Properties

 Label 920.2.a.f.1.2 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.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: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q+2.56155 q^{3} +1.00000 q^{5} +5.12311 q^{7} +3.56155 q^{9} -4.00000 q^{11} -0.561553 q^{13} +2.56155 q^{15} -3.12311 q^{17} +4.00000 q^{19} +13.1231 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} -8.56155 q^{29} +1.43845 q^{31} -10.2462 q^{33} +5.12311 q^{35} -7.12311 q^{37} -1.43845 q^{39} +0.561553 q^{41} -9.12311 q^{43} +3.56155 q^{45} -3.68466 q^{47} +19.2462 q^{49} -8.00000 q^{51} -4.24621 q^{53} -4.00000 q^{55} +10.2462 q^{57} -6.24621 q^{59} +11.1231 q^{61} +18.2462 q^{63} -0.561553 q^{65} +6.24621 q^{67} +2.56155 q^{69} +3.68466 q^{71} +16.5616 q^{73} +2.56155 q^{75} -20.4924 q^{77} -10.2462 q^{79} -7.00000 q^{81} +12.0000 q^{83} -3.12311 q^{85} -21.9309 q^{87} +10.0000 q^{89} -2.87689 q^{91} +3.68466 q^{93} +4.00000 q^{95} -16.2462 q^{97} -14.2462 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 $$2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} + 7 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39} - 3 q^{41} - 10 q^{43} + 3 q^{45} + 5 q^{47} + 22 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{55} + 4 q^{57} + 4 q^{59} + 14 q^{61} + 20 q^{63} + 3 q^{65} - 4 q^{67} + q^{69} - 5 q^{71} + 29 q^{73} + q^{75} - 8 q^{77} - 4 q^{79} - 14 q^{81} + 24 q^{83} + 2 q^{85} - 15 q^{87} + 20 q^{89} - 14 q^{91} - 5 q^{93} + 8 q^{95} - 16 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 - 8 * q^11 + 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 + 18 * q^21 + 2 * q^23 + 2 * q^25 + 7 * q^27 - 13 * q^29 + 7 * q^31 - 4 * q^33 + 2 * q^35 - 6 * q^37 - 7 * q^39 - 3 * q^41 - 10 * q^43 + 3 * q^45 + 5 * q^47 + 22 * q^49 - 16 * q^51 + 8 * q^53 - 8 * q^55 + 4 * q^57 + 4 * q^59 + 14 * q^61 + 20 * q^63 + 3 * q^65 - 4 * q^67 + q^69 - 5 * q^71 + 29 * q^73 + q^75 - 8 * q^77 - 4 * q^79 - 14 * q^81 + 24 * q^83 + 2 * q^85 - 15 * q^87 + 20 * q^89 - 14 * q^91 - 5 * q^93 + 8 * q^95 - 16 * q^97 - 12 * 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
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 5.12311 1.93635 0.968176 0.250270i $$-0.0805195\pi$$
0.968176 + 0.250270i $$0.0805195\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −0.561553 −0.155747 −0.0778734 0.996963i $$-0.524813\pi$$
−0.0778734 + 0.996963i $$0.524813\pi$$
$$14$$ 0 0
$$15$$ 2.56155 0.661390
$$16$$ 0 0
$$17$$ −3.12311 −0.757464 −0.378732 0.925506i $$-0.623640\pi$$
−0.378732 + 0.925506i $$0.623640\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 13.1231 2.86370
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.43845 0.276829
$$28$$ 0 0
$$29$$ −8.56155 −1.58984 −0.794920 0.606714i $$-0.792487\pi$$
−0.794920 + 0.606714i $$0.792487\pi$$
$$30$$ 0 0
$$31$$ 1.43845 0.258353 0.129176 0.991622i $$-0.458767\pi$$
0.129176 + 0.991622i $$0.458767\pi$$
$$32$$ 0 0
$$33$$ −10.2462 −1.78364
$$34$$ 0 0
$$35$$ 5.12311 0.865963
$$36$$ 0 0
$$37$$ −7.12311 −1.17103 −0.585516 0.810661i $$-0.699108\pi$$
−0.585516 + 0.810661i $$0.699108\pi$$
$$38$$ 0 0
$$39$$ −1.43845 −0.230336
$$40$$ 0 0
$$41$$ 0.561553 0.0876998 0.0438499 0.999038i $$-0.486038\pi$$
0.0438499 + 0.999038i $$0.486038\pi$$
$$42$$ 0 0
$$43$$ −9.12311 −1.39126 −0.695630 0.718400i $$-0.744875\pi$$
−0.695630 + 0.718400i $$0.744875\pi$$
$$44$$ 0 0
$$45$$ 3.56155 0.530925
$$46$$ 0 0
$$47$$ −3.68466 −0.537463 −0.268731 0.963215i $$-0.586604\pi$$
−0.268731 + 0.963215i $$0.586604\pi$$
$$48$$ 0 0
$$49$$ 19.2462 2.74946
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ −4.24621 −0.583262 −0.291631 0.956531i $$-0.594198\pi$$
−0.291631 + 0.956531i $$0.594198\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 10.2462 1.35714
$$58$$ 0 0
$$59$$ −6.24621 −0.813187 −0.406594 0.913609i $$-0.633284\pi$$
−0.406594 + 0.913609i $$0.633284\pi$$
$$60$$ 0 0
$$61$$ 11.1231 1.42417 0.712084 0.702094i $$-0.247752\pi$$
0.712084 + 0.702094i $$0.247752\pi$$
$$62$$ 0 0
$$63$$ 18.2462 2.29881
$$64$$ 0 0
