Newform orbit 538.8.a.c

• Label: 538.8.a.c
• Level: $538$
• Weight: $8$
• Character orbit: 538.a
• Self dual: yes
• Analytic conductor: $168.063$
• Analytic rank: $1$
• Dimension: $40$
• 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:	$1$
Dimension:	$40$
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.

$\operatorname{Tr}(f)(q) =$ 40 * q - 320 * q^2 - 109 * q^3 + 2560 * q^4 - 751 * q^5 + 872 * q^6 - 233 * q^7 - 20480 * q^8 + 29623 * q^9 + 6008 * q^10 - 12977 * q^11 - 6976 * q^12 - 4870 * q^13 + 1864 * q^14 - 9068 * q^15 + 163840 * q^16 - 37896 * q^17 - 236984 * q^18 + 36270 * q^19 - 48064 * q^20 - 39351 * q^21 + 103816 * q^22 - 254388 * q^23 + 55808 * q^24 + 621875 * q^25 + 38960 * q^26 - 314650 * q^27 - 14912 * q^28 - 77138 * q^29 + 72544 * q^30 - 170617 * q^31 - 1310720 * q^32 - 338481 * q^33 + 303168 * q^34 - 498065 * q^35 + 1895872 * q^36 - 432250 * q^37 - 290160 * q^38 - 237233 * q^39 + 384512 * q^40 - 739673 * q^41 + 314808 * q^42 - 949447 * q^43 - 830528 * q^44 - 889534 * q^45 + 2035104 * q^46 - 1712951 * q^47 - 446464 * q^48 + 5466815 * q^49 - 4975000 * q^50 - 1611137 * q^51 - 311680 * q^52 - 3150530 * q^53 + 2517200 * q^54 - 466659 * q^55 + 119296 * q^56 - 3495794 * q^57 + 617104 * q^58 - 4357669 * q^59 - 580352 * q^60 + 4667204 * q^61 + 1364936 * q^62 + 1730648 * q^63 + 10485760 * q^64 - 1915803 * q^65 + 2707848 * q^66 - 6921617 * q^67 - 2425344 * q^68 - 10298459 * q^69 + 3984520 * q^70 - 18615993 * q^71 - 15166976 * q^72 - 6531063 * q^73 + 3458000 * q^74 - 22172708 * q^75 + 2321280 * q^76 - 20594134 * q^77 + 1897864 * q^78 + 1641715 * q^79 - 3076096 * q^80 + 21032872 * q^81 + 5917384 * q^82 - 16336922 * q^83 - 2518464 * q^84 + 13126141 * q^85 + 7595576 * q^86 + 11928706 * q^87 + 6644224 * q^88 + 9041896 * q^89 + 7116272 * q^90 + 9867502 * q^91 - 16280832 * q^92 + 10660597 * q^93 + 13703608 * q^94 + 8580994 * q^95 + 3571712 * q^96 + 10979403 * q^97 - 43734520 * q^98 + 9909340 * 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			−88.0215			64.0000			−492.867			704.172			1216.54			−512.000			5560.79			3942.93
1.2	−8.00000			−86.9213			64.0000			−78.3854			695.370			51.4362			−512.000			5368.31			627.083
1.3	−8.00000			−84.1165			64.0000			317.613			672.932			132.828			−512.000			4888.58			−2540.91
1.4	−8.00000			−82.6709			64.0000			−228.141			661.368			−449.827			−512.000			4647.48			1825.13
1.5	−8.00000			−78.7674			64.0000			294.812			630.140			89.8790			−512.000			4017.31			−2358.49
1.6	−8.00000			−72.1201			64.0000			−232.395			576.961			1331.71			−512.000			3014.30			1859.16
1.7	−8.00000			−66.3983			64.0000			433.449			531.186			−1762.13			−512.000			2221.73			−3467.59
1.8	−8.00000			−61.6300			64.0000			68.6323			493.040			353.255			−512.000			1611.25			−549.058
1.9	−8.00000			−57.3727			64.0000			−154.183			458.982			−1126.39			−512.000			1104.63			1233.46
1.10	−8.00000			−54.3544			64.0000			−133.374			434.836			−588.564			−512.000			767.405			1066.99
1.11	−8.00000			−38.6224			64.0000			27.7308			308.980			1156.53			−512.000			−695.307			−221.846
1.12	−8.00000			−33.0172			64.0000			26.6415			264.137			631.905			−512.000			−1096.87			−213.132
1.13	−8.00000			−29.8214			64.0000			183.000			238.571			−278.933			−512.000			−1297.69			−1464.00
1.14	−8.00000			−29.1710			64.0000			−496.219			233.368			−305.036			−512.000			−1336.05			3969.75
1.15	−8.00000			−27.3504			64.0000			−426.693			218.803			−434.276			−512.000			−1438.96			3413.55
1.16	−8.00000			−25.6095			64.0000			548.719			204.876			−21.1122			−512.000			−1531.15			−4389.75
1.17	−8.00000			−25.5264			64.0000			439.995			204.211			−363.962			−512.000			−1535.40			−3519.96
1.18	−8.00000			−19.6652			64.0000			−242.251			157.322			−1441.33			−512.000			−1800.28			1938.01
1.19	−8.00000			−15.6593			64.0000			356.182			125.275			−732.429			−512.000			−1941.78			−2849.46
1.20	−8.00000			−14.7346			64.0000			−379.054			117.877			1370.51			−512.000			−1969.89			3032.43

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.c	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
538.8.a.c	✓	40	1.a	even	1	1	trivial