# Properties

 Label 8281.2.a.bv.1.2 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{23})$$ Defining polynomial: $$x^{4} - 24 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-4.09827$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +2.68406 q^{10} +5.79583 q^{11} +1.41421 q^{12} -3.79583 q^{15} -1.00000 q^{16} -5.51249 q^{17} -1.00000 q^{18} +2.82843 q^{19} -2.68406 q^{20} +5.79583 q^{22} +1.79583 q^{23} +4.24264 q^{24} +2.20417 q^{25} +5.65685 q^{27} -8.79583 q^{29} -3.79583 q^{30} +1.41421 q^{31} +5.00000 q^{32} -8.19654 q^{33} -5.51249 q^{34} +1.00000 q^{36} -6.79583 q^{37} +2.82843 q^{38} -8.05217 q^{40} -9.75513 q^{41} -1.79583 q^{43} -5.79583 q^{44} -2.68406 q^{45} +1.79583 q^{46} -2.82843 q^{47} +1.41421 q^{48} +2.20417 q^{50} +7.79583 q^{51} +6.59166 q^{53} +5.65685 q^{54} +15.5563 q^{55} -4.00000 q^{57} -8.79583 q^{58} -1.12548 q^{59} +3.79583 q^{60} -1.55858 q^{61} +1.41421 q^{62} +7.00000 q^{64} -8.19654 q^{66} +5.79583 q^{67} +5.51249 q^{68} -2.53969 q^{69} +6.00000 q^{71} +3.00000 q^{72} +5.80122 q^{73} -6.79583 q^{74} -3.11716 q^{75} -2.82843 q^{76} -11.7958 q^{79} -2.68406 q^{80} -5.00000 q^{81} -9.75513 q^{82} +9.89949 q^{83} -14.7958 q^{85} -1.79583 q^{86} +12.4392 q^{87} -17.3875 q^{88} -12.1504 q^{89} -2.68406 q^{90} -1.79583 q^{92} -2.00000 q^{93} -2.82843 q^{94} +7.59166 q^{95} -7.07107 q^{96} +4.24264 q^{97} -5.79583 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44} - 12 q^{46} + 28 q^{50} + 12 q^{51} - 12 q^{53} - 16 q^{57} - 16 q^{58} - 4 q^{60} + 28 q^{64} + 4 q^{67} + 24 q^{71} + 12 q^{72} - 8 q^{74} - 28 q^{79} - 20 q^{81} - 40 q^{85} + 12 q^{86} - 12 q^{88} + 12 q^{92} - 8 q^{93} - 8 q^{95} - 4 q^{99} + O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ −1.41421 −0.816497 −0.408248 0.912871i $$-0.633860\pi$$
−0.408248 + 0.912871i $$0.633860\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 2.68406 1.20035 0.600174 0.799870i $$-0.295098\pi$$
0.600174 + 0.799870i $$0.295098\pi$$
$$6$$ −1.41421 −0.577350
$$7$$ 0 0
$$8$$ −3.00000 −1.06066
$$9$$ −1.00000 −0.333333
$$10$$ 2.68406 0.848774
$$11$$ 5.79583 1.74751 0.873754 0.486367i $$-0.161678\pi$$
0.873754 + 0.486367i $$0.161678\pi$$
$$12$$ 1.41421 0.408248
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −3.79583 −0.980079
$$16$$ −1.00000 −0.250000
$$17$$ −5.51249 −1.33697 −0.668487 0.743724i $$-0.733058\pi$$
−0.668487 + 0.743724i $$0.733058\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ −2.68406 −0.600174
$$21$$ 0 0
$$22$$ 5.79583 1.23568
$$23$$ 1.79583 0.374457 0.187228 0.982316i $$-0.440050\pi$$
0.187228 + 0.982316i $$0.440050\pi$$
$$24$$ 4.24264 0.866025
$$25$$ 2.20417 0.440834
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ −8.79583 −1.63334 −0.816672 0.577101i $$-0.804184\pi$$
−0.816672 + 0.577101i $$0.804184\pi$$
$$30$$ −3.79583 −0.693021
$$31$$ 1.41421 0.254000 0.127000 0.991903i $$-0.459465\pi$$
0.127000 + 0.991903i $$0.459465\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −8.19654 −1.42684
$$34$$ −5.51249 −0.945383
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.79583 −1.11723 −0.558614 0.829428i $$-0.688667\pi$$
−0.558614 + 0.829428i $$0.688667\pi$$
$$38$$ 2.82843 0.458831
$$39$$ 0 0
$$40$$ −8.05217 −1.27316
$$41$$ −9.75513 −1.52349 −0.761747 0.647874i $$-0.775658\pi$$
−0.761747 + 0.647874i $$0.775658\pi$$
$$42$$ 0 0
$$43$$ −1.79583 −0.273862 −0.136931 0.990581i $$-0.543724\pi$$
−0.136931 + 0.990581i $$0.543724\pi$$
$$44$$ −5.79583 −0.873754
$$45$$ −2.68406 −0.400116
$$46$$ 1.79583 0.264781
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 1.41421 0.204124
$$49$$ 0 0
$$50$$ 2.20417 0.311716
$$51$$ 7.79583 1.09163
$$52$$ 0 0
$$53$$ 6.59166 0.905435 0.452717 0.891654i $$-0.350455\pi$$
0.452717 + 0.891654i $$0.350455\pi$$
$$54$$ 5.65685 0.769800
$$55$$ 15.5563 2.09762
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ −8.79583 −1.15495
$$59$$ −1.12548 −0.146524 −0.0732622 0.997313i $$-0.523341\pi$$
−0.0732622 + 0.997313i $$0.523341\pi$$
$$60$$ 3.79583 0.490040
$$61$$ −1.55858 −0.199556 −0.0997780 0.995010i $$-0.531813\pi$$
−0.0997780 + 0.995010i $$0.531813\pi$$
$$62$$ 1.41421 0.179605
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ −8.19654 −1.00892
$$67$$ 5.79583 0.708074 0.354037 0.935232i $$-0.384809\pi$$
0.354037 + 0.935232i $$0.384809\pi$$
$$68$$ 5.51249 0.668487
$$69$$ −2.53969 −0.305743
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 5.80122 0.678982 0.339491 0.940609i $$-0.389745\pi$$
0.339491 + 0.940609i $$0.389745\pi$$
$$74$$ −6.79583 −0.789999
$$75$$ −3.11716 −0.359939
$$76$$ −2.82843 −0.324443
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.7958 −1.32713 −0.663567 0.748117i $$-0.730958\pi$$
−0.663567 + 0.748117i $$0.730958\pi$$
$$80$$ −2.68406 −0.300087
$$81$$ −5.00000 −0.555556
