# Properties

 Label 1200.5.e.b.751.2 Level $1200$ Weight $5$ Character 1200.751 Analytic conductor $124.044$ 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] = [1200,5,Mod(751,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.751");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 1200.e (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: $$124.043955701$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 751.2 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.751 Dual form 1200.5.e.b.751.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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
$$3$$ 5.19615i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 76.2102i − 1.55531i −0.628691 0.777655i $$-0.716409\pi$$
0.628691 0.777655i $$-0.283591\pi$$
$$8$$ 0 0
$$9$$ −27.0000 −0.333333
$$10$$ 0 0
$$11$$ − 20.7846i − 0.171774i −0.996305 0.0858868i $$-0.972628\pi$$
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 $$-0.947841\pi$$
0.986605 0.163129i $$-0.0521588\pi$$
$$20$$ 0 0
$$21$$ 396.000 0.897959
$$22$$ 0 0
$$23$$ 748.246i 1.41445i 0.706987 + 0.707227i $$0.250054\pi$$
−0.706987 + 0.707227i $$0.749946\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 140.296i − 0.192450i
$$28$$ 0 0
$$29$$ 78.0000 0.0927467 0.0463734 0.998924i $$-0.485234\pi$$
0.0463734 + 0.998924i $$0.485234\pi$$
$$30$$ 0 0
$$31$$ 1475.71i 1.53560i 0.640692 + 0.767798i $$0.278647\pi$$
−0.640692 + 0.767798i $$0.721353\pi$$
$$32$$ 0 0
$$33$$ 108.000 0.0991736
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −530.000 −0.387144 −0.193572 0.981086i $$-0.562007\pi$$
−0.193572 + 0.981086i $$0.562007\pi$$
$$38$$ 0 0
$$39$$ 945.700i 0.621762i
$$40$$ 0 0
$$41$$ −918.000 −0.546104 −0.273052 0.961999i $$-0.588033\pi$$
−0.273052 + 0.961999i $$0.588033\pi$$
$$42$$ 0 0
$$43$$ 852.169i 0.460881i 0.973086 + 0.230441i $$0.0740167\pi$$
−0.973086 + 0.230441i $$0.925983\pi$$
$$44$$ 0 0
$$45$$ 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
$$51$$ 1278.25i 0.491447i
$$52$$ 0 0
$$53$$ 4626.00 1.64685 0.823425 0.567426i $$-0.192061\pi$$
0.823425 + 0.567426i $$0.192061\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 612.000 0.188366
$$58$$ 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 $$-0.442110\pi$$
0.180865 + 0.983508i $$0.442110\pi$$
$$62$$ 0 0
$$63$$ 2057.68i 0.518437i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 1087.73i − 0.242310i −0.992634 0.121155i $$-0.961340\pi$$
0.992634 0.121155i $$-0.0386597\pi$$
$$68$$ 0 0
$$69$$ −3888.00 −0.816635
$$70$$ 0 0
$$71$$ − 1829.05i − 0.362834i −0.983406 0.181417i $$-0.941932\pi$$
0.983406 0.181417i $$-0.0580684\pi$$
$$72$$ 0 0
$$73$$ 926.000 0.173766 0.0868831 0.996219i $$-0.472309\pi$$
0.0868831 + 0.996219i $$0.472309\pi$$
$$74$$ 0 0
$$75$$ 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
$$81$$ 729.000 0.111111
$$82$$ 0 0
$$83$$ − 11992.7i − 1.74085i −0.492301 0.870425i $$-0.663844\pi$$
0.492301 0.870425i $$-0.336156\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 405.300i 0.0535473i
$$88$$ 0 0
$$89$$ 11586.0 1.46269 0.731347 0.682005i $$-0.238892\pi$$
0.731347 + 0.682005i $$0.238892\pi$$
$$90$$ 0 0
$$91$$ − 13870.3i − 1.67495i
$$92$$ 0 0
$$93$$ −7668.00 −0.886576
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13118.0 1.39420 0.697099 0.716975i $$-0.254474\pi$$
0.697099 + 0.716975i $$0.254474\pi$$
$$98$$ 0 0
$$99$$ 561.184i 0.0572579i
$$100$$ 0 0
$$101$$ −5490.00 −0.538183 −0.269091 0.963115i $$-0.586723\pi$$
