# Properties

 Label 9075.2.a.cd.1.3 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.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: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.12489$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.12489 q^{2} +1.00000 q^{3} +2.51514 q^{4} +2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +2.51514 q^{12} +1.51514 q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516 q^{17} +2.12489 q^{18} -2.60975 q^{19} +3.64002 q^{21} +5.73463 q^{23} +1.09461 q^{24} +3.21949 q^{26} +1.00000 q^{27} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} -7.93567 q^{32} +2.45459 q^{34} +2.51514 q^{36} -0.454586 q^{37} -5.54541 q^{38} +1.51514 q^{39} -4.12489 q^{41} +7.73463 q^{42} +11.7044 q^{43} +12.1854 q^{46} +3.48486 q^{47} -2.70436 q^{48} +6.24977 q^{49} +1.15516 q^{51} +3.81078 q^{52} +12.5601 q^{53} +2.12489 q^{54} +3.98440 q^{56} -2.60975 q^{57} -13.2800 q^{58} -7.73463 q^{59} +12.0147 q^{61} +11.7190 q^{62} +3.64002 q^{63} -11.4537 q^{64} +14.2645 q^{67} +2.90539 q^{68} +5.73463 q^{69} +8.51514 q^{71} +1.09461 q^{72} -9.21949 q^{73} -0.965943 q^{74} -6.56387 q^{76} +3.21949 q^{78} -5.09461 q^{79} +1.00000 q^{81} -8.76491 q^{82} -14.7493 q^{83} +9.15516 q^{84} +24.8704 q^{86} -6.24977 q^{87} -10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} +5.51514 q^{93} +7.40493 q^{94} -7.93567 q^{96} +6.77959 q^{97} +13.2800 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 $$3 q - 2 q^{2} + 3 q^{3} + 8 q^{4} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 3 q^{21} - 6 q^{24} - 8 q^{26} + 3 q^{27} + 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 8 q^{36} - 18 q^{38} + 5 q^{39} - 4 q^{41} + 6 q^{42} + 17 q^{43} + 30 q^{46} + 10 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} + 30 q^{52} + 6 q^{53} - 2 q^{54} - 22 q^{56} + q^{57} - 24 q^{58} - 6 q^{59} + 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} - 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} - 14 q^{74} + 24 q^{76} - 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} + 6 q^{83} + 20 q^{84} + 28 q^{86} - 2 q^{87} + 2 q^{89} + 17 q^{91} - 26 q^{92} + 17 q^{93} - 2 q^{94} - 34 q^{96} - 29 q^{97} + 24 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 + 3 * q^3 + 8 * q^4 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 8 * q^12 + 5 * q^13 + 6 * q^14 + 10 * q^16 - 4 * q^17 - 2 * q^18 + q^19 + 3 * q^21 - 6 * q^24 - 8 * q^26 + 3 * q^27 + 20 * q^28 - 2 * q^29 + 17 * q^31 - 34 * q^32 + 6 * q^34 + 8 * q^36 - 18 * q^38 + 5 * q^39 - 4 * q^41 + 6 * q^42 + 17 * q^43 + 30 * q^46 + 10 * q^47 + 10 * q^48 + 2 * q^49 - 4 * q^51 + 30 * q^52 + 6 * q^53 - 2 * q^54 - 22 * q^56 + q^57 - 24 * q^58 - 6 * q^59 + 3 * q^61 - 16 * q^62 + 3 * q^63 + 34 * q^64 - 7 * q^67 + 18 * q^68 + 26 * q^71 - 6 * q^72 - 10 * q^73 - 14 * q^74 + 24 * q^76 - 8 * q^78 - 6 * q^79 + 3 * q^81 - 10 * q^82 + 6 * q^83 + 20 * q^84 + 28 * q^86 - 2 * q^87 + 2 * q^89 + 17 * q^91 - 26 * q^92 + 17 * q^93 - 2 * q^94 - 34 * q^96 - 29 * q^97 + 24 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.12489 1.50252 0.751260 0.660006i $$-0.229446\pi$$
0.751260 + 0.660006i $$0.229446\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.51514 1.25757
$$5$$ 0 0
$$6$$ 2.12489 0.867481
$$7$$ 3.64002 1.37580 0.687900 0.725806i $$-0.258533\pi$$
0.687900 + 0.725806i $$0.258533\pi$$
$$8$$ 1.09461 0.387003
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 2.51514 0.726058
$$13$$ 1.51514 0.420224 0.210112 0.977677i $$-0.432617\pi$$
0.210112 + 0.977677i $$0.432617\pi$$
$$14$$ 7.73463 2.06717
$$15$$ 0 0
$$16$$ −2.70436 −0.676089
$$17$$ 1.15516 0.280168 0.140084 0.990140i $$-0.455263\pi$$
0.140084 + 0.990140i $$0.455263\pi$$
$$18$$ 2.12489 0.500840
$$19$$ −2.60975 −0.598717 −0.299359 0.954141i $$-0.596773\pi$$
−0.299359 + 0.954141i $$0.596773\pi$$
$$20$$ 0 0
$$21$$ 3.64002 0.794318
$$22$$ 0 0
$$23$$ 5.73463 1.19575 0.597877 0.801588i $$-0.296011\pi$$
0.597877 + 0.801588i $$0.296011\pi$$
$$24$$ 1.09461 0.223436
$$25$$ 0 0
$$26$$ 3.21949 0.631395
$$27$$ 1.00000 0.192450
$$28$$ 9.15516 1.73016
$$29$$ −6.24977 −1.16055 −0.580277 0.814419i $$-0.697056\pi$$
−0.580277 + 0.814419i $$0.697056\pi$$
$$30$$ 0 0
$$31$$ 5.51514 0.990548 0.495274 0.868737i $$-0.335068\pi$$
0.495274 + 0.868737i $$0.335068\pi$$
$$32$$ −7.93567 −1.40284
$$33$$ 0 0
$$34$$ 2.45459 0.420958
$$35$$ 0 0
$$36$$ 2.51514 0.419190
$$37$$ −0.454586 −0.0747335 −0.0373667 0.999302i $$-0.511897\pi$$
−0.0373667 + 0.999302i $$0.511897\pi$$
$$38$$ −5.54541 −0.899585
$$39$$ 1.51514 0.242616
$$40$$ 0 0
$$41$$ −4.12489 −0.644199 −0.322099 0.946706i $$-0.604389\pi$$
−0.322099 + 0.946706i $$0.604389\pi$$
$$42$$ 7.73463 1.19348
$$43$$ 11.7044 1.78490 0.892449 0.451149i $$-0.148986\pi$$
0.892449 + 0.451149i $$0.148986\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 12.1854 1.79664
