Properties

 Label 1805.2.a.k.1.1 Level $1805$ Weight $2$ Character 1805.1 Self dual yes Analytic conductor $14.413$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$14.4129975648$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.17557$$ of defining polynomial Character $$\chi$$ $$=$$ 1805.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-3.07768 q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} +6.47214 q^{9} +O(q^{10})$$ $$q-3.07768 q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} +6.47214 q^{9} -5.85410 q^{11} +6.15537 q^{12} +3.07768 q^{13} +3.07768 q^{15} +4.00000 q^{16} -2.85410 q^{17} +2.00000 q^{20} +1.90211 q^{21} +5.47214 q^{23} +1.00000 q^{25} -10.6861 q^{27} +1.23607 q^{28} +3.80423 q^{29} +1.62460 q^{31} +18.0171 q^{33} +0.618034 q^{35} -12.9443 q^{36} +8.33499 q^{37} -9.47214 q^{39} +11.5842 q^{41} -7.38197 q^{43} +11.7082 q^{44} -6.47214 q^{45} +4.70820 q^{47} -12.3107 q^{48} -6.61803 q^{49} +8.78402 q^{51} -6.15537 q^{52} +5.15131 q^{53} +5.85410 q^{55} -4.25325 q^{59} -6.15537 q^{60} -7.23607 q^{61} -4.00000 q^{63} -8.00000 q^{64} -3.07768 q^{65} -7.33094 q^{67} +5.70820 q^{68} -16.8415 q^{69} -0.171513 q^{71} -11.0000 q^{73} -3.07768 q^{75} +3.61803 q^{77} -13.5923 q^{79} -4.00000 q^{80} +13.4721 q^{81} +10.7082 q^{83} -3.80423 q^{84} +2.85410 q^{85} -11.7082 q^{87} -8.22899 q^{89} -1.90211 q^{91} -10.9443 q^{92} -5.00000 q^{93} +8.33499 q^{97} -37.8885 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10})$$ 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9 $$4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100})$$ 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 + 2 * q^17 + 8 * q^20 + 4 * q^23 + 4 * q^25 - 4 * q^28 - 2 * q^35 - 16 * q^36 - 20 * q^39 - 34 * q^43 + 20 * q^44 - 8 * q^45 - 8 * q^47 - 22 * q^49 + 10 * q^55 - 20 * q^61 - 16 * q^63 - 32 * q^64 - 4 * q^68 - 44 * q^73 + 10 * q^77 - 16 * q^80 + 36 * q^81 + 16 * q^83 - 2 * q^85 - 20 * q^87 - 8 * q^92 - 20 * q^93 - 80 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −3.07768 −1.77690 −0.888451 0.458972i $$-0.848218\pi$$
−0.888451 + 0.458972i $$0.848218\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.618034 −0.233595 −0.116797 0.993156i $$-0.537263\pi$$
−0.116797 + 0.993156i $$0.537263\pi$$
$$8$$ 0 0
$$9$$ 6.47214 2.15738
$$10$$ 0 0
$$11$$ −5.85410 −1.76508 −0.882539 0.470239i $$-0.844168\pi$$
−0.882539 + 0.470239i $$0.844168\pi$$
$$12$$ 6.15537 1.77690
$$13$$ 3.07768 0.853596 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$14$$ 0 0
$$15$$ 3.07768 0.794654
$$16$$ 4.00000 1.00000
$$17$$ −2.85410 −0.692221 −0.346111 0.938194i $$-0.612498\pi$$
−0.346111 + 0.938194i $$0.612498\pi$$
$$18$$ 0 0
$$19$$ 0 0
$$20$$ 2.00000 0.447214
$$21$$ 1.90211 0.415075
$$22$$ 0 0
$$23$$ 5.47214 1.14102 0.570510 0.821291i $$-0.306746\pi$$
0.570510 + 0.821291i $$0.306746\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −10.6861 −2.05655
$$28$$ 1.23607 0.233595
$$29$$ 3.80423 0.706427 0.353214 0.935543i $$-0.385089\pi$$
0.353214 + 0.935543i $$0.385089\pi$$
$$30$$ 0 0
$$31$$ 1.62460 0.291787 0.145893 0.989300i $$-0.453394\pi$$
0.145893 + 0.989300i $$0.453394\pi$$
$$32$$ 0 0
$$33$$ 18.0171 3.13637
$$34$$ 0 0
$$35$$ 0.618034 0.104467
$$36$$ −12.9443 −2.15738
$$37$$ 8.33499 1.37026 0.685132 0.728419i $$-0.259744\pi$$
0.685132 + 0.728419i $$0.259744\pi$$
$$38$$ 0 0
$$39$$ −9.47214 −1.51676
$$40$$ 0 0
$$41$$ 11.5842 1.80915 0.904573 0.426318i $$-0.140190\pi$$
0.904573 + 0.426318i $$0.140190\pi$$
$$42$$ 0 0
$$43$$ −7.38197 −1.12574 −0.562870 0.826546i $$-0.690303\pi$$
−0.562870 + 0.826546i $$0.690303\pi$$
$$44$$ 11.7082 1.76508
$$45$$ −6.47214 −0.964809
$$46$$ 0 0
$$47$$ 4.70820 0.686762 0.343381 0.939196i $$-0.388428\pi$$
0.343381 + 0.939196i $$0.388428\pi$$
$$48$$ −12.3107 −1.77690
$$49$$ −6.61803 −0.945433
$$50$$ 0 0
$$51$$ 8.78402 1.23001
$$52$$ −6.15537 −0.853596
$$53$$ 5.15131 0.707587 0.353793 0.935324i $$-0.384892\pi$$
0.353793 + 0.935324i $$0.384892\pi$$
$$54$$ 0 0
$$55$$ 5.85410 0.789367
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.25325 −0.553727 −0.276863 0.960909i $$-0.589295\pi$$
−0.276863 + 0.960909i $$0.589295\pi$$
$$60$$ −6.15537 −0.794654
$$61$$ −7.23607 −0.926484 −0.463242 0.886232i $$-0.653314\pi$$
−0.463242 + 0.886232i $$0.653314\pi$$
$$62$$ 0 0
$$63$$ −4.00000 −0.503953
$$64$$ −8.00000 −1.00000
$$65$$ −3.07768 −0.381740
$$66$$ 0 0
$$67$$ −7.33094 −0.895617 −0.447808 0.894130i $$-0.647795\pi$$
