# Properties

 Label 768.5.b.c.127.2 Level $768$ Weight $5$ Character 768.127 Analytic conductor $79.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 768.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$79.3881316484$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.2 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 768.127 Dual form 768.5.b.c.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +O(q^{10})$$ $$q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +20.7846 q^{11} -182.000i q^{13} -218.238i q^{15} -246.000 q^{17} -117.779 q^{19} -396.000i q^{21} -748.246i q^{23} -1139.00 q^{25} -140.296 q^{27} +78.0000i q^{29} -1475.71i q^{31} -108.000 q^{33} -3200.83 q^{35} -530.000i q^{37} +945.700i q^{39} +918.000 q^{41} +852.169 q^{43} +1134.00i q^{45} -3782.80i q^{47} -3407.00 q^{49} +1278.25 q^{51} +4626.00i q^{53} +872.954i q^{55} +612.000 q^{57} +228.631 q^{59} +1346.00i q^{61} +2057.68i q^{63} +7644.00 q^{65} +1087.73 q^{67} +3888.00i q^{69} -1829.05i q^{71} +926.000 q^{73} +5918.42 q^{75} +1584.00i q^{77} +4399.41i q^{79} +729.000 q^{81} +11992.7 q^{83} -10332.0i q^{85} -405.300i q^{87} -11586.0 q^{89} +13870.3 q^{91} +7668.00i q^{93} -4946.74i q^{95} -13118.0 q^{97} +561.184 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 108 q^{9}+O(q^{10})$$ 4 * q + 108 * q^9 $$4 q + 108 q^{9} - 984 q^{17} - 4556 q^{25} - 432 q^{33} + 3672 q^{41} - 13628 q^{49} + 2448 q^{57} + 30576 q^{65} + 3704 q^{73} + 2916 q^{81} - 46344 q^{89} - 52472 q^{97}+O(q^{100})$$ 4 * q + 108 * q^9 - 984 * q^17 - 4556 * q^25 - 432 * q^33 + 3672 * q^41 - 13628 * q^49 + 2448 * q^57 + 30576 * q^65 + 3704 * q^73 + 2916 * q^81 - 46344 * q^89 - 52472 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −5.19615 −0.577350
$$4$$ 0 0
$$5$$ 42.0000i 1.68000i 0.542586 + 0.840000i $$0.317445\pi$$
−0.542586 + 0.840000i $$0.682555\pi$$
$$6$$ 0 0
$$7$$ 76.2102i 1.55531i 0.628691 + 0.777655i $$0.283591\pi$$
−0.628691 + 0.777655i $$0.716409\pi$$
$$8$$ 0 0
$$9$$ 27.0000 0.333333
$$10$$ 0 0
$$11$$ 20.7846 0.171774 0.0858868 0.996305i $$-0.472628\pi$$
0.0858868 + 0.996305i $$0.472628\pi$$
$$12$$ 0 0
$$13$$ − 182.000i − 1.07692i −0.842650 0.538462i $$-0.819006\pi$$
0.842650 0.538462i $$-0.180994\pi$$
$$14$$ 0 0
$$15$$ − 218.238i − 0.969948i
$$16$$ 0 0
$$17$$ −246.000 −0.851211 −0.425606 0.904909i $$-0.639939\pi$$
−0.425606 + 0.904909i $$0.639939\pi$$
$$18$$ 0 0
$$19$$ −117.779 −0.326259 −0.163129 0.986605i $$-0.552159\pi$$
−0.163129 + 0.986605i $$0.552159\pi$$
$$20$$ 0 0
$$21$$ − 396.000i − 0.897959i
$$22$$ 0 0
$$23$$ − 748.246i − 1.41445i −0.706987 0.707227i $$-0.749946\pi$$
0.706987 0.707227i $$-0.250054\pi$$
$$24$$ 0 0
$$25$$ −1139.00 −1.82240
$$26$$ 0 0
$$27$$ −140.296 −0.192450
$$28$$ 0 0
$$29$$ 78.0000i 0.0927467i 0.998924 + 0.0463734i $$0.0147664\pi$$
−0.998924 + 0.0463734i $$0.985234\pi$$
$$30$$ 0 0
$$31$$ − 1475.71i − 1.53560i −0.640692 0.767798i $$-0.721353\pi$$
0.640692 0.767798i $$-0.278647\pi$$
$$32$$ 0 0
$$33$$ −108.000 −0.0991736
$$34$$ 0 0
$$35$$ −3200.83 −2.61292
$$36$$ 0 0
$$37$$ − 530.000i − 0.387144i −0.981086 0.193572i $$-0.937993\pi$$
0.981086 0.193572i $$-0.0620073\pi$$
$$38$$ 0 0
$$39$$ 945.700i 0.621762i
$$40$$ 0 0
$$41$$ 918.000 0.546104 0.273052 0.961999i $$-0.411967\pi$$
0.273052 + 0.961999i $$0.411967\pi$$
$$42$$ 0 0
$$43$$ 852.169 0.460881 0.230441 0.973086i $$-0.425983\pi$$
0.230441 + 0.973086i $$0.425983\pi$$
$$44$$ 0 0
$$45$$ 1134.00i 0.560000i
$$46$$ 0 0
$$47$$ − 3782.80i − 1.71245i −0.516604 0.856224i $$-0.672804\pi$$
0.516604 0.856224i $$-0.327196\pi$$
$$48$$ 0 0
$$49$$ −3407.00 −1.41899
$$50$$ 0 0
$$51$$ 1278.25 0.491447
$$52$$ 0 0
$$53$$ 4626.00i 1.64685i 0.567426 + 0.823425i $$0.307939\pi$$
−0.567426 + 0.823425i $$0.692061\pi$$
$$54$$ 0 0
$$55$$ 872.954i 0.288580i
$$56$$ 0 0
$$57$$ 612.000 0.188366
$$58$$ 0 0
$$59$$ 228.631 0.0656796 0.0328398 0.999461i $$-0.489545\pi$$
0.0328398 + 0.999461i $$0.489545\pi$$
$$60$$ 0 0
$$61$$ 1346.00i 0.361731i 0.983508 + 0.180865i $$0.0578898\pi$$
−0.983508 + 0.180865i $$0.942110\pi$$
$$62$$ 0 0
$$63$$ 2057.68i 0.518437i
$$64$$ 0 0
$$65$$ 7644.00 1.80923
$$66$$ 0 0
$$67$$ 1087.73 0.242310 0.121155 0.992634i $$-0.461340\pi$$
0.121155 + 0.992634i $$0.461340\pi$$
$$68$$ 0 0
$$69$$ 3888.00i 0.816635i
$$70$$ 0 0
$$71$$ − 1829.05i − 0.362834i −0.983406 0.181417i $$-0.941932\pi$$
0.983406 0.181417i $$-0.0580684\pi$$
$$72$$ 0 0
