# Properties

 Label 6080.2.a.cc.1.3 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.78165$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.296842 q^{3} +1.00000 q^{5} -3.56331 q^{7} -2.91188 q^{9} +O(q^{10})$$ $$q-0.296842 q^{3} +1.00000 q^{5} -3.56331 q^{7} -2.91188 q^{9} -5.56331 q^{11} -5.26647 q^{13} -0.296842 q^{15} +1.40632 q^{17} -1.00000 q^{19} +1.05774 q^{21} -6.96962 q^{23} +1.00000 q^{25} +1.75489 q^{27} -1.40632 q^{29} +1.75489 q^{31} +1.65142 q^{33} -3.56331 q^{35} -3.61504 q^{37} +1.56331 q^{39} +4.34858 q^{41} +3.56331 q^{43} -2.91188 q^{45} -8.26046 q^{47} +5.69716 q^{49} -0.417453 q^{51} +7.61504 q^{53} -5.56331 q^{55} +0.296842 q^{57} -9.47519 q^{59} -9.21473 q^{61} +10.3759 q^{63} -5.26647 q^{65} +4.76090 q^{67} +2.06888 q^{69} -14.0689 q^{71} +6.59368 q^{73} -0.296842 q^{75} +19.8238 q^{77} +5.47519 q^{79} +8.21473 q^{81} +4.15699 q^{83} +1.40632 q^{85} +0.417453 q^{87} -9.23009 q^{89} +18.7660 q^{91} -0.520926 q^{93} -1.00000 q^{95} +11.5116 q^{97} +16.1997 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9 $$4 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 4 q^{19} + 4 q^{21} - 8 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} + 6 q^{37} - 12 q^{39} + 16 q^{41} - 4 q^{43} + 8 q^{45} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 4 q^{55} + 2 q^{57} - 20 q^{61} + 20 q^{63} - 2 q^{65} + 18 q^{67} - 28 q^{69} - 20 q^{71} + 28 q^{73} - 2 q^{75} + 40 q^{77} - 16 q^{79} + 16 q^{81} + 4 q^{85} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 - 4 * q^19 + 4 * q^21 - 8 * q^23 + 4 * q^25 + 4 * q^27 - 4 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 + 6 * q^37 - 12 * q^39 + 16 * q^41 - 4 * q^43 + 8 * q^45 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 4 * q^55 + 2 * q^57 - 20 * q^61 + 20 * q^63 - 2 * q^65 + 18 * q^67 - 28 * q^69 - 20 * q^71 + 28 * q^73 - 2 * q^75 + 40 * q^77 - 16 * q^79 + 16 * q^81 + 4 * q^85 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 4 * q^95 + 30 * q^97 + 4 * 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
$$3$$ −0.296842 −0.171382 −0.0856908 0.996322i $$-0.527310\pi$$
−0.0856908 + 0.996322i $$0.527310\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.56331 −1.34680 −0.673402 0.739277i $$-0.735168\pi$$
−0.673402 + 0.739277i $$0.735168\pi$$
$$8$$ 0 0
$$9$$ −2.91188 −0.970628
$$10$$ 0 0
$$11$$ −5.56331 −1.67740 −0.838700 0.544594i $$-0.816684\pi$$
−0.838700 + 0.544594i $$0.816684\pi$$
$$12$$ 0 0
$$13$$ −5.26647 −1.46065 −0.730327 0.683097i $$-0.760632\pi$$
−0.730327 + 0.683097i $$0.760632\pi$$
$$14$$ 0 0
$$15$$ −0.296842 −0.0766442
$$16$$ 0 0
$$17$$ 1.40632 0.341082 0.170541 0.985351i $$-0.445448\pi$$
0.170541 + 0.985351i $$0.445448\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 1.05774 0.230817
$$22$$ 0 0
$$23$$ −6.96962 −1.45327 −0.726633 0.687025i $$-0.758916\pi$$
−0.726633 + 0.687025i $$0.758916\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.75489 0.337730
$$28$$ 0 0
$$29$$ −1.40632 −0.261146 −0.130573 0.991439i $$-0.541682\pi$$
−0.130573 + 0.991439i $$0.541682\pi$$
$$30$$ 0 0
$$31$$ 1.75489 0.315188 0.157594 0.987504i $$-0.449626\pi$$
0.157594 + 0.987504i $$0.449626\pi$$
$$32$$ 0 0
$$33$$ 1.65142 0.287476
$$34$$ 0 0
$$35$$ −3.56331 −0.602309
$$36$$ 0 0
$$37$$ −3.61504 −0.594309 −0.297155 0.954829i $$-0.596038\pi$$
−0.297155 + 0.954829i $$0.596038\pi$$
$$38$$ 0 0
$$39$$ 1.56331 0.250329
$$40$$ 0 0
$$41$$ 4.34858 0.679134 0.339567 0.940582i $$-0.389720\pi$$
0.339567 + 0.940582i $$0.389720\pi$$
$$42$$ 0 0
$$43$$ 3.56331 0.543399 0.271700 0.962382i $$-0.412414\pi$$
0.271700 + 0.962382i $$0.412414\pi$$
$$44$$ 0 0
$$45$$ −2.91188 −0.434078
$$46$$ 0 0
$$47$$ −8.26046 −1.20491 −0.602456 0.798152i $$-0.705811\pi$$
−0.602456 + 0.798152i $$0.705811\pi$$
$$48$$ 0 0
$$49$$ 5.69716 0.813879
$$50$$ 0 0
$$51$$ −0.417453 −0.0584552
$$52$$ 0 0
$$53$$ 7.61504 1.04601 0.523003 0.852331i $$-0.324812\pi$$
0.523003 + 0.852331i $$0.324812\pi$$
$$54$$ 0 0
$$55$$ −5.56331 −0.750156
$$56$$ 0 0
$$57$$ 0.296842 0.0393177
$$58$$ 0 0
$$59$$ −9.47519 −1.23356 −0.616782 0.787134i $$-0.711564\pi$$
−0.616782 + 0.787134i $$0.711564\pi$$
$$60$$ 0 0
$$61$$ −9.21473 −1.17983 −0.589913 0.807467i $$-0.700838\pi$$
−0.589913 + 0.807467i $$0.700838\pi$$
$$62$$ 0 0
$$63$$ 10.3759 1.30725
