Properties

 Label 729.2.a.c.1.1 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 1.1 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 $$-0.745200\pi$$
−0.696364 + 0.717689i $$0.745200\pi$$
$$3$$ 0 0
$$4$$ 1.87939 0.939693
$$5$$ 3.70167 1.65544 0.827718 0.561145i $$-0.189639\pi$$
0.827718 + 0.561145i $$0.189639\pi$$
$$6$$ 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 $$-0.606518\pi$$
−0.328425 + 0.944530i $$0.606518\pi$$
$$12$$ 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 $$-0.615838\pi$$
−0.355936 + 0.934510i $$0.615838\pi$$
$$18$$ 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 $$-0.482767\pi$$
0.0541126 + 0.998535i $$0.482767\pi$$
$$24$$ 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 $$-0.605127\pi$$
−0.324294 + 0.945956i $$0.605127\pi$$
$$30$$ 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 $$-0.437494\pi$$
0.195108 + 0.980782i $$0.437494\pi$$
$$42$$ 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 $$-0.505515\pi$$
−0.0173262 + 0.999850i $$0.505515\pi$$
$$48$$ 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 $$-0.396135\pi$$
0.320543 + 0.947234i $$0.396135\pi$$
$$54$$ 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 $$-0.833667\pi$$
−0.866549 + 0.499092i $$0.833667\pi$$
$$60$$ 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 $$-0.477257\pi$$
0.0713891 + 0.997449i $$0.477257\pi$$
$$72$$ 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 $$-0.713024\pi$$
−0.620385 + 0.784297i $$0.713024\pi$$
$$84$$ 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 $$-0.488200\pi$$
0.0370621 + 0.999313i $$0.488200\pi$$
$$90$$ 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 $$-0.425245\pi$$
0.232696 + 0.972550i $$0.425245\pi$$
$$102$$ 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 $$-0.309898\pi$$
0.562350 + 0.826900i $$0.309898\pi$$
$$108$$ 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 $$-0.429109\pi$$
0.220873 + 0.975303i $$0.429109\pi$$
$$114$$ 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 $$-0.350685\pi$$
0.452072 + 0.891981i $$0.350685\pi$$
$$132$$ 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 $$-0.196596\pi$$
0.815257 + 0.579100i $$0.196596\pi$$
$$138$$ 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 $$-0.280939\pi$$
0.635149 + 0.772390i $$0.280939\pi$$
$$150$$ 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 $$-0.547320\pi$$
−0.148113 + 0.988971i $$0.547320\pi$$
$$168$$ 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 $$-0.586104\pi$$
−0.267218 + 0.963636i $$0.586104\pi$$
$$174$$ 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 $$-0.680672\pi$$
−0.537607 + 0.843195i $$0.680672\pi$$
$$180$$ 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 $$-0.338587\pi$$
0.485639 + 0.874159i $$0.338587\pi$$
$$192$$ 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 $$-0.792612\pi$$
−0.795158 + 0.606402i $$0.792612\pi$$
$$198$$ 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 $$-0.615194\pi$$
−0.354046 + 0.935228i $$0.615194\pi$$
$$228$$ 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 $$-0.362980\pi$$
0.417290 + 0.908774i $$0.362980\pi$$
$$234$$ 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 $$-0.338587\pi$$
0.485638 + 0.874160i $$0.338587\pi$$
$$240$$ 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 $$-0.415480\pi$$
0.262419 + 0.964954i $$0.415480\pi$$
$$252$$ 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 $$-0.794676\pi$$
−0.799074 + 0.601233i $$0.794676\pi$$
$$258$$ 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 $$-0.167140\pi$$
0.865281 + 0.501287i $$0.167140\pi$$
$$264$$ 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 $$-0.871631\pi$$
−0.919778 + 0.392439i $$0.871631\pi$$
$$270$$ 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 $$-0.516597\pi$$
−0.0521175 + 0.998641i $$0.516597\pi$$
$$282$$ 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 $$-0.648008\pi$$
−0.448404 + 0.893831i $$0.648008\pi$$
$$294$$ 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 $$-0.349891\pi$$
0.454295 + 0.890852i $$0.349891\pi$$
$$312$$ 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 $$-0.348544\pi$$
0.458061 + 0.888921i $$0.348544\pi$$
$$318$$ 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 $$-0.712125\pi$$
−0.618168 + 0.786046i $$0.712125\pi$$
$$348$$ 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 $$-0.481908\pi$$
0.0568073 + 0.998385i $$0.481908\pi$$
$$354$$ 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 $$-0.720749\pi$$
−0.639234 + 0.769012i $$0.720749\pi$$
$$360$$ 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 $$-0.505467\pi$$
−0.0171752 + 0.999852i $$0.505467\pi$$
$$384$$ 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 $$-0.465563\pi$$
0.107978 + 0.994153i $$0.465563\pi$$
$$390$$ 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 $$-0.876726\pi$$
−0.925941 + 0.377667i $$0.876726\pi$$
$$402$$ 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 $$-0.331653\pi$$
0.504565 + 0.863374i $$0.331653\pi$$
$$420$$ 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 $$-0.480200\pi$$
0.0621636 + 0.998066i $$0.480200\pi$$
$$432$$ 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 $$-0.515741\pi$$
−0.0494303 + 0.998778i $$0.515741\pi$$
$$444$$ 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 $$-0.580089\pi$$
−0.248962 + 0.968513i $$0.580089\pi$$
$$450$$ 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 $$-0.135800\pi$$
0.910366 + 0.413804i $$0.135800\pi$$
