# Properties

 Label 9200.2.a.cz.1.3 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.58319$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} +O(q^{10})$$ $$q-1.58319 q^{3} -2.84620 q^{7} -0.493499 q^{9} -1.98637 q^{11} -4.69204 q^{13} -3.16675 q^{17} +6.16110 q^{19} +4.50609 q^{21} +1.00000 q^{23} +5.53088 q^{27} -6.61816 q^{29} +8.29732 q^{31} +3.14481 q^{33} +1.71181 q^{37} +7.42840 q^{39} +6.72440 q^{41} -0.177968 q^{43} +11.4749 q^{47} +1.10086 q^{49} +5.01358 q^{51} -6.18762 q^{53} -9.75422 q^{57} +7.61559 q^{59} +11.8577 q^{61} +1.40460 q^{63} +3.07554 q^{67} -1.58319 q^{69} +1.96195 q^{71} -4.94732 q^{73} +5.65361 q^{77} -8.53182 q^{79} -7.27596 q^{81} +3.49429 q^{83} +10.4778 q^{87} +7.07506 q^{89} +13.3545 q^{91} -13.1363 q^{93} +7.00705 q^{97} +0.980272 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q - 3q^{3} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$7q - 3q^{3} - 4q^{7} + 2q^{9} + 7q^{11} - 7q^{13} + 7q^{19} - 6q^{21} + 7q^{23} - 11q^{29} + 10q^{31} - 19q^{33} - 19q^{37} + 24q^{39} - 16q^{41} - 6q^{43} - 6q^{47} - 17q^{49} + 7q^{51} - 15q^{53} - 8q^{57} + 11q^{59} + 5q^{61} - 13q^{63} - 9q^{67} - 3q^{69} + 14q^{71} - 10q^{73} - 6q^{77} + 32q^{79} - 5q^{81} - q^{83} - 10q^{87} - 24q^{89} + 7q^{91} - 26q^{93} + 7q^{97} + 61q^{99} + O(q^{100})$$

## 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
$$3$$ −1.58319 −0.914057 −0.457029 0.889452i $$-0.651086\pi$$
−0.457029 + 0.889452i $$0.651086\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.84620 −1.07576 −0.537881 0.843020i $$-0.680775\pi$$
−0.537881 + 0.843020i $$0.680775\pi$$
$$8$$ 0 0
$$9$$ −0.493499 −0.164500
$$10$$ 0 0
$$11$$ −1.98637 −0.598913 −0.299457 0.954110i $$-0.596805\pi$$
−0.299457 + 0.954110i $$0.596805\pi$$
$$12$$ 0 0
$$13$$ −4.69204 −1.30134 −0.650668 0.759362i $$-0.725511\pi$$
−0.650668 + 0.759362i $$0.725511\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.16675 −0.768050 −0.384025 0.923323i $$-0.625462\pi$$
−0.384025 + 0.923323i $$0.625462\pi$$
$$18$$ 0 0
$$19$$ 6.16110 1.41345 0.706727 0.707486i $$-0.250171\pi$$
0.706727 + 0.707486i $$0.250171\pi$$
$$20$$ 0 0
$$21$$ 4.50609 0.983309
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.53088 1.06442
$$28$$ 0 0
$$29$$ −6.61816 −1.22896 −0.614481 0.788932i $$-0.710635\pi$$
−0.614481 + 0.788932i $$0.710635\pi$$
$$30$$ 0 0
$$31$$ 8.29732 1.49024 0.745121 0.666929i $$-0.232392\pi$$
0.745121 + 0.666929i $$0.232392\pi$$
$$32$$ 0 0
$$33$$ 3.14481 0.547441
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.71181 0.281420 0.140710 0.990051i $$-0.455061\pi$$
0.140710 + 0.990051i $$0.455061\pi$$
$$38$$ 0 0
$$39$$ 7.42840 1.18950
$$40$$ 0 0
$$41$$ 6.72440 1.05018 0.525088 0.851048i $$-0.324033\pi$$
0.525088 + 0.851048i $$0.324033\pi$$
$$42$$ 0 0
$$43$$ −0.177968 −0.0271399 −0.0135700 0.999908i $$-0.504320\pi$$
−0.0135700 + 0.999908i $$0.504320\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.4749 1.67378 0.836891 0.547369i $$-0.184371\pi$$
0.836891 + 0.547369i $$0.184371\pi$$
$$48$$ 0 0
$$49$$ 1.10086 0.157266
$$50$$ 0 0
$$51$$ 5.01358 0.702042
$$52$$ 0 0
$$53$$ −6.18762 −0.849934 −0.424967 0.905209i $$-0.639714\pi$$
−0.424967 + 0.905209i $$0.639714\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −9.75422 −1.29198
$$58$$ 0 0
$$59$$ 7.61559 0.991465 0.495733 0.868475i $$-0.334900\pi$$
0.495733 + 0.868475i $$0.334900\pi$$
$$60$$ 0 0
$$61$$ 11.8577 1.51822 0.759109 0.650964i $$-0.225635\pi$$
0.759109 + 0.650964i $$0.225635\pi$$
$$62$$ 0 0
$$63$$ 1.40460 0.176963
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.07554 0.375737 0.187868 0.982194i $$-0.439842\pi$$
0.187868 + 0.982194i $$0.439842\pi$$
$$68$$ 0 0
$$69$$ −1.58319 −0.190594
$$70$$ 0 0
$$71$$ 1.96195 0.232840 0.116420 0.993200i $$-0.462858\pi$$
0.116420 + 0.993200i $$0.462858\pi$$
$$72$$ 0 0
$$73$$ −4.94732 −0.579041 −0.289520 0.957172i $$-0.593496\pi$$
−0.289520 + 0.957172i $$0.593496\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.65361 0.644289
$$78$$ 0 0
$$79$$ −8.53182 −0.959905 −0.479952 0.877295i $$-0.659346\pi$$
