# Properties

 Label 1200.5.j.b.799.3 Level $1200$ Weight $5$ Character 1200.799 Analytic conductor $124.044$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,5,Mod(799,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.799");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 1200.j (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: $$124.043955701$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.3 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.799 Dual form 1200.5.j.b.799.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} +O(q^{10})$$ $$q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} -20.7846i q^{11} -182.000i q^{13} +246.000i q^{17} +117.779i q^{19} +396.000 q^{21} +748.246 q^{23} +140.296 q^{27} -78.0000 q^{29} +1475.71i q^{31} -108.000i q^{33} -530.000i q^{37} -945.700i q^{39} -918.000 q^{41} +852.169 q^{43} +3782.80 q^{47} +3407.00 q^{49} +1278.25i q^{51} -4626.00i q^{53} +612.000i q^{57} +228.631i q^{59} +1346.00 q^{61} +2057.68 q^{63} +1087.73 q^{67} +3888.00 q^{69} -1829.05i q^{71} -926.000i q^{73} -1584.00i q^{77} +4399.41i q^{79} +729.000 q^{81} -11992.7 q^{83} -405.300 q^{87} -11586.0 q^{89} -13870.3i q^{91} +7668.00i q^{93} +13118.0i q^{97} -561.184i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 108 q^{9}+O(q^{10})$$ 4 * q + 108 * q^9 $$4 q + 108 q^{9} + 1584 q^{21} - 312 q^{29} - 3672 q^{41} + 13628 q^{49} + 5384 q^{61} + 15552 q^{69} + 2916 q^{81} - 46344 q^{89}+O(q^{100})$$ 4 * q + 108 * q^9 + 1584 * q^21 - 312 * q^29 - 3672 * q^41 + 13628 * q^49 + 5384 * q^61 + 15552 * q^69 + 2916 * q^81 - 46344 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 5.19615 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 76.2102 1.55531 0.777655 0.628691i $$-0.216409\pi$$
0.777655 + 0.628691i $$0.216409\pi$$
$$8$$ 0 0
$$9$$ 27.0000 0.333333
$$10$$ 0 0
$$11$$ − 20.7846i − 0.171774i −0.996305 0.0858868i $$-0.972628\pi$$
0.996305 0.0858868i $$-0.0273723\pi$$
$$12$$ 0 0
$$13$$ − 182.000i − 1.07692i −0.842650 0.538462i $$-0.819006\pi$$
0.842650 0.538462i $$-0.180994\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 246.000i 0.851211i 0.904909 + 0.425606i $$0.139939\pi$$
−0.904909 + 0.425606i $$0.860061\pi$$
$$18$$ 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
$$23$$ 748.246 1.41445 0.707227 0.706987i $$-0.249946\pi$$
0.707227 + 0.706987i $$0.249946\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 140.296 0.192450
$$28$$ 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
$$33$$ − 108.000i − 0.0991736i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 530.000i − 0.387144i −0.981086 0.193572i $$-0.937993\pi$$
0.981086 0.193572i $$-0.0620073\pi$$
$$38$$ 0 0
$$39$$ − 945.700i − 0.621762i
$$40$$ 0 0
$$41$$ −918.000 −0.546104 −0.273052 0.961999i $$-0.588033\pi$$
−0.273052 + 0.961999i $$0.588033\pi$$
$$42$$ 0 0
$$43$$ 852.169 0.460881 0.230441 0.973086i $$-0.425983\pi$$
0.230441 + 0.973086i $$0.425983\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3782.80 1.71245 0.856224 0.516604i $$-0.172804\pi$$
0.856224 + 0.516604i $$0.172804\pi$$
$$48$$ 0 0
$$49$$ 3407.00 1.41899
$$50$$ 0 0
$$51$$ 1278.25i 0.491447i
$$52$$ 0 0
$$53$$ − 4626.00i − 1.64685i −0.567426 0.823425i $$-0.692061\pi$$
0.567426 0.823425i $$-0.307939\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 612.000i 0.188366i
$$58$$ 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
$$63$$ 2057.68 0.518437
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1087.73 0.242310 0.121155 0.992634i $$-0.461340\pi$$
0.121155 + 0.992634i $$0.461340\pi$$
$$68$$ 0 0
$$69$$ 3888.00 0.816635
$$70$$ 0 0
$$71$$ − 1829.05i − 0.362834i −0.983406 0.181417i $$-0.941932\pi$$
