# Properties

 Label 5070.2.b.w.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-0.445042i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.w.1351.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.44504i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -3.44504i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.24698i q^{11} -1.00000 q^{12} -3.44504 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.78986 q^{17} -1.00000i q^{18} +6.26875i q^{19} +1.00000i q^{20} -3.44504i q^{21} -4.24698 q^{22} +1.30798 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.44504i q^{28} -9.14675 q^{29} -1.00000 q^{30} +3.75302i q^{31} -1.00000i q^{32} -4.24698i q^{33} +6.78986i q^{34} -3.44504 q^{35} -1.00000 q^{36} -6.82908i q^{37} +6.26875 q^{38} +1.00000 q^{40} +4.26875i q^{41} -3.44504 q^{42} -3.07069 q^{43} +4.24698i q^{44} -1.00000i q^{45} -1.30798i q^{46} -7.76809i q^{47} +1.00000 q^{48} -4.86831 q^{49} +1.00000i q^{50} -6.78986 q^{51} -8.93900 q^{53} -1.00000i q^{54} -4.24698 q^{55} +3.44504 q^{56} +6.26875i q^{57} +9.14675i q^{58} +10.3327i q^{59} +1.00000i q^{60} +2.53319 q^{61} +3.75302 q^{62} -3.44504i q^{63} -1.00000 q^{64} -4.24698 q^{66} -0.0760644i q^{67} +6.78986 q^{68} +1.30798 q^{69} +3.44504i q^{70} +0.374354i q^{71} +1.00000i q^{72} +16.7114i q^{73} -6.82908 q^{74} -1.00000 q^{75} -6.26875i q^{76} -14.6310 q^{77} -1.33513 q^{79} -1.00000i q^{80} +1.00000 q^{81} +4.26875 q^{82} -0.740939i q^{83} +3.44504i q^{84} +6.78986i q^{85} +3.07069i q^{86} -9.14675 q^{87} +4.24698 q^{88} -13.3274i q^{89} -1.00000 q^{90} -1.30798 q^{92} +3.75302i q^{93} -7.76809 q^{94} +6.26875 q^{95} -1.00000i q^{96} -13.1903i q^{97} +4.86831i q^{98} -4.24698i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} - 6q^{12} - 20q^{14} + 6q^{16} + 6q^{17} - 16q^{22} + 18q^{23} - 6q^{25} + 6q^{27} - 6q^{30} - 20q^{35} - 6q^{36} + 22q^{38} + 6q^{40} - 20q^{42} + 6q^{43} + 6q^{48} - 34q^{49} + 6q^{51} - 34q^{53} - 16q^{55} + 20q^{56} + 22q^{61} + 32q^{62} - 6q^{64} - 16q^{66} - 6q^{68} + 18q^{69} - 20q^{74} - 6q^{75} - 58q^{77} - 6q^{79} + 6q^{81} + 10q^{82} + 16q^{88} - 6q^{90} - 18q^{92} - 6q^{94} + 22q^{95} + O(q^{100})$$

## 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
$$7$$ − 3.44504i − 1.30210i −0.759033 0.651052i $$-0.774328\pi$$
0.759033 0.651052i $$-0.225672\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ − 4.24698i − 1.28051i −0.768161 0.640256i $$-0.778828\pi$$
0.768161 0.640256i $$-0.221172\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −3.44504 −0.920726
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −6.78986 −1.64678 −0.823391 0.567474i $$-0.807921\pi$$
−0.823391 + 0.567474i $$0.807921\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 6.26875i 1.43815i 0.694933 + 0.719075i $$0.255434\pi$$
−0.694933 + 0.719075i $$0.744566\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 3.44504i − 0.751770i
$$22$$ −4.24698 −0.905459
$$23$$ 1.30798 0.272732 0.136366 0.990658i $$-0.456458\pi$$
0.136366 + 0.990658i $$0.456458\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 3.44504i 0.651052i
$$29$$ −9.14675 −1.69851 −0.849255 0.527984i $$-0.822948\pi$$
−0.849255 + 0.527984i $$0.822948\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 3.75302i 0.674062i 0.941493 + 0.337031i $$0.109423\pi$$
−0.941493 + 0.337031i $$0.890577\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.24698i − 0.739304i
$$34$$ 6.78986i 1.16445i
$$35$$ −3.44504 −0.582318
$$36$$ −1.00000 −0.166667
$$37$$ − 6.82908i − 1.12269i −0.827580 0.561347i $$-0.810283\pi$$
0.827580 0.561347i $$-0.189717\pi$$
$$38$$ 6.26875 1.01693
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 4.26875i 0.666667i 0.942809 + 0.333333i $$0.108173\pi$$
−0.942809 + 0.333333i $$0.891827\pi$$
$$42$$ −3.44504 −0.531582
$$43$$ −3.07069 −0.468275 −0.234138 0.972203i $$-0.575227\pi$$
−0.234138 + 0.972203i $$0.575227\pi$$
$$44$$ 4.24698i 0.640256i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 1.30798i − 0.192851i
$$47$$ − 7.76809i − 1.13309i −0.824030 0.566546i $$-0.808279\pi$$
0.824030 0.566546i $$-0.191721\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −4.86831 −0.695473
$$50$$ 1.00000i 0.141421i
$$51$$ −6.78986 −0.950770
$$52$$ 0 0
$$53$$ −8.93900 −1.22787 −0.613933 0.789358i $$-0.710414\pi$$
−0.613933 + 0.789358i $$0.710414\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ −4.24698 −0.572663
$$56$$ 3.44504 0.460363
$$57$$ 6.26875i 0.830316i
$$58$$ 9.14675i 1.20103i
$$59$$ 10.3327i 1.34521i 0.740003 + 0.672604i $$0.234824\pi$$
−0.740003 + 0.672604i $$0.765176\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 2.53319 0.324341 0.162171 0.986763i $$-0.448150\pi$$
0.162171 + 0.986763i $$0.448150\pi$$
$$62$$ 3.75302 0.476634
$$63$$ − 3.44504i − 0.434034i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.24698 −0.522767
$$67$$ − 0.0760644i − 0.00929275i −0.999989 0.00464637i $$-0.998521\pi$$
0.999989 0.00464637i $$-0.00147899\pi$$
$$68$$ 6.78986 0.823391
$$69$$ 1.30798 0.157462
$$70$$ 3.44504i 0.411761i
$$71$$ 0.374354i 0.0444277i 0.999753 + 0.0222138i $$0.00707147\pi$$
−0.999753 + 0.0222138i $$0.992929\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 16.7114i 1.95592i 0.208789 + 0.977961i $$0.433048\pi$$
