# Properties

 Label 3744.2.a.z.1.2 Level $3744$ Weight $2$ Character 3744.1 Self dual yes Analytic conductor $29.896$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3744.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: $$29.8959905168$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1248) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 3744.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.622216 q^{5} +4.42864 q^{7} +O(q^{10})$$ $$q-0.622216 q^{5} +4.42864 q^{7} -5.80642 q^{11} +1.00000 q^{13} -2.00000 q^{17} -4.42864 q^{19} +8.85728 q^{23} -4.61285 q^{25} -2.00000 q^{29} -7.18421 q^{31} -2.75557 q^{35} +0.755569 q^{37} -3.37778 q^{41} -7.61285 q^{43} -1.80642 q^{47} +12.6128 q^{49} -4.75557 q^{53} +3.61285 q^{55} -11.0509 q^{59} -8.10171 q^{61} -0.622216 q^{65} +8.04149 q^{67} +7.05086 q^{71} +7.24443 q^{73} -25.7146 q^{77} -12.0000 q^{79} +3.05086 q^{83} +1.24443 q^{85} +1.86665 q^{89} +4.42864 q^{91} +2.75557 q^{95} -0.755569 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{5}+O(q^{10})$$ 3 * q - 2 * q^5 $$3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100})$$ 3 * q - 2 * q^5 - 4 * q^11 + 3 * q^13 - 6 * q^17 + 13 * q^25 - 6 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^37 - 10 * q^41 + 4 * q^43 + 8 * q^47 + 11 * q^49 - 14 * q^53 - 16 * q^55 - 20 * q^59 + 2 * q^61 - 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 - 24 * q^77 - 36 * q^79 - 4 * q^83 + 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −0.622216 −0.278263 −0.139132 0.990274i $$-0.544431\pi$$
−0.139132 + 0.990274i $$0.544431\pi$$
$$6$$ 0 0
$$7$$ 4.42864 1.67387 0.836934 0.547304i $$-0.184346\pi$$
0.836934 + 0.547304i $$0.184346\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.80642 −1.75070 −0.875351 0.483487i $$-0.839370\pi$$
−0.875351 + 0.483487i $$0.839370\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −4.42864 −1.01600 −0.508000 0.861357i $$-0.669615\pi$$
−0.508000 + 0.861357i $$0.669615\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.85728 1.84687 0.923435 0.383754i $$-0.125369\pi$$
0.923435 + 0.383754i $$0.125369\pi$$
$$24$$ 0 0
$$25$$ −4.61285 −0.922570
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −7.18421 −1.29032 −0.645161 0.764047i $$-0.723210\pi$$
−0.645161 + 0.764047i $$0.723210\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.75557 −0.465776
$$36$$ 0 0
$$37$$ 0.755569 0.124215 0.0621074 0.998069i $$-0.480218\pi$$
0.0621074 + 0.998069i $$0.480218\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.37778 −0.527521 −0.263761 0.964588i $$-0.584963\pi$$
−0.263761 + 0.964588i $$0.584963\pi$$
$$42$$ 0 0
$$43$$ −7.61285 −1.16095 −0.580474 0.814279i $$-0.697133\pi$$
−0.580474 + 0.814279i $$0.697133\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.80642 −0.263494 −0.131747 0.991283i $$-0.542059\pi$$
−0.131747 + 0.991283i $$0.542059\pi$$
$$48$$ 0 0
$$49$$ 12.6128 1.80184
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −4.75557 −0.653228 −0.326614 0.945158i $$-0.605908\pi$$
−0.326614 + 0.945158i $$0.605908\pi$$
$$54$$ 0 0
$$55$$ 3.61285 0.487156
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −11.0509 −1.43870 −0.719349 0.694648i $$-0.755560\pi$$
−0.719349 + 0.694648i $$0.755560\pi$$
$$60$$ 0 0
$$61$$ −8.10171 −1.03732 −0.518659 0.854981i $$-0.673569\pi$$
−0.518659 + 0.854981i $$0.673569\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.622216 −0.0771764
$$66$$ 0 0
$$67$$ 8.04149 0.982424 0.491212 0.871040i $$-0.336554\pi$$
0.491212 + 0.871040i $$0.336554\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.05086 0.836783 0.418391 0.908267i $$-0.362594\pi$$
0.418391 + 0.908267i $$0.362594\pi$$
$$72$$ 0 0
$$73$$ 7.24443 0.847897 0.423948 0.905686i $$-0.360644\pi$$
0.423948 + 0.905686i $$0.360644\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −25.7146 −2.93045