$$65$$ −0.561553 −0.0696521
$$66$$ 0 0
$$67$$ 6.24621 0.763096 0.381548 0.924349i $$-0.375391\pi$$
0.381548 + 0.924349i $$0.375391\pi$$
$$68$$ 0 0
$$69$$ 2.56155 0.308375
$$70$$ 0 0
$$71$$ 3.68466 0.437289 0.218644 0.975805i $$-0.429837\pi$$
0.218644 + 0.975805i $$0.429837\pi$$
$$72$$ 0 0
$$73$$ 16.5616 1.93838 0.969192 0.246308i $$-0.0792175\pi$$
0.969192 + 0.246308i $$0.0792175\pi$$
$$74$$ 0 0
$$75$$ 2.56155 0.295783
$$76$$ 0 0
$$77$$ −20.4924 −2.33533
$$78$$ 0 0
$$79$$ −10.2462 −1.15279 −0.576394 0.817172i $$-0.695541\pi$$
−0.576394 + 0.817172i $$0.695541\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −3.12311 −0.338748
$$86$$ 0 0
$$87$$ −21.9309 −2.35124
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −2.87689 −0.301580
$$92$$ 0 0
$$93$$ 3.68466 0.382081
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −16.2462 −1.64955 −0.824776 0.565459i $$-0.808699\pi$$
−0.824776 + 0.565459i $$0.808699\pi$$
$$98$$ 0 0
$$99$$ −14.2462 −1.43180
$$100$$ 0 0
$$101$$ 0.246211 0.0244989 0.0122495 0.999925i $$-0.496101\pi$$
0.0122495 + 0.999925i $$0.496101\pi$$
$$102$$ 0 0
$$103$$ −2.24621 −0.221326 −0.110663 0.993858i $$-0.535297\pi$$
−0.110663 + 0.993858i $$0.535297\pi$$
$$104$$ 0 0
$$105$$ 13.1231 1.28068
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 8.24621 0.789844 0.394922 0.918715i $$-0.370772\pi$$
0.394922 + 0.918715i $$0.370772\pi$$
$$110$$ 0 0
$$111$$ −18.2462 −1.73185
$$112$$ 0 0
$$113$$ 20.2462 1.90460 0.952302 0.305158i $$-0.0987093\pi$$
0.952302 + 0.305158i $$0.0987093\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 1.43845 0.129700
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.8078 −1.49145 −0.745724 0.666255i $$-0.767896\pi$$
−0.745724 + 0.666255i $$0.767896\pi$$
$$128$$ 0 0
$$129$$ −23.3693 −2.05755
$$130$$ 0 0
$$131$$ 9.93087 0.867664 0.433832 0.900994i $$-0.357161\pi$$
0.433832 + 0.900994i $$0.357161\pi$$
$$132$$ 0 0
$$133$$ 20.4924 1.77692
$$134$$ 0 0
$$135$$ 1.43845 0.123802
$$136$$ 0 0
$$137$$ 15.1231 1.29205 0.646027 0.763315i $$-0.276429\pi$$
0.646027 + 0.763315i $$0.276429\pi$$
$$138$$ 0 0
$$139$$ 0.315342 0.0267469 0.0133735 0.999911i $$-0.495743\pi$$
0.0133735 + 0.999911i $$0.495743\pi$$
$$140$$ 0 0
$$141$$ −9.43845 −0.794861
$$142$$ 0 0
$$143$$ 2.24621 0.187838
$$144$$ 0 0
$$145$$ −8.56155 −0.710998
$$146$$ 0 0
$$147$$ 49.3002 4.06621
$$148$$ 0 0
$$149$$ −4.24621 −0.347863 −0.173932 0.984758i $$-0.555647\pi$$
−0.173932 + 0.984758i $$0.555647\pi$$
$$150$$ 0 0
$$151$$ −16.8078 −1.36780 −0.683898 0.729577i $$-0.739717\pi$$
−0.683898 + 0.729577i $$0.739717\pi$$
$$152$$ 0 0
$$153$$ −11.1231 −0.899250
$$154$$ 0 0
$$155$$ 1.43845 0.115539
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −10.8769 −0.862594
$$160$$ 0 0
$$161$$ 5.12311 0.403757
$$162$$ 0 0
$$163$$ −0.315342 −0.0246995 −0.0123497 0.999924i $$-0.503931\pi$$
−0.0123497 + 0.999924i $$0.503931\pi$$
$$164$$ 0 0
$$165$$ −10.2462 −0.797666
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −12.6847 −0.975743
$$170$$ 0 0
$$171$$ 14.2462 1.08944
$$172$$ 0 0
$$173$$ 10.4924 0.797724 0.398862 0.917011i $$-0.369405\pi$$
0.398862 + 0.917011i $$0.369405\pi$$
$$174$$ 0 0
$$175$$ 5.12311 0.387270
$$176$$ 0 0
$$177$$ −16.0000 −1.20263
$$178$$ 0 0
$$179$$ −7.68466 −0.574378 −0.287189 0.957874i $$-0.592721\pi$$
−0.287189 + 0.957874i $$0.592721\pi$$
$$180$$ 0 0
$$181$$ 3.12311 0.232139 0.116069 0.993241i $$-0.462971\pi$$
0.116069 + 0.993241i $$0.462971\pi$$
$$182$$ 0 0
$$183$$ 28.4924 2.10622
$$184$$ 0 0
$$185$$ −7.12311 −0.523701
$$186$$ 0 0
$$187$$ 12.4924 0.913536
$$188$$ 0 0
$$189$$ 7.36932 0.536039
$$190$$ 0 0
$$191$$ −2.87689 −0.208165 −0.104082 0.994569i $$-0.533191\pi$$
−0.104082 + 0.994569i $$0.533191\pi$$
$$192$$ 0 0
$$193$$ 24.5616 1.76798 0.883990 0.467507i $$-0.154848\pi$$
0.883990 + 0.467507i $$0.154848\pi$$
$$194$$ 0 0
$$195$$ −1.43845 −0.103009
$$196$$ 0 0
$$197$$ −11.4384 −0.814956 −0.407478 0.913215i $$-0.633592\pi$$
−0.407478 + 0.913215i $$0.633592\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ −43.8617 −3.07849
