Properties

 Label 48.5.g.a.31.2 Level $48$ Weight $5$ Character 48.31 Analytic conductor $4.962$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,5,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 48.g (of order $$2$$, degree $$1$$, 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: $$4.96175822802$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 31.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 48.31 Dual form 48.5.g.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.19615i q^{3} -42.0000 q^{5} -76.2102i q^{7} -27.0000 q^{9} +O(q^{10})$$ $$q+5.19615i q^{3} -42.0000 q^{5} -76.2102i q^{7} -27.0000 q^{9} +20.7846i q^{11} -182.000 q^{13} -218.238i q^{15} -246.000 q^{17} +117.779i q^{19} +396.000 q^{21} +748.246i q^{23} +1139.00 q^{25} -140.296i q^{27} +78.0000 q^{29} -1475.71i q^{31} -108.000 q^{33} +3200.83i q^{35} +530.000 q^{37} -945.700i q^{39} -918.000 q^{41} +852.169i q^{43} +1134.00 q^{45} -3782.80i q^{47} -3407.00 q^{49} -1278.25i q^{51} -4626.00 q^{53} -872.954i q^{55} -612.000 q^{57} +228.631i q^{59} +1346.00 q^{61} +2057.68i q^{63} +7644.00 q^{65} -1087.73i q^{67} -3888.00 q^{69} +1829.05i q^{71} -926.000 q^{73} +5918.42i q^{75} +1584.00 q^{77} +4399.41i q^{79} +729.000 q^{81} -11992.7i q^{83} +10332.0 q^{85} +405.300i q^{87} +11586.0 q^{89} +13870.3i q^{91} +7668.00 q^{93} -4946.74i q^{95} -13118.0 q^{97} -561.184i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 84 q^{5} - 54 q^{9}+O(q^{10})$$ 2 * q - 84 * q^5 - 54 * q^9 $$2 q - 84 q^{5} - 54 q^{9} - 364 q^{13} - 492 q^{17} + 792 q^{21} + 2278 q^{25} + 156 q^{29} - 216 q^{33} + 1060 q^{37} - 1836 q^{41} + 2268 q^{45} - 6814 q^{49} - 9252 q^{53} - 1224 q^{57} + 2692 q^{61} + 15288 q^{65} - 7776 q^{69} - 1852 q^{73} + 3168 q^{77} + 1458 q^{81} + 20664 q^{85} + 23172 q^{89} + 15336 q^{93} - 26236 q^{97}+O(q^{100})$$ 2 * q - 84 * q^5 - 54 * q^9 - 364 * q^13 - 492 * q^17 + 792 * q^21 + 2278 * q^25 + 156 * q^29 - 216 * q^33 + 1060 * q^37 - 1836 * q^41 + 2268 * q^45 - 6814 * q^49 - 9252 * q^53 - 1224 * q^57 + 2692 * q^61 + 15288 * q^65 - 7776 * q^69 - 1852 * q^73 + 3168 * q^77 + 1458 * q^81 + 20664 * q^85 + 23172 * q^89 + 15336 * q^93 - 26236 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$31$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$3$$ 5.19615i 0.577350i
$$4$$ 0 0
$$5$$ −42.0000 −1.68000 −0.840000 0.542586i $$-0.817445\pi$$
−0.840000 + 0.542586i $$0.817445\pi$$
$$6$$ 0 0
$$7$$ − 76.2102i − 1.55531i −0.628691 0.777655i $$-0.716409\pi$$
0.628691 0.777655i $$-0.283591\pi$$
$$8$$ 0 0
$$9$$ −27.0000 −0.333333
$$10$$ 0 0
$$11$$ 20.7846i 0.171774i 0.996305 + 0.0858868i $$0.0273723\pi$$
−0.996305 + 0.0858868i $$0.972628\pi$$
$$12$$ 0 0
$$13$$ −182.000 −1.07692 −0.538462 0.842650i $$-0.680994\pi$$
−0.538462 + 0.842650i $$0.680994\pi$$
$$14$$ 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
$$19$$ 117.779i 0.326259i 0.986605 + 0.163129i $$0.0521588\pi$$
−0.986605 + 0.163129i $$0.947841\pi$$
$$20$$ 0 0
$$21$$ 396.000 0.897959
$$22$$ 0 0
$$23$$ 748.246i 1.41445i 0.706987 + 0.707227i $$0.250054\pi$$
−0.706987 + 0.707227i $$0.749946\pi$$
$$24$$ 0 0
$$25$$ 1139.00 1.82240
$$26$$ 0 0
$$27$$ − 140.296i − 0.192450i
$$28$$ 0 0
$$29$$ 78.0000 0.0927467 0.0463734 0.998924i $$-0.485234\pi$$
0.0463734 + 0.998924i $$0.485234\pi$$
$$30$$ 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
$$35$$ 3200.83i 2.61292i
$$36$$ 0 0
$$37$$ 530.000 0.387144 0.193572 0.981086i $$-0.437993\pi$$
0.193572 + 0.981086i $$0.437993\pi$$
$$38$$ 0 0
$$39$$ − 945.700i − 0.621762i
$$40$$ 0 0
$$41$$ −918.000 −0.546104 −0.273052 0.961999i $$-0.588033\pi$$
−0.273052 + 0.961999i $$0.588033\pi$$
$$42$$ 0 0
$$43$$ 852.169i 0.460881i 0.973086 + 0.230441i $$0.0740167\pi$$
−0.973086 + 0.230441i $$0.925983\pi$$
$$44$$ 0 0
$$45$$ 1134.00 0.560000
$$46$$ 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
$$51$$ − 1278.25i − 0.491447i
$$52$$ 0 0
$$53$$ −4626.00 −1.64685 −0.823425 0.567426i $$-0.807939\pi$$
