# Properties

 Label 425.2.b.b.324.1 Level $425$ Weight $2$ Character 425.324 Analytic conductor $3.394$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 425.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.39364208590$$ 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 17) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 324.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 425.324 Dual form 425.2.b.b.324.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +1.00000 q^{34} +3.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{46} -9.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{62} +12.0000i q^{63} -7.00000 q^{64} +4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -9.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +9.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{92} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 24 q^{79} + 18 q^{81} - 8 q^{86} - 20 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^14 - 2 * q^16 + 8 * q^19 + 4 * q^26 - 12 * q^29 + 8 * q^31 + 2 * q^34 + 6 * q^36 - 12 * q^41 - 8 * q^46 - 18 * q^49 + 24 * q^56 + 24 * q^59 - 20 * q^61 - 14 * q^64 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 24 * q^79 + 18 * q^81 - 8 * q^86 - 20 * q^89 - 16 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 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
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 1.00000i 0.242536i
$$18$$ − 3.00000i − 0.707107i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 4.00000i 0.755929i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 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
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 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
$$58$$ 6.00000i 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 12.0000i 1.51186i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ − 9.00000i − 1.06066i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 6.00000i 0.662589i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 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
$$92$$ − 4.00000i − 0.417029i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 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
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000i 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 10.0000i 0.905357i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 12.0000 1.06904
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 4.00000i 0.335673i
$$143$$ 0 0
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ − 12.0000i − 0.973329i
$$153$$ 3.00000i 0.242536i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ − 9.00000i − 0.707107i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 4.00000i − 0.309529i −0.987951 0.154765i $$-0.950538\pi$$
0.987951 0.154765i $$-0.0494619\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 22.0000i − 1.67263i −0.548250 0.836315i $$-0.684706\pi$$
0.548250 0.836315i $$-0.315294\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 10.0000i 0.749532i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000i 0.703598i
$$203$$ − 24.0000i − 1.68447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ − 12.0000i − 0.834058i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 6.00000i 0.406371i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ − 24.0000i − 1.60716i −0.595198 0.803579i $$-0.702926\pi$$
0.595198 0.803579i $$-0.297074\pi$$
$$224$$ 20.0000 1.33631
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 18.0000i 1.18176i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 4.00000i 0.259281i
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ − 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 $$0.376365\pi$$
$$252$$ 12.0000i 0.755929i
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ − 16.0000i − 0.988483i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 0 0
$$268$$ 4.00000i 0.244339i
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.0000i 0.841178i 0.907251 + 0.420589i $$0.138177\pi$$
−0.907251 + 0.420589i $$0.861823\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ − 15.0000i − 0.883883i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6.00000i 0.351123i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ − 10.0000i − 0.579284i
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 16.0000i 0.920697i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 3.00000 0.171499
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ − 10.0000i − 0.561656i −0.959758 0.280828i $$-0.909391\pi$$
