# Properties

 Label 1617.2.a.k.1.1 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - 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 $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1617.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -2.23607 q^{5} +2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -2.23607 q^{5} +2.61803 q^{6} -7.47214 q^{8} +1.00000 q^{9} +5.85410 q^{10} -1.00000 q^{11} -4.85410 q^{12} +3.47214 q^{13} +2.23607 q^{15} +9.85410 q^{16} +6.00000 q^{17} -2.61803 q^{18} -4.23607 q^{19} -10.8541 q^{20} +2.61803 q^{22} -2.47214 q^{23} +7.47214 q^{24} -9.09017 q^{26} -1.00000 q^{27} -10.2361 q^{29} -5.85410 q^{30} +8.47214 q^{31} -10.8541 q^{32} +1.00000 q^{33} -15.7082 q^{34} +4.85410 q^{36} -0.527864 q^{37} +11.0902 q^{38} -3.47214 q^{39} +16.7082 q^{40} +6.00000 q^{41} -0.472136 q^{43} -4.85410 q^{44} -2.23607 q^{45} +6.47214 q^{46} +11.4721 q^{47} -9.85410 q^{48} -6.00000 q^{51} +16.8541 q^{52} -12.9443 q^{53} +2.61803 q^{54} +2.23607 q^{55} +4.23607 q^{57} +26.7984 q^{58} -11.9443 q^{59} +10.8541 q^{60} -6.94427 q^{61} -22.1803 q^{62} +8.70820 q^{64} -7.76393 q^{65} -2.61803 q^{66} +0.708204 q^{67} +29.1246 q^{68} +2.47214 q^{69} +4.47214 q^{71} -7.47214 q^{72} -1.00000 q^{73} +1.38197 q^{74} -20.5623 q^{76} +9.09017 q^{78} -2.47214 q^{79} -22.0344 q^{80} +1.00000 q^{81} -15.7082 q^{82} +12.4721 q^{83} -13.4164 q^{85} +1.23607 q^{86} +10.2361 q^{87} +7.47214 q^{88} -13.4164 q^{89} +5.85410 q^{90} -12.0000 q^{92} -8.47214 q^{93} -30.0344 q^{94} +9.47214 q^{95} +10.8541 q^{96} -6.94427 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{8} + 2 q^{9} + 5 q^{10} - 2 q^{11} - 3 q^{12} - 2 q^{13} + 13 q^{16} + 12 q^{17} - 3 q^{18} - 4 q^{19} - 15 q^{20} + 3 q^{22} + 4 q^{23} + 6 q^{24} - 7 q^{26} - 2 q^{27} - 16 q^{29} - 5 q^{30} + 8 q^{31} - 15 q^{32} + 2 q^{33} - 18 q^{34} + 3 q^{36} - 10 q^{37} + 11 q^{38} + 2 q^{39} + 20 q^{40} + 12 q^{41} + 8 q^{43} - 3 q^{44} + 4 q^{46} + 14 q^{47} - 13 q^{48} - 12 q^{51} + 27 q^{52} - 8 q^{53} + 3 q^{54} + 4 q^{57} + 29 q^{58} - 6 q^{59} + 15 q^{60} + 4 q^{61} - 22 q^{62} + 4 q^{64} - 20 q^{65} - 3 q^{66} - 12 q^{67} + 18 q^{68} - 4 q^{69} - 6 q^{72} - 2 q^{73} + 5 q^{74} - 21 q^{76} + 7 q^{78} + 4 q^{79} - 15 q^{80} + 2 q^{81} - 18 q^{82} + 16 q^{83} - 2 q^{86} + 16 q^{87} + 6 q^{88} + 5 q^{90} - 24 q^{92} - 8 q^{93} - 31 q^{94} + 10 q^{95} + 15 q^{96} + 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 3 * q^6 - 6 * q^8 + 2 * q^9 + 5 * q^10 - 2 * q^11 - 3 * q^12 - 2 * q^13 + 13 * q^16 + 12 * q^17 - 3 * q^18 - 4 * q^19 - 15 * q^20 + 3 * q^22 + 4 * q^23 + 6 * q^24 - 7 * q^26 - 2 * q^27 - 16 * q^29 - 5 * q^30 + 8 * q^31 - 15 * q^32 + 2 * q^33 - 18 * q^34 + 3 * q^36 - 10 * q^37 + 11 * q^38 + 2 * q^39 + 20 * q^40 + 12 * q^41 + 8 * q^43 - 3 * q^44 + 4 * q^46 + 14 * q^47 - 13 * q^48 - 12 * q^51 + 27 * q^52 - 8 * q^53 + 3 * q^54 + 4 * q^57 + 29 * q^58 - 6 * q^59 + 15 * q^60 + 4 * q^61 - 22 * q^62 + 4 * q^64 - 20 * q^65 - 3 * q^66 - 12 * q^67 + 18 * q^68 - 4 * q^69 - 6 * q^72 - 2 * q^73 + 5 * q^74 - 21 * q^76 + 7 * q^78 + 4 * q^79 - 15 * q^80 + 2 * q^81 - 18 * q^82 + 16 * q^83 - 2 * q^86 + 16 * q^87 + 6 * q^88 + 5 * q^90 - 24 * q^92 - 8 * q^93 - 31 * q^94 + 10 * q^95 + 15 * q^96 + 4 * q^97 - 2 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.61803 −1.85123 −0.925615 0.378467i $$-0.876451\pi$$
