# Properties

 Label 2496.4.a.bp.1.3 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 2496.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} +25.8785 q^{11} -13.0000 q^{13} +34.2966 q^{15} -20.3276 q^{17} +154.712 q^{19} +33.7629 q^{21} +180.418 q^{23} +5.69520 q^{25} +27.0000 q^{27} +20.4522 q^{29} -266.424 q^{31} +77.6355 q^{33} +128.661 q^{35} -115.984 q^{37} -39.0000 q^{39} +391.184 q^{41} +151.407 q^{43} +102.890 q^{45} +467.365 q^{47} -216.341 q^{49} -60.9828 q^{51} -79.9842 q^{53} +295.848 q^{55} +464.136 q^{57} -873.710 q^{59} +187.068 q^{61} +101.289 q^{63} -148.619 q^{65} -609.204 q^{67} +541.255 q^{69} -248.038 q^{71} +852.765 q^{73} +17.0856 q^{75} +291.244 q^{77} +331.221 q^{79} +81.0000 q^{81} -435.432 q^{83} -232.389 q^{85} +61.3566 q^{87} +259.233 q^{89} -146.306 q^{91} -799.273 q^{93} +1768.70 q^{95} +1225.17 q^{97} +232.907 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 9 q^{3} - 4 q^{5} - 30 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q + 9 * q^3 - 4 * q^5 - 30 * q^7 + 27 * q^9 $$3 q + 9 q^{3} - 4 q^{5} - 30 q^{7} + 27 q^{9} - 16 q^{11} - 39 q^{13} - 12 q^{15} - 146 q^{17} + 94 q^{19} - 90 q^{21} + 48 q^{23} + 145 q^{25} + 81 q^{27} + 2 q^{29} - 302 q^{31} - 48 q^{33} + 80 q^{35} - 374 q^{37} - 117 q^{39} + 480 q^{41} - 260 q^{43} - 36 q^{45} + 24 q^{47} + 447 q^{49} - 438 q^{51} + 678 q^{53} + 1552 q^{55} + 282 q^{57} - 1788 q^{59} - 230 q^{61} - 270 q^{63} + 52 q^{65} + 74 q^{67} + 144 q^{69} + 948 q^{71} - 222 q^{73} + 435 q^{75} - 112 q^{77} + 24 q^{79} + 243 q^{81} - 796 q^{83} + 248 q^{85} + 6 q^{87} + 1436 q^{89} + 390 q^{91} - 906 q^{93} + 4032 q^{95} + 3242 q^{97} - 144 q^{99}+O(q^{100})$$ 3 * q + 9 * q^3 - 4 * q^5 - 30 * q^7 + 27 * q^9 - 16 * q^11 - 39 * q^13 - 12 * q^15 - 146 * q^17 + 94 * q^19 - 90 * q^21 + 48 * q^23 + 145 * q^25 + 81 * q^27 + 2 * q^29 - 302 * q^31 - 48 * q^33 + 80 * q^35 - 374 * q^37 - 117 * q^39 + 480 * q^41 - 260 * q^43 - 36 * q^45 + 24 * q^47 + 447 * q^49 - 438 * q^51 + 678 * q^53 + 1552 * q^55 + 282 * q^57 - 1788 * q^59 - 230 * q^61 - 270 * q^63 + 52 * q^65 + 74 * q^67 + 144 * q^69 + 948 * q^71 - 222 * q^73 + 435 * q^75 - 112 * q^77 + 24 * q^79 + 243 * q^81 - 796 * q^83 + 248 * q^85 + 6 * q^87 + 1436 * q^89 + 390 * q^91 - 906 * q^93 + 4032 * q^95 + 3242 * q^97 - 144 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 11.4322 1.02253 0.511264 0.859424i $$-0.329178\pi$$
0.511264 + 0.859424i $$0.329178\pi$$
$$6$$ 0 0
$$7$$ 11.2543 0.607675 0.303838 0.952724i $$-0.401732\pi$$
0.303838 + 0.952724i $$0.401732\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 25.8785 0.709333 0.354666 0.934993i $$-0.384594\pi$$
0.354666 + 0.934993i $$0.384594\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 34.2966 0.590356
$$16$$ 0 0
$$17$$ −20.3276 −0.290010 −0.145005 0.989431i $$-0.546320\pi$$
−0.145005 + 0.989431i $$0.546320\pi$$
$$18$$ 0 0
$$19$$ 154.712 1.86807 0.934035 0.357181i $$-0.116262\pi$$
0.934035 + 0.357181i $$0.116262\pi$$
$$20$$ 0 0
$$21$$ 33.7629 0.350841
$$22$$ 0 0
$$23$$ 180.418 1.63565 0.817823 0.575471i $$-0.195181\pi$$
0.817823 + 0.575471i $$0.195181\pi$$
$$24$$ 0 0
$$25$$ 5.69520 0.0455616
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 20.4522 0.130961 0.0654806 0.997854i $$-0.479142\pi$$
0.0654806 + 0.997854i $$0.479142\pi$$
$$30$$ 0 0
$$31$$ −266.424 −1.54359 −0.771794 0.635873i $$-0.780640\pi$$
−0.771794 + 0.635873i $$0.780640\pi$$
$$32$$ 0 0
$$33$$ 77.6355 0.409534
$$34$$ 0 0
$$35$$ 128.661 0.621364
$$36$$ 0 0
$$37$$ −115.984 −0.515340 −0.257670 0.966233i $$-0.582955\pi$$
−0.257670 + 0.966233i $$0.582955\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ 391.184 1.49006 0.745032 0.667029i $$-0.232434\pi$$
0.745032 + 0.667029i $$0.232434\pi$$
$$42$$ 0 0
$$43$$ 151.407 0.536963 0.268482 0.963285i $$-0.413478\pi$$
0.268482 + 0.963285i $$0.413478\pi$$
$$44$$ 0 0
$$45$$ 102.890 0.340842
$$46$$ 0 0
$$47$$ 467.365 1.45047 0.725236 0.688500i $$-0.241731\pi$$
0.725236 + 0.688500i $$0.241731\pi$$
$$48$$ 0 0
$$49$$ −216.341 −0.630731
$$50$$ 0 0
$$51$$ −60.9828 −0.167437
$$52$$ 0 0
$$53$$ −79.9842 −0.207296 −0.103648 0.994614i $$-0.533051\pi$$
−0.103648 + 0.994614i $$0.533051\pi$$
$$54$$ 0 0
$$55$$ 295.848 0.725312
$$56$$ 0 0
$$57$$ 464.136 1.07853
$$58$$ 0 0
