# Properties

 Label 7942.2.a.bc.1.3 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.4171892853$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-2.16425$$ of defining polynomial Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.16425 q^{3} +1.00000 q^{4} +4.16425 q^{5} -2.16425 q^{6} -0.683969 q^{7} -1.00000 q^{8} +1.68397 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.16425 q^{3} +1.00000 q^{4} +4.16425 q^{5} -2.16425 q^{6} -0.683969 q^{7} -1.00000 q^{8} +1.68397 q^{9} -4.16425 q^{10} -1.00000 q^{11} +2.16425 q^{12} -2.68397 q^{13} +0.683969 q^{14} +9.01247 q^{15} +1.00000 q^{16} +3.36794 q^{17} -1.68397 q^{18} +4.16425 q^{20} -1.48028 q^{21} +1.00000 q^{22} -1.36794 q^{23} -2.16425 q^{24} +12.3410 q^{25} +2.68397 q^{26} -2.84822 q^{27} -0.683969 q^{28} -2.79631 q^{29} -9.01247 q^{30} -5.01247 q^{31} -1.00000 q^{32} -2.16425 q^{33} -3.36794 q^{34} -2.84822 q^{35} +1.68397 q^{36} +11.6964 q^{37} -5.80877 q^{39} -4.16425 q^{40} +7.01247 q^{41} +1.48028 q^{42} +7.86068 q^{43} -1.00000 q^{44} +7.01247 q^{45} +1.36794 q^{46} +2.96056 q^{47} +2.16425 q^{48} -6.53219 q^{49} -12.3410 q^{50} +7.28905 q^{51} -2.68397 q^{52} +10.3285 q^{53} +2.84822 q^{54} -4.16425 q^{55} +0.683969 q^{56} +2.79631 q^{58} +9.92112 q^{59} +9.01247 q^{60} +2.00000 q^{61} +5.01247 q^{62} -1.15178 q^{63} +1.00000 q^{64} -11.1767 q^{65} +2.16425 q^{66} +0.683969 q^{67} +3.36794 q^{68} -2.96056 q^{69} +2.84822 q^{70} -3.53219 q^{71} -1.68397 q^{72} +0.407381 q^{73} -11.6964 q^{74} +26.7089 q^{75} +0.683969 q^{77} +5.80877 q^{78} +7.28905 q^{79} +4.16425 q^{80} -11.2162 q^{81} -7.01247 q^{82} -12.1892 q^{83} -1.48028 q^{84} +14.0249 q^{85} -7.86068 q^{86} -6.05191 q^{87} +1.00000 q^{88} +16.6819 q^{89} -7.01247 q^{90} +1.83575 q^{91} -1.36794 q^{92} -10.8482 q^{93} -2.96056 q^{94} -2.16425 q^{96} +11.6964 q^{97} +6.53219 q^{98} -1.68397 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} - 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 - 3 * q^8 + 2 * q^9 $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} - 3 q^{11} - q^{12} - 5 q^{13} - q^{14} + 9 q^{15} + 3 q^{16} + 4 q^{17} - 2 q^{18} + 5 q^{20} + 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} + 2 q^{27} + q^{28} - 7 q^{29} - 9 q^{30} + 3 q^{31} - 3 q^{32} + q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} + 14 q^{37} + 2 q^{39} - 5 q^{40} + 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} - 2 q^{46} - q^{48} - 6 q^{49} - 4 q^{50} - 2 q^{51} - 5 q^{52} + 16 q^{53} - 2 q^{54} - 5 q^{55} - q^{56} + 7 q^{58} + 12 q^{59} + 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} - 8 q^{65} - q^{66} - q^{67} + 4 q^{68} - 2 q^{70} + 3 q^{71} - 2 q^{72} + 4 q^{73} - 14 q^{74} + 41 q^{75} - q^{77} - 2 q^{78} - 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} + 5 q^{86} - 9 q^{87} + 3 q^{88} - 16 q^{89} - 3 q^{90} + 13 q^{91} + 2 q^{92} - 22 q^{93} + q^{96} + 14 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 - 3 * q^8 + 2 * q^9 - 5 * q^10 - 3 * q^11 - q^12 - 5 * q^13 - q^14 + 9 * q^15 + 3 * q^16 + 4 * q^17 - 2 * q^18 + 5 * q^20 + 3 * q^22 + 2 * q^23 + q^24 + 4 * q^25 + 5 * q^26 + 2 * q^27 + q^28 - 7 * q^29 - 9 * q^30 + 3 * q^31 - 3 * q^32 + q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 + 14 * q^37 + 2 * q^39 - 5 * q^40 + 3 * q^41 - 5 * q^43 - 3 * q^44 + 3 * q^45 - 2 * q^46 - q^48 - 6 * q^49 - 4 * q^50 - 2 * q^51 - 5 * q^52 + 16 * q^53 - 2 * q^54 - 5 * q^55 - q^56 + 7 * q^58 + 12 * q^59 + 9 * q^60 + 6 * q^61 - 3 * q^62 - 14 * q^63 + 3 * q^64 - 8 * q^65 - q^66 - q^67 + 4 * q^68 - 2 * q^70 + 3 * q^71 - 2 * q^72 + 4 * q^73 - 14 * q^74 + 41 * q^75 - q^77 - 2 * q^78 - 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 + 7 * q^83 + 6 * q^85 + 5 * q^86 - 9 * q^87 + 3 * q^88 - 16 * q^89 - 3 * q^90 + 13 * q^91 + 2 * q^92 - 22 * q^93 + q^96 + 14 * q^97 + 6 * q^98 - 2 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 2.16425 1.24953 0.624765 0.780813i $$-0.285195\pi$$
0.624765 + 0.780813i $$0.285195\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 4.16425 1.86231 0.931154 0.364626i $$-0.118803\pi$$
0.931154 + 0.364626i $$0.118803\pi$$
$$6$$ −2.16425 −0.883551
$$7$$ −0.683969 −0.258516 −0.129258 0.991611i $$-0.541260\pi$$
−0.129258 + 0.991611i $$0.541260\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.68397 0.561323
$$10$$ −4.16425 −1.31685
$$11$$ −1.00000 −0.301511
$$12$$ 2.16425 0.624765
$$13$$ −2.68397 −0.744399 −0.372200 0.928153i $$-0.621396\pi$$
−0.372200 + 0.928153i $$0.621396\pi$$
$$14$$ 0.683969 0.182798
$$15$$ 9.01247 2.32701
$$16$$ 1.00000 0.250000
$$17$$ 3.36794 0.816845 0.408423 0.912793i $$-0.366079\pi$$
0.408423 + 0.912793i $$0.366079\pi$$
$$18$$ −1.68397 −0.396915
$$19$$ 0 0
$$20$$ 4.16425 0.931154
$$21$$ −1.48028 −0.323023
$$22$$ 1.00000 0.213201
$$23$$ −1.36794 −0.285235 −0.142617 0.989778i $$-0.545552\pi$$
−0.142617 + 0.989778i $$0.545552\pi$$
$$24$$ −2.16425 −0.441775
$$25$$ 12.3410 2.46819
$$26$$ 2.68397 0.526370
$$27$$ −2.84822 −0.548140
$$28$$ −0.683969 −0.129258
$$29$$ −2.79631 −0.519262 −0.259631 0.965708i $$-0.583601\pi$$
−0.259631 + 0.965708i $$0.583601\pi$$
$$30$$ −9.01247 −1.64544
$$31$$ −5.01247 −0.900265 −0.450133 0.892962i $$-0.648623\pi$$