$$64$$ 0 0
$$65$$ −5.26647 −0.653225
$$66$$ 0 0
$$67$$ 4.76090 0.581636 0.290818 0.956778i $$-0.406073\pi$$
0.290818 + 0.956778i $$0.406073\pi$$
$$68$$ 0 0
$$69$$ 2.06888 0.249063
$$70$$ 0 0
$$71$$ −14.0689 −1.66967 −0.834834 0.550502i $$-0.814437\pi$$
−0.834834 + 0.550502i $$0.814437\pi$$
$$72$$ 0 0
$$73$$ 6.59368 0.771732 0.385866 0.922555i $$-0.373903\pi$$
0.385866 + 0.922555i $$0.373903\pi$$
$$74$$ 0 0
$$75$$ −0.296842 −0.0342763
$$76$$ 0 0
$$77$$ 19.8238 2.25913
$$78$$ 0 0
$$79$$ 5.47519 0.616007 0.308004 0.951385i $$-0.400339\pi$$
0.308004 + 0.951385i $$0.400339\pi$$
$$80$$ 0 0
$$81$$ 8.21473 0.912748
$$82$$ 0 0
$$83$$ 4.15699 0.456289 0.228144 0.973627i $$-0.426734\pi$$
0.228144 + 0.973627i $$0.426734\pi$$
$$84$$ 0 0
$$85$$ 1.40632 0.152536
$$86$$ 0 0
$$87$$ 0.417453 0.0447557
$$88$$ 0 0
$$89$$ −9.23009 −0.978387 −0.489194 0.872175i $$-0.662709\pi$$
−0.489194 + 0.872175i $$0.662709\pi$$
$$90$$ 0 0
$$91$$ 18.7660 1.96721
$$92$$ 0 0
$$93$$ −0.520926 −0.0540175
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 11.5116 1.16882 0.584411 0.811457i $$-0.301325\pi$$
0.584411 + 0.811457i $$0.301325\pi$$
$$98$$ 0 0
$$99$$ 16.1997 1.62813
$$100$$ 0 0
$$101$$ 11.8511 1.17923 0.589616 0.807684i $$-0.299279\pi$$
0.589616 + 0.807684i $$0.299279\pi$$
$$102$$ 0 0
$$103$$ 1.35458 0.133471 0.0667354 0.997771i $$-0.478742\pi$$
0.0667354 + 0.997771i $$0.478742\pi$$
$$104$$ 0 0
$$105$$ 1.05774 0.103225
$$106$$ 0 0
$$107$$ 7.06287 0.682794 0.341397 0.939919i $$-0.389100\pi$$
0.341397 + 0.939919i $$0.389100\pi$$
$$108$$ 0 0
$$109$$ −10.1844 −0.975484 −0.487742 0.872988i $$-0.662179\pi$$
−0.487742 + 0.872988i $$0.662179\pi$$
$$110$$ 0 0
$$111$$ 1.07310 0.101854
$$112$$ 0 0
$$113$$ 1.86015 0.174988 0.0874940 0.996165i $$-0.472114\pi$$
0.0874940 + 0.996165i $$0.472114\pi$$
$$114$$ 0 0
$$115$$ −6.96962 −0.649921
$$116$$ 0 0
$$117$$ 15.3353 1.41775
$$118$$ 0 0
$$119$$ −5.01114 −0.459370
$$120$$ 0 0
$$121$$ 19.9504 1.81367
$$122$$ 0 0
$$123$$ −1.29084 −0.116391
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −4.89053 −0.433964 −0.216982 0.976176i $$-0.569621\pi$$
−0.216982 + 0.976176i $$0.569621\pi$$
$$128$$ 0 0
$$129$$ −1.05774 −0.0931287
$$130$$ 0 0
$$131$$ −2.81263 −0.245741 −0.122870 0.992423i $$-0.539210\pi$$
−0.122870 + 0.992423i $$0.539210\pi$$
$$132$$ 0 0
$$133$$ 3.56331 0.308978
$$134$$ 0 0
$$135$$ 1.75489 0.151037
$$136$$ 0 0
$$137$$ 9.23009 0.788579 0.394290 0.918986i $$-0.370991\pi$$
0.394290 + 0.918986i $$0.370991\pi$$
$$138$$ 0 0
$$139$$ 3.67878 0.312030 0.156015 0.987755i $$-0.450135\pi$$
0.156015 + 0.987755i $$0.450135\pi$$
$$140$$ 0 0
$$141$$ 2.45205 0.206500
$$142$$ 0 0
$$143$$ 29.2990 2.45010
$$144$$ 0 0
$$145$$ −1.40632 −0.116788
$$146$$ 0 0
$$147$$ −1.69115 −0.139484
$$148$$ 0 0
$$149$$ −7.09925 −0.581593 −0.290797 0.956785i $$-0.593920\pi$$
−0.290797 + 0.956785i $$0.593920\pi$$
$$150$$ 0 0
$$151$$ −18.3567 −1.49385 −0.746924 0.664910i $$-0.768470\pi$$
−0.746924 + 0.664910i $$0.768470\pi$$
$$152$$ 0 0
$$153$$ −4.09503 −0.331064
$$154$$ 0 0
$$155$$ 1.75489 0.140957
$$156$$ 0 0
$$157$$ −17.2301 −1.37511 −0.687555 0.726132i $$-0.741316\pi$$
−0.687555 + 0.726132i $$0.741316\pi$$
$$158$$ 0 0
$$159$$ −2.26046 −0.179266
$$160$$ 0 0
$$161$$ 24.8349 1.95726
$$162$$ 0 0
$$163$$ −10.8662 −0.851103 −0.425551 0.904934i $$-0.639920\pi$$
−0.425551 + 0.904934i $$0.639920\pi$$
$$164$$ 0 0
$$165$$ 1.65142 0.128563
$$166$$ 0 0
$$167$$ 2.82977 0.218974 0.109487 0.993988i $$-0.465079\pi$$
0.109487 + 0.993988i $$0.465079\pi$$
$$168$$ 0 0
$$169$$ 14.7357 1.13351
$$170$$ 0 0
$$171$$ 2.91188 0.222677
$$172$$ 0 0
$$173$$ 9.26647 0.704516 0.352258 0.935903i $$-0.385414\pi$$
0.352258 + 0.935903i $$0.385414\pi$$
$$174$$ 0 0
$$175$$ −3.56331 −0.269361
$$176$$ 0 0
$$177$$ 2.81263 0.211410
$$178$$ 0 0
$$179$$ 3.59067 0.268379 0.134190 0.990956i $$-0.457157\pi$$
0.134190 + 0.990956i $$0.457157\pi$$
$$180$$ 0 0
$$181$$ 19.7630 1.46897 0.734487 0.678623i $$-0.237423\pi$$
0.734487 + 0.678623i $$0.237423\pi$$
$$182$$ 0 0
$$183$$ 2.73532 0.202200
$$184$$ 0 0
$$185$$ −3.61504 −0.265783
$$186$$ 0 0
$$187$$ −7.82377 −0.572131
$$188$$ 0 0
$$189$$ −6.25323 −0.454855
$$190$$ 0 0
$$191$$ 8.31398 0.601579 0.300789 0.953691i $$-0.402750\pi$$
0.300789 + 0.953691i $$0.402750\pi$$
$$192$$ 0 0
