# Properties

 Label 768.4.a.u.1.2 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.9792.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 2x + 7$$ x^4 - 2*x^3 - 7*x^2 + 2*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.63019$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.67763 q^{5} +33.0917 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.67763 q^{5} +33.0917 q^{7} +9.00000 q^{9} -34.6274 q^{11} -82.2421 q^{13} +17.0329 q^{15} +97.8823 q^{17} +55.8823 q^{19} -99.2750 q^{21} -130.418 q^{23} -92.7645 q^{25} -27.0000 q^{27} +147.451 q^{29} +101.223 q^{31} +103.882 q^{33} -187.882 q^{35} +184.439 q^{37} +246.726 q^{39} -237.411 q^{41} -199.882 q^{43} -51.0987 q^{45} -334.813 q^{47} +752.058 q^{49} -293.647 q^{51} +102.030 q^{53} +196.602 q^{55} -167.647 q^{57} -105.961 q^{59} +717.803 q^{61} +297.825 q^{63} +466.940 q^{65} -316.471 q^{67} +391.255 q^{69} -800.045 q^{71} -301.058 q^{73} +278.294 q^{75} -1145.88 q^{77} -42.8329 q^{79} +81.0000 q^{81} -1236.67 q^{83} -555.739 q^{85} -442.354 q^{87} -1325.29 q^{89} -2721.53 q^{91} -303.670 q^{93} -317.279 q^{95} -505.765 q^{97} -311.647 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 36 * q^9 $$4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100})$$ 4 * q - 12 * q^3 + 36 * q^9 - 48 * q^11 + 120 * q^17 - 48 * q^19 + 172 * q^25 - 108 * q^27 + 144 * q^33 - 480 * q^35 + 408 * q^41 - 528 * q^43 + 836 * q^49 - 360 * q^51 + 144 * q^57 - 1872 * q^59 - 576 * q^65 - 2352 * q^67 + 968 * q^73 - 516 * q^75 + 324 * q^81 - 3408 * q^83 - 3672 * q^89 - 5184 * q^91 - 1480 * q^97 - 432 * 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
$$5$$ −5.67763 −0.507823 −0.253911 0.967227i $$-0.581717\pi$$
−0.253911 + 0.967227i $$0.581717\pi$$
$$6$$ 0 0
$$7$$ 33.0917 1.78678 0.893391 0.449280i $$-0.148320\pi$$
0.893391 + 0.449280i $$0.148320\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −34.6274 −0.949142 −0.474571 0.880217i $$-0.657397\pi$$
−0.474571 + 0.880217i $$0.657397\pi$$
$$12$$ 0 0
$$13$$ −82.2421 −1.75460 −0.877302 0.479939i $$-0.840659\pi$$
−0.877302 + 0.479939i $$0.840659\pi$$
$$14$$ 0 0
$$15$$ 17.0329 0.293192
$$16$$ 0 0
$$17$$ 97.8823 1.39647 0.698233 0.715870i $$-0.253970\pi$$
0.698233 + 0.715870i $$0.253970\pi$$
$$18$$ 0 0
$$19$$ 55.8823 0.674751 0.337375 0.941370i $$-0.390461\pi$$
0.337375 + 0.941370i $$0.390461\pi$$
$$20$$ 0 0
$$21$$ −99.2750 −1.03160
$$22$$ 0 0
$$23$$ −130.418 −1.18235 −0.591176 0.806542i $$-0.701336\pi$$
−0.591176 + 0.806542i $$0.701336\pi$$
$$24$$ 0 0
$$25$$ −92.7645 −0.742116
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 147.451 0.944173 0.472086 0.881552i $$-0.343501\pi$$
0.472086 + 0.881552i $$0.343501\pi$$
$$30$$ 0 0
$$31$$ 101.223 0.586459 0.293230 0.956042i $$-0.405270\pi$$
0.293230 + 0.956042i $$0.405270\pi$$
$$32$$ 0 0
$$33$$ 103.882 0.547987
$$34$$ 0 0
$$35$$ −187.882 −0.907368
$$36$$ 0 0
$$37$$ 184.439 0.819504 0.409752 0.912197i $$-0.365615\pi$$
0.409752 + 0.912197i $$0.365615\pi$$
$$38$$ 0 0
$$39$$ 246.726 1.01302
$$40$$ 0 0
$$41$$ −237.411 −0.904327 −0.452164 0.891935i $$-0.649348\pi$$
−0.452164 + 0.891935i $$0.649348\pi$$
$$42$$ 0 0
$$43$$ −199.882 −0.708878 −0.354439 0.935079i $$-0.615328\pi$$
−0.354439 + 0.935079i $$0.615328\pi$$
$$44$$ 0 0
$$45$$ −51.0987 −0.169274
$$46$$ 0 0
$$47$$ −334.813 −1.03910 −0.519548 0.854441i $$-0.673900\pi$$
−0.519548 + 0.854441i $$0.673900\pi$$
$$48$$ 0 0
$$49$$ 752.058 2.19259
$$50$$ 0 0
$$51$$ −293.647 −0.806250
$$52$$ 0 0
$$53$$ 102.030 0.264433 0.132216 0.991221i $$-0.457791\pi$$
0.132216 + 0.991221i $$0.457791\pi$$
$$54$$ 0 0
$$55$$ 196.602 0.481996
$$56$$ 0 0
$$57$$ −167.647 −0.389568
$$58$$ 0 0
$$59$$ −105.961 −0.233813 −0.116907 0.993143i $$-0.537298\pi$$
−0.116907 + 0.993143i $$0.537298\pi$$
$$60$$ 0 0
$$61$$ 717.803 1.50664 0.753321 0.657653i $$-0.228451\pi$$
0.753321 + 0.657653i $$0.228451\pi$$
$$62$$ 0 0
$$63$$ 297.825 0.595594
$$64$$ 0 0
$$65$$ 466.940 0.891028
$$66$$ 0 0
$$67$$ −316.471 −0.577061 −0.288530 0.957471i $$-0.593167\pi$$
−0.288530 + 0.957471i $$0.593167\pi$$
$$68$$ 0 0
$$69$$ 391.255 0.682632
$$70$$ 0 0
$$71$$ −800.045 −1.33729 −0.668647 0.743580i $$-0.733126\pi$$
−0.668647 + 0.743580i $$0.733126\pi$$
$$72$$ 0 0
$$73$$ −301.058 −0.482687 −0.241344 0.970440i $$-0.577588\pi$$