−0.447808 + 0.894130i $$0.647795\pi$$
$$68$$ 5.70820 0.692221
$$69$$ −16.8415 −2.02748
$$70$$ 0 0
$$71$$ −0.171513 −0.0203549 −0.0101774 0.999948i $$-0.503240\pi$$
−0.0101774 + 0.999948i $$0.503240\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 0 0
$$75$$ −3.07768 −0.355380
$$76$$ 0 0
$$77$$ 3.61803 0.412313
$$78$$ 0 0
$$79$$ −13.5923 −1.52925 −0.764627 0.644473i $$-0.777077\pi$$
−0.764627 + 0.644473i $$0.777077\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 13.4721 1.49690
$$82$$ 0 0
$$83$$ 10.7082 1.17538 0.587689 0.809087i $$-0.300038\pi$$
0.587689 + 0.809087i $$0.300038\pi$$
$$84$$ −3.80423 −0.415075
$$85$$ 2.85410 0.309571
$$86$$ 0 0
$$87$$ −11.7082 −1.25525
$$88$$ 0 0
$$89$$ −8.22899 −0.872272 −0.436136 0.899881i $$-0.643653\pi$$
−0.436136 + 0.899881i $$0.643653\pi$$
$$90$$ 0 0
$$91$$ −1.90211 −0.199396
$$92$$ −10.9443 −1.14102
$$93$$ −5.00000 −0.518476
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.33499 0.846290 0.423145 0.906062i $$-0.360926\pi$$
0.423145 + 0.906062i $$0.360926\pi$$
$$98$$ 0 0
$$99$$ −37.8885 −3.80794
$$100$$ −2.00000 −0.200000
$$101$$ −1.90983 −0.190035 −0.0950176 0.995476i $$-0.530291\pi$$
−0.0950176 + 0.995476i $$0.530291\pi$$
$$102$$ 0 0
$$103$$ 9.06154 0.892860 0.446430 0.894819i $$-0.352695\pi$$
0.446430 + 0.894819i $$0.352695\pi$$
$$104$$ 0 0
$$105$$ −1.90211 −0.185627
$$106$$ 0 0
$$107$$ −2.45714 −0.237541 −0.118770 0.992922i $$-0.537895\pi$$
−0.118770 + 0.992922i $$0.537895\pi$$
$$108$$ 21.3723 2.05655
$$109$$ −16.2865 −1.55996 −0.779981 0.625804i $$-0.784771\pi$$
−0.779981 + 0.625804i $$0.784771\pi$$
$$110$$ 0 0
$$111$$ −25.6525 −2.43483
$$112$$ −2.47214 −0.233595
$$113$$ 2.80017 0.263418 0.131709 0.991288i $$-0.457954\pi$$
0.131709 + 0.991288i $$0.457954\pi$$
$$114$$ 0 0
$$115$$ −5.47214 −0.510279
$$116$$ −7.60845 −0.706427
$$117$$ 19.9192 1.84153
$$118$$ 0 0
$$119$$ 1.76393 0.161699
$$120$$ 0 0
$$121$$ 23.2705 2.11550
$$122$$ 0 0
$$123$$ −35.6525 −3.21468
$$124$$ −3.24920 −0.291787
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 0.620541 0.0550641 0.0275321 0.999621i $$-0.491235\pi$$
0.0275321 + 0.999621i $$0.491235\pi$$
$$128$$ 0 0
$$129$$ 22.7194 2.00033
$$130$$ 0 0
$$131$$ 5.23607 0.457477 0.228739 0.973488i $$-0.426540\pi$$
0.228739 + 0.973488i $$0.426540\pi$$
$$132$$ −36.0341 −3.13637
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 10.6861 0.919716
$$136$$ 0 0
$$137$$ 14.5623 1.24414 0.622071 0.782961i $$-0.286292\pi$$
0.622071 + 0.782961i $$0.286292\pi$$
$$138$$ 0 0
$$139$$ −0.381966 −0.0323979 −0.0161990 0.999869i $$-0.505157\pi$$
−0.0161990 + 0.999869i $$0.505157\pi$$
$$140$$ −1.23607 −0.104467
$$141$$ −14.4904 −1.22031
$$142$$ 0 0
$$143$$ −18.0171 −1.50666
$$144$$ 25.8885 2.15738
$$145$$ −3.80423 −0.315924
$$146$$ 0 0
$$147$$ 20.3682 1.67994
$$148$$ −16.6700 −1.37026
$$149$$ −3.76393 −0.308353 −0.154177 0.988043i $$-0.549272\pi$$
−0.154177 + 0.988043i $$0.549272\pi$$
$$150$$ 0 0
$$151$$ 3.35520 0.273042 0.136521 0.990637i $$-0.456408\pi$$
0.136521 + 0.990637i $$0.456408\pi$$
$$152$$ 0 0
$$153$$ −18.4721 −1.49338
$$154$$ 0 0
$$155$$ −1.62460 −0.130491
$$156$$ 18.9443 1.51676
$$157$$ 15.0344 1.19988 0.599940 0.800045i $$-0.295191\pi$$
0.599940 + 0.800045i $$0.295191\pi$$
$$158$$ 0 0
$$159$$ −15.8541 −1.25731
$$160$$ 0 0
$$161$$ −3.38197 −0.266536
$$162$$ 0 0
$$163$$ −12.0902 −0.946975 −0.473488 0.880800i $$-0.657005\pi$$
−0.473488 + 0.880800i $$0.657005\pi$$
$$164$$ −23.1684 −1.80915
$$165$$ −18.0171 −1.40263
$$166$$ 0 0
$$167$$ 2.45714 0.190139 0.0950697 0.995471i $$-0.469693\pi$$
0.0950697 + 0.995471i $$0.469693\pi$$
$$168$$ 0 0
$$169$$ −3.52786 −0.271374
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 14.7639 1.12574
$$173$$ 6.43288 0.489083 0.244541 0.969639i $$-0.421363\pi$$
0.244541 + 0.969639i $$0.421363\pi$$
$$174$$ 0 0
$$175$$ −0.618034 −0.0467190
$$176$$ −23.4164 −1.76508
$$177$$ 13.0902 0.983917
$$178$$ 0 0
$$179$$ −5.42882 −0.405769 −0.202885 0.979203i $$-0.565032\pi$$
−0.202885 + 0.979203i $$0.565032\pi$$
$$180$$ 12.9443 0.964809
$$181$$ −17.7396 −1.31857 −0.659286 0.751893i $$-0.729141\pi$$
−0.659286 + 0.751893i $$0.729141\pi$$
$$182$$ 0 0
$$183$$ 22.2703 1.64627
$$184$$ 0 0
$$185$$ −8.33499 −0.612801
$$186$$ 0 0
$$187$$ 16.7082 1.22182
$$188$$ −9.41641 −0.686762
$$189$$ 6.60440 0.480399
$$190$$ 0 0
$$191$$ 6.09017 0.440669 0.220335 0.975424i $$-0.429285\pi$$
0.220335 + 0.975424i $$0.429285\pi$$
$$192$$ 24.6215 1.77690
$$193$$ 23.4459 1.68767 0.843836 0.536601i $$-0.180292\pi$$
