# Properties

 Label 1216.2.a.w.1.1 Level $1216$ Weight $2$ Character 1216.1 Self dual yes Analytic conductor $9.710$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$9.70980888579$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.69353$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +0.644529 q^{11} +3.94860 q^{13} +6.77410 q^{15} +5.56802 q^{17} -1.00000 q^{19} +0.463560 q^{21} -4.46356 q^{23} +1.99353 q^{25} -1.43845 q^{27} +3.94860 q^{29} -5.48504 q^{31} -1.65100 q^{33} +0.478577 q^{35} -7.48504 q^{37} -10.1146 q^{39} +8.41216 q^{41} +5.76763 q^{43} -9.41863 q^{45} -11.2527 q^{47} -6.96725 q^{49} -14.2628 q^{51} -7.58667 q^{53} -1.70448 q^{55} +2.56155 q^{57} -11.8506 q^{59} +2.47858 q^{61} -0.644529 q^{63} -10.4422 q^{65} +8.97372 q^{67} +11.4336 q^{69} -7.63806 q^{71} +10.6911 q^{73} -5.10654 q^{75} -0.116639 q^{77} +10.6081 q^{79} -7.00000 q^{81} -5.65100 q^{83} -14.7248 q^{85} -10.1146 q^{87} -17.1733 q^{89} -0.714573 q^{91} +14.0502 q^{93} +2.64453 q^{95} +3.28906 q^{97} +2.29552 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 $$4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37} - 12 q^{39} - 2 q^{41} - 3 q^{43} + 7 q^{45} - 3 q^{47} + 7 q^{49} + 6 q^{51} - 4 q^{53} - 35 q^{55} + 2 q^{57} - 20 q^{59} + 3 q^{61} + 7 q^{63} - 12 q^{65} - 8 q^{67} + 4 q^{69} - 30 q^{71} + 9 q^{73} - 34 q^{75} + 7 q^{77} + 10 q^{79} - 28 q^{81} - 4 q^{83} - 19 q^{85} - 12 q^{87} - 16 q^{89} - 10 q^{91} + 20 q^{93} + q^{95} - 6 q^{97} - 19 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ −2.64453 −1.18267 −0.591335 0.806426i $$-0.701399\pi$$
−0.591335 + 0.806426i $$0.701399\pi$$
$$6$$ 0 0
$$7$$ −0.180969 −0.0683997 −0.0341998 0.999415i $$-0.510888\pi$$
−0.0341998 + 0.999415i $$0.510888\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 0.644529 0.194333 0.0971664 0.995268i $$-0.469022\pi$$
0.0971664 + 0.995268i $$0.469022\pi$$
$$12$$ 0 0
$$13$$ 3.94860 1.09515 0.547573 0.836758i $$-0.315552\pi$$
0.547573 + 0.836758i $$0.315552\pi$$
$$14$$ 0 0
$$15$$ 6.77410 1.74907
$$16$$ 0 0
$$17$$ 5.56802 1.35044 0.675221 0.737615i $$-0.264048\pi$$
0.675221 + 0.737615i $$0.264048\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.463560 0.101157
$$22$$ 0 0
$$23$$ −4.46356 −0.930717 −0.465358 0.885122i $$-0.654075\pi$$
−0.465358 + 0.885122i $$0.654075\pi$$
$$24$$ 0 0
$$25$$ 1.99353 0.398707
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ 3.94860 0.733237 0.366619 0.930371i $$-0.380515\pi$$
0.366619 + 0.930371i $$0.380515\pi$$
$$30$$ 0 0
$$31$$ −5.48504 −0.985143 −0.492571 0.870272i $$-0.663943\pi$$
−0.492571 + 0.870272i $$0.663943\pi$$
$$32$$ 0 0
$$33$$ −1.65100 −0.287401
$$34$$ 0 0
$$35$$ 0.478577 0.0808942
$$36$$ 0 0
$$37$$ −7.48504 −1.23053 −0.615267 0.788319i $$-0.710952\pi$$
−0.615267 + 0.788319i $$0.710952\pi$$
$$38$$ 0 0
$$39$$ −10.1146 −1.61963
$$40$$ 0 0
$$41$$ 8.41216 1.31376 0.656880 0.753995i $$-0.271876\pi$$
0.656880 + 0.753995i $$0.271876\pi$$
$$42$$ 0 0
$$43$$ 5.76763 0.879556 0.439778 0.898107i $$-0.355057\pi$$
0.439778 + 0.898107i $$0.355057\pi$$
$$44$$ 0 0
$$45$$ −9.41863 −1.40405
$$46$$ 0 0
$$47$$ −11.2527 −1.64137 −0.820686 0.571380i $$-0.806408\pi$$
−0.820686 + 0.571380i $$0.806408\pi$$
$$48$$ 0 0
$$49$$ −6.96725 −0.995321
$$50$$ 0 0
$$51$$ −14.2628 −1.99719
$$52$$ 0 0
$$53$$ −7.58667 −1.04211 −0.521054 0.853523i $$-0.674461\pi$$
−0.521054 + 0.853523i $$0.674461\pi$$
$$54$$ 0 0
$$55$$ −1.70448 −0.229831
$$56$$ 0 0
$$57$$ 2.56155 0.339286
$$58$$ 0 0
$$59$$ −11.8506 −1.54282 −0.771409 0.636340i $$-0.780448\pi$$
−0.771409 + 0.636340i $$0.780448\pi$$
$$60$$ 0 0
$$61$$ 2.47858 0.317349 0.158675 0.987331i $$-0.449278\pi$$
0.158675 + 0.987331i $$0.449278\pi$$
$$62$$ 0 0
$$63$$ −0.644529 −0.0812030
$$64$$ 0 0
$$65$$ −10.4422 −1.29519
$$66$$ 0 0
$$67$$ 8.97372 1.09631 0.548157 0.836375i $$-0.315330\pi$$
0.548157 + 0.836375i $$0.315330\pi$$
$$68$$ 0 0
$$69$$ 11.4336 1.37645
$$70$$ 0 0
$$71$$ −7.63806 −0.906471 −0.453236 0.891391i $$-0.649730\pi$$
−0.453236 + 0.891391i $$0.649730\pi$$
$$72$$ 0 0
$$73$$ 10.6911 1.25130 0.625651 0.780103i $$-0.284834\pi$$
0.625651 + 0.780103i $$0.284834\pi$$
$$74$$ 0 0
