# Properties

 Label 1728.3.g.n.703.3 Level $1728$ Weight $3$ Character 1728.703 Analytic conductor $47.085$ 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] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$47.0845896815$$ 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: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 703.3 Root $$-1.27597 - 0.609843i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.703 Dual form 1728.3.g.n.703.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.22512 q^{5} -6.57221i q^{7} +O(q^{10})$$ $$q-3.22512 q^{5} -6.57221i q^{7} -13.8903i q^{11} +19.0865 q^{13} -23.0865 q^{17} -18.8425i q^{19} -13.5947i q^{23} -14.5986 q^{25} +0.752110 q^{29} +46.4433i q^{31} +21.1962i q^{35} +2.68178 q^{37} +34.1405 q^{41} -20.9127i q^{43} +15.4086i q^{47} +5.80609 q^{49} -46.8189 q^{53} +44.7980i q^{55} -40.4126i q^{59} +105.543 q^{61} -61.5562 q^{65} -27.9369i q^{67} -24.0130i q^{71} -120.117 q^{73} -91.2901 q^{77} +95.6394i q^{79} +115.446i q^{83} +74.4567 q^{85} -169.589 q^{89} -125.440i q^{91} +60.7694i q^{95} -93.1142 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5}+O(q^{10})$$ 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 $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$-0.844455\pi$$
0.882963 0.469443i $$-0.155545\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 13.8903i − 1.26276i −0.775475 0.631379i $$-0.782489\pi$$
0.775475 0.631379i $$-0.217511\pi$$
$$12$$ 0 0
$$13$$ 19.0865 1.46819 0.734095 0.679046i $$-0.237606\pi$$
0.734095 + 0.679046i $$0.237606\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −23.0865 −1.35803 −0.679014 0.734125i $$-0.737593\pi$$
−0.679014 + 0.734125i $$0.737593\pi$$
$$18$$ 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 $$0.269507\pi$$
−0.662473 + 0.749085i $$0.730493\pi$$
$$32$$ 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 $$-0.488462\pi$$
0.0362403 + 0.999343i $$0.488462\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 34.1405 0.832694 0.416347 0.909206i $$-0.363310\pi$$
0.416347 + 0.909206i $$0.363310\pi$$
$$42$$ 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 $$-0.888733\pi$$
0.939525 0.342480i $$-0.111267\pi$$
$$60$$ 0 0
$$61$$ 105.543 1.73022 0.865108 0.501586i $$-0.167250\pi$$
0.865108 + 0.501586i $$0.167250\pi$$
$$62$$ 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 $$0.206952\pi$$
−0.795988 + 0.605312i $$0.793048\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 115.446i 1.39091i 0.718568 + 0.695457i $$0.244798\pi$$
−0.718568 + 0.695457i $$0.755202\pi$$
$$84$$ 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 $$-0.901755\pi$$
−0.952745 + 0.303770i $$0.901755\pi$$
$$90$$ 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 $$-0.880774\pi$$
0.930669 0.365861i $$-0.119226\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 69.4279i 0.648859i 0.945910 + 0.324429i $$0.105172\pi$$
−0.945910 + 0.324429i $$0.894828\pi$$
$$108$$ 0 0
$$109$$ −60.4669 −0.554742 −0.277371 0.960763i $$-0.589463\pi$$
−0.277371 + 0.960763i $$0.589463\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −140.698 −1.24512 −0.622559 0.782573i $$-0.713907\pi$$
−0.622559 + 0.782573i $$0.713907\pi$$
$$114$$ 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 $$-0.915066\pi$$
0.964612 0.263672i $$-0.0849335\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 255.148i − 1.94770i −0.227196 0.973849i $$-0.572956\pi$$
0.227196 0.973849i $$-0.427044\pi$$
$$132$$ 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 $$-0.468355\pi$$
0.0992505 + 0.995062i $$0.468355\pi$$
$$138$$ 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 $$0.310888\pi$$
−0.559774 + 0.828645i $$0.689112\pi$$
$$152$$ 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 $$-0.518433\pi$$
−0.0578757 + 0.998324i $$0.518433\pi$$
$$158$$ 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 $$0.287188\pi$$
−0.619864 + 0.784709i $$0.712812\pi$$
$$180$$ 0 0
$$181$$ 295.508 1.63264 0.816320 0.577600i $$-0.196011\pi$$
0.816320 + 0.577600i $$0.196011\pi$$
