# Properties

 Label 2736.2.a.bc.1.2 Level $2736$ Weight $2$ Character 2736.1 Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 8x + 10$$ x^3 - x^2 - 8*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1368) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.31955$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.319551 q^{5} +2.61968 q^{7} +O(q^{10})$$ $$q+0.319551 q^{5} +2.61968 q^{7} -4.31955 q^{11} +6.51757 q^{13} -4.19802 q^{17} +1.00000 q^{19} +0.639102 q^{23} -4.89789 q^{25} +7.87847 q^{29} +6.00000 q^{31} +0.837122 q^{35} +4.00000 q^{37} +12.5176 q^{41} +1.38032 q^{43} -7.55892 q^{47} -0.137255 q^{49} +13.1567 q^{53} -1.38032 q^{55} -13.0351 q^{59} -5.89789 q^{61} +2.08270 q^{65} -11.7569 q^{67} +11.7569 q^{71} +15.1373 q^{73} -11.3159 q^{77} -0.517571 q^{79} -11.8785 q^{83} -1.34148 q^{85} -3.87847 q^{89} +17.0740 q^{91} +0.319551 q^{95} +11.2782 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{5} + 2 q^{7}+O(q^{10})$$ 3 * q - 2 * q^5 + 2 * q^7 $$3 q - 2 q^{5} + 2 q^{7} - 10 q^{11} - 4 q^{13} + 8 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} + 6 q^{29} + 18 q^{31} - 24 q^{35} + 12 q^{37} + 14 q^{41} + 10 q^{43} - 8 q^{47} + 29 q^{49} + 10 q^{53} - 10 q^{55} + 8 q^{59} + 24 q^{65} + 16 q^{73} + 16 q^{77} + 22 q^{79} - 18 q^{83} - 10 q^{85} + 6 q^{89} + 4 q^{91} - 2 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q - 2 * q^5 + 2 * q^7 - 10 * q^11 - 4 * q^13 + 8 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 + 6 * q^29 + 18 * q^31 - 24 * q^35 + 12 * q^37 + 14 * q^41 + 10 * q^43 - 8 * q^47 + 29 * q^49 + 10 * q^53 - 10 * q^55 + 8 * q^59 + 24 * q^65 + 16 * q^73 + 16 * q^77 + 22 * q^79 - 18 * q^83 - 10 * q^85 + 6 * q^89 + 4 * q^91 - 2 * q^95 + 22 * 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
$$5$$ 0.319551 0.142907 0.0714537 0.997444i $$-0.477236\pi$$
0.0714537 + 0.997444i $$0.477236\pi$$
$$6$$ 0 0
$$7$$ 2.61968 0.990148 0.495074 0.868851i $$-0.335141\pi$$
0.495074 + 0.868851i $$0.335141\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.31955 −1.30239 −0.651197 0.758909i $$-0.725733\pi$$
−0.651197 + 0.758909i $$0.725733\pi$$
$$12$$ 0 0
$$13$$ 6.51757 1.80765 0.903825 0.427903i $$-0.140748\pi$$
0.903825 + 0.427903i $$0.140748\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.19802 −1.01817 −0.509085 0.860716i $$-0.670016\pi$$
−0.509085 + 0.860716i $$0.670016\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.639102 0.133262 0.0666309 0.997778i $$-0.478775\pi$$
0.0666309 + 0.997778i $$0.478775\pi$$
$$24$$ 0 0
$$25$$ −4.89789 −0.979577
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.87847 1.46300 0.731498 0.681844i $$-0.238822\pi$$
0.731498 + 0.681844i $$0.238822\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.837122 0.141499
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 12.5176 1.95492 0.977458 0.211129i $$-0.0677142\pi$$
0.977458 + 0.211129i $$0.0677142\pi$$
$$42$$ 0 0
$$43$$ 1.38032 0.210496 0.105248 0.994446i $$-0.466436\pi$$
0.105248 + 0.994446i $$0.466436\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.55892 −1.10258 −0.551291 0.834313i $$-0.685865\pi$$
−0.551291 + 0.834313i $$0.685865\pi$$
$$48$$ 0 0
$$49$$ −0.137255 −0.0196079
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 13.1567 1.80721 0.903604 0.428369i $$-0.140912\pi$$
0.903604 + 0.428369i $$0.140912\pi$$
$$54$$ 0 0
$$55$$ −1.38032 −0.186122
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −13.0351 −1.69703 −0.848516 0.529171i $$-0.822503\pi$$
−0.848516 + 0.529171i $$0.822503\pi$$
$$60$$ 0 0
$$61$$ −5.89789 −0.755147 −0.377574 0.925980i $$-0.623241\pi$$
−0.377574 + 0.925980i $$0.623241\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.08270 0.258327
$$66$$ 0 0
$$67$$ −11.7569 −1.43634 −0.718169 0.695868i $$-0.755020\pi$$
−0.718169 + 0.695868i $$0.755020\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.7569 1.39529 0.697646 0.716443i $$-0.254231\pi$$
