# Properties

 Label 882.6.a.s.1.1 Level $882$ Weight $6$ Character 882.1 Self dual yes Analytic conductor $141.459$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 882.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: $$141.458529075$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 882.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} +26.0000 q^{5} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} +26.0000 q^{5} +64.0000 q^{8} +104.000 q^{10} -664.000 q^{11} -318.000 q^{13} +256.000 q^{16} +1582.00 q^{17} -236.000 q^{19} +416.000 q^{20} -2656.00 q^{22} -2212.00 q^{23} -2449.00 q^{25} -1272.00 q^{26} +4954.00 q^{29} +7128.00 q^{31} +1024.00 q^{32} +6328.00 q^{34} +4358.00 q^{37} -944.000 q^{38} +1664.00 q^{40} +10542.0 q^{41} -8452.00 q^{43} -10624.0 q^{44} -8848.00 q^{46} +5352.00 q^{47} -9796.00 q^{50} -5088.00 q^{52} +33354.0 q^{53} -17264.0 q^{55} +19816.0 q^{58} -15436.0 q^{59} +36762.0 q^{61} +28512.0 q^{62} +4096.00 q^{64} -8268.00 q^{65} +40972.0 q^{67} +25312.0 q^{68} +9092.00 q^{71} +73454.0 q^{73} +17432.0 q^{74} -3776.00 q^{76} +89400.0 q^{79} +6656.00 q^{80} +42168.0 q^{82} -6428.00 q^{83} +41132.0 q^{85} -33808.0 q^{86} -42496.0 q^{88} -122658. q^{89} -35392.0 q^{92} +21408.0 q^{94} -6136.00 q^{95} -21370.0 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ 26.0000 0.465102 0.232551 0.972584i $$-0.425293\pi$$
0.232551 + 0.972584i $$0.425293\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ 104.000 0.328877
$$11$$ −664.000 −1.65457 −0.827287 0.561779i $$-0.810117\pi$$
−0.827287 + 0.561779i $$0.810117\pi$$
$$12$$ 0 0
$$13$$ −318.000 −0.521878 −0.260939 0.965355i $$-0.584032\pi$$
−0.260939 + 0.965355i $$0.584032\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 1582.00 1.32765 0.663826 0.747887i $$-0.268932\pi$$
0.663826 + 0.747887i $$0.268932\pi$$
$$18$$ 0 0
$$19$$ −236.000 −0.149978 −0.0749891 0.997184i $$-0.523892\pi$$
−0.0749891 + 0.997184i $$0.523892\pi$$
$$20$$ 416.000 0.232551
$$21$$ 0 0
$$22$$ −2656.00 −1.16996
$$23$$ −2212.00 −0.871898 −0.435949 0.899971i $$-0.643587\pi$$
−0.435949 + 0.899971i $$0.643587\pi$$
$$24$$ 0 0
$$25$$ −2449.00 −0.783680
$$26$$ −1272.00 −0.369023
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4954.00 1.09386 0.546929 0.837179i $$-0.315797\pi$$
0.546929 + 0.837179i $$0.315797\pi$$
$$30$$ 0 0
$$31$$ 7128.00 1.33218 0.666091 0.745871i $$-0.267966\pi$$
0.666091 + 0.745871i $$0.267966\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ 6328.00 0.938792
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4358.00 0.523339 0.261669 0.965158i $$-0.415727\pi$$
0.261669 + 0.965158i $$0.415727\pi$$
$$38$$ −944.000 −0.106051
$$39$$ 0 0
$$40$$ 1664.00 0.164438
$$41$$ 10542.0 0.979407 0.489704 0.871889i $$-0.337105\pi$$
0.489704 + 0.871889i $$0.337105\pi$$
$$42$$ 0 0
$$43$$ −8452.00 −0.697089 −0.348545 0.937292i $$-0.613324\pi$$
−0.348545 + 0.937292i $$0.613324\pi$$
$$44$$ −10624.0 −0.827287
$$45$$ 0 0
$$46$$ −8848.00 −0.616525
$$47$$ 5352.00 0.353404 0.176702 0.984264i $$-0.443457\pi$$
0.176702 + 0.984264i $$0.443457\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −9796.00 −0.554145
$$51$$ 0 0
$$52$$ −5088.00 −0.260939
$$53$$ 33354.0 1.63102 0.815508 0.578746i $$-0.196458\pi$$
0.815508 + 0.578746i $$0.196458\pi$$
$$54$$ 0 0
$$55$$ −17264.0 −0.769546
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 19816.0 0.773475
$$59$$ −15436.0 −0.577304 −0.288652 0.957434i $$-0.593207\pi$$
−0.288652 + 0.957434i $$0.593207\pi$$
$$60$$ 0 0
$$61$$ 36762.0 1.26495 0.632477 0.774579i $$-0.282038\pi$$
0.632477 + 0.774579i $$0.282038\pi$$
$$62$$ 28512.0 0.941995
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ −8268.00 −0.242726
$$66$$ 0 0
$$67$$ 40972.0 1.11506 0.557532 0.830155i $$-0.311748\pi$$
0.557532 + 0.830155i $$0.311748\pi$$
$$68$$ 25312.0 0.663826
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9092.00 0.214049 0.107025 0.994256i $$-0.465868\pi$$
0.107025 + 0.994256i $$0.465868\pi$$
$$72$$ 0 0
$$73$$ 73454.0 1.61327 0.806637 0.591047i $$-0.201285\pi$$
0.806637 + 0.591047i $$0.201285\pi$$
$$74$$ 17432.0 0.370056
$$75$$ 0 0
$$76$$ −3776.00 −0.0749891
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 89400.0 1.61165 0.805823 0.592156i $$-0.201723\pi$$
0.805823 + 0.592156i $$0.201723\pi$$
$$80$$ 6656.00 0.116276
$$81$$ 0 0
$$82$$ 42168.0 0.692546
$$83$$ −6428.00 −0.102419 −0.0512095 0.998688i $$-0.516308\pi$$
−0.0512095 + 0.998688i $$0.516308\pi$$
$$84$$ 0 0
$$85$$ 41132.0 0.617494
$$86$$ −33808.0 −0.492916
$$87$$ 0 0
$$88$$ −42496.0 −0.584980
$$89$$ −122658. −1.64142 −0.820712 0.571342i $$-0.806423\pi$$