−0.925615 + 0.378467i $$0.876451\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 4.85410 2.42705
$$5$$ −2.23607 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$6$$ 2.61803 1.06881
$$7$$ 0 0
$$8$$ −7.47214 −2.64180
$$9$$ 1.00000 0.333333
$$10$$ 5.85410 1.85123
$$11$$ −1.00000 −0.301511
$$12$$ −4.85410 −1.40126
$$13$$ 3.47214 0.962997 0.481499 0.876447i $$-0.340093\pi$$
0.481499 + 0.876447i $$0.340093\pi$$
$$14$$ 0 0
$$15$$ 2.23607 0.577350
$$16$$ 9.85410 2.46353
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −2.61803 −0.617077
$$19$$ −4.23607 −0.971821 −0.485910 0.874009i $$-0.661512\pi$$
−0.485910 + 0.874009i $$0.661512\pi$$
$$20$$ −10.8541 −2.42705
$$21$$ 0 0
$$22$$ 2.61803 0.558167
$$23$$ −2.47214 −0.515476 −0.257738 0.966215i $$-0.582977\pi$$
−0.257738 + 0.966215i $$0.582977\pi$$
$$24$$ 7.47214 1.52524
$$25$$ 0 0
$$26$$ −9.09017 −1.78273
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −10.2361 −1.90079 −0.950395 0.311045i $$-0.899321\pi$$
−0.950395 + 0.311045i $$0.899321\pi$$
$$30$$ −5.85410 −1.06881
$$31$$ 8.47214 1.52164 0.760820 0.648963i $$-0.224797\pi$$
0.760820 + 0.648963i $$0.224797\pi$$
$$32$$ −10.8541 −1.91875
$$33$$ 1.00000 0.174078
$$34$$ −15.7082 −2.69393
$$35$$ 0 0
$$36$$ 4.85410 0.809017
$$37$$ −0.527864 −0.0867803 −0.0433902 0.999058i $$-0.513816\pi$$
−0.0433902 + 0.999058i $$0.513816\pi$$
$$38$$ 11.0902 1.79906
$$39$$ −3.47214 −0.555987
$$40$$ 16.7082 2.64180
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −0.472136 −0.0720001 −0.0360000 0.999352i $$-0.511462\pi$$
−0.0360000 + 0.999352i $$0.511462\pi$$
$$44$$ −4.85410 −0.731783
$$45$$ −2.23607 −0.333333
$$46$$ 6.47214 0.954264
$$47$$ 11.4721 1.67338 0.836692 0.547674i $$-0.184487\pi$$
0.836692 + 0.547674i $$0.184487\pi$$
$$48$$ −9.85410 −1.42232
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 16.8541 2.33724
$$53$$ −12.9443 −1.77803 −0.889016 0.457876i $$-0.848610\pi$$
−0.889016 + 0.457876i $$0.848610\pi$$
$$54$$ 2.61803 0.356269
$$55$$ 2.23607 0.301511
$$56$$ 0 0
$$57$$ 4.23607 0.561081
$$58$$ 26.7984 3.51880
$$59$$ −11.9443 −1.55501 −0.777506 0.628876i $$-0.783515\pi$$
−0.777506 + 0.628876i $$0.783515\pi$$
$$60$$ 10.8541 1.40126
$$61$$ −6.94427 −0.889123 −0.444561 0.895748i $$-0.646640\pi$$
−0.444561 + 0.895748i $$0.646640\pi$$
$$62$$ −22.1803 −2.81691
$$63$$ 0 0
$$64$$ 8.70820 1.08853
$$65$$ −7.76393 −0.962997
$$66$$ −2.61803 −0.322258
$$67$$ 0.708204 0.0865209 0.0432604 0.999064i $$-0.486225\pi$$
0.0432604 + 0.999064i $$0.486225\pi$$
$$68$$ 29.1246 3.53188
$$69$$ 2.47214 0.297610
$$70$$ 0 0
$$71$$ 4.47214 0.530745 0.265372 0.964146i $$-0.414505\pi$$
0.265372 + 0.964146i $$0.414505\pi$$
$$72$$ −7.47214 −0.880600
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 1.38197 0.160650
$$75$$ 0 0
$$76$$ −20.5623 −2.35866
$$77$$ 0 0
$$78$$ 9.09017 1.02926
$$79$$ −2.47214 −0.278137 −0.139069 0.990283i $$-0.544411\pi$$
−0.139069 + 0.990283i $$0.544411\pi$$
$$80$$ −22.0344 −2.46353
$$81$$ 1.00000 0.111111
$$82$$ −15.7082 −1.73468
$$83$$ 12.4721 1.36899 0.684497 0.729015i $$-0.260022\pi$$
