# Properties

 Label 504.6.a.l.1.1 Level $504$ Weight $6$ Character 504.1 Self dual yes Analytic conductor $80.833$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 504.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: $$80.8334451857$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.54138$$ of defining polynomial Character $$\chi$$ $$=$$ 504.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-84.8276 q^{5} -49.0000 q^{7} +O(q^{10})$$ $$q-84.8276 q^{5} -49.0000 q^{7} -634.124 q^{11} +895.242 q^{13} +2057.46 q^{17} +2451.95 q^{19} -569.186 q^{23} +4070.73 q^{25} +1471.78 q^{29} -2006.65 q^{31} +4156.55 q^{35} +4860.54 q^{37} -17228.4 q^{41} -15481.3 q^{43} +5006.40 q^{47} +2401.00 q^{49} -19560.3 q^{53} +53791.3 q^{55} +14515.6 q^{59} +3572.67 q^{61} -75941.3 q^{65} -41480.7 q^{67} +9247.05 q^{71} -41350.0 q^{73} +31072.1 q^{77} -37962.2 q^{79} +79211.1 q^{83} -174530. q^{85} -92538.5 q^{89} -43866.9 q^{91} -207993. q^{95} +175419. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 48 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 48 * q^5 - 98 * q^7 $$2 q - 48 q^{5} - 98 q^{7} - 368 q^{11} - 156 q^{13} + 3312 q^{17} + 3736 q^{19} - 1552 q^{23} + 2302 q^{25} - 1728 q^{29} - 3624 q^{31} + 2352 q^{35} + 6996 q^{37} - 26160 q^{41} - 30184 q^{43} + 11424 q^{47} + 4802 q^{49} + 17376 q^{53} + 63592 q^{55} - 15008 q^{59} + 35564 q^{61} - 114656 q^{65} - 70504 q^{67} - 40752 q^{71} - 53892 q^{73} + 18032 q^{77} - 9744 q^{79} - 31360 q^{83} - 128328 q^{85} - 35952 q^{89} + 7644 q^{91} - 160704 q^{95} + 66652 q^{97}+O(q^{100})$$ 2 * q - 48 * q^5 - 98 * q^7 - 368 * q^11 - 156 * q^13 + 3312 * q^17 + 3736 * q^19 - 1552 * q^23 + 2302 * q^25 - 1728 * q^29 - 3624 * q^31 + 2352 * q^35 + 6996 * q^37 - 26160 * q^41 - 30184 * q^43 + 11424 * q^47 + 4802 * q^49 + 17376 * q^53 + 63592 * q^55 - 15008 * q^59 + 35564 * q^61 - 114656 * q^65 - 70504 * q^67 - 40752 * q^71 - 53892 * q^73 + 18032 * q^77 - 9744 * q^79 - 31360 * q^83 - 128328 * q^85 - 35952 * q^89 + 7644 * q^91 - 160704 * q^95 + 66652 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −84.8276 −1.51744 −0.758721 0.651415i $$-0.774176\pi$$
−0.758721 + 0.651415i $$0.774176\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −634.124 −1.58013 −0.790065 0.613023i $$-0.789953\pi$$
−0.790065 + 0.613023i $$0.789953\pi$$
$$12$$ 0 0
$$13$$ 895.242 1.46920 0.734602 0.678498i $$-0.237369\pi$$
0.734602 + 0.678498i $$0.237369\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2057.46 1.72667 0.863335 0.504630i $$-0.168371\pi$$
0.863335 + 0.504630i $$0.168371\pi$$
$$18$$ 0 0
$$19$$ 2451.95 1.55821 0.779106 0.626892i $$-0.215673\pi$$
0.779106 + 0.626892i $$0.215673\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −569.186 −0.224354 −0.112177 0.993688i $$-0.535782\pi$$
−0.112177 + 0.993688i $$0.535782\pi$$
$$24$$ 0 0
$$25$$ 4070.73 1.30263
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1471.78 0.324974 0.162487 0.986711i $$-0.448049\pi$$
0.162487 + 0.986711i $$0.448049\pi$$
$$30$$ 0 0
$$31$$ −2006.65 −0.375031 −0.187515 0.982262i $$-0.560043\pi$$
−0.187515 + 0.982262i $$0.560043\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4156.55 0.573539
$$36$$ 0 0
$$37$$ 4860.54 0.583687 0.291844 0.956466i $$-0.405731\pi$$
0.291844 + 0.956466i $$0.405731\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −17228.4 −1.60061 −0.800307 0.599591i $$-0.795330\pi$$
−0.800307 + 0.599591i $$0.795330\pi$$
$$42$$ 0 0
$$43$$ −15481.3 −1.27684 −0.638420 0.769689i $$-0.720412\pi$$
−0.638420 + 0.769689i $$0.720412\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5006.40 0.330583 0.165292 0.986245i $$-0.447143\pi$$
0.165292 + 0.986245i $$0.447143\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −19560.3 −0.956504 −0.478252 0.878223i $$-0.658729\pi$$
−0.478252 + 0.878223i $$0.658729\pi$$
$$54$$ 0 0
$$55$$ 53791.3 2.39776
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 14515.6 0.542881 0.271441 0.962455i $$-0.412500\pi$$
0.271441 + 0.962455i $$0.412500\pi$$
$$60$$ 0 0
$$61$$ 3572.67 0.122933 0.0614664 0.998109i $$-0.480422\pi$$