−0.208789 + 0.977961i $$0.566952\pi$$
$$74$$ −6.82908 −0.793865
$$75$$ −1.00000 −0.115470
$$76$$ − 6.26875i − 0.719075i
$$77$$ −14.6310 −1.66736
$$78$$ 0 0
$$79$$ −1.33513 −0.150213 −0.0751067 0.997176i $$-0.523930\pi$$
−0.0751067 + 0.997176i $$0.523930\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 4.26875 0.471405
$$83$$ − 0.740939i − 0.0813286i −0.999173 0.0406643i $$-0.987053\pi$$
0.999173 0.0406643i $$-0.0129474\pi$$
$$84$$ 3.44504i 0.375885i
$$85$$ 6.78986i 0.736463i
$$86$$ 3.07069i 0.331121i
$$87$$ −9.14675 −0.980635
$$88$$ 4.24698 0.452730
$$89$$ − 13.3274i − 1.41270i −0.707864 0.706348i $$-0.750341\pi$$
0.707864 0.706348i $$-0.249659\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −1.30798 −0.136366
$$93$$ 3.75302i 0.389170i
$$94$$ −7.76809 −0.801217
$$95$$ 6.26875 0.643160
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 13.1903i − 1.33927i −0.742690 0.669636i $$-0.766450\pi$$
0.742690 0.669636i $$-0.233550\pi$$
$$98$$ 4.86831i 0.491774i
$$99$$ − 4.24698i − 0.426838i
$$100$$ 1.00000 0.100000
$$101$$ 15.0422 1.49676 0.748378 0.663272i $$-0.230833\pi$$
0.748378 + 0.663272i $$0.230833\pi$$
$$102$$ 6.78986i 0.672296i
$$103$$ 8.92154 0.879066 0.439533 0.898227i $$-0.355144\pi$$
0.439533 + 0.898227i $$0.355144\pi$$
$$104$$ 0 0
$$105$$ −3.44504 −0.336202
$$106$$ 8.93900i 0.868233i
$$107$$ 14.6746 1.41864 0.709322 0.704885i $$-0.249001\pi$$
0.709322 + 0.704885i $$0.249001\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 9.66786i 0.926013i 0.886355 + 0.463006i $$0.153229\pi$$
−0.886355 + 0.463006i $$0.846771\pi$$
$$110$$ 4.24698i 0.404934i
$$111$$ − 6.82908i − 0.648188i
$$112$$ − 3.44504i − 0.325526i
$$113$$ −11.3448 −1.06723 −0.533615 0.845727i $$-0.679167\pi$$
−0.533615 + 0.845727i $$0.679167\pi$$
$$114$$ 6.26875 0.587122
$$115$$ − 1.30798i − 0.121970i
$$116$$ 9.14675 0.849255
$$117$$ 0 0
$$118$$ 10.3327 0.951205
$$119$$ 23.3913i 2.14428i
$$120$$ 1.00000 0.0912871
$$121$$ −7.03684 −0.639712
$$122$$ − 2.53319i − 0.229344i
$$123$$ 4.26875i 0.384900i
$$124$$ − 3.75302i − 0.337031i
$$125$$ 1.00000i 0.0894427i
$$126$$ −3.44504 −0.306909
$$127$$ 2.54288 0.225644 0.112822 0.993615i $$-0.464011\pi$$
0.112822 + 0.993615i $$0.464011\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −3.07069 −0.270359
$$130$$ 0 0
$$131$$ −10.8509 −0.948044 −0.474022 0.880513i $$-0.657198\pi$$
−0.474022 + 0.880513i $$0.657198\pi$$
$$132$$ 4.24698i 0.369652i
$$133$$ 21.5961 1.87262
$$134$$ −0.0760644 −0.00657096
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 6.78986i − 0.582225i
$$137$$ 18.6407i 1.59258i 0.604913 + 0.796292i $$0.293208\pi$$
−0.604913 + 0.796292i $$0.706792\pi$$
$$138$$ − 1.30798i − 0.111343i
$$139$$ 3.53750 0.300047 0.150023 0.988682i $$-0.452065\pi$$
0.150023 + 0.988682i $$0.452065\pi$$
$$140$$ 3.44504 0.291159
$$141$$ − 7.76809i − 0.654191i
$$142$$ 0.374354 0.0314151
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 9.14675i 0.759596i
$$146$$ 16.7114 1.38305
$$147$$ −4.86831 −0.401532
$$148$$ 6.82908i 0.561347i
$$149$$ − 16.6843i − 1.36683i −0.730031 0.683414i $$-0.760495\pi$$
0.730031 0.683414i $$-0.239505\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 18.3327i − 1.49190i −0.666004 0.745948i $$-0.731997\pi$$
0.666004 0.745948i $$-0.268003\pi$$
$$152$$ −6.26875 −0.508463
$$153$$ −6.78986 −0.548927
$$154$$ 14.6310i 1.17900i
$$155$$ 3.75302 0.301450
$$156$$ 0 0
$$157$$ −18.6015 −1.48456 −0.742280 0.670090i $$-0.766256\pi$$
−0.742280 + 0.670090i $$0.766256\pi$$
$$158$$ 1.33513i 0.106217i
$$159$$ −8.93900 −0.708909
$$160$$ −1.00000 −0.0790569
$$161$$ − 4.50604i − 0.355126i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 18.8944i − 1.47992i −0.672649 0.739962i $$-0.734844\pi$$
0.672649 0.739962i $$-0.265156\pi$$
$$164$$ − 4.26875i − 0.333333i
$$165$$ −4.24698 −0.330627
$$166$$ −0.740939 −0.0575080
$$167$$ 4.24698i 0.328641i 0.986407 + 0.164321i $$0.0525432\pi$$
−0.986407 + 0.164321i $$0.947457\pi$$
$$168$$ 3.44504 0.265791
$$169$$ 0 0
$$170$$ 6.78986 0.520758
$$171$$ 6.26875i 0.479383i
$$172$$ 3.07069 0.234138
$$173$$ −1.16852 −0.0888411 −0.0444206 0.999013i $$-0.514144\pi$$
−0.0444206 + 0.999013i $$0.514144\pi$$
$$174$$ 9.14675i 0.693413i
$$175$$ 3.44504i 0.260421i
$$176$$ − 4.24698i − 0.320128i
$$177$$ 10.3327i 0.776656i
$$178$$ −13.3274 −0.998928
$$179$$ 10.1836 0.761157 0.380579 0.924749i $$-0.375725\pi$$
0.380579 + 0.924749i $$0.375725\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −10.3351 −0.768204 −0.384102 0.923291i $$-0.625489\pi$$
−0.384102 + 0.923291i $$0.625489\pi$$
$$182$$ 0 0
$$183$$ 2.53319 0.187259
$$184$$ 1.30798i 0.0964255i
$$185$$ −6.82908 −0.502084
$$186$$ 3.75302 0.275185
$$187$$ 28.8364i 2.10872i
$$188$$ 7.76809i 0.566546i
$$189$$ − 3.44504i − 0.250590i
$$190$$ − 6.26875i − 0.454783i
$$191$$ −9.43296 −0.682545 −0.341273 0.939964i $$-0.610858\pi$$
−0.341273 + 0.939964i $$0.610858\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ − 22.9191i − 1.64976i −0.565310 0.824878i $$-0.691244\pi$$
0.565310 0.824878i $$-0.308756\pi$$
$$194$$ −13.1903 −0.947008
$$195$$ 0 0
$$196$$ 4.86831 0.347737
$$197$$ − 14.3502i − 1.02241i −0.859459 0.511204i $$-0.829200\pi$$
