# Properties

 Label 9576.2.a.bg.1.2 Level $9576$ Weight $2$ Character 9576.1 Self dual yes Analytic conductor $76.465$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 9576.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.23607 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+1.23607 q^{5} -1.00000 q^{7} +5.23607 q^{11} -0.763932 q^{13} -7.23607 q^{17} +1.00000 q^{19} +3.23607 q^{23} -3.47214 q^{25} -8.47214 q^{29} +0.472136 q^{31} -1.23607 q^{35} +8.94427 q^{37} -2.00000 q^{41} -4.00000 q^{43} +6.47214 q^{47} +1.00000 q^{49} +0.472136 q^{53} +6.47214 q^{55} +8.00000 q^{59} +8.47214 q^{61} -0.944272 q^{65} +0.763932 q^{67} +1.52786 q^{71} +4.47214 q^{73} -5.23607 q^{77} +15.7082 q^{79} -2.00000 q^{83} -8.94427 q^{85} +10.9443 q^{89} +0.763932 q^{91} +1.23607 q^{95} +1.23607 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^7 $$2 q - 2 q^{5} - 2 q^{7} + 6 q^{11} - 6 q^{13} - 10 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{25} - 8 q^{29} - 8 q^{31} + 2 q^{35} - 4 q^{41} - 8 q^{43} + 4 q^{47} + 2 q^{49} - 8 q^{53} + 4 q^{55} + 16 q^{59} + 8 q^{61} + 16 q^{65} + 6 q^{67} + 12 q^{71} - 6 q^{77} + 18 q^{79} - 4 q^{83} + 4 q^{89} + 6 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^7 + 6 * q^11 - 6 * q^13 - 10 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^25 - 8 * q^29 - 8 * q^31 + 2 * q^35 - 4 * q^41 - 8 * q^43 + 4 * q^47 + 2 * q^49 - 8 * q^53 + 4 * q^55 + 16 * q^59 + 8 * q^61 + 16 * q^65 + 6 * q^67 + 12 * q^71 - 6 * q^77 + 18 * q^79 - 4 * q^83 + 4 * q^89 + 6 * q^91 - 2 * q^95 - 2 * 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
$$5$$ 1.23607 0.552786 0.276393 0.961045i $$-0.410861\pi$$
0.276393 + 0.961045i $$0.410861\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.23607 1.57873 0.789367 0.613922i $$-0.210409\pi$$
0.789367 + 0.613922i $$0.210409\pi$$
$$12$$ 0 0
$$13$$ −0.763932 −0.211877 −0.105938 0.994373i $$-0.533785\pi$$
−0.105938 + 0.994373i $$0.533785\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −7.23607 −1.75500 −0.877502 0.479573i $$-0.840792\pi$$
−0.877502 + 0.479573i $$0.840792\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.23607 0.674767 0.337383 0.941367i $$-0.390458\pi$$
0.337383 + 0.941367i $$0.390458\pi$$
$$24$$ 0 0
$$25$$ −3.47214 −0.694427
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.47214 −1.57324 −0.786618 0.617440i $$-0.788170\pi$$
−0.786618 + 0.617440i $$0.788170\pi$$
$$30$$ 0 0
$$31$$ 0.472136 0.0847981 0.0423991 0.999101i $$-0.486500\pi$$
0.0423991 + 0.999101i $$0.486500\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.23607 −0.208934
$$36$$ 0 0
$$37$$ 8.94427 1.47043 0.735215 0.677834i $$-0.237081\pi$$
0.735215 + 0.677834i $$0.237081\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.47214 0.944058 0.472029 0.881583i $$-0.343522\pi$$
0.472029 + 0.881583i $$0.343522\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0.472136 0.0648529 0.0324264 0.999474i $$-0.489677\pi$$
0.0324264 + 0.999474i $$0.489677\pi$$
$$54$$ 0 0
$$55$$ 6.47214 0.872703
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 8.47214 1.08475 0.542373 0.840138i $$-0.317526\pi$$
0.542373 + 0.840138i $$0.317526\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.944272 −0.117123
$$66$$ 0 0
$$67$$ 0.763932 0.0933292 0.0466646 0.998911i $$-0.485141\pi$$
0.0466646 + 0.998911i $$0.485141\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.52786 0.181324 0.0906621 0.995882i $$-0.471102\pi$$
0.0906621 + 0.995882i $$0.471102\pi$$
$$72$$ 0 0
$$73$$ 4.47214 0.523424 0.261712 0.965146i $$-0.415713\pi$$
