# Properties

 Label 1280.4.d.a.641.2 Level $1280$ Weight $4$ Character 1280.641 Analytic conductor $75.522$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$75.5224448073$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 641.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1280.641 Dual form 1280.4.d.a.641.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +16.0000i q^{11} +58.0000i q^{13} -30.0000 q^{15} -70.0000 q^{17} -4.00000i q^{19} -204.000i q^{21} -134.000 q^{23} -25.0000 q^{25} +108.000i q^{27} -242.000i q^{29} -100.000 q^{31} -96.0000 q^{33} -170.000i q^{35} +438.000i q^{37} -348.000 q^{39} +138.000 q^{41} +178.000i q^{43} -45.0000i q^{45} -22.0000 q^{47} +813.000 q^{49} -420.000i q^{51} -162.000i q^{53} -80.0000 q^{55} +24.0000 q^{57} -268.000i q^{59} +250.000i q^{61} +306.000 q^{63} -290.000 q^{65} -422.000i q^{67} -804.000i q^{69} -852.000 q^{71} -306.000 q^{73} -150.000i q^{75} -544.000i q^{77} +456.000 q^{79} -891.000 q^{81} -434.000i q^{83} -350.000i q^{85} +1452.00 q^{87} +726.000 q^{89} -1972.00i q^{91} -600.000i q^{93} +20.0000 q^{95} +1378.00 q^{97} -144.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 68 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q - 68 * q^7 - 18 * q^9 $$2 q - 68 q^{7} - 18 q^{9} - 60 q^{15} - 140 q^{17} - 268 q^{23} - 50 q^{25} - 200 q^{31} - 192 q^{33} - 696 q^{39} + 276 q^{41} - 44 q^{47} + 1626 q^{49} - 160 q^{55} + 48 q^{57} + 612 q^{63} - 580 q^{65} - 1704 q^{71} - 612 q^{73} + 912 q^{79} - 1782 q^{81} + 2904 q^{87} + 1452 q^{89} + 40 q^{95} + 2756 q^{97}+O(q^{100})$$ 2 * q - 68 * q^7 - 18 * q^9 - 60 * q^15 - 140 * q^17 - 268 * q^23 - 50 * q^25 - 200 * q^31 - 192 * q^33 - 696 * q^39 + 276 * q^41 - 44 * q^47 + 1626 * q^49 - 160 * q^55 + 48 * q^57 + 612 * q^63 - 580 * q^65 - 1704 * q^71 - 612 * q^73 + 912 * q^79 - 1782 * q^81 + 2904 * q^87 + 1452 * q^89 + 40 * q^95 + 2756 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 6.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 5.00000i 0.447214i
$$6$$ 0 0
$$7$$ −34.0000 −1.83583 −0.917914 0.396780i $$-0.870128\pi$$
−0.917914 + 0.396780i $$0.870128\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 16.0000i 0.438562i 0.975662 + 0.219281i $$0.0703711\pi$$
−0.975662 + 0.219281i $$0.929629\pi$$
$$12$$ 0 0
$$13$$ 58.0000i 1.23741i 0.785624 + 0.618704i $$0.212342\pi$$
−0.785624 + 0.618704i $$0.787658\pi$$
$$14$$ 0 0
$$15$$ −30.0000 −0.516398
$$16$$ 0 0
$$17$$ −70.0000 −0.998676 −0.499338 0.866407i $$-0.666423\pi$$
−0.499338 + 0.866407i $$0.666423\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.0482980i −0.999708 0.0241490i $$-0.992312\pi$$
0.999708 0.0241490i $$-0.00768762\pi$$
$$20$$ 0 0
$$21$$ − 204.000i − 2.11983i
$$22$$ 0 0
$$23$$ −134.000 −1.21482 −0.607412 0.794387i $$-0.707792\pi$$
−0.607412 + 0.794387i $$0.707792\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 108.000i 0.769800i
$$28$$ 0 0
$$29$$ − 242.000i − 1.54960i −0.632209 0.774798i $$-0.717852\pi$$
0.632209 0.774798i $$-0.282148\pi$$
$$30$$ 0 0
$$31$$ −100.000 −0.579372 −0.289686 0.957122i $$-0.593551\pi$$
−0.289686 + 0.957122i $$0.593551\pi$$
$$32$$ 0 0
$$33$$ −96.0000 −0.506408
$$34$$ 0 0
$$35$$ − 170.000i − 0.821007i
$$36$$ 0 0
$$37$$ 438.000i 1.94613i 0.230534 + 0.973064i $$0.425953\pi$$
−0.230534 + 0.973064i $$0.574047\pi$$
$$38$$ 0 0
$$39$$ −348.000 −1.42884
$$40$$ 0 0
$$41$$ 138.000 0.525658 0.262829 0.964842i $$-0.415344\pi$$
0.262829 + 0.964842i $$0.415344\pi$$
$$42$$ 0 0
$$43$$ 178.000i 0.631273i 0.948880 + 0.315637i $$0.102218\pi$$
−0.948880 + 0.315637i $$0.897782\pi$$
$$44$$ 0 0
$$45$$ − 45.0000i − 0.149071i
$$46$$ 0 0
$$47$$ −22.0000 −0.0682772 −0.0341386 0.999417i $$-0.510869\pi$$
−0.0341386 + 0.999417i $$0.510869\pi$$
$$48$$ 0 0
$$49$$ 813.000 2.37026
$$50$$ 0 0
$$51$$ − 420.000i − 1.15317i
$$52$$ 0 0
$$53$$ − 162.000i − 0.419857i −0.977717 0.209928i $$-0.932677\pi$$
0.977717 0.209928i $$-0.0673231\pi$$
$$54$$ 0 0
$$55$$ −80.0000 −0.196131
$$56$$ 0 0
$$57$$ 24.0000 0.0557698
$$58$$ 0 0
$$59$$ − 268.000i − 0.591367i −0.955286 0.295683i $$-0.904453\pi$$
0.955286 0.295683i $$-0.0955473\pi$$
$$60$$ 0 0
$$61$$ 250.000i 0.524741i 0.964967 + 0.262371i $$0.0845043\pi$$
−0.964967 + 0.262371i $$0.915496\pi$$
$$62$$ 0 0
$$63$$ 306.000 0.611942
$$64$$ 0 0
$$65$$ −290.000 −0.553386
$$66$$ 0 0
$$67$$ − 422.000i − 0.769485i −0.923024 0.384743i $$-0.874290\pi$$
0.923024 0.384743i $$-0.125710\pi$$
$$68$$ 0 0
