# Properties

 Label 2400.2.f.a.1249.1 Level 2400 Weight 2 Character 2400.1249 Analytic conductor 19.164 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1640964851$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.1 Root $$-1.00000i$$ of $$x^{2} + 1$$ Character $$\chi$$ $$=$$ 2400.1249 Dual form 2400.2.f.a.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -2.00000i q^{13} +6.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} +1.00000i q^{27} -2.00000 q^{29} -4.00000 q^{31} +4.00000i q^{33} +2.00000i q^{37} -2.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +8.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} +10.0000i q^{53} +4.00000i q^{57} -4.00000 q^{59} +6.00000 q^{61} +4.00000i q^{63} +4.00000i q^{67} +16.0000 q^{71} -6.00000i q^{73} +16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +2.00000i q^{87} -10.0000 q^{89} -8.00000 q^{91} +4.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 8q^{11} - 8q^{19} - 8q^{21} - 4q^{29} - 8q^{31} - 4q^{39} + 4q^{41} - 18q^{49} + 12q^{51} - 8q^{59} + 12q^{61} + 32q^{71} + 8q^{79} + 2q^{81} - 20q^{89} - 16q^{91} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$1951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 4.00000i 0.696311i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 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
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 4.00000i 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 0 0
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 16.0000i 1.82337i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 12.0000i 1.18240i 0.806527 + 0.591198i $$0.201345\pi$$
−0.806527 + 0.591198i $$0.798655\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 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
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 20.0000i − 1.77471i −0.461084 0.887357i $$-0.652539\pi$$
0.461084 0.887357i $$-0.347461\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ 8.00000i 0.668994i
$$144$$ 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
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 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 $$0.869342\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.00000i 0.300658i
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 6.00000i − 0.443533i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 18.0000i 1.29567i 0.761781 + 0.647834i $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 8.00000i 0.561490i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ − 16.0000i − 1.09630i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 8.00000i 0.484182i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 4.00000i − 0.232104i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 24.0000i − 1.33540i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 14.0000i 0.774202i
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2.00000i − 0.108947i −0.998515 0.0544735i $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 20.0000i 1.07366i 0.843692 + 0.536828i $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 24.0000i − 1.27021i
$$358$$ 0 0
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 12.0000i − 0.626395i −0.949688 0.313197i $$-0.898600\pi$$
0.949688 0.313197i $$-0.101400\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 0 0
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −20.0000 −1.02463
$$382$$ 0 0
$$383$$ − 32.0000i − 1.63512i −0.575841 0.817562i $$-0.695325\pi$$
0.575841 0.817562i $$-0.304675\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 4.00000i 0.201773i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 0 0
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ 16.0000i 0.787309i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 0 0
$$423$$ − 8.00000i − 0.388973i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 24.0000i − 1.16144i
$$428$$ 0 0
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 18.0000i 0.851371i
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 0 0
$$453$$ 12.0000i 0.563809i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 10.0000i − 0.457869i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 4.00000i − 0.181257i −0.995885 0.0906287i $$-0.971112\pi$$
0.995885 0.0906287i $$-0.0288876\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 64.0000i − 2.87079i
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 0 0
$$513$$ − 4.00000i − 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 24.0000i − 1.04546i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.0000i 0.517838i
$$538$$ 0 0
