# Properties

 Label 2496.4.a.u.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$147.268767374$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{43})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 43$$ x^2 - 43 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-6.55744$$ of defining polynomial Character $$\chi$$ $$=$$ 2496.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -19.1149 q^{5} +35.1149 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -19.1149 q^{5} +35.1149 q^{7} +9.00000 q^{9} -26.0000 q^{11} +13.0000 q^{13} +57.3446 q^{15} -36.2298 q^{17} +95.5744 q^{19} -105.345 q^{21} -161.379 q^{23} +240.379 q^{25} -27.0000 q^{27} +91.3785 q^{29} -266.723 q^{31} +78.0000 q^{33} -671.217 q^{35} +149.608 q^{37} -39.0000 q^{39} -77.8041 q^{41} -183.608 q^{43} -172.034 q^{45} -60.6893 q^{47} +890.055 q^{49} +108.689 q^{51} -281.540 q^{53} +496.987 q^{55} -286.723 q^{57} +542.527 q^{59} -65.0810 q^{61} +316.034 q^{63} -248.493 q^{65} +1033.94 q^{67} +484.136 q^{69} +1041.81 q^{71} +483.311 q^{73} -721.136 q^{75} -912.987 q^{77} -1337.05 q^{79} +81.0000 q^{81} -812.825 q^{83} +692.527 q^{85} -274.136 q^{87} +936.885 q^{89} +456.493 q^{91} +800.169 q^{93} -1826.89 q^{95} -954.068 q^{97} -234.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 12 q^{5} + 44 q^{7} + 18 q^{9} - 52 q^{11} + 26 q^{13} + 36 q^{15} - 20 q^{17} + 60 q^{19} - 132 q^{21} - 8 q^{23} + 166 q^{25} - 54 q^{27} - 132 q^{29} - 140 q^{31} + 156 q^{33} - 608 q^{35} - 68 q^{37} - 78 q^{39} + 28 q^{41} - 108 q^{45} + 36 q^{47} + 626 q^{49} + 60 q^{51} - 668 q^{53} + 312 q^{55} - 180 q^{57} + 508 q^{59} - 340 q^{61} + 396 q^{63} - 156 q^{65} + 940 q^{67} + 24 q^{69} + 300 q^{71} + 1124 q^{73} - 498 q^{75} - 1144 q^{77} - 1520 q^{79} + 162 q^{81} - 524 q^{83} + 808 q^{85} + 396 q^{87} + 1900 q^{89} + 572 q^{91} + 420 q^{93} - 2080 q^{95} - 1436 q^{97} - 468 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 12 * q^5 + 44 * q^7 + 18 * q^9 - 52 * q^11 + 26 * q^13 + 36 * q^15 - 20 * q^17 + 60 * q^19 - 132 * q^21 - 8 * q^23 + 166 * q^25 - 54 * q^27 - 132 * q^29 - 140 * q^31 + 156 * q^33 - 608 * q^35 - 68 * q^37 - 78 * q^39 + 28 * q^41 - 108 * q^45 + 36 * q^47 + 626 * q^49 + 60 * q^51 - 668 * q^53 + 312 * q^55 - 180 * q^57 + 508 * q^59 - 340 * q^61 + 396 * q^63 - 156 * q^65 + 940 * q^67 + 24 * q^69 + 300 * q^71 + 1124 * q^73 - 498 * q^75 - 1144 * q^77 - 1520 * q^79 + 162 * q^81 - 524 * q^83 + 808 * q^85 + 396 * q^87 + 1900 * q^89 + 572 * q^91 + 420 * q^93 - 2080 * q^95 - 1436 * q^97 - 468 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −19.1149 −1.70969 −0.854843 0.518886i $$-0.826347\pi$$
−0.854843 + 0.518886i $$0.826347\pi$$
$$6$$ 0 0
$$7$$ 35.1149 1.89603 0.948013 0.318233i $$-0.103089\pi$$
0.948013 + 0.318233i $$0.103089\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −26.0000 −0.712663 −0.356332 0.934360i $$-0.615973\pi$$
−0.356332 + 0.934360i $$0.615973\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ 57.3446 0.987088
$$16$$ 0 0
$$17$$ −36.2298 −0.516883 −0.258441 0.966027i $$-0.583209\pi$$
−0.258441 + 0.966027i $$0.583209\pi$$
$$18$$ 0 0
$$19$$ 95.5744 1.15401 0.577007 0.816739i $$-0.304221\pi$$
0.577007 + 0.816739i $$0.304221\pi$$
$$20$$ 0 0
$$21$$ −105.345 −1.09467
$$22$$ 0 0
$$23$$ −161.379 −1.46303 −0.731516 0.681824i $$-0.761187\pi$$
−0.731516 + 0.681824i $$0.761187\pi$$
$$24$$ 0 0
$$25$$ 240.379 1.92303
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 91.3785 0.585123 0.292561 0.956247i $$-0.405492\pi$$
0.292561 + 0.956247i $$0.405492\pi$$
$$30$$ 0 0
$$31$$ −266.723 −1.54532 −0.772660 0.634821i $$-0.781074\pi$$
−0.772660 + 0.634821i $$0.781074\pi$$
$$32$$ 0 0
$$33$$ 78.0000 0.411456
$$34$$ 0 0
$$35$$ −671.217 −3.24161
$$36$$ 0 0
$$37$$ 149.608 0.664742 0.332371 0.943149i $$-0.392151\pi$$
0.332371 + 0.943149i $$0.392151\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ −77.8041 −0.296365 −0.148183 0.988960i $$-0.547342\pi$$
−0.148183 + 0.988960i $$0.547342\pi$$
$$42$$ 0 0
$$43$$ −183.608 −0.651163 −0.325581 0.945514i $$-0.605560\pi$$
−0.325581 + 0.945514i $$0.605560\pi$$
$$44$$ 0 0
$$45$$ −172.034 −0.569896
$$46$$ 0 0
$$47$$ −60.6893 −0.188350 −0.0941749 0.995556i $$-0.530021\pi$$
−0.0941749 + 0.995556i $$0.530021\pi$$
$$48$$ 0 0
$$49$$ 890.055 2.59491
