# Properties

 Label 5775.2.a.bp.1.2 Level $5775$ Weight $2$ Character 5775.1 Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5775.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1136071673$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 5775.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.46260 q^{2} -1.00000 q^{3} +0.139194 q^{4} +1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.46260 q^{2} -1.00000 q^{3} +0.139194 q^{4} +1.46260 q^{6} +1.00000 q^{7} +2.72161 q^{8} +1.00000 q^{9} -1.00000 q^{11} -0.139194 q^{12} -5.04502 q^{13} -1.46260 q^{14} -4.25901 q^{16} +6.36842 q^{17} -1.46260 q^{18} -5.32340 q^{19} -1.00000 q^{21} +1.46260 q^{22} -4.92520 q^{23} -2.72161 q^{24} +7.37883 q^{26} -1.00000 q^{27} +0.139194 q^{28} +5.04502 q^{29} -7.57201 q^{31} +0.786003 q^{32} +1.00000 q^{33} -9.31444 q^{34} +0.139194 q^{36} -4.24860 q^{37} +7.78600 q^{38} +5.04502 q^{39} -0.646809 q^{41} +1.46260 q^{42} +10.5180 q^{43} -0.139194 q^{44} +7.20359 q^{46} -0.526989 q^{47} +4.25901 q^{48} +1.00000 q^{49} -6.36842 q^{51} -0.702237 q^{52} -3.72161 q^{53} +1.46260 q^{54} +2.72161 q^{56} +5.32340 q^{57} -7.37883 q^{58} +7.97021 q^{59} -2.00000 q^{61} +11.0748 q^{62} +1.00000 q^{63} +7.36842 q^{64} -1.46260 q^{66} -8.76663 q^{67} +0.886447 q^{68} +4.92520 q^{69} -11.4432 q^{71} +2.72161 q^{72} +13.0450 q^{73} +6.21400 q^{74} -0.740987 q^{76} -1.00000 q^{77} -7.37883 q^{78} +11.4432 q^{79} +1.00000 q^{81} +0.946021 q^{82} -13.1648 q^{83} -0.139194 q^{84} -15.3836 q^{86} -5.04502 q^{87} -2.72161 q^{88} +11.8504 q^{89} -5.04502 q^{91} -0.685559 q^{92} +7.57201 q^{93} +0.770774 q^{94} -0.786003 q^{96} +1.87122 q^{97} -1.46260 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 3 * q^3 + 6 * q^4 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} + 4 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 3 q^{21} + 2 q^{22} - 10 q^{23} + 3 q^{24} - q^{26} - 3 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} - 8 q^{32} + 3 q^{33} - 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 14 q^{41} + 2 q^{42} + 14 q^{43} - 6 q^{44} + 28 q^{46} + 4 q^{48} + 3 q^{49} + 8 q^{51} + 29 q^{52} + 2 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} - 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 2 q^{66} + 4 q^{67} - 42 q^{68} + 10 q^{69} - 12 q^{71} - 3 q^{72} + 20 q^{73} + 29 q^{74} - 11 q^{76} - 3 q^{77} + q^{78} + 12 q^{79} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{84} + 24 q^{86} + 4 q^{87} + 3 q^{88} + 26 q^{89} + 4 q^{91} - 26 q^{92} + 2 q^{93} + 35 q^{94} + 8 q^{96} + 4 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 3 * q^3 + 6 * q^4 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^11 - 6 * q^12 + 4 * q^13 - 2 * q^14 - 4 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 3 * q^21 + 2 * q^22 - 10 * q^23 + 3 * q^24 - q^26 - 3 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 - 8 * q^32 + 3 * q^33 - 4 * q^34 + 6 * q^36 + 13 * q^38 - 4 * q^39 + 14 * q^41 + 2 * q^42 + 14 * q^43 - 6 * q^44 + 28 * q^46 + 4 * q^48 + 3 * q^49 + 8 * q^51 + 29 * q^52 + 2 * q^54 - 3 * q^56 + 8 * q^57 + q^58 - 6 * q^61 + 38 * q^62 + 3 * q^63 - 5 * q^64 - 2 * q^66 + 4 * q^67 - 42 * q^68 + 10 * q^69 - 12 * q^71 - 3 * q^72 + 20 * q^73 + 29 * q^74 - 11 * q^76 - 3 * q^77 + q^78 + 12 * q^79 + 3 * q^81 + 6 * q^82 - 6 * q^83 - 6 * q^84 + 24 * q^86 + 4 * q^87 + 3 * q^88 + 26 * q^89 + 4 * q^91 - 26 * q^92 + 2 * q^93 + 35 * q^94 + 8 * q^96 + 4 * q^97 - 2 * q^98 - 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.46260 −1.03421 −0.517107 0.855921i $$-0.672991\pi$$
−0.517107 + 0.855921i $$0.672991\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.139194 0.0695971
$$5$$ 0 0
$$6$$ 1.46260 0.597103
$$7$$ 1.00000 0.377964
$$8$$ 2.72161 0.962235
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −0.139194 −0.0401819
$$13$$ −5.04502 −1.39924 −0.699618 0.714517i $$-0.746646\pi$$
−0.699618 + 0.714517i $$0.746646\pi$$
$$14$$ −1.46260 −0.390896
$$15$$ 0 0
$$16$$ −4.25901 −1.06475
$$17$$ 6.36842 1.54457 0.772284 0.635277i $$-0.219114\pi$$
0.772284 + 0.635277i $$0.219114\pi$$
$$18$$ −1.46260 −0.344738
$$19$$ −5.32340 −1.22127 −0.610636 0.791911i $$-0.709086\pi$$
−0.610636 + 0.791911i $$0.709086\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 1.46260 0.311827
$$23$$ −4.92520 −1.02697 −0.513487 0.858097i $$-0.671647\pi$$
−0.513487 + 0.858097i $$0.671647\pi$$
$$24$$ −2.72161 −0.555547
$$25$$ 0 0
$$26$$ 7.37883 1.44711
$$27$$ −1.00000 −0.192450
$$28$$ 0.139194 0.0263052
$$29$$ 5.04502 0.936836 0.468418 0.883507i $$-0.344824\pi$$
0.468418 + 0.883507i $$0.344824\pi$$
$$30$$ 0 0
$$31$$ −7.57201 −1.35997 −0.679986 0.733225i $$-0.738014\pi$$
−0.679986 + 0.733225i $$0.738014\pi$$
$$32$$ 0.786003 0.138947
$$33$$ 1.00000 0.174078
$$34$$ −9.31444 −1.59741
$$35$$ 0 0
$$36$$ 0.139194 0.0231990
$$37$$ −4.24860 −0.698466 −0.349233 0.937036i $$-0.613558\pi$$
−0.349233 + 0.937036i $$0.613558\pi$$
