# Properties

 Label 7800.2.a.bn.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 $1$

# Learn more

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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.12489$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +2.12489 q^{7} +1.00000 q^{9} +3.15516 q^{11} -1.00000 q^{13} -0.515138 q^{17} +6.73463 q^{19} +2.12489 q^{21} +4.24977 q^{23} +1.00000 q^{27} +0.0302761 q^{29} +7.15516 q^{31} +3.15516 q^{33} -5.76491 q^{37} -1.00000 q^{39} -3.76491 q^{41} +3.28005 q^{43} +4.60975 q^{47} -2.48486 q^{49} -0.515138 q^{51} -0.280047 q^{53} +6.73463 q^{57} +11.3444 q^{59} -5.01468 q^{61} +2.12489 q^{63} -15.1093 q^{67} +4.24977 q^{69} +2.23509 q^{71} +2.96972 q^{73} +6.70436 q^{77} -8.98440 q^{79} +1.00000 q^{81} -5.40493 q^{83} +0.0302761 q^{87} -15.5298 q^{89} -2.12489 q^{91} +7.15516 q^{93} -0.909172 q^{97} +3.15516 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * 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
$$7$$ 2.12489 0.803131 0.401566 0.915830i $$-0.368466\pi$$
0.401566 + 0.915830i $$0.368466\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.15516 0.951317 0.475658 0.879630i $$-0.342210\pi$$
0.475658 + 0.879630i $$0.342210\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.515138 −0.124939 −0.0624697 0.998047i $$-0.519898\pi$$
−0.0624697 + 0.998047i $$0.519898\pi$$
$$18$$ 0 0
$$19$$ 6.73463 1.54503 0.772515 0.634996i $$-0.218998\pi$$
0.772515 + 0.634996i $$0.218998\pi$$
$$20$$ 0 0
$$21$$ 2.12489 0.463688
$$22$$ 0 0
$$23$$ 4.24977 0.886138 0.443069 0.896487i $$-0.353890\pi$$
0.443069 + 0.896487i $$0.353890\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 0.0302761 0.00562213 0.00281106 0.999996i $$-0.499105\pi$$
0.00281106 + 0.999996i $$0.499105\pi$$
$$30$$ 0 0
$$31$$ 7.15516 1.28510 0.642552 0.766242i $$-0.277875\pi$$
0.642552 + 0.766242i $$0.277875\pi$$
$$32$$ 0 0
$$33$$ 3.15516 0.549243
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.76491 −0.947745 −0.473873 0.880593i $$-0.657144\pi$$
−0.473873 + 0.880593i $$0.657144\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −3.76491 −0.587980 −0.293990 0.955808i $$-0.594983\pi$$
−0.293990 + 0.955808i $$0.594983\pi$$
$$42$$ 0 0
$$43$$ 3.28005 0.500202 0.250101 0.968220i $$-0.419536\pi$$
0.250101 + 0.968220i $$0.419536\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.60975 0.672401 0.336200 0.941790i $$-0.390858\pi$$
0.336200 + 0.941790i $$0.390858\pi$$
$$48$$ 0 0
$$49$$ −2.48486 −0.354980
$$50$$ 0 0
$$51$$ −0.515138 −0.0721338
$$52$$ 0 0
$$53$$ −0.280047 −0.0384674 −0.0192337 0.999815i $$-0.506123\pi$$
−0.0192337 + 0.999815i $$0.506123\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.73463 0.892024
$$58$$ 0 0
$$59$$ 11.3444 1.47691 0.738456 0.674301i $$-0.235555\pi$$
0.738456 + 0.674301i $$0.235555\pi$$
$$60$$ 0 0
$$61$$ −5.01468 −0.642064 −0.321032 0.947068i $$-0.604030\pi$$
−0.321032 + 0.947068i $$0.604030\pi$$
$$62$$ 0 0
$$63$$ 2.12489 0.267710
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −15.1093 −1.84589 −0.922947 0.384928i $$-0.874226\pi$$
−0.922947 + 0.384928i $$0.874226\pi$$
$$68$$ 0 0
$$69$$ 4.24977 0.511612
$$70$$ 0 0
$$71$$ 2.23509 0.265257 0.132628 0.991166i $$-0.457658\pi$$
0.132628 + 0.991166i $$0.457658\pi$$
$$72$$ 0 0
$$73$$ 2.96972 0.347580 0.173790 0.984783i $$-0.444399\pi$$
0.173790 + 0.984783i $$0.444399\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.70436 0.764032
$$78$$ 0 0
$$79$$ −8.98440 −1.01082 −0.505412 0.862878i $$-0.668660\pi$$
−0.505412 + 0.862878i $$0.668660\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −5.40493 −0.593268 −0.296634 0.954991i $$-0.595864\pi$$
−0.296634 + 0.954991i $$0.595864\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0.0302761 0.00324594
$$88$$ 0 0
$$89$$ −15.5298 −1.64616 −0.823079 0.567927i $$-0.807745\pi$$
−0.823079 + 0.567927i $$0.807745\pi$$
$$90$$ 0 0
$$91$$ −2.12489 −0.222749
$$92$$ 0 0