−0.269091 + 0.963115i $$0.586723\pi$$
$$102$$ 0 0
$$103$$ 5701.91i 0.537460i 0.963216 + 0.268730i $$0.0866039\pi$$
−0.963216 + 0.268730i $$0.913396\pi$$
$$104$$ 0 0
$$105$$ 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
$$111$$ − 2753.96i − 0.223518i
$$112$$ 0 0
$$113$$ −1842.00 −0.144256 −0.0721278 0.997395i $$-0.522979\pi$$
−0.0721278 + 0.997395i $$0.522979\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4914.00 −0.358974
$$118$$ 0 0
$$119$$ − 18747.7i − 1.32390i
$$120$$ 0 0
$$121$$ 14209.0 0.970494
$$122$$ 0 0
$$123$$ − 4770.07i − 0.315293i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 394.908i 0.0244843i 0.999925 + 0.0122422i $$0.00389690\pi$$
−0.999925 + 0.0122422i $$0.996103\pi$$
$$128$$ 0 0
$$129$$ −4428.00 −0.266090
$$130$$ 0 0
$$131$$ − 353.338i − 0.0205896i −0.999947 0.0102948i $$-0.996723\pi$$
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 $$0.111074\pi$$
−0.939733 + 0.341910i $$0.888926\pi$$
$$140$$ 0 0
$$141$$ 19656.0 0.988683
$$142$$ 0 0
$$143$$ − 3782.80i − 0.184987i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 17703.3i − 0.819255i
$$148$$ 0 0
$$149$$ 438.000 0.0197288 0.00986442 0.999951i $$-0.496860\pi$$
0.00986442 + 0.999951i $$0.496860\pi$$
$$150$$ 0 0
$$151$$ − 28052.3i − 1.23031i −0.788406 0.615155i $$-0.789093\pi$$
0.788406 0.615155i $$-0.210907\pi$$
$$152$$ 0 0
$$153$$ −6642.00 −0.283737
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −19346.0 −0.784859 −0.392430 0.919782i $$-0.628365\pi$$
−0.392430 + 0.919782i $$0.628365\pi$$
$$158$$ 0 0
$$159$$ 24037.4i 0.950809i
$$160$$ 0 0
$$161$$ 57024.0 2.19992
$$162$$ 0 0
$$163$$ 36255.3i 1.36457i 0.731086 + 0.682286i $$0.239014\pi$$
−0.731086 + 0.682286i $$0.760986\pi$$
$$164$$ 0 0
$$165$$ 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
$$171$$ 3180.05i 0.108753i
$$172$$ 0 0
$$173$$ 34410.0 1.14972 0.574861 0.818251i $$-0.305056\pi$$
0.574861 + 0.818251i $$0.305056\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1188.00 0.0379201
$$178$$ 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 $$-0.422949\pi$$
0.239706 + 0.970846i $$0.422949\pi$$
$$182$$ 0 0
$$183$$ 6994.02i 0.208845i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 5113.01i − 0.146216i
$$188$$ 0 0
$$189$$ −10692.0 −0.299320
$$190$$ 0 0
$$191$$ 2660.43i 0.0729265i 0.999335 + 0.0364632i $$0.0116092\pi$$
−0.999335 + 0.0364632i $$0.988391\pi$$
$$192$$ 0 0
$$193$$ 26782.0 0.718999 0.359500 0.933145i $$-0.382947\pi$$
0.359500 + 0.933145i $$0.382947\pi$$
$$194$$ 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
$$201$$ 5652.00 0.139898
$$202$$ 0 0
$$203$$ − 5944.40i − 0.144250i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 20202.6i − 0.471485i
$$208$$ 0 0
$$209$$ −2448.00 −0.0560427
$$210$$ 0 0
$$211$$ 23895.4i 0.536721i 0.963319 + 0.268361i $$0.0864819\pi$$
−0.963319 + 0.268361i $$0.913518\pi$$
$$212$$ 0 0
$$213$$ 9504.00 0.209482
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 112464. 2.38833
$$218$$ 0 0
$$219$$ 4811.64i 0.100324i
$$220$$ 0 0
$$221$$ 44772.0 0.916689
$$222$$ 0 0
$$223$$ − 852.169i − 0.0171363i −0.999963 0.00856813i $$-0.997273\pi$$
0.999963 0.00856813i $$-0.00272735\pi$$
$$224$$ 0 0
$$225$$ 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
$$231$$ − 8230.71i − 0.154246i
$$232$$ 0 0
$$233$$ −48738.0 −0.897751 −0.448875 0.893594i $$-0.648175\pi$$
−0.448875 + 0.893594i $$0.648175\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 22860.0 0.406986
$$238$$ 0 0