$$47$$ 3.48486 0.508319 0.254160 0.967162i $$-0.418201\pi$$
0.254160 + 0.967162i $$0.418201\pi$$
$$48$$ −2.70436 −0.390340
$$49$$ 6.24977 0.892824
$$50$$ 0 0
$$51$$ 1.15516 0.161755
$$52$$ 3.81078 0.528460
$$53$$ 12.5601 1.72526 0.862631 0.505834i $$-0.168815\pi$$
0.862631 + 0.505834i $$0.168815\pi$$
$$54$$ 2.12489 0.289160
$$55$$ 0 0
$$56$$ 3.98440 0.532438
$$57$$ −2.60975 −0.345669
$$58$$ −13.2800 −1.74376
$$59$$ −7.73463 −1.00696 −0.503482 0.864006i $$-0.667948\pi$$
−0.503482 + 0.864006i $$0.667948\pi$$
$$60$$ 0 0
$$61$$ 12.0147 1.53832 0.769161 0.639055i $$-0.220674\pi$$
0.769161 + 0.639055i $$0.220674\pi$$
$$62$$ 11.7190 1.48832
$$63$$ 3.64002 0.458600
$$64$$ −11.4537 −1.43171
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.2645 1.74268 0.871340 0.490680i $$-0.163251\pi$$
0.871340 + 0.490680i $$0.163251\pi$$
$$68$$ 2.90539 0.352330
$$69$$ 5.73463 0.690369
$$70$$ 0 0
$$71$$ 8.51514 1.01056 0.505280 0.862955i $$-0.331389\pi$$
0.505280 + 0.862955i $$0.331389\pi$$
$$72$$ 1.09461 0.129001
$$73$$ −9.21949 −1.07906 −0.539530 0.841966i $$-0.681398\pi$$
−0.539530 + 0.841966i $$0.681398\pi$$
$$74$$ −0.965943 −0.112289
$$75$$ 0 0
$$76$$ −6.56387 −0.752928
$$77$$ 0 0
$$78$$ 3.21949 0.364536
$$79$$ −5.09461 −0.573188 −0.286594 0.958052i $$-0.592523\pi$$
−0.286594 + 0.958052i $$0.592523\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −8.76491 −0.967922
$$83$$ −14.7493 −1.61895 −0.809474 0.587156i $$-0.800247\pi$$
−0.809474 + 0.587156i $$0.800247\pi$$
$$84$$ 9.15516 0.998910
$$85$$ 0 0
$$86$$ 24.8704 2.68185
$$87$$ −6.24977 −0.670046
$$88$$ 0 0
$$89$$ −10.4995 −1.11295 −0.556475 0.830865i $$-0.687846\pi$$
−0.556475 + 0.830865i $$0.687846\pi$$
$$90$$ 0 0
$$91$$ 5.51514 0.578144
$$92$$ 14.4234 1.50374
$$93$$ 5.51514 0.571893
$$94$$ 7.40493 0.763760
$$95$$ 0 0
$$96$$ −7.93567 −0.809931
$$97$$ 6.77959 0.688363 0.344181 0.938903i $$-0.388156\pi$$
0.344181 + 0.938903i $$0.388156\pi$$
$$98$$ 13.2800 1.34149
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.40493 −0.736818 −0.368409 0.929664i $$-0.620097\pi$$
−0.368409 + 0.929664i $$0.620097\pi$$
$$102$$ 2.45459 0.243040
$$103$$ 16.4995 1.62575 0.812874 0.582439i $$-0.197902\pi$$
0.812874 + 0.582439i $$0.197902\pi$$
$$104$$ 1.65848 0.162628
$$105$$ 0 0
$$106$$ 26.6888 2.59224
$$107$$ 3.93945 0.380841 0.190420 0.981703i $$-0.439015\pi$$
0.190420 + 0.981703i $$0.439015\pi$$
$$108$$ 2.51514 0.242019
$$109$$ −6.73463 −0.645061 −0.322530 0.946559i $$-0.604533\pi$$
−0.322530 + 0.946559i $$0.604533\pi$$
$$110$$ 0 0
$$111$$ −0.454586 −0.0431474
$$112$$ −9.84392 −0.930163
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −5.54541 −0.519376
$$115$$ 0 0
$$116$$ −15.7190 −1.45948
$$117$$ 1.51514 0.140075
$$118$$ −16.4352 −1.51298
$$119$$ 4.20482 0.385455
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 25.5298 2.31136
$$123$$ −4.12489 −0.371928
$$124$$ 13.8713 1.24568
$$125$$ 0 0
$$126$$ 7.73463 0.689056
$$127$$ 8.06433 0.715594 0.357797 0.933799i $$-0.383528\pi$$
0.357797 + 0.933799i $$0.383528\pi$$
$$128$$ −8.46640 −0.748331
$$129$$ 11.7044 1.03051
$$130$$ 0 0
$$131$$ 12.8099 1.11920 0.559602 0.828762i $$-0.310954\pi$$
0.559602 + 0.828762i $$0.310954\pi$$
$$132$$ 0 0
$$133$$ −9.49954 −0.823715
$$134$$ 30.3103 2.61841
$$135$$ 0 0
$$136$$ 1.26445 0.108426
$$137$$ −22.8099 −1.94878 −0.974389 0.224868i $$-0.927805\pi$$
−0.974389 + 0.224868i $$0.927805\pi$$
$$138$$ 12.1854 1.03729
$$139$$ −7.59037 −0.643807 −0.321903 0.946773i $$-0.604323\pi$$
−0.321903 + 0.946773i $$0.604323\pi$$
$$140$$ 0 0
$$141$$ 3.48486 0.293478
$$142$$ 18.0937 1.51839
$$143$$ 0 0
$$144$$ −2.70436 −0.225363
$$145$$ 0 0
$$146$$ −19.5904 −1.62131
$$147$$ 6.24977 0.515472
$$148$$ −1.14335 −0.0939825
$$149$$ 1.81456 0.148655 0.0743274 0.997234i $$-0.476319\pi$$
0.0743274 + 0.997234i $$0.476319\pi$$
$$150$$ 0 0
$$151$$ −24.3250 −1.97954 −0.989770 0.142670i $$-0.954431\pi$$
−0.989770 + 0.142670i $$0.954431\pi$$
$$152$$ −2.85665 −0.231705
$$153$$ 1.15516 0.0933893
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 3.81078 0.305107
$$157$$ −9.76491 −0.779325 −0.389662 0.920958i $$-0.627408\pi$$
−0.389662 + 0.920958i $$0.627408\pi$$
$$158$$ −10.8255 −0.861227
$$159$$ 12.5601 0.996080
$$160$$ 0 0
$$161$$ 20.8742 1.64512
$$162$$ 2.12489 0.166947
$$163$$ 6.98440 0.547061 0.273530 0.961863i $$-0.411809\pi$$
0.273530 + 0.961863i $$0.411809\pi$$
$$164$$ −10.3747 −0.810125
$$165$$ 0 0
$$166$$ −31.3406 −2.43250
$$167$$ −6.31032 −0.488307 −0.244154 0.969737i $$-0.578510\pi$$
−0.244154 + 0.969737i $$0.578510\pi$$
$$168$$ 3.98440 0.307403
$$169$$ −10.7044 −0.823412
$$170$$ 0 0
$$171$$ −2.60975 −0.199572
$$172$$ 29.4381 2.24463
$$173$$ −12.8448 −0.976575 −0.488287 0.872683i $$-0.662378\pi$$
−0.488287 + 0.872683i $$0.662378\pi$$
$$174$$ −13.2800 −1.00676
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −7.73463 −0.581371