−0.479952 + 0.877295i $$0.659346\pi$$
$$80$$ 0 0
$$81$$ −7.27596 −0.808440
$$82$$ 0 0
$$83$$ 3.49429 0.383548 0.191774 0.981439i $$-0.438576\pi$$
0.191774 + 0.981439i $$0.438576\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.4778 1.12334
$$88$$ 0 0
$$89$$ 7.07506 0.749954 0.374977 0.927034i $$-0.377651\pi$$
0.374977 + 0.927034i $$0.377651\pi$$
$$90$$ 0 0
$$91$$ 13.3545 1.39993
$$92$$ 0 0
$$93$$ −13.1363 −1.36217
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.00705 0.711458 0.355729 0.934589i $$-0.384233\pi$$
0.355729 + 0.934589i $$0.384233\pi$$
$$98$$ 0 0
$$99$$ 0.980272 0.0985210
$$100$$ 0 0
$$101$$ 7.22362 0.718777 0.359389 0.933188i $$-0.382985\pi$$
0.359389 + 0.933188i $$0.382985\pi$$
$$102$$ 0 0
$$103$$ −12.8178 −1.26297 −0.631487 0.775386i $$-0.717555\pi$$
−0.631487 + 0.775386i $$0.717555\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.6841 −1.12954 −0.564772 0.825247i $$-0.691036\pi$$
−0.564772 + 0.825247i $$0.691036\pi$$
$$108$$ 0 0
$$109$$ −13.6019 −1.30283 −0.651413 0.758723i $$-0.725824\pi$$
−0.651413 + 0.758723i $$0.725824\pi$$
$$110$$ 0 0
$$111$$ −2.71013 −0.257234
$$112$$ 0 0
$$113$$ −1.44526 −0.135959 −0.0679795 0.997687i $$-0.521655\pi$$
−0.0679795 + 0.997687i $$0.521655\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.31552 0.214069
$$118$$ 0 0
$$119$$ 9.01322 0.826240
$$120$$ 0 0
$$121$$ −7.05433 −0.641303
$$122$$ 0 0
$$123$$ −10.6460 −0.959920
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.12328 −0.809561 −0.404780 0.914414i $$-0.632652\pi$$
−0.404780 + 0.914414i $$0.632652\pi$$
$$128$$ 0 0
$$129$$ 0.281758 0.0248074
$$130$$ 0 0
$$131$$ −9.33180 −0.815323 −0.407661 0.913133i $$-0.633656\pi$$
−0.407661 + 0.913133i $$0.633656\pi$$
$$132$$ 0 0
$$133$$ −17.5357 −1.52054
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.0013 1.79426 0.897129 0.441768i $$-0.145649\pi$$
0.897129 + 0.441768i $$0.145649\pi$$
$$138$$ 0 0
$$139$$ 16.1299 1.36812 0.684061 0.729425i $$-0.260212\pi$$
0.684061 + 0.729425i $$0.260212\pi$$
$$140$$ 0 0
$$141$$ −18.1669 −1.52993
$$142$$ 0 0
$$143$$ 9.32012 0.779388
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.74288 −0.143750
$$148$$ 0 0
$$149$$ −9.57227 −0.784191 −0.392096 0.919925i $$-0.628250\pi$$
−0.392096 + 0.919925i $$0.628250\pi$$
$$150$$ 0 0
$$151$$ −3.04120 −0.247489 −0.123745 0.992314i $$-0.539490\pi$$
−0.123745 + 0.992314i $$0.539490\pi$$
$$152$$ 0 0
$$153$$ 1.56279 0.126344
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.4786 0.995900 0.497950 0.867206i $$-0.334086\pi$$
0.497950 + 0.867206i $$0.334086\pi$$
$$158$$ 0 0
$$159$$ 9.79619 0.776889
$$160$$ 0 0
$$161$$ −2.84620 −0.224312
$$162$$ 0 0
$$163$$ −16.3939 −1.28407 −0.642034 0.766676i $$-0.721909\pi$$
−0.642034 + 0.766676i $$0.721909\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.13265 −0.629323 −0.314662 0.949204i $$-0.601891\pi$$
−0.314662 + 0.949204i $$0.601891\pi$$
$$168$$ 0 0
$$169$$ 9.01521 0.693477
$$170$$ 0 0
$$171$$ −3.04050 −0.232513
$$172$$ 0 0
$$173$$ 10.5256 0.800250 0.400125 0.916461i $$-0.368967\pi$$
0.400125 + 0.916461i $$0.368967\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0570 −0.906256
$$178$$ 0 0
$$179$$ −20.5421 −1.53539 −0.767695 0.640816i $$-0.778596\pi$$
−0.767695 + 0.640816i $$0.778596\pi$$
$$180$$ 0 0
$$181$$ −22.0754 −1.64085 −0.820424 0.571756i $$-0.806263\pi$$
−0.820424 + 0.571756i $$0.806263\pi$$
$$182$$ 0 0
$$183$$ −18.7730 −1.38774
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.29034 0.459995
$$188$$ 0 0
$$189$$ −15.7420 −1.14506
$$190$$ 0 0
$$191$$ 4.50910 0.326267 0.163133 0.986604i $$-0.447840\pi$$
0.163133 + 0.986604i $$0.447840\pi$$
$$192$$ 0 0
$$193$$ −21.1534 −1.52266 −0.761329 0.648366i $$-0.775453\pi$$
−0.761329 + 0.648366i $$0.775453\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.1475 −0.722980 −0.361490 0.932376i $$-0.617732\pi$$
−0.361490 + 0.932376i $$0.617732\pi$$
$$198$$ 0 0
$$199$$ −6.57598 −0.466159 −0.233079 0.972458i $$-0.574880\pi$$
−0.233079 + 0.972458i $$0.574880\pi$$
$$200$$ 0 0
$$201$$ −4.86917 −0.343445
$$202$$ 0 0
$$203$$ 18.8366 1.32207
