# Properties

 Label 1840.2.a.u.1.1 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 Defining polynomial: $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.32973$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{3} +1.00000 q^{5} -4.06562 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q-1.56155 q^{3} +1.00000 q^{5} -4.06562 q^{7} -0.561553 q^{9} -2.65945 q^{11} -5.91023 q^{13} -1.56155 q^{15} -3.40617 q^{17} +2.65945 q^{19} +6.34868 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56155 q^{27} +5.84461 q^{29} +9.31640 q^{31} +4.15288 q^{33} -4.06562 q^{35} -4.18059 q^{37} +9.22914 q^{39} +2.15539 q^{41} -2.34868 q^{43} -0.561553 q^{45} -0.242644 q^{47} +9.52927 q^{49} +5.31891 q^{51} +9.03585 q^{53} -2.65945 q^{55} -4.15288 q^{57} -14.4143 q^{59} +2.46365 q^{61} +2.28306 q^{63} -5.91023 q^{65} +7.75485 q^{67} -1.56155 q^{69} -12.4395 q^{71} -2.08977 q^{73} -1.56155 q^{75} +10.8123 q^{77} +4.15288 q^{79} -7.00000 q^{81} +12.5293 q^{83} -3.40617 q^{85} -9.12667 q^{87} -9.35682 q^{89} +24.0288 q^{91} -14.5481 q^{93} +2.65945 q^{95} +3.47179 q^{97} +1.49342 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} - 4q^{11} + 2q^{15} - q^{17} + 4q^{19} + 10q^{21} + 4q^{23} + 4q^{25} + 14q^{27} + 19q^{29} + q^{31} - 2q^{33} + 3q^{35} - 3q^{37} + 13q^{41} + 6q^{43} + 6q^{45} - 6q^{47} + 9q^{49} + 8q^{51} + 19q^{53} - 4q^{55} + 2q^{57} - 23q^{59} + 13q^{63} + 3q^{67} + 2q^{69} + 3q^{71} - 32q^{73} + 2q^{75} + 18q^{77} - 2q^{79} - 28q^{81} + 21q^{83} - q^{85} + 18q^{87} + 40q^{91} - 8q^{93} + 4q^{95} - 18q^{97} - 6q^{99} + O(q^{100})$$

## 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
$$3$$ −1.56155 −0.901563 −0.450781 0.892634i $$-0.648855\pi$$
−0.450781 + 0.892634i $$0.648855\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.06562 −1.53666 −0.768330 0.640054i $$-0.778912\pi$$
−0.768330 + 0.640054i $$0.778912\pi$$
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −2.65945 −0.801856 −0.400928 0.916110i $$-0.631312\pi$$
−0.400928 + 0.916110i $$0.631312\pi$$
$$12$$ 0 0
$$13$$ −5.91023 −1.63920 −0.819602 0.572933i $$-0.805805\pi$$
−0.819602 + 0.572933i $$0.805805\pi$$
$$14$$ 0 0
$$15$$ −1.56155 −0.403191
$$16$$ 0 0
$$17$$ −3.40617 −0.826117 −0.413058 0.910705i $$-0.635539\pi$$
−0.413058 + 0.910705i $$0.635539\pi$$
$$18$$ 0 0
$$19$$ 2.65945 0.610121 0.305060 0.952333i $$-0.401323\pi$$
0.305060 + 0.952333i $$0.401323\pi$$
$$20$$ 0 0
$$21$$ 6.34868 1.38540
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.56155 1.07032
$$28$$ 0 0
$$29$$ 5.84461 1.08532 0.542659 0.839953i $$-0.317418\pi$$
0.542659 + 0.839953i $$0.317418\pi$$
$$30$$ 0 0
$$31$$ 9.31640 1.67327 0.836637 0.547757i $$-0.184518\pi$$
0.836637 + 0.547757i $$0.184518\pi$$
$$32$$ 0 0
$$33$$ 4.15288 0.722923
$$34$$ 0 0
$$35$$ −4.06562 −0.687215
$$36$$ 0 0
$$37$$ −4.18059 −0.687285 −0.343642 0.939101i $$-0.611661\pi$$
−0.343642 + 0.939101i $$0.611661\pi$$
$$38$$ 0 0
$$39$$ 9.22914 1.47785
$$40$$ 0 0
$$41$$ 2.15539 0.336615 0.168307 0.985735i $$-0.446170\pi$$
0.168307 + 0.985735i $$0.446170\pi$$
$$42$$ 0 0
$$43$$ −2.34868 −0.358171 −0.179085 0.983834i $$-0.557314\pi$$
−0.179085 + 0.983834i $$0.557314\pi$$
$$44$$ 0 0
$$45$$ −0.561553 −0.0837114
$$46$$ 0 0
$$47$$ −0.242644 −0.0353933 −0.0176966 0.999843i $$-0.505633\pi$$
−0.0176966 + 0.999843i $$0.505633\pi$$
$$48$$ 0 0
$$49$$ 9.52927 1.36132
$$50$$ 0 0
$$51$$ 5.31891 0.744796
$$52$$ 0 0
$$53$$ 9.03585 1.24117 0.620585 0.784140i $$-0.286895\pi$$
0.620585 + 0.784140i $$0.286895\pi$$
$$54$$ 0 0
$$55$$ −2.65945 −0.358601
$$56$$ 0 0
$$57$$ −4.15288 −0.550062
$$58$$ 0 0
$$59$$ −14.4143 −1.87658 −0.938291 0.345846i $$-0.887592\pi$$
−0.938291 + 0.345846i $$0.887592\pi$$
$$60$$ 0 0
$$61$$ 2.46365 0.315438 0.157719 0.987484i $$-0.449586\pi$$
0.157719 + 0.987484i $$0.449586\pi$$
$$62$$ 0 0
$$63$$ 2.28306 0.287639
$$64$$ 0 0
$$65$$ −5.91023 −0.733074
$$66$$ 0 0
$$67$$ 7.75485 0.947405 0.473703 0.880685i $$-0.342917\pi$$
0.473703 + 0.880685i $$0.342917\pi$$
$$68$$ 0 0
$$69$$ −1.56155 −0.187989
$$70$$ 0 0
$$71$$ −12.4395 −1.47630 −0.738149 0.674638i $$-0.764300\pi$$
−0.738149 + 0.674638i $$0.764300\pi$$
$$72$$ 0 0
$$73$$ −2.08977 −0.244589 −0.122294 0.992494i $$-0.539025\pi$$
