# Properties

 Label 4275.2.a.x.1.1 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$34.1360468641$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} -1.82843 q^{4} -0.585786 q^{7} +1.58579 q^{8} +O(q^{10})$$ $$q-0.414214 q^{2} -1.82843 q^{4} -0.585786 q^{7} +1.58579 q^{8} -1.41421 q^{11} -5.41421 q^{13} +0.242641 q^{14} +3.00000 q^{16} -1.17157 q^{17} +1.00000 q^{19} +0.585786 q^{22} +7.65685 q^{23} +2.24264 q^{26} +1.07107 q^{28} +9.07107 q^{29} +6.48528 q^{31} -4.41421 q^{32} +0.485281 q^{34} -11.0711 q^{37} -0.414214 q^{38} +7.41421 q^{41} -0.585786 q^{43} +2.58579 q^{44} -3.17157 q^{46} +0.343146 q^{47} -6.65685 q^{49} +9.89949 q^{52} +4.00000 q^{53} -0.928932 q^{56} -3.75736 q^{58} -8.48528 q^{59} +5.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} -12.0000 q^{67} +2.14214 q^{68} +4.48528 q^{71} +2.00000 q^{73} +4.58579 q^{74} -1.82843 q^{76} +0.828427 q^{77} -11.3137 q^{79} -3.07107 q^{82} -10.4853 q^{83} +0.242641 q^{86} -2.24264 q^{88} -10.7279 q^{89} +3.17157 q^{91} -14.0000 q^{92} -0.142136 q^{94} +4.24264 q^{97} +2.75736 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8} - 8 q^{13} - 8 q^{14} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} - 4 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 2 q^{38} + 12 q^{41} - 4 q^{43} + 8 q^{44} - 12 q^{46} + 12 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{56} - 16 q^{58} - 28 q^{62} - 14 q^{64} - 24 q^{67} - 24 q^{68} - 8 q^{71} + 4 q^{73} + 12 q^{74} + 2 q^{76} - 4 q^{77} + 8 q^{82} - 4 q^{83} - 8 q^{86} + 4 q^{88} + 4 q^{89} + 12 q^{91} - 28 q^{92} + 28 q^{94} + 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 6 * q^8 - 8 * q^13 - 8 * q^14 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 - 4 * q^26 - 12 * q^28 + 4 * q^29 - 4 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 2 * q^38 + 12 * q^41 - 4 * q^43 + 8 * q^44 - 12 * q^46 + 12 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^56 - 16 * q^58 - 28 * q^62 - 14 * q^64 - 24 * q^67 - 24 * q^68 - 8 * q^71 + 4 * q^73 + 12 * q^74 + 2 * q^76 - 4 * q^77 + 8 * q^82 - 4 * q^83 - 8 * q^86 + 4 * q^88 + 4 * q^89 + 12 * q^91 - 28 * q^92 + 28 * q^94 + 14 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 0 0
$$4$$ −1.82843 −0.914214
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.585786 −0.221406 −0.110703 0.993854i $$-0.535310\pi$$
−0.110703 + 0.993854i $$0.535310\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.41421 −0.426401 −0.213201 0.977008i $$-0.568389\pi$$
−0.213201 + 0.977008i $$0.568389\pi$$
$$12$$ 0 0
$$13$$ −5.41421 −1.50163 −0.750816 0.660511i $$-0.770340\pi$$
−0.750816 + 0.660511i $$0.770340\pi$$
$$14$$ 0.242641 0.0648485
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0.585786 0.124890
$$23$$ 7.65685 1.59656 0.798282 0.602284i $$-0.205742\pi$$
0.798282 + 0.602284i $$0.205742\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.24264 0.439818
$$27$$ 0 0
$$28$$ 1.07107 0.202413
$$29$$ 9.07107 1.68446 0.842228 0.539122i $$-0.181244\pi$$
0.842228 + 0.539122i $$0.181244\pi$$
$$30$$ 0 0
$$31$$ 6.48528 1.16479 0.582395 0.812906i $$-0.302116\pi$$
0.582395 + 0.812906i $$0.302116\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 0 0
$$34$$ 0.485281 0.0832251
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −11.0711 −1.82007 −0.910036 0.414529i $$-0.863946\pi$$
−0.910036 + 0.414529i $$0.863946\pi$$
$$38$$ −0.414214 −0.0671943
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.41421 1.15791 0.578953 0.815361i $$-0.303462\pi$$
0.578953 + 0.815361i $$0.303462\pi$$
$$42$$ 0 0
$$43$$ −0.585786 −0.0893316 −0.0446658 0.999002i $$-0.514222\pi$$
−0.0446658 + 0.999002i $$0.514222\pi$$
$$44$$ 2.58579 0.389822
$$45$$ 0 0
$$46$$ −3.17157 −0.467623
$$47$$ 0.343146 0.0500530 0.0250265 0.999687i $$-0.492033\pi$$
0.0250265 + 0.999687i $$0.492033\pi$$
$$48$$ 0 0
$$49$$ −6.65685 −0.950979
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 9.89949 1.37281
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −0.928932 −0.124134
$$57$$ 0 0
$$58$$ −3.75736 −0.493365
$$59$$ −8.48528 −1.10469 −0.552345 0.833616i $$-0.686267\pi$$
−0.552345 + 0.833616i $$0.686267\pi$$
$$60$$ 0 0
$$61$$ 5.65685 0.724286 0.362143 0.932123i $$-0.382045\pi$$
0.362143 + 0.932123i $$0.382045\pi$$
$$62$$ −2.68629 −0.341159
$$63$$ 0 0
$$64$$ −4.17157 −0.521447
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 2.14214 0.259772
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.48528 0.532305 0.266152 0.963931i $$-0.414248\pi$$
