# Properties

 Label 825.2.c.f.199.5 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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, 1, 0]))

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.5 Root $$1.32001 + 1.32001i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.f.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.12489i q^{2} +1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} +3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+2.12489i q^{2} +1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} +3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.51514i q^{12} -1.51514i q^{13} -7.73463 q^{14} -2.70436 q^{16} +1.15516i q^{17} -2.12489i q^{18} -2.60975 q^{19} -3.64002 q^{21} +2.12489i q^{22} +5.73463i q^{23} +1.09461 q^{24} +3.21949 q^{26} -1.00000i q^{27} -9.15516i q^{28} -6.24977 q^{29} +5.51514 q^{31} -7.93567i q^{32} +1.00000i q^{33} -2.45459 q^{34} +2.51514 q^{36} +0.454586i q^{37} -5.54541i q^{38} +1.51514 q^{39} +4.12489 q^{41} -7.73463i q^{42} -11.7044i q^{43} -2.51514 q^{44} -12.1854 q^{46} -3.48486i q^{47} -2.70436i q^{48} -6.24977 q^{49} -1.15516 q^{51} +3.81078i q^{52} +12.5601i q^{53} +2.12489 q^{54} +3.98440 q^{56} -2.60975i q^{57} -13.2800i q^{58} +7.73463 q^{59} -12.0147 q^{61} +11.7190i q^{62} -3.64002i q^{63} +11.4537 q^{64} -2.12489 q^{66} -14.2645i q^{67} -2.90539i q^{68} -5.73463 q^{69} +8.51514 q^{71} +1.09461i q^{72} +9.21949i q^{73} -0.965943 q^{74} +6.56387 q^{76} +3.64002i q^{77} +3.21949i q^{78} -5.09461 q^{79} +1.00000 q^{81} +8.76491i q^{82} +14.7493i q^{83} +9.15516 q^{84} +24.8704 q^{86} -6.24977i q^{87} -1.09461i q^{88} +10.4995 q^{89} +5.51514 q^{91} -14.4234i q^{92} +5.51514i q^{93} +7.40493 q^{94} +7.93567 q^{96} -6.77959i q^{97} -13.2800i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.12489i 1.50252i 0.660006 + 0.751260i $$0.270554\pi$$
−0.660006 + 0.751260i $$0.729446\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.51514 −1.25757
$$5$$ 0 0
$$6$$ −2.12489 −0.867481
$$7$$ 3.64002i 1.37580i 0.725806 + 0.687900i $$0.241467\pi$$
−0.725806 + 0.687900i $$0.758533\pi$$
$$8$$ − 1.09461i − 0.387003i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 2.51514i − 0.726058i
$$13$$ − 1.51514i − 0.420224i −0.977677 0.210112i $$-0.932617\pi$$
0.977677 0.210112i $$-0.0673828\pi$$
$$14$$ −7.73463 −2.06717
$$15$$ 0 0
$$16$$ −2.70436 −0.676089
$$17$$ 1.15516i 0.280168i 0.990140 + 0.140084i $$0.0447372\pi$$
−0.990140 + 0.140084i $$0.955263\pi$$
$$18$$ − 2.12489i − 0.500840i
$$19$$ −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
$$22$$ 2.12489i 0.453027i
$$23$$ 5.73463i 1.19575i 0.801588 + 0.597877i $$0.203989\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$24$$ 1.09461 0.223436
$$25$$ 0 0
$$26$$ 3.21949 0.631395
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 9.15516i − 1.73016i
$$29$$ −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 $$0.335068\pi$$
$$32$$ − 7.93567i − 1.40284i
$$33$$ 1.00000i 0.174078i
$$34$$ −2.45459 −0.420958
$$35$$ 0 0
$$36$$ 2.51514 0.419190
$$37$$ 0.454586i 0.0747335i 0.999302 + 0.0373667i $$0.0118970\pi$$
−0.999302 + 0.0373667i $$0.988103\pi$$
$$38$$ − 5.54541i − 0.899585i
$$39$$ 1.51514 0.242616
$$40$$ 0 0
$$41$$ 4.12489 0.644199 0.322099 0.946706i $$-0.395611\pi$$
0.322099 + 0.946706i $$0.395611\pi$$
$$42$$ − 7.73463i − 1.19348i
$$43$$ − 11.7044i − 1.78490i −0.451149 0.892449i $$-0.648986\pi$$
0.451149 0.892449i $$-0.351014\pi$$
$$44$$ −2.51514 −0.379171
$$45$$ 0 0
$$46$$ −12.1854 −1.79664
$$47$$ − 3.48486i − 0.508319i −0.967162 0.254160i $$-0.918201\pi$$
0.967162 0.254160i $$-0.0817989\pi$$
$$48$$ − 2.70436i − 0.390340i
$$49$$ −6.24977 −0.892824
$$50$$ 0 0
$$51$$ −1.15516 −0.161755
$$52$$ 3.81078i 0.528460i
$$53$$ 12.5601i 1.72526i 0.505834 + 0.862631i $$0.331185\pi$$
−0.505834 + 0.862631i $$0.668815\pi$$
$$54$$ 2.12489 0.289160
$$55$$ 0 0
$$56$$ 3.98440 0.532438
$$57$$ − 2.60975i − 0.345669i
$$58$$ − 13.2800i − 1.74376i
$$59$$ 7.73463 1.00696 0.503482 0.864006i $$-0.332052\pi$$
0.503482 + 0.864006i $$0.332052\pi$$
$$60$$ 0 0
$$61$$ −12.0147 −1.53832 −0.769161 0.639055i $$-0.779326\pi$$
−0.769161 + 0.639055i $$0.779326\pi$$
$$62$$ 11.7190i 1.48832i
$$63$$ − 3.64002i − 0.458600i
$$64$$ 11.4537 1.43171
$$65$$ 0 0
$$66$$ −2.12489 −0.261555
$$67$$ − 14.2645i − 1.74268i −0.490680 0.871340i $$-0.663251\pi$$
0.490680 0.871340i $$-0.336749\pi$$
$$68$$ − 2.90539i − 0.352330i
$$69$$ −5.73463 −0.690369
$$70$$ 0 0
$$71$$ 8.51514 1.01056 0.505280 0.862955i $$-0.331389\pi$$
0.505280 + 0.862955i $$0.331389\pi$$
$$72$$ 1.09461i 0.129001i
$$73$$ 9.21949i 1.07906i 0.841966 + 0.539530i $$0.181398\pi$$
−0.841966 + 0.539530i $$0.818602\pi$$
$$74$$ −0.965943 −0.112289