$$182$$ 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 $$-0.722189\pi$$
0.642708 0.766111i $$-0.277811\pi$$
$$200$$ 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 $$-0.933933\pi$$
0.978537 0.206070i $$-0.0660673\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 31.5125i − 0.138822i −0.997588 0.0694108i $$-0.977888\pi$$
0.997588 0.0694108i $$-0.0221119\pi$$
$$228$$ 0 0
$$229$$ −242.515 −1.05902 −0.529509 0.848304i $$-0.677624\pi$$
−0.529509 + 0.848304i $$0.677624\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −402.297 −1.72660 −0.863298 0.504695i $$-0.831605\pi$$
−0.863298 + 0.504695i $$0.831605\pi$$
$$234$$ 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 $$-0.985739\pi$$
0.998997 0.0447859i $$-0.0142606\pi$$
$$252$$ 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 $$-0.175716\pi$$
0.851463 + 0.524415i $$0.175716\pi$$
$$258$$ 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 $$-0.864192\pi$$
0.910355 0.413828i $$-0.135808\pi$$
$$272$$ 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 $$-0.470398\pi$$
0.0928625 + 0.995679i $$0.470398\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −40.1606 −0.142920 −0.0714602 0.997443i $$-0.522766\pi$$
−0.0714602 + 0.997443i $$0.522766\pi$$
$$282$$ 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 $$-0.754430\pi$$
0.716879 0.697198i $$-0.245570\pi$$
$$348$$ 0 0
$$349$$ −22.1213 −0.0633849 −0.0316924 0.999498i $$-0.510090\pi$$
−0.0316924 + 0.999498i $$0.510090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 459.068 1.30048 0.650238 0.759731i $$-0.274669\pi$$
0.650238 + 0.759731i $$0.274669\pi$$
$$354$$ 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 $$0.0393638\pi$$
−0.992363 + 0.123350i $$0.960636\pi$$
$$368$$ 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 $$-0.596634\pi$$
−0.298942 + 0.954271i $$0.596634\pi$$
$$374$$ 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 $$-0.497731\pi$$
0.00712906 + 0.999975i $$0.497731\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 250.097 0.623683 0.311842 0.950134i $$-0.399054\pi$$
0.311842 + 0.950134i $$0.399054\pi$$
$$402$$ 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 $$0.175762\pi$$
−0.851387 + 0.524539i $$0.824238\pi$$
$$420$$ 0 0
$$421$$ −65.4392 −0.155438 −0.0777188 0.996975i $$-0.524764\pi$$
−0.0777188 + 0.996975i $$0.524764\pi$$
$$422$$ 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 $$0.322390\pi$$
−0.529472 + 0.848328i $$0.677610\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 320.933i − 0.724455i −0.932090 0.362227i $$-0.882016\pi$$
0.932090 0.362227i $$-0.117984\pi$$
$$444$$ 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 $$-0.337147\pi$$
0.489589 + 0.871953i $$0.337147\pi$$
$$450$$ 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 $$-0.779321\pi$$
0.769152 0.639066i $$-0.220679\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 50.8002i − 0.108780i −0.998520 0.0543899i $$-0.982679\pi$$
0.998520 0.0543899i $$-0.0173214\pi$$
$$468$$ 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 $$0.109425\pi$$
−0.941491 + 0.337039i $$0.890575\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 558.319i − 1.13711i −0.822647 0.568553i $$-0.807504\pi$$
0.822647 0.568553i $$-0.192496\pi$$
$$492$$ 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 $$-0.258583\pi$$
0.687785 + 0.725914i $$0.258583\pi$$
$$522$$ 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 $$-0.465618\pi$$
0.107804 + 0.994172i $$0.465618\pi$$
$$542$$ 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 $$0.127541\pi$$
−0.920796 + 0.390045i $$0.872459\pi$$
$$564$$ 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 $$-0.383870\pi$$
0.356794 + 0.934183i $$0.383870\pi$$
$$570$$ 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 $$0.0413847\pi$$
−0.991560 + 0.129648i $$0.958615\pi$$
$$588$$ 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 $$-0.197666\pi$$
0.813305 + 0.581838i $$0.197666\pi$$
$$594$$ 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 $$0.131539\pi$$
