# Properties

 Label 6592.2.a.h.1.1 Level $6592$ Weight $2$ Character 6592.1 Self dual yes Analytic conductor $52.637$ 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] = [6592,2,Mod(1,6592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6592, 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("6592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6592 = 2^{6} \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6592.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: $$52.6373850124$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 6592.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +0.381966 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +0.381966 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.61803 q^{11} +4.85410 q^{13} -0.381966 q^{15} -5.61803 q^{17} +5.85410 q^{19} -1.00000 q^{21} -4.47214 q^{23} -4.85410 q^{25} +5.00000 q^{27} +5.23607 q^{29} +6.70820 q^{31} +2.61803 q^{33} +0.381966 q^{35} -6.70820 q^{37} -4.85410 q^{39} +8.94427 q^{41} -8.70820 q^{43} -0.763932 q^{45} -4.09017 q^{47} -6.00000 q^{49} +5.61803 q^{51} -1.09017 q^{53} -1.00000 q^{55} -5.85410 q^{57} +6.38197 q^{59} -4.14590 q^{61} -2.00000 q^{63} +1.85410 q^{65} +14.4164 q^{67} +4.47214 q^{69} +4.09017 q^{71} -10.8541 q^{73} +4.85410 q^{75} -2.61803 q^{77} +6.56231 q^{79} +1.00000 q^{81} -6.32624 q^{83} -2.14590 q^{85} -5.23607 q^{87} -2.29180 q^{89} +4.85410 q^{91} -6.70820 q^{93} +2.23607 q^{95} -1.70820 q^{97} +5.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 - 4 * q^9 $$2 q - 2 q^{3} + 3 q^{5} + 2 q^{7} - 4 q^{9} - 3 q^{11} + 3 q^{13} - 3 q^{15} - 9 q^{17} + 5 q^{19} - 2 q^{21} - 3 q^{25} + 10 q^{27} + 6 q^{29} + 3 q^{33} + 3 q^{35} - 3 q^{39} - 4 q^{43} - 6 q^{45} + 3 q^{47} - 12 q^{49} + 9 q^{51} + 9 q^{53} - 2 q^{55} - 5 q^{57} + 15 q^{59} - 15 q^{61} - 4 q^{63} - 3 q^{65} + 2 q^{67} - 3 q^{71} - 15 q^{73} + 3 q^{75} - 3 q^{77} - 7 q^{79} + 2 q^{81} + 3 q^{83} - 11 q^{85} - 6 q^{87} - 18 q^{89} + 3 q^{91} + 10 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 3 * q^5 + 2 * q^7 - 4 * q^9 - 3 * q^11 + 3 * q^13 - 3 * q^15 - 9 * q^17 + 5 * q^19 - 2 * q^21 - 3 * q^25 + 10 * q^27 + 6 * q^29 + 3 * q^33 + 3 * q^35 - 3 * q^39 - 4 * q^43 - 6 * q^45 + 3 * q^47 - 12 * q^49 + 9 * q^51 + 9 * q^53 - 2 * q^55 - 5 * q^57 + 15 * q^59 - 15 * q^61 - 4 * q^63 - 3 * q^65 + 2 * q^67 - 3 * q^71 - 15 * q^73 + 3 * q^75 - 3 * q^77 - 7 * q^79 + 2 * q^81 + 3 * q^83 - 11 * q^85 - 6 * q^87 - 18 * q^89 + 3 * q^91 + 10 * q^97 + 6 * 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
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 0.381966 0.170820 0.0854102 0.996346i $$-0.472780\pi$$
0.0854102 + 0.996346i $$0.472780\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −2.61803 −0.789367 −0.394683 0.918817i $$-0.629146\pi$$
−0.394683 + 0.918817i $$0.629146\pi$$
$$12$$ 0 0
$$13$$ 4.85410 1.34629 0.673143 0.739512i $$-0.264944\pi$$
0.673143 + 0.739512i $$0.264944\pi$$
$$14$$ 0 0
$$15$$ −0.381966 −0.0986232
$$16$$ 0 0
$$17$$ −5.61803 −1.36257 −0.681287 0.732017i $$-0.738579\pi$$
−0.681287 + 0.732017i $$0.738579\pi$$
$$18$$ 0 0
$$19$$ 5.85410 1.34302 0.671512 0.740994i $$-0.265645\pi$$
0.671512 + 0.740994i $$0.265645\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 0 0
$$25$$ −4.85410 −0.970820
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 5.23607 0.972313 0.486157 0.873872i $$-0.338398\pi$$
0.486157 + 0.873872i $$0.338398\pi$$
$$30$$ 0 0
$$31$$ 6.70820 1.20483 0.602414 0.798183i $$-0.294205\pi$$
0.602414 + 0.798183i $$0.294205\pi$$
$$32$$ 0 0
$$33$$ 2.61803 0.455741
$$34$$ 0 0
$$35$$ 0.381966 0.0645640
$$36$$ 0 0
$$37$$ −6.70820 −1.10282 −0.551411 0.834234i $$-0.685910\pi$$
−0.551411 + 0.834234i $$0.685910\pi$$
$$38$$ 0 0
$$39$$ −4.85410 −0.777278
$$40$$ 0 0
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ 0 0
$$43$$ −8.70820 −1.32799 −0.663994 0.747738i $$-0.731140\pi$$
−0.663994 + 0.747738i $$0.731140\pi$$
$$44$$ 0 0
$$45$$ −0.763932 −0.113880
$$46$$ 0 0
$$47$$ −4.09017 −0.596613 −0.298306 0.954470i $$-0.596422\pi$$
−0.298306 + 0.954470i $$0.596422\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 5.61803 0.786682
$$52$$ 0 0