$$69$$ − 804.000i − 1.40276i
$$70$$ 0 0
$$71$$ −852.000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −306.000 −0.490611 −0.245305 0.969446i $$-0.578888\pi$$
−0.245305 + 0.969446i $$0.578888\pi$$
$$74$$ 0 0
$$75$$ − 150.000i − 0.230940i
$$76$$ 0 0
$$77$$ − 544.000i − 0.805124i
$$78$$ 0 0
$$79$$ 456.000 0.649418 0.324709 0.945814i $$-0.394734\pi$$
0.324709 + 0.945814i $$0.394734\pi$$
$$80$$ 0 0
$$81$$ −891.000 −1.22222
$$82$$ 0 0
$$83$$ − 434.000i − 0.573948i −0.957938 0.286974i $$-0.907351\pi$$
0.957938 0.286974i $$-0.0926493\pi$$
$$84$$ 0 0
$$85$$ − 350.000i − 0.446622i
$$86$$ 0 0
$$87$$ 1452.00 1.78932
$$88$$ 0 0
$$89$$ 726.000 0.864672 0.432336 0.901712i $$-0.357689\pi$$
0.432336 + 0.901712i $$0.357689\pi$$
$$90$$ 0 0
$$91$$ − 1972.00i − 2.27167i
$$92$$ 0 0
$$93$$ − 600.000i − 0.669001i
$$94$$ 0 0
$$95$$ 20.0000 0.0215995
$$96$$ 0 0
$$97$$ 1378.00 1.44242 0.721210 0.692717i $$-0.243586\pi$$
0.721210 + 0.692717i $$0.243586\pi$$
$$98$$ 0 0
$$99$$ − 144.000i − 0.146187i
$$100$$ 0 0
$$101$$ − 126.000i − 0.124133i −0.998072 0.0620667i $$-0.980231\pi$$
0.998072 0.0620667i $$-0.0197691\pi$$
$$102$$ 0 0
$$103$$ −1262.00 −1.20727 −0.603634 0.797262i $$-0.706281\pi$$
−0.603634 + 0.797262i $$0.706281\pi$$
$$104$$ 0 0
$$105$$ 1020.00 0.948017
$$106$$ 0 0
$$107$$ 510.000i 0.460781i 0.973098 + 0.230390i $$0.0740003\pi$$
−0.973098 + 0.230390i $$0.926000\pi$$
$$108$$ 0 0
$$109$$ 26.0000i 0.0228472i 0.999935 + 0.0114236i $$0.00363633\pi$$
−0.999935 + 0.0114236i $$0.996364\pi$$
$$110$$ 0 0
$$111$$ −2628.00 −2.24720
$$112$$ 0 0
$$113$$ 1242.00 1.03396 0.516980 0.855997i $$-0.327056\pi$$
0.516980 + 0.855997i $$0.327056\pi$$
$$114$$ 0 0
$$115$$ − 670.000i − 0.543285i
$$116$$ 0 0
$$117$$ − 522.000i − 0.412469i
$$118$$ 0 0
$$119$$ 2380.00 1.83340
$$120$$ 0 0
$$121$$ 1075.00 0.807663
$$122$$ 0 0
$$123$$ 828.000i 0.606978i
$$124$$ 0 0
$$125$$ − 125.000i − 0.0894427i
$$126$$ 0 0
$$127$$ 978.000 0.683334 0.341667 0.939821i $$-0.389008\pi$$
0.341667 + 0.939821i $$0.389008\pi$$
$$128$$ 0 0
$$129$$ −1068.00 −0.728931
$$130$$ 0 0
$$131$$ 912.000i 0.608258i 0.952631 + 0.304129i $$0.0983654\pi$$
−0.952631 + 0.304129i $$0.901635\pi$$
$$132$$ 0 0
$$133$$ 136.000i 0.0886669i
$$134$$ 0 0
$$135$$ −540.000 −0.344265
$$136$$ 0 0
$$137$$ 926.000 0.577471 0.288735 0.957409i $$-0.406765\pi$$
0.288735 + 0.957409i $$0.406765\pi$$
$$138$$ 0 0
$$139$$ 516.000i 0.314867i 0.987530 + 0.157434i $$0.0503220\pi$$
−0.987530 + 0.157434i $$0.949678\pi$$
$$140$$ 0 0
$$141$$ − 132.000i − 0.0788398i
$$142$$ 0 0
$$143$$ −928.000 −0.542680
$$144$$ 0 0
$$145$$ 1210.00 0.693000
$$146$$ 0 0
$$147$$ 4878.00i 2.73694i
$$148$$ 0 0
$$149$$ 958.000i 0.526728i 0.964697 + 0.263364i $$0.0848320\pi$$
−0.964697 + 0.263364i $$0.915168\pi$$
$$150$$ 0 0
$$151$$ 332.000 0.178926 0.0894628 0.995990i $$-0.471485\pi$$
0.0894628 + 0.995990i $$0.471485\pi$$
$$152$$ 0 0
$$153$$ 630.000 0.332892
$$154$$ 0 0
$$155$$ − 500.000i − 0.259103i
$$156$$ 0 0
$$157$$ − 1022.00i − 0.519519i −0.965673 0.259759i $$-0.916357\pi$$
0.965673 0.259759i $$-0.0836433\pi$$
$$158$$ 0 0
$$159$$ 972.000 0.484809
$$160$$ 0 0
$$161$$ 4556.00 2.23021
$$162$$ 0 0
$$163$$ 926.000i 0.444969i 0.974936 + 0.222484i $$0.0714166\pi$$
−0.974936 + 0.222484i $$0.928583\pi$$
$$164$$ 0 0
$$165$$ − 480.000i − 0.226472i
$$166$$ 0 0
$$167$$ 654.000 0.303042 0.151521 0.988454i $$-0.451583\pi$$
0.151521 + 0.988454i $$0.451583\pi$$
$$168$$ 0 0
$$169$$ −1167.00 −0.531179
$$170$$ 0 0
$$171$$ 36.0000i 0.0160993i
$$172$$ 0 0
$$173$$ − 1294.00i − 0.568676i −0.958724 0.284338i $$-0.908226\pi$$
0.958724 0.284338i $$-0.0917738\pi$$
$$174$$ 0 0
$$175$$ 850.000 0.367165
$$176$$ 0 0
$$177$$ 1608.00 0.682851
$$178$$ 0 0
$$179$$ 2836.00i 1.18420i 0.805863 + 0.592102i $$0.201702\pi$$
−0.805863 + 0.592102i $$0.798298\pi$$
$$180$$ 0 0
$$181$$ − 1742.00i − 0.715369i −0.933842 0.357685i $$-0.883566\pi$$
0.933842 0.357685i $$-0.116434\pi$$
$$182$$ 0 0
$$183$$ −1500.00 −0.605919
$$184$$ 0 0
$$185$$ −2190.00 −0.870335
$$186$$ 0 0
$$187$$ − 1120.00i − 0.437981i
$$188$$ 0 0
$$189$$ − 3672.00i − 1.41322i
$$190$$ 0 0
$$191$$ −4460.00 −1.68960 −0.844802 0.535079i $$-0.820282\pi$$
−0.844802 + 0.535079i $$0.820282\pi$$
$$192$$ 0 0
$$193$$ −3782.00 −1.41054 −0.705270 0.708939i $$-0.749174\pi$$
−0.705270 + 0.708939i $$0.749174\pi$$
$$194$$ 0 0
$$195$$ − 1740.00i − 0.638995i
$$196$$ 0 0
$$197$$ − 4474.00i − 1.61807i −0.587762 0.809034i $$-0.699991\pi$$