$$239$$ 71000.2i 1.24298i 0.783422 + 0.621490i $$0.213472\pi$$
−0.783422 + 0.621490i $$0.786528\pi$$
$$240$$ 0 0
$$241$$ 73138.0 1.25924 0.629621 0.776903i $$-0.283210\pi$$
0.629621 + 0.776903i $$0.283210\pi$$
$$242$$ 0 0
$$243$$ 3788.00i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 21435.9i − 0.351356i
$$248$$ 0 0
$$249$$ 62316.0 1.00508
$$250$$ 0 0
$$251$$ − 91888.8i − 1.45853i −0.684232 0.729264i $$-0.739862\pi$$
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
$$261$$ −2106.00 −0.0309156
$$262$$ 0 0
$$263$$ − 78191.7i − 1.13044i −0.824939 0.565222i $$-0.808790\pi$$
0.824939 0.565222i $$-0.191210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 60202.6i 0.844487i
$$268$$ 0 0
$$269$$ −71538.0 −0.988626 −0.494313 0.869284i $$-0.664580\pi$$
−0.494313 + 0.869284i $$0.664580\pi$$
$$270$$ 0 0
$$271$$ − 108198.i − 1.47326i −0.676296 0.736630i $$-0.736416\pi$$
0.676296 0.736630i $$-0.263584\pi$$
$$272$$ 0 0
$$273$$ 72072.0 0.967033
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 120518. 1.57070 0.785348 0.619054i $$-0.212484\pi$$
0.785348 + 0.619054i $$0.212484\pi$$
$$278$$ 0 0
$$279$$ − 39844.1i − 0.511865i
$$280$$ 0 0
$$281$$ −3054.00 −0.0386773 −0.0193387 0.999813i $$-0.506156\pi$$
−0.0193387 + 0.999813i $$0.506156\pi$$
$$282$$ 0 0
$$283$$ − 132959.i − 1.66014i −0.557657 0.830071i $$-0.688300\pi$$
0.557657 0.830071i $$-0.311700\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 69961.0i 0.849361i
$$288$$ 0 0
$$289$$ −23005.0 −0.275440
$$290$$ 0 0
$$291$$ 68163.1i 0.804940i
$$292$$ 0 0
$$293$$ −151662. −1.76661 −0.883307 0.468795i $$-0.844688\pi$$
−0.883307 + 0.468795i $$0.844688\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2916.00 −0.0330579
$$298$$ 0 0
$$299$$ 136181.i 1.52326i
$$300$$ 0 0
$$301$$ 64944.0 0.716813
$$302$$ 0 0
$$303$$ − 28526.9i − 0.310720i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5424.78i 0.0575580i 0.999586 + 0.0287790i $$0.00916190\pi$$
−0.999586 + 0.0287790i $$0.990838\pi$$
$$308$$ 0 0
$$309$$ −29628.0 −0.310303
$$310$$ 0 0
$$311$$ − 141127.i − 1.45912i −0.683917 0.729560i $$-0.739725\pi$$
0.683917 0.729560i $$-0.260275\pi$$
$$312$$ 0 0
$$313$$ 128686. 1.31354 0.656769 0.754092i $$-0.271923\pi$$
0.656769 + 0.754092i $$0.271923\pi$$
$$314$$ 0 0
$$315$$ 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
$$321$$ 52380.0 0.508341
$$322$$ 0 0
$$323$$ − 28973.7i − 0.277715i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 84001.0i − 0.785577i
$$328$$ 0 0
$$329$$ −288288. −2.66339
$$330$$ 0 0
$$331$$ 57026.0i 0.520496i 0.965542 + 0.260248i $$0.0838043\pi$$
−0.965542 + 0.260248i $$0.916196\pi$$
$$332$$ 0 0
$$333$$ 14310.0 0.129048
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −98674.0 −0.868846 −0.434423 0.900709i $$-0.643048\pi$$
−0.434423 + 0.900709i $$0.643048\pi$$
$$338$$ 0 0
$$339$$ − 9571.31i − 0.0832860i
$$340$$ 0 0
$$341$$ 30672.0 0.263775
$$342$$ 0 0
$$343$$ 76667.5i 0.651663i
$$344$$ 0 0
$$345$$ 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
$$351$$ − 25533.9i − 0.207254i
$$352$$ 0 0
$$353$$ 4302.00 0.0345240 0.0172620 0.999851i $$-0.494505\pi$$
0.0172620 + 0.999851i $$0.494505\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 97416.0 0.764353
$$358$$ 0 0
$$359$$ − 185232.i − 1.43724i −0.695405 0.718618i $$-0.744775\pi$$
0.695405 0.718618i $$-0.255225\pi$$
$$360$$ 0 0
$$361$$ 116449. 0.893555
$$362$$ 0 0
$$363$$ 73832.1i 0.560315i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 182690.i − 1.35638i −0.734885 0.678191i $$-0.762764\pi$$