0.959758 0.280828i $$-0.0906090\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 16.0000i − 0.891645i
$$323$$ 4.00000i 0.222566i
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ −24.0000 −1.32924
$$327$$ 0 0
$$328$$ 18.0000i 0.993884i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −4.00000 −0.218870
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 12.0000i − 0.648886i
$$343$$ − 8.00000i − 0.431959i
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ 0 0
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000i 0.105118i
$$363$$ 0 0
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.0000i 1.46159i 0.682598 + 0.730794i $$0.260850\pi$$
−0.682598 + 0.730794i $$0.739150\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 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
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000i 0.818631i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 12.0000i − 0.609994i
$$388$$ 2.00000i 0.101535i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 27.0000i 1.36371i
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ − 20.0000i − 1.00251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 8.00000i − 0.394132i
$$413$$ 48.0000i 2.36193i
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −8.00000 −0.390826 −0.195413 0.980721i $$-0.562605\pi$$
−0.195413 + 0.980721i $$0.562605\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ 0 0
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 40.0000i − 1.93574i
$$428$$ 8.00000i 0.386695i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ − 16.0000i − 0.765384i
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ −27.0000 −1.28571
$$442$$ 2.00000i 0.0951303i
$$443$$ − 28.0000i − 1.33032i −0.746701 0.665160i $$-0.768363\pi$$
0.746701 0.665160i $$-0.231637\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ − 28.0000i − 1.32288i
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 14.0000i 0.658505i
$$453$$ 0 0
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 36.0000i − 1.65703i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ − 18.0000i − 0.824163i
$$478$$ − 16.0000i − 0.731823i
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ − 18.0000i − 0.819878i
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ 30.0000i 1.35804i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ − 6.00000i − 0.270226i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ − 16.0000i − 0.717698i
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 36.0000 1.60357
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000i 0.354943i
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 11.0000i 0.486136i
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ − 8.00000i − 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ 36.0000i 1.57417i 0.616844 + 0.787085i $$0.288411\pi$$
−0.616844 + 0.787085i $$0.711589\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 4.00000i 0.174243i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 36.0000 1.56227
$$532$$ 16.0000i 0.693688i
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 22.0000i 0.948487i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ 0 0
$$544$$ 5.00000 0.214373
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −30.0000 −1.28037
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ − 48.0000i − 2.04117i
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ − 12.0000i − 0.508001i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000i 0.253095i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 36.0000i 1.51186i
$$568$$ 12.0000i 0.503509i
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ − 14.0000i − 0.582828i −0.956597 0.291414i $$-0.905874\pi$$
0.956597 0.291414i $$-0.0941257\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ − 8.00000i − 0.327144i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ 12.0000i 0.488678i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.0000i 0.811775i 0.913923 + 0.405887i $$0.133038\pi$$
−0.913923 + 0.405887i $$0.866962\pi$$
$$608$$ − 20.0000i − 0.811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 3.00000i 0.121268i
$$613$$ 26.0000i 1.05013i 0.851062 + 0.525065i $$0.175959\pi$$
−0.851062 + 0.525065i $$0.824041\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 0 0
$$619$$ 48.0000 1.92928 0.964641 0.263566i $$-0.0848986\pi$$
0.964641 + 0.263566i $$0.0848986\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 28.0000i − 1.12270i
