# Properties

 Label 5070.2.b.i.1351.1 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.i.1351.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000i q^{18} +4.00000i q^{19} +1.00000i q^{20} -4.00000i q^{21} -8.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -4.00000 q^{42} -12.0000 q^{43} -1.00000i q^{45} +8.00000i q^{46} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +10.0000 q^{53} -1.00000i q^{54} +4.00000 q^{56} +4.00000i q^{57} -2.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -2.00000 q^{68} -8.00000 q^{69} +4.00000i q^{70} -16.0000i q^{71} +1.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000i q^{83} +4.00000i q^{84} -2.00000i q^{85} +12.0000i q^{86} +2.00000 q^{87} +14.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} +4.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} - 8q^{14} + 2q^{16} + 4q^{17} - 16q^{23} - 2q^{25} + 2q^{27} + 4q^{29} - 2q^{30} - 8q^{35} - 2q^{36} + 8q^{38} + 2q^{40} - 8q^{42} - 24q^{43} + 2q^{48} - 18q^{49} + 4q^{51} + 20q^{53} + 8q^{56} - 20q^{61} - 16q^{62} - 2q^{64} - 4q^{68} - 16q^{69} - 4q^{74} - 2q^{75} - 16q^{79} + 2q^{81} - 12q^{82} + 4q^{87} - 2q^{90} + 16q^{92} + 8q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −4.00000 −1.06904
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 4.00000i − 0.872872i
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 4.00000i 0.755929i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 8.00000i − 1.43684i −0.695608 0.718421i $$-0.744865\pi$$
0.695608 0.718421i $$-0.255135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ − 2.00000i − 0.342997i
$$35$$ −4.00000 −0.676123
$$36$$ −1.00000 −0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 0 0
$$45$$ − 1.00000i − 0.149071i
$$46$$ 8.00000i 1.17954i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −9.00000 −1.28571
$$50$$ 1.00000i 0.141421i
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 4.00000i 0.529813i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ − 4.00000i − 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 4.00000i 0.478091i
$$71$$ − 16.0000i − 1.89885i −0.313993 0.949425i $$-0.601667\pi$$
0.313993 0.949425i $$-0.398333\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ − 4.00000i − 0.458831i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 4.00000i 0.436436i
$$85$$ − 2.00000i − 0.216930i
$$86$$ 12.0000i 1.29399i
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 14.0000i 1.48400i 0.670402 + 0.741999i $$0.266122\pi$$
−0.670402 + 0.741999i $$0.733878\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ − 10.0000i − 0.971286i
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000i 1.34096i 0.741929 + 0.670478i $$0.233911\pi$$
−0.741929 + 0.670478i $$0.766089\pi$$
$$110$$ 0 0
$$111$$ − 2.00000i − 0.189832i
$$112$$ − 4.00000i − 0.377964i
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 8.00000i 0.746004i
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 8.00000i − 0.733359i
$$120$$ 1.00000 0.0912871
$$121$$ 11.0000 1.00000
$$122$$ 10.0000i 0.905357i
$$123$$ − 6.00000i − 0.541002i
$$124$$ 8.00000i 0.718421i
$$125$$ 1.00000i 0.0894427i
$$126$$ −4.00000 −0.356348
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ −4.00000 −0.345547
$$135$$ − 1.00000i − 0.0860663i
$$136$$ 2.00000i 0.171499i
$$137$$ − 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 $$0.822307\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 2.00000i − 0.166091i
$$146$$ 6.00000 0.496564
$$147$$ −9.00000 −0.742307
$$148$$ 2.00000i 0.164399i
$$149$$ 10.0000i 0.819232i 0.912258 + 0.409616i $$0.134337\pi$$
−0.912258 + 0.409616i $$0.865663\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 8.00000i 0.651031i 0.945537 + 0.325515i $$0.105538\pi$$