$$50$$ 0 0
$$51$$ 108.689 0.298422
$$52$$ 0 0
$$53$$ −281.540 −0.729671 −0.364835 0.931072i $$-0.618875\pi$$
−0.364835 + 0.931072i $$0.618875\pi$$
$$54$$ 0 0
$$55$$ 496.987 1.21843
$$56$$ 0 0
$$57$$ −286.723 −0.666270
$$58$$ 0 0
$$59$$ 542.527 1.19714 0.598568 0.801072i $$-0.295737\pi$$
0.598568 + 0.801072i $$0.295737\pi$$
$$60$$ 0 0
$$61$$ −65.0810 −0.136603 −0.0683014 0.997665i $$-0.521758\pi$$
−0.0683014 + 0.997665i $$0.521758\pi$$
$$62$$ 0 0
$$63$$ 316.034 0.632008
$$64$$ 0 0
$$65$$ −248.493 −0.474182
$$66$$ 0 0
$$67$$ 1033.94 1.88531 0.942656 0.333767i $$-0.108320\pi$$
0.942656 + 0.333767i $$0.108320\pi$$
$$68$$ 0 0
$$69$$ 484.136 0.844682
$$70$$ 0 0
$$71$$ 1041.81 1.74141 0.870706 0.491803i $$-0.163662\pi$$
0.870706 + 0.491803i $$0.163662\pi$$
$$72$$ 0 0
$$73$$ 483.311 0.774894 0.387447 0.921892i $$-0.373357\pi$$
0.387447 + 0.921892i $$0.373357\pi$$
$$74$$ 0 0
$$75$$ −721.136 −1.11026
$$76$$ 0 0
$$77$$ −912.987 −1.35123
$$78$$ 0 0
$$79$$ −1337.05 −1.90418 −0.952091 0.305815i $$-0.901071\pi$$
−0.952091 + 0.305815i $$0.901071\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −812.825 −1.07493 −0.537465 0.843286i $$-0.680618\pi$$
−0.537465 + 0.843286i $$0.680618\pi$$
$$84$$ 0 0
$$85$$ 692.527 0.883707
$$86$$ 0 0
$$87$$ −274.136 −0.337821
$$88$$ 0 0
$$89$$ 936.885 1.11584 0.557919 0.829895i $$-0.311600\pi$$
0.557919 + 0.829895i $$0.311600\pi$$
$$90$$ 0 0
$$91$$ 456.493 0.525863
$$92$$ 0 0
$$93$$ 800.169 0.892190
$$94$$ 0 0
$$95$$ −1826.89 −1.97300
$$96$$ 0 0
$$97$$ −954.068 −0.998669 −0.499335 0.866409i $$-0.666422\pi$$
−0.499335 + 0.866409i $$0.666422\pi$$
$$98$$ 0 0
$$99$$ −234.000 −0.237554
$$100$$ 0 0
$$101$$ −1401.15 −1.38039 −0.690196 0.723623i $$-0.742476\pi$$
−0.690196 + 0.723623i $$0.742476\pi$$
$$102$$ 0 0
$$103$$ 954.960 0.913544 0.456772 0.889584i $$-0.349005\pi$$
0.456772 + 0.889584i $$0.349005\pi$$
$$104$$ 0 0
$$105$$ 2013.65 1.87154
$$106$$ 0 0
$$107$$ 924.595 0.835364 0.417682 0.908593i $$-0.362843\pi$$
0.417682 + 0.908593i $$0.362843\pi$$
$$108$$ 0 0
$$109$$ −172.893 −0.151928 −0.0759638 0.997111i $$-0.524203\pi$$
−0.0759638 + 0.997111i $$0.524203\pi$$
$$110$$ 0 0
$$111$$ −448.825 −0.383789
$$112$$ 0 0
$$113$$ 1241.81 1.03380 0.516902 0.856045i $$-0.327085\pi$$
0.516902 + 0.856045i $$0.327085\pi$$
$$114$$ 0 0
$$115$$ 3084.73 2.50133
$$116$$ 0 0
$$117$$ 117.000 0.0924500
$$118$$ 0 0
$$119$$ −1272.20 −0.980023
$$120$$ 0 0
$$121$$ −655.000 −0.492111
$$122$$ 0 0
$$123$$ 233.412 0.171106
$$124$$ 0 0
$$125$$ −2205.45 −1.57809
$$126$$ 0 0
$$127$$ −1079.93 −0.754555 −0.377278 0.926100i $$-0.623140\pi$$
−0.377278 + 0.926100i $$0.623140\pi$$
$$128$$ 0 0
$$129$$ 550.825 0.375949
$$130$$ 0 0
$$131$$ 918.026 0.612277 0.306139 0.951987i $$-0.400963\pi$$
0.306139 + 0.951987i $$0.400963\pi$$
$$132$$ 0 0
$$133$$ 3356.08 2.18804
$$134$$ 0 0
$$135$$ 516.102 0.329029
$$136$$ 0 0
$$137$$ 1311.06 0.817603 0.408801 0.912623i $$-0.365947\pi$$
0.408801 + 0.912623i $$0.365947\pi$$
$$138$$ 0 0
$$139$$ 227.593 0.138879 0.0694396 0.997586i $$-0.477879\pi$$
0.0694396 + 0.997586i $$0.477879\pi$$
$$140$$ 0 0
$$141$$ 182.068 0.108744
$$142$$ 0 0
$$143$$ −338.000 −0.197657
$$144$$ 0 0
$$145$$ −1746.69 −1.00038
$$146$$ 0 0
$$147$$ −2670.16 −1.49817
$$148$$ 0 0
$$149$$ −1325.48 −0.728776 −0.364388 0.931247i $$-0.618722\pi$$
−0.364388 + 0.931247i $$0.618722\pi$$
$$150$$ 0 0
$$151$$ 252.426 0.136040 0.0680202 0.997684i $$-0.478332\pi$$
0.0680202 + 0.997684i $$0.478332\pi$$
$$152$$ 0 0
$$153$$ −326.068 −0.172294
$$154$$ 0 0
$$155$$ 5098.38 2.64201
$$156$$ 0 0
$$157$$ 2268.62 1.15322 0.576611 0.817019i $$-0.304375\pi$$
0.576611 + 0.817019i $$0.304375\pi$$
$$158$$ 0 0
$$159$$ 844.621 0.421276
$$160$$ 0 0
$$161$$ −5666.79 −2.77395
$$162$$ 0 0
$$163$$ 1083.45 0.520630 0.260315 0.965524i $$-0.416174\pi$$
0.260315 + 0.965524i $$0.416174\pi$$
$$164$$ 0 0
$$165$$ −1490.96 −0.703461
$$166$$ 0 0
$$167$$ −796.041 −0.368859 −0.184430 0.982846i $$-0.559044\pi$$
−0.184430 + 0.982846i $$0.559044\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 860.169 0.384671
$$172$$ 0 0
$$173$$ 1.09598 0.000481654 0 0.000240827 1.00000i $$-0.499923\pi$$
0.000240827 1.00000i $$0.499923\pi$$
$$174$$ 0 0
$$175$$ 8440.86 3.64611
$$176$$ 0 0
$$177$$ −1627.58 −0.691167
$$178$$ 0 0
