# Properties

 Label 3344.2.a.ba.1.5 Level $3344$ Weight $2$ Character 3344.1 Self dual yes Analytic conductor $26.702$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3344.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: $$26.7019744359$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$0.456669$$ of defining polynomial Character $$\chi$$ $$=$$ 3344.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +O(q^{10})$$ $$q+0.835165 q^{3} +0.221953 q^{5} -4.69915 q^{7} -2.30250 q^{9} +1.00000 q^{11} +5.89016 q^{13} +0.185367 q^{15} +7.06513 q^{17} -1.00000 q^{19} -3.92457 q^{21} -1.06348 q^{23} -4.95074 q^{25} -4.42846 q^{27} -7.62662 q^{29} -0.901295 q^{31} +0.835165 q^{33} -1.04299 q^{35} -2.71758 q^{37} +4.91925 q^{39} +0.788714 q^{41} -0.714571 q^{43} -0.511046 q^{45} -3.96368 q^{47} +15.0820 q^{49} +5.90055 q^{51} -9.69714 q^{53} +0.221953 q^{55} -0.835165 q^{57} +7.33476 q^{59} +8.15179 q^{61} +10.8198 q^{63} +1.30734 q^{65} -7.86697 q^{67} -0.888178 q^{69} +3.13400 q^{71} -6.49076 q^{73} -4.13468 q^{75} -4.69915 q^{77} -12.0148 q^{79} +3.20900 q^{81} -16.3902 q^{83} +1.56812 q^{85} -6.36948 q^{87} -12.9487 q^{89} -27.6788 q^{91} -0.752730 q^{93} -0.221953 q^{95} +8.14414 q^{97} -2.30250 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * 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
$$3$$ 0.835165 0.482183 0.241091 0.970502i $$-0.422495\pi$$
0.241091 + 0.970502i $$0.422495\pi$$
$$4$$ 0 0
$$5$$ 0.221953 0.0992603 0.0496301 0.998768i $$-0.484196\pi$$
0.0496301 + 0.998768i $$0.484196\pi$$
$$6$$ 0 0
$$7$$ −4.69915 −1.77611 −0.888057 0.459734i $$-0.847945\pi$$
−0.888057 + 0.459734i $$0.847945\pi$$
$$8$$ 0 0
$$9$$ −2.30250 −0.767500
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.89016 1.63364 0.816818 0.576895i $$-0.195736\pi$$
0.816818 + 0.576895i $$0.195736\pi$$
$$14$$ 0 0
$$15$$ 0.185367 0.0478616
$$16$$ 0 0
$$17$$ 7.06513 1.71355 0.856773 0.515694i $$-0.172466\pi$$
0.856773 + 0.515694i $$0.172466\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −3.92457 −0.856411
$$22$$ 0 0
$$23$$ −1.06348 −0.221750 −0.110875 0.993834i $$-0.535365\pi$$
−0.110875 + 0.993834i $$0.535365\pi$$
$$24$$ 0 0
$$25$$ −4.95074 −0.990147
$$26$$ 0 0
$$27$$ −4.42846 −0.852258
$$28$$ 0 0
$$29$$ −7.62662 −1.41623 −0.708114 0.706098i $$-0.750454\pi$$
−0.708114 + 0.706098i $$0.750454\pi$$
$$30$$ 0 0
$$31$$ −0.901295 −0.161877 −0.0809387 0.996719i $$-0.525792\pi$$
−0.0809387 + 0.996719i $$0.525792\pi$$
$$32$$ 0 0
$$33$$ 0.835165 0.145384
$$34$$ 0 0
$$35$$ −1.04299 −0.176297
$$36$$ 0 0
$$37$$ −2.71758 −0.446768 −0.223384 0.974731i $$-0.571710\pi$$
−0.223384 + 0.974731i $$0.571710\pi$$
$$38$$ 0 0
$$39$$ 4.91925 0.787711
$$40$$ 0 0
$$41$$ 0.788714 0.123176 0.0615882 0.998102i $$-0.480383\pi$$
0.0615882 + 0.998102i $$0.480383\pi$$
$$42$$ 0 0
$$43$$ −0.714571 −0.108971 −0.0544855 0.998515i $$-0.517352\pi$$
−0.0544855 + 0.998515i $$0.517352\pi$$
$$44$$ 0 0
$$45$$ −0.511046 −0.0761823
$$46$$ 0 0
$$47$$ −3.96368 −0.578163 −0.289081 0.957305i $$-0.593350\pi$$
−0.289081 + 0.957305i $$0.593350\pi$$
$$48$$ 0 0
$$49$$ 15.0820 2.15458
$$50$$ 0 0
$$51$$ 5.90055 0.826242
$$52$$ 0 0
$$53$$ −9.69714 −1.33200 −0.666002 0.745950i $$-0.731996\pi$$
−0.666002 + 0.745950i $$0.731996\pi$$
$$54$$ 0 0
$$55$$ 0.221953 0.0299281
$$56$$ 0 0
$$57$$ −0.835165 −0.110620
$$58$$ 0 0
$$59$$ 7.33476 0.954904 0.477452 0.878658i $$-0.341560\pi$$
0.477452 + 0.878658i $$0.341560\pi$$
$$60$$ 0 0
$$61$$ 8.15179 1.04373 0.521865 0.853028i $$-0.325236\pi$$
0.521865 + 0.853028i $$0.325236\pi$$
$$62$$ 0 0
$$63$$ 10.8198 1.36317
$$64$$ 0 0
$$65$$ 1.30734 0.162155
$$66$$ 0 0
$$67$$ −7.86697 −0.961104 −0.480552 0.876966i $$-0.659564\pi$$
−0.480552 + 0.876966i $$0.659564\pi$$
$$68$$ 0 0
$$69$$ −0.888178 −0.106924
$$70$$ 0 0
$$71$$ 3.13400 0.371937 0.185969 0.982556i $$-0.440458\pi$$
0.185969 + 0.982556i $$0.440458\pi$$
$$72$$ 0 0
$$73$$ −6.49076 −0.759687 −0.379843 0.925051i $$-0.624022\pi$$
−0.379843 + 0.925051i $$0.624022\pi$$
$$74$$ 0 0
$$75$$ −4.13468 −0.477432
$$76$$ 0 0
$$77$$ −4.69915 −0.535518
$$78$$ 0 0
$$79$$ −12.0148 −1.35177 −0.675884 0.737008i $$-0.736238\pi$$
