# Properties

 Label 7800.2.a.bj.1.3 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ 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] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.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: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 1.3 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{9} +1.50466 q^{11} -1.00000 q^{13} +2.72666 q^{17} +0.726656 q^{19} +4.72666 q^{23} -1.00000 q^{27} -7.55602 q^{29} -3.00933 q^{31} -1.50466 q^{33} +5.00933 q^{37} +1.00000 q^{39} +5.78734 q^{41} -2.72666 q^{43} +10.2313 q^{47} -7.00000 q^{49} -2.72666 q^{51} +7.55602 q^{53} -0.726656 q^{57} -12.5140 q^{59} +6.28267 q^{61} +12.5653 q^{67} -4.72666 q^{69} +4.77801 q^{71} -12.0187 q^{73} +5.27334 q^{79} +1.00000 q^{81} -7.78734 q^{83} +7.55602 q^{87} +1.78734 q^{89} +3.00933 q^{93} +6.00000 q^{97} +1.50466 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{9} - 6 q^{11} - 3 q^{13} + 4 q^{17} - 2 q^{19} + 10 q^{23} - 3 q^{27} - 10 q^{29} + 12 q^{31} + 6 q^{33} - 6 q^{37} + 3 q^{39} - 10 q^{41} - 4 q^{43} + 16 q^{47} - 21 q^{49} - 4 q^{51} + 10 q^{53} + 2 q^{57} - 6 q^{59} + 2 q^{61} + 4 q^{67} - 10 q^{69} + 8 q^{71} + 6 q^{73} + 20 q^{79} + 3 q^{81} + 4 q^{83} + 10 q^{87} - 22 q^{89} - 12 q^{93} + 18 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^9 - 6 * q^11 - 3 * q^13 + 4 * q^17 - 2 * q^19 + 10 * q^23 - 3 * q^27 - 10 * q^29 + 12 * q^31 + 6 * q^33 - 6 * q^37 + 3 * q^39 - 10 * q^41 - 4 * q^43 + 16 * q^47 - 21 * q^49 - 4 * q^51 + 10 * q^53 + 2 * q^57 - 6 * q^59 + 2 * q^61 + 4 * q^67 - 10 * q^69 + 8 * q^71 + 6 * q^73 + 20 * q^79 + 3 * q^81 + 4 * q^83 + 10 * q^87 - 22 * q^89 - 12 * q^93 + 18 * q^97 - 6 * q^99

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.50466 0.453673 0.226837 0.973933i $$-0.427162\pi$$
0.226837 + 0.973933i $$0.427162\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.72666 0.661311 0.330656 0.943751i $$-0.392730\pi$$
0.330656 + 0.943751i $$0.392730\pi$$
$$18$$ 0 0
$$19$$ 0.726656 0.166706 0.0833532 0.996520i $$-0.473437\pi$$
0.0833532 + 0.996520i $$0.473437\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.72666 0.985576 0.492788 0.870149i $$-0.335978\pi$$
0.492788 + 0.870149i $$0.335978\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −7.55602 −1.40312 −0.701558 0.712612i $$-0.747512\pi$$
−0.701558 + 0.712612i $$0.747512\pi$$
$$30$$ 0 0
$$31$$ −3.00933 −0.540491 −0.270246 0.962791i $$-0.587105\pi$$
−0.270246 + 0.962791i $$0.587105\pi$$
$$32$$ 0 0
$$33$$ −1.50466 −0.261928
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.00933 0.823529 0.411764 0.911290i $$-0.364913\pi$$
0.411764 + 0.911290i $$0.364913\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 5.78734 0.903830 0.451915 0.892061i $$-0.350741\pi$$
0.451915 + 0.892061i $$0.350741\pi$$
$$42$$ 0 0
$$43$$ −2.72666 −0.415811 −0.207906 0.978149i $$-0.566665\pi$$
−0.207906 + 0.978149i $$0.566665\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.2313 1.49239 0.746196 0.665727i $$-0.231878\pi$$
0.746196 + 0.665727i $$0.231878\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −2.72666 −0.381808
$$52$$ 0 0
$$53$$ 7.55602 1.03790 0.518949 0.854805i $$-0.326323\pi$$
0.518949 + 0.854805i $$0.326323\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.726656 −0.0962480
$$58$$ 0 0
$$59$$ −12.5140 −1.62918 −0.814592 0.580035i $$-0.803039\pi$$
−0.814592 + 0.580035i $$0.803039\pi$$
$$60$$ 0 0
$$61$$ 6.28267 0.804414 0.402207 0.915549i $$-0.368243\pi$$
0.402207 + 0.915549i $$0.368243\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.5653 1.53510 0.767551 0.640988i $$-0.221475\pi$$
0.767551 + 0.640988i $$0.221475\pi$$
$$68$$ 0 0
$$69$$ −4.72666 −0.569023
$$70$$ 0 0
$$71$$ 4.77801 0.567045 0.283523 0.958966i $$-0.408497\pi$$
0.283523 + 0.958966i $$0.408497\pi$$
$$72$$ 0 0
