# Properties

 Label 576.5.g.d.127.2 Level $576$ Weight $5$ Character 576.127 Analytic conductor $59.541$ 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] = [576,5,Mod(127,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, 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("576.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 576.g (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: $$59.5410987363$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 576.127 Dual form 576.5.g.d.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-42.0000 q^{5} +76.2102i q^{7} +O(q^{10})$$ $$q-42.0000 q^{5} +76.2102i q^{7} -20.7846i q^{11} +182.000 q^{13} +246.000 q^{17} +117.779i q^{19} +748.246i q^{23} +1139.00 q^{25} +78.0000 q^{29} +1475.71i q^{31} -3200.83i q^{35} -530.000 q^{37} +918.000 q^{41} +852.169i q^{43} -3782.80i q^{47} -3407.00 q^{49} -4626.00 q^{53} +872.954i q^{55} -228.631i q^{59} -1346.00 q^{61} -7644.00 q^{65} -1087.73i q^{67} +1829.05i q^{71} -926.000 q^{73} +1584.00 q^{77} -4399.41i q^{79} +11992.7i q^{83} -10332.0 q^{85} -11586.0 q^{89} +13870.3i q^{91} -4946.74i q^{95} -13118.0 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 84 q^{5}+O(q^{10})$$ 2 * q - 84 * q^5 $$2 q - 84 q^{5} + 364 q^{13} + 492 q^{17} + 2278 q^{25} + 156 q^{29} - 1060 q^{37} + 1836 q^{41} - 6814 q^{49} - 9252 q^{53} - 2692 q^{61} - 15288 q^{65} - 1852 q^{73} + 3168 q^{77} - 20664 q^{85} - 23172 q^{89} - 26236 q^{97}+O(q^{100})$$ 2 * q - 84 * q^5 + 364 * q^13 + 492 * q^17 + 2278 * q^25 + 156 * q^29 - 1060 * q^37 + 1836 * q^41 - 6814 * q^49 - 9252 * q^53 - 2692 * q^61 - 15288 * q^65 - 1852 * q^73 + 3168 * q^77 - 20664 * q^85 - 23172 * q^89 - 26236 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 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 $$0.283591\pi$$
−0.628691 + 0.777655i $$0.716409\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 20.7846i − 0.171774i −0.996305 0.0858868i $$-0.972628\pi$$
0.996305 0.0858868i $$-0.0273723\pi$$
$$12$$ 0 0
$$13$$ 182.000 1.07692 0.538462 0.842650i $$-0.319006\pi$$
0.538462 + 0.842650i $$0.319006\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 246.000 0.851211 0.425606 0.904909i $$-0.360061\pi$$
0.425606 + 0.904909i $$0.360061\pi$$
$$18$$ 0 0
$$19$$ 117.779i 0.326259i 0.986605 + 0.163129i $$0.0521588\pi$$
−0.986605 + 0.163129i $$0.947841\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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
$$27$$ 0 0
$$28$$ 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 $$0.278647\pi$$
−0.640692 + 0.767798i $$0.721353\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 3200.83i − 2.61292i
$$36$$ 0 0
$$37$$ −530.000 −0.387144 −0.193572 0.981086i $$-0.562007\pi$$
−0.193572 + 0.981086i $$0.562007\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 918.000 0.546104 0.273052 0.961999i $$-0.411967\pi$$
0.273052 + 0.961999i $$0.411967\pi$$
$$42$$ 0 0
$$43$$ 852.169i 0.460881i 0.973086 + 0.230441i $$0.0740167\pi$$
−0.973086 + 0.230441i $$0.925983\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 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
$$51$$ 0 0
$$52$$ 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
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 228.631i − 0.0656796i −0.999461 0.0328398i $$-0.989545\pi$$
0.999461 0.0328398i $$-0.0104551\pi$$
$$60$$ 0 0
$$61$$ −1346.00 −0.361731 −0.180865 0.983508i $$-0.557890\pi$$
−0.180865 + 0.983508i $$0.557890\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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
$$69$$ 0 0
$$70$$ 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
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1584.00 0.267161
$$78$$ 0 0
$$79$$ − 4399.41i − 0.704921i −0.935827 0.352460i $$-0.885345\pi$$
0.935827 0.352460i $$-0.114655\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 11992.7i 1.74085i 0.492301 + 0.870425i $$0.336156\pi$$
−0.492301 + 0.870425i $$0.663844\pi$$
$$84$$ 0 0
$$85$$ −10332.0 −1.43003
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −11586.0 −1.46269 −0.731347 0.682005i $$-0.761108\pi$$
