# Properties

 Label 864.3.g.c.703.4 Level $864$ Weight $3$ Character 864.703 Analytic conductor $23.542$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(703,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("864.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.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: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 703.4 Root $$-1.27597 - 0.609843i$$ of defining polynomial Character $$\chi$$ $$=$$ 864.703 Dual form 864.3.g.c.703.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.22512 q^{5} +6.57221i q^{7} +O(q^{10})$$ $$q-3.22512 q^{5} +6.57221i q^{7} +13.8903i q^{11} -19.0865 q^{13} +23.0865 q^{17} -18.8425i q^{19} -13.5947i q^{23} -14.5986 q^{25} +0.752110 q^{29} -46.4433i q^{31} -21.1962i q^{35} -2.68178 q^{37} -34.1405 q^{41} -20.9127i q^{43} +15.4086i q^{47} +5.80609 q^{49} -46.8189 q^{53} -44.7980i q^{55} +40.4126i q^{59} -105.543 q^{61} +61.5562 q^{65} -27.9369i q^{67} -24.0130i q^{71} -120.117 q^{73} -91.2901 q^{77} -95.6394i q^{79} -115.446i q^{83} -74.4567 q^{85} +169.589 q^{89} -125.440i q^{91} +60.7694i q^{95} -93.1142 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5}+O(q^{10})$$ 8 * q + 8 * q^5 $$8 q + 8 q^{5} - 16 q^{13} + 48 q^{17} + 48 q^{25} - 32 q^{29} - 96 q^{37} - 128 q^{41} + 168 q^{53} + 32 q^{61} - 112 q^{65} + 24 q^{73} - 440 q^{77} + 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100})$$ 8 * q + 8 * q^5 - 16 * q^13 + 48 * q^17 + 48 * q^25 - 32 * q^29 - 96 * q^37 - 128 * q^41 + 168 * q^53 + 32 * q^61 - 112 * q^65 + 24 * q^73 - 440 * q^77 + 144 * q^85 + 624 * q^89 - 136 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$5$$ −3.22512 −0.645024 −0.322512 0.946565i $$-0.604527\pi$$
−0.322512 + 0.946565i $$0.604527\pi$$
$$6$$ 0 0
$$7$$ 6.57221i 0.938887i 0.882963 + 0.469443i $$0.155545\pi$$
−0.882963 + 0.469443i $$0.844455\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 13.8903i 1.26276i 0.775475 + 0.631379i $$0.217511\pi$$
−0.775475 + 0.631379i $$0.782489\pi$$
$$12$$ 0 0
$$13$$ −19.0865 −1.46819 −0.734095 0.679046i $$-0.762394\pi$$
−0.734095 + 0.679046i $$0.762394\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 23.0865 1.35803 0.679014 0.734125i $$-0.262407\pi$$
0.679014 + 0.734125i $$0.262407\pi$$
$$18$$ 0 0
$$19$$ − 18.8425i − 0.991713i −0.868405 0.495856i $$-0.834854\pi$$
0.868405 0.495856i $$-0.165146\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 13.5947i − 0.591072i −0.955332 0.295536i $$-0.904502\pi$$
0.955332 0.295536i $$-0.0954982\pi$$
$$24$$ 0 0
$$25$$ −14.5986 −0.583944
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.752110 0.0259348 0.0129674 0.999916i $$-0.495872\pi$$
0.0129674 + 0.999916i $$0.495872\pi$$
$$30$$ 0 0
$$31$$ − 46.4433i − 1.49817i −0.662473 0.749085i $$-0.730493\pi$$
0.662473 0.749085i $$-0.269507\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 21.1962i − 0.605604i
$$36$$ 0 0
$$37$$ −2.68178 −0.0724807 −0.0362403 0.999343i $$-0.511538\pi$$
−0.0362403 + 0.999343i $$0.511538\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −34.1405 −0.832694 −0.416347 0.909206i $$-0.636690\pi$$
−0.416347 + 0.909206i $$0.636690\pi$$
$$42$$ 0 0
$$43$$ − 20.9127i − 0.486341i −0.969984 0.243171i $$-0.921812\pi$$
0.969984 0.243171i $$-0.0781875\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 15.4086i 0.327844i 0.986473 + 0.163922i $$0.0524145\pi$$
−0.986473 + 0.163922i $$0.947586\pi$$
$$48$$ 0 0
$$49$$ 5.80609 0.118492
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −46.8189 −0.883376 −0.441688 0.897169i $$-0.645620\pi$$
−0.441688 + 0.897169i $$0.645620\pi$$
$$54$$ 0 0
$$55$$ − 44.7980i − 0.814508i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 40.4126i 0.684959i 0.939525 + 0.342480i $$0.111267\pi$$
−0.939525 + 0.342480i $$0.888733\pi$$
$$60$$ 0 0
$$61$$ −105.543 −1.73022 −0.865108 0.501586i $$-0.832750\pi$$
−0.865108 + 0.501586i $$0.832750\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 61.5562 0.947018
$$66$$ 0 0
$$67$$ − 27.9369i − 0.416969i −0.978026 0.208485i $$-0.933147\pi$$
