Properties

 Label 4400.2.b.b.4049.1 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 4049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.b.4049.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +8.00000 q^{29} +7.00000 q^{31} -3.00000i q^{33} -1.00000i q^{37} +4.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -12.0000i q^{57} -1.00000 q^{59} +4.00000 q^{61} -12.0000i q^{63} +5.00000i q^{67} +3.00000 q^{69} -3.00000 q^{71} -16.0000i q^{73} +2.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} -24.0000i q^{87} -15.0000 q^{89} -21.0000i q^{93} -7.00000i q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 2 * q^11 + 8 * q^19 + 12 * q^21 + 16 * q^29 + 14 * q^31 + 8 * q^41 + 6 * q^49 - 36 * q^51 - 2 * q^59 + 8 * q^61 + 6 * q^69 - 6 * q^71 + 4 * q^79 + 18 * q^81 - 30 * q^89 - 12 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 3.00000i − 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ 1.00000i 0.208514i 0.994550 + 0.104257i $$0.0332465\pi$$
−0.994550 + 0.104257i $$0.966753\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 9.00000i 1.73205i
$$28$$ 0 0
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 0 0
$$45$$ 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
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −18.0000 −2.52050
$$52$$ 0 0
$$53$$ − 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
$$57$$ − 12.0000i − 1.58944i
$$58$$ 0 0
$$59$$ −1.00000 −0.130189 −0.0650945 0.997879i $$-0.520735\pi$$
−0.0650945 + 0.997879i $$0.520735\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ − 12.0000i − 1.51186i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000i 0.610847i 0.952217 + 0.305424i $$0.0987981\pi$$
−0.952217 + 0.305424i $$0.901202\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ − 16.0000i − 1.87266i −0.351123 0.936329i $$-0.614200\pi$$
0.351123 0.936329i $$-0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000i 0.227921i
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 2.00000i − 0.219529i −0.993958 0.109764i $$-0.964990\pi$$
0.993958 0.109764i $$-0.0350096\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 24.0000i − 2.57307i
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 21.0000i − 2.17760i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.00000i − 0.710742i −0.934725 0.355371i $$-0.884354\pi$$
0.934725 0.355371i $$-0.115646\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 2.00000i − 0.193347i −0.995316 0.0966736i $$-0.969180\pi$$
0.995316 0.0966736i $$-0.0308203\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ 7.00000i 0.658505i 0.944242 + 0.329252i $$0.106797\pi$$
−0.944242 + 0.329252i $$0.893203\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 0 0
$$129$$ 18.0000 1.58481
$$130$$ 0 0
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 0 0
$$133$$ 8.00000i 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 15.0000i − 1.28154i −0.767734 0.640768i $$-0.778616\pi$$
0.767734 0.640768i $$-0.221384\pi$$
$$138$$ 0 0
$$139$$ −22.0000 −1.86602 −0.933008 0.359856i $$-0.882826\pi$$
−0.933008 + 0.359856i $$0.882826\pi$$
$$140$$ 0 0
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 9.00000i − 0.742307i
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ 36.0000i 2.91043i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 11.0000i − 0.877896i −0.898513 0.438948i $$-0.855351\pi$$
0.898513 0.438948i $$-0.144649\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −24.0000 −1.83533
$$172$$ 0 0
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.00000i 0.225494i
$$178$$ 0 0
$$179$$ −5.00000 −0.373718 −0.186859 0.982387i $$-0.559831\pi$$