$$82$$ −9.75513 −1.07727
$$83$$ 9.89949 1.08661 0.543305 0.839535i $$-0.317173\pi$$
0.543305 + 0.839535i $$0.317173\pi$$
$$84$$ 0 0
$$85$$ −14.7958 −1.60483
$$86$$ −1.79583 −0.193649
$$87$$ 12.4392 1.33362
$$88$$ −17.3875 −1.85351
$$89$$ −12.1504 −1.28794 −0.643972 0.765049i $$-0.722715\pi$$
−0.643972 + 0.765049i $$0.722715\pi$$
$$90$$ −2.68406 −0.282925
$$91$$ 0 0
$$92$$ −1.79583 −0.187228
$$93$$ −2.00000 −0.207390
$$94$$ −2.82843 −0.291730
$$95$$ 7.59166 0.778888
$$96$$ −7.07107 −0.721688
$$97$$ 4.24264 0.430775 0.215387 0.976529i $$-0.430899\pi$$
0.215387 + 0.976529i $$0.430899\pi$$
$$98$$ 0 0
$$99$$ −5.79583 −0.582503
$$100$$ −2.20417 −0.220417
$$101$$ 2.97280 0.295804 0.147902 0.989002i $$-0.452748\pi$$
0.147902 + 0.989002i $$0.452748\pi$$
$$102$$ 7.79583 0.771902
$$103$$ 8.19654 0.807629 0.403815 0.914841i $$-0.367684\pi$$
0.403815 + 0.914841i $$0.367684\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 6.59166 0.640239
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −5.65685 −0.544331
$$109$$ −17.5917 −1.68498 −0.842488 0.538715i $$-0.818910\pi$$
−0.842488 + 0.538715i $$0.818910\pi$$
$$110$$ 15.5563 1.48324
$$111$$ 9.61076 0.912213
$$112$$ 0 0
$$113$$ −16.5917 −1.56081 −0.780406 0.625273i $$-0.784988\pi$$
−0.780406 + 0.625273i $$0.784988\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 4.82012 0.449478
$$116$$ 8.79583 0.816672
$$117$$ 0 0
$$118$$ −1.12548 −0.103608
$$119$$ 0 0
$$120$$ 11.3875 1.03953
$$121$$ 22.5917 2.05379
$$122$$ −1.55858 −0.141107
$$123$$ 13.7958 1.24393
$$124$$ −1.41421 −0.127000
$$125$$ −7.50417 −0.671194
$$126$$ 0 0
$$127$$ 7.59166 0.673651 0.336826 0.941567i $$-0.390647\pi$$
0.336826 + 0.941567i $$0.390647\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 2.53969 0.223607
$$130$$ 0 0
$$131$$ 8.19654 0.716135 0.358068 0.933696i $$-0.383436\pi$$
0.358068 + 0.933696i $$0.383436\pi$$
$$132$$ 8.19654 0.713418
$$133$$ 0 0
$$134$$ 5.79583 0.500684
$$135$$ 15.1833 1.30677
$$136$$ 16.5375 1.41808
$$137$$ −9.20417 −0.786365 −0.393183 0.919460i $$-0.628626\pi$$
−0.393183 + 0.919460i $$0.628626\pi$$
$$138$$ −2.53969 −0.216193
$$139$$ 15.2676 1.29498 0.647491 0.762073i $$-0.275818\pi$$
0.647491 + 0.762073i $$0.275818\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −23.6085 −1.96058
$$146$$ 5.80122 0.480113
$$147$$ 0 0
$$148$$ 6.79583 0.558614
$$149$$ 4.59166 0.376164 0.188082 0.982153i $$-0.439773\pi$$
0.188082 + 0.982153i $$0.439773\pi$$
$$150$$ −3.11716 −0.254515
$$151$$ −17.5917 −1.43159 −0.715795 0.698311i $$-0.753935\pi$$
−0.715795 + 0.698311i $$0.753935\pi$$
$$152$$ −8.48528 −0.688247
$$153$$ 5.51249 0.445658
$$154$$ 0 0
$$155$$ 3.79583 0.304889
$$156$$ 0 0
$$157$$ 6.92670 0.552811 0.276405 0.961041i $$-0.410857\pi$$
0.276405 + 0.961041i $$0.410857\pi$$
$$158$$ −11.7958 −0.938426
$$159$$ −9.32202 −0.739284
$$160$$ 13.4203 1.06097
$$161$$ 0 0
$$162$$ −5.00000 −0.392837
$$163$$ −13.3875 −1.04859 −0.524295 0.851537i $$-0.675671\pi$$
−0.524295 + 0.851537i $$0.675671\pi$$
$$164$$ 9.75513 0.761747
$$165$$ −22.0000 −1.71270
$$166$$ 9.89949 0.768350
$$167$$ −15.2676 −1.18144 −0.590722 0.806875i $$-0.701157\pi$$
−0.590722 + 0.806875i $$0.701157\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −14.7958 −1.13479
$$171$$ −2.82843 −0.216295
$$172$$ 1.79583 0.136931
$$173$$ −9.32202 −0.708740 −0.354370 0.935105i $$-0.615305\pi$$
−0.354370 + 0.935105i $$0.615305\pi$$
$$174$$ 12.4392 0.943012
$$175$$ 0 0
$$176$$ −5.79583 −0.436877
$$177$$ 1.59166 0.119637
$$178$$ −12.1504 −0.910714
$$179$$ −0.408337 −0.0305205 −0.0152603 0.999884i $$-0.504858\pi$$
−0.0152603 + 0.999884i $$0.504858\pi$$
$$180$$ 2.68406 0.200058
$$181$$ 16.5375 1.22922 0.614610 0.788831i $$-0.289314\pi$$
0.614610 + 0.788831i $$0.289314\pi$$
$$182$$ 0 0
$$183$$ 2.20417 0.162937
$$184$$ −5.38749 −0.397171
$$185$$ −18.2404 −1.34106
$$186$$ −2.00000 −0.146647
$$187$$ −31.9494 −2.33637
$$188$$ 2.82843 0.206284
$$189$$ 0 0
$$190$$ 7.59166 0.550757
$$191$$ −25.7958 −1.86652 −0.933260 0.359200i $$-0.883049\pi$$
−0.933260 + 0.359200i $$0.883049\pi$$
$$192$$ −9.89949 −0.714435
$$193$$ 3.40834 0.245337 0.122669 0.992448i $$-0.460855\pi$$
0.122669 + 0.992448i $$0.460855\pi$$
$$194$$ 4.24264 0.304604
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ −5.79583 −0.411892
$$199$$ 22.0499 1.56308 0.781539 0.623856i $$-0.214435\pi$$
0.781539 + 0.623856i $$0.214435\pi$$
$$200$$ −6.61251 −0.467575
$$201$$ −8.19654 −0.578140
$$202$$ 2.97280 0.209165
$$203$$ 0 0
$$204$$ −7.79583 −0.545817
$$205$$ −26.1833 −1.82872
$$206$$ 8.19654 0.571080
$$207$$ −1.79583 −0.124819
$$208$$ 0 0
$$209$$ 16.3931 1.13393
$$210$$ 0 0
$$211$$ −1.79583 −0.123630 −0.0618151 0.998088i $$-0.519689\pi$$
−0.0618151 + 0.998088i $$0.519689\pi$$
$$212$$ −6.59166 −0.452717
$$213$$ −8.48528 −0.581402
$$214$$ −6.00000 −0.410152