−0.823425 + 0.567426i $$0.807939\pi$$
$$54$$ 0 0
$$55$$ − 872.954i − 0.288580i
$$56$$ 0 0
$$57$$ −612.000 −0.188366
$$58$$ 0 0
$$59$$ 228.631i 0.0656796i 0.999461 + 0.0328398i $$0.0104551\pi$$
−0.999461 + 0.0328398i $$0.989545\pi$$
$$60$$ 0 0
$$61$$ 1346.00 0.361731 0.180865 0.983508i $$-0.442110\pi$$
0.180865 + 0.983508i $$0.442110\pi$$
$$62$$ 0 0
$$63$$ 2057.68i 0.518437i
$$64$$ 0 0
$$65$$ 7644.00 1.80923
$$66$$ 0 0
$$67$$ − 1087.73i − 0.242310i −0.992634 0.121155i $$-0.961340\pi$$
0.992634 0.121155i $$-0.0386597\pi$$
$$68$$ 0 0
$$69$$ −3888.00 −0.816635
$$70$$ 0 0
$$71$$ 1829.05i 0.362834i 0.983406 + 0.181417i $$0.0580684\pi$$
−0.983406 + 0.181417i $$0.941932\pi$$
$$72$$ 0 0
$$73$$ −926.000 −0.173766 −0.0868831 0.996219i $$-0.527691\pi$$
−0.0868831 + 0.996219i $$0.527691\pi$$
$$74$$ 0 0
$$75$$ 5918.42i 1.05216i
$$76$$ 0 0
$$77$$ 1584.00 0.267161
$$78$$ 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
$$83$$ − 11992.7i − 1.74085i −0.492301 0.870425i $$-0.663844\pi$$
0.492301 0.870425i $$-0.336156\pi$$
$$84$$ 0 0
$$85$$ 10332.0 1.43003
$$86$$ 0 0
$$87$$ 405.300i 0.0535473i
$$88$$ 0 0
$$89$$ 11586.0 1.46269 0.731347 0.682005i $$-0.238892\pi$$
0.731347 + 0.682005i $$0.238892\pi$$
$$90$$ 0 0
$$91$$ 13870.3i 1.67495i
$$92$$ 0 0
$$93$$ 7668.00 0.886576
$$94$$ 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
$$99$$ − 561.184i − 0.0572579i
$$100$$ 0 0
$$101$$ −5490.00 −0.538183 −0.269091 0.963115i $$-0.586723\pi$$
−0.269091 + 0.963115i $$0.586723\pi$$
$$102$$ 0 0
$$103$$ 5701.91i 0.537460i 0.963216 + 0.268730i $$0.0866039\pi$$
−0.963216 + 0.268730i $$0.913396\pi$$
$$104$$ 0 0
$$105$$ −16632.0 −1.50857
$$106$$ 0 0
$$107$$ − 10080.5i − 0.880473i −0.897882 0.440237i $$-0.854895\pi$$
0.897882 0.440237i $$-0.145105\pi$$
$$108$$ 0 0
$$109$$ −16166.0 −1.36066 −0.680330 0.732906i $$-0.738164\pi$$
−0.680330 + 0.732906i $$0.738164\pi$$
$$110$$ 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
$$115$$ − 31426.3i − 2.37628i
$$116$$ 0 0
$$117$$ 4914.00 0.358974
$$118$$ 0 0
$$119$$ 18747.7i 1.32390i
$$120$$ 0 0
$$121$$ 14209.0 0.970494
$$122$$ 0 0
$$123$$ − 4770.07i − 0.315293i
$$124$$ 0 0
$$125$$ −21588.0 −1.38163
$$126$$ 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
$$131$$ 353.338i 0.0205896i 0.999947 + 0.0102948i $$0.00327700\pi$$
−0.999947 + 0.0102948i $$0.996723\pi$$
$$132$$ 0 0
$$133$$ 8976.00 0.507434
$$134$$ 0 0
$$135$$ 5892.44i 0.323316i
$$136$$ 0 0
$$137$$ −13254.0 −0.706164 −0.353082 0.935592i $$-0.614866\pi$$
−0.353082 + 0.935592i $$0.614866\pi$$
$$138$$ 0 0
$$139$$ − 13212.1i − 0.683820i −0.939733 0.341910i $$-0.888926\pi$$
0.939733 0.341910i $$-0.111074\pi$$
$$140$$ 0 0
$$141$$ 19656.0 0.988683
$$142$$ 0 0
$$143$$ − 3782.80i − 0.184987i
$$144$$ 0 0
$$145$$ −3276.00 −0.155815
$$146$$ 0 0
$$147$$ − 17703.3i − 0.819255i
$$148$$ 0 0
$$149$$ 438.000 0.0197288 0.00986442 0.999951i $$-0.496860\pi$$
0.00986442 + 0.999951i $$0.496860\pi$$
$$150$$ 0 0
$$151$$ 28052.3i 1.23031i 0.788406 + 0.615155i $$0.210907\pi$$
−0.788406 + 0.615155i $$0.789093\pi$$
$$152$$ 0 0
$$153$$ 6642.00 0.283737
$$154$$ 0 0
$$155$$ 61979.7i 2.57980i
$$156$$ 0 0
$$157$$ 19346.0 0.784859 0.392430 0.919782i $$-0.371635\pi$$
0.392430 + 0.919782i $$0.371635\pi$$
$$158$$ 0 0
$$159$$ − 24037.4i − 0.950809i
$$160$$ 0 0
$$161$$ 57024.0 2.19992
$$162$$ 0 0
$$163$$ 36255.3i 1.36457i 0.731086 + 0.682286i $$0.239014\pi$$
−0.731086 + 0.682286i $$0.760986\pi$$
$$164$$ 0 0
$$165$$ 4536.00 0.166612
$$166$$ 0 0
$$167$$ 18747.7i 0.672226i 0.941822 + 0.336113i $$0.109112\pi$$
−0.941822 + 0.336113i $$0.890888\pi$$
$$168$$ 0 0
$$169$$ 4563.00 0.159763
$$170$$ 0 0
$$171$$ − 3180.05i − 0.108753i
$$172$$ 0 0
$$173$$ −34410.0 −1.14972 −0.574861 0.818251i $$-0.694944\pi$$
−0.574861 + 0.818251i $$0.694944\pi$$
$$174$$ 0 0
$$175$$ − 86803.5i − 2.83440i
$$176$$ 0 0
$$177$$ −1188.00 −0.0379201
$$178$$ 0 0
$$179$$ − 16856.3i − 0.526086i −0.964784 0.263043i $$-0.915274\pi$$
