# Properties

 Label 729.2.a.c.1.6 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{36})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$1.96962$$ of defining polynomial Character $$\chi$$ $$=$$ 729.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.96962 q^{2} +1.87939 q^{4} -3.70167 q^{5} -2.34730 q^{7} -0.237565 q^{8} +O(q^{10})$$ $$q+1.96962 q^{2} +1.87939 q^{4} -3.70167 q^{5} -2.34730 q^{7} -0.237565 q^{8} -7.29086 q^{10} +2.17853 q^{11} -4.71688 q^{13} -4.62327 q^{14} -4.22668 q^{16} +2.93512 q^{17} -6.22668 q^{19} -6.95686 q^{20} +4.29086 q^{22} -0.519030 q^{23} +8.70233 q^{25} -9.29044 q^{26} -4.41147 q^{28} +3.49276 q^{29} -4.30541 q^{31} -7.84981 q^{32} +5.78106 q^{34} +8.68891 q^{35} +2.41147 q^{37} -12.2642 q^{38} +0.879385 q^{40} -2.49860 q^{41} +1.06418 q^{43} +4.09429 q^{44} -1.02229 q^{46} +0.237565 q^{47} -1.49020 q^{49} +17.1403 q^{50} -8.86484 q^{52} -4.66717 q^{53} -8.06418 q^{55} +0.557635 q^{56} +6.87939 q^{58} +13.3122 q^{59} -3.67499 q^{61} -8.48000 q^{62} -7.00774 q^{64} +17.4603 q^{65} +14.2986 q^{67} +5.51622 q^{68} +17.1138 q^{70} -1.20307 q^{71} -4.68004 q^{73} +4.74968 q^{74} -11.7023 q^{76} -5.11365 q^{77} -12.8007 q^{79} +15.6458 q^{80} -4.92127 q^{82} +11.3040 q^{83} -10.8648 q^{85} +2.09602 q^{86} -0.517541 q^{88} -0.699287 q^{89} +11.0719 q^{91} -0.975457 q^{92} +0.467911 q^{94} +23.0491 q^{95} -7.08647 q^{97} -2.93512 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{7}+O(q^{10})$$ 6 * q - 12 * q^7 $$6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100})$$ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.96962 1.39273 0.696364 0.717689i $$-0.254800\pi$$
0.696364 + 0.717689i $$0.254800\pi$$
$$3$$ 0 0
$$4$$ 1.87939 0.939693
$$5$$ −3.70167 −1.65544 −0.827718 0.561145i $$-0.810361\pi$$
−0.827718 + 0.561145i $$0.810361\pi$$
$$6$$ 0 0
$$7$$ −2.34730 −0.887195 −0.443597 0.896226i $$-0.646298\pi$$
−0.443597 + 0.896226i $$0.646298\pi$$
$$8$$ −0.237565 −0.0839918
$$9$$ 0 0
$$10$$ −7.29086 −2.30557
$$11$$ 2.17853 0.656850 0.328425 0.944530i $$-0.393482\pi$$
0.328425 + 0.944530i $$0.393482\pi$$
$$12$$ 0 0
$$13$$ −4.71688 −1.30823 −0.654114 0.756396i $$-0.726958\pi$$
−0.654114 + 0.756396i $$0.726958\pi$$
$$14$$ −4.62327 −1.23562
$$15$$ 0 0
$$16$$ −4.22668 −1.05667
$$17$$ 2.93512 0.711871 0.355936 0.934510i $$-0.384162\pi$$
0.355936 + 0.934510i $$0.384162\pi$$
$$18$$ 0 0
$$19$$ −6.22668 −1.42850 −0.714249 0.699891i $$-0.753232\pi$$
−0.714249 + 0.699891i $$0.753232\pi$$
$$20$$ −6.95686 −1.55560
$$21$$ 0 0
$$22$$ 4.29086 0.914814
$$23$$ −0.519030 −0.108225 −0.0541126 0.998535i $$-0.517233\pi$$
−0.0541126 + 0.998535i $$0.517233\pi$$
$$24$$ 0 0
$$25$$ 8.70233 1.74047
$$26$$ −9.29044 −1.82201
$$27$$ 0 0
$$28$$ −4.41147 −0.833690
$$29$$ 3.49276 0.648588 0.324294 0.945956i $$-0.394873\pi$$
0.324294 + 0.945956i $$0.394873\pi$$
$$30$$ 0 0
$$31$$ −4.30541 −0.773274 −0.386637 0.922232i $$-0.626363\pi$$
−0.386637 + 0.922232i $$0.626363\pi$$
$$32$$ −7.84981 −1.38766
$$33$$ 0 0
$$34$$ 5.78106 0.991443
$$35$$ 8.68891 1.46869
$$36$$ 0 0
$$37$$ 2.41147 0.396444 0.198222 0.980157i $$-0.436483\pi$$
0.198222 + 0.980157i $$0.436483\pi$$
$$38$$ −12.2642 −1.98951
$$39$$ 0 0
$$40$$ 0.879385 0.139043
$$41$$ −2.49860 −0.390215 −0.195108 0.980782i $$-0.562506\pi$$
−0.195108 + 0.980782i $$0.562506\pi$$
$$42$$ 0 0
$$43$$ 1.06418 0.162286 0.0811428 0.996702i $$-0.474143\pi$$
0.0811428 + 0.996702i $$0.474143\pi$$
$$44$$ 4.09429 0.617237
$$45$$ 0 0
$$46$$ −1.02229 −0.150728
$$47$$ 0.237565 0.0346524 0.0173262 0.999850i $$-0.494485\pi$$
0.0173262 + 0.999850i $$0.494485\pi$$
$$48$$ 0 0
$$49$$ −1.49020 −0.212886
$$50$$ 17.1403 2.42400
$$51$$ 0 0
$$52$$ −8.86484 −1.22933
$$53$$ −4.66717 −0.641085 −0.320543 0.947234i $$-0.603865\pi$$
−0.320543 + 0.947234i $$0.603865\pi$$
$$54$$ 0 0
$$55$$ −8.06418 −1.08737
$$56$$ 0.557635 0.0745171
$$57$$ 0 0
$$58$$ 6.87939 0.903308
$$59$$ 13.3122 1.73310 0.866549 0.499092i $$-0.166333\pi$$
0.866549 + 0.499092i $$0.166333\pi$$
$$60$$ 0 0
$$61$$ −3.67499 −0.470535 −0.235267 0.971931i $$-0.575597\pi$$
−0.235267 + 0.971931i $$0.575597\pi$$
$$62$$ −8.48000 −1.07696
$$63$$ 0 0
$$64$$ −7.00774 −0.875968
$$65$$ 17.4603 2.16569
$$66$$ 0 0
$$67$$ 14.2986 1.74685 0.873426 0.486957i $$-0.161893\pi$$
0.873426 + 0.486957i $$0.161893\pi$$
$$68$$ 5.51622 0.668940
$$69$$ 0 0
$$70$$ 17.1138 2.04549
$$71$$ −1.20307 −0.142778 −0.0713891 0.997449i $$-0.522743\pi$$
−0.0713891 + 0.997449i $$0.522743\pi$$
$$72$$ 0 0
$$73$$ −4.68004 −0.547758 −0.273879 0.961764i $$-0.588307\pi$$
−0.273879 + 0.961764i $$0.588307\pi$$