$$53$$ −1.09017 −0.149746 −0.0748732 0.997193i $$-0.523855\pi$$
−0.0748732 + 0.997193i $$0.523855\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ −5.85410 −0.775395
$$58$$ 0 0
$$59$$ 6.38197 0.830861 0.415431 0.909625i $$-0.363631\pi$$
0.415431 + 0.909625i $$0.363631\pi$$
$$60$$ 0 0
$$61$$ −4.14590 −0.530828 −0.265414 0.964135i $$-0.585509\pi$$
−0.265414 + 0.964135i $$0.585509\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ 1.85410 0.229973
$$66$$ 0 0
$$67$$ 14.4164 1.76124 0.880622 0.473819i $$-0.157125\pi$$
0.880622 + 0.473819i $$0.157125\pi$$
$$68$$ 0 0
$$69$$ 4.47214 0.538382
$$70$$ 0 0
$$71$$ 4.09017 0.485414 0.242707 0.970100i $$-0.421965\pi$$
0.242707 + 0.970100i $$0.421965\pi$$
$$72$$ 0 0
$$73$$ −10.8541 −1.27038 −0.635188 0.772357i $$-0.719077\pi$$
−0.635188 + 0.772357i $$0.719077\pi$$
$$74$$ 0 0
$$75$$ 4.85410 0.560503
$$76$$ 0 0
$$77$$ −2.61803 −0.298353
$$78$$ 0 0
$$79$$ 6.56231 0.738317 0.369159 0.929366i $$-0.379646\pi$$
0.369159 + 0.929366i $$0.379646\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.32624 −0.694395 −0.347197 0.937792i $$-0.612867\pi$$
−0.347197 + 0.937792i $$0.612867\pi$$
$$84$$ 0 0
$$85$$ −2.14590 −0.232755
$$86$$ 0 0
$$87$$ −5.23607 −0.561365
$$88$$ 0 0
$$89$$ −2.29180 −0.242930 −0.121465 0.992596i $$-0.538759\pi$$
−0.121465 + 0.992596i $$0.538759\pi$$
$$90$$ 0 0
$$91$$ 4.85410 0.508848
$$92$$ 0 0
$$93$$ −6.70820 −0.695608
$$94$$ 0 0
$$95$$ 2.23607 0.229416
$$96$$ 0 0
$$97$$ −1.70820 −0.173442 −0.0867209 0.996233i $$-0.527639\pi$$
−0.0867209 + 0.996233i $$0.527639\pi$$
$$98$$ 0 0
$$99$$ 5.23607 0.526245
$$100$$ 0 0
$$101$$ −1.90983 −0.190035 −0.0950176 0.995476i $$-0.530291\pi$$
−0.0950176 + 0.995476i $$0.530291\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329
$$104$$ 0 0
$$105$$ −0.381966 −0.0372761
$$106$$ 0 0
$$107$$ −7.09017 −0.685433 −0.342716 0.939439i $$-0.611347\pi$$
−0.342716 + 0.939439i $$0.611347\pi$$
$$108$$ 0 0
$$109$$ 9.56231 0.915903 0.457951 0.888977i $$-0.348583\pi$$
0.457951 + 0.888977i $$0.348583\pi$$
$$110$$ 0 0
$$111$$ 6.70820 0.636715
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ −1.70820 −0.159291
$$116$$ 0 0
$$117$$ −9.70820 −0.897524
$$118$$ 0 0
$$119$$ −5.61803 −0.515004
$$120$$ 0 0
$$121$$ −4.14590 −0.376900
$$122$$ 0 0
$$123$$ −8.94427 −0.806478
$$124$$ 0 0
$$125$$ −3.76393 −0.336656
$$126$$ 0 0
$$127$$ −15.2705 −1.35504 −0.677519 0.735505i $$-0.736945\pi$$
−0.677519 + 0.735505i $$0.736945\pi$$
$$128$$ 0 0
$$129$$ 8.70820 0.766715
$$130$$ 0 0
$$131$$ −2.23607 −0.195366 −0.0976831 0.995218i $$-0.531143\pi$$
−0.0976831 + 0.995218i $$0.531143\pi$$
$$132$$ 0 0
$$133$$ 5.85410 0.507615
$$134$$ 0 0
$$135$$ 1.90983 0.164372
$$136$$ 0 0
$$137$$ −12.7082 −1.08574 −0.542868 0.839818i $$-0.682661\pi$$
−0.542868 + 0.839818i $$0.682661\pi$$
$$138$$ 0 0
$$139$$ −11.1459 −0.945383 −0.472691 0.881228i $$-0.656717\pi$$
−0.472691 + 0.881228i $$0.656717\pi$$
$$140$$ 0 0
$$141$$ 4.09017 0.344454
$$142$$ 0 0
$$143$$ −12.7082 −1.06271
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 6.00000 0.494872
$$148$$ 0 0
$$149$$ 7.47214 0.612141 0.306071 0.952009i $$-0.400986\pi$$
0.306071 + 0.952009i $$0.400986\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ 11.2361 0.908382
$$154$$ 0 0
$$155$$ 2.56231 0.205809
$$156$$ 0 0
$$157$$ −16.7082 −1.33346 −0.666730 0.745299i $$-0.732307\pi$$
−0.666730 + 0.745299i $$0.732307\pi$$
$$158$$ 0 0
$$159$$ 1.09017 0.0864561
$$160$$ 0 0
$$161$$ −4.47214 −0.352454
$$162$$ 0 0
$$163$$ 2.70820 0.212123 0.106061 0.994360i $$-0.466176\pi$$
0.106061 + 0.994360i $$0.466176\pi$$
$$164$$ 0 0
$$165$$ 1.00000 0.0778499
$$166$$ 0 0
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ 10.5623 0.812485
$$170$$ 0 0
$$171$$ −11.7082 −0.895349
$$172$$ 0 0
$$173$$ −16.0344 −1.21908 −0.609538 0.792757i $$-0.708645\pi$$
−0.609538 + 0.792757i $$0.708645\pi$$
$$174$$ 0 0
$$175$$ −4.85410 −0.366936
$$176$$ 0 0
$$177$$ −6.38197 −0.479698
$$178$$ 0 0
$$179$$ −7.85410 −0.587043 −0.293522 0.955952i $$-0.594827\pi$$
−0.293522 + 0.955952i $$0.594827\pi$$
$$180$$ 0 0
$$181$$ −3.85410 −0.286473 −0.143237 0.989688i $$-0.545751\pi$$
−0.143237 + 0.989688i $$0.545751\pi$$
$$182$$ 0 0
$$183$$ 4.14590 0.306474
$$184$$ 0 0
$$185$$ −2.56231 −0.188384