$$59$$ −873.710 −1.92792 −0.963960 0.266045i $$-0.914283\pi$$
−0.963960 + 0.266045i $$0.914283\pi$$
$$60$$ 0 0
$$61$$ 187.068 0.392649 0.196325 0.980539i $$-0.437099\pi$$
0.196325 + 0.980539i $$0.437099\pi$$
$$62$$ 0 0
$$63$$ 101.289 0.202558
$$64$$ 0 0
$$65$$ −148.619 −0.283598
$$66$$ 0 0
$$67$$ −609.204 −1.11084 −0.555418 0.831571i $$-0.687442\pi$$
−0.555418 + 0.831571i $$0.687442\pi$$
$$68$$ 0 0
$$69$$ 541.255 0.944340
$$70$$ 0 0
$$71$$ −248.038 −0.414601 −0.207301 0.978277i $$-0.566468\pi$$
−0.207301 + 0.978277i $$0.566468\pi$$
$$72$$ 0 0
$$73$$ 852.765 1.36724 0.683621 0.729838i $$-0.260404\pi$$
0.683621 + 0.729838i $$0.260404\pi$$
$$74$$ 0 0
$$75$$ 17.0856 0.0263050
$$76$$ 0 0
$$77$$ 291.244 0.431044
$$78$$ 0 0
$$79$$ 331.221 0.471712 0.235856 0.971788i $$-0.424211\pi$$
0.235856 + 0.971788i $$0.424211\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −435.432 −0.575842 −0.287921 0.957654i $$-0.592964\pi$$
−0.287921 + 0.957654i $$0.592964\pi$$
$$84$$ 0 0
$$85$$ −232.389 −0.296543
$$86$$ 0 0
$$87$$ 61.3566 0.0756105
$$88$$ 0 0
$$89$$ 259.233 0.308749 0.154375 0.988012i $$-0.450664\pi$$
0.154375 + 0.988012i $$0.450664\pi$$
$$90$$ 0 0
$$91$$ −146.306 −0.168539
$$92$$ 0 0
$$93$$ −799.273 −0.891191
$$94$$ 0 0
$$95$$ 1768.70 1.91015
$$96$$ 0 0
$$97$$ 1225.17 1.28245 0.641223 0.767355i $$-0.278428\pi$$
0.641223 + 0.767355i $$0.278428\pi$$
$$98$$ 0 0
$$99$$ 232.907 0.236444
$$100$$ 0 0
$$101$$ −645.416 −0.635855 −0.317927 0.948115i $$-0.602987\pi$$
−0.317927 + 0.948115i $$0.602987\pi$$
$$102$$ 0 0
$$103$$ 511.137 0.488969 0.244484 0.969653i $$-0.421381\pi$$
0.244484 + 0.969653i $$0.421381\pi$$
$$104$$ 0 0
$$105$$ 385.984 0.358745
$$106$$ 0 0
$$107$$ 608.195 0.549499 0.274750 0.961516i $$-0.411405\pi$$
0.274750 + 0.961516i $$0.411405\pi$$
$$108$$ 0 0
$$109$$ 1300.04 1.14239 0.571197 0.820813i $$-0.306479\pi$$
0.571197 + 0.820813i $$0.306479\pi$$
$$110$$ 0 0
$$111$$ −347.951 −0.297532
$$112$$ 0 0
$$113$$ 42.1953 0.0351274 0.0175637 0.999846i $$-0.494409\pi$$
0.0175637 + 0.999846i $$0.494409\pi$$
$$114$$ 0 0
$$115$$ 2062.58 1.67249
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −228.773 −0.176232
$$120$$ 0 0
$$121$$ −661.303 −0.496847
$$122$$ 0 0
$$123$$ 1173.55 0.860289
$$124$$ 0 0
$$125$$ −1363.92 −0.975939
$$126$$ 0 0
$$127$$ 311.018 0.217310 0.108655 0.994080i $$-0.465346\pi$$
0.108655 + 0.994080i $$0.465346\pi$$
$$128$$ 0 0
$$129$$ 454.222 0.310016
$$130$$ 0 0
$$131$$ 2000.98 1.33456 0.667278 0.744809i $$-0.267459\pi$$
0.667278 + 0.744809i $$0.267459\pi$$
$$132$$ 0 0
$$133$$ 1741.17 1.13518
$$134$$ 0 0
$$135$$ 308.669 0.196785
$$136$$ 0 0
$$137$$ 1038.53 0.647644 0.323822 0.946118i $$-0.395032\pi$$
0.323822 + 0.946118i $$0.395032\pi$$
$$138$$ 0 0
$$139$$ −2858.46 −1.74426 −0.872128 0.489277i $$-0.837261\pi$$
−0.872128 + 0.489277i $$0.837261\pi$$
$$140$$ 0 0
$$141$$ 1402.09 0.837430
$$142$$ 0 0
$$143$$ −336.421 −0.196734
$$144$$ 0 0
$$145$$ 233.814 0.133911
$$146$$ 0 0
$$147$$ −649.022 −0.364153
$$148$$ 0 0
$$149$$ −743.479 −0.408780 −0.204390 0.978890i $$-0.565521\pi$$
−0.204390 + 0.978890i $$0.565521\pi$$
$$150$$ 0 0
$$151$$ −2277.24 −1.22728 −0.613640 0.789586i $$-0.710295\pi$$
−0.613640 + 0.789586i $$0.710295\pi$$
$$152$$ 0 0
$$153$$ −182.948 −0.0966700
$$154$$ 0 0
$$155$$ −3045.82 −1.57836
$$156$$ 0 0
$$157$$ −3173.51 −1.61321 −0.806605 0.591091i $$-0.798697\pi$$
−0.806605 + 0.591091i $$0.798697\pi$$
$$158$$ 0 0
$$159$$ −239.953 −0.119682
$$160$$ 0 0
$$161$$ 2030.48 0.993941
$$162$$ 0 0
$$163$$ −2314.65 −1.11225 −0.556126 0.831098i $$-0.687713\pi$$
−0.556126 + 0.831098i $$0.687713\pi$$
$$164$$ 0 0
$$165$$ 887.545 0.418759
$$166$$ 0 0
$$167$$ 2665.65 1.23517 0.617587 0.786502i $$-0.288110\pi$$
0.617587 + 0.786502i $$0.288110\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 1392.41 0.622690
$$172$$ 0 0
$$173$$ 165.243 0.0726198 0.0363099 0.999341i $$-0.488440\pi$$
0.0363099 + 0.999341i $$0.488440\pi$$
$$174$$ 0 0
$$175$$ 64.0954 0.0276866
$$176$$ 0 0
$$177$$ −2621.13 −1.11309
$$178$$ 0 0
$$179$$ 712.339 0.297446 0.148723 0.988879i $$-0.452484\pi$$
0.148723 + 0.988879i $$0.452484\pi$$
$$180$$ 0 0
$$181$$ −2206.53 −0.906133 −0.453066 0.891477i $$-0.649670\pi$$
−0.453066 + 0.891477i $$0.649670\pi$$
$$182$$ 0 0
$$183$$ 561.204 0.226696
$$184$$ 0 0
$$185$$ −1325.95 −0.526949
