# Properties

 Label 435.4.a.c.1.1 Level $435$ Weight $4$ Character 435.1 Self dual yes Analytic conductor $25.666$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,4,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, 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("435.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$25.6658308525$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 435.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} -15.0000 q^{6} +16.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} +5.00000 q^{5} -15.0000 q^{6} +16.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +25.0000 q^{10} -44.0000 q^{11} -51.0000 q^{12} +78.0000 q^{13} +80.0000 q^{14} -15.0000 q^{15} +89.0000 q^{16} +18.0000 q^{17} +45.0000 q^{18} -28.0000 q^{19} +85.0000 q^{20} -48.0000 q^{21} -220.000 q^{22} +184.000 q^{23} -135.000 q^{24} +25.0000 q^{25} +390.000 q^{26} -27.0000 q^{27} +272.000 q^{28} +29.0000 q^{29} -75.0000 q^{30} -224.000 q^{31} +85.0000 q^{32} +132.000 q^{33} +90.0000 q^{34} +80.0000 q^{35} +153.000 q^{36} +254.000 q^{37} -140.000 q^{38} -234.000 q^{39} +225.000 q^{40} -78.0000 q^{41} -240.000 q^{42} -260.000 q^{43} -748.000 q^{44} +45.0000 q^{45} +920.000 q^{46} +312.000 q^{47} -267.000 q^{48} -87.0000 q^{49} +125.000 q^{50} -54.0000 q^{51} +1326.00 q^{52} +574.000 q^{53} -135.000 q^{54} -220.000 q^{55} +720.000 q^{56} +84.0000 q^{57} +145.000 q^{58} +180.000 q^{59} -255.000 q^{60} -610.000 q^{61} -1120.00 q^{62} +144.000 q^{63} -287.000 q^{64} +390.000 q^{65} +660.000 q^{66} -340.000 q^{67} +306.000 q^{68} -552.000 q^{69} +400.000 q^{70} +296.000 q^{71} +405.000 q^{72} +394.000 q^{73} +1270.00 q^{74} -75.0000 q^{75} -476.000 q^{76} -704.000 q^{77} -1170.00 q^{78} -960.000 q^{79} +445.000 q^{80} +81.0000 q^{81} -390.000 q^{82} -908.000 q^{83} -816.000 q^{84} +90.0000 q^{85} -1300.00 q^{86} -87.0000 q^{87} -1980.00 q^{88} -990.000 q^{89} +225.000 q^{90} +1248.00 q^{91} +3128.00 q^{92} +672.000 q^{93} +1560.00 q^{94} -140.000 q^{95} -255.000 q^{96} +1234.00 q^{97} -435.000 q^{98} -396.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 17.0000 2.12500
$$5$$ 5.00000 0.447214
$$6$$ −15.0000 −1.02062
$$7$$ 16.0000 0.863919 0.431959 0.901893i $$-0.357822\pi$$
0.431959 + 0.901893i $$0.357822\pi$$
$$8$$ 45.0000 1.98874
$$9$$ 9.00000 0.333333
$$10$$ 25.0000 0.790569
$$11$$ −44.0000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −51.0000 −1.22687
$$13$$ 78.0000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 80.0000 1.52721
$$15$$ −15.0000 −0.258199
$$16$$ 89.0000 1.39062
$$17$$ 18.0000 0.256802 0.128401 0.991722i $$-0.459015\pi$$
0.128401 + 0.991722i $$0.459015\pi$$
$$18$$ 45.0000 0.589256
$$19$$ −28.0000 −0.338086 −0.169043 0.985609i $$-0.554068\pi$$
−0.169043 + 0.985609i $$0.554068\pi$$
$$20$$ 85.0000 0.950329
$$21$$ −48.0000 −0.498784
$$22$$ −220.000 −2.13201
$$23$$ 184.000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ −135.000 −1.14820
$$25$$ 25.0000 0.200000
$$26$$ 390.000 2.94174
$$27$$ −27.0000 −0.192450
$$28$$ 272.000 1.83583
$$29$$ 29.0000 0.185695
$$30$$ −75.0000 −0.456435
$$31$$ −224.000 −1.29779 −0.648897 0.760877i $$-0.724769\pi$$
−0.648897 + 0.760877i $$0.724769\pi$$
$$32$$ 85.0000 0.469563
$$33$$ 132.000 0.696311
$$34$$ 90.0000 0.453967
$$35$$ 80.0000 0.386356
$$36$$ 153.000 0.708333
$$37$$ 254.000 1.12858 0.564288 0.825578i $$-0.309151\pi$$
0.564288 + 0.825578i $$0.309151\pi$$
$$38$$ −140.000 −0.597658
$$39$$ −234.000 −0.960769
$$40$$ 225.000 0.889391
$$41$$ −78.0000 −0.297111 −0.148556 0.988904i $$-0.547462\pi$$
−0.148556 + 0.988904i $$0.547462\pi$$
$$42$$ −240.000 −0.881733
$$43$$ −260.000 −0.922084 −0.461042 0.887378i $$-0.652524\pi$$
−0.461042 + 0.887378i $$0.652524\pi$$
$$44$$ −748.000 −2.56285
$$45$$ 45.0000 0.149071
$$46$$ 920.000 2.94884
$$47$$ 312.000 0.968295 0.484148 0.874986i $$-0.339130\pi$$
0.484148 + 0.874986i $$0.339130\pi$$
$$48$$ −267.000 −0.802878
$$49$$ −87.0000 −0.253644
$$50$$ 125.000 0.353553
$$51$$ −54.0000 −0.148265
$$52$$ 1326.00 3.53621
$$53$$ 574.000 1.48764 0.743820 0.668380i $$-0.233012\pi$$
0.743820 + 0.668380i $$0.233012\pi$$
$$54$$ −135.000 −0.340207
$$55$$ −220.000 −0.539360
$$56$$ 720.000 1.71811
$$57$$ 84.0000 0.195194
$$58$$ 145.000 0.328266
$$59$$ 180.000 0.397187 0.198593 0.980082i $$-0.436363\pi$$
0.198593 + 0.980082i $$0.436363\pi$$
$$60$$ −255.000 −0.548673
$$61$$ −610.000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −1120.00 −2.29420
$$63$$ 144.000 0.287973
$$64$$ −287.000 −0.560547
$$65$$ 390.000 0.744208
$$66$$ 660.000 1.23091
$$67$$ −340.000 −0.619964 −0.309982 0.950742i $$-0.600323\pi$$
−0.309982 + 0.950742i $$0.600323\pi$$
$$68$$ 306.000 0.545705
$$69$$ −552.000 −0.963087
$$70$$ 400.000 0.682988
$$71$$ 296.000 0.494771 0.247385 0.968917i $$-0.420429\pi$$
0.247385 + 0.968917i $$0.420429\pi$$
$$72$$ 405.000 0.662913
$$73$$ 394.000 0.631702 0.315851 0.948809i $$-0.397710\pi$$
0.315851 + 0.948809i $$0.397710\pi$$
$$74$$ 1270.00 1.99506
$$75$$ −75.0000 −0.115470
$$76$$ −476.000 −0.718433
