# Properties

 Label 1008.6.a.bj.1.1 Level $1008$ Weight $6$ Character 1008.1 Self dual yes Analytic conductor $161.667$ 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] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1008.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: $$161.666890371$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{114})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 114$$ x^2 - 114 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 504) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-10.6771$$ of defining polynomial Character $$\chi$$ $$=$$ 1008.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-78.0625 q^{5} +49.0000 q^{7} +O(q^{10})$$ $$q-78.0625 q^{5} +49.0000 q^{7} -746.437 q^{11} +9.75012 q^{13} -1754.69 q^{17} +603.250 q^{19} -3175.31 q^{23} +2968.75 q^{25} -3227.50 q^{29} -7881.75 q^{31} -3825.06 q^{35} +12362.7 q^{37} -3699.56 q^{41} -16915.0 q^{43} +658.130 q^{47} +2401.00 q^{49} -27891.1 q^{53} +58268.7 q^{55} +7456.38 q^{59} -43705.7 q^{61} -761.119 q^{65} +28643.5 q^{67} -9244.93 q^{71} -29816.5 q^{73} -36575.4 q^{77} -32928.5 q^{79} +40524.5 q^{83} +136975. q^{85} -41351.3 q^{89} +477.756 q^{91} -47091.2 q^{95} -110780. q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 28 q^{5} + 98 q^{7}+O(q^{10})$$ 2 * q - 28 * q^5 + 98 * q^7 $$2 q - 28 q^{5} + 98 q^{7} - 596 q^{11} + 532 q^{13} - 2100 q^{17} + 2744 q^{19} - 3660 q^{23} + 2350 q^{25} + 720 q^{29} - 3976 q^{31} - 1372 q^{35} + 6788 q^{37} + 4004 q^{41} - 19480 q^{43} + 21560 q^{47} + 4802 q^{49} - 57576 q^{53} + 65800 q^{55} + 22344 q^{59} - 38724 q^{61} + 25384 q^{65} - 7288 q^{67} + 3932 q^{71} + 19292 q^{73} - 29204 q^{77} - 15632 q^{79} + 22624 q^{83} + 119688 q^{85} - 112812 q^{89} + 26068 q^{91} + 60080 q^{95} - 36036 q^{97}+O(q^{100})$$ 2 * q - 28 * q^5 + 98 * q^7 - 596 * q^11 + 532 * q^13 - 2100 * q^17 + 2744 * q^19 - 3660 * q^23 + 2350 * q^25 + 720 * q^29 - 3976 * q^31 - 1372 * q^35 + 6788 * q^37 + 4004 * q^41 - 19480 * q^43 + 21560 * q^47 + 4802 * q^49 - 57576 * q^53 + 65800 * q^55 + 22344 * q^59 - 38724 * q^61 + 25384 * q^65 - 7288 * q^67 + 3932 * q^71 + 19292 * q^73 - 29204 * q^77 - 15632 * q^79 + 22624 * q^83 + 119688 * q^85 - 112812 * q^89 + 26068 * q^91 + 60080 * q^95 - 36036 * 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
$$5$$ −78.0625 −1.39642 −0.698212 0.715891i $$-0.746021\pi$$
−0.698212 + 0.715891i $$0.746021\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −746.437 −1.85999 −0.929997 0.367567i $$-0.880191\pi$$
−0.929997 + 0.367567i $$0.880191\pi$$
$$12$$ 0 0
$$13$$ 9.75012 0.0160012 0.00800058 0.999968i $$-0.497453\pi$$
0.00800058 + 0.999968i $$0.497453\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1754.69 −1.47257 −0.736287 0.676669i $$-0.763423\pi$$
−0.736287 + 0.676669i $$0.763423\pi$$
$$18$$ 0 0
$$19$$ 603.250 0.383366 0.191683 0.981457i $$-0.438605\pi$$
0.191683 + 0.981457i $$0.438605\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3175.31 −1.25160 −0.625802 0.779982i $$-0.715228\pi$$
−0.625802 + 0.779982i $$0.715228\pi$$
$$24$$ 0 0
$$25$$ 2968.75 0.950000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3227.50 −0.712641 −0.356321 0.934364i $$-0.615969\pi$$
−0.356321 + 0.934364i $$0.615969\pi$$
$$30$$ 0 0
$$31$$ −7881.75 −1.47305 −0.736526 0.676409i $$-0.763535\pi$$
−0.736526 + 0.676409i $$0.763535\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3825.06 −0.527799
$$36$$ 0 0
$$37$$ 12362.7 1.48460 0.742302 0.670065i $$-0.233734\pi$$
0.742302 + 0.670065i $$0.233734\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3699.56 −0.343709 −0.171854 0.985122i $$-0.554976\pi$$
−0.171854 + 0.985122i $$0.554976\pi$$
$$42$$ 0 0
$$43$$ −16915.0 −1.39509 −0.697543 0.716543i $$-0.745723\pi$$
−0.697543 + 0.716543i $$0.745723\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 658.130 0.0434577 0.0217289 0.999764i $$-0.493083\pi$$
0.0217289 + 0.999764i $$0.493083\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −27891.1 −1.36388 −0.681940 0.731408i $$-0.738864\pi$$
−0.681940 + 0.731408i $$0.738864\pi$$
$$54$$ 0 0
$$55$$ 58268.7 2.59734
