# Properties

 Label 2304.4.a.cb.1.3 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.9792.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 2x + 7$$ x^4 - 2*x^3 - 7*x^2 + 2*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.21597$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.67763 q^{5} +33.0917 q^{7} +O(q^{10})$$ $$q+5.67763 q^{5} +33.0917 q^{7} +34.6274 q^{11} -82.2421 q^{13} -97.8823 q^{17} +55.8823 q^{19} +130.418 q^{23} -92.7645 q^{25} -147.451 q^{29} +101.223 q^{31} +187.882 q^{35} +184.439 q^{37} +237.411 q^{41} -199.882 q^{43} +334.813 q^{47} +752.058 q^{49} -102.030 q^{53} +196.602 q^{55} +105.961 q^{59} +717.803 q^{61} -466.940 q^{65} -316.471 q^{67} +800.045 q^{71} -301.058 q^{73} +1145.88 q^{77} -42.8329 q^{79} +1236.67 q^{83} -555.739 q^{85} +1325.29 q^{89} -2721.53 q^{91} +317.279 q^{95} -505.765 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 48 q^{11} - 120 q^{17} - 48 q^{19} + 172 q^{25} + 480 q^{35} - 408 q^{41} - 528 q^{43} + 836 q^{49} + 1872 q^{59} + 576 q^{65} - 2352 q^{67} + 968 q^{73} + 3408 q^{83} + 3672 q^{89} - 5184 q^{91} - 1480 q^{97}+O(q^{100})$$ 4 * q + 48 * q^11 - 120 * q^17 - 48 * q^19 + 172 * q^25 + 480 * q^35 - 408 * q^41 - 528 * q^43 + 836 * q^49 + 1872 * q^59 + 576 * q^65 - 2352 * q^67 + 968 * q^73 + 3408 * q^83 + 3672 * q^89 - 5184 * q^91 - 1480 * q^97

## 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
$$5$$ 5.67763 0.507823 0.253911 0.967227i $$-0.418283\pi$$
0.253911 + 0.967227i $$0.418283\pi$$
$$6$$ 0 0
$$7$$ 33.0917 1.78678 0.893391 0.449280i $$-0.148320\pi$$
0.893391 + 0.449280i $$0.148320\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 34.6274 0.949142 0.474571 0.880217i $$-0.342603\pi$$
0.474571 + 0.880217i $$0.342603\pi$$
$$12$$ 0 0
$$13$$ −82.2421 −1.75460 −0.877302 0.479939i $$-0.840659\pi$$
−0.877302 + 0.479939i $$0.840659\pi$$
$$14$$ 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
$$19$$ 55.8823 0.674751 0.337375 0.941370i $$-0.390461\pi$$
0.337375 + 0.941370i $$0.390461\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 130.418 1.18235 0.591176 0.806542i $$-0.298664\pi$$
0.591176 + 0.806542i $$0.298664\pi$$
$$24$$ 0 0
$$25$$ −92.7645 −0.742116
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −147.451 −0.944173 −0.472086 0.881552i $$-0.656499\pi$$
−0.472086 + 0.881552i $$0.656499\pi$$
$$30$$ 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
$$35$$ 187.882 0.907368
$$36$$ 0 0
$$37$$ 184.439 0.819504 0.409752 0.912197i $$-0.365615\pi$$
0.409752 + 0.912197i $$0.365615\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 237.411 0.904327 0.452164 0.891935i $$-0.350652\pi$$
0.452164 + 0.891935i $$0.350652\pi$$
$$42$$ 0 0
$$43$$ −199.882 −0.708878 −0.354439 0.935079i $$-0.615328\pi$$
−0.354439 + 0.935079i $$0.615328\pi$$
$$44$$ 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
$$53$$ −102.030 −0.264433 −0.132216 0.991221i $$-0.542209\pi$$
−0.132216 + 0.991221i $$0.542209\pi$$
$$54$$ 0 0
$$55$$ 196.602 0.481996
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 105.961 0.233813 0.116907 0.993143i $$-0.462702\pi$$
0.116907 + 0.993143i $$0.462702\pi$$
$$60$$ 0 0
$$61$$ 717.803 1.50664 0.753321 0.657653i $$-0.228451\pi$$
0.753321 + 0.657653i $$0.228451\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −466.940 −0.891028