0.964784 0.263043i $$-0.0847261\pi$$
$$180$$ 0 0
$$181$$ 15706.0 0.479411 0.239706 0.970846i $$-0.422949\pi$$
0.239706 + 0.970846i $$0.422949\pi$$
$$182$$ 0 0
$$183$$ 6994.02i 0.208845i
$$184$$ 0 0
$$185$$ −22260.0 −0.650402
$$186$$ 0 0
$$187$$ − 5113.01i − 0.146216i
$$188$$ 0 0
$$189$$ −10692.0 −0.299320
$$190$$ 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
$$195$$ 39719.4i 1.04456i
$$196$$ 0 0
$$197$$ −52482.0 −1.35232 −0.676158 0.736757i $$-0.736356\pi$$
−0.676158 + 0.736757i $$0.736356\pi$$
$$198$$ 0 0
$$199$$ 23077.8i 0.582759i 0.956608 + 0.291380i $$0.0941143\pi$$
−0.956608 + 0.291380i $$0.905886\pi$$
$$200$$ 0 0
$$201$$ 5652.00 0.139898
$$202$$ 0 0
$$203$$ − 5944.40i − 0.144250i
$$204$$ 0 0
$$205$$ 38556.0 0.917454
$$206$$ 0 0
$$207$$ − 20202.6i − 0.471485i
$$208$$ 0 0
$$209$$ −2448.00 −0.0560427
$$210$$ 0 0
$$211$$ − 23895.4i − 0.536721i −0.963319 0.268361i $$-0.913518\pi$$
0.963319 0.268361i $$-0.0864819\pi$$
$$212$$ 0 0
$$213$$ −9504.00 −0.209482
$$214$$ 0 0
$$215$$ − 35791.1i − 0.774280i
$$216$$ 0 0
$$217$$ −112464. −2.38833
$$218$$ 0 0
$$219$$ − 4811.64i − 0.100324i
$$220$$ 0 0
$$221$$ 44772.0 0.916689
$$222$$ 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
$$227$$ − 76175.6i − 1.47831i −0.673538 0.739153i $$-0.735226\pi$$
0.673538 0.739153i $$-0.264774\pi$$
$$228$$ 0 0
$$229$$ −48470.0 −0.924277 −0.462138 0.886808i $$-0.652918\pi$$
−0.462138 + 0.886808i $$0.652918\pi$$
$$230$$ 0 0
$$231$$ 8230.71i 0.154246i
$$232$$ 0 0
$$233$$ 48738.0 0.897751 0.448875 0.893594i $$-0.351825\pi$$
0.448875 + 0.893594i $$0.351825\pi$$
$$234$$ 0 0
$$235$$ 158878.i 2.87691i
$$236$$ 0 0
$$237$$ −22860.0 −0.406986
$$238$$ 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
$$243$$ 3788.00i 0.0641500i
$$244$$ 0 0
$$245$$ 143094. 2.38391
$$246$$ 0 0
$$247$$ − 21435.9i − 0.351356i
$$248$$ 0 0
$$249$$ 62316.0 1.00508
$$250$$ 0 0
$$251$$ 91888.8i 1.45853i 0.684232 + 0.729264i $$0.260138\pi$$
−0.684232 + 0.729264i $$0.739862\pi$$
$$252$$ 0 0
$$253$$ −15552.0 −0.242966
$$254$$ 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
$$259$$ − 40391.4i − 0.602129i
$$260$$ 0 0
$$261$$ −2106.00 −0.0309156
$$262$$ 0 0
$$263$$ − 78191.7i − 1.13044i −0.824939 0.565222i $$-0.808790\pi$$
0.824939 0.565222i $$-0.191210\pi$$
$$264$$ 0 0
$$265$$ 194292. 2.76671
$$266$$ 0 0
$$267$$ 60202.6i 0.844487i
$$268$$ 0 0
$$269$$ −71538.0 −0.988626 −0.494313 0.869284i $$-0.664580\pi$$
−0.494313 + 0.869284i $$0.664580\pi$$
$$270$$ 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
$$275$$ 23673.7i 0.313040i
$$276$$ 0 0
$$277$$ −120518. −1.57070 −0.785348 0.619054i $$-0.787516\pi$$
−0.785348 + 0.619054i $$0.787516\pi$$
$$278$$ 0 0
$$279$$ 39844.1i 0.511865i
$$280$$ 0 0
$$281$$ −3054.00 −0.0386773 −0.0193387 0.999813i $$-0.506156\pi$$
−0.0193387 + 0.999813i $$0.506156\pi$$
$$282$$ 0 0
$$283$$ − 132959.i − 1.66014i −0.557657 0.830071i $$-0.688300\pi$$
0.557657 0.830071i $$-0.311700\pi$$
$$284$$ 0 0
$$285$$ 25704.0 0.316454
$$286$$ 0 0
$$287$$ 69961.0i 0.849361i
$$288$$ 0 0
$$289$$ −23005.0 −0.275440
$$290$$ 0 0
$$291$$ − 68163.1i − 0.804940i
$$292$$ 0 0
$$293$$ 151662. 1.76661 0.883307 0.468795i $$-0.155312\pi$$
0.883307 + 0.468795i $$0.155312\pi$$
$$294$$ 0 0
$$295$$ − 9602.49i − 0.110342i
$$296$$ 0 0
$$297$$ 2916.00 0.0330579
$$298$$ 0 0
$$299$$ − 136181.i − 1.52326i
$$300$$ 0 0
$$301$$ 64944.0 0.716813
$$302$$ 0 0
$$303$$ − 28526.9i − 0.310720i
$$304$$ 0 0
$$305$$ −56532.0 −0.607708
$$306$$ 0 0
$$307$$ 5424.78i 0.0575580i 0.999586 + 0.0287790i $$0.00916190\pi$$
−0.999586 + 0.0287790i $$0.990838\pi$$
$$308$$ 0 0
$$309$$ −29628.0 −0.310303
$$310$$ 0 0
$$311$$ 141127.i 1.45912i 0.683917 + 0.729560i $$0.260275\pi$$
−0.683917 + 0.729560i $$0.739725\pi$$
$$312$$ 0 0
$$313$$ −128686. −1.31354 −0.656769 0.754092i $$-0.728077\pi$$
−0.656769 + 0.754092i $$0.728077\pi$$
$$314$$ 0 0
$$315$$ − 86422.4i − 0.870974i
$$316$$ 0 0
$$317$$ −73986.0 −0.736260 −0.368130 0.929774i $$-0.620002\pi$$