$$73$$ −12.0187 −1.40668 −0.703339 0.710855i $$-0.748308\pi$$
−0.703339 + 0.710855i $$0.748308\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.27334 0.593297 0.296649 0.954987i $$-0.404131\pi$$
0.296649 + 0.954987i $$0.404131\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.78734 −0.854771 −0.427386 0.904069i $$-0.640565\pi$$
−0.427386 + 0.904069i $$0.640565\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7.55602 0.810090
$$88$$ 0 0
$$89$$ 1.78734 0.189457 0.0947286 0.995503i $$-0.469802\pi$$
0.0947286 + 0.995503i $$0.469802\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.00933 0.312053
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 1.50466 0.151224
$$100$$ 0 0
$$101$$ −2.99067 −0.297583 −0.148791 0.988869i $$-0.547538\pi$$
−0.148791 + 0.988869i $$0.547538\pi$$
$$102$$ 0 0
$$103$$ 0.443984 0.0437471 0.0218735 0.999761i $$-0.493037\pi$$
0.0218735 + 0.999761i $$0.493037\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.00933 −0.677617 −0.338809 0.940855i $$-0.610024\pi$$
−0.338809 + 0.940855i $$0.610024\pi$$
$$108$$ 0 0
$$109$$ −13.8387 −1.32551 −0.662753 0.748838i $$-0.730612\pi$$
−0.662753 + 0.748838i $$0.730612\pi$$
$$110$$ 0 0
$$111$$ −5.00933 −0.475464
$$112$$ 0 0
$$113$$ 4.28267 0.402880 0.201440 0.979501i $$-0.435438\pi$$
0.201440 + 0.979501i $$0.435438\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.73599 −0.794180
$$122$$ 0 0
$$123$$ −5.78734 −0.521827
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.71733 0.507331 0.253665 0.967292i $$-0.418364\pi$$
0.253665 + 0.967292i $$0.418364\pi$$
$$128$$ 0 0
$$129$$ 2.72666 0.240069
$$130$$ 0 0
$$131$$ 5.55602 0.485431 0.242716 0.970097i $$-0.421962\pi$$
0.242716 + 0.970097i $$0.421962\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.67531 0.570310 0.285155 0.958481i $$-0.407955\pi$$
0.285155 + 0.958481i $$0.407955\pi$$
$$138$$ 0 0
$$139$$ 19.4720 1.65159 0.825795 0.563970i $$-0.190727\pi$$
0.825795 + 0.563970i $$0.190727\pi$$
$$140$$ 0 0
$$141$$ −10.2313 −0.861633
$$142$$ 0 0
$$143$$ −1.50466 −0.125826
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 7.00000 0.577350
$$148$$ 0 0
$$149$$ −14.5140 −1.18903 −0.594516 0.804084i $$-0.702656\pi$$
−0.594516 + 0.804084i $$0.702656\pi$$
$$150$$ 0 0
$$151$$ −4.46264 −0.363165 −0.181582 0.983376i $$-0.558122\pi$$
−0.181582 + 0.983376i $$0.558122\pi$$
$$152$$ 0 0
$$153$$ 2.72666 0.220437
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.30133 −0.662518 −0.331259 0.943540i $$-0.607473\pi$$
−0.331259 + 0.943540i $$0.607473\pi$$
$$158$$ 0 0
$$159$$ −7.55602 −0.599231
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0093 0.862317 0.431159 0.902276i $$-0.358105\pi$$
0.431159 + 0.902276i $$0.358105\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.76868 0.446394 0.223197 0.974773i $$-0.428351\pi$$
0.223197 + 0.974773i $$0.428351\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0.726656 0.0555688
$$172$$ 0 0
$$173$$ −4.90663 −0.373044 −0.186522 0.982451i $$-0.559722\pi$$
−0.186522 + 0.982451i $$0.559722\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.5140 0.940609
$$178$$ 0 0
$$179$$ 9.45331 0.706574 0.353287 0.935515i $$-0.385064\pi$$
0.353287 + 0.935515i $$0.385064\pi$$
$$180$$ 0 0
$$181$$ 17.4720 1.29868 0.649341 0.760498i $$-0.275045\pi$$
0.649341 + 0.760498i $$0.275045\pi$$
$$182$$ 0 0
$$183$$ −6.28267 −0.464428
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.10270 0.300019
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.1120 −0.804038 −0.402019 0.915631i $$-0.631691\pi$$
−0.402019 + 0.915631i $$0.631691\pi$$
$$192$$ 0 0
$$193$$ −6.10270 −0.439282 −0.219641 0.975581i $$-0.570489\pi$$
−0.219641 + 0.975581i $$0.570489\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −21.3620 −1.52198 −0.760990 0.648764i $$-0.775286\pi$$
−0.760990 + 0.648764i $$0.775286\pi$$
$$198$$ 0 0