$$186$$ 0 0
$$187$$ 14.7082 1.07557
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −5.61803 −0.406507 −0.203253 0.979126i $$-0.565152\pi$$
−0.203253 + 0.979126i $$0.565152\pi$$
$$192$$ 0 0
$$193$$ 20.1246 1.44860 0.724301 0.689484i $$-0.242163\pi$$
0.724301 + 0.689484i $$0.242163\pi$$
$$194$$ 0 0
$$195$$ −1.85410 −0.132775
$$196$$ 0 0
$$197$$ 16.4164 1.16962 0.584810 0.811170i $$-0.301169\pi$$
0.584810 + 0.811170i $$0.301169\pi$$
$$198$$ 0 0
$$199$$ 23.4164 1.65995 0.829973 0.557804i $$-0.188356\pi$$
0.829973 + 0.557804i $$0.188356\pi$$
$$200$$ 0 0
$$201$$ −14.4164 −1.01686
$$202$$ 0 0
$$203$$ 5.23607 0.367500
$$204$$ 0 0
$$205$$ 3.41641 0.238612
$$206$$ 0 0
$$207$$ 8.94427 0.621670
$$208$$ 0 0
$$209$$ −15.3262 −1.06014
$$210$$ 0 0
$$211$$ 14.8541 1.02260 0.511299 0.859403i $$-0.329164\pi$$
0.511299 + 0.859403i $$0.329164\pi$$
$$212$$ 0 0
$$213$$ −4.09017 −0.280254
$$214$$ 0 0
$$215$$ −3.32624 −0.226848
$$216$$ 0 0
$$217$$ 6.70820 0.455383
$$218$$ 0 0
$$219$$ 10.8541 0.733452
$$220$$ 0 0
$$221$$ −27.2705 −1.83441
$$222$$ 0 0
$$223$$ −7.70820 −0.516180 −0.258090 0.966121i $$-0.583093\pi$$
−0.258090 + 0.966121i $$0.583093\pi$$
$$224$$ 0 0
$$225$$ 9.70820 0.647214
$$226$$ 0 0
$$227$$ 2.94427 0.195418 0.0977091 0.995215i $$-0.468849\pi$$
0.0977091 + 0.995215i $$0.468849\pi$$
$$228$$ 0 0
$$229$$ −6.70820 −0.443291 −0.221645 0.975127i $$-0.571143\pi$$
−0.221645 + 0.975127i $$0.571143\pi$$
$$230$$ 0 0
$$231$$ 2.61803 0.172254
$$232$$ 0 0
$$233$$ −26.8885 −1.76153 −0.880764 0.473556i $$-0.842970\pi$$
−0.880764 + 0.473556i $$0.842970\pi$$
$$234$$ 0 0
$$235$$ −1.56231 −0.101914
$$236$$ 0 0
$$237$$ −6.56231 −0.426268
$$238$$ 0 0
$$239$$ −8.67376 −0.561059 −0.280530 0.959845i $$-0.590510\pi$$
−0.280530 + 0.959845i $$0.590510\pi$$
$$240$$ 0 0
$$241$$ −8.27051 −0.532750 −0.266375 0.963869i $$-0.585826\pi$$
−0.266375 + 0.963869i $$0.585826\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ −2.29180 −0.146417
$$246$$ 0 0
$$247$$ 28.4164 1.80809
$$248$$ 0 0
$$249$$ 6.32624 0.400909
$$250$$ 0 0
$$251$$ 11.2361 0.709214 0.354607 0.935015i $$-0.384615\pi$$
0.354607 + 0.935015i $$0.384615\pi$$
$$252$$ 0 0
$$253$$ 11.7082 0.736088
$$254$$ 0 0
$$255$$ 2.14590 0.134381
$$256$$ 0 0
$$257$$ −13.4721 −0.840369 −0.420184 0.907439i $$-0.638035\pi$$
−0.420184 + 0.907439i $$0.638035\pi$$
$$258$$ 0 0
$$259$$ −6.70820 −0.416828
$$260$$ 0 0
$$261$$ −10.4721 −0.648209
$$262$$ 0 0
$$263$$ 20.6180 1.27136 0.635681 0.771952i $$-0.280719\pi$$
0.635681 + 0.771952i $$0.280719\pi$$
$$264$$ 0 0
$$265$$ −0.416408 −0.0255797
$$266$$ 0 0
$$267$$ 2.29180 0.140256
$$268$$ 0 0
$$269$$ −12.3262 −0.751544 −0.375772 0.926712i $$-0.622622\pi$$
−0.375772 + 0.926712i $$0.622622\pi$$
$$270$$ 0 0
$$271$$ −1.00000 −0.0607457 −0.0303728 0.999539i $$-0.509669\pi$$
−0.0303728 + 0.999539i $$0.509669\pi$$
$$272$$ 0 0
$$273$$ −4.85410 −0.293784
$$274$$ 0 0
$$275$$ 12.7082 0.766334
$$276$$ 0 0
$$277$$ −4.70820 −0.282889 −0.141444 0.989946i $$-0.545175\pi$$
−0.141444 + 0.989946i $$0.545175\pi$$
$$278$$ 0 0
$$279$$ −13.4164 −0.803219
$$280$$ 0 0
$$281$$ −31.4721 −1.87747 −0.938735 0.344640i $$-0.888001\pi$$
−0.938735 + 0.344640i $$0.888001\pi$$
$$282$$ 0 0
$$283$$ −19.7082 −1.17153 −0.585766 0.810481i $$-0.699206\pi$$
−0.585766 + 0.810481i $$0.699206\pi$$
$$284$$ 0 0
$$285$$ −2.23607 −0.132453
$$286$$ 0 0
$$287$$ 8.94427 0.527964
$$288$$ 0 0
$$289$$ 14.5623 0.856606
$$290$$ 0 0
$$291$$ 1.70820 0.100137
$$292$$ 0 0
$$293$$ 21.6525 1.26495 0.632476 0.774580i $$-0.282039\pi$$
0.632476 + 0.774580i $$0.282039\pi$$
$$294$$ 0 0
$$295$$ 2.43769 0.141928
$$296$$ 0 0
$$297$$ −13.0902 −0.759569
$$298$$ 0 0
$$299$$ −21.7082 −1.25542
$$300$$ 0 0
$$301$$ −8.70820 −0.501933
$$302$$ 0 0
$$303$$ 1.90983 0.109717
$$304$$ 0 0
$$305$$ −1.58359 −0.0906762
$$306$$ 0 0
$$307$$ −2.85410 −0.162892 −0.0814461 0.996678i $$-0.525954\pi$$
−0.0814461 + 0.996678i $$0.525954\pi$$
$$308$$ 0 0
$$309$$ −1.00000 −0.0568880
$$310$$ 0 0
$$311$$ −2.88854 −0.163794 −0.0818971 0.996641i $$-0.526098\pi$$
−0.0818971 + 0.996641i $$0.526098\pi$$
$$312$$ 0 0
$$313$$ 16.2918 0.920867 0.460433 0.887694i $$-0.347694\pi$$
0.460433 + 0.887694i $$0.347694\pi$$
$$314$$ 0 0
