# Properties

 Label 9360.2.a.cy.1.1 Level $9360$ Weight $2$ Character 9360.1 Self dual yes Analytic conductor $74.740$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.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: $$74.7399762919$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.940.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 7x - 4$$ x^3 - 7*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1560) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.602705$$ of defining polynomial Character $$\chi$$ $$=$$ 9360.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -3.43134 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -3.43134 q^{7} -2.63675 q^{11} -1.00000 q^{13} -7.84216 q^{17} -0.794590 q^{19} -3.43134 q^{23} +1.00000 q^{25} +4.06808 q^{29} -9.27349 q^{31} +3.43134 q^{35} -0.636747 q^{37} +4.63675 q^{41} -2.41082 q^{43} +5.27349 q^{47} +4.77407 q^{49} -11.4994 q^{53} +2.63675 q^{55} -12.1362 q^{59} -11.4994 q^{61} +1.00000 q^{65} +6.41082 q^{67} -10.6367 q^{71} +16.0681 q^{73} +9.04757 q^{77} -10.6367 q^{79} -16.1362 q^{83} +7.84216 q^{85} -8.63675 q^{89} +3.43134 q^{91} +0.794590 q^{95} +12.2940 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - q^7 $$3 q - 3 q^{5} - q^{7} + 5 q^{11} - 3 q^{13} - 7 q^{17} - 6 q^{19} - q^{23} + 3 q^{25} - 10 q^{29} - 2 q^{31} + q^{35} + 11 q^{37} + q^{41} - 10 q^{47} + 20 q^{49} - 3 q^{53} - 5 q^{55} + 8 q^{59} - 3 q^{61} + 3 q^{65} + 12 q^{67} - 19 q^{71} + 26 q^{73} + 7 q^{77} - 19 q^{79} - 4 q^{83} + 7 q^{85} - 13 q^{89} + q^{91} + 6 q^{95} + 9 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - q^7 + 5 * q^11 - 3 * q^13 - 7 * q^17 - 6 * q^19 - q^23 + 3 * q^25 - 10 * q^29 - 2 * q^31 + q^35 + 11 * q^37 + q^41 - 10 * q^47 + 20 * q^49 - 3 * q^53 - 5 * q^55 + 8 * q^59 - 3 * q^61 + 3 * q^65 + 12 * q^67 - 19 * q^71 + 26 * q^73 + 7 * q^77 - 19 * q^79 - 4 * q^83 + 7 * q^85 - 13 * q^89 + q^91 + 6 * q^95 + 9 * 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.43134 −1.29692 −0.648462 0.761247i $$-0.724587\pi$$
−0.648462 + 0.761247i $$0.724587\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.63675 −0.795009 −0.397505 0.917600i $$-0.630124\pi$$
−0.397505 + 0.917600i $$0.630124\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −7.84216 −1.90200 −0.951001 0.309187i $$-0.899943\pi$$
−0.951001 + 0.309187i $$0.899943\pi$$
$$18$$ 0 0
$$19$$ −0.794590 −0.182291 −0.0911457 0.995838i $$-0.529053\pi$$
−0.0911457 + 0.995838i $$0.529053\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.43134 −0.715483 −0.357742 0.933821i $$-0.616453\pi$$
−0.357742 + 0.933821i $$0.616453\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.06808 0.755424 0.377712 0.925923i $$-0.376711\pi$$
0.377712 + 0.925923i $$0.376711\pi$$
$$30$$ 0 0
$$31$$ −9.27349 −1.66557 −0.832784 0.553598i $$-0.813255\pi$$
−0.832784 + 0.553598i $$0.813255\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.43134 0.580002
$$36$$ 0 0
$$37$$ −0.636747 −0.104681 −0.0523403 0.998629i $$-0.516668\pi$$
−0.0523403 + 0.998629i $$0.516668\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.63675 0.724138 0.362069 0.932151i $$-0.382070\pi$$
0.362069 + 0.932151i $$0.382070\pi$$
$$42$$ 0 0
$$43$$ −2.41082 −0.367647 −0.183823 0.982959i $$-0.558847\pi$$
−0.183823 + 0.982959i $$0.558847\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.27349 0.769218 0.384609 0.923080i $$-0.374336\pi$$
0.384609 + 0.923080i $$0.374336\pi$$
$$48$$ 0 0
$$49$$ 4.77407 0.682010
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −11.4994 −1.57957 −0.789783 0.613386i $$-0.789807\pi$$
−0.789783 + 0.613386i $$0.789807\pi$$
$$54$$ 0 0
$$55$$ 2.63675 0.355539
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −12.1362 −1.57999 −0.789997 0.613110i $$-0.789918\pi$$
−0.789997 + 0.613110i $$0.789918\pi$$
$$60$$ 0 0
$$61$$ −11.4994 −1.47235 −0.736175 0.676791i $$-0.763370\pi$$
