# Properties

 Label 4900.2.a.be.1.1 Level $4900$ Weight $2$ Character 4900.1 Self dual yes Analytic conductor $39.127$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.1266969904$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11x^{2} + 16$$ x^4 - 11*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.04547$$ of defining polynomial Character $$\chi$$ $$=$$ 4900.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} +O(q^{10})$$ $$q-1.73205 q^{3} -5.27492 q^{11} -2.62685 q^{13} -0.418627 q^{17} +3.27492 q^{19} +7.82300 q^{23} +5.19615 q^{27} +4.27492 q^{29} +3.27492 q^{31} +9.13642 q^{33} +9.97368 q^{37} +4.54983 q^{39} -3.72508 q^{41} +2.15068 q^{43} -6.50958 q^{47} +0.725083 q^{51} +5.67232 q^{53} -5.67232 q^{57} -3.27492 q^{59} -13.5498 q^{61} -3.52165 q^{67} -13.5498 q^{69} -4.54983 q^{71} -6.50958 q^{73} +7.27492 q^{79} -9.00000 q^{81} +7.40437 q^{83} -7.40437 q^{87} -7.00000 q^{89} -5.67232 q^{93} +6.92820 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 6 q^{11} - 2 q^{19} + 2 q^{29} - 2 q^{31} - 12 q^{39} - 30 q^{41} + 18 q^{51} + 2 q^{59} - 24 q^{61} - 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} - 28 q^{89}+O(q^{100})$$ 4 * q - 6 * q^11 - 2 * q^19 + 2 * q^29 - 2 * q^31 - 12 * q^39 - 30 * q^41 + 18 * q^51 + 2 * q^59 - 24 * q^61 - 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 - 28 * q^89

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.73205 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.27492 −1.59045 −0.795224 0.606316i $$-0.792647\pi$$
−0.795224 + 0.606316i $$0.792647\pi$$
$$12$$ 0 0
$$13$$ −2.62685 −0.728557 −0.364278 0.931290i $$-0.618684\pi$$
−0.364278 + 0.931290i $$0.618684\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.418627 −0.101532 −0.0507659 0.998711i $$-0.516166\pi$$
−0.0507659 + 0.998711i $$0.516166\pi$$
$$18$$ 0 0
$$19$$ 3.27492 0.751318 0.375659 0.926758i $$-0.377416\pi$$
0.375659 + 0.926758i $$0.377416\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.82300 1.63121 0.815604 0.578610i $$-0.196405\pi$$
0.815604 + 0.578610i $$0.196405\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.19615 1.00000
$$28$$ 0 0
$$29$$ 4.27492 0.793832 0.396916 0.917855i $$-0.370080\pi$$
0.396916 + 0.917855i $$0.370080\pi$$
$$30$$ 0 0
$$31$$ 3.27492 0.588192 0.294096 0.955776i $$-0.404981\pi$$
0.294096 + 0.955776i $$0.404981\pi$$
$$32$$ 0 0
$$33$$ 9.13642 1.59045
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.97368 1.63966 0.819831 0.572605i $$-0.194067\pi$$
0.819831 + 0.572605i $$0.194067\pi$$
$$38$$ 0 0
$$39$$ 4.54983 0.728557
$$40$$ 0 0
$$41$$ −3.72508 −0.581760 −0.290880 0.956760i $$-0.593948\pi$$
−0.290880 + 0.956760i $$0.593948\pi$$
$$42$$ 0 0
$$43$$ 2.15068 0.327975 0.163988 0.986462i $$-0.447564\pi$$
0.163988 + 0.986462i $$0.447564\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.50958 −0.949519 −0.474760 0.880115i $$-0.657465\pi$$
−0.474760 + 0.880115i $$0.657465\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.725083 0.101532
$$52$$ 0 0
$$53$$ 5.67232 0.779153 0.389577 0.920994i $$-0.372621\pi$$
0.389577 + 0.920994i $$0.372621\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.67232 −0.751318
$$58$$ 0 0
$$59$$ −3.27492 −0.426358 −0.213179 0.977013i $$-0.568382\pi$$
−0.213179 + 0.977013i $$0.568382\pi$$
$$60$$ 0 0
$$61$$ −13.5498 −1.73488 −0.867439 0.497543i $$-0.834236\pi$$
−0.867439 + 0.497543i $$0.834236\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.52165 −0.430237 −0.215119 0.976588i $$-0.569014\pi$$
−0.215119 + 0.976588i $$0.569014\pi$$
$$68$$ 0 0
$$69$$ −13.5498 −1.63121
$$70$$ 0 0
$$71$$ −4.54983 −0.539966 −0.269983 0.962865i $$-0.587018\pi$$
−0.269983 + 0.962865i $$0.587018\pi$$
$$72$$ 0 0
$$73$$ −6.50958 −0.761888 −0.380944 0.924598i $$-0.624401\pi$$
−0.380944 + 0.924598i $$0.624401\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 7.27492 0.818492 0.409246 0.912424i $$-0.365792\pi$$
