# Properties

 Label 1900.2.c.a.1749.1 Level $1900$ Weight $2$ Character 1900.1749 Analytic conductor $15.172$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1749.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1749 Dual form 1900.2.c.a.1749.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} -6.00000i q^{13} +2.00000i q^{17} +1.00000 q^{19} +4.00000 q^{21} +2.00000i q^{23} -4.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} -10.0000i q^{37} -12.0000 q^{39} -10.0000 q^{41} -6.00000i q^{43} -6.00000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} -2.00000i q^{57} +4.00000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -2.00000i q^{67} +4.00000 q^{69} +12.0000 q^{71} +6.00000i q^{73} -8.00000 q^{79} -11.0000 q^{81} +2.00000i q^{83} -4.00000i q^{87} -2.00000 q^{89} +12.0000 q^{91} -8.00000i q^{93} -18.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} + 8 q^{59} + 4 q^{61} + 8 q^{69} + 24 q^{71} - 16 q^{79} - 22 q^{81} - 4 q^{89} + 24 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^19 + 8 * q^21 + 4 * q^29 + 8 * q^31 - 24 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 + 8 * q^59 + 4 * q^61 + 8 * q^69 + 24 * q^71 - 16 * q^79 - 22 * q^81 - 4 * q^89 + 24 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 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
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ 2.00000i 0.417029i 0.978019 + 0.208514i $$0.0668628\pi$$
−0.978019 + 0.208514i $$0.933137\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ −12.0000 −1.92154
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 2.00000i − 0.244339i −0.992509 0.122169i $$-0.961015\pi$$
0.992509 0.122169i $$-0.0389851\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 2.00000i 0.219529i 0.993958 + 0.109764i $$0.0350096\pi$$
−0.993958 + 0.109764i $$0.964990\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 4.00000i − 0.428845i
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 6.00000i 0.591198i 0.955312 + 0.295599i $$0.0955191\pi$$
−0.955312 + 0.295599i $$0.904481\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 18.0000i − 1.74013i −0.492941 0.870063i $$-0.664078\pi$$
0.492941 0.870063i $$-0.335922\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −20.0000 −1.89832
$$112$$ 0 0
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 20.0000i 1.80334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000i 0.532414i 0.963916 + 0.266207i $$0.0857705\pi$$
−0.963916 + 0.266207i $$0.914230\pi$$
$$128$$ 0 0
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 6.00000i − 0.494872i
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.0000i 1.08335i 0.840587 + 0.541676i $$0.182210\pi$$
−0.840587 + 0.541676i $$0.817790\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 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
$$177$$ − 8.00000i − 0.601317i
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ − 22.0000i − 1.58359i −0.610784 0.791797i $$-0.709146\pi$$
0.610784 0.791797i $$-0.290854\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 4.00000i 0.280745i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ − 24.0000i − 1.64445i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.00000i 0.543075i
$$218$$ 0 0
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 14.0000i 0.929213i 0.885517 + 0.464606i $$0.153804\pi$$
−0.885517 + 0.464606i $$0.846196\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 6.00000i − 0.381771i
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 16.0000 1.00991 0.504956 0.863145i $$-0.331509\pi$$
0.504956 + 0.863145i $$0.331509\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 26.0000i 1.60323i 0.597841 + 0.801614i $$0.296025\pi$$
−0.597841 + 0.801614i $$0.703975\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 4.00000i 0.244796i
$$268$$ 0 0
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 0 0
$$273$$ − 24.0000i − 1.45255i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 14.0000i − 0.841178i −0.907251 0.420589i $$-0.861823\pi$$
0.907251 0.420589i $$-0.138177\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 20.0000i − 1.18056i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −36.0000 −2.11036
$$292$$ 0 0
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 20.0000i 1.14897i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −36.0000 −2.00932