$$75$$ −5.10654 −0.589653
$$76$$ 0 0
$$77$$ −0.116639 −0.0132923
$$78$$ 0 0
$$79$$ 10.6081 1.19351 0.596755 0.802424i $$-0.296456\pi$$
0.596755 + 0.802424i $$0.296456\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −5.65100 −0.620277 −0.310139 0.950691i $$-0.600375\pi$$
−0.310139 + 0.950691i $$0.600375\pi$$
$$84$$ 0 0
$$85$$ −14.7248 −1.59713
$$86$$ 0 0
$$87$$ −10.1146 −1.08439
$$88$$ 0 0
$$89$$ −17.1733 −1.82037 −0.910185 0.414202i $$-0.864061\pi$$
−0.910185 + 0.414202i $$0.864061\pi$$
$$90$$ 0 0
$$91$$ −0.714573 −0.0749076
$$92$$ 0 0
$$93$$ 14.0502 1.45694
$$94$$ 0 0
$$95$$ 2.64453 0.271323
$$96$$ 0 0
$$97$$ 3.28906 0.333953 0.166977 0.985961i $$-0.446600\pi$$
0.166977 + 0.985961i $$0.446600\pi$$
$$98$$ 0 0
$$99$$ 2.29552 0.230709
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ 14.4122 1.42007 0.710036 0.704165i $$-0.248678\pi$$
0.710036 + 0.704165i $$0.248678\pi$$
$$104$$ 0 0
$$105$$ −1.22590 −0.119636
$$106$$ 0 0
$$107$$ −9.69759 −0.937501 −0.468751 0.883330i $$-0.655296\pi$$
−0.468751 + 0.883330i $$0.655296\pi$$
$$108$$ 0 0
$$109$$ −19.4839 −1.86622 −0.933108 0.359596i $$-0.882915\pi$$
−0.933108 + 0.359596i $$0.882915\pi$$
$$110$$ 0 0
$$111$$ 19.1733 1.81985
$$112$$ 0 0
$$113$$ −7.28906 −0.685697 −0.342848 0.939391i $$-0.611392\pi$$
−0.342848 + 0.939391i $$0.611392\pi$$
$$114$$ 0 0
$$115$$ 11.8040 1.10073
$$116$$ 0 0
$$117$$ 14.0632 1.30014
$$118$$ 0 0
$$119$$ −1.00764 −0.0923699
$$120$$ 0 0
$$121$$ −10.5846 −0.962235
$$122$$ 0 0
$$123$$ −21.5482 −1.94294
$$124$$ 0 0
$$125$$ 7.95069 0.711131
$$126$$ 0 0
$$127$$ −0.595216 −0.0528169 −0.0264084 0.999651i $$-0.508407\pi$$
−0.0264084 + 0.999651i $$0.508407\pi$$
$$128$$ 0 0
$$129$$ −14.7741 −1.30079
$$130$$ 0 0
$$131$$ 15.7806 1.37875 0.689377 0.724403i $$-0.257884\pi$$
0.689377 + 0.724403i $$0.257884\pi$$
$$132$$ 0 0
$$133$$ 0.180969 0.0156920
$$134$$ 0 0
$$135$$ 3.80402 0.327398
$$136$$ 0 0
$$137$$ −15.9802 −1.36528 −0.682640 0.730755i $$-0.739168\pi$$
−0.682640 + 0.730755i $$0.739168\pi$$
$$138$$ 0 0
$$139$$ −4.47858 −0.379868 −0.189934 0.981797i $$-0.560827\pi$$
−0.189934 + 0.981797i $$0.560827\pi$$
$$140$$ 0 0
$$141$$ 28.8243 2.42745
$$142$$ 0 0
$$143$$ 2.54499 0.212823
$$144$$ 0 0
$$145$$ −10.4422 −0.867177
$$146$$ 0 0
$$147$$ 17.8470 1.47199
$$148$$ 0 0
$$149$$ −10.3758 −0.850017 −0.425009 0.905189i $$-0.639729\pi$$
−0.425009 + 0.905189i $$0.639729\pi$$
$$150$$ 0 0
$$151$$ −16.6584 −1.35564 −0.677820 0.735228i $$-0.737075\pi$$
−0.677820 + 0.735228i $$0.737075\pi$$
$$152$$ 0 0
$$153$$ 19.8308 1.60322
$$154$$ 0 0
$$155$$ 14.5054 1.16510
$$156$$ 0 0
$$157$$ 11.0203 0.879517 0.439758 0.898116i $$-0.355064\pi$$
0.439758 + 0.898116i $$0.355064\pi$$
$$158$$ 0 0
$$159$$ 19.4336 1.54119
$$160$$ 0 0
$$161$$ 0.807764 0.0636607
$$162$$ 0 0
$$163$$ 7.80402 0.611258 0.305629 0.952151i $$-0.401133\pi$$
0.305629 + 0.952151i $$0.401133\pi$$
$$164$$ 0 0
$$165$$ 4.36610 0.339901
$$166$$ 0 0
$$167$$ −10.2462 −0.792876 −0.396438 0.918062i $$-0.629754\pi$$
−0.396438 + 0.918062i $$0.629754\pi$$
$$168$$ 0 0
$$169$$ 2.59147 0.199344
$$170$$ 0 0
$$171$$ −3.56155 −0.272359
$$172$$ 0 0
$$173$$ 1.80402 0.137157 0.0685784 0.997646i $$-0.478154\pi$$
0.0685784 + 0.997646i $$0.478154\pi$$
$$174$$ 0 0
$$175$$ −0.360767 −0.0272714
$$176$$ 0 0
$$177$$ 30.3560 2.28169
$$178$$ 0 0
$$179$$ −1.75379 −0.131084 −0.0655422 0.997850i $$-0.520878\pi$$
−0.0655422 + 0.997850i $$0.520878\pi$$
$$180$$ 0 0
$$181$$ −6.52789 −0.485214 −0.242607 0.970125i $$-0.578003\pi$$
−0.242607 + 0.970125i $$0.578003\pi$$
$$182$$ 0 0
$$183$$ −6.34900 −0.469332
$$184$$ 0 0
$$185$$ 19.7944 1.45531
$$186$$ 0 0
$$187$$ 3.58875 0.262435
$$188$$ 0 0
$$189$$ 0.260314 0.0189350
$$190$$ 0 0
$$191$$ −7.45709 −0.539576 −0.269788 0.962920i $$-0.586954\pi$$
−0.269788 + 0.962920i $$0.586954\pi$$
$$192$$ 0 0
$$193$$ 19.0203 1.36911 0.684556 0.728960i $$-0.259996\pi$$
0.684556 + 0.728960i $$0.259996\pi$$
$$194$$ 0 0
$$195$$ 26.7482 1.91548
$$196$$ 0 0
$$197$$ −0.153020 −0.0109022 −0.00545112 0.999985i $$-0.501735\pi$$
−0.00545112 + 0.999985i $$0.501735\pi$$
$$198$$ 0 0
$$199$$ 21.3672 1.51468 0.757341 0.653019i $$-0.226498\pi$$
