# Properties

 Label 400.9.h.b.399.3 Level $400$ Weight $9$ Character 400.399 Analytic conductor $162.951$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,9,Mod(399,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("400.399");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 400.h (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$162.951444024$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 17x^{2} + 81$$ x^4 - 17*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 399.3 Root $$-2.95804 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.399 Dual form 400.9.h.b.399.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} +O(q^{10})$$ $$q+141.986 q^{3} +2555.75 q^{7} +13599.0 q^{9} -19168.1i q^{11} -27710.0i q^{13} -50370.0i q^{17} +108619. i q^{19} +362880. q^{21} +176347. q^{23} +999297. q^{27} -54978.0 q^{29} -1.17564e6i q^{31} -2.72160e6i q^{33} -793730. i q^{37} -3.93443e6i q^{39} -75582.0 q^{41} +499648. q^{43} -2.86755e6 q^{47} +767039. q^{49} -7.15183e6i q^{51} +1.11662e7i q^{53} +1.54224e7i q^{57} -2.18325e7i q^{59} -2.38266e7 q^{61} +3.47556e7 q^{63} -7.49473e6 q^{67} +2.50387e7 q^{69} +1.00824e7i q^{71} +6.51661e6i q^{73} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -7.80610e6 q^{87} -8.67958e7 q^{89} -7.08197e7i q^{91} -1.66925e8i q^{93} +4.66703e7i q^{97} -2.60667e8i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 54396 q^{9}+O(q^{10})$$ 4 * q + 54396 * q^9 $$4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100})$$ 4 * q + 54396 * q^9 + 1451520 * q^21 - 219912 * q^29 - 302328 * q^41 + 3068156 * q^49 - 95306488 * q^61 + 100154880 * q^69 + 210652164 * q^81 - 347183112 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$3$$ 141.986 1.75291 0.876456 0.481481i $$-0.159901\pi$$
0.876456 + 0.481481i $$0.159901\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2555.75 1.06445 0.532225 0.846603i $$-0.321356\pi$$
0.532225 + 0.846603i $$0.321356\pi$$
$$8$$ 0 0
$$9$$ 13599.0 2.07270
$$10$$ 0 0
$$11$$ − 19168.1i − 1.30921i −0.755972 0.654603i $$-0.772836\pi$$
0.755972 0.654603i $$-0.227164\pi$$
$$12$$ 0 0
$$13$$ − 27710.0i − 0.970204i −0.874458 0.485102i $$-0.838782\pi$$
0.874458 0.485102i $$-0.161218\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 50370.0i − 0.603082i −0.953453 0.301541i $$-0.902499\pi$$
0.953453 0.301541i $$-0.0975010\pi$$
$$18$$ 0 0
$$19$$ 108619.i 0.833474i 0.909027 + 0.416737i $$0.136826\pi$$
−0.909027 + 0.416737i $$0.863174\pi$$
$$20$$ 0 0
$$21$$ 362880. 1.86589
$$22$$ 0 0
$$23$$ 176347. 0.630167 0.315083 0.949064i $$-0.397968\pi$$
0.315083 + 0.949064i $$0.397968\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 999297. 1.88035
$$28$$ 0 0
$$29$$ −54978.0 −0.0777315 −0.0388657 0.999244i $$-0.512374\pi$$
−0.0388657 + 0.999244i $$0.512374\pi$$
$$30$$ 0 0
$$31$$ − 1.17564e6i − 1.27300i −0.771276 0.636501i $$-0.780381\pi$$
0.771276 0.636501i $$-0.219619\pi$$
$$32$$ 0 0
$$33$$ − 2.72160e6i − 2.29493i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 793730.i − 0.423512i −0.977323 0.211756i $$-0.932082\pi$$
0.977323 0.211756i $$-0.0679182\pi$$
$$38$$ 0 0
$$39$$ − 3.93443e6i − 1.70068i
$$40$$ 0 0
$$41$$ −75582.0 −0.0267475 −0.0133737 0.999911i $$-0.504257\pi$$
−0.0133737 + 0.999911i $$0.504257\pi$$
$$42$$ 0 0
$$43$$ 499648. 0.146147 0.0730736 0.997327i $$-0.476719\pi$$
0.0730736 + 0.997327i $$0.476719\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.86755e6 −0.587651 −0.293825 0.955859i $$-0.594928\pi$$
−0.293825 + 0.955859i $$0.594928\pi$$
$$48$$ 0 0
$$49$$ 767039. 0.133056
$$50$$ 0 0
$$51$$ − 7.15183e6i − 1.05715i
$$52$$ 0 0
$$53$$ 1.11662e7i 1.41515i 0.706639 + 0.707575i $$0.250211\pi$$
−0.706639 + 0.707575i $$0.749789\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.54224e7i 1.46101i
$$58$$ 0 0
$$59$$ − 2.18325e7i − 1.80175i −0.434078 0.900875i $$-0.642926\pi$$
0.434078 0.900875i $$-0.357074\pi$$
$$60$$ 0 0
$$61$$ −2.38266e7 −1.72085 −0.860425 0.509577i $$-0.829802\pi$$
−0.860425 + 0.509577i $$0.829802\pi$$
$$62$$ 0 0
$$63$$ 3.47556e7 2.20629
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.49473e6 −0.371926 −0.185963 0.982557i $$-0.559540\pi$$
−0.185963 + 0.982557i $$0.559540\pi$$
$$68$$ 0 0
$$69$$ 2.50387e7 1.10463
$$70$$ 0 0
$$71$$ 1.00824e7i 0.396763i 0.980125 + 0.198382i $$0.0635685\pi$$
−0.980125 + 0.198382i $$0.936431\pi$$
$$72$$ 0 0
