# Properties

 Label 2200.2.a.l.1.1 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} -4.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} -4.56155 q^{7} +3.56155 q^{9} -1.00000 q^{11} +1.12311 q^{13} +7.68466 q^{17} +1.43845 q^{19} +11.6847 q^{21} -1.12311 q^{23} -1.43845 q^{27} +8.56155 q^{29} -1.43845 q^{31} +2.56155 q^{33} -7.43845 q^{37} -2.87689 q^{39} -12.2462 q^{41} +3.12311 q^{43} +11.3693 q^{47} +13.8078 q^{49} -19.6847 q^{51} -9.68466 q^{53} -3.68466 q^{57} +1.12311 q^{59} -12.5616 q^{61} -16.2462 q^{63} +2.87689 q^{69} +3.68466 q^{71} -1.12311 q^{73} +4.56155 q^{77} -11.3693 q^{79} -7.00000 q^{81} +6.00000 q^{83} -21.9309 q^{87} +9.68466 q^{89} -5.12311 q^{91} +3.68466 q^{93} +4.87689 q^{97} -3.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 5 * q^7 + 3 * q^9 $$2 q - q^{3} - 5 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{17} + 7 q^{19} + 11 q^{21} + 6 q^{23} - 7 q^{27} + 13 q^{29} - 7 q^{31} + q^{33} - 19 q^{37} - 14 q^{39} - 8 q^{41} - 2 q^{43} - 2 q^{47} + 7 q^{49} - 27 q^{51} - 7 q^{53} + 5 q^{57} - 6 q^{59} - 21 q^{61} - 16 q^{63} + 14 q^{69} - 5 q^{71} + 6 q^{73} + 5 q^{77} + 2 q^{79} - 14 q^{81} + 12 q^{83} - 15 q^{87} + 7 q^{89} - 2 q^{91} - 5 q^{93} + 18 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q - q^3 - 5 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^17 + 7 * q^19 + 11 * q^21 + 6 * q^23 - 7 * q^27 + 13 * q^29 - 7 * q^31 + q^33 - 19 * q^37 - 14 * q^39 - 8 * q^41 - 2 * q^43 - 2 * q^47 + 7 * q^49 - 27 * q^51 - 7 * q^53 + 5 * q^57 - 6 * q^59 - 21 * q^61 - 16 * q^63 + 14 * q^69 - 5 * q^71 + 6 * q^73 + 5 * q^77 + 2 * q^79 - 14 * q^81 + 12 * q^83 - 15 * q^87 + 7 * q^89 - 2 * q^91 - 5 * q^93 + 18 * q^97 - 3 * 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
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.56155 −1.72410 −0.862052 0.506819i $$-0.830821\pi$$
−0.862052 + 0.506819i $$0.830821\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 1.12311 0.311493 0.155747 0.987797i $$-0.450222\pi$$
0.155747 + 0.987797i $$0.450222\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.68466 1.86380 0.931902 0.362711i $$-0.118149\pi$$
0.931902 + 0.362711i $$0.118149\pi$$
$$18$$ 0 0
$$19$$ 1.43845 0.330002 0.165001 0.986293i $$-0.447237\pi$$
0.165001 + 0.986293i $$0.447237\pi$$
$$20$$ 0 0
$$21$$ 11.6847 2.54980
$$22$$ 0 0
$$23$$ −1.12311 −0.234184 −0.117092 0.993121i $$-0.537357\pi$$
−0.117092 + 0.993121i $$0.537357\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ 8.56155 1.58984 0.794920 0.606714i $$-0.207513\pi$$
0.794920 + 0.606714i $$0.207513\pi$$
$$30$$ 0 0
$$31$$ −1.43845 −0.258353 −0.129176 0.991622i $$-0.541233\pi$$
−0.129176 + 0.991622i $$0.541233\pi$$
$$32$$ 0 0
$$33$$ 2.56155 0.445909
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.43845 −1.22287 −0.611437 0.791293i $$-0.709408\pi$$
−0.611437 + 0.791293i $$0.709408\pi$$
$$38$$ 0 0
$$39$$ −2.87689 −0.460672
$$40$$ 0 0
$$41$$ −12.2462 −1.91254 −0.956268 0.292490i $$-0.905516\pi$$
−0.956268 + 0.292490i $$0.905516\pi$$
$$42$$ 0 0
$$43$$ 3.12311 0.476269 0.238135 0.971232i $$-0.423464\pi$$
0.238135 + 0.971232i $$0.423464\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.3693 1.65839 0.829193 0.558963i $$-0.188801\pi$$
0.829193 + 0.558963i $$0.188801\pi$$
$$48$$ 0 0
$$49$$ 13.8078 1.97254
$$50$$ 0 0
$$51$$ −19.6847 −2.75640
$$52$$ 0 0
$$53$$ −9.68466 −1.33029 −0.665145 0.746714i $$-0.731630\pi$$
−0.665145 + 0.746714i $$0.731630\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.68466 −0.488045
$$58$$ 0 0
$$59$$ 1.12311 0.146216 0.0731079 0.997324i $$-0.476708\pi$$
0.0731079 + 0.997324i $$0.476708\pi$$
$$60$$ 0 0
$$61$$ −12.5616 −1.60834 −0.804171 0.594398i $$-0.797390\pi$$
−0.804171 + 0.594398i $$0.797390\pi$$
$$62$$ 0 0
$$63$$ −16.2462 −2.04683
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 2.87689 0.346337
$$70$$ 0 0
$$71$$ 3.68466 0.437289 0.218644 0.975805i $$-0.429837\pi$$