0.757341 + 0.653019i $$0.226498\pi$$
$$200$$ 0 0
$$201$$ −22.9866 −1.62135
$$202$$ 0 0
$$203$$ −0.714573 −0.0501532
$$204$$ 0 0
$$205$$ −22.2462 −1.55374
$$206$$ 0 0
$$207$$ −15.8972 −1.10493
$$208$$ 0 0
$$209$$ −0.644529 −0.0445830
$$210$$ 0 0
$$211$$ −6.82070 −0.469556 −0.234778 0.972049i $$-0.575436\pi$$
−0.234778 + 0.972049i $$0.575436\pi$$
$$212$$ 0 0
$$213$$ 19.5653 1.34059
$$214$$ 0 0
$$215$$ −15.2527 −1.04022
$$216$$ 0 0
$$217$$ 0.992620 0.0673835
$$218$$ 0 0
$$219$$ −27.3859 −1.85057
$$220$$ 0 0
$$221$$ 21.9859 1.47893
$$222$$ 0 0
$$223$$ −8.42510 −0.564186 −0.282093 0.959387i $$-0.591029\pi$$
−0.282093 + 0.959387i $$0.591029\pi$$
$$224$$ 0 0
$$225$$ 7.10008 0.473338
$$226$$ 0 0
$$227$$ −23.3859 −1.55218 −0.776088 0.630625i $$-0.782799\pi$$
−0.776088 + 0.630625i $$0.782799\pi$$
$$228$$ 0 0
$$229$$ 3.37567 0.223070 0.111535 0.993760i $$-0.464423\pi$$
0.111535 + 0.993760i $$0.464423\pi$$
$$230$$ 0 0
$$231$$ 0.298778 0.0196582
$$232$$ 0 0
$$233$$ −15.8308 −1.03711 −0.518555 0.855044i $$-0.673530\pi$$
−0.518555 + 0.855044i $$0.673530\pi$$
$$234$$ 0 0
$$235$$ 29.7580 1.94120
$$236$$ 0 0
$$237$$ −27.1733 −1.76510
$$238$$ 0 0
$$239$$ 2.39715 0.155059 0.0775293 0.996990i $$-0.475297\pi$$
0.0775293 + 0.996990i $$0.475297\pi$$
$$240$$ 0 0
$$241$$ 5.44208 0.350555 0.175278 0.984519i $$-0.443918\pi$$
0.175278 + 0.984519i $$0.443918\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ 18.4251 1.17714
$$246$$ 0 0
$$247$$ −3.94860 −0.251244
$$248$$ 0 0
$$249$$ 14.4753 0.917336
$$250$$ 0 0
$$251$$ −12.8405 −0.810486 −0.405243 0.914209i $$-0.632813\pi$$
−0.405243 + 0.914209i $$0.632813\pi$$
$$252$$ 0 0
$$253$$ −2.87689 −0.180869
$$254$$ 0 0
$$255$$ 37.7183 2.36201
$$256$$ 0 0
$$257$$ −19.1863 −1.19681 −0.598403 0.801195i $$-0.704198\pi$$
−0.598403 + 0.801195i $$0.704198\pi$$
$$258$$ 0 0
$$259$$ 1.35456 0.0841681
$$260$$ 0 0
$$261$$ 14.0632 0.870488
$$262$$ 0 0
$$263$$ 20.2098 1.24619 0.623096 0.782146i $$-0.285875\pi$$
0.623096 + 0.782146i $$0.285875\pi$$
$$264$$ 0 0
$$265$$ 20.0632 1.23247
$$266$$ 0 0
$$267$$ 43.9904 2.69217
$$268$$ 0 0
$$269$$ −16.4422 −1.00250 −0.501249 0.865303i $$-0.667126\pi$$
−0.501249 + 0.865303i $$0.667126\pi$$
$$270$$ 0 0
$$271$$ 8.02886 0.487719 0.243859 0.969811i $$-0.421586\pi$$
0.243859 + 0.969811i $$0.421586\pi$$
$$272$$ 0 0
$$273$$ 1.83042 0.110782
$$274$$ 0 0
$$275$$ 1.28489 0.0774818
$$276$$ 0 0
$$277$$ −9.68738 −0.582058 −0.291029 0.956714i $$-0.593998\pi$$
−0.291029 + 0.956714i $$0.593998\pi$$
$$278$$ 0 0
$$279$$ −19.5353 −1.16955
$$280$$ 0 0
$$281$$ −8.15302 −0.486368 −0.243184 0.969980i $$-0.578192\pi$$
−0.243184 + 0.969980i $$0.578192\pi$$
$$282$$ 0 0
$$283$$ −27.9742 −1.66289 −0.831447 0.555604i $$-0.812487\pi$$
−0.831447 + 0.555604i $$0.812487\pi$$
$$284$$ 0 0
$$285$$ −6.77410 −0.401263
$$286$$ 0 0
$$287$$ −1.52234 −0.0898607
$$288$$ 0 0
$$289$$ 14.0028 0.823696
$$290$$ 0 0
$$291$$ −8.42510 −0.493888
$$292$$ 0 0
$$293$$ 12.5438 0.732818 0.366409 0.930454i $$-0.380587\pi$$
0.366409 + 0.930454i $$0.380587\pi$$
$$294$$ 0 0
$$295$$ 31.3393 1.82464
$$296$$ 0 0
$$297$$ −0.927121 −0.0537970
$$298$$ 0 0
$$299$$ −17.6248 −1.01927
$$300$$ 0 0
$$301$$ −1.04376 −0.0601614
$$302$$ 0 0
$$303$$ 0.630683 0.0362318
$$304$$ 0 0
$$305$$ −6.55467 −0.375319
$$306$$ 0 0
$$307$$ 12.9572 0.739504 0.369752 0.929131i $$-0.379443\pi$$
0.369752 + 0.929131i $$0.379443\pi$$
$$308$$ 0 0
$$309$$ −36.9175 −2.10016
$$310$$ 0 0
$$311$$ 31.8467 1.80586 0.902931 0.429786i $$-0.141411\pi$$
0.902931 + 0.429786i $$0.141411\pi$$
$$312$$ 0 0
$$313$$ 16.2628 0.919226 0.459613 0.888119i $$-0.347988\pi$$
0.459613 + 0.888119i $$0.347988\pi$$
$$314$$ 0 0
$$315$$ 1.70448 0.0960363
$$316$$ 0 0
$$317$$ 6.88866 0.386905 0.193453 0.981110i $$-0.438031\pi$$
0.193453 + 0.981110i $$0.438031\pi$$
$$318$$ 0 0
$$319$$ 2.54499 0.142492
$$320$$ 0 0
$$321$$ 24.8409 1.38648
$$322$$ 0 0
$$323$$ −5.56802 −0.309813
$$324$$ 0 0
$$325$$ 7.87167 0.436642
$$326$$ 0 0
$$327$$ 49.9090 2.75997
$$328$$ 0 0
$$329$$ 2.03638 0.112269
$$330$$ 0 0
$$331$$ −25.6976 −1.41247 −0.706234 0.707979i $$-0.749607\pi$$
−0.706234 + 0.707979i $$0.749607\pi$$
$$332$$ 0 0
$$333$$ −26.6584 −1.46087