−0.450133 + 0.892962i $$0.648623\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.16425 −0.376747
$$34$$ −3.36794 −0.577597
$$35$$ −2.84822 −0.481437
$$36$$ 1.68397 0.280662
$$37$$ 11.6964 1.92288 0.961441 0.275011i $$-0.0886816\pi$$
0.961441 + 0.275011i $$0.0886816\pi$$
$$38$$ 0 0
$$39$$ −5.80877 −0.930148
$$40$$ −4.16425 −0.658425
$$41$$ 7.01247 1.09516 0.547582 0.836752i $$-0.315549\pi$$
0.547582 + 0.836752i $$0.315549\pi$$
$$42$$ 1.48028 0.228412
$$43$$ 7.86068 1.19874 0.599371 0.800471i $$-0.295417\pi$$
0.599371 + 0.800471i $$0.295417\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 7.01247 1.04536
$$46$$ 1.36794 0.201691
$$47$$ 2.96056 0.431842 0.215921 0.976411i $$-0.430725\pi$$
0.215921 + 0.976411i $$0.430725\pi$$
$$48$$ 2.16425 0.312382
$$49$$ −6.53219 −0.933169
$$50$$ −12.3410 −1.74528
$$51$$ 7.28905 1.02067
$$52$$ −2.68397 −0.372200
$$53$$ 10.3285 1.41873 0.709364 0.704842i $$-0.248982\pi$$
0.709364 + 0.704842i $$0.248982\pi$$
$$54$$ 2.84822 0.387593
$$55$$ −4.16425 −0.561507
$$56$$ 0.683969 0.0913992
$$57$$ 0 0
$$58$$ 2.79631 0.367173
$$59$$ 9.92112 1.29162 0.645810 0.763499i $$-0.276520\pi$$
0.645810 + 0.763499i $$0.276520\pi$$
$$60$$ 9.01247 1.16350
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 5.01247 0.636584
$$63$$ −1.15178 −0.145111
$$64$$ 1.00000 0.125000
$$65$$ −11.1767 −1.38630
$$66$$ 2.16425 0.266401
$$67$$ 0.683969 0.0835601 0.0417801 0.999127i $$-0.486697\pi$$
0.0417801 + 0.999127i $$0.486697\pi$$
$$68$$ 3.36794 0.408423
$$69$$ −2.96056 −0.356409
$$70$$ 2.84822 0.340427
$$71$$ −3.53219 −0.419193 −0.209597 0.977788i $$-0.567215\pi$$
−0.209597 + 0.977788i $$0.567215\pi$$
$$72$$ −1.68397 −0.198458
$$73$$ 0.407381 0.0476803 0.0238402 0.999716i $$-0.492411\pi$$
0.0238402 + 0.999716i $$0.492411\pi$$
$$74$$ −11.6964 −1.35968
$$75$$ 26.7089 3.08408
$$76$$ 0 0
$$77$$ 0.683969 0.0779455
$$78$$ 5.80877 0.657714
$$79$$ 7.28905 0.820083 0.410041 0.912067i $$-0.365514\pi$$
0.410041 + 0.912067i $$0.365514\pi$$
$$80$$ 4.16425 0.465577
$$81$$ −11.2162 −1.24624
$$82$$ −7.01247 −0.774397
$$83$$ −12.1892 −1.33794 −0.668968 0.743291i $$-0.733264\pi$$
−0.668968 + 0.743291i $$0.733264\pi$$
$$84$$ −1.48028 −0.161512
$$85$$ 14.0249 1.52122
$$86$$ −7.86068 −0.847639
$$87$$ −6.05191 −0.648833
$$88$$ 1.00000 0.106600
$$89$$ 16.6819 1.76828 0.884140 0.467222i $$-0.154745\pi$$
0.884140 + 0.467222i $$0.154745\pi$$
$$90$$ −7.01247 −0.739179
$$91$$ 1.83575 0.192439
$$92$$ −1.36794 −0.142617
$$93$$ −10.8482 −1.12491
$$94$$ −2.96056 −0.305358
$$95$$ 0 0
$$96$$ −2.16425 −0.220888
$$97$$ 11.6964 1.18759 0.593796 0.804615i $$-0.297628\pi$$
0.593796 + 0.804615i $$0.297628\pi$$
$$98$$ 6.53219 0.659850
$$99$$ −1.68397 −0.169245
$$100$$ 12.3410 1.23410
$$101$$ 3.59262 0.357479 0.178739 0.983896i $$-0.442798\pi$$
0.178739 + 0.983896i $$0.442798\pi$$
$$102$$ −7.28905 −0.721724
$$103$$ 4.46781 0.440227 0.220113 0.975474i $$-0.429357\pi$$
0.220113 + 0.975474i $$0.429357\pi$$
$$104$$ 2.68397 0.263185
$$105$$ −6.16425 −0.601569
$$106$$ −10.3285 −1.00319
$$107$$ −9.69643 −0.937390 −0.468695 0.883360i $$-0.655276\pi$$
−0.468695 + 0.883360i $$0.655276\pi$$
$$108$$ −2.84822 −0.274070
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 4.16425 0.397045
$$111$$ 25.3140 2.40270
$$112$$ −0.683969 −0.0646290
$$113$$ −10.4323 −0.981389 −0.490695 0.871332i $$-0.663257\pi$$
−0.490695 + 0.871332i $$0.663257\pi$$
$$114$$ 0 0
$$115$$ −5.69643 −0.531195
$$116$$ −2.79631 −0.259631
$$117$$ −4.51972 −0.417848
$$118$$ −9.92112 −0.913313
$$119$$ −2.30357 −0.211168
$$120$$ −9.01247 −0.822722
$$121$$ 1.00000 0.0909091
$$122$$ −2.00000 −0.181071
$$123$$ 15.1767 1.36844
$$124$$ −5.01247 −0.450133
$$125$$ 30.5696 2.73423
$$126$$ 1.15178 0.102609
$$127$$ −15.2891 −1.35668 −0.678342 0.734746i $$-0.737301\pi$$
−0.678342 + 0.734746i $$0.737301\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 17.0125 1.49786
$$130$$ 11.1767 0.980263
$$131$$ 3.64453 0.318424 0.159212 0.987244i $$-0.449105\pi$$
0.159212 + 0.987244i $$0.449105\pi$$
$$132$$ −2.16425 −0.188374
$$133$$ 0 0
$$134$$ −0.683969 −0.0590859
$$135$$ −11.8607 −1.02080
$$136$$ −3.36794 −0.288798
$$137$$ 14.6840 1.25454 0.627268 0.778803i $$-0.284173\pi$$
0.627268 + 0.778803i $$0.284173\pi$$
$$138$$ 2.96056 0.252019
$$139$$ −18.5966 −1.57734 −0.788670 0.614817i $$-0.789230\pi$$
−0.788670 + 0.614817i $$0.789230\pi$$
$$140$$ −2.84822 −0.240718
$$141$$ 6.40738 0.539599
$$142$$ 3.53219 0.296414
$$143$$ 2.68397 0.224445
$$144$$ 1.68397 0.140331
$$145$$ −11.6445 −0.967025
$$146$$ −0.407381 −0.0337151
$$147$$ −14.1373 −1.16602
$$148$$ 11.6964 0.961441
$$149$$ 4.30357 0.352562 0.176281 0.984340i $$-0.443593\pi$$
0.176281 + 0.984340i $$0.443593\pi$$
$$150$$ −26.7089 −2.18077
$$151$$ 21.6964 1.76563 0.882815 0.469720i $$-0.155645\pi$$
0.882815 + 0.469720i $$0.155645\pi$$
$$152$$ 0 0
$$153$$ 5.67150 0.458514
$$154$$ −0.683969 −0.0551158
$$155$$ −20.8731 −1.67657
$$156$$ −5.80877 −0.465074
$$157$$ −10.9001 −0.869925 −0.434962 0.900449i $$-0.643238\pi$$
−0.434962 + 0.900449i $$0.643238\pi$$
$$158$$ −7.28905 −0.579886
$$159$$ 22.3534 1.77274
$$160$$ −4.16425 −0.329213
$$161$$ 0.935628 0.0737378
$$162$$ 11.2162 0.881224
$$163$$ 19.7214 1.54470 0.772348 0.635199i $$-0.219082\pi$$
