# Properties

 Label 6525.2.a.bn.1.5 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.331312.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 7x^{3} - 4x^{2} + 7x + 4$$ x^5 - 7*x^3 - 4*x^2 + 7*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$2.70559$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.88448 q^{2} +1.55125 q^{4} +3.97545 q^{7} -0.845662 q^{8} +O(q^{10})$$ $$q+1.88448 q^{2} +1.55125 q^{4} +3.97545 q^{7} -0.845662 q^{8} +5.05034 q^{11} -1.97545 q^{13} +7.49164 q^{14} -4.69613 q^{16} -4.61461 q^{17} +4.64986 q^{19} +9.51724 q^{22} -1.20650 q^{23} -3.72269 q^{26} +6.16691 q^{28} +1.00000 q^{29} +6.45387 q^{31} -7.15841 q^{32} -8.69613 q^{34} +4.35033 q^{37} +8.76255 q^{38} +11.2474 q^{41} +7.53991 q^{43} +7.83433 q^{44} -2.27362 q^{46} -6.70001 q^{47} +8.80420 q^{49} -3.06441 q^{52} -3.70001 q^{53} -3.36188 q^{56} +1.88448 q^{58} +12.9218 q^{59} -7.28614 q^{61} +12.1622 q^{62} -4.09760 q^{64} -13.4019 q^{67} -7.15841 q^{68} -6.06103 q^{71} -6.53640 q^{73} +8.19809 q^{74} +7.21309 q^{76} +20.0774 q^{77} +5.30104 q^{79} +21.1954 q^{82} +7.97193 q^{83} +14.2088 q^{86} -4.27088 q^{88} -12.0927 q^{89} -7.85330 q^{91} -1.87158 q^{92} -12.6260 q^{94} +10.1052 q^{97} +16.5913 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} + 8 q^{4} + 4 q^{7} - 18 q^{8}+O(q^{10})$$ 5 * q - 2 * q^2 + 8 * q^4 + 4 * q^7 - 18 * q^8 $$5 q - 2 q^{2} + 8 q^{4} + 4 q^{7} - 18 q^{8} + 5 q^{11} + 6 q^{13} + 8 q^{14} + 14 q^{16} - 14 q^{17} + 2 q^{19} + 8 q^{22} - 13 q^{23} - 12 q^{26} - 4 q^{28} + 5 q^{29} + 2 q^{31} - 18 q^{32} - 6 q^{34} + 17 q^{37} - 14 q^{38} + 19 q^{41} + 7 q^{43} + 22 q^{44} + 30 q^{46} - 18 q^{47} + 9 q^{49} + 20 q^{52} - 3 q^{53} + 16 q^{56} - 2 q^{58} + 22 q^{59} - 8 q^{61} + 4 q^{62} + 8 q^{64} + 10 q^{67} - 18 q^{68} + 18 q^{71} + 19 q^{73} - 2 q^{74} + 52 q^{76} - 8 q^{77} + 16 q^{79} + 22 q^{82} + 3 q^{83} - 26 q^{88} - 2 q^{89} - 36 q^{91} - 48 q^{92} + 2 q^{94} + 5 q^{97} + 6 q^{98}+O(q^{100})$$ 5 * q - 2 * q^2 + 8 * q^4 + 4 * q^7 - 18 * q^8 + 5 * q^11 + 6 * q^13 + 8 * q^14 + 14 * q^16 - 14 * q^17 + 2 * q^19 + 8 * q^22 - 13 * q^23 - 12 * q^26 - 4 * q^28 + 5 * q^29 + 2 * q^31 - 18 * q^32 - 6 * q^34 + 17 * q^37 - 14 * q^38 + 19 * q^41 + 7 * q^43 + 22 * q^44 + 30 * q^46 - 18 * q^47 + 9 * q^49 + 20 * q^52 - 3 * q^53 + 16 * q^56 - 2 * q^58 + 22 * q^59 - 8 * q^61 + 4 * q^62 + 8 * q^64 + 10 * q^67 - 18 * q^68 + 18 * q^71 + 19 * q^73 - 2 * q^74 + 52 * q^76 - 8 * q^77 + 16 * q^79 + 22 * q^82 + 3 * q^83 - 26 * q^88 - 2 * q^89 - 36 * q^91 - 48 * q^92 + 2 * q^94 + 5 * q^97 + 6 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.88448 1.33253 0.666263 0.745717i $$-0.267893\pi$$
0.666263 + 0.745717i $$0.267893\pi$$
$$3$$ 0 0
$$4$$ 1.55125 0.775624
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.97545 1.50258 0.751289 0.659973i $$-0.229432\pi$$
0.751289 + 0.659973i $$0.229432\pi$$
$$8$$ −0.845662 −0.298987
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.05034 1.52273 0.761367 0.648321i $$-0.224529\pi$$
0.761367 + 0.648321i $$0.224529\pi$$
$$12$$ 0 0
$$13$$ −1.97545 −0.547891 −0.273946 0.961745i $$-0.588329\pi$$
−0.273946 + 0.961745i $$0.588329\pi$$
$$14$$ 7.49164 2.00222
$$15$$ 0 0
$$16$$ −4.69613 −1.17403
$$17$$ −4.61461 −1.11921 −0.559604 0.828760i $$-0.689047\pi$$
−0.559604 + 0.828760i $$0.689047\pi$$
$$18$$ 0 0
$$19$$ 4.64986 1.06675 0.533376 0.845879i $$-0.320923\pi$$
0.533376 + 0.845879i $$0.320923\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 9.51724 2.02908
$$23$$ −1.20650 −0.251572 −0.125786 0.992057i $$-0.540145\pi$$
−0.125786 + 0.992057i $$0.540145\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3.72269 −0.730079
$$27$$ 0 0
$$28$$ 6.16691 1.16544
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 6.45387 1.15915 0.579575 0.814919i $$-0.303219\pi$$
0.579575 + 0.814919i $$0.303219\pi$$
$$32$$ −7.15841 −1.26544
$$33$$ 0 0
$$34$$ −8.69613 −1.49137
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.35033 0.715189 0.357595 0.933877i $$-0.383597\pi$$
0.357595 + 0.933877i $$0.383597\pi$$
$$38$$ 8.76255 1.42147
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.2474 1.75654 0.878272 0.478161i $$-0.158696\pi$$
0.878272 + 0.478161i $$0.158696\pi$$
$$42$$ 0 0
$$43$$ 7.53991 1.14983 0.574913 0.818215i $$-0.305036\pi$$
0.574913 + 0.818215i $$0.305036\pi$$
$$44$$ 7.83433 1.18107
$$45$$ 0 0
$$46$$ −2.27362 −0.335226
$$47$$ −6.70001 −0.977297 −0.488648 0.872481i $$-0.662510\pi$$
−0.488648 + 0.872481i $$0.662510\pi$$
$$48$$ 0 0
$$49$$ 8.80420 1.25774
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.06441 −0.424958
$$53$$ −3.70001 −0.508235 −0.254118 0.967173i $$-0.581785\pi$$
