# Properties

 Label 192.3.g.b.127.1 Level $192$ Weight $3$ Character 192.127 Analytic conductor $5.232$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 192.127 Dual form 192.3.g.b.127.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +2.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +2.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} -6.92820i q^{11} -2.00000 q^{13} -3.46410i q^{15} +10.0000 q^{17} -20.7846i q^{19} -12.0000 q^{21} -27.7128i q^{23} -21.0000 q^{25} +5.19615i q^{27} +26.0000 q^{29} +6.92820i q^{31} -12.0000 q^{33} -13.8564i q^{35} -26.0000 q^{37} +3.46410i q^{39} +58.0000 q^{41} +48.4974i q^{43} -6.00000 q^{45} +69.2820i q^{47} +1.00000 q^{49} -17.3205i q^{51} +74.0000 q^{53} -13.8564i q^{55} -36.0000 q^{57} +90.0666i q^{59} -26.0000 q^{61} +20.7846i q^{63} -4.00000 q^{65} -6.92820i q^{67} -48.0000 q^{69} -46.0000 q^{73} +36.3731i q^{75} -48.0000 q^{77} +117.779i q^{79} +9.00000 q^{81} -48.4974i q^{83} +20.0000 q^{85} -45.0333i q^{87} +82.0000 q^{89} +13.8564i q^{91} +12.0000 q^{93} -41.5692i q^{95} +2.00000 q^{97} +20.7846i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^5 - 6 * q^9 $$2 q + 4 q^{5} - 6 q^{9} - 4 q^{13} + 20 q^{17} - 24 q^{21} - 42 q^{25} + 52 q^{29} - 24 q^{33} - 52 q^{37} + 116 q^{41} - 12 q^{45} + 2 q^{49} + 148 q^{53} - 72 q^{57} - 52 q^{61} - 8 q^{65} - 96 q^{69} - 92 q^{73} - 96 q^{77} + 18 q^{81} + 40 q^{85} + 164 q^{89} + 24 q^{93} + 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 - 6 * q^9 - 4 * q^13 + 20 * q^17 - 24 * q^21 - 42 * q^25 + 52 * q^29 - 24 * q^33 - 52 * q^37 + 116 * q^41 - 12 * q^45 + 2 * q^49 + 148 * q^53 - 72 * q^57 - 52 * q^61 - 8 * q^65 - 96 * q^69 - 92 * q^73 - 96 * q^77 + 18 * q^81 + 40 * q^85 + 164 * q^89 + 24 * q^93 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.73205i − 0.577350i
$$4$$ 0 0
$$5$$ 2.00000 0.400000 0.200000 0.979796i $$-0.435906\pi$$
0.200000 + 0.979796i $$0.435906\pi$$
$$6$$ 0 0
$$7$$ − 6.92820i − 0.989743i −0.868966 0.494872i $$-0.835215\pi$$
0.868966 0.494872i $$-0.164785\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −0.333333
$$10$$ 0 0
$$11$$ − 6.92820i − 0.629837i −0.949119 0.314918i $$-0.898023\pi$$
0.949119 0.314918i $$-0.101977\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.153846 −0.0769231 0.997037i $$-0.524510\pi$$
−0.0769231 + 0.997037i $$0.524510\pi$$
$$14$$ 0 0
$$15$$ − 3.46410i − 0.230940i
$$16$$ 0 0
$$17$$ 10.0000 0.588235 0.294118 0.955769i $$-0.404974\pi$$
0.294118 + 0.955769i $$0.404974\pi$$
$$18$$ 0 0
$$19$$ − 20.7846i − 1.09393i −0.837157 0.546963i $$-0.815784\pi$$
0.837157 0.546963i $$-0.184216\pi$$
$$20$$ 0 0
$$21$$ −12.0000 −0.571429
$$22$$ 0 0
$$23$$ − 27.7128i − 1.20490i −0.798155 0.602452i $$-0.794190\pi$$
0.798155 0.602452i $$-0.205810\pi$$
$$24$$ 0 0
$$25$$ −21.0000 −0.840000
$$26$$ 0 0
$$27$$ 5.19615i 0.192450i
$$28$$ 0 0
$$29$$ 26.0000 0.896552 0.448276 0.893895i $$-0.352038\pi$$
0.448276 + 0.893895i $$0.352038\pi$$
$$30$$ 0 0
$$31$$ 6.92820i 0.223490i 0.993737 + 0.111745i $$0.0356441\pi$$
−0.993737 + 0.111745i $$0.964356\pi$$
$$32$$ 0 0
$$33$$ −12.0000 −0.363636
$$34$$ 0 0
$$35$$ − 13.8564i − 0.395897i
$$36$$ 0 0
$$37$$ −26.0000 −0.702703 −0.351351 0.936244i $$-0.614278\pi$$
−0.351351 + 0.936244i $$0.614278\pi$$
$$38$$ 0 0
$$39$$ 3.46410i 0.0888231i
$$40$$ 0 0
$$41$$ 58.0000 1.41463 0.707317 0.706896i $$-0.249905\pi$$
0.707317 + 0.706896i $$0.249905\pi$$
$$42$$ 0 0
$$43$$ 48.4974i 1.12785i 0.825827 + 0.563924i $$0.190709\pi$$
−0.825827 + 0.563924i $$0.809291\pi$$
$$44$$ 0 0
$$45$$ −6.00000 −0.133333
$$46$$ 0 0
$$47$$ 69.2820i 1.47409i 0.675846 + 0.737043i $$0.263778\pi$$
−0.675846 + 0.737043i $$0.736222\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.0204082
$$50$$ 0 0
$$51$$ − 17.3205i − 0.339618i
$$52$$ 0 0
$$53$$ 74.0000 1.39623 0.698113 0.715987i $$-0.254023\pi$$
0.698113 + 0.715987i $$0.254023\pi$$
$$54$$ 0 0
$$55$$ − 13.8564i − 0.251935i
$$56$$ 0 0
$$57$$ −36.0000 −0.631579
$$58$$ 0 0
$$59$$ 90.0666i 1.52655i 0.646072 + 0.763277i $$0.276411\pi$$
−0.646072 + 0.763277i $$0.723589\pi$$
$$60$$ 0 0
$$61$$ −26.0000 −0.426230 −0.213115 0.977027i $$-0.568361\pi$$
−0.213115 + 0.977027i $$0.568361\pi$$
$$62$$ 0 0
$$63$$ 20.7846i 0.329914i
$$64$$ 0 0
$$65$$ −4.00000 −0.0615385
$$66$$ 0 0
$$67$$ − 6.92820i − 0.103406i −0.998663 0.0517030i $$-0.983535\pi$$
0.998663 0.0517030i $$-0.0164649\pi$$
$$68$$ 0 0
$$69$$ −48.0000 −0.695652
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −46.0000 −0.630137 −0.315068 0.949069i $$-0.602027\pi$$
−0.315068 + 0.949069i $$0.602027\pi$$
