# Properties

 Label 3850.2.c.d.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$1$$ 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 154) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.d.1849.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} -4.00000 q^{19} -2.00000 q^{21} +1.00000i q^{22} +4.00000i q^{23} +2.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} +8.00000 q^{39} -2.00000i q^{42} -4.00000i q^{43} -1.00000 q^{44} -4.00000 q^{46} -10.0000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +4.00000i q^{52} -14.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000i q^{57} -2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} -10.0000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000i q^{77} +8.00000i q^{78} -16.0000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +2.00000 q^{84} +4.00000 q^{86} -4.00000i q^{87} -1.00000i q^{88} -10.0000 q^{89} +4.00000 q^{91} -4.00000i q^{92} -20.0000i q^{93} +10.0000 q^{94} -2.00000 q^{96} -6.00000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 4 q^{24} + 8 q^{26} - 4 q^{29} - 20 q^{31} + 2 q^{36} + 16 q^{39} - 2 q^{44} - 8 q^{46} - 2 q^{49} - 8 q^{54} + 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} + 8 q^{91} + 20 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 2 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 4 * q^24 + 8 * q^26 - 4 * q^29 - 20 * q^31 + 2 * q^36 + 16 * q^39 - 2 * q^44 - 8 * q^46 - 2 * q^49 - 8 * q^54 + 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 - 4 * q^66 - 16 * q^69 - 8 * q^71 - 12 * q^74 + 8 * q^76 - 32 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 20 * q^89 + 8 * q^91 + 20 * q^94 - 4 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 1.00000i 0.377964i
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 1.00000i 0.213201i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 4.00000i 0.769800i
$$28$$ − 1.00000i − 0.188982i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 2.00000i 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.00000i 0.554700i
$$53$$ − 14.0000i − 1.92305i −0.274721 0.961524i $$-0.588586\pi$$
0.274721 0.961524i $$-0.411414\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ − 8.00000i − 1.05963i
$$58$$ − 2.00000i − 0.262613i
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ − 10.0000i − 1.27000i
$$63$$ − 1.00000i − 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 1.00000i 0.113961i
$$78$$ 8.00000i 0.905822i
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 4.00000i − 0.428845i
$$88$$ − 1.00000i − 0.106600i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 20.0000i − 2.07390i
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 2.00000i 0.197066i 0.995134 + 0.0985329i $$0.0314150\pi$$
−0.995134 + 0.0985329i $$0.968585\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ 1.00000i 0.0944911i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 4.00000i 0.369800i
$$118$$ − 10.0000i − 0.920575i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 8.00000i − 0.724286i
$$123$$ 0 0
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ − 4.00000i − 0.346844i
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 20.0000 1.68430
$$142$$ − 4.00000i − 0.335673i
$$143$$ − 4.00000i − 0.334497i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ − 2.00000i − 0.164957i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ −8.00000 −0.640513
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ 28.0000 2.22054
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ − 11.0000i − 0.864242i
$$163$$ 24.0000i 1.87983i 0.341415 + 0.939913i $$0.389094\pi$$
−0.341415 + 0.939913i $$0.610906\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ 4.00000i 0.304114i 0.988372 + 0.152057i $$0.0485898\pi$$
−0.988372 + 0.152057i $$0.951410\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ − 20.0000i − 1.50329i
$$178$$ − 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
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ − 16.0000i − 1.18275i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 20.0000 1.46647
$$187$$ 0 0
$$188$$ 10.0000i 0.729325i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ − 1.00000i − 0.0710669i
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 12.0000i 0.844317i
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.00000 −0.139347
$$207$$ − 4.00000i − 0.278019i
$$208$$ − 4.00000i − 0.277350i
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 14.0000i 0.961524i
$$213$$ − 8.00000i − 0.548151i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ − 10.0000i − 0.678844i