−0.736175 + 0.676791i $$0.763370\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 6.41082 0.783206 0.391603 0.920134i $$-0.371921\pi$$
0.391603 + 0.920134i $$0.371921\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.6367 −1.26235 −0.631175 0.775641i $$-0.717427\pi$$
−0.631175 + 0.775641i $$0.717427\pi$$
$$72$$ 0 0
$$73$$ 16.0681 1.88063 0.940313 0.340310i $$-0.110532\pi$$
0.940313 + 0.340310i $$0.110532\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 9.04757 1.03107
$$78$$ 0 0
$$79$$ −10.6367 −1.19673 −0.598364 0.801225i $$-0.704182\pi$$
−0.598364 + 0.801225i $$0.704182\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −16.1362 −1.77117 −0.885587 0.464473i $$-0.846244\pi$$
−0.885587 + 0.464473i $$0.846244\pi$$
$$84$$ 0 0
$$85$$ 7.84216 0.850601
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −8.63675 −0.915493 −0.457747 0.889083i $$-0.651343\pi$$
−0.457747 + 0.889083i $$0.651343\pi$$
$$90$$ 0 0
$$91$$ 3.43134 0.359702
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0.794590 0.0815232
$$96$$ 0 0
$$97$$ 12.2940 1.24827 0.624134 0.781317i $$-0.285452\pi$$
0.624134 + 0.781317i $$0.285452\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −5.34158 −0.531507 −0.265753 0.964041i $$-0.585621\pi$$
−0.265753 + 0.964041i $$0.585621\pi$$
$$102$$ 0 0
$$103$$ 10.4108 1.02581 0.512904 0.858446i $$-0.328570\pi$$
0.512904 + 0.858446i $$0.328570\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.04757 0.487967 0.243983 0.969779i $$-0.421546\pi$$
0.243983 + 0.969779i $$0.421546\pi$$
$$108$$ 0 0
$$109$$ 6.79459 0.650804 0.325402 0.945576i $$-0.394500\pi$$
0.325402 + 0.945576i $$0.394500\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 8.06808 0.758981 0.379491 0.925196i $$-0.376099\pi$$
0.379491 + 0.925196i $$0.376099\pi$$
$$114$$ 0 0
$$115$$ 3.43134 0.319974
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 26.9091 2.46675
$$120$$ 0 0
$$121$$ −4.04757 −0.367961
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 15.6843 1.39176 0.695879 0.718159i $$-0.255015\pi$$
0.695879 + 0.718159i $$0.255015\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.9308 1.12977 0.564883 0.825171i $$-0.308921\pi$$
0.564883 + 0.825171i $$0.308921\pi$$
$$132$$ 0 0
$$133$$ 2.72651 0.236418
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ −19.9102 −1.68876 −0.844382 0.535741i $$-0.820032\pi$$
−0.844382 + 0.535741i $$0.820032\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 2.63675 0.220496
$$144$$ 0 0
$$145$$ −4.06808 −0.337836
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −8.63675 −0.707550 −0.353775 0.935331i $$-0.615102\pi$$
−0.353775 + 0.935331i $$0.615102\pi$$
$$150$$ 0 0
$$151$$ 17.7253 1.44247 0.721234 0.692691i $$-0.243575\pi$$
0.721234 + 0.692691i $$0.243575\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 9.27349 0.744865
$$156$$ 0 0
$$157$$ 20.5470 1.63983 0.819914 0.572487i $$-0.194021\pi$$
0.819914 + 0.572487i $$0.194021\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 11.7741 0.927927
$$162$$ 0 0
$$163$$ 4.22593 0.331000 0.165500 0.986210i $$-0.447076\pi$$
0.165500 + 0.986210i $$0.447076\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.4108 −1.11514 −0.557571 0.830129i $$-0.688267\pi$$
−0.557571 + 0.830129i $$0.688267\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ −3.43134 −0.259385
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −20.9308 −1.56444 −0.782219 0.623003i $$-0.785912\pi$$
−0.782219 + 0.623003i $$0.785912\pi$$
$$180$$ 0 0
$$181$$ −5.08860 −0.378233 −0.189116 0.981955i $$-0.560562\pi$$
−0.189116 + 0.981955i $$0.560562\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0.636747 0.0468146
$$186$$ 0 0
$$187$$ 20.6778 1.51211
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.7253 −0.993131 −0.496566 0.867999i $$-0.665406\pi$$
−0.496566 + 0.867999i $$0.665406\pi$$
$$192$$ 0 0
$$193$$ −5.11565 −0.368233 −0.184116 0.982904i $$-0.558942\pi$$