$$74$$ 0 0
$$75$$ 36.3731i 0.484974i
$$76$$ 0 0
$$77$$ −48.0000 −0.623377
$$78$$ 0 0
$$79$$ 117.779i 1.49088i 0.666573 + 0.745440i $$0.267760\pi$$
−0.666573 + 0.745440i $$0.732240\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ − 48.4974i − 0.584306i −0.956372 0.292153i $$-0.905628\pi$$
0.956372 0.292153i $$-0.0943717\pi$$
$$84$$ 0 0
$$85$$ 20.0000 0.235294
$$86$$ 0 0
$$87$$ − 45.0333i − 0.517624i
$$88$$ 0 0
$$89$$ 82.0000 0.921348 0.460674 0.887569i $$-0.347608\pi$$
0.460674 + 0.887569i $$0.347608\pi$$
$$90$$ 0 0
$$91$$ 13.8564i 0.152268i
$$92$$ 0 0
$$93$$ 12.0000 0.129032
$$94$$ 0 0
$$95$$ − 41.5692i − 0.437571i
$$96$$ 0 0
$$97$$ 2.00000 0.0206186 0.0103093 0.999947i $$-0.496718\pi$$
0.0103093 + 0.999947i $$0.496718\pi$$
$$98$$ 0 0
$$99$$ 20.7846i 0.209946i
$$100$$ 0 0
$$101$$ 74.0000 0.732673 0.366337 0.930482i $$-0.380612\pi$$
0.366337 + 0.930482i $$0.380612\pi$$
$$102$$ 0 0
$$103$$ − 76.2102i − 0.739905i −0.929051 0.369953i $$-0.879374\pi$$
0.929051 0.369953i $$-0.120626\pi$$
$$104$$ 0 0
$$105$$ −24.0000 −0.228571
$$106$$ 0 0
$$107$$ − 20.7846i − 0.194249i −0.995272 0.0971243i $$-0.969036\pi$$
0.995272 0.0971243i $$-0.0309645\pi$$
$$108$$ 0 0
$$109$$ 46.0000 0.422018 0.211009 0.977484i $$-0.432325\pi$$
0.211009 + 0.977484i $$0.432325\pi$$
$$110$$ 0 0
$$111$$ 45.0333i 0.405706i
$$112$$ 0 0
$$113$$ −110.000 −0.973451 −0.486726 0.873555i $$-0.661809\pi$$
−0.486726 + 0.873555i $$0.661809\pi$$
$$114$$ 0 0
$$115$$ − 55.4256i − 0.481962i
$$116$$ 0 0
$$117$$ 6.00000 0.0512821
$$118$$ 0 0
$$119$$ − 69.2820i − 0.582202i
$$120$$ 0 0
$$121$$ 73.0000 0.603306
$$122$$ 0 0
$$123$$ − 100.459i − 0.816739i
$$124$$ 0 0
$$125$$ −92.0000 −0.736000
$$126$$ 0 0
$$127$$ − 145.492i − 1.14561i −0.819692 0.572804i $$-0.805856\pi$$
0.819692 0.572804i $$-0.194144\pi$$
$$128$$ 0 0
$$129$$ 84.0000 0.651163
$$130$$ 0 0
$$131$$ − 117.779i − 0.899080i −0.893260 0.449540i $$-0.851588\pi$$
0.893260 0.449540i $$-0.148412\pi$$
$$132$$ 0 0
$$133$$ −144.000 −1.08271
$$134$$ 0 0
$$135$$ 10.3923i 0.0769800i
$$136$$ 0 0
$$137$$ 10.0000 0.0729927 0.0364964 0.999334i $$-0.488380\pi$$
0.0364964 + 0.999334i $$0.488380\pi$$
$$138$$ 0 0
$$139$$ − 48.4974i − 0.348902i −0.984666 0.174451i $$-0.944185\pi$$
0.984666 0.174451i $$-0.0558151\pi$$
$$140$$ 0 0
$$141$$ 120.000 0.851064
$$142$$ 0 0
$$143$$ 13.8564i 0.0968979i
$$144$$ 0 0
$$145$$ 52.0000 0.358621
$$146$$ 0 0
$$147$$ − 1.73205i − 0.0117827i
$$148$$ 0 0
$$149$$ 2.00000 0.0134228 0.00671141 0.999977i $$-0.497864\pi$$
0.00671141 + 0.999977i $$0.497864\pi$$
$$150$$ 0 0
$$151$$ − 90.0666i − 0.596468i −0.954493 0.298234i $$-0.903602\pi$$
0.954493 0.298234i $$-0.0963975\pi$$
$$152$$ 0 0
$$153$$ −30.0000 −0.196078
$$154$$ 0 0
$$155$$ 13.8564i 0.0893962i
$$156$$ 0 0
$$157$$ 214.000 1.36306 0.681529 0.731791i $$-0.261315\pi$$
0.681529 + 0.731791i $$0.261315\pi$$
$$158$$ 0 0
$$159$$ − 128.172i − 0.806112i
$$160$$ 0 0
$$161$$ −192.000 −1.19255
$$162$$ 0 0
$$163$$ − 20.7846i − 0.127513i −0.997965 0.0637565i $$-0.979692\pi$$
0.997965 0.0637565i $$-0.0203081\pi$$
$$164$$ 0 0
$$165$$ −24.0000 −0.145455
$$166$$ 0 0
$$167$$ 96.9948i 0.580807i 0.956904 + 0.290404i $$0.0937896\pi$$
−0.956904 + 0.290404i $$0.906210\pi$$
$$168$$ 0 0
$$169$$ −165.000 −0.976331
$$170$$ 0 0
$$171$$ 62.3538i 0.364642i
$$172$$ 0 0
$$173$$ −334.000 −1.93064 −0.965318 0.261077i $$-0.915922\pi$$
−0.965318 + 0.261077i $$0.915922\pi$$
$$174$$ 0 0
$$175$$ 145.492i 0.831384i
$$176$$ 0 0
$$177$$ 156.000 0.881356
$$178$$ 0 0
$$179$$ 187.061i 1.04504i 0.852628 + 0.522518i $$0.175007\pi$$
−0.852628 + 0.522518i $$0.824993\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.0110497 −0.00552486 0.999985i $$-0.501759\pi$$
−0.00552486 + 0.999985i $$0.501759\pi$$
$$182$$ 0 0
$$183$$ 45.0333i 0.246084i
$$184$$ 0 0
$$185$$ −52.0000 −0.281081
$$186$$ 0 0
$$187$$ − 69.2820i − 0.370492i
$$188$$ 0 0
$$189$$ 36.0000 0.190476
$$190$$ 0 0
$$191$$ − 221.703i − 1.16075i −0.814351 0.580373i $$-0.802907\pi$$
0.814351 0.580373i $$-0.197093\pi$$
$$192$$ 0 0
$$193$$ 290.000 1.50259 0.751295 0.659966i $$-0.229429\pi$$
0.751295 + 0.659966i $$0.229429\pi$$
$$194$$ 0 0
$$195$$ 6.92820i 0.0355292i
$$196$$ 0 0
$$197$$ 26.0000 0.131980 0.0659898 0.997820i $$-0.478980\pi$$
0.0659898 + 0.997820i $$0.478980\pi$$
$$198$$ 0 0
$$199$$ 394.908i 1.98446i 0.124416 + 0.992230i $$0.460294\pi$$
−0.124416 + 0.992230i $$0.539706\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.0597015
$$202$$ 0 0
$$203$$ − 180.133i − 0.887356i
$$204$$ 0 0
$$205$$ 116.000 0.565854
$$206$$ 0 0
$$207$$ 83.1384i 0.401635i
$$208$$ 0 0