0.684497 + 0.729015i $$0.260022\pi$$
$$84$$ 0 0
$$85$$ −13.4164 −1.45521
$$86$$ 1.23607 0.133289
$$87$$ 10.2361 1.09742
$$88$$ 7.47214 0.796532
$$89$$ −13.4164 −1.42214 −0.711068 0.703123i $$-0.751788\pi$$
−0.711068 + 0.703123i $$0.751788\pi$$
$$90$$ 5.85410 0.617077
$$91$$ 0 0
$$92$$ −12.0000 −1.25109
$$93$$ −8.47214 −0.878520
$$94$$ −30.0344 −3.09782
$$95$$ 9.47214 0.971821
$$96$$ 10.8541 1.10779
$$97$$ −6.94427 −0.705084 −0.352542 0.935796i $$-0.614683\pi$$
−0.352542 + 0.935796i $$0.614683\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 3.52786 0.351036 0.175518 0.984476i $$-0.443840\pi$$
0.175518 + 0.984476i $$0.443840\pi$$
$$102$$ 15.7082 1.55534
$$103$$ −7.41641 −0.730760 −0.365380 0.930858i $$-0.619061\pi$$
−0.365380 + 0.930858i $$0.619061\pi$$
$$104$$ −25.9443 −2.54405
$$105$$ 0 0
$$106$$ 33.8885 3.29155
$$107$$ 19.9443 1.92809 0.964043 0.265747i $$-0.0856186\pi$$
0.964043 + 0.265747i $$0.0856186\pi$$
$$108$$ −4.85410 −0.467086
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −5.85410 −0.558167
$$111$$ 0.527864 0.0501026
$$112$$ 0 0
$$113$$ 6.47214 0.608847 0.304424 0.952537i $$-0.401536\pi$$
0.304424 + 0.952537i $$0.401536\pi$$
$$114$$ −11.0902 −1.03869
$$115$$ 5.52786 0.515476
$$116$$ −49.6869 −4.61331
$$117$$ 3.47214 0.320999
$$118$$ 31.2705 2.87868
$$119$$ 0 0
$$120$$ −16.7082 −1.52524
$$121$$ 1.00000 0.0909091
$$122$$ 18.1803 1.64597
$$123$$ −6.00000 −0.541002
$$124$$ 41.1246 3.69310
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.09017 −0.0963583
$$129$$ 0.472136 0.0415693
$$130$$ 20.3262 1.78273
$$131$$ −4.47214 −0.390732 −0.195366 0.980730i $$-0.562590\pi$$
−0.195366 + 0.980730i $$0.562590\pi$$
$$132$$ 4.85410 0.422495
$$133$$ 0 0
$$134$$ −1.85410 −0.160170
$$135$$ 2.23607 0.192450
$$136$$ −44.8328 −3.84438
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ −6.47214 −0.550945
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −11.4721 −0.966128
$$142$$ −11.7082 −0.982531
$$143$$ −3.47214 −0.290355
$$144$$ 9.85410 0.821175
$$145$$ 22.8885 1.90079
$$146$$ 2.61803 0.216670
$$147$$ 0 0
$$148$$ −2.56231 −0.210620
$$149$$ 15.6525 1.28230 0.641150 0.767415i $$-0.278457\pi$$
0.641150 + 0.767415i $$0.278457\pi$$
$$150$$ 0 0
$$151$$ −6.94427 −0.565117 −0.282558 0.959250i $$-0.591183\pi$$
−0.282558 + 0.959250i $$0.591183\pi$$
$$152$$ 31.6525 2.56735
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −18.9443 −1.52164
$$156$$ −16.8541 −1.34941
$$157$$ 3.41641 0.272659 0.136330 0.990664i $$-0.456469\pi$$
0.136330 + 0.990664i $$0.456469\pi$$
$$158$$ 6.47214 0.514895
$$159$$ 12.9443 1.02655
$$160$$ 24.2705 1.91875
$$161$$ 0 0
$$162$$ −2.61803 −0.205692
$$163$$ −18.1246 −1.41963 −0.709815 0.704389i $$-0.751221\pi$$
−0.709815 + 0.704389i $$0.751221\pi$$
$$164$$ 29.1246 2.27425
$$165$$ −2.23607 −0.174078
$$166$$ −32.6525 −2.53432
$$167$$ 18.4721 1.42942 0.714708 0.699423i $$-0.246559\pi$$
0.714708 + 0.699423i $$0.246559\pi$$
$$168$$ 0 0
$$169$$ −0.944272 −0.0726363
$$170$$ 35.1246 2.69393
$$171$$ −4.23607 −0.323940
$$172$$ −2.29180 −0.174748
$$173$$ 5.52786 0.420276 0.210138 0.977672i $$-0.432609\pi$$
0.210138 + 0.977672i $$0.432609\pi$$
$$174$$ −26.7984 −2.03158
$$175$$ 0 0
