# Properties

 Label 144.5.g.f.127.2 Level $144$ Weight $5$ Character 144.127 Analytic conductor $14.885$ 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] = [144,5,Mod(127,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, 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("144.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 144.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: $$14.8852746841$$ 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 $$\chi$$ $$=$$ 144.127 Dual form 144.5.g.f.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 $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$-0.182555\pi$$
0.840000 + 0.542586i $$0.182555\pi$$
$$6$$ 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 $$0.0273723\pi$$
−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
$$15$$ 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 $$-0.947841\pi$$
0.986605 0.163129i $$-0.0521588\pi$$
$$20$$ 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 $$-0.514766\pi$$
−0.0463734 + 0.998924i $$0.514766\pi$$
$$30$$ 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 $$-0.437993\pi$$
0.193572 + 0.981086i $$0.437993\pi$$
$$38$$ 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 $$-0.925983\pi$$
0.973086 0.230441i $$-0.0740167\pi$$
$$44$$ 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 $$-0.192061\pi$$
0.823425 + 0.567426i $$0.192061\pi$$
$$54$$ 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 $$0.0104551\pi$$
−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
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −7644.00 −1.80923
$$66$$ 0 0
$$67$$ 1087.73i 0.242310i 0.992634 + 0.121155i $$0.0386597\pi$$
−0.992634 + 0.121155i $$0.961340\pi$$
$$68$$ 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 $$-0.663844\pi$$
0.492301 0.870425i $$-0.336156\pi$$
$$84$$ 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 $$-0.413277\pi$$
0.269091 + 0.963115i $$0.413277\pi$$
$$102$$ 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 $$-0.854895\pi$$
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
$$111$$ 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 $$0.00327700\pi$$
−0.999947 + 0.0102948i $$0.996723\pi$$
$$132$$ 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 $$0.111074\pi$$
−0.939733 + 0.341910i $$0.888926\pi$$
$$140$$ 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 $$-0.503140\pi$$
−0.00986442 + 0.999951i $$0.503140\pi$$
$$150$$ 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 $$-0.371635\pi$$
0.392430 + 0.919782i $$0.371635\pi$$
$$158$$ 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 $$-0.760986\pi$$
0.731086 0.682286i $$-0.239014\pi$$
$$164$$ 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 $$-0.305056\pi$$
0.574861 + 0.818251i $$0.305056\pi$$
$$174$$ 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 $$-0.915274\pi$$
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
$$183$$ 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 $$-0.263644\pi$$
0.676158 + 0.736757i $$0.263644\pi$$
$$198$$ 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 $$0.0864819\pi$$
−0.963319 + 0.268361i $$0.913518\pi$$
$$212$$ 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 $$-0.735226\pi$$
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
$$231$$ 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 $$0.260138\pi$$
−0.684232 + 0.729264i $$0.739862\pi$$
$$252$$ 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 $$-0.335420\pi$$
0.494313 + 0.869284i $$0.335420\pi$$
$$270$$ 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 $$-0.787516\pi$$
−0.785348 + 0.619054i $$0.787516\pi$$
$$278$$ 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 $$0.311700\pi$$
−0.557657 + 0.830071i $$0.688300\pi$$
$$284$$ 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 $$-0.844688\pi$$
−0.883307 + 0.468795i $$0.844688\pi$$
$$294$$ 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 $$-0.990838\pi$$
0.999586 0.0287790i $$-0.00916190\pi$$
$$308$$ 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 $$-0.379998\pi$$
0.368130 + 0.929774i $$0.379998\pi$$
$$318$$ 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 $$0.0838043\pi$$
−0.965542 + 0.260248i $$0.916196\pi$$
$$332$$ 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 $$-0.924033\pi$$
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
$$351$$ 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 $$-0.316356\pi$$
0.545458 + 0.838138i $$0.316356\pi$$
$$374$$ 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 $$0.0399999\pi$$
−0.992115 + 0.125333i $$0.960000\pi$$
$$380$$ 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 $$-0.613622\pi$$
−0.349423 + 0.936965i $$0.613622\pi$$
$$390$$ 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 $$-0.528245\pi$$
−0.0886180 + 0.996066i $$0.528245\pi$$
$$398$$ 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 $$0.0218554\pi$$
−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
$$423$$ 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 $$-0.958977\pi$$
0.991707 0.128521i $$-0.0410230\pi$$
$$444$$ 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 $$-0.357987\pi$$
0.431491 + 0.902117i $$0.357987\pi$$
$$462$$ 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 $$0.0503705\pi$$
−0.987506 + 0.157584i $$0.949629\pi$$
$$468$$ 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 $$0.156678\pi$$
−0.881286 + 0.472583i $$0.843322\pi$$
$$492$$ 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 $$-0.833442\pi$$