0.859459 0.511204i $$-0.170800\pi$$
$$198$$ −4.24698 −0.301820
$$199$$ −14.2524 −1.01032 −0.505161 0.863025i $$-0.668567\pi$$
−0.505161 + 0.863025i $$0.668567\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 0.0760644i − 0.00536517i
$$202$$ − 15.0422i − 1.05837i
$$203$$ 31.5109i 2.21163i
$$204$$ 6.78986 0.475385
$$205$$ 4.26875 0.298142
$$206$$ − 8.92154i − 0.621593i
$$207$$ 1.30798 0.0909108
$$208$$ 0 0
$$209$$ 26.6233 1.84157
$$210$$ 3.44504i 0.237730i
$$211$$ 0.396125 0.0272703 0.0136352 0.999907i $$-0.495660\pi$$
0.0136352 + 0.999907i $$0.495660\pi$$
$$212$$ 8.93900 0.613933
$$213$$ 0.374354i 0.0256503i
$$214$$ − 14.6746i − 1.00313i
$$215$$ 3.07069i 0.209419i
$$216$$ 1.00000i 0.0680414i
$$217$$ 12.9293 0.877699
$$218$$ 9.66786 0.654790
$$219$$ 16.7114i 1.12925i
$$220$$ 4.24698 0.286331
$$221$$ 0 0
$$222$$ −6.82908 −0.458338
$$223$$ 1.49635i 0.100203i 0.998744 + 0.0501016i $$0.0159545\pi$$
−0.998744 + 0.0501016i $$0.984045\pi$$
$$224$$ −3.44504 −0.230182
$$225$$ −1.00000 −0.0666667
$$226$$ 11.3448i 0.754646i
$$227$$ 25.2150i 1.67358i 0.547523 + 0.836791i $$0.315571\pi$$
−0.547523 + 0.836791i $$0.684429\pi$$
$$228$$ − 6.26875i − 0.415158i
$$229$$ 22.3666i 1.47803i 0.673691 + 0.739013i $$0.264708\pi$$
−0.673691 + 0.739013i $$0.735292\pi$$
$$230$$ −1.30798 −0.0862456
$$231$$ −14.6310 −0.962651
$$232$$ − 9.14675i − 0.600514i
$$233$$ −14.0858 −0.922788 −0.461394 0.887195i $$-0.652651\pi$$
−0.461394 + 0.887195i $$0.652651\pi$$
$$234$$ 0 0
$$235$$ −7.76809 −0.506734
$$236$$ − 10.3327i − 0.672604i
$$237$$ −1.33513 −0.0867257
$$238$$ 23.3913 1.51624
$$239$$ 5.38106i 0.348072i 0.984739 + 0.174036i $$0.0556809\pi$$
−0.984739 + 0.174036i $$0.944319\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ 29.9420i 1.92873i 0.264570 + 0.964366i $$0.414770\pi$$
−0.264570 + 0.964366i $$0.585230\pi$$
$$242$$ 7.03684i 0.452345i
$$243$$ 1.00000 0.0641500
$$244$$ −2.53319 −0.162171
$$245$$ 4.86831i 0.311025i
$$246$$ 4.26875 0.272166
$$247$$ 0 0
$$248$$ −3.75302 −0.238317
$$249$$ − 0.740939i − 0.0469551i
$$250$$ 1.00000 0.0632456
$$251$$ 8.73556 0.551384 0.275692 0.961246i $$-0.411093\pi$$
0.275692 + 0.961246i $$0.411093\pi$$
$$252$$ 3.44504i 0.217017i
$$253$$ − 5.55496i − 0.349237i
$$254$$ − 2.54288i − 0.159554i
$$255$$ 6.78986i 0.425197i
$$256$$ 1.00000 0.0625000
$$257$$ −21.2717 −1.32689 −0.663447 0.748223i $$-0.730907\pi$$
−0.663447 + 0.748223i $$0.730907\pi$$
$$258$$ 3.07069i 0.191173i
$$259$$ −23.5265 −1.46186
$$260$$ 0 0
$$261$$ −9.14675 −0.566170
$$262$$ 10.8509i 0.670368i
$$263$$ −22.2325 −1.37091 −0.685457 0.728113i $$-0.740398\pi$$
−0.685457 + 0.728113i $$0.740398\pi$$
$$264$$ 4.24698 0.261384
$$265$$ 8.93900i 0.549118i
$$266$$ − 21.5961i − 1.32414i
$$267$$ − 13.3274i − 0.815621i
$$268$$ 0.0760644i 0.00464637i
$$269$$ −10.2241 −0.623377 −0.311689 0.950184i $$-0.600895\pi$$
−0.311689 + 0.950184i $$0.600895\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 0.716185i − 0.0435051i −0.999763 0.0217526i $$-0.993075\pi$$
0.999763 0.0217526i $$-0.00692460\pi$$
$$272$$ −6.78986 −0.411696
$$273$$ 0 0
$$274$$ 18.6407 1.12613
$$275$$ 4.24698i 0.256103i
$$276$$ −1.30798 −0.0787311
$$277$$ 22.8713 1.37420 0.687102 0.726561i $$-0.258883\pi$$
0.687102 + 0.726561i $$0.258883\pi$$
$$278$$ − 3.53750i − 0.212165i
$$279$$ 3.75302i 0.224687i
$$280$$ − 3.44504i − 0.205881i
$$281$$ − 4.13036i − 0.246397i −0.992382 0.123198i $$-0.960685\pi$$
0.992382 0.123198i $$-0.0393151\pi$$
$$282$$ −7.76809 −0.462583
$$283$$ −16.2959 −0.968691 −0.484345 0.874877i $$-0.660942\pi$$
−0.484345 + 0.874877i $$0.660942\pi$$
$$284$$ − 0.374354i − 0.0222138i
$$285$$ 6.26875 0.371329
$$286$$ 0 0
$$287$$ 14.7060 0.868069
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 29.1021 1.71189
$$290$$ 9.14675 0.537116
$$291$$ − 13.1903i − 0.773229i
$$292$$ − 16.7114i − 0.977961i
$$293$$ − 7.72587i − 0.451350i −0.974203 0.225675i $$-0.927541\pi$$
0.974203 0.225675i $$-0.0724588\pi$$
$$294$$ 4.86831i 0.283926i
$$295$$ 10.3327 0.601595
$$296$$ 6.82908 0.396932
$$297$$ − 4.24698i − 0.246435i
$$298$$ −16.6843 −0.966493
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 10.5786i 0.609743i
$$302$$ −18.3327 −1.05493
$$303$$ 15.0422 0.864153
$$304$$ 6.26875i 0.359537i
$$305$$ − 2.53319i − 0.145050i
$$306$$ 6.78986i 0.388150i
$$307$$ 5.13036i 0.292805i 0.989225 + 0.146403i $$0.0467695\pi$$
−0.989225 + 0.146403i $$0.953231\pi$$
$$308$$ 14.6310 0.833680
$$309$$ 8.92154 0.507529
$$310$$ − 3.75302i − 0.213157i
$$311$$ −33.1987 −1.88252 −0.941261 0.337679i $$-0.890358\pi$$
−0.941261 + 0.337679i $$0.890358\pi$$
$$312$$ 0 0
$$313$$ 0.0814412 0.00460333 0.00230167 0.999997i $$-0.499267\pi$$
0.00230167 + 0.999997i $$0.499267\pi$$
$$314$$ 18.6015i 1.04974i
$$315$$ −3.44504 −0.194106
$$316$$ 1.33513 0.0751067
$$317$$ − 19.4969i − 1.09506i −0.836787 0.547529i $$-0.815569\pi$$
0.836787 0.547529i $$-0.184431\pi$$
$$318$$ 8.93900i 0.501274i
$$319$$ 38.8461i 2.17496i
$$320$$ 1.00000i 0.0559017i
$$321$$ 14.6746 0.819054
$$322$$ −4.50604 −0.251112
$$323$$ − 42.5639i − 2.36832i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −18.8944 −1.04646
$$327$$ 9.66786i 0.534634i
$$328$$ −4.26875 −0.235702
$$329$$ −26.7614 −1.47540
$$330$$ 4.24698i 0.233789i