$$215$$ −4.82012 −0.328729
$$216$$ −16.9706 −1.15470
$$217$$ 0 0
$$218$$ −17.5917 −1.19146
$$219$$ −8.20417 −0.554386
$$220$$ −15.5563 −1.04881
$$221$$ 0 0
$$222$$ 9.61076 0.645032
$$223$$ 5.65685 0.378811 0.189405 0.981899i $$-0.439344\pi$$
0.189405 + 0.981899i $$0.439344\pi$$
$$224$$ 0 0
$$225$$ −2.20417 −0.146945
$$226$$ −16.5917 −1.10366
$$227$$ −20.9245 −1.38881 −0.694403 0.719587i $$-0.744331\pi$$
−0.694403 + 0.719587i $$0.744331\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −12.7279 −0.841085 −0.420542 0.907273i $$-0.638160\pi$$
−0.420542 + 0.907273i $$0.638160\pi$$
$$230$$ 4.82012 0.317829
$$231$$ 0 0
$$232$$ 26.3875 1.73242
$$233$$ −21.1833 −1.38777 −0.693883 0.720088i $$-0.744101\pi$$
−0.693883 + 0.720088i $$0.744101\pi$$
$$234$$ 0 0
$$235$$ −7.59166 −0.495225
$$236$$ 1.12548 0.0732622
$$237$$ 16.6818 1.08360
$$238$$ 0 0
$$239$$ −19.7958 −1.28049 −0.640243 0.768172i $$-0.721166\pi$$
−0.640243 + 0.768172i $$0.721166\pi$$
$$240$$ 3.79583 0.245020
$$241$$ −4.38701 −0.282592 −0.141296 0.989967i $$-0.545127\pi$$
−0.141296 + 0.989967i $$0.545127\pi$$
$$242$$ 22.5917 1.45225
$$243$$ −9.89949 −0.635053
$$244$$ 1.55858 0.0997780
$$245$$ 0 0
$$246$$ 13.7958 0.879590
$$247$$ 0 0
$$248$$ −4.24264 −0.269408
$$249$$ −14.0000 −0.887214
$$250$$ −7.50417 −0.474606
$$251$$ 3.11716 0.196754 0.0983769 0.995149i $$-0.468635\pi$$
0.0983769 + 0.995149i $$0.468635\pi$$
$$252$$ 0 0
$$253$$ 10.4083 0.654367
$$254$$ 7.59166 0.476343
$$255$$ 20.9245 1.31034
$$256$$ −17.0000 −1.06250
$$257$$ −15.4120 −0.961373 −0.480686 0.876893i $$-0.659612\pi$$
−0.480686 + 0.876893i $$0.659612\pi$$
$$258$$ 2.53969 0.158114
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 8.79583 0.544448
$$262$$ 8.19654 0.506384
$$263$$ −25.3875 −1.56546 −0.782730 0.622361i $$-0.786173\pi$$
−0.782730 + 0.622361i $$0.786173\pi$$
$$264$$ 24.5896 1.51339
$$265$$ 17.6924 1.08684
$$266$$ 0 0
$$267$$ 17.1833 1.05160
$$268$$ −5.79583 −0.354037
$$269$$ 1.12548 0.0686215 0.0343107 0.999411i $$-0.489076\pi$$
0.0343107 + 0.999411i $$0.489076\pi$$
$$270$$ 15.1833 0.924028
$$271$$ 23.1754 1.40781 0.703903 0.710296i $$-0.251439\pi$$
0.703903 + 0.710296i $$0.251439\pi$$
$$272$$ 5.51249 0.334244
$$273$$ 0 0
$$274$$ −9.20417 −0.556044
$$275$$ 12.7750 0.770361
$$276$$ 2.53969 0.152871
$$277$$ −6.18333 −0.371520 −0.185760 0.982595i $$-0.559475\pi$$
−0.185760 + 0.982595i $$0.559475\pi$$
$$278$$ 15.2676 0.915690
$$279$$ −1.41421 −0.0846668
$$280$$ 0 0
$$281$$ 15.2042 0.907005 0.453502 0.891255i $$-0.350174\pi$$
0.453502 + 0.891255i $$0.350174\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 29.4097 1.74823 0.874114 0.485721i $$-0.161443\pi$$
0.874114 + 0.485721i $$0.161443\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ −10.7362 −0.635960
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ 13.3875 0.787500
$$290$$ −23.6085 −1.38634
$$291$$ −6.00000 −0.351726
$$292$$ −5.80122 −0.339491
$$293$$ 9.75513 0.569901 0.284950 0.958542i $$-0.408023\pi$$
0.284950 + 0.958542i $$0.408023\pi$$
$$294$$ 0 0
$$295$$ −3.02084 −0.175880
$$296$$ 20.3875 1.18500
$$297$$ 32.7862 1.90245
$$298$$ 4.59166 0.265988
$$299$$ 0 0
$$300$$ 3.11716 0.179970
$$301$$ 0 0
$$302$$ −17.5917 −1.01229
$$303$$ −4.20417 −0.241523
$$304$$ −2.82843 −0.162221
$$305$$ −4.18333 −0.239537
$$306$$ 5.51249 0.315128
$$307$$ −34.2004 −1.95192 −0.975960 0.217951i $$-0.930063\pi$$
−0.975960 + 0.217951i $$0.930063\pi$$
$$308$$ 0 0
$$309$$ −11.5917 −0.659427
$$310$$ 3.79583 0.215589
$$311$$ −11.8617 −0.672616 −0.336308 0.941752i $$-0.609178\pi$$
−0.336308 + 0.941752i $$0.609178\pi$$
$$312$$ 0 0
$$313$$ −1.70295 −0.0962565 −0.0481283 0.998841i $$-0.515326\pi$$
−0.0481283 + 0.998841i $$0.515326\pi$$
$$314$$ 6.92670 0.390896
$$315$$ 0 0
$$316$$ 11.7958 0.663567
$$317$$ −12.5917 −0.707218 −0.353609 0.935393i $$-0.615046\pi$$
−0.353609 + 0.935393i $$0.615046\pi$$
$$318$$ −9.32202 −0.522753
$$319$$ −50.9792 −2.85428
$$320$$ 18.7884 1.05030
$$321$$ 8.48528 0.473602
$$322$$ 0 0
$$323$$ −15.5917 −0.867543
$$324$$ 5.00000 0.277778
$$325$$ 0 0
$$326$$ −13.3875 −0.741465
$$327$$ 24.8784 1.37578
$$328$$ 29.2654 1.61591
$$329$$ 0 0
$$330$$ −22.0000 −1.21106
$$331$$ 0.612505 0.0336663 0.0168332 0.999858i $$-0.494642\pi$$
0.0168332 + 0.999858i $$0.494642\pi$$
$$332$$ −9.89949 −0.543305
$$333$$ 6.79583 0.372409
$$334$$ −15.2676 −0.835407
$$335$$ 15.5563 0.849934
$$336$$ 0 0
$$337$$ −29.9792 −1.63307 −0.816534 0.577297i $$-0.804108\pi$$
−0.816534 + 0.577297i $$0.804108\pi$$
$$338$$ 0 0
$$339$$ 23.4642 1.27440
$$340$$ 14.7958 0.802417
$$341$$ 8.19654 0.443868
$$342$$ −2.82843 −0.152944
$$343$$ 0 0
$$344$$ 5.38749 0.290474
$$345$$ −6.81667 −0.366997
$$346$$ −9.32202 −0.501155
$$347$$ 14.9792 0.804123 0.402062 0.915613i $$-0.368294\pi$$
0.402062 + 0.915613i $$0.368294\pi$$