$$462$$ 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 $$-0.203122\pi$$
0.803214 + 0.595691i $$0.203122\pi$$
$$468$$ 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 $$-0.452691\pi$$
0.148079 + 0.988976i $$0.452691\pi$$
$$480$$ 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 $$-0.298415\pi$$
0.591806 + 0.806081i $$0.298415\pi$$
$$492$$ 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 $$-0.338241\pi$$
0.486589 + 0.873631i $$0.338241\pi$$
$$504$$ 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 $$-0.723045\pi$$
−0.644765 + 0.764381i $$0.723045\pi$$
$$510$$ 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 $$-0.403016\pi$$
0.299991 + 0.953942i $$0.403016\pi$$
$$522$$ 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 $$-0.127953\pi$$
0.920290 + 0.391237i $$0.127953\pi$$
$$558$$ 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 $$-0.264277\pi$$
0.674691 + 0.738100i $$0.264277\pi$$
$$564$$ 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 $$-0.454664\pi$$
0.141947 + 0.989874i $$0.454664\pi$$
$$570$$ 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 $$-0.499147\pi$$
0.00267998 + 0.999996i $$0.499147\pi$$
$$588$$ 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 $$-0.681285\pi$$
−0.539230 + 0.842158i $$0.681285\pi$$
$$594$$ 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 $$-0.690643\pi$$
−0.563754 + 0.825943i $$0.690643\pi$$
$$600$$ 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 $$-0.660832\pi$$
−0.484042 + 0.875045i $$0.660832\pi$$
$$618$$ 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 $$-0.376386\pi$$
0.378657 + 0.925537i $$0.376386\pi$$
$$642$$ 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 $$-0.555159\pi$$
−0.172420 + 0.985024i $$0.555159\pi$$
$$648$$ 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 $$-0.721881\pi$$
−0.641965 + 0.766734i $$0.721881\pi$$
$$654$$ 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 $$-0.618344\pi$$
−0.363282 + 0.931679i $$0.618344\pi$$
$$660$$ 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 $$-0.708306\pi$$
−0.608692 + 0.793406i $$0.708306\pi$$
$$678$$ 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 $$-0.312304\pi$$
0.556083 + 0.831127i $$0.312304\pi$$
$$684$$ 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 $$-0.339016\pi$$
0.484461 + 0.874813i $$0.339016\pi$$
$$702$$ 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 $$-0.759771\pi$$
−0.728476 + 0.685072i $$0.759771\pi$$
$$720$$ 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 $$-0.678155\pi$$
−0.530924 + 0.847419i $$0.678155\pi$$
$$744$$ 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 $$-0.559950\pi$$
−0.187226 + 0.982317i $$0.559950\pi$$
$$762$$ 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 $$-0.780814\pi$$
−0.772141 + 0.635452i $$0.780814\pi$$
$$774$$ 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 $$-0.565110\pi$$
−0.203126 + 0.979153i $$0.565110\pi$$
$$798$$ 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 $$-0.770988\pi$$
−0.752161 + 0.658979i $$0.770988\pi$$
$$810$$ 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 $$-0.657745\pi$$
−0.475535 + 0.879697i $$0.657745\pi$$
$$822$$ 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 $$-0.538991\pi$$
−0.122186 + 0.992507i $$0.538991\pi$$
$$828$$ 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 $$-0.257998\pi$$
0.689118 + 0.724649i $$0.257998\pi$$
$$840$$ 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 $$-0.456180\pi$$
0.137230 + 0.990539i $$0.456180\pi$$
$$858$$ 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 $$-0.555997\pi$$
−0.175015 + 0.984566i $$0.555997\pi$$
$$864$$ 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 $$-0.729821\pi$$
−0.660890 + 0.750483i $$0.729821\pi$$
$$882$$ 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 $$-0.625361\pi$$
−0.383732 + 0.923444i $$0.625361\pi$$
$$888$$ 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 $$-0.837887\pi$$
−0.873089 + 0.487561i $$0.837887\pi$$
$$912$$ 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 $$-0.595144\pi$$
−0.294472 + 0.955660i $$0.595144\pi$$
$$930$$ 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 $$-0.337300\pi$$
0.489169 + 0.872189i $$0.337300\pi$$
$$942$$ 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 $$-0.765611\pi$$
−0.740922 + 0.671591i $$0.765611\pi$$
$$948$$ 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 $$-0.431440\pi$$
0.213726 + 0.976894i $$0.431440\pi$$
$$954$$ 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 $$-0.359726\pi$$
0.426559 + 0.904460i $$0.359726\pi$$
$$972$$ 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 $$-0.574008\pi$$
−0.230414 + 0.973093i $$0.574008\pi$$
$$978$$ 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 $$-0.259560\pi$$
0.685554 + 0.728022i $$0.259560\pi$$
$$984$$ 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.1.1 6
3.2 odd 2 inner 729.2.a.c.1.6 yes 6
9.2 odd 6 729.2.c.c.244.1 12
9.4 even 3 729.2.c.c.487.6 12
9.5 odd 6 729.2.c.c.487.1 12
9.7 even 3 729.2.c.c.244.6 12
27.2 odd 18 729.2.e.r.568.1 12
27.4 even 9 729.2.e.q.406.1 12
27.5 odd 18 729.2.e.m.649.2 12
27.7 even 9 729.2.e.q.325.1 12
27.11 odd 18 729.2.e.m.82.2 12
27.13 even 9 729.2.e.r.163.2 12
27.14 odd 18 729.2.e.r.163.1 12
27.16 even 9 729.2.e.m.82.1 12
27.20 odd 18 729.2.e.q.325.2 12
27.22 even 9 729.2.e.m.649.1 12
27.23 odd 18 729.2.e.q.406.2 12
27.25 even 9 729.2.e.r.568.2 12

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