0.587762 0.809034i $$-0.300009\pi$$
$$198$$ 0 0
$$199$$ 3608.00 1.28525 0.642624 0.766182i $$-0.277846\pi$$
0.642624 + 0.766182i $$0.277846\pi$$
$$200$$ 0 0
$$201$$ 2532.00 0.888525
$$202$$ 0 0
$$203$$ 8228.00i 2.84479i
$$204$$ 0 0
$$205$$ 690.000i 0.235081i
$$206$$ 0 0
$$207$$ 1206.00 0.404941
$$208$$ 0 0
$$209$$ 64.0000 0.0211817
$$210$$ 0 0
$$211$$ 256.000i 0.0835250i 0.999128 + 0.0417625i $$0.0132973\pi$$
−0.999128 + 0.0417625i $$0.986703\pi$$
$$212$$ 0 0
$$213$$ − 5112.00i − 1.64445i
$$214$$ 0 0
$$215$$ −890.000 −0.282314
$$216$$ 0 0
$$217$$ 3400.00 1.06363
$$218$$ 0 0
$$219$$ − 1836.00i − 0.566509i
$$220$$ 0 0
$$221$$ − 4060.00i − 1.23577i
$$222$$ 0 0
$$223$$ 5158.00 1.54890 0.774451 0.632634i $$-0.218026\pi$$
0.774451 + 0.632634i $$0.218026\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 2226.00i 0.650858i 0.945566 + 0.325429i $$0.105509\pi$$
−0.945566 + 0.325429i $$0.894491\pi$$
$$228$$ 0 0
$$229$$ − 2086.00i − 0.601951i −0.953632 0.300975i $$-0.902688\pi$$
0.953632 0.300975i $$-0.0973122\pi$$
$$230$$ 0 0
$$231$$ 3264.00 0.929677
$$232$$ 0 0
$$233$$ 5718.00 1.60772 0.803860 0.594819i $$-0.202776\pi$$
0.803860 + 0.594819i $$0.202776\pi$$
$$234$$ 0 0
$$235$$ − 110.000i − 0.0305345i
$$236$$ 0 0
$$237$$ 2736.00i 0.749883i
$$238$$ 0 0
$$239$$ 3624.00 0.980825 0.490412 0.871491i $$-0.336846\pi$$
0.490412 + 0.871491i $$0.336846\pi$$
$$240$$ 0 0
$$241$$ −82.0000 −0.0219174 −0.0109587 0.999940i $$-0.503488\pi$$
−0.0109587 + 0.999940i $$0.503488\pi$$
$$242$$ 0 0
$$243$$ − 2430.00i − 0.641500i
$$244$$ 0 0
$$245$$ 4065.00i 1.06001i
$$246$$ 0 0
$$247$$ 232.000 0.0597644
$$248$$ 0 0
$$249$$ 2604.00 0.662738
$$250$$ 0 0
$$251$$ − 5040.00i − 1.26742i −0.773571 0.633709i $$-0.781532\pi$$
0.773571 0.633709i $$-0.218468\pi$$
$$252$$ 0 0
$$253$$ − 2144.00i − 0.532775i
$$254$$ 0 0
$$255$$ 2100.00 0.515714
$$256$$ 0 0
$$257$$ −2310.00 −0.560676 −0.280338 0.959901i $$-0.590447\pi$$
−0.280338 + 0.959901i $$0.590447\pi$$
$$258$$ 0 0
$$259$$ − 14892.0i − 3.57276i
$$260$$ 0 0
$$261$$ 2178.00i 0.516532i
$$262$$ 0 0
$$263$$ −4110.00 −0.963625 −0.481813 0.876274i $$-0.660021\pi$$
−0.481813 + 0.876274i $$0.660021\pi$$
$$264$$ 0 0
$$265$$ 810.000 0.187766
$$266$$ 0 0
$$267$$ 4356.00i 0.998438i
$$268$$ 0 0
$$269$$ 746.000i 0.169087i 0.996420 + 0.0845435i $$0.0269432\pi$$
−0.996420 + 0.0845435i $$0.973057\pi$$
$$270$$ 0 0
$$271$$ 4596.00 1.03021 0.515105 0.857127i $$-0.327753\pi$$
0.515105 + 0.857127i $$0.327753\pi$$
$$272$$ 0 0
$$273$$ 11832.0 2.62310
$$274$$ 0 0
$$275$$ − 400.000i − 0.0877124i
$$276$$ 0 0
$$277$$ 2206.00i 0.478504i 0.970957 + 0.239252i $$0.0769023\pi$$
−0.970957 + 0.239252i $$0.923098\pi$$
$$278$$ 0 0
$$279$$ 900.000 0.193124
$$280$$ 0 0
$$281$$ −8278.00 −1.75738 −0.878691 0.477392i $$-0.841582\pi$$
−0.878691 + 0.477392i $$0.841582\pi$$
$$282$$ 0 0
$$283$$ 1178.00i 0.247438i 0.992317 + 0.123719i $$0.0394821\pi$$
−0.992317 + 0.123719i $$0.960518\pi$$
$$284$$ 0 0
$$285$$ 120.000i 0.0249410i
$$286$$ 0 0
$$287$$ −4692.00 −0.965017
$$288$$ 0 0
$$289$$ −13.0000 −0.00264604
$$290$$ 0 0
$$291$$ 8268.00i 1.66556i
$$292$$ 0 0
$$293$$ − 106.000i − 0.0211351i −0.999944 0.0105676i $$-0.996636\pi$$
0.999944 0.0105676i $$-0.00336382\pi$$
$$294$$ 0 0
$$295$$ 1340.00 0.264467
$$296$$ 0 0
$$297$$ −1728.00 −0.337605
$$298$$ 0 0
$$299$$ − 7772.00i − 1.50323i
$$300$$ 0 0
$$301$$ − 6052.00i − 1.15891i
$$302$$ 0 0
$$303$$ 756.000 0.143337
$$304$$ 0 0
$$305$$ −1250.00 −0.234671
$$306$$ 0 0
$$307$$ − 8134.00i − 1.51216i −0.654482 0.756078i $$-0.727113\pi$$
0.654482 0.756078i $$-0.272887\pi$$
$$308$$ 0 0
$$309$$ − 7572.00i − 1.39403i
$$310$$ 0 0
$$311$$ −4396.00 −0.801525 −0.400763 0.916182i $$-0.631255\pi$$
−0.400763 + 0.916182i $$0.631255\pi$$
$$312$$ 0 0
$$313$$ −4826.00 −0.871507 −0.435753 0.900066i $$-0.643518\pi$$
−0.435753 + 0.900066i $$0.643518\pi$$
$$314$$ 0 0
$$315$$ 1530.00i 0.273669i
$$316$$ 0 0
$$317$$ 7026.00i 1.24486i 0.782677 + 0.622428i $$0.213854\pi$$
−0.782677 + 0.622428i $$0.786146\pi$$
$$318$$ 0 0
$$319$$ 3872.00 0.679594
$$320$$ 0 0
$$321$$ −3060.00 −0.532064
$$322$$ 0 0
$$323$$ 280.000i 0.0482341i
$$324$$ 0 0
$$325$$ − 1450.00i − 0.247482i
$$326$$ 0 0
$$327$$ −156.000 −0.0263817
$$328$$ 0 0
$$329$$ 748.000 0.125345
$$330$$ 0 0
$$331$$ 8808.00i 1.46263i 0.682038 + 0.731316i $$0.261094\pi$$
−0.682038 + 0.731316i $$0.738906\pi$$
$$332$$ 0 0
$$333$$ − 3942.00i − 0.648710i
$$334$$ 0 0
$$335$$ 2110.00 0.344124
$$336$$ 0 0