$$348$$ −12.4392 −0.666810
$$349$$ −13.0167 −0.696766 −0.348383 0.937352i $$-0.613269\pi$$
−0.348383 + 0.937352i $$0.613269\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 28.9792 1.54459
$$353$$ 15.1232 0.804929 0.402464 0.915436i $$-0.368154\pi$$
0.402464 + 0.915436i $$0.368154\pi$$
$$354$$ 1.59166 0.0845959
$$355$$ 16.1043 0.854730
$$356$$ 12.1504 0.643972
$$357$$ 0 0
$$358$$ −0.408337 −0.0215813
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 8.05217 0.424387
$$361$$ −11.0000 −0.578947
$$362$$ 16.5375 0.869189
$$363$$ −31.9494 −1.67691
$$364$$ 0 0
$$365$$ 15.5708 0.815014
$$366$$ 2.20417 0.115214
$$367$$ −21.2132 −1.10732 −0.553660 0.832743i $$-0.686769\pi$$
−0.553660 + 0.832743i $$0.686769\pi$$
$$368$$ −1.79583 −0.0936142
$$369$$ 9.75513 0.507832
$$370$$ −18.2404 −0.948274
$$371$$ 0 0
$$372$$ 2.00000 0.103695
$$373$$ −12.5917 −0.651972 −0.325986 0.945375i $$-0.605696\pi$$
−0.325986 + 0.945375i $$0.605696\pi$$
$$374$$ −31.9494 −1.65207
$$375$$ 10.6125 0.548027
$$376$$ 8.48528 0.437595
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −3.38749 −0.174004 −0.0870020 0.996208i $$-0.527729\pi$$
−0.0870020 + 0.996208i $$0.527729\pi$$
$$380$$ −7.59166 −0.389444
$$381$$ −10.7362 −0.550034
$$382$$ −25.7958 −1.31983
$$383$$ 14.9789 0.765385 0.382692 0.923876i $$-0.374997\pi$$
0.382692 + 0.923876i $$0.374997\pi$$
$$384$$ 4.24264 0.216506
$$385$$ 0 0
$$386$$ 3.40834 0.173480
$$387$$ 1.79583 0.0912872
$$388$$ −4.24264 −0.215387
$$389$$ 28.3875 1.43930 0.719652 0.694335i $$-0.244302\pi$$
0.719652 + 0.694335i $$0.244302\pi$$
$$390$$ 0 0
$$391$$ −9.89949 −0.500639
$$392$$ 0 0
$$393$$ −11.5917 −0.584722
$$394$$ −8.00000 −0.403034
$$395$$ −31.6607 −1.59302
$$396$$ 5.79583 0.291251
$$397$$ −7.07107 −0.354887 −0.177443 0.984131i $$-0.556783\pi$$
−0.177443 + 0.984131i $$0.556783\pi$$
$$398$$ 22.0499 1.10526
$$399$$ 0 0
$$400$$ −2.20417 −0.110208
$$401$$ 24.3875 1.21785 0.608927 0.793227i $$-0.291600\pi$$
0.608927 + 0.793227i $$0.291600\pi$$
$$402$$ −8.19654 −0.408806
$$403$$ 0 0
$$404$$ −2.97280 −0.147902
$$405$$ −13.4203 −0.666860
$$406$$ 0 0
$$407$$ −39.3875 −1.95237
$$408$$ −23.3875 −1.15785
$$409$$ 28.6879 1.41853 0.709263 0.704944i $$-0.249028\pi$$
0.709263 + 0.704944i $$0.249028\pi$$
$$410$$ −26.1833 −1.29310
$$411$$ 13.0167 0.642064
$$412$$ −8.19654 −0.403815
$$413$$ 0 0
$$414$$ −1.79583 −0.0882603
$$415$$ 26.5708 1.30431
$$416$$ 0 0
$$417$$ −21.5917 −1.05735
$$418$$ 16.3931 0.801812
$$419$$ −28.5435 −1.39444 −0.697221 0.716856i $$-0.745581\pi$$
−0.697221 + 0.716856i $$0.745581\pi$$
$$420$$ 0 0
$$421$$ −6.59166 −0.321258 −0.160629 0.987015i $$-0.551352\pi$$
−0.160629 + 0.987015i $$0.551352\pi$$
$$422$$ −1.79583 −0.0874197
$$423$$ 2.82843 0.137523
$$424$$ −19.7750 −0.960358
$$425$$ −12.1504 −0.589383
$$426$$ −8.48528 −0.411113
$$427$$ 0 0
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −4.82012 −0.232447
$$431$$ −5.18333 −0.249672 −0.124836 0.992177i $$-0.539840\pi$$
−0.124836 + 0.992177i $$0.539840\pi$$
$$432$$ −5.65685 −0.272166
$$433$$ −13.4203 −0.644938 −0.322469 0.946580i $$-0.604513\pi$$
−0.322469 + 0.946580i $$0.604513\pi$$
$$434$$ 0 0
$$435$$ 33.3875 1.60081
$$436$$ 17.5917 0.842488
$$437$$ 5.07938 0.242980
$$438$$ −8.20417 −0.392010
$$439$$ −34.2004 −1.63230 −0.816148 0.577843i $$-0.803894\pi$$
−0.816148 + 0.577843i $$0.803894\pi$$
$$440$$ −46.6690 −2.22486
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10.0000 0.475114 0.237557 0.971374i $$-0.423653\pi$$
0.237557 + 0.971374i $$0.423653\pi$$
$$444$$ −9.61076 −0.456106
$$445$$ −32.6125 −1.54598
$$446$$ 5.65685 0.267860
$$447$$ −6.49359 −0.307136
$$448$$ 0 0
$$449$$ 4.40834 0.208042 0.104021 0.994575i $$-0.466829\pi$$
0.104021 + 0.994575i $$0.466829\pi$$
$$450$$ −2.20417 −0.103905
$$451$$ −56.5391 −2.66232
$$452$$ 16.5917 0.780406
$$453$$ 24.8784 1.16889
$$454$$ −20.9245 −0.982034
$$455$$ 0 0
$$456$$ 12.0000 0.561951
$$457$$ −14.5917 −0.682569 −0.341285 0.939960i $$-0.610862\pi$$
−0.341285 + 0.939960i $$0.610862\pi$$
$$458$$ −12.7279 −0.594737
$$459$$ −31.1833 −1.45551
$$460$$ −4.82012 −0.224739
$$461$$ −30.9683 −1.44234 −0.721169 0.692759i $$-0.756395\pi$$
−0.721169 + 0.692759i $$0.756395\pi$$
$$462$$ 0 0
$$463$$ 11.3875 0.529222 0.264611 0.964355i $$-0.414756\pi$$
0.264611 + 0.964355i $$0.414756\pi$$
$$464$$ 8.79583 0.408336
$$465$$ −5.36812 −0.248940
$$466$$ −21.1833 −0.981299
$$467$$ 4.82012 0.223048 0.111524 0.993762i $$-0.464427\pi$$
0.111524 + 0.993762i $$0.464427\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −7.59166 −0.350177
$$471$$ −9.79583 −0.451368
$$472$$ 3.37643 0.155413
$$473$$ −10.4083 −0.478576
$$474$$ 16.6818 0.766222
$$475$$ 6.23433 0.286051
$$476$$ 0 0
$$477$$ −6.59166 −0.301812
$$478$$ −19.7958 −0.905440
$$479$$ 7.35981 0.336278 0.168139 0.985763i $$-0.446224\pi$$