0.266152 + 0.963931i $$0.414248\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 4.58579 0.533087
$$75$$ 0 0
$$76$$ −1.82843 −0.209735
$$77$$ 0.828427 0.0944080
$$78$$ 0 0
$$79$$ −11.3137 −1.27289 −0.636446 0.771321i $$-0.719596\pi$$
−0.636446 + 0.771321i $$0.719596\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −3.07107 −0.339143
$$83$$ −10.4853 −1.15091 −0.575455 0.817834i $$-0.695175\pi$$
−0.575455 + 0.817834i $$0.695175\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0.242641 0.0261646
$$87$$ 0 0
$$88$$ −2.24264 −0.239066
$$89$$ −10.7279 −1.13716 −0.568579 0.822629i $$-0.692507\pi$$
−0.568579 + 0.822629i $$0.692507\pi$$
$$90$$ 0 0
$$91$$ 3.17157 0.332471
$$92$$ −14.0000 −1.45960
$$93$$ 0 0
$$94$$ −0.142136 −0.0146602
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.24264 0.430775 0.215387 0.976529i $$-0.430899\pi$$
0.215387 + 0.976529i $$0.430899\pi$$
$$98$$ 2.75736 0.278535
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.82843 0.480446 0.240223 0.970718i $$-0.422779\pi$$
0.240223 + 0.970718i $$0.422779\pi$$
$$102$$ 0 0
$$103$$ 1.65685 0.163255 0.0816274 0.996663i $$-0.473988\pi$$
0.0816274 + 0.996663i $$0.473988\pi$$
$$104$$ −8.58579 −0.841906
$$105$$ 0 0
$$106$$ −1.65685 −0.160928
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 3.17157 0.303782 0.151891 0.988397i $$-0.451464\pi$$
0.151891 + 0.988397i $$0.451464\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.75736 −0.166055
$$113$$ 12.4853 1.17452 0.587258 0.809400i $$-0.300207\pi$$
0.587258 + 0.809400i $$0.300207\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −16.5858 −1.53995
$$117$$ 0 0
$$118$$ 3.51472 0.323556
$$119$$ 0.686292 0.0629122
$$120$$ 0 0
$$121$$ −9.00000 −0.818182
$$122$$ −2.34315 −0.212138
$$123$$ 0 0
$$124$$ −11.8579 −1.06487
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 10.5563 0.933058
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.7279 −1.46153 −0.730763 0.682632i $$-0.760835\pi$$
−0.730763 + 0.682632i $$0.760835\pi$$
$$132$$ 0 0
$$133$$ −0.585786 −0.0507941
$$134$$ 4.97056 0.429391
$$135$$ 0 0
$$136$$ −1.85786 −0.159311
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 1.17157 0.0993715 0.0496858 0.998765i $$-0.484178\pi$$
0.0496858 + 0.998765i $$0.484178\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1.85786 −0.155909
$$143$$ 7.65685 0.640298
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −0.828427 −0.0685611
$$147$$ 0 0
$$148$$ 20.2426 1.66393
$$149$$ 3.65685 0.299581 0.149791 0.988718i $$-0.452140\pi$$
0.149791 + 0.988718i $$0.452140\pi$$
$$150$$ 0 0
$$151$$ 17.7990 1.44846 0.724231 0.689558i $$-0.242195\pi$$
0.724231 + 0.689558i $$0.242195\pi$$
$$152$$ 1.58579 0.128624
$$153$$ 0 0
$$154$$ −0.343146 −0.0276515
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 21.7990 1.73975 0.869874 0.493273i $$-0.164200\pi$$
0.869874 + 0.493273i $$0.164200\pi$$
$$158$$ 4.68629 0.372821
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.48528 −0.353490
$$162$$ 0 0
$$163$$ −7.89949 −0.618736 −0.309368 0.950942i $$-0.600118\pi$$
−0.309368 + 0.950942i $$0.600118\pi$$
$$164$$ −13.5563 −1.05857
$$165$$ 0 0
$$166$$ 4.34315 0.337093
$$167$$ 10.0000 0.773823 0.386912 0.922117i $$-0.373542\pi$$
0.386912 + 0.922117i $$0.373542\pi$$
$$168$$ 0 0
$$169$$ 16.3137 1.25490
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.07107 0.0816682
$$173$$ −6.14214 −0.466978 −0.233489 0.972359i $$-0.575014\pi$$
−0.233489 + 0.972359i $$0.575014\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.24264 −0.319801
$$177$$ 0 0
$$178$$ 4.44365 0.333066
$$179$$ −17.1716 −1.28346 −0.641732 0.766929i $$-0.721784\pi$$
−0.641732 + 0.766929i $$0.721784\pi$$
$$180$$ 0 0
$$181$$ −19.1716 −1.42501 −0.712506 0.701666i $$-0.752440\pi$$
−0.712506 + 0.701666i $$0.752440\pi$$
$$182$$ −1.31371 −0.0973786
$$183$$ 0 0
$$184$$ 12.1421 0.895130
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.65685 0.121161
$$188$$ −0.627417 −0.0457591
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.89949 −0.137443 −0.0687213 0.997636i $$-0.521892\pi$$
−0.0687213 + 0.997636i $$0.521892\pi$$
$$192$$ 0 0
$$193$$ 15.0711 1.08484 0.542420 0.840108i $$-0.317508\pi$$
0.542420 + 0.840108i $$0.317508\pi$$
$$194$$ −1.75736 −0.126171
$$195$$ 0 0
$$196$$ 12.1716 0.869398
$$197$$ −14.8284 −1.05648 −0.528241 0.849095i $$-0.677148\pi$$
−0.528241 + 0.849095i $$0.677148\pi$$
$$198$$ 0 0
$$199$$ −16.4853 −1.16861 −0.584305 0.811534i $$-0.698633\pi$$
−0.584305 + 0.811534i $$0.698633\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −2.00000 −0.140720
