# Properties

 Label 825.2.a.i.1.3 Level $825$ Weight $2$ Character 825.1 Self dual yes Analytic conductor $6.588$ 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] = [825,2,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$6.58765816676$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.12489$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.12489 q^{2} -1.00000 q^{3} +2.51514 q^{4} -2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.12489 q^{2} -1.00000 q^{3} +2.51514 q^{4} -2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +1.00000 q^{11} -2.51514 q^{12} +1.51514 q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516 q^{17} +2.12489 q^{18} +2.60975 q^{19} -3.64002 q^{21} +2.12489 q^{22} -5.73463 q^{23} -1.09461 q^{24} +3.21949 q^{26} -1.00000 q^{27} +9.15516 q^{28} +6.24977 q^{29} +5.51514 q^{31} -7.93567 q^{32} -1.00000 q^{33} +2.45459 q^{34} +2.51514 q^{36} +0.454586 q^{37} +5.54541 q^{38} -1.51514 q^{39} +4.12489 q^{41} -7.73463 q^{42} +11.7044 q^{43} +2.51514 q^{44} -12.1854 q^{46} -3.48486 q^{47} +2.70436 q^{48} +6.24977 q^{49} -1.15516 q^{51} +3.81078 q^{52} -12.5601 q^{53} -2.12489 q^{54} +3.98440 q^{56} -2.60975 q^{57} +13.2800 q^{58} -7.73463 q^{59} -12.0147 q^{61} +11.7190 q^{62} +3.64002 q^{63} -11.4537 q^{64} -2.12489 q^{66} -14.2645 q^{67} +2.90539 q^{68} +5.73463 q^{69} +8.51514 q^{71} +1.09461 q^{72} -9.21949 q^{73} +0.965943 q^{74} +6.56387 q^{76} +3.64002 q^{77} -3.21949 q^{78} +5.09461 q^{79} +1.00000 q^{81} +8.76491 q^{82} -14.7493 q^{83} -9.15516 q^{84} +24.8704 q^{86} -6.24977 q^{87} +1.09461 q^{88} -10.4995 q^{89} +5.51514 q^{91} -14.4234 q^{92} -5.51514 q^{93} -7.40493 q^{94} +7.93567 q^{96} -6.77959 q^{97} +13.2800 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 3 q^{3} + 8 q^{4} + 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 3 * q^3 + 8 * q^4 + 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 $$3 q - 2 q^{2} - 3 q^{3} + 8 q^{4} + 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 3 q^{11} - 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} - q^{19} - 3 q^{21} - 2 q^{22} + 6 q^{24} - 8 q^{26} - 3 q^{27} + 20 q^{28} + 2 q^{29} + 17 q^{31} - 34 q^{32} - 3 q^{33} + 6 q^{34} + 8 q^{36} + 18 q^{38} - 5 q^{39} + 4 q^{41} - 6 q^{42} + 17 q^{43} + 8 q^{44} - 30 q^{46} - 10 q^{47} - 10 q^{48} + 2 q^{49} + 4 q^{51} + 30 q^{52} - 6 q^{53} + 2 q^{54} - 22 q^{56} + q^{57} + 24 q^{58} - 6 q^{59} - 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} + 2 q^{66} + 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} + 14 q^{74} - 24 q^{76} + 3 q^{77} + 8 q^{78} + 6 q^{79} + 3 q^{81} + 10 q^{82} + 6 q^{83} - 20 q^{84} + 28 q^{86} - 2 q^{87} - 6 q^{88} + 2 q^{89} + 17 q^{91} + 26 q^{92} - 17 q^{93} + 2 q^{94} + 34 q^{96} + 29 q^{97} + 24 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 3 * q^3 + 8 * q^4 + 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 3 * q^11 - 8 * q^12 + 5 * q^13 + 6 * q^14 + 10 * q^16 - 4 * q^17 - 2 * q^18 - q^19 - 3 * q^21 - 2 * q^22 + 6 * q^24 - 8 * q^26 - 3 * q^27 + 20 * q^28 + 2 * q^29 + 17 * q^31 - 34 * q^32 - 3 * q^33 + 6 * q^34 + 8 * q^36 + 18 * q^38 - 5 * q^39 + 4 * q^41 - 6 * q^42 + 17 * q^43 + 8 * q^44 - 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 + 2 * q^66 + 7 * q^67 + 18 * q^68 + 26 * q^71 - 6 * q^72 - 10 * q^73 + 14 * q^74 - 24 * q^76 + 3 * q^77 + 8 * q^78 + 6 * q^79 + 3 * q^81 + 10 * q^82 + 6 * q^83 - 20 * q^84 + 28 * q^86 - 2 * q^87 - 6 * q^88 + 2 * q^89 + 17 * q^91 + 26 * q^92 - 17 * q^93 + 2 * q^94 + 34 * q^96 + 29 * q^97 + 24 * q^98 + 3 * q^99