$$337$$ 5602.00 0.905520 0.452760 0.891632i $$-0.350439\pi$$
0.452760 + 0.891632i $$0.350439\pi$$
$$338$$ 0 0
$$339$$ 7452.00i 1.19391i
$$340$$ 0 0
$$341$$ − 1600.00i − 0.254090i
$$342$$ 0 0
$$343$$ −15980.0 −2.51557
$$344$$ 0 0
$$345$$ 4020.00 0.627332
$$346$$ 0 0
$$347$$ − 6634.00i − 1.02632i −0.858294 0.513158i $$-0.828475\pi$$
0.858294 0.513158i $$-0.171525\pi$$
$$348$$ 0 0
$$349$$ 3198.00i 0.490501i 0.969460 + 0.245251i $$0.0788703\pi$$
−0.969460 + 0.245251i $$0.921130\pi$$
$$350$$ 0 0
$$351$$ −6264.00 −0.952557
$$352$$ 0 0
$$353$$ −5230.00 −0.788569 −0.394284 0.918988i $$-0.629008\pi$$
−0.394284 + 0.918988i $$0.629008\pi$$
$$354$$ 0 0
$$355$$ − 4260.00i − 0.636894i
$$356$$ 0 0
$$357$$ 14280.0i 2.11702i
$$358$$ 0 0
$$359$$ −312.000 −0.0458683 −0.0229342 0.999737i $$-0.507301\pi$$
−0.0229342 + 0.999737i $$0.507301\pi$$
$$360$$ 0 0
$$361$$ 6843.00 0.997667
$$362$$ 0 0
$$363$$ 6450.00i 0.932609i
$$364$$ 0 0
$$365$$ − 1530.00i − 0.219408i
$$366$$ 0 0
$$367$$ −10790.0 −1.53470 −0.767348 0.641231i $$-0.778424\pi$$
−0.767348 + 0.641231i $$0.778424\pi$$
$$368$$ 0 0
$$369$$ −1242.00 −0.175219
$$370$$ 0 0
$$371$$ 5508.00i 0.770785i
$$372$$ 0 0
$$373$$ 4190.00i 0.581635i 0.956778 + 0.290818i $$0.0939273\pi$$
−0.956778 + 0.290818i $$0.906073\pi$$
$$374$$ 0 0
$$375$$ 750.000 0.103280
$$376$$ 0 0
$$377$$ 14036.0 1.91748
$$378$$ 0 0
$$379$$ − 6980.00i − 0.946012i −0.881059 0.473006i $$-0.843169\pi$$
0.881059 0.473006i $$-0.156831\pi$$
$$380$$ 0 0
$$381$$ 5868.00i 0.789047i
$$382$$ 0 0
$$383$$ −13962.0 −1.86273 −0.931364 0.364089i $$-0.881380\pi$$
−0.931364 + 0.364089i $$0.881380\pi$$
$$384$$ 0 0
$$385$$ 2720.00 0.360062
$$386$$ 0 0
$$387$$ − 1602.00i − 0.210424i
$$388$$ 0 0
$$389$$ − 3810.00i − 0.496593i −0.968684 0.248296i $$-0.920129\pi$$
0.968684 0.248296i $$-0.0798707\pi$$
$$390$$ 0 0
$$391$$ 9380.00 1.21321
$$392$$ 0 0
$$393$$ −5472.00 −0.702356
$$394$$ 0 0
$$395$$ 2280.00i 0.290428i
$$396$$ 0 0
$$397$$ − 9158.00i − 1.15775i −0.815416 0.578875i $$-0.803492\pi$$
0.815416 0.578875i $$-0.196508\pi$$
$$398$$ 0 0
$$399$$ −816.000 −0.102384
$$400$$ 0 0
$$401$$ 4866.00 0.605976 0.302988 0.952994i $$-0.402016\pi$$
0.302988 + 0.952994i $$0.402016\pi$$
$$402$$ 0 0
$$403$$ − 5800.00i − 0.716920i
$$404$$ 0 0
$$405$$ − 4455.00i − 0.546594i
$$406$$ 0 0
$$407$$ −7008.00 −0.853498
$$408$$ 0 0
$$409$$ −13486.0 −1.63042 −0.815208 0.579169i $$-0.803377\pi$$
−0.815208 + 0.579169i $$0.803377\pi$$
$$410$$ 0 0
$$411$$ 5556.00i 0.666806i
$$412$$ 0 0
$$413$$ 9112.00i 1.08565i
$$414$$ 0 0
$$415$$ 2170.00 0.256677
$$416$$ 0 0
$$417$$ −3096.00 −0.363577
$$418$$ 0 0
$$419$$ − 5628.00i − 0.656195i −0.944644 0.328098i $$-0.893593\pi$$
0.944644 0.328098i $$-0.106407\pi$$
$$420$$ 0 0
$$421$$ − 7938.00i − 0.918942i −0.888193 0.459471i $$-0.848039\pi$$
0.888193 0.459471i $$-0.151961\pi$$
$$422$$ 0 0
$$423$$ 198.000 0.0227591
$$424$$ 0 0
$$425$$ 1750.00 0.199735
$$426$$ 0 0
$$427$$ − 8500.00i − 0.963334i
$$428$$ 0 0
$$429$$ − 5568.00i − 0.626633i
$$430$$ 0 0
$$431$$ −1916.00 −0.214131 −0.107066 0.994252i $$-0.534145\pi$$
−0.107066 + 0.994252i $$0.534145\pi$$
$$432$$ 0 0
$$433$$ −16510.0 −1.83238 −0.916189 0.400746i $$-0.868751\pi$$
−0.916189 + 0.400746i $$0.868751\pi$$
$$434$$ 0 0
$$435$$ 7260.00i 0.800208i
$$436$$ 0 0
$$437$$ 536.000i 0.0586736i
$$438$$ 0 0
$$439$$ −1256.00 −0.136550 −0.0682752 0.997667i $$-0.521750\pi$$
−0.0682752 + 0.997667i $$0.521750\pi$$
$$440$$ 0 0
$$441$$ −7317.00 −0.790087
$$442$$ 0 0
$$443$$ − 12222.0i − 1.31080i −0.755282 0.655400i $$-0.772500\pi$$
0.755282 0.655400i $$-0.227500\pi$$
$$444$$ 0 0
$$445$$ 3630.00i 0.386693i
$$446$$ 0 0
$$447$$ −5748.00 −0.608213
$$448$$ 0 0
$$449$$ −5946.00 −0.624965 −0.312482 0.949924i $$-0.601160\pi$$
−0.312482 + 0.949924i $$0.601160\pi$$
$$450$$ 0 0
$$451$$ 2208.00i 0.230534i
$$452$$ 0 0
$$453$$ 1992.00i 0.206606i
$$454$$ 0 0
$$455$$ 9860.00 1.01592
$$456$$ 0 0
$$457$$ −1258.00 −0.128768 −0.0643838 0.997925i $$-0.520508\pi$$
−0.0643838 + 0.997925i $$0.520508\pi$$
$$458$$ 0 0
$$459$$ − 7560.00i − 0.768781i
$$460$$ 0 0
$$461$$ 16422.0i 1.65911i 0.558426 + 0.829554i $$0.311405\pi$$
−0.558426 + 0.829554i $$0.688595\pi$$
$$462$$ 0 0
$$463$$ −2658.00 −0.266799 −0.133399 0.991062i $$-0.542589\pi$$
−0.133399 + 0.991062i $$0.542589\pi$$
$$464$$ 0 0
$$465$$ 3000.00 0.299186
$$466$$ 0 0
$$467$$ − 3686.00i − 0.365241i −0.983183 0.182621i $$-0.941542\pi$$
0.983183 0.182621i $$-0.0584580\pi$$
$$468$$ 0 0