−0.254118 + 0.967173i $$0.581785\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.36188 −0.449251
$$57$$ 0 0
$$58$$ 1.88448 0.247444
$$59$$ 12.9218 1.68227 0.841137 0.540823i $$-0.181887\pi$$
0.841137 + 0.540823i $$0.181887\pi$$
$$60$$ 0 0
$$61$$ −7.28614 −0.932894 −0.466447 0.884549i $$-0.654466\pi$$
−0.466447 + 0.884549i $$0.654466\pi$$
$$62$$ 12.1622 1.54460
$$63$$ 0 0
$$64$$ −4.09760 −0.512200
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.4019 −1.63730 −0.818651 0.574291i $$-0.805278\pi$$
−0.818651 + 0.574291i $$0.805278\pi$$
$$68$$ −7.15841 −0.868085
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.06103 −0.719312 −0.359656 0.933085i $$-0.617106\pi$$
−0.359656 + 0.933085i $$0.617106\pi$$
$$72$$ 0 0
$$73$$ −6.53640 −0.765027 −0.382514 0.923950i $$-0.624942\pi$$
−0.382514 + 0.923950i $$0.624942\pi$$
$$74$$ 8.19809 0.953008
$$75$$ 0 0
$$76$$ 7.21309 0.827398
$$77$$ 20.0774 2.28803
$$78$$ 0 0
$$79$$ 5.30104 0.596413 0.298207 0.954501i $$-0.403612\pi$$
0.298207 + 0.954501i $$0.403612\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 21.1954 2.34064
$$83$$ 7.97193 0.875033 0.437517 0.899210i $$-0.355858\pi$$
0.437517 + 0.899210i $$0.355858\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 14.2088 1.53217
$$87$$ 0 0
$$88$$ −4.27088 −0.455277
$$89$$ −12.0927 −1.28183 −0.640913 0.767614i $$-0.721444\pi$$
−0.640913 + 0.767614i $$0.721444\pi$$
$$90$$ 0 0
$$91$$ −7.85330 −0.823250
$$92$$ −1.87158 −0.195126
$$93$$ 0 0
$$94$$ −12.6260 −1.30227
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.1052 1.02603 0.513016 0.858379i $$-0.328528\pi$$
0.513016 + 0.858379i $$0.328528\pi$$
$$98$$ 16.5913 1.67597
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.44029 0.342321 0.171161 0.985243i $$-0.445248\pi$$
0.171161 + 0.985243i $$0.445248\pi$$
$$102$$ 0 0
$$103$$ 6.91278 0.681136 0.340568 0.940220i $$-0.389381\pi$$
0.340568 + 0.940220i $$0.389381\pi$$
$$104$$ 1.67056 0.163812
$$105$$ 0 0
$$106$$ −6.97258 −0.677237
$$107$$ 9.11319 0.881006 0.440503 0.897751i $$-0.354800\pi$$
0.440503 + 0.897751i $$0.354800\pi$$
$$108$$ 0 0
$$109$$ 16.7429 1.60368 0.801839 0.597540i $$-0.203855\pi$$
0.801839 + 0.597540i $$0.203855\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −18.6692 −1.76407
$$113$$ 7.15846 0.673411 0.336706 0.941610i $$-0.390687\pi$$
0.336706 + 0.941610i $$0.390687\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.55125 0.144030
$$117$$ 0 0
$$118$$ 24.3508 2.24167
$$119$$ −18.3452 −1.68170
$$120$$ 0 0
$$121$$ 14.5059 1.31872
$$122$$ −13.7305 −1.24311
$$123$$ 0 0
$$124$$ 10.0116 0.899064
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.05015 0.625600 0.312800 0.949819i $$-0.398733\pi$$
0.312800 + 0.949819i $$0.398733\pi$$
$$128$$ 6.59500 0.582921
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −15.5744 −1.36074 −0.680370 0.732869i $$-0.738181\pi$$
−0.680370 + 0.732869i $$0.738181\pi$$
$$132$$ 0 0
$$133$$ 18.4853 1.60288
$$134$$ −25.2556 −2.18175
$$135$$ 0 0
$$136$$ 3.90240 0.334628
$$137$$ −18.9602 −1.61988 −0.809941 0.586512i $$-0.800501\pi$$
−0.809941 + 0.586512i $$0.800501\pi$$
$$138$$ 0 0
$$139$$ −5.78101 −0.490339 −0.245170 0.969480i $$-0.578844\pi$$
−0.245170 + 0.969480i $$0.578844\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −11.4219 −0.958502
$$143$$ −9.97669 −0.834292
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −12.3177 −1.01942
$$147$$ 0 0
$$148$$ 6.74844 0.554718
$$149$$ −0.928816 −0.0760916 −0.0380458 0.999276i $$-0.512113\pi$$
−0.0380458 + 0.999276i $$0.512113\pi$$
$$150$$ 0 0
$$151$$ 1.48920 0.121189 0.0605947 0.998162i $$-0.480700\pi$$
0.0605947 + 0.998162i $$0.480700\pi$$
$$152$$ −3.93221 −0.318944
$$153$$ 0 0
$$154$$ 37.8353 3.04885
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.3479 0.985466 0.492733 0.870180i $$-0.335998\pi$$
0.492733 + 0.870180i $$0.335998\pi$$
$$158$$ 9.98968 0.794736
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.79637 −0.378007
$$162$$ 0 0
$$163$$ −13.9493 −1.09259 −0.546295 0.837593i $$-0.683962\pi$$
−0.546295 + 0.837593i $$0.683962\pi$$
$$164$$ 17.4475 1.36242
$$165$$ 0 0
$$166$$ 15.0229 1.16600
$$167$$ 11.2844 0.873217 0.436608 0.899652i $$-0.356180\pi$$
0.436608 + 0.899652i $$0.356180\pi$$
$$168$$ 0 0
$$169$$ −9.09760 −0.699815
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 11.6963 0.891833
$$173$$ −4.72255 −0.359049 −0.179524 0.983754i $$-0.557456\pi$$
−0.179524 + 0.983754i $$0.557456\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −23.7170 −1.78774
$$177$$ 0 0
$$178$$ −22.7884 −1.70807
$$179$$ −9.36954 −0.700312 −0.350156 0.936691i $$-0.613871\pi$$
−0.350156 + 0.936691i $$0.613871\pi$$
$$180$$ 0 0
$$181$$ 12.9621 0.963462 0.481731 0.876319i $$-0.340008\pi$$