0.0614664 + 0.998109i $$0.480422\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −75941.3 −2.22943
$$66$$ 0 0
$$67$$ −41480.7 −1.12891 −0.564455 0.825464i $$-0.690914\pi$$
−0.564455 + 0.825464i $$0.690914\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9247.05 0.217700 0.108850 0.994058i $$-0.465283\pi$$
0.108850 + 0.994058i $$0.465283\pi$$
$$72$$ 0 0
$$73$$ −41350.0 −0.908172 −0.454086 0.890958i $$-0.650034\pi$$
−0.454086 + 0.890958i $$0.650034\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 31072.1 0.597233
$$78$$ 0 0
$$79$$ −37962.2 −0.684359 −0.342179 0.939635i $$-0.611165\pi$$
−0.342179 + 0.939635i $$0.611165\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 79211.1 1.26209 0.631046 0.775746i $$-0.282626\pi$$
0.631046 + 0.775746i $$0.282626\pi$$
$$84$$ 0 0
$$85$$ −174530. −2.62012
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −92538.5 −1.23836 −0.619181 0.785248i $$-0.712535\pi$$
−0.619181 + 0.785248i $$0.712535\pi$$
$$90$$ 0 0
$$91$$ −43866.9 −0.555307
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −207993. −2.36450
$$96$$ 0 0
$$97$$ 175419. 1.89299 0.946495 0.322720i $$-0.104597\pi$$
0.946495 + 0.322720i $$0.104597\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 56163.2 0.547834 0.273917 0.961753i $$-0.411681\pi$$
0.273917 + 0.961753i $$0.411681\pi$$
$$102$$ 0 0
$$103$$ −158376. −1.47095 −0.735474 0.677553i $$-0.763040\pi$$
−0.735474 + 0.677553i $$0.763040\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −23496.4 −0.198400 −0.0992001 0.995068i $$-0.531628\pi$$
−0.0992001 + 0.995068i $$0.531628\pi$$
$$108$$ 0 0
$$109$$ 205342. 1.65543 0.827715 0.561148i $$-0.189640\pi$$
0.827715 + 0.561148i $$0.189640\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −185993. −1.37025 −0.685125 0.728426i $$-0.740252\pi$$
−0.685125 + 0.728426i $$0.740252\pi$$
$$114$$ 0 0
$$115$$ 48282.7 0.340445
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −100816. −0.652620
$$120$$ 0 0
$$121$$ 241063. 1.49681
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −80223.7 −0.459227
$$126$$ 0 0
$$127$$ 348363. 1.91656 0.958281 0.285828i $$-0.0922685\pi$$
0.958281 + 0.285828i $$0.0922685\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −294849. −1.50114 −0.750571 0.660790i $$-0.770221\pi$$
−0.750571 + 0.660790i $$0.770221\pi$$
$$132$$ 0 0
$$133$$ −120145. −0.588949
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 119607. 0.544446 0.272223 0.962234i $$-0.412241\pi$$
0.272223 + 0.962234i $$0.412241\pi$$
$$138$$ 0 0
$$139$$ 145630. 0.639312 0.319656 0.947534i $$-0.396433\pi$$
0.319656 + 0.947534i $$0.396433\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −567695. −2.32153
$$144$$ 0 0
$$145$$ −124848. −0.493129
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 49750.8 0.183584 0.0917919 0.995778i $$-0.470741\pi$$
0.0917919 + 0.995778i $$0.470741\pi$$
$$150$$ 0 0
$$151$$ −214254. −0.764691 −0.382346 0.924019i $$-0.624884\pi$$
−0.382346 + 0.924019i $$0.624884\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 170219. 0.569088
$$156$$ 0 0
$$157$$ −197114. −0.638217 −0.319108 0.947718i $$-0.603383\pi$$
−0.319108 + 0.947718i $$0.603383\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 27890.1 0.0847980
$$162$$ 0 0
$$163$$ 103013. 0.303686 0.151843 0.988405i $$-0.451479\pi$$
0.151843 + 0.988405i $$0.451479\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −231396. −0.642044 −0.321022 0.947072i $$-0.604026\pi$$
−0.321022 + 0.947072i $$0.604026\pi$$
$$168$$ 0 0
$$169$$ 430165. 1.15856
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −207599. −0.527365 −0.263682 0.964610i $$-0.584937\pi$$
−0.263682 + 0.964610i $$0.584937\pi$$
$$174$$ 0 0
$$175$$ −199466. −0.492349
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −713483. −1.66438 −0.832188 0.554494i $$-0.812912\pi$$
−0.832188 + 0.554494i $$0.812912\pi$$
$$180$$ 0 0
$$181$$ 241894. 0.548818 0.274409 0.961613i $$-0.411518\pi$$
0.274409 + 0.961613i $$0.411518\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −412308. −0.885712
$$186$$ 0 0
$$187$$ −1.30469e6 −2.72836