−0.820712 + 0.571342i $$0.806423\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −35392.0 −0.435949
$$93$$ 0 0
$$94$$ 21408.0 0.249894
$$95$$ −6136.00 −0.0697552
$$96$$ 0 0
$$97$$ −21370.0 −0.230608 −0.115304 0.993330i $$-0.536784\pi$$
−0.115304 + 0.993330i $$0.536784\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −39184.0 −0.391840
$$101$$ −36814.0 −0.359095 −0.179548 0.983749i $$-0.557463\pi$$
−0.179548 + 0.983749i $$0.557463\pi$$
$$102$$ 0 0
$$103$$ −104528. −0.970822 −0.485411 0.874286i $$-0.661330\pi$$
−0.485411 + 0.874286i $$0.661330\pi$$
$$104$$ −20352.0 −0.184512
$$105$$ 0 0
$$106$$ 133416. 1.15330
$$107$$ −214440. −1.81070 −0.905350 0.424667i $$-0.860391\pi$$
−0.905350 + 0.424667i $$0.860391\pi$$
$$108$$ 0 0
$$109$$ 28798.0 0.232165 0.116082 0.993240i $$-0.462966\pi$$
0.116082 + 0.993240i $$0.462966\pi$$
$$110$$ −69056.0 −0.544151
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 56014.0 0.412668 0.206334 0.978482i $$-0.433847\pi$$
0.206334 + 0.978482i $$0.433847\pi$$
$$114$$ 0 0
$$115$$ −57512.0 −0.405521
$$116$$ 79264.0 0.546929
$$117$$ 0 0
$$118$$ −61744.0 −0.408216
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 279845. 1.73762
$$122$$ 147048. 0.894457
$$123$$ 0 0
$$124$$ 114048. 0.666091
$$125$$ −144924. −0.829593
$$126$$ 0 0
$$127$$ 185400. 1.02000 0.510000 0.860174i $$-0.329645\pi$$
0.510000 + 0.860174i $$0.329645\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ −33072.0 −0.171634
$$131$$ 64532.0 0.328547 0.164273 0.986415i $$-0.447472\pi$$
0.164273 + 0.986415i $$0.447472\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 163888. 0.788470
$$135$$ 0 0
$$136$$ 101248. 0.469396
$$137$$ −152930. −0.696131 −0.348066 0.937470i $$-0.613161\pi$$
−0.348066 + 0.937470i $$0.613161\pi$$
$$138$$ 0 0
$$139$$ 343460. 1.50778 0.753892 0.656998i $$-0.228174\pi$$
0.753892 + 0.656998i $$0.228174\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 36368.0 0.151356
$$143$$ 211152. 0.863486
$$144$$ 0 0
$$145$$ 128804. 0.508756
$$146$$ 293816. 1.14076
$$147$$ 0 0
$$148$$ 69728.0 0.261669
$$149$$ 174858. 0.645238 0.322619 0.946529i $$-0.395437\pi$$
0.322619 + 0.946529i $$0.395437\pi$$
$$150$$ 0 0
$$151$$ −452552. −1.61520 −0.807600 0.589731i $$-0.799234\pi$$
−0.807600 + 0.589731i $$0.799234\pi$$
$$152$$ −15104.0 −0.0530253
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 185328. 0.619601
$$156$$ 0 0
$$157$$ 499066. 1.61588 0.807940 0.589265i $$-0.200583\pi$$
0.807940 + 0.589265i $$0.200583\pi$$
$$158$$ 357600. 1.13961
$$159$$ 0 0
$$160$$ 26624.0 0.0822192
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −475588. −1.40204 −0.701022 0.713139i $$-0.747273\pi$$
−0.701022 + 0.713139i $$0.747273\pi$$
$$164$$ 168672. 0.489704
$$165$$ 0 0
$$166$$ −25712.0 −0.0724212
$$167$$ 120224. 0.333580 0.166790 0.985992i $$-0.446660\pi$$
0.166790 + 0.985992i $$0.446660\pi$$
$$168$$ 0 0
$$169$$ −270169. −0.727644
$$170$$ 164528. 0.436634
$$171$$ 0 0
$$172$$ −135232. −0.348545
$$173$$ 508874. 1.29269 0.646346 0.763045i $$-0.276296\pi$$
0.646346 + 0.763045i $$0.276296\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −169984. −0.413644
$$177$$ 0 0
$$178$$ −490632. −1.16066
$$179$$ −487560. −1.13735 −0.568677 0.822561i $$-0.692544\pi$$
−0.568677 + 0.822561i $$0.692544\pi$$
$$180$$ 0 0
$$181$$ 544410. 1.23518 0.617589 0.786501i $$-0.288109\pi$$
0.617589 + 0.786501i $$0.288109\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −141568. −0.308262
$$185$$ 113308. 0.243406
$$186$$ 0 0
$$187$$ −1.05045e6 −2.19670
$$188$$ 85632.0 0.176702
$$189$$ 0 0
$$190$$ −24544.0 −0.0493243
$$191$$ −376404. −0.746570 −0.373285 0.927717i $$-0.621769\pi$$
−0.373285 + 0.927717i $$0.621769\pi$$
$$192$$ 0 0
$$193$$ 844946. 1.63281 0.816405 0.577480i $$-0.195964\pi$$
0.816405 + 0.577480i $$0.195964\pi$$
$$194$$ −85480.0 −0.163065
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 492794. 0.904690 0.452345 0.891843i $$-0.350588\pi$$
0.452345 + 0.891843i $$0.350588\pi$$
$$198$$ 0 0
$$199$$ 914776. 1.63750 0.818751 0.574148i $$-0.194667\pi$$
0.818751 + 0.574148i $$0.194667\pi$$
$$200$$ −156736. −0.277073
$$201$$ 0 0
$$202$$ −147256. −0.253919
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 274092. 0.455524
$$206$$ −418112. −0.686475
$$207$$ 0 0
$$208$$ −81408.0 −0.130469
$$209$$ 156704. 0.248150
$$210$$ 0 0
$$211$$ 311780. 0.482106 0.241053 0.970512i $$-0.422507\pi$$
0.241053 + 0.970512i $$0.422507\pi$$
$$212$$ 533664. 0.815508
$$213$$ 0 0
$$214$$ −857760. −1.28036
$$215$$ −219752. −0.324218
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 115192. 0.164165
$$219$$ 0 0
$$220$$ −276224. −0.384773