$$186$$ 0 0
$$187$$ −526.048 −0.205714
$$188$$ 0 0
$$189$$ 303.866 0.116947
$$190$$ 0 0
$$191$$ −1470.64 −0.557129 −0.278565 0.960417i $$-0.589859\pi$$
−0.278565 + 0.960417i $$0.589859\pi$$
$$192$$ 0 0
$$193$$ 369.560 0.137832 0.0689158 0.997622i $$-0.478046\pi$$
0.0689158 + 0.997622i $$0.478046\pi$$
$$194$$ 0 0
$$195$$ −445.856 −0.163735
$$196$$ 0 0
$$197$$ 4273.41 1.54552 0.772761 0.634697i $$-0.218875\pi$$
0.772761 + 0.634697i $$0.218875\pi$$
$$198$$ 0 0
$$199$$ −4154.31 −1.47985 −0.739927 0.672687i $$-0.765140\pi$$
−0.739927 + 0.672687i $$0.765140\pi$$
$$200$$ 0 0
$$201$$ −1827.61 −0.641342
$$202$$ 0 0
$$203$$ 230.175 0.0795819
$$204$$ 0 0
$$205$$ 4472.09 1.52363
$$206$$ 0 0
$$207$$ 1623.77 0.545215
$$208$$ 0 0
$$209$$ 4003.71 1.32508
$$210$$ 0 0
$$211$$ 1231.59 0.401830 0.200915 0.979609i $$-0.435608\pi$$
0.200915 + 0.979609i $$0.435608\pi$$
$$212$$ 0 0
$$213$$ −744.114 −0.239370
$$214$$ 0 0
$$215$$ 1730.92 0.549059
$$216$$ 0 0
$$217$$ −2998.42 −0.938000
$$218$$ 0 0
$$219$$ 2558.30 0.789377
$$220$$ 0 0
$$221$$ 264.259 0.0804343
$$222$$ 0 0
$$223$$ 2187.24 0.656809 0.328404 0.944537i $$-0.393489\pi$$
0.328404 + 0.944537i $$0.393489\pi$$
$$224$$ 0 0
$$225$$ 51.2568 0.0151872
$$226$$ 0 0
$$227$$ 4138.67 1.21010 0.605051 0.796187i $$-0.293153\pi$$
0.605051 + 0.796187i $$0.293153\pi$$
$$228$$ 0 0
$$229$$ 835.354 0.241056 0.120528 0.992710i $$-0.461541\pi$$
0.120528 + 0.992710i $$0.461541\pi$$
$$230$$ 0 0
$$231$$ 873.733 0.248863
$$232$$ 0 0
$$233$$ 3685.51 1.03625 0.518124 0.855305i $$-0.326630\pi$$
0.518124 + 0.855305i $$0.326630\pi$$
$$234$$ 0 0
$$235$$ 5343.01 1.48315
$$236$$ 0 0
$$237$$ 993.662 0.272343
$$238$$ 0 0
$$239$$ −3026.21 −0.819034 −0.409517 0.912303i $$-0.634303\pi$$
−0.409517 + 0.912303i $$0.634303\pi$$
$$240$$ 0 0
$$241$$ 3265.58 0.872839 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −2473.25 −0.644940
$$246$$ 0 0
$$247$$ −2011.25 −0.518110
$$248$$ 0 0
$$249$$ −1306.30 −0.332463
$$250$$ 0 0
$$251$$ −6363.16 −1.60016 −0.800078 0.599897i $$-0.795208\pi$$
−0.800078 + 0.599897i $$0.795208\pi$$
$$252$$ 0 0
$$253$$ 4668.96 1.16022
$$254$$ 0 0
$$255$$ −697.168 −0.171209
$$256$$ 0 0
$$257$$ −6085.36 −1.47702 −0.738511 0.674242i $$-0.764471\pi$$
−0.738511 + 0.674242i $$0.764471\pi$$
$$258$$ 0 0
$$259$$ −1305.31 −0.313159
$$260$$ 0 0
$$261$$ 184.070 0.0436538
$$262$$ 0 0
$$263$$ −123.227 −0.0288916 −0.0144458 0.999896i $$-0.504598\pi$$
−0.0144458 + 0.999896i $$0.504598\pi$$
$$264$$ 0 0
$$265$$ −914.395 −0.211965
$$266$$ 0 0
$$267$$ 777.700 0.178256
$$268$$ 0 0
$$269$$ 1935.79 0.438763 0.219381 0.975639i $$-0.429596\pi$$
0.219381 + 0.975639i $$0.429596\pi$$
$$270$$ 0 0
$$271$$ 4612.69 1.03395 0.516976 0.856000i $$-0.327058\pi$$
0.516976 + 0.856000i $$0.327058\pi$$
$$272$$ 0 0
$$273$$ −438.918 −0.0973059
$$274$$ 0 0
$$275$$ 147.383 0.0323183
$$276$$ 0 0
$$277$$ 5834.30 1.26552 0.632761 0.774347i $$-0.281922\pi$$
0.632761 + 0.774347i $$0.281922\pi$$
$$278$$ 0 0
$$279$$ −2397.82 −0.514529
$$280$$ 0 0
$$281$$ 4691.91 0.996071 0.498036 0.867157i $$-0.334055\pi$$
0.498036 + 0.867157i $$0.334055\pi$$
$$282$$ 0 0
$$283$$ 3465.60 0.727945 0.363973 0.931410i $$-0.381420\pi$$
0.363973 + 0.931410i $$0.381420\pi$$
$$284$$ 0 0
$$285$$ 5306.09 1.10283
$$286$$ 0 0
$$287$$ 4402.50 0.905475
$$288$$ 0 0
$$289$$ −4499.79 −0.915894
$$290$$ 0 0
$$291$$ 3675.51 0.740420
$$292$$ 0 0
$$293$$ −2677.31 −0.533822 −0.266911 0.963721i $$-0.586003\pi$$
−0.266911 + 0.963721i $$0.586003\pi$$
$$294$$ 0 0
$$295$$ −9988.43 −1.97135
$$296$$ 0 0
$$297$$ 698.720 0.136511
$$298$$ 0 0
$$299$$ −2345.44 −0.453646
$$300$$ 0 0
$$301$$ 1703.98 0.326299
$$302$$ 0 0
$$303$$ −1936.25 −0.367111
$$304$$ 0 0
$$305$$ 2138.60 0.401494
$$306$$ 0 0
$$307$$ 471.915 0.0877316 0.0438658 0.999037i $$-0.486033\pi$$
0.0438658 + 0.999037i $$0.486033\pi$$
$$308$$ 0 0
$$309$$ 1533.41 0.282306
$$310$$ 0 0
$$311$$ 1518.52 0.276872 0.138436 0.990371i $$-0.455793\pi$$
0.138436 + 0.990371i $$0.455793\pi$$
$$312$$ 0 0
$$313$$ 4049.86 0.731348 0.365674 0.930743i $$-0.380839\pi$$
0.365674 + 0.930743i $$0.380839\pi$$
$$314$$ 0 0
$$315$$ 1157.95 0.207121
$$316$$ 0 0
$$317$$ −3253.96 −0.576532 −0.288266 0.957550i $$-0.593079\pi$$
−0.288266 + 0.957550i $$0.593079\pi$$
$$318$$ 0 0
$$319$$ 529.272 0.0928951
$$320$$ 0 0