0.168139 + 0.985763i $$0.446224\pi$$
$$480$$ −18.9792 −0.866276
$$481$$ 0 0
$$482$$ −4.38701 −0.199823
$$483$$ 0 0
$$484$$ −22.5917 −1.02689
$$485$$ 11.3875 0.517079
$$486$$ −9.89949 −0.449050
$$487$$ 19.5917 0.887783 0.443891 0.896081i $$-0.353598\pi$$
0.443891 + 0.896081i $$0.353598\pi$$
$$488$$ 4.67575 0.211661
$$489$$ 18.9328 0.856170
$$490$$ 0 0
$$491$$ −9.59166 −0.432866 −0.216433 0.976298i $$-0.569442\pi$$
−0.216433 + 0.976298i $$0.569442\pi$$
$$492$$ −13.7958 −0.621964
$$493$$ 48.4869 2.18374
$$494$$ 0 0
$$495$$ −15.5563 −0.699206
$$496$$ −1.41421 −0.0635001
$$497$$ 0 0
$$498$$ −14.0000 −0.627355
$$499$$ 16.2042 0.725398 0.362699 0.931906i $$-0.381855\pi$$
0.362699 + 0.931906i $$0.381855\pi$$
$$500$$ 7.50417 0.335597
$$501$$ 21.5917 0.964644
$$502$$ 3.11716 0.139126
$$503$$ 28.5435 1.27269 0.636347 0.771403i $$-0.280445\pi$$
0.636347 + 0.771403i $$0.280445\pi$$
$$504$$ 0 0
$$505$$ 7.97916 0.355068
$$506$$ 10.4083 0.462707
$$507$$ 0 0
$$508$$ −7.59166 −0.336826
$$509$$ −26.4370 −1.17180 −0.585899 0.810384i $$-0.699258\pi$$
−0.585899 + 0.810384i $$0.699258\pi$$
$$510$$ 20.9245 0.926551
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 16.0000 0.706417
$$514$$ −15.4120 −0.679793
$$515$$ 22.0000 0.969436
$$516$$ −2.53969 −0.111804
$$517$$ −16.3931 −0.720967
$$518$$ 0 0
$$519$$ 13.1833 0.578684
$$520$$ 0 0
$$521$$ −25.0227 −1.09627 −0.548133 0.836391i $$-0.684661\pi$$
−0.548133 + 0.836391i $$0.684661\pi$$
$$522$$ 8.79583 0.384983
$$523$$ −28.5730 −1.24941 −0.624705 0.780861i $$-0.714781\pi$$
−0.624705 + 0.780861i $$0.714781\pi$$
$$524$$ −8.19654 −0.358068
$$525$$ 0 0
$$526$$ −25.3875 −1.10695
$$527$$ −7.79583 −0.339592
$$528$$ 8.19654 0.356709
$$529$$ −19.7750 −0.859782
$$530$$ 17.6924 0.768509
$$531$$ 1.12548 0.0488415
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 17.1833 0.743595
$$535$$ −16.1043 −0.696252
$$536$$ −17.3875 −0.751025
$$537$$ 0.577476 0.0249199
$$538$$ 1.12548 0.0485227
$$539$$ 0 0
$$540$$ −15.1833 −0.653386
$$541$$ −12.5917 −0.541358 −0.270679 0.962670i $$-0.587248\pi$$
−0.270679 + 0.962670i $$0.587248\pi$$
$$542$$ 23.1754 0.995469
$$543$$ −23.3875 −1.00365
$$544$$ −27.5624 −1.18173
$$545$$ −47.2170 −2.02256
$$546$$ 0 0
$$547$$ −36.9792 −1.58111 −0.790557 0.612388i $$-0.790209\pi$$
−0.790557 + 0.612388i $$0.790209\pi$$
$$548$$ 9.20417 0.393183
$$549$$ 1.55858 0.0665187
$$550$$ 12.7750 0.544727
$$551$$ −24.8784 −1.05985
$$552$$ 7.61907 0.324289
$$553$$ 0 0
$$554$$ −6.18333 −0.262704
$$555$$ 25.7958 1.09497
$$556$$ −15.2676 −0.647491
$$557$$ 20.5917 0.872497 0.436248 0.899826i $$-0.356307\pi$$
0.436248 + 0.899826i $$0.356307\pi$$
$$558$$ −1.41421 −0.0598684
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 45.1833 1.90764
$$562$$ 15.2042 0.641349
$$563$$ 1.12548 0.0474331 0.0237166 0.999719i $$-0.492450\pi$$
0.0237166 + 0.999719i $$0.492450\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ −44.5330 −1.87352
$$566$$ 29.4097 1.23618
$$567$$ 0 0
$$568$$ −18.0000 −0.755263
$$569$$ 6.40834 0.268651 0.134326 0.990937i $$-0.457113\pi$$
0.134326 + 0.990937i $$0.457113\pi$$
$$570$$ −10.7362 −0.449691
$$571$$ 15.1833 0.635402 0.317701 0.948191i $$-0.397089\pi$$
0.317701 + 0.948191i $$0.397089\pi$$
$$572$$ 0 0
$$573$$ 36.4808 1.52401
$$574$$ 0 0
$$575$$ 3.95832 0.165073
$$576$$ −7.00000 −0.291667
$$577$$ 9.17765 0.382071 0.191035 0.981583i $$-0.438816\pi$$
0.191035 + 0.981583i $$0.438816\pi$$
$$578$$ 13.3875 0.556846
$$579$$ −4.82012 −0.200317
$$580$$ 23.6085 0.980291
$$581$$ 0 0
$$582$$ −6.00000 −0.248708
$$583$$ 38.2042 1.58225
$$584$$ −17.4037 −0.720169
$$585$$ 0 0
$$586$$ 9.75513 0.402981
$$587$$ −3.11716 −0.128659 −0.0643296 0.997929i $$-0.520491\pi$$
−0.0643296 + 0.997929i $$0.520491\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ −3.02084 −0.124366
$$591$$ 11.3137 0.465384
$$592$$ 6.79583 0.279307
$$593$$ 1.55858 0.0640033 0.0320017 0.999488i $$-0.489812\pi$$
0.0320017 + 0.999488i $$0.489812\pi$$
$$594$$ 32.7862 1.34523
$$595$$ 0 0
$$596$$ −4.59166 −0.188082
$$597$$ −31.1833 −1.27625
$$598$$ 0 0
$$599$$ −14.4083 −0.588709 −0.294354 0.955696i $$-0.595105\pi$$
−0.294354 + 0.955696i $$0.595105\pi$$
$$600$$ 9.35149 0.381773
$$601$$ 30.9683 1.26322 0.631612 0.775284i $$-0.282393\pi$$
0.631612 + 0.775284i $$0.282393\pi$$
$$602$$ 0 0
$$603$$ −5.79583 −0.236025
$$604$$ 17.5917 0.715795
$$605$$ 60.6373 2.46526
$$606$$ −4.20417 −0.170783
$$607$$ 35.9033 1.45727 0.728636 0.684901i $$-0.240155\pi$$
0.728636 + 0.684901i $$0.240155\pi$$
$$608$$ 14.1421 0.573539
$$609$$ 0 0
$$610$$ −4.18333 −0.169378
$$611$$ 0 0
$$612$$ −5.51249 −0.222829
$$613$$ −5.97916 −0.241496 −0.120748 0.992683i $$-0.538529\pi$$
−0.120748 + 0.992683i $$0.538529\pi$$
$$614$$ −34.2004 −1.38022
$$615$$ 37.0288 1.49315
$$616$$ 0 0