$$77$$ −704.000 −1.04193
$$78$$ −1170.00 −1.69842
$$79$$ −960.000 −1.36720 −0.683598 0.729859i $$-0.739586\pi$$
−0.683598 + 0.729859i $$0.739586\pi$$
$$80$$ 445.000 0.621906
$$81$$ 81.0000 0.111111
$$82$$ −390.000 −0.525223
$$83$$ −908.000 −1.20079 −0.600397 0.799702i $$-0.704991\pi$$
−0.600397 + 0.799702i $$0.704991\pi$$
$$84$$ −816.000 −1.05992
$$85$$ 90.0000 0.114846
$$86$$ −1300.00 −1.63003
$$87$$ −87.0000 −0.107211
$$88$$ −1980.00 −2.39851
$$89$$ −990.000 −1.17910 −0.589549 0.807732i $$-0.700695\pi$$
−0.589549 + 0.807732i $$0.700695\pi$$
$$90$$ 225.000 0.263523
$$91$$ 1248.00 1.43765
$$92$$ 3128.00 3.54475
$$93$$ 672.000 0.749281
$$94$$ 1560.00 1.71172
$$95$$ −140.000 −0.151197
$$96$$ −255.000 −0.271102
$$97$$ 1234.00 1.29169 0.645844 0.763469i $$-0.276506\pi$$
0.645844 + 0.763469i $$0.276506\pi$$
$$98$$ −435.000 −0.448384
$$99$$ −396.000 −0.402015
$$100$$ 425.000 0.425000
$$101$$ 1022.00 1.00686 0.503430 0.864036i $$-0.332071\pi$$
0.503430 + 0.864036i $$0.332071\pi$$
$$102$$ −270.000 −0.262098
$$103$$ −1248.00 −1.19387 −0.596937 0.802288i $$-0.703616\pi$$
−0.596937 + 0.802288i $$0.703616\pi$$
$$104$$ 3510.00 3.30946
$$105$$ −240.000 −0.223063
$$106$$ 2870.00 2.62980
$$107$$ −116.000 −0.104805 −0.0524025 0.998626i $$-0.516688\pi$$
−0.0524025 + 0.998626i $$0.516688\pi$$
$$108$$ −459.000 −0.408956
$$109$$ −826.000 −0.725839 −0.362920 0.931820i $$-0.618220\pi$$
−0.362920 + 0.931820i $$0.618220\pi$$
$$110$$ −1100.00 −0.953463
$$111$$ −762.000 −0.651584
$$112$$ 1424.00 1.20139
$$113$$ −2206.00 −1.83649 −0.918243 0.396016i $$-0.870392\pi$$
−0.918243 + 0.396016i $$0.870392\pi$$
$$114$$ 420.000 0.345058
$$115$$ 920.000 0.746004
$$116$$ 493.000 0.394603
$$117$$ 702.000 0.554700
$$118$$ 900.000 0.702133
$$119$$ 288.000 0.221856
$$120$$ −675.000 −0.513490
$$121$$ 605.000 0.454545
$$122$$ −3050.00 −2.26339
$$123$$ 234.000 0.171537
$$124$$ −3808.00 −2.75781
$$125$$ 125.000 0.0894427
$$126$$ 720.000 0.509069
$$127$$ −2056.00 −1.43654 −0.718270 0.695765i $$-0.755066\pi$$
−0.718270 + 0.695765i $$0.755066\pi$$
$$128$$ −2115.00 −1.46048
$$129$$ 780.000 0.532366
$$130$$ 1950.00 1.31559
$$131$$ 12.0000 0.00800340 0.00400170 0.999992i $$-0.498726\pi$$
0.00400170 + 0.999992i $$0.498726\pi$$
$$132$$ 2244.00 1.47966
$$133$$ −448.000 −0.292079
$$134$$ −1700.00 −1.09595
$$135$$ −135.000 −0.0860663
$$136$$ 810.000 0.510713
$$137$$ −2758.00 −1.71994 −0.859970 0.510344i $$-0.829518\pi$$
−0.859970 + 0.510344i $$0.829518\pi$$
$$138$$ −2760.00 −1.70251
$$139$$ −1436.00 −0.876258 −0.438129 0.898912i $$-0.644359\pi$$
−0.438129 + 0.898912i $$0.644359\pi$$
$$140$$ 1360.00 0.821007
$$141$$ −936.000 −0.559046
$$142$$ 1480.00 0.874640
$$143$$ −3432.00 −2.00698
$$144$$ 801.000 0.463542
$$145$$ 145.000 0.0830455
$$146$$ 1970.00 1.11670
$$147$$ 261.000 0.146442
$$148$$ 4318.00 2.39823
$$149$$ −498.000 −0.273810 −0.136905 0.990584i $$-0.543716\pi$$
−0.136905 + 0.990584i $$0.543716\pi$$
$$150$$ −375.000 −0.204124
$$151$$ −2696.00 −1.45296 −0.726481 0.687186i $$-0.758846\pi$$
−0.726481 + 0.687186i $$0.758846\pi$$
$$152$$ −1260.00 −0.672365
$$153$$ 162.000 0.0856008
$$154$$ −3520.00 −1.84188
$$155$$ −1120.00 −0.580391
$$156$$ −3978.00 −2.04163
$$157$$ 534.000 0.271451 0.135726 0.990746i $$-0.456663\pi$$
0.135726 + 0.990746i $$0.456663\pi$$
$$158$$ −4800.00 −2.41688
$$159$$ −1722.00 −0.858890
$$160$$ 425.000 0.209995
$$161$$ 2944.00 1.44112
$$162$$ 405.000 0.196419
$$163$$ 1380.00 0.663128 0.331564 0.943433i $$-0.392424\pi$$
0.331564 + 0.943433i $$0.392424\pi$$
$$164$$ −1326.00 −0.631361
$$165$$ 660.000 0.311400
$$166$$ −4540.00 −2.12272
$$167$$ 2616.00 1.21217 0.606084 0.795400i $$-0.292739\pi$$
0.606084 + 0.795400i $$0.292739\pi$$
$$168$$ −2160.00 −0.991950
$$169$$ 3887.00 1.76923
$$170$$ 450.000 0.203020
$$171$$ −252.000 −0.112695
$$172$$ −4420.00 −1.95943
$$173$$ −330.000 −0.145026 −0.0725128 0.997367i $$-0.523102\pi$$
−0.0725128 + 0.997367i $$0.523102\pi$$
$$174$$ −435.000 −0.189525
$$175$$ 400.000 0.172784
$$176$$ −3916.00 −1.67716
$$177$$ −540.000 −0.229316
$$178$$ −4950.00 −2.08437
$$179$$ −372.000 −0.155333 −0.0776664 0.996979i $$-0.524747\pi$$
−0.0776664 + 0.996979i $$0.524747\pi$$
$$180$$ 765.000 0.316776
$$181$$ −1010.00 −0.414766 −0.207383 0.978260i $$-0.566495\pi$$
−0.207383 + 0.978260i $$0.566495\pi$$
$$182$$ 6240.00 2.54143
$$183$$ 1830.00 0.739221
$$184$$ 8280.00 3.31744
$$185$$ 1270.00 0.504715
$$186$$ 3360.00 1.32455
$$187$$ −792.000 −0.309715
$$188$$ 5304.00 2.05763
$$189$$ −432.000 −0.166261
$$190$$ −700.000 −0.267281
$$191$$ 2008.00 0.760700 0.380350 0.924843i $$-0.375803\pi$$
0.380350 + 0.924843i $$0.375803\pi$$
$$192$$ 861.000 0.323632
$$193$$ 2578.00 0.961495 0.480747 0.876859i $$-0.340365\pi$$
0.480747 + 0.876859i $$0.340365\pi$$
$$194$$ 6170.00 2.28340
$$195$$ −1170.00 −0.429669
$$196$$ −1479.00 −0.538994
$$197$$ 526.000 0.190233 0.0951166 0.995466i $$-0.469678\pi$$
0.0951166 + 0.995466i $$0.469678\pi$$
$$198$$ −1980.00 −0.710669
$$199$$ 4440.00 1.58162 0.790812 0.612059i $$-0.209658\pi$$