−0.368130 + 0.929774i $$0.620002\pi$$
$$318$$ 0 0
$$319$$ 1621.20i 0.0159314i
$$320$$ 0 0
$$321$$ 52380.0 0.508341
$$322$$ 0 0
$$323$$ − 28973.7i − 0.277715i
$$324$$ 0 0
$$325$$ −207298. −1.96258
$$326$$ 0 0
$$327$$ − 84001.0i − 0.785577i
$$328$$ 0 0
$$329$$ −288288. −2.66339
$$330$$ 0 0
$$331$$ − 57026.0i − 0.520496i −0.965542 0.260248i $$-0.916196\pi$$
0.965542 0.260248i $$-0.0838043\pi$$
$$332$$ 0 0
$$333$$ −14310.0 −0.129048
$$334$$ 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
$$339$$ 9571.31i 0.0832860i
$$340$$ 0 0
$$341$$ 30672.0 0.263775
$$342$$ 0 0
$$343$$ 76667.5i 0.651663i
$$344$$ 0 0
$$345$$ 163296. 1.37195
$$346$$ 0 0
$$347$$ − 56929.0i − 0.472797i −0.971656 0.236399i $$-0.924033\pi$$
0.971656 0.236399i $$-0.0759671\pi$$
$$348$$ 0 0
$$349$$ 181346. 1.48887 0.744436 0.667694i $$-0.232719\pi$$
0.744436 + 0.667694i $$0.232719\pi$$
$$350$$ 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
$$355$$ − 76819.9i − 0.609561i
$$356$$ 0 0
$$357$$ −97416.0 −0.764353
$$358$$ 0 0
$$359$$ 185232.i 1.43724i 0.695405 + 0.718618i $$0.255225\pi$$
−0.695405 + 0.718618i $$0.744775\pi$$
$$360$$ 0 0
$$361$$ 116449. 0.893555
$$362$$ 0 0
$$363$$ 73832.1i 0.560315i
$$364$$ 0 0
$$365$$ 38892.0 0.291927
$$366$$ 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.i 2.56136i
$$372$$ 0 0
$$373$$ 151778. 1.09092 0.545458 0.838138i $$-0.316356\pi$$
0.545458 + 0.838138i $$0.316356\pi$$
$$374$$ 0 0
$$375$$ − 112175.i − 0.797686i
$$376$$ 0 0
$$377$$ −14196.0 −0.0998811
$$378$$ 0 0
$$379$$ − 36005.9i − 0.250666i −0.992115 0.125333i $$-0.960000\pi$$
0.992115 0.125333i $$-0.0399999\pi$$
$$380$$ 0 0
$$381$$ −2052.00 −0.0141360
$$382$$ 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
$$387$$ − 23008.6i − 0.153627i
$$388$$ 0 0
$$389$$ 105750. 0.698846 0.349423 0.936965i $$-0.386378\pi$$
0.349423 + 0.936965i $$0.386378\pi$$
$$390$$ 0 0
$$391$$ − 184069.i − 1.20400i
$$392$$ 0 0
$$393$$ −1836.00 −0.0118874
$$394$$ 0 0
$$395$$ − 184775.i − 1.18427i
$$396$$ 0 0
$$397$$ −27934.0 −0.177236 −0.0886180 0.996066i $$-0.528245\pi$$
−0.0886180 + 0.996066i $$0.528245\pi$$
$$398$$ 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.i 1.65372i
$$404$$ 0 0
$$405$$ −30618.0 −0.186667
$$406$$ 0 0
$$407$$ 11015.8i 0.0665011i
$$408$$ 0 0
$$409$$ −20270.0 −0.121173 −0.0605867 0.998163i $$-0.519297\pi$$
−0.0605867 + 0.998163i $$0.519297\pi$$
$$410$$ 0 0
$$411$$ − 68869.8i − 0.407704i
$$412$$ 0 0
$$413$$ 17424.0 0.102152
$$414$$ 0 0
$$415$$ 503694.i 2.92463i
$$416$$ 0 0
$$417$$ 68652.0 0.394804
$$418$$ 0 0
$$419$$ 24089.4i 0.137214i 0.997644 + 0.0686068i $$0.0218554\pi$$
−0.997644 + 0.0686068i $$0.978145\pi$$
$$420$$ 0 0
$$421$$ 116698. 0.658414 0.329207 0.944258i $$-0.393219\pi$$
0.329207 + 0.944258i $$0.393219\pi$$
$$422$$ 0 0
$$423$$ 102136.i 0.570816i
$$424$$ 0 0
$$425$$ −280194. −1.55125
$$426$$ 0 0
$$427$$ − 102579.i − 0.562604i
$$428$$ 0 0
$$429$$ 19656.0 0.106802
$$430$$ 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
$$435$$ − 17022.6i − 0.0899595i
$$436$$ 0 0
$$437$$ −88128.0 −0.461478
$$438$$ 0 0
$$439$$ − 146469.i − 0.760006i −0.924985 0.380003i $$-0.875923\pi$$
0.924985 0.380003i $$-0.124077\pi$$
$$440$$ 0 0
$$441$$ 91989.0 0.472997
$$442$$ 0 0
$$443$$ − 50444.2i − 0.257042i −0.991707 0.128521i $$-0.958977\pi$$
0.991707 0.128521i $$-0.0410230\pi$$
$$444$$ 0 0
$$445$$ −486612. −2.45733
$$446$$ 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
$$451$$ − 19080.3i − 0.0938062i
$$452$$ 0 0
$$453$$ −145764. −0.710320
$$454$$ 0 0
$$455$$ − 582551.i − 2.81392i
$$456$$ 0 0
$$457$$ 284338. 1.36145 0.680726 0.732538i $$-0.261664\pi$$
0.680726 + 0.732538i $$0.261664\pi$$
$$458$$ 0 0
$$459$$ 34512.8i 0.163816i
$$460$$ 0 0
$$461$$ −183402. −0.862983 −0.431491 0.902117i $$-0.642013\pi$$
−0.431491 + 0.902117i $$0.642013\pi$$
$$462$$ 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
$$467$$ 68734.7i 0.315168i 0.987506 + 0.157584i $$0.0503705\pi$$