0.983406 0.181417i $$-0.0580684\pi$$
$$72$$ 0 0
$$73$$ − 926.000i − 0.173766i −0.996219 0.0868831i $$-0.972309\pi$$
0.996219 0.0868831i $$-0.0276907\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1584.00i − 0.267161i
$$78$$ 0 0
$$79$$ 4399.41i 0.704921i 0.935827 + 0.352460i $$0.114655\pi$$
−0.935827 + 0.352460i $$0.885345\pi$$
$$80$$ 0 0
$$81$$ 729.000 0.111111
$$82$$ 0 0
$$83$$ −11992.7 −1.74085 −0.870425 0.492301i $$-0.836156\pi$$
−0.870425 + 0.492301i $$0.836156\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −405.300 −0.0535473
$$88$$ 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
$$93$$ 7668.00i 0.886576i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13118.0i 1.39420i 0.716975 + 0.697099i $$0.245526\pi$$
−0.716975 + 0.697099i $$0.754474\pi$$
$$98$$ 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
$$103$$ 5701.91 0.537460 0.268730 0.963216i $$-0.413396\pi$$
0.268730 + 0.963216i $$0.413396\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10080.5 0.880473 0.440237 0.897882i $$-0.354895\pi$$
0.440237 + 0.897882i $$0.354895\pi$$
$$108$$ 0 0
$$109$$ 16166.0 1.36066 0.680330 0.732906i $$-0.261836\pi$$
0.680330 + 0.732906i $$0.261836\pi$$
$$110$$ 0 0
$$111$$ − 2753.96i − 0.223518i
$$112$$ 0 0
$$113$$ 1842.00i 0.144256i 0.997395 + 0.0721278i $$0.0229789\pi$$
−0.997395 + 0.0721278i $$0.977021\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4914.00i − 0.358974i
$$118$$ 0 0
$$119$$ 18747.7i 1.32390i
$$120$$ 0 0
$$121$$ 14209.0 0.970494
$$122$$ 0 0
$$123$$ −4770.07 −0.315293
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −394.908 −0.0244843 −0.0122422 0.999925i $$-0.503897\pi$$
−0.0122422 + 0.999925i $$0.503897\pi$$
$$128$$ 0 0
$$129$$ 4428.00 0.266090
$$130$$ 0 0
$$131$$ − 353.338i − 0.0205896i −0.999947 0.0102948i $$-0.996723\pi$$
0.999947 0.0102948i $$-0.00327700\pi$$
$$132$$ 0 0
$$133$$ 8976.00i 0.507434i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 13254.0i 0.706164i 0.935592 + 0.353082i $$0.114866\pi$$
−0.935592 + 0.353082i $$0.885134\pi$$
$$138$$ 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
$$143$$ −3782.80 −0.184987
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 17703.3 0.819255
$$148$$ 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
$$153$$ 6642.00i 0.283737i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 19346.0i − 0.784859i −0.919782 0.392430i $$-0.871635\pi$$
0.919782 0.392430i $$-0.128365\pi$$
$$158$$ 0 0
$$159$$ − 24037.4i − 0.950809i
$$160$$ 0 0
$$161$$ 57024.0 2.19992
$$162$$ 0 0
$$163$$ 36255.3 1.36457 0.682286 0.731086i $$-0.260986\pi$$
0.682286 + 0.731086i $$0.260986\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −18747.7 −0.672226 −0.336113 0.941822i $$-0.609112\pi$$
−0.336113 + 0.941822i $$0.609112\pi$$
$$168$$ 0 0
$$169$$ −4563.00 −0.159763
$$170$$ 0 0
$$171$$ 3180.05i 0.108753i
$$172$$ 0 0
$$173$$ − 34410.0i − 1.14972i −0.818251 0.574861i $$-0.805056\pi$$
0.818251 0.574861i $$-0.194944\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1188.00i 0.0379201i
$$178$$ 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
$$183$$ 6994.02 0.208845
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5113.01 0.146216
$$188$$ 0 0
$$189$$ 10692.0 0.299320
$$190$$ 0 0
$$191$$ 2660.43i 0.0729265i 0.999335 + 0.0364632i $$0.0116092\pi$$
−0.999335 + 0.0364632i $$0.988391\pi$$
$$192$$ 0 0
$$193$$ − 26782.0i − 0.718999i −0.933145 0.359500i $$-0.882947\pi$$
0.933145 0.359500i $$-0.117053\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 52482.0i 1.35232i 0.736757 + 0.676158i $$0.236356\pi$$
−0.736757 + 0.676158i $$0.763644\pi$$
$$198$$ 0 0
