Newform orbit 169.10.a.h

• Label: 169.10.a.h
• Level: $169$
• Weight: $10$
• Character orbit: 169.a
• Self dual: yes
• Analytic conductor: $87.041$
• Analytic rank: $0$
• Dimension: $27$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$87.0410563117$
Analytic rank:	$0$
Dimension:	$27$
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.

$\operatorname{Tr}(f)(q) =$ 27 * q + 65 * q^2 + q^3 + 7169 * q^4 + 3238 * q^5 + 8490 * q^6 + 17378 * q^7 + 54204 * q^8 + 191118 * q^9 + 11697 * q^10 + 164171 * q^11 - 181941 * q^12 - 77651 * q^14 + 614110 * q^15 + 3012565 * q^16 + 105667 * q^17 + 584849 * q^18 + 2140031 * q^19 + 3722382 * q^20 + 2953252 * q^21 + 3789362 * q^22 + 3586420 * q^23 + 6334338 * q^24 + 10278737 * q^25 - 4332980 * q^27 + 5293616 * q^28 - 4203616 * q^29 - 34788079 * q^30 + 18634294 * q^31 + 38778128 * q^32 + 15170356 * q^33 + 10546914 * q^34 + 16260796 * q^35 + 112071987 * q^36 + 7214770 * q^37 - 60128169 * q^38 - 46270051 * q^40 + 67544533 * q^41 - 31249012 * q^42 - 55585357 * q^43 + 246583161 * q^44 + 172236942 * q^45 + 168715272 * q^46 + 175569972 * q^47 - 48573948 * q^48 + 144680057 * q^49 + 3455750 * q^50 + 122191382 * q^51 + 105504478 * q^53 - 359394409 * q^54 - 2471426 * q^55 + 260765426 * q^56 - 307372146 * q^57 - 777873375 * q^58 + 267562255 * q^59 - 745460915 * q^60 - 371334848 * q^61 + 52175323 * q^62 + 431551952 * q^63 + 642079732 * q^64 + 973549503 * q^66 + 213848603 * q^67 - 278629116 * q^68 - 201174310 * q^69 - 122562066 * q^70 + 970709202 * q^71 + 773619646 * q^72 + 271733895 * q^73 - 1068711482 * q^74 + 1236853937 * q^75 + 2585634844 * q^76 - 244285066 * q^77 + 1052001020 * q^79 + 3293699123 * q^80 - 456846001 * q^81 - 3832425 * q^82 + 1491498691 * q^83 + 5153327787 * q^84 + 2713007958 * q^85 + 3421126005 * q^86 - 1622618562 * q^87 + 6574070508 * q^88 + 4616501291 * q^89 - 2822436128 * q^90 + 3684890562 * q^92 + 3324407374 * q^93 - 5582614303 * q^94 - 2892286998 * q^95 + 12902380548 * q^96 + 3944038157 * q^97 + 7822213576 * q^98 + 5866875443 * 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	−43.3737			−225.369			1369.27			1637.42			9775.06			−1956.78			−37183.1			31108.1			−71021.0
1.2	−36.8678			240.621			847.232			1970.86			−8871.14			9117.25			−12359.3			38215.3			−72661.1
1.3	−36.1368			103.192			793.865			926.370			−3729.02			4023.64			−10185.7			−9034.43			−33476.0
1.4	−35.7320			−157.443			764.774			−2363.03			5625.74			−3406.89			−9032.12			5105.19			84435.6
1.5	−35.3614			−118.041			738.428			−111.377			4174.09			884.005			−8006.80			−5749.36			3938.43
1.6	−30.3833			82.6182			411.145			−433.016			−2510.21			409.354			3064.31			−12857.2			13156.5
1.7	−22.4110			83.2634			−9.74613			−970.434			−1866.02			−11431.9			11692.9			−12750.2			21748.4
1.8	−14.8244			−139.228			−292.237			1387.80			2063.98			−2985.58			11922.3			−298.471			−20573.4
1.9	−12.3426			−225.156			−359.660			−2236.39			2779.01			11037.4			10758.6			31012.0			27602.8
1.10	−8.72715			168.263			−435.837			−932.577			−1468.46			−1292.90			8271.92			8629.37			8138.74
1.11	−6.13624			−213.333			−474.347			517.983			1309.07			2961.86			6052.46			25828.1			−3178.47
1.12	−4.82532			−93.8913			−488.716			1259.17			453.056			−5921.56			4828.78			−10867.4			−6075.91
1.13	−4.74730			109.518			−489.463			−1538.18			−519.914			10378.9			4754.24			−7688.85			7302.18
1.14	−2.76873			273.564			−504.334			1902.60			−757.424			4322.02			2813.96			55154.1			−5267.78
1.15	4.33003			54.1364			−493.251			2633.10			234.412			−3591.16			−4352.76			−16752.3			11401.4
1.16	11.1269			−140.546			−388.192			−992.121			−1563.84			10567.0			−10016.3			70.0974			−11039.2
1.17	13.4402			117.761			−331.361			−2291.59			1582.73			−5059.42			−11334.9			−5815.41			−30799.5
1.18	22.4400			166.502			−8.44752			2749.81			3736.30			4995.16			−11678.8			8039.89			61705.7
1.19	24.5615			−5.41917			91.2697			−546.263			−133.103			−9850.09			−10333.8			−19653.6			−13417.1
1.20	25.3541			−120.880			130.831			−1004.42			−3064.81			−3164.79			−9664.20			−5071.03			−25466.3

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	169.10.a.h	yes	27
13.b	even	2	1		169.10.a.g	✓	27

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
169.10.a.g	✓	27	13.b	even	2	1
169.10.a.h	yes	27	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^27 - 65*T2^26 - 8384*T2^25 + 596247*T2^24 + 29229748*T2^23 - 2361550222*T2^22 - 53996240413*T2^21 + 5294275132460*T2^20 + 54337090812364*T2^19 - 7403091602940400*T2^18 - 23932190605804256*T2^17 + 6717105149081750848*T2^16 - 5405663953923256448*T2^15 - 4000199111251115413632*T2^14 + 11388458142156894598656*T2^13 + 1556425044725397760652800*T2^12 - 5117458365383754934632448*T2^11 - 390044847791349130552025088*T2^10 + 911492179909119501984890880*T2^9 + 61137147234941880773704810496*T2^8 - 20830567640687925267420676096*T2^7 - 5526943039768087230852242604032*T2^6 - 11756060840323292757052148940800*T2^5 + 231072080682670183918161098702848*T2^4 + 1068147772728068766230127241592832*T2^3 - 1536346070886501735795496119697408*T2^2 - 15862784203370629825899923513540608*T2 - 22691985360121658314885689296027648