# Properties

 Label 3380.2.f.i.3041.6 Level $3380$ Weight $2$ Character 3380.3041 Analytic conductor $26.989$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.f (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: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3041.6 Root $$-1.27597 - 0.609843i$$ of defining polynomial Character $$\chi$$ $$=$$ 3380.3041 Dual form 3380.2.f.i.3041.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.60020 q^{3} +1.00000i q^{5} -4.33225i q^{7} -0.439374 q^{9} +O(q^{10})$$ $$q+1.60020 q^{3} +1.00000i q^{5} -4.33225i q^{7} -0.439374 q^{9} +1.73205i q^{11} +1.60020i q^{15} -7.50367 q^{17} -5.37182i q^{19} -6.93244i q^{21} +1.16082 q^{23} -1.00000 q^{25} -5.50367 q^{27} -2.02473 q^{29} +7.86488i q^{31} +2.77162i q^{33} +4.33225 q^{35} -9.53264i q^{37} +7.73205i q^{41} -4.18555 q^{43} -0.439374i q^{45} +3.46410i q^{47} -11.7684 q^{49} -12.0073 q^{51} -12.6471 q^{53} -1.73205 q^{55} -8.59596i q^{57} -6.34709i q^{59} +3.70308 q^{61} +1.90348i q^{63} -5.26045i q^{67} +1.85754 q^{69} +12.5143i q^{71} -5.23898i q^{73} -1.60020 q^{75} +7.50367 q^{77} -8.16719 q^{79} -7.48883 q^{81} -0.456760i q^{83} -7.50367i q^{85} -3.23996 q^{87} -13.2753i q^{89} +12.5854i q^{93} +5.37182 q^{95} +2.81021i q^{97} -0.761018i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} + 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^3 + 8 * q^9 $$8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100})$$ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1691$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.60020 0.923873 0.461937 0.886913i $$-0.347155\pi$$
0.461937 + 0.886913i $$0.347155\pi$$
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ − 4.33225i − 1.63744i −0.574196 0.818718i $$-0.694685\pi$$
0.574196 0.818718i $$-0.305315\pi$$
$$8$$ 0 0
$$9$$ −0.439374 −0.146458
$$10$$ 0 0
$$11$$ 1.73205i 0.522233i 0.965307 + 0.261116i $$0.0840907\pi$$
−0.965307 + 0.261116i $$0.915909\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 1.60020i 0.413169i
$$16$$ 0 0
$$17$$ −7.50367 −1.81991 −0.909954 0.414710i $$-0.863883\pi$$
−0.909954 + 0.414710i $$0.863883\pi$$
$$18$$ 0 0
$$19$$ − 5.37182i − 1.23238i −0.787598 0.616190i $$-0.788676\pi$$
0.787598 0.616190i $$-0.211324\pi$$
$$20$$ 0 0
$$21$$ − 6.93244i − 1.51278i
$$22$$ 0 0
$$23$$ 1.16082 0.242048 0.121024 0.992650i $$-0.461382\pi$$
0.121024 + 0.992650i $$0.461382\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.50367 −1.05918
$$28$$ 0 0
$$29$$ −2.02473 −0.375983 −0.187991 0.982171i $$-0.560198\pi$$
−0.187991 + 0.982171i $$0.560198\pi$$
$$30$$ 0 0
$$31$$ 7.86488i 1.41257i 0.707925 + 0.706287i $$0.249631\pi$$
−0.707925 + 0.706287i $$0.750369\pi$$
$$32$$ 0 0
$$33$$ 2.77162i 0.482477i
$$34$$ 0 0
$$35$$ 4.33225 0.732283
$$36$$ 0 0
$$37$$ − 9.53264i − 1.56716i −0.621293 0.783578i $$-0.713392\pi$$
0.621293 0.783578i $$-0.286608\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.73205i 1.20754i 0.797157 + 0.603772i $$0.206336\pi$$
−0.797157 + 0.603772i $$0.793664\pi$$
$$42$$ 0 0
$$43$$ −4.18555 −0.638290 −0.319145 0.947706i $$-0.603396\pi$$
−0.319145 + 0.947706i $$0.603396\pi$$
$$44$$ 0 0
$$45$$ − 0.439374i − 0.0654980i
$$46$$ 0 0
$$47$$ 3.46410i 0.505291i 0.967559 + 0.252646i $$0.0813007\pi$$
−0.967559 + 0.252646i $$0.918699\pi$$
$$48$$ 0 0
$$49$$ −11.7684 −1.68119
$$50$$ 0 0
$$51$$ −12.0073 −1.68136
$$52$$ 0 0
$$53$$ −12.6471 −1.73721 −0.868607 0.495502i $$-0.834984\pi$$
−0.868607 + 0.495502i $$0.834984\pi$$
$$54$$ 0 0
$$55$$ −1.73205 −0.233550
$$56$$ 0 0
$$57$$ − 8.59596i − 1.13856i
$$58$$ 0 0
$$59$$ − 6.34709i − 0.826320i −0.910658 0.413160i $$-0.864425\pi$$
0.910658 0.413160i $$-0.135575\pi$$
$$60$$ 0 0
$$61$$ 3.70308 0.474131 0.237066 0.971494i $$-0.423814\pi$$
0.237066 + 0.971494i $$0.423814\pi$$
$$62$$ 0 0
$$63$$ 1.90348i 0.239815i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 5.26045i − 0.642666i −0.946966 0.321333i $$-0.895869\pi$$
0.946966 0.321333i $$-0.104131\pi$$
$$68$$ 0 0
$$69$$ 1.85754 0.223622
$$70$$ 0 0
$$71$$ 12.5143i 1.48517i 0.669751 + 0.742586i $$0.266401\pi$$
−0.669751 + 0.742586i $$0.733599\pi$$
$$72$$ 0 0
$$73$$ − 5.23898i − 0.613177i −0.951842 0.306588i $$-0.900813\pi$$
0.951842 0.306588i $$-0.0991875\pi$$
$$74$$ 0 0
$$75$$ −1.60020 −0.184775
$$76$$ 0 0
$$77$$ 7.50367 0.855123
$$78$$ 0 0