−0.987506 + 0.157584i $$0.949629\pi$$
$$468$$ 0 0
$$469$$ −82896.0 −0.376867
$$470$$ 0 0
$$471$$ 100525.i 0.453139i
$$472$$ 0 0
$$473$$ −17712.0 −0.0791672
$$474$$ 0 0
$$475$$ 134151.i 0.594574i
$$476$$ 0 0
$$477$$ 124902. 0.548950
$$478$$ 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.i 1.27012i
$$484$$ 0 0
$$485$$ 550956. 2.34225
$$486$$ 0 0
$$487$$ − 271108.i − 1.14310i −0.820568 0.571549i $$-0.806343\pi$$
0.820568 0.571549i $$-0.193657\pi$$
$$488$$ 0 0
$$489$$ −188388. −0.787835
$$490$$ 0 0
$$491$$ 227862.i 0.945166i 0.881286 + 0.472583i $$0.156678\pi$$
−0.881286 + 0.472583i $$0.843322\pi$$
$$492$$ 0 0
$$493$$ −19188.0 −0.0789470
$$494$$ 0 0
$$495$$ 23569.7i 0.0961932i
$$496$$ 0 0
$$497$$ 139392. 0.564320
$$498$$ 0 0
$$499$$ 248854.i 0.999410i 0.866196 + 0.499705i $$0.166558\pi$$
−0.866196 + 0.499705i $$0.833442\pi$$
$$500$$ 0 0
$$501$$ −97416.0 −0.388110
$$502$$ 0 0
$$503$$ − 446537.i − 1.76490i −0.470403 0.882452i $$-0.655891\pi$$
0.470403 0.882452i $$-0.344109\pi$$
$$504$$ 0 0
$$505$$ 230580. 0.904147
$$506$$ 0 0
$$507$$ 23710.0i 0.0922394i
$$508$$ 0 0
$$509$$ −39330.0 −0.151806 −0.0759029 0.997115i $$-0.524184\pi$$
−0.0759029 + 0.997115i $$0.524184\pi$$
$$510$$ 0 0
$$511$$ 70570.7i 0.270260i
$$512$$ 0 0
$$513$$ 16524.0 0.0627886
$$514$$ 0 0
$$515$$ − 239480.i − 0.902933i
$$516$$ 0 0
$$517$$ 78624.0 0.294154
$$518$$ 0 0
$$519$$ − 178800.i − 0.663792i
$$520$$ 0 0
$$521$$ −464598. −1.71160 −0.855799 0.517308i $$-0.826934\pi$$
−0.855799 + 0.517308i $$0.826934\pi$$
$$522$$ 0 0
$$523$$ 135509.i 0.495409i 0.968836 + 0.247704i $$0.0796762\pi$$
−0.968836 + 0.247704i $$0.920324\pi$$
$$524$$ 0 0
$$525$$ 451044. 1.63644
$$526$$ 0 0
$$527$$ 363024.i 1.30712i
$$528$$ 0 0
$$529$$ −280031. −1.00068
$$530$$ 0 0
$$531$$ − 6173.03i − 0.0218932i
$$532$$ 0 0
$$533$$ 167076. 0.588111
$$534$$ 0 0
$$535$$ 423382.i 1.47919i
$$536$$ 0 0
$$537$$ 87588.0 0.303736
$$538$$ 0 0
$$539$$ − 70813.2i − 0.243745i
$$540$$ 0 0
$$541$$ 360442. 1.23152 0.615759 0.787934i $$-0.288849\pi$$
0.615759 + 0.787934i $$0.288849\pi$$
$$542$$ 0 0
$$543$$ 81610.8i 0.276788i
$$544$$ 0 0
$$545$$ 678972. 2.28591
$$546$$ 0 0
$$547$$ 261644.i 0.874451i 0.899352 + 0.437225i $$0.144039\pi$$
−0.899352 + 0.437225i $$0.855961\pi$$
$$548$$ 0 0
$$549$$ −36342.0 −0.120577
$$550$$ 0 0
$$551$$ 9186.80i 0.0302594i
$$552$$ 0 0
$$553$$ 335280. 1.09637
$$554$$ 0 0
$$555$$ − 115666.i − 0.375510i
$$556$$ 0 0
$$557$$ −233274. −0.751893 −0.375946 0.926641i $$-0.622682\pi$$
−0.375946 + 0.926641i $$0.622682\pi$$
$$558$$ 0 0
$$559$$ − 155095.i − 0.496333i
$$560$$ 0 0
$$561$$ 26568.0 0.0844176
$$562$$ 0 0
$$563$$ − 419704.i − 1.32412i −0.749453 0.662058i $$-0.769683\pi$$
0.749453 0.662058i $$-0.230317\pi$$
$$564$$ 0 0
$$565$$ −77364.0 −0.242349
$$566$$ 0 0
$$567$$ − 55557.3i − 0.172812i
$$568$$ 0 0
$$569$$ 470058. 1.45187 0.725934 0.687765i $$-0.241408\pi$$
0.725934 + 0.687765i $$0.241408\pi$$
$$570$$ 0 0
$$571$$ − 320381.i − 0.982640i −0.870979 0.491320i $$-0.836515\pi$$
0.870979 0.491320i $$-0.163485\pi$$
$$572$$ 0 0
$$573$$ 13824.0 0.0421041
$$574$$ 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.i − 0.415114i
$$580$$ 0 0
$$581$$ −913968. −2.70756
$$582$$ 0 0
$$583$$ − 96149.6i − 0.282885i
$$584$$ 0 0
$$585$$ −206388. −0.603077
$$586$$ 0 0
$$587$$ 114128.i 0.331220i 0.986191 + 0.165610i $$0.0529594\pi$$
−0.986191 + 0.165610i $$0.947041\pi$$
$$588$$ 0 0
$$589$$ 173808. 0.501002
$$590$$ 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.i − 2.22415i
$$596$$ 0 0
$$597$$ −119916. −0.336456
$$598$$ 0 0
$$599$$ 519782.i 1.44866i 0.689452 + 0.724331i $$0.257851\pi$$
−0.689452 + 0.724331i $$0.742149\pi$$
$$600$$ 0 0
$$601$$ −627742. −1.73793 −0.868965 0.494874i $$-0.835214\pi$$
−0.868965 + 0.494874i $$0.835214\pi$$
$$602$$ 0 0
$$603$$ 29368.7i 0.0807699i
$$604$$ 0 0
$$605$$ −596778. −1.63043
$$606$$ 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.i 1.84418i
$$612$$ 0 0