0.218644 + 0.975805i $$0.429837\pi$$
$$72$$ 0 0
$$73$$ −1.12311 −0.131450 −0.0657248 0.997838i $$-0.520936\pi$$
−0.0657248 + 0.997838i $$0.520936\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.56155 0.519837
$$78$$ 0 0
$$79$$ −11.3693 −1.27915 −0.639574 0.768729i $$-0.720889\pi$$
−0.639574 + 0.768729i $$0.720889\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −21.9309 −2.35124
$$88$$ 0 0
$$89$$ 9.68466 1.02657 0.513286 0.858218i $$-0.328428\pi$$
0.513286 + 0.858218i $$0.328428\pi$$
$$90$$ 0 0
$$91$$ −5.12311 −0.537047
$$92$$ 0 0
$$93$$ 3.68466 0.382081
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.87689 0.495174 0.247587 0.968866i $$-0.420362\pi$$
0.247587 + 0.968866i $$0.420362\pi$$
$$98$$ 0 0
$$99$$ −3.56155 −0.357950
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.12311 −0.688617 −0.344308 0.938857i $$-0.611887\pi$$
−0.344308 + 0.938857i $$0.611887\pi$$
$$108$$ 0 0
$$109$$ −4.24621 −0.406713 −0.203357 0.979105i $$-0.565185\pi$$
−0.203357 + 0.979105i $$0.565185\pi$$
$$110$$ 0 0
$$111$$ 19.0540 1.80852
$$112$$ 0 0
$$113$$ −0.876894 −0.0824913 −0.0412456 0.999149i $$-0.513133\pi$$
−0.0412456 + 0.999149i $$0.513133\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ −35.0540 −3.21339
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 31.3693 2.82848
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.87689 −0.432754 −0.216377 0.976310i $$-0.569424\pi$$
−0.216377 + 0.976310i $$0.569424\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −16.8078 −1.46850 −0.734251 0.678879i $$-0.762466\pi$$
−0.734251 + 0.678879i $$0.762466\pi$$
$$132$$ 0 0
$$133$$ −6.56155 −0.568959
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 7.12311 0.608568 0.304284 0.952581i $$-0.401583\pi$$
0.304284 + 0.952581i $$0.401583\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −29.1231 −2.45261
$$142$$ 0 0
$$143$$ −1.12311 −0.0939188
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −35.3693 −2.91721
$$148$$ 0 0
$$149$$ 1.68466 0.138013 0.0690063 0.997616i $$-0.478017\pi$$
0.0690063 + 0.997616i $$0.478017\pi$$
$$150$$ 0 0
$$151$$ 6.87689 0.559634 0.279817 0.960053i $$-0.409726\pi$$
0.279817 + 0.960053i $$0.409726\pi$$
$$152$$ 0 0
$$153$$ 27.3693 2.21268
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −21.6847 −1.73062 −0.865312 0.501233i $$-0.832880\pi$$
−0.865312 + 0.501233i $$0.832880\pi$$
$$158$$ 0 0
$$159$$ 24.8078 1.96738
$$160$$ 0 0
$$161$$ 5.12311 0.403757
$$162$$ 0 0
$$163$$ −20.8078 −1.62979 −0.814895 0.579609i $$-0.803205\pi$$
−0.814895 + 0.579609i $$0.803205\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.68466 0.130363 0.0651814 0.997873i $$-0.479237\pi$$
0.0651814 + 0.997873i $$0.479237\pi$$
$$168$$ 0 0
$$169$$ −11.7386 −0.902972
$$170$$ 0 0
$$171$$ 5.12311 0.391774
$$172$$ 0 0
$$173$$ −10.8769 −0.826955 −0.413477 0.910514i $$-0.635686\pi$$
−0.413477 + 0.910514i $$0.635686\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2.87689 −0.216241
$$178$$ 0 0
$$179$$ 14.2462 1.06481 0.532406 0.846489i $$-0.321288\pi$$
0.532406 + 0.846489i $$0.321288\pi$$
$$180$$ 0 0
$$181$$ −8.24621 −0.612936 −0.306468 0.951881i $$-0.599147\pi$$
−0.306468 + 0.951881i $$0.599147\pi$$
$$182$$ 0 0
$$183$$ 32.1771 2.37860
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −7.68466 −0.561958
$$188$$ 0 0
$$189$$ 6.56155 0.477283
$$190$$ 0 0
$$191$$ 10.2462 0.741390 0.370695 0.928755i $$-0.379120\pi$$
0.370695 + 0.928755i $$0.379120\pi$$
$$192$$ 0 0
$$193$$ 21.4384 1.54317 0.771587 0.636124i $$-0.219463\pi$$
0.771587 + 0.636124i $$0.219463\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −22.7386 −1.62006 −0.810030 0.586388i $$-0.800549\pi$$
−0.810030 + 0.586388i $$0.800549\pi$$
$$198$$ 0 0
$$199$$ −4.31534 −0.305906 −0.152953 0.988233i $$-0.548878\pi$$
−0.152953 + 0.988233i $$0.548878\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −39.0540 −2.74105