−0.186859 + 0.982387i $$0.559831\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 0 0
$$183$$ − 12.0000i − 0.887066i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.00000i − 0.438763i
$$188$$ 0 0
$$189$$ −18.0000 −1.30931
$$190$$ 0 0
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ 0 0
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 15.0000 1.05802
$$202$$ 0 0
$$203$$ 16.0000i 1.12298i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 9.00000i 0.616670i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 14.0000i 0.950382i
$$218$$ 0 0
$$219$$ −48.0000 −3.24354
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 29.0000i 1.94198i 0.239113 + 0.970992i $$0.423143\pi$$
−0.239113 + 0.970992i $$0.576857\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 18.0000i − 1.19470i −0.801980 0.597351i $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ 0 0
$$229$$ 21.0000 1.38772 0.693860 0.720110i $$-0.255909\pi$$
0.693860 + 0.720110i $$0.255909\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 16.0000i 1.04819i 0.851658 + 0.524097i $$0.175597\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 6.00000i − 0.389742i
$$238$$ 0 0
$$239$$ −2.00000 −0.129369 −0.0646846 0.997906i $$-0.520604\pi$$
−0.0646846 + 0.997906i $$0.520604\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 13.0000 0.820553 0.410276 0.911961i $$-0.365432\pi$$
0.410276 + 0.911961i $$0.365432\pi$$
$$252$$ 0 0
$$253$$ 1.00000i 0.0628695i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 10.0000i − 0.623783i −0.950118 0.311891i $$-0.899037\pi$$
0.950118 0.311891i $$-0.100963\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −48.0000 −2.97113
$$262$$ 0 0
$$263$$ 14.0000i 0.863277i 0.902047 + 0.431638i $$0.142064\pi$$
−0.902047 + 0.431638i $$0.857936\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 45.0000i 2.75396i
$$268$$ 0 0
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ −42.0000 −2.51447
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.00000i 0.472225i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −21.0000 −1.23104
$$292$$ 0 0
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 9.00000i 0.522233i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ 30.0000i 1.72345i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ −48.0000 −2.73062
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 9.00000i 0.508710i 0.967111 + 0.254355i $$0.0818632\pi$$
−0.967111 + 0.254355i $$0.918137\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 15.0000i − 0.842484i −0.906948 0.421242i $$-0.861594\pi$$
0.906948 0.421242i $$-0.138406\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 42.0000i − 2.32261i
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 35.0000 1.92377 0.961887 0.273447i $$-0.0881639\pi$$
0.961887 + 0.273447i $$0.0881639\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 10.0000i − 0.544735i −0.962193 0.272367i $$-0.912193\pi$$
0.962193 0.272367i $$-0.0878066\pi$$
$$338$$ 0 0
$$339$$ 21.0000 1.14056
$$340$$ 0 0
$$341$$ 7.00000 0.379071
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 3.00000i − 0.159674i −0.996808 0.0798369i $$-0.974560\pi$$
0.996808 0.0798369i $$-0.0254400\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 36.0000i − 1.90532i
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ − 3.00000i − 0.157459i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 33.0000i − 1.72259i −0.508109 0.861293i $$-0.669655\pi$$
0.508109 0.861293i $$-0.330345\pi$$
$$368$$ 0 0
$$369$$ −24.0000 −1.24939
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 1.00000i 0.0510976i 0.999674 + 0.0255488i $$0.00813332\pi$$