$$203$$ −5.31371 −0.372949
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −0.686292 −0.0478162
$$207$$ 0 0
$$208$$ −16.2426 −1.12622
$$209$$ −1.41421 −0.0978232
$$210$$ 0 0
$$211$$ −15.3137 −1.05424 −0.527120 0.849791i $$-0.676728\pi$$
−0.527120 + 0.849791i $$0.676728\pi$$
$$212$$ −7.31371 −0.502308
$$213$$ 0 0
$$214$$ −3.31371 −0.226520
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.79899 −0.257892
$$218$$ −1.31371 −0.0889756
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.34315 0.426686
$$222$$ 0 0
$$223$$ −6.34315 −0.424768 −0.212384 0.977186i $$-0.568123\pi$$
−0.212384 + 0.977186i $$0.568123\pi$$
$$224$$ 2.58579 0.172770
$$225$$ 0 0
$$226$$ −5.17157 −0.344008
$$227$$ −18.9706 −1.25912 −0.629560 0.776952i $$-0.716765\pi$$
−0.629560 + 0.776952i $$0.716765\pi$$
$$228$$ 0 0
$$229$$ −1.65685 −0.109488 −0.0547440 0.998500i $$-0.517434\pi$$
−0.0547440 + 0.998500i $$0.517434\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 14.3848 0.944407
$$233$$ −8.34315 −0.546578 −0.273289 0.961932i $$-0.588111\pi$$
−0.273289 + 0.961932i $$0.588111\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 15.5147 1.00992
$$237$$ 0 0
$$238$$ −0.284271 −0.0184266
$$239$$ −2.58579 −0.167261 −0.0836303 0.996497i $$-0.526651\pi$$
−0.0836303 + 0.996497i $$0.526651\pi$$
$$240$$ 0 0
$$241$$ −14.9706 −0.964339 −0.482169 0.876078i $$-0.660151\pi$$
−0.482169 + 0.876078i $$0.660151\pi$$
$$242$$ 3.72792 0.239640
$$243$$ 0 0
$$244$$ −10.3431 −0.662152
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.41421 −0.344498
$$248$$ 10.2843 0.653052
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.9289 −0.816067 −0.408033 0.912967i $$-0.633785\pi$$
−0.408033 + 0.912967i $$0.633785\pi$$
$$252$$ 0 0
$$253$$ −10.8284 −0.680777
$$254$$ 3.31371 0.207921
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ 1.17157 0.0730807 0.0365404 0.999332i $$-0.488366\pi$$
0.0365404 + 0.999332i $$0.488366\pi$$
$$258$$ 0 0
$$259$$ 6.48528 0.402976
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6.92893 0.428071
$$263$$ −32.1421 −1.98197 −0.990984 0.133977i $$-0.957225\pi$$
−0.990984 + 0.133977i $$0.957225\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0.242641 0.0148773
$$267$$ 0 0
$$268$$ 21.9411 1.34027
$$269$$ −8.38478 −0.511229 −0.255614 0.966779i $$-0.582278\pi$$
−0.255614 + 0.966779i $$0.582278\pi$$
$$270$$ 0 0
$$271$$ −30.8284 −1.87269 −0.936347 0.351076i $$-0.885816\pi$$
−0.936347 + 0.351076i $$0.885816\pi$$
$$272$$ −3.51472 −0.213111
$$273$$ 0 0
$$274$$ 5.79899 0.350330
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.9706 −0.659157 −0.329579 0.944128i $$-0.606907\pi$$
−0.329579 + 0.944128i $$0.606907\pi$$
$$278$$ −0.485281 −0.0291052
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.7279 0.878594 0.439297 0.898342i $$-0.355228\pi$$
0.439297 + 0.898342i $$0.355228\pi$$
$$282$$ 0 0
$$283$$ −6.24264 −0.371086 −0.185543 0.982636i $$-0.559404\pi$$
−0.185543 + 0.982636i $$0.559404\pi$$
$$284$$ −8.20101 −0.486640
$$285$$ 0 0
$$286$$ −3.17157 −0.187539
$$287$$ −4.34315 −0.256368
$$288$$ 0 0
$$289$$ −15.6274 −0.919260
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.65685 −0.214001
$$293$$ −31.7990 −1.85772 −0.928858 0.370435i $$-0.879209\pi$$
−0.928858 + 0.370435i $$0.879209\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −17.5563 −1.02044
$$297$$ 0 0
$$298$$ −1.51472 −0.0877453
$$299$$ −41.4558 −2.39745
$$300$$ 0 0
$$301$$ 0.343146 0.0197786
$$302$$ −7.37258 −0.424244
$$303$$ 0 0
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −7.79899 −0.445112 −0.222556 0.974920i $$-0.571440\pi$$
−0.222556 + 0.974920i $$0.571440\pi$$
$$308$$ −1.51472 −0.0863091
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 32.2426 1.82831 0.914156 0.405362i $$-0.132855\pi$$
0.914156 + 0.405362i $$0.132855\pi$$
$$312$$ 0 0
$$313$$ 9.51472 0.537804 0.268902 0.963168i $$-0.413339\pi$$
0.268902 + 0.963168i $$0.413339\pi$$
$$314$$ −9.02944 −0.509561
$$315$$ 0 0
$$316$$ 20.6863 1.16369
$$317$$ 11.3137 0.635441 0.317721 0.948184i $$-0.397083\pi$$
0.317721 + 0.948184i $$0.397083\pi$$
$$318$$ 0 0
$$319$$ −12.8284 −0.718254
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1.85786 0.103535
$$323$$ −1.17157 −0.0651881
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 3.27208 0.181224
$$327$$ 0 0
$$328$$ 11.7574 0.649192
$$329$$ −0.201010 −0.0110820
$$330$$ 0 0
$$331$$ 7.17157 0.394185 0.197093 0.980385i $$-0.436850\pi$$
0.197093 + 0.980385i $$0.436850\pi$$
$$332$$ 19.1716 1.05218