$$331$$ 23.8049i 1.30844i 0.756306 + 0.654218i $$0.227002\pi$$
−0.756306 + 0.654218i $$0.772998\pi$$
$$332$$ 0.740939i 0.0406643i
$$333$$ − 6.82908i − 0.374232i
$$334$$ 4.24698 0.232384
$$335$$ −0.0760644 −0.00415584
$$336$$ − 3.44504i − 0.187942i
$$337$$ −21.2064 −1.15519 −0.577594 0.816324i $$-0.696008\pi$$
−0.577594 + 0.816324i $$0.696008\pi$$
$$338$$ 0 0
$$339$$ −11.3448 −0.616166
$$340$$ − 6.78986i − 0.368232i
$$341$$ 15.9390 0.863145
$$342$$ 6.26875 0.338975
$$343$$ − 7.34375i − 0.396525i
$$344$$ − 3.07069i − 0.165560i
$$345$$ − 1.30798i − 0.0704192i
$$346$$ 1.16852i 0.0628201i
$$347$$ −16.5773 −0.889917 −0.444959 0.895551i $$-0.646782\pi$$
−0.444959 + 0.895551i $$0.646782\pi$$
$$348$$ 9.14675 0.490317
$$349$$ − 2.51035i − 0.134376i −0.997740 0.0671880i $$-0.978597\pi$$
0.997740 0.0671880i $$-0.0214027\pi$$
$$350$$ 3.44504 0.184145
$$351$$ 0 0
$$352$$ −4.24698 −0.226365
$$353$$ − 20.8412i − 1.10926i −0.832096 0.554632i $$-0.812859\pi$$
0.832096 0.554632i $$-0.187141\pi$$
$$354$$ 10.3327 0.549179
$$355$$ 0.374354 0.0198687
$$356$$ 13.3274i 0.706348i
$$357$$ 23.3913i 1.23800i
$$358$$ − 10.1836i − 0.538219i
$$359$$ − 2.42566i − 0.128022i −0.997949 0.0640108i $$-0.979611\pi$$
0.997949 0.0640108i $$-0.0203892\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −20.2972 −1.06827
$$362$$ 10.3351i 0.543202i
$$363$$ −7.03684 −0.369338
$$364$$ 0 0
$$365$$ 16.7114 0.874715
$$366$$ − 2.53319i − 0.132412i
$$367$$ 2.66248 0.138980 0.0694902 0.997583i $$-0.477863\pi$$
0.0694902 + 0.997583i $$0.477863\pi$$
$$368$$ 1.30798 0.0681831
$$369$$ 4.26875i 0.222222i
$$370$$ 6.82908i 0.355027i
$$371$$ 30.7952i 1.59881i
$$372$$ − 3.75302i − 0.194585i
$$373$$ 5.87800 0.304351 0.152176 0.988353i $$-0.451372\pi$$
0.152176 + 0.988353i $$0.451372\pi$$
$$374$$ 28.8364 1.49109
$$375$$ 1.00000i 0.0516398i
$$376$$ 7.76809 0.400608
$$377$$ 0 0
$$378$$ −3.44504 −0.177194
$$379$$ 36.1540i 1.85711i 0.371196 + 0.928554i $$0.378948\pi$$
−0.371196 + 0.928554i $$0.621052\pi$$
$$380$$ −6.26875 −0.321580
$$381$$ 2.54288 0.130276
$$382$$ 9.43296i 0.482632i
$$383$$ − 8.98361i − 0.459041i −0.973304 0.229520i $$-0.926284\pi$$
0.973304 0.229520i $$-0.0737158\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 14.6310i 0.745666i
$$386$$ −22.9191 −1.16655
$$387$$ −3.07069 −0.156092
$$388$$ 13.1903i 0.669636i
$$389$$ −0.655186 −0.0332192 −0.0166096 0.999862i $$-0.505287\pi$$
−0.0166096 + 0.999862i $$0.505287\pi$$
$$390$$ 0 0
$$391$$ −8.88099 −0.449131
$$392$$ − 4.86831i − 0.245887i
$$393$$ −10.8509 −0.547353
$$394$$ −14.3502 −0.722952
$$395$$ 1.33513i 0.0671775i
$$396$$ 4.24698i 0.213419i
$$397$$ − 0.271143i − 0.0136083i −0.999977 0.00680413i $$-0.997834\pi$$
0.999977 0.00680413i $$-0.00216584\pi$$
$$398$$ 14.2524i 0.714406i
$$399$$ 21.5961 1.08116
$$400$$ −1.00000 −0.0500000
$$401$$ − 0.344814i − 0.0172192i −0.999963 0.00860960i $$-0.997259\pi$$
0.999963 0.00860960i $$-0.00274056\pi$$
$$402$$ −0.0760644 −0.00379375
$$403$$ 0 0
$$404$$ −15.0422 −0.748378
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 31.5109 1.56386
$$407$$ −29.0030 −1.43762
$$408$$ − 6.78986i − 0.336148i
$$409$$ 27.0388i 1.33698i 0.743721 + 0.668490i $$0.233059\pi$$
−0.743721 + 0.668490i $$0.766941\pi$$
$$410$$ − 4.26875i − 0.210819i
$$411$$ 18.6407i 0.919478i
$$412$$ −8.92154 −0.439533
$$413$$ 35.5967 1.75160
$$414$$ − 1.30798i − 0.0642836i
$$415$$ −0.740939 −0.0363713
$$416$$ 0 0
$$417$$ 3.53750 0.173232
$$418$$ − 26.6233i − 1.30219i
$$419$$ 5.86533 0.286540 0.143270 0.989684i $$-0.454238\pi$$
0.143270 + 0.989684i $$0.454238\pi$$
$$420$$ 3.44504 0.168101
$$421$$ − 28.9476i − 1.41082i −0.708799 0.705410i $$-0.750763\pi$$
0.708799 0.705410i $$-0.249237\pi$$
$$422$$ − 0.396125i − 0.0192830i
$$423$$ − 7.76809i − 0.377697i
$$424$$ − 8.93900i − 0.434116i
$$425$$ 6.78986 0.329356
$$426$$ 0.374354 0.0181375
$$427$$ − 8.72694i − 0.422326i
$$428$$ −14.6746 −0.709322
$$429$$ 0 0
$$430$$ 3.07069 0.148082
$$431$$ 4.87023i 0.234591i 0.993097 + 0.117295i $$0.0374224\pi$$
−0.993097 + 0.117295i $$0.962578\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 36.9788 1.77709 0.888544 0.458791i $$-0.151717\pi$$
0.888544 + 0.458791i $$0.151717\pi$$
$$434$$ − 12.9293i − 0.620627i
$$435$$ 9.14675i 0.438553i
$$436$$ − 9.66786i − 0.463006i
$$437$$ 8.19939i 0.392230i
$$438$$ 16.7114 0.798502
$$439$$ −22.8799 −1.09200 −0.546000 0.837785i $$-0.683850\pi$$
−0.546000 + 0.837785i $$0.683850\pi$$
$$440$$ − 4.24698i − 0.202467i
$$441$$ −4.86831 −0.231824
$$442$$ 0 0
$$443$$ −10.5942 −0.503345 −0.251673 0.967812i $$-0.580981\pi$$
−0.251673 + 0.967812i $$0.580981\pi$$
$$444$$ 6.82908i 0.324094i
$$445$$ −13.3274 −0.631777
$$446$$ 1.49635 0.0708543
$$447$$ − 16.6843i − 0.789138i
$$448$$ 3.44504i 0.162763i
$$449$$ − 19.6485i − 0.927269i −0.886027 0.463635i $$-0.846545\pi$$
0.886027 0.463635i $$-0.153455\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 18.1293 0.853675
$$452$$ 11.3448 0.533615
$$453$$ − 18.3327i − 0.861347i
$$454$$ 25.2150 1.18340
$$455$$ 0 0
$$456$$ −6.26875 −0.293561
$$457$$ − 18.4239i − 0.861832i −0.902392 0.430916i $$-0.858191\pi$$
0.902392 0.430916i $$-0.141809\pi$$
$$458$$ 22.3666 1.04512
$$459$$ −6.78986 −0.316923
$$460$$ 1.30798i 0.0609848i