$$179$$ −3250.06 −1.35710 −0.678549 0.734555i $$-0.737391\pi$$
−0.678549 + 0.734555i $$0.737391\pi$$
$$180$$ 0 0
$$181$$ −2980.84 −1.22411 −0.612055 0.790815i $$-0.709657\pi$$
−0.612055 + 0.790815i $$0.709657\pi$$
$$182$$ 0 0
$$183$$ 195.243 0.0788676
$$184$$ 0 0
$$185$$ −2859.74 −1.13650
$$186$$ 0 0
$$187$$ 941.974 0.368363
$$188$$ 0 0
$$189$$ −948.102 −0.364890
$$190$$ 0 0
$$191$$ 612.595 0.232072 0.116036 0.993245i $$-0.462981\pi$$
0.116036 + 0.993245i $$0.462981\pi$$
$$192$$ 0 0
$$193$$ −4185.81 −1.56115 −0.780573 0.625064i $$-0.785073\pi$$
−0.780573 + 0.625064i $$0.785073\pi$$
$$194$$ 0 0
$$195$$ 745.480 0.273769
$$196$$ 0 0
$$197$$ −2.06029 −0.000745124 0 −0.000372562 1.00000i $$-0.500119\pi$$
−0.000372562 1.00000i $$0.500119\pi$$
$$198$$ 0 0
$$199$$ 1492.39 0.531622 0.265811 0.964025i $$-0.414360\pi$$
0.265811 + 0.964025i $$0.414360\pi$$
$$200$$ 0 0
$$201$$ −3101.82 −1.08848
$$202$$ 0 0
$$203$$ 3208.75 1.10941
$$204$$ 0 0
$$205$$ 1487.22 0.506691
$$206$$ 0 0
$$207$$ −1452.41 −0.487678
$$208$$ 0 0
$$209$$ −2484.93 −0.822423
$$210$$ 0 0
$$211$$ −2748.27 −0.896677 −0.448338 0.893864i $$-0.647984\pi$$
−0.448338 + 0.893864i $$0.647984\pi$$
$$212$$ 0 0
$$213$$ −3125.43 −1.00541
$$214$$ 0 0
$$215$$ 3509.65 1.11328
$$216$$ 0 0
$$217$$ −9365.95 −2.92996
$$218$$ 0 0
$$219$$ −1449.93 −0.447385
$$220$$ 0 0
$$221$$ −470.987 −0.143357
$$222$$ 0 0
$$223$$ 186.655 0.0560510 0.0280255 0.999607i $$-0.491078\pi$$
0.0280255 + 0.999607i $$0.491078\pi$$
$$224$$ 0 0
$$225$$ 2163.41 0.641009
$$226$$ 0 0
$$227$$ −3932.57 −1.14984 −0.574920 0.818210i $$-0.694967\pi$$
−0.574920 + 0.818210i $$0.694967\pi$$
$$228$$ 0 0
$$229$$ −5951.00 −1.71726 −0.858632 0.512593i $$-0.828685\pi$$
−0.858632 + 0.512593i $$0.828685\pi$$
$$230$$ 0 0
$$231$$ 2738.96 0.780131
$$232$$ 0 0
$$233$$ 6548.94 1.84135 0.920676 0.390327i $$-0.127638\pi$$
0.920676 + 0.390327i $$0.127638\pi$$
$$234$$ 0 0
$$235$$ 1160.07 0.322019
$$236$$ 0 0
$$237$$ 4011.16 1.09938
$$238$$ 0 0
$$239$$ −3462.35 −0.937076 −0.468538 0.883443i $$-0.655219\pi$$
−0.468538 + 0.883443i $$0.655219\pi$$
$$240$$ 0 0
$$241$$ 4581.96 1.22469 0.612345 0.790591i $$-0.290226\pi$$
0.612345 + 0.790591i $$0.290226\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −17013.3 −4.43649
$$246$$ 0 0
$$247$$ 1242.47 0.320066
$$248$$ 0 0
$$249$$ 2438.47 0.620611
$$250$$ 0 0
$$251$$ −4746.76 −1.19368 −0.596838 0.802362i $$-0.703576\pi$$
−0.596838 + 0.802362i $$0.703576\pi$$
$$252$$ 0 0
$$253$$ 4195.84 1.04265
$$254$$ 0 0
$$255$$ −2077.58 −0.510209
$$256$$ 0 0
$$257$$ −3373.64 −0.818840 −0.409420 0.912346i $$-0.634269\pi$$
−0.409420 + 0.912346i $$0.634269\pi$$
$$258$$ 0 0
$$259$$ 5253.48 1.26037
$$260$$ 0 0
$$261$$ 822.407 0.195041
$$262$$ 0 0
$$263$$ 3527.19 0.826981 0.413491 0.910508i $$-0.364309\pi$$
0.413491 + 0.910508i $$0.364309\pi$$
$$264$$ 0 0
$$265$$ 5381.61 1.24751
$$266$$ 0 0
$$267$$ −2810.66 −0.644230
$$268$$ 0 0
$$269$$ −1144.30 −0.259365 −0.129682 0.991556i $$-0.541396\pi$$
−0.129682 + 0.991556i $$0.541396\pi$$
$$270$$ 0 0
$$271$$ 2254.25 0.505298 0.252649 0.967558i $$-0.418698\pi$$
0.252649 + 0.967558i $$0.418698\pi$$
$$272$$ 0 0
$$273$$ −1369.48 −0.303607
$$274$$ 0 0
$$275$$ −6249.84 −1.37047
$$276$$ 0 0
$$277$$ −7101.62 −1.54042 −0.770208 0.637793i $$-0.779848\pi$$
−0.770208 + 0.637793i $$0.779848\pi$$
$$278$$ 0 0
$$279$$ −2400.51 −0.515106
$$280$$ 0 0
$$281$$ −3831.81 −0.813475 −0.406738 0.913545i $$-0.633334\pi$$
−0.406738 + 0.913545i $$0.633334\pi$$
$$282$$ 0 0
$$283$$ 3032.56 0.636985 0.318493 0.947925i $$-0.396823\pi$$
0.318493 + 0.947925i $$0.396823\pi$$
$$284$$ 0 0
$$285$$ 5480.68 1.13911
$$286$$ 0 0
$$287$$ −2732.08 −0.561915
$$288$$ 0 0
$$289$$ −3600.40 −0.732832
$$290$$ 0 0
$$291$$ 2862.20 0.576582
$$292$$ 0 0
$$293$$ 2296.44 0.457882 0.228941 0.973440i $$-0.426474\pi$$
0.228941 + 0.973440i $$0.426474\pi$$
$$294$$ 0 0
$$295$$ −10370.3 −2.04673
$$296$$ 0 0
$$297$$ 702.000 0.137152
$$298$$ 0 0
$$299$$ −2097.92 −0.405772
$$300$$ 0 0
$$301$$ −6447.38 −1.23462
$$302$$ 0 0
$$303$$ 4203.45 0.796969
$$304$$ 0 0
$$305$$ 1244.01 0.233548
$$306$$ 0 0
$$307$$ −5174.26 −0.961925 −0.480962 0.876741i $$-0.659713\pi$$
−0.480962 + 0.876741i $$0.659713\pi$$
$$308$$ 0 0
$$309$$ −2864.88 −0.527435
$$310$$ 0 0
$$311$$ −7481.54 −1.36411 −0.682057 0.731299i $$-0.738914\pi$$