$$221$$ −503076. −0.692872
$$222$$ 0 0
$$223$$ 1.28776e6 1.73409 0.867047 0.498226i $$-0.166015\pi$$
0.867047 + 0.498226i $$0.166015\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 224056. 0.291800
$$227$$ 1.28905e6 1.66037 0.830187 0.557485i $$-0.188234\pi$$
0.830187 + 0.557485i $$0.188234\pi$$
$$228$$ 0 0
$$229$$ −678214. −0.854630 −0.427315 0.904103i $$-0.640540\pi$$
−0.427315 + 0.904103i $$0.640540\pi$$
$$230$$ −230048. −0.286747
$$231$$ 0 0
$$232$$ 317056. 0.386737
$$233$$ 1.11731e6 1.34829 0.674146 0.738598i $$-0.264512\pi$$
0.674146 + 0.738598i $$0.264512\pi$$
$$234$$ 0 0
$$235$$ 139152. 0.164369
$$236$$ −246976. −0.288652
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.26196e6 1.42906 0.714528 0.699606i $$-0.246641\pi$$
0.714528 + 0.699606i $$0.246641\pi$$
$$240$$ 0 0
$$241$$ −948218. −1.05164 −0.525818 0.850597i $$-0.676241\pi$$
−0.525818 + 0.850597i $$0.676241\pi$$
$$242$$ 1.11938e6 1.22868
$$243$$ 0 0
$$244$$ 588192. 0.632477
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 75048.0 0.0782703
$$248$$ 456192. 0.470997
$$249$$ 0 0
$$250$$ −579696. −0.586611
$$251$$ −486396. −0.487310 −0.243655 0.969862i $$-0.578347\pi$$
−0.243655 + 0.969862i $$0.578347\pi$$
$$252$$ 0 0
$$253$$ 1.46877e6 1.44262
$$254$$ 741600. 0.721249
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −1.03910e6 −0.981349 −0.490675 0.871343i $$-0.663250\pi$$
−0.490675 + 0.871343i $$0.663250\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −132288. −0.121363
$$261$$ 0 0
$$262$$ 258128. 0.232317
$$263$$ −1.35104e6 −1.20443 −0.602213 0.798335i $$-0.705714\pi$$
−0.602213 + 0.798335i $$0.705714\pi$$
$$264$$ 0 0
$$265$$ 867204. 0.758589
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 655552. 0.557532
$$269$$ −1.11811e6 −0.942115 −0.471057 0.882103i $$-0.656128\pi$$
−0.471057 + 0.882103i $$0.656128\pi$$
$$270$$ 0 0
$$271$$ 190104. 0.157242 0.0786209 0.996905i $$-0.474948\pi$$
0.0786209 + 0.996905i $$0.474948\pi$$
$$272$$ 404992. 0.331913
$$273$$ 0 0
$$274$$ −611720. −0.492239
$$275$$ 1.62614e6 1.29666
$$276$$ 0 0
$$277$$ −200506. −0.157010 −0.0785051 0.996914i $$-0.525015\pi$$
−0.0785051 + 0.996914i $$0.525015\pi$$
$$278$$ 1.37384e6 1.06616
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.09237e6 −0.825285 −0.412643 0.910893i $$-0.635394\pi$$
−0.412643 + 0.910893i $$0.635394\pi$$
$$282$$ 0 0
$$283$$ −1.81258e6 −1.34534 −0.672669 0.739944i $$-0.734852\pi$$
−0.672669 + 0.739944i $$0.734852\pi$$
$$284$$ 145472. 0.107025
$$285$$ 0 0
$$286$$ 844608. 0.610577
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.08287e6 0.762659
$$290$$ 515216. 0.359745
$$291$$ 0 0
$$292$$ 1.17526e6 0.806637
$$293$$ 2.10031e6 1.42927 0.714634 0.699499i $$-0.246593\pi$$
0.714634 + 0.699499i $$0.246593\pi$$
$$294$$ 0 0
$$295$$ −401336. −0.268505
$$296$$ 278912. 0.185028
$$297$$ 0 0
$$298$$ 699432. 0.456252
$$299$$ 703416. 0.455024
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −1.81021e6 −1.14212
$$303$$ 0 0
$$304$$ −60416.0 −0.0374945
$$305$$ 955812. 0.588333
$$306$$ 0 0
$$307$$ 1.64104e6 0.993743 0.496872 0.867824i $$-0.334482\pi$$
0.496872 + 0.867824i $$0.334482\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 741312. 0.438124
$$311$$ −945232. −0.554163 −0.277081 0.960846i $$-0.589367\pi$$
−0.277081 + 0.960846i $$0.589367\pi$$
$$312$$ 0 0
$$313$$ −415354. −0.239639 −0.119820 0.992796i $$-0.538232\pi$$
−0.119820 + 0.992796i $$0.538232\pi$$
$$314$$ 1.99626e6 1.14260
$$315$$ 0 0
$$316$$ 1.43040e6 0.805823
$$317$$ −1.18481e6 −0.662220 −0.331110 0.943592i $$-0.607423\pi$$
−0.331110 + 0.943592i $$0.607423\pi$$
$$318$$ 0 0
$$319$$ −3.28946e6 −1.80987
$$320$$ 106496. 0.0581378
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −373352. −0.199119
$$324$$ 0 0
$$325$$ 778782. 0.408985
$$326$$ −1.90235e6 −0.991395
$$327$$ 0 0
$$328$$ 674688. 0.346273
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.37155e6 0.688083 0.344042 0.938954i $$-0.388204\pi$$
0.344042 + 0.938954i $$0.388204\pi$$
$$332$$ −102848. −0.0512095
$$333$$ 0 0
$$334$$ 480896. 0.235877
$$335$$ 1.06527e6 0.518619
$$336$$ 0 0
$$337$$ 963522. 0.462154 0.231077 0.972935i $$-0.425775\pi$$
0.231077 + 0.972935i $$0.425775\pi$$
$$338$$ −1.08068e6 −0.514522
$$339$$ 0 0
$$340$$ 658112. 0.308747
$$341$$ −4.73299e6 −2.20419
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −540928. −0.246458
$$345$$ 0 0
$$346$$ 2.03550e6 0.914071
$$347$$ −2.57731e6 −1.14906 −0.574531 0.818483i $$-0.694815\pi$$
−0.574531 + 0.818483i $$0.694815\pi$$
$$348$$ 0 0
$$349$$ 3.06751e6 1.34810 0.674051 0.738684i $$-0.264553\pi$$
