# Properties

 Label 2304.3.b.l.127.4 Level $2304$ Weight $3$ Character 2304.127 Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.127 Dual form 2304.3.b.l.127.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{5} +6.92820i q^{7} +O(q^{10})$$ $$q+2.00000i q^{5} +6.92820i q^{7} +6.92820 q^{11} -2.00000i q^{13} -10.0000 q^{17} +20.7846 q^{19} -27.7128i q^{23} +21.0000 q^{25} -26.0000i q^{29} +6.92820i q^{31} -13.8564 q^{35} +26.0000i q^{37} +58.0000 q^{41} +48.4974 q^{43} -69.2820i q^{47} +1.00000 q^{49} +74.0000i q^{53} +13.8564i q^{55} -90.0666 q^{59} -26.0000i q^{61} +4.00000 q^{65} +6.92820 q^{67} +46.0000 q^{73} +48.0000i q^{77} +117.779i q^{79} -48.4974 q^{83} -20.0000i q^{85} +82.0000 q^{89} +13.8564 q^{91} +41.5692i q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 40q^{17} + 84q^{25} + 232q^{41} + 4q^{49} + 16q^{65} + 184q^{73} + 328q^{89} + 8q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$5$$ 2.00000i 0.400000i 0.979796 + 0.200000i $$0.0640942\pi$$
−0.979796 + 0.200000i $$0.935906\pi$$
$$6$$ 0 0
$$7$$ 6.92820i 0.989743i 0.868966 + 0.494872i $$0.164785\pi$$
−0.868966 + 0.494872i $$0.835215\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.92820 0.629837 0.314918 0.949119i $$-0.398023\pi$$
0.314918 + 0.949119i $$0.398023\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.153846i −0.997037 0.0769231i $$-0.975490\pi$$
0.997037 0.0769231i $$-0.0245096\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −10.0000 −0.588235 −0.294118 0.955769i $$-0.595026\pi$$
−0.294118 + 0.955769i $$0.595026\pi$$
$$18$$ 0 0
$$19$$ 20.7846 1.09393 0.546963 0.837157i $$-0.315784\pi$$
0.546963 + 0.837157i $$0.315784\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 27.7128i − 1.20490i −0.798155 0.602452i $$-0.794190\pi$$
0.798155 0.602452i $$-0.205810\pi$$
$$24$$ 0 0
$$25$$ 21.0000 0.840000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 26.0000i − 0.896552i −0.893895 0.448276i $$-0.852038\pi$$
0.893895 0.448276i $$-0.147962\pi$$
$$30$$ 0 0
$$31$$ 6.92820i 0.223490i 0.993737 + 0.111745i $$0.0356441\pi$$
−0.993737 + 0.111745i $$0.964356\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −13.8564 −0.395897
$$36$$ 0 0
$$37$$ 26.0000i 0.702703i 0.936244 + 0.351351i $$0.114278\pi$$
−0.936244 + 0.351351i $$0.885722\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 58.0000 1.41463 0.707317 0.706896i $$-0.249905\pi$$
0.707317 + 0.706896i $$0.249905\pi$$
$$42$$ 0 0
$$43$$ 48.4974 1.12785 0.563924 0.825827i $$-0.309291\pi$$
0.563924 + 0.825827i $$0.309291\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 69.2820i − 1.47409i −0.675846 0.737043i $$-0.736222\pi$$
0.675846 0.737043i $$-0.263778\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.0204082
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 74.0000i 1.39623i 0.715987 + 0.698113i $$0.245977\pi$$
−0.715987 + 0.698113i $$0.754023\pi$$
$$54$$ 0 0
$$55$$ 13.8564i 0.251935i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −90.0666 −1.52655 −0.763277 0.646072i $$-0.776411\pi$$
−0.763277 + 0.646072i $$0.776411\pi$$
$$60$$ 0 0
$$61$$ − 26.0000i − 0.426230i −0.977027 0.213115i $$-0.931639\pi$$
0.977027 0.213115i $$-0.0683608\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.0615385
$$66$$ 0 0
$$67$$ 6.92820 0.103406 0.0517030 0.998663i $$-0.483535\pi$$
0.0517030 + 0.998663i $$0.483535\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 46.0000 0.630137 0.315068 0.949069i $$-0.397973\pi$$
0.315068 + 0.949069i $$0.397973\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 48.0000i 0.623377i
$$78$$ 0 0
$$79$$ 117.779i 1.49088i 0.666573 + 0.745440i $$0.267760\pi$$
−0.666573 + 0.745440i $$0.732240\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −48.4974 −0.584306 −0.292153 0.956372i $$-0.594372\pi$$
−0.292153 + 0.956372i $$0.594372\pi$$
$$84$$ 0 0
$$85$$ − 20.0000i − 0.235294i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 82.0000 0.921348 0.460674 0.887569i $$-0.347608\pi$$
0.460674 + 0.887569i $$0.347608\pi$$
$$90$$ 0 0
$$91$$ 13.8564 0.152268
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 41.5692i 0.437571i
$$96$$ 0 0
$$97$$ 2.00000 0.0206186 0.0103093 0.999947i $$-0.496718\pi$$