−0.682057 + 0.731299i $$0.738914\pi$$
$$312$$ 0 0
$$313$$ −4158.97 −0.751051 −0.375526 0.926812i $$-0.622538\pi$$
−0.375526 + 0.926812i $$0.622538\pi$$
$$314$$ 0 0
$$315$$ −6040.95 −1.08054
$$316$$ 0 0
$$317$$ 370.535 0.0656508 0.0328254 0.999461i $$-0.489549\pi$$
0.0328254 + 0.999461i $$0.489549\pi$$
$$318$$ 0 0
$$319$$ −2375.84 −0.416996
$$320$$ 0 0
$$321$$ −2773.79 −0.482298
$$322$$ 0 0
$$323$$ −3462.64 −0.596490
$$324$$ 0 0
$$325$$ 3124.92 0.533352
$$326$$ 0 0
$$327$$ 518.678 0.0877154
$$328$$ 0 0
$$329$$ −2131.10 −0.357116
$$330$$ 0 0
$$331$$ −1820.84 −0.302364 −0.151182 0.988506i $$-0.548308\pi$$
−0.151182 + 0.988506i $$0.548308\pi$$
$$332$$ 0 0
$$333$$ 1346.47 0.221581
$$334$$ 0 0
$$335$$ −19763.6 −3.22329
$$336$$ 0 0
$$337$$ −11838.2 −1.91356 −0.956781 0.290809i $$-0.906076\pi$$
−0.956781 + 0.290809i $$0.906076\pi$$
$$338$$ 0 0
$$339$$ −3725.43 −0.596867
$$340$$ 0 0
$$341$$ 6934.80 1.10129
$$342$$ 0 0
$$343$$ 19209.8 3.02399
$$344$$ 0 0
$$345$$ −9254.19 −1.44414
$$346$$ 0 0
$$347$$ −10603.5 −1.64043 −0.820214 0.572057i $$-0.806145\pi$$
−0.820214 + 0.572057i $$0.806145\pi$$
$$348$$ 0 0
$$349$$ 6398.72 0.981421 0.490710 0.871323i $$-0.336737\pi$$
0.490710 + 0.871323i $$0.336737\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ −7031.27 −1.06016 −0.530080 0.847948i $$-0.677838\pi$$
−0.530080 + 0.847948i $$0.677838\pi$$
$$354$$ 0 0
$$355$$ −19914.1 −2.97727
$$356$$ 0 0
$$357$$ 3816.61 0.565816
$$358$$ 0 0
$$359$$ 5053.75 0.742972 0.371486 0.928439i $$-0.378848\pi$$
0.371486 + 0.928439i $$0.378848\pi$$
$$360$$ 0 0
$$361$$ 2275.46 0.331749
$$362$$ 0 0
$$363$$ 1965.00 0.284121
$$364$$ 0 0
$$365$$ −9238.43 −1.32483
$$366$$ 0 0
$$367$$ −81.1902 −0.0115479 −0.00577397 0.999983i $$-0.501838\pi$$
−0.00577397 + 0.999983i $$0.501838\pi$$
$$368$$ 0 0
$$369$$ −700.237 −0.0987883
$$370$$ 0 0
$$371$$ −9886.26 −1.38347
$$372$$ 0 0
$$373$$ −8295.79 −1.15158 −0.575790 0.817597i $$-0.695305\pi$$
−0.575790 + 0.817597i $$0.695305\pi$$
$$374$$ 0 0
$$375$$ 6616.34 0.911110
$$376$$ 0 0
$$377$$ 1187.92 0.162284
$$378$$ 0 0
$$379$$ 6592.90 0.893548 0.446774 0.894647i $$-0.352573\pi$$
0.446774 + 0.894647i $$0.352573\pi$$
$$380$$ 0 0
$$381$$ 3239.80 0.435643
$$382$$ 0 0
$$383$$ −7472.96 −0.996999 −0.498499 0.866890i $$-0.666115\pi$$
−0.498499 + 0.866890i $$0.666115\pi$$
$$384$$ 0 0
$$385$$ 17451.6 2.31018
$$386$$ 0 0
$$387$$ −1652.47 −0.217054
$$388$$ 0 0
$$389$$ 3175.88 0.413943 0.206971 0.978347i $$-0.433639\pi$$
0.206971 + 0.978347i $$0.433639\pi$$
$$390$$ 0 0
$$391$$ 5846.70 0.756216
$$392$$ 0 0
$$393$$ −2754.08 −0.353499
$$394$$ 0 0
$$395$$ 25557.6 3.25555
$$396$$ 0 0
$$397$$ −11394.6 −1.44050 −0.720251 0.693713i $$-0.755973\pi$$
−0.720251 + 0.693713i $$0.755973\pi$$
$$398$$ 0 0
$$399$$ −10068.2 −1.26327
$$400$$ 0 0
$$401$$ 13046.7 1.62474 0.812370 0.583142i $$-0.198177\pi$$
0.812370 + 0.583142i $$0.198177\pi$$
$$402$$ 0 0
$$403$$ −3467.40 −0.428594
$$404$$ 0 0
$$405$$ −1548.31 −0.189965
$$406$$ 0 0
$$407$$ −3889.82 −0.473737
$$408$$ 0 0
$$409$$ −2888.77 −0.349243 −0.174622 0.984636i $$-0.555870\pi$$
−0.174622 + 0.984636i $$0.555870\pi$$
$$410$$ 0 0
$$411$$ −3933.19 −0.472043
$$412$$ 0 0
$$413$$ 19050.8 2.26980
$$414$$ 0 0
$$415$$ 15537.0 1.83779
$$416$$ 0 0
$$417$$ −682.780 −0.0801819
$$418$$ 0 0
$$419$$ 14260.8 1.66273 0.831367 0.555724i $$-0.187559\pi$$
0.831367 + 0.555724i $$0.187559\pi$$
$$420$$ 0 0
$$421$$ −890.723 −0.103114 −0.0515572 0.998670i $$-0.516418\pi$$
−0.0515572 + 0.998670i $$0.516418\pi$$
$$422$$ 0 0
$$423$$ −546.203 −0.0627833
$$424$$ 0 0
$$425$$ −8708.85 −0.993980
$$426$$ 0 0
$$427$$ −2285.31 −0.259002
$$428$$ 0 0
$$429$$ 1014.00 0.114117
$$430$$ 0 0
$$431$$ −1323.12 −0.147871 −0.0739353 0.997263i $$-0.523556\pi$$
−0.0739353 + 0.997263i $$0.523556\pi$$
$$432$$ 0 0
$$433$$ −9399.47 −1.04321 −0.521605 0.853187i $$-0.674666\pi$$
−0.521605 + 0.853187i $$0.674666\pi$$
$$434$$ 0 0
$$435$$ 5240.07 0.577568
$$436$$ 0 0
$$437$$ −15423.7 −1.68836
$$438$$ 0 0
$$439$$ −6115.04 −0.664817 −0.332409 0.943135i $$-0.607861\pi$$
−0.332409 + 0.943135i $$0.607861\pi$$
$$440$$ 0 0
$$441$$ 8010.49 0.864970
$$442$$ 0 0
$$443$$ 12931.1 1.38685 0.693425 0.720529i $$-0.256101\pi$$
0.693425 + 0.720529i $$0.256101\pi$$
$$444$$ 0 0
$$445$$ −17908.4 −1.90773
$$446$$ 0 0
$$447$$ 3976.44 0.420759