$$74$$ 4.74968 0.552139
$$75$$ 0 0
$$76$$ −11.7023 −1.34235
$$77$$ −5.11365 −0.582754
$$78$$ 0 0
$$79$$ −12.8007 −1.44019 −0.720093 0.693877i $$-0.755901\pi$$
−0.720093 + 0.693877i $$0.755901\pi$$
$$80$$ 15.6458 1.74925
$$81$$ 0 0
$$82$$ −4.92127 −0.543464
$$83$$ 11.3040 1.24077 0.620385 0.784297i $$-0.286976\pi$$
0.620385 + 0.784297i $$0.286976\pi$$
$$84$$ 0 0
$$85$$ −10.8648 −1.17846
$$86$$ 2.09602 0.226020
$$87$$ 0 0
$$88$$ −0.517541 −0.0551701
$$89$$ −0.699287 −0.0741242 −0.0370621 0.999313i $$-0.511800\pi$$
−0.0370621 + 0.999313i $$0.511800\pi$$
$$90$$ 0 0
$$91$$ 11.0719 1.16065
$$92$$ −0.975457 −0.101698
$$93$$ 0 0
$$94$$ 0.467911 0.0482613
$$95$$ 23.0491 2.36479
$$96$$ 0 0
$$97$$ −7.08647 −0.719522 −0.359761 0.933045i $$-0.617142\pi$$
−0.359761 + 0.933045i $$0.617142\pi$$
$$98$$ −2.93512 −0.296492
$$99$$ 0 0
$$100$$ 16.3550 1.63550
$$101$$ −4.67712 −0.465391 −0.232696 0.972550i $$-0.574755\pi$$
−0.232696 + 0.972550i $$0.574755\pi$$
$$102$$ 0 0
$$103$$ −13.6236 −1.34237 −0.671187 0.741288i $$-0.734215\pi$$
−0.671187 + 0.741288i $$0.734215\pi$$
$$104$$ 1.12056 0.109880
$$105$$ 0 0
$$106$$ −9.19253 −0.892858
$$107$$ −11.6340 −1.12470 −0.562350 0.826900i $$-0.690102\pi$$
−0.562350 + 0.826900i $$0.690102\pi$$
$$108$$ 0 0
$$109$$ 14.6040 1.39881 0.699405 0.714725i $$-0.253448\pi$$
0.699405 + 0.714725i $$0.253448\pi$$
$$110$$ −15.8833 −1.51442
$$111$$ 0 0
$$112$$ 9.92127 0.937472
$$113$$ −4.69583 −0.441746 −0.220873 0.975303i $$-0.570891\pi$$
−0.220873 + 0.975303i $$0.570891\pi$$
$$114$$ 0 0
$$115$$ 1.92127 0.179160
$$116$$ 6.56423 0.609474
$$117$$ 0 0
$$118$$ 26.2199 2.41374
$$119$$ −6.88960 −0.631568
$$120$$ 0 0
$$121$$ −6.25402 −0.568548
$$122$$ −7.23832 −0.655327
$$123$$ 0 0
$$124$$ −8.09152 −0.726640
$$125$$ −13.7048 −1.22579
$$126$$ 0 0
$$127$$ 6.09152 0.540535 0.270267 0.962785i $$-0.412888\pi$$
0.270267 + 0.962785i $$0.412888\pi$$
$$128$$ 1.89706 0.167678
$$129$$ 0 0
$$130$$ 34.3901 3.01621
$$131$$ −10.3484 −0.904144 −0.452072 0.891981i $$-0.649315\pi$$
−0.452072 + 0.891981i $$0.649315\pi$$
$$132$$ 0 0
$$133$$ 14.6159 1.26736
$$134$$ 28.1627 2.43289
$$135$$ 0 0
$$136$$ −0.697281 −0.0597914
$$137$$ −19.0847 −1.63051 −0.815257 0.579100i $$-0.803404\pi$$
−0.815257 + 0.579100i $$0.803404\pi$$
$$138$$ 0 0
$$139$$ −23.3037 −1.97659 −0.988295 0.152555i $$-0.951250\pi$$
−0.988295 + 0.152555i $$0.951250\pi$$
$$140$$ 16.3298 1.38012
$$141$$ 0 0
$$142$$ −2.36959 −0.198851
$$143$$ −10.2759 −0.859310
$$144$$ 0 0
$$145$$ −12.9290 −1.07370
$$146$$ −9.21789 −0.762878
$$147$$ 0 0
$$148$$ 4.53209 0.372535
$$149$$ −15.5060 −1.27030 −0.635149 0.772390i $$-0.719061\pi$$
−0.635149 + 0.772390i $$0.719061\pi$$
$$150$$ 0 0
$$151$$ −5.90167 −0.480271 −0.240136 0.970739i $$-0.577192\pi$$
−0.240136 + 0.970739i $$0.577192\pi$$
$$152$$ 1.47924 0.119982
$$153$$ 0 0
$$154$$ −10.0719 −0.811618
$$155$$ 15.9372 1.28011
$$156$$ 0 0
$$157$$ −1.21213 −0.0967388 −0.0483694 0.998830i $$-0.515402\pi$$
−0.0483694 + 0.998830i $$0.515402\pi$$
$$158$$ −25.2124 −2.00579
$$159$$ 0 0
$$160$$ 29.0574 2.29719
$$161$$ 1.21832 0.0960168
$$162$$ 0 0
$$163$$ 2.77332 0.217223 0.108612 0.994084i $$-0.465360\pi$$
0.108612 + 0.994084i $$0.465360\pi$$
$$164$$ −4.69583 −0.366682
$$165$$ 0 0
$$166$$ 22.2645 1.72806
$$167$$ 3.82807 0.296225 0.148113 0.988971i $$-0.452680\pi$$
0.148113 + 0.988971i $$0.452680\pi$$
$$168$$ 0 0
$$169$$ 9.24897 0.711459
$$170$$ −21.3996 −1.64127
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 7.02941 0.534436 0.267218 0.963636i $$-0.413896\pi$$
0.267218 + 0.963636i $$0.413896\pi$$
$$174$$ 0 0
$$175$$ −20.4270 −1.54413
$$176$$ −9.20794 −0.694074
$$177$$ 0 0
$$178$$ −1.37733 −0.103235
$$179$$ 14.3854 1.07521 0.537607 0.843195i $$-0.319328\pi$$
0.537607 + 0.843195i $$0.319328\pi$$
$$180$$ 0 0
$$181$$ 13.2003 0.981169 0.490584 0.871394i $$-0.336783\pi$$
0.490584 + 0.871394i $$0.336783\pi$$
$$182$$ 21.8074 1.61647
$$183$$ 0 0
$$184$$ 0.123303 0.00909003
$$185$$ −8.92647 −0.656287
$$186$$ 0 0
$$187$$ 6.39424 0.467593
$$188$$ 0.446476 0.0325626
$$189$$ 0 0
$$190$$ 45.3979 3.29351
$$191$$ −13.4233 −0.971279 −0.485639 0.874159i $$-0.661413\pi$$
−0.485639 + 0.874159i $$0.661413\pi$$
$$192$$ 0 0
$$193$$ 15.0051 1.08009 0.540044 0.841637i $$-0.318408\pi$$
0.540044 + 0.841637i $$0.318408\pi$$
$$194$$ −13.9576 −1.00210
$$195$$ 0 0
$$196$$ −2.80066 −0.200047
$$197$$ 22.3212 1.59032 0.795158 0.606402i $$-0.207388\pi$$
0.795158 + 0.606402i $$0.207388\pi$$
$$198$$ 0 0
$$199$$ −9.10101 −0.645154 −0.322577 0.946543i $$-0.604549\pi$$
−0.322577 + 0.946543i $$0.604549\pi$$
$$200$$ −2.06737 −0.146185