0.674051 + 0.738684i $$0.264553\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −679936. −0.292490
$$353$$ −3.10144e6 −1.32473 −0.662364 0.749182i $$-0.730447\pi$$
−0.662364 + 0.749182i $$0.730447\pi$$
$$354$$ 0 0
$$355$$ 236392. 0.0995547
$$356$$ −1.96253e6 −0.820712
$$357$$ 0 0
$$358$$ −1.95024e6 −0.804230
$$359$$ 327508. 0.134118 0.0670588 0.997749i $$-0.478638\pi$$
0.0670588 + 0.997749i $$0.478638\pi$$
$$360$$ 0 0
$$361$$ −2.42040e6 −0.977507
$$362$$ 2.17764e6 0.873403
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.90980e6 0.750337
$$366$$ 0 0
$$367$$ 2.86739e6 1.11128 0.555638 0.831424i $$-0.312474\pi$$
0.555638 + 0.831424i $$0.312474\pi$$
$$368$$ −566272. −0.217974
$$369$$ 0 0
$$370$$ 453232. 0.172114
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.58029e6 1.33244 0.666218 0.745757i $$-0.267912\pi$$
0.666218 + 0.745757i $$0.267912\pi$$
$$374$$ −4.20179e6 −1.55330
$$375$$ 0 0
$$376$$ 342528. 0.124947
$$377$$ −1.57537e6 −0.570860
$$378$$ 0 0
$$379$$ 1.64235e6 0.587310 0.293655 0.955912i $$-0.405128\pi$$
0.293655 + 0.955912i $$0.405128\pi$$
$$380$$ −98176.0 −0.0348776
$$381$$ 0 0
$$382$$ −1.50562e6 −0.527905
$$383$$ −2.05698e6 −0.716527 −0.358263 0.933621i $$-0.616631\pi$$
−0.358263 + 0.933621i $$0.616631\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.37978e6 1.15457
$$387$$ 0 0
$$388$$ −341920. −0.115304
$$389$$ −616142. −0.206446 −0.103223 0.994658i $$-0.532916\pi$$
−0.103223 + 0.994658i $$0.532916\pi$$
$$390$$ 0 0
$$391$$ −3.49938e6 −1.15758
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 1.97118e6 0.639713
$$395$$ 2.32440e6 0.749580
$$396$$ 0 0
$$397$$ −2.19212e6 −0.698052 −0.349026 0.937113i $$-0.613487\pi$$
−0.349026 + 0.937113i $$0.613487\pi$$
$$398$$ 3.65910e6 1.15789
$$399$$ 0 0
$$400$$ −626944. −0.195920
$$401$$ −3.28454e6 −1.02003 −0.510015 0.860165i $$-0.670360\pi$$
−0.510015 + 0.860165i $$0.670360\pi$$
$$402$$ 0 0
$$403$$ −2.26670e6 −0.695236
$$404$$ −589024. −0.179548
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.89371e6 −0.865903
$$408$$ 0 0
$$409$$ 3.61219e6 1.06773 0.533866 0.845569i $$-0.320739\pi$$
0.533866 + 0.845569i $$0.320739\pi$$
$$410$$ 1.09637e6 0.322104
$$411$$ 0 0
$$412$$ −1.67245e6 −0.485411
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −167128. −0.0476353
$$416$$ −325632. −0.0922558
$$417$$ 0 0
$$418$$ 626816. 0.175469
$$419$$ 5.41489e6 1.50680 0.753398 0.657564i $$-0.228413\pi$$
0.753398 + 0.657564i $$0.228413\pi$$
$$420$$ 0 0
$$421$$ 3.60629e6 0.991644 0.495822 0.868424i $$-0.334867\pi$$
0.495822 + 0.868424i $$0.334867\pi$$
$$422$$ 1.24712e6 0.340900
$$423$$ 0 0
$$424$$ 2.13466e6 0.576651
$$425$$ −3.87432e6 −1.04045
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.43104e6 −0.905350
$$429$$ 0 0
$$430$$ −879008. −0.229257
$$431$$ 2.78214e6 0.721416 0.360708 0.932679i $$-0.382535\pi$$
0.360708 + 0.932679i $$0.382535\pi$$
$$432$$ 0 0
$$433$$ −6.27619e6 −1.60871 −0.804353 0.594152i $$-0.797488\pi$$
−0.804353 + 0.594152i $$0.797488\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 460768. 0.116082
$$437$$ 522032. 0.130766
$$438$$ 0 0
$$439$$ −641592. −0.158890 −0.0794452 0.996839i $$-0.525315\pi$$
−0.0794452 + 0.996839i $$0.525315\pi$$
$$440$$ −1.10490e6 −0.272076
$$441$$ 0 0
$$442$$ −2.01230e6 −0.489934
$$443$$ −6.05546e6 −1.46601 −0.733006 0.680222i $$-0.761883\pi$$
−0.733006 + 0.680222i $$0.761883\pi$$
$$444$$ 0 0
$$445$$ −3.18911e6 −0.763430
$$446$$ 5.15104e6 1.22619
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.16681e6 1.20950 0.604752 0.796414i $$-0.293272\pi$$
0.604752 + 0.796414i $$0.293272\pi$$
$$450$$ 0 0
$$451$$ −6.99989e6 −1.62050
$$452$$ 896224. 0.206334
$$453$$ 0 0
$$454$$ 5.15621e6 1.17406
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −227798. −0.0510222 −0.0255111 0.999675i $$-0.508121\pi$$
−0.0255111 + 0.999675i $$0.508121\pi$$
$$458$$ −2.71286e6 −0.604315
$$459$$ 0 0
$$460$$ −920192. −0.202761
$$461$$ 585146. 0.128237 0.0641183 0.997942i $$-0.479577\pi$$
0.0641183 + 0.997942i $$0.479577\pi$$
$$462$$ 0 0
$$463$$ −3.41454e6 −0.740251 −0.370126 0.928982i $$-0.620685\pi$$
−0.370126 + 0.928982i $$0.620685\pi$$
$$464$$ 1.26822e6 0.273465
$$465$$ 0 0
$$466$$ 4.46924e6 0.953386
$$467$$ 716300. 0.151986 0.0759929 0.997108i $$-0.475787\pi$$
0.0759929 + 0.997108i $$0.475787\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 556608. 0.116226
$$471$$ 0 0
$$472$$ −987904. −0.204108
$$473$$ 5.61213e6 1.15339
$$474$$ 0 0
$$475$$ 577964. 0.117535
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 5.04782e6 1.01050
$$479$$ 5.24092e6 1.04368 0.521842 0.853042i $$-0.325245\pi$$
0.521842 + 0.853042i $$0.325245\pi$$
$$480$$ 0 0