## 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
$$11$$ 1.00000 0.301511
$$12$$ −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 $$-0.403227\pi$$
0.299359 + 0.954141i $$0.403227\pi$$
$$20$$ 0 0
$$21$$ −3.64002 −0.794318
$$22$$ 2.12489 0.453027
$$23$$ −5.73463 −1.19575 −0.597877 0.801588i $$-0.703989\pi$$
−0.597877 + 0.801588i $$0.703989\pi$$
$$24$$ −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 $$-0.302944\pi$$
0.580277 + 0.814419i $$0.302944\pi$$
$$30$$ 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
$$33$$ −1.00000 −0.174078
$$34$$ 2.45459 0.420958
$$35$$ 0 0
$$36$$ 2.51514 0.419190
$$37$$ 0.454586 0.0747335 0.0373667 0.999302i $$-0.488103\pi$$
0.0373667 + 0.999302i $$0.488103\pi$$
$$38$$ 5.54541 0.899585
$$39$$ −1.51514 −0.242616
$$40$$ 0 0
$$41$$ 4.12489 0.644199 0.322099 0.946706i $$-0.395611\pi$$
0.322099 + 0.946706i $$0.395611\pi$$
$$42$$ −7.73463 −1.19348
$$43$$ 11.7044 1.78490 0.892449 0.451149i $$-0.148986\pi$$
0.892449 + 0.451149i $$0.148986\pi$$
$$44$$ 2.51514 0.379171
$$45$$ 0 0
$$46$$ −12.1854 −1.79664
$$47$$ −3.48486 −0.508319 −0.254160 0.967162i $$-0.581799\pi$$
−0.254160 + 0.967162i $$0.581799\pi$$
$$48$$ 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 $$-0.831185\pi$$
−0.862631 + 0.505834i $$0.831185\pi$$
$$54$$ −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 $$-0.779326\pi$$
−0.769161 + 0.639055i $$0.779326\pi$$
$$62$$ 11.7190 1.48832
$$63$$ 3.64002 0.458600
$$64$$ −11.4537 −1.43171
$$65$$ 0 0
$$66$$ −2.12489 −0.261555
$$67$$ −14.2645 −1.74268 −0.871340 0.490680i $$-0.836749\pi$$
−0.871340 + 0.490680i $$0.836749\pi$$
$$68$$ 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
$$77$$ 3.64002 0.414819
$$78$$ −3.21949 −0.364536
$$79$$ 5.09461 0.573188 0.286594 0.958052i $$-0.407477\pi$$
0.286594 + 0.958052i $$0.407477\pi$$
$$80$$ 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
$$88$$ 1.09461 0.116686
$$89$$ −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 $$-0.611844\pi$$
−0.344181 + 0.938903i $$0.611844\pi$$
$$98$$ 13.2800 1.34149
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 7.40493 0.736818 0.368409 0.929664i $$-0.379903\pi$$
0.368409 + 0.929664i $$0.379903\pi$$
$$102$$ −2.45459 −0.243040
$$103$$ −16.4995 −1.62575 −0.812874 0.582439i $$-0.802098\pi$$
−0.812874 + 0.582439i $$0.802098\pi$$
$$104$$ 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 $$-0.395467\pi$$
0.322530 + 0.946559i $$0.395467\pi$$
$$110$$ 0 0
$$111$$ −0.454586 −0.0431474
$$112$$ −9.84392 −0.930163
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −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
$$121$$ 1.00000 0.0909091
$$122$$ −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 $$-0.689046\pi$$
−0.559602 + 0.828762i $$0.689046\pi$$
$$132$$ −2.51514 −0.218915
$$133$$ 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 $$-0.0721952\pi$$
0.974389 + 0.224868i $$0.0721952\pi$$
$$138$$ 12.1854 1.03729
$$139$$ 7.59037 0.643807 0.321903 0.946773i $$-0.395677\pi$$
0.321903 + 0.946773i $$0.395677\pi$$
$$140$$ 0 0
$$141$$ 3.48486 0.293478
$$142$$ 18.0937 1.51839
$$143$$ 1.51514 0.126702
$$144$$ −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 $$-0.523681\pi$$
−0.0743274 + 0.997234i $$0.523681\pi$$
$$150$$ 0 0
$$151$$ 24.3250 1.97954 0.989770 0.142670i $$-0.0455687\pi$$