0.481731 + 0.876319i $$0.340008\pi$$
$$182$$ −14.7994 −1.09700
$$183$$ 0 0
$$184$$ 1.02029 0.0752167
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −23.3053 −1.70426
$$188$$ −10.3934 −0.758015
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2.69237 −0.194813 −0.0974066 0.995245i $$-0.531055\pi$$
−0.0974066 + 0.995245i $$0.531055\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 19.0431 1.36721
$$195$$ 0 0
$$196$$ 13.6575 0.975536
$$197$$ −10.2277 −0.728695 −0.364347 0.931263i $$-0.618708\pi$$
−0.364347 + 0.931263i $$0.618708\pi$$
$$198$$ 0 0
$$199$$ −3.39810 −0.240885 −0.120442 0.992720i $$-0.538431\pi$$
−0.120442 + 0.992720i $$0.538431\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.48314 0.456152
$$203$$ 3.97545 0.279022
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.0270 0.907632
$$207$$ 0 0
$$208$$ 9.27696 0.643241
$$209$$ 23.4834 1.62438
$$210$$ 0 0
$$211$$ 16.5171 1.13708 0.568540 0.822655i $$-0.307508\pi$$
0.568540 + 0.822655i $$0.307508\pi$$
$$212$$ −5.73963 −0.394200
$$213$$ 0 0
$$214$$ 17.1736 1.17396
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 25.6570 1.74171
$$218$$ 31.5516 2.13694
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.11593 0.613204
$$222$$ 0 0
$$223$$ 3.48895 0.233637 0.116819 0.993153i $$-0.462730\pi$$
0.116819 + 0.993153i $$0.462730\pi$$
$$224$$ −28.4579 −1.90142
$$225$$ 0 0
$$226$$ 13.4899 0.897338
$$227$$ −24.1163 −1.60066 −0.800328 0.599563i $$-0.795341\pi$$
−0.800328 + 0.599563i $$0.795341\pi$$
$$228$$ 0 0
$$229$$ −25.7198 −1.69961 −0.849805 0.527097i $$-0.823280\pi$$
−0.849805 + 0.527097i $$0.823280\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −0.845662 −0.0555204
$$233$$ 19.5181 1.27867 0.639337 0.768927i $$-0.279209\pi$$
0.639337 + 0.768927i $$0.279209\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 20.0449 1.30481
$$237$$ 0 0
$$238$$ −34.5710 −2.24091
$$239$$ −4.35357 −0.281609 −0.140805 0.990037i $$-0.544969\pi$$
−0.140805 + 0.990037i $$0.544969\pi$$
$$240$$ 0 0
$$241$$ 13.3619 0.860716 0.430358 0.902658i $$-0.358387\pi$$
0.430358 + 0.902658i $$0.358387\pi$$
$$242$$ 27.3360 1.75723
$$243$$ 0 0
$$244$$ −11.3026 −0.723575
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.18556 −0.584463
$$248$$ −5.45779 −0.346570
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5.34451 0.337343 0.168671 0.985672i $$-0.446052\pi$$
0.168671 + 0.985672i $$0.446052\pi$$
$$252$$ 0 0
$$253$$ −6.09322 −0.383078
$$254$$ 13.2858 0.833627
$$255$$ 0 0
$$256$$ 20.6233 1.28896
$$257$$ −11.5812 −0.722418 −0.361209 0.932485i $$-0.617636\pi$$
−0.361209 + 0.932485i $$0.617636\pi$$
$$258$$ 0 0
$$259$$ 17.2945 1.07463
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −29.3496 −1.81322
$$263$$ −15.2957 −0.943173 −0.471587 0.881820i $$-0.656319\pi$$
−0.471587 + 0.881820i $$0.656319\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 34.8351 2.13588
$$267$$ 0 0
$$268$$ −20.7897 −1.26993
$$269$$ −6.70294 −0.408685 −0.204343 0.978899i $$-0.565506\pi$$
−0.204343 + 0.978899i $$0.565506\pi$$
$$270$$ 0 0
$$271$$ 0.201885 0.0122637 0.00613183 0.999981i $$-0.498048\pi$$
0.00613183 + 0.999981i $$0.498048\pi$$
$$272$$ 21.6708 1.31399
$$273$$ 0 0
$$274$$ −35.7301 −2.15853
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.4754 0.809659 0.404829 0.914392i $$-0.367331\pi$$
0.404829 + 0.914392i $$0.367331\pi$$
$$278$$ −10.8942 −0.653390
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 24.5512 1.46460 0.732301 0.680981i $$-0.238446\pi$$
0.732301 + 0.680981i $$0.238446\pi$$
$$282$$ 0 0
$$283$$ −10.7258 −0.637582 −0.318791 0.947825i $$-0.603277\pi$$
−0.318791 + 0.947825i $$0.603277\pi$$
$$284$$ −9.40217 −0.557916
$$285$$ 0 0
$$286$$ −18.8008 −1.11172
$$287$$ 44.7134 2.63935
$$288$$ 0 0
$$289$$ 4.29465 0.252627
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −10.1396 −0.593374
$$293$$ −26.8072 −1.56609 −0.783046 0.621964i $$-0.786335\pi$$
−0.783046 + 0.621964i $$0.786335\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −3.67890 −0.213832
$$297$$ 0 0
$$298$$ −1.75033 −0.101394
$$299$$ 2.38338 0.137834
$$300$$ 0 0
$$301$$ 29.9745 1.72770
$$302$$ 2.80636 0.161488
$$303$$ 0 0
$$304$$ −21.8363 −1.25240
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 17.2029 0.981819 0.490909 0.871211i $$-0.336665\pi$$
0.490909 + 0.871211i $$0.336665\pi$$
$$308$$ 31.1450 1.77465
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.23228 0.296696 0.148348 0.988935i $$-0.452604\pi$$
0.148348 + 0.988935i $$0.452604\pi$$
$$312$$ 0 0
$$313$$ 21.0916 1.19217 0.596083 0.802923i $$-0.296723\pi$$
0.596083 + 0.802923i $$0.296723\pi$$
$$314$$ 23.2692 1.31316
$$315$$ 0 0
$$316$$ 8.22323 0.462593
