# Properties

 Label 1089.4.a.s.1.2 Level $1089$ Weight $4$ Character 1089.1 Self dual yes Analytic conductor $64.253$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1089.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: $$64.2530799963$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.19615 q^{2} +19.0000 q^{4} -9.00000 q^{5} -24.2487 q^{7} +57.1577 q^{8} +O(q^{10})$$ $$q+5.19615 q^{2} +19.0000 q^{4} -9.00000 q^{5} -24.2487 q^{7} +57.1577 q^{8} -46.7654 q^{10} +71.0141 q^{13} -126.000 q^{14} +145.000 q^{16} -88.3346 q^{17} -145.492 q^{19} -171.000 q^{20} -90.0000 q^{23} -44.0000 q^{25} +369.000 q^{26} -460.726 q^{28} +88.3346 q^{29} -188.000 q^{31} +296.181 q^{32} -459.000 q^{34} +218.238 q^{35} +133.000 q^{37} -756.000 q^{38} -514.419 q^{40} -36.3731 q^{41} -72.7461 q^{43} -467.654 q^{46} -72.0000 q^{47} +245.000 q^{49} -228.631 q^{50} +1349.27 q^{52} +45.0000 q^{53} -1386.00 q^{56} +459.000 q^{58} -378.000 q^{59} +623.538 q^{61} -976.877 q^{62} +379.000 q^{64} -639.127 q^{65} -386.000 q^{67} -1678.36 q^{68} +1134.00 q^{70} +198.000 q^{71} +76.2102 q^{73} +691.088 q^{74} -2764.35 q^{76} -152.420 q^{79} -1305.00 q^{80} -189.000 q^{82} -1247.08 q^{83} +795.011 q^{85} -378.000 q^{86} -45.0000 q^{89} -1722.00 q^{91} -1710.00 q^{92} -374.123 q^{94} +1309.43 q^{95} +89.0000 q^{97} +1273.06 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 38 q^{4} - 18 q^{5}+O(q^{10})$$ 2 * q + 38 * q^4 - 18 * q^5 $$2 q + 38 q^{4} - 18 q^{5} - 252 q^{14} + 290 q^{16} - 342 q^{20} - 180 q^{23} - 88 q^{25} + 738 q^{26} - 376 q^{31} - 918 q^{34} + 266 q^{37} - 1512 q^{38} - 144 q^{47} + 490 q^{49} + 90 q^{53} - 2772 q^{56} + 918 q^{58} - 756 q^{59} + 758 q^{64} - 772 q^{67} + 2268 q^{70} + 396 q^{71} - 2610 q^{80} - 378 q^{82} - 756 q^{86} - 90 q^{89} - 3444 q^{91} - 3420 q^{92} + 178 q^{97}+O(q^{100})$$ 2 * q + 38 * q^4 - 18 * q^5 - 252 * q^14 + 290 * q^16 - 342 * q^20 - 180 * q^23 - 88 * q^25 + 738 * q^26 - 376 * q^31 - 918 * q^34 + 266 * q^37 - 1512 * q^38 - 144 * q^47 + 490 * q^49 + 90 * q^53 - 2772 * q^56 + 918 * q^58 - 756 * q^59 + 758 * q^64 - 772 * q^67 + 2268 * q^70 + 396 * q^71 - 2610 * q^80 - 378 * q^82 - 756 * q^86 - 90 * q^89 - 3444 * q^91 - 3420 * q^92 + 178 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.19615 1.83712 0.918559 0.395285i $$-0.129354\pi$$
0.918559 + 0.395285i $$0.129354\pi$$
$$3$$ 0 0
$$4$$ 19.0000 2.37500
$$5$$ −9.00000 −0.804984 −0.402492 0.915423i $$-0.631856\pi$$
−0.402492 + 0.915423i $$0.631856\pi$$
$$6$$ 0 0
$$7$$ −24.2487 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 57.1577 2.52604
$$9$$ 0 0
$$10$$ −46.7654 −1.47885
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 71.0141 1.51506 0.757529 0.652801i $$-0.226406\pi$$
0.757529 + 0.652801i $$0.226406\pi$$
$$14$$ −126.000 −2.40535
$$15$$ 0 0
$$16$$ 145.000 2.26562
$$17$$ −88.3346 −1.26025 −0.630126 0.776493i $$-0.716997\pi$$
−0.630126 + 0.776493i $$0.716997\pi$$
$$18$$ 0 0
$$19$$ −145.492 −1.75675 −0.878374 0.477974i $$-0.841371\pi$$
−0.878374 + 0.477974i $$0.841371\pi$$
$$20$$ −171.000 −1.91184
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −90.0000 −0.815926 −0.407963 0.912998i $$-0.633761\pi$$
−0.407963 + 0.912998i $$0.633761\pi$$
$$24$$ 0 0
$$25$$ −44.0000 −0.352000
$$26$$ 369.000 2.78334
$$27$$ 0 0
$$28$$ −460.726 −3.10960
$$29$$ 88.3346 0.565632 0.282816 0.959174i $$-0.408731\pi$$
0.282816 + 0.959174i $$0.408731\pi$$
$$30$$ 0 0
$$31$$ −188.000 −1.08922 −0.544610 0.838690i $$-0.683322\pi$$
−0.544610 + 0.838690i $$0.683322\pi$$
$$32$$ 296.181 1.63618
$$33$$ 0 0
$$34$$ −459.000 −2.31523
$$35$$ 218.238 1.05397
$$36$$ 0 0
$$37$$ 133.000 0.590948 0.295474 0.955351i $$-0.404522\pi$$
0.295474 + 0.955351i $$0.404522\pi$$
$$38$$ −756.000 −3.22735
$$39$$ 0 0
$$40$$ −514.419 −2.03342
$$41$$ −36.3731 −0.138549 −0.0692746 0.997598i $$-0.522068\pi$$
−0.0692746 + 0.997598i $$0.522068\pi$$
$$42$$ 0 0
$$43$$ −72.7461 −0.257993 −0.128996 0.991645i $$-0.541176\pi$$
−0.128996 + 0.991645i $$0.541176\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −467.654 −1.49895
$$47$$ −72.0000 −0.223453 −0.111726 0.993739i $$-0.535638\pi$$
−0.111726 + 0.993739i $$0.535638\pi$$
$$48$$ 0 0
$$49$$ 245.000 0.714286
$$50$$ −228.631 −0.646665
$$51$$ 0 0
$$52$$ 1349.27 3.59826
$$53$$ 45.0000 0.116627 0.0583134 0.998298i $$-0.481428\pi$$
0.0583134 + 0.998298i $$0.481428\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1386.00 −3.30736
$$57$$ 0 0
$$58$$ 459.000 1.03913
$$59$$ −378.000 −0.834092 −0.417046 0.908885i $$-0.636935\pi$$
