# Properties

 Label 1323.4.a.be.1.5 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 40x^{4} + 453x^{2} - 1278$$ x^6 - 40*x^4 + 453*x^2 - 1278 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$3.68757$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +O(q^{10})$$ $$q+3.68757 q^{2} +5.59820 q^{4} +11.8719 q^{5} -8.85681 q^{8} +43.7785 q^{10} +22.2104 q^{11} -75.1040 q^{13} -77.4457 q^{16} -74.1097 q^{17} +7.41977 q^{19} +66.4612 q^{20} +81.9023 q^{22} -205.772 q^{23} +15.9417 q^{25} -276.952 q^{26} +148.304 q^{29} +164.438 q^{31} -214.732 q^{32} -273.285 q^{34} -205.239 q^{37} +27.3610 q^{38} -105.147 q^{40} -83.9896 q^{41} -368.677 q^{43} +124.338 q^{44} -758.800 q^{46} -98.3339 q^{47} +58.7864 q^{50} -420.448 q^{52} -293.897 q^{53} +263.679 q^{55} +546.883 q^{58} -509.272 q^{59} +696.384 q^{61} +606.376 q^{62} -172.276 q^{64} -891.627 q^{65} +370.879 q^{67} -414.881 q^{68} +121.723 q^{71} +682.103 q^{73} -756.832 q^{74} +41.5374 q^{76} -669.950 q^{79} -919.427 q^{80} -309.718 q^{82} +1182.06 q^{83} -879.823 q^{85} -1359.52 q^{86} -196.713 q^{88} +598.651 q^{89} -1151.95 q^{92} -362.614 q^{94} +88.0867 q^{95} +1037.20 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 32 q^{4}+O(q^{10})$$ 6 * q + 32 * q^4 $$6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100})$$ 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.68757 1.30375 0.651877 0.758325i $$-0.273982\pi$$
0.651877 + 0.758325i $$0.273982\pi$$
$$3$$ 0 0
$$4$$ 5.59820 0.699775
$$5$$ 11.8719 1.06185 0.530927 0.847418i $$-0.321844\pi$$
0.530927 + 0.847418i $$0.321844\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −8.85681 −0.391419
$$9$$ 0 0
$$10$$ 43.7785 1.38440
$$11$$ 22.2104 0.608788 0.304394 0.952546i $$-0.401546\pi$$
0.304394 + 0.952546i $$0.401546\pi$$
$$12$$ 0 0
$$13$$ −75.1040 −1.60232 −0.801158 0.598453i $$-0.795782\pi$$
−0.801158 + 0.598453i $$0.795782\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −77.4457 −1.21009
$$17$$ −74.1097 −1.05731 −0.528654 0.848837i $$-0.677303\pi$$
−0.528654 + 0.848837i $$0.677303\pi$$
$$18$$ 0 0
$$19$$ 7.41977 0.0895901 0.0447951 0.998996i $$-0.485737\pi$$
0.0447951 + 0.998996i $$0.485737\pi$$
$$20$$ 66.4612 0.743059
$$21$$ 0 0
$$22$$ 81.9023 0.793711
$$23$$ −205.772 −1.86550 −0.932749 0.360526i $$-0.882597\pi$$
−0.932749 + 0.360526i $$0.882597\pi$$
$$24$$ 0 0
$$25$$ 15.9417 0.127534
$$26$$ −276.952 −2.08903
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 148.304 0.949636 0.474818 0.880084i $$-0.342514\pi$$
0.474818 + 0.880084i $$0.342514\pi$$
$$30$$ 0 0
$$31$$ 164.438 0.952705 0.476353 0.879254i $$-0.341959\pi$$
0.476353 + 0.879254i $$0.341959\pi$$
$$32$$ −214.732 −1.18624
$$33$$ 0 0
$$34$$ −273.285 −1.37847
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −205.239 −0.911919 −0.455960 0.890001i $$-0.650704\pi$$
−0.455960 + 0.890001i $$0.650704\pi$$
$$38$$ 27.3610 0.116803
$$39$$ 0 0
$$40$$ −105.147 −0.415630
$$41$$ −83.9896 −0.319926 −0.159963 0.987123i $$-0.551137\pi$$
−0.159963 + 0.987123i $$0.551137\pi$$
$$42$$ 0 0
$$43$$ −368.677 −1.30751 −0.653753 0.756708i $$-0.726806\pi$$
−0.653753 + 0.756708i $$0.726806\pi$$
$$44$$ 124.338 0.426015
$$45$$ 0 0
$$46$$ −758.800 −2.43215
$$47$$ −98.3339 −0.305180 −0.152590 0.988290i $$-0.548761\pi$$
−0.152590 + 0.988290i $$0.548761\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 58.7864 0.166273
$$51$$ 0 0
$$52$$ −420.448 −1.12126
$$53$$ −293.897 −0.761696 −0.380848 0.924638i $$-0.624368\pi$$
−0.380848 + 0.924638i $$0.624368\pi$$
$$54$$ 0 0
$$55$$ 263.679 0.646444
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 546.883 1.23809
$$59$$ −509.272 −1.12376 −0.561878 0.827220i $$-0.689921\pi$$
−0.561878 + 0.827220i $$0.689921\pi$$
$$60$$ 0 0
$$61$$ 696.384 1.46169 0.730843 0.682546i $$-0.239127\pi$$
0.730843 + 0.682546i $$0.239127\pi$$
$$62$$ 606.376 1.24209
$$63$$ 0 0
$$64$$ −172.276 −0.336477
$$65$$ −891.627 −1.70143
$$66$$ 0 0
$$67$$ 370.879 0.676270 0.338135 0.941098i $$-0.390204\pi$$
0.338135 + 0.941098i $$0.390204\pi$$
$$68$$ −414.881 −0.739879
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 121.723 0.203462 0.101731 0.994812i $$-0.467562\pi$$
0.101731 + 0.994812i $$0.467562\pi$$
$$72$$ 0 0
$$73$$ 682.103 1.09362 0.546809 0.837257i $$-0.315842\pi$$
0.546809 + 0.837257i $$0.315842\pi$$
$$74$$ −756.832 −1.18892
$$75$$ 0 0
$$76$$ 41.5374 0.0626929
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −669.950 −0.954118 −0.477059 0.878871i $$-0.658297\pi$$
−0.477059 + 0.878871i $$0.658297\pi$$
$$80$$ −919.427 −1.28494