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.05086 0.334875 0.167437 0.985883i $$-0.446451\pi$$
0.167437 + 0.985883i $$0.446451\pi$$
$$84$$ 0 0
$$85$$ 1.24443 0.134978
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1.86665 0.197864 0.0989321 0.995094i $$-0.468457\pi$$
0.0989321 + 0.995094i $$0.468457\pi$$
$$90$$ 0 0
$$91$$ 4.42864 0.464248
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.75557 0.282715
$$96$$ 0 0
$$97$$ −0.755569 −0.0767164 −0.0383582 0.999264i $$-0.512213\pi$$
−0.0383582 + 0.999264i $$0.512213\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 15.7146 1.56366 0.781828 0.623494i $$-0.214287\pi$$
0.781828 + 0.623494i $$0.214287\pi$$
$$102$$ 0 0
$$103$$ −18.1017 −1.78361 −0.891807 0.452416i $$-0.850562\pi$$
−0.891807 + 0.452416i $$0.850562\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −15.3461 −1.48357 −0.741784 0.670639i $$-0.766020\pi$$
−0.741784 + 0.670639i $$0.766020\pi$$
$$108$$ 0 0
$$109$$ 15.7146 1.50518 0.752591 0.658488i $$-0.228804\pi$$
0.752591 + 0.658488i $$0.228804\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −14.4701 −1.36124 −0.680618 0.732639i $$-0.738288\pi$$
−0.680618 + 0.732639i $$0.738288\pi$$
$$114$$ 0 0
$$115$$ −5.51114 −0.513916
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −8.85728 −0.811945
$$120$$ 0 0
$$121$$ 22.7146 2.06496
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 5.98126 0.534981
$$126$$ 0 0
$$127$$ −20.8573 −1.85078 −0.925392 0.379011i $$-0.876264\pi$$
−0.925392 + 0.379011i $$0.876264\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.3684 0.905893 0.452946 0.891538i $$-0.350373\pi$$
0.452946 + 0.891538i $$0.350373\pi$$
$$132$$ 0 0
$$133$$ −19.6128 −1.70065
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.8479 1.69572 0.847861 0.530219i $$-0.177890\pi$$
0.847861 + 0.530219i $$0.177890\pi$$
$$138$$ 0 0
$$139$$ 6.75557 0.573000 0.286500 0.958080i $$-0.407508\pi$$
0.286500 + 0.958080i $$0.407508\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −5.80642 −0.485558
$$144$$ 0 0
$$145$$ 1.24443 0.103344
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.13335 0.174771 0.0873855 0.996175i $$-0.472149\pi$$
0.0873855 + 0.996175i $$0.472149\pi$$
$$150$$ 0 0
$$151$$ 1.67307 0.136153 0.0680763 0.997680i $$-0.478314\pi$$
0.0680763 + 0.997680i $$0.478314\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.47013 0.359049
$$156$$ 0 0
$$157$$ 15.1240 1.20703 0.603513 0.797353i $$-0.293767\pi$$
0.603513 + 0.797353i $$0.293767\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 39.2257 3.09142
$$162$$ 0 0
$$163$$ −15.7748 −1.23558 −0.617788 0.786345i $$-0.711971\pi$$
−0.617788 + 0.786345i $$0.711971\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.31756 −0.566250 −0.283125 0.959083i $$-0.591371\pi$$
−0.283125 + 0.959083i $$0.591371\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −7.51114 −0.571061 −0.285531 0.958370i $$-0.592170\pi$$
−0.285531 + 0.958370i $$0.592170\pi$$
$$174$$ 0 0
$$175$$ −20.4286 −1.54426
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −14.7556 −1.10288 −0.551441 0.834214i $$-0.685922\pi$$
−0.551441 + 0.834214i $$0.685922\pi$$
$$180$$ 0 0
$$181$$ 4.10171 0.304878 0.152439 0.988313i $$-0.451287\pi$$
0.152439 + 0.988313i $$0.451287\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −0.470127 −0.0345644
$$186$$ 0 0
$$187$$ 11.6128 0.849216
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0.266706 0.0192982 0.00964909 0.999953i $$-0.496929\pi$$
0.00964909 + 0.999953i $$0.496929\pi$$
$$192$$ 0 0
$$193$$ −26.2034 −1.88616 −0.943082 0.332561i $$-0.892087\pi$$
−0.943082 + 0.332561i $$0.892087\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −15.8479 −1.12912 −0.564558 0.825393i $$-0.690954\pi$$
−0.564558 + 0.825393i $$0.690954\pi$$
$$198$$ 0 0
$$199$$ 10.3684 0.734998 0.367499 0.930024i $$-0.380214\pi$$
0.367499 + 0.930024i $$0.380214\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −8.85728 −0.621659