$$204$$ 0 0
$$205$$ 0.561553 0.0392205
$$206$$ 0 0
$$207$$ 3.56155 0.247545
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −14.2462 −0.980750 −0.490375 0.871512i $$-0.663140\pi$$
−0.490375 + 0.871512i $$0.663140\pi$$
$$212$$ 0 0
$$213$$ 9.43845 0.646712
$$214$$ 0 0
$$215$$ −9.12311 −0.622191
$$216$$ 0 0
$$217$$ 7.36932 0.500262
$$218$$ 0 0
$$219$$ 42.4233 2.86670
$$220$$ 0 0
$$221$$ 1.75379 0.117973
$$222$$ 0 0
$$223$$ −20.4924 −1.37227 −0.686137 0.727472i $$-0.740695\pi$$
−0.686137 + 0.727472i $$0.740695\pi$$
$$224$$ 0 0
$$225$$ 3.56155 0.237437
$$226$$ 0 0
$$227$$ 9.12311 0.605522 0.302761 0.953067i $$-0.402092\pi$$
0.302761 + 0.953067i $$0.402092\pi$$
$$228$$ 0 0
$$229$$ 0.246211 0.0162701 0.00813505 0.999967i $$-0.497411\pi$$
0.00813505 + 0.999967i $$0.497411\pi$$
$$230$$ 0 0
$$231$$ −52.4924 −3.45375
$$232$$ 0 0
$$233$$ −12.5616 −0.822935 −0.411467 0.911425i $$-0.634984\pi$$
−0.411467 + 0.911425i $$0.634984\pi$$
$$234$$ 0 0
$$235$$ −3.68466 −0.240361
$$236$$ 0 0
$$237$$ −26.2462 −1.70487
$$238$$ 0 0
$$239$$ −8.80776 −0.569727 −0.284863 0.958568i $$-0.591948\pi$$
−0.284863 + 0.958568i $$0.591948\pi$$
$$240$$ 0 0
$$241$$ −18.4924 −1.19120 −0.595601 0.803281i $$-0.703086\pi$$
−0.595601 + 0.803281i $$0.703086\pi$$
$$242$$ 0 0
$$243$$ −22.2462 −1.42710
$$244$$ 0 0
$$245$$ 19.2462 1.22960
$$246$$ 0 0
$$247$$ −2.24621 −0.142923
$$248$$ 0 0
$$249$$ 30.7386 1.94798
$$250$$ 0 0
$$251$$ 6.24621 0.394257 0.197129 0.980378i $$-0.436838\pi$$
0.197129 + 0.980378i $$0.436838\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ −8.00000 −0.500979
$$256$$ 0 0
$$257$$ −9.05398 −0.564771 −0.282386 0.959301i $$-0.591126\pi$$
−0.282386 + 0.959301i $$0.591126\pi$$
$$258$$ 0 0
$$259$$ −36.4924 −2.26753
$$260$$ 0 0
$$261$$ −30.4924 −1.88743
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −4.24621 −0.260843
$$266$$ 0 0
$$267$$ 25.6155 1.56764
$$268$$ 0 0
$$269$$ −23.3002 −1.42064 −0.710319 0.703880i $$-0.751449\pi$$
−0.710319 + 0.703880i $$0.751449\pi$$
$$270$$ 0 0
$$271$$ −2.24621 −0.136448 −0.0682238 0.997670i $$-0.521733\pi$$
−0.0682238 + 0.997670i $$0.521733\pi$$
$$272$$ 0 0
$$273$$ −7.36932 −0.446011
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −11.4384 −0.687270 −0.343635 0.939103i $$-0.611658\pi$$
−0.343635 + 0.939103i $$0.611658\pi$$
$$278$$ 0 0
$$279$$ 5.12311 0.306712
$$280$$ 0 0
$$281$$ 30.4924 1.81903 0.909513 0.415676i $$-0.136455\pi$$
0.909513 + 0.415676i $$0.136455\pi$$
$$282$$ 0 0
$$283$$ 22.8769 1.35989 0.679945 0.733263i $$-0.262004\pi$$
0.679945 + 0.733263i $$0.262004\pi$$
$$284$$ 0 0
$$285$$ 10.2462 0.606933
$$286$$ 0 0
$$287$$ 2.87689 0.169818
$$288$$ 0 0
$$289$$ −7.24621 −0.426248
$$290$$ 0 0
$$291$$ −41.6155 −2.43955
$$292$$ 0 0
$$293$$ −12.8769 −0.752276 −0.376138 0.926564i $$-0.622748\pi$$
−0.376138 + 0.926564i $$0.622748\pi$$
$$294$$ 0 0
$$295$$ −6.24621 −0.363668
$$296$$ 0 0
$$297$$ −5.75379 −0.333869
$$298$$ 0 0
$$299$$ −0.561553 −0.0324754
$$300$$ 0 0
$$301$$ −46.7386 −2.69397
$$302$$ 0 0
$$303$$ 0.630683 0.0362318
$$304$$ 0 0
$$305$$ 11.1231 0.636907
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ −5.75379 −0.327322
$$310$$ 0 0
$$311$$ 13.9309 0.789947 0.394974 0.918692i $$-0.370754\pi$$
0.394974 + 0.918692i $$0.370754\pi$$
$$312$$ 0 0
$$313$$ 15.1231 0.854808 0.427404 0.904061i $$-0.359428\pi$$
0.427404 + 0.904061i $$0.359428\pi$$
$$314$$ 0 0
$$315$$ 18.2462 1.02806
$$316$$ 0 0
$$317$$ 20.7386 1.16480 0.582399 0.812903i $$-0.302114\pi$$
0.582399 + 0.812903i $$0.302114\pi$$
$$318$$ 0 0
$$319$$ 34.2462 1.91742
$$320$$ 0 0
$$321$$ −30.7386 −1.71566
$$322$$ 0 0
$$323$$ −12.4924 −0.695097
$$324$$ 0 0
$$325$$ −0.561553 −0.0311493
$$326$$ 0 0
$$327$$ 21.1231 1.16811
$$328$$ 0 0
$$329$$ −18.8769 −1.04072
$$330$$ 0 0
$$331$$ 10.5616 0.580515 0.290258 0.956949i $$-0.406259\pi$$
0.290258 + 0.956949i $$0.406259\pi$$
$$332$$ 0 0
$$333$$ −25.3693 −1.39023
$$334$$ 0 0
$$335$$ 6.24621 0.341267
$$336$$ 0 0
$$337$$ 15.7538 0.858164 0.429082 0.903266i $$-0.358837\pi$$