$$79$$ −8.16719 −0.918880 −0.459440 0.888209i $$-0.651950\pi$$
−0.459440 + 0.888209i $$0.651950\pi$$
$$80$$ 0 0
$$81$$ −7.48883 −0.832092
$$82$$ 0 0
$$83$$ − 0.456760i − 0.0501359i −0.999686 0.0250679i $$-0.992020\pi$$
0.999686 0.0250679i $$-0.00798021\pi$$
$$84$$ 0 0
$$85$$ − 7.50367i − 0.813887i
$$86$$ 0 0
$$87$$ −3.23996 −0.347360
$$88$$ 0 0
$$89$$ − 13.2753i − 1.40718i −0.710607 0.703589i $$-0.751580\pi$$
0.710607 0.703589i $$-0.248420\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 12.5854i 1.30504i
$$94$$ 0 0
$$95$$ 5.37182 0.551137
$$96$$ 0 0
$$97$$ 2.81021i 0.285334i 0.989771 + 0.142667i $$0.0455678\pi$$
−0.989771 + 0.142667i $$0.954432\pi$$
$$98$$ 0 0
$$99$$ − 0.761018i − 0.0764852i
$$100$$ 0 0
$$101$$ 6.63977 0.660681 0.330341 0.943862i $$-0.392836\pi$$
0.330341 + 0.943862i $$0.392836\pi$$
$$102$$ 0 0
$$103$$ 7.37605 0.726784 0.363392 0.931636i $$-0.381619\pi$$
0.363392 + 0.931636i $$0.381619\pi$$
$$104$$ 0 0
$$105$$ 6.93244 0.676537
$$106$$ 0 0
$$107$$ −7.91336 −0.765014 −0.382507 0.923953i $$-0.624939\pi$$
−0.382507 + 0.923953i $$0.624939\pi$$
$$108$$ 0 0
$$109$$ − 9.18301i − 0.879572i −0.898102 0.439786i $$-0.855054\pi$$
0.898102 0.439786i $$-0.144946\pi$$
$$110$$ 0 0
$$111$$ − 15.2541i − 1.44785i
$$112$$ 0 0
$$113$$ −5.77586 −0.543347 −0.271674 0.962389i $$-0.587577\pi$$
−0.271674 + 0.962389i $$0.587577\pi$$
$$114$$ 0 0
$$115$$ 1.16082i 0.108247i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 32.5078i 2.97998i
$$120$$ 0 0
$$121$$ 8.00000 0.727273
$$122$$ 0 0
$$123$$ 12.3728i 1.11562i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −18.9678 −1.68312 −0.841559 0.540165i $$-0.818362\pi$$
−0.841559 + 0.540165i $$0.818362\pi$$
$$128$$ 0 0
$$129$$ −6.69770 −0.589699
$$130$$ 0 0
$$131$$ −1.34637 −0.117633 −0.0588165 0.998269i $$-0.518733\pi$$
−0.0588165 + 0.998269i $$0.518733\pi$$
$$132$$ 0 0
$$133$$ −23.2720 −2.01794
$$134$$ 0 0
$$135$$ − 5.50367i − 0.473681i
$$136$$ 0 0
$$137$$ 12.3728i 1.05708i 0.848909 + 0.528540i $$0.177260\pi$$
−0.848909 + 0.528540i $$0.822740\pi$$
$$138$$ 0 0
$$139$$ 8.76836 0.743723 0.371861 0.928288i $$-0.378720\pi$$
0.371861 + 0.928288i $$0.378720\pi$$
$$140$$ 0 0
$$141$$ 5.54324i 0.466825i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 2.02473i − 0.168144i
$$146$$ 0 0
$$147$$ −18.8317 −1.55321
$$148$$ 0 0
$$149$$ − 4.73939i − 0.388266i −0.980975 0.194133i $$-0.937811\pi$$
0.980975 0.194133i $$-0.0621894\pi$$
$$150$$ 0 0
$$151$$ − 1.56063i − 0.127002i −0.997982 0.0635010i $$-0.979773\pi$$
0.997982 0.0635010i $$-0.0202266\pi$$
$$152$$ 0 0
$$153$$ 3.29692 0.266540
$$154$$ 0 0
$$155$$ −7.86488 −0.631723
$$156$$ 0 0
$$157$$ −9.15332 −0.730515 −0.365257 0.930907i $$-0.619019\pi$$
−0.365257 + 0.930907i $$0.619019\pi$$
$$158$$ 0 0
$$159$$ −20.2378 −1.60497
$$160$$ 0 0
$$161$$ − 5.02897i − 0.396338i
$$162$$ 0 0
$$163$$ − 2.64303i − 0.207018i −0.994629 0.103509i $$-0.966993\pi$$
0.994629 0.103509i $$-0.0330070\pi$$
$$164$$ 0 0
$$165$$ −2.77162 −0.215770
$$166$$ 0 0
$$167$$ 8.90869i 0.689375i 0.938717 + 0.344688i $$0.112015\pi$$
−0.938717 + 0.344688i $$0.887985\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 2.36023i 0.180492i
$$172$$ 0 0
$$173$$ −13.8079 −1.04980 −0.524899 0.851165i $$-0.675897\pi$$
−0.524899 + 0.851165i $$0.675897\pi$$
$$174$$ 0 0
$$175$$ 4.33225i 0.327487i
$$176$$ 0 0
$$177$$ − 10.1566i − 0.763416i
$$178$$ 0 0
$$179$$ 24.3502 1.82002 0.910009 0.414588i $$-0.136074\pi$$
0.910009 + 0.414588i $$0.136074\pi$$
$$180$$ 0 0
$$181$$ 24.8336 1.84587 0.922935 0.384956i $$-0.125783\pi$$
0.922935 + 0.384956i $$0.125783\pi$$
$$182$$ 0 0
$$183$$ 5.92566 0.438037
$$184$$ 0 0
$$185$$ 9.53264 0.700853
$$186$$ 0 0
$$187$$ − 12.9967i − 0.950416i
$$188$$ 0 0
$$189$$ 23.8433i 1.73434i
$$190$$ 0 0
$$191$$ 5.96875 0.431884 0.215942 0.976406i $$-0.430718\pi$$
0.215942 + 0.976406i $$0.430718\pi$$
$$192$$ 0 0
$$193$$ − 9.05272i − 0.651629i −0.945434 0.325814i $$-0.894362\pi$$
0.945434 0.325814i $$-0.105638\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 11.3890i − 0.811436i −0.913998 0.405718i $$-0.867022\pi$$
0.913998 0.405718i $$-0.132978\pi$$
$$198$$ 0 0
$$199$$ −10.7773 −0.763980 −0.381990 0.924166i $$-0.624761\pi$$
−0.381990 + 0.924166i $$0.624761\pi$$
$$200$$ 0 0
$$201$$ − 8.41775i − 0.593742i
$$202$$ 0 0
$$203$$ 8.77162i 0.615647i