$$178$$ −22.3103 −1.67223
$$179$$ 13.4849 1.00791 0.503953 0.863731i $$-0.331878\pi$$
0.503953 + 0.863731i $$0.331878\pi$$
$$180$$ 0 0
$$181$$ −23.0899 −1.71626 −0.858130 0.513433i $$-0.828374\pi$$
−0.858130 + 0.513433i $$0.828374\pi$$
$$182$$ 11.7190 0.868673
$$183$$ 12.0147 0.888151
$$184$$ 6.27718 0.462760
$$185$$ 0 0
$$186$$ 11.7190 0.859281
$$187$$ 0 0
$$188$$ 8.76491 0.639247
$$189$$ 3.64002 0.264773
$$190$$ 0 0
$$191$$ −7.98440 −0.577731 −0.288866 0.957370i $$-0.593278\pi$$
−0.288866 + 0.957370i $$0.593278\pi$$
$$192$$ −11.4537 −0.826597
$$193$$ 11.7649 0.846857 0.423428 0.905930i $$-0.360827\pi$$
0.423428 + 0.905930i $$0.360827\pi$$
$$194$$ 14.4058 1.03428
$$195$$ 0 0
$$196$$ 15.7190 1.12279
$$197$$ −3.81456 −0.271776 −0.135888 0.990724i $$-0.543389\pi$$
−0.135888 + 0.990724i $$0.543389\pi$$
$$198$$ 0 0
$$199$$ −12.0752 −0.855990 −0.427995 0.903781i $$-0.640780\pi$$
−0.427995 + 0.903781i $$0.640780\pi$$
$$200$$ 0 0
$$201$$ 14.2645 1.00614
$$202$$ −15.7346 −1.10708
$$203$$ −22.7493 −1.59669
$$204$$ 2.90539 0.203418
$$205$$ 0 0
$$206$$ 35.0596 2.44272
$$207$$ 5.73463 0.398585
$$208$$ −4.09747 −0.284109
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −10.2645 −0.706634 −0.353317 0.935504i $$-0.614946\pi$$
−0.353317 + 0.935504i $$0.614946\pi$$
$$212$$ 31.5904 2.16964
$$213$$ 8.51514 0.583448
$$214$$ 8.37088 0.572221
$$215$$ 0 0
$$216$$ 1.09461 0.0744787
$$217$$ 20.0752 1.36280
$$218$$ −14.3103 −0.969217
$$219$$ −9.21949 −0.622996
$$220$$ 0 0
$$221$$ 1.75023 0.117733
$$222$$ −0.965943 −0.0648298
$$223$$ −12.9239 −0.865445 −0.432723 0.901527i $$-0.642447\pi$$
−0.432723 + 0.901527i $$0.642447\pi$$
$$224$$ −28.8860 −1.93003
$$225$$ 0 0
$$226$$ 12.7493 0.848072
$$227$$ 22.8099 1.51394 0.756972 0.653447i $$-0.226678\pi$$
0.756972 + 0.653447i $$0.226678\pi$$
$$228$$ −6.56387 −0.434703
$$229$$ 14.7796 0.976663 0.488331 0.872658i $$-0.337606\pi$$
0.488331 + 0.872658i $$0.337606\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.84106 −0.449137
$$233$$ 4.96594 0.325330 0.162665 0.986681i $$-0.447991\pi$$
0.162665 + 0.986681i $$0.447991\pi$$
$$234$$ 3.21949 0.210465
$$235$$ 0 0
$$236$$ −19.4537 −1.26633
$$237$$ −5.09461 −0.330930
$$238$$ 8.93475 0.579154
$$239$$ 14.9991 0.970210 0.485105 0.874456i $$-0.338781\pi$$
0.485105 + 0.874456i $$0.338781\pi$$
$$240$$ 0 0
$$241$$ −5.04496 −0.324974 −0.162487 0.986711i $$-0.551952\pi$$
−0.162487 + 0.986711i $$0.551952\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 30.2186 1.93455
$$245$$ 0 0
$$246$$ −8.76491 −0.558830
$$247$$ −3.95413 −0.251595
$$248$$ 6.03692 0.383345
$$249$$ −14.7493 −0.934700
$$250$$ 0 0
$$251$$ −3.03028 −0.191269 −0.0956347 0.995417i $$-0.530488\pi$$
−0.0956347 + 0.995417i $$0.530488\pi$$
$$252$$ 9.15516 0.576721
$$253$$ 0 0
$$254$$ 17.1358 1.07519
$$255$$ 0 0
$$256$$ 4.91721 0.307325
$$257$$ 13.6509 0.851521 0.425761 0.904836i $$-0.360007\pi$$
0.425761 + 0.904836i $$0.360007\pi$$
$$258$$ 24.8704 1.54836
$$259$$ −1.65470 −0.102818
$$260$$ 0 0
$$261$$ −6.24977 −0.386851
$$262$$ 27.2195 1.68163
$$263$$ 12.5601 0.774489 0.387244 0.921977i $$-0.373427\pi$$
0.387244 + 0.921977i $$0.373427\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −20.1854 −1.23765
$$267$$ −10.4995 −0.642562
$$268$$ 35.8771 2.19154
$$269$$ −24.6888 −1.50530 −0.752650 0.658421i $$-0.771225\pi$$
−0.752650 + 0.658421i $$0.771225\pi$$
$$270$$ 0 0
$$271$$ −7.56479 −0.459528 −0.229764 0.973246i $$-0.573795\pi$$
−0.229764 + 0.973246i $$0.573795\pi$$
$$272$$ −3.12397 −0.189418
$$273$$ 5.51514 0.333791
$$274$$ −48.4683 −2.92808
$$275$$ 0 0
$$276$$ 14.4234 0.868186
$$277$$ 1.92477 0.115648 0.0578241 0.998327i $$-0.481584\pi$$
0.0578241 + 0.998327i $$0.481584\pi$$
$$278$$ −16.1287 −0.967333
$$279$$ 5.51514 0.330183
$$280$$ 0 0
$$281$$ 1.87511 0.111860 0.0559300 0.998435i $$-0.482188\pi$$
0.0559300 + 0.998435i $$0.482188\pi$$
$$282$$ 7.40493 0.440957
$$283$$ 30.1396 1.79161 0.895806 0.444446i $$-0.146599\pi$$
0.895806 + 0.444446i $$0.146599\pi$$
$$284$$ 21.4167 1.27085
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −15.0147 −0.886289
$$288$$ −7.93567 −0.467614
$$289$$ −15.6656 −0.921506
$$290$$ 0 0
$$291$$ 6.77959 0.397427
$$292$$ −23.1883 −1.35699
$$293$$ −29.1552 −1.70326 −0.851631 0.524141i $$-0.824386\pi$$
−0.851631 + 0.524141i $$0.824386\pi$$
$$294$$ 13.2800 0.774508
$$295$$ 0 0
$$296$$ −0.497594 −0.0289221
$$297$$ 0 0
$$298$$ 3.85574 0.223357
$$299$$ 8.68876 0.502484
$$300$$ 0 0
$$301$$ 42.6041 2.45566
$$302$$ −51.6878 −2.97430
$$303$$ −7.40493 −0.425402
$$304$$ 7.05769 0.404786
$$305$$ 0 0
$$306$$ 2.45459 0.140319
$$307$$ 27.8548 1.58976 0.794879 0.606768i $$-0.207534\pi$$
0.794879 + 0.606768i $$0.207534\pi$$
$$308$$ 0 0
$$309$$ 16.4995 0.938626
$$310$$ 0 0
$$311$$ 23.9083 1.35571 0.677856 0.735194i $$-0.262909\pi$$
0.677856 + 0.735194i $$0.262909\pi$$
$$312$$ 1.65848 0.0938932
