# Properties

 Label 1216.4.a.o.1.2 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ 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] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.37228 q^{3} -11.8614 q^{5} -26.4891 q^{7} +1.86141 q^{9} +O(q^{10})$$ $$q+5.37228 q^{3} -11.8614 q^{5} -26.4891 q^{7} +1.86141 q^{9} +49.8614 q^{11} +49.0951 q^{13} -63.7228 q^{15} +17.2337 q^{17} +19.0000 q^{19} -142.307 q^{21} +166.965 q^{23} +15.6930 q^{25} -135.052 q^{27} +109.198 q^{29} -273.783 q^{31} +267.870 q^{33} +314.198 q^{35} -167.022 q^{37} +263.753 q^{39} +15.1684 q^{41} -413.557 q^{43} -22.0789 q^{45} -161.438 q^{47} +358.674 q^{49} +92.5842 q^{51} +490.791 q^{53} -591.426 q^{55} +102.073 q^{57} +335.970 q^{59} -725.149 q^{61} -49.3070 q^{63} -582.337 q^{65} -497.709 q^{67} +896.981 q^{69} -798.554 q^{71} -311.174 q^{73} +84.3070 q^{75} -1320.79 q^{77} -665.723 q^{79} -775.793 q^{81} +372.440 q^{83} -204.416 q^{85} +586.644 q^{87} -673.783 q^{89} -1300.49 q^{91} -1470.84 q^{93} -225.367 q^{95} -960.505 q^{97} +92.8124 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} + 5 q^{5} - 30 q^{7} - 25 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 + 5 * q^5 - 30 * q^7 - 25 * q^9 $$2 q + 5 q^{3} + 5 q^{5} - 30 q^{7} - 25 q^{9} + 71 q^{11} + 35 q^{13} - 70 q^{15} + 38 q^{19} - 141 q^{21} - 5 q^{23} + 175 q^{25} - 115 q^{27} - 155 q^{29} - 88 q^{31} + 260 q^{33} + 255 q^{35} - 380 q^{37} + 269 q^{39} - 142 q^{41} - 155 q^{43} - 475 q^{45} - 455 q^{47} + 28 q^{49} + 99 q^{51} + 275 q^{53} - 235 q^{55} + 95 q^{57} + 873 q^{59} - 445 q^{61} + 45 q^{63} - 820 q^{65} - 645 q^{67} + 961 q^{69} - 1712 q^{71} - 990 q^{73} + 25 q^{75} - 1395 q^{77} - 1274 q^{79} - 58 q^{81} + 90 q^{83} - 495 q^{85} + 685 q^{87} - 888 q^{89} - 1251 q^{91} - 1540 q^{93} + 95 q^{95} + 710 q^{97} - 475 q^{99}+O(q^{100})$$ 2 * q + 5 * q^3 + 5 * q^5 - 30 * q^7 - 25 * q^9 + 71 * q^11 + 35 * q^13 - 70 * q^15 + 38 * q^19 - 141 * q^21 - 5 * q^23 + 175 * q^25 - 115 * q^27 - 155 * q^29 - 88 * q^31 + 260 * q^33 + 255 * q^35 - 380 * q^37 + 269 * q^39 - 142 * q^41 - 155 * q^43 - 475 * q^45 - 455 * q^47 + 28 * q^49 + 99 * q^51 + 275 * q^53 - 235 * q^55 + 95 * q^57 + 873 * q^59 - 445 * q^61 + 45 * q^63 - 820 * q^65 - 645 * q^67 + 961 * q^69 - 1712 * q^71 - 990 * q^73 + 25 * q^75 - 1395 * q^77 - 1274 * q^79 - 58 * q^81 + 90 * q^83 - 495 * q^85 + 685 * q^87 - 888 * q^89 - 1251 * q^91 - 1540 * q^93 + 95 * q^95 + 710 * q^97 - 475 * 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
$$3$$ 5.37228 1.03390 0.516948 0.856017i $$-0.327068\pi$$
0.516948 + 0.856017i $$0.327068\pi$$
$$4$$ 0 0
$$5$$ −11.8614 −1.06092 −0.530458 0.847711i $$-0.677980\pi$$
−0.530458 + 0.847711i $$0.677980\pi$$
$$6$$ 0 0
$$7$$ −26.4891 −1.43028 −0.715139 0.698982i $$-0.753637\pi$$
−0.715139 + 0.698982i $$0.753637\pi$$
$$8$$ 0 0
$$9$$ 1.86141 0.0689410
$$10$$ 0 0
$$11$$ 49.8614 1.36671 0.683354 0.730088i $$-0.260521\pi$$
0.683354 + 0.730088i $$0.260521\pi$$
$$12$$ 0 0
$$13$$ 49.0951 1.04743 0.523713 0.851895i $$-0.324547\pi$$
0.523713 + 0.851895i $$0.324547\pi$$
$$14$$ 0 0
$$15$$ −63.7228 −1.09688
$$16$$ 0 0
$$17$$ 17.2337 0.245870 0.122935 0.992415i $$-0.460769\pi$$
0.122935 + 0.992415i $$0.460769\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ −142.307 −1.47876
$$22$$ 0 0
$$23$$ 166.965 1.51368 0.756838 0.653603i $$-0.226743\pi$$
0.756838 + 0.653603i $$0.226743\pi$$
$$24$$ 0 0
$$25$$ 15.6930 0.125544
$$26$$ 0 0
$$27$$ −135.052 −0.962618
$$28$$ 0 0
$$29$$ 109.198 0.699228 0.349614 0.936894i $$-0.386313\pi$$
0.349614 + 0.936894i $$0.386313\pi$$
$$30$$ 0 0
$$31$$ −273.783 −1.58622 −0.793110 0.609079i $$-0.791539\pi$$
−0.793110 + 0.609079i $$0.791539\pi$$
$$32$$ 0 0
$$33$$ 267.870 1.41303
$$34$$ 0 0
$$35$$ 314.198 1.51741
$$36$$ 0 0
$$37$$ −167.022 −0.742114 −0.371057 0.928610i $$-0.621005\pi$$
−0.371057 + 0.928610i $$0.621005\pi$$
$$38$$ 0 0
$$39$$ 263.753 1.08293
$$40$$ 0 0
$$41$$ 15.1684 0.0577783 0.0288892 0.999583i $$-0.490803\pi$$
0.0288892 + 0.999583i $$0.490803\pi$$
$$42$$ 0 0
$$43$$ −413.557 −1.46667 −0.733335 0.679867i $$-0.762037\pi$$
−0.733335 + 0.679867i $$0.762037\pi$$
$$44$$ 0 0
$$45$$ −22.0789 −0.0731406
$$46$$ 0 0
$$47$$ −161.438 −0.501023 −0.250512 0.968114i $$-0.580599\pi$$
−0.250512 + 0.968114i $$0.580599\pi$$
$$48$$ 0 0