0.734885 0.678191i $$-0.237236\pi$$
$$368$$ 0 0
$$369$$ 24786.0 0.182035
$$370$$ 0 0
$$371$$ − 352549.i − 2.56136i
$$372$$ 0 0
$$373$$ −151778. −1.09092 −0.545458 0.838138i $$-0.683644\pi$$
−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 $$0.0399999\pi$$
−0.992115 + 0.125333i $$0.960000\pi$$
$$380$$ 0 0
$$381$$ −2052.00 −0.0141360
$$382$$ 0 0
$$383$$ 65346.8i 0.445479i 0.974878 + 0.222739i $$0.0714999\pi$$
−0.974878 + 0.222739i $$0.928500\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 23008.6i − 0.153627i
$$388$$ 0 0
$$389$$ 105750. 0.698846 0.349423 0.936965i $$-0.386378\pi$$
0.349423 + 0.936965i $$0.386378\pi$$
$$390$$ 0 0
$$391$$ 184069.i 1.20400i
$$392$$ 0 0
$$393$$ 1836.00 0.0118874
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 27934.0 0.177236 0.0886180 0.996066i $$-0.471755\pi$$
0.0886180 + 0.996066i $$0.471755\pi$$
$$398$$ 0 0
$$399$$ − 46640.7i − 0.292967i
$$400$$ 0 0
$$401$$ 237882. 1.47936 0.739678 0.672961i $$-0.234978\pi$$
0.739678 + 0.672961i $$0.234978\pi$$
$$402$$ 0 0
$$403$$ 268579.i 1.65372i
$$404$$ 0 0
$$405$$ 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
$$411$$ 68869.8i 0.407704i
$$412$$ 0 0
$$413$$ −17424.0 −0.102152
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −68652.0 −0.394804
$$418$$ 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 $$-0.393219\pi$$
0.329207 + 0.944258i $$0.393219\pi$$
$$422$$ 0 0
$$423$$ 102136.i 0.570816i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 102579.i − 0.562604i
$$428$$ 0 0
$$429$$ 19656.0 0.106802
$$430$$ 0 0
$$431$$ 355542.i 1.91397i 0.290132 + 0.956986i $$0.406301\pi$$
−0.290132 + 0.956986i $$0.593699\pi$$
$$432$$ 0 0
$$433$$ 199726. 1.06527 0.532634 0.846346i $$-0.321202\pi$$
0.532634 + 0.846346i $$0.321202\pi$$
$$434$$ 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
$$441$$ 91989.0 0.472997
$$442$$ 0 0
$$443$$ − 50444.2i − 0.257042i −0.991707 0.128521i $$-0.958977\pi$$
0.991707 0.128521i $$-0.0410230\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 2275.91i 0.0113905i
$$448$$ 0 0
$$449$$ 149994. 0.744014 0.372007 0.928230i $$-0.378670\pi$$
0.372007 + 0.928230i $$0.378670\pi$$
$$450$$ 0 0
$$451$$ 19080.3i 0.0938062i
$$452$$ 0 0
$$453$$ 145764. 0.710320
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −284338. −1.36145 −0.680726 0.732538i $$-0.738336\pi$$
−0.680726 + 0.732538i $$0.738336\pi$$
$$458$$ 0 0
$$459$$ − 34512.8i − 0.163816i
$$460$$ 0 0
$$461$$ −183402. −0.862983 −0.431491 0.902117i $$-0.642013\pi$$
−0.431491 + 0.902117i $$0.642013\pi$$
$$462$$ 0 0
$$463$$ − 172422.i − 0.804324i −0.915568 0.402162i $$-0.868259\pi$$
0.915568 0.402162i $$-0.131741\pi$$
$$464$$ 0 0
$$465$$ 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
$$471$$ − 100525.i − 0.453139i
$$472$$ 0 0
$$473$$ 17712.0 0.0791672
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −124902. −0.548950
$$478$$ 0 0
$$479$$ − 249956.i − 1.08941i −0.838627 0.544706i $$-0.816641\pi$$
0.838627 0.544706i $$-0.183359\pi$$
$$480$$ 0 0
$$481$$ −96460.0 −0.416924
$$482$$ 0 0
$$483$$ 296305.i 1.27012i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 271108.i − 1.14310i −0.820568 0.571549i $$-0.806343\pi$$
0.820568 0.571549i $$-0.193657\pi$$
$$488$$ 0 0
$$489$$ −188388. −0.787835
$$490$$ 0 0
$$491$$ − 227862.i − 0.945166i −0.881286 0.472583i $$-0.843322\pi$$
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 $$-0.833442\pi$$
0.866196 0.499705i $$-0.166558\pi$$