$$204$$ 0 0
$$205$$ −7.73205 −0.540030
$$206$$ 0 0
$$207$$ −0.510035 −0.0354499
$$208$$ 0 0
$$209$$ 9.30426 0.643589
$$210$$ 0 0
$$211$$ −3.22512 −0.222026 −0.111013 0.993819i $$-0.535410\pi$$
−0.111013 + 0.993819i $$0.535410\pi$$
$$212$$ 0 0
$$213$$ 20.0253i 1.37211i
$$214$$ 0 0
$$215$$ − 4.18555i − 0.285452i
$$216$$ 0 0
$$217$$ 34.0726 2.31300
$$218$$ 0 0
$$219$$ − 8.38340i − 0.566497i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 16.9794i − 1.13702i −0.822676 0.568511i $$-0.807520\pi$$
0.822676 0.568511i $$-0.192480\pi$$
$$224$$ 0 0
$$225$$ 0.439374 0.0292916
$$226$$ 0 0
$$227$$ − 2.40052i − 0.159328i −0.996822 0.0796641i $$-0.974615\pi$$
0.996822 0.0796641i $$-0.0253848\pi$$
$$228$$ 0 0
$$229$$ − 7.35671i − 0.486145i −0.970008 0.243073i $$-0.921845\pi$$
0.970008 0.243073i $$-0.0781553\pi$$
$$230$$ 0 0
$$231$$ 12.0073 0.790025
$$232$$ 0 0
$$233$$ −3.37410 −0.221045 −0.110522 0.993874i $$-0.535252\pi$$
−0.110522 + 0.993874i $$0.535252\pi$$
$$234$$ 0 0
$$235$$ −3.46410 −0.225973
$$236$$ 0 0
$$237$$ −13.0691 −0.848929
$$238$$ 0 0
$$239$$ − 23.7807i − 1.53824i −0.639103 0.769121i $$-0.720694\pi$$
0.639103 0.769121i $$-0.279306\pi$$
$$240$$ 0 0
$$241$$ − 22.3212i − 1.43784i −0.695095 0.718918i $$-0.744638\pi$$
0.695095 0.718918i $$-0.255362\pi$$
$$242$$ 0 0
$$243$$ 4.52742 0.290434
$$244$$ 0 0
$$245$$ − 11.7684i − 0.751853i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 0.730905i − 0.0463192i
$$250$$ 0 0
$$251$$ 4.62238 0.291762 0.145881 0.989302i $$-0.453398\pi$$
0.145881 + 0.989302i $$0.453398\pi$$
$$252$$ 0 0
$$253$$ 2.01060i 0.126406i
$$254$$ 0 0
$$255$$ − 12.0073i − 0.751929i
$$256$$ 0 0
$$257$$ −21.8639 −1.36383 −0.681916 0.731430i $$-0.738853\pi$$
−0.681916 + 0.731430i $$0.738853\pi$$
$$258$$ 0 0
$$259$$ −41.2977 −2.56612
$$260$$ 0 0
$$261$$ 0.889612 0.0550656
$$262$$ 0 0
$$263$$ −20.1790 −1.24429 −0.622146 0.782901i $$-0.713739\pi$$
−0.622146 + 0.782901i $$0.713739\pi$$
$$264$$ 0 0
$$265$$ − 12.6471i − 0.776906i
$$266$$ 0 0
$$267$$ − 21.2431i − 1.30005i
$$268$$ 0 0
$$269$$ 7.28687 0.444288 0.222144 0.975014i $$-0.428694\pi$$
0.222144 + 0.975014i $$0.428694\pi$$
$$270$$ 0 0
$$271$$ − 4.89189i − 0.297161i −0.988900 0.148581i $$-0.952530\pi$$
0.988900 0.148581i $$-0.0474705\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 1.73205i − 0.104447i
$$276$$ 0 0
$$277$$ −8.80059 −0.528776 −0.264388 0.964416i $$-0.585170\pi$$
−0.264388 + 0.964416i $$0.585170\pi$$
$$278$$ 0 0
$$279$$ − 3.45562i − 0.206883i
$$280$$ 0 0
$$281$$ 12.6085i 0.752161i 0.926587 + 0.376081i $$0.122728\pi$$
−0.926587 + 0.376081i $$0.877272\pi$$
$$282$$ 0 0
$$283$$ 20.9009 1.24243 0.621216 0.783640i $$-0.286639\pi$$
0.621216 + 0.783640i $$0.286639\pi$$
$$284$$ 0 0
$$285$$ 8.59596 0.509181
$$286$$ 0 0
$$287$$ 33.4972 1.97727
$$288$$ 0 0
$$289$$ 39.3051 2.31206
$$290$$ 0 0
$$291$$ 4.49689i 0.263612i
$$292$$ 0 0
$$293$$ 0.692481i 0.0404552i 0.999795 + 0.0202276i $$0.00643908\pi$$
−0.999795 + 0.0202276i $$0.993561\pi$$
$$294$$ 0 0
$$295$$ 6.34709 0.369542
$$296$$ 0 0
$$297$$ − 9.53264i − 0.553140i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 18.1328i 1.04516i
$$302$$ 0 0
$$303$$ 10.6249 0.610386
$$304$$ 0 0
$$305$$ 3.70308i 0.212038i
$$306$$ 0 0
$$307$$ − 8.83168i − 0.504051i −0.967721 0.252025i $$-0.918903\pi$$
0.967721 0.252025i $$-0.0810966\pi$$
$$308$$ 0 0
$$309$$ 11.8031 0.671457
$$310$$ 0 0
$$311$$ −11.3958 −0.646198 −0.323099 0.946365i $$-0.604725\pi$$
−0.323099 + 0.946365i $$0.604725\pi$$
$$312$$ 0 0
$$313$$ −3.38496 −0.191329 −0.0956647 0.995414i $$-0.530498\pi$$
−0.0956647 + 0.995414i $$0.530498\pi$$
$$314$$ 0 0
$$315$$ −1.90348 −0.107249
$$316$$ 0 0
$$317$$ 0.557104i 0.0312901i 0.999878 + 0.0156450i $$0.00498017\pi$$
−0.999878 + 0.0156450i $$0.995020\pi$$
$$318$$ 0 0
$$319$$ − 3.50693i − 0.196350i
$$320$$ 0 0
$$321$$ −12.6629 −0.706776
$$322$$ 0 0
$$323$$ 40.3083i 2.24282i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 14.6946i − 0.812614i
$$328$$ 0 0
$$329$$ 15.0073 0.827382
$$330$$ 0 0
$$331$$ − 3.96369i − 0.217864i −0.994049 0.108932i $$-0.965257\pi$$
0.994049 0.108932i $$-0.0347431\pi$$
$$332$$ 0 0
$$333$$ 4.18839i 0.229522i
$$334$$ 0 0
$$335$$ 5.26045 0.287409
$$336$$ 0 0
$$337$$ 16.7243 0.911030 0.455515 0.890228i $$-0.349455\pi$$
0.455515 + 0.890228i $$0.349455\pi$$
$$338$$ 0 0
