# Properties

 Label 1521.4.a.bc.1.8 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992$$ x^8 - 52*x^6 + 805*x^4 - 4210*x^2 + 4992 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$5.39589$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} -6.42494 q^{7} +70.7705 q^{8} +O(q^{10})$$ $$q+5.39589 q^{2} +21.1156 q^{4} +13.0421 q^{5} -6.42494 q^{7} +70.7705 q^{8} +70.3740 q^{10} +26.2056 q^{11} -34.6683 q^{14} +212.945 q^{16} -123.877 q^{17} +109.667 q^{19} +275.393 q^{20} +141.402 q^{22} -63.4094 q^{23} +45.0976 q^{25} -135.667 q^{28} +225.410 q^{29} +200.732 q^{31} +582.863 q^{32} -668.426 q^{34} -83.7950 q^{35} +252.509 q^{37} +591.749 q^{38} +922.999 q^{40} +227.423 q^{41} -384.032 q^{43} +553.348 q^{44} -342.150 q^{46} +34.6646 q^{47} -301.720 q^{49} +243.342 q^{50} -61.0601 q^{53} +341.777 q^{55} -454.696 q^{56} +1216.29 q^{58} -80.5562 q^{59} -26.1383 q^{61} +1083.13 q^{62} +1441.50 q^{64} -931.510 q^{67} -2615.74 q^{68} -452.149 q^{70} +427.608 q^{71} +108.518 q^{73} +1362.51 q^{74} +2315.68 q^{76} -168.369 q^{77} +384.590 q^{79} +2777.26 q^{80} +1227.15 q^{82} -85.9758 q^{83} -1615.62 q^{85} -2072.19 q^{86} +1854.58 q^{88} -495.903 q^{89} -1338.93 q^{92} +187.046 q^{94} +1430.29 q^{95} +190.857 q^{97} -1628.05 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{4} - 22 q^{7}+O(q^{10})$$ 8 * q + 40 * q^4 - 22 * q^7 $$8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100})$$ 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.39589 1.90774 0.953868 0.300228i $$-0.0970626\pi$$
0.953868 + 0.300228i $$0.0970626\pi$$
$$3$$ 0 0
$$4$$ 21.1156 2.63945
$$5$$ 13.0421 1.16653 0.583263 0.812284i $$-0.301776\pi$$
0.583263 + 0.812284i $$0.301776\pi$$
$$6$$ 0 0
$$7$$ −6.42494 −0.346914 −0.173457 0.984841i $$-0.555494\pi$$
−0.173457 + 0.984841i $$0.555494\pi$$
$$8$$ 70.7705 3.12764
$$9$$ 0 0
$$10$$ 70.3740 2.22542
$$11$$ 26.2056 0.718298 0.359149 0.933280i $$-0.383067\pi$$
0.359149 + 0.933280i $$0.383067\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −34.6683 −0.661820
$$15$$ 0 0
$$16$$ 212.945 3.32726
$$17$$ −123.877 −1.76733 −0.883663 0.468124i $$-0.844930\pi$$
−0.883663 + 0.468124i $$0.844930\pi$$
$$18$$ 0 0
$$19$$ 109.667 1.32417 0.662086 0.749428i $$-0.269671\pi$$
0.662086 + 0.749428i $$0.269671\pi$$
$$20$$ 275.393 3.07899
$$21$$ 0 0
$$22$$ 141.402 1.37032
$$23$$ −63.4094 −0.574860 −0.287430 0.957802i $$-0.592801\pi$$
−0.287430 + 0.957802i $$0.592801\pi$$
$$24$$ 0 0
$$25$$ 45.0976 0.360781
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −135.667 −0.915664
$$29$$ 225.410 1.44337 0.721683 0.692224i $$-0.243369\pi$$
0.721683 + 0.692224i $$0.243369\pi$$
$$30$$ 0 0
$$31$$ 200.732 1.16298 0.581491 0.813553i $$-0.302470\pi$$
0.581491 + 0.813553i $$0.302470\pi$$
$$32$$ 582.863 3.21989
$$33$$ 0 0
$$34$$ −668.426 −3.37159
$$35$$ −83.7950 −0.404684
$$36$$ 0 0
$$37$$ 252.509 1.12195 0.560976 0.827832i $$-0.310426\pi$$
0.560976 + 0.827832i $$0.310426\pi$$
$$38$$ 591.749 2.52617
$$39$$ 0 0
$$40$$ 922.999 3.64847
$$41$$ 227.423 0.866282 0.433141 0.901326i $$-0.357405\pi$$
0.433141 + 0.901326i $$0.357405\pi$$
$$42$$ 0 0
$$43$$ −384.032 −1.36196 −0.680980 0.732302i $$-0.738446\pi$$
−0.680980 + 0.732302i $$0.738446\pi$$
$$44$$ 553.348 1.89592
$$45$$ 0 0
$$46$$ −342.150 −1.09668
$$47$$ 34.6646 0.107582 0.0537910 0.998552i $$-0.482870\pi$$
0.0537910 + 0.998552i $$0.482870\pi$$
$$48$$ 0 0
$$49$$ −301.720 −0.879651
$$50$$ 243.342 0.688274
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −61.0601 −0.158250 −0.0791251 0.996865i $$-0.525213\pi$$
−0.0791251 + 0.996865i $$0.525213\pi$$
$$54$$ 0 0
$$55$$ 341.777 0.837913
$$56$$ −454.696 −1.08502
$$57$$ 0 0
$$58$$ 1216.29 2.75356
$$59$$ −80.5562 −0.177755 −0.0888774 0.996043i $$-0.528328\pi$$
−0.0888774 + 0.996043i $$0.528328\pi$$
$$60$$ 0 0
$$61$$ −26.1383 −0.0548634 −0.0274317 0.999624i $$-0.508733\pi$$
−0.0274317 + 0.999624i $$0.508733\pi$$
$$62$$ 1083.13 2.21866
$$63$$ 0 0
$$64$$ 1441.50 2.81544
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −931.510 −1.69854 −0.849269 0.527960i $$-0.822957\pi$$
−0.849269 + 0.527960i $$0.822957\pi$$
$$68$$ −2615.74 −4.66477
$$69$$ 0 0
$$70$$ −452.149 −0.772030
$$71$$ 427.608 0.714757 0.357379 0.933960i $$-0.383671\pi$$
0.357379 + 0.933960i $$0.383671\pi$$
$$72$$ 0 0
$$73$$ 108.518 0.173987 0.0869935 0.996209i $$-0.472274\pi$$
0.0869935 + 0.996209i $$0.472274\pi$$
$$74$$ 1362.51 2.14039
$$75$$ 0 0
$$76$$ 2315.68 3.49509
$$77$$ −168.369 −0.249188
$$78$$ 0 0
