# Properties

 Label 2320.2.a.m.1.3 Level $2320$ Weight $2$ Character 2320.1 Self dual yes Analytic conductor $18.525$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.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: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 580) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.571993$$ of defining polynomial Character $$\chi$$ $$=$$ 2320.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.67282 q^{3} -1.00000 q^{5} +4.67282 q^{7} +4.14399 q^{9} +O(q^{10})$$ $$q+2.67282 q^{3} -1.00000 q^{5} +4.67282 q^{7} +4.14399 q^{9} +0.672824 q^{11} -1.14399 q^{13} -2.67282 q^{15} +3.52884 q^{17} -5.52884 q^{19} +12.4896 q^{21} +3.81681 q^{23} +1.00000 q^{25} +3.05767 q^{27} -1.00000 q^{29} +1.52884 q^{31} +1.79834 q^{33} -4.67282 q^{35} +7.16246 q^{37} -3.05767 q^{39} +2.85601 q^{41} -8.96080 q^{43} -4.14399 q^{45} +6.67282 q^{47} +14.8353 q^{49} +9.43196 q^{51} +10.4896 q^{53} -0.672824 q^{55} -14.7776 q^{57} -10.7776 q^{59} -14.4896 q^{61} +19.3641 q^{63} +1.14399 q^{65} +7.81681 q^{67} +10.2017 q^{69} -4.48963 q^{71} -4.96080 q^{73} +2.67282 q^{75} +3.14399 q^{77} -2.38485 q^{79} -4.25934 q^{81} -14.0185 q^{83} -3.52884 q^{85} -2.67282 q^{87} -1.63362 q^{89} -5.34565 q^{91} +4.08631 q^{93} +5.52884 q^{95} -9.32718 q^{97} +2.78817 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 11 * q^9 $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 11 q^{9} - 8 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} + 16 q^{21} + 3 q^{25} - 8 q^{27} - 3 q^{29} - 4 q^{31} + 24 q^{33} - 4 q^{35} - 10 q^{37} + 8 q^{39} + 10 q^{41} - 14 q^{43} - 11 q^{45} + 10 q^{47} + 3 q^{49} + 24 q^{51} + 10 q^{53} + 8 q^{55} - 20 q^{57} - 8 q^{59} - 22 q^{61} + 8 q^{63} + 2 q^{65} + 12 q^{67} + 12 q^{69} + 8 q^{71} - 2 q^{73} - 2 q^{75} + 8 q^{77} + 23 q^{81} - 12 q^{83} - 2 q^{85} + 2 q^{87} + 18 q^{89} + 4 q^{91} + 28 q^{93} + 8 q^{95} - 38 q^{97} - 36 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 11 * q^9 - 8 * q^11 - 2 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 + 16 * q^21 + 3 * q^25 - 8 * q^27 - 3 * q^29 - 4 * q^31 + 24 * q^33 - 4 * q^35 - 10 * q^37 + 8 * q^39 + 10 * q^41 - 14 * q^43 - 11 * q^45 + 10 * q^47 + 3 * q^49 + 24 * q^51 + 10 * q^53 + 8 * q^55 - 20 * q^57 - 8 * q^59 - 22 * q^61 + 8 * q^63 + 2 * q^65 + 12 * q^67 + 12 * q^69 + 8 * q^71 - 2 * q^73 - 2 * q^75 + 8 * q^77 + 23 * q^81 - 12 * q^83 - 2 * q^85 + 2 * q^87 + 18 * q^89 + 4 * q^91 + 28 * q^93 + 8 * q^95 - 38 * q^97 - 36 * 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
$$3$$ 2.67282 1.54316 0.771578 0.636135i $$-0.219468\pi$$
0.771578 + 0.636135i $$0.219468\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.67282 1.76616 0.883081 0.469221i $$-0.155465\pi$$
0.883081 + 0.469221i $$0.155465\pi$$
$$8$$ 0 0
$$9$$ 4.14399 1.38133
$$10$$ 0 0
$$11$$ 0.672824 0.202864 0.101432 0.994842i $$-0.467658\pi$$
0.101432 + 0.994842i $$0.467658\pi$$
$$12$$ 0 0
$$13$$ −1.14399 −0.317285 −0.158642 0.987336i $$-0.550712\pi$$
−0.158642 + 0.987336i $$0.550712\pi$$
$$14$$ 0 0
$$15$$ −2.67282 −0.690120
$$16$$ 0 0
$$17$$ 3.52884 0.855869 0.427934 0.903810i $$-0.359241\pi$$
0.427934 + 0.903810i $$0.359241\pi$$
$$18$$ 0 0
$$19$$ −5.52884 −1.26840 −0.634201 0.773168i $$-0.718671\pi$$
−0.634201 + 0.773168i $$0.718671\pi$$
$$20$$ 0 0
$$21$$ 12.4896 2.72546
$$22$$ 0 0
$$23$$ 3.81681 0.795860 0.397930 0.917416i $$-0.369729\pi$$
0.397930 + 0.917416i $$0.369729\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.05767 0.588450
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 1.52884 0.274587 0.137294 0.990530i $$-0.456160\pi$$
0.137294 + 0.990530i $$0.456160\pi$$
$$32$$ 0 0
$$33$$ 1.79834 0.313051
$$34$$ 0 0
$$35$$ −4.67282 −0.789851
$$36$$ 0 0
$$37$$ 7.16246 1.17750 0.588750 0.808315i $$-0.299620\pi$$
0.588750 + 0.808315i $$0.299620\pi$$
$$38$$ 0 0
$$39$$ −3.05767 −0.489620
$$40$$ 0 0
$$41$$ 2.85601 0.446034 0.223017 0.974815i $$-0.428409\pi$$
0.223017 + 0.974815i $$0.428409\pi$$
$$42$$ 0 0
$$43$$ −8.96080 −1.36651 −0.683254 0.730180i $$-0.739436\pi$$
−0.683254 + 0.730180i $$0.739436\pi$$
$$44$$ 0 0
$$45$$ −4.14399 −0.617749
$$46$$ 0 0
$$47$$ 6.67282 0.973331 0.486666 0.873588i $$-0.338213\pi$$
0.486666 + 0.873588i $$0.338213\pi$$
$$48$$ 0 0
$$49$$ 14.8353 2.11933