0.429082 + 0.903266i $$0.358837\pi$$
$$338$$ 0 0
$$339$$ 51.8617 2.81674
$$340$$ 0 0
$$341$$ −5.75379 −0.311585
$$342$$ 0 0
$$343$$ 62.7386 3.38757
$$344$$ 0 0
$$345$$ 2.56155 0.137909
$$346$$ 0 0
$$347$$ −16.4924 −0.885360 −0.442680 0.896680i $$-0.645972\pi$$
−0.442680 + 0.896680i $$0.645972\pi$$
$$348$$ 0 0
$$349$$ −2.80776 −0.150296 −0.0751481 0.997172i $$-0.523943\pi$$
−0.0751481 + 0.997172i $$0.523943\pi$$
$$350$$ 0 0
$$351$$ −0.807764 −0.0431153
$$352$$ 0 0
$$353$$ 34.8078 1.85263 0.926315 0.376750i $$-0.122958\pi$$
0.926315 + 0.376750i $$0.122958\pi$$
$$354$$ 0 0
$$355$$ 3.68466 0.195561
$$356$$ 0 0
$$357$$ −40.9848 −2.16915
$$358$$ 0 0
$$359$$ −13.7538 −0.725897 −0.362949 0.931809i $$-0.618230\pi$$
−0.362949 + 0.931809i $$0.618230\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 12.8078 0.672233
$$364$$ 0 0
$$365$$ 16.5616 0.866871
$$366$$ 0 0
$$367$$ 20.4924 1.06970 0.534848 0.844948i $$-0.320369\pi$$
0.534848 + 0.844948i $$0.320369\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −21.7538 −1.12940
$$372$$ 0 0
$$373$$ −24.7386 −1.28092 −0.640459 0.767992i $$-0.721256\pi$$
−0.640459 + 0.767992i $$0.721256\pi$$
$$374$$ 0 0
$$375$$ 2.56155 0.132278
$$376$$ 0 0
$$377$$ 4.80776 0.247612
$$378$$ 0 0
$$379$$ −14.2462 −0.731779 −0.365889 0.930658i $$-0.619235\pi$$
−0.365889 + 0.930658i $$0.619235\pi$$
$$380$$ 0 0
$$381$$ −43.0540 −2.20572
$$382$$ 0 0
$$383$$ 33.6155 1.71767 0.858837 0.512250i $$-0.171188\pi$$
0.858837 + 0.512250i $$0.171188\pi$$
$$384$$ 0 0
$$385$$ −20.4924 −1.04439
$$386$$ 0 0
$$387$$ −32.4924 −1.65168
$$388$$ 0 0
$$389$$ 7.61553 0.386123 0.193061 0.981187i $$-0.438158\pi$$
0.193061 + 0.981187i $$0.438158\pi$$
$$390$$ 0 0
$$391$$ −3.12311 −0.157942
$$392$$ 0 0
$$393$$ 25.4384 1.28320
$$394$$ 0 0
$$395$$ −10.2462 −0.515543
$$396$$ 0 0
$$397$$ 3.93087 0.197285 0.0986423 0.995123i $$-0.468550\pi$$
0.0986423 + 0.995123i $$0.468550\pi$$
$$398$$ 0 0
$$399$$ 52.4924 2.62791
$$400$$ 0 0
$$401$$ −28.7386 −1.43514 −0.717569 0.696487i $$-0.754745\pi$$
−0.717569 + 0.696487i $$0.754745\pi$$
$$402$$ 0 0
$$403$$ −0.807764 −0.0402376
$$404$$ 0 0
$$405$$ −7.00000 −0.347833
$$406$$ 0 0
$$407$$ 28.4924 1.41232
$$408$$ 0 0
$$409$$ −2.31534 −0.114486 −0.0572431 0.998360i $$-0.518231\pi$$
−0.0572431 + 0.998360i $$0.518231\pi$$
$$410$$ 0 0
$$411$$ 38.7386 1.91084
$$412$$ 0 0
$$413$$ −32.0000 −1.57462
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0.807764 0.0395564
$$418$$ 0 0
$$419$$ −9.12311 −0.445693 −0.222846 0.974854i $$-0.571535\pi$$
−0.222846 + 0.974854i $$0.571535\pi$$
$$420$$ 0 0
$$421$$ 26.4924 1.29116 0.645581 0.763692i $$-0.276615\pi$$
0.645581 + 0.763692i $$0.276615\pi$$
$$422$$ 0 0
$$423$$ −13.1231 −0.638067
$$424$$ 0 0
$$425$$ −3.12311 −0.151493
$$426$$ 0 0
$$427$$ 56.9848 2.75769
$$428$$ 0 0
$$429$$ 5.75379 0.277796
$$430$$ 0 0
$$431$$ −33.6155 −1.61920 −0.809602 0.586980i $$-0.800317\pi$$
−0.809602 + 0.586980i $$0.800317\pi$$
$$432$$ 0 0
$$433$$ 24.7386 1.18886 0.594431 0.804146i $$-0.297377\pi$$
0.594431 + 0.804146i $$0.297377\pi$$
$$434$$ 0 0
$$435$$ −21.9309 −1.05150
$$436$$ 0 0
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 21.3002 1.01660 0.508301 0.861179i $$-0.330274\pi$$
0.508301 + 0.861179i $$0.330274\pi$$
$$440$$ 0 0
$$441$$ 68.5464 3.26411
$$442$$ 0 0
$$443$$ −15.6847 −0.745201 −0.372600 0.927992i $$-0.621534\pi$$
−0.372600 + 0.927992i $$0.621534\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ −10.8769 −0.514459
$$448$$ 0 0
$$449$$ 16.7386 0.789945 0.394972 0.918693i $$-0.370754\pi$$
0.394972 + 0.918693i $$0.370754\pi$$
$$450$$ 0 0
$$451$$ −2.24621 −0.105770
$$452$$ 0 0
$$453$$ −43.0540 −2.02285
$$454$$ 0 0
$$455$$ −2.87689 −0.134871
$$456$$ 0 0
$$457$$ −0.876894 −0.0410194 −0.0205097 0.999790i $$-0.506529\pi$$
−0.0205097 + 0.999790i $$0.506529\pi$$
$$458$$ 0 0
$$459$$ −4.49242 −0.209688
$$460$$ 0 0
$$461$$ 23.4384 1.09164 0.545819 0.837903i $$-0.316219\pi$$
0.545819 + 0.837903i $$0.316219\pi$$
$$462$$ 0 0
$$463$$ 10.2462 0.476182 0.238091 0.971243i $$-0.423478\pi$$
