Newform orbit 311.10.a.a

• Label: 311.10.a.a
• Level: $311$
• Weight: $10$
• Character orbit: 311.a
• Self dual: yes
• Analytic conductor: $160.176$
• Analytic rank: $1$
• Dimension: $107$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$311$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	311.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$160.176145047$
Analytic rank:	$1$
Dimension:	$107$
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.

$\operatorname{Tr}(f)(q) =$ 107 * q - 49 * q^2 - 251 * q^3 + 24831 * q^4 - 3013 * q^5 - 9786 * q^6 - 25917 * q^7 - 41559 * q^8 + 580870 * q^9 - 151606 * q^10 - 197170 * q^11 - 243835 * q^12 - 451437 * q^13 - 664654 * q^14 - 540384 * q^15 + 5314363 * q^16 - 1268740 * q^17 - 1292714 * q^18 - 3222276 * q^19 - 1472775 * q^20 - 4487636 * q^21 - 6900467 * q^22 - 2704408 * q^23 - 7619473 * q^24 + 25312898 * q^25 - 2942061 * q^26 - 5740538 * q^27 - 16637983 * q^28 - 24291882 * q^29 - 22137972 * q^30 - 23152134 * q^31 - 13028752 * q^32 - 27297788 * q^33 - 37858895 * q^34 - 10110514 * q^35 + 101047281 * q^36 - 66952912 * q^37 - 24523729 * q^38 - 80907118 * q^39 - 123537133 * q^40 - 84754470 * q^41 - 70742551 * q^42 - 111198356 * q^43 - 118745639 * q^44 - 80591785 * q^45 - 275634048 * q^46 - 5158461 * q^47 - 146601472 * q^48 + 267032978 * q^49 - 164158956 * q^50 - 320124042 * q^51 - 260572449 * q^52 - 100296737 * q^53 - 329649961 * q^54 - 157018700 * q^55 - 430736674 * q^56 - 474627972 * q^57 - 433448197 * q^58 - 289696574 * q^59 - 206583652 * q^60 - 972343166 * q^61 - 444379925 * q^62 - 683073607 * q^63 + 1020968335 * q^64 - 916880382 * q^65 - 289795172 * q^66 - 577727085 * q^67 - 876550148 * q^68 - 1343007998 * q^69 - 282939266 * q^70 - 393510288 * q^71 - 1257758867 * q^72 - 1745152893 * q^73 - 686935445 * q^74 - 350128857 * q^75 - 2315619458 * q^76 - 864631772 * q^77 - 3282577646 * q^78 - 3586360439 * q^79 - 7415839593 * q^80 + 348647211 * q^81 - 3506706527 * q^82 - 1490124749 * q^83 - 7915972739 * q^84 - 3425697804 * q^85 - 638412534 * q^86 - 1329249884 * q^87 - 5747413017 * q^88 - 2648208151 * q^89 - 190576650 * q^90 - 3490922250 * q^91 + 1424892446 * q^92 - 398488276 * q^93 + 1989324213 * q^94 - 737433392 * q^95 + 2357657014 * q^96 - 3661632036 * q^97 + 13671555315 * q^98 - 1262875708 * 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	−44.5278			−55.2296			1470.73			−690.991			2459.25			2049.63			−42690.2			−16632.7			30768.3
1.2	−43.5905			−219.543			1388.13			−897.333			9570.00			5057.90			−38191.1			28516.2			39115.2
1.3	−42.6069			242.746			1303.35			−1603.97			−10342.7			365.343			−33716.8			39242.6			68340.3
1.4	−42.0612			−170.126			1257.14			1585.05			7155.68			−5967.85			−31341.6			9259.73			−66669.1
1.5	−41.7690			−46.4893			1232.65			2237.07			1941.81			2415.31			−30101.0			−17521.7			−93440.2
1.6	−41.4202			−106.385			1203.63			1625.64			4406.47			1796.44			−28647.5			−8365.31			−67334.4
1.7	−40.4398			162.289			1123.38			1517.99			−6562.95			4627.84			−24723.9			6654.85			−61387.1
1.8	−39.9322			−52.4632			1082.58			−2352.44			2094.97			−1232.89			−22784.5			−16930.6			93937.9
1.9	−39.5238			168.701			1050.13			−936.297			−6667.70			6534.76			−21269.0			8776.95			37006.0
1.10	−38.2934			165.104			954.387			−2210.36			−6322.40			−8697.27			−16940.5			7576.34			84642.1
1.11	−38.2239			−174.737			949.069			−1891.58			6679.13			62.6736			−16706.5			10849.9			72303.5
1.12	−38.1590			42.8105			944.110			1930.82			−1633.61			8892.78			−16488.9			−17850.3			−73678.1
1.13	−36.7147			268.893			835.971			−443.367			−9872.32			−5286.21			−11894.5			52620.3			16278.1
1.14	−35.6137			−35.0200			756.334			−202.742			1247.19			−1713.33			−8701.64			−18456.6			7220.39
1.15	−34.1914			261.106			657.055			2198.16			−8927.59			−6416.42			−4959.63			48493.3			−75158.1
1.16	−34.1654			24.5007			655.276			−156.610			−837.077			7082.52			−4895.07			−19082.7			5350.65
1.17	−33.8875			12.6214			636.365			2326.34			−427.707			−8895.77			−4214.41			−19523.7			−78833.8
1.18	−33.6420			170.794			619.781			272.801			−5745.83			3002.29			−3625.96			9487.49			−9177.55
1.19	−33.4737			−43.8242			608.486			−757.345			1466.96			−6617.76			−3229.75			−17762.4			25351.1
1.20	−33.2237			−183.672			591.816			1174.29			6102.26			−10870.6			−2651.77			14052.3			−39014.3

Atkin-Lehner signs

$p$	Sign
$311$	$+1$

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	311.10.a.a	✓	107

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
311.10.a.a	✓	107	1.a	even	1	1	trivial