# Properties

 Label 3332.2.a.j.1.1 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.30278 q^{3} -1.30278 q^{5} +2.30278 q^{9} +O(q^{10})$$ $$q-2.30278 q^{3} -1.30278 q^{5} +2.30278 q^{9} +0.605551 q^{13} +3.00000 q^{15} -1.00000 q^{17} +0.605551 q^{19} -3.30278 q^{25} +1.60555 q^{27} -0.697224 q^{31} +4.60555 q^{37} -1.39445 q^{39} +6.90833 q^{41} +3.69722 q^{43} -3.00000 q^{45} +2.60555 q^{47} +2.30278 q^{51} -7.30278 q^{53} -1.39445 q^{57} +5.21110 q^{59} -2.90833 q^{61} -0.788897 q^{65} -5.30278 q^{67} +13.8167 q^{71} -2.90833 q^{73} +7.60555 q^{75} +5.39445 q^{79} -10.6056 q^{81} -6.00000 q^{83} +1.30278 q^{85} +9.39445 q^{89} +1.60555 q^{93} -0.788897 q^{95} +2.69722 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} + q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 + q^9 $$2 q - q^{3} + q^{5} + q^{9} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{19} - 3 q^{25} - 4 q^{27} - 5 q^{31} + 2 q^{37} - 10 q^{39} + 3 q^{41} + 11 q^{43} - 6 q^{45} - 2 q^{47} + q^{51} - 11 q^{53} - 10 q^{57} - 4 q^{59} + 5 q^{61} - 16 q^{65} - 7 q^{67} + 6 q^{71} + 5 q^{73} + 8 q^{75} + 18 q^{79} - 14 q^{81} - 12 q^{83} - q^{85} + 26 q^{89} - 4 q^{93} - 16 q^{95} + 9 q^{97}+O(q^{100})$$ 2 * q - q^3 + q^5 + q^9 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^19 - 3 * q^25 - 4 * q^27 - 5 * q^31 + 2 * q^37 - 10 * q^39 + 3 * q^41 + 11 * q^43 - 6 * q^45 - 2 * q^47 + q^51 - 11 * q^53 - 10 * q^57 - 4 * q^59 + 5 * q^61 - 16 * q^65 - 7 * q^67 + 6 * q^71 + 5 * q^73 + 8 * q^75 + 18 * q^79 - 14 * q^81 - 12 * q^83 - q^85 + 26 * q^89 - 4 * q^93 - 16 * q^95 + 9 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.30278 −1.32951 −0.664754 0.747062i $$-0.731464\pi$$
−0.664754 + 0.747062i $$0.731464\pi$$
$$4$$ 0 0
$$5$$ −1.30278 −0.582619 −0.291309 0.956629i $$-0.594091\pi$$
−0.291309 + 0.956629i $$0.594091\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 0.605551 0.167950 0.0839749 0.996468i $$-0.473238\pi$$
0.0839749 + 0.996468i $$0.473238\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 0.605551 0.138923 0.0694615 0.997585i $$-0.477872\pi$$
0.0694615 + 0.997585i $$0.477872\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −3.30278 −0.660555
$$26$$ 0 0
$$27$$ 1.60555 0.308988
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −0.697224 −0.125225 −0.0626126 0.998038i $$-0.519943\pi$$
−0.0626126 + 0.998038i $$0.519943\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.60555 0.757148 0.378574 0.925571i $$-0.376415\pi$$
0.378574 + 0.925571i $$0.376415\pi$$
$$38$$ 0 0
$$39$$ −1.39445 −0.223290
$$40$$ 0 0
$$41$$ 6.90833 1.07890 0.539450 0.842018i $$-0.318632\pi$$
0.539450 + 0.842018i $$0.318632\pi$$
$$42$$ 0 0
$$43$$ 3.69722 0.563821 0.281911 0.959441i $$-0.409032\pi$$
0.281911 + 0.959441i $$0.409032\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ 2.60555 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.30278 0.322453
$$52$$ 0 0
$$53$$ −7.30278 −1.00311 −0.501557 0.865125i $$-0.667239\pi$$
−0.501557 + 0.865125i $$0.667239\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.39445 −0.184699
$$58$$ 0 0
$$59$$ 5.21110 0.678428 0.339214 0.940709i $$-0.389839\pi$$
0.339214 + 0.940709i $$0.389839\pi$$
$$60$$ 0 0
$$61$$ −2.90833 −0.372373 −0.186187 0.982514i $$-0.559613\pi$$
−0.186187 + 0.982514i $$0.559613\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.788897 −0.0978507
$$66$$ 0 0
$$67$$ −5.30278 −0.647837 −0.323919 0.946085i $$-0.605000\pi$$
−0.323919 + 0.946085i $$0.605000\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.8167 1.63974 0.819868 0.572553i $$-0.194047\pi$$
0.819868 + 0.572553i $$0.194047\pi$$
$$72$$ 0 0
$$73$$ −2.90833 −0.340394 −0.170197 0.985410i $$-0.554440\pi$$
−0.170197 + 0.985410i $$0.554440\pi$$