0.0103093 + 0.999947i $$0.496718\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 74.0000i 0.732673i 0.930482 + 0.366337i $$0.119388\pi$$
−0.930482 + 0.366337i $$0.880612\pi$$
$$102$$ 0 0
$$103$$ 76.2102i 0.739905i 0.929051 + 0.369953i $$0.120626\pi$$
−0.929051 + 0.369953i $$0.879374\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 20.7846 0.194249 0.0971243 0.995272i $$-0.469036\pi$$
0.0971243 + 0.995272i $$0.469036\pi$$
$$108$$ 0 0
$$109$$ 46.0000i 0.422018i 0.977484 + 0.211009i $$0.0676750\pi$$
−0.977484 + 0.211009i $$0.932325\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 110.000 0.973451 0.486726 0.873555i $$-0.338191\pi$$
0.486726 + 0.873555i $$0.338191\pi$$
$$114$$ 0 0
$$115$$ 55.4256 0.481962
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 69.2820i − 0.582202i
$$120$$ 0 0
$$121$$ −73.0000 −0.603306
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 92.0000i 0.736000i
$$126$$ 0 0
$$127$$ − 145.492i − 1.14561i −0.819692 0.572804i $$-0.805856\pi$$
0.819692 0.572804i $$-0.194144\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −117.779 −0.899080 −0.449540 0.893260i $$-0.648412\pi$$
−0.449540 + 0.893260i $$0.648412\pi$$
$$132$$ 0 0
$$133$$ 144.000i 1.08271i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000 0.0729927 0.0364964 0.999334i $$-0.488380\pi$$
0.0364964 + 0.999334i $$0.488380\pi$$
$$138$$ 0 0
$$139$$ −48.4974 −0.348902 −0.174451 0.984666i $$-0.555815\pi$$
−0.174451 + 0.984666i $$0.555815\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 13.8564i − 0.0968979i
$$144$$ 0 0
$$145$$ 52.0000 0.358621
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.00000i 0.0134228i 0.999977 + 0.00671141i $$0.00213632\pi$$
−0.999977 + 0.00671141i $$0.997864\pi$$
$$150$$ 0 0
$$151$$ 90.0666i 0.596468i 0.954493 + 0.298234i $$0.0963975\pi$$
−0.954493 + 0.298234i $$0.903602\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −13.8564 −0.0893962
$$156$$ 0 0
$$157$$ 214.000i 1.36306i 0.731791 + 0.681529i $$0.238685\pi$$
−0.731791 + 0.681529i $$0.761315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 192.000 1.19255
$$162$$ 0 0
$$163$$ 20.7846 0.127513 0.0637565 0.997965i $$-0.479692\pi$$
0.0637565 + 0.997965i $$0.479692\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 96.9948i 0.580807i 0.956904 + 0.290404i $$0.0937896\pi$$
−0.956904 + 0.290404i $$0.906210\pi$$
$$168$$ 0 0
$$169$$ 165.000 0.976331
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 334.000i 1.93064i 0.261077 + 0.965318i $$0.415922\pi$$
−0.261077 + 0.965318i $$0.584078\pi$$
$$174$$ 0 0
$$175$$ 145.492i 0.831384i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 187.061 1.04504 0.522518 0.852628i $$-0.324993\pi$$
0.522518 + 0.852628i $$0.324993\pi$$
$$180$$ 0 0
$$181$$ 2.00000i 0.0110497i 0.999985 + 0.00552486i $$0.00175863\pi$$
−0.999985 + 0.00552486i $$0.998241\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −52.0000 −0.281081
$$186$$ 0 0
$$187$$ −69.2820 −0.370492
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 221.703i 1.16075i 0.814351 + 0.580373i $$0.197093\pi$$
−0.814351 + 0.580373i $$0.802907\pi$$
$$192$$ 0 0
$$193$$ 290.000 1.50259 0.751295 0.659966i $$-0.229429\pi$$
0.751295 + 0.659966i $$0.229429\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.0000i 0.131980i 0.997820 + 0.0659898i $$0.0210205\pi$$
−0.997820 + 0.0659898i $$0.978980\pi$$
$$198$$ 0 0
$$199$$ − 394.908i − 1.98446i −0.124416 0.992230i $$-0.539706\pi$$
0.124416 0.992230i $$-0.460294\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 180.133 0.887356
$$204$$ 0 0
$$205$$ 116.000i 0.565854i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 144.000 0.688995
$$210$$ 0 0
$$211$$ −242.487 −1.14923 −0.574614 0.818425i $$-0.694848\pi$$
−0.574614 + 0.818425i $$0.694848\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 96.9948i 0.451139i
$$216$$ 0 0
$$217$$ −48.0000 −0.221198
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.0000i 0.0904977i
$$222$$ 0 0
$$223$$ − 339.482i − 1.52234i −0.648552 0.761170i $$-0.724625\pi$$
0.648552 0.761170i $$-0.275375\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 284.056 1.25135 0.625675 0.780084i $$-0.284824\pi$$
0.625675 + 0.780084i $$0.284824\pi$$
$$228$$ 0 0
$$229$$ − 142.000i − 0.620087i −0.950722 0.310044i $$-0.899656\pi$$