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −1.43845 −0.0994995
$$210$$ 0 0
$$211$$ 19.6847 1.35515 0.677574 0.735455i $$-0.263031\pi$$
0.677574 + 0.735455i $$0.263031\pi$$
$$212$$ 0 0
$$213$$ −9.43845 −0.646712
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6.56155 0.445427
$$218$$ 0 0
$$219$$ 2.87689 0.194403
$$220$$ 0 0
$$221$$ 8.63068 0.580563
$$222$$ 0 0
$$223$$ −25.1231 −1.68237 −0.841184 0.540749i $$-0.818141\pi$$
−0.841184 + 0.540749i $$0.818141\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −8.24621 −0.547320 −0.273660 0.961826i $$-0.588234\pi$$
−0.273660 + 0.961826i $$0.588234\pi$$
$$228$$ 0 0
$$229$$ 16.2462 1.07358 0.536790 0.843716i $$-0.319637\pi$$
0.536790 + 0.843716i $$0.319637\pi$$
$$230$$ 0 0
$$231$$ −11.6847 −0.768794
$$232$$ 0 0
$$233$$ −27.0540 −1.77236 −0.886182 0.463336i $$-0.846652\pi$$
−0.886182 + 0.463336i $$0.846652\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 29.1231 1.89175
$$238$$ 0 0
$$239$$ −21.6155 −1.39819 −0.699096 0.715028i $$-0.746414\pi$$
−0.699096 + 0.715028i $$0.746414\pi$$
$$240$$ 0 0
$$241$$ 18.4924 1.19120 0.595601 0.803281i $$-0.296914\pi$$
0.595601 + 0.803281i $$0.296914\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.61553 0.102794
$$248$$ 0 0
$$249$$ −15.3693 −0.973991
$$250$$ 0 0
$$251$$ −27.3693 −1.72754 −0.863768 0.503890i $$-0.831902\pi$$
−0.863768 + 0.503890i $$0.831902\pi$$
$$252$$ 0 0
$$253$$ 1.12311 0.0706090
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.49242 −0.404986 −0.202493 0.979284i $$-0.564904\pi$$
−0.202493 + 0.979284i $$0.564904\pi$$
$$258$$ 0 0
$$259$$ 33.9309 2.10836
$$260$$ 0 0
$$261$$ 30.4924 1.88743
$$262$$ 0 0
$$263$$ 29.0540 1.79154 0.895772 0.444513i $$-0.146623\pi$$
0.895772 + 0.444513i $$0.146623\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −24.8078 −1.51821
$$268$$ 0 0
$$269$$ 0.876894 0.0534652 0.0267326 0.999643i $$-0.491490\pi$$
0.0267326 + 0.999643i $$0.491490\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ 13.1231 0.794246
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.2462 1.33665 0.668323 0.743872i $$-0.267013\pi$$
0.668323 + 0.743872i $$0.267013\pi$$
$$278$$ 0 0
$$279$$ −5.12311 −0.306712
$$280$$ 0 0
$$281$$ 8.24621 0.491928 0.245964 0.969279i $$-0.420896\pi$$
0.245964 + 0.969279i $$0.420896\pi$$
$$282$$ 0 0
$$283$$ 6.00000 0.356663 0.178331 0.983970i $$-0.442930\pi$$
0.178331 + 0.983970i $$0.442930\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 55.8617 3.29741
$$288$$ 0 0
$$289$$ 42.0540 2.47376
$$290$$ 0 0
$$291$$ −12.4924 −0.732319
$$292$$ 0 0
$$293$$ −17.1231 −1.00034 −0.500171 0.865927i $$-0.666730\pi$$
−0.500171 + 0.865927i $$0.666730\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.43845 0.0834672
$$298$$ 0 0
$$299$$ −1.26137 −0.0729467
$$300$$ 0 0
$$301$$ −14.2462 −0.821138
$$302$$ 0 0
$$303$$ 5.12311 0.294315
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.43845 −0.535205 −0.267603 0.963529i $$-0.586231\pi$$
−0.267603 + 0.963529i $$0.586231\pi$$
$$312$$ 0 0
$$313$$ −24.7386 −1.39831 −0.699155 0.714970i $$-0.746440\pi$$
−0.699155 + 0.714970i $$0.746440\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.31534 −0.354705 −0.177352 0.984147i $$-0.556753\pi$$
−0.177352 + 0.984147i $$0.556753\pi$$
$$318$$ 0 0
$$319$$ −8.56155 −0.479355
$$320$$ 0 0
$$321$$ 18.2462 1.01840
$$322$$ 0 0
$$323$$ 11.0540 0.615060
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.8769 0.601494
$$328$$ 0 0
$$329$$ −51.8617 −2.85923
$$330$$ 0 0
$$331$$ −18.7386 −1.02997 −0.514984 0.857200i $$-0.672202\pi$$
−0.514984 + 0.857200i $$0.672202\pi$$
$$332$$ 0 0
$$333$$ −26.4924 −1.45178
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.80776 0.261896 0.130948 0.991389i $$-0.458198\pi$$
0.130948 + 0.991389i $$0.458198\pi$$
$$338$$ 0 0
$$339$$ 2.24621 0.121997
$$340$$ 0 0
$$341$$ 1.43845 0.0778963
$$342$$ 0 0
$$343$$ −31.0540 −1.67676