$$333$$ 0 0
$$334$$ −4.14214 −0.226648
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −28.2426 −1.53847 −0.769237 0.638963i $$-0.779364\pi$$
−0.769237 + 0.638963i $$0.779364\pi$$
$$338$$ −6.75736 −0.367552
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −9.17157 −0.496669
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ −0.928932 −0.0500847
$$345$$ 0 0
$$346$$ 2.54416 0.136775
$$347$$ 10.4853 0.562879 0.281440 0.959579i $$-0.409188\pi$$
0.281440 + 0.959579i $$0.409188\pi$$
$$348$$ 0 0
$$349$$ 29.3137 1.56913 0.784563 0.620049i $$-0.212887\pi$$
0.784563 + 0.620049i $$0.212887\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6.24264 0.332734
$$353$$ 3.65685 0.194635 0.0973174 0.995253i $$-0.468974\pi$$
0.0973174 + 0.995253i $$0.468974\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 19.6152 1.03960
$$357$$ 0 0
$$358$$ 7.11270 0.375918
$$359$$ −9.89949 −0.522475 −0.261238 0.965275i $$-0.584131\pi$$
−0.261238 + 0.965275i $$0.584131\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 7.94113 0.417376
$$363$$ 0 0
$$364$$ −5.79899 −0.303950
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.4142 1.01341 0.506707 0.862118i $$-0.330863\pi$$
0.506707 + 0.862118i $$0.330863\pi$$
$$368$$ 22.9706 1.19742
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.34315 −0.121650
$$372$$ 0 0
$$373$$ 9.89949 0.512576 0.256288 0.966600i $$-0.417500\pi$$
0.256288 + 0.966600i $$0.417500\pi$$
$$374$$ −0.686292 −0.0354873
$$375$$ 0 0
$$376$$ 0.544156 0.0280627
$$377$$ −49.1127 −2.52943
$$378$$ 0 0
$$379$$ 24.1421 1.24010 0.620049 0.784563i $$-0.287113\pi$$
0.620049 + 0.784563i $$0.287113\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0.786797 0.0402560
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6.24264 −0.317742
$$387$$ 0 0
$$388$$ −7.75736 −0.393820
$$389$$ −14.9706 −0.759038 −0.379519 0.925184i $$-0.623910\pi$$
−0.379519 + 0.925184i $$0.623910\pi$$
$$390$$ 0 0
$$391$$ −8.97056 −0.453661
$$392$$ −10.5563 −0.533176
$$393$$ 0 0
$$394$$ 6.14214 0.309436
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −35.6569 −1.78957 −0.894783 0.446501i $$-0.852670\pi$$
−0.894783 + 0.446501i $$0.852670\pi$$
$$398$$ 6.82843 0.342278
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0416 −1.50021 −0.750104 0.661320i $$-0.769996\pi$$
−0.750104 + 0.661320i $$0.769996\pi$$
$$402$$ 0 0
$$403$$ −35.1127 −1.74909
$$404$$ −8.82843 −0.439231
$$405$$ 0 0
$$406$$ 2.20101 0.109234
$$407$$ 15.6569 0.776081
$$408$$ 0 0
$$409$$ 9.51472 0.470473 0.235236 0.971938i $$-0.424414\pi$$
0.235236 + 0.971938i $$0.424414\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −3.02944 −0.149250
$$413$$ 4.97056 0.244585
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 23.8995 1.17177
$$417$$ 0 0
$$418$$ 0.585786 0.0286518
$$419$$ 34.8701 1.70351 0.851757 0.523937i $$-0.175537\pi$$
0.851757 + 0.523937i $$0.175537\pi$$
$$420$$ 0 0
$$421$$ −14.6863 −0.715766 −0.357883 0.933766i $$-0.616501\pi$$
−0.357883 + 0.933766i $$0.616501\pi$$
$$422$$ 6.34315 0.308780
$$423$$ 0 0
$$424$$ 6.34315 0.308050
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3.31371 −0.160362
$$428$$ −14.6274 −0.707043
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −3.51472 −0.169298 −0.0846490 0.996411i $$-0.526977\pi$$
−0.0846490 + 0.996411i $$0.526977\pi$$
$$432$$ 0 0
$$433$$ −0.928932 −0.0446416 −0.0223208 0.999751i $$-0.507106\pi$$
−0.0223208 + 0.999751i $$0.507106\pi$$
$$434$$ 1.57359 0.0755349
$$435$$ 0 0
$$436$$ −5.79899 −0.277721
$$437$$ 7.65685 0.366277
$$438$$ 0 0
$$439$$ 0.970563 0.0463224 0.0231612 0.999732i $$-0.492627\pi$$
0.0231612 + 0.999732i $$0.492627\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −2.62742 −0.124973
$$443$$ −1.31371 −0.0624162 −0.0312081 0.999513i $$-0.509935\pi$$
−0.0312081 + 0.999513i $$0.509935\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.62742 0.124412
$$447$$ 0 0
$$448$$ 2.44365 0.115452
$$449$$ −7.89949 −0.372800 −0.186400 0.982474i $$-0.559682\pi$$
−0.186400 + 0.982474i $$0.559682\pi$$
$$450$$ 0 0
$$451$$ −10.4853 −0.493733
$$452$$ −22.8284 −1.07376
$$453$$ 0 0
$$454$$ 7.85786 0.368788
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −28.8284 −1.34854 −0.674268 0.738486i $$-0.735541\pi$$
−0.674268 + 0.738486i $$0.735541\pi$$
$$458$$ 0.686292 0.0320683
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −10.6863 −0.497710 −0.248855 0.968541i $$-0.580054\pi$$
−0.248855 + 0.968541i $$0.580054\pi$$
$$462$$ 0 0
$$463$$ 8.10051 0.376462 0.188231 0.982125i $$-0.439725\pi$$