$$218$$ 14.0000i 0.948200i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 12.0000i − 0.805387i
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 8.00000i 0.529813i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 2.00000i 0.131306i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ − 32.0000i − 2.07862i
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.0000i 1.01806i
$$248$$ 10.0000i 0.635001i
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ −26.0000 −1.64111 −0.820553 0.571571i $$-0.806334\pi$$
−0.820553 + 0.571571i $$0.806334\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ 4.00000i 0.251478i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 8.00000i 0.494242i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ − 20.0000i − 1.22398i
$$268$$ 8.00000i 0.488678i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 0 0
$$273$$ 8.00000i 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 20.0000i 1.19098i
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ − 4.00000i − 0.234082i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 4.00000i 0.232104i
$$298$$ − 22.0000i − 1.27443i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 16.0000i 0.920697i
$$303$$ 24.0000i 1.37876i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ − 8.00000i − 0.452911i
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 28.0000i 1.57016i
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ − 4.00000i − 0.222911i
$$323$$ 0 0
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −24.0000 −1.32924
$$327$$ 28.0000i 1.54840i
$$328$$ 0 0
$$329$$ 10.0000 0.551318
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 34.0000i 1.85210i 0.377403 + 0.926049i $$0.376817\pi$$
−0.377403 + 0.926049i $$0.623183\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 28.0000 1.52075
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 4.00000i 0.216295i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −4.00000 −0.215041
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ −32.0000 −1.71292 −0.856460 0.516213i $$-0.827341\pi$$
−0.856460 + 0.516213i $$0.827341\pi$$
$$350$$ 0 0
$$351$$ 16.0000 0.854017
$$352$$ 1.00000i 0.0533002i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 20.0000 1.06299
$$355$$ 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
$$362$$ 14.0000i 0.735824i
$$363$$ 2.00000i 0.104973i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 16.0000 0.836333
$$367$$ − 18.0000i − 0.939592i −0.882775 0.469796i $$-0.844327\pi$$
0.882775 0.469796i $$-0.155673\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 14.0000 0.726844
$$372$$ 20.0000i 1.03695i
$$373$$ − 34.0000i − 1.76045i −0.474554 0.880227i $$-0.657390\pi$$
0.474554 0.880227i $$-0.342610\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −10.0000 −0.515711
$$377$$ 8.00000i 0.412021i
$$378$$ − 4.00000i − 0.205738i
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −32.0000 −1.63941
$$382$$ 8.00000i 0.409316i
$$383$$ 14.0000i 0.715367i 0.933843 + 0.357683i $$0.116433\pi$$
−0.933843 + 0.357683i $$0.883567\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 4.00000i 0.203331i
$$388$$ 6.00000i 0.304604i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.00000i 0.0505076i
$$393$$ 16.0000i 0.807093i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ 14.0000i 0.701757i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 16.0000i 0.798007i
$$403$$ 40.0000i 1.99254i
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 6.00000i 0.297409i
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ − 2.00000i − 0.0985329i
$$413$$ − 10.0000i − 0.492068i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ − 40.0000i − 1.95881i
$$418$$ − 4.00000i − 0.195646i
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 10.0000i 0.486217i
$$424$$ −14.0000 −0.679900
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ − 8.00000i − 0.387147i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 10.0000i − 0.480569i −0.970702 0.240285i $$-0.922759\pi$$
0.970702 0.240285i $$-0.0772408\pi$$
$$434$$ 10.0000 0.480015
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ − 16.0000i − 0.765384i
$$438$$ − 8.00000i − 0.382255i
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 12.0000 0.569495
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ − 44.0000i − 2.08113i
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 14.0000i 0.658505i
$$453$$ 32.0000i 1.50349i
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −8.00000 −0.374634
$$457$$ 38.0000i 1.77757i 0.458329 + 0.888783i $$0.348448\pi$$