$$93$$ 7.15516 0.741956
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.909172 −0.0923124 −0.0461562 0.998934i $$-0.514697\pi$$
−0.0461562 + 0.998934i $$0.514697\pi$$
$$98$$ 0 0
$$99$$ 3.15516 0.317106
$$100$$ 0 0
$$101$$ 16.0450 1.59653 0.798266 0.602305i $$-0.205751\pi$$
0.798266 + 0.602305i $$0.205751\pi$$
$$102$$ 0 0
$$103$$ 10.5601 1.04052 0.520258 0.854009i $$-0.325836\pi$$
0.520258 + 0.854009i $$0.325836\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.0450 1.06776 0.533878 0.845561i $$-0.320734\pi$$
0.533878 + 0.845561i $$0.320734\pi$$
$$108$$ 0 0
$$109$$ 0.734633 0.0703651 0.0351825 0.999381i $$-0.488799\pi$$
0.0351825 + 0.999381i $$0.488799\pi$$
$$110$$ 0 0
$$111$$ −5.76491 −0.547181
$$112$$ 0 0
$$113$$ 7.93945 0.746880 0.373440 0.927654i $$-0.378178\pi$$
0.373440 + 0.927654i $$0.378178\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −1.09461 −0.100343
$$120$$ 0 0
$$121$$ −1.04496 −0.0949960
$$122$$ 0 0
$$123$$ −3.76491 −0.339470
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −3.51514 −0.311918 −0.155959 0.987764i $$-0.549847\pi$$
−0.155959 + 0.987764i $$0.549847\pi$$
$$128$$ 0 0
$$129$$ 3.28005 0.288792
$$130$$ 0 0
$$131$$ 7.76491 0.678423 0.339212 0.940710i $$-0.389840\pi$$
0.339212 + 0.940710i $$0.389840\pi$$
$$132$$ 0 0
$$133$$ 14.3103 1.24086
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.2039 −1.55526 −0.777632 0.628720i $$-0.783579\pi$$
−0.777632 + 0.628720i $$0.783579\pi$$
$$138$$ 0 0
$$139$$ −12.7493 −1.08138 −0.540691 0.841221i $$-0.681837\pi$$
−0.540691 + 0.841221i $$0.681837\pi$$
$$140$$ 0 0
$$141$$ 4.60975 0.388211
$$142$$ 0 0
$$143$$ −3.15516 −0.263848
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2.48486 −0.204948
$$148$$ 0 0
$$149$$ −9.52982 −0.780713 −0.390357 0.920664i $$-0.627648\pi$$
−0.390357 + 0.920664i $$0.627648\pi$$
$$150$$ 0 0
$$151$$ 18.6244 1.51563 0.757817 0.652467i $$-0.226266\pi$$
0.757817 + 0.652467i $$0.226266\pi$$
$$152$$ 0 0
$$153$$ −0.515138 −0.0416464
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.6741 0.931693 0.465847 0.884866i $$-0.345750\pi$$
0.465847 + 0.884866i $$0.345750\pi$$
$$158$$ 0 0
$$159$$ −0.280047 −0.0222092
$$160$$ 0 0
$$161$$ 9.03028 0.711685
$$162$$ 0 0
$$163$$ −1.03028 −0.0806975 −0.0403487 0.999186i $$-0.512847\pi$$
−0.0403487 + 0.999186i $$0.512847\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.7346 −0.830671 −0.415335 0.909668i $$-0.636336\pi$$
−0.415335 + 0.909668i $$0.636336\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 6.73463 0.515010
$$172$$ 0 0
$$173$$ −3.74931 −0.285055 −0.142527 0.989791i $$-0.545523\pi$$
−0.142527 + 0.989791i $$0.545523\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.3444 0.852696
$$178$$ 0 0
$$179$$ 10.4390 0.780247 0.390123 0.920763i $$-0.372432\pi$$
0.390123 + 0.920763i $$0.372432\pi$$
$$180$$ 0 0
$$181$$ 5.17454 0.384620 0.192310 0.981334i $$-0.438402\pi$$
0.192310 + 0.981334i $$0.438402\pi$$
$$182$$ 0 0
$$183$$ −5.01468 −0.370696
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.62534 −0.118857
$$188$$ 0 0
$$189$$ 2.12489 0.154563
$$190$$ 0 0
$$191$$ 7.15894 0.518003 0.259001 0.965877i $$-0.416607\pi$$
0.259001 + 0.965877i $$0.416607\pi$$
$$192$$ 0 0
$$193$$ −14.5601 −1.04806 −0.524029 0.851700i $$-0.675572\pi$$
−0.524029 + 0.851700i $$0.675572\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.5601 1.03736 0.518682 0.854967i $$-0.326423\pi$$
0.518682 + 0.854967i $$0.326423\pi$$
$$198$$ 0 0
$$199$$ 17.1055 1.21258 0.606289 0.795245i $$-0.292658\pi$$
0.606289 + 0.795245i $$0.292658\pi$$
$$200$$ 0 0
$$201$$ −15.1093 −1.06573
$$202$$ 0 0
$$203$$ 0.0643332 0.00451531
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.24977 0.295379
$$208$$ 0 0
$$209$$ 21.2489 1.46981
$$210$$ 0 0
$$211$$ 14.1892 0.976826 0.488413 0.872613i $$-0.337576\pi$$
0.488413 + 0.872613i $$0.337576\pi$$
$$212$$ 0 0
$$213$$ 2.23509 0.153146
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 15.2039 1.03211