$$313$$ −28.3094 −1.60014 −0.800071 0.599905i $$-0.795205\pi$$
−0.800071 + 0.599905i $$0.795205\pi$$
$$314$$ −20.7493 −1.17095
$$315$$ 0 0
$$316$$ −12.8136 −0.720824
$$317$$ 8.80986 0.494811 0.247406 0.968912i $$-0.420422\pi$$
0.247406 + 0.968912i $$0.420422\pi$$
$$318$$ 26.6888 1.49663
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 3.93945 0.219879
$$322$$ 44.3553 2.47182
$$323$$ −3.01468 −0.167741
$$324$$ 2.51514 0.139730
$$325$$ 0 0
$$326$$ 14.8411 0.821970
$$327$$ −6.73463 −0.372426
$$328$$ −4.51514 −0.249307
$$329$$ 12.6850 0.699346
$$330$$ 0 0
$$331$$ 32.2498 1.77261 0.886304 0.463104i $$-0.153264\pi$$
0.886304 + 0.463104i $$0.153264\pi$$
$$332$$ −37.0966 −2.03594
$$333$$ −0.454586 −0.0249112
$$334$$ −13.4087 −0.733692
$$335$$ 0 0
$$336$$ −9.84392 −0.537030
$$337$$ 28.9844 1.57888 0.789441 0.613827i $$-0.210371\pi$$
0.789441 + 0.613827i $$0.210371\pi$$
$$338$$ −22.7455 −1.23719
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −5.54541 −0.299862
$$343$$ −2.73085 −0.147452
$$344$$ 12.8117 0.690760
$$345$$ 0 0
$$346$$ −27.2938 −1.46732
$$347$$ −35.7190 −1.91750 −0.958749 0.284253i $$-0.908254\pi$$
−0.958749 + 0.284253i $$0.908254\pi$$
$$348$$ −15.7190 −0.842629
$$349$$ 23.2800 1.24615 0.623076 0.782161i $$-0.285883\pi$$
0.623076 + 0.782161i $$0.285883\pi$$
$$350$$ 0 0
$$351$$ 1.51514 0.0808721
$$352$$ 0 0
$$353$$ 9.75023 0.518952 0.259476 0.965750i $$-0.416450\pi$$
0.259476 + 0.965750i $$0.416450\pi$$
$$354$$ −16.4352 −0.873521
$$355$$ 0 0
$$356$$ −26.4078 −1.39961
$$357$$ 4.20482 0.222542
$$358$$ 28.6538 1.51440
$$359$$ −33.9007 −1.78921 −0.894605 0.446858i $$-0.852543\pi$$
−0.894605 + 0.446858i $$0.852543\pi$$
$$360$$ 0 0
$$361$$ −12.1892 −0.641538
$$362$$ −49.0634 −2.57872
$$363$$ 0 0
$$364$$ 13.8713 0.727055
$$365$$ 0 0
$$366$$ 25.5298 1.33446
$$367$$ −1.88601 −0.0984491 −0.0492245 0.998788i $$-0.515675\pi$$
−0.0492245 + 0.998788i $$0.515675\pi$$
$$368$$ −15.5085 −0.808436
$$369$$ −4.12489 −0.214733
$$370$$ 0 0
$$371$$ 45.7190 2.37361
$$372$$ 13.8713 0.719195
$$373$$ −16.3250 −0.845277 −0.422638 0.906298i $$-0.638896\pi$$
−0.422638 + 0.906298i $$0.638896\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 3.81456 0.196721
$$377$$ −9.46927 −0.487692
$$378$$ 7.73463 0.397827
$$379$$ −26.0440 −1.33779 −0.668896 0.743356i $$-0.733233\pi$$
−0.668896 + 0.743356i $$0.733233\pi$$
$$380$$ 0 0
$$381$$ 8.06433 0.413148
$$382$$ −16.9659 −0.868053
$$383$$ −12.4702 −0.637197 −0.318598 0.947890i $$-0.603212\pi$$
−0.318598 + 0.947890i $$0.603212\pi$$
$$384$$ −8.46640 −0.432049
$$385$$ 0 0
$$386$$ 24.9991 1.27242
$$387$$ 11.7044 0.594966
$$388$$ 17.0516 0.865664
$$389$$ −18.0899 −0.917195 −0.458597 0.888644i $$-0.651648\pi$$
−0.458597 + 0.888644i $$0.651648\pi$$
$$390$$ 0 0
$$391$$ 6.62443 0.335012
$$392$$ 6.84106 0.345526
$$393$$ 12.8099 0.646172
$$394$$ −8.10551 −0.408350
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 15.2342 0.764581 0.382291 0.924042i $$-0.375135\pi$$
0.382291 + 0.924042i $$0.375135\pi$$
$$398$$ −25.6585 −1.28614
$$399$$ −9.49954 −0.475572
$$400$$ 0 0
$$401$$ 2.74931 0.137294 0.0686471 0.997641i $$-0.478132\pi$$
0.0686471 + 0.997641i $$0.478132\pi$$
$$402$$ 30.3103 1.51174
$$403$$ 8.35620 0.416252
$$404$$ −18.6244 −0.926600
$$405$$ 0 0
$$406$$ −48.3397 −2.39906
$$407$$ 0 0
$$408$$ 1.26445 0.0625996
$$409$$ 3.98532 0.197061 0.0985307 0.995134i $$-0.468586\pi$$
0.0985307 + 0.995134i $$0.468586\pi$$
$$410$$ 0 0
$$411$$ −22.8099 −1.12513
$$412$$ 41.4986 2.04449
$$413$$ −28.1542 −1.38538
$$414$$ 12.1854 0.598882
$$415$$ 0 0
$$416$$ −12.0236 −0.589507
$$417$$ −7.59037 −0.371702
$$418$$ 0 0
$$419$$ −5.13578 −0.250899 −0.125450 0.992100i $$-0.540037\pi$$
−0.125450 + 0.992100i $$0.540037\pi$$
$$420$$ 0 0
$$421$$ −8.94657 −0.436029 −0.218014 0.975946i $$-0.569958\pi$$
−0.218014 + 0.975946i $$0.569958\pi$$
$$422$$ −21.8108 −1.06173
$$423$$ 3.48486 0.169440
$$424$$ 13.7484 0.667681
$$425$$ 0 0
$$426$$ 18.0937 0.876642
$$427$$ 43.7337 2.11642
$$428$$ 9.90826 0.478934
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −22.7493 −1.09580 −0.547898 0.836545i $$-0.684572\pi$$
−0.547898 + 0.836545i $$0.684572\pi$$
$$432$$ −2.70436 −0.130113
$$433$$ −7.58325 −0.364428 −0.182214 0.983259i $$-0.558326\pi$$
−0.182214 + 0.983259i $$0.558326\pi$$
$$434$$ 42.6576 2.04763
$$435$$ 0 0
$$436$$ −16.9385 −0.811209
$$437$$ −14.9659 −0.715918
$$438$$ −19.5904 −0.936064
$$439$$ −17.3903 −0.829991 −0.414996 0.909823i $$-0.636217\pi$$
−0.414996 + 0.909823i $$0.636217\pi$$
$$440$$ 0 0
$$441$$ 6.24977 0.297608
$$442$$ 3.71904 0.176897
$$443$$ −10.6438 −0.505702 −0.252851 0.967505i $$-0.581368\pi$$
−0.252851 + 0.967505i $$0.581368\pi$$
$$444$$ −1.14335 −0.0542608
$$445$$ 0 0
$$446$$ −27.4617 −1.30035
$$447$$ 1.81456 0.0858259
$$448$$ −41.6916 −1.96974
$$449$$ −26.9310 −1.27095 −0.635476 0.772121i $$-0.719196\pi$$