$$38$$ 7.78600 1.26306
$$39$$ 5.04502 0.807849
$$40$$ 0 0
$$41$$ −0.646809 −0.101015 −0.0505073 0.998724i $$-0.516084\pi$$
−0.0505073 + 0.998724i $$0.516084\pi$$
$$42$$ 1.46260 0.225684
$$43$$ 10.5180 1.60398 0.801992 0.597335i $$-0.203774\pi$$
0.801992 + 0.597335i $$0.203774\pi$$
$$44$$ −0.139194 −0.0209843
$$45$$ 0 0
$$46$$ 7.20359 1.06211
$$47$$ −0.526989 −0.0768693 −0.0384347 0.999261i $$-0.512237\pi$$
−0.0384347 + 0.999261i $$0.512237\pi$$
$$48$$ 4.25901 0.614736
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.36842 −0.891757
$$52$$ −0.702237 −0.0973827
$$53$$ −3.72161 −0.511203 −0.255601 0.966782i $$-0.582273\pi$$
−0.255601 + 0.966782i $$0.582273\pi$$
$$54$$ 1.46260 0.199034
$$55$$ 0 0
$$56$$ 2.72161 0.363691
$$57$$ 5.32340 0.705102
$$58$$ −7.37883 −0.968888
$$59$$ 7.97021 1.03763 0.518817 0.854886i $$-0.326373\pi$$
0.518817 + 0.854886i $$0.326373\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 11.0748 1.40650
$$63$$ 1.00000 0.125988
$$64$$ 7.36842 0.921053
$$65$$ 0 0
$$66$$ −1.46260 −0.180033
$$67$$ −8.76663 −1.07101 −0.535507 0.844531i $$-0.679879\pi$$
−0.535507 + 0.844531i $$0.679879\pi$$
$$68$$ 0.886447 0.107497
$$69$$ 4.92520 0.592924
$$70$$ 0 0
$$71$$ −11.4432 −1.35806 −0.679030 0.734110i $$-0.737600\pi$$
−0.679030 + 0.734110i $$0.737600\pi$$
$$72$$ 2.72161 0.320745
$$73$$ 13.0450 1.52680 0.763402 0.645924i $$-0.223528\pi$$
0.763402 + 0.645924i $$0.223528\pi$$
$$74$$ 6.21400 0.722363
$$75$$ 0 0
$$76$$ −0.740987 −0.0849970
$$77$$ −1.00000 −0.113961
$$78$$ −7.37883 −0.835488
$$79$$ 11.4432 1.28746 0.643732 0.765251i $$-0.277385\pi$$
0.643732 + 0.765251i $$0.277385\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0.946021 0.104471
$$83$$ −13.1648 −1.44503 −0.722514 0.691356i $$-0.757014\pi$$
−0.722514 + 0.691356i $$0.757014\pi$$
$$84$$ −0.139194 −0.0151873
$$85$$ 0 0
$$86$$ −15.3836 −1.65886
$$87$$ −5.04502 −0.540882
$$88$$ −2.72161 −0.290125
$$89$$ 11.8504 1.25614 0.628070 0.778157i $$-0.283845\pi$$
0.628070 + 0.778157i $$0.283845\pi$$
$$90$$ 0 0
$$91$$ −5.04502 −0.528861
$$92$$ −0.685559 −0.0714744
$$93$$ 7.57201 0.785180
$$94$$ 0.770774 0.0794993
$$95$$ 0 0
$$96$$ −0.786003 −0.0802211
$$97$$ 1.87122 0.189993 0.0949967 0.995478i $$-0.469716\pi$$
0.0949967 + 0.995478i $$0.469716\pi$$
$$98$$ −1.46260 −0.147745
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 4.51803 0.449560 0.224780 0.974409i $$-0.427834\pi$$
0.224780 + 0.974409i $$0.427834\pi$$
$$102$$ 9.31444 0.922267
$$103$$ 10.6468 1.04906 0.524531 0.851392i $$-0.324241\pi$$
0.524531 + 0.851392i $$0.324241\pi$$
$$104$$ −13.7306 −1.34639
$$105$$ 0 0
$$106$$ 5.44322 0.528693
$$107$$ −15.9702 −1.54390 −0.771949 0.635684i $$-0.780718\pi$$
−0.771949 + 0.635684i $$0.780718\pi$$
$$108$$ −0.139194 −0.0133940
$$109$$ 12.7756 1.22368 0.611840 0.790982i $$-0.290430\pi$$
0.611840 + 0.790982i $$0.290430\pi$$
$$110$$ 0 0
$$111$$ 4.24860 0.403259
$$112$$ −4.25901 −0.402439
$$113$$ −18.7368 −1.76261 −0.881307 0.472544i $$-0.843336\pi$$
−0.881307 + 0.472544i $$0.843336\pi$$
$$114$$ −7.78600 −0.729226
$$115$$ 0 0
$$116$$ 0.702237 0.0652010
$$117$$ −5.04502 −0.466412
$$118$$ −11.6572 −1.07313
$$119$$ 6.36842 0.583792
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.92520 0.264835
$$123$$ 0.646809 0.0583208
$$124$$ −1.05398 −0.0946501
$$125$$ 0 0
$$126$$ −1.46260 −0.130299
$$127$$ 2.27839 0.202174 0.101087 0.994878i $$-0.467768\pi$$
0.101087 + 0.994878i $$0.467768\pi$$
$$128$$ −12.3490 −1.09151
$$129$$ −10.5180 −0.926061
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0.139194 0.0121153
$$133$$ −5.32340 −0.461598
$$134$$ 12.8221 1.10766
$$135$$ 0 0
$$136$$ 17.3324 1.48624
$$137$$ 4.77559 0.408006 0.204003 0.978970i $$-0.434605\pi$$
0.204003 + 0.978970i $$0.434605\pi$$
$$138$$ −7.20359 −0.613210
$$139$$ −15.4432 −1.30988 −0.654939 0.755682i $$-0.727306\pi$$
−0.654939 + 0.755682i $$0.727306\pi$$
$$140$$ 0 0
$$141$$ 0.526989 0.0443805
$$142$$ 16.7368 1.40452
$$143$$ 5.04502 0.421885
$$144$$ −4.25901 −0.354918
$$145$$ 0 0
$$146$$ −19.0796 −1.57904
$$147$$ −1.00000 −0.0824786
$$148$$ −0.591380 −0.0486112
$$149$$ −9.84143 −0.806241 −0.403121 0.915147i $$-0.632075\pi$$
−0.403121 + 0.915147i $$0.632075\pi$$
$$150$$ 0 0
$$151$$ −4.12878 −0.335996 −0.167998 0.985787i $$-0.553730\pi$$
−0.167998 + 0.985787i $$0.553730\pi$$
$$152$$ −14.4882 −1.17515
$$153$$ 6.36842 0.514856
$$154$$ 1.46260 0.117860
$$155$$ 0 0
$$156$$ 0.702237 0.0562239
$$157$$ 0.946021 0.0755007 0.0377504 0.999287i $$-0.487981\pi$$
0.0377504 + 0.999287i $$0.487981\pi$$
$$158$$ −16.7368 −1.33151
$$159$$ 3.72161 0.295143
$$160$$ 0 0
$$161$$ −4.92520 −0.388160
$$162$$ −1.46260 −0.114913
$$163$$ −8.76663 −0.686655 −0.343328 0.939216i $$-0.611554\pi$$
−0.343328 + 0.939216i $$0.611554\pi$$
$$164$$ −0.0900320 −0.00703032
$$165$$ 0 0
$$166$$ 19.2549 1.49447
$$167$$ 24.3684 1.88568 0.942842 0.333239i $$-0.108142\pi$$
0.942842 + 0.333239i $$0.108142\pi$$
$$168$$ −2.72161 −0.209977