$$176$$ −9.85410 −0.742781
$$177$$ 11.9443 0.897786
$$178$$ 35.1246 2.63270
$$179$$ −5.52786 −0.413172 −0.206586 0.978428i $$-0.566235\pi$$
−0.206586 + 0.978428i $$0.566235\pi$$
$$180$$ −10.8541 −0.809017
$$181$$ 0.472136 0.0350936 0.0175468 0.999846i $$-0.494414\pi$$
0.0175468 + 0.999846i $$0.494414\pi$$
$$182$$ 0 0
$$183$$ 6.94427 0.513335
$$184$$ 18.4721 1.36178
$$185$$ 1.18034 0.0867803
$$186$$ 22.1803 1.62634
$$187$$ −6.00000 −0.438763
$$188$$ 55.6869 4.06139
$$189$$ 0 0
$$190$$ −24.7984 −1.79906
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −8.70820 −0.628460
$$193$$ 25.8885 1.86350 0.931749 0.363103i $$-0.118283\pi$$
0.931749 + 0.363103i $$0.118283\pi$$
$$194$$ 18.1803 1.30527
$$195$$ 7.76393 0.555987
$$196$$ 0 0
$$197$$ −8.47214 −0.603615 −0.301807 0.953369i $$-0.597590\pi$$
−0.301807 + 0.953369i $$0.597590\pi$$
$$198$$ 2.61803 0.186056
$$199$$ 24.4721 1.73478 0.867392 0.497626i $$-0.165795\pi$$
0.867392 + 0.497626i $$0.165795\pi$$
$$200$$ 0 0
$$201$$ −0.708204 −0.0499529
$$202$$ −9.23607 −0.649847
$$203$$ 0 0
$$204$$ −29.1246 −2.03913
$$205$$ −13.4164 −0.937043
$$206$$ 19.4164 1.35281
$$207$$ −2.47214 −0.171825
$$208$$ 34.2148 2.37237
$$209$$ 4.23607 0.293015
$$210$$ 0 0
$$211$$ 18.9443 1.30418 0.652089 0.758143i $$-0.273893\pi$$
0.652089 + 0.758143i $$0.273893\pi$$
$$212$$ −62.8328 −4.31538
$$213$$ −4.47214 −0.306426
$$214$$ −52.2148 −3.56933
$$215$$ 1.05573 0.0720001
$$216$$ 7.47214 0.508414
$$217$$ 0 0
$$218$$ −15.7082 −1.06389
$$219$$ 1.00000 0.0675737
$$220$$ 10.8541 0.731783
$$221$$ 20.8328 1.40137
$$222$$ −1.38197 −0.0927515
$$223$$ 18.0000 1.20537 0.602685 0.797980i $$-0.294098\pi$$
0.602685 + 0.797980i $$0.294098\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −16.9443 −1.12712
$$227$$ 26.8328 1.78096 0.890478 0.455026i $$-0.150370\pi$$
0.890478 + 0.455026i $$0.150370\pi$$
$$228$$ 20.5623 1.36177
$$229$$ 15.4164 1.01874 0.509372 0.860546i $$-0.329878\pi$$
0.509372 + 0.860546i $$0.329878\pi$$
$$230$$ −14.4721 −0.954264
$$231$$ 0 0
$$232$$ 76.4853 5.02151
$$233$$ 13.4164 0.878938 0.439469 0.898258i $$-0.355167\pi$$
0.439469 + 0.898258i $$0.355167\pi$$
$$234$$ −9.09017 −0.594243
$$235$$ −25.6525 −1.67338
$$236$$ −57.9787 −3.77409
$$237$$ 2.47214 0.160582
$$238$$ 0 0
$$239$$ 17.3607 1.12297 0.561485 0.827487i $$-0.310230\pi$$
0.561485 + 0.827487i $$0.310230\pi$$
$$240$$ 22.0344 1.42232
$$241$$ −19.0000 −1.22390 −0.611949 0.790897i $$-0.709614\pi$$
−0.611949 + 0.790897i $$0.709614\pi$$
$$242$$ −2.61803 −0.168294
$$243$$ −1.00000 −0.0641500
$$244$$ −33.7082 −2.15795
$$245$$ 0 0
$$246$$ 15.7082 1.00152
$$247$$ −14.7082 −0.935861
$$248$$ −63.3050 −4.01987
$$249$$ −12.4721 −0.790390
$$250$$ −29.2705 −1.85123
$$251$$ 3.94427 0.248960 0.124480 0.992222i $$-0.460274\pi$$
0.124480 + 0.992222i $$0.460274\pi$$
$$252$$ 0 0
$$253$$ 2.47214 0.155422
$$254$$ −41.8885 −2.62832
$$255$$ 13.4164 0.840168
$$256$$ −14.5623 −0.910144
$$257$$ −7.76393 −0.484301 −0.242150 0.970239i $$-0.577853\pi$$
−0.242150 + 0.970239i $$0.577853\pi$$
$$258$$ −1.23607 −0.0769542
$$259$$ 0 0
$$260$$ −37.6869 −2.33724
$$261$$ −10.2361 −0.633597
$$262$$ 11.7082 0.723335
$$263$$ −4.41641 −0.272327 −0.136164 0.990686i $$-0.543477\pi$$