$$204$$ 0 0
$$205$$ 2.10171 0.146790
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 25.7146 1.77871
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.73683 0.323049
$$216$$ 0 0
$$217$$ −31.8163 −2.15983
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ −4.69535 −0.314424 −0.157212 0.987565i $$-0.550251\pi$$
−0.157212 + 0.987565i $$0.550251\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5.80642 0.385386 0.192693 0.981259i $$-0.438278\pi$$
0.192693 + 0.981259i $$0.438278\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7.24443 −0.474598 −0.237299 0.971437i $$-0.576262\pi$$
−0.237299 + 0.971437i $$0.576262\pi$$
$$234$$ 0 0
$$235$$ 1.12399 0.0733207
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.6637 −0.689778 −0.344889 0.938644i $$-0.612083\pi$$
−0.344889 + 0.938644i $$0.612083\pi$$
$$240$$ 0 0
$$241$$ −5.73329 −0.369314 −0.184657 0.982803i $$-0.559117\pi$$
−0.184657 + 0.982803i $$0.559117\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −7.84791 −0.501385
$$246$$ 0 0
$$247$$ −4.42864 −0.281788
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −23.6128 −1.49043 −0.745215 0.666824i $$-0.767653\pi$$
−0.745215 + 0.666824i $$0.767653\pi$$
$$252$$ 0 0
$$253$$ −51.4291 −3.23332
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 11.2444 0.701408 0.350704 0.936486i $$-0.385942\pi$$
0.350704 + 0.936486i $$0.385942\pi$$
$$258$$ 0 0
$$259$$ 3.34614 0.207919
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −4.38715 −0.270523 −0.135262 0.990810i $$-0.543188\pi$$
−0.135262 + 0.990810i $$0.543188\pi$$
$$264$$ 0 0
$$265$$ 2.95899 0.181769
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −7.51114 −0.457962 −0.228981 0.973431i $$-0.573539\pi$$
−0.228981 + 0.973431i $$0.573539\pi$$
$$270$$ 0 0
$$271$$ −6.06022 −0.368132 −0.184066 0.982914i $$-0.558926\pi$$
−0.184066 + 0.982914i $$0.558926\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 26.7841 1.61514
$$276$$ 0 0
$$277$$ 0.488863 0.0293729 0.0146865 0.999892i $$-0.495325\pi$$
0.0146865 + 0.999892i $$0.495325\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.0923 −0.781024 −0.390512 0.920598i $$-0.627702\pi$$
−0.390512 + 0.920598i $$0.627702\pi$$
$$282$$ 0 0
$$283$$ 23.3461 1.38778 0.693892 0.720079i $$-0.255894\pi$$
0.693892 + 0.720079i $$0.255894\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −14.9590 −0.883001
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −13.0923 −0.764863 −0.382431 0.923984i $$-0.624913\pi$$
−0.382431 + 0.923984i $$0.624913\pi$$
$$294$$ 0 0
$$295$$ 6.87601 0.400337
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8.85728 0.512230
$$300$$ 0 0
$$301$$ −33.7146 −1.94327
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 5.04101 0.288647
$$306$$ 0 0
$$307$$ −32.0415 −1.82870 −0.914352 0.404920i $$-0.867299\pi$$
−0.914352 + 0.404920i $$0.867299\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0830 −1.36562 −0.682810 0.730596i $$-0.739242\pi$$
−0.682810 + 0.730596i $$0.739242\pi$$
$$312$$ 0 0
$$313$$ 27.7146 1.56652 0.783260 0.621695i $$-0.213556\pi$$
0.783260 + 0.621695i $$0.213556\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.0923 0.960002 0.480001 0.877268i $$-0.340636\pi$$
0.480001 + 0.877268i $$0.340636\pi$$
$$318$$ 0 0
$$319$$ 11.6128 0.650195
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.85728 0.492832
$$324$$ 0 0
$$325$$ −4.61285 −0.255875
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −7.18421 −0.394880 −0.197440 0.980315i $$-0.563263\pi$$
−0.197440 + 0.980315i $$0.563263\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −5.00354 −0.273373
$$336$$ 0 0
$$337$$ 0.285442 0.0155490 0.00777451 0.999970i $$-0.497525\pi$$
0.00777451 + 0.999970i $$0.497525\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 41.7146 2.25897
$$342$$ 0 0
$$343$$ 24.8573 1.34217
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.2034 1.29931 0.649654 0.760230i $$-0.274914\pi$$