$$469$$ 14348.0i 1.41264i
$$470$$ 0 0
$$471$$ 6132.00 0.599889
$$472$$ 0 0
$$473$$ −2848.00 −0.276852
$$474$$ 0 0
$$475$$ 100.000i 0.00965961i
$$476$$ 0 0
$$477$$ 1458.00i 0.139952i
$$478$$ 0 0
$$479$$ −88.0000 −0.00839420 −0.00419710 0.999991i $$-0.501336\pi$$
−0.00419710 + 0.999991i $$0.501336\pi$$
$$480$$ 0 0
$$481$$ −25404.0 −2.40816
$$482$$ 0 0
$$483$$ 27336.0i 2.57522i
$$484$$ 0 0
$$485$$ 6890.00i 0.645070i
$$486$$ 0 0
$$487$$ −14714.0 −1.36911 −0.684553 0.728963i $$-0.740003\pi$$
−0.684553 + 0.728963i $$0.740003\pi$$
$$488$$ 0 0
$$489$$ −5556.00 −0.513806
$$490$$ 0 0
$$491$$ − 7344.00i − 0.675010i −0.941324 0.337505i $$-0.890417\pi$$
0.941324 0.337505i $$-0.109583\pi$$
$$492$$ 0 0
$$493$$ 16940.0i 1.54754i
$$494$$ 0 0
$$495$$ 720.000 0.0653770
$$496$$ 0 0
$$497$$ 28968.0 2.61447
$$498$$ 0 0
$$499$$ − 1604.00i − 0.143898i −0.997408 0.0719488i $$-0.977078\pi$$
0.997408 0.0719488i $$-0.0229218\pi$$
$$500$$ 0 0
$$501$$ 3924.00i 0.349923i
$$502$$ 0 0
$$503$$ 14802.0 1.31210 0.656052 0.754715i $$-0.272225\pi$$
0.656052 + 0.754715i $$0.272225\pi$$
$$504$$ 0 0
$$505$$ 630.000 0.0555141
$$506$$ 0 0
$$507$$ − 7002.00i − 0.613353i
$$508$$ 0 0
$$509$$ − 22514.0i − 1.96054i −0.197660 0.980271i $$-0.563334\pi$$
0.197660 0.980271i $$-0.436666\pi$$
$$510$$ 0 0
$$511$$ 10404.0 0.900677
$$512$$ 0 0
$$513$$ 432.000 0.0371799
$$514$$ 0 0
$$515$$ − 6310.00i − 0.539906i
$$516$$ 0 0
$$517$$ − 352.000i − 0.0299438i
$$518$$ 0 0
$$519$$ 7764.00 0.656651
$$520$$ 0 0
$$521$$ 6710.00 0.564243 0.282121 0.959379i $$-0.408962\pi$$
0.282121 + 0.959379i $$0.408962\pi$$
$$522$$ 0 0
$$523$$ 7930.00i 0.663011i 0.943453 + 0.331505i $$0.107557\pi$$
−0.943453 + 0.331505i $$0.892443\pi$$
$$524$$ 0 0
$$525$$ 5100.00i 0.423966i
$$526$$ 0 0
$$527$$ 7000.00 0.578605
$$528$$ 0 0
$$529$$ 5789.00 0.475795
$$530$$ 0 0
$$531$$ 2412.00i 0.197122i
$$532$$ 0 0
$$533$$ 8004.00i 0.650454i
$$534$$ 0 0
$$535$$ −2550.00 −0.206068
$$536$$ 0 0
$$537$$ −17016.0 −1.36740
$$538$$ 0 0
$$539$$ 13008.0i 1.03951i
$$540$$ 0 0
$$541$$ 4918.00i 0.390834i 0.980720 + 0.195417i $$0.0626061\pi$$
−0.980720 + 0.195417i $$0.937394\pi$$
$$542$$ 0 0
$$543$$ 10452.0 0.826037
$$544$$ 0 0
$$545$$ −130.000 −0.0102176
$$546$$ 0 0
$$547$$ 3922.00i 0.306568i 0.988182 + 0.153284i $$0.0489849\pi$$
−0.988182 + 0.153284i $$0.951015\pi$$
$$548$$ 0 0
$$549$$ − 2250.00i − 0.174914i
$$550$$ 0 0
$$551$$ −968.000 −0.0748424
$$552$$ 0 0
$$553$$ −15504.0 −1.19222
$$554$$ 0 0
$$555$$ − 13140.0i − 1.00498i
$$556$$ 0 0
$$557$$ 17786.0i 1.35299i 0.736446 + 0.676496i $$0.236503\pi$$
−0.736446 + 0.676496i $$0.763497\pi$$
$$558$$ 0 0
$$559$$ −10324.0 −0.781143
$$560$$ 0 0
$$561$$ 6720.00 0.505737
$$562$$ 0 0
$$563$$ − 20266.0i − 1.51707i −0.651633 0.758535i $$-0.725916\pi$$
0.651633 0.758535i $$-0.274084\pi$$
$$564$$ 0 0
$$565$$ 6210.00i 0.462401i
$$566$$ 0 0
$$567$$ 30294.0 2.24379
$$568$$ 0 0
$$569$$ −13358.0 −0.984177 −0.492088 0.870545i $$-0.663766\pi$$
−0.492088 + 0.870545i $$0.663766\pi$$
$$570$$ 0 0
$$571$$ 16360.0i 1.19903i 0.800364 + 0.599514i $$0.204639\pi$$
−0.800364 + 0.599514i $$0.795361\pi$$
$$572$$ 0 0
$$573$$ − 26760.0i − 1.95099i
$$574$$ 0 0
$$575$$ 3350.00 0.242965
$$576$$ 0 0
$$577$$ −15574.0 −1.12366 −0.561832 0.827251i $$-0.689903\pi$$
−0.561832 + 0.827251i $$0.689903\pi$$
$$578$$ 0 0
$$579$$ − 22692.0i − 1.62875i
$$580$$ 0 0
$$581$$ 14756.0i 1.05367i
$$582$$ 0 0
$$583$$ 2592.00 0.184133
$$584$$ 0 0
$$585$$ 2610.00 0.184462
$$586$$ 0 0
$$587$$ 6654.00i 0.467870i 0.972252 + 0.233935i $$0.0751604\pi$$
−0.972252 + 0.233935i $$0.924840\pi$$
$$588$$ 0 0
$$589$$ 400.000i 0.0279825i
$$590$$ 0 0
$$591$$ 26844.0 1.86838
$$592$$ 0 0
$$593$$ −17742.0 −1.22863 −0.614314 0.789062i $$-0.710567\pi$$
−0.614314 + 0.789062i $$0.710567\pi$$
$$594$$ 0 0
$$595$$ 11900.0i 0.819920i
$$596$$ 0 0
$$597$$ 21648.0i 1.48408i
$$598$$ 0 0
$$599$$ 15840.0 1.08048 0.540238 0.841512i $$-0.318334\pi$$
0.540238 + 0.841512i $$0.318334\pi$$
$$600$$ 0 0
$$601$$ 3002.00 0.203751 0.101875 0.994797i $$-0.467516\pi$$
0.101875 + 0.994797i $$0.467516\pi$$
$$602$$ 0 0
$$603$$ 3798.00i 0.256495i
$$604$$ 0 0
$$605$$ 5375.00i 0.361198i
$$606$$ 0 0
$$607$$ 23610.0 1.57875 0.789374 0.613912i $$-0.210405\pi$$
0.789374 + 0.613912i $$0.210405\pi$$
$$608$$ 0 0
$$609$$ −49368.0 −3.28488
$$610$$ 0 0
$$611$$ − 1276.00i − 0.0844868i
$$612$$ 0 0
$$613$$ − 23850.0i − 1.57144i −0.618583 0.785720i $$-0.712293\pi$$
0.618583 0.785720i $$-0.287707\pi$$