0.188231 + 0.982125i $$0.439725\pi$$
$$464$$ 27.2132 1.26334
$$465$$ 0 0
$$466$$ 3.45584 0.160089
$$467$$ 16.3431 0.756271 0.378135 0.925750i $$-0.376565\pi$$
0.378135 + 0.925750i $$0.376565\pi$$
$$468$$ 0 0
$$469$$ 7.02944 0.324589
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −13.4558 −0.619355
$$473$$ 0.828427 0.0380911
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.25483 −0.0575152
$$477$$ 0 0
$$478$$ 1.07107 0.0489895
$$479$$ 38.1838 1.74466 0.872330 0.488917i $$-0.162608\pi$$
0.872330 + 0.488917i $$0.162608\pi$$
$$480$$ 0 0
$$481$$ 59.9411 2.73308
$$482$$ 6.20101 0.282448
$$483$$ 0 0
$$484$$ 16.4558 0.747993
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.82843 0.128168 0.0640841 0.997944i $$-0.479587\pi$$
0.0640841 + 0.997944i $$0.479587\pi$$
$$488$$ 8.97056 0.406078
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −21.8995 −0.988310 −0.494155 0.869374i $$-0.664523\pi$$
−0.494155 + 0.869374i $$0.664523\pi$$
$$492$$ 0 0
$$493$$ −10.6274 −0.478635
$$494$$ 2.24264 0.100901
$$495$$ 0 0
$$496$$ 19.4558 0.873593
$$497$$ −2.62742 −0.117856
$$498$$ 0 0
$$499$$ −23.7990 −1.06539 −0.532695 0.846308i $$-0.678821\pi$$
−0.532695 + 0.846308i $$0.678821\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 5.35534 0.239020
$$503$$ 0.828427 0.0369377 0.0184689 0.999829i $$-0.494121\pi$$
0.0184689 + 0.999829i $$0.494121\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 4.48528 0.199395
$$507$$ 0 0
$$508$$ 14.6274 0.648987
$$509$$ −32.3848 −1.43543 −0.717715 0.696337i $$-0.754812\pi$$
−0.717715 + 0.696337i $$0.754812\pi$$
$$510$$ 0 0
$$511$$ −1.17157 −0.0518273
$$512$$ −22.7574 −1.00574
$$513$$ 0 0
$$514$$ −0.485281 −0.0214048
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −0.485281 −0.0213427
$$518$$ −2.68629 −0.118029
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 24.3848 1.06832 0.534158 0.845385i $$-0.320629\pi$$
0.534158 + 0.845385i $$0.320629\pi$$
$$522$$ 0 0
$$523$$ 15.7990 0.690842 0.345421 0.938448i $$-0.387736\pi$$
0.345421 + 0.938448i $$0.387736\pi$$
$$524$$ 30.5858 1.33615
$$525$$ 0 0
$$526$$ 13.3137 0.580505
$$527$$ −7.59798 −0.330973
$$528$$ 0 0
$$529$$ 35.6274 1.54902
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 1.07107 0.0464367
$$533$$ −40.1421 −1.73875
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −19.0294 −0.821946
$$537$$ 0 0
$$538$$ 3.47309 0.149735
$$539$$ 9.41421 0.405499
$$540$$ 0 0
$$541$$ −32.6274 −1.40276 −0.701381 0.712786i $$-0.747433\pi$$
−0.701381 + 0.712786i $$0.747433\pi$$
$$542$$ 12.7696 0.548499
$$543$$ 0 0
$$544$$ 5.17157 0.221729
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 34.1421 1.45981 0.729906 0.683547i $$-0.239564\pi$$
0.729906 + 0.683547i $$0.239564\pi$$
$$548$$ 25.5980 1.09349
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9.07107 0.386440
$$552$$ 0 0
$$553$$ 6.62742 0.281826
$$554$$ 4.54416 0.193063
$$555$$ 0 0
$$556$$ −2.14214 −0.0908468
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ 0 0
$$559$$ 3.17157 0.134143
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.10051 −0.257334
$$563$$ 42.2843 1.78207 0.891035 0.453935i $$-0.149980\pi$$
0.891035 + 0.453935i $$0.149980\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2.58579 0.108689
$$567$$ 0 0
$$568$$ 7.11270 0.298442
$$569$$ 6.72792 0.282049 0.141025 0.990006i $$-0.454960\pi$$
0.141025 + 0.990006i $$0.454960\pi$$
$$570$$ 0 0
$$571$$ 19.7990 0.828562 0.414281 0.910149i $$-0.364033\pi$$
0.414281 + 0.910149i $$0.364033\pi$$
$$572$$ −14.0000 −0.585369
$$573$$ 0 0
$$574$$ 1.79899 0.0750884
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 37.7990 1.57359 0.786796 0.617213i $$-0.211738\pi$$
0.786796 + 0.617213i $$0.211738\pi$$
$$578$$ 6.47309 0.269245
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 6.14214 0.254819
$$582$$ 0 0
$$583$$ −5.65685 −0.234283
$$584$$ 3.17157 0.131241
$$585$$ 0 0
$$586$$ 13.1716 0.544113
$$587$$ 11.6569 0.481130 0.240565 0.970633i $$-0.422667\pi$$
0.240565 + 0.970633i $$0.422667\pi$$
$$588$$ 0 0
$$589$$ 6.48528 0.267221
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −33.2132 −1.36505
$$593$$ −29.3137 −1.20377 −0.601885 0.798583i $$-0.705583\pi$$
−0.601885 + 0.798583i $$0.705583\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.68629 −0.273881
$$597$$ 0 0
$$598$$ 17.1716 0.702198
$$599$$ 21.9411 0.896490 0.448245 0.893911i $$-0.352049\pi$$
0.448245 + 0.893911i $$0.352049\pi$$
$$600$$ 0 0
$$601$$ −28.8284 −1.17594 −0.587968 0.808884i $$-0.700072\pi$$
−0.587968 + 0.808884i $$0.700072\pi$$