0.649654 + 0.760230i $$0.274914\pi$$
$$348$$ 0 0
$$349$$ −12.7556 −0.682790 −0.341395 0.939920i $$-0.610899\pi$$
−0.341395 + 0.939920i $$0.610899\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −21.0923 −1.12263 −0.561316 0.827602i $$-0.689705\pi$$
−0.561316 + 0.827602i $$0.689705\pi$$
$$354$$ 0 0
$$355$$ −4.38715 −0.232846
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.1338 0.798733 0.399366 0.916791i $$-0.369230\pi$$
0.399366 + 0.916791i $$0.369230\pi$$
$$360$$ 0 0
$$361$$ 0.612848 0.0322551
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.50760 −0.235938
$$366$$ 0 0
$$367$$ −18.9590 −0.989651 −0.494826 0.868992i $$-0.664768\pi$$
−0.494826 + 0.868992i $$0.664768\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −21.0607 −1.09342
$$372$$ 0 0
$$373$$ 2.97773 0.154181 0.0770904 0.997024i $$-0.475437\pi$$
0.0770904 + 0.997024i $$0.475437\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ −14.1432 −0.726487 −0.363244 0.931694i $$-0.618331\pi$$
−0.363244 + 0.931694i $$0.618331\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 25.8064 1.31865 0.659323 0.751860i $$-0.270843\pi$$
0.659323 + 0.751860i $$0.270843\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 12.9590 0.657047 0.328523 0.944496i $$-0.393449\pi$$
0.328523 + 0.944496i $$0.393449\pi$$
$$390$$ 0 0
$$391$$ −17.7146 −0.895864
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 7.46659 0.375685
$$396$$ 0 0
$$397$$ 3.24443 0.162833 0.0814167 0.996680i $$-0.474056\pi$$
0.0814167 + 0.996680i $$0.474056\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.6035 1.52826 0.764132 0.645059i $$-0.223167\pi$$
0.764132 + 0.645059i $$0.223167\pi$$
$$402$$ 0 0
$$403$$ −7.18421 −0.357871
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.38715 −0.217463
$$408$$ 0 0
$$409$$ −8.75557 −0.432935 −0.216468 0.976290i $$-0.569454\pi$$
−0.216468 + 0.976290i $$0.569454\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −48.9403 −2.40819
$$414$$ 0 0
$$415$$ −1.89829 −0.0931834
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 23.8796 1.16659 0.583296 0.812259i $$-0.301763\pi$$
0.583296 + 0.812259i $$0.301763\pi$$
$$420$$ 0 0
$$421$$ −8.28544 −0.403808 −0.201904 0.979405i $$-0.564713\pi$$
−0.201904 + 0.979405i $$0.564713\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 9.22570 0.447512
$$426$$ 0 0
$$427$$ −35.8796 −1.73633
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 14.0098 0.674830 0.337415 0.941356i $$-0.390447\pi$$
0.337415 + 0.941356i $$0.390447\pi$$
$$432$$ 0 0
$$433$$ 27.1240 1.30350 0.651748 0.758436i $$-0.274036\pi$$
0.651748 + 0.758436i $$0.274036\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −39.2257 −1.87642
$$438$$ 0 0
$$439$$ 4.59057 0.219096 0.109548 0.993982i $$-0.465060\pi$$
0.109548 + 0.993982i $$0.465060\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10.3684 0.492618 0.246309 0.969191i $$-0.420782\pi$$
0.246309 + 0.969191i $$0.420782\pi$$
$$444$$ 0 0
$$445$$ −1.16146 −0.0550583
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.3368 −0.487823 −0.243911 0.969798i $$-0.578431\pi$$
−0.243911 + 0.969798i $$0.578431\pi$$
$$450$$ 0 0
$$451$$ 19.6128 0.923533
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −2.75557 −0.129183
$$456$$ 0 0
$$457$$ 5.52987 0.258677 0.129338 0.991601i $$-0.458715\pi$$
0.129338 + 0.991601i $$0.458715\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 17.8666 0.832133 0.416066 0.909334i $$-0.363408\pi$$
0.416066 + 0.909334i $$0.363408\pi$$
$$462$$ 0 0
$$463$$ 38.6766 1.79745 0.898727 0.438508i $$-0.144493\pi$$
0.898727 + 0.438508i $$0.144493\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10.1017 −0.467451 −0.233726 0.972303i $$-0.575092\pi$$
−0.233726 + 0.972303i $$0.575092\pi$$
$$468$$ 0 0
$$469$$ 35.6128 1.64445
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 44.2034 2.03248
$$474$$ 0 0