$$50$$ 0 0
$$51$$ 9.43196 1.32074
$$52$$ 0 0
$$53$$ 10.4896 1.44086 0.720431 0.693527i $$-0.243944\pi$$
0.720431 + 0.693527i $$0.243944\pi$$
$$54$$ 0 0
$$55$$ −0.672824 −0.0907235
$$56$$ 0 0
$$57$$ −14.7776 −1.95734
$$58$$ 0 0
$$59$$ −10.7776 −1.40312 −0.701562 0.712608i $$-0.747514\pi$$
−0.701562 + 0.712608i $$0.747514\pi$$
$$60$$ 0 0
$$61$$ −14.4896 −1.85521 −0.927604 0.373566i $$-0.878135\pi$$
−0.927604 + 0.373566i $$0.878135\pi$$
$$62$$ 0 0
$$63$$ 19.3641 2.43965
$$64$$ 0 0
$$65$$ 1.14399 0.141894
$$66$$ 0 0
$$67$$ 7.81681 0.954975 0.477488 0.878638i $$-0.341548\pi$$
0.477488 + 0.878638i $$0.341548\pi$$
$$68$$ 0 0
$$69$$ 10.2017 1.22814
$$70$$ 0 0
$$71$$ −4.48963 −0.532822 −0.266411 0.963860i $$-0.585838\pi$$
−0.266411 + 0.963860i $$0.585838\pi$$
$$72$$ 0 0
$$73$$ −4.96080 −0.580617 −0.290309 0.956933i $$-0.593758\pi$$
−0.290309 + 0.956933i $$0.593758\pi$$
$$74$$ 0 0
$$75$$ 2.67282 0.308631
$$76$$ 0 0
$$77$$ 3.14399 0.358291
$$78$$ 0 0
$$79$$ −2.38485 −0.268317 −0.134158 0.990960i $$-0.542833\pi$$
−0.134158 + 0.990960i $$0.542833\pi$$
$$80$$ 0 0
$$81$$ −4.25934 −0.473259
$$82$$ 0 0
$$83$$ −14.0185 −1.53873 −0.769364 0.638811i $$-0.779427\pi$$
−0.769364 + 0.638811i $$0.779427\pi$$
$$84$$ 0 0
$$85$$ −3.52884 −0.382756
$$86$$ 0 0
$$87$$ −2.67282 −0.286557
$$88$$ 0 0
$$89$$ −1.63362 −0.173163 −0.0865817 0.996245i $$-0.527594\pi$$
−0.0865817 + 0.996245i $$0.527594\pi$$
$$90$$ 0 0
$$91$$ −5.34565 −0.560376
$$92$$ 0 0
$$93$$ 4.08631 0.423731
$$94$$ 0 0
$$95$$ 5.52884 0.567247
$$96$$ 0 0
$$97$$ −9.32718 −0.947031 −0.473516 0.880785i $$-0.657015\pi$$
−0.473516 + 0.880785i $$0.657015\pi$$
$$98$$ 0 0
$$99$$ 2.78817 0.280222
$$100$$ 0 0
$$101$$ −16.3249 −1.62439 −0.812195 0.583386i $$-0.801727\pi$$
−0.812195 + 0.583386i $$0.801727\pi$$
$$102$$ 0 0
$$103$$ 13.5288 1.33304 0.666518 0.745489i $$-0.267784\pi$$
0.666518 + 0.745489i $$0.267784\pi$$
$$104$$ 0 0
$$105$$ −12.4896 −1.21886
$$106$$ 0 0
$$107$$ −0.672824 −0.0650443 −0.0325222 0.999471i $$-0.510354\pi$$
−0.0325222 + 0.999471i $$0.510354\pi$$
$$108$$ 0 0
$$109$$ 13.5473 1.29760 0.648798 0.760960i $$-0.275272\pi$$
0.648798 + 0.760960i $$0.275272\pi$$
$$110$$ 0 0
$$111$$ 19.1440 1.81707
$$112$$ 0 0
$$113$$ 20.0185 1.88318 0.941590 0.336762i $$-0.109332\pi$$
0.941590 + 0.336762i $$0.109332\pi$$
$$114$$ 0 0
$$115$$ −3.81681 −0.355919
$$116$$ 0 0
$$117$$ −4.74066 −0.438275
$$118$$ 0 0
$$119$$ 16.4896 1.51160
$$120$$ 0 0
$$121$$ −10.5473 −0.958846
$$122$$ 0 0
$$123$$ 7.63362 0.688300
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 3.03920 0.269686 0.134843 0.990867i $$-0.456947\pi$$
0.134843 + 0.990867i $$0.456947\pi$$
$$128$$ 0 0
$$129$$ −23.9506 −2.10874
$$130$$ 0 0
$$131$$ 3.81681 0.333476 0.166738 0.986001i $$-0.446677\pi$$
0.166738 + 0.986001i $$0.446677\pi$$
$$132$$ 0 0
$$133$$ −25.8353 −2.24020
$$134$$ 0 0
$$135$$ −3.05767 −0.263163
$$136$$ 0 0
$$137$$ 16.1048 1.37592 0.687962 0.725746i $$-0.258505\pi$$
0.687962 + 0.725746i $$0.258505\pi$$
$$138$$ 0 0
$$139$$ −3.05767 −0.259349 −0.129674 0.991557i $$-0.541393\pi$$
−0.129674 + 0.991557i $$0.541393\pi$$
$$140$$ 0 0
$$141$$ 17.8353 1.50200
$$142$$ 0 0
$$143$$ −0.769701 −0.0643657
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 39.6521 3.27045
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −6.77761 −0.551554 −0.275777 0.961222i $$-0.588935\pi$$
−0.275777 + 0.961222i $$0.588935\pi$$
$$152$$ 0 0
$$153$$ 14.6235 1.18224
$$154$$ 0 0
$$155$$ −1.52884 −0.122799
$$156$$ 0 0
$$157$$ 8.96080 0.715149 0.357575 0.933885i $$-0.383604\pi$$
0.357575 + 0.933885i $$0.383604\pi$$
$$158$$ 0 0
$$159$$ 28.0369 2.22347
$$160$$ 0 0
$$161$$ 17.8353 1.40562
$$162$$ 0 0
$$163$$ 6.30644 0.493959 0.246979 0.969021i $$-0.420562\pi$$
0.246979 + 0.969021i $$0.420562\pi$$
$$164$$ 0 0
$$165$$ −1.79834 −0.140001
$$166$$ 0 0
$$167$$ −12.7961 −0.990190 −0.495095 0.868839i $$-0.664867\pi$$
−0.495095 + 0.868839i $$0.664867\pi$$
$$168$$ 0 0
$$169$$ −11.6913 −0.899330
$$170$$ 0 0
$$171$$ −22.9114 −1.75208
$$172$$ 0 0
$$173$$ 18.1233 1.37789 0.688943 0.724816i $$-0.258075\pi$$
0.688943 + 0.724816i $$0.258075\pi$$
$$174$$ 0 0
$$175$$ 4.67282 0.353232
$$176$$ 0 0
$$177$$ −28.8066 −2.16524
$$178$$ 0 0
$$179$$ −6.28797 −0.469985 −0.234993 0.971997i $$-0.575507\pi$$
−0.234993 + 0.971997i $$0.575507\pi$$