−0.731347 + 0.682005i $$0.761108\pi$$
$$90$$ 0 0
$$91$$ 13870.3i 1.67495i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 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 $$-0.913396\pi$$
0.963216 0.268730i $$-0.0866039\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10080.5i 0.880473i 0.897882 + 0.440237i $$0.145105\pi$$
−0.897882 + 0.440237i $$0.854895\pi$$
$$108$$ 0 0
$$109$$ 16166.0 1.36066 0.680330 0.732906i $$-0.261836\pi$$
0.680330 + 0.732906i $$0.261836\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1842.00 −0.144256 −0.0721278 0.997395i $$-0.522979\pi$$
−0.0721278 + 0.997395i $$0.522979\pi$$
$$114$$ 0 0
$$115$$ − 31426.3i − 2.37628i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 18747.7i 1.32390i
$$120$$ 0 0
$$121$$ 14209.0 0.970494
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −21588.0 −1.38163
$$126$$ 0 0
$$127$$ − 394.908i − 0.0244843i −0.999925 0.0122422i $$-0.996103\pi$$
0.999925 0.0122422i $$-0.00389690\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 353.338i − 0.0205896i −0.999947 0.0102948i $$-0.996723\pi$$
0.999947 0.0102948i $$-0.00327700\pi$$
$$132$$ 0 0
$$133$$ −8976.00 −0.507434
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 13254.0 0.706164 0.353082 0.935592i $$-0.385134\pi$$
0.353082 + 0.935592i $$0.385134\pi$$
$$138$$ 0 0
$$139$$ − 13212.1i − 0.683820i −0.939733 0.341910i $$-0.888926\pi$$
0.939733 0.341910i $$-0.111074\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 3782.80i − 0.184987i
$$144$$ 0 0
$$145$$ −3276.00 −0.155815
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 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 $$-0.789093\pi$$
0.788406 0.615155i $$-0.210907\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 61979.7i − 2.57980i
$$156$$ 0 0
$$157$$ −19346.0 −0.784859 −0.392430 0.919782i $$-0.628365\pi$$
−0.392430 + 0.919782i $$0.628365\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 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
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ 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
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 16856.3i 0.526086i 0.964784 + 0.263043i $$0.0847261\pi$$
−0.964784 + 0.263043i $$0.915274\pi$$
$$180$$ 0 0
$$181$$ −15706.0 −0.479411 −0.239706 0.970846i $$-0.577051\pi$$
−0.239706 + 0.970846i $$0.577051\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 22260.0 0.650402
$$186$$ 0 0
$$187$$ − 5113.01i − 0.146216i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$195$$ 0 0
$$196$$ 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 $$-0.905886\pi$$
0.956608 0.291380i $$-0.0941143\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 5944.40i 0.144250i
$$204$$ 0 0
$$205$$ −38556.0 −0.917454
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 35791.1i − 0.774280i
$$216$$ 0 0
$$217$$ −112464. −2.38833
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 44772.0 0.916689
$$222$$ 0 0
$$223$$ 852.169i 0.0171363i 0.999963 + 0.00856813i $$0.00272735\pi$$
−0.999963 + 0.00856813i $$0.997273\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 76175.6i 1.47831i 0.673538 + 0.739153i $$0.264774\pi$$
−0.673538 + 0.739153i $$0.735226\pi$$
$$228$$ 0 0
$$229$$ 48470.0 0.924277 0.462138 0.886808i $$-0.347082\pi$$
0.462138 + 0.886808i $$0.347082\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 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.i 2.87691i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 143094. 2.38391
$$246$$ 0 0
$$247$$ 21435.9i 0.351356i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 91888.8i − 1.45853i −0.684232 0.729264i $$-0.739862\pi$$
0.684232 0.729264i $$-0.260138\pi$$
$$252$$ 0 0
$$253$$ 15552.0 0.242966
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 48894.0 0.740269 0.370134 0.928978i $$-0.379312\pi$$
0.370134 + 0.928978i $$0.379312\pi$$
$$258$$ 0 0
$$259$$ − 40391.4i − 0.602129i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 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
