Newform orbit 197.8.a.b

• Label: 197.8.a.b
• Level: $197$
• Weight: $8$
• Character orbit: 197.a
• Self dual: yes
• Analytic conductor: $61.540$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$197$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	197.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$61.5398500204$
Analytic rank:	$0$
Dimension:	$60$
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.

$\operatorname{Tr}(f)(q) =$ 60 * q + 16 * q^2 + 296 * q^3 + 4224 * q^4 + 554 * q^5 + 1200 * q^6 + 4959 * q^7 + 2571 * q^8 + 47384 * q^9 + 16237 * q^10 + 12452 * q^11 + 38656 * q^12 + 36460 * q^13 - 567 * q^14 + 55139 * q^15 + 319488 * q^16 + 13657 * q^17 + 75985 * q^18 + 179089 * q^19 + 78379 * q^20 + 71390 * q^21 + 220057 * q^22 + 159428 * q^23 + 288645 * q^24 + 1250470 * q^25 + 245262 * q^26 + 789647 * q^27 + 775366 * q^28 + 111693 * q^29 + 200635 * q^30 + 1035260 * q^31 + 688764 * q^32 + 535514 * q^33 + 1430866 * q^34 + 698672 * q^35 + 4315642 * q^36 + 1226630 * q^37 + 133058 * q^38 + 1107523 * q^39 + 2427846 * q^40 + 1766062 * q^41 + 693163 * q^42 + 3240035 * q^43 + 2056578 * q^44 + 1462797 * q^45 + 1088877 * q^46 + 3015185 * q^47 + 4531442 * q^48 + 8513327 * q^49 - 8803339 * q^50 - 1200965 * q^51 + 2403172 * q^52 - 2021580 * q^53 - 9268097 * q^54 + 2538479 * q^55 - 13776768 * q^56 - 638299 * q^57 + 3745679 * q^58 + 6845183 * q^59 + 5842749 * q^60 + 6328462 * q^61 + 7784379 * q^62 + 11502133 * q^63 + 28157685 * q^64 + 14369523 * q^65 + 25244790 * q^66 + 20888346 * q^67 + 18152427 * q^68 + 7874901 * q^69 + 23377047 * q^70 + 7085107 * q^71 + 50177539 * q^72 + 21946913 * q^73 + 32718747 * q^74 + 44992638 * q^75 + 52601230 * q^76 + 22159370 * q^77 + 86884893 * q^78 + 9667191 * q^79 + 50245998 * q^80 + 56936064 * q^81 + 55357649 * q^82 + 57432744 * q^83 + 65898868 * q^84 + 32071274 * q^85 + 91529136 * q^86 + 38121559 * q^87 + 95846511 * q^88 + 46839682 * q^89 + 136287359 * q^90 + 81150673 * q^91 + 99073310 * q^92 + 81689463 * q^93 + 55616387 * q^94 + 30995799 * q^95 + 156893033 * q^96 + 72697122 * q^97 + 69757306 * q^98 + 64628303 * 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	−21.9779			−46.0918			355.028			−508.529			1013.00			1271.38			−4989.61			−62.5423			11176.4
1.2	−21.3411			−55.6399			327.443			134.694			1187.42			−274.833			−4256.33			908.798			−2874.52
1.3	−21.1053			19.7965			317.434			534.839			−417.811			1641.16			−3998.06			−1795.10			−11288.0
1.4	−20.8533			−0.827001			306.860			−12.3329			17.2457			246.396			−3729.81			−2186.32			257.181
1.5	−20.7354			83.9180			301.958			−512.036			−1740.07			308.678			−3607.08			4855.22			10617.3
1.6	−20.1726			−28.7735			278.935			480.748			580.436			−1059.65			−3044.76			−1359.09			−9697.95
1.7	−18.5263			88.4348			215.222			377.477			−1638.36			879.714			−1615.90			5633.71			−6993.24
1.8	−17.1767			23.0763			167.038			−418.802			−396.373			419.875			−670.545			−1654.49			7193.63
1.9	−16.5276			69.6348			145.163			−93.2649			−1150.90			825.610			−283.663			2662.01			1541.45
1.10	−16.3073			22.4490			137.928			281.170			−366.082			−559.636			−161.904			−1683.04			−4585.12
1.11	−16.0510			−45.8139			129.635			−397.516			735.360			−164.041			−26.2505			−88.0854			6380.54
1.12	−15.5881			10.3776			114.990			315.670			−161.768			403.171			202.797			−2079.30			−4920.71
1.13	−15.5057			−44.3590			112.428			72.8541			687.820			−1088.37			241.454			−219.276			−1129.66
1.14	−14.9461			56.4076			95.3854			−196.598			−843.073			−1497.90			487.460			994.822			2938.38
1.15	−14.5260			−80.0909			83.0049			−103.281			1163.40			−773.736			653.600			4227.56			1500.26
1.16	−12.9041			78.9621			38.5156			−356.709			−1018.93			−513.898			1154.72			4048.01			4603.01
1.17	−11.4263			−79.0210			2.56123			−431.704			902.921			742.367			1433.31			4057.32			4932.80
1.18	−9.99164			15.9423			−28.1670			−216.554			−159.289			−103.542			1560.37			−1932.84			2163.73
1.19	−8.57214			−8.03142			−54.5184			3.16930			68.8464			1453.35			1564.57			−2122.50			−27.1677
1.20	−8.25431			−14.1669			−59.8663			11.5658			116.938			−664.362			1550.71			−1986.30			−95.4680

Atkin-Lehner signs

$p$	Sign
$197$	$+1$

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	197.8.a.b	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
197.8.a.b	✓	60	1.a	even	1	1	trivial