$$180$$ 0 0
$$181$$ 4.56804 0.339540 0.169770 0.985484i $$-0.445698\pi$$
0.169770 + 0.985484i $$0.445698\pi$$
$$182$$ 0 0
$$183$$ −38.7282 −2.86287
$$184$$ 0 0
$$185$$ −7.16246 −0.526594
$$186$$ 0 0
$$187$$ 2.37429 0.173625
$$188$$ 0 0
$$189$$ 14.2880 1.03930
$$190$$ 0 0
$$191$$ −21.6521 −1.56669 −0.783345 0.621587i $$-0.786488\pi$$
−0.783345 + 0.621587i $$0.786488\pi$$
$$192$$ 0 0
$$193$$ −13.2409 −0.953098 −0.476549 0.879148i $$-0.658113\pi$$
−0.476549 + 0.879148i $$0.658113\pi$$
$$194$$ 0 0
$$195$$ 3.05767 0.218965
$$196$$ 0 0
$$197$$ 7.34565 0.523356 0.261678 0.965155i $$-0.415724\pi$$
0.261678 + 0.965155i $$0.415724\pi$$
$$198$$ 0 0
$$199$$ −18.7776 −1.33111 −0.665555 0.746349i $$-0.731805\pi$$
−0.665555 + 0.746349i $$0.731805\pi$$
$$200$$ 0 0
$$201$$ 20.8930 1.47368
$$202$$ 0 0
$$203$$ −4.67282 −0.327968
$$204$$ 0 0
$$205$$ −2.85601 −0.199473
$$206$$ 0 0
$$207$$ 15.8168 1.09934
$$208$$ 0 0
$$209$$ −3.71993 −0.257313
$$210$$ 0 0
$$211$$ −17.6521 −1.21522 −0.607610 0.794235i $$-0.707872\pi$$
−0.607610 + 0.794235i $$0.707872\pi$$
$$212$$ 0 0
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ 8.96080 0.611121
$$216$$ 0 0
$$217$$ 7.14399 0.484965
$$218$$ 0 0
$$219$$ −13.2593 −0.895983
$$220$$ 0 0
$$221$$ −4.03694 −0.271554
$$222$$ 0 0
$$223$$ −7.45043 −0.498918 −0.249459 0.968385i $$-0.580253\pi$$
−0.249459 + 0.968385i $$0.580253\pi$$
$$224$$ 0 0
$$225$$ 4.14399 0.276266
$$226$$ 0 0
$$227$$ 24.2201 1.60755 0.803773 0.594936i $$-0.202822\pi$$
0.803773 + 0.594936i $$0.202822\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 8.40332 0.552898
$$232$$ 0 0
$$233$$ −0.287973 −0.0188657 −0.00943287 0.999956i $$-0.503003\pi$$
−0.00943287 + 0.999956i $$0.503003\pi$$
$$234$$ 0 0
$$235$$ −6.67282 −0.435287
$$236$$ 0 0
$$237$$ −6.37429 −0.414054
$$238$$ 0 0
$$239$$ −7.51037 −0.485805 −0.242903 0.970051i $$-0.578100\pi$$
−0.242903 + 0.970051i $$0.578100\pi$$
$$240$$ 0 0
$$241$$ −21.6706 −1.39592 −0.697962 0.716135i $$-0.745909\pi$$
−0.697962 + 0.716135i $$0.745909\pi$$
$$242$$ 0 0
$$243$$ −20.5575 −1.31876
$$244$$ 0 0
$$245$$ −14.8353 −0.947791
$$246$$ 0 0
$$247$$ 6.32492 0.402445
$$248$$ 0 0
$$249$$ −37.4689 −2.37450
$$250$$ 0 0
$$251$$ 12.6728 0.799902 0.399951 0.916537i $$-0.369027\pi$$
0.399951 + 0.916537i $$0.369027\pi$$
$$252$$ 0 0
$$253$$ 2.56804 0.161451
$$254$$ 0 0
$$255$$ −9.43196 −0.590652
$$256$$ 0 0
$$257$$ 16.9714 1.05864 0.529322 0.848421i $$-0.322446\pi$$
0.529322 + 0.848421i $$0.322446\pi$$
$$258$$ 0 0
$$259$$ 33.4689 2.07966
$$260$$ 0 0
$$261$$ −4.14399 −0.256506
$$262$$ 0 0
$$263$$ −5.90312 −0.364002 −0.182001 0.983298i $$-0.558257\pi$$
−0.182001 + 0.983298i $$0.558257\pi$$
$$264$$ 0 0
$$265$$ −10.4896 −0.644373
$$266$$ 0 0
$$267$$ −4.36638 −0.267218
$$268$$ 0 0
$$269$$ −14.8560 −0.905787 −0.452894 0.891565i $$-0.649608\pi$$
−0.452894 + 0.891565i $$0.649608\pi$$
$$270$$ 0 0
$$271$$ −22.5944 −1.37251 −0.686257 0.727360i $$-0.740747\pi$$
−0.686257 + 0.727360i $$0.740747\pi$$
$$272$$ 0 0
$$273$$ −14.2880 −0.864747
$$274$$ 0 0
$$275$$ 0.672824 0.0405728
$$276$$ 0 0
$$277$$ 27.4689 1.65045 0.825223 0.564807i $$-0.191049\pi$$
0.825223 + 0.564807i $$0.191049\pi$$
$$278$$ 0 0
$$279$$ 6.33548 0.379295
$$280$$ 0 0
$$281$$ 17.5473 1.04678 0.523392 0.852092i $$-0.324666\pi$$
0.523392 + 0.852092i $$0.324666\pi$$
$$282$$ 0 0
$$283$$ −26.5081 −1.57574 −0.787872 0.615839i $$-0.788817\pi$$
−0.787872 + 0.615839i $$0.788817\pi$$
$$284$$ 0 0
$$285$$ 14.7776 0.875350
$$286$$ 0 0
$$287$$ 13.3456 0.787769
$$288$$ 0 0
$$289$$ −4.54731 −0.267489
$$290$$ 0 0
$$291$$ −24.9299 −1.46142
$$292$$ 0 0
$$293$$ −22.7098 −1.32672 −0.663359 0.748301i $$-0.730870\pi$$
−0.663359 + 0.748301i $$0.730870\pi$$
$$294$$ 0 0
$$295$$ 10.7776 0.627497
$$296$$ 0 0
$$297$$ 2.05728 0.119375
$$298$$ 0 0
$$299$$ −4.36638 −0.252514
$$300$$ 0 0
$$301$$ −41.8722 −2.41347
$$302$$ 0 0
$$303$$ −43.6336 −2.50669
$$304$$ 0 0
$$305$$ 14.4896 0.829674
$$306$$ 0 0
$$307$$ −20.9608 −1.19630 −0.598148 0.801386i $$-0.704096\pi$$
−0.598148 + 0.801386i $$0.704096\pi$$
$$308$$ 0 0
$$309$$ 36.1602 2.05708
$$310$$ 0 0
$$311$$ 27.4874 1.55867 0.779333 0.626610i $$-0.215558\pi$$
0.779333 + 0.626610i $$0.215558\pi$$
$$312$$ 0 0
$$313$$ −18.8560 −1.06580 −0.532902 0.846177i $$-0.678899\pi$$