$$617$$ −24.3875 −0.981804 −0.490902 0.871215i $$-0.663333\pi$$
−0.490902 + 0.871215i $$0.663333\pi$$
$$618$$ −11.5917 −0.466285
$$619$$ −33.9116 −1.36302 −0.681512 0.731807i $$-0.738677\pi$$
−0.681512 + 0.731807i $$0.738677\pi$$
$$620$$ −3.79583 −0.152444
$$621$$ 10.1588 0.407657
$$622$$ −11.8617 −0.475611
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.1625 −1.24650
$$626$$ −1.70295 −0.0680636
$$627$$ −23.1833 −0.925853
$$628$$ −6.92670 −0.276405
$$629$$ 37.4619 1.49370
$$630$$ 0 0
$$631$$ −20.4083 −0.812443 −0.406222 0.913775i $$-0.633154\pi$$
−0.406222 + 0.913775i $$0.633154\pi$$
$$632$$ 35.3875 1.40764
$$633$$ 2.53969 0.100944
$$634$$ −12.5917 −0.500079
$$635$$ 20.3765 0.808615
$$636$$ 9.32202 0.369642
$$637$$ 0 0
$$638$$ −50.9792 −2.01828
$$639$$ −6.00000 −0.237356
$$640$$ −8.05217 −0.318290
$$641$$ 4.79583 0.189424 0.0947120 0.995505i $$-0.469807\pi$$
0.0947120 + 0.995505i $$0.469807\pi$$
$$642$$ 8.48528 0.334887
$$643$$ −5.65685 −0.223085 −0.111542 0.993760i $$-0.535579\pi$$
−0.111542 + 0.993760i $$0.535579\pi$$
$$644$$ 0 0
$$645$$ 6.81667 0.268406
$$646$$ −15.5917 −0.613446
$$647$$ 1.96221 0.0771426 0.0385713 0.999256i $$-0.487719\pi$$
0.0385713 + 0.999256i $$0.487719\pi$$
$$648$$ 15.0000 0.589256
$$649$$ −6.52307 −0.256053
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 13.3875 0.524295
$$653$$ 25.1833 0.985500 0.492750 0.870171i $$-0.335992\pi$$
0.492750 + 0.870171i $$0.335992\pi$$
$$654$$ 24.8784 0.972821
$$655$$ 22.0000 0.859611
$$656$$ 9.75513 0.380874
$$657$$ −5.80122 −0.226327
$$658$$ 0 0
$$659$$ −37.5917 −1.46436 −0.732182 0.681109i $$-0.761498\pi$$
−0.732182 + 0.681109i $$0.761498\pi$$
$$660$$ 22.0000 0.856349
$$661$$ −27.8512 −1.08328 −0.541642 0.840609i $$-0.682197\pi$$
−0.541642 + 0.840609i $$0.682197\pi$$
$$662$$ 0.612505 0.0238057
$$663$$ 0 0
$$664$$ −29.6985 −1.15252
$$665$$ 0 0
$$666$$ 6.79583 0.263333
$$667$$ −15.7958 −0.611617
$$668$$ 15.2676 0.590722
$$669$$ −8.00000 −0.309298
$$670$$ 15.5563 0.600994
$$671$$ −9.03328 −0.348726
$$672$$ 0 0
$$673$$ 23.9792 0.924329 0.462164 0.886794i $$-0.347073\pi$$
0.462164 + 0.886794i $$0.347073\pi$$
$$674$$ −29.9792 −1.15475
$$675$$ 12.4687 0.479919
$$676$$ 0 0
$$677$$ −34.7779 −1.33662 −0.668311 0.743882i $$-0.732982\pi$$
−0.668311 + 0.743882i $$0.732982\pi$$
$$678$$ 23.4642 0.901135
$$679$$ 0 0
$$680$$ 44.3875 1.70218
$$681$$ 29.5917 1.13395
$$682$$ 8.19654 0.313862
$$683$$ 50.7750 1.94285 0.971425 0.237345i $$-0.0762771\pi$$
0.971425 + 0.237345i $$0.0762771\pi$$
$$684$$ 2.82843 0.108148
$$685$$ −24.7045 −0.943911
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 1.79583 0.0684654
$$689$$ 0 0
$$690$$ −6.81667 −0.259506
$$691$$ −11.0250 −0.419410 −0.209705 0.977765i $$-0.567250\pi$$
−0.209705 + 0.977765i $$0.567250\pi$$
$$692$$ 9.32202 0.354370
$$693$$ 0 0
$$694$$ 14.9792 0.568601
$$695$$ 40.9792 1.55443
$$696$$ −37.3176 −1.41452
$$697$$ 53.7750 2.03687
$$698$$ −13.0167 −0.492688
$$699$$ 29.9577 1.13311
$$700$$ 0 0
$$701$$ −10.4083 −0.393117 −0.196559 0.980492i $$-0.562977\pi$$
−0.196559 + 0.980492i $$0.562977\pi$$
$$702$$ 0 0
$$703$$ −19.2215 −0.724953
$$704$$ 40.5708 1.52907
$$705$$ 10.7362 0.404350
$$706$$ 15.1232 0.569171
$$707$$ 0 0
$$708$$ −1.59166 −0.0598184
$$709$$ −18.7958 −0.705892 −0.352946 0.935644i $$-0.614820\pi$$
−0.352946 + 0.935644i $$0.614820\pi$$
$$710$$ 16.1043 0.604385
$$711$$ 11.7958 0.442378
$$712$$ 36.4513 1.36607
$$713$$ 2.53969 0.0951121
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0.408337 0.0152603
$$717$$ 27.9955 1.04551
$$718$$ −4.00000 −0.149279
$$719$$ −29.1210 −1.08603 −0.543015 0.839723i $$-0.682717\pi$$
−0.543015 + 0.839723i $$0.682717\pi$$
$$720$$ 2.68406 0.100029
$$721$$ 0 0
$$722$$ −11.0000 −0.409378
$$723$$ 6.20417 0.230736
$$724$$ −16.5375 −0.614610
$$725$$ −19.3875 −0.720033
$$726$$ −31.9494 −1.18575
$$727$$ 35.3259 1.31016 0.655082 0.755558i $$-0.272634\pi$$
0.655082 + 0.755558i $$0.272634\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 15.5708 0.576302
$$731$$ 9.89949 0.366146
$$732$$ −2.20417 −0.0814684
$$733$$ −0.692369 −0.0255732 −0.0127866 0.999918i $$-0.504070\pi$$
−0.0127866 + 0.999918i $$0.504070\pi$$
$$734$$ −21.2132 −0.782994
$$735$$ 0 0
$$736$$ 8.97916 0.330976
$$737$$ 33.5917 1.23736
$$738$$ 9.75513 0.359091
$$739$$ 15.1833 0.558528 0.279264 0.960214i $$-0.409910\pi$$
0.279264 + 0.960214i $$0.409910\pi$$
$$740$$ 18.2404 0.670531
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.18333 0.336904 0.168452 0.985710i $$-0.446123\pi$$
0.168452 + 0.985710i $$0.446123\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 12.3243 0.451527
$$746$$ −12.5917 −0.461014
$$747$$ −9.89949 −0.362204
$$748$$ 31.9494 1.16819
$$749$$ 0 0
$$750$$ 10.6125 0.387514
$$751$$ 1.79583 0.0655308 0.0327654 0.999463i $$-0.489569\pi$$
0.0327654 + 0.999463i $$0.489569\pi$$