$$74$$ 0 0
$$75$$ 7.60555 0.878213
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.39445 0.606923 0.303461 0.952844i $$-0.401858\pi$$
0.303461 + 0.952844i $$0.401858\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 1.30278 0.141306
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.39445 0.995810 0.497905 0.867232i $$-0.334103\pi$$
0.497905 + 0.867232i $$0.334103\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1.60555 0.166488
$$94$$ 0 0
$$95$$ −0.788897 −0.0809392
$$96$$ 0 0
$$97$$ 2.69722 0.273862 0.136931 0.990581i $$-0.456276\pi$$
0.136931 + 0.990581i $$0.456276\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.81665 0.180764 0.0903819 0.995907i $$-0.471191\pi$$
0.0903819 + 0.995907i $$0.471191\pi$$
$$102$$ 0 0
$$103$$ −7.21110 −0.710531 −0.355266 0.934765i $$-0.615610\pi$$
−0.355266 + 0.934765i $$0.615610\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.60555 −0.831930 −0.415965 0.909381i $$-0.636556\pi$$
−0.415965 + 0.909381i $$0.636556\pi$$
$$108$$ 0 0
$$109$$ −20.4222 −1.95609 −0.978046 0.208388i $$-0.933178\pi$$
−0.978046 + 0.208388i $$0.933178\pi$$
$$110$$ 0 0
$$111$$ −10.6056 −1.00663
$$112$$ 0 0
$$113$$ −2.60555 −0.245110 −0.122555 0.992462i $$-0.539109\pi$$
−0.122555 + 0.992462i $$0.539109\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.39445 0.128917
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −15.9083 −1.43441
$$124$$ 0 0
$$125$$ 10.8167 0.967471
$$126$$ 0 0
$$127$$ 4.09167 0.363077 0.181539 0.983384i $$-0.441892\pi$$
0.181539 + 0.983384i $$0.441892\pi$$
$$128$$ 0 0
$$129$$ −8.51388 −0.749605
$$130$$ 0 0
$$131$$ 6.78890 0.593149 0.296574 0.955010i $$-0.404156\pi$$
0.296574 + 0.955010i $$0.404156\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.09167 −0.180023
$$136$$ 0 0
$$137$$ −16.3028 −1.39284 −0.696420 0.717634i $$-0.745225\pi$$
−0.696420 + 0.717634i $$0.745225\pi$$
$$138$$ 0 0
$$139$$ −17.9083 −1.51896 −0.759482 0.650528i $$-0.774548\pi$$
−0.759482 + 0.650528i $$0.774548\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.7250 −1.20632 −0.603159 0.797621i $$-0.706091\pi$$
−0.603159 + 0.797621i $$0.706091\pi$$
$$150$$ 0 0
$$151$$ 14.1194 1.14902 0.574511 0.818497i $$-0.305192\pi$$
0.574511 + 0.818497i $$0.305192\pi$$
$$152$$ 0 0
$$153$$ −2.30278 −0.186168
$$154$$ 0 0
$$155$$ 0.908327 0.0729586
$$156$$ 0 0
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ 0 0
$$159$$ 16.8167 1.33365
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 5.39445 0.422526 0.211263 0.977429i $$-0.432242\pi$$
0.211263 + 0.977429i $$0.432242\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.72498 −0.675159 −0.337580 0.941297i $$-0.609608\pi$$
−0.337580 + 0.941297i $$0.609608\pi$$
$$168$$ 0 0
$$169$$ −12.6333 −0.971793
$$170$$ 0 0
$$171$$ 1.39445 0.106636
$$172$$ 0 0
$$173$$ −13.6972 −1.04138 −0.520690 0.853746i $$-0.674325\pi$$
−0.520690 + 0.853746i $$0.674325\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ −0.908327 −0.0678915 −0.0339458 0.999424i $$-0.510807\pi$$
−0.0339458 + 0.999424i $$0.510807\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 6.69722 0.495073
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.7250 −0.848390 −0.424195 0.905571i $$-0.639443\pi$$
−0.424195 + 0.905571i $$0.639443\pi$$
$$192$$ 0 0
$$193$$ 9.81665 0.706618 0.353309 0.935507i $$-0.385056\pi$$
0.353309 + 0.935507i $$0.385056\pi$$
$$194$$ 0 0
$$195$$ 1.81665 0.130093
$$196$$ 0 0
$$197$$ 2.60555 0.185638 0.0928189 0.995683i $$-0.470412\pi$$
0.0928189 + 0.995683i $$0.470412\pi$$
$$198$$ 0 0
$$199$$ 4.11943 0.292019 0.146009 0.989283i $$-0.453357\pi$$
0.146009 + 0.989283i $$0.453357\pi$$
$$200$$ 0 0
$$201$$ 12.2111 0.861305
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 15.0278 1.03455 0.517277 0.855818i $$-0.326946\pi$$
