# Properties

 Label 7938.2.a.bg.1.2 Level $7938$ Weight $2$ Character 7938.1 Self dual yes Analytic conductor $63.385$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7938 = 2 \cdot 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7938.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: $$63.3852491245$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1134) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 7938.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{8} +0.267949 q^{10} +6.19615 q^{11} +6.46410 q^{13} +1.00000 q^{16} -7.00000 q^{17} -0.732051 q^{19} -0.267949 q^{20} -6.19615 q^{22} -4.19615 q^{23} -4.92820 q^{25} -6.46410 q^{26} +1.53590 q^{29} -8.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} +10.6603 q^{37} +0.732051 q^{38} +0.267949 q^{40} -2.53590 q^{41} -1.46410 q^{43} +6.19615 q^{44} +4.19615 q^{46} -4.73205 q^{47} +4.92820 q^{50} +6.46410 q^{52} -9.46410 q^{53} -1.66025 q^{55} -1.53590 q^{58} -4.19615 q^{59} -3.92820 q^{61} +8.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.73205 q^{67} -7.00000 q^{68} +6.53590 q^{71} -8.26795 q^{73} -10.6603 q^{74} -0.732051 q^{76} -9.12436 q^{79} -0.267949 q^{80} +2.53590 q^{82} -16.5885 q^{83} +1.87564 q^{85} +1.46410 q^{86} -6.19615 q^{88} -9.92820 q^{89} -4.19615 q^{92} +4.73205 q^{94} +0.196152 q^{95} -10.9282 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 4 q^{37} - 2 q^{38} + 4 q^{40} - 12 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{47} - 4 q^{50} + 6 q^{52} - 12 q^{53} + 14 q^{55} - 10 q^{58} + 2 q^{59} + 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 20 q^{71} - 20 q^{73} - 4 q^{74} + 2 q^{76} + 6 q^{79} - 4 q^{80} + 12 q^{82} - 2 q^{83} + 28 q^{85} - 4 q^{86} - 2 q^{88} - 6 q^{89} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 + 2 * q^11 + 6 * q^13 + 2 * q^16 - 14 * q^17 + 2 * q^19 - 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 - 6 * q^26 + 10 * q^29 - 6 * q^31 - 2 * q^32 + 14 * q^34 + 4 * q^37 - 2 * q^38 + 4 * q^40 - 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^47 - 4 * q^50 + 6 * q^52 - 12 * q^53 + 14 * q^55 - 10 * q^58 + 2 * q^59 + 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 20 * q^71 - 20 * q^73 - 4 * q^74 + 2 * q^76 + 6 * q^79 - 4 * q^80 + 12 * q^82 - 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 - 6 * q^89 + 2 * q^92 + 6 * q^94 - 10 * q^95 - 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −0.267949 −0.119831 −0.0599153 0.998203i $$-0.519083\pi$$
−0.0599153 + 0.998203i $$0.519083\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0.267949 0.0847330
$$11$$ 6.19615 1.86821 0.934105 0.356998i $$-0.116200\pi$$
0.934105 + 0.356998i $$0.116200\pi$$
$$12$$ 0 0
$$13$$ 6.46410 1.79282 0.896410 0.443227i $$-0.146166\pi$$
0.896410 + 0.443227i $$0.146166\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 0 0
$$19$$ −0.732051 −0.167944 −0.0839720 0.996468i $$-0.526761\pi$$
−0.0839720 + 0.996468i $$0.526761\pi$$
$$20$$ −0.267949 −0.0599153
$$21$$ 0 0
$$22$$ −6.19615 −1.32102
$$23$$ −4.19615 −0.874958 −0.437479 0.899229i $$-0.644129\pi$$
−0.437479 + 0.899229i $$0.644129\pi$$
$$24$$ 0 0
$$25$$ −4.92820 −0.985641
$$26$$ −6.46410 −1.26771
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.53590 0.285209 0.142605 0.989780i $$-0.454452\pi$$
0.142605 + 0.989780i $$0.454452\pi$$
$$30$$ 0 0
$$31$$ −8.19615 −1.47207 −0.736036 0.676942i $$-0.763305\pi$$
−0.736036 + 0.676942i $$0.763305\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 7.00000 1.20049
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.6603 1.75253 0.876267 0.481825i $$-0.160026\pi$$
0.876267 + 0.481825i $$0.160026\pi$$
$$38$$ 0.732051 0.118754
$$39$$ 0 0
$$40$$ 0.267949 0.0423665
$$41$$ −2.53590 −0.396041 −0.198020 0.980198i $$-0.563451\pi$$
−0.198020 + 0.980198i $$0.563451\pi$$
$$42$$ 0 0
$$43$$ −1.46410 −0.223273 −0.111637 0.993749i $$-0.535609\pi$$
−0.111637 + 0.993749i $$0.535609\pi$$
$$44$$ 6.19615 0.934105
$$45$$ 0 0
$$46$$ 4.19615 0.618689
$$47$$ −4.73205 −0.690241 −0.345120 0.938558i $$-0.612162\pi$$
−0.345120 + 0.938558i $$0.612162\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 4.92820 0.696953
$$51$$ 0 0
$$52$$ 6.46410 0.896410
$$53$$ −9.46410 −1.29999 −0.649997 0.759937i $$-0.725230\pi$$
−0.649997 + 0.759937i $$0.725230\pi$$
$$54$$ 0 0
$$55$$ −1.66025 −0.223869
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −1.53590 −0.201673
$$59$$ −4.19615 −0.546293 −0.273146 0.961973i $$-0.588064\pi$$
−0.273146 + 0.961973i $$0.588064\pi$$
$$60$$ 0 0
$$61$$ −3.92820 −0.502955 −0.251477 0.967863i $$-0.580916\pi$$