$$218$$ 0 0
$$219$$ 2.96972 0.200675
$$220$$ 0 0
$$221$$ 0.515138 0.0346519
$$222$$ 0 0
$$223$$ 4.06055 0.271915 0.135957 0.990715i $$-0.456589\pi$$
0.135957 + 0.990715i $$0.456589\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −14.6850 −0.974676 −0.487338 0.873213i $$-0.662032\pi$$
−0.487338 + 0.873213i $$0.662032\pi$$
$$228$$ 0 0
$$229$$ −24.3250 −1.60744 −0.803721 0.595007i $$-0.797149\pi$$
−0.803721 + 0.595007i $$0.797149\pi$$
$$230$$ 0 0
$$231$$ 6.70436 0.441114
$$232$$ 0 0
$$233$$ 13.6509 0.894302 0.447151 0.894459i $$-0.352439\pi$$
0.447151 + 0.894459i $$0.352439\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.98440 −0.583600
$$238$$ 0 0
$$239$$ 15.0341 0.972472 0.486236 0.873827i $$-0.338369\pi$$
0.486236 + 0.873827i $$0.338369\pi$$
$$240$$ 0 0
$$241$$ 14.5601 0.937898 0.468949 0.883225i $$-0.344633\pi$$
0.468949 + 0.883225i $$0.344633\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.73463 −0.428514
$$248$$ 0 0
$$249$$ −5.40493 −0.342524
$$250$$ 0 0
$$251$$ −9.20482 −0.581003 −0.290501 0.956875i $$-0.593822\pi$$
−0.290501 + 0.956875i $$0.593822\pi$$
$$252$$ 0 0
$$253$$ 13.4087 0.842999
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −13.7952 −0.860520 −0.430260 0.902705i $$-0.641578\pi$$
−0.430260 + 0.902705i $$0.641578\pi$$
$$258$$ 0 0
$$259$$ −12.2498 −0.761164
$$260$$ 0 0
$$261$$ 0.0302761 0.00187404
$$262$$ 0 0
$$263$$ 3.03028 0.186855 0.0934274 0.995626i $$-0.470218\pi$$
0.0934274 + 0.995626i $$0.470218\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −15.5298 −0.950409
$$268$$ 0 0
$$269$$ −0.469266 −0.0286116 −0.0143058 0.999898i $$-0.504554\pi$$
−0.0143058 + 0.999898i $$0.504554\pi$$
$$270$$ 0 0
$$271$$ −2.87420 −0.174595 −0.0872975 0.996182i $$-0.527823\pi$$
−0.0872975 + 0.996182i $$0.527823\pi$$
$$272$$ 0 0
$$273$$ −2.12489 −0.128604
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.87890 −0.112892 −0.0564459 0.998406i $$-0.517977\pi$$
−0.0564459 + 0.998406i $$0.517977\pi$$
$$278$$ 0 0
$$279$$ 7.15516 0.428368
$$280$$ 0 0
$$281$$ 23.2342 1.38603 0.693017 0.720921i $$-0.256281\pi$$
0.693017 + 0.720921i $$0.256281\pi$$
$$282$$ 0 0
$$283$$ −26.8099 −1.59368 −0.796841 0.604190i $$-0.793497\pi$$
−0.796841 + 0.604190i $$0.793497\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.00000 −0.472225
$$288$$ 0 0
$$289$$ −16.7346 −0.984390
$$290$$ 0 0
$$291$$ −0.909172 −0.0532966
$$292$$ 0 0
$$293$$ 10.4702 0.611675 0.305837 0.952084i $$-0.401064\pi$$
0.305837 + 0.952084i $$0.401064\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.15516 0.183081
$$298$$ 0 0
$$299$$ −4.24977 −0.245771
$$300$$ 0 0
$$301$$ 6.96972 0.401728
$$302$$ 0 0
$$303$$ 16.0450 0.921759
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −26.3856 −1.50590 −0.752952 0.658076i $$-0.771371\pi$$
−0.752952 + 0.658076i $$0.771371\pi$$
$$308$$ 0 0
$$309$$ 10.5601 0.600743
$$310$$ 0 0
$$311$$ −24.2791 −1.37674 −0.688372 0.725358i $$-0.741674\pi$$
−0.688372 + 0.725358i $$0.741674\pi$$
$$312$$ 0 0
$$313$$ −14.2800 −0.807156 −0.403578 0.914945i $$-0.632234\pi$$
−0.403578 + 0.914945i $$0.632234\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.5298 −1.20924 −0.604618 0.796516i $$-0.706674\pi$$
−0.604618 + 0.796516i $$0.706674\pi$$
$$318$$ 0 0
$$319$$ 0.0955260 0.00534843
$$320$$ 0 0
$$321$$ 11.0450 0.616469
$$322$$ 0 0
$$323$$ −3.46927 −0.193035
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0.734633 0.0406253
$$328$$ 0 0
$$329$$ 9.79518 0.540026
$$330$$ 0 0
$$331$$ 31.5904 1.73636 0.868182 0.496246i $$-0.165289\pi$$
0.868182 + 0.496246i $$0.165289\pi$$
$$332$$ 0 0
$$333$$ −5.76491 −0.315915
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7.42431 −0.404428 −0.202214 0.979341i $$-0.564814\pi$$
−0.202214 + 0.979341i $$0.564814\pi$$
$$338$$ 0 0
$$339$$ 7.93945 0.431212
$$340$$ 0 0
$$341$$ 22.5757 1.22254
$$342$$ 0 0
$$343$$ −20.1542 −1.08823
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 13.9541 0.749097 0.374548 0.927207i $$-0.377798\pi$$