0.772348 + 0.635199i $$0.219082\pi$$
$$164$$ 7.01247 0.547582
$$165$$ −9.01247 −0.701619
$$166$$ 12.1892 0.946064
$$167$$ −7.77532 −0.601672 −0.300836 0.953676i $$-0.597266\pi$$
−0.300836 + 0.953676i $$0.597266\pi$$
$$168$$ 1.48028 0.114206
$$169$$ −5.79631 −0.445870
$$170$$ −14.0249 −1.07566
$$171$$ 0 0
$$172$$ 7.86068 0.599371
$$173$$ −15.7818 −1.19987 −0.599934 0.800050i $$-0.704806\pi$$
−0.599934 + 0.800050i $$0.704806\pi$$
$$174$$ 6.05191 0.458794
$$175$$ −8.44084 −0.638067
$$176$$ −1.00000 −0.0753778
$$177$$ 21.4718 1.61392
$$178$$ −16.6819 −1.25036
$$179$$ 7.97302 0.595932 0.297966 0.954577i $$-0.403692\pi$$
0.297966 + 0.954577i $$0.403692\pi$$
$$180$$ 7.01247 0.522678
$$181$$ 2.55318 0.189776 0.0948881 0.995488i $$-0.469751\pi$$
0.0948881 + 0.995488i $$0.469751\pi$$
$$182$$ −1.83575 −0.136075
$$183$$ 4.32850 0.319972
$$184$$ 1.36794 0.100846
$$185$$ 48.7069 3.58100
$$186$$ 10.8482 0.795430
$$187$$ −3.36794 −0.246288
$$188$$ 2.96056 0.215921
$$189$$ 1.94809 0.141703
$$190$$ 0 0
$$191$$ 8.22468 0.595117 0.297559 0.954704i $$-0.403828\pi$$
0.297559 + 0.954704i $$0.403828\pi$$
$$192$$ 2.16425 0.156191
$$193$$ 24.2141 1.74297 0.871485 0.490422i $$-0.163158\pi$$
0.871485 + 0.490422i $$0.163158\pi$$
$$194$$ −11.6964 −0.839755
$$195$$ −24.1892 −1.73222
$$196$$ −6.53219 −0.466585
$$197$$ 26.6570 1.89923 0.949616 0.313416i $$-0.101473\pi$$
0.949616 + 0.313416i $$0.101473\pi$$
$$198$$ 1.68397 0.119674
$$199$$ −7.06437 −0.500780 −0.250390 0.968145i $$-0.580559\pi$$
−0.250390 + 0.968145i $$0.580559\pi$$
$$200$$ −12.3410 −0.872638
$$201$$ 1.48028 0.104411
$$202$$ −3.59262 −0.252776
$$203$$ 1.91259 0.134237
$$204$$ 7.28905 0.510336
$$205$$ 29.2016 2.03953
$$206$$ −4.46781 −0.311287
$$207$$ −2.30357 −0.160109
$$208$$ −2.68397 −0.186100
$$209$$ 0 0
$$210$$ 6.16425 0.425374
$$211$$ 11.0644 0.761703 0.380851 0.924636i $$-0.375631\pi$$
0.380851 + 0.924636i $$0.375631\pi$$
$$212$$ 10.3285 0.709364
$$213$$ −7.64453 −0.523794
$$214$$ 9.69643 0.662835
$$215$$ 32.7338 2.23243
$$216$$ 2.84822 0.193797
$$217$$ 3.42837 0.232733
$$218$$ 6.00000 0.406371
$$219$$ 0.881673 0.0595779
$$220$$ −4.16425 −0.280754
$$221$$ −9.03944 −0.608059
$$222$$ −25.3140 −1.69896
$$223$$ 14.7359 0.986787 0.493394 0.869806i $$-0.335756\pi$$
0.493394 + 0.869806i $$0.335756\pi$$
$$224$$ 0.683969 0.0456996
$$225$$ 20.7818 1.38545
$$226$$ 10.4323 0.693947
$$227$$ −21.8002 −1.44693 −0.723467 0.690359i $$-0.757452\pi$$
−0.723467 + 0.690359i $$0.757452\pi$$
$$228$$ 0 0
$$229$$ 17.7483 1.17284 0.586422 0.810006i $$-0.300536\pi$$
0.586422 + 0.810006i $$0.300536\pi$$
$$230$$ 5.69643 0.375612
$$231$$ 1.48028 0.0973952
$$232$$ 2.79631 0.183587
$$233$$ −20.3534 −1.33340 −0.666699 0.745327i $$-0.732293\pi$$
−0.666699 + 0.745327i $$0.732293\pi$$
$$234$$ 4.51972 0.295463
$$235$$ 12.3285 0.804222
$$236$$ 9.92112 0.645810
$$237$$ 15.7753 1.02472
$$238$$ 2.30357 0.149318
$$239$$ 3.75687 0.243012 0.121506 0.992591i $$-0.461228\pi$$
0.121506 + 0.992591i $$0.461228\pi$$
$$240$$ 9.01247 0.581752
$$241$$ 9.14974 0.589386 0.294693 0.955592i $$-0.404783\pi$$
0.294693 + 0.955592i $$0.404783\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −15.7299 −1.00907
$$244$$ 2.00000 0.128037
$$245$$ −27.2016 −1.73785
$$246$$ −15.1767 −0.967632
$$247$$ 0 0
$$248$$ 5.01247 0.318292
$$249$$ −26.3804 −1.67179
$$250$$ −30.5696 −1.93339
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −1.15178 −0.0725555
$$253$$ 1.36794 0.0860015
$$254$$ 15.2891 0.959321
$$255$$ 30.3534 1.90081
$$256$$ 1.00000 0.0625000
$$257$$ −23.2102 −1.44781 −0.723905 0.689899i $$-0.757655\pi$$
−0.723905 + 0.689899i $$0.757655\pi$$
$$258$$ −17.0125 −1.05915
$$259$$ −8.00000 −0.497096
$$260$$ −11.1767 −0.693150
$$261$$ −4.70890 −0.291474
$$262$$ −3.64453 −0.225160
$$263$$ −22.2141 −1.36978 −0.684890 0.728646i $$-0.740150\pi$$
−0.684890 + 0.728646i $$0.740150\pi$$
$$264$$ 2.16425 0.133200
$$265$$ 43.0104 2.64211
$$266$$ 0 0
$$267$$ 36.1038 2.20952
$$268$$ 0.683969 0.0417801
$$269$$ 18.9855 1.15757 0.578783 0.815482i $$-0.303528\pi$$
0.578783 + 0.815482i $$0.303528\pi$$
$$270$$ 11.8607 0.721818
$$271$$ 20.7338 1.25949 0.629745 0.776802i $$-0.283159\pi$$
0.629745 + 0.776802i $$0.283159\pi$$
$$272$$ 3.36794 0.204211
$$273$$ 3.97302 0.240458
$$274$$ −14.6840 −0.887091
$$275$$ −12.3410 −0.744188
$$276$$ −2.96056 −0.178205
$$277$$ 21.8422 1.31237 0.656186 0.754599i $$-0.272169\pi$$
0.656186 + 0.754599i $$0.272169\pi$$
$$278$$ 18.5966 1.11535
$$279$$ −8.44084 −0.505340
$$280$$ 2.84822 0.170214
$$281$$ −0.164248 −0.00979821 −0.00489911 0.999988i $$-0.501559\pi$$
−0.00489911 + 0.999988i $$0.501559\pi$$
$$282$$ −6.40738 −0.381554
$$283$$ 4.30152 0.255699 0.127849 0.991794i $$-0.459193\pi$$
0.127849 + 0.991794i $$0.459193\pi$$
$$284$$ −3.53219 −0.209597
$$285$$ 0 0
$$286$$ −2.68397 −0.158706
$$287$$ −4.79631 −0.283117
$$288$$ −1.68397 −0.0992288
$$289$$ −5.65699 −0.332764
$$290$$ 11.6445 0.683790
$$291$$ 25.3140 1.48393
$$292$$ 0.407381 0.0238402
$$293$$ −32.5511 −1.90166 −0.950829 0.309717i $$-0.899766\pi$$
−0.950829 + 0.309717i $$0.899766\pi$$
$$294$$ 14.1373 0.824502
$$295$$ 41.3140 2.40539
$$296$$ −11.6964 −0.679841
$$297$$ 2.84822 0.165270
$$298$$ −4.30357 −0.249299
$$299$$ 3.67150 0.212329
$$300$$ 26.7089 1.54204