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 7456.38 0.278867 0.139434 0.990231i $$-0.455472\pi$$
0.139434 + 0.990231i $$0.455472\pi$$
$$60$$ 0 0
$$61$$ −43705.7 −1.50388 −0.751941 0.659230i $$-0.770882\pi$$
−0.751941 + 0.659230i $$0.770882\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −761.119 −0.0223444
$$66$$ 0 0
$$67$$ 28643.5 0.779541 0.389770 0.920912i $$-0.372554\pi$$
0.389770 + 0.920912i $$0.372554\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9244.93 −0.217650 −0.108825 0.994061i $$-0.534709\pi$$
−0.108825 + 0.994061i $$0.534709\pi$$
$$72$$ 0 0
$$73$$ −29816.5 −0.654861 −0.327431 0.944875i $$-0.606183\pi$$
−0.327431 + 0.944875i $$0.606183\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −36575.4 −0.703012
$$78$$ 0 0
$$79$$ −32928.5 −0.593614 −0.296807 0.954938i $$-0.595922\pi$$
−0.296807 + 0.954938i $$0.595922\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 40524.5 0.645687 0.322844 0.946452i $$-0.395361\pi$$
0.322844 + 0.946452i $$0.395361\pi$$
$$84$$ 0 0
$$85$$ 136975. 2.05634
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −41351.3 −0.553368 −0.276684 0.960961i $$-0.589236\pi$$
−0.276684 + 0.960961i $$0.589236\pi$$
$$90$$ 0 0
$$91$$ 477.756 0.00604787
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −47091.2 −0.535341
$$96$$ 0 0
$$97$$ −110780. −1.19546 −0.597728 0.801699i $$-0.703930\pi$$
−0.597728 + 0.801699i $$0.703930\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −80973.4 −0.789840 −0.394920 0.918716i $$-0.629228\pi$$
−0.394920 + 0.918716i $$0.629228\pi$$
$$102$$ 0 0
$$103$$ −104902. −0.974298 −0.487149 0.873319i $$-0.661963\pi$$
−0.487149 + 0.873319i $$0.661963\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 193878. 1.63708 0.818540 0.574450i $$-0.194784\pi$$
0.818540 + 0.574450i $$0.194784\pi$$
$$108$$ 0 0
$$109$$ −154948. −1.24917 −0.624584 0.780957i $$-0.714732\pi$$
−0.624584 + 0.780957i $$0.714732\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −91578.5 −0.674679 −0.337340 0.941383i $$-0.609527\pi$$
−0.337340 + 0.941383i $$0.609527\pi$$
$$114$$ 0 0
$$115$$ 247873. 1.74777
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −85979.7 −0.556581
$$120$$ 0 0
$$121$$ 396118. 2.45958
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 12197.3 0.0698216
$$126$$ 0 0
$$127$$ −110234. −0.606465 −0.303233 0.952917i $$-0.598066\pi$$
−0.303233 + 0.952917i $$0.598066\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 361365. 1.83979 0.919895 0.392166i $$-0.128274\pi$$
0.919895 + 0.392166i $$0.128274\pi$$
$$132$$ 0 0
$$133$$ 29559.3 0.144899
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −293644. −1.33665 −0.668327 0.743867i $$-0.732989\pi$$
−0.668327 + 0.743867i $$0.732989\pi$$
$$138$$ 0 0
$$139$$ −92541.4 −0.406255 −0.203128 0.979152i $$-0.565111\pi$$
−0.203128 + 0.979152i $$0.565111\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −7277.85 −0.0297621
$$144$$ 0 0
$$145$$ 251946. 0.995149
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −13179.0 −0.0486313 −0.0243157 0.999704i $$-0.507741\pi$$
−0.0243157 + 0.999704i $$0.507741\pi$$
$$150$$ 0 0
$$151$$ 304869. 1.08811 0.544053 0.839051i $$-0.316889\pi$$
0.544053 + 0.839051i $$0.316889\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 615269. 2.05701
$$156$$ 0 0
$$157$$ −64050.2 −0.207382 −0.103691 0.994610i $$-0.533065\pi$$
−0.103691 + 0.994610i $$0.533065\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −155590. −0.473062
$$162$$ 0 0
$$163$$ −617219. −1.81958 −0.909788 0.415072i $$-0.863756\pi$$
−0.909788 + 0.415072i $$0.863756\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 494693. 1.37260 0.686301 0.727318i $$-0.259233\pi$$
0.686301 + 0.727318i $$0.259233\pi$$
$$168$$ 0 0
$$169$$ −371198. −0.999744
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 701459. 1.78191 0.890957 0.454088i $$-0.150035\pi$$
0.890957 + 0.454088i $$0.150035\pi$$
$$174$$ 0 0
$$175$$ 145469. 0.359066
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 197449. 0.460598 0.230299 0.973120i $$-0.426029\pi$$
0.230299 + 0.973120i $$0.426029\pi$$
$$180$$ 0 0
$$181$$ 869311. 1.97233 0.986163 0.165777i $$-0.0530133\pi$$
