# Properties

 Label 784.2.a.m.1.1 Level $784$ Weight $2$ Character 784.1 Self dual yes Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} +O(q^{10})$$ $$q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} -4.00000 q^{11} -4.24264 q^{13} +4.00000 q^{15} -1.41421 q^{17} +2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +8.00000 q^{29} +11.3137 q^{33} -8.00000 q^{37} +12.0000 q^{39} +7.07107 q^{41} +4.00000 q^{43} -7.07107 q^{45} +5.65685 q^{47} +4.00000 q^{51} +10.0000 q^{53} +5.65685 q^{55} -8.00000 q^{57} +14.1421 q^{59} +7.07107 q^{61} +6.00000 q^{65} -11.3137 q^{69} +7.07107 q^{73} +8.48528 q^{75} -8.00000 q^{79} +1.00000 q^{81} -14.1421 q^{83} +2.00000 q^{85} -22.6274 q^{87} -7.07107 q^{89} -4.00000 q^{95} +1.41421 q^{97} -20.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{9}+O(q^{10})$$ 2 * q + 10 * q^9 $$2 q + 10 q^{9} - 8 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} + 16 q^{29} - 16 q^{37} + 24 q^{39} + 8 q^{43} + 8 q^{51} + 20 q^{53} - 16 q^{57} + 12 q^{65} - 16 q^{79} + 2 q^{81} + 4 q^{85} - 8 q^{95} - 40 q^{99}+O(q^{100})$$ 2 * q + 10 * q^9 - 8 * q^11 + 8 * q^15 + 8 * q^23 - 6 * q^25 + 16 * q^29 - 16 * q^37 + 24 * q^39 + 8 * q^43 + 8 * q^51 + 20 * q^53 - 16 * q^57 + 12 * q^65 - 16 * q^79 + 2 * q^81 + 4 * q^85 - 8 * q^95 - 40 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.82843 −1.63299 −0.816497 0.577350i $$-0.804087\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ −1.41421 −0.632456 −0.316228 0.948683i $$-0.602416\pi$$
−0.316228 + 0.948683i $$0.602416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 5.00000 1.66667
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −4.24264 −1.17670 −0.588348 0.808608i $$-0.700222\pi$$
−0.588348 + 0.808608i $$0.700222\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 0 0
$$17$$ −1.41421 −0.342997 −0.171499 0.985184i $$-0.554861\pi$$
−0.171499 + 0.985184i $$0.554861\pi$$
$$18$$ 0 0
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 11.3137 1.96946
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 12.0000 1.92154
$$40$$ 0 0
$$41$$ 7.07107 1.10432 0.552158 0.833740i $$-0.313805\pi$$
0.552158 + 0.833740i $$0.313805\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ −7.07107 −1.05409
$$46$$ 0 0
$$47$$ 5.65685 0.825137 0.412568 0.910927i $$-0.364632\pi$$
0.412568 + 0.910927i $$0.364632\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 5.65685 0.762770
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ 14.1421 1.84115 0.920575 0.390567i $$-0.127721\pi$$
0.920575 + 0.390567i $$0.127721\pi$$
$$60$$ 0 0
$$61$$ 7.07107 0.905357 0.452679 0.891674i $$-0.350468\pi$$
0.452679 + 0.891674i $$0.350468\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ −11.3137 −1.36201
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 7.07107 0.827606 0.413803 0.910366i $$-0.364200\pi$$
0.413803 + 0.910366i $$0.364200\pi$$
$$74$$ 0 0
$$75$$ 8.48528 0.979796
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −14.1421 −1.55230 −0.776151 0.630548i $$-0.782830\pi$$
−0.776151 + 0.630548i $$0.782830\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ −22.6274 −2.42591
$$88$$ 0 0