0.843836 + 0.536601i $$0.180292\pi$$
$$194$$ 0 0
$$195$$ 9.47214 0.678314
$$196$$ 13.2361 0.945433
$$197$$ −8.65248 −0.616463 −0.308232 0.951311i $$-0.599737\pi$$
−0.308232 + 0.951311i $$0.599737\pi$$
$$198$$ 0 0
$$199$$ −7.56231 −0.536078 −0.268039 0.963408i $$-0.586376\pi$$
−0.268039 + 0.963408i $$0.586376\pi$$
$$200$$ 0 0
$$201$$ 22.5623 1.59142
$$202$$ 0 0
$$203$$ −2.35114 −0.165018
$$204$$ −17.5680 −1.23001
$$205$$ −11.5842 −0.809075
$$206$$ 0 0
$$207$$ 35.4164 2.46161
$$208$$ 12.3107 0.853596
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.78402 −0.604717 −0.302359 0.953194i $$-0.597774\pi$$
−0.302359 + 0.953194i $$0.597774\pi$$
$$212$$ −10.3026 −0.707587
$$213$$ 0.527864 0.0361686
$$214$$ 0 0
$$215$$ 7.38197 0.503446
$$216$$ 0 0
$$217$$ −1.00406 −0.0681598
$$218$$ 0 0
$$219$$ 33.8545 2.28768
$$220$$ −11.7082 −0.789367
$$221$$ −8.78402 −0.590877
$$222$$ 0 0
$$223$$ −12.5882 −0.842971 −0.421486 0.906835i $$-0.638491\pi$$
−0.421486 + 0.906835i $$0.638491\pi$$
$$224$$ 0 0
$$225$$ 6.47214 0.431476
$$226$$ 0 0
$$227$$ 9.61657 0.638274 0.319137 0.947709i $$-0.396607\pi$$
0.319137 + 0.947709i $$0.396607\pi$$
$$228$$ 0 0
$$229$$ 1.85410 0.122523 0.0612613 0.998122i $$-0.480488\pi$$
0.0612613 + 0.998122i $$0.480488\pi$$
$$230$$ 0 0
$$231$$ −11.1352 −0.732640
$$232$$ 0 0
$$233$$ −4.94427 −0.323910 −0.161955 0.986798i $$-0.551780\pi$$
−0.161955 + 0.986798i $$0.551780\pi$$
$$234$$ 0 0
$$235$$ −4.70820 −0.307129
$$236$$ 8.50651 0.553727
$$237$$ 41.8328 2.71733
$$238$$ 0 0
$$239$$ −13.9443 −0.901980 −0.450990 0.892529i $$-0.648929\pi$$
−0.450990 + 0.892529i $$0.648929\pi$$
$$240$$ 12.3107 0.794654
$$241$$ −3.18368 −0.205079 −0.102540 0.994729i $$-0.532697\pi$$
−0.102540 + 0.994729i $$0.532697\pi$$
$$242$$ 0 0
$$243$$ −9.40456 −0.603303
$$244$$ 14.4721 0.926484
$$245$$ 6.61803 0.422811
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −32.9565 −2.08853
$$250$$ 0 0
$$251$$ 12.8541 0.811344 0.405672 0.914019i $$-0.367038\pi$$
0.405672 + 0.914019i $$0.367038\pi$$
$$252$$ 8.00000 0.503953
$$253$$ −32.0344 −2.01399
$$254$$ 0 0
$$255$$ −8.78402 −0.550077
$$256$$ 16.0000 1.00000
$$257$$ −5.36331 −0.334554 −0.167277 0.985910i $$-0.553497\pi$$
−0.167277 + 0.985910i $$0.553497\pi$$
$$258$$ 0 0
$$259$$ −5.15131 −0.320087
$$260$$ 6.15537 0.381740
$$261$$ 24.6215 1.52403
$$262$$ 0 0
$$263$$ 22.0902 1.36214 0.681069 0.732219i $$-0.261515\pi$$
0.681069 + 0.732219i $$0.261515\pi$$
$$264$$ 0 0
$$265$$ −5.15131 −0.316442
$$266$$ 0 0
$$267$$ 25.3262 1.54994
$$268$$ 14.6619 0.895617
$$269$$ −22.3763 −1.36431 −0.682154 0.731208i $$-0.738957\pi$$
−0.682154 + 0.731208i $$0.738957\pi$$
$$270$$ 0 0
$$271$$ 8.41641 0.511260 0.255630 0.966775i $$-0.417717\pi$$
0.255630 + 0.966775i $$0.417717\pi$$
$$272$$ −11.4164 −0.692221
$$273$$ 5.85410 0.354306
$$274$$ 0 0
$$275$$ −5.85410 −0.353016
$$276$$ 33.6830 2.02748
$$277$$ −19.3607 −1.16327 −0.581635 0.813450i $$-0.697587\pi$$
−0.581635 + 0.813450i $$0.697587\pi$$
$$278$$ 0 0
$$279$$ 10.5146 0.629494
$$280$$ 0 0
$$281$$ 15.6659 0.934551 0.467276 0.884112i $$-0.345236\pi$$
0.467276 + 0.884112i $$0.345236\pi$$
$$282$$ 0 0
$$283$$ −24.0000 −1.42665 −0.713326 0.700832i $$-0.752812\pi$$
−0.713326 + 0.700832i $$0.752812\pi$$
$$284$$ 0.343027 0.0203549
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.15942 −0.422607
$$288$$ 0 0
$$289$$ −8.85410 −0.520830
$$290$$ 0 0
$$291$$ −25.6525 −1.50377
$$292$$ 22.0000 1.28745
$$293$$ 22.5478 1.31726 0.658629 0.752467i $$-0.271136\pi$$
0.658629 + 0.752467i $$0.271136\pi$$
$$294$$ 0 0
$$295$$ 4.25325 0.247634
$$296$$ 0 0
$$297$$ 62.5577 3.62997
$$298$$ 0 0
$$299$$ 16.8415 0.973969
$$300$$ 6.15537 0.355380
$$301$$ 4.56231 0.262967
$$302$$ 0 0
$$303$$ 5.87785 0.337674
$$304$$ 0 0
$$305$$ 7.23607 0.414336
$$306$$ 0 0
$$307$$ −12.2047 −0.696561 −0.348280 0.937390i $$-0.613234\pi$$
−0.348280 + 0.937390i $$0.613234\pi$$
$$308$$ −7.23607 −0.412313
$$309$$ −27.8885 −1.58652
$$310$$ 0 0
$$311$$ −27.0344 −1.53298 −0.766491 0.642255i $$-0.777999\pi$$
−0.766491 + 0.642255i $$0.777999\pi$$
$$312$$ 0 0
$$313$$ 13.6738 0.772887 0.386443 0.922313i $$-0.373703\pi$$
0.386443 + 0.922313i $$0.373703\pi$$
$$314$$ 0 0
$$315$$ 4.00000 0.225374
$$316$$ 27.1846 1.52925
$$317$$ −34.4095 −1.93263 −0.966316 0.257357i $$-0.917148\pi$$
−0.966316 + 0.257357i $$0.917148\pi$$
$$318$$ 0 0
$$319$$ −22.2703 −1.24690
$$320$$ 8.00000 0.447214
$$321$$ 7.56231 0.422087
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −26.9443 −1.49690