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 14.0000 0.751559 0.375780 0.926709i $$-0.377375\pi$$
0.375780 + 0.926709i $$0.377375\pi$$
$$348$$ 0 0
$$349$$ 14.4924 0.775762 0.387881 0.921710i $$-0.373207\pi$$
0.387881 + 0.921710i $$0.373207\pi$$
$$350$$ 0 0
$$351$$ −1.61553 −0.0862305
$$352$$ 0 0
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 89.7926 4.75233
$$358$$ 0 0
$$359$$ 18.7386 0.988987 0.494494 0.869181i $$-0.335354\pi$$
0.494494 + 0.869181i $$0.335354\pi$$
$$360$$ 0 0
$$361$$ −16.9309 −0.891098
$$362$$ 0 0
$$363$$ −2.56155 −0.134447
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −16.4924 −0.860897 −0.430449 0.902615i $$-0.641645\pi$$
−0.430449 + 0.902615i $$0.641645\pi$$
$$368$$ 0 0
$$369$$ −43.6155 −2.27053
$$370$$ 0 0
$$371$$ 44.1771 2.29356
$$372$$ 0 0
$$373$$ −23.3693 −1.21002 −0.605009 0.796219i $$-0.706830\pi$$
−0.605009 + 0.796219i $$0.706830\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.61553 0.495225
$$378$$ 0 0
$$379$$ −14.8769 −0.764175 −0.382087 0.924126i $$-0.624795\pi$$
−0.382087 + 0.924126i $$0.624795\pi$$
$$380$$ 0 0
$$381$$ 12.4924 0.640006
$$382$$ 0 0
$$383$$ 0.630683 0.0322264 0.0161132 0.999870i $$-0.494871\pi$$
0.0161132 + 0.999870i $$0.494871\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.1231 0.565419
$$388$$ 0 0
$$389$$ −14.4924 −0.734795 −0.367397 0.930064i $$-0.619751\pi$$
−0.367397 + 0.930064i $$0.619751\pi$$
$$390$$ 0 0
$$391$$ −8.63068 −0.436472
$$392$$ 0 0
$$393$$ 43.0540 2.17179
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 16.8078 0.841441
$$400$$ 0 0
$$401$$ 30.8078 1.53847 0.769233 0.638968i $$-0.220638\pi$$
0.769233 + 0.638968i $$0.220638\pi$$
$$402$$ 0 0
$$403$$ −1.61553 −0.0804752
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.43845 0.368710
$$408$$ 0 0
$$409$$ 17.3693 0.858857 0.429429 0.903101i $$-0.358715\pi$$
0.429429 + 0.903101i $$0.358715\pi$$
$$410$$ 0 0
$$411$$ −18.2462 −0.900019
$$412$$ 0 0
$$413$$ −5.12311 −0.252092
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 10.2462 0.501759
$$418$$ 0 0
$$419$$ 0.492423 0.0240564 0.0120282 0.999928i $$-0.496171\pi$$
0.0120282 + 0.999928i $$0.496171\pi$$
$$420$$ 0 0
$$421$$ 27.6155 1.34590 0.672949 0.739689i $$-0.265027\pi$$
0.672949 + 0.739689i $$0.265027\pi$$
$$422$$ 0 0
$$423$$ 40.4924 1.96881
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 57.3002 2.77295
$$428$$ 0 0
$$429$$ 2.87689 0.138898
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 4.87689 0.234369 0.117184 0.993110i $$-0.462613\pi$$
0.117184 + 0.993110i $$0.462613\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.61553 −0.0772812
$$438$$ 0 0
$$439$$ 10.2462 0.489025 0.244512 0.969646i $$-0.421372\pi$$
0.244512 + 0.969646i $$0.421372\pi$$
$$440$$ 0 0
$$441$$ 49.1771 2.34177
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.31534 −0.204109
$$448$$ 0 0
$$449$$ 0.246211 0.0116194 0.00580971 0.999983i $$-0.498151\pi$$
0.00580971 + 0.999983i $$0.498151\pi$$
$$450$$ 0 0
$$451$$ 12.2462 0.576652
$$452$$ 0 0
$$453$$ −17.6155 −0.827650
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.68466 −0.172361 −0.0861805 0.996280i $$-0.527466\pi$$
−0.0861805 + 0.996280i $$0.527466\pi$$
$$458$$ 0 0
$$459$$ −11.0540 −0.515955
$$460$$ 0 0
$$461$$ −21.0540 −0.980581 −0.490291 0.871559i $$-0.663109\pi$$
−0.490291 + 0.871559i $$0.663109\pi$$
$$462$$ 0 0
$$463$$ 27.3693 1.27196 0.635980 0.771706i $$-0.280596\pi$$
0.635980 + 0.771706i $$0.280596\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −32.8078 −1.51816 −0.759081 0.650996i $$-0.774351\pi$$
−0.759081 + 0.650996i $$0.774351\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 55.5464 2.55944
$$472$$ 0 0
$$473$$ −3.12311 −0.143601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −34.4924 −1.57930
$$478$$ 0 0
$$479$$ 14.2462 0.650926 0.325463 0.945555i $$-0.394480\pi$$
0.325463 + 0.945555i $$0.394480\pi$$
$$480$$ 0 0
$$481$$ −8.35416 −0.380917