$$75$$ 0 0
$$76$$ 6.56387 0.752928
$$77$$ 3.64002i 0.414819i
$$78$$ 3.21949i 0.364536i
$$79$$ −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
$$82$$ 8.76491i 0.967922i
$$83$$ 14.7493i 1.61895i 0.587156 + 0.809474i $$0.300247\pi$$
−0.587156 + 0.809474i $$0.699753\pi$$
$$84$$ 9.15516 0.998910
$$85$$ 0 0
$$86$$ 24.8704 2.68185
$$87$$ − 6.24977i − 0.670046i
$$88$$ − 1.09461i − 0.116686i
$$89$$ 10.4995 1.11295 0.556475 0.830865i $$-0.312154\pi$$
0.556475 + 0.830865i $$0.312154\pi$$
$$90$$ 0 0
$$91$$ 5.51514 0.578144
$$92$$ − 14.4234i − 1.50374i
$$93$$ 5.51514i 0.571893i
$$94$$ 7.40493 0.763760
$$95$$ 0 0
$$96$$ 7.93567 0.809931
$$97$$ − 6.77959i − 0.688363i −0.938903 0.344181i $$-0.888156\pi$$
0.938903 0.344181i $$-0.111844\pi$$
$$98$$ − 13.2800i − 1.34149i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 7.40493 0.736818 0.368409 0.929664i $$-0.379903\pi$$
0.368409 + 0.929664i $$0.379903\pi$$
$$102$$ − 2.45459i − 0.243040i
$$103$$ 16.4995i 1.62575i 0.582439 + 0.812874i $$0.302098\pi$$
−0.582439 + 0.812874i $$0.697902\pi$$
$$104$$ −1.65848 −0.162628
$$105$$ 0 0
$$106$$ −26.6888 −2.59224
$$107$$ 3.93945i 0.380841i 0.981703 + 0.190420i $$0.0609851\pi$$
−0.981703 + 0.190420i $$0.939015\pi$$
$$108$$ 2.51514i 0.242019i
$$109$$ −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
$$112$$ − 9.84392i − 0.930163i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 5.54541 0.519376
$$115$$ 0 0
$$116$$ 15.7190 1.45948
$$117$$ 1.51514i 0.140075i
$$118$$ 16.4352i 1.51298i
$$119$$ −4.20482 −0.385455
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 25.5298i − 2.31136i
$$123$$ 4.12489i 0.371928i
$$124$$ −13.8713 −1.24568
$$125$$ 0 0
$$126$$ 7.73463 0.689056
$$127$$ 8.06433i 0.715594i 0.933799 + 0.357797i $$0.116472\pi$$
−0.933799 + 0.357797i $$0.883528\pi$$
$$128$$ 8.46640i 0.748331i
$$129$$ 11.7044 1.03051
$$130$$ 0 0
$$131$$ −12.8099 −1.11920 −0.559602 0.828762i $$-0.689046\pi$$
−0.559602 + 0.828762i $$0.689046\pi$$
$$132$$ − 2.51514i − 0.218915i
$$133$$ − 9.49954i − 0.823715i
$$134$$ 30.3103 2.61841
$$135$$ 0 0
$$136$$ 1.26445 0.108426
$$137$$ 22.8099i 1.94878i 0.224868 + 0.974389i $$0.427805\pi$$
−0.224868 + 0.974389i $$0.572195\pi$$
$$138$$ − 12.1854i − 1.03729i
$$139$$ −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
$$142$$ 18.0937i 1.51839i
$$143$$ − 1.51514i − 0.126702i
$$144$$ 2.70436 0.225363
$$145$$ 0 0
$$146$$ −19.5904 −1.62131
$$147$$ − 6.24977i − 0.515472i
$$148$$ − 1.14335i − 0.0939825i
$$149$$ 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 $$-0.0455687\pi$$
0.989770 + 0.142670i $$0.0455687\pi$$
$$152$$ 2.85665i 0.231705i
$$153$$ − 1.15516i − 0.0933893i
$$154$$ −7.73463 −0.623274
$$155$$ 0 0
$$156$$ −3.81078 −0.305107
$$157$$ 9.76491i 0.779325i 0.920958 + 0.389662i $$0.127408\pi$$
−0.920958 + 0.389662i $$0.872592\pi$$
$$158$$ − 10.8255i − 0.861227i
$$159$$ −12.5601 −0.996080
$$160$$ 0 0
$$161$$ −20.8742 −1.64512
$$162$$ 2.12489i 0.166947i
$$163$$ 6.98440i 0.547061i 0.961863 + 0.273530i $$0.0881914\pi$$
−0.961863 + 0.273530i $$0.911809\pi$$
$$164$$ −10.3747 −0.810125
$$165$$ 0 0
$$166$$ −31.3406 −2.43250
$$167$$ − 6.31032i − 0.488307i −0.969737 0.244154i $$-0.921490\pi$$
0.969737 0.244154i $$-0.0785102\pi$$
$$168$$ 3.98440i 0.307403i
$$169$$ 10.7044 0.823412
$$170$$ 0 0
$$171$$ 2.60975 0.199572
$$172$$ 29.4381i 2.24463i
$$173$$ 12.8448i 0.976575i 0.872683 + 0.488287i $$0.162378\pi$$
−0.872683 + 0.488287i $$0.837622\pi$$
$$174$$ 13.2800 1.00676
$$175$$ 0 0
$$176$$ −2.70436 −0.203849
$$177$$ 7.73463i 0.581371i
$$178$$ 22.3103i 1.67223i
$$179$$ −13.4849 −1.00791 −0.503953 0.863731i $$-0.668122\pi$$
−0.503953 + 0.863731i $$0.668122\pi$$
$$180$$ 0 0
$$181$$ −23.0899 −1.71626 −0.858130 0.513433i $$-0.828374\pi$$
−0.858130 + 0.513433i $$0.828374\pi$$
$$182$$ 11.7190i 0.868673i
$$183$$ − 12.0147i − 0.888151i
$$184$$ 6.27718 0.462760
$$185$$ 0 0
$$186$$ −11.7190 −0.859281
$$187$$ 1.15516i 0.0844738i
$$188$$ 8.76491i 0.639247i
$$189$$ 3.64002 0.264773
$$190$$ 0 0
$$191$$ −7.98440 −0.577731 −0.288866 0.957370i $$-0.593278\pi$$
−0.288866 + 0.957370i $$0.593278\pi$$
$$192$$ 11.4537i 0.826597i
$$193$$ − 11.7649i − 0.846857i −0.905930 0.423428i $$-0.860827\pi$$
0.905930 0.423428i $$-0.139173\pi$$
$$194$$ 14.4058 1.03428
$$195$$ 0 0
$$196$$ 15.7190 1.12279
$$197$$ − 3.81456i − 0.271776i −0.990724 0.135888i $$-0.956611\pi$$
0.990724 0.135888i $$-0.0433888\pi$$
$$198$$ − 2.12489i − 0.151009i
$$199$$ 12.0752 0.855990 0.427995 0.903781i $$-0.359220\pi$$
0.427995 + 0.903781i $$0.359220\pi$$
$$200$$ 0 0
$$201$$ 14.2645 1.00614
$$202$$ 15.7346i 1.10708i
$$203$$ − 22.7493i − 1.59669i
$$204$$ 2.90539 0.203418
$$205$$ 0 0
$$206$$ −35.0596 −2.44272