$$475$$ 20.4286 0.937330
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 30.8671 1.41035 0.705177 0.709031i $$-0.250867\pi$$
0.705177 + 0.709031i $$0.250867\pi$$
$$480$$ 0 0
$$481$$ 0.755569 0.0344510
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0.470127 0.0213474
$$486$$ 0 0
$$487$$ 25.7560 1.16712 0.583559 0.812071i $$-0.301660\pi$$
0.583559 + 0.812071i $$0.301660\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −27.8163 −1.25533 −0.627665 0.778483i $$-0.715989\pi$$
−0.627665 + 0.778483i $$0.715989\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 31.2257 1.40066
$$498$$ 0 0
$$499$$ 0.815792 0.0365199 0.0182599 0.999833i $$-0.494187\pi$$
0.0182599 + 0.999833i $$0.494187\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −26.8385 −1.19667 −0.598336 0.801245i $$-0.704171\pi$$
−0.598336 + 0.801245i $$0.704171\pi$$
$$504$$ 0 0
$$505$$ −9.77784 −0.435108
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.60348 0.292694 0.146347 0.989233i $$-0.453248\pi$$
0.146347 + 0.989233i $$0.453248\pi$$
$$510$$ 0 0
$$511$$ 32.0830 1.41927
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 11.2632 0.496314
$$516$$ 0 0
$$517$$ 10.4889 0.461300
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −35.9813 −1.57637 −0.788184 0.615440i $$-0.788978\pi$$
−0.788184 + 0.615440i $$0.788978\pi$$
$$522$$ 0 0
$$523$$ −34.9590 −1.52865 −0.764325 0.644831i $$-0.776928\pi$$
−0.764325 + 0.644831i $$0.776928\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.3684 0.625898
$$528$$ 0 0
$$529$$ 55.4514 2.41093
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.37778 −0.146308
$$534$$ 0 0
$$535$$ 9.54861 0.412822
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −73.2355 −3.15448
$$540$$ 0 0
$$541$$ 19.5111 0.838849 0.419425 0.907790i $$-0.362232\pi$$
0.419425 + 0.907790i $$0.362232\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −9.77784 −0.418837
$$546$$ 0 0
$$547$$ −4.26671 −0.182431 −0.0912156 0.995831i $$-0.529075\pi$$
−0.0912156 + 0.995831i $$0.529075\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.85728 0.377333
$$552$$ 0 0
$$553$$ −53.1437 −2.25990
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −5.86665 −0.248578 −0.124289 0.992246i $$-0.539665\pi$$
−0.124289 + 0.992246i $$0.539665\pi$$
$$558$$ 0 0
$$559$$ −7.61285 −0.321989
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 27.4924 1.15867 0.579333 0.815091i $$-0.303313\pi$$
0.579333 + 0.815091i $$0.303313\pi$$
$$564$$ 0 0
$$565$$ 9.00354 0.378782
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 13.4924 0.565631 0.282815 0.959174i $$-0.408732\pi$$
0.282815 + 0.959174i $$0.408732\pi$$
$$570$$ 0 0
$$571$$ 23.8796 0.999328 0.499664 0.866219i $$-0.333457\pi$$
0.499664 + 0.866219i $$0.333457\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −40.8573 −1.70387
$$576$$ 0 0
$$577$$ 32.4514 1.35097 0.675485 0.737374i $$-0.263935\pi$$
0.675485 + 0.737374i $$0.263935\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13.5111 0.560536
$$582$$ 0 0
$$583$$ 27.6128 1.14361
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 17.3363 0.715546 0.357773 0.933809i $$-0.383536\pi$$
0.357773 + 0.933809i $$0.383536\pi$$
$$588$$ 0 0
$$589$$ 31.8163 1.31097
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 18.8069 0.772307 0.386153 0.922435i $$-0.373804\pi$$
0.386153 + 0.922435i $$0.373804\pi$$
$$594$$ 0 0
$$595$$ 5.51114 0.225935
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −31.8163 −1.29998 −0.649989 0.759944i $$-0.725226\pi$$
−0.649989 + 0.759944i $$0.725226\pi$$
$$600$$ 0 0
$$601$$ −4.28544 −0.174807 −0.0874034 0.996173i $$-0.527857\pi$$
−0.0874034 + 0.996173i $$0.527857\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −14.1334 −0.574603
$$606$$ 0 0
$$607$$ 17.2444 0.699930 0.349965 0.936763i $$-0.386193\pi$$
0.349965 + 0.936763i $$0.386193\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.80642 −0.0730801