$$461$$ − 22.2446i − 1.03603i −0.855370 0.518017i $$-0.826670\pi$$
0.855370 0.518017i $$-0.173330\pi$$
$$462$$ 14.6310i 0.680697i
$$463$$ 8.68532i 0.403641i 0.979423 + 0.201820i $$0.0646857\pi$$
−0.979423 + 0.201820i $$0.935314\pi$$
$$464$$ −9.14675 −0.424627
$$465$$ 3.75302 0.174042
$$466$$ 14.0858i 0.652510i
$$467$$ −19.6136 −0.907608 −0.453804 0.891102i $$-0.649933\pi$$
−0.453804 + 0.891102i $$0.649933\pi$$
$$468$$ 0 0
$$469$$ −0.262045 −0.0121001
$$470$$ 7.76809i 0.358315i
$$471$$ −18.6015 −0.857111
$$472$$ −10.3327 −0.475603
$$473$$ 13.0411i 0.599633i
$$474$$ 1.33513i 0.0613243i
$$475$$ − 6.26875i − 0.287630i
$$476$$ − 23.3913i − 1.07214i
$$477$$ −8.93900 −0.409289
$$478$$ 5.38106 0.246124
$$479$$ − 31.4698i − 1.43789i −0.695066 0.718946i $$-0.744625\pi$$
0.695066 0.718946i $$-0.255375\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 29.9420 1.36382
$$483$$ − 4.50604i − 0.205032i
$$484$$ 7.03684 0.319856
$$485$$ −13.1903 −0.598940
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 13.2644i 0.601069i 0.953771 + 0.300535i $$0.0971651\pi$$
−0.953771 + 0.300535i $$0.902835\pi$$
$$488$$ 2.53319i 0.114672i
$$489$$ − 18.8944i − 0.854434i
$$490$$ 4.86831 0.219928
$$491$$ −11.7259 −0.529181 −0.264591 0.964361i $$-0.585237\pi$$
−0.264591 + 0.964361i $$0.585237\pi$$
$$492$$ − 4.26875i − 0.192450i
$$493$$ 62.1051 2.79707
$$494$$ 0 0
$$495$$ −4.24698 −0.190888
$$496$$ 3.75302i 0.168516i
$$497$$ 1.28967 0.0578494
$$498$$ −0.740939 −0.0332023
$$499$$ − 26.7633i − 1.19809i −0.800715 0.599045i $$-0.795547\pi$$
0.800715 0.599045i $$-0.204453\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 4.24698i 0.189741i
$$502$$ − 8.73556i − 0.389887i
$$503$$ 17.2239 0.767975 0.383987 0.923338i $$-0.374551\pi$$
0.383987 + 0.923338i $$0.374551\pi$$
$$504$$ 3.44504 0.153454
$$505$$ − 15.0422i − 0.669370i
$$506$$ −5.55496 −0.246948
$$507$$ 0 0
$$508$$ −2.54288 −0.112822
$$509$$ − 28.2737i − 1.25321i −0.779338 0.626604i $$-0.784444\pi$$
0.779338 0.626604i $$-0.215556\pi$$
$$510$$ 6.78986 0.300660
$$511$$ 57.5715 2.54681
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 6.26875i 0.276772i
$$514$$ 21.2717i 0.938256i
$$515$$ − 8.92154i − 0.393130i
$$516$$ 3.07069 0.135179
$$517$$ −32.9909 −1.45094
$$518$$ 23.5265i 1.03369i
$$519$$ −1.16852 −0.0512924
$$520$$ 0 0
$$521$$ 1.47889 0.0647915 0.0323958 0.999475i $$-0.489686\pi$$
0.0323958 + 0.999475i $$0.489686\pi$$
$$522$$ 9.14675i 0.400342i
$$523$$ −28.2198 −1.23397 −0.616984 0.786976i $$-0.711646\pi$$
−0.616984 + 0.786976i $$0.711646\pi$$
$$524$$ 10.8509 0.474022
$$525$$ 3.44504i 0.150354i
$$526$$ 22.2325i 0.969383i
$$527$$ − 25.4825i − 1.11003i
$$528$$ − 4.24698i − 0.184826i
$$529$$ −21.2892 −0.925617
$$530$$ 8.93900 0.388285
$$531$$ 10.3327i 0.448402i
$$532$$ −21.5961 −0.936310
$$533$$ 0 0
$$534$$ −13.3274 −0.576731
$$535$$ − 14.6746i − 0.634437i
$$536$$ 0.0760644 0.00328548
$$537$$ 10.1836 0.439454
$$538$$ 10.2241i 0.440794i
$$539$$ 20.6756i 0.890562i
$$540$$ 1.00000i 0.0430331i
$$541$$ − 0.245915i − 0.0105727i −0.999986 0.00528635i $$-0.998317\pi$$
0.999986 0.00528635i $$-0.00168270\pi$$
$$542$$ −0.716185 −0.0307628
$$543$$ −10.3351 −0.443523
$$544$$ 6.78986i 0.291113i
$$545$$ 9.66786 0.414126
$$546$$ 0 0
$$547$$ −13.6407 −0.583235 −0.291617 0.956535i $$-0.594193\pi$$
−0.291617 + 0.956535i $$0.594193\pi$$
$$548$$ − 18.6407i − 0.796292i
$$549$$ 2.53319 0.108114
$$550$$ 4.24698 0.181092
$$551$$ − 57.3387i − 2.44271i
$$552$$ 1.30798i 0.0556713i
$$553$$ 4.59956i 0.195593i
$$554$$ − 22.8713i − 0.971708i
$$555$$ −6.82908 −0.289879
$$556$$ −3.53750 −0.150023
$$557$$ − 14.0248i − 0.594248i −0.954839 0.297124i $$-0.903973\pi$$
0.954839 0.297124i $$-0.0960275\pi$$
$$558$$ 3.75302 0.158878
$$559$$ 0 0
$$560$$ −3.44504 −0.145580
$$561$$ 28.8364i 1.21747i
$$562$$ −4.13036 −0.174229
$$563$$ 3.26981 0.137806 0.0689031 0.997623i $$-0.478050\pi$$
0.0689031 + 0.997623i $$0.478050\pi$$
$$564$$ 7.76809i 0.327095i
$$565$$ 11.3448i 0.477280i
$$566$$ 16.2959i 0.684968i
$$567$$ − 3.44504i − 0.144678i
$$568$$ −0.374354 −0.0157076
$$569$$ −4.10752 −0.172196 −0.0860982 0.996287i $$-0.527440\pi$$
−0.0860982 + 0.996287i $$0.527440\pi$$
$$570$$ − 6.26875i − 0.262569i
$$571$$ −9.18465 −0.384366 −0.192183 0.981359i $$-0.561557\pi$$
−0.192183 + 0.981359i $$0.561557\pi$$
$$572$$ 0 0
$$573$$ −9.43296 −0.394068
$$574$$ − 14.7060i − 0.613817i
$$575$$ −1.30798 −0.0545465
$$576$$ −1.00000 −0.0416667
$$577$$ 28.0000i 1.16566i 0.812596 + 0.582828i $$0.198054\pi$$
−0.812596 + 0.582828i $$0.801946\pi$$
$$578$$ − 29.1021i − 1.21049i
$$579$$ − 22.9191i − 0.952487i
$$580$$ − 9.14675i − 0.379798i
$$581$$ −2.55257 −0.105898
$$582$$ −13.1903 −0.546755
$$583$$ 37.9638i 1.57230i
$$584$$ −16.7114 −0.691523
$$585$$ 0 0
$$586$$ −7.72587 −0.319153
$$587$$ − 47.6765i − 1.96782i −0.178668 0.983910i $$-0.557179\pi$$
0.178668 0.983910i $$-0.442821\pi$$
$$588$$ 4.86831 0.200766
$$589$$ −23.5267 −0.969403
$$590$$ − 10.3327i − 0.425392i
$$591$$ − 14.3502i − 0.590288i
$$592$$ − 6.82908i − 0.280674i
$$593$$ 16.7681i 0.688583i 0.938863 + 0.344291i $$0.111881\pi$$
−0.938863 + 0.344291i $$0.888119\pi$$
$$594$$ −4.24698 −0.174256
$$595$$ 23.3913 0.958951
$$596$$ 16.6843i 0.683414i