$$169$$ 12.4522 0.957860
$$170$$ 0 0
$$171$$ −5.32340 −0.407091
$$172$$ 1.46405 0.111633
$$173$$ −12.3476 −0.938770 −0.469385 0.882994i $$-0.655524\pi$$
−0.469385 + 0.882994i $$0.655524\pi$$
$$174$$ 7.37883 0.559388
$$175$$ 0 0
$$176$$ 4.25901 0.321035
$$177$$ −7.97021 −0.599078
$$178$$ −17.3324 −1.29912
$$179$$ −5.59283 −0.418028 −0.209014 0.977913i $$-0.567025\pi$$
−0.209014 + 0.977913i $$0.567025\pi$$
$$180$$ 0 0
$$181$$ 13.5720 1.00880 0.504400 0.863470i $$-0.331714\pi$$
0.504400 + 0.863470i $$0.331714\pi$$
$$182$$ 7.37883 0.546955
$$183$$ 2.00000 0.147844
$$184$$ −13.4045 −0.988191
$$185$$ 0 0
$$186$$ −11.0748 −0.812044
$$187$$ −6.36842 −0.465705
$$188$$ −0.0733538 −0.00534988
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −9.42240 −0.681781 −0.340890 0.940103i $$-0.610729\pi$$
−0.340890 + 0.940103i $$0.610729\pi$$
$$192$$ −7.36842 −0.531770
$$193$$ 10.1288 0.729086 0.364543 0.931187i $$-0.381225\pi$$
0.364543 + 0.931187i $$0.381225\pi$$
$$194$$ −2.73684 −0.196494
$$195$$ 0 0
$$196$$ 0.139194 0.00994244
$$197$$ 2.25756 0.160845 0.0804224 0.996761i $$-0.474373\pi$$
0.0804224 + 0.996761i $$0.474373\pi$$
$$198$$ 1.46260 0.103942
$$199$$ 3.07480 0.217967 0.108984 0.994044i $$-0.465240\pi$$
0.108984 + 0.994044i $$0.465240\pi$$
$$200$$ 0 0
$$201$$ 8.76663 0.618350
$$202$$ −6.60806 −0.464941
$$203$$ 5.04502 0.354091
$$204$$ −0.886447 −0.0620637
$$205$$ 0 0
$$206$$ −15.5720 −1.08495
$$207$$ −4.92520 −0.342325
$$208$$ 21.4868 1.48984
$$209$$ 5.32340 0.368228
$$210$$ 0 0
$$211$$ −14.6468 −1.00833 −0.504164 0.863608i $$-0.668199\pi$$
−0.504164 + 0.863608i $$0.668199\pi$$
$$212$$ −0.518027 −0.0355782
$$213$$ 11.4432 0.784077
$$214$$ 23.3580 1.59672
$$215$$ 0 0
$$216$$ −2.72161 −0.185182
$$217$$ −7.57201 −0.514021
$$218$$ −18.6856 −1.26555
$$219$$ −13.0450 −0.881500
$$220$$ 0 0
$$221$$ −32.1288 −2.16122
$$222$$ −6.21400 −0.417056
$$223$$ 1.90997 0.127901 0.0639505 0.997953i $$-0.479630\pi$$
0.0639505 + 0.997953i $$0.479630\pi$$
$$224$$ 0.786003 0.0525170
$$225$$ 0 0
$$226$$ 27.4045 1.82292
$$227$$ 3.20359 0.212629 0.106315 0.994333i $$-0.466095\pi$$
0.106315 + 0.994333i $$0.466095\pi$$
$$228$$ 0.740987 0.0490730
$$229$$ −18.3088 −1.20988 −0.604941 0.796270i $$-0.706803\pi$$
−0.604941 + 0.796270i $$0.706803\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 13.7306 0.901456
$$233$$ 16.5872 1.08667 0.543333 0.839517i $$-0.317162\pi$$
0.543333 + 0.839517i $$0.317162\pi$$
$$234$$ 7.37883 0.482369
$$235$$ 0 0
$$236$$ 1.10941 0.0722162
$$237$$ −11.4432 −0.743317
$$238$$ −9.31444 −0.603766
$$239$$ 2.91623 0.188635 0.0943177 0.995542i $$-0.469933\pi$$
0.0943177 + 0.995542i $$0.469933\pi$$
$$240$$ 0 0
$$241$$ −6.09899 −0.392871 −0.196435 0.980517i $$-0.562937\pi$$
−0.196435 + 0.980517i $$0.562937\pi$$
$$242$$ −1.46260 −0.0940194
$$243$$ −1.00000 −0.0641500
$$244$$ −0.278388 −0.0178220
$$245$$ 0 0
$$246$$ −0.946021 −0.0603161
$$247$$ 26.8567 1.70885
$$248$$ −20.6081 −1.30861
$$249$$ 13.1648 0.834288
$$250$$ 0 0
$$251$$ 1.62262 0.102419 0.0512093 0.998688i $$-0.483692\pi$$
0.0512093 + 0.998688i $$0.483692\pi$$
$$252$$ 0.139194 0.00876841
$$253$$ 4.92520 0.309644
$$254$$ −3.33237 −0.209091
$$255$$ 0 0
$$256$$ 3.32485 0.207803
$$257$$ 6.89541 0.430124 0.215062 0.976600i $$-0.431005\pi$$
0.215062 + 0.976600i $$0.431005\pi$$
$$258$$ 15.3836 0.957744
$$259$$ −4.24860 −0.263995
$$260$$ 0 0
$$261$$ 5.04502 0.312279
$$262$$ −5.85039 −0.361439
$$263$$ −5.08377 −0.313478 −0.156739 0.987640i $$-0.550098\pi$$
−0.156739 + 0.987640i $$0.550098\pi$$
$$264$$ 2.72161 0.167504
$$265$$ 0 0
$$266$$ 7.78600 0.477390
$$267$$ −11.8504 −0.725232
$$268$$ −1.22026 −0.0745394
$$269$$ −0.886447 −0.0540476 −0.0270238 0.999635i $$-0.508603\pi$$
−0.0270238 + 0.999635i $$0.508603\pi$$
$$270$$ 0 0
$$271$$ −25.3234 −1.53829 −0.769144 0.639076i $$-0.779317\pi$$
−0.769144 + 0.639076i $$0.779317\pi$$
$$272$$ −27.1232 −1.64458
$$273$$ 5.04502 0.305338
$$274$$ −6.98477 −0.421965
$$275$$ 0 0
$$276$$ 0.685559 0.0412658
$$277$$ 24.8269 1.49170 0.745851 0.666113i $$-0.232043\pi$$
0.745851 + 0.666113i $$0.232043\pi$$
$$278$$ 22.5872 1.35469
$$279$$ −7.57201 −0.453324
$$280$$ 0 0
$$281$$ 1.90101 0.113404 0.0567022 0.998391i $$-0.481941\pi$$
0.0567022 + 0.998391i $$0.481941\pi$$
$$282$$ −0.770774 −0.0458989
$$283$$ 22.3178 1.32666 0.663328 0.748329i $$-0.269143\pi$$
0.663328 + 0.748329i $$0.269143\pi$$
$$284$$ −1.59283 −0.0945171
$$285$$ 0 0
$$286$$ −7.37883 −0.436320
$$287$$ −0.646809 −0.0381799
$$288$$ 0.786003 0.0463157
$$289$$ 23.5568 1.38569
$$290$$ 0 0
$$291$$ −1.87122 −0.109693
$$292$$ 1.81579 0.106261
$$293$$ −12.0900 −0.706307 −0.353154 0.935565i $$-0.614891\pi$$
−0.353154 + 0.935565i $$0.614891\pi$$
$$294$$ 1.46260 0.0853005
$$295$$ 0 0
$$296$$ −11.5630 −0.672088
$$297$$ 1.00000 0.0580259
$$298$$ 14.3941 0.833826
$$299$$ 24.8477 1.43698
$$300$$ 0 0
$$301$$ 10.5180 0.606249
$$302$$ 6.03875 0.347491
$$303$$ −4.51803 −0.259554
$$304$$ 22.6724 1.30035