0.978026 0.208485i $$-0.0668531\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 24.0130i − 0.338212i −0.985598 0.169106i $$-0.945912\pi$$
0.985598 0.169106i $$-0.0540880\pi$$
$$72$$ 0 0
$$73$$ −120.117 −1.64545 −0.822723 0.568443i $$-0.807546\pi$$
−0.822723 + 0.568443i $$0.807546\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −91.2901 −1.18559
$$78$$ 0 0
$$79$$ − 95.6394i − 1.21062i −0.795988 0.605312i $$-0.793048\pi$$
0.795988 0.605312i $$-0.206952\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 115.446i − 1.39091i −0.718568 0.695457i $$-0.755202\pi$$
0.718568 0.695457i $$-0.244798\pi$$
$$84$$ 0 0
$$85$$ −74.4567 −0.875961
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 169.589 1.90549 0.952745 0.303770i $$-0.0982455\pi$$
0.952745 + 0.303770i $$0.0982455\pi$$
$$90$$ 0 0
$$91$$ − 125.440i − 1.37846i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 60.7694i 0.639678i
$$96$$ 0 0
$$97$$ −93.1142 −0.959940 −0.479970 0.877285i $$-0.659352\pi$$
−0.479970 + 0.877285i $$0.659352\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −159.090 −1.57514 −0.787572 0.616222i $$-0.788662\pi$$
−0.787572 + 0.616222i $$0.788662\pi$$
$$102$$ 0 0
$$103$$ 75.3674i 0.731722i 0.930669 + 0.365861i $$0.119226\pi$$
−0.930669 + 0.365861i $$0.880774\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 69.4279i − 0.648859i −0.945910 0.324429i $$-0.894828\pi$$
0.945910 0.324429i $$-0.105172\pi$$
$$108$$ 0 0
$$109$$ 60.4669 0.554742 0.277371 0.960763i $$-0.410537\pi$$
0.277371 + 0.960763i $$0.410537\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 140.698 1.24512 0.622559 0.782573i $$-0.286093\pi$$
0.622559 + 0.782573i $$0.286093\pi$$
$$114$$ 0 0
$$115$$ 43.8444i 0.381255i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 151.729i 1.27503i
$$120$$ 0 0
$$121$$ −71.9412 −0.594556
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 127.710 1.02168
$$126$$ 0 0
$$127$$ 66.9726i 0.527343i 0.964612 + 0.263672i $$0.0849335\pi$$
−0.964612 + 0.263672i $$0.915066\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 255.148i 1.94770i 0.227196 + 0.973849i $$0.427044\pi$$
−0.227196 + 0.973849i $$0.572956\pi$$
$$132$$ 0 0
$$133$$ 123.837 0.931106
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −27.1946 −0.198501 −0.0992505 0.995062i $$-0.531645\pi$$
−0.0992505 + 0.995062i $$0.531645\pi$$
$$138$$ 0 0
$$139$$ − 203.956i − 1.46731i −0.679521 0.733656i $$-0.737812\pi$$
0.679521 0.733656i $$-0.262188\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 265.117i − 1.85397i
$$144$$ 0 0
$$145$$ −2.42564 −0.0167286
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −251.507 −1.68797 −0.843984 0.536369i $$-0.819796\pi$$
−0.843984 + 0.536369i $$0.819796\pi$$
$$150$$ 0 0
$$151$$ − 250.251i − 1.65729i −0.559774 0.828645i $$-0.689112\pi$$
0.559774 0.828645i $$-0.310888\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 149.785i 0.966356i
$$156$$ 0 0
$$157$$ 18.1730 0.115751 0.0578757 0.998324i $$-0.481567\pi$$
0.0578757 + 0.998324i $$0.481567\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 89.3469 0.554950
$$162$$ 0 0
$$163$$ 50.0180i 0.306859i 0.988160 + 0.153429i $$0.0490318\pi$$
−0.988160 + 0.153429i $$0.950968\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 84.8389i 0.508017i 0.967202 + 0.254009i $$0.0817492\pi$$
−0.967202 + 0.254009i $$0.918251\pi$$
$$168$$ 0 0
$$169$$ 195.294 1.15558
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −144.703 −0.836433 −0.418216 0.908347i $$-0.637345\pi$$
−0.418216 + 0.908347i $$0.637345\pi$$
$$174$$ 0 0
$$175$$ − 95.9451i − 0.548258i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 280.926i − 1.56942i −0.619864 0.784709i $$-0.712812\pi$$
0.619864 0.784709i $$-0.287188\pi$$
$$180$$ 0 0
$$181$$ −295.508 −1.63264 −0.816320 0.577600i $$-0.803989\pi$$
−0.816320 + 0.577600i $$0.803989\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.64907 0.0467518
$$186$$ 0 0
$$187$$ 320.679i 1.71486i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 295.982i − 1.54965i −0.632179 0.774823i $$-0.717839\pi$$
0.632179 0.774823i $$-0.282161\pi$$
$$192$$ 0 0
$$193$$ −18.0202 −0.0933688 −0.0466844 0.998910i $$-0.514866\pi$$