−0.184116 + 0.982904i $$0.558942\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 20.9988 1.49611 0.748053 0.663639i $$-0.230989\pi$$
0.748053 + 0.663639i $$0.230989\pi$$
$$198$$ 0 0
$$199$$ −9.58918 −0.679759 −0.339879 0.940469i $$-0.610386\pi$$
−0.339879 + 0.940469i $$0.610386\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −13.9590 −0.979727
$$204$$ 0 0
$$205$$ −4.63675 −0.323844
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2.09513 0.144923
$$210$$ 0 0
$$211$$ 8.82164 0.607307 0.303653 0.952783i $$-0.401794\pi$$
0.303653 + 0.952783i $$0.401794\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 2.41082 0.164417
$$216$$ 0 0
$$217$$ 31.8205 2.16011
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.84216 0.527521
$$222$$ 0 0
$$223$$ 9.93192 0.665090 0.332545 0.943087i $$-0.392093\pi$$
0.332545 + 0.943087i $$0.392093\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 9.27349 0.615503 0.307752 0.951467i $$-0.400423\pi$$
0.307752 + 0.951467i $$0.400423\pi$$
$$228$$ 0 0
$$229$$ 12.0681 0.797481 0.398741 0.917064i $$-0.369447\pi$$
0.398741 + 0.917064i $$0.369447\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.1157 −0.859235 −0.429617 0.903011i $$-0.641352\pi$$
−0.429617 + 0.903011i $$0.641352\pi$$
$$234$$ 0 0
$$235$$ −5.27349 −0.344005
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −22.7729 −1.47306 −0.736529 0.676406i $$-0.763536\pi$$
−0.736529 + 0.676406i $$0.763536\pi$$
$$240$$ 0 0
$$241$$ 6.82164 0.439420 0.219710 0.975565i $$-0.429489\pi$$
0.219710 + 0.975565i $$0.429489\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.77407 −0.305004
$$246$$ 0 0
$$247$$ 0.794590 0.0505586
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 9.24644 0.583630 0.291815 0.956475i $$-0.405741\pi$$
0.291815 + 0.956475i $$0.405741\pi$$
$$252$$ 0 0
$$253$$ 9.04757 0.568816
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 11.2464 0.701534 0.350767 0.936463i $$-0.385921\pi$$
0.350767 + 0.936463i $$0.385921\pi$$
$$258$$ 0 0
$$259$$ 2.18489 0.135763
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −22.0681 −1.36078 −0.680388 0.732852i $$-0.738189\pi$$
−0.680388 + 0.732852i $$0.738189\pi$$
$$264$$ 0 0
$$265$$ 11.4994 0.706404
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 13.2054 0.805148 0.402574 0.915387i $$-0.368116\pi$$
0.402574 + 0.915387i $$0.368116\pi$$
$$270$$ 0 0
$$271$$ −14.0951 −0.856218 −0.428109 0.903727i $$-0.640820\pi$$
−0.428109 + 0.903727i $$0.640820\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.63675 −0.159002
$$276$$ 0 0
$$277$$ −30.2723 −1.81889 −0.909444 0.415826i $$-0.863493\pi$$
−0.909444 + 0.415826i $$0.863493\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.13617 −0.127433 −0.0637165 0.997968i $$-0.520295\pi$$
−0.0637165 + 0.997968i $$0.520295\pi$$
$$282$$ 0 0
$$283$$ −0.136167 −0.00809431 −0.00404715 0.999992i $$-0.501288\pi$$
−0.00404715 + 0.999992i $$0.501288\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −15.9102 −0.939152
$$288$$ 0 0
$$289$$ 44.4994 2.61761
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 15.2735 0.892287 0.446144 0.894961i $$-0.352797\pi$$
0.446144 + 0.894961i $$0.352797\pi$$
$$294$$ 0 0
$$295$$ 12.1362 0.706595
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.43134 0.198439
$$300$$ 0 0
$$301$$ 8.27233 0.476809
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 11.4994 0.658455
$$306$$ 0 0
$$307$$ −7.91024 −0.451461 −0.225731 0.974190i $$-0.572477\pi$$
−0.225731 + 0.974190i $$0.572477\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.54699 0.144426 0.0722132 0.997389i $$-0.476994\pi$$
0.0722132 + 0.997389i $$0.476994\pi$$
$$312$$ 0 0
$$313$$ 19.5892 1.10725 0.553623 0.832767i $$-0.313245\pi$$
0.553623 + 0.832767i $$0.313245\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 23.2735 1.30717 0.653585 0.756853i $$-0.273264\pi$$
0.653585 + 0.756853i $$0.273264\pi$$
$$318$$ 0 0
$$319$$ −10.7265 −0.600569
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.23130 0.346719