$$500$$ 0 0
$$501$$ −97416.0 −0.388110
$$502$$ 0 0
$$503$$ − 446537.i − 1.76490i −0.470403 0.882452i $$-0.655891\pi$$
0.470403 0.882452i $$-0.344109\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23710.0i 0.0922394i
$$508$$ 0 0
$$509$$ −39330.0 −0.151806 −0.0759029 0.997115i $$-0.524184\pi$$
−0.0759029 + 0.997115i $$0.524184\pi$$
$$510$$ 0 0
$$511$$ − 70570.7i − 0.270260i
$$512$$ 0 0
$$513$$ −16524.0 −0.0627886
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −78624.0 −0.294154
$$518$$ 0 0
$$519$$ 178800.i 0.663792i
$$520$$ 0 0
$$521$$ −464598. −1.71160 −0.855799 0.517308i $$-0.826934\pi$$
−0.855799 + 0.517308i $$0.826934\pi$$
$$522$$ 0 0
$$523$$ 135509.i 0.495409i 0.968836 + 0.247704i $$0.0796762\pi$$
−0.968836 + 0.247704i $$0.920324\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 363024.i 1.30712i
$$528$$ 0 0
$$529$$ −280031. −1.00068
$$530$$ 0 0
$$531$$ 6173.03i 0.0218932i
$$532$$ 0 0
$$533$$ −167076. −0.588111
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −87588.0 −0.303736
$$538$$ 0 0
$$539$$ 70813.2i 0.243745i
$$540$$ 0 0
$$541$$ 360442. 1.23152 0.615759 0.787934i $$-0.288849\pi$$
0.615759 + 0.787934i $$0.288849\pi$$
$$542$$ 0 0
$$543$$ 81610.8i 0.276788i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 261644.i 0.874451i 0.899352 + 0.437225i $$0.144039\pi$$
−0.899352 + 0.437225i $$0.855961\pi$$
$$548$$ 0 0
$$549$$ −36342.0 −0.120577
$$550$$ 0 0
$$551$$ − 9186.80i − 0.0302594i
$$552$$ 0 0
$$553$$ −335280. −1.09637
$$554$$ 0 0
$$555$$ 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
$$561$$ 26568.0 0.0844176
$$562$$ 0 0
$$563$$ − 419704.i − 1.32412i −0.749453 0.662058i $$-0.769683\pi$$
0.749453 0.662058i $$-0.230317\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 55557.3i − 0.172812i
$$568$$ 0 0
$$569$$ 470058. 1.45187 0.725934 0.687765i $$-0.241408\pi$$
0.725934 + 0.687765i $$0.241408\pi$$
$$570$$ 0 0
$$571$$ 320381.i 0.982640i 0.870979 + 0.491320i $$0.163485\pi$$
−0.870979 + 0.491320i $$0.836515\pi$$
$$572$$ 0 0
$$573$$ −13824.0 −0.0421041
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 341038. 1.02436 0.512178 0.858879i $$-0.328839\pi$$
0.512178 + 0.858879i $$0.328839\pi$$
$$578$$ 0 0
$$579$$ 139163.i 0.415114i
$$580$$ 0 0
$$581$$ −913968. −2.70756
$$582$$ 0 0
$$583$$ − 96149.6i − 0.282885i
$$584$$ 0 0
$$585$$ 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
$$591$$ 272704.i 0.780760i
$$592$$ 0 0
$$593$$ 96846.0 0.275405 0.137703 0.990474i $$-0.456028\pi$$
0.137703 + 0.990474i $$0.456028\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 119916. 0.336456
$$598$$ 0 0
$$599$$ − 519782.i − 1.44866i −0.689452 0.724331i $$-0.742149\pi$$
0.689452 0.724331i $$-0.257851\pi$$
$$600$$ 0 0
$$601$$ −627742. −1.73793 −0.868965 0.494874i $$-0.835214\pi$$
−0.868965 + 0.494874i $$0.835214\pi$$
$$602$$ 0 0
$$603$$ 29368.7i 0.0807699i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 133195.i 0.361501i 0.983529 + 0.180751i $$0.0578527\pi$$
−0.983529 + 0.180751i $$0.942147\pi$$
$$608$$ 0 0
$$609$$ 30888.0 0.0832828
$$610$$ 0 0
$$611$$ − 688469.i − 1.84418i
$$612$$ 0 0
$$613$$ −247202. −0.657856 −0.328928 0.944355i $$-0.606687\pi$$
−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 $$0.327156\pi$$
−0.516712 + 0.856160i $$0.672844\pi$$
$$620$$ 0 0
$$621$$ 104976. 0.272212
$$622$$ 0 0
$$623$$ − 882972.i − 2.27494i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 12720.2i − 0.0323563i
$$628$$ 0 0
$$629$$ −130380. −0.329541
$$630$$ 0 0