$$193$$ −22.2514 −1.60169 −0.800847 0.598869i $$-0.795617\pi$$
−0.800847 + 0.598869i $$0.795617\pi$$
$$194$$ 0 0
$$195$$ 1.56331 0.111951
$$196$$ 0 0
$$197$$ 8.81263 0.627874 0.313937 0.949444i $$-0.398352\pi$$
0.313937 + 0.949444i $$0.398352\pi$$
$$198$$ 0 0
$$199$$ 21.0659 1.49332 0.746660 0.665206i $$-0.231656\pi$$
0.746660 + 0.665206i $$0.231656\pi$$
$$200$$ 0 0
$$201$$ −1.41323 −0.0996818
$$202$$ 0 0
$$203$$ 5.01114 0.351713
$$204$$ 0 0
$$205$$ 4.34858 0.303718
$$206$$ 0 0
$$207$$ 20.2947 1.41058
$$208$$ 0 0
$$209$$ 5.56331 0.384822
$$210$$ 0 0
$$211$$ −5.34556 −0.368004 −0.184002 0.982926i $$-0.558905\pi$$
−0.184002 + 0.982926i $$0.558905\pi$$
$$212$$ 0 0
$$213$$ 4.17623 0.286151
$$214$$ 0 0
$$215$$ 3.56331 0.243016
$$216$$ 0 0
$$217$$ −6.25323 −0.424497
$$218$$ 0 0
$$219$$ −1.95728 −0.132261
$$220$$ 0 0
$$221$$ −7.40632 −0.498203
$$222$$ 0 0
$$223$$ −3.10947 −0.208226 −0.104113 0.994565i $$-0.533200\pi$$
−0.104113 + 0.994565i $$0.533200\pi$$
$$224$$ 0 0
$$225$$ −2.91188 −0.194126
$$226$$ 0 0
$$227$$ −14.4692 −0.960354 −0.480177 0.877172i $$-0.659428\pi$$
−0.480177 + 0.877172i $$0.659428\pi$$
$$228$$ 0 0
$$229$$ 5.21473 0.344599 0.172299 0.985045i $$-0.444880\pi$$
0.172299 + 0.985045i $$0.444880\pi$$
$$230$$ 0 0
$$231$$ −5.88452 −0.387173
$$232$$ 0 0
$$233$$ −3.18737 −0.208811 −0.104406 0.994535i $$-0.533294\pi$$
−0.104406 + 0.994535i $$0.533294\pi$$
$$234$$ 0 0
$$235$$ −8.26046 −0.538853
$$236$$ 0 0
$$237$$ −1.62527 −0.105572
$$238$$ 0 0
$$239$$ 16.5209 1.06865 0.534325 0.845279i $$-0.320566\pi$$
0.534325 + 0.845279i $$0.320566\pi$$
$$240$$ 0 0
$$241$$ −12.2271 −0.787615 −0.393807 0.919193i $$-0.628842\pi$$
−0.393807 + 0.919193i $$0.628842\pi$$
$$242$$ 0 0
$$243$$ −7.70316 −0.494158
$$244$$ 0 0
$$245$$ 5.69716 0.363978
$$246$$ 0 0
$$247$$ 5.26647 0.335097
$$248$$ 0 0
$$249$$ −1.23397 −0.0781996
$$250$$ 0 0
$$251$$ −4.52093 −0.285358 −0.142679 0.989769i $$-0.545572\pi$$
−0.142679 + 0.989769i $$0.545572\pi$$
$$252$$ 0 0
$$253$$ 38.7742 2.43771
$$254$$ 0 0
$$255$$ −0.417453 −0.0261420
$$256$$ 0 0
$$257$$ 15.9290 0.993625 0.496813 0.867858i $$-0.334504\pi$$
0.496813 + 0.867858i $$0.334504\pi$$
$$258$$ 0 0
$$259$$ 12.8815 0.800418
$$260$$ 0 0
$$261$$ 4.09503 0.253476
$$262$$ 0 0
$$263$$ 0.854147 0.0526689 0.0263345 0.999653i $$-0.491617\pi$$
0.0263345 + 0.999653i $$0.491617\pi$$
$$264$$ 0 0
$$265$$ 7.61504 0.467788
$$266$$ 0 0
$$267$$ 2.73988 0.167678
$$268$$ 0 0
$$269$$ −10.3913 −0.633569 −0.316784 0.948498i $$-0.602603\pi$$
−0.316784 + 0.948498i $$0.602603\pi$$
$$270$$ 0 0
$$271$$ −8.19971 −0.498097 −0.249048 0.968491i $$-0.580118\pi$$
−0.249048 + 0.968491i $$0.580118\pi$$
$$272$$ 0 0
$$273$$ −5.57054 −0.337145
$$274$$ 0 0
$$275$$ −5.56331 −0.335480
$$276$$ 0 0
$$277$$ −14.6484 −0.880137 −0.440069 0.897964i $$-0.645046\pi$$
−0.440069 + 0.897964i $$0.645046\pi$$
$$278$$ 0 0
$$279$$ −5.11005 −0.305931
$$280$$ 0 0
$$281$$ −31.6129 −1.88587 −0.942935 0.332977i $$-0.891947\pi$$
−0.942935 + 0.332977i $$0.891947\pi$$
$$282$$ 0 0
$$283$$ −24.7326 −1.47020 −0.735101 0.677957i $$-0.762865\pi$$
−0.735101 + 0.677957i $$0.762865\pi$$
$$284$$ 0 0
$$285$$ 0.296842 0.0175834
$$286$$ 0 0
$$287$$ −15.4953 −0.914660
$$288$$ 0 0
$$289$$ −15.0223 −0.883663
$$290$$ 0 0
$$291$$ −3.41712 −0.200315
$$292$$ 0 0
$$293$$ −30.2857 −1.76931 −0.884655 0.466245i $$-0.845606\pi$$
−0.884655 + 0.466245i $$0.845606\pi$$
$$294$$ 0 0
$$295$$ −9.47519 −0.551667
$$296$$ 0 0
$$297$$ −9.76302 −0.566508
$$298$$ 0 0
$$299$$ 36.7053 2.12272
$$300$$ 0 0
$$301$$ −12.6972 −0.731852
$$302$$ 0 0
$$303$$ −3.51791 −0.202099
$$304$$ 0 0
$$305$$ −9.21473 −0.527634
$$306$$ 0 0
$$307$$ 23.0629 1.31627 0.658134 0.752901i $$-0.271346\pi$$
0.658134 + 0.752901i $$0.271346\pi$$
$$308$$ 0 0
$$309$$ −0.402096 −0.0228744
$$310$$ 0 0
$$311$$ −10.3152 −0.584921 −0.292460 0.956278i $$-0.594474\pi$$
−0.292460 + 0.956278i $$0.594474\pi$$
$$312$$ 0 0
$$313$$ −4.95038 −0.279812 −0.139906 0.990165i $$-0.544680\pi$$
−0.139906 + 0.990165i $$0.544680\pi$$
$$314$$ 0 0
$$315$$ 10.3759 0.584618
$$316$$ 0 0
$$317$$ 14.2433 0.799985 0.399992 0.916518i $$-0.369013\pi$$
0.399992 + 0.916518i $$0.369013\pi$$
$$318$$ 0 0
$$319$$ 7.82377 0.438047
$$320$$ 0 0
$$321$$ −2.09656 −0.117018
$$322$$ 0 0
$$323$$ −1.40632 −0.0782495
$$324$$ 0 0
$$325$$ −5.26647 −0.292131