$$623$$ − 40.0000i − 1.60257i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 36.0000i 1.43200i
$$633$$ 0 0
$$634$$ −10.0000 −0.397151
$$635$$ 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 $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ 8.00000i 0.314512i 0.987558 + 0.157256i $$0.0502649\pi$$
−0.987558 + 0.157256i $$0.949735\pi$$
$$648$$ − 27.0000i − 1.06066i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 24.0000i − 0.939913i
$$653$$ − 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
$$657$$ 18.0000i 0.702247i
$$658$$ 0 0
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 24.0000i 0.929284i
$$668$$ − 4.00000i − 0.154765i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 40.0000i 1.53056i 0.643699 + 0.765279i $$0.277399\pi$$
−0.643699 + 0.765279i $$0.722601\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ 0 0
$$694$$ 32.0000 1.21470
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 6.00000i − 0.227266i
$$698$$ − 18.0000i − 0.681310i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ − 40.0000i − 1.50435i
$$708$$ 0 0
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ −36.0000 −1.35011
$$712$$ 30.0000i 1.12430i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −4.00000 −0.149175 −0.0745874 0.997214i $$-0.523764\pi$$
−0.0745874 + 0.997214i $$0.523764\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 40.0000i 1.48352i 0.670667 + 0.741759i $$0.266008\pi$$
−0.670667 + 0.741759i $$0.733992\pi$$
$$728$$ 24.0000i 0.889499i
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 50.0000i 1.84679i 0.383849 + 0.923396i $$0.374598\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ 0 0
$$738$$ 18.0000i 0.662589i
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 24.0000i − 0.881068i
$$743$$ − 12.0000i − 0.440237i −0.975473 0.220119i $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000i 0.799604i 0.916602 + 0.399802i $$0.130921\pi$$
−0.916602 + 0.399802i $$0.869079\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ − 24.0000i − 0.868858i
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 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
$$771$$ 0 0
$$772$$ − 2.00000i − 0.0719816i
$$773$$ 26.0000i 0.935155i 0.883952 + 0.467578i $$0.154873\pi$$
−0.883952 + 0.467578i $$0.845127\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ 6.00000i 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 4.00000i − 0.143040i
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ − 50.0000i − 1.77109i −0.464553 0.885545i $$-0.653785\pi$$
0.464553 0.885545i $$-0.346215\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 14.0000i 0.494357i
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ 30.0000i 1.05540i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 40.0000 1.40459 0.702295 0.711886i $$-0.252159\pi$$
0.702295 + 0.711886i $$0.252159\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ 26.0000i 0.909069i
$$819$$ −24.0000 −0.838628
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ − 20.0000i − 0.697156i −0.937280 0.348578i $$-0.886665\pi$$
0.937280 0.348578i $$-0.113335\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ − 12.0000i − 0.417029i
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 14.0000i − 0.485363i
$$833$$ − 9.00000i − 0.311832i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 8.00000i 0.276355i
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 22.0000i − 0.758170i
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ 6.00000i 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ 0 0
$$856$$ 24.0000 0.820303
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 12.0000i − 0.408722i
$$863$$ − 16.0000i − 0.544646i −0.962206 0.272323i $$-0.912208\pi$$
0.962206 0.272323i $$-0.0877920\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ 16.0000i 0.543075i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 18.0000i 0.609557i
$$873$$ 6.00000i 0.203069i
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6.00000i 0.202606i 0.994856 + 0.101303i $$0.0323011\pi$$
−0.994856 + 0.101303i $$0.967699\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 27.0000i 0.909137i
$$883$$ 12.0000i 0.403832i 0.979403 + 0.201916i $$0.0647168\pi$$
−0.979403 + 0.201916i $$0.935283\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ −28.0000 −0.940678
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 0 0
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 24.0000i − 0.803579i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 12.0000 0.400892