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 1.41421 0.143592 0.0717958 0.997419i $$-0.477127\pi$$
0.0717958 + 0.997419i $$0.477127\pi$$
$$98$$ 0 0
$$99$$ −20.0000 −2.01008
$$100$$ 0 0
$$101$$ 12.7279 1.26648 0.633238 0.773957i $$-0.281726\pi$$
0.633238 + 0.773957i $$0.281726\pi$$
$$102$$ 0 0
$$103$$ −11.3137 −1.11477 −0.557386 0.830253i $$-0.688196\pi$$
−0.557386 + 0.830253i $$0.688196\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ 0 0
$$111$$ 22.6274 2.14770
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −5.65685 −0.527504
$$116$$ 0 0
$$117$$ −21.2132 −1.96116
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −20.0000 −1.80334
$$124$$ 0 0
$$125$$ 11.3137 1.01193
$$126$$ 0 0
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 0 0
$$129$$ −11.3137 −0.996116
$$130$$ 0 0
$$131$$ −8.48528 −0.741362 −0.370681 0.928760i $$-0.620876\pi$$
−0.370681 + 0.928760i $$0.620876\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 8.00000 0.688530
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 2.82843 0.239904 0.119952 0.992780i $$-0.461726\pi$$
0.119952 + 0.992780i $$0.461726\pi$$
$$140$$ 0 0
$$141$$ −16.0000 −1.34744
$$142$$ 0 0
$$143$$ 16.9706 1.41915
$$144$$ 0 0
$$145$$ −11.3137 −0.939552
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ −7.07107 −0.571662
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −7.07107 −0.564333 −0.282166 0.959366i $$-0.591053\pi$$
−0.282166 + 0.959366i $$0.591053\pi$$
$$158$$ 0 0
$$159$$ −28.2843 −2.24309
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −16.0000 −1.24560
$$166$$ 0 0
$$167$$ −5.65685 −0.437741 −0.218870 0.975754i $$-0.570237\pi$$
−0.218870 + 0.975754i $$0.570237\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 14.1421 1.08148
$$172$$ 0 0
$$173$$ 4.24264 0.322562 0.161281 0.986909i $$-0.448437\pi$$
0.161281 + 0.986909i $$0.448437\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −40.0000 −3.00658
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 21.2132 1.57676 0.788382 0.615185i $$-0.210919\pi$$
0.788382 + 0.615185i $$0.210919\pi$$
$$182$$ 0 0
$$183$$ −20.0000 −1.47844
$$184$$ 0 0
$$185$$ 11.3137 0.831800
$$186$$ 0 0
$$187$$ 5.65685 0.413670
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 0 0
$$195$$ −16.9706 −1.21529
$$196$$ 0 0
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.0000 −0.698430
$$206$$ 0 0
$$207$$ 20.0000 1.39010
$$208$$ 0 0
$$209$$ −11.3137 −0.782586
$$210$$ 0 0
$$211$$ −24.0000 −1.65223 −0.826114 0.563503i $$-0.809453\pi$$
−0.826114 + 0.563503i $$0.809453\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.65685 −0.385794
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −20.0000 −1.35147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ 16.9706 1.13643 0.568216 0.822879i $$-0.307634\pi$$
0.568216 + 0.822879i $$0.307634\pi$$
$$224$$ 0 0
$$225$$ −15.0000 −1.00000
$$226$$ 0 0
$$227$$ 8.48528 0.563188 0.281594 0.959534i $$-0.409137\pi$$
0.281594 + 0.959534i $$0.409137\pi$$
$$228$$ 0 0
$$229$$ 21.2132 1.40181 0.700904 0.713256i $$-0.252780\pi$$
0.700904 + 0.713256i $$0.252780\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 0 0