0.697646 + 0.716443i $$0.254231\pi$$
$$72$$ 0 0
$$73$$ 15.1373 1.77168 0.885841 0.463989i $$-0.153582\pi$$
0.885841 + 0.463989i $$0.153582\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −11.3159 −1.28956
$$78$$ 0 0
$$79$$ −0.517571 −0.0582313 −0.0291157 0.999576i $$-0.509269\pi$$
−0.0291157 + 0.999576i $$0.509269\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −11.8785 −1.30383 −0.651916 0.758291i $$-0.726034\pi$$
−0.651916 + 0.758291i $$0.726034\pi$$
$$84$$ 0 0
$$85$$ −1.34148 −0.145504
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.87847 −0.411117 −0.205558 0.978645i $$-0.565901\pi$$
−0.205558 + 0.978645i $$0.565901\pi$$
$$90$$ 0 0
$$91$$ 17.0740 1.78984
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0.319551 0.0327852
$$96$$ 0 0
$$97$$ 11.2782 1.14513 0.572564 0.819860i $$-0.305949\pi$$
0.572564 + 0.819860i $$0.305949\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.51757 −0.449515 −0.224758 0.974415i $$-0.572159\pi$$
−0.224758 + 0.974415i $$0.572159\pi$$
$$102$$ 0 0
$$103$$ 19.0351 1.87559 0.937794 0.347192i $$-0.112865\pi$$
0.937794 + 0.347192i $$0.112865\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.11784 0.688107 0.344054 0.938950i $$-0.388200\pi$$
0.344054 + 0.938950i $$0.388200\pi$$
$$108$$ 0 0
$$109$$ −5.27820 −0.505560 −0.252780 0.967524i $$-0.581345\pi$$
−0.252780 + 0.967524i $$0.581345\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.39973 −0.131676 −0.0658379 0.997830i $$-0.520972\pi$$
−0.0658379 + 0.997830i $$0.520972\pi$$
$$114$$ 0 0
$$115$$ 0.204225 0.0190441
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −10.9975 −1.00814
$$120$$ 0 0
$$121$$ 7.65852 0.696229
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3.16288 −0.282896
$$126$$ 0 0
$$127$$ 9.75694 0.865788 0.432894 0.901445i $$-0.357492\pi$$
0.432894 + 0.901445i $$0.357492\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 19.4374 1.69825 0.849126 0.528190i $$-0.177129\pi$$
0.849126 + 0.528190i $$0.177129\pi$$
$$132$$ 0 0
$$133$$ 2.61968 0.227155
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.9198 0.932943 0.466471 0.884536i $$-0.345525\pi$$
0.466471 + 0.884536i $$0.345525\pi$$
$$138$$ 0 0
$$139$$ 6.61968 0.561474 0.280737 0.959785i $$-0.409421\pi$$
0.280737 + 0.959785i $$0.409421\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −28.1530 −2.35427
$$144$$ 0 0
$$145$$ 2.51757 0.209073
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −0.958652 −0.0785359 −0.0392679 0.999229i $$-0.512503\pi$$
−0.0392679 + 0.999229i $$0.512503\pi$$
$$150$$ 0 0
$$151$$ −0.517571 −0.0421194 −0.0210597 0.999778i $$-0.506704\pi$$
−0.0210597 + 0.999778i $$0.506704\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.91730 0.154002
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.67424 0.131949
$$162$$ 0 0
$$163$$ −17.0351 −1.33430 −0.667148 0.744925i $$-0.732485\pi$$
−0.667148 + 0.744925i $$0.732485\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.6391 0.978043 0.489022 0.872272i $$-0.337354\pi$$
0.489022 + 0.872272i $$0.337354\pi$$
$$168$$ 0 0
$$169$$ 29.4787 2.26760
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.60027 0.197695 0.0988473 0.995103i $$-0.468484\pi$$
0.0988473 + 0.995103i $$0.468484\pi$$
$$174$$ 0 0
$$175$$ −12.8309 −0.969926
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −19.1178 −1.42893 −0.714467 0.699669i $$-0.753331\pi$$
−0.714467 + 0.699669i $$0.753331\pi$$
$$180$$ 0 0
$$181$$ −22.2745 −1.65565 −0.827826 0.560985i $$-0.810422\pi$$
−0.827826 + 0.560985i $$0.810422\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.27820 0.0939754
$$186$$ 0 0
$$187$$ 18.1336 1.32606
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.55892 0.546944 0.273472 0.961880i $$-0.411828\pi$$
0.273472 + 0.961880i $$0.411828\pi$$
$$192$$ 0 0
$$193$$ −0.517571 −0.0372556 −0.0186278 0.999826i $$-0.505930\pi$$