−0.251477 + 0.967863i $$0.580916\pi$$
$$62$$ 8.19615 1.04091
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.73205 −0.214834
$$66$$ 0 0
$$67$$ −6.73205 −0.822451 −0.411225 0.911534i $$-0.634899\pi$$
−0.411225 + 0.911534i $$0.634899\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.53590 0.775668 0.387834 0.921729i $$-0.373223\pi$$
0.387834 + 0.921729i $$0.373223\pi$$
$$72$$ 0 0
$$73$$ −8.26795 −0.967690 −0.483845 0.875154i $$-0.660760\pi$$
−0.483845 + 0.875154i $$0.660760\pi$$
$$74$$ −10.6603 −1.23923
$$75$$ 0 0
$$76$$ −0.732051 −0.0839720
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −9.12436 −1.02657 −0.513285 0.858218i $$-0.671572\pi$$
−0.513285 + 0.858218i $$0.671572\pi$$
$$80$$ −0.267949 −0.0299576
$$81$$ 0 0
$$82$$ 2.53590 0.280043
$$83$$ −16.5885 −1.82082 −0.910410 0.413707i $$-0.864234\pi$$
−0.910410 + 0.413707i $$0.864234\pi$$
$$84$$ 0 0
$$85$$ 1.87564 0.203442
$$86$$ 1.46410 0.157878
$$87$$ 0 0
$$88$$ −6.19615 −0.660512
$$89$$ −9.92820 −1.05239 −0.526194 0.850365i $$-0.676381\pi$$
−0.526194 + 0.850365i $$0.676381\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −4.19615 −0.437479
$$93$$ 0 0
$$94$$ 4.73205 0.488074
$$95$$ 0.196152 0.0201248
$$96$$ 0 0
$$97$$ −10.9282 −1.10959 −0.554795 0.831987i $$-0.687203\pi$$
−0.554795 + 0.831987i $$0.687203\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −4.92820 −0.492820
$$101$$ −8.92820 −0.888389 −0.444195 0.895930i $$-0.646510\pi$$
−0.444195 + 0.895930i $$0.646510\pi$$
$$102$$ 0 0
$$103$$ 8.39230 0.826918 0.413459 0.910523i $$-0.364320\pi$$
0.413459 + 0.910523i $$0.364320\pi$$
$$104$$ −6.46410 −0.633857
$$105$$ 0 0
$$106$$ 9.46410 0.919235
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 3.19615 0.306136 0.153068 0.988216i $$-0.451085\pi$$
0.153068 + 0.988216i $$0.451085\pi$$
$$110$$ 1.66025 0.158299
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −5.73205 −0.539226 −0.269613 0.962969i $$-0.586896\pi$$
−0.269613 + 0.962969i $$0.586896\pi$$
$$114$$ 0 0
$$115$$ 1.12436 0.104847
$$116$$ 1.53590 0.142605
$$117$$ 0 0
$$118$$ 4.19615 0.386287
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 27.3923 2.49021
$$122$$ 3.92820 0.355643
$$123$$ 0 0
$$124$$ −8.19615 −0.736036
$$125$$ 2.66025 0.237940
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 1.73205 0.151911
$$131$$ −10.5359 −0.920526 −0.460263 0.887783i $$-0.652245\pi$$
−0.460263 + 0.887783i $$0.652245\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 6.73205 0.581561
$$135$$ 0 0
$$136$$ 7.00000 0.600245
$$137$$ −8.26795 −0.706379 −0.353189 0.935552i $$-0.614903\pi$$
−0.353189 + 0.935552i $$0.614903\pi$$
$$138$$ 0 0
$$139$$ 3.26795 0.277184 0.138592 0.990350i $$-0.455742\pi$$
0.138592 + 0.990350i $$0.455742\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −6.53590 −0.548480
$$143$$ 40.0526 3.34936
$$144$$ 0 0
$$145$$ −0.411543 −0.0341768
$$146$$ 8.26795 0.684260
$$147$$ 0 0
$$148$$ 10.6603 0.876267
$$149$$ 9.00000 0.737309 0.368654 0.929567i $$-0.379819\pi$$
0.368654 + 0.929567i $$0.379819\pi$$
$$150$$ 0 0
$$151$$ −5.80385 −0.472310 −0.236155 0.971715i $$-0.575887\pi$$
−0.236155 + 0.971715i $$0.575887\pi$$
$$152$$ 0.732051 0.0593772
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2.19615 0.176399
$$156$$ 0 0
$$157$$ 1.00000 0.0798087 0.0399043 0.999204i $$-0.487295\pi$$
0.0399043 + 0.999204i $$0.487295\pi$$
$$158$$ 9.12436 0.725895
$$159$$ 0 0
$$160$$ 0.267949 0.0211832
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −13.4641 −1.05459 −0.527295 0.849682i $$-0.676794\pi$$
−0.527295 + 0.849682i $$0.676794\pi$$
$$164$$ −2.53590 −0.198020
$$165$$ 0 0
$$166$$ 16.5885 1.28751
$$167$$ 1.80385 0.139586 0.0697930 0.997561i $$-0.477766\pi$$
0.0697930 + 0.997561i $$0.477766\pi$$
$$168$$ 0 0
$$169$$ 28.7846 2.21420
$$170$$ −1.87564 −0.143855
$$171$$ 0 0
$$172$$ −1.46410 −0.111637
$$173$$ −6.26795 −0.476543 −0.238272 0.971199i $$-0.576581\pi$$
−0.238272 + 0.971199i $$0.576581\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.19615 0.467053
$$177$$ 0 0
$$178$$ 9.92820 0.744150
$$179$$ −2.19615 −0.164148 −0.0820741 0.996626i $$-0.526154\pi$$
−0.0820741 + 0.996626i $$0.526154\pi$$
$$180$$ 0 0
$$181$$ 16.3923 1.21843 0.609215 0.793005i $$-0.291485\pi$$
0.609215 + 0.793005i $$0.291485\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4.19615 0.309344
$$185$$ −2.85641 −0.210007
$$186$$ 0 0
$$187$$ −43.3731 −3.17175
$$188$$ −4.73205 −0.345120
$$189$$ 0 0
$$190$$ −0.196152 −0.0142304
$$191$$ −5.66025 −0.409562 −0.204781 0.978808i $$-0.565648\pi$$
−0.204781 + 0.978808i $$0.565648\pi$$