$$199$$ 8.38538 0.594423 0.297212 0.954812i $$-0.403943\pi$$
0.297212 + 0.954812i $$0.403943\pi$$
$$200$$ 0 0
$$201$$ −12.5653 −0.886291
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.72666 0.328525
$$208$$ 0 0
$$209$$ 1.09337 0.0756303
$$210$$ 0 0
$$211$$ −1.27334 −0.0876606 −0.0438303 0.999039i $$-0.513956\pi$$
−0.0438303 + 0.999039i $$0.513956\pi$$
$$212$$ 0 0
$$213$$ −4.77801 −0.327384
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 12.0187 0.812146
$$220$$ 0 0
$$221$$ −2.72666 −0.183415
$$222$$ 0 0
$$223$$ 16.4626 1.10242 0.551210 0.834367i $$-0.314166\pi$$
0.551210 + 0.834367i $$0.314166\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.2500 0.813060 0.406530 0.913638i $$-0.366739\pi$$
0.406530 + 0.913638i $$0.366739\pi$$
$$228$$ 0 0
$$229$$ −1.27334 −0.0841449 −0.0420725 0.999115i $$-0.513396\pi$$
−0.0420725 + 0.999115i $$0.513396\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.3013 1.19896 0.599480 0.800390i $$-0.295374\pi$$
0.599480 + 0.800390i $$0.295374\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −5.27334 −0.342540
$$238$$ 0 0
$$239$$ 6.23132 0.403071 0.201535 0.979481i $$-0.435407\pi$$
0.201535 + 0.979481i $$0.435407\pi$$
$$240$$ 0 0
$$241$$ 19.5560 1.25971 0.629857 0.776711i $$-0.283114\pi$$
0.629857 + 0.776711i $$0.283114\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.726656 −0.0462360
$$248$$ 0 0
$$249$$ 7.78734 0.493502
$$250$$ 0 0
$$251$$ 26.4813 1.67148 0.835742 0.549122i $$-0.185038\pi$$
0.835742 + 0.549122i $$0.185038\pi$$
$$252$$ 0 0
$$253$$ 7.11203 0.447130
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.8294 0.925030 0.462515 0.886611i $$-0.346947\pi$$
0.462515 + 0.886611i $$0.346947\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −7.55602 −0.467706
$$262$$ 0 0
$$263$$ 22.9507 1.41520 0.707601 0.706612i $$-0.249777\pi$$
0.707601 + 0.706612i $$0.249777\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.78734 −0.109383
$$268$$ 0 0
$$269$$ −27.9160 −1.70207 −0.851033 0.525112i $$-0.824023\pi$$
−0.851033 + 0.525112i $$0.824023\pi$$
$$270$$ 0 0
$$271$$ −5.65872 −0.343743 −0.171871 0.985119i $$-0.554981\pi$$
−0.171871 + 0.985119i $$0.554981\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.73599 0.464810 0.232405 0.972619i $$-0.425340\pi$$
0.232405 + 0.972619i $$0.425340\pi$$
$$278$$ 0 0
$$279$$ −3.00933 −0.180164
$$280$$ 0 0
$$281$$ 5.11929 0.305391 0.152696 0.988273i $$-0.451205\pi$$
0.152696 + 0.988273i $$0.451205\pi$$
$$282$$ 0 0
$$283$$ 6.90663 0.410556 0.205278 0.978704i $$-0.434190\pi$$
0.205278 + 0.978704i $$0.434190\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −9.56534 −0.562667
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ 0 0
$$293$$ 26.3527 1.53954 0.769770 0.638321i $$-0.220371\pi$$
0.769770 + 0.638321i $$0.220371\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1.50466 −0.0873095
$$298$$ 0 0
$$299$$ −4.72666 −0.273350
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.99067 0.171810
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ −0.443984 −0.0252574
$$310$$ 0 0
$$311$$ −11.8973 −0.674634 −0.337317 0.941391i $$-0.609519\pi$$
−0.337317 + 0.941391i $$0.609519\pi$$
$$312$$ 0 0
$$313$$ 17.7360 1.00250 0.501249 0.865303i $$-0.332874\pi$$
0.501249 + 0.865303i $$0.332874\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.32469 −0.0744023 −0.0372011 0.999308i $$-0.511844\pi$$
−0.0372011 + 0.999308i $$0.511844\pi$$
$$318$$ 0 0
$$319$$ −11.3693 −0.636557
$$320$$ 0 0
$$321$$ 7.00933 0.391223
$$322$$ 0 0
$$323$$ 1.98134 0.110245
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 13.8387 0.765281
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −3.73599 −0.205348 −0.102674 0.994715i $$-0.532740\pi$$
−0.102674 + 0.994715i $$0.532740\pi$$
$$332$$ 0 0
$$333$$ 5.00933 0.274510
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.3947 1.27439 0.637195 0.770702i $$-0.280094\pi$$