$$207$$ − 5.73463i − 0.398585i
$$208$$ 4.09747i 0.284109i
$$209$$ −2.60975 −0.180520
$$210$$ 0 0
$$211$$ 10.2645 0.706634 0.353317 0.935504i $$-0.385054\pi$$
0.353317 + 0.935504i $$0.385054\pi$$
$$212$$ − 31.5904i − 2.16964i
$$213$$ 8.51514i 0.583448i
$$214$$ −8.37088 −0.572221
$$215$$ 0 0
$$216$$ −1.09461 −0.0744787
$$217$$ 20.0752i 1.36280i
$$218$$ − 14.3103i − 0.969217i
$$219$$ −9.21949 −0.622996
$$220$$ 0 0
$$221$$ 1.75023 0.117733
$$222$$ − 0.965943i − 0.0648298i
$$223$$ − 12.9239i − 0.865445i −0.901527 0.432723i $$-0.857553\pi$$
0.901527 0.432723i $$-0.142447\pi$$
$$224$$ 28.8860 1.93003
$$225$$ 0 0
$$226$$ −12.7493 −0.848072
$$227$$ 22.8099i 1.51394i 0.653447 + 0.756972i $$0.273322\pi$$
−0.653447 + 0.756972i $$0.726678\pi$$
$$228$$ 6.56387i 0.434703i
$$229$$ −14.7796 −0.976663 −0.488331 0.872658i $$-0.662394\pi$$
−0.488331 + 0.872658i $$0.662394\pi$$
$$230$$ 0 0
$$231$$ −3.64002 −0.239496
$$232$$ 6.84106i 0.449137i
$$233$$ − 4.96594i − 0.325330i −0.986681 0.162665i $$-0.947991\pi$$
0.986681 0.162665i $$-0.0520089\pi$$
$$234$$ −3.21949 −0.210465
$$235$$ 0 0
$$236$$ −19.4537 −1.26633
$$237$$ − 5.09461i − 0.330930i
$$238$$ − 8.93475i − 0.579154i
$$239$$ 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 $$-0.448048\pi$$
0.162487 + 0.986711i $$0.448048\pi$$
$$242$$ 2.12489i 0.136593i
$$243$$ 1.00000i 0.0641500i
$$244$$ 30.2186 1.93455
$$245$$ 0 0
$$246$$ −8.76491 −0.558830
$$247$$ 3.95413i 0.251595i
$$248$$ − 6.03692i − 0.383345i
$$249$$ −14.7493 −0.934700
$$250$$ 0 0
$$251$$ −3.03028 −0.191269 −0.0956347 0.995417i $$-0.530488\pi$$
−0.0956347 + 0.995417i $$0.530488\pi$$
$$252$$ 9.15516i 0.576721i
$$253$$ 5.73463i 0.360533i
$$254$$ −17.1358 −1.07519
$$255$$ 0 0
$$256$$ 4.91721 0.307325
$$257$$ − 13.6509i − 0.851521i −0.904836 0.425761i $$-0.860007\pi$$
0.904836 0.425761i $$-0.139993\pi$$
$$258$$ 24.8704i 1.54836i
$$259$$ −1.65470 −0.102818
$$260$$ 0 0
$$261$$ 6.24977 0.386851
$$262$$ − 27.2195i − 1.68163i
$$263$$ − 12.5601i − 0.774489i −0.921977 0.387244i $$-0.873427\pi$$
0.921977 0.387244i $$-0.126573\pi$$
$$264$$ 1.09461 0.0673685
$$265$$ 0 0
$$266$$ 20.1854 1.23765
$$267$$ 10.4995i 0.642562i
$$268$$ 35.8771i 2.19154i
$$269$$ 24.6888 1.50530 0.752650 0.658421i $$-0.228775\pi$$
0.752650 + 0.658421i $$0.228775\pi$$
$$270$$ 0 0
$$271$$ 7.56479 0.459528 0.229764 0.973246i $$-0.426205\pi$$
0.229764 + 0.973246i $$0.426205\pi$$
$$272$$ − 3.12397i − 0.189418i
$$273$$ 5.51514i 0.333791i
$$274$$ −48.4683 −2.92808
$$275$$ 0 0
$$276$$ 14.4234 0.868186
$$277$$ 1.92477i 0.115648i 0.998327 + 0.0578241i $$0.0184162\pi$$
−0.998327 + 0.0578241i $$0.981584\pi$$
$$278$$ − 16.1287i − 0.967333i
$$279$$ −5.51514 −0.330183
$$280$$ 0 0
$$281$$ −1.87511 −0.111860 −0.0559300 0.998435i $$-0.517812\pi$$
−0.0559300 + 0.998435i $$0.517812\pi$$
$$282$$ 7.40493i 0.440957i
$$283$$ − 30.1396i − 1.79161i −0.444446 0.895806i $$-0.646599\pi$$
0.444446 0.895806i $$-0.353401\pi$$
$$284$$ −21.4167 −1.27085
$$285$$ 0 0
$$286$$ 3.21949 0.190373
$$287$$ 15.0147i 0.886289i
$$288$$ 7.93567i 0.467614i
$$289$$ 15.6656 0.921506
$$290$$ 0 0
$$291$$ 6.77959 0.397427
$$292$$ − 23.1883i − 1.35699i
$$293$$ 29.1552i 1.70326i 0.524141 + 0.851631i $$0.324386\pi$$
−0.524141 + 0.851631i $$0.675614\pi$$
$$294$$ 13.2800 0.774508
$$295$$ 0 0
$$296$$ 0.497594 0.0289221
$$297$$ − 1.00000i − 0.0580259i
$$298$$ 3.85574i 0.223357i
$$299$$ 8.68876 0.502484
$$300$$ 0 0
$$301$$ 42.6041 2.45566
$$302$$ 51.6878i 2.97430i
$$303$$ 7.40493i 0.425402i
$$304$$ 7.05769 0.404786
$$305$$ 0 0
$$306$$ 2.45459 0.140319
$$307$$ 27.8548i 1.58976i 0.606768 + 0.794879i $$0.292466\pi$$
−0.606768 + 0.794879i $$0.707534\pi$$
$$308$$ − 9.15516i − 0.521664i
$$309$$ −16.4995 −0.938626
$$310$$ 0 0
$$311$$ 23.9083 1.35571 0.677856 0.735194i $$-0.262909\pi$$
0.677856 + 0.735194i $$0.262909\pi$$
$$312$$ − 1.65848i − 0.0938932i
$$313$$ − 28.3094i − 1.60014i −0.599905 0.800071i $$-0.704795\pi$$
0.599905 0.800071i $$-0.295205\pi$$
$$314$$ −20.7493 −1.17095
$$315$$ 0 0
$$316$$ 12.8136 0.720824
$$317$$ − 8.80986i − 0.494811i −0.968912 0.247406i $$-0.920422\pi$$
0.968912 0.247406i $$-0.0795780\pi$$
$$318$$ − 26.6888i − 1.49663i
$$319$$ −6.24977 −0.349920
$$320$$ 0 0
$$321$$ −3.93945 −0.219879
$$322$$ − 44.3553i − 2.47182i
$$323$$ − 3.01468i − 0.167741i
$$324$$ −2.51514 −0.139730
$$325$$ 0 0
$$326$$ −14.8411 −0.821970
$$327$$ − 6.73463i − 0.372426i
$$328$$ − 4.51514i − 0.249307i
$$329$$ 12.6850 0.699346
$$330$$ 0 0
$$331$$ 32.2498 1.77261 0.886304 0.463104i $$-0.153264\pi$$
0.886304 + 0.463104i $$0.153264\pi$$
$$332$$ − 37.0966i − 2.03594i
$$333$$ − 0.454586i − 0.0249112i
$$334$$ 13.4087 0.733692
$$335$$ 0 0
$$336$$ 9.84392 0.537030
$$337$$ 28.9844i 1.57888i 0.613827 + 0.789441i $$0.289629\pi$$