0.790812 + 0.612059i $$0.209658\pi$$
$$200$$ 1125.00 0.397748
$$201$$ 1020.00 0.357937
$$202$$ 5110.00 1.77989
$$203$$ 464.000 0.160426
$$204$$ −918.000 −0.315063
$$205$$ −390.000 −0.132872
$$206$$ −6240.00 −2.11049
$$207$$ 1656.00 0.556038
$$208$$ 6942.00 2.31414
$$209$$ 1232.00 0.407747
$$210$$ −1200.00 −0.394323
$$211$$ 308.000 0.100491 0.0502455 0.998737i $$-0.484000\pi$$
0.0502455 + 0.998737i $$0.484000\pi$$
$$212$$ 9758.00 3.16124
$$213$$ −888.000 −0.285656
$$214$$ −580.000 −0.185271
$$215$$ −1300.00 −0.412369
$$216$$ −1215.00 −0.382733
$$217$$ −3584.00 −1.12119
$$218$$ −4130.00 −1.28311
$$219$$ −1182.00 −0.364713
$$220$$ −3740.00 −1.14614
$$221$$ 1404.00 0.427345
$$222$$ −3810.00 −1.15185
$$223$$ 4120.00 1.23720 0.618600 0.785706i $$-0.287700\pi$$
0.618600 + 0.785706i $$0.287700\pi$$
$$224$$ 1360.00 0.405664
$$225$$ 225.000 0.0666667
$$226$$ −11030.0 −3.24648
$$227$$ 4932.00 1.44206 0.721032 0.692902i $$-0.243668\pi$$
0.721032 + 0.692902i $$0.243668\pi$$
$$228$$ 1428.00 0.414788
$$229$$ −3050.00 −0.880130 −0.440065 0.897966i $$-0.645045\pi$$
−0.440065 + 0.897966i $$0.645045\pi$$
$$230$$ 4600.00 1.31876
$$231$$ 2112.00 0.601556
$$232$$ 1305.00 0.369299
$$233$$ 82.0000 0.0230558 0.0115279 0.999934i $$-0.496330\pi$$
0.0115279 + 0.999934i $$0.496330\pi$$
$$234$$ 3510.00 0.980581
$$235$$ 1560.00 0.433035
$$236$$ 3060.00 0.844021
$$237$$ 2880.00 0.789351
$$238$$ 1440.00 0.392190
$$239$$ −5104.00 −1.38138 −0.690691 0.723150i $$-0.742694\pi$$
−0.690691 + 0.723150i $$0.742694\pi$$
$$240$$ −1335.00 −0.359058
$$241$$ −2158.00 −0.576801 −0.288400 0.957510i $$-0.593123\pi$$
−0.288400 + 0.957510i $$0.593123\pi$$
$$242$$ 3025.00 0.803530
$$243$$ −243.000 −0.0641500
$$244$$ −10370.0 −2.72078
$$245$$ −435.000 −0.113433
$$246$$ 1170.00 0.303238
$$247$$ −2184.00 −0.562610
$$248$$ −10080.0 −2.58097
$$249$$ 2724.00 0.693279
$$250$$ 625.000 0.158114
$$251$$ 6116.00 1.53800 0.769001 0.639248i $$-0.220754\pi$$
0.769001 + 0.639248i $$0.220754\pi$$
$$252$$ 2448.00 0.611942
$$253$$ −8096.00 −2.01182
$$254$$ −10280.0 −2.53947
$$255$$ −270.000 −0.0663061
$$256$$ −8279.00 −2.02124
$$257$$ 3418.00 0.829607 0.414803 0.909911i $$-0.363850\pi$$
0.414803 + 0.909911i $$0.363850\pi$$
$$258$$ 3900.00 0.941098
$$259$$ 4064.00 0.974999
$$260$$ 6630.00 1.58144
$$261$$ 261.000 0.0618984
$$262$$ 60.0000 0.0141481
$$263$$ 7440.00 1.74437 0.872186 0.489174i $$-0.162702\pi$$
0.872186 + 0.489174i $$0.162702\pi$$
$$264$$ 5940.00 1.38478
$$265$$ 2870.00 0.665293
$$266$$ −2240.00 −0.516328
$$267$$ 2970.00 0.680753
$$268$$ −5780.00 −1.31742
$$269$$ 6582.00 1.49186 0.745932 0.666022i $$-0.232004\pi$$
0.745932 + 0.666022i $$0.232004\pi$$
$$270$$ −675.000 −0.152145
$$271$$ −5504.00 −1.23374 −0.616871 0.787064i $$-0.711600\pi$$
−0.616871 + 0.787064i $$0.711600\pi$$
$$272$$ 1602.00 0.357116
$$273$$ −3744.00 −0.830026
$$274$$ −13790.0 −3.04045
$$275$$ −1100.00 −0.241209
$$276$$ −9384.00 −2.04656
$$277$$ 3718.00 0.806473 0.403236 0.915096i $$-0.367885\pi$$
0.403236 + 0.915096i $$0.367885\pi$$
$$278$$ −7180.00 −1.54902
$$279$$ −2016.00 −0.432598
$$280$$ 3600.00 0.768361
$$281$$ 1754.00 0.372366 0.186183 0.982515i $$-0.440388\pi$$
0.186183 + 0.982515i $$0.440388\pi$$
$$282$$ −4680.00 −0.988262
$$283$$ 3572.00 0.750295 0.375147 0.926965i $$-0.377592\pi$$
0.375147 + 0.926965i $$0.377592\pi$$
$$284$$ 5032.00 1.05139
$$285$$ 420.000 0.0872935
$$286$$ −17160.0 −3.54787
$$287$$ −1248.00 −0.256680
$$288$$ 765.000 0.156521
$$289$$ −4589.00 −0.934053
$$290$$ 725.000 0.146805
$$291$$ −3702.00 −0.745756
$$292$$ 6698.00 1.34237
$$293$$ 126.000 0.0251229 0.0125614 0.999921i $$-0.496001\pi$$
0.0125614 + 0.999921i $$0.496001\pi$$
$$294$$ 1305.00 0.258875
$$295$$ 900.000 0.177627
$$296$$ 11430.0 2.24444
$$297$$ 1188.00 0.232104
$$298$$ −2490.00 −0.484033
$$299$$ 14352.0 2.77591
$$300$$ −1275.00 −0.245374
$$301$$ −4160.00 −0.796606
$$302$$ −13480.0 −2.56850
$$303$$ −3066.00 −0.581311
$$304$$ −2492.00 −0.470151
$$305$$ −3050.00 −0.572598
$$306$$ 810.000 0.151322
$$307$$ −2412.00 −0.448404 −0.224202 0.974543i $$-0.571978\pi$$
−0.224202 + 0.974543i $$0.571978\pi$$
$$308$$ −11968.0 −2.21409
$$309$$ 3744.00 0.689284
$$310$$ −5600.00 −1.02600
$$311$$ −2928.00 −0.533864 −0.266932 0.963715i $$-0.586010\pi$$
−0.266932 + 0.963715i $$0.586010\pi$$
$$312$$ −10530.0 −1.91072
$$313$$ 2874.00 0.519003 0.259502 0.965743i $$-0.416442\pi$$
0.259502 + 0.965743i $$0.416442\pi$$
$$314$$ 2670.00 0.479862
$$315$$ 720.000 0.128785
$$316$$ −16320.0 −2.90529
$$317$$ −26.0000 −0.00460664 −0.00230332 0.999997i $$-0.500733\pi$$
−0.00230332 + 0.999997i $$0.500733\pi$$
$$318$$ −8610.00 −1.51832
$$319$$ −1276.00 −0.223957
$$320$$ −1435.00 −0.250684
$$321$$ 348.000 0.0605092
$$322$$ 14720.0 2.54756
$$323$$ −504.000 −0.0868214
$$324$$ 1377.00 0.236111
$$325$$ 1950.00 0.332820
$$326$$ 6900.00 1.17226
$$327$$ 2478.00 0.419063
$$328$$ −3510.00 −0.590876
$$329$$ 4992.00 0.836528
$$330$$ 3300.00 0.550482
$$331$$ 9340.00 1.55098 0.775488 0.631363i $$-0.217504\pi$$
0.775488 + 0.631363i $$0.217504\pi$$
$$332$$ −15436.0 −2.55169