−0.0466844 + 0.998910i $$0.514866\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 119.020 0.604162 0.302081 0.953282i $$-0.402319\pi$$
0.302081 + 0.953282i $$0.402319\pi$$
$$198$$ 0 0
$$199$$ 304.912i 1.53222i 0.642708 + 0.766111i $$0.277811\pi$$
−0.642708 + 0.766111i $$0.722189\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4.94302i 0.0243499i
$$204$$ 0 0
$$205$$ 110.107 0.537108
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 261.729 1.25229
$$210$$ 0 0
$$211$$ 282.143i 1.33717i 0.743635 + 0.668585i $$0.233100\pi$$
−0.743635 + 0.668585i $$0.766900\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 67.4459i 0.313702i
$$216$$ 0 0
$$217$$ 305.235 1.40661
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −440.640 −1.99384
$$222$$ 0 0
$$223$$ 91.9070i 0.412139i 0.978537 + 0.206070i $$0.0660673\pi$$
−0.978537 + 0.206070i $$0.933933\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 31.5125i 0.138822i 0.997588 + 0.0694108i $$0.0221119\pi$$
−0.997588 + 0.0694108i $$0.977888\pi$$
$$228$$ 0 0
$$229$$ 242.515 1.05902 0.529509 0.848304i $$-0.322376\pi$$
0.529509 + 0.848304i $$0.322376\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 402.297 1.72660 0.863298 0.504695i $$-0.168395\pi$$
0.863298 + 0.504695i $$0.168395\pi$$
$$234$$ 0 0
$$235$$ − 49.6947i − 0.211467i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 167.034i 0.698886i 0.936958 + 0.349443i $$0.113629\pi$$
−0.936958 + 0.349443i $$0.886371\pi$$
$$240$$ 0 0
$$241$$ −21.4458 −0.0889868 −0.0444934 0.999010i $$-0.514167\pi$$
−0.0444934 + 0.999010i $$0.514167\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −18.7253 −0.0764299
$$246$$ 0 0
$$247$$ 359.638i 1.45602i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 22.4825i 0.0895718i 0.998997 + 0.0447859i $$0.0142606\pi$$
−0.998997 + 0.0447859i $$0.985739\pi$$
$$252$$ 0 0
$$253$$ 188.834 0.746380
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −437.652 −1.70293 −0.851463 0.524415i $$-0.824284\pi$$
−0.851463 + 0.524415i $$0.824284\pi$$
$$258$$ 0 0
$$259$$ − 17.6252i − 0.0680511i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 266.194i 1.01214i 0.862491 + 0.506072i $$0.168903\pi$$
−0.862491 + 0.506072i $$0.831097\pi$$
$$264$$ 0 0
$$265$$ 150.997 0.569799
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −220.671 −0.820338 −0.410169 0.912010i $$-0.634530\pi$$
−0.410169 + 0.912010i $$0.634530\pi$$
$$270$$ 0 0
$$271$$ 224.295i 0.827656i 0.910355 + 0.413828i $$0.135808\pi$$
−0.910355 + 0.413828i $$0.864192\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 202.779i − 0.737380i
$$276$$ 0 0
$$277$$ −51.4458 −0.185725 −0.0928625 0.995679i $$-0.529602\pi$$
−0.0928625 + 0.995679i $$0.529602\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 40.1606 0.142920 0.0714602 0.997443i $$-0.477234\pi$$
0.0714602 + 0.997443i $$0.477234\pi$$
$$282$$ 0 0
$$283$$ 190.774i 0.674112i 0.941485 + 0.337056i $$0.109431\pi$$
−0.941485 + 0.337056i $$0.890569\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 224.378i − 0.781806i
$$288$$ 0 0
$$289$$ 243.986 0.844241
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 115.872 0.395467 0.197733 0.980256i $$-0.436642\pi$$
0.197733 + 0.980256i $$0.436642\pi$$
$$294$$ 0 0
$$295$$ − 130.335i − 0.441815i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 259.474i 0.867806i
$$300$$ 0 0
$$301$$ 137.442 0.456619
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 340.389 1.11603
$$306$$ 0 0
$$307$$ − 497.554i − 1.62070i −0.585949 0.810348i $$-0.699278\pi$$
0.585949 0.810348i $$-0.300722\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 358.447i 1.15256i 0.817252 + 0.576281i $$0.195497\pi$$
−0.817252 + 0.576281i $$0.804503\pi$$
$$312$$ 0 0
$$313$$ 32.9519 0.105278 0.0526388 0.998614i $$-0.483237\pi$$
0.0526388 + 0.998614i $$0.483237\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 216.101 0.681707 0.340853 0.940117i $$-0.389284\pi$$
0.340853 + 0.940117i $$0.389284\pi$$
$$318$$ 0 0
$$319$$ 10.4471i 0.0327494i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 435.008i − 1.34677i
$$324$$ 0 0
$$325$$ 278.636 0.857342