$$81$$ 0 0
$$82$$ −309.718 −0.417105
$$83$$ 1182.06 1.56323 0.781616 0.623759i $$-0.214396\pi$$
0.781616 + 0.623759i $$0.214396\pi$$
$$84$$ 0 0
$$85$$ −879.823 −1.12271
$$86$$ −1359.52 −1.70467
$$87$$ 0 0
$$88$$ −196.713 −0.238291
$$89$$ 598.651 0.712999 0.356500 0.934295i $$-0.383970\pi$$
0.356500 + 0.934295i $$0.383970\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1151.95 −1.30543
$$93$$ 0 0
$$94$$ −362.614 −0.397880
$$95$$ 88.0867 0.0951316
$$96$$ 0 0
$$97$$ 1037.20 1.08569 0.542846 0.839833i $$-0.317347\pi$$
0.542846 + 0.839833i $$0.317347\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 89.2451 0.0892451
$$101$$ 137.285 0.135251 0.0676254 0.997711i $$-0.478458\pi$$
0.0676254 + 0.997711i $$0.478458\pi$$
$$102$$ 0 0
$$103$$ 327.237 0.313045 0.156522 0.987674i $$-0.449972\pi$$
0.156522 + 0.987674i $$0.449972\pi$$
$$104$$ 665.182 0.627177
$$105$$ 0 0
$$106$$ −1083.77 −0.993064
$$107$$ −1051.42 −0.949947 −0.474973 0.880000i $$-0.657542\pi$$
−0.474973 + 0.880000i $$0.657542\pi$$
$$108$$ 0 0
$$109$$ 255.171 0.224229 0.112114 0.993695i $$-0.464238\pi$$
0.112114 + 0.993695i $$0.464238\pi$$
$$110$$ 972.335 0.842805
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1990.49 −1.65708 −0.828538 0.559933i $$-0.810827\pi$$
−0.828538 + 0.559933i $$0.810827\pi$$
$$114$$ 0 0
$$115$$ −2442.90 −1.98089
$$116$$ 830.238 0.664532
$$117$$ 0 0
$$118$$ −1877.98 −1.46510
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −837.700 −0.629377
$$122$$ 2567.97 1.90568
$$123$$ 0 0
$$124$$ 920.555 0.666680
$$125$$ −1294.73 −0.926432
$$126$$ 0 0
$$127$$ 536.451 0.374822 0.187411 0.982282i $$-0.439990\pi$$
0.187411 + 0.982282i $$0.439990\pi$$
$$128$$ 1082.58 0.747558
$$129$$ 0 0
$$130$$ −3287.94 −2.21824
$$131$$ −2560.09 −1.70745 −0.853727 0.520721i $$-0.825663\pi$$
−0.853727 + 0.520721i $$0.825663\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1367.64 0.881690
$$135$$ 0 0
$$136$$ 656.376 0.413851
$$137$$ −2266.25 −1.41327 −0.706637 0.707577i $$-0.749788\pi$$
−0.706637 + 0.707577i $$0.749788\pi$$
$$138$$ 0 0
$$139$$ −1837.27 −1.12112 −0.560559 0.828114i $$-0.689414\pi$$
−0.560559 + 0.828114i $$0.689414\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 448.861 0.265265
$$143$$ −1668.09 −0.975472
$$144$$ 0 0
$$145$$ 1760.65 1.00837
$$146$$ 2515.31 1.42581
$$147$$ 0 0
$$148$$ −1148.97 −0.638138
$$149$$ 626.201 0.344298 0.172149 0.985071i $$-0.444929\pi$$
0.172149 + 0.985071i $$0.444929\pi$$
$$150$$ 0 0
$$151$$ −1204.68 −0.649243 −0.324621 0.945844i $$-0.605237\pi$$
−0.324621 + 0.945844i $$0.605237\pi$$
$$152$$ −65.7155 −0.0350673
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1952.19 1.01163
$$156$$ 0 0
$$157$$ −2976.32 −1.51297 −0.756486 0.654010i $$-0.773085\pi$$
−0.756486 + 0.654010i $$0.773085\pi$$
$$158$$ −2470.49 −1.24394
$$159$$ 0 0
$$160$$ −2549.28 −1.25961
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 816.520 0.392360 0.196180 0.980568i $$-0.437146\pi$$
0.196180 + 0.980568i $$0.437146\pi$$
$$164$$ −470.191 −0.223876
$$165$$ 0 0
$$166$$ 4358.95 2.03807
$$167$$ 1597.72 0.740331 0.370166 0.928966i $$-0.379301\pi$$
0.370166 + 0.928966i $$0.379301\pi$$
$$168$$ 0 0
$$169$$ 3443.62 1.56742
$$170$$ −3244.41 −1.46374
$$171$$ 0 0
$$172$$ −2063.93 −0.914960
$$173$$ 1330.40 0.584675 0.292337 0.956315i $$-0.405567\pi$$
0.292337 + 0.956315i $$0.405567\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1720.10 −0.736689
$$177$$ 0 0
$$178$$ 2207.57 0.929576
$$179$$ 4088.95 1.70739 0.853694 0.520775i $$-0.174357\pi$$
0.853694 + 0.520775i $$0.174357\pi$$
$$180$$ 0 0
$$181$$ 1223.86 0.502591 0.251295 0.967910i $$-0.419143\pi$$
0.251295 + 0.967910i $$0.419143\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1822.48 0.730192
$$185$$ −2436.57 −0.968325
$$186$$ 0 0
$$187$$ −1646.00 −0.643677
$$188$$ −550.493 −0.213558
$$189$$ 0 0
$$190$$ 324.826 0.124028
$$191$$ 1418.72 0.537460 0.268730 0.963216i $$-0.413396\pi$$
0.268730 + 0.963216i $$0.413396\pi$$
$$192$$ 0 0
$$193$$ −3519.66 −1.31270 −0.656348 0.754458i $$-0.727900\pi$$
−0.656348 + 0.754458i $$0.727900\pi$$
$$194$$ 3824.76 1.41547
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −689.885 −0.249504 −0.124752 0.992188i $$-0.539813\pi$$
−0.124752 + 0.992188i $$0.539813\pi$$
$$198$$ 0 0
$$199$$ 511.903 0.182351 0.0911754 0.995835i $$-0.470938\pi$$
0.0911754 + 0.995835i $$0.470938\pi$$
$$200$$ −141.193 −0.0499192
$$201$$ 0 0
$$202$$ 506.248 0.176334
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −997.115 −0.339715
$$206$$ 1206.71 0.408133