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −18.0951 −0.997617
$$330$$ 0 0
$$331$$ 20.4789 1.12562 0.562811 0.826586i $$-0.309720\pi$$
0.562811 + 0.826586i $$0.309720\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −6.41082 −0.350260
$$336$$ 0 0
$$337$$ −25.6843 −1.39911 −0.699557 0.714577i $$-0.746619\pi$$
−0.699557 + 0.714577i $$0.746619\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.4519 1.32414
$$342$$ 0 0
$$343$$ 7.63791 0.412408
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 9.81511 0.526903 0.263451 0.964673i $$-0.415139\pi$$
0.263451 + 0.964673i $$0.415139\pi$$
$$348$$ 0 0
$$349$$ 12.5199 0.670177 0.335088 0.942187i $$-0.391234\pi$$
0.335088 + 0.942187i $$0.391234\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 10.6367 0.564540
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −28.1362 −1.48497 −0.742485 0.669863i $$-0.766353\pi$$
−0.742485 + 0.669863i $$0.766353\pi$$
$$360$$ 0 0
$$361$$ −18.3686 −0.966770
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −16.0681 −0.841042
$$366$$ 0 0
$$367$$ −2.41082 −0.125844 −0.0629219 0.998018i $$-0.520042\pi$$
−0.0629219 + 0.998018i $$0.520042\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 39.4584 2.04858
$$372$$ 0 0
$$373$$ −5.54815 −0.287272 −0.143636 0.989631i $$-0.545879\pi$$
−0.143636 + 0.989631i $$0.545879\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.06808 −0.209517
$$378$$ 0 0
$$379$$ 14.5199 0.745839 0.372920 0.927864i $$-0.378357\pi$$
0.372920 + 0.927864i $$0.378357\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −22.8627 −1.16823 −0.584114 0.811672i $$-0.698558\pi$$
−0.584114 + 0.811672i $$0.698558\pi$$
$$384$$ 0 0
$$385$$ −9.04757 −0.461107
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −5.34158 −0.270829 −0.135414 0.990789i $$-0.543237\pi$$
−0.135414 + 0.990789i $$0.543237\pi$$
$$390$$ 0 0
$$391$$ 26.9091 1.36085
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 10.6367 0.535193
$$396$$ 0 0
$$397$$ −21.4584 −1.07697 −0.538483 0.842637i $$-0.681002\pi$$
−0.538483 + 0.842637i $$0.681002\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 17.3145 0.864646 0.432323 0.901719i $$-0.357694\pi$$
0.432323 + 0.901719i $$0.357694\pi$$
$$402$$ 0 0
$$403$$ 9.27349 0.461946
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.67894 0.0832220
$$408$$ 0 0
$$409$$ 16.4108 0.811463 0.405731 0.913992i $$-0.367017\pi$$
0.405731 + 0.913992i $$0.367017\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 41.6433 2.04913
$$414$$ 0 0
$$415$$ 16.1362 0.792093
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 25.7524 1.25809 0.629043 0.777370i $$-0.283447\pi$$
0.629043 + 0.777370i $$0.283447\pi$$
$$420$$ 0 0
$$421$$ 13.4777 0.656865 0.328433 0.944527i $$-0.393480\pi$$
0.328433 + 0.944527i $$0.393480\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −7.84216 −0.380400
$$426$$ 0 0
$$427$$ 39.4584 1.90953
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.2723 −1.16916 −0.584579 0.811337i $$-0.698740\pi$$
−0.584579 + 0.811337i $$0.698740\pi$$
$$432$$ 0 0
$$433$$ 3.13733 0.150770 0.0753851 0.997154i $$-0.475981\pi$$
0.0753851 + 0.997154i $$0.475981\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.72651 0.130426
$$438$$ 0 0
$$439$$ −8.59571 −0.410251 −0.205125 0.978736i $$-0.565760\pi$$
−0.205125 + 0.978736i $$0.565760\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0.911399 0.0433019 0.0216509 0.999766i $$-0.493108\pi$$
0.0216509 + 0.999766i $$0.493108\pi$$
$$444$$ 0 0
$$445$$ 8.63675 0.409421
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −38.8139 −1.83174 −0.915872 0.401471i $$-0.868499\pi$$
−0.915872 + 0.401471i $$0.868499\pi$$
$$450$$ 0 0
$$451$$ −12.2259 −0.575696
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.43134 −0.160864
$$456$$ 0 0
$$457$$ −6.25298 −0.292502 −0.146251 0.989248i $$-0.546721\pi$$
−0.146251 + 0.989248i $$0.546721\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 27.9513 1.30182 0.650910 0.759155i $$-0.274387\pi$$