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −220340. −0.437029 −0.218515 0.975834i $$-0.570121\pi$$
−0.218515 + 0.975834i $$0.570121\pi$$
$$192$$ 0 0
$$193$$ −736048. −1.42237 −0.711185 0.703005i $$-0.751841\pi$$
−0.711185 + 0.703005i $$0.751841\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −586422. −1.07658 −0.538288 0.842761i $$-0.680929\pi$$
−0.538288 + 0.842761i $$0.680929\pi$$
$$198$$ 0 0
$$199$$ −680230. −1.21765 −0.608826 0.793304i $$-0.708359\pi$$
−0.608826 + 0.793304i $$0.708359\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −72117.3 −0.122828
$$204$$ 0 0
$$205$$ 1.46145e6 2.42884
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.55484e6 −2.46218
$$210$$ 0 0
$$211$$ 142662. 0.220598 0.110299 0.993898i $$-0.464819\pi$$
0.110299 + 0.993898i $$0.464819\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.31324e6 1.93753
$$216$$ 0 0
$$217$$ 98325.8 0.141748
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.84193e6 2.53683
$$222$$ 0 0
$$223$$ −1.41803e6 −1.90952 −0.954760 0.297377i $$-0.903888\pi$$
−0.954760 + 0.297377i $$0.903888\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 75557.1 0.0973219 0.0486610 0.998815i $$-0.484505\pi$$
0.0486610 + 0.998815i $$0.484505\pi$$
$$228$$ 0 0
$$229$$ 1.29149e6 1.62743 0.813714 0.581265i $$-0.197442\pi$$
0.813714 + 0.581265i $$0.197442\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −112997. −0.136357 −0.0681783 0.997673i $$-0.521719\pi$$
−0.0681783 + 0.997673i $$0.521719\pi$$
$$234$$ 0 0
$$235$$ −424681. −0.501641
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −473209. −0.535869 −0.267935 0.963437i $$-0.586341\pi$$
−0.267935 + 0.963437i $$0.586341\pi$$
$$240$$ 0 0
$$241$$ −1.39122e6 −1.54296 −0.771478 0.636256i $$-0.780482\pi$$
−0.771478 + 0.636256i $$0.780482\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −203671. −0.216778
$$246$$ 0 0
$$247$$ 2.19508e6 2.28933
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 776017. 0.777476 0.388738 0.921348i $$-0.372911\pi$$
0.388738 + 0.921348i $$0.372911\pi$$
$$252$$ 0 0
$$253$$ 360935. 0.354509
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −955090. −0.902011 −0.451005 0.892521i $$-0.648934\pi$$
−0.451005 + 0.892521i $$0.648934\pi$$
$$258$$ 0 0
$$259$$ −238166. −0.220613
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 190821. 0.170113 0.0850566 0.996376i $$-0.472893\pi$$
0.0850566 + 0.996376i $$0.472893\pi$$
$$264$$ 0 0
$$265$$ 1.65926e6 1.45144
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −224114. −0.188837 −0.0944186 0.995533i $$-0.530099\pi$$
−0.0944186 + 0.995533i $$0.530099\pi$$
$$270$$ 0 0
$$271$$ 687404. 0.568577 0.284288 0.958739i $$-0.408243\pi$$
0.284288 + 0.958739i $$0.408243\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.58135e6 −2.05833
$$276$$ 0 0
$$277$$ −1.14588e6 −0.897302 −0.448651 0.893707i $$-0.648095\pi$$
−0.448651 + 0.893707i $$0.648095\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −755908. −0.571088 −0.285544 0.958366i $$-0.592174\pi$$
−0.285544 + 0.958366i $$0.592174\pi$$
$$282$$ 0 0
$$283$$ −1.92464e6 −1.42851 −0.714257 0.699884i $$-0.753235\pi$$
−0.714257 + 0.699884i $$0.753235\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 844194. 0.604975
$$288$$ 0 0
$$289$$ 2.81329e6 1.98139
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1.46185e6 0.994795 0.497398 0.867523i $$-0.334289\pi$$
0.497398 + 0.867523i $$0.334289\pi$$
$$294$$ 0 0
$$295$$ −1.23132e6 −0.823791
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −509559. −0.329622
$$300$$ 0 0
$$301$$ 758584. 0.482600
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −303061. −0.186544
$$306$$ 0 0
$$307$$ 1.06474e6 0.644756 0.322378 0.946611i $$-0.395518\pi$$
0.322378 + 0.946611i $$0.395518\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.59214e6 1.51970 0.759850 0.650098i $$-0.225272\pi$$
0.759850 + 0.650098i $$0.225272\pi$$
$$312$$ 0 0
$$313$$ −2.14280e6 −1.23629 −0.618147 0.786063i $$-0.712116\pi$$
−0.618147 + 0.786063i $$0.712116\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.02229e6 1.68923 0.844613 0.535377i $$-0.179830\pi$$