−0.122294 + 0.992494i $$0.539025\pi$$
$$74$$ 0 0
$$75$$ −1.56155 −0.180313
$$76$$ 0 0
$$77$$ 10.8123 1.23218
$$78$$ 0 0
$$79$$ 4.15288 0.467235 0.233618 0.972329i $$-0.424944\pi$$
0.233618 + 0.972329i $$0.424944\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 12.5293 1.37527 0.687633 0.726058i $$-0.258650\pi$$
0.687633 + 0.726058i $$0.258650\pi$$
$$84$$ 0 0
$$85$$ −3.40617 −0.369451
$$86$$ 0 0
$$87$$ −9.12667 −0.978482
$$88$$ 0 0
$$89$$ −9.35682 −0.991821 −0.495910 0.868374i $$-0.665166\pi$$
−0.495910 + 0.868374i $$0.665166\pi$$
$$90$$ 0 0
$$91$$ 24.0288 2.51890
$$92$$ 0 0
$$93$$ −14.5481 −1.50856
$$94$$ 0 0
$$95$$ 2.65945 0.272854
$$96$$ 0 0
$$97$$ 3.47179 0.352507 0.176253 0.984345i $$-0.443602\pi$$
0.176253 + 0.984345i $$0.443602\pi$$
$$98$$ 0 0
$$99$$ 1.49342 0.150095
$$100$$ 0 0
$$101$$ 9.21036 0.916465 0.458233 0.888832i $$-0.348483\pi$$
0.458233 + 0.888832i $$0.348483\pi$$
$$102$$ 0 0
$$103$$ −4.69736 −0.462845 −0.231422 0.972853i $$-0.574338\pi$$
−0.231422 + 0.972853i $$0.574338\pi$$
$$104$$ 0 0
$$105$$ 6.34868 0.619568
$$106$$ 0 0
$$107$$ 0.376394 0.0363873 0.0181937 0.999834i $$-0.494208\pi$$
0.0181937 + 0.999834i $$0.494208\pi$$
$$108$$ 0 0
$$109$$ 19.0369 1.82340 0.911702 0.410851i $$-0.134768\pi$$
0.911702 + 0.410851i $$0.134768\pi$$
$$110$$ 0 0
$$111$$ 6.52821 0.619631
$$112$$ 0 0
$$113$$ 2.94252 0.276809 0.138404 0.990376i $$-0.455803\pi$$
0.138404 + 0.990376i $$0.455803\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 3.31891 0.306833
$$118$$ 0 0
$$119$$ 13.8482 1.26946
$$120$$ 0 0
$$121$$ −3.92730 −0.357028
$$122$$ 0 0
$$123$$ −3.36575 −0.303479
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −6.00357 −0.532730 −0.266365 0.963872i $$-0.585823\pi$$
−0.266365 + 0.963872i $$0.585823\pi$$
$$128$$ 0 0
$$129$$ 3.66759 0.322913
$$130$$ 0 0
$$131$$ 19.9606 1.74397 0.871985 0.489533i $$-0.162833\pi$$
0.871985 + 0.489533i $$0.162833\pi$$
$$132$$ 0 0
$$133$$ −10.8123 −0.937548
$$134$$ 0 0
$$135$$ 5.56155 0.478662
$$136$$ 0 0
$$137$$ −11.7609 −1.00480 −0.502402 0.864634i $$-0.667550\pi$$
−0.502402 + 0.864634i $$0.667550\pi$$
$$138$$ 0 0
$$139$$ −12.3891 −1.05083 −0.525415 0.850846i $$-0.676090\pi$$
−0.525415 + 0.850846i $$0.676090\pi$$
$$140$$ 0 0
$$141$$ 0.378902 0.0319093
$$142$$ 0 0
$$143$$ 15.7180 1.31441
$$144$$ 0 0
$$145$$ 5.84461 0.485369
$$146$$ 0 0
$$147$$ −14.8805 −1.22732
$$148$$ 0 0
$$149$$ −2.13124 −0.174598 −0.0872991 0.996182i $$-0.527824\pi$$
−0.0872991 + 0.996182i $$0.527824\pi$$
$$150$$ 0 0
$$151$$ 0.787129 0.0640556 0.0320278 0.999487i $$-0.489803\pi$$
0.0320278 + 0.999487i $$0.489803\pi$$
$$152$$ 0 0
$$153$$ 1.91274 0.154636
$$154$$ 0 0
$$155$$ 9.31640 0.748311
$$156$$ 0 0
$$157$$ −9.18873 −0.733340 −0.366670 0.930351i $$-0.619502\pi$$
−0.366670 + 0.930351i $$0.619502\pi$$
$$158$$ 0 0
$$159$$ −14.1100 −1.11899
$$160$$ 0 0
$$161$$ −4.06562 −0.320416
$$162$$ 0 0
$$163$$ 17.6928 1.38581 0.692903 0.721031i $$-0.256331\pi$$
0.692903 + 0.721031i $$0.256331\pi$$
$$164$$ 0 0
$$165$$ 4.15288 0.323301
$$166$$ 0 0
$$167$$ −14.2462 −1.10240 −0.551202 0.834372i $$-0.685831\pi$$
−0.551202 + 0.834372i $$0.685831\pi$$
$$168$$ 0 0
$$169$$ 21.9309 1.68699
$$170$$ 0 0
$$171$$ −1.49342 −0.114205
$$172$$ 0 0
$$173$$ −1.20394 −0.0915338 −0.0457669 0.998952i $$-0.514573\pi$$
−0.0457669 + 0.998952i $$0.514573\pi$$
$$174$$ 0 0
$$175$$ −4.06562 −0.307332
$$176$$ 0 0
$$177$$ 22.5087 1.69186
$$178$$ 0 0
$$179$$ −7.42495 −0.554967 −0.277483 0.960730i $$-0.589500\pi$$
−0.277483 + 0.960730i $$0.589500\pi$$
$$180$$ 0 0
$$181$$ 24.0450 1.78725 0.893627 0.448810i $$-0.148152\pi$$
0.893627 + 0.448810i $$0.148152\pi$$
$$182$$ 0 0
$$183$$ −3.84712 −0.284387
$$184$$ 0 0
$$185$$ −4.18059 −0.307363
$$186$$ 0 0
$$187$$ 9.05854 0.662426
$$188$$ 0 0
$$189$$ −22.6112 −1.64472
$$190$$ 0 0
$$191$$ 4.87689 0.352880 0.176440 0.984311i $$-0.443542\pi$$
0.176440 + 0.984311i $$0.443542\pi$$
$$192$$ 0 0
$$193$$ 10.2714 0.739353 0.369676 0.929161i $$-0.379469\pi$$
0.369676 + 0.929161i $$0.379469\pi$$
$$194$$ 0 0
$$195$$ 9.22914 0.660913
$$196$$ 0 0
$$197$$ 4.28557 0.305334 0.152667 0.988278i $$-0.451214\pi$$
0.152667 + 0.988278i $$0.451214\pi$$
$$198$$ 0 0