$$333$$ 2286.00 0.376192
$$334$$ 13080.0 2.14283
$$335$$ −1700.00 −0.277256
$$336$$ −4272.00 −0.693621
$$337$$ −10302.0 −1.66524 −0.832620 0.553845i $$-0.813160\pi$$
−0.832620 + 0.553845i $$0.813160\pi$$
$$338$$ 19435.0 3.12759
$$339$$ 6618.00 1.06030
$$340$$ 1530.00 0.244047
$$341$$ 9856.00 1.56520
$$342$$ −1260.00 −0.199219
$$343$$ −6880.00 −1.08305
$$344$$ −11700.0 −1.83378
$$345$$ −2760.00 −0.430706
$$346$$ −1650.00 −0.256372
$$347$$ −8020.00 −1.24074 −0.620369 0.784310i $$-0.713017\pi$$
−0.620369 + 0.784310i $$0.713017\pi$$
$$348$$ −1479.00 −0.227824
$$349$$ −1306.00 −0.200311 −0.100156 0.994972i $$-0.531934\pi$$
−0.100156 + 0.994972i $$0.531934\pi$$
$$350$$ 2000.00 0.305441
$$351$$ −2106.00 −0.320256
$$352$$ −3740.00 −0.566314
$$353$$ 5658.00 0.853102 0.426551 0.904464i $$-0.359728\pi$$
0.426551 + 0.904464i $$0.359728\pi$$
$$354$$ −2700.00 −0.405377
$$355$$ 1480.00 0.221268
$$356$$ −16830.0 −2.50558
$$357$$ −864.000 −0.128089
$$358$$ −1860.00 −0.274592
$$359$$ −12240.0 −1.79945 −0.899725 0.436457i $$-0.856233\pi$$
−0.899725 + 0.436457i $$0.856233\pi$$
$$360$$ 2025.00 0.296464
$$361$$ −6075.00 −0.885698
$$362$$ −5050.00 −0.733210
$$363$$ −1815.00 −0.262432
$$364$$ 21216.0 3.05500
$$365$$ 1970.00 0.282506
$$366$$ 9150.00 1.30677
$$367$$ 8984.00 1.27782 0.638911 0.769280i $$-0.279385\pi$$
0.638911 + 0.769280i $$0.279385\pi$$
$$368$$ 16376.0 2.31972
$$369$$ −702.000 −0.0990370
$$370$$ 6350.00 0.892218
$$371$$ 9184.00 1.28520
$$372$$ 11424.0 1.59222
$$373$$ −938.000 −0.130209 −0.0651043 0.997878i $$-0.520738\pi$$
−0.0651043 + 0.997878i $$0.520738\pi$$
$$374$$ −3960.00 −0.547505
$$375$$ −375.000 −0.0516398
$$376$$ 14040.0 1.92569
$$377$$ 2262.00 0.309016
$$378$$ −2160.00 −0.293911
$$379$$ −7796.00 −1.05661 −0.528303 0.849056i $$-0.677171\pi$$
−0.528303 + 0.849056i $$0.677171\pi$$
$$380$$ −2380.00 −0.321293
$$381$$ 6168.00 0.829386
$$382$$ 10040.0 1.34474
$$383$$ −6944.00 −0.926428 −0.463214 0.886247i $$-0.653304\pi$$
−0.463214 + 0.886247i $$0.653304\pi$$
$$384$$ 6345.00 0.843208
$$385$$ −3520.00 −0.465963
$$386$$ 12890.0 1.69970
$$387$$ −2340.00 −0.307361
$$388$$ 20978.0 2.74484
$$389$$ 2126.00 0.277101 0.138551 0.990355i $$-0.455756\pi$$
0.138551 + 0.990355i $$0.455756\pi$$
$$390$$ −5850.00 −0.759555
$$391$$ 3312.00 0.428376
$$392$$ −3915.00 −0.504432
$$393$$ −36.0000 −0.00462076
$$394$$ 2630.00 0.336288
$$395$$ −4800.00 −0.611428
$$396$$ −6732.00 −0.854282
$$397$$ −3346.00 −0.423000 −0.211500 0.977378i $$-0.567835\pi$$
−0.211500 + 0.977378i $$0.567835\pi$$
$$398$$ 22200.0 2.79594
$$399$$ 1344.00 0.168632
$$400$$ 2225.00 0.278125
$$401$$ 12850.0 1.60025 0.800123 0.599836i $$-0.204768\pi$$
0.800123 + 0.599836i $$0.204768\pi$$
$$402$$ 5100.00 0.632748
$$403$$ −17472.0 −2.15966
$$404$$ 17374.0 2.13958
$$405$$ 405.000 0.0496904
$$406$$ 2320.00 0.283595
$$407$$ −11176.0 −1.36111
$$408$$ −2430.00 −0.294860
$$409$$ 6122.00 0.740131 0.370065 0.929006i $$-0.379335\pi$$
0.370065 + 0.929006i $$0.379335\pi$$
$$410$$ −1950.00 −0.234887
$$411$$ 8274.00 0.993008
$$412$$ −21216.0 −2.53698
$$413$$ 2880.00 0.343137
$$414$$ 8280.00 0.982946
$$415$$ −4540.00 −0.537012
$$416$$ 6630.00 0.781400
$$417$$ 4308.00 0.505908
$$418$$ 6160.00 0.720803
$$419$$ 1372.00 0.159968 0.0799840 0.996796i $$-0.474513\pi$$
0.0799840 + 0.996796i $$0.474513\pi$$
$$420$$ −4080.00 −0.474009
$$421$$ 12150.0 1.40654 0.703272 0.710921i $$-0.251722\pi$$
0.703272 + 0.710921i $$0.251722\pi$$
$$422$$ 1540.00 0.177645
$$423$$ 2808.00 0.322765
$$424$$ 25830.0 2.95853
$$425$$ 450.000 0.0513605
$$426$$ −4440.00 −0.504973
$$427$$ −9760.00 −1.10613
$$428$$ −1972.00 −0.222711
$$429$$ 10296.0 1.15873
$$430$$ −6500.00 −0.728972
$$431$$ 6288.00 0.702743 0.351372 0.936236i $$-0.385715\pi$$
0.351372 + 0.936236i $$0.385715\pi$$
$$432$$ −2403.00 −0.267626
$$433$$ 15650.0 1.73693 0.868465 0.495750i $$-0.165107\pi$$
0.868465 + 0.495750i $$0.165107\pi$$
$$434$$ −17920.0 −1.98200
$$435$$ −435.000 −0.0479463
$$436$$ −14042.0 −1.54241
$$437$$ −5152.00 −0.563967
$$438$$ −5910.00 −0.644728
$$439$$ −14520.0 −1.57859 −0.789296 0.614013i $$-0.789554\pi$$
−0.789296 + 0.614013i $$0.789554\pi$$
$$440$$ −9900.00 −1.07265
$$441$$ −783.000 −0.0845481
$$442$$ 7020.00 0.755446
$$443$$ 7372.00 0.790642 0.395321 0.918543i $$-0.370633\pi$$
0.395321 + 0.918543i $$0.370633\pi$$
$$444$$ −12954.0 −1.38462
$$445$$ −4950.00 −0.527309
$$446$$ 20600.0 2.18708
$$447$$ 1494.00 0.158085
$$448$$ −4592.00 −0.484267
$$449$$ 10666.0 1.12107 0.560534 0.828131i $$-0.310596\pi$$
0.560534 + 0.828131i $$0.310596\pi$$
$$450$$ 1125.00 0.117851
$$451$$ 3432.00 0.358329
$$452$$ −37502.0 −3.90253
$$453$$ 8088.00 0.838868
$$454$$ 24660.0 2.54923
$$455$$ 6240.00 0.642936
$$456$$ 3780.00 0.388190
$$457$$ −8006.00 −0.819486 −0.409743 0.912201i $$-0.634382\pi$$
−0.409743 + 0.912201i $$0.634382\pi$$
$$458$$ −15250.0 −1.55586
$$459$$ −486.000 −0.0494217
$$460$$ 15640.0 1.58526
$$461$$ 1254.00 0.126691 0.0633456 0.997992i $$-0.479823\pi$$
0.0633456 + 0.997992i $$0.479823\pi$$
$$462$$ 10560.0 1.06341