$$192$$ 0 0
$$193$$ 18.8564 1.35731 0.678657 0.734455i $$-0.262562\pi$$
0.678657 + 0.734455i $$0.262562\pi$$
$$194$$ 10.9282 0.784599
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 15.7846 1.12461 0.562303 0.826931i $$-0.309915\pi$$
0.562303 + 0.826931i $$0.309915\pi$$
$$198$$ 0 0
$$199$$ 19.1244 1.35569 0.677845 0.735205i $$-0.262914\pi$$
0.677845 + 0.735205i $$0.262914\pi$$
$$200$$ 4.92820 0.348477
$$201$$ 0 0
$$202$$ 8.92820 0.628186
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0.679492 0.0474578
$$206$$ −8.39230 −0.584720
$$207$$ 0 0
$$208$$ 6.46410 0.448205
$$209$$ −4.53590 −0.313755
$$210$$ 0 0
$$211$$ −17.2679 −1.18877 −0.594387 0.804179i $$-0.702605\pi$$
−0.594387 + 0.804179i $$0.702605\pi$$
$$212$$ −9.46410 −0.649997
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0.392305 0.0267550
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −3.19615 −0.216471
$$219$$ 0 0
$$220$$ −1.66025 −0.111934
$$221$$ −45.2487 −3.04376
$$222$$ 0 0
$$223$$ 25.4641 1.70520 0.852601 0.522562i $$-0.175024\pi$$
0.852601 + 0.522562i $$0.175024\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 5.73205 0.381290
$$227$$ 18.9282 1.25631 0.628154 0.778089i $$-0.283811\pi$$
0.628154 + 0.778089i $$0.283811\pi$$
$$228$$ 0 0
$$229$$ 2.46410 0.162832 0.0814162 0.996680i $$-0.474056\pi$$
0.0814162 + 0.996680i $$0.474056\pi$$
$$230$$ −1.12436 −0.0741378
$$231$$ 0 0
$$232$$ −1.53590 −0.100837
$$233$$ −2.80385 −0.183686 −0.0918431 0.995773i $$-0.529276\pi$$
−0.0918431 + 0.995773i $$0.529276\pi$$
$$234$$ 0 0
$$235$$ 1.26795 0.0827119
$$236$$ −4.19615 −0.273146
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.0526 −0.650246 −0.325123 0.945672i $$-0.605406\pi$$
−0.325123 + 0.945672i $$0.605406\pi$$
$$240$$ 0 0
$$241$$ 14.2679 0.919079 0.459540 0.888157i $$-0.348014\pi$$
0.459540 + 0.888157i $$0.348014\pi$$
$$242$$ −27.3923 −1.76084
$$243$$ 0 0
$$244$$ −3.92820 −0.251477
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.73205 −0.301093
$$248$$ 8.19615 0.520456
$$249$$ 0 0
$$250$$ −2.66025 −0.168249
$$251$$ 22.0526 1.39195 0.695973 0.718068i $$-0.254973\pi$$
0.695973 + 0.718068i $$0.254973\pi$$
$$252$$ 0 0
$$253$$ −26.0000 −1.63461
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.46410 −0.403220 −0.201610 0.979466i $$-0.564617\pi$$
−0.201610 + 0.979466i $$0.564617\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1.73205 −0.107417
$$261$$ 0 0
$$262$$ 10.5359 0.650910
$$263$$ −6.33975 −0.390925 −0.195463 0.980711i $$-0.562621\pi$$
−0.195463 + 0.980711i $$0.562621\pi$$
$$264$$ 0 0
$$265$$ 2.53590 0.155779
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −6.73205 −0.411225
$$269$$ −5.58846 −0.340734 −0.170367 0.985381i $$-0.554495\pi$$
−0.170367 + 0.985381i $$0.554495\pi$$
$$270$$ 0 0
$$271$$ 19.5167 1.18555 0.592776 0.805367i $$-0.298032\pi$$
0.592776 + 0.805367i $$0.298032\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 0 0
$$274$$ 8.26795 0.499485
$$275$$ −30.5359 −1.84138
$$276$$ 0 0
$$277$$ −18.7846 −1.12866 −0.564329 0.825550i $$-0.690865\pi$$
−0.564329 + 0.825550i $$0.690865\pi$$
$$278$$ −3.26795 −0.195999
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.1962 −0.787216 −0.393608 0.919278i $$-0.628773\pi$$
−0.393608 + 0.919278i $$0.628773\pi$$
$$282$$ 0 0
$$283$$ −15.3205 −0.910710 −0.455355 0.890310i $$-0.650488\pi$$
−0.455355 + 0.890310i $$0.650488\pi$$
$$284$$ 6.53590 0.387834
$$285$$ 0 0
$$286$$ −40.0526 −2.36836
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0.411543 0.0241666
$$291$$ 0 0
$$292$$ −8.26795 −0.483845
$$293$$ −3.33975 −0.195110 −0.0975550 0.995230i $$-0.531102\pi$$
−0.0975550 + 0.995230i $$0.531102\pi$$
$$294$$ 0 0
$$295$$ 1.12436 0.0654625
$$296$$ −10.6603 −0.619615
$$297$$ 0 0
$$298$$ −9.00000 −0.521356
$$299$$ −27.1244 −1.56864
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 5.80385 0.333974
$$303$$ 0 0
$$304$$ −0.732051 −0.0419860
$$305$$ 1.05256 0.0602693
$$306$$ 0 0
$$307$$ 21.8564 1.24741 0.623706 0.781659i $$-0.285626\pi$$
0.623706 + 0.781659i $$0.285626\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2.19615 −0.124733
$$311$$ −10.1962 −0.578171 −0.289085 0.957303i $$-0.593351\pi$$
−0.289085 + 0.957303i $$0.593351\pi$$
$$312$$ 0 0
$$313$$ −25.5885 −1.44635 −0.723173 0.690667i $$-0.757317\pi$$
−0.723173 + 0.690667i $$0.757317\pi$$
$$314$$ −1.00000 −0.0564333
$$315$$ 0 0
$$316$$ −9.12436 −0.513285
$$317$$ −31.3923 −1.76317 −0.881584 0.472028i $$-0.843522\pi$$
−0.881584 + 0.472028i $$0.843522\pi$$
$$318$$ 0 0
$$319$$ 9.51666 0.532831
$$320$$ −0.267949 −0.0149788