0.409246 + 0.912424i $$0.365792\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 7.40437 0.812736 0.406368 0.913710i $$-0.366795\pi$$
0.406368 + 0.913710i $$0.366795\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −7.40437 −0.793832
$$88$$ 0 0
$$89$$ −7.00000 −0.741999 −0.370999 0.928633i $$-0.620985\pi$$
−0.370999 + 0.928633i $$0.620985\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.67232 −0.588192
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.92820 0.703452 0.351726 0.936103i $$-0.385595\pi$$
0.351726 + 0.936103i $$0.385595\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −13.5498 −1.34826 −0.674129 0.738613i $$-0.735481\pi$$
−0.674129 + 0.738613i $$0.735481\pi$$
$$102$$ 0 0
$$103$$ −11.2871 −1.11215 −0.556076 0.831132i $$-0.687694\pi$$
−0.556076 + 0.831132i $$0.687694\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −3.52165 −0.340450 −0.170225 0.985405i $$-0.554450\pi$$
−0.170225 + 0.985405i $$0.554450\pi$$
$$108$$ 0 0
$$109$$ −11.5498 −1.10627 −0.553137 0.833090i $$-0.686569\pi$$
−0.553137 + 0.833090i $$0.686569\pi$$
$$110$$ 0 0
$$111$$ −17.2749 −1.63966
$$112$$ 0 0
$$113$$ −4.30136 −0.404637 −0.202319 0.979320i $$-0.564848\pi$$
−0.202319 + 0.979320i $$0.564848\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 16.8248 1.52952
$$122$$ 0 0
$$123$$ 6.45203 0.581760
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 15.6460 1.38836 0.694179 0.719802i $$-0.255768\pi$$
0.694179 + 0.719802i $$0.255768\pi$$
$$128$$ 0 0
$$129$$ −3.72508 −0.327975
$$130$$ 0 0
$$131$$ −10.7251 −0.937055 −0.468527 0.883449i $$-0.655215\pi$$
−0.468527 + 0.883449i $$0.655215\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.3183 1.82135 0.910674 0.413126i $$-0.135563\pi$$
0.910674 + 0.413126i $$0.135563\pi$$
$$138$$ 0 0
$$139$$ 13.0997 1.11110 0.555550 0.831483i $$-0.312508\pi$$
0.555550 + 0.831483i $$0.312508\pi$$
$$140$$ 0 0
$$141$$ 11.2749 0.949519
$$142$$ 0 0
$$143$$ 13.8564 1.15873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −7.54983 −0.618507 −0.309253 0.950980i $$-0.600079\pi$$
−0.309253 + 0.950980i $$0.600079\pi$$
$$150$$ 0 0
$$151$$ −12.7251 −1.03555 −0.517776 0.855516i $$-0.673240\pi$$
−0.517776 + 0.855516i $$0.673240\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.20822 0.176235 0.0881176 0.996110i $$-0.471915\pi$$
0.0881176 + 0.996110i $$0.471915\pi$$
$$158$$ 0 0
$$159$$ −9.82475 −0.779153
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −5.67232 −0.444291 −0.222145 0.975014i $$-0.571306\pi$$
−0.222145 + 0.975014i $$0.571306\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0.476171 0.0368472 0.0184236 0.999830i $$-0.494135\pi$$
0.0184236 + 0.999830i $$0.494135\pi$$
$$168$$ 0 0
$$169$$ −6.09967 −0.469205
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −20.4811 −1.55715 −0.778573 0.627553i $$-0.784056\pi$$
−0.778573 + 0.627553i $$0.784056\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.67232 0.426358
$$178$$ 0 0
$$179$$ −7.27492 −0.543753 −0.271876 0.962332i $$-0.587644\pi$$
−0.271876 + 0.962332i $$0.587644\pi$$
$$180$$ 0 0
$$181$$ −24.2749 −1.80434 −0.902170 0.431380i $$-0.858027\pi$$
−0.902170 + 0.431380i $$0.858027\pi$$
$$182$$ 0 0
$$183$$ 23.4690 1.73488
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.20822 0.161481
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0.175248 0.0126805 0.00634026 0.999980i $$-0.497982\pi$$
0.00634026 + 0.999980i $$0.497982\pi$$
$$192$$ 0 0
$$193$$ −21.3183 −1.53453 −0.767263 0.641332i $$-0.778382\pi$$
−0.767263 + 0.641332i $$0.778382\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −8.60271 −0.612918 −0.306459 0.951884i $$-0.599144\pi$$
−0.306459 + 0.951884i $$0.599144\pi$$
$$198$$ 0 0
$$199$$ 17.2749 1.22459 0.612293 0.790631i $$-0.290247\pi$$
0.612293 + 0.790631i $$0.290247\pi$$
$$200$$ 0 0
$$201$$ 6.09967 0.430237
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −17.2749 −1.19493