$$267$$ 0 0
$$268$$ 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 $$-0.736416\pi$$
0.676296 0.736630i $$-0.263584\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 23673.7i − 0.313040i
$$276$$ 0 0
$$277$$ 120518. 1.57070 0.785348 0.619054i $$-0.212484\pi$$
0.785348 + 0.619054i $$0.212484\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 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.i − 1.66014i −0.557657 0.830071i $$-0.688300\pi$$
0.557657 0.830071i $$-0.311700\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 69961.0i 0.849361i
$$288$$ 0 0
$$289$$ −23005.0 −0.275440
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 136181.i 1.52326i
$$300$$ 0 0
$$301$$ −64944.0 −0.716813
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 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
$$309$$ 0 0
$$310$$ 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
$$315$$ 0 0
$$316$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 28973.7i 0.277715i
$$324$$ 0 0
$$325$$ 207298. 1.96258
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 30672.0 0.263775
$$342$$ 0 0
$$343$$ − 76667.5i − 0.651663i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 56929.0i 0.472797i 0.971656 + 0.236399i $$0.0759671\pi$$
−0.971656 + 0.236399i $$0.924033\pi$$
$$348$$ 0 0
$$349$$ −181346. −1.48887 −0.744436 0.667694i $$-0.767281\pi$$
−0.744436 + 0.667694i $$0.767281\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4302.00 0.0345240 0.0172620 0.999851i $$-0.494505\pi$$
0.0172620 + 0.999851i $$0.494505\pi$$
$$354$$ 0 0
$$355$$ − 76819.9i − 0.609561i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 38892.0 0.291927
$$366$$ 0 0
$$367$$ 182690.i 1.35638i 0.734885 + 0.678191i $$0.237236\pi$$
−0.734885 + 0.678191i $$0.762764\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 352549.i − 2.56136i
$$372$$ 0 0
$$373$$ −151778. −1.09092 −0.545458 0.838138i $$-0.683644\pi$$
−0.545458 + 0.838138i $$0.683644\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 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
$$387$$ 0 0
$$388$$ 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
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 184775.i 1.18427i
$$396$$ 0 0
$$397$$ 27934.0 0.177236 0.0886180 0.996066i $$-0.471755\pi$$
0.0886180 + 0.996066i $$0.471755\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −237882. −1.47936 −0.739678 0.672961i $$-0.765022\pi$$
−0.739678 + 0.672961i $$0.765022\pi$$
$$402$$ 0 0
$$403$$ 268579.i 1.65372i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 17424.0 0.102152
$$414$$ 0 0
$$415$$ − 503694.i − 2.92463i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 24089.4i − 0.137214i −0.997644 0.0686068i $$-0.978145\pi$$
0.997644 0.0686068i $$-0.0218554\pi$$
$$420$$ 0 0
$$421$$ −116698. −0.658414 −0.329207 0.944258i $$-0.606781\pi$$
−0.329207 + 0.944258i $$0.606781\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 280194. 1.55125
$$426$$ 0 0
$$427$$ − 102579.i − 0.562604i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −88128.0 −0.461478
$$438$$ 0 0
$$439$$ 146469.i 0.760006i 0.924985 + 0.380003i $$0.124077\pi$$
−0.924985 + 0.380003i $$0.875923\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 50444.2i 0.257042i 0.991707 + 0.128521i $$0.0410230\pi$$
−0.991707 + 0.128521i $$0.958977\pi$$
$$444$$ 0 0
$$445$$ 486612. 2.45733
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −149994. −0.744014 −0.372007 0.928230i $$-0.621330\pi$$
−0.372007 + 0.928230i $$0.621330\pi$$
$$450$$ 0 0
$$451$$ − 19080.3i − 0.0938062i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 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 $$0.131741\pi$$
−0.915568 + 0.402162i $$0.868259\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 68734.7i − 0.315168i −0.987506 0.157584i $$-0.949629\pi$$
0.987506 0.157584i $$-0.0503705\pi$$
$$468$$ 0 0
$$469$$ 82896.0 0.376867
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 17712.0 0.0791672
$$474$$ 0 0
$$475$$ 134151.i 0.594574i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 550956. 2.34225