−0.675884 + 0.737008i $$0.736238\pi$$
$$80$$ 0 0
$$81$$ 3.20900 0.356556
$$82$$ 0 0
$$83$$ −16.3902 −1.79906 −0.899528 0.436863i $$-0.856089\pi$$
−0.899528 + 0.436863i $$0.856089\pi$$
$$84$$ 0 0
$$85$$ 1.56812 0.170087
$$86$$ 0 0
$$87$$ −6.36948 −0.682880
$$88$$ 0 0
$$89$$ −12.9487 −1.37256 −0.686281 0.727336i $$-0.740758\pi$$
−0.686281 + 0.727336i $$0.740758\pi$$
$$90$$ 0 0
$$91$$ −27.6788 −2.90152
$$92$$ 0 0
$$93$$ −0.752730 −0.0780545
$$94$$ 0 0
$$95$$ −0.221953 −0.0227719
$$96$$ 0 0
$$97$$ 8.14414 0.826912 0.413456 0.910524i $$-0.364321\pi$$
0.413456 + 0.910524i $$0.364321\pi$$
$$98$$ 0 0
$$99$$ −2.30250 −0.231410
$$100$$ 0 0
$$101$$ −15.3713 −1.52950 −0.764750 0.644327i $$-0.777138\pi$$
−0.764750 + 0.644327i $$0.777138\pi$$
$$102$$ 0 0
$$103$$ 8.83682 0.870718 0.435359 0.900257i $$-0.356622\pi$$
0.435359 + 0.900257i $$0.356622\pi$$
$$104$$ 0 0
$$105$$ −0.871069 −0.0850076
$$106$$ 0 0
$$107$$ 9.04360 0.874277 0.437139 0.899394i $$-0.355992\pi$$
0.437139 + 0.899394i $$0.355992\pi$$
$$108$$ 0 0
$$109$$ −18.5950 −1.78107 −0.890537 0.454911i $$-0.849671\pi$$
−0.890537 + 0.454911i $$0.849671\pi$$
$$110$$ 0 0
$$111$$ −2.26963 −0.215424
$$112$$ 0 0
$$113$$ −4.42423 −0.416196 −0.208098 0.978108i $$-0.566727\pi$$
−0.208098 + 0.978108i $$0.566727\pi$$
$$114$$ 0 0
$$115$$ −0.236042 −0.0220110
$$116$$ 0 0
$$117$$ −13.5621 −1.25382
$$118$$ 0 0
$$119$$ −33.2001 −3.04345
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0.658706 0.0593935
$$124$$ 0 0
$$125$$ −2.20859 −0.197543
$$126$$ 0 0
$$127$$ 0.338657 0.0300510 0.0150255 0.999887i $$-0.495217\pi$$
0.0150255 + 0.999887i $$0.495217\pi$$
$$128$$ 0 0
$$129$$ −0.596784 −0.0525439
$$130$$ 0 0
$$131$$ 4.55211 0.397720 0.198860 0.980028i $$-0.436276\pi$$
0.198860 + 0.980028i $$0.436276\pi$$
$$132$$ 0 0
$$133$$ 4.69915 0.407468
$$134$$ 0 0
$$135$$ −0.982909 −0.0845953
$$136$$ 0 0
$$137$$ −0.654263 −0.0558974 −0.0279487 0.999609i $$-0.508898\pi$$
−0.0279487 + 0.999609i $$0.508898\pi$$
$$138$$ 0 0
$$139$$ −11.0479 −0.937072 −0.468536 0.883444i $$-0.655218\pi$$
−0.468536 + 0.883444i $$0.655218\pi$$
$$140$$ 0 0
$$141$$ −3.31033 −0.278780
$$142$$ 0 0
$$143$$ 5.89016 0.492560
$$144$$ 0 0
$$145$$ −1.69275 −0.140575
$$146$$ 0 0
$$147$$ 12.5960 1.03890
$$148$$ 0 0
$$149$$ −7.57743 −0.620767 −0.310384 0.950611i $$-0.600457\pi$$
−0.310384 + 0.950611i $$0.600457\pi$$
$$150$$ 0 0
$$151$$ 6.95296 0.565824 0.282912 0.959146i $$-0.408700\pi$$
0.282912 + 0.959146i $$0.408700\pi$$
$$152$$ 0 0
$$153$$ −16.2675 −1.31515
$$154$$ 0 0
$$155$$ −0.200045 −0.0160680
$$156$$ 0 0
$$157$$ −14.7427 −1.17659 −0.588297 0.808645i $$-0.700201\pi$$
−0.588297 + 0.808645i $$0.700201\pi$$
$$158$$ 0 0
$$159$$ −8.09871 −0.642270
$$160$$ 0 0
$$161$$ 4.99744 0.393853
$$162$$ 0 0
$$163$$ 4.94015 0.386942 0.193471 0.981106i $$-0.438025\pi$$
0.193471 + 0.981106i $$0.438025\pi$$
$$164$$ 0 0
$$165$$ 0.185367 0.0144308
$$166$$ 0 0
$$167$$ −22.6032 −1.74909 −0.874544 0.484946i $$-0.838839\pi$$
−0.874544 + 0.484946i $$0.838839\pi$$
$$168$$ 0 0
$$169$$ 21.6940 1.66877
$$170$$ 0 0
$$171$$ 2.30250 0.176077
$$172$$ 0 0
$$173$$ −8.18620 −0.622385 −0.311193 0.950347i $$-0.600728\pi$$
−0.311193 + 0.950347i $$0.600728\pi$$
$$174$$ 0 0
$$175$$ 23.2643 1.75861
$$176$$ 0 0
$$177$$ 6.12573 0.460438
$$178$$ 0 0
$$179$$ 14.0830 1.05261 0.526305 0.850296i $$-0.323577\pi$$
0.526305 + 0.850296i $$0.323577\pi$$
$$180$$ 0 0
$$181$$ −14.3260 −1.06484 −0.532422 0.846479i $$-0.678718\pi$$
−0.532422 + 0.846479i $$0.678718\pi$$
$$182$$ 0 0
$$183$$ 6.80809 0.503268
$$184$$ 0 0
$$185$$ −0.603175 −0.0443463
$$186$$ 0 0
$$187$$ 7.06513 0.516653
$$188$$ 0 0
$$189$$ 20.8100 1.51371
$$190$$ 0 0
$$191$$ −0.394967 −0.0285788 −0.0142894 0.999898i $$-0.504549\pi$$
−0.0142894 + 0.999898i $$0.504549\pi$$
$$192$$ 0 0
$$193$$ 9.03423 0.650298 0.325149 0.945663i $$-0.394586\pi$$
0.325149 + 0.945663i $$0.394586\pi$$
$$194$$ 0 0
$$195$$ 1.09184 0.0781884
$$196$$ 0 0
$$197$$ 7.85313 0.559512 0.279756 0.960071i $$-0.409746\pi$$
0.279756 + 0.960071i $$0.409746\pi$$
$$198$$ 0 0
$$199$$ −7.81540 −0.554019 −0.277009 0.960867i $$-0.589343\pi$$
−0.277009 + 0.960867i $$0.589343\pi$$
$$200$$ 0 0
$$201$$ −6.57022 −0.463427