$$326$$ 0 0
$$327$$ 3.02314 0.167180
$$328$$ 0 0
$$329$$ 29.4346 1.62278
$$330$$ 0 0
$$331$$ −2.04272 −0.112278 −0.0561390 0.998423i $$-0.517879\pi$$
−0.0561390 + 0.998423i $$0.517879\pi$$
$$332$$ 0 0
$$333$$ 10.5266 0.576854
$$334$$ 0 0
$$335$$ 4.76090 0.260116
$$336$$ 0 0
$$337$$ 34.6951 1.88996 0.944980 0.327128i $$-0.106081\pi$$
0.944980 + 0.327128i $$0.106081\pi$$
$$338$$ 0 0
$$339$$ −0.552170 −0.0299898
$$340$$ 0 0
$$341$$ −9.76302 −0.528697
$$342$$ 0 0
$$343$$ 4.64243 0.250668
$$344$$ 0 0
$$345$$ 2.06888 0.111385
$$346$$ 0 0
$$347$$ −7.35280 −0.394719 −0.197359 0.980331i $$-0.563237\pi$$
−0.197359 + 0.980331i $$0.563237\pi$$
$$348$$ 0 0
$$349$$ 31.7630 1.70024 0.850118 0.526593i $$-0.176531\pi$$
0.850118 + 0.526593i $$0.176531\pi$$
$$350$$ 0 0
$$351$$ −9.24209 −0.493306
$$352$$ 0 0
$$353$$ −7.52179 −0.400345 −0.200172 0.979761i $$-0.564150\pi$$
−0.200172 + 0.979761i $$0.564150\pi$$
$$354$$ 0 0
$$355$$ −14.0689 −0.746698
$$356$$ 0 0
$$357$$ 1.48751 0.0787276
$$358$$ 0 0
$$359$$ −30.3982 −1.60436 −0.802178 0.597085i $$-0.796326\pi$$
−0.802178 + 0.597085i $$0.796326\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −5.92211 −0.310830
$$364$$ 0 0
$$365$$ 6.59368 0.345129
$$366$$ 0 0
$$367$$ 3.91577 0.204401 0.102201 0.994764i $$-0.467412\pi$$
0.102201 + 0.994764i $$0.467412\pi$$
$$368$$ 0 0
$$369$$ −12.6626 −0.659186
$$370$$ 0 0
$$371$$ −27.1347 −1.40877
$$372$$ 0 0
$$373$$ −26.7759 −1.38641 −0.693203 0.720743i $$-0.743801\pi$$
−0.693203 + 0.720743i $$0.743801\pi$$
$$374$$ 0 0
$$375$$ −0.296842 −0.0153288
$$376$$ 0 0
$$377$$ 7.40632 0.381445
$$378$$ 0 0
$$379$$ −14.9504 −0.767950 −0.383975 0.923344i $$-0.625445\pi$$
−0.383975 + 0.923344i $$0.625445\pi$$
$$380$$ 0 0
$$381$$ 1.45171 0.0743735
$$382$$ 0 0
$$383$$ −27.9910 −1.43027 −0.715136 0.698985i $$-0.753635\pi$$
−0.715136 + 0.698985i $$0.753635\pi$$
$$384$$ 0 0
$$385$$ 19.8238 1.01031
$$386$$ 0 0
$$387$$ −10.3759 −0.527439
$$388$$ 0 0
$$389$$ 35.2036 1.78489 0.892447 0.451152i $$-0.148987\pi$$
0.892447 + 0.451152i $$0.148987\pi$$
$$390$$ 0 0
$$391$$ −9.80150 −0.495683
$$392$$ 0 0
$$393$$ 0.834907 0.0421155
$$394$$ 0 0
$$395$$ 5.47519 0.275487
$$396$$ 0 0
$$397$$ 35.9735 1.80546 0.902730 0.430208i $$-0.141560\pi$$
0.902730 + 0.430208i $$0.141560\pi$$
$$398$$ 0 0
$$399$$ −1.05774 −0.0529532
$$400$$ 0 0
$$401$$ 23.2421 1.16065 0.580327 0.814383i $$-0.302925\pi$$
0.580327 + 0.814383i $$0.302925\pi$$
$$402$$ 0 0
$$403$$ −9.24209 −0.460381
$$404$$ 0 0
$$405$$ 8.21473 0.408193
$$406$$ 0 0
$$407$$ 20.1116 0.996895
$$408$$ 0 0
$$409$$ −31.8926 −1.57699 −0.788495 0.615041i $$-0.789139\pi$$
−0.788495 + 0.615041i $$0.789139\pi$$
$$410$$ 0 0
$$411$$ −2.73988 −0.135148
$$412$$ 0 0
$$413$$ 33.7630 1.66137
$$414$$ 0 0
$$415$$ 4.15699 0.204059
$$416$$ 0 0
$$417$$ −1.09202 −0.0534763
$$418$$ 0 0
$$419$$ −31.8238 −1.55469 −0.777346 0.629073i $$-0.783435\pi$$
−0.777346 + 0.629073i $$0.783435\pi$$
$$420$$ 0 0
$$421$$ −0.348578 −0.0169887 −0.00849433 0.999964i $$-0.502704\pi$$
−0.00849433 + 0.999964i $$0.502704\pi$$
$$422$$ 0 0
$$423$$ 24.0535 1.16952
$$424$$ 0 0
$$425$$ 1.40632 0.0682164
$$426$$ 0 0
$$427$$ 32.8349 1.58899
$$428$$ 0 0
$$429$$ −8.69716 −0.419903
$$430$$ 0 0
$$431$$ 29.2764 1.41019 0.705097 0.709111i $$-0.250904\pi$$
0.705097 + 0.709111i $$0.250904\pi$$
$$432$$ 0 0
$$433$$ −0.883290 −0.0424482 −0.0212241 0.999775i $$-0.506756\pi$$
−0.0212241 + 0.999775i $$0.506756\pi$$
$$434$$ 0 0
$$435$$ 0.417453 0.0200154
$$436$$ 0 0
$$437$$ 6.96962 0.333402
$$438$$ 0 0
$$439$$ −13.8584 −0.661424 −0.330712 0.943732i $$-0.607289\pi$$
−0.330712 + 0.943732i $$0.607289\pi$$
$$440$$ 0 0
$$441$$ −16.5895 −0.789974
$$442$$ 0 0
$$443$$ 13.5753 0.644982 0.322491 0.946572i $$-0.395480\pi$$
0.322491 + 0.946572i $$0.395480\pi$$
$$444$$ 0 0
$$445$$ −9.23009 −0.437548
$$446$$ 0 0
$$447$$ 2.10735 0.0996745
$$448$$ 0 0
$$449$$ −15.6334 −0.737785 −0.368893 0.929472i $$-0.620263\pi$$
−0.368893 + 0.929472i $$0.620263\pi$$
$$450$$ 0 0
$$451$$ −24.1925 −1.13918
$$452$$ 0 0
$$453$$ 5.44904 0.256018
$$454$$ 0 0
$$455$$ 18.7660 0.879765
$$456$$ 0 0
$$457$$ 5.30284 0.248057 0.124028 0.992279i $$-0.460419\pi$$
0.124028 + 0.992279i $$0.460419\pi$$
$$458$$ 0 0
$$459$$ 2.46794 0.115193
$$460$$ 0 0
$$461$$ −0.374734 −0.0174531 −0.00872656 0.999962i $$-0.502778\pi$$