$$486$$ 0 0
$$487$$ 271108.i 1.14310i 0.820568 + 0.571549i $$0.193657\pi$$
−0.820568 + 0.571549i $$0.806343\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 227862.i − 0.945166i −0.881286 0.472583i $$-0.843322\pi$$
0.881286 0.472583i $$-0.156678\pi$$
$$492$$ 0 0
$$493$$ 19188.0 0.0789470
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 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
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ 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
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 239480.i 0.902933i
$$516$$ 0 0
$$517$$ −78624.0 −0.294154
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 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.i 0.495409i 0.968836 + 0.247704i $$0.0796762\pi$$
−0.968836 + 0.247704i $$0.920324\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 363024.i 1.30712i
$$528$$ 0 0
$$529$$ −280031. −1.00068
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 167076. 0.588111
$$534$$ 0 0
$$535$$ − 423382.i − 1.47919i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 70813.2i 0.243745i
$$540$$ 0 0
$$541$$ −360442. −1.23152 −0.615759 0.787934i $$-0.711151\pi$$
−0.615759 + 0.787934i $$0.711151\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9186.80i 0.0302594i
$$552$$ 0 0
$$553$$ 335280. 1.09637
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 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
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 419704.i 1.32412i 0.749453 + 0.662058i $$0.230317\pi$$
−0.749453 + 0.662058i $$0.769683\pi$$
$$564$$ 0 0
$$565$$ 77364.0 0.242349
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 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.i − 0.982640i −0.870979 0.491320i $$-0.836515\pi$$
0.870979 0.491320i $$-0.163485\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −913968. −2.70756
$$582$$ 0 0
$$583$$ 96149.6i 0.282885i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 114128.i − 0.331220i −0.986191 0.165610i $$-0.947041\pi$$
0.986191 0.165610i $$-0.0529594\pi$$
$$588$$ 0 0
$$589$$ −173808. −0.501002
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 96846.0 0.275405 0.137703 0.990474i $$-0.456028\pi$$
0.137703 + 0.990474i $$0.456028\pi$$
$$594$$ 0 0
$$595$$ − 787404.i − 2.22415i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −596778. −1.63043
$$606$$ 0 0
$$607$$ − 133195.i − 0.361501i −0.983529 0.180751i $$-0.942147\pi$$
0.983529 0.180751i $$-0.0578527\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 688469.i − 1.84418i
$$612$$ 0 0
$$613$$ −247202. −0.657856 −0.328928 0.944355i $$-0.606687\pi$$
−0.328928 + 0.944355i $$0.606687\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 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.i − 1.71232i −0.516712 0.856160i $$-0.672844\pi$$
0.516712 0.856160i $$-0.327156\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 882972.i − 2.27494i
$$624$$ 0 0
$$625$$ 194821. 0.498742
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −130380. −0.329541
$$630$$ 0 0
$$631$$ 417736.i 1.04916i 0.851360 + 0.524582i $$0.175778\pi$$
−0.851360 + 0.524582i $$0.824222\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16586.1i 0.0411337i
$$636$$ 0 0
$$637$$ −620074. −1.52815
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 152214. 0.370458 0.185229 0.982695i $$-0.440697\pi$$
0.185229 + 0.982695i $$0.440697\pi$$
$$642$$ 0 0
$$643$$ 714138.i 1.72727i 0.504117 + 0.863635i $$0.331818\pi$$
−0.504117 + 0.863635i $$0.668182\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 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
$$651$$ 0 0
$$652$$ 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
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 253884.i 0.584608i 0.956326 + 0.292304i $$0.0944219\pi$$
−0.956326 + 0.292304i $$0.905578\pi$$
$$660$$ 0 0
$$661$$ 722158. 1.65283 0.826417 0.563058i $$-0.190375\pi$$
0.826417 + 0.563058i $$0.190375\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 376992. 0.852489
$$666$$ 0 0