−0.136164 + 0.990686i $$0.543477\pi$$
$$264$$ −7.47214 −0.459878
$$265$$ 28.9443 1.77803
$$266$$ 0 0
$$267$$ 13.4164 0.821071
$$268$$ 3.43769 0.209991
$$269$$ −16.4721 −1.00432 −0.502162 0.864774i $$-0.667462\pi$$
−0.502162 + 0.864774i $$0.667462\pi$$
$$270$$ −5.85410 −0.356269
$$271$$ 5.29180 0.321454 0.160727 0.986999i $$-0.448616\pi$$
0.160727 + 0.986999i $$0.448616\pi$$
$$272$$ 59.1246 3.58496
$$273$$ 0 0
$$274$$ −15.7082 −0.948967
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ −18.9443 −1.13825 −0.569125 0.822251i $$-0.692718\pi$$
−0.569125 + 0.822251i $$0.692718\pi$$
$$278$$ −20.9443 −1.25615
$$279$$ 8.47214 0.507214
$$280$$ 0 0
$$281$$ 5.18034 0.309033 0.154517 0.987990i $$-0.450618\pi$$
0.154517 + 0.987990i $$0.450618\pi$$
$$282$$ 30.0344 1.78852
$$283$$ 4.70820 0.279874 0.139937 0.990160i $$-0.455310\pi$$
0.139937 + 0.990160i $$0.455310\pi$$
$$284$$ 21.7082 1.28814
$$285$$ −9.47214 −0.561081
$$286$$ 9.09017 0.537513
$$287$$ 0 0
$$288$$ −10.8541 −0.639584
$$289$$ 19.0000 1.11765
$$290$$ −59.9230 −3.51880
$$291$$ 6.94427 0.407080
$$292$$ −4.85410 −0.284065
$$293$$ −9.88854 −0.577695 −0.288847 0.957375i $$-0.593272\pi$$
−0.288847 + 0.957375i $$0.593272\pi$$
$$294$$ 0 0
$$295$$ 26.7082 1.55501
$$296$$ 3.94427 0.229256
$$297$$ 1.00000 0.0580259
$$298$$ −40.9787 −2.37383
$$299$$ −8.58359 −0.496402
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 18.1803 1.04616
$$303$$ −3.52786 −0.202670
$$304$$ −41.7426 −2.39410
$$305$$ 15.5279 0.889123
$$306$$ −15.7082 −0.897978
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 0 0
$$309$$ 7.41641 0.421905
$$310$$ 49.5967 2.81691
$$311$$ −12.9443 −0.734002 −0.367001 0.930220i $$-0.619615\pi$$
−0.367001 + 0.930220i $$0.619615\pi$$
$$312$$ 25.9443 1.46881
$$313$$ 26.3607 1.48999 0.744997 0.667068i $$-0.232451\pi$$
0.744997 + 0.667068i $$0.232451\pi$$
$$314$$ −8.94427 −0.504754
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 26.9443 1.51334 0.756671 0.653796i $$-0.226825\pi$$
0.756671 + 0.653796i $$0.226825\pi$$
$$318$$ −33.8885 −1.90038
$$319$$ 10.2361 0.573110
$$320$$ −19.4721 −1.08853
$$321$$ −19.9443 −1.11318
$$322$$ 0 0
$$323$$ −25.4164 −1.41421
$$324$$ 4.85410 0.269672
$$325$$ 0 0
$$326$$ 47.4508 2.62806
$$327$$ −6.00000 −0.331801
$$328$$ −44.8328 −2.47548
$$329$$ 0 0
$$330$$ 5.85410 0.322258
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 60.5410 3.32262
$$333$$ −0.527864 −0.0289268
$$334$$ −48.3607 −2.64618
$$335$$ −1.58359 −0.0865209
$$336$$ 0 0
$$337$$ 11.4164 0.621891 0.310946 0.950428i $$-0.399354\pi$$
0.310946 + 0.950428i $$0.399354\pi$$
$$338$$ 2.47214 0.134466
$$339$$ −6.47214 −0.351518
$$340$$ −65.1246 −3.53188
$$341$$ −8.47214 −0.458792
$$342$$ 11.0902 0.599688
$$343$$ 0 0
$$344$$ 3.52786 0.190210
$$345$$ −5.52786 −0.297610
$$346$$ −14.4721 −0.778027
$$347$$ 0.944272 0.0506912 0.0253456 0.999679i $$-0.491931\pi$$
0.0253456 + 0.999679i $$0.491931\pi$$
$$348$$ 49.6869 2.66350
$$349$$ −2.41641 −0.129347 −0.0646737 0.997906i $$-0.520601\pi$$
−0.0646737 + 0.997906i $$0.520601\pi$$
$$350$$ 0 0
$$351$$ −3.47214 −0.185329
$$352$$ 10.8541 0.578526
$$353$$ −4.23607 −0.225463 −0.112732 0.993625i $$-0.535960\pi$$
−0.112732 + 0.993625i $$0.535960\pi$$