0.517277 + 0.855818i $$0.326946\pi$$
$$212$$ 0 0
$$213$$ −31.8167 −2.18004
$$214$$ 0 0
$$215$$ −4.81665 −0.328493
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 6.69722 0.452556
$$220$$ 0 0
$$221$$ −0.605551 −0.0407338
$$222$$ 0 0
$$223$$ −27.0278 −1.80991 −0.904956 0.425505i $$-0.860097\pi$$
−0.904956 + 0.425505i $$0.860097\pi$$
$$224$$ 0 0
$$225$$ −7.60555 −0.507037
$$226$$ 0 0
$$227$$ 12.5139 0.830575 0.415288 0.909690i $$-0.363681\pi$$
0.415288 + 0.909690i $$0.363681\pi$$
$$228$$ 0 0
$$229$$ −25.2111 −1.66600 −0.832998 0.553276i $$-0.813378\pi$$
−0.832998 + 0.553276i $$0.813378\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 7.81665 0.512086 0.256043 0.966665i $$-0.417581\pi$$
0.256043 + 0.966665i $$0.417581\pi$$
$$234$$ 0 0
$$235$$ −3.39445 −0.221429
$$236$$ 0 0
$$237$$ −12.4222 −0.806909
$$238$$ 0 0
$$239$$ 7.69722 0.497892 0.248946 0.968517i $$-0.419916\pi$$
0.248946 + 0.968517i $$0.419916\pi$$
$$240$$ 0 0
$$241$$ −26.5139 −1.70791 −0.853955 0.520348i $$-0.825802\pi$$
−0.853955 + 0.520348i $$0.825802\pi$$
$$242$$ 0 0
$$243$$ 19.6056 1.25770
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.366692 0.0233321
$$248$$ 0 0
$$249$$ 13.8167 0.875595
$$250$$ 0 0
$$251$$ −25.8167 −1.62953 −0.814766 0.579789i $$-0.803135\pi$$
−0.814766 + 0.579789i $$0.803135\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −3.00000 −0.187867
$$256$$ 0 0
$$257$$ 11.2111 0.699329 0.349665 0.936875i $$-0.386296\pi$$
0.349665 + 0.936875i $$0.386296\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 10.4222 0.642661 0.321330 0.946967i $$-0.395870\pi$$
0.321330 + 0.946967i $$0.395870\pi$$
$$264$$ 0 0
$$265$$ 9.51388 0.584433
$$266$$ 0 0
$$267$$ −21.6333 −1.32394
$$268$$ 0 0
$$269$$ −0.788897 −0.0480999 −0.0240500 0.999711i $$-0.507656\pi$$
−0.0240500 + 0.999711i $$0.507656\pi$$
$$270$$ 0 0
$$271$$ −13.2111 −0.802517 −0.401259 0.915965i $$-0.631427\pi$$
−0.401259 + 0.915965i $$0.631427\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.42221 0.385873 0.192936 0.981211i $$-0.438199\pi$$
0.192936 + 0.981211i $$0.438199\pi$$
$$278$$ 0 0
$$279$$ −1.60555 −0.0961218
$$280$$ 0 0
$$281$$ 7.30278 0.435647 0.217824 0.975988i $$-0.430104\pi$$
0.217824 + 0.975988i $$0.430104\pi$$
$$282$$ 0 0
$$283$$ −8.51388 −0.506098 −0.253049 0.967454i $$-0.581433\pi$$
−0.253049 + 0.967454i $$0.581433\pi$$
$$284$$ 0 0
$$285$$ 1.81665 0.107609
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −6.21110 −0.364101
$$292$$ 0 0
$$293$$ −21.6333 −1.26383 −0.631916 0.775037i $$-0.717731\pi$$
−0.631916 + 0.775037i $$0.717731\pi$$
$$294$$ 0 0
$$295$$ −6.78890 −0.395265
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4.18335 −0.240327
$$304$$ 0 0
$$305$$ 3.78890 0.216952
$$306$$ 0 0
$$307$$ −13.2111 −0.753997 −0.376999 0.926214i $$-0.623044\pi$$
−0.376999 + 0.926214i $$0.623044\pi$$
$$308$$ 0 0
$$309$$ 16.6056 0.944657
$$310$$ 0 0
$$311$$ 27.1194 1.53780 0.768901 0.639368i $$-0.220804\pi$$
0.768901 + 0.639368i $$0.220804\pi$$
$$312$$ 0 0
$$313$$ 6.33053 0.357823 0.178911 0.983865i $$-0.442742\pi$$
0.178911 + 0.983865i $$0.442742\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 19.8167 1.10606
$$322$$ 0 0
$$323$$ −0.605551 −0.0336938
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 47.0278 2.60064
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 9.30278 0.511327 0.255663 0.966766i $$-0.417706\pi$$
0.255663 + 0.966766i $$0.417706\pi$$
$$332$$ 0 0
$$333$$ 10.6056 0.581181
$$334$$ 0 0
$$335$$ 6.90833 0.377442
$$336$$ 0 0
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 0 0
$$349$$ −14.7889 −0.791632 −0.395816 0.918330i $$-0.629538\pi$$
−0.395816 + 0.918330i $$0.629538\pi$$
$$350$$ 0 0
$$351$$ 0.972244 0.0518945
$$352$$ 0 0