$$199$$ −4.02164 −0.285086 −0.142543 0.989789i $$-0.545528\pi$$
−0.142543 + 0.989789i $$0.545528\pi$$
$$200$$ 0 0
$$201$$ −12.1096 −0.854146
$$202$$ 0 0
$$203$$ −23.7620 −1.66776
$$204$$ 0 0
$$205$$ 2.15539 0.150539
$$206$$ 0 0
$$207$$ −0.561553 −0.0390306
$$208$$ 0 0
$$209$$ −7.07270 −0.489229
$$210$$ 0 0
$$211$$ −15.3416 −1.05616 −0.528080 0.849195i $$-0.677088\pi$$
−0.528080 + 0.849195i $$0.677088\pi$$
$$212$$ 0 0
$$213$$ 19.4249 1.33098
$$214$$ 0 0
$$215$$ −2.34868 −0.160179
$$216$$ 0 0
$$217$$ −37.8770 −2.57126
$$218$$ 0 0
$$219$$ 3.26328 0.220512
$$220$$ 0 0
$$221$$ 20.1312 1.35417
$$222$$ 0 0
$$223$$ 17.6801 1.18395 0.591973 0.805958i $$-0.298349\pi$$
0.591973 + 0.805958i $$0.298349\pi$$
$$224$$ 0 0
$$225$$ −0.561553 −0.0374369
$$226$$ 0 0
$$227$$ −8.63817 −0.573335 −0.286668 0.958030i $$-0.592548\pi$$
−0.286668 + 0.958030i $$0.592548\pi$$
$$228$$ 0 0
$$229$$ −3.34055 −0.220749 −0.110375 0.993890i $$-0.535205\pi$$
−0.110375 + 0.993890i $$0.535205\pi$$
$$230$$ 0 0
$$231$$ −16.8840 −1.11089
$$232$$ 0 0
$$233$$ 8.23728 0.539642 0.269821 0.962910i $$-0.413035\pi$$
0.269821 + 0.962910i $$0.413035\pi$$
$$234$$ 0 0
$$235$$ −0.242644 −0.0158284
$$236$$ 0 0
$$237$$ −6.48494 −0.421242
$$238$$ 0 0
$$239$$ 2.86525 0.185338 0.0926688 0.995697i $$-0.470460\pi$$
0.0926688 + 0.995697i $$0.470460\pi$$
$$240$$ 0 0
$$241$$ 4.68109 0.301536 0.150768 0.988569i $$-0.451825\pi$$
0.150768 + 0.988569i $$0.451825\pi$$
$$242$$ 0 0
$$243$$ −5.75379 −0.369106
$$244$$ 0 0
$$245$$ 9.52927 0.608803
$$246$$ 0 0
$$247$$ −15.7180 −1.00011
$$248$$ 0 0
$$249$$ −19.5651 −1.23989
$$250$$ 0 0
$$251$$ 18.5824 1.17291 0.586455 0.809982i $$-0.300523\pi$$
0.586455 + 0.809982i $$0.300523\pi$$
$$252$$ 0 0
$$253$$ −2.65945 −0.167198
$$254$$ 0 0
$$255$$ 5.31891 0.333083
$$256$$ 0 0
$$257$$ −28.7226 −1.79166 −0.895832 0.444392i $$-0.853420\pi$$
−0.895832 + 0.444392i $$0.853420\pi$$
$$258$$ 0 0
$$259$$ 16.9967 1.05612
$$260$$ 0 0
$$261$$ −3.28206 −0.203154
$$262$$ 0 0
$$263$$ 9.20500 0.567604 0.283802 0.958883i $$-0.408404\pi$$
0.283802 + 0.958883i $$0.408404\pi$$
$$264$$ 0 0
$$265$$ 9.03585 0.555068
$$266$$ 0 0
$$267$$ 14.6112 0.894189
$$268$$ 0 0
$$269$$ 16.9194 1.03160 0.515798 0.856710i $$-0.327496\pi$$
0.515798 + 0.856710i $$0.327496\pi$$
$$270$$ 0 0
$$271$$ −1.15082 −0.0699072 −0.0349536 0.999389i $$-0.511128\pi$$
−0.0349536 + 0.999389i $$0.511128\pi$$
$$272$$ 0 0
$$273$$ −37.5222 −2.27095
$$274$$ 0 0
$$275$$ −2.65945 −0.160371
$$276$$ 0 0
$$277$$ 2.57505 0.154720 0.0773600 0.997003i $$-0.475351\pi$$
0.0773600 + 0.997003i $$0.475351\pi$$
$$278$$ 0 0
$$279$$ −5.23165 −0.313211
$$280$$ 0 0
$$281$$ 27.4718 1.63883 0.819415 0.573201i $$-0.194299\pi$$
0.819415 + 0.573201i $$0.194299\pi$$
$$282$$ 0 0
$$283$$ 2.36389 0.140519 0.0702595 0.997529i $$-0.477617\pi$$
0.0702595 + 0.997529i $$0.477617\pi$$
$$284$$ 0 0
$$285$$ −4.15288 −0.245995
$$286$$ 0 0
$$287$$ −8.76298 −0.517263
$$288$$ 0 0
$$289$$ −5.39803 −0.317531
$$290$$ 0 0
$$291$$ −5.42138 −0.317807
$$292$$ 0 0
$$293$$ 34.1198 1.99330 0.996650 0.0817846i $$-0.0260619\pi$$
0.996650 + 0.0817846i $$0.0260619\pi$$
$$294$$ 0 0
$$295$$ −14.4143 −0.839233
$$296$$ 0 0
$$297$$ −14.7907 −0.858243
$$298$$ 0 0
$$299$$ −5.91023 −0.341798
$$300$$ 0 0
$$301$$ 9.54885 0.550386
$$302$$ 0 0
$$303$$ −14.3825 −0.826251
$$304$$ 0 0
$$305$$ 2.46365 0.141068
$$306$$ 0 0
$$307$$ −1.49342 −0.0852342 −0.0426171 0.999091i $$-0.513570\pi$$
−0.0426171 + 0.999091i $$0.513570\pi$$
$$308$$ 0 0
$$309$$ 7.33518 0.417284
$$310$$ 0 0
$$311$$ 14.1060 0.799880 0.399940 0.916541i $$-0.369031\pi$$
0.399940 + 0.916541i $$0.369031\pi$$
$$312$$ 0 0
$$313$$ −32.8563 −1.85715 −0.928574 0.371146i $$-0.878965\pi$$
−0.928574 + 0.371146i $$0.878965\pi$$
$$314$$ 0 0
$$315$$ 2.28306 0.128636
$$316$$ 0 0
$$317$$ 12.5824 0.706698 0.353349 0.935492i $$-0.385043\pi$$
0.353349 + 0.935492i $$0.385043\pi$$
$$318$$ 0 0
$$319$$ −15.5435 −0.870268
$$320$$ 0 0
$$321$$ −0.587758 −0.0328055
$$322$$ 0 0
$$323$$ −9.05854 −0.504031
$$324$$ 0 0
$$325$$ −5.91023 −0.327841
$$326$$ 0 0
$$327$$ −29.7271 −1.64391
$$328$$ 0 0
$$329$$ 0.986499 0.0543874
$$330$$ 0 0
$$331$$ −1.57677 −0.0866669 −0.0433334 0.999061i $$-0.513798\pi$$
−0.0433334 + 0.999061i $$0.513798\pi$$