$$481$$ −1.38584e6 −0.273119
$$482$$ −3.79287e6 −0.743619
$$483$$ 0 0
$$484$$ 4.47752e6 0.868809
$$485$$ −555620. −0.107256
$$486$$ 0 0
$$487$$ 1.11702e6 0.213421 0.106710 0.994290i $$-0.465968\pi$$
0.106710 + 0.994290i $$0.465968\pi$$
$$488$$ 2.35277e6 0.447229
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.34458e6 −0.251699 −0.125850 0.992049i $$-0.540166\pi$$
−0.125850 + 0.992049i $$0.540166\pi$$
$$492$$ 0 0
$$493$$ 7.83723e6 1.45226
$$494$$ 300192. 0.0553454
$$495$$ 0 0
$$496$$ 1.82477e6 0.333045
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −6.54648e6 −1.17695 −0.588473 0.808517i $$-0.700271\pi$$
−0.588473 + 0.808517i $$0.700271\pi$$
$$500$$ −2.31878e6 −0.414797
$$501$$ 0 0
$$502$$ −1.94558e6 −0.344580
$$503$$ −8.22050e6 −1.44870 −0.724350 0.689432i $$-0.757860\pi$$
−0.724350 + 0.689432i $$0.757860\pi$$
$$504$$ 0 0
$$505$$ −957164. −0.167016
$$506$$ 5.87507e6 1.02009
$$507$$ 0 0
$$508$$ 2.96640e6 0.510000
$$509$$ −5.11045e6 −0.874308 −0.437154 0.899387i $$-0.644013\pi$$
−0.437154 + 0.899387i $$0.644013\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ −4.15639e6 −0.693919
$$515$$ −2.71773e6 −0.451531
$$516$$ 0 0
$$517$$ −3.55373e6 −0.584733
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −529152. −0.0858168
$$521$$ 9.69999e6 1.56559 0.782793 0.622282i $$-0.213794\pi$$
0.782793 + 0.622282i $$0.213794\pi$$
$$522$$ 0 0
$$523$$ 3.17295e6 0.507234 0.253617 0.967305i $$-0.418380\pi$$
0.253617 + 0.967305i $$0.418380\pi$$
$$524$$ 1.03251e6 0.164273
$$525$$ 0 0
$$526$$ −5.40418e6 −0.851658
$$527$$ 1.12765e7 1.76867
$$528$$ 0 0
$$529$$ −1.54340e6 −0.239794
$$530$$ 3.46882e6 0.536403
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.35236e6 −0.511131
$$534$$ 0 0
$$535$$ −5.57544e6 −0.842160
$$536$$ 2.62221e6 0.394235
$$537$$ 0 0
$$538$$ −4.47244e6 −0.666176
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6.62575e6 −0.973289 −0.486644 0.873600i $$-0.661779\pi$$
−0.486644 + 0.873600i $$0.661779\pi$$
$$542$$ 760416. 0.111187
$$543$$ 0 0
$$544$$ 1.61997e6 0.234698
$$545$$ 748748. 0.107980
$$546$$ 0 0
$$547$$ 3.84707e6 0.549745 0.274873 0.961481i $$-0.411364\pi$$
0.274873 + 0.961481i $$0.411364\pi$$
$$548$$ −2.44688e6 −0.348066
$$549$$ 0 0
$$550$$ 6.50454e6 0.916875
$$551$$ −1.16914e6 −0.164055
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −802024. −0.111023
$$555$$ 0 0
$$556$$ 5.49536e6 0.753892
$$557$$ −5.00176e6 −0.683101 −0.341550 0.939863i $$-0.610952\pi$$
−0.341550 + 0.939863i $$0.610952\pi$$
$$558$$ 0 0
$$559$$ 2.68774e6 0.363795
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −4.36948e6 −0.583565
$$563$$ 2.27772e6 0.302852 0.151426 0.988469i $$-0.451614\pi$$
0.151426 + 0.988469i $$0.451614\pi$$
$$564$$ 0 0
$$565$$ 1.45636e6 0.191933
$$566$$ −7.25032e6 −0.951297
$$567$$ 0 0
$$568$$ 581888. 0.0756778
$$569$$ −8.86979e6 −1.14850 −0.574252 0.818678i $$-0.694707\pi$$
−0.574252 + 0.818678i $$0.694707\pi$$
$$570$$ 0 0
$$571$$ 1.40102e7 1.79826 0.899132 0.437678i $$-0.144199\pi$$
0.899132 + 0.437678i $$0.144199\pi$$
$$572$$ 3.37843e6 0.431743
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5.41719e6 0.683289
$$576$$ 0 0
$$577$$ −8.75327e6 −1.09454 −0.547269 0.836957i $$-0.684332\pi$$
−0.547269 + 0.836957i $$0.684332\pi$$
$$578$$ 4.33147e6 0.539281
$$579$$ 0 0
$$580$$ 2.06086e6 0.254378
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2.21471e7 −2.69864
$$584$$ 4.70106e6 0.570379
$$585$$ 0 0
$$586$$ 8.40122e6 1.01064
$$587$$ −1.06117e7 −1.27113 −0.635564 0.772048i $$-0.719232\pi$$
−0.635564 + 0.772048i $$0.719232\pi$$
$$588$$ 0 0
$$589$$ −1.68221e6 −0.199798
$$590$$ −1.60534e6 −0.189862
$$591$$ 0 0
$$592$$ 1.11565e6 0.130835
$$593$$ 1.88552e6 0.220188 0.110094 0.993921i $$-0.464885\pi$$
0.110094 + 0.993921i $$0.464885\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.79773e6 0.322619
$$597$$ 0 0
$$598$$ 2.81366e6 0.321751
$$599$$ −1.27256e7 −1.44915 −0.724573 0.689198i $$-0.757963\pi$$
−0.724573 + 0.689198i $$0.757963\pi$$
$$600$$ 0 0
$$601$$ −7.18846e6 −0.811801 −0.405900 0.913917i $$-0.633042\pi$$
−0.405900 + 0.913917i $$0.633042\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −7.24083e6 −0.807600
$$605$$ 7.27597e6 0.808170
$$606$$ 0 0
$$607$$ −1.08494e7 −1.19519 −0.597593 0.801800i $$-0.703876\pi$$
−0.597593 + 0.801800i $$0.703876\pi$$
$$608$$ −241664. −0.0265126
$$609$$ 0 0
$$610$$ 3.82325e6 0.416014
$$611$$ −1.70194e6 −0.184434
$$612$$ 0 0
$$613$$ −4.90511e6 −0.527227 −0.263614 0.964628i $$-0.584914\pi$$
−0.263614 + 0.964628i $$0.584914\pi$$
$$614$$ 6.56418e6 0.702683
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.58445e6 −0.273310 −0.136655 0.990619i $$-0.543635\pi$$