−0.532902 + 0.846177i $$0.678899\pi$$
$$314$$ 0 0
$$315$$ −19.3641 −1.09104
$$316$$ 0 0
$$317$$ 29.4504 1.65410 0.827050 0.562128i $$-0.190017\pi$$
0.827050 + 0.562128i $$0.190017\pi$$
$$318$$ 0 0
$$319$$ −0.672824 −0.0376709
$$320$$ 0 0
$$321$$ −1.79834 −0.100374
$$322$$ 0 0
$$323$$ −19.5104 −1.08559
$$324$$ 0 0
$$325$$ −1.14399 −0.0634570
$$326$$ 0 0
$$327$$ 36.2096 2.00239
$$328$$ 0 0
$$329$$ 31.1809 1.71906
$$330$$ 0 0
$$331$$ −3.60724 −0.198272 −0.0991360 0.995074i $$-0.531608\pi$$
−0.0991360 + 0.995074i $$0.531608\pi$$
$$332$$ 0 0
$$333$$ 29.6811 1.62652
$$334$$ 0 0
$$335$$ −7.81681 −0.427078
$$336$$ 0 0
$$337$$ −18.3064 −0.997216 −0.498608 0.866828i $$-0.666155\pi$$
−0.498608 + 0.866828i $$0.666155\pi$$
$$338$$ 0 0
$$339$$ 53.5058 2.90604
$$340$$ 0 0
$$341$$ 1.02864 0.0557039
$$342$$ 0 0
$$343$$ 36.6129 1.97691
$$344$$ 0 0
$$345$$ −10.2017 −0.549239
$$346$$ 0 0
$$347$$ −6.38485 −0.342757 −0.171378 0.985205i $$-0.554822\pi$$
−0.171378 + 0.985205i $$0.554822\pi$$
$$348$$ 0 0
$$349$$ 17.2672 0.924294 0.462147 0.886803i $$-0.347079\pi$$
0.462147 + 0.886803i $$0.347079\pi$$
$$350$$ 0 0
$$351$$ −3.49794 −0.186706
$$352$$ 0 0
$$353$$ −9.91369 −0.527652 −0.263826 0.964570i $$-0.584985\pi$$
−0.263826 + 0.964570i $$0.584985\pi$$
$$354$$ 0 0
$$355$$ 4.48963 0.238285
$$356$$ 0 0
$$357$$ 44.0739 2.33264
$$358$$ 0 0
$$359$$ 3.81681 0.201444 0.100722 0.994915i $$-0.467885\pi$$
0.100722 + 0.994915i $$0.467885\pi$$
$$360$$ 0 0
$$361$$ 11.5680 0.608844
$$362$$ 0 0
$$363$$ −28.1911 −1.47965
$$364$$ 0 0
$$365$$ 4.96080 0.259660
$$366$$ 0 0
$$367$$ 7.98153 0.416632 0.208316 0.978062i $$-0.433202\pi$$
0.208316 + 0.978062i $$0.433202\pi$$
$$368$$ 0 0
$$369$$ 11.8353 0.616120
$$370$$ 0 0
$$371$$ 49.0162 2.54479
$$372$$ 0 0
$$373$$ −26.0369 −1.34814 −0.674071 0.738667i $$-0.735456\pi$$
−0.674071 + 0.738667i $$0.735456\pi$$
$$374$$ 0 0
$$375$$ −2.67282 −0.138024
$$376$$ 0 0
$$377$$ 1.14399 0.0589183
$$378$$ 0 0
$$379$$ −28.8824 −1.48359 −0.741794 0.670627i $$-0.766025\pi$$
−0.741794 + 0.670627i $$0.766025\pi$$
$$380$$ 0 0
$$381$$ 8.12325 0.416167
$$382$$ 0 0
$$383$$ −0.0968776 −0.00495021 −0.00247511 0.999997i $$-0.500788\pi$$
−0.00247511 + 0.999997i $$0.500788\pi$$
$$384$$ 0 0
$$385$$ −3.14399 −0.160232
$$386$$ 0 0
$$387$$ −37.1334 −1.88760
$$388$$ 0 0
$$389$$ −0.740665 −0.0375532 −0.0187766 0.999824i $$-0.505977\pi$$
−0.0187766 + 0.999824i $$0.505977\pi$$
$$390$$ 0 0
$$391$$ 13.4689 0.681152
$$392$$ 0 0
$$393$$ 10.2017 0.514606
$$394$$ 0 0
$$395$$ 2.38485 0.119995
$$396$$ 0 0
$$397$$ −23.9216 −1.20059 −0.600295 0.799779i $$-0.704950\pi$$
−0.600295 + 0.799779i $$0.704950\pi$$
$$398$$ 0 0
$$399$$ −69.0532 −3.45698
$$400$$ 0 0
$$401$$ −5.54731 −0.277019 −0.138510 0.990361i $$-0.544231\pi$$
−0.138510 + 0.990361i $$0.544231\pi$$
$$402$$ 0 0
$$403$$ −1.74897 −0.0871224
$$404$$ 0 0
$$405$$ 4.25934 0.211648
$$406$$ 0 0
$$407$$ 4.81907 0.238872
$$408$$ 0 0
$$409$$ −34.8930 −1.72535 −0.862673 0.505762i $$-0.831211\pi$$
−0.862673 + 0.505762i $$0.831211\pi$$
$$410$$ 0 0
$$411$$ 43.0452 2.12327
$$412$$ 0 0
$$413$$ −50.3619 −2.47814
$$414$$ 0 0
$$415$$ 14.0185 0.688140
$$416$$ 0 0
$$417$$ −8.17262 −0.400215
$$418$$ 0 0
$$419$$ −12.1233 −0.592260 −0.296130 0.955148i $$-0.595696\pi$$
−0.296130 + 0.955148i $$0.595696\pi$$
$$420$$ 0 0
$$421$$ 19.0656 0.929200 0.464600 0.885521i $$-0.346198\pi$$
0.464600 + 0.885521i $$0.346198\pi$$
$$422$$ 0 0
$$423$$ 27.6521 1.34449
$$424$$ 0 0
$$425$$ 3.52884 0.171174
$$426$$ 0 0
$$427$$ −67.7075 −3.27660
$$428$$ 0 0
$$429$$ −2.05728 −0.0993262
$$430$$ 0 0
$$431$$ −2.69129 −0.129635 −0.0648176 0.997897i $$-0.520647\pi$$
−0.0648176 + 0.997897i $$0.520647\pi$$
$$432$$ 0 0
$$433$$ 17.4874 0.840390 0.420195 0.907434i $$-0.361962\pi$$
0.420195 + 0.907434i $$0.361962\pi$$
$$434$$ 0 0
$$435$$ 2.67282 0.128152
$$436$$ 0 0
$$437$$ −21.1025 −1.00947
$$438$$ 0 0
$$439$$ 4.45269 0.212515 0.106258 0.994339i $$-0.466113\pi$$
0.106258 + 0.994339i $$0.466113\pi$$
$$440$$ 0 0
$$441$$ 61.4772 2.92749
$$442$$ 0 0
$$443$$ −11.6151 −0.551852 −0.275926 0.961179i $$-0.588985\pi$$
−0.275926 + 0.961179i $$0.588985\pi$$
$$444$$ 0 0
$$445$$ 1.63362 0.0774410
$$446$$ 0 0
$$447$$ 48.1108 2.27556
$$448$$ 0 0
$$449$$ −9.91369 −0.467856 −0.233928 0.972254i $$-0.575158\pi$$
−0.233928 + 0.972254i $$0.575158\pi$$
