# Properties

 Label 84.6.a.c.1.1 Level $84$ Weight $6$ Character 84.1 Self dual yes Analytic conductor $13.472$ Analytic rank $0$ 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] = [84,6,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("84.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 84.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: $$13.4722408643$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5569})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1392$$ x^2 - x - 1392 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$37.8129$$ of defining polynomial Character $$\chi$$ $$=$$ 84.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-9.00000 q^{3} -77.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q-9.00000 q^{3} -77.6257 q^{5} -49.0000 q^{7} +81.0000 q^{9} +477.380 q^{11} -63.7544 q^{13} +698.632 q^{15} +1037.63 q^{17} -667.018 q^{19} +441.000 q^{21} +3251.63 q^{23} +2900.75 q^{25} -729.000 q^{27} +2300.97 q^{29} +3717.05 q^{31} -4296.42 q^{33} +3803.66 q^{35} +12245.9 q^{37} +573.790 q^{39} -1829.65 q^{41} -20794.2 q^{43} -6287.68 q^{45} -4283.37 q^{47} +2401.00 q^{49} -9338.63 q^{51} +25718.4 q^{53} -37057.0 q^{55} +6003.16 q^{57} -2838.71 q^{59} +16803.2 q^{61} -3969.00 q^{63} +4948.98 q^{65} -62535.1 q^{67} -29264.6 q^{69} +72301.0 q^{71} -55676.9 q^{73} -26106.8 q^{75} -23391.6 q^{77} -3989.19 q^{79} +6561.00 q^{81} -46092.2 q^{83} -80546.5 q^{85} -20708.7 q^{87} +135385. q^{89} +3123.97 q^{91} -33453.5 q^{93} +51777.7 q^{95} +142878. q^{97} +38667.8 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9}+O(q^{10})$$ 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 $$2 q - 18 q^{3} - 6 q^{5} - 98 q^{7} + 162 q^{9} - 90 q^{11} + 768 q^{13} + 54 q^{15} + 1926 q^{17} + 2248 q^{19} + 882 q^{21} + 6354 q^{23} + 4906 q^{25} - 1458 q^{27} + 10572 q^{29} - 3312 q^{31} + 810 q^{33} + 294 q^{35} + 2104 q^{37} - 6912 q^{39} + 1266 q^{41} - 5768 q^{43} - 486 q^{45} + 15612 q^{47} + 4802 q^{49} - 17334 q^{51} + 16512 q^{53} - 77696 q^{55} - 20232 q^{57} - 13140 q^{59} - 5796 q^{61} - 7938 q^{63} + 64524 q^{65} - 56116 q^{67} - 57186 q^{69} + 11022 q^{71} - 85384 q^{73} - 44154 q^{75} + 4410 q^{77} - 19620 q^{79} + 13122 q^{81} - 44424 q^{83} - 16916 q^{85} - 95148 q^{87} + 211218 q^{89} - 37632 q^{91} + 29808 q^{93} + 260568 q^{95} + 44864 q^{97} - 7290 q^{99}+O(q^{100})$$ 2 * q - 18 * q^3 - 6 * q^5 - 98 * q^7 + 162 * q^9 - 90 * q^11 + 768 * q^13 + 54 * q^15 + 1926 * q^17 + 2248 * q^19 + 882 * q^21 + 6354 * q^23 + 4906 * q^25 - 1458 * q^27 + 10572 * q^29 - 3312 * q^31 + 810 * q^33 + 294 * q^35 + 2104 * q^37 - 6912 * q^39 + 1266 * q^41 - 5768 * q^43 - 486 * q^45 + 15612 * q^47 + 4802 * q^49 - 17334 * q^51 + 16512 * q^53 - 77696 * q^55 - 20232 * q^57 - 13140 * q^59 - 5796 * q^61 - 7938 * q^63 + 64524 * q^65 - 56116 * q^67 - 57186 * q^69 + 11022 * q^71 - 85384 * q^73 - 44154 * q^75 + 4410 * q^77 - 19620 * q^79 + 13122 * q^81 - 44424 * q^83 - 16916 * q^85 - 95148 * q^87 + 211218 * q^89 - 37632 * q^91 + 29808 * q^93 + 260568 * q^95 + 44864 * q^97 - 7290 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −9.00000 −0.577350
$$4$$ 0 0
$$5$$ −77.6257 −1.38861 −0.694306 0.719680i $$-0.744288\pi$$
−0.694306 + 0.719680i $$0.744288\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 477.380 1.18955 0.594775 0.803892i $$-0.297241\pi$$
0.594775 + 0.803892i $$0.297241\pi$$
$$12$$ 0 0
$$13$$ −63.7544 −0.104629 −0.0523145 0.998631i $$-0.516660\pi$$
−0.0523145 + 0.998631i $$0.516660\pi$$
$$14$$ 0 0
$$15$$ 698.632 0.801715
$$16$$ 0 0
$$17$$ 1037.63 0.870800 0.435400 0.900237i $$-0.356607\pi$$
0.435400 + 0.900237i $$0.356607\pi$$
$$18$$ 0 0
$$19$$ −667.018 −0.423890 −0.211945 0.977282i $$-0.567980\pi$$
−0.211945 + 0.977282i $$0.567980\pi$$
$$20$$ 0 0
$$21$$ 441.000 0.218218
$$22$$ 0 0
$$23$$ 3251.63 1.28168 0.640842 0.767673i $$-0.278585\pi$$
0.640842 + 0.767673i $$0.278585\pi$$
$$24$$ 0 0
$$25$$ 2900.75 0.928241
$$26$$ 0 0
$$27$$ −729.000 −0.192450
$$28$$ 0 0
$$29$$ 2300.97 0.508061 0.254031 0.967196i $$-0.418244\pi$$
0.254031 + 0.967196i $$0.418244\pi$$
$$30$$ 0 0
$$31$$ 3717.05 0.694696 0.347348 0.937736i $$-0.387082\pi$$
0.347348 + 0.937736i $$0.387082\pi$$
$$32$$ 0 0
$$33$$ −4296.42 −0.686787
$$34$$ 0 0
$$35$$ 3803.66 0.524846
$$36$$ 0 0
$$37$$ 12245.9 1.47057 0.735284 0.677759i $$-0.237049\pi$$
0.735284 + 0.677759i $$0.237049\pi$$
$$38$$ 0 0
$$39$$ 573.790 0.0604075
$$40$$ 0 0