$$305$$ 0 0
$$306$$ −9.31444 −0.532471
$$307$$ 13.5928 0.775784 0.387892 0.921705i $$-0.373203\pi$$
0.387892 + 0.921705i $$0.373203\pi$$
$$308$$ −0.139194 −0.00793132
$$309$$ −10.6468 −0.605676
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 13.7306 0.777341
$$313$$ −14.9252 −0.843622 −0.421811 0.906684i $$-0.638605\pi$$
−0.421811 + 0.906684i $$0.638605\pi$$
$$314$$ −1.38365 −0.0780838
$$315$$ 0 0
$$316$$ 1.59283 0.0896037
$$317$$ −3.97918 −0.223493 −0.111746 0.993737i $$-0.535644\pi$$
−0.111746 + 0.993737i $$0.535644\pi$$
$$318$$ −5.44322 −0.305241
$$319$$ −5.04502 −0.282467
$$320$$ 0 0
$$321$$ 15.9702 0.891370
$$322$$ 7.20359 0.401440
$$323$$ −33.9017 −1.88634
$$324$$ 0.139194 0.00773301
$$325$$ 0 0
$$326$$ 12.8221 0.710148
$$327$$ −12.7756 −0.706492
$$328$$ −1.76036 −0.0971997
$$329$$ −0.526989 −0.0290539
$$330$$ 0 0
$$331$$ 23.4432 1.28856 0.644278 0.764791i $$-0.277158\pi$$
0.644278 + 0.764791i $$0.277158\pi$$
$$332$$ −1.83247 −0.100570
$$333$$ −4.24860 −0.232822
$$334$$ −35.6412 −1.95020
$$335$$ 0 0
$$336$$ 4.25901 0.232348
$$337$$ −11.1648 −0.608187 −0.304094 0.952642i $$-0.598354\pi$$
−0.304094 + 0.952642i $$0.598354\pi$$
$$338$$ −18.2125 −0.990632
$$339$$ 18.7368 1.01765
$$340$$ 0 0
$$341$$ 7.57201 0.410047
$$342$$ 7.78600 0.421019
$$343$$ 1.00000 0.0539949
$$344$$ 28.6260 1.54341
$$345$$ 0 0
$$346$$ 18.0596 0.970889
$$347$$ 22.5872 1.21255 0.606273 0.795256i $$-0.292664\pi$$
0.606273 + 0.795256i $$0.292664\pi$$
$$348$$ −0.702237 −0.0376438
$$349$$ 27.9315 1.49514 0.747568 0.664185i $$-0.231221\pi$$
0.747568 + 0.664185i $$0.231221\pi$$
$$350$$ 0 0
$$351$$ 5.04502 0.269283
$$352$$ −0.786003 −0.0418941
$$353$$ 16.5478 0.880751 0.440376 0.897814i $$-0.354845\pi$$
0.440376 + 0.897814i $$0.354845\pi$$
$$354$$ 11.6572 0.619574
$$355$$ 0 0
$$356$$ 1.64951 0.0874236
$$357$$ −6.36842 −0.337053
$$358$$ 8.18006 0.432330
$$359$$ 22.0305 1.16272 0.581362 0.813645i $$-0.302520\pi$$
0.581362 + 0.813645i $$0.302520\pi$$
$$360$$ 0 0
$$361$$ 9.33863 0.491507
$$362$$ −19.8504 −1.04331
$$363$$ −1.00000 −0.0524864
$$364$$ −0.702237 −0.0368072
$$365$$ 0 0
$$366$$ −2.92520 −0.152902
$$367$$ 19.3836 1.01182 0.505909 0.862587i $$-0.331157\pi$$
0.505909 + 0.862587i $$0.331157\pi$$
$$368$$ 20.9765 1.09347
$$369$$ −0.646809 −0.0336715
$$370$$ 0 0
$$371$$ −3.72161 −0.193216
$$372$$ 1.05398 0.0546463
$$373$$ −29.2549 −1.51476 −0.757380 0.652975i $$-0.773521\pi$$
−0.757380 + 0.652975i $$0.773521\pi$$
$$374$$ 9.31444 0.481638
$$375$$ 0 0
$$376$$ −1.43426 −0.0739663
$$377$$ −25.4522 −1.31085
$$378$$ 1.46260 0.0752279
$$379$$ 12.5270 0.643468 0.321734 0.946830i $$-0.395734\pi$$
0.321734 + 0.946830i $$0.395734\pi$$
$$380$$ 0 0
$$381$$ −2.27839 −0.116725
$$382$$ 13.7812 0.705107
$$383$$ 17.5928 0.898952 0.449476 0.893293i $$-0.351611\pi$$
0.449476 + 0.893293i $$0.351611\pi$$
$$384$$ 12.3490 0.630185
$$385$$ 0 0
$$386$$ −14.8143 −0.754030
$$387$$ 10.5180 0.534661
$$388$$ 0.260463 0.0132230
$$389$$ 20.0900 1.01861 0.509303 0.860588i $$-0.329903\pi$$
0.509303 + 0.860588i $$0.329903\pi$$
$$390$$ 0 0
$$391$$ −31.3657 −1.58623
$$392$$ 2.72161 0.137462
$$393$$ −4.00000 −0.201773
$$394$$ −3.30191 −0.166348
$$395$$ 0 0
$$396$$ −0.139194 −0.00699477
$$397$$ 35.1053 1.76188 0.880941 0.473226i $$-0.156910\pi$$
0.880941 + 0.473226i $$0.156910\pi$$
$$398$$ −4.49720 −0.225424
$$399$$ 5.32340 0.266504
$$400$$ 0 0
$$401$$ 9.57201 0.478003 0.239002 0.971019i $$-0.423180\pi$$
0.239002 + 0.971019i $$0.423180\pi$$
$$402$$ −12.8221 −0.639506
$$403$$ 38.2009 1.90292
$$404$$ 0.628883 0.0312881
$$405$$ 0 0
$$406$$ −7.37883 −0.366205
$$407$$ 4.24860 0.210595
$$408$$ −17.3324 −0.858080
$$409$$ 38.1801 1.88788 0.943941 0.330113i $$-0.107087\pi$$
0.943941 + 0.330113i $$0.107087\pi$$
$$410$$ 0 0
$$411$$ −4.77559 −0.235563
$$412$$ 1.48197 0.0730116
$$413$$ 7.97021 0.392189
$$414$$ 7.20359 0.354037
$$415$$ 0 0
$$416$$ −3.96540 −0.194420
$$417$$ 15.4432 0.756258
$$418$$ −7.78600 −0.380826
$$419$$ 7.17380 0.350463 0.175231 0.984527i $$-0.443933\pi$$
0.175231 + 0.984527i $$0.443933\pi$$
$$420$$ 0 0
$$421$$ 15.1530 0.738511 0.369255 0.929328i $$-0.379613\pi$$
0.369255 + 0.929328i $$0.379613\pi$$
$$422$$ 21.4224 1.04283
$$423$$ −0.526989 −0.0256231
$$424$$ −10.1288 −0.491897
$$425$$ 0 0
$$426$$ −16.7368 −0.810903
$$427$$ −2.00000 −0.0967868
$$428$$ −2.22296 −0.107451
$$429$$ −5.04502 −0.243576
$$430$$ 0 0
$$431$$ 5.56304 0.267962 0.133981 0.990984i $$-0.457224\pi$$
0.133981 + 0.990984i $$0.457224\pi$$
$$432$$ 4.25901 0.204912
$$433$$ 25.6412 1.23224 0.616119 0.787653i $$-0.288704\pi$$
0.616119 + 0.787653i $$0.288704\pi$$
$$434$$ 11.0748 0.531608
$$435$$ 0 0
$$436$$ 1.77829 0.0851645
$$437$$ 26.2188 1.25422
$$438$$ 19.0796 0.911659
$$439$$ 23.6710 1.12976 0.564878 0.825175i $$-0.308923\pi$$
0.564878 + 0.825175i $$0.308923\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 46.9915 2.23516
$$443$$ 18.0305 0.856653 0.428326 0.903624i $$-0.359103\pi$$