$$354$$ −31.2705 −1.66201
$$355$$ −10.0000 −0.530745
$$356$$ −65.1246 −3.45160
$$357$$ 0 0
$$358$$ 14.4721 0.764876
$$359$$ −20.9443 −1.10540 −0.552698 0.833381i $$-0.686402\pi$$
−0.552698 + 0.833381i $$0.686402\pi$$
$$360$$ 16.7082 0.880600
$$361$$ −1.05573 −0.0555646
$$362$$ −1.23607 −0.0649663
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 2.23607 0.117041
$$366$$ −18.1803 −0.950302
$$367$$ −25.4164 −1.32673 −0.663363 0.748298i $$-0.730871\pi$$
−0.663363 + 0.748298i $$0.730871\pi$$
$$368$$ −24.3607 −1.26989
$$369$$ 6.00000 0.312348
$$370$$ −3.09017 −0.160650
$$371$$ 0 0
$$372$$ −41.1246 −2.13221
$$373$$ −16.9443 −0.877341 −0.438671 0.898648i $$-0.644550\pi$$
−0.438671 + 0.898648i $$0.644550\pi$$
$$374$$ 15.7082 0.812252
$$375$$ −11.1803 −0.577350
$$376$$ −85.7214 −4.42074
$$377$$ −35.5410 −1.83046
$$378$$ 0 0
$$379$$ −25.6525 −1.31768 −0.658840 0.752283i $$-0.728952\pi$$
−0.658840 + 0.752283i $$0.728952\pi$$
$$380$$ 45.9787 2.35866
$$381$$ −16.0000 −0.819705
$$382$$ −31.4164 −1.60740
$$383$$ 8.94427 0.457031 0.228515 0.973540i $$-0.426613\pi$$
0.228515 + 0.973540i $$0.426613\pi$$
$$384$$ 1.09017 0.0556325
$$385$$ 0 0
$$386$$ −67.7771 −3.44976
$$387$$ −0.472136 −0.0240000
$$388$$ −33.7082 −1.71127
$$389$$ 20.8328 1.05627 0.528133 0.849162i $$-0.322892\pi$$
0.528133 + 0.849162i $$0.322892\pi$$
$$390$$ −20.3262 −1.02926
$$391$$ −14.8328 −0.750128
$$392$$ 0 0
$$393$$ 4.47214 0.225589
$$394$$ 22.1803 1.11743
$$395$$ 5.52786 0.278137
$$396$$ −4.85410 −0.243928
$$397$$ 9.41641 0.472596 0.236298 0.971681i $$-0.424066\pi$$
0.236298 + 0.971681i $$0.424066\pi$$
$$398$$ −64.0689 −3.21148
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.47214 −0.423078 −0.211539 0.977370i $$-0.567848\pi$$
−0.211539 + 0.977370i $$0.567848\pi$$
$$402$$ 1.85410 0.0924742
$$403$$ 29.4164 1.46534
$$404$$ 17.1246 0.851981
$$405$$ −2.23607 −0.111111
$$406$$ 0 0
$$407$$ 0.527864 0.0261652
$$408$$ 44.8328 2.21956
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 35.1246 1.73468
$$411$$ −6.00000 −0.295958
$$412$$ −36.0000 −1.77359
$$413$$ 0 0
$$414$$ 6.47214 0.318088
$$415$$ −27.8885 −1.36899
$$416$$ −37.6869 −1.84775
$$417$$ −8.00000 −0.391762
$$418$$ −11.0902 −0.542438
$$419$$ 3.00000 0.146560 0.0732798 0.997311i $$-0.476653\pi$$
0.0732798 + 0.997311i $$0.476653\pi$$
$$420$$ 0 0
$$421$$ 29.4721 1.43638 0.718192 0.695845i $$-0.244970\pi$$
0.718192 + 0.695845i $$0.244970\pi$$
$$422$$ −49.5967 −2.41433
$$423$$ 11.4721 0.557794
$$424$$ 96.7214 4.69720
$$425$$ 0 0
$$426$$ 11.7082 0.567264
$$427$$ 0 0
$$428$$ 96.8115 4.67956
$$429$$ 3.47214 0.167636
$$430$$ −2.76393 −0.133289
$$431$$ −17.3607 −0.836235 −0.418117 0.908393i $$-0.637310\pi$$
−0.418117 + 0.908393i $$0.637310\pi$$
$$432$$ −9.85410 −0.474106
$$433$$ −12.5836 −0.604729 −0.302364 0.953192i $$-0.597776\pi$$
−0.302364 + 0.953192i $$0.597776\pi$$
$$434$$ 0 0
$$435$$ −22.8885 −1.09742
$$436$$ 29.1246 1.39482
$$437$$ 10.4721 0.500950
$$438$$ −2.61803 −0.125095
$$439$$ 8.12461 0.387767 0.193883 0.981025i $$-0.437892\pi$$
0.193883 + 0.981025i $$0.437892\pi$$
$$440$$ −16.7082 −0.796532
$$441$$ 0 0
$$442$$ −54.5410 −2.59425
$$443$$ 9.05573 0.430251 0.215125 0.976586i $$-0.430984\pi$$