−0.613827 + 0.789441i $$0.710371\pi$$
$$338$$ 22.7455i 1.23719i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 5.51514 0.298661
$$342$$ 5.54541i 0.299862i
$$343$$ 2.73085i 0.147452i
$$344$$ −12.8117 −0.690760
$$345$$ 0 0
$$346$$ −27.2938 −1.46732
$$347$$ − 35.7190i − 1.91750i −0.284253 0.958749i $$-0.591746\pi$$
0.284253 0.958749i $$-0.408254\pi$$
$$348$$ 15.7190i 0.842629i
$$349$$ 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
$$352$$ − 7.93567i − 0.422972i
$$353$$ 9.75023i 0.518952i 0.965750 + 0.259476i $$0.0835499\pi$$
−0.965750 + 0.259476i $$0.916450\pi$$
$$354$$ −16.4352 −0.873521
$$355$$ 0 0
$$356$$ −26.4078 −1.39961
$$357$$ − 4.20482i − 0.222542i
$$358$$ − 28.6538i − 1.51440i
$$359$$ −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
$$362$$ − 49.0634i − 2.57872i
$$363$$ 1.00000i 0.0524864i
$$364$$ −13.8713 −0.727055
$$365$$ 0 0
$$366$$ 25.5298 1.33446
$$367$$ 1.88601i 0.0984491i 0.998788 + 0.0492245i $$0.0156750\pi$$
−0.998788 + 0.0492245i $$0.984325\pi$$
$$368$$ − 15.5085i − 0.808436i
$$369$$ −4.12489 −0.214733
$$370$$ 0 0
$$371$$ −45.7190 −2.37361
$$372$$ − 13.8713i − 0.719195i
$$373$$ 16.3250i 0.845277i 0.906298 + 0.422638i $$0.138896\pi$$
−0.906298 + 0.422638i $$0.861104\pi$$
$$374$$ −2.45459 −0.126924
$$375$$ 0 0
$$376$$ −3.81456 −0.196721
$$377$$ 9.46927i 0.487692i
$$378$$ 7.73463i 0.397827i
$$379$$ 26.0440 1.33779 0.668896 0.743356i $$-0.266767\pi$$
0.668896 + 0.743356i $$0.266767\pi$$
$$380$$ 0 0
$$381$$ −8.06433 −0.413148
$$382$$ − 16.9659i − 0.868053i
$$383$$ − 12.4702i − 0.637197i −0.947890 0.318598i $$-0.896788\pi$$
0.947890 0.318598i $$-0.103212\pi$$
$$384$$ −8.46640 −0.432049
$$385$$ 0 0
$$386$$ 24.9991 1.27242
$$387$$ 11.7044i 0.594966i
$$388$$ 17.0516i 0.865664i
$$389$$ 18.0899 0.917195 0.458597 0.888644i $$-0.348352\pi$$
0.458597 + 0.888644i $$0.348352\pi$$
$$390$$ 0 0
$$391$$ −6.62443 −0.335012
$$392$$ 6.84106i 0.345526i
$$393$$ − 12.8099i − 0.646172i
$$394$$ 8.10551 0.408350
$$395$$ 0 0
$$396$$ 2.51514 0.126390
$$397$$ − 15.2342i − 0.764581i −0.924042 0.382291i $$-0.875135\pi$$
0.924042 0.382291i $$-0.124865\pi$$
$$398$$ 25.6585i 1.28614i
$$399$$ 9.49954 0.475572
$$400$$ 0 0
$$401$$ 2.74931 0.137294 0.0686471 0.997641i $$-0.478132\pi$$
0.0686471 + 0.997641i $$0.478132\pi$$
$$402$$ 30.3103i 1.51174i
$$403$$ − 8.35620i − 0.416252i
$$404$$ −18.6244 −0.926600
$$405$$ 0 0
$$406$$ 48.3397 2.39906
$$407$$ 0.454586i 0.0225330i
$$408$$ 1.26445i 0.0625996i
$$409$$ 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
$$412$$ − 41.4986i − 2.04449i
$$413$$ 28.1542i 1.38538i
$$414$$ 12.1854 0.598882
$$415$$ 0 0
$$416$$ −12.0236 −0.589507
$$417$$ − 7.59037i − 0.371702i
$$418$$ − 5.54541i − 0.271235i
$$419$$ 5.13578 0.250899 0.125450 0.992100i $$-0.459963\pi$$
0.125450 + 0.992100i $$0.459963\pi$$
$$420$$ 0 0
$$421$$ −8.94657 −0.436029 −0.218014 0.975946i $$-0.569958\pi$$
−0.218014 + 0.975946i $$0.569958\pi$$
$$422$$ 21.8108i 1.06173i
$$423$$ 3.48486i 0.169440i
$$424$$ 13.7484 0.667681
$$425$$ 0 0
$$426$$ −18.0937 −0.876642
$$427$$ − 43.7337i − 2.11642i
$$428$$ − 9.90826i − 0.478934i
$$429$$ 1.51514 0.0731516
$$430$$ 0 0
$$431$$ 22.7493 1.09580 0.547898 0.836545i $$-0.315428\pi$$
0.547898 + 0.836545i $$0.315428\pi$$
$$432$$ 2.70436i 0.130113i
$$433$$ − 7.58325i − 0.364428i −0.983259 0.182214i $$-0.941674\pi$$
0.983259 0.182214i $$-0.0583263\pi$$
$$434$$ −42.6576 −2.04763
$$435$$ 0 0
$$436$$ 16.9385 0.811209
$$437$$ − 14.9659i − 0.715918i
$$438$$ − 19.5904i − 0.936064i
$$439$$ −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
$$442$$ 3.71904i 0.176897i
$$443$$ − 10.6438i − 0.505702i −0.967505 0.252851i $$-0.918632\pi$$
0.967505 0.252851i $$-0.0813683\pi$$
$$444$$ 1.14335 0.0542608
$$445$$ 0 0
$$446$$ 27.4617 1.30035
$$447$$ 1.81456i 0.0858259i
$$448$$ 41.6916i 1.96974i
$$449$$ 26.9310 1.27095 0.635476 0.772121i $$-0.280804\pi$$
0.635476 + 0.772121i $$0.280804\pi$$
$$450$$ 0 0
$$451$$ 4.12489 0.194233
$$452$$ − 15.0908i − 0.709813i
$$453$$ 24.3250i 1.14289i
$$454$$ −48.4683 −2.27473
$$455$$ 0 0
$$456$$ −2.85665 −0.133775
$$457$$ − 15.7796i − 0.738138i −0.929402 0.369069i $$-0.879677\pi$$
0.929402 0.369069i $$-0.120323\pi$$
$$458$$ − 31.4049i − 1.46746i
$$459$$ 1.15516 0.0539183
$$460$$ 0 0
$$461$$ 8.18922 0.381410 0.190705 0.981647i $$-0.438923\pi$$
0.190705 + 0.981647i $$0.438923\pi$$
$$462$$ − 7.73463i − 0.359848i
$$463$$ − 16.0899i − 0.747762i −0.927477 0.373881i $$-0.878027\pi$$
0.927477 0.373881i $$-0.121973\pi$$
$$464$$ 16.9016 0.784638
$$465$$ 0 0
$$466$$ 10.5521 0.488815
$$467$$ − 29.4693i − 1.36367i −0.731504 0.681837i $$-0.761181\pi$$
0.731504 0.681837i $$-0.238819\pi$$
$$468$$ − 3.81078i − 0.176153i
