# Properties

 Label 1617.2.a.r.1.1 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 1617.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.86081 q^{2} +1.00000 q^{3} +1.46260 q^{4} -1.32340 q^{5} -1.86081 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.86081 q^{2} +1.00000 q^{3} +1.46260 q^{4} -1.32340 q^{5} -1.86081 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.46260 q^{10} +1.00000 q^{11} +1.46260 q^{12} -0.398207 q^{13} -1.32340 q^{15} -4.78600 q^{16} +6.64681 q^{17} -1.86081 q^{18} -1.32340 q^{19} -1.93561 q^{20} -1.86081 q^{22} +1.00000 q^{24} -3.24860 q^{25} +0.740987 q^{26} +1.00000 q^{27} +1.47301 q^{29} +2.46260 q^{30} +4.79641 q^{31} +6.90582 q^{32} +1.00000 q^{33} -12.3684 q^{34} +1.46260 q^{36} -1.60179 q^{37} +2.46260 q^{38} -0.398207 q^{39} -1.32340 q^{40} -4.79641 q^{41} -4.79641 q^{43} +1.46260 q^{44} -1.32340 q^{45} -3.04502 q^{47} -4.78600 q^{48} +6.04502 q^{50} +6.64681 q^{51} -0.582418 q^{52} +8.64681 q^{53} -1.86081 q^{54} -1.32340 q^{55} -1.32340 q^{57} -2.74099 q^{58} +13.6918 q^{59} -1.93561 q^{60} +3.20359 q^{61} -8.92520 q^{62} -3.27839 q^{64} +0.526989 q^{65} -1.86081 q^{66} -10.6170 q^{67} +9.72161 q^{68} +15.9404 q^{71} +1.00000 q^{72} -1.45219 q^{73} +2.98062 q^{74} -3.24860 q^{75} -1.93561 q^{76} +0.740987 q^{78} +3.44322 q^{79} +6.33382 q^{80} +1.00000 q^{81} +8.92520 q^{82} +0.796415 q^{83} -8.79641 q^{85} +8.92520 q^{86} +1.47301 q^{87} +1.00000 q^{88} +10.6468 q^{89} +2.46260 q^{90} +4.79641 q^{93} +5.66618 q^{94} +1.75140 q^{95} +6.90582 q^{96} +12.2396 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 + 5 * q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 + 4 * q^15 - 4 * q^16 + 4 * q^17 + 4 * q^19 - 5 * q^20 + 3 * q^24 + 3 * q^25 + 11 * q^26 + 3 * q^27 + 6 * q^29 + 5 * q^30 + 8 * q^31 - 4 * q^32 + 3 * q^33 - 10 * q^34 + 2 * q^36 - 8 * q^37 + 5 * q^38 + 2 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 2 * q^44 + 4 * q^45 + 10 * q^47 - 4 * q^48 - q^50 + 4 * q^51 + 15 * q^52 + 10 * q^53 + 4 * q^55 + 4 * q^57 - 17 * q^58 + 6 * q^59 - 5 * q^60 + 16 * q^61 - 22 * q^62 - 21 * q^64 + 8 * q^67 + 18 * q^68 + 3 * q^72 + 2 * q^73 - 11 * q^74 + 3 * q^75 - 5 * q^76 + 11 * q^78 - 12 * q^79 + 15 * q^80 + 3 * q^81 + 22 * q^82 - 4 * q^83 - 20 * q^85 + 22 * q^86 + 6 * q^87 + 3 * q^88 + 16 * q^89 + 5 * q^90 + 8 * q^93 + 21 * q^94 + 18 * q^95 - 4 * q^96 + 8 * q^97 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.86081 −1.31579 −0.657894 0.753110i $$-0.728553\pi$$
−0.657894 + 0.753110i $$0.728553\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 1.46260 0.731299
$$5$$ −1.32340 −0.591844 −0.295922 0.955212i $$-0.595627\pi$$
−0.295922 + 0.955212i $$0.595627\pi$$
$$6$$ −1.86081 −0.759671
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.46260 0.778742
$$11$$ 1.00000 0.301511
$$12$$ 1.46260 0.422216
$$13$$ −0.398207 −0.110443 −0.0552214 0.998474i $$-0.517586\pi$$
−0.0552214 + 0.998474i $$0.517586\pi$$
$$14$$ 0 0
$$15$$ −1.32340 −0.341702
$$16$$ −4.78600 −1.19650
$$17$$ 6.64681 1.61209 0.806044 0.591856i $$-0.201604\pi$$
0.806044 + 0.591856i $$0.201604\pi$$
$$18$$ −1.86081 −0.438596
$$19$$ −1.32340 −0.303610 −0.151805 0.988410i $$-0.548509\pi$$
−0.151805 + 0.988410i $$0.548509\pi$$
$$20$$ −1.93561 −0.432815
$$21$$ 0 0
$$22$$ −1.86081 −0.396725
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −3.24860 −0.649720
$$26$$ 0.740987 0.145319
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 1.47301 0.273531 0.136766 0.990603i $$-0.456329\pi$$
0.136766 + 0.990603i $$0.456329\pi$$
$$30$$ 2.46260 0.449607
$$31$$ 4.79641 0.861462 0.430731 0.902480i $$-0.358256\pi$$
0.430731 + 0.902480i $$0.358256\pi$$
$$32$$ 6.90582 1.22079
$$33$$ 1.00000 0.174078
$$34$$ −12.3684 −2.12117
$$35$$ 0 0
$$36$$ 1.46260 0.243766
$$37$$ −1.60179 −0.263333 −0.131667 0.991294i $$-0.542033\pi$$
−0.131667 + 0.991294i $$0.542033\pi$$
$$38$$ 2.46260 0.399486
$$39$$ −0.398207 −0.0637642
$$40$$ −1.32340 −0.209249
$$41$$ −4.79641 −0.749074 −0.374537 0.927212i $$-0.622198\pi$$
−0.374537 + 0.927212i $$0.622198\pi$$
$$42$$ 0 0
$$43$$ −4.79641 −0.731446 −0.365723 0.930724i $$-0.619178\pi$$
−0.365723 + 0.930724i $$0.619178\pi$$
$$44$$ 1.46260 0.220495
$$45$$ −1.32340 −0.197281
$$46$$ 0 0
$$47$$ −3.04502 −0.444161 −0.222081 0.975028i $$-0.571285\pi$$
−0.222081 + 0.975028i $$0.571285\pi$$
$$48$$ −4.78600 −0.690800
$$49$$ 0 0
$$50$$ 6.04502 0.854894
$$51$$ 6.64681 0.930739
$$52$$ −0.582418 −0.0807668
$$53$$ 8.64681 1.18773 0.593865 0.804565i $$-0.297601\pi$$
0.593865 + 0.804565i $$0.297601\pi$$
$$54$$ −1.86081 −0.253224
$$55$$ −1.32340 −0.178448
$$56$$ 0 0
$$57$$ −1.32340 −0.175289
$$58$$ −2.74099 −0.359909
$$59$$ 13.6918 1.78252 0.891262 0.453489i $$-0.149821\pi$$
0.891262 + 0.453489i $$0.149821\pi$$
$$60$$ −1.93561 −0.249886
$$61$$ 3.20359 0.410177 0.205089 0.978743i $$-0.434252\pi$$
0.205089 + 0.978743i $$0.434252\pi$$