$$49$$ 358.674 1.04570
$$50$$ 0 0
$$51$$ 92.5842 0.254204
$$52$$ 0 0
$$53$$ 490.791 1.27199 0.635993 0.771695i $$-0.280591\pi$$
0.635993 + 0.771695i $$0.280591\pi$$
$$54$$ 0 0
$$55$$ −591.426 −1.44996
$$56$$ 0 0
$$57$$ 102.073 0.237192
$$58$$ 0 0
$$59$$ 335.970 0.741349 0.370674 0.928763i $$-0.379127\pi$$
0.370674 + 0.928763i $$0.379127\pi$$
$$60$$ 0 0
$$61$$ −725.149 −1.52206 −0.761032 0.648715i $$-0.775307\pi$$
−0.761032 + 0.648715i $$0.775307\pi$$
$$62$$ 0 0
$$63$$ −49.3070 −0.0986048
$$64$$ 0 0
$$65$$ −582.337 −1.11123
$$66$$ 0 0
$$67$$ −497.709 −0.907535 −0.453768 0.891120i $$-0.649920\pi$$
−0.453768 + 0.891120i $$0.649920\pi$$
$$68$$ 0 0
$$69$$ 896.981 1.56498
$$70$$ 0 0
$$71$$ −798.554 −1.33480 −0.667401 0.744698i $$-0.732593\pi$$
−0.667401 + 0.744698i $$0.732593\pi$$
$$72$$ 0 0
$$73$$ −311.174 −0.498906 −0.249453 0.968387i $$-0.580251\pi$$
−0.249453 + 0.968387i $$0.580251\pi$$
$$74$$ 0 0
$$75$$ 84.3070 0.129799
$$76$$ 0 0
$$77$$ −1320.79 −1.95477
$$78$$ 0 0
$$79$$ −665.723 −0.948097 −0.474049 0.880499i $$-0.657208\pi$$
−0.474049 + 0.880499i $$0.657208\pi$$
$$80$$ 0 0
$$81$$ −775.793 −1.06419
$$82$$ 0 0
$$83$$ 372.440 0.492537 0.246269 0.969202i $$-0.420795\pi$$
0.246269 + 0.969202i $$0.420795\pi$$
$$84$$ 0 0
$$85$$ −204.416 −0.260847
$$86$$ 0 0
$$87$$ 586.644 0.722929
$$88$$ 0 0
$$89$$ −673.783 −0.802481 −0.401240 0.915973i $$-0.631421\pi$$
−0.401240 + 0.915973i $$0.631421\pi$$
$$90$$ 0 0
$$91$$ −1300.49 −1.49811
$$92$$ 0 0
$$93$$ −1470.84 −1.63999
$$94$$ 0 0
$$95$$ −225.367 −0.243391
$$96$$ 0 0
$$97$$ −960.505 −1.00541 −0.502704 0.864459i $$-0.667661\pi$$
−0.502704 + 0.864459i $$0.667661\pi$$
$$98$$ 0 0
$$99$$ 92.8124 0.0942221
$$100$$ 0 0
$$101$$ −1889.68 −1.86169 −0.930845 0.365415i $$-0.880927\pi$$
−0.930845 + 0.365415i $$0.880927\pi$$
$$102$$ 0 0
$$103$$ 760.217 0.727247 0.363624 0.931546i $$-0.381539\pi$$
0.363624 + 0.931546i $$0.381539\pi$$
$$104$$ 0 0
$$105$$ 1687.96 1.56884
$$106$$ 0 0
$$107$$ −681.730 −0.615938 −0.307969 0.951396i $$-0.599649\pi$$
−0.307969 + 0.951396i $$0.599649\pi$$
$$108$$ 0 0
$$109$$ 1310.76 1.15182 0.575910 0.817513i $$-0.304648\pi$$
0.575910 + 0.817513i $$0.304648\pi$$
$$110$$ 0 0
$$111$$ −897.288 −0.767268
$$112$$ 0 0
$$113$$ −713.413 −0.593914 −0.296957 0.954891i $$-0.595972\pi$$
−0.296957 + 0.954891i $$0.595972\pi$$
$$114$$ 0 0
$$115$$ −1980.43 −1.60588
$$116$$ 0 0
$$117$$ 91.3859 0.0722105
$$118$$ 0 0
$$119$$ −456.505 −0.351662
$$120$$ 0 0
$$121$$ 1155.16 0.867889
$$122$$ 0 0
$$123$$ 81.4891 0.0597368
$$124$$ 0 0
$$125$$ 1296.54 0.927725
$$126$$ 0 0
$$127$$ −107.847 −0.0753535 −0.0376768 0.999290i $$-0.511996\pi$$
−0.0376768 + 0.999290i $$0.511996\pi$$
$$128$$ 0 0
$$129$$ −2221.74 −1.51638
$$130$$ 0 0
$$131$$ −373.247 −0.248937 −0.124469 0.992224i $$-0.539723\pi$$
−0.124469 + 0.992224i $$0.539723\pi$$
$$132$$ 0 0
$$133$$ −503.293 −0.328128
$$134$$ 0 0
$$135$$ 1601.90 1.02126
$$136$$ 0 0
$$137$$ −3118.25 −1.94460 −0.972301 0.233732i $$-0.924906\pi$$
−0.972301 + 0.233732i $$0.924906\pi$$
$$138$$ 0 0
$$139$$ 56.3774 0.0344019 0.0172010 0.999852i $$-0.494524\pi$$
0.0172010 + 0.999852i $$0.494524\pi$$
$$140$$ 0 0
$$141$$ −867.288 −0.518006
$$142$$ 0 0
$$143$$ 2447.95 1.43152
$$144$$ 0 0
$$145$$ −1295.25 −0.741823
$$146$$ 0 0
$$147$$ 1926.90 1.08114
$$148$$ 0 0
$$149$$ 1810.51 0.995457 0.497728 0.867333i $$-0.334168\pi$$
0.497728 + 0.867333i $$0.334168\pi$$
$$150$$ 0 0
$$151$$ 2894.72 1.56006 0.780029 0.625744i $$-0.215204\pi$$
0.780029 + 0.625744i $$0.215204\pi$$
$$152$$ 0 0
$$153$$ 32.0789 0.0169505
$$154$$ 0 0
$$155$$ 3247.45 1.68285
$$156$$ 0 0
$$157$$ −1381.71 −0.702371 −0.351185 0.936306i $$-0.614221\pi$$
−0.351185 + 0.936306i $$0.614221\pi$$
$$158$$ 0 0
$$159$$ 2636.67 1.31510
$$160$$ 0 0
$$161$$ −4422.75 −2.16498
$$162$$ 0 0
$$163$$ 3740.83 1.79757 0.898785 0.438389i $$-0.144451\pi$$
0.898785 + 0.438389i $$0.144451\pi$$
$$164$$ 0 0
$$165$$ −3177.31 −1.49911
$$166$$ 0 0
$$167$$ 3085.17 1.42957 0.714783 0.699347i $$-0.246526\pi$$
0.714783 + 0.699347i $$0.246526\pi$$
$$168$$ 0 0
$$169$$ 213.328 0.0970998
$$170$$ 0 0
$$171$$ 35.3667 0.0158161
$$172$$ 0 0
$$173$$ −293.000 −0.128765 −0.0643827 0.997925i $$-0.520508\pi$$
−0.0643827 + 0.997925i $$0.520508\pi$$
$$174$$ 0 0
$$175$$ −415.693 −0.179562
$$176$$ 0 0
$$177$$ 1804.93 0.766478
$$178$$ 0 0