$$209$$ −144.000 −0.688995
$$210$$ 0 0
$$211$$ 242.487i 1.14923i 0.818425 + 0.574614i $$0.194848\pi$$
−0.818425 + 0.574614i $$0.805152\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 96.9948i 0.451139i
$$216$$ 0 0
$$217$$ 48.0000 0.221198
$$218$$ 0 0
$$219$$ 79.6743i 0.363810i
$$220$$ 0 0
$$221$$ −20.0000 −0.0904977
$$222$$ 0 0
$$223$$ − 339.482i − 1.52234i −0.648552 0.761170i $$-0.724625\pi$$
0.648552 0.761170i $$-0.275375\pi$$
$$224$$ 0 0
$$225$$ 63.0000 0.280000
$$226$$ 0 0
$$227$$ 284.056i 1.25135i 0.780084 + 0.625675i $$0.215176\pi$$
−0.780084 + 0.625675i $$0.784824\pi$$
$$228$$ 0 0
$$229$$ 142.000 0.620087 0.310044 0.950722i $$-0.399656\pi$$
0.310044 + 0.950722i $$0.399656\pi$$
$$230$$ 0 0
$$231$$ 83.1384i 0.359907i
$$232$$ 0 0
$$233$$ 82.0000 0.351931 0.175966 0.984396i $$-0.443695\pi$$
0.175966 + 0.984396i $$0.443695\pi$$
$$234$$ 0 0
$$235$$ 138.564i 0.589634i
$$236$$ 0 0
$$237$$ 204.000 0.860759
$$238$$ 0 0
$$239$$ 387.979i 1.62334i 0.584113 + 0.811672i $$0.301442\pi$$
−0.584113 + 0.811672i $$0.698558\pi$$
$$240$$ 0 0
$$241$$ −46.0000 −0.190871 −0.0954357 0.995436i $$-0.530424\pi$$
−0.0954357 + 0.995436i $$0.530424\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 0.0641500i
$$244$$ 0 0
$$245$$ 2.00000 0.00816327
$$246$$ 0 0
$$247$$ 41.5692i 0.168296i
$$248$$ 0 0
$$249$$ −84.0000 −0.337349
$$250$$ 0 0
$$251$$ − 145.492i − 0.579650i −0.957080 0.289825i $$-0.906403\pi$$
0.957080 0.289825i $$-0.0935972\pi$$
$$252$$ 0 0
$$253$$ −192.000 −0.758893
$$254$$ 0 0
$$255$$ − 34.6410i − 0.135847i
$$256$$ 0 0
$$257$$ −254.000 −0.988327 −0.494163 0.869369i $$-0.664526\pi$$
−0.494163 + 0.869369i $$0.664526\pi$$
$$258$$ 0 0
$$259$$ 180.133i 0.695495i
$$260$$ 0 0
$$261$$ −78.0000 −0.298851
$$262$$ 0 0
$$263$$ 152.420i 0.579546i 0.957095 + 0.289773i $$0.0935797\pi$$
−0.957095 + 0.289773i $$0.906420\pi$$
$$264$$ 0 0
$$265$$ 148.000 0.558491
$$266$$ 0 0
$$267$$ − 142.028i − 0.531941i
$$268$$ 0 0
$$269$$ −262.000 −0.973978 −0.486989 0.873408i $$-0.661905\pi$$
−0.486989 + 0.873408i $$0.661905\pi$$
$$270$$ 0 0
$$271$$ 20.7846i 0.0766960i 0.999264 + 0.0383480i $$0.0122095\pi$$
−0.999264 + 0.0383480i $$0.987790\pi$$
$$272$$ 0 0
$$273$$ 24.0000 0.0879121
$$274$$ 0 0
$$275$$ 145.492i 0.529063i
$$276$$ 0 0
$$277$$ −290.000 −1.04693 −0.523466 0.852047i $$-0.675361\pi$$
−0.523466 + 0.852047i $$0.675361\pi$$
$$278$$ 0 0
$$279$$ − 20.7846i − 0.0744968i
$$280$$ 0 0
$$281$$ 226.000 0.804270 0.402135 0.915580i $$-0.368268\pi$$
0.402135 + 0.915580i $$0.368268\pi$$
$$282$$ 0 0
$$283$$ − 297.913i − 1.05270i −0.850269 0.526348i $$-0.823561\pi$$
0.850269 0.526348i $$-0.176439\pi$$
$$284$$ 0 0
$$285$$ −72.0000 −0.252632
$$286$$ 0 0
$$287$$ − 401.836i − 1.40012i
$$288$$ 0 0
$$289$$ −189.000 −0.653979
$$290$$ 0 0
$$291$$ − 3.46410i − 0.0119041i
$$292$$ 0 0
$$293$$ 362.000 1.23549 0.617747 0.786377i $$-0.288045\pi$$
0.617747 + 0.786377i $$0.288045\pi$$
$$294$$ 0 0
$$295$$ 180.133i 0.610621i
$$296$$ 0 0
$$297$$ 36.0000 0.121212
$$298$$ 0 0
$$299$$ 55.4256i 0.185370i
$$300$$ 0 0
$$301$$ 336.000 1.11628
$$302$$ 0 0
$$303$$ − 128.172i − 0.423009i
$$304$$ 0 0
$$305$$ −52.0000 −0.170492
$$306$$ 0 0
$$307$$ − 145.492i − 0.473916i −0.971520 0.236958i $$-0.923850\pi$$
0.971520 0.236958i $$-0.0761504\pi$$
$$308$$ 0 0
$$309$$ −132.000 −0.427184
$$310$$ 0 0
$$311$$ − 235.559i − 0.757424i −0.925515 0.378712i $$-0.876367\pi$$
0.925515 0.378712i $$-0.123633\pi$$
$$312$$ 0 0
$$313$$ −478.000 −1.52716 −0.763578 0.645715i $$-0.776559\pi$$
−0.763578 + 0.645715i $$0.776559\pi$$
$$314$$ 0 0
$$315$$ 41.5692i 0.131966i
$$316$$ 0 0
$$317$$ 170.000 0.536278 0.268139 0.963380i $$-0.413591\pi$$
0.268139 + 0.963380i $$0.413591\pi$$
$$318$$ 0 0
$$319$$ − 180.133i − 0.564681i
$$320$$ 0 0
$$321$$ −36.0000 −0.112150
$$322$$ 0 0
$$323$$ − 207.846i − 0.643486i
$$324$$ 0 0
$$325$$ 42.0000 0.129231
$$326$$ 0 0
$$327$$ − 79.6743i − 0.243652i
$$328$$ 0 0
$$329$$ 480.000 1.45897
$$330$$ 0 0
$$331$$ − 408.764i − 1.23494i −0.786596 0.617468i $$-0.788158\pi$$
0.786596 0.617468i $$-0.211842\pi$$
$$332$$ 0 0
$$333$$ 78.0000 0.234234
$$334$$ 0 0
$$335$$ − 13.8564i − 0.0413624i
$$336$$ 0 0
$$337$$ 338.000 1.00297 0.501484 0.865167i $$-0.332788\pi$$
0.501484 + 0.865167i $$0.332788\pi$$
$$338$$ 0 0
$$339$$ 190.526i 0.562022i
$$340$$ 0 0
$$341$$ 48.0000 0.140762
$$342$$ 0 0
$$343$$ − 346.410i − 1.00994i
$$344$$ 0 0
$$345$$ −96.0000 −0.278261
$$346$$ 0 0
$$347$$ − 200.918i − 0.579014i −0.957176 0.289507i $$-0.906509\pi$$
0.957176 0.289507i $$-0.0934914\pi$$
$$348$$ 0 0
$$349$$ −506.000 −1.44986 −0.724928 0.688824i $$-0.758127\pi$$