0.986163 + 0.165777i $$0.0530133\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −965066. −2.07314
$$186$$ 0 0
$$187$$ 1.30976e6 2.73898
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 119524. 0.237068 0.118534 0.992950i $$-0.462181\pi$$
0.118534 + 0.992950i $$0.462181\pi$$
$$192$$ 0 0
$$193$$ 170916. 0.330285 0.165143 0.986270i $$-0.447192\pi$$
0.165143 + 0.986270i $$0.447192\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 715090. 1.31279 0.656394 0.754418i $$-0.272081\pi$$
0.656394 + 0.754418i $$0.272081\pi$$
$$198$$ 0 0
$$199$$ 715708. 1.28116 0.640580 0.767892i $$-0.278694\pi$$
0.640580 + 0.767892i $$0.278694\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −158147. −0.269353
$$204$$ 0 0
$$205$$ 288797. 0.479963
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −450289. −0.713059
$$210$$ 0 0
$$211$$ 880061. 1.36084 0.680420 0.732823i $$-0.261798\pi$$
0.680420 + 0.732823i $$0.261798\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.32043e6 1.94813
$$216$$ 0 0
$$217$$ −386206. −0.556762
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −17108.4 −0.0235629
$$222$$ 0 0
$$223$$ −514501. −0.692826 −0.346413 0.938082i $$-0.612600\pi$$
−0.346413 + 0.938082i $$0.612600\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −873242. −1.12479 −0.562393 0.826870i $$-0.690119\pi$$
−0.562393 + 0.826870i $$0.690119\pi$$
$$228$$ 0 0
$$229$$ −1.11239e6 −1.40174 −0.700869 0.713290i $$-0.747204\pi$$
−0.700869 + 0.713290i $$0.747204\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.21568e6 −1.46700 −0.733499 0.679691i $$-0.762114\pi$$
−0.733499 + 0.679691i $$0.762114\pi$$
$$234$$ 0 0
$$235$$ −51375.2 −0.0606854
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −462246. −0.523454 −0.261727 0.965142i $$-0.584292\pi$$
−0.261727 + 0.965142i $$0.584292\pi$$
$$240$$ 0 0
$$241$$ −247817. −0.274845 −0.137423 0.990513i $$-0.543882\pi$$
−0.137423 + 0.990513i $$0.543882\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −187428. −0.199489
$$246$$ 0 0
$$247$$ 5881.76 0.00613430
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.25628e6 −1.25864 −0.629322 0.777145i $$-0.716667\pi$$
−0.629322 + 0.777145i $$0.716667\pi$$
$$252$$ 0 0
$$253$$ 2.37017e6 2.32798
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −330237. −0.311884 −0.155942 0.987766i $$-0.549841\pi$$
−0.155942 + 0.987766i $$0.549841\pi$$
$$258$$ 0 0
$$259$$ 605775. 0.561128
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 187729. 0.167356 0.0836781 0.996493i $$-0.473333\pi$$
0.0836781 + 0.996493i $$0.473333\pi$$
$$264$$ 0 0
$$265$$ 2.17725e6 1.90456
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −1.02334e6 −0.862264 −0.431132 0.902289i $$-0.641886\pi$$
−0.431132 + 0.902289i $$0.641886\pi$$
$$270$$ 0 0
$$271$$ 649145. 0.536931 0.268466 0.963289i $$-0.413483\pi$$
0.268466 + 0.963289i $$0.413483\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.21599e6 −1.76699
$$276$$ 0 0
$$277$$ 1.40211e6 1.09795 0.548973 0.835840i $$-0.315019\pi$$
0.548973 + 0.835840i $$0.315019\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.63255e6 −1.23339 −0.616697 0.787201i $$-0.711530\pi$$
−0.616697 + 0.787201i $$0.711530\pi$$
$$282$$ 0 0
$$283$$ −621056. −0.460962 −0.230481 0.973077i $$-0.574030\pi$$
−0.230481 + 0.973077i $$0.574030\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −181278. −0.129910
$$288$$ 0 0
$$289$$ 1.65907e6 1.16848
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −115908. −0.0788759 −0.0394380 0.999222i $$-0.512557\pi$$
−0.0394380 + 0.999222i $$0.512557\pi$$
$$294$$ 0 0
$$295$$ −582063. −0.389417
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −30959.7 −0.0200271
$$300$$ 0 0
$$301$$ −828835. −0.527293
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 3.41178e6 2.10006
$$306$$ 0 0
$$307$$ −765157. −0.463345 −0.231673 0.972794i $$-0.574420\pi$$
−0.231673 + 0.972794i $$0.574420\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −179636. −0.105315 −0.0526576 0.998613i $$-0.516769\pi$$
−0.0526576 + 0.998613i $$0.516769\pi$$
$$312$$ 0 0
$$313$$ 895716. 0.516785 0.258392 0.966040i $$-0.416807\pi$$