$$41$$ −1829.65 −0.169984 −0.0849920 0.996382i $$-0.527086\pi$$
−0.0849920 + 0.996382i $$0.527086\pi$$
$$42$$ 0 0
$$43$$ −20794.2 −1.71503 −0.857513 0.514463i $$-0.827991\pi$$
−0.857513 + 0.514463i $$0.827991\pi$$
$$44$$ 0 0
$$45$$ −6287.68 −0.462870
$$46$$ 0 0
$$47$$ −4283.37 −0.282840 −0.141420 0.989950i $$-0.545167\pi$$
−0.141420 + 0.989950i $$0.545167\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −9338.63 −0.502757
$$52$$ 0 0
$$53$$ 25718.4 1.25764 0.628818 0.777553i $$-0.283539\pi$$
0.628818 + 0.777553i $$0.283539\pi$$
$$54$$ 0 0
$$55$$ −37057.0 −1.65182
$$56$$ 0 0
$$57$$ 6003.16 0.244733
$$58$$ 0 0
$$59$$ −2838.71 −0.106167 −0.0530837 0.998590i $$-0.516905\pi$$
−0.0530837 + 0.998590i $$0.516905\pi$$
$$60$$ 0 0
$$61$$ 16803.2 0.578186 0.289093 0.957301i $$-0.406646\pi$$
0.289093 + 0.957301i $$0.406646\pi$$
$$62$$ 0 0
$$63$$ −3969.00 −0.125988
$$64$$ 0 0
$$65$$ 4948.98 0.145289
$$66$$ 0 0
$$67$$ −62535.1 −1.70191 −0.850955 0.525238i $$-0.823976\pi$$
−0.850955 + 0.525238i $$0.823976\pi$$
$$68$$ 0 0
$$69$$ −29264.6 −0.739981
$$70$$ 0 0
$$71$$ 72301.0 1.70215 0.851077 0.525042i $$-0.175950\pi$$
0.851077 + 0.525042i $$0.175950\pi$$
$$72$$ 0 0
$$73$$ −55676.9 −1.22283 −0.611417 0.791308i $$-0.709400\pi$$
−0.611417 + 0.791308i $$0.709400\pi$$
$$74$$ 0 0
$$75$$ −26106.8 −0.535920
$$76$$ 0 0
$$77$$ −23391.6 −0.449608
$$78$$ 0 0
$$79$$ −3989.19 −0.0719146 −0.0359573 0.999353i $$-0.511448\pi$$
−0.0359573 + 0.999353i $$0.511448\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 0 0
$$83$$ −46092.2 −0.734400 −0.367200 0.930142i $$-0.619684\pi$$
−0.367200 + 0.930142i $$0.619684\pi$$
$$84$$ 0 0
$$85$$ −80546.5 −1.20920
$$86$$ 0 0
$$87$$ −20708.7 −0.293329
$$88$$ 0 0
$$89$$ 135385. 1.81173 0.905867 0.423562i $$-0.139220\pi$$
0.905867 + 0.423562i $$0.139220\pi$$
$$90$$ 0 0
$$91$$ 3123.97 0.0395460
$$92$$ 0 0
$$93$$ −33453.5 −0.401083
$$94$$ 0 0
$$95$$ 51777.7 0.588619
$$96$$ 0 0
$$97$$ 142878. 1.54183 0.770914 0.636939i $$-0.219800\pi$$
0.770914 + 0.636939i $$0.219800\pi$$
$$98$$ 0 0
$$99$$ 38667.8 0.396517
$$100$$ 0 0
$$101$$ 44467.1 0.433746 0.216873 0.976200i $$-0.430414\pi$$
0.216873 + 0.976200i $$0.430414\pi$$
$$102$$ 0 0
$$103$$ 202619. 1.88186 0.940931 0.338598i $$-0.109953\pi$$
0.940931 + 0.338598i $$0.109953\pi$$
$$104$$ 0 0
$$105$$ −34232.9 −0.303020
$$106$$ 0 0
$$107$$ 99525.9 0.840382 0.420191 0.907436i $$-0.361963\pi$$
0.420191 + 0.907436i $$0.361963\pi$$
$$108$$ 0 0
$$109$$ 220930. 1.78110 0.890551 0.454883i $$-0.150319\pi$$
0.890551 + 0.454883i $$0.150319\pi$$
$$110$$ 0 0
$$111$$ −110213. −0.849033
$$112$$ 0 0
$$113$$ 29623.1 0.218240 0.109120 0.994029i $$-0.465197\pi$$
0.109120 + 0.994029i $$0.465197\pi$$
$$114$$ 0 0
$$115$$ −252410. −1.77976
$$116$$ 0 0
$$117$$ −5164.11 −0.0348763
$$118$$ 0 0
$$119$$ −50843.7 −0.329131
$$120$$ 0 0
$$121$$ 66840.8 0.415029
$$122$$ 0 0
$$123$$ 16466.8 0.0981403
$$124$$ 0 0
$$125$$ 17407.2 0.0996448
$$126$$ 0 0
$$127$$ −264132. −1.45315 −0.726577 0.687086i $$-0.758890\pi$$
−0.726577 + 0.687086i $$0.758890\pi$$
$$128$$ 0 0
$$129$$ 187148. 0.990170
$$130$$ 0 0
$$131$$ −77850.1 −0.396352 −0.198176 0.980166i $$-0.563502\pi$$
−0.198176 + 0.980166i $$0.563502\pi$$
$$132$$ 0 0
$$133$$ 32683.9 0.160215
$$134$$ 0 0
$$135$$ 56589.2 0.267238
$$136$$ 0 0
$$137$$ 372403. 1.69516 0.847581 0.530666i $$-0.178058\pi$$
0.847581 + 0.530666i $$0.178058\pi$$
$$138$$ 0 0
$$139$$ −274550. −1.20527 −0.602636 0.798016i $$-0.705883\pi$$
−0.602636 + 0.798016i $$0.705883\pi$$
$$140$$ 0 0
$$141$$ 38550.3 0.163298
$$142$$ 0 0
$$143$$ −30435.1 −0.124461
$$144$$ 0 0
$$145$$ −178615. −0.705500
$$146$$ 0 0
$$147$$ −21609.0 −0.0824786
$$148$$ 0 0
$$149$$ −312982. −1.15492 −0.577462 0.816417i $$-0.695957\pi$$
−0.577462 + 0.816417i $$0.695957\pi$$
$$150$$ 0 0
$$151$$ 432095. 1.54219 0.771093 0.636723i $$-0.219710\pi$$
0.771093 + 0.636723i $$0.219710\pi$$
$$152$$ 0 0
$$153$$ 84047.7 0.290267
$$154$$ 0 0
$$155$$ −288539. −0.964662
$$156$$ 0 0
$$157$$ −84603.7 −0.273930 −0.136965 0.990576i $$-0.543735\pi$$
−0.136965 + 0.990576i $$0.543735\pi$$
$$158$$ 0 0
$$159$$ −231466. −0.726096
$$160$$ 0 0
$$161$$ −159330. −0.484431
$$162$$ 0 0
$$163$$ 306303. 0.902989 0.451494 0.892274i $$-0.350891\pi$$
0.451494 + 0.892274i $$0.350891\pi$$
$$164$$ 0 0
$$165$$ 333513. 0.953680
$$166$$ 0 0
$$167$$ −606514. −1.68287 −0.841433 0.540362i $$-0.818287\pi$$
−0.841433 + 0.540362i $$0.818287\pi$$
$$168$$ 0 0
$$169$$ −367228. −0.989053
$$170$$ 0 0
$$171$$ −54028.4 −0.141297