0.650910 + 0.759155i $$0.274387\pi$$
$$462$$ 0 0
$$463$$ 11.8832 0.552259 0.276129 0.961120i $$-0.410948\pi$$
0.276129 + 0.961120i $$0.410948\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −28.8680 −1.33585 −0.667927 0.744227i $$-0.732818\pi$$
−0.667927 + 0.744227i $$0.732818\pi$$
$$468$$ 0 0
$$469$$ −21.9977 −1.01576
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6.35672 0.292282
$$474$$ 0 0
$$475$$ −0.794590 −0.0364583
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −12.0464 −0.550414 −0.275207 0.961385i $$-0.588746\pi$$
−0.275207 + 0.961385i $$0.588746\pi$$
$$480$$ 0 0
$$481$$ 0.636747 0.0290332
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12.2940 −0.558242
$$486$$ 0 0
$$487$$ −11.2518 −0.509869 −0.254934 0.966958i $$-0.582054\pi$$
−0.254934 + 0.966958i $$0.582054\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 31.0259 1.40018 0.700089 0.714055i $$-0.253143\pi$$
0.700089 + 0.714055i $$0.253143\pi$$
$$492$$ 0 0
$$493$$ −31.9025 −1.43682
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 36.4983 1.63717
$$498$$ 0 0
$$499$$ 5.61623 0.251417 0.125708 0.992067i $$-0.459880\pi$$
0.125708 + 0.992067i $$0.459880\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −3.52110 −0.156998 −0.0784990 0.996914i $$-0.525013\pi$$
−0.0784990 + 0.996914i $$0.525013\pi$$
$$504$$ 0 0
$$505$$ 5.34158 0.237697
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.7741 0.610525 0.305263 0.952268i $$-0.401256\pi$$
0.305263 + 0.952268i $$0.401256\pi$$
$$510$$ 0 0
$$511$$ −55.1350 −2.43903
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −10.4108 −0.458756
$$516$$ 0 0
$$517$$ −13.9049 −0.611535
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.3145 −1.10905 −0.554525 0.832167i $$-0.687100\pi$$
−0.554525 + 0.832167i $$0.687100\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 72.7242 3.16792
$$528$$ 0 0
$$529$$ −11.2259 −0.488084
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4.63675 −0.200840
$$534$$ 0 0
$$535$$ −5.04757 −0.218225
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12.5880 −0.542204
$$540$$ 0 0
$$541$$ −26.6151 −1.14427 −0.572136 0.820159i $$-0.693885\pi$$
−0.572136 + 0.820159i $$0.693885\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6.79459 −0.291048
$$546$$ 0 0
$$547$$ 17.7253 0.757881 0.378941 0.925421i $$-0.376288\pi$$
0.378941 + 0.925421i $$0.376288\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.23246 −0.137707
$$552$$ 0 0
$$553$$ 36.4983 1.55206
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −30.2723 −1.28268 −0.641340 0.767257i $$-0.721621\pi$$
−0.641340 + 0.767257i $$0.721621\pi$$
$$558$$ 0 0
$$559$$ 2.41082 0.101967
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.6367 0.954025 0.477013 0.878896i $$-0.341720\pi$$
0.477013 + 0.878896i $$0.341720\pi$$
$$564$$ 0 0
$$565$$ −8.06808 −0.339427
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −20.0410 −0.840164 −0.420082 0.907486i $$-0.637999\pi$$
−0.420082 + 0.907486i $$0.637999\pi$$
$$570$$ 0 0
$$571$$ −20.3621 −0.852127 −0.426064 0.904693i $$-0.640100\pi$$
−0.426064 + 0.904693i $$0.640100\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3.43134 −0.143097
$$576$$ 0 0
$$577$$ 2.70483 0.112604 0.0563018 0.998414i $$-0.482069\pi$$
0.0563018 + 0.998414i $$0.482069\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 55.3686 2.29708
$$582$$ 0 0
$$583$$ 30.3211 1.25577
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 33.2735 1.37334 0.686672 0.726967i $$-0.259071\pi$$
0.686672 + 0.726967i $$0.259071\pi$$
$$588$$ 0 0
$$589$$ 7.36863 0.303619
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 28.2313 1.15932 0.579660 0.814858i $$-0.303185\pi$$
0.579660 + 0.814858i $$0.303185\pi$$
$$594$$ 0 0
$$595$$ −26.9091 −1.10316
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 29.0940 1.18875 0.594374 0.804189i $$-0.297400\pi$$
0.594374 + 0.804189i $$0.297400\pi$$