$$237$$ 22.6274 1.46981
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −12.7279 −0.819878 −0.409939 0.912113i $$-0.634450\pi$$
−0.409939 + 0.912113i $$0.634450\pi$$
$$242$$ 0 0
$$243$$ 14.1421 0.907218
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 0 0
$$249$$ 40.0000 2.53490
$$250$$ 0 0
$$251$$ 19.7990 1.24970 0.624851 0.780744i $$-0.285160\pi$$
0.624851 + 0.780744i $$0.285160\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 0 0
$$255$$ −5.65685 −0.354246
$$256$$ 0 0
$$257$$ −21.2132 −1.32324 −0.661622 0.749838i $$-0.730131\pi$$
−0.661622 + 0.749838i $$0.730131\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 40.0000 2.47594
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ −14.1421 −0.868744
$$266$$ 0 0
$$267$$ 20.0000 1.22398
$$268$$ 0 0
$$269$$ 18.3848 1.12094 0.560470 0.828175i $$-0.310621\pi$$
0.560470 + 0.828175i $$0.310621\pi$$
$$270$$ 0 0
$$271$$ 28.2843 1.71815 0.859074 0.511852i $$-0.171040\pi$$
0.859074 + 0.511852i $$0.171040\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ −2.82843 −0.168133 −0.0840663 0.996460i $$-0.526791\pi$$
−0.0840663 + 0.996460i $$0.526791\pi$$
$$284$$ 0 0
$$285$$ 11.3137 0.670166
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ 0 0
$$293$$ −32.5269 −1.90024 −0.950121 0.311881i $$-0.899041\pi$$
−0.950121 + 0.311881i $$0.899041\pi$$
$$294$$ 0 0
$$295$$ −20.0000 −1.16445
$$296$$ 0 0
$$297$$ 22.6274 1.31298
$$298$$ 0 0
$$299$$ −16.9706 −0.981433
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −36.0000 −2.06815
$$304$$ 0 0
$$305$$ −10.0000 −0.572598
$$306$$ 0 0
$$307$$ −19.7990 −1.12999 −0.564994 0.825095i $$-0.691122\pi$$
−0.564994 + 0.825095i $$0.691122\pi$$
$$308$$ 0 0
$$309$$ 32.0000 1.82042
$$310$$ 0 0
$$311$$ −22.6274 −1.28308 −0.641542 0.767088i $$-0.721705\pi$$
−0.641542 + 0.767088i $$0.721705\pi$$
$$312$$ 0 0
$$313$$ −4.24264 −0.239808 −0.119904 0.992785i $$-0.538259\pi$$
−0.119904 + 0.992785i $$0.538259\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ −32.0000 −1.79166
$$320$$ 0 0
$$321$$ 22.6274 1.26294
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ 12.7279 0.706018
$$326$$ 0 0
$$327$$ 22.6274 1.25130
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ −40.0000 −2.19199
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ −16.9706 −0.921714
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 4.24264 0.227103 0.113552 0.993532i $$-0.463777\pi$$
0.113552 + 0.993532i $$0.463777\pi$$
$$350$$ 0 0
$$351$$ 24.0000 1.28103
$$352$$ 0 0
$$353$$ −9.89949 −0.526897 −0.263448 0.964673i $$-0.584860\pi$$
−0.263448 + 0.964673i $$0.584860\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ −14.1421 −0.742270
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ 5.65685 0.295285 0.147643 0.989041i $$-0.452831\pi$$
0.147643 + 0.989041i $$0.452831\pi$$
$$368$$ 0 0
$$369$$ 35.3553 1.84053
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ −32.0000 −1.65247
$$376$$ 0 0
$$377$$ −33.9411 −1.74806
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −56.5685 −2.89809
$$382$$ 0 0
$$383$$ −5.65685 −0.289052 −0.144526 0.989501i $$-0.546166\pi$$