$$597$$ −14.2524 −0.583310
$$598$$ 0 0
$$599$$ 44.0157 1.79843 0.899215 0.437506i $$-0.144138\pi$$
0.899215 + 0.437506i $$0.144138\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ 39.1299 1.59614 0.798071 0.602564i $$-0.205854\pi$$
0.798071 + 0.602564i $$0.205854\pi$$
$$602$$ 10.5786 0.431153
$$603$$ − 0.0760644i − 0.00309758i
$$604$$ 18.3327i 0.745948i
$$605$$ 7.03684i 0.286088i
$$606$$ − 15.0422i − 0.611048i
$$607$$ 6.33273 0.257038 0.128519 0.991707i $$-0.458978\pi$$
0.128519 + 0.991707i $$0.458978\pi$$
$$608$$ 6.26875 0.254231
$$609$$ 31.5109i 1.27689i
$$610$$ −2.53319 −0.102566
$$611$$ 0 0
$$612$$ 6.78986 0.274464
$$613$$ 29.1129i 1.17586i 0.808912 + 0.587929i $$0.200057\pi$$
−0.808912 + 0.587929i $$0.799943\pi$$
$$614$$ 5.13036 0.207044
$$615$$ 4.26875 0.172133
$$616$$ − 14.6310i − 0.589501i
$$617$$ − 27.9933i − 1.12697i −0.826127 0.563484i $$-0.809461\pi$$
0.826127 0.563484i $$-0.190539\pi$$
$$618$$ − 8.92154i − 0.358877i
$$619$$ − 23.0084i − 0.924784i −0.886676 0.462392i $$-0.846991\pi$$
0.886676 0.462392i $$-0.153009\pi$$
$$620$$ −3.75302 −0.150725
$$621$$ 1.30798 0.0524874
$$622$$ 33.1987i 1.33114i
$$623$$ −45.9133 −1.83948
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 0.0814412i − 0.00325505i
$$627$$ 26.6233 1.06323
$$628$$ 18.6015 0.742280
$$629$$ 46.3685i 1.84883i
$$630$$ 3.44504i 0.137254i
$$631$$ 6.18731i 0.246313i 0.992387 + 0.123156i $$0.0393017\pi$$
−0.992387 + 0.123156i $$0.960698\pi$$
$$632$$ − 1.33513i − 0.0531084i
$$633$$ 0.396125 0.0157445
$$634$$ −19.4969 −0.774323
$$635$$ − 2.54288i − 0.100911i
$$636$$ 8.93900 0.354454
$$637$$ 0 0
$$638$$ 38.8461 1.53793
$$639$$ 0.374354i 0.0148092i
$$640$$ 1.00000 0.0395285
$$641$$ −39.1159 −1.54498 −0.772492 0.635024i $$-0.780990\pi$$
−0.772492 + 0.635024i $$0.780990\pi$$
$$642$$ − 14.6746i − 0.579159i
$$643$$ 8.78209i 0.346332i 0.984893 + 0.173166i $$0.0553997\pi$$
−0.984893 + 0.173166i $$0.944600\pi$$
$$644$$ 4.50604i 0.177563i
$$645$$ 3.07069i 0.120908i
$$646$$ −42.5639 −1.67465
$$647$$ 30.1333 1.18466 0.592332 0.805694i $$-0.298207\pi$$
0.592332 + 0.805694i $$0.298207\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 43.8829 1.72255
$$650$$ 0 0
$$651$$ 12.9293 0.506740
$$652$$ 18.8944i 0.739962i
$$653$$ −27.0858 −1.05995 −0.529974 0.848014i $$-0.677798\pi$$
−0.529974 + 0.848014i $$0.677798\pi$$
$$654$$ 9.66786 0.378043
$$655$$ 10.8509i 0.423978i
$$656$$ 4.26875i 0.166667i
$$657$$ 16.7114i 0.651974i
$$658$$ 26.7614i 1.04327i
$$659$$ 5.69681 0.221916 0.110958 0.993825i $$-0.464608\pi$$
0.110958 + 0.993825i $$0.464608\pi$$
$$660$$ 4.24698 0.165313
$$661$$ 17.1008i 0.665145i 0.943078 + 0.332572i $$0.107917\pi$$
−0.943078 + 0.332572i $$0.892083\pi$$
$$662$$ 23.8049 0.925205
$$663$$ 0 0
$$664$$ 0.740939 0.0287540
$$665$$ − 21.5961i − 0.837461i
$$666$$ −6.82908 −0.264622
$$667$$ −11.9638 −0.463238
$$668$$ − 4.24698i − 0.164321i
$$669$$ 1.49635i 0.0578523i
$$670$$ 0.0760644i 0.00293862i
$$671$$ − 10.7584i − 0.415323i
$$672$$ −3.44504 −0.132895
$$673$$ 43.1957 1.66507 0.832535 0.553972i $$-0.186889\pi$$
0.832535 + 0.553972i $$0.186889\pi$$
$$674$$ 21.2064i 0.816841i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −7.78746 −0.299297 −0.149648 0.988739i $$-0.547814\pi$$
−0.149648 + 0.988739i $$0.547814\pi$$
$$678$$ 11.3448i 0.435695i
$$679$$ −45.4411 −1.74387
$$680$$ −6.78986 −0.260379
$$681$$ 25.2150i 0.966243i
$$682$$ − 15.9390i − 0.610336i
$$683$$ − 30.9138i − 1.18288i −0.806348 0.591441i $$-0.798559\pi$$
0.806348 0.591441i $$-0.201441\pi$$
$$684$$ − 6.26875i − 0.239692i
$$685$$ 18.6407 0.712225
$$686$$ −7.34375 −0.280386
$$687$$ 22.3666i 0.853338i
$$688$$ −3.07069 −0.117069
$$689$$ 0 0
$$690$$ −1.30798 −0.0497939
$$691$$ − 6.20237i − 0.235949i −0.993017 0.117975i $$-0.962360\pi$$
0.993017 0.117975i $$-0.0376402\pi$$
$$692$$ 1.16852 0.0444206
$$693$$ −14.6310 −0.555787
$$694$$ 16.5773i 0.629266i
$$695$$ − 3.53750i − 0.134185i
$$696$$ − 9.14675i − 0.346707i
$$697$$ − 28.9842i − 1.09785i
$$698$$ −2.51035 −0.0950182
$$699$$ −14.0858 −0.532772
$$700$$ − 3.44504i − 0.130210i
$$701$$ −26.3220 −0.994167 −0.497084 0.867703i $$-0.665596\pi$$
−0.497084 + 0.867703i $$0.665596\pi$$
$$702$$ 0 0
$$703$$ 42.8098 1.61460
$$704$$ 4.24698i 0.160064i
$$705$$ −7.76809 −0.292563
$$706$$ −20.8412 −0.784368
$$707$$ − 51.8211i − 1.94893i
$$708$$ − 10.3327i − 0.388328i
$$709$$ − 33.7429i − 1.26724i −0.773645 0.633620i $$-0.781568\pi$$
0.773645 0.633620i $$-0.218432\pi$$
$$710$$ − 0.374354i − 0.0140493i
$$711$$ −1.33513 −0.0500711
$$712$$ 13.3274 0.499464
$$713$$ 4.90887i 0.183839i
$$714$$ 23.3913 0.875399
$$715$$ 0 0
$$716$$ −10.1836 −0.380579
$$717$$ 5.38106i 0.200959i
$$718$$ −2.42566 −0.0905250
$$719$$ 9.50173 0.354355 0.177177 0.984179i $$-0.443303\pi$$
0.177177 + 0.984179i $$0.443303\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 30.7351i − 1.14463i
$$722$$ 20.2972i 0.755384i
$$723$$ 29.9420i 1.11355i
$$724$$ 10.3351 0.384102
$$725$$ 9.14675 0.339702
$$726$$ 7.03684i 0.261161i
$$727$$ 12.2631 0.454814 0.227407 0.973800i $$-0.426975\pi$$
0.227407 + 0.973800i $$0.426975\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 16.7114i − 0.618517i
$$731$$ 20.8495 0.771148
$$732$$ −2.53319 −0.0936293