$$322$$ 0 0
$$323$$ 2.00000i 0.111283i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.0000i 1.10600i
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 16.0000 0.879440 0.439720 0.898135i $$-0.355078\pi$$
0.439720 + 0.898135i $$0.355078\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 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
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 6.00000i − 0.322097i −0.986947 0.161048i $$-0.948512\pi$$
0.986947 0.161048i $$-0.0514875\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −24.0000 −1.28103
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000i 0.423405i
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 22.0000i 1.15470i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.0000i 1.35719i 0.734513 + 0.678594i $$0.237411\pi$$
−0.734513 + 0.678594i $$0.762589\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ 6.00000i 0.306586i 0.988181 + 0.153293i $$0.0489878\pi$$
−0.988181 + 0.153293i $$0.951012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 6.00000i 0.304997i
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ − 32.0000i − 1.61419i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 10.0000i 0.501886i 0.968002 + 0.250943i $$0.0807406\pi$$
−0.968002 + 0.250943i $$0.919259\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ − 24.0000i − 1.19553i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 24.0000i 1.17529i
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 6.00000i 0.291730i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ 18.0000i 0.865025i 0.901628 + 0.432512i $$0.142373\pi$$
−0.901628 + 0.432512i $$0.857627\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.00000i 0.0956730i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 30.0000i − 1.42534i −0.701498 0.712672i $$-0.747485\pi$$
0.701498 0.712672i $$-0.252515\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 20.0000i 0.945968i
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 24.0000i − 1.12762i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 38.0000i − 1.77757i −0.458329 0.888783i $$-0.651552\pi$$
0.458329 0.888783i $$-0.348448\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ 26.0000i 1.20832i 0.796862 + 0.604161i $$0.206492\pi$$
−0.796862 + 0.604161i $$0.793508\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 10.0000i 0.462745i 0.972865 + 0.231372i $$0.0743216\pi$$
−0.972865 + 0.231372i $$0.925678\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 36.0000 1.65879
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 0 0
$$483$$ 8.00000i 0.364013i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 26.0000i − 1.17817i −0.808070 0.589086i $$-0.799488\pi$$
0.808070 0.589086i $$-0.200512\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 32.0000 1.44414 0.722070 0.691820i $$-0.243191\pi$$
0.722070 + 0.691820i $$0.243191\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000i 1.07655i
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 28.0000 1.25095
$$502$$ 0 0
$$503$$ 26.0000i 1.15928i 0.814872 + 0.579641i $$0.196807\pi$$
−0.814872 + 0.579641i $$0.803193\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 46.0000i 2.04293i
$$508$$ 0 0
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ 0 0
$$513$$ − 4.00000i − 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ − 26.0000i − 1.13690i −0.822718 0.568450i $$-0.807543\pi$$
0.822718 0.568450i $$-0.192457\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000i 0.348485i
$$528$$ 0 0
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 60.0000i 2.59889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 24.0000i − 1.03568i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 20.0000i 0.858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 2.00000i − 0.0855138i −0.999086 0.0427569i $$-0.986386\pi$$
0.999086 0.0427569i $$-0.0136141\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ − 16.0000i − 0.680389i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 26.0000i 1.10166i 0.834619 + 0.550828i $$0.185688\pi$$
−0.834619 + 0.550828i $$0.814312\pi$$
$$558$$ 0 0
$$559$$ −36.0000 −1.52264
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 38.0000i 1.60151i 0.598993 + 0.800755i $$0.295568\pi$$