$$469$$ 51.9229 2.39758
$$470$$ 0 0
$$471$$ −9.76491 −0.449943
$$472$$ − 8.46640i − 0.389698i
$$473$$ − 11.7044i − 0.538167i
$$474$$ 10.8255 0.497230
$$475$$ 0 0
$$476$$ 10.5757 0.484736
$$477$$ − 12.5601i − 0.575087i
$$478$$ 31.8713i 1.45776i
$$479$$ −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
$$482$$ 10.7200i 0.488280i
$$483$$ − 20.8742i − 0.949809i
$$484$$ −2.51514 −0.114324
$$485$$ 0 0
$$486$$ −2.12489 −0.0963868
$$487$$ − 35.8936i − 1.62649i −0.581919 0.813247i $$-0.697698\pi$$
0.581919 0.813247i $$-0.302302\pi$$
$$488$$ 13.1514i 0.595335i
$$489$$ −6.98440 −0.315846
$$490$$ 0 0
$$491$$ −7.15894 −0.323079 −0.161539 0.986866i $$-0.551646\pi$$
−0.161539 + 0.986866i $$0.551646\pi$$
$$492$$ − 10.3747i − 0.467726i
$$493$$ − 7.21949i − 0.325150i
$$494$$ −8.40207 −0.378027
$$495$$ 0 0
$$496$$ −14.9149 −0.669699
$$497$$ 30.9953i 1.39033i
$$498$$ − 31.3406i − 1.40441i
$$499$$ −27.0743 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$500$$ 0 0
$$501$$ 6.31032 0.281924
$$502$$ − 6.43899i − 0.287386i
$$503$$ − 26.9991i − 1.20383i −0.798560 0.601915i $$-0.794405\pi$$
0.798560 0.601915i $$-0.205595\pi$$
$$504$$ −3.98440 −0.177479
$$505$$ 0 0
$$506$$ −12.1854 −0.541709
$$507$$ 10.7044i 0.475397i
$$508$$ − 20.2829i − 0.899909i
$$509$$ −15.5904 −0.691031 −0.345515 0.938413i $$-0.612296\pi$$
−0.345515 + 0.938413i $$0.612296\pi$$
$$510$$ 0 0
$$511$$ −33.5592 −1.48457
$$512$$ 27.3813i 1.21009i
$$513$$ 2.60975i 0.115223i
$$514$$ 29.0066 1.27943
$$515$$ 0 0
$$516$$ −29.4381 −1.29594
$$517$$ − 3.48486i − 0.153264i
$$518$$ − 3.51605i − 0.154487i
$$519$$ −12.8448 −0.563826
$$520$$ 0 0
$$521$$ 11.1589 0.488882 0.244441 0.969664i $$-0.421396\pi$$
0.244441 + 0.969664i $$0.421396\pi$$
$$522$$ 13.2800i 0.581252i
$$523$$ 10.5786i 0.462568i 0.972886 + 0.231284i $$0.0742926\pi$$
−0.972886 + 0.231284i $$0.925707\pi$$
$$524$$ 32.2186 1.40748
$$525$$ 0 0
$$526$$ 26.6888 1.16369
$$527$$ 6.37088i 0.277520i
$$528$$ − 2.70436i − 0.117692i
$$529$$ −9.88601 −0.429827
$$530$$ 0 0
$$531$$ −7.73463 −0.335654
$$532$$ 23.8927i 1.03588i
$$533$$ − 6.24977i − 0.270708i
$$534$$ −22.3103 −0.965462
$$535$$ 0 0
$$536$$ −15.6140 −0.674422
$$537$$ − 13.4849i − 0.581915i
$$538$$ 52.4608i 2.26175i
$$539$$ −6.24977 −0.269197
$$540$$ 0 0
$$541$$ −11.2947 −0.485598 −0.242799 0.970077i $$-0.578066\pi$$
−0.242799 + 0.970077i $$0.578066\pi$$
$$542$$ 16.0743i 0.690451i
$$543$$ − 23.0899i − 0.990883i
$$544$$ 9.16698 0.393031
$$545$$ 0 0
$$546$$ −11.7190 −0.501528
$$547$$ 6.09369i 0.260547i 0.991478 + 0.130274i $$0.0415856\pi$$
−0.991478 + 0.130274i $$0.958414\pi$$
$$548$$ − 57.3700i − 2.45072i
$$549$$ 12.0147 0.512774
$$550$$ 0 0
$$551$$ 16.3103 0.694843
$$552$$ 6.27718i 0.267175i
$$553$$ − 18.5445i − 0.788592i
$$554$$ −4.08991 −0.173764
$$555$$ 0 0
$$556$$ 19.0908 0.809631
$$557$$ − 5.90069i − 0.250020i −0.992155 0.125010i $$-0.960104\pi$$
0.992155 0.125010i $$-0.0398964\pi$$
$$558$$ − 11.7190i − 0.496106i
$$559$$ −17.7337 −0.750056
$$560$$ 0 0
$$561$$ −1.15516 −0.0487710
$$562$$ − 3.98440i − 0.168072i
$$563$$ − 3.03028i − 0.127711i −0.997959 0.0638555i $$-0.979660\pi$$
0.997959 0.0638555i $$-0.0203397\pi$$
$$564$$ −8.76491 −0.369069
$$565$$ 0 0
$$566$$ 64.0431 2.69193
$$567$$ 3.64002i 0.152867i
$$568$$ − 9.32075i − 0.391090i
$$569$$ −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 $$-0.309819\pi$$
0.562553 + 0.826761i $$0.309819\pi$$
$$572$$ 3.81078i 0.159337i
$$573$$ − 7.98440i − 0.333553i
$$574$$ −31.9045 −1.33167
$$575$$ 0 0
$$576$$ −11.4537 −0.477236
$$577$$ 2.03028i 0.0845215i 0.999107 + 0.0422607i $$0.0134560\pi$$
−0.999107 + 0.0422607i $$0.986544\pi$$
$$578$$ 33.2876i 1.38458i
$$579$$ 11.7649 0.488933
$$580$$ 0 0
$$581$$ −53.6878 −2.22735
$$582$$ 14.4058i 0.597142i
$$583$$ 12.5601i 0.520186i
$$584$$ 10.0917 0.417599
$$585$$ 0 0
$$586$$ −61.9514 −2.55919
$$587$$ 21.8245i 0.900795i 0.892828 + 0.450398i $$0.148718\pi$$
−0.892828 + 0.450398i $$0.851282\pi$$
$$588$$ 15.7190i 0.648242i
$$589$$ −14.3931 −0.593058
$$590$$ 0 0
$$591$$ 3.81456 0.156910
$$592$$ − 1.22936i − 0.0505265i
$$593$$ − 8.06811i − 0.331318i −0.986183 0.165659i $$-0.947025\pi$$
0.986183 0.165659i $$-0.0529751\pi$$
$$594$$ 2.12489 0.0871851
$$595$$ 0 0
$$596$$ −4.56387 −0.186944
$$597$$ 12.0752i 0.494206i
$$598$$ 18.4626i 0.754993i
$$599$$ −7.61353 −0.311080 −0.155540 0.987830i $$-0.549712\pi$$
−0.155540 + 0.987830i $$0.549712\pi$$
$$600$$ 0 0
$$601$$ 3.57569 0.145855 0.0729277 0.997337i $$-0.476766\pi$$
0.0729277 + 0.997337i $$0.476766\pi$$
$$602$$ 90.5289i 3.68968i
$$603$$ 14.2645i 0.580893i
$$604$$ −61.1807 −2.48941
$$605$$ 0 0
$$606$$ −15.7346 −0.639176
$$607$$ 17.5298i 0.711513i 0.934579 + 0.355757i $$0.115777\pi$$