$$612$$ 0 0
$$613$$ 23.9813 0.968594 0.484297 0.874904i $$-0.339075\pi$$
0.484297 + 0.874904i $$0.339075\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.40006 −0.257657 −0.128828 0.991667i $$-0.541122\pi$$
−0.128828 + 0.991667i $$0.541122\pi$$
$$618$$ 0 0
$$619$$ −8.04149 −0.323215 −0.161607 0.986855i $$-0.551668\pi$$
−0.161607 + 0.986855i $$0.551668\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 8.26671 0.331199
$$624$$ 0 0
$$625$$ 19.3426 0.773704
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −1.51114 −0.0602530
$$630$$ 0 0
$$631$$ 23.1842 0.922949 0.461474 0.887154i $$-0.347321\pi$$
0.461474 + 0.887154i $$0.347321\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 12.9777 0.515005
$$636$$ 0 0
$$637$$ 12.6128 0.499739
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.51114 0.138682 0.0693408 0.997593i $$-0.477910\pi$$
0.0693408 + 0.997593i $$0.477910\pi$$
$$642$$ 0 0
$$643$$ 49.7560 1.96219 0.981093 0.193535i $$-0.0619952\pi$$
0.981093 + 0.193535i $$0.0619952\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.9403 0.980503 0.490251 0.871581i $$-0.336905\pi$$
0.490251 + 0.871581i $$0.336905\pi$$
$$648$$ 0 0
$$649$$ 64.1659 2.51873
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.7146 −0.458426 −0.229213 0.973376i $$-0.573615\pi$$
−0.229213 + 0.973376i $$0.573615\pi$$
$$654$$ 0 0
$$655$$ −6.45139 −0.252077
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.51114 0.0588656 0.0294328 0.999567i $$-0.490630\pi$$
0.0294328 + 0.999567i $$0.490630\pi$$
$$660$$ 0 0
$$661$$ −24.1847 −0.940675 −0.470338 0.882487i $$-0.655868\pi$$
−0.470338 + 0.882487i $$0.655868\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.2034 0.473228
$$666$$ 0 0
$$667$$ −17.7146 −0.685910
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 47.0420 1.81603
$$672$$ 0 0
$$673$$ −8.83854 −0.340701 −0.170350 0.985384i $$-0.554490\pi$$
−0.170350 + 0.985384i $$0.554490\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −39.2444 −1.50829 −0.754143 0.656710i $$-0.771947\pi$$
−0.754143 + 0.656710i $$0.771947\pi$$
$$678$$ 0 0
$$679$$ −3.34614 −0.128413
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 30.1561 1.15389 0.576946 0.816783i $$-0.304244\pi$$
0.576946 + 0.816783i $$0.304244\pi$$
$$684$$ 0 0
$$685$$ −12.3497 −0.471857
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −4.75557 −0.181173
$$690$$ 0 0
$$691$$ 21.5526 0.819900 0.409950 0.912108i $$-0.365546\pi$$
0.409950 + 0.912108i $$0.365546\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.20342 −0.159445
$$696$$ 0 0
$$697$$ 6.75557 0.255885
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0.755569 0.0285374 0.0142687 0.999898i $$-0.495458\pi$$
0.0142687 + 0.999898i $$0.495458\pi$$
$$702$$ 0 0
$$703$$ −3.34614 −0.126202
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 69.5941 2.61736
$$708$$ 0 0
$$709$$ −1.22570 −0.0460320 −0.0230160 0.999735i $$-0.507327\pi$$
−0.0230160 + 0.999735i $$0.507327\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −63.6325 −2.38306
$$714$$ 0 0
$$715$$ 3.61285 0.135113
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −11.3461 −0.423140 −0.211570 0.977363i $$-0.567858\pi$$
−0.211570 + 0.977363i $$0.567858\pi$$
$$720$$ 0 0
$$721$$ −80.1659 −2.98554
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 9.22570 0.342634
$$726$$ 0 0
$$727$$ −49.0607 −1.81956 −0.909780 0.415090i $$-0.863750\pi$$
−0.909780 + 0.415090i $$0.863750\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 15.2257 0.563143
$$732$$ 0 0
$$733$$ −53.4291 −1.97345 −0.986725 0.162402i $$-0.948076\pi$$
−0.986725 + 0.162402i $$0.948076\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −46.6923 −1.71993
$$738$$ 0 0
$$739$$ −21.0192 −0.773204 −0.386602 0.922247i $$-0.626351\pi$$
−0.386602 + 0.922247i $$0.626351\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4.56199 0.167363 0.0836816 0.996493i $$-0.473332\pi$$