0.215125 + 0.976586i $$0.430984\pi$$
$$444$$ 2.56231 0.121602
$$445$$ 30.0000 1.42214
$$446$$ −47.1246 −2.23142
$$447$$ −15.6525 −0.740337
$$448$$ 0 0
$$449$$ −25.4164 −1.19947 −0.599737 0.800197i $$-0.704728\pi$$
−0.599737 + 0.800197i $$0.704728\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 31.4164 1.47770
$$453$$ 6.94427 0.326270
$$454$$ −70.2492 −3.29696
$$455$$ 0 0
$$456$$ −31.6525 −1.48226
$$457$$ −11.4164 −0.534037 −0.267019 0.963691i $$-0.586038\pi$$
−0.267019 + 0.963691i $$0.586038\pi$$
$$458$$ −40.3607 −1.88593
$$459$$ −6.00000 −0.280056
$$460$$ 26.8328 1.25109
$$461$$ 16.4721 0.767184 0.383592 0.923503i $$-0.374687\pi$$
0.383592 + 0.923503i $$0.374687\pi$$
$$462$$ 0 0
$$463$$ −2.70820 −0.125861 −0.0629305 0.998018i $$-0.520045\pi$$
−0.0629305 + 0.998018i $$0.520045\pi$$
$$464$$ −100.867 −4.68264
$$465$$ 18.9443 0.878520
$$466$$ −35.1246 −1.62712
$$467$$ 23.8328 1.10285 0.551426 0.834224i $$-0.314084\pi$$
0.551426 + 0.834224i $$0.314084\pi$$
$$468$$ 16.8541 0.779081
$$469$$ 0 0
$$470$$ 67.1591 3.09782
$$471$$ −3.41641 −0.157420
$$472$$ 89.2492 4.10803
$$473$$ 0.472136 0.0217088
$$474$$ −6.47214 −0.297275
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.9443 −0.592677
$$478$$ −45.4508 −2.07887
$$479$$ 21.8885 1.00011 0.500057 0.865993i $$-0.333313\pi$$
0.500057 + 0.865993i $$0.333313\pi$$
$$480$$ −24.2705 −1.10779
$$481$$ −1.83282 −0.0835692
$$482$$ 49.7426 2.26572
$$483$$ 0 0
$$484$$ 4.85410 0.220641
$$485$$ 15.5279 0.705084
$$486$$ 2.61803 0.118756
$$487$$ −21.8885 −0.991865 −0.495932 0.868361i $$-0.665174\pi$$
−0.495932 + 0.868361i $$0.665174\pi$$
$$488$$ 51.8885 2.34888
$$489$$ 18.1246 0.819623
$$490$$ 0 0
$$491$$ −10.8885 −0.491393 −0.245697 0.969347i $$-0.579017\pi$$
−0.245697 + 0.969347i $$0.579017\pi$$
$$492$$ −29.1246 −1.31304
$$493$$ −61.4164 −2.76606
$$494$$ 38.5066 1.73249
$$495$$ 2.23607 0.100504
$$496$$ 83.4853 3.74860
$$497$$ 0 0
$$498$$ 32.6525 1.46319
$$499$$ 22.1246 0.990434 0.495217 0.868769i $$-0.335089\pi$$
0.495217 + 0.868769i $$0.335089\pi$$
$$500$$ 54.2705 2.42705
$$501$$ −18.4721 −0.825274
$$502$$ −10.3262 −0.460883
$$503$$ 10.5836 0.471899 0.235950 0.971765i $$-0.424180\pi$$
0.235950 + 0.971765i $$0.424180\pi$$
$$504$$ 0 0
$$505$$ −7.88854 −0.351036
$$506$$ −6.47214 −0.287722
$$507$$ 0.944272 0.0419366
$$508$$ 77.6656 3.44586
$$509$$ 35.5279 1.57474 0.787372 0.616478i $$-0.211441\pi$$
0.787372 + 0.616478i $$0.211441\pi$$
$$510$$ −35.1246 −1.55534
$$511$$ 0 0
$$512$$ 40.3050 1.78124
$$513$$ 4.23607 0.187027
$$514$$ 20.3262 0.896552
$$515$$ 16.5836 0.730760
$$516$$ 2.29180 0.100891
$$517$$ −11.4721 −0.504544
$$518$$ 0 0
$$519$$ −5.52786 −0.242646
$$520$$ 58.0132 2.54405
$$521$$ −23.7639 −1.04112 −0.520558 0.853826i $$-0.674276\pi$$
−0.520558 + 0.853826i $$0.674276\pi$$
$$522$$ 26.7984 1.17293
$$523$$ 3.29180 0.143940 0.0719701 0.997407i $$-0.477071\pi$$
0.0719701 + 0.997407i $$0.477071\pi$$
$$524$$ −21.7082 −0.948327
$$525$$ 0 0
$$526$$ 11.5623 0.504140
$$527$$ 50.8328 2.21431
$$528$$ 9.85410 0.428845
$$529$$ −16.8885 −0.734285
$$530$$ −75.7771 −3.29155
$$531$$ −11.9443 −0.518337
$$532$$ 0 0
$$533$$ 20.8328 0.902369
$$534$$ −35.1246 −1.51999
$$535$$ −44.5967 −1.92809