$$733$$ 1.52217i 0.0562227i 0.999605 + 0.0281113i $$0.00894930\pi$$
−0.999605 + 0.0281113i $$0.991051\pi$$
$$734$$ − 2.66248i − 0.0982740i
$$735$$ 4.86831i 0.179570i
$$736$$ − 1.30798i − 0.0482127i
$$737$$ −0.323044 −0.0118995
$$738$$ 4.26875 0.157135
$$739$$ − 51.1075i − 1.88002i −0.341146 0.940010i $$-0.610815\pi$$
0.341146 0.940010i $$-0.389185\pi$$
$$740$$ 6.82908 0.251042
$$741$$ 0 0
$$742$$ 30.7952 1.13053
$$743$$ − 9.08144i − 0.333166i −0.986027 0.166583i $$-0.946727\pi$$
0.986027 0.166583i $$-0.0532733\pi$$
$$744$$ −3.75302 −0.137592
$$745$$ −16.6843 −0.611264
$$746$$ − 5.87800i − 0.215209i
$$747$$ − 0.740939i − 0.0271095i
$$748$$ − 28.8364i − 1.05436i
$$749$$ − 50.5545i − 1.84722i
$$750$$ 1.00000 0.0365148
$$751$$ −31.7090 −1.15708 −0.578539 0.815655i $$-0.696377\pi$$
−0.578539 + 0.815655i $$0.696377\pi$$
$$752$$ − 7.76809i − 0.283273i
$$753$$ 8.73556 0.318342
$$754$$ 0 0
$$755$$ −18.3327 −0.667196
$$756$$ 3.44504i 0.125295i
$$757$$ −8.49934 −0.308914 −0.154457 0.988000i $$-0.549363\pi$$
−0.154457 + 0.988000i $$0.549363\pi$$
$$758$$ 36.1540 1.31317
$$759$$ − 5.55496i − 0.201632i
$$760$$ 6.26875i 0.227391i
$$761$$ 16.4198i 0.595218i 0.954688 + 0.297609i $$0.0961891\pi$$
−0.954688 + 0.297609i $$0.903811\pi$$
$$762$$ − 2.54288i − 0.0921187i
$$763$$ 33.3062 1.20576
$$764$$ 9.43296 0.341273
$$765$$ 6.78986i 0.245488i
$$766$$ −8.98361 −0.324591
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 27.1011i − 0.977290i −0.872483 0.488645i $$-0.837491\pi$$
0.872483 0.488645i $$-0.162509\pi$$
$$770$$ 14.6310 0.527265
$$771$$ −21.2717 −0.766083
$$772$$ 22.9191i 0.824878i
$$773$$ 12.6297i 0.454259i 0.973865 + 0.227129i $$0.0729340\pi$$
−0.973865 + 0.227129i $$0.927066\pi$$
$$774$$ 3.07069i 0.110374i
$$775$$ − 3.75302i − 0.134812i
$$776$$ 13.1903 0.473504
$$777$$ −23.5265 −0.844008
$$778$$ 0.655186i 0.0234895i
$$779$$ −26.7597 −0.958767
$$780$$ 0 0
$$781$$ 1.58987 0.0568902
$$782$$ 8.88099i 0.317583i
$$783$$ −9.14675 −0.326878
$$784$$ −4.86831 −0.173868
$$785$$ 18.6015i 0.663915i
$$786$$ 10.8509i 0.387037i
$$787$$ − 3.02954i − 0.107991i −0.998541 0.0539957i $$-0.982804\pi$$
0.998541 0.0539957i $$-0.0171957\pi$$
$$788$$ 14.3502i 0.511204i
$$789$$ −22.2325 −0.791498
$$790$$ 1.33513 0.0475016
$$791$$ 39.0834i 1.38964i
$$792$$ 4.24698 0.150910
$$793$$ 0 0
$$794$$ −0.271143 −0.00962250
$$795$$ 8.93900i 0.317034i
$$796$$ 14.2524 0.505161
$$797$$ −40.2218 −1.42473 −0.712364 0.701810i $$-0.752375\pi$$
−0.712364 + 0.701810i $$0.752375\pi$$
$$798$$ − 21.5961i − 0.764494i
$$799$$ 52.7442i 1.86596i
$$800$$ 1.00000i 0.0353553i
$$801$$ − 13.3274i − 0.470899i
$$802$$ −0.344814 −0.0121758
$$803$$ 70.9730 2.50458
$$804$$ 0.0760644i 0.00268259i
$$805$$ −4.50604 −0.158817
$$806$$ 0 0
$$807$$ −10.2241 −0.359907
$$808$$ 15.0422i 0.529183i
$$809$$ 33.7918 1.18806 0.594028 0.804445i $$-0.297537\pi$$
0.594028 + 0.804445i $$0.297537\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 33.6534i 1.18173i 0.806770 + 0.590865i $$0.201213\pi$$
−0.806770 + 0.590865i $$0.798787\pi$$
$$812$$ − 31.5109i − 1.10582i
$$813$$ − 0.716185i − 0.0251177i
$$814$$ 29.0030i 1.01655i
$$815$$ −18.8944 −0.661842
$$816$$ −6.78986 −0.237693
$$817$$ − 19.2494i − 0.673450i
$$818$$ 27.0388 0.945388
$$819$$ 0 0
$$820$$ −4.26875 −0.149071
$$821$$ 12.6939i 0.443022i 0.975158 + 0.221511i $$0.0710988\pi$$
−0.975158 + 0.221511i $$0.928901\pi$$
$$822$$ 18.6407 0.650169
$$823$$ 8.79523 0.306583 0.153291 0.988181i $$-0.451013\pi$$
0.153291 + 0.988181i $$0.451013\pi$$
$$824$$ 8.92154i 0.310797i
$$825$$ 4.24698i 0.147861i
$$826$$ − 35.5967i − 1.23857i
$$827$$ − 32.1758i − 1.11886i −0.828877 0.559431i $$-0.811020\pi$$
0.828877 0.559431i $$-0.188980\pi$$
$$828$$ −1.30798 −0.0454554
$$829$$ −21.1812 −0.735653 −0.367827 0.929894i $$-0.619898\pi$$
−0.367827 + 0.929894i $$0.619898\pi$$
$$830$$ 0.740939i 0.0257184i
$$831$$ 22.8713 0.793397
$$832$$ 0 0
$$833$$ 33.0551 1.14529
$$834$$ − 3.53750i − 0.122494i
$$835$$ 4.24698 0.146973
$$836$$ −26.6233 −0.920784
$$837$$ 3.75302i 0.129723i
$$838$$ − 5.86533i − 0.202614i
$$839$$ − 26.5931i − 0.918097i −0.888411 0.459048i $$-0.848190\pi$$
0.888411 0.459048i $$-0.151810\pi$$
$$840$$ − 3.44504i − 0.118865i
$$841$$ 54.6631 1.88493
$$842$$ −28.9476 −0.997601
$$843$$ − 4.13036i − 0.142257i
$$844$$ −0.396125 −0.0136352
$$845$$ 0 0
$$846$$ −7.76809 −0.267072
$$847$$ 24.2422i 0.832972i
$$848$$ −8.93900 −0.306967
$$849$$ −16.2959 −0.559274
$$850$$ − 6.78986i − 0.232890i
$$851$$ − 8.93230i − 0.306195i
$$852$$ − 0.374354i − 0.0128252i
$$853$$ 32.2457i 1.10407i 0.833821 + 0.552035i $$0.186149\pi$$
−0.833821 + 0.552035i $$0.813851\pi$$
$$854$$ −8.72694 −0.298630
$$855$$ 6.26875 0.214387
$$856$$ 14.6746i 0.501566i
$$857$$ 43.0176 1.46945 0.734726 0.678364i $$-0.237311\pi$$
0.734726 + 0.678364i $$0.237311\pi$$
$$858$$ 0 0
$$859$$ −31.1997 −1.06452 −0.532260 0.846581i $$-0.678657\pi$$
−0.532260 + 0.846581i $$0.678657\pi$$
$$860$$ − 3.07069i − 0.104710i
$$861$$ 14.7060 0.501180
$$862$$ 4.87023 0.165881
$$863$$ − 34.5424i − 1.17584i −0.808920 0.587919i $$-0.799948\pi$$
0.808920 0.587919i $$-0.200052\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 1.16852i 0.0397309i
$$866$$ − 36.9788i − 1.25659i
$$867$$ 29.1021 0.988361