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.12436 0.285127
$$324$$ 0 0
$$325$$ −31.8564 −1.76708
$$326$$ 13.4641 0.745708
$$327$$ 0 0
$$328$$ 2.53590 0.140022
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.3923 0.681143 0.340571 0.940219i $$-0.389380\pi$$
0.340571 + 0.940219i $$0.389380\pi$$
$$332$$ −16.5885 −0.910410
$$333$$ 0 0
$$334$$ −1.80385 −0.0987021
$$335$$ 1.80385 0.0985547
$$336$$ 0 0
$$337$$ −16.3923 −0.892946 −0.446473 0.894797i $$-0.647320\pi$$
−0.446473 + 0.894797i $$0.647320\pi$$
$$338$$ −28.7846 −1.56568
$$339$$ 0 0
$$340$$ 1.87564 0.101721
$$341$$ −50.7846 −2.75014
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 1.46410 0.0789391
$$345$$ 0 0
$$346$$ 6.26795 0.336967
$$347$$ 21.4641 1.15225 0.576127 0.817360i $$-0.304563\pi$$
0.576127 + 0.817360i $$0.304563\pi$$
$$348$$ 0 0
$$349$$ −1.46410 −0.0783716 −0.0391858 0.999232i $$-0.512476\pi$$
−0.0391858 + 0.999232i $$0.512476\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −6.19615 −0.330256
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −1.75129 −0.0929488
$$356$$ −9.92820 −0.526194
$$357$$ 0 0
$$358$$ 2.19615 0.116070
$$359$$ −10.9282 −0.576769 −0.288384 0.957515i $$-0.593118\pi$$
−0.288384 + 0.957515i $$0.593118\pi$$
$$360$$ 0 0
$$361$$ −18.4641 −0.971795
$$362$$ −16.3923 −0.861560
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.21539 0.115959
$$366$$ 0 0
$$367$$ −11.1244 −0.580687 −0.290343 0.956923i $$-0.593770\pi$$
−0.290343 + 0.956923i $$0.593770\pi$$
$$368$$ −4.19615 −0.218740
$$369$$ 0 0
$$370$$ 2.85641 0.148498
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.14359 −0.318103 −0.159052 0.987270i $$-0.550844\pi$$
−0.159052 + 0.987270i $$0.550844\pi$$
$$374$$ 43.3731 2.24277
$$375$$ 0 0
$$376$$ 4.73205 0.244037
$$377$$ 9.92820 0.511328
$$378$$ 0 0
$$379$$ −27.5167 −1.41344 −0.706718 0.707495i $$-0.749825\pi$$
−0.706718 + 0.707495i $$0.749825\pi$$
$$380$$ 0.196152 0.0100624
$$381$$ 0 0
$$382$$ 5.66025 0.289604
$$383$$ −19.7128 −1.00728 −0.503639 0.863914i $$-0.668006\pi$$
−0.503639 + 0.863914i $$0.668006\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.8564 −0.959766
$$387$$ 0 0
$$388$$ −10.9282 −0.554795
$$389$$ −13.4641 −0.682657 −0.341329 0.939944i $$-0.610877\pi$$
−0.341329 + 0.939944i $$0.610877\pi$$
$$390$$ 0 0
$$391$$ 29.3731 1.48546
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −15.7846 −0.795217
$$395$$ 2.44486 0.123014
$$396$$ 0 0
$$397$$ 21.0000 1.05396 0.526980 0.849878i $$-0.323324\pi$$
0.526980 + 0.849878i $$0.323324\pi$$
$$398$$ −19.1244 −0.958617
$$399$$ 0 0
$$400$$ −4.92820 −0.246410
$$401$$ 10.5167 0.525177 0.262588 0.964908i $$-0.415424\pi$$
0.262588 + 0.964908i $$0.415424\pi$$
$$402$$ 0 0
$$403$$ −52.9808 −2.63916
$$404$$ −8.92820 −0.444195
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 66.0526 3.27410
$$408$$ 0 0
$$409$$ −17.3397 −0.857395 −0.428698 0.903448i $$-0.641027\pi$$
−0.428698 + 0.903448i $$0.641027\pi$$
$$410$$ −0.679492 −0.0335577
$$411$$ 0 0
$$412$$ 8.39230 0.413459
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4.44486 0.218190
$$416$$ −6.46410 −0.316929
$$417$$ 0 0
$$418$$ 4.53590 0.221858
$$419$$ 9.46410 0.462352 0.231176 0.972912i $$-0.425743\pi$$
0.231176 + 0.972912i $$0.425743\pi$$
$$420$$ 0 0
$$421$$ 0.124356 0.00606072 0.00303036 0.999995i $$-0.499035\pi$$
0.00303036 + 0.999995i $$0.499035\pi$$
$$422$$ 17.2679 0.840591
$$423$$ 0 0
$$424$$ 9.46410 0.459617
$$425$$ 34.4974 1.67337
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −0.392305 −0.0189186
$$431$$ −14.5359 −0.700170 −0.350085 0.936718i $$-0.613847\pi$$
−0.350085 + 0.936718i $$0.613847\pi$$
$$432$$ 0 0
$$433$$ −15.7321 −0.756034 −0.378017 0.925799i $$-0.623394\pi$$
−0.378017 + 0.925799i $$0.623394\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 3.19615 0.153068
$$437$$ 3.07180 0.146944
$$438$$ 0 0
$$439$$ −23.3205 −1.11303 −0.556514 0.830839i $$-0.687861\pi$$
−0.556514 + 0.830839i $$0.687861\pi$$
$$440$$ 1.66025 0.0791495
$$441$$ 0 0
$$442$$ 45.2487 2.15226
$$443$$ −15.2679 −0.725402 −0.362701 0.931906i $$-0.618145\pi$$
−0.362701 + 0.931906i $$0.618145\pi$$
$$444$$ 0 0
$$445$$ 2.66025 0.126108
$$446$$ −25.4641 −1.20576
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15.8564 0.748310 0.374155 0.927366i $$-0.377933\pi$$
0.374155 + 0.927366i $$0.377933\pi$$
$$450$$ 0 0
$$451$$ −15.7128 −0.739887
$$452$$ −5.73205 −0.269613
$$453$$ 0 0
$$454$$ −18.9282 −0.888345
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.85641 −0.320729 −0.160365 0.987058i $$-0.551267\pi$$
−0.160365 + 0.987058i $$0.551267\pi$$