$$353$$ −27.6333 −1.47077 −0.735386 0.677648i $$-0.762999\pi$$
−0.735386 + 0.677648i $$0.762999\pi$$
$$354$$ 0 0
$$355$$ −18.0000 −0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3.90833 −0.206274 −0.103137 0.994667i $$-0.532888\pi$$
−0.103137 + 0.994667i $$0.532888\pi$$
$$360$$ 0 0
$$361$$ −18.6333 −0.980700
$$362$$ 0 0
$$363$$ 25.3305 1.32951
$$364$$ 0 0
$$365$$ 3.78890 0.198320
$$366$$ 0 0
$$367$$ 33.3305 1.73984 0.869920 0.493193i $$-0.164170\pi$$
0.869920 + 0.493193i $$0.164170\pi$$
$$368$$ 0 0
$$369$$ 15.9083 0.828154
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 15.9361 0.825139 0.412570 0.910926i $$-0.364631\pi$$
0.412570 + 0.910926i $$0.364631\pi$$
$$374$$ 0 0
$$375$$ −24.9083 −1.28626
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −21.2111 −1.08954 −0.544771 0.838585i $$-0.683383\pi$$
−0.544771 + 0.838585i $$0.683383\pi$$
$$380$$ 0 0
$$381$$ −9.42221 −0.482714
$$382$$ 0 0
$$383$$ 10.4222 0.532550 0.266275 0.963897i $$-0.414207\pi$$
0.266275 + 0.963897i $$0.414207\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.51388 0.432785
$$388$$ 0 0
$$389$$ 15.1194 0.766586 0.383293 0.923627i $$-0.374790\pi$$
0.383293 + 0.923627i $$0.374790\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −15.6333 −0.788596
$$394$$ 0 0
$$395$$ −7.02776 −0.353605
$$396$$ 0 0
$$397$$ −24.9361 −1.25151 −0.625753 0.780021i $$-0.715208\pi$$
−0.625753 + 0.780021i $$0.715208\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9.39445 −0.469136 −0.234568 0.972100i $$-0.575368\pi$$
−0.234568 + 0.972100i $$0.575368\pi$$
$$402$$ 0 0
$$403$$ −0.422205 −0.0210315
$$404$$ 0 0
$$405$$ 13.8167 0.686555
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.3944 0.958994 0.479497 0.877544i $$-0.340819\pi$$
0.479497 + 0.877544i $$0.340819\pi$$
$$410$$ 0 0
$$411$$ 37.5416 1.85179
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 7.81665 0.383704
$$416$$ 0 0
$$417$$ 41.2389 2.01948
$$418$$ 0 0
$$419$$ −0.513878 −0.0251046 −0.0125523 0.999921i $$-0.503996\pi$$
−0.0125523 + 0.999921i $$0.503996\pi$$
$$420$$ 0 0
$$421$$ 15.9361 0.776677 0.388339 0.921517i $$-0.373049\pi$$
0.388339 + 0.921517i $$0.373049\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 3.30278 0.160208
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 27.6333 1.33105 0.665525 0.746376i $$-0.268208\pi$$
0.665525 + 0.746376i $$0.268208\pi$$
$$432$$ 0 0
$$433$$ 17.8167 0.856214 0.428107 0.903728i $$-0.359181\pi$$
0.428107 + 0.903728i $$0.359181\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −2.90833 −0.138807 −0.0694034 0.997589i $$-0.522110\pi$$
−0.0694034 + 0.997589i $$0.522110\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6.78890 −0.322550 −0.161275 0.986909i $$-0.551561\pi$$
−0.161275 + 0.986909i $$0.551561\pi$$
$$444$$ 0 0
$$445$$ −12.2389 −0.580178
$$446$$ 0 0
$$447$$ 33.9083 1.60381
$$448$$ 0 0
$$449$$ 18.2389 0.860745 0.430372 0.902651i $$-0.358382\pi$$
0.430372 + 0.902651i $$0.358382\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −32.5139 −1.52764
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13.9083 −0.650604 −0.325302 0.945610i $$-0.605466\pi$$
−0.325302 + 0.945610i $$0.605466\pi$$
$$458$$ 0 0
$$459$$ −1.60555 −0.0749407
$$460$$ 0 0
$$461$$ 10.1833 0.474286 0.237143 0.971475i $$-0.423789\pi$$
0.237143 + 0.971475i $$0.423789\pi$$
$$462$$ 0 0
$$463$$ −20.3028 −0.943550 −0.471775 0.881719i $$-0.656387\pi$$
−0.471775 + 0.881719i $$0.656387\pi$$
$$464$$ 0 0
$$465$$ −2.09167 −0.0969990
$$466$$ 0 0
$$467$$ −29.4500 −1.36278 −0.681391 0.731920i $$-0.738625\pi$$
−0.681391 + 0.731920i $$0.738625\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 16.6056 0.765143
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ −16.8167 −0.769982
$$478$$ 0 0