$$62$$ −8.92520 −1.13350
$$63$$ 0 0
$$64$$ −3.27839 −0.409799
$$65$$ 0.526989 0.0653650
$$66$$ −1.86081 −0.229049
$$67$$ −10.6170 −1.29708 −0.648538 0.761182i $$-0.724619\pi$$
−0.648538 + 0.761182i $$0.724619\pi$$
$$68$$ 9.72161 1.17892
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 15.9404 1.89178 0.945890 0.324487i $$-0.105192\pi$$
0.945890 + 0.324487i $$0.105192\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −1.45219 −0.169966 −0.0849828 0.996382i $$-0.527084\pi$$
−0.0849828 + 0.996382i $$0.527084\pi$$
$$74$$ 2.98062 0.346491
$$75$$ −3.24860 −0.375116
$$76$$ −1.93561 −0.222030
$$77$$ 0 0
$$78$$ 0.740987 0.0839002
$$79$$ 3.44322 0.387393 0.193696 0.981062i $$-0.437952\pi$$
0.193696 + 0.981062i $$0.437952\pi$$
$$80$$ 6.33382 0.708142
$$81$$ 1.00000 0.111111
$$82$$ 8.92520 0.985623
$$83$$ 0.796415 0.0874179 0.0437089 0.999044i $$-0.486083\pi$$
0.0437089 + 0.999044i $$0.486083\pi$$
$$84$$ 0 0
$$85$$ −8.79641 −0.954105
$$86$$ 8.92520 0.962429
$$87$$ 1.47301 0.157923
$$88$$ 1.00000 0.106600
$$89$$ 10.6468 1.12856 0.564280 0.825584i $$-0.309154\pi$$
0.564280 + 0.825584i $$0.309154\pi$$
$$90$$ 2.46260 0.259581
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.79641 0.497365
$$94$$ 5.66618 0.584422
$$95$$ 1.75140 0.179690
$$96$$ 6.90582 0.704822
$$97$$ 12.2396 1.24275 0.621373 0.783515i $$-0.286575\pi$$
0.621373 + 0.783515i $$0.286575\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ −4.75140 −0.475140
$$101$$ −16.4972 −1.64153 −0.820766 0.571264i $$-0.806453\pi$$
−0.820766 + 0.571264i $$0.806453\pi$$
$$102$$ −12.3684 −1.22466
$$103$$ 13.8504 1.36472 0.682360 0.731016i $$-0.260954\pi$$
0.682360 + 0.731016i $$0.260954\pi$$
$$104$$ −0.398207 −0.0390475
$$105$$ 0 0
$$106$$ −16.0900 −1.56280
$$107$$ −5.69182 −0.550249 −0.275125 0.961409i $$-0.588719\pi$$
−0.275125 + 0.961409i $$0.588719\pi$$
$$108$$ 1.46260 0.140739
$$109$$ −4.14961 −0.397460 −0.198730 0.980054i $$-0.563682\pi$$
−0.198730 + 0.980054i $$0.563682\pi$$
$$110$$ 2.46260 0.234800
$$111$$ −1.60179 −0.152035
$$112$$ 0 0
$$113$$ 0.946021 0.0889942 0.0444971 0.999010i $$-0.485831\pi$$
0.0444971 + 0.999010i $$0.485831\pi$$
$$114$$ 2.46260 0.230643
$$115$$ 0 0
$$116$$ 2.15442 0.200033
$$117$$ −0.398207 −0.0368143
$$118$$ −25.4778 −2.34542
$$119$$ 0 0
$$120$$ −1.32340 −0.120810
$$121$$ 1.00000 0.0909091
$$122$$ −5.96125 −0.539706
$$123$$ −4.79641 −0.432478
$$124$$ 7.01523 0.629986
$$125$$ 10.9162 0.976378
$$126$$ 0 0
$$127$$ −4.55678 −0.404349 −0.202174 0.979350i $$-0.564801\pi$$
−0.202174 + 0.979350i $$0.564801\pi$$
$$128$$ −7.71120 −0.681580
$$129$$ −4.79641 −0.422301
$$130$$ −0.980625 −0.0860065
$$131$$ −1.05398 −0.0920866 −0.0460433 0.998939i $$-0.514661\pi$$
−0.0460433 + 0.998939i $$0.514661\pi$$
$$132$$ 1.46260 0.127303
$$133$$ 0 0
$$134$$ 19.7562 1.70668
$$135$$ −1.32340 −0.113901
$$136$$ 6.64681 0.569959
$$137$$ 1.44322 0.123303 0.0616514 0.998098i $$-0.480363\pi$$
0.0616514 + 0.998098i $$0.480363\pi$$
$$138$$ 0 0
$$139$$ 18.5872 1.57655 0.788274 0.615324i $$-0.210975\pi$$
0.788274 + 0.615324i $$0.210975\pi$$
$$140$$ 0 0
$$141$$ −3.04502 −0.256437
$$142$$ −29.6620 −2.48918
$$143$$ −0.398207 −0.0332998
$$144$$ −4.78600 −0.398834
$$145$$ −1.94939 −0.161888
$$146$$ 2.70224 0.223639
$$147$$ 0 0
$$148$$ −2.34278 −0.192575
$$149$$ 3.88018 0.317877 0.158938 0.987289i $$-0.449193\pi$$
0.158938 + 0.987289i $$0.449193\pi$$
$$150$$ 6.04502 0.493573
$$151$$ 10.6468 0.866425 0.433212 0.901292i $$-0.357380\pi$$
0.433212 + 0.901292i $$0.357380\pi$$
$$152$$ −1.32340 −0.107342
$$153$$ 6.64681 0.537363
$$154$$ 0 0
$$155$$ −6.34760 −0.509851
$$156$$ −0.582418 −0.0466307
$$157$$ 16.7368 1.33575 0.667873 0.744276i $$-0.267205\pi$$
0.667873 + 0.744276i $$0.267205\pi$$
$$158$$ −6.40717 −0.509727
$$159$$ 8.64681 0.685737
$$160$$ −9.13919 −0.722517
$$161$$ 0 0
$$162$$ −1.86081 −0.146199
$$163$$ 2.67660 0.209647 0.104824 0.994491i $$-0.466572\pi$$
0.104824 + 0.994491i $$0.466572\pi$$
$$164$$ −7.01523 −0.547797
$$165$$ −1.32340 −0.103027
$$166$$ −1.48197 −0.115023
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −12.8414 −0.987802
$$170$$ 16.3684 1.25540
$$171$$ −1.32340 −0.101203
$$172$$ −7.01523 −0.534906
$$173$$ 12.7368 0.968364 0.484182 0.874967i $$-0.339117\pi$$
0.484182 + 0.874967i $$0.339117\pi$$
$$174$$ −2.74099 −0.207794
$$175$$ 0 0
$$176$$ −4.78600 −0.360759
$$177$$ 13.6918 1.02914
$$178$$ −19.8116 −1.48495
$$179$$ 5.85039 0.437279 0.218639 0.975806i $$-0.429838\pi$$
0.218639 + 0.975806i $$0.429838\pi$$
$$180$$ −1.93561 −0.144272
$$181$$ 16.7964 1.24847 0.624234 0.781238i $$-0.285411\pi$$
0.624234 + 0.781238i $$0.285411\pi$$
$$182$$ 0 0
$$183$$ 3.20359 0.236816
$$184$$ 0 0
$$185$$ 2.11982 0.155852
$$186$$ −8.92520 −0.654427
$$187$$ 6.64681 0.486063
$$188$$ −4.45364 −0.324815
$$189$$ 0 0
$$190$$ −3.25901 −0.236434
$$191$$ −16.7368 −1.21104 −0.605518 0.795832i $$-0.707034\pi$$
