# Properties

 Label 475.2.a.a.1.1 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $1$ 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] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.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: $$3.79289409601$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} +1.00000 q^{19} +4.00000 q^{22} -6.00000 q^{23} +2.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} -4.00000 q^{34} +3.00000 q^{36} -10.0000 q^{37} -1.00000 q^{38} -10.0000 q^{41} +2.00000 q^{43} +4.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} -3.00000 q^{49} +2.00000 q^{52} +10.0000 q^{53} +6.00000 q^{56} +6.00000 q^{58} +2.00000 q^{61} +4.00000 q^{62} -6.00000 q^{63} +7.00000 q^{64} +8.00000 q^{67} -4.00000 q^{68} +4.00000 q^{71} -9.00000 q^{72} +4.00000 q^{73} +10.0000 q^{74} -1.00000 q^{76} -8.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +10.0000 q^{82} -18.0000 q^{83} -2.00000 q^{86} -12.0000 q^{88} -2.00000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +6.00000 q^{94} +6.00000 q^{97} +3.00000 q^{98} +12.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 3.00000 1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 6.00000 0.801784
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −6.00000 −0.755929
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ −9.00000 −1.06066
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 10.0000 1.10432
$$83$$ −18.0000 −1.97576 −0.987878 0.155230i $$-0.950388\pi$$
−0.987878 + 0.155230i $$0.950388\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ −12.0000 −1.27920
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 6.00000 0.534522
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 12.0000 1.02899
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ 8.00000 0.668994
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 3.00000 0.243332
$$153$$ −12.0000 −0.970143
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ −9.00000 −0.707107
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 18.0000 1.39707
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −3.00000 −0.229416
$$172$$ −2.00000 −0.152499
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ −18.0000 −1.32698
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −16.0000 −1.17004
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ −12.0000 −0.842235
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 18.0000 1.25109
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ −6.00000 −0.406371
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −10.0000 −0.668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −18.0000 −1.18176
$$233$$ 16.0000 1.04819 0.524097 0.851658i $$-0.324403\pi$$
0.524097 + 0.851658i $$0.324403\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −8.00000 −0.518563
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ −12.0000 −0.762001
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 6.00000 0.377964
$$253$$ 24.0000 1.50887
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ 12.0000 0.741362
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ −20.0000 −1.18056
$$288$$ 15.0000 0.883883
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −30.0000 −1.74371
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −24.0000 −1.38104
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 12.0000 0.685994
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 8.00000 0.455842
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −24.0000 −1.35656 −0.678280 0.734803i $$-0.737274\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ 8.00000 0.451466
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 4.00000 0.222566
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −10.0000 −0.553849
$$327$$ 0 0
$$328$$ −30.0000 −1.65647
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 18.0000 0.987878
$$333$$ 30.0000 1.64399
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 3.00000 0.162221
$$343$$ −20.0000 −1.07990
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.0000 1.06600
$$353$$ 4.00000 0.212899 0.106449 0.994318i $$-0.466052\pi$$
0.106449 + 0.994318i $$0.466052\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 18.0000 0.946059
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 30.0000 1.56174
$$370$$ 0 0
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 16.0000 0.827340
$$375$$ 0 0
$$376$$ −18.0000 −0.928279
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ −6.00000 −0.304997
$$388$$ −6.00000 −0.304604
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ −12.0000 −0.603023
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ −18.0000 −0.884652
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 18.0000 0.875190
