Newform orbit 169.8.a.h

• Label: 169.8.a.h
• Level: $169$
• Weight: $8$
• Character orbit: 169.a
• Self dual: yes
• Analytic conductor: $52.793$
• Analytic rank: $1$
• Dimension: $21$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$52.7930693068$
Analytic rank:	$1$
Dimension:	$21$
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.

$\operatorname{Tr}(f)(q) =$ 21 * q - 31 * q^2 - 26 * q^3 + 1409 * q^4 - 680 * q^5 - 1470 * q^6 - 2929 * q^7 - 4716 * q^8 + 15465 * q^9 - 5167 * q^10 - 14824 * q^11 + 21795 * q^12 - 179 * q^14 - 36398 * q^15 + 113205 * q^16 + 45016 * q^17 - 72967 * q^18 - 119316 * q^19 - 131130 * q^20 - 94178 * q^21 - 274174 * q^22 - 128105 * q^23 - 316902 * q^24 + 366537 * q^25 - 106643 * q^27 - 170352 * q^28 - 62329 * q^29 + 9761 * q^30 - 467367 * q^31 - 677464 * q^32 - 587690 * q^33 - 578614 * q^34 + 848860 * q^35 - 274245 * q^36 - 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 - 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 + 263025 * q^44 + 1938345 * q^45 + 3170392 * q^46 - 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 - 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 - 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 - 5590305 * q^57 - 7510959 * q^58 - 10746233 * q^59 - 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 - 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 - 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 - 28670666 * q^70 - 23267025 * q^71 - 49774322 * q^72 - 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 - 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 - 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 - 21455789 * q^83 - 60768429 * q^84 - 26216249 * q^85 - 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 - 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 - 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 - 66667956 * q^96 - 10916609 * q^97 - 41738840 * q^98 - 37605493 * 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	−21.6353			79.0768			340.087			120.676			−1710.85			−240.114			−4588.56			4066.15			−2610.86
1.2	−21.5931			28.3750			338.261			−473.706			−612.705			99.1966			−4540.19			−1381.86			10228.8
1.3	−19.6237			−72.4635			257.091			413.760			1422.00			1005.88			−2533.24			3063.96			−8119.51
1.4	−17.7683			58.1116			187.714			−15.5333			−1032.55			−1688.45			−1061.02			1189.96			276.002
1.5	−15.3948			−46.3939			108.998			−391.457			714.223			−1742.39			292.525			−34.6060			6026.39
1.6	−13.1620			−21.4559			45.2387			344.184			282.403			168.957			1089.31			−1726.64			−4530.16
1.7	−11.8381			−26.1495			12.1398			422.131			309.560			1215.99			1371.56			−1503.20			−4997.21
1.8	−10.5911			−6.51000			−15.8290			−397.099			68.9480			1187.35			1523.30			−2144.62			4205.71
1.9	−5.77104			82.1608			−94.6950			17.6329			−474.154			−888.558			1285.18			4563.40			−101.761
1.10	−3.19657			−57.0706			−117.782			−431.499			182.430			−225.139			785.660			1070.05			1379.32
1.11	−2.91821			44.2582			−119.484			−197.474			−129.155			303.075			722.210			−228.212			576.269
1.12	−1.25436			8.41428			−126.427			307.776			−10.5545			−83.6974			319.143			−2116.20			−386.063
1.13	2.90053			−90.5117			−119.587			−240.795			−262.532			−1391.07			−718.132			6005.38			−698.433
1.14	6.93821			−64.5145			−79.8613			33.5113			−447.615			−160.658			−1442.18			1975.12			232.508
1.15	8.58294			18.1259			−54.3331			37.6136			155.574			711.860			−1564.95			−1858.45			322.835
1.16	8.99747			87.5208			−47.0455			−190.033			787.466			−44.6495			−1574.97			5472.89			−1709.81
1.17	13.2906			38.7978			48.6390			274.097			515.644			−774.181			−1054.75			−681.732			3642.90
1.18	15.0231			−85.3884			97.6921			439.379			−1282.79			−536.518			−455.317			5104.17			6600.82
1.19	16.7260			−13.7721			151.758			−0.583029			−230.352			274.333			397.369			−1997.33			−9.75172
1.20	20.4423			−14.6497			289.887			−478.639			−299.473			1197.76			3309.33			−1972.39			−9784.47

Atkin-Lehner signs

$p$	Sign
$13$	$-1$

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	169.8.a.h	✓	21
13.b	even	2	1		169.8.a.i	yes	21
13.d	odd	4	2		169.8.b.f		42

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.8.a.h	✓	21	1.a	even	1	1	trivial
169.8.a.i	yes	21	13.b	even	2	1
169.8.b.f		42	13.d	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^21 + 31*T2^20 - 1568*T2^19 - 54321*T2^18 + 947492*T2^17 + 39220914*T2^16 - 264006077*T2^15 - 15146495220*T2^14 + 25361989564*T2^13 + 3402270784328*T2^12 + 3685037378032*T2^11 - 452991529756544*T2^10 - 1205808926314368*T2^9 + 34866523470570752*T2^8 + 124434751468174592*T2^7 - 1431482843960505344*T2^6 - 5958983572887843840*T2^5 + 25076671497098250240*T2^4 + 125069178493300281344*T2^3 - 54792691948098420736*T2^2 - 698740375987929153536*T2 - 618039895230250221568