$$667$$ 58363.2i 0.131186i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 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
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 23715.2i − 0.0508377i −0.999677 0.0254189i $$-0.991908\pi$$
0.999677 0.0254189i $$-0.00809195\pi$$
$$684$$ 0 0
$$685$$ −556668. −1.18636
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 554908.i 1.14882i
$$696$$ 0 0
$$697$$ 225828. 0.464849
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 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
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 418394.i − 0.837041i
$$708$$ 0 0
$$709$$ −533002. −1.06032 −0.530159 0.847898i $$-0.677868\pi$$
−0.530159 + 0.847898i $$0.677868\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.10419e6 −2.17203
$$714$$ 0 0
$$715$$ 158878.i 0.310778i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 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
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 88842.0 0.169022
$$726$$ 0 0
$$727$$ − 755791.i − 1.42999i −0.699130 0.714995i $$-0.746429\pi$$
0.699130 0.714995i $$-0.253571\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 209634.i 0.392307i
$$732$$ 0 0
$$733$$ 832982. 1.55034 0.775171 0.631751i $$-0.217664\pi$$
0.775171 + 0.631751i $$0.217664\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 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
$$741$$ 0 0
$$742$$ 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
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −768240. −1.36941
$$750$$ 0 0
$$751$$ 937060.i 1.66145i 0.556682 + 0.830726i $$0.312074\pi$$
−0.556682 + 0.830726i $$0.687926\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.17820e6i 2.06692i
$$756$$ 0 0
$$757$$ −295786. −0.516162 −0.258081 0.966123i $$-0.583090\pi$$
−0.258081 + 0.966123i $$0.583090\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.02615e6 1.77191 0.885955 0.463772i $$-0.153504\pi$$
0.885955 + 0.463772i $$0.153504\pi$$
$$762$$ 0 0
$$763$$ 1.23201e6i 2.11625i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 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
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 108122.i 0.178171i
$$780$$ 0 0
$$781$$ 38016.0 0.0623253
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 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
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 140379.i − 0.224362i
$$792$$ 0 0
$$793$$ −244972. −0.389556
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 19246.5i 0.0298484i
$$804$$ 0 0
$$805$$ 2.39501e6 3.69586
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 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.i 1.39907i 0.714599 + 0.699534i $$0.246609\pi$$
−0.714599 + 0.699534i $$0.753391\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1.52272e6i − 2.29248i
$$816$$ 0 0
$$817$$ −100368. −0.150367
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 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 $$-0.904401\pi$$
0.955238 0.295840i $$-0.0955995\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 17272.0i − 0.0252541i −0.999920 0.0126270i $$-0.995981\pi$$
0.999920 0.0126270i $$-0.00401942\pi$$
$$828$$ 0 0
$$829$$ 15686.0 0.0228246 0.0114123 0.999935i $$-0.496367\pi$$
0.0114123 + 0.999935i $$0.496367\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −838122. −1.20786
$$834$$ 0 0
$$835$$ − 787404.i − 1.12934i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −191646. −0.268402
$$846$$ 0 0
$$847$$ 1.08287e6i 1.50942i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 396570.i − 0.547597i
$$852$$ 0 0
$$853$$ −345938. −0.475445 −0.237722 0.971333i $$-0.576401\pi$$
−0.237722 + 0.971333i $$0.576401\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 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.i 0.707983i 0.935249 + 0.353992i $$0.115176\pi$$
−0.935249 + 0.353992i $$0.884824\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −91440.0 −0.121087
$$870$$ 0 0
$$871$$ − 197966.i − 0.260949i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 1.64523e6i − 2.14887i