−0.241344 + 0.970440i $$0.577588\pi$$
$$74$$ 0 0
$$75$$ 278.294 0.428461
$$76$$ 0 0
$$77$$ −1145.88 −1.69591
$$78$$ 0 0
$$79$$ −42.8329 −0.0610010 −0.0305005 0.999535i $$-0.509710\pi$$
−0.0305005 + 0.999535i $$0.509710\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1236.67 −1.63544 −0.817721 0.575614i $$-0.804763\pi$$
−0.817721 + 0.575614i $$0.804763\pi$$
$$84$$ 0 0
$$85$$ −555.739 −0.709158
$$86$$ 0 0
$$87$$ −442.354 −0.545119
$$88$$ 0 0
$$89$$ −1325.29 −1.57844 −0.789218 0.614113i $$-0.789514\pi$$
−0.789218 + 0.614113i $$0.789514\pi$$
$$90$$ 0 0
$$91$$ −2721.53 −3.13509
$$92$$ 0 0
$$93$$ −303.670 −0.338592
$$94$$ 0 0
$$95$$ −317.279 −0.342654
$$96$$ 0 0
$$97$$ −505.765 −0.529408 −0.264704 0.964330i $$-0.585274\pi$$
−0.264704 + 0.964330i $$0.585274\pi$$
$$98$$ 0 0
$$99$$ −311.647 −0.316381
$$100$$ 0 0
$$101$$ −1281.97 −1.26298 −0.631491 0.775383i $$-0.717557\pi$$
−0.631491 + 0.775383i $$0.717557\pi$$
$$102$$ 0 0
$$103$$ −161.562 −0.154555 −0.0772775 0.997010i $$-0.524623\pi$$
−0.0772775 + 0.997010i $$0.524623\pi$$
$$104$$ 0 0
$$105$$ 563.647 0.523869
$$106$$ 0 0
$$107$$ −481.726 −0.435235 −0.217618 0.976034i $$-0.569829\pi$$
−0.217618 + 0.976034i $$0.569829\pi$$
$$108$$ 0 0
$$109$$ −286.637 −0.251879 −0.125940 0.992038i $$-0.540195\pi$$
−0.125940 + 0.992038i $$0.540195\pi$$
$$110$$ 0 0
$$111$$ −553.318 −0.473141
$$112$$ 0 0
$$113$$ −1397.29 −1.16324 −0.581621 0.813460i $$-0.697581\pi$$
−0.581621 + 0.813460i $$0.697581\pi$$
$$114$$ 0 0
$$115$$ 740.468 0.600426
$$116$$ 0 0
$$117$$ −740.179 −0.584868
$$118$$ 0 0
$$119$$ 3239.09 2.49518
$$120$$ 0 0
$$121$$ −131.942 −0.0991300
$$122$$ 0 0
$$123$$ 712.234 0.522113
$$124$$ 0 0
$$125$$ 1236.39 0.884686
$$126$$ 0 0
$$127$$ 443.829 0.310106 0.155053 0.987906i $$-0.450445\pi$$
0.155053 + 0.987906i $$0.450445\pi$$
$$128$$ 0 0
$$129$$ 599.647 0.409271
$$130$$ 0 0
$$131$$ −1365.57 −0.910765 −0.455382 0.890296i $$-0.650497\pi$$
−0.455382 + 0.890296i $$0.650497\pi$$
$$132$$ 0 0
$$133$$ 1849.24 1.20563
$$134$$ 0 0
$$135$$ 153.296 0.0977305
$$136$$ 0 0
$$137$$ 1223.06 0.762721 0.381360 0.924426i $$-0.375456\pi$$
0.381360 + 0.924426i $$0.375456\pi$$
$$138$$ 0 0
$$139$$ −2176.70 −1.32824 −0.664121 0.747625i $$-0.731194\pi$$
−0.664121 + 0.747625i $$0.731194\pi$$
$$140$$ 0 0
$$141$$ 1004.44 0.599922
$$142$$ 0 0
$$143$$ 2847.83 1.66537
$$144$$ 0 0
$$145$$ −837.174 −0.479473
$$146$$ 0 0
$$147$$ −2256.17 −1.26589
$$148$$ 0 0
$$149$$ 2875.88 1.58122 0.790610 0.612320i $$-0.209764\pi$$
0.790610 + 0.612320i $$0.209764\pi$$
$$150$$ 0 0
$$151$$ −2330.03 −1.25573 −0.627863 0.778323i $$-0.716070\pi$$
−0.627863 + 0.778323i $$0.716070\pi$$
$$152$$ 0 0
$$153$$ 880.940 0.465489
$$154$$ 0 0
$$155$$ −574.708 −0.297817
$$156$$ 0 0
$$157$$ −2447.31 −1.24405 −0.622027 0.782996i $$-0.713690\pi$$
−0.622027 + 0.782996i $$0.713690\pi$$
$$158$$ 0 0
$$159$$ −306.091 −0.152670
$$160$$ 0 0
$$161$$ −4315.76 −2.11261
$$162$$ 0 0
$$163$$ 1436.82 0.690432 0.345216 0.938523i $$-0.387806\pi$$
0.345216 + 0.938523i $$0.387806\pi$$
$$164$$ 0 0
$$165$$ −589.805 −0.278280
$$166$$ 0 0
$$167$$ −1769.42 −0.819889 −0.409945 0.912110i $$-0.634452\pi$$
−0.409945 + 0.912110i $$0.634452\pi$$
$$168$$ 0 0
$$169$$ 4566.76 2.07863
$$170$$ 0 0
$$171$$ 502.940 0.224917
$$172$$ 0 0
$$173$$ 3897.86 1.71300 0.856499 0.516148i $$-0.172635\pi$$
0.856499 + 0.516148i $$0.172635\pi$$
$$174$$ 0 0
$$175$$ −3069.73 −1.32600
$$176$$ 0 0
$$177$$ 317.884 0.134992
$$178$$ 0 0
$$179$$ −4146.74 −1.73152 −0.865760 0.500460i $$-0.833164\pi$$
−0.865760 + 0.500460i $$0.833164\pi$$
$$180$$ 0 0
$$181$$ −1104.22 −0.453457 −0.226728 0.973958i $$-0.572803\pi$$
−0.226728 + 0.973958i $$0.572803\pi$$
$$182$$ 0 0
$$183$$ −2153.41 −0.869860
$$184$$ 0 0
$$185$$ −1047.18 −0.416163
$$186$$ 0 0
$$187$$ −3389.41 −1.32544
$$188$$ 0 0
$$189$$ −893.475 −0.343866
$$190$$ 0 0
$$191$$ −35.0686 −0.0132852 −0.00664260 0.999978i $$-0.502114\pi$$
−0.00664260 + 0.999978i $$0.502114\pi$$
$$192$$ 0 0
$$193$$ 615.884 0.229701 0.114851 0.993383i $$-0.463361\pi$$
0.114851 + 0.993383i $$0.463361\pi$$
$$194$$ 0 0
$$195$$ −1400.82 −0.514435
$$196$$ 0 0
$$197$$ −589.638 −0.213249 −0.106624 0.994299i $$-0.534004\pi$$
−0.106624 + 0.994299i $$0.534004\pi$$
$$198$$ 0 0