$$79$$ 384.590 0.547718 0.273859 0.961770i $$-0.411700\pi$$
0.273859 + 0.961770i $$0.411700\pi$$
$$80$$ 2777.26 3.88133
$$81$$ 0 0
$$82$$ 1227.15 1.65264
$$83$$ −85.9758 −0.113700 −0.0568498 0.998383i $$-0.518106\pi$$
−0.0568498 + 0.998383i $$0.518106\pi$$
$$84$$ 0 0
$$85$$ −1615.62 −2.06163
$$86$$ −2072.19 −2.59826
$$87$$ 0 0
$$88$$ 1854.58 2.24658
$$89$$ −495.903 −0.590625 −0.295313 0.955401i $$-0.595424\pi$$
−0.295313 + 0.955401i $$0.595424\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1338.93 −1.51732
$$93$$ 0 0
$$94$$ 187.046 0.205238
$$95$$ 1430.29 1.54468
$$96$$ 0 0
$$97$$ 190.857 0.199780 0.0998898 0.994999i $$-0.468151\pi$$
0.0998898 + 0.994999i $$0.468151\pi$$
$$98$$ −1628.05 −1.67814
$$99$$ 0 0
$$100$$ 952.264 0.952264
$$101$$ −850.980 −0.838373 −0.419186 0.907900i $$-0.637685\pi$$
−0.419186 + 0.907900i $$0.637685\pi$$
$$102$$ 0 0
$$103$$ −1309.50 −1.25270 −0.626352 0.779540i $$-0.715453\pi$$
−0.626352 + 0.779540i $$0.715453\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −329.474 −0.301899
$$107$$ −962.302 −0.869432 −0.434716 0.900567i $$-0.643151\pi$$
−0.434716 + 0.900567i $$0.643151\pi$$
$$108$$ 0 0
$$109$$ 891.973 0.783812 0.391906 0.920005i $$-0.371816\pi$$
0.391906 + 0.920005i $$0.371816\pi$$
$$110$$ 1844.19 1.59852
$$111$$ 0 0
$$112$$ −1368.16 −1.15427
$$113$$ −263.839 −0.219645 −0.109823 0.993951i $$-0.535028\pi$$
−0.109823 + 0.993951i $$0.535028\pi$$
$$114$$ 0 0
$$115$$ −826.995 −0.670588
$$116$$ 4759.68 3.80970
$$117$$ 0 0
$$118$$ −434.673 −0.339109
$$119$$ 795.901 0.613110
$$120$$ 0 0
$$121$$ −644.267 −0.484047
$$122$$ −141.039 −0.104665
$$123$$ 0 0
$$124$$ 4238.57 3.06964
$$125$$ −1042.10 −0.745665
$$126$$ 0 0
$$127$$ −1543.97 −1.07878 −0.539391 0.842056i $$-0.681345\pi$$
−0.539391 + 0.842056i $$0.681345\pi$$
$$128$$ 3115.30 2.15122
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 776.790 0.518080 0.259040 0.965867i $$-0.416594\pi$$
0.259040 + 0.965867i $$0.416594\pi$$
$$132$$ 0 0
$$133$$ −704.602 −0.459374
$$134$$ −5026.33 −3.24036
$$135$$ 0 0
$$136$$ −8766.82 −5.52756
$$137$$ 2434.71 1.51833 0.759166 0.650897i $$-0.225607\pi$$
0.759166 + 0.650897i $$0.225607\pi$$
$$138$$ 0 0
$$139$$ 1251.40 0.763615 0.381807 0.924242i $$-0.375302\pi$$
0.381807 + 0.924242i $$0.375302\pi$$
$$140$$ −1769.38 −1.06814
$$141$$ 0 0
$$142$$ 2307.33 1.36357
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 2939.83 1.68372
$$146$$ 585.550 0.331921
$$147$$ 0 0
$$148$$ 5331.88 2.96134
$$149$$ 783.619 0.430850 0.215425 0.976520i $$-0.430886\pi$$
0.215425 + 0.976520i $$0.430886\pi$$
$$150$$ 0 0
$$151$$ 162.220 0.0874255 0.0437128 0.999044i $$-0.486081\pi$$
0.0437128 + 0.999044i $$0.486081\pi$$
$$152$$ 7761.16 4.14154
$$153$$ 0 0
$$154$$ −908.503 −0.475385
$$155$$ 2617.97 1.35665
$$156$$ 0 0
$$157$$ −2355.76 −1.19752 −0.598758 0.800930i $$-0.704339\pi$$
−0.598758 + 0.800930i $$0.704339\pi$$
$$158$$ 2075.20 1.04490
$$159$$ 0 0
$$160$$ 7601.78 3.75608
$$161$$ 407.402 0.199427
$$162$$ 0 0
$$163$$ 1574.29 0.756488 0.378244 0.925706i $$-0.376528\pi$$
0.378244 + 0.925706i $$0.376528\pi$$
$$164$$ 4802.19 2.28651
$$165$$ 0 0
$$166$$ −463.916 −0.216909
$$167$$ 3574.56 1.65633 0.828166 0.560483i $$-0.189384\pi$$
0.828166 + 0.560483i $$0.189384\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −8717.70 −3.93304
$$171$$ 0 0
$$172$$ −8109.07 −3.59483
$$173$$ 1.44108 0.000633314 0 0.000316657 1.00000i $$-0.499899\pi$$
0.000316657 1.00000i $$0.499899\pi$$
$$174$$ 0 0
$$175$$ −289.749 −0.125160
$$176$$ 5580.34 2.38997
$$177$$ 0 0
$$178$$ −2675.84 −1.12676
$$179$$ −2082.28 −0.869478 −0.434739 0.900556i $$-0.643159\pi$$
−0.434739 + 0.900556i $$0.643159\pi$$
$$180$$ 0 0
$$181$$ −464.500 −0.190751 −0.0953756 0.995441i $$-0.530405\pi$$
−0.0953756 + 0.995441i $$0.530405\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4487.51 −1.79796
$$185$$ 3293.26 1.30878
$$186$$ 0 0
$$187$$ −3246.26 −1.26947
$$188$$ 731.965 0.283957
$$189$$ 0 0
$$190$$ 7717.68 2.94684
$$191$$ −4866.28 −1.84352 −0.921758 0.387765i $$-0.873247\pi$$
−0.921758 + 0.387765i $$0.873247\pi$$
$$192$$ 0 0
$$193$$ −856.339 −0.319381 −0.159691 0.987167i $$-0.551050\pi$$
−0.159691 + 0.987167i $$0.551050\pi$$
$$194$$ 1029.84 0.381127
$$195$$ 0 0
$$196$$ −6371.01 −2.32180
$$197$$ −1571.93 −0.568504 −0.284252 0.958750i $$-0.591745\pi$$
−0.284252 + 0.958750i $$0.591745\pi$$
$$198$$ 0 0
$$199$$ 3126.61 1.11377 0.556883 0.830591i $$-0.311997\pi$$
0.556883 + 0.830591i $$0.311997\pi$$
$$200$$ 3191.58 1.12839
$$201$$ 0 0
$$202$$ −4591.79 −1.59939