−0.417046 + 0.908885i $$0.636935\pi$$
$$60$$ 0 0
$$61$$ 623.538 1.30879 0.654393 0.756155i $$-0.272924\pi$$
0.654393 + 0.756155i $$0.272924\pi$$
$$62$$ −976.877 −2.00102
$$63$$ 0 0
$$64$$ 379.000 0.740234
$$65$$ −639.127 −1.21960
$$66$$ 0 0
$$67$$ −386.000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$68$$ −1678.36 −2.99310
$$69$$ 0 0
$$70$$ 1134.00 1.93627
$$71$$ 198.000 0.330962 0.165481 0.986213i $$-0.447082\pi$$
0.165481 + 0.986213i $$0.447082\pi$$
$$72$$ 0 0
$$73$$ 76.2102 0.122188 0.0610941 0.998132i $$-0.480541\pi$$
0.0610941 + 0.998132i $$0.480541\pi$$
$$74$$ 691.088 1.08564
$$75$$ 0 0
$$76$$ −2764.35 −4.17228
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −152.420 −0.217071 −0.108536 0.994093i $$-0.534616\pi$$
−0.108536 + 0.994093i $$0.534616\pi$$
$$80$$ −1305.00 −1.82379
$$81$$ 0 0
$$82$$ −189.000 −0.254531
$$83$$ −1247.08 −1.64921 −0.824605 0.565709i $$-0.808603\pi$$
−0.824605 + 0.565709i $$0.808603\pi$$
$$84$$ 0 0
$$85$$ 795.011 1.01448
$$86$$ −378.000 −0.473963
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −45.0000 −0.0535954 −0.0267977 0.999641i $$-0.508531\pi$$
−0.0267977 + 0.999641i $$0.508531\pi$$
$$90$$ 0 0
$$91$$ −1722.00 −1.98368
$$92$$ −1710.00 −1.93782
$$93$$ 0 0
$$94$$ −374.123 −0.410509
$$95$$ 1309.43 1.41416
$$96$$ 0 0
$$97$$ 89.0000 0.0931606 0.0465803 0.998915i $$-0.485168\pi$$
0.0465803 + 0.998915i $$0.485168\pi$$
$$98$$ 1273.06 1.31223
$$99$$ 0 0
$$100$$ −836.000 −0.836000
$$101$$ −1288.65 −1.26955 −0.634777 0.772695i $$-0.718908\pi$$
−0.634777 + 0.772695i $$0.718908\pi$$
$$102$$ 0 0
$$103$$ 1358.00 1.29910 0.649552 0.760317i $$-0.274956\pi$$
0.649552 + 0.760317i $$0.274956\pi$$
$$104$$ 4059.00 3.82709
$$105$$ 0 0
$$106$$ 233.827 0.214257
$$107$$ −1299.04 −1.17367 −0.586835 0.809706i $$-0.699626\pi$$
−0.586835 + 0.809706i $$0.699626\pi$$
$$108$$ 0 0
$$109$$ 1203.78 1.05781 0.528903 0.848683i $$-0.322604\pi$$
0.528903 + 0.848683i $$0.322604\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3516.06 −2.96640
$$113$$ −711.000 −0.591905 −0.295952 0.955203i $$-0.595637\pi$$
−0.295952 + 0.955203i $$0.595637\pi$$
$$114$$ 0 0
$$115$$ 810.000 0.656808
$$116$$ 1678.36 1.34338
$$117$$ 0 0
$$118$$ −1964.15 −1.53232
$$119$$ 2142.00 1.65006
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 3240.00 2.40439
$$123$$ 0 0
$$124$$ −3572.00 −2.58690
$$125$$ 1521.00 1.08834
$$126$$ 0 0
$$127$$ 651.251 0.455033 0.227516 0.973774i $$-0.426939\pi$$
0.227516 + 0.973774i $$0.426939\pi$$
$$128$$ −400.104 −0.276285
$$129$$ 0 0
$$130$$ −3321.00 −2.24055
$$131$$ −2608.47 −1.73972 −0.869859 0.493301i $$-0.835790\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$132$$ 0 0
$$133$$ 3528.00 2.30012
$$134$$ −2005.71 −1.29304
$$135$$ 0 0
$$136$$ −5049.00 −3.18344
$$137$$ −2394.00 −1.49294 −0.746472 0.665417i $$-0.768254\pi$$
−0.746472 + 0.665417i $$0.768254\pi$$
$$138$$ 0 0
$$139$$ 183.597 0.112033 0.0560163 0.998430i $$-0.482160\pi$$
0.0560163 + 0.998430i $$0.482160\pi$$
$$140$$ 4146.53 2.50318
$$141$$ 0 0
$$142$$ 1028.84 0.608015
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −795.011 −0.455325
$$146$$ 396.000 0.224474
$$147$$ 0 0
$$148$$ 2527.00 1.40350
$$149$$ 3611.33 1.98558 0.992790 0.119869i $$-0.0382473\pi$$
0.992790 + 0.119869i $$0.0382473\pi$$
$$150$$ 0 0
$$151$$ 1357.93 0.731832 0.365916 0.930648i $$-0.380756\pi$$
0.365916 + 0.930648i $$0.380756\pi$$
$$152$$ −8316.00 −4.43761
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1692.00 0.876805
$$156$$ 0 0
$$157$$ 1306.00 0.663886 0.331943 0.943299i $$-0.392296\pi$$
0.331943 + 0.943299i $$0.392296\pi$$
$$158$$ −792.000 −0.398786
$$159$$ 0 0
$$160$$ −2665.63 −1.31710
$$161$$ 2182.38 1.06830
$$162$$ 0 0
$$163$$ 2546.00 1.22342 0.611712 0.791081i $$-0.290481\pi$$
0.611712 + 0.791081i $$0.290481\pi$$
$$164$$ −691.088 −0.329054
$$165$$ 0 0
$$166$$ −6480.00 −3.02979
$$167$$ 675.500 0.313004 0.156502 0.987678i $$-0.449978\pi$$
0.156502 + 0.987678i $$0.449978\pi$$
$$168$$ 0 0
$$169$$ 2846.00 1.29540
$$170$$ 4131.00 1.86372
$$171$$ 0 0
$$172$$ −1382.18 −0.612732
$$173$$ 3180.05 1.39754 0.698770 0.715347i $$-0.253731\pi$$
0.698770 + 0.715347i $$0.253731\pi$$
$$174$$ 0 0
$$175$$ 1066.94 0.460876
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −233.827 −0.0984610
$$179$$ 2556.00 1.06729 0.533644 0.845709i $$-0.320822\pi$$
0.533644 + 0.845709i $$0.320822\pi$$
$$180$$ 0 0
$$181$$ −775.000 −0.318261 −0.159131 0.987258i $$-0.550869\pi$$
−0.159131 + 0.987258i $$0.550869\pi$$
$$182$$ −8947.77 −3.64425
$$183$$ 0 0
$$184$$ −5144.19 −2.06106
$$185$$ −1197.00 −0.475704