$$613$$ 247202. 0.657856 0.328928 0.944355i $$-0.393313\pi$$
0.328928 + 0.944355i $$0.393313\pi$$
$$614$$ 0 0
$$615$$ 200343.i 0.529692i
$$616$$ 0 0
$$617$$ −31758.0 −0.0834224 −0.0417112 0.999130i $$-0.513281\pi$$
−0.0417112 + 0.999130i $$0.513281\pi$$
$$618$$ 0 0
$$619$$ − 656094.i − 1.71232i −0.516712 0.856160i $$-0.672844\pi$$
0.516712 0.856160i $$-0.327156\pi$$
$$620$$ 0 0
$$621$$ 104976. 0.272212
$$622$$ 0 0
$$623$$ − 882972.i − 2.27494i
$$624$$ 0 0
$$625$$ 194821. 0.498742
$$626$$ 0 0
$$627$$ − 12720.2i − 0.0323563i
$$628$$ 0 0
$$629$$ −130380. −0.329541
$$630$$ 0 0
$$631$$ − 417736.i − 1.04916i −0.851360 0.524582i $$-0.824222\pi$$
0.851360 0.524582i $$-0.175778\pi$$
$$632$$ 0 0
$$633$$ 124164. 0.309876
$$634$$ 0 0
$$635$$ − 16586.1i − 0.0411337i
$$636$$ 0 0
$$637$$ 620074. 1.52815
$$638$$ 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.i 1.72727i 0.504117 + 0.863635i $$0.331818\pi$$
−0.504117 + 0.863635i $$0.668182\pi$$
$$644$$ 0 0
$$645$$ 185976. 0.447031
$$646$$ 0 0
$$647$$ 259558.i 0.620049i 0.950729 + 0.310025i $$0.100337\pi$$
−0.950729 + 0.310025i $$0.899663\pi$$
$$648$$ 0 0
$$649$$ −4752.00 −0.0112820
$$650$$ 0 0
$$651$$ − 584380.i − 1.37890i
$$652$$ 0 0
$$653$$ −330714. −0.775579 −0.387790 0.921748i $$-0.626761\pi$$
−0.387790 + 0.921748i $$0.626761\pi$$
$$654$$ 0 0
$$655$$ − 14840.2i − 0.0345906i
$$656$$ 0 0
$$657$$ 25002.0 0.0579221
$$658$$ 0 0
$$659$$ − 253884.i − 0.584608i −0.956326 0.292304i $$-0.905578\pi$$
0.956326 0.292304i $$-0.0944219\pi$$
$$660$$ 0 0
$$661$$ −722158. −1.65283 −0.826417 0.563058i $$-0.809625\pi$$
−0.826417 + 0.563058i $$0.809625\pi$$
$$662$$ 0 0
$$663$$ 232642.i 0.529251i
$$664$$ 0 0
$$665$$ −376992. −0.852489
$$666$$ 0 0
$$667$$ 58363.2i 0.131186i
$$668$$ 0 0
$$669$$ 4428.00 0.00989362
$$670$$ 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.i − 0.350721i
$$676$$ 0 0
$$677$$ 609030. 1.32881 0.664403 0.747375i $$-0.268686\pi$$
0.664403 + 0.747375i $$0.268686\pi$$
$$678$$ 0 0
$$679$$ 999726.i 2.16841i
$$680$$ 0 0
$$681$$ 395820. 0.853500
$$682$$ 0 0
$$683$$ 23715.2i 0.0508377i 0.999677 + 0.0254189i $$0.00809195\pi$$
−0.999677 + 0.0254189i $$0.991908\pi$$
$$684$$ 0 0
$$685$$ 556668. 1.18636
$$686$$ 0 0
$$687$$ − 251858.i − 0.533631i
$$688$$ 0 0
$$689$$ 841932. 1.77353
$$690$$ 0 0
$$691$$ − 431842.i − 0.904417i −0.891912 0.452208i $$-0.850636\pi$$
0.891912 0.452208i $$-0.149364\pi$$
$$692$$ 0 0
$$693$$ −42768.0 −0.0890538
$$694$$ 0 0
$$695$$ 554908.i 1.14882i
$$696$$ 0 0
$$697$$ 225828. 0.464849
$$698$$ 0 0
$$699$$ 253250.i 0.518317i
$$700$$ 0 0
$$701$$ 44958.0 0.0914894 0.0457447 0.998953i $$-0.485434\pi$$
0.0457447 + 0.998953i $$0.485434\pi$$
$$702$$ 0 0
$$703$$ 62423.1i 0.126309i
$$704$$ 0 0
$$705$$ −825552. −1.66099
$$706$$ 0 0
$$707$$ 418394.i 0.837041i
$$708$$ 0 0
$$709$$ 533002. 1.06032 0.530159 0.847898i $$-0.322132\pi$$
0.530159 + 0.847898i $$0.322132\pi$$
$$710$$ 0 0
$$711$$ − 118784.i − 0.234974i
$$712$$ 0 0
$$713$$ 1.10419e6 2.17203
$$714$$ 0 0
$$715$$ 158878.i 0.310778i
$$716$$ 0 0
$$717$$ 368928. 0.717634
$$718$$ 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.i 0.727023i
$$724$$ 0 0
$$725$$ 88842.0 0.169022
$$726$$ 0 0
$$727$$ 755791.i 1.42999i 0.699130 + 0.714995i $$0.253571\pi$$
−0.699130 + 0.714995i $$0.746429\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −0.0370370
$$730$$ 0 0
$$731$$ − 209634.i − 0.392307i
$$732$$ 0 0
$$733$$ −832982. −1.55034 −0.775171 0.631751i $$-0.782336\pi$$
−0.775171 + 0.631751i $$0.782336\pi$$
$$734$$ 0 0
$$735$$ 743538.i 1.37635i
$$736$$ 0 0
$$737$$ 22608.0 0.0416224
$$738$$ 0 0
$$739$$ − 698093.i − 1.27827i −0.769093 0.639137i $$-0.779292\pi$$
0.769093 0.639137i $$-0.220708\pi$$
$$740$$ 0 0
$$741$$ 111384. 0.202855
$$742$$ 0 0
$$743$$ − 461044.i − 0.835151i −0.908642 0.417575i $$-0.862880\pi$$
0.908642 0.417575i $$-0.137120\pi$$
$$744$$ 0 0
$$745$$ −18396.0 −0.0331445
$$746$$ 0 0
$$747$$ 323803.i 0.580284i
$$748$$ 0 0
$$749$$ −768240. −1.36941