$$203$$ −1448.25 −0.500724
$$204$$ 0 0
$$205$$ 2966.09 1.01054
$$206$$ −7065.90 −2.38983
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2873.88 0.951150
$$210$$ 0 0
$$211$$ 123.392 0.0402590 0.0201295 0.999797i $$-0.493592\pi$$
0.0201295 + 0.999797i $$0.493592\pi$$
$$212$$ −1289.32 −0.417694
$$213$$ 0 0
$$214$$ −5192.48 −1.65865
$$215$$ −5008.60 −1.58876
$$216$$ 0 0
$$217$$ −1289.69 −0.403455
$$218$$ 4812.99 1.49531
$$219$$ 0 0
$$220$$ 7216.84 2.21163
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2503.69 −0.751836 −0.375918 0.926653i $$-0.622673\pi$$
−0.375918 + 0.926653i $$0.622673\pi$$
$$224$$ −3744.86 −1.11703
$$225$$ 0 0
$$226$$ −1423.65 −0.419025
$$227$$ 579.270 0.169372 0.0846861 0.996408i $$-0.473011\pi$$
0.0846861 + 0.996408i $$0.473011\pi$$
$$228$$ 0 0
$$229$$ −768.922 −0.221886 −0.110943 0.993827i $$-0.535387\pi$$
−0.110943 + 0.993827i $$0.535387\pi$$
$$230$$ −4462.37 −1.27930
$$231$$ 0 0
$$232$$ 15952.4 4.51433
$$233$$ 845.695 0.237783 0.118891 0.992907i $$-0.462066\pi$$
0.118891 + 0.992907i $$0.462066\pi$$
$$234$$ 0 0
$$235$$ 452.101 0.125497
$$236$$ −1701.00 −0.469175
$$237$$ 0 0
$$238$$ 4294.59 1.16965
$$239$$ −6552.78 −1.77349 −0.886744 0.462260i $$-0.847039\pi$$
−0.886744 + 0.462260i $$0.847039\pi$$
$$240$$ 0 0
$$241$$ 5176.75 1.38367 0.691834 0.722056i $$-0.256803\pi$$
0.691834 + 0.722056i $$0.256803\pi$$
$$242$$ −3476.39 −0.923434
$$243$$ 0 0
$$244$$ −551.927 −0.144809
$$245$$ −3935.08 −1.02613
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 14205.9 3.63739
$$249$$ 0 0
$$250$$ −5623.05 −1.42253
$$251$$ 4296.82 1.08053 0.540264 0.841495i $$-0.318324\pi$$
0.540264 + 0.841495i $$0.318324\pi$$
$$252$$ 0 0
$$253$$ −1661.68 −0.412921
$$254$$ −8331.10 −2.05803
$$255$$ 0 0
$$256$$ 5277.77 1.28852
$$257$$ −1383.94 −0.335906 −0.167953 0.985795i $$-0.553716\pi$$
−0.167953 + 0.985795i $$0.553716\pi$$
$$258$$ 0 0
$$259$$ −1622.35 −0.389221
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4191.47 0.988359
$$263$$ −2478.68 −0.581147 −0.290574 0.956853i $$-0.593846\pi$$
−0.290574 + 0.956853i $$0.593846\pi$$
$$264$$ 0 0
$$265$$ −796.355 −0.184603
$$266$$ −3801.95 −0.876364
$$267$$ 0 0
$$268$$ −19669.4 −4.48321
$$269$$ −2452.64 −0.555911 −0.277956 0.960594i $$-0.589657\pi$$
−0.277956 + 0.960594i $$0.589657\pi$$
$$270$$ 0 0
$$271$$ −5980.74 −1.34060 −0.670302 0.742088i $$-0.733836\pi$$
−0.670302 + 0.742088i $$0.733836\pi$$
$$272$$ −26378.9 −5.88036
$$273$$ 0 0
$$274$$ 13137.4 2.89658
$$275$$ 1181.81 0.259148
$$276$$ 0 0
$$277$$ −7377.99 −1.60036 −0.800182 0.599758i $$-0.795264\pi$$
−0.800182 + 0.599758i $$0.795264\pi$$
$$278$$ 6752.42 1.45677
$$279$$ 0 0
$$280$$ −5930.22 −1.26571
$$281$$ −5937.17 −1.26043 −0.630217 0.776419i $$-0.717034\pi$$
−0.630217 + 0.776419i $$0.717034\pi$$
$$282$$ 0 0
$$283$$ 1486.52 0.312241 0.156121 0.987738i $$-0.450101\pi$$
0.156121 + 0.987738i $$0.450101\pi$$
$$284$$ 9029.22 1.88657
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1461.18 −0.300526
$$288$$ 0 0
$$289$$ 10432.5 2.12344
$$290$$ 15863.0 3.21210
$$291$$ 0 0
$$292$$ 2291.42 0.459231
$$293$$ 3580.05 0.713818 0.356909 0.934139i $$-0.383831\pi$$
0.356909 + 0.934139i $$0.383831\pi$$
$$294$$ 0 0
$$295$$ −1050.63 −0.207355
$$296$$ 17870.2 3.50906
$$297$$ 0 0
$$298$$ 4228.32 0.821947
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 2467.38 0.472483
$$302$$ 875.320 0.166785
$$303$$ 0 0
$$304$$ 23352.9 4.40587
$$305$$ −340.900 −0.0639995
$$306$$ 0 0
$$307$$ −415.013 −0.0771533 −0.0385767 0.999256i $$-0.512282\pi$$
−0.0385767 + 0.999256i $$0.512282\pi$$
$$308$$ −3555.23 −0.657720
$$309$$ 0 0
$$310$$ 14126.3 2.58812
$$311$$ −3009.05 −0.548642 −0.274321 0.961638i $$-0.588453\pi$$
−0.274321 + 0.961638i $$0.588453\pi$$
$$312$$ 0 0
$$313$$ 3760.84 0.679154 0.339577 0.940578i $$-0.389716\pi$$
0.339577 + 0.940578i $$0.389716\pi$$
$$314$$ −12711.4 −2.28454
$$315$$ 0 0
$$316$$ 8120.86 1.44568
$$317$$ −2772.04 −0.491146 −0.245573 0.969378i $$-0.578976\pi$$
−0.245573 + 0.969378i $$0.578976\pi$$
$$318$$ 0 0
$$319$$ 5907.01 1.03677
$$320$$ 18800.3 3.28428
$$321$$ 0 0
$$322$$ 2198.29 0.380454
$$323$$ −13585.2 −2.34024
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8494.67 1.44318
$$327$$ 0 0
$$328$$ 16094.9 2.70942
$$329$$ −222.718 −0.0373217
$$330$$ 0 0
$$331$$ 6558.16 1.08903 0.544515 0.838751i $$-0.316714\pi$$
0.544515 + 0.838751i $$0.316714\pi$$
$$332$$ −1815.43 −0.300105
$$333$$ 0 0
$$334$$ 19287.9 3.15984
$$335$$ −12148.9 −1.98139
$$336$$ 0 0