0.258392 + 0.966040i $$0.416807\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −228258. −0.127579 −0.0637893 0.997963i $$-0.520319\pi$$
−0.0637893 + 0.997963i $$0.520319\pi$$
$$318$$ 0 0
$$319$$ 2.40913e6 1.32551
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.05852e6 −0.564535
$$324$$ 0 0
$$325$$ 28945.7 0.0152011
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 32248.4 0.0164255
$$330$$ 0 0
$$331$$ 3.01591e6 1.51303 0.756516 0.653975i $$-0.226900\pi$$
0.756516 + 0.653975i $$0.226900\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −2.23598e6 −1.08857
$$336$$ 0 0
$$337$$ −1.10209e6 −0.528619 −0.264310 0.964438i $$-0.585144\pi$$
−0.264310 + 0.964438i $$0.585144\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 5.88323e6 2.73987
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.90191e6 1.29378 0.646891 0.762583i $$-0.276069\pi$$
0.646891 + 0.762583i $$0.276069\pi$$
$$348$$ 0 0
$$349$$ −2.23892e6 −0.983954 −0.491977 0.870608i $$-0.663726\pi$$
−0.491977 + 0.870608i $$0.663726\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2.05842e6 0.879217 0.439609 0.898189i $$-0.355117\pi$$
0.439609 + 0.898189i $$0.355117\pi$$
$$354$$ 0 0
$$355$$ 721682. 0.303931
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 1.96387e6 0.804222 0.402111 0.915591i $$-0.368277\pi$$
0.402111 + 0.915591i $$0.368277\pi$$
$$360$$ 0 0
$$361$$ −2.11219e6 −0.853031
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.32755e6 0.914464
$$366$$ 0 0
$$367$$ 3.55132e6 1.37634 0.688169 0.725550i $$-0.258415\pi$$
0.688169 + 0.725550i $$0.258415\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.36667e6 −0.515498
$$372$$ 0 0
$$373$$ 4.69562e6 1.74751 0.873757 0.486363i $$-0.161677\pi$$
0.873757 + 0.486363i $$0.161677\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −31468.5 −0.0114031
$$378$$ 0 0
$$379$$ 957580. 0.342434 0.171217 0.985233i $$-0.445230\pi$$
0.171217 + 0.985233i $$0.445230\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 1.79644e6 0.625772 0.312886 0.949791i $$-0.398704\pi$$
0.312886 + 0.949791i $$0.398704\pi$$
$$384$$ 0 0
$$385$$ 2.85517e6 0.981702
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 4.36239e6 1.46167 0.730836 0.682553i $$-0.239130\pi$$
0.730836 + 0.682553i $$0.239130\pi$$
$$390$$ 0 0
$$391$$ 5.57168e6 1.84308
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2.57048e6 0.828937
$$396$$ 0 0
$$397$$ 3.84239e6 1.22356 0.611780 0.791028i $$-0.290454\pi$$
0.611780 + 0.791028i $$0.290454\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.66668e6 1.13871 0.569354 0.822093i $$-0.307194\pi$$
0.569354 + 0.822093i $$0.307194\pi$$
$$402$$ 0 0
$$403$$ −76848.0 −0.0235706
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.22801e6 −2.76135
$$408$$ 0 0
$$409$$ 2.90345e6 0.858236 0.429118 0.903248i $$-0.358824\pi$$
0.429118 + 0.903248i $$0.358824\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 365362. 0.105402
$$414$$ 0 0
$$415$$ −3.16344e6 −0.901653
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −3.94202e6 −1.09694 −0.548470 0.836170i $$-0.684790\pi$$
−0.548470 + 0.836170i $$0.684790\pi$$
$$420$$ 0 0
$$421$$ −526504. −0.144776 −0.0723880 0.997377i $$-0.523062\pi$$
−0.0723880 + 0.997377i $$0.523062\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −5.20923e6 −1.39895
$$426$$ 0 0
$$427$$ −2.14158e6 −0.568414
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 387004. 0.100351 0.0501756 0.998740i $$-0.484022\pi$$
0.0501756 + 0.998740i $$0.484022\pi$$
$$432$$ 0 0
$$433$$ −2.77123e6 −0.710318 −0.355159 0.934806i $$-0.615573\pi$$
−0.355159 + 0.934806i $$0.615573\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.91551e6 −0.479822
$$438$$ 0 0
$$439$$ −3.01147e6 −0.745791 −0.372895 0.927873i $$-0.621635\pi$$
−0.372895 + 0.927873i $$0.621635\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.30598e6 0.800370 0.400185 0.916434i $$-0.368946\pi$$
0.400185 + 0.916434i $$0.368946\pi$$
$$444$$ 0 0
$$445$$ 3.22799e6 0.772737
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.93077e6 −0.686066 −0.343033 0.939323i $$-0.611454\pi$$
−0.343033 + 0.939323i $$0.611454\pi$$
$$450$$ 0 0
$$451$$ 2.76149e6 0.639296
