# Properties

 Label 2475.2.c.q.199.5 Level $2475$ Weight $2$ Character 2475.199 Analytic conductor $19.763$ 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] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 825) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.5 Root $$1.32001 + 1.32001i$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.199 Dual form 2475.2.c.q.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$0.729446\pi$$
$$3$$ 0 0
$$4$$ −2.51514 −1.25757
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.64002i − 1.37580i −0.725806 0.687900i $$-0.758533\pi$$
0.725806 0.687900i $$-0.241467\pi$$
$$8$$ − 1.09461i − 0.387003i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 1.51514i 0.420224i 0.977677 + 0.210112i $$0.0673828\pi$$
−0.977677 + 0.210112i $$0.932617\pi$$
$$14$$ 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 $$0.955263\pi$$
$$18$$ 0 0
$$19$$ −2.60975 −0.598717 −0.299359 0.954141i $$-0.596773\pi$$
−0.299359 + 0.954141i $$0.596773\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 2.12489i − 0.453027i
$$23$$ 5.73463i 1.19575i 0.801588 + 0.597877i $$0.203989\pi$$
−0.801588 + 0.597877i $$0.796011\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3.21949 −0.631395
$$27$$ 0 0
$$28$$ 9.15516i 1.73016i
$$29$$ 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 $$0.335068\pi$$
$$32$$ − 7.93567i − 1.40284i
$$33$$ 0 0
$$34$$ −2.45459 −0.420958
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 0.454586i − 0.0747335i −0.999302 0.0373667i $$-0.988103\pi$$
0.999302 0.0373667i $$-0.0118970\pi$$
$$38$$ − 5.54541i − 0.899585i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.12489 −0.644199 −0.322099 0.946706i $$-0.604389\pi$$
−0.322099 + 0.946706i $$0.604389\pi$$
$$42$$ 0 0
$$43$$ 11.7044i 1.78490i 0.451149 + 0.892449i $$0.351014\pi$$
−0.451149 + 0.892449i $$0.648986\pi$$
$$44$$ 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 $$-0.0817989\pi$$
$$48$$ 0 0
$$49$$ −6.24977 −0.892824
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 3.81078i − 0.528460i
$$53$$ 12.5601i 1.72526i 0.505834 + 0.862631i $$0.331185\pi$$
−0.505834 + 0.862631i $$0.668815\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.98440 −0.532438
$$57$$ 0 0
$$58$$ 13.2800i 1.74376i
$$59$$ −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 $$0.779326\pi$$
$$62$$ 11.7190i 1.48832i
$$63$$ 0 0
$$64$$ 11.4537 1.43171
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.2645i 1.74268i 0.490680 + 0.871340i $$0.336749\pi$$
−0.490680 + 0.871340i $$0.663251\pi$$
$$68$$ − 2.90539i − 0.352330i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.51514 −1.01056 −0.505280 0.862955i $$-0.668611\pi$$
−0.505280 + 0.862955i $$0.668611\pi$$
$$72$$ 0 0
$$73$$ − 9.21949i − 1.07906i −0.841966 0.539530i $$-0.818602\pi$$
0.841966 0.539530i $$-0.181398\pi$$
$$74$$ 0.965943 0.112289
$$75$$ 0 0
$$76$$ 6.56387 0.752928
$$77$$ 3.64002i 0.414819i
$$78$$ 0 0
$$79$$ −5.09461 −0.573188 −0.286594 0.958052i $$-0.592523\pi$$
−0.286594 + 0.958052i $$0.592523\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 8.76491i − 0.967922i
$$83$$ 14.7493i 1.61895i 0.587156 + 0.809474i $$0.300247\pi$$
−0.587156 + 0.809474i $$0.699753\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −24.8704 −2.68185
$$87$$ 0 0
$$88$$ 1.09461i 0.116686i
$$89$$ −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
$$92$$ − 14.4234i − 1.50374i
$$93$$ 0 0
$$94$$ 7.40493 0.763760
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.77959i 0.688363i 0.938903 + 0.344181i $$0.111844\pi$$
−0.938903 + 0.344181i $$0.888156\pi$$
$$98$$ − 13.2800i − 1.34149i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.40493 −0.736818 −0.368409 0.929664i $$-0.620097\pi$$
−0.368409 + 0.929664i $$0.620097\pi$$
$$102$$ 0 0