$$315$$ −0.763932 −0.0430427
$$316$$ 0 0
$$317$$ −28.4164 −1.59602 −0.798012 0.602641i $$-0.794115\pi$$
−0.798012 + 0.602641i $$0.794115\pi$$
$$318$$ 0 0
$$319$$ −13.7082 −0.767512
$$320$$ 0 0
$$321$$ 7.09017 0.395735
$$322$$ 0 0
$$323$$ −32.8885 −1.82997
$$324$$ 0 0
$$325$$ −23.5623 −1.30700
$$326$$ 0 0
$$327$$ −9.56231 −0.528797
$$328$$ 0 0
$$329$$ −4.09017 −0.225498
$$330$$ 0 0
$$331$$ −16.1459 −0.887459 −0.443729 0.896161i $$-0.646345\pi$$
−0.443729 + 0.896161i $$0.646345\pi$$
$$332$$ 0 0
$$333$$ 13.4164 0.735215
$$334$$ 0 0
$$335$$ 5.50658 0.300856
$$336$$ 0 0
$$337$$ −2.43769 −0.132790 −0.0663948 0.997793i $$-0.521150\pi$$
−0.0663948 + 0.997793i $$0.521150\pi$$
$$338$$ 0 0
$$339$$ 15.0000 0.814688
$$340$$ 0 0
$$341$$ −17.5623 −0.951052
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 1.70820 0.0919666
$$346$$ 0 0
$$347$$ −1.47214 −0.0790284 −0.0395142 0.999219i $$-0.512581\pi$$
−0.0395142 + 0.999219i $$0.512581\pi$$
$$348$$ 0 0
$$349$$ −15.4164 −0.825221 −0.412611 0.910907i $$-0.635383\pi$$
−0.412611 + 0.910907i $$0.635383\pi$$
$$350$$ 0 0
$$351$$ 24.2705 1.29546
$$352$$ 0 0
$$353$$ −25.0344 −1.33245 −0.666224 0.745751i $$-0.732091\pi$$
−0.666224 + 0.745751i $$0.732091\pi$$
$$354$$ 0 0
$$355$$ 1.56231 0.0829186
$$356$$ 0 0
$$357$$ 5.61803 0.297338
$$358$$ 0 0
$$359$$ −14.6738 −0.774452 −0.387226 0.921985i $$-0.626567\pi$$
−0.387226 + 0.921985i $$0.626567\pi$$
$$360$$ 0 0
$$361$$ 15.2705 0.803711
$$362$$ 0 0
$$363$$ 4.14590 0.217603
$$364$$ 0 0
$$365$$ −4.14590 −0.217006
$$366$$ 0 0
$$367$$ −16.4377 −0.858041 −0.429020 0.903295i $$-0.641141\pi$$
−0.429020 + 0.903295i $$0.641141\pi$$
$$368$$ 0 0
$$369$$ −17.8885 −0.931240
$$370$$ 0 0
$$371$$ −1.09017 −0.0565988
$$372$$ 0 0
$$373$$ 22.6869 1.17468 0.587342 0.809339i $$-0.300174\pi$$
0.587342 + 0.809339i $$0.300174\pi$$
$$374$$ 0 0
$$375$$ 3.76393 0.194369
$$376$$ 0 0
$$377$$ 25.4164 1.30901
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ 15.2705 0.782332
$$382$$ 0 0
$$383$$ −0.819660 −0.0418827 −0.0209413 0.999781i $$-0.506666\pi$$
−0.0209413 + 0.999781i $$0.506666\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 0 0
$$387$$ 17.4164 0.885326
$$388$$ 0 0
$$389$$ −7.41641 −0.376027 −0.188013 0.982166i $$-0.560205\pi$$
−0.188013 + 0.982166i $$0.560205\pi$$
$$390$$ 0 0
$$391$$ 25.1246 1.27061
$$392$$ 0 0
$$393$$ 2.23607 0.112795
$$394$$ 0 0
$$395$$ 2.50658 0.126120
$$396$$ 0 0
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 0 0
$$399$$ −5.85410 −0.293072
$$400$$ 0 0
$$401$$ −23.8885 −1.19294 −0.596468 0.802637i $$-0.703430\pi$$
−0.596468 + 0.802637i $$0.703430\pi$$
$$402$$ 0 0
$$403$$ 32.5623 1.62204
$$404$$ 0 0
$$405$$ 0.381966 0.0189800
$$406$$ 0 0
$$407$$ 17.5623 0.870531
$$408$$ 0 0
$$409$$ 36.7082 1.81510 0.907552 0.419940i $$-0.137949\pi$$
0.907552 + 0.419940i $$0.137949\pi$$
$$410$$ 0 0
$$411$$ 12.7082 0.626849
$$412$$ 0 0
$$413$$ 6.38197 0.314036
$$414$$ 0 0
$$415$$ −2.41641 −0.118617
$$416$$ 0 0
$$417$$ 11.1459 0.545817
$$418$$ 0 0
$$419$$ −4.09017 −0.199818 −0.0999089 0.994997i $$-0.531855\pi$$
−0.0999089 + 0.994997i $$0.531855\pi$$
$$420$$ 0 0
$$421$$ −3.00000 −0.146211 −0.0731055 0.997324i $$-0.523291\pi$$
−0.0731055 + 0.997324i $$0.523291\pi$$
$$422$$ 0 0
$$423$$ 8.18034 0.397742
$$424$$ 0 0
$$425$$ 27.2705 1.32281
$$426$$ 0 0
$$427$$ −4.14590 −0.200634
$$428$$ 0 0
$$429$$ 12.7082 0.613558
$$430$$ 0 0
$$431$$ −34.3607 −1.65510 −0.827548 0.561395i $$-0.810265\pi$$
−0.827548 + 0.561395i $$0.810265\pi$$
$$432$$ 0 0
$$433$$ 14.4164 0.692808 0.346404 0.938085i $$-0.387403\pi$$
0.346404 + 0.938085i $$0.387403\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 0 0
$$437$$ −26.1803 −1.25238
$$438$$ 0 0
$$439$$ −29.5623 −1.41093 −0.705466 0.708744i $$-0.749262\pi$$
−0.705466 + 0.708744i $$0.749262\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ −15.4377 −0.733467 −0.366733 0.930326i $$-0.619524\pi$$
−0.366733 + 0.930326i $$0.619524\pi$$
$$444$$ 0 0
$$445$$ −0.875388 −0.0414974
$$446$$ 0 0
$$447$$ −7.47214 −0.353420
$$448$$ 0 0
$$449$$ −13.3607 −0.630529 −0.315265 0.949004i $$-0.602093\pi$$
−0.315265 + 0.949004i $$0.602093\pi$$
$$450$$ 0 0
$$451$$ −23.4164 −1.10264
$$452$$ 0 0
$$453$$ −19.0000 −0.892698
$$454$$ 0 0