$$73$$ 926.000 0.173766 0.0868831 0.996219i $$-0.472309\pi$$
0.0868831 + 0.996219i $$0.472309\pi$$
$$74$$ 0 0
$$75$$ 5918.42 1.05216
$$76$$ 0 0
$$77$$ 1584.00i 0.267161i
$$78$$ 0 0
$$79$$ 4399.41i 0.704921i 0.935827 + 0.352460i $$0.114655\pi$$
−0.935827 + 0.352460i $$0.885345\pi$$
$$80$$ 0 0
$$81$$ 729.000 0.111111
$$82$$ 0 0
$$83$$ 11992.7 1.74085 0.870425 0.492301i $$-0.163844\pi$$
0.870425 + 0.492301i $$0.163844\pi$$
$$84$$ 0 0
$$85$$ − 10332.0i − 1.43003i
$$86$$ 0 0
$$87$$ − 405.300i − 0.0535473i
$$88$$ 0 0
$$89$$ −11586.0 −1.46269 −0.731347 0.682005i $$-0.761108\pi$$
−0.731347 + 0.682005i $$0.761108\pi$$
$$90$$ 0 0
$$91$$ 13870.3 1.67495
$$92$$ 0 0
$$93$$ 7668.00i 0.886576i
$$94$$ 0 0
$$95$$ − 4946.74i − 0.548115i
$$96$$ 0 0
$$97$$ −13118.0 −1.39420 −0.697099 0.716975i $$-0.745526\pi$$
−0.697099 + 0.716975i $$0.745526\pi$$
$$98$$ 0 0
$$99$$ 561.184 0.0572579
$$100$$ 0 0
$$101$$ 5490.00i 0.538183i 0.963115 + 0.269091i $$0.0867233\pi$$
−0.963115 + 0.269091i $$0.913277\pi$$
$$102$$ 0 0
$$103$$ − 5701.91i − 0.537460i −0.963216 0.268730i $$-0.913396\pi$$
0.963216 0.268730i $$-0.0866039\pi$$
$$104$$ 0 0
$$105$$ 16632.0 1.50857
$$106$$ 0 0
$$107$$ −10080.5 −0.880473 −0.440237 0.897882i $$-0.645105\pi$$
−0.440237 + 0.897882i $$0.645105\pi$$
$$108$$ 0 0
$$109$$ − 16166.0i − 1.36066i −0.732906 0.680330i $$-0.761836\pi$$
0.732906 0.680330i $$-0.238164\pi$$
$$110$$ 0 0
$$111$$ 2753.96i 0.223518i
$$112$$ 0 0
$$113$$ 1842.00 0.144256 0.0721278 0.997395i $$-0.477021\pi$$
0.0721278 + 0.997395i $$0.477021\pi$$
$$114$$ 0 0
$$115$$ 31426.3 2.37628
$$116$$ 0 0
$$117$$ − 4914.00i − 0.358974i
$$118$$ 0 0
$$119$$ − 18747.7i − 1.32390i
$$120$$ 0 0
$$121$$ −14209.0 −0.970494
$$122$$ 0 0
$$123$$ −4770.07 −0.315293
$$124$$ 0 0
$$125$$ − 21588.0i − 1.38163i
$$126$$ 0 0
$$127$$ 394.908i 0.0244843i 0.999925 + 0.0122422i $$0.00389690\pi$$
−0.999925 + 0.0122422i $$0.996103\pi$$
$$128$$ 0 0
$$129$$ −4428.00 −0.266090
$$130$$ 0 0
$$131$$ −353.338 −0.0205896 −0.0102948 0.999947i $$-0.503277\pi$$
−0.0102948 + 0.999947i $$0.503277\pi$$
$$132$$ 0 0
$$133$$ − 8976.00i − 0.507434i
$$134$$ 0 0
$$135$$ − 5892.44i − 0.323316i
$$136$$ 0 0
$$137$$ 13254.0 0.706164 0.353082 0.935592i $$-0.385134\pi$$
0.353082 + 0.935592i $$0.385134\pi$$
$$138$$ 0 0
$$139$$ −13212.1 −0.683820 −0.341910 0.939733i $$-0.611074\pi$$
−0.341910 + 0.939733i $$0.611074\pi$$
$$140$$ 0 0
$$141$$ 19656.0i 0.988683i
$$142$$ 0 0
$$143$$ − 3782.80i − 0.184987i
$$144$$ 0 0
$$145$$ −3276.00 −0.155815
$$146$$ 0 0
$$147$$ 17703.3 0.819255
$$148$$ 0 0
$$149$$ − 438.000i − 0.0197288i −0.999951 0.00986442i $$-0.996860\pi$$
0.999951 0.00986442i $$-0.00313999\pi$$
$$150$$ 0 0
$$151$$ − 28052.3i − 1.23031i −0.788406 0.615155i $$-0.789093\pi$$
0.788406 0.615155i $$-0.210907\pi$$
$$152$$ 0 0
$$153$$ −6642.00 −0.283737
$$154$$ 0 0
$$155$$ 61979.7 2.57980
$$156$$ 0 0
$$157$$ 19346.0i 0.784859i 0.919782 + 0.392430i $$0.128365\pi$$
−0.919782 + 0.392430i $$0.871635\pi$$
$$158$$ 0 0
$$159$$ − 24037.4i − 0.950809i
$$160$$ 0 0
$$161$$ 57024.0 2.19992
$$162$$ 0 0
$$163$$ −36255.3 −1.36457 −0.682286 0.731086i $$-0.739014\pi$$
−0.682286 + 0.731086i $$0.739014\pi$$
$$164$$ 0 0
$$165$$ − 4536.00i − 0.166612i
$$166$$ 0 0
$$167$$ − 18747.7i − 0.672226i −0.941822 0.336113i $$-0.890888\pi$$
0.941822 0.336113i $$-0.109112\pi$$
$$168$$ 0 0
$$169$$ −4563.00 −0.159763
$$170$$ 0 0
$$171$$ −3180.05 −0.108753
$$172$$ 0 0
$$173$$ − 34410.0i − 1.14972i −0.818251 0.574861i $$-0.805056\pi$$
0.818251 0.574861i $$-0.194944\pi$$
$$174$$ 0 0
$$175$$ − 86803.5i − 2.83440i
$$176$$ 0 0
$$177$$ −1188.00 −0.0379201
$$178$$ 0 0
$$179$$ 16856.3 0.526086 0.263043 0.964784i $$-0.415274\pi$$
0.263043 + 0.964784i $$0.415274\pi$$
$$180$$ 0 0
$$181$$ − 15706.0i − 0.479411i −0.970846 0.239706i $$-0.922949\pi$$
0.970846 0.239706i $$-0.0770510\pi$$
$$182$$ 0 0
$$183$$ − 6994.02i − 0.208845i
$$184$$ 0 0
$$185$$ 22260.0 0.650402
$$186$$ 0 0
$$187$$ −5113.01 −0.146216
$$188$$ 0 0
$$189$$ − 10692.0i − 0.299320i
$$190$$ 0 0
$$191$$ − 2660.43i − 0.0729265i −0.999335 0.0364632i $$-0.988391\pi$$
0.999335 0.0364632i $$-0.0116092\pi$$
$$192$$ 0 0
$$193$$ −26782.0 −0.718999 −0.359500 0.933145i $$-0.617053\pi$$
−0.359500 + 0.933145i $$0.617053\pi$$
$$194$$ 0 0
$$195$$ −39719.4 −1.04456
$$196$$ 0 0
$$197$$ 52482.0i 1.35232i 0.736757 + 0.676158i $$0.236356\pi$$
−0.736757 + 0.676158i $$0.763644\pi$$
$$198$$ 0 0