0.261712 + 0.965146i $$0.415713\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.23607 −0.596705
$$78$$ 0 0
$$79$$ 15.7082 1.76731 0.883656 0.468138i $$-0.155075\pi$$
0.883656 + 0.468138i $$0.155075\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ −8.94427 −0.970143
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.9443 1.16009 0.580045 0.814584i $$-0.303035\pi$$
0.580045 + 0.814584i $$0.303035\pi$$
$$90$$ 0 0
$$91$$ 0.763932 0.0800818
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.23607 0.126818
$$96$$ 0 0
$$97$$ 1.23607 0.125504 0.0627518 0.998029i $$-0.480012\pi$$
0.0627518 + 0.998029i $$0.480012\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.1803 1.80901 0.904506 0.426461i $$-0.140240\pi$$
0.904506 + 0.426461i $$0.140240\pi$$
$$102$$ 0 0
$$103$$ −7.52786 −0.741742 −0.370871 0.928684i $$-0.620941\pi$$
−0.370871 + 0.928684i $$0.620941\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.4164 −1.10367 −0.551833 0.833955i $$-0.686071\pi$$
−0.551833 + 0.833955i $$0.686071\pi$$
$$108$$ 0 0
$$109$$ 8.94427 0.856706 0.428353 0.903612i $$-0.359094\pi$$
0.428353 + 0.903612i $$0.359094\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.94427 0.653262 0.326631 0.945152i $$-0.394087\pi$$
0.326631 + 0.945152i $$0.394087\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 7.23607 0.663329
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −10.4721 −0.936656
$$126$$ 0 0
$$127$$ −6.76393 −0.600202 −0.300101 0.953907i $$-0.597020\pi$$
−0.300101 + 0.953907i $$0.597020\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.472136 0.0412507 0.0206254 0.999787i $$-0.493434\pi$$
0.0206254 + 0.999787i $$0.493434\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.47214 0.552952 0.276476 0.961021i $$-0.410833\pi$$
0.276476 + 0.961021i $$0.410833\pi$$
$$138$$ 0 0
$$139$$ −8.94427 −0.758643 −0.379322 0.925265i $$-0.623843\pi$$
−0.379322 + 0.925265i $$0.623843\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ −10.4721 −0.869664
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 16.4721 1.34945 0.674725 0.738069i $$-0.264262\pi$$
0.674725 + 0.738069i $$0.264262\pi$$
$$150$$ 0 0
$$151$$ −0.291796 −0.0237460 −0.0118730 0.999930i $$-0.503779\pi$$
−0.0118730 + 0.999930i $$0.503779\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0.583592 0.0468752
$$156$$ 0 0
$$157$$ −16.4721 −1.31462 −0.657310 0.753620i $$-0.728306\pi$$
−0.657310 + 0.753620i $$0.728306\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.23607 −0.255038
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.9443 1.31119 0.655594 0.755114i $$-0.272418\pi$$
0.655594 + 0.755114i $$0.272418\pi$$
$$168$$ 0 0
$$169$$ −12.4164 −0.955108
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −19.8885 −1.51210 −0.756049 0.654515i $$-0.772873\pi$$
−0.756049 + 0.654515i $$0.772873\pi$$
$$174$$ 0 0
$$175$$ 3.47214 0.262469
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 12.7639 0.948736 0.474368 0.880327i $$-0.342677\pi$$
0.474368 + 0.880327i $$0.342677\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 11.0557 0.812833
$$186$$ 0 0
$$187$$ −37.8885 −2.77068
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.6525 −0.770786 −0.385393 0.922753i $$-0.625934\pi$$
−0.385393 + 0.922753i $$0.625934\pi$$
$$192$$ 0 0
$$193$$ 18.9443 1.36364 0.681819 0.731521i $$-0.261189\pi$$
0.681819 + 0.731521i $$0.261189\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 17.4164 1.24087 0.620434 0.784259i $$-0.286957\pi$$
0.620434 + 0.784259i $$0.286957\pi$$
$$198$$ 0 0
$$199$$ −14.4721 −1.02590 −0.512951 0.858418i $$-0.671448\pi$$