$$103$$ − 16.4995i − 1.62575i −0.582439 0.812874i $$-0.697902\pi$$
0.582439 0.812874i $$-0.302098\pi$$
$$104$$ 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 $$0.939015\pi$$
$$108$$ 0 0
$$109$$ −6.73463 −0.645061 −0.322530 0.946559i $$-0.604533\pi$$
−0.322530 + 0.946559i $$0.604533\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 9.84392i 0.930163i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −15.7190 −1.45948
$$117$$ 0 0
$$118$$ − 16.4352i − 1.51298i
$$119$$ 4.20482 0.385455
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 25.5298i − 2.31136i
$$123$$ 0 0
$$124$$ −13.8713 −1.24568
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.06433i − 0.715594i −0.933799 0.357797i $$-0.883528\pi$$
0.933799 0.357797i $$-0.116472\pi$$
$$128$$ 8.46640i 0.748331i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.8099 1.11920 0.559602 0.828762i $$-0.310954\pi$$
0.559602 + 0.828762i $$0.310954\pi$$
$$132$$ 0 0
$$133$$ 9.49954i 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 $$0.572195\pi$$
$$138$$ 0 0
$$139$$ −7.59037 −0.643807 −0.321903 0.946773i $$-0.604323\pi$$
−0.321903 + 0.946773i $$0.604323\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 18.0937i − 1.51839i
$$143$$ − 1.51514i − 0.126702i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 19.5904 1.62131
$$147$$ 0 0
$$148$$ 1.14335i 0.0939825i
$$149$$ −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 $$0.0455687\pi$$
$$152$$ 2.85665i 0.231705i
$$153$$ 0 0
$$154$$ −7.73463 −0.623274
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 9.76491i − 0.779325i −0.920958 0.389662i $$-0.872592\pi$$
0.920958 0.389662i $$-0.127408\pi$$
$$158$$ − 10.8255i − 0.861227i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 20.8742 1.64512
$$162$$ 0 0
$$163$$ − 6.98440i − 0.547061i −0.961863 0.273530i $$-0.911809\pi$$
0.961863 0.273530i $$-0.0881914\pi$$
$$164$$ 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 $$-0.0785102\pi$$
$$168$$ 0 0
$$169$$ 10.7044 0.823412
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 29.4381i − 2.24463i
$$173$$ 12.8448i 0.976575i 0.872683 + 0.488287i $$0.162378\pi$$
−0.872683 + 0.488287i $$0.837622\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.70436 0.203849
$$177$$ 0 0
$$178$$ − 22.3103i − 1.67223i
$$179$$ 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 $$0.828374\pi$$
$$182$$ 11.7190i 0.868673i
$$183$$ 0 0
$$184$$ 6.27718 0.462760
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 1.15516i − 0.0844738i
$$188$$ 8.76491i 0.639247i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.98440 0.577731 0.288866 0.957370i $$-0.406722\pi$$
0.288866 + 0.957370i $$0.406722\pi$$
$$192$$ 0 0
$$193$$ 11.7649i 0.846857i 0.905930 + 0.423428i $$0.139173\pi$$
−0.905930 + 0.423428i $$0.860827\pi$$
$$194$$ −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 $$-0.0433888\pi$$
$$198$$ 0 0
$$199$$ 12.0752 0.855990 0.427995 0.903781i $$-0.359220\pi$$
0.427995 + 0.903781i $$0.359220\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 15.7346i − 1.10708i
$$203$$ − 22.7493i − 1.59669i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 35.0596 2.44272
$$207$$ 0 0
$$208$$ − 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
$$213$$ 0 0
$$214$$ −8.37088 −0.572221
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 20.0752i − 1.36280i
$$218$$ − 14.3103i − 0.969217i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.75023 −0.117733
$$222$$ 0 0
$$223$$ 12.9239i 0.865445i 0.901527 + 0.432723i $$0.142447\pi$$
−0.901527 + 0.432723i $$0.857553\pi$$
$$224$$ −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 $$0.726678\pi$$
$$228$$ 0 0
$$229$$ −14.7796 −0.976663 −0.488331 0.872658i $$-0.662394\pi$$
−0.488331 + 0.872658i $$0.662394\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.84106i − 0.449137i