−0.724928 + 0.688824i $$0.758127\pi$$
$$350$$ 0 0
$$351$$ − 10.3923i − 0.0296077i
$$352$$ 0 0
$$353$$ 178.000 0.504249 0.252125 0.967695i $$-0.418871\pi$$
0.252125 + 0.967695i $$0.418871\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −120.000 −0.336134
$$358$$ 0 0
$$359$$ 166.277i 0.463167i 0.972815 + 0.231583i $$0.0743906\pi$$
−0.972815 + 0.231583i $$0.925609\pi$$
$$360$$ 0 0
$$361$$ −71.0000 −0.196676
$$362$$ 0 0
$$363$$ − 126.440i − 0.348319i
$$364$$ 0 0
$$365$$ −92.0000 −0.252055
$$366$$ 0 0
$$367$$ 200.918i 0.547460i 0.961807 + 0.273730i $$0.0882575\pi$$
−0.961807 + 0.273730i $$0.911742\pi$$
$$368$$ 0 0
$$369$$ −174.000 −0.471545
$$370$$ 0 0
$$371$$ − 512.687i − 1.38191i
$$372$$ 0 0
$$373$$ 310.000 0.831099 0.415550 0.909571i $$-0.363589\pi$$
0.415550 + 0.909571i $$0.363589\pi$$
$$374$$ 0 0
$$375$$ 159.349i 0.424930i
$$376$$ 0 0
$$377$$ −52.0000 −0.137931
$$378$$ 0 0
$$379$$ 436.477i 1.15165i 0.817572 + 0.575827i $$0.195320\pi$$
−0.817572 + 0.575827i $$0.804680\pi$$
$$380$$ 0 0
$$381$$ −252.000 −0.661417
$$382$$ 0 0
$$383$$ − 609.682i − 1.59186i −0.605390 0.795929i $$-0.706983\pi$$
0.605390 0.795929i $$-0.293017\pi$$
$$384$$ 0 0
$$385$$ −96.0000 −0.249351
$$386$$ 0 0
$$387$$ − 145.492i − 0.375949i
$$388$$ 0 0
$$389$$ 578.000 1.48586 0.742931 0.669368i $$-0.233435\pi$$
0.742931 + 0.669368i $$0.233435\pi$$
$$390$$ 0 0
$$391$$ − 277.128i − 0.708768i
$$392$$ 0 0
$$393$$ −204.000 −0.519084
$$394$$ 0 0
$$395$$ 235.559i 0.596352i
$$396$$ 0 0
$$397$$ −26.0000 −0.0654912 −0.0327456 0.999464i $$-0.510425\pi$$
−0.0327456 + 0.999464i $$0.510425\pi$$
$$398$$ 0 0
$$399$$ 249.415i 0.625101i
$$400$$ 0 0
$$401$$ 250.000 0.623441 0.311721 0.950174i $$-0.399095\pi$$
0.311721 + 0.950174i $$0.399095\pi$$
$$402$$ 0 0
$$403$$ − 13.8564i − 0.0343831i
$$404$$ 0 0
$$405$$ 18.0000 0.0444444
$$406$$ 0 0
$$407$$ 180.133i 0.442588i
$$408$$ 0 0
$$409$$ 290.000 0.709046 0.354523 0.935047i $$-0.384643\pi$$
0.354523 + 0.935047i $$0.384643\pi$$
$$410$$ 0 0
$$411$$ − 17.3205i − 0.0421424i
$$412$$ 0 0
$$413$$ 624.000 1.51090
$$414$$ 0 0
$$415$$ − 96.9948i − 0.233723i
$$416$$ 0 0
$$417$$ −84.0000 −0.201439
$$418$$ 0 0
$$419$$ 339.482i 0.810219i 0.914268 + 0.405110i $$0.132767\pi$$
−0.914268 + 0.405110i $$0.867233\pi$$
$$420$$ 0 0
$$421$$ −674.000 −1.60095 −0.800475 0.599366i $$-0.795419\pi$$
−0.800475 + 0.599366i $$0.795419\pi$$
$$422$$ 0 0
$$423$$ − 207.846i − 0.491362i
$$424$$ 0 0
$$425$$ −210.000 −0.494118
$$426$$ 0 0
$$427$$ 180.133i 0.421858i
$$428$$ 0 0
$$429$$ 24.0000 0.0559441
$$430$$ 0 0
$$431$$ 540.400i 1.25383i 0.779088 + 0.626914i $$0.215682\pi$$
−0.779088 + 0.626914i $$0.784318\pi$$
$$432$$ 0 0
$$433$$ −334.000 −0.771363 −0.385681 0.922632i $$-0.626034\pi$$
−0.385681 + 0.922632i $$0.626034\pi$$
$$434$$ 0 0
$$435$$ − 90.0666i − 0.207050i
$$436$$ 0 0
$$437$$ −576.000 −1.31808
$$438$$ 0 0
$$439$$ 117.779i 0.268290i 0.990962 + 0.134145i $$0.0428288\pi$$
−0.990962 + 0.134145i $$0.957171\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.00680272
$$442$$ 0 0
$$443$$ 76.2102i 0.172032i 0.996294 + 0.0860161i $$0.0274136\pi$$
−0.996294 + 0.0860161i $$0.972586\pi$$
$$444$$ 0 0
$$445$$ 164.000 0.368539
$$446$$ 0 0
$$447$$ − 3.46410i − 0.00774967i
$$448$$ 0 0
$$449$$ 394.000 0.877506 0.438753 0.898608i $$-0.355420\pi$$
0.438753 + 0.898608i $$0.355420\pi$$
$$450$$ 0 0
$$451$$ − 401.836i − 0.890988i
$$452$$ 0 0
$$453$$ −156.000 −0.344371
$$454$$ 0 0
$$455$$ 27.7128i 0.0609073i
$$456$$ 0 0
$$457$$ −478.000 −1.04595 −0.522976 0.852347i $$-0.675178\pi$$
−0.522976 + 0.852347i $$0.675178\pi$$
$$458$$ 0 0
$$459$$ 51.9615i 0.113206i
$$460$$ 0 0
$$461$$ −142.000 −0.308026 −0.154013 0.988069i $$-0.549220\pi$$
−0.154013 + 0.988069i $$0.549220\pi$$
$$462$$ 0 0
$$463$$ 630.466i 1.36170i 0.732423 + 0.680849i $$0.238389\pi$$
−0.732423 + 0.680849i $$0.761611\pi$$
$$464$$ 0 0
$$465$$ 24.0000 0.0516129
$$466$$ 0 0
$$467$$ − 20.7846i − 0.0445067i −0.999752 0.0222533i $$-0.992916\pi$$
0.999752 0.0222533i $$-0.00708404\pi$$
$$468$$ 0 0
$$469$$ −48.0000 −0.102345
$$470$$ 0 0
$$471$$ − 370.659i − 0.786962i
$$472$$ 0 0
$$473$$ 336.000 0.710359
$$474$$ 0 0
$$475$$ 436.477i 0.918899i
$$476$$ 0 0
$$477$$ −222.000 −0.465409
$$478$$ 0 0
$$479$$ 734.390i 1.53317i 0.642141 + 0.766586i $$0.278046\pi$$
−0.642141 + 0.766586i $$0.721954\pi$$
$$480$$ 0 0
$$481$$ 52.0000 0.108108
$$482$$ 0 0
$$483$$ 332.554i 0.688517i
$$484$$ 0 0
$$485$$ 4.00000 0.00824742
$$486$$ 0 0
$$487$$ 103.923i 0.213394i 0.994292 + 0.106697i $$0.0340275\pi$$
−0.994292 + 0.106697i $$0.965972\pi$$
$$488$$ 0 0