$$424$$ 30.0000 1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 8.00000 0.380521
$$443$$ −2.00000 −0.0950229 −0.0475114 0.998871i $$-0.515129\pi$$
−0.0475114 + 0.998871i $$0.515129\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 14.0000 0.661438
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ 2.00000 0.0929479 0.0464739 0.998920i $$-0.485202\pi$$
0.0464739 + 0.998920i $$0.485202\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −16.0000 −0.741186
$$467$$ −38.0000 −1.75843 −0.879215 0.476425i $$-0.841932\pi$$
−0.879215 + 0.476425i $$0.841932\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ −30.0000 −1.37361
$$478$$ 16.0000 0.731823
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ −24.0000 −1.08091
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.00000 −0.178529
$$503$$ 10.0000 0.445878 0.222939 0.974832i $$-0.428435\pi$$
0.222939 + 0.974832i $$0.428435\pi$$
$$504$$ −18.0000 −0.801784
$$505$$ 0 0
$$506$$ −24.0000 −1.06693
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 20.0000 0.878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ −44.0000 −1.92399 −0.961993 0.273075i $$-0.911959\pi$$
−0.961993 + 0.273075i $$0.911959\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 24.0000 1.03664
$$537$$ 0 0
$$538$$ 18.0000 0.776035
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 0 0
$$544$$ −20.0000 −0.857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ −16.0000 −0.677942 −0.338971 0.940797i $$-0.610079\pi$$
−0.338971 + 0.940797i $$0.610079\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 18.0000 0.755929
$$568$$ 12.0000 0.503509
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ 24.0000 0.999133 0.499567 0.866276i $$-0.333493\pi$$
0.499567 + 0.866276i $$0.333493\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ −8.00000 −0.328521 −0.164260 0.986417i $$-0.552524\pi$$
−0.164260 + 0.986417i $$0.552524\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ −12.0000 −0.490716
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ −24.0000 −0.977356
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −44.0000 −1.78590 −0.892952 0.450151i $$-0.851370\pi$$
−0.892952 + 0.450151i $$0.851370\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 12.0000 0.485071
$$613$$ −24.0000 −0.969351 −0.484675 0.874694i $$-0.661062\pi$$
−0.484675 + 0.874694i $$0.661062\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ −24.0000 −0.966988
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 24.0000 0.959233
$$627$$ 0 0
$$628$$ 8.00000 0.319235
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 0 0
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ −24.0000 −0.950169
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ 38.0000 1.49857 0.749287 0.662246i $$-0.230396\pi$$
0.749287 + 0.662246i $$0.230396\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 30.0000 1.17942 0.589711 0.807614i $$-0.299242\pi$$
0.589711 + 0.807614i $$0.299242\pi$$
$$648$$ 27.0000 1.06066
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ 16.0000 0.626128 0.313064 0.949732i $$-0.398644\pi$$
0.313064 + 0.949732i $$0.398644\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ −12.0000 −0.468165
$$658$$ 12.0000 0.467809
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −54.0000 −2.09561
$$665$$ 0 0
$$666$$ −30.0000 −1.16248
$$667$$ 36.0000 1.39393
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −2.00000 −0.0762493
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 24.0000 0.911685
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −40.0000 −1.51511
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ −4.00000 −0.150542
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ −6.00000 −0.224860
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.00000 0.0741759 0.0370879 0.999312i $$-0.488192\pi$$
0.0370879 + 0.999312i $$0.488192\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −12.0000 −0.443230 −0.221615 0.975134i $$-0.571133\pi$$
−0.221615 + 0.975134i $$0.571133\pi$$
$$734$$ 22.0000 0.812035
$$735$$ 0 0
$$736$$ 30.0000 1.10581
$$737$$ −32.0000 −1.17874
$$738$$ −30.0000 −1.10432
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −20.0000 −0.734223
$$743$$ −4.00000 −0.146746 −0.0733729 0.997305i $$-0.523376\pi$$
−0.0733729 + 0.997305i $$0.523376\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ 54.0000 1.97576
$$748$$ 16.0000 0.585018
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −4.00000 −0.145382 −0.0726912 0.997354i $$-0.523159\pi$$
−0.0726912 + 0.997354i $$0.523159\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 12.0000 0.434429
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000 0.935760