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −37294.8 −0.00844539
$$456$$ 0 0
$$457$$ −7.55328e6 −1.69178 −0.845892 0.533354i $$-0.820931\pi$$
−0.845892 + 0.533354i $$0.820931\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7.54017e6 −1.65245 −0.826226 0.563339i $$-0.809517\pi$$
−0.826226 + 0.563339i $$0.809517\pi$$
$$462$$ 0 0
$$463$$ 2.69250e6 0.583719 0.291859 0.956461i $$-0.405726\pi$$
0.291859 + 0.956461i $$0.405726\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.28377e6 −1.33330 −0.666651 0.745370i $$-0.732273\pi$$
−0.666651 + 0.745370i $$0.732273\pi$$
$$468$$ 0 0
$$469$$ 1.40353e6 0.294639
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1.26260e7 2.59485
$$474$$ 0 0
$$475$$ 1.79090e6 0.364198
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −3.63273e6 −0.723426 −0.361713 0.932290i $$-0.617808\pi$$
−0.361713 + 0.932290i $$0.617808\pi$$
$$480$$ 0 0
$$481$$ 120538. 0.0237554
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 8.64780e6 1.66936
$$486$$ 0 0
$$487$$ −626847. −0.119767 −0.0598837 0.998205i $$-0.519073\pi$$
−0.0598837 + 0.998205i $$0.519073\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.21381e6 0.601612 0.300806 0.953685i $$-0.402744\pi$$
0.300806 + 0.953685i $$0.402744\pi$$
$$492$$ 0 0
$$493$$ 5.66325e6 1.04942
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −453002. −0.0822638
$$498$$ 0 0
$$499$$ 1.64079e6 0.294987 0.147494 0.989063i $$-0.452879\pi$$
0.147494 + 0.989063i $$0.452879\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6.47878e6 −1.14176 −0.570878 0.821035i $$-0.693397\pi$$
−0.570878 + 0.821035i $$0.693397\pi$$
$$504$$ 0 0
$$505$$ 6.32099e6 1.10295
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −5.55190e6 −0.949833 −0.474916 0.880031i $$-0.657522\pi$$
−0.474916 + 0.880031i $$0.657522\pi$$
$$510$$ 0 0
$$511$$ −1.46101e6 −0.247514
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 8.18893e6 1.36053
$$516$$ 0 0
$$517$$ −491253. −0.0808311
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −7.57348e6 −1.22237 −0.611183 0.791490i $$-0.709306\pi$$
−0.611183 + 0.791490i $$0.709306\pi$$
$$522$$ 0 0
$$523$$ −6.16189e6 −0.985053 −0.492526 0.870298i $$-0.663926\pi$$
−0.492526 + 0.870298i $$0.663926\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.38300e7 2.16918
$$528$$ 0 0
$$529$$ 3.64626e6 0.566512
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −36071.2 −0.00549974
$$534$$ 0 0
$$535$$ −1.51346e7 −2.28606
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.79220e6 −0.265713
$$540$$ 0 0
$$541$$ −9.51400e6 −1.39756 −0.698779 0.715338i $$-0.746273\pi$$
−0.698779 + 0.715338i $$0.746273\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1.20957e7 1.74437
$$546$$ 0 0
$$547$$ −590098. −0.0843249 −0.0421624 0.999111i $$-0.513425\pi$$
−0.0421624 + 0.999111i $$0.513425\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.94699e6 −0.273202
$$552$$ 0 0
$$553$$ −1.61350e6 −0.224365
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −3.71272e6 −0.507054 −0.253527 0.967328i $$-0.581591\pi$$
−0.253527 + 0.967328i $$0.581591\pi$$
$$558$$ 0 0
$$559$$ −164923. −0.0223230
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 9.24074e6 1.22867 0.614336 0.789045i $$-0.289424\pi$$
0.614336 + 0.789045i $$0.289424\pi$$
$$564$$ 0 0
$$565$$ 7.14884e6 0.942138
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.73509e6 −0.872093 −0.436047 0.899924i $$-0.643622\pi$$
−0.436047 + 0.899924i $$0.643622\pi$$
$$570$$ 0 0
$$571$$ −3.66513e6 −0.470435 −0.235217 0.971943i $$-0.575580\pi$$
−0.235217 + 0.971943i $$0.575580\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9.42670e6 −1.18902
$$576$$ 0 0
$$577$$ −8.73504e6 −1.09226 −0.546129 0.837701i $$-0.683899\pi$$
−0.546129 + 0.837701i $$0.683899\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1.98570e6 0.244047
$$582$$ 0 0
$$583$$ 2.08190e7 2.53681
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −8.06478e6 −0.966045 −0.483022 0.875608i $$-0.660461\pi$$
−0.483022 + 0.875608i $$0.660461\pi$$
$$588$$ 0 0
$$589$$ −4.75467e6 −0.564718
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.19273e6 −0.723178 −0.361589 0.932338i $$-0.617766\pi$$