$$199$$ 4813.82 1.71479 0.857394 0.514661i $$-0.172082\pi$$
0.857394 + 0.514661i $$0.172082\pi$$
$$200$$ 0 0
$$201$$ 949.413 0.333166
$$202$$ 0 0
$$203$$ 4879.41 1.68703
$$204$$ 0 0
$$205$$ 1347.93 0.459238
$$206$$ 0 0
$$207$$ −1173.77 −0.394118
$$208$$ 0 0
$$209$$ −1935.06 −0.640434
$$210$$ 0 0
$$211$$ 3294.35 1.07484 0.537422 0.843313i $$-0.319398\pi$$
0.537422 + 0.843313i $$0.319398\pi$$
$$212$$ 0 0
$$213$$ 2400.13 0.772087
$$214$$ 0 0
$$215$$ 1134.86 0.359984
$$216$$ 0 0
$$217$$ 3349.65 1.04787
$$218$$ 0 0
$$219$$ 903.174 0.278680
$$220$$ 0 0
$$221$$ −8050.04 −2.45025
$$222$$ 0 0
$$223$$ 1572.78 0.472294 0.236147 0.971717i $$-0.424115\pi$$
0.236147 + 0.971717i $$0.424115\pi$$
$$224$$ 0 0
$$225$$ −834.881 −0.247372
$$226$$ 0 0
$$227$$ 1492.51 0.436394 0.218197 0.975905i $$-0.429982\pi$$
0.218197 + 0.975905i $$0.429982\pi$$
$$228$$ 0 0
$$229$$ −6163.31 −1.77853 −0.889265 0.457393i $$-0.848783\pi$$
−0.889265 + 0.457393i $$0.848783\pi$$
$$230$$ 0 0
$$231$$ 3437.64 0.979134
$$232$$ 0 0
$$233$$ −2540.35 −0.714266 −0.357133 0.934054i $$-0.616246\pi$$
−0.357133 + 0.934054i $$0.616246\pi$$
$$234$$ 0 0
$$235$$ 1900.95 0.527677
$$236$$ 0 0
$$237$$ 128.499 0.0352190
$$238$$ 0 0
$$239$$ −1339.25 −0.362465 −0.181232 0.983440i $$-0.558009\pi$$
−0.181232 + 0.983440i $$0.558009\pi$$
$$240$$ 0 0
$$241$$ 2542.71 0.679627 0.339814 0.940493i $$-0.389636\pi$$
0.339814 + 0.940493i $$0.389636\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −4269.91 −1.11345
$$246$$ 0 0
$$247$$ −4595.87 −1.18392
$$248$$ 0 0
$$249$$ 3710.00 0.944223
$$250$$ 0 0
$$251$$ −4701.45 −1.18228 −0.591141 0.806568i $$-0.701322\pi$$
−0.591141 + 0.806568i $$0.701322\pi$$
$$252$$ 0 0
$$253$$ 4516.05 1.12222
$$254$$ 0 0
$$255$$ 1667.22 0.409432
$$256$$ 0 0
$$257$$ −863.884 −0.209679 −0.104840 0.994489i $$-0.533433\pi$$
−0.104840 + 0.994489i $$0.533433\pi$$
$$258$$ 0 0
$$259$$ 6103.41 1.46428
$$260$$ 0 0
$$261$$ 1327.06 0.314724
$$262$$ 0 0
$$263$$ −7091.36 −1.66263 −0.831315 0.555801i $$-0.812412\pi$$
−0.831315 + 0.555801i $$0.812412\pi$$
$$264$$ 0 0
$$265$$ −579.290 −0.134285
$$266$$ 0 0
$$267$$ 3975.88 0.911311
$$268$$ 0 0
$$269$$ −395.596 −0.0896650 −0.0448325 0.998995i $$-0.514275\pi$$
−0.0448325 + 0.998995i $$0.514275\pi$$
$$270$$ 0 0
$$271$$ 5532.21 1.24007 0.620033 0.784576i $$-0.287119\pi$$
0.620033 + 0.784576i $$0.287119\pi$$
$$272$$ 0 0
$$273$$ 8164.58 1.81005
$$274$$ 0 0
$$275$$ 3212.20 0.704373
$$276$$ 0 0
$$277$$ 3830.31 0.830834 0.415417 0.909631i $$-0.363636\pi$$
0.415417 + 0.909631i $$0.363636\pi$$
$$278$$ 0 0
$$279$$ 911.009 0.195486
$$280$$ 0 0
$$281$$ −5283.17 −1.12159 −0.560797 0.827954i $$-0.689505\pi$$
−0.560797 + 0.827954i $$0.689505\pi$$
$$282$$ 0 0
$$283$$ −4639.76 −0.974577 −0.487288 0.873241i $$-0.662014\pi$$
−0.487288 + 0.873241i $$0.662014\pi$$
$$284$$ 0 0
$$285$$ 951.836 0.197831
$$286$$ 0 0
$$287$$ −7856.33 −1.61584
$$288$$ 0 0
$$289$$ 4667.94 0.950119
$$290$$ 0 0
$$291$$ 1517.29 0.305654
$$292$$ 0 0
$$293$$ 6022.12 1.20074 0.600369 0.799723i $$-0.295020\pi$$
0.600369 + 0.799723i $$0.295020\pi$$
$$294$$ 0 0
$$295$$ 601.609 0.118736
$$296$$ 0 0
$$297$$ 934.940 0.182662
$$298$$ 0 0
$$299$$ 10725.9 2.07456
$$300$$ 0 0
$$301$$ −6614.44 −1.26661
$$302$$ 0 0
$$303$$ 3845.92 0.729183
$$304$$ 0 0
$$305$$ −4075.42 −0.765107
$$306$$ 0 0
$$307$$ −2998.82 −0.557497 −0.278749 0.960364i $$-0.589920\pi$$
−0.278749 + 0.960364i $$0.589920\pi$$
$$308$$ 0 0
$$309$$ 484.685 0.0892323
$$310$$ 0 0
$$311$$ 2403.97 0.438318 0.219159 0.975689i $$-0.429669\pi$$
0.219159 + 0.975689i $$0.429669\pi$$
$$312$$ 0 0
$$313$$ −5845.75 −1.05566 −0.527830 0.849350i $$-0.676994\pi$$
−0.527830 + 0.849350i $$0.676994\pi$$
$$314$$ 0 0
$$315$$ −1690.94 −0.302456
$$316$$ 0 0
$$317$$ 8917.38 1.57997 0.789984 0.613127i $$-0.210089\pi$$
0.789984 + 0.613127i $$0.210089\pi$$
$$318$$ 0 0
$$319$$ −5105.86 −0.896154
$$320$$ 0 0
$$321$$ 1445.18 0.251283
$$322$$ 0 0
$$323$$ 5469.88 0.942267
$$324$$ 0 0
$$325$$ 7629.15 1.30212
$$326$$ 0 0
$$327$$ 859.910 0.145422
$$328$$ 0 0
$$329$$ −11079.5 −1.85664
$$330$$ 0 0
$$331$$ 9011.29 1.49639 0.748196 0.663478i $$-0.230920\pi$$
0.748196 + 0.663478i $$0.230920\pi$$
$$332$$ 0 0
$$333$$ 1659.96 0.273168
$$334$$ 0 0
$$335$$ 1796.81 0.293045
$$336$$ 0 0
$$337$$ 4516.35 0.730033 0.365017 0.931001i $$-0.381063\pi$$
