# Properties

 Label 1728.3.g.m.703.5 Level $1728$ Weight $3$ Character 1728.703 Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 63*x^4 - 74*x^3 + 70*x^2 - 38*x + 13 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 703.5 Root $$0.500000 + 2.19293i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.703 Dual form 1728.3.g.m.703.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.13244 q^{5} -4.02516i q^{7} +O(q^{10})$$ $$q+5.13244 q^{5} -4.02516i q^{7} -4.30344i q^{11} +18.4862 q^{13} +23.5751 q^{17} +21.7259i q^{19} +30.7461i q^{23} +1.34199 q^{25} -12.6840 q^{29} +24.5298i q^{31} -20.6589i q^{35} -18.2314 q^{37} +38.0990 q^{41} -34.9724i q^{43} -29.6295i q^{47} +32.7981 q^{49} -39.3043 q^{53} -22.0872i q^{55} -65.3429i q^{59} +29.8541 q^{61} +94.8794 q^{65} -11.8634i q^{67} +140.651i q^{71} +119.285 q^{73} -17.3220 q^{77} -9.18859i q^{79} -113.180i q^{83} +120.998 q^{85} +7.88346 q^{89} -74.4099i q^{91} +111.507i q^{95} +55.5080 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5}+O(q^{10})$$ 8 * q + 8 * q^5 $$8 q + 8 q^{5} - 8 q^{13} - 24 q^{17} + 24 q^{25} - 128 q^{29} - 24 q^{37} - 160 q^{41} - 144 q^{49} - 48 q^{53} + 136 q^{61} + 280 q^{65} + 72 q^{73} + 520 q^{77} - 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100})$$ 8 * q + 8 * q^5 - 8 * q^13 - 24 * q^17 + 24 * q^25 - 128 * q^29 - 24 * q^37 - 160 * q^41 - 144 * q^49 - 48 * q^53 + 136 * q^61 + 280 * q^65 + 72 * q^73 + 520 * q^77 - 96 * q^85 + 168 * q^89 + 104 * 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
$$5$$ 5.13244 1.02649 0.513244 0.858242i $$-0.328443\pi$$
0.513244 + 0.858242i $$0.328443\pi$$
$$6$$ 0 0
$$7$$ − 4.02516i − 0.575023i −0.957777 0.287511i $$-0.907172\pi$$
0.957777 0.287511i $$-0.0928279\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 4.30344i − 0.391222i −0.980682 0.195611i $$-0.937331\pi$$
0.980682 0.195611i $$-0.0626689\pi$$
$$12$$ 0 0
$$13$$ 18.4862 1.42202 0.711008 0.703184i $$-0.248239\pi$$
0.711008 + 0.703184i $$0.248239\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 23.5751 1.38677 0.693384 0.720568i $$-0.256119\pi$$
0.693384 + 0.720568i $$0.256119\pi$$
$$18$$ 0 0
$$19$$ 21.7259i 1.14347i 0.820438 + 0.571735i $$0.193729\pi$$
−0.820438 + 0.571735i $$0.806271\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 30.7461i 1.33679i 0.743809 + 0.668393i $$0.233017\pi$$
−0.743809 + 0.668393i $$0.766983\pi$$
$$24$$ 0 0
$$25$$ 1.34199 0.0536794
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −12.6840 −0.437378 −0.218689 0.975795i $$-0.570178\pi$$
−0.218689 + 0.975795i $$0.570178\pi$$
$$30$$ 0 0
$$31$$ 24.5298i 0.791283i 0.918405 + 0.395642i $$0.129478\pi$$
−0.918405 + 0.395642i $$0.870522\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 20.6589i − 0.590254i
$$36$$ 0 0
$$37$$ −18.2314 −0.492740 −0.246370 0.969176i $$-0.579238\pi$$
−0.246370 + 0.969176i $$0.579238\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 38.0990 0.929244 0.464622 0.885509i $$-0.346190\pi$$
0.464622 + 0.885509i $$0.346190\pi$$
$$42$$ 0 0
$$43$$ − 34.9724i − 0.813312i −0.913581 0.406656i $$-0.866695\pi$$
0.913581 0.406656i $$-0.133305\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 29.6295i − 0.630415i −0.949023 0.315208i $$-0.897926\pi$$
0.949023 0.315208i $$-0.102074\pi$$
$$48$$ 0 0
$$49$$ 32.7981 0.669349
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −39.3043 −0.741591 −0.370796 0.928715i $$-0.620915\pi$$
−0.370796 + 0.928715i $$0.620915\pi$$
$$54$$ 0 0
$$55$$ − 22.0872i − 0.401585i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 65.3429i − 1.10751i −0.832681 0.553753i $$-0.813195\pi$$
0.832681 0.553753i $$-0.186805\pi$$
$$60$$ 0 0
$$61$$ 29.8541 0.489412 0.244706 0.969597i $$-0.421309\pi$$
0.244706 + 0.969597i $$0.421309\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 94.8794 1.45968
$$66$$ 0 0
$$67$$ − 11.8634i − 0.177066i −0.996073 0.0885329i $$-0.971782\pi$$
0.996073 0.0885329i $$-0.0282178\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 140.651i 1.98100i 0.137529 + 0.990498i $$0.456084\pi$$
−0.137529 + 0.990498i $$0.543916\pi$$