$$66$$ 0 0
$$67$$ −316.471 −0.577061 −0.288530 0.957471i $$-0.593167\pi$$
−0.288530 + 0.957471i $$0.593167\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 800.045 1.33729 0.668647 0.743580i $$-0.266874\pi$$
0.668647 + 0.743580i $$0.266874\pi$$
$$72$$ 0 0
$$73$$ −301.058 −0.482687 −0.241344 0.970440i $$-0.577588\pi$$
−0.241344 + 0.970440i $$0.577588\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1145.88 1.69591
$$78$$ 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
$$83$$ 1236.67 1.63544 0.817721 0.575614i $$-0.195237\pi$$
0.817721 + 0.575614i $$0.195237\pi$$
$$84$$ 0 0
$$85$$ −555.739 −0.709158
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1325.29 1.57844 0.789218 0.614113i $$-0.210486\pi$$
0.789218 + 0.614113i $$0.210486\pi$$
$$90$$ 0 0
$$91$$ −2721.53 −3.13509
$$92$$ 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
$$101$$ 1281.97 1.26298 0.631491 0.775383i $$-0.282443\pi$$
0.631491 + 0.775383i $$0.282443\pi$$
$$102$$ 0 0
$$103$$ −161.562 −0.154555 −0.0772775 0.997010i $$-0.524623\pi$$
−0.0772775 + 0.997010i $$0.524623\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 481.726 0.435235 0.217618 0.976034i $$-0.430171\pi$$
0.217618 + 0.976034i $$0.430171\pi$$
$$108$$ 0 0
$$109$$ −286.637 −0.251879 −0.125940 0.992038i $$-0.540195\pi$$
−0.125940 + 0.992038i $$0.540195\pi$$
$$110$$ 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
$$115$$ 740.468 0.600426
$$116$$ 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
$$125$$ −1236.39 −0.884686
$$126$$ 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
$$131$$ 1365.57 0.910765 0.455382 0.890296i $$-0.349503\pi$$
0.455382 + 0.890296i $$0.349503\pi$$
$$132$$ 0 0
$$133$$ 1849.24 1.20563
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1223.06 −0.762721 −0.381360 0.924426i $$-0.624544\pi$$
−0.381360 + 0.924426i $$0.624544\pi$$
$$138$$ 0 0
$$139$$ −2176.70 −1.32824 −0.664121 0.747625i $$-0.731194\pi$$
−0.664121 + 0.747625i $$0.731194\pi$$
$$140$$ 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
$$149$$ −2875.88 −1.58122 −0.790610 0.612320i $$-0.790236\pi$$
−0.790610 + 0.612320i $$0.790236\pi$$
$$150$$ 0 0
$$151$$ −2330.03 −1.25573 −0.627863 0.778323i $$-0.716070\pi$$
−0.627863 + 0.778323i $$0.716070\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 574.708 0.297817
$$156$$ 0 0
$$157$$ −2447.31 −1.24405 −0.622027 0.782996i $$-0.713690\pi$$
−0.622027 + 0.782996i $$0.713690\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4315.76 2.11261
$$162$$ 0 0
$$163$$ 1436.82 0.690432 0.345216 0.938523i $$-0.387806\pi$$
0.345216 + 0.938523i $$0.387806\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1769.42 0.819889 0.409945 0.912110i $$-0.365548\pi$$
0.409945 + 0.912110i $$0.365548\pi$$
$$168$$ 0 0
$$169$$ 4566.76 2.07863
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3897.86 −1.71300 −0.856499 0.516148i $$-0.827365\pi$$
−0.856499 + 0.516148i $$0.827365\pi$$
$$174$$ 0 0
$$175$$ −3069.73 −1.32600
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4146.74 1.73152 0.865760 0.500460i $$-0.166836\pi$$
0.865760 + 0.500460i $$0.166836\pi$$
$$180$$ 0 0
$$181$$ −1104.22 −0.453457 −0.226728 0.973958i $$-0.572803\pi$$
−0.226728 + 0.973958i $$0.572803\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1047.18 0.416163
$$186$$ 0 0