−0.945537 + 0.325515i $$0.894462\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 10.0000 0.793052
$$160$$ −1.00000 −0.0790569
$$161$$ 32.0000i 2.52195i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 6.00000i 0.468521i
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 4.00000 0.308607
$$169$$ 0 0
$$170$$ −2.00000 −0.153393
$$171$$ 4.00000i 0.305888i
$$172$$ 12.0000 0.914991
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ − 2.00000i − 0.151620i
$$175$$ 4.00000i 0.302372i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ − 8.00000i − 0.589768i
$$185$$ −2.00000 −0.147043
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 4.00000i − 0.290957i
$$190$$ − 4.00000i − 0.290191i
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 18.0000i − 1.29567i −0.761781 0.647834i $$-0.775675\pi$$
0.761781 0.647834i $$-0.224325\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 4.00000i − 0.282138i
$$202$$ 10.0000i 0.703598i
$$203$$ − 8.00000i − 0.561490i
$$204$$ −2.00000 −0.140028
$$205$$ −6.00000 −0.419058
$$206$$ − 4.00000i − 0.278693i
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 4.00000i 0.276026i
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ − 16.0000i − 1.09630i
$$214$$ 12.0000i 0.820303i
$$215$$ 12.0000i 0.818393i
$$216$$ 1.00000i 0.0680414i
$$217$$ −32.0000 −2.17230
$$218$$ 14.0000 0.948200
$$219$$ 6.00000i 0.405442i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ 12.0000i 0.803579i 0.915732 + 0.401790i $$0.131612\pi$$
−0.915732 + 0.401790i $$0.868388\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ −1.00000 −0.0666667
$$226$$ 10.0000i 0.665190i
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ −8.00000 −0.518563
$$239$$ 24.0000i 1.55243i 0.630468 + 0.776215i $$0.282863\pi$$
−0.630468 + 0.776215i $$0.717137\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 2.00000i − 0.128831i −0.997923 0.0644157i $$-0.979482\pi$$
0.997923 0.0644157i $$-0.0205183\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 1.00000 0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 9.00000i 0.574989i
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ − 4.00000i − 0.253490i
$$250$$ 1.00000 0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 0 0
$$254$$ − 12.0000i − 0.752947i
$$255$$ − 2.00000i − 0.125245i
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 12.0000i 0.747087i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 8.00000i − 0.494242i
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ − 10.0000i − 0.614295i
$$266$$ − 16.0000i − 0.981023i
$$267$$ 14.0000i 0.856786i
$$268$$ 4.00000i 0.244339i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 32.0000i 1.94386i 0.235267 + 0.971931i $$0.424404\pi$$
−0.235267 + 0.971931i $$0.575596\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ − 8.00000i − 0.478947i
$$280$$ − 4.00000i − 0.239046i
$$281$$ − 18.0000i − 1.07379i −0.843649 0.536895i $$-0.819597\pi$$
0.843649 0.536895i $$-0.180403\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 16.0000i 0.949425i
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ − 6.00000i − 0.351726i
$$292$$ − 6.00000i − 0.351123i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 9.00000i 0.524891i
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 48.0000i 2.76667i
$$302$$ 8.00000 0.460348
$$303$$ −10.0000 −0.574485
$$304$$ 4.00000i 0.229416i
$$305$$ 10.0000i 0.572598i
$$306$$ − 2.00000i − 0.114332i
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 8.00000i 0.454369i
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ − 18.0000i − 1.01580i
$$315$$ −4.00000 −0.225374
$$316$$ 8.00000 0.450035
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ 0 0
$$320$$ 1.00000i 0.0559017i
$$321$$ −12.0000 −0.669775
$$322$$ 32.0000 1.78329