$$72$$ 0 0
$$73$$ 119.285 1.63404 0.817022 0.576607i $$-0.195624\pi$$
0.817022 + 0.576607i $$0.195624\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −17.3220 −0.224961
$$78$$ 0 0
$$79$$ − 9.18859i − 0.116311i −0.998308 0.0581556i $$-0.981478\pi$$
0.998308 0.0581556i $$-0.0185220\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 113.180i − 1.36362i −0.731529 0.681810i $$-0.761193\pi$$
0.731529 0.681810i $$-0.238807\pi$$
$$84$$ 0 0
$$85$$ 120.998 1.42350
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.88346 0.0885782 0.0442891 0.999019i $$-0.485898\pi$$
0.0442891 + 0.999019i $$0.485898\pi$$
$$90$$ 0 0
$$91$$ − 74.4099i − 0.817691i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 111.507i 1.17376i
$$96$$ 0 0
$$97$$ 55.5080 0.572248 0.286124 0.958193i $$-0.407633\pi$$
0.286124 + 0.958193i $$0.407633\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −195.186 −1.93253 −0.966265 0.257550i $$-0.917085\pi$$
−0.966265 + 0.257550i $$0.917085\pi$$
$$102$$ 0 0
$$103$$ 29.3579i 0.285028i 0.989793 + 0.142514i $$0.0455186\pi$$
−0.989793 + 0.142514i $$0.954481\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 133.256i 1.24538i 0.782468 + 0.622691i $$0.213961\pi$$
−0.782468 + 0.622691i $$0.786039\pi$$
$$108$$ 0 0
$$109$$ 198.485 1.82096 0.910481 0.413552i $$-0.135712\pi$$
0.910481 + 0.413552i $$0.135712\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −178.745 −1.58182 −0.790908 0.611934i $$-0.790392\pi$$
−0.790908 + 0.611934i $$0.790392\pi$$
$$114$$ 0 0
$$115$$ 157.802i 1.37220i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 94.8934i − 0.797424i
$$120$$ 0 0
$$121$$ 102.480 0.846946
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −121.423 −0.971388
$$126$$ 0 0
$$127$$ − 172.213i − 1.35601i −0.735058 0.678004i $$-0.762845\pi$$
0.735058 0.678004i $$-0.237155\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 206.300i − 1.57481i −0.616435 0.787406i $$-0.711424\pi$$
0.616435 0.787406i $$-0.288576\pi$$
$$132$$ 0 0
$$133$$ 87.4503 0.657521
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 85.1089 0.621233 0.310616 0.950535i $$-0.399465\pi$$
0.310616 + 0.950535i $$0.399465\pi$$
$$138$$ 0 0
$$139$$ 150.326i 1.08148i 0.841189 + 0.540742i $$0.181856\pi$$
−0.841189 + 0.540742i $$0.818144\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 79.5542i − 0.556323i
$$144$$ 0 0
$$145$$ −65.0998 −0.448964
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 111.133 0.745861 0.372931 0.927859i $$-0.378353\pi$$
0.372931 + 0.927859i $$0.378353\pi$$
$$150$$ 0 0
$$151$$ 166.295i 1.10129i 0.834739 + 0.550646i $$0.185619\pi$$
−0.834739 + 0.550646i $$0.814381\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 125.898i 0.812243i
$$156$$ 0 0
$$157$$ −50.1560 −0.319465 −0.159732 0.987160i $$-0.551063\pi$$
−0.159732 + 0.987160i $$0.551063\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 123.758 0.768682
$$162$$ 0 0
$$163$$ − 265.994i − 1.63187i −0.578145 0.815934i $$-0.696223\pi$$
0.578145 0.815934i $$-0.303777\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 223.651i 1.33923i 0.742708 + 0.669615i $$0.233541\pi$$
−0.742708 + 0.669615i $$0.766459\pi$$
$$168$$ 0 0
$$169$$ 172.740 1.02213
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −55.4735 −0.320656 −0.160328 0.987064i $$-0.551255\pi$$
−0.160328 + 0.987064i $$0.551255\pi$$
$$174$$ 0 0
$$175$$ − 5.40171i − 0.0308669i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 10.7562i − 0.0600907i −0.999549 0.0300453i $$-0.990435\pi$$
0.999549 0.0300453i $$-0.00956517\pi$$
$$180$$ 0 0
$$181$$ 36.3638 0.200905 0.100453 0.994942i $$-0.467971\pi$$
0.100453 + 0.994942i $$0.467971\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −93.5715 −0.505792
$$186$$ 0 0
$$187$$ − 101.454i − 0.542534i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 214.445i − 1.12275i −0.827563 0.561373i $$-0.810273\pi$$
0.827563 0.561373i $$-0.189727\pi$$
$$192$$ 0 0
$$193$$ 280.031 1.45094 0.725470 0.688254i $$-0.241622\pi$$
0.725470 + 0.688254i $$0.241622\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 254.078 1.28974 0.644869 0.764293i $$-0.276912\pi$$