−0.512951 + 0.858418i $$0.671448\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 8.47214 0.594627
$$204$$ 0 0
$$205$$ −2.47214 −0.172661
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 5.23607 0.362186
$$210$$ 0 0
$$211$$ −16.1803 −1.11390 −0.556950 0.830546i $$-0.688029\pi$$
−0.556950 + 0.830546i $$0.688029\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.94427 −0.337197
$$216$$ 0 0
$$217$$ −0.472136 −0.0320507
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.52786 0.371844
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −15.4164 −1.02322 −0.511611 0.859217i $$-0.670951\pi$$
−0.511611 + 0.859217i $$0.670951\pi$$
$$228$$ 0 0
$$229$$ −1.05573 −0.0697645 −0.0348822 0.999391i $$-0.511106\pi$$
−0.0348822 + 0.999391i $$0.511106\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.3607 1.59592 0.797961 0.602710i $$-0.205912\pi$$
0.797961 + 0.602710i $$0.205912\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 27.2361 1.76175 0.880877 0.473345i $$-0.156953\pi$$
0.880877 + 0.473345i $$0.156953\pi$$
$$240$$ 0 0
$$241$$ 2.76393 0.178041 0.0890203 0.996030i $$-0.471626\pi$$
0.0890203 + 0.996030i $$0.471626\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.23607 0.0789695
$$246$$ 0 0
$$247$$ −0.763932 −0.0486078
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −8.47214 −0.534756 −0.267378 0.963592i $$-0.586157\pi$$
−0.267378 + 0.963592i $$0.586157\pi$$
$$252$$ 0 0
$$253$$ 16.9443 1.06528
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 24.4721 1.52653 0.763265 0.646086i $$-0.223595\pi$$
0.763265 + 0.646086i $$0.223595\pi$$
$$258$$ 0 0
$$259$$ −8.94427 −0.555770
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −8.18034 −0.504421 −0.252211 0.967672i $$-0.581158\pi$$
−0.252211 + 0.967672i $$0.581158\pi$$
$$264$$ 0 0
$$265$$ 0.583592 0.0358498
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.9443 0.667284 0.333642 0.942700i $$-0.391722\pi$$
0.333642 + 0.942700i $$0.391722\pi$$
$$270$$ 0 0
$$271$$ −24.9443 −1.51526 −0.757628 0.652686i $$-0.773642\pi$$
−0.757628 + 0.652686i $$0.773642\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −18.1803 −1.09632
$$276$$ 0 0
$$277$$ −11.8885 −0.714313 −0.357157 0.934044i $$-0.616254\pi$$
−0.357157 + 0.934044i $$0.616254\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 9.05573 0.540219 0.270110 0.962830i $$-0.412940\pi$$
0.270110 + 0.962830i $$0.412940\pi$$
$$282$$ 0 0
$$283$$ −7.05573 −0.419419 −0.209710 0.977764i $$-0.567252\pi$$
−0.209710 + 0.977764i $$0.567252\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ 35.3607 2.08004
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 0 0
$$295$$ 9.88854 0.575733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.47214 −0.142967
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 10.4721 0.599633
$$306$$ 0 0
$$307$$ 20.3607 1.16205 0.581023 0.813887i $$-0.302653\pi$$
0.581023 + 0.813887i $$0.302653\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.9443 0.734002 0.367001 0.930220i $$-0.380385\pi$$
0.367001 + 0.930220i $$0.380385\pi$$
$$312$$ 0 0
$$313$$ 22.9443 1.29689 0.648443 0.761263i $$-0.275420\pi$$
0.648443 + 0.761263i $$0.275420\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 0 0
$$319$$ −44.3607 −2.48372
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −7.23607 −0.402626
$$324$$ 0 0
$$325$$ 2.65248 0.147133
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −6.47214 −0.356820
$$330$$ 0 0
$$331$$ 19.2361 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0.944272 0.0515911
$$336$$ 0 0
$$337$$ −31.3050 −1.70529 −0.852645 0.522491i $$-0.825003\pi$$