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −0.493499 −0.0343005
$$208$$ 0 0
$$209$$ −12.2382 −0.846536
$$210$$ 0 0
$$211$$ 26.5357 1.82680 0.913398 0.407069i $$-0.133449\pi$$
0.913398 + 0.407069i $$0.133449\pi$$
$$212$$ 0 0
$$213$$ −3.10614 −0.212829
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −23.6158 −1.60315
$$218$$ 0 0
$$219$$ 7.83257 0.529276
$$220$$ 0 0
$$221$$ 14.8585 0.999492
$$222$$ 0 0
$$223$$ −20.9231 −1.40111 −0.700557 0.713597i $$-0.747065\pi$$
−0.700557 + 0.713597i $$0.747065\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −9.32581 −0.618975 −0.309488 0.950903i $$-0.600158\pi$$
−0.309488 + 0.950903i $$0.600158\pi$$
$$228$$ 0 0
$$229$$ 29.5189 1.95066 0.975332 0.220742i $$-0.0708479\pi$$
0.975332 + 0.220742i $$0.0708479\pi$$
$$230$$ 0 0
$$231$$ −8.95076 −0.588917
$$232$$ 0 0
$$233$$ 28.9849 1.89886 0.949431 0.313974i $$-0.101661\pi$$
0.949431 + 0.313974i $$0.101661\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 13.5075 0.877408
$$238$$ 0 0
$$239$$ 9.62209 0.622401 0.311201 0.950344i $$-0.399269\pi$$
0.311201 + 0.950344i $$0.399269\pi$$
$$240$$ 0 0
$$241$$ 15.5745 1.00324 0.501621 0.865088i $$-0.332737\pi$$
0.501621 + 0.865088i $$0.332737\pi$$
$$242$$ 0 0
$$243$$ −5.07340 −0.325459
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −28.9081 −1.83938
$$248$$ 0 0
$$249$$ −5.53214 −0.350585
$$250$$ 0 0
$$251$$ 22.8066 1.43954 0.719770 0.694213i $$-0.244247\pi$$
0.719770 + 0.694213i $$0.244247\pi$$
$$252$$ 0 0
$$253$$ −1.98637 −0.124882
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −11.5591 −0.721038 −0.360519 0.932752i $$-0.617400\pi$$
−0.360519 + 0.932752i $$0.617400\pi$$
$$258$$ 0 0
$$259$$ −4.87216 −0.302742
$$260$$ 0 0
$$261$$ 3.26606 0.202164
$$262$$ 0 0
$$263$$ 6.57534 0.405453 0.202726 0.979235i $$-0.435020\pi$$
0.202726 + 0.979235i $$0.435020\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −11.2012 −0.685501
$$268$$ 0 0
$$269$$ −16.6536 −1.01539 −0.507694 0.861537i $$-0.669502\pi$$
−0.507694 + 0.861537i $$0.669502\pi$$
$$270$$ 0 0
$$271$$ 23.8675 1.44985 0.724925 0.688828i $$-0.241874\pi$$
0.724925 + 0.688828i $$0.241874\pi$$
$$272$$ 0 0
$$273$$ −21.1427 −1.27962
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5.65369 −0.339697 −0.169849 0.985470i $$-0.554328\pi$$
−0.169849 + 0.985470i $$0.554328\pi$$
$$278$$ 0 0
$$279$$ −4.09472 −0.245144
$$280$$ 0 0
$$281$$ −31.7408 −1.89350 −0.946750 0.321971i $$-0.895655\pi$$
−0.946750 + 0.321971i $$0.895655\pi$$
$$282$$ 0 0
$$283$$ −0.998092 −0.0593304 −0.0296652 0.999560i $$-0.509444\pi$$
−0.0296652 + 0.999560i $$0.509444\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −19.1390 −1.12974
$$288$$ 0 0
$$289$$ −6.97168 −0.410099
$$290$$ 0 0
$$291$$ −11.0935 −0.650314
$$292$$ 0 0
$$293$$ −13.8947 −0.811739 −0.405869 0.913931i $$-0.633031\pi$$
−0.405869 + 0.913931i $$0.633031\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −10.9864 −0.637495
$$298$$ 0 0
$$299$$ −4.69204 −0.271347
$$300$$ 0 0
$$301$$ 0.506534 0.0291961
$$302$$ 0 0
$$303$$ −11.4364 −0.657003
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3.66633 0.209249 0.104624 0.994512i $$-0.466636\pi$$
0.104624 + 0.994512i $$0.466636\pi$$
$$308$$ 0 0
$$309$$ 20.2930 1.15443
$$310$$ 0 0
$$311$$ 14.4210 0.817741 0.408870 0.912593i $$-0.365923\pi$$
0.408870 + 0.912593i $$0.365923\pi$$
$$312$$ 0 0
$$313$$ 10.5046 0.593754 0.296877 0.954916i $$-0.404055\pi$$
0.296877 + 0.954916i $$0.404055\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −32.0818 −1.80189 −0.900946 0.433931i $$-0.857126\pi$$
−0.900946 + 0.433931i $$0.857126\pi$$
$$318$$ 0 0
$$319$$ 13.1461 0.736042
$$320$$ 0 0
$$321$$ 18.4982 1.03247
$$322$$ 0 0
$$323$$ −19.5107 −1.08560
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 21.5344 1.19086
$$328$$ 0 0
$$329$$ −32.6598 −1.80059
$$330$$ 0 0
$$331$$ −9.71307 −0.533879 −0.266939 0.963713i $$-0.586012\pi$$
−0.266939 + 0.963713i $$0.586012\pi$$
$$332$$ 0 0
$$333$$ −0.844778 −0.0462935
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −0.131832 −0.00718136 −0.00359068 0.999994i $$-0.501143\pi$$