0.0836816 + 0.996493i $$0.473332\pi$$
$$744$$ 0 0
$$745$$ −1.32741 −0.0486324
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −67.9625 −2.48330
$$750$$ 0 0
$$751$$ −21.8983 −0.799080 −0.399540 0.916716i $$-0.630830\pi$$
−0.399540 + 0.916716i $$0.630830\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.04101 −0.0378863
$$756$$ 0 0
$$757$$ −36.8385 −1.33892 −0.669460 0.742848i $$-0.733474\pi$$
−0.669460 + 0.742848i $$0.733474\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −38.8069 −1.40675 −0.703375 0.710819i $$-0.748324\pi$$
−0.703375 + 0.710819i $$0.748324\pi$$
$$762$$ 0 0
$$763$$ 69.5941 2.51948
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −11.0509 −0.399023
$$768$$ 0 0
$$769$$ 24.6923 0.890426 0.445213 0.895425i $$-0.353128\pi$$
0.445213 + 0.895425i $$0.353128\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −7.58120 −0.272677 −0.136338 0.990662i $$-0.543533\pi$$
−0.136338 + 0.990662i $$0.543533\pi$$
$$774$$ 0 0
$$775$$ 33.1397 1.19041
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 14.9590 0.535961
$$780$$ 0 0
$$781$$ −40.9403 −1.46496
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −9.41038 −0.335871
$$786$$ 0 0
$$787$$ −13.0192 −0.464085 −0.232042 0.972706i $$-0.574541\pi$$
−0.232042 + 0.972706i $$0.574541\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −64.0830 −2.27853
$$792$$ 0 0
$$793$$ −8.10171 −0.287700
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −23.2444 −0.823360 −0.411680 0.911328i $$-0.635058\pi$$
−0.411680 + 0.911328i $$0.635058\pi$$
$$798$$ 0 0
$$799$$ 3.61285 0.127813
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −42.0642 −1.48441
$$804$$ 0 0
$$805$$ −24.4068 −0.860228
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 36.9590 1.29941 0.649704 0.760187i $$-0.274893\pi$$
0.649704 + 0.760187i $$0.274893\pi$$
$$810$$ 0 0
$$811$$ 8.54909 0.300199 0.150099 0.988671i $$-0.452041\pi$$
0.150099 + 0.988671i $$0.452041\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 9.81532 0.343816
$$816$$ 0 0
$$817$$ 33.7146 1.17952
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.07007 −0.0722459 −0.0361229 0.999347i $$-0.511501\pi$$
−0.0361229 + 0.999347i $$0.511501\pi$$
$$822$$ 0 0
$$823$$ −40.5531 −1.41359 −0.706796 0.707417i $$-0.749860\pi$$
−0.706796 + 0.707417i $$0.749860\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.4987 −1.26918 −0.634592 0.772847i $$-0.718832\pi$$
−0.634592 + 0.772847i $$0.718832\pi$$
$$828$$ 0 0
$$829$$ 40.8385 1.41838 0.709191 0.705017i $$-0.249061\pi$$
0.709191 + 0.705017i $$0.249061\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −25.2257 −0.874019
$$834$$ 0 0
$$835$$ 4.55310 0.157567
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1.72345 0.0595001 0.0297500 0.999557i $$-0.490529\pi$$
0.0297500 + 0.999557i $$0.490529\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −0.622216 −0.0214049
$$846$$ 0 0
$$847$$ 100.595 3.45647
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6.69228 0.229409
$$852$$ 0 0
$$853$$ 29.4924 1.00980 0.504900 0.863178i $$-0.331529\pi$$
0.504900 + 0.863178i $$0.331529\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −8.01874 −0.273915 −0.136957 0.990577i $$-0.543732\pi$$
−0.136957 + 0.990577i $$0.543732\pi$$
$$858$$ 0 0
$$859$$ 55.4291 1.89122 0.945609 0.325307i $$-0.105468\pi$$
0.945609 + 0.325307i $$0.105468\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −25.6227 −0.872207 −0.436103 0.899897i $$-0.643642\pi$$
−0.436103 + 0.899897i $$0.643642\pi$$
$$864$$ 0 0
$$865$$ 4.67355 0.158905
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 69.6771 2.36363
$$870$$ 0 0
$$871$$ 8.04149 0.272475
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 26.4889 0.895487
$$876$$ 0 0
$$877$$ 28.1847 0.951729 0.475865 0.879519i $$-0.342135\pi$$
0.475865 + 0.879519i $$0.342135\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 9.52987 0.321070 0.160535 0.987030i $$-0.448678\pi$$