−0.144526 + 0.989501i $$0.546166\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 20.0000 1.01666
$$388$$ 0 0
$$389$$ 8.00000 0.405616 0.202808 0.979219i $$-0.434993\pi$$
0.202808 + 0.979219i $$0.434993\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ 24.0000 1.21064
$$394$$ 0 0
$$395$$ 11.3137 0.569254
$$396$$ 0 0
$$397$$ −15.5563 −0.780751 −0.390375 0.920656i $$-0.627655\pi$$
−0.390375 + 0.920656i $$0.627655\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.41421 −0.0702728
$$406$$ 0 0
$$407$$ 32.0000 1.58618
$$408$$ 0 0
$$409$$ −38.1838 −1.88807 −0.944033 0.329851i $$-0.893001\pi$$
−0.944033 + 0.329851i $$0.893001\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 20.0000 0.981761
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 14.1421 0.690889 0.345444 0.938439i $$-0.387728\pi$$
0.345444 + 0.938439i $$0.387728\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 28.2843 1.37523
$$424$$ 0 0
$$425$$ 4.24264 0.205798
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −48.0000 −2.31746
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 21.2132 1.01944 0.509721 0.860340i $$-0.329749\pi$$
0.509721 + 0.860340i $$0.329749\pi$$
$$434$$ 0 0
$$435$$ 32.0000 1.53428
$$436$$ 0 0
$$437$$ 11.3137 0.541208
$$438$$ 0 0
$$439$$ −16.9706 −0.809961 −0.404980 0.914325i $$-0.632722\pi$$
−0.404980 + 0.914325i $$0.632722\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ −28.2843 −1.33780
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −28.2843 −1.33185
$$452$$ 0 0
$$453$$ −11.3137 −0.531564
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 30.0000 1.40334 0.701670 0.712502i $$-0.252438\pi$$
0.701670 + 0.712502i $$0.252438\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −7.07107 −0.329332 −0.164666 0.986349i $$-0.552655\pi$$
−0.164666 + 0.986349i $$0.552655\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −19.7990 −0.916188 −0.458094 0.888904i $$-0.651468\pi$$
−0.458094 + 0.888904i $$0.651468\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −8.48528 −0.389331
$$476$$ 0 0
$$477$$ 50.0000 2.28934
$$478$$ 0 0
$$479$$ −11.3137 −0.516937 −0.258468 0.966020i $$-0.583218\pi$$
−0.258468 + 0.966020i $$0.583218\pi$$
$$480$$ 0 0
$$481$$ 33.9411 1.54758
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ 11.3137 0.511624
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ −11.3137 −0.509544
$$494$$ 0 0
$$495$$ 28.2843 1.27128
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 0 0
$$503$$ −39.5980 −1.76559 −0.882793 0.469762i $$-0.844340\pi$$
−0.882793 + 0.469762i $$0.844340\pi$$
$$504$$ 0 0
$$505$$ −18.0000 −0.800989
$$506$$ 0 0
$$507$$ −14.1421 −0.628074
$$508$$ 0 0
$$509$$ 18.3848 0.814891 0.407445 0.913230i $$-0.366420\pi$$
0.407445 + 0.913230i $$0.366420\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −16.0000 −0.706417
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ −22.6274 −0.995153
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 41.0122 1.79678 0.898388 0.439202i $$-0.144739\pi$$
0.898388 + 0.439202i $$0.144739\pi$$
$$522$$ 0 0
$$523$$ 42.4264 1.85518 0.927589 0.373603i $$-0.121878\pi$$