$$334$$ 0 0
$$335$$ −23.7313 −1.29658
$$336$$ 0 0
$$337$$ 7.78382 0.424012 0.212006 0.977268i $$-0.432000\pi$$
0.212006 + 0.977268i $$0.432000\pi$$
$$338$$ 0 0
$$339$$ 18.6713 1.01409
$$340$$ 0 0
$$341$$ −3.53527 −0.191446
$$342$$ 0 0
$$343$$ 2.52763 0.136479
$$344$$ 0 0
$$345$$ −30.2366 −1.62788
$$346$$ 0 0
$$347$$ 32.3758 1.73802 0.869012 0.494792i $$-0.164756\pi$$
0.869012 + 0.494792i $$0.164756\pi$$
$$348$$ 0 0
$$349$$ −25.0567 −1.34125 −0.670627 0.741795i $$-0.733975\pi$$
−0.670627 + 0.741795i $$0.733975\pi$$
$$350$$ 0 0
$$351$$ −5.67986 −0.303168
$$352$$ 0 0
$$353$$ −2.70731 −0.144096 −0.0720478 0.997401i $$-0.522953\pi$$
−0.0720478 + 0.997401i $$0.522953\pi$$
$$354$$ 0 0
$$355$$ 20.1991 1.07206
$$356$$ 0 0
$$357$$ 2.58111 0.136607
$$358$$ 0 0
$$359$$ −27.6862 −1.46122 −0.730611 0.682794i $$-0.760765\pi$$
−0.730611 + 0.682794i $$0.760765\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 27.1130 1.42306
$$364$$ 0 0
$$365$$ −28.2730 −1.47988
$$366$$ 0 0
$$367$$ 15.3020 0.798757 0.399379 0.916786i $$-0.369226\pi$$
0.399379 + 0.916786i $$0.369226\pi$$
$$368$$ 0 0
$$369$$ 29.9604 1.55967
$$370$$ 0 0
$$371$$ 1.37295 0.0712799
$$372$$ 0 0
$$373$$ −27.2506 −1.41098 −0.705491 0.708719i $$-0.749274\pi$$
−0.705491 + 0.708719i $$0.749274\pi$$
$$374$$ 0 0
$$375$$ −20.3661 −1.05170
$$376$$ 0 0
$$377$$ 15.5915 0.803001
$$378$$ 0 0
$$379$$ −30.0968 −1.54597 −0.772985 0.634424i $$-0.781237\pi$$
−0.772985 + 0.634424i $$0.781237\pi$$
$$380$$ 0 0
$$381$$ 1.52468 0.0781116
$$382$$ 0 0
$$383$$ 37.8146 1.93224 0.966118 0.258100i $$-0.0830966\pi$$
0.966118 + 0.258100i $$0.0830966\pi$$
$$384$$ 0 0
$$385$$ 0.308457 0.0157204
$$386$$ 0 0
$$387$$ 20.5417 1.04420
$$388$$ 0 0
$$389$$ −2.44128 −0.123778 −0.0618890 0.998083i $$-0.519712\pi$$
−0.0618890 + 0.998083i $$0.519712\pi$$
$$390$$ 0 0
$$391$$ −24.8532 −1.25688
$$392$$ 0 0
$$393$$ −40.4228 −2.03906
$$394$$ 0 0
$$395$$ −28.0536 −1.41153
$$396$$ 0 0
$$397$$ −16.4260 −0.824398 −0.412199 0.911094i $$-0.635239\pi$$
−0.412199 + 0.911094i $$0.635239\pi$$
$$398$$ 0 0
$$399$$ −0.463560 −0.0232071
$$400$$ 0 0
$$401$$ −30.0932 −1.50278 −0.751391 0.659857i $$-0.770617\pi$$
−0.751391 + 0.659857i $$0.770617\pi$$
$$402$$ 0 0
$$403$$ −21.6583 −1.07887
$$404$$ 0 0
$$405$$ 18.5117 0.919854
$$406$$ 0 0
$$407$$ −4.82433 −0.239133
$$408$$ 0 0
$$409$$ −39.4454 −1.95045 −0.975225 0.221216i $$-0.928998\pi$$
−0.975225 + 0.221216i $$0.928998\pi$$
$$410$$ 0 0
$$411$$ 40.9341 2.01913
$$412$$ 0 0
$$413$$ 2.14459 0.105528
$$414$$ 0 0
$$415$$ 14.9442 0.733583
$$416$$ 0 0
$$417$$ 11.4721 0.561792
$$418$$ 0 0
$$419$$ −13.3522 −0.652298 −0.326149 0.945318i $$-0.605751\pi$$
−0.326149 + 0.945318i $$0.605751\pi$$
$$420$$ 0 0
$$421$$ −21.8587 −1.06533 −0.532665 0.846326i $$-0.678809\pi$$
−0.532665 + 0.846326i $$0.678809\pi$$
$$422$$ 0 0
$$423$$ −40.0770 −1.94861
$$424$$ 0 0
$$425$$ 11.1000 0.538431
$$426$$ 0 0
$$427$$ −0.448544 −0.0217066
$$428$$ 0 0
$$429$$ −6.51912 −0.314746
$$430$$ 0 0
$$431$$ −14.3490 −0.691167 −0.345584 0.938388i $$-0.612319\pi$$
−0.345584 + 0.938388i $$0.612319\pi$$
$$432$$ 0 0
$$433$$ 19.2536 0.925269 0.462634 0.886549i $$-0.346904\pi$$
0.462634 + 0.886549i $$0.346904\pi$$
$$434$$ 0 0
$$435$$ 26.7482 1.28248
$$436$$ 0 0
$$437$$ 4.46356 0.213521
$$438$$ 0 0
$$439$$ −19.3020 −0.921234 −0.460617 0.887599i $$-0.652372\pi$$
−0.460617 + 0.887599i $$0.652372\pi$$
$$440$$ 0 0
$$441$$ −24.8142 −1.18163
$$442$$ 0 0
$$443$$ −4.61866 −0.219439 −0.109720 0.993963i $$-0.534995\pi$$
−0.109720 + 0.993963i $$0.534995\pi$$
$$444$$ 0 0
$$445$$ 45.4154 2.15290
$$446$$ 0 0
$$447$$ 26.5781 1.25710
$$448$$ 0 0
$$449$$ 8.54082 0.403066 0.201533 0.979482i $$-0.435408\pi$$
0.201533 + 0.979482i $$0.435408\pi$$
$$450$$ 0 0
$$451$$ 5.42188 0.255307
$$452$$ 0 0
$$453$$ 42.6713 2.00487
$$454$$ 0 0
$$455$$ 1.88971 0.0885909
$$456$$ 0 0
$$457$$ 25.7340 1.20378 0.601892 0.798577i $$-0.294414\pi$$
0.601892 + 0.798577i $$0.294414\pi$$
$$458$$ 0 0
$$459$$ −8.00930 −0.373842
$$460$$ 0 0
$$461$$ 12.3629 0.575795 0.287898 0.957661i $$-0.407044\pi$$
0.287898 + 0.957661i $$0.407044\pi$$
$$462$$ 0 0
$$463$$ −21.1069 −0.980922 −0.490461 0.871463i $$-0.663172\pi$$