$$325$$ 3.07768 0.170719
$$326$$ 0 0
$$327$$ 50.1246 2.77190
$$328$$ 0 0
$$329$$ −2.90983 −0.160424
$$330$$ 0 0
$$331$$ 23.5519 1.29453 0.647265 0.762265i $$-0.275913\pi$$
0.647265 + 0.762265i $$0.275913\pi$$
$$332$$ −21.4164 −1.17538
$$333$$ 53.9452 2.95618
$$334$$ 0 0
$$335$$ 7.33094 0.400532
$$336$$ 7.60845 0.415075
$$337$$ 0.171513 0.00934293 0.00467147 0.999989i $$-0.498513\pi$$
0.00467147 + 0.999989i $$0.498513\pi$$
$$338$$ 0 0
$$339$$ −8.61803 −0.468067
$$340$$ −5.70820 −0.309571
$$341$$ −9.51057 −0.515026
$$342$$ 0 0
$$343$$ 8.41641 0.454443
$$344$$ 0 0
$$345$$ 16.8415 0.906716
$$346$$ 0 0
$$347$$ −24.7082 −1.32641 −0.663203 0.748440i $$-0.730803\pi$$
−0.663203 + 0.748440i $$0.730803\pi$$
$$348$$ 23.4164 1.25525
$$349$$ 10.1246 0.541958 0.270979 0.962585i $$-0.412653\pi$$
0.270979 + 0.962585i $$0.412653\pi$$
$$350$$ 0 0
$$351$$ −32.8885 −1.75546
$$352$$ 0 0
$$353$$ 15.3820 0.818699 0.409350 0.912378i $$-0.365756\pi$$
0.409350 + 0.912378i $$0.365756\pi$$
$$354$$ 0 0
$$355$$ 0.171513 0.00910299
$$356$$ 16.4580 0.872272
$$357$$ −5.42882 −0.287324
$$358$$ 0 0
$$359$$ −6.43769 −0.339768 −0.169884 0.985464i $$-0.554339\pi$$
−0.169884 + 0.985464i $$0.554339\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 0 0
$$363$$ −71.6193 −3.75904
$$364$$ 3.80423 0.199396
$$365$$ 11.0000 0.575766
$$366$$ 0 0
$$367$$ −13.6525 −0.712653 −0.356327 0.934361i $$-0.615971\pi$$
−0.356327 + 0.934361i $$0.615971\pi$$
$$368$$ 21.8885 1.14102
$$369$$ 74.9745 3.90301
$$370$$ 0 0
$$371$$ −3.18368 −0.165289
$$372$$ 10.0000 0.518476
$$373$$ −34.9241 −1.80830 −0.904150 0.427214i $$-0.859495\pi$$
−0.904150 + 0.427214i $$0.859495\pi$$
$$374$$ 0 0
$$375$$ 3.07768 0.158931
$$376$$ 0 0
$$377$$ 11.7082 0.603003
$$378$$ 0 0
$$379$$ 30.7768 1.58090 0.790450 0.612527i $$-0.209847\pi$$
0.790450 + 0.612527i $$0.209847\pi$$
$$380$$ 0 0
$$381$$ −1.90983 −0.0978436
$$382$$ 0 0
$$383$$ −21.2008 −1.08331 −0.541654 0.840601i $$-0.682202\pi$$
−0.541654 + 0.840601i $$0.682202\pi$$
$$384$$ 0 0
$$385$$ −3.61803 −0.184392
$$386$$ 0 0
$$387$$ −47.7771 −2.42865
$$388$$ −16.6700 −0.846290
$$389$$ 7.43769 0.377106 0.188553 0.982063i $$-0.439620\pi$$
0.188553 + 0.982063i $$0.439620\pi$$
$$390$$ 0 0
$$391$$ −15.6180 −0.789838
$$392$$ 0 0
$$393$$ −16.1150 −0.812892
$$394$$ 0 0
$$395$$ 13.5923 0.683903
$$396$$ 75.7771 3.80794
$$397$$ −16.4721 −0.826713 −0.413356 0.910569i $$-0.635644\pi$$
−0.413356 + 0.910569i $$0.635644\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 8.16348 0.407665 0.203832 0.979006i $$-0.434660\pi$$
0.203832 + 0.979006i $$0.434660\pi$$
$$402$$ 0 0
$$403$$ 5.00000 0.249068
$$404$$ 3.81966 0.190035
$$405$$ −13.4721 −0.669436
$$406$$ 0 0
$$407$$ −48.7939 −2.41862
$$408$$ 0 0
$$409$$ −4.25325 −0.210310 −0.105155 0.994456i $$-0.533534\pi$$
−0.105155 + 0.994456i $$0.533534\pi$$
$$410$$ 0 0
$$411$$ −44.8182 −2.21072
$$412$$ −18.1231 −0.892860
$$413$$ 2.62866 0.129348
$$414$$ 0 0
$$415$$ −10.7082 −0.525645
$$416$$ 0 0
$$417$$ 1.17557 0.0575679
$$418$$ 0 0
$$419$$ −23.2918 −1.13788 −0.568939 0.822379i $$-0.692646\pi$$
−0.568939 + 0.822379i $$0.692646\pi$$
$$420$$ 3.80423 0.185627
$$421$$ 16.8415 0.820805 0.410402 0.911905i $$-0.365388\pi$$
0.410402 + 0.911905i $$0.365388\pi$$
$$422$$ 0 0
$$423$$ 30.4721 1.48161
$$424$$ 0 0
$$425$$ −2.85410 −0.138444
$$426$$ 0 0
$$427$$ 4.47214 0.216422
$$428$$ 4.91428 0.237541
$$429$$ 55.4508 2.67719
$$430$$ 0 0
$$431$$ −5.77185 −0.278020 −0.139010 0.990291i $$-0.544392\pi$$
−0.139010 + 0.990291i $$0.544392\pi$$
$$432$$ −42.7445 −2.05655
$$433$$ −6.04937 −0.290714 −0.145357 0.989379i $$-0.546433\pi$$
−0.145357 + 0.989379i $$0.546433\pi$$
$$434$$ 0 0
$$435$$ 11.7082 0.561365
$$436$$ 32.5729 1.55996
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −16.4985 −0.787429 −0.393715 0.919233i $$-0.628810\pi$$
−0.393715 + 0.919233i $$0.628810\pi$$
$$440$$ 0 0
$$441$$ −42.8328 −2.03966
$$442$$ 0 0
$$443$$ 17.9443 0.852558 0.426279 0.904592i $$-0.359824\pi$$
0.426279 + 0.904592i $$0.359824\pi$$
$$444$$ 51.3050 2.43483
$$445$$ 8.22899 0.390092
$$446$$ 0 0
$$447$$ 11.5842 0.547913
$$448$$ 4.94427 0.233595
$$449$$ −0.514540 −0.0242827 −0.0121413 0.999926i $$-0.503865\pi$$
−0.0121413 + 0.999926i $$0.503865\pi$$
$$450$$ 0 0
$$451$$ −67.8150 −3.19329
$$452$$ −5.60034 −0.263418
$$453$$ −10.3262 −0.485169
$$454$$ 0 0
$$455$$ 1.90211 0.0891724
$$456$$ 0 0
$$457$$ −12.6525 −0.591858 −0.295929 0.955210i $$-0.595629\pi$$