$$301$$ −5.37646 −0.309894
$$302$$ −21.6964 −1.24849
$$303$$ 7.77532 0.446680
$$304$$ 0 0
$$305$$ 8.32850 0.476888
$$306$$ −5.67150 −0.324218
$$307$$ −22.2496 −1.26985 −0.634926 0.772573i $$-0.718970\pi$$
−0.634926 + 0.772573i $$0.718970\pi$$
$$308$$ 0.683969 0.0389728
$$309$$ 9.66946 0.550076
$$310$$ 20.8731 1.18552
$$311$$ −20.9855 −1.18998 −0.594989 0.803734i $$-0.702844\pi$$
−0.594989 + 0.803734i $$0.702844\pi$$
$$312$$ 5.80877 0.328857
$$313$$ −11.4533 −0.647379 −0.323689 0.946163i $$-0.604923\pi$$
−0.323689 + 0.946163i $$0.604923\pi$$
$$314$$ 10.9001 0.615130
$$315$$ −4.79631 −0.270241
$$316$$ 7.28905 0.410041
$$317$$ −10.7109 −0.601587 −0.300793 0.953689i $$-0.597251\pi$$
−0.300793 + 0.953689i $$0.597251\pi$$
$$318$$ −22.3534 −1.25352
$$319$$ 2.79631 0.156563
$$320$$ 4.16425 0.232789
$$321$$ −20.9855 −1.17130
$$322$$ −0.935628 −0.0521405
$$323$$ 0 0
$$324$$ −11.2162 −0.623120
$$325$$ −33.1228 −1.83732
$$326$$ −19.7214 −1.09227
$$327$$ −12.9855 −0.718099
$$328$$ −7.01247 −0.387199
$$329$$ −2.02493 −0.111638
$$330$$ 9.01247 0.496120
$$331$$ 6.77138 0.372189 0.186094 0.982532i $$-0.440417\pi$$
0.186094 + 0.982532i $$0.440417\pi$$
$$332$$ −12.1892 −0.668968
$$333$$ 19.6964 1.07936
$$334$$ 7.77532 0.425447
$$335$$ 2.84822 0.155615
$$336$$ −1.48028 −0.0807558
$$337$$ −21.0374 −1.14598 −0.572990 0.819562i $$-0.694217\pi$$
−0.572990 + 0.819562i $$0.694217\pi$$
$$338$$ 5.79631 0.315278
$$339$$ −22.5781 −1.22627
$$340$$ 14.0249 0.760609
$$341$$ 5.01247 0.271440
$$342$$ 0 0
$$343$$ 9.25560 0.499755
$$344$$ −7.86068 −0.423820
$$345$$ −12.3285 −0.663744
$$346$$ 15.7818 0.848435
$$347$$ −29.9710 −1.60893 −0.804463 0.594003i $$-0.797547\pi$$
−0.804463 + 0.594003i $$0.797547\pi$$
$$348$$ −6.05191 −0.324416
$$349$$ −22.6570 −1.21280 −0.606400 0.795159i $$-0.707387\pi$$
−0.606400 + 0.795159i $$0.707387\pi$$
$$350$$ 8.44084 0.451182
$$351$$ 7.64453 0.408035
$$352$$ 1.00000 0.0533002
$$353$$ −1.18524 −0.0630839 −0.0315419 0.999502i $$-0.510042\pi$$
−0.0315419 + 0.999502i $$0.510042\pi$$
$$354$$ −21.4718 −1.14121
$$355$$ −14.7089 −0.780667
$$356$$ 16.6819 0.884140
$$357$$ −4.98549 −0.263860
$$358$$ −7.97302 −0.421387
$$359$$ 19.3659 1.02209 0.511046 0.859553i $$-0.329258\pi$$
0.511046 + 0.859553i $$0.329258\pi$$
$$360$$ −7.01247 −0.369589
$$361$$ 0 0
$$362$$ −2.55318 −0.134192
$$363$$ 2.16425 0.113594
$$364$$ 1.83575 0.0962196
$$365$$ 1.69643 0.0887954
$$366$$ −4.32850 −0.226254
$$367$$ −18.3036 −0.955438 −0.477719 0.878513i $$-0.658536\pi$$
−0.477719 + 0.878513i $$0.658536\pi$$
$$368$$ −1.36794 −0.0713087
$$369$$ 11.8088 0.614740
$$370$$ −48.7069 −2.53215
$$371$$ −7.06437 −0.366764
$$372$$ −10.8482 −0.562454
$$373$$ −6.46781 −0.334891 −0.167445 0.985881i $$-0.553552\pi$$
−0.167445 + 0.985881i $$0.553552\pi$$
$$374$$ 3.36794 0.174152
$$375$$ 66.1602 3.41650
$$376$$ −2.96056 −0.152679
$$377$$ 7.50521 0.386538
$$378$$ −1.94809 −0.100199
$$379$$ −31.9191 −1.63957 −0.819786 0.572670i $$-0.805908\pi$$
−0.819786 + 0.572670i $$0.805908\pi$$
$$380$$ 0 0
$$381$$ −33.0893 −1.69522
$$382$$ −8.22468 −0.420811
$$383$$ −5.66946 −0.289696 −0.144848 0.989454i $$-0.546269\pi$$
−0.144848 + 0.989454i $$0.546269\pi$$
$$384$$ −2.16425 −0.110444
$$385$$ 2.84822 0.145159
$$386$$ −24.2141 −1.23247
$$387$$ 13.2371 0.672882
$$388$$ 11.6964 0.593796
$$389$$ 20.1642 1.02237 0.511184 0.859471i $$-0.329207\pi$$
0.511184 + 0.859471i $$0.329207\pi$$
$$390$$ 24.1892 1.22487
$$391$$ −4.60713 −0.232993
$$392$$ 6.53219 0.329925
$$393$$ 7.88766 0.397880
$$394$$ −26.6570 −1.34296
$$395$$ 30.3534 1.52725
$$396$$ −1.68397 −0.0846226
$$397$$ −15.8272 −0.794346 −0.397173 0.917744i $$-0.630009\pi$$
−0.397173 + 0.917744i $$0.630009\pi$$
$$398$$ 7.06437 0.354105
$$399$$ 0 0
$$400$$ 12.3410 0.617048
$$401$$ −9.06437 −0.452653 −0.226327 0.974051i $$-0.572672\pi$$
−0.226327 + 0.974051i $$0.572672\pi$$
$$402$$ −1.48028 −0.0738296
$$403$$ 13.4533 0.670157
$$404$$ 3.59262 0.178739
$$405$$ −46.7069 −2.32088
$$406$$ −1.91259 −0.0949202
$$407$$ −11.6964 −0.579771
$$408$$ −7.28905 −0.360862
$$409$$ −2.90012 −0.143402 −0.0717010 0.997426i $$-0.522843\pi$$
−0.0717010 + 0.997426i $$0.522843\pi$$
$$410$$ −29.2016 −1.44217
$$411$$ 31.7797 1.56758
$$412$$ 4.46781 0.220113
$$413$$ −6.78574 −0.333904
$$414$$ 2.30357 0.113214
$$415$$ −50.7588 −2.49165
$$416$$ 2.68397 0.131592
$$417$$ −40.2476 −1.97093
$$418$$ 0 0
$$419$$ 19.7753 0.966088 0.483044 0.875596i $$-0.339531\pi$$
0.483044 + 0.875596i $$0.339531\pi$$
$$420$$ −6.16425 −0.300785
$$421$$ −14.9855 −0.730348 −0.365174 0.930939i $$-0.618991\pi$$
−0.365174 + 0.930939i $$0.618991\pi$$
$$422$$ −11.0644 −0.538605
$$423$$ 4.98549 0.242403
$$424$$ −10.3285 −0.501596
$$425$$ 41.5636 2.01613
$$426$$ 7.64453 0.370379
$$427$$ −1.36794 −0.0661992
$$428$$ −9.69643 −0.468695
$$429$$ 5.80877 0.280450
$$430$$ −32.7338 −1.57857
$$431$$ −39.4427 −1.89989 −0.949945 0.312418i $$-0.898861\pi$$
−0.949945 + 0.312418i $$0.898861\pi$$
$$432$$ −2.84822 −0.137035
$$433$$ 1.23919 0.0595518 0.0297759 0.999557i $$-0.490521\pi$$
0.0297759 + 0.999557i $$0.490521\pi$$
$$434$$ −3.42837 −0.164567
$$435$$ −25.2016 −1.20833
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ −0.881673 −0.0421280
$$439$$ 11.1852 0.533842 0.266921 0.963718i $$-0.413994\pi$$