−0.00872656 + 0.999962i $$0.502778\pi$$
$$462$$ 0 0
$$463$$ 6.65564 0.309314 0.154657 0.987968i $$-0.450573\pi$$
0.154657 + 0.987968i $$0.450573\pi$$
$$464$$ 0 0
$$465$$ −0.520926 −0.0241574
$$466$$ 0 0
$$467$$ 0.854147 0.0395252 0.0197626 0.999805i $$-0.493709\pi$$
0.0197626 + 0.999805i $$0.493709\pi$$
$$468$$ 0 0
$$469$$ −16.9645 −0.783349
$$470$$ 0 0
$$471$$ 5.11461 0.235669
$$472$$ 0 0
$$473$$ −19.8238 −0.911498
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −22.1741 −1.01528
$$478$$ 0 0
$$479$$ −17.0731 −0.780090 −0.390045 0.920796i $$-0.627540\pi$$
−0.390045 + 0.920796i $$0.627540\pi$$
$$480$$ 0 0
$$481$$ 19.0385 0.868081
$$482$$ 0 0
$$483$$ −7.37204 −0.335439
$$484$$ 0 0
$$485$$ 11.5116 0.522713
$$486$$ 0 0
$$487$$ 12.8259 0.581197 0.290598 0.956845i $$-0.406146\pi$$
0.290598 + 0.956845i $$0.406146\pi$$
$$488$$ 0 0
$$489$$ 3.22553 0.145863
$$490$$ 0 0
$$491$$ 10.4054 0.469591 0.234796 0.972045i $$-0.424558\pi$$
0.234796 + 0.972045i $$0.424558\pi$$
$$492$$ 0 0
$$493$$ −1.97773 −0.0890723
$$494$$ 0 0
$$495$$ 16.1997 0.728123
$$496$$ 0 0
$$497$$ 50.1317 2.24872
$$498$$ 0 0
$$499$$ −36.1612 −1.61880 −0.809399 0.587258i $$-0.800207\pi$$
−0.809399 + 0.587258i $$0.800207\pi$$
$$500$$ 0 0
$$501$$ −0.839995 −0.0375282
$$502$$ 0 0
$$503$$ 33.2536 1.48270 0.741352 0.671117i $$-0.234185\pi$$
0.741352 + 0.671117i $$0.234185\pi$$
$$504$$ 0 0
$$505$$ 11.8511 0.527368
$$506$$ 0 0
$$507$$ −4.37416 −0.194263
$$508$$ 0 0
$$509$$ −7.29084 −0.323161 −0.161580 0.986860i $$-0.551659\pi$$
−0.161580 + 0.986860i $$0.551659\pi$$
$$510$$ 0 0
$$511$$ −23.4953 −1.03937
$$512$$ 0 0
$$513$$ −1.75489 −0.0774805
$$514$$ 0 0
$$515$$ 1.35458 0.0596899
$$516$$ 0 0
$$517$$ 45.9555 2.02112
$$518$$ 0 0
$$519$$ −2.75067 −0.120741
$$520$$ 0 0
$$521$$ −9.87849 −0.432785 −0.216392 0.976306i $$-0.569429\pi$$
−0.216392 + 0.976306i $$0.569429\pi$$
$$522$$ 0 0
$$523$$ 11.1925 0.489414 0.244707 0.969597i $$-0.421308\pi$$
0.244707 + 0.969597i $$0.421308\pi$$
$$524$$ 0 0
$$525$$ 1.05774 0.0461635
$$526$$ 0 0
$$527$$ 2.46794 0.107505
$$528$$ 0 0
$$529$$ 25.5756 1.11198
$$530$$ 0 0
$$531$$ 27.5907 1.19733
$$532$$ 0 0
$$533$$ −22.9016 −0.991980
$$534$$ 0 0
$$535$$ 7.06287 0.305355
$$536$$ 0 0
$$537$$ −1.06586 −0.0459953
$$538$$ 0 0
$$539$$ −31.6950 −1.36520
$$540$$ 0 0
$$541$$ −2.22587 −0.0956974 −0.0478487 0.998855i $$-0.515237\pi$$
−0.0478487 + 0.998855i $$0.515237\pi$$
$$542$$ 0 0
$$543$$ −5.86649 −0.251755
$$544$$ 0 0
$$545$$ −10.1844 −0.436250
$$546$$ 0 0
$$547$$ 34.9675 1.49510 0.747552 0.664204i $$-0.231229\pi$$
0.747552 + 0.664204i $$0.231229\pi$$
$$548$$ 0 0
$$549$$ 26.8322 1.14517
$$550$$ 0 0
$$551$$ 1.40632 0.0599111
$$552$$ 0 0
$$553$$ −19.5098 −0.829641
$$554$$ 0 0
$$555$$ 1.07310 0.0455504
$$556$$ 0 0
$$557$$ 8.88539 0.376486 0.188243 0.982122i $$-0.439721\pi$$
0.188243 + 0.982122i $$0.439721\pi$$
$$558$$ 0 0
$$559$$ −18.7660 −0.793719
$$560$$ 0 0
$$561$$ 2.32242 0.0980527
$$562$$ 0 0
$$563$$ 29.6767 1.25072 0.625362 0.780335i $$-0.284952\pi$$
0.625362 + 0.780335i $$0.284952\pi$$
$$564$$ 0 0
$$565$$ 1.86015 0.0782570
$$566$$ 0 0
$$567$$ −29.2716 −1.22929
$$568$$ 0 0
$$569$$ −22.1152 −0.927116 −0.463558 0.886067i $$-0.653427\pi$$
−0.463558 + 0.886067i $$0.653427\pi$$
$$570$$ 0 0
$$571$$ 33.2656 1.39212 0.696060 0.717983i $$-0.254935\pi$$
0.696060 + 0.717983i $$0.254935\pi$$
$$572$$ 0 0
$$573$$ −2.46794 −0.103100
$$574$$ 0 0
$$575$$ −6.96962 −0.290653
$$576$$ 0 0
$$577$$ 0.934140 0.0388887 0.0194444 0.999811i $$-0.493810\pi$$
0.0194444 + 0.999811i $$0.493810\pi$$
$$578$$ 0 0
$$579$$ 6.60516 0.274501
$$580$$ 0 0
$$581$$ −14.8126 −0.614532
$$582$$ 0 0
$$583$$ −42.3648 −1.75457
$$584$$ 0 0
$$585$$ 15.3353 0.634038
$$586$$ 0 0
$$587$$ −1.57531 −0.0650200 −0.0325100 0.999471i $$-0.510350\pi$$
−0.0325100 + 0.999471i $$0.510350\pi$$
$$588$$ 0 0
$$589$$ −1.75489 −0.0723092
$$590$$ 0 0
$$591$$ −2.61596 −0.107606
$$592$$ 0 0
$$593$$ 26.0162 1.06836 0.534180 0.845371i $$-0.320621\pi$$
0.534180 + 0.845371i $$0.320621\pi$$
$$594$$ 0 0
$$595$$ −5.01114 −0.205437
$$596$$ 0 0
$$597$$ −6.25323 −0.255928
$$598$$ 0 0
$$599$$ 19.5359 0.798217 0.399109 0.916904i $$-0.369320\pi$$
0.399109 + 0.916904i $$0.369320\pi$$
$$600$$ 0 0
$$601$$ 43.5299 1.77562 0.887811 0.460208i $$-0.152225\pi$$
0.887811 + 0.460208i $$0.152225\pi$$
