Properties

 Label 2450.2.c.k.99.1 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 99.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.k.99.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} -3.00000i q^{18} -4.00000i q^{22} +6.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +2.00000 q^{34} -3.00000 q^{36} +10.0000i q^{37} -2.00000 q^{41} +4.00000i q^{43} -4.00000 q^{44} +8.00000i q^{47} -6.00000i q^{52} -2.00000i q^{53} +6.00000i q^{58} -8.00000 q^{59} +14.0000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} -2.00000i q^{68} -16.0000 q^{71} +3.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} +8.00000 q^{79} +9.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} +4.00000i q^{88} +10.0000 q^{89} +8.00000 q^{94} +2.00000i q^{97} +12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 6q^{9} + 8q^{11} + 2q^{16} + 12q^{26} - 12q^{29} - 16q^{31} + 4q^{34} - 6q^{36} - 4q^{41} - 8q^{44} - 16q^{59} + 28q^{61} - 2q^{64} - 32q^{71} + 20q^{74} + 16q^{79} + 18q^{81} + 8q^{86} + 20q^{89} + 16q^{94} + 24q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000i 0.353553i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ − 3.00000i − 0.707107i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 6.00000i − 0.832050i
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 2.00000i 0.220863i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000i 0.426401i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 0 0
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 18.0000i 1.66410i
$$118$$ 8.00000i 0.736460i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 14.0000i − 1.26750i
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ 24.0000i 2.00698i
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ − 10.0000i − 0.821995i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 9.00000i − 0.707107i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 4.00000i − 0.304997i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 14.0000i − 0.997459i −0.866758 0.498729i $$-0.833800\pi$$
0.866758 0.498729i $$-0.166200\pi$$
$$198$$ − 12.0000i − 0.852803i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 6.00000i 0.406371i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 8.00000i 0.530979i 0.964114 + 0.265489i $$0.0855335\pi$$
−0.964114 + 0.265489i $$0.914466\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 18.0000 1.17670
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.0000i − 1.37232i −0.727450 0.686161i $$-0.759294\pi$$
0.727450 0.686161i $$-0.240706\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ − 16.0000i − 0.988483i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ − 16.0000i − 0.959616i
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ − 32.0000i − 1.90220i −0.308879 0.951101i $$-0.599954\pi$$
0.308879 0.951101i $$-0.400046\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 0 0
$$288$$ − 3.00000i − 0.176777i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000i 0.117041i
$$293$$ − 10.0000i − 0.584206i −0.956387 0.292103i $$-0.905645\pi$$
0.956387 0.292103i $$-0.0943550\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 8.00000i − 0.460348i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 8.00000i − 0.456584i −0.973593 0.228292i $$-0.926686\pi$$
0.973593 0.228292i $$-0.0733141\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 22.0000i − 1.23564i −0.786318 0.617822i $$-0.788015\pi$$
0.786318 0.617822i $$-0.211985\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ − 2.00000i − 0.110432i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000i 0.439057i
$$333$$ 30.0000i 1.64399i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 14.0000i − 0.735824i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000i 0.835193i 0.908633 + 0.417597i $$0.137127\pi$$
−0.908633 + 0.417597i $$0.862873\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ − 36.0000i − 1.85409i
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 24.0000i − 1.22795i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 12.0000i 0.609994i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −14.0000 −0.705310
$$395$$ 0 0
$$396$$ −12.0000 −0.603023
$$397$$ 10.0000i 0.501886i 0.968002 + 0.250943i $$0.0807406\pi$$