$$455$$ 1.85410 0.0869216
$$456$$ 0 0
$$457$$ −9.85410 −0.460955 −0.230478 0.973078i $$-0.574029\pi$$
−0.230478 + 0.973078i $$0.574029\pi$$
$$458$$ 0 0
$$459$$ −28.0902 −1.31114
$$460$$ 0 0
$$461$$ −12.2148 −0.568899 −0.284450 0.958691i $$-0.591811\pi$$
−0.284450 + 0.958691i $$0.591811\pi$$
$$462$$ 0 0
$$463$$ 21.4164 0.995305 0.497652 0.867377i $$-0.334196\pi$$
0.497652 + 0.867377i $$0.334196\pi$$
$$464$$ 0 0
$$465$$ −2.56231 −0.118824
$$466$$ 0 0
$$467$$ −3.65248 −0.169016 −0.0845082 0.996423i $$-0.526932\pi$$
−0.0845082 + 0.996423i $$0.526932\pi$$
$$468$$ 0 0
$$469$$ 14.4164 0.665688
$$470$$ 0 0
$$471$$ 16.7082 0.769873
$$472$$ 0 0
$$473$$ 22.7984 1.04827
$$474$$ 0 0
$$475$$ −28.4164 −1.30383
$$476$$ 0 0
$$477$$ 2.18034 0.0998309
$$478$$ 0 0
$$479$$ −8.18034 −0.373769 −0.186885 0.982382i $$-0.559839\pi$$
−0.186885 + 0.982382i $$0.559839\pi$$
$$480$$ 0 0
$$481$$ −32.5623 −1.48471
$$482$$ 0 0
$$483$$ 4.47214 0.203489
$$484$$ 0 0
$$485$$ −0.652476 −0.0296274
$$486$$ 0 0
$$487$$ 23.0000 1.04223 0.521115 0.853487i $$-0.325516\pi$$
0.521115 + 0.853487i $$0.325516\pi$$
$$488$$ 0 0
$$489$$ −2.70820 −0.122469
$$490$$ 0 0
$$491$$ 35.7771 1.61460 0.807299 0.590143i $$-0.200929\pi$$
0.807299 + 0.590143i $$0.200929\pi$$
$$492$$ 0 0
$$493$$ −29.4164 −1.32485
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ 4.09017 0.183469
$$498$$ 0 0
$$499$$ 13.2705 0.594070 0.297035 0.954867i $$-0.404002\pi$$
0.297035 + 0.954867i $$0.404002\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ 0 0
$$503$$ −13.3607 −0.595723 −0.297862 0.954609i $$-0.596273\pi$$
−0.297862 + 0.954609i $$0.596273\pi$$
$$504$$ 0 0
$$505$$ −0.729490 −0.0324619
$$506$$ 0 0
$$507$$ −10.5623 −0.469088
$$508$$ 0 0
$$509$$ −26.6180 −1.17982 −0.589912 0.807468i $$-0.700837\pi$$
−0.589912 + 0.807468i $$0.700837\pi$$
$$510$$ 0 0
$$511$$ −10.8541 −0.480157
$$512$$ 0 0
$$513$$ 29.2705 1.29232
$$514$$ 0 0
$$515$$ 0.381966 0.0168314
$$516$$ 0 0
$$517$$ 10.7082 0.470946
$$518$$ 0 0
$$519$$ 16.0344 0.703834
$$520$$ 0 0
$$521$$ 17.1803 0.752684 0.376342 0.926481i $$-0.377182\pi$$
0.376342 + 0.926481i $$0.377182\pi$$
$$522$$ 0 0
$$523$$ −10.5836 −0.462788 −0.231394 0.972860i $$-0.574329\pi$$
−0.231394 + 0.972860i $$0.574329\pi$$
$$524$$ 0 0
$$525$$ 4.85410 0.211850
$$526$$ 0 0
$$527$$ −37.6869 −1.64167
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ −12.7639 −0.553907
$$532$$ 0 0
$$533$$ 43.4164 1.88057
$$534$$ 0 0
$$535$$ −2.70820 −0.117086
$$536$$ 0 0
$$537$$ 7.85410 0.338930
$$538$$ 0 0
$$539$$ 15.7082 0.676600
$$540$$ 0 0
$$541$$ −9.14590 −0.393213 −0.196606 0.980482i $$-0.562992\pi$$
−0.196606 + 0.980482i $$0.562992\pi$$
$$542$$ 0 0
$$543$$ 3.85410 0.165395
$$544$$ 0 0
$$545$$ 3.65248 0.156455
$$546$$ 0 0
$$547$$ −2.27051 −0.0970800 −0.0485400 0.998821i $$-0.515457\pi$$
−0.0485400 + 0.998821i $$0.515457\pi$$
$$548$$ 0 0
$$549$$ 8.29180 0.353885
$$550$$ 0 0
$$551$$ 30.6525 1.30584
$$552$$ 0 0
$$553$$ 6.56231 0.279058
$$554$$ 0 0
$$555$$ 2.56231 0.108764
$$556$$ 0 0
$$557$$ 31.6869 1.34262 0.671309 0.741178i $$-0.265732\pi$$
0.671309 + 0.741178i $$0.265732\pi$$
$$558$$ 0 0
$$559$$ −42.2705 −1.78785
$$560$$ 0 0
$$561$$ −14.7082 −0.620981
$$562$$ 0 0
$$563$$ −1.20163 −0.0506425 −0.0253213 0.999679i $$-0.508061\pi$$
−0.0253213 + 0.999679i $$0.508061\pi$$
$$564$$ 0 0
$$565$$ −5.72949 −0.241041
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −27.1591 −1.13857 −0.569283 0.822141i $$-0.692779\pi$$
−0.569283 + 0.822141i $$0.692779\pi$$
$$570$$ 0 0
$$571$$ −22.5623 −0.944203 −0.472102 0.881544i $$-0.656504\pi$$
−0.472102 + 0.881544i $$0.656504\pi$$
$$572$$ 0 0
$$573$$ 5.61803 0.234697
$$574$$ 0 0
$$575$$ 21.7082 0.905295
$$576$$ 0 0
$$577$$ −36.8328 −1.53337 −0.766685 0.642023i $$-0.778095\pi$$
−0.766685 + 0.642023i $$0.778095\pi$$
$$578$$ 0 0
$$579$$ −20.1246 −0.836350
$$580$$ 0 0
$$581$$ −6.32624 −0.262457
$$582$$ 0 0
$$583$$ 2.85410 0.118205
$$584$$ 0 0
$$585$$ −3.70820 −0.153315
$$586$$ 0 0
$$587$$ −47.0689 −1.94274 −0.971370 0.237570i $$-0.923649\pi$$
−0.971370 + 0.237570i $$0.923649\pi$$
$$588$$ 0 0
$$589$$ 39.2705 1.61811
$$590$$ 0 0
$$591$$ −16.4164 −0.675281
$$592$$ 0 0
$$593$$ 2.18034 0.0895358 0.0447679 0.998997i $$-0.485745\pi$$