−0.605518 + 0.795832i $$0.707034\pi$$
$$192$$ −3.27839 −0.236597
$$193$$ 9.94043 0.715527 0.357764 0.933812i $$-0.383539\pi$$
0.357764 + 0.933812i $$0.383539\pi$$
$$194$$ −22.7756 −1.63519
$$195$$ 0.526989 0.0377385
$$196$$ 0 0
$$197$$ 15.8504 1.12929 0.564647 0.825333i $$-0.309012\pi$$
0.564647 + 0.825333i $$0.309012\pi$$
$$198$$ −1.86081 −0.132242
$$199$$ −6.38924 −0.452922 −0.226461 0.974020i $$-0.572716\pi$$
−0.226461 + 0.974020i $$0.572716\pi$$
$$200$$ −3.24860 −0.229711
$$201$$ −10.6170 −0.748867
$$202$$ 30.6981 2.15991
$$203$$ 0 0
$$204$$ 9.72161 0.680649
$$205$$ 6.34760 0.443335
$$206$$ −25.7729 −1.79568
$$207$$ 0 0
$$208$$ 1.90582 0.132145
$$209$$ −1.32340 −0.0915418
$$210$$ 0 0
$$211$$ −26.0900 −1.79611 −0.898056 0.439881i $$-0.855021\pi$$
−0.898056 + 0.439881i $$0.855021\pi$$
$$212$$ 12.6468 0.868586
$$213$$ 15.9404 1.09222
$$214$$ 10.5914 0.724012
$$215$$ 6.34760 0.432902
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 7.72161 0.522974
$$219$$ −1.45219 −0.0981297
$$220$$ −1.93561 −0.130499
$$221$$ −2.64681 −0.178044
$$222$$ 2.98062 0.200046
$$223$$ 18.0900 1.21140 0.605699 0.795694i $$-0.292894\pi$$
0.605699 + 0.795694i $$0.292894\pi$$
$$224$$ 0 0
$$225$$ −3.24860 −0.216573
$$226$$ −1.76036 −0.117098
$$227$$ 28.1801 1.87038 0.935188 0.354151i $$-0.115230\pi$$
0.935188 + 0.354151i $$0.115230\pi$$
$$228$$ −1.93561 −0.128189
$$229$$ −8.00000 −0.528655 −0.264327 0.964433i $$-0.585150\pi$$
−0.264327 + 0.964433i $$0.585150\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.47301 0.0967079
$$233$$ −21.4432 −1.40479 −0.702396 0.711786i $$-0.747886\pi$$
−0.702396 + 0.711786i $$0.747886\pi$$
$$234$$ 0.740987 0.0484398
$$235$$ 4.02979 0.262874
$$236$$ 20.0256 1.30356
$$237$$ 3.44322 0.223661
$$238$$ 0 0
$$239$$ 11.5422 0.746604 0.373302 0.927710i $$-0.378226\pi$$
0.373302 + 0.927710i $$0.378226\pi$$
$$240$$ 6.33382 0.408846
$$241$$ −12.9550 −0.834504 −0.417252 0.908791i $$-0.637007\pi$$
−0.417252 + 0.908791i $$0.637007\pi$$
$$242$$ −1.86081 −0.119617
$$243$$ 1.00000 0.0641500
$$244$$ 4.68556 0.299962
$$245$$ 0 0
$$246$$ 8.92520 0.569050
$$247$$ 0.526989 0.0335315
$$248$$ 4.79641 0.304573
$$249$$ 0.796415 0.0504707
$$250$$ −20.3130 −1.28471
$$251$$ 28.8954 1.82386 0.911931 0.410343i $$-0.134591\pi$$
0.911931 + 0.410343i $$0.134591\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.47928 0.532037
$$255$$ −8.79641 −0.550853
$$256$$ 20.9058 1.30661
$$257$$ 12.2278 0.762748 0.381374 0.924421i $$-0.375451\pi$$
0.381374 + 0.924421i $$0.375451\pi$$
$$258$$ 8.92520 0.555658
$$259$$ 0 0
$$260$$ 0.770774 0.0478014
$$261$$ 1.47301 0.0911771
$$262$$ 1.96125 0.121166
$$263$$ −0.954984 −0.0588868 −0.0294434 0.999566i $$-0.509373\pi$$
−0.0294434 + 0.999566i $$0.509373\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −11.4432 −0.702952
$$266$$ 0 0
$$267$$ 10.6468 0.651574
$$268$$ −15.5284 −0.948550
$$269$$ −20.2396 −1.23403 −0.617016 0.786950i $$-0.711659\pi$$
−0.617016 + 0.786950i $$0.711659\pi$$
$$270$$ 2.46260 0.149869
$$271$$ −16.4674 −1.00032 −0.500162 0.865932i $$-0.666726\pi$$
−0.500162 + 0.865932i $$0.666726\pi$$
$$272$$ −31.8116 −1.92886
$$273$$ 0 0
$$274$$ −2.68556 −0.162241
$$275$$ −3.24860 −0.195898
$$276$$ 0 0
$$277$$ 18.1801 1.09233 0.546167 0.837676i $$-0.316086\pi$$
0.546167 + 0.837676i $$0.316086\pi$$
$$278$$ −34.5872 −2.07440
$$279$$ 4.79641 0.287154
$$280$$ 0 0
$$281$$ 24.3178 1.45068 0.725339 0.688391i $$-0.241683\pi$$
0.725339 + 0.688391i $$0.241683\pi$$
$$282$$ 5.66618 0.337416
$$283$$ −21.5035 −1.27825 −0.639124 0.769103i $$-0.720703\pi$$
−0.639124 + 0.769103i $$0.720703\pi$$
$$284$$ 23.3144 1.38346
$$285$$ 1.75140 0.103744
$$286$$ 0.740987 0.0438155
$$287$$ 0 0
$$288$$ 6.90582 0.406929
$$289$$ 27.1801 1.59883
$$290$$ 3.62743 0.213010
$$291$$ 12.2396 0.717500
$$292$$ −2.12397 −0.124296
$$293$$ 5.59283 0.326737 0.163368 0.986565i $$-0.447764\pi$$
0.163368 + 0.986565i $$0.447764\pi$$
$$294$$ 0 0
$$295$$ −18.1198 −1.05498
$$296$$ −1.60179 −0.0931023
$$297$$ 1.00000 0.0580259
$$298$$ −7.22026 −0.418259
$$299$$ 0 0
$$300$$ −4.75140 −0.274322
$$301$$ 0 0
$$302$$ −19.8116 −1.14003
$$303$$ −16.4972 −0.947740
$$304$$ 6.33382 0.363269
$$305$$ −4.23964 −0.242761
$$306$$ −12.3684 −0.707056
$$307$$ −10.5872 −0.604245 −0.302123 0.953269i $$-0.597695\pi$$
−0.302123 + 0.953269i $$0.597695\pi$$
$$308$$ 0 0
$$309$$ 13.8504 0.787921
$$310$$ 11.8116 0.670856
$$311$$ 5.59283 0.317140 0.158570 0.987348i $$-0.449312\pi$$
0.158570 + 0.987348i $$0.449312\pi$$
$$312$$ −0.398207 −0.0225441
$$313$$ 1.90997 0.107958 0.0539789 0.998542i $$-0.482810\pi$$
0.0539789 + 0.998542i $$0.482810\pi$$
$$314$$ −31.1440 −1.75756
$$315$$ 0 0
$$316$$ 5.03605 0.283300
$$317$$ 9.44322 0.530384 0.265192 0.964196i $$-0.414565\pi$$
0.265192 + 0.964196i $$0.414565\pi$$
$$318$$ −16.0900 −0.902284
$$319$$ 1.47301 0.0824728