$$210$$ 0 0
$$211$$ 25.6495 1.76578 0.882892 0.469576i $$-0.155593\pi$$
0.882892 + 0.469576i $$0.155593\pi$$
$$212$$ 0 0
$$213$$ 7.88054 0.539966
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 11.2749 0.761888
$$220$$ 0 0
$$221$$ 1.09967 0.0739717
$$222$$ 0 0
$$223$$ 8.71780 0.583787 0.291893 0.956451i $$-0.405715\pi$$
0.291893 + 0.956451i $$0.405715\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 19.5287 1.29617 0.648084 0.761569i $$-0.275571\pi$$
0.648084 + 0.761569i $$0.275571\pi$$
$$228$$ 0 0
$$229$$ 3.27492 0.216413 0.108206 0.994128i $$-0.465489\pi$$
0.108206 + 0.994128i $$0.465489\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.2750 −0.935189 −0.467594 0.883943i $$-0.654879\pi$$
−0.467594 + 0.883943i $$0.654879\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −12.6005 −0.818492
$$238$$ 0 0
$$239$$ 0.549834 0.0355658 0.0177829 0.999842i $$-0.494339\pi$$
0.0177829 + 0.999842i $$0.494339\pi$$
$$240$$ 0 0
$$241$$ 9.82475 0.632868 0.316434 0.948615i $$-0.397514\pi$$
0.316434 + 0.948615i $$0.397514\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −8.60271 −0.547377
$$248$$ 0 0
$$249$$ −12.8248 −0.812736
$$250$$ 0 0
$$251$$ −20.5498 −1.29709 −0.648547 0.761175i $$-0.724623\pi$$
−0.648547 + 0.761175i $$0.724623\pi$$
$$252$$ 0 0
$$253$$ −41.2657 −2.59435
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 11.6482 0.726594 0.363297 0.931673i $$-0.381651\pi$$
0.363297 + 0.931673i $$0.381651\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −0.779710 −0.0480790 −0.0240395 0.999711i $$-0.507653\pi$$
−0.0240395 + 0.999711i $$0.507653\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 12.1244 0.741999
$$268$$ 0 0
$$269$$ −14.4502 −0.881042 −0.440521 0.897742i $$-0.645206\pi$$
−0.440521 + 0.897742i $$0.645206\pi$$
$$270$$ 0 0
$$271$$ 9.82475 0.596811 0.298406 0.954439i $$-0.403545\pi$$
0.298406 + 0.954439i $$0.403545\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.2750 −0.857704 −0.428852 0.903375i $$-0.641082\pi$$
−0.428852 + 0.903375i $$0.641082\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −10.9260 −0.649484 −0.324742 0.945803i $$-0.605278\pi$$
−0.324742 + 0.945803i $$0.605278\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.8248 −0.989691
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 0 0
$$293$$ −6.92820 −0.404750 −0.202375 0.979308i $$-0.564866\pi$$
−0.202375 + 0.979308i $$0.564866\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −27.4093 −1.59045
$$298$$ 0 0
$$299$$ −20.5498 −1.18843
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 23.4690 1.34826
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −26.5145 −1.51326 −0.756631 0.653843i $$-0.773156\pi$$
−0.756631 + 0.653843i $$0.773156\pi$$
$$308$$ 0 0
$$309$$ 19.5498 1.11215
$$310$$ 0 0
$$311$$ −9.82475 −0.557111 −0.278555 0.960420i $$-0.589856\pi$$
−0.278555 + 0.960420i $$0.589856\pi$$
$$312$$ 0 0
$$313$$ 33.5002 1.89354 0.946772 0.321904i $$-0.104323\pi$$
0.946772 + 0.321904i $$0.104323\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −25.6197 −1.43894 −0.719472 0.694521i $$-0.755616\pi$$
−0.719472 + 0.694521i $$0.755616\pi$$
$$318$$ 0 0
$$319$$ −22.5498 −1.26255
$$320$$ 0 0
$$321$$ 6.09967 0.340450
$$322$$ 0 0
$$323$$ −1.37097 −0.0762827
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.0049 1.10627
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 17.8248 0.979737 0.489868 0.871796i $$-0.337045\pi$$
0.489868 + 0.871796i $$0.337045\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.30136 0.234310 0.117155 0.993114i $$-0.462623\pi$$
0.117155 + 0.993114i $$0.462623\pi$$
$$338$$ 0 0
$$339$$ 7.45017 0.404637
$$340$$ 0 0
$$341$$ −17.2749 −0.935489
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.1244 −0.650870 −0.325435 0.945564i $$-0.605511\pi$$
−0.325435 + 0.945564i $$0.605511\pi$$
$$348$$ 0 0
$$349$$ −3.72508 −0.199399 −0.0996996 0.995018i $$-0.531788\pi$$