$$332$$ 0 0
$$333$$ 2.34762 0.128649
$$334$$ 0 0
$$335$$ 7.75485 0.423693
$$336$$ 0 0
$$337$$ −15.4718 −0.842802 −0.421401 0.906874i $$-0.638461\pi$$
−0.421401 + 0.906874i $$0.638461\pi$$
$$338$$ 0 0
$$339$$ −4.59489 −0.249560
$$340$$ 0 0
$$341$$ −24.7765 −1.34172
$$342$$ 0 0
$$343$$ −10.2831 −0.555233
$$344$$ 0 0
$$345$$ −1.56155 −0.0840712
$$346$$ 0 0
$$347$$ 9.00312 0.483313 0.241656 0.970362i $$-0.422309\pi$$
0.241656 + 0.970362i $$0.422309\pi$$
$$348$$ 0 0
$$349$$ 1.10398 0.0590945 0.0295473 0.999563i $$-0.490593\pi$$
0.0295473 + 0.999563i $$0.490593\pi$$
$$350$$ 0 0
$$351$$ −32.8701 −1.75448
$$352$$ 0 0
$$353$$ 1.96566 0.104621 0.0523107 0.998631i $$-0.483341\pi$$
0.0523107 + 0.998631i $$0.483341\pi$$
$$354$$ 0 0
$$355$$ −12.4395 −0.660220
$$356$$ 0 0
$$357$$ −21.6247 −1.14450
$$358$$ 0 0
$$359$$ 2.60403 0.137435 0.0687177 0.997636i $$-0.478109\pi$$
0.0687177 + 0.997636i $$0.478109\pi$$
$$360$$ 0 0
$$361$$ −11.9273 −0.627753
$$362$$ 0 0
$$363$$ 6.13269 0.321883
$$364$$ 0 0
$$365$$ −2.08977 −0.109383
$$366$$ 0 0
$$367$$ −6.98042 −0.364375 −0.182188 0.983264i $$-0.558318\pi$$
−0.182188 + 0.983264i $$0.558318\pi$$
$$368$$ 0 0
$$369$$ −1.21036 −0.0630090
$$370$$ 0 0
$$371$$ −36.7363 −1.90726
$$372$$ 0 0
$$373$$ 24.3220 1.25935 0.629673 0.776860i $$-0.283189\pi$$
0.629673 + 0.776860i $$0.283189\pi$$
$$374$$ 0 0
$$375$$ −1.56155 −0.0806382
$$376$$ 0 0
$$377$$ −34.5430 −1.77906
$$378$$ 0 0
$$379$$ 24.6541 1.26640 0.633198 0.773990i $$-0.281742\pi$$
0.633198 + 0.773990i $$0.281742\pi$$
$$380$$ 0 0
$$381$$ 9.37489 0.480290
$$382$$ 0 0
$$383$$ 4.99292 0.255126 0.127563 0.991830i $$-0.459284\pi$$
0.127563 + 0.991830i $$0.459284\pi$$
$$384$$ 0 0
$$385$$ 10.8123 0.551048
$$386$$ 0 0
$$387$$ 1.31891 0.0670439
$$388$$ 0 0
$$389$$ −32.0234 −1.62365 −0.811826 0.583900i $$-0.801526\pi$$
−0.811826 + 0.583900i $$0.801526\pi$$
$$390$$ 0 0
$$391$$ −3.40617 −0.172257
$$392$$ 0 0
$$393$$ −31.1696 −1.57230
$$394$$ 0 0
$$395$$ 4.15288 0.208954
$$396$$ 0 0
$$397$$ −17.3441 −0.870476 −0.435238 0.900315i $$-0.643336\pi$$
−0.435238 + 0.900315i $$0.643336\pi$$
$$398$$ 0 0
$$399$$ 16.8840 0.845259
$$400$$ 0 0
$$401$$ −21.3352 −1.06543 −0.532714 0.846295i $$-0.678828\pi$$
−0.532714 + 0.846295i $$0.678828\pi$$
$$402$$ 0 0
$$403$$ −55.0621 −2.74284
$$404$$ 0 0
$$405$$ −7.00000 −0.347833
$$406$$ 0 0
$$407$$ 11.1181 0.551103
$$408$$ 0 0
$$409$$ −12.5328 −0.619709 −0.309855 0.950784i $$-0.600280\pi$$
−0.309855 + 0.950784i $$0.600280\pi$$
$$410$$ 0 0
$$411$$ 18.3653 0.905894
$$412$$ 0 0
$$413$$ 58.6031 2.88367
$$414$$ 0 0
$$415$$ 12.5293 0.615038
$$416$$ 0 0
$$417$$ 19.3462 0.947389
$$418$$ 0 0
$$419$$ −12.0288 −0.587644 −0.293822 0.955860i $$-0.594927\pi$$
−0.293822 + 0.955860i $$0.594927\pi$$
$$420$$ 0 0
$$421$$ −38.3721 −1.87014 −0.935071 0.354462i $$-0.884664\pi$$
−0.935071 + 0.354462i $$0.884664\pi$$
$$422$$ 0 0
$$423$$ 0.136257 0.00662507
$$424$$ 0 0
$$425$$ −3.40617 −0.165223
$$426$$ 0 0
$$427$$ −10.0163 −0.484721
$$428$$ 0 0
$$429$$ −24.5445 −1.18502
$$430$$ 0 0
$$431$$ 29.4789 1.41995 0.709975 0.704227i $$-0.248706\pi$$
0.709975 + 0.704227i $$0.248706\pi$$
$$432$$ 0 0
$$433$$ −12.0531 −0.579236 −0.289618 0.957142i $$-0.593528\pi$$
−0.289618 + 0.957142i $$0.593528\pi$$
$$434$$ 0 0
$$435$$ −9.12667 −0.437590
$$436$$ 0 0
$$437$$ 2.65945 0.127219
$$438$$ 0 0
$$439$$ −16.1656 −0.771541 −0.385771 0.922595i $$-0.626064\pi$$
−0.385771 + 0.922595i $$0.626064\pi$$
$$440$$ 0 0
$$441$$ −5.35119 −0.254819
$$442$$ 0 0
$$443$$ −19.3729 −0.920433 −0.460217 0.887807i $$-0.652228\pi$$
−0.460217 + 0.887807i $$0.652228\pi$$
$$444$$ 0 0
$$445$$ −9.35682 −0.443556
$$446$$ 0 0
$$447$$ 3.32805 0.157411
$$448$$ 0 0
$$449$$ 21.4728 1.01337 0.506683 0.862132i $$-0.330871\pi$$
0.506683 + 0.862132i $$0.330871\pi$$
$$450$$ 0 0
$$451$$ −5.73215 −0.269916
$$452$$ 0 0
$$453$$ −1.22914 −0.0577502
$$454$$ 0 0
$$455$$ 24.0288 1.12649
$$456$$ 0 0
$$457$$ −5.95602 −0.278611 −0.139305 0.990249i $$-0.544487\pi$$
−0.139305 + 0.990249i $$0.544487\pi$$
$$458$$ 0 0
$$459$$ −18.9436 −0.884210
$$460$$ 0 0
$$461$$ 34.6918 1.61576 0.807879 0.589348i $$-0.200615\pi$$
0.807879 + 0.589348i $$0.200615\pi$$
$$462$$ 0 0
$$463$$ 30.6399 1.42396 0.711979 0.702200i $$-0.247799\pi$$