−0.852645 + 0.522491i $$0.825003\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.47214 0.133874
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.70820 0.199067 0.0995334 0.995034i $$-0.468265\pi$$
0.0995334 + 0.995034i $$0.468265\pi$$
$$348$$ 0 0
$$349$$ −24.8328 −1.32927 −0.664635 0.747168i $$-0.731413\pi$$
−0.664635 + 0.747168i $$0.731413\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 9.70820 0.516716 0.258358 0.966049i $$-0.416819\pi$$
0.258358 + 0.966049i $$0.416819\pi$$
$$354$$ 0 0
$$355$$ 1.88854 0.100233
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.70820 0.301267 0.150634 0.988590i $$-0.451869\pi$$
0.150634 + 0.988590i $$0.451869\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5.52786 0.289342
$$366$$ 0 0
$$367$$ 16.9443 0.884484 0.442242 0.896896i $$-0.354183\pi$$
0.442242 + 0.896896i $$0.354183\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.472136 −0.0245121
$$372$$ 0 0
$$373$$ −28.3607 −1.46846 −0.734230 0.678901i $$-0.762457\pi$$
−0.734230 + 0.678901i $$0.762457\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.47214 0.333332
$$378$$ 0 0
$$379$$ 32.1803 1.65299 0.826497 0.562942i $$-0.190331\pi$$
0.826497 + 0.562942i $$0.190331\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.05573 0.156140 0.0780702 0.996948i $$-0.475124\pi$$
0.0780702 + 0.996948i $$0.475124\pi$$
$$384$$ 0 0
$$385$$ −6.47214 −0.329851
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −23.4164 −1.18422
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 19.4164 0.976946
$$396$$ 0 0
$$397$$ −34.3607 −1.72451 −0.862257 0.506472i $$-0.830949\pi$$
−0.862257 + 0.506472i $$0.830949\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.05573 −0.252471 −0.126236 0.992000i $$-0.540290\pi$$
−0.126236 + 0.992000i $$0.540290\pi$$
$$402$$ 0 0
$$403$$ −0.360680 −0.0179667
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 46.8328 2.32142
$$408$$ 0 0
$$409$$ 17.2361 0.852269 0.426134 0.904660i $$-0.359875\pi$$
0.426134 + 0.904660i $$0.359875\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ −2.47214 −0.121352
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 17.4164 0.850847 0.425424 0.904994i $$-0.360125\pi$$
0.425424 + 0.904994i $$0.360125\pi$$
$$420$$ 0 0
$$421$$ 17.5279 0.854256 0.427128 0.904191i $$-0.359525\pi$$
0.427128 + 0.904191i $$0.359525\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 25.1246 1.21872
$$426$$ 0 0
$$427$$ −8.47214 −0.409995
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11.0557 −0.532536 −0.266268 0.963899i $$-0.585791\pi$$
−0.266268 + 0.963899i $$0.585791\pi$$
$$432$$ 0 0
$$433$$ 21.2361 1.02054 0.510270 0.860014i $$-0.329545\pi$$
0.510270 + 0.860014i $$0.329545\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.23607 0.154802
$$438$$ 0 0
$$439$$ 2.58359 0.123308 0.0616541 0.998098i $$-0.480362\pi$$
0.0616541 + 0.998098i $$0.480362\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.1803 0.863774 0.431887 0.901928i $$-0.357848\pi$$
0.431887 + 0.901928i $$0.357848\pi$$
$$444$$ 0 0
$$445$$ 13.5279 0.641282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 17.0557 0.804910 0.402455 0.915440i $$-0.368157\pi$$
0.402455 + 0.915440i $$0.368157\pi$$
$$450$$ 0 0
$$451$$ −10.4721 −0.493114
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0.944272 0.0442681
$$456$$ 0 0
$$457$$ 9.41641 0.440481 0.220240 0.975446i $$-0.429316\pi$$
0.220240 + 0.975446i $$0.429316\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.76393 −0.315028 −0.157514 0.987517i $$-0.550348\pi$$
−0.157514 + 0.987517i $$0.550348\pi$$
$$462$$ 0 0
$$463$$ 5.52786 0.256902 0.128451 0.991716i $$-0.459000\pi$$