0.238091 + 0.971243i $$0.423478\pi$$
$$464$$ 0 0
$$465$$ 3.68466 0.170872
$$466$$ 0 0
$$467$$ 35.3693 1.63670 0.818348 0.574722i $$-0.194890\pi$$
0.818348 + 0.574722i $$0.194890\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ −5.12311 −0.236060
$$472$$ 0 0
$$473$$ 36.4924 1.67792
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −15.1231 −0.692439
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ 13.1231 0.597122
$$484$$ 0 0
$$485$$ −16.2462 −0.737702
$$486$$ 0 0
$$487$$ −3.05398 −0.138389 −0.0691944 0.997603i $$-0.522043\pi$$
−0.0691944 + 0.997603i $$0.522043\pi$$
$$488$$ 0 0
$$489$$ −0.807764 −0.0365284
$$490$$ 0 0
$$491$$ 10.5616 0.476636 0.238318 0.971187i $$-0.423404\pi$$
0.238318 + 0.971187i $$0.423404\pi$$
$$492$$ 0 0
$$493$$ 26.7386 1.20425
$$494$$ 0 0
$$495$$ −14.2462 −0.640320
$$496$$ 0 0
$$497$$ 18.8769 0.846744
$$498$$ 0 0
$$499$$ 0.946025 0.0423499 0.0211749 0.999776i $$-0.493259\pi$$
0.0211749 + 0.999776i $$0.493259\pi$$
$$500$$ 0 0
$$501$$ −20.4924 −0.915534
$$502$$ 0 0
$$503$$ 25.6155 1.14214 0.571070 0.820901i $$-0.306528\pi$$
0.571070 + 0.820901i $$0.306528\pi$$
$$504$$ 0 0
$$505$$ 0.246211 0.0109563
$$506$$ 0 0
$$507$$ −32.4924 −1.44304
$$508$$ 0 0
$$509$$ −8.56155 −0.379484 −0.189742 0.981834i $$-0.560765\pi$$
−0.189742 + 0.981834i $$0.560765\pi$$
$$510$$ 0 0
$$511$$ 84.8466 3.75339
$$512$$ 0 0
$$513$$ 5.75379 0.254036
$$514$$ 0 0
$$515$$ −2.24621 −0.0989799
$$516$$ 0 0
$$517$$ 14.7386 0.648204
$$518$$ 0 0
$$519$$ 26.8769 1.17976
$$520$$ 0 0
$$521$$ −17.8617 −0.782537 −0.391269 0.920277i $$-0.627964\pi$$
−0.391269 + 0.920277i $$0.627964\pi$$
$$522$$ 0 0
$$523$$ −23.8617 −1.04340 −0.521701 0.853129i $$-0.674702\pi$$
−0.521701 + 0.853129i $$0.674702\pi$$
$$524$$ 0 0
$$525$$ 13.1231 0.572739
$$526$$ 0 0
$$527$$ −4.49242 −0.195693
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −22.2462 −0.965403
$$532$$ 0 0
$$533$$ −0.315342 −0.0136590
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ −19.6847 −0.849456
$$538$$ 0 0
$$539$$ −76.9848 −3.31597
$$540$$ 0 0
$$541$$ 33.6847 1.44822 0.724108 0.689686i $$-0.242252\pi$$
0.724108 + 0.689686i $$0.242252\pi$$
$$542$$ 0 0
$$543$$ 8.00000 0.343313
$$544$$ 0 0
$$545$$ 8.24621 0.353229
$$546$$ 0 0
$$547$$ 0.946025 0.0404491 0.0202245 0.999795i $$-0.493562\pi$$
0.0202245 + 0.999795i $$0.493562\pi$$
$$548$$ 0 0
$$549$$ 39.6155 1.69075
$$550$$ 0 0
$$551$$ −34.2462 −1.45894
$$552$$ 0 0
$$553$$ −52.4924 −2.23220
$$554$$ 0 0
$$555$$ −18.2462 −0.774509
$$556$$ 0 0
$$557$$ −7.75379 −0.328539 −0.164269 0.986416i $$-0.552527\pi$$
−0.164269 + 0.986416i $$0.552527\pi$$
$$558$$ 0 0
$$559$$ 5.12311 0.216684
$$560$$ 0 0
$$561$$ 32.0000 1.35104
$$562$$ 0 0
$$563$$ −39.2311 −1.65339 −0.826696 0.562649i $$-0.809782\pi$$
−0.826696 + 0.562649i $$0.809782\pi$$
$$564$$ 0 0
$$565$$ 20.2462 0.851765
$$566$$ 0 0
$$567$$ −35.8617 −1.50605
$$568$$ 0 0
$$569$$ −20.7386 −0.869409 −0.434704 0.900573i $$-0.643147\pi$$
−0.434704 + 0.900573i $$0.643147\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ −7.36932 −0.307858
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −32.4233 −1.34980 −0.674900 0.737910i $$-0.735813\pi$$
−0.674900 + 0.737910i $$0.735813\pi$$
$$578$$ 0 0
$$579$$ 62.9157 2.61469
$$580$$ 0 0
$$581$$ 61.4773 2.55051
$$582$$ 0 0
$$583$$ 16.9848 0.703440
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ −36.1771 −1.49319 −0.746594 0.665280i $$-0.768312\pi$$
−0.746594 + 0.665280i $$0.768312\pi$$
$$588$$ 0 0
$$589$$ 5.75379 0.237081
$$590$$ 0 0
$$591$$ −29.3002 −1.20525
$$592$$ 0 0
$$593$$ 26.9848 1.10813 0.554067 0.832472i $$-0.313075\pi$$
0.554067 + 0.832472i $$0.313075\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ 0 0
$$597$$ 61.4773 2.51610
$$598$$ 0 0
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ 0 0
$$601$$ 18.1771 0.741459 0.370729 0.928741i $$-0.379108\pi$$
0.370729 + 0.928741i $$0.379108\pi$$
$$602$$ 0 0
$$603$$ 22.2462 0.905936
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ −112.354 −4.55282
$$610$$ 0 0
$$611$$ 2.06913 0.0837081