0.428326 + 0.903624i $$0.359103\pi$$
$$444$$ 0.591380 0.0280657
$$445$$ 0 0
$$446$$ −2.79352 −0.132277
$$447$$ 9.84143 0.465484
$$448$$ 7.36842 0.348125
$$449$$ −34.9765 −1.65064 −0.825321 0.564664i $$-0.809006\pi$$
−0.825321 + 0.564664i $$0.809006\pi$$
$$450$$ 0 0
$$451$$ 0.646809 0.0304570
$$452$$ −2.60806 −0.122673
$$453$$ 4.12878 0.193987
$$454$$ −4.68556 −0.219904
$$455$$ 0 0
$$456$$ 14.4882 0.678474
$$457$$ −4.53595 −0.212183 −0.106091 0.994356i $$-0.533834\pi$$
−0.106091 + 0.994356i $$0.533834\pi$$
$$458$$ 26.7785 1.25128
$$459$$ −6.36842 −0.297252
$$460$$ 0 0
$$461$$ −2.79641 −0.130242 −0.0651210 0.997877i $$-0.520743\pi$$
−0.0651210 + 0.997877i $$0.520743\pi$$
$$462$$ −1.46260 −0.0680462
$$463$$ −38.3595 −1.78272 −0.891358 0.453301i $$-0.850246\pi$$
−0.891358 + 0.453301i $$0.850246\pi$$
$$464$$ −21.4868 −0.997499
$$465$$ 0 0
$$466$$ −24.2605 −1.12384
$$467$$ 20.4674 0.947119 0.473560 0.880762i $$-0.342969\pi$$
0.473560 + 0.880762i $$0.342969\pi$$
$$468$$ −0.702237 −0.0324609
$$469$$ −8.76663 −0.404805
$$470$$ 0 0
$$471$$ −0.946021 −0.0435904
$$472$$ 21.6918 0.998447
$$473$$ −10.5180 −0.483619
$$474$$ 16.7368 0.768749
$$475$$ 0 0
$$476$$ 0.886447 0.0406302
$$477$$ −3.72161 −0.170401
$$478$$ −4.26528 −0.195089
$$479$$ −11.6137 −0.530641 −0.265321 0.964160i $$-0.585478\pi$$
−0.265321 + 0.964160i $$0.585478\pi$$
$$480$$ 0 0
$$481$$ 21.4343 0.977318
$$482$$ 8.92038 0.406312
$$483$$ 4.92520 0.224104
$$484$$ 0.139194 0.00632701
$$485$$ 0 0
$$486$$ 1.46260 0.0663448
$$487$$ −32.4793 −1.47178 −0.735888 0.677103i $$-0.763235\pi$$
−0.735888 + 0.677103i $$0.763235\pi$$
$$488$$ −5.44322 −0.246403
$$489$$ 8.76663 0.396441
$$490$$ 0 0
$$491$$ 26.6766 1.20390 0.601949 0.798535i $$-0.294391\pi$$
0.601949 + 0.798535i $$0.294391\pi$$
$$492$$ 0.0900320 0.00405895
$$493$$ 32.1288 1.44701
$$494$$ −39.2805 −1.76731
$$495$$ 0 0
$$496$$ 32.2493 1.44804
$$497$$ −11.4432 −0.513299
$$498$$ −19.2549 −0.862831
$$499$$ −41.2459 −1.84642 −0.923210 0.384296i $$-0.874444\pi$$
−0.923210 + 0.384296i $$0.874444\pi$$
$$500$$ 0 0
$$501$$ −24.3684 −1.08870
$$502$$ −2.37324 −0.105923
$$503$$ −30.5180 −1.36073 −0.680366 0.732873i $$-0.738179\pi$$
−0.680366 + 0.732873i $$0.738179\pi$$
$$504$$ 2.72161 0.121230
$$505$$ 0 0
$$506$$ −7.20359 −0.320238
$$507$$ −12.4522 −0.553021
$$508$$ 0.317138 0.0140707
$$509$$ −18.9944 −0.841912 −0.420956 0.907081i $$-0.638305\pi$$
−0.420956 + 0.907081i $$0.638305\pi$$
$$510$$ 0 0
$$511$$ 13.0450 0.577078
$$512$$ 19.8352 0.876599
$$513$$ 5.32340 0.235034
$$514$$ −10.0852 −0.444840
$$515$$ 0 0
$$516$$ −1.46405 −0.0644511
$$517$$ 0.526989 0.0231770
$$518$$ 6.21400 0.273027
$$519$$ 12.3476 0.541999
$$520$$ 0 0
$$521$$ 25.2430 1.10592 0.552958 0.833209i $$-0.313499\pi$$
0.552958 + 0.833209i $$0.313499\pi$$
$$522$$ −7.37883 −0.322963
$$523$$ 2.93416 0.128302 0.0641509 0.997940i $$-0.479566\pi$$
0.0641509 + 0.997940i $$0.479566\pi$$
$$524$$ 0.556777 0.0243229
$$525$$ 0 0
$$526$$ 7.43551 0.324204
$$527$$ −48.2217 −2.10057
$$528$$ −4.25901 −0.185350
$$529$$ 1.25756 0.0546767
$$530$$ 0 0
$$531$$ 7.97021 0.345878
$$532$$ −0.740987 −0.0321258
$$533$$ 3.26316 0.141343
$$534$$ 17.3324 0.750045
$$535$$ 0 0
$$536$$ −23.8594 −1.03057
$$537$$ 5.59283 0.241348
$$538$$ 1.29652 0.0558968
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ 8.90437 0.382829 0.191414 0.981509i $$-0.438693\pi$$
0.191414 + 0.981509i $$0.438693\pi$$
$$542$$ 37.0380 1.59092
$$543$$ −13.5720 −0.582430
$$544$$ 5.00560 0.214613
$$545$$ 0 0
$$546$$ −7.37883 −0.315785
$$547$$ 29.4737 1.26020 0.630102 0.776513i $$-0.283013\pi$$
0.630102 + 0.776513i $$0.283013\pi$$
$$548$$ 0.664734 0.0283960
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −26.8567 −1.14413
$$552$$ 13.4045 0.570532
$$553$$ 11.4432 0.486615
$$554$$ −36.3117 −1.54274
$$555$$ 0 0
$$556$$ −2.14961 −0.0911636
$$557$$ −14.8954 −0.631139 −0.315569 0.948903i $$-0.602196\pi$$
−0.315569 + 0.948903i $$0.602196\pi$$
$$558$$ 11.0748 0.468834
$$559$$ −53.0636 −2.24435
$$560$$ 0 0
$$561$$ 6.36842 0.268875
$$562$$ −2.78041 −0.117284
$$563$$ 7.81164 0.329222 0.164611 0.986359i $$-0.447363\pi$$
0.164611 + 0.986359i $$0.447363\pi$$
$$564$$ 0.0733538 0.00308875
$$565$$ 0 0
$$566$$ −32.6420 −1.37205
$$567$$ 1.00000 0.0419961
$$568$$ −31.1440 −1.30677
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 25.5512 1.06928 0.534642 0.845079i $$-0.320447\pi$$
0.534642 + 0.845079i $$0.320447\pi$$
$$572$$ 0.702237 0.0293620
$$573$$ 9.42240 0.393626
$$574$$ 0.946021 0.0394862
$$575$$ 0 0
$$576$$ 7.36842 0.307018
$$577$$ 29.5124 1.22862 0.614309 0.789065i $$-0.289435\pi$$
0.614309 + 0.789065i $$0.289435\pi$$
$$578$$ −34.4541 −1.43310
$$579$$ −10.1288 −0.420938
$$580$$ 0 0
$$581$$ −13.1648 −0.546169
$$582$$ 2.73684 0.113446
$$583$$ 3.72161 0.154133
$$584$$ 35.5035 1.46914
$$585$$ 0 0
$$586$$ 17.6829 0.730472
$$587$$ −31.3955 −1.29583 −0.647916 0.761712i $$-0.724359\pi$$
−0.647916 + 0.761712i $$0.724359\pi$$
