# Properties

 Label 1152.4.d.p.577.3 Level $1152$ Weight $4$ Character 1152.577 Analytic conductor $67.970$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(577,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1152.d (of order $$2$$, degree $$1$$, 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: $$67.9702003266$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.1534132224.8 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1$$ x^8 + 18*x^6 + 107*x^4 + 210*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{26}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 577.3 Root $$2.21597i$$ of defining polynomial Character $$\chi$$ $$=$$ 1152.577 Dual form 1152.4.d.p.577.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.67763i q^{5} -33.0917 q^{7} +O(q^{10})$$ $$q-5.67763i q^{5} -33.0917 q^{7} -34.6274i q^{11} -82.2421i q^{13} -97.8823 q^{17} +55.8823i q^{19} -130.418 q^{23} +92.7645 q^{25} -147.451i q^{29} +101.223 q^{31} +187.882i q^{35} -184.439i q^{37} -237.411 q^{41} +199.882i q^{43} +334.813 q^{47} +752.058 q^{49} +102.030i q^{53} -196.602 q^{55} -105.961i q^{59} +717.803i q^{61} -466.940 q^{65} -316.471i q^{67} -800.045 q^{71} +301.058 q^{73} +1145.88i q^{77} -42.8329 q^{79} +1236.67i q^{83} +555.739i q^{85} -1325.29 q^{89} +2721.53i q^{91} +317.279 q^{95} -505.765 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 240 q^{17} - 344 q^{25} + 816 q^{41} + 1672 q^{49} + 1152 q^{65} - 1936 q^{73} - 7344 q^{89} - 2960 q^{97}+O(q^{100})$$ 8 * q - 240 * q^17 - 344 * q^25 + 816 * q^41 + 1672 * q^49 + 1152 * q^65 - 1936 * q^73 - 7344 * q^89 - 2960 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 5.67763i − 0.507823i −0.967227 0.253911i $$-0.918283\pi$$
0.967227 0.253911i $$-0.0817172\pi$$
$$6$$ 0 0
$$7$$ −33.0917 −1.78678 −0.893391 0.449280i $$-0.851680\pi$$
−0.893391 + 0.449280i $$0.851680\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 34.6274i − 0.949142i −0.880217 0.474571i $$-0.842603\pi$$
0.880217 0.474571i $$-0.157397\pi$$
$$12$$ 0 0
$$13$$ − 82.2421i − 1.75460i −0.479939 0.877302i $$-0.659341\pi$$
0.479939 0.877302i $$-0.340659\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −97.8823 −1.39647 −0.698233 0.715870i $$-0.746030\pi$$
−0.698233 + 0.715870i $$0.746030\pi$$
$$18$$ 0 0
$$19$$ 55.8823i 0.674751i 0.941370 + 0.337375i $$0.109539\pi$$
−0.941370 + 0.337375i $$0.890461\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −130.418 −1.18235 −0.591176 0.806542i $$-0.701336\pi$$
−0.591176 + 0.806542i $$0.701336\pi$$
$$24$$ 0 0
$$25$$ 92.7645 0.742116
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 147.451i − 0.944173i −0.881552 0.472086i $$-0.843501\pi$$
0.881552 0.472086i $$-0.156499\pi$$
$$30$$ 0 0
$$31$$ 101.223 0.586459 0.293230 0.956042i $$-0.405270\pi$$
0.293230 + 0.956042i $$0.405270\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 187.882i 0.907368i
$$36$$ 0 0
$$37$$ − 184.439i − 0.819504i −0.912197 0.409752i $$-0.865615\pi$$
0.912197 0.409752i $$-0.134385\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −237.411 −0.904327 −0.452164 0.891935i $$-0.649348\pi$$
−0.452164 + 0.891935i $$0.649348\pi$$
$$42$$ 0 0
$$43$$ 199.882i 0.708878i 0.935079 + 0.354439i $$0.115328\pi$$
−0.935079 + 0.354439i $$0.884672\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 334.813 1.03910 0.519548 0.854441i $$-0.326100\pi$$
0.519548 + 0.854441i $$0.326100\pi$$
$$48$$ 0 0
$$49$$ 752.058 2.19259
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 102.030i 0.264433i 0.991221 + 0.132216i $$0.0422094\pi$$
−0.991221 + 0.132216i $$0.957791\pi$$
$$54$$ 0 0
$$55$$ −196.602 −0.481996
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 105.961i − 0.233813i −0.993143 0.116907i $$-0.962702\pi$$
0.993143 0.116907i $$-0.0372978\pi$$
$$60$$ 0 0
$$61$$ 717.803i 1.50664i 0.657653 + 0.753321i $$0.271549\pi$$
−0.657653 + 0.753321i $$0.728451\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −466.940 −0.891028
$$66$$ 0 0
$$67$$ − 316.471i − 0.577061i −0.957471 0.288530i $$-0.906833\pi$$
0.957471 0.288530i $$-0.0931667\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −800.045 −1.33729 −0.668647 0.743580i $$-0.733126\pi$$