−0.0996996 + 0.995018i $$0.531788\pi$$
$$350$$ 0 0
$$351$$ −13.6495 −0.728557
$$352$$ 0 0
$$353$$ −8.18408 −0.435595 −0.217797 0.975994i $$-0.569887\pi$$
−0.217797 + 0.975994i $$0.569887\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 36.3746 1.91978 0.959889 0.280382i $$-0.0904610\pi$$
0.959889 + 0.280382i $$0.0904610\pi$$
$$360$$ 0 0
$$361$$ −8.27492 −0.435522
$$362$$ 0 0
$$363$$ −29.1413 −1.52952
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6.03341 0.314941 0.157471 0.987524i $$-0.449666\pi$$
0.157471 + 0.987524i $$0.449666\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9.97368 −0.516417 −0.258209 0.966089i $$-0.583132\pi$$
−0.258209 + 0.966089i $$0.583132\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −11.2296 −0.578352
$$378$$ 0 0
$$379$$ −21.6495 −1.11206 −0.556030 0.831162i $$-0.687676\pi$$
−0.556030 + 0.831162i $$0.687676\pi$$
$$380$$ 0 0
$$381$$ −27.0997 −1.38836
$$382$$ 0 0
$$383$$ 6.14849 0.314173 0.157087 0.987585i $$-0.449790\pi$$
0.157087 + 0.987585i $$0.449790\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 32.3746 1.64146 0.820728 0.571319i $$-0.193568\pi$$
0.820728 + 0.571319i $$0.193568\pi$$
$$390$$ 0 0
$$391$$ −3.27492 −0.165620
$$392$$ 0 0
$$393$$ 18.5764 0.937055
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.8109 −0.542585 −0.271293 0.962497i $$-0.587451\pi$$
−0.271293 + 0.962497i $$0.587451\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 0 0
$$403$$ −8.60271 −0.428532
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −52.6103 −2.60780
$$408$$ 0 0
$$409$$ −20.0997 −0.993865 −0.496932 0.867789i $$-0.665540\pi$$
−0.496932 + 0.867789i $$0.665540\pi$$
$$410$$ 0 0
$$411$$ −36.9244 −1.82135
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −22.6893 −1.11110
$$418$$ 0 0
$$419$$ −13.0997 −0.639961 −0.319980 0.947424i $$-0.603676\pi$$
−0.319980 + 0.947424i $$0.603676\pi$$
$$420$$ 0 0
$$421$$ −4.27492 −0.208347 −0.104173 0.994559i $$-0.533220\pi$$
−0.104173 + 0.994559i $$0.533220\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −18.3746 −0.885073 −0.442536 0.896751i $$-0.645921\pi$$
−0.442536 + 0.896751i $$0.645921\pi$$
$$432$$ 0 0
$$433$$ 18.1578 0.872606 0.436303 0.899800i $$-0.356288\pi$$
0.436303 + 0.899800i $$0.356288\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 25.6197 1.22556
$$438$$ 0 0
$$439$$ 23.8248 1.13709 0.568547 0.822651i $$-0.307506\pi$$
0.568547 + 0.822651i $$0.307506\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.1244 0.576046 0.288023 0.957624i $$-0.407002\pi$$
0.288023 + 0.957624i $$0.407002\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 13.0767 0.618507
$$448$$ 0 0
$$449$$ −3.17525 −0.149849 −0.0749246 0.997189i $$-0.523872\pi$$
−0.0749246 + 0.997189i $$0.523872\pi$$
$$450$$ 0 0
$$451$$ 19.6495 0.925259
$$452$$ 0 0
$$453$$ 22.0405 1.03555
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.37097 0.0641312 0.0320656 0.999486i $$-0.489791\pi$$
0.0320656 + 0.999486i $$0.489791\pi$$
$$458$$ 0 0
$$459$$ −2.17525 −0.101532
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ −2.15068 −0.0999505 −0.0499752 0.998750i $$-0.515914\pi$$
−0.0499752 + 0.998750i $$0.515914\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −15.7035 −0.726673 −0.363337 0.931658i $$-0.618363\pi$$
−0.363337 + 0.931658i $$0.618363\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −3.82475 −0.176235
$$472$$ 0 0
$$473$$ −11.3446 −0.521627
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9.82475 0.448904 0.224452 0.974485i $$-0.427941\pi$$
0.224452 + 0.974485i $$0.427941\pi$$
$$480$$ 0 0
$$481$$ −26.1993 −1.19459
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.93039 −0.132789 −0.0663943 0.997793i $$-0.521150\pi$$
−0.0663943 + 0.997793i $$0.521150\pi$$
$$488$$ 0 0
$$489$$ 9.82475 0.444291
$$490$$ 0 0
$$491$$ −28.5498 −1.28844 −0.644218 0.764842i $$-0.722817\pi$$