$$489$$ −36.0000 −0.0736196
$$490$$ 0 0
$$491$$ 921.451i 1.87668i 0.345711 + 0.938341i $$0.387638\pi$$
−0.345711 + 0.938341i $$0.612362\pi$$
$$492$$ 0 0
$$493$$ 260.000 0.527383
$$494$$ 0 0
$$495$$ 41.5692i 0.0839782i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 76.2102i 0.152726i 0.997080 + 0.0763630i $$0.0243308\pi$$
−0.997080 + 0.0763630i $$0.975669\pi$$
$$500$$ 0 0
$$501$$ 168.000 0.335329
$$502$$ 0 0
$$503$$ − 581.969i − 1.15700i −0.815684 0.578498i $$-0.803639\pi$$
0.815684 0.578498i $$-0.196361\pi$$
$$504$$ 0 0
$$505$$ 148.000 0.293069
$$506$$ 0 0
$$507$$ 285.788i 0.563685i
$$508$$ 0 0
$$509$$ 842.000 1.65422 0.827112 0.562037i $$-0.189982\pi$$
0.827112 + 0.562037i $$0.189982\pi$$
$$510$$ 0 0
$$511$$ 318.697i 0.623674i
$$512$$ 0 0
$$513$$ 108.000 0.210526
$$514$$ 0 0
$$515$$ − 152.420i − 0.295962i
$$516$$ 0 0
$$517$$ 480.000 0.928433
$$518$$ 0 0
$$519$$ 578.505i 1.11465i
$$520$$ 0 0
$$521$$ −326.000 −0.625720 −0.312860 0.949799i $$-0.601287\pi$$
−0.312860 + 0.949799i $$0.601287\pi$$
$$522$$ 0 0
$$523$$ − 311.769i − 0.596117i −0.954548 0.298058i $$-0.903661\pi$$
0.954548 0.298058i $$-0.0963390\pi$$
$$524$$ 0 0
$$525$$ 252.000 0.480000
$$526$$ 0 0
$$527$$ 69.2820i 0.131465i
$$528$$ 0 0
$$529$$ −239.000 −0.451796
$$530$$ 0 0
$$531$$ − 270.200i − 0.508851i
$$532$$ 0 0
$$533$$ −116.000 −0.217636
$$534$$ 0 0
$$535$$ − 41.5692i − 0.0776995i
$$536$$ 0 0
$$537$$ 324.000 0.603352
$$538$$ 0 0
$$539$$ − 6.92820i − 0.0128538i
$$540$$ 0 0
$$541$$ −530.000 −0.979667 −0.489834 0.871816i $$-0.662942\pi$$
−0.489834 + 0.871816i $$0.662942\pi$$
$$542$$ 0 0
$$543$$ 3.46410i 0.00637956i
$$544$$ 0 0
$$545$$ 92.0000 0.168807
$$546$$ 0 0
$$547$$ 339.482i 0.620625i 0.950635 + 0.310313i $$0.100434\pi$$
−0.950635 + 0.310313i $$0.899566\pi$$
$$548$$ 0 0
$$549$$ 78.0000 0.142077
$$550$$ 0 0
$$551$$ − 540.400i − 0.980762i
$$552$$ 0 0
$$553$$ 816.000 1.47559
$$554$$ 0 0
$$555$$ 90.0666i 0.162282i
$$556$$ 0 0
$$557$$ −766.000 −1.37522 −0.687612 0.726078i $$-0.741341\pi$$
−0.687612 + 0.726078i $$0.741341\pi$$
$$558$$ 0 0
$$559$$ − 96.9948i − 0.173515i
$$560$$ 0 0
$$561$$ −120.000 −0.213904
$$562$$ 0 0
$$563$$ − 491.902i − 0.873717i −0.899530 0.436858i $$-0.856091\pi$$
0.899530 0.436858i $$-0.143909\pi$$
$$564$$ 0 0
$$565$$ −220.000 −0.389381
$$566$$ 0 0
$$567$$ − 62.3538i − 0.109971i
$$568$$ 0 0
$$569$$ −422.000 −0.741652 −0.370826 0.928702i $$-0.620925\pi$$
−0.370826 + 0.928702i $$0.620925\pi$$
$$570$$ 0 0
$$571$$ 284.056i 0.497472i 0.968571 + 0.248736i $$0.0800151\pi$$
−0.968571 + 0.248736i $$0.919985\pi$$
$$572$$ 0 0
$$573$$ −384.000 −0.670157
$$574$$ 0 0
$$575$$ 581.969i 1.01212i
$$576$$ 0 0
$$577$$ −46.0000 −0.0797227 −0.0398614 0.999205i $$-0.512692\pi$$
−0.0398614 + 0.999205i $$0.512692\pi$$
$$578$$ 0 0
$$579$$ − 502.295i − 0.867521i
$$580$$ 0 0
$$581$$ −336.000 −0.578313
$$582$$ 0 0
$$583$$ − 512.687i − 0.879395i
$$584$$ 0 0
$$585$$ 12.0000 0.0205128
$$586$$ 0 0
$$587$$ − 630.466i − 1.07405i −0.843567 0.537024i $$-0.819548\pi$$
0.843567 0.537024i $$-0.180452\pi$$
$$588$$ 0 0
$$589$$ 144.000 0.244482
$$590$$ 0 0
$$591$$ − 45.0333i − 0.0761985i
$$592$$ 0 0
$$593$$ 82.0000 0.138280 0.0691400 0.997607i $$-0.477974\pi$$
0.0691400 + 0.997607i $$0.477974\pi$$
$$594$$ 0 0
$$595$$ − 138.564i − 0.232881i
$$596$$ 0 0
$$597$$ 684.000 1.14573
$$598$$ 0 0
$$599$$ 55.4256i 0.0925303i 0.998929 + 0.0462651i $$0.0147319\pi$$
−0.998929 + 0.0462651i $$0.985268\pi$$
$$600$$ 0 0
$$601$$ −334.000 −0.555740 −0.277870 0.960619i $$-0.589629\pi$$
−0.277870 + 0.960619i $$0.589629\pi$$
$$602$$ 0 0
$$603$$ 20.7846i 0.0344687i
$$604$$ 0 0
$$605$$ 146.000 0.241322
$$606$$ 0 0
$$607$$ − 367.195i − 0.604934i −0.953160 0.302467i $$-0.902190\pi$$
0.953160 0.302467i $$-0.0978102\pi$$
$$608$$ 0 0
$$609$$ −312.000 −0.512315
$$610$$ 0 0
$$611$$ − 138.564i − 0.226782i
$$612$$ 0 0
$$613$$ 214.000 0.349103 0.174551 0.984648i $$-0.444152\pi$$
0.174551 + 0.984648i $$0.444152\pi$$
$$614$$ 0 0
$$615$$ − 200.918i − 0.326696i
$$616$$ 0 0
$$617$$ −1118.00 −1.81199 −0.905997 0.423285i $$-0.860877\pi$$
−0.905997 + 0.423285i $$0.860877\pi$$
$$618$$ 0 0
$$619$$ 672.036i 1.08568i 0.839836 + 0.542840i $$0.182651\pi$$
−0.839836 + 0.542840i $$0.817349\pi$$
$$620$$ 0 0
$$621$$ 144.000 0.231884
$$622$$ 0 0
$$623$$ − 568.113i − 0.911898i
$$624$$ 0 0
$$625$$ 341.000 0.545600
$$626$$ 0 0
$$627$$ 249.415i 0.397792i
$$628$$ 0 0
$$629$$ −260.000 −0.413355
$$630$$ 0 0
$$631$$ − 145.492i − 0.230574i −0.993332 0.115287i $$-0.963221\pi$$
0.993332 0.115287i $$-0.0367788\pi$$
$$632$$ 0 0
$$633$$ 420.000 0.663507