$$233$$ − 4.96594i − 0.325330i −0.986681 0.162665i $$-0.947991\pi$$
0.986681 0.162665i $$-0.0520089\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 19.4537 1.26633
$$237$$ 0 0
$$238$$ 8.93475i 0.579154i
$$239$$ −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 $$0.448048\pi$$
$$242$$ 2.12489i 0.136593i
$$243$$ 0 0
$$244$$ 30.2186 1.93455
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3.95413i − 0.251595i
$$248$$ − 6.03692i − 0.383345i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3.03028 0.191269 0.0956347 0.995417i $$-0.469512\pi$$
0.0956347 + 0.995417i $$0.469512\pi$$
$$252$$ 0 0
$$253$$ − 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 $$-0.139993\pi$$
$$258$$ 0 0
$$259$$ −1.65470 −0.102818
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 27.2195i 1.68163i
$$263$$ − 12.5601i − 0.774489i −0.921977 0.387244i $$-0.873427\pi$$
0.921977 0.387244i $$-0.126573\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −20.1854 −1.23765
$$267$$ 0 0
$$268$$ − 35.8771i − 2.19154i
$$269$$ −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 $$0.426205\pi$$
$$272$$ − 3.12397i − 0.189418i
$$273$$ 0 0
$$274$$ −48.4683 −2.92808
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1.92477i − 0.115648i −0.998327 0.0578241i $$-0.981584\pi$$
0.998327 0.0578241i $$-0.0184162\pi$$
$$278$$ − 16.1287i − 0.967333i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.87511 0.111860 0.0559300 0.998435i $$-0.482188\pi$$
0.0559300 + 0.998435i $$0.482188\pi$$
$$282$$ 0 0
$$283$$ 30.1396i 1.79161i 0.444446 + 0.895806i $$0.353401\pi$$
−0.444446 + 0.895806i $$0.646599\pi$$
$$284$$ 21.4167 1.27085
$$285$$ 0 0
$$286$$ 3.21949 0.190373
$$287$$ 15.0147i 0.886289i
$$288$$ 0 0
$$289$$ 15.6656 0.921506
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 23.1883i 1.35699i
$$293$$ 29.1552i 1.70326i 0.524141 + 0.851631i $$0.324386\pi$$
−0.524141 + 0.851631i $$0.675614\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −0.497594 −0.0289221
$$297$$ 0 0
$$298$$ − 3.85574i − 0.223357i
$$299$$ −8.68876 −0.502484
$$300$$ 0 0
$$301$$ 42.6041 2.45566
$$302$$ 51.6878i 2.97430i
$$303$$ 0 0
$$304$$ 7.05769 0.404786
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 27.8548i − 1.58976i −0.606768 0.794879i $$-0.707534\pi$$
0.606768 0.794879i $$-0.292466\pi$$
$$308$$ − 9.15516i − 0.521664i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −23.9083 −1.35571 −0.677856 0.735194i $$-0.737091\pi$$
−0.677856 + 0.735194i $$0.737091\pi$$
$$312$$ 0 0
$$313$$ 28.3094i 1.60014i 0.599905 + 0.800071i $$0.295205\pi$$
−0.599905 + 0.800071i $$0.704795\pi$$
$$314$$ 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 $$-0.0795780\pi$$
$$318$$ 0 0
$$319$$ −6.24977 −0.349920
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 44.3553i 2.47182i
$$323$$ − 3.01468i − 0.167741i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 14.8411 0.821970
$$327$$ 0 0
$$328$$ 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 0
$$334$$ 13.4087 0.733692
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 28.9844i − 1.57888i −0.613827 0.789441i $$-0.710371\pi$$
0.613827 0.789441i $$-0.289629\pi$$
$$338$$ 22.7455i 1.23719i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −5.51514 −0.298661
$$342$$ 0 0
$$343$$ − 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 $$-0.408254\pi$$
$$348$$ 0 0
$$349$$ 23.2800 1.24615 0.623076 0.782161i $$-0.285883\pi$$
0.623076 + 0.782161i $$0.285883\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 7.93567i 0.422972i
$$353$$ 9.75023i 0.518952i 0.965750 + 0.259476i $$0.0835499\pi$$
−0.965750 + 0.259476i $$0.916450\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 26.4078 1.39961
$$357$$ 0 0
$$358$$ 28.6538i 1.51440i
$$359$$ 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