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −1368.00 −0.530700
$$189$$ 0 0
$$190$$ 6804.00 2.59797
$$191$$ −2628.00 −0.995578 −0.497789 0.867298i $$-0.665855\pi$$
−0.497789 + 0.867298i $$0.665855\pi$$
$$192$$ 0 0
$$193$$ 2790.33 1.04069 0.520344 0.853957i $$-0.325804\pi$$
0.520344 + 0.853957i $$0.325804\pi$$
$$194$$ 462.458 0.171147
$$195$$ 0 0
$$196$$ 4655.00 1.69643
$$197$$ 3590.54 1.29856 0.649278 0.760551i $$-0.275071\pi$$
0.649278 + 0.760551i $$0.275071\pi$$
$$198$$ 0 0
$$199$$ 1618.00 0.576367 0.288183 0.957575i $$-0.406949\pi$$
0.288183 + 0.957575i $$0.406949\pi$$
$$200$$ −2514.94 −0.889165
$$201$$ 0 0
$$202$$ −6696.00 −2.33232
$$203$$ −2142.00 −0.740586
$$204$$ 0 0
$$205$$ 327.358 0.111530
$$206$$ 7056.37 2.38661
$$207$$ 0 0
$$208$$ 10297.0 3.43255
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1860.22 0.606934 0.303467 0.952842i $$-0.401856\pi$$
0.303467 + 0.952842i $$0.401856\pi$$
$$212$$ 855.000 0.276989
$$213$$ 0 0
$$214$$ −6750.00 −2.15617
$$215$$ 654.715 0.207680
$$216$$ 0 0
$$217$$ 4558.76 1.42612
$$218$$ 6255.00 1.94331
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6273.00 −1.90936
$$222$$ 0 0
$$223$$ −964.000 −0.289481 −0.144740 0.989470i $$-0.546235\pi$$
−0.144740 + 0.989470i $$0.546235\pi$$
$$224$$ −7182.00 −2.14227
$$225$$ 0 0
$$226$$ −3694.46 −1.08740
$$227$$ 2857.88 0.835614 0.417807 0.908536i $$-0.362799\pi$$
0.417807 + 0.908536i $$0.362799\pi$$
$$228$$ 0 0
$$229$$ −5519.00 −1.59260 −0.796301 0.604901i $$-0.793213\pi$$
−0.796301 + 0.604901i $$0.793213\pi$$
$$230$$ 4208.88 1.20663
$$231$$ 0 0
$$232$$ 5049.00 1.42881
$$233$$ 3611.33 1.01539 0.507695 0.861537i $$-0.330498\pi$$
0.507695 + 0.861537i $$0.330498\pi$$
$$234$$ 0 0
$$235$$ 648.000 0.179876
$$236$$ −7182.00 −1.98097
$$237$$ 0 0
$$238$$ 11130.2 3.03135
$$239$$ −3263.18 −0.883171 −0.441585 0.897219i $$-0.645584\pi$$
−0.441585 + 0.897219i $$0.645584\pi$$
$$240$$ 0 0
$$241$$ 3457.17 0.924050 0.462025 0.886867i $$-0.347123\pi$$
0.462025 + 0.886867i $$0.347123\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 11847.2 3.10836
$$245$$ −2205.00 −0.574989
$$246$$ 0 0
$$247$$ −10332.0 −2.66158
$$248$$ −10745.6 −2.75141
$$249$$ 0 0
$$250$$ 7903.35 1.99941
$$251$$ −6714.00 −1.68838 −0.844191 0.536042i $$-0.819919\pi$$
−0.844191 + 0.536042i $$0.819919\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 3384.00 0.835949
$$255$$ 0 0
$$256$$ −5111.00 −1.24780
$$257$$ −1341.00 −0.325484 −0.162742 0.986669i $$-0.552034\pi$$
−0.162742 + 0.986669i $$0.552034\pi$$
$$258$$ 0 0
$$259$$ −3225.08 −0.773732
$$260$$ −12143.4 −2.89655
$$261$$ 0 0
$$262$$ −13554.0 −3.19606
$$263$$ −956.092 −0.224164 −0.112082 0.993699i $$-0.535752\pi$$
−0.112082 + 0.993699i $$0.535752\pi$$
$$264$$ 0 0
$$265$$ −405.000 −0.0938828
$$266$$ 18332.0 4.22560
$$267$$ 0 0
$$268$$ −7334.00 −1.67162
$$269$$ −3087.00 −0.699694 −0.349847 0.936807i $$-0.613766\pi$$
−0.349847 + 0.936807i $$0.613766\pi$$
$$270$$ 0 0
$$271$$ 4035.68 0.904613 0.452306 0.891863i $$-0.350601\pi$$
0.452306 + 0.891863i $$0.350601\pi$$
$$272$$ −12808.5 −2.85526
$$273$$ 0 0
$$274$$ −12439.6 −2.74271
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3791.46 −0.822407 −0.411203 0.911544i $$-0.634891\pi$$
−0.411203 + 0.911544i $$0.634891\pi$$
$$278$$ 954.000 0.205817
$$279$$ 0 0
$$280$$ 12474.0 2.66237
$$281$$ 2722.78 0.578034 0.289017 0.957324i $$-0.406672\pi$$
0.289017 + 0.957324i $$0.406672\pi$$
$$282$$ 0 0
$$283$$ −3238.94 −0.680335 −0.340167 0.940365i $$-0.610484\pi$$
−0.340167 + 0.940365i $$0.610484\pi$$
$$284$$ 3762.00 0.786034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 882.000 0.181404
$$288$$ 0 0
$$289$$ 2890.00 0.588235
$$290$$ −4131.00 −0.836485
$$291$$ 0 0
$$292$$ 1447.99 0.290197
$$293$$ −4338.79 −0.865101 −0.432551 0.901610i $$-0.642386\pi$$
−0.432551 + 0.901610i $$0.642386\pi$$
$$294$$ 0 0
$$295$$ 3402.00 0.671431
$$296$$ 7601.97 1.49276
$$297$$ 0 0
$$298$$ 18765.0 3.64774
$$299$$ −6391.27 −1.23618
$$300$$ 0 0
$$301$$ 1764.00 0.337792
$$302$$ 7056.00 1.34446
$$303$$ 0 0
$$304$$ −21096.4 −3.98013
$$305$$ −5611.84 −1.05355
$$306$$ 0 0
$$307$$ −5781.59 −1.07483 −0.537415 0.843318i $$-0.680599\pi$$
−0.537415 + 0.843318i $$0.680599\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8791.89 1.61079
$$311$$ −5220.00 −0.951765 −0.475883 0.879509i $$-0.657871\pi$$
−0.475883 + 0.879509i $$0.657871\pi$$
$$312$$ 0 0
$$313$$ −3977.00 −0.718190 −0.359095 0.933301i $$-0.616915\pi$$
−0.359095 + 0.933301i $$0.616915\pi$$
$$314$$ 6786.18 1.21964
$$315$$ 0 0
$$316$$ −2895.99 −0.515545