$$179$$ 2996.32 1.25115 0.625574 0.780165i $$-0.284865\pi$$
0.625574 + 0.780165i $$0.284865\pi$$
$$180$$ 0 0
$$181$$ 3265.41 1.34097 0.670487 0.741922i $$-0.266085\pi$$
0.670487 + 0.741922i $$0.266085\pi$$
$$182$$ 0 0
$$183$$ −3895.71 −1.57365
$$184$$ 0 0
$$185$$ 1981.11 0.787321
$$186$$ 0 0
$$187$$ 859.296 0.336032
$$188$$ 0 0
$$189$$ 3577.40 1.37681
$$190$$ 0 0
$$191$$ −1502.27 −0.569113 −0.284556 0.958659i $$-0.591846\pi$$
−0.284556 + 0.958659i $$0.591846\pi$$
$$192$$ 0 0
$$193$$ −4103.30 −1.53037 −0.765186 0.643809i $$-0.777353\pi$$
−0.765186 + 0.643809i $$0.777353\pi$$
$$194$$ 0 0
$$195$$ −3128.48 −1.14890
$$196$$ 0 0
$$197$$ −5420.95 −1.96054 −0.980271 0.197658i $$-0.936666\pi$$
−0.980271 + 0.197658i $$0.936666\pi$$
$$198$$ 0 0
$$199$$ 1666.05 0.593484 0.296742 0.954958i $$-0.404100\pi$$
0.296742 + 0.954958i $$0.404100\pi$$
$$200$$ 0 0
$$201$$ −2673.83 −0.938297
$$202$$ 0 0
$$203$$ −2892.57 −1.00009
$$204$$ 0 0
$$205$$ −179.919 −0.0612980
$$206$$ 0 0
$$207$$ 310.789 0.104354
$$208$$ 0 0
$$209$$ 947.367 0.313544
$$210$$ 0 0
$$211$$ −5865.90 −1.91386 −0.956931 0.290314i $$-0.906240\pi$$
−0.956931 + 0.290314i $$0.906240\pi$$
$$212$$ 0 0
$$213$$ −4290.06 −1.38005
$$214$$ 0 0
$$215$$ 4905.37 1.55602
$$216$$ 0 0
$$217$$ 7252.26 2.26873
$$218$$ 0 0
$$219$$ −1671.71 −0.515817
$$220$$ 0 0
$$221$$ 846.090 0.257530
$$222$$ 0 0
$$223$$ −5402.94 −1.62246 −0.811229 0.584729i $$-0.801201\pi$$
−0.811229 + 0.584729i $$0.801201\pi$$
$$224$$ 0 0
$$225$$ 29.2110 0.00865511
$$226$$ 0 0
$$227$$ 2571.34 0.751833 0.375916 0.926654i $$-0.377328\pi$$
0.375916 + 0.926654i $$0.377328\pi$$
$$228$$ 0 0
$$229$$ −3256.08 −0.939597 −0.469799 0.882774i $$-0.655673\pi$$
−0.469799 + 0.882774i $$0.655673\pi$$
$$230$$ 0 0
$$231$$ −7095.63 −2.02103
$$232$$ 0 0
$$233$$ 1205.58 0.338972 0.169486 0.985533i $$-0.445789\pi$$
0.169486 + 0.985533i $$0.445789\pi$$
$$234$$ 0 0
$$235$$ 1914.88 0.531544
$$236$$ 0 0
$$237$$ −3576.45 −0.980234
$$238$$ 0 0
$$239$$ −3404.52 −0.921424 −0.460712 0.887550i $$-0.652406\pi$$
−0.460712 + 0.887550i $$0.652406\pi$$
$$240$$ 0 0
$$241$$ 3301.51 0.882443 0.441221 0.897398i $$-0.354545\pi$$
0.441221 + 0.897398i $$0.354545\pi$$
$$242$$ 0 0
$$243$$ −521.386 −0.137642
$$244$$ 0 0
$$245$$ −4254.38 −1.10940
$$246$$ 0 0
$$247$$ 932.807 0.240296
$$248$$ 0 0
$$249$$ 2000.85 0.509233
$$250$$ 0 0
$$251$$ 5330.81 1.34055 0.670275 0.742113i $$-0.266176\pi$$
0.670275 + 0.742113i $$0.266176\pi$$
$$252$$ 0 0
$$253$$ 8325.09 2.06875
$$254$$ 0 0
$$255$$ −1098.18 −0.269689
$$256$$ 0 0
$$257$$ 3812.14 0.925272 0.462636 0.886548i $$-0.346904\pi$$
0.462636 + 0.886548i $$0.346904\pi$$
$$258$$ 0 0
$$259$$ 4424.26 1.06143
$$260$$ 0 0
$$261$$ 203.262 0.0482055
$$262$$ 0 0
$$263$$ 899.159 0.210816 0.105408 0.994429i $$-0.466385\pi$$
0.105408 + 0.994429i $$0.466385\pi$$
$$264$$ 0 0
$$265$$ −5821.47 −1.34947
$$266$$ 0 0
$$267$$ −3619.75 −0.829682
$$268$$ 0 0
$$269$$ −2645.77 −0.599685 −0.299842 0.953989i $$-0.596934\pi$$
−0.299842 + 0.953989i $$0.596934\pi$$
$$270$$ 0 0
$$271$$ 516.529 0.115782 0.0578909 0.998323i $$-0.481562\pi$$
0.0578909 + 0.998323i $$0.481562\pi$$
$$272$$ 0 0
$$273$$ −6986.58 −1.54889
$$274$$ 0 0
$$275$$ 782.473 0.171582
$$276$$ 0 0
$$277$$ −2365.14 −0.513023 −0.256512 0.966541i $$-0.582573\pi$$
−0.256512 + 0.966541i $$0.582573\pi$$
$$278$$ 0 0
$$279$$ −509.621 −0.109356
$$280$$ 0 0
$$281$$ −1526.54 −0.324077 −0.162038 0.986784i $$-0.551807\pi$$
−0.162038 + 0.986784i $$0.551807\pi$$
$$282$$ 0 0
$$283$$ −4764.01 −1.00067 −0.500337 0.865831i $$-0.666791\pi$$
−0.500337 + 0.865831i $$0.666791\pi$$
$$284$$ 0 0
$$285$$ −1210.73 −0.251641
$$286$$ 0 0
$$287$$ −401.799 −0.0826391
$$288$$ 0 0
$$289$$ −4616.00 −0.939548
$$290$$ 0 0
$$291$$ −5160.10 −1.03949
$$292$$ 0 0
$$293$$ −3189.07 −0.635862 −0.317931 0.948114i $$-0.602988\pi$$
−0.317931 + 0.948114i $$0.602988\pi$$
$$294$$ 0 0
$$295$$ −3985.08 −0.786509
$$296$$ 0 0
$$297$$ −6733.86 −1.31562
$$298$$ 0 0
$$299$$ 8197.14 1.58546
$$300$$ 0 0
$$301$$ 10954.8 2.09775
$$302$$ 0 0
$$303$$ −10151.9 −1.92479
$$304$$ 0 0
$$305$$ 8601.29 1.61478
$$306$$ 0 0
$$307$$ −5998.56 −1.11517 −0.557583 0.830121i $$-0.688271\pi$$
−0.557583 + 0.830121i $$0.688271\pi$$
$$308$$ 0 0
$$309$$ 4084.10 0.751898
$$310$$ 0 0
$$311$$ 5114.96 0.932614 0.466307 0.884623i $$-0.345584\pi$$
