# Properties

 Label 4275.2.a.bo.1.1 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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 Root $$1.28734$$ of defining polynomial Character $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.63010 q^{2} +4.91744 q^{4} +0.574672 q^{7} -7.67316 q^{8} +O(q^{10})$$ $$q-2.63010 q^{2} +4.91744 q^{4} +0.574672 q^{7} -7.67316 q^{8} -2.57467 q^{11} +0.468387 q^{13} -1.51145 q^{14} +10.3463 q^{16} -4.08612 q^{17} +1.00000 q^{19} +6.77165 q^{22} +1.51145 q^{23} -1.23191 q^{26} +2.82591 q^{28} +4.08612 q^{29} -9.92099 q^{31} -11.8656 q^{32} +10.7469 q^{34} +8.30326 q^{37} -2.63010 q^{38} +1.83488 q^{41} +0.574672 q^{43} -12.6608 q^{44} -3.97526 q^{46} +7.09508 q^{47} -6.66975 q^{49} +2.30326 q^{52} +4.30326 q^{53} -4.40955 q^{56} -10.7469 q^{58} +2.68553 q^{59} +12.4095 q^{61} +26.0932 q^{62} +10.5150 q^{64} +2.70570 q^{67} -20.0932 q^{68} +7.40058 q^{71} -12.0861 q^{73} -21.8384 q^{74} +4.91744 q^{76} -1.47959 q^{77} -6.68553 q^{79} -4.82591 q^{82} -6.66079 q^{83} -1.51145 q^{86} +19.7559 q^{88} -14.6065 q^{89} +0.269169 q^{91} +7.43244 q^{92} -18.6608 q^{94} -17.4526 q^{97} +17.5421 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 12 * q^8 $$4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 12 q^{8} - 4 q^{11} - 2 q^{13} + 8 q^{14} + 4 q^{16} + 4 q^{17} + 4 q^{19} - 4 q^{22} - 8 q^{23} - 4 q^{26} + 8 q^{28} - 4 q^{29} + 4 q^{31} - 6 q^{32} - 4 q^{34} + 6 q^{37} - 2 q^{38} - 16 q^{41} - 4 q^{43} - 24 q^{44} - 12 q^{47} + 20 q^{49} - 18 q^{52} - 10 q^{53} + 12 q^{56} + 4 q^{58} + 20 q^{61} + 20 q^{62} - 4 q^{64} + 18 q^{67} + 4 q^{68} + 20 q^{71} - 28 q^{73} - 32 q^{74} + 8 q^{76} - 40 q^{77} - 16 q^{79} - 16 q^{82} + 8 q^{86} + 12 q^{88} - 4 q^{89} - 36 q^{91} - 28 q^{92} - 48 q^{94} - 30 q^{97} + 38 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 12 * q^8 - 4 * q^11 - 2 * q^13 + 8 * q^14 + 4 * q^16 + 4 * q^17 + 4 * q^19 - 4 * q^22 - 8 * q^23 - 4 * q^26 + 8 * q^28 - 4 * q^29 + 4 * q^31 - 6 * q^32 - 4 * q^34 + 6 * q^37 - 2 * q^38 - 16 * q^41 - 4 * q^43 - 24 * q^44 - 12 * q^47 + 20 * q^49 - 18 * q^52 - 10 * q^53 + 12 * q^56 + 4 * q^58 + 20 * q^61 + 20 * q^62 - 4 * q^64 + 18 * q^67 + 4 * q^68 + 20 * q^71 - 28 * q^73 - 32 * q^74 + 8 * q^76 - 40 * q^77 - 16 * q^79 - 16 * q^82 + 8 * q^86 + 12 * q^88 - 4 * q^89 - 36 * q^91 - 28 * q^92 - 48 * q^94 - 30 * q^97 + 38 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.63010 −1.85976 −0.929882 0.367859i $$-0.880091\pi$$
−0.929882 + 0.367859i $$0.880091\pi$$
$$3$$ 0 0
$$4$$ 4.91744 2.45872
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.574672 0.217205 0.108603 0.994085i $$-0.465362\pi$$
0.108603 + 0.994085i $$0.465362\pi$$
$$8$$ −7.67316 −2.71287
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.57467 −0.776293 −0.388146 0.921598i $$-0.626884\pi$$
−0.388146 + 0.921598i $$0.626884\pi$$
$$12$$ 0 0
$$13$$ 0.468387 0.129907 0.0649536 0.997888i $$-0.479310\pi$$
0.0649536 + 0.997888i $$0.479310\pi$$
$$14$$ −1.51145 −0.403951
$$15$$ 0 0
$$16$$ 10.3463 2.58658
$$17$$ −4.08612 −0.991029 −0.495514 0.868600i $$-0.665020\pi$$
−0.495514 + 0.868600i $$0.665020\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.77165 1.44372
$$23$$ 1.51145 0.315158 0.157579 0.987506i $$-0.449631\pi$$
0.157579 + 0.987506i $$0.449631\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.23191 −0.241596
$$27$$ 0 0
$$28$$ 2.82591 0.534047
$$29$$ 4.08612 0.758773 0.379386 0.925238i $$-0.376135\pi$$
0.379386 + 0.925238i $$0.376135\pi$$
$$30$$ 0 0
$$31$$ −9.92099 −1.78186 −0.890931 0.454138i $$-0.849947\pi$$
−0.890931 + 0.454138i $$0.849947\pi$$
$$32$$ −11.8656 −2.09755
$$33$$ 0 0
$$34$$ 10.7469 1.84308
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.30326 1.36505 0.682524 0.730863i $$-0.260882\pi$$
0.682524 + 0.730863i $$0.260882\pi$$
$$38$$ −2.63010 −0.426659
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.83488 0.286560 0.143280 0.989682i $$-0.454235\pi$$
0.143280 + 0.989682i $$0.454235\pi$$
$$42$$ 0 0
$$43$$ 0.574672 0.0876366 0.0438183 0.999040i $$-0.486048\pi$$
0.0438183 + 0.999040i $$0.486048\pi$$
$$44$$ −12.6608 −1.90869
$$45$$ 0 0
$$46$$ −3.97526 −0.586119
$$47$$ 7.09508 1.03492 0.517462 0.855706i $$-0.326877\pi$$
0.517462 + 0.855706i $$0.326877\pi$$
$$48$$ 0 0
$$49$$ −6.66975 −0.952822
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.30326 0.319405
$$53$$ 4.30326 0.591099 0.295549 0.955327i $$-0.404497\pi$$
0.295549 + 0.955327i $$0.404497\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.40955 −0.589251
$$57$$ 0 0
$$58$$ −10.7469 −1.41114
$$59$$ 2.68553 0.349627 0.174813 0.984602i $$-0.444068\pi$$
0.174813 + 0.984602i $$0.444068\pi$$
$$60$$ 0 0