$$482$$ 0 0
$$483$$ −13.1231 −0.597122
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 30.2462 1.37059 0.685293 0.728267i $$-0.259674\pi$$
0.685293 + 0.728267i $$0.259674\pi$$
$$488$$ 0 0
$$489$$ 53.3002 2.41032
$$490$$ 0 0
$$491$$ −16.3153 −0.736301 −0.368151 0.929766i $$-0.620009\pi$$
−0.368151 + 0.929766i $$0.620009\pi$$
$$492$$ 0 0
$$493$$ 65.7926 2.96315
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −16.8078 −0.753931
$$498$$ 0 0
$$499$$ −18.7386 −0.838856 −0.419428 0.907789i $$-0.637769\pi$$
−0.419428 + 0.907789i $$0.637769\pi$$
$$500$$ 0 0
$$501$$ −4.31534 −0.192795
$$502$$ 0 0
$$503$$ 3.12311 0.139252 0.0696262 0.997573i $$-0.477819\pi$$
0.0696262 + 0.997573i $$0.477819\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 30.0691 1.33542
$$508$$ 0 0
$$509$$ −0.384472 −0.0170414 −0.00852071 0.999964i $$-0.502712\pi$$
−0.00852071 + 0.999964i $$0.502712\pi$$
$$510$$ 0 0
$$511$$ 5.12311 0.226633
$$512$$ 0 0
$$513$$ −2.06913 −0.0913543
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −11.3693 −0.500022
$$518$$ 0 0
$$519$$ 27.8617 1.22299
$$520$$ 0 0
$$521$$ −30.4924 −1.33590 −0.667949 0.744207i $$-0.732827\pi$$
−0.667949 + 0.744207i $$0.732827\pi$$
$$522$$ 0 0
$$523$$ −24.8769 −1.08779 −0.543895 0.839153i $$-0.683051\pi$$
−0.543895 + 0.839153i $$0.683051\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.0540 −0.481519
$$528$$ 0 0
$$529$$ −21.7386 −0.945158
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −13.7538 −0.595743
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −36.4924 −1.57476
$$538$$ 0 0
$$539$$ −13.8078 −0.594743
$$540$$ 0 0
$$541$$ −7.93087 −0.340975 −0.170487 0.985360i $$-0.554534\pi$$
−0.170487 + 0.985360i $$0.554534\pi$$
$$542$$ 0 0
$$543$$ 21.1231 0.906479
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10.6307 0.454535 0.227268 0.973832i $$-0.427021\pi$$
0.227268 + 0.973832i $$0.427021\pi$$
$$548$$ 0 0
$$549$$ −44.7386 −1.90940
$$550$$ 0 0
$$551$$ 12.3153 0.524651
$$552$$ 0 0
$$553$$ 51.8617 2.20539
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16.6307 −0.704665 −0.352332 0.935875i $$-0.614611\pi$$
−0.352332 + 0.935875i $$0.614611\pi$$
$$558$$ 0 0
$$559$$ 3.50758 0.148355
$$560$$ 0 0
$$561$$ 19.6847 0.831087
$$562$$ 0 0
$$563$$ 42.9848 1.81160 0.905798 0.423711i $$-0.139273\pi$$
0.905798 + 0.423711i $$0.139273\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 31.9309 1.34097
$$568$$ 0 0
$$569$$ 3.61553 0.151571 0.0757854 0.997124i $$-0.475854\pi$$
0.0757854 + 0.997124i $$0.475854\pi$$
$$570$$ 0 0
$$571$$ 25.3002 1.05878 0.529390 0.848379i $$-0.322421\pi$$
0.529390 + 0.848379i $$0.322421\pi$$
$$572$$ 0 0
$$573$$ −26.2462 −1.09645
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15.7538 0.655839 0.327919 0.944706i $$-0.393653\pi$$
0.327919 + 0.944706i $$0.393653\pi$$
$$578$$ 0 0
$$579$$ −54.9157 −2.28222
$$580$$ 0 0
$$581$$ −27.3693 −1.13547
$$582$$ 0 0
$$583$$ 9.68466 0.401098
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 3.05398 0.126051 0.0630255 0.998012i $$-0.479925\pi$$
0.0630255 + 0.998012i $$0.479925\pi$$
$$588$$ 0 0
$$589$$ −2.06913 −0.0852570
$$590$$ 0 0
$$591$$ 58.2462 2.39593
$$592$$ 0 0
$$593$$ −21.6155 −0.887643 −0.443822 0.896115i $$-0.646378\pi$$
−0.443822 + 0.896115i $$0.646378\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.0540 0.452409
$$598$$ 0 0
$$599$$ −29.9309 −1.22294 −0.611471 0.791267i $$-0.709422\pi$$
−0.611471 + 0.791267i $$0.709422\pi$$
$$600$$ 0 0
$$601$$ −26.9848 −1.10073 −0.550367 0.834923i $$-0.685512\pi$$
−0.550367 + 0.834923i $$0.685512\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.8078 0.601029 0.300514 0.953777i $$-0.402842\pi$$
0.300514 + 0.953777i $$0.402842\pi$$
$$608$$ 0 0
$$609$$ 100.039 4.05378
$$610$$ 0 0
$$611$$ 12.7689 0.516576
$$612$$ 0 0
$$613$$ 2.87689 0.116197 0.0580983 0.998311i $$-0.481496\pi$$