$$479$$ 3.51388 0.160553 0.0802766 0.996773i $$-0.474420\pi$$
0.0802766 + 0.996773i $$0.474420\pi$$
$$480$$ 0 0
$$481$$ 2.78890 0.127163
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −3.51388 −0.159557
$$486$$ 0 0
$$487$$ −9.21110 −0.417395 −0.208697 0.977980i $$-0.566922\pi$$
−0.208697 + 0.977980i $$0.566922\pi$$
$$488$$ 0 0
$$489$$ −12.4222 −0.561752
$$490$$ 0 0
$$491$$ 14.3305 0.646728 0.323364 0.946275i $$-0.395186\pi$$
0.323364 + 0.946275i $$0.395186\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 30.4222 1.36188 0.680942 0.732337i $$-0.261570\pi$$
0.680942 + 0.732337i $$0.261570\pi$$
$$500$$ 0 0
$$501$$ 20.0917 0.897630
$$502$$ 0 0
$$503$$ 40.1472 1.79007 0.895037 0.445991i $$-0.147149\pi$$
0.895037 + 0.445991i $$0.147149\pi$$
$$504$$ 0 0
$$505$$ −2.36669 −0.105316
$$506$$ 0 0
$$507$$ 29.0917 1.29201
$$508$$ 0 0
$$509$$ −25.0278 −1.10934 −0.554668 0.832072i $$-0.687155\pi$$
−0.554668 + 0.832072i $$0.687155\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0.972244 0.0429256
$$514$$ 0 0
$$515$$ 9.39445 0.413969
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 31.5416 1.38452
$$520$$ 0 0
$$521$$ 36.3583 1.59289 0.796443 0.604714i $$-0.206713\pi$$
0.796443 + 0.604714i $$0.206713\pi$$
$$522$$ 0 0
$$523$$ −25.2111 −1.10240 −0.551202 0.834372i $$-0.685831\pi$$
−0.551202 + 0.834372i $$0.685831\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0.697224 0.0303716
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 4.18335 0.181201
$$534$$ 0 0
$$535$$ 11.2111 0.484698
$$536$$ 0 0
$$537$$ 2.09167 0.0902624
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.0555 −1.29219 −0.646094 0.763258i $$-0.723598\pi$$
−0.646094 + 0.763258i $$0.723598\pi$$
$$542$$ 0 0
$$543$$ 32.2389 1.38350
$$544$$ 0 0
$$545$$ 26.6056 1.13966
$$546$$ 0 0
$$547$$ −16.7889 −0.717841 −0.358921 0.933368i $$-0.616855\pi$$
−0.358921 + 0.933368i $$0.616855\pi$$
$$548$$ 0 0
$$549$$ −6.69722 −0.285831
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 13.8167 0.586484
$$556$$ 0 0
$$557$$ −9.63331 −0.408176 −0.204088 0.978953i $$-0.565423\pi$$
−0.204088 + 0.978953i $$0.565423\pi$$
$$558$$ 0 0
$$559$$ 2.23886 0.0946936
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −15.6333 −0.658865 −0.329433 0.944179i $$-0.606857\pi$$
−0.329433 + 0.944179i $$0.606857\pi$$
$$564$$ 0 0
$$565$$ 3.39445 0.142806
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 30.1194 1.26267 0.631336 0.775509i $$-0.282507\pi$$
0.631336 + 0.775509i $$0.282507\pi$$
$$570$$ 0 0
$$571$$ 27.0278 1.13108 0.565538 0.824722i $$-0.308668\pi$$
0.565538 + 0.824722i $$0.308668\pi$$
$$572$$ 0 0
$$573$$ 27.0000 1.12794
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 22.2389 0.925816 0.462908 0.886406i $$-0.346806\pi$$
0.462908 + 0.886406i $$0.346806\pi$$
$$578$$ 0 0
$$579$$ −22.6056 −0.939455
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1.81665 −0.0751094
$$586$$ 0 0
$$587$$ 3.39445 0.140104 0.0700519 0.997543i $$-0.477683\pi$$
0.0700519 + 0.997543i $$0.477683\pi$$
$$588$$ 0 0
$$589$$ −0.422205 −0.0173967
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −22.1833 −0.910961 −0.455480 0.890246i $$-0.650532\pi$$
−0.455480 + 0.890246i $$0.650532\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9.48612 −0.388241
$$598$$ 0 0
$$599$$ −38.9638 −1.59202 −0.796010 0.605284i $$-0.793060\pi$$
−0.796010 + 0.605284i $$0.793060\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ −12.2111 −0.497275
$$604$$ 0 0
$$605$$ 14.3305 0.582619
$$606$$ 0 0
$$607$$ 3.48612 0.141497 0.0707487 0.997494i $$-0.477461\pi$$
0.0707487 + 0.997494i $$0.477461\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.57779 0.0638307
$$612$$ 0 0
$$613$$ −32.1472 −1.29841 −0.649206 0.760612i $$-0.724899\pi$$
−0.649206 + 0.760612i $$0.724899\pi$$