−0.968002 + 0.250943i $$0.919259\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ − 48.0000i − 2.39105i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 16.0000i 0.788263i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 24.0000i 1.16692i
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 12.0000i 0.570782i
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 0 0
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 40.0000i − 1.85098i −0.378773 0.925490i $$-0.623654\pi$$
0.378773 0.925490i $$-0.376346\pi$$
$$468$$ − 18.0000i − 0.832050i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 8.00000i − 0.368230i
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 16.0000i 0.731823i
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ − 8.00000i − 0.354943i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −24.0000 −1.04151
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 6.00000i 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12.0000i − 0.513083i −0.966533 0.256541i $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 42.0000 1.79252
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ − 22.0000i − 0.932170i −0.884740 0.466085i $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ 24.0000i 1.01600i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 26.0000i − 1.09674i
$$563$$ 16.0000i 0.674320i 0.941447 + 0.337160i $$0.109466\pi$$
−0.941447 + 0.337160i $$0.890534\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −32.0000 −1.34506
$$567$$ 0 0
$$568$$ − 16.0000i − 0.671345i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ − 24.0000i − 1.00349i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 8.00000i − 0.331326i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000i 0.410997i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 0 0
$$603$$ 36.0000i 1.46603i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ − 6.00000i − 0.242536i
$$613$$ 6.00000i 0.242338i 0.992632 + 0.121169i $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ 0 0
$$628$$ − 10.0000i − 0.399043i
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 0 0
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 24.0000i 0.950169i
$$639$$ −48.0000 −1.89885
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 32.0000i 1.26196i 0.775800 + 0.630978i $$0.217346\pi$$
−0.775800 + 0.630978i $$0.782654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ 9.00000i 0.353553i
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ − 6.00000i − 0.234082i
$$658$$ 0 0
$$659$$ −28.0000 −1.09073 −0.545363 0.838200i $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 0 0
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ 30.0000 1.16248
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 32.0000i 1.22534i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 4.00000i − 0.151511i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 24.0000 0.900070
$$712$$ 10.0000i 0.374766i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 8.00000i 0.298557i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000i 0.707107i
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000i 1.76810i
$$738$$ 6.00000i 0.220863i
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ − 24.0000i − 0.878114i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 22.0000i − 0.799604i −0.916602 0.399802i $$-0.869079\pi$$
0.916602 0.399802i $$-0.130921\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ − 48.0000i − 1.73318i
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 14.0000i 0.501924i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 8.00000i − 0.285169i −0.989783 0.142585i $$-0.954459\pi$$
0.989783 0.142585i $$-0.0455413\pi$$
$$788$$ 14.0000i 0.498729i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 12.0000i 0.426401i
$$793$$ 84.0000i 2.98293i
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 14.0000i 0.494357i
$$803$$ − 8.00000i − 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ 0 0
$$808$$ 6.00000i 0.211079i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 30.0000i 1.04893i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ − 8.00000i − 0.278862i −0.990232 0.139431i $$-0.955473\pi$$
0.990232 0.139431i $$-0.0445274\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 6.00000i − 0.208013i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 24.0000i − 0.829066i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000i 0.344623i