0.637195 + 0.770702i $$0.280094\pi$$
$$338$$ 0 0
$$339$$ −4.28267 −0.232603
$$340$$ 0 0
$$341$$ −4.52803 −0.245207
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.5840 −1.64184 −0.820918 0.571047i $$-0.806538\pi$$
−0.820918 + 0.571047i $$0.806538\pi$$
$$348$$ 0 0
$$349$$ 1.37605 0.0736581 0.0368290 0.999322i $$-0.488274\pi$$
0.0368290 + 0.999322i $$0.488274\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −19.9087 −1.05963 −0.529817 0.848112i $$-0.677739\pi$$
−0.529817 + 0.848112i $$0.677739\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 26.6940 1.40885 0.704427 0.709777i $$-0.251204\pi$$
0.704427 + 0.709777i $$0.251204\pi$$
$$360$$ 0 0
$$361$$ −18.4720 −0.972209
$$362$$ 0 0
$$363$$ 8.73599 0.458520
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.12136 −0.423932 −0.211966 0.977277i $$-0.567987\pi$$
−0.211966 + 0.977277i $$0.567987\pi$$
$$368$$ 0 0
$$369$$ 5.78734 0.301277
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9.47197 −0.490440 −0.245220 0.969467i $$-0.578860\pi$$
−0.245220 + 0.969467i $$0.578860\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.55602 0.389155
$$378$$ 0 0
$$379$$ 16.7267 0.859191 0.429595 0.903022i $$-0.358656\pi$$
0.429595 + 0.903022i $$0.358656\pi$$
$$380$$ 0 0
$$381$$ −5.71733 −0.292908
$$382$$ 0 0
$$383$$ −19.6846 −1.00584 −0.502919 0.864334i $$-0.667741\pi$$
−0.502919 + 0.864334i $$0.667741\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.72666 −0.138604
$$388$$ 0 0
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 12.8880 0.651773
$$392$$ 0 0
$$393$$ −5.55602 −0.280264
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 3.35061 0.168162 0.0840812 0.996459i $$-0.473205\pi$$
0.0840812 + 0.996459i $$0.473205\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −36.3713 −1.81630 −0.908149 0.418647i $$-0.862504\pi$$
−0.908149 + 0.418647i $$0.862504\pi$$
$$402$$ 0 0
$$403$$ 3.00933 0.149905
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.53736 0.373613
$$408$$ 0 0
$$409$$ 35.5933 1.75998 0.879988 0.474995i $$-0.157550\pi$$
0.879988 + 0.474995i $$0.157550\pi$$
$$410$$ 0 0
$$411$$ −6.67531 −0.329269
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −19.4720 −0.953546
$$418$$ 0 0
$$419$$ −13.9160 −0.679839 −0.339919 0.940455i $$-0.610400\pi$$
−0.339919 + 0.940455i $$0.610400\pi$$
$$420$$ 0 0
$$421$$ 4.70800 0.229454 0.114727 0.993397i $$-0.463401\pi$$
0.114727 + 0.993397i $$0.463401\pi$$
$$422$$ 0 0
$$423$$ 10.2313 0.497464
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 1.50466 0.0726459
$$430$$ 0 0
$$431$$ 7.89004 0.380050 0.190025 0.981779i $$-0.439143\pi$$
0.190025 + 0.981779i $$0.439143\pi$$
$$432$$ 0 0
$$433$$ 2.30133 0.110595 0.0552974 0.998470i $$-0.482389\pi$$
0.0552974 + 0.998470i $$0.482389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.43466 0.164302
$$438$$ 0 0
$$439$$ 41.1307 1.96306 0.981530 0.191307i $$-0.0612725\pi$$
0.981530 + 0.191307i $$0.0612725\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ 29.5933 1.40602 0.703011 0.711179i $$-0.251839\pi$$
0.703011 + 0.711179i $$0.251839\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.5140 0.686488
$$448$$ 0 0
$$449$$ −11.4461 −0.540173 −0.270086 0.962836i $$-0.587052\pi$$
−0.270086 + 0.962836i $$0.587052\pi$$
$$450$$ 0 0
$$451$$ 8.70800 0.410044
$$452$$ 0 0
$$453$$ 4.46264 0.209673
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.5467 0.774021 0.387011 0.922075i $$-0.373508\pi$$
0.387011 + 0.922075i $$0.373508\pi$$
$$458$$ 0 0
$$459$$ −2.72666 −0.127269
$$460$$ 0 0
$$461$$ 3.50466 0.163228 0.0816142 0.996664i $$-0.473992\pi$$
0.0816142 + 0.996664i $$0.473992\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.44398 0.298192 0.149096 0.988823i $$-0.452364\pi$$
0.149096 + 0.988823i $$0.452364\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 8.30133 0.382505