0.0580983 + 0.998311i $$0.481496\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.6155 1.11176 0.555880 0.831263i $$-0.312382\pi$$
0.555880 + 0.831263i $$0.312382\pi$$
$$618$$ 0 0
$$619$$ −34.7386 −1.39626 −0.698132 0.715969i $$-0.745985\pi$$
−0.698132 + 0.715969i $$0.745985\pi$$
$$620$$ 0 0
$$621$$ 1.61553 0.0648289
$$622$$ 0 0
$$623$$ −44.1771 −1.76992
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3.68466 0.147151
$$628$$ 0 0
$$629$$ −57.1619 −2.27920
$$630$$ 0 0
$$631$$ −32.8078 −1.30606 −0.653028 0.757334i $$-0.726502\pi$$
−0.653028 + 0.757334i $$0.726502\pi$$
$$632$$ 0 0
$$633$$ −50.4233 −2.00415
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 15.5076 0.614433
$$638$$ 0 0
$$639$$ 13.1231 0.519142
$$640$$ 0 0
$$641$$ 20.5616 0.812133 0.406066 0.913844i $$-0.366900\pi$$
0.406066 + 0.913844i $$0.366900\pi$$
$$642$$ 0 0
$$643$$ −25.3002 −0.997742 −0.498871 0.866676i $$-0.666252\pi$$
−0.498871 + 0.866676i $$0.666252\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −45.1231 −1.77397 −0.886986 0.461796i $$-0.847205\pi$$
−0.886986 + 0.461796i $$0.847205\pi$$
$$648$$ 0 0
$$649$$ −1.12311 −0.0440858
$$650$$ 0 0
$$651$$ −16.8078 −0.658748
$$652$$ 0 0
$$653$$ −28.5616 −1.11770 −0.558850 0.829269i $$-0.688757\pi$$
−0.558850 + 0.829269i $$0.688757\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ −33.9309 −1.32176 −0.660880 0.750492i $$-0.729817\pi$$
−0.660880 + 0.750492i $$0.729817\pi$$
$$660$$ 0 0
$$661$$ 2.49242 0.0969440 0.0484720 0.998825i $$-0.484565\pi$$
0.0484720 + 0.998825i $$0.484565\pi$$
$$662$$ 0 0
$$663$$ −22.1080 −0.858602
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.61553 −0.372315
$$668$$ 0 0
$$669$$ 64.3542 2.48808
$$670$$ 0 0
$$671$$ 12.5616 0.484933
$$672$$ 0 0
$$673$$ 41.9309 1.61632 0.808158 0.588966i $$-0.200465\pi$$
0.808158 + 0.588966i $$0.200465\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −46.2462 −1.77739 −0.888693 0.458502i $$-0.848386\pi$$
−0.888693 + 0.458502i $$0.848386\pi$$
$$678$$ 0 0
$$679$$ −22.2462 −0.853731
$$680$$ 0 0
$$681$$ 21.1231 0.809439
$$682$$ 0 0
$$683$$ −40.1771 −1.53733 −0.768667 0.639650i $$-0.779079\pi$$
−0.768667 + 0.639650i $$0.779079\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −41.6155 −1.58773
$$688$$ 0 0
$$689$$ −10.8769 −0.414377
$$690$$ 0 0
$$691$$ 5.61553 0.213625 0.106812 0.994279i $$-0.465936\pi$$
0.106812 + 0.994279i $$0.465936\pi$$
$$692$$ 0 0
$$693$$ 16.2462 0.617143
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −94.1080 −3.56459
$$698$$ 0 0
$$699$$ 69.3002 2.62117
$$700$$ 0 0
$$701$$ −20.5616 −0.776599 −0.388300 0.921533i $$-0.626937\pi$$
−0.388300 + 0.921533i $$0.626937\pi$$
$$702$$ 0 0
$$703$$ −10.6998 −0.403551
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 9.12311 0.343110
$$708$$ 0 0
$$709$$ −32.2462 −1.21103 −0.605516 0.795833i $$-0.707033\pi$$
−0.605516 + 0.795833i $$0.707033\pi$$
$$710$$ 0 0
$$711$$ −40.4924 −1.51858
$$712$$ 0 0
$$713$$ 1.61553 0.0605020
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 55.3693 2.06781
$$718$$ 0 0
$$719$$ −48.1771 −1.79670 −0.898351 0.439278i $$-0.855234\pi$$
−0.898351 + 0.439278i $$0.855234\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −47.3693 −1.76168
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 10.8769 0.403402 0.201701 0.979447i $$-0.435353\pi$$
0.201701 + 0.979447i $$0.435353\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 13.7538 0.508008 0.254004 0.967203i $$-0.418252\pi$$
0.254004 + 0.967203i $$0.418252\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 22.2462 0.818340 0.409170 0.912458i $$-0.365818\pi$$
0.409170 + 0.912458i $$0.365818\pi$$
$$740$$ 0 0
$$741$$ −4.13826 −0.152023
$$742$$ 0 0
$$743$$ −20.5616 −0.754330 −0.377165 0.926146i $$-0.623101\pi$$
−0.377165 + 0.926146i $$0.623101\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 21.3693 0.781862
$$748$$ 0 0
$$749$$ 32.4924 1.18725
$$750$$ 0 0
$$751$$ −31.1922 −1.13822 −0.569110 0.822261i $$-0.692712\pi$$