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −101.269 −0.307808
$$330$$ 0 0
$$331$$ 578.213i 1.74687i 0.486944 + 0.873433i $$0.338112\pi$$
−0.486944 + 0.873433i $$0.661888\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 90.0999i 0.268955i
$$336$$ 0 0
$$337$$ 358.117 1.06266 0.531331 0.847164i $$-0.321692\pi$$
0.531331 + 0.847164i $$0.321692\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 645.113 1.89183
$$342$$ 0 0
$$343$$ 360.197i 1.05014i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 483.855i 1.39440i 0.716879 + 0.697198i $$0.245570\pi$$
−0.716879 + 0.697198i $$0.754430\pi$$
$$348$$ 0 0
$$349$$ 22.1213 0.0633849 0.0316924 0.999498i $$-0.489910\pi$$
0.0316924 + 0.999498i $$0.489910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −459.068 −1.30048 −0.650238 0.759731i $$-0.725331\pi$$
−0.650238 + 0.759731i $$0.725331\pi$$
$$354$$ 0 0
$$355$$ 77.4449i 0.218155i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 414.323i − 1.15410i −0.816708 0.577052i $$-0.804203\pi$$
0.816708 0.577052i $$-0.195797\pi$$
$$360$$ 0 0
$$361$$ 5.95856 0.0165057
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 387.393 1.06135
$$366$$ 0 0
$$367$$ − 90.5390i − 0.246700i −0.992363 0.123350i $$-0.960636\pi$$
0.992363 0.123350i $$-0.0393638\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 307.704i − 0.829390i
$$372$$ 0 0
$$373$$ 223.011 0.597884 0.298942 0.954271i $$-0.403366\pi$$
0.298942 + 0.954271i $$0.403366\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −14.3551 −0.0380773
$$378$$ 0 0
$$379$$ 118.162i 0.311774i 0.987775 + 0.155887i $$0.0498236\pi$$
−0.987775 + 0.155887i $$0.950176\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 19.6418i 0.0512840i 0.999671 + 0.0256420i $$0.00816300\pi$$
−0.999671 + 0.0256420i $$0.991837\pi$$
$$384$$ 0 0
$$385$$ 294.422 0.764731
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −352.392 −0.905892 −0.452946 0.891538i $$-0.649627\pi$$
−0.452946 + 0.891538i $$0.649627\pi$$
$$390$$ 0 0
$$391$$ − 313.853i − 0.802692i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 308.448i 0.780882i
$$396$$ 0 0
$$397$$ −5.66047 −0.0142581 −0.00712906 0.999975i $$-0.502269\pi$$
−0.00712906 + 0.999975i $$0.502269\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −250.097 −0.623683 −0.311842 0.950134i $$-0.600946\pi$$
−0.311842 + 0.950134i $$0.600946\pi$$
$$402$$ 0 0
$$403$$ 886.439i 2.19960i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 37.2509i − 0.0915255i
$$408$$ 0 0
$$409$$ −643.885 −1.57429 −0.787145 0.616767i $$-0.788442\pi$$
−0.787145 + 0.616767i $$0.788442\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −265.600 −0.643099
$$414$$ 0 0
$$415$$ 372.327i 0.897173i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 439.563i − 1.04908i −0.851387 0.524539i $$-0.824238\pi$$
0.851387 0.524539i $$-0.175762\pi$$
$$420$$ 0 0
$$421$$ 65.4392 0.155438 0.0777188 0.996975i $$-0.475236\pi$$
0.0777188 + 0.996975i $$0.475236\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −337.030 −0.793013
$$426$$ 0 0
$$427$$ − 693.651i − 1.62448i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 299.824i − 0.695646i −0.937560 0.347823i $$-0.886921\pi$$
0.937560 0.347823i $$-0.113079\pi$$
$$432$$ 0 0
$$433$$ −699.217 −1.61482 −0.807410 0.589991i $$-0.799131\pi$$
−0.807410 + 0.589991i $$0.799131\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −256.158 −0.586174
$$438$$ 0 0
$$439$$ − 744.832i − 1.69666i −0.529472 0.848328i $$-0.677610\pi$$
0.529472 0.848328i $$-0.322390\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 320.933i 0.724455i 0.932090 + 0.362227i $$0.117984\pi$$
−0.932090 + 0.362227i $$0.882016\pi$$
$$444$$ 0 0
$$445$$ −546.944 −1.22909
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −439.651 −0.979179 −0.489589 0.871953i $$-0.662853\pi$$
−0.489589 + 0.871953i $$0.662853\pi$$
$$450$$ 0 0
$$451$$ − 474.222i − 1.05149i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 404.560i 0.889143i
$$456$$ 0 0
$$457$$ −732.044 −1.60185 −0.800923 0.598767i $$-0.795658\pi$$
−0.800923 + 0.598767i $$0.795658\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 381.075 0.826627 0.413314 0.910589i $$-0.364371\pi$$