$$320$$ 4.33863 0.242537
$$321$$ −5.69182 −0.317687
$$322$$ 0 0
$$323$$ −8.79641 −0.489446
$$324$$ 1.46260 0.0812555
$$325$$ 1.29362 0.0717570
$$326$$ −4.98062 −0.275851
$$327$$ −4.14961 −0.229474
$$328$$ −4.79641 −0.264838
$$329$$ 0 0
$$330$$ 2.46260 0.135562
$$331$$ −20.4793 −1.12564 −0.562821 0.826579i $$-0.690284\pi$$
−0.562821 + 0.826579i $$0.690284\pi$$
$$332$$ 1.16484 0.0639286
$$333$$ −1.60179 −0.0877777
$$334$$ −29.7729 −1.62910
$$335$$ 14.0506 0.767667
$$336$$ 0 0
$$337$$ 17.3836 0.946948 0.473474 0.880808i $$-0.343000\pi$$
0.473474 + 0.880808i $$0.343000\pi$$
$$338$$ 23.8954 1.29974
$$339$$ 0.946021 0.0513808
$$340$$ −12.8656 −0.697736
$$341$$ 4.79641 0.259740
$$342$$ 2.46260 0.133162
$$343$$ 0 0
$$344$$ −4.79641 −0.258605
$$345$$ 0 0
$$346$$ −23.7008 −1.27416
$$347$$ −30.5872 −1.64201 −0.821004 0.570922i $$-0.806586\pi$$
−0.821004 + 0.570922i $$0.806586\pi$$
$$348$$ 2.15442 0.115489
$$349$$ −32.8775 −1.75989 −0.879946 0.475074i $$-0.842421\pi$$
−0.879946 + 0.475074i $$0.842421\pi$$
$$350$$ 0 0
$$351$$ −0.398207 −0.0212547
$$352$$ 6.90582 0.368082
$$353$$ −33.0063 −1.75675 −0.878373 0.477976i $$-0.841371\pi$$
−0.878373 + 0.477976i $$0.841371\pi$$
$$354$$ −25.4778 −1.35413
$$355$$ −21.0956 −1.11964
$$356$$ 15.5720 0.825315
$$357$$ 0 0
$$358$$ −10.8864 −0.575367
$$359$$ −19.7008 −1.03977 −0.519884 0.854237i $$-0.674025\pi$$
−0.519884 + 0.854237i $$0.674025\pi$$
$$360$$ −1.32340 −0.0697495
$$361$$ −17.2486 −0.907821
$$362$$ −31.2549 −1.64272
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 1.92183 0.100593
$$366$$ −5.96125 −0.311600
$$367$$ 31.1261 1.62477 0.812384 0.583123i $$-0.198169\pi$$
0.812384 + 0.583123i $$0.198169\pi$$
$$368$$ 0 0
$$369$$ −4.79641 −0.249691
$$370$$ −3.94457 −0.205069
$$371$$ 0 0
$$372$$ 7.01523 0.363723
$$373$$ −8.94602 −0.463207 −0.231604 0.972810i $$-0.574397\pi$$
−0.231604 + 0.972810i $$0.574397\pi$$
$$374$$ −12.3684 −0.639556
$$375$$ 10.9162 0.563712
$$376$$ −3.04502 −0.157035
$$377$$ −0.586564 −0.0302096
$$378$$ 0 0
$$379$$ 4.20985 0.216246 0.108123 0.994138i $$-0.465516\pi$$
0.108123 + 0.994138i $$0.465516\pi$$
$$380$$ 2.56159 0.131407
$$381$$ −4.55678 −0.233451
$$382$$ 31.1440 1.59347
$$383$$ −14.5872 −0.745373 −0.372686 0.927957i $$-0.621563\pi$$
−0.372686 + 0.927957i $$0.621563\pi$$
$$384$$ −7.71120 −0.393511
$$385$$ 0 0
$$386$$ −18.4972 −0.941483
$$387$$ −4.79641 −0.243815
$$388$$ 17.9017 0.908820
$$389$$ −30.4793 −1.54536 −0.772680 0.634795i $$-0.781084\pi$$
−0.772680 + 0.634795i $$0.781084\pi$$
$$390$$ −0.980625 −0.0496559
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −1.05398 −0.0531662
$$394$$ −29.4945 −1.48591
$$395$$ −4.55678 −0.229276
$$396$$ 1.46260 0.0734983
$$397$$ 10.9044 0.547275 0.273637 0.961833i $$-0.411773\pi$$
0.273637 + 0.961833i $$0.411773\pi$$
$$398$$ 11.8891 0.595949
$$399$$ 0 0
$$400$$ 15.5478 0.777391
$$401$$ 12.4072 0.619585 0.309792 0.950804i $$-0.399740\pi$$
0.309792 + 0.950804i $$0.399740\pi$$
$$402$$ 19.7562 0.985350
$$403$$ −1.90997 −0.0951423
$$404$$ −24.1288 −1.20045
$$405$$ −1.32340 −0.0657605
$$406$$ 0 0
$$407$$ −1.60179 −0.0793979
$$408$$ 6.64681 0.329066
$$409$$ −31.3836 −1.55182 −0.775911 0.630843i $$-0.782709\pi$$
−0.775911 + 0.630843i $$0.782709\pi$$
$$410$$ −11.8116 −0.583336
$$411$$ 1.44322 0.0711890
$$412$$ 20.2576 0.998019
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.05398 −0.0517378
$$416$$ −2.74995 −0.134827
$$417$$ 18.5872 0.910221
$$418$$ 2.46260 0.120450
$$419$$ −30.1711 −1.47395 −0.736977 0.675917i $$-0.763748\pi$$
−0.736977 + 0.675917i $$0.763748\pi$$
$$420$$ 0 0
$$421$$ −21.2847 −1.03735 −0.518675 0.854971i $$-0.673575\pi$$
−0.518675 + 0.854971i $$0.673575\pi$$
$$422$$ 48.5485 2.36330
$$423$$ −3.04502 −0.148054
$$424$$ 8.64681 0.419926
$$425$$ −21.5928 −1.04741
$$426$$ −29.6620 −1.43713
$$427$$ 0 0
$$428$$ −8.32485 −0.402397
$$429$$ −0.398207 −0.0192256
$$430$$ −11.8116 −0.569608
$$431$$ −14.7279 −0.709417 −0.354708 0.934977i $$-0.615420\pi$$
−0.354708 + 0.934977i $$0.615420\pi$$
$$432$$ −4.78600 −0.230267
$$433$$ 14.1496 0.679987 0.339993 0.940428i $$-0.389575\pi$$
0.339993 + 0.940428i $$0.389575\pi$$
$$434$$ 0 0
$$435$$ −1.94939 −0.0934660
$$436$$ −6.06921 −0.290662
$$437$$ 0 0
$$438$$ 2.70224 0.129118
$$439$$ 26.0007 1.24094 0.620472 0.784228i $$-0.286941\pi$$
0.620472 + 0.784228i $$0.286941\pi$$
$$440$$ −1.32340 −0.0630908
$$441$$ 0 0
$$442$$ 4.92520 0.234268
$$443$$ −32.9765 −1.56676 −0.783380 0.621543i $$-0.786506\pi$$
−0.783380 + 0.621543i $$0.786506\pi$$
$$444$$ −2.34278 −0.111183
$$445$$ −14.0900 −0.667932
$$446$$ −33.6620 −1.59394
$$447$$ 3.88018 0.183526
$$448$$ 0 0
$$449$$ 4.96395 0.234263 0.117132 0.993116i $$-0.462630\pi$$
0.117132 + 0.993116i $$0.462630\pi$$
$$450$$ 6.04502 0.284965
$$451$$ −4.79641 −0.225854
$$452$$ 1.38365 0.0650814
$$453$$ 10.6468 0.500231
$$454$$ −52.4376 −2.46102
$$455$$ 0 0
$$456$$ −1.32340 −0.0619741