$$61$$ 12.4095 1.58888 0.794440 0.607343i $$-0.207765\pi$$
0.794440 + 0.607343i $$0.207765\pi$$
$$62$$ 26.0932 3.31384
$$63$$ 0 0
$$64$$ 10.5150 1.31438
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.70570 0.330554 0.165277 0.986247i $$-0.447148\pi$$
0.165277 + 0.986247i $$0.447148\pi$$
$$68$$ −20.0932 −2.43666
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.40058 0.878288 0.439144 0.898417i $$-0.355282\pi$$
0.439144 + 0.898417i $$0.355282\pi$$
$$72$$ 0 0
$$73$$ −12.0861 −1.41457 −0.707286 0.706927i $$-0.750081\pi$$
−0.707286 + 0.706927i $$0.750081\pi$$
$$74$$ −21.8384 −2.53867
$$75$$ 0 0
$$76$$ 4.91744 0.564069
$$77$$ −1.47959 −0.168615
$$78$$ 0 0
$$79$$ −6.68553 −0.752181 −0.376091 0.926583i $$-0.622732\pi$$
−0.376091 + 0.926583i $$0.622732\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −4.82591 −0.532933
$$83$$ −6.66079 −0.731117 −0.365558 0.930788i $$-0.619122\pi$$
−0.365558 + 0.930788i $$0.619122\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.51145 −0.162983
$$87$$ 0 0
$$88$$ 19.7559 2.10598
$$89$$ −14.6065 −1.54829 −0.774144 0.633009i $$-0.781820\pi$$
−0.774144 + 0.633009i $$0.781820\pi$$
$$90$$ 0 0
$$91$$ 0.269169 0.0282165
$$92$$ 7.43244 0.774885
$$93$$ 0 0
$$94$$ −18.6608 −1.92471
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.4526 −1.77204 −0.886022 0.463643i $$-0.846542\pi$$
−0.886022 + 0.463643i $$0.846542\pi$$
$$98$$ 17.5421 1.77202
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.2831 −1.42122 −0.710611 0.703586i $$-0.751581\pi$$
−0.710611 + 0.703586i $$0.751581\pi$$
$$102$$ 0 0
$$103$$ −4.79182 −0.472152 −0.236076 0.971735i $$-0.575861\pi$$
−0.236076 + 0.971735i $$0.575861\pi$$
$$104$$ −3.59401 −0.352421
$$105$$ 0 0
$$106$$ −11.3180 −1.09930
$$107$$ 9.22611 0.891922 0.445961 0.895052i $$-0.352862\pi$$
0.445961 + 0.895052i $$0.352862\pi$$
$$108$$ 0 0
$$109$$ 4.89810 0.469153 0.234577 0.972098i $$-0.424630\pi$$
0.234577 + 0.972098i $$0.424630\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.94574 0.561819
$$113$$ 1.61773 0.152183 0.0760916 0.997101i $$-0.475756\pi$$
0.0760916 + 0.997101i $$0.475756\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 20.0932 1.86561
$$117$$ 0 0
$$118$$ −7.06323 −0.650223
$$119$$ −2.34818 −0.215257
$$120$$ 0 0
$$121$$ −4.37107 −0.397370
$$122$$ −32.6384 −2.95494
$$123$$ 0 0
$$124$$ −48.7859 −4.38110
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.1292 1.16503 0.582513 0.812821i $$-0.302070\pi$$
0.582513 + 0.812821i $$0.302070\pi$$
$$128$$ −3.92440 −0.346871
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.17223 0.714011 0.357006 0.934102i $$-0.383798\pi$$
0.357006 + 0.934102i $$0.383798\pi$$
$$132$$ 0 0
$$133$$ 0.574672 0.0498304
$$134$$ −7.11627 −0.614752
$$135$$ 0 0
$$136$$ 31.3534 2.68853
$$137$$ −14.6065 −1.24792 −0.623960 0.781456i $$-0.714477\pi$$
−0.623960 + 0.781456i $$0.714477\pi$$
$$138$$ 0 0
$$139$$ −2.07219 −0.175761 −0.0878804 0.996131i $$-0.528009\pi$$
−0.0878804 + 0.996131i $$0.528009\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −19.4643 −1.63341
$$143$$ −1.20594 −0.100846
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 31.7877 2.63077
$$147$$ 0 0
$$148$$ 40.8308 3.35627
$$149$$ −8.91203 −0.730102 −0.365051 0.930988i $$-0.618948\pi$$
−0.365051 + 0.930988i $$0.618948\pi$$
$$150$$ 0 0
$$151$$ 11.4572 0.932372 0.466186 0.884687i $$-0.345628\pi$$
0.466186 + 0.884687i $$0.345628\pi$$
$$152$$ −7.67316 −0.622376
$$153$$ 0 0
$$154$$ 3.89148 0.313584
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.60653 0.527258 0.263629 0.964624i $$-0.415081\pi$$
0.263629 + 0.964624i $$0.415081\pi$$
$$158$$ 17.5836 1.39888
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0.868585 0.0684541
$$162$$ 0 0
$$163$$ −20.2444 −1.58567 −0.792833 0.609439i $$-0.791395\pi$$
−0.792833 + 0.609439i $$0.791395\pi$$
$$164$$ 9.02289 0.704569
$$165$$ 0 0
$$166$$ 17.5186 1.35970
$$167$$ −5.89372 −0.456069 −0.228035 0.973653i $$-0.573230\pi$$
−0.228035 + 0.973653i $$0.573230\pi$$
$$168$$ 0 0
$$169$$ −12.7806 −0.983124
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.82591 0.215474
$$173$$ −3.53161 −0.268504 −0.134252 0.990947i $$-0.542863\pi$$
−0.134252 + 0.990947i $$0.542863\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −26.6384 −2.00794
$$177$$ 0 0
$$178$$ 38.4167 2.87945
$$179$$ −7.18801 −0.537257 −0.268629 0.963244i $$-0.586570\pi$$
−0.268629 + 0.963244i $$0.586570\pi$$
$$180$$ 0 0
$$181$$ 15.5433 1.15532 0.577662 0.816276i $$-0.303965\pi$$
0.577662 + 0.816276i $$0.303965\pi$$
$$182$$ −0.707941 −0.0524761
$$183$$ 0 0
$$184$$ −11.5976 −0.854984
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 10.5204 0.769329
$$188$$ 34.8896 2.54459