−0.644218 + 0.764842i $$0.722817\pi$$
$$492$$ 0 0
$$493$$ −1.78959 −0.0805993
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.62541 0.0727635 0.0363818 0.999338i $$-0.488417\pi$$
0.0363818 + 0.999338i $$0.488417\pi$$
$$500$$ 0 0
$$501$$ −0.824752 −0.0368472
$$502$$ 0 0
$$503$$ −31.7682 −1.41647 −0.708236 0.705975i $$-0.750509\pi$$
−0.708236 + 0.705975i $$0.750509\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 10.5649 0.469205
$$508$$ 0 0
$$509$$ −14.4502 −0.640492 −0.320246 0.947334i $$-0.603766\pi$$
−0.320246 + 0.947334i $$0.603766\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 17.0170 0.751318
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 34.3375 1.51016
$$518$$ 0 0
$$519$$ 35.4743 1.55715
$$520$$ 0 0
$$521$$ 9.82475 0.430430 0.215215 0.976567i $$-0.430955\pi$$
0.215215 + 0.976567i $$0.430955\pi$$
$$522$$ 0 0
$$523$$ −7.34683 −0.321254 −0.160627 0.987015i $$-0.551352\pi$$
−0.160627 + 0.987015i $$0.551352\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.37097 −0.0597203
$$528$$ 0 0
$$529$$ 38.1993 1.66084
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 9.78523 0.423845
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.6005 0.543753
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.5498 −0.754526 −0.377263 0.926106i $$-0.623135\pi$$
−0.377263 + 0.926106i $$0.623135\pi$$
$$542$$ 0 0
$$543$$ 42.0454 1.80434
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.5386 0.878168 0.439084 0.898446i $$-0.355303\pi$$
0.439084 + 0.898446i $$0.355303\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 14.0000 0.596420
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −9.97368 −0.422598 −0.211299 0.977421i $$-0.567769\pi$$
−0.211299 + 0.977421i $$0.567769\pi$$
$$558$$ 0 0
$$559$$ −5.64950 −0.238949
$$560$$ 0 0
$$561$$ −3.82475 −0.161481
$$562$$ 0 0
$$563$$ 22.6317 0.953814 0.476907 0.878954i $$-0.341758\pi$$
0.476907 + 0.878954i $$0.341758\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8.37459 0.351081 0.175540 0.984472i $$-0.443833\pi$$
0.175540 + 0.984472i $$0.443833\pi$$
$$570$$ 0 0
$$571$$ 7.27492 0.304446 0.152223 0.988346i $$-0.451357\pi$$
0.152223 + 0.988346i $$0.451357\pi$$
$$572$$ 0 0
$$573$$ −0.303539 −0.0126805
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 3.88273 0.161640 0.0808200 0.996729i $$-0.474246\pi$$
0.0808200 + 0.996729i $$0.474246\pi$$
$$578$$ 0 0
$$579$$ 36.9244 1.53453
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −29.9210 −1.23920
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −20.8997 −0.862623 −0.431311 0.902203i $$-0.641949\pi$$
−0.431311 + 0.902203i $$0.641949\pi$$
$$588$$ 0 0
$$589$$ 10.7251 0.441919
$$590$$ 0 0
$$591$$ 14.9003 0.612918
$$592$$ 0 0
$$593$$ −33.3851 −1.37096 −0.685482 0.728090i $$-0.740408\pi$$
−0.685482 + 0.728090i $$0.740408\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −29.9210 −1.22459
$$598$$ 0 0
$$599$$ 5.27492 0.215527 0.107764 0.994177i $$-0.465631\pi$$
0.107764 + 0.994177i $$0.465631\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −11.4022 −0.462801 −0.231400 0.972859i $$-0.574331\pi$$
−0.231400 + 0.972859i $$0.574331\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 17.0997 0.691779
$$612$$ 0 0
$$613$$ 28.3616 1.14551 0.572757 0.819725i $$-0.305874\pi$$
0.572757 + 0.819725i $$0.305874\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 31.2920 1.25977 0.629884 0.776689i $$-0.283102\pi$$
0.629884 + 0.776689i $$0.283102\pi$$
$$618$$ 0 0
$$619$$ −8.92442 −0.358703 −0.179351 0.983785i $$-0.557400\pi$$
−0.179351 + 0.983785i $$0.557400\pi$$
$$620$$ 0 0
$$621$$ 40.6495 1.63121
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 29.9210 1.19493
$$628$$ 0 0
$$629$$ −4.17525 −0.166478
$$630$$ 0 0
$$631$$ 33.0997 1.31768 0.658839 0.752284i $$-0.271048\pi$$
0.658839 + 0.752284i $$0.271048\pi$$
$$632$$ 0 0
$$633$$ −44.4262 −1.76578