$$202$$ 0 0
$$203$$ 35.8386 2.51538
$$204$$ 0 0
$$205$$ 0.175057 0.0122265
$$206$$ 0 0
$$207$$ 2.44865 0.170193
$$208$$ 0 0
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 21.5955 1.48670 0.743348 0.668905i $$-0.233237\pi$$
0.743348 + 0.668905i $$0.233237\pi$$
$$212$$ 0 0
$$213$$ 2.61741 0.179342
$$214$$ 0 0
$$215$$ −0.158601 −0.0108165
$$216$$ 0 0
$$217$$ 4.23533 0.287513
$$218$$ 0 0
$$219$$ −5.42086 −0.366308
$$220$$ 0 0
$$221$$ 41.6147 2.79931
$$222$$ 0 0
$$223$$ 6.95854 0.465979 0.232989 0.972479i $$-0.425149\pi$$
0.232989 + 0.972479i $$0.425149\pi$$
$$224$$ 0 0
$$225$$ 11.3991 0.759938
$$226$$ 0 0
$$227$$ −7.67819 −0.509619 −0.254810 0.966991i $$-0.582013\pi$$
−0.254810 + 0.966991i $$0.582013\pi$$
$$228$$ 0 0
$$229$$ 5.83925 0.385869 0.192934 0.981212i $$-0.438200\pi$$
0.192934 + 0.981212i $$0.438200\pi$$
$$230$$ 0 0
$$231$$ −3.92457 −0.258218
$$232$$ 0 0
$$233$$ −21.8431 −1.43099 −0.715494 0.698619i $$-0.753798\pi$$
−0.715494 + 0.698619i $$0.753798\pi$$
$$234$$ 0 0
$$235$$ −0.879751 −0.0573886
$$236$$ 0 0
$$237$$ −10.0343 −0.651799
$$238$$ 0 0
$$239$$ −4.21004 −0.272325 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$240$$ 0 0
$$241$$ −17.4276 −1.12261 −0.561304 0.827610i $$-0.689700\pi$$
−0.561304 + 0.827610i $$0.689700\pi$$
$$242$$ 0 0
$$243$$ 15.9654 1.02418
$$244$$ 0 0
$$245$$ 3.34750 0.213864
$$246$$ 0 0
$$247$$ −5.89016 −0.374782
$$248$$ 0 0
$$249$$ −13.6885 −0.867474
$$250$$ 0 0
$$251$$ −2.14594 −0.135451 −0.0677254 0.997704i $$-0.521574\pi$$
−0.0677254 + 0.997704i $$0.521574\pi$$
$$252$$ 0 0
$$253$$ −1.06348 −0.0668602
$$254$$ 0 0
$$255$$ 1.30964 0.0820130
$$256$$ 0 0
$$257$$ −5.23899 −0.326799 −0.163399 0.986560i $$-0.552246\pi$$
−0.163399 + 0.986560i $$0.552246\pi$$
$$258$$ 0 0
$$259$$ 12.7703 0.793510
$$260$$ 0 0
$$261$$ 17.5603 1.08695
$$262$$ 0 0
$$263$$ −12.2597 −0.755967 −0.377983 0.925812i $$-0.623382\pi$$
−0.377983 + 0.925812i $$0.623382\pi$$
$$264$$ 0 0
$$265$$ −2.15231 −0.132215
$$266$$ 0 0
$$267$$ −10.8143 −0.661826
$$268$$ 0 0
$$269$$ −9.37273 −0.571465 −0.285733 0.958309i $$-0.592237\pi$$
−0.285733 + 0.958309i $$0.592237\pi$$
$$270$$ 0 0
$$271$$ 13.3131 0.808716 0.404358 0.914601i $$-0.367495\pi$$
0.404358 + 0.914601i $$0.367495\pi$$
$$272$$ 0 0
$$273$$ −23.1163 −1.39906
$$274$$ 0 0
$$275$$ −4.95074 −0.298541
$$276$$ 0 0
$$277$$ −14.5641 −0.875073 −0.437537 0.899201i $$-0.644149\pi$$
−0.437537 + 0.899201i $$0.644149\pi$$
$$278$$ 0 0
$$279$$ 2.07523 0.124241
$$280$$ 0 0
$$281$$ 20.2411 1.20748 0.603741 0.797180i $$-0.293676\pi$$
0.603741 + 0.797180i $$0.293676\pi$$
$$282$$ 0 0
$$283$$ −14.6230 −0.869248 −0.434624 0.900612i $$-0.643119\pi$$
−0.434624 + 0.900612i $$0.643119\pi$$
$$284$$ 0 0
$$285$$ −0.185367 −0.0109802
$$286$$ 0 0
$$287$$ −3.70629 −0.218775
$$288$$ 0 0
$$289$$ 32.9161 1.93624
$$290$$ 0 0
$$291$$ 6.80170 0.398723
$$292$$ 0 0
$$293$$ 27.6524 1.61547 0.807737 0.589543i $$-0.200692\pi$$
0.807737 + 0.589543i $$0.200692\pi$$
$$294$$ 0 0
$$295$$ 1.62797 0.0947841
$$296$$ 0 0
$$297$$ −4.42846 −0.256965
$$298$$ 0 0
$$299$$ −6.26404 −0.362259
$$300$$ 0 0
$$301$$ 3.35788 0.193545
$$302$$ 0 0
$$303$$ −12.8376 −0.737499
$$304$$ 0 0
$$305$$ 1.80931 0.103601
$$306$$ 0 0
$$307$$ 0.756065 0.0431509 0.0215755 0.999767i $$-0.493132\pi$$
0.0215755 + 0.999767i $$0.493132\pi$$
$$308$$ 0 0
$$309$$ 7.38020 0.419845
$$310$$ 0 0
$$311$$ −1.13352 −0.0642761 −0.0321381 0.999483i $$-0.510232\pi$$
−0.0321381 + 0.999483i $$0.510232\pi$$
$$312$$ 0 0
$$313$$ −10.9012 −0.616172 −0.308086 0.951359i $$-0.599688\pi$$
−0.308086 + 0.951359i $$0.599688\pi$$
$$314$$ 0 0
$$315$$ 2.40148 0.135308
$$316$$ 0 0
$$317$$ 8.32326 0.467481 0.233741 0.972299i $$-0.424903\pi$$
0.233741 + 0.972299i $$0.424903\pi$$
$$318$$ 0 0
$$319$$ −7.62662 −0.427009
$$320$$ 0 0
$$321$$ 7.55289 0.421561
$$322$$ 0 0
$$323$$ −7.06513 −0.393114
$$324$$ 0 0
$$325$$ −29.1606 −1.61754
$$326$$ 0 0
$$327$$ −15.5299 −0.858803
$$328$$ 0 0
$$329$$ 18.6260 1.02688
$$330$$ 0 0
$$331$$ 2.47466 0.136020 0.0680098 0.997685i $$-0.478335\pi$$
0.0680098 + 0.997685i $$0.478335\pi$$
$$332$$ 0 0
$$333$$ 6.25723 0.342894
$$334$$ 0 0
$$335$$ −1.74610 −0.0953994
$$336$$ 0 0