$$602$$ −0.142136 −0.00579302
$$603$$ 0 0
$$604$$ −32.5442 −1.32420
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.14214 0.0869466 0.0434733 0.999055i $$-0.486158\pi$$
0.0434733 + 0.999055i $$0.486158\pi$$
$$608$$ −4.41421 −0.179020
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.85786 −0.0751611
$$612$$ 0 0
$$613$$ −25.1127 −1.01429 −0.507146 0.861860i $$-0.669300\pi$$
−0.507146 + 0.861860i $$0.669300\pi$$
$$614$$ 3.23045 0.130370
$$615$$ 0 0
$$616$$ 1.31371 0.0529308
$$617$$ 10.1421 0.408307 0.204154 0.978939i $$-0.434556\pi$$
0.204154 + 0.978939i $$0.434556\pi$$
$$618$$ 0 0
$$619$$ −43.7990 −1.76043 −0.880215 0.474575i $$-0.842602\pi$$
−0.880215 + 0.474575i $$0.842602\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −13.3553 −0.535500
$$623$$ 6.28427 0.251774
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −3.94113 −0.157519
$$627$$ 0 0
$$628$$ −39.8579 −1.59050
$$629$$ 12.9706 0.517170
$$630$$ 0 0
$$631$$ 22.6274 0.900783 0.450392 0.892831i $$-0.351284\pi$$
0.450392 + 0.892831i $$0.351284\pi$$
$$632$$ −17.9411 −0.713660
$$633$$ 0 0
$$634$$ −4.68629 −0.186116
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 36.0416 1.42802
$$638$$ 5.31371 0.210372
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.58579 0.339118 0.169559 0.985520i $$-0.445766\pi$$
0.169559 + 0.985520i $$0.445766\pi$$
$$642$$ 0 0
$$643$$ −6.04163 −0.238259 −0.119129 0.992879i $$-0.538010\pi$$
−0.119129 + 0.992879i $$0.538010\pi$$
$$644$$ 8.20101 0.323165
$$645$$ 0 0
$$646$$ 0.485281 0.0190931
$$647$$ 45.1127 1.77356 0.886782 0.462189i $$-0.152936\pi$$
0.886782 + 0.462189i $$0.152936\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 14.4437 0.565657
$$653$$ 42.4264 1.66027 0.830137 0.557560i $$-0.188262\pi$$
0.830137 + 0.557560i $$0.188262\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 22.2426 0.868429
$$657$$ 0 0
$$658$$ 0.0832611 0.00324586
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 18.4853 0.718994 0.359497 0.933146i $$-0.382948\pi$$
0.359497 + 0.933146i $$0.382948\pi$$
$$662$$ −2.97056 −0.115454
$$663$$ 0 0
$$664$$ −16.6274 −0.645269
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 69.4558 2.68934
$$668$$ −18.2843 −0.707440
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −21.8995 −0.844163 −0.422082 0.906558i $$-0.638700\pi$$
−0.422082 + 0.906558i $$0.638700\pi$$
$$674$$ 11.6985 0.450609
$$675$$ 0 0
$$676$$ −29.8284 −1.14725
$$677$$ −44.9706 −1.72836 −0.864180 0.503184i $$-0.832162\pi$$
−0.864180 + 0.503184i $$0.832162\pi$$
$$678$$ 0 0
$$679$$ −2.48528 −0.0953763
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3.79899 0.145471
$$683$$ 5.65685 0.216454 0.108227 0.994126i $$-0.465483\pi$$
0.108227 + 0.994126i $$0.465483\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −3.31371 −0.126518
$$687$$ 0 0
$$688$$ −1.75736 −0.0669987
$$689$$ −21.6569 −0.825060
$$690$$ 0 0
$$691$$ −23.1127 −0.879248 −0.439624 0.898182i $$-0.644888\pi$$
−0.439624 + 0.898182i $$0.644888\pi$$
$$692$$ 11.2304 0.426918
$$693$$ 0 0
$$694$$ −4.34315 −0.164864
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.68629 −0.329017
$$698$$ −12.1421 −0.459587
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0.343146 0.0129604 0.00648022 0.999979i $$-0.497937\pi$$
0.00648022 + 0.999979i $$0.497937\pi$$
$$702$$ 0 0
$$703$$ −11.0711 −0.417553
$$704$$ 5.89949 0.222346
$$705$$ 0 0
$$706$$ −1.51472 −0.0570072
$$707$$ −2.82843 −0.106374
$$708$$ 0 0
$$709$$ −35.3137 −1.32623 −0.663117 0.748516i $$-0.730767\pi$$
−0.663117 + 0.748516i $$0.730767\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −17.0122 −0.637559
$$713$$ 49.6569 1.85966
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 31.3970 1.17336
$$717$$ 0 0
$$718$$ 4.10051 0.153029
$$719$$ 16.4437 0.613245 0.306622 0.951831i $$-0.400801\pi$$
0.306622 + 0.951831i $$0.400801\pi$$
$$720$$ 0 0
$$721$$ −0.970563 −0.0361456
$$722$$ −0.414214 −0.0154154
$$723$$ 0 0
$$724$$ 35.0538 1.30277
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.58579 0.170077 0.0850387 0.996378i $$-0.472899\pi$$
0.0850387 + 0.996378i $$0.472899\pi$$
$$728$$ 5.02944 0.186403
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0.686292 0.0253834
$$732$$ 0 0
$$733$$ 1.31371 0.0485229 0.0242615 0.999706i $$-0.492277\pi$$
0.0242615 + 0.999706i $$0.492277\pi$$
$$734$$ −8.04163 −0.296822
$$735$$ 0 0
$$736$$ −33.7990 −1.24585
$$737$$ 16.9706 0.625119
$$738$$ 0 0
$$739$$ −9.65685 −0.355233 −0.177617 0.984100i $$-0.556839\pi$$