$$321$$ 1824.59 0.317254
$$322$$ 0 0
$$323$$ −3144.92 −0.541759
$$324$$ 0 0
$$325$$ −74.0375 −0.0126365
$$326$$ 0 0
$$327$$ 3900.11 0.659561
$$328$$ 0 0
$$329$$ 5259.86 0.881415
$$330$$ 0 0
$$331$$ −3422.45 −0.568322 −0.284161 0.958777i $$-0.591715\pi$$
−0.284161 + 0.958777i $$0.591715\pi$$
$$332$$ 0 0
$$333$$ −1043.85 −0.171780
$$334$$ 0 0
$$335$$ −6964.54 −1.13586
$$336$$ 0 0
$$337$$ −9301.67 −1.50354 −0.751772 0.659423i $$-0.770801\pi$$
−0.751772 + 0.659423i $$0.770801\pi$$
$$338$$ 0 0
$$339$$ 126.586 0.0202808
$$340$$ 0 0
$$341$$ −6894.66 −1.09492
$$342$$ 0 0
$$343$$ −6294.99 −0.990955
$$344$$ 0 0
$$345$$ 6187.74 0.965613
$$346$$ 0 0
$$347$$ 216.898 0.0335554 0.0167777 0.999859i $$-0.494659\pi$$
0.0167777 + 0.999859i $$0.494659\pi$$
$$348$$ 0 0
$$349$$ 4809.84 0.737721 0.368861 0.929485i $$-0.379748\pi$$
0.368861 + 0.929485i $$0.379748\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ −2859.64 −0.431170 −0.215585 0.976485i $$-0.569166\pi$$
−0.215585 + 0.976485i $$0.569166\pi$$
$$354$$ 0 0
$$355$$ −2835.62 −0.423941
$$356$$ 0 0
$$357$$ −686.319 −0.101747
$$358$$ 0 0
$$359$$ −3686.04 −0.541899 −0.270949 0.962594i $$-0.587338\pi$$
−0.270949 + 0.962594i $$0.587338\pi$$
$$360$$ 0 0
$$361$$ 17076.8 2.48969
$$362$$ 0 0
$$363$$ −1983.91 −0.286855
$$364$$ 0 0
$$365$$ 9748.98 1.39804
$$366$$ 0 0
$$367$$ 3470.59 0.493633 0.246816 0.969062i $$-0.420616\pi$$
0.246816 + 0.969062i $$0.420616\pi$$
$$368$$ 0 0
$$369$$ 3520.65 0.496688
$$370$$ 0 0
$$371$$ −900.166 −0.125968
$$372$$ 0 0
$$373$$ 11963.4 1.66070 0.830352 0.557240i $$-0.188140\pi$$
0.830352 + 0.557240i $$0.188140\pi$$
$$374$$ 0 0
$$375$$ −4091.75 −0.563459
$$376$$ 0 0
$$377$$ −265.879 −0.0363221
$$378$$ 0 0
$$379$$ 345.604 0.0468403 0.0234202 0.999726i $$-0.492544\pi$$
0.0234202 + 0.999726i $$0.492544\pi$$
$$380$$ 0 0
$$381$$ 933.055 0.125464
$$382$$ 0 0
$$383$$ 3386.40 0.451793 0.225897 0.974151i $$-0.427469\pi$$
0.225897 + 0.974151i $$0.427469\pi$$
$$384$$ 0 0
$$385$$ 3329.56 0.440754
$$386$$ 0 0
$$387$$ 1362.67 0.178988
$$388$$ 0 0
$$389$$ 1629.88 0.212438 0.106219 0.994343i $$-0.466126\pi$$
0.106219 + 0.994343i $$0.466126\pi$$
$$390$$ 0 0
$$391$$ −3667.47 −0.474353
$$392$$ 0 0
$$393$$ 6002.95 0.770506
$$394$$ 0 0
$$395$$ 3786.58 0.482338
$$396$$ 0 0
$$397$$ −7938.94 −1.00364 −0.501819 0.864973i $$-0.667336\pi$$
−0.501819 + 0.864973i $$0.667336\pi$$
$$398$$ 0 0
$$399$$ 5223.52 0.655396
$$400$$ 0 0
$$401$$ 214.402 0.0267001 0.0133500 0.999911i $$-0.495750\pi$$
0.0133500 + 0.999911i $$0.495750\pi$$
$$402$$ 0 0
$$403$$ 3463.52 0.428114
$$404$$ 0 0
$$405$$ 926.008 0.113614
$$406$$ 0 0
$$407$$ −3001.48 −0.365548
$$408$$ 0 0
$$409$$ −4783.73 −0.578338 −0.289169 0.957278i $$-0.593379\pi$$
−0.289169 + 0.957278i $$0.593379\pi$$
$$410$$ 0 0
$$411$$ 3115.58 0.373917
$$412$$ 0 0
$$413$$ −9832.99 −1.17155
$$414$$ 0 0
$$415$$ −4977.95 −0.588815
$$416$$ 0 0
$$417$$ −8575.39 −1.00705
$$418$$ 0 0
$$419$$ −9903.67 −1.15472 −0.577358 0.816491i $$-0.695916\pi$$
−0.577358 + 0.816491i $$0.695916\pi$$
$$420$$ 0 0
$$421$$ 12120.6 1.40314 0.701572 0.712598i $$-0.252482\pi$$
0.701572 + 0.712598i $$0.252482\pi$$
$$422$$ 0 0
$$423$$ 4206.28 0.483491
$$424$$ 0 0
$$425$$ −115.770 −0.0132133
$$426$$ 0 0
$$427$$ 2105.32 0.238603
$$428$$ 0 0
$$429$$ −1009.26 −0.113584
$$430$$ 0 0
$$431$$ 13672.6 1.52805 0.764023 0.645189i $$-0.223221\pi$$
0.764023 + 0.645189i $$0.223221\pi$$
$$432$$ 0 0
$$433$$ 7113.10 0.789455 0.394727 0.918798i $$-0.370839\pi$$
0.394727 + 0.918798i $$0.370839\pi$$
$$434$$ 0 0
$$435$$ 701.441 0.0773138
$$436$$ 0 0
$$437$$ 27912.9 3.05550
$$438$$ 0 0
$$439$$ 6022.04 0.654707 0.327353 0.944902i $$-0.393843\pi$$
0.327353 + 0.944902i $$0.393843\pi$$
$$440$$ 0 0
$$441$$ −1947.07 −0.210244
$$442$$ 0 0
$$443$$ −12994.4 −1.39364 −0.696821 0.717245i $$-0.745403\pi$$
−0.696821 + 0.717245i $$0.745403\pi$$
$$444$$ 0 0
$$445$$ 2963.61 0.315704
$$446$$ 0 0
$$447$$ −2230.44 −0.236009
$$448$$ 0 0
$$449$$ 10984.3 1.15452 0.577260 0.816560i $$-0.304122\pi$$
0.577260 + 0.816560i $$0.304122\pi$$
$$450$$ 0 0
$$451$$ 10123.2 1.05695
$$452$$ 0 0
$$453$$ −6831.72 −0.708570
$$454$$ 0 0
$$455$$ −1672.60 −0.172335
$$456$$ 0 0
$$457$$ 9834.10 1.00661 0.503304 0.864109i $$-0.332118\pi$$
0.503304 + 0.864109i $$0.332118\pi$$
$$458$$ 0 0
$$459$$ −548.845 −0.0558124