−0.0186278 + 0.999826i $$0.505930\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.4824 0.818089 0.409045 0.912514i $$-0.365862\pi$$
0.409045 + 0.912514i $$0.365862\pi$$
$$198$$ 0 0
$$199$$ 1.38032 0.0978480 0.0489240 0.998803i $$-0.484421\pi$$
0.0489240 + 0.998803i $$0.484421\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.6391 1.44858
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.31955 −0.298790
$$210$$ 0 0
$$211$$ −2.72180 −0.187376 −0.0936881 0.995602i $$-0.529866\pi$$
−0.0936881 + 0.995602i $$0.529866\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0.441081 0.0300815
$$216$$ 0 0
$$217$$ 15.7181 1.06701
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −27.3609 −1.84049
$$222$$ 0 0
$$223$$ 9.79577 0.655974 0.327987 0.944682i $$-0.393630\pi$$
0.327987 + 0.944682i $$0.393630\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.6391 0.838887 0.419443 0.907782i $$-0.362225\pi$$
0.419443 + 0.907782i $$0.362225\pi$$
$$228$$ 0 0
$$229$$ −11.1373 −0.735971 −0.367985 0.929832i $$-0.619952\pi$$
−0.367985 + 0.929832i $$0.619952\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −21.8723 −1.43290 −0.716450 0.697639i $$-0.754234\pi$$
−0.716450 + 0.697639i $$0.754234\pi$$
$$234$$ 0 0
$$235$$ −2.41546 −0.157567
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 25.8723 1.67354 0.836769 0.547556i $$-0.184442\pi$$
0.836769 + 0.547556i $$0.184442\pi$$
$$240$$ 0 0
$$241$$ 9.79577 0.631001 0.315501 0.948925i $$-0.397828\pi$$
0.315501 + 0.948925i $$0.397828\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.0438601 −0.00280212
$$246$$ 0 0
$$247$$ 6.51757 0.414703
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3.28441 0.207310 0.103655 0.994613i $$-0.466946\pi$$
0.103655 + 0.994613i $$0.466946\pi$$
$$252$$ 0 0
$$253$$ −2.76063 −0.173559
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.5176 −0.780825 −0.390412 0.920640i $$-0.627668\pi$$
−0.390412 + 0.920640i $$0.627668\pi$$
$$258$$ 0 0
$$259$$ 10.4787 0.651117
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −24.1980 −1.49211 −0.746057 0.665882i $$-0.768055\pi$$
−0.746057 + 0.665882i $$0.768055\pi$$
$$264$$ 0 0
$$265$$ 4.20423 0.258264
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 11.8785 0.724243 0.362122 0.932131i $$-0.382053\pi$$
0.362122 + 0.932131i $$0.382053\pi$$
$$270$$ 0 0
$$271$$ 6.55641 0.398273 0.199137 0.979972i $$-0.436186\pi$$
0.199137 + 0.979972i $$0.436186\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 21.1567 1.27580
$$276$$ 0 0
$$277$$ −16.6585 −1.00091 −0.500457 0.865762i $$-0.666835\pi$$
−0.500457 + 0.865762i $$0.666835\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.3572 −0.617859 −0.308930 0.951085i $$-0.599971\pi$$
−0.308930 + 0.951085i $$0.599971\pi$$
$$282$$ 0 0
$$283$$ 30.4155 1.80801 0.904006 0.427520i $$-0.140613\pi$$
0.904006 + 0.427520i $$0.140613\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 32.7921 1.93566
$$288$$ 0 0
$$289$$ 0.623376 0.0366692
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 12.2745 0.717085 0.358542 0.933513i $$-0.383274\pi$$
0.358542 + 0.933513i $$0.383274\pi$$
$$294$$ 0 0
$$295$$ −4.16539 −0.242518
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.16539 0.240891
$$300$$ 0 0
$$301$$ 3.61599 0.208422
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1.88467 −0.107916
$$306$$ 0 0
$$307$$ −2.72180 −0.155341 −0.0776706 0.996979i $$-0.524748\pi$$
−0.0776706 + 0.996979i $$0.524748\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −0.837122 −0.0474688 −0.0237344 0.999718i $$-0.507556\pi$$
−0.0237344 + 0.999718i $$0.507556\pi$$
$$312$$ 0 0
$$313$$ 12.5564 0.709730 0.354865 0.934918i $$-0.384527\pi$$
0.354865 + 0.934918i $$0.384527\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0.517571 0.0290697 0.0145349 0.999894i $$-0.495373\pi$$
0.0145349 + 0.999894i $$0.495373\pi$$
$$318$$ 0 0
$$319$$ −34.0315 −1.90540
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.19802 −0.233584
$$324$$ 0 0
$$325$$ −31.9223 −1.77073