$$201$$ 0 0
$$202$$ −9.21213 −0.648163
$$203$$ −8.19853 −0.575424
$$204$$ 0 0
$$205$$ 9.24897 0.645976
$$206$$ −26.8333 −1.86956
$$207$$ 0 0
$$208$$ 19.9368 1.38237
$$209$$ −13.5650 −0.938310
$$210$$ 0 0
$$211$$ −5.97771 −0.411523 −0.205761 0.978602i $$-0.565967\pi$$
−0.205761 + 0.978602i $$0.565967\pi$$
$$212$$ −8.77141 −0.602423
$$213$$ 0 0
$$214$$ −22.9145 −1.56640
$$215$$ −3.93923 −0.268653
$$216$$ 0 0
$$217$$ 10.1061 0.686045
$$218$$ 28.7643 1.94816
$$219$$ 0 0
$$220$$ −15.1557 −1.02180
$$221$$ −13.8446 −0.931290
$$222$$ 0 0
$$223$$ −8.90167 −0.596100 −0.298050 0.954550i $$-0.596336\pi$$
−0.298050 + 0.954550i $$0.596336\pi$$
$$224$$ 18.4258 1.23113
$$225$$ 0 0
$$226$$ −9.24897 −0.615232
$$227$$ 10.6685 0.708092 0.354046 0.935228i $$-0.384806\pi$$
0.354046 + 0.935228i $$0.384806\pi$$
$$228$$ 0 0
$$229$$ −8.27126 −0.546580 −0.273290 0.961932i $$-0.588112\pi$$
−0.273290 + 0.961932i $$0.588112\pi$$
$$230$$ 3.78417 0.249521
$$231$$ 0 0
$$232$$ −0.829755 −0.0544761
$$233$$ −12.7393 −0.834579 −0.417290 0.908774i $$-0.637020\pi$$
−0.417290 + 0.908774i $$0.637020\pi$$
$$234$$ 0 0
$$235$$ −0.879385 −0.0573648
$$236$$ 25.0187 1.62858
$$237$$ 0 0
$$238$$ −13.5699 −0.879603
$$239$$ −15.0156 −0.971277 −0.485638 0.874160i $$-0.661413\pi$$
−0.485638 + 0.874160i $$0.661413\pi$$
$$240$$ 0 0
$$241$$ 0.795607 0.0512496 0.0256248 0.999672i $$-0.491842\pi$$
0.0256248 + 0.999672i $$0.491842\pi$$
$$242$$ −12.3180 −0.791832
$$243$$ 0 0
$$244$$ −6.90673 −0.442158
$$245$$ 5.51622 0.352419
$$246$$ 0 0
$$247$$ 29.3705 1.86880
$$248$$ 1.02281 0.0649487
$$249$$ 0 0
$$250$$ −26.9932 −1.70720
$$251$$ −8.31499 −0.524837 −0.262419 0.964954i $$-0.584520\pi$$
−0.262419 + 0.964954i $$0.584520\pi$$
$$252$$ 0 0
$$253$$ −1.13072 −0.0710877
$$254$$ 11.9980 0.752818
$$255$$ 0 0
$$256$$ 17.7520 1.10950
$$257$$ 25.6202 1.59815 0.799074 0.601233i $$-0.205324\pi$$
0.799074 + 0.601233i $$0.205324\pi$$
$$258$$ 0 0
$$259$$ −5.66044 −0.351723
$$260$$ 32.8147 2.03508
$$261$$ 0 0
$$262$$ −20.3824 −1.25923
$$263$$ −28.0650 −1.73056 −0.865281 0.501287i $$-0.832860\pi$$
−0.865281 + 0.501287i $$0.832860\pi$$
$$264$$ 0 0
$$265$$ 17.2763 1.06128
$$266$$ 28.7876 1.76508
$$267$$ 0 0
$$268$$ 26.8726 1.64150
$$269$$ 30.1710 1.83956 0.919778 0.392439i $$-0.128369\pi$$
0.919778 + 0.392439i $$0.128369\pi$$
$$270$$ 0 0
$$271$$ −19.0000 −1.15417 −0.577084 0.816685i $$-0.695809\pi$$
−0.577084 + 0.816685i $$0.695809\pi$$
$$272$$ −12.4058 −0.752213
$$273$$ 0 0
$$274$$ −37.5895 −2.27086
$$275$$ 18.9583 1.14323
$$276$$ 0 0
$$277$$ 21.1215 1.26907 0.634535 0.772894i $$-0.281191\pi$$
0.634535 + 0.772894i $$0.281191\pi$$
$$278$$ −45.8992 −2.75285
$$279$$ 0 0
$$280$$ −2.06418 −0.123358
$$281$$ 1.74730 0.104235 0.0521175 0.998641i $$-0.483403\pi$$
0.0521175 + 0.998641i $$0.483403\pi$$
$$282$$ 0 0
$$283$$ −7.28817 −0.433237 −0.216618 0.976256i $$-0.569503\pi$$
−0.216618 + 0.976256i $$0.569503\pi$$
$$284$$ −2.26103 −0.134168
$$285$$ 0 0
$$286$$ −20.2395 −1.19679
$$287$$ 5.86495 0.346197
$$288$$ 0 0
$$289$$ −8.38507 −0.493239
$$290$$ −25.4652 −1.49537
$$291$$ 0 0
$$292$$ −8.79561 −0.514724
$$293$$ 15.3509 0.896809 0.448404 0.893831i $$-0.351992\pi$$
0.448404 + 0.893831i $$0.351992\pi$$
$$294$$ 0 0
$$295$$ −49.2772 −2.86903
$$296$$ −0.572881 −0.0332980
$$297$$ 0 0
$$298$$ −30.5408 −1.76918
$$299$$ 2.44820 0.141583
$$300$$ 0 0
$$301$$ −2.49794 −0.143979
$$302$$ −11.6240 −0.668888
$$303$$ 0 0
$$304$$ 26.3182 1.50945
$$305$$ 13.6036 0.778940
$$306$$ 0 0
$$307$$ 16.7638 0.956762 0.478381 0.878152i $$-0.341224\pi$$
0.478381 + 0.878152i $$0.341224\pi$$
$$308$$ −9.61051 −0.547610
$$309$$ 0 0
$$310$$ 31.3901 1.78284
$$311$$ −16.0231 −0.908589 −0.454295 0.890852i $$-0.650109\pi$$
−0.454295 + 0.890852i $$0.650109\pi$$
$$312$$ 0 0
$$313$$ 34.4056 1.94472 0.972360 0.233488i $$-0.0750138\pi$$
0.972360 + 0.233488i $$0.0750138\pi$$
$$314$$ −2.38744 −0.134731
$$315$$ 0 0
$$316$$ −24.0574 −1.35333
$$317$$ −16.3111 −0.916123 −0.458061 0.888921i $$-0.651456\pi$$
−0.458061 + 0.888921i $$0.651456\pi$$
$$318$$ 0 0
$$319$$ 7.60906 0.426026
$$320$$ 25.9403 1.45011
$$321$$ 0 0
$$322$$ 2.39961 0.133725
$$323$$ −18.2761 −1.01691
$$324$$ 0 0
$$325$$ −41.0479 −2.27693
$$326$$ 5.46237 0.302533
$$327$$ 0 0
$$328$$ 0.593578 0.0327749
$$329$$ −0.557635 −0.0307434
$$330$$ 0 0
$$331$$ −31.5800 −1.73579 −0.867896 0.496746i $$-0.834528\pi$$
−0.867896 + 0.496746i $$0.834528\pi$$
$$332$$ 21.2445 1.16594
$$333$$ 0 0
$$334$$ 7.53983 0.412561
$$335$$ −52.9286 −2.89180
$$336$$ 0 0
$$337$$ 6.00505 0.327116 0.163558 0.986534i $$-0.447703\pi$$
0.163558 + 0.986534i $$0.447703\pi$$