0.128451 + 0.991716i $$0.459000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −27.8885 −1.29053 −0.645264 0.763960i $$-0.723253\pi$$
−0.645264 + 0.763960i $$0.723253\pi$$
$$468$$ 0 0
$$469$$ −0.763932 −0.0352751
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −20.9443 −0.963019
$$474$$ 0 0
$$475$$ −3.47214 −0.159313
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −6.83282 −0.311550
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.52786 0.0693767
$$486$$ 0 0
$$487$$ −23.7082 −1.07432 −0.537161 0.843480i $$-0.680503\pi$$
−0.537161 + 0.843480i $$0.680503\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0.875388 0.0395057 0.0197529 0.999805i $$-0.493712\pi$$
0.0197529 + 0.999805i $$0.493712\pi$$
$$492$$ 0 0
$$493$$ 61.3050 2.76104
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1.52786 −0.0685341
$$498$$ 0 0
$$499$$ −5.52786 −0.247461 −0.123731 0.992316i $$-0.539486\pi$$
−0.123731 + 0.992316i $$0.539486\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 39.7771 1.77357 0.886786 0.462180i $$-0.152932\pi$$
0.886786 + 0.462180i $$0.152932\pi$$
$$504$$ 0 0
$$505$$ 22.4721 0.999997
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −43.3050 −1.91946 −0.959729 0.280927i $$-0.909358\pi$$
−0.959729 + 0.280927i $$0.909358\pi$$
$$510$$ 0 0
$$511$$ −4.47214 −0.197836
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −9.30495 −0.410025
$$516$$ 0 0
$$517$$ 33.8885 1.49042
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −13.4164 −0.587784 −0.293892 0.955839i $$-0.594951\pi$$
−0.293892 + 0.955839i $$0.594951\pi$$
$$522$$ 0 0
$$523$$ 29.3050 1.28142 0.640708 0.767785i $$-0.278641\pi$$
0.640708 + 0.767785i $$0.278641\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3.41641 −0.148821
$$528$$ 0 0
$$529$$ −12.5279 −0.544690
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1.52786 0.0661791
$$534$$ 0 0
$$535$$ −14.1115 −0.610091
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 5.23607 0.225533
$$540$$ 0 0
$$541$$ 2.94427 0.126584 0.0632921 0.997995i $$-0.479840\pi$$
0.0632921 + 0.997995i $$0.479840\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 11.0557 0.473575
$$546$$ 0 0
$$547$$ 9.70820 0.415093 0.207546 0.978225i $$-0.433452\pi$$
0.207546 + 0.978225i $$0.433452\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.47214 −0.360925
$$552$$ 0 0
$$553$$ −15.7082 −0.667981
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 6.58359 0.278956 0.139478 0.990225i $$-0.455458\pi$$
0.139478 + 0.990225i $$0.455458\pi$$
$$558$$ 0 0
$$559$$ 3.05573 0.129244
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ 0 0
$$565$$ 8.58359 0.361114
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 16.8328 0.705668 0.352834 0.935686i $$-0.385218\pi$$
0.352834 + 0.935686i $$0.385218\pi$$
$$570$$ 0 0
$$571$$ 34.4721 1.44261 0.721307 0.692615i $$-0.243542\pi$$
0.721307 + 0.692615i $$0.243542\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −11.2361 −0.468576
$$576$$ 0 0
$$577$$ −37.4164 −1.55767 −0.778833 0.627232i $$-0.784188\pi$$
−0.778833 + 0.627232i $$0.784188\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.00000 0.0829740
$$582$$ 0 0
$$583$$ 2.47214 0.102385
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −14.0000 −0.577842 −0.288921 0.957353i $$-0.593296\pi$$
−0.288921 + 0.957353i $$0.593296\pi$$
$$588$$ 0 0
$$589$$ 0.472136 0.0194540
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 41.1246 1.68879 0.844393 0.535725i $$-0.179962\pi$$
0.844393 + 0.535725i $$0.179962\pi$$
$$594$$ 0 0
$$595$$ 8.94427 0.366679
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 22.4721 0.918187 0.459093 0.888388i $$-0.348174\pi$$
