Newform orbit 538.8.a.d

• Label: 538.8.a.d
• Level: $538$
• Weight: $8$
• Character orbit: 538.a
• Self dual: yes
• Analytic conductor: $168.063$
• Analytic rank: $0$
• Dimension: $43$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$538 = 2 \cdot 269$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$168.063143710$
Analytic rank:	$0$
Dimension:	$43$
Twist minimal:	yes
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.

$\operatorname{Tr}(f)(q) =$ 43 * q + 344 * q^2 + 123 * q^3 + 2752 * q^4 + 1249 * q^5 + 984 * q^6 + 2292 * q^7 + 22016 * q^8 + 38034 * q^9 + 9992 * q^10 + 9236 * q^11 + 7872 * q^12 + 20734 * q^13 + 18336 * q^14 + 45412 * q^15 + 176128 * q^16 + 73650 * q^17 + 304272 * q^18 + 29172 * q^19 + 79936 * q^20 + 121999 * q^21 + 73888 * q^22 + 292888 * q^23 + 62976 * q^24 + 922452 * q^25 + 165872 * q^26 + 361146 * q^27 + 146688 * q^28 + 564477 * q^29 + 363296 * q^30 + 430638 * q^31 + 1409024 * q^32 + 443001 * q^33 + 589200 * q^34 + 933371 * q^35 + 2434176 * q^36 + 1358862 * q^37 + 233376 * q^38 + 1743349 * q^39 + 639488 * q^40 + 1869846 * q^41 + 975992 * q^42 + 1398712 * q^43 + 591104 * q^44 + 3495454 * q^45 + 2343104 * q^46 + 2157317 * q^47 + 503808 * q^48 + 7543237 * q^49 + 7379616 * q^50 + 230971 * q^51 + 1326976 * q^52 + 5925672 * q^53 + 2889168 * q^54 + 1107773 * q^55 + 1173504 * q^56 + 9117130 * q^57 + 4515816 * q^58 + 2165139 * q^59 + 2906368 * q^60 + 5047648 * q^61 + 3445104 * q^62 + 5691679 * q^63 + 11272192 * q^64 + 12361273 * q^65 + 3544008 * q^66 + 4415992 * q^67 + 4713600 * q^68 - 7538249 * q^69 + 7466968 * q^70 + 6241518 * q^71 + 19473408 * q^72 + 3473436 * q^73 + 10870896 * q^74 - 5305402 * q^75 + 1867008 * q^76 - 3658634 * q^77 + 13946792 * q^78 + 15661969 * q^79 + 5115904 * q^80 + 38608119 * q^81 + 14958768 * q^82 + 12126950 * q^83 + 7807936 * q^84 + 9453663 * q^85 + 11189696 * q^86 + 41966326 * q^87 + 4728832 * q^88 + 29273561 * q^89 + 27963632 * q^90 + 35476972 * q^91 + 18744832 * q^92 + 44382745 * q^93 + 17258536 * q^94 + 62134118 * q^95 + 4030464 * q^96 + 58118870 * q^97 + 60345896 * q^98 + 81437781 * q^99

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	8.00000			−86.9882			64.0000			−404.737			−695.905			−311.659			512.000			5379.94			−3237.89
1.2	8.00000			−85.3227			64.0000			−39.4502			−682.582			−1573.59			512.000			5092.96			−315.602
1.3	8.00000			−83.8072			64.0000			491.541			−670.458			−774.992			512.000			4836.65			3932.33
1.4	8.00000			−78.2289			64.0000			−435.660			−625.831			1442.05			512.000			3932.76			−3485.28
1.5	8.00000			−69.4454			64.0000			377.122			−555.563			−302.336			512.000			2635.66			3016.97
1.6	8.00000			−68.6733			64.0000			−92.2308			−549.387			−211.395			512.000			2529.03			−737.847
1.7	8.00000			−61.4104			64.0000			414.437			−491.283			1630.21			512.000			1584.24			3315.49
1.8	8.00000			−61.0758			64.0000			308.233			−488.606			−1086.67			512.000			1543.25			2465.87
1.9	8.00000			−56.8922			64.0000			−193.797			−455.137			485.218			512.000			1049.72			−1550.38
1.10	8.00000			−53.6923			64.0000			−214.599			−429.538			380.530			512.000			695.859			−1716.79
1.11	8.00000			−52.6428			64.0000			173.825			−421.143			701.621			512.000			584.267			1390.60
1.12	8.00000			−47.1657			64.0000			395.078			−377.326			1404.80			512.000			37.6026			3160.63
1.13	8.00000			−42.5136			64.0000			−385.072			−340.109			1083.70			512.000			−379.592			−3080.57
1.14	8.00000			−32.2549			64.0000			−211.678			−258.039			−876.962			512.000			−1146.62			−1693.43
1.15	8.00000			−30.6882			64.0000			83.2051			−245.506			52.0613			512.000			−1245.23			665.641
1.16	8.00000			−20.8703			64.0000			−401.676			−166.963			184.574			512.000			−1751.43			−3213.41
1.17	8.00000			−20.0835			64.0000			112.777			−160.668			−1635.26			512.000			−1783.65			902.217
1.18	8.00000			−16.9346			64.0000			−19.7537			−135.477			920.984			512.000			−1900.22			−158.030
1.19	8.00000			−15.5956			64.0000			406.109			−124.765			774.181			512.000			−1943.78			3248.87
1.20	8.00000			2.36922			64.0000			414.136			18.9538			−1162.79			512.000			−2181.39			3313.08

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$269$	$-1$

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	538.8.a.d	✓	43

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
538.8.a.d	✓	43	1.a	even	1	1	trivial