$$463$$ −4584.00 −0.460122 −0.230061 0.973176i $$-0.573893\pi$$
−0.230061 + 0.973176i $$0.573893\pi$$
$$464$$ 2581.00 0.258233
$$465$$ 3360.00 0.335089
$$466$$ 410.000 0.0407573
$$467$$ 11588.0 1.14824 0.574121 0.818771i $$-0.305344\pi$$
0.574121 + 0.818771i $$0.305344\pi$$
$$468$$ 11934.0 1.17874
$$469$$ −5440.00 −0.535599
$$470$$ 7800.00 0.765505
$$471$$ −1602.00 −0.156722
$$472$$ 8100.00 0.789900
$$473$$ 11440.0 1.11208
$$474$$ 14400.0 1.39539
$$475$$ −700.000 −0.0676173
$$476$$ 4896.00 0.471445
$$477$$ 5166.00 0.495880
$$478$$ −25520.0 −2.44196
$$479$$ −18200.0 −1.73607 −0.868037 0.496500i $$-0.834618\pi$$
−0.868037 + 0.496500i $$0.834618\pi$$
$$480$$ −1275.00 −0.121241
$$481$$ 19812.0 1.87807
$$482$$ −10790.0 −1.01965
$$483$$ −8832.00 −0.832029
$$484$$ 10285.0 0.965909
$$485$$ 6170.00 0.577660
$$486$$ −1215.00 −0.113402
$$487$$ 3824.00 0.355815 0.177908 0.984047i $$-0.443067\pi$$
0.177908 + 0.984047i $$0.443067\pi$$
$$488$$ −27450.0 −2.54632
$$489$$ −4140.00 −0.382857
$$490$$ −2175.00 −0.200523
$$491$$ −3100.00 −0.284931 −0.142465 0.989800i $$-0.545503\pi$$
−0.142465 + 0.989800i $$0.545503\pi$$
$$492$$ 3978.00 0.364516
$$493$$ 522.000 0.0476870
$$494$$ −10920.0 −0.994563
$$495$$ −1980.00 −0.179787
$$496$$ −19936.0 −1.80474
$$497$$ 4736.00 0.427442
$$498$$ 13620.0 1.22556
$$499$$ 19740.0 1.77091 0.885455 0.464726i $$-0.153847\pi$$
0.885455 + 0.464726i $$0.153847\pi$$
$$500$$ 2125.00 0.190066
$$501$$ −7848.00 −0.699846
$$502$$ 30580.0 2.71883
$$503$$ −6720.00 −0.595686 −0.297843 0.954615i $$-0.596267\pi$$
−0.297843 + 0.954615i $$0.596267\pi$$
$$504$$ 6480.00 0.572703
$$505$$ 5110.00 0.450281
$$506$$ −40480.0 −3.55643
$$507$$ −11661.0 −1.02147
$$508$$ −34952.0 −3.05265
$$509$$ 10886.0 0.947964 0.473982 0.880535i $$-0.342816\pi$$
0.473982 + 0.880535i $$0.342816\pi$$
$$510$$ −1350.00 −0.117214
$$511$$ 6304.00 0.545739
$$512$$ −24475.0 −2.11260
$$513$$ 756.000 0.0650647
$$514$$ 17090.0 1.46655
$$515$$ −6240.00 −0.533917
$$516$$ 13260.0 1.13128
$$517$$ −13728.0 −1.16781
$$518$$ 20320.0 1.72357
$$519$$ 990.000 0.0837306
$$520$$ 17550.0 1.48004
$$521$$ 22522.0 1.89387 0.946935 0.321424i $$-0.104161\pi$$
0.946935 + 0.321424i $$0.104161\pi$$
$$522$$ 1305.00 0.109422
$$523$$ −13212.0 −1.10463 −0.552314 0.833636i $$-0.686255\pi$$
−0.552314 + 0.833636i $$0.686255\pi$$
$$524$$ 204.000 0.0170072
$$525$$ −1200.00 −0.0997567
$$526$$ 37200.0 3.08364
$$527$$ −4032.00 −0.333276
$$528$$ 11748.0 0.968307
$$529$$ 21689.0 1.78261
$$530$$ 14350.0 1.17608
$$531$$ 1620.00 0.132396
$$532$$ −7616.00 −0.620668
$$533$$ −6084.00 −0.494423
$$534$$ 14850.0 1.20341
$$535$$ −580.000 −0.0468703
$$536$$ −15300.0 −1.23295
$$537$$ 1116.00 0.0896815
$$538$$ 32910.0 2.63727
$$539$$ 3828.00 0.305907
$$540$$ −2295.00 −0.182891
$$541$$ −4642.00 −0.368900 −0.184450 0.982842i $$-0.559050\pi$$
−0.184450 + 0.982842i $$0.559050\pi$$
$$542$$ −27520.0 −2.18097
$$543$$ 3030.00 0.239465
$$544$$ 1530.00 0.120585
$$545$$ −4130.00 −0.324605
$$546$$ −18720.0 −1.46729
$$547$$ 4060.00 0.317355 0.158677 0.987330i $$-0.449277\pi$$
0.158677 + 0.987330i $$0.449277\pi$$
$$548$$ −46886.0 −3.65487
$$549$$ −5490.00 −0.426790
$$550$$ −5500.00 −0.426401
$$551$$ −812.000 −0.0627811
$$552$$ −24840.0 −1.91533
$$553$$ −15360.0 −1.18115
$$554$$ 18590.0 1.42566
$$555$$ −3810.00 −0.291397
$$556$$ −24412.0 −1.86205
$$557$$ −1386.00 −0.105434 −0.0527170 0.998609i $$-0.516788\pi$$
−0.0527170 + 0.998609i $$0.516788\pi$$
$$558$$ −10080.0 −0.764732
$$559$$ −20280.0 −1.53444
$$560$$ 7120.00 0.537277
$$561$$ 2376.00 0.178814
$$562$$ 8770.00 0.658256
$$563$$ 2452.00 0.183551 0.0917757 0.995780i $$-0.470746\pi$$
0.0917757 + 0.995780i $$0.470746\pi$$
$$564$$ −15912.0 −1.18797
$$565$$ −11030.0 −0.821302
$$566$$ 17860.0 1.32635
$$567$$ 1296.00 0.0959910
$$568$$ 13320.0 0.983970
$$569$$ −20862.0 −1.53705 −0.768524 0.639821i $$-0.779009\pi$$
−0.768524 + 0.639821i $$0.779009\pi$$
$$570$$ 2100.00 0.154315
$$571$$ −9420.00 −0.690394 −0.345197 0.938530i $$-0.612188\pi$$
−0.345197 + 0.938530i $$0.612188\pi$$
$$572$$ −58344.0 −4.26483
$$573$$ −6024.00 −0.439191
$$574$$ −6240.00 −0.453750
$$575$$ 4600.00 0.333623
$$576$$ −2583.00 −0.186849
$$577$$ 13202.0 0.952524 0.476262 0.879303i $$-0.341991\pi$$
0.476262 + 0.879303i $$0.341991\pi$$
$$578$$ −22945.0 −1.65119
$$579$$ −7734.00 −0.555119
$$580$$ 2465.00 0.176472
$$581$$ −14528.0 −1.03739
$$582$$ −18510.0 −1.31832
$$583$$ −25256.0 −1.79416
$$584$$ 17730.0 1.25629
$$585$$ 3510.00 0.248069
$$586$$ 630.000 0.0444114
$$587$$ −8708.00 −0.612296 −0.306148 0.951984i $$-0.599040\pi$$
−0.306148 + 0.951984i $$0.599040\pi$$
$$588$$ 4437.00 0.311188
$$589$$ 6272.00 0.438766
$$590$$ 4500.00 0.314004
$$591$$ −1578.00 −0.109831
$$592$$ 22606.0 1.56943
$$593$$ −4390.00 −0.304006 −0.152003 0.988380i $$-0.548572\pi$$
−0.152003 + 0.988380i $$0.548572\pi$$
$$594$$ 5940.00 0.410305
$$595$$ 1440.00 0.0992172
$$596$$ −8466.00 −0.581847
$$597$$ −13320.0 −0.913151
$$598$$ 71760.0 4.90716
$$599$$ −20256.0 −1.38170 −0.690850 0.722999i $$-0.742763\pi$$
−0.690850 + 0.722999i $$0.742763\pi$$