$$317$$ −28.6493 −1.60910 −0.804551 0.593883i $$-0.797594\pi$$
−0.804551 + 0.593883i $$0.797594\pi$$
$$318$$ 0 0
$$319$$ 5.05034 0.282765
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −9.03865 −0.503704
$$323$$ −21.4573 −1.19392
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −26.2870 −1.45590
$$327$$ 0 0
$$328$$ −9.51147 −0.525183
$$329$$ −26.6356 −1.46847
$$330$$ 0 0
$$331$$ −30.3560 −1.66852 −0.834260 0.551372i $$-0.814105\pi$$
−0.834260 + 0.551372i $$0.814105\pi$$
$$332$$ 12.3664 0.678697
$$333$$ 0 0
$$334$$ 21.2653 1.16358
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.7033 1.12778 0.563890 0.825850i $$-0.309304\pi$$
0.563890 + 0.825850i $$0.309304\pi$$
$$338$$ −17.1442 −0.932522
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 32.5942 1.76508
$$342$$ 0 0
$$343$$ 7.17250 0.387279
$$344$$ −6.37621 −0.343782
$$345$$ 0 0
$$346$$ −8.89953 −0.478442
$$347$$ −13.6308 −0.731737 −0.365869 0.930667i $$-0.619228\pi$$
−0.365869 + 0.930667i $$0.619228\pi$$
$$348$$ 0 0
$$349$$ 9.39303 0.502797 0.251399 0.967884i $$-0.419110\pi$$
0.251399 + 0.967884i $$0.419110\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −36.1524 −1.92693
$$353$$ 10.7225 0.570701 0.285351 0.958423i $$-0.407890\pi$$
0.285351 + 0.958423i $$0.407890\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −18.7588 −0.994215
$$357$$ 0 0
$$358$$ −17.6567 −0.933184
$$359$$ 5.62043 0.296635 0.148317 0.988940i $$-0.452614\pi$$
0.148317 + 0.988940i $$0.452614\pi$$
$$360$$ 0 0
$$361$$ 2.62120 0.137958
$$362$$ 24.4267 1.28384
$$363$$ 0 0
$$364$$ −12.1824 −0.638532
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 12.4996 0.652476 0.326238 0.945288i $$-0.394219\pi$$
0.326238 + 0.945288i $$0.394219\pi$$
$$368$$ 5.66587 0.295354
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −14.7092 −0.763664
$$372$$ 0 0
$$373$$ 20.5248 1.06274 0.531368 0.847141i $$-0.321678\pi$$
0.531368 + 0.847141i $$0.321678\pi$$
$$374$$ −43.9184 −2.27096
$$375$$ 0 0
$$376$$ 5.66594 0.292199
$$377$$ −1.97545 −0.101741
$$378$$ 0 0
$$379$$ 18.6409 0.957516 0.478758 0.877947i $$-0.341087\pi$$
0.478758 + 0.877947i $$0.341087\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −5.07371 −0.259594
$$383$$ −16.9354 −0.865356 −0.432678 0.901548i $$-0.642431\pi$$
−0.432678 + 0.901548i $$0.642431\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 26.3827 1.34284
$$387$$ 0 0
$$388$$ 15.6757 0.795815
$$389$$ −23.2880 −1.18075 −0.590374 0.807130i $$-0.701020\pi$$
−0.590374 + 0.807130i $$0.701020\pi$$
$$390$$ 0 0
$$391$$ 5.56752 0.281562
$$392$$ −7.44537 −0.376048
$$393$$ 0 0
$$394$$ −19.2739 −0.971005
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.20635 0.411865 0.205933 0.978566i $$-0.433977\pi$$
0.205933 + 0.978566i $$0.433977\pi$$
$$398$$ −6.40363 −0.320985
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 20.0238 0.999942 0.499971 0.866042i $$-0.333344\pi$$
0.499971 + 0.866042i $$0.333344\pi$$
$$402$$ 0 0
$$403$$ −12.7493 −0.635088
$$404$$ 5.33674 0.265513
$$405$$ 0 0
$$406$$ 7.49164 0.371804
$$407$$ 21.9706 1.08904
$$408$$ 0 0
$$409$$ 27.2928 1.34954 0.674771 0.738027i $$-0.264242\pi$$
0.674771 + 0.738027i $$0.264242\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 10.7234 0.528306
$$413$$ 51.3699 2.52775
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 14.1411 0.693323
$$417$$ 0 0
$$418$$ 44.2538 2.16453
$$419$$ 34.5488 1.68782 0.843909 0.536486i $$-0.180249\pi$$
0.843909 + 0.536486i $$0.180249\pi$$
$$420$$ 0 0
$$421$$ −26.1250 −1.27325 −0.636626 0.771173i $$-0.719671\pi$$
−0.636626 + 0.771173i $$0.719671\pi$$
$$422$$ 31.1260 1.51519
$$423$$ 0 0
$$424$$ 3.12896 0.151956
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −28.9657 −1.40175
$$428$$ 14.1368 0.683329
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −30.4575 −1.46709 −0.733544 0.679642i $$-0.762135\pi$$
−0.733544 + 0.679642i $$0.762135\pi$$
$$432$$ 0 0
$$433$$ 15.9802 0.767957 0.383979 0.923342i $$-0.374554\pi$$
0.383979 + 0.923342i $$0.374554\pi$$
$$434$$ 48.3501 2.32088
$$435$$ 0 0
$$436$$ 25.9724 1.24385
$$437$$ −5.61005 −0.268365
$$438$$ 0 0
$$439$$ 19.2547 0.918976 0.459488 0.888184i $$-0.348033\pi$$
0.459488 + 0.888184i $$0.348033\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 17.1788 0.817110
$$443$$ −12.2199 −0.580585 −0.290293 0.956938i $$-0.593753\pi$$
−0.290293 + 0.956938i $$0.593753\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 6.57484 0.311328
$$447$$ 0 0
$$448$$ −16.2898 −0.769621
$$449$$ 14.4104 0.680069 0.340035 0.940413i $$-0.389561\pi$$
0.340035 + 0.940413i $$0.389561\pi$$
$$450$$ 0 0
$$451$$ 56.8030 2.67475
$$452$$ 11.1046 0.522314
$$453$$ 0 0
$$454$$ −45.4466 −2.13291