$$614$$ 0 0
$$615$$ −4140.00 −0.271449
$$616$$ 0 0
$$617$$ 5334.00 0.348037 0.174018 0.984742i $$-0.444325\pi$$
0.174018 + 0.984742i $$0.444325\pi$$
$$618$$ 0 0
$$619$$ − 2164.00i − 0.140515i −0.997529 0.0702573i $$-0.977618\pi$$
0.997529 0.0702573i $$-0.0223820\pi$$
$$620$$ 0 0
$$621$$ − 14472.0i − 0.935171i
$$622$$ 0 0
$$623$$ −24684.0 −1.58739
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 384.000i 0.0244585i
$$628$$ 0 0
$$629$$ − 30660.0i − 1.94355i
$$630$$ 0 0
$$631$$ −25220.0 −1.59111 −0.795557 0.605879i $$-0.792821\pi$$
−0.795557 + 0.605879i $$0.792821\pi$$
$$632$$ 0 0
$$633$$ −1536.00 −0.0964463
$$634$$ 0 0
$$635$$ 4890.00i 0.305596i
$$636$$ 0 0
$$637$$ 47154.0i 2.93298i
$$638$$ 0 0
$$639$$ 7668.00 0.474713
$$640$$ 0 0
$$641$$ −12306.0 −0.758280 −0.379140 0.925339i $$-0.623780\pi$$
−0.379140 + 0.925339i $$0.623780\pi$$
$$642$$ 0 0
$$643$$ 27414.0i 1.68134i 0.541547 + 0.840671i $$0.317839\pi$$
−0.541547 + 0.840671i $$0.682161\pi$$
$$644$$ 0 0
$$645$$ − 5340.00i − 0.325988i
$$646$$ 0 0
$$647$$ −21834.0 −1.32671 −0.663356 0.748304i $$-0.730869\pi$$
−0.663356 + 0.748304i $$0.730869\pi$$
$$648$$ 0 0
$$649$$ 4288.00 0.259351
$$650$$ 0 0
$$651$$ 20400.0i 1.22817i
$$652$$ 0 0
$$653$$ − 23998.0i − 1.43815i −0.694931 0.719077i $$-0.744565\pi$$
0.694931 0.719077i $$-0.255435\pi$$
$$654$$ 0 0
$$655$$ −4560.00 −0.272021
$$656$$ 0 0
$$657$$ 2754.00 0.163537
$$658$$ 0 0
$$659$$ 32004.0i 1.89180i 0.324452 + 0.945902i $$0.394820\pi$$
−0.324452 + 0.945902i $$0.605180\pi$$
$$660$$ 0 0
$$661$$ 8526.00i 0.501699i 0.968026 + 0.250849i $$0.0807099\pi$$
−0.968026 + 0.250849i $$0.919290\pi$$
$$662$$ 0 0
$$663$$ 24360.0 1.42694
$$664$$ 0 0
$$665$$ −680.000 −0.0396530
$$666$$ 0 0
$$667$$ 32428.0i 1.88248i
$$668$$ 0 0
$$669$$ 30948.0i 1.78852i
$$670$$ 0 0
$$671$$ −4000.00 −0.230132
$$672$$ 0 0
$$673$$ 8178.00 0.468408 0.234204 0.972187i $$-0.424752\pi$$
0.234204 + 0.972187i $$0.424752\pi$$
$$674$$ 0 0
$$675$$ − 2700.00i − 0.153960i
$$676$$ 0 0
$$677$$ 16646.0i 0.944989i 0.881334 + 0.472495i $$0.156646\pi$$
−0.881334 + 0.472495i $$0.843354\pi$$
$$678$$ 0 0
$$679$$ −46852.0 −2.64803
$$680$$ 0 0
$$681$$ −13356.0 −0.751546
$$682$$ 0 0
$$683$$ − 22446.0i − 1.25750i −0.777608 0.628750i $$-0.783567\pi$$
0.777608 0.628750i $$-0.216433\pi$$
$$684$$ 0 0
$$685$$ 4630.00i 0.258253i
$$686$$ 0 0
$$687$$ 12516.0 0.695073
$$688$$ 0 0
$$689$$ 9396.00 0.519534
$$690$$ 0 0
$$691$$ − 35336.0i − 1.94536i −0.232147 0.972681i $$-0.574575\pi$$
0.232147 0.972681i $$-0.425425\pi$$
$$692$$ 0 0
$$693$$ 4896.00i 0.268375i
$$694$$ 0 0
$$695$$ −2580.00 −0.140813
$$696$$ 0 0
$$697$$ −9660.00 −0.524962
$$698$$ 0 0
$$699$$ 34308.0i 1.85643i
$$700$$ 0 0
$$701$$ 3482.00i 0.187608i 0.995591 + 0.0938041i $$0.0299027\pi$$
−0.995591 + 0.0938041i $$0.970097\pi$$
$$702$$ 0 0
$$703$$ 1752.00 0.0939942
$$704$$ 0 0
$$705$$ 660.000 0.0352582
$$706$$ 0 0
$$707$$ 4284.00i 0.227887i
$$708$$ 0 0
$$709$$ 19402.0i 1.02773i 0.857872 + 0.513863i $$0.171786\pi$$
−0.857872 + 0.513863i $$0.828214\pi$$
$$710$$ 0 0
$$711$$ −4104.00 −0.216473
$$712$$ 0 0
$$713$$ 13400.0 0.703834
$$714$$ 0 0
$$715$$ − 4640.00i − 0.242694i
$$716$$ 0 0
$$717$$ 21744.0i 1.13256i
$$718$$ 0 0
$$719$$ 9896.00 0.513294 0.256647 0.966505i $$-0.417382\pi$$
0.256647 + 0.966505i $$0.417382\pi$$
$$720$$ 0 0
$$721$$ 42908.0 2.21633
$$722$$ 0 0
$$723$$ − 492.000i − 0.0253080i
$$724$$ 0 0
$$725$$ 6050.00i 0.309919i
$$726$$ 0 0
$$727$$ 494.000 0.0252014 0.0126007 0.999921i $$-0.495989\pi$$
0.0126007 + 0.999921i $$0.495989\pi$$
$$728$$ 0 0
$$729$$ −9477.00 −0.481481
$$730$$ 0 0
$$731$$ − 12460.0i − 0.630437i
$$732$$ 0 0
$$733$$ 9282.00i 0.467720i 0.972270 + 0.233860i $$0.0751357\pi$$
−0.972270 + 0.233860i $$0.924864\pi$$
$$734$$ 0 0
$$735$$ −24390.0 −1.22400
$$736$$ 0 0
$$737$$ 6752.00 0.337467
$$738$$ 0 0
$$739$$ 3252.00i 0.161877i 0.996719 + 0.0809383i $$0.0257917\pi$$
−0.996719 + 0.0809383i $$0.974208\pi$$
$$740$$ 0 0
$$741$$ 1392.00i 0.0690100i
$$742$$ 0 0
$$743$$ −4710.00 −0.232561 −0.116281 0.993216i $$-0.537097\pi$$
−0.116281 + 0.993216i $$0.537097\pi$$
$$744$$ 0 0
$$745$$ −4790.00 −0.235560
$$746$$ 0 0
$$747$$ 3906.00i 0.191316i
$$748$$ 0 0
$$749$$ − 17340.0i − 0.845914i
$$750$$ 0 0
$$751$$ −25764.0 −1.25185 −0.625927 0.779882i $$-0.715279\pi$$
−0.625927 + 0.779882i $$0.715279\pi$$
$$752$$ 0 0
$$753$$ 30240.0 1.46349
$$754$$ 0 0
$$755$$ 1660.00i 0.0800180i
$$756$$ 0 0
$$757$$ 30094.0i 1.44489i 0.691426 + 0.722447i $$0.256983\pi$$