0.844613 + 0.535377i $$0.179830\pi$$
$$318$$ 0 0
$$319$$ −933292. −0.513501
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.04478e6 2.69052
$$324$$ 0 0
$$325$$ 3.64428e6 1.91383
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −245314. −0.124949
$$330$$ 0 0
$$331$$ −1.03801e6 −0.520751 −0.260376 0.965507i $$-0.583846\pi$$
−0.260376 + 0.965507i $$0.583846\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 3.51871e6 1.71306
$$336$$ 0 0
$$337$$ 2.23059e6 1.06991 0.534953 0.844882i $$-0.320329\pi$$
0.534953 + 0.844882i $$0.320329\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.27246e6 0.592598
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2.57483e6 −1.14795 −0.573977 0.818872i $$-0.694600\pi$$
−0.573977 + 0.818872i $$0.694600\pi$$
$$348$$ 0 0
$$349$$ 311098. 0.136721 0.0683604 0.997661i $$-0.478223\pi$$
0.0683604 + 0.997661i $$0.478223\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3.77642e6 −1.61303 −0.806517 0.591212i $$-0.798650\pi$$
−0.806517 + 0.591212i $$0.798650\pi$$
$$354$$ 0 0
$$355$$ −784406. −0.330347
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 996020. 0.407880 0.203940 0.978983i $$-0.434625\pi$$
0.203940 + 0.978983i $$0.434625\pi$$
$$360$$ 0 0
$$361$$ 3.53594e6 1.42803
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.50762e6 1.37810
$$366$$ 0 0
$$367$$ −3.83928e6 −1.48794 −0.743969 0.668214i $$-0.767059\pi$$
−0.743969 + 0.668214i $$0.767059\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 958457. 0.361525
$$372$$ 0 0
$$373$$ −5.05517e6 −1.88132 −0.940662 0.339344i $$-0.889795\pi$$
−0.940662 + 0.339344i $$0.889795\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.31760e6 0.477453
$$378$$ 0 0
$$379$$ −3.46825e6 −1.24026 −0.620129 0.784500i $$-0.712920\pi$$
−0.620129 + 0.784500i $$0.712920\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3.58445e6 −1.24861 −0.624304 0.781182i $$-0.714617\pi$$
−0.624304 + 0.781182i $$0.714617\pi$$
$$384$$ 0 0
$$385$$ −2.63577e6 −0.906267
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 3.86098e6 1.29367 0.646834 0.762630i $$-0.276092\pi$$
0.646834 + 0.762630i $$0.276092\pi$$
$$390$$ 0 0
$$391$$ −1.17108e6 −0.387386
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3.22025e6 1.03848
$$396$$ 0 0
$$397$$ −1.20136e6 −0.382559 −0.191280 0.981536i $$-0.561264\pi$$
−0.191280 + 0.981536i $$0.561264\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.88850e6 1.20760 0.603798 0.797138i $$-0.293653\pi$$
0.603798 + 0.797138i $$0.293653\pi$$
$$402$$ 0 0
$$403$$ −1.79644e6 −0.550997
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.08219e6 −0.922301
$$408$$ 0 0
$$409$$ 2.63640e6 0.779296 0.389648 0.920964i $$-0.372597\pi$$
0.389648 + 0.920964i $$0.372597\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −711264. −0.205190
$$414$$ 0 0
$$415$$ −6.71929e6 −1.91515
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −432053. −0.120227 −0.0601134 0.998192i $$-0.519146\pi$$
−0.0601134 + 0.998192i $$0.519146\pi$$
$$420$$ 0 0
$$421$$ 3.31215e6 0.910763 0.455381 0.890296i $$-0.349503\pi$$
0.455381 + 0.890296i $$0.349503\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8.37537e6 2.24922
$$426$$ 0 0
$$427$$ −175061. −0.0464642
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5.30508e6 −1.37562 −0.687811 0.725890i $$-0.741428\pi$$
−0.687811 + 0.725890i $$0.741428\pi$$
$$432$$ 0 0
$$433$$ −2.86645e6 −0.734723 −0.367362 0.930078i $$-0.619739\pi$$
−0.367362 + 0.930078i $$0.619739\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.39561e6 −0.349592
$$438$$ 0 0
$$439$$ 758493. 0.187841 0.0939205 0.995580i $$-0.470060\pi$$
0.0939205 + 0.995580i $$0.470060\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.02230e6 0.731693 0.365846 0.930675i $$-0.380780\pi$$
0.365846 + 0.930675i $$0.380780\pi$$
$$444$$ 0 0
$$445$$ 7.84982e6 1.87914
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.55957e6 1.30144 0.650721 0.759317i $$-0.274467\pi$$
0.650721 + 0.759317i $$0.274467\pi$$
$$450$$ 0 0
$$451$$ 1.09250e7 2.52918
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 3.72112e6 0.842646
$$456$$ 0 0