$$337$$ −30.3242 −1.65187 −0.825933 0.563768i $$-0.809351\pi$$
−0.825933 + 0.563768i $$0.809351\pi$$
$$338$$ 0 0
$$339$$ −3.69496 −0.200683
$$340$$ 0 0
$$341$$ −0.901295 −0.0488079
$$342$$ 0 0
$$343$$ −37.9788 −2.05066
$$344$$ 0 0
$$345$$ −0.197134 −0.0106133
$$346$$ 0 0
$$347$$ 10.4332 0.560081 0.280041 0.959988i $$-0.409652\pi$$
0.280041 + 0.959988i $$0.409652\pi$$
$$348$$ 0 0
$$349$$ 4.89049 0.261782 0.130891 0.991397i $$-0.458216\pi$$
0.130891 + 0.991397i $$0.458216\pi$$
$$350$$ 0 0
$$351$$ −26.0843 −1.39228
$$352$$ 0 0
$$353$$ 34.1254 1.81631 0.908155 0.418634i $$-0.137491\pi$$
0.908155 + 0.418634i $$0.137491\pi$$
$$354$$ 0 0
$$355$$ 0.695600 0.0369186
$$356$$ 0 0
$$357$$ −27.7276 −1.46750
$$358$$ 0 0
$$359$$ −10.5366 −0.556103 −0.278051 0.960566i $$-0.589689\pi$$
−0.278051 + 0.960566i $$0.589689\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0.835165 0.0438348
$$364$$ 0 0
$$365$$ −1.44064 −0.0754067
$$366$$ 0 0
$$367$$ −31.1277 −1.62485 −0.812427 0.583062i $$-0.801854\pi$$
−0.812427 + 0.583062i $$0.801854\pi$$
$$368$$ 0 0
$$369$$ −1.81601 −0.0945378
$$370$$ 0 0
$$371$$ 45.5684 2.36579
$$372$$ 0 0
$$373$$ 28.9881 1.50095 0.750473 0.660901i $$-0.229826\pi$$
0.750473 + 0.660901i $$0.229826\pi$$
$$374$$ 0 0
$$375$$ −1.84454 −0.0952516
$$376$$ 0 0
$$377$$ −44.9220 −2.31360
$$378$$ 0 0
$$379$$ 26.4204 1.35712 0.678561 0.734544i $$-0.262604\pi$$
0.678561 + 0.734544i $$0.262604\pi$$
$$380$$ 0 0
$$381$$ 0.282835 0.0144901
$$382$$ 0 0
$$383$$ 13.6280 0.696360 0.348180 0.937428i $$-0.386800\pi$$
0.348180 + 0.937428i $$0.386800\pi$$
$$384$$ 0 0
$$385$$ −1.04299 −0.0531557
$$386$$ 0 0
$$387$$ 1.64530 0.0836353
$$388$$ 0 0
$$389$$ −15.7277 −0.797428 −0.398714 0.917075i $$-0.630543\pi$$
−0.398714 + 0.917075i $$0.630543\pi$$
$$390$$ 0 0
$$391$$ −7.51360 −0.379979
$$392$$ 0 0
$$393$$ 3.80176 0.191774
$$394$$ 0 0
$$395$$ −2.66671 −0.134177
$$396$$ 0 0
$$397$$ 16.8523 0.845790 0.422895 0.906179i $$-0.361014\pi$$
0.422895 + 0.906179i $$0.361014\pi$$
$$398$$ 0 0
$$399$$ 3.92457 0.196474
$$400$$ 0 0
$$401$$ 30.3744 1.51682 0.758412 0.651775i $$-0.225975\pi$$
0.758412 + 0.651775i $$0.225975\pi$$
$$402$$ 0 0
$$403$$ −5.30877 −0.264449
$$404$$ 0 0
$$405$$ 0.712247 0.0353919
$$406$$ 0 0
$$407$$ −2.71758 −0.134706
$$408$$ 0 0
$$409$$ −1.31844 −0.0651925 −0.0325962 0.999469i $$-0.510378\pi$$
−0.0325962 + 0.999469i $$0.510378\pi$$
$$410$$ 0 0
$$411$$ −0.546417 −0.0269528
$$412$$ 0 0
$$413$$ −34.4672 −1.69602
$$414$$ 0 0
$$415$$ −3.63785 −0.178575
$$416$$ 0 0
$$417$$ −9.22683 −0.451840
$$418$$ 0 0
$$419$$ 30.3257 1.48151 0.740753 0.671778i $$-0.234469\pi$$
0.740753 + 0.671778i $$0.234469\pi$$
$$420$$ 0 0
$$421$$ 8.27204 0.403154 0.201577 0.979473i $$-0.435393\pi$$
0.201577 + 0.979473i $$0.435393\pi$$
$$422$$ 0 0
$$423$$ 9.12638 0.443740
$$424$$ 0 0
$$425$$ −34.9776 −1.69666
$$426$$ 0 0
$$427$$ −38.3065 −1.85378
$$428$$ 0 0
$$429$$ 4.91925 0.237504
$$430$$ 0 0
$$431$$ −22.9574 −1.10582 −0.552909 0.833241i $$-0.686482\pi$$
−0.552909 + 0.833241i $$0.686482\pi$$
$$432$$ 0 0
$$433$$ 11.4207 0.548846 0.274423 0.961609i $$-0.411513\pi$$
0.274423 + 0.961609i $$0.411513\pi$$
$$434$$ 0 0
$$435$$ −1.41372 −0.0677829
$$436$$ 0 0
$$437$$ 1.06348 0.0508730
$$438$$ 0 0
$$439$$ −5.43642 −0.259466 −0.129733 0.991549i $$-0.541412\pi$$
−0.129733 + 0.991549i $$0.541412\pi$$
$$440$$ 0 0
$$441$$ −34.7264 −1.65364
$$442$$ 0 0
$$443$$ 34.7494 1.65100 0.825498 0.564405i $$-0.190894\pi$$
0.825498 + 0.564405i $$0.190894\pi$$
$$444$$ 0 0
$$445$$ −2.87401 −0.136241
$$446$$ 0 0
$$447$$ −6.32840 −0.299323
$$448$$ 0 0
$$449$$ −11.3505 −0.535662 −0.267831 0.963466i $$-0.586307\pi$$
−0.267831 + 0.963466i $$0.586307\pi$$
$$450$$ 0 0
$$451$$ 0.788714 0.0371391
$$452$$ 0 0
$$453$$ 5.80687 0.272830
$$454$$ 0 0
$$455$$ −6.14338 −0.288006
$$456$$ 0 0
$$457$$ 6.89453 0.322513 0.161256 0.986913i $$-0.448445\pi$$
0.161256 + 0.986913i $$0.448445\pi$$
$$458$$ 0 0
$$459$$ −31.2877 −1.46038
$$460$$ 0 0
$$461$$ 13.5620 0.631643 0.315822 0.948819i $$-0.397720\pi$$
0.315822 + 0.948819i $$0.397720\pi$$
$$462$$ 0 0
$$463$$ 18.6923 0.868704 0.434352 0.900743i $$-0.356977\pi$$
0.434352 + 0.900743i $$0.356977\pi$$