$$600$$ 0 0
$$601$$ 46.0464 1.87827 0.939136 0.343545i $$-0.111628\pi$$
0.939136 + 0.343545i $$0.111628\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 4.04757 0.164557
$$606$$ 0 0
$$607$$ −24.1362 −0.979657 −0.489828 0.871819i $$-0.662941\pi$$
−0.489828 + 0.871819i $$0.662941\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.27349 −0.213343
$$612$$ 0 0
$$613$$ 3.73304 0.150776 0.0753880 0.997154i $$-0.475980\pi$$
0.0753880 + 0.997154i $$0.475980\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.3686 0.538201 0.269100 0.963112i $$-0.413274\pi$$
0.269100 + 0.963112i $$0.413274\pi$$
$$618$$ 0 0
$$619$$ 21.8886 0.879776 0.439888 0.898053i $$-0.355018\pi$$
0.439888 + 0.898053i $$0.355018\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 29.6356 1.18732
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4.99347 0.199103
$$630$$ 0 0
$$631$$ −21.5892 −0.859452 −0.429726 0.902959i $$-0.641390\pi$$
−0.429726 + 0.902959i $$0.641390\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −15.6843 −0.622413
$$636$$ 0 0
$$637$$ −4.77407 −0.189156
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 5.54815 0.219139 0.109569 0.993979i $$-0.465053\pi$$
0.109569 + 0.993979i $$0.465053\pi$$
$$642$$ 0 0
$$643$$ 34.6367 1.36594 0.682970 0.730446i $$-0.260688\pi$$
0.682970 + 0.730446i $$0.260688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 29.9783 1.17857 0.589285 0.807925i $$-0.299410\pi$$
0.589285 + 0.807925i $$0.299410\pi$$
$$648$$ 0 0
$$649$$ 32.0000 1.25611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −32.6832 −1.27899 −0.639495 0.768795i $$-0.720857\pi$$
−0.639495 + 0.768795i $$0.720857\pi$$
$$654$$ 0 0
$$655$$ −12.9308 −0.505247
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 23.9296 0.932165 0.466082 0.884741i $$-0.345665\pi$$
0.466082 + 0.884741i $$0.345665\pi$$
$$660$$ 0 0
$$661$$ −34.1632 −1.32880 −0.664398 0.747379i $$-0.731312\pi$$
−0.664398 + 0.747379i $$0.731312\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.72651 −0.105729
$$666$$ 0 0
$$667$$ −13.9590 −0.540493
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 30.3211 1.17053
$$672$$ 0 0
$$673$$ −6.90371 −0.266118 −0.133059 0.991108i $$-0.542480\pi$$
−0.133059 + 0.991108i $$0.542480\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −49.2789 −1.89394 −0.946970 0.321321i $$-0.895873\pi$$
−0.946970 + 0.321321i $$0.895873\pi$$
$$678$$ 0 0
$$679$$ −42.1849 −1.61891
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 3.04219 0.116406 0.0582031 0.998305i $$-0.481463\pi$$
0.0582031 + 0.998305i $$0.481463\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 11.4994 0.438093
$$690$$ 0 0
$$691$$ −13.8886 −0.528346 −0.264173 0.964475i $$-0.585099\pi$$
−0.264173 + 0.964475i $$0.585099\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 19.9102 0.755238
$$696$$ 0 0
$$697$$ −36.3621 −1.37731
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5.20541 0.196606 0.0983028 0.995157i $$-0.468659\pi$$
0.0983028 + 0.995157i $$0.468659\pi$$
$$702$$ 0 0
$$703$$ 0.505953 0.0190824
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.3288 0.689324
$$708$$ 0 0
$$709$$ −19.7524 −0.741817 −0.370908 0.928669i $$-0.620954\pi$$
−0.370908 + 0.928669i $$0.620954\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 31.8205 1.19169
$$714$$ 0 0
$$715$$ −2.63675 −0.0986087
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −4.13617 −0.154253 −0.0771265 0.997021i $$-0.524575\pi$$
−0.0771265 + 0.997021i $$0.524575\pi$$
$$720$$ 0 0
$$721$$ −35.7230 −1.33040
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4.06808 0.151085
$$726$$ 0 0
$$727$$ −6.09513 −0.226056 −0.113028 0.993592i $$-0.536055\pi$$
−0.113028 + 0.993592i $$0.536055\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 18.9060 0.699265
$$732$$ 0 0
$$733$$ −40.0054 −1.47763 −0.738816 0.673907i $$-0.764615\pi$$
−0.738816 + 0.673907i $$0.764615\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.9037 −0.622656