$$460$$ 0 0
$$461$$ −3401.42 −0.343644 −0.171822 0.985128i $$-0.554965\pi$$
−0.171822 + 0.985128i $$0.554965\pi$$
$$462$$ 0 0
$$463$$ −1739.42 −0.174596 −0.0872979 0.996182i $$-0.527823\pi$$
−0.0872979 + 0.996182i $$0.527823\pi$$
$$464$$ 0 0
$$465$$ −9137.45 −0.911267
$$466$$ 0 0
$$467$$ −7958.82 −0.788630 −0.394315 0.918975i $$-0.629018\pi$$
−0.394315 + 0.918975i $$0.629018\pi$$
$$468$$ 0 0
$$469$$ −6856.16 −0.675028
$$470$$ 0 0
$$471$$ −9520.54 −0.931387
$$472$$ 0 0
$$473$$ 3918.20 0.380886
$$474$$ 0 0
$$475$$ 881.114 0.0851122
$$476$$ 0 0
$$477$$ −719.858 −0.0690986
$$478$$ 0 0
$$479$$ −8431.98 −0.804315 −0.402158 0.915570i $$-0.631740\pi$$
−0.402158 + 0.915570i $$0.631740\pi$$
$$480$$ 0 0
$$481$$ 1507.79 0.142930
$$482$$ 0 0
$$483$$ 6091.45 0.573852
$$484$$ 0 0
$$485$$ 14006.4 1.31133
$$486$$ 0 0
$$487$$ 11684.7 1.08723 0.543617 0.839334i $$-0.317055\pi$$
0.543617 + 0.839334i $$0.317055\pi$$
$$488$$ 0 0
$$489$$ −6943.94 −0.642159
$$490$$ 0 0
$$491$$ −3954.70 −0.363489 −0.181745 0.983346i $$-0.558174\pi$$
−0.181745 + 0.983346i $$0.558174\pi$$
$$492$$ 0 0
$$493$$ −415.744 −0.0379801
$$494$$ 0 0
$$495$$ 2662.63 0.241771
$$496$$ 0 0
$$497$$ −2791.49 −0.251943
$$498$$ 0 0
$$499$$ −5690.37 −0.510493 −0.255246 0.966876i $$-0.582157\pi$$
−0.255246 + 0.966876i $$0.582157\pi$$
$$500$$ 0 0
$$501$$ 7996.95 0.713128
$$502$$ 0 0
$$503$$ −10859.1 −0.962595 −0.481298 0.876557i $$-0.659834\pi$$
−0.481298 + 0.876557i $$0.659834\pi$$
$$504$$ 0 0
$$505$$ −7378.53 −0.650178
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ 18558.6 1.61610 0.808049 0.589115i $$-0.200524\pi$$
0.808049 + 0.589115i $$0.200524\pi$$
$$510$$ 0 0
$$511$$ 9597.27 0.830838
$$512$$ 0 0
$$513$$ 4177.22 0.359510
$$514$$ 0 0
$$515$$ 5843.42 0.499984
$$516$$ 0 0
$$517$$ 12094.7 1.02887
$$518$$ 0 0
$$519$$ 495.730 0.0419271
$$520$$ 0 0
$$521$$ 17297.5 1.45454 0.727271 0.686350i $$-0.240788\pi$$
0.727271 + 0.686350i $$0.240788\pi$$
$$522$$ 0 0
$$523$$ −5016.11 −0.419386 −0.209693 0.977767i $$-0.567247\pi$$
−0.209693 + 0.977767i $$0.567247\pi$$
$$524$$ 0 0
$$525$$ 192.286 0.0159849
$$526$$ 0 0
$$527$$ 5415.77 0.447656
$$528$$ 0 0
$$529$$ 20383.8 1.67533
$$530$$ 0 0
$$531$$ −7863.39 −0.642640
$$532$$ 0 0
$$533$$ −5085.39 −0.413269
$$534$$ 0 0
$$535$$ 6953.01 0.561878
$$536$$ 0 0
$$537$$ 2137.02 0.171730
$$538$$ 0 0
$$539$$ −5598.57 −0.447398
$$540$$ 0 0
$$541$$ −17642.3 −1.40204 −0.701018 0.713144i $$-0.747271\pi$$
−0.701018 + 0.713144i $$0.747271\pi$$
$$542$$ 0 0
$$543$$ −6619.59 −0.523156
$$544$$ 0 0
$$545$$ 14862.3 1.16813
$$546$$ 0 0
$$547$$ −18414.9 −1.43943 −0.719713 0.694271i $$-0.755727\pi$$
−0.719713 + 0.694271i $$0.755727\pi$$
$$548$$ 0 0
$$549$$ 1683.61 0.130883
$$550$$ 0 0
$$551$$ 3164.20 0.244645
$$552$$ 0 0
$$553$$ 3727.66 0.286648
$$554$$ 0 0
$$555$$ −3977.84 −0.304234
$$556$$ 0 0
$$557$$ −8179.15 −0.622193 −0.311096 0.950378i $$-0.600696\pi$$
−0.311096 + 0.950378i $$0.600696\pi$$
$$558$$ 0 0
$$559$$ −1968.30 −0.148927
$$560$$ 0 0
$$561$$ −1578.14 −0.118769
$$562$$ 0 0
$$563$$ −1880.07 −0.140738 −0.0703690 0.997521i $$-0.522418\pi$$
−0.0703690 + 0.997521i $$0.522418\pi$$
$$564$$ 0 0
$$565$$ 482.385 0.0359187
$$566$$ 0 0
$$567$$ 911.598 0.0675194
$$568$$ 0 0
$$569$$ 10118.3 0.745485 0.372743 0.927935i $$-0.378417\pi$$
0.372743 + 0.927935i $$0.378417\pi$$
$$570$$ 0 0
$$571$$ 23428.9 1.71711 0.858555 0.512721i $$-0.171362\pi$$
0.858555 + 0.512721i $$0.171362\pi$$
$$572$$ 0 0
$$573$$ −4411.92 −0.321659
$$574$$ 0 0
$$575$$ 1027.52 0.0745225
$$576$$ 0 0
$$577$$ 20508.1 1.47966 0.739831 0.672793i $$-0.234906\pi$$
0.739831 + 0.672793i $$0.234906\pi$$
$$578$$ 0 0
$$579$$ 1108.68 0.0795771
$$580$$ 0 0
$$581$$ −4900.49 −0.349925
$$582$$ 0 0
$$583$$ −2069.87 −0.147042
$$584$$ 0 0
$$585$$ −1337.57 −0.0945327
$$586$$ 0 0
$$587$$ −5968.43 −0.419665 −0.209833 0.977737i $$-0.567292\pi$$
−0.209833 + 0.977737i $$0.567292\pi$$
$$588$$ 0 0
$$589$$ −41219.0 −2.88353
$$590$$ 0 0
$$591$$ 12820.2 0.892308
$$592$$ 0 0
$$593$$ −14659.5 −1.01517 −0.507584 0.861602i $$-0.669461\pi$$
−0.507584 + 0.861602i $$0.669461\pi$$
$$594$$ 0 0
$$595$$ −2615.38 −0.180202
$$596$$ 0 0
$$597$$ −12462.9 −0.854394
$$598$$ 0 0
$$599$$ −23635.9 −1.61225 −0.806125 0.591746i $$-0.798439\pi$$
−0.806125 + 0.591746i $$0.798439\pi$$
$$600$$ 0 0