$$631$$ 417736.i 1.04916i 0.851360 + 0.524582i $$0.175778\pi$$
−0.851360 + 0.524582i $$0.824222\pi$$
$$632$$ 0 0
$$633$$ −124164. −0.309876
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −620074. −1.52815
$$638$$ 0 0
$$639$$ 49384.2i 0.120945i
$$640$$ 0 0
$$641$$ −152214. −0.370458 −0.185229 0.982695i $$-0.559303\pi$$
−0.185229 + 0.982695i $$0.559303\pi$$
$$642$$ 0 0
$$643$$ 714138.i 1.72727i 0.504117 + 0.863635i $$0.331818\pi$$
−0.504117 + 0.863635i $$0.668182\pi$$
$$644$$ 0 0
$$645$$ 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
$$651$$ 584380.i 1.37890i
$$652$$ 0 0
$$653$$ 330714. 0.775579 0.387790 0.921748i $$-0.373239\pi$$
0.387790 + 0.921748i $$0.373239\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −25002.0 −0.0579221
$$658$$ 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 $$-0.809625\pi$$
−0.826417 + 0.563058i $$0.809625\pi$$
$$662$$ 0 0
$$663$$ 232642.i 0.529251i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 58363.2i 0.131186i
$$668$$ 0 0
$$669$$ 4428.00 0.00989362
$$670$$ 0 0
$$671$$ − 27976.1i − 0.0621358i
$$672$$ 0 0
$$673$$ 552910. 1.22074 0.610372 0.792115i $$-0.291020\pi$$
0.610372 + 0.792115i $$0.291020\pi$$
$$674$$ 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
$$681$$ 395820. 0.853500
$$682$$ 0 0
$$683$$ 23715.2i 0.0508377i 0.999677 + 0.0254189i $$0.00809195\pi$$
−0.999677 + 0.0254189i $$0.991908\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 251858.i − 0.533631i
$$688$$ 0 0
$$689$$ 841932. 1.77353
$$690$$ 0 0
$$691$$ 431842.i 0.904417i 0.891912 + 0.452208i $$0.149364\pi$$
−0.891912 + 0.452208i $$0.850636\pi$$
$$692$$ 0 0
$$693$$ 42768.0 0.0890538
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −225828. −0.464849
$$698$$ 0 0
$$699$$ − 253250.i − 0.518317i
$$700$$ 0 0
$$701$$ 44958.0 0.0914894 0.0457447 0.998953i $$-0.485434\pi$$
0.0457447 + 0.998953i $$0.485434\pi$$
$$702$$ 0 0
$$703$$ 62423.1i 0.126309i
$$704$$ 0 0
$$705$$ 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
$$711$$ 118784.i 0.234974i
$$712$$ 0 0
$$713$$ −1.10419e6 −2.17203
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −368928. −0.717634
$$718$$ 0 0
$$719$$ 292107.i 0.565046i 0.959260 + 0.282523i $$0.0911714\pi$$
−0.959260 + 0.282523i $$0.908829\pi$$
$$720$$ 0 0
$$721$$ 434544. 0.835917
$$722$$ 0 0
$$723$$ 380036.i 0.727023i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 755791.i 1.42999i 0.699130 + 0.714995i $$0.253571\pi$$
−0.699130 + 0.714995i $$0.746429\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −0.0370370
$$730$$ 0 0
$$731$$ 209634.i 0.392307i
$$732$$ 0 0
$$733$$ 832982. 1.55034 0.775171 0.631751i $$-0.217664\pi$$
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 $$0.220708\pi$$
−0.769093 + 0.639137i $$0.779292\pi$$
$$740$$ 0 0
$$741$$ 111384. 0.202855
$$742$$ 0 0
$$743$$ − 461044.i − 0.835151i −0.908642 0.417575i $$-0.862880\pi$$
0.908642 0.417575i $$-0.137120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 323803.i 0.580284i
$$748$$ 0 0
$$749$$ −768240. −1.36941
$$750$$ 0 0
$$751$$ 937060.i 1.66145i 0.556682 + 0.830726i $$0.312074\pi$$
−0.556682 + 0.830726i $$0.687926\pi$$
$$752$$ 0 0
$$753$$ 477468. 0.842082
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −295786. −0.516162 −0.258081 0.966123i $$-0.583090\pi$$
−0.258081 + 0.966123i $$0.583090\pi$$
$$758$$ 0 0
$$759$$ 80810.6i 0.140276i
$$760$$ 0 0
$$761$$ −1.02615e6 −1.77191 −0.885955 0.463772i $$-0.846496\pi$$
−0.885955 + 0.463772i $$0.846496\pi$$