0.644869 + 0.764293i $$0.276912\pi$$
$$198$$ 0 0
$$199$$ 150.197i 0.754757i 0.926059 + 0.377379i $$0.123174\pi$$
−0.926059 + 0.377379i $$0.876826\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 51.0550i 0.251503i
$$204$$ 0 0
$$205$$ 195.541 0.953858
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 93.4962 0.447350
$$210$$ 0 0
$$211$$ 294.457i 1.39553i 0.716326 + 0.697765i $$0.245822\pi$$
−0.716326 + 0.697765i $$0.754178\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 179.494i − 0.834855i
$$216$$ 0 0
$$217$$ 98.7363 0.455006
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 435.813 1.97201
$$222$$ 0 0
$$223$$ − 90.5784i − 0.406181i −0.979160 0.203091i $$-0.934901\pi$$
0.979160 0.203091i $$-0.0650986\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 401.033i 1.76667i 0.468746 + 0.883333i $$0.344706\pi$$
−0.468746 + 0.883333i $$0.655294\pi$$
$$228$$ 0 0
$$229$$ 155.539 0.679209 0.339605 0.940568i $$-0.389707\pi$$
0.339605 + 0.940568i $$0.389707\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −238.595 −1.02401 −0.512007 0.858981i $$-0.671098\pi$$
−0.512007 + 0.858981i $$0.671098\pi$$
$$234$$ 0 0
$$235$$ − 152.072i − 0.647114i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 61.3314i − 0.256617i −0.991734 0.128308i $$-0.959045\pi$$
0.991734 0.128308i $$-0.0409547\pi$$
$$240$$ 0 0
$$241$$ 270.776 1.12355 0.561776 0.827290i $$-0.310118\pi$$
0.561776 + 0.827290i $$0.310118\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 168.334 0.687079
$$246$$ 0 0
$$247$$ 401.630i 1.62603i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 229.191i − 0.913110i −0.889695 0.456555i $$-0.849083\pi$$
0.889695 0.456555i $$-0.150917\pi$$
$$252$$ 0 0
$$253$$ 132.314 0.522979
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −366.617 −1.42653 −0.713263 0.700897i $$-0.752783\pi$$
−0.713263 + 0.700897i $$0.752783\pi$$
$$258$$ 0 0
$$259$$ 73.3842i 0.283337i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 159.531i − 0.606581i −0.952898 0.303290i $$-0.901915\pi$$
0.952898 0.303290i $$-0.0980852\pi$$
$$264$$ 0 0
$$265$$ −201.727 −0.761235
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −91.2480 −0.339212 −0.169606 0.985512i $$-0.554249\pi$$
−0.169606 + 0.985512i $$0.554249\pi$$
$$270$$ 0 0
$$271$$ − 506.145i − 1.86769i −0.357675 0.933846i $$-0.616430\pi$$
0.357675 0.933846i $$-0.383570\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 5.77515i − 0.0210006i
$$276$$ 0 0
$$277$$ 28.7143 0.103662 0.0518308 0.998656i $$-0.483494\pi$$
0.0518308 + 0.998656i $$0.483494\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 194.297 0.691447 0.345724 0.938336i $$-0.387633\pi$$
0.345724 + 0.938336i $$0.387633\pi$$
$$282$$ 0 0
$$283$$ − 212.089i − 0.749432i −0.927140 0.374716i $$-0.877740\pi$$
0.927140 0.374716i $$-0.122260\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 153.355i − 0.534336i
$$288$$ 0 0
$$289$$ 266.784 0.923127
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 118.777 0.405382 0.202691 0.979243i $$-0.435031\pi$$
0.202691 + 0.979243i $$0.435031\pi$$
$$294$$ 0 0
$$295$$ − 335.369i − 1.13684i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 568.378i 1.90093i
$$300$$ 0 0
$$301$$ −140.769 −0.467673
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 153.225 0.502376
$$306$$ 0 0
$$307$$ 118.198i 0.385011i 0.981296 + 0.192506i $$0.0616614\pi$$
−0.981296 + 0.192506i $$0.938339\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 57.5386i 0.185012i 0.995712 + 0.0925059i $$0.0294877\pi$$
−0.995712 + 0.0925059i $$0.970512\pi$$
$$312$$ 0 0
$$313$$ −401.012 −1.28119 −0.640594 0.767879i $$-0.721312\pi$$
−0.640594 + 0.767879i $$0.721312\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.34020 −0.0263098 −0.0131549 0.999913i $$-0.504187\pi$$
−0.0131549 + 0.999913i $$0.504187\pi$$
$$318$$ 0 0
$$319$$ 54.5847i 0.171112i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 512.190i 1.58573i
$$324$$ 0 0
$$325$$ 24.8082 0.0763330
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −119.264 −0.362503