−0.177617 + 0.984100i $$0.556839\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.970563 0.0356305
$$743$$ −15.3137 −0.561805 −0.280903 0.959736i $$-0.590634\pi$$
−0.280903 + 0.959736i $$0.590634\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −4.10051 −0.150130
$$747$$ 0 0
$$748$$ −3.02944 −0.110767
$$749$$ −4.68629 −0.171233
$$750$$ 0 0
$$751$$ −37.1127 −1.35426 −0.677131 0.735863i $$-0.736777\pi$$
−0.677131 + 0.735863i $$0.736777\pi$$
$$752$$ 1.02944 0.0375397
$$753$$ 0 0
$$754$$ 20.3431 0.740854
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −16.8284 −0.611640 −0.305820 0.952089i $$-0.598931\pi$$
−0.305820 + 0.952089i $$0.598931\pi$$
$$758$$ −10.0000 −0.363216
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.2843 0.372805 0.186402 0.982474i $$-0.440317\pi$$
0.186402 + 0.982474i $$0.440317\pi$$
$$762$$ 0 0
$$763$$ −1.85786 −0.0672592
$$764$$ 3.47309 0.125652
$$765$$ 0 0
$$766$$ −4.97056 −0.179594
$$767$$ 45.9411 1.65884
$$768$$ 0 0
$$769$$ −35.6569 −1.28582 −0.642910 0.765942i $$-0.722273\pi$$
−0.642910 + 0.765942i $$0.722273\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −27.5563 −0.991775
$$773$$ −32.9706 −1.18587 −0.592934 0.805251i $$-0.702031\pi$$
−0.592934 + 0.805251i $$0.702031\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.72792 0.241518
$$777$$ 0 0
$$778$$ 6.20101 0.222317
$$779$$ 7.41421 0.265642
$$780$$ 0 0
$$781$$ −6.34315 −0.226976
$$782$$ 3.71573 0.132874
$$783$$ 0 0
$$784$$ −19.9706 −0.713234
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −13.4558 −0.479649 −0.239825 0.970816i $$-0.577090\pi$$
−0.239825 + 0.970816i $$0.577090\pi$$
$$788$$ 27.1127 0.965850
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7.31371 −0.260046
$$792$$ 0 0
$$793$$ −30.6274 −1.08761
$$794$$ 14.7696 0.524152
$$795$$ 0 0
$$796$$ 30.1421 1.06836
$$797$$ 10.8284 0.383563 0.191781 0.981438i $$-0.438574\pi$$
0.191781 + 0.981438i $$0.438574\pi$$
$$798$$ 0 0
$$799$$ −0.402020 −0.0142225
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.4437 0.439401
$$803$$ −2.82843 −0.0998130
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 14.5442 0.512296
$$807$$ 0 0
$$808$$ 7.65685 0.269367
$$809$$ −49.3137 −1.73378 −0.866889 0.498502i $$-0.833884\pi$$
−0.866889 + 0.498502i $$0.833884\pi$$
$$810$$ 0 0
$$811$$ −15.3137 −0.537737 −0.268869 0.963177i $$-0.586650\pi$$
−0.268869 + 0.963177i $$0.586650\pi$$
$$812$$ 9.71573 0.340955
$$813$$ 0 0
$$814$$ −6.48528 −0.227309
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −0.585786 −0.0204941
$$818$$ −3.94113 −0.137798
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −51.4558 −1.79582 −0.897911 0.440178i $$-0.854915\pi$$
−0.897911 + 0.440178i $$0.854915\pi$$
$$822$$ 0 0
$$823$$ 2.72792 0.0950894 0.0475447 0.998869i $$-0.484860\pi$$
0.0475447 + 0.998869i $$0.484860\pi$$
$$824$$ 2.62742 0.0915304
$$825$$ 0 0
$$826$$ −2.05887 −0.0716374
$$827$$ 48.6274 1.69094 0.845470 0.534022i $$-0.179320\pi$$
0.845470 + 0.534022i $$0.179320\pi$$
$$828$$ 0 0
$$829$$ −38.4853 −1.33665 −0.668325 0.743870i $$-0.732988\pi$$
−0.668325 + 0.743870i $$0.732988\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 22.5858 0.783021
$$833$$ 7.79899 0.270219
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.58579 0.0894313
$$837$$ 0 0
$$838$$ −14.4437 −0.498948
$$839$$ −27.1127 −0.936034 −0.468017 0.883719i $$-0.655031\pi$$
−0.468017 + 0.883719i $$0.655031\pi$$
$$840$$ 0 0
$$841$$ 53.2843 1.83739
$$842$$ 6.08326 0.209643
$$843$$ 0 0
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.27208 0.181151
$$848$$ 12.0000 0.412082
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −84.7696 −2.90586
$$852$$ 0 0
$$853$$ 9.51472 0.325778 0.162889 0.986644i $$-0.447919\pi$$
0.162889 + 0.986644i $$0.447919\pi$$
$$854$$ 1.37258 0.0469688
$$855$$ 0 0
$$856$$ 12.6863 0.433609
$$857$$ 37.9411 1.29604 0.648022 0.761622i $$-0.275596\pi$$
0.648022 + 0.761622i $$0.275596\pi$$
$$858$$ 0 0
$$859$$ 25.9411 0.885100 0.442550 0.896744i $$-0.354074\pi$$
0.442550 + 0.896744i $$0.354074\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 1.45584 0.0495862
$$863$$ −31.3137 −1.06593 −0.532966 0.846137i $$-0.678922\pi$$
−0.532966 + 0.846137i $$0.678922\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0.384776 0.0130752
$$867$$ 0 0
$$868$$ 6.94618 0.235769
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 64.9706 2.20144
$$872$$ 5.02944 0.170318
$$873$$ 0 0
$$874$$ −3.17157 −0.107280
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17.8995 −0.604423 −0.302211 0.953241i $$-0.597725\pi$$