$$457$$ −22.1980 −1.03838 −0.519189 0.854659i $$-0.673766\pi$$
−0.519189 + 0.854659i $$0.673766\pi$$
$$458$$ 14.8864 0.695598
$$459$$ 6.64681 0.310246
$$460$$ 0 0
$$461$$ −12.5389 −0.583993 −0.291996 0.956419i $$-0.594319\pi$$
−0.291996 + 0.956419i $$0.594319\pi$$
$$462$$ 0 0
$$463$$ 17.5214 0.814288 0.407144 0.913364i $$-0.366525\pi$$
0.407144 + 0.913364i $$0.366525\pi$$
$$464$$ −7.04983 −0.327280
$$465$$ −6.34760 −0.294363
$$466$$ 39.9017 1.84841
$$467$$ −11.7819 −0.545199 −0.272600 0.962128i $$-0.587883\pi$$
−0.272600 + 0.962128i $$0.587883\pi$$
$$468$$ −0.582418 −0.0269223
$$469$$ 0 0
$$470$$ −7.49865 −0.345887
$$471$$ 16.7368 0.771193
$$472$$ 13.6918 0.630217
$$473$$ −4.79641 −0.220539
$$474$$ −6.40717 −0.294291
$$475$$ 4.29921 0.197261
$$476$$ 0 0
$$477$$ 8.64681 0.395910
$$478$$ −21.4778 −0.982373
$$479$$ −5.03605 −0.230103 −0.115052 0.993360i $$-0.536703\pi$$
−0.115052 + 0.993360i $$0.536703\pi$$
$$480$$ −9.13919 −0.417145
$$481$$ 0.637846 0.0290833
$$482$$ 24.1067 1.09803
$$483$$ 0 0
$$484$$ 1.46260 0.0664817
$$485$$ −16.1980 −0.735513
$$486$$ −1.86081 −0.0844079
$$487$$ 20.1801 0.914446 0.457223 0.889352i $$-0.348844\pi$$
0.457223 + 0.889352i $$0.348844\pi$$
$$488$$ 3.20359 0.145019
$$489$$ 2.67660 0.121040
$$490$$ 0 0
$$491$$ −25.3926 −1.14595 −0.572976 0.819572i $$-0.694211\pi$$
−0.572976 + 0.819572i $$0.694211\pi$$
$$492$$ −7.01523 −0.316271
$$493$$ 9.79082 0.440956
$$494$$ −0.980625 −0.0441204
$$495$$ −1.32340 −0.0594826
$$496$$ −22.9557 −1.03074
$$497$$ 0 0
$$498$$ −1.48197 −0.0664088
$$499$$ 14.9162 0.667742 0.333871 0.942619i $$-0.391645\pi$$
0.333871 + 0.942619i $$0.391645\pi$$
$$500$$ 15.9661 0.714024
$$501$$ 16.0000 0.714827
$$502$$ −53.7687 −2.39982
$$503$$ −28.9765 −1.29200 −0.645999 0.763339i $$-0.723559\pi$$
−0.645999 + 0.763339i $$0.723559\pi$$
$$504$$ 0 0
$$505$$ 21.8325 0.971532
$$506$$ 0 0
$$507$$ −12.8414 −0.570308
$$508$$ −6.66473 −0.295700
$$509$$ 1.05398 0.0467168 0.0233584 0.999727i $$-0.492564\pi$$
0.0233584 + 0.999727i $$0.492564\pi$$
$$510$$ 16.3684 0.724806
$$511$$ 0 0
$$512$$ −23.4793 −1.03765
$$513$$ −1.32340 −0.0584297
$$514$$ −22.7535 −1.00361
$$515$$ −18.3297 −0.807702
$$516$$ −7.01523 −0.308828
$$517$$ −3.04502 −0.133920
$$518$$ 0 0
$$519$$ 12.7368 0.559085
$$520$$ 0.526989 0.0231100
$$521$$ 14.2999 0.626489 0.313245 0.949672i $$-0.398584\pi$$
0.313245 + 0.949672i $$0.398584\pi$$
$$522$$ −2.74099 −0.119970
$$523$$ −34.8567 −1.52418 −0.762088 0.647474i $$-0.775825\pi$$
−0.762088 + 0.647474i $$0.775825\pi$$
$$524$$ −1.54155 −0.0673428
$$525$$ 0 0
$$526$$ 1.77704 0.0774826
$$527$$ 31.8809 1.38875
$$528$$ −4.78600 −0.208284
$$529$$ −23.0000 −1.00000
$$530$$ 21.2936 0.924936
$$531$$ 13.6918 0.594175
$$532$$ 0 0
$$533$$ 1.90997 0.0827299
$$534$$ −19.8116 −0.857334
$$535$$ 7.53258 0.325662
$$536$$ −10.6170 −0.458585
$$537$$ 5.85039 0.252463
$$538$$ 37.6620 1.62373
$$539$$ 0 0
$$540$$ −1.93561 −0.0832953
$$541$$ 39.2161 1.68603 0.843016 0.537888i $$-0.180778\pi$$
0.843016 + 0.537888i $$0.180778\pi$$
$$542$$ 30.6427 1.31622
$$543$$ 16.7964 0.720803
$$544$$ 45.9017 1.96802
$$545$$ 5.49161 0.235235
$$546$$ 0 0
$$547$$ −38.5277 −1.64732 −0.823662 0.567081i $$-0.808073\pi$$
−0.823662 + 0.567081i $$0.808073\pi$$
$$548$$ 2.11086 0.0901713
$$549$$ 3.20359 0.136726
$$550$$ 6.04502 0.257760
$$551$$ −1.94939 −0.0830467
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −33.8296 −1.43728
$$555$$ 2.11982 0.0899813
$$556$$ 27.1857 1.15293
$$557$$ 43.7610 1.85421 0.927107 0.374796i $$-0.122287\pi$$
0.927107 + 0.374796i $$0.122287\pi$$
$$558$$ −8.92520 −0.377834
$$559$$ 1.90997 0.0807830
$$560$$ 0 0
$$561$$ 6.64681 0.280628
$$562$$ −45.2507 −1.90879
$$563$$ −19.9821 −0.842144 −0.421072 0.907027i $$-0.638346\pi$$
−0.421072 + 0.907027i $$0.638346\pi$$
$$564$$ −4.45364 −0.187532
$$565$$ −1.25197 −0.0526707
$$566$$ 40.0138 1.68190
$$567$$ 0 0
$$568$$ 15.9404 0.668845
$$569$$ 31.7312 1.33024 0.665121 0.746735i $$-0.268380\pi$$
0.665121 + 0.746735i $$0.268380\pi$$
$$570$$ −3.25901 −0.136505
$$571$$ −10.9460 −0.458077 −0.229038 0.973417i $$-0.573558\pi$$
−0.229038 + 0.973417i $$0.573558\pi$$
$$572$$ −0.582418 −0.0243521
$$573$$ −16.7368 −0.699192
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −3.27839 −0.136600
$$577$$ −3.24523 −0.135101 −0.0675504 0.997716i $$-0.521518\pi$$
−0.0675504 + 0.997716i $$0.521518\pi$$
$$578$$ −50.5768 −2.10372
$$579$$ 9.94043 0.413110
$$580$$ −2.85117 −0.118388
$$581$$ 0 0
$$582$$ −22.7756 −0.944079
$$583$$ 8.64681 0.358114
$$584$$ −1.45219 −0.0600919
$$585$$ 0.526989 0.0217883
$$586$$ −10.4072 −0.429916
$$587$$ 26.7875 1.10564 0.552818 0.833302i $$-0.313552\pi$$
0.552818 + 0.833302i $$0.313552\pi$$
$$588$$ 0 0
$$589$$ −6.34760 −0.261548
$$590$$ 33.7175 1.38813
$$591$$ 15.8504 0.651998
$$592$$ 7.66618 0.315078
$$593$$ 32.2396 1.32392 0.661962 0.749538i $$-0.269724\pi$$