$$172$$ 0 0
$$173$$ −288481. −0.732828 −0.366414 0.930452i $$-0.619415\pi$$
−0.366414 + 0.930452i $$0.619415\pi$$
$$174$$ 0 0
$$175$$ −142137. −0.350842
$$176$$ 0 0
$$177$$ 25548.4 0.0612958
$$178$$ 0 0
$$179$$ 148858. 0.347248 0.173624 0.984812i $$-0.444452\pi$$
0.173624 + 0.984812i $$0.444452\pi$$
$$180$$ 0 0
$$181$$ 93377.8 0.211859 0.105930 0.994374i $$-0.466218\pi$$
0.105930 + 0.994374i $$0.466218\pi$$
$$182$$ 0 0
$$183$$ −151229. −0.333816
$$184$$ 0 0
$$185$$ −950594. −2.04205
$$186$$ 0 0
$$187$$ 495342. 1.03586
$$188$$ 0 0
$$189$$ 35721.0 0.0727393
$$190$$ 0 0
$$191$$ 246915. 0.489738 0.244869 0.969556i $$-0.421255\pi$$
0.244869 + 0.969556i $$0.421255\pi$$
$$192$$ 0 0
$$193$$ 481437. 0.930349 0.465175 0.885219i $$-0.345992\pi$$
0.465175 + 0.885219i $$0.345992\pi$$
$$194$$ 0 0
$$195$$ −44540.8 −0.0838826
$$196$$ 0 0
$$197$$ −548236. −1.00647 −0.503237 0.864149i $$-0.667858\pi$$
−0.503237 + 0.864149i $$0.667858\pi$$
$$198$$ 0 0
$$199$$ 158179. 0.283150 0.141575 0.989928i $$-0.454783\pi$$
0.141575 + 0.989928i $$0.454783\pi$$
$$200$$ 0 0
$$201$$ 562816. 0.982599
$$202$$ 0 0
$$203$$ −112748. −0.192029
$$204$$ 0 0
$$205$$ 142028. 0.236042
$$206$$ 0 0
$$207$$ 263382. 0.427228
$$208$$ 0 0
$$209$$ −318421. −0.504238
$$210$$ 0 0
$$211$$ 283510. 0.438392 0.219196 0.975681i $$-0.429657\pi$$
0.219196 + 0.975681i $$0.429657\pi$$
$$212$$ 0 0
$$213$$ −650709. −0.982739
$$214$$ 0 0
$$215$$ 1.61416e6 2.38150
$$216$$ 0 0
$$217$$ −182136. −0.262570
$$218$$ 0 0
$$219$$ 501092. 0.706004
$$220$$ 0 0
$$221$$ −66153.2 −0.0911109
$$222$$ 0 0
$$223$$ −651135. −0.876817 −0.438409 0.898776i $$-0.644458\pi$$
−0.438409 + 0.898776i $$0.644458\pi$$
$$224$$ 0 0
$$225$$ 234961. 0.309414
$$226$$ 0 0
$$227$$ −378294. −0.487264 −0.243632 0.969868i $$-0.578339\pi$$
−0.243632 + 0.969868i $$0.578339\pi$$
$$228$$ 0 0
$$229$$ −22332.8 −0.0281420 −0.0140710 0.999901i $$-0.504479\pi$$
−0.0140710 + 0.999901i $$0.504479\pi$$
$$230$$ 0 0
$$231$$ 210525. 0.259581
$$232$$ 0 0
$$233$$ −908940. −1.09685 −0.548423 0.836201i $$-0.684771\pi$$
−0.548423 + 0.836201i $$0.684771\pi$$
$$234$$ 0 0
$$235$$ 332500. 0.392755
$$236$$ 0 0
$$237$$ 35902.7 0.0415199
$$238$$ 0 0
$$239$$ 1.05363e6 1.19315 0.596573 0.802559i $$-0.296529\pi$$
0.596573 + 0.802559i $$0.296529\pi$$
$$240$$ 0 0
$$241$$ 1.05233e6 1.16710 0.583550 0.812077i $$-0.301663\pi$$
0.583550 + 0.812077i $$0.301663\pi$$
$$242$$ 0 0
$$243$$ −59049.0 −0.0641500
$$244$$ 0 0
$$245$$ −186379. −0.198373
$$246$$ 0 0
$$247$$ 42525.3 0.0443512
$$248$$ 0 0
$$249$$ 414830. 0.424006
$$250$$ 0 0
$$251$$ 972876. 0.974705 0.487352 0.873205i $$-0.337963\pi$$
0.487352 + 0.873205i $$0.337963\pi$$
$$252$$ 0 0
$$253$$ 1.55226e6 1.52463
$$254$$ 0 0
$$255$$ 724918. 0.698134
$$256$$ 0 0
$$257$$ 1.77948e6 1.68058 0.840290 0.542137i $$-0.182385\pi$$
0.840290 + 0.542137i $$0.182385\pi$$
$$258$$ 0 0
$$259$$ −600047. −0.555822
$$260$$ 0 0
$$261$$ 186379. 0.169354
$$262$$ 0 0
$$263$$ 21959.6 0.0195765 0.00978827 0.999952i $$-0.496884\pi$$
0.00978827 + 0.999952i $$0.496884\pi$$
$$264$$ 0 0
$$265$$ −1.99641e6 −1.74637
$$266$$ 0 0
$$267$$ −1.21846e6 −1.04601
$$268$$ 0 0
$$269$$ −1.69111e6 −1.42492 −0.712461 0.701712i $$-0.752419\pi$$
−0.712461 + 0.701712i $$0.752419\pi$$
$$270$$ 0 0
$$271$$ −467863. −0.386986 −0.193493 0.981102i $$-0.561982\pi$$
−0.193493 + 0.981102i $$0.561982\pi$$
$$272$$ 0 0
$$273$$ −28115.7 −0.0228319
$$274$$ 0 0
$$275$$ 1.38476e6 1.10419
$$276$$ 0 0
$$277$$ 906169. 0.709594 0.354797 0.934943i $$-0.384550\pi$$
0.354797 + 0.934943i $$0.384550\pi$$
$$278$$ 0 0
$$279$$ 301081. 0.231565
$$280$$ 0 0
$$281$$ −781388. −0.590338 −0.295169 0.955445i $$-0.595376\pi$$
−0.295169 + 0.955445i $$0.595376\pi$$
$$282$$ 0 0
$$283$$ −1.49843e6 −1.11217 −0.556085 0.831125i $$-0.687697\pi$$
−0.556085 + 0.831125i $$0.687697\pi$$
$$284$$ 0 0
$$285$$ −466000. −0.339839
$$286$$ 0 0
$$287$$ 89652.8 0.0642479
$$288$$ 0 0
$$289$$ −343190. −0.241707
$$290$$ 0 0
$$291$$ −1.28590e6 −0.890175
$$292$$ 0 0
$$293$$ −1.56070e6 −1.06206 −0.531031 0.847352i $$-0.678195\pi$$
−0.531031 + 0.847352i $$0.678195\pi$$
$$294$$ 0 0
$$295$$ 220357. 0.147425
$$296$$ 0 0
$$297$$ −348010. −0.228929
$$298$$ 0 0
$$299$$ −207305. −0.134101
$$300$$ 0 0
$$301$$ 1.01891e6 0.648219
$$302$$ 0 0
$$303$$ −400204. −0.250423
$$304$$ 0 0
$$305$$ −1.30436e6 −0.802875
$$306$$ 0 0
$$307$$ −889308. −0.538525 −0.269263 0.963067i $$-0.586780\pi$$
−0.269263 + 0.963067i $$0.586780\pi$$
$$308$$ 0 0