0.365017 + 0.931001i $$0.381063\pi$$
$$338$$ 0 0
$$339$$ 4191.88 0.671598
$$340$$ 0 0
$$341$$ −3505.10 −0.556633
$$342$$ 0 0
$$343$$ 13536.4 2.13090
$$344$$ 0 0
$$345$$ −2221.40 −0.346656
$$346$$ 0 0
$$347$$ 3651.48 0.564904 0.282452 0.959281i $$-0.408852\pi$$
0.282452 + 0.959281i $$0.408852\pi$$
$$348$$ 0 0
$$349$$ 3250.36 0.498532 0.249266 0.968435i $$-0.419811\pi$$
0.249266 + 0.968435i $$0.419811\pi$$
$$350$$ 0 0
$$351$$ 2220.54 0.337674
$$352$$ 0 0
$$353$$ 592.345 0.0893125 0.0446563 0.999002i $$-0.485781\pi$$
0.0446563 + 0.999002i $$0.485781\pi$$
$$354$$ 0 0
$$355$$ 4542.36 0.679108
$$356$$ 0 0
$$357$$ −9717.26 −1.44059
$$358$$ 0 0
$$359$$ 3443.48 0.506239 0.253120 0.967435i $$-0.418543\pi$$
0.253120 + 0.967435i $$0.418543\pi$$
$$360$$ 0 0
$$361$$ −3736.17 −0.544711
$$362$$ 0 0
$$363$$ 395.826 0.0572327
$$364$$ 0 0
$$365$$ 1709.30 0.245120
$$366$$ 0 0
$$367$$ 3297.39 0.468998 0.234499 0.972116i $$-0.424655\pi$$
0.234499 + 0.972116i $$0.424655\pi$$
$$368$$ 0 0
$$369$$ −2136.70 −0.301442
$$370$$ 0 0
$$371$$ 3376.35 0.472483
$$372$$ 0 0
$$373$$ 50.1819 0.00696601 0.00348300 0.999994i $$-0.498891\pi$$
0.00348300 + 0.999994i $$0.498891\pi$$
$$374$$ 0 0
$$375$$ −3709.16 −0.510774
$$376$$ 0 0
$$377$$ −12126.7 −1.65665
$$378$$ 0 0
$$379$$ 6769.28 0.917453 0.458726 0.888578i $$-0.348306\pi$$
0.458726 + 0.888578i $$0.348306\pi$$
$$380$$ 0 0
$$381$$ −1331.49 −0.179040
$$382$$ 0 0
$$383$$ 4208.46 0.561468 0.280734 0.959786i $$-0.409422\pi$$
0.280734 + 0.959786i $$0.409422\pi$$
$$384$$ 0 0
$$385$$ 6505.88 0.861221
$$386$$ 0 0
$$387$$ −1798.94 −0.236293
$$388$$ 0 0
$$389$$ 2490.47 0.324607 0.162303 0.986741i $$-0.448108\pi$$
0.162303 + 0.986741i $$0.448108\pi$$
$$390$$ 0 0
$$391$$ −12765.6 −1.65112
$$392$$ 0 0
$$393$$ 4096.70 0.525830
$$394$$ 0 0
$$395$$ 243.190 0.0309777
$$396$$ 0 0
$$397$$ −7905.51 −0.999411 −0.499706 0.866195i $$-0.666558\pi$$
−0.499706 + 0.866195i $$0.666558\pi$$
$$398$$ 0 0
$$399$$ −5547.71 −0.696072
$$400$$ 0 0
$$401$$ −10389.4 −1.29382 −0.646911 0.762566i $$-0.723939\pi$$
−0.646911 + 0.762566i $$0.723939\pi$$
$$402$$ 0 0
$$403$$ −8324.81 −1.02900
$$404$$ 0 0
$$405$$ −459.888 −0.0564248
$$406$$ 0 0
$$407$$ −6386.66 −0.777826
$$408$$ 0 0
$$409$$ 13448.1 1.62583 0.812917 0.582379i $$-0.197878\pi$$
0.812917 + 0.582379i $$0.197878\pi$$
$$410$$ 0 0
$$411$$ −3669.17 −0.440357
$$412$$ 0 0
$$413$$ −3506.44 −0.417773
$$414$$ 0 0
$$415$$ 7021.33 0.830515
$$416$$ 0 0
$$417$$ 6530.11 0.766860
$$418$$ 0 0
$$419$$ 5191.34 0.605283 0.302641 0.953105i $$-0.402132\pi$$
0.302641 + 0.953105i $$0.402132\pi$$
$$420$$ 0 0
$$421$$ −5061.52 −0.585946 −0.292973 0.956121i $$-0.594645\pi$$
−0.292973 + 0.956121i $$0.594645\pi$$
$$422$$ 0 0
$$423$$ −3013.32 −0.346365
$$424$$ 0 0
$$425$$ −9080.00 −1.03634
$$426$$ 0 0
$$427$$ 23753.3 2.69204
$$428$$ 0 0
$$429$$ −8543.49 −0.961501
$$430$$ 0 0
$$431$$ −3005.64 −0.335909 −0.167954 0.985795i $$-0.553716\pi$$
−0.167954 + 0.985795i $$0.553716\pi$$
$$432$$ 0 0
$$433$$ 5895.88 0.654360 0.327180 0.944962i $$-0.393902\pi$$
0.327180 + 0.944962i $$0.393902\pi$$
$$434$$ 0 0
$$435$$ 2511.52 0.276824
$$436$$ 0 0
$$437$$ −7288.07 −0.797794
$$438$$ 0 0
$$439$$ −11556.8 −1.25644 −0.628218 0.778038i $$-0.716215\pi$$
−0.628218 + 0.778038i $$0.716215\pi$$
$$440$$ 0 0
$$441$$ 6768.52 0.730863
$$442$$ 0 0
$$443$$ −14007.8 −1.50233 −0.751163 0.660117i $$-0.770507\pi$$
−0.751163 + 0.660117i $$0.770507\pi$$
$$444$$ 0 0
$$445$$ 7524.53 0.801566
$$446$$ 0 0
$$447$$ −8627.65 −0.912917
$$448$$ 0 0
$$449$$ 3783.06 0.397625 0.198813 0.980038i $$-0.436291\pi$$
0.198813 + 0.980038i $$0.436291\pi$$
$$450$$ 0 0
$$451$$ 8220.94 0.858335
$$452$$ 0 0
$$453$$ 6990.08 0.724994
$$454$$ 0 0
$$455$$ 15451.8 1.59207
$$456$$ 0 0
$$457$$ 1545.53 0.158198 0.0790992 0.996867i $$-0.474796\pi$$
0.0790992 + 0.996867i $$0.474796\pi$$
$$458$$ 0 0
$$459$$ −2642.82 −0.268750
$$460$$ 0 0
$$461$$ −12730.2 −1.28613 −0.643065 0.765811i $$-0.722338\pi$$
−0.643065 + 0.765811i $$0.722338\pi$$
$$462$$ 0 0
$$463$$ −19656.4 −1.97303 −0.986513 0.163682i $$-0.947663\pi$$
−0.986513 + 0.163682i $$0.947663\pi$$
$$464$$ 0 0
$$465$$ 1724.12 0.171945
$$466$$ 0 0
$$467$$ 6016.26 0.596144 0.298072 0.954543i $$-0.403656\pi$$
0.298072 + 0.954543i $$0.403656\pi$$