$$317$$ −9918.00 −1.75726 −0.878628 0.477506i $$-0.841541\pi$$
−0.878628 + 0.477506i $$0.841541\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −3411.00 −0.595877
$$321$$ 0 0
$$322$$ 11340.0 1.96259
$$323$$ 12852.0 2.21395
$$324$$ 0 0
$$325$$ −3124.62 −0.533301
$$326$$ 13229.4 2.24757
$$327$$ 0 0
$$328$$ −2079.00 −0.349980
$$329$$ 1745.91 0.292568
$$330$$ 0 0
$$331$$ 8756.00 1.45400 0.726999 0.686639i $$-0.240915\pi$$
0.726999 + 0.686639i $$0.240915\pi$$
$$332$$ −23694.5 −3.91687
$$333$$ 0 0
$$334$$ 3510.00 0.575026
$$335$$ 3474.00 0.566582
$$336$$ 0 0
$$337$$ 9919.45 1.60340 0.801702 0.597724i $$-0.203928\pi$$
0.801702 + 0.597724i $$0.203928\pi$$
$$338$$ 14788.2 2.37981
$$339$$ 0 0
$$340$$ 15105.2 2.40940
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 2376.37 0.374088
$$344$$ −4158.00 −0.651699
$$345$$ 0 0
$$346$$ 16524.0 2.56744
$$347$$ 1330.22 0.205792 0.102896 0.994692i $$-0.467189\pi$$
0.102896 + 0.994692i $$0.467189\pi$$
$$348$$ 0 0
$$349$$ 2842.30 0.435944 0.217972 0.975955i $$-0.430056\pi$$
0.217972 + 0.975955i $$0.430056\pi$$
$$350$$ 5544.00 0.846684
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1431.00 0.215763 0.107882 0.994164i $$-0.465593\pi$$
0.107882 + 0.994164i $$0.465593\pi$$
$$354$$ 0 0
$$355$$ −1782.00 −0.266419
$$356$$ −855.000 −0.127289
$$357$$ 0 0
$$358$$ 13281.4 1.96073
$$359$$ 5393.61 0.792935 0.396467 0.918049i $$-0.370236\pi$$
0.396467 + 0.918049i $$0.370236\pi$$
$$360$$ 0 0
$$361$$ 14309.0 2.08616
$$362$$ −4027.02 −0.584683
$$363$$ 0 0
$$364$$ −32718.0 −4.71123
$$365$$ −685.892 −0.0983595
$$366$$ 0 0
$$367$$ 3554.00 0.505497 0.252748 0.967532i $$-0.418666\pi$$
0.252748 + 0.967532i $$0.418666\pi$$
$$368$$ −13050.0 −1.84858
$$369$$ 0 0
$$370$$ −6219.79 −0.873924
$$371$$ −1091.19 −0.152700
$$372$$ 0 0
$$373$$ −5577.20 −0.774200 −0.387100 0.922038i $$-0.626523\pi$$
−0.387100 + 0.922038i $$0.626523\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −4115.35 −0.564450
$$377$$ 6273.00 0.856965
$$378$$ 0 0
$$379$$ −13186.0 −1.78712 −0.893561 0.448942i $$-0.851801\pi$$
−0.893561 + 0.448942i $$0.851801\pi$$
$$380$$ 24879.2 3.35862
$$381$$ 0 0
$$382$$ −13655.5 −1.82899
$$383$$ 3330.00 0.444269 0.222135 0.975016i $$-0.428698\pi$$
0.222135 + 0.975016i $$0.428698\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14499.0 1.91186
$$387$$ 0 0
$$388$$ 1691.00 0.221256
$$389$$ −3537.00 −0.461010 −0.230505 0.973071i $$-0.574038\pi$$
−0.230505 + 0.973071i $$0.574038\pi$$
$$390$$ 0 0
$$391$$ 7950.11 1.02827
$$392$$ 14003.6 1.80431
$$393$$ 0 0
$$394$$ 18657.0 2.38560
$$395$$ 1371.78 0.174739
$$396$$ 0 0
$$397$$ −4501.00 −0.569014 −0.284507 0.958674i $$-0.591830\pi$$
−0.284507 + 0.958674i $$0.591830\pi$$
$$398$$ 8407.37 1.05885
$$399$$ 0 0
$$400$$ −6380.00 −0.797500
$$401$$ −3303.00 −0.411332 −0.205666 0.978622i $$-0.565936\pi$$
−0.205666 + 0.978622i $$0.565936\pi$$
$$402$$ 0 0
$$403$$ −13350.6 −1.65023
$$404$$ −24484.3 −3.01519
$$405$$ 0 0
$$406$$ −11130.2 −1.36054
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −9576.51 −1.15777 −0.578885 0.815409i $$-0.696512\pi$$
−0.578885 + 0.815409i $$0.696512\pi$$
$$410$$ 1701.00 0.204894
$$411$$ 0 0
$$412$$ 25802.0 3.08537
$$413$$ 9166.01 1.09208
$$414$$ 0 0
$$415$$ 11223.7 1.32759
$$416$$ 21033.0 2.47891
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 6750.00 0.787015 0.393507 0.919322i $$-0.371262\pi$$
0.393507 + 0.919322i $$0.371262\pi$$
$$420$$ 0 0
$$421$$ −13481.0 −1.56063 −0.780313 0.625389i $$-0.784940\pi$$
−0.780313 + 0.625389i $$0.784940\pi$$
$$422$$ 9666.00 1.11501
$$423$$ 0 0
$$424$$ 2572.10 0.294604
$$425$$ 3886.72 0.443609
$$426$$ 0 0
$$427$$ −15120.0 −1.71360
$$428$$ −24681.7 −2.78747
$$429$$ 0 0
$$430$$ 3402.00 0.381533
$$431$$ 1382.18 0.154471 0.0772356 0.997013i $$-0.475391\pi$$
0.0772356 + 0.997013i $$0.475391\pi$$
$$432$$ 0 0
$$433$$ −1531.00 −0.169920 −0.0849598 0.996384i $$-0.527076\pi$$
−0.0849598 + 0.996384i $$0.527076\pi$$
$$434$$ 23688.0 2.61995
$$435$$ 0 0
$$436$$ 22871.7 2.51229
$$437$$ 13094.3 1.43338
$$438$$ 0 0
$$439$$ 1919.11 0.208643 0.104321 0.994544i $$-0.466733\pi$$
0.104321 + 0.994544i $$0.466733\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −32595.5 −3.50771
$$443$$ −14598.0 −1.56563 −0.782813 0.622258i $$-0.786216\pi$$
−0.782813 + 0.622258i $$0.786216\pi$$
$$444$$ 0 0
$$445$$ 405.000 0.0431435
$$446$$ −5009.09 −0.531810
$$447$$ 0 0
$$448$$ −9190.26 −0.969194
$$449$$ 12591.0 1.32340 0.661699 0.749769i $$-0.269836\pi$$
0.661699 + 0.749769i $$0.269836\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −13509.0 −1.40577