$$199$$ − 23077.8i − 0.582759i −0.956608 0.291380i $$-0.905886\pi$$
0.956608 0.291380i $$-0.0941143\pi$$
$$200$$ 0 0
$$201$$ −5652.00 −0.139898
$$202$$ 0 0
$$203$$ −5944.40 −0.144250
$$204$$ 0 0
$$205$$ 38556.0i 0.917454i
$$206$$ 0 0
$$207$$ − 20202.6i − 0.471485i
$$208$$ 0 0
$$209$$ −2448.00 −0.0560427
$$210$$ 0 0
$$211$$ 23895.4 0.536721 0.268361 0.963319i $$-0.413518\pi$$
0.268361 + 0.963319i $$0.413518\pi$$
$$212$$ 0 0
$$213$$ 9504.00i 0.209482i
$$214$$ 0 0
$$215$$ 35791.1i 0.774280i
$$216$$ 0 0
$$217$$ 112464. 2.38833
$$218$$ 0 0
$$219$$ −4811.64 −0.100324
$$220$$ 0 0
$$221$$ 44772.0i 0.916689i
$$222$$ 0 0
$$223$$ − 852.169i − 0.0171363i −0.999963 0.00856813i $$-0.997273\pi$$
0.999963 0.00856813i $$-0.00272735\pi$$
$$224$$ 0 0
$$225$$ −30753.0 −0.607467
$$226$$ 0 0
$$227$$ 76175.6 1.47831 0.739153 0.673538i $$-0.235226\pi$$
0.739153 + 0.673538i $$0.235226\pi$$
$$228$$ 0 0
$$229$$ 48470.0i 0.924277i 0.886808 + 0.462138i $$0.152918\pi$$
−0.886808 + 0.462138i $$0.847082\pi$$
$$230$$ 0 0
$$231$$ − 8230.71i − 0.154246i
$$232$$ 0 0
$$233$$ −48738.0 −0.897751 −0.448875 0.893594i $$-0.648175\pi$$
−0.448875 + 0.893594i $$0.648175\pi$$
$$234$$ 0 0
$$235$$ 158878. 2.87691
$$236$$ 0 0
$$237$$ − 22860.0i − 0.406986i
$$238$$ 0 0
$$239$$ − 71000.2i − 1.24298i −0.783422 0.621490i $$-0.786528\pi$$
0.783422 0.621490i $$-0.213472\pi$$
$$240$$ 0 0
$$241$$ 73138.0 1.25924 0.629621 0.776903i $$-0.283210\pi$$
0.629621 + 0.776903i $$0.283210\pi$$
$$242$$ 0 0
$$243$$ −3788.00 −0.0641500
$$244$$ 0 0
$$245$$ − 143094.i − 2.38391i
$$246$$ 0 0
$$247$$ 21435.9i 0.351356i
$$248$$ 0 0
$$249$$ −62316.0 −1.00508
$$250$$ 0 0
$$251$$ 91888.8 1.45853 0.729264 0.684232i $$-0.239862\pi$$
0.729264 + 0.684232i $$0.239862\pi$$
$$252$$ 0 0
$$253$$ − 15552.0i − 0.242966i
$$254$$ 0 0
$$255$$ 53686.6i 0.825631i
$$256$$ 0 0
$$257$$ −48894.0 −0.740269 −0.370134 0.928978i $$-0.620688\pi$$
−0.370134 + 0.928978i $$0.620688\pi$$
$$258$$ 0 0
$$259$$ 40391.4 0.602129
$$260$$ 0 0
$$261$$ 2106.00i 0.0309156i
$$262$$ 0 0
$$263$$ 78191.7i 1.13044i 0.824939 + 0.565222i $$0.191210\pi$$
−0.824939 + 0.565222i $$0.808790\pi$$
$$264$$ 0 0
$$265$$ −194292. −2.76671
$$266$$ 0 0
$$267$$ 60202.6 0.844487
$$268$$ 0 0
$$269$$ − 71538.0i − 0.988626i −0.869284 0.494313i $$-0.835420\pi$$
0.869284 0.494313i $$-0.164580\pi$$
$$270$$ 0 0
$$271$$ 108198.i 1.47326i 0.676296 + 0.736630i $$0.263584\pi$$
−0.676296 + 0.736630i $$0.736416\pi$$
$$272$$ 0 0
$$273$$ −72072.0 −0.967033
$$274$$ 0 0
$$275$$ −23673.7 −0.313040
$$276$$ 0 0
$$277$$ 120518.i 1.57070i 0.619054 + 0.785348i $$0.287516\pi$$
−0.619054 + 0.785348i $$0.712484\pi$$
$$278$$ 0 0
$$279$$ − 39844.1i − 0.511865i
$$280$$ 0 0
$$281$$ 3054.00 0.0386773 0.0193387 0.999813i $$-0.493844\pi$$
0.0193387 + 0.999813i $$0.493844\pi$$
$$282$$ 0 0
$$283$$ −132959. −1.66014 −0.830071 0.557657i $$-0.811700\pi$$
−0.830071 + 0.557657i $$0.811700\pi$$
$$284$$ 0 0
$$285$$ 25704.0i 0.316454i
$$286$$ 0 0
$$287$$ 69961.0i 0.849361i
$$288$$ 0 0
$$289$$ −23005.0 −0.275440
$$290$$ 0 0
$$291$$ 68163.1 0.804940
$$292$$ 0 0
$$293$$ − 151662.i − 1.76661i −0.468795 0.883307i $$-0.655312\pi$$
0.468795 0.883307i $$-0.344688\pi$$
$$294$$ 0 0
$$295$$ 9602.49i 0.110342i
$$296$$ 0 0
$$297$$ −2916.00 −0.0330579
$$298$$ 0 0
$$299$$ −136181. −1.52326
$$300$$ 0 0
$$301$$ 64944.0i 0.716813i
$$302$$ 0 0
$$303$$ − 28526.9i − 0.310720i
$$304$$ 0 0
$$305$$ −56532.0 −0.607708
$$306$$ 0 0
$$307$$ −5424.78 −0.0575580 −0.0287790 0.999586i $$-0.509162\pi$$
−0.0287790 + 0.999586i $$0.509162\pi$$
$$308$$ 0 0
$$309$$ 29628.0i 0.310303i
$$310$$ 0 0
$$311$$ − 141127.i − 1.45912i −0.683917 0.729560i $$-0.739725\pi$$
0.683917 0.729560i $$-0.260275\pi$$
$$312$$ 0 0
$$313$$ 128686. 1.31354 0.656769 0.754092i $$-0.271923\pi$$
0.656769 + 0.754092i $$0.271923\pi$$
$$314$$ 0 0
$$315$$ −86422.4 −0.870974
$$316$$ 0 0
$$317$$ − 73986.0i − 0.736260i −0.929774 0.368130i $$-0.879998\pi$$
0.929774 0.368130i $$-0.120002\pi$$
$$318$$ 0 0
$$319$$ 1621.20i 0.0159314i
$$320$$ 0 0
$$321$$ 52380.0 0.508341
$$322$$ 0 0
$$323$$ 28973.7 0.277715
$$324$$ 0 0
$$325$$ 207298.i 1.96258i
$$326$$ 0 0
$$327$$ 84001.0i 0.785577i
$$328$$ 0 0
$$329$$ 288288. 2.66339
$$330$$ 0 0
$$331$$ −57026.0 −0.520496 −0.260248 0.965542i $$-0.583804\pi$$
−0.260248 + 0.965542i $$0.583804\pi$$
$$332$$ 0 0
$$333$$ − 14310.0i − 0.129048i
$$334$$ 0 0
$$335$$ 45684.6i 0.407080i
$$336$$ 0 0