$$362$$ − 49.0634i − 2.57872i
$$363$$ 0 0
$$364$$ −13.8713 −0.727055
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 1.88601i − 0.0984491i −0.998788 0.0492245i $$-0.984325\pi$$
0.998788 0.0492245i $$-0.0156750\pi$$
$$368$$ − 15.5085i − 0.808436i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 45.7190 2.37361
$$372$$ 0 0
$$373$$ − 16.3250i − 0.845277i −0.906298 0.422638i $$-0.861104\pi$$
0.906298 0.422638i $$-0.138896\pi$$
$$374$$ 2.45459 0.126924
$$375$$ 0 0
$$376$$ −3.81456 −0.196721
$$377$$ 9.46927i 0.487692i
$$378$$ 0 0
$$379$$ 26.0440 1.33779 0.668896 0.743356i $$-0.266767\pi$$
0.668896 + 0.743356i $$0.266767\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.9659i 0.868053i
$$383$$ − 12.4702i − 0.637197i −0.947890 0.318598i $$-0.896788\pi$$
0.947890 0.318598i $$-0.103212\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −24.9991 −1.27242
$$387$$ 0 0
$$388$$ − 17.0516i − 0.865664i
$$389$$ −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
$$392$$ 6.84106i 0.345526i
$$393$$ 0 0
$$394$$ 8.10551 0.408350
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 15.2342i 0.764581i 0.924042 + 0.382291i $$0.124865\pi$$
−0.924042 + 0.382291i $$0.875135\pi$$
$$398$$ 25.6585i 1.28614i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.74931 −0.137294 −0.0686471 0.997641i $$-0.521868\pi$$
−0.0686471 + 0.997641i $$0.521868\pi$$
$$402$$ 0 0
$$403$$ 8.35620i 0.416252i
$$404$$ 18.6244 0.926600
$$405$$ 0 0
$$406$$ 48.3397 2.39906
$$407$$ 0.454586i 0.0225330i
$$408$$ 0 0
$$409$$ 3.98532 0.197061 0.0985307 0.995134i $$-0.468586\pi$$
0.0985307 + 0.995134i $$0.468586\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 41.4986i 2.04449i
$$413$$ 28.1542i 1.38538i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 12.0236 0.589507
$$417$$ 0 0
$$418$$ 5.54541i 0.271235i
$$419$$ −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 $$0.569958\pi$$
$$422$$ 21.8108i 1.06173i
$$423$$ 0 0
$$424$$ 13.7484 0.667681
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 43.7337i 2.11642i
$$428$$ − 9.90826i − 0.478934i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −22.7493 −1.09580 −0.547898 0.836545i $$-0.684572\pi$$
−0.547898 + 0.836545i $$0.684572\pi$$
$$432$$ 0 0
$$433$$ 7.58325i 0.364428i 0.983259 + 0.182214i $$0.0583263\pi$$
−0.983259 + 0.182214i $$0.941674\pi$$
$$434$$ 42.6576 2.04763
$$435$$ 0 0
$$436$$ 16.9385 0.811209
$$437$$ − 14.9659i − 0.715918i
$$438$$ 0 0
$$439$$ −17.3903 −0.829991 −0.414996 0.909823i $$-0.636217\pi$$
−0.414996 + 0.909823i $$0.636217\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 3.71904i − 0.176897i
$$443$$ − 10.6438i − 0.505702i −0.967505 0.252851i $$-0.918632\pi$$
0.967505 0.252851i $$-0.0813683\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −27.4617 −1.30035
$$447$$ 0 0
$$448$$ − 41.6916i − 1.96974i
$$449$$ −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
$$452$$ − 15.0908i − 0.709813i
$$453$$ 0 0
$$454$$ −48.4683 −2.27473
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15.7796i 0.738138i 0.929402 + 0.369069i $$0.120323\pi$$
−0.929402 + 0.369069i $$0.879677\pi$$
$$458$$ − 31.4049i − 1.46746i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8.18922 −0.381410 −0.190705 0.981647i $$-0.561077\pi$$
−0.190705 + 0.981647i $$0.561077\pi$$
$$462$$ 0 0
$$463$$ 16.0899i 0.747762i 0.927477 + 0.373881i $$0.121973\pi$$
−0.927477 + 0.373881i $$0.878027\pi$$
$$464$$ −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 $$-0.238819\pi$$
$$468$$ 0 0
$$469$$ 51.9229 2.39758
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 8.46640i 0.389698i
$$473$$ − 11.7044i − 0.538167i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −10.5757 −0.484736
$$477$$ 0 0
$$478$$ − 31.8713i − 1.45776i