$$614$$ 0 0
$$615$$ 20.7250 0.835712
$$616$$ 0 0
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 0 0
$$619$$ 0.366692 0.0147386 0.00736930 0.999973i $$-0.497654\pi$$
0.00736930 + 0.999973i $$0.497654\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 2.42221 0.0968882
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.60555 −0.183635
$$630$$ 0 0
$$631$$ 17.1194 0.681514 0.340757 0.940151i $$-0.389317\pi$$
0.340757 + 0.940151i $$0.389317\pi$$
$$632$$ 0 0
$$633$$ −34.6056 −1.37545
$$634$$ 0 0
$$635$$ −5.33053 −0.211536
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 31.8167 1.25865
$$640$$ 0 0
$$641$$ 44.8444 1.77125 0.885624 0.464403i $$-0.153731\pi$$
0.885624 + 0.464403i $$0.153731\pi$$
$$642$$ 0 0
$$643$$ 23.1472 0.912836 0.456418 0.889766i $$-0.349132\pi$$
0.456418 + 0.889766i $$0.349132\pi$$
$$644$$ 0 0
$$645$$ 11.0917 0.436734
$$646$$ 0 0
$$647$$ 10.4222 0.409739 0.204870 0.978789i $$-0.434323\pi$$
0.204870 + 0.978789i $$0.434323\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −44.0555 −1.72403 −0.862013 0.506887i $$-0.830796\pi$$
−0.862013 + 0.506887i $$0.830796\pi$$
$$654$$ 0 0
$$655$$ −8.84441 −0.345580
$$656$$ 0 0
$$657$$ −6.69722 −0.261284
$$658$$ 0 0
$$659$$ 34.6972 1.35161 0.675806 0.737080i $$-0.263796\pi$$
0.675806 + 0.737080i $$0.263796\pi$$
$$660$$ 0 0
$$661$$ 35.8167 1.39311 0.696553 0.717505i $$-0.254716\pi$$
0.696553 + 0.717505i $$0.254716\pi$$
$$662$$ 0 0
$$663$$ 1.39445 0.0541559
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 62.2389 2.40629
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −41.8167 −1.61191 −0.805957 0.591974i $$-0.798349\pi$$
−0.805957 + 0.591974i $$0.798349\pi$$
$$674$$ 0 0
$$675$$ −5.30278 −0.204104
$$676$$ 0 0
$$677$$ −19.5778 −0.752436 −0.376218 0.926531i $$-0.622776\pi$$
−0.376218 + 0.926531i $$0.622776\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −28.8167 −1.10426
$$682$$ 0 0
$$683$$ −23.4500 −0.897288 −0.448644 0.893711i $$-0.648093\pi$$
−0.448644 + 0.893711i $$0.648093\pi$$
$$684$$ 0 0
$$685$$ 21.2389 0.811495
$$686$$ 0 0
$$687$$ 58.0555 2.21496
$$688$$ 0 0
$$689$$ −4.42221 −0.168473
$$690$$ 0 0
$$691$$ −47.5139 −1.80751 −0.903757 0.428047i $$-0.859202\pi$$
−0.903757 + 0.428047i $$0.859202\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 23.3305 0.884978
$$696$$ 0 0
$$697$$ −6.90833 −0.261672
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ 38.8444 1.46713 0.733567 0.679618i $$-0.237854\pi$$
0.733567 + 0.679618i $$0.237854\pi$$
$$702$$ 0 0
$$703$$ 2.78890 0.105185
$$704$$ 0 0
$$705$$ 7.81665 0.294392
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 33.8167 1.27001 0.635006 0.772508i $$-0.280998\pi$$
0.635006 + 0.772508i $$0.280998\pi$$
$$710$$ 0 0
$$711$$ 12.4222 0.465869
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −17.7250 −0.661952
$$718$$ 0 0
$$719$$ 25.6972 0.958345 0.479172 0.877721i $$-0.340937\pi$$
0.479172 + 0.877721i $$0.340937\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 61.0555 2.27068
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −6.42221 −0.238186 −0.119093 0.992883i $$-0.537999\pi$$
−0.119093 + 0.992883i $$0.537999\pi$$
$$728$$ 0 0
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ −3.69722 −0.136747
$$732$$ 0 0
$$733$$ −20.2389 −0.747539 −0.373770 0.927522i $$-0.621935\pi$$
−0.373770 + 0.927522i $$0.621935\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 50.3583 1.85246 0.926230 0.376959i $$-0.123030\pi$$
0.926230 + 0.376959i $$0.123030\pi$$
$$740$$ 0 0
$$741$$ −0.844410 −0.0310202
$$742$$ 0 0
$$743$$ −19.8167 −0.727003 −0.363501 0.931594i $$-0.618419\pi$$
−0.363501 + 0.931594i $$0.618419\pi$$
$$744$$ 0 0
$$745$$ 19.1833 0.702823
$$746$$ 0 0
$$747$$ −13.8167 −0.505525