−0.136655 + 0.990619i $$0.543635\pi$$
$$618$$ 0 0
$$619$$ 4.99336e6 0.523801 0.261901 0.965095i $$-0.415651\pi$$
0.261901 + 0.965095i $$0.415651\pi$$
$$620$$ 2.96525e6 0.309800
$$621$$ 0 0
$$622$$ −3.78093e6 −0.391852
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 3.88510e6 0.397834
$$626$$ −1.66142e6 −0.169450
$$627$$ 0 0
$$628$$ 7.98506e6 0.807940
$$629$$ 6.89436e6 0.694812
$$630$$ 0 0
$$631$$ −1.18219e7 −1.18199 −0.590997 0.806674i $$-0.701265\pi$$
−0.590997 + 0.806674i $$0.701265\pi$$
$$632$$ 5.72160e6 0.569803
$$633$$ 0 0
$$634$$ −4.73926e6 −0.468260
$$635$$ 4.82040e6 0.474404
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.31578e7 −1.27977
$$639$$ 0 0
$$640$$ 425984. 0.0411096
$$641$$ 5.47007e6 0.525833 0.262916 0.964819i $$-0.415316\pi$$
0.262916 + 0.964819i $$0.415316\pi$$
$$642$$ 0 0
$$643$$ −9.64934e6 −0.920386 −0.460193 0.887819i $$-0.652220\pi$$
−0.460193 + 0.887819i $$0.652220\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −1.49341e6 −0.140798
$$647$$ 292368. 0.0274580 0.0137290 0.999906i $$-0.495630\pi$$
0.0137290 + 0.999906i $$0.495630\pi$$
$$648$$ 0 0
$$649$$ 1.02495e7 0.955193
$$650$$ 3.11513e6 0.289196
$$651$$ 0 0
$$652$$ −7.60941e6 −0.701022
$$653$$ −6.94081e6 −0.636982 −0.318491 0.947926i $$-0.603176\pi$$
−0.318491 + 0.947926i $$0.603176\pi$$
$$654$$ 0 0
$$655$$ 1.67783e6 0.152808
$$656$$ 2.69875e6 0.244852
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.32912e7 1.19221 0.596104 0.802908i $$-0.296715\pi$$
0.596104 + 0.802908i $$0.296715\pi$$
$$660$$ 0 0
$$661$$ −2.05219e6 −0.182690 −0.0913448 0.995819i $$-0.529117\pi$$
−0.0913448 + 0.995819i $$0.529117\pi$$
$$662$$ 5.48619e6 0.486548
$$663$$ 0 0
$$664$$ −411392. −0.0362106
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.09582e7 −0.953732
$$668$$ 1.92358e6 0.166790
$$669$$ 0 0
$$670$$ 4.26109e6 0.366719
$$671$$ −2.44100e7 −2.09296
$$672$$ 0 0
$$673$$ −1.57039e7 −1.33650 −0.668252 0.743935i $$-0.732957\pi$$
−0.668252 + 0.743935i $$0.732957\pi$$
$$674$$ 3.85409e6 0.326792
$$675$$ 0 0
$$676$$ −4.32270e6 −0.363822
$$677$$ −969534. −0.0813002 −0.0406501 0.999173i $$-0.512943\pi$$
−0.0406501 + 0.999173i $$0.512943\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 2.63245e6 0.218317
$$681$$ 0 0
$$682$$ −1.89320e7 −1.55860
$$683$$ 1.49908e7 1.22962 0.614812 0.788673i $$-0.289232\pi$$
0.614812 + 0.788673i $$0.289232\pi$$
$$684$$ 0 0
$$685$$ −3.97618e6 −0.323772
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −2.16371e6 −0.174272
$$689$$ −1.06066e7 −0.851191
$$690$$ 0 0
$$691$$ 7.16038e6 0.570481 0.285240 0.958456i $$-0.407927\pi$$
0.285240 + 0.958456i $$0.407927\pi$$
$$692$$ 8.14198e6 0.646346
$$693$$ 0 0
$$694$$ −1.03092e7 −0.812509
$$695$$ 8.92996e6 0.701274
$$696$$ 0 0
$$697$$ 1.66774e7 1.30031
$$698$$ 1.22701e7 0.953253
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 91834.0 0.00705844 0.00352922 0.999994i $$-0.498877\pi$$
0.00352922 + 0.999994i $$0.498877\pi$$
$$702$$ 0 0
$$703$$ −1.02849e6 −0.0784894
$$704$$ −2.71974e6 −0.206822
$$705$$ 0 0
$$706$$ −1.24058e7 −0.936725
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.20981e7 1.65097 0.825487 0.564422i $$-0.190901\pi$$
0.825487 + 0.564422i $$0.190901\pi$$
$$710$$ 945568. 0.0703958
$$711$$ 0 0
$$712$$ −7.85011e6 −0.580331
$$713$$ −1.57671e7 −1.16153
$$714$$ 0 0
$$715$$ 5.48995e6 0.401609
$$716$$ −7.80096e6 −0.568677
$$717$$ 0 0
$$718$$ 1.31003e6 0.0948355
$$719$$ 1.58388e7 1.14262 0.571308 0.820736i $$-0.306436\pi$$
0.571308 + 0.820736i $$0.306436\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −9.68161e6 −0.691202
$$723$$ 0 0
$$724$$ 8.71056e6 0.617589
$$725$$ −1.21323e7 −0.857235
$$726$$ 0 0
$$727$$ −6.31418e6 −0.443078 −0.221539 0.975151i $$-0.571108\pi$$
−0.221539 + 0.975151i $$0.571108\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 7.63922e6 0.530569
$$731$$ −1.33711e7 −0.925492
$$732$$ 0 0
$$733$$ −6.93003e6 −0.476404 −0.238202 0.971216i $$-0.576558\pi$$
−0.238202 + 0.971216i $$0.576558\pi$$
$$734$$ 1.14696e7 0.785791
$$735$$ 0 0
$$736$$ −2.26509e6 −0.154131
$$737$$ −2.72054e7 −1.84496
$$738$$ 0 0
$$739$$ 1.42331e7 0.958714 0.479357 0.877620i $$-0.340870\pi$$
0.479357 + 0.877620i $$0.340870\pi$$
$$740$$ 1.81293e6 0.121703
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 5.94460e6 0.395048 0.197524 0.980298i $$-0.436710\pi$$
0.197524 + 0.980298i $$0.436710\pi$$
$$744$$ 0 0
$$745$$ 4.54631e6 0.300102
$$746$$ 1.43212e7 0.942175
$$747$$ 0 0
$$748$$ −1.68072e7 −1.09835
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −682752. −0.0441736 −0.0220868 0.999756i $$-0.507031\pi$$