$$450$$ 0 0
$$451$$ 1.92159 0.0904843
$$452$$ 0 0
$$453$$ −18.1153 −0.851133
$$454$$ 0 0
$$455$$ 5.34565 0.250608
$$456$$ 0 0
$$457$$ 32.6050 1.52520 0.762598 0.646872i $$-0.223923\pi$$
0.762598 + 0.646872i $$0.223923\pi$$
$$458$$ 0 0
$$459$$ 10.7900 0.503636
$$460$$ 0 0
$$461$$ 5.23030 0.243599 0.121800 0.992555i $$-0.461133\pi$$
0.121800 + 0.992555i $$0.461133\pi$$
$$462$$ 0 0
$$463$$ 35.0841 1.63049 0.815247 0.579113i $$-0.196601\pi$$
0.815247 + 0.579113i $$0.196601\pi$$
$$464$$ 0 0
$$465$$ −4.08631 −0.189498
$$466$$ 0 0
$$467$$ −21.3641 −0.988614 −0.494307 0.869288i $$-0.664578\pi$$
−0.494307 + 0.869288i $$0.664578\pi$$
$$468$$ 0 0
$$469$$ 36.5266 1.68664
$$470$$ 0 0
$$471$$ 23.9506 1.10359
$$472$$ 0 0
$$473$$ −6.02904 −0.277215
$$474$$ 0 0
$$475$$ −5.52884 −0.253680
$$476$$ 0 0
$$477$$ 43.4689 1.99030
$$478$$ 0 0
$$479$$ −5.03920 −0.230247 −0.115124 0.993351i $$-0.536726\pi$$
−0.115124 + 0.993351i $$0.536726\pi$$
$$480$$ 0 0
$$481$$ −8.19376 −0.373603
$$482$$ 0 0
$$483$$ 47.6706 2.16909
$$484$$ 0 0
$$485$$ 9.32718 0.423525
$$486$$ 0 0
$$487$$ −28.8824 −1.30879 −0.654393 0.756155i $$-0.727076\pi$$
−0.654393 + 0.756155i $$0.727076\pi$$
$$488$$ 0 0
$$489$$ 16.8560 0.762255
$$490$$ 0 0
$$491$$ −5.16246 −0.232978 −0.116489 0.993192i $$-0.537164\pi$$
−0.116489 + 0.993192i $$0.537164\pi$$
$$492$$ 0 0
$$493$$ −3.52884 −0.158931
$$494$$ 0 0
$$495$$ −2.78817 −0.125319
$$496$$ 0 0
$$497$$ −20.9793 −0.941049
$$498$$ 0 0
$$499$$ 27.0946 1.21292 0.606461 0.795113i $$-0.292589\pi$$
0.606461 + 0.795113i $$0.292589\pi$$
$$500$$ 0 0
$$501$$ −34.2017 −1.52802
$$502$$ 0 0
$$503$$ −25.7305 −1.14727 −0.573633 0.819112i $$-0.694466\pi$$
−0.573633 + 0.819112i $$0.694466\pi$$
$$504$$ 0 0
$$505$$ 16.3249 0.726449
$$506$$ 0 0
$$507$$ −31.2488 −1.38781
$$508$$ 0 0
$$509$$ 11.8353 0.524590 0.262295 0.964988i $$-0.415521\pi$$
0.262295 + 0.964988i $$0.415521\pi$$
$$510$$ 0 0
$$511$$ −23.1809 −1.02546
$$512$$ 0 0
$$513$$ −16.9054 −0.746391
$$514$$ 0 0
$$515$$ −13.5288 −0.596152
$$516$$ 0 0
$$517$$ 4.48963 0.197454
$$518$$ 0 0
$$519$$ 48.4403 2.12629
$$520$$ 0 0
$$521$$ −0.164719 −0.00721646 −0.00360823 0.999993i $$-0.501149\pi$$
−0.00360823 + 0.999993i $$0.501149\pi$$
$$522$$ 0 0
$$523$$ 11.6072 0.507549 0.253775 0.967263i $$-0.418328\pi$$
0.253775 + 0.967263i $$0.418328\pi$$
$$524$$ 0 0
$$525$$ 12.4896 0.545092
$$526$$ 0 0
$$527$$ 5.39502 0.235011
$$528$$ 0 0
$$529$$ −8.43196 −0.366607
$$530$$ 0 0
$$531$$ −44.6623 −1.93818
$$532$$ 0 0
$$533$$ −3.26724 −0.141520
$$534$$ 0 0
$$535$$ 0.672824 0.0290887
$$536$$ 0 0
$$537$$ −16.8066 −0.725260
$$538$$ 0 0
$$539$$ 9.98153 0.429935
$$540$$ 0 0
$$541$$ −35.5922 −1.53023 −0.765113 0.643896i $$-0.777317\pi$$
−0.765113 + 0.643896i $$0.777317\pi$$
$$542$$ 0 0
$$543$$ 12.2096 0.523963
$$544$$ 0 0
$$545$$ −13.5473 −0.580303
$$546$$ 0 0
$$547$$ 44.3803 1.89757 0.948783 0.315929i $$-0.102316\pi$$
0.948783 + 0.315929i $$0.102316\pi$$
$$548$$ 0 0
$$549$$ −60.0448 −2.56265
$$550$$ 0 0
$$551$$ 5.52884 0.235536
$$552$$ 0 0
$$553$$ −11.1440 −0.473891
$$554$$ 0 0
$$555$$ −19.1440 −0.812617
$$556$$ 0 0
$$557$$ −26.0739 −1.10479 −0.552393 0.833584i $$-0.686285\pi$$
−0.552393 + 0.833584i $$0.686285\pi$$
$$558$$ 0 0
$$559$$ 10.2510 0.433572
$$560$$ 0 0
$$561$$ 6.34605 0.267930
$$562$$ 0 0
$$563$$ 16.4218 0.692096 0.346048 0.938217i $$-0.387523\pi$$
0.346048 + 0.938217i $$0.387523\pi$$
$$564$$ 0 0
$$565$$ −20.0185 −0.842183
$$566$$ 0 0
$$567$$ −19.9031 −0.835853
$$568$$ 0 0
$$569$$ −7.79834 −0.326923 −0.163462 0.986550i $$-0.552266\pi$$
−0.163462 + 0.986550i $$0.552266\pi$$
$$570$$ 0 0
$$571$$ 33.3826 1.39702 0.698509 0.715601i $$-0.253847\pi$$
0.698509 + 0.715601i $$0.253847\pi$$
$$572$$ 0 0
$$573$$ −57.8722 −2.41765
$$574$$ 0 0
$$575$$ 3.81681 0.159172
$$576$$ 0 0
$$577$$ −8.10478 −0.337407 −0.168703 0.985667i $$-0.553958\pi$$
−0.168703 + 0.985667i $$0.553958\pi$$
$$578$$ 0 0
$$579$$ −35.3905 −1.47078
$$580$$ 0 0
$$581$$ −65.5058 −2.71764
$$582$$ 0 0
$$583$$ 7.05767 0.292299
$$584$$ 0 0
$$585$$ 4.74066 0.196002
$$586$$ 0 0
$$587$$ −28.5865 −1.17989 −0.589946 0.807443i $$-0.700851\pi$$
−0.589946 + 0.807443i $$0.700851\pi$$
$$588$$ 0 0
$$589$$ −8.45269 −0.348287
$$590$$ 0 0
$$591$$ 19.6336 0.807619
$$592$$ 0 0
$$593$$ 22.4896 0.923539 0.461769 0.887000i $$-0.347215\pi$$