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.5395 −0.820463 −0.410232 0.911981i $$-0.634552\pi$$
−0.410232 + 0.911981i $$0.634552\pi$$
$$458$$ −48.4683 −2.26477
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.02126 −0.280438 −0.140219 0.990121i $$-0.544781\pi$$
−0.140219 + 0.990121i $$0.544781\pi$$
$$462$$ 0 0
$$463$$ −23.0161 −1.06965 −0.534824 0.844964i $$-0.679622\pi$$
−0.534824 + 0.844964i $$0.679622\pi$$
$$464$$ −4.69613 −0.218012
$$465$$ 0 0
$$466$$ 36.7814 1.70386
$$467$$ −9.45324 −0.437444 −0.218722 0.975787i $$-0.570189\pi$$
−0.218722 + 0.975787i $$0.570189\pi$$
$$468$$ 0 0
$$469$$ −53.2786 −2.46018
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −10.9275 −0.502977
$$473$$ 38.0791 1.75088
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −28.4579 −1.30437
$$477$$ 0 0
$$478$$ −8.20420 −0.375252
$$479$$ −40.2452 −1.83885 −0.919424 0.393268i $$-0.871344\pi$$
−0.919424 + 0.393268i $$0.871344\pi$$
$$480$$ 0 0
$$481$$ −8.59385 −0.391846
$$482$$ 25.1802 1.14693
$$483$$ 0 0
$$484$$ 22.5023 1.02283
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −27.3621 −1.23989 −0.619947 0.784644i $$-0.712846\pi$$
−0.619947 + 0.784644i $$0.712846\pi$$
$$488$$ 6.16161 0.278923
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 16.1708 0.729778 0.364889 0.931051i $$-0.381107\pi$$
0.364889 + 0.931051i $$0.381107\pi$$
$$492$$ 0 0
$$493$$ −4.61461 −0.207832
$$494$$ −17.3100 −0.778812
$$495$$ 0 0
$$496$$ −30.3082 −1.36088
$$497$$ −24.0953 −1.08082
$$498$$ 0 0
$$499$$ −24.2809 −1.08696 −0.543482 0.839421i $$-0.682894\pi$$
−0.543482 + 0.839421i $$0.682894\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 10.0716 0.449518
$$503$$ 24.4131 1.08853 0.544263 0.838915i $$-0.316809\pi$$
0.544263 + 0.838915i $$0.316809\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −11.4825 −0.510461
$$507$$ 0 0
$$508$$ 10.9365 0.485230
$$509$$ −33.6547 −1.49172 −0.745859 0.666103i $$-0.767961\pi$$
−0.745859 + 0.666103i $$0.767961\pi$$
$$510$$ 0 0
$$511$$ −25.9851 −1.14951
$$512$$ 25.6741 1.13465
$$513$$ 0 0
$$514$$ −21.8246 −0.962641
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −33.8373 −1.48816
$$518$$ 32.5911 1.43197
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −8.83636 −0.387128 −0.193564 0.981088i $$-0.562005\pi$$
−0.193564 + 0.981088i $$0.562005\pi$$
$$522$$ 0 0
$$523$$ 9.71669 0.424881 0.212441 0.977174i $$-0.431859\pi$$
0.212441 + 0.977174i $$0.431859\pi$$
$$524$$ −24.1597 −1.05542
$$525$$ 0 0
$$526$$ −28.8244 −1.25680
$$527$$ −29.7821 −1.29733
$$528$$ 0 0
$$529$$ −21.5444 −0.936711
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 28.6753 1.24323
$$533$$ −22.2186 −0.962395
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 11.3335 0.489531
$$537$$ 0 0
$$538$$ −12.6315 −0.544584
$$539$$ 44.4642 1.91521
$$540$$ 0 0
$$541$$ −11.4439 −0.492012 −0.246006 0.969268i $$-0.579118\pi$$
−0.246006 + 0.969268i $$0.579118\pi$$
$$542$$ 0.380448 0.0163416
$$543$$ 0 0
$$544$$ 33.0333 1.41629
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −15.6211 −0.667908 −0.333954 0.942589i $$-0.608383\pi$$
−0.333954 + 0.942589i $$0.608383\pi$$
$$548$$ −29.4120 −1.25642
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.64986 0.198091
$$552$$ 0 0
$$553$$ 21.0740 0.896158
$$554$$ 25.3941 1.07889
$$555$$ 0 0
$$556$$ −8.96779 −0.380319
$$557$$ 18.9932 0.804766 0.402383 0.915471i $$-0.368182\pi$$
0.402383 + 0.915471i $$0.368182\pi$$
$$558$$ 0 0
$$559$$ −14.8947 −0.629979
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 46.2661 1.95162
$$563$$ −20.7203 −0.873257 −0.436629 0.899642i $$-0.643828\pi$$
−0.436629 + 0.899642i $$0.643828\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −20.2125 −0.849595
$$567$$ 0 0
$$568$$ 5.12558 0.215065
$$569$$ 23.0163 0.964892 0.482446 0.875926i $$-0.339748\pi$$
0.482446 + 0.875926i $$0.339748\pi$$
$$570$$ 0 0
$$571$$ −18.1927 −0.761341 −0.380670 0.924711i $$-0.624307\pi$$
−0.380670 + 0.924711i $$0.624307\pi$$
$$572$$ −15.4763 −0.647097
$$573$$ 0 0
$$574$$ 84.2612 3.51700
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.4435 −0.434771 −0.217385 0.976086i $$-0.569753\pi$$
−0.217385 + 0.976086i $$0.569753\pi$$
$$578$$ 8.09317 0.336631
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 31.6920 1.31481
$$582$$ 0 0
$$583$$ −18.6863 −0.773907
$$584$$ 5.52758 0.228733
$$585$$ 0 0
$$586$$ −50.5175 −2.08686
$$587$$ −20.7562 −0.856700 −0.428350 0.903613i $$-0.640905\pi$$
−0.428350 + 0.903613i $$0.640905\pi$$
$$588$$ 0 0
$$589$$ 30.0096 1.23652
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −20.4297 −0.839655
$$593$$ −2.43875 −0.100147 −0.0500737 0.998746i $$-0.515946\pi$$
−0.0500737 + 0.998746i $$0.515946\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.44082 −0.0590185