0.413314 + 0.910589i $$0.364371\pi$$
$$462$$ 0 0
$$463$$ 591.775i 1.27813i 0.769152 + 0.639066i $$0.220679\pi$$
−0.769152 + 0.639066i $$0.779321\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 50.8002i 0.108780i 0.998520 + 0.0543899i $$0.0173214\pi$$
−0.998520 + 0.0543899i $$0.982679\pi$$
$$468$$ 0 0
$$469$$ 183.607 0.391487
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 290.484 0.614131
$$474$$ 0 0
$$475$$ 275.075i 0.579105i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 844.826i − 1.76373i −0.471503 0.881865i $$-0.656288\pi$$
0.471503 0.881865i $$-0.343712\pi$$
$$480$$ 0 0
$$481$$ 51.1858 0.106415
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 300.304 0.619184
$$486$$ 0 0
$$487$$ − 328.276i − 0.674077i −0.941491 0.337039i $$-0.890575\pi$$
0.941491 0.337039i $$-0.109425\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 558.319i 1.13711i 0.822647 + 0.568553i $$0.192496\pi$$
−0.822647 + 0.568553i $$0.807504\pi$$
$$492$$ 0 0
$$493$$ 17.3636 0.0352202
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 157.819 0.317543
$$498$$ 0 0
$$499$$ − 507.059i − 1.01615i −0.861313 0.508075i $$-0.830357\pi$$
0.861313 0.508075i $$-0.169643\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 884.817i 1.75908i 0.475825 + 0.879540i $$0.342150\pi$$
−0.475825 + 0.879540i $$0.657850\pi$$
$$504$$ 0 0
$$505$$ 513.083 1.01601
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 398.001 0.781928 0.390964 0.920406i $$-0.372142\pi$$
0.390964 + 0.920406i $$0.372142\pi$$
$$510$$ 0 0
$$511$$ − 789.437i − 1.54489i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 243.069i − 0.471978i
$$516$$ 0 0
$$517$$ −214.031 −0.413987
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −716.672 −1.37557 −0.687785 0.725914i $$-0.741417\pi$$
−0.687785 + 0.725914i $$0.741417\pi$$
$$522$$ 0 0
$$523$$ − 706.115i − 1.35012i −0.737761 0.675062i $$-0.764117\pi$$
0.737761 0.675062i $$-0.235883\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1072.21i − 2.03456i
$$528$$ 0 0
$$529$$ 344.185 0.650634
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 651.621 1.22255
$$534$$ 0 0
$$535$$ 223.913i 0.418529i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 80.6485i 0.149626i
$$540$$ 0 0
$$541$$ −116.644 −0.215609 −0.107804 0.994172i $$-0.534382\pi$$
−0.107804 + 0.994172i $$0.534382\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −195.013 −0.357822
$$546$$ 0 0
$$547$$ − 422.278i − 0.771989i −0.922501 0.385995i $$-0.873858\pi$$
0.922501 0.385995i $$-0.126142\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 14.1717i − 0.0257199i
$$552$$ 0 0
$$553$$ 628.562 1.13664
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 686.288 1.23212 0.616058 0.787701i $$-0.288729\pi$$
0.616058 + 0.787701i $$0.288729\pi$$
$$558$$ 0 0
$$559$$ 399.149i 0.714042i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 439.191i − 0.780090i −0.920796 0.390045i $$-0.872459\pi$$
0.920796 0.390045i $$-0.127541\pi$$
$$564$$ 0 0
$$565$$ −453.769 −0.803130
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −406.032 −0.713589 −0.356794 0.934183i $$-0.616130\pi$$
−0.356794 + 0.934183i $$0.616130\pi$$
$$570$$ 0 0
$$571$$ 166.816i 0.292148i 0.989274 + 0.146074i $$0.0466637\pi$$
−0.989274 + 0.146074i $$0.953336\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 198.463i 0.345153i
$$576$$ 0 0
$$577$$ −897.895 −1.55614 −0.778072 0.628175i $$-0.783802\pi$$
−0.778072 + 0.628175i $$0.783802\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 758.734 1.30591
$$582$$ 0 0
$$583$$ − 650.330i − 1.11549i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 152.207i − 0.259296i −0.991560 0.129648i $$-0.958615\pi$$
0.991560 0.129648i $$-0.0413847\pi$$
$$588$$ 0 0
$$589$$ −875.110 −1.48576
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −964.580 −1.62661 −0.813305 0.581838i $$-0.802334\pi$$
−0.813305 + 0.581838i $$0.802334\pi$$
$$594$$ 0 0
$$595$$ − 489.345i − 0.822428i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 983.578i 1.64203i 0.570905 + 0.821016i $$0.306593\pi$$
−0.570905 + 0.821016i $$0.693407\pi$$
$$600$$ 0 0