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ 10.0000i 0.429141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ − 16.0000i − 0.680389i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.0000i 0.932170i 0.884740 + 0.466085i $$0.154336\pi$$
−0.884740 + 0.466085i $$0.845664\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ 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
$$567$$ − 4.00000i − 0.167984i
$$568$$ 0 0
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 18.0000i − 0.749350i −0.927156 0.374675i $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ 0 0
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ −48.0000 −1.99138
$$582$$ 0 0
$$583$$ − 40.0000i − 1.65663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 0 0
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 4.00000i − 0.163709i
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 20.0000i − 0.811775i −0.913923 0.405887i $$-0.866962\pi$$
0.913923 0.405887i $$-0.133038\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ − 18.0000i − 0.727013i −0.931592 0.363507i $$-0.881579\pi$$
0.931592 0.363507i $$-0.118421\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 0 0
$$619$$ 12.0000 0.482321 0.241160 0.970485i $$-0.422472\pi$$
0.241160 + 0.970485i $$0.422472\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 40.0000i 1.60257i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 0 0
$$633$$ 20.0000i 0.794929i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 0 0
$$643$$ − 12.0000i − 0.473234i −0.971603 0.236617i $$-0.923961\pi$$
0.971603 0.236617i $$-0.0760386\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 48.0000i 1.88707i 0.331266 + 0.943537i $$0.392524\pi$$
−0.331266 + 0.943537i $$0.607476\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 0 0
$$653$$ 10.0000i 0.391330i 0.980671 + 0.195665i $$0.0626866\pi$$
−0.980671 + 0.195665i $$0.937313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 0 0
$$663$$ − 12.0000i − 0.466041i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 14.0000i 0.538064i 0.963131 + 0.269032i $$0.0867037\pi$$
−0.963131 + 0.269032i $$0.913296\pi$$
$$678$$ 0 0
$$679$$ 56.0000 2.14908
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 10.0000i − 0.381524i
$$688$$ 0 0
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 52.0000 1.97817 0.989087 0.147335i $$-0.0470696\pi$$
0.989087 + 0.147335i $$0.0470696\pi$$
$$692$$ 0 0
$$693$$ − 16.0000i − 0.607790i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000i 0.454532i
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 0 0
$$723$$ 14.0000i 0.520666i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.0000i 0.445055i 0.974926 + 0.222528i $$0.0714308\pi$$
−0.974926 + 0.222528i $$0.928569\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 16.0000i − 0.589368i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 6.00000i − 0.218074i −0.994038 0.109037i $$-0.965223\pi$$
0.994038 0.109037i $$-0.0347767\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 56.0000i 2.02734i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 0 0
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 8.00000i − 0.286998i
$$778$$ 0 0
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ 0 0
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 12.0000i 0.427754i 0.976861 + 0.213877i $$0.0686091\pi$$
−0.976861 + 0.213877i $$0.931391\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 0 0
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 24.0000i 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000i 0.633630i
$$808$$ 0 0
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 12.0000i 0.420858i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ 0 0
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 26.0000 0.907406 0.453703 0.891153i $$-0.350103\pi$$
0.453703 + 0.891153i $$0.350103\pi$$
$$822$$ 0 0
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 20.0000i − 0.695468i −0.937593 0.347734i $$-0.886951\pi$$
0.937593 0.347734i $$-0.113049\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 0 0
$$833$$ − 54.0000i − 1.87099i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 4.00000i − 0.138260i
$$838$$ 0 0
$$839$$ 32.0000 1.10476 0.552381 0.833592i $$-0.313719\pi$$
0.552381 + 0.833592i $$0.313719\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 6.00000i 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 20.0000i − 0.687208i
$$848$$ 0 0
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 18.0000i − 0.616308i −0.951336 0.308154i $$-0.900289\pi$$
0.951336 0.308154i $$-0.0997113\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 14.0000i 0.478231i 0.970991 + 0.239115i $$0.0768574\pi$$
