Newform orbit 538.8.a.b

• Label: 538.8.a.b
• Level: $538$
• Weight: $8$
• Character orbit: 538.a
• Self dual: yes
• Analytic conductor: $168.063$
• Analytic rank: $0$
• Dimension: $37$
• 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:	$37$
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.

$\operatorname{Tr}(f)(q) =$ 37 * q - 296 * q^2 + 80 * q^3 + 2368 * q^4 + 624 * q^5 - 640 * q^6 + 453 * q^7 - 18944 * q^8 + 26707 * q^9 - 4992 * q^10 + 12312 * q^11 + 5120 * q^12 - 2673 * q^13 - 3624 * q^14 + 38182 * q^15 + 151552 * q^16 + 21060 * q^17 - 213656 * q^18 + 98001 * q^19 + 39936 * q^20 - 39351 * q^21 - 98496 * q^22 + 207958 * q^23 - 40960 * q^24 + 465625 * q^25 + 21384 * q^26 + 236474 * q^27 + 28992 * q^28 + 93585 * q^29 - 305456 * q^30 - 111035 * q^31 - 1212416 * q^32 + 380259 * q^33 - 168480 * q^34 + 702435 * q^35 + 1709248 * q^36 + 124933 * q^37 - 784008 * q^38 - 118595 * q^39 - 319488 * q^40 - 50463 * q^41 + 314808 * q^42 + 879214 * q^43 + 787968 * q^44 + 1388591 * q^45 - 1663664 * q^46 + 2024677 * q^47 + 327680 * q^48 + 2996186 * q^49 - 3725000 * q^50 + 1572487 * q^51 - 171072 * q^52 + 4144443 * q^53 - 1891792 * q^54 - 799409 * q^55 - 231936 * q^56 + 948838 * q^57 - 748680 * q^58 + 3652112 * q^59 + 2443648 * q^60 - 2823169 * q^61 + 888280 * q^62 + 7731776 * q^63 + 9699328 * q^64 + 7421447 * q^65 - 3042072 * q^66 + 6612718 * q^67 + 1347840 * q^68 + 14375831 * q^69 - 5619480 * q^70 + 17942161 * q^71 - 13673984 * q^72 + 7096865 * q^73 - 999464 * q^74 + 18244751 * q^75 + 6272064 * q^76 + 20035604 * q^77 + 948760 * q^78 - 697521 * q^79 + 2555904 * q^80 + 24254129 * q^81 + 403704 * q^82 + 12083041 * q^83 - 2518464 * q^84 - 7886425 * q^85 - 7033712 * q^86 - 12543280 * q^87 - 6303744 * q^88 + 13232596 * q^89 - 11108728 * q^90 + 94454 * q^91 + 13309312 * q^92 + 9950813 * q^93 - 16197416 * q^94 - 7443768 * q^95 - 2621440 * q^96 - 21427311 * q^97 - 23969488 * q^98 - 31108161 * 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			−86.3900			64.0000			−4.83256			691.120			−1234.41			−512.000			5276.23			38.6605
1.2	−8.00000			−84.1077			64.0000			439.841			672.862			1264.47			−512.000			4887.11			−3518.73
1.3	−8.00000			−76.7774			64.0000			−436.364			614.219			−815.525			−512.000			3707.77			3490.91
1.4	−8.00000			−67.8298			64.0000			289.818			542.639			−87.4120			−512.000			2413.89			−2318.55
1.5	−8.00000			−66.6431			64.0000			−38.8553			533.145			952.805			−512.000			2254.30			310.843
1.6	−8.00000			−65.8915			64.0000			−67.1672			527.132			1448.62			−512.000			2154.68			537.337
1.7	−8.00000			−63.0323			64.0000			−242.543			504.259			−1045.70			−512.000			1786.08			1940.34
1.8	−8.00000			−53.5974			64.0000			422.327			428.779			1358.76			−512.000			685.680			−3378.61
1.9	−8.00000			−52.3112			64.0000			−409.466			418.490			440.021			−512.000			549.463			3275.73
1.10	−8.00000			−48.7222			64.0000			165.782			389.778			−1320.62			−512.000			186.857			−1326.26
1.11	−8.00000			−47.2845			64.0000			259.043			378.276			−285.965			−512.000			48.8225			−2072.34
1.12	−8.00000			−37.8586			64.0000			−407.058			302.869			521.075			−512.000			−753.728			3256.47
1.13	−8.00000			−25.6095			64.0000			−255.788			204.876			−1147.04			−512.000			−1531.16			2046.31
1.14	−8.00000			−24.3629			64.0000			−116.636			194.903			1267.01			−512.000			−1593.45			933.090
1.15	−8.00000			−18.4868			64.0000			117.318			147.895			−512.012			−512.000			−1845.24			−938.543
1.16	−8.00000			−2.40564			64.0000			307.971			19.2451			1505.23			−512.000			−2181.21			−2463.77
1.17	−8.00000			1.29627			64.0000			−505.522			−10.3702			316.327			−512.000			−2185.32			4044.18
1.18	−8.00000			1.56345			64.0000			206.112			−12.5076			−1184.30			−512.000			−2184.56			−1648.90
1.19	−8.00000			2.39663			64.0000			−162.235			−19.1730			677.883			−512.000			−2181.26			1297.88
1.20	−8.00000			3.68486			64.0000			−24.6615			−29.4789			748.149			−512.000			−2173.42			197.292

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.b	✓	37

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
538.8.a.b	✓	37	1.a	even	1	1	trivial