$$458$$ −2.46410 −0.115140
$$459$$ 0 0
$$460$$ 1.12436 0.0524234
$$461$$ 6.78461 0.315991 0.157995 0.987440i $$-0.449497\pi$$
0.157995 + 0.987440i $$0.449497\pi$$
$$462$$ 0 0
$$463$$ −1.41154 −0.0656000 −0.0328000 0.999462i $$-0.510442\pi$$
−0.0328000 + 0.999462i $$0.510442\pi$$
$$464$$ 1.53590 0.0713023
$$465$$ 0 0
$$466$$ 2.80385 0.129886
$$467$$ 16.5885 0.767622 0.383811 0.923412i $$-0.374611\pi$$
0.383811 + 0.923412i $$0.374611\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −1.26795 −0.0584861
$$471$$ 0 0
$$472$$ 4.19615 0.193144
$$473$$ −9.07180 −0.417122
$$474$$ 0 0
$$475$$ 3.60770 0.165532
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 10.0526 0.459793
$$479$$ 21.5167 0.983121 0.491561 0.870843i $$-0.336427\pi$$
0.491561 + 0.870843i $$0.336427\pi$$
$$480$$ 0 0
$$481$$ 68.9090 3.14198
$$482$$ −14.2679 −0.649887
$$483$$ 0 0
$$484$$ 27.3923 1.24510
$$485$$ 2.92820 0.132963
$$486$$ 0 0
$$487$$ 2.58846 0.117294 0.0586471 0.998279i $$-0.481321\pi$$
0.0586471 + 0.998279i $$0.481321\pi$$
$$488$$ 3.92820 0.177821
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 26.5359 1.19755 0.598774 0.800918i $$-0.295655\pi$$
0.598774 + 0.800918i $$0.295655\pi$$
$$492$$ 0 0
$$493$$ −10.7513 −0.484214
$$494$$ 4.73205 0.212905
$$495$$ 0 0
$$496$$ −8.19615 −0.368018
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 19.8038 0.886542 0.443271 0.896388i $$-0.353818\pi$$
0.443271 + 0.896388i $$0.353818\pi$$
$$500$$ 2.66025 0.118970
$$501$$ 0 0
$$502$$ −22.0526 −0.984254
$$503$$ −40.0526 −1.78586 −0.892928 0.450200i $$-0.851353\pi$$
−0.892928 + 0.450200i $$0.851353\pi$$
$$504$$ 0 0
$$505$$ 2.39230 0.106456
$$506$$ 26.0000 1.15584
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ −31.8564 −1.41201 −0.706005 0.708207i $$-0.749504\pi$$
−0.706005 + 0.708207i $$0.749504\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 6.46410 0.285119
$$515$$ −2.24871 −0.0990901
$$516$$ 0 0
$$517$$ −29.3205 −1.28951
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1.73205 0.0759555
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −33.1769 −1.45073 −0.725363 0.688367i $$-0.758328\pi$$
−0.725363 + 0.688367i $$0.758328\pi$$
$$524$$ −10.5359 −0.460263
$$525$$ 0 0
$$526$$ 6.33975 0.276426
$$527$$ 57.3731 2.49921
$$528$$ 0 0
$$529$$ −5.39230 −0.234448
$$530$$ −2.53590 −0.110152
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −16.3923 −0.710030
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.73205 0.290780
$$537$$ 0 0
$$538$$ 5.58846 0.240936
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 3.33975 0.143587 0.0717934 0.997420i $$-0.477128\pi$$
0.0717934 + 0.997420i $$0.477128\pi$$
$$542$$ −19.5167 −0.838312
$$543$$ 0 0
$$544$$ 7.00000 0.300123
$$545$$ −0.856406 −0.0366844
$$546$$ 0 0
$$547$$ −22.7321 −0.971952 −0.485976 0.873972i $$-0.661536\pi$$
−0.485976 + 0.873972i $$0.661536\pi$$
$$548$$ −8.26795 −0.353189
$$549$$ 0 0
$$550$$ 30.5359 1.30206
$$551$$ −1.12436 −0.0478992
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 18.7846 0.798082
$$555$$ 0 0
$$556$$ 3.26795 0.138592
$$557$$ 23.9282 1.01387 0.506935 0.861984i $$-0.330778\pi$$
0.506935 + 0.861984i $$0.330778\pi$$
$$558$$ 0 0
$$559$$ −9.46410 −0.400289
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 13.1962 0.556646
$$563$$ −19.7128 −0.830796 −0.415398 0.909640i $$-0.636358\pi$$
−0.415398 + 0.909640i $$0.636358\pi$$
$$564$$ 0 0
$$565$$ 1.53590 0.0646157
$$566$$ 15.3205 0.643969
$$567$$ 0 0
$$568$$ −6.53590 −0.274240
$$569$$ 23.1962 0.972433 0.486217 0.873838i $$-0.338377\pi$$
0.486217 + 0.873838i $$0.338377\pi$$
$$570$$ 0 0
$$571$$ −22.7321 −0.951307 −0.475653 0.879633i $$-0.657788\pi$$
−0.475653 + 0.879633i $$0.657788\pi$$
$$572$$ 40.0526 1.67468
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 20.6795 0.862394
$$576$$ 0 0
$$577$$ −24.6603 −1.02662 −0.513310 0.858203i $$-0.671581\pi$$
−0.513310 + 0.858203i $$0.671581\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ 0 0
$$580$$ −0.411543 −0.0170884
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −58.6410 −2.42866
$$584$$ 8.26795 0.342130
$$585$$ 0 0
$$586$$ 3.33975 0.137964
$$587$$ −14.7321 −0.608057 −0.304028 0.952663i $$-0.598332\pi$$
−0.304028 + 0.952663i $$0.598332\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ −1.12436 −0.0462890
$$591$$ 0 0
$$592$$ 10.6603 0.438134
$$593$$ 22.1769 0.910697 0.455348 0.890313i $$-0.349515\pi$$
0.455348 + 0.890313i $$0.349515\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 9.00000 0.368654
$$597$$ 0 0
$$598$$ 27.1244 1.10920
$$599$$ −15.1244 −0.617964 −0.308982 0.951068i $$-0.599988\pi$$