0.374548 + 0.927207i $$0.377798\pi$$
$$348$$ 0 0
$$349$$ −4.06055 −0.217356 −0.108678 0.994077i $$-0.534662\pi$$
−0.108678 + 0.994077i $$0.534662\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ 24.3856 1.29791 0.648956 0.760826i $$-0.275206\pi$$
0.648956 + 0.760826i $$0.275206\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1.09461 −0.0579329
$$358$$ 0 0
$$359$$ −3.76869 −0.198904 −0.0994519 0.995042i $$-0.531709\pi$$
−0.0994519 + 0.995042i $$0.531709\pi$$
$$360$$ 0 0
$$361$$ 26.3553 1.38712
$$362$$ 0 0
$$363$$ −1.04496 −0.0548460
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −19.2654 −1.00564 −0.502822 0.864390i $$-0.667705\pi$$
−0.502822 + 0.864390i $$0.667705\pi$$
$$368$$ 0 0
$$369$$ −3.76491 −0.195993
$$370$$ 0 0
$$371$$ −0.595068 −0.0308944
$$372$$ 0 0
$$373$$ 14.3553 0.743288 0.371644 0.928375i $$-0.378794\pi$$
0.371644 + 0.928375i $$0.378794\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.0302761 −0.00155930
$$378$$ 0 0
$$379$$ 33.1845 1.70457 0.852287 0.523074i $$-0.175215\pi$$
0.852287 + 0.523074i $$0.175215\pi$$
$$380$$ 0 0
$$381$$ −3.51514 −0.180086
$$382$$ 0 0
$$383$$ 30.8851 1.57815 0.789077 0.614294i $$-0.210559\pi$$
0.789077 + 0.614294i $$0.210559\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.28005 0.166734
$$388$$ 0 0
$$389$$ 0.295643 0.0149897 0.00749486 0.999972i $$-0.497614\pi$$
0.00749486 + 0.999972i $$0.497614\pi$$
$$390$$ 0 0
$$391$$ −2.18922 −0.110714
$$392$$ 0 0
$$393$$ 7.76491 0.391688
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.3250 −0.819328 −0.409664 0.912236i $$-0.634354\pi$$
−0.409664 + 0.912236i $$0.634354\pi$$
$$398$$ 0 0
$$399$$ 14.3103 0.716412
$$400$$ 0 0
$$401$$ 28.4390 1.42018 0.710088 0.704113i $$-0.248655\pi$$
0.710088 + 0.704113i $$0.248655\pi$$
$$402$$ 0 0
$$403$$ −7.15516 −0.356424
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −18.1892 −0.901606
$$408$$ 0 0
$$409$$ −16.3784 −0.809862 −0.404931 0.914347i $$-0.632704\pi$$
−0.404931 + 0.914347i $$0.632704\pi$$
$$410$$ 0 0
$$411$$ −18.2039 −0.897932
$$412$$ 0 0
$$413$$ 24.1055 1.18615
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −12.7493 −0.624337
$$418$$ 0 0
$$419$$ 31.4158 1.53476 0.767382 0.641190i $$-0.221559\pi$$
0.767382 + 0.641190i $$0.221559\pi$$
$$420$$ 0 0
$$421$$ 23.5298 1.14677 0.573387 0.819285i $$-0.305629\pi$$
0.573387 + 0.819285i $$0.305629\pi$$
$$422$$ 0 0
$$423$$ 4.60975 0.224134
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −10.6556 −0.515662
$$428$$ 0 0
$$429$$ −3.15516 −0.152333
$$430$$ 0 0
$$431$$ −19.3553 −0.932311 −0.466155 0.884703i $$-0.654361\pi$$
−0.466155 + 0.884703i $$0.654361\pi$$
$$432$$ 0 0
$$433$$ −19.7044 −0.946931 −0.473465 0.880812i $$-0.656997\pi$$
−0.473465 + 0.880812i $$0.656997\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 28.6206 1.36911
$$438$$ 0 0
$$439$$ 8.98440 0.428802 0.214401 0.976746i $$-0.431220\pi$$
0.214401 + 0.976746i $$0.431220\pi$$
$$440$$ 0 0
$$441$$ −2.48486 −0.118327
$$442$$ 0 0
$$443$$ 2.92385 0.138916 0.0694582 0.997585i $$-0.477873\pi$$
0.0694582 + 0.997585i $$0.477873\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −9.52982 −0.450745
$$448$$ 0 0
$$449$$ −17.3553 −0.819046 −0.409523 0.912300i $$-0.634305\pi$$
−0.409523 + 0.912300i $$0.634305\pi$$
$$450$$ 0 0
$$451$$ −11.8789 −0.559355
$$452$$ 0 0
$$453$$ 18.6244 0.875052
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −7.05964 −0.330236 −0.165118 0.986274i $$-0.552800\pi$$
−0.165118 + 0.986274i $$0.552800\pi$$
$$458$$ 0 0
$$459$$ −0.515138 −0.0240446
$$460$$ 0 0
$$461$$ −1.52982 −0.0712507 −0.0356254 0.999365i $$-0.511342\pi$$
−0.0356254 + 0.999365i $$0.511342\pi$$
$$462$$ 0 0
$$463$$ 27.3132 1.26935 0.634676 0.772779i $$-0.281134\pi$$
0.634676 + 0.772779i $$0.281134\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.29564 −0.291328 −0.145664 0.989334i $$-0.546532\pi$$
−0.145664 + 0.989334i $$0.546532\pi$$