−0.999674 + 0.0255488i $$0.991867\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 36.0000i − 1.82998i
$$388$$ 0 0
$$389$$ −13.0000 −0.659126 −0.329563 0.944134i $$-0.606901\pi$$
−0.329563 + 0.944134i $$0.606901\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 0 0
$$393$$ − 6.00000i − 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 26.0000i − 1.30490i −0.757831 0.652451i $$-0.773741\pi$$
0.757831 0.652451i $$-0.226259\pi$$
$$398$$ 0 0
$$399$$ 24.0000 1.20150
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1.00000i − 0.0495682i
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ −45.0000 −2.21969
$$412$$ 0 0
$$413$$ − 2.00000i − 0.0984136i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 66.0000i 3.23203i
$$418$$ 0 0
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 0 0
$$423$$ − 48.0000i − 2.33384i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 26.0000 1.25238 0.626188 0.779672i $$-0.284614\pi$$
0.626188 + 0.779672i $$0.284614\pi$$
$$432$$ 0 0
$$433$$ − 13.0000i − 0.624740i −0.949960 0.312370i $$-0.898877\pi$$
0.949960 0.312370i $$-0.101123\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.00000i 0.191346i
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ − 9.00000i − 0.427603i −0.976877 0.213801i $$-0.931415\pi$$
0.976877 0.213801i $$-0.0685846\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 54.0000i 2.55411i
$$448$$ 0 0
$$449$$ 21.0000 0.991051 0.495526 0.868593i $$-0.334975\pi$$
0.495526 + 0.868593i $$0.334975\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 0 0
$$453$$ − 54.0000i − 2.53714i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.0000i 0.561336i 0.959805 + 0.280668i $$0.0905560\pi$$
−0.959805 + 0.280668i $$0.909444\pi$$
$$458$$ 0 0
$$459$$ 54.0000 2.52050
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ 27.0000i 1.25480i 0.778699 + 0.627398i $$0.215880\pi$$
−0.778699 + 0.627398i $$0.784120\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 33.0000i 1.52706i 0.645774 + 0.763529i $$0.276535\pi$$
−0.645774 + 0.763529i $$0.723465\pi$$
$$468$$ 0 0
$$469$$ −10.0000 −0.461757
$$470$$ 0 0
$$471$$ −33.0000 −1.52056
$$472$$ 0 0
$$473$$ 6.00000i 0.275880i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 9.00000i − 0.407829i −0.978989 0.203914i $$-0.934634\pi$$
0.978989 0.203914i $$-0.0653664\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 0 0
$$493$$ − 48.0000i − 2.16181i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ −48.0000 −2.14448
$$502$$ 0 0
$$503$$ − 30.0000i − 1.33763i −0.743427 0.668817i $$-0.766801\pi$$
0.743427 0.668817i $$-0.233199\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 39.0000i − 1.73205i
$$508$$ 0 0
$$509$$ 13.0000 0.576215 0.288107 0.957598i $$-0.406974\pi$$
0.288107 + 0.957598i $$0.406974\pi$$
$$510$$ 0 0
$$511$$ 32.0000 1.41560
$$512$$ 0 0
$$513$$ 36.0000i 1.58944i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.00000i 0.351840i
$$518$$ 0 0
$$519$$ −54.0000 −2.37034
$$520$$ 0 0
$$521$$ 37.0000 1.62100 0.810500 0.585739i $$-0.199196\pi$$
0.810500 + 0.585739i $$0.199196\pi$$
$$522$$ 0 0
$$523$$ 44.0000i 1.92399i 0.273075 + 0.961993i $$0.411959\pi$$
−0.273075 + 0.961993i $$0.588041\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 42.0000i − 1.82955i
$$528$$ 0 0
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 15.0000i 0.647298i
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ 0 0
$$543$$ 15.0000i 0.643712i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ −24.0000 −1.02430
$$550$$ 0 0
$$551$$ 32.0000 1.36325
$$552$$ 0 0
$$553$$ 4.00000i 0.170097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 42.0000i 1.77960i 0.456354 + 0.889799i $$0.349155\pi$$