$$187$$ −3389.41 −1.32544
$$188$$ 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
$$197$$ 589.638 0.213249 0.106624 0.994299i $$-0.465996\pi$$
0.106624 + 0.994299i $$0.465996\pi$$
$$198$$ 0 0
$$199$$ 4813.82 1.71479 0.857394 0.514661i $$-0.172082\pi$$
0.857394 + 0.514661i $$0.172082\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −4879.41 −1.68703
$$204$$ 0 0
$$205$$ 1347.93 0.459238
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1935.06 0.640434
$$210$$ 0 0
$$211$$ 3294.35 1.07484 0.537422 0.843313i $$-0.319398\pi$$
0.537422 + 0.843313i $$0.319398\pi$$
$$212$$ 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
$$221$$ 8050.04 2.45025
$$222$$ 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
$$227$$ −1492.51 −0.436394 −0.218197 0.975905i $$-0.570018\pi$$
−0.218197 + 0.975905i $$0.570018\pi$$
$$228$$ 0 0
$$229$$ −6163.31 −1.77853 −0.889265 0.457393i $$-0.848783\pi$$
−0.889265 + 0.457393i $$0.848783\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2540.35 0.714266 0.357133 0.934054i $$-0.383754\pi$$
0.357133 + 0.934054i $$0.383754\pi$$
$$234$$ 0 0
$$235$$ 1900.95 0.527677
$$236$$ 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
$$245$$ 4269.91 1.11345
$$246$$ 0 0
$$247$$ −4595.87 −1.18392
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4701.45 1.18228 0.591141 0.806568i $$-0.298678\pi$$
0.591141 + 0.806568i $$0.298678\pi$$
$$252$$ 0 0
$$253$$ 4516.05 1.12222
$$254$$ 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
$$259$$ 6103.41 1.46428
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 7091.36 1.66263 0.831315 0.555801i $$-0.187588\pi$$
0.831315 + 0.555801i $$0.187588\pi$$
$$264$$ 0 0
$$265$$ −579.290 −0.134285
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 395.596 0.0896650 0.0448325 0.998995i $$-0.485725\pi$$
0.0448325 + 0.998995i $$0.485725\pi$$
$$270$$ 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
$$275$$ −3212.20 −0.704373
$$276$$ 0 0
$$277$$ 3830.31 0.830834 0.415417 0.909631i $$-0.363636\pi$$
0.415417 + 0.909631i $$0.363636\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5283.17 1.12159 0.560797 0.827954i $$-0.310495\pi$$
0.560797 + 0.827954i $$0.310495\pi$$
$$282$$ 0 0
$$283$$ −4639.76 −0.974577 −0.487288 0.873241i $$-0.662014\pi$$
−0.487288 + 0.873241i $$0.662014\pi$$
$$284$$ 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
$$293$$ −6022.12 −1.20074 −0.600369 0.799723i $$-0.704980\pi$$
−0.600369 + 0.799723i $$0.704980\pi$$
$$294$$ 0 0
$$295$$ 601.609 0.118736
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −10725.9 −2.07456
$$300$$ 0 0
$$301$$ −6614.44 −1.26661
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4075.42 0.765107
$$306$$ 0 0
$$307$$ −2998.82 −0.557497 −0.278749 0.960364i $$-0.589920\pi$$
−0.278749 + 0.960364i $$0.589920\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2403.97 −0.438318 −0.219159 0.975689i $$-0.570331\pi$$
−0.219159 + 0.975689i $$0.570331\pi$$
$$312$$ 0 0
$$313$$ −5845.75 −1.05566 −0.527830 0.849350i $$-0.676994\pi$$
−0.527830 + 0.849350i $$0.676994\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8917.38 −1.57997 −0.789984 0.613127i $$-0.789911\pi$$
−0.789984 + 0.613127i $$0.789911\pi$$
$$318$$ 0 0
$$319$$ −5105.86 −0.896154
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5469.88 −0.942267
$$324$$ 0 0