−0.490461 + 0.871463i $$0.663172\pi$$
$$464$$ 0 0
$$465$$ −37.1562 −1.72308
$$466$$ 0 0
$$467$$ −25.2729 −1.16949 −0.584745 0.811218i $$-0.698805\pi$$
−0.584745 + 0.811218i $$0.698805\pi$$
$$468$$ 0 0
$$469$$ −1.62396 −0.0749875
$$470$$ 0 0
$$471$$ −28.2291 −1.30073
$$472$$ 0 0
$$473$$ 3.71741 0.170927
$$474$$ 0 0
$$475$$ −1.99353 −0.0914696
$$476$$ 0 0
$$477$$ −27.0203 −1.23718
$$478$$ 0 0
$$479$$ 3.27613 0.149690 0.0748450 0.997195i $$-0.476154\pi$$
0.0748450 + 0.997195i $$0.476154\pi$$
$$480$$ 0 0
$$481$$ −29.5555 −1.34761
$$482$$ 0 0
$$483$$ −2.06913 −0.0941487
$$484$$ 0 0
$$485$$ −8.69801 −0.394956
$$486$$ 0 0
$$487$$ −14.0202 −0.635316 −0.317658 0.948205i $$-0.602896\pi$$
−0.317658 + 0.948205i $$0.602896\pi$$
$$488$$ 0 0
$$489$$ −19.9904 −0.903997
$$490$$ 0 0
$$491$$ 21.6487 0.976990 0.488495 0.872567i $$-0.337546\pi$$
0.488495 + 0.872567i $$0.337546\pi$$
$$492$$ 0 0
$$493$$ 21.9859 0.990195
$$494$$ 0 0
$$495$$ −6.07058 −0.272852
$$496$$ 0 0
$$497$$ 1.38225 0.0620023
$$498$$ 0 0
$$499$$ 34.4187 1.54079 0.770397 0.637564i $$-0.220058\pi$$
0.770397 + 0.637564i $$0.220058\pi$$
$$500$$ 0 0
$$501$$ 26.2462 1.17259
$$502$$ 0 0
$$503$$ −14.3178 −0.638399 −0.319200 0.947687i $$-0.603414\pi$$
−0.319200 + 0.947687i $$0.603414\pi$$
$$504$$ 0 0
$$505$$ 0.651113 0.0289741
$$506$$ 0 0
$$507$$ −6.63818 −0.294812
$$508$$ 0 0
$$509$$ −17.3393 −0.768550 −0.384275 0.923219i $$-0.625549\pi$$
−0.384275 + 0.923219i $$0.625549\pi$$
$$510$$ 0 0
$$511$$ −1.93476 −0.0855886
$$512$$ 0 0
$$513$$ 1.43845 0.0635090
$$514$$ 0 0
$$515$$ −38.1134 −1.67948
$$516$$ 0 0
$$517$$ −7.25268 −0.318972
$$518$$ 0 0
$$519$$ −4.62108 −0.202843
$$520$$ 0 0
$$521$$ 30.5856 1.33998 0.669990 0.742370i $$-0.266298\pi$$
0.669990 + 0.742370i $$0.266298\pi$$
$$522$$ 0 0
$$523$$ −22.7275 −0.993804 −0.496902 0.867807i $$-0.665529\pi$$
−0.496902 + 0.867807i $$0.665529\pi$$
$$524$$ 0 0
$$525$$ 0.924124 0.0403321
$$526$$ 0 0
$$527$$ −30.5408 −1.33038
$$528$$ 0 0
$$529$$ −3.07663 −0.133766
$$530$$ 0 0
$$531$$ −42.2066 −1.83161
$$532$$ 0 0
$$533$$ 33.2163 1.43876
$$534$$ 0 0
$$535$$ 25.6456 1.10875
$$536$$ 0 0
$$537$$ 4.49242 0.193862
$$538$$ 0 0
$$539$$ −4.49060 −0.193424
$$540$$ 0 0
$$541$$ 31.9913 1.37541 0.687707 0.725988i $$-0.258617\pi$$
0.687707 + 0.725988i $$0.258617\pi$$
$$542$$ 0 0
$$543$$ 16.7215 0.717590
$$544$$ 0 0
$$545$$ 51.5257 2.20712
$$546$$ 0 0
$$547$$ 16.1005 0.688406 0.344203 0.938895i $$-0.388149\pi$$
0.344203 + 0.938895i $$0.388149\pi$$
$$548$$ 0 0
$$549$$ 8.82758 0.376752
$$550$$ 0 0
$$551$$ −3.94860 −0.168216
$$552$$ 0 0
$$553$$ −1.91974 −0.0816357
$$554$$ 0 0
$$555$$ −50.7044 −2.15228
$$556$$ 0 0
$$557$$ 14.3126 0.606445 0.303223 0.952920i $$-0.401937\pi$$
0.303223 + 0.952920i $$0.401937\pi$$
$$558$$ 0 0
$$559$$ 22.7741 0.963242
$$560$$ 0 0
$$561$$ −9.19277 −0.388119
$$562$$ 0 0
$$563$$ −13.9143 −0.586418 −0.293209 0.956048i $$-0.594723\pi$$
−0.293209 + 0.956048i $$0.594723\pi$$
$$564$$ 0 0
$$565$$ 19.2761 0.810953
$$566$$ 0 0
$$567$$ 1.26678 0.0531998
$$568$$ 0 0
$$569$$ −7.71415 −0.323394 −0.161697 0.986840i $$-0.551697\pi$$
−0.161697 + 0.986840i $$0.551697\pi$$
$$570$$ 0 0
$$571$$ 9.42188 0.394294 0.197147 0.980374i $$-0.436832\pi$$
0.197147 + 0.980374i $$0.436832\pi$$
$$572$$ 0 0
$$573$$ 19.1017 0.797987
$$574$$ 0 0
$$575$$ −8.89826 −0.371083
$$576$$ 0 0
$$577$$ −10.3851 −0.432337 −0.216168 0.976356i $$-0.569356\pi$$
−0.216168 + 0.976356i $$0.569356\pi$$
$$578$$ 0 0
$$579$$ −48.7215 −2.02480
$$580$$ 0 0
$$581$$ 1.02265 0.0424268
$$582$$ 0 0
$$583$$ −4.88983 −0.202516
$$584$$ 0 0
$$585$$ −37.1904 −1.53764
$$586$$ 0 0
$$587$$ 38.8866 1.60502 0.802510 0.596638i $$-0.203497\pi$$
0.802510 + 0.596638i $$0.203497\pi$$
$$588$$ 0 0
$$589$$ 5.48504 0.226007
$$590$$ 0 0
$$591$$ 0.391969 0.0161235
$$592$$ 0 0
$$593$$ 19.1304 0.785590 0.392795 0.919626i $$-0.371508\pi$$
0.392795 + 0.919626i $$0.371508\pi$$
$$594$$ 0 0
$$595$$ 2.66472 0.109243
$$596$$ 0 0
$$597$$ −54.7333 −2.24008
$$598$$ 0 0
$$599$$ −27.5353 −1.12506 −0.562530 0.826777i $$-0.690172\pi$$
−0.562530 + 0.826777i $$0.690172\pi$$
$$600$$ 0 0
$$601$$ −5.73669 −0.234004 −0.117002 0.993132i $$-0.537328\pi$$