$$457$$ 1.48117e6 0.331752 0.165876 0.986147i $$-0.446955\pi$$
0.165876 + 0.986147i $$0.446955\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12329.0 0.00270193 0.00135097 0.999999i $$-0.499570\pi$$
0.00135097 + 0.999999i $$0.499570\pi$$
$$462$$ 0 0
$$463$$ −1.27201e6 −0.275765 −0.137882 0.990449i $$-0.544030\pi$$
−0.137882 + 0.990449i $$0.544030\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −7.15829e6 −1.51886 −0.759429 0.650590i $$-0.774521\pi$$
−0.759429 + 0.650590i $$0.774521\pi$$
$$468$$ 0 0
$$469$$ 2.03256e6 0.426688
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.81707e6 2.01757
$$474$$ 0 0
$$475$$ 9.98120e6 2.02978
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −5.01478e6 −0.998649 −0.499325 0.866415i $$-0.666418\pi$$
−0.499325 + 0.866415i $$0.666418\pi$$
$$480$$ 0 0
$$481$$ 4.35136e6 0.857555
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.48804e7 −2.87250
$$486$$ 0 0
$$487$$ 6.46712e6 1.23563 0.617815 0.786323i $$-0.288018\pi$$
0.617815 + 0.786323i $$0.288018\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6.89564e6 1.29084 0.645418 0.763829i $$-0.276683\pi$$
0.645418 + 0.763829i $$0.276683\pi$$
$$492$$ 0 0
$$493$$ 3.02813e6 0.561123
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −453106. −0.0822827
$$498$$ 0 0
$$499$$ −1.00230e7 −1.80196 −0.900979 0.433862i $$-0.857151\pi$$
−0.900979 + 0.433862i $$0.857151\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 1.02505e7 1.80644 0.903220 0.429178i $$-0.141197\pi$$
0.903220 + 0.429178i $$0.141197\pi$$
$$504$$ 0 0
$$505$$ −4.76419e6 −0.831306
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1.95058e6 0.333710 0.166855 0.985981i $$-0.446639\pi$$
0.166855 + 0.985981i $$0.446639\pi$$
$$510$$ 0 0
$$511$$ 2.02615e6 0.343257
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1.34347e7 2.23208
$$516$$ 0 0
$$517$$ −3.17468e6 −0.522364
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −5.02692e6 −0.811349 −0.405674 0.914018i $$-0.632963\pi$$
−0.405674 + 0.914018i $$0.632963\pi$$
$$522$$ 0 0
$$523$$ −356811. −0.0570405 −0.0285203 0.999593i $$-0.509080\pi$$
−0.0285203 + 0.999593i $$0.509080\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.12860e6 −0.647555
$$528$$ 0 0
$$529$$ −6.11237e6 −0.949665
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −1.54236e7 −2.35163
$$534$$ 0 0
$$535$$ 1.99314e6 0.301061
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.52253e6 −0.225733
$$540$$ 0 0
$$541$$ −1.16485e7 −1.71111 −0.855555 0.517712i $$-0.826784\pi$$
−0.855555 + 0.517712i $$0.826784\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1.74187e7 −2.51202
$$546$$ 0 0
$$547$$ −2.56657e6 −0.366763 −0.183382 0.983042i $$-0.558704\pi$$
−0.183382 + 0.983042i $$0.558704\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.60873e6 0.506378
$$552$$ 0 0
$$553$$ 1.86015e6 0.258663
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.19382e6 1.25562 0.627809 0.778367i $$-0.283952\pi$$
0.627809 + 0.778367i $$0.283952\pi$$
$$558$$ 0 0
$$559$$ −1.38595e7 −1.87594
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −2.73412e6 −0.363535 −0.181768 0.983342i $$-0.558182\pi$$
−0.181768 + 0.983342i $$0.558182\pi$$
$$564$$ 0 0
$$565$$ 1.57773e7 2.07927
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.51619e6 −0.714264 −0.357132 0.934054i $$-0.616245\pi$$
−0.357132 + 0.934054i $$0.616245\pi$$
$$570$$ 0 0
$$571$$ −1.15557e7 −1.48322 −0.741612 0.670829i $$-0.765938\pi$$
−0.741612 + 0.670829i $$0.765938\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.31700e6 −0.292251
$$576$$ 0 0
$$577$$ −9.36955e6 −1.17160 −0.585800 0.810456i $$-0.699220\pi$$
−0.585800 + 0.810456i $$0.699220\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.88134e6 −0.477026
$$582$$ 0 0
$$583$$ 1.24037e7 1.51140
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.61313e6 −0.432801 −0.216400 0.976305i $$-0.569432\pi$$
−0.216400 + 0.976305i $$0.569432\pi$$
$$588$$ 0 0
$$589$$ −4.92019e6 −0.584378
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 9.65554e6 1.12756 0.563780 0.825925i $$-0.309346\pi$$
0.563780 + 0.825925i $$0.309346\pi$$