−0.295929 + 0.955210i $$0.595629\pi$$
$$458$$ 0 0
$$459$$ 30.4993 1.42359
$$460$$ 10.9443 0.510279
$$461$$ −6.41641 −0.298842 −0.149421 0.988774i $$-0.547741\pi$$
−0.149421 + 0.988774i $$0.547741\pi$$
$$462$$ 0 0
$$463$$ −17.2918 −0.803618 −0.401809 0.915724i $$-0.631618\pi$$
−0.401809 + 0.915724i $$0.631618\pi$$
$$464$$ 15.2169 0.706427
$$465$$ 5.00000 0.231869
$$466$$ 0 0
$$467$$ 10.9443 0.506441 0.253220 0.967409i $$-0.418510\pi$$
0.253220 + 0.967409i $$0.418510\pi$$
$$468$$ −39.8384 −1.84153
$$469$$ 4.53077 0.209211
$$470$$ 0 0
$$471$$ −46.2713 −2.13207
$$472$$ 0 0
$$473$$ 43.2148 1.98702
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.52786 −0.161699
$$477$$ 33.3400 1.52653
$$478$$ 0 0
$$479$$ 25.1459 1.14895 0.574473 0.818524i $$-0.305207\pi$$
0.574473 + 0.818524i $$0.305207\pi$$
$$480$$ 0 0
$$481$$ 25.6525 1.16965
$$482$$ 0 0
$$483$$ 10.4086 0.473609
$$484$$ −46.5410 −2.11550
$$485$$ −8.33499 −0.378473
$$486$$ 0 0
$$487$$ 19.9192 0.902624 0.451312 0.892366i $$-0.350956\pi$$
0.451312 + 0.892366i $$0.350956\pi$$
$$488$$ 0 0
$$489$$ 37.2097 1.68268
$$490$$ 0 0
$$491$$ −30.6180 −1.38177 −0.690886 0.722963i $$-0.742779\pi$$
−0.690886 + 0.722963i $$0.742779\pi$$
$$492$$ 71.3050 3.21468
$$493$$ −10.8576 −0.489004
$$494$$ 0 0
$$495$$ 37.8885 1.70296
$$496$$ 6.49839 0.291787
$$497$$ 0.106001 0.00475480
$$498$$ 0 0
$$499$$ 16.1246 0.721837 0.360918 0.932597i $$-0.382463\pi$$
0.360918 + 0.932597i $$0.382463\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ −7.56231 −0.337859
$$502$$ 0 0
$$503$$ 25.1459 1.12120 0.560600 0.828087i $$-0.310571\pi$$
0.560600 + 0.828087i $$0.310571\pi$$
$$504$$ 0 0
$$505$$ 1.90983 0.0849863
$$506$$ 0 0
$$507$$ 10.8576 0.482205
$$508$$ −1.24108 −0.0550641
$$509$$ −22.2048 −0.984211 −0.492106 0.870536i $$-0.663773\pi$$
−0.492106 + 0.870536i $$0.663773\pi$$
$$510$$ 0 0
$$511$$ 6.79837 0.300742
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −9.06154 −0.399299
$$516$$ −45.4387 −2.00033
$$517$$ −27.5623 −1.21219
$$518$$ 0 0
$$519$$ −19.7984 −0.869052
$$520$$ 0 0
$$521$$ 41.7405 1.82868 0.914342 0.404943i $$-0.132709\pi$$
0.914342 + 0.404943i $$0.132709\pi$$
$$522$$ 0 0
$$523$$ 23.1684 1.01308 0.506541 0.862216i $$-0.330924\pi$$
0.506541 + 0.862216i $$0.330924\pi$$
$$524$$ −10.4721 −0.457477
$$525$$ 1.90211 0.0830150
$$526$$ 0 0
$$527$$ −4.63677 −0.201981
$$528$$ 72.0683 3.13637
$$529$$ 6.94427 0.301925
$$530$$ 0 0
$$531$$ −27.5276 −1.19460
$$532$$ 0 0
$$533$$ 35.6525 1.54428
$$534$$ 0 0
$$535$$ 2.45714 0.106232
$$536$$ 0 0
$$537$$ 16.7082 0.721012
$$538$$ 0 0
$$539$$ 38.7426 1.66876
$$540$$ −21.3723 −0.919716
$$541$$ −28.9443 −1.24441 −0.622206 0.782854i $$-0.713763\pi$$
−0.622206 + 0.782854i $$0.713763\pi$$
$$542$$ 0 0
$$543$$ 54.5967 2.34297
$$544$$ 0 0
$$545$$ 16.2865 0.697636
$$546$$ 0 0
$$547$$ −26.9726 −1.15327 −0.576633 0.817003i $$-0.695634\pi$$
−0.576633 + 0.817003i $$0.695634\pi$$
$$548$$ −29.1246 −1.24414
$$549$$ −46.8328 −1.99878
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.40051 0.357226
$$554$$ 0 0
$$555$$ 25.6525 1.08889
$$556$$ 0.763932 0.0323979
$$557$$ −34.3607 −1.45591 −0.727954 0.685626i $$-0.759529\pi$$
−0.727954 + 0.685626i $$0.759529\pi$$
$$558$$ 0 0
$$559$$ −22.7194 −0.960926
$$560$$ 2.47214 0.104467
$$561$$ −51.4226 −2.17106
$$562$$ 0 0
$$563$$ −11.5842 −0.488215 −0.244108 0.969748i $$-0.578495\pi$$
−0.244108 + 0.969748i $$0.578495\pi$$
$$564$$ 28.9807 1.22031
$$565$$ −2.80017 −0.117804
$$566$$ 0 0
$$567$$ −8.32624 −0.349669
$$568$$ 0 0
$$569$$ −36.5892 −1.53390 −0.766949 0.641708i $$-0.778226\pi$$
−0.766949 + 0.641708i $$0.778226\pi$$
$$570$$ 0 0
$$571$$ −38.5410 −1.61289 −0.806446 0.591308i $$-0.798612\pi$$
−0.806446 + 0.591308i $$0.798612\pi$$
$$572$$ 36.0341 1.50666
$$573$$ −18.7436 −0.783026
$$574$$ 0 0
$$575$$ 5.47214 0.228204
$$576$$ −51.7771 −2.15738
$$577$$ −12.1246 −0.504754 −0.252377 0.967629i $$-0.581212\pi$$
−0.252377 + 0.967629i $$0.581212\pi$$
$$578$$ 0 0
$$579$$ −72.1591 −2.99883
$$580$$ 7.60845 0.315924
$$581$$ −6.61803 −0.274562
$$582$$ 0 0
$$583$$ −30.1563 −1.24895
$$584$$ 0 0
$$585$$ −19.9192 −0.823557
$$586$$ 0 0
$$587$$ −19.8885 −0.820888 −0.410444 0.911886i $$-0.634626\pi$$
−0.410444 + 0.911886i $$0.634626\pi$$
$$588$$ −40.7364 −1.67994
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 26.6296 1.09539
$$592$$ 33.3400 1.37026
$$593$$ −24.2148 −0.994382 −0.497191 0.867641i $$-0.665635\pi$$
−0.497191 + 0.867641i $$0.665635\pi$$
$$594$$ 0 0
$$595$$ −1.76393 −0.0723142