$$472$$ 0 0
$$473$$ −4.10270 −0.188642
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 7.55602 0.345966
$$478$$ 0 0
$$479$$ 22.2313 1.01577 0.507887 0.861423i $$-0.330427\pi$$
0.507887 + 0.861423i $$0.330427\pi$$
$$480$$ 0 0
$$481$$ −5.00933 −0.228406
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −13.9160 −0.630592 −0.315296 0.948993i $$-0.602104\pi$$
−0.315296 + 0.948993i $$0.602104\pi$$
$$488$$ 0 0
$$489$$ −11.0093 −0.497859
$$490$$ 0 0
$$491$$ −28.1400 −1.26994 −0.634971 0.772536i $$-0.718988\pi$$
−0.634971 + 0.772536i $$0.718988\pi$$
$$492$$ 0 0
$$493$$ −20.6027 −0.927897
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 38.8480 1.73908 0.869538 0.493866i $$-0.164417\pi$$
0.869538 + 0.493866i $$0.164417\pi$$
$$500$$ 0 0
$$501$$ −5.76868 −0.257726
$$502$$ 0 0
$$503$$ 8.19863 0.365559 0.182779 0.983154i $$-0.441491\pi$$
0.182779 + 0.983154i $$0.441491\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 16.0700 0.712291 0.356145 0.934431i $$-0.384091\pi$$
0.356145 + 0.934431i $$0.384091\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −0.726656 −0.0320827
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 15.3947 0.677058
$$518$$ 0 0
$$519$$ 4.90663 0.215377
$$520$$ 0 0
$$521$$ −23.0280 −1.00887 −0.504437 0.863448i $$-0.668300\pi$$
−0.504437 + 0.863448i $$0.668300\pi$$
$$522$$ 0 0
$$523$$ 5.47875 0.239569 0.119784 0.992800i $$-0.461780\pi$$
0.119784 + 0.992800i $$0.461780\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.20541 −0.357433
$$528$$ 0 0
$$529$$ −0.658719 −0.0286399
$$530$$ 0 0
$$531$$ −12.5140 −0.543061
$$532$$ 0 0
$$533$$ −5.78734 −0.250677
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −9.45331 −0.407941
$$538$$ 0 0
$$539$$ −10.5327 −0.453673
$$540$$ 0 0
$$541$$ 17.8387 0.766945 0.383473 0.923552i $$-0.374728\pi$$
0.383473 + 0.923552i $$0.374728\pi$$
$$542$$ 0 0
$$543$$ −17.4720 −0.749794
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 29.1053 1.24445 0.622225 0.782838i $$-0.286229\pi$$
0.622225 + 0.782838i $$0.286229\pi$$
$$548$$ 0 0
$$549$$ 6.28267 0.268138
$$550$$ 0 0
$$551$$ −5.49063 −0.233909
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 44.0114 1.86482 0.932411 0.361399i $$-0.117701\pi$$
0.932411 + 0.361399i $$0.117701\pi$$
$$558$$ 0 0
$$559$$ 2.72666 0.115325
$$560$$ 0 0
$$561$$ −4.10270 −0.173216
$$562$$ 0 0
$$563$$ 7.43466 0.313333 0.156667 0.987652i $$-0.449925\pi$$
0.156667 + 0.987652i $$0.449925\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 8.48130 0.355555 0.177777 0.984071i $$-0.443109\pi$$
0.177777 + 0.984071i $$0.443109\pi$$
$$570$$ 0 0
$$571$$ 40.7826 1.70670 0.853350 0.521339i $$-0.174567\pi$$
0.853350 + 0.521339i $$0.174567\pi$$
$$572$$ 0 0
$$573$$ 11.1120 0.464212
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.57467 0.0655545 0.0327773 0.999463i $$-0.489565\pi$$
0.0327773 + 0.999463i $$0.489565\pi$$
$$578$$ 0 0
$$579$$ 6.10270 0.253620
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 11.3693 0.470867
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27.8247 1.14845 0.574223 0.818699i $$-0.305304\pi$$
0.574223 + 0.818699i $$0.305304\pi$$
$$588$$ 0 0
$$589$$ −2.18675 −0.0901034
$$590$$ 0 0
$$591$$ 21.3620 0.878716
$$592$$ 0 0
$$593$$ 32.6940 1.34258 0.671290 0.741195i $$-0.265740\pi$$
0.671290 + 0.741195i $$0.265740\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −8.38538 −0.343191
$$598$$ 0 0
$$599$$ −32.1400 −1.31321 −0.656603 0.754237i $$-0.728007\pi$$
−0.656603 + 0.754237i $$0.728007\pi$$
$$600$$ 0 0
$$601$$ 40.8667 1.66699 0.833493 0.552530i $$-0.186337\pi$$
0.833493 + 0.552530i $$0.186337\pi$$
$$602$$ 0 0
$$603$$ 12.5653 0.511700
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.9907 0.608453 0.304226 0.952600i $$-0.401602\pi$$
0.304226 + 0.952600i $$0.401602\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10.2313 −0.413915