$$897$$ 0 0
$$898$$ 34.0000i 1.13459i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 6.00000 0.199889
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 42.0000 1.39690
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ −4.00000 −0.132526 −0.0662630 0.997802i $$-0.521108\pi$$
−0.0662630 + 0.997802i $$0.521108\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 64.0000i 2.11347i
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 2.00000i 0.0658665i
$$923$$ − 8.00000i − 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ − 24.0000i − 0.788263i
$$928$$ 30.0000i 0.984798i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ 10.0000i 0.326686i 0.986569 + 0.163343i $$0.0522277\pi$$
−0.986569 + 0.163343i $$0.947772\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ 24.0000i 0.781548i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 32.0000i 1.03986i 0.854209 + 0.519930i $$0.174042\pi$$
−0.854209 + 0.519930i $$0.825958\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 12.0000i 0.388922i
$$953$$ 22.0000i 0.712650i 0.934362 + 0.356325i $$0.115970\pi$$
−0.934362 + 0.356325i $$0.884030\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 36.0000i 1.16311i
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 4.00000i − 0.128965i
$$963$$ 24.0000i 0.773389i
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 33.0000i 1.06066i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 32.0000i 1.02587i
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ − 20.0000i − 0.638226i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −6.00000 −0.191079
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −12.0000 −0.381193 −0.190596 0.981669i $$-0.561042\pi$$
−0.190596 + 0.981669i $$0.561042\pi$$
$$992$$ − 20.0000i − 0.635001i
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.0000i 1.45683i 0.685134 + 0.728417i $$0.259744\pi$$
−0.685134 + 0.728417i $$0.740256\pi$$
$$998$$ − 40.0000i − 1.26618i
$$999$$ 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 425.2.b.b.324.1 2
5.2 odd 4 425.2.a.d.1.1 1
5.3 odd 4 17.2.a.a.1.1 1
5.4 even 2 inner 425.2.b.b.324.2 2
15.2 even 4 3825.2.a.d.1.1 1
15.8 even 4 153.2.a.c.1.1 1
20.3 even 4 272.2.a.b.1.1 1
20.7 even 4 6800.2.a.n.1.1 1
35.3 even 12 833.2.e.a.324.1 2
35.13 even 4 833.2.a.a.1.1 1
35.18 odd 12 833.2.e.b.324.1 2
35.23 odd 12 833.2.e.b.18.1 2
35.33 even 12 833.2.e.a.18.1 2
40.3 even 4 1088.2.a.h.1.1 1
40.13 odd 4 1088.2.a.i.1.1 1
55.43 even 4 2057.2.a.e.1.1 1
60.23 odd 4 2448.2.a.o.1.1 1
65.38 odd 4 2873.2.a.c.1.1 1
85.3 even 16 289.2.d.d.179.2 8
85.8 odd 8 289.2.c.a.251.1 4
85.13 odd 4 289.2.b.a.288.1 2
85.23 even 16 289.2.d.d.155.2 8
85.28 even 16 289.2.d.d.155.1 8
85.33 odd 4 289.2.a.a.1.1 1
85.38 odd 4 289.2.b.a.288.2 2
85.43 odd 8 289.2.c.a.251.2 4
85.48 even 16 289.2.d.d.179.1 8
85.53 odd 8 289.2.c.a.38.2 4
85.58 even 16 289.2.d.d.134.2 8
85.63 even 16 289.2.d.d.110.1 8
85.67 odd 4 7225.2.a.g.1.1 1
85.73 even 16 289.2.d.d.110.2 8
85.78 even 16 289.2.d.d.134.1 8
85.83 odd 8 289.2.c.a.38.1 4
95.18 even 4 6137.2.a.b.1.1 1
105.83 odd 4 7497.2.a.l.1.1 1
115.68 even 4 8993.2.a.a.1.1 1
120.53 even 4 9792.2.a.n.1.1 1
120.83 odd 4 9792.2.a.i.1.1 1
255.203 even 4 2601.2.a.g.1.1 1
340.203 even 4 4624.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 1 5.3 odd 4
153.2.a.c.1.1 1 15.8 even 4
272.2.a.b.1.1 1 20.3 even 4
289.2.a.a.1.1 1 85.33 odd 4
289.2.b.a.288.1 2 85.13 odd 4
289.2.b.a.288.2 2 85.38 odd 4
289.2.c.a.38.1 4 85.83 odd 8
289.2.c.a.38.2 4 85.53 odd 8
289.2.c.a.251.1 4 85.8 odd 8
289.2.c.a.251.2 4 85.43 odd 8
289.2.d.d.110.1 8 85.63 even 16
289.2.d.d.110.2 8 85.73 even 16
289.2.d.d.134.1 8 85.78 even 16
289.2.d.d.134.2 8 85.58 even 16
289.2.d.d.155.1 8 85.28 even 16
289.2.d.d.155.2 8 85.23 even 16
289.2.d.d.179.1 8 85.48 even 16
289.2.d.d.179.2 8 85.3 even 16
425.2.a.d.1.1 1 5.2 odd 4
425.2.b.b.324.1 2 1.1 even 1 trivial
425.2.b.b.324.2 2 5.4 even 2 inner
833.2.a.a.1.1 1 35.13 even 4
833.2.e.a.18.1 2 35.33 even 12
833.2.e.a.324.1 2 35.3 even 12
833.2.e.b.18.1 2 35.23 odd 12
833.2.e.b.324.1 2 35.18 odd 12
1088.2.a.h.1.1 1 40.3 even 4
1088.2.a.i.1.1 1 40.13 odd 4
2057.2.a.e.1.1 1 55.43 even 4
2448.2.a.o.1.1 1 60.23 odd 4
2601.2.a.g.1.1 1 255.203 even 4
2873.2.a.c.1.1 1 65.38 odd 4
3825.2.a.d.1.1 1 15.2 even 4
4624.2.a.d.1.1 1 340.203 even 4
6137.2.a.b.1.1 1 95.18 even 4
6800.2.a.n.1.1 1 20.7 even 4
7225.2.a.g.1.1 1 85.67 odd 4
7497.2.a.l.1.1 1 105.83 odd 4
8993.2.a.a.1.1 1 115.68 even 4
9792.2.a.i.1.1 1 120.83 odd 4
9792.2.a.n.1.1 1 120.53 even 4