$$596$$ 7.52786 0.308353
$$597$$ 23.2744 0.952557
$$598$$ 0 0
$$599$$ −27.2501 −1.11341 −0.556705 0.830710i $$-0.687935\pi$$
−0.556705 + 0.830710i $$0.687935\pi$$
$$600$$ 0 0
$$601$$ 36.0341 1.46986 0.734932 0.678141i $$-0.237214\pi$$
0.734932 + 0.678141i $$0.237214\pi$$
$$602$$ 0 0
$$603$$ −47.4468 −1.93218
$$604$$ −6.71040 −0.273042
$$605$$ −23.2705 −0.946081
$$606$$ 0 0
$$607$$ −42.5325 −1.72634 −0.863171 0.504911i $$-0.831525\pi$$
−0.863171 + 0.504911i $$0.831525\pi$$
$$608$$ 0 0
$$609$$ 7.23607 0.293220
$$610$$ 0 0
$$611$$ 14.4904 0.586217
$$612$$ 36.9443 1.49338
$$613$$ −23.2705 −0.939887 −0.469944 0.882696i $$-0.655726\pi$$
−0.469944 + 0.882696i $$0.655726\pi$$
$$614$$ 0 0
$$615$$ 35.6525 1.43765
$$616$$ 0 0
$$617$$ 0.888544 0.0357714 0.0178857 0.999840i $$-0.494306\pi$$
0.0178857 + 0.999840i $$0.494306\pi$$
$$618$$ 0 0
$$619$$ −20.9098 −0.840437 −0.420219 0.907423i $$-0.638047\pi$$
−0.420219 + 0.907423i $$0.638047\pi$$
$$620$$ 3.24920 0.130491
$$621$$ −58.4760 −2.34656
$$622$$ 0 0
$$623$$ 5.08580 0.203758
$$624$$ −37.8885 −1.51676
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −30.0689 −1.19988
$$629$$ −23.7889 −0.948527
$$630$$ 0 0
$$631$$ −24.4721 −0.974220 −0.487110 0.873341i $$-0.661949\pi$$
−0.487110 + 0.873341i $$0.661949\pi$$
$$632$$ 0 0
$$633$$ 27.0344 1.07452
$$634$$ 0 0
$$635$$ −0.620541 −0.0246254
$$636$$ 31.7082 1.25731
$$637$$ −20.3682 −0.807018
$$638$$ 0 0
$$639$$ −1.11006 −0.0439132
$$640$$ 0 0
$$641$$ −19.3642 −0.764838 −0.382419 0.923989i $$-0.624909\pi$$
−0.382419 + 0.923989i $$0.624909\pi$$
$$642$$ 0 0
$$643$$ 14.9443 0.589345 0.294672 0.955598i $$-0.404790\pi$$
0.294672 + 0.955598i $$0.404790\pi$$
$$644$$ 6.76393 0.266536
$$645$$ −22.7194 −0.894574
$$646$$ 0 0
$$647$$ 45.3607 1.78331 0.891656 0.452713i $$-0.149544\pi$$
0.891656 + 0.452713i $$0.149544\pi$$
$$648$$ 0 0
$$649$$ 24.8990 0.977371
$$650$$ 0 0
$$651$$ 3.09017 0.121113
$$652$$ 24.1803 0.946975
$$653$$ −9.32624 −0.364964 −0.182482 0.983209i $$-0.558413\pi$$
−0.182482 + 0.983209i $$0.558413\pi$$
$$654$$ 0 0
$$655$$ −5.23607 −0.204590
$$656$$ 46.3368 1.80915
$$657$$ −71.1935 −2.77752
$$658$$ 0 0
$$659$$ 29.6013 1.15310 0.576551 0.817061i $$-0.304398\pi$$
0.576551 + 0.817061i $$0.304398\pi$$
$$660$$ 36.0341 1.40263
$$661$$ 20.0907 0.781438 0.390719 0.920510i $$-0.372226\pi$$
0.390719 + 0.920510i $$0.372226\pi$$
$$662$$ 0 0
$$663$$ 27.0344 1.04993
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.8172 0.806047
$$668$$ −4.91428 −0.190139
$$669$$ 38.7426 1.49788
$$670$$ 0 0
$$671$$ 42.3607 1.63532
$$672$$ 0 0
$$673$$ −0.661030 −0.0254808 −0.0127404 0.999919i $$-0.504056\pi$$
−0.0127404 + 0.999919i $$0.504056\pi$$
$$674$$ 0 0
$$675$$ −10.6861 −0.411310
$$676$$ 7.05573 0.271374
$$677$$ 4.04125 0.155318 0.0776590 0.996980i $$-0.475255\pi$$
0.0776590 + 0.996980i $$0.475255\pi$$
$$678$$ 0 0
$$679$$ −5.15131 −0.197689
$$680$$ 0 0
$$681$$ −29.5967 −1.13415
$$682$$ 0 0
$$683$$ 26.8011 1.02552 0.512758 0.858533i $$-0.328624\pi$$
0.512758 + 0.858533i $$0.328624\pi$$
$$684$$ 0 0
$$685$$ −14.5623 −0.556397
$$686$$ 0 0
$$687$$ −5.70634 −0.217710
$$688$$ −29.5279 −1.12574
$$689$$ 15.8541 0.603993
$$690$$ 0 0
$$691$$ −28.9443 −1.10109 −0.550546 0.834805i $$-0.685580\pi$$
−0.550546 + 0.834805i $$0.685580\pi$$
$$692$$ −12.8658 −0.489083
$$693$$ 23.4164 0.889516
$$694$$ 0 0
$$695$$ 0.381966 0.0144888
$$696$$ 0 0
$$697$$ −33.0625 −1.25233
$$698$$ 0 0
$$699$$ 15.2169 0.575556
$$700$$ 1.23607 0.0467190
$$701$$ 25.0000 0.944237 0.472118 0.881535i $$-0.343489\pi$$
0.472118 + 0.881535i $$0.343489\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 46.8328 1.76508
$$705$$ 14.4904 0.545739
$$706$$ 0 0
$$707$$ 1.18034 0.0443913
$$708$$ −26.1803 −0.983917
$$709$$ −25.5279 −0.958719 −0.479360 0.877619i $$-0.659131\pi$$
−0.479360 + 0.877619i $$0.659131\pi$$
$$710$$ 0 0
$$711$$ −87.9713 −3.29918
$$712$$ 0 0
$$713$$ 8.89002 0.332934
$$714$$ 0 0
$$715$$ 18.0171 0.673800
$$716$$ 10.8576 0.405769
$$717$$ 42.9161 1.60273
$$718$$ 0 0
$$719$$ 8.23607 0.307154 0.153577 0.988137i $$-0.450921\pi$$
0.153577 + 0.988137i $$0.450921\pi$$
$$720$$ −25.8885 −0.964809
$$721$$ −5.60034 −0.208567
$$722$$ 0 0
$$723$$ 9.79837 0.364405
$$724$$ 35.4791 1.31857
$$725$$ 3.80423 0.141285
$$726$$ 0 0
$$727$$ 21.0689 0.781402 0.390701 0.920518i $$-0.372233\pi$$
0.390701 + 0.920518i $$0.372233\pi$$
$$728$$ 0 0
$$729$$ −11.4721 −0.424894
$$730$$ 0 0
$$731$$ 21.0689 0.779261
$$732$$ −44.5407 −1.64627