−0.361589 + 0.932338i $$0.617766\pi$$
$$594$$ 0 0
$$595$$ 6.71179e6 0.777223
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 2.30764e6 0.262786 0.131393 0.991330i $$-0.458055\pi$$
0.131393 + 0.991330i $$0.458055\pi$$
$$600$$ 0 0
$$601$$ −1.09721e7 −1.23909 −0.619545 0.784961i $$-0.712683\pi$$
−0.619545 + 0.784961i $$0.712683\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −3.09219e7 −3.43461
$$606$$ 0 0
$$607$$ 1.32918e7 1.46424 0.732121 0.681175i $$-0.238531\pi$$
0.732121 + 0.681175i $$0.238531\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6416.85 0.000695374 0
$$612$$ 0 0
$$613$$ −1.17502e7 −1.26298 −0.631488 0.775385i $$-0.717556\pi$$
−0.631488 + 0.775385i $$0.717556\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.26344e6 −0.239362 −0.119681 0.992812i $$-0.538187\pi$$
−0.119681 + 0.992812i $$0.538187\pi$$
$$618$$ 0 0
$$619$$ −1.23588e7 −1.29643 −0.648215 0.761457i $$-0.724484\pi$$
−0.648215 + 0.761457i $$0.724484\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2.02621e6 −0.209154
$$624$$ 0 0
$$625$$ −1.02295e7 −1.04750
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −2.16928e7 −2.18619
$$630$$ 0 0
$$631$$ 2.74069e6 0.274023 0.137012 0.990569i $$-0.456250\pi$$
0.137012 + 0.990569i $$0.456250\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 8.60513e6 0.846883
$$636$$ 0 0
$$637$$ 23410.0 0.00228588
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.49344e7 1.43563 0.717816 0.696232i $$-0.245142\pi$$
0.717816 + 0.696232i $$0.245142\pi$$
$$642$$ 0 0
$$643$$ −1.12339e7 −1.07153 −0.535764 0.844368i $$-0.679976\pi$$
−0.535764 + 0.844368i $$0.679976\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −5.29688e6 −0.497461 −0.248731 0.968573i $$-0.580013\pi$$
−0.248731 + 0.968573i $$0.580013\pi$$
$$648$$ 0 0
$$649$$ −5.56572e6 −0.518692
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.08138e7 −0.992418 −0.496209 0.868203i $$-0.665275\pi$$
−0.496209 + 0.868203i $$0.665275\pi$$
$$654$$ 0 0
$$655$$ −2.82091e7 −2.56913
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 417354. 0.0374362 0.0187181 0.999825i $$-0.494042\pi$$
0.0187181 + 0.999825i $$0.494042\pi$$
$$660$$ 0 0
$$661$$ 6.80352e6 0.605661 0.302831 0.953044i $$-0.402068\pi$$
0.302831 + 0.953044i $$0.402068\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.30747e6 −0.202340
$$666$$ 0 0
$$667$$ 1.02483e7 0.891944
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.26236e7 2.79721
$$672$$ 0 0
$$673$$ −1.43365e7 −1.22013 −0.610066 0.792351i $$-0.708857\pi$$
−0.610066 + 0.792351i $$0.708857\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −4.09804e6 −0.343641 −0.171820 0.985128i $$-0.554965\pi$$
−0.171820 + 0.985128i $$0.554965\pi$$
$$678$$ 0 0
$$679$$ −5.42824e6 −0.451840
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −8.16379e6 −0.669638 −0.334819 0.942282i $$-0.608675\pi$$
−0.334819 + 0.942282i $$0.608675\pi$$
$$684$$ 0 0
$$685$$ 2.29225e7 1.86654
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −271942. −0.0218237
$$690$$ 0 0
$$691$$ −9.40335e6 −0.749182 −0.374591 0.927190i $$-0.622217\pi$$
−0.374591 + 0.927190i $$0.622217\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 7.22401e6 0.567305
$$696$$ 0 0
$$697$$ 6.49157e6 0.506137
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.76733e7 1.35838 0.679192 0.733961i $$-0.262330\pi$$
0.679192 + 0.733961i $$0.262330\pi$$
$$702$$ 0 0
$$703$$ 7.45783e6 0.569147
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3.96770e6 −0.298531
$$708$$ 0 0
$$709$$ 1.61349e7 1.20546 0.602728 0.797947i $$-0.294080\pi$$
0.602728 + 0.797947i $$0.294080\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 2.50270e7 1.84368
$$714$$ 0 0
$$715$$ 568127. 0.0415605
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.10160e7 1.51610 0.758051 0.652195i $$-0.226152\pi$$
0.758051 + 0.652195i $$0.226152\pi$$
$$720$$ 0 0
$$721$$ −5.14021e6 −0.368250
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −9.58163e6 −0.677009
$$726$$ 0 0
$$727$$ −2.51729e7 −1.76643 −0.883217 0.468964i $$-0.844627\pi$$