0.459093 + 0.888388i $$0.348174\pi$$
$$600$$ 0 0
$$601$$ −4.87539 −0.198871 −0.0994356 0.995044i $$-0.531704\pi$$
−0.0994356 + 0.995044i $$0.531704\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 20.2918 0.824979
$$606$$ 0 0
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.94427 −0.200024
$$612$$ 0 0
$$613$$ −23.8885 −0.964849 −0.482425 0.875938i $$-0.660244\pi$$
−0.482425 + 0.875938i $$0.660244\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.4721 −0.743660 −0.371830 0.928301i $$-0.621270\pi$$
−0.371830 + 0.928301i $$0.621270\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −10.9443 −0.438473
$$624$$ 0 0
$$625$$ 4.41641 0.176656
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −64.7214 −2.58061
$$630$$ 0 0
$$631$$ −12.3607 −0.492071 −0.246035 0.969261i $$-0.579128\pi$$
−0.246035 + 0.969261i $$0.579128\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −8.36068 −0.331783
$$636$$ 0 0
$$637$$ −0.763932 −0.0302681
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 17.8885 0.705455 0.352728 0.935726i $$-0.385254\pi$$
0.352728 + 0.935726i $$0.385254\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.94427 0.351636 0.175818 0.984423i $$-0.443743\pi$$
0.175818 + 0.984423i $$0.443743\pi$$
$$648$$ 0 0
$$649$$ 41.8885 1.64427
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −23.8885 −0.934831 −0.467415 0.884038i $$-0.654815\pi$$
−0.467415 + 0.884038i $$0.654815\pi$$
$$654$$ 0 0
$$655$$ 0.583592 0.0228028
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.41641 −0.133084 −0.0665422 0.997784i $$-0.521197\pi$$
−0.0665422 + 0.997784i $$0.521197\pi$$
$$660$$ 0 0
$$661$$ 9.70820 0.377605 0.188803 0.982015i $$-0.439539\pi$$
0.188803 + 0.982015i $$0.439539\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.23607 −0.0479327
$$666$$ 0 0
$$667$$ −27.4164 −1.06157
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 44.3607 1.71253
$$672$$ 0 0
$$673$$ 2.58359 0.0995902 0.0497951 0.998759i $$-0.484143\pi$$
0.0497951 + 0.998759i $$0.484143\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −6.58359 −0.253028 −0.126514 0.991965i $$-0.540379\pi$$
−0.126514 + 0.991965i $$0.540379\pi$$
$$678$$ 0 0
$$679$$ −1.23607 −0.0474359
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 5.88854 0.225319 0.112659 0.993634i $$-0.464063\pi$$
0.112659 + 0.993634i $$0.464063\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −0.360680 −0.0137408
$$690$$ 0 0
$$691$$ −8.94427 −0.340256 −0.170128 0.985422i $$-0.554418\pi$$
−0.170128 + 0.985422i $$0.554418\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −11.0557 −0.419368
$$696$$ 0 0
$$697$$ 14.4721 0.548171
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20.8328 0.786845 0.393422 0.919358i $$-0.371291\pi$$
0.393422 + 0.919358i $$0.371291\pi$$
$$702$$ 0 0
$$703$$ 8.94427 0.337340
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −18.1803 −0.683742
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.52786 0.0572190
$$714$$ 0 0
$$715$$ −4.94427 −0.184905
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 7.52786 0.280352
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 29.4164 1.09250
$$726$$ 0 0
$$727$$ −17.8885 −0.663449 −0.331725 0.943376i $$-0.607631\pi$$
−0.331725 + 0.943376i $$0.607631\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 28.9443 1.07054
$$732$$ 0 0
$$733$$ 18.9443 0.699723 0.349861 0.936802i $$-0.386229\pi$$
0.349861 + 0.936802i $$0.386229\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000 0.147342
$$738$$ 0 0
$$739$$ −52.3607 −1.92612 −0.963059 0.269289i $$-0.913211\pi$$