$$597$$ 0 0
$$598$$ 4.49141 0.183668
$$599$$ 10.9851 0.448840 0.224420 0.974493i $$-0.427951\pi$$
0.224420 + 0.974493i $$0.427951\pi$$
$$600$$ 0 0
$$601$$ 16.7056 0.681435 0.340717 0.940166i $$-0.389330\pi$$
0.340717 + 0.940166i $$0.389330\pi$$
$$602$$ 56.4863 2.30221
$$603$$ 0 0
$$604$$ 2.31012 0.0939975
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −47.0785 −1.91086 −0.955429 0.295222i $$-0.904606\pi$$
−0.955429 + 0.295222i $$0.904606\pi$$
$$608$$ −33.2856 −1.34991
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 13.2355 0.535452
$$612$$ 0 0
$$613$$ −0.285402 −0.0115273 −0.00576364 0.999983i $$-0.501835\pi$$
−0.00576364 + 0.999983i $$0.501835\pi$$
$$614$$ 32.4184 1.30830
$$615$$ 0 0
$$616$$ −16.9787 −0.684089
$$617$$ −45.4926 −1.83146 −0.915731 0.401792i $$-0.868387\pi$$
−0.915731 + 0.401792i $$0.868387\pi$$
$$618$$ 0 0
$$619$$ −37.4525 −1.50534 −0.752672 0.658396i $$-0.771235\pi$$
−0.752672 + 0.658396i $$0.771235\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 9.86011 0.395354
$$623$$ −48.0740 −1.92604
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 39.7466 1.58859
$$627$$ 0 0
$$628$$ 19.1546 0.764352
$$629$$ −20.0751 −0.800446
$$630$$ 0 0
$$631$$ −12.1278 −0.482799 −0.241400 0.970426i $$-0.577606\pi$$
−0.241400 + 0.970426i $$0.577606\pi$$
$$632$$ −4.48288 −0.178320
$$633$$ 0 0
$$634$$ −53.9888 −2.14417
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −17.3923 −0.689106
$$638$$ 9.51724 0.376791
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.98414 −0.0783687 −0.0391843 0.999232i $$-0.512476\pi$$
−0.0391843 + 0.999232i $$0.512476\pi$$
$$642$$ 0 0
$$643$$ −38.6749 −1.52519 −0.762595 0.646876i $$-0.776075\pi$$
−0.762595 + 0.646876i $$0.776075\pi$$
$$644$$ −7.44037 −0.293191
$$645$$ 0 0
$$646$$ −40.4358 −1.59092
$$647$$ −27.6995 −1.08898 −0.544489 0.838768i $$-0.683277\pi$$
−0.544489 + 0.838768i $$0.683277\pi$$
$$648$$ 0 0
$$649$$ 65.2594 2.56165
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −21.6388 −0.847440
$$653$$ −34.8231 −1.36273 −0.681366 0.731942i $$-0.738614\pi$$
−0.681366 + 0.731942i $$0.738614\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −52.8191 −2.06224
$$657$$ 0 0
$$658$$ −50.1940 −1.95677
$$659$$ −7.37091 −0.287130 −0.143565 0.989641i $$-0.545857\pi$$
−0.143565 + 0.989641i $$0.545857\pi$$
$$660$$ 0 0
$$661$$ 25.7621 1.00203 0.501015 0.865439i $$-0.332960\pi$$
0.501015 + 0.865439i $$0.332960\pi$$
$$662$$ −57.2052 −2.22334
$$663$$ 0 0
$$664$$ −6.74156 −0.261623
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.20650 −0.0467158
$$668$$ 17.5050 0.677288
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −36.7974 −1.42055
$$672$$ 0 0
$$673$$ −39.0838 −1.50657 −0.753285 0.657694i $$-0.771532\pi$$
−0.753285 + 0.657694i $$0.771532\pi$$
$$674$$ 39.0149 1.50280
$$675$$ 0 0
$$676$$ −14.1126 −0.542794
$$677$$ 40.1309 1.54236 0.771178 0.636619i $$-0.219668\pi$$
0.771178 + 0.636619i $$0.219668\pi$$
$$678$$ 0 0
$$679$$ 40.1729 1.54169
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 61.4230 2.35201
$$683$$ 21.7747 0.833185 0.416592 0.909093i $$-0.363224\pi$$
0.416592 + 0.909093i $$0.363224\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13.5164 0.516059
$$687$$ 0 0
$$688$$ −35.4084 −1.34993
$$689$$ 7.30918 0.278458
$$690$$ 0 0
$$691$$ 27.2866 1.03803 0.519016 0.854765i $$-0.326299\pi$$
0.519016 + 0.854765i $$0.326299\pi$$
$$692$$ −7.32585 −0.278487
$$693$$ 0 0
$$694$$ −25.6868 −0.975058
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −51.9023 −1.96594
$$698$$ 17.7009 0.669990
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 9.80694 0.370403 0.185202 0.982701i $$-0.440706\pi$$
0.185202 + 0.982701i $$0.440706\pi$$
$$702$$ 0 0
$$703$$ 20.2284 0.762929
$$704$$ −20.6943 −0.779944
$$705$$ 0 0
$$706$$ 20.2063 0.760474
$$707$$ 13.6767 0.514365
$$708$$ 0 0
$$709$$ 34.6952 1.30301 0.651503 0.758646i $$-0.274139\pi$$
0.651503 + 0.758646i $$0.274139\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.2263 0.383249
$$713$$ −7.78659 −0.291610
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −14.5345 −0.543179
$$717$$ 0 0
$$718$$ 10.5916 0.395273
$$719$$ 0.619226 0.0230932 0.0115466 0.999933i $$-0.496325\pi$$
0.0115466 + 0.999933i $$0.496325\pi$$
$$720$$ 0 0
$$721$$ 27.4814 1.02346
$$722$$ 4.93959 0.183833
$$723$$ 0 0
$$724$$ 20.1074 0.747284
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.82966 0.216210 0.108105 0.994139i $$-0.465522\pi$$
0.108105 + 0.994139i $$0.465522\pi$$
$$728$$ 6.64123 0.246141
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −34.7938 −1.28689
$$732$$ 0 0
$$733$$ 4.63787 0.171304 0.0856519 0.996325i $$-0.472703\pi$$