$$73$$ 6.51661e6i 0.229472i 0.993396 + 0.114736i $$0.0366023\pi$$
−0.993396 + 0.114736i $$0.963398\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 4.89888e7i − 1.39359i
$$78$$ 0 0
$$79$$ − 4.87892e7i − 1.25261i −0.779579 0.626304i $$-0.784567\pi$$
0.779579 0.626304i $$-0.215433\pi$$
$$80$$ 0 0
$$81$$ 5.26630e7 1.22339
$$82$$ 0 0
$$83$$ 7.34483e7 1.54764 0.773819 0.633407i $$-0.218344\pi$$
0.773819 + 0.633407i $$0.218344\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −7.80610e6 −0.136256
$$88$$ 0 0
$$89$$ −8.67958e7 −1.38337 −0.691685 0.722199i $$-0.743131\pi$$
−0.691685 + 0.722199i $$0.743131\pi$$
$$90$$ 0 0
$$91$$ − 7.08197e7i − 1.03273i
$$92$$ 0 0
$$93$$ − 1.66925e8i − 2.23146i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.66703e7i 0.527173i 0.964636 + 0.263587i $$0.0849055\pi$$
−0.964636 + 0.263587i $$0.915095\pi$$
$$98$$ 0 0
$$99$$ − 2.60667e8i − 2.71360i
$$100$$ 0 0
$$101$$ 6.59910e7 0.634161 0.317080 0.948399i $$-0.397297\pi$$
0.317080 + 0.948399i $$0.397297\pi$$
$$102$$ 0 0
$$103$$ 1.64884e8 1.46497 0.732486 0.680782i $$-0.238360\pi$$
0.732486 + 0.680782i $$0.238360\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1.27326e8 −0.971364 −0.485682 0.874136i $$-0.661429\pi$$
−0.485682 + 0.874136i $$0.661429\pi$$
$$108$$ 0 0
$$109$$ 1.56119e8 1.10598 0.552992 0.833186i $$-0.313486\pi$$
0.552992 + 0.833186i $$0.313486\pi$$
$$110$$ 0 0
$$111$$ − 1.12698e8i − 0.742380i
$$112$$ 0 0
$$113$$ 2.36346e8i 1.44955i 0.688984 + 0.724776i $$0.258057\pi$$
−0.688984 + 0.724776i $$0.741943\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 3.76828e8i − 2.01094i
$$118$$ 0 0
$$119$$ − 1.28733e8i − 0.641951i
$$120$$ 0 0
$$121$$ −1.53057e8 −0.714023
$$122$$ 0 0
$$123$$ −1.07316e7 −0.0468860
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 3.67741e8 1.41360 0.706802 0.707412i $$-0.250137\pi$$
0.706802 + 0.707412i $$0.250137\pi$$
$$128$$ 0 0
$$129$$ 7.09430e7 0.256183
$$130$$ 0 0
$$131$$ − 2.16350e8i − 0.734636i −0.930095 0.367318i $$-0.880276\pi$$
0.930095 0.367318i $$-0.119724\pi$$
$$132$$ 0 0
$$133$$ 2.77603e8i 0.887193i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.86442e8i 1.09699i 0.836155 + 0.548494i $$0.184799\pi$$
−0.836155 + 0.548494i $$0.815201\pi$$
$$138$$ 0 0
$$139$$ 3.51077e8i 0.940465i 0.882543 + 0.470232i $$0.155830\pi$$
−0.882543 + 0.470232i $$0.844170\pi$$
$$140$$ 0 0
$$141$$ −4.07151e8 −1.03010
$$142$$ 0 0
$$143$$ −5.31148e8 −1.27020
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.08909e8 0.233235
$$148$$ 0 0
$$149$$ 4.54099e8 0.921308 0.460654 0.887580i $$-0.347615\pi$$
0.460654 + 0.887580i $$0.347615\pi$$
$$150$$ 0 0
$$151$$ 6.60188e8i 1.26987i 0.772565 + 0.634936i $$0.218973\pi$$
−0.772565 + 0.634936i $$0.781027\pi$$
$$152$$ 0 0
$$153$$ − 6.84982e8i − 1.25001i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 4.35318e8i − 0.716486i −0.933628 0.358243i $$-0.883376\pi$$
0.933628 0.358243i $$-0.116624\pi$$
$$158$$ 0 0
$$159$$ 1.58544e9i 2.48063i
$$160$$ 0 0
$$161$$ 4.50697e8 0.670782
$$162$$ 0 0
$$163$$ 2.44065e8 0.345744 0.172872 0.984944i $$-0.444695\pi$$
0.172872 + 0.984944i $$0.444695\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.71351e8 −0.863145 −0.431573 0.902078i $$-0.642041\pi$$
−0.431573 + 0.902078i $$0.642041\pi$$
$$168$$ 0 0
$$169$$ 4.78866e7 0.0587040
$$170$$ 0 0
$$171$$ 1.47711e9i 1.72754i
$$172$$ 0 0
$$173$$ − 1.76764e9i − 1.97337i −0.162644 0.986685i $$-0.552002\pi$$
0.162644 0.986685i $$-0.447998\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 3.09990e9i − 3.15831i
$$178$$ 0 0
$$179$$ 7.56967e8i 0.737335i 0.929561 + 0.368668i $$0.120186\pi$$
−0.929561 + 0.368668i $$0.879814\pi$$
$$180$$ 0 0
$$181$$ 6.27094e8 0.584277 0.292138 0.956376i $$-0.405633\pi$$
0.292138 + 0.956376i $$0.405633\pi$$
$$182$$ 0 0
$$183$$ −3.38304e9 −3.01650
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −9.65497e8 −0.789559
$$188$$ 0 0
$$189$$ 2.55395e9 2.00154
$$190$$ 0 0
$$191$$ − 1.07924e9i − 0.810933i −0.914110 0.405466i $$-0.867109\pi$$
0.914110 0.405466i $$-0.132891\pi$$
$$192$$ 0 0
$$193$$ − 2.96757e7i − 0.0213881i −0.999943 0.0106940i $$-0.996596\pi$$
0.999943 0.0106940i $$-0.00340408\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.12484e9i 0.746837i 0.927663 + 0.373419i $$0.121814\pi$$
−0.927663 + 0.373419i $$0.878186\pi$$
$$198$$ 0 0
$$199$$ 1.04718e9i 0.667742i 0.942619 + 0.333871i $$0.108355\pi$$