$$330$$ 0 0
$$331$$ − 253.990i − 0.767340i −0.923470 0.383670i $$-0.874660\pi$$
0.923470 0.383670i $$-0.125340\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 60.8883i − 0.181756i
$$336$$ 0 0
$$337$$ −620.332 −1.84075 −0.920374 0.391040i $$-0.872115\pi$$
−0.920374 + 0.391040i $$0.872115\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 105.562 0.309567
$$342$$ 0 0
$$343$$ − 329.250i − 0.959914i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 496.172i − 1.42989i −0.699181 0.714945i $$-0.746452\pi$$
0.699181 0.714945i $$-0.253548\pi$$
$$348$$ 0 0
$$349$$ −63.4267 −0.181738 −0.0908692 0.995863i $$-0.528965\pi$$
−0.0908692 + 0.995863i $$0.528965\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −439.982 −1.24641 −0.623204 0.782059i $$-0.714170\pi$$
−0.623204 + 0.782059i $$0.714170\pi$$
$$354$$ 0 0
$$355$$ 721.882i 2.03347i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 489.550i − 1.36365i −0.731516 0.681824i $$-0.761187\pi$$
0.731516 0.681824i $$-0.238813\pi$$
$$360$$ 0 0
$$361$$ −111.016 −0.307524
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 612.224 1.67733
$$366$$ 0 0
$$367$$ 393.063i 1.07102i 0.844530 + 0.535509i $$0.179880\pi$$
−0.844530 + 0.535509i $$0.820120\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 158.206i 0.426432i
$$372$$ 0 0
$$373$$ −11.4757 −0.0307658 −0.0153829 0.999882i $$-0.504897\pi$$
−0.0153829 + 0.999882i $$0.504897\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −234.478 −0.621959
$$378$$ 0 0
$$379$$ − 661.804i − 1.74618i −0.487556 0.873092i $$-0.662112\pi$$
0.487556 0.873092i $$-0.337888\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 95.8558i 0.250276i 0.992139 + 0.125138i $$0.0399374\pi$$
−0.992139 + 0.125138i $$0.960063\pi$$
$$384$$ 0 0
$$385$$ −88.9043 −0.230920
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −471.609 −1.21236 −0.606182 0.795326i $$-0.707300\pi$$
−0.606182 + 0.795326i $$0.707300\pi$$
$$390$$ 0 0
$$391$$ 724.840i 1.85381i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 47.1599i − 0.119392i
$$396$$ 0 0
$$397$$ −718.221 −1.80912 −0.904560 0.426346i $$-0.859801\pi$$
−0.904560 + 0.426346i $$0.859801\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −225.675 −0.562780 −0.281390 0.959593i $$-0.590796\pi$$
−0.281390 + 0.959593i $$0.590796\pi$$
$$402$$ 0 0
$$403$$ 453.462i 1.12522i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 78.4576i 0.192770i
$$408$$ 0 0
$$409$$ 588.917 1.43990 0.719948 0.694028i $$-0.244166\pi$$
0.719948 + 0.694028i $$0.244166\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −263.015 −0.636841
$$414$$ 0 0
$$415$$ − 580.892i − 1.39974i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 121.906i − 0.290944i −0.989362 0.145472i $$-0.953530\pi$$
0.989362 0.145472i $$-0.0464701\pi$$
$$420$$ 0 0
$$421$$ −289.293 −0.687157 −0.343578 0.939124i $$-0.611639\pi$$
−0.343578 + 0.939124i $$0.611639\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 31.6374 0.0744410
$$426$$ 0 0
$$427$$ − 120.168i − 0.281423i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 337.402i 0.782835i 0.920213 + 0.391418i $$0.128015\pi$$
−0.920213 + 0.391418i $$0.871985\pi$$
$$432$$ 0 0
$$433$$ −7.18171 −0.0165859 −0.00829297 0.999966i $$-0.502640\pi$$
−0.00829297 + 0.999966i $$0.502640\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −667.987 −1.52857
$$438$$ 0 0
$$439$$ 780.145i 1.77710i 0.458783 + 0.888548i $$0.348285\pi$$
−0.458783 + 0.888548i $$0.651715\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 478.844i − 1.08091i −0.841372 0.540456i $$-0.818252\pi$$
0.841372 0.540456i $$-0.181748\pi$$
$$444$$ 0 0
$$445$$ 40.4614 0.0909245
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 55.5835 0.123794 0.0618970 0.998083i $$-0.480285\pi$$
0.0618970 + 0.998083i $$0.480285\pi$$
$$450$$ 0 0
$$451$$ − 163.957i − 0.363540i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 381.905i − 0.839351i
$$456$$ 0 0
$$457$$ 31.6281 0.0692080 0.0346040 0.999401i $$-0.488983\pi$$
0.0346040 + 0.999401i $$0.488983\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −498.508 −1.08136 −0.540681 0.841228i $$-0.681833\pi$$