$$738$$ 0 0
$$739$$ −12.0270 −0.442422 −0.221211 0.975226i $$-0.571001\pi$$
−0.221211 + 0.975226i $$0.571001\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10.0410 0.368370 0.184185 0.982892i $$-0.441035\pi$$
0.184185 + 0.982892i $$0.441035\pi$$
$$744$$ 0 0
$$745$$ 8.63675 0.316426
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −17.3199 −0.632855
$$750$$ 0 0
$$751$$ −12.6778 −0.462619 −0.231309 0.972880i $$-0.574301\pi$$
−0.231309 + 0.972880i $$0.574301\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −17.7253 −0.645091
$$756$$ 0 0
$$757$$ 13.6302 0.495399 0.247699 0.968837i $$-0.420326\pi$$
0.247699 + 0.968837i $$0.420326\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 11.8615 0.429980 0.214990 0.976616i $$-0.431028\pi$$
0.214990 + 0.976616i $$0.431028\pi$$
$$762$$ 0 0
$$763$$ −23.3145 −0.844043
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.1362 0.438212
$$768$$ 0 0
$$769$$ 20.9988 0.757238 0.378619 0.925553i $$-0.376399\pi$$
0.378619 + 0.925553i $$0.376399\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −14.4519 −0.519797 −0.259899 0.965636i $$-0.583689\pi$$
−0.259899 + 0.965636i $$0.583689\pi$$
$$774$$ 0 0
$$775$$ −9.27349 −0.333114
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −3.68431 −0.132004
$$780$$ 0 0
$$781$$ 28.0464 1.00358
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −20.5470 −0.733353
$$786$$ 0 0
$$787$$ 10.5470 0.375959 0.187980 0.982173i $$-0.439806\pi$$
0.187980 + 0.982173i $$0.439806\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −27.6843 −0.984341
$$792$$ 0 0
$$793$$ 11.4994 0.408356
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −40.7729 −1.44425 −0.722125 0.691762i $$-0.756835\pi$$
−0.722125 + 0.691762i $$0.756835\pi$$
$$798$$ 0 0
$$799$$ −41.3556 −1.46305
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −42.3675 −1.49512
$$804$$ 0 0
$$805$$ −11.7741 −0.414982
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −20.2747 −0.712819 −0.356409 0.934330i $$-0.615999\pi$$
−0.356409 + 0.934330i $$0.615999\pi$$
$$810$$ 0 0
$$811$$ 15.0259 0.527630 0.263815 0.964573i $$-0.415019\pi$$
0.263815 + 0.964573i $$0.415019\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −4.22593 −0.148028
$$816$$ 0 0
$$817$$ 1.91561 0.0670188
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.36209 0.0824377 0.0412188 0.999150i $$-0.486876\pi$$
0.0412188 + 0.999150i $$0.486876\pi$$
$$822$$ 0 0
$$823$$ 16.1362 0.562471 0.281236 0.959639i $$-0.409256\pi$$
0.281236 + 0.959639i $$0.409256\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24.5880 0.855009 0.427505 0.904013i $$-0.359393\pi$$
0.427505 + 0.904013i $$0.359393\pi$$
$$828$$ 0 0
$$829$$ −22.8216 −0.792628 −0.396314 0.918115i $$-0.629711\pi$$
−0.396314 + 0.918115i $$0.629711\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −37.4390 −1.29719
$$834$$ 0 0
$$835$$ 14.4108 0.498707
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 46.3211 1.59918 0.799590 0.600546i $$-0.205050\pi$$
0.799590 + 0.600546i $$0.205050\pi$$
$$840$$ 0 0
$$841$$ −12.4507 −0.429334
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 13.8886 0.477217
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2.18489 0.0748972
$$852$$ 0 0
$$853$$ 47.1837 1.61554 0.807770 0.589498i $$-0.200674\pi$$
0.807770 + 0.589498i $$0.200674\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 11.8422 0.404520 0.202260 0.979332i $$-0.435171\pi$$
0.202260 + 0.979332i $$0.435171\pi$$
$$858$$ 0 0
$$859$$ −10.3211 −0.352150 −0.176075 0.984377i $$-0.556340\pi$$
−0.176075 + 0.984377i $$0.556340\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1.13733 0.0387150 0.0193575 0.999813i $$-0.493838\pi$$
0.0193575 + 0.999813i $$0.493838\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 28.0464 0.951409
$$870$$ 0 0
$$871$$ −6.41082 −0.217222
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.43134 0.116000
$$876$$ 0 0