0.989770 + 0.142670i $$0.0455687\pi$$
$$152$$ 2.85665 0.231705
$$153$$ 1.15516 0.0933893
$$154$$ 7.73463 0.623274
$$155$$ 0 0
$$156$$ −3.81078 −0.305107
$$157$$ 9.76491 0.779325 0.389662 0.920958i $$-0.372592\pi$$
0.389662 + 0.920958i $$0.372592\pi$$
$$158$$ 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 $$-0.588191\pi$$
−0.273530 + 0.961863i $$0.588191\pi$$
$$164$$ 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
$$176$$ −2.70436 −0.203849
$$177$$ 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
$$187$$ 1.15516 0.0844738
$$188$$ −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 $$0.543389\pi$$
$$198$$ 2.12489 0.151009
$$199$$ −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
$$209$$ 2.60975 0.180520
$$210$$ 0 0
$$211$$ 10.2645 0.706634 0.353317 0.935504i $$-0.385054\pi$$
0.353317 + 0.935504i $$0.385054\pi$$
$$212$$ −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 $$-0.357553\pi$$
0.432723 + 0.901527i $$0.357553\pi$$
$$224$$ −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
$$231$$ −3.64002 −0.239496
$$232$$ 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 $$-0.661219\pi$$
−0.485105 + 0.874456i $$0.661219\pi$$
$$240$$ 0 0
$$241$$ 5.04496 0.324974 0.162487 0.986711i $$-0.448048\pi$$
0.162487 + 0.986711i $$0.448048\pi$$
$$242$$ 2.12489 0.136593
$$243$$ −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
$$253$$ −5.73463 −0.360533
$$254$$ 17.1358 1.07519
$$255$$ 0 0
$$256$$ 4.91721 0.307325
$$257$$ −13.6509 −0.851521 −0.425761 0.904836i $$-0.639993\pi$$
−0.425761 + 0.904836i $$0.639993\pi$$
$$258$$ −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 $$0.373427\pi$$
$$264$$ −1.09461 −0.0673685
$$265$$ 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 $$-0.426205\pi$$
0.229764 + 0.973246i $$0.426205\pi$$
$$272$$ −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 $$-0.517812\pi$$
−0.0559300 + 0.998435i $$0.517812\pi$$
$$282$$ 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
$$286$$ 3.21949 0.190373
$$287$$ 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
$$297$$ −1.00000 −0.0580259
$$298$$ −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 $$0.207534\pi$$
$$308$$ 9.15516 0.521664
$$309$$ 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 $$-0.204795\pi$$
0.800071 + 0.599905i $$0.204795\pi$$
$$314$$ 20.7493 1.17095
$$315$$ 0 0
$$316$$ 12.8136 0.720824
$$317$$ −8.80986 −0.494811 −0.247406 0.968912i $$-0.579578\pi$$
−0.247406 + 0.968912i $$0.579578\pi$$
$$318$$ 26.6888 1.49663
$$319$$ 6.24977 0.349920
$$320$$ 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
$$341$$ 5.51514 0.298661
$$342$$ 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 $$-0.714117\pi$$
−0.623076 + 0.782161i $$0.714117\pi$$
$$350$$ 0 0
$$351$$ −1.51514 −0.0808721
$$352$$ −7.93567 −0.422972
$$353$$ −9.75023 −0.518952 −0.259476 0.965750i $$-0.583550\pi$$
−0.259476 + 0.965750i $$0.583550\pi$$
$$354$$ 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 $$-0.147457\pi$$
0.894605 + 0.446858i $$0.147457\pi$$
$$360$$ 0 0
$$361$$ −12.1892 −0.641538
$$362$$ −49.0634 −2.57872
$$363$$ −1.00000 −0.0524864
$$364$$ 13.8713 0.727055
$$365$$ 0 0
$$366$$ 25.5298 1.33446
$$367$$ 1.88601 0.0984491 0.0492245 0.998788i $$-0.484325\pi$$
0.0492245 + 0.998788i $$0.484325\pi$$
$$368$$ 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 $$0.638896\pi$$
$$374$$ 2.45459 0.126924