$$453$$ 0 0
$$454$$ 14850.0 1.53512
$$455$$ 15498.0 1.59683
$$456$$ 0 0
$$457$$ −16778.4 −1.71742 −0.858708 0.512465i $$-0.828733\pi$$
−0.858708 + 0.512465i $$0.828733\pi$$
$$458$$ −28677.6 −2.92580
$$459$$ 0 0
$$460$$ 15390.0 1.55992
$$461$$ 14138.7 1.42843 0.714215 0.699926i $$-0.246784\pi$$
0.714215 + 0.699926i $$0.246784\pi$$
$$462$$ 0 0
$$463$$ 6902.00 0.692793 0.346396 0.938088i $$-0.387405\pi$$
0.346396 + 0.938088i $$0.387405\pi$$
$$464$$ 12808.5 1.28151
$$465$$ 0 0
$$466$$ 18765.0 1.86539
$$467$$ −15894.0 −1.57492 −0.787459 0.616367i $$-0.788604\pi$$
−0.787459 + 0.616367i $$0.788604\pi$$
$$468$$ 0 0
$$469$$ 9360.00 0.921545
$$470$$ 3367.11 0.330453
$$471$$ 0 0
$$472$$ −21605.6 −2.10695
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 6401.66 0.618375
$$476$$ 40698.0 3.91889
$$477$$ 0 0
$$478$$ −16956.0 −1.62249
$$479$$ −9716.81 −0.926873 −0.463436 0.886130i $$-0.653384\pi$$
−0.463436 + 0.886130i $$0.653384\pi$$
$$480$$ 0 0
$$481$$ 9444.87 0.895320
$$482$$ 17964.0 1.69759
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −801.000 −0.0749929
$$486$$ 0 0
$$487$$ −7622.00 −0.709211 −0.354606 0.935016i $$-0.615385\pi$$
−0.354606 + 0.935016i $$0.615385\pi$$
$$488$$ 35640.0 3.30604
$$489$$ 0 0
$$490$$ −11457.5 −1.05632
$$491$$ 14902.6 1.36974 0.684871 0.728664i $$-0.259859\pi$$
0.684871 + 0.728664i $$0.259859\pi$$
$$492$$ 0 0
$$493$$ −7803.00 −0.712839
$$494$$ −53686.6 −4.88963
$$495$$ 0 0
$$496$$ −27260.0 −2.46776
$$497$$ −4801.24 −0.433331
$$498$$ 0 0
$$499$$ 13006.0 1.16679 0.583395 0.812188i $$-0.301724\pi$$
0.583395 + 0.812188i $$0.301724\pi$$
$$500$$ 28899.0 2.58481
$$501$$ 0 0
$$502$$ −34887.0 −3.10176
$$503$$ −14185.5 −1.25746 −0.628728 0.777626i $$-0.716424\pi$$
−0.628728 + 0.777626i $$0.716424\pi$$
$$504$$ 0 0
$$505$$ 11597.8 1.02197
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 12373.8 1.08070
$$509$$ −18234.0 −1.58783 −0.793917 0.608026i $$-0.791962\pi$$
−0.793917 + 0.608026i $$0.791962\pi$$
$$510$$ 0 0
$$511$$ −1848.00 −0.159982
$$512$$ −23356.7 −2.01607
$$513$$ 0 0
$$514$$ −6968.04 −0.597952
$$515$$ −12222.0 −1.04576
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −16758.0 −1.42144
$$519$$ 0 0
$$520$$ −36531.0 −3.08075
$$521$$ −20502.0 −1.72401 −0.862005 0.506900i $$-0.830791\pi$$
−0.862005 + 0.506900i $$0.830791\pi$$
$$522$$ 0 0
$$523$$ 10600.2 0.886257 0.443128 0.896458i $$-0.353869\pi$$
0.443128 + 0.896458i $$0.353869\pi$$
$$524$$ −49560.9 −4.13183
$$525$$ 0 0
$$526$$ −4968.00 −0.411816
$$527$$ 16606.9 1.37269
$$528$$ 0 0
$$529$$ −4067.00 −0.334265
$$530$$ −2104.44 −0.172474
$$531$$ 0 0
$$532$$ 67032.0 5.46279
$$533$$ −2583.00 −0.209910
$$534$$ 0 0
$$535$$ 11691.3 0.944787
$$536$$ −22062.9 −1.77793
$$537$$ 0 0
$$538$$ −16040.5 −1.28542
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6082.96 −0.483414 −0.241707 0.970349i $$-0.577707\pi$$
−0.241707 + 0.970349i $$0.577707\pi$$
$$542$$ 20970.0 1.66188
$$543$$ 0 0
$$544$$ −26163.0 −2.06200
$$545$$ −10834.0 −0.851517
$$546$$ 0 0
$$547$$ −19378.2 −1.51472 −0.757360 0.652998i $$-0.773511\pi$$
−0.757360 + 0.652998i $$0.773511\pi$$
$$548$$ −45486.0 −3.54574
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12852.0 −0.993673
$$552$$ 0 0
$$553$$ 3696.00 0.284213
$$554$$ −19701.0 −1.51086
$$555$$ 0 0
$$556$$ 3488.35 0.266077
$$557$$ −14216.7 −1.08147 −0.540736 0.841192i $$-0.681854\pi$$
−0.540736 + 0.841192i $$0.681854\pi$$
$$558$$ 0 0
$$559$$ −5166.00 −0.390874
$$560$$ 31644.6 2.38791
$$561$$ 0 0
$$562$$ 14148.0 1.06192
$$563$$ −2130.42 −0.159479 −0.0797394 0.996816i $$-0.525409\pi$$
−0.0797394 + 0.996816i $$0.525409\pi$$
$$564$$ 0 0
$$565$$ 6399.00 0.476474
$$566$$ −16830.0 −1.24985
$$567$$ 0 0
$$568$$ 11317.2 0.836021
$$569$$ 17251.2 1.27102 0.635509 0.772094i $$-0.280790\pi$$
0.635509 + 0.772094i $$0.280790\pi$$
$$570$$ 0 0
$$571$$ −9079.41 −0.665432 −0.332716 0.943027i $$-0.607965\pi$$
−0.332716 + 0.943027i $$0.607965\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 4583.01 0.333260
$$575$$ 3960.00 0.287206
$$576$$ 0 0
$$577$$ −7427.00 −0.535858 −0.267929 0.963439i $$-0.586339\pi$$
−0.267929 + 0.963439i $$0.586339\pi$$
$$578$$ 15016.9 1.08066
$$579$$ 0 0
$$580$$ −15105.2 −1.08140
$$581$$ 30240.0 2.15932
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 4356.00 0.308652
$$585$$ 0 0
$$586$$ −22545.0 −1.58929
$$587$$ 9972.00 0.701173 0.350586 0.936530i $$-0.385982\pi$$
0.350586 + 0.936530i $$0.385982\pi$$
$$588$$ 0 0
$$589$$ 27352.5 1.91348
$$590$$ 17677.3 1.23350
$$591$$ 0 0
$$592$$ 19285.0 1.33887