$$338$$ 18.2169 0.990870
$$339$$ 0 0
$$340$$ −20.4192 −1.10739
$$341$$ −9.37944 −0.507925
$$342$$ 0 0
$$343$$ 19.9290 1.07607
$$344$$ −0.252811 −0.0136307
$$345$$ 0 0
$$346$$ 13.8452 0.744325
$$347$$ 23.0304 1.23634 0.618168 0.786046i $$-0.287875\pi$$
0.618168 + 0.786046i $$0.287875\pi$$
$$348$$ 0 0
$$349$$ −11.5662 −0.619126 −0.309563 0.950879i $$-0.600183\pi$$
−0.309563 + 0.950879i $$0.600183\pi$$
$$350$$ −40.2332 −2.15056
$$351$$ 0 0
$$352$$ −17.1010 −0.911487
$$353$$ −2.13463 −0.113615 −0.0568073 0.998385i $$-0.518092\pi$$
−0.0568073 + 0.998385i $$0.518092\pi$$
$$354$$ 0 0
$$355$$ 4.45336 0.236360
$$356$$ −1.31423 −0.0696540
$$357$$ 0 0
$$358$$ 28.3337 1.49748
$$359$$ 24.2235 1.27847 0.639234 0.769012i $$-0.279251\pi$$
0.639234 + 0.769012i $$0.279251\pi$$
$$360$$ 0 0
$$361$$ 19.7716 1.04061
$$362$$ 25.9995 1.36650
$$363$$ 0 0
$$364$$ 20.8084 1.09066
$$365$$ 17.3240 0.906778
$$366$$ 0 0
$$367$$ −2.83481 −0.147976 −0.0739879 0.997259i $$-0.523573\pi$$
−0.0739879 + 0.997259i $$0.523573\pi$$
$$368$$ 2.19377 0.114358
$$369$$ 0 0
$$370$$ −17.5817 −0.914030
$$371$$ 10.9552 0.568767
$$372$$ 0 0
$$373$$ −28.8334 −1.49294 −0.746468 0.665421i $$-0.768252\pi$$
−0.746468 + 0.665421i $$0.768252\pi$$
$$374$$ 12.5942 0.651230
$$375$$ 0 0
$$376$$ −0.0564370 −0.00291052
$$377$$ −16.4749 −0.848501
$$378$$ 0 0
$$379$$ 32.1985 1.65393 0.826963 0.562256i $$-0.190066\pi$$
0.826963 + 0.562256i $$0.190066\pi$$
$$380$$ 43.3181 2.22217
$$381$$ 0 0
$$382$$ −26.4388 −1.35273
$$383$$ 0.672250 0.0343504 0.0171752 0.999852i $$-0.494533\pi$$
0.0171752 + 0.999852i $$0.494533\pi$$
$$384$$ 0 0
$$385$$ 18.9290 0.964712
$$386$$ 29.5542 1.50427
$$387$$ 0 0
$$388$$ −13.3182 −0.676129
$$389$$ −4.25930 −0.215955 −0.107978 0.994153i $$-0.534437\pi$$
−0.107978 + 0.994153i $$0.534437\pi$$
$$390$$ 0 0
$$391$$ −1.52341 −0.0770424
$$392$$ 0.354019 0.0178807
$$393$$ 0 0
$$394$$ 43.9641 2.21488
$$395$$ 47.3838 2.38414
$$396$$ 0 0
$$397$$ −8.86484 −0.444913 −0.222457 0.974943i $$-0.571408\pi$$
−0.222457 + 0.974943i $$0.571408\pi$$
$$398$$ −17.9255 −0.898524
$$399$$ 0 0
$$400$$ −36.7820 −1.83910
$$401$$ 37.0839 1.85188 0.925941 0.377667i $$-0.123274\pi$$
0.925941 + 0.377667i $$0.123274\pi$$
$$402$$ 0 0
$$403$$ 20.3081 1.01162
$$404$$ −8.79012 −0.437325
$$405$$ 0 0
$$406$$ −16.1480 −0.801410
$$407$$ 5.25346 0.260404
$$408$$ 0 0
$$409$$ −3.38919 −0.167584 −0.0837922 0.996483i $$-0.526703\pi$$
−0.0837922 + 0.996483i $$0.526703\pi$$
$$410$$ 18.2169 0.899669
$$411$$ 0 0
$$412$$ −25.6040 −1.26142
$$413$$ −31.2476 −1.53760
$$414$$ 0 0
$$415$$ −41.8435 −2.05402
$$416$$ 37.0266 1.81538
$$417$$ 0 0
$$418$$ −26.7178 −1.30681
$$419$$ −20.6564 −1.00913 −0.504565 0.863374i $$-0.668347\pi$$
−0.504565 + 0.863374i $$0.668347\pi$$
$$420$$ 0 0
$$421$$ 27.5330 1.34188 0.670939 0.741513i $$-0.265891\pi$$
0.670939 + 0.741513i $$0.265891\pi$$
$$422$$ −11.7738 −0.573139
$$423$$ 0 0
$$424$$ 1.10876 0.0538459
$$425$$ 25.5424 1.23899
$$426$$ 0 0
$$427$$ 8.62630 0.417456
$$428$$ −21.8647 −1.05687
$$429$$ 0 0
$$430$$ −7.75877 −0.374161
$$431$$ −2.58110 −0.124327 −0.0621636 0.998066i $$-0.519800\pi$$
−0.0621636 + 0.998066i $$0.519800\pi$$
$$432$$ 0 0
$$433$$ −27.0137 −1.29820 −0.649098 0.760704i $$-0.724854\pi$$
−0.649098 + 0.760704i $$0.724854\pi$$
$$434$$ 19.9051 0.955474
$$435$$ 0 0
$$436$$ 27.4466 1.31445
$$437$$ 3.23183 0.154599
$$438$$ 0 0
$$439$$ 11.7101 0.558891 0.279446 0.960162i $$-0.409849\pi$$
0.279446 + 0.960162i $$0.409849\pi$$
$$440$$ 1.91576 0.0913305
$$441$$ 0 0
$$442$$ −27.2686 −1.29703
$$443$$ 2.08077 0.0988606 0.0494303 0.998778i $$-0.484259\pi$$
0.0494303 + 0.998778i $$0.484259\pi$$
$$444$$ 0 0
$$445$$ 2.58853 0.122708
$$446$$ −17.5329 −0.830206
$$447$$ 0 0
$$448$$ 16.4492 0.777154
$$449$$ 10.5508 0.497924 0.248962 0.968513i $$-0.419911\pi$$
0.248962 + 0.968513i $$0.419911\pi$$
$$450$$ 0 0
$$451$$ −5.44326 −0.256313
$$452$$ −8.82526 −0.415106
$$453$$ 0 0
$$454$$ 21.0128 0.986179
$$455$$ −40.9845 −1.92139
$$456$$ 0 0
$$457$$ −7.90941 −0.369987 −0.184993 0.982740i $$-0.559226\pi$$
−0.184993 + 0.982740i $$0.559226\pi$$
$$458$$ −16.2912 −0.761238
$$459$$ 0 0
$$460$$ 3.61081 0.168355
$$461$$ −39.0928 −1.82073 −0.910366 0.413804i $$-0.864200\pi$$
−0.910366 + 0.413804i $$0.864200\pi$$
$$462$$ 0 0
$$463$$ 23.6928 1.10110 0.550550 0.834802i $$-0.314418\pi$$
0.550550 + 0.834802i $$0.314418\pi$$
$$464$$ −14.7628 −0.685344
$$465$$ 0 0
$$466$$ −25.0915 −1.16234
$$467$$ −34.7152 −1.60643 −0.803214 0.595691i $$-0.796878\pi$$
−0.803214 + 0.595691i $$0.796878\pi$$
$$468$$ 0 0
$$469$$ −33.5631 −1.54980