$$199$$ 23077.8i 0.582759i 0.956608 + 0.291380i $$0.0941143\pi$$
−0.956608 + 0.291380i $$0.905886\pi$$
$$200$$ 0 0
$$201$$ 5652.00 0.139898
$$202$$ 0 0
$$203$$ −5944.40 −0.144250
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 20202.6 0.471485
$$208$$ 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
$$213$$ − 9504.00i − 0.209482i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 112464.i 2.38833i
$$218$$ 0 0
$$219$$ − 4811.64i − 0.100324i
$$220$$ 0 0
$$221$$ 44772.0 0.916689
$$222$$ 0 0
$$223$$ −852.169 −0.0171363 −0.00856813 0.999963i $$-0.502727\pi$$
−0.00856813 + 0.999963i $$0.502727\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 76175.6 1.47831 0.739153 0.673538i $$-0.235226\pi$$
0.739153 + 0.673538i $$0.235226\pi$$
$$228$$ 0 0
$$229$$ 48470.0 0.924277 0.462138 0.886808i $$-0.347082\pi$$
0.462138 + 0.886808i $$0.347082\pi$$
$$230$$ 0 0
$$231$$ − 8230.71i − 0.154246i
$$232$$ 0 0
$$233$$ 48738.0i 0.897751i 0.893594 + 0.448875i $$0.148175\pi$$
−0.893594 + 0.448875i $$0.851825\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 22860.0i 0.406986i
$$238$$ 0 0
$$239$$ − 71000.2i − 1.24298i −0.783422 0.621490i $$-0.786528\pi$$
0.783422 0.621490i $$-0.213472\pi$$
$$240$$ 0 0
$$241$$ 73138.0 1.25924 0.629621 0.776903i $$-0.283210\pi$$
0.629621 + 0.776903i $$0.283210\pi$$
$$242$$ 0 0
$$243$$ 3788.00 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 21435.9 0.351356
$$248$$ 0 0
$$249$$ −62316.0 −1.00508
$$250$$ 0 0
$$251$$ − 91888.8i − 1.45853i −0.684232 0.729264i $$-0.739862\pi$$
0.684232 0.729264i $$-0.260138\pi$$
$$252$$ 0 0
$$253$$ − 15552.0i − 0.242966i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 48894.0i 0.740269i 0.928978 + 0.370134i $$0.120688\pi$$
−0.928978 + 0.370134i $$0.879312\pi$$
$$258$$ 0 0
$$259$$ − 40391.4i − 0.602129i
$$260$$ 0 0
$$261$$ −2106.00 −0.0309156
$$262$$ 0 0
$$263$$ −78191.7 −1.13044 −0.565222 0.824939i $$-0.691210\pi$$
−0.565222 + 0.824939i $$0.691210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −60202.6 −0.844487
$$268$$ 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
$$273$$ − 72072.0i − 0.967033i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 120518.i 1.57070i 0.619054 + 0.785348i $$0.287516\pi$$
−0.619054 + 0.785348i $$0.712484\pi$$
$$278$$ 0 0
$$279$$ 39844.1i 0.511865i
$$280$$ 0 0
$$281$$ −3054.00 −0.0386773 −0.0193387 0.999813i $$-0.506156\pi$$
−0.0193387 + 0.999813i $$0.506156\pi$$
$$282$$ 0 0
$$283$$ −132959. −1.66014 −0.830071 0.557657i $$-0.811700\pi$$
−0.830071 + 0.557657i $$0.811700\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −69961.0 −0.849361
$$288$$ 0 0
$$289$$ 23005.0 0.275440
$$290$$ 0 0
$$291$$ 68163.1i 0.804940i
$$292$$ 0 0
$$293$$ 151662.i 1.76661i 0.468795 + 0.883307i $$0.344688\pi$$
−0.468795 + 0.883307i $$0.655312\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2916.00i − 0.0330579i
$$298$$ 0 0
$$299$$ − 136181.i − 1.52326i
$$300$$ 0 0
$$301$$ 64944.0 0.716813
$$302$$ 0 0
$$303$$ −28526.9 −0.310720
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5424.78 −0.0575580 −0.0287790 0.999586i $$-0.509162\pi$$
−0.0287790 + 0.999586i $$0.509162\pi$$
$$308$$ 0 0
$$309$$ 29628.0 0.310303
$$310$$ 0 0
$$311$$ − 141127.i − 1.45912i −0.683917 0.729560i $$-0.739725\pi$$
0.683917 0.729560i $$-0.260275\pi$$
$$312$$ 0 0
$$313$$ − 128686.i − 1.31354i −0.754092 0.656769i $$-0.771923\pi$$
0.754092 0.656769i $$-0.228077\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 73986.0i 0.736260i 0.929774 + 0.368130i $$0.120002\pi$$
−0.929774 + 0.368130i $$0.879998\pi$$
$$318$$ 0 0
$$319$$ 1621.20i 0.0159314i
$$320$$ 0 0
$$321$$ 52380.0 0.508341
$$322$$ 0 0
$$323$$ −28973.7 −0.277715