$$375$$ 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 $$-0.396788\pi$$
0.318598 + 0.947890i $$0.396788\pi$$
$$384$$ 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
$$396$$ 2.51514 0.126390
$$397$$ −15.2342 −0.764581 −0.382291 0.924042i $$-0.624865\pi$$
−0.382291 + 0.924042i $$0.624865\pi$$
$$398$$ −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
$$407$$ 0.454586 0.0225330
$$408$$ −1.26445 −0.0625996
$$409$$ −3.98532 −0.197061 −0.0985307 0.995134i $$-0.531414\pi$$
−0.0985307 + 0.995134i $$0.531414\pi$$
$$410$$ 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
$$418$$ 5.54541 0.271235
$$419$$ −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
$$429$$ −1.51514 −0.0731516
$$430$$ 0 0
$$431$$ 22.7493 1.09580 0.547898 0.836545i $$-0.315428\pi$$
0.547898 + 0.836545i $$0.315428\pi$$
$$432$$ 2.70436 0.130113
$$433$$ 7.58325 0.364428 0.182214 0.983259i $$-0.441674\pi$$
0.182214 + 0.983259i $$0.441674\pi$$
$$434$$ 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 $$-0.363783\pi$$
0.414996 + 0.909823i $$0.363783\pi$$
$$440$$ 0 0
$$441$$ 6.24977 0.297608
$$442$$ 3.71904 0.176897
$$443$$ 10.6438 0.505702 0.252851 0.967505i $$-0.418632\pi$$
0.252851 + 0.967505i $$0.418632\pi$$
$$444$$ −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
$$451$$ 4.12489 0.194233
$$452$$ −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 $$-0.438923\pi$$
0.190705 + 0.981647i $$0.438923\pi$$
$$462$$ −7.73463 −0.359848
$$463$$ 16.0899 0.747762 0.373881 0.927477i $$-0.378027\pi$$
0.373881 + 0.927477i $$0.378027\pi$$
$$464$$ −16.9016 −0.784638
$$465$$ 0 0
$$466$$ 10.5521 0.488815
$$467$$ −29.4693 −1.36367 −0.681837 0.731504i $$-0.738819\pi$$
−0.681837 + 0.731504i $$0.738819\pi$$
$$468$$ 3.81078 0.176153
$$469$$ −51.9229 −2.39758
$$470$$ 0 0
$$471$$ −9.76491 −0.449943
$$472$$ −8.46640 −0.389698
$$473$$ 11.7044 0.538167
$$474$$ −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 $$-0.236688\pi$$
0.736052 + 0.676925i $$0.236688\pi$$
$$480$$ 0 0
$$481$$ 0.688760 0.0314048
$$482$$ 10.7200 0.488280
$$483$$ 20.8742 0.949809
$$484$$ 2.51514 0.114324
$$485$$ 0 0
$$486$$ −2.12489 −0.0963868
$$487$$ −35.8936 −1.62649 −0.813247 0.581919i $$-0.802302\pi$$
−0.813247 + 0.581919i $$0.802302\pi$$
$$488$$ −13.1514 −0.595335
$$489$$ 6.98440 0.315846
$$490$$ 0 0
$$491$$ −7.15894 −0.323079 −0.161539 0.986866i $$-0.551646\pi$$
−0.161539 + 0.986866i $$0.551646\pi$$
$$492$$ −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
$$506$$ −12.1854 −0.541709
$$507$$ 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
$$517$$ −3.48486 −0.153264
$$518$$ 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
$$528$$ 2.70436 0.117692
$$529$$ 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
$$539$$ 6.24977 0.269197
$$540$$ 0 0
$$541$$ −11.2947 −0.485598 −0.242799 0.970077i $$-0.578066\pi$$
−0.242799 + 0.970077i $$0.578066\pi$$
$$542$$ 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
$$561$$ −1.15516 −0.0487710
$$562$$ −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 $$-0.409340\pi$$
0.280982 + 0.959713i $$0.409340\pi$$
$$570$$ 0 0
$$571$$ 26.8851 1.12511 0.562553 0.826761i $$-0.309819\pi$$
0.562553 + 0.826761i $$0.309819\pi$$
$$572$$ 3.81078 0.159337
$$573$$ 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 $$-0.486544\pi$$
0.0422607 + 0.999107i $$0.486544\pi$$
$$578$$ −33.2876 −1.38458
$$579$$ −11.7649 −0.488933
$$580$$ 0 0
$$581$$ −53.6878 −2.22735
$$582$$ 14.4058 0.597142