$$323$$ 8.00000i 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 14.0000i 0.774202i
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −16.0000 −0.875481
$$335$$ −4.00000 −0.218543
$$336$$ − 4.00000i − 0.218218i
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ −10.0000 −0.543125
$$340$$ 2.00000i 0.108465i
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ 8.00000i 0.431959i
$$344$$ − 12.0000i − 0.646997i
$$345$$ 8.00000i 0.430706i
$$346$$ 2.00000i 0.107521i
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ − 14.0000i − 0.749403i −0.927146 0.374701i $$-0.877745\pi$$
0.927146 0.374701i $$-0.122255\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ − 14.0000i − 0.741999i
$$357$$ − 8.00000i − 0.423405i
$$358$$ 16.0000i 0.845626i
$$359$$ − 24.0000i − 1.26667i −0.773877 0.633336i $$-0.781685\pi$$
0.773877 0.633336i $$-0.218315\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 3.00000 0.157895
$$362$$ − 2.00000i − 0.105118i
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 10.0000i 0.522708i
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ − 6.00000i − 0.312348i
$$370$$ 2.00000i 0.103975i
$$371$$ − 40.0000i − 2.07670i
$$372$$ 8.00000i 0.414781i
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ 1.00000i 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −4.00000 −0.205738
$$379$$ 20.0000i 1.02733i 0.857991 + 0.513665i $$0.171713\pi$$
−0.857991 + 0.513665i $$0.828287\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 12.0000 0.614779
$$382$$ − 8.00000i − 0.409316i
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ −12.0000 −0.609994
$$388$$ 6.00000i 0.304604i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 9.00000i − 0.454569i
$$393$$ 8.00000 0.403547
$$394$$ −6.00000 −0.302276
$$395$$ 8.00000i 0.402524i
$$396$$ 0 0
$$397$$ − 10.0000i − 0.501886i −0.968002 0.250943i $$-0.919259\pi$$
0.968002 0.250943i $$-0.0807406\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 16.0000 0.801002
$$400$$ −1.00000 −0.0500000
$$401$$ − 26.0000i − 1.29838i −0.760627 0.649189i $$-0.775108\pi$$
0.760627 0.649189i $$-0.224892\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 2.00000i 0.0990148i
$$409$$ 10.0000i 0.494468i 0.968956 + 0.247234i $$0.0795217\pi$$
−0.968956 + 0.247234i $$0.920478\pi$$
$$410$$ 6.00000i 0.296319i
$$411$$ − 6.00000i − 0.295958i
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 8.00000i 0.393179i
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ −20.0000 −0.979404
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 4.00000 0.195180
$$421$$ 30.0000i 1.46211i 0.682318 + 0.731055i $$0.260972\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ 0 0
$$424$$ 10.0000i 0.485643i
$$425$$ −2.00000 −0.0970143
$$426$$ −16.0000 −0.775203
$$427$$ 40.0000i 1.93574i
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 12.0000 0.578691
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −10.0000 −0.480569 −0.240285 0.970702i $$-0.577241\pi$$
−0.240285 + 0.970702i $$0.577241\pi$$
$$434$$ 32.0000i 1.53605i
$$435$$ − 2.00000i − 0.0958927i
$$436$$ − 14.0000i − 0.670478i
$$437$$ − 32.0000i − 1.53077i
$$438$$ 6.00000 0.286691
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 2.00000i 0.0949158i
$$445$$ 14.0000 0.663664
$$446$$ 12.0000 0.568216
$$447$$ 10.0000i 0.472984i
$$448$$ 4.00000i 0.188982i
$$449$$ 6.00000i 0.283158i 0.989927 + 0.141579i $$0.0452178\pi$$
−0.989927 + 0.141579i $$0.954782\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 0 0
$$452$$ 10.0000 0.470360
$$453$$ 8.00000i 0.375873i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 2.00000 0.0933520
$$460$$ − 8.00000i − 0.373002i
$$461$$ 2.00000i 0.0931493i 0.998915 + 0.0465746i $$0.0148305\pi$$
−0.998915 + 0.0465746i $$0.985169\pi$$