−0.635476 + 0.772121i $$0.719196\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 15.0908 0.709813
$$453$$ −24.3250 −1.14289
$$454$$ 48.4683 2.27473
$$455$$ 0 0
$$456$$ −2.85665 −0.133775
$$457$$ −15.7796 −0.738138 −0.369069 0.929402i $$-0.620323\pi$$
−0.369069 + 0.929402i $$0.620323\pi$$
$$458$$ 31.4049 1.46746
$$459$$ 1.15516 0.0539183
$$460$$ 0 0
$$461$$ −8.18922 −0.381410 −0.190705 0.981647i $$-0.561077\pi$$
−0.190705 + 0.981647i $$0.561077\pi$$
$$462$$ 0 0
$$463$$ −16.0899 −0.747762 −0.373881 0.927477i $$-0.621973\pi$$
−0.373881 + 0.927477i $$0.621973\pi$$
$$464$$ 16.9016 0.784638
$$465$$ 0 0
$$466$$ 10.5521 0.488815
$$467$$ 29.4693 1.36367 0.681837 0.731504i $$-0.261181\pi$$
0.681837 + 0.731504i $$0.261181\pi$$
$$468$$ 3.81078 0.176153
$$469$$ 51.9229 2.39758
$$470$$ 0 0
$$471$$ −9.76491 −0.449943
$$472$$ −8.46640 −0.389698
$$473$$ 0 0
$$474$$ −10.8255 −0.497230
$$475$$ 0 0
$$476$$ 10.5757 0.484736
$$477$$ 12.5601 0.575087
$$478$$ 31.8713 1.45776
$$479$$ −32.2186 −1.47210 −0.736052 0.676925i $$-0.763312\pi$$
−0.736052 + 0.676925i $$0.763312\pi$$
$$480$$ 0 0
$$481$$ −0.688760 −0.0314048
$$482$$ −10.7200 −0.488280
$$483$$ 20.8742 0.949809
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 2.12489 0.0963868
$$487$$ 35.8936 1.62649 0.813247 0.581919i $$-0.197698\pi$$
0.813247 + 0.581919i $$0.197698\pi$$
$$488$$ 13.1514 0.595335
$$489$$ 6.98440 0.315846
$$490$$ 0 0
$$491$$ 7.15894 0.323079 0.161539 0.986866i $$-0.448354\pi$$
0.161539 + 0.986866i $$0.448354\pi$$
$$492$$ −10.3747 −0.467726
$$493$$ −7.21949 −0.325150
$$494$$ −8.40207 −0.378027
$$495$$ 0 0
$$496$$ −14.9149 −0.669699
$$497$$ 30.9953 1.39033
$$498$$ −31.3406 −1.40441
$$499$$ 27.0743 1.21201 0.606006 0.795460i $$-0.292771\pi$$
0.606006 + 0.795460i $$0.292771\pi$$
$$500$$ 0 0
$$501$$ −6.31032 −0.281924
$$502$$ −6.43899 −0.287386
$$503$$ 26.9991 1.20383 0.601915 0.798560i $$-0.294405\pi$$
0.601915 + 0.798560i $$0.294405\pi$$
$$504$$ 3.98440 0.177479
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −10.7044 −0.475397
$$508$$ 20.2829 0.899909
$$509$$ 15.5904 0.691031 0.345515 0.938413i $$-0.387704\pi$$
0.345515 + 0.938413i $$0.387704\pi$$
$$510$$ 0 0
$$511$$ −33.5592 −1.48457
$$512$$ 27.3813 1.21009
$$513$$ −2.60975 −0.115223
$$514$$ 29.0066 1.27943
$$515$$ 0 0
$$516$$ 29.4381 1.29594
$$517$$ 0 0
$$518$$ −3.51605 −0.154487
$$519$$ −12.8448 −0.563826
$$520$$ 0 0
$$521$$ 11.1589 0.488882 0.244441 0.969664i $$-0.421396\pi$$
0.244441 + 0.969664i $$0.421396\pi$$
$$522$$ −13.2800 −0.581252
$$523$$ −10.5786 −0.462568 −0.231284 0.972886i $$-0.574293\pi$$
−0.231284 + 0.972886i $$0.574293\pi$$
$$524$$ 32.2186 1.40748
$$525$$ 0 0
$$526$$ 26.6888 1.16369
$$527$$ 6.37088 0.277520
$$528$$ 0 0
$$529$$ 9.88601 0.429827
$$530$$ 0 0
$$531$$ −7.73463 −0.335654
$$532$$ −23.8927 −1.03588
$$533$$ −6.24977 −0.270708
$$534$$ −22.3103 −0.965462
$$535$$ 0 0
$$536$$ 15.6140 0.674422
$$537$$ 13.4849 0.581915
$$538$$ −52.4608 −2.26175
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11.2947 0.485598 0.242799 0.970077i $$-0.421934\pi$$
0.242799 + 0.970077i $$0.421934\pi$$
$$542$$ −16.0743 −0.690451
$$543$$ −23.0899 −0.990883
$$544$$ −9.16698 −0.393031
$$545$$ 0 0
$$546$$ 11.7190 0.501528
$$547$$ 6.09369 0.260547 0.130274 0.991478i $$-0.458414\pi$$
0.130274 + 0.991478i $$0.458414\pi$$
$$548$$ −57.3700 −2.45072
$$549$$ 12.0147 0.512774
$$550$$ 0 0
$$551$$ 16.3103 0.694843
$$552$$ 6.27718 0.267175
$$553$$ −18.5445 −0.788592
$$554$$ 4.08991 0.173764
$$555$$ 0 0
$$556$$ −19.0908 −0.809631
$$557$$ −5.90069 −0.250020 −0.125010 0.992155i $$-0.539896\pi$$
−0.125010 + 0.992155i $$0.539896\pi$$
$$558$$ 11.7190 0.496106
$$559$$ 17.7337 0.750056
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3.98440 0.168072
$$563$$ 3.03028 0.127711 0.0638555 0.997959i $$-0.479660\pi$$
0.0638555 + 0.997959i $$0.479660\pi$$
$$564$$ 8.76491 0.369069
$$565$$ 0 0
$$566$$ 64.0431 2.69193
$$567$$ 3.64002 0.152867
$$568$$ 9.32075 0.391090
$$569$$ −13.4049 −0.561964 −0.280982 0.959713i $$-0.590660\pi$$
−0.280982 + 0.959713i $$0.590660\pi$$
$$570$$ 0 0
$$571$$ −26.8851 −1.12511 −0.562553 0.826761i $$-0.690181\pi$$
−0.562553 + 0.826761i $$0.690181\pi$$
$$572$$ 0 0
$$573$$ −7.98440 −0.333553
$$574$$ −31.9045 −1.33167
$$575$$ 0 0
$$576$$ −11.4537 −0.477236
$$577$$ −2.03028 −0.0845215 −0.0422607 0.999107i $$-0.513456\pi$$
−0.0422607 + 0.999107i $$0.513456\pi$$
$$578$$ −33.2876 −1.38458
$$579$$ 11.7649 0.488933
$$580$$ 0 0
$$581$$ −53.6878 −2.22735
$$582$$ 14.4058 0.597142
$$583$$ 0 0
$$584$$ −10.0917 −0.417599
$$585$$ 0 0
$$586$$ −61.9514 −2.55919
$$587$$ −21.8245 −0.900795 −0.450398 0.892828i $$-0.648718\pi$$
−0.450398 + 0.892828i $$0.648718\pi$$
$$588$$ 15.7190 0.648242
$$589$$ −14.3931 −0.593058
$$590$$ 0 0
$$591$$ −3.81456 −0.156910
$$592$$ 1.22936 0.0505265
$$593$$ 8.06811 0.331318 0.165659 0.986183i $$-0.447025\pi$$