$$207$$ 0 0
$$208$$ 5816.49 1.93895
$$209$$ 164.796 0.0545414
$$210$$ 0 0
$$211$$ −4487.25 −1.46405 −0.732027 0.681276i $$-0.761425\pi$$
−0.732027 + 0.681276i $$0.761425\pi$$
$$212$$ −1645.30 −0.533016
$$213$$ 0 0
$$214$$ −3877.18 −1.23850
$$215$$ −4376.89 −1.38838
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 940.961 0.292339
$$219$$ 0 0
$$220$$ 1476.13 0.452366
$$221$$ 5565.94 1.69414
$$222$$ 0 0
$$223$$ −779.975 −0.234220 −0.117110 0.993119i $$-0.537363\pi$$
−0.117110 + 0.993119i $$0.537363\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7340.08 −2.16042
$$227$$ 733.716 0.214531 0.107265 0.994230i $$-0.465791\pi$$
0.107265 + 0.994230i $$0.465791\pi$$
$$228$$ 0 0
$$229$$ 4774.12 1.37765 0.688827 0.724926i $$-0.258126\pi$$
0.688827 + 0.724926i $$0.258126\pi$$
$$230$$ −9008.39 −2.58259
$$231$$ 0 0
$$232$$ −1313.50 −0.371706
$$233$$ 6104.15 1.71629 0.858146 0.513406i $$-0.171617\pi$$
0.858146 + 0.513406i $$0.171617\pi$$
$$234$$ 0 0
$$235$$ −1167.41 −0.324057
$$236$$ −2851.01 −0.786377
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6750.43 1.82698 0.913491 0.406859i $$-0.133376\pi$$
0.913491 + 0.406859i $$0.133376\pi$$
$$240$$ 0 0
$$241$$ −1720.38 −0.459833 −0.229916 0.973210i $$-0.573845\pi$$
−0.229916 + 0.973210i $$0.573845\pi$$
$$242$$ −3089.08 −0.820552
$$243$$ 0 0
$$244$$ 3898.50 1.02285
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −557.255 −0.143552
$$248$$ −1456.39 −0.372907
$$249$$ 0 0
$$250$$ −4774.40 −1.20784
$$251$$ −5719.78 −1.43836 −0.719181 0.694822i $$-0.755483\pi$$
−0.719181 + 0.694822i $$0.755483\pi$$
$$252$$ 0 0
$$253$$ −4570.27 −1.13569
$$254$$ 1978.20 0.488675
$$255$$ 0 0
$$256$$ 5370.30 1.31111
$$257$$ 812.156 0.197124 0.0985621 0.995131i $$-0.468576\pi$$
0.0985621 + 0.995131i $$0.468576\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −4991.51 −1.19062
$$261$$ 0 0
$$262$$ −9440.54 −2.22610
$$263$$ −6976.18 −1.63563 −0.817813 0.575484i $$-0.804814\pi$$
−0.817813 + 0.575484i $$0.804814\pi$$
$$264$$ 0 0
$$265$$ −3489.11 −0.808809
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2076.26 0.473237
$$269$$ 265.535 0.0601857 0.0300928 0.999547i $$-0.490420\pi$$
0.0300928 + 0.999547i $$0.490420\pi$$
$$270$$ 0 0
$$271$$ −8109.83 −1.81785 −0.908925 0.416960i $$-0.863096\pi$$
−0.908925 + 0.416960i $$0.863096\pi$$
$$272$$ 5739.48 1.27944
$$273$$ 0 0
$$274$$ −8356.95 −1.84256
$$275$$ 354.072 0.0776412
$$276$$ 0 0
$$277$$ −5941.40 −1.28875 −0.644375 0.764709i $$-0.722883\pi$$
−0.644375 + 0.764709i $$0.722883\pi$$
$$278$$ −6775.08 −1.46166
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5686.26 1.20717 0.603584 0.797300i $$-0.293739\pi$$
0.603584 + 0.797300i $$0.293739\pi$$
$$282$$ 0 0
$$283$$ 3582.77 0.752558 0.376279 0.926506i $$-0.377203\pi$$
0.376279 + 0.926506i $$0.377203\pi$$
$$284$$ 681.428 0.142378
$$285$$ 0 0
$$286$$ −6151.19 −1.27178
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 579.254 0.117902
$$290$$ 6492.54 1.31467
$$291$$ 0 0
$$292$$ 3818.55 0.765287
$$293$$ 140.120 0.0279382 0.0139691 0.999902i $$-0.495553\pi$$
0.0139691 + 0.999902i $$0.495553\pi$$
$$294$$ 0 0
$$295$$ −6046.03 −1.19327
$$296$$ 1817.76 0.356943
$$297$$ 0 0
$$298$$ 2309.16 0.448880
$$299$$ 15454.3 2.98912
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4442.35 −0.846453
$$303$$ 0 0
$$304$$ −574.630 −0.108412
$$305$$ 8267.39 1.55210
$$306$$ 0 0
$$307$$ −2318.11 −0.430949 −0.215475 0.976509i $$-0.569130\pi$$
−0.215475 + 0.976509i $$0.569130\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 7198.83 1.31892
$$311$$ 5860.43 1.06853 0.534267 0.845316i $$-0.320588\pi$$
0.534267 + 0.845316i $$0.320588\pi$$
$$312$$ 0 0
$$313$$ −401.620 −0.0725268 −0.0362634 0.999342i $$-0.511546\pi$$
−0.0362634 + 0.999342i $$0.511546\pi$$
$$314$$ −10975.4 −1.97254
$$315$$ 0 0
$$316$$ −3750.52 −0.667668
$$317$$ 9058.74 1.60501 0.802507 0.596643i $$-0.203499\pi$$
0.802507 + 0.596643i $$0.203499\pi$$
$$318$$ 0 0
$$319$$ 3293.89 0.578127
$$320$$ −2045.24 −0.357289
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −549.877 −0.0947244
$$324$$ 0 0
$$325$$ −1197.29 −0.204350
$$326$$ 3010.98 0.511542
$$327$$ 0 0
$$328$$ 743.879 0.125225
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2357.68 −0.391510 −0.195755 0.980653i $$-0.562716\pi$$
−0.195755 + 0.980653i $$0.562716\pi$$
$$332$$ 6617.43 1.09391
$$333$$ 0 0
$$334$$ 5891.71 0.965210
$$335$$ 4403.03 0.718100
$$336$$ 0 0
$$337$$ 4774.30 0.771729 0.385865 0.922555i $$-0.373903\pi$$
0.385865 + 0.922555i $$0.373903\pi$$
$$338$$ 12698.6 2.04353