$$309$$ −1.82357e6 −1.08649
$$310$$ 0 0
$$311$$ 2.44826e6 1.43535 0.717674 0.696380i $$-0.245207\pi$$
0.717674 + 0.696380i $$0.245207\pi$$
$$312$$ 0 0
$$313$$ −2.73102e6 −1.57567 −0.787834 0.615887i $$-0.788798\pi$$
−0.787834 + 0.615887i $$0.788798\pi$$
$$314$$ 0 0
$$315$$ 308097. 0.174949
$$316$$ 0 0
$$317$$ −249603. −0.139509 −0.0697545 0.997564i $$-0.522222\pi$$
−0.0697545 + 0.997564i $$0.522222\pi$$
$$318$$ 0 0
$$319$$ 1.09844e6 0.604364
$$320$$ 0 0
$$321$$ −895733. −0.485195
$$322$$ 0 0
$$323$$ −692115. −0.369124
$$324$$ 0 0
$$325$$ −184936. −0.0971209
$$326$$ 0 0
$$327$$ −1.98837e6 −1.02832
$$328$$ 0 0
$$329$$ 209885. 0.106903
$$330$$ 0 0
$$331$$ 646617. 0.324397 0.162199 0.986758i $$-0.448142\pi$$
0.162199 + 0.986758i $$0.448142\pi$$
$$332$$ 0 0
$$333$$ 991915. 0.490189
$$334$$ 0 0
$$335$$ 4.85433e6 2.36329
$$336$$ 0 0
$$337$$ −1.02782e6 −0.492994 −0.246497 0.969144i $$-0.579280\pi$$
−0.246497 + 0.969144i $$0.579280\pi$$
$$338$$ 0 0
$$339$$ −266608. −0.126001
$$340$$ 0 0
$$341$$ 1.77445e6 0.826375
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ 2.27169e6 1.02755
$$346$$ 0 0
$$347$$ −1.64916e6 −0.735256 −0.367628 0.929973i $$-0.619830\pi$$
−0.367628 + 0.929973i $$0.619830\pi$$
$$348$$ 0 0
$$349$$ 2.21201e6 0.972128 0.486064 0.873923i $$-0.338432\pi$$
0.486064 + 0.873923i $$0.338432\pi$$
$$350$$ 0 0
$$351$$ 46477.0 0.0201358
$$352$$ 0 0
$$353$$ 1.10158e6 0.470520 0.235260 0.971932i $$-0.424406\pi$$
0.235260 + 0.971932i $$0.424406\pi$$
$$354$$ 0 0
$$355$$ −5.61242e6 −2.36363
$$356$$ 0 0
$$357$$ 457593. 0.190024
$$358$$ 0 0
$$359$$ −1.29940e6 −0.532116 −0.266058 0.963957i $$-0.585721\pi$$
−0.266058 + 0.963957i $$0.585721\pi$$
$$360$$ 0 0
$$361$$ −2.03119e6 −0.820317
$$362$$ 0 0
$$363$$ −601567. −0.239617
$$364$$ 0 0
$$365$$ 4.32196e6 1.69804
$$366$$ 0 0
$$367$$ 1.55669e6 0.603305 0.301652 0.953418i $$-0.402462\pi$$
0.301652 + 0.953418i $$0.402462\pi$$
$$368$$ 0 0
$$369$$ −148202. −0.0566614
$$370$$ 0 0
$$371$$ −1.26020e6 −0.475341
$$372$$ 0 0
$$373$$ −3.12660e6 −1.16359 −0.581796 0.813335i $$-0.697650\pi$$
−0.581796 + 0.813335i $$0.697650\pi$$
$$374$$ 0 0
$$375$$ −156665. −0.0575299
$$376$$ 0 0
$$377$$ −146697. −0.0531579
$$378$$ 0 0
$$379$$ −2.96497e6 −1.06029 −0.530143 0.847908i $$-0.677862\pi$$
−0.530143 + 0.847908i $$0.677862\pi$$
$$380$$ 0 0
$$381$$ 2.37719e6 0.838978
$$382$$ 0 0
$$383$$ 2.74774e6 0.957149 0.478574 0.878047i $$-0.341154\pi$$
0.478574 + 0.878047i $$0.341154\pi$$
$$384$$ 0 0
$$385$$ 1.81579e6 0.624330
$$386$$ 0 0
$$387$$ −1.68433e6 −0.571675
$$388$$ 0 0
$$389$$ −611343. −0.204838 −0.102419 0.994741i $$-0.532658\pi$$
−0.102419 + 0.994741i $$0.532658\pi$$
$$390$$ 0 0
$$391$$ 3.37397e6 1.11609
$$392$$ 0 0
$$393$$ 700651. 0.228834
$$394$$ 0 0
$$395$$ 309664. 0.0998615
$$396$$ 0 0
$$397$$ 2.16794e6 0.690352 0.345176 0.938538i $$-0.387819\pi$$
0.345176 + 0.938538i $$0.387819\pi$$
$$398$$ 0 0
$$399$$ −294155. −0.0925004
$$400$$ 0 0
$$401$$ −3.14644e6 −0.977143 −0.488572 0.872524i $$-0.662482\pi$$
−0.488572 + 0.872524i $$0.662482\pi$$
$$402$$ 0 0
$$403$$ −236978. −0.0726852
$$404$$ 0 0
$$405$$ −509302. −0.154290
$$406$$ 0 0
$$407$$ 5.84593e6 1.74931
$$408$$ 0 0
$$409$$ 5.58165e6 1.64989 0.824943 0.565215i $$-0.191207\pi$$
0.824943 + 0.565215i $$0.191207\pi$$
$$410$$ 0 0
$$411$$ −3.35162e6 −0.978703
$$412$$ 0 0
$$413$$ 139097. 0.0401275
$$414$$ 0 0
$$415$$ 3.57794e6 1.01980
$$416$$ 0 0
$$417$$ 2.47095e6 0.695864
$$418$$ 0 0
$$419$$ 2.25054e6 0.626257 0.313128 0.949711i $$-0.398623\pi$$
0.313128 + 0.949711i $$0.398623\pi$$
$$420$$ 0 0
$$421$$ 3.45914e6 0.951180 0.475590 0.879667i $$-0.342235\pi$$
0.475590 + 0.879667i $$0.342235\pi$$
$$422$$ 0 0
$$423$$ −346953. −0.0942800
$$424$$ 0 0
$$425$$ 3.00990e6 0.808313
$$426$$ 0 0
$$427$$ −823356. −0.218534
$$428$$ 0 0
$$429$$ 273916. 0.0718578
$$430$$ 0 0
$$431$$ −6.55926e6 −1.70083 −0.850417 0.526109i $$-0.823650\pi$$
−0.850417 + 0.526109i $$0.823650\pi$$
$$432$$ 0 0
$$433$$ 5.05669e6 1.29612 0.648062 0.761587i $$-0.275580\pi$$
0.648062 + 0.761587i $$0.275580\pi$$
$$434$$ 0 0
$$435$$ 1.60753e6 0.407320
$$436$$ 0 0
$$437$$ −2.16889e6 −0.543293
$$438$$ 0 0
$$439$$ 4.23225e6 1.04812 0.524059 0.851682i $$-0.324417\pi$$
0.524059 + 0.851682i $$0.324417\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 0 0
$$443$$ −6.05525e6 −1.46596 −0.732981 0.680249i $$-0.761872\pi$$
−0.732981 + 0.680249i $$0.761872\pi$$
$$444$$ 0 0
$$445$$ −1.05093e7 −2.51579
$$446$$ 0 0
$$447$$ 2.81684e6 0.666796