$$752$$ 2.82843 0.103142
$$753$$ −4.40834 −0.160649
$$754$$ 0 0
$$755$$ −47.2170 −1.71840
$$756$$ 0 0
$$757$$ −25.1833 −0.915304 −0.457652 0.889132i $$-0.651309\pi$$
−0.457652 + 0.889132i $$0.651309\pi$$
$$758$$ −3.38749 −0.123039
$$759$$ −14.7196 −0.534288
$$760$$ −22.7750 −0.826136
$$761$$ −9.32202 −0.337923 −0.168961 0.985623i $$-0.554041\pi$$
−0.168961 + 0.985623i $$0.554041\pi$$
$$762$$ −10.7362 −0.388933
$$763$$ 0 0
$$764$$ 25.7958 0.933260
$$765$$ 14.7958 0.534944
$$766$$ 14.9789 0.541209
$$767$$ 0 0
$$768$$ 24.0416 0.867528
$$769$$ −4.24264 −0.152994 −0.0764968 0.997070i $$-0.524373\pi$$
−0.0764968 + 0.997070i $$0.524373\pi$$
$$770$$ 0 0
$$771$$ 21.7958 0.784958
$$772$$ −3.40834 −0.122669
$$773$$ −7.64854 −0.275099 −0.137549 0.990495i $$-0.543923\pi$$
−0.137549 + 0.990495i $$0.543923\pi$$
$$774$$ 1.79583 0.0645498
$$775$$ 3.11716 0.111972
$$776$$ −12.7279 −0.456906
$$777$$ 0 0
$$778$$ 28.3875 1.01774
$$779$$ −27.5917 −0.988574
$$780$$ 0 0
$$781$$ 34.7750 1.24435
$$782$$ −9.89949 −0.354005
$$783$$ −49.7567 −1.77816
$$784$$ 0 0
$$785$$ 18.5917 0.663565
$$786$$ −11.5917 −0.413461
$$787$$ −25.4264 −0.906352 −0.453176 0.891421i $$-0.649709\pi$$
−0.453176 + 0.891421i $$0.649709\pi$$
$$788$$ 8.00000 0.284988
$$789$$ 35.9033 1.27819
$$790$$ −31.6607 −1.12644
$$791$$ 0 0
$$792$$ 17.3875 0.617838
$$793$$ 0 0
$$794$$ −7.07107 −0.250943
$$795$$ −25.0208 −0.887398
$$796$$ −22.0499 −0.781539
$$797$$ 26.2926 0.931331 0.465666 0.884961i $$-0.345815\pi$$
0.465666 + 0.884961i $$0.345815\pi$$
$$798$$ 0 0
$$799$$ 15.5917 0.551593
$$800$$ 11.0208 0.389646
$$801$$ 12.1504 0.429315
$$802$$ 24.3875 0.861152
$$803$$ 33.6229 1.18653
$$804$$ 8.19654 0.289070
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.59166 −0.0560292
$$808$$ −8.91839 −0.313748
$$809$$ 39.3667 1.38406 0.692029 0.721870i $$-0.256717\pi$$
0.692029 + 0.721870i $$0.256717\pi$$
$$810$$ −13.4203 −0.471541
$$811$$ 38.4725 1.35095 0.675476 0.737382i $$-0.263938\pi$$
0.675476 + 0.737382i $$0.263938\pi$$
$$812$$ 0 0
$$813$$ −32.7750 −1.14947
$$814$$ −39.3875 −1.38053
$$815$$ −35.9328 −1.25867
$$816$$ −7.79583 −0.272909
$$817$$ −5.07938 −0.177705
$$818$$ 28.6879 1.00305
$$819$$ 0 0
$$820$$ 26.1833 0.914361
$$821$$ −40.3667 −1.40881 −0.704403 0.709800i $$-0.748785\pi$$
−0.704403 + 0.709800i $$0.748785\pi$$
$$822$$ 13.0167 0.454008
$$823$$ 36.7750 1.28190 0.640948 0.767584i $$-0.278542\pi$$
0.640948 + 0.767584i $$0.278542\pi$$
$$824$$ −24.5896 −0.856620
$$825$$ −18.0666 −0.628997
$$826$$ 0 0
$$827$$ 26.9792 0.938157 0.469079 0.883156i $$-0.344586\pi$$
0.469079 + 0.883156i $$0.344586\pi$$
$$828$$ 1.79583 0.0624095
$$829$$ −20.2026 −0.701666 −0.350833 0.936438i $$-0.614102\pi$$
−0.350833 + 0.936438i $$0.614102\pi$$
$$830$$ 26.5708 0.922287
$$831$$ 8.74454 0.303345
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −21.5917 −0.747658
$$835$$ −40.9792 −1.41814
$$836$$ −16.3931 −0.566967
$$837$$ 8.00000 0.276520
$$838$$ −28.5435 −0.986020
$$839$$ −5.65685 −0.195296 −0.0976481 0.995221i $$-0.531132\pi$$
−0.0976481 + 0.995221i $$0.531132\pi$$
$$840$$ 0 0
$$841$$ 48.3667 1.66782
$$842$$ −6.59166 −0.227164
$$843$$ −21.5019 −0.740566
$$844$$ 1.79583 0.0618151
$$845$$ 0 0
$$846$$ 2.82843 0.0972433
$$847$$ 0 0
$$848$$ −6.59166 −0.226359
$$849$$ −41.5917 −1.42742
$$850$$ −12.1504 −0.416757
$$851$$ −12.2042 −0.418354
$$852$$ 8.48528 0.290701
$$853$$ 13.1610 0.450625 0.225313 0.974287i $$-0.427660\pi$$
0.225313 + 0.974287i $$0.427660\pi$$
$$854$$ 0 0
$$855$$ −7.59166 −0.259629
$$856$$ 18.0000 0.615227
$$857$$ 37.7507 1.28954 0.644769 0.764377i $$-0.276954\pi$$
0.644769 + 0.764377i $$0.276954\pi$$
$$858$$ 0 0
$$859$$ −23.7529 −0.810438 −0.405219 0.914220i $$-0.632805\pi$$
−0.405219 + 0.914220i $$0.632805\pi$$
$$860$$ 4.82012 0.164365
$$861$$ 0 0
$$862$$ −5.18333 −0.176545
$$863$$ 17.7958 0.605777 0.302889 0.953026i $$-0.402049\pi$$
0.302889 + 0.953026i $$0.402049\pi$$
$$864$$ 28.2843 0.962250
$$865$$ −25.0208 −0.850734
$$866$$ −13.4203 −0.456040
$$867$$ −18.9328 −0.642991
$$868$$ 0 0
$$869$$ −68.3667 −2.31918
$$870$$ 33.3875 1.13194
$$871$$ 0 0
$$872$$ 52.7750 1.78719
$$873$$ −4.24264 −0.143592
$$874$$ 5.07938 0.171813
$$875$$ 0 0
$$876$$ 8.20417 0.277193
$$877$$ −4.79583 −0.161944 −0.0809719 0.996716i $$-0.525802\pi$$
−0.0809719 + 0.996716i $$0.525802\pi$$
$$878$$ −34.2004 −1.15421
$$879$$ −13.7958 −0.465322
$$880$$ −15.5563 −0.524404
$$881$$ 23.3198 0.785664 0.392832 0.919610i $$-0.371495\pi$$
0.392832 + 0.919610i $$0.371495\pi$$
$$882$$ 0 0
$$883$$ −16.6125 −0.559055 −0.279528 0.960138i $$-0.590178\pi$$
−0.279528 + 0.960138i $$0.590178\pi$$
$$884$$ 0 0
$$885$$ 4.27212 0.143606
$$886$$ 10.0000 0.335957
$$887$$ 28.5730 0.959388 0.479694 0.877436i $$-0.340748\pi$$
0.479694 + 0.877436i $$0.340748\pi$$
$$888$$ −28.8323 −0.967548