$$601$$ −443.392 −0.737757 −0.368878 0.929478i $$-0.620258\pi$$
−0.368878 + 0.929478i $$0.620258\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 232.019 0.383503
$$606$$ 0 0
$$607$$ − 487.520i − 0.803163i −0.915823 0.401582i $$-0.868461\pi$$
0.915823 0.401582i $$-0.131539\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 294.097i − 0.481337i
$$612$$ 0 0
$$613$$ −909.815 −1.48420 −0.742100 0.670289i $$-0.766170\pi$$
−0.742100 + 0.670289i $$0.766170\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −481.557 −0.780481 −0.390241 0.920713i $$-0.627608\pi$$
−0.390241 + 0.920713i $$0.627608\pi$$
$$618$$ 0 0
$$619$$ − 350.103i − 0.565594i −0.959180 0.282797i $$-0.908738\pi$$
0.959180 0.282797i $$-0.0912623\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1114.57i 1.78904i
$$624$$ 0 0
$$625$$ −46.9156 −0.0750649
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −61.9130 −0.0984308
$$630$$ 0 0
$$631$$ 274.658i 0.435274i 0.976030 + 0.217637i $$0.0698350\pi$$
−0.976030 + 0.217637i $$0.930165\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 215.995i − 0.340149i
$$636$$ 0 0
$$637$$ −110.818 −0.173968
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 383.085 0.597636 0.298818 0.954310i $$-0.403408\pi$$
0.298818 + 0.954310i $$0.403408\pi$$
$$642$$ 0 0
$$643$$ − 177.134i − 0.275480i −0.990468 0.137740i $$-0.956016\pi$$
0.990468 0.137740i $$-0.0439838\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 877.305i 1.35596i 0.735081 + 0.677979i $$0.237144\pi$$
−0.735081 + 0.677979i $$0.762856\pi$$
$$648$$ 0 0
$$649$$ −561.344 −0.864937
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.3934 0.0542012 0.0271006 0.999633i $$-0.491373\pi$$
0.0271006 + 0.999633i $$0.491373\pi$$
$$654$$ 0 0
$$655$$ − 822.884i − 1.25631i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 556.337i 0.844214i 0.906546 + 0.422107i $$0.138709\pi$$
−0.906546 + 0.422107i $$0.861291\pi$$
$$660$$ 0 0
$$661$$ 318.875 0.482413 0.241206 0.970474i $$-0.422457\pi$$
0.241206 + 0.970474i $$0.422457\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −399.389 −0.600586
$$666$$ 0 0
$$667$$ − 10.2247i − 0.0153293i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 1466.03i − 2.18484i
$$672$$ 0 0
$$673$$ 135.295 0.201033 0.100517 0.994935i $$-0.467950\pi$$
0.100517 + 0.994935i $$0.467950\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −123.989 −0.183145 −0.0915723 0.995798i $$-0.529189\pi$$
−0.0915723 + 0.995798i $$0.529189\pi$$
$$678$$ 0 0
$$679$$ − 611.966i − 0.901275i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 609.347i 0.892163i 0.894992 + 0.446081i $$0.147181\pi$$
−0.894992 + 0.446081i $$0.852819\pi$$
$$684$$ 0 0
$$685$$ 87.7059 0.128038
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 893.609 1.29696
$$690$$ 0 0
$$691$$ 353.682i 0.511840i 0.966698 + 0.255920i $$0.0823784\pi$$
−0.966698 + 0.255920i $$0.917622\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 657.784i 0.946452i
$$696$$ 0 0
$$697$$ −788.183 −1.13082
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −85.2186 −0.121567 −0.0607836 0.998151i $$-0.519360\pi$$
−0.0607836 + 0.998151i $$0.519360\pi$$
$$702$$ 0 0
$$703$$ 50.5316i 0.0718800i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 1045.57i − 1.47888i
$$708$$ 0 0
$$709$$ −701.943 −0.990046 −0.495023 0.868880i $$-0.664840\pi$$
−0.495023 + 0.868880i $$0.664840\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −631.381 −0.885527
$$714$$ 0 0
$$715$$ 855.036i 1.19585i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 993.471i − 1.38174i −0.722979 0.690870i $$-0.757228\pi$$
0.722979 0.690870i $$-0.242772\pi$$
$$720$$ 0 0
$$721$$ −495.330 −0.687004
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −10.9798 −0.0151445
$$726$$ 0 0
$$727$$ 485.145i 0.667325i 0.942693 + 0.333662i $$0.108285\pi$$
−0.942693 + 0.333662i $$0.891715\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 482.800i − 0.660465i
$$732$$ 0 0
$$733$$ 706.167 0.963394 0.481697 0.876338i $$-0.340021\pi$$
0.481697 + 0.876338i $$0.340021\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 388.053 0.526531
$$738$$ 0 0