$$601$$ −11527.0 −0.782356 −0.391178 0.920315i $$-0.627932\pi$$
−0.391178 + 0.920315i $$0.627932\pi$$
$$602$$ 0 0
$$603$$ −5482.83 −0.370279
$$604$$ 0 0
$$605$$ −7560.15 −0.508039
$$606$$ 0 0
$$607$$ 5098.56 0.340930 0.170465 0.985364i $$-0.445473\pi$$
0.170465 + 0.985364i $$0.445473\pi$$
$$608$$ 0 0
$$609$$ 690.525 0.0459466
$$610$$ 0 0
$$611$$ −6075.74 −0.402288
$$612$$ 0 0
$$613$$ −1516.39 −0.0999128 −0.0499564 0.998751i $$-0.515908\pi$$
−0.0499564 + 0.998751i $$0.515908\pi$$
$$614$$ 0 0
$$615$$ 13416.3 0.879668
$$616$$ 0 0
$$617$$ 18539.3 1.20966 0.604832 0.796353i $$-0.293240\pi$$
0.604832 + 0.796353i $$0.293240\pi$$
$$618$$ 0 0
$$619$$ 25684.9 1.66779 0.833897 0.551920i $$-0.186105\pi$$
0.833897 + 0.551920i $$0.186105\pi$$
$$620$$ 0 0
$$621$$ 4871.30 0.314780
$$622$$ 0 0
$$623$$ 2917.49 0.187619
$$624$$ 0 0
$$625$$ −16304.5 −1.04349
$$626$$ 0 0
$$627$$ 12011.1 0.765038
$$628$$ 0 0
$$629$$ 2357.67 0.149454
$$630$$ 0 0
$$631$$ 22410.9 1.41389 0.706945 0.707269i $$-0.250073\pi$$
0.706945 + 0.707269i $$0.250073\pi$$
$$632$$ 0 0
$$633$$ 3694.77 0.231997
$$634$$ 0 0
$$635$$ 3555.62 0.222206
$$636$$ 0 0
$$637$$ 2812.43 0.174933
$$638$$ 0 0
$$639$$ −2232.34 −0.138200
$$640$$ 0 0
$$641$$ 6827.81 0.420721 0.210361 0.977624i $$-0.432536\pi$$
0.210361 + 0.977624i $$0.432536\pi$$
$$642$$ 0 0
$$643$$ −23264.3 −1.42684 −0.713418 0.700738i $$-0.752854\pi$$
−0.713418 + 0.700738i $$0.752854\pi$$
$$644$$ 0 0
$$645$$ 5192.76 0.316999
$$646$$ 0 0
$$647$$ −14745.9 −0.896014 −0.448007 0.894030i $$-0.647866\pi$$
−0.448007 + 0.894030i $$0.647866\pi$$
$$648$$ 0 0
$$649$$ −22610.3 −1.36754
$$650$$ 0 0
$$651$$ −8995.26 −0.541554
$$652$$ 0 0
$$653$$ −10909.0 −0.653755 −0.326878 0.945067i $$-0.605997\pi$$
−0.326878 + 0.945067i $$0.605997\pi$$
$$654$$ 0 0
$$655$$ 22875.7 1.36462
$$656$$ 0 0
$$657$$ 7674.89 0.455747
$$658$$ 0 0
$$659$$ −4182.99 −0.247263 −0.123631 0.992328i $$-0.539454\pi$$
−0.123631 + 0.992328i $$0.539454\pi$$
$$660$$ 0 0
$$661$$ −2224.23 −0.130881 −0.0654406 0.997856i $$-0.520845\pi$$
−0.0654406 + 0.997856i $$0.520845\pi$$
$$662$$ 0 0
$$663$$ 792.776 0.0464387
$$664$$ 0 0
$$665$$ 19905.4 1.16075
$$666$$ 0 0
$$667$$ 3689.95 0.214206
$$668$$ 0 0
$$669$$ 6561.72 0.379209
$$670$$ 0 0
$$671$$ 4841.04 0.278519
$$672$$ 0 0
$$673$$ −24152.5 −1.38337 −0.691687 0.722197i $$-0.743132\pi$$
−0.691687 + 0.722197i $$0.743132\pi$$
$$674$$ 0 0
$$675$$ 153.770 0.00876833
$$676$$ 0 0
$$677$$ 15310.7 0.869187 0.434593 0.900627i $$-0.356892\pi$$
0.434593 + 0.900627i $$0.356892\pi$$
$$678$$ 0 0
$$679$$ 13788.4 0.779310
$$680$$ 0 0
$$681$$ 12416.0 0.698652
$$682$$ 0 0
$$683$$ 11399.6 0.638646 0.319323 0.947646i $$-0.396545\pi$$
0.319323 + 0.947646i $$0.396545\pi$$
$$684$$ 0 0
$$685$$ 11872.6 0.662233
$$686$$ 0 0
$$687$$ 2506.06 0.139174
$$688$$ 0 0
$$689$$ 1039.79 0.0574935
$$690$$ 0 0
$$691$$ −3323.23 −0.182955 −0.0914773 0.995807i $$-0.529159\pi$$
−0.0914773 + 0.995807i $$0.529159\pi$$
$$692$$ 0 0
$$693$$ 2621.20 0.143681
$$694$$ 0 0
$$695$$ −32678.5 −1.78355
$$696$$ 0 0
$$697$$ −7951.82 −0.432133
$$698$$ 0 0
$$699$$ 11056.5 0.598279
$$700$$ 0 0
$$701$$ 12670.4 0.682673 0.341336 0.939941i $$-0.389120\pi$$
0.341336 + 0.939941i $$0.389120\pi$$
$$702$$ 0 0
$$703$$ −17944.0 −0.962692
$$704$$ 0 0
$$705$$ 16029.0 0.856295
$$706$$ 0 0
$$707$$ −7263.71 −0.386393
$$708$$ 0 0
$$709$$ −13075.2 −0.692594 −0.346297 0.938125i $$-0.612561\pi$$
−0.346297 + 0.938125i $$0.612561\pi$$
$$710$$ 0 0
$$711$$ 2980.99 0.157237
$$712$$ 0 0
$$713$$ −48067.8 −2.52476
$$714$$ 0 0
$$715$$ −3846.03 −0.201165
$$716$$ 0 0
$$717$$ −9078.62 −0.472869
$$718$$ 0 0
$$719$$ 2988.41 0.155005 0.0775026 0.996992i $$-0.475305\pi$$
0.0775026 + 0.996992i $$0.475305\pi$$
$$720$$ 0 0
$$721$$ 5752.48 0.297134
$$722$$ 0 0
$$723$$ 9796.73 0.503934
$$724$$ 0 0
$$725$$ 116.479 0.00596680
$$726$$ 0 0
$$727$$ 5507.46 0.280963 0.140482 0.990083i $$-0.455135\pi$$
0.140482 + 0.990083i $$0.455135\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −3077.75 −0.155725
$$732$$ 0 0
$$733$$ 36585.2 1.84353 0.921764 0.387751i $$-0.126748\pi$$
0.921764 + 0.387751i $$0.126748\pi$$
$$734$$ 0 0
$$735$$ −7419.75 −0.372356
$$736$$ 0 0
$$737$$ −15765.3 −0.787953
$$738$$ 0 0
$$739$$ 6425.89 0.319865 0.159933 0.987128i $$-0.448872\pi$$
0.159933 + 0.987128i $$0.448872\pi$$