$$337$$ 98674.0 0.868846 0.434423 0.900709i $$-0.356952\pi$$
0.434423 + 0.900709i $$0.356952\pi$$
$$338$$ 0 0
$$339$$ −9571.31 −0.0832860
$$340$$ 0 0
$$341$$ − 30672.0i − 0.263775i
$$342$$ 0 0
$$343$$ − 76667.5i − 0.651663i
$$344$$ 0 0
$$345$$ −163296. −1.37195
$$346$$ 0 0
$$347$$ −56929.0 −0.472797 −0.236399 0.971656i $$-0.575967\pi$$
−0.236399 + 0.971656i $$0.575967\pi$$
$$348$$ 0 0
$$349$$ 181346.i 1.48887i 0.667694 + 0.744436i $$0.267281\pi$$
−0.667694 + 0.744436i $$0.732719\pi$$
$$350$$ 0 0
$$351$$ 25533.9i 0.207254i
$$352$$ 0 0
$$353$$ −4302.00 −0.0345240 −0.0172620 0.999851i $$-0.505495\pi$$
−0.0172620 + 0.999851i $$0.505495\pi$$
$$354$$ 0 0
$$355$$ 76819.9 0.609561
$$356$$ 0 0
$$357$$ 97416.0i 0.764353i
$$358$$ 0 0
$$359$$ − 185232.i − 1.43724i −0.695405 0.718618i $$-0.744775\pi$$
0.695405 0.718618i $$-0.255225\pi$$
$$360$$ 0 0
$$361$$ −116449. −0.893555
$$362$$ 0 0
$$363$$ 73832.1 0.560315
$$364$$ 0 0
$$365$$ 38892.0i 0.291927i
$$366$$ 0 0
$$367$$ − 182690.i − 1.35638i −0.734885 0.678191i $$-0.762764\pi$$
0.734885 0.678191i $$-0.237236\pi$$
$$368$$ 0 0
$$369$$ 24786.0 0.182035
$$370$$ 0 0
$$371$$ −352549. −2.56136
$$372$$ 0 0
$$373$$ − 151778.i − 1.09092i −0.838138 0.545458i $$-0.816356\pi$$
0.838138 0.545458i $$-0.183644\pi$$
$$374$$ 0 0
$$375$$ 112175.i 0.797686i
$$376$$ 0 0
$$377$$ 14196.0 0.0998811
$$378$$ 0 0
$$379$$ −36005.9 −0.250666 −0.125333 0.992115i $$-0.540000\pi$$
−0.125333 + 0.992115i $$0.540000\pi$$
$$380$$ 0 0
$$381$$ − 2052.00i − 0.0141360i
$$382$$ 0 0
$$383$$ 65346.8i 0.445479i 0.974878 + 0.222739i $$0.0714999\pi$$
−0.974878 + 0.222739i $$0.928500\pi$$
$$384$$ 0 0
$$385$$ −66528.0 −0.448831
$$386$$ 0 0
$$387$$ 23008.6 0.153627
$$388$$ 0 0
$$389$$ − 105750.i − 0.698846i −0.936965 0.349423i $$-0.886378\pi$$
0.936965 0.349423i $$-0.113622\pi$$
$$390$$ 0 0
$$391$$ 184069.i 1.20400i
$$392$$ 0 0
$$393$$ 1836.00 0.0118874
$$394$$ 0 0
$$395$$ −184775. −1.18427
$$396$$ 0 0
$$397$$ − 27934.0i − 0.177236i −0.996066 0.0886180i $$-0.971755\pi$$
0.996066 0.0886180i $$-0.0282450\pi$$
$$398$$ 0 0
$$399$$ 46640.7i 0.292967i
$$400$$ 0 0
$$401$$ 237882. 1.47936 0.739678 0.672961i $$-0.234978\pi$$
0.739678 + 0.672961i $$0.234978\pi$$
$$402$$ 0 0
$$403$$ −268579. −1.65372
$$404$$ 0 0
$$405$$ 30618.0i 0.186667i
$$406$$ 0 0
$$407$$ − 11015.8i − 0.0665011i
$$408$$ 0 0
$$409$$ 20270.0 0.121173 0.0605867 0.998163i $$-0.480703\pi$$
0.0605867 + 0.998163i $$0.480703\pi$$
$$410$$ 0 0
$$411$$ −68869.8 −0.407704
$$412$$ 0 0
$$413$$ 17424.0i 0.102152i
$$414$$ 0 0
$$415$$ 503694.i 2.92463i
$$416$$ 0 0
$$417$$ 68652.0 0.394804
$$418$$ 0 0
$$419$$ −24089.4 −0.137214 −0.0686068 0.997644i $$-0.521855\pi$$
−0.0686068 + 0.997644i $$0.521855\pi$$
$$420$$ 0 0
$$421$$ − 116698.i − 0.658414i −0.944258 0.329207i $$-0.893219\pi$$
0.944258 0.329207i $$-0.106781\pi$$
$$422$$ 0 0
$$423$$ − 102136.i − 0.570816i
$$424$$ 0 0
$$425$$ 280194. 1.55125
$$426$$ 0 0
$$427$$ −102579. −0.562604
$$428$$ 0 0
$$429$$ 19656.0i 0.106802i
$$430$$ 0 0
$$431$$ − 355542.i − 1.91397i −0.290132 0.956986i $$-0.593699\pi$$
0.290132 0.956986i $$-0.406301\pi$$
$$432$$ 0 0
$$433$$ −199726. −1.06527 −0.532634 0.846346i $$-0.678798\pi$$
−0.532634 + 0.846346i $$0.678798\pi$$
$$434$$ 0 0
$$435$$ 17022.6 0.0899595
$$436$$ 0 0
$$437$$ 88128.0i 0.461478i
$$438$$ 0 0
$$439$$ 146469.i 0.760006i 0.924985 + 0.380003i $$0.124077\pi$$
−0.924985 + 0.380003i $$0.875923\pi$$
$$440$$ 0 0
$$441$$ −91989.0 −0.472997
$$442$$ 0 0
$$443$$ −50444.2 −0.257042 −0.128521 0.991707i $$-0.541023\pi$$
−0.128521 + 0.991707i $$0.541023\pi$$
$$444$$ 0 0
$$445$$ − 486612.i − 2.45733i
$$446$$ 0 0
$$447$$ 2275.91i 0.0113905i
$$448$$ 0 0
$$449$$ 149994. 0.744014 0.372007 0.928230i $$-0.378670\pi$$
0.372007 + 0.928230i $$0.378670\pi$$
$$450$$ 0 0
$$451$$ 19080.3 0.0938062
$$452$$ 0 0
$$453$$ 145764.i 0.710320i
$$454$$ 0 0
$$455$$ 582551.i 2.81392i
$$456$$ 0 0
$$457$$ −284338. −1.36145 −0.680726 0.732538i $$-0.738336\pi$$
−0.680726 + 0.732538i $$0.738336\pi$$
$$458$$ 0 0
$$459$$ 34512.8 0.163816
$$460$$ 0 0
$$461$$ − 183402.i − 0.862983i −0.902117 0.431491i $$-0.857987\pi$$
0.902117 0.431491i $$-0.142013\pi$$
$$462$$ 0 0
$$463$$ − 172422.i − 0.804324i −0.915568 0.402162i $$-0.868259\pi$$
0.915568 0.402162i $$-0.131741\pi$$
$$464$$ 0 0
$$465$$ −322056. −1.48945
$$466$$ 0 0
$$467$$ −68734.7 −0.315168 −0.157584 0.987506i $$-0.550371\pi$$
−0.157584 + 0.987506i $$0.550371\pi$$