0.661962 + 0.749538i $$0.269724\pi$$
$$594$$ −1.86081 −0.0763498
$$595$$ 0 0
$$596$$ 5.67515 0.232463
$$597$$ −6.38924 −0.261494
$$598$$ 0 0
$$599$$ −7.14401 −0.291896 −0.145948 0.989292i $$-0.546623\pi$$
−0.145948 + 0.989292i $$0.546623\pi$$
$$600$$ −3.24860 −0.132624
$$601$$ −21.7514 −0.887258 −0.443629 0.896211i $$-0.646309\pi$$
−0.443629 + 0.896211i $$0.646309\pi$$
$$602$$ 0 0
$$603$$ −10.6170 −0.432359
$$604$$ 15.5720 0.633616
$$605$$ −1.32340 −0.0538040
$$606$$ 30.6981 1.24702
$$607$$ 39.4134 1.59974 0.799871 0.600172i $$-0.204901\pi$$
0.799871 + 0.600172i $$0.204901\pi$$
$$608$$ −9.13919 −0.370643
$$609$$ 0 0
$$610$$ 7.88914 0.319422
$$611$$ 1.21255 0.0490544
$$612$$ 9.72161 0.392973
$$613$$ −22.1980 −0.896568 −0.448284 0.893891i $$-0.647965\pi$$
−0.448284 + 0.893891i $$0.647965\pi$$
$$614$$ 19.7008 0.795059
$$615$$ 6.34760 0.255960
$$616$$ 0 0
$$617$$ −37.4432 −1.50741 −0.753704 0.657214i $$-0.771735\pi$$
−0.753704 + 0.657214i $$0.771735\pi$$
$$618$$ −25.7729 −1.03674
$$619$$ 15.7008 0.631068 0.315534 0.948914i $$-0.397816\pi$$
0.315534 + 0.948914i $$0.397816\pi$$
$$620$$ −9.28398 −0.372854
$$621$$ 0 0
$$622$$ −10.4072 −0.417290
$$623$$ 0 0
$$624$$ 1.90582 0.0762939
$$625$$ 1.79641 0.0718566
$$626$$ −3.55408 −0.142050
$$627$$ −1.32340 −0.0528517
$$628$$ 24.4793 0.976829
$$629$$ −10.6468 −0.424516
$$630$$ 0 0
$$631$$ 35.0665 1.39598 0.697988 0.716110i $$-0.254079\pi$$
0.697988 + 0.716110i $$0.254079\pi$$
$$632$$ 3.44322 0.136964
$$633$$ −26.0900 −1.03699
$$634$$ −17.5720 −0.697873
$$635$$ 6.03046 0.239311
$$636$$ 12.6468 0.501479
$$637$$ 0 0
$$638$$ −2.74099 −0.108517
$$639$$ 15.9404 0.630593
$$640$$ 10.2050 0.403389
$$641$$ −30.1801 −1.19204 −0.596020 0.802969i $$-0.703252\pi$$
−0.596020 + 0.802969i $$0.703252\pi$$
$$642$$ 10.5914 0.418008
$$643$$ 36.7964 1.45111 0.725554 0.688165i $$-0.241583\pi$$
0.725554 + 0.688165i $$0.241583\pi$$
$$644$$ 0 0
$$645$$ 6.34760 0.249936
$$646$$ 16.3684 0.644007
$$647$$ 33.4522 1.31514 0.657571 0.753393i $$-0.271584\pi$$
0.657571 + 0.753393i $$0.271584\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 13.6918 0.537451
$$650$$ −2.40717 −0.0944170
$$651$$ 0 0
$$652$$ 3.91478 0.153315
$$653$$ 5.50280 0.215341 0.107671 0.994187i $$-0.465661\pi$$
0.107671 + 0.994187i $$0.465661\pi$$
$$654$$ 7.72161 0.301939
$$655$$ 1.39484 0.0545009
$$656$$ 22.9557 0.896268
$$657$$ −1.45219 −0.0566552
$$658$$ 0 0
$$659$$ 33.1946 1.29308 0.646539 0.762881i $$-0.276216\pi$$
0.646539 + 0.762881i $$0.276216\pi$$
$$660$$ −1.93561 −0.0753435
$$661$$ −43.3241 −1.68511 −0.842556 0.538609i $$-0.818950\pi$$
−0.842556 + 0.538609i $$0.818950\pi$$
$$662$$ 38.1080 1.48111
$$663$$ −2.64681 −0.102794
$$664$$ 0.796415 0.0309069
$$665$$ 0 0
$$666$$ 2.98062 0.115497
$$667$$ 0 0
$$668$$ 23.4016 0.905434
$$669$$ 18.0900 0.699401
$$670$$ −26.1455 −1.01009
$$671$$ 3.20359 0.123673
$$672$$ 0 0
$$673$$ −28.0721 −1.08210 −0.541050 0.840990i $$-0.681973\pi$$
−0.541050 + 0.840990i $$0.681973\pi$$
$$674$$ −32.3476 −1.24598
$$675$$ −3.24860 −0.125039
$$676$$ −18.7819 −0.722379
$$677$$ −22.8448 −0.877997 −0.438998 0.898488i $$-0.644667\pi$$
−0.438998 + 0.898488i $$0.644667\pi$$
$$678$$ −1.76036 −0.0676063
$$679$$ 0 0
$$680$$ −8.79641 −0.337327
$$681$$ 28.1801 1.07986
$$682$$ −8.92520 −0.341763
$$683$$ 11.4016 0.436269 0.218135 0.975919i $$-0.430003\pi$$
0.218135 + 0.975919i $$0.430003\pi$$
$$684$$ −1.93561 −0.0740099
$$685$$ −1.90997 −0.0729761
$$686$$ 0 0
$$687$$ −8.00000 −0.305219
$$688$$ 22.9557 0.875176
$$689$$ −3.44322 −0.131176
$$690$$ 0 0
$$691$$ 11.9404 0.454235 0.227118 0.973867i $$-0.427070\pi$$
0.227118 + 0.973867i $$0.427070\pi$$
$$692$$ 18.6289 0.708164
$$693$$ 0 0
$$694$$ 56.9169 2.16054
$$695$$ −24.5984 −0.933071
$$696$$ 1.47301 0.0558343
$$697$$ −31.8809 −1.20757
$$698$$ 61.1786 2.31564
$$699$$ −21.4432 −0.811057
$$700$$ 0 0
$$701$$ 17.1440 0.647520 0.323760 0.946139i $$-0.395053\pi$$
0.323760 + 0.946139i $$0.395053\pi$$
$$702$$ 0.740987 0.0279667
$$703$$ 2.11982 0.0799505
$$704$$ −3.27839 −0.123559
$$705$$ 4.02979 0.151771
$$706$$ 61.4183 2.31151
$$707$$ 0 0
$$708$$ 20.0256 0.752610
$$709$$ −50.7583 −1.90627 −0.953135 0.302546i $$-0.902163\pi$$
−0.953135 + 0.302546i $$0.902163\pi$$
$$710$$ 39.2549 1.47321
$$711$$ 3.44322 0.129131
$$712$$ 10.6468 0.399006
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0.526989 0.0197083
$$716$$ 8.55678 0.319782
$$717$$ 11.5422 0.431052
$$718$$ 36.6593 1.36811
$$719$$ −4.35656 −0.162472 −0.0812361 0.996695i $$-0.525887\pi$$
−0.0812361 + 0.996695i $$0.525887\pi$$
$$720$$ 6.33382 0.236047
$$721$$ 0 0
$$722$$ 32.0963 1.19450
$$723$$ −12.9550 −0.481801
$$724$$ 24.5664 0.913003
$$725$$ −4.78522 −0.177719
$$726$$ −1.86081 −0.0690610
$$727$$ 33.1745 1.23037 0.615186 0.788382i $$-0.289081\pi$$
0.615186 + 0.788382i $$0.289081\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −3.57615 −0.132359