−0.883217 + 0.468964i $$0.844627\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.96805e7 2.05437
$$732$$ 0 0
$$733$$ −1.80996e7 −1.24425 −0.622127 0.782916i $$-0.713731\pi$$
−0.622127 + 0.782916i $$0.713731\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.13806e7 −1.44994
$$738$$ 0 0
$$739$$ 9.69580e6 0.653089 0.326545 0.945182i $$-0.394116\pi$$
0.326545 + 0.945182i $$0.394116\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4.72447e6 0.313965 0.156982 0.987601i $$-0.449823\pi$$
0.156982 + 0.987601i $$0.449823\pi$$
$$744$$ 0 0
$$745$$ 1.02878e6 0.0679099
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 9.50004e6 0.618758
$$750$$ 0 0
$$751$$ −3.00055e7 −1.94134 −0.970668 0.240422i $$-0.922714\pi$$
−0.970668 + 0.240422i $$0.922714\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −2.37989e7 −1.51946
$$756$$ 0 0
$$757$$ 907379. 0.0575505 0.0287752 0.999586i $$-0.490839\pi$$
0.0287752 + 0.999586i $$0.490839\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.22057e7 0.764014 0.382007 0.924159i $$-0.375233\pi$$
0.382007 + 0.924159i $$0.375233\pi$$
$$762$$ 0 0
$$763$$ −7.59248e6 −0.472141
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 72700.6 0.00446220
$$768$$ 0 0
$$769$$ −2.85670e7 −1.74200 −0.871000 0.491283i $$-0.836528\pi$$
−0.871000 + 0.491283i $$0.836528\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −905880. −0.0545283 −0.0272642 0.999628i $$-0.508680\pi$$
−0.0272642 + 0.999628i $$0.508680\pi$$
$$774$$ 0 0
$$775$$ −2.33989e7 −1.39940
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.23176e6 −0.131766
$$780$$ 0 0
$$781$$ 6.90076e6 0.404827
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.99992e6 0.289593
$$786$$ 0 0
$$787$$ 1.25311e7 0.721192 0.360596 0.932722i $$-0.382573\pi$$
0.360596 + 0.932722i $$0.382573\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.48735e6 −0.255005
$$792$$ 0 0
$$793$$ −426136. −0.0240639
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3.14513e6 −0.175385 −0.0876924 0.996148i $$-0.527949\pi$$
−0.0876924 + 0.996148i $$0.527949\pi$$
$$798$$ 0 0
$$799$$ −1.15481e6 −0.0639947
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2.22561e7 1.21804
$$804$$ 0 0
$$805$$ 1.21458e7 0.660595
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 3.26907e6 0.175611 0.0878057 0.996138i $$-0.472015\pi$$
0.0878057 + 0.996138i $$0.472015\pi$$
$$810$$ 0 0
$$811$$ −566320. −0.0302350 −0.0151175 0.999886i $$-0.504812\pi$$
−0.0151175 + 0.999886i $$0.504812\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 4.81817e7 2.54090
$$816$$ 0 0
$$817$$ −1.02040e7 −0.534828
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.98337e6 −0.102694 −0.0513471 0.998681i $$-0.516351\pi$$
−0.0513471 + 0.998681i $$0.516351\pi$$
$$822$$ 0 0
$$823$$ 3.14336e7 1.61769 0.808844 0.588024i $$-0.200094\pi$$
0.808844 + 0.588024i $$0.200094\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.44099e7 −0.732651 −0.366325 0.930487i $$-0.619384\pi$$
−0.366325 + 0.930487i $$0.619384\pi$$
$$828$$ 0 0
$$829$$ −3.60154e7 −1.82013 −0.910063 0.414469i $$-0.863967\pi$$
−0.910063 + 0.414469i $$0.863967\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4.21300e6 −0.210368
$$834$$ 0 0
$$835$$ −3.86170e7 −1.91673
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −2.05418e7 −1.00747 −0.503737 0.863857i $$-0.668042\pi$$
−0.503737 + 0.863857i $$0.668042\pi$$
$$840$$ 0 0
$$841$$ −1.00944e7 −0.492142
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 2.89766e7 1.39607
$$846$$ 0 0
$$847$$ 1.94098e7 0.929633
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3.92556e7 −1.85814
$$852$$ 0 0
$$853$$ −3.21606e7 −1.51339 −0.756695 0.653768i $$-0.773187\pi$$
−0.756695 + 0.653768i $$0.773187\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −2.87973e7 −1.33937 −0.669683 0.742647i $$-0.733570\pi$$
−0.669683 + 0.742647i $$0.733570\pi$$
$$858$$ 0 0
$$859$$ 2.72233e7 1.25880 0.629401 0.777081i $$-0.283301\pi$$
0.629401 + 0.777081i $$0.283301\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56328.3 −0.00257454 −0.00128727 0.999999i $$-0.500410\pi$$
−0.00128727 + 0.999999i $$0.500410\pi$$
$$864$$ 0 0
$$865$$ −5.47576e7 −2.48831