$$634$$ 0 0
$$635$$ − 290.985i − 0.458243i
$$636$$ 0 0
$$637$$ −2.00000 −0.00313972
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10.0000 0.0156006 0.00780031 0.999970i $$-0.497517\pi$$
0.00780031 + 0.999970i $$0.497517\pi$$
$$642$$ 0 0
$$643$$ − 1212.44i − 1.88559i −0.333370 0.942796i $$-0.608186\pi$$
0.333370 0.942796i $$-0.391814\pi$$
$$644$$ 0 0
$$645$$ 168.000 0.260465
$$646$$ 0 0
$$647$$ − 332.554i − 0.513993i −0.966412 0.256997i $$-0.917267\pi$$
0.966412 0.256997i $$-0.0827330\pi$$
$$648$$ 0 0
$$649$$ 624.000 0.961479
$$650$$ 0 0
$$651$$ − 83.1384i − 0.127709i
$$652$$ 0 0
$$653$$ −670.000 −1.02603 −0.513017 0.858379i $$-0.671472\pi$$
−0.513017 + 0.858379i $$0.671472\pi$$
$$654$$ 0 0
$$655$$ − 235.559i − 0.359632i
$$656$$ 0 0
$$657$$ 138.000 0.210046
$$658$$ 0 0
$$659$$ 824.456i 1.25107i 0.780195 + 0.625536i $$0.215120\pi$$
−0.780195 + 0.625536i $$0.784880\pi$$
$$660$$ 0 0
$$661$$ 1222.00 1.84871 0.924357 0.381529i $$-0.124602\pi$$
0.924357 + 0.381529i $$0.124602\pi$$
$$662$$ 0 0
$$663$$ 34.6410i 0.0522489i
$$664$$ 0 0
$$665$$ −288.000 −0.433083
$$666$$ 0 0
$$667$$ − 720.533i − 1.08026i
$$668$$ 0 0
$$669$$ −588.000 −0.878924
$$670$$ 0 0
$$671$$ 180.133i 0.268455i
$$672$$ 0 0
$$673$$ −334.000 −0.496285 −0.248143 0.968724i $$-0.579820\pi$$
−0.248143 + 0.968724i $$0.579820\pi$$
$$674$$ 0 0
$$675$$ − 109.119i − 0.161658i
$$676$$ 0 0
$$677$$ −1006.00 −1.48597 −0.742984 0.669309i $$-0.766590\pi$$
−0.742984 + 0.669309i $$0.766590\pi$$
$$678$$ 0 0
$$679$$ − 13.8564i − 0.0204071i
$$680$$ 0 0
$$681$$ 492.000 0.722467
$$682$$ 0 0
$$683$$ 187.061i 0.273882i 0.990579 + 0.136941i $$0.0437271\pi$$
−0.990579 + 0.136941i $$0.956273\pi$$
$$684$$ 0 0
$$685$$ 20.0000 0.0291971
$$686$$ 0 0
$$687$$ − 245.951i − 0.358008i
$$688$$ 0 0
$$689$$ −148.000 −0.214804
$$690$$ 0 0
$$691$$ − 990.733i − 1.43377i −0.697193 0.716884i $$-0.745568\pi$$
0.697193 0.716884i $$-0.254432\pi$$
$$692$$ 0 0
$$693$$ 144.000 0.207792
$$694$$ 0 0
$$695$$ − 96.9948i − 0.139561i
$$696$$ 0 0
$$697$$ 580.000 0.832138
$$698$$ 0 0
$$699$$ − 142.028i − 0.203188i
$$700$$ 0 0
$$701$$ 1034.00 1.47504 0.737518 0.675328i $$-0.235998\pi$$
0.737518 + 0.675328i $$0.235998\pi$$
$$702$$ 0 0
$$703$$ 540.400i 0.768705i
$$704$$ 0 0
$$705$$ 240.000 0.340426
$$706$$ 0 0
$$707$$ − 512.687i − 0.725158i
$$708$$ 0 0
$$709$$ −530.000 −0.747532 −0.373766 0.927523i $$-0.621934\pi$$
−0.373766 + 0.927523i $$0.621934\pi$$
$$710$$ 0 0
$$711$$ − 353.338i − 0.496960i
$$712$$ 0 0
$$713$$ 192.000 0.269285
$$714$$ 0 0
$$715$$ 27.7128i 0.0387592i
$$716$$ 0 0
$$717$$ 672.000 0.937238
$$718$$ 0 0
$$719$$ − 706.677i − 0.982861i −0.870917 0.491430i $$-0.836474\pi$$
0.870917 0.491430i $$-0.163526\pi$$
$$720$$ 0 0
$$721$$ −528.000 −0.732316
$$722$$ 0 0
$$723$$ 79.6743i 0.110200i
$$724$$ 0 0
$$725$$ −546.000 −0.753103
$$726$$ 0 0
$$727$$ 242.487i 0.333545i 0.985995 + 0.166772i $$0.0533345\pi$$
−0.985995 + 0.166772i $$0.946665\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −0.0370370
$$730$$ 0 0
$$731$$ 484.974i 0.663439i
$$732$$ 0 0
$$733$$ −194.000 −0.264666 −0.132333 0.991205i $$-0.542247\pi$$
−0.132333 + 0.991205i $$0.542247\pi$$
$$734$$ 0 0
$$735$$ − 3.46410i − 0.00471306i
$$736$$ 0 0
$$737$$ −48.0000 −0.0651289
$$738$$ 0 0
$$739$$ 1351.00i 1.82815i 0.405550 + 0.914073i $$0.367080\pi$$
−0.405550 + 0.914073i $$0.632920\pi$$
$$740$$ 0 0
$$741$$ 72.0000 0.0971660
$$742$$ 0 0
$$743$$ 678.964i 0.913814i 0.889514 + 0.456907i $$0.151043\pi$$
−0.889514 + 0.456907i $$0.848957\pi$$
$$744$$ 0 0
$$745$$ 4.00000 0.00536913
$$746$$ 0 0
$$747$$ 145.492i 0.194769i
$$748$$ 0 0
$$749$$ −144.000 −0.192256
$$750$$ 0 0
$$751$$ − 658.179i − 0.876404i −0.898877 0.438202i $$-0.855615\pi$$
0.898877 0.438202i $$-0.144385\pi$$
$$752$$ 0 0
$$753$$ −252.000 −0.334661
$$754$$ 0 0
$$755$$ − 180.133i − 0.238587i
$$756$$ 0 0
$$757$$ 1006.00 1.32893 0.664465 0.747319i $$-0.268659\pi$$
0.664465 + 0.747319i $$0.268659\pi$$
$$758$$ 0 0
$$759$$ 332.554i 0.438147i
$$760$$ 0 0
$$761$$ −758.000 −0.996058 −0.498029 0.867160i $$-0.665943\pi$$
−0.498029 + 0.867160i $$0.665943\pi$$
$$762$$ 0 0
$$763$$ − 318.697i − 0.417690i
$$764$$ 0 0
$$765$$ −60.0000 −0.0784314
$$766$$ 0 0
$$767$$ − 180.133i − 0.234854i
$$768$$ 0 0
$$769$$ 2.00000 0.00260078 0.00130039 0.999999i $$-0.499586\pi$$
0.00130039 + 0.999999i $$0.499586\pi$$
$$770$$ 0 0
$$771$$ 439.941i 0.570611i
$$772$$ 0 0
$$773$$ −262.000 −0.338939 −0.169470 0.985535i $$-0.554205\pi$$
−0.169470 + 0.985535i $$0.554205\pi$$
$$774$$ 0 0
$$775$$ − 145.492i − 0.187732i
$$776$$ 0 0
$$777$$ 312.000 0.401544
$$778$$ 0 0