$$468$$ 0 0
$$469$$ 82896.0i 0.376867i
$$470$$ 0 0
$$471$$ − 100525.i − 0.453139i
$$472$$ 0 0
$$473$$ 17712.0 0.0791672
$$474$$ 0 0
$$475$$ 134151. 0.594574
$$476$$ 0 0
$$477$$ 124902.i 0.548950i
$$478$$ 0 0
$$479$$ 249956.i 1.08941i 0.838627 + 0.544706i $$0.183359\pi$$
−0.838627 + 0.544706i $$0.816641\pi$$
$$480$$ 0 0
$$481$$ −96460.0 −0.416924
$$482$$ 0 0
$$483$$ −296305. −1.27012
$$484$$ 0 0
$$485$$ − 550956.i − 2.34225i
$$486$$ 0 0
$$487$$ 271108.i 1.14310i 0.820568 + 0.571549i $$0.193657\pi$$
−0.820568 + 0.571549i $$0.806343\pi$$
$$488$$ 0 0
$$489$$ 188388. 0.787835
$$490$$ 0 0
$$491$$ 227862. 0.945166 0.472583 0.881286i $$-0.343322\pi$$
0.472583 + 0.881286i $$0.343322\pi$$
$$492$$ 0 0
$$493$$ − 19188.0i − 0.0789470i
$$494$$ 0 0
$$495$$ 23569.7i 0.0961932i
$$496$$ 0 0
$$497$$ 139392. 0.564320
$$498$$ 0 0
$$499$$ −248854. −0.999410 −0.499705 0.866196i $$-0.666558\pi$$
−0.499705 + 0.866196i $$0.666558\pi$$
$$500$$ 0 0
$$501$$ 97416.0i 0.388110i
$$502$$ 0 0
$$503$$ 446537.i 1.76490i 0.470403 + 0.882452i $$0.344109\pi$$
−0.470403 + 0.882452i $$0.655891\pi$$
$$504$$ 0 0
$$505$$ −230580. −0.904147
$$506$$ 0 0
$$507$$ 23710.0 0.0922394
$$508$$ 0 0
$$509$$ − 39330.0i − 0.151806i −0.997115 0.0759029i $$-0.975816\pi$$
0.997115 0.0759029i $$-0.0241839\pi$$
$$510$$ 0 0
$$511$$ 70570.7i 0.270260i
$$512$$ 0 0
$$513$$ 16524.0 0.0627886
$$514$$ 0 0
$$515$$ 239480. 0.902933
$$516$$ 0 0
$$517$$ − 78624.0i − 0.294154i
$$518$$ 0 0
$$519$$ 178800.i 0.663792i
$$520$$ 0 0
$$521$$ 464598. 1.71160 0.855799 0.517308i $$-0.173066\pi$$
0.855799 + 0.517308i $$0.173066\pi$$
$$522$$ 0 0
$$523$$ 135509. 0.495409 0.247704 0.968836i $$-0.420324\pi$$
0.247704 + 0.968836i $$0.420324\pi$$
$$524$$ 0 0
$$525$$ 451044.i 1.63644i
$$526$$ 0 0
$$527$$ 363024.i 1.30712i
$$528$$ 0 0
$$529$$ −280031. −1.00068
$$530$$ 0 0
$$531$$ 6173.03 0.0218932
$$532$$ 0 0
$$533$$ − 167076.i − 0.588111i
$$534$$ 0 0
$$535$$ − 423382.i − 1.47919i
$$536$$ 0 0
$$537$$ −87588.0 −0.303736
$$538$$ 0 0
$$539$$ −70813.2 −0.243745
$$540$$ 0 0
$$541$$ 360442.i 1.23152i 0.787934 + 0.615759i $$0.211151\pi$$
−0.787934 + 0.615759i $$0.788849\pi$$
$$542$$ 0 0
$$543$$ 81610.8i 0.276788i
$$544$$ 0 0
$$545$$ 678972. 2.28591
$$546$$ 0 0
$$547$$ −261644. −0.874451 −0.437225 0.899352i $$-0.644039\pi$$
−0.437225 + 0.899352i $$0.644039\pi$$
$$548$$ 0 0
$$549$$ 36342.0i 0.120577i
$$550$$ 0 0
$$551$$ − 9186.80i − 0.0302594i
$$552$$ 0 0
$$553$$ −335280. −1.09637
$$554$$ 0 0
$$555$$ −115666. −0.375510
$$556$$ 0 0
$$557$$ − 233274.i − 0.751893i −0.926641 0.375946i $$-0.877318\pi$$
0.926641 0.375946i $$-0.122682\pi$$
$$558$$ 0 0
$$559$$ − 155095.i − 0.496333i
$$560$$ 0 0
$$561$$ 26568.0 0.0844176
$$562$$ 0 0
$$563$$ 419704. 1.32412 0.662058 0.749453i $$-0.269683\pi$$
0.662058 + 0.749453i $$0.269683\pi$$
$$564$$ 0 0
$$565$$ 77364.0i 0.242349i
$$566$$ 0 0
$$567$$ 55557.3i 0.172812i
$$568$$ 0 0
$$569$$ −470058. −1.45187 −0.725934 0.687765i $$-0.758592\pi$$
−0.725934 + 0.687765i $$0.758592\pi$$
$$570$$ 0 0
$$571$$ −320381. −0.982640 −0.491320 0.870979i $$-0.663485\pi$$
−0.491320 + 0.870979i $$0.663485\pi$$
$$572$$ 0 0
$$573$$ 13824.0i 0.0421041i
$$574$$ 0 0
$$575$$ 852252.i 2.57770i
$$576$$ 0 0
$$577$$ −341038. −1.02436 −0.512178 0.858879i $$-0.671161\pi$$
−0.512178 + 0.858879i $$0.671161\pi$$
$$578$$ 0 0
$$579$$ 139163. 0.415114
$$580$$ 0 0
$$581$$ 913968.i 2.70756i
$$582$$ 0 0
$$583$$ 96149.6i 0.282885i
$$584$$ 0 0
$$585$$ 206388. 0.603077
$$586$$ 0 0
$$587$$ 114128. 0.331220 0.165610 0.986191i $$-0.447041\pi$$
0.165610 + 0.986191i $$0.447041\pi$$
$$588$$ 0 0
$$589$$ 173808.i 0.501002i
$$590$$ 0 0
$$591$$ − 272704.i − 0.780760i
$$592$$ 0 0
$$593$$ −96846.0 −0.275405 −0.137703 0.990474i $$-0.543972\pi$$
−0.137703 + 0.990474i $$0.543972\pi$$
$$594$$ 0 0
$$595$$ 787404. 2.22415
$$596$$ 0 0
$$597$$ 119916.i 0.336456i
$$598$$ 0 0
$$599$$ − 519782.i − 1.44866i −0.689452 0.724331i $$-0.742149\pi$$
0.689452 0.724331i $$-0.257851\pi$$
$$600$$ 0 0
$$601$$ 627742. 1.73793 0.868965 0.494874i $$-0.164786\pi$$
0.868965 + 0.494874i $$0.164786\pi$$
$$602$$ 0 0
$$603$$ 29368.7 0.0807699
$$604$$ 0 0
$$605$$ − 596778.i − 1.63043i
$$606$$ 0 0
$$607$$ 133195.i 0.361501i 0.983529 + 0.180751i $$0.0578527\pi$$
−0.983529 + 0.180751i $$0.942147\pi$$
$$608$$ 0 0
$$609$$ 30888.0 0.0832828
$$610$$ 0 0
$$611$$ −688469. −1.84418
$$612$$ 0 0
$$613$$ − 247202.i − 0.657856i −0.944355 0.328928i $$-0.893313\pi$$