0.266921 + 0.963718i $$0.413994\pi$$
$$440$$ 4.16425 0.198523
$$441$$ −11.0000 −0.523810
$$442$$ 9.03944 0.429962
$$443$$ 23.8422 1.13278 0.566389 0.824138i $$-0.308340\pi$$
0.566389 + 0.824138i $$0.308340\pi$$
$$444$$ 25.3140 1.20135
$$445$$ 69.4677 3.29308
$$446$$ −14.7359 −0.697764
$$447$$ 9.31398 0.440536
$$448$$ −0.683969 −0.0323145
$$449$$ −20.9066 −0.986644 −0.493322 0.869847i $$-0.664217\pi$$
−0.493322 + 0.869847i $$0.664217\pi$$
$$450$$ −20.7818 −0.979663
$$451$$ −7.01247 −0.330204
$$452$$ −10.4323 −0.490695
$$453$$ 46.9565 2.20621
$$454$$ 21.8002 1.02314
$$455$$ 7.64453 0.358381
$$456$$ 0 0
$$457$$ −4.18270 −0.195658 −0.0978292 0.995203i $$-0.531190\pi$$
−0.0978292 + 0.995203i $$0.531190\pi$$
$$458$$ −17.7483 −0.829326
$$459$$ −9.59262 −0.447745
$$460$$ −5.69643 −0.265598
$$461$$ −31.5387 −1.46890 −0.734451 0.678662i $$-0.762560\pi$$
−0.734451 + 0.678662i $$0.762560\pi$$
$$462$$ −1.48028 −0.0688688
$$463$$ −4.60713 −0.214112 −0.107056 0.994253i $$-0.534142\pi$$
−0.107056 + 0.994253i $$0.534142\pi$$
$$464$$ −2.79631 −0.129815
$$465$$ −45.1747 −2.09492
$$466$$ 20.3534 0.942854
$$467$$ −3.94605 −0.182601 −0.0913006 0.995823i $$-0.529102\pi$$
−0.0913006 + 0.995823i $$0.529102\pi$$
$$468$$ −4.51972 −0.208924
$$469$$ −0.467814 −0.0216016
$$470$$ −12.3285 −0.568671
$$471$$ −23.5906 −1.08700
$$472$$ −9.92112 −0.456656
$$473$$ −7.86068 −0.361435
$$474$$ −15.7753 −0.724584
$$475$$ 0 0
$$476$$ −2.30357 −0.105584
$$477$$ 17.3929 0.796365
$$478$$ −3.75687 −0.171835
$$479$$ −16.6840 −0.762310 −0.381155 0.924511i $$-0.624474\pi$$
−0.381155 + 0.924511i $$0.624474\pi$$
$$480$$ −9.01247 −0.411361
$$481$$ −31.3929 −1.43139
$$482$$ −9.14974 −0.416759
$$483$$ 2.02493 0.0921375
$$484$$ 1.00000 0.0454545
$$485$$ 48.7069 2.21166
$$486$$ 15.7299 0.713522
$$487$$ 14.4388 0.654284 0.327142 0.944975i $$-0.393914\pi$$
0.327142 + 0.944975i $$0.393914\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 42.6819 1.93014
$$490$$ 27.2016 1.22884
$$491$$ 6.38040 0.287944 0.143972 0.989582i $$-0.454013\pi$$
0.143972 + 0.989582i $$0.454013\pi$$
$$492$$ 15.1767 0.684219
$$493$$ −9.41780 −0.424156
$$494$$ 0 0
$$495$$ −7.01247 −0.315187
$$496$$ −5.01247 −0.225066
$$497$$ 2.41591 0.108368
$$498$$ 26.3804 1.18213
$$499$$ −31.8252 −1.42469 −0.712345 0.701829i $$-0.752367\pi$$
−0.712345 + 0.701829i $$0.752367\pi$$
$$500$$ 30.5696 1.36711
$$501$$ −16.8277 −0.751807
$$502$$ 12.0000 0.535586
$$503$$ 24.5716 1.09559 0.547797 0.836611i $$-0.315466\pi$$
0.547797 + 0.836611i $$0.315466\pi$$
$$504$$ 1.15178 0.0513045
$$505$$ 14.9606 0.665736
$$506$$ −1.36794 −0.0608123
$$507$$ −12.5447 −0.557128
$$508$$ −15.2891 −0.678342
$$509$$ 5.56769 0.246783 0.123392 0.992358i $$-0.460623\pi$$
0.123392 + 0.992358i $$0.460623\pi$$
$$510$$ −30.3534 −1.34407
$$511$$ −0.278636 −0.0123261
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 23.2102 1.02376
$$515$$ 18.6051 0.819838
$$516$$ 17.0125 0.748932
$$517$$ −2.96056 −0.130205
$$518$$ 8.00000 0.351500
$$519$$ −34.1557 −1.49927
$$520$$ 11.1767 0.490131
$$521$$ 10.8148 0.473803 0.236902 0.971534i $$-0.423868\pi$$
0.236902 + 0.971534i $$0.423868\pi$$
$$522$$ 4.70890 0.206103
$$523$$ 18.4074 0.804899 0.402449 0.915442i $$-0.368159\pi$$
0.402449 + 0.915442i $$0.368159\pi$$
$$524$$ 3.64453 0.159212
$$525$$ −18.2681 −0.797284
$$526$$ 22.2141 0.968581
$$527$$ −16.8817 −0.735377
$$528$$ −2.16425 −0.0941868
$$529$$ −21.1287 −0.918641
$$530$$ −43.0104 −1.86825
$$531$$ 16.7069 0.725016
$$532$$ 0 0
$$533$$ −18.8212 −0.815238
$$534$$ −36.1038 −1.56236
$$535$$ −40.3784 −1.74571
$$536$$ −0.683969 −0.0295430
$$537$$ 17.2556 0.744634
$$538$$ −18.9855 −0.818522
$$539$$ 6.53219 0.281361
$$540$$ −11.8607 −0.510402
$$541$$ 33.0644 1.42155 0.710774 0.703420i $$-0.248345\pi$$
0.710774 + 0.703420i $$0.248345\pi$$
$$542$$ −20.7338 −0.890594
$$543$$ 5.52571 0.237131
$$544$$ −3.36794 −0.144399
$$545$$ −24.9855 −1.07026
$$546$$ −3.97302 −0.170030
$$547$$ −34.1287 −1.45924 −0.729620 0.683853i $$-0.760303\pi$$
−0.729620 + 0.683853i $$0.760303\pi$$
$$548$$ 14.6840 0.627268
$$549$$ 3.36794 0.143740
$$550$$ 12.3410 0.526220
$$551$$ 0 0
$$552$$ 2.96056 0.126010
$$553$$ −4.98549 −0.212004
$$554$$ −21.8422 −0.927987
$$555$$ 105.414 4.47456
$$556$$ −18.5966 −0.788670
$$557$$ 20.5112 0.869087 0.434544 0.900651i $$-0.356910\pi$$
0.434544 + 0.900651i $$0.356910\pi$$
$$558$$ 8.44084 0.357329
$$559$$ −21.0978 −0.892343
$$560$$ −2.84822 −0.120359
$$561$$ −7.28905 −0.307744
$$562$$ 0.164248 0.00692838
$$563$$ 27.0104 1.13835 0.569177 0.822215i $$-0.307262\pi$$
0.569177 + 0.822215i $$0.307262\pi$$
$$564$$ 6.40738 0.269799
$$565$$ −43.4427 −1.82765
$$566$$ −4.30152 −0.180806
$$567$$ 7.67150 0.322173
$$568$$ 3.53219 0.148207
$$569$$ −7.56564 −0.317168 −0.158584 0.987345i $$-0.550693\pi$$
−0.158584 + 0.987345i $$0.550693\pi$$
$$570$$ 0 0
$$571$$ 42.2226 1.76696 0.883481 0.468467i $$-0.155193\pi$$
0.883481 + 0.468467i $$0.155193\pi$$
$$572$$ 2.68397 0.112222
$$573$$ 17.8002 0.743616
$$574$$ 4.79631 0.200194
$$575$$ −16.8817 −0.704014
$$576$$ 1.68397 0.0701654
$$577$$ −25.9191 −1.07902 −0.539512 0.841978i $$-0.681391\pi$$
−0.539512 + 0.841978i $$0.681391\pi$$
$$578$$ 5.65699 0.235300
$$579$$ 52.4053 2.17789