$$479$$ 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
$$482$$ 10.7200i 0.488280i
$$483$$ 0 0
$$484$$ −2.51514 −0.114324
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 35.8936i 1.62649i 0.581919 + 0.813247i $$0.302302\pi$$
−0.581919 + 0.813247i $$0.697698\pi$$
$$488$$ 13.1514i 0.595335i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.15894 0.323079 0.161539 0.986866i $$-0.448354\pi$$
0.161539 + 0.986866i $$0.448354\pi$$
$$492$$ 0 0
$$493$$ 7.21949i 0.325150i
$$494$$ 8.40207 0.378027
$$495$$ 0 0
$$496$$ −14.9149 −0.669699
$$497$$ 30.9953i 1.39033i
$$498$$ 0 0
$$499$$ −27.0743 −1.21201 −0.606006 0.795460i $$-0.707229\pi$$
−0.606006 + 0.795460i $$0.707229\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 6.43899i 0.287386i
$$503$$ − 26.9991i − 1.20383i −0.798560 0.601915i $$-0.794405\pi$$
0.798560 0.601915i $$-0.205595\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 12.1854 0.541709
$$507$$ 0 0
$$508$$ 20.2829i 0.899909i
$$509$$ 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
$$512$$ 27.3813i 1.21009i
$$513$$ 0 0
$$514$$ 29.0066 1.27943
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.48486i 0.153264i
$$518$$ − 3.51605i − 0.154487i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11.1589 −0.488882 −0.244441 0.969664i $$-0.578604\pi$$
−0.244441 + 0.969664i $$0.578604\pi$$
$$522$$ 0 0
$$523$$ − 10.5786i − 0.462568i −0.972886 0.231284i $$-0.925707\pi$$
0.972886 0.231284i $$-0.0742926\pi$$
$$524$$ −32.2186 −1.40748
$$525$$ 0 0
$$526$$ 26.6888 1.16369
$$527$$ 6.37088i 0.277520i
$$528$$ 0 0
$$529$$ −9.88601 −0.429827
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 23.8927i − 1.03588i
$$533$$ − 6.24977i − 0.270708i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 15.6140 0.674422
$$537$$ 0 0
$$538$$ − 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
$$543$$ 0 0
$$544$$ 9.16698 0.393031
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.09369i − 0.260547i −0.991478 0.130274i $$-0.958414\pi$$
0.991478 0.130274i $$-0.0415856\pi$$
$$548$$ − 57.3700i − 2.45072i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −16.3103 −0.694843
$$552$$ 0 0
$$553$$ 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 $$-0.0398964\pi$$
$$558$$ 0 0
$$559$$ −17.7337 −0.750056
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3.98440i 0.168072i
$$563$$ − 3.03028i − 0.127711i −0.997959 0.0638555i $$-0.979660\pi$$
0.997959 0.0638555i $$-0.0203397\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −64.0431 −2.69193
$$567$$ 0 0
$$568$$ 9.32075i 0.391090i
$$569$$ 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 $$0.309819\pi$$
$$572$$ 3.81078i 0.159337i
$$573$$ 0 0
$$574$$ −31.9045 −1.33167
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.03028i − 0.0845215i −0.999107 0.0422607i $$-0.986544\pi$$
0.999107 0.0422607i $$-0.0134560\pi$$
$$578$$ 33.2876i 1.38458i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 53.6878 2.22735
$$582$$ 0 0
$$583$$ − 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 $$0.851282\pi$$
$$588$$ 0 0
$$589$$ −14.3931 −0.593058
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.22936i 0.0505265i
$$593$$ − 8.06811i − 0.331318i −0.986183 0.165659i $$-0.947025\pi$$
0.986183 0.165659i $$-0.0529751\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.56387 0.186944
$$597$$ 0 0
$$598$$ − 18.4626i − 0.754993i
$$599$$ 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 $$0.476766\pi$$
$$602$$ 90.5289i 3.68968i
$$603$$ 0 0
$$604$$ −61.1807 −2.48941
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 17.5298i − 0.711513i −0.934579 0.355757i $$-0.884223\pi$$
0.934579 0.355757i $$-0.115777\pi$$
$$608$$ 20.7101i 0.839905i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 5.28005 0.213608
$$612$$ 0 0
$$613$$ 12.5601i 0.507297i 0.967296 + 0.253649i $$0.0816307\pi$$