$$468$$ 0 0
$$469$$ −10472.6 −1.03108
$$470$$ 0 0
$$471$$ 7341.92 0.718255
$$472$$ 0 0
$$473$$ 6921.41 0.672826
$$474$$ 0 0
$$475$$ −5183.89 −0.500743
$$476$$ 0 0
$$477$$ 918.272 0.0881442
$$478$$ 0 0
$$479$$ 13890.6 1.32501 0.662505 0.749057i $$-0.269493\pi$$
0.662505 + 0.749057i $$0.269493\pi$$
$$480$$ 0 0
$$481$$ −15168.7 −1.43791
$$482$$ 0 0
$$483$$ 12947.3 1.21971
$$484$$ 0 0
$$485$$ 2871.54 0.268846
$$486$$ 0 0
$$487$$ 9314.23 0.866669 0.433335 0.901233i $$-0.357337\pi$$
0.433335 + 0.901233i $$0.357337\pi$$
$$488$$ 0 0
$$489$$ −4310.46 −0.398621
$$490$$ 0 0
$$491$$ 7253.34 0.666677 0.333339 0.942807i $$-0.391825\pi$$
0.333339 + 0.942807i $$0.391825\pi$$
$$492$$ 0 0
$$493$$ 14432.9 1.31851
$$494$$ 0 0
$$495$$ 1769.42 0.160665
$$496$$ 0 0
$$497$$ −26474.8 −2.38945
$$498$$ 0 0
$$499$$ −16208.2 −1.45407 −0.727034 0.686602i $$-0.759102\pi$$
−0.727034 + 0.686602i $$0.759102\pi$$
$$500$$ 0 0
$$501$$ 5308.25 0.473363
$$502$$ 0 0
$$503$$ −2182.04 −0.193425 −0.0967123 0.995312i $$-0.530833\pi$$
−0.0967123 + 0.995312i $$0.530833\pi$$
$$504$$ 0 0
$$505$$ 7278.58 0.641371
$$506$$ 0 0
$$507$$ −13700.3 −1.20010
$$508$$ 0 0
$$509$$ −1795.97 −0.156395 −0.0781974 0.996938i $$-0.524916\pi$$
−0.0781974 + 0.996938i $$0.524916\pi$$
$$510$$ 0 0
$$511$$ −9962.51 −0.862457
$$512$$ 0 0
$$513$$ −1508.82 −0.129856
$$514$$ 0 0
$$515$$ 917.288 0.0784865
$$516$$ 0 0
$$517$$ 11593.7 0.986249
$$518$$ 0 0
$$519$$ −11693.6 −0.989000
$$520$$ 0 0
$$521$$ 20946.1 1.76135 0.880676 0.473718i $$-0.157088\pi$$
0.880676 + 0.473718i $$0.157088\pi$$
$$522$$ 0 0
$$523$$ 5363.64 0.448443 0.224221 0.974538i $$-0.428016\pi$$
0.224221 + 0.974538i $$0.428016\pi$$
$$524$$ 0 0
$$525$$ 9209.19 0.765566
$$526$$ 0 0
$$527$$ 9907.96 0.818970
$$528$$ 0 0
$$529$$ 4841.96 0.397958
$$530$$ 0 0
$$531$$ −953.652 −0.0779378
$$532$$ 0 0
$$533$$ 19525.2 1.58674
$$534$$ 0 0
$$535$$ 2735.06 0.221022
$$536$$ 0 0
$$537$$ 12440.2 0.999693
$$538$$ 0 0
$$539$$ −26041.8 −2.08108
$$540$$ 0 0
$$541$$ 3431.20 0.272678 0.136339 0.990662i $$-0.456466\pi$$
0.136339 + 0.990662i $$0.456466\pi$$
$$542$$ 0 0
$$543$$ 3312.65 0.261803
$$544$$ 0 0
$$545$$ 1627.42 0.127910
$$546$$ 0 0
$$547$$ −19449.0 −1.52026 −0.760128 0.649773i $$-0.774864\pi$$
−0.760128 + 0.649773i $$0.774864\pi$$
$$548$$ 0 0
$$549$$ 6460.22 0.502214
$$550$$ 0 0
$$551$$ 8239.91 0.637082
$$552$$ 0 0
$$553$$ −1417.41 −0.108996
$$554$$ 0 0
$$555$$ 3141.54 0.240272
$$556$$ 0 0
$$557$$ 9898.44 0.752981 0.376491 0.926420i $$-0.377131\pi$$
0.376491 + 0.926420i $$0.377131\pi$$
$$558$$ 0 0
$$559$$ 16438.7 1.24380
$$560$$ 0 0
$$561$$ 10168.2 0.765246
$$562$$ 0 0
$$563$$ −13809.8 −1.03378 −0.516888 0.856053i $$-0.672909\pi$$
−0.516888 + 0.856053i $$0.672909\pi$$
$$564$$ 0 0
$$565$$ 7933.32 0.590721
$$566$$ 0 0
$$567$$ 2680.42 0.198531
$$568$$ 0 0
$$569$$ 123.053 0.00906615 0.00453308 0.999990i $$-0.498557\pi$$
0.00453308 + 0.999990i $$0.498557\pi$$
$$570$$ 0 0
$$571$$ −2540.72 −0.186210 −0.0931049 0.995656i $$-0.529679\pi$$
−0.0931049 + 0.995656i $$0.529679\pi$$
$$572$$ 0 0
$$573$$ 105.206 0.00767022
$$574$$ 0 0
$$575$$ 12098.2 0.877443
$$576$$ 0 0
$$577$$ −15618.2 −1.12685 −0.563427 0.826166i $$-0.690517\pi$$
−0.563427 + 0.826166i $$0.690517\pi$$
$$578$$ 0 0
$$579$$ −1847.65 −0.132618
$$580$$ 0 0
$$581$$ −40923.3 −2.92218
$$582$$ 0 0
$$583$$ −3533.04 −0.250984
$$584$$ 0 0
$$585$$ 4202.46 0.297009
$$586$$ 0 0
$$587$$ 1809.56 0.127238 0.0636188 0.997974i $$-0.479736\pi$$
0.0636188 + 0.997974i $$0.479736\pi$$
$$588$$ 0 0
$$589$$ 5656.58 0.395714
$$590$$ 0 0
$$591$$ 1768.91 0.123119
$$592$$ 0 0
$$593$$ 898.229 0.0622021 0.0311010 0.999516i $$-0.490099\pi$$
0.0311010 + 0.999516i $$0.490099\pi$$
$$594$$ 0 0
$$595$$ −18390.3 −1.26711
$$596$$ 0 0
$$597$$ −14441.5 −0.990033
$$598$$ 0 0
$$599$$ −22990.9 −1.56825 −0.784125 0.620603i $$-0.786888\pi$$
−0.784125 + 0.620603i $$0.786888\pi$$
$$600$$ 0 0
$$601$$ 26893.7 1.82532 0.912661 0.408717i $$-0.134024\pi$$
0.912661 + 0.408717i $$0.134024\pi$$
$$602$$ 0 0
$$603$$ −2848.24 −0.192354
$$604$$ 0 0
$$605$$ 749.118 0.0503405
$$606$$ 0 0
$$607$$ 6306.93 0.421730 0.210865 0.977515i $$-0.432372\pi$$
0.210865 + 0.977515i $$0.432372\pi$$
$$608$$ 0 0
$$609$$ −14638.2 −0.974008
$$610$$ 0 0
$$611$$ 27535.7 1.82320
$$612$$ 0 0