0.950722 0.310044i $$-0.100344\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 82.0000 0.351931 0.175966 0.984396i $$-0.443695\pi$$
0.175966 + 0.984396i $$0.443695\pi$$
$$234$$ 0 0
$$235$$ 138.564 0.589634
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 387.979i − 1.62334i −0.584113 0.811672i $$-0.698558\pi$$
0.584113 0.811672i $$-0.301442\pi$$
$$240$$ 0 0
$$241$$ −46.0000 −0.190871 −0.0954357 0.995436i $$-0.530424\pi$$
−0.0954357 + 0.995436i $$0.530424\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.00000i 0.00816327i
$$246$$ 0 0
$$247$$ − 41.5692i − 0.168296i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 145.492 0.579650 0.289825 0.957080i $$-0.406403\pi$$
0.289825 + 0.957080i $$0.406403\pi$$
$$252$$ 0 0
$$253$$ − 192.000i − 0.758893i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 254.000 0.988327 0.494163 0.869369i $$-0.335474\pi$$
0.494163 + 0.869369i $$0.335474\pi$$
$$258$$ 0 0
$$259$$ −180.133 −0.695495
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 152.420i 0.579546i 0.957095 + 0.289773i $$0.0935797\pi$$
−0.957095 + 0.289773i $$0.906420\pi$$
$$264$$ 0 0
$$265$$ −148.000 −0.558491
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 262.000i 0.973978i 0.873408 + 0.486989i $$0.161905\pi$$
−0.873408 + 0.486989i $$0.838095\pi$$
$$270$$ 0 0
$$271$$ 20.7846i 0.0766960i 0.999264 + 0.0383480i $$0.0122095\pi$$
−0.999264 + 0.0383480i $$0.987790\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 145.492 0.529063
$$276$$ 0 0
$$277$$ 290.000i 1.04693i 0.852047 + 0.523466i $$0.175361\pi$$
−0.852047 + 0.523466i $$0.824639\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 226.000 0.804270 0.402135 0.915580i $$-0.368268\pi$$
0.402135 + 0.915580i $$0.368268\pi$$
$$282$$ 0 0
$$283$$ −297.913 −1.05270 −0.526348 0.850269i $$-0.676439\pi$$
−0.526348 + 0.850269i $$0.676439\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 401.836i 1.40012i
$$288$$ 0 0
$$289$$ −189.000 −0.653979
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 362.000i 1.23549i 0.786377 + 0.617747i $$0.211955\pi$$
−0.786377 + 0.617747i $$0.788045\pi$$
$$294$$ 0 0
$$295$$ − 180.133i − 0.610621i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −55.4256 −0.185370
$$300$$ 0 0
$$301$$ 336.000i 1.11628i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 52.0000 0.170492
$$306$$ 0 0
$$307$$ 145.492 0.473916 0.236958 0.971520i $$-0.423850\pi$$
0.236958 + 0.971520i $$0.423850\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 235.559i − 0.757424i −0.925515 0.378712i $$-0.876367\pi$$
0.925515 0.378712i $$-0.123633\pi$$
$$312$$ 0 0
$$313$$ 478.000 1.52716 0.763578 0.645715i $$-0.223441\pi$$
0.763578 + 0.645715i $$0.223441\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 170.000i − 0.536278i −0.963380 0.268139i $$-0.913591\pi$$
0.963380 0.268139i $$-0.0864086\pi$$
$$318$$ 0 0
$$319$$ − 180.133i − 0.564681i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −207.846 −0.643486
$$324$$ 0 0
$$325$$ − 42.0000i − 0.129231i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 480.000 1.45897
$$330$$ 0 0
$$331$$ −408.764 −1.23494 −0.617468 0.786596i $$-0.711842\pi$$
−0.617468 + 0.786596i $$0.711842\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 13.8564i 0.0413624i
$$336$$ 0 0
$$337$$ 338.000 1.00297 0.501484 0.865167i $$-0.332788\pi$$
0.501484 + 0.865167i $$0.332788\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 48.0000i 0.140762i
$$342$$ 0 0
$$343$$ 346.410i 1.00994i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 200.918 0.579014 0.289507 0.957176i $$-0.406509\pi$$
0.289507 + 0.957176i $$0.406509\pi$$
$$348$$ 0 0
$$349$$ − 506.000i − 1.44986i −0.688824 0.724928i $$-0.741873\pi$$
0.688824 0.724928i $$-0.258127\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −178.000 −0.504249 −0.252125 0.967695i $$-0.581129\pi$$
−0.252125 + 0.967695i $$0.581129\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 166.277i 0.463167i 0.972815 + 0.231583i $$0.0743906\pi$$
−0.972815 + 0.231583i $$0.925609\pi$$
$$360$$ 0 0
$$361$$ 71.0000 0.196676
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 92.0000i 0.252055i
$$366$$ 0 0
$$367$$ 200.918i 0.547460i 0.961807 + 0.273730i $$0.0882575\pi$$