$$464$$ 0 0
$$465$$ −0.167071 −0.00774771
$$466$$ 0 0
$$467$$ 14.7156 0.680957 0.340479 0.940252i $$-0.389411\pi$$
0.340479 + 0.940252i $$0.389411\pi$$
$$468$$ 0 0
$$469$$ 36.9681 1.70703
$$470$$ 0 0
$$471$$ −12.3126 −0.567333
$$472$$ 0 0
$$473$$ −0.714571 −0.0328560
$$474$$ 0 0
$$475$$ 4.95074 0.227155
$$476$$ 0 0
$$477$$ 22.3277 1.02231
$$478$$ 0 0
$$479$$ 16.3467 0.746901 0.373450 0.927650i $$-0.378175\pi$$
0.373450 + 0.927650i $$0.378175\pi$$
$$480$$ 0 0
$$481$$ −16.0070 −0.729856
$$482$$ 0 0
$$483$$ 4.17368 0.189909
$$484$$ 0 0
$$485$$ 1.80762 0.0820796
$$486$$ 0 0
$$487$$ 24.8625 1.12663 0.563314 0.826243i $$-0.309526\pi$$
0.563314 + 0.826243i $$0.309526\pi$$
$$488$$ 0 0
$$489$$ 4.12584 0.186577
$$490$$ 0 0
$$491$$ −17.5025 −0.789878 −0.394939 0.918707i $$-0.629234\pi$$
−0.394939 + 0.918707i $$0.629234\pi$$
$$492$$ 0 0
$$493$$ −53.8830 −2.42677
$$494$$ 0 0
$$495$$ −0.511046 −0.0229698
$$496$$ 0 0
$$497$$ −14.7271 −0.660603
$$498$$ 0 0
$$499$$ 3.13006 0.140121 0.0700603 0.997543i $$-0.477681\pi$$
0.0700603 + 0.997543i $$0.477681\pi$$
$$500$$ 0 0
$$501$$ −18.8774 −0.843380
$$502$$ 0 0
$$503$$ 8.98965 0.400829 0.200414 0.979711i $$-0.435771\pi$$
0.200414 + 0.979711i $$0.435771\pi$$
$$504$$ 0 0
$$505$$ −3.41170 −0.151819
$$506$$ 0 0
$$507$$ 18.1180 0.804650
$$508$$ 0 0
$$509$$ 23.6416 1.04790 0.523949 0.851750i $$-0.324458\pi$$
0.523949 + 0.851750i $$0.324458\pi$$
$$510$$ 0 0
$$511$$ 30.5011 1.34929
$$512$$ 0 0
$$513$$ 4.42846 0.195521
$$514$$ 0 0
$$515$$ 1.96136 0.0864277
$$516$$ 0 0
$$517$$ −3.96368 −0.174323
$$518$$ 0 0
$$519$$ −6.83683 −0.300103
$$520$$ 0 0
$$521$$ 36.0036 1.57735 0.788673 0.614813i $$-0.210768\pi$$
0.788673 + 0.614813i $$0.210768\pi$$
$$522$$ 0 0
$$523$$ −7.08081 −0.309622 −0.154811 0.987944i $$-0.549477\pi$$
−0.154811 + 0.987944i $$0.549477\pi$$
$$524$$ 0 0
$$525$$ 19.4295 0.847973
$$526$$ 0 0
$$527$$ −6.36777 −0.277384
$$528$$ 0 0
$$529$$ −21.8690 −0.950827
$$530$$ 0 0
$$531$$ −16.8883 −0.732889
$$532$$ 0 0
$$533$$ 4.64565 0.201225
$$534$$ 0 0
$$535$$ 2.00725 0.0867810
$$536$$ 0 0
$$537$$ 11.7616 0.507550
$$538$$ 0 0
$$539$$ 15.0820 0.649630
$$540$$ 0 0
$$541$$ −4.23609 −0.182124 −0.0910619 0.995845i $$-0.529026\pi$$
−0.0910619 + 0.995845i $$0.529026\pi$$
$$542$$ 0 0
$$543$$ −11.9646 −0.513449
$$544$$ 0 0
$$545$$ −4.12720 −0.176790
$$546$$ 0 0
$$547$$ −9.53317 −0.407609 −0.203805 0.979012i $$-0.565331\pi$$
−0.203805 + 0.979012i $$0.565331\pi$$
$$548$$ 0 0
$$549$$ −18.7695 −0.801063
$$550$$ 0 0
$$551$$ 7.62662 0.324905
$$552$$ 0 0
$$553$$ 56.4593 2.40089
$$554$$ 0 0
$$555$$ −0.503750 −0.0213830
$$556$$ 0 0
$$557$$ 2.85948 0.121160 0.0605800 0.998163i $$-0.480705\pi$$
0.0605800 + 0.998163i $$0.480705\pi$$
$$558$$ 0 0
$$559$$ −4.20894 −0.178019
$$560$$ 0 0
$$561$$ 5.90055 0.249121
$$562$$ 0 0
$$563$$ −12.8091 −0.539841 −0.269920 0.962883i $$-0.586997\pi$$
−0.269920 + 0.962883i $$0.586997\pi$$
$$564$$ 0 0
$$565$$ −0.981969 −0.0413118
$$566$$ 0 0
$$567$$ −15.0796 −0.633284
$$568$$ 0 0
$$569$$ −10.5273 −0.441327 −0.220664 0.975350i $$-0.570822\pi$$
−0.220664 + 0.975350i $$0.570822\pi$$
$$570$$ 0 0
$$571$$ 24.9055 1.04226 0.521132 0.853476i $$-0.325510\pi$$
0.521132 + 0.853476i $$0.325510\pi$$
$$572$$ 0 0
$$573$$ −0.329863 −0.0137802
$$574$$ 0 0
$$575$$ 5.26499 0.219565
$$576$$ 0 0
$$577$$ −26.7799 −1.11486 −0.557430 0.830224i $$-0.688213\pi$$
−0.557430 + 0.830224i $$0.688213\pi$$
$$578$$ 0 0
$$579$$ 7.54507 0.313562
$$580$$ 0 0
$$581$$ 77.0200 3.19533
$$582$$ 0 0
$$583$$ −9.69714 −0.401615
$$584$$ 0 0
$$585$$ −3.01014 −0.124454
$$586$$ 0 0
$$587$$ −18.7991 −0.775920 −0.387960 0.921676i $$-0.626820\pi$$
−0.387960 + 0.921676i $$0.626820\pi$$
$$588$$ 0 0
$$589$$ 0.901295 0.0371372
$$590$$ 0 0
$$591$$ 6.55866 0.269787
$$592$$ 0 0
$$593$$ −3.68216 −0.151208 −0.0756041 0.997138i $$-0.524089\pi$$
−0.0756041 + 0.997138i $$0.524089\pi$$
$$594$$ 0 0
$$595$$ −7.36886 −0.302094
$$596$$ 0 0
$$597$$ −6.52715 −0.267138
$$598$$ 0 0
$$599$$ −43.2659 −1.76780 −0.883899 0.467677i $$-0.845091\pi$$
−0.883899 + 0.467677i $$0.845091\pi$$
$$600$$ 0 0
$$601$$ −14.4072 −0.587683 −0.293842 0.955854i $$-0.594934\pi$$
−0.293842 + 0.955854i $$0.594934\pi$$