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.09967 −0.0829319 −0.0414660 0.999140i $$-0.513203\pi$$
−0.0414660 + 0.999140i $$0.513203\pi$$
$$642$$ 0 0
$$643$$ 31.4071 1.23857 0.619287 0.785164i $$-0.287422\pi$$
0.619287 + 0.785164i $$0.287422\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −26.9331 −1.05885 −0.529425 0.848357i $$-0.677592\pi$$
−0.529425 + 0.848357i $$0.677592\pi$$
$$648$$ 0 0
$$649$$ 17.2749 0.678100
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −28.3616 −1.10988 −0.554938 0.831892i $$-0.687258\pi$$
−0.554938 + 0.831892i $$0.687258\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −40.5498 −1.57960 −0.789799 0.613366i $$-0.789815\pi$$
−0.789799 + 0.613366i $$0.789815\pi$$
$$660$$ 0 0
$$661$$ −0.450166 −0.0175094 −0.00875471 0.999962i $$-0.502787\pi$$
−0.00875471 + 0.999962i $$0.502787\pi$$
$$662$$ 0 0
$$663$$ −1.90468 −0.0739717
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 33.4427 1.29491
$$668$$ 0 0
$$669$$ −15.0997 −0.583787
$$670$$ 0 0
$$671$$ 71.4743 2.75923
$$672$$ 0 0
$$673$$ 31.2920 1.20622 0.603109 0.797659i $$-0.293928\pi$$
0.603109 + 0.797659i $$0.293928\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −46.4043 −1.78346 −0.891731 0.452566i $$-0.850509\pi$$
−0.891731 + 0.452566i $$0.850509\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −33.8248 −1.29617
$$682$$ 0 0
$$683$$ −19.1676 −0.733430 −0.366715 0.930333i $$-0.619518\pi$$
−0.366715 + 0.930333i $$0.619518\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −5.67232 −0.216413
$$688$$ 0 0
$$689$$ −14.9003 −0.567657
$$690$$ 0 0
$$691$$ 30.3746 1.15550 0.577752 0.816212i $$-0.303930\pi$$
0.577752 + 0.816212i $$0.303930\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.55942 0.0590672
$$698$$ 0 0
$$699$$ 24.7251 0.935189
$$700$$ 0 0
$$701$$ 8.82475 0.333306 0.166653 0.986016i $$-0.446704\pi$$
0.166653 + 0.986016i $$0.446704\pi$$
$$702$$ 0 0
$$703$$ 32.6630 1.23191
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.4502 −0.392464 −0.196232 0.980557i $$-0.562871\pi$$
−0.196232 + 0.980557i $$0.562871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 25.6197 0.959465
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −0.952341 −0.0355658
$$718$$ 0 0
$$719$$ −30.3746 −1.13278 −0.566390 0.824137i $$-0.691661\pi$$
−0.566390 + 0.824137i $$0.691661\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −17.0170 −0.632868
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −3.10302 −0.115085 −0.0575423 0.998343i $$-0.518326\pi$$
−0.0575423 + 0.998343i $$0.518326\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −0.900331 −0.0332999
$$732$$ 0 0
$$733$$ −37.6865 −1.39198 −0.695991 0.718050i $$-0.745035\pi$$
−0.695991 + 0.718050i $$0.745035\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.5764 0.684270
$$738$$ 0 0
$$739$$ −20.9244 −0.769717 −0.384859 0.922976i $$-0.625750\pi$$
−0.384859 + 0.922976i $$0.625750\pi$$
$$740$$ 0 0
$$741$$ 14.9003 0.547377
$$742$$ 0 0
$$743$$ 6.45203 0.236702 0.118351 0.992972i $$-0.462239\pi$$
0.118351 + 0.992972i $$0.462239\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14.7251 −0.537326 −0.268663 0.963234i $$-0.586582\pi$$
−0.268663 + 0.963234i $$0.586582\pi$$
$$752$$ 0 0
$$753$$ 35.5934 1.29709
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −35.5934 −1.29366 −0.646831 0.762633i $$-0.723906\pi$$
−0.646831 + 0.762633i $$0.723906\pi$$
$$758$$ 0 0
$$759$$ 71.4743 2.59435
$$760$$ 0 0
$$761$$ 22.9244 0.831010 0.415505 0.909591i $$-0.363605\pi$$
0.415505 + 0.909591i $$0.363605\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.60271 0.310626
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −20.1752 −0.726594
$$772$$ 0 0
$$773$$ 40.3133 1.44997 0.724985 0.688765i $$-0.241847\pi$$
0.724985 + 0.688765i $$0.241847\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −12.1993 −0.437087
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ 22.2131 0.793832