−0.668647 + 0.743580i $$0.733126\pi$$
$$72$$ 0 0
$$73$$ 301.058 0.482687 0.241344 0.970440i $$-0.422412\pi$$
0.241344 + 0.970440i $$0.422412\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1145.88i 1.69591i
$$78$$ 0 0
$$79$$ −42.8329 −0.0610010 −0.0305005 0.999535i $$-0.509710\pi$$
−0.0305005 + 0.999535i $$0.509710\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1236.67i 1.63544i 0.575614 + 0.817721i $$0.304763\pi$$
−0.575614 + 0.817721i $$0.695237\pi$$
$$84$$ 0 0
$$85$$ 555.739i 0.709158i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1325.29 −1.57844 −0.789218 0.614113i $$-0.789514\pi$$
−0.789218 + 0.614113i $$0.789514\pi$$
$$90$$ 0 0
$$91$$ 2721.53i 3.13509i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 317.279 0.342654
$$96$$ 0 0
$$97$$ −505.765 −0.529408 −0.264704 0.964330i $$-0.585274\pi$$
−0.264704 + 0.964330i $$0.585274\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 1281.97i − 1.26298i −0.775383 0.631491i $$-0.782443\pi$$
0.775383 0.631491i $$-0.217557\pi$$
$$102$$ 0 0
$$103$$ 161.562 0.154555 0.0772775 0.997010i $$-0.475377\pi$$
0.0772775 + 0.997010i $$0.475377\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 481.726i − 0.435235i −0.976034 0.217618i $$-0.930171\pi$$
0.976034 0.217618i $$-0.0698286\pi$$
$$108$$ 0 0
$$109$$ − 286.637i − 0.251879i −0.992038 0.125940i $$-0.959805\pi$$
0.992038 0.125940i $$-0.0401945\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1397.29 1.16324 0.581621 0.813460i $$-0.302419\pi$$
0.581621 + 0.813460i $$0.302419\pi$$
$$114$$ 0 0
$$115$$ 740.468i 0.600426i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3239.09 2.49518
$$120$$ 0 0
$$121$$ 131.942 0.0991300
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 1236.39i − 0.884686i
$$126$$ 0 0
$$127$$ 443.829 0.310106 0.155053 0.987906i $$-0.450445\pi$$
0.155053 + 0.987906i $$0.450445\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1365.57i 0.910765i 0.890296 + 0.455382i $$0.150497\pi$$
−0.890296 + 0.455382i $$0.849503\pi$$
$$132$$ 0 0
$$133$$ − 1849.24i − 1.20563i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1223.06 0.762721 0.381360 0.924426i $$-0.375456\pi$$
0.381360 + 0.924426i $$0.375456\pi$$
$$138$$ 0 0
$$139$$ 2176.70i 1.32824i 0.747625 + 0.664121i $$0.231194\pi$$
−0.747625 + 0.664121i $$0.768806\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2847.83 −1.66537
$$144$$ 0 0
$$145$$ −837.174 −0.479473
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2875.88i 1.58122i 0.612320 + 0.790610i $$0.290236\pi$$
−0.612320 + 0.790610i $$0.709764\pi$$
$$150$$ 0 0
$$151$$ 2330.03 1.25573 0.627863 0.778323i $$-0.283930\pi$$
0.627863 + 0.778323i $$0.283930\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 574.708i − 0.297817i
$$156$$ 0 0
$$157$$ − 2447.31i − 1.24405i −0.782996 0.622027i $$-0.786310\pi$$
0.782996 0.622027i $$-0.213690\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4315.76 2.11261
$$162$$ 0 0
$$163$$ 1436.82i 0.690432i 0.938523 + 0.345216i $$0.112194\pi$$
−0.938523 + 0.345216i $$0.887806\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1769.42 −0.819889 −0.409945 0.912110i $$-0.634452\pi$$
−0.409945 + 0.912110i $$0.634452\pi$$
$$168$$ 0 0
$$169$$ −4566.76 −2.07863
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 3897.86i − 1.71300i −0.516148 0.856499i $$-0.672635\pi$$
0.516148 0.856499i $$-0.327365\pi$$
$$174$$ 0 0
$$175$$ −3069.73 −1.32600
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4146.74i 1.73152i 0.500460 + 0.865760i $$0.333164\pi$$
−0.500460 + 0.865760i $$0.666836\pi$$
$$180$$ 0 0
$$181$$ 1104.22i 0.453457i 0.973958 + 0.226728i $$0.0728030\pi$$
−0.973958 + 0.226728i $$0.927197\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1047.18 −0.416163
$$186$$ 0 0
$$187$$ 3389.41i 1.32544i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 35.0686 0.0132852 0.00664260 0.999978i $$-0.497886\pi$$
0.00664260 + 0.999978i $$0.497886\pi$$
$$192$$ 0 0
$$193$$ 615.884 0.229701 0.114851 0.993383i $$-0.463361\pi$$