$$600$$ −3375.00 −0.229640
$$601$$ 9610.00 0.652246 0.326123 0.945327i $$-0.394258\pi$$
0.326123 + 0.945327i $$0.394258\pi$$
$$602$$ −20800.0 −1.40821
$$603$$ −3060.00 −0.206655
$$604$$ −45832.0 −3.08755
$$605$$ 3025.00 0.203279
$$606$$ −15330.0 −1.02762
$$607$$ 10376.0 0.693820 0.346910 0.937898i $$-0.387231\pi$$
0.346910 + 0.937898i $$0.387231\pi$$
$$608$$ −2380.00 −0.158753
$$609$$ −1392.00 −0.0926218
$$610$$ −15250.0 −1.01222
$$611$$ 24336.0 1.61134
$$612$$ 2754.00 0.181902
$$613$$ 6822.00 0.449491 0.224746 0.974417i $$-0.427845\pi$$
0.224746 + 0.974417i $$0.427845\pi$$
$$614$$ −12060.0 −0.792674
$$615$$ 1170.00 0.0767137
$$616$$ −31680.0 −2.07212
$$617$$ −20070.0 −1.30954 −0.654771 0.755827i $$-0.727235\pi$$
−0.654771 + 0.755827i $$0.727235\pi$$
$$618$$ 18720.0 1.21849
$$619$$ 19228.0 1.24853 0.624264 0.781214i $$-0.285399\pi$$
0.624264 + 0.781214i $$0.285399\pi$$
$$620$$ −19040.0 −1.23333
$$621$$ −4968.00 −0.321029
$$622$$ −14640.0 −0.943747
$$623$$ −15840.0 −1.01865
$$624$$ −20826.0 −1.33607
$$625$$ 625.000 0.0400000
$$626$$ 14370.0 0.917477
$$627$$ −3696.00 −0.235413
$$628$$ 9078.00 0.576834
$$629$$ 4572.00 0.289821
$$630$$ 3600.00 0.227663
$$631$$ 6552.00 0.413361 0.206681 0.978408i $$-0.433734\pi$$
0.206681 + 0.978408i $$0.433734\pi$$
$$632$$ −43200.0 −2.71899
$$633$$ −924.000 −0.0580185
$$634$$ −130.000 −0.00814347
$$635$$ −10280.0 −0.642440
$$636$$ −29274.0 −1.82514
$$637$$ −6786.00 −0.422090
$$638$$ −6380.00 −0.395904
$$639$$ 2664.00 0.164924
$$640$$ −10575.0 −0.653146
$$641$$ −14422.0 −0.888666 −0.444333 0.895862i $$-0.646559\pi$$
−0.444333 + 0.895862i $$0.646559\pi$$
$$642$$ 1740.00 0.106966
$$643$$ −6212.00 −0.380991 −0.190496 0.981688i $$-0.561009\pi$$
−0.190496 + 0.981688i $$0.561009\pi$$
$$644$$ 50048.0 3.06237
$$645$$ 3900.00 0.238081
$$646$$ −2520.00 −0.153480
$$647$$ 22024.0 1.33826 0.669129 0.743146i $$-0.266667\pi$$
0.669129 + 0.743146i $$0.266667\pi$$
$$648$$ 3645.00 0.220971
$$649$$ −7920.00 −0.479025
$$650$$ 9750.00 0.588348
$$651$$ 10752.0 0.647318
$$652$$ 23460.0 1.40915
$$653$$ 16630.0 0.996604 0.498302 0.867004i $$-0.333957\pi$$
0.498302 + 0.867004i $$0.333957\pi$$
$$654$$ 12390.0 0.740806
$$655$$ 60.0000 0.00357923
$$656$$ −6942.00 −0.413170
$$657$$ 3546.00 0.210567
$$658$$ 24960.0 1.47879
$$659$$ −24468.0 −1.44634 −0.723170 0.690670i $$-0.757316\pi$$
−0.723170 + 0.690670i $$0.757316\pi$$
$$660$$ 11220.0 0.661724
$$661$$ −10226.0 −0.601733 −0.300866 0.953666i $$-0.597276\pi$$
−0.300866 + 0.953666i $$0.597276\pi$$
$$662$$ 46700.0 2.74176
$$663$$ −4212.00 −0.246728
$$664$$ −40860.0 −2.38807
$$665$$ −2240.00 −0.130622
$$666$$ 11430.0 0.665020
$$667$$ 5336.00 0.309761
$$668$$ 44472.0 2.57586
$$669$$ −12360.0 −0.714298
$$670$$ −8500.00 −0.490125
$$671$$ 26840.0 1.54418
$$672$$ −4080.00 −0.234210
$$673$$ 13458.0 0.770829 0.385414 0.922744i $$-0.374059\pi$$
0.385414 + 0.922744i $$0.374059\pi$$
$$674$$ −51510.0 −2.94376
$$675$$ −675.000 −0.0384900
$$676$$ 66079.0 3.75962
$$677$$ 22174.0 1.25881 0.629406 0.777076i $$-0.283298\pi$$
0.629406 + 0.777076i $$0.283298\pi$$
$$678$$ 33090.0 1.87436
$$679$$ 19744.0 1.11591
$$680$$ 4050.00 0.228398
$$681$$ −14796.0 −0.832576
$$682$$ 49280.0 2.76690
$$683$$ −2404.00 −0.134680 −0.0673400 0.997730i $$-0.521451\pi$$
−0.0673400 + 0.997730i $$0.521451\pi$$
$$684$$ −4284.00 −0.239478
$$685$$ −13790.0 −0.769181
$$686$$ −34400.0 −1.91457
$$687$$ 9150.00 0.508143
$$688$$ −23140.0 −1.28227
$$689$$ 44772.0 2.47558
$$690$$ −13800.0 −0.761387
$$691$$ −5956.00 −0.327897 −0.163949 0.986469i $$-0.552423\pi$$
−0.163949 + 0.986469i $$0.552423\pi$$
$$692$$ −5610.00 −0.308179
$$693$$ −6336.00 −0.347308
$$694$$ −40100.0 −2.19334
$$695$$ −7180.00 −0.391875
$$696$$ −3915.00 −0.213215
$$697$$ −1404.00 −0.0762988
$$698$$ −6530.00 −0.354103
$$699$$ −246.000 −0.0133113
$$700$$ 6800.00 0.367165
$$701$$ −14586.0 −0.785885 −0.392943 0.919563i $$-0.628543\pi$$
−0.392943 + 0.919563i $$0.628543\pi$$
$$702$$ −10530.0 −0.566139
$$703$$ −7112.00 −0.381556
$$704$$ 12628.0 0.676045
$$705$$ −4680.00 −0.250013
$$706$$ 28290.0 1.50809
$$707$$ 16352.0 0.869845
$$708$$ −9180.00 −0.487296
$$709$$ −4370.00 −0.231479 −0.115740 0.993280i $$-0.536924\pi$$
−0.115740 + 0.993280i $$0.536924\pi$$
$$710$$ 7400.00 0.391151
$$711$$ −8640.00 −0.455732
$$712$$ −44550.0 −2.34492
$$713$$ −41216.0 −2.16487
$$714$$ −4320.00 −0.226431
$$715$$ −17160.0 −0.897549
$$716$$ −6324.00 −0.330082
$$717$$ 15312.0 0.797541
$$718$$ −61200.0 −3.18101
$$719$$ 144.000 0.00746912 0.00373456 0.999993i $$-0.498811\pi$$
0.00373456 + 0.999993i $$0.498811\pi$$
$$720$$ 4005.00 0.207302
$$721$$ −19968.0 −1.03141
$$722$$ −30375.0 −1.56571
$$723$$ 6474.00 0.333016
$$724$$ −17170.0 −0.881378
$$725$$ 725.000 0.0371391
$$726$$ −9075.00 −0.463919
$$727$$ −9632.00 −0.491377 −0.245689 0.969349i $$-0.579014\pi$$
−0.245689 + 0.969349i $$0.579014\pi$$
$$728$$ 56160.0 2.85910
$$729$$ 729.000 0.0370370
$$730$$ 9850.00 0.499404
$$731$$ −4680.00 −0.236794
$$732$$ 31110.0 1.57085
$$733$$ −19306.0 −0.972829 −0.486414 0.873728i $$-0.661695\pi$$