$$470$$ −1.73205 −0.0798935
$$471$$ 0 0
$$472$$ −3.16250 −0.145566
$$473$$ 2.31834 0.106597
$$474$$ 0 0
$$475$$ −54.1867 −2.48625
$$476$$ −12.9482 −0.593480
$$477$$ 0 0
$$478$$ −29.5749 −1.35272
$$479$$ −6.48173 −0.296158 −0.148079 0.988976i $$-0.547309\pi$$
−0.148079 + 0.988976i $$0.547309\pi$$
$$480$$ 0 0
$$481$$ −11.3746 −0.518639
$$482$$ 1.56704 0.0713767
$$483$$ 0 0
$$484$$ −11.7537 −0.534260
$$485$$ 26.2317 1.19112
$$486$$ 0 0
$$487$$ 19.9828 0.905505 0.452753 0.891636i $$-0.350442\pi$$
0.452753 + 0.891636i $$0.350442\pi$$
$$488$$ 0.873048 0.0395210
$$489$$ 0 0
$$490$$ 10.8648 0.490823
$$491$$ −26.2271 −1.18361 −0.591806 0.806081i $$-0.701585\pi$$
−0.591806 + 0.806081i $$0.701585\pi$$
$$492$$ 0 0
$$493$$ 10.2517 0.461711
$$494$$ 57.8486 2.60273
$$495$$ 0 0
$$496$$ 18.1976 0.817096
$$497$$ 2.82396 0.126672
$$498$$ 0 0
$$499$$ 6.31727 0.282800 0.141400 0.989953i $$-0.454840\pi$$
0.141400 + 0.989953i $$0.454840\pi$$
$$500$$ −25.7566 −1.15187
$$501$$ 0 0
$$502$$ −16.3773 −0.730956
$$503$$ −21.8261 −0.973179 −0.486589 0.873631i $$-0.661759\pi$$
−0.486589 + 0.873631i $$0.661759\pi$$
$$504$$ 0 0
$$505$$ 17.3131 0.770425
$$506$$ −2.22708 −0.0990059
$$507$$ 0 0
$$508$$ 11.4483 0.507937
$$509$$ 29.0931 1.28953 0.644765 0.764381i $$-0.276955\pi$$
0.644765 + 0.764381i $$0.276955\pi$$
$$510$$ 0 0
$$511$$ 10.9855 0.485968
$$512$$ 31.1704 1.37755
$$513$$ 0 0
$$514$$ 50.4620 2.22579
$$515$$ 50.4301 2.22221
$$516$$ 0 0
$$517$$ 0.517541 0.0227614
$$518$$ −11.1489 −0.489855
$$519$$ 0 0
$$520$$ −4.14796 −0.181900
$$521$$ −13.6949 −0.599982 −0.299991 0.953942i $$-0.596984\pi$$
−0.299991 + 0.953942i $$0.596984\pi$$
$$522$$ 0 0
$$523$$ 13.1506 0.575038 0.287519 0.957775i $$-0.407170\pi$$
0.287519 + 0.957775i $$0.407170\pi$$
$$524$$ −19.4486 −0.849618
$$525$$ 0 0
$$526$$ −55.2772 −2.41020
$$527$$ −12.6369 −0.550472
$$528$$ 0 0
$$529$$ −22.7306 −0.988287
$$530$$ 34.0277 1.47807
$$531$$ 0 0
$$532$$ 27.4688 1.19093
$$533$$ 11.7856 0.510490
$$534$$ 0 0
$$535$$ 43.0651 1.86187
$$536$$ −3.39684 −0.146721
$$537$$ 0 0
$$538$$ 59.4252 2.56200
$$539$$ −3.24644 −0.139834
$$540$$ 0 0
$$541$$ 6.26083 0.269174 0.134587 0.990902i $$-0.457029\pi$$
0.134587 + 0.990902i $$0.457029\pi$$
$$542$$ −37.4227 −1.60744
$$543$$ 0 0
$$544$$ −23.0401 −0.987838
$$545$$ −54.0592 −2.31564
$$546$$ 0 0
$$547$$ −31.3783 −1.34164 −0.670819 0.741621i $$-0.734057\pi$$
−0.670819 + 0.741621i $$0.734057\pi$$
$$548$$ −35.8674 −1.53218
$$549$$ 0 0
$$550$$ 37.3405 1.59220
$$551$$ −21.7483 −0.926508
$$552$$ 0 0
$$553$$ 30.0469 1.27773
$$554$$ 41.6013 1.76747
$$555$$ 0 0
$$556$$ −43.7965 −1.85739
$$557$$ −43.4392 −1.84058 −0.920290 0.391237i $$-0.872047\pi$$
−0.920290 + 0.391237i $$0.872047\pi$$
$$558$$ 0 0
$$559$$ −5.01960 −0.212306
$$560$$ −36.7252 −1.55192
$$561$$ 0 0
$$562$$ 3.44150 0.145171
$$563$$ −32.0176 −1.34938 −0.674691 0.738100i $$-0.735723\pi$$
−0.674691 + 0.738100i $$0.735723\pi$$
$$564$$ 0 0
$$565$$ 17.3824 0.731282
$$566$$ −14.3549 −0.603381
$$567$$ 0 0
$$568$$ 0.285807 0.0119922
$$569$$ −6.77194 −0.283895 −0.141947 0.989874i $$-0.545336\pi$$
−0.141947 + 0.989874i $$0.545336\pi$$
$$570$$ 0 0
$$571$$ −21.1530 −0.885226 −0.442613 0.896713i $$-0.645948\pi$$
−0.442613 + 0.896713i $$0.645948\pi$$
$$572$$ −19.3123 −0.807487
$$573$$ 0 0
$$574$$ 11.5517 0.482158
$$575$$ −4.51677 −0.188362
$$576$$ 0 0
$$577$$ 11.9162 0.496079 0.248039 0.968750i $$-0.420214\pi$$
0.248039 + 0.968750i $$0.420214\pi$$
$$578$$ −16.5154 −0.686948
$$579$$ 0 0
$$580$$ −24.2986 −1.00894
$$581$$ −26.5337 −1.10081
$$582$$ 0 0
$$583$$ −10.1676 −0.421097
$$584$$ 1.11181 0.0460072
$$585$$ 0 0
$$586$$ 30.2354 1.24901
$$587$$ −0.129862 −0.00535996 −0.00267998 0.999996i $$-0.500853\pi$$
−0.00267998 + 0.999996i $$0.500853\pi$$
$$588$$ 0 0
$$589$$ 26.8084 1.10462
$$590$$ −97.0572 −3.99578
$$591$$ 0 0
$$592$$ −10.1925 −0.418911
$$593$$ 26.2622 1.07846 0.539230 0.842158i $$-0.318715\pi$$
0.539230 + 0.842158i $$0.318715\pi$$
$$594$$ 0 0
$$595$$ 25.5030 1.04552
$$596$$ −29.1417 −1.19369
$$597$$ 0 0
$$598$$ 4.82201 0.197187
$$599$$ 27.5952 1.12751 0.563754 0.825943i $$-0.309357\pi$$
0.563754 + 0.825943i $$0.309357\pi$$
$$600$$ 0 0
$$601$$ 1.33275 0.0543639 0.0271820 0.999631i $$-0.491347\pi$$
0.0271820 + 0.999631i $$0.491347\pi$$
$$602$$ −4.91998 −0.200524
$$603$$ 0 0
$$604$$ −11.0915 −0.451308
$$605$$ 23.1503 0.941194
$$606$$ 0 0
$$607$$ −12.5645 −0.509977 −0.254988 0.966944i $$-0.582072\pi$$
−0.254988 + 0.966944i $$0.582072\pi$$
$$608$$ 48.8783 1.98228
$$609$$ 0 0
$$610$$ 26.7939 1.08485
$$611$$ −1.12056 −0.0453332
$$612$$ 0 0