$$750$$ 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
$$755$$ − 1.17820e6i − 2.06692i
$$756$$ 0 0
$$757$$ 295786. 0.516162 0.258081 0.966123i $$-0.416910\pi$$
0.258081 + 0.966123i $$0.416910\pi$$
$$758$$ 0 0
$$759$$ − 80810.6i − 0.140276i
$$760$$ 0 0
$$761$$ −1.02615e6 −1.77191 −0.885955 0.463772i $$-0.846496\pi$$
−0.885955 + 0.463772i $$0.846496\pi$$
$$762$$ 0 0
$$763$$ 1.23201e6i 2.11625i
$$764$$ 0 0
$$765$$ −278964. −0.476678
$$766$$ 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.i − 0.427394i
$$772$$ 0 0
$$773$$ 1.02608e6 1.71720 0.858601 0.512644i $$-0.171334\pi$$
0.858601 + 0.512644i $$0.171334\pi$$
$$774$$ 0 0
$$775$$ − 1.68083e6i − 2.79847i
$$776$$ 0 0
$$777$$ 209880. 0.347639
$$778$$ 0 0
$$779$$ − 108122.i − 0.178171i
$$780$$ 0 0
$$781$$ −38016.0 −0.0623253
$$782$$ 0 0
$$783$$ − 10943.1i − 0.0178491i
$$784$$ 0 0
$$785$$ −812532. −1.31856
$$786$$ 0 0
$$787$$ 850042.i 1.37243i 0.727398 + 0.686216i $$0.240730\pi$$
−0.727398 + 0.686216i $$0.759270\pi$$
$$788$$ 0 0
$$789$$ 406296. 0.652662
$$790$$ 0 0
$$791$$ − 140379.i − 0.224362i
$$792$$ 0 0
$$793$$ −244972. −0.389556
$$794$$ 0 0
$$795$$ 1.00957e6i 1.59736i
$$796$$ 0 0
$$797$$ 761478. 1.19878 0.599392 0.800456i $$-0.295409\pi$$
0.599392 + 0.800456i $$0.295409\pi$$
$$798$$ 0 0
$$799$$ 930569.i 1.45766i
$$800$$ 0 0
$$801$$ −312822. −0.487565
$$802$$ 0 0
$$803$$ − 19246.5i − 0.0298484i
$$804$$ 0 0
$$805$$ −2.39501e6 −3.69586
$$806$$ 0 0
$$807$$ − 371722.i − 0.570784i
$$808$$ 0 0
$$809$$ 247674. 0.378428 0.189214 0.981936i $$-0.439406\pi$$
0.189214 + 0.981936i $$0.439406\pi$$
$$810$$ 0 0
$$811$$ 920197.i 1.39907i 0.714599 + 0.699534i $$0.246609\pi$$
−0.714599 + 0.699534i $$0.753391\pi$$
$$812$$ 0 0
$$813$$ −562212. −0.850588
$$814$$ 0 0
$$815$$ − 1.52272e6i − 2.29248i
$$816$$ 0 0
$$817$$ −100368. −0.150367
$$818$$ 0 0
$$819$$ − 374497.i − 0.558317i
$$820$$ 0 0
$$821$$ −250242. −0.371256 −0.185628 0.982620i $$-0.559432\pi$$
−0.185628 + 0.982620i $$0.559432\pi$$
$$822$$ 0 0
$$823$$ 400762.i 0.591680i 0.955238 + 0.295840i $$0.0955995\pi$$
−0.955238 + 0.295840i $$0.904401\pi$$
$$824$$ 0 0
$$825$$ −123012. −0.180734
$$826$$ 0 0
$$827$$ 17272.0i 0.0252541i 0.999920 + 0.0126270i $$0.00401942\pi$$
−0.999920 + 0.0126270i $$0.995981\pi$$
$$828$$ 0 0
$$829$$ −15686.0 −0.0228246 −0.0114123 0.999935i $$-0.503633\pi$$
−0.0114123 + 0.999935i $$0.503633\pi$$
$$830$$ 0 0
$$831$$ − 626230.i − 0.906842i
$$832$$ 0 0
$$833$$ 838122. 1.20786
$$834$$ 0 0
$$835$$ − 787404.i − 1.12934i
$$836$$ 0 0
$$837$$ −207036. −0.295525
$$838$$ 0 0
$$839$$ 115479.i 0.164051i 0.996630 + 0.0820257i $$0.0261390\pi$$
−0.996630 + 0.0820257i $$0.973861\pi$$
$$840$$ 0 0
$$841$$ −701197. −0.991398
$$842$$ 0 0
$$843$$ − 15869.0i − 0.0223304i
$$844$$ 0 0
$$845$$ −191646. −0.268402
$$846$$ 0 0
$$847$$ − 1.08287e6i − 1.50942i
$$848$$ 0 0
$$849$$ 690876. 0.958484
$$850$$ 0 0
$$851$$ 396570.i 0.547597i
$$852$$ 0 0
$$853$$ 345938. 0.475445 0.237722 0.971333i $$-0.423599\pi$$
0.237722 + 0.971333i $$0.423599\pi$$
$$854$$ 0 0
$$855$$ 133562.i 0.182705i
$$856$$ 0 0
$$857$$ −267990. −0.364886 −0.182443 0.983216i $$-0.558400\pi$$
−0.182443 + 0.983216i $$0.558400\pi$$
$$858$$ 0 0
$$859$$ 522407.i 0.707983i 0.935249 + 0.353992i $$0.115176\pi$$
−0.935249 + 0.353992i $$0.884824\pi$$
$$860$$ 0 0
$$861$$ −363528. −0.490379
$$862$$ 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.i − 0.159025i
$$868$$ 0 0
$$869$$ −91440.0 −0.121087
$$870$$ 0 0
$$871$$ 197966.i 0.260949i
$$872$$ 0 0
$$873$$ 354186. 0.464732
$$874$$ 0 0
$$875$$ 1.64523e6i 2.14887i
$$876$$ 0 0
$$877$$ 1.11629e6 1.45137 0.725685 0.688028i $$-0.241523\pi$$
0.725685 + 0.688028i $$0.241523\pi$$
$$878$$ 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.i − 0.729662i −0.931074 0.364831i $$-0.881127\pi$$
0.931074 0.364831i $$-0.118873\pi$$
$$884$$ 0 0
$$885$$ 49896.0 0.0637058
$$886$$ 0 0
$$887$$ 1.09015e6i 1.38561i 0.721126 + 0.692804i $$0.243625\pi$$
−0.721126 + 0.692804i $$0.756375\pi$$