−0.934579 + 0.355757i $$0.884223\pi$$
$$608$$ 20.7101i 0.839905i
$$609$$ 22.7493 0.921849
$$610$$ 0 0
$$611$$ −5.28005 −0.213608
$$612$$ 2.90539i 0.117443i
$$613$$ − 12.5601i − 0.507297i −0.967296 0.253649i $$-0.918369\pi$$
0.967296 0.253649i $$-0.0816307\pi$$
$$614$$ −59.1883 −2.38865
$$615$$ 0 0
$$616$$ 3.98440 0.160536
$$617$$ 15.9612i 0.642576i 0.946982 + 0.321288i $$0.104116\pi$$
−0.946982 + 0.321288i $$0.895884\pi$$
$$618$$ − 35.0596i − 1.41031i
$$619$$ 9.23417 0.371153 0.185576 0.982630i $$-0.440585\pi$$
0.185576 + 0.982630i $$0.440585\pi$$
$$620$$ 0 0
$$621$$ 5.73463 0.230123
$$622$$ 50.8023i 2.03699i
$$623$$ 38.2186i 1.53119i
$$624$$ −4.09747 −0.164030
$$625$$ 0 0
$$626$$ 60.1542 2.40425
$$627$$ − 2.60975i − 0.104223i
$$628$$ − 24.5601i − 0.980054i
$$629$$ −0.525120 −0.0209379
$$630$$ 0 0
$$631$$ 29.2342 1.16379 0.581897 0.813262i $$-0.302311\pi$$
0.581897 + 0.813262i $$0.302311\pi$$
$$632$$ 5.57661i 0.221826i
$$633$$ 10.2645i 0.407975i
$$634$$ 18.7200 0.743464
$$635$$ 0 0
$$636$$ 31.5904 1.25264
$$637$$ 9.46927i 0.375186i
$$638$$ − 13.2800i − 0.525762i
$$639$$ −8.51514 −0.336854
$$640$$ 0 0
$$641$$ 13.9612 0.551436 0.275718 0.961239i $$-0.411084\pi$$
0.275718 + 0.961239i $$0.411084\pi$$
$$642$$ − 8.37088i − 0.330372i
$$643$$ − 12.6206i − 0.497710i −0.968541 0.248855i $$-0.919946\pi$$
0.968541 0.248855i $$-0.0800542\pi$$
$$644$$ 52.5015 2.06885
$$645$$ 0 0
$$646$$ 6.40585 0.252035
$$647$$ 29.6429i 1.16538i 0.812694 + 0.582691i $$0.198000\pi$$
−0.812694 + 0.582691i $$0.802000\pi$$
$$648$$ − 1.09461i − 0.0430003i
$$649$$ 7.73463 0.303611
$$650$$ 0 0
$$651$$ −20.0752 −0.786810
$$652$$ − 17.5667i − 0.687967i
$$653$$ − 9.90069i − 0.387444i −0.981056 0.193722i $$-0.937944\pi$$
0.981056 0.193722i $$-0.0620560\pi$$
$$654$$ 14.3103 0.559578
$$655$$ 0 0
$$656$$ −11.1552 −0.435536
$$657$$ − 9.21949i − 0.359687i
$$658$$ 26.9541i 1.05078i
$$659$$ 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 $$0.674913\pi$$
$$662$$ 68.5271i 2.66338i
$$663$$ 1.75023i 0.0679733i
$$664$$ 16.1447 0.626537
$$665$$ 0 0
$$666$$ 0.965943 0.0374295
$$667$$ − 35.8401i − 1.38774i
$$668$$ 15.8713i 0.614080i
$$669$$ 12.9239 0.499665
$$670$$ 0 0
$$671$$ −12.0147 −0.463822
$$672$$ 28.8860i 1.11430i
$$673$$ 3.81834i 0.147186i 0.997288 + 0.0735932i $$0.0234466\pi$$
−0.997288 + 0.0735932i $$0.976553\pi$$
$$674$$ −61.5885 −2.37230
$$675$$ 0 0
$$676$$ −26.9229 −1.03550
$$677$$ − 15.6897i − 0.603003i −0.953466 0.301502i $$-0.902512\pi$$
0.953466 0.301502i $$-0.0974879\pi$$
$$678$$ − 12.7493i − 0.489634i
$$679$$ 24.6779 0.947049
$$680$$ 0 0
$$681$$ −22.8099 −0.874076
$$682$$ 11.7190i 0.448745i
$$683$$ 15.6353i 0.598269i 0.954211 + 0.299135i $$0.0966979\pi$$
−0.954211 + 0.299135i $$0.903302\pi$$
$$684$$ −6.56387 −0.250976
$$685$$ 0 0
$$686$$ −5.80275 −0.221550
$$687$$ − 14.7796i − 0.563876i
$$688$$ 31.6528i 1.20675i
$$689$$ 19.0303 0.724996
$$690$$ 0 0
$$691$$ −31.4305 −1.19567 −0.597836 0.801618i $$-0.703973\pi$$
−0.597836 + 0.801618i $$0.703973\pi$$
$$692$$ − 32.3065i − 1.22811i
$$693$$ − 3.64002i − 0.138273i
$$694$$ 75.8989 2.88108
$$695$$ 0 0
$$696$$ −6.84106 −0.259310
$$697$$ 4.76491i 0.180484i
$$698$$ 49.4674i 1.87237i
$$699$$ 4.96594 0.187829
$$700$$ 0 0
$$701$$ 3.24507 0.122565 0.0612824 0.998120i $$-0.480481\pi$$
0.0612824 + 0.998120i $$0.480481\pi$$
$$702$$ − 3.21949i − 0.121512i
$$703$$ − 1.18635i − 0.0447442i
$$704$$ 11.4537 0.431676
$$705$$ 0 0
$$706$$ −20.7181 −0.779737
$$707$$ 26.9541i 1.01371i
$$708$$ − 19.4537i − 0.731114i
$$709$$ 26.7190 1.00345 0.501727 0.865026i $$-0.332698\pi$$
0.501727 + 0.865026i $$0.332698\pi$$
$$710$$ 0 0
$$711$$ 5.09461 0.191063
$$712$$ − 11.4929i − 0.430714i
$$713$$ 31.6273i 1.18445i
$$714$$ 8.93475 0.334375
$$715$$ 0 0
$$716$$ 33.9163 1.26751
$$717$$ 14.9991i 0.560151i
$$718$$ − 72.0351i − 2.68833i
$$719$$ 6.78807 0.253152 0.126576 0.991957i $$-0.459601\pi$$
0.126576 + 0.991957i $$0.459601\pi$$
$$720$$ 0 0
$$721$$ −60.0587 −2.23670
$$722$$ − 25.9007i − 0.963924i
$$723$$ 5.04496i 0.187624i
$$724$$ 58.0743 2.15831
$$725$$ 0 0
$$726$$ −2.12489 −0.0788619
$$727$$ − 19.9154i − 0.738620i −0.929306 0.369310i $$-0.879594\pi$$
0.929306 0.369310i $$-0.120406\pi$$
$$728$$ − 6.03692i − 0.223743i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 13.5204 0.500071
$$732$$ 30.2186i 1.11691i
$$733$$ 31.3388i 1.15752i 0.815497 + 0.578762i $$0.196464\pi$$
−0.815497 + 0.578762i $$0.803536\pi$$
$$734$$ −4.00756 −0.147922
$$735$$ 0 0
$$736$$ 45.5081 1.67745
$$737$$ − 14.2645i − 0.525438i
$$738$$ − 8.76491i − 0.322641i
$$739$$ 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
$$742$$ − 97.1477i − 3.56640i
$$743$$ 6.74931i 0.247608i 0.992307 + 0.123804i $$0.0395095\pi$$