0.114851 + 0.993383i $$0.463361\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 589.638i − 0.213249i −0.994299 0.106624i $$-0.965996\pi$$
0.994299 0.106624i $$-0.0340042\pi$$
$$198$$ 0 0
$$199$$ −4813.82 −1.71479 −0.857394 0.514661i $$-0.827918\pi$$
−0.857394 + 0.514661i $$0.827918\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4879.41i 1.68703i
$$204$$ 0 0
$$205$$ 1347.93i 0.459238i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1935.06 0.640434
$$210$$ 0 0
$$211$$ 3294.35i 1.07484i 0.843313 + 0.537422i $$0.180602\pi$$
−0.843313 + 0.537422i $$0.819398\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1134.86 0.359984
$$216$$ 0 0
$$217$$ −3349.65 −1.04787
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8050.04i 2.45025i
$$222$$ 0 0
$$223$$ 1572.78 0.472294 0.236147 0.971717i $$-0.424115\pi$$
0.236147 + 0.971717i $$0.424115\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 1492.51i − 0.436394i −0.975905 0.218197i $$-0.929982\pi$$
0.975905 0.218197i $$-0.0700176\pi$$
$$228$$ 0 0
$$229$$ 6163.31i 1.77853i 0.457393 + 0.889265i $$0.348783\pi$$
−0.457393 + 0.889265i $$0.651217\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2540.35 −0.714266 −0.357133 0.934054i $$-0.616246\pi$$
−0.357133 + 0.934054i $$0.616246\pi$$
$$234$$ 0 0
$$235$$ − 1900.95i − 0.527677i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1339.25 0.362465 0.181232 0.983440i $$-0.441991\pi$$
0.181232 + 0.983440i $$0.441991\pi$$
$$240$$ 0 0
$$241$$ 2542.71 0.679627 0.339814 0.940493i $$-0.389636\pi$$
0.339814 + 0.940493i $$0.389636\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 4269.91i − 1.11345i
$$246$$ 0 0
$$247$$ 4595.87 1.18392
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 4701.45i − 1.18228i −0.806568 0.591141i $$-0.798678\pi$$
0.806568 0.591141i $$-0.201322\pi$$
$$252$$ 0 0
$$253$$ 4516.05i 1.12222i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 863.884 0.209679 0.104840 0.994489i $$-0.466567\pi$$
0.104840 + 0.994489i $$0.466567\pi$$
$$258$$ 0 0
$$259$$ 6103.41i 1.46428i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7091.36 −1.66263 −0.831315 0.555801i $$-0.812412\pi$$
−0.831315 + 0.555801i $$0.812412\pi$$
$$264$$ 0 0
$$265$$ 579.290 0.134285
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 395.596i 0.0896650i 0.998995 + 0.0448325i $$0.0142754\pi$$
−0.998995 + 0.0448325i $$0.985725\pi$$
$$270$$ 0 0
$$271$$ 5532.21 1.24007 0.620033 0.784576i $$-0.287119\pi$$
0.620033 + 0.784576i $$0.287119\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3212.20i − 0.704373i
$$276$$ 0 0
$$277$$ − 3830.31i − 0.830834i −0.909631 0.415417i $$-0.863636\pi$$
0.909631 0.415417i $$-0.136364\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5283.17 −1.12159 −0.560797 0.827954i $$-0.689505\pi$$
−0.560797 + 0.827954i $$0.689505\pi$$
$$282$$ 0 0
$$283$$ 4639.76i 0.974577i 0.873241 + 0.487288i $$0.162014\pi$$
−0.873241 + 0.487288i $$0.837986\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 7856.33 1.61584
$$288$$ 0 0
$$289$$ 4667.94 0.950119
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 6022.12i 1.20074i 0.799723 + 0.600369i $$0.204980\pi$$
−0.799723 + 0.600369i $$0.795020\pi$$
$$294$$ 0 0
$$295$$ −601.609 −0.118736
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 10725.9i 2.07456i
$$300$$ 0 0
$$301$$ − 6614.44i − 1.26661i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4075.42 0.765107
$$306$$ 0 0
$$307$$ − 2998.82i − 0.557497i −0.960364 0.278749i $$-0.910080\pi$$
0.960364 0.278749i $$-0.0899196\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2403.97 0.438318 0.219159 0.975689i $$-0.429669\pi$$
0.219159 + 0.975689i $$0.429669\pi$$
$$312$$ 0 0
$$313$$ 5845.75 1.05566 0.527830 0.849350i $$-0.323006\pi$$
0.527830 + 0.849350i $$0.323006\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8917.38i − 1.57997i −0.613127 0.789984i $$-0.710089\pi$$
0.613127 0.789984i $$-0.289911\pi$$
$$318$$ 0 0
$$319$$ −5105.86 −0.896154
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 5469.88i − 0.942267i
$$324$$ 0 0
$$325$$ − 7629.15i − 1.30212i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −11079.5 −1.85664
$$330$$ 0 0