$$588$$ −0.139194 −0.00574027
$$589$$ 40.3088 1.66090
$$590$$ 0 0
$$591$$ −2.25756 −0.0928638
$$592$$ 18.0948 0.743694
$$593$$ 7.90997 0.324823 0.162412 0.986723i $$-0.448073\pi$$
0.162412 + 0.986723i $$0.448073\pi$$
$$594$$ −1.46260 −0.0600111
$$595$$ 0 0
$$596$$ −1.36987 −0.0561120
$$597$$ −3.07480 −0.125843
$$598$$ −36.3422 −1.48614
$$599$$ 27.4432 1.12130 0.560650 0.828053i $$-0.310551\pi$$
0.560650 + 0.828053i $$0.310551\pi$$
$$600$$ 0 0
$$601$$ 31.9910 1.30494 0.652471 0.757814i $$-0.273732\pi$$
0.652471 + 0.757814i $$0.273732\pi$$
$$602$$ −15.3836 −0.626991
$$603$$ −8.76663 −0.357005
$$604$$ −0.574702 −0.0233843
$$605$$ 0 0
$$606$$ 6.60806 0.268434
$$607$$ −7.41344 −0.300902 −0.150451 0.988617i $$-0.548073\pi$$
−0.150451 + 0.988617i $$0.548073\pi$$
$$608$$ −4.18421 −0.169692
$$609$$ −5.04502 −0.204434
$$610$$ 0 0
$$611$$ 2.65867 0.107558
$$612$$ 0.886447 0.0358325
$$613$$ 33.9917 1.37291 0.686456 0.727171i $$-0.259165\pi$$
0.686456 + 0.727171i $$0.259165\pi$$
$$614$$ −19.8809 −0.802326
$$615$$ 0 0
$$616$$ −2.72161 −0.109657
$$617$$ 44.0305 1.77260 0.886300 0.463112i $$-0.153267\pi$$
0.886300 + 0.463112i $$0.153267\pi$$
$$618$$ 15.5720 0.626398
$$619$$ 40.0096 1.60812 0.804061 0.594546i $$-0.202668\pi$$
0.804061 + 0.594546i $$0.202668\pi$$
$$620$$ 0 0
$$621$$ 4.92520 0.197641
$$622$$ −11.7008 −0.469159
$$623$$ 11.8504 0.474776
$$624$$ −21.4868 −0.860160
$$625$$ 0 0
$$626$$ 21.8296 0.872485
$$627$$ −5.32340 −0.212596
$$628$$ 0.131681 0.00525463
$$629$$ −27.0569 −1.07883
$$630$$ 0 0
$$631$$ 28.5568 1.13683 0.568414 0.822743i $$-0.307557\pi$$
0.568414 + 0.822743i $$0.307557\pi$$
$$632$$ 31.1440 1.23884
$$633$$ 14.6468 0.582158
$$634$$ 5.81994 0.231139
$$635$$ 0 0
$$636$$ 0.518027 0.0205411
$$637$$ −5.04502 −0.199891
$$638$$ 7.37883 0.292131
$$639$$ −11.4432 −0.452687
$$640$$ 0 0
$$641$$ −31.1053 −1.22858 −0.614292 0.789079i $$-0.710558\pi$$
−0.614292 + 0.789079i $$0.710558\pi$$
$$642$$ −23.3580 −0.921867
$$643$$ 5.48197 0.216188 0.108094 0.994141i $$-0.465525\pi$$
0.108094 + 0.994141i $$0.465525\pi$$
$$644$$ −0.685559 −0.0270148
$$645$$ 0 0
$$646$$ 49.5845 1.95088
$$647$$ 9.26383 0.364199 0.182099 0.983280i $$-0.441711\pi$$
0.182099 + 0.983280i $$0.441711\pi$$
$$648$$ 2.72161 0.106915
$$649$$ −7.97021 −0.312858
$$650$$ 0 0
$$651$$ 7.57201 0.296770
$$652$$ −1.22026 −0.0477892
$$653$$ 29.9821 1.17329 0.586645 0.809844i $$-0.300449\pi$$
0.586645 + 0.809844i $$0.300449\pi$$
$$654$$ 18.6856 0.730663
$$655$$ 0 0
$$656$$ 2.75477 0.107556
$$657$$ 13.0450 0.508935
$$658$$ 0.770774 0.0300479
$$659$$ 23.9702 0.933747 0.466873 0.884324i $$-0.345380\pi$$
0.466873 + 0.884324i $$0.345380\pi$$
$$660$$ 0 0
$$661$$ −40.4585 −1.57365 −0.786826 0.617175i $$-0.788277\pi$$
−0.786826 + 0.617175i $$0.788277\pi$$
$$662$$ −34.2880 −1.33264
$$663$$ 32.1288 1.24778
$$664$$ −35.8296 −1.39046
$$665$$ 0 0
$$666$$ 6.21400 0.240788
$$667$$ −24.8477 −0.962107
$$668$$ 3.39194 0.131238
$$669$$ −1.90997 −0.0738436
$$670$$ 0 0
$$671$$ 2.00000 0.0772091
$$672$$ −0.786003 −0.0303207
$$673$$ −21.8712 −0.843073 −0.421537 0.906811i $$-0.638509\pi$$
−0.421537 + 0.906811i $$0.638509\pi$$
$$674$$ 16.3297 0.628995
$$675$$ 0 0
$$676$$ 1.73327 0.0666643
$$677$$ 1.26316 0.0485472 0.0242736 0.999705i $$-0.492273\pi$$
0.0242736 + 0.999705i $$0.492273\pi$$
$$678$$ −27.4045 −1.05246
$$679$$ 1.87122 0.0718108
$$680$$ 0 0
$$681$$ −3.20359 −0.122762
$$682$$ −11.0748 −0.424076
$$683$$ −37.6441 −1.44041 −0.720206 0.693760i $$-0.755953\pi$$
−0.720206 + 0.693760i $$0.755953\pi$$
$$684$$ −0.740987 −0.0283323
$$685$$ 0 0
$$686$$ −1.46260 −0.0558423
$$687$$ 18.3088 0.698526
$$688$$ −44.7964 −1.70785
$$689$$ 18.7756 0.715293
$$690$$ 0 0
$$691$$ 14.3892 0.547393 0.273696 0.961816i $$-0.411754\pi$$
0.273696 + 0.961816i $$0.411754\pi$$
$$692$$ −1.71871 −0.0653357
$$693$$ −1.00000 −0.0379869
$$694$$ −33.0361 −1.25403
$$695$$ 0 0
$$696$$ −13.7306 −0.520456
$$697$$ −4.11915 −0.156024
$$698$$ −40.8525 −1.54629
$$699$$ −16.5872 −0.627387
$$700$$ 0 0
$$701$$ −39.2936 −1.48410 −0.742050 0.670345i $$-0.766146\pi$$
−0.742050 + 0.670345i $$0.766146\pi$$
$$702$$ −7.37883 −0.278496
$$703$$ 22.6170 0.853017
$$704$$ −7.36842 −0.277708
$$705$$ 0 0
$$706$$ −24.2028 −0.910885
$$707$$ 4.51803 0.169918
$$708$$ −1.10941 −0.0416941
$$709$$ −49.2430 −1.84936 −0.924680 0.380745i $$-0.875667\pi$$
−0.924680 + 0.380745i $$0.875667\pi$$
$$710$$ 0 0
$$711$$ 11.4432 0.429154
$$712$$ 32.2522 1.20870
$$713$$ 37.2936 1.39666
$$714$$ 9.31444 0.348584
$$715$$ 0 0
$$716$$ −0.778489 −0.0290935
$$717$$ −2.91623 −0.108909
$$718$$ −32.2217 −1.20250
$$719$$ −7.41344 −0.276475 −0.138237 0.990399i $$-0.544144\pi$$
−0.138237 + 0.990399i $$0.544144\pi$$
$$720$$ 0 0
$$721$$ 10.6468 0.396508
$$722$$ −13.6587 −0.508323
$$723$$ 6.09899 0.226824
$$724$$ 1.88914 0.0702095
$$725$$ 0 0
$$726$$ 1.46260 0.0542821
$$727$$ −18.9557 −0.703026 −0.351513 0.936183i $$-0.614333\pi$$
−0.351513 + 0.936183i $$0.614333\pi$$