$$468$$ 0 0
$$469$$ −32.1055 −1.48249
$$470$$ 0 0
$$471$$ 11.6741 0.537913
$$472$$ 0 0
$$473$$ 10.3491 0.475851
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −0.280047 −0.0128225
$$478$$ 0 0
$$479$$ 26.9806 1.23278 0.616388 0.787443i $$-0.288595\pi$$
0.616388 + 0.787443i $$0.288595\pi$$
$$480$$ 0 0
$$481$$ 5.76491 0.262857
$$482$$ 0 0
$$483$$ 9.03028 0.410892
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −11.8751 −0.538113 −0.269056 0.963124i $$-0.586712\pi$$
−0.269056 + 0.963124i $$0.586712\pi$$
$$488$$ 0 0
$$489$$ −1.03028 −0.0465907
$$490$$ 0 0
$$491$$ −41.8695 −1.88954 −0.944772 0.327728i $$-0.893717\pi$$
−0.944772 + 0.327728i $$0.893717\pi$$
$$492$$ 0 0
$$493$$ −0.0155964 −0.000702425 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.74931 0.213036
$$498$$ 0 0
$$499$$ 30.5786 1.36888 0.684442 0.729067i $$-0.260046\pi$$
0.684442 + 0.729067i $$0.260046\pi$$
$$500$$ 0 0
$$501$$ −10.7346 −0.479588
$$502$$ 0 0
$$503$$ 32.9991 1.47136 0.735678 0.677331i $$-0.236864\pi$$
0.735678 + 0.677331i $$0.236864\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ 41.6197 1.84476 0.922381 0.386281i $$-0.126241\pi$$
0.922381 + 0.386281i $$0.126241\pi$$
$$510$$ 0 0
$$511$$ 6.31032 0.279152
$$512$$ 0 0
$$513$$ 6.73463 0.297341
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.5445 0.639666
$$518$$ 0 0
$$519$$ −3.74931 −0.164577
$$520$$ 0 0
$$521$$ 6.47018 0.283464 0.141732 0.989905i $$-0.454733\pi$$
0.141732 + 0.989905i $$0.454733\pi$$
$$522$$ 0 0
$$523$$ −37.8401 −1.65463 −0.827317 0.561735i $$-0.810134\pi$$
−0.827317 + 0.561735i $$0.810134\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3.68590 −0.160560
$$528$$ 0 0
$$529$$ −4.93945 −0.214759
$$530$$ 0 0
$$531$$ 11.3444 0.492304
$$532$$ 0 0
$$533$$ 3.76491 0.163076
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10.4390 0.450476
$$538$$ 0 0
$$539$$ −7.84014 −0.337699
$$540$$ 0 0
$$541$$ 6.37844 0.274230 0.137115 0.990555i $$-0.456217\pi$$
0.137115 + 0.990555i $$0.456217\pi$$
$$542$$ 0 0
$$543$$ 5.17454 0.222061
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10.0899 0.431413 0.215707 0.976458i $$-0.430794\pi$$
0.215707 + 0.976458i $$0.430794\pi$$
$$548$$ 0 0
$$549$$ −5.01468 −0.214021
$$550$$ 0 0
$$551$$ 0.203898 0.00868636
$$552$$ 0 0
$$553$$ −19.0908 −0.811825
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.96972 0.210574 0.105287 0.994442i $$-0.466424\pi$$
0.105287 + 0.994442i $$0.466424\pi$$
$$558$$ 0 0
$$559$$ −3.28005 −0.138731
$$560$$ 0 0
$$561$$ −1.62534 −0.0686221
$$562$$ 0 0
$$563$$ −14.8027 −0.623861 −0.311931 0.950105i $$-0.600976\pi$$
−0.311931 + 0.950105i $$0.600976\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.12489 0.0892368
$$568$$ 0 0
$$569$$ −3.85574 −0.161641 −0.0808205 0.996729i $$-0.525754\pi$$
−0.0808205 + 0.996729i $$0.525754\pi$$
$$570$$ 0 0
$$571$$ 24.6282 1.03066 0.515329 0.856992i $$-0.327670\pi$$
0.515329 + 0.856992i $$0.327670\pi$$
$$572$$ 0 0
$$573$$ 7.15894 0.299069
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.02936 0.251006 0.125503 0.992093i $$-0.459946\pi$$
0.125503 + 0.992093i $$0.459946\pi$$
$$578$$ 0 0
$$579$$ −14.5601 −0.605097
$$580$$ 0 0
$$581$$ −11.4849 −0.476472
$$582$$ 0 0
$$583$$ −0.883593 −0.0365947
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.0861 0.581397 0.290698 0.956815i $$-0.406112\pi$$
0.290698 + 0.956815i $$0.406112\pi$$
$$588$$ 0 0
$$589$$ 48.1874 1.98553
$$590$$ 0 0
$$591$$ 14.5601 0.598922
$$592$$ 0 0
$$593$$ −28.2333 −1.15940 −0.579700 0.814830i $$-0.696830\pi$$
−0.579700 + 0.814830i $$0.696830\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.1055 0.700082
$$598$$ 0 0
$$599$$ −23.0596 −0.942191 −0.471096 0.882082i $$-0.656141\pi$$
−0.471096 + 0.882082i $$0.656141\pi$$
$$600$$ 0 0
$$601$$ 14.1901 0.578828 0.289414 0.957204i $$-0.406540\pi$$
0.289414 + 0.957204i $$0.406540\pi$$
$$602$$ 0 0
$$603$$ −15.1093 −0.615298