$$339$$ 0 0
$$340$$ −4925.43 −0.785643
$$341$$ 3652.22 0.579996
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 3265.30 0.511783
$$345$$ 0 0
$$346$$ 4905.96 0.762272
$$347$$ 7684.47 1.18883 0.594415 0.804159i $$-0.297384\pi$$
0.594415 + 0.804159i $$0.297384\pi$$
$$348$$ 0 0
$$349$$ 3771.66 0.578488 0.289244 0.957255i $$-0.406596\pi$$
0.289244 + 0.957255i $$0.406596\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4769.28 −0.722170
$$353$$ −7924.16 −1.19479 −0.597394 0.801948i $$-0.703797\pi$$
−0.597394 + 0.801948i $$0.703797\pi$$
$$354$$ 0 0
$$355$$ 1445.08 0.216047
$$356$$ 3351.37 0.498939
$$357$$ 0 0
$$358$$ 15078.3 2.22601
$$359$$ −9078.25 −1.33463 −0.667314 0.744776i $$-0.732556\pi$$
−0.667314 + 0.744776i $$0.732556\pi$$
$$360$$ 0 0
$$361$$ −6803.95 −0.991974
$$362$$ 4513.08 0.655255
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 8097.85 1.16126
$$366$$ 0 0
$$367$$ 2091.98 0.297549 0.148775 0.988871i $$-0.452467\pi$$
0.148775 + 0.988871i $$0.452467\pi$$
$$368$$ 15936.2 2.25742
$$369$$ 0 0
$$370$$ −8985.03 −1.26246
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9157.84 −1.27125 −0.635623 0.771999i $$-0.719257\pi$$
−0.635623 + 0.771999i $$0.719257\pi$$
$$374$$ −6069.76 −0.839197
$$375$$ 0 0
$$376$$ 870.925 0.119453
$$377$$ −11138.3 −1.52162
$$378$$ 0 0
$$379$$ −604.558 −0.0819368 −0.0409684 0.999160i $$-0.513044\pi$$
−0.0409684 + 0.999160i $$0.513044\pi$$
$$380$$ 493.127 0.0665708
$$381$$ 0 0
$$382$$ 5231.63 0.700716
$$383$$ 2872.17 0.383188 0.191594 0.981474i $$-0.438634\pi$$
0.191594 + 0.981474i $$0.438634\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −12979.0 −1.71143
$$387$$ 0 0
$$388$$ 5806.47 0.759740
$$389$$ −2832.57 −0.369196 −0.184598 0.982814i $$-0.559098\pi$$
−0.184598 + 0.982814i $$0.559098\pi$$
$$390$$ 0 0
$$391$$ 15249.7 1.97241
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2544.00 −0.325292
$$395$$ −7953.58 −1.01313
$$396$$ 0 0
$$397$$ 1788.13 0.226055 0.113027 0.993592i $$-0.463945\pi$$
0.113027 + 0.993592i $$0.463945\pi$$
$$398$$ 1887.68 0.237741
$$399$$ 0 0
$$400$$ −1234.62 −0.154328
$$401$$ 6715.05 0.836243 0.418122 0.908391i $$-0.362689\pi$$
0.418122 + 0.908391i $$0.362689\pi$$
$$402$$ 0 0
$$403$$ −12349.9 −1.52654
$$404$$ 768.548 0.0946452
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4558.42 −0.555166
$$408$$ 0 0
$$409$$ 14483.0 1.75095 0.875477 0.483259i $$-0.160547\pi$$
0.875477 + 0.483259i $$0.160547\pi$$
$$410$$ −3676.93 −0.442904
$$411$$ 0 0
$$412$$ 1831.94 0.219061
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 14033.3 1.65992
$$416$$ 16127.3 1.90073
$$417$$ 0 0
$$418$$ 607.696 0.0711086
$$419$$ 4513.58 0.526260 0.263130 0.964760i $$-0.415245\pi$$
0.263130 + 0.964760i $$0.415245\pi$$
$$420$$ 0 0
$$421$$ 8684.07 1.00531 0.502655 0.864487i $$-0.332357\pi$$
0.502655 + 0.864487i $$0.332357\pi$$
$$422$$ −16547.1 −1.90877
$$423$$ 0 0
$$424$$ 2602.99 0.298142
$$425$$ −1181.44 −0.134843
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −5886.04 −0.664749
$$429$$ 0 0
$$430$$ −16140.1 −1.81011
$$431$$ −10770.7 −1.20373 −0.601863 0.798599i $$-0.705575\pi$$
−0.601863 + 0.798599i $$0.705575\pi$$
$$432$$ 0 0
$$433$$ −1704.52 −0.189178 −0.0945890 0.995516i $$-0.530154\pi$$
−0.0945890 + 0.995516i $$0.530154\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 1428.50 0.156910
$$437$$ −1526.78 −0.167130
$$438$$ 0 0
$$439$$ −4516.83 −0.491063 −0.245531 0.969389i $$-0.578962\pi$$
−0.245531 + 0.969389i $$0.578962\pi$$
$$440$$ −2335.35 −0.253031
$$441$$ 0 0
$$442$$ 20524.8 2.20875
$$443$$ 13103.6 1.40536 0.702678 0.711508i $$-0.251987\pi$$
0.702678 + 0.711508i $$0.251987\pi$$
$$444$$ 0 0
$$445$$ 7107.12 0.757101
$$446$$ −2876.22 −0.305365
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1429.10 0.150208 0.0751040 0.997176i $$-0.476071\pi$$
0.0751040 + 0.997176i $$0.476071\pi$$
$$450$$ 0 0
$$451$$ −1865.44 −0.194767
$$452$$ −11143.2 −1.15958
$$453$$ 0 0
$$454$$ 2705.63 0.279695
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −16338.3 −1.67237 −0.836186 0.548447i $$-0.815219\pi$$
−0.836186 + 0.548447i $$0.815219\pi$$
$$458$$ 17604.9 1.79612
$$459$$ 0 0
$$460$$ −13675.9 −1.38618
$$461$$ 13542.4 1.36818 0.684089 0.729398i $$-0.260200\pi$$
0.684089 + 0.729398i $$0.260200\pi$$
$$462$$ 0 0
$$463$$ 1851.49 0.185844 0.0929222 0.995673i $$-0.470379\pi$$
0.0929222 + 0.995673i $$0.470379\pi$$
$$464$$ −11485.5 −1.14914
$$465$$ 0 0
$$466$$ 22509.5 2.23762
$$467$$ 79.3624 0.00786393 0.00393196 0.999992i $$-0.498748\pi$$