$$594$$ 0 0
$$595$$ 8.55195e6 0.990314
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −7.36706e6 −0.838932 −0.419466 0.907771i $$-0.637783\pi$$
−0.419466 + 0.907771i $$0.637783\pi$$
$$600$$ 0 0
$$601$$ −9.04647e6 −1.02163 −0.510814 0.859691i $$-0.670656\pi$$
−0.510814 + 0.859691i $$0.670656\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.04488e7 −2.27132
$$606$$ 0 0
$$607$$ 1.07723e7 1.18669 0.593345 0.804948i $$-0.297807\pi$$
0.593345 + 0.804948i $$0.297807\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.48194e6 0.485694
$$612$$ 0 0
$$613$$ 5.64815e6 0.607093 0.303547 0.952817i $$-0.401829\pi$$
0.303547 + 0.952817i $$0.401829\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.21062e7 1.28025 0.640123 0.768272i $$-0.278883\pi$$
0.640123 + 0.768272i $$0.278883\pi$$
$$618$$ 0 0
$$619$$ −3.22164e6 −0.337948 −0.168974 0.985620i $$-0.554045\pi$$
−0.168974 + 0.985620i $$0.554045\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4.53439e6 0.468057
$$624$$ 0 0
$$625$$ −5.91583e6 −0.605781
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.00004e7 1.00784
$$630$$ 0 0
$$631$$ −8.59962e6 −0.859816 −0.429908 0.902873i $$-0.641454\pi$$
−0.429908 + 0.902873i $$0.641454\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2.95508e7 −2.90827
$$636$$ 0 0
$$637$$ 2.14948e6 0.209886
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −8.89346e6 −0.854920 −0.427460 0.904034i $$-0.640592\pi$$
−0.427460 + 0.904034i $$0.640592\pi$$
$$642$$ 0 0
$$643$$ −6.57628e6 −0.627268 −0.313634 0.949544i $$-0.601546\pi$$
−0.313634 + 0.949544i $$0.601546\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1.16128e7 −1.09063 −0.545313 0.838232i $$-0.683589\pi$$
−0.545313 + 0.838232i $$0.683589\pi$$
$$648$$ 0 0
$$649$$ −9.20470e6 −0.857823
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7.03391e6 −0.645527 −0.322763 0.946480i $$-0.604612\pi$$
−0.322763 + 0.946480i $$0.604612\pi$$
$$654$$ 0 0
$$655$$ 2.50114e7 2.27790
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.10193e7 0.988422 0.494211 0.869342i $$-0.335457\pi$$
0.494211 + 0.869342i $$0.335457\pi$$
$$660$$ 0 0
$$661$$ 3.37812e6 0.300726 0.150363 0.988631i $$-0.451956\pi$$
0.150363 + 0.988631i $$0.451956\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.01916e7 0.893696
$$666$$ 0 0
$$667$$ −837717. −0.0729093
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2.26552e6 −0.194250
$$672$$ 0 0
$$673$$ −3.46735e6 −0.295094 −0.147547 0.989055i $$-0.547138\pi$$
−0.147547 + 0.989055i $$0.547138\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −1.81353e7 −1.52074 −0.760369 0.649492i $$-0.774982\pi$$
−0.760369 + 0.649492i $$0.774982\pi$$
$$678$$ 0 0
$$679$$ −8.59555e6 −0.715483
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.60726e7 −1.31836 −0.659182 0.751983i $$-0.729097\pi$$
−0.659182 + 0.751983i $$0.729097\pi$$
$$684$$ 0 0
$$685$$ −1.01460e7 −0.826166
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1.75112e7 −1.40530
$$690$$ 0 0
$$691$$ 779272. 0.0620860 0.0310430 0.999518i $$-0.490117\pi$$
0.0310430 + 0.999518i $$0.490117\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.23534e7 −0.970120
$$696$$ 0 0
$$697$$ −3.54469e7 −2.76373
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 616669. 0.0473977 0.0236989 0.999719i $$-0.492456\pi$$
0.0236989 + 0.999719i $$0.492456\pi$$
$$702$$ 0 0
$$703$$ 1.19178e7 0.909509
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.75200e6 −0.207062
$$708$$ 0 0
$$709$$ 1.94867e7 1.45587 0.727935 0.685646i $$-0.240480\pi$$
0.727935 + 0.685646i $$0.240480\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.14216e6 0.0841398
$$714$$ 0 0
$$715$$ 4.81562e7 3.52279
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 6.07365e6 0.438155 0.219078 0.975707i $$-0.429695\pi$$
0.219078 + 0.975707i $$0.429695\pi$$
$$720$$ 0 0
$$721$$ 7.76044e6 0.555966
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5.99122e6 0.423321
$$726$$ 0 0
$$727$$ 9.38803e6 0.658777 0.329389 0.944194i $$-0.393157\pi$$
0.329389 + 0.944194i $$0.393157\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.18522e7 −2.20468