$$448$$ 0 0
$$449$$ −2050.73 −0.215546 −0.107773 0.994176i $$-0.534372\pi$$
−0.107773 + 0.994176i $$0.534372\pi$$
$$450$$ 0 0
$$451$$ 2022.91 0.211208
$$452$$ 0 0
$$453$$ −757.277 −0.0785430
$$454$$ 0 0
$$455$$ −8725.82 −0.899061
$$456$$ 0 0
$$457$$ −11923.7 −1.22049 −0.610247 0.792211i $$-0.708930\pi$$
−0.610247 + 0.792211i $$0.708930\pi$$
$$458$$ 0 0
$$459$$ 978.203 0.0994741
$$460$$ 0 0
$$461$$ −1324.14 −0.133777 −0.0668886 0.997760i $$-0.521307\pi$$
−0.0668886 + 0.997760i $$0.521307\pi$$
$$462$$ 0 0
$$463$$ −6840.65 −0.686635 −0.343318 0.939219i $$-0.611551\pi$$
−0.343318 + 0.939219i $$0.611551\pi$$
$$464$$ 0 0
$$465$$ −15295.1 −1.52537
$$466$$ 0 0
$$467$$ 9041.73 0.895935 0.447967 0.894050i $$-0.352148\pi$$
0.447967 + 0.894050i $$0.352148\pi$$
$$468$$ 0 0
$$469$$ 36306.7 3.57460
$$470$$ 0 0
$$471$$ −6805.86 −0.665812
$$472$$ 0 0
$$473$$ 4773.82 0.464060
$$474$$ 0 0
$$475$$ 22974.0 2.21920
$$476$$ 0 0
$$477$$ −2533.86 −0.243224
$$478$$ 0 0
$$479$$ 4650.71 0.443625 0.221812 0.975089i $$-0.428803\pi$$
0.221812 + 0.975089i $$0.428803\pi$$
$$480$$ 0 0
$$481$$ 1944.91 0.184366
$$482$$ 0 0
$$483$$ 17000.4 1.60154
$$484$$ 0 0
$$485$$ 18236.9 1.70741
$$486$$ 0 0
$$487$$ −19632.4 −1.82675 −0.913375 0.407120i $$-0.866533\pi$$
−0.913375 + 0.407120i $$0.866533\pi$$
$$488$$ 0 0
$$489$$ −3250.36 −0.300586
$$490$$ 0 0
$$491$$ 14472.0 1.33017 0.665084 0.746769i $$-0.268396\pi$$
0.665084 + 0.746769i $$0.268396\pi$$
$$492$$ 0 0
$$493$$ −3310.62 −0.302440
$$494$$ 0 0
$$495$$ 4472.88 0.406144
$$496$$ 0 0
$$497$$ 36583.1 3.30176
$$498$$ 0 0
$$499$$ 7065.91 0.633895 0.316947 0.948443i $$-0.397342\pi$$
0.316947 + 0.948443i $$0.397342\pi$$
$$500$$ 0 0
$$501$$ 2388.12 0.212961
$$502$$ 0 0
$$503$$ 11955.1 1.05974 0.529871 0.848078i $$-0.322240\pi$$
0.529871 + 0.848078i $$0.322240\pi$$
$$504$$ 0 0
$$505$$ 26782.8 2.36004
$$506$$ 0 0
$$507$$ −507.000 −0.0444116
$$508$$ 0 0
$$509$$ 8321.15 0.724614 0.362307 0.932059i $$-0.381989\pi$$
0.362307 + 0.932059i $$0.381989\pi$$
$$510$$ 0 0
$$511$$ 16971.4 1.46922
$$512$$ 0 0
$$513$$ −2580.51 −0.222090
$$514$$ 0 0
$$515$$ −18254.0 −1.56187
$$516$$ 0 0
$$517$$ 1577.92 0.134230
$$518$$ 0 0
$$519$$ −3.28795 −0.000278083 0
$$520$$ 0 0
$$521$$ −20130.2 −1.69274 −0.846372 0.532592i $$-0.821218\pi$$
−0.846372 + 0.532592i $$0.821218\pi$$
$$522$$ 0 0
$$523$$ −8810.52 −0.736630 −0.368315 0.929701i $$-0.620065\pi$$
−0.368315 + 0.929701i $$0.620065\pi$$
$$524$$ 0 0
$$525$$ −25322.6 −2.10508
$$526$$ 0 0
$$527$$ 9663.31 0.798749
$$528$$ 0 0
$$529$$ 13876.0 1.14046
$$530$$ 0 0
$$531$$ 4882.75 0.399045
$$532$$ 0 0
$$533$$ −1011.45 −0.0821969
$$534$$ 0 0
$$535$$ −17673.5 −1.42821
$$536$$ 0 0
$$537$$ 9750.17 0.783521
$$538$$ 0 0
$$539$$ −23141.4 −1.84930
$$540$$ 0 0
$$541$$ −7765.62 −0.617135 −0.308567 0.951202i $$-0.599850\pi$$
−0.308567 + 0.951202i $$0.599850\pi$$
$$542$$ 0 0
$$543$$ 8942.52 0.706741
$$544$$ 0 0
$$545$$ 3304.82 0.259749
$$546$$ 0 0
$$547$$ −19782.4 −1.54631 −0.773156 0.634216i $$-0.781323\pi$$
−0.773156 + 0.634216i $$0.781323\pi$$
$$548$$ 0 0
$$549$$ −585.729 −0.0455342
$$550$$ 0 0
$$551$$ 8733.45 0.675240
$$552$$ 0 0
$$553$$ −46950.5 −3.61038
$$554$$ 0 0
$$555$$ 8579.23 0.656159
$$556$$ 0 0
$$557$$ 10153.9 0.772413 0.386206 0.922412i $$-0.373785\pi$$
0.386206 + 0.922412i $$0.373785\pi$$
$$558$$ 0 0
$$559$$ −2386.91 −0.180600
$$560$$ 0 0
$$561$$ −2825.92 −0.212675
$$562$$ 0 0
$$563$$ −18388.2 −1.37650 −0.688252 0.725472i $$-0.741622\pi$$
−0.688252 + 0.725472i $$0.741622\pi$$
$$564$$ 0 0
$$565$$ −23737.1 −1.76748
$$566$$ 0 0
$$567$$ 2844.31 0.210669
$$568$$ 0 0
$$569$$ 16365.0 1.20573 0.602863 0.797845i $$-0.294027\pi$$
0.602863 + 0.797845i $$0.294027\pi$$
$$570$$ 0 0
$$571$$ −544.241 −0.0398875 −0.0199437 0.999801i $$-0.506349\pi$$
−0.0199437 + 0.999801i $$0.506349\pi$$
$$572$$ 0 0
$$573$$ −1837.79 −0.133987
$$574$$ 0 0
$$575$$ −38791.9 −2.81345
$$576$$ 0 0
$$577$$ −798.738 −0.0576289 −0.0288145 0.999585i $$-0.509173\pi$$
−0.0288145 + 0.999585i $$0.509173\pi$$
$$578$$ 0 0
$$579$$ 12557.4 0.901328
$$580$$ 0 0
$$581$$ −28542.2 −2.03809
$$582$$ 0 0
$$583$$ 7320.05 0.520010
$$584$$ 0 0
$$585$$ −2236.44 −0.158061
$$586$$ 0 0
$$587$$ −9187.53 −0.646013 −0.323007 0.946397i $$-0.604694\pi$$
−0.323007 + 0.946397i $$0.604694\pi$$
$$588$$ 0 0