−0.00359068 + 0.999994i $$0.501143\pi$$
$$338$$ 0 0
$$339$$ 2.28813 0.124274
$$340$$ 0 0
$$341$$ −16.4815 −0.892526
$$342$$ 0 0
$$343$$ 16.7901 0.906582
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 13.1981 0.708512 0.354256 0.935148i $$-0.384734\pi$$
0.354256 + 0.935148i $$0.384734\pi$$
$$348$$ 0 0
$$349$$ −34.0451 −1.82239 −0.911197 0.411971i $$-0.864841\pi$$
−0.911197 + 0.411971i $$0.864841\pi$$
$$350$$ 0 0
$$351$$ −25.9511 −1.38517
$$352$$ 0 0
$$353$$ 10.8593 0.577984 0.288992 0.957331i $$-0.406680\pi$$
0.288992 + 0.957331i $$0.406680\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −14.2697 −0.755231
$$358$$ 0 0
$$359$$ 20.3091 1.07188 0.535938 0.844257i $$-0.319958\pi$$
0.535938 + 0.844257i $$0.319958\pi$$
$$360$$ 0 0
$$361$$ 18.9592 0.997853
$$362$$ 0 0
$$363$$ 11.1684 0.586188
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 18.6731 0.974727 0.487364 0.873199i $$-0.337959\pi$$
0.487364 + 0.873199i $$0.337959\pi$$
$$368$$ 0 0
$$369$$ −3.31848 −0.172753
$$370$$ 0 0
$$371$$ 17.6112 0.914328
$$372$$ 0 0
$$373$$ 9.51979 0.492916 0.246458 0.969153i $$-0.420733\pi$$
0.246458 + 0.969153i $$0.420733\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 31.0527 1.59929
$$378$$ 0 0
$$379$$ −18.8304 −0.967251 −0.483625 0.875275i $$-0.660680\pi$$
−0.483625 + 0.875275i $$0.660680\pi$$
$$380$$ 0 0
$$381$$ 14.4439 0.739985
$$382$$ 0 0
$$383$$ 10.9401 0.559015 0.279507 0.960144i $$-0.409829\pi$$
0.279507 + 0.960144i $$0.409829\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.0878272 0.00446451
$$388$$ 0 0
$$389$$ 22.2830 1.12979 0.564896 0.825162i $$-0.308916\pi$$
0.564896 + 0.825162i $$0.308916\pi$$
$$390$$ 0 0
$$391$$ −3.16675 −0.160150
$$392$$ 0 0
$$393$$ 14.7740 0.745252
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.8388 −0.543983 −0.271991 0.962300i $$-0.587682\pi$$
−0.271991 + 0.962300i $$0.587682\pi$$
$$398$$ 0 0
$$399$$ 27.7625 1.38986
$$400$$ 0 0
$$401$$ −2.71329 −0.135495 −0.0677476 0.997702i $$-0.521581\pi$$
−0.0677476 + 0.997702i $$0.521581\pi$$
$$402$$ 0 0
$$403$$ −38.9313 −1.93931
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3.40029 −0.168546
$$408$$ 0 0
$$409$$ 21.6271 1.06939 0.534696 0.845044i $$-0.320426\pi$$
0.534696 + 0.845044i $$0.320426\pi$$
$$410$$ 0 0
$$411$$ −33.2490 −1.64005
$$412$$ 0 0
$$413$$ −21.6755 −1.06658
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −25.5368 −1.25054
$$418$$ 0 0
$$419$$ −4.00587 −0.195700 −0.0978498 0.995201i $$-0.531196\pi$$
−0.0978498 + 0.995201i $$0.531196\pi$$
$$420$$ 0 0
$$421$$ 25.4251 1.23914 0.619572 0.784940i $$-0.287306\pi$$
0.619572 + 0.784940i $$0.287306\pi$$
$$422$$ 0 0
$$423$$ −5.66284 −0.275337
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −33.7493 −1.63324
$$428$$ 0 0
$$429$$ −14.7556 −0.712405
$$430$$ 0 0
$$431$$ −1.07679 −0.0518673 −0.0259336 0.999664i $$-0.508256\pi$$
−0.0259336 + 0.999664i $$0.508256\pi$$
$$432$$ 0 0
$$433$$ −3.93871 −0.189283 −0.0946413 0.995511i $$-0.530170\pi$$
−0.0946413 + 0.995511i $$0.530170\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.16110 0.294726
$$438$$ 0 0
$$439$$ −13.1539 −0.627802 −0.313901 0.949456i $$-0.601636\pi$$
−0.313901 + 0.949456i $$0.601636\pi$$
$$440$$ 0 0
$$441$$ −0.543274 −0.0258702
$$442$$ 0 0
$$443$$ 8.95394 0.425415 0.212707 0.977116i $$-0.431772\pi$$
0.212707 + 0.977116i $$0.431772\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 15.1548 0.716795
$$448$$ 0 0
$$449$$ 8.23267 0.388524 0.194262 0.980950i $$-0.437769\pi$$
0.194262 + 0.980950i $$0.437769\pi$$
$$450$$ 0 0
$$451$$ −13.3571 −0.628964
$$452$$ 0 0
$$453$$ 4.81480 0.226219
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 36.3473 1.70026 0.850128 0.526576i $$-0.176524\pi$$
0.850128 + 0.526576i $$0.176524\pi$$
$$458$$ 0 0
$$459$$ −17.5149 −0.817527
$$460$$ 0 0
$$461$$ −15.2421 −0.709894 −0.354947 0.934886i $$-0.615501\pi$$
−0.354947 + 0.934886i $$0.615501\pi$$
$$462$$ 0 0
$$463$$ 5.57424 0.259057 0.129528 0.991576i $$-0.458654\pi$$
0.129528 + 0.991576i $$0.458654\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.21488 0.287590 0.143795 0.989607i $$-0.454069\pi$$