$$325$$ 7629.15 1.30212
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 11079.5 1.85664
$$330$$ 0 0
$$331$$ 9011.29 1.49639 0.748196 0.663478i $$-0.230920\pi$$
0.748196 + 0.663478i $$0.230920\pi$$
$$332$$ 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
$$341$$ 3505.10 0.556633
$$342$$ 0 0
$$343$$ 13536.4 2.13090
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3651.48 −0.564904 −0.282452 0.959281i $$-0.591148\pi$$
−0.282452 + 0.959281i $$0.591148\pi$$
$$348$$ 0 0
$$349$$ 3250.36 0.498532 0.249266 0.968435i $$-0.419811\pi$$
0.249266 + 0.968435i $$0.419811\pi$$
$$350$$ 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
$$355$$ 4542.36 0.679108
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3443.48 −0.506239 −0.253120 0.967435i $$-0.581457\pi$$
−0.253120 + 0.967435i $$0.581457\pi$$
$$360$$ 0 0
$$361$$ −3736.17 −0.544711
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1709.30 −0.245120
$$366$$ 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
$$371$$ −3376.35 −0.472483
$$372$$ 0 0
$$373$$ 50.1819 0.00696601 0.00348300 0.999994i $$-0.498891\pi$$
0.00348300 + 0.999994i $$0.498891\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12126.7 1.65665
$$378$$ 0 0
$$379$$ 6769.28 0.917453 0.458726 0.888578i $$-0.348306\pi$$
0.458726 + 0.888578i $$0.348306\pi$$
$$380$$ 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
$$389$$ −2490.47 −0.324607 −0.162303 0.986741i $$-0.551892\pi$$
−0.162303 + 0.986741i $$0.551892\pi$$
$$390$$ 0 0
$$391$$ −12765.6 −1.65112
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −243.190 −0.0309777
$$396$$ 0 0
$$397$$ −7905.51 −0.999411 −0.499706 0.866195i $$-0.666558\pi$$
−0.499706 + 0.866195i $$0.666558\pi$$
$$398$$ 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
$$403$$ −8324.81 −1.02900
$$404$$ 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 $$-0.197878\pi$$
0.812917 + 0.582379i $$0.197878\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3506.44 0.417773
$$414$$ 0 0
$$415$$ 7021.33 0.830515
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −5191.34 −0.605283 −0.302641 0.953105i $$-0.597868\pi$$
−0.302641 + 0.953105i $$0.597868\pi$$
$$420$$ 0 0
$$421$$ −5061.52 −0.585946 −0.292973 0.956121i $$-0.594645\pi$$
−0.292973 + 0.956121i $$0.594645\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 9080.00 1.03634
$$426$$ 0 0
$$427$$ 23753.3 2.69204
$$428$$ 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
$$437$$ 7288.07 0.797794
$$438$$ 0 0
$$439$$ −11556.8 −1.25644 −0.628218 0.778038i $$-0.716215\pi$$
−0.628218 + 0.778038i $$0.716215\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14007.8 1.50233 0.751163 0.660117i $$-0.229493\pi$$
0.751163 + 0.660117i $$0.229493\pi$$
$$444$$ 0 0
$$445$$ 7524.53 0.801566
$$446$$ 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
$$451$$ 8220.94 0.858335
$$452$$ 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 $$-0.474796\pi$$
0.0790992 + 0.996867i $$0.474796\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12730.2 1.28613 0.643065 0.765811i $$-0.277662\pi$$
0.643065 + 0.765811i $$0.277662\pi$$
$$462$$ 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
$$467$$ −6016.26 −0.596144 −0.298072 0.954543i $$-0.596344\pi$$
−0.298072 + 0.954543i $$0.596344\pi$$
$$468$$ 0 0
$$469$$ −10472.6 −1.03108