−0.456354 + 0.889799i $$0.650845\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 18.0000i 0.755929i
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ − 27.0000i − 1.12794i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 23.0000i − 0.957503i −0.877951 0.478751i $$-0.841090\pi$$
0.877951 0.478751i $$-0.158910\pi$$
$$578$$ 0 0
$$579$$ −12.0000 −0.498703
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ − 2.00000i − 0.0828315i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ 28.0000 1.15372
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ 4.00000i 0.164260i 0.996622 + 0.0821302i $$0.0261723\pi$$
−0.996622 + 0.0821302i $$0.973828\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 24.0000i − 0.982255i
$$598$$ 0 0
$$599$$ 48.0000 1.96123 0.980613 0.195952i $$-0.0627798\pi$$
0.980613 + 0.195952i $$0.0627798\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ − 30.0000i − 1.22169i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18.0000i 0.730597i 0.930890 + 0.365299i $$0.119033\pi$$
−0.930890 + 0.365299i $$0.880967\pi$$
$$608$$ 0 0
$$609$$ 48.0000 1.94506
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −3.00000 −0.120580 −0.0602901 0.998181i $$-0.519203\pi$$
−0.0602901 + 0.998181i $$0.519203\pi$$
$$620$$ 0 0
$$621$$ −9.00000 −0.361158
$$622$$ 0 0
$$623$$ − 30.0000i − 1.20192i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 12.0000i − 0.479234i
$$628$$ 0 0
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −9.00000 −0.358284 −0.179142 0.983823i $$-0.557332\pi$$
−0.179142 + 0.983823i $$0.557332\pi$$
$$632$$ 0 0
$$633$$ 60.0000i 2.38479i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 18.0000 0.712069
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ 7.00000i 0.276053i 0.990429 + 0.138027i $$0.0440759\pi$$
−0.990429 + 0.138027i $$0.955924\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 15.0000i − 0.589711i −0.955542 0.294855i $$-0.904729\pi$$
0.955542 0.294855i $$-0.0952715\pi$$
$$648$$ 0 0
$$649$$ −1.00000 −0.0392534
$$650$$ 0 0
$$651$$ 42.0000 1.64611
$$652$$ 0 0
$$653$$ − 11.0000i − 0.430463i −0.976563 0.215232i $$-0.930949\pi$$
0.976563 0.215232i $$-0.0690506\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 96.0000i 3.74532i
$$658$$ 0 0
$$659$$ 22.0000 0.856998 0.428499 0.903542i $$-0.359042\pi$$
0.428499 + 0.903542i $$0.359042\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000i 0.309761i
$$668$$ 0 0
$$669$$ 87.0000 3.36361
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 30.0000i − 1.15299i −0.817099 0.576497i $$-0.804419\pi$$
0.817099 0.576497i $$-0.195581\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ −54.0000 −2.06928
$$682$$ 0 0
$$683$$ 8.00000i 0.306111i 0.988218 + 0.153056i $$0.0489114\pi$$
−0.988218 + 0.153056i $$0.951089\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 63.0000i − 2.40360i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 29.0000 1.10321 0.551606 0.834105i $$-0.314015\pi$$
0.551606 + 0.834105i $$0.314015\pi$$
$$692$$ 0 0
$$693$$ − 12.0000i − 0.455842i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 24.0000i − 0.909065i
$$698$$ 0 0
$$699$$ 48.0000 1.81553
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 20.0000i − 0.752177i
$$708$$ 0 0
$$709$$ −19.0000 −0.713560 −0.356780 0.934188i $$-0.616125\pi$$
−0.356780 + 0.934188i $$0.616125\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 0 0
$$713$$ 7.00000i 0.262152i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000i 0.224074i
$$718$$ 0 0
$$719$$ −23.0000 −0.857755 −0.428878 0.903363i $$-0.641091\pi$$
−0.428878 + 0.903363i $$0.641091\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 35.0000i 1.29808i 0.760755 + 0.649039i $$0.224829\pi$$