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 84001.0 0.785577
$$328$$ 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
$$333$$ − 14310.0i − 0.129048i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 98674.0i − 0.868846i −0.900709 0.434423i $$-0.856952\pi$$
0.900709 0.434423i $$-0.143048\pi$$
$$338$$ 0 0
$$339$$ 9571.31i 0.0832860i
$$340$$ 0 0
$$341$$ 30672.0 0.263775
$$342$$ 0 0
$$343$$ 76667.5 0.651663
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 56929.0 0.472797 0.236399 0.971656i $$-0.424033\pi$$
0.236399 + 0.971656i $$0.424033\pi$$
$$348$$ 0 0
$$349$$ −181346. −1.48887 −0.744436 0.667694i $$-0.767281\pi$$
−0.744436 + 0.667694i $$0.767281\pi$$
$$350$$ 0 0
$$351$$ − 25533.9i − 0.207254i
$$352$$ 0 0
$$353$$ − 4302.00i − 0.0345240i −0.999851 0.0172620i $$-0.994505\pi$$
0.999851 0.0172620i $$-0.00549494\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 97416.0i 0.764353i
$$358$$ 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
$$363$$ 73832.1 0.560315
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 182690. 1.35638 0.678191 0.734885i $$-0.262764\pi$$
0.678191 + 0.734885i $$0.262764\pi$$
$$368$$ 0 0
$$369$$ −24786.0 −0.182035
$$370$$ 0 0
$$371$$ − 352549.i − 2.56136i
$$372$$ 0 0
$$373$$ 151778.i 1.09092i 0.838138 + 0.545458i $$0.183644\pi$$
−0.838138 + 0.545458i $$0.816356\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 14196.0i 0.0998811i
$$378$$ 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
$$383$$ 65346.8 0.445479 0.222739 0.974878i $$-0.428500\pi$$
0.222739 + 0.974878i $$0.428500\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 23008.6 0.153627
$$388$$ 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
$$393$$ − 1836.00i − 0.0118874i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 27934.0i 0.177236i 0.996066 + 0.0886180i $$0.0282450\pi$$
−0.996066 + 0.0886180i $$0.971755\pi$$
$$398$$ 0 0
$$399$$ 46640.7i 0.292967i
$$400$$ 0 0
$$401$$ 237882. 1.47936 0.739678 0.672961i $$-0.234978\pi$$
0.739678 + 0.672961i $$0.234978\pi$$
$$402$$ 0 0
$$403$$ 268579. 1.65372
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −11015.8 −0.0665011
$$408$$ 0 0
$$409$$ 20270.0 0.121173 0.0605867 0.998163i $$-0.480703\pi$$
0.0605867 + 0.998163i $$0.480703\pi$$
$$410$$ 0 0
$$411$$ 68869.8i 0.407704i
$$412$$ 0 0
$$413$$ 17424.0i 0.102152i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 68652.0i − 0.394804i
$$418$$ 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. 0.570816
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 102579. 0.562604
$$428$$ 0 0
$$429$$ −19656.0 −0.106802
$$430$$ 0 0
$$431$$ 355542.i 1.91397i 0.290132 + 0.956986i $$0.406301\pi$$
−0.290132 + 0.956986i $$0.593699\pi$$
$$432$$ 0 0
$$433$$ − 199726.i − 1.06527i −0.846346 0.532634i $$-0.821202\pi$$
0.846346 0.532634i $$-0.178798\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 88128.0i 0.461478i
$$438$$ 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
$$443$$ −50444.2 −0.257042 −0.128521 0.991707i $$-0.541023\pi$$
−0.128521 + 0.991707i $$0.541023\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −2275.91 −0.0113905
$$448$$ 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
$$453$$ − 145764.i − 0.710320i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 284338.i − 1.36145i −0.732538 0.680726i $$-0.761664\pi$$
0.732538 0.680726i $$-0.238336\pi$$
$$458$$ 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. −0.804324 −0.402162 0.915568i $$-0.631741\pi$$
−0.402162 + 0.915568i $$0.631741\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −68734.7 −0.315168 −0.157584 0.987506i $$-0.550371\pi$$
−0.157584 + 0.987506i $$0.550371\pi$$
$$468$$ 0 0