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 11.3944 0.415789 0.207895 0.978151i $$-0.433339\pi$$
0.207895 + 0.978151i $$0.433339\pi$$
$$752$$ 0 0
$$753$$ 59.4500 2.16648
$$754$$ 0 0
$$755$$ −18.3944 −0.669443
$$756$$ 0 0
$$757$$ −13.2750 −0.482489 −0.241244 0.970464i $$-0.577556\pi$$
−0.241244 + 0.970464i $$0.577556\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −16.1833 −0.586646 −0.293323 0.956013i $$-0.594761\pi$$
−0.293323 + 0.956013i $$0.594761\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 3.00000 0.108465
$$766$$ 0 0
$$767$$ 3.15559 0.113942
$$768$$ 0 0
$$769$$ −33.2666 −1.19962 −0.599812 0.800141i $$-0.704758\pi$$
−0.599812 + 0.800141i $$0.704758\pi$$
$$770$$ 0 0
$$771$$ −25.8167 −0.929764
$$772$$ 0 0
$$773$$ −4.18335 −0.150465 −0.0752323 0.997166i $$-0.523970\pi$$
−0.0752323 + 0.997166i $$0.523970\pi$$
$$774$$ 0 0
$$775$$ 2.30278 0.0827181
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.18335 0.149884
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9.39445 0.335302
$$786$$ 0 0
$$787$$ 26.4222 0.941850 0.470925 0.882173i $$-0.343920\pi$$
0.470925 + 0.882173i $$0.343920\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −1.76114 −0.0625400
$$794$$ 0 0
$$795$$ −21.9083 −0.777008
$$796$$ 0 0
$$797$$ 28.4222 1.00677 0.503383 0.864063i $$-0.332088\pi$$
0.503383 + 0.864063i $$0.332088\pi$$
$$798$$ 0 0
$$799$$ −2.60555 −0.0921778
$$800$$ 0 0
$$801$$ 21.6333 0.764375
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.81665 0.0639492
$$808$$ 0 0
$$809$$ 33.3944 1.17409 0.587043 0.809556i $$-0.300292\pi$$
0.587043 + 0.809556i $$0.300292\pi$$
$$810$$ 0 0
$$811$$ −14.9083 −0.523502 −0.261751 0.965135i $$-0.584300\pi$$
−0.261751 + 0.965135i $$0.584300\pi$$
$$812$$ 0 0
$$813$$ 30.4222 1.06695
$$814$$ 0 0
$$815$$ −7.02776 −0.246172
$$816$$ 0 0
$$817$$ 2.23886 0.0783278
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.39445 −0.118467 −0.0592335 0.998244i $$-0.518866\pi$$
−0.0592335 + 0.998244i $$0.518866\pi$$
$$822$$ 0 0
$$823$$ −49.8722 −1.73843 −0.869217 0.494430i $$-0.835377\pi$$
−0.869217 + 0.494430i $$0.835377\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15.6333 −0.543623 −0.271812 0.962350i $$-0.587623\pi$$
−0.271812 + 0.962350i $$0.587623\pi$$
$$828$$ 0 0
$$829$$ 19.3944 0.673597 0.336799 0.941577i $$-0.390656\pi$$
0.336799 + 0.941577i $$0.390656\pi$$
$$830$$ 0 0
$$831$$ −14.7889 −0.513021
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 11.3667 0.393361
$$836$$ 0 0
$$837$$ −1.11943 −0.0386931
$$838$$ 0 0
$$839$$ 10.4222 0.359814 0.179907 0.983684i $$-0.442420\pi$$
0.179907 + 0.983684i $$0.442420\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ −16.8167 −0.579196
$$844$$ 0 0
$$845$$ 16.4584 0.566185
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 19.6056 0.672861
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −12.4222 −0.425328 −0.212664 0.977125i $$-0.568214\pi$$
−0.212664 + 0.977125i $$0.568214\pi$$
$$854$$ 0 0
$$855$$ −1.81665 −0.0621282
$$856$$ 0 0
$$857$$ −19.5416 −0.667530 −0.333765 0.942656i $$-0.608319\pi$$
−0.333765 + 0.942656i $$0.608319\pi$$
$$858$$ 0 0
$$859$$ 27.4500 0.936581 0.468290 0.883575i $$-0.344870\pi$$
0.468290 + 0.883575i $$0.344870\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 8.09167 0.275444 0.137722 0.990471i $$-0.456022\pi$$
0.137722 + 0.990471i $$0.456022\pi$$
$$864$$ 0 0
$$865$$ 17.8444 0.606728
$$866$$ 0 0
$$867$$ −2.30278 −0.0782064
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −3.21110 −0.108804
$$872$$ 0 0
$$873$$ 6.21110 0.210214
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −25.3944 −0.857510 −0.428755 0.903421i $$-0.641048\pi$$
−0.428755 + 0.903421i $$0.641048\pi$$
$$878$$ 0 0
$$879$$ 49.8167 1.68027
$$880$$ 0 0
