048*T2^2 - 2512*T2 - 641

T5^24 - 80*T5^22 - 32*T5^21 + 2764*T5^20 + 2112*T5^19 - 53768*T5^18 - 58768*T5^17 + 644164*T5^16 + 899408*T5^15 - 4885952*T5^14 - 8289520*T5^13 + 23162884*T5^12 + 47392080*T5^11 - 64210016*T5^10 - 166759584*T5^9 + 82550895*T5^8 + 344051424*T5^7 + 18386984*T5^6 - 368377152*T5^5 - 155145924*T5^4 + 146224512*T5^3 + 101741776*T5^2 + 3712*T5 - 7106564

T7^24 - 16*T7^23 + 16*T7^22 + 1008*T7^21 - 4640*T7^20 - 20672*T7^19 + 172616*T7^18 + 58688*T7^17 - 2837346*T7^16 + 3578160*T7^15 + 23229648*T7^14 - 53575792*T7^13 - 89619088*T7^12 + 319244352*T7^11 + 122705528*T7^10 - 915892480*T7^9 + 49295137*T7^8 + 1336775584*T7^7 - 199537952*T7^6 - 907200512*T7^5 + 69281920*T7^4 + 164993024*T7^3 - 23672832*T7^2 - 6258688*T7 + 1052672

T11^24 + 16*T11^23 - 32*T11^22 - 1792*T11^21 - 5212*T11^20 + 76272*T11^19 + 423544*T11^18 - 1412256*T11^17 - 13759546*T11^16 + 3960064*T11^15 + 239360784*T11^14 + 295295552*T11^13 - 2340745696*T11^12 - 5533841728*T11^11 + 11996895536*T11^10 + 45885515360*T11^9 - 20795518468*T11^8 - 199052493440*T11^7 - 67229324672*T11^6 + 444046627840*T11^5 + 338862427904*T11^4 - 459663040512*T11^3 - 465492873216*T11^2 + 174287912960*T11 + 206235170816

T13^24 - 32*T13^23 + 336*T13^22 - 192*T13^21 - 21728*T13^20 + 132224*T13^19 + 241872*T13^18 - 4917824*T13^17 + 9690512*T13^16 + 69496576*T13^15 - 315739744*T13^14 - 244088320*T13^13 + 3733325960*T13^12 - 3904134016*T13^11 - 19916046272*T13^10 + 46423006144*T13^9 + 33753163680*T13^8 - 188431164416*T13^7 + 75972777760*T13^6 + 297177877504*T13^5 - 276493267328*T13^4 - 148228906240*T13^3 + 215756749056*T13^2 + 10919695744*T13 - 45247233016