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.45791e7 1.10412
$$870$$ 0 0
$$871$$ 279277. 0.0124736
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 597669. 0.0263901
$$876$$ 0 0
$$877$$ 2.32650e7 1.02142 0.510710 0.859753i $$-0.329383\pi$$
0.510710 + 0.859753i $$0.329383\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 3.92252e7 1.70265 0.851326 0.524638i $$-0.175799\pi$$
0.851326 + 0.524638i $$0.175799\pi$$
$$882$$ 0 0
$$883$$ −2.86152e7 −1.23508 −0.617540 0.786540i $$-0.711870\pi$$
−0.617540 + 0.786540i $$0.711870\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5.81794e6 −0.248290 −0.124145 0.992264i $$-0.539619\pi$$
−0.124145 + 0.992264i $$0.539619\pi$$
$$888$$ 0 0
$$889$$ −5.40146e6 −0.229222
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 397017. 0.0166602
$$894$$ 0 0
$$895$$ −1.54134e7 −0.643191
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 2.54383e7 1.04976
$$900$$ 0 0
$$901$$ 4.89402e7 2.00842
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6.78606e7 −2.75420
$$906$$ 0 0
$$907$$ −9.55467e6 −0.385654 −0.192827 0.981233i $$-0.561766\pi$$
−0.192827 + 0.981233i $$0.561766\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.36257e6 −0.214081 −0.107040 0.994255i $$-0.534137\pi$$
−0.107040 + 0.994255i $$0.534137\pi$$
$$912$$ 0 0
$$913$$ −3.02490e7 −1.20097
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.77069e7 0.695375
$$918$$ 0 0
$$919$$ −1.97667e6 −0.0772050 −0.0386025 0.999255i $$-0.512291\pi$$
−0.0386025 + 0.999255i $$0.512291\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −90139.2 −0.00348265
$$924$$ 0 0
$$925$$ 3.67019e7 1.41037
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −1.09537e7 −0.416412 −0.208206 0.978085i $$-0.566762\pi$$
−0.208206 + 0.978085i $$0.566762\pi$$
$$930$$ 0 0
$$931$$ 1.44840e6 0.0547666
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −1.02243e8 −3.82478
$$936$$ 0 0
$$937$$ −6.80229e6 −0.253108 −0.126554 0.991960i $$-0.540392\pi$$
−0.126554 + 0.991960i $$0.540392\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −3.00214e6 −0.110524 −0.0552620 0.998472i $$-0.517599\pi$$
−0.0552620 + 0.998472i $$0.517599\pi$$
$$942$$ 0 0
$$943$$ 1.17473e7 0.430187
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.85814e7 −1.76033 −0.880166 0.474665i $$-0.842569\pi$$
−0.880166 + 0.474665i $$0.842569\pi$$
$$948$$ 0 0
$$949$$ −290714. −0.0104785
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 3.24058e7 1.15582 0.577911 0.816100i $$-0.303868\pi$$
0.577911 + 0.816100i $$0.303868\pi$$
$$954$$ 0 0
$$955$$ −9.33036e6 −0.331047
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1.43885e7 −0.505208
$$960$$ 0 0
$$961$$ 3.34928e7 1.16988
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −1.33421e7 −0.461218
$$966$$ 0 0
$$967$$ −2.37554e7 −0.816950 −0.408475 0.912769i $$-0.633939\pi$$
−0.408475 + 0.912769i $$0.633939\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3.27649e7 1.11522 0.557610 0.830103i $$-0.311718\pi$$
0.557610 + 0.830103i $$0.311718\pi$$
$$972$$ 0 0
$$973$$ −4.53453e6 −0.153550
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −5.09726e7 −1.70844 −0.854222 0.519908i $$-0.825966\pi$$
−0.854222 + 0.519908i $$0.825966\pi$$
$$978$$ 0 0
$$979$$ 3.08662e7 1.02926
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −3.63735e7 −1.20061 −0.600304 0.799772i $$-0.704954\pi$$
−0.600304 + 0.799772i $$0.704954\pi$$
$$984$$ 0 0
$$985$$ −5.58217e7 −1.83321
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 5.37104e7 1.74609
$$990$$ 0 0
$$991$$ −7.46245e6 −0.241378 −0.120689 0.992690i $$-0.538510\pi$$
−0.120689 + 0.992690i $$0.538510\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −5.58699e7 −1.78904
$$996$$ 0 0
$$997$$ −373648. −0.0119049 −0.00595243 0.999982i $$-0.501895\pi$$
−0.00595243 + 0.999982i $$0.501895\pi$$
$$998$$ 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 1008.6.a.bj.1.1 2
3.2 odd 2 1008.6.a.br.1.2 2
4.3 odd 2 504.6.a.n.1.1 2
12.11 even 2 504.6.a.q.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.n.1.1 2 4.3 odd 2
504.6.a.q.1.2 yes 2 12.11 even 2
1008.6.a.bj.1.1 2 1.1 even 1 trivial
1008.6.a.br.1.2 2 3.2 odd 2