−0.942619 + 0.333871i $$0.891645\pi$$
$$200$$ 0 0
$$201$$ −1.06415e9 −0.651954
$$202$$ 0 0
$$203$$ −1.40510e8 −0.0827413
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 2.39814e9 1.30615
$$208$$ 0 0
$$209$$ 2.08202e9 1.09119
$$210$$ 0 0
$$211$$ 2.77676e9i 1.40090i 0.713699 + 0.700452i $$0.247018\pi$$
−0.713699 + 0.700452i $$0.752982\pi$$
$$212$$ 0 0
$$213$$ 1.43156e9i 0.695491i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 3.00465e9i − 1.35505i
$$218$$ 0 0
$$219$$ 9.25267e8i 0.402245i
$$220$$ 0 0
$$221$$ −1.39575e9 −0.585113
$$222$$ 0 0
$$223$$ 2.27822e9 0.921248 0.460624 0.887595i $$-0.347626\pi$$
0.460624 + 0.887595i $$0.347626\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.48863e9 0.560639 0.280319 0.959907i $$-0.409560\pi$$
0.280319 + 0.959907i $$0.409560\pi$$
$$228$$ 0 0
$$229$$ −1.69447e9 −0.616157 −0.308079 0.951361i $$-0.599686\pi$$
−0.308079 + 0.951361i $$0.599686\pi$$
$$230$$ 0 0
$$231$$ − 6.95572e9i − 2.44284i
$$232$$ 0 0
$$233$$ 5.21423e8i 0.176916i 0.996080 + 0.0884580i $$0.0281939\pi$$
−0.996080 + 0.0884580i $$0.971806\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 6.92738e9i − 2.19571i
$$238$$ 0 0
$$239$$ − 4.40690e9i − 1.35065i −0.737522 0.675323i $$-0.764004\pi$$
0.737522 0.675323i $$-0.235996\pi$$
$$240$$ 0 0
$$241$$ −1.62148e9 −0.480666 −0.240333 0.970691i $$-0.577257\pi$$
−0.240333 + 0.970691i $$0.577257\pi$$
$$242$$ 0 0
$$243$$ 9.21023e8 0.264147
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00984e9 0.808640
$$248$$ 0 0
$$249$$ 1.04286e10 2.71287
$$250$$ 0 0
$$251$$ 1.31321e8i 0.0330855i 0.999863 + 0.0165428i $$0.00526597\pi$$
−0.999863 + 0.0165428i $$0.994734\pi$$
$$252$$ 0 0
$$253$$ − 3.38023e9i − 0.825019i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 5.27789e9i − 1.20984i −0.796287 0.604920i $$-0.793205\pi$$
0.796287 0.604920i $$-0.206795\pi$$
$$258$$ 0 0
$$259$$ − 2.02857e9i − 0.450808i
$$260$$ 0 0
$$261$$ −7.47646e8 −0.161114
$$262$$ 0 0
$$263$$ 7.38745e9 1.54409 0.772044 0.635570i $$-0.219235\pi$$
0.772044 + 0.635570i $$0.219235\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.23238e10 −2.42493
$$268$$ 0 0
$$269$$ −3.46450e8 −0.0661655 −0.0330828 0.999453i $$-0.510532\pi$$
−0.0330828 + 0.999453i $$0.510532\pi$$
$$270$$ 0 0
$$271$$ − 6.51715e6i − 0.00120832i −1.00000 0.000604158i $$-0.999808\pi$$
1.00000 0.000604158i $$-0.000192310\pi$$
$$272$$ 0 0
$$273$$ − 1.00554e10i − 1.81029i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.15061e9i − 0.365293i −0.983179 0.182647i $$-0.941534\pi$$
0.983179 0.182647i $$-0.0584664\pi$$
$$278$$ 0 0
$$279$$ − 1.59876e10i − 2.63855i
$$280$$ 0 0
$$281$$ 1.04256e10 1.67215 0.836074 0.548616i $$-0.184845\pi$$
0.836074 + 0.548616i $$0.184845\pi$$
$$282$$ 0 0
$$283$$ 1.28042e9 0.199622 0.0998108 0.995006i $$-0.468176\pi$$
0.0998108 + 0.995006i $$0.468176\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.93168e8 −0.0284714
$$288$$ 0 0
$$289$$ 4.43862e9 0.636292
$$290$$ 0 0
$$291$$ 6.62652e9i 0.924089i
$$292$$ 0 0
$$293$$ − 2.13786e9i − 0.290074i −0.989426 0.145037i $$-0.953670\pi$$
0.989426 0.145037i $$-0.0463301\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1.91546e10i − 2.46177i
$$298$$ 0 0
$$299$$ − 4.88656e9i − 0.611390i
$$300$$ 0 0
$$301$$ 1.27697e9 0.155567
$$302$$ 0 0
$$303$$ 9.36980e9 1.11163
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.45140e9 −0.613698 −0.306849 0.951758i $$-0.599275\pi$$
−0.306849 + 0.951758i $$0.599275\pi$$
$$308$$ 0 0
$$309$$ 2.34112e10 2.56797
$$310$$ 0 0
$$311$$ − 1.07550e10i − 1.14965i −0.818275 0.574827i $$-0.805069\pi$$
0.818275 0.574827i $$-0.194931\pi$$
$$312$$ 0 0
$$313$$ − 2.99804e8i − 0.0312364i −0.999878 0.0156182i $$-0.995028\pi$$
0.999878 0.0156182i $$-0.00497163\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 5.31172e9i − 0.526015i −0.964794 0.263007i $$-0.915286\pi$$
0.964794 0.263007i $$-0.0847144\pi$$
$$318$$ 0 0
$$319$$ 1.05382e9i 0.101767i
$$320$$ 0 0
$$321$$ −1.80785e10 −1.70272
$$322$$ 0 0
$$323$$ 5.47115e9 0.502653
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.21667e10 1.93869
$$328$$ 0 0
$$329$$ −7.32872e9 −0.625525
$$330$$ 0 0
$$331$$ 1.01004e10i 0.841446i 0.907189 + 0.420723i $$0.138224\pi$$
−0.907189 + 0.420723i $$0.861776\pi$$
$$332$$ 0 0
$$333$$ − 1.07939e10i − 0.877815i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.84359e10i 1.42937i 0.699448 + 0.714684i $$0.253429\pi$$