0.944355 0.328928i $$-0.106687\pi$$
$$614$$ 0 0
$$615$$ − 200343.i − 0.529692i
$$616$$ 0 0
$$617$$ 31758.0 0.0834224 0.0417112 0.999130i $$-0.486719\pi$$
0.0417112 + 0.999130i $$0.486719\pi$$
$$618$$ 0 0
$$619$$ −656094. −1.71232 −0.856160 0.516712i $$-0.827156\pi$$
−0.856160 + 0.516712i $$0.827156\pi$$
$$620$$ 0 0
$$621$$ 104976.i 0.272212i
$$622$$ 0 0
$$623$$ − 882972.i − 2.27494i
$$624$$ 0 0
$$625$$ 194821. 0.498742
$$626$$ 0 0
$$627$$ 12720.2 0.0323563
$$628$$ 0 0
$$629$$ 130380.i 0.329541i
$$630$$ 0 0
$$631$$ 417736.i 1.04916i 0.851360 + 0.524582i $$0.175778\pi$$
−0.851360 + 0.524582i $$0.824222\pi$$
$$632$$ 0 0
$$633$$ −124164. −0.309876
$$634$$ 0 0
$$635$$ −16586.1 −0.0411337
$$636$$ 0 0
$$637$$ 620074.i 1.52815i
$$638$$ 0 0
$$639$$ − 49384.2i − 0.120945i
$$640$$ 0 0
$$641$$ −152214. −0.370458 −0.185229 0.982695i $$-0.559303\pi$$
−0.185229 + 0.982695i $$0.559303\pi$$
$$642$$ 0 0
$$643$$ −714138. −1.72727 −0.863635 0.504117i $$-0.831818\pi$$
−0.863635 + 0.504117i $$0.831818\pi$$
$$644$$ 0 0
$$645$$ − 185976.i − 0.447031i
$$646$$ 0 0
$$647$$ − 259558.i − 0.620049i −0.950729 0.310025i $$-0.899663\pi$$
0.950729 0.310025i $$-0.100337\pi$$
$$648$$ 0 0
$$649$$ 4752.00 0.0112820
$$650$$ 0 0
$$651$$ −584380. −1.37890
$$652$$ 0 0
$$653$$ − 330714.i − 0.775579i −0.921748 0.387790i $$-0.873239\pi$$
0.921748 0.387790i $$-0.126761\pi$$
$$654$$ 0 0
$$655$$ − 14840.2i − 0.0345906i
$$656$$ 0 0
$$657$$ 25002.0 0.0579221
$$658$$ 0 0
$$659$$ 253884. 0.584608 0.292304 0.956326i $$-0.405578\pi$$
0.292304 + 0.956326i $$0.405578\pi$$
$$660$$ 0 0
$$661$$ 722158.i 1.65283i 0.563058 + 0.826417i $$0.309625\pi$$
−0.563058 + 0.826417i $$0.690375\pi$$
$$662$$ 0 0
$$663$$ − 232642.i − 0.529251i
$$664$$ 0 0
$$665$$ 376992. 0.852489
$$666$$ 0 0
$$667$$ 58363.2 0.131186
$$668$$ 0 0
$$669$$ 4428.00i 0.00989362i
$$670$$ 0 0
$$671$$ 27976.1i 0.0621358i
$$672$$ 0 0
$$673$$ −552910. −1.22074 −0.610372 0.792115i $$-0.708980\pi$$
−0.610372 + 0.792115i $$0.708980\pi$$
$$674$$ 0 0
$$675$$ 159797. 0.350721
$$676$$ 0 0
$$677$$ − 609030.i − 1.32881i −0.747375 0.664403i $$-0.768686\pi$$
0.747375 0.664403i $$-0.231314\pi$$
$$678$$ 0 0
$$679$$ − 999726.i − 2.16841i
$$680$$ 0 0
$$681$$ −395820. −0.853500
$$682$$ 0 0
$$683$$ 23715.2 0.0508377 0.0254189 0.999677i $$-0.491908\pi$$
0.0254189 + 0.999677i $$0.491908\pi$$
$$684$$ 0 0
$$685$$ 556668.i 1.18636i
$$686$$ 0 0
$$687$$ − 251858.i − 0.533631i
$$688$$ 0 0
$$689$$ 841932. 1.77353
$$690$$ 0 0
$$691$$ 431842. 0.904417 0.452208 0.891912i $$-0.350636\pi$$
0.452208 + 0.891912i $$0.350636\pi$$
$$692$$ 0 0
$$693$$ 42768.0i 0.0890538i
$$694$$ 0 0
$$695$$ − 554908.i − 1.14882i
$$696$$ 0 0
$$697$$ −225828. −0.464849
$$698$$ 0 0
$$699$$ 253250. 0.518317
$$700$$ 0 0
$$701$$ 44958.0i 0.0914894i 0.998953 + 0.0457447i $$0.0145661\pi$$
−0.998953 + 0.0457447i $$0.985434\pi$$
$$702$$ 0 0
$$703$$ 62423.1i 0.126309i
$$704$$ 0 0
$$705$$ −825552. −1.66099
$$706$$ 0 0
$$707$$ −418394. −0.837041
$$708$$ 0 0
$$709$$ − 533002.i − 1.06032i −0.847898 0.530159i $$-0.822132\pi$$
0.847898 0.530159i $$-0.177868\pi$$
$$710$$ 0 0
$$711$$ 118784.i 0.234974i
$$712$$ 0 0
$$713$$ −1.10419e6 −2.17203
$$714$$ 0 0
$$715$$ 158878. 0.310778
$$716$$ 0 0
$$717$$ 368928.i 0.717634i
$$718$$ 0 0
$$719$$ − 292107.i − 0.565046i −0.959260 0.282523i $$-0.908829\pi$$
0.959260 0.282523i $$-0.0911714\pi$$
$$720$$ 0 0
$$721$$ 434544. 0.835917
$$722$$ 0 0
$$723$$ −380036. −0.727023
$$724$$ 0 0
$$725$$ − 88842.0i − 0.169022i
$$726$$ 0 0
$$727$$ − 755791.i − 1.42999i −0.699130 0.714995i $$-0.746429\pi$$
0.699130 0.714995i $$-0.253571\pi$$
$$728$$ 0 0
$$729$$ 19683.0 0.0370370
$$730$$ 0 0
$$731$$ −209634. −0.392307
$$732$$ 0 0
$$733$$ − 832982.i − 1.55034i −0.631751 0.775171i $$-0.717664\pi$$
0.631751 0.775171i $$-0.282336\pi$$
$$734$$ 0 0
$$735$$ 743538.i 1.37635i
$$736$$ 0 0
$$737$$ 22608.0 0.0416224
$$738$$ 0 0
$$739$$ 698093. 1.27827 0.639137 0.769093i $$-0.279292\pi$$
0.639137 + 0.769093i $$0.279292\pi$$
$$740$$ 0 0
$$741$$ − 111384.i − 0.202855i
$$742$$ 0 0
$$743$$ 461044.i 0.835151i 0.908642 + 0.417575i $$0.137120\pi$$
−0.908642 + 0.417575i $$0.862880\pi$$
$$744$$ 0 0
$$745$$ 18396.0 0.0331445
$$746$$ 0 0
$$747$$ 323803. 0.580284
$$748$$ 0 0
$$749$$ − 768240.i − 1.36941i
$$750$$ 0 0
$$751$$ − 937060.i − 1.66145i −0.556682 0.830726i $$-0.687926\pi$$
0.556682 0.830726i $$-0.312074\pi$$
$$752$$ 0 0