$$612$$ 0 0
$$613$$ 13.5747 0.548276 0.274138 0.961690i $$-0.411608\pi$$
0.274138 + 0.961690i $$0.411608\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4.69396 −0.188972 −0.0944859 0.995526i $$-0.530121\pi$$
−0.0944859 + 0.995526i $$0.530121\pi$$
$$618$$ 0 0
$$619$$ −20.8667 −0.838702 −0.419351 0.907824i $$-0.637742\pi$$
−0.419351 + 0.907824i $$0.637742\pi$$
$$620$$ 0 0
$$621$$ −4.72666 −0.189674
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −1.09337 −0.0436652
$$628$$ 0 0
$$629$$ 13.6587 0.544609
$$630$$ 0 0
$$631$$ 45.9533 1.82937 0.914685 0.404167i $$-0.132438\pi$$
0.914685 + 0.404167i $$0.132438\pi$$
$$632$$ 0 0
$$633$$ 1.27334 0.0506109
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.00000 0.277350
$$638$$ 0 0
$$639$$ 4.77801 0.189015
$$640$$ 0 0
$$641$$ 20.0187 0.790689 0.395345 0.918533i $$-0.370625\pi$$
0.395345 + 0.918533i $$0.370625\pi$$
$$642$$ 0 0
$$643$$ −26.5840 −1.04837 −0.524185 0.851604i $$-0.675630\pi$$
−0.524185 + 0.851604i $$0.675630\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.39470 0.369344 0.184672 0.982800i $$-0.440878\pi$$
0.184672 + 0.982800i $$0.440878\pi$$
$$648$$ 0 0
$$649$$ −18.8294 −0.739117
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.00000 0.0782660 0.0391330 0.999234i $$-0.487540\pi$$
0.0391330 + 0.999234i $$0.487540\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −12.0187 −0.468892
$$658$$ 0 0
$$659$$ 47.6774 1.85725 0.928623 0.371024i $$-0.120993\pi$$
0.928623 + 0.371024i $$0.120993\pi$$
$$660$$ 0 0
$$661$$ −22.0959 −0.859432 −0.429716 0.902964i $$-0.641386\pi$$
−0.429716 + 0.902964i $$0.641386\pi$$
$$662$$ 0 0
$$663$$ 2.72666 0.105895
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −35.7147 −1.38288
$$668$$ 0 0
$$669$$ −16.4626 −0.636482
$$670$$ 0 0
$$671$$ 9.45331 0.364941
$$672$$ 0 0
$$673$$ −48.6027 −1.87349 −0.936747 0.350006i $$-0.886180\pi$$
−0.936747 + 0.350006i $$0.886180\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10.4253 0.400678 0.200339 0.979727i $$-0.435796\pi$$
0.200339 + 0.979727i $$0.435796\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.2500 −0.469420
$$682$$ 0 0
$$683$$ −9.13795 −0.349654 −0.174827 0.984599i $$-0.555937\pi$$
−0.174827 + 0.984599i $$0.555937\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 1.27334 0.0485811
$$688$$ 0 0
$$689$$ −7.55602 −0.287861
$$690$$ 0 0
$$691$$ 33.7173 1.28267 0.641334 0.767262i $$-0.278381\pi$$
0.641334 + 0.767262i $$0.278381\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15.7801 0.597713
$$698$$ 0 0
$$699$$ −18.3013 −0.692220
$$700$$ 0 0
$$701$$ −19.9813 −0.754685 −0.377342 0.926074i $$-0.623162\pi$$
−0.377342 + 0.926074i $$0.623162\pi$$
$$702$$ 0 0
$$703$$ 3.64006 0.137288
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −43.3293 −1.62727 −0.813633 0.581378i $$-0.802514\pi$$
−0.813633 + 0.581378i $$0.802514\pi$$
$$710$$ 0 0
$$711$$ 5.27334 0.197766
$$712$$ 0 0
$$713$$ −14.2241 −0.532695
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.23132 −0.232713
$$718$$ 0 0
$$719$$ −34.6867 −1.29360 −0.646798 0.762661i $$-0.723892\pi$$
−0.646798 + 0.762661i $$0.723892\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −19.5560 −0.727296
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −5.39470 −0.200078 −0.100039 0.994983i $$-0.531897\pi$$
−0.100039 + 0.994983i $$0.531897\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −7.43466 −0.274981
$$732$$ 0 0
$$733$$ −9.11203 −0.336561 −0.168280 0.985739i $$-0.553821\pi$$
−0.168280 + 0.985739i $$0.553821\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.9066 0.696435
$$738$$ 0 0
$$739$$ −4.82936 −0.177651 −0.0888254 0.996047i $$-0.528311\pi$$
−0.0888254 + 0.996047i $$0.528311\pi$$
$$740$$ 0 0
$$741$$ 0.726656 0.0266944
$$742$$ 0 0
$$743$$ −50.4087 −1.84931 −0.924657 0.380801i $$-0.875648\pi$$