$$613$$ −13.9982 −0.565384 −0.282692 0.959211i $$-0.591227\pi$$
−0.282692 + 0.959211i $$0.591227\pi$$
$$614$$ 33.0183 1.33251
$$615$$ 0 0
$$616$$ 1.21482 0.0489466
$$617$$ 24.0467 0.968084 0.484042 0.875045i $$-0.339168\pi$$
0.484042 + 0.875045i $$0.339168\pi$$
$$618$$ 0 0
$$619$$ 6.87164 0.276195 0.138097 0.990419i $$-0.455901\pi$$
0.138097 + 0.990419i $$0.455901\pi$$
$$620$$ 29.9521 1.20291
$$621$$ 0 0
$$622$$ −31.5594 −1.26542
$$623$$ 1.64143 0.0657626
$$624$$ 0 0
$$625$$ 7.21894 0.288758
$$626$$ 67.7658 2.70847
$$627$$ 0 0
$$628$$ −2.27807 −0.0909047
$$629$$ 7.07797 0.282217
$$630$$ 0 0
$$631$$ −35.3773 −1.40835 −0.704175 0.710027i $$-0.748683\pi$$
−0.704175 + 0.710027i $$0.748683\pi$$
$$632$$ 3.04098 0.120964
$$633$$ 0 0
$$634$$ −32.1266 −1.27591
$$635$$ −22.5488 −0.894821
$$636$$ 0 0
$$637$$ 7.02910 0.278503
$$638$$ 14.9869 0.593338
$$639$$ 0 0
$$640$$ −7.02229 −0.277580
$$641$$ −19.1737 −0.757314 −0.378657 0.925537i $$-0.623614\pi$$
−0.378657 + 0.925537i $$0.623614\pi$$
$$642$$ 0 0
$$643$$ 19.3764 0.764130 0.382065 0.924135i $$-0.375213\pi$$
0.382065 + 0.924135i $$0.375213\pi$$
$$644$$ 2.28969 0.0902263
$$645$$ 0 0
$$646$$ −35.9968 −1.41628
$$647$$ 8.77141 0.344840 0.172420 0.985024i $$-0.444841\pi$$
0.172420 + 0.985024i $$0.444841\pi$$
$$648$$ 0 0
$$649$$ 29.0009 1.13839
$$650$$ −80.8485 −3.17114
$$651$$ 0 0
$$652$$ 5.21213 0.204123
$$653$$ 32.8094 1.28393 0.641965 0.766734i $$-0.278119\pi$$
0.641965 + 0.766734i $$0.278119\pi$$
$$654$$ 0 0
$$655$$ 38.3063 1.49675
$$656$$ 10.5608 0.412329
$$657$$ 0 0
$$658$$ −1.09833 −0.0428172
$$659$$ 18.6516 0.726563 0.363282 0.931679i $$-0.381656\pi$$
0.363282 + 0.931679i $$0.381656\pi$$
$$660$$ 0 0
$$661$$ 36.4074 1.41608 0.708041 0.706171i $$-0.249579\pi$$
0.708041 + 0.706171i $$0.249579\pi$$
$$662$$ −62.2004 −2.41749
$$663$$ 0 0
$$664$$ −2.68542 −0.104215
$$665$$ −54.1031 −2.09803
$$666$$ 0 0
$$667$$ −1.81284 −0.0701936
$$668$$ 7.19442 0.278361
$$669$$ 0 0
$$670$$ −104.249 −4.02749
$$671$$ −8.00607 −0.309071
$$672$$ 0 0
$$673$$ 39.3901 1.51838 0.759189 0.650871i $$-0.225596\pi$$
0.759189 + 0.650871i $$0.225596\pi$$
$$674$$ 11.8276 0.455584
$$675$$ 0 0
$$676$$ 17.3824 0.668553
$$677$$ 31.6754 1.21738 0.608692 0.793406i $$-0.291694\pi$$
0.608692 + 0.793406i $$0.291694\pi$$
$$678$$ 0 0
$$679$$ 16.6340 0.638356
$$680$$ 2.58110 0.0989807
$$681$$ 0 0
$$682$$ −18.4739 −0.707402
$$683$$ −29.0656 −1.11217 −0.556083 0.831127i $$-0.687696\pi$$
−0.556083 + 0.831127i $$0.687696\pi$$
$$684$$ 0 0
$$685$$ 70.6451 2.69921
$$686$$ 39.2525 1.49867
$$687$$ 0 0
$$688$$ −4.49794 −0.171482
$$689$$ 22.0145 0.838685
$$690$$ 0 0
$$691$$ 5.35740 0.203805 0.101903 0.994794i $$-0.467507\pi$$
0.101903 + 0.994794i $$0.467507\pi$$
$$692$$ 13.2110 0.502206
$$693$$ 0 0
$$694$$ 45.3610 1.72188
$$695$$ 86.2623 3.27212
$$696$$ 0 0
$$697$$ −7.33368 −0.277783
$$698$$ −22.7810 −0.862275
$$699$$ 0 0
$$700$$ −38.3901 −1.45101
$$701$$ −25.6536 −0.968922 −0.484461 0.874813i $$-0.660984\pi$$
−0.484461 + 0.874813i $$0.660984\pi$$
$$702$$ 0 0
$$703$$ −15.0155 −0.566320
$$704$$ −15.2665 −0.575380
$$705$$ 0 0
$$706$$ −4.20439 −0.158234
$$707$$ 10.9786 0.412893
$$708$$ 0 0
$$709$$ −4.69047 −0.176154 −0.0880772 0.996114i $$-0.528072\pi$$
−0.0880772 + 0.996114i $$0.528072\pi$$
$$710$$ 8.77141 0.329185
$$711$$ 0 0
$$712$$ 0.166126 0.00622583
$$713$$ 2.23463 0.0836877
$$714$$ 0 0
$$715$$ 38.0378 1.42253
$$716$$ 27.0357 1.01037
$$717$$ 0 0
$$718$$ 47.7110 1.78056
$$719$$ 39.0669 1.45695 0.728476 0.685072i $$-0.240229\pi$$
0.728476 + 0.685072i $$0.240229\pi$$
$$720$$ 0 0
$$721$$ 31.9786 1.19095
$$722$$ 38.9424 1.44929
$$723$$ 0 0
$$724$$ 24.8084 0.921997
$$725$$ 30.3951 1.12885
$$726$$ 0 0
$$727$$ −10.8253 −0.401489 −0.200744 0.979644i $$-0.564336\pi$$
−0.200744 + 0.979644i $$0.564336\pi$$
$$728$$ −2.63030 −0.0974853
$$729$$ 0 0
$$730$$ 34.1215 1.26290
$$731$$ 3.12349 0.115526
$$732$$ 0 0
$$733$$ −3.41241 −0.126040 −0.0630201 0.998012i $$-0.520073\pi$$
−0.0630201 + 0.998012i $$0.520073\pi$$
$$734$$ −5.58348 −0.206090
$$735$$ 0 0
$$736$$ 4.07428 0.150180
$$737$$ 31.1499 1.14742
$$738$$ 0 0
$$739$$ 26.3010 0.967497 0.483748 0.875207i $$-0.339275\pi$$
0.483748 + 0.875207i $$0.339275\pi$$
$$740$$ −16.7763 −0.616708
$$741$$ 0 0
$$742$$ 21.5776 0.792139
$$743$$ 28.9439 1.06185 0.530924 0.847419i $$-0.321845\pi$$
0.530924 + 0.847419i $$0.321845\pi$$
$$744$$ 0 0
$$745$$ 57.3979 2.10289
$$746$$ −56.7907 −2.07925
$$747$$ 0 0
$$748$$ 12.0172 0.439394
$$749$$ 27.3084 0.997827
$$750$$ 0 0
$$751$$ −19.4483 −0.709679 −0.354839 0.934927i $$-0.615464\pi$$
−0.354839 + 0.934927i $$0.615464\pi$$