−0.486414 + 0.873728i $$0.661695\pi$$
$$734$$ 44920.0 2.25889
$$735$$ 1305.00 0.0654907
$$736$$ 15640.0 0.783285
$$737$$ 14960.0 0.747705
$$738$$ −3510.00 −0.175074
$$739$$ −36540.0 −1.81887 −0.909435 0.415845i $$-0.863486\pi$$
−0.909435 + 0.415845i $$0.863486\pi$$
$$740$$ 21590.0 1.07252
$$741$$ 6552.00 0.324823
$$742$$ 45920.0 2.27194
$$743$$ 5408.00 0.267026 0.133513 0.991047i $$-0.457374\pi$$
0.133513 + 0.991047i $$0.457374\pi$$
$$744$$ 30240.0 1.49012
$$745$$ −2490.00 −0.122452
$$746$$ −4690.00 −0.230178
$$747$$ −8172.00 −0.400265
$$748$$ −13464.0 −0.658145
$$749$$ −1856.00 −0.0905431
$$750$$ −1875.00 −0.0912871
$$751$$ −13952.0 −0.677917 −0.338959 0.940801i $$-0.610075\pi$$
−0.338959 + 0.940801i $$0.610075\pi$$
$$752$$ 27768.0 1.34654
$$753$$ −18348.0 −0.887966
$$754$$ 11310.0 0.546268
$$755$$ −13480.0 −0.649785
$$756$$ −7344.00 −0.353305
$$757$$ −4274.00 −0.205206 −0.102603 0.994722i $$-0.532717\pi$$
−0.102603 + 0.994722i $$0.532717\pi$$
$$758$$ −38980.0 −1.86783
$$759$$ 24288.0 1.16153
$$760$$ −6300.00 −0.300691
$$761$$ −230.000 −0.0109560 −0.00547799 0.999985i $$-0.501744\pi$$
−0.00547799 + 0.999985i $$0.501744\pi$$
$$762$$ 30840.0 1.46616
$$763$$ −13216.0 −0.627066
$$764$$ 34136.0 1.61649
$$765$$ 810.000 0.0382818
$$766$$ −34720.0 −1.63771
$$767$$ 14040.0 0.660958
$$768$$ 24837.0 1.16696
$$769$$ −7854.00 −0.368300 −0.184150 0.982898i $$-0.558953\pi$$
−0.184150 + 0.982898i $$0.558953\pi$$
$$770$$ −17600.0 −0.823714
$$771$$ −10254.0 −0.478974
$$772$$ 43826.0 2.04318
$$773$$ 19550.0 0.909657 0.454828 0.890579i $$-0.349701\pi$$
0.454828 + 0.890579i $$0.349701\pi$$
$$774$$ −11700.0 −0.543343
$$775$$ −5600.00 −0.259559
$$776$$ 55530.0 2.56883
$$777$$ −12192.0 −0.562916
$$778$$ 10630.0 0.489851
$$779$$ 2184.00 0.100449
$$780$$ −19890.0 −0.913046
$$781$$ −13024.0 −0.596716
$$782$$ 16560.0 0.757269
$$783$$ −783.000 −0.0357371
$$784$$ −7743.00 −0.352724
$$785$$ 2670.00 0.121397
$$786$$ −180.000 −0.00816843
$$787$$ 27228.0 1.23326 0.616629 0.787254i $$-0.288498\pi$$
0.616629 + 0.787254i $$0.288498\pi$$
$$788$$ 8942.00 0.404246
$$789$$ −22320.0 −1.00711
$$790$$ −24000.0 −1.08086
$$791$$ −35296.0 −1.58658
$$792$$ −17820.0 −0.799503
$$793$$ −47580.0 −2.13066
$$794$$ −16730.0 −0.747765
$$795$$ −8610.00 −0.384107
$$796$$ 75480.0 3.36095
$$797$$ −3386.00 −0.150487 −0.0752436 0.997165i $$-0.523973\pi$$
−0.0752436 + 0.997165i $$0.523973\pi$$
$$798$$ 6720.00 0.298102
$$799$$ 5616.00 0.248661
$$800$$ 2125.00 0.0939126
$$801$$ −8910.00 −0.393033
$$802$$ 64250.0 2.82886
$$803$$ −17336.0 −0.761861
$$804$$ 17340.0 0.760615
$$805$$ 14720.0 0.644487
$$806$$ −87360.0 −3.81777
$$807$$ −19746.0 −0.861329
$$808$$ 45990.0 2.00238
$$809$$ 26994.0 1.17313 0.586563 0.809904i $$-0.300481\pi$$
0.586563 + 0.809904i $$0.300481\pi$$
$$810$$ 2025.00 0.0878410
$$811$$ 8356.00 0.361799 0.180899 0.983502i $$-0.442099\pi$$
0.180899 + 0.983502i $$0.442099\pi$$
$$812$$ 7888.00 0.340905
$$813$$ 16512.0 0.712302
$$814$$ −55880.0 −2.40613
$$815$$ 6900.00 0.296560
$$816$$ −4806.00 −0.206181
$$817$$ 7280.00 0.311744
$$818$$ 30610.0 1.30838
$$819$$ 11232.0 0.479216
$$820$$ −6630.00 −0.282353
$$821$$ 9838.00 0.418208 0.209104 0.977893i $$-0.432945\pi$$
0.209104 + 0.977893i $$0.432945\pi$$
$$822$$ 41370.0 1.75541
$$823$$ −29552.0 −1.25166 −0.625831 0.779959i $$-0.715240\pi$$
−0.625831 + 0.779959i $$0.715240\pi$$
$$824$$ −56160.0 −2.37430
$$825$$ 3300.00 0.139262
$$826$$ 14400.0 0.606586
$$827$$ 18556.0 0.780236 0.390118 0.920765i $$-0.372434\pi$$
0.390118 + 0.920765i $$0.372434\pi$$
$$828$$ 28152.0 1.18158
$$829$$ 7966.00 0.333740 0.166870 0.985979i $$-0.446634\pi$$
0.166870 + 0.985979i $$0.446634\pi$$
$$830$$ −22700.0 −0.949311
$$831$$ −11154.0 −0.465617
$$832$$ −22386.0 −0.932806
$$833$$ −1566.00 −0.0651365
$$834$$ 21540.0 0.894328
$$835$$ 13080.0 0.542098
$$836$$ 20944.0 0.866463
$$837$$ 6048.00 0.249760
$$838$$ 6860.00 0.282786
$$839$$ −42048.0 −1.73022 −0.865112 0.501578i $$-0.832753\pi$$
−0.865112 + 0.501578i $$0.832753\pi$$
$$840$$ −10800.0 −0.443614
$$841$$ 841.000 0.0344828
$$842$$ 60750.0 2.48644
$$843$$ −5262.00 −0.214986
$$844$$ 5236.00 0.213543
$$845$$ 19435.0 0.791224
$$846$$ 14040.0 0.570573
$$847$$ 9680.00 0.392690
$$848$$ 51086.0 2.06875
$$849$$ −10716.0 −0.433183
$$850$$ 2250.00 0.0907934
$$851$$ 46736.0 1.88260
$$852$$ −15096.0 −0.607019
$$853$$ 29950.0 1.20219 0.601095 0.799177i $$-0.294731\pi$$
0.601095 + 0.799177i $$0.294731\pi$$
$$854$$ −48800.0 −1.95539
$$855$$ −1260.00 −0.0503989
$$856$$ −5220.00 −0.208430
$$857$$ −31454.0 −1.25373 −0.626866 0.779127i $$-0.715663\pi$$
−0.626866 + 0.779127i $$0.715663\pi$$
$$858$$ 51480.0 2.04837
$$859$$ −22036.0 −0.875272 −0.437636 0.899152i $$-0.644184\pi$$
−0.437636 + 0.899152i $$0.644184\pi$$
$$860$$ −22100.0 −0.876283
$$861$$ 3744.00 0.148194
$$862$$ 31440.0 1.24229
$$863$$ 14048.0 0.554113 0.277056 0.960854i $$-0.410641\pi$$
0.277056 + 0.960854i $$0.410641\pi$$
$$864$$ −2295.00 −0.0903675
$$865$$ −1650.00 −0.0648574
$$866$$ 78250.0 3.07049
$$867$$ 13767.0 0.539275