0.866196 0.499705i $$-0.166558\pi$$
$$500$$ 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 $$-0.475816\pi$$
0.0759029 + 0.997115i $$0.475816\pi$$
$$510$$ 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 $$-0.920324\pi$$
0.968836 0.247704i $$-0.0796762\pi$$
$$524$$ 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 $$-0.288849\pi$$
0.615759 + 0.787934i $$0.288849\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −678972. −2.28591
$$546$$ 0 0
$$547$$ − 261644.i − 0.874451i −0.899352 0.437225i $$-0.855961\pi$$
0.899352 0.437225i $$-0.144039\pi$$
$$548$$ 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 $$-0.377318\pi$$
0.375946 + 0.926641i $$0.377318\pi$$
$$558$$ 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 $$-0.769683\pi$$
0.749453 0.662058i $$-0.230317\pi$$
$$564$$ 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 $$0.163485\pi$$
−0.870979 + 0.491320i $$0.836515\pi$$
$$572$$ 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 $$0.0529594\pi$$
−0.986191 + 0.165610i $$0.947041\pi$$
$$588$$ 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 $$-0.393313\pi$$
0.328928 + 0.944355i $$0.393313\pi$$
$$614$$ 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 $$0.327156\pi$$
−0.516712 + 0.856160i $$0.672844\pi$$
$$620$$ 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 $$-0.668182\pi$$
0.504117 0.863635i $$-0.331818\pi$$
$$644$$ 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 $$-0.373239\pi$$
0.387790 + 0.921748i $$0.373239\pi$$
$$654$$ 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 $$-0.905578\pi$$
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
$$663$$ 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 $$-0.731314\pi$$
−0.664403 + 0.747375i $$0.731314\pi$$
$$678$$ 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 $$0.00809195\pi$$
−0.999677 + 0.0254189i $$0.991908\pi$$
$$684$$ 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 $$0.149364\pi$$
−0.891912 + 0.452208i $$0.850636\pi$$
$$692$$ 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 $$-0.514566\pi$$
−0.0457447 + 0.998953i $$0.514566\pi$$
$$702$$ 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 $$-0.322132\pi$$
0.530159 + 0.847898i $$0.322132\pi$$
$$710$$ 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 $$-0.782336\pi$$
−0.775171 + 0.631751i $$0.782336\pi$$
$$734$$ 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 $$0.220708\pi$$
−0.769093 + 0.639137i $$0.779292\pi$$
$$740$$ 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 $$-0.416910\pi$$
0.258081 + 0.966123i $$0.416910\pi$$
$$758$$ 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 $$-0.828666\pi$$
−0.858601 + 0.512644i $$0.828666\pi$$
$$774$$ 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 $$-0.759270\pi$$
0.727398 0.686216i $$-0.240730\pi$$
$$788$$ 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 $$-0.704591\pi$$
−0.599392 + 0.800456i $$0.704591\pi$$
$$798$$ 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 $$-0.753391\pi$$
0.714599 0.699534i $$-0.246609\pi$$
$$812$$ 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 $$-0.440568\pi$$
0.185628 + 0.982620i $$0.440568\pi$$
$$822$$ 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 $$0.00401942\pi$$
−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
$$831$$ 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 $$-0.423599\pi$$
0.237722 + 0.971333i $$0.423599\pi$$
$$854$$ 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 $$-0.884824\pi$$
0.935249 0.353992i $$-0.115176\pi$$
$$860$$ 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 $$-0.241523\pi$$
0.725685 + 0.688028i $$0.241523\pi$$
$$878$$ 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 $$0.118873\pi$$
−0.931074 + 0.364831i $$0.881127\pi$$
$$884$$ 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 $$0.187992\pi$$
−0.830610 + 0.556855i $$0.812008\pi$$
$$908$$ 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 $$-0.551341\pi$$
−0.160594 + 0.987021i $$0.551341\pi$$
$$942$$ 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 $$0.186781\pi$$
−0.832722 + 0.553691i $$0.813219\pi$$
$$948$$ 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 $$0.228243\pi$$
−0.753749 + 0.657162i $$0.771757\pi$$
$$972$$ 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 $$-0.586875\pi$$
−0.269550 + 0.962986i $$0.586875\pi$$
$$998$$ 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 144.5.g.f.127.2 2
3.2 odd 2 48.5.g.a.31.1 2
4.3 odd 2 inner 144.5.g.f.127.1 2
8.3 odd 2 576.5.g.d.127.1 2
8.5 even 2 576.5.g.d.127.2 2
12.11 even 2 48.5.g.a.31.2 yes 2
15.2 even 4 1200.5.j.b.799.1 4
15.8 even 4 1200.5.j.b.799.3 4
15.14 odd 2 1200.5.e.b.751.2 2
24.5 odd 2 192.5.g.b.127.2 2
24.11 even 2 192.5.g.b.127.1 2
48.5 odd 4 768.5.b.c.127.1 4
48.11 even 4 768.5.b.c.127.3 4
48.29 odd 4 768.5.b.c.127.4 4
48.35 even 4 768.5.b.c.127.2 4
60.23 odd 4 1200.5.j.b.799.2 4
60.47 odd 4 1200.5.j.b.799.4 4
60.59 even 2 1200.5.e.b.751.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 3.2 odd 2
48.5.g.a.31.2 yes 2 12.11 even 2
144.5.g.f.127.1 2 4.3 odd 2 inner
144.5.g.f.127.2 2 1.1 even 1 trivial
192.5.g.b.127.1 2 24.11 even 2
192.5.g.b.127.2 2 24.5 odd 2
576.5.g.d.127.1 2 8.3 odd 2
576.5.g.d.127.2 2 8.5 even 2
768.5.b.c.127.1 4 48.5 odd 4
768.5.b.c.127.2 4 48.35 even 4
768.5.b.c.127.3 4 48.11 even 4
768.5.b.c.127.4 4 48.29 odd 4
1200.5.e.b.751.1 2 60.59 even 2
1200.5.e.b.751.2 2 15.14 odd 2
1200.5.j.b.799.1 4 15.2 even 4
1200.5.j.b.799.2 4 60.23 odd 4
1200.5.j.b.799.3 4 15.8 even 4
1200.5.j.b.799.4 4 60.47 odd 4