$$733$$ −25.7984 −0.952885 −0.476442 0.879206i $$-0.658074\pi$$
−0.476442 + 0.879206i $$0.658074\pi$$
$$734$$ 0 0
$$735$$ −20.3682 −0.751293
$$736$$ 0 0
$$737$$ 42.9161 1.58083
$$738$$ 0 0
$$739$$ −38.4164 −1.41317 −0.706585 0.707628i $$-0.749765\pi$$
−0.706585 + 0.707628i $$0.749765\pi$$
$$740$$ 16.6700 0.612801
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −38.4508 −1.41062 −0.705312 0.708897i $$-0.749193\pi$$
−0.705312 + 0.708897i $$0.749193\pi$$
$$744$$ 0 0
$$745$$ 3.76393 0.137900
$$746$$ 0 0
$$747$$ 69.3050 2.53574
$$748$$ −33.4164 −1.22182
$$749$$ 1.51860 0.0554883
$$750$$ 0 0
$$751$$ 6.53888 0.238607 0.119304 0.992858i $$-0.461934\pi$$
0.119304 + 0.992858i $$0.461934\pi$$
$$752$$ 18.8328 0.686762
$$753$$ −39.5609 −1.44168
$$754$$ 0 0
$$755$$ −3.35520 −0.122108
$$756$$ −13.2088 −0.480399
$$757$$ 52.7771 1.91822 0.959108 0.283041i $$-0.0913431\pi$$
0.959108 + 0.283041i $$0.0913431\pi$$
$$758$$ 0 0
$$759$$ 98.5919 3.57866
$$760$$ 0 0
$$761$$ −39.0689 −1.41625 −0.708123 0.706089i $$-0.750458\pi$$
−0.708123 + 0.706089i $$0.750458\pi$$
$$762$$ 0 0
$$763$$ 10.0656 0.364399
$$764$$ −12.1803 −0.440669
$$765$$ 18.4721 0.667861
$$766$$ 0 0
$$767$$ −13.0902 −0.472659
$$768$$ −49.2429 −1.77690
$$769$$ 18.3607 0.662103 0.331052 0.943613i $$-0.392597\pi$$
0.331052 + 0.943613i $$0.392597\pi$$
$$770$$ 0 0
$$771$$ 16.5066 0.594470
$$772$$ −46.8918 −1.68767
$$773$$ 21.3723 0.768707 0.384354 0.923186i $$-0.374424\pi$$
0.384354 + 0.923186i $$0.374424\pi$$
$$774$$ 0 0
$$775$$ 1.62460 0.0583573
$$776$$ 0 0
$$777$$ 15.8541 0.568763
$$778$$ 0 0
$$779$$ 0 0
$$780$$ −18.9443 −0.678314
$$781$$ 1.00406 0.0359280
$$782$$ 0 0
$$783$$ −40.6525 −1.45280
$$784$$ −26.4721 −0.945433
$$785$$ −15.0344 −0.536602
$$786$$ 0 0
$$787$$ 22.0583 0.786294 0.393147 0.919476i $$-0.371386\pi$$
0.393147 + 0.919476i $$0.371386\pi$$
$$788$$ 17.3050 0.616463
$$789$$ −67.9866 −2.42039
$$790$$ 0 0
$$791$$ −1.73060 −0.0615330
$$792$$ 0 0
$$793$$ −22.2703 −0.790843
$$794$$ 0 0
$$795$$ 15.8541 0.562287
$$796$$ 15.1246 0.536078
$$797$$ −12.4822 −0.442144 −0.221072 0.975258i $$-0.570956\pi$$
−0.221072 + 0.975258i $$0.570956\pi$$
$$798$$ 0 0
$$799$$ −13.4377 −0.475391
$$800$$ 0 0
$$801$$ −53.2592 −1.88182
$$802$$ 0 0
$$803$$ 64.3951 2.27245
$$804$$ −45.1246 −1.59142
$$805$$ 3.38197 0.119199
$$806$$ 0 0
$$807$$ 68.8673 2.42424
$$808$$ 0 0
$$809$$ 42.2361 1.48494 0.742471 0.669879i $$-0.233654\pi$$
0.742471 + 0.669879i $$0.233654\pi$$
$$810$$ 0 0
$$811$$ 31.9524 1.12200 0.561000 0.827816i $$-0.310417\pi$$
0.561000 + 0.827816i $$0.310417\pi$$
$$812$$ 4.70228 0.165018
$$813$$ −25.9030 −0.908459
$$814$$ 0 0
$$815$$ 12.0902 0.423500
$$816$$ 35.1361 1.23001
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −12.3107 −0.430172
$$820$$ 23.1684 0.809075
$$821$$ 17.5967 0.614131 0.307065 0.951688i $$-0.400653\pi$$
0.307065 + 0.951688i $$0.400653\pi$$
$$822$$ 0 0
$$823$$ 18.8885 0.658413 0.329207 0.944258i $$-0.393219\pi$$
0.329207 + 0.944258i $$0.393219\pi$$
$$824$$ 0 0
$$825$$ 18.0171 0.627274
$$826$$ 0 0
$$827$$ 25.0705 0.871787 0.435893 0.899998i $$-0.356433\pi$$
0.435893 + 0.899998i $$0.356433\pi$$
$$828$$ −70.8328 −2.46161
$$829$$ 32.0584 1.11343 0.556717 0.830702i $$-0.312061\pi$$
0.556717 + 0.830702i $$0.312061\pi$$
$$830$$ 0 0
$$831$$ 59.5860 2.06702
$$832$$ −24.6215 −0.853596
$$833$$ 18.8885 0.654449
$$834$$ 0 0
$$835$$ −2.45714 −0.0850329
$$836$$ 0 0
$$837$$ −17.3607 −0.600073
$$838$$ 0 0
$$839$$ 24.7930 0.855949 0.427974 0.903791i $$-0.359227\pi$$
0.427974 + 0.903791i $$0.359227\pi$$
$$840$$ 0 0
$$841$$ −14.5279 −0.500961
$$842$$ 0 0
$$843$$ −48.2148 −1.66061
$$844$$ 17.5680 0.604717
$$845$$ 3.52786 0.121362
$$846$$ 0 0
$$847$$ −14.3820 −0.494170
$$848$$ 20.6052 0.707587
$$849$$ 73.8644 2.53502
$$850$$ 0 0
$$851$$ 45.6102 1.56350
$$852$$ −1.05573 −0.0361686
$$853$$ −30.2705 −1.03644 −0.518221 0.855247i $$-0.673406\pi$$
−0.518221 + 0.855247i $$0.673406\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −47.1693 −1.61127 −0.805636 0.592410i $$-0.798176\pi$$
−0.805636 + 0.592410i $$0.798176\pi$$
$$858$$ 0 0
$$859$$ 6.83282 0.233133 0.116566 0.993183i $$-0.462811\pi$$
0.116566 + 0.993183i $$0.462811\pi$$
$$860$$ −14.7639 −0.503446
$$861$$ 22.0344 0.750932
$$862$$ 0 0
$$863$$ −9.02105 −0.307080 −0.153540 0.988142i $$-0.549067\pi$$
−0.153540 + 0.988142i $$0.549067\pi$$
$$864$$ 0 0
$$865$$ −6.43288 −0.218725
$$866$$ 0 0
$$867$$ 27.2501 0.925463
$$868$$ 2.00811 0.0681598
$$869$$ 79.5707 2.69925