−0.961807 + 0.273730i $$0.911742\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −512.687 −1.38191
$$372$$ 0 0
$$373$$ − 310.000i − 0.831099i −0.909571 0.415550i $$-0.863589\pi$$
0.909571 0.415550i $$-0.136411\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −52.0000 −0.137931
$$378$$ 0 0
$$379$$ 436.477 1.15165 0.575827 0.817572i $$-0.304680\pi$$
0.575827 + 0.817572i $$0.304680\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 609.682i 1.59186i 0.605390 + 0.795929i $$0.293017\pi$$
−0.605390 + 0.795929i $$0.706983\pi$$
$$384$$ 0 0
$$385$$ −96.0000 −0.249351
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 578.000i 1.48586i 0.669368 + 0.742931i $$0.266565\pi$$
−0.669368 + 0.742931i $$0.733435\pi$$
$$390$$ 0 0
$$391$$ 277.128i 0.708768i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −235.559 −0.596352
$$396$$ 0 0
$$397$$ − 26.0000i − 0.0654912i −0.999464 0.0327456i $$-0.989575\pi$$
0.999464 0.0327456i $$-0.0104251\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −250.000 −0.623441 −0.311721 0.950174i $$-0.600905\pi$$
−0.311721 + 0.950174i $$0.600905\pi$$
$$402$$ 0 0
$$403$$ 13.8564 0.0343831
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 180.133i 0.442588i
$$408$$ 0 0
$$409$$ −290.000 −0.709046 −0.354523 0.935047i $$-0.615357\pi$$
−0.354523 + 0.935047i $$0.615357\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 624.000i − 1.51090i
$$414$$ 0 0
$$415$$ − 96.9948i − 0.233723i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 339.482 0.810219 0.405110 0.914268i $$-0.367233\pi$$
0.405110 + 0.914268i $$0.367233\pi$$
$$420$$ 0 0
$$421$$ 674.000i 1.60095i 0.599366 + 0.800475i $$0.295419\pi$$
−0.599366 + 0.800475i $$0.704581\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −210.000 −0.494118
$$426$$ 0 0
$$427$$ 180.133 0.421858
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 540.400i − 1.25383i −0.779088 0.626914i $$-0.784318\pi$$
0.779088 0.626914i $$-0.215682\pi$$
$$432$$ 0 0
$$433$$ −334.000 −0.771363 −0.385681 0.922632i $$-0.626034\pi$$
−0.385681 + 0.922632i $$0.626034\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 576.000i − 1.31808i
$$438$$ 0 0
$$439$$ − 117.779i − 0.268290i −0.990962 0.134145i $$-0.957171\pi$$
0.990962 0.134145i $$-0.0428288\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −76.2102 −0.172032 −0.0860161 0.996294i $$-0.527414\pi$$
−0.0860161 + 0.996294i $$0.527414\pi$$
$$444$$ 0 0
$$445$$ 164.000i 0.368539i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −394.000 −0.877506 −0.438753 0.898608i $$-0.644580\pi$$
−0.438753 + 0.898608i $$0.644580\pi$$
$$450$$ 0 0
$$451$$ 401.836 0.890988
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 27.7128i 0.0609073i
$$456$$ 0 0
$$457$$ 478.000 1.04595 0.522976 0.852347i $$-0.324822\pi$$
0.522976 + 0.852347i $$0.324822\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 142.000i 0.308026i 0.988069 + 0.154013i $$0.0492198\pi$$
−0.988069 + 0.154013i $$0.950780\pi$$
$$462$$ 0 0
$$463$$ 630.466i 1.36170i 0.732423 + 0.680849i $$0.238389\pi$$
−0.732423 + 0.680849i $$0.761611\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −20.7846 −0.0445067 −0.0222533 0.999752i $$-0.507084\pi$$
−0.0222533 + 0.999752i $$0.507084\pi$$
$$468$$ 0 0
$$469$$ 48.0000i 0.102345i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 336.000 0.710359
$$474$$ 0 0
$$475$$ 436.477 0.918899
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 734.390i − 1.53317i −0.642141 0.766586i $$-0.721954\pi$$
0.642141 0.766586i $$-0.278046\pi$$
$$480$$ 0 0
$$481$$ 52.0000 0.108108
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.00000i 0.00824742i
$$486$$ 0 0
$$487$$ − 103.923i − 0.213394i −0.994292 0.106697i $$-0.965972\pi$$
0.994292 0.106697i $$-0.0340275\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −921.451 −1.87668 −0.938341 0.345711i $$-0.887638\pi$$
−0.938341 + 0.345711i $$0.887638\pi$$
$$492$$ 0 0
$$493$$ 260.000i 0.527383i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −76.2102 −0.152726 −0.0763630 0.997080i $$-0.524331\pi$$
−0.0763630 + 0.997080i $$0.524331\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 581.969i − 1.15700i −0.815684 0.578498i $$-0.803639\pi$$
0.815684 0.578498i $$-0.196361\pi$$
$$504$$ 0 0