0.00393196 + 0.999992i $$0.498748\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4304.91 −0.422491
$$471$$ 0 0
$$472$$ 4510.53 0.439860
$$473$$ −8188.45 −0.795994
$$474$$ 0 0
$$475$$ 118.284 0.0114258
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 24892.7 2.38194
$$479$$ 1895.96 0.180853 0.0904263 0.995903i $$-0.471177\pi$$
0.0904263 + 0.995903i $$0.471177\pi$$
$$480$$ 0 0
$$481$$ 15414.2 1.46118
$$482$$ −6344.04 −0.599509
$$483$$ 0 0
$$484$$ −4689.62 −0.440422
$$485$$ 12313.6 1.15285
$$486$$ 0 0
$$487$$ 3897.51 0.362655 0.181327 0.983423i $$-0.441961\pi$$
0.181327 + 0.983423i $$0.441961\pi$$
$$488$$ −6167.74 −0.572132
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 13409.7 1.23253 0.616266 0.787538i $$-0.288645\pi$$
0.616266 + 0.787538i $$0.288645\pi$$
$$492$$ 0 0
$$493$$ −10990.8 −1.00406
$$494$$ −2054.92 −0.187156
$$495$$ 0 0
$$496$$ −12735.0 −1.15286
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4044.94 0.362878 0.181439 0.983402i $$-0.441924\pi$$
0.181439 + 0.983402i $$0.441924\pi$$
$$500$$ −7248.15 −0.648294
$$501$$ 0 0
$$502$$ −21092.1 −1.87527
$$503$$ 15977.7 1.41632 0.708160 0.706052i $$-0.249526\pi$$
0.708160 + 0.706052i $$0.249526\pi$$
$$504$$ 0 0
$$505$$ 1629.83 0.143617
$$506$$ −16853.2 −1.48067
$$507$$ 0 0
$$508$$ 3003.16 0.262291
$$509$$ −3949.42 −0.343919 −0.171960 0.985104i $$-0.555010\pi$$
−0.171960 + 0.985104i $$0.555010\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11142.7 0.961805
$$513$$ 0 0
$$514$$ 2994.89 0.257002
$$515$$ 3884.92 0.332408
$$516$$ 0 0
$$517$$ −2184.03 −0.185790
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 7896.97 0.665971
$$521$$ −19894.6 −1.67294 −0.836468 0.548016i $$-0.815383\pi$$
−0.836468 + 0.548016i $$0.815383\pi$$
$$522$$ 0 0
$$523$$ −20668.8 −1.72807 −0.864037 0.503428i $$-0.832072\pi$$
−0.864037 + 0.503428i $$0.832072\pi$$
$$524$$ −14331.9 −1.19483
$$525$$ 0 0
$$526$$ −25725.2 −2.13245
$$527$$ −12186.4 −1.00730
$$528$$ 0 0
$$529$$ 30175.2 2.48008
$$530$$ −12866.4 −1.05449
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 6307.95 0.512623
$$534$$ 0 0
$$535$$ −12482.3 −1.00870
$$536$$ −3284.80 −0.264705
$$537$$ 0 0
$$538$$ 979.179 0.0784673
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9452.74 −0.751211 −0.375605 0.926780i $$-0.622565\pi$$
−0.375605 + 0.926780i $$0.622565\pi$$
$$542$$ −29905.6 −2.37003
$$543$$ 0 0
$$544$$ 15913.8 1.25422
$$545$$ 3029.36 0.238098
$$546$$ 0 0
$$547$$ 15550.8 1.21555 0.607773 0.794111i $$-0.292063\pi$$
0.607773 + 0.794111i $$0.292063\pi$$
$$548$$ −12686.9 −0.988974
$$549$$ 0 0
$$550$$ 1305.67 0.101225
$$551$$ 1100.38 0.0850780
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −21909.3 −1.68021
$$555$$ 0 0
$$556$$ −10285.4 −0.784531
$$557$$ −15865.0 −1.20686 −0.603432 0.797414i $$-0.706201\pi$$
−0.603432 + 0.797414i $$0.706201\pi$$
$$558$$ 0 0
$$559$$ 27689.1 2.09504
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20968.5 1.57385
$$563$$ −198.200 −0.0148369 −0.00741843 0.999972i $$-0.502361\pi$$
−0.00741843 + 0.999972i $$0.502361\pi$$
$$564$$ 0 0
$$565$$ −23630.9 −1.75957
$$566$$ 13211.7 0.981150
$$567$$ 0 0
$$568$$ −1078.07 −0.0796390
$$569$$ −18915.2 −1.39361 −0.696807 0.717259i $$-0.745396\pi$$
−0.696807 + 0.717259i $$0.745396\pi$$
$$570$$ 0 0
$$571$$ 1678.21 0.122996 0.0614981 0.998107i $$-0.480412\pi$$
0.0614981 + 0.998107i $$0.480412\pi$$
$$572$$ −9338.29 −0.682611
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3280.37 −0.237914
$$576$$ 0 0
$$577$$ −9694.82 −0.699481 −0.349741 0.936847i $$-0.613730\pi$$
−0.349741 + 0.936847i $$0.613730\pi$$
$$578$$ 2136.04 0.153716
$$579$$ 0 0
$$580$$ 9856.49 0.705636
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6527.56 −0.463711
$$584$$ −6041.25 −0.428063
$$585$$ 0 0
$$586$$ 516.702 0.0364245
$$587$$ 19044.8 1.33912 0.669559 0.742759i $$-0.266483\pi$$
0.669559 + 0.742759i $$0.266483\pi$$
$$588$$ 0 0
$$589$$ 1220.09 0.0853530
$$590$$ −22295.2 −1.55572
$$591$$ 0 0
$$592$$ 15894.9 1.10350
$$593$$ −6298.22 −0.436150 −0.218075 0.975932i $$-0.569978\pi$$
−0.218075 + 0.975932i $$0.569978\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3505.60 0.240931
$$597$$ 0 0
$$598$$ 56988.9 3.89708
$$599$$ 987.205 0.0673391 0.0336695 0.999433i $$-0.489281\pi$$
0.0336695 + 0.999433i $$0.489281\pi$$
$$600$$ 0 0
$$601$$ 20530.0 1.39341 0.696703 0.717360i $$-0.254650\pi$$
0.696703 + 0.717360i $$0.254650\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −6744.05 −0.454324
$$605$$ −9945.08 −0.668306
$$606$$ 0 0
$$607$$ −2185.99 −0.146172 −0.0730861 0.997326i $$-0.523285\pi$$