$$580$$ −11.6445 −0.483513
$$581$$ 8.33702 0.345878
$$582$$ −25.3140 −1.04930
$$583$$ −10.3285 −0.427763
$$584$$ −0.407381 −0.0168575
$$585$$ −18.8212 −0.778162
$$586$$ 32.5511 1.34467
$$587$$ 20.6570 0.852605 0.426303 0.904581i $$-0.359816\pi$$
0.426303 + 0.904581i $$0.359816\pi$$
$$588$$ −14.1373 −0.583011
$$589$$ 0 0
$$590$$ −41.3140 −1.70087
$$591$$ 57.6923 2.37315
$$592$$ 11.6964 0.480720
$$593$$ −28.1287 −1.15511 −0.577555 0.816352i $$-0.695993\pi$$
−0.577555 + 0.816352i $$0.695993\pi$$
$$594$$ −2.84822 −0.116864
$$595$$ −9.59262 −0.393259
$$596$$ 4.30357 0.176281
$$597$$ −15.2891 −0.625739
$$598$$ −3.67150 −0.150139
$$599$$ 26.3179 1.07532 0.537661 0.843161i $$-0.319308\pi$$
0.537661 + 0.843161i $$0.319308\pi$$
$$600$$ −26.7089 −1.09039
$$601$$ −38.4053 −1.56659 −0.783293 0.621653i $$-0.786462\pi$$
−0.783293 + 0.621653i $$0.786462\pi$$
$$602$$ 5.37646 0.219128
$$603$$ 1.15178 0.0469042
$$604$$ 21.6964 0.882815
$$605$$ 4.16425 0.169301
$$606$$ −7.77532 −0.315851
$$607$$ 33.2600 1.34998 0.674991 0.737826i $$-0.264147\pi$$
0.674991 + 0.737826i $$0.264147\pi$$
$$608$$ 0 0
$$609$$ 4.13932 0.167734
$$610$$ −8.32850 −0.337211
$$611$$ −7.94605 −0.321463
$$612$$ 5.67150 0.229257
$$613$$ 17.5506 0.708864 0.354432 0.935082i $$-0.384674\pi$$
0.354432 + 0.935082i $$0.384674\pi$$
$$614$$ 22.2496 0.897921
$$615$$ 63.1996 2.54845
$$616$$ −0.683969 −0.0275579
$$617$$ 14.6840 0.591154 0.295577 0.955319i $$-0.404488\pi$$
0.295577 + 0.955319i $$0.404488\pi$$
$$618$$ −9.66946 −0.388963
$$619$$ 11.1183 0.446883 0.223442 0.974717i $$-0.428271\pi$$
0.223442 + 0.974717i $$0.428271\pi$$
$$620$$ −20.8731 −0.838286
$$621$$ 3.89619 0.156349
$$622$$ 20.9855 0.841441
$$623$$ −11.4099 −0.457129
$$624$$ −5.80877 −0.232537
$$625$$ 65.5945 2.62378
$$626$$ 11.4533 0.457766
$$627$$ 0 0
$$628$$ −10.9001 −0.434962
$$629$$ 39.3929 1.57070
$$630$$ 4.79631 0.191090
$$631$$ −46.0748 −1.83421 −0.917104 0.398648i $$-0.869480\pi$$
−0.917104 + 0.398648i $$0.869480\pi$$
$$632$$ −7.28905 −0.289943
$$633$$ 23.9460 0.951770
$$634$$ 10.7109 0.425386
$$635$$ −63.6674 −2.52656
$$636$$ 22.3534 0.886371
$$637$$ 17.5322 0.694651
$$638$$ −2.79631 −0.110707
$$639$$ −5.94809 −0.235303
$$640$$ −4.16425 −0.164606
$$641$$ 11.3181 0.447037 0.223519 0.974700i $$-0.428246\pi$$
0.223519 + 0.974700i $$0.428246\pi$$
$$642$$ 20.9855 0.828231
$$643$$ −30.6280 −1.20785 −0.603925 0.797041i $$-0.706397\pi$$
−0.603925 + 0.797041i $$0.706397\pi$$
$$644$$ 0.935628 0.0368689
$$645$$ 70.8441 2.78948
$$646$$ 0 0
$$647$$ −31.2720 −1.22943 −0.614715 0.788750i $$-0.710729\pi$$
−0.614715 + 0.788750i $$0.710729\pi$$
$$648$$ 11.2162 0.440612
$$649$$ −9.92112 −0.389438
$$650$$ 33.1228 1.29918
$$651$$ 7.41985 0.290807
$$652$$ 19.7214 0.772348
$$653$$ −2.09389 −0.0819402 −0.0409701 0.999160i $$-0.513045\pi$$
−0.0409701 + 0.999160i $$0.513045\pi$$
$$654$$ 12.9855 0.507773
$$655$$ 15.1767 0.593003
$$656$$ 7.01247 0.273791
$$657$$ 0.686016 0.0267641
$$658$$ 2.02493 0.0789400
$$659$$ −24.7608 −0.964544 −0.482272 0.876022i $$-0.660188\pi$$
−0.482272 + 0.876022i $$0.660188\pi$$
$$660$$ −9.01247 −0.350810
$$661$$ −36.4033 −1.41592 −0.707962 0.706251i $$-0.750385\pi$$
−0.707962 + 0.706251i $$0.750385\pi$$
$$662$$ −6.77138 −0.263177
$$663$$ −19.5636 −0.759787
$$664$$ 12.1892 0.473032
$$665$$ 0 0
$$666$$ −19.6964 −0.763221
$$667$$ 3.82518 0.148112
$$668$$ −7.77532 −0.300836
$$669$$ 31.8921 1.23302
$$670$$ −2.84822 −0.110036
$$671$$ −2.00000 −0.0772091
$$672$$ 1.48028 0.0571030
$$673$$ 16.1728 0.623415 0.311707 0.950178i $$-0.399099\pi$$
0.311707 + 0.950178i $$0.399099\pi$$
$$674$$ 21.0374 0.810330
$$675$$ −35.1497 −1.35291
$$676$$ −5.79631 −0.222935
$$677$$ −5.32456 −0.204639 −0.102320 0.994752i $$-0.532626\pi$$
−0.102320 + 0.994752i $$0.532626\pi$$
$$678$$ 22.5781 0.867107
$$679$$ −8.00000 −0.307012
$$680$$ −14.0249 −0.537832
$$681$$ −47.1811 −1.80799
$$682$$ −5.01247 −0.191937
$$683$$ −49.1562 −1.88091 −0.940455 0.339918i $$-0.889601\pi$$
−0.940455 + 0.339918i $$0.889601\pi$$
$$684$$ 0 0
$$685$$ 61.1477 2.33633
$$686$$ −9.25560 −0.353380
$$687$$ 38.4118 1.46550
$$688$$ 7.86068 0.299686
$$689$$ −27.7214 −1.05610
$$690$$ 12.3285 0.469338
$$691$$ −6.51120 −0.247698 −0.123849 0.992301i $$-0.539524\pi$$
−0.123849 + 0.992301i $$0.539524\pi$$
$$692$$ −15.7818 −0.599934
$$693$$ 1.15178 0.0437526
$$694$$ 29.9710 1.13768
$$695$$ −77.4407 −2.93749
$$696$$ 6.05191 0.229397
$$697$$ 23.6175 0.894578
$$698$$ 22.6570 0.857580
$$699$$ −44.0499 −1.66612
$$700$$ −8.44084 −0.319034
$$701$$ −5.77532 −0.218131 −0.109065 0.994035i $$-0.534786\pi$$
−0.109065 + 0.994035i $$0.534786\pi$$
$$702$$ −7.64453 −0.288524
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 26.6819 1.00490
$$706$$ 1.18524 0.0446070
$$707$$ −2.45724 −0.0924140
$$708$$ 21.4718 0.806958
$$709$$ −35.6155 −1.33757 −0.668784 0.743457i $$-0.733185\pi$$
−0.668784 + 0.743457i $$0.733185\pi$$
$$710$$ 14.7089 0.552015
$$711$$ 12.2745 0.460331
$$712$$ −16.6819 −0.625181
$$713$$ 6.85674 0.256787
$$714$$ 4.98549 0.186577
$$715$$ 11.1767 0.417985
$$716$$ 7.97302 0.297966
$$717$$ 8.13079 0.303650
$$718$$ −19.3659 −0.722729
$$719$$ −8.93563 −0.333243 −0.166621 0.986021i $$-0.553286\pi$$
−0.166621 + 0.986021i $$0.553286\pi$$
$$720$$ 7.01247 0.261339