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −43.9229 −1.78278 −0.891388 0.453240i $$-0.850268\pi$$
−0.891388 + 0.453240i $$0.850268\pi$$
$$608$$ 0 0
$$609$$ 0.0643332 0.00260691
$$610$$ 0 0
$$611$$ −4.60975 −0.186490
$$612$$ 0 0
$$613$$ 1.62156 0.0654943 0.0327472 0.999464i $$-0.489574\pi$$
0.0327472 + 0.999464i $$0.489574\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.3250 −0.496186 −0.248093 0.968736i $$-0.579804\pi$$
−0.248093 + 0.968736i $$0.579804\pi$$
$$618$$ 0 0
$$619$$ −5.15138 −0.207051 −0.103526 0.994627i $$-0.533012\pi$$
−0.103526 + 0.994627i $$0.533012\pi$$
$$620$$ 0 0
$$621$$ 4.24977 0.170537
$$622$$ 0 0
$$623$$ −32.9991 −1.32208
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 21.2489 0.848597
$$628$$ 0 0
$$629$$ 2.96972 0.118411
$$630$$ 0 0
$$631$$ 1.97064 0.0784500 0.0392250 0.999230i $$-0.487511\pi$$
0.0392250 + 0.999230i $$0.487511\pi$$
$$632$$ 0 0
$$633$$ 14.1892 0.563971
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.48486 0.0984538
$$638$$ 0 0
$$639$$ 2.23509 0.0884188
$$640$$ 0 0
$$641$$ 4.33348 0.171162 0.0855811 0.996331i $$-0.472725\pi$$
0.0855811 + 0.996331i $$0.472725\pi$$
$$642$$ 0 0
$$643$$ −6.85574 −0.270364 −0.135182 0.990821i $$-0.543162\pi$$
−0.135182 + 0.990821i $$0.543162\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −5.93945 −0.233504 −0.116752 0.993161i $$-0.537248\pi$$
−0.116752 + 0.993161i $$0.537248\pi$$
$$648$$ 0 0
$$649$$ 35.7934 1.40501
$$650$$ 0 0
$$651$$ 15.2039 0.595888
$$652$$ 0 0
$$653$$ 9.61353 0.376206 0.188103 0.982149i $$-0.439766\pi$$
0.188103 + 0.982149i $$0.439766\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.96972 0.115860
$$658$$ 0 0
$$659$$ 0.455503 0.0177439 0.00887193 0.999961i $$-0.497176\pi$$
0.00887193 + 0.999961i $$0.497176\pi$$
$$660$$ 0 0
$$661$$ −29.2635 −1.13822 −0.569110 0.822262i $$-0.692712\pi$$
−0.569110 + 0.822262i $$0.692712\pi$$
$$662$$ 0 0
$$663$$ 0.515138 0.0200063
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.128666 0.00498199
$$668$$ 0 0
$$669$$ 4.06055 0.156990
$$670$$ 0 0
$$671$$ −15.8221 −0.610806
$$672$$ 0 0
$$673$$ −24.7796 −0.955183 −0.477591 0.878582i $$-0.658490\pi$$
−0.477591 + 0.878582i $$0.658490\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 7.93945 0.305138 0.152569 0.988293i $$-0.451245\pi$$
0.152569 + 0.988293i $$0.451245\pi$$
$$678$$ 0 0
$$679$$ −1.93189 −0.0741390
$$680$$ 0 0
$$681$$ −14.6850 −0.562730
$$682$$ 0 0
$$683$$ 7.53360 0.288265 0.144133 0.989558i $$-0.453961\pi$$
0.144133 + 0.989558i $$0.453961\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −24.3250 −0.928057
$$688$$ 0 0
$$689$$ 0.280047 0.0106689
$$690$$ 0 0
$$691$$ −51.4490 −1.95721 −0.978606 0.205745i $$-0.934038\pi$$
−0.978606 + 0.205745i $$0.934038\pi$$
$$692$$ 0 0
$$693$$ 6.70436 0.254677
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.93945 0.0734618
$$698$$ 0 0
$$699$$ 13.6509 0.516325
$$700$$ 0 0
$$701$$ −5.35620 −0.202301 −0.101150 0.994871i $$-0.532252\pi$$
−0.101150 + 0.994871i $$0.532252\pi$$
$$702$$ 0 0
$$703$$ −38.8245 −1.46430
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 34.0937 1.28223
$$708$$ 0 0
$$709$$ 0.969724 0.0364187 0.0182094 0.999834i $$-0.494203\pi$$
0.0182094 + 0.999834i $$0.494203\pi$$
$$710$$ 0 0
$$711$$ −8.98440 −0.336941
$$712$$ 0 0
$$713$$ 30.4078 1.13878
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 15.0341 0.561457
$$718$$ 0 0
$$719$$ 23.5592 0.878609 0.439305 0.898338i $$-0.355225\pi$$
0.439305 + 0.898338i $$0.355225\pi$$
$$720$$ 0 0
$$721$$ 22.4390 0.835672
$$722$$ 0 0
$$723$$ 14.5601 0.541496
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −22.3179 −0.827725 −0.413862 0.910340i $$-0.635820\pi$$
−0.413862 + 0.910340i $$0.635820\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −1.68968 −0.0624950
$$732$$ 0 0
$$733$$ 46.7034 1.72503 0.862515 0.506031i $$-0.168888\pi$$
0.862515 + 0.506031i $$0.168888\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −47.6722 −1.75603
$$738$$ 0 0