$$752$$ −1.00411 −0.0366161
$$753$$ 0 0
$$754$$ −32.4492 −1.18173
$$755$$ 21.8460 0.795058
$$756$$ 0 0
$$757$$ 6.59627 0.239745 0.119873 0.992789i $$-0.461751\pi$$
0.119873 + 0.992789i $$0.461751\pi$$
$$758$$ 63.4187 2.30347
$$759$$ 0 0
$$760$$ −5.47565 −0.198623
$$761$$ 10.3297 0.374451 0.187226 0.982317i $$-0.440050\pi$$
0.187226 + 0.982317i $$0.440050\pi$$
$$762$$ 0 0
$$763$$ −34.2799 −1.24102
$$764$$ −25.2276 −0.912703
$$765$$ 0 0
$$766$$ 1.32407 0.0478407
$$767$$ −62.7920 −2.26729
$$768$$ 0 0
$$769$$ 44.8590 1.61766 0.808828 0.588046i $$-0.200102\pi$$
0.808828 + 0.588046i $$0.200102\pi$$
$$770$$ 37.2829 1.34358
$$771$$ 0 0
$$772$$ 28.2003 1.01495
$$773$$ 42.9355 1.54428 0.772141 0.635452i $$-0.219186\pi$$
0.772141 + 0.635452i $$0.219186\pi$$
$$774$$ 0 0
$$775$$ −37.4671 −1.34586
$$776$$ 1.68349 0.0604339
$$777$$ 0 0
$$778$$ −8.38919 −0.300767
$$779$$ 15.5580 0.557422
$$780$$ 0 0
$$781$$ −2.62092 −0.0937839
$$782$$ −3.00054 −0.107299
$$783$$ 0 0
$$784$$ 6.29860 0.224950
$$785$$ 4.48691 0.160145
$$786$$ 0 0
$$787$$ 0.477407 0.0170177 0.00850885 0.999964i $$-0.497292\pi$$
0.00850885 + 0.999964i $$0.497292\pi$$
$$788$$ 41.9501 1.49441
$$789$$ 0 0
$$790$$ 93.3278 3.32045
$$791$$ 11.0225 0.391915
$$792$$ 0 0
$$793$$ 17.3345 0.615566
$$794$$ −17.4603 −0.619644
$$795$$ 0 0
$$796$$ −17.1043 −0.606246
$$797$$ 11.4690 0.406252 0.203126 0.979153i $$-0.434890\pi$$
0.203126 + 0.979153i $$0.434890\pi$$
$$798$$ 0 0
$$799$$ 0.697281 0.0246680
$$800$$ −68.3116 −2.41518
$$801$$ 0 0
$$802$$ 73.0411 2.57917
$$803$$ −10.1956 −0.359795
$$804$$ 0 0
$$805$$ −4.50980 −0.158950
$$806$$ 39.9991 1.40891
$$807$$ 0 0
$$808$$ 1.11112 0.0390890
$$809$$ 42.7873 1.50432 0.752161 0.658979i $$-0.229012\pi$$
0.752161 + 0.658979i $$0.229012\pi$$
$$810$$ 0 0
$$811$$ −31.2098 −1.09592 −0.547962 0.836503i $$-0.684596\pi$$
−0.547962 + 0.836503i $$0.684596\pi$$
$$812$$ −15.4082 −0.540722
$$813$$ 0 0
$$814$$ 10.3473 0.362673
$$815$$ −10.2659 −0.359599
$$816$$ 0 0
$$817$$ −6.62630 −0.231825
$$818$$ −6.67539 −0.233400
$$819$$ 0 0
$$820$$ 17.3824 0.607019
$$821$$ 27.2511 0.951070 0.475535 0.879697i $$-0.342255\pi$$
0.475535 + 0.879697i $$0.342255\pi$$
$$822$$ 0 0
$$823$$ −14.4406 −0.503367 −0.251683 0.967810i $$-0.580984\pi$$
−0.251683 + 0.967810i $$0.580984\pi$$
$$824$$ 3.23649 0.112748
$$825$$ 0 0
$$826$$ −61.5458 −2.14145
$$827$$ 7.02757 0.244373 0.122186 0.992507i $$-0.461009\pi$$
0.122186 + 0.992507i $$0.461009\pi$$
$$828$$ 0 0
$$829$$ −42.0806 −1.46152 −0.730760 0.682635i $$-0.760834\pi$$
−0.730760 + 0.682635i $$0.760834\pi$$
$$830$$ −82.4156 −2.86069
$$831$$ 0 0
$$832$$ 33.0547 1.14596
$$833$$ −4.37392 −0.151547
$$834$$ 0 0
$$835$$ −14.1702 −0.490382
$$836$$ −25.4938 −0.881723
$$837$$ 0 0
$$838$$ −40.6851 −1.40544
$$839$$ −39.9213 −1.37824 −0.689118 0.724649i $$-0.742002\pi$$
−0.689118 + 0.724649i $$0.742002\pi$$
$$840$$ 0 0
$$841$$ −16.8007 −0.579333
$$842$$ 54.2295 1.86887
$$843$$ 0 0
$$844$$ −11.2344 −0.386705
$$845$$ −34.2366 −1.17777
$$846$$ 0 0
$$847$$ 14.6800 0.504412
$$848$$ 19.7266 0.677416
$$849$$ 0 0
$$850$$ 50.3087 1.72557
$$851$$ −1.25163 −0.0429052
$$852$$ 0 0
$$853$$ −23.4810 −0.803975 −0.401988 0.915645i $$-0.631680\pi$$
−0.401988 + 0.915645i $$0.631680\pi$$
$$854$$ 16.9905 0.581402
$$855$$ 0 0
$$856$$ 2.76382 0.0944655
$$857$$ −8.03472 −0.274461 −0.137230 0.990539i $$-0.543820\pi$$
−0.137230 + 0.990539i $$0.543820\pi$$
$$858$$ 0 0
$$859$$ −24.2959 −0.828966 −0.414483 0.910057i $$-0.636038\pi$$
−0.414483 + 0.910057i $$0.636038\pi$$
$$860$$ −7.40333 −0.252452
$$861$$ 0 0
$$862$$ −5.08378 −0.173154
$$863$$ 10.2828 0.350029 0.175015 0.984566i $$-0.444003\pi$$
0.175015 + 0.984566i $$0.444003\pi$$
$$864$$ 0 0
$$865$$ −26.0205 −0.884725
$$866$$ −53.2067 −1.80804
$$867$$ 0 0
$$868$$ 18.9932 0.644671
$$869$$ −27.8866 −0.945987
$$870$$ 0 0
$$871$$ −67.4448 −2.28528
$$872$$ −3.46940 −0.117489
$$873$$ 0 0
$$874$$ 6.36547 0.215315
$$875$$ 32.1692 1.08752
$$876$$ 0 0
$$877$$ −26.2276 −0.885644 −0.442822 0.896610i $$-0.646023\pi$$
−0.442822 + 0.896610i $$0.646023\pi$$
$$878$$ 23.0643 0.778384
$$879$$ 0 0
$$880$$ 34.0847 1.14900
$$881$$ 39.2326 1.32178 0.660890 0.750483i $$-0.270179\pi$$
0.660890 + 0.750483i $$0.270179\pi$$
$$882$$ 0 0
$$883$$ 7.79830 0.262434 0.131217 0.991354i $$-0.458112\pi$$
0.131217 + 0.991354i $$0.458112\pi$$
$$884$$ −26.0194 −0.875126
$$885$$ 0 0
$$886$$ 4.09833 0.137686
$$887$$ 22.8571 0.767465 0.383732 0.923444i $$-0.374639\pi$$
0.383732 + 0.923444i $$0.374639\pi$$
$$888$$ 0 0
$$889$$ −14.2986 −0.479560
$$890$$ 5.09840 0.170899
$$891$$ 0 0