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −6921.41 −0.672826
$$474$$ 0 0
$$475$$ −5183.89 −0.500743
$$476$$ 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
$$485$$ −2871.54 −0.268846
$$486$$ 0 0
$$487$$ 9314.23 0.866669 0.433335 0.901233i $$-0.357337\pi$$
0.433335 + 0.901233i $$0.357337\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7253.34 −0.666677 −0.333339 0.942807i $$-0.608175\pi$$
−0.333339 + 0.942807i $$0.608175\pi$$
$$492$$ 0 0
$$493$$ 14432.9 1.31851
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 26474.8 2.38945
$$498$$ 0 0
$$499$$ −16208.2 −1.45407 −0.727034 0.686602i $$-0.759102\pi$$
−0.727034 + 0.686602i $$0.759102\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2182.04 0.193425 0.0967123 0.995312i $$-0.469167\pi$$
0.0967123 + 0.995312i $$0.469167\pi$$
$$504$$ 0 0
$$505$$ 7278.58 0.641371
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1795.97 0.156395 0.0781974 0.996938i $$-0.475084\pi$$
0.0781974 + 0.996938i $$0.475084\pi$$
$$510$$ 0 0
$$511$$ −9962.51 −0.862457
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −917.288 −0.0784865
$$516$$ 0 0
$$517$$ 11593.7 0.986249
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20946.1 −1.76135 −0.880676 0.473718i $$-0.842912\pi$$
−0.880676 + 0.473718i $$0.842912\pi$$
$$522$$ 0 0
$$523$$ 5363.64 0.448443 0.224221 0.974538i $$-0.428016\pi$$
0.224221 + 0.974538i $$0.428016\pi$$
$$524$$ 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
$$533$$ −19525.2 −1.58674
$$534$$ 0 0
$$535$$ 2735.06 0.221022
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 26041.8 2.08108
$$540$$ 0 0
$$541$$ 3431.20 0.272678 0.136339 0.990662i $$-0.456466\pi$$
0.136339 + 0.990662i $$0.456466\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1627.42 −0.127910
$$546$$ 0 0
$$547$$ −19449.0 −1.52026 −0.760128 0.649773i $$-0.774864\pi$$
−0.760128 + 0.649773i $$0.774864\pi$$
$$548$$ 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
$$557$$ −9898.44 −0.752981 −0.376491 0.926420i $$-0.622869\pi$$
−0.376491 + 0.926420i $$0.622869\pi$$
$$558$$ 0 0
$$559$$ 16438.7 1.24380
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 13809.8 1.03378 0.516888 0.856053i $$-0.327091\pi$$
0.516888 + 0.856053i $$0.327091\pi$$
$$564$$ 0 0
$$565$$ 7933.32 0.590721
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −123.053 −0.00906615 −0.00453308 0.999990i $$-0.501443\pi$$
−0.00453308 + 0.999990i $$0.501443\pi$$
$$570$$ 0 0
$$571$$ −2540.72 −0.186210 −0.0931049 0.995656i $$-0.529679\pi$$
−0.0931049 + 0.995656i $$0.529679\pi$$
$$572$$ 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
$$581$$ 40923.3 2.92218
$$582$$ 0 0
$$583$$ −3533.04 −0.250984
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1809.56 −0.127238 −0.0636188 0.997974i $$-0.520264\pi$$
−0.0636188 + 0.997974i $$0.520264\pi$$
$$588$$ 0 0
$$589$$ 5656.58 0.395714
$$590$$ 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
$$595$$ −18390.3 −1.26711
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 22990.9 1.56825 0.784125 0.620603i $$-0.213112\pi$$
0.784125 + 0.620603i $$0.213112\pi$$
$$600$$ 0 0
$$601$$ 26893.7 1.82532 0.912661 0.408717i $$-0.134024\pi$$
0.912661 + 0.408717i $$0.134024\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −749.118 −0.0503405
$$606$$ 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