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.3216 −0.963915 −0.481958 0.876194i $$-0.660074\pi$$
−0.481958 + 0.876194i $$0.660074\pi$$
$$192$$ 0 0
$$193$$ −18.9959 −1.36736 −0.683678 0.729784i $$-0.739621\pi$$
−0.683678 + 0.729784i $$0.739621\pi$$
$$194$$ 45.9021 3.29558
$$195$$ 0 0
$$196$$ −32.7981 −2.34272
$$197$$ 2.17223 0.154765 0.0773826 0.997001i $$-0.475344\pi$$
0.0773826 + 0.997001i $$0.475344\pi$$
$$198$$ 0 0
$$199$$ −1.87355 −0.132812 −0.0664061 0.997793i $$-0.521153\pi$$
−0.0664061 + 0.997793i $$0.521153\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 37.5660 2.64313
$$203$$ 2.34818 0.164810
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 12.6030 0.878091
$$207$$ 0 0
$$208$$ 4.84608 0.336015
$$209$$ −2.57467 −0.178094
$$210$$ 0 0
$$211$$ −17.1090 −1.17783 −0.588916 0.808194i $$-0.700445\pi$$
−0.588916 + 0.808194i $$0.700445\pi$$
$$212$$ 21.1610 1.45335
$$213$$ 0 0
$$214$$ −24.2656 −1.65876
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −5.70131 −0.387030
$$218$$ −12.8825 −0.872514
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.91388 −0.128742
$$222$$ 0 0
$$223$$ −5.12918 −0.343475 −0.171737 0.985143i $$-0.554938\pi$$
−0.171737 + 0.985143i $$0.554938\pi$$
$$224$$ −6.81880 −0.455600
$$225$$ 0 0
$$226$$ −4.25480 −0.283025
$$227$$ −7.31223 −0.485330 −0.242665 0.970110i $$-0.578022\pi$$
−0.242665 + 0.970110i $$0.578022\pi$$
$$228$$ 0 0
$$229$$ −8.40955 −0.555719 −0.277859 0.960622i $$-0.589625\pi$$
−0.277859 + 0.960622i $$0.589625\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −31.3534 −2.05845
$$233$$ −14.1722 −0.928454 −0.464227 0.885716i $$-0.653668\pi$$
−0.464227 + 0.885716i $$0.653668\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 13.2059 0.859634
$$237$$ 0 0
$$238$$ 6.17594 0.400327
$$239$$ 14.1902 0.917885 0.458943 0.888466i $$-0.348228\pi$$
0.458943 + 0.888466i $$0.348228\pi$$
$$240$$ 0 0
$$241$$ 27.8807 1.79595 0.897976 0.440045i $$-0.145038\pi$$
0.897976 + 0.440045i $$0.145038\pi$$
$$242$$ 11.4964 0.739013
$$243$$ 0 0
$$244$$ 61.0232 3.90661
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.468387 0.0298027
$$248$$ 76.1254 4.83397
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 26.1902 1.65311 0.826554 0.562857i $$-0.190298\pi$$
0.826554 + 0.562857i $$0.190298\pi$$
$$252$$ 0 0
$$253$$ −3.89148 −0.244655
$$254$$ −34.5311 −2.16667
$$255$$ 0 0
$$256$$ −10.7084 −0.669276
$$257$$ 9.01831 0.562547 0.281273 0.959628i $$-0.409243\pi$$
0.281273 + 0.959628i $$0.409243\pi$$
$$258$$ 0 0
$$259$$ 4.77165 0.296496
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −21.4938 −1.32789
$$263$$ −9.00896 −0.555517 −0.277758 0.960651i $$-0.589591\pi$$
−0.277758 + 0.960651i $$0.589591\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1.51145 −0.0926727
$$267$$ 0 0
$$268$$ 13.3051 0.812739
$$269$$ 30.6136 1.86655 0.933273 0.359167i $$-0.116939\pi$$
0.933273 + 0.359167i $$0.116939\pi$$
$$270$$ 0 0
$$271$$ 24.1180 1.46506 0.732531 0.680733i $$-0.238339\pi$$
0.732531 + 0.680733i $$0.238339\pi$$
$$272$$ −42.2763 −2.56338
$$273$$ 0 0
$$274$$ 38.4167 2.32084
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.56075 −0.274029 −0.137014 0.990569i $$-0.543751\pi$$
−0.137014 + 0.990569i $$0.543751\pi$$
$$278$$ 5.45007 0.326874
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.11563 0.364828 0.182414 0.983222i $$-0.441609\pi$$
0.182414 + 0.983222i $$0.441609\pi$$
$$282$$ 0 0
$$283$$ 19.0547 1.13269 0.566344 0.824169i $$-0.308358\pi$$
0.566344 + 0.824169i $$0.308358\pi$$
$$284$$ 36.3919 2.15946
$$285$$ 0 0
$$286$$ 3.17175 0.187550
$$287$$ 1.05445 0.0622423
$$288$$ 0 0
$$289$$ −0.303649 −0.0178617
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −59.4327 −3.47804
$$293$$ −6.43887 −0.376163 −0.188081 0.982153i $$-0.560227\pi$$
−0.188081 + 0.982153i $$0.560227\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −63.7123 −3.70320
$$297$$ 0 0
$$298$$ 23.4395 1.35782
$$299$$ 0.707941 0.0409413
$$300$$ 0 0
$$301$$ 0.330247 0.0190351
$$302$$ −30.1336 −1.73399
$$303$$ 0 0
$$304$$ 10.3463 0.593402
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6.77389 0.386606 0.193303 0.981139i $$-0.438080\pi$$
0.193303 + 0.981139i $$0.438080\pi$$
$$308$$ −7.27580 −0.414577
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.6205 −1.16928 −0.584639 0.811293i $$-0.698764\pi$$
−0.584639 + 0.811293i $$0.698764\pi$$
$$312$$ 0 0
$$313$$ −19.3711 −1.09492 −0.547459 0.836833i $$-0.684405\pi$$
−0.547459 + 0.836833i $$0.684405\pi$$
$$314$$ −17.3758 −0.980575
$$315$$ 0 0
$$316$$ −32.8757 −1.84940
$$317$$ 3.37360 0.189480 0.0947401 0.995502i $$-0.469798\pi$$
0.0947401 + 0.995502i $$0.469798\pi$$
$$318$$ 0 0
$$319$$ −10.5204 −0.589030
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −2.28447 −0.127308