$$593$$ −21922.6 −1.51813 −0.759066 0.651014i $$-0.774344\pi$$
−0.759066 + 0.651014i $$0.774344\pi$$
$$594$$ 0 0
$$595$$ −19278.0 −1.32827
$$596$$ 68615.2 4.71575
$$597$$ 0 0
$$598$$ −33210.0 −2.27100
$$599$$ −576.000 −0.0392900 −0.0196450 0.999807i $$-0.506254\pi$$
−0.0196450 + 0.999807i $$0.506254\pi$$
$$600$$ 0 0
$$601$$ 7740.54 0.525363 0.262681 0.964883i $$-0.415393\pi$$
0.262681 + 0.964883i $$0.415393\pi$$
$$602$$ 9166.01 0.620563
$$603$$ 0 0
$$604$$ 25800.6 1.73810
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14500.7 0.969632 0.484816 0.874616i $$-0.338887\pi$$
0.484816 + 0.874616i $$0.338887\pi$$
$$608$$ −43092.0 −2.87436
$$609$$ 0 0
$$610$$ −29160.0 −1.93550
$$611$$ −5113.01 −0.338544
$$612$$ 0 0
$$613$$ −5298.34 −0.349100 −0.174550 0.984648i $$-0.555847\pi$$
−0.174550 + 0.984648i $$0.555847\pi$$
$$614$$ −30042.0 −1.97459
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6939.00 −0.452761 −0.226381 0.974039i $$-0.572689\pi$$
−0.226381 + 0.974039i $$0.572689\pi$$
$$618$$ 0 0
$$619$$ 1286.00 0.0835036 0.0417518 0.999128i $$-0.486706\pi$$
0.0417518 + 0.999128i $$0.486706\pi$$
$$620$$ 32148.0 2.08241
$$621$$ 0 0
$$622$$ −27123.9 −1.74850
$$623$$ 1091.19 0.0701728
$$624$$ 0 0
$$625$$ −8189.00 −0.524096
$$626$$ −20665.1 −1.31940
$$627$$ 0 0
$$628$$ 24814.0 1.57673
$$629$$ −11748.5 −0.744743
$$630$$ 0 0
$$631$$ −15110.0 −0.953280 −0.476640 0.879099i $$-0.658145\pi$$
−0.476640 + 0.879099i $$0.658145\pi$$
$$632$$ −8712.00 −0.548330
$$633$$ 0 0
$$634$$ −51535.4 −3.22829
$$635$$ −5861.26 −0.366294
$$636$$ 0 0
$$637$$ 17398.5 1.08218
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 3600.93 0.222405
$$641$$ −13293.0 −0.819098 −0.409549 0.912288i $$-0.634314\pi$$
−0.409549 + 0.912288i $$0.634314\pi$$
$$642$$ 0 0
$$643$$ 20528.0 1.25901 0.629506 0.776995i $$-0.283257\pi$$
0.629506 + 0.776995i $$0.283257\pi$$
$$644$$ 41465.3 2.53721
$$645$$ 0 0
$$646$$ 66781.0 4.06728
$$647$$ 11124.0 0.675934 0.337967 0.941158i $$-0.390261\pi$$
0.337967 + 0.941158i $$0.390261\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −16236.0 −0.979736
$$651$$ 0 0
$$652$$ 48374.0 2.90563
$$653$$ −5562.00 −0.333320 −0.166660 0.986014i $$-0.553298\pi$$
−0.166660 + 0.986014i $$0.553298\pi$$
$$654$$ 0 0
$$655$$ 23476.2 1.40045
$$656$$ −5274.09 −0.313901
$$657$$ 0 0
$$658$$ 9072.00 0.537482
$$659$$ 16378.3 0.968144 0.484072 0.875028i $$-0.339157\pi$$
0.484072 + 0.875028i $$0.339157\pi$$
$$660$$ 0 0
$$661$$ −28385.0 −1.67027 −0.835135 0.550045i $$-0.814611\pi$$
−0.835135 + 0.550045i $$0.814611\pi$$
$$662$$ 45497.5 2.67116
$$663$$ 0 0
$$664$$ −71280.0 −4.16596
$$665$$ −31752.0 −1.85156
$$666$$ 0 0
$$667$$ −7950.11 −0.461514
$$668$$ 12834.5 0.743386
$$669$$ 0 0
$$670$$ 18051.4 1.04088
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 6948.99 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$674$$ 51543.0 2.94564
$$675$$ 0 0
$$676$$ 54074.0 3.07658
$$677$$ −3923.10 −0.222713 −0.111357 0.993781i $$-0.535520\pi$$
−0.111357 + 0.993781i $$0.535520\pi$$
$$678$$ 0 0
$$679$$ −2158.14 −0.121976
$$680$$ 45441.0 2.56262
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −2034.00 −0.113951 −0.0569757 0.998376i $$-0.518146\pi$$
−0.0569757 + 0.998376i $$0.518146\pi$$
$$684$$ 0 0
$$685$$ 21546.0 1.20180
$$686$$ 12348.0 0.687243
$$687$$ 0 0
$$688$$ −10548.2 −0.584514
$$689$$ 3195.63 0.176697
$$690$$ 0 0
$$691$$ 4598.00 0.253135 0.126567 0.991958i $$-0.459604\pi$$
0.126567 + 0.991958i $$0.459604\pi$$
$$692$$ 60420.9 3.31916
$$693$$ 0 0
$$694$$ 6912.00 0.378063
$$695$$ −1652.38 −0.0901845
$$696$$ 0 0
$$697$$ 3213.00 0.174607
$$698$$ 14769.0 0.800881
$$699$$ 0 0
$$700$$ 20271.9 1.09458
$$701$$ −5949.59 −0.320561 −0.160280 0.987072i $$-0.551240\pi$$
−0.160280 + 0.987072i $$0.551240\pi$$
$$702$$ 0 0
$$703$$ −19350.5 −1.03815
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 7435.69 0.396382
$$707$$ 31248.0 1.66224
$$708$$ 0 0
$$709$$ 24814.0 1.31440 0.657200 0.753716i $$-0.271741\pi$$
0.657200 + 0.753716i $$0.271741\pi$$
$$710$$ −9259.54 −0.489443
$$711$$ 0 0
$$712$$ −2572.10 −0.135384
$$713$$ 16920.0 0.888722
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 48564.0 2.53481
$$717$$ 0 0
$$718$$ 28026.0 1.45671
$$719$$ −26442.0 −1.37152 −0.685758 0.727829i $$-0.740529\pi$$
−0.685758 + 0.727829i $$0.740529\pi$$
$$720$$ 0 0
$$721$$ −32929.7 −1.70093
$$722$$ 74351.7 3.83253
$$723$$ 0 0
$$724$$ −14725.0 −0.755871
$$725$$ −3886.72 −0.199102
$$726$$ 0 0
$$727$$ −28370.0 −1.44730 −0.723649 0.690169i $$-0.757536\pi$$
−0.723649 + 0.690169i $$0.757536\pi$$