−0.699448 + 0.714684i $$0.746571\pi$$
$$338$$ 0 0
$$339$$ 3.35578e10i 2.54094i
$$340$$ 0 0
$$341$$ −2.25348e10 −1.66662
$$342$$ 0 0
$$343$$ −1.27730e10 −0.922820
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.27822e10 −0.881636 −0.440818 0.897597i $$-0.645312\pi$$
−0.440818 + 0.897597i $$0.645312\pi$$
$$348$$ 0 0
$$349$$ 6.39381e8 0.0430981 0.0215490 0.999768i $$-0.493140\pi$$
0.0215490 + 0.999768i $$0.493140\pi$$
$$350$$ 0 0
$$351$$ − 2.76905e10i − 1.82433i
$$352$$ 0 0
$$353$$ 2.59837e10i 1.67341i 0.547653 + 0.836705i $$0.315521\pi$$
−0.547653 + 0.836705i $$0.684479\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 1.82783e10i − 1.12528i
$$358$$ 0 0
$$359$$ − 2.22541e10i − 1.33978i −0.742461 0.669889i $$-0.766342\pi$$
0.742461 0.669889i $$-0.233658\pi$$
$$360$$ 0 0
$$361$$ 5.18543e9 0.305320
$$362$$ 0 0
$$363$$ −2.17320e10 −1.25162
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.86229e10 1.02656 0.513280 0.858221i $$-0.328430\pi$$
0.513280 + 0.858221i $$0.328430\pi$$
$$368$$ 0 0
$$369$$ −1.02784e9 −0.0554396
$$370$$ 0 0
$$371$$ 2.85380e10i 1.50636i
$$372$$ 0 0
$$373$$ 1.19680e10i 0.618283i 0.951016 + 0.309141i $$0.100042\pi$$
−0.951016 + 0.309141i $$0.899958\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.52344e9i 0.0754154i
$$378$$ 0 0
$$379$$ 2.30787e10i 1.11855i 0.828982 + 0.559275i $$0.188920\pi$$
−0.828982 + 0.559275i $$0.811080\pi$$
$$380$$ 0 0
$$381$$ 5.22141e10 2.47792
$$382$$ 0 0
$$383$$ −1.43419e10 −0.666518 −0.333259 0.942835i $$-0.608148\pi$$
−0.333259 + 0.942835i $$0.608148\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 6.79472e9 0.302920
$$388$$ 0 0
$$389$$ 2.73457e10 1.19424 0.597119 0.802152i $$-0.296312\pi$$
0.597119 + 0.802152i $$0.296312\pi$$
$$390$$ 0 0
$$391$$ − 8.88257e9i − 0.380042i
$$392$$ 0 0
$$393$$ − 3.07187e10i − 1.28775i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.99456e10i − 1.60808i −0.594576 0.804039i $$-0.702680\pi$$
0.594576 0.804039i $$-0.297320\pi$$
$$398$$ 0 0
$$399$$ 3.94157e10i 1.55517i
$$400$$ 0 0
$$401$$ −2.12767e10 −0.822863 −0.411431 0.911441i $$-0.634971\pi$$
−0.411431 + 0.911441i $$0.634971\pi$$
$$402$$ 0 0
$$403$$ −3.25771e10 −1.23507
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.52143e10 −0.554465
$$408$$ 0 0
$$409$$ −1.14283e10 −0.408404 −0.204202 0.978929i $$-0.565460\pi$$
−0.204202 + 0.978929i $$0.565460\pi$$
$$410$$ 0 0
$$411$$ 5.48693e10i 1.92292i
$$412$$ 0 0
$$413$$ − 5.57982e10i − 1.91788i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.98479e10i 1.64855i
$$418$$ 0 0
$$419$$ 1.10009e10i 0.356922i 0.983947 + 0.178461i $$0.0571119\pi$$
−0.983947 + 0.178461i $$0.942888\pi$$
$$420$$ 0 0
$$421$$ 2.28766e10 0.728220 0.364110 0.931356i $$-0.381373\pi$$
0.364110 + 0.931356i $$0.381373\pi$$
$$422$$ 0 0
$$423$$ −3.89958e10 −1.21802
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.08948e10 −1.83176
$$428$$ 0 0
$$429$$ −7.54155e10 −2.22655
$$430$$ 0 0
$$431$$ 9.55108e8i 0.0276786i 0.999904 + 0.0138393i $$0.00440532\pi$$
−0.999904 + 0.0138393i $$0.995595\pi$$
$$432$$ 0 0
$$433$$ 3.82225e10i 1.08735i 0.839297 + 0.543673i $$0.182967\pi$$
−0.839297 + 0.543673i $$0.817033\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.91546e10i 0.525228i
$$438$$ 0 0
$$439$$ 6.40288e10i 1.72392i 0.506976 + 0.861960i $$0.330763\pi$$
−0.506976 + 0.861960i $$0.669237\pi$$
$$440$$ 0 0
$$441$$ 1.04310e10 0.275785
$$442$$ 0 0
$$443$$ 7.47659e10 1.94128 0.970641 0.240533i $$-0.0773222\pi$$
0.970641 + 0.240533i $$0.0773222\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.44756e10 1.61497
$$448$$ 0 0
$$449$$ −2.51987e10 −0.620001 −0.310000 0.950736i $$-0.600329\pi$$
−0.310000 + 0.950736i $$0.600329\pi$$
$$450$$ 0 0
$$451$$ 1.44876e9i 0.0350180i
$$452$$ 0 0
$$453$$ 9.37373e10i 2.22597i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.66828e9i − 0.107027i −0.998567 0.0535133i $$-0.982958\pi$$
0.998567 0.0535133i $$-0.0170420\pi$$
$$458$$ 0 0
$$459$$ − 5.03346e10i − 1.13401i
$$460$$ 0 0
$$461$$ −3.88096e10 −0.859281 −0.429641 0.903000i $$-0.641360\pi$$
−0.429641 + 0.903000i $$0.641360\pi$$
$$462$$ 0 0
$$463$$ −3.23432e10 −0.703817 −0.351908 0.936034i $$-0.614467\pi$$
−0.351908 + 0.936034i $$0.614467\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 2.31902e10 0.487570 0.243785 0.969829i $$-0.421611\pi$$
0.243785 + 0.969829i $$0.421611\pi$$
$$468$$ 0 0
$$469$$ −1.91546e10 −0.395897