$$762$$ 0 0
$$763$$ 1.23201e6i 2.11625i
$$764$$ 0 0
$$765$$ 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
$$771$$ 254061.i 0.427394i
$$772$$ 0 0
$$773$$ −1.02608e6 −1.71720 −0.858601 0.512644i $$-0.828666\pi$$
−0.858601 + 0.512644i $$0.828666\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −209880. −0.347639
$$778$$ 0 0
$$779$$ 108122.i 0.178171i
$$780$$ 0 0
$$781$$ −38016.0 −0.0623253
$$782$$ 0 0
$$783$$ − 10943.1i − 0.0178491i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 850042.i 1.37243i 0.727398 + 0.686216i $$0.240730\pi$$
−0.727398 + 0.686216i $$0.759270\pi$$
$$788$$ 0 0
$$789$$ 406296. 0.652662
$$790$$ 0 0
$$791$$ 140379.i 0.224362i
$$792$$ 0 0
$$793$$ 244972. 0.389556
$$794$$ 0 0
$$795$$ 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
$$801$$ −312822. −0.487565
$$802$$ 0 0
$$803$$ − 19246.5i − 0.0298484i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 371722.i − 0.570784i
$$808$$ 0 0
$$809$$ 247674. 0.378428 0.189214 0.981936i $$-0.439406\pi$$
0.189214 + 0.981936i $$0.439406\pi$$
$$810$$ 0 0
$$811$$ − 920197.i − 1.39907i −0.714599 0.699534i $$-0.753391\pi$$
0.714599 0.699534i $$-0.246609\pi$$
$$812$$ 0 0
$$813$$ 562212. 0.850588
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 100368. 0.150367
$$818$$ 0 0
$$819$$ 374497.i 0.558317i
$$820$$ 0 0
$$821$$ −250242. −0.371256 −0.185628 0.982620i $$-0.559432\pi$$
−0.185628 + 0.982620i $$0.559432\pi$$
$$822$$ 0 0
$$823$$ 400762.i 0.591680i 0.955238 + 0.295840i $$0.0955995\pi$$
−0.955238 + 0.295840i $$0.904401\pi$$
$$824$$ 0 0
$$825$$ 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
$$831$$ 626230.i 0.906842i
$$832$$ 0 0
$$833$$ −838122. −1.20786
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 207036. 0.295525
$$838$$ 0 0
$$839$$ − 115479.i − 0.164051i −0.996630 0.0820257i $$-0.973861\pi$$
0.996630 0.0820257i $$-0.0261390\pi$$
$$840$$ 0 0
$$841$$ −701197. −0.991398
$$842$$ 0 0
$$843$$ − 15869.0i − 0.0223304i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.08287e6i − 1.50942i
$$848$$ 0 0
$$849$$ 690876. 0.958484
$$850$$ 0 0
$$851$$ − 396570.i − 0.547597i
$$852$$ 0 0
$$853$$ −345938. −0.475445 −0.237722 0.971333i $$-0.576401\pi$$
−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 $$-0.884824\pi$$
0.935249 0.353992i $$-0.115176\pi$$
$$860$$ 0 0
$$861$$ −363528. −0.490379
$$862$$ 0 0
$$863$$ − 826895.i − 1.11027i −0.831760 0.555135i $$-0.812667\pi$$
0.831760 0.555135i $$-0.187333\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 119537.i − 0.159025i
$$868$$ 0 0
$$869$$ −91440.0 −0.121087
$$870$$ 0 0
$$871$$ − 197966.i − 0.260949i
$$872$$ 0 0
$$873$$ −354186. −0.464732
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.11629e6 −1.45137 −0.725685 0.688028i $$-0.758477\pi$$
−0.725685 + 0.688028i $$0.758477\pi$$
$$878$$ 0 0
$$879$$ − 788059.i − 1.01995i
$$880$$ 0 0
$$881$$ 19170.0 0.0246985 0.0123492 0.999924i $$-0.496069\pi$$
0.0123492 + 0.999924i $$0.496069\pi$$
$$882$$ 0 0
$$883$$ − 568909.i − 0.729662i −0.931074 0.364831i $$-0.881127\pi$$
0.931074 0.364831i $$-0.118873\pi$$
$$884$$ 0 0
$$885$$ 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
$$891$$ − 15152.0i − 0.0190860i
$$892$$ 0 0
$$893$$ −445536. −0.558702
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −707616. −0.879453
$$898$$ 0 0
$$899$$ 115105.i 0.142421i
$$900$$ 0 0
$$901$$ 1.13800e6 1.40182
$$902$$ 0 0
$$903$$ 337459.i 0.413852i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 916193.i − 1.11371i −0.830610 0.556855i $$-0.812008\pi$$