$$602$$ 0 0
$$603$$ −13.8632 −0.564552
$$604$$ 0 0
$$605$$ 19.9504 0.811098
$$606$$ 0 0
$$607$$ −12.5847 −0.510796 −0.255398 0.966836i $$-0.582206\pi$$
−0.255398 + 0.966836i $$0.582206\pi$$
$$608$$ 0 0
$$609$$ −1.48751 −0.0602771
$$610$$ 0 0
$$611$$ 43.5034 1.75996
$$612$$ 0 0
$$613$$ 18.2412 0.736756 0.368378 0.929676i $$-0.379913\pi$$
0.368378 + 0.929676i $$0.379913\pi$$
$$614$$ 0 0
$$615$$ −1.29084 −0.0520517
$$616$$ 0 0
$$617$$ 26.7314 1.07617 0.538084 0.842892i $$-0.319148\pi$$
0.538084 + 0.842892i $$0.319148\pi$$
$$618$$ 0 0
$$619$$ −2.92690 −0.117642 −0.0588211 0.998269i $$-0.518734\pi$$
−0.0588211 + 0.998269i $$0.518734\pi$$
$$620$$ 0 0
$$621$$ −12.2310 −0.490811
$$622$$ 0 0
$$623$$ 32.8896 1.31770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −1.65142 −0.0659514
$$628$$ 0 0
$$629$$ −5.08389 −0.202708
$$630$$ 0 0
$$631$$ −13.2493 −0.527447 −0.263724 0.964598i $$-0.584951\pi$$
−0.263724 + 0.964598i $$0.584951\pi$$
$$632$$ 0 0
$$633$$ 1.58679 0.0630691
$$634$$ 0 0
$$635$$ −4.89053 −0.194075
$$636$$ 0 0
$$637$$ −30.0039 −1.18880
$$638$$ 0 0
$$639$$ 40.9669 1.62063
$$640$$ 0 0
$$641$$ 6.93026 0.273729 0.136864 0.990590i $$-0.456298\pi$$
0.136864 + 0.990590i $$0.456298\pi$$
$$642$$ 0 0
$$643$$ −14.4794 −0.571012 −0.285506 0.958377i $$-0.592162\pi$$
−0.285506 + 0.958377i $$0.592162\pi$$
$$644$$ 0 0
$$645$$ −1.05774 −0.0416484
$$646$$ 0 0
$$647$$ 6.35549 0.249860 0.124930 0.992166i $$-0.460129\pi$$
0.124930 + 0.992166i $$0.460129\pi$$
$$648$$ 0 0
$$649$$ 52.7134 2.06918
$$650$$ 0 0
$$651$$ 1.85622 0.0727510
$$652$$ 0 0
$$653$$ 34.4030 1.34629 0.673146 0.739509i $$-0.264942\pi$$
0.673146 + 0.739509i $$0.264942\pi$$
$$654$$ 0 0
$$655$$ −2.81263 −0.109899
$$656$$ 0 0
$$657$$ −19.2000 −0.749065
$$658$$ 0 0
$$659$$ 19.6214 0.764341 0.382170 0.924092i $$-0.375177\pi$$
0.382170 + 0.924092i $$0.375177\pi$$
$$660$$ 0 0
$$661$$ −39.8054 −1.54825 −0.774126 0.633032i $$-0.781810\pi$$
−0.774126 + 0.633032i $$0.781810\pi$$
$$662$$ 0 0
$$663$$ 2.19850 0.0853828
$$664$$ 0 0
$$665$$ 3.56331 0.138179
$$666$$ 0 0
$$667$$ 9.80150 0.379515
$$668$$ 0 0
$$669$$ 0.923022 0.0356861
$$670$$ 0 0
$$671$$ 51.2644 1.97904
$$672$$ 0 0
$$673$$ −8.90374 −0.343214 −0.171607 0.985166i $$-0.554896\pi$$
−0.171607 + 0.985166i $$0.554896\pi$$
$$674$$ 0 0
$$675$$ 1.75489 0.0675459
$$676$$ 0 0
$$677$$ −22.2695 −0.855886 −0.427943 0.903806i $$-0.640762\pi$$
−0.427943 + 0.903806i $$0.640762\pi$$
$$678$$ 0 0
$$679$$ −41.0193 −1.57417
$$680$$ 0 0
$$681$$ 4.29506 0.164587
$$682$$ 0 0
$$683$$ −15.4054 −0.589472 −0.294736 0.955579i $$-0.595232\pi$$
−0.294736 + 0.955579i $$0.595232\pi$$
$$684$$ 0 0
$$685$$ 9.23009 0.352663
$$686$$ 0 0
$$687$$ −1.54795 −0.0590580
$$688$$ 0 0
$$689$$ −40.1044 −1.52785
$$690$$ 0 0
$$691$$ 17.8773 0.680084 0.340042 0.940410i $$-0.389559\pi$$
0.340042 + 0.940410i $$0.389559\pi$$
$$692$$ 0 0
$$693$$ −57.7245 −2.19277
$$694$$ 0 0
$$695$$ 3.67878 0.139544
$$696$$ 0 0
$$697$$ 6.11548 0.231640
$$698$$ 0 0
$$699$$ 0.946144 0.0357864
$$700$$ 0 0
$$701$$ −35.3609 −1.33556 −0.667782 0.744357i $$-0.732756\pi$$
−0.667782 + 0.744357i $$0.732756\pi$$
$$702$$ 0 0
$$703$$ 3.61504 0.136344
$$704$$ 0 0
$$705$$ 2.45205 0.0923496
$$706$$ 0 0
$$707$$ −42.2292 −1.58819
$$708$$ 0 0
$$709$$ −6.90410 −0.259289 −0.129644 0.991561i $$-0.541384\pi$$
−0.129644 + 0.991561i $$0.541384\pi$$
$$710$$ 0 0
$$711$$ −15.9431 −0.597914
$$712$$ 0 0
$$713$$ −12.2310 −0.458053
$$714$$ 0 0
$$715$$ 29.2990 1.09572
$$716$$ 0 0
$$717$$ −4.90410 −0.183147
$$718$$ 0 0
$$719$$ −38.9431 −1.45233 −0.726167 0.687518i $$-0.758700\pi$$
−0.726167 + 0.687518i $$0.758700\pi$$
$$720$$ 0 0
$$721$$ −4.82678 −0.179759
$$722$$ 0 0
$$723$$ 3.62951 0.134983
$$724$$ 0 0
$$725$$ −1.40632 −0.0522293
$$726$$ 0 0
$$727$$ −6.18347 −0.229332 −0.114666 0.993404i $$-0.536580\pi$$
−0.114666 + 0.993404i $$0.536580\pi$$
$$728$$ 0 0
$$729$$ −22.3576 −0.828058
$$730$$ 0 0
$$731$$ 5.01114 0.185344
$$732$$ 0 0
$$733$$ 0.688715 0.0254383 0.0127191 0.999919i $$-0.495951\pi$$
0.0127191 + 0.999919i $$0.495951\pi$$
$$734$$ 0 0
$$735$$ −1.69115 −0.0623792
$$736$$ 0 0
$$737$$ −26.4863 −0.975636
$$738$$ 0 0
$$739$$ 26.2532 0.965741 0.482870 0.875692i $$-0.339594\pi$$
0.482870 + 0.875692i $$0.339594\pi$$
$$740$$ 0 0
$$741$$ −1.56331 −0.0574295
$$742$$ 0 0