$$470$$ 0 0
$$471$$ − 6.18090e10i − 1.25594i
$$472$$ 0 0
$$473$$ − 9.57731e9i − 0.191337i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.51849e11i 2.93318i
$$478$$ 0 0
$$479$$ − 4.83542e9i − 0.0918528i −0.998945 0.0459264i $$-0.985376\pi$$
0.998945 0.0459264i $$-0.0146240\pi$$
$$480$$ 0 0
$$481$$ −2.19943e10 −0.410893
$$482$$ 0 0
$$483$$ 6.39926e10 1.17582
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3.03878e10 −0.540236 −0.270118 0.962827i $$-0.587063\pi$$
−0.270118 + 0.962827i $$0.587063\pi$$
$$488$$ 0 0
$$489$$ 3.46538e10 0.606059
$$490$$ 0 0
$$491$$ 5.56483e10i 0.957472i 0.877959 + 0.478736i $$0.158905\pi$$
−0.877959 + 0.478736i $$0.841095\pi$$
$$492$$ 0 0
$$493$$ 2.76924e9i 0.0468784i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.57681e10i 0.422335i
$$498$$ 0 0
$$499$$ − 7.88458e10i − 1.27168i −0.771822 0.635838i $$-0.780655\pi$$
0.771822 0.635838i $$-0.219345\pi$$
$$500$$ 0 0
$$501$$ −9.53224e10 −1.51302
$$502$$ 0 0
$$503$$ 4.41092e10 0.689061 0.344530 0.938775i $$-0.388038\pi$$
0.344530 + 0.938775i $$0.388038\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 6.79923e9 0.102903
$$508$$ 0 0
$$509$$ −1.05927e10 −0.157811 −0.0789055 0.996882i $$-0.525143\pi$$
−0.0789055 + 0.996882i $$0.525143\pi$$
$$510$$ 0 0
$$511$$ 1.66548e10i 0.244262i
$$512$$ 0 0
$$513$$ 1.08543e11i 1.56723i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5.49654e10i 0.769356i
$$518$$ 0 0
$$519$$ − 2.50979e11i − 3.45914i
$$520$$ 0 0
$$521$$ 1.24958e11 1.69595 0.847973 0.530039i $$-0.177823\pi$$
0.847973 + 0.530039i $$0.177823\pi$$
$$522$$ 0 0
$$523$$ 2.80408e10 0.374786 0.187393 0.982285i $$-0.439996\pi$$
0.187393 + 0.982285i $$0.439996\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.92172e10 −0.767724
$$528$$ 0 0
$$529$$ −4.72129e10 −0.602890
$$530$$ 0 0
$$531$$ − 2.96900e11i − 3.73449i
$$532$$ 0 0
$$533$$ 2.09438e9i 0.0259505i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 1.07479e11i 1.29248i
$$538$$ 0 0
$$539$$ − 1.47027e10i − 0.174197i
$$540$$ 0 0
$$541$$ 1.44659e11 1.68871 0.844356 0.535782i $$-0.179983\pi$$
0.844356 + 0.535782i $$0.179983\pi$$
$$542$$ 0 0
$$543$$ 8.90386e10 1.02419
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.03774e10 −0.115915 −0.0579573 0.998319i $$-0.518459\pi$$
−0.0579573 + 0.998319i $$0.518459\pi$$
$$548$$ 0 0
$$549$$ −3.24018e11 −3.56681
$$550$$ 0 0
$$551$$ − 5.97167e9i − 0.0647872i
$$552$$ 0 0
$$553$$ − 1.24693e11i − 1.33334i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.47312e9i 0.0568610i 0.999596 + 0.0284305i $$0.00905093\pi$$
−0.999596 + 0.0284305i $$0.990949\pi$$
$$558$$ 0 0
$$559$$ − 1.38453e10i − 0.141793i
$$560$$ 0 0
$$561$$ −1.37087e11 −1.38403
$$562$$ 0 0
$$563$$ −4.36118e10 −0.434081 −0.217040 0.976163i $$-0.569640\pi$$
−0.217040 + 0.976163i $$0.569640\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.34593e11 1.30224
$$568$$ 0 0
$$569$$ −1.27822e10 −0.121943 −0.0609716 0.998140i $$-0.519420\pi$$
−0.0609716 + 0.998140i $$0.519420\pi$$
$$570$$ 0 0
$$571$$ 7.59455e10i 0.714427i 0.934023 + 0.357213i $$0.116273\pi$$
−0.934023 + 0.357213i $$0.883727\pi$$
$$572$$ 0 0
$$573$$ − 1.53237e11i − 1.42149i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.13827e10i − 0.192912i −0.995337 0.0964560i $$-0.969249\pi$$
0.995337 0.0964560i $$-0.0307507\pi$$
$$578$$ 0 0
$$579$$ − 4.21353e9i − 0.0374914i
$$580$$ 0 0
$$581$$ 1.87715e11 1.64738
$$582$$ 0 0
$$583$$ 2.14035e11 1.85272
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −6.07298e10 −0.511504 −0.255752 0.966742i $$-0.582323\pi$$
−0.255752 + 0.966742i $$0.582323\pi$$
$$588$$ 0 0
$$589$$ 1.27697e11 1.06101
$$590$$ 0 0
$$591$$ 1.59711e11i 1.30914i
$$592$$ 0 0
$$593$$ 1.15978e11i 0.937899i 0.883225 + 0.468949i $$0.155367\pi$$
−0.883225 + 0.468949i $$0.844633\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1.48685e11i 1.17049i
$$598$$ 0 0
$$599$$ 2.40647e11i 1.86927i 0.355607 + 0.934636i $$0.384274\pi$$
−0.355607 + 0.934636i $$0.615726\pi$$
$$600$$ 0 0
$$601$$ −1.92942e11 −1.47887 −0.739434 0.673229i $$-0.764907\pi$$
−0.739434 + 0.673229i $$0.764907\pi$$
$$602$$ 0 0
$$603$$ −1.01921e11 −0.770892
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.62042e11 −1.19364 −0.596819 0.802376i $$-0.703569\pi$$
−0.596819 + 0.802376i $$0.703569\pi$$
$$608$$ 0 0
$$609$$ −1.99504e10 −0.145038
$$610$$ 0 0
$$611$$ 7.94597e10i 0.570141i