$$448$$ 0 0
$$449$$ −299186. −0.0700368 −0.0350184 0.999387i $$-0.511149\pi$$
−0.0350184 + 0.999387i $$0.511149\pi$$
$$450$$ 0 0
$$451$$ −873438. −0.202204
$$452$$ 0 0
$$453$$ −3.88885e6 −0.890381
$$454$$ 0 0
$$455$$ −242500. −0.0549140
$$456$$ 0 0
$$457$$ 3.75177e6 0.840322 0.420161 0.907450i $$-0.361974\pi$$
0.420161 + 0.907450i $$0.361974\pi$$
$$458$$ 0 0
$$459$$ −756429. −0.167586
$$460$$ 0 0
$$461$$ 6.94525e6 1.52207 0.761036 0.648709i $$-0.224691\pi$$
0.761036 + 0.648709i $$0.224691\pi$$
$$462$$ 0 0
$$463$$ −9.13226e6 −1.97982 −0.989910 0.141697i $$-0.954744\pi$$
−0.989910 + 0.141697i $$0.954744\pi$$
$$464$$ 0 0
$$465$$ 2.59685e6 0.556948
$$466$$ 0 0
$$467$$ −424136. −0.0899940 −0.0449970 0.998987i $$-0.514328\pi$$
−0.0449970 + 0.998987i $$0.514328\pi$$
$$468$$ 0 0
$$469$$ 3.06422e6 0.643262
$$470$$ 0 0
$$471$$ 761433. 0.158154
$$472$$ 0 0
$$473$$ −9.92673e6 −2.04011
$$474$$ 0 0
$$475$$ −1.93485e6 −0.393472
$$476$$ 0 0
$$477$$ 2.08319e6 0.419212
$$478$$ 0 0
$$479$$ −7.78605e6 −1.55052 −0.775262 0.631640i $$-0.782382\pi$$
−0.775262 + 0.631640i $$0.782382\pi$$
$$480$$ 0 0
$$481$$ −780727. −0.153864
$$482$$ 0 0
$$483$$ 1.43397e6 0.279686
$$484$$ 0 0
$$485$$ −1.10910e7 −2.14100
$$486$$ 0 0
$$487$$ 2.32259e6 0.443763 0.221881 0.975074i $$-0.428780\pi$$
0.221881 + 0.975074i $$0.428780\pi$$
$$488$$ 0 0
$$489$$ −2.75673e6 −0.521341
$$490$$ 0 0
$$491$$ 6.01036e6 1.12512 0.562558 0.826758i $$-0.309817\pi$$
0.562558 + 0.826758i $$0.309817\pi$$
$$492$$ 0 0
$$493$$ 2.38755e6 0.442420
$$494$$ 0 0
$$495$$ −3.00162e6 −0.550607
$$496$$ 0 0
$$497$$ −3.54275e6 −0.643353
$$498$$ 0 0
$$499$$ 3.37385e6 0.606562 0.303281 0.952901i $$-0.401918\pi$$
0.303281 + 0.952901i $$0.401918\pi$$
$$500$$ 0 0
$$501$$ 5.45862e6 0.971603
$$502$$ 0 0
$$503$$ −1.22068e6 −0.215120 −0.107560 0.994199i $$-0.534304\pi$$
−0.107560 + 0.994199i $$0.534304\pi$$
$$504$$ 0 0
$$505$$ −3.45179e6 −0.602304
$$506$$ 0 0
$$507$$ 3.30506e6 0.571030
$$508$$ 0 0
$$509$$ 1.56897e6 0.268423 0.134212 0.990953i $$-0.457150\pi$$
0.134212 + 0.990953i $$0.457150\pi$$
$$510$$ 0 0
$$511$$ 2.72817e6 0.462188
$$512$$ 0 0
$$513$$ 486256. 0.0815777
$$514$$ 0 0
$$515$$ −1.57285e7 −2.61318
$$516$$ 0 0
$$517$$ −2.04480e6 −0.336452
$$518$$ 0 0
$$519$$ 2.59633e6 0.423098
$$520$$ 0 0
$$521$$ 1.06779e7 1.72342 0.861708 0.507404i $$-0.169395\pi$$
0.861708 + 0.507404i $$0.169395\pi$$
$$522$$ 0 0
$$523$$ −1.21007e7 −1.93444 −0.967219 0.253943i $$-0.918272\pi$$
−0.967219 + 0.253943i $$0.918272\pi$$
$$524$$ 0 0
$$525$$ 1.27923e6 0.202559
$$526$$ 0 0
$$527$$ 3.85691e6 0.604941
$$528$$ 0 0
$$529$$ 4.13673e6 0.642714
$$530$$ 0 0
$$531$$ −229936. −0.0353892
$$532$$ 0 0
$$533$$ 116648. 0.0177852
$$534$$ 0 0
$$535$$ −7.72577e6 −1.16696
$$536$$ 0 0
$$537$$ −1.33972e6 −0.200484
$$538$$ 0 0
$$539$$ 1.14619e6 0.169936
$$540$$ 0 0
$$541$$ −3.19805e6 −0.469778 −0.234889 0.972022i $$-0.575473\pi$$
−0.234889 + 0.972022i $$0.575473\pi$$
$$542$$ 0 0
$$543$$ −840400. −0.122317
$$544$$ 0 0
$$545$$ −1.71499e7 −2.47326
$$546$$ 0 0
$$547$$ 3.97811e6 0.568471 0.284235 0.958755i $$-0.408260\pi$$
0.284235 + 0.958755i $$0.408260\pi$$
$$548$$ 0 0
$$549$$ 1.36106e6 0.192729
$$550$$ 0 0
$$551$$ −1.53479e6 −0.215362
$$552$$ 0 0
$$553$$ 195470. 0.0271812
$$554$$ 0 0
$$555$$ 8.55534e6 1.17898
$$556$$ 0 0
$$557$$ 1.14113e7 1.55847 0.779236 0.626731i $$-0.215608\pi$$
0.779236 + 0.626731i $$0.215608\pi$$
$$558$$ 0 0
$$559$$ 1.32572e6 0.179441
$$560$$ 0 0
$$561$$ −4.45808e6 −0.598054
$$562$$ 0 0
$$563$$ −6.46973e6 −0.860231 −0.430115 0.902774i $$-0.641527\pi$$
−0.430115 + 0.902774i $$0.641527\pi$$
$$564$$ 0 0
$$565$$ −2.29952e6 −0.303051
$$566$$ 0 0
$$567$$ −321489. −0.0419961
$$568$$ 0 0
$$569$$ 717028. 0.0928444 0.0464222 0.998922i $$-0.485218\pi$$
0.0464222 + 0.998922i $$0.485218\pi$$
$$570$$ 0 0
$$571$$ 6.00286e6 0.770491 0.385246 0.922814i $$-0.374117\pi$$
0.385246 + 0.922814i $$0.374117\pi$$
$$572$$ 0 0
$$573$$ −2.22223e6 −0.282750
$$574$$ 0 0
$$575$$ 9.43217e6 1.18971
$$576$$ 0 0
$$577$$ 1.55712e7 1.94707 0.973537 0.228531i $$-0.0733922\pi$$
0.973537 + 0.228531i $$0.0733922\pi$$
$$578$$ 0 0
$$579$$ −4.33293e6 −0.537137
$$580$$ 0 0
$$581$$ 2.25852e6 0.277577
$$582$$ 0 0
$$583$$ 1.22775e7 1.49602
$$584$$ 0 0
$$585$$ 400868. 0.0484296
$$586$$ 0 0
$$587$$ 7.84621e6 0.939863 0.469931 0.882703i $$-0.344279\pi$$
0.469931 + 0.882703i $$0.344279\pi$$
$$588$$ 0 0
$$589$$ −2.47934e6 −0.294475
$$590$$ 0 0
$$591$$ 4.93413e6 0.581088
$$592$$ 0 0