$$613$$ −2612.97 −0.172164 −0.0860822 0.996288i $$-0.527435\pi$$
−0.0860822 + 0.996288i $$0.527435\pi$$
$$614$$ 0 0
$$615$$ −4043.80 −0.265141
$$616$$ 0 0
$$617$$ 2803.88 0.182949 0.0914747 0.995807i $$-0.470842\pi$$
0.0914747 + 0.995807i $$0.470842\pi$$
$$618$$ 0 0
$$619$$ 10547.1 0.684849 0.342425 0.939545i $$-0.388752\pi$$
0.342425 + 0.939545i $$0.388752\pi$$
$$620$$ 0 0
$$621$$ 3521.30 0.227544
$$622$$ 0 0
$$623$$ −43856.2 −2.82032
$$624$$ 0 0
$$625$$ 4575.82 0.292852
$$626$$ 0 0
$$627$$ 5805.17 0.369755
$$628$$ 0 0
$$629$$ 18053.3 1.14441
$$630$$ 0 0
$$631$$ 14161.4 0.893431 0.446716 0.894676i $$-0.352594\pi$$
0.446716 + 0.894676i $$0.352594\pi$$
$$632$$ 0 0
$$633$$ −9883.04 −0.620562
$$634$$ 0 0
$$635$$ −2519.90 −0.157479
$$636$$ 0 0
$$637$$ −61850.8 −3.84713
$$638$$ 0 0
$$639$$ −7200.40 −0.445764
$$640$$ 0 0
$$641$$ 5622.61 0.346458 0.173229 0.984882i $$-0.444580\pi$$
0.173229 + 0.984882i $$0.444580\pi$$
$$642$$ 0 0
$$643$$ −29438.7 −1.80552 −0.902759 0.430146i $$-0.858462\pi$$
−0.902759 + 0.430146i $$0.858462\pi$$
$$644$$ 0 0
$$645$$ −3404.57 −0.207837
$$646$$ 0 0
$$647$$ 4607.39 0.279962 0.139981 0.990154i $$-0.455296\pi$$
0.139981 + 0.990154i $$0.455296\pi$$
$$648$$ 0 0
$$649$$ 3669.17 0.221922
$$650$$ 0 0
$$651$$ −10048.9 −0.604991
$$652$$ 0 0
$$653$$ 16634.1 0.996850 0.498425 0.866933i $$-0.333912\pi$$
0.498425 + 0.866933i $$0.333912\pi$$
$$654$$ 0 0
$$655$$ 7753.19 0.462507
$$656$$ 0 0
$$657$$ −2709.52 −0.160896
$$658$$ 0 0
$$659$$ −20619.9 −1.21887 −0.609437 0.792835i $$-0.708604\pi$$
−0.609437 + 0.792835i $$0.708604\pi$$
$$660$$ 0 0
$$661$$ 881.112 0.0518476 0.0259238 0.999664i $$-0.491747\pi$$
0.0259238 + 0.999664i $$0.491747\pi$$
$$662$$ 0 0
$$663$$ 24150.1 1.41465
$$664$$ 0 0
$$665$$ −10499.3 −0.612248
$$666$$ 0 0
$$667$$ −19230.4 −1.11635
$$668$$ 0 0
$$669$$ −4718.35 −0.272679
$$670$$ 0 0
$$671$$ −24855.6 −1.43002
$$672$$ 0 0
$$673$$ 8943.86 0.512274 0.256137 0.966641i $$-0.417550\pi$$
0.256137 + 0.966641i $$0.417550\pi$$
$$674$$ 0 0
$$675$$ 2504.64 0.142820
$$676$$ 0 0
$$677$$ 21748.7 1.23467 0.617333 0.786702i $$-0.288213\pi$$
0.617333 + 0.786702i $$0.288213\pi$$
$$678$$ 0 0
$$679$$ −16736.6 −0.945937
$$680$$ 0 0
$$681$$ −4477.53 −0.251952
$$682$$ 0 0
$$683$$ −7922.37 −0.443838 −0.221919 0.975065i $$-0.571232\pi$$
−0.221919 + 0.975065i $$0.571232\pi$$
$$684$$ 0 0
$$685$$ −6944.06 −0.387327
$$686$$ 0 0
$$687$$ 18489.9 1.02683
$$688$$ 0 0
$$689$$ −8391.18 −0.463975
$$690$$ 0 0
$$691$$ 177.517 0.00977288 0.00488644 0.999988i $$-0.498445\pi$$
0.00488644 + 0.999988i $$0.498445\pi$$
$$692$$ 0 0
$$693$$ −10312.9 −0.565303
$$694$$ 0 0
$$695$$ 12358.5 0.674511
$$696$$ 0 0
$$697$$ −23238.3 −1.26286
$$698$$ 0 0
$$699$$ 7621.05 0.412382
$$700$$ 0 0
$$701$$ −767.980 −0.0413783 −0.0206891 0.999786i $$-0.506586\pi$$
−0.0206891 + 0.999786i $$0.506586\pi$$
$$702$$ 0 0
$$703$$ 10306.9 0.552961
$$704$$ 0 0
$$705$$ −5702.84 −0.304654
$$706$$ 0 0
$$707$$ −42422.7 −2.25667
$$708$$ 0 0
$$709$$ −31634.2 −1.67566 −0.837832 0.545928i $$-0.816177\pi$$
−0.837832 + 0.545928i $$0.816177\pi$$
$$710$$ 0 0
$$711$$ −385.496 −0.0203337
$$712$$ 0 0
$$713$$ −13201.4 −0.693401
$$714$$ 0 0
$$715$$ −16168.9 −0.845712
$$716$$ 0 0
$$717$$ 4017.76 0.209269
$$718$$ 0 0
$$719$$ 17351.7 0.900011 0.450005 0.893026i $$-0.351422\pi$$
0.450005 + 0.893026i $$0.351422\pi$$
$$720$$ 0 0
$$721$$ −5346.35 −0.276156
$$722$$ 0 0
$$723$$ −7628.12 −0.392383
$$724$$ 0 0
$$725$$ −13678.2 −0.700686
$$726$$ 0 0
$$727$$ 16364.6 0.834842 0.417421 0.908713i $$-0.362934\pi$$
0.417421 + 0.908713i $$0.362934\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −19564.9 −0.989925
$$732$$ 0 0
$$733$$ 23552.7 1.18682 0.593410 0.804900i $$-0.297781\pi$$
0.593410 + 0.804900i $$0.297781\pi$$
$$734$$ 0 0
$$735$$ 12809.7 0.642849
$$736$$ 0 0
$$737$$ 10958.6 0.547713
$$738$$ 0 0
$$739$$ 3519.05 0.175169 0.0875847 0.996157i $$-0.472085\pi$$
0.0875847 + 0.996157i $$0.472085\pi$$
$$740$$ 0 0
$$741$$ 13787.6 0.683537
$$742$$ 0 0
$$743$$ −20752.4 −1.02467 −0.512336 0.858785i $$-0.671220\pi$$
−0.512336 + 0.858785i $$0.671220\pi$$
$$744$$ 0 0
$$745$$ −16328.2 −0.802979
$$746$$ 0 0
$$747$$ −11130.0 −0.545148
$$748$$ 0 0
$$749$$ −15941.1 −0.777671
$$750$$ 0 0
$$751$$ −3679.08 −0.178764 −0.0893818 0.995997i $$-0.528489\pi$$