$$739$$ 414.250i 0.560554i 0.959919 + 0.280277i $$0.0904264\pi$$
−0.959919 + 0.280277i $$0.909574\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 1386.38i − 1.86592i −0.359978 0.932961i $$-0.617216\pi$$
0.359978 0.932961i $$-0.382784\pi$$
$$744$$ 0 0
$$745$$ 811.141 1.08878
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 456.294 0.609205
$$750$$ 0 0
$$751$$ − 601.365i − 0.800752i −0.916351 0.400376i $$-0.868879\pi$$
0.916351 0.400376i $$-0.131121\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 807.089i 1.06899i
$$756$$ 0 0
$$757$$ −254.294 −0.335924 −0.167962 0.985794i $$-0.553719\pi$$
−0.167962 + 0.985794i $$0.553719\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −466.017 −0.612375 −0.306187 0.951971i $$-0.599053\pi$$
−0.306187 + 0.951971i $$0.599053\pi$$
$$762$$ 0 0
$$763$$ 397.401i 0.520840i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 771.334i − 1.00565i
$$768$$ 0 0
$$769$$ −365.822 −0.475712 −0.237856 0.971300i $$-0.576445\pi$$
−0.237856 + 0.971300i $$0.576445\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −1426.70 −1.84566 −0.922830 0.385207i $$-0.874130\pi$$
−0.922830 + 0.385207i $$0.874130\pi$$
$$774$$ 0 0
$$775$$ 678.007i 0.874848i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 643.293i 0.825794i
$$780$$ 0 0
$$781$$ 333.549 0.427079
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −58.6100 −0.0746624
$$786$$ 0 0
$$787$$ − 410.337i − 0.521394i −0.965421 0.260697i $$-0.916048\pi$$
0.965421 0.260697i $$-0.0839523\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 924.698i 1.16902i
$$792$$ 0 0
$$793$$ 2014.45 2.54029
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 738.968 0.927187 0.463594 0.886048i $$-0.346560\pi$$
0.463594 + 0.886048i $$0.346560\pi$$
$$798$$ 0 0
$$799$$ 355.731i 0.445221i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 1668.47i − 2.07780i
$$804$$ 0 0
$$805$$ −288.154 −0.357956
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −130.444 −0.161241 −0.0806204 0.996745i $$-0.525690\pi$$
−0.0806204 + 0.996745i $$0.525690\pi$$
$$810$$ 0 0
$$811$$ − 1558.44i − 1.92163i −0.277187 0.960816i $$-0.589402\pi$$
0.277187 0.960816i $$-0.410598\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 161.314i − 0.197931i
$$816$$ 0 0
$$817$$ −394.048 −0.482311
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 55.0345 0.0670335 0.0335168 0.999438i $$-0.489329\pi$$
0.0335168 + 0.999438i $$0.489329\pi$$
$$822$$ 0 0
$$823$$ 293.058i 0.356085i 0.984023 + 0.178043i $$0.0569765\pi$$
−0.984023 + 0.178043i $$0.943023\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1118.55i 1.35253i 0.736657 + 0.676267i $$0.236403\pi$$
−0.736657 + 0.676267i $$0.763597\pi$$
$$828$$ 0 0
$$829$$ 1069.30 1.28986 0.644932 0.764240i $$-0.276885\pi$$
0.644932 + 0.764240i $$0.276885\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 134.042 0.160915
$$834$$ 0 0
$$835$$ − 273.616i − 0.327683i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 41.6050i − 0.0495888i −0.999693 0.0247944i $$-0.992107\pi$$
0.999693 0.0247944i $$-0.00789311\pi$$
$$840$$ 0 0
$$841$$ −840.434 −0.999327
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −629.846 −0.745379
$$846$$ 0 0
$$847$$ − 472.813i − 0.558220i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 36.4579i 0.0428413i
$$852$$ 0 0
$$853$$ 1302.63 1.52711 0.763555 0.645743i $$-0.223452\pi$$
0.763555 + 0.645743i $$0.223452\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 734.683 0.857273 0.428636 0.903477i $$-0.358994\pi$$
0.428636 + 0.903477i $$0.358994\pi$$
$$858$$ 0 0
$$859$$ − 445.296i − 0.518388i −0.965825 0.259194i $$-0.916543\pi$$
0.965825 0.259194i $$-0.0834570\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1338.23i 1.55067i 0.631550 + 0.775335i $$0.282419\pi$$
−0.631550 + 0.775335i $$0.717581\pi$$
$$864$$ 0 0
$$865$$ 466.684 0.539519
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1328.46 1.52873
$$870$$ 0 0
$$871$$ 533.218i 0.612190i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 839.338i 0.959244i
$$876$$ 0 0
$$877$$ −329.803 −0.376058 −0.188029 0.982163i $$-0.560210\pi$$