$$589$$ −25491.9 −1.78332
$$590$$ 0 0
$$591$$ 6.18086 0.000430197 0
$$592$$ 0 0
$$593$$ −4371.80 −0.302746 −0.151373 0.988477i $$-0.548369\pi$$
−0.151373 + 0.988477i $$0.548369\pi$$
$$594$$ 0 0
$$595$$ 24318.0 1.67553
$$596$$ 0 0
$$597$$ −4477.18 −0.306932
$$598$$ 0 0
$$599$$ −6780.78 −0.462530 −0.231265 0.972891i $$-0.574286\pi$$
−0.231265 + 0.972891i $$0.574286\pi$$
$$600$$ 0 0
$$601$$ −21348.5 −1.44895 −0.724477 0.689299i $$-0.757919\pi$$
−0.724477 + 0.689299i $$0.757919\pi$$
$$602$$ 0 0
$$603$$ 9305.46 0.628437
$$604$$ 0 0
$$605$$ 12520.2 0.841356
$$606$$ 0 0
$$607$$ 11161.8 0.746362 0.373181 0.927758i $$-0.378267\pi$$
0.373181 + 0.927758i $$0.378267\pi$$
$$608$$ 0 0
$$609$$ −9626.24 −0.640517
$$610$$ 0 0
$$611$$ −788.960 −0.0522388
$$612$$ 0 0
$$613$$ −12407.8 −0.817528 −0.408764 0.912640i $$-0.634040\pi$$
−0.408764 + 0.912640i $$0.634040\pi$$
$$614$$ 0 0
$$615$$ −4461.65 −0.292538
$$616$$ 0 0
$$617$$ 5035.01 0.328528 0.164264 0.986416i $$-0.447475\pi$$
0.164264 + 0.986416i $$0.447475\pi$$
$$618$$ 0 0
$$619$$ 1623.67 0.105430 0.0527148 0.998610i $$-0.483213\pi$$
0.0527148 + 0.998610i $$0.483213\pi$$
$$620$$ 0 0
$$621$$ 4357.22 0.281561
$$622$$ 0 0
$$623$$ 32898.6 2.11566
$$624$$ 0 0
$$625$$ 12109.5 0.775009
$$626$$ 0 0
$$627$$ 7454.80 0.474826
$$628$$ 0 0
$$629$$ −5420.27 −0.343594
$$630$$ 0 0
$$631$$ 24809.0 1.56519 0.782593 0.622534i $$-0.213897\pi$$
0.782593 + 0.622534i $$0.213897\pi$$
$$632$$ 0 0
$$633$$ 8244.81 0.517697
$$634$$ 0 0
$$635$$ 20642.8 1.29005
$$636$$ 0 0
$$637$$ 11570.7 0.719699
$$638$$ 0 0
$$639$$ 9376.30 0.580471
$$640$$ 0 0
$$641$$ 18076.6 1.11386 0.556928 0.830561i $$-0.311980\pi$$
0.556928 + 0.830561i $$0.311980\pi$$
$$642$$ 0 0
$$643$$ −19954.2 −1.22382 −0.611910 0.790928i $$-0.709598\pi$$
−0.611910 + 0.790928i $$0.709598\pi$$
$$644$$ 0 0
$$645$$ −10528.9 −0.642755
$$646$$ 0 0
$$647$$ −11206.1 −0.680921 −0.340461 0.940259i $$-0.610583\pi$$
−0.340461 + 0.940259i $$0.610583\pi$$
$$648$$ 0 0
$$649$$ −14105.7 −0.853155
$$650$$ 0 0
$$651$$ 28097.9 1.69162
$$652$$ 0 0
$$653$$ −2456.10 −0.147189 −0.0735947 0.997288i $$-0.523447\pi$$
−0.0735947 + 0.997288i $$0.523447\pi$$
$$654$$ 0 0
$$655$$ −17548.0 −1.04680
$$656$$ 0 0
$$657$$ 4349.80 0.258298
$$658$$ 0 0
$$659$$ 23280.8 1.37616 0.688081 0.725633i $$-0.258453\pi$$
0.688081 + 0.725633i $$0.258453\pi$$
$$660$$ 0 0
$$661$$ 6999.49 0.411874 0.205937 0.978565i $$-0.433976\pi$$
0.205937 + 0.978565i $$0.433976\pi$$
$$662$$ 0 0
$$663$$ 1412.96 0.0827675
$$664$$ 0 0
$$665$$ −64151.1 −3.74086
$$666$$ 0 0
$$667$$ −14746.5 −0.856054
$$668$$ 0 0
$$669$$ −559.966 −0.0323610
$$670$$ 0 0
$$671$$ 1692.11 0.0973517
$$672$$ 0 0
$$673$$ 33166.1 1.89964 0.949820 0.312798i $$-0.101266\pi$$
0.949820 + 0.312798i $$0.101266\pi$$
$$674$$ 0 0
$$675$$ −6490.22 −0.370087
$$676$$ 0 0
$$677$$ −19998.6 −1.13532 −0.567658 0.823265i $$-0.692150\pi$$
−0.567658 + 0.823265i $$0.692150\pi$$
$$678$$ 0 0
$$679$$ −33502.0 −1.89350
$$680$$ 0 0
$$681$$ 11797.7 0.663861
$$682$$ 0 0
$$683$$ −22857.9 −1.28058 −0.640289 0.768134i $$-0.721185\pi$$
−0.640289 + 0.768134i $$0.721185\pi$$
$$684$$ 0 0
$$685$$ −25060.8 −1.39784
$$686$$ 0 0
$$687$$ 17853.0 0.991462
$$688$$ 0 0
$$689$$ −3660.03 −0.202374
$$690$$ 0 0
$$691$$ 2866.56 0.157813 0.0789066 0.996882i $$-0.474857\pi$$
0.0789066 + 0.996882i $$0.474857\pi$$
$$692$$ 0 0
$$693$$ −8216.88 −0.450409
$$694$$ 0 0
$$695$$ −4350.42 −0.237440
$$696$$ 0 0
$$697$$ 2818.82 0.153186
$$698$$ 0 0
$$699$$ −19646.8 −1.06311
$$700$$ 0 0
$$701$$ −30788.1 −1.65885 −0.829424 0.558620i $$-0.811331\pi$$
−0.829424 + 0.558620i $$0.811331\pi$$
$$702$$ 0 0
$$703$$ 14298.7 0.767121
$$704$$ 0 0
$$705$$ −3480.20 −0.185918
$$706$$ 0 0
$$707$$ −49201.2 −2.61726
$$708$$ 0 0
$$709$$ 4389.07 0.232489 0.116245 0.993221i $$-0.462914\pi$$
0.116245 + 0.993221i $$0.462914\pi$$
$$710$$ 0 0
$$711$$ −12033.5 −0.634727
$$712$$ 0 0
$$713$$ 43043.4 2.26085
$$714$$ 0 0
$$715$$ 6460.83 0.337932
$$716$$ 0 0
$$717$$ 10387.1 0.541021
$$718$$ 0 0
$$719$$ −8729.09 −0.452768 −0.226384 0.974038i $$-0.572690\pi$$
−0.226384 + 0.974038i $$0.572690\pi$$
$$720$$ 0 0
$$721$$ 33533.3 1.73210
$$722$$ 0 0
$$723$$ −13745.9 −0.707075
$$724$$ 0 0
$$725$$ 21965.4 1.12521
$$726$$ 0 0
$$727$$ 1471.81 0.0750845 0.0375423 0.999295i $$-0.488047\pi$$