0.143795 + 0.989607i $$0.454069\pi$$
$$468$$ 0 0
$$469$$ −8.75360 −0.404204
$$470$$ 0 0
$$471$$ −19.7560 −0.910309
$$472$$ 0 0
$$473$$ 0.353511 0.0162545
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3.05358 0.139814
$$478$$ 0 0
$$479$$ 34.4168 1.57254 0.786272 0.617880i $$-0.212008\pi$$
0.786272 + 0.617880i $$0.212008\pi$$
$$480$$ 0 0
$$481$$ −8.03189 −0.366223
$$482$$ 0 0
$$483$$ 4.50609 0.205034
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −22.7752 −1.03204 −0.516021 0.856576i $$-0.672587\pi$$
−0.516021 + 0.856576i $$0.672587\pi$$
$$488$$ 0 0
$$489$$ 25.9547 1.17371
$$490$$ 0 0
$$491$$ −3.16911 −0.143020 −0.0715099 0.997440i $$-0.522782\pi$$
−0.0715099 + 0.997440i $$0.522782\pi$$
$$492$$ 0 0
$$493$$ 20.9581 0.943905
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −5.58410 −0.250481
$$498$$ 0 0
$$499$$ 14.0809 0.630346 0.315173 0.949034i $$-0.397937\pi$$
0.315173 + 0.949034i $$0.397937\pi$$
$$500$$ 0 0
$$501$$ 12.8756 0.575237
$$502$$ 0 0
$$503$$ −21.4946 −0.958399 −0.479199 0.877706i $$-0.659073\pi$$
−0.479199 + 0.877706i $$0.659073\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −14.2728 −0.633878
$$508$$ 0 0
$$509$$ −34.0952 −1.51124 −0.755622 0.655008i $$-0.772665\pi$$
−0.755622 + 0.655008i $$0.772665\pi$$
$$510$$ 0 0
$$511$$ 14.0811 0.622910
$$512$$ 0 0
$$513$$ 34.0764 1.50451
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −22.7933 −1.00245
$$518$$ 0 0
$$519$$ −16.6641 −0.731474
$$520$$ 0 0
$$521$$ −14.1367 −0.619339 −0.309669 0.950844i $$-0.600218\pi$$
−0.309669 + 0.950844i $$0.600218\pi$$
$$522$$ 0 0
$$523$$ 11.2020 0.489828 0.244914 0.969545i $$-0.421240\pi$$
0.244914 + 0.969545i $$0.421240\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −26.2756 −1.14458
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −3.75829 −0.163096
$$532$$ 0 0
$$533$$ −31.5511 −1.36663
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 32.5221 1.40343
$$538$$ 0 0
$$539$$ −2.18672 −0.0941887
$$540$$ 0 0
$$541$$ −33.6704 −1.44760 −0.723802 0.690008i $$-0.757607\pi$$
−0.723802 + 0.690008i $$0.757607\pi$$
$$542$$ 0 0
$$543$$ 34.9496 1.49983
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −23.1062 −0.987949 −0.493975 0.869476i $$-0.664456\pi$$
−0.493975 + 0.869476i $$0.664456\pi$$
$$548$$ 0 0
$$549$$ −5.85174 −0.249746
$$550$$ 0 0
$$551$$ −40.7752 −1.73708
$$552$$ 0 0
$$553$$ 24.2833 1.03263
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −3.48937 −0.147849 −0.0739246 0.997264i $$-0.523552\pi$$
−0.0739246 + 0.997264i $$0.523552\pi$$
$$558$$ 0 0
$$559$$ 0.835034 0.0353182
$$560$$ 0 0
$$561$$ −9.95883 −0.420462
$$562$$ 0 0
$$563$$ −38.8593 −1.63772 −0.818862 0.573991i $$-0.805394\pi$$
−0.818862 + 0.573991i $$0.805394\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 20.7089 0.869690
$$568$$ 0 0
$$569$$ −23.2055 −0.972823 −0.486412 0.873730i $$-0.661694\pi$$
−0.486412 + 0.873730i $$0.661694\pi$$
$$570$$ 0 0
$$571$$ 5.39836 0.225914 0.112957 0.993600i $$-0.463968\pi$$
0.112957 + 0.993600i $$0.463968\pi$$
$$572$$ 0 0
$$573$$ −7.13877 −0.298226
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.1082 0.545701 0.272850 0.962056i $$-0.412034\pi$$
0.272850 + 0.962056i $$0.412034\pi$$
$$578$$ 0 0
$$579$$ 33.4900 1.39180
$$580$$ 0 0
$$581$$ −9.94546 −0.412607
$$582$$ 0 0
$$583$$ 12.2909 0.509037
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.9646 0.535108 0.267554 0.963543i $$-0.413785\pi$$
0.267554 + 0.963543i $$0.413785\pi$$
$$588$$ 0 0
$$589$$ 51.1206 2.10639
$$590$$ 0 0
$$591$$ 16.0655 0.660845
$$592$$ 0 0
$$593$$ 17.0225 0.699030 0.349515 0.936931i $$-0.386346\pi$$
0.349515 + 0.936931i $$0.386346\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10.4110 0.426096
$$598$$ 0 0
$$599$$ 5.98111 0.244381 0.122191 0.992507i $$-0.461008\pi$$
0.122191 + 0.992507i $$0.461008\pi$$
$$600$$ 0 0
$$601$$ −21.3680 −0.871617 −0.435809 0.900039i $$-0.643538\pi$$
−0.435809 + 0.900039i $$0.643538\pi$$
$$602$$ 0 0
$$603$$ −1.51778 −0.0618086
