# Properties

 Label 2925.2.c.e.2224.2 Level $2925$ Weight $2$ Character 2925.2224 Analytic conductor $23.356$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2224.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2925.2224 Dual form 2925.2.c.e.2224.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} +3.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} +3.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} -4.00000i q^{22} -1.00000 q^{26} +4.00000i q^{28} -10.0000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +2.00000i q^{37} -6.00000 q^{41} -12.0000i q^{43} -4.00000 q^{44} -9.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} -12.0000 q^{56} -10.0000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -7.00000 q^{64} +8.00000i q^{67} +2.00000i q^{68} +2.00000i q^{73} -2.00000 q^{74} -16.0000i q^{77} -8.00000 q^{79} -6.00000i q^{82} -4.00000i q^{83} +12.0000 q^{86} -12.0000i q^{88} -2.00000 q^{89} -4.00000 q^{91} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} - 8 q^{11} - 8 q^{14} - 2 q^{16} - 2 q^{26} - 20 q^{29} + 8 q^{31} - 4 q^{34} - 12 q^{41} - 8 q^{44} - 18 q^{49} - 24 q^{56} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 4 q^{74} - 16 q^{79} + 24 q^{86} - 4 q^{89} - 8 q^{91}+O(q^{100})$$ 2 * q + 2 * q^4 - 8 * q^11 - 8 * q^14 - 2 * q^16 - 2 * q^26 - 20 * q^29 + 8 * q^31 - 4 * q^34 - 12 * q^41 - 8 * q^44 - 18 * q^49 - 24 * q^56 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 4 * q^74 - 16 * q^79 + 24 * q^86 - 4 * q^89 - 8 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ 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
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 4.00000i 0.755929i
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 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
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −12.0000 −1.60357
$$57$$ 0 0
$$58$$ − 10.0000i − 1.31306i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 16.0000i − 1.82337i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 0 0
$$88$$ − 12.0000i − 1.27920i
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ 0 0
$$118$$ 12.0000i 1.10469i
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 12.0000i − 0.914991i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ − 2.00000i − 0.149906i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ 18.0000i 1.29567i 0.761781 + 0.647834i $$0.224325\pi$$
−0.761781 + 0.647834i $$0.775675\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 18.0000i 1.26648i
$$203$$ − 40.0000i − 2.80745i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 2.00000i 0.135457i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 30.0000i − 1.96960i
$$233$$ 14.0000i 0.917170i 0.888650 + 0.458585i $$0.151644\pi$$
−0.888650 + 0.458585i $$0.848356\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ − 8.00000i − 0.518563i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 12.0000i 0.762001i
$$249$$ 0 0
$$250$$ 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
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 26.0000i 1.62184i 0.585160 + 0.810918i $$0.301032\pi$$
−0.585160 + 0.810918i $$0.698968\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 4.00000i − 0.247121i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 8.00000i 0.488678i
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 12.0000i 0.713326i 0.934233 + 0.356663i $$0.116086\pi$$
−0.934233 + 0.356663i $$0.883914\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ − 24.0000i − 1.41668i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000i 0.117041i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ − 6.00000i − 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 48.0000 2.76667
$$302$$ 4.00000i 0.230174i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ − 16.0000i − 0.911685i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 26.0000i 1.46031i 0.683284 + 0.730153i $$0.260551\pi$$
−0.683284 + 0.730153i $$0.739449\pi$$
$$318$$ 0 0
$$319$$ 40.0000 2.23957
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ − 18.0000i − 0.993884i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 18.0000i − 0.980522i −0.871576 0.490261i $$-0.836901\pi$$
0.871576 0.490261i $$-0.163099\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 36.0000 1.94099
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 20.0000i − 1.06600i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 4.00000i 0.211407i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 10.0000i − 0.525588i
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 10.0000i − 0.515026i
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 8.00000i − 0.409316i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ 0 0
$$388$$ − 10.0000i − 0.507673i
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 27.0000i − 1.36371i
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 38.0000i − 1.90717i −0.301131 0.953583i $$-0.597364\pi$$
0.301131 0.953583i $$-0.402636\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 40.0000 1.98517
$$407$$ − 8.00000i − 0.396545i
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 48.0000i 2.36193i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ 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
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 2.00000i − 0.0951303i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ − 28.0000i − 1.32288i
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 6.00000i 0.282216i