$$868$$ −12.9293 −0.438849
$$869$$ 5.67025i 0.192350i
$$870$$ 9.14675 0.310104
$$871$$ 0 0
$$872$$ −9.66786 −0.327395
$$873$$ − 13.1903i − 0.446424i
$$874$$ 8.19939 0.277349
$$875$$ 3.44504 0.116464
$$876$$ − 16.7114i − 0.564626i
$$877$$ 2.12093i 0.0716188i 0.999359 + 0.0358094i $$0.0114009\pi$$
−0.999359 + 0.0358094i $$0.988599\pi$$
$$878$$ 22.8799i 0.772160i
$$879$$ − 7.72587i − 0.260587i
$$880$$ −4.24698 −0.143166
$$881$$ −27.3489 −0.921407 −0.460703 0.887554i $$-0.652403\pi$$
−0.460703 + 0.887554i $$0.652403\pi$$
$$882$$ 4.86831i 0.163925i
$$883$$ 48.2959 1.62529 0.812643 0.582762i $$-0.198028\pi$$
0.812643 + 0.582762i $$0.198028\pi$$
$$884$$ 0 0
$$885$$ 10.3327 0.347331
$$886$$ 10.5942i 0.355919i
$$887$$ 25.1943 0.845943 0.422972 0.906143i $$-0.360987\pi$$
0.422972 + 0.906143i $$0.360987\pi$$
$$888$$ 6.82908 0.229169
$$889$$ − 8.76032i − 0.293812i
$$890$$ 13.3274i 0.446734i
$$891$$ − 4.24698i − 0.142279i
$$892$$ − 1.49635i − 0.0501016i
$$893$$ 48.6962 1.62956
$$894$$ −16.6843 −0.558005
$$895$$ − 10.1836i − 0.340400i
$$896$$ 3.44504 0.115091
$$897$$ 0 0
$$898$$ −19.6485 −0.655678
$$899$$ − 34.3279i − 1.14490i
$$900$$ 1.00000 0.0333333
$$901$$ 60.6945 2.02203
$$902$$ − 18.1293i − 0.603639i
$$903$$ 10.5786i 0.352035i
$$904$$ − 11.3448i − 0.377323i
$$905$$ 10.3351i 0.343551i
$$906$$ −18.3327 −0.609064
$$907$$ 20.0949 0.667239 0.333619 0.942708i $$-0.391730\pi$$
0.333619 + 0.942708i $$0.391730\pi$$
$$908$$ − 25.2150i − 0.836791i
$$909$$ 15.0422 0.498919
$$910$$ 0 0
$$911$$ −7.44563 −0.246685 −0.123342 0.992364i $$-0.539361\pi$$
−0.123342 + 0.992364i $$0.539361\pi$$
$$912$$ 6.26875i 0.207579i
$$913$$ −3.14675 −0.104142
$$914$$ −18.4239 −0.609407
$$915$$ − 2.53319i − 0.0837446i
$$916$$ − 22.3666i − 0.739013i
$$917$$ 37.3817i 1.23445i
$$918$$ 6.78986i 0.224099i
$$919$$ −14.2765 −0.470939 −0.235469 0.971882i $$-0.575663\pi$$
−0.235469 + 0.971882i $$0.575663\pi$$
$$920$$ 1.30798 0.0431228
$$921$$ 5.13036i 0.169051i
$$922$$ −22.2446 −0.732586
$$923$$ 0 0
$$924$$ 14.6310 0.481325
$$925$$ 6.82908i 0.224539i
$$926$$ 8.68532 0.285417
$$927$$ 8.92154 0.293022
$$928$$ 9.14675i 0.300257i
$$929$$ 30.9148i 1.01428i 0.861863 + 0.507141i $$0.169298\pi$$
−0.861863 + 0.507141i $$0.830702\pi$$
$$930$$ − 3.75302i − 0.123066i
$$931$$ − 30.5182i − 1.00019i
$$932$$ 14.0858 0.461394
$$933$$ −33.1987 −1.08688
$$934$$ 19.6136i 0.641775i
$$935$$ 28.8364 0.943050
$$936$$ 0 0
$$937$$ −16.7187 −0.546176 −0.273088 0.961989i $$-0.588045\pi$$
−0.273088 + 0.961989i $$0.588045\pi$$
$$938$$ 0.262045i 0.00855608i
$$939$$ 0.0814412 0.00265773
$$940$$ 7.76809 0.253367
$$941$$ 36.9033i 1.20301i 0.798867 + 0.601507i $$0.205433\pi$$
−0.798867 + 0.601507i $$0.794567\pi$$
$$942$$ 18.6015i 0.606069i
$$943$$ 5.58343i 0.181822i
$$944$$ 10.3327i 0.336302i
$$945$$ −3.44504 −0.112067
$$946$$ 13.0411 0.424004
$$947$$ 39.1661i 1.27273i 0.771389 + 0.636364i $$0.219562\pi$$
−0.771389 + 0.636364i $$0.780438\pi$$
$$948$$ 1.33513 0.0433629
$$949$$ 0 0
$$950$$ −6.26875 −0.203385
$$951$$ − 19.4969i − 0.632232i
$$952$$ −23.3913 −0.758118
$$953$$ 19.6200 0.635554 0.317777 0.948165i $$-0.397064\pi$$
0.317777 + 0.948165i $$0.397064\pi$$
$$954$$ 8.93900i 0.289411i
$$955$$ 9.43296i 0.305243i
$$956$$ − 5.38106i − 0.174036i
$$957$$ 38.8461i 1.25572i
$$958$$ −31.4698 −1.01674
$$959$$ 64.2180 2.07371
$$960$$ 1.00000i 0.0322749i
$$961$$ 16.9148 0.545640
$$962$$ 0 0
$$963$$ 14.6746 0.472881
$$964$$ − 29.9420i − 0.964366i
$$965$$ −22.9191 −0.737794
$$966$$ −4.50604 −0.144979
$$967$$ − 43.1564i − 1.38782i −0.720063 0.693909i $$-0.755887\pi$$
0.720063 0.693909i $$-0.244113\pi$$
$$968$$ − 7.03684i − 0.226172i
$$969$$ − 42.5639i − 1.36735i
$$970$$ 13.1903i 0.423515i
$$971$$ 39.5032 1.26772 0.633859 0.773449i $$-0.281470\pi$$
0.633859 + 0.773449i $$0.281470\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ − 12.1868i − 0.390692i
$$974$$ 13.2644 0.425020
$$975$$ 0 0
$$976$$ 2.53319 0.0810854
$$977$$ − 3.84117i − 0.122890i −0.998110 0.0614449i $$-0.980429\pi$$
0.998110 0.0614449i $$-0.0195708\pi$$
$$978$$ −18.8944 −0.604176
$$979$$ −56.6010 −1.80898
$$980$$ − 4.86831i − 0.155513i
$$981$$ 9.66786i 0.308671i
$$982$$ 11.7259i 0.374188i
$$983$$ 32.4300i 1.03436i 0.855878 + 0.517178i $$0.173017\pi$$
−0.855878 + 0.517178i $$0.826983\pi$$
$$984$$ −4.26875 −0.136083
$$985$$ −14.3502 −0.457235
$$986$$ − 62.1051i − 1.97783i
$$987$$ −26.7614 −0.851824
$$988$$ 0 0
$$989$$ −4.01639 −0.127714
$$990$$ 4.24698i 0.134978i
$$991$$ −61.9807 −1.96888 −0.984442 0.175712i $$-0.943777\pi$$
−0.984442 + 0.175712i $$0.943777\pi$$
$$992$$ 3.75302 0.119159
$$993$$ 23.8049i 0.755426i
$$994$$ − 1.28967i − 0.0409057i
$$995$$ 14.2524i 0.451830i
$$996$$ 0.740939i 0.0234775i
$$997$$ 28.9162 0.915784 0.457892 0.889008i $$-0.348605\pi$$
0.457892 + 0.889008i $$0.348605\pi$$
$$998$$ −26.7633 −0.847177
$$999$$ − 6.82908i − 0.216063i
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.w.1351.2 6
13.5 odd 4 5070.2.a.bo.1.2 3
13.8 odd 4 5070.2.a.bx.1.2 yes 3
13.12 even 2 inner 5070.2.b.w.1351.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.2 3 13.5 odd 4
5070.2.a.bx.1.2 yes 3 13.8 odd 4
5070.2.b.w.1351.2 6 1.1 even 1 trivial
5070.2.b.w.1351.5 6 13.12 even 2 inner