$$462$$ 0 0
$$463$$ − 20.0000i − 0.929479i −0.885448 0.464739i $$-0.846148\pi$$
0.885448 0.464739i $$-0.153852\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ −8.00000 −0.370991
$$466$$ − 26.0000i − 1.20443i
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 8.00000i 0.367452i
$$475$$ − 4.00000i − 0.183533i
$$476$$ 8.00000i 0.366679i
$$477$$ 10.0000 0.457869
$$478$$ 24.0000 1.09773
$$479$$ − 8.00000i − 0.365529i −0.983157 0.182765i $$-0.941495\pi$$
0.983157 0.182765i $$-0.0585046\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −2.00000 −0.0910975
$$483$$ 32.0000i 1.45605i
$$484$$ −11.0000 −0.500000
$$485$$ −6.00000 −0.272446
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 20.0000i 0.904431i
$$490$$ 9.00000 0.406579
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 8.00000i − 0.359211i
$$497$$ −64.0000 −2.87079
$$498$$ −4.00000 −0.179244
$$499$$ − 44.0000i − 1.96971i −0.173379 0.984855i $$-0.555468\pi$$
0.173379 0.984855i $$-0.444532\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 16.0000i − 0.714827i
$$502$$ 0 0
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 10.0000i 0.444994i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ − 38.0000i − 1.68432i −0.539227 0.842160i $$-0.681284\pi$$
0.539227 0.842160i $$-0.318716\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ 24.0000 1.06170
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 22.0000i 0.970378i
$$515$$ − 4.00000i − 0.176261i
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 8.00000i 0.351500i
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ 2.00000 0.0876216 0.0438108 0.999040i $$-0.486050\pi$$
0.0438108 + 0.999040i $$0.486050\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 4.00000i 0.174574i
$$526$$ − 24.0000i − 1.04645i
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −10.0000 −0.434372
$$531$$ 0 0
$$532$$ −16.0000 −0.693688
$$533$$ 0 0
$$534$$ 14.0000 0.605839
$$535$$ 12.0000i 0.518805i
$$536$$ 4.00000 0.172774
$$537$$ −16.0000 −0.690451
$$538$$ 14.0000i 0.603583i
$$539$$ 0 0
$$540$$ 1.00000i 0.0430331i
$$541$$ 2.00000i 0.0859867i 0.999075 + 0.0429934i $$0.0136894\pi$$
−0.999075 + 0.0429934i $$0.986311\pi$$
$$542$$ 32.0000 1.37452
$$543$$ 2.00000 0.0858282
$$544$$ − 2.00000i − 0.0857493i
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 8.00000i 0.340811i
$$552$$ − 8.00000i − 0.340503i
$$553$$ 32.0000i 1.36078i
$$554$$ 18.0000i 0.764747i
$$555$$ −2.00000 −0.0848953
$$556$$ 20.0000 0.848189
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ 10.0000i 0.420703i
$$566$$ 20.0000i 0.840663i
$$567$$ − 4.00000i − 0.167984i
$$568$$ 16.0000 0.671345
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ − 4.00000i − 0.167542i
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ 24.0000i 1.00174i
$$575$$ 8.00000 0.333623
$$576$$ −1.00000 −0.0416667
$$577$$ − 22.0000i − 0.915872i −0.888985 0.457936i $$-0.848589\pi$$
0.888985 0.457936i $$-0.151411\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ − 18.0000i − 0.748054i
$$580$$ 2.00000i 0.0830455i
$$581$$ −16.0000 −0.663792
$$582$$ −6.00000 −0.248708
$$583$$ 0 0
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ − 6.00000i − 0.246807i
$$592$$ − 2.00000i − 0.0821995i
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 0 0
$$595$$ −8.00000 −0.327968
$$596$$ − 10.0000i − 0.409616i
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 48.0000 1.95633
$$603$$ − 4.00000i − 0.162893i
$$604$$ − 8.00000i − 0.325515i
$$605$$ − 11.0000i − 0.447214i
$$606$$ 10.0000i 0.406222i
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 4.00000 0.162221
$$609$$ − 8.00000i − 0.324176i
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 12.0000 0.484281