0.466307 + 0.884623i $$0.345584\pi$$
$$312$$ 0 0
$$313$$ 2131.63 0.384942 0.192471 0.981303i $$-0.438350\pi$$
0.192471 + 0.981303i $$0.438350\pi$$
$$314$$ 0 0
$$315$$ 584.851 0.104611
$$316$$ 0 0
$$317$$ −4457.03 −0.789691 −0.394845 0.918748i $$-0.629202\pi$$
−0.394845 + 0.918748i $$0.629202\pi$$
$$318$$ 0 0
$$319$$ 5444.78 0.955640
$$320$$ 0 0
$$321$$ −3662.45 −0.636816
$$322$$ 0 0
$$323$$ 327.440 0.0564064
$$324$$ 0 0
$$325$$ 770.448 0.131498
$$326$$ 0 0
$$327$$ 7041.79 1.19086
$$328$$ 0 0
$$329$$ 4276.34 0.716602
$$330$$ 0 0
$$331$$ 4143.68 0.688088 0.344044 0.938954i $$-0.388203\pi$$
0.344044 + 0.938954i $$0.388203\pi$$
$$332$$ 0 0
$$333$$ −310.895 −0.0511620
$$334$$ 0 0
$$335$$ 5903.53 0.962819
$$336$$ 0 0
$$337$$ −5148.60 −0.832231 −0.416115 0.909312i $$-0.636609\pi$$
−0.416115 + 0.909312i $$0.636609\pi$$
$$338$$ 0 0
$$339$$ −3832.66 −0.614045
$$340$$ 0 0
$$341$$ −13651.2 −2.16790
$$342$$ 0 0
$$343$$ −415.184 −0.0653581
$$344$$ 0 0
$$345$$ −10639.5 −1.66032
$$346$$ 0 0
$$347$$ 5873.56 0.908672 0.454336 0.890830i $$-0.349877\pi$$
0.454336 + 0.890830i $$0.349877\pi$$
$$348$$ 0 0
$$349$$ −5167.58 −0.792590 −0.396295 0.918123i $$-0.629704\pi$$
−0.396295 + 0.918123i $$0.629704\pi$$
$$350$$ 0 0
$$351$$ −6630.37 −1.00827
$$352$$ 0 0
$$353$$ 7191.17 1.08427 0.542135 0.840291i $$-0.317616\pi$$
0.542135 + 0.840291i $$0.317616\pi$$
$$354$$ 0 0
$$355$$ 9471.98 1.41611
$$356$$ 0 0
$$357$$ −2452.47 −0.363582
$$358$$ 0 0
$$359$$ −7049.62 −1.03639 −0.518196 0.855262i $$-0.673396\pi$$
−0.518196 + 0.855262i $$0.673396\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 6205.84 0.897307
$$364$$ 0 0
$$365$$ 3690.96 0.529298
$$366$$ 0 0
$$367$$ 6935.53 0.986463 0.493231 0.869898i $$-0.335816\pi$$
0.493231 + 0.869898i $$0.335816\pi$$
$$368$$ 0 0
$$369$$ 28.2346 0.00398330
$$370$$ 0 0
$$371$$ −13000.6 −1.81929
$$372$$ 0 0
$$373$$ 6346.92 0.881048 0.440524 0.897741i $$-0.354793\pi$$
0.440524 + 0.897741i $$0.354793\pi$$
$$374$$ 0 0
$$375$$ 6965.35 0.959171
$$376$$ 0 0
$$377$$ 5361.10 0.732389
$$378$$ 0 0
$$379$$ −11363.0 −1.54004 −0.770022 0.638017i $$-0.779755\pi$$
−0.770022 + 0.638017i $$0.779755\pi$$
$$380$$ 0 0
$$381$$ −579.386 −0.0779077
$$382$$ 0 0
$$383$$ −3848.09 −0.513390 −0.256695 0.966493i $$-0.582634\pi$$
−0.256695 + 0.966493i $$0.582634\pi$$
$$384$$ 0 0
$$385$$ 15666.4 2.07385
$$386$$ 0 0
$$387$$ −769.798 −0.101114
$$388$$ 0 0
$$389$$ −1936.75 −0.252435 −0.126217 0.992003i $$-0.540284\pi$$
−0.126217 + 0.992003i $$0.540284\pi$$
$$390$$ 0 0
$$391$$ 2877.42 0.372167
$$392$$ 0 0
$$393$$ −2005.19 −0.257375
$$394$$ 0 0
$$395$$ 7896.41 1.00585
$$396$$ 0 0
$$397$$ −3851.40 −0.486892 −0.243446 0.969914i $$-0.578278\pi$$
−0.243446 + 0.969914i $$0.578278\pi$$
$$398$$ 0 0
$$399$$ −2703.83 −0.339251
$$400$$ 0 0
$$401$$ −1381.47 −0.172038 −0.0860189 0.996294i $$-0.527415\pi$$
−0.0860189 + 0.996294i $$0.527415\pi$$
$$402$$ 0 0
$$403$$ −13441.4 −1.66145
$$404$$ 0 0
$$405$$ 9202.00 1.12901
$$406$$ 0 0
$$407$$ −8327.94 −1.01425
$$408$$ 0 0
$$409$$ 368.878 0.0445962 0.0222981 0.999751i $$-0.492902\pi$$
0.0222981 + 0.999751i $$0.492902\pi$$
$$410$$ 0 0
$$411$$ −16752.1 −2.01052
$$412$$ 0 0
$$413$$ −8899.56 −1.06034
$$414$$ 0 0
$$415$$ −4417.66 −0.522541
$$416$$ 0 0
$$417$$ 302.875 0.0355680
$$418$$ 0 0
$$419$$ 1814.85 0.211602 0.105801 0.994387i $$-0.466259\pi$$
0.105801 + 0.994387i $$0.466259\pi$$
$$420$$ 0 0
$$421$$ −4614.01 −0.534141 −0.267070 0.963677i $$-0.586056\pi$$
−0.267070 + 0.963677i $$0.586056\pi$$
$$422$$ 0 0
$$423$$ −300.501 −0.0345410
$$424$$ 0 0
$$425$$ 270.448 0.0308674
$$426$$ 0 0
$$427$$ 19208.6 2.17697
$$428$$ 0 0
$$429$$ 13151.1 1.48005
$$430$$ 0 0
$$431$$ −2739.14 −0.306125 −0.153062 0.988217i $$-0.548914\pi$$
−0.153062 + 0.988217i $$0.548914\pi$$
$$432$$ 0 0
$$433$$ −10827.9 −1.20174 −0.600872 0.799345i $$-0.705180\pi$$
−0.600872 + 0.799345i $$0.705180\pi$$
$$434$$ 0 0
$$435$$ −6958.42 −0.766967
$$436$$ 0 0
$$437$$ 3172.33 0.347261
$$438$$ 0 0
$$439$$ −10359.5 −1.12627 −0.563137 0.826364i $$-0.690406\pi$$
−0.563137 + 0.826364i $$0.690406\pi$$
$$440$$ 0 0
$$441$$ 667.638 0.0720913
$$442$$ 0 0
$$443$$ −8040.36 −0.862322 −0.431161 0.902275i $$-0.641896\pi$$
−0.431161 + 0.902275i $$0.641896\pi$$
$$444$$ 0 0
$$445$$ 7992.01 0.851365
$$446$$ 0 0
$$447$$ 9726.59 1.02920
$$448$$ 0 0
$$449$$ −14071.8 −1.47904 −0.739522 0.673132i $$-0.764948\pi$$