$$602$$ 0 0
$$603$$ 18.1137 0.737647
$$604$$ 0 0
$$605$$ 0.221953 0.00902366
$$606$$ 0 0
$$607$$ 6.06496 0.246169 0.123084 0.992396i $$-0.460721\pi$$
0.123084 + 0.992396i $$0.460721\pi$$
$$608$$ 0 0
$$609$$ 29.9312 1.21287
$$610$$ 0 0
$$611$$ −23.3467 −0.944508
$$612$$ 0 0
$$613$$ −3.51855 −0.142113 −0.0710565 0.997472i $$-0.522637\pi$$
−0.0710565 + 0.997472i $$0.522637\pi$$
$$614$$ 0 0
$$615$$ 0.146202 0.00589542
$$616$$ 0 0
$$617$$ 7.17642 0.288912 0.144456 0.989511i $$-0.453857\pi$$
0.144456 + 0.989511i $$0.453857\pi$$
$$618$$ 0 0
$$619$$ 8.27421 0.332568 0.166284 0.986078i $$-0.446823\pi$$
0.166284 + 0.986078i $$0.446823\pi$$
$$620$$ 0 0
$$621$$ 4.70956 0.188988
$$622$$ 0 0
$$623$$ 60.8480 2.43783
$$624$$ 0 0
$$625$$ 24.2635 0.970539
$$626$$ 0 0
$$627$$ −0.835165 −0.0333533
$$628$$ 0 0
$$629$$ −19.2001 −0.765557
$$630$$ 0 0
$$631$$ 9.09560 0.362090 0.181045 0.983475i $$-0.442052\pi$$
0.181045 + 0.983475i $$0.442052\pi$$
$$632$$ 0 0
$$633$$ 18.0358 0.716859
$$634$$ 0 0
$$635$$ 0.0751660 0.00298287
$$636$$ 0 0
$$637$$ 88.8356 3.51980
$$638$$ 0 0
$$639$$ −7.21604 −0.285462
$$640$$ 0 0
$$641$$ 16.4386 0.649285 0.324642 0.945837i $$-0.394756\pi$$
0.324642 + 0.945837i $$0.394756\pi$$
$$642$$ 0 0
$$643$$ −7.04594 −0.277865 −0.138932 0.990302i $$-0.544367\pi$$
−0.138932 + 0.990302i $$0.544367\pi$$
$$644$$ 0 0
$$645$$ −0.132458 −0.00521553
$$646$$ 0 0
$$647$$ −18.1024 −0.711679 −0.355840 0.934547i $$-0.615805\pi$$
−0.355840 + 0.934547i $$0.615805\pi$$
$$648$$ 0 0
$$649$$ 7.33476 0.287914
$$650$$ 0 0
$$651$$ 3.53719 0.138634
$$652$$ 0 0
$$653$$ 37.2985 1.45960 0.729802 0.683659i $$-0.239612\pi$$
0.729802 + 0.683659i $$0.239612\pi$$
$$654$$ 0 0
$$655$$ 1.01035 0.0394778
$$656$$ 0 0
$$657$$ 14.9450 0.583059
$$658$$ 0 0
$$659$$ −13.2223 −0.515068 −0.257534 0.966269i $$-0.582910\pi$$
−0.257534 + 0.966269i $$0.582910\pi$$
$$660$$ 0 0
$$661$$ −17.6212 −0.685385 −0.342692 0.939448i $$-0.611339\pi$$
−0.342692 + 0.939448i $$0.611339\pi$$
$$662$$ 0 0
$$663$$ 34.7552 1.34978
$$664$$ 0 0
$$665$$ 1.04299 0.0404454
$$666$$ 0 0
$$667$$ 8.11073 0.314049
$$668$$ 0 0
$$669$$ 5.81153 0.224687
$$670$$ 0 0
$$671$$ 8.15179 0.314696
$$672$$ 0 0
$$673$$ 32.2335 1.24251 0.621255 0.783609i $$-0.286623\pi$$
0.621255 + 0.783609i $$0.286623\pi$$
$$674$$ 0 0
$$675$$ 21.9241 0.843861
$$676$$ 0 0
$$677$$ −10.2788 −0.395045 −0.197523 0.980298i $$-0.563290\pi$$
−0.197523 + 0.980298i $$0.563290\pi$$
$$678$$ 0 0
$$679$$ −38.2706 −1.46869
$$680$$ 0 0
$$681$$ −6.41255 −0.245730
$$682$$ 0 0
$$683$$ −25.4018 −0.971973 −0.485986 0.873966i $$-0.661540\pi$$
−0.485986 + 0.873966i $$0.661540\pi$$
$$684$$ 0 0
$$685$$ −0.145215 −0.00554840
$$686$$ 0 0
$$687$$ 4.87674 0.186059
$$688$$ 0 0
$$689$$ −57.1177 −2.17601
$$690$$ 0 0
$$691$$ 2.58092 0.0981828 0.0490914 0.998794i $$-0.484367\pi$$
0.0490914 + 0.998794i $$0.484367\pi$$
$$692$$ 0 0
$$693$$ 10.8198 0.411010
$$694$$ 0 0
$$695$$ −2.45212 −0.0930141
$$696$$ 0 0
$$697$$ 5.57236 0.211068
$$698$$ 0 0
$$699$$ −18.2426 −0.689998
$$700$$ 0 0
$$701$$ −30.0612 −1.13540 −0.567698 0.823237i $$-0.692166\pi$$
−0.567698 + 0.823237i $$0.692166\pi$$
$$702$$ 0 0
$$703$$ 2.71758 0.102496
$$704$$ 0 0
$$705$$ −0.734737 −0.0276718
$$706$$ 0 0
$$707$$ 72.2321 2.71657
$$708$$ 0 0
$$709$$ −6.90683 −0.259391 −0.129696 0.991554i $$-0.541400\pi$$
−0.129696 + 0.991554i $$0.541400\pi$$
$$710$$ 0 0
$$711$$ 27.6640 1.03748
$$712$$ 0 0
$$713$$ 0.958506 0.0358963
$$714$$ 0 0
$$715$$ 1.30734 0.0488916
$$716$$ 0 0
$$717$$ −3.51608 −0.131310
$$718$$ 0 0
$$719$$ −13.1525 −0.490506 −0.245253 0.969459i $$-0.578871\pi$$
−0.245253 + 0.969459i $$0.578871\pi$$
$$720$$ 0 0
$$721$$ −41.5256 −1.54649
$$722$$ 0 0
$$723$$ −14.5549 −0.541302
$$724$$ 0 0
$$725$$ 37.7574 1.40227
$$726$$ 0 0
$$727$$ 46.8434 1.73733 0.868663 0.495404i $$-0.164980\pi$$
0.868663 + 0.495404i $$0.164980\pi$$
$$728$$ 0 0
$$729$$ 3.70675 0.137287
$$730$$ 0 0
$$731$$ −5.04854 −0.186727
$$732$$ 0 0
$$733$$ −31.9162 −1.17885 −0.589426 0.807822i $$-0.700646\pi$$
−0.589426 + 0.807822i $$0.700646\pi$$
$$734$$ 0 0
$$735$$ 2.79571 0.103121
$$736$$ 0 0
$$737$$ −7.86697 −0.289784
$$738$$ 0 0