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 28.0000i 1.12542i 0.826656 + 0.562708i $$0.190240\pi$$
−0.826656 + 0.562708i $$0.809760\pi$$
$$620$$ 8.00000 0.321288
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 56.0000 2.24359
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 18.0000i − 0.719425i
$$627$$ 0 0
$$628$$ −18.0000 −0.718278
$$629$$ − 4.00000i − 0.159490i
$$630$$ 4.00000i 0.159364i
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ −12.0000 −0.476957
$$634$$ −14.0000 −0.556011
$$635$$ − 12.0000i − 0.476205i
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 16.0000i − 0.632950i
$$640$$ 1.00000 0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ − 32.0000i − 1.26098i
$$645$$ 12.0000i 0.472500i
$$646$$ 8.00000 0.314756
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −32.0000 −1.25418
$$652$$ − 20.0000i − 0.783260i
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 14.0000 0.547443
$$655$$ − 8.00000i − 0.312586i
$$656$$ − 6.00000i − 0.234261i
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 10.0000i 0.388955i 0.980907 + 0.194477i $$0.0623011\pi$$
−0.980907 + 0.194477i $$0.937699\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ − 16.0000i − 0.620453i
$$666$$ −2.00000 −0.0774984
$$667$$ −16.0000 −0.619522
$$668$$ 16.0000i 0.619059i
$$669$$ 12.0000i 0.463947i
$$670$$ 4.00000i 0.154533i
$$671$$ 0 0
$$672$$ −4.00000 −0.154303
$$673$$ 38.0000 1.46479 0.732396 0.680879i $$-0.238402\pi$$
0.732396 + 0.680879i $$0.238402\pi$$
$$674$$ − 22.0000i − 0.847408i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 26.0000 0.999261 0.499631 0.866239i $$-0.333469\pi$$
0.499631 + 0.866239i $$0.333469\pi$$
$$678$$ 10.0000i 0.384048i
$$679$$ −24.0000 −0.921035
$$680$$ 2.00000 0.0766965
$$681$$ − 12.0000i − 0.459841i
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ − 4.00000i − 0.152944i
$$685$$ −6.00000 −0.229248
$$686$$ 8.00000 0.305441
$$687$$ − 14.0000i − 0.534133i
$$688$$ −12.0000 −0.457496
$$689$$ 0 0
$$690$$ 8.00000 0.304555
$$691$$ 20.0000i 0.760836i 0.924815 + 0.380418i $$0.124220\pi$$
−0.924815 + 0.380418i $$0.875780\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ 4.00000i 0.151838i
$$695$$ 20.0000i 0.758643i
$$696$$ 2.00000i 0.0758098i
$$697$$ − 12.0000i − 0.454532i
$$698$$ −14.0000 −0.529908
$$699$$ 26.0000 0.983410
$$700$$ − 4.00000i − 0.151186i
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ 40.0000i 1.50435i
$$708$$ 0 0
$$709$$ 34.0000i 1.27690i 0.769665 + 0.638448i $$0.220423\pi$$
−0.769665 + 0.638448i $$0.779577\pi$$
$$710$$ 16.0000i 0.600469i
$$711$$ −8.00000 −0.300023
$$712$$ −14.0000 −0.524672
$$713$$ 64.0000i 2.39682i
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 16.0000 0.597948
$$717$$ 24.0000i 0.896296i
$$718$$ −24.0000 −0.895672
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 16.0000i − 0.595871i
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −2.00000 −0.0743294
$$725$$ −2.00000 −0.0742781
$$726$$ − 11.0000i − 0.408248i
$$727$$ −52.0000 −1.92857 −0.964287 0.264861i $$-0.914674\pi$$
−0.964287 + 0.264861i $$0.914674\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 6.00000i − 0.222070i
$$731$$ −24.0000 −0.887672
$$732$$ 10.0000 0.369611
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ − 4.00000i − 0.147643i
$$735$$ 9.00000i 0.331970i
$$736$$ 8.00000i 0.294884i
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ − 4.00000i − 0.147142i −0.997290 0.0735712i $$-0.976560\pi$$
0.997290 0.0735712i $$-0.0234396\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ −40.0000 −1.46845
$$743$$ − 40.0000i − 1.46746i −0.679442 0.733729i $$-0.737778\pi$$
0.679442 0.733729i $$-0.262222\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 10.0000 0.366372
$$746$$ 22.0000i 0.805477i
$$747$$ − 4.00000i − 0.146352i
$$748$$ 0 0