$$739$$ −29.2985 −1.07776 −0.538882 0.842382i $$-0.681153\pi$$
−0.538882 + 0.842382i $$0.681153\pi$$
$$740$$ 0 0
$$741$$ −6.73463 −0.247403
$$742$$ 0 0
$$743$$ 51.0175 1.87165 0.935826 0.352462i $$-0.114655\pi$$
0.935826 + 0.352462i $$0.114655\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −5.40493 −0.197756
$$748$$ 0 0
$$749$$ 23.4693 0.857548
$$750$$ 0 0
$$751$$ 4.36376 0.159236 0.0796179 0.996825i $$-0.474630\pi$$
0.0796179 + 0.996825i $$0.474630\pi$$
$$752$$ 0 0
$$753$$ −9.20482 −0.335442
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −42.6344 −1.54957 −0.774787 0.632222i $$-0.782143\pi$$
−0.774787 + 0.632222i $$0.782143\pi$$
$$758$$ 0 0
$$759$$ 13.4087 0.486705
$$760$$ 0 0
$$761$$ −3.73372 −0.135347 −0.0676736 0.997708i $$-0.521558\pi$$
−0.0676736 + 0.997708i $$0.521558\pi$$
$$762$$ 0 0
$$763$$ 1.56101 0.0565124
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −11.3444 −0.409622
$$768$$ 0 0
$$769$$ −20.3784 −0.734865 −0.367433 0.930050i $$-0.619763\pi$$
−0.367433 + 0.930050i $$0.619763\pi$$
$$770$$ 0 0
$$771$$ −13.7952 −0.496821
$$772$$ 0 0
$$773$$ −14.5289 −0.522568 −0.261284 0.965262i $$-0.584146\pi$$
−0.261284 + 0.965262i $$0.584146\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −12.2498 −0.439458
$$778$$ 0 0
$$779$$ −25.3553 −0.908447
$$780$$ 0 0
$$781$$ 7.05207 0.252343
$$782$$ 0 0
$$783$$ 0.0302761 0.00108198
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −5.00470 −0.178398 −0.0891991 0.996014i $$-0.528431\pi$$
−0.0891991 + 0.996014i $$0.528431\pi$$
$$788$$ 0 0
$$789$$ 3.03028 0.107881
$$790$$ 0 0
$$791$$ 16.8704 0.599843
$$792$$ 0 0
$$793$$ 5.01468 0.178076
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 48.3846 1.71387 0.856936 0.515423i $$-0.172365\pi$$
0.856936 + 0.515423i $$0.172365\pi$$
$$798$$ 0 0
$$799$$ −2.37466 −0.0840093
$$800$$ 0 0
$$801$$ −15.5298 −0.548719
$$802$$ 0 0
$$803$$ 9.36996 0.330659
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −0.469266 −0.0165189
$$808$$ 0 0
$$809$$ 9.18074 0.322778 0.161389 0.986891i $$-0.448403\pi$$
0.161389 + 0.986891i $$0.448403\pi$$
$$810$$ 0 0
$$811$$ 30.2224 1.06125 0.530625 0.847607i $$-0.321957\pi$$
0.530625 + 0.847607i $$0.321957\pi$$
$$812$$ 0 0
$$813$$ −2.87420 −0.100803
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 22.0899 0.772828
$$818$$ 0 0
$$819$$ −2.12489 −0.0742495
$$820$$ 0 0
$$821$$ 31.6197 1.10354 0.551768 0.833998i $$-0.313953\pi$$
0.551768 + 0.833998i $$0.313953\pi$$
$$822$$ 0 0
$$823$$ −31.4158 −1.09509 −0.547544 0.836777i $$-0.684437\pi$$
−0.547544 + 0.836777i $$0.684437\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −11.2838 −0.392377 −0.196189 0.980566i $$-0.562856\pi$$
−0.196189 + 0.980566i $$0.562856\pi$$
$$828$$ 0 0
$$829$$ −4.83302 −0.167858 −0.0839289 0.996472i $$-0.526747\pi$$
−0.0839289 + 0.996472i $$0.526747\pi$$
$$830$$ 0 0
$$831$$ −1.87890 −0.0651782
$$832$$ 0 0
$$833$$ 1.28005 0.0443510
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.15516 0.247319
$$838$$ 0 0
$$839$$ −15.4693 −0.534058 −0.267029 0.963688i $$-0.586042\pi$$
−0.267029 + 0.963688i $$0.586042\pi$$
$$840$$ 0 0
$$841$$ −28.9991 −0.999968
$$842$$ 0 0
$$843$$ 23.2342 0.800227
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.22041 −0.0762942
$$848$$ 0 0
$$849$$ −26.8099 −0.920112
$$850$$ 0 0
$$851$$ −24.4995 −0.839833
$$852$$ 0 0
$$853$$ −41.2635 −1.41284 −0.706418 0.707795i $$-0.749690\pi$$
−0.706418 + 0.707795i $$0.749690\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 9.87890 0.337457 0.168728 0.985663i $$-0.446034\pi$$
0.168728 + 0.985663i $$0.446034\pi$$
$$858$$ 0 0
$$859$$ −20.7200 −0.706956 −0.353478 0.935443i $$-0.615001\pi$$
−0.353478 + 0.935443i $$0.615001\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ −1.91915 −0.0653288 −0.0326644 0.999466i $$-0.510399\pi$$
−0.0326644 + 0.999466i $$0.510399\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −16.7346 −0.568338
$$868$$ 0 0