−0.992307 + 0.123804i $$0.960491\pi$$
$$744$$ 6.03692 0.221324
$$745$$ 0 0
$$746$$ −34.6888 −1.27005
$$747$$ − 14.7493i − 0.539649i
$$748$$ − 2.90539i − 0.106232i
$$749$$ −14.3397 −0.523961
$$750$$ 0 0
$$751$$ 22.4390 0.818810 0.409405 0.912353i $$-0.365736\pi$$
0.409405 + 0.912353i $$0.365736\pi$$
$$752$$ 9.42431i 0.343669i
$$753$$ − 3.03028i − 0.110429i
$$754$$ −20.1211 −0.732767
$$755$$ 0 0
$$756$$ −9.15516 −0.332970
$$757$$ 25.4158i 0.923754i 0.886944 + 0.461877i $$0.152824\pi$$
−0.886944 + 0.461877i $$0.847176\pi$$
$$758$$ 55.3406i 2.01006i
$$759$$ −5.73463 −0.208154
$$760$$ 0 0
$$761$$ −30.7493 −1.11466 −0.557331 0.830291i $$-0.688174\pi$$
−0.557331 + 0.830291i $$0.688174\pi$$
$$762$$ − 17.1358i − 0.620764i
$$763$$ − 24.5142i − 0.887474i
$$764$$ 20.0819 0.726537
$$765$$ 0 0
$$766$$ 26.4977 0.957401
$$767$$ − 11.7190i − 0.423150i
$$768$$ 4.91721i 0.177434i
$$769$$ 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
$$772$$ 29.5904i 1.06498i
$$773$$ − 48.7787i − 1.75445i −0.480082 0.877223i $$-0.659393\pi$$
0.480082 0.877223i $$-0.340607\pi$$
$$774$$ −24.8704 −0.893949
$$775$$ 0 0
$$776$$ −7.42100 −0.266398
$$777$$ − 1.65470i − 0.0593621i
$$778$$ 38.4390i 1.37810i
$$779$$ −10.7649 −0.385693
$$780$$ 0 0
$$781$$ 8.51514 0.304696
$$782$$ − 14.0761i − 0.503362i
$$783$$ 6.24977i 0.223349i
$$784$$ 16.9016 0.603629
$$785$$ 0 0
$$786$$ 27.2195 0.970887
$$787$$ − 46.2001i − 1.64686i −0.567421 0.823428i $$-0.692059\pi$$
0.567421 0.823428i $$-0.307941\pi$$
$$788$$ 9.59415i 0.341777i
$$789$$ 12.5601 0.447151
$$790$$ 0 0
$$791$$ −21.8401 −0.776546
$$792$$ 1.09461i 0.0388952i
$$793$$ 18.2039i 0.646439i
$$794$$ 32.3709 1.14880
$$795$$ 0 0
$$796$$ −30.3709 −1.07647
$$797$$ 36.3784i 1.28859i 0.764777 + 0.644295i $$0.222849\pi$$
−0.764777 + 0.644295i $$0.777151\pi$$
$$798$$ 20.1854i 0.714557i
$$799$$ 4.02558 0.142415
$$800$$ 0 0
$$801$$ −10.4995 −0.370983
$$802$$ 5.84197i 0.206287i
$$803$$ 9.21949i 0.325349i
$$804$$ −35.8771 −1.26529
$$805$$ 0 0
$$806$$ 17.7560 0.625427
$$807$$ 24.6888i 0.869086i
$$808$$ − 8.10551i − 0.285151i
$$809$$ −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 $$-0.575541\pi$$
−0.235098 + 0.971972i $$0.575541\pi$$
$$812$$ 57.2177i 2.00795i
$$813$$ 7.56479i 0.265309i
$$814$$ −0.965943 −0.0338563
$$815$$ 0 0
$$816$$ 3.12397 0.109361
$$817$$ 30.5454i 1.06865i
$$818$$ 8.46835i 0.296089i
$$819$$ −5.51514 −0.192715
$$820$$ 0 0
$$821$$ 32.0975 1.12021 0.560105 0.828422i $$-0.310761\pi$$
0.560105 + 0.828422i $$0.310761\pi$$
$$822$$ − 48.4683i − 1.69053i
$$823$$ − 16.7952i − 0.585443i −0.956198 0.292722i $$-0.905439\pi$$
0.956198 0.292722i $$-0.0945609\pi$$
$$824$$ 18.0606 0.629169
$$825$$ 0 0
$$826$$ −59.8245 −2.08156
$$827$$ − 45.5904i − 1.58533i −0.609656 0.792666i $$-0.708692\pi$$
0.609656 0.792666i $$-0.291308\pi$$
$$828$$ 14.4234i 0.501248i
$$829$$ −12.9385 −0.449374 −0.224687 0.974431i $$-0.572136\pi$$
−0.224687 + 0.974431i $$0.572136\pi$$
$$830$$ 0 0
$$831$$ −1.92477 −0.0667695
$$832$$ − 17.3539i − 0.601638i
$$833$$ − 7.21949i − 0.250141i
$$834$$ 16.1287 0.558490
$$835$$ 0 0
$$836$$ 6.56387 0.227016
$$837$$ − 5.51514i − 0.190631i
$$838$$ 10.9130i 0.376982i
$$839$$ 1.59037 0.0549057 0.0274528 0.999623i $$-0.491260\pi$$
0.0274528 + 0.999623i $$0.491260\pi$$
$$840$$ 0 0
$$841$$ 10.0596 0.346884
$$842$$ − 19.0104i − 0.655143i
$$843$$ − 1.87511i − 0.0645824i
$$844$$ −25.8165 −0.888641
$$845$$ 0 0
$$846$$ −7.40493 −0.254587
$$847$$ 3.64002i 0.125073i
$$848$$ − 33.9670i − 1.16643i
$$849$$ 30.1396 1.03439
$$850$$ 0 0
$$851$$ −2.60688 −0.0893628
$$852$$ − 21.4167i − 0.733726i
$$853$$ − 10.5161i − 0.360063i −0.983661 0.180031i $$-0.942380\pi$$
0.983661 0.180031i $$-0.0576199\pi$$
$$854$$ 92.9291 3.17997
$$855$$ 0 0
$$856$$ 4.31216 0.147386
$$857$$ − 57.4637i − 1.96292i −0.191665 0.981460i $$-0.561389\pi$$
0.191665 0.981460i $$-0.438611\pi$$
$$858$$ 3.21949i 0.109912i
$$859$$ −32.7181 −1.11633 −0.558164 0.829731i $$-0.688494\pi$$
−0.558164 + 0.829731i $$0.688494\pi$$
$$860$$ 0 0
$$861$$ −15.0147 −0.511699
$$862$$ 48.3397i 1.64646i
$$863$$ 43.1807i 1.46989i 0.678127 + 0.734945i $$0.262792\pi$$
−0.678127 + 0.734945i $$0.737208\pi$$
$$864$$ −7.93567 −0.269977
$$865$$ 0 0
$$866$$ 16.1135 0.547560
$$867$$ 15.6656i 0.532032i
$$868$$ − 50.4920i − 1.71381i
$$869$$ −5.09461 −0.172823
$$870$$ 0 0
$$871$$ −21.6126 −0.732315
$$872$$ 7.37179i 0.249640i
$$873$$ 6.77959i 0.229454i
$$874$$ 31.8009 1.07568
$$875$$ 0 0
$$876$$ 23.1883 0.783460
$$877$$ − 16.0752i − 0.542822i −0.962464 0.271411i $$-0.912510\pi$$
0.962464 0.271411i $$-0.0874903\pi$$
$$878$$ − 36.9523i − 1.24708i
$$879$$ −29.1552 −0.983379
$$880$$ 0 0
$$881$$ 31.2876 1.05411 0.527053 0.849832i $$-0.323297\pi$$