$$339$$ −9.24251 −0.501984
$$340$$ 0 0
$$341$$ −13.6224 −0.737693
$$342$$ 0 0
$$343$$ 20.6577i 1.11541i
$$344$$ 0 0
$$345$$ 1.85754i 0.100007i
$$346$$ 0 0
$$347$$ 7.58085 0.406962 0.203481 0.979079i $$-0.434775\pi$$
0.203481 + 0.979079i $$0.434775\pi$$
$$348$$ 0 0
$$349$$ 27.3356i 1.46324i 0.681712 + 0.731621i $$0.261236\pi$$
−0.681712 + 0.731621i $$0.738764\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.0921i 1.38874i 0.719616 + 0.694372i $$0.244318\pi$$
−0.719616 + 0.694372i $$0.755682\pi$$
$$354$$ 0 0
$$355$$ −12.5143 −0.664189
$$356$$ 0 0
$$357$$ 52.0188i 2.75313i
$$358$$ 0 0
$$359$$ − 0.694176i − 0.0366372i −0.999832 0.0183186i $$-0.994169\pi$$
0.999832 0.0183186i $$-0.00583132\pi$$
$$360$$ 0 0
$$361$$ −9.85641 −0.518758
$$362$$ 0 0
$$363$$ 12.8016 0.671908
$$364$$ 0 0
$$365$$ 5.23898 0.274221
$$366$$ 0 0
$$367$$ 22.8218 1.19129 0.595644 0.803249i $$-0.296897\pi$$
0.595644 + 0.803249i $$0.296897\pi$$
$$368$$ 0 0
$$369$$ − 3.39726i − 0.176854i
$$370$$ 0 0
$$371$$ 54.7904i 2.84458i
$$372$$ 0 0
$$373$$ −7.88129 −0.408078 −0.204039 0.978963i $$-0.565407\pi$$
−0.204039 + 0.978963i $$0.565407\pi$$
$$374$$ 0 0
$$375$$ − 1.60020i − 0.0826338i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 31.1426i 1.59969i 0.600209 + 0.799843i $$0.295084\pi$$
−0.600209 + 0.799843i $$0.704916\pi$$
$$380$$ 0 0
$$381$$ −30.3521 −1.55499
$$382$$ 0 0
$$383$$ − 34.5123i − 1.76349i −0.471723 0.881747i $$-0.656368\pi$$
0.471723 0.881747i $$-0.343632\pi$$
$$384$$ 0 0
$$385$$ 7.50367i 0.382422i
$$386$$ 0 0
$$387$$ 1.83902 0.0934827
$$388$$ 0 0
$$389$$ −6.18414 −0.313548 −0.156774 0.987634i $$-0.550109\pi$$
−0.156774 + 0.987634i $$0.550109\pi$$
$$390$$ 0 0
$$391$$ −8.71043 −0.440505
$$392$$ 0 0
$$393$$ −2.15446 −0.108678
$$394$$ 0 0
$$395$$ − 8.16719i − 0.410936i
$$396$$ 0 0
$$397$$ − 22.6563i − 1.13709i −0.822654 0.568543i $$-0.807507\pi$$
0.822654 0.568543i $$-0.192493\pi$$
$$398$$ 0 0
$$399$$ −37.2398 −1.86432
$$400$$ 0 0
$$401$$ 3.87803i 0.193660i 0.995301 + 0.0968298i $$0.0308703\pi$$
−0.995301 + 0.0968298i $$0.969130\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 7.48883i − 0.372123i
$$406$$ 0 0
$$407$$ 16.5110 0.818421
$$408$$ 0 0
$$409$$ − 13.8606i − 0.685365i −0.939451 0.342682i $$-0.888665\pi$$
0.939451 0.342682i $$-0.111335\pi$$
$$410$$ 0 0
$$411$$ 19.7989i 0.976607i
$$412$$ 0 0
$$413$$ −27.4972 −1.35305
$$414$$ 0 0
$$415$$ 0.456760 0.0224214
$$416$$ 0 0
$$417$$ 14.0311 0.687106
$$418$$ 0 0
$$419$$ −30.3502 −1.48270 −0.741352 0.671116i $$-0.765815\pi$$
−0.741352 + 0.671116i $$0.765815\pi$$
$$420$$ 0 0
$$421$$ − 0.608516i − 0.0296573i −0.999890 0.0148286i $$-0.995280\pi$$
0.999890 0.0148286i $$-0.00472027\pi$$
$$422$$ 0 0
$$423$$ − 1.52204i − 0.0740039i
$$424$$ 0 0
$$425$$ 7.50367 0.363982
$$426$$ 0 0
$$427$$ − 16.0427i − 0.776359i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 39.3544i − 1.89564i −0.318812 0.947818i $$-0.603284\pi$$
0.318812 0.947818i $$-0.396716\pi$$
$$432$$ 0 0
$$433$$ −6.42949 −0.308981 −0.154491 0.987994i $$-0.549374\pi$$
−0.154491 + 0.987994i $$0.549374\pi$$
$$434$$ 0 0
$$435$$ − 3.23996i − 0.155344i
$$436$$ 0 0
$$437$$ − 6.23572i − 0.298295i
$$438$$ 0 0
$$439$$ −1.47144 −0.0702282 −0.0351141 0.999383i $$-0.511179\pi$$
−0.0351141 + 0.999383i $$0.511179\pi$$
$$440$$ 0 0
$$441$$ 5.17071 0.246224
$$442$$ 0 0
$$443$$ −16.9193 −0.803860 −0.401930 0.915670i $$-0.631660\pi$$
−0.401930 + 0.915670i $$0.631660\pi$$
$$444$$ 0 0
$$445$$ 13.2753 0.629309
$$446$$ 0 0
$$447$$ − 7.58396i − 0.358709i
$$448$$ 0 0
$$449$$ 9.11701i 0.430258i 0.976586 + 0.215129i $$0.0690173\pi$$
−0.976586 + 0.215129i $$0.930983\pi$$
$$450$$ 0 0
$$451$$ −13.3923 −0.630619
$$452$$ 0 0
$$453$$ − 2.49731i − 0.117334i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 17.5562i − 0.821246i −0.911805 0.410623i $$-0.865311\pi$$
0.911805 0.410623i $$-0.134689\pi$$
$$458$$ 0 0
$$459$$ 41.2977 1.92761
$$460$$ 0 0
$$461$$ 15.3544i 0.715127i 0.933889 + 0.357564i $$0.116392\pi$$
−0.933889 + 0.357564i $$0.883608\pi$$
$$462$$ 0 0
$$463$$ 18.8614i 0.876562i 0.898838 + 0.438281i $$0.144412\pi$$
−0.898838 + 0.438281i $$0.855588\pi$$
$$464$$ 0 0
$$465$$ −12.5854 −0.583632
$$466$$ 0 0
$$467$$ 32.9302 1.52383 0.761913 0.647679i $$-0.224260\pi$$
0.761913 + 0.647679i $$0.224260\pi$$
$$468$$ 0 0
$$469$$ −22.7896 −1.05232