$$779$$ − 1205.51i − 1.54751i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 135.100i 0.172541i
$$784$$ 0 0
$$785$$ 428.000 0.545223
$$786$$ 0 0
$$787$$ 1447.99i 1.83989i 0.392046 + 0.919946i $$0.371767\pi$$
−0.392046 + 0.919946i $$0.628233\pi$$
$$788$$ 0 0
$$789$$ 264.000 0.334601
$$790$$ 0 0
$$791$$ 762.102i 0.963467i
$$792$$ 0 0
$$793$$ 52.0000 0.0655738
$$794$$ 0 0
$$795$$ − 256.344i − 0.322445i
$$796$$ 0 0
$$797$$ 866.000 1.08657 0.543287 0.839547i $$-0.317179\pi$$
0.543287 + 0.839547i $$0.317179\pi$$
$$798$$ 0 0
$$799$$ 692.820i 0.867109i
$$800$$ 0 0
$$801$$ −246.000 −0.307116
$$802$$ 0 0
$$803$$ 318.697i 0.396883i
$$804$$ 0 0
$$805$$ −384.000 −0.477019
$$806$$ 0 0
$$807$$ 453.797i 0.562326i
$$808$$ 0 0
$$809$$ 10.0000 0.0123609 0.00618047 0.999981i $$-0.498033\pi$$
0.00618047 + 0.999981i $$0.498033\pi$$
$$810$$ 0 0
$$811$$ 436.477i 0.538196i 0.963113 + 0.269098i $$0.0867255\pi$$
−0.963113 + 0.269098i $$0.913274\pi$$
$$812$$ 0 0
$$813$$ 36.0000 0.0442804
$$814$$ 0 0
$$815$$ − 41.5692i − 0.0510052i
$$816$$ 0 0
$$817$$ 1008.00 1.23378
$$818$$ 0 0
$$819$$ − 41.5692i − 0.0507561i
$$820$$ 0 0
$$821$$ −838.000 −1.02071 −0.510353 0.859965i $$-0.670485\pi$$
−0.510353 + 0.859965i $$0.670485\pi$$
$$822$$ 0 0
$$823$$ 879.882i 1.06912i 0.845132 + 0.534558i $$0.179522\pi$$
−0.845132 + 0.534558i $$0.820478\pi$$
$$824$$ 0 0
$$825$$ 252.000 0.305455
$$826$$ 0 0
$$827$$ 727.461i 0.879639i 0.898086 + 0.439819i $$0.144958\pi$$
−0.898086 + 0.439819i $$0.855042\pi$$
$$828$$ 0 0
$$829$$ −1298.00 −1.56574 −0.782871 0.622184i $$-0.786246\pi$$
−0.782871 + 0.622184i $$0.786246\pi$$
$$830$$ 0 0
$$831$$ 502.295i 0.604446i
$$832$$ 0 0
$$833$$ 10.0000 0.0120048
$$834$$ 0 0
$$835$$ 193.990i 0.232323i
$$836$$ 0 0
$$837$$ −36.0000 −0.0430108
$$838$$ 0 0
$$839$$ 193.990i 0.231215i 0.993295 + 0.115608i $$0.0368815\pi$$
−0.993295 + 0.115608i $$0.963118\pi$$
$$840$$ 0 0
$$841$$ −165.000 −0.196195
$$842$$ 0 0
$$843$$ − 391.443i − 0.464346i
$$844$$ 0 0
$$845$$ −330.000 −0.390533
$$846$$ 0 0
$$847$$ − 505.759i − 0.597118i
$$848$$ 0 0
$$849$$ −516.000 −0.607774
$$850$$ 0 0
$$851$$ 720.533i 0.846690i
$$852$$ 0 0
$$853$$ −506.000 −0.593200 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$854$$ 0 0
$$855$$ 124.708i 0.145857i
$$856$$ 0 0
$$857$$ −998.000 −1.16453 −0.582264 0.813000i $$-0.697833\pi$$
−0.582264 + 0.813000i $$0.697833\pi$$
$$858$$ 0 0
$$859$$ − 505.759i − 0.588776i −0.955686 0.294388i $$-0.904884\pi$$
0.955686 0.294388i $$-0.0951158\pi$$
$$860$$ 0 0
$$861$$ −696.000 −0.808362
$$862$$ 0 0
$$863$$ 166.277i 0.192673i 0.995349 + 0.0963365i $$0.0307125\pi$$
−0.995349 + 0.0963365i $$0.969287\pi$$
$$864$$ 0 0
$$865$$ −668.000 −0.772254
$$866$$ 0 0
$$867$$ 327.358i 0.377575i
$$868$$ 0 0
$$869$$ 816.000 0.939010
$$870$$ 0 0
$$871$$ 13.8564i 0.0159086i
$$872$$ 0 0
$$873$$ −6.00000 −0.00687285
$$874$$ 0 0
$$875$$ 637.395i 0.728451i
$$876$$ 0 0
$$877$$ 646.000 0.736602 0.368301 0.929707i $$-0.379940\pi$$
0.368301 + 0.929707i $$0.379940\pi$$
$$878$$ 0 0
$$879$$ − 627.002i − 0.713313i
$$880$$ 0 0
$$881$$ 898.000 1.01930 0.509648 0.860383i $$-0.329776\pi$$
0.509648 + 0.860383i $$0.329776\pi$$
$$882$$ 0 0
$$883$$ 727.461i 0.823852i 0.911217 + 0.411926i $$0.135144\pi$$
−0.911217 + 0.411926i $$0.864856\pi$$
$$884$$ 0 0
$$885$$ 312.000 0.352542
$$886$$ 0 0
$$887$$ 845.241i 0.952921i 0.879196 + 0.476460i $$0.158080\pi$$
−0.879196 + 0.476460i $$0.841920\pi$$
$$888$$ 0 0
$$889$$ −1008.00 −1.13386
$$890$$ 0 0
$$891$$ − 62.3538i − 0.0699819i
$$892$$ 0 0
$$893$$ 1440.00 1.61254
$$894$$ 0 0
$$895$$ 374.123i 0.418014i
$$896$$ 0 0
$$897$$ 96.0000 0.107023
$$898$$ 0 0
$$899$$ 180.133i 0.200371i
$$900$$ 0 0
$$901$$ 740.000 0.821310
$$902$$ 0 0
$$903$$ − 581.969i − 0.644484i
$$904$$ 0 0
$$905$$ −4.00000 −0.00441989
$$906$$ 0 0
$$907$$ − 1364.86i − 1.50480i −0.658705 0.752401i $$-0.728895\pi$$
0.658705 0.752401i $$-0.271105\pi$$
$$908$$ 0 0
$$909$$ −222.000 −0.244224
$$910$$ 0 0
$$911$$ − 387.979i − 0.425883i −0.977065 0.212941i $$-0.931696\pi$$
0.977065 0.212941i $$-0.0683044\pi$$
$$912$$ 0 0
$$913$$ −336.000 −0.368018
$$914$$ 0 0
$$915$$ 90.0666i 0.0984335i
$$916$$ 0 0
$$917$$ −816.000 −0.889858
$$918$$ 0 0
$$919$$ 602.754i 0.655880i 0.944699 + 0.327940i $$0.106354\pi$$
−0.944699 + 0.327940i $$0.893646\pi$$
$$920$$ 0 0
$$921$$ −252.000 −0.273616
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 546.000 0.590270
$$926$$ 0 0
$$927$$ 228.631i 0.246635i
$$928$$ 0 0
$$929$$ 1594.00 1.71582 0.857912 0.513797i $$-0.171762\pi$$
0.857912 + 0.513797i $$0.171762\pi$$
$$930$$ 0 0