0.461769 + 0.887000i $$0.347215\pi$$
$$594$$ 0 0
$$595$$ −16.4896 −0.676009
$$596$$ 0 0
$$597$$ −50.1892 −2.05411
$$598$$ 0 0
$$599$$ 31.7305 1.29647 0.648237 0.761439i $$-0.275507\pi$$
0.648237 + 0.761439i $$0.275507\pi$$
$$600$$ 0 0
$$601$$ 4.56804 0.186334 0.0931671 0.995650i $$-0.470301\pi$$
0.0931671 + 0.995650i $$0.470301\pi$$
$$602$$ 0 0
$$603$$ 32.3928 1.31914
$$604$$ 0 0
$$605$$ 10.5473 0.428809
$$606$$ 0 0
$$607$$ 12.7882 0.519056 0.259528 0.965736i $$-0.416433\pi$$
0.259528 + 0.965736i $$0.416433\pi$$
$$608$$ 0 0
$$609$$ −12.4896 −0.506106
$$610$$ 0 0
$$611$$ −7.63362 −0.308823
$$612$$ 0 0
$$613$$ −6.97927 −0.281890 −0.140945 0.990017i $$-0.545014\pi$$
−0.140945 + 0.990017i $$0.545014\pi$$
$$614$$ 0 0
$$615$$ −7.63362 −0.307817
$$616$$ 0 0
$$617$$ 30.9193 1.24477 0.622383 0.782713i $$-0.286165\pi$$
0.622383 + 0.782713i $$0.286165\pi$$
$$618$$ 0 0
$$619$$ 20.7098 0.832396 0.416198 0.909274i $$-0.363362\pi$$
0.416198 + 0.909274i $$0.363362\pi$$
$$620$$ 0 0
$$621$$ 11.6706 0.468324
$$622$$ 0 0
$$623$$ −7.63362 −0.305835
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −9.94272 −0.397074
$$628$$ 0 0
$$629$$ 25.2751 1.00779
$$630$$ 0 0
$$631$$ −9.67508 −0.385159 −0.192580 0.981281i $$-0.561685\pi$$
−0.192580 + 0.981281i $$0.561685\pi$$
$$632$$ 0 0
$$633$$ −47.1809 −1.87527
$$634$$ 0 0
$$635$$ −3.03920 −0.120607
$$636$$ 0 0
$$637$$ −16.9714 −0.672430
$$638$$ 0 0
$$639$$ −18.6050 −0.736002
$$640$$ 0 0
$$641$$ 23.5058 0.928425 0.464213 0.885724i $$-0.346337\pi$$
0.464213 + 0.885724i $$0.346337\pi$$
$$642$$ 0 0
$$643$$ −40.6235 −1.60203 −0.801016 0.598643i $$-0.795707\pi$$
−0.801016 + 0.598643i $$0.795707\pi$$
$$644$$ 0 0
$$645$$ 23.9506 0.943055
$$646$$ 0 0
$$647$$ 10.2280 0.402106 0.201053 0.979580i $$-0.435564\pi$$
0.201053 + 0.979580i $$0.435564\pi$$
$$648$$ 0 0
$$649$$ −7.25143 −0.284644
$$650$$ 0 0
$$651$$ 19.0946 0.748377
$$652$$ 0 0
$$653$$ −28.2280 −1.10465 −0.552324 0.833629i $$-0.686259\pi$$
−0.552324 + 0.833629i $$0.686259\pi$$
$$654$$ 0 0
$$655$$ −3.81681 −0.149135
$$656$$ 0 0
$$657$$ −20.5575 −0.802023
$$658$$ 0 0
$$659$$ 15.0471 0.586152 0.293076 0.956089i $$-0.405321\pi$$
0.293076 + 0.956089i $$0.405321\pi$$
$$660$$ 0 0
$$661$$ 13.8767 0.539743 0.269871 0.962896i $$-0.413019\pi$$
0.269871 + 0.962896i $$0.413019\pi$$
$$662$$ 0 0
$$663$$ −10.7900 −0.419050
$$664$$ 0 0
$$665$$ 25.8353 1.00185
$$666$$ 0 0
$$667$$ −3.81681 −0.147787
$$668$$ 0 0
$$669$$ −19.9137 −0.769908
$$670$$ 0 0
$$671$$ −9.74897 −0.376355
$$672$$ 0 0
$$673$$ 15.3456 0.591531 0.295766 0.955261i $$-0.404425\pi$$
0.295766 + 0.955261i $$0.404425\pi$$
$$674$$ 0 0
$$675$$ 3.05767 0.117690
$$676$$ 0 0
$$677$$ −50.5530 −1.94291 −0.971454 0.237228i $$-0.923761\pi$$
−0.971454 + 0.237228i $$0.923761\pi$$
$$678$$ 0 0
$$679$$ −43.5843 −1.67261
$$680$$ 0 0
$$681$$ 64.7361 2.48069
$$682$$ 0 0
$$683$$ −41.6027 −1.59188 −0.795942 0.605373i $$-0.793024\pi$$
−0.795942 + 0.605373i $$0.793024\pi$$
$$684$$ 0 0
$$685$$ −16.1048 −0.615332
$$686$$ 0 0
$$687$$ 5.34565 0.203949
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 41.6627 1.58492 0.792461 0.609922i $$-0.208799\pi$$
0.792461 + 0.609922i $$0.208799\pi$$
$$692$$ 0 0
$$693$$ 13.0286 0.494917
$$694$$ 0 0
$$695$$ 3.05767 0.115984
$$696$$ 0 0
$$697$$ 10.0784 0.381747
$$698$$ 0 0
$$699$$ −0.769701 −0.0291128
$$700$$ 0 0
$$701$$ −44.8145 −1.69262 −0.846311 0.532689i $$-0.821182\pi$$
−0.846311 + 0.532689i $$0.821182\pi$$
$$702$$ 0 0
$$703$$ −39.6001 −1.49354
$$704$$ 0 0
$$705$$ −17.8353 −0.671716
$$706$$ 0 0
$$707$$ −76.2835 −2.86893
$$708$$ 0 0
$$709$$ 23.4320 0.880006 0.440003 0.897996i $$-0.354977\pi$$
0.440003 + 0.897996i $$0.354977\pi$$
$$710$$ 0 0
$$711$$ −9.88279 −0.370634
$$712$$ 0 0
$$713$$ 5.83528 0.218533
$$714$$ 0 0
$$715$$ 0.769701 0.0287852
$$716$$ 0 0
$$717$$ −20.0739 −0.749673
$$718$$ 0 0
$$719$$ −32.3328 −1.20581 −0.602905 0.797813i $$-0.705990\pi$$
−0.602905 + 0.797813i $$0.705990\pi$$
$$720$$ 0 0
$$721$$ 63.2179 2.35436
$$722$$ 0 0
$$723$$ −57.9216 −2.15413
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −4.01847 −0.149037 −0.0745184 0.997220i $$-0.523742\pi$$
−0.0745184 + 0.997220i $$0.523742\pi$$
$$728$$ 0 0
$$729$$ −42.1685 −1.56180
$$730$$ 0 0
$$731$$ −31.6212 −1.16955
$$732$$ 0 0