$$877$$ −54.5447 −1.84184 −0.920921 0.389749i $$-0.872562\pi$$
−0.920921 + 0.389749i $$0.872562\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −48.9555 −1.64935 −0.824676 0.565605i $$-0.808643\pi$$
−0.824676 + 0.565605i $$0.808643\pi$$
$$882$$ 0 0
$$883$$ −27.3686 −0.921028 −0.460514 0.887653i $$-0.652335\pi$$
−0.460514 + 0.887653i $$0.652335\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.5265 −0.722788 −0.361394 0.932413i $$-0.617699\pi$$
−0.361394 + 0.932413i $$0.617699\pi$$
$$888$$ 0 0
$$889$$ −53.8182 −1.80500
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −4.19027 −0.140222
$$894$$ 0 0
$$895$$ 20.9308 0.699638
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −37.7253 −1.25821
$$900$$ 0 0
$$901$$ 90.1803 3.00434
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.08860 0.169151
$$906$$ 0 0
$$907$$ 39.6843 1.31770 0.658848 0.752276i $$-0.271044\pi$$
0.658848 + 0.752276i $$0.271044\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −4.13617 −0.137037 −0.0685187 0.997650i $$-0.521827\pi$$
−0.0685187 + 0.997650i $$0.521827\pi$$
$$912$$ 0 0
$$913$$ 42.5470 1.40810
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −44.3698 −1.46522
$$918$$ 0 0
$$919$$ −17.6789 −0.583174 −0.291587 0.956544i $$-0.594183\pi$$
−0.291587 + 0.956544i $$0.594183\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.6367 0.350113
$$924$$ 0 0
$$925$$ −0.636747 −0.0209361
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −27.8151 −0.912584 −0.456292 0.889830i $$-0.650823\pi$$
−0.456292 + 0.889830i $$0.650823\pi$$
$$930$$ 0 0
$$931$$ −3.79343 −0.124325
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −20.6778 −0.676236
$$936$$ 0 0
$$937$$ −54.5447 −1.78190 −0.890948 0.454105i $$-0.849959\pi$$
−0.890948 + 0.454105i $$0.849959\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 12.1849 0.397216 0.198608 0.980079i $$-0.436358\pi$$
0.198608 + 0.980079i $$0.436358\pi$$
$$942$$ 0 0
$$943$$ −15.9102 −0.518109
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 39.5048 1.28373 0.641867 0.766816i $$-0.278160\pi$$
0.641867 + 0.766816i $$0.278160\pi$$
$$948$$ 0 0
$$949$$ −16.0681 −0.521592
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −19.0205 −0.616135 −0.308067 0.951365i $$-0.599682\pi$$
−0.308067 + 0.951365i $$0.599682\pi$$
$$954$$ 0 0
$$955$$ 13.7253 0.444142
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6.86267 0.221607
$$960$$ 0 0
$$961$$ 54.9977 1.77412
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 5.11565 0.164679
$$966$$ 0 0
$$967$$ 38.3404 1.23294 0.616472 0.787377i $$-0.288561\pi$$
0.616472 + 0.787377i $$0.288561\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 15.7114 0.504202 0.252101 0.967701i $$-0.418879\pi$$
0.252101 + 0.967701i $$0.418879\pi$$
$$972$$ 0 0
$$973$$ 68.3187 2.19020
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −53.9566 −1.72623 −0.863113 0.505011i $$-0.831489\pi$$
−0.863113 + 0.505011i $$0.831489\pi$$
$$978$$ 0 0
$$979$$ 22.7729 0.727825
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −45.0399 −1.43655 −0.718274 0.695760i $$-0.755068\pi$$
−0.718274 + 0.695760i $$0.755068\pi$$
$$984$$ 0 0
$$985$$ −20.9988 −0.669079
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.27233 0.263045
$$990$$ 0 0
$$991$$ −7.63791 −0.242626 −0.121313 0.992614i $$-0.538710\pi$$
−0.121313 + 0.992614i $$0.538710\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 9.58918 0.303997
$$996$$ 0 0
$$997$$ −16.0951 −0.509738 −0.254869 0.966976i $$-0.582032\pi$$
−0.254869 + 0.966976i $$0.582032\pi$$
$$998$$ 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 9360.2.a.cy.1.1 3
3.2 odd 2 3120.2.a.bi.1.1 3
4.3 odd 2 4680.2.a.bh.1.3 3
12.11 even 2 1560.2.a.q.1.3 3
60.59 even 2 7800.2.a.bi.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.q.1.3 3 12.11 even 2
3120.2.a.bi.1.1 3 3.2 odd 2
4680.2.a.bh.1.3 3 4.3 odd 2
7800.2.a.bi.1.1 3 60.59 even 2
9360.2.a.cy.1.1 3 1.1 even 1 trivial