−0.458329 + 0.888783i $$0.651552\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 32.0000 1.49039 0.745194 0.666847i $$-0.232357\pi$$
0.745194 + 0.666847i $$0.232357\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ 12.0000i 0.557687i 0.960337 + 0.278844i $$0.0899511\pi$$
−0.960337 + 0.278844i $$0.910049\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 14.0000i − 0.647843i −0.946084 0.323921i $$-0.894999\pi$$
0.946084 0.323921i $$-0.105001\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ 10.0000i 0.460287i
$$473$$ − 4.00000i − 0.183920i
$$474$$ 32.0000 1.46981
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 14.0000i 0.641016i
$$478$$ 8.00000i 0.365911i
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ 8.00000i 0.364390i
$$483$$ − 8.00000i − 0.364013i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ − 12.0000i − 0.543772i −0.962329 0.271886i $$-0.912353\pi$$
0.962329 0.271886i $$-0.0876473\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ −48.0000 −2.17064
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ − 4.00000i − 0.179425i
$$498$$ − 8.00000i − 0.358489i
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ − 26.0000i − 1.16044i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ − 6.00000i − 0.266469i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ − 16.0000i − 0.706417i
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ − 10.0000i − 0.439799i
$$518$$ − 6.00000i − 0.263625i
$$519$$ −8.00000 −0.351161
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 2.00000i 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ 4.00000i 0.173422i
$$533$$ 0 0
$$534$$ 20.0000 0.865485
$$535$$ 0 0
$$536$$ −8.00000 −0.345547
$$537$$ − 24.0000i − 1.03568i
$$538$$ − 14.0000i − 0.603583i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ − 28.0000i − 1.20270i
$$543$$ 28.0000i 1.20160i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 8.00000i 0.340503i
$$553$$ − 16.0000i − 0.680389i
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 10.0000i 0.423334i
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000i 1.26547i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 11.0000i − 0.461957i
$$568$$ 4.00000i 0.167836i
$$569$$ −14.0000 −0.586911 −0.293455 0.955973i $$-0.594805\pi$$
−0.293455 + 0.955973i $$0.594805\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 12.0000 0.498703
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 12.0000i 0.497416i
$$583$$ − 14.0000i − 0.579821i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 42.0000i 1.73353i 0.498721 + 0.866763i $$0.333803\pi$$
−0.498721 + 0.866763i $$0.666197\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 40.0000 1.64817
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 6.00000i 0.246598i
$$593$$ 12.0000i 0.492781i 0.969171 + 0.246390i $$0.0792446\pi$$
−0.969171 + 0.246390i $$0.920755\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 28.0000i 1.14596i
$$598$$ 16.0000i 0.654289i
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −24.0000 −0.978980 −0.489490 0.872009i $$-0.662817\pi$$
−0.489490 + 0.872009i $$0.662817\pi$$
$$602$$ 4.00000i 0.163028i
$$603$$ 8.00000i 0.325785i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −24.0000 −0.974933
$$607$$ 24.0000i 0.974130i 0.873366 + 0.487065i $$0.161933\pi$$
−0.873366 + 0.487065i $$0.838067\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ −40.0000 −1.61823
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ − 38.0000i − 1.52982i −0.644136 0.764911i $$-0.722783\pi$$
0.644136 0.764911i $$-0.277217\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ −16.0000 −0.642058
$$622$$ − 6.00000i − 0.240578i
$$623$$ − 10.0000i − 0.400642i
$$624$$ 8.00000 0.320256
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ − 8.00000i − 0.319489i
$$628$$ 10.0000i 0.399043i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ − 8.00000i − 0.317971i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −28.0000 −1.11027
$$637$$ 4.00000i 0.158486i
$$638$$ − 2.00000i − 0.0791808i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 24.0000i − 0.947204i
$$643$$ 22.0000i 0.867595i 0.901010 + 0.433798i $$0.142827\pi$$
−0.901010 + 0.433798i $$0.857173\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ −10.0000 −0.392534
$$650$$ 0 0
$$651$$ 20.0000 0.783862
$$652$$ − 24.0000i − 0.939913i
$$653$$ 46.0000i 1.80012i 0.435767 + 0.900060i $$0.356477\pi$$
−0.435767 + 0.900060i $$0.643523\pi$$
$$654$$ −28.0000 −1.09489
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 4.00000i − 0.156055i
$$658$$ 10.0000i 0.389841i