$$868$$ −60928.0 −2.38252
$$869$$ 42240.0 1.64890
$$870$$ −2175.00 −0.0847579
$$871$$ −26520.0 −1.03168
$$872$$ −37170.0 −1.44350
$$873$$ 11106.0 0.430563
$$874$$ −25760.0 −0.996962
$$875$$ 2000.00 0.0772712
$$876$$ −20094.0 −0.775015
$$877$$ −39922.0 −1.53714 −0.768569 0.639767i $$-0.779031\pi$$
−0.768569 + 0.639767i $$0.779031\pi$$
$$878$$ −72600.0 −2.79058
$$879$$ −378.000 −0.0145047
$$880$$ −19580.0 −0.750047
$$881$$ 38730.0 1.48110 0.740549 0.672003i $$-0.234566\pi$$
0.740549 + 0.672003i $$0.234566\pi$$
$$882$$ −3915.00 −0.149461
$$883$$ −32948.0 −1.25571 −0.627853 0.778332i $$-0.716066\pi$$
−0.627853 + 0.778332i $$0.716066\pi$$
$$884$$ 23868.0 0.908108
$$885$$ −2700.00 −0.102553
$$886$$ 36860.0 1.39767
$$887$$ −21680.0 −0.820680 −0.410340 0.911933i $$-0.634590\pi$$
−0.410340 + 0.911933i $$0.634590\pi$$
$$888$$ −34290.0 −1.29583
$$889$$ −32896.0 −1.24105
$$890$$ −24750.0 −0.932159
$$891$$ −3564.00 −0.134005
$$892$$ 70040.0 2.62905
$$893$$ −8736.00 −0.327367
$$894$$ 7470.00 0.279457
$$895$$ −1860.00 −0.0694670
$$896$$ −33840.0 −1.26174
$$897$$ −43056.0 −1.60267
$$898$$ 53330.0 1.98179
$$899$$ −6496.00 −0.240994
$$900$$ 3825.00 0.141667
$$901$$ 10332.0 0.382030
$$902$$ 17160.0 0.633443
$$903$$ 12480.0 0.459921
$$904$$ −99270.0 −3.65229
$$905$$ −5050.00 −0.185489
$$906$$ 40440.0 1.48292
$$907$$ 2236.00 0.0818580 0.0409290 0.999162i $$-0.486968\pi$$
0.0409290 + 0.999162i $$0.486968\pi$$
$$908$$ 83844.0 3.06438
$$909$$ 9198.00 0.335620
$$910$$ 31200.0 1.13656
$$911$$ 35816.0 1.30257 0.651283 0.758835i $$-0.274231\pi$$
0.651283 + 0.758835i $$0.274231\pi$$
$$912$$ 7476.00 0.271442
$$913$$ 39952.0 1.44821
$$914$$ −40030.0 −1.44866
$$915$$ 9150.00 0.330590
$$916$$ −51850.0 −1.87028
$$917$$ 192.000 0.00691428
$$918$$ −2430.00 −0.0873660
$$919$$ 39704.0 1.42515 0.712576 0.701595i $$-0.247529\pi$$
0.712576 + 0.701595i $$0.247529\pi$$
$$920$$ 41400.0 1.48361
$$921$$ 7236.00 0.258886
$$922$$ 6270.00 0.223960
$$923$$ 23088.0 0.823349
$$924$$ 35904.0 1.27831
$$925$$ 6350.00 0.225715
$$926$$ −22920.0 −0.813389
$$927$$ −11232.0 −0.397958
$$928$$ 2465.00 0.0871957
$$929$$ −19534.0 −0.689871 −0.344935 0.938626i $$-0.612099\pi$$
−0.344935 + 0.938626i $$0.612099\pi$$
$$930$$ 16800.0 0.592359
$$931$$ 2436.00 0.0857537
$$932$$ 1394.00 0.0489935
$$933$$ 8784.00 0.308226
$$934$$ 57940.0 2.02982
$$935$$ −3960.00 −0.138509
$$936$$ 31590.0 1.10315
$$937$$ −8678.00 −0.302559 −0.151280 0.988491i $$-0.548339\pi$$
−0.151280 + 0.988491i $$0.548339\pi$$
$$938$$ −27200.0 −0.946814
$$939$$ −8622.00 −0.299647
$$940$$ 26520.0 0.920199
$$941$$ −7050.00 −0.244233 −0.122117 0.992516i $$-0.538968\pi$$
−0.122117 + 0.992516i $$0.538968\pi$$
$$942$$ −8010.00 −0.277049
$$943$$ −14352.0 −0.495616
$$944$$ 16020.0 0.552337
$$945$$ −2160.00 −0.0743543
$$946$$ 57200.0 1.96589
$$947$$ 23396.0 0.802817 0.401409 0.915899i $$-0.368521\pi$$
0.401409 + 0.915899i $$0.368521\pi$$
$$948$$ 48960.0 1.67737
$$949$$ 30732.0 1.05121
$$950$$ −3500.00 −0.119532
$$951$$ 78.0000 0.00265965
$$952$$ 12960.0 0.441214
$$953$$ −36126.0 −1.22795 −0.613975 0.789326i $$-0.710430\pi$$
−0.613975 + 0.789326i $$0.710430\pi$$
$$954$$ 25830.0 0.876601
$$955$$ 10040.0 0.340196
$$956$$ −86768.0 −2.93544
$$957$$ 3828.00 0.129302
$$958$$ −91000.0 −3.06897
$$959$$ −44128.0 −1.48589
$$960$$ 4305.00 0.144733
$$961$$ 20385.0 0.684267
$$962$$ 99060.0 3.31998
$$963$$ −1044.00 −0.0349350
$$964$$ −36686.0 −1.22570
$$965$$ 12890.0 0.429994
$$966$$ −44160.0 −1.47083
$$967$$ 38624.0 1.28445 0.642225 0.766516i $$-0.278011\pi$$
0.642225 + 0.766516i $$0.278011\pi$$
$$968$$ 27225.0 0.903972
$$969$$ 1512.00 0.0501264
$$970$$ 30850.0 1.02117
$$971$$ −7292.00 −0.241000 −0.120500 0.992713i $$-0.538450\pi$$
−0.120500 + 0.992713i $$0.538450\pi$$
$$972$$ −4131.00 −0.136319
$$973$$ −22976.0 −0.757016
$$974$$ 19120.0 0.628998
$$975$$ −5850.00 −0.192154
$$976$$ −54290.0 −1.78051
$$977$$ −26838.0 −0.878837 −0.439418 0.898282i $$-0.644815\pi$$
−0.439418 + 0.898282i $$0.644815\pi$$
$$978$$ −20700.0 −0.676803
$$979$$ 43560.0 1.42205
$$980$$ −7395.00 −0.241046
$$981$$ −7434.00 −0.241946
$$982$$ −15500.0 −0.503691
$$983$$ 20192.0 0.655163 0.327581 0.944823i $$-0.393766\pi$$
0.327581 + 0.944823i $$0.393766\pi$$
$$984$$ 10530.0 0.341142
$$985$$ 2630.00 0.0850749
$$986$$ 2610.00 0.0842995
$$987$$ −14976.0 −0.482970
$$988$$ −37128.0 −1.19555
$$989$$ −47840.0 −1.53814
$$990$$ −9900.00 −0.317821
$$991$$ 34400.0 1.10268 0.551338 0.834282i $$-0.314117\pi$$
0.551338 + 0.834282i $$0.314117\pi$$
$$992$$ −19040.0 −0.609396
$$993$$ −28020.0 −0.895456
$$994$$ 23680.0 0.755618
$$995$$ 22200.0 0.707324
$$996$$ 46308.0 1.47322
$$997$$ 58430.0 1.85606 0.928032 0.372499i $$-0.121499\pi$$
0.928032 + 0.372499i $$0.121499\pi$$
$$998$$ 98700.0 3.13056
$$999$$ −6858.00 −0.217195
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 435.4.a.c.1.1 1
3.2 odd 2 1305.4.a.a.1.1 1
5.4 even 2 2175.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.c.1.1 1 1.1 even 1 trivial
1305.4.a.a.1.1 1 3.2 odd 2
2175.4.a.a.1.1 1 5.4 even 2