$$469$$ 82896.0 0.376867
$$470$$ 0 0
$$471$$ − 100525.i − 0.453139i
$$472$$ 0 0
$$473$$ − 17712.0i − 0.0791672i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 124902.i − 0.548950i
$$478$$ 0 0
$$479$$ 249956.i 1.08941i 0.838627 + 0.544706i $$0.183359\pi$$
−0.838627 + 0.544706i $$0.816641\pi$$
$$480$$ 0 0
$$481$$ −96460.0 −0.416924
$$482$$ 0 0
$$483$$ 296305. 1.27012
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 271108. 1.14310 0.571549 0.820568i $$-0.306343\pi$$
0.571549 + 0.820568i $$0.306343\pi$$
$$488$$ 0 0
$$489$$ 188388. 0.787835
$$490$$ 0 0
$$491$$ − 227862.i − 0.945166i −0.881286 0.472583i $$-0.843322\pi$$
0.881286 0.472583i $$-0.156678\pi$$
$$492$$ 0 0
$$493$$ − 19188.0i − 0.0789470i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 139392.i − 0.564320i
$$498$$ 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. −1.76490 −0.882452 0.470403i $$-0.844109\pi$$
−0.882452 + 0.470403i $$0.844109\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −23710.0 −0.0922394
$$508$$ 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
$$513$$ 16524.0i 0.0627886i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 78624.0i − 0.294154i
$$518$$ 0 0
$$519$$ − 178800.i − 0.663792i
$$520$$ 0 0
$$521$$ −464598. −1.71160 −0.855799 0.517308i $$-0.826934\pi$$
−0.855799 + 0.517308i $$0.826934\pi$$
$$522$$ 0 0
$$523$$ 135509. 0.495409 0.247704 0.968836i $$-0.420324\pi$$
0.247704 + 0.968836i $$0.420324\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −363024. −1.30712
$$528$$ 0 0
$$529$$ 280031. 1.00068
$$530$$ 0 0
$$531$$ 6173.03i 0.0218932i
$$532$$ 0 0
$$533$$ 167076.i 0.588111i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 87588.0i − 0.303736i
$$538$$ 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
$$543$$ 81610.8 0.276788
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −261644. −0.874451 −0.437225 0.899352i $$-0.644039\pi$$
−0.437225 + 0.899352i $$0.644039\pi$$
$$548$$ 0 0
$$549$$ 36342.0 0.120577
$$550$$ 0 0
$$551$$ − 9186.80i − 0.0302594i
$$552$$ 0 0
$$553$$ 335280.i 1.09637i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 233274.i 0.751893i 0.926641 + 0.375946i $$0.122682\pi$$
−0.926641 + 0.375946i $$0.877318\pi$$
$$558$$ 0 0
$$559$$ − 155095.i − 0.496333i
$$560$$ 0 0
$$561$$ 26568.0 0.0844176
$$562$$ 0 0
$$563$$ −419704. −1.32412 −0.662058 0.749453i $$-0.730317\pi$$
−0.662058 + 0.749453i $$0.730317\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 55557.3 0.172812
$$568$$ 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
$$573$$ 13824.0i 0.0421041i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 341038.i 1.02436i 0.858879 + 0.512178i $$0.171161\pi$$
−0.858879 + 0.512178i $$0.828839\pi$$
$$578$$ 0 0
$$579$$ − 139163.i − 0.415114i
$$580$$ 0 0
$$581$$ −913968. −2.70756
$$582$$ 0 0
$$583$$ −96149.6 −0.282885
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −114128. −0.331220 −0.165610 0.986191i $$-0.552959\pi$$
−0.165610 + 0.986191i $$0.552959\pi$$
$$588$$ 0 0
$$589$$ −173808. −0.501002
$$590$$ 0 0
$$591$$ 272704.i 0.780760i
$$592$$ 0 0
$$593$$ − 96846.0i − 0.275405i −0.990474 0.137703i $$-0.956028\pi$$
0.990474 0.137703i $$-0.0439718\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 119916.i 0.336456i
$$598$$ 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
$$603$$ 29368.7 0.0807699
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −133195. −0.361501 −0.180751 0.983529i $$-0.557853\pi$$
−0.180751 + 0.983529i $$0.557853\pi$$
$$608$$ 0 0
$$609$$ −30888.0 −0.0832828
$$610$$ 0 0
$$611$$ − 688469.i − 1.84418i
$$612$$ 0 0
$$613$$ 247202.i 0.657856i 0.944355 + 0.328928i $$0.106687\pi$$