$$612$$ 0 0
$$613$$ 1.76424e11i 1.24944i 0.780847 + 0.624722i $$0.214788\pi$$
−0.780847 + 0.624722i $$0.785212\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.84986e10i 0.679656i 0.940488 + 0.339828i $$0.110369\pi$$
−0.940488 + 0.339828i $$0.889631\pi$$
$$618$$ 0 0
$$619$$ − 1.28596e10i − 0.0875923i −0.999040 0.0437961i $$-0.986055\pi$$
0.999040 0.0437961i $$-0.0139452\pi$$
$$620$$ 0 0
$$621$$ 1.76223e11 1.18494
$$622$$ 0 0
$$623$$ −2.21828e11 −1.47253
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 2.95618e11 1.91276
$$628$$ 0 0
$$629$$ −3.99802e10 −0.255413
$$630$$ 0 0
$$631$$ 1.73463e11i 1.09418i 0.837073 + 0.547091i $$0.184265\pi$$
−0.837073 + 0.547091i $$0.815735\pi$$
$$632$$ 0 0
$$633$$ 3.94261e11i 2.45566i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.12547e10i − 0.129091i
$$638$$ 0 0
$$639$$ 1.37111e11i 0.822372i
$$640$$ 0 0
$$641$$ −1.13903e11 −0.674690 −0.337345 0.941381i $$-0.609529\pi$$
−0.337345 + 0.941381i $$0.609529\pi$$
$$642$$ 0 0
$$643$$ −7.04067e10 −0.411879 −0.205940 0.978565i $$-0.566025\pi$$
−0.205940 + 0.978565i $$0.566025\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1.99175e11 −1.13663 −0.568314 0.822812i $$-0.692404\pi$$
−0.568314 + 0.822812i $$0.692404\pi$$
$$648$$ 0 0
$$649$$ −4.18487e11 −2.35886
$$650$$ 0 0
$$651$$ − 4.26617e11i − 2.37528i
$$652$$ 0 0
$$653$$ 6.49972e9i 0.0357472i 0.999840 + 0.0178736i $$0.00568964\pi$$
−0.999840 + 0.0178736i $$0.994310\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.86194e10i 0.475628i
$$658$$ 0 0
$$659$$ 2.20982e11i 1.17170i 0.810421 + 0.585848i $$0.199239\pi$$
−0.810421 + 0.585848i $$0.800761\pi$$
$$660$$ 0 0
$$661$$ −2.69549e11 −1.41199 −0.705995 0.708217i $$-0.749500\pi$$
−0.705995 + 0.708217i $$0.749500\pi$$
$$662$$ 0 0
$$663$$ −1.98177e11 −1.02565
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.69518e9 −0.0489838
$$668$$ 0 0
$$669$$ 3.23476e11 1.61487
$$670$$ 0 0
$$671$$ 4.56711e11i 2.25295i
$$672$$ 0 0
$$673$$ 9.44470e10i 0.460392i 0.973144 + 0.230196i $$0.0739368\pi$$
−0.973144 + 0.230196i $$0.926063\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 8.02735e10i − 0.382136i −0.981577 0.191068i $$-0.938805\pi$$
0.981577 0.191068i $$-0.0611950\pi$$
$$678$$ 0 0
$$679$$ 1.19277e11i 0.561150i
$$680$$ 0 0
$$681$$ 2.11364e11 0.982751
$$682$$ 0 0
$$683$$ −3.00783e11 −1.38220 −0.691099 0.722760i $$-0.742873\pi$$
−0.691099 + 0.722760i $$0.742873\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2.40591e11 −1.08007
$$688$$ 0 0
$$689$$ 3.09416e11 1.37298
$$690$$ 0 0
$$691$$ 3.06208e11i 1.34309i 0.740964 + 0.671544i $$0.234369\pi$$
−0.740964 + 0.671544i $$0.765631\pi$$
$$692$$ 0 0
$$693$$ − 6.66199e11i − 2.88849i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3.80707e9i 0.0161309i
$$698$$ 0 0
$$699$$ 7.40348e10i 0.310118i
$$700$$ 0 0
$$701$$ −2.73603e11 −1.13305 −0.566524 0.824045i $$-0.691712\pi$$
−0.566524 + 0.824045i $$0.691712\pi$$
$$702$$ 0 0
$$703$$ 8.62143e10 0.352987
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.68656e11 0.675033
$$708$$ 0 0
$$709$$ 1.76662e11 0.699129 0.349564 0.936912i $$-0.386330\pi$$
0.349564 + 0.936912i $$0.386330\pi$$
$$710$$ 0 0
$$711$$ − 6.63484e11i − 2.59628i
$$712$$ 0 0
$$713$$ − 2.07321e11i − 0.802203i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 6.25718e11i − 2.36756i
$$718$$ 0 0
$$719$$ 2.25510e11i 0.843821i 0.906637 + 0.421911i $$0.138640\pi$$
−0.906637 + 0.421911i $$0.861360\pi$$
$$720$$ 0 0
$$721$$ 4.21402e11 1.55939
$$722$$ 0 0
$$723$$ −2.30227e11 −0.842565
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.87080e11 1.02770 0.513849 0.857881i $$-0.328219\pi$$
0.513849 + 0.857881i $$0.328219\pi$$
$$728$$ 0 0
$$729$$ −2.14750e11 −0.760366
$$730$$ 0 0
$$731$$ − 2.51673e10i − 0.0881388i
$$732$$ 0 0
$$733$$ − 2.94176e11i − 1.01904i −0.860459 0.509520i $$-0.829823\pi$$
0.860459 0.509520i $$-0.170177\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.43660e11i 0.486928i
$$738$$ 0 0
$$739$$ 9.69888e10i 0.325195i 0.986692 + 0.162598i $$0.0519872\pi$$
−0.986692 + 0.162598i $$0.948013\pi$$
$$740$$ 0 0
$$741$$ 4.27355e11 1.41748
$$742$$ 0 0
$$743$$ 1.01567e11 0.333271 0.166635 0.986019i $$-0.446710\pi$$
0.166635 + 0.986019i $$0.446710\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 9.98824e11 3.20779
$$748$$ 0 0
$$749$$ −3.25413e11 −1.03397
$$750$$ 0 0
$$751$$ 4.17899e11i 1.31375i 0.754001 + 0.656873i $$0.228121\pi$$