0.527053 + 0.849832i $$0.323297\pi$$
$$882$$ 13.2800i 0.447162i
$$883$$ − 24.7640i − 0.833375i −0.909050 0.416687i $$-0.863191\pi$$
0.909050 0.416687i $$-0.136809\pi$$
$$884$$ −4.40207 −0.148058
$$885$$ 0 0
$$886$$ 22.6169 0.759828
$$887$$ − 26.3085i − 0.883353i −0.897174 0.441676i $$-0.854384\pi$$
0.897174 0.441676i $$-0.145616\pi$$
$$888$$ 0.497594i 0.0166982i
$$889$$ −29.3544 −0.984514
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 32.5053i 1.08836i
$$893$$ 9.09461i 0.304339i
$$894$$ −3.85574 −0.128955
$$895$$ 0 0
$$896$$ −30.8179 −1.02955
$$897$$ 8.68876i 0.290109i
$$898$$ 57.2252i 1.90963i
$$899$$ −34.4683 −1.14958
$$900$$ 0 0
$$901$$ −14.5089 −0.483363
$$902$$ 8.76491i 0.291840i
$$903$$ 42.6041i 1.41778i
$$904$$ 6.56766 0.218437
$$905$$ 0 0
$$906$$ −51.6878 −1.71721
$$907$$ − 55.9301i − 1.85713i −0.371174 0.928563i $$-0.621045\pi$$
0.371174 0.928563i $$-0.378955\pi$$
$$908$$ − 57.3700i − 1.90389i
$$909$$ −7.40493 −0.245606
$$910$$ 0 0
$$911$$ −15.3931 −0.509997 −0.254998 0.966941i $$-0.582075\pi$$
−0.254998 + 0.966941i $$0.582075\pi$$
$$912$$ 7.05769i 0.233703i
$$913$$ 14.7493i 0.488131i
$$914$$ 33.5298 1.10907
$$915$$ 0 0
$$916$$ 37.1727 1.22822
$$917$$ − 46.6282i − 1.53980i
$$918$$ 2.45459i 0.0810134i
$$919$$ 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
$$922$$ 17.4012i 0.573076i
$$923$$ − 12.9016i − 0.424662i
$$924$$ 9.15516 0.301183
$$925$$ 0 0
$$926$$ 34.1892 1.12353
$$927$$ − 16.4995i − 0.541916i
$$928$$ 49.5961i 1.62807i
$$929$$ −14.8099 −0.485896 −0.242948 0.970039i $$-0.578114\pi$$
−0.242948 + 0.970039i $$0.578114\pi$$
$$930$$ 0 0
$$931$$ 16.3103 0.534549
$$932$$ 12.4900i 0.409125i
$$933$$ 23.9083i 0.782721i
$$934$$ 62.6188 2.04895
$$935$$ 0 0
$$936$$ 1.65848 0.0542093
$$937$$ 27.2654i 0.890721i 0.895351 + 0.445360i $$0.146924\pi$$
−0.895351 + 0.445360i $$0.853076\pi$$
$$938$$ 110.330i 3.60241i
$$939$$ 28.3094 0.923843
$$940$$ 0 0
$$941$$ −49.5630 −1.61571 −0.807853 0.589384i $$-0.799371\pi$$
−0.807853 + 0.589384i $$0.799371\pi$$
$$942$$ − 20.7493i − 0.676049i
$$943$$ 23.6547i 0.770303i
$$944$$ −20.9172 −0.680797
$$945$$ 0 0
$$946$$ 24.8704 0.808607
$$947$$ − 13.9844i − 0.454432i −0.973844 0.227216i $$-0.927038\pi$$
0.973844 0.227216i $$-0.0729623\pi$$
$$948$$ 12.8136i 0.416168i
$$949$$ 13.9688 0.453447
$$950$$ 0 0
$$951$$ 8.80986 0.285679
$$952$$ 4.60263i 0.149172i
$$953$$ − 17.1240i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894561\pi$$
$$954$$ 26.6888 0.864081
$$955$$ 0 0
$$956$$ −37.7248 −1.22011
$$957$$ − 6.24977i − 0.202026i
$$958$$ − 68.4608i − 2.21187i
$$959$$ −83.0284 −2.68113
$$960$$ 0 0
$$961$$ −0.583252 −0.0188146
$$962$$ 1.46354i 0.0471863i
$$963$$ − 3.93945i − 0.126947i
$$964$$ −12.6888 −0.408677
$$965$$ 0 0
$$966$$ 44.3553 1.42711
$$967$$ 1.90826i 0.0613653i 0.999529 + 0.0306827i $$0.00976813\pi$$
−0.999529 + 0.0306827i $$0.990232\pi$$
$$968$$ − 1.09461i − 0.0351821i
$$969$$ 3.01468 0.0968455
$$970$$ 0 0
$$971$$ 31.3856 1.00721 0.503605 0.863934i $$-0.332007\pi$$
0.503605 + 0.863934i $$0.332007\pi$$
$$972$$ − 2.51514i − 0.0806731i
$$973$$ − 27.6291i − 0.885749i
$$974$$ 76.2697 2.44384
$$975$$ 0 0
$$976$$ 32.4920 1.04004
$$977$$ 41.4693i 1.32672i 0.748301 + 0.663360i $$0.230870\pi$$
−0.748301 + 0.663360i $$0.769130\pi$$
$$978$$ − 14.8411i − 0.474565i
$$979$$ 10.4995 0.335567
$$980$$ 0 0
$$981$$ 6.73463 0.215020
$$982$$ − 15.2119i − 0.485432i
$$983$$ 6.23601i 0.198898i 0.995043 + 0.0994489i $$0.0317080\pi$$
−0.995043 + 0.0994489i $$0.968292\pi$$
$$984$$ 4.51514 0.143937
$$985$$ 0 0
$$986$$ 15.3406 0.488544
$$987$$ 12.6850i 0.403767i
$$988$$ − 9.94518i − 0.316398i
$$989$$ 67.1202 2.13430
$$990$$ 0 0
$$991$$ 27.2048 0.864189 0.432095 0.901828i $$-0.357775\pi$$
0.432095 + 0.901828i $$0.357775\pi$$
$$992$$ − 43.7663i − 1.38958i
$$993$$ 32.2498i 1.02342i
$$994$$ −65.8615 −2.08900
$$995$$ 0 0
$$996$$ 37.0966 1.17545
$$997$$ 28.0606i 0.888687i 0.895857 + 0.444343i $$0.146563\pi$$
−0.895857 + 0.444343i $$0.853437\pi$$
$$998$$ − 57.5298i − 1.82107i
$$999$$ 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.c.f.199.5 6
3.2 odd 2 2475.2.c.q.199.2 6
5.2 odd 4 825.2.a.m.1.1 yes 3
5.3 odd 4 825.2.a.i.1.3 3
5.4 even 2 inner 825.2.c.f.199.2 6
15.2 even 4 2475.2.a.z.1.3 3
15.8 even 4 2475.2.a.bd.1.1 3
15.14 odd 2 2475.2.c.q.199.5 6
55.32 even 4 9075.2.a.cd.1.3 3
55.43 even 4 9075.2.a.cj.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 5.3 odd 4
825.2.a.m.1.1 yes 3 5.2 odd 4
825.2.c.f.199.2 6 5.4 even 2 inner
825.2.c.f.199.5 6 1.1 even 1 trivial
2475.2.a.z.1.3 3 15.2 even 4
2475.2.a.bd.1.1 3 15.8 even 4
2475.2.c.q.199.2 6 3.2 odd 2
2475.2.c.q.199.5 6 15.14 odd 2
9075.2.a.cd.1.3 3 55.32 even 4
9075.2.a.cj.1.1 3 55.43 even 4