$$728$$ −98425.5 −5.01084
$$729$$ 0 0
$$730$$ −3564.00 −0.180698
$$731$$ 6426.00 0.325136
$$732$$ 0 0
$$733$$ 20332.5 1.02456 0.512278 0.858820i $$-0.328802\pi$$
0.512278 + 0.858820i $$0.328802\pi$$
$$734$$ 18467.1 0.928657
$$735$$ 0 0
$$736$$ −26656.3 −1.33500
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −9048.23 −0.450399 −0.225199 0.974313i $$-0.572303\pi$$
−0.225199 + 0.974313i $$0.572303\pi$$
$$740$$ −22743.0 −1.12980
$$741$$ 0 0
$$742$$ −5670.00 −0.280529
$$743$$ 9758.37 0.481830 0.240915 0.970546i $$-0.422552\pi$$
0.240915 + 0.970546i $$0.422552\pi$$
$$744$$ 0 0
$$745$$ −32501.9 −1.59836
$$746$$ −28980.0 −1.42230
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 31500.0 1.53670
$$750$$ 0 0
$$751$$ −34010.0 −1.65252 −0.826260 0.563289i $$-0.809536\pi$$
−0.826260 + 0.563289i $$0.809536\pi$$
$$752$$ −10440.0 −0.506260
$$753$$ 0 0
$$754$$ 32595.5 1.57435
$$755$$ −12221.4 −0.589113
$$756$$ 0 0
$$757$$ −5345.00 −0.256628 −0.128314 0.991734i $$-0.540957\pi$$
−0.128314 + 0.991734i $$0.540957\pi$$
$$758$$ −68516.5 −3.28315
$$759$$ 0 0
$$760$$ 74844.0 3.57221
$$761$$ −22629.2 −1.07794 −0.538968 0.842326i $$-0.681186\pi$$
−0.538968 + 0.842326i $$0.681186\pi$$
$$762$$ 0 0
$$763$$ −29190.0 −1.38499
$$764$$ −49932.0 −2.36450
$$765$$ 0 0
$$766$$ 17303.2 0.816174
$$767$$ −26843.3 −1.26370
$$768$$ 0 0
$$769$$ 34961.4 1.63946 0.819728 0.572753i $$-0.194124\pi$$
0.819728 + 0.572753i $$0.194124\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 53016.3 2.47163
$$773$$ −29610.0 −1.37775 −0.688873 0.724882i $$-0.741894\pi$$
−0.688873 + 0.724882i $$0.741894\pi$$
$$774$$ 0 0
$$775$$ 8272.00 0.383405
$$776$$ 5087.03 0.235327
$$777$$ 0 0
$$778$$ −18378.8 −0.846930
$$779$$ 5292.00 0.243396
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 41310.0 1.88906
$$783$$ 0 0
$$784$$ 35525.0 1.61830
$$785$$ −11754.0 −0.534418
$$786$$ 0 0
$$787$$ 5785.05 0.262026 0.131013 0.991381i $$-0.458177\pi$$
0.131013 + 0.991381i $$0.458177\pi$$
$$788$$ 68220.3 3.08407
$$789$$ 0 0
$$790$$ 7128.00 0.321016
$$791$$ 17240.8 0.774985
$$792$$ 0 0
$$793$$ 44280.0 1.98289
$$794$$ −23387.9 −1.04535
$$795$$ 0 0
$$796$$ 30742.0 1.36887
$$797$$ 4626.00 0.205598 0.102799 0.994702i $$-0.467220\pi$$
0.102799 + 0.994702i $$0.467220\pi$$
$$798$$ 0 0
$$799$$ 6360.09 0.281607
$$800$$ −13032.0 −0.575936
$$801$$ 0 0
$$802$$ −17162.9 −0.755664
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −19641.5 −0.859963
$$806$$ −69372.0 −3.03167
$$807$$ 0 0
$$808$$ −73656.0 −3.20694
$$809$$ 31239.3 1.35762 0.678810 0.734314i $$-0.262496\pi$$
0.678810 + 0.734314i $$0.262496\pi$$
$$810$$ 0 0
$$811$$ −44340.5 −1.91986 −0.959929 0.280242i $$-0.909585\pi$$
−0.959929 + 0.280242i $$0.909585\pi$$
$$812$$ −40698.0 −1.75889
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −22914.0 −0.984837
$$816$$ 0 0
$$817$$ 10584.0 0.453228
$$818$$ −49761.0 −2.12696
$$819$$ 0 0
$$820$$ 6219.79 0.264884
$$821$$ −38493.1 −1.63632 −0.818160 0.574991i $$-0.805006\pi$$
−0.818160 + 0.574991i $$0.805006\pi$$
$$822$$ 0 0
$$823$$ 8678.00 0.367553 0.183776 0.982968i $$-0.441168\pi$$
0.183776 + 0.982968i $$0.441168\pi$$
$$824$$ 77620.1 3.28158
$$825$$ 0 0
$$826$$ 47628.0 2.00628
$$827$$ −27238.2 −1.14530 −0.572652 0.819799i $$-0.694085\pi$$
−0.572652 + 0.819799i $$0.694085\pi$$
$$828$$ 0 0
$$829$$ −19789.0 −0.829072 −0.414536 0.910033i $$-0.636056\pi$$
−0.414536 + 0.910033i $$0.636056\pi$$
$$830$$ 58320.0 2.43894
$$831$$ 0 0
$$832$$ 26914.3 1.12150
$$833$$ −21642.0 −0.900180
$$834$$ 0 0
$$835$$ −6079.50 −0.251964
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 35074.0 1.44584
$$839$$ 1800.00 0.0740678 0.0370339 0.999314i $$-0.488209\pi$$
0.0370339 + 0.999314i $$0.488209\pi$$
$$840$$ 0 0
$$841$$ −16586.0 −0.680061
$$842$$ −70049.3 −2.86705
$$843$$ 0 0
$$844$$ 35344.2 1.44147
$$845$$ −25614.0 −1.04278
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6525.00 0.264233
$$849$$ 0 0
$$850$$ 20196.0 0.814961
$$851$$ −11970.0 −0.482170
$$852$$ 0 0
$$853$$ −33513.5 −1.34523 −0.672614 0.739994i $$-0.734828\pi$$
−0.672614 + 0.739994i $$0.734828\pi$$
$$854$$ −78565.8 −3.14809
$$855$$ 0 0
$$856$$ −74250.0 −2.96473
$$857$$ 26687.4 1.06374 0.531870 0.846826i $$-0.321489\pi$$
0.531870 + 0.846826i $$0.321489\pi$$
$$858$$ 0 0
$$859$$ 46694.0 1.85469 0.927345 0.374207i $$-0.122085\pi$$
0.927345 + 0.374207i $$0.122085\pi$$
$$860$$ 12439.6 0.493240
$$861$$ 0 0
$$862$$ 7182.00 0.283782
$$863$$ 36018.0 1.42070 0.710352 0.703847i $$-0.248536\pi$$
0.710352 + 0.703847i $$0.248536\pi$$
$$864$$ 0 0
$$865$$ −28620.4 −1.12500