−0.967296 + 0.253649i $$0.918369\pi$$
$$614$$ 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 $$0.895884\pi$$
$$618$$ 0 0
$$619$$ 9.23417 0.371153 0.185576 0.982630i $$-0.440585\pi$$
0.185576 + 0.982630i $$0.440585\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 50.8023i − 2.03699i
$$623$$ 38.2186i 1.53119i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −60.1542 −2.40425
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 18.7200 0.743464
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 9.46927i − 0.375186i
$$638$$ − 13.2800i − 0.525762i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −13.9612 −0.551436 −0.275718 0.961239i $$-0.588916\pi$$
−0.275718 + 0.961239i $$0.588916\pi$$
$$642$$ 0 0
$$643$$ 12.6206i 0.497710i 0.968541 + 0.248855i $$0.0800542\pi$$
−0.968541 + 0.248855i $$0.919946\pi$$
$$644$$ −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 $$0.802000\pi$$
$$648$$ 0 0
$$649$$ 7.73463 0.303611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 17.5667i 0.687967i
$$653$$ − 9.90069i − 0.387444i −0.981056 0.193722i $$-0.937944\pi$$
0.981056 0.193722i $$-0.0620560\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 11.1552 0.435536
$$657$$ 0 0
$$658$$ − 26.9541i − 1.05078i
$$659$$ −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 $$0.674913\pi$$
$$662$$ 68.5271i 2.66338i
$$663$$ 0 0
$$664$$ 16.1447 0.626537
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 35.8401i 1.38774i
$$668$$ 15.8713i 0.614080i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 12.0147 0.463822
$$672$$ 0 0
$$673$$ − 3.81834i − 0.147186i −0.997288 0.0735932i $$-0.976553\pi$$
0.997288 0.0735932i $$-0.0234466\pi$$
$$674$$ 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 $$-0.0974879\pi$$
$$678$$ 0 0
$$679$$ 24.6779 0.947049
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 11.7190i − 0.448745i
$$683$$ 15.6353i 0.598269i 0.954211 + 0.299135i $$0.0966979\pi$$
−0.954211 + 0.299135i $$0.903302\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5.80275 0.221550
$$687$$ 0 0
$$688$$ − 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
$$693$$ 0 0
$$694$$ 75.8989 2.88108
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 4.76491i − 0.180484i
$$698$$ 49.4674i 1.87237i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3.24507 −0.122565 −0.0612824 0.998120i $$-0.519519\pi$$
−0.0612824 + 0.998120i $$0.519519\pi$$
$$702$$ 0 0
$$703$$ 1.18635i 0.0447442i
$$704$$ −11.4537 −0.431676
$$705$$ 0 0
$$706$$ −20.7181 −0.779737
$$707$$ 26.9541i 1.01371i
$$708$$ 0 0
$$709$$ 26.7190 1.00345 0.501727 0.865026i $$-0.332698\pi$$
0.501727 + 0.865026i $$0.332698\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 11.4929i 0.430714i
$$713$$ 31.6273i 1.18445i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −33.9163 −1.26751
$$717$$ 0 0
$$718$$ 72.0351i 2.68833i
$$719$$ −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
$$722$$ − 25.9007i − 0.963924i
$$723$$ 0 0
$$724$$ 58.0743 2.15831
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 19.9154i 0.738620i 0.929306 + 0.369310i $$0.120406\pi$$
−0.929306 + 0.369310i $$0.879594\pi$$
$$728$$ − 6.03692i − 0.223743i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −13.5204 −0.500071
$$732$$ 0 0
$$733$$ − 31.3388i − 1.15752i −0.815497 0.578762i $$-0.803536\pi$$
0.815497 0.578762i $$-0.196464\pi$$
$$734$$ 4.00756 0.147922
$$735$$ 0 0
$$736$$ 45.5081 1.67745
$$737$$ − 14.2645i − 0.525438i
$$738$$ 0 0
$$739$$ 2.25355 0.0828982 0.0414491 0.999141i $$-0.486803\pi$$
0.0414491 + 0.999141i $$0.486803\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 97.1477i 3.56640i
$$743$$ 6.74931i 0.247608i 0.992307 + 0.123804i $$0.0395095\pi$$
−0.992307 + 0.123804i $$0.960491\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 34.6888 1.27005