$$743$$ −32.5688 −1.19483 −0.597416 0.801931i $$-0.703806\pi$$
−0.597416 + 0.801931i $$0.703806\pi$$
$$744$$ 0 0
$$745$$ −7.09925 −0.260096
$$746$$ 0 0
$$747$$ −12.1047 −0.442887
$$748$$ 0 0
$$749$$ −25.1672 −0.919589
$$750$$ 0 0
$$751$$ 45.4833 1.65971 0.829855 0.557979i $$-0.188423\pi$$
0.829855 + 0.557979i $$0.188423\pi$$
$$752$$ 0 0
$$753$$ 1.34200 0.0489052
$$754$$ 0 0
$$755$$ −18.3567 −0.668069
$$756$$ 0 0
$$757$$ −2.74947 −0.0999311 −0.0499656 0.998751i $$-0.515911\pi$$
−0.0499656 + 0.998751i $$0.515911\pi$$
$$758$$ 0 0
$$759$$ −11.5098 −0.417779
$$760$$ 0 0
$$761$$ −33.2978 −1.20704 −0.603521 0.797347i $$-0.706236\pi$$
−0.603521 + 0.797347i $$0.706236\pi$$
$$762$$ 0 0
$$763$$ 36.2900 1.31379
$$764$$ 0 0
$$765$$ −4.09503 −0.148056
$$766$$ 0 0
$$767$$ 49.9008 1.80181
$$768$$ 0 0
$$769$$ −19.1540 −0.690710 −0.345355 0.938472i $$-0.612241\pi$$
−0.345355 + 0.938472i $$0.612241\pi$$
$$770$$ 0 0
$$771$$ −4.72840 −0.170289
$$772$$ 0 0
$$773$$ −0.569309 −0.0204766 −0.0102383 0.999948i $$-0.503259\pi$$
−0.0102383 + 0.999948i $$0.503259\pi$$
$$774$$ 0 0
$$775$$ 1.75489 0.0630377
$$776$$ 0 0
$$777$$ −3.82377 −0.137177
$$778$$ 0 0
$$779$$ −4.34858 −0.155804
$$780$$ 0 0
$$781$$ 78.2695 2.80070
$$782$$ 0 0
$$783$$ −2.46794 −0.0881969
$$784$$ 0 0
$$785$$ −17.2301 −0.614968
$$786$$ 0 0
$$787$$ −15.9991 −0.570307 −0.285153 0.958482i $$-0.592044\pi$$
−0.285153 + 0.958482i $$0.592044\pi$$
$$788$$ 0 0
$$789$$ −0.253546 −0.00902649
$$790$$ 0 0
$$791$$ −6.62828 −0.235675
$$792$$ 0 0
$$793$$ 48.5290 1.72332
$$794$$ 0 0
$$795$$ −2.26046 −0.0801704
$$796$$ 0 0
$$797$$ 35.7528 1.26643 0.633214 0.773976i $$-0.281735\pi$$
0.633214 + 0.773976i $$0.281735\pi$$
$$798$$ 0 0
$$799$$ −11.6168 −0.410974
$$800$$ 0 0
$$801$$ 26.8769 0.949650
$$802$$ 0 0
$$803$$ −36.6827 −1.29450
$$804$$ 0 0
$$805$$ 24.8349 0.875315
$$806$$ 0 0
$$807$$ 3.08457 0.108582
$$808$$ 0 0
$$809$$ 23.2036 0.815796 0.407898 0.913028i $$-0.366262\pi$$
0.407898 + 0.913028i $$0.366262\pi$$
$$810$$ 0 0
$$811$$ 21.7549 0.763918 0.381959 0.924179i $$-0.375250\pi$$
0.381959 + 0.924179i $$0.375250\pi$$
$$812$$ 0 0
$$813$$ 2.43402 0.0853647
$$814$$ 0 0
$$815$$ −10.8662 −0.380625
$$816$$ 0 0
$$817$$ −3.56331 −0.124664
$$818$$ 0 0
$$819$$ −54.6445 −1.90943
$$820$$ 0 0
$$821$$ −52.2532 −1.82365 −0.911825 0.410579i $$-0.865327\pi$$
−0.911825 + 0.410579i $$0.865327\pi$$
$$822$$ 0 0
$$823$$ −9.44783 −0.329331 −0.164665 0.986349i $$-0.552654\pi$$
−0.164665 + 0.986349i $$0.552654\pi$$
$$824$$ 0 0
$$825$$ 1.65142 0.0574951
$$826$$ 0 0
$$827$$ 50.2216 1.74638 0.873188 0.487383i $$-0.162048\pi$$
0.873188 + 0.487383i $$0.162048\pi$$
$$828$$ 0 0
$$829$$ −44.2066 −1.53536 −0.767680 0.640834i $$-0.778589\pi$$
−0.767680 + 0.640834i $$0.778589\pi$$
$$830$$ 0 0
$$831$$ 4.34826 0.150839
$$832$$ 0 0
$$833$$ 8.01200 0.277599
$$834$$ 0 0
$$835$$ 2.82977 0.0979283
$$836$$ 0 0
$$837$$ 3.07965 0.106448
$$838$$ 0 0
$$839$$ −30.4033 −1.04964 −0.524819 0.851214i $$-0.675867\pi$$
−0.524819 + 0.851214i $$0.675867\pi$$
$$840$$ 0 0
$$841$$ −27.0223 −0.931803
$$842$$ 0 0
$$843$$ 9.38404 0.323204
$$844$$ 0 0
$$845$$ 14.7357 0.506922
$$846$$ 0 0
$$847$$ −71.0893 −2.44266
$$848$$ 0 0
$$849$$ 7.34168 0.251966
$$850$$ 0 0
$$851$$ 25.1955 0.863690
$$852$$ 0 0
$$853$$ −3.93925 −0.134877 −0.0674386 0.997723i $$-0.521483\pi$$
−0.0674386 + 0.997723i $$0.521483\pi$$
$$854$$ 0 0
$$855$$ 2.91188 0.0995844
$$856$$ 0 0
$$857$$ 27.4388 0.937292 0.468646 0.883386i $$-0.344742\pi$$
0.468646 + 0.883386i $$0.344742\pi$$
$$858$$ 0 0
$$859$$ −32.4517 −1.10724 −0.553619 0.832770i $$-0.686754\pi$$
−0.553619 + 0.832770i $$0.686754\pi$$
$$860$$ 0 0
$$861$$ 4.59966 0.156756
$$862$$ 0 0
$$863$$ −2.10861 −0.0717778 −0.0358889 0.999356i $$-0.511426\pi$$
−0.0358889 + 0.999356i $$0.511426\pi$$
$$864$$ 0 0
$$865$$ 9.26647 0.315069
$$866$$ 0 0
$$867$$ 4.45924 0.151444
$$868$$ 0 0
$$869$$ −30.4602 −1.03329
$$870$$ 0 0
$$871$$ −25.0731 −0.849569
$$872$$ 0 0
$$873$$ −33.5204 −1.13449
$$874$$ 0 0
$$875$$ −3.56331 −0.120462
$$876$$ 0 0
$$877$$ 37.6613 1.27173 0.635866 0.771799i $$-0.280643\pi$$
0.635866 + 0.771799i $$0.280643\pi$$
$$878$$ 0 0
$$879$$ 8.99007 0.303227
$$880$$ 0 0
$$881$$ −39.8818 −1.34365 −0.671827 0.740708i $$-0.734490\pi$$
−0.671827 + 0.740708i $$0.734490\pi$$
$$882$$ 0 0