$$931$$ − 20.7846i − 0.0223250i
$$932$$ 0 0
$$933$$ −408.000 −0.437299
$$934$$ 0 0
$$935$$ − 138.564i − 0.148197i
$$936$$ 0 0
$$937$$ 674.000 0.719317 0.359658 0.933084i $$-0.382893\pi$$
0.359658 + 0.933084i $$0.382893\pi$$
$$938$$ 0 0
$$939$$ 827.920i 0.881704i
$$940$$ 0 0
$$941$$ −430.000 −0.456961 −0.228480 0.973549i $$-0.573376\pi$$
−0.228480 + 0.973549i $$0.573376\pi$$
$$942$$ 0 0
$$943$$ − 1607.34i − 1.70450i
$$944$$ 0 0
$$945$$ 72.0000 0.0761905
$$946$$ 0 0
$$947$$ 76.2102i 0.0804754i 0.999190 + 0.0402377i $$0.0128115\pi$$
−0.999190 + 0.0402377i $$0.987188\pi$$
$$948$$ 0 0
$$949$$ 92.0000 0.0969442
$$950$$ 0 0
$$951$$ − 294.449i − 0.309620i
$$952$$ 0 0
$$953$$ 730.000 0.766002 0.383001 0.923748i $$-0.374891\pi$$
0.383001 + 0.923748i $$0.374891\pi$$
$$954$$ 0 0
$$955$$ − 443.405i − 0.464298i
$$956$$ 0 0
$$957$$ −312.000 −0.326019
$$958$$ 0 0
$$959$$ − 69.2820i − 0.0722440i
$$960$$ 0 0
$$961$$ 913.000 0.950052
$$962$$ 0 0
$$963$$ 62.3538i 0.0647496i
$$964$$ 0 0
$$965$$ 580.000 0.601036
$$966$$ 0 0
$$967$$ 921.451i 0.952897i 0.879202 + 0.476448i $$0.158076\pi$$
−0.879202 + 0.476448i $$0.841924\pi$$
$$968$$ 0 0
$$969$$ −360.000 −0.371517
$$970$$ 0 0
$$971$$ − 1475.71i − 1.51978i −0.650051 0.759890i $$-0.725253\pi$$
0.650051 0.759890i $$-0.274747\pi$$
$$972$$ 0 0
$$973$$ −336.000 −0.345324
$$974$$ 0 0
$$975$$ − 72.7461i − 0.0746114i
$$976$$ 0 0
$$977$$ 346.000 0.354145 0.177073 0.984198i $$-0.443337\pi$$
0.177073 + 0.984198i $$0.443337\pi$$
$$978$$ 0 0
$$979$$ − 568.113i − 0.580299i
$$980$$ 0 0
$$981$$ −138.000 −0.140673
$$982$$ 0 0
$$983$$ 734.390i 0.747090i 0.927612 + 0.373545i $$0.121858\pi$$
−0.927612 + 0.373545i $$0.878142\pi$$
$$984$$ 0 0
$$985$$ 52.0000 0.0527919
$$986$$ 0 0
$$987$$ − 831.384i − 0.842335i
$$988$$ 0 0
$$989$$ 1344.00 1.35895
$$990$$ 0 0
$$991$$ 976.877i 0.985748i 0.870101 + 0.492874i $$0.164054\pi$$
−0.870101 + 0.492874i $$0.835946\pi$$
$$992$$ 0 0
$$993$$ −708.000 −0.712991
$$994$$ 0 0
$$995$$ 789.815i 0.793784i
$$996$$ 0 0
$$997$$ −458.000 −0.459378 −0.229689 0.973264i $$-0.573771\pi$$
−0.229689 + 0.973264i $$0.573771\pi$$
$$998$$ 0 0
$$999$$ − 135.100i − 0.135235i
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 192.3.g.b.127.1 2
3.2 odd 2 576.3.g.e.127.1 2
4.3 odd 2 inner 192.3.g.b.127.2 2
8.3 odd 2 12.3.d.a.7.2 yes 2
8.5 even 2 12.3.d.a.7.1 2
12.11 even 2 576.3.g.e.127.2 2
16.3 odd 4 768.3.b.c.127.1 4
16.5 even 4 768.3.b.c.127.2 4
16.11 odd 4 768.3.b.c.127.4 4
16.13 even 4 768.3.b.c.127.3 4
24.5 odd 2 36.3.d.c.19.2 2
24.11 even 2 36.3.d.c.19.1 2
40.3 even 4 300.3.f.a.199.4 4
40.13 odd 4 300.3.f.a.199.2 4
40.19 odd 2 300.3.c.b.151.1 2
40.27 even 4 300.3.f.a.199.1 4
40.29 even 2 300.3.c.b.151.2 2
40.37 odd 4 300.3.f.a.199.3 4
48.5 odd 4 2304.3.b.l.127.2 4
48.11 even 4 2304.3.b.l.127.1 4
48.29 odd 4 2304.3.b.l.127.4 4
48.35 even 4 2304.3.b.l.127.3 4
56.13 odd 2 588.3.g.b.295.1 2
56.27 even 2 588.3.g.b.295.2 2
72.5 odd 6 324.3.f.a.55.1 2
72.11 even 6 324.3.f.a.271.1 2
72.13 even 6 324.3.f.j.55.1 2
72.29 odd 6 324.3.f.g.271.1 2
72.43 odd 6 324.3.f.j.271.1 2
72.59 even 6 324.3.f.g.55.1 2
72.61 even 6 324.3.f.d.271.1 2
72.67 odd 6 324.3.f.d.55.1 2
120.29 odd 2 900.3.c.e.451.1 2
120.53 even 4 900.3.f.c.199.3 4
120.59 even 2 900.3.c.e.451.2 2
120.77 even 4 900.3.f.c.199.2 4
120.83 odd 4 900.3.f.c.199.1 4
120.107 odd 4 900.3.f.c.199.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.d.a.7.1 2 8.5 even 2
12.3.d.a.7.2 yes 2 8.3 odd 2
36.3.d.c.19.1 2 24.11 even 2
36.3.d.c.19.2 2 24.5 odd 2
192.3.g.b.127.1 2 1.1 even 1 trivial
192.3.g.b.127.2 2 4.3 odd 2 inner
300.3.c.b.151.1 2 40.19 odd 2
300.3.c.b.151.2 2 40.29 even 2
300.3.f.a.199.1 4 40.27 even 4
300.3.f.a.199.2 4 40.13 odd 4
300.3.f.a.199.3 4 40.37 odd 4
300.3.f.a.199.4 4 40.3 even 4
324.3.f.a.55.1 2 72.5 odd 6
324.3.f.a.271.1 2 72.11 even 6
324.3.f.d.55.1 2 72.67 odd 6
324.3.f.d.271.1 2 72.61 even 6
324.3.f.g.55.1 2 72.59 even 6
324.3.f.g.271.1 2 72.29 odd 6
324.3.f.j.55.1 2 72.13 even 6
324.3.f.j.271.1 2 72.43 odd 6
576.3.g.e.127.1 2 3.2 odd 2
576.3.g.e.127.2 2 12.11 even 2
588.3.g.b.295.1 2 56.13 odd 2
588.3.g.b.295.2 2 56.27 even 2
768.3.b.c.127.1 4 16.3 odd 4
768.3.b.c.127.2 4 16.5 even 4
768.3.b.c.127.3 4 16.13 even 4
768.3.b.c.127.4 4 16.11 odd 4
900.3.c.e.451.1 2 120.29 odd 2
900.3.c.e.451.2 2 120.59 even 2
900.3.f.c.199.1 4 120.83 odd 4
900.3.f.c.199.2 4 120.77 even 4
900.3.f.c.199.3 4 120.53 even 4
900.3.f.c.199.4 4 120.107 odd 4
2304.3.b.l.127.1 4 48.11 even 4
2304.3.b.l.127.2 4 48.5 odd 4
2304.3.b.l.127.3 4 48.35 even 4
2304.3.b.l.127.4 4 48.29 odd 4