$$728$$ −13.7306 −0.508889
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 66.9832 2.47746
$$732$$ 0.278388 0.0102895
$$733$$ 3.59283 0.132704 0.0663521 0.997796i $$-0.478864\pi$$
0.0663521 + 0.997796i $$0.478864\pi$$
$$734$$ −28.3505 −1.04644
$$735$$ 0 0
$$736$$ −3.87122 −0.142695
$$737$$ 8.76663 0.322923
$$738$$ 0.946021 0.0348235
$$739$$ −26.7756 −0.984956 −0.492478 0.870325i $$-0.663909\pi$$
−0.492478 + 0.870325i $$0.663909\pi$$
$$740$$ 0 0
$$741$$ −26.8567 −0.986604
$$742$$ 5.44322 0.199827
$$743$$ 33.8027 1.24010 0.620050 0.784562i $$-0.287112\pi$$
0.620050 + 0.784562i $$0.287112\pi$$
$$744$$ 20.6081 0.755528
$$745$$ 0 0
$$746$$ 42.7881 1.56658
$$747$$ −13.1648 −0.481676
$$748$$ −0.886447 −0.0324117
$$749$$ −15.9702 −0.583539
$$750$$ 0 0
$$751$$ 35.3955 1.29160 0.645800 0.763506i $$-0.276524\pi$$
0.645800 + 0.763506i $$0.276524\pi$$
$$752$$ 2.24445 0.0818468
$$753$$ −1.62262 −0.0591314
$$754$$ 37.2263 1.35570
$$755$$ 0 0
$$756$$ −0.139194 −0.00506244
$$757$$ 29.3442 1.06653 0.533267 0.845947i $$-0.320964\pi$$
0.533267 + 0.845947i $$0.320964\pi$$
$$758$$ −18.3220 −0.665483
$$759$$ −4.92520 −0.178773
$$760$$ 0 0
$$761$$ 12.9044 0.467783 0.233892 0.972263i $$-0.424854\pi$$
0.233892 + 0.972263i $$0.424854\pi$$
$$762$$ 3.33237 0.120719
$$763$$ 12.7756 0.462507
$$764$$ −1.31154 −0.0474500
$$765$$ 0 0
$$766$$ −25.7312 −0.929708
$$767$$ −40.2099 −1.45189
$$768$$ −3.32485 −0.119975
$$769$$ 9.78186 0.352743 0.176371 0.984324i $$-0.443564\pi$$
0.176371 + 0.984324i $$0.443564\pi$$
$$770$$ 0 0
$$771$$ −6.89541 −0.248332
$$772$$ 1.40987 0.0507422
$$773$$ −29.7223 −1.06904 −0.534518 0.845157i $$-0.679507\pi$$
−0.534518 + 0.845157i $$0.679507\pi$$
$$774$$ −15.3836 −0.552954
$$775$$ 0 0
$$776$$ 5.09273 0.182818
$$777$$ 4.24860 0.152418
$$778$$ −29.3836 −1.05345
$$779$$ 3.44322 0.123366
$$780$$ 0 0
$$781$$ 11.4432 0.409471
$$782$$ 45.8755 1.64050
$$783$$ −5.04502 −0.180294
$$784$$ −4.25901 −0.152108
$$785$$ 0 0
$$786$$ 5.85039 0.208677
$$787$$ −17.0242 −0.606847 −0.303423 0.952856i $$-0.598130\pi$$
−0.303423 + 0.952856i $$0.598130\pi$$
$$788$$ 0.314240 0.0111943
$$789$$ 5.08377 0.180987
$$790$$ 0 0
$$791$$ −18.7368 −0.666205
$$792$$ −2.72161 −0.0967083
$$793$$ 10.0900 0.358308
$$794$$ −51.3449 −1.82216
$$795$$ 0 0
$$796$$ 0.427995 0.0151699
$$797$$ 36.4287 1.29037 0.645185 0.764027i $$-0.276780\pi$$
0.645185 + 0.764027i $$0.276780\pi$$
$$798$$ −7.78600 −0.275622
$$799$$ −3.35609 −0.118730
$$800$$ 0 0
$$801$$ 11.8504 0.418713
$$802$$ −14.0000 −0.494357
$$803$$ −13.0450 −0.460349
$$804$$ 1.22026 0.0430354
$$805$$ 0 0
$$806$$ −55.8726 −1.96803
$$807$$ 0.886447 0.0312044
$$808$$ 12.2963 0.432583
$$809$$ 44.4882 1.56412 0.782062 0.623201i $$-0.214168\pi$$
0.782062 + 0.623201i $$0.214168\pi$$
$$810$$ 0 0
$$811$$ 7.65307 0.268736 0.134368 0.990932i $$-0.457100\pi$$
0.134368 + 0.990932i $$0.457100\pi$$
$$812$$ 0.702237 0.0246437
$$813$$ 25.3234 0.888131
$$814$$ −6.21400 −0.217800
$$815$$ 0 0
$$816$$ 27.1232 0.949501
$$817$$ −55.9917 −1.95890
$$818$$ −55.8421 −1.95247
$$819$$ −5.04502 −0.176287
$$820$$ 0 0
$$821$$ −44.3691 −1.54849 −0.774246 0.632885i $$-0.781871\pi$$
−0.774246 + 0.632885i $$0.781871\pi$$
$$822$$ 6.98477 0.243622
$$823$$ −6.61702 −0.230655 −0.115327 0.993328i $$-0.536792\pi$$
−0.115327 + 0.993328i $$0.536792\pi$$
$$824$$ 28.9765 1.00944
$$825$$ 0 0
$$826$$ −11.6572 −0.405607
$$827$$ −39.7126 −1.38094 −0.690472 0.723359i $$-0.742597\pi$$
−0.690472 + 0.723359i $$0.742597\pi$$
$$828$$ −0.685559 −0.0238248
$$829$$ −3.90997 −0.135799 −0.0678994 0.997692i $$-0.521630\pi$$
−0.0678994 + 0.997692i $$0.521630\pi$$
$$830$$ 0 0
$$831$$ −24.8269 −0.861235
$$832$$ −37.1738 −1.28877
$$833$$ 6.36842 0.220653
$$834$$ −22.5872 −0.782132
$$835$$ 0 0
$$836$$ 0.740987 0.0256276
$$837$$ 7.57201 0.261727
$$838$$ −10.4924 −0.362453
$$839$$ −9.58097 −0.330772 −0.165386 0.986229i $$-0.552887\pi$$
−0.165386 + 0.986229i $$0.552887\pi$$
$$840$$ 0 0
$$841$$ −3.54781 −0.122338
$$842$$ −22.1627 −0.763778
$$843$$ −1.90101 −0.0654741
$$844$$ −2.03875 −0.0701767
$$845$$ 0 0
$$846$$ 0.770774 0.0264998
$$847$$ 1.00000 0.0343604
$$848$$ 15.8504 0.544305
$$849$$ −22.3178 −0.765945
$$850$$ 0 0
$$851$$ 20.9252 0.717307
$$852$$ 1.59283 0.0545694
$$853$$ 14.5568 0.498415 0.249207 0.968450i $$-0.419830\pi$$
0.249207 + 0.968450i $$0.419830\pi$$
$$854$$ 2.92520 0.100098
$$855$$ 0 0
$$856$$ −43.4647 −1.48559
$$857$$ −10.4793 −0.357965 −0.178983 0.983852i $$-0.557281\pi$$
−0.178983 + 0.983852i $$0.557281\pi$$
$$858$$ 7.37883 0.251909
$$859$$ 2.88645 0.0984843 0.0492421 0.998787i $$-0.484319\pi$$
0.0492421 + 0.998787i $$0.484319\pi$$
$$860$$ 0 0
$$861$$ 0.646809 0.0220432
$$862$$ −8.13650 −0.277130
$$863$$ 20.5485 0.699479 0.349739 0.936847i $$-0.386270\pi$$
0.349739 + 0.936847i $$0.386270\pi$$
$$864$$ −0.786003 −0.0267404
$$865$$ 0 0
$$866$$ −37.5028 −1.27440
$$867$$ −23.5568 −0.800030
$$868$$ −1.05398 −0.0357744
$$869$$ −11.4432 −0.388185
$$870$$ 0 0