−0.691426 + 0.722447i $$0.743017\pi$$
$$758$$ 0 0
$$759$$ 12864.0 0.615196
$$760$$ 0 0
$$761$$ −22362.0 −1.06521 −0.532603 0.846365i $$-0.678786\pi$$
−0.532603 + 0.846365i $$0.678786\pi$$
$$762$$ 0 0
$$763$$ − 884.000i − 0.0419436i
$$764$$ 0 0
$$765$$ 3150.00i 0.148874i
$$766$$ 0 0
$$767$$ 15544.0 0.731762
$$768$$ 0 0
$$769$$ −30398.0 −1.42546 −0.712731 0.701438i $$-0.752542\pi$$
−0.712731 + 0.701438i $$0.752542\pi$$
$$770$$ 0 0
$$771$$ − 13860.0i − 0.647413i
$$772$$ 0 0
$$773$$ − 1290.00i − 0.0600234i −0.999550 0.0300117i $$-0.990446\pi$$
0.999550 0.0300117i $$-0.00955445\pi$$
$$774$$ 0 0
$$775$$ 2500.00 0.115874
$$776$$ 0 0
$$777$$ 89352.0 4.12546
$$778$$ 0 0
$$779$$ − 552.000i − 0.0253883i
$$780$$ 0 0
$$781$$ − 13632.0i − 0.624573i
$$782$$ 0 0
$$783$$ 26136.0 1.19288
$$784$$ 0 0
$$785$$ 5110.00 0.232336
$$786$$ 0 0
$$787$$ − 14.0000i 0 0.000634112i −1.00000 0.000317056i $$-0.999899\pi$$
1.00000 0.000317056i $$-0.000100922\pi$$
$$788$$ 0 0
$$789$$ − 24660.0i − 1.11270i
$$790$$ 0 0
$$791$$ −42228.0 −1.89817
$$792$$ 0 0
$$793$$ −14500.0 −0.649319
$$794$$ 0 0
$$795$$ 4860.00i 0.216813i
$$796$$ 0 0
$$797$$ − 38814.0i − 1.72505i −0.506017 0.862523i $$-0.668883\pi$$
0.506017 0.862523i $$-0.331117\pi$$
$$798$$ 0 0
$$799$$ 1540.00 0.0681868
$$800$$ 0 0
$$801$$ −6534.00 −0.288224
$$802$$ 0 0
$$803$$ − 4896.00i − 0.215163i
$$804$$ 0 0
$$805$$ 22780.0i 0.997378i
$$806$$ 0 0
$$807$$ −4476.00 −0.195245
$$808$$ 0 0
$$809$$ −27402.0 −1.19086 −0.595428 0.803408i $$-0.703018\pi$$
−0.595428 + 0.803408i $$0.703018\pi$$
$$810$$ 0 0
$$811$$ − 28576.0i − 1.23729i −0.785672 0.618643i $$-0.787683\pi$$
0.785672 0.618643i $$-0.212317\pi$$
$$812$$ 0 0
$$813$$ 27576.0i 1.18958i
$$814$$ 0 0
$$815$$ −4630.00 −0.198996
$$816$$ 0 0
$$817$$ 712.000 0.0304893
$$818$$ 0 0
$$819$$ 17748.0i 0.757223i
$$820$$ 0 0
$$821$$ − 31762.0i − 1.35018i −0.737733 0.675092i $$-0.764104\pi$$
0.737733 0.675092i $$-0.235896\pi$$
$$822$$ 0 0
$$823$$ 20506.0 0.868523 0.434261 0.900787i $$-0.357009\pi$$
0.434261 + 0.900787i $$0.357009\pi$$
$$824$$ 0 0
$$825$$ 2400.00 0.101282
$$826$$ 0 0
$$827$$ 13014.0i 0.547208i 0.961842 + 0.273604i $$0.0882158\pi$$
−0.961842 + 0.273604i $$0.911784\pi$$
$$828$$ 0 0
$$829$$ − 22790.0i − 0.954800i −0.878686 0.477400i $$-0.841579\pi$$
0.878686 0.477400i $$-0.158421\pi$$
$$830$$ 0 0
$$831$$ −13236.0 −0.552529
$$832$$ 0 0
$$833$$ −56910.0 −2.36712
$$834$$ 0 0
$$835$$ 3270.00i 0.135525i
$$836$$ 0 0
$$837$$ − 10800.0i − 0.446001i
$$838$$ 0 0
$$839$$ 23696.0 0.975062 0.487531 0.873106i $$-0.337898\pi$$
0.487531 + 0.873106i $$0.337898\pi$$
$$840$$ 0 0
$$841$$ −34175.0 −1.40125
$$842$$ 0 0
$$843$$ − 49668.0i − 2.02925i
$$844$$ 0 0
$$845$$ − 5835.00i − 0.237550i
$$846$$ 0 0
$$847$$ −36550.0 −1.48273
$$848$$ 0 0
$$849$$ −7068.00 −0.285716
$$850$$ 0 0
$$851$$ − 58692.0i − 2.36420i
$$852$$ 0 0
$$853$$ − 5306.00i − 0.212982i −0.994314 0.106491i $$-0.966038\pi$$
0.994314 0.106491i $$-0.0339616\pi$$
$$854$$ 0 0
$$855$$ −180.000 −0.00719985
$$856$$ 0 0
$$857$$ 21054.0 0.839196 0.419598 0.907710i $$-0.362171\pi$$
0.419598 + 0.907710i $$0.362171\pi$$
$$858$$ 0 0
$$859$$ 7364.00i 0.292499i 0.989248 + 0.146249i $$0.0467202\pi$$
−0.989248 + 0.146249i $$0.953280\pi$$
$$860$$ 0 0
$$861$$ − 28152.0i − 1.11431i
$$862$$ 0 0
$$863$$ −17226.0 −0.679467 −0.339733 0.940522i $$-0.610337\pi$$
−0.339733 + 0.940522i $$0.610337\pi$$
$$864$$ 0 0
$$865$$ 6470.00 0.254320
$$866$$ 0 0
$$867$$ − 78.0000i − 0.00305539i
$$868$$ 0 0
$$869$$ 7296.00i 0.284810i
$$870$$ 0 0
$$871$$ 24476.0 0.952167
$$872$$ 0 0
$$873$$ −12402.0 −0.480807
$$874$$ 0 0
$$875$$ 4250.00i 0.164201i
$$876$$ 0 0
$$877$$ 21202.0i 0.816352i 0.912903 + 0.408176i $$0.133835\pi$$
−0.912903 + 0.408176i $$0.866165\pi$$
$$878$$ 0 0
$$879$$ 636.000 0.0244047
$$880$$ 0 0
$$881$$ −29490.0 −1.12774 −0.563872 0.825862i $$-0.690689\pi$$
−0.563872 + 0.825862i $$0.690689\pi$$
$$882$$ 0 0
$$883$$ − 2570.00i − 0.0979472i −0.998800 0.0489736i $$-0.984405\pi$$
0.998800 0.0489736i $$-0.0155950\pi$$
$$884$$ 0 0
$$885$$ 8040.00i 0.305380i
$$886$$ 0 0
$$887$$ 36334.0 1.37540 0.687698 0.725997i $$-0.258621\pi$$
0.687698 + 0.725997i $$0.258621\pi$$
$$888$$ 0 0
$$889$$ −33252.0 −1.25448
$$890$$ 0 0
$$891$$ − 14256.0i − 0.536020i
$$892$$ 0 0
$$893$$ 88.0000i 0.00329766i
$$894$$ 0 0
$$895$$ −14180.0 −0.529592
$$896$$ 0 0
$$897$$ 46632.0 1.73578
$$898$$ 0 0
$$899$$ 24200.0i 0.897792i
$$900$$ 0 0