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.00000i − 0.0935561i −0.998905 0.0467780i $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 38.0000 1.76984 0.884918 0.465746i $$-0.154214\pi$$
0.884918 + 0.465746i $$0.154214\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 0 0
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 36.0000i 1.65703i
$$473$$ 48.0000i 2.20704i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ 0 0
$$478$$ − 24.0000i − 1.09773i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 12.0000i − 0.543772i −0.962329 0.271886i $$-0.912353\pi$$
0.962329 0.271886i $$-0.0876473\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 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
$$493$$ − 20.0000i − 0.900755i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ 8.00000i 0.356702i 0.983967 + 0.178351i $$0.0570763\pi$$
−0.983967 + 0.178351i $$0.942924\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ −26.0000 −1.14681
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ − 8.00000i − 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 44.0000i 1.92399i 0.273075 + 0.961993i $$0.411959\pi$$
−0.273075 + 0.961993i $$0.588041\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 8.00000i 0.348485i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 6.00000i − 0.259889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ 0 0
$$538$$ 22.0000i 0.948487i
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ − 12.0000i − 0.515444i
$$543$$ 0 0
$$544$$ −10.0000 −0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 32.0000i − 1.36078i
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000i 0.421825i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 46.0000i 1.91501i 0.288425 + 0.957503i $$0.406868\pi$$
−0.288425 + 0.957503i $$0.593132\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 16.0000 0.663792
$$582$$ 0 0
$$583$$ 24.0000i 0.993978i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 26.0000i 1.06769i 0.845582 + 0.533846i $$0.179254\pi$$
−0.845582 + 0.533846i $$0.820746\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 48.0000i 1.95633i
$$603$$ 0 0
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.0000i 0.649420i 0.945814 + 0.324710i $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 48.0000 1.93398
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 0 0
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 8.00000i − 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ − 24.0000i − 0.954669i
$$633$$ 0 0
$$634$$ −26.0000 −1.03259
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 9.00000i − 0.356593i
$$638$$ 40.0000i 1.58362i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.00000i 0.313304i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ − 16.0000i − 0.621858i
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 8.00000i − 0.309529i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 0 0
$$679$$ 40.0000 1.53506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 16.0000i − 0.612672i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ 12.0000i 0.457496i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −24.0000 −0.913003 −0.456502 0.889723i $$-0.650898\pi$$
−0.456502 + 0.889723i $$0.650898\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 12.0000i − 0.454532i
$$698$$ 26.0000i 0.984115i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 72.0000i 2.70784i
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6.00000i − 0.224860i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 24.0000i 0.895672i
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 19.0000i − 0.707107i
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 30.0000i 1.10808i 0.832492 + 0.554038i $$0.186914\pi$$
−0.832492 + 0.554038i $$0.813086\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 32.0000i − 1.17874i
$$738$$ 0 0
$$739$$ −32.0000 −1.17714 −0.588570 0.808447i $$-0.700309\pi$$
−0.588570 + 0.808447i $$0.700309\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 24.0000i 0.881068i
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ 0 0
$$748$$ − 8.00000i − 0.292509i
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 10.0000 0.364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 22.0000i − 0.799604i −0.916602 0.399802i $$-0.869079\pi$$
0.916602 0.399802i $$-0.130921\pi$$
$$758$$ 24.0000i 0.871719i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 8.00000i 0.289619i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 12.0000i 0.433295i
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18.0000i 0.647834i
$$773$$ − 10.0000i − 0.359675i −0.983696 0.179838i $$-0.942443\pi$$
0.983696 0.179838i $$-0.0575572\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 30.0000 1.07694
$$777$$ 0 0
$$778$$ 22.0000i 0.788738i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ − 2.00000i − 0.0710221i
$$794$$ 38.0000 1.34857
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 46.0000i 1.62940i 0.579880 + 0.814702i $$0.303099\pi$$
−0.579880 + 0.814702i $$0.696901\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 22.0000i − 0.776847i
$$803$$ − 8.00000i − 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 54.0000i 1.89971i
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ − 40.0000i − 1.40372i
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ − 34.0000i − 1.18878i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ 4.00000i 0.139094i 0.997579 + 0.0695468i $$0.0221553\pi$$