$$323$$ −4.08612 −0.227358
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 53.2449 2.94896
$$327$$ 0 0
$$328$$ −14.0793 −0.777399
$$329$$ 4.07734 0.224791
$$330$$ 0 0
$$331$$ −32.7788 −1.80168 −0.900842 0.434148i $$-0.857050\pi$$
−0.900842 + 0.434148i $$0.857050\pi$$
$$332$$ −32.7540 −1.79761
$$333$$ 0 0
$$334$$ 15.5011 0.848181
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8.74915 −0.476596 −0.238298 0.971192i $$-0.576590\pi$$
−0.238298 + 0.971192i $$0.576590\pi$$
$$338$$ 33.6143 1.82838
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 25.5433 1.38325
$$342$$ 0 0
$$343$$ −7.85562 −0.424164
$$344$$ −4.40955 −0.237747
$$345$$ 0 0
$$346$$ 9.28850 0.499353
$$347$$ −18.5028 −0.993281 −0.496640 0.867956i $$-0.665433\pi$$
−0.496640 + 0.867956i $$0.665433\pi$$
$$348$$ 0 0
$$349$$ 3.54330 0.189668 0.0948342 0.995493i $$-0.469768\pi$$
0.0948342 + 0.995493i $$0.469768\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 30.5499 1.62832
$$353$$ −3.41140 −0.181571 −0.0907853 0.995870i $$-0.528938\pi$$
−0.0907853 + 0.995870i $$0.528938\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −71.8267 −3.80681
$$357$$ 0 0
$$358$$ 18.9052 0.999172
$$359$$ 1.70609 0.0900438 0.0450219 0.998986i $$-0.485664\pi$$
0.0450219 + 0.998986i $$0.485664\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −40.8805 −2.14863
$$363$$ 0 0
$$364$$ 1.32362 0.0693765
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −37.6155 −1.96351 −0.981756 0.190144i $$-0.939105\pi$$
−0.981756 + 0.190144i $$0.939105\pi$$
$$368$$ 15.6379 0.815182
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2.47296 0.128390
$$372$$ 0 0
$$373$$ −13.4031 −0.693987 −0.346994 0.937868i $$-0.612797\pi$$
−0.346994 + 0.937868i $$0.612797\pi$$
$$374$$ −27.6698 −1.43077
$$375$$ 0 0
$$376$$ −54.4417 −2.80762
$$377$$ 1.91388 0.0985700
$$378$$ 0 0
$$379$$ −9.37107 −0.481359 −0.240680 0.970605i $$-0.577370\pi$$
−0.240680 + 0.970605i $$0.577370\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 35.0371 1.79265
$$383$$ −2.09917 −0.107263 −0.0536314 0.998561i $$-0.517080\pi$$
−0.0536314 + 0.998561i $$0.517080\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 49.9612 2.54296
$$387$$ 0 0
$$388$$ −85.8221 −4.35696
$$389$$ −1.07238 −0.0543718 −0.0271859 0.999630i $$-0.508655\pi$$
−0.0271859 + 0.999630i $$0.508655\pi$$
$$390$$ 0 0
$$391$$ −6.17594 −0.312331
$$392$$ 51.1781 2.58488
$$393$$ 0 0
$$394$$ −5.71320 −0.287827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 23.5341 1.18114 0.590572 0.806985i $$-0.298902\pi$$
0.590572 + 0.806985i $$0.298902\pi$$
$$398$$ 4.92762 0.246999
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.6469 −0.931180 −0.465590 0.885001i $$-0.654158\pi$$
−0.465590 + 0.885001i $$0.654158\pi$$
$$402$$ 0 0
$$403$$ −4.64686 −0.231477
$$404$$ −70.2362 −3.49438
$$405$$ 0 0
$$406$$ −6.17594 −0.306507
$$407$$ −21.3782 −1.05968
$$408$$ 0 0
$$409$$ −6.88017 −0.340203 −0.170101 0.985427i $$-0.554410\pi$$
−0.170101 + 0.985427i $$0.554410\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −23.5635 −1.16089
$$413$$ 1.54330 0.0759408
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.55767 −0.272487
$$417$$ 0 0
$$418$$ 6.77165 0.331212
$$419$$ −13.4796 −0.658521 −0.329261 0.944239i $$-0.606799\pi$$
−0.329261 + 0.944239i $$0.606799\pi$$
$$420$$ 0 0
$$421$$ −5.83488 −0.284374 −0.142187 0.989840i $$-0.545414\pi$$
−0.142187 + 0.989840i $$0.545414\pi$$
$$422$$ 44.9984 2.19049
$$423$$ 0 0
$$424$$ −33.0196 −1.60357
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 7.13142 0.345113
$$428$$ 45.3688 2.19298
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −29.2039 −1.40670 −0.703351 0.710843i $$-0.748314\pi$$
−0.703351 + 0.710843i $$0.748314\pi$$
$$432$$ 0 0
$$433$$ 12.5229 0.601814 0.300907 0.953653i $$-0.402711\pi$$
0.300907 + 0.953653i $$0.402711\pi$$
$$434$$ 14.9950 0.719785
$$435$$ 0 0
$$436$$ 24.0861 1.15352
$$437$$ 1.51145 0.0723022
$$438$$ 0 0
$$439$$ 15.6769 0.748216 0.374108 0.927385i $$-0.377949\pi$$
0.374108 + 0.927385i $$0.377949\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 5.03371 0.239429
$$443$$ −29.8281 −1.41717 −0.708587 0.705624i $$-0.750667\pi$$
−0.708587 + 0.705624i $$0.750667\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 13.4903 0.638782
$$447$$ 0 0
$$448$$ 6.04267 0.285489
$$449$$ −29.9668 −1.41422 −0.707110 0.707104i $$-0.750001\pi$$
−0.707110 + 0.707104i $$0.750001\pi$$
$$450$$ 0 0
$$451$$ −4.72420 −0.222454
$$452$$ 7.95509 0.374176
$$453$$ 0 0
$$454$$ 19.2319 0.902598
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.6698 −0.826556 −0.413278 0.910605i $$-0.635616\pi$$
−0.413278 + 0.910605i $$0.635616\pi$$
$$458$$ 22.1180 1.03350