$$470$$ 0 0
$$471$$ −14.6471 −0.674903
$$472$$ 0 0
$$473$$ − 7.24958i − 0.333336i
$$474$$ 0 0
$$475$$ 5.37182i 0.246476i
$$476$$ 0 0
$$477$$ 5.55681 0.254429
$$478$$ 0 0
$$479$$ − 17.2476i − 0.788061i −0.919097 0.394031i $$-0.871080\pi$$
0.919097 0.394031i $$-0.128920\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 8.04733i − 0.366166i
$$484$$ 0 0
$$485$$ −2.81021 −0.127605
$$486$$ 0 0
$$487$$ 17.9744i 0.814498i 0.913317 + 0.407249i $$0.133512\pi$$
−0.913317 + 0.407249i $$0.866488\pi$$
$$488$$ 0 0
$$489$$ − 4.22936i − 0.191258i
$$490$$ 0 0
$$491$$ −26.1953 −1.18218 −0.591089 0.806607i $$-0.701302\pi$$
−0.591089 + 0.806607i $$0.701302\pi$$
$$492$$ 0 0
$$493$$ 15.1929 0.684253
$$494$$ 0 0
$$495$$ 0.761018 0.0342052
$$496$$ 0 0
$$497$$ 54.2149 2.43187
$$498$$ 0 0
$$499$$ 18.3428i 0.821139i 0.911829 + 0.410569i $$0.134670\pi$$
−0.911829 + 0.410569i $$0.865330\pi$$
$$500$$ 0 0
$$501$$ 14.2557i 0.636896i
$$502$$ 0 0
$$503$$ −14.2562 −0.635653 −0.317827 0.948149i $$-0.602953\pi$$
−0.317827 + 0.948149i $$0.602953\pi$$
$$504$$ 0 0
$$505$$ 6.63977i 0.295466i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 31.7981i 1.40943i 0.709491 + 0.704714i $$0.248925\pi$$
−0.709491 + 0.704714i $$0.751075\pi$$
$$510$$ 0 0
$$511$$ −22.6966 −1.00404
$$512$$ 0 0
$$513$$ 29.5647i 1.30531i
$$514$$ 0 0
$$515$$ 7.37605i 0.325028i
$$516$$ 0 0
$$517$$ −6.00000 −0.263880
$$518$$ 0 0
$$519$$ −22.0954 −0.969880
$$520$$ 0 0
$$521$$ −45.1676 −1.97883 −0.989414 0.145122i $$-0.953642\pi$$
−0.989414 + 0.145122i $$0.953642\pi$$
$$522$$ 0 0
$$523$$ 26.1146 1.14191 0.570955 0.820981i $$-0.306573\pi$$
0.570955 + 0.820981i $$0.306573\pi$$
$$524$$ 0 0
$$525$$ 6.93244i 0.302557i
$$526$$ 0 0
$$527$$ − 59.0155i − 2.57076i
$$528$$ 0 0
$$529$$ −21.6525 −0.941413
$$530$$ 0 0
$$531$$ 2.78874i 0.121021i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 7.91336i − 0.342124i
$$536$$ 0 0
$$537$$ 38.9651 1.68147
$$538$$ 0 0
$$539$$ − 20.3834i − 0.877975i
$$540$$ 0 0
$$541$$ − 31.4750i − 1.35321i −0.736344 0.676607i $$-0.763450\pi$$
0.736344 0.676607i $$-0.236550\pi$$
$$542$$ 0 0
$$543$$ 39.7387 1.70535
$$544$$ 0 0
$$545$$ 9.18301 0.393357
$$546$$ 0 0
$$547$$ −5.70391 −0.243881 −0.121941 0.992537i $$-0.538912\pi$$
−0.121941 + 0.992537i $$0.538912\pi$$
$$548$$ 0 0
$$549$$ −1.62704 −0.0694403
$$550$$ 0 0
$$551$$ 10.8765i 0.463353i
$$552$$ 0 0
$$553$$ 35.3823i 1.50461i
$$554$$ 0 0
$$555$$ 15.2541 0.647500
$$556$$ 0 0
$$557$$ 20.8720i 0.884374i 0.896923 + 0.442187i $$0.145797\pi$$
−0.896923 + 0.442187i $$0.854203\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 20.7973i − 0.878064i
$$562$$ 0 0
$$563$$ 33.2315 1.40054 0.700270 0.713878i $$-0.253063\pi$$
0.700270 + 0.713878i $$0.253063\pi$$
$$564$$ 0 0
$$565$$ − 5.77586i − 0.242992i
$$566$$ 0 0
$$567$$ 32.4435i 1.36250i
$$568$$ 0 0
$$569$$ 40.0671 1.67970 0.839851 0.542817i $$-0.182642\pi$$
0.839851 + 0.542817i $$0.182642\pi$$
$$570$$ 0 0
$$571$$ −4.53590 −0.189821 −0.0949107 0.995486i $$-0.530257\pi$$
−0.0949107 + 0.995486i $$0.530257\pi$$
$$572$$ 0 0
$$573$$ 9.55117 0.399006
$$574$$ 0 0
$$575$$ −1.16082 −0.0484096
$$576$$ 0 0
$$577$$ 35.2706i 1.46834i 0.678968 + 0.734168i $$0.262427\pi$$
−0.678968 + 0.734168i $$0.737573\pi$$
$$578$$ 0 0
$$579$$ − 14.4861i − 0.602023i
$$580$$ 0 0
$$581$$ −1.97879 −0.0820942
$$582$$ 0 0
$$583$$ − 21.9054i − 0.907230i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 41.1663i − 1.69912i −0.527496 0.849558i $$-0.676869\pi$$
0.527496 0.849558i $$-0.323131\pi$$
$$588$$ 0 0
$$589$$ 42.2487 1.74083
$$590$$ 0 0
$$591$$ − 18.2247i − 0.749664i
$$592$$ 0 0
$$593$$ − 9.39726i − 0.385899i −0.981209 0.192950i $$-0.938195\pi$$
0.981209 0.192950i $$-0.0618054\pi$$
$$594$$ 0 0
$$595$$ −32.5078 −1.33269
$$596$$ 0 0
$$597$$ −17.2457 −0.705821
$$598$$ 0 0
$$599$$ 21.7881 0.890239 0.445119 0.895471i $$-0.353161\pi$$
0.445119 + 0.895471i $$0.353161\pi$$
$$600$$ 0 0
$$601$$ 25.0962 1.02370 0.511848 0.859076i $$-0.328961\pi$$
0.511848 + 0.859076i $$0.328961\pi$$
$$602$$ 0 0
$$603$$ 2.31130i 0.0941236i
$$604$$ 0 0
$$605$$ 8.00000i 0.325246i
$$606$$ 0 0
$$607$$ −33.9775 −1.37910 −0.689552 0.724236i $$-0.742193\pi$$
−0.689552 + 0.724236i $$0.742193\pi$$
$$608$$ 0 0
$$609$$ 14.0363i 0.568780i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 21.4470i 0.866235i 0.901337 + 0.433118i $$0.142587\pi$$