0.711979 + 0.702200i $$0.247799\pi$$
$$464$$ 0 0
$$465$$ −14.5481 −0.674650
$$466$$ 0 0
$$467$$ 8.89547 0.411633 0.205817 0.978591i $$-0.434015\pi$$
0.205817 + 0.978591i $$0.434015\pi$$
$$468$$ 0 0
$$469$$ −31.5283 −1.45584
$$470$$ 0 0
$$471$$ 14.3487 0.661152
$$472$$ 0 0
$$473$$ 6.24621 0.287201
$$474$$ 0 0
$$475$$ 2.65945 0.122024
$$476$$ 0 0
$$477$$ −5.07411 −0.232327
$$478$$ 0 0
$$479$$ 9.13224 0.417263 0.208631 0.977994i $$-0.433099\pi$$
0.208631 + 0.977994i $$0.433099\pi$$
$$480$$ 0 0
$$481$$ 24.7083 1.12660
$$482$$ 0 0
$$483$$ 6.34868 0.288875
$$484$$ 0 0
$$485$$ 3.47179 0.157646
$$486$$ 0 0
$$487$$ −14.2085 −0.643849 −0.321924 0.946765i $$-0.604330\pi$$
−0.321924 + 0.946765i $$0.604330\pi$$
$$488$$ 0 0
$$489$$ −27.6282 −1.24939
$$490$$ 0 0
$$491$$ 23.1760 1.04592 0.522960 0.852357i $$-0.324828\pi$$
0.522960 + 0.852357i $$0.324828\pi$$
$$492$$ 0 0
$$493$$ −19.9077 −0.896599
$$494$$ 0 0
$$495$$ 1.49342 0.0671244
$$496$$ 0 0
$$497$$ 50.5743 2.26857
$$498$$ 0 0
$$499$$ 2.19831 0.0984099 0.0492050 0.998789i $$-0.484331\pi$$
0.0492050 + 0.998789i $$0.484331\pi$$
$$500$$ 0 0
$$501$$ 22.2462 0.993887
$$502$$ 0 0
$$503$$ 44.0461 1.96392 0.981959 0.189092i $$-0.0605545\pi$$
0.981959 + 0.189092i $$0.0605545\pi$$
$$504$$ 0 0
$$505$$ 9.21036 0.409856
$$506$$ 0 0
$$507$$ −34.2462 −1.52093
$$508$$ 0 0
$$509$$ −9.09254 −0.403020 −0.201510 0.979486i $$-0.564585\pi$$
−0.201510 + 0.979486i $$0.564585\pi$$
$$510$$ 0 0
$$511$$ 8.49619 0.375850
$$512$$ 0 0
$$513$$ 14.7907 0.653025
$$514$$ 0 0
$$515$$ −4.69736 −0.206991
$$516$$ 0 0
$$517$$ 0.645301 0.0283803
$$518$$ 0 0
$$519$$ 1.88001 0.0825235
$$520$$ 0 0
$$521$$ −17.1231 −0.750177 −0.375088 0.926989i $$-0.622388\pi$$
−0.375088 + 0.926989i $$0.622388\pi$$
$$522$$ 0 0
$$523$$ 26.6112 1.16362 0.581812 0.813323i $$-0.302344\pi$$
0.581812 + 0.813323i $$0.302344\pi$$
$$524$$ 0 0
$$525$$ 6.34868 0.277079
$$526$$ 0 0
$$527$$ −31.7332 −1.38232
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 8.09439 0.351267
$$532$$ 0 0
$$533$$ −12.7388 −0.551780
$$534$$ 0 0
$$535$$ 0.376394 0.0162729
$$536$$ 0 0
$$537$$ 11.5944 0.500337
$$538$$ 0 0
$$539$$ −25.3427 −1.09159
$$540$$ 0 0
$$541$$ 31.5941 1.35834 0.679168 0.733983i $$-0.262341\pi$$
0.679168 + 0.733983i $$0.262341\pi$$
$$542$$ 0 0
$$543$$ −37.5476 −1.61132
$$544$$ 0 0
$$545$$ 19.0369 0.815452
$$546$$ 0 0
$$547$$ −22.7009 −0.970622 −0.485311 0.874342i $$-0.661294\pi$$
−0.485311 + 0.874342i $$0.661294\pi$$
$$548$$ 0 0
$$549$$ −1.38347 −0.0590451
$$550$$ 0 0
$$551$$ 15.5435 0.662175
$$552$$ 0 0
$$553$$ −16.8840 −0.717982
$$554$$ 0 0
$$555$$ 6.52821 0.277107
$$556$$ 0 0
$$557$$ 35.1292 1.48847 0.744236 0.667917i $$-0.232814\pi$$
0.744236 + 0.667917i $$0.232814\pi$$
$$558$$ 0 0
$$559$$ 13.8813 0.587115
$$560$$ 0 0
$$561$$ −14.1454 −0.597219
$$562$$ 0 0
$$563$$ −32.0511 −1.35079 −0.675397 0.737455i $$-0.736028\pi$$
−0.675397 + 0.737455i $$0.736028\pi$$
$$564$$ 0 0
$$565$$ 2.94252 0.123793
$$566$$ 0 0
$$567$$ 28.4593 1.19518
$$568$$ 0 0
$$569$$ −22.0575 −0.924700 −0.462350 0.886697i $$-0.652994\pi$$
−0.462350 + 0.886697i $$0.652994\pi$$
$$570$$ 0 0
$$571$$ −8.49242 −0.355397 −0.177698 0.984085i $$-0.556865\pi$$
−0.177698 + 0.984085i $$0.556865\pi$$
$$572$$ 0 0
$$573$$ −7.61553 −0.318143
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 27.1554 1.13050 0.565248 0.824921i $$-0.308780\pi$$
0.565248 + 0.824921i $$0.308780\pi$$
$$578$$ 0 0
$$579$$ −16.0394 −0.666573
$$580$$ 0 0
$$581$$ −50.9393 −2.11332
$$582$$ 0 0
$$583$$ −24.0304 −0.995239
$$584$$ 0 0
$$585$$ 3.31891 0.137220
$$586$$ 0 0
$$587$$ 30.8322 1.27258 0.636290 0.771450i $$-0.280468\pi$$
0.636290 + 0.771450i $$0.280468\pi$$
$$588$$ 0 0
$$589$$ 24.7765 1.02090
$$590$$ 0 0
$$591$$ −6.69214 −0.275278
$$592$$ 0 0
$$593$$ −27.7559 −1.13980 −0.569899 0.821715i $$-0.693018\pi$$
−0.569899 + 0.821715i $$0.693018\pi$$
$$594$$ 0 0
$$595$$ 13.8482 0.567720
$$596$$ 0 0
$$597$$ 6.28000 0.257023
$$598$$ 0 0
$$599$$ 30.7712 1.25728 0.628638 0.777698i $$-0.283613\pi$$
0.628638 + 0.777698i $$0.283613\pi$$
$$600$$ 0 0
$$601$$ −17.9830 −0.733541 −0.366771 0.930311i $$-0.619537\pi$$
−0.366771 + 0.930311i $$0.619537\pi$$
$$602$$ 0 0