$$869$$ −28.3472 −0.961614
$$870$$ 0 0
$$871$$ 15.1093 0.511959
$$872$$ 0 0
$$873$$ −0.909172 −0.0307708
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −18.3250 −0.618791 −0.309396 0.950933i $$-0.600127\pi$$
−0.309396 + 0.950933i $$0.600127\pi$$
$$878$$ 0 0
$$879$$ 10.4702 0.353150
$$880$$ 0 0
$$881$$ 12.7649 0.430061 0.215030 0.976607i $$-0.431015\pi$$
0.215030 + 0.976607i $$0.431015\pi$$
$$882$$ 0 0
$$883$$ −11.8789 −0.399757 −0.199878 0.979821i $$-0.564055\pi$$
−0.199878 + 0.979821i $$0.564055\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −40.7493 −1.36823 −0.684114 0.729375i $$-0.739811\pi$$
−0.684114 + 0.729375i $$0.739811\pi$$
$$888$$ 0 0
$$889$$ −7.46927 −0.250511
$$890$$ 0 0
$$891$$ 3.15516 0.105702
$$892$$ 0 0
$$893$$ 31.0450 1.03888
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −4.24977 −0.141896
$$898$$ 0 0
$$899$$ 0.216630 0.00722503
$$900$$ 0 0
$$901$$ 0.144263 0.00480609
$$902$$ 0 0
$$903$$ 6.96972 0.231938
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.5289 1.08010 0.540052 0.841632i $$-0.318404\pi$$
0.540052 + 0.841632i $$0.318404\pi$$
$$908$$ 0 0
$$909$$ 16.0450 0.532178
$$910$$ 0 0
$$911$$ −22.2810 −0.738201 −0.369101 0.929389i $$-0.620334\pi$$
−0.369101 + 0.929389i $$0.620334\pi$$
$$912$$ 0 0
$$913$$ −17.0534 −0.564386
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.4995 0.544863
$$918$$ 0 0
$$919$$ 26.9239 0.888136 0.444068 0.895993i $$-0.353535\pi$$
0.444068 + 0.895993i $$0.353535\pi$$
$$920$$ 0 0
$$921$$ −26.3856 −0.869434
$$922$$ 0 0
$$923$$ −2.23509 −0.0735689
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 10.5601 0.346839
$$928$$ 0 0
$$929$$ −36.4755 −1.19672 −0.598361 0.801227i $$-0.704181\pi$$
−0.598361 + 0.801227i $$0.704181\pi$$
$$930$$ 0 0
$$931$$ −16.7346 −0.548455
$$932$$ 0 0
$$933$$ −24.2791 −0.794863
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.26445 0.237319 0.118660 0.992935i $$-0.462140\pi$$
0.118660 + 0.992935i $$0.462140\pi$$
$$938$$ 0 0
$$939$$ −14.2800 −0.466012
$$940$$ 0 0
$$941$$ −8.08991 −0.263724 −0.131862 0.991268i $$-0.542096\pi$$
−0.131862 + 0.991268i $$0.542096\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −48.9036 −1.58915 −0.794576 0.607165i $$-0.792307\pi$$
−0.794576 + 0.607165i $$0.792307\pi$$
$$948$$ 0 0
$$949$$ −2.96972 −0.0964013
$$950$$ 0 0
$$951$$ −21.5298 −0.698152
$$952$$ 0 0
$$953$$ 6.58325 0.213252 0.106626 0.994299i $$-0.465995\pi$$
0.106626 + 0.994299i $$0.465995\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0.0955260 0.00308792
$$958$$ 0 0
$$959$$ −38.6812 −1.24908
$$960$$ 0 0
$$961$$ 20.1963 0.651495
$$962$$ 0 0
$$963$$ 11.0450 0.355919
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −9.58659 −0.308284 −0.154142 0.988049i $$-0.549261\pi$$
−0.154142 + 0.988049i $$0.549261\pi$$
$$968$$ 0 0
$$969$$ −3.46927 −0.111449
$$970$$ 0 0
$$971$$ −36.1358 −1.15965 −0.579826 0.814740i $$-0.696880\pi$$
−0.579826 + 0.814740i $$0.696880\pi$$
$$972$$ 0 0
$$973$$ −27.0908 −0.868492
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27.0303 0.864775 0.432388 0.901688i $$-0.357671\pi$$
0.432388 + 0.901688i $$0.357671\pi$$
$$978$$ 0 0
$$979$$ −48.9991 −1.56602
$$980$$ 0 0
$$981$$ 0.734633 0.0234550
$$982$$ 0 0
$$983$$ −49.9338 −1.59264 −0.796321 0.604874i $$-0.793223\pi$$
−0.796321 + 0.604874i $$0.793223\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 9.79518 0.311784
$$988$$ 0 0
$$989$$ 13.9394 0.443249
$$990$$ 0 0
$$991$$ 20.4849 0.650723 0.325362 0.945590i $$-0.394514\pi$$
0.325362 + 0.945590i $$0.394514\pi$$
$$992$$ 0 0
$$993$$ 31.5904 1.00249
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 40.9541 1.29703 0.648515 0.761202i $$-0.275390\pi$$
0.648515 + 0.761202i $$0.275390\pi$$
$$998$$ 0 0
$$999$$ −5.76491 −0.182394
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.bn.1.3 yes 3
5.4 even 2 7800.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bm.1.1 3 5.4 even 2
7800.2.a.bn.1.3 yes 3 1.1 even 1 trivial