−0.569110 + 0.822261i $$0.692712\pi$$
$$752$$ 0 0
$$753$$ 70.1080 2.55488
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 0 0
$$759$$ −2.87689 −0.104425
$$760$$ 0 0
$$761$$ 7.75379 0.281075 0.140537 0.990075i $$-0.455117\pi$$
0.140537 + 0.990075i $$0.455117\pi$$
$$762$$ 0 0
$$763$$ 19.3693 0.701216
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.26137 0.0455453
$$768$$ 0 0
$$769$$ 3.75379 0.135365 0.0676825 0.997707i $$-0.478439\pi$$
0.0676825 + 0.997707i $$0.478439\pi$$
$$770$$ 0 0
$$771$$ 16.6307 0.598939
$$772$$ 0 0
$$773$$ −43.9309 −1.58008 −0.790042 0.613053i $$-0.789941\pi$$
−0.790042 + 0.613053i $$0.789941\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −86.9157 −3.11808
$$778$$ 0 0
$$779$$ −17.6155 −0.631142
$$780$$ 0 0
$$781$$ −3.68466 −0.131847
$$782$$ 0 0
$$783$$ −12.3153 −0.440114
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17.8617 0.636702 0.318351 0.947973i $$-0.396871\pi$$
0.318351 + 0.947973i $$0.396871\pi$$
$$788$$ 0 0
$$789$$ −74.4233 −2.64954
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ −14.1080 −0.500988
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −20.2462 −0.717158 −0.358579 0.933499i $$-0.616739\pi$$
−0.358579 + 0.933499i $$0.616739\pi$$
$$798$$ 0 0
$$799$$ 87.3693 3.09090
$$800$$ 0 0
$$801$$ 34.4924 1.21873
$$802$$ 0 0
$$803$$ 1.12311 0.0396335
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −2.24621 −0.0790704
$$808$$ 0 0
$$809$$ −40.7386 −1.43229 −0.716147 0.697949i $$-0.754096\pi$$
−0.716147 + 0.697949i $$0.754096\pi$$
$$810$$ 0 0
$$811$$ −1.93087 −0.0678020 −0.0339010 0.999425i $$-0.510793\pi$$
−0.0339010 + 0.999425i $$0.510793\pi$$
$$812$$ 0 0
$$813$$ 10.2462 0.359350
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.49242 0.157170
$$818$$ 0 0
$$819$$ −18.2462 −0.637574
$$820$$ 0 0
$$821$$ 14.4924 0.505789 0.252895 0.967494i $$-0.418617\pi$$
0.252895 + 0.967494i $$0.418617\pi$$
$$822$$ 0 0
$$823$$ −12.4924 −0.435458 −0.217729 0.976009i $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −29.8617 −1.03839 −0.519197 0.854654i $$-0.673769\pi$$
−0.519197 + 0.854654i $$0.673769\pi$$
$$828$$ 0 0
$$829$$ −32.8769 −1.14186 −0.570931 0.820998i $$-0.693418\pi$$
−0.570931 + 0.820998i $$0.693418\pi$$
$$830$$ 0 0
$$831$$ −56.9848 −1.97678
$$832$$ 0 0
$$833$$ 106.108 3.67642
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.06913 0.0715196
$$838$$ 0 0
$$839$$ 19.5076 0.673476 0.336738 0.941598i $$-0.390676\pi$$
0.336738 + 0.941598i $$0.390676\pi$$
$$840$$ 0 0
$$841$$ 44.3002 1.52759
$$842$$ 0 0
$$843$$ −21.1231 −0.727518
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4.56155 −0.156737
$$848$$ 0 0
$$849$$ −15.3693 −0.527474
$$850$$ 0 0
$$851$$ 8.35416 0.286377
$$852$$ 0 0
$$853$$ 37.1231 1.27107 0.635535 0.772072i $$-0.280779\pi$$
0.635535 + 0.772072i $$0.280779\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12.8078 −0.437505 −0.218752 0.975780i $$-0.570199\pi$$
−0.218752 + 0.975780i $$0.570199\pi$$
$$858$$ 0 0
$$859$$ 23.8617 0.814152 0.407076 0.913394i $$-0.366548\pi$$
0.407076 + 0.913394i $$0.366548\pi$$
$$860$$ 0 0
$$861$$ −143.093 −4.87659
$$862$$ 0 0
$$863$$ −29.1231 −0.991362 −0.495681 0.868505i $$-0.665081\pi$$
−0.495681 + 0.868505i $$0.665081\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −107.723 −3.65848
$$868$$ 0 0
$$869$$ 11.3693 0.385678
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 17.3693 0.587862
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 25.6155 0.864975 0.432487 0.901640i $$-0.357636\pi$$
0.432487 + 0.901640i $$0.357636\pi$$
$$878$$ 0 0
$$879$$ 43.8617 1.47942
$$880$$ 0 0
$$881$$ 5.50758 0.185555 0.0927775 0.995687i $$-0.470425\pi$$
0.0927775 + 0.995687i $$0.470425\pi$$
$$882$$ 0 0
$$883$$ 27.5464 0.927010 0.463505 0.886094i $$-0.346592\pi$$
0.463505 + 0.886094i $$0.346592\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 27.1231 0.910705 0.455352 0.890311i $$-0.349513\pi$$