$$901$$ 11340.0i 0.419301i
$$902$$ 0 0
$$903$$ 36312.0 1.33819
$$904$$ 0 0
$$905$$ 8710.00 0.319923
$$906$$ 0 0
$$907$$ − 12474.0i − 0.456662i −0.973584 0.228331i $$-0.926673\pi$$
0.973584 0.228331i $$-0.0733268\pi$$
$$908$$ 0 0
$$909$$ 1134.00i 0.0413778i
$$910$$ 0 0
$$911$$ 41132.0 1.49590 0.747949 0.663756i $$-0.231039\pi$$
0.747949 + 0.663756i $$0.231039\pi$$
$$912$$ 0 0
$$913$$ 6944.00 0.251712
$$914$$ 0 0
$$915$$ − 7500.00i − 0.270975i
$$916$$ 0 0
$$917$$ − 31008.0i − 1.11666i
$$918$$ 0 0
$$919$$ −38416.0 −1.37892 −0.689460 0.724324i $$-0.742152\pi$$
−0.689460 + 0.724324i $$0.742152\pi$$
$$920$$ 0 0
$$921$$ 48804.0 1.74609
$$922$$ 0 0
$$923$$ − 49416.0i − 1.76224i
$$924$$ 0 0
$$925$$ − 10950.0i − 0.389226i
$$926$$ 0 0
$$927$$ 11358.0 0.402423
$$928$$ 0 0
$$929$$ 41302.0 1.45864 0.729319 0.684174i $$-0.239837\pi$$
0.729319 + 0.684174i $$0.239837\pi$$
$$930$$ 0 0
$$931$$ − 3252.00i − 0.114479i
$$932$$ 0 0
$$933$$ − 26376.0i − 0.925521i
$$934$$ 0 0
$$935$$ 5600.00 0.195871
$$936$$ 0 0
$$937$$ 26150.0 0.911722 0.455861 0.890051i $$-0.349331\pi$$
0.455861 + 0.890051i $$0.349331\pi$$
$$938$$ 0 0
$$939$$ − 28956.0i − 1.00633i
$$940$$ 0 0
$$941$$ 35254.0i 1.22130i 0.791899 + 0.610652i $$0.209093\pi$$
−0.791899 + 0.610652i $$0.790907\pi$$
$$942$$ 0 0
$$943$$ −18492.0 −0.638582
$$944$$ 0 0
$$945$$ 18360.0 0.632011
$$946$$ 0 0
$$947$$ − 18550.0i − 0.636530i −0.948002 0.318265i $$-0.896900\pi$$
0.948002 0.318265i $$-0.103100\pi$$
$$948$$ 0 0
$$949$$ − 17748.0i − 0.607086i
$$950$$ 0 0
$$951$$ −42156.0 −1.43744
$$952$$ 0 0
$$953$$ −17322.0 −0.588788 −0.294394 0.955684i $$-0.595118\pi$$
−0.294394 + 0.955684i $$0.595118\pi$$
$$954$$ 0 0
$$955$$ − 22300.0i − 0.755614i
$$956$$ 0 0
$$957$$ 23232.0i 0.784727i
$$958$$ 0 0
$$959$$ −31484.0 −1.06014
$$960$$ 0 0
$$961$$ −19791.0 −0.664328
$$962$$ 0 0
$$963$$ − 4590.00i − 0.153594i
$$964$$ 0 0
$$965$$ − 18910.0i − 0.630813i
$$966$$ 0 0
$$967$$ 35190.0 1.17025 0.585126 0.810942i $$-0.301045\pi$$
0.585126 + 0.810942i $$0.301045\pi$$
$$968$$ 0 0
$$969$$ −1680.00 −0.0556960
$$970$$ 0 0
$$971$$ − 40696.0i − 1.34500i −0.740096 0.672501i $$-0.765220\pi$$
0.740096 0.672501i $$-0.234780\pi$$
$$972$$ 0 0
$$973$$ − 17544.0i − 0.578042i
$$974$$ 0 0
$$975$$ 8700.00 0.285767
$$976$$ 0 0
$$977$$ 44306.0 1.45084 0.725422 0.688304i $$-0.241645\pi$$
0.725422 + 0.688304i $$0.241645\pi$$
$$978$$ 0 0
$$979$$ 11616.0i 0.379212i
$$980$$ 0 0
$$981$$ − 234.000i − 0.00761574i
$$982$$ 0 0
$$983$$ −18798.0 −0.609932 −0.304966 0.952363i $$-0.598645\pi$$
−0.304966 + 0.952363i $$0.598645\pi$$
$$984$$ 0 0
$$985$$ 22370.0 0.723622
$$986$$ 0 0
$$987$$ 4488.00i 0.144736i
$$988$$ 0 0
$$989$$ − 23852.0i − 0.766885i
$$990$$ 0 0
$$991$$ −2468.00 −0.0791106 −0.0395553 0.999217i $$-0.512594\pi$$
−0.0395553 + 0.999217i $$0.512594\pi$$
$$992$$ 0 0
$$993$$ −52848.0 −1.68890
$$994$$ 0 0
$$995$$ 18040.0i 0.574780i
$$996$$ 0 0
$$997$$ 61086.0i 1.94043i 0.242237 + 0.970217i $$0.422119\pi$$
−0.242237 + 0.970217i $$0.577881\pi$$
$$998$$ 0 0
$$999$$ −47304.0 −1.49813
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 1280.4.d.a.641.2 2
4.3 odd 2 1280.4.d.p.641.1 2
8.3 odd 2 1280.4.d.p.641.2 2
8.5 even 2 inner 1280.4.d.a.641.1 2
16.3 odd 4 320.4.a.l.1.1 1
16.5 even 4 80.4.a.e.1.1 1
16.11 odd 4 40.4.a.a.1.1 1
16.13 even 4 320.4.a.c.1.1 1
48.5 odd 4 720.4.a.bd.1.1 1
48.11 even 4 360.4.a.h.1.1 1
80.19 odd 4 1600.4.a.j.1.1 1
80.27 even 4 200.4.c.c.49.2 2
80.29 even 4 1600.4.a.br.1.1 1
80.37 odd 4 400.4.c.f.49.1 2
80.43 even 4 200.4.c.c.49.1 2
80.53 odd 4 400.4.c.f.49.2 2
80.59 odd 4 200.4.a.i.1.1 1
80.69 even 4 400.4.a.e.1.1 1
112.27 even 4 1960.4.a.h.1.1 1
240.59 even 4 1800.4.a.bi.1.1 1
240.107 odd 4 1800.4.f.j.649.1 2
240.203 odd 4 1800.4.f.j.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 16.11 odd 4
80.4.a.e.1.1 1 16.5 even 4
200.4.a.i.1.1 1 80.59 odd 4
200.4.c.c.49.1 2 80.43 even 4
200.4.c.c.49.2 2 80.27 even 4
320.4.a.c.1.1 1 16.13 even 4
320.4.a.l.1.1 1 16.3 odd 4
360.4.a.h.1.1 1 48.11 even 4
400.4.a.e.1.1 1 80.69 even 4
400.4.c.f.49.1 2 80.37 odd 4
400.4.c.f.49.2 2 80.53 odd 4
720.4.a.bd.1.1 1 48.5 odd 4
1280.4.d.a.641.1 2 8.5 even 2 inner
1280.4.d.a.641.2 2 1.1 even 1 trivial
1280.4.d.p.641.1 2 4.3 odd 2
1280.4.d.p.641.2 2 8.3 odd 2
1600.4.a.j.1.1 1 80.19 odd 4
1600.4.a.br.1.1 1 80.29 even 4
1800.4.a.bi.1.1 1 240.59 even 4
1800.4.f.j.649.1 2 240.107 odd 4
1800.4.f.j.649.2 2 240.203 odd 4
1960.4.a.h.1.1 1 112.27 even 4