$$331$$ − 9011.29i − 1.49639i −0.663478 0.748196i $$-0.730920\pi$$
0.663478 0.748196i $$-0.269080\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −1796.81 −0.293045
$$336$$ 0 0
$$337$$ 4516.35 0.730033 0.365017 0.931001i $$-0.381063\pi$$
0.365017 + 0.931001i $$0.381063\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 3505.10i − 0.556633i
$$342$$ 0 0
$$343$$ −13536.4 −2.13090
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3651.48i 0.564904i 0.959281 + 0.282452i $$0.0911478\pi$$
−0.959281 + 0.282452i $$0.908852\pi$$
$$348$$ 0 0
$$349$$ 3250.36i 0.498532i 0.968435 + 0.249266i $$0.0801894\pi$$
−0.968435 + 0.249266i $$0.919811\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −592.345 −0.0893125 −0.0446563 0.999002i $$-0.514219\pi$$
−0.0446563 + 0.999002i $$0.514219\pi$$
$$354$$ 0 0
$$355$$ 4542.36i 0.679108i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3443.48 0.506239 0.253120 0.967435i $$-0.418543\pi$$
0.253120 + 0.967435i $$0.418543\pi$$
$$360$$ 0 0
$$361$$ 3736.17 0.544711
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 1709.30i − 0.245120i
$$366$$ 0 0
$$367$$ 3297.39 0.468998 0.234499 0.972116i $$-0.424655\pi$$
0.234499 + 0.972116i $$0.424655\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 3376.35i − 0.472483i
$$372$$ 0 0
$$373$$ − 50.1819i − 0.00696601i −0.999994 0.00348300i $$-0.998891\pi$$
0.999994 0.00348300i $$-0.00110868\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12126.7 −1.65665
$$378$$ 0 0
$$379$$ − 6769.28i − 0.917453i −0.888578 0.458726i $$-0.848306\pi$$
0.888578 0.458726i $$-0.151694\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −4208.46 −0.561468 −0.280734 0.959786i $$-0.590578\pi$$
−0.280734 + 0.959786i $$0.590578\pi$$
$$384$$ 0 0
$$385$$ 6505.88 0.861221
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2490.47i 0.324607i 0.986741 + 0.162303i $$0.0518923\pi$$
−0.986741 + 0.162303i $$0.948108\pi$$
$$390$$ 0 0
$$391$$ 12765.6 1.65112
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 243.190i 0.0309777i
$$396$$ 0 0
$$397$$ − 7905.51i − 0.999411i −0.866195 0.499706i $$-0.833442\pi$$
0.866195 0.499706i $$-0.166558\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10389.4 1.29382 0.646911 0.762566i $$-0.276061\pi$$
0.646911 + 0.762566i $$0.276061\pi$$
$$402$$ 0 0
$$403$$ − 8324.81i − 1.02900i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6386.66 −0.777826
$$408$$ 0 0
$$409$$ −13448.1 −1.62583 −0.812917 0.582379i $$-0.802122\pi$$
−0.812917 + 0.582379i $$0.802122\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3506.44i 0.417773i
$$414$$ 0 0
$$415$$ 7021.33 0.830515
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 5191.34i − 0.605283i −0.953105 0.302641i $$-0.902132\pi$$
0.953105 0.302641i $$-0.0978684\pi$$
$$420$$ 0 0
$$421$$ 5061.52i 0.585946i 0.956121 + 0.292973i $$0.0946447\pi$$
−0.956121 + 0.292973i $$0.905355\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −9080.00 −1.03634
$$426$$ 0 0
$$427$$ − 23753.3i − 2.69204i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3005.64 0.335909 0.167954 0.985795i $$-0.446284\pi$$
0.167954 + 0.985795i $$0.446284\pi$$
$$432$$ 0 0
$$433$$ 5895.88 0.654360 0.327180 0.944962i $$-0.393902\pi$$
0.327180 + 0.944962i $$0.393902\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 7288.07i − 0.797794i
$$438$$ 0 0
$$439$$ 11556.8 1.25644 0.628218 0.778038i $$-0.283785\pi$$
0.628218 + 0.778038i $$0.283785\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 14007.8i − 1.50233i −0.660117 0.751163i $$-0.729493\pi$$
0.660117 0.751163i $$-0.270507\pi$$
$$444$$ 0 0
$$445$$ 7524.53i 0.801566i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3783.06 −0.397625 −0.198813 0.980038i $$-0.563709\pi$$
−0.198813 + 0.980038i $$0.563709\pi$$
$$450$$ 0 0
$$451$$ 8220.94i 0.858335i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 15451.8 1.59207
$$456$$ 0 0
$$457$$ −1545.53 −0.158198 −0.0790992 0.996867i $$-0.525204\pi$$
−0.0790992 + 0.996867i $$0.525204\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12730.2i 1.28613i 0.765811 + 0.643065i $$0.222338\pi$$
−0.765811 + 0.643065i $$0.777662\pi$$