$$876$$ 0 0
$$877$$ −1.11629e6 −1.45137 −0.725685 0.688028i $$-0.758477\pi$$
−0.725685 + 0.688028i $$0.758477\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −19170.0 −0.0246985 −0.0123492 0.999924i $$-0.503931\pi$$
−0.0123492 + 0.999924i $$0.503931\pi$$
$$882$$ 0 0
$$883$$ − 568909.i − 0.729662i −0.931074 0.364831i $$-0.881127\pi$$
0.931074 0.364831i $$-0.118873\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 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
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 445536. 0.558702
$$894$$ 0 0
$$895$$ − 707965.i − 0.883824i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 115105.i 0.142421i
$$900$$ 0 0
$$901$$ −1.13800e6 −1.40182
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 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
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 26928.0 0.0320233
$$918$$ 0 0
$$919$$ − 97084.9i − 0.114953i −0.998347 0.0574766i $$-0.981695\pi$$
0.998347 0.0574766i $$-0.0183054\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 332886.i 0.390744i
$$924$$ 0 0
$$925$$ −603670. −0.705531
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.27882e6 1.48176 0.740881 0.671636i $$-0.234408\pi$$
0.740881 + 0.671636i $$0.234408\pi$$
$$930$$ 0 0
$$931$$ − 401275.i − 0.462959i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 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
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 993109.i − 1.10738i −0.832722 0.553691i $$-0.813219\pi$$
0.832722 0.553691i $$-0.186781\pi$$
$$948$$ 0 0
$$949$$ −168532. −0.187133
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 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.i 0.122516i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1.01009e6i 1.09831i
$$960$$ 0 0
$$961$$ −1.25419e6 −1.35805
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1.12484e6 1.20792
$$966$$ 0 0
$$967$$ − 575810.i − 0.615781i −0.951422 0.307890i $$-0.900377\pi$$
0.951422 0.307890i $$-0.0996230\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 1.23920e6i − 1.31432i −0.753749 0.657162i $$-0.771757\pi$$
0.753749 0.657162i $$-0.228243\pi$$
$$972$$ 0 0
$$973$$ 1.00690e6 1.06355
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1.04074e6 1.09032 0.545160 0.838332i $$-0.316469\pi$$
0.545160 + 0.838332i $$0.316469\pi$$
$$978$$ 0 0
$$979$$ 240810.i 0.251252i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 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
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −637632. −0.651895
$$990$$ 0 0
$$991$$ 616007.i 0.627247i 0.949547 + 0.313623i $$0.101543\pi$$
−0.949547 + 0.313623i $$0.898457\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 969269.i 0.979035i
$$996$$ 0 0
$$997$$ 535870. 0.539100 0.269550 0.962986i $$-0.413125\pi$$
0.269550 + 0.962986i $$0.413125\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 576.5.g.d.127.2 2
3.2 odd 2 192.5.g.b.127.2 2
4.3 odd 2 inner 576.5.g.d.127.1 2
8.3 odd 2 144.5.g.f.127.1 2
8.5 even 2 144.5.g.f.127.2 2
12.11 even 2 192.5.g.b.127.1 2
24.5 odd 2 48.5.g.a.31.1 2
24.11 even 2 48.5.g.a.31.2 yes 2
48.5 odd 4 768.5.b.c.127.4 4
48.11 even 4 768.5.b.c.127.2 4
48.29 odd 4 768.5.b.c.127.1 4
48.35 even 4 768.5.b.c.127.3 4
120.29 odd 2 1200.5.e.b.751.2 2
120.53 even 4 1200.5.j.b.799.3 4
120.59 even 2 1200.5.e.b.751.1 2
120.77 even 4 1200.5.j.b.799.1 4
120.83 odd 4 1200.5.j.b.799.2 4
120.107 odd 4 1200.5.j.b.799.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 24.5 odd 2
48.5.g.a.31.2 yes 2 24.11 even 2
144.5.g.f.127.1 2 8.3 odd 2
144.5.g.f.127.2 2 8.5 even 2
192.5.g.b.127.1 2 12.11 even 2
192.5.g.b.127.2 2 3.2 odd 2
576.5.g.d.127.1 2 4.3 odd 2 inner
576.5.g.d.127.2 2 1.1 even 1 trivial
768.5.b.c.127.1 4 48.29 odd 4
768.5.b.c.127.2 4 48.11 even 4
768.5.b.c.127.3 4 48.35 even 4
768.5.b.c.127.4 4 48.5 odd 4
1200.5.e.b.751.1 2 120.59 even 2
1200.5.e.b.751.2 2 120.29 odd 2
1200.5.j.b.799.1 4 120.77 even 4
1200.5.j.b.799.2 4 120.83 odd 4
1200.5.j.b.799.3 4 120.53 even 4
1200.5.j.b.799.4 4 120.107 odd 4