$$612$$ 0 0
$$613$$ 3.12311 0.126141 0.0630705 0.998009i $$-0.479911\pi$$
0.0630705 + 0.998009i $$0.479911\pi$$
$$614$$ 0 0
$$615$$ 1.43845 0.0580038
$$616$$ 0 0
$$617$$ 35.6155 1.43383 0.716914 0.697162i $$-0.245554\pi$$
0.716914 + 0.697162i $$0.245554\pi$$
$$618$$ 0 0
$$619$$ −28.9848 −1.16500 −0.582500 0.812831i $$-0.697925\pi$$
−0.582500 + 0.812831i $$0.697925\pi$$
$$620$$ 0 0
$$621$$ 1.43845 0.0577229
$$622$$ 0 0
$$623$$ 51.2311 2.05253
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −40.9848 −1.63678
$$628$$ 0 0
$$629$$ 22.2462 0.887015
$$630$$ 0 0
$$631$$ −31.3693 −1.24879 −0.624396 0.781108i $$-0.714655\pi$$
−0.624396 + 0.781108i $$0.714655\pi$$
$$632$$ 0 0
$$633$$ −36.4924 −1.45044
$$634$$ 0 0
$$635$$ −16.8078 −0.666996
$$636$$ 0 0
$$637$$ −10.8078 −0.428219
$$638$$ 0 0
$$639$$ 13.1231 0.519142
$$640$$ 0 0
$$641$$ 23.7538 0.938218 0.469109 0.883140i $$-0.344575\pi$$
0.469109 + 0.883140i $$0.344575\pi$$
$$642$$ 0 0
$$643$$ 44.3542 1.74916 0.874579 0.484884i $$-0.161138\pi$$
0.874579 + 0.484884i $$0.161138\pi$$
$$644$$ 0 0
$$645$$ −23.3693 −0.920166
$$646$$ 0 0
$$647$$ −1.43845 −0.0565512 −0.0282756 0.999600i $$-0.509002\pi$$
−0.0282756 + 0.999600i $$0.509002\pi$$
$$648$$ 0 0
$$649$$ 24.9848 0.980741
$$650$$ 0 0
$$651$$ 18.8769 0.739844
$$652$$ 0 0
$$653$$ −35.7926 −1.40067 −0.700337 0.713813i $$-0.746967\pi$$
−0.700337 + 0.713813i $$0.746967\pi$$
$$654$$ 0 0
$$655$$ 9.93087 0.388031
$$656$$ 0 0
$$657$$ 58.9848 2.30122
$$658$$ 0 0
$$659$$ 14.2462 0.554954 0.277477 0.960732i $$-0.410502\pi$$
0.277477 + 0.960732i $$0.410502\pi$$
$$660$$ 0 0
$$661$$ 10.4924 0.408108 0.204054 0.978960i $$-0.434588\pi$$
0.204054 + 0.978960i $$0.434588\pi$$
$$662$$ 0 0
$$663$$ 4.49242 0.174471
$$664$$ 0 0
$$665$$ 20.4924 0.794662
$$666$$ 0 0
$$667$$ −8.56155 −0.331505
$$668$$ 0 0
$$669$$ −52.4924 −2.02947
$$670$$ 0 0
$$671$$ −44.4924 −1.71761
$$672$$ 0 0
$$673$$ −4.56155 −0.175835 −0.0879175 0.996128i $$-0.528021\pi$$
−0.0879175 + 0.996128i $$0.528021\pi$$
$$674$$ 0 0
$$675$$ 1.43845 0.0553659
$$676$$ 0 0
$$677$$ 20.7386 0.797050 0.398525 0.917157i $$-0.369522\pi$$
0.398525 + 0.917157i $$0.369522\pi$$
$$678$$ 0 0
$$679$$ −83.2311 −3.19411
$$680$$ 0 0
$$681$$ 23.3693 0.895514
$$682$$ 0 0
$$683$$ −18.5616 −0.710238 −0.355119 0.934821i $$-0.615560\pi$$
−0.355119 + 0.934821i $$0.615560\pi$$
$$684$$ 0 0
$$685$$ 15.1231 0.577824
$$686$$ 0 0
$$687$$ 0.630683 0.0240621
$$688$$ 0 0
$$689$$ 2.38447 0.0908411
$$690$$ 0 0
$$691$$ 6.24621 0.237617 0.118809 0.992917i $$-0.462093\pi$$
0.118809 + 0.992917i $$0.462093\pi$$
$$692$$ 0 0
$$693$$ −72.9848 −2.77247
$$694$$ 0 0
$$695$$ 0.315342 0.0119616
$$696$$ 0 0
$$697$$ −1.75379 −0.0664295
$$698$$ 0 0
$$699$$ −32.1771 −1.21705
$$700$$ 0 0
$$701$$ 38.9848 1.47244 0.736219 0.676744i $$-0.236610\pi$$
0.736219 + 0.676744i $$0.236610\pi$$
$$702$$ 0 0
$$703$$ −28.4924 −1.07461
$$704$$ 0 0
$$705$$ −9.43845 −0.355472
$$706$$ 0 0
$$707$$ 1.26137 0.0474386
$$708$$ 0 0
$$709$$ −40.7386 −1.52997 −0.764986 0.644047i $$-0.777254\pi$$
−0.764986 + 0.644047i $$0.777254\pi$$
$$710$$ 0 0
$$711$$ −36.4924 −1.36857
$$712$$ 0 0
$$713$$ 1.43845 0.0538703
$$714$$ 0 0
$$715$$ 2.24621 0.0840035
$$716$$ 0 0
$$717$$ −22.5616 −0.842577
$$718$$ 0 0
$$719$$ 38.7386 1.44471 0.722354 0.691524i $$-0.243060\pi$$
0.722354 + 0.691524i $$0.243060\pi$$
$$720$$ 0 0
$$721$$ −11.5076 −0.428565
$$722$$ 0 0
$$723$$ −47.3693 −1.76168
$$724$$ 0 0
$$725$$ −8.56155 −0.317968
$$726$$ 0 0
$$727$$ 37.1231 1.37682 0.688410 0.725322i $$-0.258309\pi$$
0.688410 + 0.725322i $$0.258309\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 28.4924 1.05383
$$732$$ 0 0
$$733$$ 11.1231 0.410841 0.205421 0.978674i $$-0.434144\pi$$
0.205421 + 0.978674i $$0.434144\pi$$
$$734$$ 0 0
$$735$$ 49.3002 1.81846
$$736$$ 0 0
$$737$$ −24.9848 −0.920329
$$738$$ 0 0
$$739$$ 27.5464 1.01331 0.506655 0.862149i $$-0.330882\pi$$
0.506655 + 0.862149i $$0.330882\pi$$
$$740$$ 0 0
$$741$$ −5.75379 −0.211371
$$742$$ 0 0
$$743$$ −13.7538 −0.504578 −0.252289 0.967652i $$-0.581183\pi$$