−0.739522 + 0.673132i $$0.764948\pi$$
$$450$$ 0 0
$$451$$ 756.320 0.0789661
$$452$$ 0 0
$$453$$ 15551.2 1.61294
$$454$$ 0 0
$$455$$ 15425.6 1.58937
$$456$$ 0 0
$$457$$ −4372.74 −0.447589 −0.223795 0.974636i $$-0.571845\pi$$
−0.223795 + 0.974636i $$0.571845\pi$$
$$458$$ 0 0
$$459$$ −2327.44 −0.236679
$$460$$ 0 0
$$461$$ 17819.4 1.80029 0.900146 0.435589i $$-0.143460\pi$$
0.900146 + 0.435589i $$0.143460\pi$$
$$462$$ 0 0
$$463$$ 7114.51 0.714124 0.357062 0.934081i $$-0.383778\pi$$
0.357062 + 0.934081i $$0.383778\pi$$
$$464$$ 0 0
$$465$$ 17446.2 1.73989
$$466$$ 0 0
$$467$$ 8208.41 0.813361 0.406681 0.913570i $$-0.366686\pi$$
0.406681 + 0.913570i $$0.366686\pi$$
$$468$$ 0 0
$$469$$ 13183.9 1.29803
$$470$$ 0 0
$$471$$ −7422.92 −0.726178
$$472$$ 0 0
$$473$$ −20620.5 −2.00451
$$474$$ 0 0
$$475$$ 298.166 0.0288017
$$476$$ 0 0
$$477$$ 913.561 0.0876920
$$478$$ 0 0
$$479$$ −11889.5 −1.13413 −0.567064 0.823674i $$-0.691921\pi$$
−0.567064 + 0.823674i $$0.691921\pi$$
$$480$$ 0 0
$$481$$ −8199.95 −0.777309
$$482$$ 0 0
$$483$$ −23760.2 −2.23836
$$484$$ 0 0
$$485$$ 11392.9 1.06665
$$486$$ 0 0
$$487$$ 13922.4 1.29545 0.647725 0.761875i $$-0.275721\pi$$
0.647725 + 0.761875i $$0.275721\pi$$
$$488$$ 0 0
$$489$$ 20096.8 1.85850
$$490$$ 0 0
$$491$$ −2718.22 −0.249840 −0.124920 0.992167i $$-0.539867\pi$$
−0.124920 + 0.992167i $$0.539867\pi$$
$$492$$ 0 0
$$493$$ 1881.89 0.171919
$$494$$ 0 0
$$495$$ −1100.89 −0.0999618
$$496$$ 0 0
$$497$$ 21153.0 1.90914
$$498$$ 0 0
$$499$$ 7830.51 0.702489 0.351244 0.936284i $$-0.385759\pi$$
0.351244 + 0.936284i $$0.385759\pi$$
$$500$$ 0 0
$$501$$ 16574.4 1.47802
$$502$$ 0 0
$$503$$ −6554.35 −0.581002 −0.290501 0.956875i $$-0.593822\pi$$
−0.290501 + 0.956875i $$0.593822\pi$$
$$504$$ 0 0
$$505$$ 22414.3 1.97510
$$506$$ 0 0
$$507$$ 1146.06 0.100391
$$508$$ 0 0
$$509$$ −12611.4 −1.09821 −0.549105 0.835753i $$-0.685031\pi$$
−0.549105 + 0.835753i $$0.685031\pi$$
$$510$$ 0 0
$$511$$ 8242.73 0.713575
$$512$$ 0 0
$$513$$ −2565.98 −0.220840
$$514$$ 0 0
$$515$$ −9017.25 −0.771548
$$516$$ 0 0
$$517$$ −8049.50 −0.684752
$$518$$ 0 0
$$519$$ −1574.08 −0.133130
$$520$$ 0 0
$$521$$ 1695.01 0.142533 0.0712666 0.997457i $$-0.477296\pi$$
0.0712666 + 0.997457i $$0.477296\pi$$
$$522$$ 0 0
$$523$$ 21001.4 1.75589 0.877944 0.478764i $$-0.158915\pi$$
0.877944 + 0.478764i $$0.158915\pi$$
$$524$$ 0 0
$$525$$ −2233.22 −0.185649
$$526$$ 0 0
$$527$$ −4718.28 −0.390003
$$528$$ 0 0
$$529$$ 15710.2 1.29121
$$530$$ 0 0
$$531$$ 625.377 0.0511093
$$532$$ 0 0
$$533$$ 744.696 0.0605185
$$534$$ 0 0
$$535$$ 8086.28 0.653459
$$536$$ 0 0
$$537$$ 16097.1 1.29356
$$538$$ 0 0
$$539$$ 17884.0 1.42916
$$540$$ 0 0
$$541$$ 555.994 0.0441849 0.0220925 0.999756i $$-0.492967\pi$$
0.0220925 + 0.999756i $$0.492967\pi$$
$$542$$ 0 0
$$543$$ 17542.7 1.38643
$$544$$ 0 0
$$545$$ −15547.5 −1.22198
$$546$$ 0 0
$$547$$ 2987.93 0.233555 0.116778 0.993158i $$-0.462744\pi$$
0.116778 + 0.993158i $$0.462744\pi$$
$$548$$ 0 0
$$549$$ −1349.80 −0.104933
$$550$$ 0 0
$$551$$ 2074.77 0.160414
$$552$$ 0 0
$$553$$ 17634.4 1.35604
$$554$$ 0 0
$$555$$ 10643.1 0.814008
$$556$$ 0 0
$$557$$ 4357.38 0.331469 0.165735 0.986170i $$-0.447001\pi$$
0.165735 + 0.986170i $$0.447001\pi$$
$$558$$ 0 0
$$559$$ −20303.6 −1.53623
$$560$$ 0 0
$$561$$ 4616.38 0.347422
$$562$$ 0 0
$$563$$ −1669.67 −0.124988 −0.0624941 0.998045i $$-0.519905\pi$$
−0.0624941 + 0.998045i $$0.519905\pi$$
$$564$$ 0 0
$$565$$ 8462.08 0.630093
$$566$$ 0 0
$$567$$ 20550.1 1.52209
$$568$$ 0 0
$$569$$ −20440.1 −1.50596 −0.752982 0.658041i $$-0.771385\pi$$
−0.752982 + 0.658041i $$0.771385\pi$$
$$570$$ 0 0
$$571$$ 6920.98 0.507240 0.253620 0.967304i $$-0.418379\pi$$
0.253620 + 0.967304i $$0.418379\pi$$
$$572$$ 0 0
$$573$$ −8070.62 −0.588403
$$574$$ 0 0
$$575$$ 2620.17 0.190032
$$576$$ 0 0
$$577$$ −5021.59 −0.362307 −0.181154 0.983455i $$-0.557983\pi$$
−0.181154 + 0.983455i $$0.557983\pi$$
$$578$$ 0 0
$$579$$ −22044.1 −1.58225
$$580$$ 0 0
$$581$$ −9865.61 −0.704466
$$582$$ 0 0
$$583$$ 24471.5 1.73843
$$584$$ 0 0
$$585$$ −1083.97 −0.0766093
$$586$$ 0 0
$$587$$ 6784.80 0.477068 0.238534 0.971134i $$-0.423333\pi$$
0.238534 + 0.971134i $$0.423333\pi$$
$$588$$ 0 0
$$589$$ −5201.87 −0.363904
$$590$$ 0 0
$$591$$ −29122.9 −2.02700
$$592$$ 0 0
$$593$$ 22058.5 1.52754 0.763771 0.645487i $$-0.223346\pi$$