−0.308982 + 0.951068i $$0.599988\pi$$
$$600$$ 0 0
$$601$$ −19.1962 −0.783027 −0.391514 0.920172i $$-0.628048\pi$$
−0.391514 + 0.920172i $$0.628048\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −5.80385 −0.236155
$$605$$ −7.33975 −0.298403
$$606$$ 0 0
$$607$$ 24.5885 0.998015 0.499007 0.866598i $$-0.333698\pi$$
0.499007 + 0.866598i $$0.333698\pi$$
$$608$$ 0.732051 0.0296886
$$609$$ 0 0
$$610$$ −1.05256 −0.0426169
$$611$$ −30.5885 −1.23748
$$612$$ 0 0
$$613$$ 26.7846 1.08182 0.540910 0.841080i $$-0.318080\pi$$
0.540910 + 0.841080i $$0.318080\pi$$
$$614$$ −21.8564 −0.882053
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11.9808 0.482327 0.241164 0.970484i $$-0.422471\pi$$
0.241164 + 0.970484i $$0.422471\pi$$
$$618$$ 0 0
$$619$$ 23.7128 0.953098 0.476549 0.879148i $$-0.341887\pi$$
0.476549 + 0.879148i $$0.341887\pi$$
$$620$$ 2.19615 0.0881996
$$621$$ 0 0
$$622$$ 10.1962 0.408828
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 23.9282 0.957128
$$626$$ 25.5885 1.02272
$$627$$ 0 0
$$628$$ 1.00000 0.0399043
$$629$$ −74.6218 −2.97537
$$630$$ 0 0
$$631$$ −3.66025 −0.145712 −0.0728562 0.997342i $$-0.523211\pi$$
−0.0728562 + 0.997342i $$0.523211\pi$$
$$632$$ 9.12436 0.362947
$$633$$ 0 0
$$634$$ 31.3923 1.24675
$$635$$ −3.21539 −0.127599
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −9.51666 −0.376768
$$639$$ 0 0
$$640$$ 0.267949 0.0105916
$$641$$ 39.4449 1.55798 0.778989 0.627037i $$-0.215733\pi$$
0.778989 + 0.627037i $$0.215733\pi$$
$$642$$ 0 0
$$643$$ 9.41154 0.371155 0.185578 0.982630i $$-0.440584\pi$$
0.185578 + 0.982630i $$0.440584\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −5.12436 −0.201615
$$647$$ 4.39230 0.172679 0.0863397 0.996266i $$-0.472483\pi$$
0.0863397 + 0.996266i $$0.472483\pi$$
$$648$$ 0 0
$$649$$ −26.0000 −1.02059
$$650$$ 31.8564 1.24951
$$651$$ 0 0
$$652$$ −13.4641 −0.527295
$$653$$ −30.2487 −1.18372 −0.591862 0.806039i $$-0.701607\pi$$
−0.591862 + 0.806039i $$0.701607\pi$$
$$654$$ 0 0
$$655$$ 2.82309 0.110307
$$656$$ −2.53590 −0.0990102
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.3923 −1.41764 −0.708821 0.705388i $$-0.750773\pi$$
−0.708821 + 0.705388i $$0.750773\pi$$
$$660$$ 0 0
$$661$$ 12.8564 0.500056 0.250028 0.968239i $$-0.419560\pi$$
0.250028 + 0.968239i $$0.419560\pi$$
$$662$$ −12.3923 −0.481641
$$663$$ 0 0
$$664$$ 16.5885 0.643757
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.44486 −0.249546
$$668$$ 1.80385 0.0697930
$$669$$ 0 0
$$670$$ −1.80385 −0.0696887
$$671$$ −24.3397 −0.939625
$$672$$ 0 0
$$673$$ −18.3205 −0.706204 −0.353102 0.935585i $$-0.614873\pi$$
−0.353102 + 0.935585i $$0.614873\pi$$
$$674$$ 16.3923 0.631408
$$675$$ 0 0
$$676$$ 28.7846 1.10710
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −1.87564 −0.0719277
$$681$$ 0 0
$$682$$ 50.7846 1.94464
$$683$$ −1.85641 −0.0710334 −0.0355167 0.999369i $$-0.511308\pi$$
−0.0355167 + 0.999369i $$0.511308\pi$$
$$684$$ 0 0
$$685$$ 2.21539 0.0846457
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −1.46410 −0.0558184
$$689$$ −61.1769 −2.33065
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −6.26795 −0.238272
$$693$$ 0 0
$$694$$ −21.4641 −0.814766
$$695$$ −0.875644 −0.0332151
$$696$$ 0 0
$$697$$ 17.7513 0.672378
$$698$$ 1.46410 0.0554171
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 27.3923 1.03459 0.517297 0.855806i $$-0.326938\pi$$
0.517297 + 0.855806i $$0.326938\pi$$
$$702$$ 0 0
$$703$$ −7.80385 −0.294328
$$704$$ 6.19615 0.233526
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −3.87564 −0.145553 −0.0727764 0.997348i $$-0.523186\pi$$
−0.0727764 + 0.997348i $$0.523186\pi$$
$$710$$ 1.75129 0.0657247
$$711$$ 0 0
$$712$$ 9.92820 0.372075
$$713$$ 34.3923 1.28800
$$714$$ 0 0
$$715$$ −10.7321 −0.401356
$$716$$ −2.19615 −0.0820741
$$717$$ 0 0
$$718$$ 10.9282 0.407837
$$719$$ 9.46410 0.352951 0.176476 0.984305i $$-0.443530\pi$$
0.176476 + 0.984305i $$0.443530\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 18.4641 0.687163
$$723$$ 0 0
$$724$$ 16.3923 0.609215
$$725$$ −7.56922 −0.281114
$$726$$ 0 0
$$727$$ −51.3205 −1.90337 −0.951686 0.307072i $$-0.900651\pi$$
−0.951686 + 0.307072i $$0.900651\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2.21539 −0.0819953
$$731$$ 10.2487 0.379062
$$732$$ 0 0
$$733$$ −47.3205 −1.74782 −0.873911 0.486085i $$-0.838424\pi$$
−0.873911 + 0.486085i $$0.838424\pi$$
$$734$$ 11.1244 0.410607
$$735$$ 0 0
$$736$$ 4.19615 0.154672
$$737$$ −41.7128 −1.53651
$$738$$ 0 0
$$739$$ −13.2679 −0.488069 −0.244035 0.969767i $$-0.578471\pi$$
−0.244035 + 0.969767i $$0.578471\pi$$