0.455352 + 0.890311i $$0.349513\pi$$
$$888$$ 0 0
$$889$$ 22.2462 0.746114
$$890$$ 0 0
$$891$$ 7.00000 0.234509
$$892$$ 0 0
$$893$$ 16.3542 0.547271
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3.23106 0.107882
$$898$$ 0 0
$$899$$ −12.3153 −0.410740
$$900$$ 0 0
$$901$$ −74.4233 −2.47940
$$902$$ 0 0
$$903$$ 36.4924 1.21439
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 43.6847 1.45053 0.725263 0.688472i $$-0.241718\pi$$
0.725263 + 0.688472i $$0.241718\pi$$
$$908$$ 0 0
$$909$$ −7.12311 −0.236259
$$910$$ 0 0
$$911$$ 41.4384 1.37292 0.686459 0.727169i $$-0.259164\pi$$
0.686459 + 0.727169i $$0.259164\pi$$
$$912$$ 0 0
$$913$$ −6.00000 −0.198571
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 76.6695 2.53185
$$918$$ 0 0
$$919$$ −35.2311 −1.16217 −0.581083 0.813845i $$-0.697371\pi$$
−0.581083 + 0.813845i $$0.697371\pi$$
$$920$$ 0 0
$$921$$ 46.1080 1.51931
$$922$$ 0 0
$$923$$ 4.13826 0.136213
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 33.5464 1.10062 0.550311 0.834960i $$-0.314509\pi$$
0.550311 + 0.834960i $$0.314509\pi$$
$$930$$ 0 0
$$931$$ 19.8617 0.650942
$$932$$ 0 0
$$933$$ 24.1771 0.791522
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 49.6155 1.62087 0.810434 0.585829i $$-0.199231\pi$$
0.810434 + 0.585829i $$0.199231\pi$$
$$938$$ 0 0
$$939$$ 63.3693 2.06798
$$940$$ 0 0
$$941$$ 9.05398 0.295151 0.147576 0.989051i $$-0.452853\pi$$
0.147576 + 0.989051i $$0.452853\pi$$
$$942$$ 0 0
$$943$$ 13.7538 0.447885
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.9460 −0.420689 −0.210345 0.977627i $$-0.567459\pi$$
−0.210345 + 0.977627i $$0.567459\pi$$
$$948$$ 0 0
$$949$$ −1.26137 −0.0409457
$$950$$ 0 0
$$951$$ 16.1771 0.524578
$$952$$ 0 0
$$953$$ 6.56155 0.212550 0.106275 0.994337i $$-0.466108\pi$$
0.106275 + 0.994337i $$0.466108\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 21.9309 0.708924
$$958$$ 0 0
$$959$$ −32.4924 −1.04924
$$960$$ 0 0
$$961$$ −28.9309 −0.933254
$$962$$ 0 0
$$963$$ −25.3693 −0.817515
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.5616 0.403952 0.201976 0.979390i $$-0.435264\pi$$
0.201976 + 0.979390i $$0.435264\pi$$
$$968$$ 0 0
$$969$$ −28.3153 −0.909620
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 18.2462 0.584947
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 21.8617 0.699419 0.349710 0.936858i $$-0.386280\pi$$
0.349710 + 0.936858i $$0.386280\pi$$
$$978$$ 0 0
$$979$$ −9.68466 −0.309523
$$980$$ 0 0
$$981$$ −15.1231 −0.482844
$$982$$ 0 0
$$983$$ 16.6307 0.530436 0.265218 0.964188i $$-0.414556\pi$$
0.265218 + 0.964188i $$0.414556\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 132.847 4.22855
$$988$$ 0 0
$$989$$ −3.50758 −0.111534
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 48.0000 1.52323
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −1.26137 −0.0399479 −0.0199739 0.999801i $$-0.506358\pi$$
−0.0199739 + 0.999801i $$0.506358\pi$$
$$998$$ 0 0
$$999$$ 10.6998 0.338527
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 2200.2.a.l.1.1 2
4.3 odd 2 4400.2.a.bt.1.2 2
5.2 odd 4 2200.2.b.f.1849.4 4
5.3 odd 4 2200.2.b.f.1849.1 4
5.4 even 2 440.2.a.g.1.2 2
15.14 odd 2 3960.2.a.bf.1.2 2
20.3 even 4 4400.2.b.w.4049.4 4
20.7 even 4 4400.2.b.w.4049.1 4
20.19 odd 2 880.2.a.k.1.1 2
40.19 odd 2 3520.2.a.br.1.2 2
40.29 even 2 3520.2.a.bm.1.1 2
55.54 odd 2 4840.2.a.m.1.2 2
60.59 even 2 7920.2.a.by.1.1 2
220.219 even 2 9680.2.a.bm.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.2 2 5.4 even 2
880.2.a.k.1.1 2 20.19 odd 2
2200.2.a.l.1.1 2 1.1 even 1 trivial
2200.2.b.f.1849.1 4 5.3 odd 4
2200.2.b.f.1849.4 4 5.2 odd 4
3520.2.a.bm.1.1 2 40.29 even 2
3520.2.a.br.1.2 2 40.19 odd 2
3960.2.a.bf.1.2 2 15.14 odd 2
4400.2.a.bt.1.2 2 4.3 odd 2
4400.2.b.w.4049.1 4 20.7 even 4
4400.2.b.w.4049.4 4 20.3 even 4
4840.2.a.m.1.2 2 55.54 odd 2
7920.2.a.by.1.1 2 60.59 even 2
9680.2.a.bm.1.1 2 220.219 even 2