$$889$$ 0 0
$$890$$ −32.6125 −1.09317
$$891$$ −28.9792 −0.970838
$$892$$ −5.65685 −0.189405
$$893$$ −8.00000 −0.267710
$$894$$ −6.49359 −0.217178
$$895$$ −1.09600 −0.0366352
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 4.40834 0.147108
$$899$$ −12.4392 −0.414870
$$900$$ 2.20417 0.0734723
$$901$$ −36.3364 −1.21054
$$902$$ −56.5391 −1.88255
$$903$$ 0 0
$$904$$ 49.7750 1.65549
$$905$$ 44.3875 1.47549
$$906$$ 24.8784 0.826528
$$907$$ 49.3875 1.63988 0.819942 0.572446i $$-0.194005\pi$$
0.819942 + 0.572446i $$0.194005\pi$$
$$908$$ 20.9245 0.694403
$$909$$ −2.97280 −0.0986014
$$910$$ 0 0
$$911$$ −43.1833 −1.43073 −0.715364 0.698752i $$-0.753739\pi$$
−0.715364 + 0.698752i $$0.753739\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 57.3758 1.89886
$$914$$ −14.5917 −0.482649
$$915$$ 5.91612 0.195581
$$916$$ 12.7279 0.420542
$$917$$ 0 0
$$918$$ −31.1833 −1.02920
$$919$$ 10.0000 0.329870 0.164935 0.986304i $$-0.447259\pi$$
0.164935 + 0.986304i $$0.447259\pi$$
$$920$$ −14.4603 −0.476744
$$921$$ 48.3667 1.59374
$$922$$ −30.9683 −1.01989
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −14.9792 −0.492512
$$926$$ 11.3875 0.374216
$$927$$ −8.19654 −0.269210
$$928$$ −43.9792 −1.44369
$$929$$ 28.1399 0.923240 0.461620 0.887078i $$-0.347268\pi$$
0.461620 + 0.887078i $$0.347268\pi$$
$$930$$ −5.36812 −0.176027
$$931$$ 0 0
$$932$$ 21.1833 0.693883
$$933$$ 16.7750 0.549188
$$934$$ 4.82012 0.157719
$$935$$ −85.7541 −2.80446
$$936$$ 0 0
$$937$$ −6.63796 −0.216853 −0.108426 0.994104i $$-0.534581\pi$$
−0.108426 + 0.994104i $$0.534581\pi$$
$$938$$ 0 0
$$939$$ 2.40834 0.0785931
$$940$$ 7.59166 0.247613
$$941$$ 23.4642 0.764910 0.382455 0.923974i $$-0.375079\pi$$
0.382455 + 0.923974i $$0.375079\pi$$
$$942$$ −9.79583 −0.319165
$$943$$ −17.5186 −0.570483
$$944$$ 1.12548 0.0366311
$$945$$ 0 0
$$946$$ −10.4083 −0.338404
$$947$$ 28.2042 0.916512 0.458256 0.888820i $$-0.348474\pi$$
0.458256 + 0.888820i $$0.348474\pi$$
$$948$$ −16.6818 −0.541800
$$949$$ 0 0
$$950$$ 6.23433 0.202268
$$951$$ 17.8073 0.577441
$$952$$ 0 0
$$953$$ −4.81667 −0.156027 −0.0780137 0.996952i $$-0.524858\pi$$
−0.0780137 + 0.996952i $$0.524858\pi$$
$$954$$ −6.59166 −0.213413
$$955$$ −69.2375 −2.24047
$$956$$ 19.7958 0.640243
$$957$$ 72.0954 2.33051
$$958$$ 7.35981 0.237785
$$959$$ 0 0
$$960$$ −26.5708 −0.857570
$$961$$ −29.0000 −0.935484
$$962$$ 0 0
$$963$$ 6.00000 0.193347
$$964$$ 4.38701 0.141296
$$965$$ 9.14817 0.294490
$$966$$ 0 0
$$967$$ 11.3875 0.366197 0.183099 0.983095i $$-0.441387\pi$$
0.183099 + 0.983095i $$0.441387\pi$$
$$968$$ −67.7750 −2.17837
$$969$$ 22.0499 0.708346
$$970$$ 11.3875 0.365630
$$971$$ 22.6274 0.726148 0.363074 0.931760i $$-0.381727\pi$$
0.363074 + 0.931760i $$0.381727\pi$$
$$972$$ 9.89949 0.317526
$$973$$ 0 0
$$974$$ 19.5917 0.627757
$$975$$ 0 0
$$976$$ 1.55858 0.0498890
$$977$$ 0.387495 0.0123970 0.00619852 0.999981i $$-0.498027\pi$$
0.00619852 + 0.999981i $$0.498027\pi$$
$$978$$ 18.9328 0.605403
$$979$$ −70.4219 −2.25069
$$980$$ 0 0
$$981$$ 17.5917 0.561659
$$982$$ −9.59166 −0.306082
$$983$$ −41.5602 −1.32556 −0.662782 0.748812i $$-0.730624\pi$$
−0.662782 + 0.748812i $$0.730624\pi$$
$$984$$ −41.3875 −1.31939
$$985$$ −21.4725 −0.684170
$$986$$ 48.4869 1.54414
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.22501 −0.102549
$$990$$ −15.5563 −0.494413
$$991$$ 18.2042 0.578274 0.289137 0.957288i $$-0.406632\pi$$
0.289137 + 0.957288i $$0.406632\pi$$
$$992$$ 7.07107 0.224507
$$993$$ −0.866213 −0.0274885
$$994$$ 0 0
$$995$$ 59.1833 1.87624
$$996$$ 14.0000 0.443607
$$997$$ 5.51249 0.174582 0.0872911 0.996183i $$-0.472179\pi$$
0.0872911 + 0.996183i $$0.472179\pi$$
$$998$$ 16.2042 0.512934
$$999$$ −38.4430 −1.21628
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 8281.2.a.bv.1.2 4
7.6 odd 2 inner 8281.2.a.bv.1.3 4
13.3 even 3 637.2.f.g.295.4 yes 8
13.9 even 3 637.2.f.g.393.4 yes 8
13.12 even 2 8281.2.a.bn.1.1 4
91.3 odd 6 637.2.g.h.373.4 8
91.9 even 3 637.2.g.h.263.1 8
91.16 even 3 637.2.h.k.165.3 8
91.48 odd 6 637.2.f.g.393.1 yes 8
91.55 odd 6 637.2.f.g.295.1 8
91.61 odd 6 637.2.g.h.263.4 8
91.68 odd 6 637.2.h.k.165.2 8
91.74 even 3 637.2.h.k.471.3 8
91.81 even 3 637.2.g.h.373.1 8
91.87 odd 6 637.2.h.k.471.2 8
91.90 odd 2 8281.2.a.bn.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.f.g.295.1 8 91.55 odd 6
637.2.f.g.295.4 yes 8 13.3 even 3
637.2.f.g.393.1 yes 8 91.48 odd 6
637.2.f.g.393.4 yes 8 13.9 even 3
637.2.g.h.263.1 8 91.9 even 3
637.2.g.h.263.4 8 91.61 odd 6
637.2.g.h.373.1 8 91.81 even 3
637.2.g.h.373.4 8 91.3 odd 6
637.2.h.k.165.2 8 91.68 odd 6
637.2.h.k.165.3 8 91.16 even 3
637.2.h.k.471.2 8 91.87 odd 6
637.2.h.k.471.3 8 91.74 even 3
8281.2.a.bn.1.1 4 13.12 even 2
8281.2.a.bn.1.4 4 91.90 odd 2
8281.2.a.bv.1.2 4 1.1 even 1 trivial
8281.2.a.bv.1.3 4 7.6 odd 2 inner