0.0447679 + 0.998997i $$0.485745\pi$$
$$594$$ 0 0
$$595$$ −2.14590 −0.0879732
$$596$$ 0 0
$$597$$ −23.4164 −0.958370
$$598$$ 0 0
$$599$$ 35.4508 1.44848 0.724241 0.689547i $$-0.242190\pi$$
0.724241 + 0.689547i $$0.242190\pi$$
$$600$$ 0 0
$$601$$ 3.56231 0.145309 0.0726547 0.997357i $$-0.476853\pi$$
0.0726547 + 0.997357i $$0.476853\pi$$
$$602$$ 0 0
$$603$$ −28.8328 −1.17416
$$604$$ 0 0
$$605$$ −1.58359 −0.0643822
$$606$$ 0 0
$$607$$ −5.70820 −0.231689 −0.115844 0.993267i $$-0.536957\pi$$
−0.115844 + 0.993267i $$0.536957\pi$$
$$608$$ 0 0
$$609$$ −5.23607 −0.212176
$$610$$ 0 0
$$611$$ −19.8541 −0.803211
$$612$$ 0 0
$$613$$ −24.4164 −0.986169 −0.493085 0.869981i $$-0.664131\pi$$
−0.493085 + 0.869981i $$0.664131\pi$$
$$614$$ 0 0
$$615$$ −3.41641 −0.137763
$$616$$ 0 0
$$617$$ 3.27051 0.131666 0.0658329 0.997831i $$-0.479030\pi$$
0.0658329 + 0.997831i $$0.479030\pi$$
$$618$$ 0 0
$$619$$ −28.6869 −1.15302 −0.576512 0.817088i $$-0.695587\pi$$
−0.576512 + 0.817088i $$0.695587\pi$$
$$620$$ 0 0
$$621$$ −22.3607 −0.897303
$$622$$ 0 0
$$623$$ −2.29180 −0.0918189
$$624$$ 0 0
$$625$$ 22.8328 0.913313
$$626$$ 0 0
$$627$$ 15.3262 0.612071
$$628$$ 0 0
$$629$$ 37.6869 1.50268
$$630$$ 0 0
$$631$$ −42.2705 −1.68276 −0.841381 0.540442i $$-0.818257\pi$$
−0.841381 + 0.540442i $$0.818257\pi$$
$$632$$ 0 0
$$633$$ −14.8541 −0.590398
$$634$$ 0 0
$$635$$ −5.83282 −0.231468
$$636$$ 0 0
$$637$$ −29.1246 −1.15396
$$638$$ 0 0
$$639$$ −8.18034 −0.323609
$$640$$ 0 0
$$641$$ −15.0000 −0.592464 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$642$$ 0 0
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ 0 0
$$645$$ 3.32624 0.130970
$$646$$ 0 0
$$647$$ −1.25735 −0.0494317 −0.0247158 0.999695i $$-0.507868\pi$$
−0.0247158 + 0.999695i $$0.507868\pi$$
$$648$$ 0 0
$$649$$ −16.7082 −0.655854
$$650$$ 0 0
$$651$$ −6.70820 −0.262915
$$652$$ 0 0
$$653$$ 5.23607 0.204903 0.102452 0.994738i $$-0.467331\pi$$
0.102452 + 0.994738i $$0.467331\pi$$
$$654$$ 0 0
$$655$$ −0.854102 −0.0333725
$$656$$ 0 0
$$657$$ 21.7082 0.846918
$$658$$ 0 0
$$659$$ 36.5967 1.42561 0.712803 0.701364i $$-0.247425\pi$$
0.712803 + 0.701364i $$0.247425\pi$$
$$660$$ 0 0
$$661$$ 31.5623 1.22763 0.613816 0.789449i $$-0.289634\pi$$
0.613816 + 0.789449i $$0.289634\pi$$
$$662$$ 0 0
$$663$$ 27.2705 1.05910
$$664$$ 0 0
$$665$$ 2.23607 0.0867110
$$666$$ 0 0
$$667$$ −23.4164 −0.906687
$$668$$ 0 0
$$669$$ 7.70820 0.298016
$$670$$ 0 0
$$671$$ 10.8541 0.419018
$$672$$ 0 0
$$673$$ −24.2918 −0.936380 −0.468190 0.883628i $$-0.655094\pi$$
−0.468190 + 0.883628i $$0.655094\pi$$
$$674$$ 0 0
$$675$$ −24.2705 −0.934172
$$676$$ 0 0
$$677$$ −40.0344 −1.53865 −0.769324 0.638858i $$-0.779407\pi$$
−0.769324 + 0.638858i $$0.779407\pi$$
$$678$$ 0 0
$$679$$ −1.70820 −0.0655549
$$680$$ 0 0
$$681$$ −2.94427 −0.112825
$$682$$ 0 0
$$683$$ 47.2361 1.80744 0.903719 0.428126i $$-0.140826\pi$$
0.903719 + 0.428126i $$0.140826\pi$$
$$684$$ 0 0
$$685$$ −4.85410 −0.185466
$$686$$ 0 0
$$687$$ 6.70820 0.255934
$$688$$ 0 0
$$689$$ −5.29180 −0.201601
$$690$$ 0 0
$$691$$ 7.85410 0.298784 0.149392 0.988778i $$-0.452268\pi$$
0.149392 + 0.988778i $$0.452268\pi$$
$$692$$ 0 0
$$693$$ 5.23607 0.198902
$$694$$ 0 0
$$695$$ −4.25735 −0.161491
$$696$$ 0 0
$$697$$ −50.2492 −1.90333
$$698$$ 0 0
$$699$$ 26.8885 1.01702
$$700$$ 0 0
$$701$$ 25.7984 0.974391 0.487196 0.873293i $$-0.338020\pi$$
0.487196 + 0.873293i $$0.338020\pi$$
$$702$$ 0 0
$$703$$ −39.2705 −1.48112
$$704$$ 0 0
$$705$$ 1.56231 0.0588398
$$706$$ 0 0
$$707$$ −1.90983 −0.0718266
$$708$$ 0 0
$$709$$ 1.02129 0.0383552 0.0191776 0.999816i $$-0.493895\pi$$
0.0191776 + 0.999816i $$0.493895\pi$$
$$710$$ 0 0
$$711$$ −13.1246 −0.492211
$$712$$ 0 0
$$713$$ −30.0000 −1.12351
$$714$$ 0 0
$$715$$ −4.85410 −0.181533
$$716$$ 0 0
$$717$$ 8.67376 0.323928
$$718$$ 0 0
$$719$$ −39.3262 −1.46662 −0.733311 0.679894i $$-0.762026\pi$$
−0.733311 + 0.679894i $$0.762026\pi$$
$$720$$ 0 0
$$721$$ 1.00000 0.0372419
$$722$$ 0 0
$$723$$ 8.27051 0.307584
$$724$$ 0 0
$$725$$ −25.4164 −0.943942
$$726$$ 0 0
$$727$$ 4.72949 0.175407 0.0877035 0.996147i $$-0.472047\pi$$
0.0877035 + 0.996147i $$0.472047\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 48.9230 1.80948
$$732$$ 0 0