−0.944355 + 0.328928i $$0.893313\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 31758.0i 0.0834224i 0.999130 + 0.0417112i $$0.0132809\pi$$
−0.999130 + 0.0417112i $$0.986719\pi$$
$$618$$ 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. −2.27494
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 12720.2 0.0323563
$$628$$ 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
$$633$$ 124164.i 0.309876i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 620074.i − 1.52815i
$$638$$ 0 0
$$639$$ − 49384.2i − 0.120945i
$$640$$ 0 0
$$641$$ −152214. −0.370458 −0.185229 0.982695i $$-0.559303\pi$$
−0.185229 + 0.982695i $$0.559303\pi$$
$$642$$ 0 0
$$643$$ 714138. 1.72727 0.863635 0.504117i $$-0.168182\pi$$
0.863635 + 0.504117i $$0.168182\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −259558. −0.620049 −0.310025 0.950729i $$-0.600337\pi$$
−0.310025 + 0.950729i $$0.600337\pi$$
$$648$$ 0 0
$$649$$ 4752.00 0.0112820
$$650$$ 0 0
$$651$$ 584380.i 1.37890i
$$652$$ 0 0
$$653$$ − 330714.i − 0.775579i −0.921748 0.387790i $$-0.873239\pi$$
0.921748 0.387790i $$-0.126761\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 25002.0i − 0.0579221i
$$658$$ 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. 0.529251
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −58363.2 −0.131186
$$668$$ 0 0
$$669$$ −4428.00 −0.00989362
$$670$$ 0 0
$$671$$ − 27976.1i − 0.0621358i
$$672$$ 0 0
$$673$$ − 552910.i − 1.22074i −0.792115 0.610372i $$-0.791020\pi$$
0.792115 0.610372i $$-0.208980\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 609030.i − 1.32881i −0.747375 0.664403i $$-0.768686\pi$$
0.747375 0.664403i $$-0.231314\pi$$
$$678$$ 0 0
$$679$$ 999726.i 2.16841i
$$680$$ 0 0
$$681$$ 395820. 0.853500
$$682$$ 0 0
$$683$$ 23715.2 0.0508377 0.0254189 0.999677i $$-0.491908\pi$$
0.0254189 + 0.999677i $$0.491908\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 251858. 0.533631
$$688$$ 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
$$693$$ − 42768.0i − 0.0890538i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 225828.i − 0.464849i
$$698$$ 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
$$703$$ 62423.1 0.126309
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −418394. −0.837041
$$708$$ 0 0
$$709$$ −533002. −1.06032 −0.530159 0.847898i $$-0.677868\pi$$
−0.530159 + 0.847898i $$0.677868\pi$$
$$710$$ 0 0
$$711$$ 118784.i 0.234974i
$$712$$ 0 0
$$713$$ 1.10419e6i 2.17203i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 368928.i − 0.717634i
$$718$$ 0 0
$$719$$ − 292107.i − 0.565046i −0.959260 0.282523i $$-0.908829\pi$$
0.959260 0.282523i $$-0.0911714\pi$$
$$720$$ 0 0
$$721$$ 434544. 0.835917
$$722$$ 0 0
$$723$$ 380036. 0.727023
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −755791. −1.42999 −0.714995 0.699130i $$-0.753571\pi$$
−0.714995 + 0.699130i $$0.753571\pi$$
$$728$$ 0 0
$$729$$ 19683.0 0.0370370
$$730$$ 0 0
$$731$$ 209634.i 0.392307i
$$732$$ 0 0
$$733$$ − 832982.i − 1.55034i −0.631751 0.775171i $$-0.717664\pi$$
0.631751 0.775171i $$-0.282336\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 22608.0i − 0.0416224i
$$738$$ 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. −0.835151 −0.417575 0.908642i $$-0.637120\pi$$
−0.417575 + 0.908642i $$0.637120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −323803. −0.580284
$$748$$ 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
$$753$$ − 477468.i − 0.842082i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 295786.i − 0.516162i −0.966123 0.258081i $$-0.916910\pi$$
0.966123 0.258081i $$-0.0830901\pi$$
$$758$$ 0 0
$$759$$ − 80810.6i − 0.140276i
$$760$$ 0 0
$$761$$ −1.02615e6 −1.77191 −0.885955 0.463772i $$-0.846496\pi$$