−0.915823 + 0.401582i $$0.868461\pi$$
$$608$$ 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 $$-0.233830\pi$$
0.742100 + 0.670289i $$0.233830\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 481.557 0.780481 0.390241 0.920713i $$-0.372392\pi$$
0.390241 + 0.920713i $$0.372392\pi$$
$$618$$ 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 $$-0.930165\pi$$
0.976030 0.217637i $$-0.0698350\pi$$
$$632$$ 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 $$-0.596592\pi$$
−0.298818 + 0.954310i $$0.596592\pi$$
$$642$$ 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 $$-0.861291\pi$$
0.906546 0.422107i $$-0.138709\pi$$
$$660$$ 0 0
$$661$$ −318.875 −0.482413 −0.241206 0.970474i $$-0.577543\pi$$
−0.241206 + 0.970474i $$0.577543\pi$$
$$662$$ 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 $$-0.852819\pi$$
0.894992 0.446081i $$-0.147181\pi$$
$$684$$ 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 $$-0.335160\pi$$
0.495023 + 0.868880i $$0.335160\pi$$
$$710$$ 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 $$-0.891715\pi$$
0.942693 0.333662i $$-0.108285\pi$$
$$728$$ 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 $$-0.659979\pi$$
−0.481697 + 0.876338i $$0.659979\pi$$
$$734$$ 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 $$0.131121\pi$$
−0.916351 + 0.400376i $$0.868879\pi$$
$$752$$ 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 $$-0.446281\pi$$
0.167962 + 0.985794i $$0.446281\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 466.017 0.612375 0.306187 0.951971i $$-0.400947\pi$$
0.306187 + 0.951971i $$0.400947\pi$$
$$762$$ 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 $$-0.474310\pi$$
0.0806204 + 0.996745i $$0.474310\pi$$
$$810$$ 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 $$-0.943023\pi$$
0.984023 0.178043i $$-0.0569765\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 1118.55i − 1.35253i −0.736657 0.676267i $$-0.763597\pi$$
0.736657 0.676267i $$-0.236403\pi$$
$$828$$ 0 0
$$829$$ −1069.30 −1.28986 −0.644932 0.764240i $$-0.723115\pi$$
−0.644932 + 0.764240i $$0.723115\pi$$
$$830$$ 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 $$-0.776548\pi$$
−0.763555 + 0.645743i $$0.776548\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −734.683 −0.857273 −0.428636 0.903477i $$-0.641006\pi$$
−0.428636 + 0.903477i $$0.641006\pi$$
$$858$$ 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 $$-0.439790\pi$$
0.188029 + 0.982163i $$0.439790\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1173.32 1.33181 0.665903 0.746038i $$-0.268046\pi$$
0.665903 + 0.746038i $$0.268046\pi$$
$$882$$ 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 $$0.0275703\pi$$
−0.996251 + 0.0865063i $$0.972430\pi$$
$$920$$ 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 $$-0.406013\pi$$
0.290996 + 0.956724i $$0.406013\pi$$
$$930$$ 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 $$0.0762974\pi$$
−0.971410 + 0.237407i $$0.923703\pi$$
$$948$$ 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 $$-0.496408\pi$$
0.0112838 + 0.999936i $$0.496408\pi$$
$$954$$ 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 $$-0.752793\pi$$
0.713283 0.700876i $$-0.247207\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 938.488i − 0.966517i −0.875478 0.483258i $$-0.839453\pi$$
0.875478 0.483258i $$-0.160547\pi$$
$$972$$ 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 $$-0.688524\pi$$
−0.558243 + 0.829678i $$0.688524\pi$$
$$978$$ 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 $$-0.956039\pi$$
0.990478 0.137668i $$-0.0439607\pi$$
$$992$$ 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 $$-0.703626\pi$$
−0.596964 + 0.802268i $$0.703626\pi$$
$$998$$ 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 1728.3.g.n.703.3 8
3.2 odd 2 1728.3.g.k.703.5 8
4.3 odd 2 inner 1728.3.g.n.703.4 8
8.3 odd 2 864.3.g.a.703.6 yes 8
8.5 even 2 864.3.g.a.703.5 8
12.11 even 2 1728.3.g.k.703.6 8
24.5 odd 2 864.3.g.c.703.3 yes 8
24.11 even 2 864.3.g.c.703.4 yes 8

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