−0.0893818 + 0.995997i $$0.528489\pi$$
$$752$$ 0 0
$$753$$ 14104.3 0.682591
$$754$$ 0 0
$$755$$ 13229.0 0.637687
$$756$$ 0 0
$$757$$ 527.275 0.0253159 0.0126579 0.999920i $$-0.495971\pi$$
0.0126579 + 0.999920i $$0.495971\pi$$
$$758$$ 0 0
$$759$$ −13548.2 −0.647914
$$760$$ 0 0
$$761$$ −3231.54 −0.153933 −0.0769666 0.997034i $$-0.524523\pi$$
−0.0769666 + 0.997034i $$0.524523\pi$$
$$762$$ 0 0
$$763$$ −9485.29 −0.450053
$$764$$ 0 0
$$765$$ −5001.65 −0.236386
$$766$$ 0 0
$$767$$ 8714.48 0.410250
$$768$$ 0 0
$$769$$ 13681.7 0.641581 0.320791 0.947150i $$-0.396051\pi$$
0.320791 + 0.947150i $$0.396051\pi$$
$$770$$ 0 0
$$771$$ 2591.65 0.121058
$$772$$ 0 0
$$773$$ 9216.63 0.428847 0.214424 0.976741i $$-0.431213\pi$$
0.214424 + 0.976741i $$0.431213\pi$$
$$774$$ 0 0
$$775$$ −9389.92 −0.435221
$$776$$ 0 0
$$777$$ −18310.2 −0.845400
$$778$$ 0 0
$$779$$ −13267.1 −0.610196
$$780$$ 0 0
$$781$$ 27703.5 1.26928
$$782$$ 0 0
$$783$$ −3981.18 −0.181706
$$784$$ 0 0
$$785$$ 13894.9 0.631759
$$786$$ 0 0
$$787$$ 33169.0 1.50235 0.751174 0.660104i $$-0.229488\pi$$
0.751174 + 0.660104i $$0.229488\pi$$
$$788$$ 0 0
$$789$$ 21274.1 0.959920
$$790$$ 0 0
$$791$$ −46238.8 −2.07846
$$792$$ 0 0
$$793$$ −59033.6 −2.64356
$$794$$ 0 0
$$795$$ 1737.87 0.0775294
$$796$$ 0 0
$$797$$ 23781.9 1.05696 0.528481 0.848945i $$-0.322762\pi$$
0.528481 + 0.848945i $$0.322762\pi$$
$$798$$ 0 0
$$799$$ −32772.3 −1.45106
$$800$$ 0 0
$$801$$ −11927.6 −0.526145
$$802$$ 0 0
$$803$$ 10424.9 0.458139
$$804$$ 0 0
$$805$$ 24503.3 1.07283
$$806$$ 0 0
$$807$$ 1186.79 0.0517681
$$808$$ 0 0
$$809$$ 29842.4 1.29691 0.648456 0.761252i $$-0.275415\pi$$
0.648456 + 0.761252i $$0.275415\pi$$
$$810$$ 0 0
$$811$$ 19074.5 0.825888 0.412944 0.910756i $$-0.364500\pi$$
0.412944 + 0.910756i $$0.364500\pi$$
$$812$$ 0 0
$$813$$ −16596.6 −0.715952
$$814$$ 0 0
$$815$$ −8157.74 −0.350617
$$816$$ 0 0
$$817$$ −11169.9 −0.478316
$$818$$ 0 0
$$819$$ −24493.7 −1.04503
$$820$$ 0 0
$$821$$ 7493.97 0.318564 0.159282 0.987233i $$-0.449082\pi$$
0.159282 + 0.987233i $$0.449082\pi$$
$$822$$ 0 0
$$823$$ 1780.90 0.0754294 0.0377147 0.999289i $$-0.487992\pi$$
0.0377147 + 0.999289i $$0.487992\pi$$
$$824$$ 0 0
$$825$$ −9636.59 −0.406670
$$826$$ 0 0
$$827$$ −10860.1 −0.456640 −0.228320 0.973586i $$-0.573323\pi$$
−0.228320 + 0.973586i $$0.573323\pi$$
$$828$$ 0 0
$$829$$ 34105.1 1.42885 0.714426 0.699711i $$-0.246688\pi$$
0.714426 + 0.699711i $$0.246688\pi$$
$$830$$ 0 0
$$831$$ −11490.9 −0.479682
$$832$$ 0 0
$$833$$ 73613.1 3.06188
$$834$$ 0 0
$$835$$ 10046.1 0.416358
$$836$$ 0 0
$$837$$ −2733.03 −0.112864
$$838$$ 0 0
$$839$$ 32688.9 1.34511 0.672555 0.740047i $$-0.265197\pi$$
0.672555 + 0.740047i $$0.265197\pi$$
$$840$$ 0 0
$$841$$ −2647.12 −0.108537
$$842$$ 0 0
$$843$$ 15849.5 0.647552
$$844$$ 0 0
$$845$$ −25928.4 −1.05558
$$846$$ 0 0
$$847$$ −4366.18 −0.177124
$$848$$ 0 0
$$849$$ 13919.3 0.562672
$$850$$ 0 0
$$851$$ −24054.3 −0.968943
$$852$$ 0 0
$$853$$ −17391.2 −0.698080 −0.349040 0.937108i $$-0.613492\pi$$
−0.349040 + 0.937108i $$0.613492\pi$$
$$854$$ 0 0
$$855$$ −2855.51 −0.114218
$$856$$ 0 0
$$857$$ 4182.34 0.166705 0.0833525 0.996520i $$-0.473437\pi$$
0.0833525 + 0.996520i $$0.473437\pi$$
$$858$$ 0 0
$$859$$ −20585.0 −0.817639 −0.408820 0.912615i $$-0.634059\pi$$
−0.408820 + 0.912615i $$0.634059\pi$$
$$860$$ 0 0
$$861$$ 23569.0 0.932903
$$862$$ 0 0
$$863$$ −14159.7 −0.558518 −0.279259 0.960216i $$-0.590089\pi$$
−0.279259 + 0.960216i $$0.590089\pi$$
$$864$$ 0 0
$$865$$ −22130.6 −0.869900
$$866$$ 0 0
$$867$$ −14003.8 −0.548552
$$868$$ 0 0
$$869$$ 1483.19 0.0578986
$$870$$ 0 0
$$871$$ 26027.2 1.01251
$$872$$ 0 0
$$873$$ −4551.88 −0.176469
$$874$$ 0 0
$$875$$ 40914.1 1.58074
$$876$$ 0 0
$$877$$ 5156.38 0.198539 0.0992695 0.995061i $$-0.468349\pi$$
0.0992695 + 0.995061i $$0.468349\pi$$
$$878$$ 0 0
$$879$$ −18066.4 −0.693246
$$880$$ 0 0
$$881$$ 841.065 0.0321637 0.0160818 0.999871i $$-0.494881\pi$$
0.0160818 + 0.999871i $$0.494881\pi$$
$$882$$ 0 0
$$883$$ −7845.99 −0.299024 −0.149512 0.988760i $$-0.547770\pi$$
−0.149512 + 0.988760i $$0.547770\pi$$
$$884$$ 0 0
$$885$$ −1804.83 −0.0685521
$$886$$ 0 0
$$887$$ 38781.9 1.46806 0.734029 0.679118i $$-0.237637\pi$$
0.734029 + 0.679118i $$0.237637\pi$$
$$888$$ 0 0