$$866$$ −7955.31 −0.312162
$$867$$ 0 0
$$868$$ 86616.4 3.38704
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −27411.4 −1.06636
$$872$$ 68805.0 2.67205
$$873$$ 0 0
$$874$$ 68040.0 2.63328
$$875$$ −36882.3 −1.42497
$$876$$ 0 0
$$877$$ 9191.99 0.353924 0.176962 0.984218i $$-0.443373\pi$$
0.176962 + 0.984218i $$0.443373\pi$$
$$878$$ 9972.00 0.383301
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 40005.0 1.52986 0.764928 0.644116i $$-0.222775\pi$$
0.764928 + 0.644116i $$0.222775\pi$$
$$882$$ 0 0
$$883$$ 4492.00 0.171198 0.0855990 0.996330i $$-0.472720\pi$$
0.0855990 + 0.996330i $$0.472720\pi$$
$$884$$ −119187. −4.53472
$$885$$ 0 0
$$886$$ −75853.4 −2.87624
$$887$$ 43554.1 1.64871 0.824355 0.566074i $$-0.191538\pi$$
0.824355 + 0.566074i $$0.191538\pi$$
$$888$$ 0 0
$$889$$ −15792.0 −0.595778
$$890$$ 2104.44 0.0792596
$$891$$ 0 0
$$892$$ −18316.0 −0.687517
$$893$$ 10475.4 0.392550
$$894$$ 0 0
$$895$$ −23004.0 −0.859150
$$896$$ 9702.00 0.361742
$$897$$ 0 0
$$898$$ 65424.8 2.43124
$$899$$ −16606.9 −0.616097
$$900$$ 0 0
$$901$$ −3975.06 −0.146979
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −40639.1 −1.49517
$$905$$ 6975.00 0.256195
$$906$$ 0 0
$$907$$ 7634.00 0.279474 0.139737 0.990189i $$-0.455374\pi$$
0.139737 + 0.990189i $$0.455374\pi$$
$$908$$ 54299.8 1.98458
$$909$$ 0 0
$$910$$ 80530.0 2.93356
$$911$$ 43830.0 1.59402 0.797010 0.603966i $$-0.206414\pi$$
0.797010 + 0.603966i $$0.206414\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −87183.0 −3.15510
$$915$$ 0 0
$$916$$ −104861. −3.78243
$$917$$ 63252.0 2.27782
$$918$$ 0 0
$$919$$ 32531.4 1.16769 0.583847 0.811864i $$-0.301547\pi$$
0.583847 + 0.811864i $$0.301547\pi$$
$$920$$ 46297.7 1.65912
$$921$$ 0 0
$$922$$ 73467.0 2.62419
$$923$$ 14060.8 0.501426
$$924$$ 0 0
$$925$$ −5852.00 −0.208014
$$926$$ 35863.8 1.27274
$$927$$ 0 0
$$928$$ 26163.0 0.925477
$$929$$ −15651.0 −0.552737 −0.276368 0.961052i $$-0.589131\pi$$
−0.276368 + 0.961052i $$0.589131\pi$$
$$930$$ 0 0
$$931$$ −35645.6 −1.25482
$$932$$ 68615.2 2.41155
$$933$$ 0 0
$$934$$ −82587.6 −2.89331
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 6337.57 0.220960 0.110480 0.993878i $$-0.464761\pi$$
0.110480 + 0.993878i $$0.464761\pi$$
$$938$$ 48636.0 1.69299
$$939$$ 0 0
$$940$$ 12312.0 0.427205
$$941$$ 9846.71 0.341120 0.170560 0.985347i $$-0.445442\pi$$
0.170560 + 0.985347i $$0.445442\pi$$
$$942$$ 0 0
$$943$$ 3273.58 0.113046
$$944$$ −54810.0 −1.88974
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 738.000 0.0253239 0.0126620 0.999920i $$-0.495969\pi$$
0.0126620 + 0.999920i $$0.495969\pi$$
$$948$$ 0 0
$$949$$ 5412.00 0.185122
$$950$$ 33264.0 1.13603
$$951$$ 0 0
$$952$$ 122432. 4.16810
$$953$$ −15541.7 −0.528274 −0.264137 0.964485i $$-0.585087\pi$$
−0.264137 + 0.964485i $$0.585087\pi$$
$$954$$ 0 0
$$955$$ 23652.0 0.801425
$$956$$ −62000.5 −2.09753
$$957$$ 0 0
$$958$$ −50490.0 −1.70277
$$959$$ 58051.4 1.95472
$$960$$ 0 0
$$961$$ 5553.00 0.186399
$$962$$ 49077.0 1.64481
$$963$$ 0 0
$$964$$ 65686.3 2.19462
$$965$$ −25113.0 −0.837737
$$966$$ 0 0
$$967$$ 37692.9 1.25349 0.626743 0.779226i $$-0.284387\pi$$
0.626743 + 0.779226i $$0.284387\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −4162.12 −0.137771
$$971$$ 12402.0 0.409886 0.204943 0.978774i $$-0.434299\pi$$
0.204943 + 0.978774i $$0.434299\pi$$
$$972$$ 0 0
$$973$$ −4452.00 −0.146685
$$974$$ −39605.1 −1.30290
$$975$$ 0 0
$$976$$ 90413.1 2.96522
$$977$$ −31203.0 −1.02177 −0.510887 0.859648i $$-0.670683\pi$$
−0.510887 + 0.859648i $$0.670683\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −41895.0 −1.36560
$$981$$ 0 0
$$982$$ 77436.0 2.51638
$$983$$ 36540.0 1.18560 0.592800 0.805350i $$-0.298022\pi$$
0.592800 + 0.805350i $$0.298022\pi$$
$$984$$ 0 0
$$985$$ −32314.9 −1.04532
$$986$$ −40545.6 −1.30957
$$987$$ 0 0
$$988$$ −196308. −6.32124
$$989$$ 6547.15 0.210503
$$990$$ 0 0
$$991$$ −56888.0 −1.82352 −0.911759 0.410725i $$-0.865276\pi$$
−0.911759 + 0.410725i $$0.865276\pi$$
$$992$$ −55682.0 −1.78216
$$993$$ 0 0
$$994$$ −24948.0 −0.796079
$$995$$ −14562.0 −0.463966
$$996$$ 0 0
$$997$$ −15711.4 −0.499083 −0.249542 0.968364i $$-0.580280\pi$$
−0.249542 + 0.968364i $$0.580280\pi$$
$$998$$ 67581.2 2.14353
$$999$$ 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 1089.4.a.s.1.2 2
3.2 odd 2 363.4.a.l.1.1 2
11.10 odd 2 inner 1089.4.a.s.1.1 2
33.32 even 2 363.4.a.l.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.l.1.1 2 3.2 odd 2
363.4.a.l.1.2 yes 2 33.32 even 2
1089.4.a.s.1.1 2 11.10 odd 2 inner
1089.4.a.s.1.2 2 1.1 even 1 trivial