−0.0730861 + 0.997326i $$0.523285\pi$$
$$608$$ −1593.27 −0.106275
$$609$$ 0 0
$$610$$ 30486.6 2.02355
$$611$$ 7385.28 0.488996
$$612$$ 0 0
$$613$$ −14150.3 −0.932343 −0.466172 0.884694i $$-0.654367\pi$$
−0.466172 + 0.884694i $$0.654367\pi$$
$$614$$ −8548.19 −0.561852
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1478.16 0.0964478 0.0482239 0.998837i $$-0.484644\pi$$
0.0482239 + 0.998837i $$0.484644\pi$$
$$618$$ 0 0
$$619$$ 6480.22 0.420779 0.210389 0.977618i $$-0.432527\pi$$
0.210389 + 0.977618i $$0.432527\pi$$
$$620$$ 10928.7 0.707917
$$621$$ 0 0
$$622$$ 21610.8 1.39311
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −17363.6 −1.11127
$$626$$ −1481.00 −0.0945571
$$627$$ 0 0
$$628$$ −16662.1 −1.05874
$$629$$ 15210.2 0.964180
$$630$$ 0 0
$$631$$ −1844.94 −0.116396 −0.0581979 0.998305i $$-0.518535\pi$$
−0.0581979 + 0.998305i $$0.518535\pi$$
$$632$$ 5933.62 0.373460
$$633$$ 0 0
$$634$$ 33404.8 2.09254
$$635$$ 6368.69 0.398006
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 12146.5 0.753736
$$639$$ 0 0
$$640$$ 12852.3 0.793797
$$641$$ −13064.2 −0.804997 −0.402498 0.915421i $$-0.631858\pi$$
−0.402498 + 0.915421i $$0.631858\pi$$
$$642$$ 0 0
$$643$$ 10753.3 0.659518 0.329759 0.944065i $$-0.393032\pi$$
0.329759 + 0.944065i $$0.393032\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −2027.71 −0.123497
$$647$$ 1836.66 0.111602 0.0558010 0.998442i $$-0.482229\pi$$
0.0558010 + 0.998442i $$0.482229\pi$$
$$648$$ 0 0
$$649$$ −11311.1 −0.684130
$$650$$ −4415.09 −0.266422
$$651$$ 0 0
$$652$$ 4571.04 0.274564
$$653$$ −9412.50 −0.564073 −0.282037 0.959404i $$-0.591010\pi$$
−0.282037 + 0.959404i $$0.591010\pi$$
$$654$$ 0 0
$$655$$ −30393.2 −1.81307
$$656$$ 6504.63 0.387139
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1425.12 0.0842408 0.0421204 0.999113i $$-0.486589\pi$$
0.0421204 + 0.999113i $$0.486589\pi$$
$$660$$ 0 0
$$661$$ −13381.3 −0.787403 −0.393701 0.919238i $$-0.628806\pi$$
−0.393701 + 0.919238i $$0.628806\pi$$
$$662$$ −8694.12 −0.510433
$$663$$ 0 0
$$664$$ −10469.3 −0.611879
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −30516.9 −1.77154
$$668$$ 8944.36 0.518066
$$669$$ 0 0
$$670$$ 16236.5 0.936226
$$671$$ 15466.9 0.889857
$$672$$ 0 0
$$673$$ −14806.5 −0.848069 −0.424035 0.905646i $$-0.639387\pi$$
−0.424035 + 0.905646i $$0.639387\pi$$
$$674$$ 17605.6 1.00615
$$675$$ 0 0
$$676$$ 19278.1 1.09684
$$677$$ −15282.0 −0.867555 −0.433777 0.901020i $$-0.642820\pi$$
−0.433777 + 0.901020i $$0.642820\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 7792.42 0.439449
$$681$$ 0 0
$$682$$ 13467.8 0.756172
$$683$$ −7166.26 −0.401478 −0.200739 0.979645i $$-0.564334\pi$$
−0.200739 + 0.979645i $$0.564334\pi$$
$$684$$ 0 0
$$685$$ −26904.6 −1.50069
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 28552.5 1.58220
$$689$$ 22072.9 1.22048
$$690$$ 0 0
$$691$$ 11758.4 0.647339 0.323669 0.946170i $$-0.395083\pi$$
0.323669 + 0.946170i $$0.395083\pi$$
$$692$$ 7447.87 0.409141
$$693$$ 0 0
$$694$$ 28337.0 1.54994
$$695$$ −21811.9 −1.19046
$$696$$ 0 0
$$697$$ 6224.44 0.338261
$$698$$ 13908.3 0.754207
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3610.56 −0.194535 −0.0972674 0.995258i $$-0.531010\pi$$
−0.0972674 + 0.995258i $$0.531010\pi$$
$$702$$ 0 0
$$703$$ −1522.82 −0.0816989
$$704$$ −3826.31 −0.204843
$$705$$ 0 0
$$706$$ −29220.9 −1.55771
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 13816.1 0.731842 0.365921 0.930646i $$-0.380754\pi$$
0.365921 + 0.930646i $$0.380754\pi$$
$$710$$ 5328.83 0.281672
$$711$$ 0 0
$$712$$ −5302.14 −0.279082
$$713$$ −33836.7 −1.77727
$$714$$ 0 0
$$715$$ −19803.3 −1.03581
$$716$$ 22890.8 1.19479
$$717$$ 0 0
$$718$$ −33476.7 −1.74003
$$719$$ −382.174 −0.0198229 −0.00991147 0.999951i $$-0.503155\pi$$
−0.00991147 + 0.999951i $$0.503155\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −25090.1 −1.29329
$$723$$ 0 0
$$724$$ 6851.43 0.351701
$$725$$ 2364.23 0.121111
$$726$$ 0 0
$$727$$ −3909.70 −0.199453 −0.0997267 0.995015i $$-0.531797\pi$$
−0.0997267 + 0.995015i $$0.531797\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 29861.4 1.51400
$$731$$ 27322.6 1.38244
$$732$$ 0 0
$$733$$ −1365.99 −0.0688320 −0.0344160 0.999408i $$-0.510957\pi$$
−0.0344160 + 0.999408i $$0.510957\pi$$
$$734$$ 7714.34 0.387931
$$735$$ 0 0
$$736$$ 44186.0 2.21293
$$737$$ 8237.35 0.411705
$$738$$ 0 0
$$739$$ 17222.5 0.857293 0.428647 0.903472i $$-0.358991\pi$$
0.428647 + 0.903472i $$0.358991\pi$$
$$740$$ −13640.4 −0.677610