$$603$$ −4.35476 −0.177339
$$604$$ 0 0
$$605$$ −3.92730 −0.159668
$$606$$ 0 0
$$607$$ 18.3220 0.743668 0.371834 0.928299i $$-0.378729\pi$$
0.371834 + 0.928299i $$0.378729\pi$$
$$608$$ 0 0
$$609$$ 37.1056 1.50359
$$610$$ 0 0
$$611$$ 1.43408 0.0580168
$$612$$ 0 0
$$613$$ 35.4451 1.43162 0.715808 0.698297i $$-0.246059\pi$$
0.715808 + 0.698297i $$0.246059\pi$$
$$614$$ 0 0
$$615$$ −3.36575 −0.135720
$$616$$ 0 0
$$617$$ 43.1567 1.73742 0.868712 0.495318i $$-0.164948\pi$$
0.868712 + 0.495318i $$0.164948\pi$$
$$618$$ 0 0
$$619$$ −5.81133 −0.233577 −0.116789 0.993157i $$-0.537260\pi$$
−0.116789 + 0.993157i $$0.537260\pi$$
$$620$$ 0 0
$$621$$ 5.56155 0.223177
$$622$$ 0 0
$$623$$ 38.0413 1.52409
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 11.0444 0.441070
$$628$$ 0 0
$$629$$ 14.2398 0.567777
$$630$$ 0 0
$$631$$ 27.4626 1.09327 0.546635 0.837371i $$-0.315908\pi$$
0.546635 + 0.837371i $$0.315908\pi$$
$$632$$ 0 0
$$633$$ 23.9567 0.952194
$$634$$ 0 0
$$635$$ −6.00357 −0.238244
$$636$$ 0 0
$$637$$ −56.3202 −2.23149
$$638$$ 0 0
$$639$$ 6.98544 0.276340
$$640$$ 0 0
$$641$$ 37.7938 1.49277 0.746383 0.665517i $$-0.231789\pi$$
0.746383 + 0.665517i $$0.231789\pi$$
$$642$$ 0 0
$$643$$ −12.5343 −0.494304 −0.247152 0.968977i $$-0.579495\pi$$
−0.247152 + 0.968977i $$0.579495\pi$$
$$644$$ 0 0
$$645$$ 3.66759 0.144411
$$646$$ 0 0
$$647$$ 21.5133 0.845774 0.422887 0.906183i $$-0.361017\pi$$
0.422887 + 0.906183i $$0.361017\pi$$
$$648$$ 0 0
$$649$$ 38.3342 1.50475
$$650$$ 0 0
$$651$$ 59.1469 2.31815
$$652$$ 0 0
$$653$$ 19.9011 0.778790 0.389395 0.921071i $$-0.372684\pi$$
0.389395 + 0.921071i $$0.372684\pi$$
$$654$$ 0 0
$$655$$ 19.9606 0.779927
$$656$$ 0 0
$$657$$ 1.17351 0.0457831
$$658$$ 0 0
$$659$$ 4.95773 0.193126 0.0965628 0.995327i $$-0.469215\pi$$
0.0965628 + 0.995327i $$0.469215\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ −31.4360 −1.22087
$$664$$ 0 0
$$665$$ −10.8123 −0.419284
$$666$$ 0 0
$$667$$ 5.84461 0.226304
$$668$$ 0 0
$$669$$ −27.6084 −1.06740
$$670$$ 0 0
$$671$$ −6.55197 −0.252936
$$672$$ 0 0
$$673$$ −28.5176 −1.09927 −0.549637 0.835404i $$-0.685234\pi$$
−0.549637 + 0.835404i $$0.685234\pi$$
$$674$$ 0 0
$$675$$ 5.56155 0.214064
$$676$$ 0 0
$$677$$ 28.6214 1.10001 0.550004 0.835162i $$-0.314626\pi$$
0.550004 + 0.835162i $$0.314626\pi$$
$$678$$ 0 0
$$679$$ −14.1150 −0.541683
$$680$$ 0 0
$$681$$ 13.4890 0.516898
$$682$$ 0 0
$$683$$ −11.3983 −0.436144 −0.218072 0.975933i $$-0.569977\pi$$
−0.218072 + 0.975933i $$0.569977\pi$$
$$684$$ 0 0
$$685$$ −11.7609 −0.449362
$$686$$ 0 0
$$687$$ 5.21644 0.199020
$$688$$ 0 0
$$689$$ −53.4040 −2.03453
$$690$$ 0 0
$$691$$ −45.1898 −1.71910 −0.859550 0.511051i $$-0.829256\pi$$
−0.859550 + 0.511051i $$0.829256\pi$$
$$692$$ 0 0
$$693$$ −6.07170 −0.230645
$$694$$ 0 0
$$695$$ −12.3891 −0.469945
$$696$$ 0 0
$$697$$ −7.34160 −0.278083
$$698$$ 0 0
$$699$$ −12.8629 −0.486521
$$700$$ 0 0
$$701$$ 40.4871 1.52918 0.764588 0.644520i $$-0.222943\pi$$
0.764588 + 0.644520i $$0.222943\pi$$
$$702$$ 0 0
$$703$$ −11.1181 −0.419327
$$704$$ 0 0
$$705$$ 0.378902 0.0142703
$$706$$ 0 0
$$707$$ −37.4458 −1.40830
$$708$$ 0 0
$$709$$ −41.6443 −1.56398 −0.781992 0.623288i $$-0.785796\pi$$
−0.781992 + 0.623288i $$0.785796\pi$$
$$710$$ 0 0
$$711$$ −2.33206 −0.0874591
$$712$$ 0 0
$$713$$ 9.31640 0.348902
$$714$$ 0 0
$$715$$ 15.7180 0.587820
$$716$$ 0 0
$$717$$ −4.47424 −0.167093
$$718$$ 0 0
$$719$$ −49.2288 −1.83592 −0.917961 0.396670i $$-0.870166\pi$$
−0.917961 + 0.396670i $$0.870166\pi$$
$$720$$ 0 0
$$721$$ 19.0977 0.711235
$$722$$ 0 0
$$723$$ −7.30977 −0.271853
$$724$$ 0 0
$$725$$ 5.84461 0.217063
$$726$$ 0 0
$$727$$ −27.9581 −1.03691 −0.518455 0.855105i $$-0.673493\pi$$
−0.518455 + 0.855105i $$0.673493\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −18.6655 −0.689427 −0.344714 0.938708i $$-0.612024\pi$$
−0.344714 + 0.938708i $$0.612024\pi$$
$$734$$ 0 0
$$735$$ −14.8805 −0.548874
$$736$$ 0 0
$$737$$ −20.6237 −0.759682
$$738$$ 0 0
$$739$$ −2.50407 −0.0921136 −0.0460568 0.998939i $$-0.514666\pi$$
−0.0460568 + 0.998939i $$0.514666\pi$$
$$740$$ 0 0
$$741$$ 24.5445 0.901664
$$742$$ 0 0
$$743$$ −4.69736 −0.172330 −0.0861648 0.996281i $$-0.527461\pi$$
−0.0861648 + 0.996281i $$0.527461\pi$$