$$753$$ −477468. −0.842082
$$754$$ 0 0
$$755$$ 1.17820e6 2.06692
$$756$$ 0 0
$$757$$ − 295786.i − 0.516162i −0.966123 0.258081i $$-0.916910\pi$$
0.966123 0.258081i $$-0.0830901\pi$$
$$758$$ 0 0
$$759$$ 80810.6i 0.140276i
$$760$$ 0 0
$$761$$ 1.02615e6 1.77191 0.885955 0.463772i $$-0.153504\pi$$
0.885955 + 0.463772i $$0.153504\pi$$
$$762$$ 0 0
$$763$$ 1.23201e6 2.11625
$$764$$ 0 0
$$765$$ − 278964.i − 0.476678i
$$766$$ 0 0
$$767$$ − 41610.8i − 0.0707319i
$$768$$ 0 0
$$769$$ 362306. 0.612665 0.306332 0.951925i $$-0.400898\pi$$
0.306332 + 0.951925i $$0.400898\pi$$
$$770$$ 0 0
$$771$$ 254061. 0.427394
$$772$$ 0 0
$$773$$ − 1.02608e6i − 1.71720i −0.512644 0.858601i $$-0.671334\pi$$
0.512644 0.858601i $$-0.328666\pi$$
$$774$$ 0 0
$$775$$ 1.68083e6i 2.79847i
$$776$$ 0 0
$$777$$ −209880. −0.347639
$$778$$ 0 0
$$779$$ −108122. −0.178171
$$780$$ 0 0
$$781$$ − 38016.0i − 0.0623253i
$$782$$ 0 0
$$783$$ − 10943.1i − 0.0178491i
$$784$$ 0 0
$$785$$ −812532. −1.31856
$$786$$ 0 0
$$787$$ −850042. −1.37243 −0.686216 0.727398i $$-0.740730\pi$$
−0.686216 + 0.727398i $$0.740730\pi$$
$$788$$ 0 0
$$789$$ − 406296.i − 0.652662i
$$790$$ 0 0
$$791$$ 140379.i 0.224362i
$$792$$ 0 0
$$793$$ 244972. 0.389556
$$794$$ 0 0
$$795$$ 1.00957e6 1.59736
$$796$$ 0 0
$$797$$ 761478.i 1.19878i 0.800456 + 0.599392i $$0.204591\pi$$
−0.800456 + 0.599392i $$0.795409\pi$$
$$798$$ 0 0
$$799$$ 930569.i 1.45766i
$$800$$ 0 0
$$801$$ −312822. −0.487565
$$802$$ 0 0
$$803$$ 19246.5 0.0298484
$$804$$ 0 0
$$805$$ 2.39501e6i 3.69586i
$$806$$ 0 0
$$807$$ 371722.i 0.570784i
$$808$$ 0 0
$$809$$ −247674. −0.378428 −0.189214 0.981936i $$-0.560594\pi$$
−0.189214 + 0.981936i $$0.560594\pi$$
$$810$$ 0 0
$$811$$ 920197. 1.39907 0.699534 0.714599i $$-0.253391\pi$$
0.699534 + 0.714599i $$0.253391\pi$$
$$812$$ 0 0
$$813$$ − 562212.i − 0.850588i
$$814$$ 0 0
$$815$$ − 1.52272e6i − 2.29248i
$$816$$ 0 0
$$817$$ −100368. −0.150367
$$818$$ 0 0
$$819$$ 374497. 0.558317
$$820$$ 0 0
$$821$$ 250242.i 0.371256i 0.982620 + 0.185628i $$0.0594320\pi$$
−0.982620 + 0.185628i $$0.940568\pi$$
$$822$$ 0 0
$$823$$ − 400762.i − 0.591680i −0.955238 0.295840i $$-0.904401\pi$$
0.955238 0.295840i $$-0.0955995\pi$$
$$824$$ 0 0
$$825$$ 123012. 0.180734
$$826$$ 0 0
$$827$$ 17272.0 0.0252541 0.0126270 0.999920i $$-0.495981\pi$$
0.0126270 + 0.999920i $$0.495981\pi$$
$$828$$ 0 0
$$829$$ − 15686.0i − 0.0228246i −0.999935 0.0114123i $$-0.996367\pi$$
0.999935 0.0114123i $$-0.00363273\pi$$
$$830$$ 0 0
$$831$$ − 626230.i − 0.906842i
$$832$$ 0 0
$$833$$ 838122. 1.20786
$$834$$ 0 0
$$835$$ 787404. 1.12934
$$836$$ 0 0
$$837$$ 207036.i 0.295525i
$$838$$ 0 0
$$839$$ − 115479.i − 0.164051i −0.996630 0.0820257i $$-0.973861\pi$$
0.996630 0.0820257i $$-0.0261390\pi$$
$$840$$ 0 0
$$841$$ 701197. 0.991398
$$842$$ 0 0
$$843$$ −15869.0 −0.0223304
$$844$$ 0 0
$$845$$ − 191646.i − 0.268402i
$$846$$ 0 0
$$847$$ − 1.08287e6i − 1.50942i
$$848$$ 0 0
$$849$$ 690876. 0.958484
$$850$$ 0 0
$$851$$ −396570. −0.547597
$$852$$ 0 0
$$853$$ − 345938.i − 0.475445i −0.971333 0.237722i $$-0.923599\pi$$
0.971333 0.237722i $$-0.0764009\pi$$
$$854$$ 0 0
$$855$$ − 133562.i − 0.182705i
$$856$$ 0 0
$$857$$ 267990. 0.364886 0.182443 0.983216i $$-0.441600\pi$$
0.182443 + 0.983216i $$0.441600\pi$$
$$858$$ 0 0
$$859$$ 522407. 0.707983 0.353992 0.935249i $$-0.384824\pi$$
0.353992 + 0.935249i $$0.384824\pi$$
$$860$$ 0 0
$$861$$ − 363528.i − 0.490379i
$$862$$ 0 0
$$863$$ − 826895.i − 1.11027i −0.831760 0.555135i $$-0.812667\pi$$
0.831760 0.555135i $$-0.187333\pi$$
$$864$$ 0 0
$$865$$ 1.44522e6 1.93153
$$866$$ 0 0
$$867$$ 119537. 0.159025
$$868$$ 0 0
$$869$$ 91440.0i 0.121087i
$$870$$ 0 0
$$871$$ − 197966.i − 0.260949i
$$872$$ 0 0
$$873$$ −354186. −0.464732
$$874$$ 0 0
$$875$$ 1.64523e6 2.14887
$$876$$ 0 0
$$877$$ 1.11629e6i 1.45137i 0.688028 + 0.725685i $$0.258477\pi$$
−0.688028 + 0.725685i $$0.741523\pi$$
$$878$$ 0 0
$$879$$ 788059.i 1.01995i
$$880$$ 0 0
$$881$$ 19170.0 0.0246985 0.0123492 0.999924i $$-0.496069\pi$$
0.0123492 + 0.999924i $$0.496069\pi$$
$$882$$ 0 0
$$883$$ 568909. 0.729662 0.364831 0.931074i $$-0.381127\pi$$
0.364831 + 0.931074i $$0.381127\pi$$
$$884$$ 0 0
$$885$$ − 49896.0i − 0.0637058i
$$886$$ 0 0
$$887$$ − 1.09015e6i − 1.38561i −0.721126 0.692804i $$-0.756375\pi$$
0.721126 0.692804i $$-0.243625\pi$$
$$888$$ 0 0
$$889$$ −30096.0 −0.0380807
$$890$$ 0 0
$$891$$ 15152.0 0.0190860
$$892$$ 0 0
$$893$$ 445536.i 0.558702i