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.73205 0.0617409 0.0308705 0.999523i $$-0.490172\pi$$
0.0308705 + 0.999523i $$0.490172\pi$$
$$788$$ 0 0
$$789$$ 1.35050 0.0480790
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 35.5934 1.26396
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 46.8229 1.65855 0.829276 0.558839i $$-0.188753\pi$$
0.829276 + 0.558839i $$0.188753\pi$$
$$798$$ 0 0
$$799$$ 2.72508 0.0964065
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 34.3375 1.21174
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 25.0284 0.881042
$$808$$ 0 0
$$809$$ −17.1993 −0.604697 −0.302348 0.953198i $$-0.597771\pi$$
−0.302348 + 0.953198i $$0.597771\pi$$
$$810$$ 0 0
$$811$$ 7.45017 0.261611 0.130805 0.991408i $$-0.458244\pi$$
0.130805 + 0.991408i $$0.458244\pi$$
$$812$$ 0 0
$$813$$ −17.0170 −0.596811
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.04329 0.246414
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −20.3746 −0.711078 −0.355539 0.934661i $$-0.615703\pi$$
−0.355539 + 0.934661i $$0.615703\pi$$
$$822$$ 0 0
$$823$$ −46.1583 −1.60898 −0.804488 0.593968i $$-0.797560\pi$$
−0.804488 + 0.593968i $$0.797560\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 15.0547 0.523505 0.261752 0.965135i $$-0.415700\pi$$
0.261752 + 0.965135i $$0.415700\pi$$
$$828$$ 0 0
$$829$$ −50.9244 −1.76868 −0.884339 0.466845i $$-0.845391\pi$$
−0.884339 + 0.466845i $$0.845391\pi$$
$$830$$ 0 0
$$831$$ 24.7251 0.857704
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 17.0170 0.588192
$$838$$ 0 0
$$839$$ 41.0997 1.41892 0.709459 0.704747i $$-0.248939\pi$$
0.709459 + 0.704747i $$0.248939\pi$$
$$840$$ 0 0
$$841$$ −10.7251 −0.369830
$$842$$ 0 0
$$843$$ −10.3923 −0.357930
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 18.9244 0.649484
$$850$$ 0 0
$$851$$ 78.0241 2.67463
$$852$$ 0 0
$$853$$ −13.1342 −0.449708 −0.224854 0.974392i $$-0.572190\pi$$
−0.224854 + 0.974392i $$0.572190\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −37.6865 −1.28735 −0.643673 0.765301i $$-0.722590\pi$$
−0.643673 + 0.765301i $$0.722590\pi$$
$$858$$ 0 0
$$859$$ −2.37459 −0.0810198 −0.0405099 0.999179i $$-0.512898\pi$$
−0.0405099 + 0.999179i $$0.512898\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 16.4257 0.559138 0.279569 0.960126i $$-0.409808\pi$$
0.279569 + 0.960126i $$0.409808\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 29.1413 0.989691
$$868$$ 0 0
$$869$$ −38.3746 −1.30177
$$870$$ 0 0
$$871$$ 9.25083 0.313452
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.8777 0.772527 0.386263 0.922389i $$-0.373766\pi$$
0.386263 + 0.922389i $$0.373766\pi$$
$$878$$ 0 0
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ −43.0241 −1.44952 −0.724759 0.689002i $$-0.758049\pi$$
−0.724759 + 0.689002i $$0.758049\pi$$
$$882$$ 0 0
$$883$$ −55.5407 −1.86909 −0.934547 0.355840i $$-0.884195\pi$$
−0.934547 + 0.355840i $$0.884195\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 39.2301 1.31722 0.658609 0.752486i $$-0.271145\pi$$
0.658609 + 0.752486i $$0.271145\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 47.4743 1.59045
$$892$$ 0 0
$$893$$ −21.3183 −0.713391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 35.5934 1.18843
$$898$$ 0 0
$$899$$ 14.0000 0.466926
$$900$$ 0 0
$$901$$ −2.37459 −0.0791089
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 41.8569 1.38984 0.694918 0.719089i $$-0.255440\pi$$
0.694918 + 0.719089i $$0.255440\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −25.0997 −0.831589 −0.415795 0.909459i $$-0.636496\pi$$
−0.415795 + 0.909459i $$0.636496\pi$$
$$912$$ 0 0
$$913$$ −39.0575 −1.29261
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −46.9244 −1.54789 −0.773947 0.633250i $$-0.781720\pi$$
−0.773947 + 0.633250i $$0.781720\pi$$
$$920$$ 0 0
$$921$$ 45.9244 1.51326
$$922$$ 0 0
$$923$$ 11.9517 0.393396
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −48.0997 −1.57810 −0.789049 0.614330i $$-0.789427\pi$$