$$731$$ −31.8809 −1.17916
$$732$$ 4.68556 0.173183
$$733$$ 31.3836 1.15918 0.579591 0.814908i $$-0.303212\pi$$
0.579591 + 0.814908i $$0.303212\pi$$
$$734$$ −57.9196 −2.13785
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.6170 −0.391083
$$738$$ 8.92520 0.328541
$$739$$ 29.4737 1.08421 0.542103 0.840312i $$-0.317628\pi$$
0.542103 + 0.840312i $$0.317628\pi$$
$$740$$ 3.10044 0.113975
$$741$$ 0.526989 0.0193594
$$742$$ 0 0
$$743$$ −15.8414 −0.581166 −0.290583 0.956850i $$-0.593849\pi$$
−0.290583 + 0.956850i $$0.593849\pi$$
$$744$$ 4.79641 0.175845
$$745$$ −5.13505 −0.188134
$$746$$ 16.6468 0.609483
$$747$$ 0.796415 0.0291393
$$748$$ 9.72161 0.355457
$$749$$ 0 0
$$750$$ −20.3130 −0.741726
$$751$$ 44.1919 1.61259 0.806293 0.591516i $$-0.201470\pi$$
0.806293 + 0.591516i $$0.201470\pi$$
$$752$$ 14.5735 0.531439
$$753$$ 28.8954 1.05301
$$754$$ 1.09148 0.0397494
$$755$$ −14.0900 −0.512789
$$756$$ 0 0
$$757$$ 44.1711 1.60543 0.802713 0.596366i $$-0.203389\pi$$
0.802713 + 0.596366i $$0.203389\pi$$
$$758$$ −7.83372 −0.284533
$$759$$ 0 0
$$760$$ 1.75140 0.0635299
$$761$$ 41.0361 1.48756 0.743778 0.668427i $$-0.233032\pi$$
0.743778 + 0.668427i $$0.233032\pi$$
$$762$$ 8.47928 0.307172
$$763$$ 0 0
$$764$$ −24.4793 −0.885629
$$765$$ −8.79641 −0.318035
$$766$$ 27.1440 0.980753
$$767$$ −5.45219 −0.196867
$$768$$ 20.9058 0.754374
$$769$$ −52.8358 −1.90531 −0.952654 0.304055i $$-0.901659\pi$$
−0.952654 + 0.304055i $$0.901659\pi$$
$$770$$ 0 0
$$771$$ 12.2278 0.440373
$$772$$ 14.5389 0.523265
$$773$$ −32.0907 −1.15422 −0.577111 0.816666i $$-0.695820\pi$$
−0.577111 + 0.816666i $$0.695820\pi$$
$$774$$ 8.92520 0.320810
$$775$$ −15.5816 −0.559709
$$776$$ 12.2396 0.439377
$$777$$ 0 0
$$778$$ 56.7160 2.03337
$$779$$ 6.34760 0.227426
$$780$$ 0.770774 0.0275981
$$781$$ 15.9404 0.570393
$$782$$ 0 0
$$783$$ 1.47301 0.0526411
$$784$$ 0 0
$$785$$ −22.1496 −0.790553
$$786$$ 1.96125 0.0699555
$$787$$ −2.37738 −0.0847446 −0.0423723 0.999102i $$-0.513492\pi$$
−0.0423723 + 0.999102i $$0.513492\pi$$
$$788$$ 23.1828 0.825852
$$789$$ −0.954984 −0.0339983
$$790$$ 8.47928 0.301679
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −1.27569 −0.0453011
$$794$$ −20.2909 −0.720098
$$795$$ −11.4432 −0.405849
$$796$$ −9.34490 −0.331221
$$797$$ 20.5091 0.726468 0.363234 0.931698i $$-0.381673\pi$$
0.363234 + 0.931698i $$0.381673\pi$$
$$798$$ 0 0
$$799$$ −20.2396 −0.716027
$$800$$ −22.4343 −0.793171
$$801$$ 10.6468 0.376186
$$802$$ −23.0873 −0.815242
$$803$$ −1.45219 −0.0512465
$$804$$ −15.5284 −0.547646
$$805$$ 0 0
$$806$$ 3.55408 0.125187
$$807$$ −20.2396 −0.712469
$$808$$ −16.4972 −0.580370
$$809$$ −8.85666 −0.311384 −0.155692 0.987806i $$-0.549761\pi$$
−0.155692 + 0.987806i $$0.549761\pi$$
$$810$$ 2.46260 0.0865269
$$811$$ −8.26943 −0.290379 −0.145189 0.989404i $$-0.546379\pi$$
−0.145189 + 0.989404i $$0.546379\pi$$
$$812$$ 0 0
$$813$$ −16.4674 −0.577537
$$814$$ 2.98062 0.104471
$$815$$ −3.54222 −0.124078
$$816$$ −31.8116 −1.11363
$$817$$ 6.34760 0.222074
$$818$$ 58.3989 2.04187
$$819$$ 0 0
$$820$$ 9.28398 0.324211
$$821$$ −32.3595 −1.12935 −0.564676 0.825312i $$-0.690999\pi$$
−0.564676 + 0.825312i $$0.690999\pi$$
$$822$$ −2.68556 −0.0936696
$$823$$ −45.2459 −1.57717 −0.788587 0.614924i $$-0.789187\pi$$
−0.788587 + 0.614924i $$0.789187\pi$$
$$824$$ 13.8504 0.482501
$$825$$ −3.24860 −0.113102
$$826$$ 0 0
$$827$$ 20.8954 0.726605 0.363302 0.931671i $$-0.381649\pi$$
0.363302 + 0.931671i $$0.381649\pi$$
$$828$$ 0 0
$$829$$ −44.6952 −1.55233 −0.776164 0.630531i $$-0.782837\pi$$
−0.776164 + 0.630531i $$0.782837\pi$$
$$830$$ 1.96125 0.0680760
$$831$$ 18.1801 0.630659
$$832$$ 1.30548 0.0452593
$$833$$ 0 0
$$834$$ −34.5872 −1.19766
$$835$$ −21.1745 −0.732773
$$836$$ −1.93561 −0.0669444
$$837$$ 4.79641 0.165788
$$838$$ 56.1426 1.93941
$$839$$ −0.457782 −0.0158044 −0.00790219 0.999969i $$-0.502515\pi$$
−0.00790219 + 0.999969i $$0.502515\pi$$
$$840$$ 0 0
$$841$$ −26.8302 −0.925181
$$842$$ 39.6066 1.36493
$$843$$ 24.3178 0.837550
$$844$$ −38.1592 −1.31350
$$845$$ 16.9944 0.584625
$$846$$ 5.66618 0.194807
$$847$$ 0 0
$$848$$ −41.3836 −1.42112
$$849$$ −21.5035 −0.737997
$$850$$ 40.1801 1.37816
$$851$$ 0 0
$$852$$ 23.3144 0.798740
$$853$$ −26.0900 −0.893306 −0.446653 0.894707i $$-0.647384\pi$$
−0.446653 + 0.894707i $$0.647384\pi$$
$$854$$ 0 0
$$855$$ 1.75140 0.0598966
$$856$$ −5.69182 −0.194543
$$857$$ 6.38924 0.218252 0.109126 0.994028i $$-0.465195\pi$$
0.109126 + 0.994028i $$0.465195\pi$$
$$858$$ 0.740987 0.0252969
$$859$$ −3.26316 −0.111338 −0.0556688 0.998449i $$-0.517729\pi$$
−0.0556688 + 0.998449i $$0.517729\pi$$
$$860$$ 9.28398 0.316581
$$861$$ 0 0
$$862$$ 27.4057 0.933443
$$863$$ −12.6164 −0.429466 −0.214733 0.976673i $$-0.568888\pi$$
−0.214733 + 0.976673i $$0.568888\pi$$
$$864$$ 6.90582 0.234941
$$865$$ −16.8560 −0.573121
$$866$$ −26.3297 −0.894719
$$867$$ 27.1801 0.923083