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.5464 0.712186 0.356093 0.934450i $$-0.384109\pi$$
0.356093 + 0.934450i $$0.384109\pi$$
$$608$$ 0 0
$$609$$ −29.8220 −1.20845
$$610$$ 0 0
$$611$$ −53.8405 −2.17815
$$612$$ 0 0
$$613$$ −45.4601 −1.83612 −0.918059 0.396444i $$-0.870244\pi$$
−0.918059 + 0.396444i $$0.870244\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −25.8720 −1.04157 −0.520785 0.853688i $$-0.674360\pi$$
−0.520785 + 0.853688i $$0.674360\pi$$
$$618$$ 0 0
$$619$$ 3.20381 0.128772 0.0643859 0.997925i $$-0.479491\pi$$
0.0643859 + 0.997925i $$0.479491\pi$$
$$620$$ 0 0
$$621$$ 5.53088 0.221947
$$622$$ 0 0
$$623$$ −20.1370 −0.806773
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 19.3755 0.773783
$$628$$ 0 0
$$629$$ −5.42089 −0.216145
$$630$$ 0 0
$$631$$ −0.0248650 −0.000989858 0 −0.000494929 1.00000i $$-0.500158\pi$$
−0.000494929 1.00000i $$0.500158\pi$$
$$632$$ 0 0
$$633$$ −42.0112 −1.66980
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5.16528 −0.204656
$$638$$ 0 0
$$639$$ −0.968219 −0.0383022
$$640$$ 0 0
$$641$$ −9.77145 −0.385949 −0.192975 0.981204i $$-0.561813\pi$$
−0.192975 + 0.981204i $$0.561813\pi$$
$$642$$ 0 0
$$643$$ −36.8211 −1.45208 −0.726041 0.687651i $$-0.758642\pi$$
−0.726041 + 0.687651i $$0.758642\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 46.5570 1.83034 0.915172 0.403064i $$-0.132055\pi$$
0.915172 + 0.403064i $$0.132055\pi$$
$$648$$ 0 0
$$649$$ −15.1274 −0.593802
$$650$$ 0 0
$$651$$ 37.3884 1.46537
$$652$$ 0 0
$$653$$ −2.76029 −0.108018 −0.0540092 0.998540i $$-0.517200\pi$$
−0.0540092 + 0.998540i $$0.517200\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.44150 0.0952520
$$658$$ 0 0
$$659$$ −39.3922 −1.53450 −0.767252 0.641346i $$-0.778376\pi$$
−0.767252 + 0.641346i $$0.778376\pi$$
$$660$$ 0 0
$$661$$ −17.7641 −0.690943 −0.345472 0.938429i $$-0.612281\pi$$
−0.345472 + 0.938429i $$0.612281\pi$$
$$662$$ 0 0
$$663$$ −23.5239 −0.913593
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.61816 −0.256256
$$668$$ 0 0
$$669$$ 33.1253 1.28070
$$670$$ 0 0
$$671$$ −23.5537 −0.909280
$$672$$ 0 0
$$673$$ 23.7072 0.913847 0.456923 0.889506i $$-0.348951\pi$$
0.456923 + 0.889506i $$0.348951\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −49.3507 −1.89670 −0.948352 0.317221i $$-0.897250\pi$$
−0.948352 + 0.317221i $$0.897250\pi$$
$$678$$ 0 0
$$679$$ −19.9435 −0.765361
$$680$$ 0 0
$$681$$ 14.7646 0.565779
$$682$$ 0 0
$$683$$ 15.6461 0.598682 0.299341 0.954146i $$-0.403233\pi$$
0.299341 + 0.954146i $$0.403233\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −46.7341 −1.78302
$$688$$ 0 0
$$689$$ 29.0325 1.10605
$$690$$ 0 0
$$691$$ −3.79498 −0.144368 −0.0721839 0.997391i $$-0.522997\pi$$
−0.0721839 + 0.997391i $$0.522997\pi$$
$$692$$ 0 0
$$693$$ −2.79005 −0.105985
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −21.2945 −0.806587
$$698$$ 0 0
$$699$$ −45.8887 −1.73567
$$700$$ 0 0
$$701$$ 0.751725 0.0283923 0.0141961 0.999899i $$-0.495481\pi$$
0.0141961 + 0.999899i $$0.495481\pi$$
$$702$$ 0 0
$$703$$ 10.5467 0.397775
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −20.5599 −0.773234
$$708$$ 0 0
$$709$$ 4.52035 0.169765 0.0848827 0.996391i $$-0.472948\pi$$
0.0848827 + 0.996391i $$0.472948\pi$$
$$710$$ 0 0
$$711$$ 4.21044 0.157904
$$712$$ 0 0
$$713$$ 8.29732 0.310737
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −15.2336 −0.568910
$$718$$ 0 0
$$719$$ 36.6742 1.36772 0.683859 0.729614i $$-0.260300\pi$$
0.683859 + 0.729614i $$0.260300\pi$$
$$720$$ 0 0
$$721$$ 36.4820 1.35866
$$722$$ 0 0
$$723$$ −24.6574 −0.917020
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −33.6449 −1.24782 −0.623910 0.781496i $$-0.714457\pi$$
−0.623910 + 0.781496i $$0.714457\pi$$
$$728$$ 0 0
$$729$$ 29.8601 1.10593
$$730$$ 0 0
$$731$$ 0.563582 0.0208448
$$732$$ 0 0
$$733$$ −47.5745 −1.75720 −0.878602 0.477555i $$-0.841523\pi$$
−0.878602 + 0.477555i $$0.841523\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −6.10916 −0.225034
$$738$$ 0 0
$$739$$ −28.8914 −1.06279 −0.531395 0.847124i $$-0.678332\pi$$
−0.531395 + 0.847124i $$0.678332\pi$$
$$740$$ 0 0