−0.901337 + 0.433118i $$0.857413\pi$$
$$614$$ 0 0
$$615$$ −12.3728 −0.498919
$$616$$ 0 0
$$617$$ − 34.0743i − 1.37178i −0.727705 0.685890i $$-0.759413\pi$$
0.727705 0.685890i $$-0.240587\pi$$
$$618$$ 0 0
$$619$$ 26.9791i 1.08438i 0.840256 + 0.542191i $$0.182405\pi$$
−0.840256 + 0.542191i $$0.817595\pi$$
$$620$$ 0 0
$$621$$ −6.38878 −0.256373
$$622$$ 0 0
$$623$$ −57.5118 −2.30416
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 14.8886 0.594595
$$628$$ 0 0
$$629$$ 71.5298i 2.85208i
$$630$$ 0 0
$$631$$ 37.9738i 1.51171i 0.654737 + 0.755857i $$0.272779\pi$$
−0.654737 + 0.755857i $$0.727221\pi$$
$$632$$ 0 0
$$633$$ −5.16082 −0.205124
$$634$$ 0 0
$$635$$ − 18.9678i − 0.752713i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 5.49844i − 0.217515i
$$640$$ 0 0
$$641$$ 8.25180 0.325927 0.162963 0.986632i $$-0.447895\pi$$
0.162963 + 0.986632i $$0.447895\pi$$
$$642$$ 0 0
$$643$$ − 48.1501i − 1.89885i −0.313989 0.949427i $$-0.601666\pi$$
0.313989 0.949427i $$-0.398334\pi$$
$$644$$ 0 0
$$645$$ − 6.69770i − 0.263722i
$$646$$ 0 0
$$647$$ 2.17903 0.0856665 0.0428332 0.999082i $$-0.486362\pi$$
0.0428332 + 0.999082i $$0.486362\pi$$
$$648$$ 0 0
$$649$$ 10.9935 0.431532
$$650$$ 0 0
$$651$$ 54.5229 2.13692
$$652$$ 0 0
$$653$$ 33.8682 1.32537 0.662684 0.748900i $$-0.269417\pi$$
0.662684 + 0.748900i $$0.269417\pi$$
$$654$$ 0 0
$$655$$ − 1.34637i − 0.0526071i
$$656$$ 0 0
$$657$$ 2.30187i 0.0898046i
$$658$$ 0 0
$$659$$ 26.4491 1.03031 0.515155 0.857097i $$-0.327734\pi$$
0.515155 + 0.857097i $$0.327734\pi$$
$$660$$ 0 0
$$661$$ − 30.3985i − 1.18236i −0.806538 0.591182i $$-0.798661\pi$$
0.806538 0.591182i $$-0.201339\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 23.2720i − 0.902451i
$$666$$ 0 0
$$667$$ −2.35035 −0.0910059
$$668$$ 0 0
$$669$$ − 27.1703i − 1.05046i
$$670$$ 0 0
$$671$$ 6.41393i 0.247607i
$$672$$ 0 0
$$673$$ 36.3952 1.40293 0.701467 0.712702i $$-0.252529\pi$$
0.701467 + 0.712702i $$0.252529\pi$$
$$674$$ 0 0
$$675$$ 5.50367 0.211836
$$676$$ 0 0
$$677$$ 36.0043 1.38376 0.691880 0.722013i $$-0.256783\pi$$
0.691880 + 0.722013i $$0.256783\pi$$
$$678$$ 0 0
$$679$$ 12.1745 0.467215
$$680$$ 0 0
$$681$$ − 3.84130i − 0.147199i
$$682$$ 0 0
$$683$$ 12.8296i 0.490909i 0.969408 + 0.245455i $$0.0789372\pi$$
−0.969408 + 0.245455i $$0.921063\pi$$
$$684$$ 0 0
$$685$$ −12.3728 −0.472740
$$686$$ 0 0
$$687$$ − 11.7722i − 0.449137i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32.9792i 1.25459i 0.778783 + 0.627294i $$0.215838\pi$$
−0.778783 + 0.627294i $$0.784162\pi$$
$$692$$ 0 0
$$693$$ −3.29692 −0.125239
$$694$$ 0 0
$$695$$ 8.76836i 0.332603i
$$696$$ 0 0
$$697$$ − 58.0188i − 2.19762i
$$698$$ 0 0
$$699$$ −5.39922 −0.204217
$$700$$ 0 0
$$701$$ 40.7352 1.53855 0.769273 0.638921i $$-0.220619\pi$$
0.769273 + 0.638921i $$0.220619\pi$$
$$702$$ 0 0
$$703$$ −51.2076 −1.93133
$$704$$ 0 0
$$705$$ −5.54324 −0.208771
$$706$$ 0 0
$$707$$ − 28.7651i − 1.08182i
$$708$$ 0 0
$$709$$ 17.6591i 0.663202i 0.943420 + 0.331601i $$0.107589\pi$$
−0.943420 + 0.331601i $$0.892411\pi$$
$$710$$ 0 0
$$711$$ 3.58845 0.134577
$$712$$ 0 0
$$713$$ 9.12973i 0.341911i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 38.0537i − 1.42114i
$$718$$ 0 0
$$719$$ −49.5672 −1.84855 −0.924273 0.381733i $$-0.875327\pi$$
−0.924273 + 0.381733i $$0.875327\pi$$
$$720$$ 0 0
$$721$$ − 31.9549i − 1.19006i
$$722$$ 0 0
$$723$$ − 35.7183i − 1.32838i
$$724$$ 0 0
$$725$$ 2.02473 0.0751965
$$726$$ 0 0
$$727$$ −36.4738 −1.35274 −0.676370 0.736562i $$-0.736448\pi$$
−0.676370 + 0.736562i $$0.736448\pi$$
$$728$$ 0 0
$$729$$ 29.7112 1.10042
$$730$$ 0 0
$$731$$ 31.4070 1.16163
$$732$$ 0 0
$$733$$ 28.8491i 1.06556i 0.846252 + 0.532782i $$0.178854\pi$$
−0.846252 + 0.532782i $$0.821146\pi$$
$$734$$ 0 0
$$735$$ − 18.8317i − 0.694617i
$$736$$ 0 0
$$737$$ 9.11137 0.335621
$$738$$ 0 0
$$739$$ 19.7835i 0.727746i 0.931448 + 0.363873i $$0.118546\pi$$
−0.931448 + 0.363873i $$0.881454\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 26.4005i − 0.968541i −0.874918 0.484271i $$-0.839085\pi$$
0.874918 0.484271i $$-0.160915\pi$$
$$744$$ 0 0
$$745$$ 4.73939 0.173638
$$746$$ 0 0
$$747$$ 0.200688i 0.00734279i
$$748$$ 0 0
$$749$$ 34.2826i 1.25266i
$$750$$ 0 0
$$751$$ 13.8034 0.503694 0.251847 0.967767i $$-0.418962\pi$$
0.251847 + 0.967767i $$0.418962\pi$$
$$752$$ 0 0