0.0856519 + 0.996325i $$0.472703\pi$$
$$734$$ 23.5553 0.869441
$$735$$ 0 0
$$736$$ 8.63661 0.318350
$$737$$ −67.6841 −2.49318
$$738$$ 0 0
$$739$$ 21.0686 0.775022 0.387511 0.921865i $$-0.373335\pi$$
0.387511 + 0.921865i $$0.373335\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −27.7191 −1.01760
$$743$$ −38.5779 −1.41529 −0.707643 0.706571i $$-0.750241\pi$$
−0.707643 + 0.706571i $$0.750241\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 38.6785 1.41612
$$747$$ 0 0
$$748$$ −36.1524 −1.32186
$$749$$ 36.2290 1.32378
$$750$$ 0 0
$$751$$ −41.4212 −1.51148 −0.755740 0.654872i $$-0.772722\pi$$
−0.755740 + 0.654872i $$0.772722\pi$$
$$752$$ 31.4641 1.14738
$$753$$ 0 0
$$754$$ −3.72269 −0.135572
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.7076 −0.570901 −0.285450 0.958393i $$-0.592143\pi$$
−0.285450 + 0.958393i $$0.592143\pi$$
$$758$$ 35.1282 1.27592
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15.1038 −0.547513 −0.273756 0.961799i $$-0.588266\pi$$
−0.273756 + 0.961799i $$0.588266\pi$$
$$762$$ 0 0
$$763$$ 66.5605 2.40965
$$764$$ −4.17654 −0.151102
$$765$$ 0 0
$$766$$ −31.9143 −1.15311
$$767$$ −25.5263 −0.921702
$$768$$ 0 0
$$769$$ −10.4395 −0.376457 −0.188228 0.982125i $$-0.560275\pi$$
−0.188228 + 0.982125i $$0.560275\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 21.7175 0.781629
$$773$$ −23.0481 −0.828982 −0.414491 0.910053i $$-0.636040\pi$$
−0.414491 + 0.910053i $$0.636040\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −8.54561 −0.306770
$$777$$ 0 0
$$778$$ −43.8856 −1.57338
$$779$$ 52.2987 1.87380
$$780$$ 0 0
$$781$$ −30.6103 −1.09532
$$782$$ 10.4919 0.375188
$$783$$ 0 0
$$784$$ −41.3456 −1.47663
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −0.591462 −0.0210833 −0.0105417 0.999944i $$-0.503356\pi$$
−0.0105417 + 0.999944i $$0.503356\pi$$
$$788$$ −15.8657 −0.565193
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 28.4581 1.01185
$$792$$ 0 0
$$793$$ 14.3934 0.511124
$$794$$ 15.4647 0.548821
$$795$$ 0 0
$$796$$ −5.27129 −0.186836
$$797$$ 25.4764 0.902419 0.451210 0.892418i $$-0.350993\pi$$
0.451210 + 0.892418i $$0.350993\pi$$
$$798$$ 0 0
$$799$$ 30.9180 1.09380
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 37.7344 1.33245
$$803$$ −33.0110 −1.16493
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −24.0257 −0.846271
$$807$$ 0 0
$$808$$ −2.90932 −0.102349
$$809$$ 8.55103 0.300638 0.150319 0.988638i $$-0.451970\pi$$
0.150319 + 0.988638i $$0.451970\pi$$
$$810$$ 0 0
$$811$$ −19.9128 −0.699232 −0.349616 0.936893i $$-0.613688\pi$$
−0.349616 + 0.936893i $$0.613688\pi$$
$$812$$ 6.16691 0.216416
$$813$$ 0 0
$$814$$ 41.4031 1.45118
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 35.0595 1.22658
$$818$$ 51.4326 1.79830
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −38.1575 −1.33170 −0.665852 0.746084i $$-0.731932\pi$$
−0.665852 + 0.746084i $$0.731932\pi$$
$$822$$ 0 0
$$823$$ −27.8681 −0.971421 −0.485710 0.874120i $$-0.661439\pi$$
−0.485710 + 0.874120i $$0.661439\pi$$
$$824$$ −5.84587 −0.203651
$$825$$ 0 0
$$826$$ 96.8053 3.36829
$$827$$ 19.0910 0.663858 0.331929 0.943304i $$-0.392300\pi$$
0.331929 + 0.943304i $$0.392300\pi$$
$$828$$ 0 0
$$829$$ −20.6808 −0.718274 −0.359137 0.933285i $$-0.616929\pi$$
−0.359137 + 0.933285i $$0.616929\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.09460 0.280630
$$833$$ −40.6280 −1.40768
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 36.4285 1.25991
$$837$$ 0 0
$$838$$ 65.1063 2.24906
$$839$$ 18.3865 0.634772 0.317386 0.948296i $$-0.397195\pi$$
0.317386 + 0.948296i $$0.397195\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −49.2318 −1.69664
$$843$$ 0 0
$$844$$ 25.6221 0.881947
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 57.6675 1.98148
$$848$$ 17.3757 0.596684
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5.24866 −0.179922
$$852$$ 0 0
$$853$$ −28.4263 −0.973297 −0.486648 0.873598i $$-0.661781\pi$$
−0.486648 + 0.873598i $$0.661781\pi$$
$$854$$ −54.5851 −1.86786
$$855$$ 0 0
$$856$$ −7.70668 −0.263409
$$857$$ 22.6832 0.774842 0.387421 0.921903i $$-0.373366\pi$$
0.387421 + 0.921903i $$0.373366\pi$$
$$858$$ 0 0
$$859$$ −17.3179 −0.590878 −0.295439 0.955362i $$-0.595466\pi$$
−0.295439 + 0.955362i $$0.595466\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −57.3965 −1.95493
$$863$$ −25.9340 −0.882805 −0.441403 0.897309i $$-0.645519\pi$$
−0.441403 + 0.897309i $$0.645519\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 30.1142 1.02332
$$867$$ 0 0
$$868$$ 39.8004 1.35092
$$869$$ 26.7720 0.908179
$$870$$ 0 0
$$871$$ 26.4748 0.897064
$$872$$ −14.1588 −0.479478
$$873$$ 0 0
$$874$$ −10.5720 −0.357603