0.165659 + 0.986183i $$0.447025\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.56387 0.186944
$$597$$ −12.0752 −0.494206
$$598$$ 18.4626 0.754993
$$599$$ 7.61353 0.311080 0.155540 0.987830i $$-0.450288\pi$$
0.155540 + 0.987830i $$0.450288\pi$$
$$600$$ 0 0
$$601$$ −3.57569 −0.145855 −0.0729277 0.997337i $$-0.523234\pi$$
−0.0729277 + 0.997337i $$0.523234\pi$$
$$602$$ 90.5289 3.68968
$$603$$ 14.2645 0.580893
$$604$$ −61.1807 −2.48941
$$605$$ 0 0
$$606$$ −15.7346 −0.639176
$$607$$ 17.5298 0.711513 0.355757 0.934579i $$-0.384223\pi$$
0.355757 + 0.934579i $$0.384223\pi$$
$$608$$ 20.7101 0.839905
$$609$$ −22.7493 −0.921849
$$610$$ 0 0
$$611$$ 5.28005 0.213608
$$612$$ 2.90539 0.117443
$$613$$ 12.5601 0.507297 0.253649 0.967296i $$-0.418369\pi$$
0.253649 + 0.967296i $$0.418369\pi$$
$$614$$ 59.1883 2.38865
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −15.9612 −0.642576 −0.321288 0.946982i $$-0.604116\pi$$
−0.321288 + 0.946982i $$0.604116\pi$$
$$618$$ 35.0596 1.41031
$$619$$ −9.23417 −0.371153 −0.185576 0.982630i $$-0.559415\pi$$
−0.185576 + 0.982630i $$0.559415\pi$$
$$620$$ 0 0
$$621$$ 5.73463 0.230123
$$622$$ 50.8023 2.03699
$$623$$ −38.2186 −1.53119
$$624$$ −4.09747 −0.164030
$$625$$ 0 0
$$626$$ −60.1542 −2.40425
$$627$$ 0 0
$$628$$ −24.5601 −0.980054
$$629$$ −0.525120 −0.0209379
$$630$$ 0 0
$$631$$ 29.2342 1.16379 0.581897 0.813262i $$-0.302311\pi$$
0.581897 + 0.813262i $$0.302311\pi$$
$$632$$ −5.57661 −0.221826
$$633$$ −10.2645 −0.407975
$$634$$ 18.7200 0.743464
$$635$$ 0 0
$$636$$ 31.5904 1.25264
$$637$$ 9.46927 0.375186
$$638$$ 0 0
$$639$$ 8.51514 0.336854
$$640$$ 0 0
$$641$$ 13.9612 0.551436 0.275718 0.961239i $$-0.411084\pi$$
0.275718 + 0.961239i $$0.411084\pi$$
$$642$$ 8.37088 0.330372
$$643$$ −12.6206 −0.497710 −0.248855 0.968541i $$-0.580054\pi$$
−0.248855 + 0.968541i $$0.580054\pi$$
$$644$$ 52.5015 2.06885
$$645$$ 0 0
$$646$$ −6.40585 −0.252035
$$647$$ −29.6429 −1.16538 −0.582691 0.812694i $$-0.698000\pi$$
−0.582691 + 0.812694i $$0.698000\pi$$
$$648$$ 1.09461 0.0430003
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 20.0752 0.786810
$$652$$ 17.5667 0.687967
$$653$$ −9.90069 −0.387444 −0.193722 0.981056i $$-0.562056\pi$$
−0.193722 + 0.981056i $$0.562056\pi$$
$$654$$ −14.3103 −0.559578
$$655$$ 0 0
$$656$$ 11.1552 0.435536
$$657$$ −9.21949 −0.359687
$$658$$ 26.9541 1.05078
$$659$$ 5.28005 0.205681 0.102841 0.994698i $$-0.467207\pi$$
0.102841 + 0.994698i $$0.467207\pi$$
$$660$$ 0 0
$$661$$ −26.8548 −1.04453 −0.522266 0.852783i $$-0.674913\pi$$
−0.522266 + 0.852783i $$0.674913\pi$$
$$662$$ 68.5271 2.66338
$$663$$ 1.75023 0.0679733
$$664$$ −16.1447 −0.626537
$$665$$ 0 0
$$666$$ −0.965943 −0.0374295
$$667$$ −35.8401 −1.38774
$$668$$ −15.8713 −0.614080
$$669$$ −12.9239 −0.499665
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −28.8860 −1.11430
$$673$$ −3.81834 −0.147186 −0.0735932 0.997288i $$-0.523447\pi$$
−0.0735932 + 0.997288i $$0.523447\pi$$
$$674$$ 61.5885 2.37230
$$675$$ 0 0
$$676$$ −26.9229 −1.03550
$$677$$ −15.6897 −0.603003 −0.301502 0.953466i $$-0.597488\pi$$
−0.301502 + 0.953466i $$0.597488\pi$$
$$678$$ 12.7493 0.489634
$$679$$ 24.6779 0.947049
$$680$$ 0 0
$$681$$ 22.8099 0.874076
$$682$$ 0 0
$$683$$ 15.6353 0.598269 0.299135 0.954211i $$-0.403302\pi$$
0.299135 + 0.954211i $$0.403302\pi$$
$$684$$ −6.56387 −0.250976
$$685$$ 0 0
$$686$$ −5.80275 −0.221550
$$687$$ 14.7796 0.563876
$$688$$ −31.6528 −1.20675
$$689$$ 19.0303 0.724996
$$690$$ 0 0
$$691$$ −31.4305 −1.19567 −0.597836 0.801618i $$-0.703973\pi$$
−0.597836 + 0.801618i $$0.703973\pi$$
$$692$$ −32.3065 −1.22811
$$693$$ 0 0
$$694$$ −75.8989 −2.88108
$$695$$ 0 0
$$696$$ −6.84106 −0.259310
$$697$$ −4.76491 −0.180484
$$698$$ 49.4674 1.87237
$$699$$ 4.96594 0.187829
$$700$$ 0 0
$$701$$ −3.24507 −0.122565 −0.0612824 0.998120i $$-0.519519\pi$$
−0.0612824 + 0.998120i $$0.519519\pi$$
$$702$$ 3.21949 0.121512
$$703$$ 1.18635 0.0447442
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 20.7181 0.779737
$$707$$ −26.9541 −1.01371
$$708$$ −19.4537 −0.731114
$$709$$ −26.7190 −1.00345 −0.501727 0.865026i $$-0.667302\pi$$
−0.501727 + 0.865026i $$0.667302\pi$$
$$710$$ 0 0
$$711$$ −5.09461 −0.191063
$$712$$ −11.4929 −0.430714
$$713$$ 31.6273 1.18445
$$714$$ 8.93475 0.334375
$$715$$ 0 0
$$716$$ 33.9163 1.26751
$$717$$ 14.9991 0.560151
$$718$$ −72.0351 −2.68833
$$719$$ −6.78807 −0.253152 −0.126576 0.991957i $$-0.540399\pi$$
−0.126576 + 0.991957i $$0.540399\pi$$
$$720$$ 0 0
$$721$$ 60.0587 2.23670
$$722$$ −25.9007 −0.963924
$$723$$ −5.04496 −0.187624
$$724$$ −58.0743 −2.15831
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 19.9154 0.738620 0.369310 0.929306i $$-0.379594\pi$$
0.369310 + 0.929306i $$0.379594\pi$$
$$728$$ 6.03692 0.223743
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 13.5204 0.500071
$$732$$ 30.2186 1.11691