$$593$$ 1.33248e7 1.55605 0.778026 0.628232i $$-0.216221\pi$$
0.778026 + 0.628232i $$0.216221\pi$$
$$594$$ 0 0
$$595$$ 3.94678e6 0.457036
$$596$$ 0 0
$$597$$ −1.42361e6 −0.163477
$$598$$ 0 0
$$599$$ −7.77916e6 −0.885861 −0.442930 0.896556i $$-0.646061\pi$$
−0.442930 + 0.896556i $$0.646061\pi$$
$$600$$ 0 0
$$601$$ 8.62898e6 0.974480 0.487240 0.873268i $$-0.338004\pi$$
0.487240 + 0.873268i $$0.338004\pi$$
$$602$$ 0 0
$$603$$ −5.06534e6 −0.567304
$$604$$ 0 0
$$605$$ −5.18857e6 −0.576314
$$606$$ 0 0
$$607$$ −7.09353e6 −0.781431 −0.390715 0.920512i $$-0.627772\pi$$
−0.390715 + 0.920512i $$0.627772\pi$$
$$608$$ 0 0
$$609$$ 1.01473e6 0.110868
$$610$$ 0 0
$$611$$ 273084. 0.0295932
$$612$$ 0 0
$$613$$ 4.52640e6 0.486521 0.243261 0.969961i $$-0.421783\pi$$
0.243261 + 0.969961i $$0.421783\pi$$
$$614$$ 0 0
$$615$$ −1.27825e6 −0.136279
$$616$$ 0 0
$$617$$ −8.38009e6 −0.886208 −0.443104 0.896470i $$-0.646123\pi$$
−0.443104 + 0.896470i $$0.646123\pi$$
$$618$$ 0 0
$$619$$ −168342. −0.0176590 −0.00882949 0.999961i $$-0.502811\pi$$
−0.00882949 + 0.999961i $$0.502811\pi$$
$$620$$ 0 0
$$621$$ −2.37044e6 −0.246660
$$622$$ 0 0
$$623$$ −6.63385e6 −0.684771
$$624$$ 0 0
$$625$$ −1.04161e7 −1.06661
$$626$$ 0 0
$$627$$ 2.86579e6 0.291122
$$628$$ 0 0
$$629$$ 1.27066e7 1.28057
$$630$$ 0 0
$$631$$ 1.66208e7 1.66180 0.830899 0.556423i $$-0.187826\pi$$
0.830899 + 0.556423i $$0.187826\pi$$
$$632$$ 0 0
$$633$$ −2.55159e6 −0.253106
$$634$$ 0 0
$$635$$ 2.05034e7 2.01786
$$636$$ 0 0
$$637$$ −153074. −0.0149470
$$638$$ 0 0
$$639$$ 5.85638e6 0.567384
$$640$$ 0 0
$$641$$ 1.02473e6 0.0985068 0.0492534 0.998786i $$-0.484316\pi$$
0.0492534 + 0.998786i $$0.484316\pi$$
$$642$$ 0 0
$$643$$ −1.22962e7 −1.17286 −0.586428 0.810001i $$-0.699466\pi$$
−0.586428 + 0.810001i $$0.699466\pi$$
$$644$$ 0 0
$$645$$ −1.45275e7 −1.37496
$$646$$ 0 0
$$647$$ −2.16537e6 −0.203363 −0.101682 0.994817i $$-0.532422\pi$$
−0.101682 + 0.994817i $$0.532422\pi$$
$$648$$ 0 0
$$649$$ −1.35515e6 −0.126292
$$650$$ 0 0
$$651$$ 1.63922e6 0.151595
$$652$$ 0 0
$$653$$ −1.22501e7 −1.12424 −0.562119 0.827056i $$-0.690014\pi$$
−0.562119 + 0.827056i $$0.690014\pi$$
$$654$$ 0 0
$$655$$ 6.04317e6 0.550379
$$656$$ 0 0
$$657$$ −4.50983e6 −0.407612
$$658$$ 0 0
$$659$$ 1.18607e6 0.106389 0.0531944 0.998584i $$-0.483060\pi$$
0.0531944 + 0.998584i $$0.483060\pi$$
$$660$$ 0 0
$$661$$ −1.45382e7 −1.29421 −0.647107 0.762399i $$-0.724021\pi$$
−0.647107 + 0.762399i $$0.724021\pi$$
$$662$$ 0 0
$$663$$ 595379. 0.0526029
$$664$$ 0 0
$$665$$ −2.53711e6 −0.222477
$$666$$ 0 0
$$667$$ 7.48190e6 0.651174
$$668$$ 0 0
$$669$$ 5.86022e6 0.506231
$$670$$ 0 0
$$671$$ 8.02151e6 0.687781
$$672$$ 0 0
$$673$$ −5.99405e6 −0.510132 −0.255066 0.966924i $$-0.582097\pi$$
−0.255066 + 0.966924i $$0.582097\pi$$
$$674$$ 0 0
$$675$$ −2.11465e6 −0.178640
$$676$$ 0 0
$$677$$ 2.08389e7 1.74744 0.873721 0.486428i $$-0.161700\pi$$
0.873721 + 0.486428i $$0.161700\pi$$
$$678$$ 0 0
$$679$$ −7.00102e6 −0.582756
$$680$$ 0 0
$$681$$ 3.40464e6 0.281322
$$682$$ 0 0
$$683$$ 4.55565e6 0.373679 0.186840 0.982390i $$-0.440176\pi$$
0.186840 + 0.982390i $$0.440176\pi$$
$$684$$ 0 0
$$685$$ −2.89080e7 −2.35392
$$686$$ 0 0
$$687$$ 200995. 0.0162478
$$688$$ 0 0
$$689$$ −1.63966e6 −0.131585
$$690$$ 0 0
$$691$$ 1.68542e6 0.134281 0.0671404 0.997744i $$-0.478612\pi$$
0.0671404 + 0.997744i $$0.478612\pi$$
$$692$$ 0 0
$$693$$ −1.89472e6 −0.149869
$$694$$ 0 0
$$695$$ 2.13122e7 1.67365
$$696$$ 0 0
$$697$$ −1.89849e6 −0.148022
$$698$$ 0 0
$$699$$ 8.18046e6 0.633264
$$700$$ 0 0
$$701$$ 2.45349e6 0.188577 0.0942887 0.995545i $$-0.469942\pi$$
0.0942887 + 0.995545i $$0.469942\pi$$
$$702$$ 0 0
$$703$$ −8.16820e6 −0.623359
$$704$$ 0 0
$$705$$ −2.99250e6 −0.226757
$$706$$ 0 0
$$707$$ −2.17889e6 −0.163940
$$708$$ 0 0
$$709$$ −1.32314e7 −0.988530 −0.494265 0.869311i $$-0.664563\pi$$
−0.494265 + 0.869311i $$0.664563\pi$$
$$710$$ 0 0
$$711$$ −323125. −0.0239715
$$712$$ 0 0
$$713$$ 1.20865e7 0.890380
$$714$$ 0 0
$$715$$ 2.36255e6 0.172828
$$716$$ 0 0
$$717$$ −9.48267e6 −0.688863
$$718$$ 0 0
$$719$$ 1.35337e7 0.976324 0.488162 0.872753i $$-0.337667\pi$$
0.488162 + 0.872753i $$0.337667\pi$$
$$720$$ 0 0
$$721$$ −9.92835e6 −0.711277
$$722$$ 0 0
$$723$$ −9.47094e6 −0.673826
$$724$$ 0 0
$$725$$ 6.67455e6 0.471604
$$726$$ 0 0
$$727$$ 5.29416e6 0.371502 0.185751 0.982597i $$-0.440528\pi$$
0.185751 + 0.982597i $$0.440528\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −2.15766e7 −1.49344
$$732$$ 0 0