$$583$$ −12.5601 −0.520186
$$584$$ −10.0917 −0.417599
$$585$$ 0 0
$$586$$ −61.9514 −2.55919
$$587$$ 21.8245 0.900795 0.450398 0.892828i $$-0.351282\pi$$
0.450398 + 0.892828i $$0.351282\pi$$
$$588$$ −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 $$0.447025\pi$$
$$594$$ −2.12489 −0.0871851
$$595$$ 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 $$-0.476766\pi$$
0.0729277 + 0.997337i $$0.476766\pi$$
$$602$$ 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
$$616$$ 3.98440 0.160536
$$617$$ 15.9612 0.642576 0.321288 0.946982i $$-0.395884\pi$$
0.321288 + 0.946982i $$0.395884\pi$$
$$618$$ 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
$$627$$ −2.60975 −0.104223
$$628$$ 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
$$638$$ 13.2800 0.525762
$$639$$ 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 $$-0.419946\pi$$
0.248855 + 0.968541i $$0.419946\pi$$
$$644$$ −52.5015 −2.06885
$$645$$ 0 0
$$646$$ 6.40585 0.252035
$$647$$ 29.6429 1.16538 0.582691 0.812694i $$-0.302000\pi$$
0.582691 + 0.812694i $$0.302000\pi$$
$$648$$ 1.09461 0.0430003
$$649$$ −7.73463 −0.303611
$$650$$ 0 0
$$651$$ −20.0752 −0.786810
$$652$$ −17.5667 −0.687967
$$653$$ 9.90069 0.387444 0.193722 0.981056i $$-0.437944\pi$$
0.193722 + 0.981056i $$0.437944\pi$$
$$654$$ −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 $$-0.532793\pi$$
−0.102841 + 0.994698i $$0.532793\pi$$
$$660$$ 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
$$671$$ −12.0147 −0.463822
$$672$$ 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
$$682$$ 11.7190 0.448745
$$683$$ −15.6353 −0.598269 −0.299135 0.954211i $$-0.596698\pi$$
−0.299135 + 0.954211i $$0.596698\pi$$
$$684$$ 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
$$693$$ 3.64002 0.138273
$$694$$ −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 $$-0.480481\pi$$
0.0612824 + 0.998120i $$0.480481\pi$$
$$702$$ −3.21949 −0.121512
$$703$$ 1.18635 0.0447442
$$704$$ −11.4537 −0.431676
$$705$$ 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
$$726$$ −2.12489 −0.0788619
$$727$$ −19.9154 −0.738620 −0.369310 0.929306i $$-0.620406\pi$$
−0.369310 + 0.929306i $$0.620406\pi$$
$$728$$ 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
$$737$$ −14.2645 −0.525438
$$738$$ 8.76491 0.322641
$$739$$ −2.25355 −0.0828982 −0.0414491 0.999141i $$-0.513197\pi$$
−0.0414491 + 0.999141i $$0.513197\pi$$
$$740$$ 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
$$748$$ 2.90539 0.106232
$$749$$ 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 $$-0.347176\pi$$
0.461877 + 0.886944i $$0.347176\pi$$
$$758$$ −55.3406 −2.01006
$$759$$ 5.73463 0.208154
$$760$$ 0 0
$$761$$ −30.7493 −1.11466 −0.557331 0.830291i $$-0.688174\pi$$
−0.557331 + 0.830291i $$0.688174\pi$$
$$762$$ −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 $$-0.594926\pi$$
−0.293818 + 0.955861i $$0.594926\pi$$
$$770$$ 0 0
$$771$$ 13.6509 0.491626
$$772$$ 29.5904 1.06498
$$773$$ 48.7787 1.75445 0.877223 0.480082i $$-0.159393\pi$$
0.877223 + 0.480082i $$0.159393\pi$$
$$774$$ 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
$$781$$ 8.51514 0.304696
$$782$$ −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
$$792$$ 1.09461 0.0388952
$$793$$ −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 $$-0.277151\pi$$
0.644295 + 0.764777i $$0.277151\pi$$
$$798$$ −20.1854 −0.714557
$$799$$ −4.02558 −0.142415
$$800$$ 0 0
$$801$$ −10.4995 −0.370983