$$733$$ −31.3388 −1.15752 −0.578762 0.815497i $$-0.696464\pi$$
−0.578762 + 0.815497i $$0.696464\pi$$
$$734$$ −4.00756 −0.147922
$$735$$ 0 0
$$736$$ −45.5081 −1.67745
$$737$$ 0 0
$$738$$ −8.76491 −0.322641
$$739$$ 2.25355 0.0828982 0.0414491 0.999141i $$-0.486803\pi$$
0.0414491 + 0.999141i $$0.486803\pi$$
$$740$$ 0 0
$$741$$ −3.95413 −0.145259
$$742$$ 97.1477 3.56640
$$743$$ −6.74931 −0.247608 −0.123804 0.992307i $$-0.539509\pi$$
−0.123804 + 0.992307i $$0.539509\pi$$
$$744$$ 6.03692 0.221324
$$745$$ 0 0
$$746$$ −34.6888 −1.27005
$$747$$ −14.7493 −0.539649
$$748$$ 0 0
$$749$$ 14.3397 0.523961
$$750$$ 0 0
$$751$$ 22.4390 0.818810 0.409405 0.912353i $$-0.365736\pi$$
0.409405 + 0.912353i $$0.365736\pi$$
$$752$$ −9.42431 −0.343669
$$753$$ −3.03028 −0.110429
$$754$$ −20.1211 −0.732767
$$755$$ 0 0
$$756$$ 9.15516 0.332970
$$757$$ −25.4158 −0.923754 −0.461877 0.886944i $$-0.652824\pi$$
−0.461877 + 0.886944i $$0.652824\pi$$
$$758$$ −55.3406 −2.01006
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.7493 1.11466 0.557331 0.830291i $$-0.311826\pi$$
0.557331 + 0.830291i $$0.311826\pi$$
$$762$$ 17.1358 0.620764
$$763$$ −24.5142 −0.887474
$$764$$ −20.0819 −0.726537
$$765$$ 0 0
$$766$$ −26.4977 −0.957401
$$767$$ −11.7190 −0.423150
$$768$$ 4.91721 0.177434
$$769$$ 16.2956 0.587636 0.293818 0.955861i $$-0.405074\pi$$
0.293818 + 0.955861i $$0.405074\pi$$
$$770$$ 0 0
$$771$$ 13.6509 0.491626
$$772$$ 29.5904 1.06498
$$773$$ −48.7787 −1.75445 −0.877223 0.480082i $$-0.840607\pi$$
−0.877223 + 0.480082i $$0.840607\pi$$
$$774$$ 24.8704 0.893949
$$775$$ 0 0
$$776$$ 7.42100 0.266398
$$777$$ −1.65470 −0.0593621
$$778$$ −38.4390 −1.37810
$$779$$ 10.7649 0.385693
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 14.0761 0.503362
$$783$$ −6.24977 −0.223349
$$784$$ −16.9016 −0.603629
$$785$$ 0 0
$$786$$ 27.2195 0.970887
$$787$$ −46.2001 −1.64686 −0.823428 0.567421i $$-0.807941\pi$$
−0.823428 + 0.567421i $$0.807941\pi$$
$$788$$ −9.59415 −0.341777
$$789$$ 12.5601 0.447151
$$790$$ 0 0
$$791$$ 21.8401 0.776546
$$792$$ 0 0
$$793$$ 18.2039 0.646439
$$794$$ 32.3709 1.14880
$$795$$ 0 0
$$796$$ −30.3709 −1.07647
$$797$$ −36.3784 −1.28859 −0.644295 0.764777i $$-0.722849\pi$$
−0.644295 + 0.764777i $$0.722849\pi$$
$$798$$ −20.1854 −0.714557
$$799$$ 4.02558 0.142415
$$800$$ 0 0
$$801$$ −10.4995 −0.370983
$$802$$ 5.84197 0.206287
$$803$$ 0 0
$$804$$ 35.8771 1.26529
$$805$$ 0 0
$$806$$ 17.7560 0.625427
$$807$$ −24.6888 −0.869086
$$808$$ −8.10551 −0.285151
$$809$$ −11.3737 −0.399879 −0.199940 0.979808i $$-0.564075\pi$$
−0.199940 + 0.979808i $$0.564075\pi$$
$$810$$ 0 0
$$811$$ 13.3903 0.470195 0.235098 0.971972i $$-0.424459\pi$$
0.235098 + 0.971972i $$0.424459\pi$$
$$812$$ −57.2177 −2.00795
$$813$$ −7.56479 −0.265309
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −3.12397 −0.109361
$$817$$ −30.5454 −1.06865
$$818$$ 8.46835 0.296089
$$819$$ 5.51514 0.192715
$$820$$ 0 0
$$821$$ −32.0975 −1.12021 −0.560105 0.828422i $$-0.689239\pi$$
−0.560105 + 0.828422i $$0.689239\pi$$
$$822$$ −48.4683 −1.69053
$$823$$ −16.7952 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$824$$ 18.0606 0.629169
$$825$$ 0 0
$$826$$ −59.8245 −2.08156
$$827$$ −45.5904 −1.58533 −0.792666 0.609656i $$-0.791308\pi$$
−0.792666 + 0.609656i $$0.791308\pi$$
$$828$$ 14.4234 0.501248
$$829$$ 12.9385 0.449374 0.224687 0.974431i $$-0.427864\pi$$
0.224687 + 0.974431i $$0.427864\pi$$
$$830$$ 0 0
$$831$$ 1.92477 0.0667695
$$832$$ −17.3539 −0.601638
$$833$$ 7.21949 0.250141
$$834$$ −16.1287 −0.558490
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.51514 0.190631
$$838$$ −10.9130 −0.376982
$$839$$ −1.59037 −0.0549057 −0.0274528 0.999623i $$-0.508740\pi$$
−0.0274528 + 0.999623i $$0.508740\pi$$
$$840$$ 0 0
$$841$$ 10.0596 0.346884
$$842$$ −19.0104 −0.655143
$$843$$ 1.87511 0.0645824
$$844$$ −25.8165 −0.888641
$$845$$ 0 0
$$846$$ 7.40493 0.254587
$$847$$ 0 0
$$848$$ −33.9670 −1.16643
$$849$$ 30.1396 1.03439
$$850$$ 0 0
$$851$$ −2.60688 −0.0893628
$$852$$ 21.4167 0.733726
$$853$$ 10.5161 0.360063 0.180031 0.983661i $$-0.442380\pi$$
0.180031 + 0.983661i $$0.442380\pi$$
$$854$$ 92.9291 3.17997
$$855$$ 0 0
$$856$$ 4.31216 0.147386
$$857$$ −57.4637 −1.96292 −0.981460 0.191665i $$-0.938611\pi$$
−0.981460 + 0.191665i $$0.938611\pi$$
$$858$$ 0 0
$$859$$ 32.7181 1.11633 0.558164 0.829731i $$-0.311506\pi$$
0.558164 + 0.829731i $$0.311506\pi$$
$$860$$ 0 0
$$861$$ −15.0147 −0.511699
$$862$$ −48.3397 −1.64646
$$863$$ 43.1807 1.46989 0.734945 0.678127i $$-0.237208\pi$$
0.734945 + 0.678127i $$0.237208\pi$$
$$864$$ −7.93567 −0.269977
$$865$$ 0 0
$$866$$ −16.1135 −0.547560
$$867$$ −15.6656 −0.532032
$$868$$ 50.4920 1.71381
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 21.6126 0.732315
$$872$$ −7.37179 −0.249640
$$873$$ 6.77959 0.229454
$$874$$ −31.8009 −1.07568
$$875$$ 0 0
$$876$$ −23.1883 −0.783460