−0.117002 + 0.993132i $$0.537328\pi$$
$$602$$ 0 0
$$603$$ 31.9604 1.30153
$$604$$ 0 0
$$605$$ 27.9912 1.13801
$$606$$ 0 0
$$607$$ 25.3191 1.02767 0.513835 0.857889i $$-0.328224\pi$$
0.513835 + 0.857889i $$0.328224\pi$$
$$608$$ 0 0
$$609$$ 1.83042 0.0741722
$$610$$ 0 0
$$611$$ −44.4324 −1.79754
$$612$$ 0 0
$$613$$ −1.55872 −0.0629560 −0.0314780 0.999504i $$-0.510021\pi$$
−0.0314780 + 0.999504i $$0.510021\pi$$
$$614$$ 0 0
$$615$$ 56.9848 2.29785
$$616$$ 0 0
$$617$$ −11.6317 −0.468275 −0.234138 0.972203i $$-0.575227\pi$$
−0.234138 + 0.972203i $$0.575227\pi$$
$$618$$ 0 0
$$619$$ −15.6754 −0.630046 −0.315023 0.949084i $$-0.602012\pi$$
−0.315023 + 0.949084i $$0.602012\pi$$
$$620$$ 0 0
$$621$$ 6.42060 0.257650
$$622$$ 0 0
$$623$$ 3.10783 0.124513
$$624$$ 0 0
$$625$$ −30.9935 −1.23974
$$626$$ 0 0
$$627$$ 1.65100 0.0659344
$$628$$ 0 0
$$629$$ −41.6769 −1.66177
$$630$$ 0 0
$$631$$ −9.63171 −0.383432 −0.191716 0.981450i $$-0.561405\pi$$
−0.191716 + 0.981450i $$0.561405\pi$$
$$632$$ 0 0
$$633$$ 17.4716 0.694433
$$634$$ 0 0
$$635$$ 1.57407 0.0624649
$$636$$ 0 0
$$637$$ −27.5109 −1.09002
$$638$$ 0 0
$$639$$ −27.2034 −1.07615
$$640$$ 0 0
$$641$$ −16.5150 −0.652302 −0.326151 0.945318i $$-0.605752\pi$$
−0.326151 + 0.945318i $$0.605752\pi$$
$$642$$ 0 0
$$643$$ 18.3887 0.725180 0.362590 0.931949i $$-0.381893\pi$$
0.362590 + 0.931949i $$0.381893\pi$$
$$644$$ 0 0
$$645$$ 39.0705 1.53840
$$646$$ 0 0
$$647$$ 17.3414 0.681760 0.340880 0.940107i $$-0.389275\pi$$
0.340880 + 0.940107i $$0.389275\pi$$
$$648$$ 0 0
$$649$$ −7.63806 −0.299820
$$650$$ 0 0
$$651$$ −2.54265 −0.0996543
$$652$$ 0 0
$$653$$ 3.46886 0.135747 0.0678734 0.997694i $$-0.478379\pi$$
0.0678734 + 0.997694i $$0.478379\pi$$
$$654$$ 0 0
$$655$$ −41.7322 −1.63061
$$656$$ 0 0
$$657$$ 38.0770 1.48553
$$658$$ 0 0
$$659$$ −36.6677 −1.42837 −0.714185 0.699957i $$-0.753202\pi$$
−0.714185 + 0.699957i $$0.753202\pi$$
$$660$$ 0 0
$$661$$ 48.6960 1.89405 0.947027 0.321153i $$-0.104070\pi$$
0.947027 + 0.321153i $$0.104070\pi$$
$$662$$ 0 0
$$663$$ −56.3180 −2.18721
$$664$$ 0 0
$$665$$ −0.478577 −0.0185584
$$666$$ 0 0
$$667$$ −17.6248 −0.682436
$$668$$ 0 0
$$669$$ 21.5813 0.834382
$$670$$ 0 0
$$671$$ 1.59751 0.0616714
$$672$$ 0 0
$$673$$ 24.2535 0.934903 0.467451 0.884019i $$-0.345172\pi$$
0.467451 + 0.884019i $$0.345172\pi$$
$$674$$ 0 0
$$675$$ −2.86759 −0.110374
$$676$$ 0 0
$$677$$ −16.7471 −0.643642 −0.321821 0.946801i $$-0.604295\pi$$
−0.321821 + 0.946801i $$0.604295\pi$$
$$678$$ 0 0
$$679$$ −0.595216 −0.0228423
$$680$$ 0 0
$$681$$ 59.9042 2.29553
$$682$$ 0 0
$$683$$ −17.0299 −0.651632 −0.325816 0.945433i $$-0.605639\pi$$
−0.325816 + 0.945433i $$0.605639\pi$$
$$684$$ 0 0
$$685$$ 42.2601 1.61467
$$686$$ 0 0
$$687$$ −8.64695 −0.329902
$$688$$ 0 0
$$689$$ −29.9567 −1.14126
$$690$$ 0 0
$$691$$ −38.8811 −1.47911 −0.739554 0.673097i $$-0.764964\pi$$
−0.739554 + 0.673097i $$0.764964\pi$$
$$692$$ 0 0
$$693$$ −0.415418 −0.0157804
$$694$$ 0 0
$$695$$ 11.8437 0.449258
$$696$$ 0 0
$$697$$ 46.8391 1.77416
$$698$$ 0 0
$$699$$ 40.5514 1.53380
$$700$$ 0 0
$$701$$ 8.17888 0.308912 0.154456 0.988000i $$-0.450637\pi$$
0.154456 + 0.988000i $$0.450637\pi$$
$$702$$ 0 0
$$703$$ 7.48504 0.282304
$$704$$ 0 0
$$705$$ −76.2268 −2.87087
$$706$$ 0 0
$$707$$ 0.0445565 0.00167572
$$708$$ 0 0
$$709$$ −23.3522 −0.877011 −0.438505 0.898729i $$-0.644492\pi$$
−0.438505 + 0.898729i $$0.644492\pi$$
$$710$$ 0 0
$$711$$ 37.7815 1.41692
$$712$$ 0 0
$$713$$ 24.4828 0.916889
$$714$$ 0 0
$$715$$ −6.73030 −0.251699
$$716$$ 0 0
$$717$$ −6.14042 −0.229318
$$718$$ 0 0
$$719$$ −24.1810 −0.901798 −0.450899 0.892575i $$-0.648897\pi$$
−0.450899 + 0.892575i $$0.648897\pi$$
$$720$$ 0 0
$$721$$ −2.60815 −0.0971325
$$722$$ 0 0
$$723$$ −13.9402 −0.518441
$$724$$ 0 0
$$725$$ 7.87167 0.292347
$$726$$ 0 0
$$727$$ 10.3503 0.383870 0.191935 0.981408i $$-0.438524\pi$$
0.191935 + 0.981408i $$0.438524\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 32.1143 1.18779
$$732$$ 0 0
$$733$$ 1.81128 0.0669011 0.0334505 0.999440i $$-0.489350\pi$$
0.0334505 + 0.999440i $$0.489350\pi$$
$$734$$ 0 0
$$735$$ −47.1969 −1.74088
$$736$$ 0 0
$$737$$ 5.78382 0.213050