$$337$$ 9509.17 1.53708 0.768542 0.639799i $$-0.220983\pi$$
0.768542 + 0.639799i $$0.220983\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −34114.8 −5.44158
$$341$$ 5260.29 0.835368
$$342$$ 0 0
$$343$$ 4142.29 0.652077
$$344$$ −27178.1 −4.25973
$$345$$ 0 0
$$346$$ 7.77591 0.00120820
$$347$$ −9808.78 −1.51747 −0.758736 0.651398i $$-0.774183\pi$$
−0.758736 + 0.651398i $$0.774183\pi$$
$$348$$ 0 0
$$349$$ 794.952 0.121928 0.0609639 0.998140i $$-0.480583\pi$$
0.0609639 + 0.998140i $$0.480583\pi$$
$$350$$ −1563.46 −0.238772
$$351$$ 0 0
$$352$$ 15274.3 2.31284
$$353$$ −7115.60 −1.07288 −0.536438 0.843940i $$-0.680230\pi$$
−0.536438 + 0.843940i $$0.680230\pi$$
$$354$$ 0 0
$$355$$ 5576.93 0.833782
$$356$$ −10471.3 −1.55893
$$357$$ 0 0
$$358$$ −11235.7 −1.65873
$$359$$ −6907.19 −1.01545 −0.507726 0.861518i $$-0.669514\pi$$
−0.507726 + 0.861518i $$0.669514\pi$$
$$360$$ 0 0
$$361$$ 5167.78 0.753430
$$362$$ −2506.39 −0.363903
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1415.31 0.202960
$$366$$ 0 0
$$367$$ −459.533 −0.0653609 −0.0326804 0.999466i $$-0.510404\pi$$
−0.0326804 + 0.999466i $$0.510404\pi$$
$$368$$ −13502.7 −1.91271
$$369$$ 0 0
$$370$$ 17770.1 2.49681
$$371$$ 392.308 0.0548992
$$372$$ 0 0
$$373$$ 5359.92 0.744038 0.372019 0.928225i $$-0.378666\pi$$
0.372019 + 0.928225i $$0.378666\pi$$
$$374$$ −17516.5 −2.42181
$$375$$ 0 0
$$376$$ 2453.23 0.336478
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8594.06 1.16477 0.582384 0.812914i $$-0.302120\pi$$
0.582384 + 0.812914i $$0.302120\pi$$
$$380$$ 30201.4 4.07711
$$381$$ 0 0
$$382$$ −26257.9 −3.51694
$$383$$ 9336.49 1.24562 0.622810 0.782373i $$-0.285991\pi$$
0.622810 + 0.782373i $$0.285991\pi$$
$$384$$ 0 0
$$385$$ −2195.90 −0.290684
$$386$$ −4620.71 −0.609295
$$387$$ 0 0
$$388$$ 4030.07 0.527309
$$389$$ 12792.1 1.66732 0.833659 0.552279i $$-0.186242\pi$$
0.833659 + 0.552279i $$0.186242\pi$$
$$390$$ 0 0
$$391$$ 7854.95 1.01596
$$392$$ −21352.9 −2.75123
$$393$$ 0 0
$$394$$ −8481.96 −1.08456
$$395$$ 5015.88 0.638927
$$396$$ 0 0
$$397$$ −13384.5 −1.69206 −0.846030 0.533135i $$-0.821014\pi$$
−0.846030 + 0.533135i $$0.821014\pi$$
$$398$$ 16870.8 2.12477
$$399$$ 0 0
$$400$$ 9603.30 1.20041
$$401$$ −6357.41 −0.791706 −0.395853 0.918314i $$-0.629551\pi$$
−0.395853 + 0.918314i $$0.629551\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −17969.0 −2.21285
$$405$$ 0 0
$$406$$ −7814.58 −0.955249
$$407$$ 6617.14 0.805896
$$408$$ 0 0
$$409$$ −5301.63 −0.640950 −0.320475 0.947257i $$-0.603843\pi$$
−0.320475 + 0.947257i $$0.603843\pi$$
$$410$$ 16004.7 1.92784
$$411$$ 0 0
$$412$$ −27650.8 −3.30645
$$413$$ 517.569 0.0616656
$$414$$ 0 0
$$415$$ −1121.31 −0.132633
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 15507.1 1.81454
$$419$$ 16395.8 1.91167 0.955835 0.293905i $$-0.0949551\pi$$
0.955835 + 0.293905i $$0.0949551\pi$$
$$420$$ 0 0
$$421$$ 8484.68 0.982227 0.491114 0.871095i $$-0.336590\pi$$
0.491114 + 0.871095i $$0.336590\pi$$
$$422$$ 665.808 0.0768034
$$423$$ 0 0
$$424$$ −4321.26 −0.494950
$$425$$ −5586.55 −0.637617
$$426$$ 0 0
$$427$$ 167.937 0.0190329
$$428$$ −20319.6 −2.29483
$$429$$ 0 0
$$430$$ −27025.8 −3.03094
$$431$$ 1594.74 0.178227 0.0891136 0.996021i $$-0.471597\pi$$
0.0891136 + 0.996021i $$0.471597\pi$$
$$432$$ 0 0
$$433$$ −3387.57 −0.375973 −0.187987 0.982172i $$-0.560196\pi$$
−0.187987 + 0.982172i $$0.560196\pi$$
$$434$$ −6959.02 −0.769685
$$435$$ 0 0
$$436$$ 18834.6 2.06884
$$437$$ −6953.90 −0.761213
$$438$$ 0 0
$$439$$ −7211.54 −0.784027 −0.392014 0.919959i $$-0.628221\pi$$
−0.392014 + 0.919959i $$0.628221\pi$$
$$440$$ 24187.7 2.62069
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7500.66 −0.804441 −0.402220 0.915543i $$-0.631761\pi$$
−0.402220 + 0.915543i $$0.631761\pi$$
$$444$$ 0 0
$$445$$ −6467.65 −0.688979
$$446$$ −13509.6 −1.43430
$$447$$ 0 0
$$448$$ −9261.58 −0.976715
$$449$$ −7649.53 −0.804017 −0.402009 0.915636i $$-0.631688\pi$$
−0.402009 + 0.915636i $$0.631688\pi$$
$$450$$ 0 0
$$451$$ 5959.77 0.622249
$$452$$ −5571.13 −0.579743
$$453$$ 0 0
$$454$$ 3125.67 0.323117
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9001.79 −0.921414 −0.460707 0.887552i $$-0.652404\pi$$
−0.460707 + 0.887552i $$0.652404\pi$$
$$458$$ −4149.02 −0.423299
$$459$$ 0 0
$$460$$ −17462.5 −1.76999
$$461$$ −1347.83 −0.136171 −0.0680854 0.997679i $$-0.521689\pi$$
−0.0680854 + 0.997679i $$0.521689\pi$$
$$462$$ 0 0
$$463$$ −94.9035 −0.00952600 −0.00476300 0.999989i $$-0.501516\pi$$
−0.00476300 + 0.999989i $$0.501516\pi$$
$$464$$ 47999.9 4.80246