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −19.8020 −1.09172
$$330$$ 0 0
$$331$$ −18.3133 −1.00659 −0.503296 0.864114i $$-0.667880\pi$$
−0.503296 + 0.864114i $$0.667880\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −3.75694 −0.205264
$$336$$ 0 0
$$337$$ −3.27820 −0.178575 −0.0892876 0.996006i $$-0.528459\pi$$
−0.0892876 + 0.996006i $$0.528459\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −25.9173 −1.40350
$$342$$ 0 0
$$343$$ −18.6974 −1.00956
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −13.5978 −0.729966 −0.364983 0.931014i $$-0.618925\pi$$
−0.364983 + 0.931014i $$0.618925\pi$$
$$348$$ 0 0
$$349$$ 16.1724 0.865689 0.432844 0.901469i $$-0.357510\pi$$
0.432844 + 0.901469i $$0.357510\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.47874 −0.344828 −0.172414 0.985025i $$-0.555157\pi$$
−0.172414 + 0.985025i $$0.555157\pi$$
$$354$$ 0 0
$$355$$ 3.75694 0.199398
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.67676 −0.352386 −0.176193 0.984356i $$-0.556378\pi$$
−0.176193 + 0.984356i $$0.556378\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.83712 0.253187
$$366$$ 0 0
$$367$$ −25.0351 −1.30682 −0.653412 0.757003i $$-0.726663\pi$$
−0.653412 + 0.757003i $$0.726663\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 34.4663 1.78940
$$372$$ 0 0
$$373$$ 14.5564 0.753702 0.376851 0.926274i $$-0.377007\pi$$
0.376851 + 0.926274i $$0.377007\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 51.3485 2.64458
$$378$$ 0 0
$$379$$ 1.23937 0.0636621 0.0318310 0.999493i $$-0.489866\pi$$
0.0318310 + 0.999493i $$0.489866\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 1.67424 0.0855499 0.0427749 0.999085i $$-0.486380\pi$$
0.0427749 + 0.999085i $$0.486380\pi$$
$$384$$ 0 0
$$385$$ −3.61599 −0.184288
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −26.6329 −1.35034 −0.675171 0.737661i $$-0.735930\pi$$
−0.675171 + 0.737661i $$0.735930\pi$$
$$390$$ 0 0
$$391$$ −2.68296 −0.135683
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −0.165390 −0.00832169
$$396$$ 0 0
$$397$$ −17.8979 −0.898269 −0.449135 0.893464i $$-0.648268\pi$$
−0.449135 + 0.893464i $$0.648268\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −33.7958 −1.68768 −0.843840 0.536595i $$-0.819710\pi$$
−0.843840 + 0.536595i $$0.819710\pi$$
$$402$$ 0 0
$$403$$ 39.1054 1.94798
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −17.2782 −0.856449
$$408$$ 0 0
$$409$$ −21.5139 −1.06379 −0.531896 0.846809i $$-0.678520\pi$$
−0.531896 + 0.846809i $$0.678520\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −34.1480 −1.68031
$$414$$ 0 0
$$415$$ −3.79577 −0.186327
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −10.6003 −0.517857 −0.258928 0.965897i $$-0.583369\pi$$
−0.258928 + 0.965897i $$0.583369\pi$$
$$420$$ 0 0
$$421$$ −23.7181 −1.15595 −0.577975 0.816055i $$-0.696157\pi$$
−0.577975 + 0.816055i $$0.696157\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 20.5614 0.997376
$$426$$ 0 0
$$427$$ −15.4506 −0.747707
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 19.3609 0.932582 0.466291 0.884631i $$-0.345590\pi$$
0.466291 + 0.884631i $$0.345590\pi$$
$$432$$ 0 0
$$433$$ −13.7569 −0.661116 −0.330558 0.943786i $$-0.607237\pi$$
−0.330558 + 0.943786i $$0.607237\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.639102 0.0305724
$$438$$ 0 0
$$439$$ −20.2357 −0.965796 −0.482898 0.875677i $$-0.660416\pi$$
−0.482898 + 0.875677i $$0.660416\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6.87596 −0.326687 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$444$$ 0 0
$$445$$ −1.23937 −0.0587517
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4.76063 0.224668 0.112334 0.993670i $$-0.464167\pi$$
0.112334 + 0.993670i $$0.464167\pi$$
$$450$$ 0 0
$$451$$ −54.0703 −2.54607
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 5.45600 0.255781
$$456$$ 0 0
$$457$$ 5.82022 0.272258 0.136129 0.990691i $$-0.456534\pi$$