$$733$$ −2.25014e7 −1.54685 −0.773426 0.633886i $$-0.781459\pi$$
−0.773426 + 0.633886i $$0.781459\pi$$
$$734$$ 0 0
$$735$$ 1.67741e6 0.114531
$$736$$ 0 0
$$737$$ −2.98530e7 −2.02451
$$738$$ 0 0
$$739$$ 1.60739e7 1.08271 0.541353 0.840795i $$-0.317912\pi$$
0.541353 + 0.840795i $$0.317912\pi$$
$$740$$ 0 0
$$741$$ −382728. −0.0256062
$$742$$ 0 0
$$743$$ −2.31604e6 −0.153913 −0.0769563 0.997034i $$-0.524520\pi$$
−0.0769563 + 0.997034i $$0.524520\pi$$
$$744$$ 0 0
$$745$$ 2.42955e7 1.60374
$$746$$ 0 0
$$747$$ −3.73347e6 −0.244800
$$748$$ 0 0
$$749$$ −4.87677e6 −0.317634
$$750$$ 0 0
$$751$$ −7.77263e6 −0.502885 −0.251442 0.967872i $$-0.580905\pi$$
−0.251442 + 0.967872i $$0.580905\pi$$
$$752$$ 0 0
$$753$$ −8.75588e6 −0.562746
$$754$$ 0 0
$$755$$ −3.35417e7 −2.14150
$$756$$ 0 0
$$757$$ −1.59716e7 −1.01300 −0.506498 0.862241i $$-0.669060\pi$$
−0.506498 + 0.862241i $$0.669060\pi$$
$$758$$ 0 0
$$759$$ −1.39704e7 −0.880244
$$760$$ 0 0
$$761$$ 1.84940e7 1.15763 0.578815 0.815459i $$-0.303515\pi$$
0.578815 + 0.815459i $$0.303515\pi$$
$$762$$ 0 0
$$763$$ −1.08256e7 −0.673193
$$764$$ 0 0
$$765$$ −6.52426e6 −0.403068
$$766$$ 0 0
$$767$$ 180980. 0.0111082
$$768$$ 0 0
$$769$$ −2.33524e7 −1.42402 −0.712009 0.702170i $$-0.752214\pi$$
−0.712009 + 0.702170i $$0.752214\pi$$
$$770$$ 0 0
$$771$$ −1.60153e7 −0.970283
$$772$$ 0 0
$$773$$ 7.25262e6 0.436562 0.218281 0.975886i $$-0.429955\pi$$
0.218281 + 0.975886i $$0.429955\pi$$
$$774$$ 0 0
$$775$$ 1.07823e7 0.644845
$$776$$ 0 0
$$777$$ 5.40042e6 0.320904
$$778$$ 0 0
$$779$$ 1.22041e6 0.0720546
$$780$$ 0 0
$$781$$ 3.45151e7 2.02480
$$782$$ 0 0
$$783$$ −1.67741e6 −0.0977764
$$784$$ 0 0
$$785$$ 6.56742e6 0.380383
$$786$$ 0 0
$$787$$ 1.18935e6 0.0684497 0.0342248 0.999414i $$-0.489104\pi$$
0.0342248 + 0.999414i $$0.489104\pi$$
$$788$$ 0 0
$$789$$ −197637. −0.0113025
$$790$$ 0 0
$$791$$ −1.45153e6 −0.0824871
$$792$$ 0 0
$$793$$ −1.07128e6 −0.0604949
$$794$$ 0 0
$$795$$ 1.79677e7 1.00827
$$796$$ 0 0
$$797$$ 2.14041e7 1.19358 0.596790 0.802397i $$-0.296442\pi$$
0.596790 + 0.802397i $$0.296442\pi$$
$$798$$ 0 0
$$799$$ −4.44453e6 −0.246297
$$800$$ 0 0
$$801$$ 1.09662e7 0.603911
$$802$$ 0 0
$$803$$ −2.65790e7 −1.45462
$$804$$ 0 0
$$805$$ 1.23681e7 0.672686
$$806$$ 0 0
$$807$$ 1.52200e7 0.822679
$$808$$ 0 0
$$809$$ 9.60341e6 0.515886 0.257943 0.966160i $$-0.416955\pi$$
0.257943 + 0.966160i $$0.416955\pi$$
$$810$$ 0 0
$$811$$ 2.62263e7 1.40018 0.700091 0.714054i $$-0.253143\pi$$
0.700091 + 0.714054i $$0.253143\pi$$
$$812$$ 0 0
$$813$$ 4.21077e6 0.223427
$$814$$ 0 0
$$815$$ −2.37770e7 −1.25390
$$816$$ 0 0
$$817$$ 1.38701e7 0.726982
$$818$$ 0 0
$$819$$ 253041. 0.0131820
$$820$$ 0 0
$$821$$ 3.65685e7 1.89343 0.946715 0.322073i $$-0.104380\pi$$
0.946715 + 0.322073i $$0.104380\pi$$
$$822$$ 0 0
$$823$$ −7.73488e6 −0.398065 −0.199032 0.979993i $$-0.563780\pi$$
−0.199032 + 0.979993i $$0.563780\pi$$
$$824$$ 0 0
$$825$$ −1.24629e7 −0.637504
$$826$$ 0 0
$$827$$ −1.47172e7 −0.748277 −0.374138 0.927373i $$-0.622061\pi$$
−0.374138 + 0.927373i $$0.622061\pi$$
$$828$$ 0 0
$$829$$ −7.41886e6 −0.374931 −0.187465 0.982271i $$-0.560027\pi$$
−0.187465 + 0.982271i $$0.560027\pi$$
$$830$$ 0 0
$$831$$ −8.15552e6 −0.409684
$$832$$ 0 0
$$833$$ 2.49134e6 0.124400
$$834$$ 0 0
$$835$$ 4.70811e7 2.33685
$$836$$ 0 0
$$837$$ −2.70973e6 −0.133694
$$838$$ 0 0
$$839$$ 116538. 0.00571559 0.00285780 0.999996i $$-0.499090\pi$$
0.00285780 + 0.999996i $$0.499090\pi$$
$$840$$ 0 0
$$841$$ −1.52167e7 −0.741874
$$842$$ 0 0
$$843$$ 7.03249e6 0.340832
$$844$$ 0 0
$$845$$ 2.85064e7 1.37341
$$846$$ 0 0
$$847$$ −3.27520e6 −0.156866
$$848$$ 0 0
$$849$$ 1.34859e7 0.642112
$$850$$ 0 0
$$851$$ 3.98190e7 1.88480
$$852$$ 0 0
$$853$$ −1.91763e7 −0.902387 −0.451193 0.892426i $$-0.649002\pi$$
−0.451193 + 0.892426i $$0.649002\pi$$
$$854$$ 0 0
$$855$$ 4.19400e6 0.196206
$$856$$ 0 0
$$857$$ −1.62548e6 −0.0756012 −0.0378006 0.999285i $$-0.512035\pi$$
−0.0378006 + 0.999285i $$0.512035\pi$$
$$858$$ 0 0
$$859$$ −1.65931e7 −0.767265 −0.383632 0.923486i $$-0.625327\pi$$
−0.383632 + 0.923486i $$0.625327\pi$$
$$860$$ 0 0
$$861$$ −806875. −0.0370936
$$862$$ 0 0
$$863$$ −2.30415e7 −1.05314 −0.526568 0.850133i $$-0.676521\pi$$
−0.526568 + 0.850133i $$0.676521\pi$$
$$864$$ 0 0
$$865$$ 2.23935e7 1.01761
$$866$$ 0 0
$$867$$ 3.08871e6 0.139550
$$868$$ 0 0
$$869$$ −1.90436e6 −0.0855460
$$870$$ 0 0
$$871$$ 3.98689e6 0.178069
$$872$$ 0 0
$$873$$ 1.15731e7 0.513943
$$874$$ 0 0
$$875$$ −852954. −0.0376622