0.927589 + 0.373603i $$0.121878\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 70.7107 3.06858
$$532$$ 0 0
$$533$$ −30.0000 −1.29944
$$534$$ 0 0
$$535$$ 11.3137 0.489134
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ 0 0
$$543$$ −60.0000 −2.57485
$$544$$ 0 0
$$545$$ 11.3137 0.484626
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 35.3553 1.50893
$$550$$ 0 0
$$551$$ 22.6274 0.963960
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −32.0000 −1.35832
$$556$$ 0 0
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ −16.9706 −0.717778
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ 14.1421 0.596020 0.298010 0.954563i $$-0.403677\pi$$
0.298010 + 0.954563i $$0.403677\pi$$
$$564$$ 0 0
$$565$$ −8.48528 −0.356978
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 40.0000 1.67689 0.838444 0.544988i $$-0.183466\pi$$
0.838444 + 0.544988i $$0.183466\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ −45.2548 −1.89055
$$574$$ 0 0
$$575$$ −12.0000 −0.500435
$$576$$ 0 0
$$577$$ −12.7279 −0.529870 −0.264935 0.964266i $$-0.585351\pi$$
−0.264935 + 0.964266i $$0.585351\pi$$
$$578$$ 0 0
$$579$$ 28.2843 1.17545
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ 0 0
$$585$$ 30.0000 1.24035
$$586$$ 0 0
$$587$$ 25.4558 1.05068 0.525338 0.850894i $$-0.323939\pi$$
0.525338 + 0.850894i $$0.323939\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 28.2843 1.16346
$$592$$ 0 0
$$593$$ 9.89949 0.406524 0.203262 0.979124i $$-0.434846\pi$$
0.203262 + 0.979124i $$0.434846\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ 29.6985 1.21143 0.605713 0.795683i $$-0.292888\pi$$
0.605713 + 0.795683i $$0.292888\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −7.07107 −0.287480
$$606$$ 0 0
$$607$$ −33.9411 −1.37763 −0.688814 0.724938i $$-0.741868\pi$$
−0.688814 + 0.724938i $$0.741868\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 24.0000 0.969351 0.484675 0.874694i $$-0.338938\pi$$
0.484675 + 0.874694i $$0.338938\pi$$
$$614$$ 0 0
$$615$$ 28.2843 1.14053
$$616$$ 0 0
$$617$$ −8.00000 −0.322068 −0.161034 0.986949i $$-0.551483\pi$$
−0.161034 + 0.986949i $$0.551483\pi$$
$$618$$ 0 0
$$619$$ −31.1127 −1.25052 −0.625262 0.780415i $$-0.715008\pi$$
−0.625262 + 0.780415i $$0.715008\pi$$
$$620$$ 0 0
$$621$$ −22.6274 −0.908007
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ 32.0000 1.27796
$$628$$ 0 0
$$629$$ 11.3137 0.451107
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 0 0
$$633$$ 67.8823 2.69808
$$634$$ 0 0
$$635$$ −28.2843 −1.12243
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.00000 0.315981 0.157991 0.987441i $$-0.449498\pi$$
0.157991 + 0.987441i $$0.449498\pi$$
$$642$$ 0 0
$$643$$ 2.82843 0.111542 0.0557711 0.998444i $$-0.482238\pi$$
0.0557711 + 0.998444i $$0.482238\pi$$
$$644$$ 0 0
$$645$$ 16.0000 0.629999
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −56.5685 −2.22051
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ 35.3553 1.37934
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 21.2132 0.825098 0.412549 0.910935i $$-0.364639\pi$$
0.412549 + 0.910935i $$0.364639\pi$$
$$662$$ 0 0
$$663$$ −16.9706 −0.659082
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 32.0000 1.23904