−0.540681 + 0.841228i $$0.681833\pi$$
$$462$$ 0 0
$$463$$ − 63.5797i − 0.137321i −0.997640 0.0686606i $$-0.978127\pi$$
0.997640 0.0686606i $$-0.0218726\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 550.457i − 1.17871i −0.807875 0.589354i $$-0.799382\pi$$
0.807875 0.589354i $$-0.200618\pi$$
$$468$$ 0 0
$$469$$ −47.7521 −0.101817
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −150.502 −0.318185
$$474$$ 0 0
$$475$$ 29.1559i 0.0613808i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 760.357i 1.58738i 0.608320 + 0.793692i $$0.291844\pi$$
−0.608320 + 0.793692i $$0.708156\pi$$
$$480$$ 0 0
$$481$$ −337.029 −0.700683
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 284.892 0.587406
$$486$$ 0 0
$$487$$ − 340.198i − 0.698558i −0.937019 0.349279i $$-0.886427\pi$$
0.937019 0.349279i $$-0.113573\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 488.166i 0.994229i 0.867685 + 0.497114i $$0.165607\pi$$
−0.867685 + 0.497114i $$0.834393\pi$$
$$492$$ 0 0
$$493$$ −299.026 −0.606543
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 566.141 1.13912
$$498$$ 0 0
$$499$$ 195.282i 0.391347i 0.980669 + 0.195674i $$0.0626893\pi$$
−0.980669 + 0.195674i $$0.937311\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 167.239i 0.332483i 0.986085 + 0.166241i $$0.0531631\pi$$
−0.986085 + 0.166241i $$0.946837\pi$$
$$504$$ 0 0
$$505$$ −1001.78 −1.98372
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −374.972 −0.736683 −0.368342 0.929691i $$-0.620074\pi$$
−0.368342 + 0.929691i $$0.620074\pi$$
$$510$$ 0 0
$$511$$ − 480.142i − 0.939612i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 150.678i 0.292578i
$$516$$ 0 0
$$517$$ −127.509 −0.246632
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −49.7644 −0.0955170 −0.0477585 0.998859i $$-0.515208\pi$$
−0.0477585 + 0.998859i $$0.515208\pi$$
$$522$$ 0 0
$$523$$ − 878.700i − 1.68012i −0.542497 0.840058i $$-0.682521\pi$$
0.542497 0.840058i $$-0.317479\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 578.291i 1.09733i
$$528$$ 0 0
$$529$$ −416.320 −0.786995
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 704.306 1.32140
$$534$$ 0 0
$$535$$ 683.928i 1.27837i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 141.145i − 0.261864i
$$540$$ 0 0
$$541$$ −183.963 −0.340043 −0.170022 0.985440i $$-0.554384\pi$$
−0.170022 + 0.985440i $$0.554384\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1018.71 1.86920
$$546$$ 0 0
$$547$$ 642.639i 1.17484i 0.809281 + 0.587421i $$0.199857\pi$$
−0.809281 + 0.587421i $$0.800143\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 275.571i − 0.500129i
$$552$$ 0 0
$$553$$ −36.9855 −0.0668816
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −231.980 −0.416480 −0.208240 0.978078i $$-0.566774\pi$$
−0.208240 + 0.978078i $$0.566774\pi$$
$$558$$ 0 0
$$559$$ − 646.507i − 1.15654i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 356.868i 0.633869i 0.948447 + 0.316935i $$0.102654\pi$$
−0.948447 + 0.316935i $$0.897346\pi$$
$$564$$ 0 0
$$565$$ −917.400 −1.62372
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 440.786 0.774669 0.387334 0.921939i $$-0.373396\pi$$
0.387334 + 0.921939i $$0.373396\pi$$
$$570$$ 0 0
$$571$$ 712.649i 1.24807i 0.781396 + 0.624036i $$0.214508\pi$$
−0.781396 + 0.624036i $$0.785492\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 41.2608i 0.0717579i
$$576$$ 0 0
$$577$$ −253.401 −0.439170 −0.219585 0.975593i $$-0.570470\pi$$
−0.219585 + 0.975593i $$0.570470\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −455.569 −0.784112
$$582$$ 0 0
$$583$$ 169.144i 0.290126i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 800.800i − 1.36422i −0.731248 0.682112i $$-0.761062\pi$$
0.731248 0.682112i $$-0.238938\pi$$
$$588$$ 0 0
$$589$$ −532.932 −0.904809
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 618.665 1.04328 0.521640 0.853166i $$-0.325321\pi$$
0.521640 + 0.853166i $$0.325321\pi$$
$$594$$ 0 0
$$595$$ − 487.035i − 0.818546i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 249.660i 0.416795i 0.978044 + 0.208398i $$0.0668248\pi$$