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ −20.1211 −0.732767
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 25.4158i − 0.923754i −0.886944 0.461877i $$-0.847176\pi$$
0.886944 0.461877i $$-0.152824\pi$$
$$758$$ 55.3406i 2.01006i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.7493 1.11466 0.557331 0.830291i $$-0.311826\pi$$
0.557331 + 0.830291i $$0.311826\pi$$
$$762$$ 0 0
$$763$$ 24.5142i 0.887474i
$$764$$ −20.0819 −0.726537
$$765$$ 0 0
$$766$$ 26.4977 0.957401
$$767$$ − 11.7190i − 0.423150i
$$768$$ 0 0
$$769$$ 16.2956 0.587636 0.293818 0.955861i $$-0.405074\pi$$
0.293818 + 0.955861i $$0.405074\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 29.5904i − 1.06498i
$$773$$ − 48.7787i − 1.75445i −0.480082 0.877223i $$-0.659393\pi$$
0.480082 0.877223i $$-0.340607\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 7.42100 0.266398
$$777$$ 0 0
$$778$$ − 38.4390i − 1.37810i
$$779$$ 10.7649 0.385693
$$780$$ 0 0
$$781$$ 8.51514 0.304696
$$782$$ − 14.0761i − 0.503362i
$$783$$ 0 0
$$784$$ 16.9016 0.603629
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 46.2001i 1.64686i 0.567421 + 0.823428i $$0.307941\pi$$
−0.567421 + 0.823428i $$0.692059\pi$$
$$788$$ 9.59415i 0.341777i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 21.8401 0.776546
$$792$$ 0 0
$$793$$ − 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 $$0.777151\pi$$
$$798$$ 0 0
$$799$$ 4.02558 0.142415
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 5.84197i − 0.206287i
$$803$$ 9.21949i 0.325349i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −17.7560 −0.625427
$$807$$ 0 0
$$808$$ 8.10551i 0.285151i
$$809$$ 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 $$0.575541\pi$$
$$812$$ 57.2177i 2.00795i
$$813$$ 0 0
$$814$$ −0.965943 −0.0338563
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 30.5454i − 1.06865i
$$818$$ 8.46835i 0.296089i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −32.0975 −1.12021 −0.560105 0.828422i $$-0.689239\pi$$
−0.560105 + 0.828422i $$0.689239\pi$$
$$822$$ 0 0
$$823$$ 16.7952i 0.585443i 0.956198 + 0.292722i $$0.0945609\pi$$
−0.956198 + 0.292722i $$0.905439\pi$$
$$824$$ −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 $$-0.291308\pi$$
$$828$$ 0 0
$$829$$ −12.9385 −0.449374 −0.224687 0.974431i $$-0.572136\pi$$
−0.224687 + 0.974431i $$0.572136\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 17.3539i 0.601638i
$$833$$ − 7.21949i − 0.250141i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6.56387 −0.227016
$$837$$ 0 0
$$838$$ − 10.9130i − 0.376982i
$$839$$ −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
$$842$$ − 19.0104i − 0.655143i
$$843$$ 0 0
$$844$$ −25.8165 −0.888641
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3.64002i − 0.125073i
$$848$$ − 33.9670i − 1.16643i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2.60688 0.0893628
$$852$$ 0 0
$$853$$ 10.5161i 0.360063i 0.983661 + 0.180031i $$0.0576199\pi$$
−0.983661 + 0.180031i $$0.942380\pi$$
$$854$$ −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 $$-0.438611\pi$$
$$858$$ 0 0
$$859$$ −32.7181 −1.11633 −0.558164 0.829731i $$-0.688494\pi$$
−0.558164 + 0.829731i $$0.688494\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 48.3397i − 1.64646i
$$863$$ 43.1807i 1.46989i 0.678127 + 0.734945i $$0.262792\pi$$
−0.678127 + 0.734945i $$0.737208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.1135 −0.547560
$$867$$ 0 0
$$868$$ 50.4920i 1.71381i
$$869$$ 5.09461 0.172823
$$870$$ 0 0
$$871$$ −21.6126 −0.732315
$$872$$ 7.37179i 0.249640i
$$873$$ 0 0
$$874$$ 31.8009 1.07568
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 16.0752i 0.542822i 0.962464 + 0.271411i $$0.0874903\pi$$
−0.962464 + 0.271411i $$0.912510\pi$$