−0.302211 + 0.953241i $$0.597725\pi$$
$$878$$ −0.402020 −0.0135675
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 47.4558 1.59883 0.799414 0.600781i $$-0.205143\pi$$
0.799414 + 0.600781i $$0.205143\pi$$
$$882$$ 0 0
$$883$$ −46.5269 −1.56576 −0.782878 0.622176i $$-0.786249\pi$$
−0.782878 + 0.622176i $$0.786249\pi$$
$$884$$ −11.5980 −0.390082
$$885$$ 0 0
$$886$$ 0.544156 0.0182813
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 4.68629 0.157173
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 11.5980 0.388329
$$893$$ 0.343146 0.0114829
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −6.18377 −0.206585
$$897$$ 0 0
$$898$$ 3.27208 0.109191
$$899$$ 58.8284 1.96204
$$900$$ 0 0
$$901$$ −4.68629 −0.156123
$$902$$ 4.34315 0.144611
$$903$$ 0 0
$$904$$ 19.7990 0.658505
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −38.1421 −1.26649 −0.633244 0.773952i $$-0.718277\pi$$
−0.633244 + 0.773952i $$0.718277\pi$$
$$908$$ 34.6863 1.15111
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 23.3137 0.772418 0.386209 0.922411i $$-0.373784\pi$$
0.386209 + 0.922411i $$0.373784\pi$$
$$912$$ 0 0
$$913$$ 14.8284 0.490749
$$914$$ 11.9411 0.394977
$$915$$ 0 0
$$916$$ 3.02944 0.100095
$$917$$ 9.79899 0.323591
$$918$$ 0 0
$$919$$ −12.0000 −0.395843 −0.197922 0.980218i $$-0.563419\pi$$
−0.197922 + 0.980218i $$0.563419\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 4.42641 0.145776
$$923$$ −24.2843 −0.799327
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −3.35534 −0.110263
$$927$$ 0 0
$$928$$ −40.0416 −1.31443
$$929$$ 51.4558 1.68821 0.844106 0.536177i $$-0.180132\pi$$
0.844106 + 0.536177i $$0.180132\pi$$
$$930$$ 0 0
$$931$$ −6.65685 −0.218170
$$932$$ 15.2548 0.499689
$$933$$ 0 0
$$934$$ −6.76955 −0.221507
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.7696 0.613175 0.306587 0.951843i $$-0.400813\pi$$
0.306587 + 0.951843i $$0.400813\pi$$
$$938$$ −2.91169 −0.0950700
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 17.5563 0.572321 0.286160 0.958182i $$-0.407621\pi$$
0.286160 + 0.958182i $$0.407621\pi$$
$$942$$ 0 0
$$943$$ 56.7696 1.84867
$$944$$ −25.4558 −0.828517
$$945$$ 0 0
$$946$$ −0.343146 −0.0111566
$$947$$ 12.8284 0.416868 0.208434 0.978036i $$-0.433163\pi$$
0.208434 + 0.978036i $$0.433163\pi$$
$$948$$ 0 0
$$949$$ −10.8284 −0.351506
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 1.08831 0.0352724
$$953$$ −5.85786 −0.189755 −0.0948774 0.995489i $$-0.530246\pi$$
−0.0948774 + 0.995489i $$0.530246\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 4.72792 0.152912
$$957$$ 0 0
$$958$$ −15.8162 −0.510999
$$959$$ 8.20101 0.264824
$$960$$ 0 0
$$961$$ 11.0589 0.356738
$$962$$ −24.8284 −0.800501
$$963$$ 0 0
$$964$$ 27.3726 0.881612
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 27.8995 0.897187 0.448594 0.893736i $$-0.351925\pi$$
0.448594 + 0.893736i $$0.351925\pi$$
$$968$$ −14.2721 −0.458722
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −17.6569 −0.566635 −0.283318 0.959026i $$-0.591435\pi$$
−0.283318 + 0.959026i $$0.591435\pi$$
$$972$$ 0 0
$$973$$ −0.686292 −0.0220015
$$974$$ −1.17157 −0.0375396
$$975$$ 0 0
$$976$$ 16.9706 0.543214
$$977$$ 39.5980 1.26685 0.633426 0.773803i $$-0.281648\pi$$
0.633426 + 0.773803i $$0.281648\pi$$
$$978$$ 0 0
$$979$$ 15.1716 0.484886
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 9.07107 0.289469
$$983$$ 31.9411 1.01876 0.509382 0.860541i $$-0.329874\pi$$
0.509382 + 0.860541i $$0.329874\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 4.40202 0.140189
$$987$$ 0 0
$$988$$ 9.89949 0.314945
$$989$$ −4.48528 −0.142624
$$990$$ 0 0
$$991$$ −61.6569 −1.95859 −0.979297 0.202427i $$-0.935117\pi$$
−0.979297 + 0.202427i $$0.935117\pi$$
$$992$$ −28.6274 −0.908921
$$993$$ 0 0
$$994$$ 1.08831 0.0345192
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 50.4853 1.59888 0.799442 0.600743i $$-0.205128\pi$$
0.799442 + 0.600743i $$0.205128\pi$$
$$998$$ 9.85786 0.312045
$$999$$ 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 4275.2.a.x.1.1 2
3.2 odd 2 1425.2.a.l.1.2 2
5.4 even 2 855.2.a.e.1.2 2
15.2 even 4 1425.2.c.j.799.3 4
15.8 even 4 1425.2.c.j.799.2 4
15.14 odd 2 285.2.a.f.1.1 2
60.59 even 2 4560.2.a.bj.1.2 2
285.284 even 2 5415.2.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 15.14 odd 2
855.2.a.e.1.2 2 5.4 even 2
1425.2.a.l.1.2 2 3.2 odd 2
1425.2.c.j.799.2 4 15.8 even 4
1425.2.c.j.799.3 4 15.2 even 4
4275.2.a.x.1.1 2 1.1 even 1 trivial
4560.2.a.bj.1.2 2 60.59 even 2
5415.2.a.p.1.2 2 285.284 even 2