$$753$$ 7.39671 0.269551
$$754$$ 0 0
$$755$$ 1.56063 0.0567970
$$756$$ 0 0
$$757$$ −5.18160 −0.188328 −0.0941642 0.995557i $$-0.530018\pi$$
−0.0941642 + 0.995557i $$0.530018\pi$$
$$758$$ 0 0
$$759$$ 3.21736i 0.116783i
$$760$$ 0 0
$$761$$ − 0.00858275i 0 0.000311124i −1.00000 0.000155562i $$-0.999950\pi$$
1.00000 0.000155562i $$-4.95170e-5\pi$$
$$762$$ 0 0
$$763$$ −39.7830 −1.44024
$$764$$ 0 0
$$765$$ 3.29692i 0.119200i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 11.8483i 0.427262i 0.976914 + 0.213631i $$0.0685291\pi$$
−0.976914 + 0.213631i $$0.931471\pi$$
$$770$$ 0 0
$$771$$ −34.9865 −1.26001
$$772$$ 0 0
$$773$$ 21.0458i 0.756964i 0.925609 + 0.378482i $$0.123554\pi$$
−0.925609 + 0.378482i $$0.876446\pi$$
$$774$$ 0 0
$$775$$ − 7.86488i − 0.282515i
$$776$$ 0 0
$$777$$ −66.0845 −2.37077
$$778$$ 0 0
$$779$$ 41.5352 1.48815
$$780$$ 0 0
$$781$$ −21.6754 −0.775605
$$782$$ 0 0
$$783$$ 11.1434 0.398234
$$784$$ 0 0
$$785$$ − 9.15332i − 0.326696i
$$786$$ 0 0
$$787$$ 5.01174i 0.178649i 0.996003 + 0.0893246i $$0.0284708\pi$$
−0.996003 + 0.0893246i $$0.971529\pi$$
$$788$$ 0 0
$$789$$ −32.2904 −1.14957
$$790$$ 0 0
$$791$$ 25.0224i 0.889696i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 20.2378i − 0.717762i
$$796$$ 0 0
$$797$$ −40.6566 −1.44013 −0.720065 0.693907i $$-0.755888\pi$$
−0.720065 + 0.693907i $$0.755888\pi$$
$$798$$ 0 0
$$799$$ − 25.9935i − 0.919583i
$$800$$ 0 0
$$801$$ 5.83281i 0.206092i
$$802$$ 0 0
$$803$$ 9.07418 0.320221
$$804$$ 0 0
$$805$$ 5.02897 0.177248
$$806$$ 0 0
$$807$$ 11.6604 0.410466
$$808$$ 0 0
$$809$$ 5.41351 0.190329 0.0951644 0.995462i $$-0.469662\pi$$
0.0951644 + 0.995462i $$0.469662\pi$$
$$810$$ 0 0
$$811$$ − 0.616994i − 0.0216656i −0.999941 0.0108328i $$-0.996552\pi$$
0.999941 0.0108328i $$-0.00344825\pi$$
$$812$$ 0 0
$$813$$ − 7.82799i − 0.274540i
$$814$$ 0 0
$$815$$ 2.64303 0.0925811
$$816$$ 0 0
$$817$$ 22.4840i 0.786616i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 22.9066i − 0.799445i −0.916636 0.399723i $$-0.869106\pi$$
0.916636 0.399723i $$-0.130894\pi$$
$$822$$ 0 0
$$823$$ −9.65856 −0.336676 −0.168338 0.985729i $$-0.553840\pi$$
−0.168338 + 0.985729i $$0.553840\pi$$
$$824$$ 0 0
$$825$$ − 2.77162i − 0.0964954i
$$826$$ 0 0
$$827$$ 21.2602i 0.739289i 0.929173 + 0.369645i $$0.120521\pi$$
−0.929173 + 0.369645i $$0.879479\pi$$
$$828$$ 0 0
$$829$$ 23.9895 0.833191 0.416595 0.909092i $$-0.363223\pi$$
0.416595 + 0.909092i $$0.363223\pi$$
$$830$$ 0 0
$$831$$ −14.0827 −0.488522
$$832$$ 0 0
$$833$$ 88.3059 3.05962
$$834$$ 0 0
$$835$$ −8.90869 −0.308298
$$836$$ 0 0
$$837$$ − 43.2857i − 1.49617i
$$838$$ 0 0
$$839$$ 21.3055i 0.735548i 0.929915 + 0.367774i $$0.119880\pi$$
−0.929915 + 0.367774i $$0.880120\pi$$
$$840$$ 0 0
$$841$$ −24.9005 −0.858637
$$842$$ 0 0
$$843$$ 20.1761i 0.694902i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 34.6580i − 1.19086i
$$848$$ 0 0
$$849$$ 33.4456 1.14785
$$850$$ 0 0
$$851$$ − 11.0657i − 0.379327i
$$852$$ 0 0
$$853$$ 12.1098i 0.414631i 0.978274 + 0.207315i $$0.0664726\pi$$
−0.978274 + 0.207315i $$0.933527\pi$$
$$854$$ 0 0
$$855$$ −2.36023 −0.0807183
$$856$$ 0 0
$$857$$ 35.4491 1.21092 0.605459 0.795876i $$-0.292990\pi$$
0.605459 + 0.795876i $$0.292990\pi$$
$$858$$ 0 0
$$859$$ −47.4600 −1.61931 −0.809657 0.586904i $$-0.800347\pi$$
−0.809657 + 0.586904i $$0.800347\pi$$
$$860$$ 0 0
$$861$$ 53.6020 1.82675
$$862$$ 0 0
$$863$$ − 18.7268i − 0.637467i −0.947844 0.318733i $$-0.896743\pi$$
0.947844 0.318733i $$-0.103257\pi$$
$$864$$ 0 0
$$865$$ − 13.8079i − 0.469484i
$$866$$ 0 0
$$867$$ 62.8958 2.13605
$$868$$ 0 0
$$869$$ − 14.1460i − 0.479870i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 1.23473i − 0.0417894i
$$874$$ 0 0
$$875$$ −4.33225 −0.146457
$$876$$ 0 0
$$877$$ − 1.81517i − 0.0612938i −0.999530 0.0306469i $$-0.990243\pi$$
0.999530 0.0306469i $$-0.00975674\pi$$
$$878$$ 0 0
$$879$$ 1.10811i 0.0373755i
$$880$$ 0 0
$$881$$ 50.3437 1.69612 0.848061 0.529899i $$-0.177770\pi$$
0.848061 + 0.529899i $$0.177770\pi$$
$$882$$ 0 0
$$883$$ 4.42003 0.148746 0.0743729 0.997230i $$-0.476304\pi$$
0.0743729 + 0.997230i $$0.476304\pi$$
$$884$$ 0 0
$$885$$ 10.1566 0.341410
$$886$$ 0 0
$$887$$ 2.69134 0.0903662 0.0451831 0.998979i $$-0.485613\pi$$
0.0451831 + 0.998979i $$0.485613\pi$$
$$888$$ 0 0
$$889$$ 82.1731i 2.75600i
$$890$$ 0 0