0.136129 + 0.990691i $$0.456534\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 15.0413 0.700545 0.350273 0.936648i $$-0.386089\pi$$
0.350273 + 0.936648i $$0.386089\pi$$
$$462$$ 0 0
$$463$$ 27.6548 1.28523 0.642614 0.766190i $$-0.277850\pi$$
0.642614 + 0.766190i $$0.277850\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −34.9513 −1.61735 −0.808676 0.588254i $$-0.799815\pi$$
−0.808676 + 0.588254i $$0.799815\pi$$
$$468$$ 0 0
$$469$$ −30.7995 −1.42219
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.96234 −0.274149
$$474$$ 0 0
$$475$$ −4.89789 −0.224730
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9.67424 0.442028 0.221014 0.975271i $$-0.429063\pi$$
0.221014 + 0.975271i $$0.429063\pi$$
$$480$$ 0 0
$$481$$ 26.0703 1.18870
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.60396 0.163647
$$486$$ 0 0
$$487$$ −20.5176 −0.929740 −0.464870 0.885379i $$-0.653899\pi$$
−0.464870 + 0.885379i $$0.653899\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.9136 −1.66589 −0.832944 0.553357i $$-0.813346\pi$$
−0.832944 + 0.553357i $$0.813346\pi$$
$$492$$ 0 0
$$493$$ −33.0740 −1.48958
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 30.7995 1.38154
$$498$$ 0 0
$$499$$ −9.17609 −0.410778 −0.205389 0.978680i $$-0.565846\pi$$
−0.205389 + 0.978680i $$0.565846\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 32.1530 1.43363 0.716815 0.697263i $$-0.245599\pi$$
0.716815 + 0.697263i $$0.245599\pi$$
$$504$$ 0 0
$$505$$ −1.44359 −0.0642391
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −16.9136 −0.749683 −0.374841 0.927089i $$-0.622303\pi$$
−0.374841 + 0.927089i $$0.622303\pi$$
$$510$$ 0 0
$$511$$ 39.6548 1.75423
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 6.08270 0.268036
$$516$$ 0 0
$$517$$ 32.6511 1.43600
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10.0388 0.439809 0.219905 0.975521i $$-0.429425\pi$$
0.219905 + 0.975521i $$0.429425\pi$$
$$522$$ 0 0
$$523$$ −34.2745 −1.49872 −0.749360 0.662163i $$-0.769639\pi$$
−0.749360 + 0.662163i $$0.769639\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −25.1881 −1.09721
$$528$$ 0 0
$$529$$ −22.5915 −0.982241
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 81.5842 3.53380
$$534$$ 0 0
$$535$$ 2.27451 0.0983357
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.592882 0.0255372
$$540$$ 0 0
$$541$$ −28.1724 −1.21123 −0.605613 0.795759i $$-0.707072\pi$$
−0.605613 + 0.795759i $$0.707072\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1.68665 −0.0722483
$$546$$ 0 0
$$547$$ 7.55271 0.322931 0.161465 0.986878i $$-0.448378\pi$$
0.161465 + 0.986878i $$0.448378\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7.87847 0.335634
$$552$$ 0 0
$$553$$ −1.35587 −0.0576576
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 23.4374 0.993074 0.496537 0.868016i $$-0.334605\pi$$
0.496537 + 0.868016i $$0.334605\pi$$
$$558$$ 0 0
$$559$$ 8.99631 0.380503
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 34.3133 1.44614 0.723068 0.690777i $$-0.242732\pi$$
0.723068 + 0.690777i $$0.242732\pi$$
$$564$$ 0 0
$$565$$ −0.447286 −0.0188175
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −23.8785 −1.00104 −0.500519 0.865726i $$-0.666857\pi$$
−0.500519 + 0.865726i $$0.666857\pi$$
$$570$$ 0 0
$$571$$ 9.03514 0.378109 0.189054 0.981967i $$-0.439458\pi$$
0.189054 + 0.981967i $$0.439458\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3.13025 −0.130540
$$576$$ 0 0
$$577$$ −18.8554 −0.784959 −0.392479 0.919761i $$-0.628383\pi$$
−0.392479 + 0.919761i $$0.628383\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −31.1178 −1.29099
$$582$$ 0 0
$$583$$ −56.8309 −2.35370
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.68045 −0.151908 −0.0759542 0.997111i $$-0.524200\pi$$
−0.0759542 + 0.997111i $$0.524200\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −23.5139 −0.965599 −0.482800 0.875731i $$-0.660380\pi$$