−0.598993 + 0.800755i $$0.704432\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 22.0000i − 0.923913i
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ − 40.0000i − 1.67102i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ 0 0
$$579$$ −44.0000 −1.82858
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 4.00000 0.164538
$$592$$ 0 0
$$593$$ − 2.00000i − 0.0821302i −0.999156 0.0410651i $$-0.986925\pi$$
0.999156 0.0410651i $$-0.0130751\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000i 0.654836i
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 2.00000i 0.0814463i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22.0000i 0.892952i 0.894795 + 0.446476i $$0.147321\pi$$
−0.894795 + 0.446476i $$0.852679\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ −36.0000 −1.45640
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 34.0000i 1.36879i 0.729112 + 0.684394i $$0.239933\pi$$
−0.729112 + 0.684394i $$0.760067\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ − 4.00000i − 0.160257i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −12.0000 −0.477712 −0.238856 0.971055i $$-0.576772\pi$$
−0.238856 + 0.971055i $$0.576772\pi$$
$$632$$ 0 0
$$633$$ − 16.0000i − 0.635943i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 18.0000i − 0.713186i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 26.0000i 1.02534i 0.858586 + 0.512670i $$0.171344\pi$$
−0.858586 + 0.512670i $$0.828656\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000i 0.707653i 0.935311 + 0.353827i $$0.115120\pi$$
−0.935311 + 0.353827i $$0.884880\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 0 0
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ 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 $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 0 0
$$663$$ − 24.0000i − 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000i 0.154881i
$$668$$ 0 0
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 2.00000i − 0.0768662i −0.999261 0.0384331i $$-0.987763\pi$$
0.999261 0.0384331i $$-0.0122367\pi$$
$$678$$ 0 0
$$679$$ 36.0000 1.38155
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ 0 0
$$683$$ 30.0000i 1.14792i 0.818884 + 0.573959i $$0.194593\pi$$
−0.818884 + 0.573959i $$0.805407\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 36.0000i − 1.37349i
$$688$$ 0 0
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.0000i − 0.757554i
$$698$$ 0 0
$$699$$ 44.0000 1.66423
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 20.0000i − 0.752177i
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 48.0000i − 1.79259i
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ 0 0
$$723$$ − 28.0000i − 1.04133i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ −12.0000 −0.440831
$$742$$ 0 0
$$743$$ − 10.0000i − 0.366864i −0.983032 0.183432i $$-0.941279\pi$$
0.983032 0.183432i $$-0.0587208\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 2.00000i − 0.0731762i
$$748$$ 0 0
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ − 32.0000i − 1.16614i
$$754$$ 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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ − 20.0000i − 0.724049i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 24.0000i − 0.866590i
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −36.0000 −1.29651
$$772$$ 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
$$777$$ − 40.0000i − 1.43499i
$$778$$ 0 0
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 8.00000i − 0.285897i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14.0000i 0.499046i 0.968369 + 0.249523i $$0.0802738\pi$$
−0.968369 + 0.249523i $$0.919726\pi$$
$$788$$ 0 0
$$789$$ 52.0000 1.85125
$$790$$ 0 0
$$791$$ −20.0000 −0.711118
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.0000i 1.34603i 0.739629 + 0.673015i $$0.235001\pi$$
−0.739629 + 0.673015i $$0.764999\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 52.0000i 1.83049i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −40.0000 −1.40459 −0.702295 0.711886i $$-0.747841\pi$$
−0.702295 + 0.711886i $$0.747841\pi$$
$$812$$ 0 0
$$813$$ 56.0000i 1.96401i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 6.00000i − 0.209913i
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ − 38.0000i − 1.32460i −0.749240 0.662298i $$-0.769581\pi$$