−0.252289 + 0.967652i $$0.581183\pi$$
$$744$$ 0 0
$$745$$ −4.24621 −0.155569
$$746$$ 0 0
$$747$$ 42.7386 1.56372
$$748$$ 0 0
$$749$$ −61.4773 −2.24633
$$750$$ 0 0
$$751$$ 49.6155 1.81050 0.905248 0.424883i $$-0.139685\pi$$
0.905248 + 0.424883i $$0.139685\pi$$
$$752$$ 0 0
$$753$$ 16.0000 0.583072
$$754$$ 0 0
$$755$$ −16.8078 −0.611697
$$756$$ 0 0
$$757$$ 24.8769 0.904166 0.452083 0.891976i $$-0.350681\pi$$
0.452083 + 0.891976i $$0.350681\pi$$
$$758$$ 0 0
$$759$$ −10.2462 −0.371914
$$760$$ 0 0
$$761$$ −11.3002 −0.409631 −0.204816 0.978801i $$-0.565660\pi$$
−0.204816 + 0.978801i $$0.565660\pi$$
$$762$$ 0 0
$$763$$ 42.2462 1.52942
$$764$$ 0 0
$$765$$ −11.1231 −0.402157
$$766$$ 0 0
$$767$$ 3.50758 0.126651
$$768$$ 0 0
$$769$$ 38.4924 1.38807 0.694036 0.719940i $$-0.255831\pi$$
0.694036 + 0.719940i $$0.255831\pi$$
$$770$$ 0 0
$$771$$ −23.1922 −0.835248
$$772$$ 0 0
$$773$$ −4.24621 −0.152726 −0.0763628 0.997080i $$-0.524331\pi$$
−0.0763628 + 0.997080i $$0.524331\pi$$
$$774$$ 0 0
$$775$$ 1.43845 0.0516705
$$776$$ 0 0
$$777$$ −93.4773 −3.35348
$$778$$ 0 0
$$779$$ 2.24621 0.0804789
$$780$$ 0 0
$$781$$ −14.7386 −0.527390
$$782$$ 0 0
$$783$$ −12.3153 −0.440114
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −46.2462 −1.64850 −0.824250 0.566226i $$-0.808403\pi$$
−0.824250 + 0.566226i $$0.808403\pi$$
$$788$$ 0 0
$$789$$ −20.4924 −0.729550
$$790$$ 0 0
$$791$$ 103.723 3.68798
$$792$$ 0 0
$$793$$ −6.24621 −0.221809
$$794$$ 0 0
$$795$$ −10.8769 −0.385764
$$796$$ 0 0
$$797$$ 27.1231 0.960750 0.480375 0.877063i $$-0.340501\pi$$
0.480375 + 0.877063i $$0.340501\pi$$
$$798$$ 0 0
$$799$$ 11.5076 0.407109
$$800$$ 0 0
$$801$$ 35.6155 1.25841
$$802$$ 0 0
$$803$$ −66.2462 −2.33778
$$804$$ 0 0
$$805$$ 5.12311 0.180566
$$806$$ 0 0
$$807$$ −59.6847 −2.10100
$$808$$ 0 0
$$809$$ −10.4924 −0.368894 −0.184447 0.982842i $$-0.559049\pi$$
−0.184447 + 0.982842i $$0.559049\pi$$
$$810$$ 0 0
$$811$$ −8.31534 −0.291991 −0.145996 0.989285i $$-0.546639\pi$$
−0.145996 + 0.989285i $$0.546639\pi$$
$$812$$ 0 0
$$813$$ −5.75379 −0.201794
$$814$$ 0 0
$$815$$ −0.315342 −0.0110459
$$816$$ 0 0
$$817$$ −36.4924 −1.27671
$$818$$ 0 0
$$819$$ −10.2462 −0.358032
$$820$$ 0 0
$$821$$ −31.7538 −1.10821 −0.554107 0.832445i $$-0.686940\pi$$
−0.554107 + 0.832445i $$0.686940\pi$$
$$822$$ 0 0
$$823$$ −17.4384 −0.607866 −0.303933 0.952693i $$-0.598300\pi$$
−0.303933 + 0.952693i $$0.598300\pi$$
$$824$$ 0 0
$$825$$ −10.2462 −0.356727
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ 50.4924 1.75367 0.876837 0.480787i $$-0.159649\pi$$
0.876837 + 0.480787i $$0.159649\pi$$
$$830$$ 0 0
$$831$$ −29.3002 −1.01641
$$832$$ 0 0
$$833$$ −60.1080 −2.08262
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 2.06913 0.0715196
$$838$$ 0 0
$$839$$ 9.61553 0.331965 0.165982 0.986129i $$-0.446920\pi$$
0.165982 + 0.986129i $$0.446920\pi$$
$$840$$ 0 0
$$841$$ 44.3002 1.52759
$$842$$ 0 0
$$843$$ 78.1080 2.69018
$$844$$ 0 0
$$845$$ −12.6847 −0.436366
$$846$$ 0 0
$$847$$ 25.6155 0.880160
$$848$$ 0 0
$$849$$ 58.6004 2.01116
$$850$$ 0 0
$$851$$ −7.12311 −0.244177
$$852$$ 0 0
$$853$$ 22.9848 0.786986 0.393493 0.919328i $$-0.371267\pi$$
0.393493 + 0.919328i $$0.371267\pi$$
$$854$$ 0 0
$$855$$ 14.2462 0.487210
$$856$$ 0 0
$$857$$ 34.1771 1.16747 0.583733 0.811945i $$-0.301591\pi$$
0.583733 + 0.811945i $$0.301591\pi$$
$$858$$ 0 0
$$859$$ 38.4233 1.31099 0.655493 0.755201i $$-0.272461\pi$$
0.655493 + 0.755201i $$0.272461\pi$$
$$860$$ 0 0
$$861$$ 7.36932 0.251146
$$862$$ 0 0
$$863$$ 18.4233 0.627136 0.313568 0.949566i $$-0.398476\pi$$
0.313568 + 0.949566i $$0.398476\pi$$
$$864$$ 0 0
$$865$$ 10.4924 0.356753
$$866$$ 0 0
$$867$$ −18.5616 −0.630383
$$868$$ 0 0
$$869$$ 40.9848 1.39032
$$870$$ 0 0
$$871$$ −3.50758 −0.118850
$$872$$ 0 0
$$873$$ −57.8617 −1.95832
$$874$$ 0 0
$$875$$ 5.12311 0.173193
$$876$$ 0 0
$$877$$ −40.7386 −1.37565 −0.687823 0.725878i $$-0.741433\pi$$
−0.687823 + 0.725878i $$0.741433\pi$$
$$878$$ 0 0
$$879$$ −32.9848 −1.11255
$$880$$ 0 0
$$881$$ 24.1080 0.812217 0.406109 0.913825i $$-0.366885\pi$$