$$740$$ −2.85641 −0.105004
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −40.3923 −1.48185 −0.740925 0.671588i $$-0.765613\pi$$
−0.740925 + 0.671588i $$0.765613\pi$$
$$744$$ 0 0
$$745$$ −2.41154 −0.0883521
$$746$$ 6.14359 0.224933
$$747$$ 0 0
$$748$$ −43.3731 −1.58588
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −22.1436 −0.808031 −0.404016 0.914752i $$-0.632386\pi$$
−0.404016 + 0.914752i $$0.632386\pi$$
$$752$$ −4.73205 −0.172560
$$753$$ 0 0
$$754$$ −9.92820 −0.361564
$$755$$ 1.55514 0.0565972
$$756$$ 0 0
$$757$$ 20.7846 0.755429 0.377715 0.925922i $$-0.376710\pi$$
0.377715 + 0.925922i $$0.376710\pi$$
$$758$$ 27.5167 0.999450
$$759$$ 0 0
$$760$$ −0.196152 −0.00711520
$$761$$ −37.0000 −1.34125 −0.670624 0.741797i $$-0.733974\pi$$
−0.670624 + 0.741797i $$0.733974\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −5.66025 −0.204781
$$765$$ 0 0
$$766$$ 19.7128 0.712253
$$767$$ −27.1244 −0.979404
$$768$$ 0 0
$$769$$ 4.41154 0.159084 0.0795421 0.996832i $$-0.474654\pi$$
0.0795421 + 0.996832i $$0.474654\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18.8564 0.678657
$$773$$ 4.12436 0.148343 0.0741714 0.997246i $$-0.476369\pi$$
0.0741714 + 0.997246i $$0.476369\pi$$
$$774$$ 0 0
$$775$$ 40.3923 1.45093
$$776$$ 10.9282 0.392300
$$777$$ 0 0
$$778$$ 13.4641 0.482711
$$779$$ 1.85641 0.0665127
$$780$$ 0 0
$$781$$ 40.4974 1.44911
$$782$$ −29.3731 −1.05038
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −0.267949 −0.00956352
$$786$$ 0 0
$$787$$ 28.3923 1.01208 0.506038 0.862511i $$-0.331109\pi$$
0.506038 + 0.862511i $$0.331109\pi$$
$$788$$ 15.7846 0.562303
$$789$$ 0 0
$$790$$ −2.44486 −0.0869843
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −25.3923 −0.901707
$$794$$ −21.0000 −0.745262
$$795$$ 0 0
$$796$$ 19.1244 0.677845
$$797$$ 29.4449 1.04299 0.521495 0.853254i $$-0.325374\pi$$
0.521495 + 0.853254i $$0.325374\pi$$
$$798$$ 0 0
$$799$$ 33.1244 1.17186
$$800$$ 4.92820 0.174238
$$801$$ 0 0
$$802$$ −10.5167 −0.371356
$$803$$ −51.2295 −1.80785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 52.9808 1.86617
$$807$$ 0 0
$$808$$ 8.92820 0.314093
$$809$$ 32.1244 1.12943 0.564716 0.825285i $$-0.308986\pi$$
0.564716 + 0.825285i $$0.308986\pi$$
$$810$$ 0 0
$$811$$ −18.1962 −0.638953 −0.319477 0.947594i $$-0.603507\pi$$
−0.319477 + 0.947594i $$0.603507\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −66.0526 −2.31514
$$815$$ 3.60770 0.126372
$$816$$ 0 0
$$817$$ 1.07180 0.0374974
$$818$$ 17.3397 0.606270
$$819$$ 0 0
$$820$$ 0.679492 0.0237289
$$821$$ 25.9282 0.904901 0.452450 0.891790i $$-0.350550\pi$$
0.452450 + 0.891790i $$0.350550\pi$$
$$822$$ 0 0
$$823$$ 0.784610 0.0273498 0.0136749 0.999906i $$-0.495647\pi$$
0.0136749 + 0.999906i $$0.495647\pi$$
$$824$$ −8.39230 −0.292360
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −23.3205 −0.810934 −0.405467 0.914110i $$-0.632891\pi$$
−0.405467 + 0.914110i $$0.632891\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ −4.44486 −0.154283
$$831$$ 0 0
$$832$$ 6.46410 0.224102
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −0.483340 −0.0167267
$$836$$ −4.53590 −0.156877
$$837$$ 0 0
$$838$$ −9.46410 −0.326932
$$839$$ −1.46410 −0.0505464 −0.0252732 0.999681i $$-0.508046\pi$$
−0.0252732 + 0.999681i $$0.508046\pi$$
$$840$$ 0 0
$$841$$ −26.6410 −0.918656
$$842$$ −0.124356 −0.00428558
$$843$$ 0 0
$$844$$ −17.2679 −0.594387
$$845$$ −7.71281 −0.265329
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −9.46410 −0.324999
$$849$$ 0 0
$$850$$ −34.4974 −1.18325
$$851$$ −44.7321 −1.53339
$$852$$ 0 0
$$853$$ 49.7128 1.70213 0.851067 0.525057i $$-0.175956\pi$$
0.851067 + 0.525057i $$0.175956\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.8564 0.575804 0.287902 0.957660i $$-0.407042\pi$$
0.287902 + 0.957660i $$0.407042\pi$$
$$858$$ 0 0
$$859$$ −24.3923 −0.832255 −0.416127 0.909306i $$-0.636613\pi$$
−0.416127 + 0.909306i $$0.636613\pi$$
$$860$$ 0.392305 0.0133775
$$861$$ 0 0
$$862$$ 14.5359 0.495095
$$863$$ −7.12436 −0.242516 −0.121258 0.992621i $$-0.538693\pi$$
−0.121258 + 0.992621i $$0.538693\pi$$
$$864$$ 0 0
$$865$$ 1.67949 0.0571044
$$866$$ 15.7321 0.534597
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −56.5359 −1.91785
$$870$$ 0 0
$$871$$ −43.5167 −1.47451
$$872$$ −3.19615 −0.108235
$$873$$ 0 0
$$874$$ −3.07180 −0.103905
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36.5167 −1.23308 −0.616540 0.787324i $$-0.711466\pi$$
−0.616540 + 0.787324i $$0.711466\pi$$
$$878$$ 23.3205 0.787029
$$879$$ 0 0
$$880$$ −1.66025 −0.0559672