−0.885955 + 0.463772i $$0.846496\pi$$
$$762$$ 0 0
$$763$$ 1.23201e6 2.11625
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 41610.8 0.0707319
$$768$$ 0 0
$$769$$ −362306. −0.612665 −0.306332 0.951925i $$-0.599102\pi$$
−0.306332 + 0.951925i $$0.599102\pi$$
$$770$$ 0 0
$$771$$ 254061.i 0.427394i
$$772$$ 0 0
$$773$$ 1.02608e6i 1.71720i 0.512644 + 0.858601i $$0.328666\pi$$
−0.512644 + 0.858601i $$0.671334\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 209880.i − 0.347639i
$$778$$ 0 0
$$779$$ − 108122.i − 0.178171i
$$780$$ 0 0
$$781$$ −38016.0 −0.0623253
$$782$$ 0 0
$$783$$ −10943.1 −0.0178491
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −850042. −1.37243 −0.686216 0.727398i $$-0.740730\pi$$
−0.686216 + 0.727398i $$0.740730\pi$$
$$788$$ 0 0
$$789$$ −406296. −0.652662
$$790$$ 0 0
$$791$$ 140379.i 0.224362i
$$792$$ 0 0
$$793$$ − 244972.i − 0.389556i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 761478.i − 1.19878i −0.800456 0.599392i $$-0.795409\pi$$
0.800456 0.599392i $$-0.204591\pi$$
$$798$$ 0 0
$$799$$ 930569.i 1.45766i
$$800$$ 0 0
$$801$$ −312822. −0.487565
$$802$$ 0 0
$$803$$ −19246.5 −0.0298484
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 371722. 0.570784
$$808$$ 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
$$813$$ − 562212.i − 0.850588i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 100368.i 0.150367i
$$818$$ 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. 0.591680 0.295840 0.955238i $$-0.404401\pi$$
0.295840 + 0.955238i $$0.404401\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −17272.0 −0.0252541 −0.0126270 0.999920i $$-0.504019\pi$$
−0.0126270 + 0.999920i $$0.504019\pi$$
$$828$$ 0 0
$$829$$ 15686.0 0.0228246 0.0114123 0.999935i $$-0.496367\pi$$
0.0114123 + 0.999935i $$0.496367\pi$$
$$830$$ 0 0
$$831$$ 626230.i 0.906842i
$$832$$ 0 0
$$833$$ 838122.i 1.20786i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 207036.i 0.295525i
$$838$$ 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
$$843$$ −15869.0 −0.0223304
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.08287e6 1.50942
$$848$$ 0 0
$$849$$ −690876. −0.958484
$$850$$ 0 0
$$851$$ − 396570.i − 0.547597i
$$852$$ 0 0
$$853$$ 345938.i 0.475445i 0.971333 + 0.237722i $$0.0764009\pi$$
−0.971333 + 0.237722i $$0.923599\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 267990.i 0.364886i 0.983216 + 0.182443i $$0.0584005\pi$$
−0.983216 + 0.182443i $$0.941600\pi$$
$$858$$ 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. −1.11027 −0.555135 0.831760i $$-0.687333\pi$$
−0.555135 + 0.831760i $$0.687333\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 119537. 0.159025
$$868$$ 0 0
$$869$$ 91440.0 0.121087
$$870$$ 0 0
$$871$$ − 197966.i − 0.260949i
$$872$$ 0 0
$$873$$ 354186.i 0.464732i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 1.11629e6i − 1.45137i −0.688028 0.725685i $$-0.741523\pi$$
0.688028 0.725685i $$-0.258477\pi$$
$$878$$ 0 0
$$879$$ 788059.i 1.01995i
$$880$$ 0 0
$$881$$ 19170.0 0.0246985 0.0123492 0.999924i $$-0.496069\pi$$
0.0123492 + 0.999924i $$0.496069\pi$$
$$882$$ 0 0
$$883$$ −568909. −0.729662 −0.364831 0.931074i $$-0.618873\pi$$
−0.364831 + 0.931074i $$0.618873\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −1.09015e6 −1.38561 −0.692804 0.721126i $$-0.743625\pi$$
−0.692804 + 0.721126i $$0.743625\pi$$
$$888$$ 0 0
$$889$$ −30096.0 −0.0380807
$$890$$ 0 0
$$891$$ − 15152.0i − 0.0190860i
$$892$$ 0 0
$$893$$ 445536.i 0.558702i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 707616.i − 0.879453i
$$898$$ 0 0
$$899$$ − 115105.i − 0.142421i
$$900$$ 0 0
$$901$$ 1.13800e6 1.40182
$$902$$ 0 0
$$903$$ 337459. 0.413852