−0.188029 + 0.982163i $$0.560210\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1173.32 −1.33181 −0.665903 0.746038i $$-0.731954\pi$$
−0.665903 + 0.746038i $$0.731954\pi$$
$$882$$ 0 0
$$883$$ 410.459i 0.464846i 0.972615 + 0.232423i $$0.0746654\pi$$
−0.972615 + 0.232423i $$0.925335\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 604.280i 0.681263i 0.940197 + 0.340631i $$0.110641\pi$$
−0.940197 + 0.340631i $$0.889359\pi$$
$$888$$ 0 0
$$889$$ −440.158 −0.495116
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 290.338 0.325127
$$894$$ 0 0
$$895$$ 906.020i 1.01231i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 34.9305i − 0.0388548i
$$900$$ 0 0
$$901$$ −1080.88 −1.19965
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 953.048 1.05309
$$906$$ 0 0
$$907$$ 735.914i 0.811371i 0.914013 + 0.405686i $$0.132967\pi$$
−0.914013 + 0.405686i $$0.867033\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 586.390i − 0.643677i −0.946795 0.321838i $$-0.895699\pi$$
0.946795 0.321838i $$-0.104301\pi$$
$$912$$ 0 0
$$913$$ 1603.58 1.75639
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1676.89 −1.82867
$$918$$ 0 0
$$919$$ − 158.999i − 0.173013i −0.996251 0.0865063i $$-0.972430\pi$$
0.996251 0.0865063i $$-0.0275703\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 458.324i 0.496560i
$$924$$ 0 0
$$925$$ 39.1503 0.0423247
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −540.670 −0.581992 −0.290996 0.956724i $$-0.593987\pi$$
−0.290996 + 0.956724i $$0.593987\pi$$
$$930$$ 0 0
$$931$$ − 109.402i − 0.117510i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 1034.23i − 1.10613i
$$936$$ 0 0
$$937$$ 384.947 0.410829 0.205415 0.978675i $$-0.434146\pi$$
0.205415 + 0.978675i $$0.434146\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 39.6287 0.0421134 0.0210567 0.999778i $$-0.493297\pi$$
0.0210567 + 0.999778i $$0.493297\pi$$
$$942$$ 0 0
$$943$$ 464.128i 0.492182i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 449.648i − 0.474813i −0.971410 0.237407i $$-0.923703\pi$$
0.971410 0.237407i $$-0.0762974\pi$$
$$948$$ 0 0
$$949$$ 2292.62 2.41583
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −21.5070 −0.0225676 −0.0112838 0.999936i $$-0.503592\pi$$
−0.0112838 + 0.999936i $$0.503592\pi$$
$$954$$ 0 0
$$955$$ 954.578i 0.999558i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 178.729i − 0.186370i
$$960$$ 0 0
$$961$$ −1195.98 −1.24452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 58.1172 0.0602251
$$966$$ 0 0
$$967$$ 1355.49i 1.40175i 0.713283 + 0.700876i $$0.247207\pi$$
−0.713283 + 0.700876i $$0.752793\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 938.488i 0.966517i 0.875478 + 0.483258i $$0.160547\pi$$
−0.875478 + 0.483258i $$0.839453\pi$$
$$972$$ 0 0
$$973$$ 1340.44 1.37764
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1090.81 1.11649 0.558243 0.829678i $$-0.311476\pi$$
0.558243 + 0.829678i $$0.311476\pi$$
$$978$$ 0 0
$$979$$ 2355.64i 2.40617i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 27.7492i − 0.0282291i −0.999900 0.0141146i $$-0.995507\pi$$
0.999900 0.0141146i $$-0.00449296\pi$$
$$984$$ 0 0
$$985$$ −383.853 −0.389699
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −284.301 −0.287463
$$990$$ 0 0
$$991$$ 272.858i 0.275336i 0.990478 + 0.137668i $$0.0439607\pi$$
−0.990478 + 0.137668i $$0.956039\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 983.378i − 0.988320i
$$996$$ 0 0
$$997$$ 1190.35 1.19393 0.596964 0.802268i $$-0.296374\pi$$
0.596964 + 0.802268i $$0.296374\pi$$
$$998$$ 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 864.3.g.c.703.4 yes 8
3.2 odd 2 864.3.g.a.703.6 yes 8
4.3 odd 2 inner 864.3.g.c.703.3 yes 8
8.3 odd 2 1728.3.g.k.703.5 8
8.5 even 2 1728.3.g.k.703.6 8
12.11 even 2 864.3.g.a.703.5 8
24.5 odd 2 1728.3.g.n.703.4 8
24.11 even 2 1728.3.g.n.703.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.a.703.5 8 12.11 even 2
864.3.g.a.703.6 yes 8 3.2 odd 2
864.3.g.c.703.3 yes 8 4.3 odd 2 inner
864.3.g.c.703.4 yes 8 1.1 even 1 trivial
1728.3.g.k.703.5 8 8.3 odd 2
1728.3.g.k.703.6 8 8.5 even 2
1728.3.g.n.703.3 8 24.11 even 2
1728.3.g.n.703.4 8 24.5 odd 2