0.0375423 + 0.999295i $$0.488047\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 6652.08 0.336575
$$732$$ 0 0
$$733$$ 14082.8 0.709632 0.354816 0.934936i $$-0.384543\pi$$
0.354816 + 0.934936i $$0.384543\pi$$
$$734$$ 0 0
$$735$$ 51039.9 2.56141
$$736$$ 0 0
$$737$$ −26882.4 −1.34359
$$738$$ 0 0
$$739$$ −12529.5 −0.623687 −0.311843 0.950133i $$-0.600946\pi$$
−0.311843 + 0.950133i $$0.600946\pi$$
$$740$$ 0 0
$$741$$ −3727.40 −0.184790
$$742$$ 0 0
$$743$$ −19044.9 −0.940363 −0.470182 0.882570i $$-0.655812\pi$$
−0.470182 + 0.882570i $$0.655812\pi$$
$$744$$ 0 0
$$745$$ 25336.4 1.24598
$$746$$ 0 0
$$747$$ −7315.42 −0.358310
$$748$$ 0 0
$$749$$ 32467.0 1.58387
$$750$$ 0 0
$$751$$ −22009.9 −1.06944 −0.534722 0.845028i $$-0.679584\pi$$
−0.534722 + 0.845028i $$0.679584\pi$$
$$752$$ 0 0
$$753$$ 14240.3 0.689169
$$754$$ 0 0
$$755$$ −4825.08 −0.232587
$$756$$ 0 0
$$757$$ 8198.54 0.393634 0.196817 0.980440i $$-0.436939\pi$$
0.196817 + 0.980440i $$0.436939\pi$$
$$758$$ 0 0
$$759$$ −12587.5 −0.601974
$$760$$ 0 0
$$761$$ −31929.6 −1.52096 −0.760478 0.649363i $$-0.775036\pi$$
−0.760478 + 0.649363i $$0.775036\pi$$
$$762$$ 0 0
$$763$$ −6071.10 −0.288059
$$764$$ 0 0
$$765$$ 6232.75 0.294569
$$766$$ 0 0
$$767$$ 7052.85 0.332026
$$768$$ 0 0
$$769$$ −26725.8 −1.25326 −0.626630 0.779317i $$-0.715566\pi$$
−0.626630 + 0.779317i $$0.715566\pi$$
$$770$$ 0 0
$$771$$ 10120.9 0.472757
$$772$$ 0 0
$$773$$ −23760.8 −1.10558 −0.552792 0.833320i $$-0.686437\pi$$
−0.552792 + 0.833320i $$0.686437\pi$$
$$774$$ 0 0
$$775$$ −64114.5 −2.97169
$$776$$ 0 0
$$777$$ −15760.4 −0.727673
$$778$$ 0 0
$$779$$ −7436.08 −0.342009
$$780$$ 0 0
$$781$$ −27087.1 −1.24104
$$782$$ 0 0
$$783$$ −2467.22 −0.112607
$$784$$ 0 0
$$785$$ −43364.4 −1.97165
$$786$$ 0 0
$$787$$ 23647.2 1.07107 0.535536 0.844513i $$-0.320110\pi$$
0.535536 + 0.844513i $$0.320110\pi$$
$$788$$ 0 0
$$789$$ −10581.6 −0.477458
$$790$$ 0 0
$$791$$ 43606.1 1.96012
$$792$$ 0 0
$$793$$ −846.053 −0.0378868
$$794$$ 0 0
$$795$$ −16144.8 −0.720249
$$796$$ 0 0
$$797$$ −21067.5 −0.936324 −0.468162 0.883643i $$-0.655084\pi$$
−0.468162 + 0.883643i $$0.655084\pi$$
$$798$$ 0 0
$$799$$ 2198.76 0.0973547
$$800$$ 0 0
$$801$$ 8431.97 0.371946
$$802$$ 0 0
$$803$$ −12566.1 −0.552238
$$804$$ 0 0
$$805$$ 108320. 4.74258
$$806$$ 0 0
$$807$$ 3432.89 0.149744
$$808$$ 0 0
$$809$$ 1877.67 0.0816011 0.0408005 0.999167i $$-0.487009\pi$$
0.0408005 + 0.999167i $$0.487009\pi$$
$$810$$ 0 0
$$811$$ −2178.85 −0.0943401 −0.0471700 0.998887i $$-0.515020\pi$$
−0.0471700 + 0.998887i $$0.515020\pi$$
$$812$$ 0 0
$$813$$ −6762.75 −0.291734
$$814$$ 0 0
$$815$$ −20710.1 −0.890114
$$816$$ 0 0
$$817$$ −17548.2 −0.751451
$$818$$ 0 0
$$819$$ 4108.44 0.175288
$$820$$ 0 0
$$821$$ −11731.4 −0.498696 −0.249348 0.968414i $$-0.580216\pi$$
−0.249348 + 0.968414i $$0.580216\pi$$
$$822$$ 0 0
$$823$$ −6925.69 −0.293334 −0.146667 0.989186i $$-0.546855\pi$$
−0.146667 + 0.989186i $$0.546855\pi$$
$$824$$ 0 0
$$825$$ 18749.5 0.791242
$$826$$ 0 0
$$827$$ 35912.7 1.51004 0.755022 0.655700i $$-0.227626\pi$$
0.755022 + 0.655700i $$0.227626\pi$$
$$828$$ 0 0
$$829$$ 7107.67 0.297780 0.148890 0.988854i $$-0.452430\pi$$
0.148890 + 0.988854i $$0.452430\pi$$
$$830$$ 0 0
$$831$$ 21304.9 0.889360
$$832$$ 0 0
$$833$$ −32246.5 −1.34126
$$834$$ 0 0
$$835$$ 15216.2 0.630634
$$836$$ 0 0
$$837$$ 7201.53 0.297397
$$838$$ 0 0
$$839$$ 37844.0 1.55724 0.778618 0.627498i $$-0.215921\pi$$
0.778618 + 0.627498i $$0.215921\pi$$
$$840$$ 0 0
$$841$$ −16039.0 −0.657631
$$842$$ 0 0
$$843$$ 11495.4 0.469660
$$844$$ 0 0
$$845$$ −3230.41 −0.131514
$$846$$ 0 0
$$847$$ −23000.2 −0.933055
$$848$$ 0 0
$$849$$ −9097.67 −0.367764
$$850$$ 0 0
$$851$$ −24143.6 −0.972539
$$852$$ 0 0
$$853$$ −23280.9 −0.934492 −0.467246 0.884127i $$-0.654754\pi$$
−0.467246 + 0.884127i $$0.654754\pi$$
$$854$$ 0 0
$$855$$ −16442.0 −0.657667
$$856$$ 0 0
$$857$$ 10395.1 0.414342 0.207171 0.978305i $$-0.433574\pi$$
0.207171 + 0.978305i $$0.433574\pi$$
$$858$$ 0 0
$$859$$ 6698.89 0.266080 0.133040 0.991111i $$-0.457526\pi$$
0.133040 + 0.991111i $$0.457526\pi$$
$$860$$ 0 0
$$861$$ 8196.25 0.324422
$$862$$ 0 0
$$863$$ 31844.8 1.25609 0.628047 0.778175i $$-0.283854\pi$$
0.628047 + 0.778175i $$0.283854\pi$$
$$864$$ 0 0
$$865$$ −20.9496 −0.000823477 0
$$866$$ 0 0
$$867$$ 10801.2 0.423101
$$868$$ 0 0