−0.978044 + 0.208398i $$0.933175\pi$$
$$600$$ 0 0
$$601$$ −1137.02 −1.89188 −0.945941 0.324339i $$-0.894858\pi$$
−0.945941 + 0.324339i $$0.894858\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 525.975 0.869380
$$606$$ 0 0
$$607$$ − 520.853i − 0.858077i −0.903286 0.429038i $$-0.858852\pi$$
0.903286 0.429038i $$-0.141148\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 547.737i − 0.896460i
$$612$$ 0 0
$$613$$ −148.439 −0.242152 −0.121076 0.992643i $$-0.538635\pi$$
−0.121076 + 0.992643i $$0.538635\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1081.44 1.75273 0.876366 0.481646i $$-0.159961\pi$$
0.876366 + 0.481646i $$0.159961\pi$$
$$618$$ 0 0
$$619$$ − 306.359i − 0.494925i −0.968897 0.247463i $$-0.920403\pi$$
0.968897 0.247463i $$-0.0795967\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 31.7322i − 0.0509345i
$$624$$ 0 0
$$625$$ −656.749 −1.05080
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −429.806 −0.683316
$$630$$ 0 0
$$631$$ 807.507i 1.27973i 0.768489 + 0.639863i $$0.221009\pi$$
−0.768489 + 0.639863i $$0.778991\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 883.874i − 1.39193i
$$636$$ 0 0
$$637$$ 606.312 0.951824
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −387.743 −0.604904 −0.302452 0.953165i $$-0.597805\pi$$
−0.302452 + 0.953165i $$0.597805\pi$$
$$642$$ 0 0
$$643$$ 978.913i 1.52242i 0.648508 + 0.761208i $$0.275393\pi$$
−0.648508 + 0.761208i $$0.724607\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 786.067i 1.21494i 0.794342 + 0.607470i $$0.207816\pi$$
−0.794342 + 0.607470i $$0.792184\pi$$
$$648$$ 0 0
$$649$$ −281.199 −0.433280
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −251.026 −0.384419 −0.192209 0.981354i $$-0.561565\pi$$
−0.192209 + 0.981354i $$0.561565\pi$$
$$654$$ 0 0
$$655$$ − 1058.82i − 1.61653i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 933.805i − 1.41700i −0.705709 0.708501i $$-0.749372\pi$$
0.705709 0.708501i $$-0.250628\pi$$
$$660$$ 0 0
$$661$$ −473.319 −0.716065 −0.358033 0.933709i $$-0.616552\pi$$
−0.358033 + 0.933709i $$0.616552\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 448.834 0.674938
$$666$$ 0 0
$$667$$ − 389.982i − 0.584681i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 128.475i − 0.191469i
$$672$$ 0 0
$$673$$ 254.846 0.378671 0.189336 0.981912i $$-0.439367\pi$$
0.189336 + 0.981912i $$0.439367\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −104.575 −0.154468 −0.0772338 0.997013i $$-0.524609\pi$$
−0.0772338 + 0.997013i $$0.524609\pi$$
$$678$$ 0 0
$$679$$ − 223.429i − 0.329055i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 623.576i 0.912995i 0.889725 + 0.456497i $$0.150896\pi$$
−0.889725 + 0.456497i $$0.849104\pi$$
$$684$$ 0 0
$$685$$ 436.817 0.637689
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −726.588 −1.05455
$$690$$ 0 0
$$691$$ − 544.623i − 0.788167i −0.919075 0.394083i $$-0.871062\pi$$
0.919075 0.394083i $$-0.128938\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 771.541i 1.11013i
$$696$$ 0 0
$$697$$ 898.186 1.28865
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −411.842 −0.587507 −0.293753 0.955881i $$-0.594904\pi$$
−0.293753 + 0.955881i $$0.594904\pi$$
$$702$$ 0 0
$$703$$ − 396.093i − 0.563433i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 785.653i 1.11125i
$$708$$ 0 0
$$709$$ −679.236 −0.958020 −0.479010 0.877809i $$-0.659004\pi$$
−0.479010 + 0.877809i $$0.659004\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −754.194 −1.05778
$$714$$ 0 0
$$715$$ − 408.308i − 0.571060i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 619.766i 0.861983i 0.902356 + 0.430991i $$0.141836\pi$$
−0.902356 + 0.430991i $$0.858164\pi$$
$$720$$ 0 0
$$721$$ 118.170 0.163898
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −17.0217 −0.0234782
$$726$$ 0 0
$$727$$ 1224.73i 1.68463i 0.538982 + 0.842317i $$0.318809\pi$$
−0.538982 + 0.842317i $$0.681191\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 824.477i − 1.12788i
$$732$$ 0 0
$$733$$ −797.758 −1.08835 −0.544173 0.838973i $$-0.683156\pi$$