$$465$$ 0 0
$$466$$ 4563.28 0.453626
$$467$$ 13290.9 1.31698 0.658489 0.752591i $$-0.271196\pi$$
0.658489 + 0.752591i $$0.271196\pi$$
$$468$$ 0 0
$$469$$ 5984.90 0.589247
$$470$$ 2439.49 0.239415
$$471$$ 0 0
$$472$$ −5701.00 −0.555953
$$473$$ −10063.8 −0.978294
$$474$$ 0 0
$$475$$ 4945.70 0.477736
$$476$$ 16806.0 1.61828
$$477$$ 0 0
$$478$$ −35358.1 −3.38335
$$479$$ −3801.59 −0.362628 −0.181314 0.983425i $$-0.558035\pi$$
−0.181314 + 0.983425i $$0.558035\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 27933.2 2.63967
$$483$$ 0 0
$$484$$ −13604.1 −1.27762
$$485$$ 2489.19 0.233048
$$486$$ 0 0
$$487$$ −8747.34 −0.813922 −0.406961 0.913446i $$-0.633412\pi$$
−0.406961 + 0.913446i $$0.633412\pi$$
$$488$$ −1849.82 −0.171593
$$489$$ 0 0
$$490$$ −21233.2 −1.95759
$$491$$ 15085.4 1.38655 0.693273 0.720675i $$-0.256168\pi$$
0.693273 + 0.720675i $$0.256168\pi$$
$$492$$ 0 0
$$493$$ −27923.1 −2.55090
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 42744.7 3.86955
$$497$$ −2747.36 −0.247959
$$498$$ 0 0
$$499$$ 14593.9 1.30925 0.654623 0.755955i $$-0.272827\pi$$
0.654623 + 0.755955i $$0.272827\pi$$
$$500$$ −22004.6 −1.96815
$$501$$ 0 0
$$502$$ 23185.2 2.06136
$$503$$ −18204.9 −1.61375 −0.806876 0.590721i $$-0.798843\pi$$
−0.806876 + 0.590721i $$0.798843\pi$$
$$504$$ 0 0
$$505$$ −11098.6 −0.977983
$$506$$ −8966.25 −0.787744
$$507$$ 0 0
$$508$$ −32601.9 −2.84739
$$509$$ −7891.92 −0.687236 −0.343618 0.939109i $$-0.611653\pi$$
−0.343618 + 0.939109i $$0.611653\pi$$
$$510$$ 0 0
$$511$$ −697.221 −0.0603586
$$512$$ 3555.88 0.306932
$$513$$ 0 0
$$514$$ −7467.59 −0.640819
$$515$$ −17078.6 −1.46131
$$516$$ 0 0
$$517$$ 908.406 0.0772759
$$518$$ −8754.05 −0.742530
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 647.902 0.0544820 0.0272410 0.999629i $$-0.491328\pi$$
0.0272410 + 0.999629i $$0.491328\pi$$
$$522$$ 0 0
$$523$$ 9269.99 0.775044 0.387522 0.921860i $$-0.373331\pi$$
0.387522 + 0.921860i $$0.373331\pi$$
$$524$$ 16402.4 1.36745
$$525$$ 0 0
$$526$$ −13374.7 −1.10867
$$527$$ −24866.0 −2.05537
$$528$$ 0 0
$$529$$ −8146.25 −0.669536
$$530$$ −4297.05 −0.352173
$$531$$ 0 0
$$532$$ −14878.1 −1.21250
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −12550.5 −1.01421
$$536$$ −65923.4 −5.31242
$$537$$ 0 0
$$538$$ −13234.2 −1.06053
$$539$$ −7906.75 −0.631852
$$540$$ 0 0
$$541$$ 3844.83 0.305549 0.152775 0.988261i $$-0.451179\pi$$
0.152775 + 0.988261i $$0.451179\pi$$
$$542$$ −32271.4 −2.55752
$$543$$ 0 0
$$544$$ −72203.1 −5.69060
$$545$$ 11633.2 0.914337
$$546$$ 0 0
$$547$$ −20245.4 −1.58251 −0.791253 0.611489i $$-0.790571\pi$$
−0.791253 + 0.611489i $$0.790571\pi$$
$$548$$ 51410.5 4.00757
$$549$$ 0 0
$$550$$ 6376.91 0.494386
$$551$$ 24720.0 1.91126
$$552$$ 0 0
$$553$$ −2470.97 −0.190011
$$554$$ −39810.8 −3.05307
$$555$$ 0 0
$$556$$ 26424.1 2.01553
$$557$$ −7223.34 −0.549485 −0.274742 0.961518i $$-0.588593\pi$$
−0.274742 + 0.961518i $$0.588593\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −17843.7 −1.34649
$$561$$ 0 0
$$562$$ −32036.3 −2.40458
$$563$$ 17290.3 1.29431 0.647156 0.762358i $$-0.275958\pi$$
0.647156 + 0.762358i $$0.275958\pi$$
$$564$$ 0 0
$$565$$ −3441.03 −0.256221
$$566$$ 8021.09 0.595674
$$567$$ 0 0
$$568$$ 30262.0 2.23551
$$569$$ −17364.0 −1.27933 −0.639663 0.768655i $$-0.720926\pi$$
−0.639663 + 0.768655i $$0.720926\pi$$
$$570$$ 0 0
$$571$$ 158.149 0.0115908 0.00579540 0.999983i $$-0.498155\pi$$
0.00579540 + 0.999983i $$0.498155\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −7884.38 −0.573323
$$575$$ −2859.61 −0.207398
$$576$$ 0 0
$$577$$ 9627.04 0.694591 0.347295 0.937756i $$-0.387100\pi$$
0.347295 + 0.937756i $$0.387100\pi$$
$$578$$ 56292.4 4.05096
$$579$$ 0 0
$$580$$ 62076.4 4.44411
$$581$$ 552.389 0.0394440
$$582$$ 0 0
$$583$$ −1600.12 −0.113671
$$584$$ 7679.86 0.544169
$$585$$ 0 0
$$586$$ 19317.6 1.36178
$$587$$ 13321.2 0.936668 0.468334 0.883551i $$-0.344854\pi$$
0.468334 + 0.883551i $$0.344854\pi$$
$$588$$ 0 0
$$589$$ 22013.6 1.53999
$$590$$ −5669.06 −0.395579
$$591$$ 0 0
$$592$$ 53770.4 3.73303
$$593$$ −16723.4 −1.15809 −0.579044 0.815296i $$-0.696574\pi$$
−0.579044 + 0.815296i $$0.696574\pi$$
$$594$$ 0 0
$$595$$ 10380.3 0.715209
$$596$$ 16546.6 1.13721
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −17160.0 −1.17051 −0.585256 0.810849i $$-0.699006\pi$$
−0.585256 + 0.810849i $$0.699006\pi$$
$$600$$ 0 0
$$601$$ 11048.6 0.749885 0.374942 0.927048i $$-0.377663\pi$$
0.374942 + 0.927048i $$0.377663\pi$$
$$602$$ 13313.7 0.901373
$$603$$ 0 0