$$871$$ 44.2278 1.49860
$$872$$ 34.7702 1.17747
$$873$$ 1.87122 0.0633311
$$874$$ −38.3476 −1.29713
$$875$$ 0 0
$$876$$ −1.81579 −0.0613499
$$877$$ −59.1149 −1.99617 −0.998084 0.0618724i $$-0.980293\pi$$
−0.998084 + 0.0618724i $$0.980293\pi$$
$$878$$ −34.6212 −1.16841
$$879$$ 12.0900 0.407787
$$880$$ 0 0
$$881$$ 30.3982 1.02414 0.512071 0.858943i $$-0.328879\pi$$
0.512071 + 0.858943i $$0.328879\pi$$
$$882$$ −1.46260 −0.0492483
$$883$$ 35.6114 1.19842 0.599210 0.800592i $$-0.295481\pi$$
0.599210 + 0.800592i $$0.295481\pi$$
$$884$$ −4.47214 −0.150414
$$885$$ 0 0
$$886$$ −26.3713 −0.885962
$$887$$ −22.9736 −0.771377 −0.385689 0.922629i $$-0.626036\pi$$
−0.385689 + 0.922629i $$0.626036\pi$$
$$888$$ 11.5630 0.388030
$$889$$ 2.27839 0.0764147
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0.265856 0.00890153
$$893$$ 2.80538 0.0938784
$$894$$ −14.3941 −0.481409
$$895$$ 0 0
$$896$$ −12.3490 −0.412553
$$897$$ −24.8477 −0.829640
$$898$$ 51.1565 1.70712
$$899$$ −38.2009 −1.27407
$$900$$ 0 0
$$901$$ −23.7008 −0.789588
$$902$$ −0.946021 −0.0314991
$$903$$ −10.5180 −0.350018
$$904$$ −50.9944 −1.69605
$$905$$ 0 0
$$906$$ −6.03875 −0.200624
$$907$$ 57.1745 1.89845 0.949224 0.314602i $$-0.101871\pi$$
0.949224 + 0.314602i $$0.101871\pi$$
$$908$$ 0.445920 0.0147984
$$909$$ 4.51803 0.149853
$$910$$ 0 0
$$911$$ 6.82687 0.226184 0.113092 0.993584i $$-0.463924\pi$$
0.113092 + 0.993584i $$0.463924\pi$$
$$912$$ −22.6724 −0.750760
$$913$$ 13.1648 0.435692
$$914$$ 6.63428 0.219442
$$915$$ 0 0
$$916$$ −2.54848 −0.0842043
$$917$$ 4.00000 0.132092
$$918$$ 9.31444 0.307422
$$919$$ 12.0692 0.398126 0.199063 0.979987i $$-0.436210\pi$$
0.199063 + 0.979987i $$0.436210\pi$$
$$920$$ 0 0
$$921$$ −13.5928 −0.447899
$$922$$ 4.09003 0.134698
$$923$$ 57.7312 1.90025
$$924$$ 0.139194 0.00457915
$$925$$ 0 0
$$926$$ 56.1045 1.84371
$$927$$ 10.6468 0.349687
$$928$$ 3.96540 0.130171
$$929$$ 26.8954 0.882410 0.441205 0.897406i $$-0.354551\pi$$
0.441205 + 0.897406i $$0.354551\pi$$
$$930$$ 0 0
$$931$$ −5.32340 −0.174468
$$932$$ 2.30885 0.0756288
$$933$$ −8.00000 −0.261908
$$934$$ −29.9356 −0.979523
$$935$$ 0 0
$$936$$ −13.7306 −0.448798
$$937$$ 14.9944 0.489846 0.244923 0.969543i $$-0.421237\pi$$
0.244923 + 0.969543i $$0.421237\pi$$
$$938$$ 12.8221 0.418655
$$939$$ 14.9252 0.487065
$$940$$ 0 0
$$941$$ −30.1205 −0.981900 −0.490950 0.871188i $$-0.663350\pi$$
−0.490950 + 0.871188i $$0.663350\pi$$
$$942$$ 1.38365 0.0450817
$$943$$ 3.18566 0.103739
$$944$$ −33.9452 −1.10482
$$945$$ 0 0
$$946$$ 15.3836 0.500166
$$947$$ 17.3532 0.563903 0.281951 0.959429i $$-0.409018\pi$$
0.281951 + 0.959429i $$0.409018\pi$$
$$948$$ −1.59283 −0.0517327
$$949$$ −65.8123 −2.13636
$$950$$ 0 0
$$951$$ 3.97918 0.129034
$$952$$ 17.3324 0.561745
$$953$$ 2.14064 0.0693422 0.0346711 0.999399i $$-0.488962\pi$$
0.0346711 + 0.999399i $$0.488962\pi$$
$$954$$ 5.44322 0.176231
$$955$$ 0 0
$$956$$ 0.405923 0.0131285
$$957$$ 5.04502 0.163082
$$958$$ 16.9861 0.548796
$$959$$ 4.77559 0.154212
$$960$$ 0 0
$$961$$ 26.3353 0.849525
$$962$$ −31.3497 −1.01076
$$963$$ −15.9702 −0.514633
$$964$$ −0.848944 −0.0273427
$$965$$ 0 0
$$966$$ −7.20359 −0.231772
$$967$$ 1.53326 0.0493062 0.0246531 0.999696i $$-0.492152\pi$$
0.0246531 + 0.999696i $$0.492152\pi$$
$$968$$ 2.72161 0.0874759
$$969$$ 33.9017 1.08908
$$970$$ 0 0
$$971$$ −26.5574 −0.852269 −0.426135 0.904660i $$-0.640125\pi$$
−0.426135 + 0.904660i $$0.640125\pi$$
$$972$$ −0.139194 −0.00446465
$$973$$ −15.4432 −0.495087
$$974$$ 47.5041 1.52213
$$975$$ 0 0
$$976$$ 8.51803 0.272655
$$977$$ 55.9017 1.78845 0.894227 0.447615i $$-0.147726\pi$$
0.894227 + 0.447615i $$0.147726\pi$$
$$978$$ −12.8221 −0.410004
$$979$$ −11.8504 −0.378740
$$980$$ 0 0
$$981$$ 12.7756 0.407893
$$982$$ −39.0171 −1.24509
$$983$$ 53.0361 1.69159 0.845794 0.533510i $$-0.179127\pi$$
0.845794 + 0.533510i $$0.179127\pi$$
$$984$$ 1.76036 0.0561183
$$985$$ 0 0
$$986$$ −46.9915 −1.49651
$$987$$ 0.526989 0.0167743
$$988$$ 3.73829 0.118931
$$989$$ −51.8034 −1.64725
$$990$$ 0 0
$$991$$ 14.7362 0.468110 0.234055 0.972223i $$-0.424800\pi$$
0.234055 + 0.972223i $$0.424800\pi$$
$$992$$ −5.95162 −0.188964
$$993$$ −23.4432 −0.743948
$$994$$ 16.7368 0.530860
$$995$$ 0 0
$$996$$ 1.83247 0.0580640
$$997$$ 45.0665 1.42727 0.713635 0.700517i $$-0.247047\pi$$
0.713635 + 0.700517i $$0.247047\pi$$
$$998$$ 60.3262 1.90959
$$999$$ 4.24860 0.134420
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 5775.2.a.bp.1.2 3
5.4 even 2 231.2.a.e.1.2 3
15.14 odd 2 693.2.a.l.1.2 3
20.19 odd 2 3696.2.a.bo.1.2 3
35.34 odd 2 1617.2.a.t.1.2 3
55.54 odd 2 2541.2.a.bg.1.2 3
105.104 even 2 4851.2.a.bi.1.2 3
165.164 even 2 7623.2.a.cd.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.2 3 5.4 even 2
693.2.a.l.1.2 3 15.14 odd 2
1617.2.a.t.1.2 3 35.34 odd 2
2541.2.a.bg.1.2 3 55.54 odd 2
3696.2.a.bo.1.2 3 20.19 odd 2
4851.2.a.bi.1.2 3 105.104 even 2
5775.2.a.bp.1.2 3 1.1 even 1 trivial
7623.2.a.cd.1.2 3 165.164 even 2