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −22.3445 −1.04069 −0.520343 0.853957i $$-0.674196\pi$$
−0.520343 + 0.853957i $$0.674196\pi$$
$$462$$ 0 0
$$463$$ 6.83302 0.317557 0.158779 0.987314i $$-0.449244\pi$$
0.158779 + 0.987314i $$0.449244\pi$$
$$464$$ 42.2763 1.96263
$$465$$ 0 0
$$466$$ 37.2744 1.72670
$$467$$ 9.00896 0.416885 0.208443 0.978035i $$-0.433161\pi$$
0.208443 + 0.978035i $$0.433161\pi$$
$$468$$ 0 0
$$469$$ 1.55489 0.0717981
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −20.6065 −0.948492
$$473$$ −1.47959 −0.0680317
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −11.5470 −0.529256
$$477$$ 0 0
$$478$$ −37.3216 −1.70705
$$479$$ −9.26731 −0.423434 −0.211717 0.977331i $$-0.567906\pi$$
−0.211717 + 0.977331i $$0.567906\pi$$
$$480$$ 0 0
$$481$$ 3.88914 0.177329
$$482$$ −73.3290 −3.34004
$$483$$ 0 0
$$484$$ −21.4944 −0.977020
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.7694 1.75681 0.878405 0.477917i $$-0.158608\pi$$
0.878405 + 0.477917i $$0.158608\pi$$
$$488$$ −95.2205 −4.31043
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −21.6877 −0.978751 −0.489376 0.872073i $$-0.662775\pi$$
−0.489376 + 0.872073i $$0.662775\pi$$
$$492$$ 0 0
$$493$$ −16.6964 −0.751966
$$494$$ −1.23191 −0.0554260
$$495$$ 0 0
$$496$$ −102.646 −4.60893
$$497$$ 4.25291 0.190769
$$498$$ 0 0
$$499$$ −27.8372 −1.24616 −0.623082 0.782156i $$-0.714120\pi$$
−0.623082 + 0.782156i $$0.714120\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −68.8828 −3.07439
$$503$$ −41.2449 −1.83902 −0.919510 0.393067i $$-0.871414\pi$$
−0.919510 + 0.393067i $$0.871414\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 10.2350 0.455000
$$507$$ 0 0
$$508$$ 64.5619 2.86447
$$509$$ −0.416364 −0.0184550 −0.00922751 0.999957i $$-0.502937\pi$$
−0.00922751 + 0.999957i $$0.502937\pi$$
$$510$$ 0 0
$$511$$ −6.94555 −0.307253
$$512$$ 36.0131 1.59157
$$513$$ 0 0
$$514$$ −23.7191 −1.04620
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −18.2675 −0.803404
$$518$$ −12.5499 −0.551412
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −24.0458 −1.05346 −0.526732 0.850031i $$-0.676583\pi$$
−0.526732 + 0.850031i $$0.676583\pi$$
$$522$$ 0 0
$$523$$ 5.19736 0.227265 0.113632 0.993523i $$-0.463751\pi$$
0.113632 + 0.993523i $$0.463751\pi$$
$$524$$ 40.1865 1.75555
$$525$$ 0 0
$$526$$ 23.6945 1.03313
$$527$$ 40.5383 1.76588
$$528$$ 0 0
$$529$$ −20.7155 −0.900675
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.82591 0.122519
$$533$$ 0.859432 0.0372261
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −20.7613 −0.896751
$$537$$ 0 0
$$538$$ −80.5170 −3.47133
$$539$$ 17.1724 0.739669
$$540$$ 0 0
$$541$$ −1.93863 −0.0833481 −0.0416741 0.999131i $$-0.513269\pi$$
−0.0416741 + 0.999131i $$0.513269\pi$$
$$542$$ −63.4327 −2.72467
$$543$$ 0 0
$$544$$ 48.4841 2.07874
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.9075 0.551883 0.275941 0.961174i $$-0.411010\pi$$
0.275941 + 0.961174i $$0.411010\pi$$
$$548$$ −71.8267 −3.06828
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.08612 0.174074
$$552$$ 0 0
$$553$$ −3.84199 −0.163378
$$554$$ 11.9952 0.509628
$$555$$ 0 0
$$556$$ −10.1899 −0.432147
$$557$$ −34.1040 −1.44503 −0.722517 0.691353i $$-0.757015\pi$$
−0.722517 + 0.691353i $$0.757015\pi$$
$$558$$ 0 0
$$559$$ 0.269169 0.0113846
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −16.0847 −0.678494
$$563$$ −14.4911 −0.610727 −0.305363 0.952236i $$-0.598778\pi$$
−0.305363 + 0.952236i $$0.598778\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −50.1159 −2.10653
$$567$$ 0 0
$$568$$ −56.7859 −2.38268
$$569$$ 39.2110 1.64381 0.821906 0.569624i $$-0.192911\pi$$
0.821906 + 0.569624i $$0.192911\pi$$
$$570$$ 0 0
$$571$$ 21.9915 0.920316 0.460158 0.887837i $$-0.347793\pi$$
0.460158 + 0.887837i $$0.347793\pi$$
$$572$$ −5.93015 −0.247952
$$573$$ 0 0
$$574$$ −2.77331 −0.115756
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23.8735 −0.993869 −0.496934 0.867788i $$-0.665541\pi$$
−0.496934 + 0.867788i $$0.665541\pi$$
$$578$$ 0.798628 0.0332185
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.82777 −0.158803
$$582$$ 0 0
$$583$$ −11.0795 −0.458866
$$584$$ 92.7387 3.83756
$$585$$ 0 0
$$586$$ 16.9349 0.699574
$$587$$ 17.8281 0.735843 0.367921 0.929857i $$-0.380070\pi$$
0.367921 + 0.929857i $$0.380070\pi$$
$$588$$ 0 0
$$589$$ −9.92099 −0.408787
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 85.9082 3.53081
$$593$$ −21.2446 −0.872412 −0.436206 0.899847i $$-0.643678\pi$$
−0.436206 + 0.899847i $$0.643678\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −43.8244 −1.79512
$$597$$ 0 0
$$598$$ −1.86196 −0.0761411
$$599$$ −24.3374 −0.994397 −0.497199 0.867637i $$-0.665638\pi$$
−0.497199 + 0.867637i $$0.665638\pi$$