$$604$$ 3425.37 0.230756
$$605$$ −8402.62 −0.564653
$$606$$ 0 0
$$607$$ 21119.6 1.41222 0.706110 0.708102i $$-0.250448\pi$$
0.706110 + 0.708102i $$0.250448\pi$$
$$608$$ 63920.6 4.26369
$$609$$ 0 0
$$610$$ −1839.46 −0.122094
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −6233.83 −0.410737 −0.205369 0.978685i $$-0.565839\pi$$
−0.205369 + 0.978685i $$0.565839\pi$$
$$614$$ −2239.37 −0.147188
$$615$$ 0 0
$$616$$ −11915.6 −0.779371
$$617$$ 5598.36 0.365286 0.182643 0.983179i $$-0.441535\pi$$
0.182643 + 0.983179i $$0.441535\pi$$
$$618$$ 0 0
$$619$$ −15874.3 −1.03076 −0.515382 0.856961i $$-0.672350\pi$$
−0.515382 + 0.856961i $$0.672350\pi$$
$$620$$ 55280.1 3.58081
$$621$$ 0 0
$$622$$ −16236.5 −1.04666
$$623$$ 3186.15 0.204896
$$624$$ 0 0
$$625$$ −19228.4 −1.23062
$$626$$ 20293.1 1.29565
$$627$$ 0 0
$$628$$ −49743.3 −3.16079
$$629$$ −31280.0 −1.98285
$$630$$ 0 0
$$631$$ 11249.0 0.709693 0.354846 0.934925i $$-0.384533\pi$$
0.354846 + 0.934925i $$0.384533\pi$$
$$632$$ 27217.6 1.71307
$$633$$ 0 0
$$634$$ −14957.6 −0.936976
$$635$$ −20136.7 −1.25843
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 31873.6 1.97788
$$639$$ 0 0
$$640$$ 40630.2 2.50945
$$641$$ −23768.3 −1.46457 −0.732287 0.680996i $$-0.761547\pi$$
−0.732287 + 0.680996i $$0.761547\pi$$
$$642$$ 0 0
$$643$$ −4368.95 −0.267954 −0.133977 0.990984i $$-0.542775\pi$$
−0.133977 + 0.990984i $$0.542775\pi$$
$$644$$ 8602.54 0.526378
$$645$$ 0 0
$$646$$ −73304.0 −4.46456
$$647$$ −21305.2 −1.29458 −0.647290 0.762244i $$-0.724098\pi$$
−0.647290 + 0.762244i $$0.724098\pi$$
$$648$$ 0 0
$$649$$ −2111.02 −0.127681
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 33242.0 1.99672
$$653$$ −27958.3 −1.67549 −0.837744 0.546064i $$-0.816126\pi$$
−0.837744 + 0.546064i $$0.816126\pi$$
$$654$$ 0 0
$$655$$ 10131.0 0.604353
$$656$$ 48428.6 2.88235
$$657$$ 0 0
$$658$$ −1201.76 −0.0711999
$$659$$ 9619.46 0.568621 0.284310 0.958732i $$-0.408235\pi$$
0.284310 + 0.958732i $$0.408235\pi$$
$$660$$ 0 0
$$661$$ 4414.40 0.259759 0.129879 0.991530i $$-0.458541\pi$$
0.129879 + 0.991530i $$0.458541\pi$$
$$662$$ 35387.1 2.07758
$$663$$ 0 0
$$664$$ −6084.55 −0.355612
$$665$$ −9189.52 −0.535871
$$666$$ 0 0
$$667$$ −14293.1 −0.829733
$$668$$ 75479.0 4.37181
$$669$$ 0 0
$$670$$ −65554.1 −3.77996
$$671$$ −684.970 −0.0394083
$$672$$ 0 0
$$673$$ −18001.8 −1.03108 −0.515540 0.856865i $$-0.672409\pi$$
−0.515540 + 0.856865i $$0.672409\pi$$
$$674$$ 51310.4 2.93235
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −11192.0 −0.635369 −0.317684 0.948197i $$-0.602905\pi$$
−0.317684 + 0.948197i $$0.602905\pi$$
$$678$$ 0 0
$$679$$ −1226.25 −0.0693064
$$680$$ −114338. −6.44804
$$681$$ 0 0
$$682$$ 28383.9 1.59366
$$683$$ 35122.3 1.96767 0.983835 0.179076i $$-0.0573109\pi$$
0.983835 + 0.179076i $$0.0573109\pi$$
$$684$$ 0 0
$$685$$ 31753.9 1.77117
$$686$$ 22351.3 1.24399
$$687$$ 0 0
$$688$$ −81777.5 −4.53160
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 29534.6 1.62597 0.812987 0.582282i $$-0.197840\pi$$
0.812987 + 0.582282i $$0.197840\pi$$
$$692$$ 30.4293 0.00167160
$$693$$ 0 0
$$694$$ −52927.1 −2.89494
$$695$$ 16321.0 0.890776
$$696$$ 0 0
$$697$$ −28172.5 −1.53100
$$698$$ 4289.47 0.232606
$$699$$ 0 0
$$700$$ −6118.24 −0.330354
$$701$$ 8804.19 0.474365 0.237182 0.971465i $$-0.423776\pi$$
0.237182 + 0.971465i $$0.423776\pi$$
$$702$$ 0 0
$$703$$ 27691.8 1.48566
$$704$$ 37775.5 2.02232
$$705$$ 0 0
$$706$$ −38395.0 −2.04676
$$707$$ 5467.49 0.290843
$$708$$ 0 0
$$709$$ 27138.2 1.43752 0.718758 0.695261i $$-0.244711\pi$$
0.718758 + 0.695261i $$0.244711\pi$$
$$710$$ 30092.5 1.59064
$$711$$ 0 0
$$712$$ −35095.3 −1.84727
$$713$$ −12728.3 −0.668552
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −43968.6 −2.29495
$$717$$ 0 0
$$718$$ −37270.4 −1.93721
$$719$$ 17337.4 0.899274 0.449637 0.893211i $$-0.351553\pi$$
0.449637 + 0.893211i $$0.351553\pi$$
$$720$$ 0 0
$$721$$ 8413.44 0.434581
$$722$$ 27884.8 1.43735
$$723$$ 0 0
$$724$$ −9808.20 −0.503479
$$725$$ 10165.5 0.520739
$$726$$ 0 0
$$727$$ 25771.3 1.31473 0.657363 0.753574i $$-0.271672\pi$$
0.657363 + 0.753574i $$0.271672\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 7636.83 0.387194
$$731$$ 47572.6 2.40703
$$732$$ 0 0
$$733$$ 21784.2 1.09771 0.548853 0.835919i $$-0.315065\pi$$
0.548853 + 0.835919i $$0.315065\pi$$
$$734$$ −2479.59 −0.124691
$$735$$ 0 0
$$736$$ −36959.0 −1.85099
$$737$$ −24410.8 −1.22006
$$738$$ 0 0
$$739$$ 22353.8 1.11272 0.556360 0.830942i $$-0.312198\pi$$
0.556360 + 0.830942i $$0.312198\pi$$