$$773$$ 46.0000 1.65451 0.827253 0.561830i $$-0.189903\pi$$
0.827253 + 0.561830i $$0.189903\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 12.0000 0.427482
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 36.0000 1.27920
$$793$$ −4.00000 −0.142044
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −18.0000 −0.635602
$$803$$ −16.0000 −0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ −18.0000 −0.633238
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −24.0000 −0.842754 −0.421377 0.906886i $$-0.638453\pi$$
−0.421377 + 0.906886i $$0.638453\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ 14.0000 0.489499
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ −42.0000 −1.46403 −0.732014 0.681290i $$-0.761419\pi$$
−0.732014 + 0.681290i $$0.761419\pi$$
$$824$$ 48.0000 1.67216
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ −12.0000 −0.415775
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ −36.0000 −1.24360
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 30.0000 1.03387
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ −18.0000 −0.618853
$$847$$ 10.0000 0.343604
$$848$$ −10.0000 −0.343401
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 60.0000 2.05677
$$852$$ 0 0
$$853$$ −4.00000 −0.136957 −0.0684787 0.997653i $$-0.521815\pi$$
−0.0684787 + 0.997653i $$0.521815\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −28.0000 −0.953131 −0.476566 0.879139i $$-0.658119\pi$$
−0.476566 + 0.879139i $$0.658119\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ 8.00000 0.271538
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 18.0000 0.609557
$$873$$ −18.0000 −0.609208
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ 34.0000 1.14419 0.572096 0.820187i $$-0.306131\pi$$
0.572096 + 0.820187i $$0.306131\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 2.00000 0.0671913
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −36.0000 −1.20605
$$892$$ −16.0000 −0.535720
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 6.00000 0.200446
$$897$$ 0 0
$$898$$ −10.0000 −0.333704
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 40.0000 1.33259
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ 72.0000 2.38285
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ −24.0000 −0.792550
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −14.0000 −0.461065
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −2.00000 −0.0657241
$$927$$ −48.0000 −1.57653
$$928$$ 30.0000 0.984798
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ −16.0000 −0.524097
$$933$$ 0 0
$$934$$ 38.0000 1.24340
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ 60.0000 1.95387
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −30.0000 −0.974869 −0.487435 0.873160i $$-0.662067\pi$$
−0.487435 + 0.873160i $$0.662067\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 24.0000 0.777844
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ 30.0000 0.971286
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −20.0000 −0.644826
$$963$$ 12.0000 0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −2.00000 −0.0643157 −0.0321578 0.999483i $$-0.510238\pi$$
−0.0321578 + 0.999483i $$0.510238\pi$$
$$968$$ 15.0000 0.482118
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −14.0000 −0.447900 −0.223950 0.974601i $$-0.571895\pi$$
−0.223950 + 0.974601i $$0.571895\pi$$
$$978$$ 0 0
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ 20.0000 0.638226
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 20.0000 0.635001
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 32.0000 1.01345 0.506725 0.862108i $$-0.330856\pi$$
0.506725 + 0.862108i $$0.330856\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 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 475.2.a.a.1.1 1
3.2 odd 2 4275.2.a.p.1.1 1
4.3 odd 2 7600.2.a.i.1.1 1
5.2 odd 4 95.2.b.a.39.1 2
5.3 odd 4 95.2.b.a.39.2 yes 2
5.4 even 2 475.2.a.c.1.1 1
15.2 even 4 855.2.c.b.514.2 2
15.8 even 4 855.2.c.b.514.1 2
15.14 odd 2 4275.2.a.e.1.1 1
19.18 odd 2 9025.2.a.h.1.1 1
20.3 even 4 1520.2.d.b.609.2 2
20.7 even 4 1520.2.d.b.609.1 2
20.19 odd 2 7600.2.a.l.1.1 1
95.18 even 4 1805.2.b.c.1084.1 2
95.37 even 4 1805.2.b.c.1084.2 2
95.94 odd 2 9025.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.a.39.1 2 5.2 odd 4
95.2.b.a.39.2 yes 2 5.3 odd 4
475.2.a.a.1.1 1 1.1 even 1 trivial
475.2.a.c.1.1 1 5.4 even 2
855.2.c.b.514.1 2 15.8 even 4
855.2.c.b.514.2 2 15.2 even 4
1520.2.d.b.609.1 2 20.7 even 4
1520.2.d.b.609.2 2 20.3 even 4
1805.2.b.c.1084.1 2 95.18 even 4
1805.2.b.c.1084.2 2 95.37 even 4
4275.2.a.e.1.1 1 15.14 odd 2
4275.2.a.p.1.1 1 3.2 odd 2
7600.2.a.i.1.1 1 4.3 odd 2
7600.2.a.l.1.1 1 20.19 odd 2
9025.2.a.c.1.1 1 95.94 odd 2
9025.2.a.h.1.1 1 19.18 odd 2