0.763771 + 0.645487i $$0.223346\pi$$
$$594$$ 0 0
$$595$$ 5414.80 0.373084
$$596$$ 0 0
$$597$$ 8950.51 0.613601
$$598$$ 0 0
$$599$$ −15616.7 −1.06524 −0.532620 0.846354i $$-0.678793\pi$$
−0.532620 + 0.846354i $$0.678793\pi$$
$$600$$ 0 0
$$601$$ 26769.6 1.81690 0.908448 0.417998i $$-0.137268\pi$$
0.908448 + 0.417998i $$0.137268\pi$$
$$602$$ 0 0
$$603$$ −926.439 −0.0625664
$$604$$ 0 0
$$605$$ −13701.8 −0.920757
$$606$$ 0 0
$$607$$ 5164.26 0.345323 0.172661 0.984981i $$-0.444763\pi$$
0.172661 + 0.984981i $$0.444763\pi$$
$$608$$ 0 0
$$609$$ −15539.7 −1.03399
$$610$$ 0 0
$$611$$ −7925.79 −0.524784
$$612$$ 0 0
$$613$$ −4226.03 −0.278447 −0.139223 0.990261i $$-0.544461\pi$$
−0.139223 + 0.990261i $$0.544461\pi$$
$$614$$ 0 0
$$615$$ −966.576 −0.0633758
$$616$$ 0 0
$$617$$ 14233.5 0.928717 0.464358 0.885647i $$-0.346285\pi$$
0.464358 + 0.885647i $$0.346285\pi$$
$$618$$ 0 0
$$619$$ 3737.82 0.242707 0.121353 0.992609i $$-0.461277\pi$$
0.121353 + 0.992609i $$0.461277\pi$$
$$620$$ 0 0
$$621$$ −22548.8 −1.45709
$$622$$ 0 0
$$623$$ 17847.9 1.14777
$$624$$ 0 0
$$625$$ −17340.4 −1.10978
$$626$$ 0 0
$$627$$ 5089.52 0.324172
$$628$$ 0 0
$$629$$ −2878.40 −0.182463
$$630$$ 0 0
$$631$$ −2893.31 −0.182537 −0.0912684 0.995826i $$-0.529092\pi$$
−0.0912684 + 0.995826i $$0.529092\pi$$
$$632$$ 0 0
$$633$$ −31513.3 −1.97874
$$634$$ 0 0
$$635$$ 1279.22 0.0799438
$$636$$ 0 0
$$637$$ 17609.1 1.09529
$$638$$ 0 0
$$639$$ −1486.43 −0.0920226
$$640$$ 0 0
$$641$$ 6867.26 0.423152 0.211576 0.977362i $$-0.432140\pi$$
0.211576 + 0.977362i $$0.432140\pi$$
$$642$$ 0 0
$$643$$ −4975.32 −0.305144 −0.152572 0.988292i $$-0.548756\pi$$
−0.152572 + 0.988292i $$0.548756\pi$$
$$644$$ 0 0
$$645$$ 26353.0 1.60876
$$646$$ 0 0
$$647$$ −24033.2 −1.46035 −0.730174 0.683262i $$-0.760561\pi$$
−0.730174 + 0.683262i $$0.760561\pi$$
$$648$$ 0 0
$$649$$ 16751.9 1.01321
$$650$$ 0 0
$$651$$ 38961.2 2.34564
$$652$$ 0 0
$$653$$ 25718.4 1.54125 0.770626 0.637288i $$-0.219944\pi$$
0.770626 + 0.637288i $$0.219944\pi$$
$$654$$ 0 0
$$655$$ 4427.24 0.264102
$$656$$ 0 0
$$657$$ −579.221 −0.0343951
$$658$$ 0 0
$$659$$ 7970.84 0.471168 0.235584 0.971854i $$-0.424300\pi$$
0.235584 + 0.971854i $$0.424300\pi$$
$$660$$ 0 0
$$661$$ −14207.8 −0.836038 −0.418019 0.908438i $$-0.637275\pi$$
−0.418019 + 0.908438i $$0.637275\pi$$
$$662$$ 0 0
$$663$$ 4545.43 0.266259
$$664$$ 0 0
$$665$$ 5969.77 0.348117
$$666$$ 0 0
$$667$$ 18232.2 1.05840
$$668$$ 0 0
$$669$$ −29026.1 −1.67745
$$670$$ 0 0
$$671$$ −36157.0 −2.08021
$$672$$ 0 0
$$673$$ 17080.0 0.978287 0.489143 0.872203i $$-0.337309\pi$$
0.489143 + 0.872203i $$0.337309\pi$$
$$674$$ 0 0
$$675$$ −2119.36 −0.120851
$$676$$ 0 0
$$677$$ 16410.9 0.931641 0.465821 0.884879i $$-0.345759\pi$$
0.465821 + 0.884879i $$0.345759\pi$$
$$678$$ 0 0
$$679$$ 25442.9 1.43801
$$680$$ 0 0
$$681$$ 13814.0 0.777317
$$682$$ 0 0
$$683$$ 183.169 0.0102617 0.00513086 0.999987i $$-0.498367\pi$$
0.00513086 + 0.999987i $$0.498367\pi$$
$$684$$ 0 0
$$685$$ 36986.9 2.06306
$$686$$ 0 0
$$687$$ −17492.6 −0.971446
$$688$$ 0 0
$$689$$ 24095.4 1.33231
$$690$$ 0 0
$$691$$ −14974.0 −0.824366 −0.412183 0.911101i $$-0.635234\pi$$
−0.412183 + 0.911101i $$0.635234\pi$$
$$692$$ 0 0
$$693$$ −2458.52 −0.134764
$$694$$ 0 0
$$695$$ −668.715 −0.0364976
$$696$$ 0 0
$$697$$ 261.408 0.0142059
$$698$$ 0 0
$$699$$ 6476.74 0.350462
$$700$$ 0 0
$$701$$ −4341.08 −0.233895 −0.116947 0.993138i $$-0.537311\pi$$
−0.116947 + 0.993138i $$0.537311\pi$$
$$702$$ 0 0
$$703$$ −3173.41 −0.170253
$$704$$ 0 0
$$705$$ 10287.3 0.549561
$$706$$ 0 0
$$707$$ 50056.1 2.66273
$$708$$ 0 0
$$709$$ −23931.9 −1.26767 −0.633837 0.773467i $$-0.718521\pi$$
−0.633837 + 0.773467i $$0.718521\pi$$
$$710$$ 0 0
$$711$$ −1239.18 −0.0653627
$$712$$ 0 0
$$713$$ −45712.0 −2.40102
$$714$$ 0 0
$$715$$ −29036.1 −1.51873
$$716$$ 0 0
$$717$$ −18290.1 −0.952656
$$718$$ 0 0
$$719$$ 16258.6 0.843314 0.421657 0.906756i $$-0.361449\pi$$
0.421657 + 0.906756i $$0.361449\pi$$
$$720$$ 0 0
$$721$$ −20137.5 −1.04017
$$722$$ 0 0
$$723$$ 17736.6 0.912354
$$724$$ 0 0
$$725$$ 1713.65 0.0877837
$$726$$ 0 0
$$727$$ −2608.95 −0.133096 −0.0665479 0.997783i $$-0.521199\pi$$
−0.0665479 + 0.997783i $$0.521199\pi$$
$$728$$ 0 0
$$729$$ 18145.4 0.921881
$$730$$ 0 0
$$731$$ −7127.11 −0.360610
$$732$$ 0 0