$$462$$ 0 0
$$463$$ −19656.4 −1.97303 −0.986513 0.163682i $$-0.947663\pi$$
−0.986513 + 0.163682i $$0.947663\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6016.26i − 0.596144i −0.954543 0.298072i $$-0.903656\pi$$
0.954543 0.298072i $$-0.0963436\pi$$
$$468$$ 0 0
$$469$$ 10472.6i 1.03108i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6921.41 0.672826
$$474$$ 0 0
$$475$$ 5183.89i 0.500743i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −13890.6 −1.32501 −0.662505 0.749057i $$-0.730507\pi$$
−0.662505 + 0.749057i $$0.730507\pi$$
$$480$$ 0 0
$$481$$ −15168.7 −1.43791
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2871.54i 0.268846i
$$486$$ 0 0
$$487$$ −9314.23 −0.866669 −0.433335 0.901233i $$-0.642663\pi$$
−0.433335 + 0.901233i $$0.642663\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7253.34i 0.666677i 0.942807 + 0.333339i $$0.108175\pi$$
−0.942807 + 0.333339i $$0.891825\pi$$
$$492$$ 0 0
$$493$$ 14432.9i 1.31851i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 26474.8 2.38945
$$498$$ 0 0
$$499$$ − 16208.2i − 1.45407i −0.686602 0.727034i $$-0.740898\pi$$
0.686602 0.727034i $$-0.259102\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −2182.04 −0.193425 −0.0967123 0.995312i $$-0.530833\pi$$
−0.0967123 + 0.995312i $$0.530833\pi$$
$$504$$ 0 0
$$505$$ −7278.58 −0.641371
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1795.97i 0.156395i 0.996938 + 0.0781974i $$0.0249164\pi$$
−0.996938 + 0.0781974i $$0.975084\pi$$
$$510$$ 0 0
$$511$$ −9962.51 −0.862457
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 917.288i − 0.0784865i
$$516$$ 0 0
$$517$$ − 11593.7i − 0.986249i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20946.1 1.76135 0.880676 0.473718i $$-0.157088\pi$$
0.880676 + 0.473718i $$0.157088\pi$$
$$522$$ 0 0
$$523$$ − 5363.64i − 0.448443i −0.974538 0.224221i $$-0.928016\pi$$
0.974538 0.224221i $$-0.0719838\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9907.96 −0.818970
$$528$$ 0 0
$$529$$ 4841.96 0.397958
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 19525.2i 1.58674i
$$534$$ 0 0
$$535$$ −2735.06 −0.221022
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 26041.8i − 2.08108i
$$540$$ 0 0
$$541$$ 3431.20i 0.272678i 0.990662 + 0.136339i $$0.0435337\pi$$
−0.990662 + 0.136339i $$0.956466\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1627.42 −0.127910
$$546$$ 0 0
$$547$$ − 19449.0i − 1.52026i −0.649773 0.760128i $$-0.725136\pi$$
0.649773 0.760128i $$-0.274864\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8239.91 0.637082
$$552$$ 0 0
$$553$$ 1417.41 0.108996
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 9898.44i − 0.752981i −0.926420 0.376491i $$-0.877131\pi$$
0.926420 0.376491i $$-0.122869\pi$$
$$558$$ 0 0
$$559$$ 16438.7 1.24380
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 13809.8i 1.03378i 0.856053 + 0.516888i $$0.172909\pi$$
−0.856053 + 0.516888i $$0.827091\pi$$
$$564$$ 0 0
$$565$$ − 7933.32i − 0.590721i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 123.053 0.00906615 0.00453308 0.999990i $$-0.498557\pi$$
0.00453308 + 0.999990i $$0.498557\pi$$
$$570$$ 0 0
$$571$$ 2540.72i 0.186210i 0.995656 + 0.0931049i $$0.0296792\pi$$
−0.995656 + 0.0931049i $$0.970321\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −12098.2 −0.877443
$$576$$ 0 0
$$577$$ −15618.2 −1.12685 −0.563427 0.826166i $$-0.690517\pi$$
−0.563427 + 0.826166i $$0.690517\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 40923.3i − 2.92218i
$$582$$ 0 0
$$583$$ 3533.04 0.250984
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1809.56i 0.127238i 0.997974 + 0.0636188i $$0.0202642\pi$$
−0.997974 + 0.0636188i $$0.979736\pi$$
$$588$$ 0 0
$$589$$ 5656.58i 0.395714i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −898.229 −0.0622021 −0.0311010 0.999516i $$-0.509901\pi$$
−0.0311010 + 0.999516i $$0.509901\pi$$
$$594$$ 0 0
$$595$$ − 18390.3i − 1.26711i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −22990.9 −1.56825 −0.784125 0.620603i $$-0.786888\pi$$
−0.784125 + 0.620603i $$0.786888\pi$$
$$600$$ 0 0
$$601$$ −26893.7 −1.82532 −0.912661 0.408717i $$-0.865976\pi$$