$$744$$ 0 0
$$745$$ −2.13124 −0.0780826
$$746$$ 0 0
$$747$$ −7.03585 −0.257428
$$748$$ 0 0
$$749$$ −1.53027 −0.0559150
$$750$$ 0 0
$$751$$ 16.0720 0.586477 0.293239 0.956039i $$-0.405267\pi$$
0.293239 + 0.956039i $$0.405267\pi$$
$$752$$ 0 0
$$753$$ −29.0174 −1.05745
$$754$$ 0 0
$$755$$ 0.787129 0.0286465
$$756$$ 0 0
$$757$$ −25.8357 −0.939014 −0.469507 0.882929i $$-0.655568\pi$$
−0.469507 + 0.882929i $$0.655568\pi$$
$$758$$ 0 0
$$759$$ 4.15288 0.150740
$$760$$ 0 0
$$761$$ 15.6005 0.565518 0.282759 0.959191i $$-0.408750\pi$$
0.282759 + 0.959191i $$0.408750\pi$$
$$762$$ 0 0
$$763$$ −77.3968 −2.80195
$$764$$ 0 0
$$765$$ 1.91274 0.0691553
$$766$$ 0 0
$$767$$ 85.1919 3.07610
$$768$$ 0 0
$$769$$ −37.7722 −1.36210 −0.681050 0.732237i $$-0.738476\pi$$
−0.681050 + 0.732237i $$0.738476\pi$$
$$770$$ 0 0
$$771$$ 44.8518 1.61530
$$772$$ 0 0
$$773$$ −42.9078 −1.54329 −0.771643 0.636056i $$-0.780565\pi$$
−0.771643 + 0.636056i $$0.780565\pi$$
$$774$$ 0 0
$$775$$ 9.31640 0.334655
$$776$$ 0 0
$$777$$ −26.5412 −0.952162
$$778$$ 0 0
$$779$$ 5.73215 0.205376
$$780$$ 0 0
$$781$$ 33.0823 1.18378
$$782$$ 0 0
$$783$$ 32.5051 1.16164
$$784$$ 0 0
$$785$$ −9.18873 −0.327960
$$786$$ 0 0
$$787$$ 33.5324 1.19530 0.597650 0.801757i $$-0.296101\pi$$
0.597650 + 0.801757i $$0.296101\pi$$
$$788$$ 0 0
$$789$$ −14.3741 −0.511731
$$790$$ 0 0
$$791$$ −11.9632 −0.425361
$$792$$ 0 0
$$793$$ −14.5608 −0.517068
$$794$$ 0 0
$$795$$ −14.1100 −0.500428
$$796$$ 0 0
$$797$$ 32.7488 1.16002 0.580012 0.814608i $$-0.303048\pi$$
0.580012 + 0.814608i $$0.303048\pi$$
$$798$$ 0 0
$$799$$ 0.826486 0.0292390
$$800$$ 0 0
$$801$$ 5.25435 0.185653
$$802$$ 0 0
$$803$$ 5.55764 0.196125
$$804$$ 0 0
$$805$$ −4.06562 −0.143294
$$806$$ 0 0
$$807$$ −26.4206 −0.930049
$$808$$ 0 0
$$809$$ 28.8513 1.01436 0.507179 0.861841i $$-0.330688\pi$$
0.507179 + 0.861841i $$0.330688\pi$$
$$810$$ 0 0
$$811$$ −36.1791 −1.27042 −0.635211 0.772339i $$-0.719087\pi$$
−0.635211 + 0.772339i $$0.719087\pi$$
$$812$$ 0 0
$$813$$ 1.79706 0.0630257
$$814$$ 0 0
$$815$$ 17.6928 0.619752
$$816$$ 0 0
$$817$$ −6.24621 −0.218527
$$818$$ 0 0
$$819$$ −13.4934 −0.471498
$$820$$ 0 0
$$821$$ 5.73752 0.200241 0.100120 0.994975i $$-0.468077\pi$$
0.100120 + 0.994975i $$0.468077\pi$$
$$822$$ 0 0
$$823$$ 6.87333 0.239589 0.119795 0.992799i $$-0.461776\pi$$
0.119795 + 0.992799i $$0.461776\pi$$
$$824$$ 0 0
$$825$$ 4.15288 0.144585
$$826$$ 0 0
$$827$$ 19.7332 0.686191 0.343095 0.939301i $$-0.388525\pi$$
0.343095 + 0.939301i $$0.388525\pi$$
$$828$$ 0 0
$$829$$ −2.88833 −0.100316 −0.0501580 0.998741i $$-0.515972\pi$$
−0.0501580 + 0.998741i $$0.515972\pi$$
$$830$$ 0 0
$$831$$ −4.02108 −0.139490
$$832$$ 0 0
$$833$$ −32.4583 −1.12461
$$834$$ 0 0
$$835$$ −14.2462 −0.493010
$$836$$ 0 0
$$837$$ 51.8137 1.79094
$$838$$ 0 0
$$839$$ −24.1904 −0.835147 −0.417573 0.908643i $$-0.637119\pi$$
−0.417573 + 0.908643i $$0.637119\pi$$
$$840$$ 0 0
$$841$$ 5.15951 0.177914
$$842$$ 0 0
$$843$$ −42.8986 −1.47751
$$844$$ 0 0
$$845$$ 21.9309 0.754445
$$846$$ 0 0
$$847$$ 15.9669 0.548630
$$848$$ 0 0
$$849$$ −3.69135 −0.126687
$$850$$ 0 0
$$851$$ −4.18059 −0.143309
$$852$$ 0 0
$$853$$ 27.5381 0.942887 0.471444 0.881896i $$-0.343733\pi$$
0.471444 + 0.881896i $$0.343733\pi$$
$$854$$ 0 0
$$855$$ −1.49342 −0.0510740
$$856$$ 0 0
$$857$$ −17.3995 −0.594357 −0.297178 0.954822i $$-0.596046\pi$$
−0.297178 + 0.954822i $$0.596046\pi$$
$$858$$ 0 0
$$859$$ −1.24560 −0.0424993 −0.0212497 0.999774i $$-0.506764\pi$$
−0.0212497 + 0.999774i $$0.506764\pi$$
$$860$$ 0 0
$$861$$ 13.6839 0.466345
$$862$$ 0 0
$$863$$ 4.60383 0.156716 0.0783580 0.996925i $$-0.475032\pi$$
0.0783580 + 0.996925i $$0.475032\pi$$
$$864$$ 0 0
$$865$$ −1.20394 −0.0409352
$$866$$ 0 0
$$867$$ 8.42931 0.286274
$$868$$ 0 0
$$869$$ −11.0444 −0.374655
$$870$$ 0 0
$$871$$ −45.8330 −1.55299
$$872$$ 0 0
$$873$$ −1.94959 −0.0659837
$$874$$ 0 0
$$875$$ −4.06562 −0.137443
$$876$$ 0 0
$$877$$ 33.8276 1.14228 0.571138 0.820854i $$-0.306502\pi$$
0.571138 + 0.820854i $$0.306502\pi$$
$$878$$ 0 0
$$879$$ −53.2799 −1.79709
$$880$$ 0 0
$$881$$ 26.1134 0.879782 0.439891 0.898051i $$-0.355017\pi$$
0.439891 + 0.898051i $$0.355017\pi$$
$$882$$ 0 0
$$883$$ 54.5412 1.83546 0.917729 0.397206i $$-0.130020\pi$$