−0.963059 + 0.269289i $$0.913211\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 45.3050 1.66208 0.831039 0.556215i $$-0.187747\pi$$
0.831039 + 0.556215i $$0.187747\pi$$
$$744$$ 0 0
$$745$$ 20.3607 0.745958
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 11.4164 0.417146
$$750$$ 0 0
$$751$$ 3.12461 0.114019 0.0570094 0.998374i $$-0.481844\pi$$
0.0570094 + 0.998374i $$0.481844\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −0.360680 −0.0131265
$$756$$ 0 0
$$757$$ 19.8885 0.722861 0.361431 0.932399i $$-0.382288\pi$$
0.361431 + 0.932399i $$0.382288\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0.403252 0.0146179 0.00730894 0.999973i $$-0.497673\pi$$
0.00730894 + 0.999973i $$0.497673\pi$$
$$762$$ 0 0
$$763$$ −8.94427 −0.323804
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.11146 −0.220672
$$768$$ 0 0
$$769$$ −42.9443 −1.54861 −0.774305 0.632813i $$-0.781900\pi$$
−0.774305 + 0.632813i $$0.781900\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −27.3050 −0.982091 −0.491045 0.871134i $$-0.663385\pi$$
−0.491045 + 0.871134i $$0.663385\pi$$
$$774$$ 0 0
$$775$$ −1.63932 −0.0588861
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −20.3607 −0.726704
$$786$$ 0 0
$$787$$ −7.41641 −0.264366 −0.132183 0.991225i $$-0.542199\pi$$
−0.132183 + 0.991225i $$0.542199\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.94427 −0.246910
$$792$$ 0 0
$$793$$ −6.47214 −0.229832
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −35.3050 −1.25057 −0.625283 0.780398i $$-0.715016\pi$$
−0.625283 + 0.780398i $$0.715016\pi$$
$$798$$ 0 0
$$799$$ −46.8328 −1.65683
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 23.4164 0.826347
$$804$$ 0 0
$$805$$ −4.00000 −0.140981
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ −54.2492 −1.90495 −0.952474 0.304620i $$-0.901471\pi$$
−0.952474 + 0.304620i $$0.901471\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 14.8328 0.519571
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.3607 0.780393 0.390197 0.920732i $$-0.372407\pi$$
0.390197 + 0.920732i $$0.372407\pi$$
$$822$$ 0 0
$$823$$ −47.4164 −1.65283 −0.826416 0.563060i $$-0.809624\pi$$
−0.826416 + 0.563060i $$0.809624\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 48.9443 1.70196 0.850980 0.525199i $$-0.176009\pi$$
0.850980 + 0.525199i $$0.176009\pi$$
$$828$$ 0 0
$$829$$ 26.0689 0.905410 0.452705 0.891660i $$-0.350459\pi$$
0.452705 + 0.891660i $$0.350459\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −7.23607 −0.250715
$$834$$ 0 0
$$835$$ 20.9443 0.724806
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 34.8328 1.20256 0.601281 0.799038i $$-0.294657\pi$$
0.601281 + 0.799038i $$0.294657\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −15.3475 −0.527971
$$846$$ 0 0
$$847$$ −16.4164 −0.564074
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 28.9443 0.992197
$$852$$ 0 0
$$853$$ −7.88854 −0.270099 −0.135049 0.990839i $$-0.543119\pi$$
−0.135049 + 0.990839i $$0.543119\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.83282 0.165086 0.0825429 0.996588i $$-0.473696\pi$$
0.0825429 + 0.996588i $$0.473696\pi$$
$$858$$ 0 0
$$859$$ 51.7771 1.76661 0.883306 0.468797i $$-0.155313\pi$$
0.883306 + 0.468797i $$0.155313\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32.3607 1.10157 0.550785 0.834647i $$-0.314328\pi$$
0.550785 + 0.834647i $$0.314328\pi$$
$$864$$ 0 0
$$865$$ −24.5836 −0.835867
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 82.2492 2.79011
$$870$$ 0 0
$$871$$ −0.583592 −0.0197743
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 10.4721 0.354023