−0.924657 + 0.380801i $$0.875648\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −7.78734 −0.284924
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.88797 0.178365 0.0891823 0.996015i $$-0.471575\pi$$
0.0891823 + 0.996015i $$0.471575\pi$$
$$752$$ 0 0
$$753$$ −26.4813 −0.965032
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −32.2241 −1.17120 −0.585602 0.810599i $$-0.699142\pi$$
−0.585602 + 0.810599i $$0.699142\pi$$
$$758$$ 0 0
$$759$$ −7.11203 −0.258150
$$760$$ 0 0
$$761$$ 36.4740 1.32218 0.661091 0.750305i $$-0.270094\pi$$
0.661091 + 0.750305i $$0.270094\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.5140 0.451854
$$768$$ 0 0
$$769$$ 13.1120 0.472832 0.236416 0.971652i $$-0.424027\pi$$
0.236416 + 0.971652i $$0.424027\pi$$
$$770$$ 0 0
$$771$$ −14.8294 −0.534066
$$772$$ 0 0
$$773$$ 12.5913 0.452876 0.226438 0.974026i $$-0.427292\pi$$
0.226438 + 0.974026i $$0.427292\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.20541 0.150674
$$780$$ 0 0
$$781$$ 7.18930 0.257253
$$782$$ 0 0
$$783$$ 7.55602 0.270030
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −36.9253 −1.31624 −0.658122 0.752911i $$-0.728649\pi$$
−0.658122 + 0.752911i $$0.728649\pi$$
$$788$$ 0 0
$$789$$ −22.9507 −0.817067
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −6.28267 −0.223104
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 25.0466 0.887198 0.443599 0.896225i $$-0.353702\pi$$
0.443599 + 0.896225i $$0.353702\pi$$
$$798$$ 0 0
$$799$$ 27.8973 0.986935
$$800$$ 0 0
$$801$$ 1.78734 0.0631524
$$802$$ 0 0
$$803$$ −18.0840 −0.638172
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 27.9160 0.982688
$$808$$ 0 0
$$809$$ 11.2334 0.394945 0.197473 0.980308i $$-0.436727\pi$$
0.197473 + 0.980308i $$0.436727\pi$$
$$810$$ 0 0
$$811$$ 31.8387 1.11801 0.559004 0.829165i $$-0.311184\pi$$
0.559004 + 0.829165i $$0.311184\pi$$
$$812$$ 0 0
$$813$$ 5.65872 0.198460
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.98134 −0.0693184
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −32.1073 −1.12055 −0.560277 0.828306i $$-0.689305\pi$$
−0.560277 + 0.828306i $$0.689305\pi$$
$$822$$ 0 0
$$823$$ 46.7054 1.62805 0.814023 0.580832i $$-0.197273\pi$$
0.814023 + 0.580832i $$0.197273\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −26.6940 −0.928240 −0.464120 0.885772i $$-0.653629\pi$$
−0.464120 + 0.885772i $$0.653629\pi$$
$$828$$ 0 0
$$829$$ −28.8294 −1.00129 −0.500643 0.865654i $$-0.666903\pi$$
−0.500643 + 0.865654i $$0.666903\pi$$
$$830$$ 0 0
$$831$$ −7.73599 −0.268358
$$832$$ 0 0
$$833$$ −19.0866 −0.661311
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.00933 0.104018
$$838$$ 0 0
$$839$$ 7.68463 0.265303 0.132652 0.991163i $$-0.457651\pi$$
0.132652 + 0.991163i $$0.457651\pi$$
$$840$$ 0 0
$$841$$ 28.0934 0.968737
$$842$$ 0 0
$$843$$ −5.11929 −0.176318
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −6.90663 −0.237035
$$850$$ 0 0
$$851$$ 23.6774 0.811650
$$852$$ 0 0
$$853$$ −18.4626 −0.632149 −0.316074 0.948734i $$-0.602365\pi$$
−0.316074 + 0.948734i $$0.602365\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 32.2827 1.10276 0.551378 0.834256i $$-0.314102\pi$$
0.551378 + 0.834256i $$0.314102\pi$$
$$858$$ 0 0
$$859$$ −18.9694 −0.647227 −0.323613 0.946189i $$-0.604898\pi$$
−0.323613 + 0.946189i $$0.604898\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −16.8807 −0.574626 −0.287313 0.957837i $$-0.592762\pi$$
−0.287313 + 0.957837i $$0.592762\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 9.56534 0.324856
$$868$$ 0 0
$$869$$ 7.93461 0.269163
$$870$$ 0 0
$$871$$ −12.5653 −0.425760
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28.1214 −0.949591 −0.474795 0.880096i $$-0.657478\pi$$
−0.474795 + 0.880096i $$0.657478\pi$$
$$878$$ 0 0
$$879$$ −26.3527 −0.888854
$$880$$ 0 0
$$881$$ −49.4066 −1.66455 −0.832275 0.554363i $$-0.812962\pi$$