$$732$$ 0 0
$$733$$ 6.87253e6 0.472451 0.236226 0.971698i $$-0.424090\pi$$
0.236226 + 0.971698i $$0.424090\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.63040e7 1.78383
$$738$$ 0 0
$$739$$ 3.13208e6 0.210970 0.105485 0.994421i $$-0.466360\pi$$
0.105485 + 0.994421i $$0.466360\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8.05007e6 0.534968 0.267484 0.963562i $$-0.413808\pi$$
0.267484 + 0.963562i $$0.413808\pi$$
$$744$$ 0 0
$$745$$ −4.22024e6 −0.278578
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1.15132e6 0.0749882
$$750$$ 0 0
$$751$$ −1.51621e7 −0.980976 −0.490488 0.871448i $$-0.663181\pi$$
−0.490488 + 0.871448i $$0.663181\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.81746e7 1.16037
$$756$$ 0 0
$$757$$ 3.86111e6 0.244891 0.122445 0.992475i $$-0.460926\pi$$
0.122445 + 0.992475i $$0.460926\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.74204e7 −1.71638 −0.858188 0.513335i $$-0.828410\pi$$
−0.858188 + 0.513335i $$0.828410\pi$$
$$762$$ 0 0
$$763$$ −1.00617e7 −0.625694
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.29950e7 0.797603
$$768$$ 0 0
$$769$$ 7.96097e6 0.485456 0.242728 0.970094i $$-0.421958\pi$$
0.242728 + 0.970094i $$0.421958\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 4.09621e6 0.246566 0.123283 0.992372i $$-0.460658\pi$$
0.123283 + 0.992372i $$0.460658\pi$$
$$774$$ 0 0
$$775$$ −8.16852e6 −0.488527
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.22432e7 −2.49410
$$780$$ 0 0
$$781$$ −5.86378e6 −0.343994
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.67207e7 0.968457
$$786$$ 0 0
$$787$$ 1.43564e7 0.826244 0.413122 0.910676i $$-0.364438\pi$$
0.413122 + 0.910676i $$0.364438\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9.11364e6 0.517906
$$792$$ 0 0
$$793$$ 3.19840e6 0.180613
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.61963e7 1.46081 0.730407 0.683012i $$-0.239330\pi$$
0.730407 + 0.683012i $$0.239330\pi$$
$$798$$ 0 0
$$799$$ 1.03005e7 0.570809
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2.62210e7 1.43503
$$804$$ 0 0
$$805$$ −2.36585e6 −0.128676
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 3.31159e6 0.177896 0.0889478 0.996036i $$-0.471650\pi$$
0.0889478 + 0.996036i $$0.471650\pi$$
$$810$$ 0 0
$$811$$ 6.43440e6 0.343523 0.171761 0.985139i $$-0.445054\pi$$
0.171761 + 0.985139i $$0.445054\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −8.73839e6 −0.460826
$$816$$ 0 0
$$817$$ −3.79593e7 −1.98959
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.12076e7 −1.09808 −0.549039 0.835796i $$-0.685006\pi$$
−0.549039 + 0.835796i $$0.685006\pi$$
$$822$$ 0 0
$$823$$ 1.96206e7 1.00975 0.504875 0.863193i $$-0.331539\pi$$
0.504875 + 0.863193i $$0.331539\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.34210e6 0.474986 0.237493 0.971389i $$-0.423674\pi$$
0.237493 + 0.971389i $$0.423674\pi$$
$$828$$ 0 0
$$829$$ −8.15436e6 −0.412101 −0.206050 0.978541i $$-0.566061\pi$$
−0.206050 + 0.978541i $$0.566061\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 4.93997e6 0.246667
$$834$$ 0 0
$$835$$ 1.96288e7 0.974265
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −4.14558e6 −0.203320 −0.101660 0.994819i $$-0.532415\pi$$
−0.101660 + 0.994819i $$0.532415\pi$$
$$840$$ 0 0
$$841$$ −1.83450e7 −0.894392
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3.64899e7 −1.75805
$$846$$ 0 0
$$847$$ −1.18121e7 −0.565741
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2.76655e6 −0.130953
$$852$$ 0 0
$$853$$ 3.40559e6 0.160258 0.0801291 0.996784i $$-0.474467\pi$$
0.0801291 + 0.996784i $$0.474467\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −2.26615e6 −0.105399 −0.0526996 0.998610i $$-0.516783\pi$$
−0.0526996 + 0.998610i $$0.516783\pi$$
$$858$$ 0 0
$$859$$ −1.45799e7 −0.674174 −0.337087 0.941474i $$-0.609442\pi$$
−0.337087 + 0.941474i $$0.609442\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −3.37184e7 −1.54113 −0.770567 0.637359i $$-0.780027\pi$$
−0.770567 + 0.637359i $$0.780027\pi$$
$$864$$ 0 0
$$865$$ 1.76102e7 0.800245
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.40728e7 1.08138