−0.789049 + 0.614330i $$0.789427\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 17.0170 0.557111
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −24.3638 −0.795931 −0.397965 0.917400i $$-0.630284\pi$$
−0.397965 + 0.917400i $$0.630284\pi$$
$$938$$ 0 0
$$939$$ −58.0241 −1.89354
$$940$$ 0 0
$$941$$ −3.27492 −0.106759 −0.0533796 0.998574i $$-0.516999\pi$$
−0.0533796 + 0.998574i $$0.516999\pi$$
$$942$$ 0 0
$$943$$ −29.1413 −0.948972
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 10.5649 0.343314 0.171657 0.985157i $$-0.445088\pi$$
0.171657 + 0.985157i $$0.445088\pi$$
$$948$$ 0 0
$$949$$ 17.0997 0.555079
$$950$$ 0 0
$$951$$ 44.3746 1.43894
$$952$$ 0 0
$$953$$ −22.6893 −0.734978 −0.367489 0.930028i $$-0.619782\pi$$
−0.367489 + 0.930028i $$0.619782\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 39.0575 1.26255
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −20.2749 −0.654030
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.15068 0.0691611 0.0345806 0.999402i $$-0.488990\pi$$
0.0345806 + 0.999402i $$0.488990\pi$$
$$968$$ 0 0
$$969$$ 2.37459 0.0762827
$$970$$ 0 0
$$971$$ −36.9244 −1.18496 −0.592481 0.805585i $$-0.701851\pi$$
−0.592481 + 0.805585i $$0.701851\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 29.9210 0.957259 0.478629 0.878017i $$-0.341134\pi$$
0.478629 + 0.878017i $$0.341134\pi$$
$$978$$ 0 0
$$979$$ 36.9244 1.18011
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −45.9281 −1.46488 −0.732440 0.680832i $$-0.761618\pi$$
−0.732440 + 0.680832i $$0.761618\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.8248 0.534996
$$990$$ 0 0
$$991$$ 33.4743 1.06334 0.531672 0.846950i $$-0.321564\pi$$
0.531672 + 0.846950i $$0.321564\pi$$
$$992$$ 0 0
$$993$$ −30.8734 −0.979737
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −0.418627 −0.0132580 −0.00662902 0.999978i $$-0.502110\pi$$
−0.00662902 + 0.999978i $$0.502110\pi$$
$$998$$ 0 0
$$999$$ 51.8248 1.63966
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 4900.2.a.be.1.1 4
5.2 odd 4 980.2.e.f.589.3 4
5.3 odd 4 980.2.e.f.589.1 4
5.4 even 2 inner 4900.2.a.be.1.3 4
7.2 even 3 700.2.i.f.501.3 8
7.4 even 3 700.2.i.f.401.3 8
7.6 odd 2 4900.2.a.bf.1.3 4
35.2 odd 12 140.2.q.a.109.2 yes 4
35.3 even 12 980.2.q.g.569.1 4
35.4 even 6 700.2.i.f.401.2 8
35.9 even 6 700.2.i.f.501.2 8
35.12 even 12 980.2.q.g.949.1 4
35.13 even 4 980.2.e.c.589.4 4
35.17 even 12 980.2.q.b.569.2 4
35.18 odd 12 140.2.q.a.9.2 4
35.23 odd 12 140.2.q.b.109.2 yes 4
35.27 even 4 980.2.e.c.589.2 4
35.32 odd 12 140.2.q.b.9.1 yes 4
35.33 even 12 980.2.q.b.949.1 4
35.34 odd 2 4900.2.a.bf.1.1 4
105.2 even 12 1260.2.bm.a.109.1 4
105.23 even 12 1260.2.bm.b.109.1 4
105.32 even 12 1260.2.bm.b.289.2 4
105.53 even 12 1260.2.bm.a.289.1 4
140.23 even 12 560.2.bw.a.529.2 4
140.67 even 12 560.2.bw.a.289.1 4
140.107 even 12 560.2.bw.e.529.2 4
140.123 even 12 560.2.bw.e.289.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.q.a.9.2 4 35.18 odd 12
140.2.q.a.109.2 yes 4 35.2 odd 12
140.2.q.b.9.1 yes 4 35.32 odd 12
140.2.q.b.109.2 yes 4 35.23 odd 12
560.2.bw.a.289.1 4 140.67 even 12
560.2.bw.a.529.2 4 140.23 even 12
560.2.bw.e.289.2 4 140.123 even 12
560.2.bw.e.529.2 4 140.107 even 12
700.2.i.f.401.2 8 35.4 even 6
700.2.i.f.401.3 8 7.4 even 3
700.2.i.f.501.2 8 35.9 even 6
700.2.i.f.501.3 8 7.2 even 3
980.2.e.c.589.2 4 35.27 even 4
980.2.e.c.589.4 4 35.13 even 4
980.2.e.f.589.1 4 5.3 odd 4
980.2.e.f.589.3 4 5.2 odd 4
980.2.q.b.569.2 4 35.17 even 12
980.2.q.b.949.1 4 35.33 even 12
980.2.q.g.569.1 4 35.3 even 12
980.2.q.g.949.1 4 35.12 even 12
1260.2.bm.a.109.1 4 105.2 even 12
1260.2.bm.a.289.1 4 105.53 even 12
1260.2.bm.b.109.1 4 105.23 even 12
1260.2.bm.b.289.2 4 105.32 even 12
4900.2.a.be.1.1 4 1.1 even 1 trivial
4900.2.a.be.1.3 4 5.4 even 2 inner
4900.2.a.bf.1.1 4 35.34 odd 2
4900.2.a.bf.1.3 4 7.6 odd 2