$$611$$ −27535.7 −1.82320
$$612$$ 0 0
$$613$$ −2612.97 −0.172164 −0.0860822 0.996288i $$-0.527435\pi$$
−0.0860822 + 0.996288i $$0.527435\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2803.88 −0.182949 −0.0914747 0.995807i $$-0.529158\pi$$
−0.0914747 + 0.995807i $$0.529158\pi$$
$$618$$ 0 0
$$619$$ 10547.1 0.684849 0.342425 0.939545i $$-0.388752\pi$$
0.342425 + 0.939545i $$0.388752\pi$$
$$620$$ 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
$$629$$ −18053.3 −1.14441
$$630$$ 0 0
$$631$$ 14161.4 0.893431 0.446716 0.894676i $$-0.352594\pi$$
0.446716 + 0.894676i $$0.352594\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2519.90 0.157479
$$636$$ 0 0
$$637$$ −61850.8 −3.84713
$$638$$ 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
$$643$$ −29438.7 −1.80552 −0.902759 0.430146i $$-0.858462\pi$$
−0.902759 + 0.430146i $$0.858462\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −4607.39 −0.279962 −0.139981 0.990154i $$-0.544704\pi$$
−0.139981 + 0.990154i $$0.544704\pi$$
$$648$$ 0 0
$$649$$ 3669.17 0.221922
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −16634.1 −0.996850 −0.498425 0.866933i $$-0.666088\pi$$
−0.498425 + 0.866933i $$0.666088\pi$$
$$654$$ 0 0
$$655$$ 7753.19 0.462507
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 20619.9 1.21887 0.609437 0.792835i $$-0.291396\pi$$
0.609437 + 0.792835i $$0.291396\pi$$
$$660$$ 0 0
$$661$$ 881.112 0.0518476 0.0259238 0.999664i $$-0.491747\pi$$
0.0259238 + 0.999664i $$0.491747\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10499.3 0.612248
$$666$$ 0 0
$$667$$ −19230.4 −1.11635
$$668$$ 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
$$677$$ −21748.7 −1.23467 −0.617333 0.786702i $$-0.711787\pi$$
−0.617333 + 0.786702i $$0.711787\pi$$
$$678$$ 0 0
$$679$$ −16736.6 −0.945937
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 7922.37 0.443838 0.221919 0.975065i $$-0.428768\pi$$
0.221919 + 0.975065i $$0.428768\pi$$
$$684$$ 0 0
$$685$$ −6944.06 −0.387327
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 8391.18 0.463975
$$690$$ 0 0
$$691$$ 177.517 0.00977288 0.00488644 0.999988i $$-0.498445\pi$$
0.00488644 + 0.999988i $$0.498445\pi$$
$$692$$ 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
$$701$$ 767.980 0.0413783 0.0206891 0.999786i $$-0.493414\pi$$
0.0206891 + 0.999786i $$0.493414\pi$$
$$702$$ 0 0
$$703$$ 10306.9 0.552961
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 42422.7 2.25667
$$708$$ 0 0
$$709$$ −31634.2 −1.67566 −0.837832 0.545928i $$-0.816177\pi$$
−0.837832 + 0.545928i $$0.816177\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 13201.4 0.693401
$$714$$ 0 0
$$715$$ −16168.9 −0.845712
$$716$$ 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
$$725$$ 13678.2 0.700686
$$726$$ 0 0
$$727$$ 16364.6 0.834842 0.417421 0.908713i $$-0.362934\pi$$
0.417421 + 0.908713i $$0.362934\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 19564.9 0.989925
$$732$$ 0 0
$$733$$ 23552.7 1.18682 0.593410 0.804900i $$-0.297781\pi$$
0.593410 + 0.804900i $$0.297781\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10958.6 −0.547713
$$738$$ 0 0
$$739$$ 3519.05 0.175169 0.0875847 0.996157i $$-0.472085\pi$$
0.0875847 + 0.996157i $$0.472085\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20752.4 1.02467 0.512336 0.858785i $$-0.328780\pi$$