0.749240 0.662298i $$-0.230419\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 18.0000i − 0.625921i −0.949766 0.312961i $$-0.898679\pi$$
0.949766 0.312961i $$-0.101321\pi$$
$$828$$ 0 0
$$829$$ −42.0000 −1.45872 −0.729360 0.684130i $$-0.760182\pi$$
−0.729360 + 0.684130i $$0.760182\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ 6.00000i 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ − 44.0000i − 1.51544i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 22.0000i − 0.755929i
$$848$$ 0 0
$$849$$ −44.0000 −1.51008
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ − 2.00000i − 0.0684787i −0.999414 0.0342393i $$-0.989099\pi$$
0.999414 0.0342393i $$-0.0109009\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −40.0000 −1.36320
$$862$$ 0 0
$$863$$ 54.0000i 1.83818i 0.394046 + 0.919091i $$0.371075\pi$$
−0.394046 + 0.919091i $$0.628925\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 26.0000i − 0.883006i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ 0 0
$$873$$ 18.0000i 0.609208i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 26.0000i − 0.877958i −0.898497 0.438979i $$-0.855340\pi$$
0.898497 0.438979i $$-0.144660\pi$$
$$878$$ 0 0
$$879$$ 4.00000 0.134917
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 34.0000i 1.14419i 0.820187 + 0.572096i $$0.193869\pi$$
−0.820187 + 0.572096i $$0.806131\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 22.0000i 0.738688i 0.929293 + 0.369344i $$0.120418\pi$$
−0.929293 + 0.369344i $$0.879582\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 6.00000i − 0.200782i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 24.0000i − 0.801337i
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ − 24.0000i − 0.798670i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 10.0000i − 0.332045i −0.986122 0.166022i $$-0.946908\pi$$
0.986122 0.166022i $$-0.0530924\pi$$
$$908$$ 0 0
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 52.0000 1.72284 0.861418 0.507896i $$-0.169577\pi$$
0.861418 + 0.507896i $$0.169577\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 32.0000i 1.05673i
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 44.0000 1.44985
$$922$$ 0 0
$$923$$ − 72.0000i − 2.36991i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 6.00000i − 0.197066i
$$928$$ 0 0
$$929$$ −54.0000 −1.77168 −0.885841 0.463988i $$-0.846418\pi$$
−0.885841 + 0.463988i $$0.846418\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 46.0000i − 1.50275i −0.659873 0.751377i $$-0.729390\pi$$
0.659873 0.751377i $$-0.270610\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ − 20.0000i − 0.651290i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 22.0000i − 0.714904i −0.933932 0.357452i $$-0.883646\pi$$
0.933932 0.357452i $$-0.116354\pi$$
$$948$$ 0 0
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ 60.0000 1.94563
$$952$$ 0 0
$$953$$ − 14.0000i − 0.453504i −0.973952 0.226752i $$-0.927189\pi$$
0.973952 0.226752i $$-0.0728108\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 18.0000i 0.580042i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 18.0000i 0.578841i 0.957202 + 0.289420i $$0.0934626\pi$$
−0.957202 + 0.289420i $$0.906537\pi$$
$$968$$ 0 0
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ − 24.0000i − 0.769405i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ − 18.0000i − 0.574111i −0.957914 0.287055i $$-0.907324\pi$$
0.957914 0.287055i $$-0.0926764\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ 0 0
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ 0 0
$$993$$ − 32.0000i − 1.01549i
$$994$$ 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
$$999$$ −40.0000 −1.26554
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 1900.2.c.a.1749.1 2
5.2 odd 4 1900.2.a.a.1.1 1
5.3 odd 4 380.2.a.b.1.1 1
5.4 even 2 inner 1900.2.c.a.1749.2 2
15.8 even 4 3420.2.a.f.1.1 1
20.3 even 4 1520.2.a.b.1.1 1
20.7 even 4 7600.2.a.r.1.1 1
40.3 even 4 6080.2.a.v.1.1 1
40.13 odd 4 6080.2.a.f.1.1 1
95.18 even 4 7220.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.b.1.1 1 5.3 odd 4
1520.2.a.b.1.1 1 20.3 even 4
1900.2.a.a.1.1 1 5.2 odd 4
1900.2.c.a.1749.1 2 1.1 even 1 trivial
1900.2.c.a.1749.2 2 5.4 even 2 inner
3420.2.a.f.1.1 1 15.8 even 4
6080.2.a.f.1.1 1 40.13 odd 4
6080.2.a.v.1.1 1 40.3 even 4
7220.2.a.b.1.1 1 95.18 even 4
7600.2.a.r.1.1 1 20.7 even 4