$$739$$ −30.0302 −1.10468 −0.552339 0.833620i $$-0.686264\pi$$
−0.552339 + 0.833620i $$0.686264\pi$$
$$740$$ 0 0
$$741$$ −4.91925 −0.180713
$$742$$ 0 0
$$743$$ 42.6871 1.56604 0.783019 0.621998i $$-0.213679\pi$$
0.783019 + 0.621998i $$0.213679\pi$$
$$744$$ 0 0
$$745$$ −1.68183 −0.0616175
$$746$$ 0 0
$$747$$ 37.7384 1.38078
$$748$$ 0 0
$$749$$ −42.4972 −1.55282
$$750$$ 0 0
$$751$$ −30.7211 −1.12103 −0.560514 0.828145i $$-0.689396\pi$$
−0.560514 + 0.828145i $$0.689396\pi$$
$$752$$ 0 0
$$753$$ −1.79222 −0.0653120
$$754$$ 0 0
$$755$$ 1.54323 0.0561638
$$756$$ 0 0
$$757$$ −49.0714 −1.78353 −0.891765 0.452498i $$-0.850533\pi$$
−0.891765 + 0.452498i $$0.850533\pi$$
$$758$$ 0 0
$$759$$ −0.888178 −0.0322388
$$760$$ 0 0
$$761$$ −24.8178 −0.899645 −0.449822 0.893118i $$-0.648513\pi$$
−0.449822 + 0.893118i $$0.648513\pi$$
$$762$$ 0 0
$$763$$ 87.3806 3.16339
$$764$$ 0 0
$$765$$ −3.61061 −0.130542
$$766$$ 0 0
$$767$$ 43.2029 1.55997
$$768$$ 0 0
$$769$$ −38.9862 −1.40588 −0.702940 0.711250i $$-0.748130\pi$$
−0.702940 + 0.711250i $$0.748130\pi$$
$$770$$ 0 0
$$771$$ −4.37542 −0.157577
$$772$$ 0 0
$$773$$ 0.998099 0.0358991 0.0179496 0.999839i $$-0.494286\pi$$
0.0179496 + 0.999839i $$0.494286\pi$$
$$774$$ 0 0
$$775$$ 4.46208 0.160283
$$776$$ 0 0
$$777$$ 10.6653 0.382617
$$778$$ 0 0
$$779$$ −0.788714 −0.0282586
$$780$$ 0 0
$$781$$ 3.13400 0.112143
$$782$$ 0 0
$$783$$ 33.7742 1.20699
$$784$$ 0 0
$$785$$ −3.27218 −0.116789
$$786$$ 0 0
$$787$$ −18.7420 −0.668082 −0.334041 0.942559i $$-0.608412\pi$$
−0.334041 + 0.942559i $$0.608412\pi$$
$$788$$ 0 0
$$789$$ −10.2389 −0.364514
$$790$$ 0 0
$$791$$ 20.7901 0.739211
$$792$$ 0 0
$$793$$ 48.0153 1.70507
$$794$$ 0 0
$$795$$ −1.79753 −0.0637519
$$796$$ 0 0
$$797$$ 5.05128 0.178925 0.0894627 0.995990i $$-0.471485\pi$$
0.0894627 + 0.995990i $$0.471485\pi$$
$$798$$ 0 0
$$799$$ −28.0039 −0.990708
$$800$$ 0 0
$$801$$ 29.8144 1.05344
$$802$$ 0 0
$$803$$ −6.49076 −0.229054
$$804$$ 0 0
$$805$$ 1.10920 0.0390940
$$806$$ 0 0
$$807$$ −7.82777 −0.275551
$$808$$ 0 0
$$809$$ −4.98214 −0.175163 −0.0875813 0.996157i $$-0.527914\pi$$
−0.0875813 + 0.996157i $$0.527914\pi$$
$$810$$ 0 0
$$811$$ −25.3145 −0.888913 −0.444456 0.895800i $$-0.646603\pi$$
−0.444456 + 0.895800i $$0.646603\pi$$
$$812$$ 0 0
$$813$$ 11.1187 0.389949
$$814$$ 0 0
$$815$$ 1.09648 0.0384080
$$816$$ 0 0
$$817$$ 0.714571 0.0249997
$$818$$ 0 0
$$819$$ 63.7303 2.22692
$$820$$ 0 0
$$821$$ 5.81543 0.202960 0.101480 0.994838i $$-0.467642\pi$$
0.101480 + 0.994838i $$0.467642\pi$$
$$822$$ 0 0
$$823$$ −47.7881 −1.66579 −0.832895 0.553431i $$-0.813318\pi$$
−0.832895 + 0.553431i $$0.813318\pi$$
$$824$$ 0 0
$$825$$ −4.13468 −0.143951
$$826$$ 0 0
$$827$$ 39.0692 1.35857 0.679284 0.733875i $$-0.262290\pi$$
0.679284 + 0.733875i $$0.262290\pi$$
$$828$$ 0 0
$$829$$ −7.57440 −0.263070 −0.131535 0.991312i $$-0.541991\pi$$
−0.131535 + 0.991312i $$0.541991\pi$$
$$830$$ 0 0
$$831$$ −12.1634 −0.421945
$$832$$ 0 0
$$833$$ 106.557 3.69197
$$834$$ 0 0
$$835$$ −5.01684 −0.173615
$$836$$ 0 0
$$837$$ 3.99135 0.137961
$$838$$ 0 0
$$839$$ 28.0343 0.967850 0.483925 0.875110i $$-0.339211\pi$$
0.483925 + 0.875110i $$0.339211\pi$$
$$840$$ 0 0
$$841$$ 29.1653 1.00570
$$842$$ 0 0
$$843$$ 16.9046 0.582227
$$844$$ 0 0
$$845$$ 4.81504 0.165642
$$846$$ 0 0
$$847$$ −4.69915 −0.161465
$$848$$ 0 0
$$849$$ −12.2126 −0.419136
$$850$$ 0 0
$$851$$ 2.89008 0.0990708
$$852$$ 0 0
$$853$$ −27.9605 −0.957349 −0.478674 0.877992i $$-0.658883\pi$$
−0.478674 + 0.877992i $$0.658883\pi$$
$$854$$ 0 0
$$855$$ 0.511046 0.0174774
$$856$$ 0 0
$$857$$ 17.8805 0.610787 0.305393 0.952226i $$-0.401212\pi$$
0.305393 + 0.952226i $$0.401212\pi$$
$$858$$ 0 0
$$859$$ −7.76017 −0.264774 −0.132387 0.991198i $$-0.542264\pi$$
−0.132387 + 0.991198i $$0.542264\pi$$
$$860$$ 0 0
$$861$$ −3.09536 −0.105490
$$862$$ 0 0
$$863$$ −31.9699 −1.08827 −0.544135 0.838998i $$-0.683142\pi$$
−0.544135 + 0.838998i $$0.683142\pi$$
$$864$$ 0 0
$$865$$ −1.81695 −0.0617782
$$866$$ 0 0
$$867$$ 27.4903 0.933621
$$868$$ 0 0
$$869$$ −12.0148 −0.407574
$$870$$ 0 0
$$871$$ −46.3377 −1.57009
$$872$$ 0 0
$$873$$ −18.7519 −0.634655
$$874$$ 0 0
$$875$$ 10.3785 0.350858
$$876$$ 0 0