$$668$$ 0 0
$$669$$ −48.0000 −1.85579
$$670$$ 0 0
$$671$$ −28.2843 −1.09190
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 16.9706 0.653197
$$676$$ 0 0
$$677$$ −12.7279 −0.489174 −0.244587 0.969627i $$-0.578652\pi$$
−0.244587 + 0.969627i $$0.578652\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −60.0000 −2.28914
$$688$$ 0 0
$$689$$ −42.4264 −1.61632
$$690$$ 0 0
$$691$$ −42.4264 −1.61398 −0.806988 0.590567i $$-0.798904\pi$$
−0.806988 + 0.590567i $$0.798904\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ −10.0000 −0.378777
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 0 0
$$703$$ −22.6274 −0.853409
$$704$$ 0 0
$$705$$ 22.6274 0.852198
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ −40.0000 −1.50012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −24.0000 −0.897549
$$716$$ 0 0
$$717$$ −33.9411 −1.26755
$$718$$ 0 0
$$719$$ 39.5980 1.47676 0.738378 0.674387i $$-0.235592\pi$$
0.738378 + 0.674387i $$0.235592\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 36.0000 1.33885
$$724$$ 0 0
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ 28.2843 1.04901 0.524503 0.851409i $$-0.324251\pi$$
0.524503 + 0.851409i $$0.324251\pi$$
$$728$$ 0 0
$$729$$ −43.0000 −1.59259
$$730$$ 0 0
$$731$$ −5.65685 −0.209226
$$732$$ 0 0
$$733$$ 38.1838 1.41035 0.705175 0.709034i $$-0.250869\pi$$
0.705175 + 0.709034i $$0.250869\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 33.9411 1.24686
$$742$$ 0 0
$$743$$ 20.0000 0.733729 0.366864 0.930274i $$-0.380431\pi$$
0.366864 + 0.930274i $$0.380431\pi$$
$$744$$ 0 0
$$745$$ −14.1421 −0.518128
$$746$$ 0 0
$$747$$ −70.7107 −2.58717
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ −56.0000 −2.04075
$$754$$ 0 0
$$755$$ −5.65685 −0.205874
$$756$$ 0 0
$$757$$ −40.0000 −1.45382 −0.726912 0.686730i $$-0.759045\pi$$
−0.726912 + 0.686730i $$0.759045\pi$$
$$758$$ 0 0
$$759$$ 45.2548 1.64265
$$760$$ 0 0
$$761$$ −41.0122 −1.48669 −0.743345 0.668908i $$-0.766762\pi$$
−0.743345 + 0.668908i $$0.766762\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 10.0000 0.361551
$$766$$ 0 0
$$767$$ −60.0000 −2.16647
$$768$$ 0 0
$$769$$ −46.6690 −1.68293 −0.841464 0.540312i $$-0.818306\pi$$
−0.841464 + 0.540312i $$0.818306\pi$$
$$770$$ 0 0
$$771$$ 60.0000 2.16085
$$772$$ 0 0
$$773$$ −24.0416 −0.864717 −0.432359 0.901702i $$-0.642319\pi$$
−0.432359 + 0.901702i $$0.642319\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −45.2548 −1.61728
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 48.0833 1.71398 0.856992 0.515330i $$-0.172331\pi$$
0.856992 + 0.515330i $$0.172331\pi$$
$$788$$ 0 0
$$789$$ −67.8823 −2.41667
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −30.0000 −1.06533
$$794$$ 0 0
$$795$$ 40.0000 1.41865
$$796$$ 0 0
$$797$$ 29.6985 1.05197 0.525987 0.850493i $$-0.323696\pi$$
0.525987 + 0.850493i $$0.323696\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ −35.3553 −1.24922
$$802$$ 0 0
$$803$$ −28.2843 −0.998130
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −52.0000 −1.83049
$$808$$ 0 0
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ 8.48528 0.297959 0.148979 0.988840i $$-0.452401\pi$$