$$877$$ −16.0752 −0.542822 −0.271411 0.962464i $$-0.587490\pi$$
−0.271411 + 0.962464i $$0.587490\pi$$
$$878$$ −36.9523 −1.24708
$$879$$ −29.1552 −0.983379
$$880$$ 0 0
$$881$$ 31.2876 1.05411 0.527053 0.849832i $$-0.323297\pi$$
0.527053 + 0.849832i $$0.323297\pi$$
$$882$$ 13.2800 0.447162
$$883$$ −24.7640 −0.833375 −0.416687 0.909050i $$-0.636809\pi$$
−0.416687 + 0.909050i $$0.636809\pi$$
$$884$$ 4.40207 0.148058
$$885$$ 0 0
$$886$$ −22.6169 −0.759828
$$887$$ −26.3085 −0.883353 −0.441676 0.897174i $$-0.645616\pi$$
−0.441676 + 0.897174i $$0.645616\pi$$
$$888$$ −0.497594 −0.0166982
$$889$$ 29.3544 0.984514
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −32.5053 −1.08836
$$893$$ −9.09461 −0.304339
$$894$$ 3.85574 0.128955
$$895$$ 0 0
$$896$$ −30.8179 −1.02955
$$897$$ 8.68876 0.290109
$$898$$ −57.2252 −1.90963
$$899$$ −34.4683 −1.14958
$$900$$ 0 0
$$901$$ 14.5089 0.483363
$$902$$ 0 0
$$903$$ 42.6041 1.41778
$$904$$ 6.56766 0.218437
$$905$$ 0 0
$$906$$ −51.6878 −1.71721
$$907$$ 55.9301 1.85713 0.928563 0.371174i $$-0.121045\pi$$
0.928563 + 0.371174i $$0.121045\pi$$
$$908$$ 57.3700 1.90389
$$909$$ −7.40493 −0.245606
$$910$$ 0 0
$$911$$ −15.3931 −0.509997 −0.254998 0.966941i $$-0.582075\pi$$
−0.254998 + 0.966941i $$0.582075\pi$$
$$912$$ 7.05769 0.233703
$$913$$ 0 0
$$914$$ −33.5298 −1.10907
$$915$$ 0 0
$$916$$ 37.1727 1.22822
$$917$$ 46.6282 1.53980
$$918$$ 2.45459 0.0810134
$$919$$ 51.2598 1.69090 0.845452 0.534052i $$-0.179331\pi$$
0.845452 + 0.534052i $$0.179331\pi$$
$$920$$ 0 0
$$921$$ 27.8548 0.917848
$$922$$ −17.4012 −0.573076
$$923$$ 12.9016 0.424662
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −34.1892 −1.12353
$$927$$ 16.4995 0.541916
$$928$$ 49.5961 1.62807
$$929$$ 14.8099 0.485896 0.242948 0.970039i $$-0.421886\pi$$
0.242948 + 0.970039i $$0.421886\pi$$
$$930$$ 0 0
$$931$$ −16.3103 −0.534549
$$932$$ 12.4900 0.409125
$$933$$ 23.9083 0.782721
$$934$$ 62.6188 2.04895
$$935$$ 0 0
$$936$$ 1.65848 0.0542093
$$937$$ 27.2654 0.890721 0.445360 0.895351i $$-0.353076\pi$$
0.445360 + 0.895351i $$0.353076\pi$$
$$938$$ 110.330 3.60241
$$939$$ −28.3094 −0.923843
$$940$$ 0 0
$$941$$ 49.5630 1.61571 0.807853 0.589384i $$-0.200629\pi$$
0.807853 + 0.589384i $$0.200629\pi$$
$$942$$ −20.7493 −0.676049
$$943$$ −23.6547 −0.770303
$$944$$ 20.9172 0.680797
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13.9844 0.454432 0.227216 0.973844i $$-0.427038\pi$$
0.227216 + 0.973844i $$0.427038\pi$$
$$948$$ −12.8136 −0.416168
$$949$$ −13.9688 −0.453447
$$950$$ 0 0
$$951$$ 8.80986 0.285679
$$952$$ 4.60263 0.149172
$$953$$ 17.1240 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$954$$ 26.6888 0.864081
$$955$$ 0 0
$$956$$ 37.7248 1.22011
$$957$$ 0 0
$$958$$ −68.4608 −2.21187
$$959$$ −83.0284 −2.68113
$$960$$ 0 0
$$961$$ −0.583252 −0.0188146
$$962$$ −1.46354 −0.0471863
$$963$$ 3.93945 0.126947
$$964$$ −12.6888 −0.408677
$$965$$ 0 0
$$966$$ 44.3553 1.42711
$$967$$ 1.90826 0.0613653 0.0306827 0.999529i $$-0.490232\pi$$
0.0306827 + 0.999529i $$0.490232\pi$$
$$968$$ 0 0
$$969$$ −3.01468 −0.0968455
$$970$$ 0 0
$$971$$ 31.3856 1.00721 0.503605 0.863934i $$-0.332007\pi$$
0.503605 + 0.863934i $$0.332007\pi$$
$$972$$ 2.51514 0.0806731
$$973$$ −27.6291 −0.885749
$$974$$ 76.2697 2.44384
$$975$$ 0 0
$$976$$ −32.4920 −1.04004
$$977$$ −41.4693 −1.32672 −0.663360 0.748301i $$-0.730870\pi$$
−0.663360 + 0.748301i $$0.730870\pi$$
$$978$$ 14.8411 0.474565
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −6.73463 −0.215020
$$982$$ 15.2119 0.485432
$$983$$ 6.23601 0.198898 0.0994489 0.995043i $$-0.468292\pi$$
0.0994489 + 0.995043i $$0.468292\pi$$
$$984$$ −4.51514 −0.143937
$$985$$ 0 0
$$986$$ −15.3406 −0.488544
$$987$$ 12.6850 0.403767
$$988$$ −9.94518 −0.316398
$$989$$ 67.1202 2.13430
$$990$$ 0 0
$$991$$ 27.2048 0.864189 0.432095 0.901828i $$-0.357775\pi$$
0.432095 + 0.901828i $$0.357775\pi$$
$$992$$ −43.7663 −1.38958
$$993$$ 32.2498 1.02342
$$994$$ 65.8615 2.08900
$$995$$ 0 0
$$996$$ −37.0966 −1.17545
$$997$$ 28.0606 0.888687 0.444343 0.895857i $$-0.353437\pi$$
0.444343 + 0.895857i $$0.353437\pi$$
$$998$$ 57.5298 1.82107
$$999$$ −0.454586 −0.0143825
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 9075.2.a.cd.1.3 3
5.4 even 2 9075.2.a.cj.1.1 3
11.10 odd 2 825.2.a.m.1.1 yes 3
33.32 even 2 2475.2.a.z.1.3 3
55.32 even 4 825.2.c.f.199.2 6
55.43 even 4 825.2.c.f.199.5 6
55.54 odd 2 825.2.a.i.1.3 3
165.32 odd 4 2475.2.c.q.199.5 6
165.98 odd 4 2475.2.c.q.199.2 6
165.164 even 2 2475.2.a.bd.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 55.54 odd 2
825.2.a.m.1.1 yes 3 11.10 odd 2
825.2.c.f.199.2 6 55.32 even 4
825.2.c.f.199.5 6 55.43 even 4
2475.2.a.z.1.3 3 33.32 even 2
2475.2.a.bd.1.1 3 165.164 even 2
2475.2.c.q.199.2 6 165.98 odd 4
2475.2.c.q.199.5 6 165.32 odd 4
9075.2.a.cd.1.3 3 1.1 even 1 trivial
9075.2.a.cj.1.1 3 5.4 even 2