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 24.0000 0.825137
$$847$$ 0 0
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 46.0000i 1.57501i 0.616308 + 0.787505i $$0.288628\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 54.0000i − 1.84460i −0.386469 0.922302i $$-0.626305\pi$$
0.386469 0.922302i $$-0.373695\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −72.0000 −2.43963
$$872$$ − 6.00000i − 0.203186i
$$873$$ 6.00000i 0.203069i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ − 12.0000i − 0.403832i −0.979403 0.201916i $$-0.935283\pi$$
0.979403 0.201916i $$-0.0647168\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ − 16.0000i − 0.537227i −0.963248 0.268614i $$-0.913434\pi$$
0.963248 0.268614i $$-0.0865655\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 36.0000 1.20605
$$892$$ 16.0000i 0.535720i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 30.0000i − 1.00111i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 8.00000i 0.266371i
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ − 8.00000i − 0.265489i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ − 32.0000i − 1.05905i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10.0000i 0.329332i
$$923$$ − 96.0000i − 3.15988i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ − 48.0000i − 1.57653i
$$928$$ 6.00000i 0.196960i
$$929$$ 58.0000 1.90292 0.951459 0.307775i $$-0.0995844\pi$$
0.951459 + 0.307775i $$0.0995844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ −40.0000 −1.30884
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ 50.0000i 1.63343i 0.577042 + 0.816714i $$0.304207\pi$$
−0.577042 + 0.816714i $$0.695793\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 44.0000i − 1.42981i −0.699223 0.714904i $$-0.746470\pi$$
0.699223 0.714904i $$-0.253530\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 54.0000i − 1.74923i −0.484817 0.874616i $$-0.661114\pi$$
0.484817 0.874616i $$-0.338886\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 60.0000i 1.93448i
$$963$$ − 36.0000i − 1.16008i
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 16.0000i − 0.514525i −0.966342 0.257263i $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ − 12.0000i − 0.382935i
$$983$$ 16.0000i 0.510321i 0.966899 + 0.255160i $$0.0821283\pi$$
−0.966899 + 0.255160i $$0.917872\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 22.0000i − 0.696747i −0.937356 0.348373i $$-0.886734\pi$$
0.937356 0.348373i $$-0.113266\pi$$
$$998$$ − 12.0000i − 0.379853i
$$999$$ 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 2450.2.c.k.99.1 2
5.2 odd 4 490.2.a.h.1.1 1
5.3 odd 4 2450.2.a.l.1.1 1
5.4 even 2 inner 2450.2.c.k.99.2 2
7.6 odd 2 350.2.c.b.99.1 2
15.2 even 4 4410.2.a.b.1.1 1
20.7 even 4 3920.2.a.t.1.1 1
21.20 even 2 3150.2.g.c.2899.2 2
28.27 even 2 2800.2.g.n.449.1 2
35.2 odd 12 490.2.e.c.361.1 2
35.12 even 12 490.2.e.d.361.1 2
35.13 even 4 350.2.a.b.1.1 1
35.17 even 12 490.2.e.d.471.1 2
35.27 even 4 70.2.a.a.1.1 1
35.32 odd 12 490.2.e.c.471.1 2
35.34 odd 2 350.2.c.b.99.2 2
105.62 odd 4 630.2.a.d.1.1 1
105.83 odd 4 3150.2.a.bj.1.1 1
105.104 even 2 3150.2.g.c.2899.1 2
140.27 odd 4 560.2.a.d.1.1 1
140.83 odd 4 2800.2.a.m.1.1 1
140.139 even 2 2800.2.g.n.449.2 2
280.27 odd 4 2240.2.a.q.1.1 1
280.237 even 4 2240.2.a.n.1.1 1
385.307 odd 4 8470.2.a.j.1.1 1
420.167 even 4 5040.2.a.bm.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.a.a.1.1 1 35.27 even 4
350.2.a.b.1.1 1 35.13 even 4
350.2.c.b.99.1 2 7.6 odd 2
350.2.c.b.99.2 2 35.34 odd 2
490.2.a.h.1.1 1 5.2 odd 4
490.2.e.c.361.1 2 35.2 odd 12
490.2.e.c.471.1 2 35.32 odd 12
490.2.e.d.361.1 2 35.12 even 12
490.2.e.d.471.1 2 35.17 even 12
560.2.a.d.1.1 1 140.27 odd 4
630.2.a.d.1.1 1 105.62 odd 4
2240.2.a.n.1.1 1 280.237 even 4
2240.2.a.q.1.1 1 280.27 odd 4
2450.2.a.l.1.1 1 5.3 odd 4
2450.2.c.k.99.1 2 1.1 even 1 trivial
2450.2.c.k.99.2 2 5.4 even 2 inner
2800.2.a.m.1.1 1 140.83 odd 4
2800.2.g.n.449.1 2 28.27 even 2
2800.2.g.n.449.2 2 140.139 even 2
3150.2.a.bj.1.1 1 105.83 odd 4
3150.2.g.c.2899.1 2 105.104 even 2
3150.2.g.c.2899.2 2 21.20 even 2
3920.2.a.t.1.1 1 20.7 even 4
4410.2.a.b.1.1 1 15.2 even 4
5040.2.a.bm.1.1 1 420.167 even 4
8470.2.a.j.1.1 1 385.307 odd 4