$$869$$ 34763.4 1.35704
$$870$$ 0 0
$$871$$ 13441.2 0.522891
$$872$$ 0 0
$$873$$ −8586.61 −0.332890
$$874$$ 0 0
$$875$$ −77444.0 −2.99210
$$876$$ 0 0
$$877$$ 20114.1 0.774465 0.387233 0.921982i $$-0.373431\pi$$
0.387233 + 0.921982i $$0.373431\pi$$
$$878$$ 0 0
$$879$$ −6889.32 −0.264358
$$880$$ 0 0
$$881$$ −2596.61 −0.0992987 −0.0496493 0.998767i $$-0.515810\pi$$
−0.0496493 + 0.998767i $$0.515810\pi$$
$$882$$ 0 0
$$883$$ 46374.7 1.76742 0.883710 0.468035i $$-0.155038\pi$$
0.883710 + 0.468035i $$0.155038\pi$$
$$884$$ 0 0
$$885$$ 31111.0 1.18168
$$886$$ 0 0
$$887$$ −31863.1 −1.20615 −0.603077 0.797683i $$-0.706059\pi$$
−0.603077 + 0.797683i $$0.706059\pi$$
$$888$$ 0 0
$$889$$ −37921.7 −1.43066
$$890$$ 0 0
$$891$$ −2106.00 −0.0791848
$$892$$ 0 0
$$893$$ −5800.34 −0.217358
$$894$$ 0 0
$$895$$ 62124.4 2.32021
$$896$$ 0 0
$$897$$ 6293.76 0.234273
$$898$$ 0 0
$$899$$ −24372.8 −0.904202
$$900$$ 0 0
$$901$$ 10200.1 0.377154
$$902$$ 0 0
$$903$$ 19342.1 0.712809
$$904$$ 0 0
$$905$$ 56978.4 2.09285
$$906$$ 0 0
$$907$$ −3004.29 −0.109984 −0.0549922 0.998487i $$-0.517513\pi$$
−0.0549922 + 0.998487i $$0.517513\pi$$
$$908$$ 0 0
$$909$$ −12610.3 −0.460130
$$910$$ 0 0
$$911$$ −9985.05 −0.363139 −0.181569 0.983378i $$-0.558118\pi$$
−0.181569 + 0.983378i $$0.558118\pi$$
$$912$$ 0 0
$$913$$ 21133.4 0.766062
$$914$$ 0 0
$$915$$ −3732.04 −0.134839
$$916$$ 0 0
$$917$$ 32236.4 1.16089
$$918$$ 0 0
$$919$$ −8216.29 −0.294919 −0.147459 0.989068i $$-0.547110\pi$$
−0.147459 + 0.989068i $$0.547110\pi$$
$$920$$ 0 0
$$921$$ 15522.8 0.555367
$$922$$ 0 0
$$923$$ 13543.6 0.482981
$$924$$ 0 0
$$925$$ 35962.6 1.27832
$$926$$ 0 0
$$927$$ 8594.64 0.304515
$$928$$ 0 0
$$929$$ −17832.8 −0.629791 −0.314896 0.949126i $$-0.601970\pi$$
−0.314896 + 0.949126i $$0.601970\pi$$
$$930$$ 0 0
$$931$$ 85066.4 2.99456
$$932$$ 0 0
$$933$$ 22444.6 0.787572
$$934$$ 0 0
$$935$$ −18005.7 −0.629786
$$936$$ 0 0
$$937$$ −444.289 −0.0154902 −0.00774509 0.999970i $$-0.502465\pi$$
−0.00774509 + 0.999970i $$0.502465\pi$$
$$938$$ 0 0
$$939$$ 12476.9 0.433620
$$940$$ 0 0
$$941$$ −30762.5 −1.06571 −0.532853 0.846208i $$-0.678880\pi$$
−0.532853 + 0.846208i $$0.678880\pi$$
$$942$$ 0 0
$$943$$ 12555.9 0.433592
$$944$$ 0 0
$$945$$ 18122.8 0.623848
$$946$$ 0 0
$$947$$ 8791.45 0.301672 0.150836 0.988559i $$-0.451803\pi$$
0.150836 + 0.988559i $$0.451803\pi$$
$$948$$ 0 0
$$949$$ 6283.04 0.214917
$$950$$ 0 0
$$951$$ −1111.60 −0.0379035
$$952$$ 0 0
$$953$$ 1754.41 0.0596336 0.0298168 0.999555i $$-0.490508\pi$$
0.0298168 + 0.999555i $$0.490508\pi$$
$$954$$ 0 0
$$955$$ −11709.7 −0.396771
$$956$$ 0 0
$$957$$ 7127.52 0.240753
$$958$$ 0 0
$$959$$ 46037.8 1.55020
$$960$$ 0 0
$$961$$ 41350.2 1.38801
$$962$$ 0 0
$$963$$ 8321.36 0.278455
$$964$$ 0 0
$$965$$ 80011.3 2.66907
$$966$$ 0 0
$$967$$ 26655.9 0.886447 0.443224 0.896411i $$-0.353835\pi$$
0.443224 + 0.896411i $$0.353835\pi$$
$$968$$ 0 0
$$969$$ 10387.9 0.344384
$$970$$ 0 0
$$971$$ 42151.7 1.39311 0.696556 0.717502i $$-0.254715\pi$$
0.696556 + 0.717502i $$0.254715\pi$$
$$972$$ 0 0
$$973$$ 7991.91 0.263318
$$974$$ 0 0
$$975$$ −9374.76 −0.307931
$$976$$ 0 0
$$977$$ 14594.7 0.477919 0.238959 0.971030i $$-0.423194\pi$$
0.238959 + 0.971030i $$0.423194\pi$$
$$978$$ 0 0
$$979$$ −24359.0 −0.795217
$$980$$ 0 0
$$981$$ −1556.03 −0.0506425
$$982$$ 0 0
$$983$$ 57077.8 1.85198 0.925991 0.377546i $$-0.123232\pi$$
0.925991 + 0.377546i $$0.123232\pi$$
$$984$$ 0 0
$$985$$ 39.3821 0.00127393
$$986$$ 0 0
$$987$$ 6393.29 0.206181
$$988$$ 0 0
$$989$$ 29630.4 0.952672
$$990$$ 0 0
$$991$$ 11986.6 0.384224 0.192112 0.981373i $$-0.438466\pi$$
0.192112 + 0.981373i $$0.438466\pi$$
$$992$$ 0 0
$$993$$ 5462.52 0.174570
$$994$$ 0 0
$$995$$ −28526.9 −0.908908
$$996$$ 0 0
$$997$$ −32210.1 −1.02317 −0.511587 0.859231i $$-0.670942\pi$$
−0.511587 + 0.859231i $$0.670942\pi$$
$$998$$ 0 0
$$999$$ −4039.42 −0.127930
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2496.4.a.u.1.1 2
4.3 odd 2 2496.4.a.bd.1.1 2
8.3 odd 2 624.4.a.l.1.2 2
8.5 even 2 312.4.a.f.1.2 2
24.5 odd 2 936.4.a.c.1.1 2
24.11 even 2 1872.4.a.v.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.f.1.2 2 8.5 even 2
624.4.a.l.1.2 2 8.3 odd 2
936.4.a.c.1.1 2 24.5 odd 2
1872.4.a.v.1.1 2 24.11 even 2
2496.4.a.u.1.1 2 1.1 even 1 trivial
2496.4.a.bd.1.1 2 4.3 odd 2