0.830610 0.556855i $$-0.187992\pi$$
$$908$$ 0 0
$$909$$ 148230. 0.179394
$$910$$ 0 0
$$911$$ − 995500.i − 1.19951i −0.800183 0.599756i $$-0.795264\pi$$
0.800183 0.599756i $$-0.204736\pi$$
$$912$$ 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
$$921$$ −28188.0 −0.0332311
$$922$$ 0 0
$$923$$ − 332886.i − 0.390744i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 153952.i − 0.179153i
$$928$$ 0 0
$$929$$ −1.27882e6 −1.48176 −0.740881 0.671636i $$-0.765592\pi$$
−0.740881 + 0.671636i $$0.765592\pi$$
$$930$$ 0 0
$$931$$ 401275.i 0.462959i
$$932$$ 0 0
$$933$$ 733320. 0.842423
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 981262. 1.11765 0.558825 0.829286i $$-0.311252\pi$$
0.558825 + 0.829286i $$0.311252\pi$$
$$938$$ 0 0
$$939$$ 668672.i 0.758371i
$$940$$ 0 0
$$941$$ 284406. 0.321188 0.160594 0.987021i $$-0.448659\pi$$
0.160594 + 0.987021i $$0.448659\pi$$
$$942$$ 0 0
$$943$$ − 686890.i − 0.772438i
$$944$$ 0 0
$$945$$ 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
$$951$$ 384443.i 0.425080i
$$952$$ 0 0
$$953$$ −602922. −0.663858 −0.331929 0.943304i $$-0.607699\pi$$
−0.331929 + 0.943304i $$0.607699\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8424.00 0.00919802
$$958$$ 0 0
$$959$$ − 1.01009e6i − 1.09831i
$$960$$ 0 0
$$961$$ −1.25419e6 −1.35805
$$962$$ 0 0
$$963$$ 272174.i 0.293491i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 575810.i 0.615781i 0.951422 + 0.307890i $$0.0996230\pi$$
−0.951422 + 0.307890i $$0.900377\pi$$
$$968$$ 0 0
$$969$$ 150552. 0.160339
$$970$$ 0 0
$$971$$ − 1.23920e6i − 1.31432i −0.753749 0.657162i $$-0.771757\pi$$
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
$$981$$ 436482. 0.453553
$$982$$ 0 0
$$983$$ − 948734.i − 0.981833i −0.871207 0.490916i $$-0.836662\pi$$
0.871207 0.490916i $$-0.163338\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 1.49799e6i − 1.53771i
$$988$$ 0 0
$$989$$ −637632. −0.651895
$$990$$ 0 0
$$991$$ 616007.i 0.627247i 0.949547 + 0.313623i $$0.101543\pi$$
−0.949547 + 0.313623i $$0.898457\pi$$
$$992$$ 0 0
$$993$$ −296316. −0.300508
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 535870. 0.539100 0.269550 0.962986i $$-0.413125\pi$$
0.269550 + 0.962986i $$0.413125\pi$$
$$998$$ 0 0
$$999$$ 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.e.b.751.2 2
4.3 odd 2 inner 1200.5.e.b.751.1 2
5.2 odd 4 1200.5.j.b.799.3 4
5.3 odd 4 1200.5.j.b.799.1 4
5.4 even 2 48.5.g.a.31.1 2
15.14 odd 2 144.5.g.f.127.2 2
20.3 even 4 1200.5.j.b.799.4 4
20.7 even 4 1200.5.j.b.799.2 4
20.19 odd 2 48.5.g.a.31.2 yes 2
40.19 odd 2 192.5.g.b.127.1 2
40.29 even 2 192.5.g.b.127.2 2
60.59 even 2 144.5.g.f.127.1 2
80.19 odd 4 768.5.b.c.127.2 4
80.29 even 4 768.5.b.c.127.4 4
80.59 odd 4 768.5.b.c.127.3 4
80.69 even 4 768.5.b.c.127.1 4
120.29 odd 2 576.5.g.d.127.2 2
120.59 even 2 576.5.g.d.127.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.5.g.a.31.1 2 5.4 even 2
48.5.g.a.31.2 yes 2 20.19 odd 2
144.5.g.f.127.1 2 60.59 even 2
144.5.g.f.127.2 2 15.14 odd 2
192.5.g.b.127.1 2 40.19 odd 2
192.5.g.b.127.2 2 40.29 even 2
576.5.g.d.127.1 2 120.59 even 2
576.5.g.d.127.2 2 120.29 odd 2
768.5.b.c.127.1 4 80.69 even 4
768.5.b.c.127.2 4 80.19 odd 4
768.5.b.c.127.3 4 80.59 odd 4
768.5.b.c.127.4 4 80.29 even 4
1200.5.e.b.751.1 2 4.3 odd 2 inner
1200.5.e.b.751.2 2 1.1 even 1 trivial
1200.5.j.b.799.1 4 5.3 odd 4
1200.5.j.b.799.2 4 20.7 even 4
1200.5.j.b.799.3 4 5.2 odd 4
1200.5.j.b.799.4 4 20.3 even 4