$$749$$ 48.0000i 1.75388i
$$750$$ 1.00000 0.0365148
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 4.00000i 0.145479i
$$757$$ 50.0000 1.81728 0.908640 0.417579i $$-0.137121\pi$$
0.908640 + 0.417579i $$0.137121\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 4.00000i 0.145095i
$$761$$ − 18.0000i − 0.652499i −0.945284 0.326250i $$-0.894215\pi$$
0.945284 0.326250i $$-0.105785\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ 56.0000 2.02734
$$764$$ −8.00000 −0.289430
$$765$$ − 2.00000i − 0.0723102i
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 2.00000i 0.0721218i 0.999350 + 0.0360609i $$0.0114810\pi$$
−0.999350 + 0.0360609i $$0.988519\pi$$
$$770$$ 0 0
$$771$$ −22.0000 −0.792311
$$772$$ 18.0000i 0.647834i
$$773$$ 10.0000i 0.359675i 0.983696 + 0.179838i $$0.0575572\pi$$
−0.983696 + 0.179838i $$0.942443\pi$$
$$774$$ 12.0000i 0.431331i
$$775$$ 8.00000i 0.287368i
$$776$$ 6.00000 0.215387
$$777$$ −8.00000 −0.286998
$$778$$ − 14.0000i − 0.501924i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 16.0000i 0.572159i
$$783$$ 2.00000 0.0714742
$$784$$ −9.00000 −0.321429
$$785$$ − 18.0000i − 0.642448i
$$786$$ − 8.00000i − 0.285351i
$$787$$ − 44.0000i − 1.56843i −0.620489 0.784215i $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 8.00000 0.284627
$$791$$ 40.0000i 1.42224i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −10.0000 −0.354887
$$795$$ − 10.0000i − 0.354663i
$$796$$ −8.00000 −0.283552
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ 0 0
$$800$$ 1.00000i 0.0353553i
$$801$$ 14.0000i 0.494666i
$$802$$ −26.0000 −0.918092
$$803$$ 0 0
$$804$$ 4.00000i 0.141069i
$$805$$ 32.0000 1.12785
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ − 10.0000i − 0.351799i
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 4.00000i 0.140459i 0.997531 + 0.0702295i $$0.0223732\pi$$
−0.997531 + 0.0702295i $$0.977627\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 32.0000i 1.12229i
$$814$$ 0 0
$$815$$ 20.0000 0.700569
$$816$$ 2.00000 0.0700140
$$817$$ − 48.0000i − 1.67931i
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 2.00000i 0.0698005i 0.999391 + 0.0349002i $$0.0111113\pi$$
−0.999391 + 0.0349002i $$0.988889\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 4.00000i 0.138842i
$$831$$ −18.0000 −0.624413
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 20.0000i 0.692543i
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 24.0000i 0.829066i
$$839$$ 8.00000i 0.276191i 0.990419 + 0.138095i $$0.0440980\pi$$
−0.990419 + 0.138095i $$0.955902\pi$$
$$840$$ − 4.00000i − 0.138013i
$$841$$ −25.0000 −0.862069
$$842$$ 30.0000 1.03387
$$843$$ − 18.0000i − 0.619953i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ 10.0000 0.343401
$$849$$ −20.0000 −0.686398
$$850$$ 2.00000i 0.0685994i
$$851$$ 16.0000i 0.548473i
$$852$$ 16.0000i 0.548151i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 4.00000 0.136797
$$856$$ − 12.0000i − 0.410152i
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ − 12.0000i − 0.409197i
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 2.00000i 0.0680020i
$$866$$ 10.0000i 0.339814i
$$867$$ −13.0000 −0.441503
$$868$$ 32.0000 1.08615
$$869$$ 0 0
$$870$$ −2.00000 −0.0678064
$$871$$ 0 0
$$872$$ −14.0000 −0.474100
$$873$$ − 6.00000i − 0.203069i
$$874$$ −32.0000 −1.08242
$$875$$ 4.00000 0.135225
$$876$$ − 6.00000i − 0.202721i
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ 6.00000i 0.202375i
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 36.0000i 1.20944i
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ − 48.0000i − 1.60987i
$$890$$ − 14.0000i − 0.469281i
$$891$$ 0 0
$$892$$ − 12.0000i − 0.401790i
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ 16.0000i 0.534821i