$$876$$ 0 0
$$877$$ 1.84493e7 0.809993 0.404996 0.914318i $$-0.367273\pi$$
0.404996 + 0.914318i $$0.367273\pi$$
$$878$$ 0 0
$$879$$ 1.40463e7 0.613182
$$880$$ 0 0
$$881$$ −3.70548e7 −1.60844 −0.804219 0.594333i $$-0.797416\pi$$
−0.804219 + 0.594333i $$0.797416\pi$$
$$882$$ 0 0
$$883$$ −5.28466e6 −0.228095 −0.114047 0.993475i $$-0.536382\pi$$
−0.114047 + 0.993475i $$0.536382\pi$$
$$884$$ 0 0
$$885$$ −1.98321e6 −0.0851161
$$886$$ 0 0
$$887$$ −1.98545e7 −0.847326 −0.423663 0.905820i $$-0.639256\pi$$
−0.423663 + 0.905820i $$0.639256\pi$$
$$888$$ 0 0
$$889$$ 1.29425e7 0.549240
$$890$$ 0 0
$$891$$ 3.13209e6 0.132172
$$892$$ 0 0
$$893$$ 2.85708e6 0.119893
$$894$$ 0 0
$$895$$ −1.15552e7 −0.482193
$$896$$ 0 0
$$897$$ 1.86575e6 0.0774234
$$898$$ 0 0
$$899$$ 8.55283e6 0.352948
$$900$$ 0 0
$$901$$ 2.66861e7 1.09515
$$902$$ 0 0
$$903$$ −9.17023e6 −0.374249
$$904$$ 0 0
$$905$$ −7.24852e6 −0.294190
$$906$$ 0 0
$$907$$ −2.00483e7 −0.809207 −0.404603 0.914492i $$-0.632590\pi$$
−0.404603 + 0.914492i $$0.632590\pi$$
$$908$$ 0 0
$$909$$ 3.60183e6 0.144582
$$910$$ 0 0
$$911$$ −765753. −0.0305698 −0.0152849 0.999883i $$-0.504866\pi$$
−0.0152849 + 0.999883i $$0.504866\pi$$
$$912$$ 0 0
$$913$$ −2.20035e7 −0.873605
$$914$$ 0 0
$$915$$ 1.17392e7 0.463540
$$916$$ 0 0
$$917$$ 3.81465e6 0.149807
$$918$$ 0 0
$$919$$ −1.87846e7 −0.733692 −0.366846 0.930282i $$-0.619562\pi$$
−0.366846 + 0.930282i $$0.619562\pi$$
$$920$$ 0 0
$$921$$ 8.00377e6 0.310918
$$922$$ 0 0
$$923$$ −4.60951e6 −0.178094
$$924$$ 0 0
$$925$$ 3.55222e7 1.36504
$$926$$ 0 0
$$927$$ 1.64122e7 0.627288
$$928$$ 0 0
$$929$$ −4.56290e7 −1.73461 −0.867304 0.497778i $$-0.834149\pi$$
−0.867304 + 0.497778i $$0.834149\pi$$
$$930$$ 0 0
$$931$$ −1.60151e6 −0.0605557
$$932$$ 0 0
$$933$$ −2.20344e7 −0.828698
$$934$$ 0 0
$$935$$ −3.84513e7 −1.43841
$$936$$ 0 0
$$937$$ 6.67800e6 0.248484 0.124242 0.992252i $$-0.460350\pi$$
0.124242 + 0.992252i $$0.460350\pi$$
$$938$$ 0 0
$$939$$ 2.45792e7 0.909713
$$940$$ 0 0
$$941$$ 3.42716e7 1.26171 0.630857 0.775899i $$-0.282703\pi$$
0.630857 + 0.775899i $$0.282703\pi$$
$$942$$ 0 0
$$943$$ −5.94933e6 −0.217866
$$944$$ 0 0
$$945$$ −2.77287e6 −0.101007
$$946$$ 0 0
$$947$$ 5.28766e7 1.91597 0.957984 0.286821i $$-0.0925984\pi$$
0.957984 + 0.286821i $$0.0925984\pi$$
$$948$$ 0 0
$$949$$ 3.54965e6 0.127944
$$950$$ 0 0
$$951$$ 2.24643e6 0.0805455
$$952$$ 0 0
$$953$$ 5.33439e6 0.190262 0.0951311 0.995465i $$-0.469673\pi$$
0.0951311 + 0.995465i $$0.469673\pi$$
$$954$$ 0 0
$$955$$ −1.91669e7 −0.680056
$$956$$ 0 0
$$957$$ −9.88594e6 −0.348930
$$958$$ 0 0
$$959$$ −1.82477e7 −0.640711
$$960$$ 0 0
$$961$$ −1.48127e7 −0.517398
$$962$$ 0 0
$$963$$ 8.06160e6 0.280127
$$964$$ 0 0
$$965$$ −3.73719e7 −1.29189
$$966$$ 0 0
$$967$$ −1.93877e7 −0.666744 −0.333372 0.942795i $$-0.608187\pi$$
−0.333372 + 0.942795i $$0.608187\pi$$
$$968$$ 0 0
$$969$$ 6.22903e6 0.213114
$$970$$ 0 0
$$971$$ 1.07463e7 0.365772 0.182886 0.983134i $$-0.441456\pi$$
0.182886 + 0.983134i $$0.441456\pi$$
$$972$$ 0 0
$$973$$ 1.34530e7 0.455550
$$974$$ 0 0
$$975$$ 1.66442e6 0.0560728
$$976$$ 0 0
$$977$$ −2.41051e7 −0.807926 −0.403963 0.914775i $$-0.632368\pi$$
−0.403963 + 0.914775i $$0.632368\pi$$
$$978$$ 0 0
$$979$$ 6.46300e7 2.15515
$$980$$ 0 0
$$981$$ 1.78953e7 0.593701
$$982$$ 0 0
$$983$$ −4.05038e7 −1.33694 −0.668470 0.743739i $$-0.733050\pi$$
−0.668470 + 0.743739i $$0.733050\pi$$
$$984$$ 0 0
$$985$$ 4.25573e7 1.39760
$$986$$ 0 0
$$987$$ −1.88897e6 −0.0617207
$$988$$ 0 0
$$989$$ −6.76149e7 −2.19812
$$990$$ 0 0
$$991$$ −4.01588e7 −1.29896 −0.649482 0.760377i $$-0.725014\pi$$
−0.649482 + 0.760377i $$0.725014\pi$$
$$992$$ 0 0
$$993$$ −5.81955e6 −0.187291
$$994$$ 0 0
$$995$$ −1.22788e7 −0.393185
$$996$$ 0 0
$$997$$ −2.72276e7 −0.867505 −0.433753 0.901032i $$-0.642811\pi$$
−0.433753 + 0.901032i $$0.642811\pi$$
$$998$$ 0 0
$$999$$ −8.92723e6 −0.283011
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 84.6.a.c.1.1 2
3.2 odd 2 252.6.a.h.1.2 2
4.3 odd 2 336.6.a.x.1.1 2
7.2 even 3 588.6.i.l.361.2 4
7.3 odd 6 588.6.i.i.373.1 4
7.4 even 3 588.6.i.l.373.2 4
7.5 odd 6 588.6.i.i.361.1 4
7.6 odd 2 588.6.a.k.1.2 2
12.11 even 2 1008.6.a.bo.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.c.1.1 2 1.1 even 1 trivial
252.6.a.h.1.2 2 3.2 odd 2
336.6.a.x.1.1 2 4.3 odd 2
588.6.a.k.1.2 2 7.6 odd 2
588.6.i.i.361.1 4 7.5 odd 6
588.6.i.i.373.1 4 7.3 odd 6
588.6.i.l.361.2 4 7.2 even 3
588.6.i.l.373.2 4 7.4 even 3
1008.6.a.bo.1.2 2 12.11 even 2