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.52354 0.287819 0.143910 0.989591i $$-0.454033\pi$$
0.143910 + 0.989591i $$0.454033\pi$$
$$878$$ 36.2850 1.22456
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −48.4352 −1.63182 −0.815912 0.578177i $$-0.803764\pi$$
−0.815912 + 0.578177i $$0.803764\pi$$
$$882$$ 0 0
$$883$$ 5.87273 0.197633 0.0988166 0.995106i $$-0.468494\pi$$
0.0988166 + 0.995106i $$0.468494\pi$$
$$884$$ 14.1411 0.475616
$$885$$ 0 0
$$886$$ −23.0281 −0.773644
$$887$$ 29.9614 1.00601 0.503003 0.864285i $$-0.332229\pi$$
0.503003 + 0.864285i $$0.332229\pi$$
$$888$$ 0 0
$$889$$ 28.0275 0.940013
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 5.41222 0.181215
$$893$$ −31.1541 −1.04253
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 26.2181 0.875884
$$897$$ 0 0
$$898$$ 27.1561 0.906210
$$899$$ 6.45387 0.215249
$$900$$ 0 0
$$901$$ 17.0741 0.568821
$$902$$ 107.044 3.56417
$$903$$ 0 0
$$904$$ −6.05364 −0.201341
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −41.3847 −1.37416 −0.687078 0.726584i $$-0.741107\pi$$
−0.687078 + 0.726584i $$0.741107\pi$$
$$908$$ −37.4104 −1.24151
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −15.1439 −0.501739 −0.250870 0.968021i $$-0.580717\pi$$
−0.250870 + 0.968021i $$0.580717\pi$$
$$912$$ 0 0
$$913$$ 40.2610 1.33244
$$914$$ −33.0528 −1.09329
$$915$$ 0 0
$$916$$ −39.8977 −1.31826
$$917$$ −61.9152 −2.04462
$$918$$ 0 0
$$919$$ −1.12139 −0.0369911 −0.0184956 0.999829i $$-0.505888\pi$$
−0.0184956 + 0.999829i $$0.505888\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −11.3469 −0.373691
$$923$$ 11.9733 0.394105
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −43.3732 −1.42533
$$927$$ 0 0
$$928$$ −7.15841 −0.234986
$$929$$ −3.38813 −0.111161 −0.0555804 0.998454i $$-0.517701\pi$$
−0.0555804 + 0.998454i $$0.517701\pi$$
$$930$$ 0 0
$$931$$ 40.9383 1.34170
$$932$$ 30.2774 0.991770
$$933$$ 0 0
$$934$$ −17.8144 −0.582905
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 17.9125 0.585175 0.292587 0.956239i $$-0.405484\pi$$
0.292587 + 0.956239i $$0.405484\pi$$
$$938$$ −100.402 −3.27825
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 26.2459 0.855592 0.427796 0.903875i $$-0.359290\pi$$
0.427796 + 0.903875i $$0.359290\pi$$
$$942$$ 0 0
$$943$$ −13.5699 −0.441898
$$944$$ −60.6823 −1.97504
$$945$$ 0 0
$$946$$ 71.7591 2.33309
$$947$$ −24.9518 −0.810825 −0.405413 0.914134i $$-0.632872\pi$$
−0.405413 + 0.914134i $$0.632872\pi$$
$$948$$ 0 0
$$949$$ 12.9123 0.419152
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 15.5138 0.502805
$$953$$ −34.5021 −1.11763 −0.558817 0.829291i $$-0.688745\pi$$
−0.558817 + 0.829291i $$0.688745\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −6.75347 −0.218423
$$957$$ 0 0
$$958$$ −75.8410 −2.45031
$$959$$ −75.3754 −2.43400
$$960$$ 0 0
$$961$$ 10.6525 0.343628
$$962$$ −16.1949 −0.522145
$$963$$ 0 0
$$964$$ 20.7276 0.667592
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −34.8694 −1.12132 −0.560661 0.828045i $$-0.689453\pi$$
−0.560661 + 0.828045i $$0.689453\pi$$
$$968$$ −12.2671 −0.394279
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 50.6681 1.62602 0.813008 0.582252i $$-0.197828\pi$$
0.813008 + 0.582252i $$0.197828\pi$$
$$972$$ 0 0
$$973$$ −22.9821 −0.736773
$$974$$ −51.5632 −1.65219
$$975$$ 0 0
$$976$$ 34.2166 1.09525
$$977$$ −20.0017 −0.639911 −0.319956 0.947433i $$-0.603668\pi$$
−0.319956 + 0.947433i $$0.603668\pi$$
$$978$$ 0 0
$$979$$ −61.0723 −1.95188
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 30.4735 0.972447
$$983$$ −8.51776 −0.271674 −0.135837 0.990731i $$-0.543372\pi$$
−0.135837 + 0.990731i $$0.543372\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.69613 −0.276941
$$987$$ 0 0
$$988$$ −14.2491 −0.453324
$$989$$ −9.09689 −0.289264
$$990$$ 0 0
$$991$$ 41.0770 1.30485 0.652427 0.757852i $$-0.273751\pi$$
0.652427 + 0.757852i $$0.273751\pi$$
$$992$$ −46.1995 −1.46683
$$993$$ 0 0
$$994$$ −45.4071 −1.44022
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 28.8974 0.915189 0.457594 0.889161i $$-0.348711\pi$$
0.457594 + 0.889161i $$0.348711\pi$$
$$998$$ −45.7568 −1.44841
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6525.2.a.bn.1.5 5
3.2 odd 2 2175.2.a.y.1.1 5
5.2 odd 4 1305.2.c.i.784.8 10
5.3 odd 4 1305.2.c.i.784.3 10
5.4 even 2 6525.2.a.br.1.1 5
15.2 even 4 435.2.c.d.349.3 10
15.8 even 4 435.2.c.d.349.8 yes 10
15.14 odd 2 2175.2.a.x.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.d.349.3 10 15.2 even 4
435.2.c.d.349.8 yes 10 15.8 even 4
1305.2.c.i.784.3 10 5.3 odd 4
1305.2.c.i.784.8 10 5.2 odd 4
2175.2.a.x.1.5 5 15.14 odd 2
2175.2.a.y.1.1 5 3.2 odd 2
6525.2.a.bn.1.5 5 1.1 even 1 trivial
6525.2.a.br.1.1 5 5.4 even 2