$$740$$ 0 0
$$741$$ −6033.76 −0.299131
$$742$$ 0 0
$$743$$ −20411.0 −1.00782 −0.503908 0.863757i $$-0.668105\pi$$
−0.503908 + 0.863757i $$0.668105\pi$$
$$744$$ 0 0
$$745$$ −8499.60 −0.417988
$$746$$ 0 0
$$747$$ −3918.89 −0.191947
$$748$$ 0 0
$$749$$ 6844.81 0.333917
$$750$$ 0 0
$$751$$ 24259.5 1.17875 0.589375 0.807860i $$-0.299374\pi$$
0.589375 + 0.807860i $$0.299374\pi$$
$$752$$ 0 0
$$753$$ −19089.5 −0.923850
$$754$$ 0 0
$$755$$ −26033.9 −1.25493
$$756$$ 0 0
$$757$$ −9295.39 −0.446297 −0.223148 0.974785i $$-0.571633\pi$$
−0.223148 + 0.974785i $$0.571633\pi$$
$$758$$ 0 0
$$759$$ 14006.9 0.669851
$$760$$ 0 0
$$761$$ −21974.7 −1.04676 −0.523378 0.852101i $$-0.675328\pi$$
−0.523378 + 0.852101i $$0.675328\pi$$
$$762$$ 0 0
$$763$$ 14631.0 0.694204
$$764$$ 0 0
$$765$$ −2091.50 −0.0988476
$$766$$ 0 0
$$767$$ 11358.2 0.534709
$$768$$ 0 0
$$769$$ 22987.4 1.07795 0.538977 0.842320i $$-0.318811\pi$$
0.538977 + 0.842320i $$0.318811\pi$$
$$770$$ 0 0
$$771$$ −18256.1 −0.852759
$$772$$ 0 0
$$773$$ −31970.9 −1.48760 −0.743799 0.668404i $$-0.766978\pi$$
−0.743799 + 0.668404i $$0.766978\pi$$
$$774$$ 0 0
$$775$$ −1517.34 −0.0703283
$$776$$ 0 0
$$777$$ −3915.94 −0.180803
$$778$$ 0 0
$$779$$ 60520.8 2.78354
$$780$$ 0 0
$$781$$ −6418.85 −0.294090
$$782$$ 0 0
$$783$$ 552.209 0.0252035
$$784$$ 0 0
$$785$$ −36280.2 −1.64955
$$786$$ 0 0
$$787$$ 6087.26 0.275715 0.137857 0.990452i $$-0.455978\pi$$
0.137857 + 0.990452i $$0.455978\pi$$
$$788$$ 0 0
$$789$$ −369.680 −0.0166805
$$790$$ 0 0
$$791$$ 474.878 0.0213460
$$792$$ 0 0
$$793$$ −2431.88 −0.108901
$$794$$ 0 0
$$795$$ −2743.19 −0.122378
$$796$$ 0 0
$$797$$ −23080.0 −1.02577 −0.512883 0.858458i $$-0.671423\pi$$
−0.512883 + 0.858458i $$0.671423\pi$$
$$798$$ 0 0
$$799$$ −9500.41 −0.420651
$$800$$ 0 0
$$801$$ 2333.10 0.102916
$$802$$ 0 0
$$803$$ 22068.3 0.969829
$$804$$ 0 0
$$805$$ 23212.9 1.01633
$$806$$ 0 0
$$807$$ 5807.37 0.253320
$$808$$ 0 0
$$809$$ −32377.8 −1.40710 −0.703550 0.710646i $$-0.748403\pi$$
−0.703550 + 0.710646i $$0.748403\pi$$
$$810$$ 0 0
$$811$$ 26352.8 1.14103 0.570513 0.821288i $$-0.306744\pi$$
0.570513 + 0.821288i $$0.306744\pi$$
$$812$$ 0 0
$$813$$ 13838.1 0.596952
$$814$$ 0 0
$$815$$ −26461.5 −1.13731
$$816$$ 0 0
$$817$$ 23424.5 1.00308
$$818$$ 0 0
$$819$$ −1316.75 −0.0561796
$$820$$ 0 0
$$821$$ −35355.3 −1.50294 −0.751468 0.659770i $$-0.770654\pi$$
−0.751468 + 0.659770i $$0.770654\pi$$
$$822$$ 0 0
$$823$$ 12663.3 0.536347 0.268173 0.963371i $$-0.413580\pi$$
0.268173 + 0.963371i $$0.413580\pi$$
$$824$$ 0 0
$$825$$ 442.149 0.0186590
$$826$$ 0 0
$$827$$ −16295.2 −0.685176 −0.342588 0.939486i $$-0.611303\pi$$
−0.342588 + 0.939486i $$0.611303\pi$$
$$828$$ 0 0
$$829$$ −13638.9 −0.571411 −0.285705 0.958318i $$-0.592228\pi$$
−0.285705 + 0.958318i $$0.592228\pi$$
$$830$$ 0 0
$$831$$ 17502.9 0.730649
$$832$$ 0 0
$$833$$ 4397.69 0.182918
$$834$$ 0 0
$$835$$ 30474.2 1.26300
$$836$$ 0 0
$$837$$ −7193.46 −0.297064
$$838$$ 0 0
$$839$$ 1890.31 0.0777838 0.0388919 0.999243i $$-0.487617\pi$$
0.0388919 + 0.999243i $$0.487617\pi$$
$$840$$ 0 0
$$841$$ −23970.7 −0.982849
$$842$$ 0 0
$$843$$ 14075.7 0.575082
$$844$$ 0 0
$$845$$ 1932.04 0.0786559
$$846$$ 0 0
$$847$$ −7442.50 −0.301921
$$848$$ 0 0
$$849$$ 10396.8 0.420279
$$850$$ 0 0
$$851$$ −20925.6 −0.842914
$$852$$ 0 0
$$853$$ −1620.21 −0.0650351 −0.0325175 0.999471i $$-0.510352\pi$$
−0.0325175 + 0.999471i $$0.510352\pi$$
$$854$$ 0 0
$$855$$ 15918.3 0.636718
$$856$$ 0 0
$$857$$ −14508.4 −0.578292 −0.289146 0.957285i $$-0.593371\pi$$
−0.289146 + 0.957285i $$0.593371\pi$$
$$858$$ 0 0
$$859$$ 29639.8 1.17730 0.588648 0.808389i $$-0.299660\pi$$
0.588648 + 0.808389i $$0.299660\pi$$
$$860$$ 0 0
$$861$$ 13207.5 0.522776
$$862$$ 0 0
$$863$$ −21528.8 −0.849186 −0.424593 0.905384i $$-0.639583\pi$$
−0.424593 + 0.905384i $$0.639583\pi$$
$$864$$ 0 0
$$865$$ 1889.10 0.0742557
$$866$$ 0 0
$$867$$ −13499.4 −0.528792
$$868$$ 0 0
$$869$$ 8571.50 0.334601
$$870$$ 0 0
$$871$$ 7919.65 0.308091
$$872$$ 0 0
$$873$$ 11026.5 0.427482
$$874$$ 0 0
$$875$$ −15349.9 −0.593054
$$876$$ 0 0
$$877$$ −14865.3 −0.572366 −0.286183 0.958175i $$-0.592387\pi$$
−0.286183 + 0.958175i $$0.592387\pi$$
$$878$$ 0 0
$$879$$ −8031.92 −0.308202
$$880$$ 0 0
$$881$$ −21336.0 −0.815921 −0.407961 0.913000i $$-0.633760\pi$$