$$721$$ −3.05585 −0.113806
$$722$$ 0 0
$$723$$ 19.8023 0.736455
$$724$$ 2.55318 0.0948881
$$725$$ −34.5091 −1.28164
$$726$$ −2.16425 −0.0803228
$$727$$ −1.42189 −0.0527351 −0.0263675 0.999652i $$-0.508394\pi$$
−0.0263675 + 0.999652i $$0.508394\pi$$
$$728$$ −1.83575 −0.0680375
$$729$$ −0.394916 −0.0146265
$$730$$ −1.69643 −0.0627878
$$731$$ 26.4743 0.979187
$$732$$ 4.32850 0.159986
$$733$$ −46.7567 −1.72700 −0.863499 0.504350i $$-0.831732\pi$$
−0.863499 + 0.504350i $$0.831732\pi$$
$$734$$ 18.3036 0.675597
$$735$$ −58.8711 −2.17149
$$736$$ 1.36794 0.0504229
$$737$$ −0.683969 −0.0251943
$$738$$ −11.8088 −0.434687
$$739$$ −34.4886 −1.26869 −0.634343 0.773052i $$-0.718729\pi$$
−0.634343 + 0.773052i $$0.718729\pi$$
$$740$$ 48.7069 1.79050
$$741$$ 0 0
$$742$$ 7.06437 0.259341
$$743$$ 10.9606 0.402104 0.201052 0.979581i $$-0.435564\pi$$
0.201052 + 0.979581i $$0.435564\pi$$
$$744$$ 10.8482 0.397715
$$745$$ 17.9211 0.656579
$$746$$ 6.46781 0.236803
$$747$$ −20.5262 −0.751014
$$748$$ −3.36794 −0.123144
$$749$$ 6.63206 0.242330
$$750$$ −66.1602 −2.41583
$$751$$ −13.3140 −0.485834 −0.242917 0.970047i $$-0.578104\pi$$
−0.242917 + 0.970047i $$0.578104\pi$$
$$752$$ 2.96056 0.107960
$$753$$ −25.9710 −0.946435
$$754$$ −7.50521 −0.273324
$$755$$ 90.3493 3.28815
$$756$$ 1.94809 0.0708514
$$757$$ −8.54260 −0.310486 −0.155243 0.987876i $$-0.549616\pi$$
−0.155243 + 0.987876i $$0.549616\pi$$
$$758$$ 31.9191 1.15935
$$759$$ 2.96056 0.107461
$$760$$ 0 0
$$761$$ 32.7318 1.18653 0.593263 0.805009i $$-0.297839\pi$$
0.593263 + 0.805009i $$0.297839\pi$$
$$762$$ 33.0893 1.19870
$$763$$ 4.10381 0.148568
$$764$$ 8.22468 0.297559
$$765$$ 23.6175 0.853894
$$766$$ 5.66946 0.204846
$$767$$ −26.6280 −0.961480
$$768$$ 2.16425 0.0780956
$$769$$ 50.5491 1.82285 0.911423 0.411470i $$-0.134985\pi$$
0.911423 + 0.411470i $$0.134985\pi$$
$$770$$ −2.84822 −0.102643
$$771$$ −50.2326 −1.80908
$$772$$ 24.2141 0.871485
$$773$$ 40.7359 1.46517 0.732584 0.680677i $$-0.238314\pi$$
0.732584 + 0.680677i $$0.238314\pi$$
$$774$$ −13.2371 −0.475799
$$775$$ −61.8586 −2.22203
$$776$$ −11.6964 −0.419878
$$777$$ −17.3140 −0.621136
$$778$$ −20.1642 −0.722923
$$779$$ 0 0
$$780$$ −24.1892 −0.866112
$$781$$ 3.53219 0.126392
$$782$$ 4.60713 0.164751
$$783$$ 7.96450 0.284628
$$784$$ −6.53219 −0.233292
$$785$$ −45.3908 −1.62007
$$786$$ −7.88766 −0.281343
$$787$$ −8.53613 −0.304280 −0.152140 0.988359i $$-0.548616\pi$$
−0.152140 + 0.988359i $$0.548616\pi$$
$$788$$ 26.6570 0.949616
$$789$$ −48.0768 −1.71158
$$790$$ −30.3534 −1.07993
$$791$$ 7.13538 0.253705
$$792$$ 1.68397 0.0598372
$$793$$ −5.36794 −0.190621
$$794$$ 15.8272 0.561687
$$795$$ 93.0852 3.30139
$$796$$ −7.06437 −0.250390
$$797$$ 16.0249 0.567632 0.283816 0.958879i $$-0.408399\pi$$
0.283816 + 0.958879i $$0.408399\pi$$
$$798$$ 0 0
$$799$$ 9.97098 0.352748
$$800$$ −12.3410 −0.436319
$$801$$ 28.0918 0.992576
$$802$$ 9.06437 0.320074
$$803$$ −0.407381 −0.0143762
$$804$$ 1.48028 0.0522054
$$805$$ 3.89619 0.137322
$$806$$ −13.4533 −0.473872
$$807$$ 41.0893 1.44641
$$808$$ −3.59262 −0.126388
$$809$$ −20.1458 −0.708288 −0.354144 0.935191i $$-0.615228\pi$$
−0.354144 + 0.935191i $$0.615228\pi$$
$$810$$ 46.7069 1.64111
$$811$$ −8.27454 −0.290558 −0.145279 0.989391i $$-0.546408\pi$$
−0.145279 + 0.989391i $$0.546408\pi$$
$$812$$ 1.91259 0.0671187
$$813$$ 44.8731 1.57377
$$814$$ 11.6964 0.409960
$$815$$ 82.1247 2.87670
$$816$$ 7.28905 0.255168
$$817$$ 0 0
$$818$$ 2.90012 0.101400
$$819$$ 3.09135 0.108021
$$820$$ 29.2016 1.01977
$$821$$ −21.5137 −0.750835 −0.375417 0.926856i $$-0.622501\pi$$
−0.375417 + 0.926856i $$0.622501\pi$$
$$822$$ −31.7797 −1.10845
$$823$$ −5.26412 −0.183496 −0.0917479 0.995782i $$-0.529245\pi$$
−0.0917479 + 0.995782i $$0.529245\pi$$
$$824$$ −4.46781 −0.155644
$$825$$ −26.7089 −0.929885
$$826$$ 6.78574 0.236106
$$827$$ −16.7069 −0.580954 −0.290477 0.956882i $$-0.593814\pi$$
−0.290477 + 0.956882i $$0.593814\pi$$
$$828$$ −2.30357 −0.0800544
$$829$$ 5.55064 0.192782 0.0963908 0.995344i $$-0.469270\pi$$
0.0963908 + 0.995344i $$0.469270\pi$$
$$830$$ 50.7588 1.76186
$$831$$ 47.2720 1.63985
$$832$$ −2.68397 −0.0930499
$$833$$ −22.0000 −0.762255
$$834$$ 40.2476 1.39366
$$835$$ −32.3784 −1.12050
$$836$$ 0 0
$$837$$ 14.2766 0.493471
$$838$$ −19.7753 −0.683127
$$839$$ −15.3744 −0.530784 −0.265392 0.964141i $$-0.585501\pi$$
−0.265392 + 0.964141i $$0.585501\pi$$
$$840$$ 6.16425 0.212687
$$841$$ −21.1807 −0.730367
$$842$$ 14.9855 0.516434
$$843$$ −0.355473 −0.0122431
$$844$$ 11.0644 0.380851
$$845$$ −24.1373 −0.830347
$$846$$ −4.98549 −0.171405
$$847$$ −0.683969 −0.0235015
$$848$$ 10.3285 0.354682
$$849$$ 9.30955 0.319503
$$850$$ −41.5636 −1.42562
$$851$$ −16.0000 −0.548473
$$852$$ −7.64453 −0.261897
$$853$$ 24.8028 0.849231 0.424616 0.905374i $$-0.360409\pi$$
0.424616 + 0.905374i $$0.360409\pi$$
$$854$$ 1.36794 0.0468099
$$855$$ 0 0
$$856$$ 9.69643 0.331417
$$857$$ 45.6445 1.55919 0.779594 0.626286i $$-0.215426\pi$$
0.779594 + 0.626286i $$0.215426\pi$$
$$858$$ −5.80877 −0.198308
$$859$$ 54.8356 1.87097 0.935483 0.353371i $$-0.114965\pi$$
0.935483 + 0.353371i $$0.114965\pi$$
$$860$$ 32.7338 1.11621
$$861$$ −10.3804 −0.353763
$$862$$ 39.4427 1.34342
$$863$$ −45.8771 −1.56167 −0.780837 0.624735i $$-0.785207\pi$$