$$878$$ − 36.9523i − 1.24708i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −31.2876 −1.05411 −0.527053 0.849832i $$-0.676703\pi$$
−0.527053 + 0.849832i $$0.676703\pi$$
$$882$$ 0 0
$$883$$ 24.7640i 0.833375i 0.909050 + 0.416687i $$0.136809\pi$$
−0.909050 + 0.416687i $$0.863191\pi$$
$$884$$ 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 0
$$889$$ −29.3544 −0.984514
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 32.5053i − 1.08836i
$$893$$ 9.09461i 0.304339i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 30.8179 1.02955
$$897$$ 0 0
$$898$$ − 57.2252i − 1.90963i
$$899$$ 34.4683 1.14958
$$900$$ 0 0
$$901$$ −14.5089 −0.483363
$$902$$ 8.76491i 0.291840i
$$903$$ 0 0
$$904$$ 6.56766 0.218437
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 55.9301i 1.85713i 0.371174 + 0.928563i $$0.378955\pi$$
−0.371174 + 0.928563i $$0.621045\pi$$
$$908$$ − 57.3700i − 1.90389i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 15.3931 0.509997 0.254998 0.966941i $$-0.417925\pi$$
0.254998 + 0.966941i $$0.417925\pi$$
$$912$$ 0 0
$$913$$ − 14.7493i − 0.488131i
$$914$$ −33.5298 −1.10907
$$915$$ 0 0
$$916$$ 37.1727 1.22822
$$917$$ − 46.6282i − 1.53980i
$$918$$ 0 0
$$919$$ 51.2598 1.69090 0.845452 0.534052i $$-0.179331\pi$$
0.845452 + 0.534052i $$0.179331\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 17.4012i − 0.573076i
$$923$$ − 12.9016i − 0.424662i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −34.1892 −1.12353
$$927$$ 0 0
$$928$$ − 49.5961i − 1.62807i
$$929$$ 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
$$932$$ 12.4900i 0.409125i
$$933$$ 0 0
$$934$$ 62.6188 2.04895
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 27.2654i − 0.890721i −0.895351 0.445360i $$-0.853076\pi$$
0.895351 0.445360i $$-0.146924\pi$$
$$938$$ 110.330i 3.60241i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 49.5630 1.61571 0.807853 0.589384i $$-0.200629\pi$$
0.807853 + 0.589384i $$0.200629\pi$$
$$942$$ 0 0
$$943$$ − 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 $$-0.0729623\pi$$
$$948$$ 0 0
$$949$$ 13.9688 0.453447
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 4.60263i − 0.149172i
$$953$$ − 17.1240i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894561\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 37.7248 1.22011
$$957$$ 0 0
$$958$$ 68.4608i 2.21187i
$$959$$ 83.0284 2.68113
$$960$$ 0 0
$$961$$ −0.583252 −0.0188146
$$962$$ 1.46354i 0.0471863i
$$963$$ 0 0
$$964$$ −12.6888 −0.408677
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.90826i − 0.0613653i −0.999529 0.0306827i $$-0.990232\pi$$
0.999529 0.0306827i $$-0.00976813\pi$$
$$968$$ − 1.09461i − 0.0351821i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −31.3856 −1.00721 −0.503605 0.863934i $$-0.667993\pi$$
−0.503605 + 0.863934i $$0.667993\pi$$
$$972$$ 0 0
$$973$$ 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 $$0.769130\pi$$
$$978$$ 0 0
$$979$$ 10.4995 0.335567
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 15.2119i 0.485432i
$$983$$ 6.23601i 0.198898i 0.995043 + 0.0994489i $$0.0317080\pi$$
−0.995043 + 0.0994489i $$0.968292\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −15.3406 −0.488544
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ −65.8615 −2.08900
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 28.0606i − 0.888687i −0.895857 0.444343i $$-0.853437\pi$$
0.895857 0.444343i $$-0.146563\pi$$
$$998$$ − 57.5298i − 1.82107i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 15.2 even 4
825.2.a.m.1.1 yes 3 15.8 even 4
825.2.c.f.199.2 6 3.2 odd 2
825.2.c.f.199.5 6 15.14 odd 2
2475.2.a.z.1.3 3 5.3 odd 4
2475.2.a.bd.1.1 3 5.2 odd 4
2475.2.c.q.199.2 6 5.4 even 2 inner
2475.2.c.q.199.5 6 1.1 even 1 trivial
9075.2.a.cd.1.3 3 165.98 odd 4
9075.2.a.cj.1.1 3 165.32 odd 4