−0.482800 + 0.875731i $$0.660380\pi$$
$$594$$ 0 0
$$595$$ −3.51426 −0.144070
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6.31335 0.257956 0.128978 0.991647i $$-0.458830\pi$$
0.128978 + 0.991647i $$0.458830\pi$$
$$600$$ 0 0
$$601$$ 19.4824 0.794705 0.397352 0.917666i $$-0.369929\pi$$
0.397352 + 0.917666i $$0.369929\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 2.44729 0.0994963
$$606$$ 0 0
$$607$$ 37.7569 1.53251 0.766253 0.642538i $$-0.222119\pi$$
0.766253 + 0.642538i $$0.222119\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −49.2658 −1.99308
$$612$$ 0 0
$$613$$ −4.65852 −0.188156 −0.0940779 0.995565i $$-0.529990\pi$$
−0.0940779 + 0.995565i $$0.529990\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −36.3510 −1.46344 −0.731718 0.681607i $$-0.761281\pi$$
−0.731718 + 0.681607i $$0.761281\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −10.1604 −0.407066
$$624$$ 0 0
$$625$$ 23.4787 0.939149
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −16.7921 −0.669544
$$630$$ 0 0
$$631$$ −35.4506 −1.41127 −0.705633 0.708577i $$-0.749337\pi$$
−0.705633 + 0.708577i $$0.749337\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 3.11784 0.123728
$$636$$ 0 0
$$637$$ −0.894572 −0.0354442
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23.8785 0.943143 0.471571 0.881828i $$-0.343687\pi$$
0.471571 + 0.881828i $$0.343687\pi$$
$$642$$ 0 0
$$643$$ 3.93672 0.155249 0.0776246 0.996983i $$-0.475266\pi$$
0.0776246 + 0.996983i $$0.475266\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −15.8020 −0.621240 −0.310620 0.950534i $$-0.600537\pi$$
−0.310620 + 0.950534i $$0.600537\pi$$
$$648$$ 0 0
$$649$$ 56.3060 2.21020
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 43.2720 1.69336 0.846682 0.532099i $$-0.178597\pi$$
0.846682 + 0.532099i $$0.178597\pi$$
$$654$$ 0 0
$$655$$ 6.21123 0.242693
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −21.8396 −0.850751 −0.425376 0.905017i $$-0.639858\pi$$
−0.425376 + 0.905017i $$0.639858\pi$$
$$660$$ 0 0
$$661$$ −9.03514 −0.351426 −0.175713 0.984441i $$-0.556223\pi$$
−0.175713 + 0.984441i $$0.556223\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.837122 0.0324622
$$666$$ 0 0
$$667$$ 5.03514 0.194962
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 25.4762 0.983499
$$672$$ 0 0
$$673$$ 9.96116 0.383975 0.191987 0.981397i $$-0.438507\pi$$
0.191987 + 0.981397i $$0.438507\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.35721 0.244327 0.122164 0.992510i $$-0.461017\pi$$
0.122164 + 0.992510i $$0.461017\pi$$
$$678$$ 0 0
$$679$$ 29.5453 1.13385
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 37.1881 1.42296 0.711482 0.702704i $$-0.248024\pi$$
0.711482 + 0.702704i $$0.248024\pi$$
$$684$$ 0 0
$$685$$ 3.48944 0.133325
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 85.7496 3.26680
$$690$$ 0 0
$$691$$ −32.8942 −1.25135 −0.625677 0.780082i $$-0.715177\pi$$
−0.625677 + 0.780082i $$0.715177\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2.11533 0.0802389
$$696$$ 0 0
$$697$$ −52.5490 −1.99044
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16.4399 0.620926 0.310463 0.950585i $$-0.399516\pi$$
0.310463 + 0.950585i $$0.399516\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −11.8346 −0.445086
$$708$$ 0 0
$$709$$ −41.5139 −1.55909 −0.779543 0.626348i $$-0.784549\pi$$
−0.779543 + 0.626348i $$0.784549\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.83461 0.143607
$$714$$ 0 0
$$715$$ −8.99631 −0.336443
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −32.8371 −1.22462 −0.612309 0.790619i $$-0.709759\pi$$
−0.612309 + 0.790619i $$0.709759\pi$$
$$720$$ 0 0
$$721$$ 49.8661 1.85711
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −38.5879 −1.43312
$$726$$ 0 0
$$727$$ −4.89419 −0.181516 −0.0907578 0.995873i $$-0.528929\pi$$
−0.0907578 + 0.995873i $$0.528929\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −5.79459 −0.214321