0.160535 + 0.987030i $$0.448678\pi$$
$$882$$ 0 0
$$883$$ 17.8350 0.600196 0.300098 0.953908i $$-0.402981\pi$$
0.300098 + 0.953908i $$0.402981\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −7.73329 −0.259659 −0.129829 0.991536i $$-0.541443\pi$$
−0.129829 + 0.991536i $$0.541443\pi$$
$$888$$ 0 0
$$889$$ −92.3694 −3.09797
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ 9.18115 0.306892
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 14.3684 0.479214
$$900$$ 0 0
$$901$$ 9.51114 0.316862
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −2.55215 −0.0848363
$$906$$ 0 0
$$907$$ −36.0830 −1.19812 −0.599058 0.800706i $$-0.704458\pi$$
−0.599058 + 0.800706i $$0.704458\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 58.9215 1.95216 0.976078 0.217418i $$-0.0697636\pi$$
0.976078 + 0.217418i $$0.0697636\pi$$
$$912$$ 0 0
$$913$$ −17.7146 −0.586266
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 45.9180 1.51635
$$918$$ 0 0
$$919$$ 12.5334 0.413439 0.206720 0.978400i $$-0.433721\pi$$
0.206720 + 0.978400i $$0.433721\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 7.05086 0.232082
$$924$$ 0 0
$$925$$ −3.48532 −0.114597
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.0291 1.47736 0.738678 0.674059i $$-0.235451\pi$$
0.738678 + 0.674059i $$0.235451\pi$$
$$930$$ 0 0
$$931$$ −55.8578 −1.83066
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −7.22570 −0.236306
$$936$$ 0 0
$$937$$ 0.876015 0.0286182 0.0143091 0.999898i $$-0.495445\pi$$
0.0143091 + 0.999898i $$0.495445\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −0.888922 −0.0289780 −0.0144890 0.999895i $$-0.504612\pi$$
−0.0144890 + 0.999895i $$0.504612\pi$$
$$942$$ 0 0
$$943$$ −29.9180 −0.974263
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −43.9911 −1.42952 −0.714759 0.699370i $$-0.753464\pi$$
−0.714759 + 0.699370i $$0.753464\pi$$
$$948$$ 0 0
$$949$$ 7.24443 0.235164
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 18.4701 0.598306 0.299153 0.954205i $$-0.403296\pi$$
0.299153 + 0.954205i $$0.403296\pi$$
$$954$$ 0 0
$$955$$ −0.165949 −0.00536998
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 87.8992 2.83841
$$960$$ 0 0
$$961$$ 20.6128 0.664931
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 16.3042 0.524850
$$966$$ 0 0
$$967$$ 50.2895 1.61720 0.808600 0.588359i $$-0.200226\pi$$
0.808600 + 0.588359i $$0.200226\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.2257 1.13045 0.565223 0.824938i $$-0.308790\pi$$
0.565223 + 0.824938i $$0.308790\pi$$
$$972$$ 0 0
$$973$$ 29.9180 0.959126
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 58.8069 1.88140 0.940700 0.339240i $$-0.110170\pi$$
0.940700 + 0.339240i $$0.110170\pi$$
$$978$$ 0 0
$$979$$ −10.8385 −0.346401
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −47.2168 −1.50598 −0.752991 0.658031i $$-0.771390\pi$$
−0.752991 + 0.658031i $$0.771390\pi$$
$$984$$ 0 0
$$985$$ 9.86082 0.314192
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −67.4291 −2.14412
$$990$$ 0 0
$$991$$ −35.4924 −1.12745 −0.563727 0.825961i $$-0.690633\pi$$
−0.563727 + 0.825961i $$0.690633\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −6.45139 −0.204523
$$996$$ 0 0
$$997$$ 49.4291 1.56544 0.782718 0.622377i $$-0.213833\pi$$
0.782718 + 0.622377i $$0.213833\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 3744.2.a.z.1.2 3
3.2 odd 2 1248.2.a.p.1.2 yes 3
4.3 odd 2 3744.2.a.ba.1.2 3
8.3 odd 2 7488.2.a.cx.1.2 3
8.5 even 2 7488.2.a.cy.1.2 3
12.11 even 2 1248.2.a.o.1.2 3
24.5 odd 2 2496.2.a.bk.1.2 3
24.11 even 2 2496.2.a.bl.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.2 3 12.11 even 2
1248.2.a.p.1.2 yes 3 3.2 odd 2
2496.2.a.bk.1.2 3 24.5 odd 2
2496.2.a.bl.1.2 3 24.11 even 2
3744.2.a.z.1.2 3 1.1 even 1 trivial
3744.2.a.ba.1.2 3 4.3 odd 2
7488.2.a.cx.1.2 3 8.3 odd 2
7488.2.a.cy.1.2 3 8.5 even 2