−0.544173 + 0.838973i $$0.683156\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −51.0534 −0.0692719
$$738$$ 0 0
$$739$$ 422.067i 0.571133i 0.958359 + 0.285567i $$0.0921818\pi$$
−0.958359 + 0.285567i $$0.907818\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 805.671i − 1.08435i −0.840266 0.542174i $$-0.817601\pi$$
0.840266 0.542174i $$-0.182399\pi$$
$$744$$ 0 0
$$745$$ 570.385 0.765618
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 536.376 0.716123
$$750$$ 0 0
$$751$$ 1014.96i 1.35148i 0.737138 + 0.675742i $$0.236177\pi$$
−0.737138 + 0.675742i $$0.763823\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 853.501i 1.13046i
$$756$$ 0 0
$$757$$ 14.8738 0.0196483 0.00982416 0.999952i $$-0.496873\pi$$
0.00982416 + 0.999952i $$0.496873\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 884.373 1.16212 0.581060 0.813861i $$-0.302638\pi$$
0.581060 + 0.813861i $$0.302638\pi$$
$$762$$ 0 0
$$763$$ − 798.933i − 1.04709i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1207.94i − 1.57489i
$$768$$ 0 0
$$769$$ −775.760 −1.00879 −0.504395 0.863473i $$-0.668285\pi$$
−0.504395 + 0.863473i $$0.668285\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 794.115 1.02732 0.513658 0.857995i $$-0.328290\pi$$
0.513658 + 0.857995i $$0.328290\pi$$
$$774$$ 0 0
$$775$$ 32.9186i 0.0424756i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 827.736i 1.06256i
$$780$$ 0 0
$$781$$ 605.281 0.775008
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −257.423 −0.327927
$$786$$ 0 0
$$787$$ − 344.287i − 0.437468i −0.975785 0.218734i $$-0.929807\pi$$
0.975785 0.218734i $$-0.0701927\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 719.478i 0.909581i
$$792$$ 0 0
$$793$$ 551.890 0.695952
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −986.969 −1.23836 −0.619178 0.785251i $$-0.712534\pi$$
−0.619178 + 0.785251i $$0.712534\pi$$
$$798$$ 0 0
$$799$$ − 698.518i − 0.874240i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 513.336i − 0.639273i
$$804$$ 0 0
$$805$$ 635.180 0.789043
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −33.7831 −0.0417591 −0.0208795 0.999782i $$-0.506647\pi$$
−0.0208795 + 0.999782i $$0.506647\pi$$
$$810$$ 0 0
$$811$$ − 174.359i − 0.214993i −0.994205 0.107496i $$-0.965717\pi$$
0.994205 0.107496i $$-0.0342834\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1365.20i − 1.67509i
$$816$$ 0 0
$$817$$ 759.808 0.929997
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −826.976 −1.00728 −0.503640 0.863914i $$-0.668006\pi$$
−0.503640 + 0.863914i $$0.668006\pi$$
$$822$$ 0 0
$$823$$ 789.862i 0.959735i 0.877341 + 0.479868i $$0.159315\pi$$
−0.877341 + 0.479868i $$0.840685\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 94.0797i − 0.113760i −0.998381 0.0568801i $$-0.981885\pi$$
0.998381 0.0568801i $$-0.0181153\pi$$
$$828$$ 0 0
$$829$$ −383.996 −0.463204 −0.231602 0.972811i $$-0.574397\pi$$
−0.231602 + 0.972811i $$0.574397\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 773.217 0.928232
$$834$$ 0 0
$$835$$ 1147.88i 1.37470i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 1148.75i 1.36919i 0.728924 + 0.684594i $$0.240021\pi$$
−0.728924 + 0.684594i $$0.759979\pi$$
$$840$$ 0 0
$$841$$ −680.117 −0.808700
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 886.576 1.04920
$$846$$ 0 0
$$847$$ − 412.500i − 0.487013i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 560.543i − 0.658687i
$$852$$ 0 0
$$853$$ −165.565 −0.194097 −0.0970485 0.995280i $$-0.530940\pi$$
−0.0970485 + 0.995280i $$0.530940\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −1294.69 −1.51073 −0.755363 0.655307i $$-0.772539\pi$$
−0.755363 + 0.655307i $$0.772539\pi$$
$$858$$ 0 0
$$859$$ 14.0574i 0.0163648i 0.999967 + 0.00818241i $$0.00260457\pi$$
−0.999967 + 0.00818241i $$0.997395\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 409.653i − 0.474685i −0.971426 0.237343i $$-0.923724\pi$$
0.971426 0.237343i $$-0.0762764\pi$$
$$864$$ 0 0
$$865$$ −284.715 −0.329150
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −39.5425 −0.0455035