$$741$$ 45.7672 1.68130
$$742$$ 0 0
$$743$$ 3.61970 0.132794 0.0663969 0.997793i $$-0.478850\pi$$
0.0663969 + 0.997793i $$0.478850\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1.72443 −0.0630936
$$748$$ 0 0
$$749$$ 33.2553 1.21512
$$750$$ 0 0
$$751$$ −19.2050 −0.700801 −0.350401 0.936600i $$-0.613955\pi$$
−0.350401 + 0.936600i $$0.613955\pi$$
$$752$$ 0 0
$$753$$ −36.1073 −1.31582
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12.6638 0.460275 0.230137 0.973158i $$-0.426082\pi$$
0.230137 + 0.973158i $$0.426082\pi$$
$$758$$ 0 0
$$759$$ 3.14481 0.114149
$$760$$ 0 0
$$761$$ −38.9053 −1.41032 −0.705159 0.709050i $$-0.749124\pi$$
−0.705159 + 0.709050i $$0.749124\pi$$
$$762$$ 0 0
$$763$$ 38.7138 1.40153
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −35.7326 −1.29023
$$768$$ 0 0
$$769$$ 4.16816 0.150308 0.0751539 0.997172i $$-0.476055\pi$$
0.0751539 + 0.997172i $$0.476055\pi$$
$$770$$ 0 0
$$771$$ 18.3003 0.659070
$$772$$ 0 0
$$773$$ −26.6030 −0.956842 −0.478421 0.878131i $$-0.658791\pi$$
−0.478421 + 0.878131i $$0.658791\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 7.71358 0.276723
$$778$$ 0 0
$$779$$ 41.4297 1.48437
$$780$$ 0 0
$$781$$ −3.89716 −0.139451
$$782$$ 0 0
$$783$$ −36.6043 −1.30813
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21.7538 0.775440 0.387720 0.921777i $$-0.373263\pi$$
0.387720 + 0.921777i $$0.373263\pi$$
$$788$$ 0 0
$$789$$ −10.4100 −0.370607
$$790$$ 0 0
$$791$$ 4.11351 0.146260
$$792$$ 0 0
$$793$$ −55.6366 −1.97571
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −10.2490 −0.363039 −0.181519 0.983387i $$-0.558102\pi$$
−0.181519 + 0.983387i $$0.558102\pi$$
$$798$$ 0 0
$$799$$ −36.3381 −1.28555
$$800$$ 0 0
$$801$$ −3.49153 −0.123367
$$802$$ 0 0
$$803$$ 9.82722 0.346795
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 26.3659 0.928123
$$808$$ 0 0
$$809$$ 37.0993 1.30434 0.652170 0.758072i $$-0.273859\pi$$
0.652170 + 0.758072i $$0.273859\pi$$
$$810$$ 0 0
$$811$$ −35.8084 −1.25740 −0.628701 0.777647i $$-0.716413\pi$$
−0.628701 + 0.777647i $$0.716413\pi$$
$$812$$ 0 0
$$813$$ −37.7869 −1.32525
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.09648 −0.0383610
$$818$$ 0 0
$$819$$ −6.59042 −0.230288
$$820$$ 0 0
$$821$$ −21.5343 −0.751551 −0.375775 0.926711i $$-0.622624\pi$$
−0.375775 + 0.926711i $$0.622624\pi$$
$$822$$ 0 0
$$823$$ −41.1466 −1.43428 −0.717140 0.696929i $$-0.754549\pi$$
−0.717140 + 0.696929i $$0.754549\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −9.48319 −0.329763 −0.164881 0.986313i $$-0.552724\pi$$
−0.164881 + 0.986313i $$0.552724\pi$$
$$828$$ 0 0
$$829$$ 24.0091 0.833872 0.416936 0.908936i $$-0.363104\pi$$
0.416936 + 0.908936i $$0.363104\pi$$
$$830$$ 0 0
$$831$$ 8.95088 0.310503
$$832$$ 0 0
$$833$$ −3.48616 −0.120788
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 45.8915 1.58624
$$838$$ 0 0
$$839$$ −9.89039 −0.341454 −0.170727 0.985318i $$-0.554612\pi$$
−0.170727 + 0.985318i $$0.554612\pi$$
$$840$$ 0 0
$$841$$ 14.8001 0.510348
$$842$$ 0 0
$$843$$ 50.2519 1.73077
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 20.0781 0.689890
$$848$$ 0 0
$$849$$ 1.58017 0.0542314
$$850$$ 0 0
$$851$$ 1.71181 0.0586802
$$852$$ 0 0
$$853$$ −5.02358 −0.172004 −0.0860021 0.996295i $$-0.527409\pi$$
−0.0860021 + 0.996295i $$0.527409\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 26.8011 0.915509 0.457754 0.889079i $$-0.348654\pi$$
0.457754 + 0.889079i $$0.348654\pi$$
$$858$$ 0 0
$$859$$ −27.3018 −0.931525 −0.465763 0.884910i $$-0.654220\pi$$
−0.465763 + 0.884910i $$0.654220\pi$$
$$860$$ 0 0
$$861$$ 30.3007 1.03265
$$862$$ 0 0
$$863$$ 2.22910 0.0758796 0.0379398 0.999280i $$-0.487920\pi$$
0.0379398 + 0.999280i $$0.487920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 11.0375 0.374854
$$868$$ 0 0
$$869$$ 16.9474 0.574900
$$870$$ 0 0
$$871$$ −14.4305 −0.488960
$$872$$ 0 0
$$873$$ −3.45797 −0.117035
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.4517 1.56856 0.784280 0.620407i $$-0.213032\pi$$
0.784280 + 0.620407i $$0.213032\pi$$
$$878$$ 0 0
$$879$$ 21.9980 0.741975
$$880$$ 0 0