$$733$$ −3.49189 −0.128976 −0.0644880 0.997918i $$-0.520541\pi$$
−0.0644880 + 0.997918i $$0.520541\pi$$
$$734$$ 0 0
$$735$$ −39.6521 −1.46259
$$736$$ 0 0
$$737$$ 5.25934 0.193730
$$738$$ 0 0
$$739$$ 45.4090 1.67040 0.835198 0.549949i $$-0.185353\pi$$
0.835198 + 0.549949i $$0.185353\pi$$
$$740$$ 0 0
$$741$$ 16.9054 0.621035
$$742$$ 0 0
$$743$$ −40.6683 −1.49198 −0.745988 0.665960i $$-0.768022\pi$$
−0.745988 + 0.665960i $$0.768022\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ −58.0924 −2.12549
$$748$$ 0 0
$$749$$ −3.14399 −0.114879
$$750$$ 0 0
$$751$$ −49.0268 −1.78901 −0.894506 0.447055i $$-0.852473\pi$$
−0.894506 + 0.447055i $$0.852473\pi$$
$$752$$ 0 0
$$753$$ 33.8722 1.23437
$$754$$ 0 0
$$755$$ 6.77761 0.246662
$$756$$ 0 0
$$757$$ −51.7260 −1.88001 −0.940006 0.341157i $$-0.889181\pi$$
−0.940006 + 0.341157i $$0.889181\pi$$
$$758$$ 0 0
$$759$$ 6.86392 0.249144
$$760$$ 0 0
$$761$$ 34.2307 1.24086 0.620431 0.784261i $$-0.286958\pi$$
0.620431 + 0.784261i $$0.286958\pi$$
$$762$$ 0 0
$$763$$ 63.3042 2.29177
$$764$$ 0 0
$$765$$ −14.6235 −0.528712
$$766$$ 0 0
$$767$$ 12.3294 0.445190
$$768$$ 0 0
$$769$$ 22.5971 0.814871 0.407436 0.913234i $$-0.366423\pi$$
0.407436 + 0.913234i $$0.366423\pi$$
$$770$$ 0 0
$$771$$ 45.3615 1.63365
$$772$$ 0 0
$$773$$ 13.8538 0.498285 0.249142 0.968467i $$-0.419851\pi$$
0.249142 + 0.968467i $$0.419851\pi$$
$$774$$ 0 0
$$775$$ 1.52884 0.0549175
$$776$$ 0 0
$$777$$ 89.4565 3.20923
$$778$$ 0 0
$$779$$ −15.7904 −0.565751
$$780$$ 0 0
$$781$$ −3.02073 −0.108090
$$782$$ 0 0
$$783$$ −3.05767 −0.109272
$$784$$ 0 0
$$785$$ −8.96080 −0.319825
$$786$$ 0 0
$$787$$ −20.0969 −0.716376 −0.358188 0.933649i $$-0.616605\pi$$
−0.358188 + 0.933649i $$0.616605\pi$$
$$788$$ 0 0
$$789$$ −15.7780 −0.561712
$$790$$ 0 0
$$791$$ 93.5428 3.32600
$$792$$ 0 0
$$793$$ 16.5759 0.588629
$$794$$ 0 0
$$795$$ −28.0369 −0.994368
$$796$$ 0 0
$$797$$ 43.6890 1.54754 0.773772 0.633464i $$-0.218367\pi$$
0.773772 + 0.633464i $$0.218367\pi$$
$$798$$ 0 0
$$799$$ 23.5473 0.833044
$$800$$ 0 0
$$801$$ −6.76970 −0.239196
$$802$$ 0 0
$$803$$ −3.33774 −0.117786
$$804$$ 0 0
$$805$$ −17.8353 −0.628611
$$806$$ 0 0
$$807$$ −39.7075 −1.39777
$$808$$ 0 0
$$809$$ −18.4403 −0.648325 −0.324163 0.946001i $$-0.605082\pi$$
−0.324163 + 0.946001i $$0.605082\pi$$
$$810$$ 0 0
$$811$$ −15.0577 −0.528746 −0.264373 0.964420i $$-0.585165\pi$$
−0.264373 + 0.964420i $$0.585165\pi$$
$$812$$ 0 0
$$813$$ −60.3909 −2.11800
$$814$$ 0 0
$$815$$ −6.30644 −0.220905
$$816$$ 0 0
$$817$$ 49.5428 1.73328
$$818$$ 0 0
$$819$$ −22.1523 −0.774064
$$820$$ 0 0
$$821$$ 50.9793 1.77919 0.889594 0.456751i $$-0.150987\pi$$
0.889594 + 0.456751i $$0.150987\pi$$
$$822$$ 0 0
$$823$$ 2.84545 0.0991861 0.0495930 0.998770i $$-0.484208\pi$$
0.0495930 + 0.998770i $$0.484208\pi$$
$$824$$ 0 0
$$825$$ 1.79834 0.0626101
$$826$$ 0 0
$$827$$ 15.0761 0.524249 0.262124 0.965034i $$-0.415577\pi$$
0.262124 + 0.965034i $$0.415577\pi$$
$$828$$ 0 0
$$829$$ −34.6992 −1.20515 −0.602577 0.798061i $$-0.705859\pi$$
−0.602577 + 0.798061i $$0.705859\pi$$
$$830$$ 0 0
$$831$$ 73.4195 2.54690
$$832$$ 0 0
$$833$$ 52.3513 1.81386
$$834$$ 0 0
$$835$$ 12.7961 0.442827
$$836$$ 0 0
$$837$$ 4.67469 0.161581
$$838$$ 0 0
$$839$$ 14.7882 0.510544 0.255272 0.966869i $$-0.417835\pi$$
0.255272 + 0.966869i $$0.417835\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 46.9009 1.61535
$$844$$ 0 0
$$845$$ 11.6913 0.402193
$$846$$ 0 0
$$847$$ −49.2857 −1.69348
$$848$$ 0 0
$$849$$ −70.8515 −2.43162
$$850$$ 0 0
$$851$$ 27.3377 0.937126
$$852$$ 0 0
$$853$$ 35.7753 1.22492 0.612462 0.790500i $$-0.290179\pi$$
0.612462 + 0.790500i $$0.290179\pi$$
$$854$$ 0 0
$$855$$ 22.9114 0.783554
$$856$$ 0 0
$$857$$ −12.4112 −0.423959 −0.211980 0.977274i $$-0.567991\pi$$
−0.211980 + 0.977274i $$0.567991\pi$$
$$858$$ 0 0
$$859$$ 0.0599353 0.00204497 0.00102248 0.999999i $$-0.499675\pi$$
0.00102248 + 0.999999i $$0.499675\pi$$
$$860$$ 0 0
$$861$$ 35.6706 1.21565
$$862$$ 0 0
$$863$$ 44.5530 1.51660 0.758300 0.651906i $$-0.226030\pi$$
0.758300 + 0.651906i $$0.226030\pi$$
$$864$$ 0 0
$$865$$ −18.1233 −0.616209
$$866$$ 0 0
$$867$$ −12.1542 −0.412777
$$868$$ 0 0
$$869$$ −1.60458 −0.0544318
$$870$$ 0 0
$$871$$ −8.94233 −0.302999
$$872$$ 0 0
$$873$$ −38.6517 −1.30816
$$874$$ 0 0
$$875$$ −4.67282 −0.157970