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ − 8.00000i − 0.309761i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 28.0000 1.08254
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ 28.0000i 1.07533i
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 16.0000 0.613121
$$682$$ − 10.0000i − 0.382920i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 20.0000i 0.763048i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −56.0000 −2.13343
$$690$$ 0 0
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ − 4.00000i − 0.152057i
$$693$$ − 1.00000i − 0.0379869i
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ 0 0
$$698$$ − 32.0000i − 1.21122i
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 16.0000i 0.603881i
$$703$$ − 24.0000i − 0.905177i
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 12.0000i 0.451306i
$$708$$ 20.0000i 0.751646i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 10.0000i 0.374766i
$$713$$ − 40.0000i − 1.49801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ −2.00000 −0.0744839
$$722$$ − 3.00000i − 0.111648i
$$723$$ 16.0000i 0.595046i
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 46.0000i − 1.70605i −0.521874 0.853023i $$-0.674767\pi$$
0.521874 0.853023i $$-0.325233\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 16.0000i 0.591377i
$$733$$ − 8.00000i − 0.295487i −0.989026 0.147743i $$-0.952799\pi$$
0.989026 0.147743i $$-0.0472010\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ − 8.00000i − 0.294684i
$$738$$ 0 0
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ 0 0
$$741$$ −32.0000 −1.17555
$$742$$ 14.0000i 0.513956i
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ −20.0000 −0.733236
$$745$$ 0 0
$$746$$ 34.0000 1.24483
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ − 10.0000i − 0.364662i
$$753$$ − 52.0000i − 1.89499i
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ − 30.0000i − 1.09037i −0.838316 0.545184i $$-0.816460\pi$$
0.838316 0.545184i $$-0.183540\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ − 32.0000i − 1.15924i
$$763$$ 14.0000i 0.506834i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −14.0000 −0.505841
$$767$$ 40.0000i 1.44432i
$$768$$ 2.00000i 0.0721688i
$$769$$ −4.00000 −0.144244 −0.0721218 0.997396i $$-0.522977\pi$$
−0.0721218 + 0.997396i $$0.522977\pi$$
$$770$$ 0 0
$$771$$ 4.00000 0.144056
$$772$$ 6.00000i 0.215945i
$$773$$ − 34.0000i − 1.22290i −0.791285 0.611448i $$-0.790588\pi$$
0.791285 0.611448i $$-0.209412\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ − 12.0000i − 0.430498i
$$778$$ 18.0000i 0.645331i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 0 0
$$783$$ − 8.00000i − 0.285897i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ −16.0000 −0.570701
$$787$$ − 20.0000i − 0.712923i −0.934310 0.356462i $$-0.883983\pi$$
0.934310 0.356462i $$-0.116017\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 48.0000 1.70885
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 1.00000i 0.0355335i
$$793$$ 32.0000i 1.13635i
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ −14.0000 −0.496217
$$797$$ − 26.0000i − 0.920967i −0.887668 0.460484i $$-0.847676\pi$$
0.887668 0.460484i $$-0.152324\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 10.0000i 0.353112i
$$803$$ 4.00000i 0.141157i
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ −40.0000 −1.40894
$$807$$ − 28.0000i − 0.985647i
$$808$$ − 12.0000i − 0.422159i
$$809$$ −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 $$-0.747841\pi$$
−0.702295 + 0.711886i $$0.747841\pi$$
$$812$$ 2.00000i 0.0701862i
$$813$$ − 56.0000i − 1.96401i
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ 4.00000i 0.139857i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 10.0000 0.347945
$$827$$ 20.0000i 0.695468i 0.937593 + 0.347734i $$0.113049\pi$$
−0.937593 + 0.347734i $$0.886951\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.416275
$$832$$ 4.00000i 0.138675i
$$833$$ 0 0
$$834$$ 40.0000 1.38509
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ − 40.0000i − 1.38260i
$$838$$ 30.0000i 1.03633i
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 10.0000i 0.344623i
$$843$$ 60.0000i 2.06651i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ 1.00000i 0.0343604i
$$848$$ − 14.0000i − 0.480762i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ 8.00000i 0.274075i
$$853$$ 4.00000i 0.136957i 0.997653 + 0.0684787i $$0.0218145\pi$$
−0.997653 + 0.0684787i $$0.978185\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 12.0000i − 0.409912i −0.978771 0.204956i $$-0.934295\pi$$