$$870$$ 0 0
$$871$$ −22.5623 −0.764495
$$872$$ 0 0
$$873$$ 53.9452 1.82577
$$874$$ 0 0
$$875$$ 0.618034 0.0208934
$$876$$ −67.7090 −2.28768
$$877$$ −26.4581 −0.893426 −0.446713 0.894677i $$-0.647405\pi$$
−0.446713 + 0.894677i $$0.647405\pi$$
$$878$$ 0 0
$$879$$ −69.3951 −2.34064
$$880$$ 23.4164 0.789367
$$881$$ 29.5967 0.997140 0.498570 0.866850i $$-0.333859\pi$$
0.498570 + 0.866850i $$0.333859\pi$$
$$882$$ 0 0
$$883$$ −32.0902 −1.07992 −0.539960 0.841691i $$-0.681561\pi$$
−0.539960 + 0.841691i $$0.681561\pi$$
$$884$$ 17.5680 0.590877
$$885$$ −13.0902 −0.440021
$$886$$ 0 0
$$887$$ 58.7940 1.97411 0.987055 0.160385i $$-0.0512736\pi$$
0.987055 + 0.160385i $$0.0512736\pi$$
$$888$$ 0 0
$$889$$ −0.383516 −0.0128627
$$890$$ 0 0
$$891$$ −78.8673 −2.64215
$$892$$ 25.1765 0.842971
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 5.42882 0.181466
$$896$$ 0 0
$$897$$ −51.8328 −1.73065
$$898$$ 0 0
$$899$$ 6.18034 0.206126
$$900$$ −12.9443 −0.431476
$$901$$ −14.7024 −0.489807
$$902$$ 0 0
$$903$$ −14.0413 −0.467266
$$904$$ 0 0
$$905$$ 17.7396 0.589683
$$906$$ 0 0
$$907$$ 36.5237 1.21275 0.606374 0.795179i $$-0.292623\pi$$
0.606374 + 0.795179i $$0.292623\pi$$
$$908$$ −19.2331 −0.638274
$$909$$ −12.3607 −0.409978
$$910$$ 0 0
$$911$$ −41.0139 −1.35885 −0.679426 0.733744i $$-0.737771\pi$$
−0.679426 + 0.733744i $$0.737771\pi$$
$$912$$ 0 0
$$913$$ −62.6869 −2.07463
$$914$$ 0 0
$$915$$ −22.2703 −0.736234
$$916$$ −3.70820 −0.122523
$$917$$ −3.23607 −0.106864
$$918$$ 0 0
$$919$$ 9.59675 0.316567 0.158284 0.987394i $$-0.449404\pi$$
0.158284 + 0.987394i $$0.449404\pi$$
$$920$$ 0 0
$$921$$ 37.5623 1.23772
$$922$$ 0 0
$$923$$ −0.527864 −0.0173749
$$924$$ 22.2703 0.732640
$$925$$ 8.33499 0.274053
$$926$$ 0 0
$$927$$ 58.6475 1.92624
$$928$$ 0 0
$$929$$ −23.5410 −0.772356 −0.386178 0.922424i $$-0.626205\pi$$
−0.386178 + 0.922424i $$0.626205\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9.88854 0.323910
$$933$$ 83.2035 2.72396
$$934$$ 0 0
$$935$$ −16.7082 −0.546417
$$936$$ 0 0
$$937$$ −19.4377 −0.635002 −0.317501 0.948258i $$-0.602844\pi$$
−0.317501 + 0.948258i $$0.602844\pi$$
$$938$$ 0 0
$$939$$ −42.0835 −1.37334
$$940$$ 9.41641 0.307129
$$941$$ −4.49028 −0.146379 −0.0731895 0.997318i $$-0.523318\pi$$
−0.0731895 + 0.997318i $$0.523318\pi$$
$$942$$ 0 0
$$943$$ 63.3903 2.06427
$$944$$ −17.0130 −0.553727
$$945$$ −6.60440 −0.214841
$$946$$ 0 0
$$947$$ −38.1033 −1.23819 −0.619096 0.785315i $$-0.712501\pi$$
−0.619096 + 0.785315i $$0.712501\pi$$
$$948$$ −83.6656 −2.71733
$$949$$ −33.8545 −1.09896
$$950$$ 0 0
$$951$$ 105.902 3.43410
$$952$$ 0 0
$$953$$ −23.3804 −0.757365 −0.378682 0.925527i $$-0.623623\pi$$
−0.378682 + 0.925527i $$0.623623\pi$$
$$954$$ 0 0
$$955$$ −6.09017 −0.197073
$$956$$ 27.8885 0.901980
$$957$$ 68.5410 2.21562
$$958$$ 0 0
$$959$$ −9.00000 −0.290625
$$960$$ −24.6215 −0.794654
$$961$$ −28.3607 −0.914861
$$962$$ 0 0
$$963$$ −15.9030 −0.512466
$$964$$ 6.36737 0.205079
$$965$$ −23.4459 −0.754750
$$966$$ 0 0
$$967$$ −42.0689 −1.35284 −0.676422 0.736514i $$-0.736470\pi$$
−0.676422 + 0.736514i $$0.736470\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −21.8618 −0.701578 −0.350789 0.936454i $$-0.614087\pi$$
−0.350789 + 0.936454i $$0.614087\pi$$
$$972$$ 18.8091 0.603303
$$973$$ 0.236068 0.00756799
$$974$$ 0 0
$$975$$ −9.47214 −0.303351
$$976$$ −28.9443 −0.926484
$$977$$ −32.8505 −1.05098 −0.525490 0.850800i $$-0.676118\pi$$
−0.525490 + 0.850800i $$0.676118\pi$$
$$978$$ 0 0
$$979$$ 48.1734 1.53963
$$980$$ −13.2361 −0.422811
$$981$$ −105.408 −3.36543
$$982$$ 0 0
$$983$$ −5.81234 −0.185385 −0.0926924 0.995695i $$-0.529547\pi$$
−0.0926924 + 0.995695i $$0.529547\pi$$
$$984$$ 0 0
$$985$$ 8.65248 0.275691
$$986$$ 0 0
$$987$$ 8.95554 0.285058
$$988$$ 0 0
$$989$$ −40.3951 −1.28449
$$990$$ 0 0
$$991$$ 30.6708 0.974291 0.487146 0.873321i $$-0.338038\pi$$
0.487146 + 0.873321i $$0.338038\pi$$
$$992$$ 0 0
$$993$$ −72.4853 −2.30025
$$994$$ 0 0
$$995$$ 7.56231 0.239741
$$996$$ 65.9129 2.08853
$$997$$ 9.23607 0.292509 0.146255 0.989247i $$-0.453278\pi$$
0.146255 + 0.989247i $$0.453278\pi$$
$$998$$ 0 0
$$999$$ −89.0689 −2.81801
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 1805.2.a.k.1.1 4
5.4 even 2 9025.2.a.bi.1.4 4
19.18 odd 2 inner 1805.2.a.k.1.4 yes 4
95.94 odd 2 9025.2.a.bi.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.k.1.1 4 1.1 even 1 trivial
1805.2.a.k.1.4 yes 4 19.18 odd 2 inner
9025.2.a.bi.1.1 4 95.94 odd 2
9025.2.a.bi.1.4 4 5.4 even 2