$$868$$ 0 0
$$869$$ 3.44322 0.116803
$$870$$ 3.62743 0.122982
$$871$$ 4.22778 0.143253
$$872$$ −4.14961 −0.140523
$$873$$ 12.2396 0.414249
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −2.12397 −0.0717621
$$877$$ −22.7964 −0.769780 −0.384890 0.922962i $$-0.625761\pi$$
−0.384890 + 0.922962i $$0.625761\pi$$
$$878$$ −48.3822 −1.63282
$$879$$ 5.59283 0.188641
$$880$$ 6.33382 0.213513
$$881$$ −15.9881 −0.538654 −0.269327 0.963049i $$-0.586801\pi$$
−0.269327 + 0.963049i $$0.586801\pi$$
$$882$$ 0 0
$$883$$ 30.9162 1.04041 0.520207 0.854040i $$-0.325855\pi$$
0.520207 + 0.854040i $$0.325855\pi$$
$$884$$ −3.87122 −0.130203
$$885$$ −18.1198 −0.609091
$$886$$ 61.3628 2.06152
$$887$$ 27.7008 0.930101 0.465051 0.885284i $$-0.346036\pi$$
0.465051 + 0.885284i $$0.346036\pi$$
$$888$$ −1.60179 −0.0537526
$$889$$ 0 0
$$890$$ 26.2188 0.878857
$$891$$ 1.00000 0.0335013
$$892$$ 26.4585 0.885895
$$893$$ 4.02979 0.134852
$$894$$ −7.22026 −0.241482
$$895$$ −7.74244 −0.258801
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −9.23694 −0.308241
$$899$$ 7.06517 0.235637
$$900$$ −4.75140 −0.158380
$$901$$ 57.4737 1.91473
$$902$$ 8.92520 0.297177
$$903$$ 0 0
$$904$$ 0.946021 0.0314642
$$905$$ −22.2284 −0.738899
$$906$$ −19.8116 −0.658198
$$907$$ 17.2936 0.574225 0.287113 0.957897i $$-0.407305\pi$$
0.287113 + 0.957897i $$0.407305\pi$$
$$908$$ 41.2161 1.36780
$$909$$ −16.4972 −0.547178
$$910$$ 0 0
$$911$$ 8.85599 0.293412 0.146706 0.989180i $$-0.453133\pi$$
0.146706 + 0.989180i $$0.453133\pi$$
$$912$$ 6.33382 0.209734
$$913$$ 0.796415 0.0263575
$$914$$ 41.3061 1.36629
$$915$$ −4.23964 −0.140158
$$916$$ −11.7008 −0.386605
$$917$$ 0 0
$$918$$ −12.3684 −0.408219
$$919$$ 54.0305 1.78230 0.891150 0.453708i $$-0.149899\pi$$
0.891150 + 0.453708i $$0.149899\pi$$
$$920$$ 0 0
$$921$$ −10.5872 −0.348861
$$922$$ 23.3324 0.768411
$$923$$ −6.34760 −0.208934
$$924$$ 0 0
$$925$$ 5.20359 0.171093
$$926$$ −32.6039 −1.07143
$$927$$ 13.8504 0.454907
$$928$$ 10.1723 0.333924
$$929$$ 20.7666 0.681331 0.340665 0.940185i $$-0.389348\pi$$
0.340665 + 0.940185i $$0.389348\pi$$
$$930$$ 11.8116 0.387319
$$931$$ 0 0
$$932$$ −31.3628 −1.02732
$$933$$ 5.59283 0.183101
$$934$$ 21.9237 0.717367
$$935$$ −8.79641 −0.287674
$$936$$ −0.398207 −0.0130158
$$937$$ −3.20359 −0.104657 −0.0523283 0.998630i $$-0.516664\pi$$
−0.0523283 + 0.998630i $$0.516664\pi$$
$$938$$ 0 0
$$939$$ 1.90997 0.0623295
$$940$$ 5.89396 0.192240
$$941$$ −43.0249 −1.40257 −0.701285 0.712881i $$-0.747390\pi$$
−0.701285 + 0.712881i $$0.747390\pi$$
$$942$$ −31.1440 −1.01473
$$943$$ 0 0
$$944$$ −65.5291 −2.13279
$$945$$ 0 0
$$946$$ 8.92520 0.290183
$$947$$ 17.5749 0.571108 0.285554 0.958363i $$-0.407822\pi$$
0.285554 + 0.958363i $$0.407822\pi$$
$$948$$ 5.03605 0.163563
$$949$$ 0.578271 0.0187715
$$950$$ −8.00000 −0.259554
$$951$$ 9.44322 0.306217
$$952$$ 0 0
$$953$$ −14.7846 −0.478919 −0.239459 0.970906i $$-0.576970\pi$$
−0.239459 + 0.970906i $$0.576970\pi$$
$$954$$ −16.0900 −0.520934
$$955$$ 22.1496 0.716744
$$956$$ 16.8816 0.545991
$$957$$ 1.47301 0.0476157
$$958$$ 9.37112 0.302767
$$959$$ 0 0
$$960$$ 4.33863 0.140029
$$961$$ −7.99440 −0.257884
$$962$$ −1.18691 −0.0382674
$$963$$ −5.69182 −0.183416
$$964$$ −18.9479 −0.610272
$$965$$ −13.1552 −0.423481
$$966$$ 0 0
$$967$$ −23.1440 −0.744261 −0.372131 0.928180i $$-0.621373\pi$$
−0.372131 + 0.928180i $$0.621373\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −8.79641 −0.282582
$$970$$ 30.1413 0.967779
$$971$$ −36.5783 −1.17385 −0.586926 0.809640i $$-0.699662\pi$$
−0.586926 + 0.809640i $$0.699662\pi$$
$$972$$ 1.46260 0.0469129
$$973$$ 0 0
$$974$$ −37.5512 −1.20322
$$975$$ 1.29362 0.0414289
$$976$$ −15.3324 −0.490777
$$977$$ 54.1621 1.73280 0.866400 0.499350i $$-0.166428\pi$$
0.866400 + 0.499350i $$0.166428\pi$$
$$978$$ −4.98062 −0.159263
$$979$$ 10.6468 0.340273
$$980$$ 0 0
$$981$$ −4.14961 −0.132487
$$982$$ 47.2507 1.50783
$$983$$ −1.29362 −0.0412600 −0.0206300 0.999787i $$-0.506567\pi$$
−0.0206300 + 0.999787i $$0.506567\pi$$
$$984$$ −4.79641 −0.152904
$$985$$ −20.9765 −0.668366
$$986$$ −18.2188 −0.580205
$$987$$ 0 0
$$988$$ 0.770774 0.0245216
$$989$$ 0 0
$$990$$ 2.46260 0.0782665
$$991$$ −15.9523 −0.506741 −0.253371 0.967369i $$-0.581539\pi$$
−0.253371 + 0.967369i $$0.581539\pi$$
$$992$$ 33.1232 1.05166
$$993$$ −20.4793 −0.649890
$$994$$ 0 0
$$995$$ 8.45555 0.268059
$$996$$ 1.16484 0.0369092
$$997$$ 23.3836 0.740568 0.370284 0.928919i $$-0.379260\pi$$
0.370284 + 0.928919i $$0.379260\pi$$
$$998$$ −27.7562 −0.878608
$$999$$ −1.60179 −0.0506785
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 1617.2.a.r.1.1 yes 3
3.2 odd 2 4851.2.a.bl.1.3 3
7.6 odd 2 1617.2.a.q.1.1 3
21.20 even 2 4851.2.a.bm.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.q.1.1 3 7.6 odd 2
1617.2.a.r.1.1 yes 3 1.1 even 1 trivial
4851.2.a.bl.1.3 3 3.2 odd 2
4851.2.a.bm.1.3 3 21.20 even 2