$$738$$ 0 0
$$739$$ 17.0342 0.626612 0.313306 0.949652i $$-0.398564\pi$$
0.313306 + 0.949652i $$0.398564\pi$$
$$740$$ 0 0
$$741$$ 10.1146 0.371567
$$742$$ 0 0
$$743$$ 0.992620 0.0364157 0.0182079 0.999834i $$-0.494204\pi$$
0.0182079 + 0.999834i $$0.494204\pi$$
$$744$$ 0 0
$$745$$ 27.4391 1.00529
$$746$$ 0 0
$$747$$ −20.1263 −0.736383
$$748$$ 0 0
$$749$$ 1.75496 0.0641248
$$750$$ 0 0
$$751$$ 40.1263 1.46423 0.732115 0.681181i $$-0.238533\pi$$
0.732115 + 0.681181i $$0.238533\pi$$
$$752$$ 0 0
$$753$$ 32.8917 1.19864
$$754$$ 0 0
$$755$$ 44.0536 1.60327
$$756$$ 0 0
$$757$$ 41.8810 1.52219 0.761096 0.648640i $$-0.224662\pi$$
0.761096 + 0.648640i $$0.224662\pi$$
$$758$$ 0 0
$$759$$ 7.36932 0.267489
$$760$$ 0 0
$$761$$ 23.8142 0.863265 0.431633 0.902050i $$-0.357938\pi$$
0.431633 + 0.902050i $$0.357938\pi$$
$$762$$ 0 0
$$763$$ 3.52597 0.127649
$$764$$ 0 0
$$765$$ −52.4431 −1.89608
$$766$$ 0 0
$$767$$ −46.7934 −1.68961
$$768$$ 0 0
$$769$$ −47.5828 −1.71588 −0.857939 0.513751i $$-0.828256\pi$$
−0.857939 + 0.513751i $$0.828256\pi$$
$$770$$ 0 0
$$771$$ 49.1466 1.76997
$$772$$ 0 0
$$773$$ −33.4240 −1.20218 −0.601090 0.799182i $$-0.705267\pi$$
−0.601090 + 0.799182i $$0.705267\pi$$
$$774$$ 0 0
$$775$$ −10.9346 −0.392783
$$776$$ 0 0
$$777$$ −3.46977 −0.124477
$$778$$ 0 0
$$779$$ −8.41216 −0.301397
$$780$$ 0 0
$$781$$ −4.92295 −0.176157
$$782$$ 0 0
$$783$$ −5.67986 −0.202982
$$784$$ 0 0
$$785$$ −29.1435 −1.04018
$$786$$ 0 0
$$787$$ −0.315342 −0.0112407 −0.00562036 0.999984i $$-0.501789\pi$$
−0.00562036 + 0.999984i $$0.501789\pi$$
$$788$$ 0 0
$$789$$ −51.7685 −1.84301
$$790$$ 0 0
$$791$$ 1.31909 0.0469015
$$792$$ 0 0
$$793$$ 9.78692 0.347544
$$794$$ 0 0
$$795$$ −51.3928 −1.82272
$$796$$ 0 0
$$797$$ −11.0942 −0.392978 −0.196489 0.980506i $$-0.562954\pi$$
−0.196489 + 0.980506i $$0.562954\pi$$
$$798$$ 0 0
$$799$$ −62.6551 −2.21658
$$800$$ 0 0
$$801$$ −61.1637 −2.16111
$$802$$ 0 0
$$803$$ 6.89074 0.243169
$$804$$ 0 0
$$805$$ −2.13616 −0.0752896
$$806$$ 0 0
$$807$$ 42.1176 1.48261
$$808$$ 0 0
$$809$$ 17.4021 0.611824 0.305912 0.952060i $$-0.401039\pi$$
0.305912 + 0.952060i $$0.401039\pi$$
$$810$$ 0 0
$$811$$ 44.2703 1.55454 0.777270 0.629167i $$-0.216604\pi$$
0.777270 + 0.629167i $$0.216604\pi$$
$$812$$ 0 0
$$813$$ −20.5664 −0.721294
$$814$$ 0 0
$$815$$ −20.6379 −0.722916
$$816$$ 0 0
$$817$$ −5.76763 −0.201784
$$818$$ 0 0
$$819$$ −2.54499 −0.0889291
$$820$$ 0 0
$$821$$ −5.12402 −0.178830 −0.0894148 0.995994i $$-0.528500\pi$$
−0.0894148 + 0.995994i $$0.528500\pi$$
$$822$$ 0 0
$$823$$ 6.29435 0.219407 0.109704 0.993964i $$-0.465010\pi$$
0.109704 + 0.993964i $$0.465010\pi$$
$$824$$ 0 0
$$825$$ −3.29131 −0.114589
$$826$$ 0 0
$$827$$ 28.9996 1.00841 0.504207 0.863583i $$-0.331785\pi$$
0.504207 + 0.863583i $$0.331785\pi$$
$$828$$ 0 0
$$829$$ 11.0215 0.382792 0.191396 0.981513i $$-0.438699\pi$$
0.191396 + 0.981513i $$0.438699\pi$$
$$830$$ 0 0
$$831$$ 24.8147 0.860813
$$832$$ 0 0
$$833$$ −38.7938 −1.34412
$$834$$ 0 0
$$835$$ 27.0964 0.937710
$$836$$ 0 0
$$837$$ 7.88994 0.272716
$$838$$ 0 0
$$839$$ 14.8727 0.513464 0.256732 0.966483i $$-0.417354\pi$$
0.256732 + 0.966483i $$0.417354\pi$$
$$840$$ 0 0
$$841$$ −13.4085 −0.462363
$$842$$ 0 0
$$843$$ 20.8844 0.719297
$$844$$ 0 0
$$845$$ −6.85321 −0.235758
$$846$$ 0 0
$$847$$ 1.91548 0.0658166
$$848$$ 0 0
$$849$$ 71.6574 2.45928
$$850$$ 0 0
$$851$$ 33.4099 1.14528
$$852$$ 0 0
$$853$$ −19.8801 −0.680682 −0.340341 0.940302i $$-0.610542\pi$$
−0.340341 + 0.940302i $$0.610542\pi$$
$$854$$ 0 0
$$855$$ 9.41863 0.322110
$$856$$ 0 0
$$857$$ −23.9474 −0.818029 −0.409014 0.912528i $$-0.634127\pi$$
−0.409014 + 0.912528i $$0.634127\pi$$
$$858$$ 0 0
$$859$$ 44.5394 1.51966 0.759832 0.650119i $$-0.225281\pi$$
0.759832 + 0.650119i $$0.225281\pi$$
$$860$$ 0 0
$$861$$ 3.89955 0.132896
$$862$$ 0 0
$$863$$ −7.47937 −0.254601 −0.127300 0.991864i $$-0.540631\pi$$
−0.127300 + 0.991864i $$0.540631\pi$$
$$864$$ 0 0
$$865$$ −4.77077 −0.162211
$$866$$ 0 0
$$867$$ −35.8690 −1.21818
$$868$$ 0 0
$$869$$ 6.83726 0.231938
$$870$$ 0 0
$$871$$ 35.4336 1.20062
$$872$$ 0 0
$$873$$ 11.7142 0.396464
$$874$$ 0 0
$$875$$ −1.43882 −0.0486411
$$876$$ 0 0