−0.780837 + 0.624735i $$0.785207\pi$$
$$864$$ 2.84822 0.0968983
$$865$$ −65.7193 −2.23452
$$866$$ −1.23919 −0.0421095
$$867$$ −12.2431 −0.415799
$$868$$ 3.42837 0.116367
$$869$$ −7.28905 −0.247264
$$870$$ 25.2016 0.854416
$$871$$ −1.83575 −0.0622021
$$872$$ 6.00000 0.203186
$$873$$ 19.6964 0.666623
$$874$$ 0 0
$$875$$ −20.9087 −0.706841
$$876$$ 0.881673 0.0297890
$$877$$ 15.6609 0.528832 0.264416 0.964409i $$-0.414821\pi$$
0.264416 + 0.964409i $$0.414821\pi$$
$$878$$ −11.1852 −0.377484
$$879$$ −70.4487 −2.37618
$$880$$ −4.16425 −0.140377
$$881$$ −3.39492 −0.114378 −0.0571888 0.998363i $$-0.518214\pi$$
−0.0571888 + 0.998363i $$0.518214\pi$$
$$882$$ 11.0000 0.370389
$$883$$ −33.6425 −1.13216 −0.566080 0.824350i $$-0.691541\pi$$
−0.566080 + 0.824350i $$0.691541\pi$$
$$884$$ −9.03944 −0.304029
$$885$$ 89.4137 3.00561
$$886$$ −23.8422 −0.800995
$$887$$ 22.1287 0.743011 0.371505 0.928431i $$-0.378842\pi$$
0.371505 + 0.928431i $$0.378842\pi$$
$$888$$ −25.3140 −0.849482
$$889$$ 10.4572 0.350725
$$890$$ −69.4677 −2.32856
$$891$$ 11.2162 0.375755
$$892$$ 14.7359 0.493394
$$893$$ 0 0
$$894$$ −9.31398 −0.311506
$$895$$ 33.2016 1.10981
$$896$$ 0.683969 0.0228498
$$897$$ 7.94605 0.265311
$$898$$ 20.9066 0.697662
$$899$$ 14.0164 0.467473
$$900$$ 20.7818 0.692727
$$901$$ 34.7857 1.15888
$$902$$ 7.01247 0.233490
$$903$$ −11.6360 −0.387222
$$904$$ 10.4323 0.346973
$$905$$ 10.6321 0.353422
$$906$$ −46.9565 −1.56002
$$907$$ 21.7633 0.722640 0.361320 0.932442i $$-0.382326\pi$$
0.361320 + 0.932442i $$0.382326\pi$$
$$908$$ −21.8002 −0.723467
$$909$$ 6.04986 0.200661
$$910$$ −7.64453 −0.253414
$$911$$ −46.8356 −1.55173 −0.775866 0.630897i $$-0.782687\pi$$
−0.775866 + 0.630897i $$0.782687\pi$$
$$912$$ 0 0
$$913$$ 12.1892 0.403403
$$914$$ 4.18270 0.138351
$$915$$ 18.0249 0.595886
$$916$$ 17.7483 0.586422
$$917$$ −2.49274 −0.0823177
$$918$$ 9.59262 0.316604
$$919$$ −42.0563 −1.38731 −0.693655 0.720307i $$-0.744001\pi$$
−0.693655 + 0.720307i $$0.744001\pi$$
$$920$$ 5.69643 0.187806
$$921$$ −48.1537 −1.58672
$$922$$ 31.5387 1.03867
$$923$$ 9.48028 0.312047
$$924$$ 1.48028 0.0486976
$$925$$ 144.345 4.74604
$$926$$ 4.60713 0.151400
$$927$$ 7.52366 0.247109
$$928$$ 2.79631 0.0917934
$$929$$ 36.1018 1.18446 0.592230 0.805769i $$-0.298248\pi$$
0.592230 + 0.805769i $$0.298248\pi$$
$$930$$ 45.1747 1.48134
$$931$$ 0 0
$$932$$ −20.3534 −0.666699
$$933$$ −45.4178 −1.48691
$$934$$ 3.94605 0.129119
$$935$$ −14.0249 −0.458664
$$936$$ 4.51972 0.147732
$$937$$ 47.3638 1.54731 0.773655 0.633607i $$-0.218427\pi$$
0.773655 + 0.633607i $$0.218427\pi$$
$$938$$ 0.467814 0.0152747
$$939$$ −24.7878 −0.808919
$$940$$ 12.3285 0.402111
$$941$$ 9.34301 0.304573 0.152287 0.988336i $$-0.451336\pi$$
0.152287 + 0.988336i $$0.451336\pi$$
$$942$$ 23.5906 0.768622
$$943$$ −9.59262 −0.312379
$$944$$ 9.92112 0.322905
$$945$$ 8.11234 0.263894
$$946$$ 7.86068 0.255573
$$947$$ 22.2496 0.723015 0.361508 0.932369i $$-0.382262\pi$$
0.361508 + 0.932369i $$0.382262\pi$$
$$948$$ 15.7753 0.512359
$$949$$ −1.09340 −0.0354932
$$950$$ 0 0
$$951$$ −23.1811 −0.751700
$$952$$ 2.30357 0.0746590
$$953$$ 31.9710 1.03564 0.517821 0.855489i $$-0.326743\pi$$
0.517821 + 0.855489i $$0.326743\pi$$
$$954$$ −17.3929 −0.563115
$$955$$ 34.2496 1.10829
$$956$$ 3.75687 0.121506
$$957$$ 6.05191 0.195630
$$958$$ 16.6840 0.539035
$$959$$ −10.0434 −0.324318
$$960$$ 9.01247 0.290876
$$961$$ −5.87519 −0.189522
$$962$$ 31.3929 1.01215
$$963$$ −16.3285 −0.526178
$$964$$ 9.14974 0.294693
$$965$$ 100.834 3.24595
$$966$$ −2.02493 −0.0651511
$$967$$ 5.97098 0.192014 0.0960068 0.995381i $$-0.469393\pi$$
0.0960068 + 0.995381i $$0.469393\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −48.7069 −1.56388
$$971$$ 19.0999 0.612944 0.306472 0.951880i $$-0.400851\pi$$
0.306472 + 0.951880i $$0.400851\pi$$
$$972$$ −15.7299 −0.504536
$$973$$ 12.7195 0.407768
$$974$$ −14.4388 −0.462649
$$975$$ −71.6859 −2.29578
$$976$$ 2.00000 0.0640184
$$977$$ 10.2745 0.328712 0.164356 0.986401i $$-0.447445\pi$$
0.164356 + 0.986401i $$0.447445\pi$$
$$978$$ −42.6819 −1.36482
$$979$$ −16.6819 −0.533157
$$980$$ −27.2016 −0.868925
$$981$$ −10.1038 −0.322590
$$982$$ −6.38040 −0.203607
$$983$$ −33.0084 −1.05280 −0.526402 0.850236i $$-0.676459\pi$$
−0.526402 + 0.850236i $$0.676459\pi$$
$$984$$ −15.1767 −0.483816
$$985$$ 111.006 3.53696
$$986$$ 9.41780 0.299924
$$987$$ −4.38245 −0.139495
$$988$$ 0 0
$$989$$ −10.7529 −0.341923
$$990$$ 7.01247 0.222871
$$991$$ −1.56564 −0.0497343 −0.0248671 0.999691i $$-0.507916\pi$$
−0.0248671 + 0.999691i $$0.507916\pi$$
$$992$$ 5.01247 0.159146
$$993$$ 14.6549 0.465061
$$994$$ −2.41591 −0.0766279
$$995$$ −29.4178 −0.932607
$$996$$ −26.3804 −0.835895
$$997$$ 46.5361 1.47381 0.736907 0.675994i $$-0.236286\pi$$
0.736907 + 0.675994i $$0.236286\pi$$
$$998$$ 31.8252 1.00741
$$999$$ −33.3140 −1.05401
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 7942.2.a.bc.1.3 3
19.18 odd 2 418.2.a.h.1.1 3
57.56 even 2 3762.2.a.bd.1.1 3
76.75 even 2 3344.2.a.p.1.3 3
209.208 even 2 4598.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.h.1.1 3 19.18 odd 2
3344.2.a.p.1.3 3 76.75 even 2
3762.2.a.bd.1.1 3 57.56 even 2
4598.2.a.bm.1.1 3 209.208 even 2
7942.2.a.bc.1.3 3 1.1 even 1 trivial