−0.970991 + 0.239115i $$0.923143\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 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
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ 0 0
$$889$$ −80.0000 −2.68311
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ 0 0
$$903$$ 16.0000i 0.532447i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000i 0.664089i 0.943264 + 0.332045i $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 48.0000i 1.58857i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.0000i 0.528367i
$$918$$ 0 0
$$919$$ −36.0000 −1.18753 −0.593765 0.804638i $$-0.702359\pi$$
−0.593765 + 0.804638i $$0.702359\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ − 32.0000i − 1.05329i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 12.0000i − 0.394132i
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 36.0000 1.17985
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 58.0000 1.89075 0.945373 0.325991i $$-0.105698\pi$$
0.945373 + 0.325991i $$0.105698\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ 0 0
$$959$$ −72.0000 −2.32500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 12.0000i − 0.385894i −0.981209 0.192947i $$-0.938195\pi$$
0.981209 0.192947i $$-0.0618045\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 0 0
$$973$$ 80.0000i 2.56468i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 22.0000i 0.703842i 0.936030 + 0.351921i $$0.114471\pi$$
−0.936030 + 0.351921i $$0.885529\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 0 0
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 32.0000i − 1.01857i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 0 0
$$999$$ −2.00000 −0.0632772
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 2400.2.f.a.1249.1 2
3.2 odd 2 7200.2.f.x.6049.1 2
4.3 odd 2 2400.2.f.r.1249.2 2
5.2 odd 4 96.2.a.a.1.1 1
5.3 odd 4 2400.2.a.r.1.1 1
5.4 even 2 inner 2400.2.f.a.1249.2 2
8.3 odd 2 4800.2.f.e.3649.1 2
8.5 even 2 4800.2.f.bh.3649.2 2
12.11 even 2 7200.2.f.f.6049.2 2
15.2 even 4 288.2.a.c.1.1 1
15.8 even 4 7200.2.a.e.1.1 1
15.14 odd 2 7200.2.f.x.6049.2 2
20.3 even 4 2400.2.a.q.1.1 1
20.7 even 4 96.2.a.b.1.1 yes 1
20.19 odd 2 2400.2.f.r.1249.1 2
35.27 even 4 4704.2.a.t.1.1 1
40.3 even 4 4800.2.a.co.1.1 1
40.13 odd 4 4800.2.a.f.1.1 1
40.19 odd 2 4800.2.f.e.3649.2 2
40.27 even 4 192.2.a.a.1.1 1
40.29 even 2 4800.2.f.bh.3649.1 2
40.37 odd 4 192.2.a.c.1.1 1
45.2 even 12 2592.2.i.q.1729.1 2
45.7 odd 12 2592.2.i.b.1729.1 2
45.22 odd 12 2592.2.i.b.865.1 2
45.32 even 12 2592.2.i.q.865.1 2
60.23 odd 4 7200.2.a.bx.1.1 1
60.47 odd 4 288.2.a.b.1.1 1
60.59 even 2 7200.2.f.f.6049.1 2
80.27 even 4 768.2.d.h.385.1 2
80.37 odd 4 768.2.d.a.385.2 2
80.67 even 4 768.2.d.h.385.2 2
80.77 odd 4 768.2.d.a.385.1 2
120.77 even 4 576.2.a.h.1.1 1
120.107 odd 4 576.2.a.g.1.1 1
140.27 odd 4 4704.2.a.e.1.1 1
180.7 even 12 2592.2.i.h.1729.1 2
180.47 odd 12 2592.2.i.w.1729.1 2
180.67 even 12 2592.2.i.h.865.1 2
180.167 odd 12 2592.2.i.w.865.1 2
240.77 even 4 2304.2.d.c.1153.2 2
240.107 odd 4 2304.2.d.s.1153.1 2
240.197 even 4 2304.2.d.c.1153.1 2
240.227 odd 4 2304.2.d.s.1153.2 2
280.27 odd 4 9408.2.a.ct.1.1 1
280.237 even 4 9408.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.a.a.1.1 1 5.2 odd 4
96.2.a.b.1.1 yes 1 20.7 even 4
192.2.a.a.1.1 1 40.27 even 4
192.2.a.c.1.1 1 40.37 odd 4
288.2.a.b.1.1 1 60.47 odd 4
288.2.a.c.1.1 1 15.2 even 4
576.2.a.g.1.1 1 120.107 odd 4
576.2.a.h.1.1 1 120.77 even 4
768.2.d.a.385.1 2 80.77 odd 4
768.2.d.a.385.2 2 80.37 odd 4
768.2.d.h.385.1 2 80.27 even 4
768.2.d.h.385.2 2 80.67 even 4
2304.2.d.c.1153.1 2 240.197 even 4
2304.2.d.c.1153.2 2 240.77 even 4
2304.2.d.s.1153.1 2 240.107 odd 4
2304.2.d.s.1153.2 2 240.227 odd 4
2400.2.a.q.1.1 1 20.3 even 4
2400.2.a.r.1.1 1 5.3 odd 4
2400.2.f.a.1249.1 2 1.1 even 1 trivial
2400.2.f.a.1249.2 2 5.4 even 2 inner
2400.2.f.r.1249.1 2 20.19 odd 2
2400.2.f.r.1249.2 2 4.3 odd 2
2592.2.i.b.865.1 2 45.22 odd 12
2592.2.i.b.1729.1 2 45.7 odd 12
2592.2.i.h.865.1 2 180.67 even 12
2592.2.i.h.1729.1 2 180.7 even 12
2592.2.i.q.865.1 2 45.32 even 12
2592.2.i.q.1729.1 2 45.2 even 12
2592.2.i.w.865.1 2 180.167 odd 12
2592.2.i.w.1729.1 2 180.47 odd 12
4704.2.a.e.1.1 1 140.27 odd 4
4704.2.a.t.1.1 1 35.27 even 4
4800.2.a.f.1.1 1 40.13 odd 4
4800.2.a.co.1.1 1 40.3 even 4
4800.2.f.e.3649.1 2 8.3 odd 2
4800.2.f.e.3649.2 2 40.19 odd 2
4800.2.f.bh.3649.1 2 40.29 even 2
4800.2.f.bh.3649.2 2 8.5 even 2
7200.2.a.e.1.1 1 15.8 even 4
7200.2.a.bx.1.1 1 60.23 odd 4
7200.2.f.f.6049.1 2 60.59 even 2
7200.2.f.f.6049.2 2 12.11 even 2
7200.2.f.x.6049.1 2 3.2 odd 2
7200.2.f.x.6049.2 2 15.14 odd 2
9408.2.a.bj.1.1 1 280.237 even 4
9408.2.a.ct.1.1 1 280.27 odd 4