−0.760755 + 0.649039i $$0.775171\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 36.0000 1.33151
$$732$$ 0 0
$$733$$ − 4.00000i − 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 5.00000i 0.184177i
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 0 0
$$753$$ − 39.0000i − 1.42124i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 28.0000i 1.01367i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 0 0
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 6.00000i − 0.215249i
$$778$$ 0 0
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ −3.00000 −0.107348
$$782$$ 0 0
$$783$$ 72.0000i 2.57307i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ 0 0
$$789$$ 42.0000 1.49524
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.00000i 0.0354218i 0.999843 + 0.0177109i $$0.00563785\pi$$
−0.999843 + 0.0177109i $$0.994362\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ 90.0000 3.17999
$$802$$ 0 0
$$803$$ − 16.0000i − 0.564628i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 78.0000i 2.74573i
$$808$$ 0 0
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ 26.0000 0.912983 0.456492 0.889728i $$-0.349106\pi$$
0.456492 + 0.889728i $$0.349106\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.0000i 0.839654i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ − 31.0000i − 1.08059i −0.841475 0.540296i $$-0.818312\pi$$
0.841475 0.540296i $$-0.181688\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 16.0000i 0.556375i 0.960527 + 0.278187i $$0.0897336\pi$$
−0.960527 + 0.278187i $$0.910266\pi$$
$$828$$ 0 0
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ 0 0
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 63.0000i 2.17760i
$$838$$ 0 0
$$839$$ 21.0000 0.725001 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ − 18.0000i − 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000i 0.0687208i
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ 1.00000 0.0342796
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 24.0000i − 0.819824i −0.912125 0.409912i $$-0.865559\pi$$
0.912125 0.409912i $$-0.134441\pi$$
$$858$$ 0 0
$$859$$ 11.0000 0.375315 0.187658 0.982235i $$-0.439910\pi$$
0.187658 + 0.982235i $$0.439910\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 57.0000i 1.93582i
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 42.0000i 1.42148i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 56.0000i 1.89099i 0.325643 + 0.945493i $$0.394419\pi$$
−0.325643 + 0.945493i $$0.605581\pi$$
$$878$$ 0 0
$$879$$ −36.0000 −1.21425
$$880$$ 0 0
$$881$$ −19.0000 −0.640126 −0.320063 0.947396i $$-0.603704\pi$$
−0.320063 + 0.947396i $$0.603704\pi$$
$$882$$ 0 0
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 34.0000i 1.14161i 0.821086 + 0.570804i $$0.193368\pi$$
−0.821086 + 0.570804i $$0.806632\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 9.00000 0.301511
$$892$$ 0 0
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 56.0000 1.86770
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 36.0000i 1.19800i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 0 0
$$909$$ 60.0000 1.99007
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ − 2.00000i − 0.0661903i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4.00000i 0.132092i
$$918$$ 0 0
$$919$$ −50.0000 −1.64935 −0.824674 0.565608i $$-0.808641\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 96.0000i 3.15305i
$$928$$ 0 0
$$929$$ 22.0000 0.721797 0.360898 0.932605i $$-0.382470\pi$$
0.360898 + 0.932605i $$0.382470\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 0 0