−0.912661 + 0.408717i $$0.865976\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 749.118i − 0.0503405i
$$606$$ 0 0
$$607$$ 6306.93 0.421730 0.210865 0.977515i $$-0.432372\pi$$
0.210865 + 0.977515i $$0.432372\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 27535.7i − 1.82320i
$$612$$ 0 0
$$613$$ 2612.97i 0.172164i 0.996288 + 0.0860822i $$0.0274348\pi$$
−0.996288 + 0.0860822i $$0.972565\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2803.88 0.182949 0.0914747 0.995807i $$-0.470842\pi$$
0.0914747 + 0.995807i $$0.470842\pi$$
$$618$$ 0 0
$$619$$ − 10547.1i − 0.684849i −0.939545 0.342425i $$-0.888752\pi$$
0.939545 0.342425i $$-0.111248\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 43856.2 2.82032
$$624$$ 0 0
$$625$$ 4575.82 0.292852
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 18053.3i 1.14441i
$$630$$ 0 0
$$631$$ −14161.4 −0.893431 −0.446716 0.894676i $$-0.647406\pi$$
−0.446716 + 0.894676i $$0.647406\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 2519.90i − 0.157479i
$$636$$ 0 0
$$637$$ − 61850.8i − 3.84713i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5622.61 −0.346458 −0.173229 0.984882i $$-0.555420\pi$$
−0.173229 + 0.984882i $$0.555420\pi$$
$$642$$ 0 0
$$643$$ − 29438.7i − 1.80552i −0.430146 0.902759i $$-0.641538\pi$$
0.430146 0.902759i $$-0.358462\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 4607.39 0.279962 0.139981 0.990154i $$-0.455296\pi$$
0.139981 + 0.990154i $$0.455296\pi$$
$$648$$ 0 0
$$649$$ −3669.17 −0.221922
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 16634.1i − 0.996850i −0.866933 0.498425i $$-0.833912\pi$$
0.866933 0.498425i $$-0.166088\pi$$
$$654$$ 0 0
$$655$$ 7753.19 0.462507
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 20619.9i 1.21887i 0.792835 + 0.609437i $$0.208604\pi$$
−0.792835 + 0.609437i $$0.791396\pi$$
$$660$$ 0 0
$$661$$ − 881.112i − 0.0518476i −0.999664 0.0259238i $$-0.991747\pi$$
0.999664 0.0259238i $$-0.00825273\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −10499.3 −0.612248
$$666$$ 0 0
$$667$$ 19230.4i 1.11635i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24855.6 1.43002
$$672$$ 0 0
$$673$$ 8943.86 0.512274 0.256137 0.966641i $$-0.417550\pi$$
0.256137 + 0.966641i $$0.417550\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 21748.7i 1.23467i 0.786702 + 0.617333i $$0.211787\pi$$
−0.786702 + 0.617333i $$0.788213\pi$$
$$678$$ 0 0
$$679$$ 16736.6 0.945937
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 7922.37i − 0.443838i −0.975065 0.221919i $$-0.928768\pi$$
0.975065 0.221919i $$-0.0712320\pi$$
$$684$$ 0 0
$$685$$ − 6944.06i − 0.387327i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8391.18 0.463975
$$690$$ 0 0
$$691$$ 177.517i 0.00977288i 0.999988 + 0.00488644i $$0.00155541\pi$$
−0.999988 + 0.00488644i $$0.998445\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12358.5 0.674511
$$696$$ 0 0
$$697$$ 23238.3 1.26286
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 767.980i 0.0413783i 0.999786 + 0.0206891i $$0.00658603\pi$$
−0.999786 + 0.0206891i $$0.993414\pi$$
$$702$$ 0 0
$$703$$ 10306.9 0.552961
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 42422.7i 2.25667i
$$708$$ 0 0
$$709$$ 31634.2i 1.67566i 0.545928 + 0.837832i $$0.316177\pi$$
−0.545928 + 0.837832i $$0.683823\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −13201.4 −0.693401
$$714$$ 0 0
$$715$$ 16168.9i 0.845712i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −17351.7 −0.900011 −0.450005 0.893026i $$-0.648578\pi$$
−0.450005 + 0.893026i $$0.648578\pi$$
$$720$$ 0 0
$$721$$ −5346.35 −0.276156
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 13678.2i − 0.700686i
$$726$$ 0 0
$$727$$ −16364.6 −0.834842 −0.417421 0.908713i $$-0.637066\pi$$
−0.417421 + 0.908713i $$0.637066\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 19564.9i − 0.989925i
$$732$$ 0 0
$$733$$ 23552.7i 1.18682i 0.804900 + 0.593410i $$0.202219\pi$$
−0.804900 + 0.593410i $$0.797781\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10958.6 −0.547713
$$738$$ 0 0
$$739$$ 3519.05i 0.175169i 0.996157 + 0.0875847i $$0.0279149\pi$$