−0.832275 + 0.554363i $$0.812962\pi$$
$$882$$ 0 0
$$883$$ 5.65872 0.190431 0.0952155 0.995457i $$-0.469646\pi$$
0.0952155 + 0.995457i $$0.469646\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −28.2640 −0.949013 −0.474506 0.880252i $$-0.657373\pi$$
−0.474506 + 0.880252i $$0.657373\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.50466 0.0504082
$$892$$ 0 0
$$893$$ 7.43466 0.248791
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 4.72666 0.157818
$$898$$ 0 0
$$899$$ 22.7385 0.758373
$$900$$ 0 0
$$901$$ 20.6027 0.686374
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 14.5840 0.484254 0.242127 0.970245i $$-0.422155\pi$$
0.242127 + 0.970245i $$0.422155\pi$$
$$908$$ 0 0
$$909$$ −2.99067 −0.0991943
$$910$$ 0 0
$$911$$ 29.1680 0.966379 0.483190 0.875516i $$-0.339478\pi$$
0.483190 + 0.875516i $$0.339478\pi$$
$$912$$ 0 0
$$913$$ −11.7173 −0.387787
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 2.36672 0.0780708 0.0390354 0.999238i $$-0.487571\pi$$
0.0390354 + 0.999238i $$0.487571\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ −4.77801 −0.157270
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0.443984 0.0145824
$$928$$ 0 0
$$929$$ 31.1753 1.02283 0.511414 0.859335i $$-0.329122\pi$$
0.511414 + 0.859335i $$0.329122\pi$$
$$930$$ 0 0
$$931$$ −5.08660 −0.166706
$$932$$ 0 0
$$933$$ 11.8973 0.389500
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.46942 −0.211347 −0.105673 0.994401i $$-0.533700\pi$$
−0.105673 + 0.994401i $$0.533700\pi$$
$$938$$ 0 0
$$939$$ −17.7360 −0.578792
$$940$$ 0 0
$$941$$ −19.4020 −0.632486 −0.316243 0.948678i $$-0.602421\pi$$
−0.316243 + 0.948678i $$0.602421\pi$$
$$942$$ 0 0
$$943$$ 27.3548 0.890793
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −50.0632 −1.62684 −0.813418 0.581679i $$-0.802396\pi$$
−0.813418 + 0.581679i $$0.802396\pi$$
$$948$$ 0 0
$$949$$ 12.0187 0.390142
$$950$$ 0 0
$$951$$ 1.32469 0.0429562
$$952$$ 0 0
$$953$$ −6.30133 −0.204120 −0.102060 0.994778i $$-0.532543\pi$$
−0.102060 + 0.994778i $$0.532543\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 11.3693 0.367516
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −21.9439 −0.707869
$$962$$ 0 0
$$963$$ −7.00933 −0.225872
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 37.2334 1.19735 0.598673 0.800994i $$-0.295695\pi$$
0.598673 + 0.800994i $$0.295695\pi$$
$$968$$ 0 0
$$969$$ −1.98134 −0.0636499
$$970$$ 0 0
$$971$$ 13.4533 0.431737 0.215869 0.976422i $$-0.430742\pi$$
0.215869 + 0.976422i $$0.430742\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37.4647 1.19860 0.599301 0.800524i $$-0.295445\pi$$
0.599301 + 0.800524i $$0.295445\pi$$
$$978$$ 0 0
$$979$$ 2.68934 0.0859517
$$980$$ 0 0
$$981$$ −13.8387 −0.441835
$$982$$ 0 0
$$983$$ −29.2406 −0.932632 −0.466316 0.884618i $$-0.654419\pi$$
−0.466316 + 0.884618i $$0.654419\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −12.8880 −0.409814
$$990$$ 0 0
$$991$$ 3.11203 0.0988569 0.0494285 0.998778i $$-0.484260\pi$$
0.0494285 + 0.998778i $$0.484260\pi$$
$$992$$ 0 0
$$993$$ 3.73599 0.118558
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 42.3200 1.34029 0.670144 0.742231i $$-0.266232\pi$$
0.670144 + 0.742231i $$0.266232\pi$$
$$998$$ 0 0
$$999$$ −5.00933 −0.158488
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 7800.2.a.bj.1.3 3
5.2 odd 4 1560.2.l.c.1249.4 yes 6
5.3 odd 4 1560.2.l.c.1249.1 6
5.4 even 2 7800.2.a.bp.1.3 3
15.2 even 4 4680.2.l.e.2809.6 6
15.8 even 4 4680.2.l.e.2809.5 6
20.3 even 4 3120.2.l.m.1249.4 6
20.7 even 4 3120.2.l.m.1249.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.c.1249.1 6 5.3 odd 4
1560.2.l.c.1249.4 yes 6 5.2 odd 4
3120.2.l.m.1249.1 6 20.7 even 4
3120.2.l.m.1249.4 6 20.3 even 4
4680.2.l.e.2809.5 6 15.8 even 4
4680.2.l.e.2809.6 6 15.2 even 4
7800.2.a.bj.1.3 3 1.1 even 1 trivial
7800.2.a.bp.1.3 3 5.4 even 2