$$505$$ −148.000 −0.293069
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 842.000i − 1.65422i −0.562037 0.827112i $$-0.689982\pi$$
0.562037 0.827112i $$-0.310018\pi$$
$$510$$ 0 0
$$511$$ 318.697i 0.623674i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −152.420 −0.295962
$$516$$ 0 0
$$517$$ − 480.000i − 0.928433i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −326.000 −0.625720 −0.312860 0.949799i $$-0.601287\pi$$
−0.312860 + 0.949799i $$0.601287\pi$$
$$522$$ 0 0
$$523$$ −311.769 −0.596117 −0.298058 0.954548i $$-0.596339\pi$$
−0.298058 + 0.954548i $$0.596339\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 69.2820i − 0.131465i
$$528$$ 0 0
$$529$$ −239.000 −0.451796
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 116.000i − 0.217636i
$$534$$ 0 0
$$535$$ 41.5692i 0.0776995i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.92820 0.0128538
$$540$$ 0 0
$$541$$ − 530.000i − 0.979667i −0.871816 0.489834i $$-0.837058\pi$$
0.871816 0.489834i $$-0.162942\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −92.0000 −0.168807
$$546$$ 0 0
$$547$$ −339.482 −0.620625 −0.310313 0.950635i $$-0.600434\pi$$
−0.310313 + 0.950635i $$0.600434\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 540.400i − 0.980762i
$$552$$ 0 0
$$553$$ −816.000 −1.47559
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 766.000i 1.37522i 0.726078 + 0.687612i $$0.241341\pi$$
−0.726078 + 0.687612i $$0.758659\pi$$
$$558$$ 0 0
$$559$$ − 96.9948i − 0.173515i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −491.902 −0.873717 −0.436858 0.899530i $$-0.643909\pi$$
−0.436858 + 0.899530i $$0.643909\pi$$
$$564$$ 0 0
$$565$$ 220.000i 0.389381i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −422.000 −0.741652 −0.370826 0.928702i $$-0.620925\pi$$
−0.370826 + 0.928702i $$0.620925\pi$$
$$570$$ 0 0
$$571$$ 284.056 0.497472 0.248736 0.968571i $$-0.419985\pi$$
0.248736 + 0.968571i $$0.419985\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 581.969i − 1.01212i
$$576$$ 0 0
$$577$$ −46.0000 −0.0797227 −0.0398614 0.999205i $$-0.512692\pi$$
−0.0398614 + 0.999205i $$0.512692\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 336.000i − 0.578313i
$$582$$ 0 0
$$583$$ 512.687i 0.879395i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 630.466 1.07405 0.537024 0.843567i $$-0.319548\pi$$
0.537024 + 0.843567i $$0.319548\pi$$
$$588$$ 0 0
$$589$$ 144.000i 0.244482i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −82.0000 −0.138280 −0.0691400 0.997607i $$-0.522026\pi$$
−0.0691400 + 0.997607i $$0.522026\pi$$
$$594$$ 0 0
$$595$$ 138.564 0.232881
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 55.4256i 0.0925303i 0.998929 + 0.0462651i $$0.0147319\pi$$
−0.998929 + 0.0462651i $$0.985268\pi$$
$$600$$ 0 0
$$601$$ 334.000 0.555740 0.277870 0.960619i $$-0.410371\pi$$
0.277870 + 0.960619i $$0.410371\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 146.000i − 0.241322i
$$606$$ 0 0
$$607$$ − 367.195i − 0.604934i −0.953160 0.302467i $$-0.902190\pi$$
0.953160 0.302467i $$-0.0978102\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −138.564 −0.226782
$$612$$ 0 0
$$613$$ − 214.000i − 0.349103i −0.984648 0.174551i $$-0.944152\pi$$
0.984648 0.174551i $$-0.0558475\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1118.00 −1.81199 −0.905997 0.423285i $$-0.860877\pi$$
−0.905997 + 0.423285i $$0.860877\pi$$
$$618$$ 0 0
$$619$$ 672.036 1.08568 0.542840 0.839836i $$-0.317349\pi$$
0.542840 + 0.839836i $$0.317349\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 568.113i 0.911898i
$$624$$ 0 0
$$625$$ 341.000 0.545600
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 260.000i − 0.413355i
$$630$$ 0 0
$$631$$ 145.492i 0.230574i 0.993332 + 0.115287i $$0.0367788\pi$$
−0.993332 + 0.115287i $$0.963221\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 290.985 0.458243
$$636$$ 0 0
$$637$$ − 2.00000i − 0.00313972i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −10.0000 −0.0156006 −0.00780031 0.999970i $$-0.502483\pi$$
−0.00780031 + 0.999970i $$0.502483\pi$$
$$642$$ 0 0
$$643$$ 1212.44 1.88559 0.942796 0.333370i $$-0.108186\pi$$
0.942796 + 0.333370i $$0.108186\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 332.554i − 0.513993i −0.966412 0.256997i $$-0.917267\pi$$