$$740$$ 69539.2 3.45448
$$741$$ 0 0
$$742$$ 2116.85 0.104733
$$743$$ −18382.6 −0.907662 −0.453831 0.891088i $$-0.649943\pi$$
−0.453831 + 0.891088i $$0.649943\pi$$
$$744$$ 0 0
$$745$$ 10220.1 0.502597
$$746$$ 28921.5 1.41943
$$747$$ 0 0
$$748$$ −68546.9 −3.35070
$$749$$ 6182.74 0.301618
$$750$$ 0 0
$$751$$ 13782.8 0.669694 0.334847 0.942272i $$-0.391315\pi$$
0.334847 + 0.942272i $$0.391315\pi$$
$$752$$ 7381.64 0.357953
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 2115.69 0.101984
$$756$$ 0 0
$$757$$ 25639.5 1.23102 0.615510 0.788129i $$-0.288950\pi$$
0.615510 + 0.788129i $$0.288950\pi$$
$$758$$ 46372.6 2.22207
$$759$$ 0 0
$$760$$ 101222. 4.83121
$$761$$ 43.4757 0.00207095 0.00103548 0.999999i $$-0.499670\pi$$
0.00103548 + 0.999999i $$0.499670\pi$$
$$762$$ 0 0
$$763$$ −5730.88 −0.271916
$$764$$ −102755. −4.86588
$$765$$ 0 0
$$766$$ 50378.7 2.37631
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −20485.7 −0.960641 −0.480321 0.877093i $$-0.659480\pi$$
−0.480321 + 0.877093i $$0.659480\pi$$
$$770$$ −11848.8 −0.554548
$$771$$ 0 0
$$772$$ −18082.1 −0.842992
$$773$$ 29450.3 1.37031 0.685157 0.728395i $$-0.259734\pi$$
0.685157 + 0.728395i $$0.259734\pi$$
$$774$$ 0 0
$$775$$ 9052.51 0.419582
$$776$$ 13507.1 0.624839
$$777$$ 0 0
$$778$$ 69025.0 3.18080
$$779$$ 24940.8 1.14711
$$780$$ 0 0
$$781$$ 11205.7 0.513409
$$782$$ 42384.5 1.93819
$$783$$ 0 0
$$784$$ −64249.7 −2.92683
$$785$$ −30724.1 −1.39693
$$786$$ 0 0
$$787$$ 9751.10 0.441664 0.220832 0.975312i $$-0.429123\pi$$
0.220832 + 0.975312i $$0.429123\pi$$
$$788$$ −33192.3 −1.50054
$$789$$ 0 0
$$790$$ 27065.1 1.21890
$$791$$ 1695.15 0.0761980
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −72221.2 −3.22800
$$795$$ 0 0
$$796$$ 66020.3 2.93973
$$797$$ −37750.8 −1.67779 −0.838896 0.544291i $$-0.816799\pi$$
−0.838896 + 0.544291i $$0.816799\pi$$
$$798$$ 0 0
$$799$$ −4294.14 −0.190132
$$800$$ 26285.7 1.16167
$$801$$ 0 0
$$802$$ −34303.9 −1.51037
$$803$$ 2843.77 0.124975
$$804$$ 0 0
$$805$$ 5313.39 0.232637
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −60224.2 −2.62213
$$809$$ −6999.45 −0.304187 −0.152094 0.988366i $$-0.548602\pi$$
−0.152094 + 0.988366i $$0.548602\pi$$
$$810$$ 0 0
$$811$$ −35792.1 −1.54973 −0.774864 0.632128i $$-0.782182\pi$$
−0.774864 + 0.632128i $$0.782182\pi$$
$$812$$ −30580.6 −1.32164
$$813$$ 0 0
$$814$$ 35705.4 1.53744
$$815$$ 20532.1 0.882463
$$816$$ 0 0
$$817$$ −42115.5 −1.80347
$$818$$ −28607.0 −1.22276
$$819$$ 0 0
$$820$$ 62630.9 2.66727
$$821$$ 9776.62 0.415599 0.207799 0.978171i $$-0.433370\pi$$
0.207799 + 0.978171i $$0.433370\pi$$
$$822$$ 0 0
$$823$$ 19662.0 0.832777 0.416389 0.909187i $$-0.363296\pi$$
0.416389 + 0.909187i $$0.363296\pi$$
$$824$$ −92673.7 −3.91801
$$825$$ 0 0
$$826$$ 2792.75 0.117642
$$827$$ 5281.22 0.222063 0.111032 0.993817i $$-0.464585\pi$$
0.111032 + 0.993817i $$0.464585\pi$$
$$828$$ 0 0
$$829$$ 21735.5 0.910621 0.455310 0.890333i $$-0.349528\pi$$
0.455310 + 0.890333i $$0.349528\pi$$
$$830$$ −6050.46 −0.253030
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 37376.1 1.55463
$$834$$ 0 0
$$835$$ 46619.9 1.93215
$$836$$ 60683.8 2.51052
$$837$$ 0 0
$$838$$ 88470.2 3.64696
$$839$$ −28574.7 −1.17582 −0.587908 0.808928i $$-0.700048\pi$$
−0.587908 + 0.808928i $$0.700048\pi$$
$$840$$ 0 0
$$841$$ 26420.7 1.08330
$$842$$ 45782.4 1.87383
$$843$$ 0 0
$$844$$ 2605.49 0.106262
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4139.38 0.167923
$$848$$ −13002.4 −0.526540
$$849$$ 0 0
$$850$$ −30144.4 −1.21640
$$851$$ −16011.4 −0.644965
$$852$$ 0 0
$$853$$ 5251.39 0.210791 0.105395 0.994430i $$-0.466389\pi$$
0.105395 + 0.994430i $$0.466389\pi$$
$$854$$ 906.170 0.0363097
$$855$$ 0 0
$$856$$ −68102.6 −2.71927
$$857$$ −34851.0 −1.38913 −0.694566 0.719429i $$-0.744404\pi$$
−0.694566 + 0.719429i $$0.744404\pi$$
$$858$$ 0 0
$$859$$ 41697.5 1.65623 0.828114 0.560559i $$-0.189414\pi$$
0.828114 + 0.560559i $$0.189414\pi$$
$$860$$ −105760. −4.19346
$$861$$ 0 0
$$862$$ 8605.05 0.340010
$$863$$ 28648.9 1.13004 0.565018 0.825079i $$-0.308869\pi$$
0.565018 + 0.825079i $$0.308869\pi$$
$$864$$ 0 0
$$865$$ 18.7948 0.000738777 0
$$866$$ −18279.0 −0.717257
$$867$$ 0 0
$$868$$ −27232.6 −1.06490
$$869$$ 10078.4 0.393425
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 63125.4 2.45149
$$873$$ 0 0
$$874$$ −37522.5 −1.45219
$$875$$ 6695.42 0.258682
$$876$$ 0 0
$$877$$ −31516.8 −1.21351 −0.606754 0.794890i $$-0.707529\pi$$
−0.606754 + 0.794890i $$0.707529\pi$$
$$878$$ −38912.7 −1.49572