−0.996157 + 0.0875847i $$0.972085\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20752.4 −1.02467 −0.512336 0.858785i $$-0.671220\pi$$
−0.512336 + 0.858785i $$0.671220\pi$$
$$744$$ 0 0
$$745$$ 16328.2 0.802979
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 15941.1i 0.777671i
$$750$$ 0 0
$$751$$ −3679.08 −0.178764 −0.0893818 0.995997i $$-0.528489\pi$$
−0.0893818 + 0.995997i $$0.528489\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 13229.0i − 0.637687i
$$756$$ 0 0
$$757$$ − 527.275i − 0.0253159i −0.999920 0.0126579i $$-0.995971\pi$$
0.999920 0.0126579i $$-0.00402926\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3231.54 −0.153933 −0.0769666 0.997034i $$-0.524523\pi$$
−0.0769666 + 0.997034i $$0.524523\pi$$
$$762$$ 0 0
$$763$$ 9485.29i 0.450053i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8714.48 −0.410250
$$768$$ 0 0
$$769$$ 13681.7 0.641581 0.320791 0.947150i $$-0.396051\pi$$
0.320791 + 0.947150i $$0.396051\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 9216.63i 0.428847i 0.976741 + 0.214424i $$0.0687873\pi$$
−0.976741 + 0.214424i $$0.931213\pi$$
$$774$$ 0 0
$$775$$ 9389.92 0.435221
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 13267.1i − 0.610196i
$$780$$ 0 0
$$781$$ 27703.5i 1.26928i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −13894.9 −0.631759
$$786$$ 0 0
$$787$$ 33169.0i 1.50235i 0.660104 + 0.751174i $$0.270512\pi$$
−0.660104 + 0.751174i $$0.729488\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −46238.8 −2.07846
$$792$$ 0 0
$$793$$ 59033.6 2.64356
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 23781.9i − 1.05696i −0.848945 0.528481i $$-0.822762\pi$$
0.848945 0.528481i $$-0.177238\pi$$
$$798$$ 0 0
$$799$$ −32772.3 −1.45106
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 10424.9i − 0.458139i
$$804$$ 0 0
$$805$$ − 24503.3i − 1.07283i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 29842.4 1.29691 0.648456 0.761252i $$-0.275415\pi$$
0.648456 + 0.761252i $$0.275415\pi$$
$$810$$ 0 0
$$811$$ − 19074.5i − 0.825888i −0.910756 0.412944i $$-0.864500\pi$$
0.910756 0.412944i $$-0.135500\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8157.74 0.350617
$$816$$ 0 0
$$817$$ −11169.9 −0.478316
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 7493.97i 0.318564i 0.987233 + 0.159282i $$0.0509180\pi$$
−0.987233 + 0.159282i $$0.949082\pi$$
$$822$$ 0 0
$$823$$ −1780.90 −0.0754294 −0.0377147 0.999289i $$-0.512008\pi$$
−0.0377147 + 0.999289i $$0.512008\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 10860.1i − 0.456640i −0.973586 0.228320i $$-0.926677\pi$$
0.973586 0.228320i $$-0.0733232\pi$$
$$828$$ 0 0
$$829$$ 34105.1i 1.42885i 0.699711 + 0.714426i $$0.253312\pi$$
−0.699711 + 0.714426i $$0.746688\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −73613.1 −3.06188
$$834$$ 0 0
$$835$$ 10046.1i 0.416358i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 32688.9 1.34511 0.672555 0.740047i $$-0.265197\pi$$
0.672555 + 0.740047i $$0.265197\pi$$
$$840$$ 0 0
$$841$$ 2647.12 0.108537
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 25928.4i 1.05558i
$$846$$ 0 0
$$847$$ −4366.18 −0.177124
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24054.3i 0.968943i
$$852$$ 0 0
$$853$$ 17391.2i 0.698080i 0.937108 + 0.349040i $$0.113492\pi$$
−0.937108 + 0.349040i $$0.886508\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4182.34 0.166705 0.0833525 0.996520i $$-0.473437\pi$$
0.0833525 + 0.996520i $$0.473437\pi$$
$$858$$ 0 0
$$859$$ 20585.0i 0.817639i 0.912615 + 0.408820i $$0.134059\pi$$
−0.912615 + 0.408820i $$0.865941\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 14159.7 0.558518 0.279259 0.960216i $$-0.409911\pi$$
0.279259 + 0.960216i $$0.409911\pi$$
$$864$$ 0 0
$$865$$ −22130.6 −0.869900
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1483.19i 0.0578986i
$$870$$ 0 0
$$871$$ −26027.2 −1.01251
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 40914.1i 1.58074i
$$876$$ 0 0
$$877$$ 5156.38i 0.198539i 0.995061 + 0.0992695i $$0.0316506\pi$$