$$881$$ −30.2487 −1.01910 −0.509552 0.860440i $$-0.670189\pi$$
−0.509552 + 0.860440i $$0.670189\pi$$
$$882$$ 0 0
$$883$$ −9.66025 −0.325093 −0.162547 0.986701i $$-0.551971\pi$$
−0.162547 + 0.986701i $$0.551971\pi$$
$$884$$ −45.2487 −1.52188
$$885$$ 0 0
$$886$$ 15.2679 0.512937
$$887$$ 56.4449 1.89523 0.947617 0.319410i $$-0.103485\pi$$
0.947617 + 0.319410i $$0.103485\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −2.66025 −0.0891719
$$891$$ 0 0
$$892$$ 25.4641 0.852601
$$893$$ 3.46410 0.115922
$$894$$ 0 0
$$895$$ 0.588457 0.0196700
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −15.8564 −0.529135
$$899$$ −12.5885 −0.419849
$$900$$ 0 0
$$901$$ 66.2487 2.20706
$$902$$ 15.7128 0.523179
$$903$$ 0 0
$$904$$ 5.73205 0.190645
$$905$$ −4.39230 −0.146005
$$906$$ 0 0
$$907$$ −36.0000 −1.19536 −0.597680 0.801735i $$-0.703911\pi$$
−0.597680 + 0.801735i $$0.703911\pi$$
$$908$$ 18.9282 0.628154
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 42.2487 1.39976 0.699881 0.714259i $$-0.253236\pi$$
0.699881 + 0.714259i $$0.253236\pi$$
$$912$$ 0 0
$$913$$ −102.785 −3.40167
$$914$$ 6.85641 0.226790
$$915$$ 0 0
$$916$$ 2.46410 0.0814162
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 26.9808 0.890013 0.445007 0.895527i $$-0.353201\pi$$
0.445007 + 0.895527i $$0.353201\pi$$
$$920$$ −1.12436 −0.0370689
$$921$$ 0 0
$$922$$ −6.78461 −0.223439
$$923$$ 42.2487 1.39063
$$924$$ 0 0
$$925$$ −52.5359 −1.72737
$$926$$ 1.41154 0.0463862
$$927$$ 0 0
$$928$$ −1.53590 −0.0504183
$$929$$ 51.4974 1.68958 0.844788 0.535101i $$-0.179727\pi$$
0.844788 + 0.535101i $$0.179727\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2.80385 −0.0918431
$$933$$ 0 0
$$934$$ −16.5885 −0.542791
$$935$$ 11.6218 0.380073
$$936$$ 0 0
$$937$$ 25.8372 0.844064 0.422032 0.906581i $$-0.361317\pi$$
0.422032 + 0.906581i $$0.361317\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 1.26795 0.0413559
$$941$$ 34.1244 1.11242 0.556211 0.831041i $$-0.312255\pi$$
0.556211 + 0.831041i $$0.312255\pi$$
$$942$$ 0 0
$$943$$ 10.6410 0.346519
$$944$$ −4.19615 −0.136573
$$945$$ 0 0
$$946$$ 9.07180 0.294950
$$947$$ −46.2487 −1.50288 −0.751441 0.659801i $$-0.770641\pi$$
−0.751441 + 0.659801i $$0.770641\pi$$
$$948$$ 0 0
$$949$$ −53.4449 −1.73489
$$950$$ −3.60770 −0.117049
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −41.5885 −1.34718 −0.673591 0.739104i $$-0.735249\pi$$
−0.673591 + 0.739104i $$0.735249\pi$$
$$954$$ 0 0
$$955$$ 1.51666 0.0490780
$$956$$ −10.0526 −0.325123
$$957$$ 0 0
$$958$$ −21.5167 −0.695172
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 36.1769 1.16700
$$962$$ −68.9090 −2.22171
$$963$$ 0 0
$$964$$ 14.2679 0.459540
$$965$$ −5.05256 −0.162648
$$966$$ 0 0
$$967$$ −3.66025 −0.117706 −0.0588529 0.998267i $$-0.518744\pi$$
−0.0588529 + 0.998267i $$0.518744\pi$$
$$968$$ −27.3923 −0.880422
$$969$$ 0 0
$$970$$ −2.92820 −0.0940189
$$971$$ −8.87564 −0.284833 −0.142416 0.989807i $$-0.545487\pi$$
−0.142416 + 0.989807i $$0.545487\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −2.58846 −0.0829395
$$975$$ 0 0
$$976$$ −3.92820 −0.125739
$$977$$ 57.7128 1.84640 0.923198 0.384324i $$-0.125565\pi$$
0.923198 + 0.384324i $$0.125565\pi$$
$$978$$ 0 0
$$979$$ −61.5167 −1.96608
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −26.5359 −0.846795
$$983$$ −58.6410 −1.87036 −0.935179 0.354176i $$-0.884762\pi$$
−0.935179 + 0.354176i $$0.884762\pi$$
$$984$$ 0 0
$$985$$ −4.22947 −0.134762
$$986$$ 10.7513 0.342391
$$987$$ 0 0
$$988$$ −4.73205 −0.150547
$$989$$ 6.14359 0.195355
$$990$$ 0 0
$$991$$ −27.6603 −0.878657 −0.439328 0.898327i $$-0.644784\pi$$
−0.439328 + 0.898327i $$0.644784\pi$$
$$992$$ 8.19615 0.260228
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −5.12436 −0.162453
$$996$$ 0 0
$$997$$ −41.2487 −1.30636 −0.653180 0.757203i $$-0.726565\pi$$
−0.653180 + 0.757203i $$0.726565\pi$$
$$998$$ −19.8038 −0.626880
$$999$$ 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 7938.2.a.bg.1.2 2
3.2 odd 2 7938.2.a.bt.1.1 2
7.6 odd 2 1134.2.a.l.1.1 2
21.20 even 2 1134.2.a.m.1.2 yes 2
28.27 even 2 9072.2.a.bp.1.1 2
63.13 odd 6 1134.2.f.s.379.2 4
63.20 even 6 1134.2.f.r.757.1 4
63.34 odd 6 1134.2.f.s.757.2 4
63.41 even 6 1134.2.f.r.379.1 4
84.83 odd 2 9072.2.a.y.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 2 7.6 odd 2
1134.2.a.m.1.2 yes 2 21.20 even 2
1134.2.f.r.379.1 4 63.41 even 6
1134.2.f.r.757.1 4 63.20 even 6
1134.2.f.s.379.2 4 63.13 odd 6
1134.2.f.s.757.2 4 63.34 odd 6
7938.2.a.bg.1.2 2 1.1 even 1 trivial
7938.2.a.bt.1.1 2 3.2 odd 2
9072.2.a.y.1.2 2 84.83 odd 2
9072.2.a.bp.1.1 2 28.27 even 2