−0.754001 + 0.656873i $$0.771879\pi$$
$$752$$ 0 0
$$753$$ 1.86457e10i 0.0579960i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 1.82006e11i − 0.554244i −0.960835 0.277122i $$-0.910619\pi$$
0.960835 0.277122i $$-0.0893806\pi$$
$$758$$ 0 0
$$759$$ − 4.79945e11i − 1.44619i
$$760$$ 0 0
$$761$$ −4.27419e11 −1.27443 −0.637213 0.770687i $$-0.719913\pi$$
−0.637213 + 0.770687i $$0.719913\pi$$
$$762$$ 0 0
$$763$$ 3.99000e11 1.17727
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.04978e11 −1.74807
$$768$$ 0 0
$$769$$ 5.09969e11 1.45827 0.729136 0.684368i $$-0.239922\pi$$
0.729136 + 0.684368i $$0.239922\pi$$
$$770$$ 0 0
$$771$$ − 7.49386e11i − 2.12074i
$$772$$ 0 0
$$773$$ − 1.49408e11i − 0.418462i −0.977866 0.209231i $$-0.932904\pi$$
0.977866 0.209231i $$-0.0670961\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 2.88029e11i − 0.790227i
$$778$$ 0 0
$$779$$ − 8.20966e9i − 0.0222933i
$$780$$ 0 0
$$781$$ 1.93261e11 0.519445
$$782$$ 0 0
$$783$$ −5.49393e10 −0.146163
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −7.33252e11 −1.91141 −0.955706 0.294323i $$-0.904906\pi$$
−0.955706 + 0.294323i $$0.904906\pi$$
$$788$$ 0 0
$$789$$ 1.04891e12 2.70665
$$790$$ 0 0
$$791$$ 6.04040e11i 1.54298i
$$792$$ 0 0
$$793$$ 6.60236e11i 1.66958i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 3.02703e11i − 0.750212i −0.926982 0.375106i $$-0.877606\pi$$
0.926982 0.375106i $$-0.122394\pi$$
$$798$$ 0 0
$$799$$ 1.44438e11i 0.354401i
$$800$$ 0 0
$$801$$ −1.18034e12 −2.86732
$$802$$ 0 0
$$803$$ 1.24911e11 0.300427
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −4.91911e10 −0.115982
$$808$$ 0 0
$$809$$ 5.84316e11 1.36412 0.682062 0.731295i $$-0.261084\pi$$
0.682062 + 0.731295i $$0.261084\pi$$
$$810$$ 0 0
$$811$$ 1.21470e11i 0.280793i 0.990095 + 0.140396i $$0.0448377\pi$$
−0.990095 + 0.140396i $$0.955162\pi$$
$$812$$ 0 0
$$813$$ − 9.25344e8i − 0.00211807i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.42714e10i 0.121810i
$$818$$ 0 0
$$819$$ − 9.63078e11i − 2.14055i
$$820$$ 0 0
$$821$$ 4.52470e11 0.995903 0.497952 0.867205i $$-0.334086\pi$$
0.497952 + 0.867205i $$0.334086\pi$$
$$822$$ 0 0
$$823$$ 3.06704e11 0.668528 0.334264 0.942479i $$-0.391512\pi$$
0.334264 + 0.942479i $$0.391512\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.93276e11 −0.626982 −0.313491 0.949591i $$-0.601499\pi$$
−0.313491 + 0.949591i $$0.601499\pi$$
$$828$$ 0 0
$$829$$ −3.35532e11 −0.710421 −0.355210 0.934786i $$-0.615591\pi$$
−0.355210 + 0.934786i $$0.615591\pi$$
$$830$$ 0 0
$$831$$ − 3.05356e11i − 0.640327i
$$832$$ 0 0
$$833$$ − 3.86358e10i − 0.0802434i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1.17482e12i − 2.39369i
$$838$$ 0 0
$$839$$ − 3.42844e11i − 0.691908i −0.938252 0.345954i $$-0.887555\pi$$
0.938252 0.345954i $$-0.112445\pi$$
$$840$$ 0 0
$$841$$ −4.97224e11 −0.993958
$$842$$ 0 0
$$843$$ 1.48029e12 2.93113
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −3.91175e11 −0.760042
$$848$$ 0 0
$$849$$ 1.81802e11 0.349919
$$850$$ 0 0
$$851$$ − 1.39972e11i − 0.266883i
$$852$$ 0 0
$$853$$ − 5.08662e11i − 0.960801i −0.877049 0.480400i $$-0.840491\pi$$
0.877049 0.480400i $$-0.159509\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 6.06764e11i − 1.12486i −0.826846 0.562428i $$-0.809867\pi$$
0.826846 0.562428i $$-0.190133\pi$$
$$858$$ 0 0
$$859$$ − 9.49431e11i − 1.74378i −0.489705 0.871888i $$-0.662895\pi$$
0.489705 0.871888i $$-0.337105\pi$$
$$860$$ 0 0
$$861$$ −2.74272e10 −0.0499078
$$862$$ 0 0
$$863$$ −2.99836e10 −0.0540556 −0.0270278 0.999635i $$-0.508604\pi$$
−0.0270278 + 0.999635i $$0.508604\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 6.30222e11 1.11536
$$868$$ 0 0
$$869$$ −9.35196e11 −1.63992
$$870$$ 0 0
$$871$$ 2.07679e11i 0.360844i
$$872$$ 0 0
$$873$$ 6.34669e11i 1.09267i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.80195e11i 1.48792i 0.668223 + 0.743961i $$0.267055\pi$$
−0.668223 + 0.743961i $$0.732945\pi$$
$$878$$ 0 0
$$879$$ − 3.03546e11i − 0.508474i
$$880$$ 0 0
$$881$$ 1.04085e12 1.72776 0.863879 0.503699i $$-0.168028\pi$$
0.863879 + 0.503699i $$0.168028\pi$$
$$882$$ 0 0
$$883$$ −7.60446e11 −1.25091 −0.625454 0.780261i $$-0.715086\pi$$
−0.625454 + 0.780261i $$0.715086\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −3.34097e11 −0.539732 −0.269866 0.962898i $$-0.586979\pi$$
−0.269866 + 0.962898i $$0.586979\pi$$