$$883$$ −36.0458 −1.21304 −0.606518 0.795070i $$-0.707434\pi$$
−0.606518 + 0.795070i $$0.707434\pi$$
$$884$$ 0 0
$$885$$ 2.81263 0.0945456
$$886$$ 0 0
$$887$$ −26.2057 −0.879902 −0.439951 0.898022i $$-0.645004\pi$$
−0.439951 + 0.898022i $$0.645004\pi$$
$$888$$ 0 0
$$889$$ 17.4264 0.584464
$$890$$ 0 0
$$891$$ −45.7011 −1.53104
$$892$$ 0 0
$$893$$ 8.26046 0.276426
$$894$$ 0 0
$$895$$ 3.59067 0.120023
$$896$$ 0 0
$$897$$ −10.8957 −0.363796
$$898$$ 0 0
$$899$$ −2.46794 −0.0823103
$$900$$ 0 0
$$901$$ 10.7092 0.356774
$$902$$ 0 0
$$903$$ 3.76905 0.125426
$$904$$ 0 0
$$905$$ 19.7630 0.656945
$$906$$ 0 0
$$907$$ 22.8036 0.757182 0.378591 0.925564i $$-0.376409\pi$$
0.378591 + 0.925564i $$0.376409\pi$$
$$908$$ 0 0
$$909$$ −34.5091 −1.14460
$$910$$ 0 0
$$911$$ 4.44146 0.147152 0.0735761 0.997290i $$-0.476559\pi$$
0.0735761 + 0.997290i $$0.476559\pi$$
$$912$$ 0 0
$$913$$ −23.1266 −0.765379
$$914$$ 0 0
$$915$$ 2.73532 0.0904268
$$916$$ 0 0
$$917$$ 10.0223 0.330965
$$918$$ 0 0
$$919$$ 37.0111 1.22088 0.610442 0.792061i $$-0.290992\pi$$
0.610442 + 0.792061i $$0.290992\pi$$
$$920$$ 0 0
$$921$$ −6.84602 −0.225584
$$922$$ 0 0
$$923$$ 74.0932 2.43881
$$924$$ 0 0
$$925$$ −3.61504 −0.118862
$$926$$ 0 0
$$927$$ −3.94438 −0.129550
$$928$$ 0 0
$$929$$ −3.04185 −0.0997999 −0.0499000 0.998754i $$-0.515890\pi$$
−0.0499000 + 0.998754i $$0.515890\pi$$
$$930$$ 0 0
$$931$$ −5.69716 −0.186717
$$932$$ 0 0
$$933$$ 3.06198 0.100245
$$934$$ 0 0
$$935$$ −7.82377 −0.255865
$$936$$ 0 0
$$937$$ 38.6099 1.26133 0.630666 0.776055i $$-0.282782\pi$$
0.630666 + 0.776055i $$0.282782\pi$$
$$938$$ 0 0
$$939$$ 1.46948 0.0479547
$$940$$ 0 0
$$941$$ −2.69716 −0.0879248 −0.0439624 0.999033i $$-0.513998\pi$$
−0.0439624 + 0.999033i $$0.513998\pi$$
$$942$$ 0 0
$$943$$ −30.3080 −0.986963
$$944$$ 0 0
$$945$$ −6.25323 −0.203418
$$946$$ 0 0
$$947$$ 20.1877 0.656012 0.328006 0.944676i $$-0.393623\pi$$
0.328006 + 0.944676i $$0.393623\pi$$
$$948$$ 0 0
$$949$$ −34.7254 −1.12723
$$950$$ 0 0
$$951$$ −4.22801 −0.137103
$$952$$ 0 0
$$953$$ 5.18559 0.167978 0.0839888 0.996467i $$-0.473234\pi$$
0.0839888 + 0.996467i $$0.473234\pi$$
$$954$$ 0 0
$$955$$ 8.31398 0.269034
$$956$$ 0 0
$$957$$ −2.32242 −0.0750732
$$958$$ 0 0
$$959$$ −32.8896 −1.06206
$$960$$ 0 0
$$961$$ −27.9203 −0.900656
$$962$$ 0 0
$$963$$ −20.5663 −0.662739
$$964$$ 0 0
$$965$$ −22.2514 −0.716299
$$966$$ 0 0
$$967$$ 58.0054 1.86533 0.932665 0.360744i $$-0.117477\pi$$
0.932665 + 0.360744i $$0.117477\pi$$
$$968$$ 0 0
$$969$$ 0.417453 0.0134105
$$970$$ 0 0
$$971$$ 24.3221 0.780533 0.390267 0.920702i $$-0.372383\pi$$
0.390267 + 0.920702i $$0.372383\pi$$
$$972$$ 0 0
$$973$$ −13.1086 −0.420244
$$974$$ 0 0
$$975$$ 1.56331 0.0500659
$$976$$ 0 0
$$977$$ −36.1134 −1.15537 −0.577685 0.816260i $$-0.696044\pi$$
−0.577685 + 0.816260i $$0.696044\pi$$
$$978$$ 0 0
$$979$$ 51.3498 1.64115
$$980$$ 0 0
$$981$$ 29.6557 0.946832
$$982$$ 0 0
$$983$$ 21.1603 0.674909 0.337455 0.941342i $$-0.390434\pi$$
0.337455 + 0.941342i $$0.390434\pi$$
$$984$$ 0 0
$$985$$ 8.81263 0.280794
$$986$$ 0 0
$$987$$ −8.73741 −0.278115
$$988$$ 0 0
$$989$$ −24.8349 −0.789704
$$990$$ 0 0
$$991$$ 20.2652 0.643746 0.321873 0.946783i $$-0.395688\pi$$
0.321873 + 0.946783i $$0.395688\pi$$
$$992$$ 0 0
$$993$$ 0.606364 0.0192424
$$994$$ 0 0
$$995$$ 21.0659 0.667833
$$996$$ 0 0
$$997$$ −24.2875 −0.769193 −0.384597 0.923085i $$-0.625659\pi$$
−0.384597 + 0.923085i $$0.625659\pi$$
$$998$$ 0 0
$$999$$ −6.34402 −0.200716
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 6080.2.a.cc.1.3 4
4.3 odd 2 6080.2.a.ch.1.2 4
8.3 odd 2 1520.2.a.t.1.3 4
8.5 even 2 95.2.a.b.1.4 4
24.5 odd 2 855.2.a.m.1.1 4
40.13 odd 4 475.2.b.e.324.3 8
40.19 odd 2 7600.2.a.cf.1.2 4
40.29 even 2 475.2.a.i.1.1 4
40.37 odd 4 475.2.b.e.324.6 8
56.13 odd 2 4655.2.a.y.1.4 4
120.29 odd 2 4275.2.a.bo.1.4 4
152.37 odd 2 1805.2.a.p.1.1 4
760.189 odd 2 9025.2.a.bf.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.4 4 8.5 even 2
475.2.a.i.1.1 4 40.29 even 2
475.2.b.e.324.3 8 40.13 odd 4
475.2.b.e.324.6 8 40.37 odd 4
855.2.a.m.1.1 4 24.5 odd 2
1520.2.a.t.1.3 4 8.3 odd 2
1805.2.a.p.1.1 4 152.37 odd 2
4275.2.a.bo.1.4 4 120.29 odd 2
4655.2.a.y.1.4 4 56.13 odd 2
6080.2.a.cc.1.3 4 1.1 even 1 trivial
6080.2.a.ch.1.2 4 4.3 odd 2
7600.2.a.cf.1.2 4 40.19 odd 2
9025.2.a.bf.1.4 4 760.189 odd 2