0.512336 + 0.858785i $$0.328780\pi$$
$$744$$ 0 0
$$745$$ −16328.2 −0.802979
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 15941.1 0.777671
$$750$$ 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
$$755$$ −13229.0 −0.637687
$$756$$ 0 0
$$757$$ 527.275 0.0253159 0.0126579 0.999920i $$-0.495971\pi$$
0.0126579 + 0.999920i $$0.495971\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3231.54 0.153933 0.0769666 0.997034i $$-0.475477\pi$$
0.0769666 + 0.997034i $$0.475477\pi$$
$$762$$ 0 0
$$763$$ −9485.29 −0.450053
$$764$$ 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
$$773$$ −9216.63 −0.428847 −0.214424 0.976741i $$-0.568787\pi$$
−0.214424 + 0.976741i $$0.568787\pi$$
$$774$$ 0 0
$$775$$ −9389.92 −0.435221
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 13267.1 0.610196
$$780$$ 0 0
$$781$$ 27703.5 1.26928
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −13894.9 −0.631759
$$786$$ 0 0
$$787$$ 33169.0 1.50235 0.751174 0.660104i $$-0.229488\pi$$
0.751174 + 0.660104i $$0.229488\pi$$
$$788$$ 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
$$797$$ −23781.9 −1.05696 −0.528481 0.848945i $$-0.677238\pi$$
−0.528481 + 0.848945i $$0.677238\pi$$
$$798$$ 0 0
$$799$$ −32772.3 −1.45106
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −10424.9 −0.458139
$$804$$ 0 0
$$805$$ 24503.3 1.07283
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −29842.4 −1.29691 −0.648456 0.761252i $$-0.724585\pi$$
−0.648456 + 0.761252i $$0.724585\pi$$
$$810$$ 0 0
$$811$$ 19074.5 0.825888 0.412944 0.910756i $$-0.364500\pi$$
0.412944 + 0.910756i $$0.364500\pi$$
$$812$$ 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
$$821$$ −7493.97 −0.318564 −0.159282 0.987233i $$-0.550918\pi$$
−0.159282 + 0.987233i $$0.550918\pi$$
$$822$$ 0 0
$$823$$ 1780.90 0.0754294 0.0377147 0.999289i $$-0.487992\pi$$
0.0377147 + 0.999289i $$0.487992\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10860.1 0.456640 0.228320 0.973586i $$-0.426677\pi$$
0.228320 + 0.973586i $$0.426677\pi$$
$$828$$ 0 0
$$829$$ 34105.1 1.42885 0.714426 0.699711i $$-0.246688\pi$$
0.714426 + 0.699711i $$0.246688\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −73613.1 −3.06188
$$834$$ 0 0
$$835$$ 10046.1 0.416358
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −32688.9 −1.34511 −0.672555 0.740047i $$-0.734803\pi$$
−0.672555 + 0.740047i $$0.734803\pi$$
$$840$$ 0 0
$$841$$ −2647.12 −0.108537
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 25928.4 1.05558
$$846$$ 0 0
$$847$$ −4366.18 −0.177124
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24054.3 0.968943
$$852$$ 0 0
$$853$$ −17391.2 −0.698080 −0.349040 0.937108i $$-0.613492\pi$$
−0.349040 + 0.937108i $$0.613492\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −4182.34 −0.166705 −0.0833525 0.996520i $$-0.526563\pi$$
−0.0833525 + 0.996520i $$0.526563\pi$$
$$858$$ 0 0
$$859$$ −20585.0 −0.817639 −0.408820 0.912615i $$-0.634059\pi$$
−0.408820 + 0.912615i $$0.634059\pi$$
$$860$$ 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
$$869$$ −1483.19 −0.0578986
$$870$$ 0 0
$$871$$ 26027.2 1.01251
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −40914.1 −1.58074
$$876$$ 0 0
$$877$$ 5156.38 0.198539 0.0992695 0.995061i $$-0.468349\pi$$
0.0992695 + 0.995061i $$0.468349\pi$$