$$870$$ 0 0
$$871$$ − 219.309i − 0.251790i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 488.749i 0.558570i
$$876$$ 0 0
$$877$$ 10.0270 0.0114333 0.00571666 0.999984i $$-0.498180\pi$$
0.00571666 + 0.999984i $$0.498180\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1388.38 −1.57591 −0.787956 0.615731i $$-0.788861\pi$$
−0.787956 + 0.615731i $$0.788861\pi$$
$$882$$ 0 0
$$883$$ 1445.91i 1.63750i 0.574153 + 0.818748i $$0.305332\pi$$
−0.574153 + 0.818748i $$0.694668\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 661.924i − 0.746250i −0.927781 0.373125i $$-0.878286\pi$$
0.927781 0.373125i $$-0.121714\pi$$
$$888$$ 0 0
$$889$$ −693.185 −0.779736
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 643.729 0.720861
$$894$$ 0 0
$$895$$ − 55.2058i − 0.0616824i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 311.135i − 0.346090i
$$900$$ 0 0
$$901$$ −926.602 −1.02842
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 186.635 0.206227
$$906$$ 0 0
$$907$$ 314.099i 0.346306i 0.984895 + 0.173153i $$0.0553955\pi$$
−0.984895 + 0.173153i $$0.944605\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 620.050i 0.680626i 0.940312 + 0.340313i $$0.110533\pi$$
−0.940312 + 0.340313i $$0.889467\pi$$
$$912$$ 0 0
$$913$$ −487.065 −0.533478
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −830.391 −0.905552
$$918$$ 0 0
$$919$$ − 259.390i − 0.282253i −0.989992 0.141126i $$-0.954928\pi$$
0.989992 0.141126i $$-0.0450724\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 2600.10i 2.81701i
$$924$$ 0 0
$$925$$ −24.4662 −0.0264500
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −511.871 −0.550991 −0.275496 0.961302i $$-0.588842\pi$$
−0.275496 + 0.961302i $$0.588842\pi$$
$$930$$ 0 0
$$931$$ 712.569i 0.765380i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 520.706i − 0.556905i
$$936$$ 0 0
$$937$$ 531.153 0.566866 0.283433 0.958992i $$-0.408527\pi$$
0.283433 + 0.958992i $$0.408527\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1401.22 −1.48908 −0.744540 0.667578i $$-0.767331\pi$$
−0.744540 + 0.667578i $$0.767331\pi$$
$$942$$ 0 0
$$943$$ 1171.39i 1.24220i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 246.077i − 0.259849i −0.991524 0.129925i $$-0.958526\pi$$
0.991524 0.129925i $$-0.0414736\pi$$
$$948$$ 0 0
$$949$$ 2205.13 2.32363
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1711.93 −1.79636 −0.898178 0.439631i $$-0.855109\pi$$
−0.898178 + 0.439631i $$0.855109\pi$$
$$954$$ 0 0
$$955$$ − 1100.62i − 1.15249i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 342.577i − 0.357223i
$$960$$ 0 0
$$961$$ 359.290 0.373871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1437.25 1.48937
$$966$$ 0 0
$$967$$ − 801.048i − 0.828385i −0.910189 0.414192i $$-0.864064\pi$$
0.910189 0.414192i $$-0.135936\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 526.798i 0.542532i 0.962504 + 0.271266i $$0.0874423\pi$$
−0.962504 + 0.271266i $$0.912558\pi$$
$$972$$ 0 0
$$973$$ 605.087 0.621877
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −630.886 −0.645738 −0.322869 0.946444i $$-0.604647\pi$$
−0.322869 + 0.946444i $$0.604647\pi$$
$$978$$ 0 0
$$979$$ − 33.9260i − 0.0346537i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 1553.23i − 1.58009i −0.613051 0.790043i $$-0.710058\pi$$
0.613051 0.790043i $$-0.289942\pi$$
$$984$$ 0 0
$$985$$ 1304.04 1.32390
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1075.26 1.08722
$$990$$ 0 0
$$991$$ − 487.053i − 0.491476i −0.969336 0.245738i $$-0.920970\pi$$
0.969336 0.245738i $$-0.0790303\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 770.876i 0.774750i
$$996$$ 0 0
$$997$$ 1665.23 1.67024 0.835122 0.550065i $$-0.185397\pi$$
0.835122 + 0.550065i $$0.185397\pi$$
$$998$$ 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.m.703.5 8
3.2 odd 2 1728.3.g.j.703.3 8
4.3 odd 2 inner 1728.3.g.m.703.6 8
8.3 odd 2 864.3.g.b.703.4 yes 8
8.5 even 2 864.3.g.b.703.3 8
12.11 even 2 1728.3.g.j.703.4 8
24.5 odd 2 864.3.g.d.703.5 yes 8
24.11 even 2 864.3.g.d.703.6 yes 8

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