0.966412 0.256997i $$-0.0827330\pi$$
$$648$$ 0 0
$$649$$ −624.000 −0.961479
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 670.000i 1.02603i 0.858379 + 0.513017i $$0.171472\pi$$
−0.858379 + 0.513017i $$0.828528\pi$$
$$654$$ 0 0
$$655$$ − 235.559i − 0.359632i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 824.456 1.25107 0.625536 0.780195i $$-0.284880\pi$$
0.625536 + 0.780195i $$0.284880\pi$$
$$660$$ 0 0
$$661$$ − 1222.00i − 1.84871i −0.381529 0.924357i $$-0.624602\pi$$
0.381529 0.924357i $$-0.375398\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −288.000 −0.433083
$$666$$ 0 0
$$667$$ −720.533 −1.08026
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 180.133i − 0.268455i
$$672$$ 0 0
$$673$$ −334.000 −0.496285 −0.248143 0.968724i $$-0.579820\pi$$
−0.248143 + 0.968724i $$0.579820\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 1006.00i − 1.48597i −0.669309 0.742984i $$-0.733410\pi$$
0.669309 0.742984i $$-0.266590\pi$$
$$678$$ 0 0
$$679$$ 13.8564i 0.0204071i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −187.061 −0.273882 −0.136941 0.990579i $$-0.543727\pi$$
−0.136941 + 0.990579i $$0.543727\pi$$
$$684$$ 0 0
$$685$$ 20.0000i 0.0291971i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 148.000 0.214804
$$690$$ 0 0
$$691$$ 990.733 1.43377 0.716884 0.697193i $$-0.245568\pi$$
0.716884 + 0.697193i $$0.245568\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 96.9948i − 0.139561i
$$696$$ 0 0
$$697$$ −580.000 −0.832138
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 1034.00i − 1.47504i −0.675328 0.737518i $$-0.735998\pi$$
0.675328 0.737518i $$-0.264002\pi$$
$$702$$ 0 0
$$703$$ 540.400i 0.768705i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −512.687 −0.725158
$$708$$ 0 0
$$709$$ 530.000i 0.747532i 0.927523 + 0.373766i $$0.121934\pi$$
−0.927523 + 0.373766i $$0.878066\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 192.000 0.269285
$$714$$ 0 0
$$715$$ 27.7128 0.0387592
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 706.677i 0.982861i 0.870917 + 0.491430i $$0.163526\pi$$
−0.870917 + 0.491430i $$0.836474\pi$$
$$720$$ 0 0
$$721$$ −528.000 −0.732316
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 546.000i − 0.753103i
$$726$$ 0 0
$$727$$ − 242.487i − 0.333545i −0.985995 0.166772i $$-0.946665\pi$$
0.985995 0.166772i $$-0.0533345\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −484.974 −0.663439
$$732$$ 0 0
$$733$$ − 194.000i − 0.264666i −0.991205 0.132333i $$-0.957753\pi$$
0.991205 0.132333i $$-0.0422468\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000 0.0651289
$$738$$ 0 0
$$739$$ −1351.00 −1.82815 −0.914073 0.405550i $$-0.867080\pi$$
−0.914073 + 0.405550i $$0.867080\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 678.964i 0.913814i 0.889514 + 0.456907i $$0.151043\pi$$
−0.889514 + 0.456907i $$0.848957\pi$$
$$744$$ 0 0
$$745$$ −4.00000 −0.00536913
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 144.000i 0.192256i
$$750$$ 0 0
$$751$$ − 658.179i − 0.876404i −0.898877 0.438202i $$-0.855615\pi$$
0.898877 0.438202i $$-0.144385\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −180.133 −0.238587
$$756$$ 0 0
$$757$$ − 1006.00i − 1.32893i −0.747319 0.664465i $$-0.768659\pi$$
0.747319 0.664465i $$-0.231341\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −758.000 −0.996058 −0.498029 0.867160i $$-0.665943\pi$$
−0.498029 + 0.867160i $$0.665943\pi$$
$$762$$ 0 0
$$763$$ −318.697 −0.417690
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 180.133i 0.234854i
$$768$$ 0 0
$$769$$ 2.00000 0.00260078 0.00130039 0.999999i $$-0.499586\pi$$
0.00130039 + 0.999999i $$0.499586\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 262.000i − 0.338939i −0.985535 0.169470i $$-0.945795\pi$$
0.985535 0.169470i $$-0.0542055\pi$$
$$774$$ 0 0
$$775$$ 145.492i 0.187732i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1205.51 1.54751
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −428.000 −0.545223
$$786$$ 0 0
$$787$$ −1447.99 −1.83989 −0.919946 0.392046i $$-0.871767\pi$$
−0.919946 + 0.392046i $$0.871767\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 762.102i 0.963467i
$$792$$ 0 0
$$793$$ −52.0000 −0.0655738