$$600$$ 0 0
$$601$$ 15.7891 0.644051 0.322025 0.946731i $$-0.395636\pi$$
0.322025 + 0.946731i $$0.395636\pi$$
$$602$$ −0.868585 −0.0354009
$$603$$ 0 0
$$604$$ 56.3400 2.29244
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −7.81471 −0.317189 −0.158595 0.987344i $$-0.550696\pi$$
−0.158595 + 0.987344i $$0.550696\pi$$
$$608$$ −11.8656 −0.481212
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.32324 0.134444
$$612$$ 0 0
$$613$$ −12.9547 −0.523235 −0.261618 0.965172i $$-0.584256\pi$$
−0.261618 + 0.965172i $$0.584256\pi$$
$$614$$ −17.8160 −0.718996
$$615$$ 0 0
$$616$$ 11.3531 0.457431
$$617$$ 18.8873 0.760373 0.380187 0.924910i $$-0.375860\pi$$
0.380187 + 0.924910i $$0.375860\pi$$
$$618$$ 0 0
$$619$$ 29.2673 1.17635 0.588176 0.808733i $$-0.299846\pi$$
0.588176 + 0.808733i $$0.299846\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 54.2339 2.17458
$$623$$ −8.39396 −0.336297
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 50.9479 2.03629
$$627$$ 0 0
$$628$$ 32.4872 1.29638
$$629$$ −33.9281 −1.35280
$$630$$ 0 0
$$631$$ −5.25309 −0.209122 −0.104561 0.994518i $$-0.533344\pi$$
−0.104561 + 0.994518i $$0.533344\pi$$
$$632$$ 51.2992 2.04057
$$633$$ 0 0
$$634$$ −8.87291 −0.352388
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.12402 −0.123778
$$638$$ 27.6698 1.09546
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 13.0021 0.513554 0.256777 0.966471i $$-0.417339\pi$$
0.256777 + 0.966471i $$0.417339\pi$$
$$642$$ 0 0
$$643$$ 17.3534 0.684353 0.342176 0.939636i $$-0.388836\pi$$
0.342176 + 0.939636i $$0.388836\pi$$
$$644$$ 4.27121 0.168309
$$645$$ 0 0
$$646$$ 10.7469 0.422831
$$647$$ 12.4848 0.490830 0.245415 0.969418i $$-0.421076\pi$$
0.245415 + 0.969418i $$0.421076\pi$$
$$648$$ 0 0
$$649$$ −6.91437 −0.271413
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −99.5507 −3.89871
$$653$$ −28.3532 −1.10955 −0.554774 0.832001i $$-0.687195\pi$$
−0.554774 + 0.832001i $$0.687195\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 18.9842 0.741209
$$657$$ 0 0
$$658$$ −10.7238 −0.418058
$$659$$ −45.2202 −1.76153 −0.880764 0.473556i $$-0.842970\pi$$
−0.880764 + 0.473556i $$0.842970\pi$$
$$660$$ 0 0
$$661$$ −14.6086 −0.568207 −0.284104 0.958794i $$-0.591696\pi$$
−0.284104 + 0.958794i $$0.591696\pi$$
$$662$$ 86.2115 3.35070
$$663$$ 0 0
$$664$$ 51.1093 1.98343
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.17594 0.239133
$$668$$ −28.9820 −1.12135
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −31.9505 −1.23344
$$672$$ 0 0
$$673$$ 0.440534 0.0169813 0.00849067 0.999964i $$-0.497297\pi$$
0.00849067 + 0.999964i $$0.497297\pi$$
$$674$$ 23.0111 0.886356
$$675$$ 0 0
$$676$$ −62.8479 −2.41723
$$677$$ 32.8057 1.26083 0.630414 0.776259i $$-0.282885\pi$$
0.630414 + 0.776259i $$0.282885\pi$$
$$678$$ 0 0
$$679$$ −10.0295 −0.384898
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −67.1815 −2.57251
$$683$$ −39.6092 −1.51561 −0.757803 0.652484i $$-0.773727\pi$$
−0.757803 + 0.652484i $$0.773727\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.6611 0.788844
$$687$$ 0 0
$$688$$ 5.94574 0.226679
$$689$$ 2.01559 0.0767879
$$690$$ 0 0
$$691$$ −19.8962 −0.756889 −0.378444 0.925624i $$-0.623541\pi$$
−0.378444 + 0.925624i $$0.623541\pi$$
$$692$$ −17.3665 −0.660175
$$693$$ 0 0
$$694$$ 48.6642 1.84727
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7.49752 −0.283989
$$698$$ −9.31924 −0.352738
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 14.1251 0.533497 0.266748 0.963766i $$-0.414051\pi$$
0.266748 + 0.963766i $$0.414051\pi$$
$$702$$ 0 0
$$703$$ 8.30326 0.313163
$$704$$ −27.0727 −1.02034
$$705$$ 0 0
$$706$$ 8.97234 0.337678
$$707$$ −8.20809 −0.308697
$$708$$ 0 0
$$709$$ −41.1815 −1.54660 −0.773302 0.634038i $$-0.781396\pi$$
−0.773302 + 0.634038i $$0.781396\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 112.078 4.20031
$$713$$ −14.9950 −0.561569
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −35.3466 −1.32097
$$717$$ 0 0
$$718$$ −4.48718 −0.167460
$$719$$ 18.0227 0.672133 0.336067 0.941838i $$-0.390903\pi$$
0.336067 + 0.941838i $$0.390903\pi$$
$$720$$ 0 0
$$721$$ −2.75372 −0.102554
$$722$$ −2.63010 −0.0978823
$$723$$ 0 0
$$724$$ 76.4332 2.84062
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 21.1266 0.783544 0.391772 0.920062i $$-0.371862\pi$$
0.391772 + 0.920062i $$0.371862\pi$$
$$728$$ −2.06537 −0.0765479
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.34818 −0.0868504
$$732$$ 0 0
$$733$$ 27.6660 1.02187 0.510934 0.859620i $$-0.329300\pi$$
0.510934 + 0.859620i $$0.329300\pi$$
$$734$$ 98.9326 3.65167
$$735$$ 0 0
$$736$$ −17.9341 −0.661061
$$737$$ −6.96629 −0.256607
$$738$$ 0 0
$$739$$ −14.2987 −0.525986 −0.262993 0.964798i $$-0.584710\pi$$