0.148979 + 0.988840i $$0.452401\pi$$
$$812$$ 0 0
$$813$$ −80.0000 −2.80572
$$814$$ 0 0
$$815$$ 5.65685 0.198151
$$816$$ 0 0
$$817$$ 11.3137 0.395817
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ −33.9411 −1.18168
$$826$$ 0 0
$$827$$ −48.0000 −1.66912 −0.834562 0.550914i $$-0.814279\pi$$
−0.834562 + 0.550914i $$0.814279\pi$$
$$828$$ 0 0
$$829$$ 7.07107 0.245588 0.122794 0.992432i $$-0.460815\pi$$
0.122794 + 0.992432i $$0.460815\pi$$
$$830$$ 0 0
$$831$$ −62.2254 −2.15858
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −11.3137 −0.390593 −0.195296 0.980744i $$-0.562567\pi$$
−0.195296 + 0.980744i $$0.562567\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ −45.2548 −1.55866
$$844$$ 0 0
$$845$$ −7.07107 −0.243252
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 0 0
$$853$$ −12.7279 −0.435796 −0.217898 0.975972i $$-0.569920\pi$$
−0.217898 + 0.975972i $$0.569920\pi$$
$$854$$ 0 0
$$855$$ −20.0000 −0.683986
$$856$$ 0 0
$$857$$ −7.07107 −0.241543 −0.120772 0.992680i $$-0.538537\pi$$
−0.120772 + 0.992680i $$0.538537\pi$$
$$858$$ 0 0
$$859$$ 14.1421 0.482523 0.241262 0.970460i $$-0.422439\pi$$
0.241262 + 0.970460i $$0.422439\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 40.0000 1.36162 0.680808 0.732462i $$-0.261629\pi$$
0.680808 + 0.732462i $$0.261629\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 42.4264 1.44088
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 7.07107 0.239319
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ 0 0
$$879$$ 92.0000 3.10308
$$880$$ 0 0
$$881$$ 21.2132 0.714691 0.357345 0.933972i $$-0.383682\pi$$
0.357345 + 0.933972i $$0.383682\pi$$
$$882$$ 0 0
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ 0 0
$$885$$ 56.5685 1.90153
$$886$$ 0 0
$$887$$ 28.2843 0.949693 0.474846 0.880069i $$-0.342504\pi$$
0.474846 + 0.880069i $$0.342504\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 48.0000 1.60267
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −14.1421 −0.471143
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −30.0000 −0.997234
$$906$$ 0 0
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 0 0
$$909$$ 63.6396 2.11079
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ 56.5685 1.87215
$$914$$ 0 0
$$915$$ 28.2843 0.935049
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 56.0000 1.84526
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 24.0000 0.789115
$$926$$ 0 0
$$927$$ −56.5685 −1.85795
$$928$$ 0 0
$$929$$ −35.3553 −1.15997 −0.579986 0.814627i $$-0.696942\pi$$
−0.579986 + 0.814627i $$0.696942\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 64.0000 2.09527
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −15.5563 −0.508204 −0.254102 0.967177i $$-0.581780\pi$$
−0.254102 + 0.967177i $$0.581780\pi$$
$$938$$ 0 0
$$939$$ 12.0000 0.391605
$$940$$ 0 0
$$941$$ 7.07107 0.230510 0.115255 0.993336i $$-0.463231\pi$$
0.115255 + 0.993336i $$0.463231\pi$$
$$942$$ 0 0
$$943$$ 28.2843 0.921063
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −60.0000 −1.94974 −0.974869 0.222779i $$-0.928487\pi$$
−0.974869 + 0.222779i $$0.928487\pi$$
$$948$$ 0 0