0.406109 + 0.913825i $$0.366885\pi$$
$$882$$ 0 0
$$883$$ −40.4924 −1.36268 −0.681339 0.731968i $$-0.738602\pi$$
−0.681339 + 0.731968i $$0.738602\pi$$
$$884$$ 0 0
$$885$$ −16.0000 −0.537834
$$886$$ 0 0
$$887$$ −33.4384 −1.12275 −0.561377 0.827560i $$-0.689728\pi$$
−0.561377 + 0.827560i $$0.689728\pi$$
$$888$$ 0 0
$$889$$ −86.1080 −2.88797
$$890$$ 0 0
$$891$$ 28.0000 0.938035
$$892$$ 0 0
$$893$$ −14.7386 −0.493210
$$894$$ 0 0
$$895$$ −7.68466 −0.256870
$$896$$ 0 0
$$897$$ −1.43845 −0.0480284
$$898$$ 0 0
$$899$$ −12.3153 −0.410740
$$900$$ 0 0
$$901$$ 13.2614 0.441800
$$902$$ 0 0
$$903$$ −119.723 −3.98415
$$904$$ 0 0
$$905$$ 3.12311 0.103816
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 0 0
$$909$$ 0.876894 0.0290848
$$910$$ 0 0
$$911$$ −20.4924 −0.678944 −0.339472 0.940616i $$-0.610248\pi$$
−0.339472 + 0.940616i $$0.610248\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 28.4924 0.941930
$$916$$ 0 0
$$917$$ 50.8769 1.68010
$$918$$ 0 0
$$919$$ −35.8617 −1.18297 −0.591485 0.806316i $$-0.701458\pi$$
−0.591485 + 0.806316i $$0.701458\pi$$
$$920$$ 0 0
$$921$$ 30.7386 1.01287
$$922$$ 0 0
$$923$$ −2.06913 −0.0681063
$$924$$ 0 0
$$925$$ −7.12311 −0.234206
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 13.0540 0.428287 0.214144 0.976802i $$-0.431304\pi$$
0.214144 + 0.976802i $$0.431304\pi$$
$$930$$ 0 0
$$931$$ 76.9848 2.52308
$$932$$ 0 0
$$933$$ 35.6847 1.16826
$$934$$ 0 0
$$935$$ 12.4924 0.408546
$$936$$ 0 0
$$937$$ 41.3693 1.35148 0.675738 0.737142i $$-0.263825\pi$$
0.675738 + 0.737142i $$0.263825\pi$$
$$938$$ 0 0
$$939$$ 38.7386 1.26419
$$940$$ 0 0
$$941$$ −60.6004 −1.97552 −0.987758 0.155995i $$-0.950142\pi$$
−0.987758 + 0.155995i $$0.950142\pi$$
$$942$$ 0 0
$$943$$ 0.561553 0.0182867
$$944$$ 0 0
$$945$$ 7.36932 0.239724
$$946$$ 0 0
$$947$$ −35.1922 −1.14359 −0.571797 0.820395i $$-0.693754\pi$$
−0.571797 + 0.820395i $$0.693754\pi$$
$$948$$ 0 0
$$949$$ −9.30019 −0.301897
$$950$$ 0 0
$$951$$ 53.1231 1.72263
$$952$$ 0 0
$$953$$ 13.5076 0.437553 0.218777 0.975775i $$-0.429793\pi$$
0.218777 + 0.975775i $$0.429793\pi$$
$$954$$ 0 0
$$955$$ −2.87689 −0.0930941
$$956$$ 0 0
$$957$$ 87.7235 2.83570
$$958$$ 0 0
$$959$$ 77.4773 2.50187
$$960$$ 0 0
$$961$$ −28.9309 −0.933254
$$962$$ 0 0
$$963$$ −42.7386 −1.37723
$$964$$ 0 0
$$965$$ 24.5616 0.790664
$$966$$ 0 0
$$967$$ 0.177081 0.00569454 0.00284727 0.999996i $$-0.499094\pi$$
0.00284727 + 0.999996i $$0.499094\pi$$
$$968$$ 0 0
$$969$$ −32.0000 −1.02799
$$970$$ 0 0
$$971$$ 3.36932 0.108127 0.0540633 0.998538i $$-0.482783\pi$$
0.0540633 + 0.998538i $$0.482783\pi$$
$$972$$ 0 0
$$973$$ 1.61553 0.0517915
$$974$$ 0 0
$$975$$ −1.43845 −0.0460672
$$976$$ 0 0
$$977$$ 48.1080 1.53911 0.769555 0.638581i $$-0.220478\pi$$
0.769555 + 0.638581i $$0.220478\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 29.3693 0.937690
$$982$$ 0 0
$$983$$ 37.1231 1.18404 0.592022 0.805922i $$-0.298330\pi$$
0.592022 + 0.805922i $$0.298330\pi$$
$$984$$ 0 0
$$985$$ −11.4384 −0.364459
$$986$$ 0 0
$$987$$ −48.3542 −1.53913
$$988$$ 0 0
$$989$$ −9.12311 −0.290098
$$990$$ 0 0
$$991$$ −12.4924 −0.396835 −0.198417 0.980118i $$-0.563580\pi$$
−0.198417 + 0.980118i $$0.563580\pi$$
$$992$$ 0 0
$$993$$ 27.0540 0.858532
$$994$$ 0 0
$$995$$ 24.0000 0.760851
$$996$$ 0 0
$$997$$ −48.7386 −1.54357 −0.771784 0.635885i $$-0.780635\pi$$
−0.771784 + 0.635885i $$0.780635\pi$$
$$998$$ 0 0
$$999$$ −10.2462 −0.324176
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 920.2.a.f.1.2 2
3.2 odd 2 8280.2.a.bb.1.2 2
4.3 odd 2 1840.2.a.k.1.1 2
5.2 odd 4 4600.2.e.m.4049.1 4
5.3 odd 4 4600.2.e.m.4049.4 4
5.4 even 2 4600.2.a.r.1.1 2
8.3 odd 2 7360.2.a.bm.1.2 2
8.5 even 2 7360.2.a.bj.1.1 2
20.19 odd 2 9200.2.a.bx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.2 2 1.1 even 1 trivial
1840.2.a.k.1.1 2 4.3 odd 2
4600.2.a.r.1.1 2 5.4 even 2
4600.2.e.m.4049.1 4 5.2 odd 4
4600.2.e.m.4049.4 4 5.3 odd 4
7360.2.a.bj.1.1 2 8.5 even 2
7360.2.a.bm.1.2 2 8.3 odd 2
8280.2.a.bb.1.2 2 3.2 odd 2
9200.2.a.bx.1.2 2 20.19 odd 2