0.978771 0.204956i $$-0.0657052\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 16.0000i − 0.544962i
$$863$$ 40.0000i 1.36162i 0.732462 + 0.680808i $$0.238371\pi$$
−0.732462 + 0.680808i $$0.761629\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 10.0000 0.339814
$$867$$ 34.0000i 1.15470i
$$868$$ 10.0000i 0.339422i
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ − 14.0000i − 0.474100i
$$873$$ 6.00000i 0.203069i
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 28.0000i 0.944954i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 28.0000i 0.940148i 0.882627 + 0.470074i $$0.155773\pi$$
−0.882627 + 0.470074i $$0.844227\pi$$
$$888$$ 12.0000i 0.402694i
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ 14.0000i 0.468755i
$$893$$ 40.0000i 1.33855i
$$894$$ 44.0000 1.47158
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 32.0000i 1.06845i
$$898$$ − 6.00000i − 0.200223i
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −32.0000 −1.06313
$$907$$ − 48.0000i − 1.59381i −0.604102 0.796907i $$-0.706468\pi$$
0.604102 0.796907i $$-0.293532\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −4.00000 −0.132526 −0.0662630 0.997802i $$-0.521108\pi$$
−0.0662630 + 0.997802i $$0.521108\pi$$
$$912$$ − 8.00000i − 0.264906i
$$913$$ 4.00000i 0.132381i
$$914$$ −38.0000 −1.25693
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 8.00000i 0.264183i
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 32.0000 1.05444
$$922$$ 32.0000i 1.05386i
$$923$$ 16.0000i 0.526646i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ −12.0000 −0.394344
$$927$$ − 2.00000i − 0.0656886i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 12.0000i − 0.392862i
$$934$$ 14.0000 0.458094
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 52.0000i 1.69877i 0.527777 + 0.849383i $$0.323026\pi$$
−0.527777 + 0.849383i $$0.676974\pi$$
$$938$$ 8.00000i 0.261209i
$$939$$ 12.0000 0.391605
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 20.0000i 0.651635i
$$943$$ 0 0
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 32.0000i 1.03931i
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −36.0000 −1.16738
$$952$$ 0 0
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ − 4.00000i − 0.129302i
$$958$$ − 12.0000i − 0.387702i
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 24.0000i 0.773791i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ 24.0000i 0.771788i 0.922543 + 0.385894i $$0.126107\pi$$
−0.922543 + 0.385894i $$0.873893\pi$$
$$968$$ − 1.00000i − 0.0321412i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 30.0000 0.962746 0.481373 0.876516i $$-0.340138\pi$$
0.481373 + 0.876516i $$0.340138\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ − 20.0000i − 0.641171i
$$974$$ 12.0000 0.384505
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 22.0000i 0.703842i 0.936030 + 0.351921i $$0.114471\pi$$
−0.936030 + 0.351921i $$0.885529\pi$$
$$978$$ − 48.0000i − 1.53487i
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ − 28.0000i − 0.893516i
$$983$$ 26.0000i 0.829271i 0.909988 + 0.414636i $$0.136091\pi$$
−0.909988 + 0.414636i $$0.863909\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 20.0000i 0.636607i
$$988$$ − 16.0000i − 0.509028i
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ − 10.0000i − 0.317500i
$$993$$ − 40.0000i − 1.26936i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ 8.00000 0.253490
$$997$$ − 36.0000i − 1.14013i −0.821599 0.570066i $$-0.806918\pi$$
0.821599 0.570066i $$-0.193082\pi$$
$$998$$ 16.0000i 0.506471i
$$999$$ −24.0000 −0.759326
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 3850.2.c.d.1849.2 2
5.2 odd 4 154.2.a.b.1.1 1
5.3 odd 4 3850.2.a.o.1.1 1
5.4 even 2 inner 3850.2.c.d.1849.1 2
15.2 even 4 1386.2.a.f.1.1 1
20.7 even 4 1232.2.a.c.1.1 1
35.2 odd 12 1078.2.e.h.67.1 2
35.12 even 12 1078.2.e.l.67.1 2
35.17 even 12 1078.2.e.l.177.1 2
35.27 even 4 1078.2.a.b.1.1 1
35.32 odd 12 1078.2.e.h.177.1 2
40.27 even 4 4928.2.a.bf.1.1 1
40.37 odd 4 4928.2.a.d.1.1 1
55.32 even 4 1694.2.a.i.1.1 1
105.62 odd 4 9702.2.a.bz.1.1 1
140.27 odd 4 8624.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.b.1.1 1 5.2 odd 4
1078.2.a.b.1.1 1 35.27 even 4
1078.2.e.h.67.1 2 35.2 odd 12
1078.2.e.h.177.1 2 35.32 odd 12
1078.2.e.l.67.1 2 35.12 even 12
1078.2.e.l.177.1 2 35.17 even 12
1232.2.a.c.1.1 1 20.7 even 4
1386.2.a.f.1.1 1 15.2 even 4
1694.2.a.i.1.1 1 55.32 even 4
3850.2.a.o.1.1 1 5.3 odd 4
3850.2.c.d.1849.1 2 5.4 even 2 inner
3850.2.c.d.1849.2 2 1.1 even 1 trivial
4928.2.a.d.1.1 1 40.37 odd 4
4928.2.a.bf.1.1 1 40.27 even 4
8624.2.a.z.1.1 1 140.27 odd 4
9702.2.a.bz.1.1 1 105.62 odd 4