$$891$$ − 12.9710i − 0.434546i
$$892$$ 0 0
$$893$$ 18.6085 0.622710
$$894$$ 0 0
$$895$$ 24.3502i 0.813937i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 15.9243i − 0.531103i
$$900$$ 0 0
$$901$$ 94.8997 3.16157
$$902$$ 0 0
$$903$$ 29.0161i 0.965595i
$$904$$ 0 0
$$905$$ 24.8336i 0.825498i
$$906$$ 0 0
$$907$$ −14.4017 −0.478199 −0.239100 0.970995i $$-0.576852\pi$$
−0.239100 + 0.970995i $$0.576852\pi$$
$$908$$ 0 0
$$909$$ −2.91734 −0.0967620
$$910$$ 0 0
$$911$$ −8.88723 −0.294447 −0.147223 0.989103i $$-0.547034\pi$$
−0.147223 + 0.989103i $$0.547034\pi$$
$$912$$ 0 0
$$913$$ 0.791131 0.0261826
$$914$$ 0 0
$$915$$ 5.92566i 0.195896i
$$916$$ 0 0
$$917$$ 5.83281i 0.192617i
$$918$$ 0 0
$$919$$ −42.9747 −1.41760 −0.708802 0.705408i $$-0.750764\pi$$
−0.708802 + 0.705408i $$0.750764\pi$$
$$920$$ 0 0
$$921$$ − 14.1324i − 0.465679i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 9.53264i 0.313431i
$$926$$ 0 0
$$927$$ −3.24084 −0.106443
$$928$$ 0 0
$$929$$ 29.7378i 0.975666i 0.872937 + 0.487833i $$0.162213\pi$$
−0.872937 + 0.487833i $$0.837787\pi$$
$$930$$ 0 0
$$931$$ 63.2175i 2.07187i
$$932$$ 0 0
$$933$$ −18.2356 −0.597005
$$934$$ 0 0
$$935$$ 12.9967 0.425039
$$936$$ 0 0
$$937$$ 17.1355 0.559793 0.279896 0.960030i $$-0.409700\pi$$
0.279896 + 0.960030i $$0.409700\pi$$
$$938$$ 0 0
$$939$$ −5.41660 −0.176764
$$940$$ 0 0
$$941$$ − 23.5767i − 0.768580i −0.923212 0.384290i $$-0.874446\pi$$
0.923212 0.384290i $$-0.125554\pi$$
$$942$$ 0 0
$$943$$ 8.97553i 0.292284i
$$944$$ 0 0
$$945$$ −23.8433 −0.775621
$$946$$ 0 0
$$947$$ 17.6272i 0.572807i 0.958109 + 0.286404i $$0.0924598\pi$$
−0.958109 + 0.286404i $$0.907540\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0.891475i 0.0289080i
$$952$$ 0 0
$$953$$ 58.7631 1.90352 0.951762 0.306837i $$-0.0992707\pi$$
0.951762 + 0.306837i $$0.0992707\pi$$
$$954$$ 0 0
$$955$$ 5.96875i 0.193144i
$$956$$ 0 0
$$957$$ − 5.61178i − 0.181403i
$$958$$ 0 0
$$959$$ 53.6020 1.73090
$$960$$ 0 0
$$961$$ −30.8564 −0.995368
$$962$$ 0 0
$$963$$ 3.47692 0.112042
$$964$$ 0 0
$$965$$ 9.05272 0.291417
$$966$$ 0 0
$$967$$ − 2.92168i − 0.0939550i −0.998896 0.0469775i $$-0.985041\pi$$
0.998896 0.0469775i $$-0.0149589\pi$$
$$968$$ 0 0
$$969$$ 64.5012i 2.07208i
$$970$$ 0 0
$$971$$ −10.0139 −0.321360 −0.160680 0.987007i $$-0.551369\pi$$
−0.160680 + 0.987007i $$0.551369\pi$$
$$972$$ 0 0
$$973$$ − 37.9867i − 1.21780i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 26.5301i − 0.848772i −0.905481 0.424386i $$-0.860490\pi$$
0.905481 0.424386i $$-0.139510\pi$$
$$978$$ 0 0
$$979$$ 22.9935 0.734875
$$980$$ 0 0
$$981$$ 4.03477i 0.128820i
$$982$$ 0 0
$$983$$ − 4.69317i − 0.149689i −0.997195 0.0748445i $$-0.976154\pi$$
0.997195 0.0748445i $$-0.0238460\pi$$
$$984$$ 0 0
$$985$$ 11.3890 0.362885
$$986$$ 0 0
$$987$$ 24.0147 0.764396
$$988$$ 0 0
$$989$$ −4.85868 −0.154497
$$990$$ 0 0
$$991$$ −22.9201 −0.728081 −0.364041 0.931383i $$-0.618603\pi$$
−0.364041 + 0.931383i $$0.618603\pi$$
$$992$$ 0 0
$$993$$ − 6.34268i − 0.201279i
$$994$$ 0 0
$$995$$ − 10.7773i − 0.341662i
$$996$$ 0 0
$$997$$ 49.4426 1.56586 0.782932 0.622108i $$-0.213723\pi$$
0.782932 + 0.622108i $$0.213723\pi$$
$$998$$ 0 0
$$999$$ 52.4645i 1.65990i
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 3380.2.f.i.3041.6 8
13.5 odd 4 3380.2.a.q.1.3 4
13.8 odd 4 3380.2.a.p.1.3 4
13.9 even 3 260.2.x.a.101.2 8
13.10 even 6 260.2.x.a.121.2 yes 8
13.12 even 2 inner 3380.2.f.i.3041.5 8
39.23 odd 6 2340.2.dj.d.901.4 8
39.35 odd 6 2340.2.dj.d.361.2 8
52.23 odd 6 1040.2.da.c.641.3 8
52.35 odd 6 1040.2.da.c.881.3 8
65.9 even 6 1300.2.y.b.101.3 8
65.22 odd 12 1300.2.ba.c.49.1 8
65.23 odd 12 1300.2.ba.c.849.1 8
65.48 odd 12 1300.2.ba.b.49.4 8
65.49 even 6 1300.2.y.b.901.3 8
65.62 odd 12 1300.2.ba.b.849.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.x.a.101.2 8 13.9 even 3
260.2.x.a.121.2 yes 8 13.10 even 6
1040.2.da.c.641.3 8 52.23 odd 6
1040.2.da.c.881.3 8 52.35 odd 6
1300.2.y.b.101.3 8 65.9 even 6
1300.2.y.b.901.3 8 65.49 even 6
1300.2.ba.b.49.4 8 65.48 odd 12
1300.2.ba.b.849.4 8 65.62 odd 12
1300.2.ba.c.49.1 8 65.22 odd 12
1300.2.ba.c.849.1 8 65.23 odd 12
2340.2.dj.d.361.2 8 39.35 odd 6
2340.2.dj.d.901.4 8 39.23 odd 6
3380.2.a.p.1.3 4 13.8 odd 4
3380.2.a.q.1.3 4 13.5 odd 4
3380.2.f.i.3041.5 8 13.12 even 2 inner
3380.2.f.i.3041.6 8 1.1 even 1 trivial