$$877$$ −43.0919 −1.45511 −0.727555 0.686049i $$-0.759343\pi$$
−0.727555 + 0.686049i $$0.759343\pi$$
$$878$$ 0 0
$$879$$ −32.1317 −1.08377
$$880$$ 0 0
$$881$$ 12.4283 0.418722 0.209361 0.977838i $$-0.432862\pi$$
0.209361 + 0.977838i $$0.432862\pi$$
$$882$$ 0 0
$$883$$ −34.0244 −1.14501 −0.572507 0.819900i $$-0.694029\pi$$
−0.572507 + 0.819900i $$0.694029\pi$$
$$884$$ 0 0
$$885$$ −80.2772 −2.69849
$$886$$ 0 0
$$887$$ 45.2609 1.51971 0.759855 0.650092i $$-0.225270\pi$$
0.759855 + 0.650092i $$0.225270\pi$$
$$888$$ 0 0
$$889$$ 0.107715 0.00361266
$$890$$ 0 0
$$891$$ −4.51170 −0.151148
$$892$$ 0 0
$$893$$ 11.2527 0.376556
$$894$$ 0 0
$$895$$ 4.63795 0.155029
$$896$$ 0 0
$$897$$ 45.1469 1.50741
$$898$$ 0 0
$$899$$ −21.6583 −0.722343
$$900$$ 0 0
$$901$$ −42.2427 −1.40731
$$902$$ 0 0
$$903$$ 2.67365 0.0889734
$$904$$ 0 0
$$905$$ 17.2632 0.573848
$$906$$ 0 0
$$907$$ −24.7681 −0.822412 −0.411206 0.911542i $$-0.634892\pi$$
−0.411206 + 0.911542i $$0.634892\pi$$
$$908$$ 0 0
$$909$$ −0.876894 −0.0290848
$$910$$ 0 0
$$911$$ 12.8875 0.426981 0.213491 0.976945i $$-0.431517\pi$$
0.213491 + 0.976945i $$0.431517\pi$$
$$912$$ 0 0
$$913$$ −3.64223 −0.120540
$$914$$ 0 0
$$915$$ 16.7901 0.555064
$$916$$ 0 0
$$917$$ −2.85579 −0.0943064
$$918$$ 0 0
$$919$$ −33.0846 −1.09136 −0.545681 0.837993i $$-0.683729\pi$$
−0.545681 + 0.837993i $$0.683729\pi$$
$$920$$ 0 0
$$921$$ −33.1904 −1.09366
$$922$$ 0 0
$$923$$ −30.1597 −0.992718
$$924$$ 0 0
$$925$$ −14.9217 −0.490622
$$926$$ 0 0
$$927$$ 51.3297 1.68589
$$928$$ 0 0
$$929$$ 48.0572 1.57671 0.788353 0.615224i $$-0.210934\pi$$
0.788353 + 0.615224i $$0.210934\pi$$
$$930$$ 0 0
$$931$$ 6.96725 0.228342
$$932$$ 0 0
$$933$$ −81.5771 −2.67071
$$934$$ 0 0
$$935$$ −9.49055 −0.310374
$$936$$ 0 0
$$937$$ −4.71132 −0.153912 −0.0769560 0.997034i $$-0.524520\pi$$
−0.0769560 + 0.997034i $$0.524520\pi$$
$$938$$ 0 0
$$939$$ −41.6580 −1.35946
$$940$$ 0 0
$$941$$ 55.0727 1.79532 0.897659 0.440690i $$-0.145266\pi$$
0.897659 + 0.440690i $$0.145266\pi$$
$$942$$ 0 0
$$943$$ −37.5482 −1.22274
$$944$$ 0 0
$$945$$ −0.688407 −0.0223939
$$946$$ 0 0
$$947$$ −23.5984 −0.766846 −0.383423 0.923573i $$-0.625255\pi$$
−0.383423 + 0.923573i $$0.625255\pi$$
$$948$$ 0 0
$$949$$ 42.2150 1.37036
$$950$$ 0 0
$$951$$ −17.6457 −0.572200
$$952$$ 0 0
$$953$$ 35.4123 1.14712 0.573558 0.819165i $$-0.305563\pi$$
0.573558 + 0.819165i $$0.305563\pi$$
$$954$$ 0 0
$$955$$ 19.7205 0.638140
$$956$$ 0 0
$$957$$ −6.51912 −0.210733
$$958$$ 0 0
$$959$$ 2.89191 0.0933847
$$960$$ 0 0
$$961$$ −0.914306 −0.0294938
$$962$$ 0 0
$$963$$ −34.5385 −1.11299
$$964$$ 0 0
$$965$$ −50.2998 −1.61921
$$966$$ 0 0
$$967$$ 58.6786 1.88698 0.943488 0.331407i $$-0.107523\pi$$
0.943488 + 0.331407i $$0.107523\pi$$
$$968$$ 0 0
$$969$$ 14.2628 0.458186
$$970$$ 0 0
$$971$$ 20.7839 0.666988 0.333494 0.942752i $$-0.391772\pi$$
0.333494 + 0.942752i $$0.391772\pi$$
$$972$$ 0 0
$$973$$ 0.810482 0.0259828
$$974$$ 0 0
$$975$$ −20.1637 −0.645756
$$976$$ 0 0
$$977$$ 11.7111 0.374670 0.187335 0.982296i $$-0.440015\pi$$
0.187335 + 0.982296i $$0.440015\pi$$
$$978$$ 0 0
$$979$$ −11.0687 −0.353758
$$980$$ 0 0
$$981$$ −69.3928 −2.21554
$$982$$ 0 0
$$983$$ −55.7921 −1.77949 −0.889745 0.456457i $$-0.849118\pi$$
−0.889745 + 0.456457i $$0.849118\pi$$
$$984$$ 0 0
$$985$$ 0.404666 0.0128937
$$986$$ 0 0
$$987$$ −5.21630 −0.166037
$$988$$ 0 0
$$989$$ −25.7442 −0.818617
$$990$$ 0 0
$$991$$ −52.1167 −1.65554 −0.827771 0.561066i $$-0.810391\pi$$
−0.827771 + 0.561066i $$0.810391\pi$$
$$992$$ 0 0
$$993$$ 65.8257 2.08892
$$994$$ 0 0
$$995$$ −56.5063 −1.79137
$$996$$ 0 0
$$997$$ 35.9871 1.13972 0.569862 0.821740i $$-0.306997\pi$$
0.569862 + 0.821740i $$0.306997\pi$$
$$998$$ 0 0
$$999$$ 10.7668 0.340648
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 1216.2.a.w.1.1 4
4.3 odd 2 1216.2.a.x.1.3 4
8.3 odd 2 608.2.a.i.1.2 4
8.5 even 2 608.2.a.j.1.4 yes 4
24.5 odd 2 5472.2.a.bs.1.2 4
24.11 even 2 5472.2.a.bt.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.2 4 8.3 odd 2
608.2.a.j.1.4 yes 4 8.5 even 2
1216.2.a.w.1.1 4 1.1 even 1 trivial
1216.2.a.x.1.3 4 4.3 odd 2
5472.2.a.bs.1.2 4 24.5 odd 2
5472.2.a.bt.1.2 4 24.11 even 2