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −524.887 −0.0259168 −0.0129584 0.999916i $$-0.504125\pi$$
−0.0129584 + 0.999916i $$0.504125\pi$$
$$744$$ 0 0
$$745$$ 7434.19 0.365594
$$746$$ −33770.2 −1.65739
$$747$$ 0 0
$$748$$ −9214.66 −0.450430
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 13190.1 0.640898 0.320449 0.947266i $$-0.396166\pi$$
0.320449 + 0.947266i $$0.396166\pi$$
$$752$$ 7615.55 0.369296
$$753$$ 0 0
$$754$$ −41073.2 −1.98381
$$755$$ −14301.8 −0.689401
$$756$$ 0 0
$$757$$ −24776.1 −1.18957 −0.594784 0.803885i $$-0.702763\pi$$
−0.594784 + 0.803885i $$0.702763\pi$$
$$758$$ −2229.35 −0.106825
$$759$$ 0 0
$$760$$ −780.167 −0.0372363
$$761$$ −7116.89 −0.339010 −0.169505 0.985529i $$-0.554217\pi$$
−0.169505 + 0.985529i $$0.554217\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 7942.27 0.376101
$$765$$ 0 0
$$766$$ 10591.3 0.499584
$$767$$ 38248.4 1.80061
$$768$$ 0 0
$$769$$ 17.4695 0.000819200 0 0.000409600 1.00000i $$-0.499870\pi$$
0.000409600 1.00000i $$0.499870\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −19703.8 −0.918593
$$773$$ −34872.0 −1.62258 −0.811292 0.584640i $$-0.801236\pi$$
−0.811292 + 0.584640i $$0.801236\pi$$
$$774$$ 0 0
$$775$$ 2621.42 0.121502
$$776$$ −9186.31 −0.424960
$$777$$ 0 0
$$778$$ −10445.3 −0.481341
$$779$$ −623.183 −0.0286622
$$780$$ 0 0
$$781$$ 2703.50 0.123865
$$782$$ 56234.5 2.57154
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −35334.6 −1.60656
$$786$$ 0 0
$$787$$ −2978.09 −0.134889 −0.0674443 0.997723i $$-0.521485\pi$$
−0.0674443 + 0.997723i $$0.521485\pi$$
$$788$$ −3862.11 −0.174597
$$789$$ 0 0
$$790$$ −29329.4 −1.32088
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −52301.2 −2.34208
$$794$$ 6593.87 0.294720
$$795$$ 0 0
$$796$$ 2865.74 0.127605
$$797$$ 2764.97 0.122886 0.0614431 0.998111i $$-0.480430\pi$$
0.0614431 + 0.998111i $$0.480430\pi$$
$$798$$ 0 0
$$799$$ 7287.50 0.322670
$$800$$ −3423.21 −0.151286
$$801$$ 0 0
$$802$$ 24762.2 1.09026
$$803$$ 15149.7 0.665782
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −45541.3 −1.99023
$$807$$ 0 0
$$808$$ −1215.90 −0.0529398
$$809$$ 18307.8 0.795633 0.397817 0.917465i $$-0.369768\pi$$
0.397817 + 0.917465i $$0.369768\pi$$
$$810$$ 0 0
$$811$$ 21748.4 0.941664 0.470832 0.882223i $$-0.343954\pi$$
0.470832 + 0.882223i $$0.343954\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −16809.5 −0.723800
$$815$$ 9693.63 0.416630
$$816$$ 0 0
$$817$$ −2735.50 −0.117140
$$818$$ 53407.3 2.28281
$$819$$ 0 0
$$820$$ −5582.05 −0.237724
$$821$$ −10998.9 −0.467557 −0.233778 0.972290i $$-0.575109\pi$$
−0.233778 + 0.972290i $$0.575109\pi$$
$$822$$ 0 0
$$823$$ 7773.89 0.329260 0.164630 0.986355i $$-0.447357\pi$$
0.164630 + 0.986355i $$0.447357\pi$$
$$824$$ −2898.27 −0.122532
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 29154.8 1.22589 0.612946 0.790125i $$-0.289984\pi$$
0.612946 + 0.790125i $$0.289984\pi$$
$$828$$ 0 0
$$829$$ −22928.6 −0.960607 −0.480303 0.877102i $$-0.659473\pi$$
−0.480303 + 0.877102i $$0.659473\pi$$
$$830$$ 51748.9 2.16413
$$831$$ 0 0
$$832$$ 12938.6 0.539142
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 18968.0 0.786124
$$836$$ 922.560 0.0381667
$$837$$ 0 0
$$838$$ 16644.2 0.686114
$$839$$ −37893.1 −1.55925 −0.779627 0.626244i $$-0.784591\pi$$
−0.779627 + 0.626244i $$0.784591\pi$$
$$840$$ 0 0
$$841$$ −2394.81 −0.0981922
$$842$$ 32023.2 1.31068
$$843$$ 0 0
$$844$$ −25120.6 −1.02451
$$845$$ 40882.2 1.66437
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 22761.1 0.921720
$$849$$ 0 0
$$850$$ −4356.64 −0.175802
$$851$$ 42232.4 1.70118
$$852$$ 0 0
$$853$$ −11927.9 −0.478783 −0.239392 0.970923i $$-0.576948\pi$$
−0.239392 + 0.970923i $$0.576948\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 9312.19 0.371827
$$857$$ −34449.4 −1.37312 −0.686562 0.727071i $$-0.740881\pi$$
−0.686562 + 0.727071i $$0.740881\pi$$
$$858$$ 0 0
$$859$$ −19600.3 −0.778528 −0.389264 0.921126i $$-0.627271\pi$$
−0.389264 + 0.921126i $$0.627271\pi$$
$$860$$ −24502.7 −0.971554
$$861$$ 0 0
$$862$$ −39717.8 −1.56936
$$863$$ 14606.4 0.576140 0.288070 0.957609i $$-0.406986\pi$$
0.288070 + 0.957609i $$0.406986\pi$$
$$864$$ 0 0
$$865$$ 15794.4 0.620839
$$866$$ −6285.55 −0.246642
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −14879.8 −0.580856
$$870$$ 0 0
$$871$$ −27854.5 −1.08360
$$872$$ −2260.00 −0.0877674
$$873$$ 0 0
$$874$$ −5630.12 −0.217897
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1509.83 0.0581337 0.0290669 0.999577i $$-0.490746\pi$$
0.0290669 + 0.999577i $$0.490746\pi$$
$$878$$ −16656.1 −0.640225