−0.0220868 + 0.999756i $$0.507031\pi$$
$$752$$ 1.37011e6 0.0883510
$$753$$ 0 0
$$754$$ −6.30149e6 −0.403659
$$755$$ −1.17664e7 −0.751233
$$756$$ 0 0
$$757$$ 1.46333e7 0.928116 0.464058 0.885805i $$-0.346393\pi$$
0.464058 + 0.885805i $$0.346393\pi$$
$$758$$ 6.56939e6 0.415291
$$759$$ 0 0
$$760$$ −392704. −0.0246622
$$761$$ −1.16367e7 −0.728399 −0.364200 0.931321i $$-0.618657\pi$$
−0.364200 + 0.931321i $$0.618657\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −6.02246e6 −0.373285
$$765$$ 0 0
$$766$$ −8.22790e6 −0.506661
$$767$$ 4.90865e6 0.301282
$$768$$ 0 0
$$769$$ −1.91472e7 −1.16759 −0.583793 0.811902i $$-0.698432\pi$$
−0.583793 + 0.811902i $$0.698432\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.35191e7 0.816405
$$773$$ −5.39261e6 −0.324601 −0.162301 0.986741i $$-0.551891\pi$$
−0.162301 + 0.986741i $$0.551891\pi$$
$$774$$ 0 0
$$775$$ −1.74565e7 −1.04400
$$776$$ −1.36768e6 −0.0815324
$$777$$ 0 0
$$778$$ −2.46457e6 −0.145979
$$779$$ −2.48791e6 −0.146890
$$780$$ 0 0
$$781$$ −6.03709e6 −0.354160
$$782$$ −1.39975e7 −0.818530
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.29757e7 0.751549
$$786$$ 0 0
$$787$$ −3.04348e6 −0.175159 −0.0875796 0.996158i $$-0.527913\pi$$
−0.0875796 + 0.996158i $$0.527913\pi$$
$$788$$ 7.88470e6 0.452345
$$789$$ 0 0
$$790$$ 9.29760e6 0.530033
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −1.16903e7 −0.660151
$$794$$ −8.76847e6 −0.493597
$$795$$ 0 0
$$796$$ 1.46364e7 0.818751
$$797$$ 2.29652e7 1.28063 0.640316 0.768111i $$-0.278803\pi$$
0.640316 + 0.768111i $$0.278803\pi$$
$$798$$ 0 0
$$799$$ 8.46686e6 0.469197
$$800$$ −2.50778e6 −0.138536
$$801$$ 0 0
$$802$$ −1.31382e7 −0.721271
$$803$$ −4.87735e7 −2.66928
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −9.06682e6 −0.491606
$$807$$ 0 0
$$808$$ −2.35610e6 −0.126959
$$809$$ −1.90787e7 −1.02489 −0.512445 0.858720i $$-0.671260\pi$$
−0.512445 + 0.858720i $$0.671260\pi$$
$$810$$ 0 0
$$811$$ −1.09414e7 −0.584147 −0.292074 0.956396i $$-0.594345\pi$$
−0.292074 + 0.956396i $$0.594345\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −1.15748e7 −0.612286
$$815$$ −1.23653e7 −0.652094
$$816$$ 0 0
$$817$$ 1.99467e6 0.104548
$$818$$ 1.44488e7 0.755001
$$819$$ 0 0
$$820$$ 4.38547e6 0.227762
$$821$$ −2.12594e7 −1.10076 −0.550380 0.834914i $$-0.685517\pi$$
−0.550380 + 0.834914i $$0.685517\pi$$
$$822$$ 0 0
$$823$$ −1.42256e7 −0.732103 −0.366052 0.930595i $$-0.619291\pi$$
−0.366052 + 0.930595i $$0.619291\pi$$
$$824$$ −6.68979e6 −0.343237
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.76103e6 −0.140381 −0.0701904 0.997534i $$-0.522361\pi$$
−0.0701904 + 0.997534i $$0.522361\pi$$
$$828$$ 0 0
$$829$$ 3.82147e7 1.93127 0.965637 0.259895i $$-0.0836880\pi$$
0.965637 + 0.259895i $$0.0836880\pi$$
$$830$$ −668512. −0.0336832
$$831$$ 0 0
$$832$$ −1.30253e6 −0.0652347
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.12582e6 0.155149
$$836$$ 2.50726e6 0.124075
$$837$$ 0 0
$$838$$ 2.16596e7 1.06547
$$839$$ 1.06044e7 0.520094 0.260047 0.965596i $$-0.416262\pi$$
0.260047 + 0.965596i $$0.416262\pi$$
$$840$$ 0 0
$$841$$ 4.03097e6 0.196526
$$842$$ 1.44252e7 0.701198
$$843$$ 0 0
$$844$$ 4.98848e6 0.241053
$$845$$ −7.02439e6 −0.338429
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 8.53862e6 0.407754
$$849$$ 0 0
$$850$$ −1.54973e7 −0.735712
$$851$$ −9.63990e6 −0.456298
$$852$$ 0 0
$$853$$ 4.07009e7 1.91527 0.957637 0.287977i $$-0.0929826\pi$$
0.957637 + 0.287977i $$0.0929826\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −1.37242e7 −0.640179
$$857$$ −3.10120e7 −1.44237 −0.721187 0.692741i $$-0.756403\pi$$
−0.721187 + 0.692741i $$0.756403\pi$$
$$858$$ 0 0
$$859$$ −1.09104e7 −0.504495 −0.252247 0.967663i $$-0.581170\pi$$
−0.252247 + 0.967663i $$0.581170\pi$$
$$860$$ −3.51603e6 −0.162109
$$861$$ 0 0
$$862$$ 1.11286e7 0.510118
$$863$$ −1.04089e7 −0.475751 −0.237875 0.971296i $$-0.576451\pi$$
−0.237875 + 0.971296i $$0.576451\pi$$
$$864$$ 0 0
$$865$$ 1.32307e7 0.601234
$$866$$ −2.51048e7 −1.13753
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −5.93616e7 −2.66659
$$870$$ 0 0
$$871$$ −1.30291e7 −0.581928
$$872$$ 1.84307e6 0.0820826
$$873$$ 0 0
$$874$$ 2.08813e6 0.0924652
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.64064e7 0.720299 0.360150 0.932895i $$-0.382726\pi$$
0.360150 + 0.932895i $$0.382726\pi$$
$$878$$ −2.56637e6 −0.112352
$$879$$ 0 0
$$880$$ −4.41958e6 −0.192387
$$881$$ 1.48577e7 0.644927 0.322464 0.946582i $$-0.395489\pi$$
0.322464 + 0.946582i $$0.395489\pi$$
$$882$$ 0 0
$$883$$ −2.72018e7 −1.17407 −0.587037 0.809560i $$-0.699706\pi$$
−0.587037 + 0.809560i $$0.699706\pi$$