$$949$$ −30.0000 −0.973841
$$950$$ 0 0
$$951$$ 5.65685 0.183436
$$952$$ 0 0
$$953$$ 10.0000 0.323932 0.161966 0.986796i $$-0.448217\pi$$
0.161966 + 0.986796i $$0.448217\pi$$
$$954$$ 0 0
$$955$$ −22.6274 −0.732206
$$956$$ 0 0
$$957$$ 90.5097 2.92576
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −40.0000 −1.28898
$$964$$ 0 0
$$965$$ 14.1421 0.455251
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ 11.3137 0.363449
$$970$$ 0 0
$$971$$ 14.1421 0.453843 0.226921 0.973913i $$-0.427134\pi$$
0.226921 + 0.973913i $$0.427134\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −36.0000 −1.15292
$$976$$ 0 0
$$977$$ 48.0000 1.53566 0.767828 0.640656i $$-0.221338\pi$$
0.767828 + 0.640656i $$0.221338\pi$$
$$978$$ 0 0
$$979$$ 28.2843 0.903969
$$980$$ 0 0
$$981$$ −40.0000 −1.27710
$$982$$ 0 0
$$983$$ 45.2548 1.44341 0.721703 0.692203i $$-0.243360\pi$$
0.721703 + 0.692203i $$0.243360\pi$$
$$984$$ 0 0
$$985$$ 14.1421 0.450606
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −56.5685 −1.79515
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −1.41421 −0.0447886 −0.0223943 0.999749i $$-0.507129\pi$$
−0.0223943 + 0.999749i $$0.507129\pi$$
$$998$$ 0 0
$$999$$ 45.2548 1.43180
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 784.2.a.m.1.1 2
3.2 odd 2 7056.2.a.cr.1.2 2
4.3 odd 2 196.2.a.c.1.2 yes 2
7.2 even 3 784.2.i.l.753.2 4
7.3 odd 6 784.2.i.l.177.1 4
7.4 even 3 784.2.i.l.177.2 4
7.5 odd 6 784.2.i.l.753.1 4
7.6 odd 2 inner 784.2.a.m.1.2 2
8.3 odd 2 3136.2.a.br.1.1 2
8.5 even 2 3136.2.a.bs.1.2 2
12.11 even 2 1764.2.a.l.1.2 2
20.3 even 4 4900.2.e.p.2549.4 4
20.7 even 4 4900.2.e.p.2549.2 4
20.19 odd 2 4900.2.a.y.1.1 2
21.20 even 2 7056.2.a.cr.1.1 2
28.3 even 6 196.2.e.b.177.2 4
28.11 odd 6 196.2.e.b.177.1 4
28.19 even 6 196.2.e.b.165.2 4
28.23 odd 6 196.2.e.b.165.1 4
28.27 even 2 196.2.a.c.1.1 2
56.13 odd 2 3136.2.a.bs.1.1 2
56.27 even 2 3136.2.a.br.1.2 2
84.11 even 6 1764.2.k.l.1549.1 4
84.23 even 6 1764.2.k.l.361.1 4
84.47 odd 6 1764.2.k.l.361.2 4
84.59 odd 6 1764.2.k.l.1549.2 4
84.83 odd 2 1764.2.a.l.1.1 2
140.27 odd 4 4900.2.e.p.2549.3 4
140.83 odd 4 4900.2.e.p.2549.1 4
140.139 even 2 4900.2.a.y.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
196.2.a.c.1.1 2 28.27 even 2
196.2.a.c.1.2 yes 2 4.3 odd 2
196.2.e.b.165.1 4 28.23 odd 6
196.2.e.b.165.2 4 28.19 even 6
196.2.e.b.177.1 4 28.11 odd 6
196.2.e.b.177.2 4 28.3 even 6
784.2.a.m.1.1 2 1.1 even 1 trivial
784.2.a.m.1.2 2 7.6 odd 2 inner
784.2.i.l.177.1 4 7.3 odd 6
784.2.i.l.177.2 4 7.4 even 3
784.2.i.l.753.1 4 7.5 odd 6
784.2.i.l.753.2 4 7.2 even 3
1764.2.a.l.1.1 2 84.83 odd 2
1764.2.a.l.1.2 2 12.11 even 2
1764.2.k.l.361.1 4 84.23 even 6
1764.2.k.l.361.2 4 84.47 odd 6
1764.2.k.l.1549.1 4 84.11 even 6
1764.2.k.l.1549.2 4 84.59 odd 6
3136.2.a.br.1.1 2 8.3 odd 2
3136.2.a.br.1.2 2 56.27 even 2
3136.2.a.bs.1.1 2 56.13 odd 2
3136.2.a.bs.1.2 2 8.5 even 2
4900.2.a.y.1.1 2 20.19 odd 2
4900.2.a.y.1.2 2 140.139 even 2
4900.2.e.p.2549.1 4 140.83 odd 4
4900.2.e.p.2549.2 4 20.7 even 4
4900.2.e.p.2549.3 4 140.27 odd 4
4900.2.e.p.2549.4 4 20.3 even 4
7056.2.a.cr.1.1 2 21.20 even 2
7056.2.a.cr.1.2 2 3.2 odd 2