−0.995061 + 0.0992695i $$0.968349\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −841.065 −0.0321637 −0.0160818 0.999871i $$-0.505119\pi$$
−0.0160818 + 0.999871i $$0.505119\pi$$
$$882$$ 0 0
$$883$$ − 7845.99i − 0.299024i −0.988760 0.149512i $$-0.952230\pi$$
0.988760 0.149512i $$-0.0477703\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 38781.9 1.46806 0.734029 0.679118i $$-0.237637\pi$$
0.734029 + 0.679118i $$0.237637\pi$$
$$888$$ 0 0
$$889$$ −14687.1 −0.554092
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 18710.1i 0.701131i
$$894$$ 0 0
$$895$$ 23543.7 0.879305
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 14925.5i − 0.553719i
$$900$$ 0 0
$$901$$ − 9986.95i − 0.369271i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6269.33 0.230276
$$906$$ 0 0
$$907$$ 36431.2i 1.33371i 0.745187 + 0.666856i $$0.232360\pi$$
−0.745187 + 0.666856i $$0.767640\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −51077.9 −1.85762 −0.928808 0.370562i $$-0.879165\pi$$
−0.928808 + 0.370562i $$0.879165\pi$$
$$912$$ 0 0
$$913$$ 42822.6 1.55227
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 45188.9i − 1.62734i
$$918$$ 0 0
$$919$$ 37080.1 1.33097 0.665484 0.746412i $$-0.268225\pi$$
0.665484 + 0.746412i $$0.268225\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 65797.3i 2.34642i
$$924$$ 0 0
$$925$$ − 17109.4i − 0.608167i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −38259.5 −1.35119 −0.675595 0.737273i $$-0.736113\pi$$
−0.675595 + 0.737273i $$0.736113\pi$$
$$930$$ 0 0
$$931$$ 42026.7i 1.47945i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 19243.8 0.673091
$$936$$ 0 0
$$937$$ 4413.55 0.153879 0.0769394 0.997036i $$-0.475485\pi$$
0.0769394 + 0.997036i $$0.475485\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 31511.2i 1.09164i 0.837902 + 0.545821i $$0.183782\pi$$
−0.837902 + 0.545821i $$0.816218\pi$$
$$942$$ 0 0
$$943$$ 30962.8 1.06923
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 23561.3i 0.808490i 0.914651 + 0.404245i $$0.132466\pi$$
−0.914651 + 0.404245i $$0.867534\pi$$
$$948$$ 0 0
$$949$$ − 24759.6i − 0.846925i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −35039.7 −1.19103 −0.595513 0.803346i $$-0.703051\pi$$
−0.595513 + 0.803346i $$0.703051\pi$$
$$954$$ 0 0
$$955$$ − 199.107i − 0.00674653i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −40473.0 −1.36282
$$960$$ 0 0
$$961$$ −19544.9 −0.656066
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 3496.76i − 0.116647i
$$966$$ 0 0
$$967$$ −7975.38 −0.265223 −0.132612 0.991168i $$-0.542336\pi$$
−0.132612 + 0.991168i $$0.542336\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 21386.8i − 0.706834i −0.935466 0.353417i $$-0.885020\pi$$
0.935466 0.353417i $$-0.114980\pi$$
$$972$$ 0 0
$$973$$ − 72030.7i − 2.37328i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 40142.3 1.31450 0.657250 0.753673i $$-0.271720\pi$$
0.657250 + 0.753673i $$0.271720\pi$$
$$978$$ 0 0
$$979$$ 45891.5i 1.49816i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −9205.44 −0.298686 −0.149343 0.988785i $$-0.547716\pi$$
−0.149343 + 0.988785i $$0.547716\pi$$
$$984$$ 0 0
$$985$$ −3347.75 −0.108292
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 26068.3i − 0.838144i
$$990$$ 0 0
$$991$$ −22082.2 −0.707834 −0.353917 0.935277i $$-0.615150\pi$$
−0.353917 + 0.935277i $$0.615150\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 27331.1i 0.870808i
$$996$$ 0 0
$$997$$ − 7207.66i − 0.228956i −0.993426 0.114478i $$-0.963480\pi$$
0.993426 0.114478i $$-0.0365195\pi$$
$$998$$ 0 0
$$999$$ 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 1152.4.d.p.577.3 8
3.2 odd 2 384.4.d.f.193.3 yes 8
4.3 odd 2 inner 1152.4.d.p.577.4 8
8.3 odd 2 inner 1152.4.d.p.577.6 8
8.5 even 2 inner 1152.4.d.p.577.5 8
12.11 even 2 384.4.d.f.193.7 yes 8
16.3 odd 4 2304.4.a.cb.1.2 4
16.5 even 4 2304.4.a.cb.1.3 4
16.11 odd 4 2304.4.a.by.1.3 4
16.13 even 4 2304.4.a.by.1.2 4
24.5 odd 2 384.4.d.f.193.6 yes 8
24.11 even 2 384.4.d.f.193.2 8
48.5 odd 4 768.4.a.u.1.2 4
48.11 even 4 768.4.a.v.1.2 4
48.29 odd 4 768.4.a.v.1.3 4
48.35 even 4 768.4.a.u.1.3 4

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