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ − 16.0000i − 0.533630i
$$900$$ 1.00000 0.0333333
$$901$$ 20.0000 0.666297
$$902$$ 0 0
$$903$$ 48.0000i 1.59734i
$$904$$ − 10.0000i − 0.332595i
$$905$$ − 2.00000i − 0.0664822i
$$906$$ 8.00000 0.265782
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ 18.0000 0.595387
$$915$$ 10.0000i 0.330590i
$$916$$ 14.0000i 0.462573i
$$917$$ − 32.0000i − 1.05673i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 12.0000i 0.395413i
$$922$$ 2.00000 0.0658665
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000i 0.0657596i
$$926$$ −20.0000 −0.657241
$$927$$ 4.00000 0.131377
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 26.0000i 0.853032i 0.904480 + 0.426516i $$0.140259\pi$$
−0.904480 + 0.426516i $$0.859741\pi$$
$$930$$ 8.00000i 0.262330i
$$931$$ − 36.0000i − 1.17985i
$$932$$ −26.0000 −0.851658
$$933$$ 0 0
$$934$$ − 36.0000i − 1.17796i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.00000 −0.196011 −0.0980057 0.995186i $$-0.531246\pi$$
−0.0980057 + 0.995186i $$0.531246\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ − 30.0000i − 0.977972i −0.872292 0.488986i $$-0.837367\pi$$
0.872292 0.488986i $$-0.162633\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ 48.0000i 1.56310i
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ −4.00000 −0.129777
$$951$$ − 14.0000i − 0.453981i
$$952$$ 8.00000 0.259281
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ − 10.0000i − 0.323762i
$$955$$ − 8.00000i − 0.258874i
$$956$$ − 24.0000i − 0.776215i
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ −24.0000 −0.775000
$$960$$ 1.00000i 0.0322749i
$$961$$ −33.0000 −1.06452
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 2.00000i 0.0644157i
$$965$$ −18.0000 −0.579441
$$966$$ 32.0000 1.02958
$$967$$ − 28.0000i − 0.900419i −0.892923 0.450210i $$-0.851349\pi$$
0.892923 0.450210i $$-0.148651\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 8.00000i 0.256997i
$$970$$ 6.00000i 0.192648i
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 80.0000i 2.56468i
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 38.0000i 1.21573i 0.794041 + 0.607864i $$0.207973\pi$$
−0.794041 + 0.607864i $$0.792027\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ − 9.00000i − 0.287494i
$$981$$ 14.0000i 0.446986i
$$982$$ 0 0
$$983$$ − 56.0000i − 1.78612i −0.449935 0.893061i $$-0.648553\pi$$
0.449935 0.893061i $$-0.351447\pi$$
$$984$$ 6.00000 0.191273
$$985$$ −6.00000 −0.191176
$$986$$ − 4.00000i − 0.127386i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 96.0000 3.05262
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ − 20.0000i − 0.634681i
$$994$$ 64.0000i 2.02996i
$$995$$ − 8.00000i − 0.253617i
$$996$$ 4.00000i 0.126745i
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −44.0000 −1.39280
$$999$$ − 2.00000i − 0.0632772i
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 5070.2.b.i.1351.1 2
13.5 odd 4 390.2.a.c.1.1 1
13.8 odd 4 5070.2.a.u.1.1 1
13.12 even 2 inner 5070.2.b.i.1351.2 2
39.5 even 4 1170.2.a.n.1.1 1
52.31 even 4 3120.2.a.a.1.1 1
65.18 even 4 1950.2.e.e.1249.2 2
65.44 odd 4 1950.2.a.n.1.1 1
65.57 even 4 1950.2.e.e.1249.1 2
156.83 odd 4 9360.2.a.bc.1.1 1
195.44 even 4 5850.2.a.c.1.1 1
195.83 odd 4 5850.2.e.m.5149.1 2
195.122 odd 4 5850.2.e.m.5149.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.c.1.1 1 13.5 odd 4
1170.2.a.n.1.1 1 39.5 even 4
1950.2.a.n.1.1 1 65.44 odd 4
1950.2.e.e.1249.1 2 65.57 even 4
1950.2.e.e.1249.2 2 65.18 even 4
3120.2.a.a.1.1 1 52.31 even 4
5070.2.a.u.1.1 1 13.8 odd 4
5070.2.b.i.1351.1 2 1.1 even 1 trivial
5070.2.b.i.1351.2 2 13.12 even 2 inner
5850.2.a.c.1.1 1 195.44 even 4
5850.2.e.m.5149.1 2 195.83 odd 4
5850.2.e.m.5149.2 2 195.122 odd 4
9360.2.a.bc.1.1 1 156.83 odd 4