$$733$$ −9843.63 −0.496020 −0.248010 0.968757i $$-0.579777\pi$$
−0.248010 + 0.968757i $$0.579777\pi$$
$$734$$ 0 0
$$735$$ −22855.7 −1.14700
$$736$$ 0 0
$$737$$ −24816.5 −1.24033
$$738$$ 0 0
$$739$$ −2419.46 −0.120435 −0.0602174 0.998185i $$-0.519179\pi$$
−0.0602174 + 0.998185i $$0.519179\pi$$
$$740$$ 0 0
$$741$$ 5011.30 0.248441
$$742$$ 0 0
$$743$$ 10442.9 0.515631 0.257816 0.966194i $$-0.416997\pi$$
0.257816 + 0.966194i $$0.416997\pi$$
$$744$$ 0 0
$$745$$ −21475.2 −1.05610
$$746$$ 0 0
$$747$$ 693.262 0.0339560
$$748$$ 0 0
$$749$$ 18058.4 0.880963
$$750$$ 0 0
$$751$$ 29434.6 1.43021 0.715103 0.699019i $$-0.246380\pi$$
0.715103 + 0.699019i $$0.246380\pi$$
$$752$$ 0 0
$$753$$ 28638.6 1.38599
$$754$$ 0 0
$$755$$ −34335.4 −1.65509
$$756$$ 0 0
$$757$$ 20638.3 0.990902 0.495451 0.868636i $$-0.335003\pi$$
0.495451 + 0.868636i $$0.335003\pi$$
$$758$$ 0 0
$$759$$ 44724.7 2.13887
$$760$$ 0 0
$$761$$ 4103.09 0.195449 0.0977246 0.995213i $$-0.468844\pi$$
0.0977246 + 0.995213i $$0.468844\pi$$
$$762$$ 0 0
$$763$$ −34721.0 −1.64742
$$764$$ 0 0
$$765$$ −380.501 −0.0179831
$$766$$ 0 0
$$767$$ 16494.5 0.776508
$$768$$ 0 0
$$769$$ −14020.0 −0.657444 −0.328722 0.944427i $$-0.606618\pi$$
−0.328722 + 0.944427i $$0.606618\pi$$
$$770$$ 0 0
$$771$$ 20479.9 0.956635
$$772$$ 0 0
$$773$$ −32823.8 −1.52728 −0.763642 0.645640i $$-0.776591\pi$$
−0.763642 + 0.645640i $$0.776591\pi$$
$$774$$ 0 0
$$775$$ −4296.46 −0.199140
$$776$$ 0 0
$$777$$ 23768.4 1.09741
$$778$$ 0 0
$$779$$ 288.200 0.0132553
$$780$$ 0 0
$$781$$ −39817.0 −1.82428
$$782$$ 0 0
$$783$$ −14747.4 −0.673090
$$784$$ 0 0
$$785$$ 16389.0 0.745157
$$786$$ 0 0
$$787$$ −33903.0 −1.53559 −0.767796 0.640694i $$-0.778647\pi$$
−0.767796 + 0.640694i $$0.778647\pi$$
$$788$$ 0 0
$$789$$ 4830.54 0.217962
$$790$$ 0 0
$$791$$ 18897.7 0.849462
$$792$$ 0 0
$$793$$ −35601.3 −1.59425
$$794$$ 0 0
$$795$$ −31274.6 −1.39521
$$796$$ 0 0
$$797$$ −20830.2 −0.925778 −0.462889 0.886416i $$-0.653187\pi$$
−0.462889 + 0.886416i $$0.653187\pi$$
$$798$$ 0 0
$$799$$ −2782.16 −0.123186
$$800$$ 0 0
$$801$$ −1254.18 −0.0553238
$$802$$ 0 0
$$803$$ −15515.6 −0.681859
$$804$$ 0 0
$$805$$ 52460.0 2.29686
$$806$$ 0 0
$$807$$ −14213.8 −0.620012
$$808$$ 0 0
$$809$$ −26505.7 −1.15190 −0.575952 0.817484i $$-0.695368\pi$$
−0.575952 + 0.817484i $$0.695368\pi$$
$$810$$ 0 0
$$811$$ 10829.0 0.468876 0.234438 0.972131i $$-0.424675\pi$$
0.234438 + 0.972131i $$0.424675\pi$$
$$812$$ 0 0
$$813$$ 2774.94 0.119706
$$814$$ 0 0
$$815$$ −44371.5 −1.90707
$$816$$ 0 0
$$817$$ −7857.58 −0.336477
$$818$$ 0 0
$$819$$ −2420.73 −0.103281
$$820$$ 0 0
$$821$$ −42046.8 −1.78738 −0.893692 0.448681i $$-0.851894\pi$$
−0.893692 + 0.448681i $$0.851894\pi$$
$$822$$ 0 0
$$823$$ −12949.0 −0.548448 −0.274224 0.961666i $$-0.588421\pi$$
−0.274224 + 0.961666i $$0.588421\pi$$
$$824$$ 0 0
$$825$$ 4203.67 0.177397
$$826$$ 0 0
$$827$$ 13874.0 0.583368 0.291684 0.956515i $$-0.405784\pi$$
0.291684 + 0.956515i $$0.405784\pi$$
$$828$$ 0 0
$$829$$ 2387.20 0.100013 0.0500065 0.998749i $$-0.484076\pi$$
0.0500065 + 0.998749i $$0.484076\pi$$
$$830$$ 0 0
$$831$$ −12706.2 −0.530412
$$832$$ 0 0
$$833$$ 6181.27 0.257105
$$834$$ 0 0
$$835$$ −36594.4 −1.51665
$$836$$ 0 0
$$837$$ 36974.8 1.52692
$$838$$ 0 0
$$839$$ 47308.8 1.94670 0.973350 0.229324i $$-0.0736515\pi$$
0.973350 + 0.229324i $$0.0736515\pi$$
$$840$$ 0 0
$$841$$ −12464.7 −0.511080
$$842$$ 0 0
$$843$$ −8200.99 −0.335062
$$844$$ 0 0
$$845$$ −2530.37 −0.103015
$$846$$ 0 0
$$847$$ −30599.2 −1.24132
$$848$$ 0 0
$$849$$ −25593.6 −1.03459
$$850$$ 0 0
$$851$$ −27886.7 −1.12332
$$852$$ 0 0
$$853$$ 18211.3 0.730998 0.365499 0.930812i $$-0.380898\pi$$
0.365499 + 0.930812i $$0.380898\pi$$
$$854$$ 0 0
$$855$$ −419.499 −0.0167796
$$856$$ 0 0
$$857$$ −14824.5 −0.590893 −0.295447 0.955359i $$-0.595468\pi$$
−0.295447 + 0.955359i $$0.595468\pi$$
$$858$$ 0 0
$$859$$ 44191.1 1.75527 0.877637 0.479326i $$-0.159119\pi$$
0.877637 + 0.479326i $$0.159119\pi$$
$$860$$ 0 0
$$861$$ −2158.58 −0.0854403
$$862$$ 0 0
$$863$$ −34927.0 −1.37767 −0.688835 0.724918i $$-0.741878\pi$$
−0.688835 + 0.724918i $$0.741878\pi$$
$$864$$ 0 0
$$865$$ 3475.40 0.136609
$$866$$ 0 0
$$867$$ −24798.5 −0.971395
$$868$$ 0 0
$$869$$ −33193.9 −1.29577
$$870$$ 0 0
$$871$$ −24435.1 −0.950575
$$872$$ 0 0
$$873$$ −1787.89 −0.0693138
$$874$$ 0 0
$$875$$ −34344.1 −1.32691
$$876$$ 0 0