$$881$$ −25.6878 −0.865444 −0.432722 0.901527i $$-0.642447\pi$$
−0.432722 + 0.901527i $$0.642447\pi$$
$$882$$ 0 0
$$883$$ 48.8481 1.64387 0.821934 0.569582i $$-0.192895\pi$$
0.821934 + 0.569582i $$0.192895\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 45.1121 1.51472 0.757358 0.653000i $$-0.226490\pi$$
0.757358 + 0.653000i $$0.226490\pi$$
$$888$$ 0 0
$$889$$ 25.9667 0.870895
$$890$$ 0 0
$$891$$ 14.4528 0.484185
$$892$$ 0 0
$$893$$ 70.6979 2.36581
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 7.42840 0.248027
$$898$$ 0 0
$$899$$ −54.9130 −1.83145
$$900$$ 0 0
$$901$$ 19.5947 0.652792
$$902$$ 0 0
$$903$$ −0.801941 −0.0266869
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 15.5516 0.516382 0.258191 0.966094i $$-0.416874\pi$$
0.258191 + 0.966094i $$0.416874\pi$$
$$908$$ 0 0
$$909$$ −3.56485 −0.118239
$$910$$ 0 0
$$911$$ 30.1283 0.998195 0.499097 0.866546i $$-0.333665\pi$$
0.499097 + 0.866546i $$0.333665\pi$$
$$912$$ 0 0
$$913$$ −6.94096 −0.229712
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 26.5602 0.877094
$$918$$ 0 0
$$919$$ −34.2483 −1.12975 −0.564874 0.825177i $$-0.691075\pi$$
−0.564874 + 0.825177i $$0.691075\pi$$
$$920$$ 0 0
$$921$$ −5.80452 −0.191265
$$922$$ 0 0
$$923$$ −9.20553 −0.303004
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 6.32556 0.207759
$$928$$ 0 0
$$929$$ −54.6989 −1.79461 −0.897307 0.441408i $$-0.854479\pi$$
−0.897307 + 0.441408i $$0.854479\pi$$
$$930$$ 0 0
$$931$$ 6.78252 0.222288
$$932$$ 0 0
$$933$$ −22.8312 −0.747462
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.1635 0.397364 0.198682 0.980064i $$-0.436334\pi$$
0.198682 + 0.980064i $$0.436334\pi$$
$$938$$ 0 0
$$939$$ −16.6308 −0.542725
$$940$$ 0 0
$$941$$ −27.9048 −0.909672 −0.454836 0.890575i $$-0.650302\pi$$
−0.454836 + 0.890575i $$0.650302\pi$$
$$942$$ 0 0
$$943$$ 6.72440 0.218977
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −10.5104 −0.341541 −0.170770 0.985311i $$-0.554626\pi$$
−0.170770 + 0.985311i $$0.554626\pi$$
$$948$$ 0 0
$$949$$ 23.2130 0.753527
$$950$$ 0 0
$$951$$ 50.7916 1.64703
$$952$$ 0 0
$$953$$ −11.7005 −0.379017 −0.189508 0.981879i $$-0.560689\pi$$
−0.189508 + 0.981879i $$0.560689\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −20.8129 −0.672784
$$958$$ 0 0
$$959$$ −59.7738 −1.93020
$$960$$ 0 0
$$961$$ 37.8455 1.22082
$$962$$ 0 0
$$963$$ 5.76609 0.185810
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −11.7133 −0.376673 −0.188337 0.982105i $$-0.560310\pi$$
−0.188337 + 0.982105i $$0.560310\pi$$
$$968$$ 0 0
$$969$$ 30.8892 0.992304
$$970$$ 0 0
$$971$$ 37.8963 1.21615 0.608075 0.793879i $$-0.291942\pi$$
0.608075 + 0.793879i $$0.291942\pi$$
$$972$$ 0 0
$$973$$ −45.9090 −1.47178
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2.83500 −0.0906997 −0.0453499 0.998971i $$-0.514440\pi$$
−0.0453499 + 0.998971i $$0.514440\pi$$
$$978$$ 0 0
$$979$$ −14.0537 −0.449158
$$980$$ 0 0
$$981$$ 6.71253 0.214314
$$982$$ 0 0
$$983$$ 4.47696 0.142793 0.0713964 0.997448i $$-0.477254\pi$$
0.0713964 + 0.997448i $$0.477254\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 51.7068 1.64584
$$988$$ 0 0
$$989$$ −0.177968 −0.00565906
$$990$$ 0 0
$$991$$ 32.2466 1.02435 0.512173 0.858882i $$-0.328841\pi$$
0.512173 + 0.858882i $$0.328841\pi$$
$$992$$ 0 0
$$993$$ 15.3777 0.487996
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −27.5708 −0.873177 −0.436589 0.899661i $$-0.643813\pi$$
−0.436589 + 0.899661i $$0.643813\pi$$
$$998$$ 0 0
$$999$$ 9.46784 0.299549
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 9200.2.a.cz.1.3 7
4.3 odd 2 4600.2.a.bi.1.5 7
5.2 odd 4 1840.2.e.g.369.11 14
5.3 odd 4 1840.2.e.g.369.4 14
5.4 even 2 9200.2.a.dc.1.5 7
20.3 even 4 920.2.e.b.369.11 yes 14
20.7 even 4 920.2.e.b.369.4 14
20.19 odd 2 4600.2.a.bh.1.3 7

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.b.369.4 14 20.7 even 4
920.2.e.b.369.11 yes 14 20.3 even 4
1840.2.e.g.369.4 14 5.3 odd 4
1840.2.e.g.369.11 14 5.2 odd 4
4600.2.a.bh.1.3 7 20.19 odd 2
4600.2.a.bi.1.5 7 4.3 odd 2
9200.2.a.cz.1.3 7 1.1 even 1 trivial
9200.2.a.dc.1.5 7 5.4 even 2