−0.407961 + 0.913000i $$0.633760\pi$$
$$882$$ 0 0
$$883$$ 37538.2 1.43065 0.715323 0.698794i $$-0.246280\pi$$
0.715323 + 0.698794i $$0.246280\pi$$
$$884$$ 0 0
$$885$$ −29965.3 −1.13816
$$886$$ 0 0
$$887$$ −34575.0 −1.30881 −0.654406 0.756144i $$-0.727081\pi$$
−0.654406 + 0.756144i $$0.727081\pi$$
$$888$$ 0 0
$$889$$ 3500.29 0.132054
$$890$$ 0 0
$$891$$ 2096.16 0.0788148
$$892$$ 0 0
$$893$$ 72306.9 2.70958
$$894$$ 0 0
$$895$$ 8143.61 0.304146
$$896$$ 0 0
$$897$$ −7036.32 −0.261913
$$898$$ 0 0
$$899$$ −5448.96 −0.202150
$$900$$ 0 0
$$901$$ 1625.89 0.0601178
$$902$$ 0 0
$$903$$ 5111.95 0.188389
$$904$$ 0 0
$$905$$ −25225.5 −0.926545
$$906$$ 0 0
$$907$$ −10424.8 −0.381641 −0.190820 0.981625i $$-0.561115\pi$$
−0.190820 + 0.981625i $$0.561115\pi$$
$$908$$ 0 0
$$909$$ −5808.75 −0.211952
$$910$$ 0 0
$$911$$ −10961.8 −0.398661 −0.199331 0.979932i $$-0.563877\pi$$
−0.199331 + 0.979932i $$0.563877\pi$$
$$912$$ 0 0
$$913$$ −11268.3 −0.408464
$$914$$ 0 0
$$915$$ 6415.80 0.231803
$$916$$ 0 0
$$917$$ 22519.7 0.810976
$$918$$ 0 0
$$919$$ 10779.2 0.386914 0.193457 0.981109i $$-0.438030\pi$$
0.193457 + 0.981109i $$0.438030\pi$$
$$920$$ 0 0
$$921$$ 1415.74 0.0506519
$$922$$ 0 0
$$923$$ 3224.49 0.114990
$$924$$ 0 0
$$925$$ −660.549 −0.0234797
$$926$$ 0 0
$$927$$ 4600.23 0.162990
$$928$$ 0 0
$$929$$ 5429.07 0.191735 0.0958675 0.995394i $$-0.469437\pi$$
0.0958675 + 0.995394i $$0.469437\pi$$
$$930$$ 0 0
$$931$$ −33470.5 −1.17825
$$932$$ 0 0
$$933$$ 4555.55 0.159852
$$934$$ 0 0
$$935$$ −6013.88 −0.210348
$$936$$ 0 0
$$937$$ −21300.1 −0.742631 −0.371315 0.928507i $$-0.621093\pi$$
−0.371315 + 0.928507i $$0.621093\pi$$
$$938$$ 0 0
$$939$$ 12149.6 0.422244
$$940$$ 0 0
$$941$$ −26851.2 −0.930207 −0.465103 0.885256i $$-0.653983\pi$$
−0.465103 + 0.885256i $$0.653983\pi$$
$$942$$ 0 0
$$943$$ 70576.7 2.43722
$$944$$ 0 0
$$945$$ 3473.86 0.119582
$$946$$ 0 0
$$947$$ −8021.68 −0.275258 −0.137629 0.990484i $$-0.543948\pi$$
−0.137629 + 0.990484i $$0.543948\pi$$
$$948$$ 0 0
$$949$$ −11085.9 −0.379204
$$950$$ 0 0
$$951$$ −9761.88 −0.332861
$$952$$ 0 0
$$953$$ 35715.0 1.21398 0.606990 0.794709i $$-0.292377\pi$$
0.606990 + 0.794709i $$0.292377\pi$$
$$954$$ 0 0
$$955$$ −16812.6 −0.569680
$$956$$ 0 0
$$957$$ 1587.82 0.0536330
$$958$$ 0 0
$$959$$ 11687.9 0.393557
$$960$$ 0 0
$$961$$ 41190.9 1.38266
$$962$$ 0 0
$$963$$ 5473.76 0.183166
$$964$$ 0 0
$$965$$ 4224.88 0.140936
$$966$$ 0 0
$$967$$ −53338.8 −1.77380 −0.886898 0.461965i $$-0.847145\pi$$
−0.886898 + 0.461965i $$0.847145\pi$$
$$968$$ 0 0
$$969$$ −9434.77 −0.312785
$$970$$ 0 0
$$971$$ 23112.9 0.763882 0.381941 0.924187i $$-0.375256\pi$$
0.381941 + 0.924187i $$0.375256\pi$$
$$972$$ 0 0
$$973$$ −32170.0 −1.05994
$$974$$ 0 0
$$975$$ −222.113 −0.00729569
$$976$$ 0 0
$$977$$ −52874.6 −1.73143 −0.865715 0.500538i $$-0.833136\pi$$
−0.865715 + 0.500538i $$0.833136\pi$$
$$978$$ 0 0
$$979$$ 6708.57 0.219006
$$980$$ 0 0
$$981$$ 11700.3 0.380798
$$982$$ 0 0
$$983$$ −45173.1 −1.46572 −0.732858 0.680381i $$-0.761814\pi$$
−0.732858 + 0.680381i $$0.761814\pi$$
$$984$$ 0 0
$$985$$ 48854.5 1.58034
$$986$$ 0 0
$$987$$ 15779.6 0.508885
$$988$$ 0 0
$$989$$ 27316.7 0.878281
$$990$$ 0 0
$$991$$ −60485.6 −1.93884 −0.969418 0.245414i $$-0.921076\pi$$
−0.969418 + 0.245414i $$0.921076\pi$$
$$992$$ 0 0
$$993$$ −10267.3 −0.328121
$$994$$ 0 0
$$995$$ −47492.8 −1.51319
$$996$$ 0 0
$$997$$ 18108.1 0.575214 0.287607 0.957749i $$-0.407140\pi$$
0.287607 + 0.957749i $$0.407140\pi$$
$$998$$ 0 0
$$999$$ −3131.56 −0.0991773
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 2496.4.a.bp.1.3 3
4.3 odd 2 2496.4.a.bl.1.3 3
8.3 odd 2 39.4.a.c.1.3 3
8.5 even 2 624.4.a.t.1.1 3
24.5 odd 2 1872.4.a.bk.1.3 3
24.11 even 2 117.4.a.f.1.1 3
40.19 odd 2 975.4.a.l.1.1 3
56.27 even 2 1911.4.a.k.1.3 3
104.51 odd 2 507.4.a.h.1.1 3
104.83 even 4 507.4.b.g.337.1 6
104.99 even 4 507.4.b.g.337.6 6
312.155 even 2 1521.4.a.u.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 8.3 odd 2
117.4.a.f.1.1 3 24.11 even 2
507.4.a.h.1.1 3 104.51 odd 2
507.4.b.g.337.1 6 104.83 even 4
507.4.b.g.337.6 6 104.99 even 4
624.4.a.t.1.1 3 8.5 even 2
975.4.a.l.1.1 3 40.19 odd 2
1521.4.a.u.1.3 3 312.155 even 2
1872.4.a.bk.1.3 3 24.5 odd 2
1911.4.a.k.1.3 3 56.27 even 2
2496.4.a.bl.1.3 3 4.3 odd 2
2496.4.a.bp.1.3 3 1.1 even 1 trivial