$$877$$ −2.12224 −0.0716630 −0.0358315 0.999358i $$-0.511408\pi$$
−0.0358315 + 0.999358i $$0.511408\pi$$
$$878$$ 0 0
$$879$$ 23.0944 0.778953
$$880$$ 0 0
$$881$$ −15.1859 −0.511627 −0.255814 0.966726i $$-0.582343\pi$$
−0.255814 + 0.966726i $$0.582343\pi$$
$$882$$ 0 0
$$883$$ −29.5964 −0.995999 −0.498000 0.867177i $$-0.665932\pi$$
−0.498000 + 0.867177i $$0.665932\pi$$
$$884$$ 0 0
$$885$$ 1.35962 0.0457032
$$886$$ 0 0
$$887$$ −23.2340 −0.780121 −0.390061 0.920789i $$-0.627546\pi$$
−0.390061 + 0.920789i $$0.627546\pi$$
$$888$$ 0 0
$$889$$ −1.59140 −0.0533740
$$890$$ 0 0
$$891$$ 3.20900 0.107506
$$892$$ 0 0
$$893$$ 3.96368 0.132640
$$894$$ 0 0
$$895$$ 3.12575 0.104482
$$896$$ 0 0
$$897$$ −5.23151 −0.174675
$$898$$ 0 0
$$899$$ 6.87384 0.229255
$$900$$ 0 0
$$901$$ −68.5116 −2.28245
$$902$$ 0 0
$$903$$ 2.80438 0.0933240
$$904$$ 0 0
$$905$$ −3.17970 −0.105697
$$906$$ 0 0
$$907$$ 35.4985 1.17871 0.589354 0.807875i $$-0.299382\pi$$
0.589354 + 0.807875i $$0.299382\pi$$
$$908$$ 0 0
$$909$$ 35.3924 1.17389
$$910$$ 0 0
$$911$$ 38.5075 1.27581 0.637905 0.770115i $$-0.279801\pi$$
0.637905 + 0.770115i $$0.279801\pi$$
$$912$$ 0 0
$$913$$ −16.3902 −0.542436
$$914$$ 0 0
$$915$$ 1.51107 0.0499546
$$916$$ 0 0
$$917$$ −21.3911 −0.706395
$$918$$ 0 0
$$919$$ −20.9347 −0.690572 −0.345286 0.938497i $$-0.612218\pi$$
−0.345286 + 0.938497i $$0.612218\pi$$
$$920$$ 0 0
$$921$$ 0.631439 0.0208066
$$922$$ 0 0
$$923$$ 18.4598 0.607610
$$924$$ 0 0
$$925$$ 13.4540 0.442366
$$926$$ 0 0
$$927$$ −20.3468 −0.668276
$$928$$ 0 0
$$929$$ −0.536958 −0.0176170 −0.00880851 0.999961i $$-0.502804\pi$$
−0.00880851 + 0.999961i $$0.502804\pi$$
$$930$$ 0 0
$$931$$ −15.0820 −0.494294
$$932$$ 0 0
$$933$$ −0.946677 −0.0309928
$$934$$ 0 0
$$935$$ 1.56812 0.0512832
$$936$$ 0 0
$$937$$ 34.4984 1.12701 0.563507 0.826112i $$-0.309452\pi$$
0.563507 + 0.826112i $$0.309452\pi$$
$$938$$ 0 0
$$939$$ −9.10430 −0.297107
$$940$$ 0 0
$$941$$ 17.8959 0.583391 0.291695 0.956511i $$-0.405781\pi$$
0.291695 + 0.956511i $$0.405781\pi$$
$$942$$ 0 0
$$943$$ −0.838778 −0.0273144
$$944$$ 0 0
$$945$$ 4.61884 0.150251
$$946$$ 0 0
$$947$$ −16.0979 −0.523111 −0.261556 0.965188i $$-0.584235\pi$$
−0.261556 + 0.965188i $$0.584235\pi$$
$$948$$ 0 0
$$949$$ −38.2316 −1.24105
$$950$$ 0 0
$$951$$ 6.95130 0.225411
$$952$$ 0 0
$$953$$ 29.2828 0.948562 0.474281 0.880374i $$-0.342708\pi$$
0.474281 + 0.880374i $$0.342708\pi$$
$$954$$ 0 0
$$955$$ −0.0876640 −0.00283674
$$956$$ 0 0
$$957$$ −6.36948 −0.205896
$$958$$ 0 0
$$959$$ 3.07448 0.0992802
$$960$$ 0 0
$$961$$ −30.1877 −0.973796
$$962$$ 0 0
$$963$$ −20.8229 −0.671008
$$964$$ 0 0
$$965$$ 2.00517 0.0645488
$$966$$ 0 0
$$967$$ 39.4644 1.26909 0.634544 0.772887i $$-0.281188\pi$$
0.634544 + 0.772887i $$0.281188\pi$$
$$968$$ 0 0
$$969$$ −5.90055 −0.189553
$$970$$ 0 0
$$971$$ −27.0245 −0.867257 −0.433628 0.901092i $$-0.642767\pi$$
−0.433628 + 0.901092i $$0.642767\pi$$
$$972$$ 0 0
$$973$$ 51.9159 1.66435
$$974$$ 0 0
$$975$$ −24.3539 −0.779950
$$976$$ 0 0
$$977$$ −55.2922 −1.76895 −0.884477 0.466584i $$-0.845484\pi$$
−0.884477 + 0.466584i $$0.845484\pi$$
$$978$$ 0 0
$$979$$ −12.9487 −0.413843
$$980$$ 0 0
$$981$$ 42.8149 1.36697
$$982$$ 0 0
$$983$$ −36.0221 −1.14893 −0.574463 0.818531i $$-0.694789\pi$$
−0.574463 + 0.818531i $$0.694789\pi$$
$$984$$ 0 0
$$985$$ 1.74302 0.0555373
$$986$$ 0 0
$$987$$ 15.5557 0.495145
$$988$$ 0 0
$$989$$ 0.759929 0.0241643
$$990$$ 0 0
$$991$$ −50.7944 −1.61354 −0.806768 0.590868i $$-0.798785\pi$$
−0.806768 + 0.590868i $$0.798785\pi$$
$$992$$ 0 0
$$993$$ 2.06675 0.0655863
$$994$$ 0 0
$$995$$ −1.73465 −0.0549921
$$996$$ 0 0
$$997$$ −41.8525 −1.32548 −0.662741 0.748849i $$-0.730607\pi$$
−0.662741 + 0.748849i $$0.730607\pi$$
$$998$$ 0 0
$$999$$ 12.0347 0.380761
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 3344.2.a.ba.1.5 7
4.3 odd 2 209.2.a.d.1.4 7
12.11 even 2 1881.2.a.p.1.4 7
20.19 odd 2 5225.2.a.n.1.4 7
44.43 even 2 2299.2.a.q.1.4 7
76.75 even 2 3971.2.a.i.1.4 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.4 7 4.3 odd 2
1881.2.a.p.1.4 7 12.11 even 2
2299.2.a.q.1.4 7 44.43 even 2
3344.2.a.ba.1.5 7 1.1 even 1 trivial
3971.2.a.i.1.4 7 76.75 even 2
5225.2.a.n.1.4 7 20.19 odd 2