$$733$$ −14.7082 −0.543260 −0.271630 0.962402i $$-0.587563\pi$$
−0.271630 + 0.962402i $$0.587563\pi$$
$$734$$ 0 0
$$735$$ 2.29180 0.0845342
$$736$$ 0 0
$$737$$ −37.7426 −1.39027
$$738$$ 0 0
$$739$$ −16.8328 −0.619205 −0.309603 0.950866i $$-0.600196\pi$$
−0.309603 + 0.950866i $$0.600196\pi$$
$$740$$ 0 0
$$741$$ −28.4164 −1.04390
$$742$$ 0 0
$$743$$ −30.7082 −1.12657 −0.563287 0.826261i $$-0.690464\pi$$
−0.563287 + 0.826261i $$0.690464\pi$$
$$744$$ 0 0
$$745$$ 2.85410 0.104566
$$746$$ 0 0
$$747$$ 12.6525 0.462930
$$748$$ 0 0
$$749$$ −7.09017 −0.259069
$$750$$ 0 0
$$751$$ 44.1246 1.61013 0.805065 0.593187i $$-0.202130\pi$$
0.805065 + 0.593187i $$0.202130\pi$$
$$752$$ 0 0
$$753$$ −11.2361 −0.409465
$$754$$ 0 0
$$755$$ 7.25735 0.264122
$$756$$ 0 0
$$757$$ 21.2918 0.773863 0.386932 0.922108i $$-0.373535\pi$$
0.386932 + 0.922108i $$0.373535\pi$$
$$758$$ 0 0
$$759$$ −11.7082 −0.424981
$$760$$ 0 0
$$761$$ −7.47214 −0.270865 −0.135432 0.990787i $$-0.543242\pi$$
−0.135432 + 0.990787i $$0.543242\pi$$
$$762$$ 0 0
$$763$$ 9.56231 0.346179
$$764$$ 0 0
$$765$$ 4.29180 0.155170
$$766$$ 0 0
$$767$$ 30.9787 1.11858
$$768$$ 0 0
$$769$$ 31.3951 1.13214 0.566069 0.824358i $$-0.308464\pi$$
0.566069 + 0.824358i $$0.308464\pi$$
$$770$$ 0 0
$$771$$ 13.4721 0.485187
$$772$$ 0 0
$$773$$ 16.5279 0.594466 0.297233 0.954805i $$-0.403936\pi$$
0.297233 + 0.954805i $$0.403936\pi$$
$$774$$ 0 0
$$775$$ −32.5623 −1.16967
$$776$$ 0 0
$$777$$ 6.70820 0.240655
$$778$$ 0 0
$$779$$ 52.3607 1.87602
$$780$$ 0 0
$$781$$ −10.7082 −0.383170
$$782$$ 0 0
$$783$$ 26.1803 0.935609
$$784$$ 0 0
$$785$$ −6.38197 −0.227782
$$786$$ 0 0
$$787$$ 43.4164 1.54763 0.773814 0.633413i $$-0.218347\pi$$
0.773814 + 0.633413i $$0.218347\pi$$
$$788$$ 0 0
$$789$$ −20.6180 −0.734021
$$790$$ 0 0
$$791$$ −15.0000 −0.533339
$$792$$ 0 0
$$793$$ −20.1246 −0.714646
$$794$$ 0 0
$$795$$ 0.416408 0.0147685
$$796$$ 0 0
$$797$$ 50.1246 1.77550 0.887752 0.460321i $$-0.152266\pi$$
0.887752 + 0.460321i $$0.152266\pi$$
$$798$$ 0 0
$$799$$ 22.9787 0.812928
$$800$$ 0 0
$$801$$ 4.58359 0.161953
$$802$$ 0 0
$$803$$ 28.4164 1.00279
$$804$$ 0 0
$$805$$ −1.70820 −0.0602063
$$806$$ 0 0
$$807$$ 12.3262 0.433904
$$808$$ 0 0
$$809$$ 2.94427 0.103515 0.0517575 0.998660i $$-0.483518\pi$$
0.0517575 + 0.998660i $$0.483518\pi$$
$$810$$ 0 0
$$811$$ 21.5410 0.756408 0.378204 0.925722i $$-0.376542\pi$$
0.378204 + 0.925722i $$0.376542\pi$$
$$812$$ 0 0
$$813$$ 1.00000 0.0350715
$$814$$ 0 0
$$815$$ 1.03444 0.0362349
$$816$$ 0 0
$$817$$ −50.9787 −1.78352
$$818$$ 0 0
$$819$$ −9.70820 −0.339232
$$820$$ 0 0
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ 0 0
$$823$$ −3.43769 −0.119830 −0.0599152 0.998203i $$-0.519083\pi$$
−0.0599152 + 0.998203i $$0.519083\pi$$
$$824$$ 0 0
$$825$$ −12.7082 −0.442443
$$826$$ 0 0
$$827$$ 6.70820 0.233267 0.116634 0.993175i $$-0.462790\pi$$
0.116634 + 0.993175i $$0.462790\pi$$
$$828$$ 0 0
$$829$$ −14.2705 −0.495635 −0.247818 0.968807i $$-0.579713\pi$$
−0.247818 + 0.968807i $$0.579713\pi$$
$$830$$ 0 0
$$831$$ 4.70820 0.163326
$$832$$ 0 0
$$833$$ 33.7082 1.16792
$$834$$ 0 0
$$835$$ −3.43769 −0.118966
$$836$$ 0 0
$$837$$ 33.5410 1.15935
$$838$$ 0 0
$$839$$ −23.6180 −0.815385 −0.407693 0.913119i $$-0.633666\pi$$
−0.407693 + 0.913119i $$0.633666\pi$$
$$840$$ 0 0
$$841$$ −1.58359 −0.0546066
$$842$$ 0 0
$$843$$ 31.4721 1.08396
$$844$$ 0 0
$$845$$ 4.03444 0.138789
$$846$$ 0 0
$$847$$ −4.14590 −0.142455
$$848$$ 0 0
$$849$$ 19.7082 0.676384
$$850$$ 0 0
$$851$$ 30.0000 1.02839
$$852$$ 0 0
$$853$$ 44.2705 1.51579 0.757897 0.652375i $$-0.226227\pi$$
0.757897 + 0.652375i $$0.226227\pi$$
$$854$$ 0 0
$$855$$ −4.47214 −0.152944
$$856$$ 0 0
$$857$$ 8.23607 0.281339 0.140669 0.990057i $$-0.455075\pi$$
0.140669 + 0.990057i $$0.455075\pi$$
$$858$$ 0 0
$$859$$ 10.5623 0.360381 0.180191 0.983632i $$-0.442329\pi$$
0.180191 + 0.983632i $$0.442329\pi$$
$$860$$ 0 0
$$861$$ −8.94427 −0.304820
$$862$$ 0 0
$$863$$ 21.7082 0.738956 0.369478 0.929240i $$-0.379537\pi$$
0.369478 + 0.929240i $$0.379537\pi$$
$$864$$ 0 0
$$865$$ −6.12461 −0.208243
$$866$$ 0 0
$$867$$ −14.5623 −0.494562
$$868$$ 0 0
$$869$$ −17.1803 −0.582803
$$870$$ 0 0
$$871$$ 69.9787 2.37114
$$872$$ 0 0
$$873$$ 3.41641 0.115628