−0.262993 + 0.964798i $$0.584710\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −6.50415 −0.238775
$$743$$ 49.9438 1.83226 0.916130 0.400881i $$-0.131296\pi$$
0.916130 + 0.400881i $$0.131296\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 35.2516 1.29065
$$747$$ 0 0
$$748$$ 51.7335 1.89156
$$749$$ 5.30198 0.193730
$$750$$ 0 0
$$751$$ 9.69216 0.353672 0.176836 0.984240i $$-0.443414\pi$$
0.176836 + 0.984240i $$0.443414\pi$$
$$752$$ 73.4080 2.67691
$$753$$ 0 0
$$754$$ −5.03371 −0.183317
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −46.6889 −1.69694 −0.848469 0.529245i $$-0.822475\pi$$
−0.848469 + 0.529245i $$0.822475\pi$$
$$758$$ 24.6469 0.895214
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.7361 1.40418 0.702091 0.712087i $$-0.252250\pi$$
0.702091 + 0.712087i $$0.252250\pi$$
$$762$$ 0 0
$$763$$ 2.81480 0.101903
$$764$$ −65.5080 −2.37000
$$765$$ 0 0
$$766$$ 5.52104 0.199483
$$767$$ 1.25787 0.0454190
$$768$$ 0 0
$$769$$ −5.38666 −0.194248 −0.0971239 0.995272i $$-0.530964\pi$$
−0.0971239 + 0.995272i $$0.530964\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −93.4112 −3.36194
$$773$$ 7.20137 0.259015 0.129508 0.991578i $$-0.458660\pi$$
0.129508 + 0.991578i $$0.458660\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 133.917 4.80733
$$777$$ 0 0
$$778$$ 2.82047 0.101119
$$779$$ 1.83488 0.0657413
$$780$$ 0 0
$$781$$ −19.0541 −0.681808
$$782$$ 16.2434 0.580861
$$783$$ 0 0
$$784$$ −69.0074 −2.46455
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 33.5231 1.19497 0.597485 0.801880i $$-0.296167\pi$$
0.597485 + 0.801880i $$0.296167\pi$$
$$788$$ 10.6818 0.380524
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.929664 0.0330550
$$792$$ 0 0
$$793$$ 5.81247 0.206407
$$794$$ −61.8972 −2.19665
$$795$$ 0 0
$$796$$ −9.21305 −0.326548
$$797$$ −10.4979 −0.371855 −0.185927 0.982563i $$-0.559529\pi$$
−0.185927 + 0.982563i $$0.559529\pi$$
$$798$$ 0 0
$$799$$ −28.9913 −1.02564
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 49.0432 1.73177
$$803$$ 31.1178 1.09812
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 12.2217 0.430492
$$807$$ 0 0
$$808$$ 109.596 3.85559
$$809$$ 13.0724 0.459600 0.229800 0.973238i $$-0.426193\pi$$
0.229800 + 0.973238i $$0.426193\pi$$
$$810$$ 0 0
$$811$$ −10.0790 −0.353922 −0.176961 0.984218i $$-0.556627\pi$$
−0.176961 + 0.984218i $$0.556627\pi$$
$$812$$ 11.5470 0.405221
$$813$$ 0 0
$$814$$ 56.2268 1.97075
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0.574672 0.0201052
$$818$$ 18.0956 0.632697
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −40.2987 −1.40643 −0.703217 0.710975i $$-0.748254\pi$$
−0.703217 + 0.710975i $$0.748254\pi$$
$$822$$ 0 0
$$823$$ 5.07715 0.176978 0.0884892 0.996077i $$-0.471796\pi$$
0.0884892 + 0.996077i $$0.471796\pi$$
$$824$$ 36.7684 1.28089
$$825$$ 0 0
$$826$$ −4.05904 −0.141232
$$827$$ −42.8077 −1.48857 −0.744285 0.667862i $$-0.767209\pi$$
−0.744285 + 0.667862i $$0.767209\pi$$
$$828$$ 0 0
$$829$$ 24.2018 0.840562 0.420281 0.907394i $$-0.361932\pi$$
0.420281 + 0.907394i $$0.361932\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4.92509 0.170747
$$833$$ 27.2534 0.944274
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.6608 −0.437883
$$837$$ 0 0
$$838$$ 35.4527 1.22469
$$839$$ 8.63975 0.298277 0.149139 0.988816i $$-0.452350\pi$$
0.149139 + 0.988816i $$0.452350\pi$$
$$840$$ 0 0
$$841$$ −12.3036 −0.424264
$$842$$ 15.3463 0.528869
$$843$$ 0 0
$$844$$ −84.1325 −2.89596
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.51193 −0.0863109
$$848$$ 44.5229 1.52892
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 12.5499 0.430206
$$852$$ 0 0
$$853$$ 13.0229 0.445895 0.222948 0.974830i $$-0.428432\pi$$
0.222948 + 0.974830i $$0.428432\pi$$
$$854$$ −18.7564 −0.641829
$$855$$ 0 0
$$856$$ −70.7934 −2.41967
$$857$$ −2.82367 −0.0964548 −0.0482274 0.998836i $$-0.515357\pi$$
−0.0482274 + 0.998836i $$0.515357\pi$$
$$858$$ 0 0
$$859$$ 24.1227 0.823057 0.411529 0.911397i $$-0.364995\pi$$
0.411529 + 0.911397i $$0.364995\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 76.8092 2.61613
$$863$$ 32.7307 1.11417 0.557084 0.830456i $$-0.311920\pi$$
0.557084 + 0.830456i $$0.311920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −32.9366 −1.11923
$$867$$ 0 0
$$868$$ −28.0359 −0.951599
$$869$$ 17.2131 0.583913
$$870$$ 0 0
$$871$$ 1.26731 0.0429413
$$872$$ −37.5839 −1.27275
$$873$$ 0 0
$$874$$ −3.97526 −0.134465
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 49.5072 1.67174 0.835869 0.548929i $$-0.184964\pi$$
0.835869 + 0.548929i $$0.184964\pi$$
$$878$$ −41.2318 −1.39150
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −40.3152 −1.35826 −0.679128 0.734020i $$-0.737642\pi$$