$$876$$ 0 0
$$877$$ 29.9137 1.01011 0.505057 0.863086i $$-0.331472\pi$$
0.505057 + 0.863086i $$0.331472\pi$$
$$878$$ 0 0
$$879$$ −60.6992 −2.04733
$$880$$ 0 0
$$881$$ −32.2386 −1.08615 −0.543073 0.839685i $$-0.682739\pi$$
−0.543073 + 0.839685i $$0.682739\pi$$
$$882$$ 0 0
$$883$$ −36.7098 −1.23538 −0.617691 0.786421i $$-0.711932\pi$$
−0.617691 + 0.786421i $$0.711932\pi$$
$$884$$ 0 0
$$885$$ 28.8066 0.968325
$$886$$ 0 0
$$887$$ 2.63588 0.0885042 0.0442521 0.999020i $$-0.485910\pi$$
0.0442521 + 0.999020i $$0.485910\pi$$
$$888$$ 0 0
$$889$$ 14.2017 0.476308
$$890$$ 0 0
$$891$$ −2.86578 −0.0960073
$$892$$ 0 0
$$893$$ −36.8930 −1.23458
$$894$$ 0 0
$$895$$ 6.28797 0.210184
$$896$$ 0 0
$$897$$ −11.6706 −0.389669
$$898$$ 0 0
$$899$$ −1.52884 −0.0509896
$$900$$ 0 0
$$901$$ 37.0162 1.23319
$$902$$ 0 0
$$903$$ −111.917 −3.72437
$$904$$ 0 0
$$905$$ −4.56804 −0.151847
$$906$$ 0 0
$$907$$ −16.0185 −0.531885 −0.265942 0.963989i $$-0.585683\pi$$
−0.265942 + 0.963989i $$0.585683\pi$$
$$908$$ 0 0
$$909$$ −67.6502 −2.24382
$$910$$ 0 0
$$911$$ −37.9480 −1.25727 −0.628636 0.777700i $$-0.716387\pi$$
−0.628636 + 0.777700i $$0.716387\pi$$
$$912$$ 0 0
$$913$$ −9.43196 −0.312152
$$914$$ 0 0
$$915$$ 38.7282 1.28032
$$916$$ 0 0
$$917$$ 17.8353 0.588973
$$918$$ 0 0
$$919$$ 7.58425 0.250181 0.125091 0.992145i $$-0.460078\pi$$
0.125091 + 0.992145i $$0.460078\pi$$
$$920$$ 0 0
$$921$$ −56.0245 −1.84607
$$922$$ 0 0
$$923$$ 5.13608 0.169056
$$924$$ 0 0
$$925$$ 7.16246 0.235500
$$926$$ 0 0
$$927$$ 56.0633 1.84136
$$928$$ 0 0
$$929$$ 26.4033 0.866265 0.433132 0.901330i $$-0.357408\pi$$
0.433132 + 0.901330i $$0.357408\pi$$
$$930$$ 0 0
$$931$$ −82.0219 −2.68816
$$932$$ 0 0
$$933$$ 73.4689 2.40526
$$934$$ 0 0
$$935$$ −2.37429 −0.0776474
$$936$$ 0 0
$$937$$ 44.6050 1.45718 0.728591 0.684949i $$-0.240176\pi$$
0.728591 + 0.684949i $$0.240176\pi$$
$$938$$ 0 0
$$939$$ −50.3988 −1.64470
$$940$$ 0 0
$$941$$ −9.26724 −0.302103 −0.151052 0.988526i $$-0.548266\pi$$
−0.151052 + 0.988526i $$0.548266\pi$$
$$942$$ 0 0
$$943$$ 10.9009 0.354981
$$944$$ 0 0
$$945$$ −14.2880 −0.464788
$$946$$ 0 0
$$947$$ 28.4218 0.923584 0.461792 0.886988i $$-0.347207\pi$$
0.461792 + 0.886988i $$0.347207\pi$$
$$948$$ 0 0
$$949$$ 5.67508 0.184221
$$950$$ 0 0
$$951$$ 78.7158 2.55254
$$952$$ 0 0
$$953$$ 55.2100 1.78843 0.894213 0.447642i $$-0.147736\pi$$
0.894213 + 0.447642i $$0.147736\pi$$
$$954$$ 0 0
$$955$$ 21.6521 0.700645
$$956$$ 0 0
$$957$$ −1.79834 −0.0581320
$$958$$ 0 0
$$959$$ 75.2548 2.43010
$$960$$ 0 0
$$961$$ −28.6627 −0.924602
$$962$$ 0 0
$$963$$ −2.78817 −0.0898476
$$964$$ 0 0
$$965$$ 13.2409 0.426238
$$966$$ 0 0
$$967$$ 34.1496 1.09818 0.549089 0.835764i $$-0.314975\pi$$
0.549089 + 0.835764i $$0.314975\pi$$
$$968$$ 0 0
$$969$$ −52.1478 −1.67523
$$970$$ 0 0
$$971$$ 41.4090 1.32888 0.664438 0.747343i $$-0.268671\pi$$
0.664438 + 0.747343i $$0.268671\pi$$
$$972$$ 0 0
$$973$$ −14.2880 −0.458051
$$974$$ 0 0
$$975$$ −3.05767 −0.0979239
$$976$$ 0 0
$$977$$ −13.9506 −0.446320 −0.223160 0.974782i $$-0.571637\pi$$
−0.223160 + 0.974782i $$0.571637\pi$$
$$978$$ 0 0
$$979$$ −1.09914 −0.0351286
$$980$$ 0 0
$$981$$ 56.1399 1.79241
$$982$$ 0 0
$$983$$ 39.6521 1.26471 0.632353 0.774681i $$-0.282089\pi$$
0.632353 + 0.774681i $$0.282089\pi$$
$$984$$ 0 0
$$985$$ −7.34565 −0.234052
$$986$$ 0 0
$$987$$ 83.3411 2.65278
$$988$$ 0 0
$$989$$ −34.2017 −1.08755
$$990$$ 0 0
$$991$$ −2.93442 −0.0932149 −0.0466075 0.998913i $$-0.514841\pi$$
−0.0466075 + 0.998913i $$0.514841\pi$$
$$992$$ 0 0
$$993$$ −9.64153 −0.305965
$$994$$ 0 0
$$995$$ 18.7776 0.595290
$$996$$ 0 0
$$997$$ 12.9977 0.411643 0.205821 0.978590i $$-0.434013\pi$$
0.205821 + 0.978590i $$0.434013\pi$$
$$998$$ 0 0
$$999$$ 21.9005 0.692900
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 2320.2.a.m.1.3 3
4.3 odd 2 580.2.a.c.1.1 3
8.3 odd 2 9280.2.a.bk.1.3 3
8.5 even 2 9280.2.a.bw.1.1 3
12.11 even 2 5220.2.a.x.1.1 3
20.3 even 4 2900.2.c.f.349.2 6
20.7 even 4 2900.2.c.f.349.5 6
20.19 odd 2 2900.2.a.g.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
580.2.a.c.1.1 3 4.3 odd 2
2320.2.a.m.1.3 3 1.1 even 1 trivial
2900.2.a.g.1.3 3 20.19 odd 2
2900.2.c.f.349.2 6 20.3 even 4
2900.2.c.f.349.5 6 20.7 even 4
5220.2.a.x.1.1 3 12.11 even 2
9280.2.a.bk.1.3 3 8.3 odd 2
9280.2.a.bw.1.1 3 8.5 even 2