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 866.000i − 1.08657i −0.839547 0.543287i $$-0.817179\pi$$
0.839547 0.543287i $$-0.182821\pi$$
$$798$$ 0 0
$$799$$ 692.820i 0.867109i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 318.697 0.396883
$$804$$ 0 0
$$805$$ 384.000i 0.477019i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 10.0000 0.0123609 0.00618047 0.999981i $$-0.498033\pi$$
0.00618047 + 0.999981i $$0.498033\pi$$
$$810$$ 0 0
$$811$$ 436.477 0.538196 0.269098 0.963113i $$-0.413274\pi$$
0.269098 + 0.963113i $$0.413274\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 41.5692i 0.0510052i
$$816$$ 0 0
$$817$$ 1008.00 1.23378
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 838.000i − 1.02071i −0.859965 0.510353i $$-0.829515\pi$$
0.859965 0.510353i $$-0.170485\pi$$
$$822$$ 0 0
$$823$$ − 879.882i − 1.06912i −0.845132 0.534558i $$-0.820478\pi$$
0.845132 0.534558i $$-0.179522\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −727.461 −0.879639 −0.439819 0.898086i $$-0.644958\pi$$
−0.439819 + 0.898086i $$0.644958\pi$$
$$828$$ 0 0
$$829$$ − 1298.00i − 1.56574i −0.622184 0.782871i $$-0.713754\pi$$
0.622184 0.782871i $$-0.286246\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.0000 −0.0120048
$$834$$ 0 0
$$835$$ −193.990 −0.232323
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 193.990i 0.231215i 0.993295 + 0.115608i $$0.0368815\pi$$
−0.993295 + 0.115608i $$0.963118\pi$$
$$840$$ 0 0
$$841$$ 165.000 0.196195
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 330.000i 0.390533i
$$846$$ 0 0
$$847$$ − 505.759i − 0.597118i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 720.533 0.846690
$$852$$ 0 0
$$853$$ 506.000i 0.593200i 0.955002 + 0.296600i $$0.0958529\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −998.000 −1.16453 −0.582264 0.813000i $$-0.697833\pi$$
−0.582264 + 0.813000i $$0.697833\pi$$
$$858$$ 0 0
$$859$$ −505.759 −0.588776 −0.294388 0.955686i $$-0.595116\pi$$
−0.294388 + 0.955686i $$0.595116\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 166.277i − 0.192673i −0.995349 0.0963365i $$-0.969287\pi$$
0.995349 0.0963365i $$-0.0307125\pi$$
$$864$$ 0 0
$$865$$ −668.000 −0.772254
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 816.000i 0.939010i
$$870$$ 0 0
$$871$$ − 13.8564i − 0.0159086i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −637.395 −0.728451
$$876$$ 0 0
$$877$$ 646.000i 0.736602i 0.929707 + 0.368301i $$0.120060\pi$$
−0.929707 + 0.368301i $$0.879940\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −898.000 −1.01930 −0.509648 0.860383i $$-0.670224\pi$$
−0.509648 + 0.860383i $$0.670224\pi$$
$$882$$ 0 0
$$883$$ −727.461 −0.823852 −0.411926 0.911217i $$-0.635144\pi$$
−0.411926 + 0.911217i $$0.635144\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 845.241i 0.952921i 0.879196 + 0.476460i $$0.158080\pi$$
−0.879196 + 0.476460i $$0.841920\pi$$
$$888$$ 0 0
$$889$$ 1008.00 1.13386
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 1440.00i − 1.61254i
$$894$$ 0 0
$$895$$ 374.123i 0.418014i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 180.133 0.200371
$$900$$ 0 0
$$901$$ − 740.000i − 0.821310i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.00000 −0.00441989
$$906$$ 0 0
$$907$$ −1364.86 −1.50480 −0.752401 0.658705i $$-0.771105\pi$$
−0.752401 + 0.658705i $$0.771105\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 387.979i 0.425883i 0.977065 + 0.212941i $$0.0683044\pi$$
−0.977065 + 0.212941i $$0.931696\pi$$
$$912$$ 0 0
$$913$$ −336.000 −0.368018
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 816.000i − 0.889858i
$$918$$ 0 0
$$919$$ − 602.754i − 0.655880i −0.944699 0.327940i $$-0.893646\pi$$
0.944699 0.327940i $$-0.106354\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 546.000i 0.590270i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −1594.00 −1.71582 −0.857912 0.513797i $$-0.828238\pi$$
−0.857912 + 0.513797i $$0.828238\pi$$
$$930$$ 0 0
$$931$$ 20.7846 0.0223250
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 138.564i − 0.148197i
$$936$$ 0 0
$$937$$ −674.000 −0.719317 −0.359658 0.933084i $$-0.617107\pi$$
−0.359658 + 0.933084i $$0.617107\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 430.000i 0.456961i 0.973549 + 0.228480i $$0.0733757\pi$$