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 50.7847 1.87068
$$738$$ 0 0
$$739$$ −19.4506 −0.715502 −0.357751 0.933817i $$-0.616456\pi$$
−0.357751 + 0.933817i $$0.616456\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −23.5915 −0.865490 −0.432745 0.901516i $$-0.642455\pi$$
−0.432745 + 0.901516i $$0.642455\pi$$
$$744$$ 0 0
$$745$$ −0.306338 −0.0112234
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 18.6465 0.681328
$$750$$ 0 0
$$751$$ −4.76063 −0.173718 −0.0868590 0.996221i $$-0.527683\pi$$
−0.0868590 + 0.996221i $$0.527683\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −0.165390 −0.00601917
$$756$$ 0 0
$$757$$ −46.9330 −1.70581 −0.852905 0.522066i $$-0.825161\pi$$
−0.852905 + 0.522066i $$0.825161\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.3108 −1.20752 −0.603758 0.797167i $$-0.706331\pi$$
−0.603758 + 0.797167i $$0.706331\pi$$
$$762$$ 0 0
$$763$$ −13.8272 −0.500579
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −84.9575 −3.06764
$$768$$ 0 0
$$769$$ −14.9330 −0.538499 −0.269249 0.963070i $$-0.586776\pi$$
−0.269249 + 0.963070i $$0.586776\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −32.5176 −1.16958 −0.584788 0.811186i $$-0.698822\pi$$
−0.584788 + 0.811186i $$0.698822\pi$$
$$774$$ 0 0
$$775$$ −29.3873 −1.05562
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.5176 0.448489
$$780$$ 0 0
$$781$$ −50.7847 −1.81722
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0.639102 0.0228105
$$786$$ 0 0
$$787$$ −14.5564 −0.518880 −0.259440 0.965759i $$-0.583538\pi$$
−0.259440 + 0.965759i $$0.583538\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −3.66686 −0.130379
$$792$$ 0 0
$$793$$ −38.4399 −1.36504
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −23.2394 −0.823181 −0.411590 0.911369i $$-0.635027\pi$$
−0.411590 + 0.911369i $$0.635027\pi$$
$$798$$ 0 0
$$799$$ 31.7325 1.12262
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −65.3861 −2.30743
$$804$$ 0 0
$$805$$ 0.535006 0.0188565
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39.7896 1.39893 0.699463 0.714668i $$-0.253422\pi$$
0.699463 + 0.714668i $$0.253422\pi$$
$$810$$ 0 0
$$811$$ −39.4750 −1.38616 −0.693078 0.720862i $$-0.743746\pi$$
−0.693078 + 0.720862i $$0.743746\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −5.44359 −0.190681
$$816$$ 0 0
$$817$$ 1.38032 0.0482911
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.3196 −0.569556 −0.284778 0.958593i $$-0.591920\pi$$
−0.284778 + 0.958593i $$0.591920\pi$$
$$822$$ 0 0
$$823$$ −16.9719 −0.591602 −0.295801 0.955250i $$-0.595587\pi$$
−0.295801 + 0.955250i $$0.595587\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25.1128 −0.873258 −0.436629 0.899642i $$-0.643828\pi$$
−0.436629 + 0.899642i $$0.643828\pi$$
$$828$$ 0 0
$$829$$ −35.5915 −1.23615 −0.618073 0.786121i $$-0.712086\pi$$
−0.618073 + 0.786121i $$0.712086\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0.576201 0.0199642
$$834$$ 0 0
$$835$$ 4.03884 0.139770
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −29.8396 −1.03018 −0.515089 0.857137i $$-0.672241\pi$$
−0.515089 + 0.857137i $$0.672241\pi$$
$$840$$ 0 0
$$841$$ 33.0703 1.14035
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 9.41995 0.324056
$$846$$ 0 0
$$847$$ 20.0629 0.689369
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2.55641 0.0876325
$$852$$ 0 0
$$853$$ 14.0777 0.482010 0.241005 0.970524i $$-0.422523\pi$$
0.241005 + 0.970524i $$0.422523\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 30.9963 1.05881 0.529407 0.848368i $$-0.322415\pi$$
0.529407 + 0.848368i $$0.322415\pi$$
$$858$$ 0 0
$$859$$ 43.2464 1.47555 0.737774 0.675048i $$-0.235877\pi$$
0.737774 + 0.675048i $$0.235877\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 12.9575 0.441077 0.220539 0.975378i $$-0.429218\pi$$
0.220539 + 0.975378i $$0.429218\pi$$
$$864$$ 0 0
$$865$$ 0.830917 0.0282520
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 2.23568 0.0758401