$$881$$ 13.3028 0.448182 0.224091 0.974568i $$-0.428059\pi$$
0.224091 + 0.974568i $$0.428059\pi$$
$$882$$ 0 0
$$883$$ −54.3305 −1.82837 −0.914184 0.405299i $$-0.867167\pi$$
−0.914184 + 0.405299i $$0.867167\pi$$
$$884$$ 0 0
$$885$$ 15.6333 0.525508
$$886$$ 0 0
$$887$$ 38.0917 1.27899 0.639497 0.768794i $$-0.279143\pi$$
0.639497 + 0.768794i $$0.279143\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.57779 0.0527989
$$894$$ 0 0
$$895$$ 1.18335 0.0395549
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 7.30278 0.243291
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.2389 0.606280
$$906$$ 0 0
$$907$$ 34.8444 1.15699 0.578495 0.815686i $$-0.303640\pi$$
0.578495 + 0.815686i $$0.303640\pi$$
$$908$$ 0 0
$$909$$ 4.18335 0.138753
$$910$$ 0 0
$$911$$ 16.1833 0.536178 0.268089 0.963394i $$-0.413608\pi$$
0.268089 + 0.963394i $$0.413608\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −8.72498 −0.288439
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −15.3305 −0.505708 −0.252854 0.967505i $$-0.581369\pi$$
−0.252854 + 0.967505i $$0.581369\pi$$
$$920$$ 0 0
$$921$$ 30.4222 1.00245
$$922$$ 0 0
$$923$$ 8.36669 0.275393
$$924$$ 0 0
$$925$$ −15.2111 −0.500138
$$926$$ 0 0
$$927$$ −16.6056 −0.545398
$$928$$ 0 0
$$929$$ −7.06392 −0.231760 −0.115880 0.993263i $$-0.536969\pi$$
−0.115880 + 0.993263i $$0.536969\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −62.4500 −2.04452
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21.2111 0.692937 0.346468 0.938062i $$-0.387381\pi$$
0.346468 + 0.938062i $$0.387381\pi$$
$$938$$ 0 0
$$939$$ −14.5778 −0.475728
$$940$$ 0 0
$$941$$ 32.7250 1.06680 0.533402 0.845862i $$-0.320913\pi$$
0.533402 + 0.845862i $$0.320913\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.84441 −0.287405 −0.143702 0.989621i $$-0.545901\pi$$
−0.143702 + 0.989621i $$0.545901\pi$$
$$948$$ 0 0
$$949$$ −1.76114 −0.0571691
$$950$$ 0 0
$$951$$ 13.8167 0.448036
$$952$$ 0 0
$$953$$ −0.513878 −0.0166461 −0.00832307 0.999965i $$-0.502649\pi$$
−0.00832307 + 0.999965i $$0.502649\pi$$
$$954$$ 0 0
$$955$$ 15.2750 0.494288
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.5139 −0.984319
$$962$$ 0 0
$$963$$ −19.8167 −0.638583
$$964$$ 0 0
$$965$$ −12.7889 −0.411689
$$966$$ 0 0
$$967$$ −1.51388 −0.0486830 −0.0243415 0.999704i $$-0.507749\pi$$
−0.0243415 + 0.999704i $$0.507749\pi$$
$$968$$ 0 0
$$969$$ 1.39445 0.0447961
$$970$$ 0 0
$$971$$ 20.8444 0.668929 0.334464 0.942408i $$-0.391445\pi$$
0.334464 + 0.942408i $$0.391445\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 4.60555 0.147496
$$976$$ 0 0
$$977$$ 28.1472 0.900508 0.450254 0.892900i $$-0.351333\pi$$
0.450254 + 0.892900i $$0.351333\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −47.0278 −1.50148
$$982$$ 0 0
$$983$$ −33.9083 −1.08151 −0.540754 0.841181i $$-0.681861\pi$$
−0.540754 + 0.841181i $$0.681861\pi$$
$$984$$ 0 0
$$985$$ −3.39445 −0.108156
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 14.0000 0.444725 0.222362 0.974964i $$-0.428623\pi$$
0.222362 + 0.974964i $$0.428623\pi$$
$$992$$ 0 0
$$993$$ −21.4222 −0.679813
$$994$$ 0 0
$$995$$ −5.36669 −0.170136
$$996$$ 0 0
$$997$$ 47.1472 1.49317 0.746583 0.665292i $$-0.231693\pi$$
0.746583 + 0.665292i $$0.231693\pi$$
$$998$$ 0 0
$$999$$ 7.39445 0.233950
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 3332.2.a.j.1.1 2
7.6 odd 2 476.2.a.d.1.2 2
21.20 even 2 4284.2.a.n.1.1 2
28.27 even 2 1904.2.a.h.1.1 2
56.13 odd 2 7616.2.a.q.1.1 2
56.27 even 2 7616.2.a.v.1.2 2
119.118 odd 2 8092.2.a.k.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.d.1.2 2 7.6 odd 2
1904.2.a.h.1.1 2 28.27 even 2
3332.2.a.j.1.1 2 1.1 even 1 trivial
4284.2.a.n.1.1 2 21.20 even 2
7616.2.a.q.1.1 2 56.13 odd 2
7616.2.a.v.1.2 2 56.27 even 2
8092.2.a.k.1.1 2 119.118 odd 2