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 916193. 1.11371 0.556855 0.830610i $$-0.312008\pi$$
0.556855 + 0.830610i $$0.312008\pi$$
$$908$$ 0 0
$$909$$ −148230. −0.179394
$$910$$ 0 0
$$911$$ − 995500.i − 1.19951i −0.800183 0.599756i $$-0.795264\pi$$
0.800183 0.599756i $$-0.204736\pi$$
$$912$$ 0 0
$$913$$ 249264.i 0.299032i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 26928.0i − 0.0320233i
$$918$$ 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. −0.390744
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 153952. 0.179153
$$928$$ 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
$$933$$ − 733320.i − 0.842423i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 981262.i 1.11765i 0.829286 + 0.558825i $$0.188748\pi$$
−0.829286 + 0.558825i $$0.811252\pi$$
$$938$$ 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. −0.772438
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −993109. −1.10738 −0.553691 0.832722i $$-0.686781\pi$$
−0.553691 + 0.832722i $$0.686781\pi$$
$$948$$ 0 0
$$949$$ −168532. −0.187133
$$950$$ 0 0
$$951$$ 384443.i 0.425080i
$$952$$ 0 0
$$953$$ 602922.i 0.663858i 0.943304 + 0.331929i $$0.107699\pi$$
−0.943304 + 0.331929i $$0.892301\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8424.00i 0.00919802i
$$958$$ 0 0
$$959$$ 1.01009e6i 1.09831i
$$960$$ 0 0
$$961$$ −1.25419e6 −1.35805
$$962$$ 0 0
$$963$$ 272174. 0.293491
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −575810. −0.615781 −0.307890 0.951422i $$-0.599623\pi$$
−0.307890 + 0.951422i $$0.599623\pi$$
$$968$$ 0 0
$$969$$ −150552. −0.160339
$$970$$ 0 0
$$971$$ − 1.23920e6i − 1.31432i −0.753749 0.657162i $$-0.771757\pi$$
0.753749 0.657162i $$-0.228243\pi$$
$$972$$ 0 0
$$973$$ − 1.00690e6i − 1.06355i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1.04074e6i 1.09032i 0.838332 + 0.545160i $$0.183531\pi$$
−0.838332 + 0.545160i $$0.816469\pi$$
$$978$$ 0 0
$$979$$ 240810.i 0.251252i
$$980$$ 0 0
$$981$$ 436482. 0.453553
$$982$$ 0 0
$$983$$ −948734. −0.981833 −0.490916 0.871207i $$-0.663338\pi$$
−0.490916 + 0.871207i $$0.663338\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 1.49799e6 1.53771
$$988$$ 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
$$993$$ 296316.i 0.300508i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 535870.i 0.539100i 0.962986 + 0.269550i $$0.0868749\pi$$
−0.962986 + 0.269550i $$0.913125\pi$$
$$998$$ 0 0
$$999$$ − 74356.9i − 0.0745059i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.5.j.b.799.3 4
4.3 odd 2 inner 1200.5.j.b.799.2 4
5.2 odd 4 48.5.g.a.31.1 2
5.3 odd 4 1200.5.e.b.751.2 2
5.4 even 2 inner 1200.5.j.b.799.1 4
15.2 even 4 144.5.g.f.127.2 2
20.3 even 4 1200.5.e.b.751.1 2
20.7 even 4 48.5.g.a.31.2 yes 2
20.19 odd 2 inner 1200.5.j.b.799.4 4
40.27 even 4 192.5.g.b.127.1 2
40.37 odd 4 192.5.g.b.127.2 2
60.47 odd 4 144.5.g.f.127.1 2
80.27 even 4 768.5.b.c.127.3 4
80.37 odd 4 768.5.b.c.127.1 4
80.67 even 4 768.5.b.c.127.2 4
80.77 odd 4 768.5.b.c.127.4 4
120.77 even 4 576.5.g.d.127.2 2
120.107 odd 4 576.5.g.d.127.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 5.2 odd 4
48.5.g.a.31.2 yes 2 20.7 even 4
144.5.g.f.127.1 2 60.47 odd 4
144.5.g.f.127.2 2 15.2 even 4
192.5.g.b.127.1 2 40.27 even 4
192.5.g.b.127.2 2 40.37 odd 4
576.5.g.d.127.1 2 120.107 odd 4
576.5.g.d.127.2 2 120.77 even 4
768.5.b.c.127.1 4 80.37 odd 4
768.5.b.c.127.2 4 80.67 even 4
768.5.b.c.127.3 4 80.27 even 4
768.5.b.c.127.4 4 80.77 odd 4
1200.5.e.b.751.1 2 20.3 even 4
1200.5.e.b.751.2 2 5.3 odd 4
1200.5.j.b.799.1 4 5.4 even 2 inner
1200.5.j.b.799.2 4 4.3 odd 2 inner
1200.5.j.b.799.3 4 1.1 even 1 trivial
1200.5.j.b.799.4 4 20.19 odd 2 inner