−0.679128 + 0.734020i $$0.737642\pi$$
$$882$$ 0 0
$$883$$ 29.3347 0.987192 0.493596 0.869691i $$-0.335682\pi$$
0.493596 + 0.869691i $$0.335682\pi$$
$$884$$ −9.41140 −0.316540
$$885$$ 0 0
$$886$$ 78.4508 2.63561
$$887$$ 43.3214 1.45459 0.727295 0.686325i $$-0.240777\pi$$
0.727295 + 0.686325i $$0.240777\pi$$
$$888$$ 0 0
$$889$$ 7.54496 0.253050
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −25.2224 −0.844508
$$893$$ 7.09508 0.237428
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.25524 −0.0753424
$$897$$ 0 0
$$898$$ 78.8157 2.63011
$$899$$ −40.5383 −1.35203
$$900$$ 0 0
$$901$$ −17.5836 −0.585796
$$902$$ 12.4251 0.413712
$$903$$ 0 0
$$904$$ −12.4131 −0.412854
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −14.0731 −0.467288 −0.233644 0.972322i $$-0.575065\pi$$
−0.233644 + 0.972322i $$0.575065\pi$$
$$908$$ −35.9574 −1.19329
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0725 −0.996346 −0.498173 0.867078i $$-0.665996\pi$$
−0.498173 + 0.867078i $$0.665996\pi$$
$$912$$ 0 0
$$913$$ 17.1493 0.567560
$$914$$ 46.4733 1.53720
$$915$$ 0 0
$$916$$ −41.3534 −1.36636
$$917$$ 4.69635 0.155087
$$918$$ 0 0
$$919$$ 29.6518 0.978123 0.489062 0.872249i $$-0.337339\pi$$
0.489062 + 0.872249i $$0.337339\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 58.7682 1.93543
$$923$$ 3.46634 0.114096
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −17.9715 −0.590582
$$927$$ 0 0
$$928$$ −48.4841 −1.59157
$$929$$ −58.3803 −1.91540 −0.957698 0.287775i $$-0.907085\pi$$
−0.957698 + 0.287775i $$0.907085\pi$$
$$930$$ 0 0
$$931$$ −6.66975 −0.218592
$$932$$ −69.6911 −2.28281
$$933$$ 0 0
$$934$$ −23.6945 −0.775308
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.15850 0.103184 0.0515918 0.998668i $$-0.483571\pi$$
0.0515918 + 0.998668i $$0.483571\pi$$
$$938$$ −4.08952 −0.133528
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 9.66975 0.315225 0.157612 0.987501i $$-0.449620\pi$$
0.157612 + 0.987501i $$0.449620\pi$$
$$942$$ 0 0
$$943$$ 2.77331 0.0903116
$$944$$ 27.7854 0.904337
$$945$$ 0 0
$$946$$ 3.89148 0.126523
$$947$$ 31.3714 1.01943 0.509716 0.860343i $$-0.329750\pi$$
0.509716 + 0.860343i $$0.329750\pi$$
$$948$$ 0 0
$$949$$ −5.66098 −0.183763
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 18.0179 0.583964
$$953$$ −13.1224 −0.425075 −0.212537 0.977153i $$-0.568173\pi$$
−0.212537 + 0.977153i $$0.568173\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 69.7792 2.25682
$$957$$ 0 0
$$958$$ 24.3740 0.787488
$$959$$ −8.39396 −0.271055
$$960$$ 0 0
$$961$$ 67.4261 2.17504
$$962$$ −10.2288 −0.329791
$$963$$ 0 0
$$964$$ 137.101 4.41574
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 44.4400 1.42910 0.714548 0.699587i $$-0.246633\pi$$
0.714548 + 0.699587i $$0.246633\pi$$
$$968$$ 33.5399 1.07801
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5.69927 0.182898 0.0914491 0.995810i $$-0.470850\pi$$
0.0914491 + 0.995810i $$0.470850\pi$$
$$972$$ 0 0
$$973$$ −1.19083 −0.0381762
$$974$$ −101.968 −3.26725
$$975$$ 0 0
$$976$$ 128.393 4.10977
$$977$$ −23.9164 −0.765154 −0.382577 0.923924i $$-0.624963\pi$$
−0.382577 + 0.923924i $$0.624963\pi$$
$$978$$ 0 0
$$979$$ 37.6070 1.20193
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 57.0408 1.82025
$$983$$ −45.5302 −1.45219 −0.726095 0.687595i $$-0.758667\pi$$
−0.726095 + 0.687595i $$0.758667\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 43.9131 1.39848
$$987$$ 0 0
$$988$$ 2.30326 0.0732766
$$989$$ 0.868585 0.0276194
$$990$$ 0 0
$$991$$ 27.5521 0.875220 0.437610 0.899165i $$-0.355825\pi$$
0.437610 + 0.899165i $$0.355825\pi$$
$$992$$ 117.718 3.73756
$$993$$ 0 0
$$994$$ −11.1856 −0.354785
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −16.8557 −0.533826 −0.266913 0.963721i $$-0.586004\pi$$
−0.266913 + 0.963721i $$0.586004\pi$$
$$998$$ 73.2147 2.31757
$$999$$ 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 4275.2.a.bo.1.1 4
3.2 odd 2 475.2.a.i.1.4 4
5.4 even 2 855.2.a.m.1.4 4
12.11 even 2 7600.2.a.cf.1.4 4
15.2 even 4 475.2.b.e.324.8 8
15.8 even 4 475.2.b.e.324.1 8
15.14 odd 2 95.2.a.b.1.1 4
57.56 even 2 9025.2.a.bf.1.1 4
60.59 even 2 1520.2.a.t.1.1 4
105.104 even 2 4655.2.a.y.1.1 4
120.29 odd 2 6080.2.a.cc.1.1 4
120.59 even 2 6080.2.a.ch.1.4 4
285.284 even 2 1805.2.a.p.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.1 4 15.14 odd 2
475.2.a.i.1.4 4 3.2 odd 2
475.2.b.e.324.1 8 15.8 even 4
475.2.b.e.324.8 8 15.2 even 4
855.2.a.m.1.4 4 5.4 even 2
1520.2.a.t.1.1 4 60.59 even 2
1805.2.a.p.1.4 4 285.284 even 2
4275.2.a.bo.1.1 4 1.1 even 1 trivial
4655.2.a.y.1.1 4 105.104 even 2
6080.2.a.cc.1.1 4 120.29 odd 2
6080.2.a.ch.1.4 4 120.59 even 2
7600.2.a.cf.1.4 4 12.11 even 2
9025.2.a.bf.1.1 4 57.56 even 2