$$876$$ 0 0
$$877$$ −40.9443 −1.38259 −0.691295 0.722573i $$-0.742959\pi$$
−0.691295 + 0.722573i $$0.742959\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −35.9574 −1.21144 −0.605718 0.795679i $$-0.707114\pi$$
−0.605718 + 0.795679i $$0.707114\pi$$
$$882$$ 0 0
$$883$$ −10.4721 −0.352415 −0.176208 0.984353i $$-0.556383\pi$$
−0.176208 + 0.984353i $$0.556383\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −25.8885 −0.869252 −0.434626 0.900611i $$-0.643119\pi$$
−0.434626 + 0.900611i $$0.643119\pi$$
$$888$$ 0 0
$$889$$ 6.76393 0.226855
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.47214 0.216582
$$894$$ 0 0
$$895$$ 4.94427 0.165269
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −3.41641 −0.113817
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 15.7771 0.524448
$$906$$ 0 0
$$907$$ −37.7082 −1.25208 −0.626040 0.779791i $$-0.715325\pi$$
−0.626040 + 0.779791i $$0.715325\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −19.4164 −0.643294 −0.321647 0.946860i $$-0.604236\pi$$
−0.321647 + 0.946860i $$0.604236\pi$$
$$912$$ 0 0
$$913$$ −10.4721 −0.346577
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.472136 −0.0155913
$$918$$ 0 0
$$919$$ −5.52786 −0.182347 −0.0911737 0.995835i $$-0.529062\pi$$
−0.0911737 + 0.995835i $$0.529062\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −1.16718 −0.0384183
$$924$$ 0 0
$$925$$ −31.0557 −1.02111
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −9.34752 −0.306682 −0.153341 0.988173i $$-0.549003\pi$$
−0.153341 + 0.988173i $$0.549003\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −46.8328 −1.53160
$$936$$ 0 0
$$937$$ −0.832816 −0.0272069 −0.0136035 0.999907i $$-0.504330\pi$$
−0.0136035 + 0.999907i $$0.504330\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −27.3050 −0.890116 −0.445058 0.895502i $$-0.646817\pi$$
−0.445058 + 0.895502i $$0.646817\pi$$
$$942$$ 0 0
$$943$$ −6.47214 −0.210762
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −50.1803 −1.63064 −0.815321 0.579009i $$-0.803440\pi$$
−0.815321 + 0.579009i $$0.803440\pi$$
$$948$$ 0 0
$$949$$ −3.41641 −0.110901
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ −13.1672 −0.426080
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6.47214 −0.208996
$$960$$ 0 0
$$961$$ −30.7771 −0.992809
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 23.4164 0.753801
$$966$$ 0 0
$$967$$ −54.8328 −1.76330 −0.881652 0.471900i $$-0.843568\pi$$
−0.881652 + 0.471900i $$0.843568\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.2492 0.714012 0.357006 0.934102i $$-0.383798\pi$$
0.357006 + 0.934102i $$0.383798\pi$$
$$972$$ 0 0
$$973$$ 8.94427 0.286740
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −8.83282 −0.282587 −0.141293 0.989968i $$-0.545126\pi$$
−0.141293 + 0.989968i $$0.545126\pi$$
$$978$$ 0 0
$$979$$ 57.3050 1.83147
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −31.7771 −1.01353 −0.506766 0.862084i $$-0.669159\pi$$
−0.506766 + 0.862084i $$0.669159\pi$$
$$984$$ 0 0
$$985$$ 21.5279 0.685935
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −12.9443 −0.411604
$$990$$ 0 0
$$991$$ 22.5410 0.716039 0.358020 0.933714i $$-0.383452\pi$$
0.358020 + 0.933714i $$0.383452\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −17.8885 −0.567105
$$996$$ 0 0
$$997$$ 4.47214 0.141634 0.0708170 0.997489i $$-0.477439\pi$$
0.0708170 + 0.997489i $$0.477439\pi$$
$$998$$ 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 9576.2.a.bg.1.2 2
3.2 odd 2 9576.2.a.bp.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bg.1.2 2 1.1 even 1 trivial
9576.2.a.bp.1.1 yes 2 3.2 odd 2