$$878$$ 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
$$883$$ −7845.99 −0.299024 −0.149512 0.988760i $$-0.547770\pi$$
−0.149512 + 0.988760i $$0.547770\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −38781.9 −1.46806 −0.734029 0.679118i $$-0.762363\pi$$
−0.734029 + 0.679118i $$0.762363\pi$$
$$888$$ 0 0
$$889$$ 14687.1 0.554092
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 18710.1 0.701131
$$894$$ 0 0
$$895$$ 23543.7 0.879305
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −14925.5 −0.553719
$$900$$ 0 0
$$901$$ 9986.95 0.369271
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6269.33 −0.230276
$$906$$ 0 0
$$907$$ −36431.2 −1.33371 −0.666856 0.745187i $$-0.732360\pi$$
−0.666856 + 0.745187i $$0.732360\pi$$
$$908$$ 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
$$917$$ 45188.9 1.62734
$$918$$ 0 0
$$919$$ −37080.1 −1.33097 −0.665484 0.746412i $$-0.731775\pi$$
−0.665484 + 0.746412i $$0.731775\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −65797.3 −2.34642
$$924$$ 0 0
$$925$$ −17109.4 −0.608167
$$926$$ 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
$$931$$ 42026.7 1.47945
$$932$$ 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 $$-0.524515\pi$$
−0.0769394 + 0.997036i $$0.524515\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 31511.2 1.09164 0.545821 0.837902i $$-0.316218\pi$$
0.545821 + 0.837902i $$0.316218\pi$$
$$942$$ 0 0
$$943$$ 30962.8 1.06923
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 23561.3 0.808490 0.404245 0.914651i $$-0.367534\pi$$
0.404245 + 0.914651i $$0.367534\pi$$
$$948$$ 0 0
$$949$$ 24759.6 0.846925
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 35039.7 1.19103 0.595513 0.803346i $$-0.296949\pi$$
0.595513 + 0.803346i $$0.296949\pi$$
$$954$$ 0 0
$$955$$ 199.107 0.00674653
$$956$$ 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
$$965$$ 3496.76 0.116647
$$966$$ 0 0
$$967$$ 7975.38 0.265223 0.132612 0.991168i $$-0.457664\pi$$
0.132612 + 0.991168i $$0.457664\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21386.8 0.706834 0.353417 0.935466i $$-0.385020\pi$$
0.353417 + 0.935466i $$0.385020\pi$$
$$972$$ 0 0
$$973$$ −72030.7 −2.37328
$$974$$ 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
$$979$$ 45891.5 1.49816
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 9205.44 0.298686 0.149343 0.988785i $$-0.452284\pi$$
0.149343 + 0.988785i $$0.452284\pi$$
$$984$$ 0 0
$$985$$ 3347.75 0.108292
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −26068.3 −0.838144
$$990$$ 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
$$995$$ 27331.1 0.870808
$$996$$ 0 0
$$997$$ 7207.66 0.228956 0.114478 0.993426i $$-0.463480\pi$$
0.114478 + 0.993426i $$0.463480\pi$$
$$998$$ 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 2304.4.a.cb.1.3 4
3.2 odd 2 768.4.a.u.1.2 4
4.3 odd 2 2304.4.a.by.1.3 4
8.3 odd 2 inner 2304.4.a.cb.1.2 4
8.5 even 2 2304.4.a.by.1.2 4
12.11 even 2 768.4.a.v.1.2 4
16.3 odd 4 1152.4.d.p.577.4 8
16.5 even 4 1152.4.d.p.577.5 8
16.11 odd 4 1152.4.d.p.577.6 8
16.13 even 4 1152.4.d.p.577.3 8
24.5 odd 2 768.4.a.v.1.3 4
24.11 even 2 768.4.a.u.1.3 4
48.5 odd 4 384.4.d.f.193.6 yes 8
48.11 even 4 384.4.d.f.193.2 8
48.29 odd 4 384.4.d.f.193.3 yes 8
48.35 even 4 384.4.d.f.193.7 yes 8

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