0.917729 + 0.397206i $$0.130020\pi$$
$$884$$ 0 0
$$885$$ 22.5087 0.756621
$$886$$ 0 0
$$887$$ −22.6201 −0.759509 −0.379754 0.925087i $$-0.623991\pi$$
−0.379754 + 0.925087i $$0.623991\pi$$
$$888$$ 0 0
$$889$$ 24.4082 0.818626
$$890$$ 0 0
$$891$$ 18.6162 0.623666
$$892$$ 0 0
$$893$$ −0.645301 −0.0215942
$$894$$ 0 0
$$895$$ −7.42495 −0.248189
$$896$$ 0 0
$$897$$ 9.22914 0.308152
$$898$$ 0 0
$$899$$ 54.4508 1.81603
$$900$$ 0 0
$$901$$ −30.7776 −1.02535
$$902$$ 0 0
$$903$$ −14.9110 −0.496208
$$904$$ 0 0
$$905$$ 24.0450 0.799284
$$906$$ 0 0
$$907$$ 3.62149 0.120250 0.0601248 0.998191i $$-0.480850\pi$$
0.0601248 + 0.998191i $$0.480850\pi$$
$$908$$ 0 0
$$909$$ −5.17211 −0.171548
$$910$$ 0 0
$$911$$ −26.9241 −0.892034 −0.446017 0.895025i $$-0.647158\pi$$
−0.446017 + 0.895025i $$0.647158\pi$$
$$912$$ 0 0
$$913$$ −33.3210 −1.10277
$$914$$ 0 0
$$915$$ −3.84712 −0.127182
$$916$$ 0 0
$$917$$ −81.1524 −2.67989
$$918$$ 0 0
$$919$$ 19.5059 0.643441 0.321721 0.946835i $$-0.395739\pi$$
0.321721 + 0.946835i $$0.395739\pi$$
$$920$$ 0 0
$$921$$ 2.33206 0.0768440
$$922$$ 0 0
$$923$$ 73.5204 2.41995
$$924$$ 0 0
$$925$$ −4.18059 −0.137457
$$926$$ 0 0
$$927$$ 2.63782 0.0866373
$$928$$ 0 0
$$929$$ 27.2139 0.892860 0.446430 0.894819i $$-0.352695\pi$$
0.446430 + 0.894819i $$0.352695\pi$$
$$930$$ 0 0
$$931$$ 25.3427 0.830572
$$932$$ 0 0
$$933$$ −22.0273 −0.721142
$$934$$ 0 0
$$935$$ 9.05854 0.296246
$$936$$ 0 0
$$937$$ 3.85626 0.125978 0.0629892 0.998014i $$-0.479937\pi$$
0.0629892 + 0.998014i $$0.479937\pi$$
$$938$$ 0 0
$$939$$ 51.3069 1.67434
$$940$$ 0 0
$$941$$ −60.6471 −1.97704 −0.988519 0.151097i $$-0.951720\pi$$
−0.988519 + 0.151097i $$0.951720\pi$$
$$942$$ 0 0
$$943$$ 2.15539 0.0701890
$$944$$ 0 0
$$945$$ −22.6112 −0.735541
$$946$$ 0 0
$$947$$ 1.11954 0.0363801 0.0181901 0.999835i $$-0.494210\pi$$
0.0181901 + 0.999835i $$0.494210\pi$$
$$948$$ 0 0
$$949$$ 12.3510 0.400931
$$950$$ 0 0
$$951$$ −19.6481 −0.637132
$$952$$ 0 0
$$953$$ 14.8590 0.481331 0.240666 0.970608i $$-0.422634\pi$$
0.240666 + 0.970608i $$0.422634\pi$$
$$954$$ 0 0
$$955$$ 4.87689 0.157813
$$956$$ 0 0
$$957$$ 24.2720 0.784601
$$958$$ 0 0
$$959$$ 47.8155 1.54404
$$960$$ 0 0
$$961$$ 55.7953 1.79985
$$962$$ 0 0
$$963$$ −0.211365 −0.00681114
$$964$$ 0 0
$$965$$ 10.2714 0.330649
$$966$$ 0 0
$$967$$ −38.5718 −1.24039 −0.620193 0.784449i $$-0.712946\pi$$
−0.620193 + 0.784449i $$0.712946\pi$$
$$968$$ 0 0
$$969$$ 14.1454 0.454416
$$970$$ 0 0
$$971$$ 25.4984 0.818284 0.409142 0.912471i $$-0.365828\pi$$
0.409142 + 0.912471i $$0.365828\pi$$
$$972$$ 0 0
$$973$$ 50.3694 1.61477
$$974$$ 0 0
$$975$$ 9.22914 0.295569
$$976$$ 0 0
$$977$$ −15.8952 −0.508533 −0.254267 0.967134i $$-0.581834\pi$$
−0.254267 + 0.967134i $$0.581834\pi$$
$$978$$ 0 0
$$979$$ 24.8840 0.795297
$$980$$ 0 0
$$981$$ −10.6902 −0.341313
$$982$$ 0 0
$$983$$ 2.50587 0.0799247 0.0399624 0.999201i $$-0.487276\pi$$
0.0399624 + 0.999201i $$0.487276\pi$$
$$984$$ 0 0
$$985$$ 4.28557 0.136550
$$986$$ 0 0
$$987$$ −1.54047 −0.0490337
$$988$$ 0 0
$$989$$ −2.34868 −0.0746837
$$990$$ 0 0
$$991$$ −12.5272 −0.397938 −0.198969 0.980006i $$-0.563759\pi$$
−0.198969 + 0.980006i $$0.563759\pi$$
$$992$$ 0 0
$$993$$ 2.46220 0.0781356
$$994$$ 0 0
$$995$$ −4.02164 −0.127494
$$996$$ 0 0
$$997$$ −9.34803 −0.296055 −0.148028 0.988983i $$-0.547292\pi$$
−0.148028 + 0.988983i $$0.547292\pi$$
$$998$$ 0 0
$$999$$ −23.2506 −0.735616
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 1840.2.a.u.1.1 4
4.3 odd 2 115.2.a.c.1.2 4
5.4 even 2 9200.2.a.cl.1.4 4
8.3 odd 2 7360.2.a.cj.1.2 4
8.5 even 2 7360.2.a.cg.1.3 4
12.11 even 2 1035.2.a.o.1.3 4
20.3 even 4 575.2.b.e.24.5 8
20.7 even 4 575.2.b.e.24.4 8
20.19 odd 2 575.2.a.h.1.3 4
28.27 even 2 5635.2.a.v.1.2 4
60.59 even 2 5175.2.a.bx.1.2 4
92.91 even 2 2645.2.a.m.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.a.c.1.2 4 4.3 odd 2
575.2.a.h.1.3 4 20.19 odd 2
575.2.b.e.24.4 8 20.7 even 4
575.2.b.e.24.5 8 20.3 even 4
1035.2.a.o.1.3 4 12.11 even 2
1840.2.a.u.1.1 4 1.1 even 1 trivial
2645.2.a.m.1.2 4 92.91 even 2
5175.2.a.bx.1.2 4 60.59 even 2
5635.2.a.v.1.2 4 28.27 even 2
7360.2.a.cg.1.3 4 8.5 even 2
7360.2.a.cj.1.2 4 8.3 odd 2
9200.2.a.cl.1.4 4 5.4 even 2