$$884$$ −8.04922e6 −0.346436
$$885$$ 0 0
$$886$$ −2.42218e7 −1.03663
$$887$$ 2.71242e7 1.15757 0.578785 0.815480i $$-0.303527\pi$$
0.578785 + 0.815480i $$0.303527\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −1.27564e7 −0.539827
$$891$$ 0 0
$$892$$ 2.06042e7 0.867047
$$893$$ −1.26307e6 −0.0530029
$$894$$ 0 0
$$895$$ −1.26766e7 −0.528986
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.06673e7 0.855248
$$899$$ 3.53121e7 1.45722
$$900$$ 0 0
$$901$$ 5.27660e7 2.16542
$$902$$ −2.79996e7 −1.14587
$$903$$ 0 0
$$904$$ 3.58490e6 0.145900
$$905$$ 1.41547e7 0.574484
$$906$$ 0 0
$$907$$ −8.42269e6 −0.339964 −0.169982 0.985447i $$-0.554371\pi$$
−0.169982 + 0.985447i $$0.554371\pi$$
$$908$$ 2.06248e7 0.830187
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3.08637e7 −1.23212 −0.616060 0.787700i $$-0.711272\pi$$
−0.616060 + 0.787700i $$0.711272\pi$$
$$912$$ 0 0
$$913$$ 4.26819e6 0.169460
$$914$$ −911192. −0.0360782
$$915$$ 0 0
$$916$$ −1.08514e7 −0.427315
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.93895e6 0.192906 0.0964531 0.995338i $$-0.469250\pi$$
0.0964531 + 0.995338i $$0.469250\pi$$
$$920$$ −3.68077e6 −0.143373
$$921$$ 0 0
$$922$$ 2.34058e6 0.0906770
$$923$$ −2.89126e6 −0.111707
$$924$$ 0 0
$$925$$ −1.06727e7 −0.410130
$$926$$ −1.36581e7 −0.523437
$$927$$ 0 0
$$928$$ 5.07290e6 0.193369
$$929$$ 5.62575e6 0.213866 0.106933 0.994266i $$-0.465897\pi$$
0.106933 + 0.994266i $$0.465897\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1.78770e7 0.674146
$$933$$ 0 0
$$934$$ 2.86520e6 0.107470
$$935$$ −2.73116e7 −1.02169
$$936$$ 0 0
$$937$$ −2.60073e7 −0.967714 −0.483857 0.875147i $$-0.660764\pi$$
−0.483857 + 0.875147i $$0.660764\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 2.22643e6 0.0821845
$$941$$ 3.02160e6 0.111241 0.0556203 0.998452i $$-0.482286\pi$$
0.0556203 + 0.998452i $$0.482286\pi$$
$$942$$ 0 0
$$943$$ −2.33189e7 −0.853943
$$944$$ −3.95162e6 −0.144326
$$945$$ 0 0
$$946$$ 2.24485e7 0.815567
$$947$$ 3.48282e7 1.26199 0.630995 0.775787i $$-0.282647\pi$$
0.630995 + 0.775787i $$0.282647\pi$$
$$948$$ 0 0
$$949$$ −2.33584e7 −0.841932
$$950$$ 2.31186e6 0.0831097
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.39009e6 0.334917 0.167459 0.985879i $$-0.446444\pi$$
0.167459 + 0.985879i $$0.446444\pi$$
$$954$$ 0 0
$$955$$ −9.78650e6 −0.347232
$$956$$ 2.01913e7 0.714528
$$957$$ 0 0
$$958$$ 2.09637e7 0.737996
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 2.21792e7 0.774708
$$962$$ −5.54338e6 −0.193124
$$963$$ 0 0
$$964$$ −1.51715e7 −0.525818
$$965$$ 2.19686e7 0.759423
$$966$$ 0 0
$$967$$ 1.44768e7 0.497860 0.248930 0.968521i $$-0.419921\pi$$
0.248930 + 0.968521i $$0.419921\pi$$
$$968$$ 1.79101e7 0.614340
$$969$$ 0 0
$$970$$ −2.22248e6 −0.0758418
$$971$$ 9.24976e6 0.314834 0.157417 0.987532i $$-0.449683\pi$$
0.157417 + 0.987532i $$0.449683\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 4.46806e6 0.150911
$$975$$ 0 0
$$976$$ 9.41107e6 0.316238
$$977$$ 4.97780e7 1.66840 0.834202 0.551459i $$-0.185929\pi$$
0.834202 + 0.551459i $$0.185929\pi$$
$$978$$ 0 0
$$979$$ 8.14449e7 2.71586
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −5.37830e6 −0.177978
$$983$$ −8.95601e6 −0.295618 −0.147809 0.989016i $$-0.547222\pi$$
−0.147809 + 0.989016i $$0.547222\pi$$
$$984$$ 0 0
$$985$$ 1.28126e7 0.420773
$$986$$ 3.13489e7 1.02690
$$987$$ 0 0
$$988$$ 1.20077e6 0.0391351
$$989$$ 1.86958e7 0.607790
$$990$$ 0 0
$$991$$ 2.62400e7 0.848751 0.424376 0.905486i $$-0.360494\pi$$
0.424376 + 0.905486i $$0.360494\pi$$
$$992$$ 7.29907e6 0.235499
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.37842e7 0.761606
$$996$$ 0 0
$$997$$ −2.80506e7 −0.893727 −0.446863 0.894602i $$-0.647459\pi$$
−0.446863 + 0.894602i $$0.647459\pi$$
$$998$$ −2.61859e7 −0.832226
$$999$$ 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 882.6.a.s.1.1 1
3.2 odd 2 294.6.a.b.1.1 1
7.6 odd 2 126.6.a.i.1.1 1
21.2 odd 6 294.6.e.p.67.1 2
21.5 even 6 294.6.e.i.67.1 2
21.11 odd 6 294.6.e.p.79.1 2
21.17 even 6 294.6.e.i.79.1 2
21.20 even 2 42.6.a.d.1.1 1
28.27 even 2 1008.6.a.j.1.1 1
84.83 odd 2 336.6.a.h.1.1 1
105.62 odd 4 1050.6.g.i.799.1 2
105.83 odd 4 1050.6.g.i.799.2 2
105.104 even 2 1050.6.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.d.1.1 1 21.20 even 2
126.6.a.i.1.1 1 7.6 odd 2
294.6.a.b.1.1 1 3.2 odd 2
294.6.e.i.67.1 2 21.5 even 6
294.6.e.i.79.1 2 21.17 even 6
294.6.e.p.67.1 2 21.2 odd 6
294.6.e.p.79.1 2 21.11 odd 6
336.6.a.h.1.1 1 84.83 odd 2
882.6.a.s.1.1 1 1.1 even 1 trivial
1008.6.a.j.1.1 1 28.27 even 2
1050.6.a.k.1.1 1 105.104 even 2
1050.6.g.i.799.1 2 105.62 odd 4
1050.6.g.i.799.2 2 105.83 odd 4