$$879$$ 0 0
$$880$$ −20420.8 −0.782256
$$881$$ −38219.1 −1.46156 −0.730779 0.682614i $$-0.760843\pi$$
−0.730779 + 0.682614i $$0.760843\pi$$
$$882$$ 0 0
$$883$$ 40698.0 1.55107 0.775535 0.631304i $$-0.217480\pi$$
0.775535 + 0.631304i $$0.217480\pi$$
$$884$$ 31159.3 1.18552
$$885$$ 0 0
$$886$$ 48320.7 1.83224
$$887$$ −10228.5 −0.387193 −0.193597 0.981081i $$-0.562015\pi$$
−0.193597 + 0.981081i $$0.562015\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 26208.0 0.987074
$$891$$ 0 0
$$892$$ −4366.46 −0.163901
$$893$$ −729.615 −0.0273411
$$894$$ 0 0
$$895$$ 48543.5 1.81300
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 5269.91 0.195834
$$899$$ 24386.8 0.904723
$$900$$ 0 0
$$901$$ 21780.6 0.805347
$$902$$ −6878.94 −0.253929
$$903$$ 0 0
$$904$$ 17629.4 0.648611
$$905$$ 14529.6 0.533678
$$906$$ 0 0
$$907$$ −36302.2 −1.32899 −0.664496 0.747292i $$-0.731354\pi$$
−0.664496 + 0.747292i $$0.731354\pi$$
$$908$$ 4107.49 0.150123
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 23768.0 0.864401 0.432201 0.901777i $$-0.357737\pi$$
0.432201 + 0.901777i $$0.357737\pi$$
$$912$$ 0 0
$$913$$ 26254.0 0.951678
$$914$$ −60248.7 −2.18036
$$915$$ 0 0
$$916$$ 26726.5 0.964048
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −1364.96 −0.0489944 −0.0244972 0.999700i $$-0.507798\pi$$
−0.0244972 + 0.999700i $$0.507798\pi$$
$$920$$ 21636.3 0.775357
$$921$$ 0 0
$$922$$ 49938.4 1.78377
$$923$$ −9141.86 −0.326011
$$924$$ 0 0
$$925$$ −3271.86 −0.116301
$$926$$ 6827.50 0.242296
$$927$$ 0 0
$$928$$ −31845.8 −1.12650
$$929$$ −18850.3 −0.665725 −0.332862 0.942975i $$-0.608014\pi$$
−0.332862 + 0.942975i $$0.608014\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 34172.2 1.20102
$$933$$ 0 0
$$934$$ 292.655 0.0102526
$$935$$ −19541.2 −0.683492
$$936$$ 0 0
$$937$$ 31404.8 1.09493 0.547465 0.836828i $$-0.315593\pi$$
0.547465 + 0.836828i $$0.315593\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −6535.40 −0.226767
$$941$$ 31589.8 1.09437 0.547184 0.837013i $$-0.315700\pi$$
0.547184 + 0.837013i $$0.315700\pi$$
$$942$$ 0 0
$$943$$ 17282.7 0.596821
$$944$$ 39441.0 1.35985
$$945$$ 0 0
$$946$$ −30195.5 −1.03778
$$947$$ 12062.2 0.413907 0.206954 0.978351i $$-0.433645\pi$$
0.206954 + 0.978351i $$0.433645\pi$$
$$948$$ 0 0
$$949$$ −51228.7 −1.75232
$$950$$ 436.181 0.0148964
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2655.82 0.0902733 0.0451366 0.998981i $$-0.485628\pi$$
0.0451366 + 0.998981i $$0.485628\pi$$
$$954$$ 0 0
$$955$$ 16842.9 0.570704
$$956$$ 37790.3 1.27848
$$957$$ 0 0
$$958$$ 6991.48 0.235787
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2751.27 −0.0923524
$$962$$ 56841.2 1.90502
$$963$$ 0 0
$$964$$ −9631.06 −0.321779
$$965$$ −41785.0 −1.39389
$$966$$ 0 0
$$967$$ −57274.5 −1.90468 −0.952339 0.305041i $$-0.901330\pi$$
−0.952339 + 0.305041i $$0.901330\pi$$
$$968$$ 7419.35 0.246350
$$969$$ 0 0
$$970$$ 45407.2 1.50303
$$971$$ −25867.0 −0.854904 −0.427452 0.904038i $$-0.640589\pi$$
−0.427452 + 0.904038i $$0.640589\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 14372.3 0.472813
$$975$$ 0 0
$$976$$ −53932.0 −1.76877
$$977$$ −14650.4 −0.479741 −0.239870 0.970805i $$-0.577105\pi$$
−0.239870 + 0.970805i $$0.577105\pi$$
$$978$$ 0 0
$$979$$ 13296.3 0.434066
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 49449.4 1.60692
$$983$$ 4533.90 0.147110 0.0735548 0.997291i $$-0.476566\pi$$
0.0735548 + 0.997291i $$0.476566\pi$$
$$984$$ 0 0
$$985$$ −8190.23 −0.264937
$$986$$ −40529.4 −1.30905
$$987$$ 0 0
$$988$$ −3119.62 −0.100454
$$989$$ 75863.5 2.43915
$$990$$ 0 0
$$991$$ −4375.14 −0.140243 −0.0701216 0.997538i $$-0.522339\pi$$
−0.0701216 + 0.997538i $$0.522339\pi$$
$$992$$ −35310.1 −1.13014
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 6077.25 0.193630
$$996$$ 0 0
$$997$$ −10801.8 −0.343124 −0.171562 0.985173i $$-0.554881\pi$$
−0.171562 + 0.985173i $$0.554881\pi$$
$$998$$ 14916.0 0.473104
$$999$$ 0 0
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 1323.4.a.be.1.5 6
3.2 odd 2 inner 1323.4.a.be.1.2 6
7.3 odd 6 189.4.e.e.163.2 yes 12
7.5 odd 6 189.4.e.e.109.2 12
7.6 odd 2 1323.4.a.bd.1.5 6
21.5 even 6 189.4.e.e.109.5 yes 12
21.17 even 6 189.4.e.e.163.5 yes 12
21.20 even 2 1323.4.a.bd.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.e.109.2 12 7.5 odd 6
189.4.e.e.109.5 yes 12 21.5 even 6
189.4.e.e.163.2 yes 12 7.3 odd 6
189.4.e.e.163.5 yes 12 21.17 even 6
1323.4.a.bd.1.2 6 21.20 even 2
1323.4.a.bd.1.5 6 7.6 odd 2
1323.4.a.be.1.2 6 3.2 odd 2 inner
1323.4.a.be.1.5 6 1.1 even 1 trivial