$$870$$ 0 0
$$871$$ −76.6267 −2.59640
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −8.28574 −0.280109
$$876$$ 0 0
$$877$$ −11.7181 −0.395692 −0.197846 0.980233i $$-0.563395\pi$$
−0.197846 + 0.980233i $$0.563395\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −28.3510 −0.955169 −0.477585 0.878586i $$-0.658488\pi$$
−0.477585 + 0.878586i $$0.658488\pi$$
$$882$$ 0 0
$$883$$ −22.3378 −0.751726 −0.375863 0.926675i $$-0.622654\pi$$
−0.375863 + 0.926675i $$0.622654\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.5490 1.22720 0.613598 0.789619i $$-0.289722\pi$$
0.613598 + 0.789619i $$0.289722\pi$$
$$888$$ 0 0
$$889$$ 25.5601 0.857258
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −7.55892 −0.252950
$$894$$ 0 0
$$895$$ −6.10912 −0.204205
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 47.2708 1.57657
$$900$$ 0 0
$$901$$ −55.2320 −1.84004
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −7.11784 −0.236605
$$906$$ 0 0
$$907$$ −47.3097 −1.57089 −0.785446 0.618931i $$-0.787566\pi$$
−0.785446 + 0.618931i $$0.787566\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 37.8396 1.25368 0.626842 0.779147i $$-0.284347\pi$$
0.626842 + 0.779147i $$0.284347\pi$$
$$912$$ 0 0
$$913$$ 51.3097 1.69810
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 50.9198 1.68152
$$918$$ 0 0
$$919$$ 43.5915 1.43795 0.718976 0.695035i $$-0.244611\pi$$
0.718976 + 0.695035i $$0.244611\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 76.6267 2.52220
$$924$$ 0 0
$$925$$ −19.5915 −0.644166
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −23.7569 −0.779440 −0.389720 0.920933i $$-0.627428\pi$$
−0.389720 + 0.920933i $$0.627428\pi$$
$$930$$ 0 0
$$931$$ −0.137255 −0.00449836
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5.79459 0.189504
$$936$$ 0 0
$$937$$ −45.4894 −1.48608 −0.743038 0.669250i $$-0.766616\pi$$
−0.743038 + 0.669250i $$0.766616\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 48.7482 1.58915 0.794573 0.607168i $$-0.207695\pi$$
0.794573 + 0.607168i $$0.207695\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 5.94876 0.193309 0.0966543 0.995318i $$-0.469186\pi$$
0.0966543 + 0.995318i $$0.469186\pi$$
$$948$$ 0 0
$$949$$ 98.6581 3.20258
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 14.4473 0.467993 0.233997 0.972237i $$-0.424819\pi$$
0.233997 + 0.972237i $$0.424819\pi$$
$$954$$ 0 0
$$955$$ 2.41546 0.0781624
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 28.6065 0.923751
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −0.165390 −0.00532410
$$966$$ 0 0
$$967$$ −18.0703 −0.581101 −0.290551 0.956860i $$-0.593838\pi$$
−0.290551 + 0.956860i $$0.593838\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40.0000 1.28366 0.641831 0.766846i $$-0.278175\pi$$
0.641831 + 0.766846i $$0.278175\pi$$
$$972$$ 0 0
$$973$$ 17.3415 0.555942
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −13.1567 −0.420919 −0.210460 0.977603i $$-0.567496\pi$$
−0.210460 + 0.977603i $$0.567496\pi$$
$$978$$ 0 0
$$979$$ 16.7532 0.535436
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6.23568 0.198887 0.0994436 0.995043i $$-0.468294\pi$$
0.0994436 + 0.995043i $$0.468294\pi$$
$$984$$ 0 0
$$985$$ 3.66922 0.116911
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0.882162 0.0280511
$$990$$ 0 0
$$991$$ 47.8661 1.52052 0.760258 0.649622i $$-0.225073\pi$$
0.760258 + 0.649622i $$0.225073\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0.441081 0.0139832
$$996$$ 0 0
$$997$$ 44.9256 1.42281 0.711405 0.702783i $$-0.248059\pi$$
0.711405 + 0.702783i $$0.248059\pi$$
$$998$$ 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 2736.2.a.bc.1.2 3
3.2 odd 2 2736.2.a.be.1.2 3
4.3 odd 2 1368.2.a.m.1.2 3
12.11 even 2 1368.2.a.o.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.a.m.1.2 3 4.3 odd 2
1368.2.a.o.1.2 yes 3 12.11 even 2
2736.2.a.bc.1.2 3 1.1 even 1 trivial
2736.2.a.be.1.2 3 3.2 odd 2