$$888$$ 0 0
$$889$$ 30096.0 0.0380807
$$890$$ 0 0
$$891$$ 15152.0i 0.0190860i
$$892$$ 0 0
$$893$$ 445536. 0.558702
$$894$$ 0 0
$$895$$ 707965.i 0.883824i
$$896$$ 0 0
$$897$$ 707616. 0.879453
$$898$$ 0 0
$$899$$ − 115105.i − 0.142421i
$$900$$ 0 0
$$901$$ 1.13800e6 1.40182
$$902$$ 0 0
$$903$$ 337459.i 0.413852i
$$904$$ 0 0
$$905$$ −659652. −0.805411
$$906$$ 0 0
$$907$$ − 916193.i − 1.11371i −0.830610 0.556855i $$-0.812008\pi$$
0.830610 0.556855i $$-0.187992\pi$$
$$908$$ 0 0
$$909$$ 148230. 0.179394
$$910$$ 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.i − 0.350860i
$$916$$ 0 0
$$917$$ 26928.0 0.0320233
$$918$$ 0 0
$$919$$ 97084.9i 0.114953i 0.998347 + 0.0574766i $$0.0183054\pi$$
−0.998347 + 0.0574766i $$0.981695\pi$$
$$920$$ 0 0
$$921$$ −28188.0 −0.0332311
$$922$$ 0 0
$$923$$ − 332886.i − 0.390744i
$$924$$ 0 0
$$925$$ 603670. 0.705531
$$926$$ 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.i − 0.462959i
$$932$$ 0 0
$$933$$ −733320. −0.842423
$$934$$ 0 0
$$935$$ 214747.i 0.245642i
$$936$$ 0 0
$$937$$ −981262. −1.11765 −0.558825 0.829286i $$-0.688748\pi$$
−0.558825 + 0.829286i $$0.688748\pi$$
$$938$$ 0 0
$$939$$ − 668672.i − 0.758371i
$$940$$ 0 0
$$941$$ 284406. 0.321188 0.160594 0.987021i $$-0.448659\pi$$
0.160594 + 0.987021i $$0.448659\pi$$
$$942$$ 0 0
$$943$$ − 686890.i − 0.772438i
$$944$$ 0 0
$$945$$ 449064. 0.502857
$$946$$ 0 0
$$947$$ 993109.i 1.10738i 0.832722 + 0.553691i $$0.186781\pi$$
−0.832722 + 0.553691i $$0.813219\pi$$
$$948$$ 0 0
$$949$$ 168532. 0.187133
$$950$$ 0 0
$$951$$ − 384443.i − 0.425080i
$$952$$ 0 0
$$953$$ 602922. 0.663858 0.331929 0.943304i $$-0.392301\pi$$
0.331929 + 0.943304i $$0.392301\pi$$
$$954$$ 0 0
$$955$$ 111738.i 0.122516i
$$956$$ 0 0
$$957$$ −8424.00 −0.00919802
$$958$$ 0 0
$$959$$ 1.01009e6i 1.09831i
$$960$$ 0 0
$$961$$ −1.25419e6 −1.35805
$$962$$ 0 0
$$963$$ 272174.i 0.293491i
$$964$$ 0 0
$$965$$ 1.12484e6 1.20792
$$966$$ 0 0
$$967$$ 575810.i 0.615781i 0.951422 + 0.307890i $$0.0996230\pi$$
−0.951422 + 0.307890i $$0.900377\pi$$
$$968$$ 0 0
$$969$$ 150552. 0.160339
$$970$$ 0 0
$$971$$ 1.23920e6i 1.31432i 0.753749 + 0.657162i $$0.228243\pi$$
−0.753749 + 0.657162i $$0.771757\pi$$
$$972$$ 0 0
$$973$$ −1.00690e6 −1.06355
$$974$$ 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.i 0.251252i
$$980$$ 0 0
$$981$$ 436482. 0.453553
$$982$$ 0 0
$$983$$ − 948734.i − 0.981833i −0.871207 0.490916i $$-0.836662\pi$$
0.871207 0.490916i $$-0.163338\pi$$
$$984$$ 0 0
$$985$$ 2.20424e6 2.27189
$$986$$ 0 0
$$987$$ − 1.49799e6i − 1.53771i
$$988$$ 0 0
$$989$$ −637632. −0.651895
$$990$$ 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.i − 0.979035i
$$996$$ 0 0
$$997$$ −535870. −0.539100 −0.269550 0.962986i $$-0.586875\pi$$
−0.269550 + 0.962986i $$0.586875\pi$$
$$998$$ 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 48.5.g.a.31.2 yes 2
3.2 odd 2 144.5.g.f.127.1 2
4.3 odd 2 inner 48.5.g.a.31.1 2
5.2 odd 4 1200.5.j.b.799.4 4
5.3 odd 4 1200.5.j.b.799.2 4
5.4 even 2 1200.5.e.b.751.1 2
8.3 odd 2 192.5.g.b.127.2 2
8.5 even 2 192.5.g.b.127.1 2
12.11 even 2 144.5.g.f.127.2 2
16.3 odd 4 768.5.b.c.127.4 4
16.5 even 4 768.5.b.c.127.3 4
16.11 odd 4 768.5.b.c.127.1 4
16.13 even 4 768.5.b.c.127.2 4
20.3 even 4 1200.5.j.b.799.3 4
20.7 even 4 1200.5.j.b.799.1 4
20.19 odd 2 1200.5.e.b.751.2 2
24.5 odd 2 576.5.g.d.127.1 2
24.11 even 2 576.5.g.d.127.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 4.3 odd 2 inner
48.5.g.a.31.2 yes 2 1.1 even 1 trivial
144.5.g.f.127.1 2 3.2 odd 2
144.5.g.f.127.2 2 12.11 even 2
192.5.g.b.127.1 2 8.5 even 2
192.5.g.b.127.2 2 8.3 odd 2
576.5.g.d.127.1 2 24.5 odd 2
576.5.g.d.127.2 2 24.11 even 2
768.5.b.c.127.1 4 16.11 odd 4
768.5.b.c.127.2 4 16.13 even 4
768.5.b.c.127.3 4 16.5 even 4
768.5.b.c.127.4 4 16.3 odd 4
1200.5.e.b.751.1 2 5.4 even 2
1200.5.e.b.751.2 2 20.19 odd 2
1200.5.j.b.799.1 4 20.7 even 4
1200.5.j.b.799.2 4 5.3 odd 4
1200.5.j.b.799.3 4 20.3 even 4
1200.5.j.b.799.4 4 5.2 odd 4