$$870$$ 0 0
$$871$$ −3.71353e7 −1.65860
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.93096e6 0.173572
$$876$$ 0 0
$$877$$ 1.34267e7 0.589481 0.294740 0.955577i $$-0.404767\pi$$
0.294740 + 0.955577i $$0.404767\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.01261e6 0.260990 0.130495 0.991449i $$-0.458343\pi$$
0.130495 + 0.991449i $$0.458343\pi$$
$$882$$ 0 0
$$883$$ −4.30820e7 −1.85949 −0.929747 0.368200i $$-0.879974\pi$$
−0.929747 + 0.368200i $$0.879974\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 2.82279e7 1.20467 0.602336 0.798242i $$-0.294237\pi$$
0.602336 + 0.798242i $$0.294237\pi$$
$$888$$ 0 0
$$889$$ −1.70698e7 −0.724392
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.22754e7 0.515119
$$894$$ 0 0
$$895$$ 6.05231e7 2.52559
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −2.95335e6 −0.121875
$$900$$ 0 0
$$901$$ −4.02447e7 −1.65157
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −2.05193e7 −0.832799
$$906$$ 0 0
$$907$$ −1.56044e7 −0.629836 −0.314918 0.949119i $$-0.601977\pi$$
−0.314918 + 0.949119i $$0.601977\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −6.33384e6 −0.252855 −0.126427 0.991976i $$-0.540351\pi$$
−0.126427 + 0.991976i $$0.540351\pi$$
$$912$$ 0 0
$$913$$ −5.02297e7 −1.99427
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.44476e7 0.567378
$$918$$ 0 0
$$919$$ 2.11662e7 0.826713 0.413357 0.910569i $$-0.364356\pi$$
0.413357 + 0.910569i $$0.364356\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 8.27835e6 0.319845
$$924$$ 0 0
$$925$$ 1.97859e7 0.760330
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −3.25377e6 −0.123694 −0.0618468 0.998086i $$-0.519699\pi$$
−0.0618468 + 0.998086i $$0.519699\pi$$
$$930$$ 0 0
$$931$$ 5.88712e6 0.222602
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 1.10674e8 4.14014
$$936$$ 0 0
$$937$$ −1.00930e7 −0.375555 −0.187777 0.982212i $$-0.560128\pi$$
−0.187777 + 0.982212i $$0.560128\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −3.04029e7 −1.11929 −0.559643 0.828734i $$-0.689062\pi$$
−0.559643 + 0.828734i $$0.689062\pi$$
$$942$$ 0 0
$$943$$ 9.80619e6 0.359105
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.37126e7 1.58391 0.791957 0.610577i $$-0.209062\pi$$
0.791957 + 0.610577i $$0.209062\pi$$
$$948$$ 0 0
$$949$$ −3.70182e7 −1.33429
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.31900e7 0.827120 0.413560 0.910477i $$-0.364285\pi$$
0.413560 + 0.910477i $$0.364285\pi$$
$$954$$ 0 0
$$955$$ 1.86909e7 0.663167
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −5.86074e6 −0.205781
$$960$$ 0 0
$$961$$ −2.46025e7 −0.859352
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 6.24372e7 2.15837
$$966$$ 0 0
$$967$$ −2.78623e7 −0.958188 −0.479094 0.877764i $$-0.659035\pi$$
−0.479094 + 0.877764i $$0.659035\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3.65933e7 1.24553 0.622763 0.782410i $$-0.286010\pi$$
0.622763 + 0.782410i $$0.286010\pi$$
$$972$$ 0 0
$$973$$ −7.13586e6 −0.241637
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1.83769e7 0.615938 0.307969 0.951396i $$-0.400351\pi$$
0.307969 + 0.951396i $$0.400351\pi$$
$$978$$ 0 0
$$979$$ 5.86809e7 1.95677
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −5.09387e7 −1.68137 −0.840687 0.541522i $$-0.817848\pi$$
−0.840687 + 0.541522i $$0.817848\pi$$
$$984$$ 0 0
$$985$$ 4.97448e7 1.63364
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.81174e6 0.286465
$$990$$ 0 0
$$991$$ −946169. −0.0306044 −0.0153022 0.999883i $$-0.504871\pi$$
−0.0153022 + 0.999883i $$0.504871\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 5.77023e7 1.84772
$$996$$ 0 0
$$997$$ 8.61505e6 0.274486 0.137243 0.990537i $$-0.456176\pi$$
0.137243 + 0.990537i $$0.456176\pi$$
$$998$$ 0 0
$$999$$ 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 504.6.a.l.1.1 2
3.2 odd 2 504.6.a.r.1.2 yes 2
4.3 odd 2 1008.6.a.bh.1.1 2
12.11 even 2 1008.6.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.l.1.1 2 1.1 even 1 trivial
504.6.a.r.1.2 yes 2 3.2 odd 2
1008.6.a.bh.1.1 2 4.3 odd 2
1008.6.a.bs.1.2 2 12.11 even 2