−0.973549 + 0.228480i $$0.926624\pi$$
$$942$$ 0 0
$$943$$ − 1607.34i − 1.70450i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 76.2102 0.0804754 0.0402377 0.999190i $$-0.487188\pi$$
0.0402377 + 0.999190i $$0.487188\pi$$
$$948$$ 0 0
$$949$$ − 92.0000i − 0.0969442i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 730.000 0.766002 0.383001 0.923748i $$-0.374891\pi$$
0.383001 + 0.923748i $$0.374891\pi$$
$$954$$ 0 0
$$955$$ −443.405 −0.464298
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 69.2820i 0.0722440i
$$960$$ 0 0
$$961$$ 913.000 0.950052
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 580.000i 0.601036i
$$966$$ 0 0
$$967$$ − 921.451i − 0.952897i −0.879202 0.476448i $$-0.841924\pi$$
0.879202 0.476448i $$-0.158076\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1475.71 1.51978 0.759890 0.650051i $$-0.225253\pi$$
0.759890 + 0.650051i $$0.225253\pi$$
$$972$$ 0 0
$$973$$ − 336.000i − 0.345324i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −346.000 −0.354145 −0.177073 0.984198i $$-0.556663\pi$$
−0.177073 + 0.984198i $$0.556663\pi$$
$$978$$ 0 0
$$979$$ 568.113 0.580299
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 734.390i 0.747090i 0.927612 + 0.373545i $$0.121858\pi$$
−0.927612 + 0.373545i $$0.878142\pi$$
$$984$$ 0 0
$$985$$ −52.0000 −0.0527919
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 1344.00i − 1.35895i
$$990$$ 0 0
$$991$$ 976.877i 0.985748i 0.870101 + 0.492874i $$0.164054\pi$$
−0.870101 + 0.492874i $$0.835946\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 789.815 0.793784
$$996$$ 0 0
$$997$$ 458.000i 0.459378i 0.973264 + 0.229689i $$0.0737709\pi$$
−0.973264 + 0.229689i $$0.926229\pi$$
$$998$$ 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 2304.3.b.l.127.4 4
3.2 odd 2 768.3.b.c.127.3 4
4.3 odd 2 inner 2304.3.b.l.127.3 4
8.3 odd 2 inner 2304.3.b.l.127.1 4
8.5 even 2 inner 2304.3.b.l.127.2 4
12.11 even 2 768.3.b.c.127.1 4
16.3 odd 4 36.3.d.c.19.1 2
16.5 even 4 576.3.g.e.127.1 2
16.11 odd 4 576.3.g.e.127.2 2
16.13 even 4 36.3.d.c.19.2 2
24.5 odd 2 768.3.b.c.127.2 4
24.11 even 2 768.3.b.c.127.4 4
48.5 odd 4 192.3.g.b.127.1 2
48.11 even 4 192.3.g.b.127.2 2
48.29 odd 4 12.3.d.a.7.1 2
48.35 even 4 12.3.d.a.7.2 yes 2
80.3 even 4 900.3.f.c.199.1 4
80.13 odd 4 900.3.f.c.199.3 4
80.19 odd 4 900.3.c.e.451.2 2
80.29 even 4 900.3.c.e.451.1 2
80.67 even 4 900.3.f.c.199.4 4
80.77 odd 4 900.3.f.c.199.2 4
144.13 even 12 324.3.f.a.55.1 2
144.29 odd 12 324.3.f.d.271.1 2
144.61 even 12 324.3.f.g.271.1 2
144.67 odd 12 324.3.f.g.55.1 2
144.77 odd 12 324.3.f.j.55.1 2
144.83 even 12 324.3.f.j.271.1 2
144.115 odd 12 324.3.f.a.271.1 2
144.131 even 12 324.3.f.d.55.1 2
240.29 odd 4 300.3.c.b.151.2 2
240.77 even 4 300.3.f.a.199.3 4
240.83 odd 4 300.3.f.a.199.4 4
240.173 even 4 300.3.f.a.199.2 4
240.179 even 4 300.3.c.b.151.1 2
240.227 odd 4 300.3.f.a.199.1 4
336.83 odd 4 588.3.g.b.295.2 2
336.125 even 4 588.3.g.b.295.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.d.a.7.1 2 48.29 odd 4
12.3.d.a.7.2 yes 2 48.35 even 4
36.3.d.c.19.1 2 16.3 odd 4
36.3.d.c.19.2 2 16.13 even 4
192.3.g.b.127.1 2 48.5 odd 4
192.3.g.b.127.2 2 48.11 even 4
300.3.c.b.151.1 2 240.179 even 4
300.3.c.b.151.2 2 240.29 odd 4
300.3.f.a.199.1 4 240.227 odd 4
300.3.f.a.199.2 4 240.173 even 4
300.3.f.a.199.3 4 240.77 even 4
300.3.f.a.199.4 4 240.83 odd 4
324.3.f.a.55.1 2 144.13 even 12
324.3.f.a.271.1 2 144.115 odd 12
324.3.f.d.55.1 2 144.131 even 12
324.3.f.d.271.1 2 144.29 odd 12
324.3.f.g.55.1 2 144.67 odd 12
324.3.f.g.271.1 2 144.61 even 12
324.3.f.j.55.1 2 144.77 odd 12
324.3.f.j.271.1 2 144.83 even 12
576.3.g.e.127.1 2 16.5 even 4
576.3.g.e.127.2 2 16.11 odd 4
588.3.g.b.295.1 2 336.125 even 4
588.3.g.b.295.2 2 336.83 odd 4
768.3.b.c.127.1 4 12.11 even 2
768.3.b.c.127.2 4 24.5 odd 2
768.3.b.c.127.3 4 3.2 odd 2
768.3.b.c.127.4 4 24.11 even 2
900.3.c.e.451.1 2 80.29 even 4
900.3.c.e.451.2 2 80.19 odd 4
900.3.f.c.199.1 4 80.3 even 4
900.3.f.c.199.2 4 80.77 odd 4
900.3.f.c.199.3 4 80.13 odd 4
900.3.f.c.199.4 4 80.67 even 4
2304.3.b.l.127.1 4 8.3 odd 2 inner
2304.3.b.l.127.2 4 8.5 even 2 inner
2304.3.b.l.127.3 4 4.3 odd 2 inner
2304.3.b.l.127.4 4 1.1 even 1 trivial