$$933$$ − 36.0000i − 1.17859i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.00000i 0.130674i 0.997863 + 0.0653372i $$0.0208123\pi$$
−0.997863 + 0.0653372i $$0.979188\pi$$
$$938$$ 0 0
$$939$$ 27.0000 0.881112
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ 4.00000i 0.130258i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 47.0000i − 1.52729i −0.645634 0.763647i $$-0.723407\pi$$
0.645634 0.763647i $$-0.276593\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −45.0000 −1.45922
$$952$$ 0 0
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 24.0000i − 0.775810i
$$958$$ 0 0
$$959$$ 30.0000 0.968751
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ 0 0
$$969$$ −72.0000 −2.31297
$$970$$ 0 0
$$971$$ −53.0000 −1.70085 −0.850425 0.526096i $$-0.823655\pi$$
−0.850425 + 0.526096i $$0.823655\pi$$
$$972$$ 0 0
$$973$$ − 44.0000i − 1.41058i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 11.0000i − 0.351921i −0.984397 0.175961i $$-0.943697\pi$$
0.984397 0.175961i $$-0.0563031\pi$$
$$978$$ 0 0
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ −84.0000 −2.68191
$$982$$ 0 0
$$983$$ 25.0000i 0.797376i 0.917087 + 0.398688i $$0.130534\pi$$
−0.917087 + 0.398688i $$0.869466\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 48.0000i 1.52786i
$$988$$ 0 0
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ − 105.000i − 3.33207i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.00000i 0.0633406i 0.999498 + 0.0316703i $$0.0100827\pi$$
−0.999498 + 0.0316703i $$0.989917\pi$$
$$998$$ 0 0
$$999$$ 9.00000 0.284747
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 4400.2.b.b.4049.1 2
4.3 odd 2 2200.2.b.a.1849.2 2
5.2 odd 4 4400.2.a.a.1.1 1
5.3 odd 4 176.2.a.c.1.1 1
5.4 even 2 inner 4400.2.b.b.4049.2 2
15.8 even 4 1584.2.a.q.1.1 1
20.3 even 4 88.2.a.a.1.1 1
20.7 even 4 2200.2.a.k.1.1 1
20.19 odd 2 2200.2.b.a.1849.1 2
35.13 even 4 8624.2.a.c.1.1 1
40.3 even 4 704.2.a.l.1.1 1
40.13 odd 4 704.2.a.b.1.1 1
55.43 even 4 1936.2.a.l.1.1 1
60.23 odd 4 792.2.a.g.1.1 1
80.3 even 4 2816.2.c.i.1409.1 2
80.13 odd 4 2816.2.c.d.1409.2 2
80.43 even 4 2816.2.c.i.1409.2 2
80.53 odd 4 2816.2.c.d.1409.1 2
120.53 even 4 6336.2.a.k.1.1 1
120.83 odd 4 6336.2.a.h.1.1 1
140.83 odd 4 4312.2.a.l.1.1 1
220.3 even 20 968.2.i.j.9.1 4
220.43 odd 4 968.2.a.a.1.1 1
220.63 odd 20 968.2.i.i.9.1 4
220.83 odd 20 968.2.i.i.729.1 4
220.103 even 20 968.2.i.j.753.1 4
220.123 odd 20 968.2.i.i.81.1 4
220.163 even 20 968.2.i.j.81.1 4
220.183 odd 20 968.2.i.i.753.1 4
220.203 even 20 968.2.i.j.729.1 4
440.43 odd 4 7744.2.a.bk.1.1 1
440.373 even 4 7744.2.a.b.1.1 1
660.263 even 4 8712.2.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.a.1.1 1 20.3 even 4
176.2.a.c.1.1 1 5.3 odd 4
704.2.a.b.1.1 1 40.13 odd 4
704.2.a.l.1.1 1 40.3 even 4
792.2.a.g.1.1 1 60.23 odd 4
968.2.a.a.1.1 1 220.43 odd 4
968.2.i.i.9.1 4 220.63 odd 20
968.2.i.i.81.1 4 220.123 odd 20
968.2.i.i.729.1 4 220.83 odd 20
968.2.i.i.753.1 4 220.183 odd 20
968.2.i.j.9.1 4 220.3 even 20
968.2.i.j.81.1 4 220.163 even 20
968.2.i.j.729.1 4 220.203 even 20
968.2.i.j.753.1 4 220.103 even 20
1584.2.a.q.1.1 1 15.8 even 4
1936.2.a.l.1.1 1 55.43 even 4
2200.2.a.k.1.1 1 20.7 even 4
2200.2.b.a.1849.1 2 20.19 odd 2
2200.2.b.a.1849.2 2 4.3 odd 2
2816.2.c.d.1409.1 2 80.53 odd 4
2816.2.c.d.1409.2 2 80.13 odd 4
2816.2.c.i.1409.1 2 80.3 even 4
2816.2.c.i.1409.2 2 80.43 even 4
4312.2.a.l.1.1 1 140.83 odd 4
4400.2.a.a.1.1 1 5.2 odd 4
4400.2.b.b.4049.1 2 1.1 even 1 trivial
4400.2.b.b.4049.2 2 5.4 even 2 inner
6336.2.a.h.1.1 1 120.83 odd 4
6336.2.a.k.1.1 1 120.53 even 4
7744.2.a.b.1.1 1 440.373 even 4
7744.2.a.bk.1.1 1 440.43 odd 4
8624.2.a.c.1.1 1 35.13 even 4
8712.2.a.x.1.1 1 660.263 even 4