$$889$$ 14687.1 0.554092
$$890$$ 0 0
$$891$$ −2804.82 −0.105460
$$892$$ 0 0
$$893$$ −18710.1 −0.701131
$$894$$ 0 0
$$895$$ 23543.7 0.879305
$$896$$ 0 0
$$897$$ −32177.6 −1.19775
$$898$$ 0 0
$$899$$ 14925.5 0.553719
$$900$$ 0 0
$$901$$ 9986.95 0.369271
$$902$$ 0 0
$$903$$ 19843.3 0.731278
$$904$$ 0 0
$$905$$ 6269.33 0.230276
$$906$$ 0 0
$$907$$ −36431.2 −1.33371 −0.666856 0.745187i $$-0.732360\pi$$
−0.666856 + 0.745187i $$0.732360\pi$$
$$908$$ 0 0
$$909$$ −11537.8 −0.420994
$$910$$ 0 0
$$911$$ 51077.9 1.85762 0.928808 0.370562i $$-0.120835\pi$$
0.928808 + 0.370562i $$0.120835\pi$$
$$912$$ 0 0
$$913$$ 42822.6 1.55227
$$914$$ 0 0
$$915$$ 12226.3 0.441735
$$916$$ 0 0
$$917$$ −45188.9 −1.62734
$$918$$ 0 0
$$919$$ −37080.1 −1.33097 −0.665484 0.746412i $$-0.731775\pi$$
−0.665484 + 0.746412i $$0.731775\pi$$
$$920$$ 0 0
$$921$$ 8996.46 0.321871
$$922$$ 0 0
$$923$$ 65797.3 2.34642
$$924$$ 0 0
$$925$$ −17109.4 −0.608167
$$926$$ 0 0
$$927$$ −1454.06 −0.0515183
$$928$$ 0 0
$$929$$ 38259.5 1.35119 0.675595 0.737273i $$-0.263887\pi$$
0.675595 + 0.737273i $$0.263887\pi$$
$$930$$ 0 0
$$931$$ 42026.7 1.47945
$$932$$ 0 0
$$933$$ −7211.92 −0.253063
$$934$$ 0 0
$$935$$ 19243.8 0.673091
$$936$$ 0 0
$$937$$ −4413.55 −0.153879 −0.0769394 0.997036i $$-0.524515\pi$$
−0.0769394 + 0.997036i $$0.524515\pi$$
$$938$$ 0 0
$$939$$ 17537.3 0.609486
$$940$$ 0 0
$$941$$ −31511.2 −1.09164 −0.545821 0.837902i $$-0.683782\pi$$
−0.545821 + 0.837902i $$0.683782\pi$$
$$942$$ 0 0
$$943$$ 30962.8 1.06923
$$944$$ 0 0
$$945$$ 5072.82 0.174623
$$946$$ 0 0
$$947$$ −23561.3 −0.808490 −0.404245 0.914651i $$-0.632466\pi$$
−0.404245 + 0.914651i $$0.632466\pi$$
$$948$$ 0 0
$$949$$ 24759.6 0.846925
$$950$$ 0 0
$$951$$ −26752.1 −0.912195
$$952$$ 0 0
$$953$$ −35039.7 −1.19103 −0.595513 0.803346i $$-0.703051\pi$$
−0.595513 + 0.803346i $$0.703051\pi$$
$$954$$ 0 0
$$955$$ 199.107 0.00674653
$$956$$ 0 0
$$957$$ 15317.6 0.517395
$$958$$ 0 0
$$959$$ 40473.0 1.36282
$$960$$ 0 0
$$961$$ −19544.9 −0.656066
$$962$$ 0 0
$$963$$ −4335.53 −0.145078
$$964$$ 0 0
$$965$$ −3496.76 −0.116647
$$966$$ 0 0
$$967$$ 7975.38 0.265223 0.132612 0.991168i $$-0.457664\pi$$
0.132612 + 0.991168i $$0.457664\pi$$
$$968$$ 0 0
$$969$$ −16409.6 −0.544018
$$970$$ 0 0
$$971$$ −21386.8 −0.706834 −0.353417 0.935466i $$-0.614980\pi$$
−0.353417 + 0.935466i $$0.614980\pi$$
$$972$$ 0 0
$$973$$ −72030.7 −2.37328
$$974$$ 0 0
$$975$$ −22887.4 −0.751779
$$976$$ 0 0
$$977$$ −40142.3 −1.31450 −0.657250 0.753673i $$-0.728280\pi$$
−0.657250 + 0.753673i $$0.728280\pi$$
$$978$$ 0 0
$$979$$ 45891.5 1.49816
$$980$$ 0 0
$$981$$ −2579.73 −0.0839597
$$982$$ 0 0
$$983$$ −9205.44 −0.298686 −0.149343 0.988785i $$-0.547716\pi$$
−0.149343 + 0.988785i $$0.547716\pi$$
$$984$$ 0 0
$$985$$ 3347.75 0.108292
$$986$$ 0 0
$$987$$ 33238.6 1.07193
$$988$$ 0 0
$$989$$ 26068.3 0.838144
$$990$$ 0 0
$$991$$ −22082.2 −0.707834 −0.353917 0.935277i $$-0.615150\pi$$
−0.353917 + 0.935277i $$0.615150\pi$$
$$992$$ 0 0
$$993$$ −27033.9 −0.863942
$$994$$ 0 0
$$995$$ −27331.1 −0.870808
$$996$$ 0 0
$$997$$ 7207.66 0.228956 0.114478 0.993426i $$-0.463480\pi$$
0.114478 + 0.993426i $$0.463480\pi$$
$$998$$ 0 0
$$999$$ −4979.87 −0.157714
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 768.4.a.u.1.2 4
3.2 odd 2 2304.4.a.cb.1.3 4
4.3 odd 2 768.4.a.v.1.2 4
8.3 odd 2 inner 768.4.a.u.1.3 4
8.5 even 2 768.4.a.v.1.3 4
12.11 even 2 2304.4.a.by.1.3 4
16.3 odd 4 384.4.d.f.193.7 yes 8
16.5 even 4 384.4.d.f.193.6 yes 8
16.11 odd 4 384.4.d.f.193.2 8
16.13 even 4 384.4.d.f.193.3 yes 8
24.5 odd 2 2304.4.a.by.1.2 4
24.11 even 2 2304.4.a.cb.1.2 4
48.5 odd 4 1152.4.d.p.577.5 8
48.11 even 4 1152.4.d.p.577.6 8
48.29 odd 4 1152.4.d.p.577.3 8
48.35 even 4 1152.4.d.p.577.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.f.193.2 8 16.11 odd 4
384.4.d.f.193.3 yes 8 16.13 even 4
384.4.d.f.193.6 yes 8 16.5 even 4
384.4.d.f.193.7 yes 8 16.3 odd 4
768.4.a.u.1.2 4 1.1 even 1 trivial
768.4.a.u.1.3 4 8.3 odd 2 inner
768.4.a.v.1.2 4 4.3 odd 2
768.4.a.v.1.3 4 8.5 even 2
1152.4.d.p.577.3 8 48.29 odd 4
1152.4.d.p.577.4 8 48.35 even 4
1152.4.d.p.577.5 8 48.5 odd 4
1152.4.d.p.577.6 8 48.11 even 4
2304.4.a.by.1.2 4 24.5 odd 2
2304.4.a.by.1.3 4 12.11 even 2
2304.4.a.cb.1.2 4 24.11 even 2
2304.4.a.cb.1.3 4 3.2 odd 2