$$894$$ 0 0
$$895$$ 707965.i 0.883824i
$$896$$ 0 0
$$897$$ 707616. 0.879453
$$898$$ 0 0
$$899$$ 115105. 0.142421
$$900$$ 0 0
$$901$$ − 1.13800e6i − 1.40182i
$$902$$ 0 0
$$903$$ − 337459.i − 0.413852i
$$904$$ 0 0
$$905$$ 659652. 0.805411
$$906$$ 0 0
$$907$$ −916193. −1.11371 −0.556855 0.830610i $$-0.687992\pi$$
−0.556855 + 0.830610i $$0.687992\pi$$
$$908$$ 0 0
$$909$$ 148230.i 0.179394i
$$910$$ 0 0
$$911$$ 995500.i 1.19951i 0.800183 + 0.599756i $$0.204736\pi$$
−0.800183 + 0.599756i $$0.795264\pi$$
$$912$$ 0 0
$$913$$ 249264. 0.299032
$$914$$ 0 0
$$915$$ 293749. 0.350860
$$916$$ 0 0
$$917$$ − 26928.0i − 0.0320233i
$$918$$ 0 0
$$919$$ − 97084.9i − 0.114953i −0.998347 0.0574766i $$-0.981695\pi$$
0.998347 0.0574766i $$-0.0183054\pi$$
$$920$$ 0 0
$$921$$ 28188.0 0.0332311
$$922$$ 0 0
$$923$$ −332886. −0.390744
$$924$$ 0 0
$$925$$ 603670.i 0.705531i
$$926$$ 0 0
$$927$$ − 153952.i − 0.179153i
$$928$$ 0 0
$$929$$ −1.27882e6 −1.48176 −0.740881 0.671636i $$-0.765592\pi$$
−0.740881 + 0.671636i $$0.765592\pi$$
$$930$$ 0 0
$$931$$ 401275. 0.462959
$$932$$ 0 0
$$933$$ 733320.i 0.842423i
$$934$$ 0 0
$$935$$ − 214747.i − 0.245642i
$$936$$ 0 0
$$937$$ 981262. 1.11765 0.558825 0.829286i $$-0.311252\pi$$
0.558825 + 0.829286i $$0.311252\pi$$
$$938$$ 0 0
$$939$$ −668672. −0.758371
$$940$$ 0 0
$$941$$ 284406.i 0.321188i 0.987021 + 0.160594i $$0.0513410\pi$$
−0.987021 + 0.160594i $$0.948659\pi$$
$$942$$ 0 0
$$943$$ − 686890.i − 0.772438i
$$944$$ 0 0
$$945$$ 449064. 0.502857
$$946$$ 0 0
$$947$$ −993109. −1.10738 −0.553691 0.832722i $$-0.686781\pi$$
−0.553691 + 0.832722i $$0.686781\pi$$
$$948$$ 0 0
$$949$$ − 168532.i − 0.187133i
$$950$$ 0 0
$$951$$ 384443.i 0.425080i
$$952$$ 0 0
$$953$$ −602922. −0.663858 −0.331929 0.943304i $$-0.607699\pi$$
−0.331929 + 0.943304i $$0.607699\pi$$
$$954$$ 0 0
$$955$$ 111738. 0.122516
$$956$$ 0 0
$$957$$ − 8424.00i − 0.00919802i
$$958$$ 0 0
$$959$$ 1.01009e6i 1.09831i
$$960$$ 0 0
$$961$$ −1.25419e6 −1.35805
$$962$$ 0 0
$$963$$ −272174. −0.293491
$$964$$ 0 0
$$965$$ − 1.12484e6i − 1.20792i
$$966$$ 0 0
$$967$$ − 575810.i − 0.615781i −0.951422 0.307890i $$-0.900377\pi$$
0.951422 0.307890i $$-0.0996230\pi$$
$$968$$ 0 0
$$969$$ −150552. −0.160339
$$970$$ 0 0
$$971$$ 1.23920e6 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$972$$ 0 0
$$973$$ − 1.00690e6i − 1.06355i
$$974$$ 0 0
$$975$$ − 1.07715e6i − 1.13310i
$$976$$ 0 0
$$977$$ −1.04074e6 −1.09032 −0.545160 0.838332i $$-0.683531\pi$$
−0.545160 + 0.838332i $$0.683531\pi$$
$$978$$ 0 0
$$979$$ −240810. −0.251252
$$980$$ 0 0
$$981$$ − 436482.i − 0.453553i
$$982$$ 0 0
$$983$$ 948734.i 0.981833i 0.871207 + 0.490916i $$0.163338\pi$$
−0.871207 + 0.490916i $$0.836662\pi$$
$$984$$ 0 0
$$985$$ −2.20424e6 −2.27189
$$986$$ 0 0
$$987$$ −1.49799e6 −1.53771
$$988$$ 0 0
$$989$$ − 637632.i − 0.651895i
$$990$$ 0 0
$$991$$ − 616007.i − 0.627247i −0.949547 0.313623i $$-0.898457\pi$$
0.949547 0.313623i $$-0.101543\pi$$
$$992$$ 0 0
$$993$$ 296316. 0.300508
$$994$$ 0 0
$$995$$ 969269. 0.979035
$$996$$ 0 0
$$997$$ 535870.i 0.539100i 0.962986 + 0.269550i $$0.0868749\pi$$
−0.962986 + 0.269550i $$0.913125\pi$$
$$998$$ 0 0
$$999$$ 74356.9i 0.0745059i
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 768.5.b.c.127.2 4
4.3 odd 2 inner 768.5.b.c.127.4 4
8.3 odd 2 inner 768.5.b.c.127.1 4
8.5 even 2 inner 768.5.b.c.127.3 4
16.3 odd 4 192.5.g.b.127.2 2
16.5 even 4 48.5.g.a.31.2 yes 2
16.11 odd 4 48.5.g.a.31.1 2
16.13 even 4 192.5.g.b.127.1 2
48.5 odd 4 144.5.g.f.127.1 2
48.11 even 4 144.5.g.f.127.2 2
48.29 odd 4 576.5.g.d.127.1 2
48.35 even 4 576.5.g.d.127.2 2
80.27 even 4 1200.5.j.b.799.1 4
80.37 odd 4 1200.5.j.b.799.4 4
80.43 even 4 1200.5.j.b.799.3 4
80.53 odd 4 1200.5.j.b.799.2 4
80.59 odd 4 1200.5.e.b.751.2 2
80.69 even 4 1200.5.e.b.751.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 16.11 odd 4
48.5.g.a.31.2 yes 2 16.5 even 4
144.5.g.f.127.1 2 48.5 odd 4
144.5.g.f.127.2 2 48.11 even 4
192.5.g.b.127.1 2 16.13 even 4
192.5.g.b.127.2 2 16.3 odd 4
576.5.g.d.127.1 2 48.29 odd 4
576.5.g.d.127.2 2 48.35 even 4
768.5.b.c.127.1 4 8.3 odd 2 inner
768.5.b.c.127.2 4 1.1 even 1 trivial
768.5.b.c.127.3 4 8.5 even 2 inner
768.5.b.c.127.4 4 4.3 odd 2 inner
1200.5.e.b.751.1 2 80.69 even 4
1200.5.e.b.751.2 2 80.59 odd 4
1200.5.j.b.799.1 4 80.27 even 4
1200.5.j.b.799.2 4 80.53 odd 4
1200.5.j.b.799.3 4 80.43 even 4
1200.5.j.b.799.4 4 80.37 odd 4