$$877$$ 16914.8 0.651278 0.325639 0.945494i $$-0.394421\pi$$
0.325639 + 0.945494i $$0.394421\pi$$
$$878$$ 0 0
$$879$$ −17132.6 −0.657415
$$880$$ 0 0
$$881$$ 12634.8 0.483177 0.241588 0.970379i $$-0.422332\pi$$
0.241588 + 0.970379i $$0.422332\pi$$
$$882$$ 0 0
$$883$$ 47260.3 1.80117 0.900586 0.434678i $$-0.143138\pi$$
0.900586 + 0.434678i $$0.143138\pi$$
$$884$$ 0 0
$$885$$ −21409.0 −0.813169
$$886$$ 0 0
$$887$$ −4758.84 −0.180142 −0.0900711 0.995935i $$-0.528709\pi$$
−0.0900711 + 0.995935i $$0.528709\pi$$
$$888$$ 0 0
$$889$$ 2856.78 0.107777
$$890$$ 0 0
$$891$$ −38682.1 −1.45443
$$892$$ 0 0
$$893$$ −3067.31 −0.114943
$$894$$ 0 0
$$895$$ −35540.6 −1.32736
$$896$$ 0 0
$$897$$ 44037.4 1.63920
$$898$$ 0 0
$$899$$ −29896.6 −1.10913
$$900$$ 0 0
$$901$$ 8458.13 0.312743
$$902$$ 0 0
$$903$$ 58852.1 2.16885
$$904$$ 0 0
$$905$$ −38732.4 −1.42266
$$906$$ 0 0
$$907$$ 4111.61 0.150522 0.0752612 0.997164i $$-0.476021\pi$$
0.0752612 + 0.997164i $$0.476021\pi$$
$$908$$ 0 0
$$909$$ −3517.47 −0.128347
$$910$$ 0 0
$$911$$ −35113.1 −1.27700 −0.638501 0.769621i $$-0.720445\pi$$
−0.638501 + 0.769621i $$0.720445\pi$$
$$912$$ 0 0
$$913$$ 18570.4 0.673155
$$914$$ 0 0
$$915$$ 46208.5 1.66952
$$916$$ 0 0
$$917$$ 9887.00 0.356049
$$918$$ 0 0
$$919$$ 37509.1 1.34637 0.673184 0.739475i $$-0.264926\pi$$
0.673184 + 0.739475i $$0.264926\pi$$
$$920$$ 0 0
$$921$$ −32226.0 −1.15297
$$922$$ 0 0
$$923$$ −39205.1 −1.39811
$$924$$ 0 0
$$925$$ −2621.07 −0.0931677
$$926$$ 0 0
$$927$$ 1415.07 0.0501371
$$928$$ 0 0
$$929$$ 18393.8 0.649604 0.324802 0.945782i $$-0.394702\pi$$
0.324802 + 0.945782i $$0.394702\pi$$
$$930$$ 0 0
$$931$$ 6814.80 0.239899
$$932$$ 0 0
$$933$$ 27479.0 0.964226
$$934$$ 0 0
$$935$$ −10192.5 −0.356502
$$936$$ 0 0
$$937$$ −17106.7 −0.596427 −0.298214 0.954499i $$-0.596391\pi$$
−0.298214 + 0.954499i $$0.596391\pi$$
$$938$$ 0 0
$$939$$ 11451.7 0.397990
$$940$$ 0 0
$$941$$ 38100.7 1.31992 0.659961 0.751300i $$-0.270573\pi$$
0.659961 + 0.751300i $$0.270573\pi$$
$$942$$ 0 0
$$943$$ 2532.59 0.0874576
$$944$$ 0 0
$$945$$ −42433.0 −1.46068
$$946$$ 0 0
$$947$$ 52712.7 1.80880 0.904400 0.426686i $$-0.140319\pi$$
0.904400 + 0.426686i $$0.140319\pi$$
$$948$$ 0 0
$$949$$ −15277.1 −0.522567
$$950$$ 0 0
$$951$$ −23944.4 −0.816458
$$952$$ 0 0
$$953$$ −6744.38 −0.229246 −0.114623 0.993409i $$-0.536566\pi$$
−0.114623 + 0.993409i $$0.536566\pi$$
$$954$$ 0 0
$$955$$ 17819.0 0.603781
$$956$$ 0 0
$$957$$ 29250.9 0.988032
$$958$$ 0 0
$$959$$ 82599.8 2.78132
$$960$$ 0 0
$$961$$ 45165.9 1.51609
$$962$$ 0 0
$$963$$ −1268.98 −0.0424634
$$964$$ 0 0
$$965$$ 48670.9 1.62360
$$966$$ 0 0
$$967$$ −28449.8 −0.946104 −0.473052 0.881034i $$-0.656848\pi$$
−0.473052 + 0.881034i $$0.656848\pi$$
$$968$$ 0 0
$$969$$ 1759.10 0.0583183
$$970$$ 0 0
$$971$$ 13018.7 0.430266 0.215133 0.976585i $$-0.430981\pi$$
0.215133 + 0.976585i $$0.430981\pi$$
$$972$$ 0 0
$$973$$ −1493.39 −0.0492043
$$974$$ 0 0
$$975$$ 4139.06 0.135955
$$976$$ 0 0
$$977$$ −20043.8 −0.656355 −0.328178 0.944616i $$-0.606434\pi$$
−0.328178 + 0.944616i $$0.606434\pi$$
$$978$$ 0 0
$$979$$ −33595.7 −1.09676
$$980$$ 0 0
$$981$$ 2439.86 0.0794076
$$982$$ 0 0
$$983$$ −1823.72 −0.0591735 −0.0295868 0.999562i $$-0.509419\pi$$
−0.0295868 + 0.999562i $$0.509419\pi$$
$$984$$ 0 0
$$985$$ 64300.1 2.07997
$$986$$ 0 0
$$987$$ 22973.7 0.740892
$$988$$ 0 0
$$989$$ −69049.4 −2.22006
$$990$$ 0 0
$$991$$ −26635.4 −0.853786 −0.426893 0.904302i $$-0.640392\pi$$
−0.426893 + 0.904302i $$0.640392\pi$$
$$992$$ 0 0
$$993$$ 22261.0 0.711411
$$994$$ 0 0
$$995$$ −19761.7 −0.629637
$$996$$ 0 0
$$997$$ −50874.6 −1.61606 −0.808031 0.589139i $$-0.799467\pi$$
−0.808031 + 0.589139i $$0.799467\pi$$
$$998$$ 0 0
$$999$$ 22556.6 0.714372
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 1216.4.a.o.1.2 2
4.3 odd 2 1216.4.a.h.1.1 2
8.3 odd 2 304.4.a.f.1.2 2
8.5 even 2 76.4.a.a.1.1 2
24.5 odd 2 684.4.a.g.1.1 2
40.13 odd 4 1900.4.c.b.1749.1 4
40.29 even 2 1900.4.a.b.1.2 2
40.37 odd 4 1900.4.c.b.1749.4 4
152.37 odd 2 1444.4.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.a.a.1.1 2 8.5 even 2
304.4.a.f.1.2 2 8.3 odd 2
684.4.a.g.1.1 2 24.5 odd 2
1216.4.a.h.1.1 2 4.3 odd 2
1216.4.a.o.1.2 2 1.1 even 1 trivial
1444.4.a.d.1.2 2 152.37 odd 2
1900.4.a.b.1.2 2 40.29 even 2
1900.4.c.b.1749.1 4 40.13 odd 4
1900.4.c.b.1749.4 4 40.37 odd 4