$$892$$ −16.7297 −0.560151
$$893$$ −1.47924 −0.0495009
$$894$$ 0 0
$$895$$ −53.2499 −1.77995
$$896$$ −4.45297 −0.148763
$$897$$ 0 0
$$898$$ 20.7811 0.693473
$$899$$ −15.0377 −0.501537
$$900$$ 0 0
$$901$$ −13.6987 −0.456370
$$902$$ −10.7211 −0.356974
$$903$$ 0 0
$$904$$ 1.11556 0.0371031
$$905$$ −48.8630 −1.62426
$$906$$ 0 0
$$907$$ −37.8485 −1.25674 −0.628370 0.777915i $$-0.716278\pi$$
−0.628370 + 0.777915i $$0.716278\pi$$
$$908$$ 20.0502 0.665388
$$909$$ 0 0
$$910$$ −80.7238 −2.67597
$$911$$ 52.7045 1.74618 0.873089 0.487561i $$-0.162113\pi$$
0.873089 + 0.487561i $$0.162113\pi$$
$$912$$ 0 0
$$913$$ 24.6260 0.815001
$$914$$ −15.5785 −0.515291
$$915$$ 0 0
$$916$$ −15.5449 −0.513617
$$917$$ 24.2908 0.802152
$$918$$ 0 0
$$919$$ −49.1052 −1.61983 −0.809916 0.586545i $$-0.800488\pi$$
−0.809916 + 0.586545i $$0.800488\pi$$
$$920$$ −0.456427 −0.0150480
$$921$$ 0 0
$$922$$ −76.9977 −2.53579
$$923$$ 5.67474 0.186786
$$924$$ 0 0
$$925$$ 20.9855 0.689997
$$926$$ 46.6658 1.53353
$$927$$ 0 0
$$928$$ −27.4175 −0.900022
$$929$$ 17.9507 0.588943 0.294472 0.955660i $$-0.404856\pi$$
0.294472 + 0.955660i $$0.404856\pi$$
$$930$$ 0 0
$$931$$ 9.27900 0.304107
$$932$$ −23.9420 −0.784248
$$933$$ 0 0
$$934$$ −68.3756 −2.23732
$$935$$ −23.6693 −0.774070
$$936$$ 0 0
$$937$$ −29.7980 −0.973457 −0.486729 0.873553i $$-0.661810\pi$$
−0.486729 + 0.873553i $$0.661810\pi$$
$$938$$ −66.1063 −2.15845
$$939$$ 0 0
$$940$$ −1.65270 −0.0539052
$$941$$ −30.0112 −0.978339 −0.489169 0.872189i $$-0.662700\pi$$
−0.489169 + 0.872189i $$0.662700\pi$$
$$942$$ 0 0
$$943$$ 1.29685 0.0422311
$$944$$ −56.2663 −1.83131
$$945$$ 0 0
$$946$$ 4.56624 0.148461
$$947$$ 45.6013 1.48184 0.740922 0.671591i $$-0.234389\pi$$
0.740922 + 0.671591i $$0.234389\pi$$
$$948$$ 0 0
$$949$$ 22.0752 0.716592
$$950$$ −106.727 −3.46268
$$951$$ 0 0
$$952$$ 1.63673 0.0530466
$$953$$ −13.1957 −0.427451 −0.213726 0.976894i $$-0.568560\pi$$
−0.213726 + 0.976894i $$0.568560\pi$$
$$954$$ 0 0
$$955$$ 49.6887 1.60789
$$956$$ −28.2201 −0.912702
$$957$$ 0 0
$$958$$ −12.7665 −0.412467
$$959$$ 44.7974 1.44658
$$960$$ 0 0
$$961$$ −12.4635 −0.402047
$$962$$ −22.4037 −0.722323
$$963$$ 0 0
$$964$$ 1.49525 0.0481588
$$965$$ −55.5437 −1.78801
$$966$$ 0 0
$$967$$ −20.1548 −0.648133 −0.324067 0.946034i $$-0.605050\pi$$
−0.324067 + 0.946034i $$0.605050\pi$$
$$968$$ 1.48574 0.0477533
$$969$$ 0 0
$$970$$ 51.6664 1.65891
$$971$$ −26.5839 −0.853118 −0.426559 0.904460i $$-0.640274\pi$$
−0.426559 + 0.904460i $$0.640274\pi$$
$$972$$ 0 0
$$973$$ 54.7006 1.75362
$$974$$ 39.3584 1.26112
$$975$$ 0 0
$$976$$ 15.5330 0.497200
$$977$$ 14.4041 0.460828 0.230414 0.973093i $$-0.425992\pi$$
0.230414 + 0.973093i $$0.425992\pi$$
$$978$$ 0 0
$$979$$ −1.52341 −0.0486885
$$980$$ 10.3671 0.331165
$$981$$ 0 0
$$982$$ −51.6573 −1.64845
$$983$$ −42.9881 −1.37111 −0.685554 0.728022i $$-0.740440\pi$$
−0.685554 + 0.728022i $$0.740440\pi$$
$$984$$ 0 0
$$985$$ −82.6255 −2.63267
$$986$$ 20.1918 0.643039
$$987$$ 0 0
$$988$$ 55.1985 1.75610
$$989$$ −0.552340 −0.0175634
$$990$$ 0 0
$$991$$ −15.5794 −0.494895 −0.247447 0.968901i $$-0.579592\pi$$
−0.247447 + 0.968901i $$0.579592\pi$$
$$992$$ 33.7966 1.07304
$$993$$ 0 0
$$994$$ 5.56212 0.176420
$$995$$ 33.6889 1.06801
$$996$$ 0 0
$$997$$ 40.6186 1.28640 0.643201 0.765697i $$-0.277606\pi$$
0.643201 + 0.765697i $$0.277606\pi$$
$$998$$ 12.4426 0.393863
$$999$$ 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 729.2.a.c.1.6 yes 6
3.2 odd 2 inner 729.2.a.c.1.1 6
9.2 odd 6 729.2.c.c.244.6 12
9.4 even 3 729.2.c.c.487.1 12
9.5 odd 6 729.2.c.c.487.6 12
9.7 even 3 729.2.c.c.244.1 12
27.2 odd 18 729.2.e.r.568.2 12
27.4 even 9 729.2.e.q.406.2 12
27.5 odd 18 729.2.e.m.649.1 12
27.7 even 9 729.2.e.q.325.2 12
27.11 odd 18 729.2.e.m.82.1 12
27.13 even 9 729.2.e.r.163.1 12
27.14 odd 18 729.2.e.r.163.2 12
27.16 even 9 729.2.e.m.82.2 12
27.20 odd 18 729.2.e.q.325.1 12
27.22 even 9 729.2.e.m.649.2 12
27.23 odd 18 729.2.e.q.406.1 12
27.25 even 9 729.2.e.r.568.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.1 6 3.2 odd 2 inner
729.2.a.c.1.6 yes 6 1.1 even 1 trivial
729.2.c.c.244.1 12 9.7 even 3
729.2.c.c.244.6 12 9.2 odd 6
729.2.c.c.487.1 12 9.4 even 3
729.2.c.c.487.6 12 9.5 odd 6
729.2.e.m.82.1 12 27.11 odd 18
729.2.e.m.82.2 12 27.16 even 9
729.2.e.m.649.1 12 27.5 odd 18
729.2.e.m.649.2 12 27.22 even 9
729.2.e.q.325.1 12 27.20 odd 18
729.2.e.q.325.2 12 27.7 even 9
729.2.e.q.406.1 12 27.23 odd 18
729.2.e.q.406.2 12 27.4 even 9
729.2.e.r.163.1 12 27.13 even 9
729.2.e.r.163.2 12 27.14 odd 18
729.2.e.r.568.1 12 27.25 even 9
729.2.e.r.568.2 12 27.2 odd 18