$$879$$ 0 0
$$880$$ 72779.7 2.78796
$$881$$ 31246.1 1.19490 0.597451 0.801905i $$-0.296180\pi$$
0.597451 + 0.801905i $$0.296180\pi$$
$$882$$ 0 0
$$883$$ −16181.8 −0.616718 −0.308359 0.951270i $$-0.599780\pi$$
−0.308359 + 0.951270i $$0.599780\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −40472.8 −1.53466
$$887$$ −13289.0 −0.503046 −0.251523 0.967851i $$-0.580931\pi$$
−0.251523 + 0.967851i $$0.580931\pi$$
$$888$$ 0 0
$$889$$ 9919.92 0.374245
$$890$$ −34898.7 −1.31439
$$891$$ 0 0
$$892$$ −52867.0 −1.98444
$$893$$ 3801.55 0.142457
$$894$$ 0 0
$$895$$ −27157.4 −1.01427
$$896$$ −20015.6 −0.746288
$$897$$ 0 0
$$898$$ −41276.0 −1.53385
$$899$$ 45246.9 1.67861
$$900$$ 0 0
$$901$$ 7563.94 0.279679
$$902$$ 32158.2 1.18709
$$903$$ 0 0
$$904$$ −18672.0 −0.686971
$$905$$ −6058.07 −0.222516
$$906$$ 0 0
$$907$$ 47128.4 1.72533 0.862664 0.505778i $$-0.168795\pi$$
0.862664 + 0.505778i $$0.168795\pi$$
$$908$$ 12231.6 0.447050
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2884.69 −0.104911 −0.0524556 0.998623i $$-0.516705\pi$$
−0.0524556 + 0.998623i $$0.516705\pi$$
$$912$$ 0 0
$$913$$ −2253.05 −0.0816703
$$914$$ −48572.7 −1.75781
$$915$$ 0 0
$$916$$ −16236.3 −0.585657
$$917$$ −4990.83 −0.179729
$$918$$ 0 0
$$919$$ −25752.2 −0.924359 −0.462179 0.886787i $$-0.652932\pi$$
−0.462179 + 0.886787i $$0.652932\pi$$
$$920$$ −58526.8 −2.09736
$$921$$ 0 0
$$922$$ −7272.75 −0.259778
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 11387.5 0.404779
$$926$$ −512.089 −0.0181731
$$927$$ 0 0
$$928$$ 131383. 4.64748
$$929$$ −51394.7 −1.81508 −0.907539 0.419969i $$-0.862041\pi$$
−0.907539 + 0.419969i $$0.862041\pi$$
$$930$$ 0 0
$$931$$ −33088.6 −1.16481
$$932$$ 17857.4 0.627616
$$933$$ 0 0
$$934$$ 71716.1 2.51244
$$935$$ −42338.3 −1.48087
$$936$$ 0 0
$$937$$ 6781.86 0.236450 0.118225 0.992987i $$-0.462280\pi$$
0.118225 + 0.992987i $$0.462280\pi$$
$$938$$ 32293.9 1.12413
$$939$$ 0 0
$$940$$ 9546.39 0.331244
$$941$$ 22038.2 0.763470 0.381735 0.924272i $$-0.375327\pi$$
0.381735 + 0.924272i $$0.375327\pi$$
$$942$$ 0 0
$$943$$ −14420.8 −0.497991
$$944$$ −17154.0 −0.591436
$$945$$ 0 0
$$946$$ −54303.0 −1.86633
$$947$$ 45531.0 1.56236 0.781182 0.624304i $$-0.214617\pi$$
0.781182 + 0.624304i $$0.214617\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 26686.5 0.911393
$$951$$ 0 0
$$952$$ 56326.3 1.91759
$$953$$ 26541.2 0.902154 0.451077 0.892485i $$-0.351040\pi$$
0.451077 + 0.892485i $$0.351040\pi$$
$$954$$ 0 0
$$955$$ −63466.7 −2.15051
$$956$$ −138366. −4.68104
$$957$$ 0 0
$$958$$ −20513.0 −0.691799
$$959$$ −15642.9 −0.526731
$$960$$ 0 0
$$961$$ 10502.1 0.352527
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 109310. 3.65213
$$965$$ −11168.5 −0.372566
$$966$$ 0 0
$$967$$ 48269.6 1.60522 0.802609 0.596506i $$-0.203445\pi$$
0.802609 + 0.596506i $$0.203445\pi$$
$$968$$ −45595.1 −1.51393
$$969$$ 0 0
$$970$$ 13431.4 0.444594
$$971$$ −34161.3 −1.12903 −0.564515 0.825423i $$-0.690937\pi$$
−0.564515 + 0.825423i $$0.690937\pi$$
$$972$$ 0 0
$$973$$ −8040.18 −0.264909
$$974$$ −47199.7 −1.55275
$$975$$ 0 0
$$976$$ −5566.02 −0.182545
$$977$$ −24124.2 −0.789971 −0.394985 0.918687i $$-0.629250\pi$$
−0.394985 + 0.918687i $$0.629250\pi$$
$$978$$ 0 0
$$979$$ −12995.4 −0.424245
$$980$$ −83091.7 −2.70843
$$981$$ 0 0
$$982$$ 81399.1 2.64516
$$983$$ 19023.4 0.617246 0.308623 0.951185i $$-0.400132\pi$$
0.308623 + 0.951185i $$0.400132\pi$$
$$984$$ 0 0
$$985$$ −20501.3 −0.663175
$$986$$ −150670. −4.86644
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24351.2 0.782936
$$990$$ 0 0
$$991$$ −229.962 −0.00737131 −0.00368566 0.999993i $$-0.501173\pi$$
−0.00368566 + 0.999993i $$0.501173\pi$$
$$992$$ 116999. 3.74468
$$993$$ 0 0
$$994$$ −14824.4 −0.473041
$$995$$ 40777.7 1.29924
$$996$$ 0 0
$$997$$ 26991.1 0.857389 0.428694 0.903450i $$-0.358974\pi$$
0.428694 + 0.903450i $$0.358974\pi$$
$$998$$ 78747.2 2.49769
$$999$$ 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 1521.4.a.bc.1.8 8
3.2 odd 2 inner 1521.4.a.bc.1.1 8
13.3 even 3 117.4.g.f.100.1 yes 16
13.9 even 3 117.4.g.f.55.1 16
13.12 even 2 1521.4.a.bd.1.1 8
39.29 odd 6 117.4.g.f.100.8 yes 16
39.35 odd 6 117.4.g.f.55.8 yes 16
39.38 odd 2 1521.4.a.bd.1.8 8

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.g.f.55.1 16 13.9 even 3
117.4.g.f.55.8 yes 16 39.35 odd 6
117.4.g.f.100.1 yes 16 13.3 even 3
117.4.g.f.100.8 yes 16 39.29 odd 6
1521.4.a.bc.1.1 8 3.2 odd 2 inner
1521.4.a.bc.1.8 8 1.1 even 1 trivial
1521.4.a.bd.1.1 8 13.12 even 2
1521.4.a.bd.1.8 8 39.38 odd 2