$$874$$ 0 0
$$875$$ −3.76393 −0.127244
$$876$$ 0 0
$$877$$ 13.0000 0.438979 0.219489 0.975615i $$-0.429561\pi$$
0.219489 + 0.975615i $$0.429561\pi$$
$$878$$ 0 0
$$879$$ −21.6525 −0.730320
$$880$$ 0 0
$$881$$ 29.8885 1.00697 0.503485 0.864004i $$-0.332051\pi$$
0.503485 + 0.864004i $$0.332051\pi$$
$$882$$ 0 0
$$883$$ 5.87539 0.197723 0.0988613 0.995101i $$-0.468480\pi$$
0.0988613 + 0.995101i $$0.468480\pi$$
$$884$$ 0 0
$$885$$ −2.43769 −0.0819422
$$886$$ 0 0
$$887$$ −40.1935 −1.34957 −0.674783 0.738016i $$-0.735763\pi$$
−0.674783 + 0.738016i $$0.735763\pi$$
$$888$$ 0 0
$$889$$ −15.2705 −0.512156
$$890$$ 0 0
$$891$$ −2.61803 −0.0877074
$$892$$ 0 0
$$893$$ −23.9443 −0.801265
$$894$$ 0 0
$$895$$ −3.00000 −0.100279
$$896$$ 0 0
$$897$$ 21.7082 0.724816
$$898$$ 0 0
$$899$$ 35.1246 1.17147
$$900$$ 0 0
$$901$$ 6.12461 0.204040
$$902$$ 0 0
$$903$$ 8.70820 0.289791
$$904$$ 0 0
$$905$$ −1.47214 −0.0489355
$$906$$ 0 0
$$907$$ −7.12461 −0.236569 −0.118284 0.992980i $$-0.537739\pi$$
−0.118284 + 0.992980i $$0.537739\pi$$
$$908$$ 0 0
$$909$$ 3.81966 0.126690
$$910$$ 0 0
$$911$$ 10.0344 0.332456 0.166228 0.986087i $$-0.446841\pi$$
0.166228 + 0.986087i $$0.446841\pi$$
$$912$$ 0 0
$$913$$ 16.5623 0.548132
$$914$$ 0 0
$$915$$ 1.58359 0.0523519
$$916$$ 0 0
$$917$$ −2.23607 −0.0738415
$$918$$ 0 0
$$919$$ 27.9787 0.922933 0.461466 0.887158i $$-0.347324\pi$$
0.461466 + 0.887158i $$0.347324\pi$$
$$920$$ 0 0
$$921$$ 2.85410 0.0940459
$$922$$ 0 0
$$923$$ 19.8541 0.653506
$$924$$ 0 0
$$925$$ 32.5623 1.07064
$$926$$ 0 0
$$927$$ −2.00000 −0.0656886
$$928$$ 0 0
$$929$$ 14.9443 0.490306 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$930$$ 0 0
$$931$$ −35.1246 −1.15116
$$932$$ 0 0
$$933$$ 2.88854 0.0945667
$$934$$ 0 0
$$935$$ 5.61803 0.183729
$$936$$ 0 0
$$937$$ −11.0000 −0.359354 −0.179677 0.983726i $$-0.557505\pi$$
−0.179677 + 0.983726i $$0.557505\pi$$
$$938$$ 0 0
$$939$$ −16.2918 −0.531663
$$940$$ 0 0
$$941$$ 23.3951 0.762659 0.381330 0.924439i $$-0.375466\pi$$
0.381330 + 0.924439i $$0.375466\pi$$
$$942$$ 0 0
$$943$$ −40.0000 −1.30258
$$944$$ 0 0
$$945$$ 1.90983 0.0621268
$$946$$ 0 0
$$947$$ −41.0132 −1.33275 −0.666374 0.745617i $$-0.732155\pi$$
−0.666374 + 0.745617i $$0.732155\pi$$
$$948$$ 0 0
$$949$$ −52.6869 −1.71029
$$950$$ 0 0
$$951$$ 28.4164 0.921465
$$952$$ 0 0
$$953$$ 13.3607 0.432795 0.216397 0.976305i $$-0.430569\pi$$
0.216397 + 0.976305i $$0.430569\pi$$
$$954$$ 0 0
$$955$$ −2.14590 −0.0694396
$$956$$ 0 0
$$957$$ 13.7082 0.443123
$$958$$ 0 0
$$959$$ −12.7082 −0.410369
$$960$$ 0 0
$$961$$ 14.0000 0.451613
$$962$$ 0 0
$$963$$ 14.1803 0.456955
$$964$$ 0 0
$$965$$ 7.68692 0.247451
$$966$$ 0 0
$$967$$ 14.4164 0.463600 0.231800 0.972763i $$-0.425538\pi$$
0.231800 + 0.972763i $$0.425538\pi$$
$$968$$ 0 0
$$969$$ 32.8885 1.05653
$$970$$ 0 0
$$971$$ −53.0132 −1.70127 −0.850637 0.525754i $$-0.823783\pi$$
−0.850637 + 0.525754i $$0.823783\pi$$
$$972$$ 0 0
$$973$$ −11.1459 −0.357321
$$974$$ 0 0
$$975$$ 23.5623 0.754598
$$976$$ 0 0
$$977$$ 46.7426 1.49543 0.747715 0.664020i $$-0.231151\pi$$
0.747715 + 0.664020i $$0.231151\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −19.1246 −0.610602
$$982$$ 0 0
$$983$$ −18.6525 −0.594922 −0.297461 0.954734i $$-0.596140\pi$$
−0.297461 + 0.954734i $$0.596140\pi$$
$$984$$ 0 0
$$985$$ 6.27051 0.199795
$$986$$ 0 0
$$987$$ 4.09017 0.130192
$$988$$ 0 0
$$989$$ 38.9443 1.23836
$$990$$ 0 0
$$991$$ −24.2705 −0.770978 −0.385489 0.922712i $$-0.625967\pi$$
−0.385489 + 0.922712i $$0.625967\pi$$
$$992$$ 0 0
$$993$$ 16.1459 0.512375
$$994$$ 0 0
$$995$$ 8.94427 0.283552
$$996$$ 0 0
$$997$$ 39.2918 1.24438 0.622192 0.782865i $$-0.286242\pi$$
0.622192 + 0.782865i $$0.286242\pi$$
$$998$$ 0 0
$$999$$ −33.5410 −1.06119
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 6592.2.a.h.1.1 2
4.3 odd 2 6592.2.a.t.1.1 2
8.3 odd 2 103.2.a.a.1.1 2
8.5 even 2 1648.2.a.f.1.2 2
24.11 even 2 927.2.a.b.1.2 2
40.19 odd 2 2575.2.a.g.1.2 2
56.27 even 2 5047.2.a.a.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 8.3 odd 2
927.2.a.b.1.2 2 24.11 even 2
1648.2.a.f.1.2 2 8.5 even 2
2575.2.a.g.1.2 2 40.19 odd 2
5047.2.a.a.1.1 2 56.27 even 2
6592.2.a.h.1.1 2 1.1 even 1 trivial
6592.2.a.t.1.1 2 4.3 odd 2