$$888$$ 0 0
$$889$$ 9.39853e11 1.50471
$$890$$ 0 0
$$891$$ − 1.00945e12i − 1.60167i
$$892$$ 0 0
$$893$$ − 3.11471e11i − 0.489792i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.93823e11i − 1.07171i
$$898$$ 0 0
$$899$$ 6.46345e10i 0.0989523i
$$900$$ 0 0
$$901$$ 5.62442e11 0.853451
$$902$$ 0 0
$$903$$ 1.81312e11 0.272695
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −7.65213e11 −1.13071 −0.565357 0.824846i $$-0.691262\pi$$
−0.565357 + 0.824846i $$0.691262\pi$$
$$908$$ 0 0
$$909$$ 8.97412e11 1.31443
$$910$$ 0 0
$$911$$ 3.83541e11i 0.556851i 0.960458 + 0.278425i $$0.0898125\pi$$
−0.960458 + 0.278425i $$0.910188\pi$$
$$912$$ 0 0
$$913$$ − 1.40786e12i − 2.02618i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 5.52937e11i − 0.781984i
$$918$$ 0 0
$$919$$ 6.82775e11i 0.957229i 0.878025 + 0.478615i $$0.158861\pi$$
−0.878025 + 0.478615i $$0.841139\pi$$
$$920$$ 0 0
$$921$$ −7.74022e11 −1.07576
$$922$$ 0 0
$$923$$ 2.79384e11 0.384941
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.24226e12 3.03645
$$928$$ 0 0
$$929$$ −2.94973e11 −0.396021 −0.198011 0.980200i $$-0.563448\pi$$
−0.198011 + 0.980200i $$0.563448\pi$$
$$930$$ 0 0
$$931$$ 8.33152e10i 0.110898i
$$932$$ 0 0
$$933$$ − 1.52705e12i − 2.01524i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.03941e12i − 1.34843i −0.738533 0.674217i $$-0.764481\pi$$
0.738533 0.674217i $$-0.235519\pi$$
$$938$$ 0 0
$$939$$ − 4.25680e10i − 0.0547546i
$$940$$ 0 0
$$941$$ 1.26891e11 0.161836 0.0809178 0.996721i $$-0.474215\pi$$
0.0809178 + 0.996721i $$0.474215\pi$$
$$942$$ 0 0
$$943$$ −1.33286e10 −0.0168554
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.13064e11 0.762265 0.381133 0.924520i $$-0.375534\pi$$
0.381133 + 0.924520i $$0.375534\pi$$
$$948$$ 0 0
$$949$$ 1.80575e11 0.222635
$$950$$ 0 0
$$951$$ − 7.54189e11i − 0.922058i
$$952$$ 0 0
$$953$$ 6.58227e11i 0.798002i 0.916951 + 0.399001i $$0.130643\pi$$
−0.916951 + 0.399001i $$0.869357\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1.49628e11i 0.178388i
$$958$$ 0 0
$$959$$ 9.87647e11i 1.16769i
$$960$$ 0 0
$$961$$ −5.29246e11 −0.620532
$$962$$ 0 0
$$963$$ −1.73151e12 −2.01335
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 5.41485e11 0.619271 0.309635 0.950855i $$-0.399793\pi$$
0.309635 + 0.950855i $$0.399793\pi$$
$$968$$ 0 0
$$969$$ 7.76826e11 0.881107
$$970$$ 0 0
$$971$$ 3.54981e11i 0.399327i 0.979865 + 0.199663i $$0.0639849\pi$$
−0.979865 + 0.199663i $$0.936015\pi$$
$$972$$ 0 0
$$973$$ 8.97263e11i 1.00108i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 6.02238e11i − 0.660982i −0.943809 0.330491i $$-0.892786\pi$$
0.943809 0.330491i $$-0.107214\pi$$
$$978$$ 0 0
$$979$$ 1.66371e12i 1.81112i
$$980$$ 0 0
$$981$$ 2.12306e12 2.29238
$$982$$ 0 0
$$983$$ 1.37465e12 1.47223 0.736117 0.676854i $$-0.236657\pi$$
0.736117 + 0.676854i $$0.236657\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −1.04058e12 −1.09649
$$988$$ 0 0
$$989$$ 8.81113e10 0.0920972
$$990$$ 0 0
$$991$$ − 1.01081e12i − 1.04803i −0.851709 0.524015i $$-0.824433\pi$$
0.851709 0.524015i $$-0.175567\pi$$
$$992$$ 0 0
$$993$$ 1.43411e12i 1.47498i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3.28556e11i 0.332528i 0.986081 + 0.166264i $$0.0531704\pi$$
−0.986081 + 0.166264i $$0.946830\pi$$
$$998$$ 0 0
$$999$$ − 7.93172e11i − 0.796353i
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 400.9.h.b.399.3 4
4.3 odd 2 inner 400.9.h.b.399.2 4
5.2 odd 4 16.9.c.a.15.1 2
5.3 odd 4 400.9.b.c.351.2 2
5.4 even 2 inner 400.9.h.b.399.1 4
15.2 even 4 144.9.g.g.127.2 2
20.3 even 4 400.9.b.c.351.1 2
20.7 even 4 16.9.c.a.15.2 yes 2
20.19 odd 2 inner 400.9.h.b.399.4 4
40.27 even 4 64.9.c.d.63.1 2
40.37 odd 4 64.9.c.d.63.2 2
60.47 odd 4 144.9.g.g.127.1 2
80.27 even 4 256.9.d.f.127.3 4
80.37 odd 4 256.9.d.f.127.1 4
80.67 even 4 256.9.d.f.127.2 4
80.77 odd 4 256.9.d.f.127.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 5.2 odd 4
16.9.c.a.15.2 yes 2 20.7 even 4
64.9.c.d.63.1 2 40.27 even 4
64.9.c.d.63.2 2 40.37 odd 4
144.9.g.g.127.1 2 60.47 odd 4
144.9.g.g.127.2 2 15.2 even 4
256.9.d.f.127.1 4 80.37 odd 4
256.9.d.f.127.2 4 80.67 even 4
256.9.d.f.127.3 4 80.27 even 4
256.9.d.f.127.4 4 80.77 odd 4
400.9.b.c.351.1 2 20.3 even 4
400.9.b.c.351.2 2 5.3 odd 4
400.9.h.b.399.1 4 5.4 even 2 inner
400.9.h.b.399.2 4 4.3 odd 2 inner
400.9.h.b.399.3 4 1.1 even 1 trivial
400.9.h.b.399.4 4 20.19 odd 2 inner