−0.997579 + 0.0695468i $$0.977845\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 7.00000i − 0.242681i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 4.00000i 0.138178i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ − 10.0000i − 0.344623i
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 20.0000i 0.687208i
$$848$$ 6.00000i 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 30.0000i 1.02718i 0.858036 + 0.513590i $$0.171685\pi$$
−0.858036 + 0.513590i $$0.828315\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ − 46.0000i − 1.57133i −0.618652 0.785665i $$-0.712321\pi$$
0.618652 0.785665i $$-0.287679\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ 16.0000i 0.543075i
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 6.00000i 0.203186i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ − 32.0000i − 1.07995i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −58.0000 −1.95407 −0.977035 0.213080i $$-0.931651\pi$$
−0.977035 + 0.213080i $$0.931651\pi$$
$$882$$ 0 0
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 0 0
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.00000i 0.133930i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −12.0000 −0.400892
$$897$$ 0 0
$$898$$ 22.0000i 0.734150i
$$899$$ −40.0000 −1.33407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 24.0000i 0.799113i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 12.0000i − 0.398453i −0.979953 0.199227i $$-0.936157\pi$$
0.979953 0.199227i $$-0.0638430\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 16.0000i 0.529523i
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ − 16.0000i − 0.528367i
$$918$$ 0 0
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 38.0000i 1.25146i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ − 50.0000i − 1.64133i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.0000i 0.458585i
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 26.0000i − 0.849383i −0.905338 0.424691i $$-0.860383\pi$$
0.905338 0.424691i $$-0.139617\pi$$
$$938$$ − 32.0000i − 1.04484i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ − 60.0000i − 1.94974i −0.222779 0.974869i $$-0.571513\pi$$
0.222779 0.974869i $$-0.428487\pi$$
$$948$$ 0 0
$$949$$ −2.00000 −0.0649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 24.0000i − 0.777844i
$$953$$ 30.0000i 0.971795i 0.874016 + 0.485898i $$0.161507\pi$$
−0.874016 + 0.485898i $$0.838493\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ − 24.0000i − 0.775405i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 52.0000i 1.67221i 0.548572 + 0.836104i $$0.315172\pi$$
−0.548572 + 0.836104i $$0.684828\pi$$
$$968$$ 15.0000i 0.482118i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 0 0
$$973$$ − 48.0000i − 1.53881i
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 0 0
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000i 0.382935i
$$983$$ − 16.0000i − 0.510321i −0.966899 0.255160i $$-0.917872\pi$$
0.966899 0.255160i $$-0.0821283\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 20.0000 0.636930
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 20.0000i 0.635001i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 24.0000i 0.759707i
$$999$$ 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 2925.2.c.e.2224.2 2
3.2 odd 2 975.2.c.f.274.1 2
5.2 odd 4 117.2.a.a.1.1 1
5.3 odd 4 2925.2.a.p.1.1 1
5.4 even 2 inner 2925.2.c.e.2224.1 2
15.2 even 4 39.2.a.a.1.1 1
15.8 even 4 975.2.a.f.1.1 1
15.14 odd 2 975.2.c.f.274.2 2
20.7 even 4 1872.2.a.h.1.1 1
35.27 even 4 5733.2.a.e.1.1 1
40.27 even 4 7488.2.a.by.1.1 1
40.37 odd 4 7488.2.a.bl.1.1 1
45.2 even 12 1053.2.e.b.352.1 2
45.7 odd 12 1053.2.e.d.352.1 2
45.22 odd 12 1053.2.e.d.703.1 2
45.32 even 12 1053.2.e.b.703.1 2
60.47 odd 4 624.2.a.i.1.1 1
65.12 odd 4 1521.2.a.e.1.1 1
65.47 even 4 1521.2.b.b.1351.1 2
65.57 even 4 1521.2.b.b.1351.2 2
105.62 odd 4 1911.2.a.f.1.1 1
120.77 even 4 2496.2.a.q.1.1 1
120.107 odd 4 2496.2.a.e.1.1 1
165.32 odd 4 4719.2.a.c.1.1 1
195.2 odd 12 507.2.j.e.316.2 4
195.17 even 12 507.2.e.b.484.1 2
195.32 odd 12 507.2.j.e.361.1 4
195.47 odd 4 507.2.b.a.337.2 2
195.62 even 12 507.2.e.b.22.1 2
195.77 even 4 507.2.a.a.1.1 1
195.107 even 12 507.2.e.a.22.1 2
195.122 odd 4 507.2.b.a.337.1 2
195.137 odd 12 507.2.j.e.361.2 4
195.152 even 12 507.2.e.a.484.1 2
195.167 odd 12 507.2.j.e.316.1 4
780.467 odd 4 8112.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 15.2 even 4
117.2.a.a.1.1 1 5.2 odd 4
507.2.a.a.1.1 1 195.77 even 4
507.2.b.a.337.1 2 195.122 odd 4
507.2.b.a.337.2 2 195.47 odd 4
507.2.e.a.22.1 2 195.107 even 12
507.2.e.a.484.1 2 195.152 even 12
507.2.e.b.22.1 2 195.62 even 12
507.2.e.b.484.1 2 195.17 even 12
507.2.j.e.316.1 4 195.167 odd 12
507.2.j.e.316.2 4 195.2 odd 12
507.2.j.e.361.1 4 195.32 odd 12
507.2.j.e.361.2 4 195.137 odd 12
624.2.a.i.1.1 1 60.47 odd 4
975.2.a.f.1.1 1 15.8 even 4
975.2.c.f.274.1 2 3.2 odd 2
975.2.c.f.274.2 2 15.14 odd 2
1053.2.e.b.352.1 2 45.2 even 12
1053.2.e.b.703.1 2 45.32 even 12
1053.2.e.d.352.1 2 45.7 odd 12
1053.2.e.d.703.1 2 45.22 odd 12
1521.2.a.e.1.1 1 65.12 odd 4
1521.2.b.b.1351.1 2 65.47 even 4
1521.2.b.b.1351.2 2 65.57 even 4
1872.2.a.h.1.1 1 20.7 even 4
1911.2.a.f.1.1 1 105.62 odd 4
2496.2.a.e.1.1 1 120.107 odd 4
2496.2.a.q.1.1 1 120.77 even 4
2925.2.a.p.1.1 1 5.3 odd 4
2925.2.c.e.2224.1 2 5.4 even 2 inner
2925.2.c.e.2224.2 2 1.1 even 1 trivial
4719.2.a.c.1.1 1 165.32 odd 4
5733.2.a.e.1.1 1 35.27 even 4
7488.2.a.bl.1.1 1 40.37 odd 4
7488.2.a.by.1.1 1 40.27 even 4
8112.2.a.s.1.1 1 780.467 odd 4