$$802$$ 5.84197 0.206287
$$803$$ −9.21949 −0.325349
$$804$$ 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 $$-0.435925\pi$$
0.199940 + 0.979808i $$0.435925\pi$$
$$810$$ 0 0
$$811$$ −13.3903 −0.470195 −0.235098 0.971972i $$-0.575541\pi$$
−0.235098 + 0.971972i $$0.575541\pi$$
$$812$$ 57.2177 2.00795
$$813$$ −7.56479 −0.265309
$$814$$ 0.965943 0.0338563
$$815$$ 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 $$-0.310761\pi$$
0.560105 + 0.828422i $$0.310761\pi$$
$$822$$ −48.4683 −1.69053
$$823$$ 16.7952 0.585443 0.292722 0.956198i $$-0.405439\pi$$
0.292722 + 0.956198i $$0.405439\pi$$
$$824$$ −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
$$836$$ 6.56387 0.227016
$$837$$ −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
$$847$$ 3.64002 0.125073
$$848$$ 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 $$0.938611\pi$$
$$858$$ −3.21949 −0.109912
$$859$$ 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 $$-0.762792\pi$$
−0.734945 + 0.678127i $$0.762792\pi$$
$$864$$ 7.93567 0.269977
$$865$$ 0 0
$$866$$ 16.1135 0.547560
$$867$$ 15.6656 0.532032
$$868$$ 50.4920 1.71381
$$869$$ 5.09461 0.172823
$$870$$ 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 $$-0.363191\pi$$
0.416687 + 0.909050i $$0.363191\pi$$
$$884$$ 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
$$891$$ 1.00000 0.0335013
$$892$$ 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
$$902$$ 8.76491 0.291840
$$903$$ −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 $$-0.878955\pi$$
−0.928563 + 0.371174i $$0.878955\pi$$
$$908$$ 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
$$913$$ −14.7493 −0.488131
$$914$$ −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 $$-0.820669\pi$$
−0.845452 + 0.534052i $$0.820669\pi$$
$$920$$ 0 0
$$921$$ −27.8548 −0.917848
$$922$$ 17.4012 0.573076
$$923$$ 12.9016 0.424662
$$924$$ −9.15516 −0.301183
$$925$$ 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 $$-0.799371\pi$$
−0.807853 + 0.589384i $$0.799371\pi$$
$$942$$ −20.7493 −0.676049
$$943$$ −23.6547 −0.770303
$$944$$ 20.9172 0.680797
$$945$$ 0 0
$$946$$ 24.8704 0.808607
$$947$$ −13.9844 −0.454432 −0.227216 0.973844i $$-0.572962\pi$$
−0.227216 + 0.973844i $$0.572962\pi$$
$$948$$ −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
$$957$$ −6.24977 −0.202026
$$958$$ 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 $$0.490232\pi$$
$$968$$ 1.09461 0.0351821
$$969$$ −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 $$-0.269130\pi$$
0.663360 + 0.748301i $$0.269130\pi$$
$$978$$ 14.8411 0.474565
$$979$$ −10.4995 −0.335567
$$980$$ 0 0
$$981$$ 6.73463 0.215020
$$982$$ −15.2119 −0.485432
$$983$$ −6.23601 −0.198898 −0.0994489 0.995043i $$-0.531708\pi$$
−0.0994489 + 0.995043i $$0.531708\pi$$
$$984$$ −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 825.2.a.i.1.3 3
3.2 odd 2 2475.2.a.bd.1.1 3
5.2 odd 4 825.2.c.f.199.5 6
5.3 odd 4 825.2.c.f.199.2 6
5.4 even 2 825.2.a.m.1.1 yes 3
11.10 odd 2 9075.2.a.cj.1.1 3
15.2 even 4 2475.2.c.q.199.2 6
15.8 even 4 2475.2.c.q.199.5 6
15.14 odd 2 2475.2.a.z.1.3 3
55.54 odd 2 9075.2.a.cd.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 1.1 even 1 trivial
825.2.a.m.1.1 yes 3 5.4 even 2
825.2.c.f.199.2 6 5.3 odd 4
825.2.c.f.199.5 6 5.2 odd 4
2475.2.a.z.1.3 3 15.14 odd 2
2475.2.a.bd.1.1 3 3.2 odd 2
2475.2.c.q.199.2 6 15.2 even 4
2475.2.c.q.199.5 6 15.8 even 4
9075.2.a.cd.1.3 3 55.54 odd 2
9075.2.a.cj.1.1 3 11.10 odd 2