$$536$$ −5.29180 −0.228571
$$537$$ 5.52786 0.238545
$$538$$ 43.1246 1.85923
$$539$$ 0 0
$$540$$ 10.8541 0.467086
$$541$$ 21.4164 0.920763 0.460382 0.887721i $$-0.347713\pi$$
0.460382 + 0.887721i $$0.347713\pi$$
$$542$$ −13.8541 −0.595085
$$543$$ −0.472136 −0.0202613
$$544$$ −65.1246 −2.79219
$$545$$ −13.4164 −0.574696
$$546$$ 0 0
$$547$$ −13.4164 −0.573644 −0.286822 0.957984i $$-0.592599\pi$$
−0.286822 + 0.957984i $$0.592599\pi$$
$$548$$ 29.1246 1.24414
$$549$$ −6.94427 −0.296374
$$550$$ 0 0
$$551$$ 43.3607 1.84723
$$552$$ −18.4721 −0.786226
$$553$$ 0 0
$$554$$ 49.5967 2.10716
$$555$$ −1.18034 −0.0501026
$$556$$ 38.8328 1.64688
$$557$$ −8.12461 −0.344251 −0.172125 0.985075i $$-0.555063\pi$$
−0.172125 + 0.985075i $$0.555063\pi$$
$$558$$ −22.1803 −0.938969
$$559$$ −1.63932 −0.0693359
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ −13.5623 −0.572091
$$563$$ −4.47214 −0.188478 −0.0942390 0.995550i $$-0.530042\pi$$
−0.0942390 + 0.995550i $$0.530042\pi$$
$$564$$ −55.6869 −2.34484
$$565$$ −14.4721 −0.608847
$$566$$ −12.3262 −0.518110
$$567$$ 0 0
$$568$$ −33.4164 −1.40212
$$569$$ −12.4721 −0.522859 −0.261430 0.965223i $$-0.584194\pi$$
−0.261430 + 0.965223i $$0.584194\pi$$
$$570$$ 24.7984 1.03869
$$571$$ −10.5836 −0.442910 −0.221455 0.975171i $$-0.571081\pi$$
−0.221455 + 0.975171i $$0.571081\pi$$
$$572$$ −16.8541 −0.704705
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 8.70820 0.362842
$$577$$ 34.9443 1.45475 0.727375 0.686241i $$-0.240740\pi$$
0.727375 + 0.686241i $$0.240740\pi$$
$$578$$ −49.7426 −2.06902
$$579$$ −25.8885 −1.07589
$$580$$ 111.103 4.61331
$$581$$ 0 0
$$582$$ −18.1803 −0.753599
$$583$$ 12.9443 0.536097
$$584$$ 7.47214 0.309199
$$585$$ −7.76393 −0.320999
$$586$$ 25.8885 1.06945
$$587$$ 14.0557 0.580142 0.290071 0.957005i $$-0.406321\pi$$
0.290071 + 0.957005i $$0.406321\pi$$
$$588$$ 0 0
$$589$$ −35.8885 −1.47876
$$590$$ −69.9230 −2.87868
$$591$$ 8.47214 0.348497
$$592$$ −5.20163 −0.213786
$$593$$ −37.4164 −1.53651 −0.768254 0.640145i $$-0.778874\pi$$
−0.768254 + 0.640145i $$0.778874\pi$$
$$594$$ −2.61803 −0.107419
$$595$$ 0 0
$$596$$ 75.9787 3.11221
$$597$$ −24.4721 −1.00158
$$598$$ 22.4721 0.918954
$$599$$ 25.3050 1.03393 0.516966 0.856006i $$-0.327061\pi$$
0.516966 + 0.856006i $$0.327061\pi$$
$$600$$ 0 0
$$601$$ 5.83282 0.237926 0.118963 0.992899i $$-0.462043\pi$$
0.118963 + 0.992899i $$0.462043\pi$$
$$602$$ 0 0
$$603$$ 0.708204 0.0288403
$$604$$ −33.7082 −1.37157
$$605$$ −2.23607 −0.0909091
$$606$$ 9.23607 0.375190
$$607$$ −22.7082 −0.921698 −0.460849 0.887479i $$-0.652455\pi$$
−0.460849 + 0.887479i $$0.652455\pi$$
$$608$$ 45.9787 1.86468
$$609$$ 0 0
$$610$$ −40.6525 −1.64597
$$611$$ 39.8328 1.61146
$$612$$ 29.1246 1.17729
$$613$$ 19.4164 0.784221 0.392111 0.919918i $$-0.371745\pi$$
0.392111 + 0.919918i $$0.371745\pi$$
$$614$$ −41.8885 −1.69048
$$615$$ 13.4164 0.541002
$$616$$ 0 0
$$617$$ −40.2492 −1.62037 −0.810186 0.586172i $$-0.800634\pi$$
−0.810186 + 0.586172i $$0.800634\pi$$
$$618$$ −19.4164 −0.781042
$$619$$ 27.4164 1.10196 0.550979 0.834519i $$-0.314254\pi$$
0.550979 + 0.834519i $$0.314254\pi$$
$$620$$ −91.9574 −3.69310
$$621$$ 2.47214 0.0992034
$$622$$ 33.8885 1.35881
$$623$$ 0 0
$$624$$ −34.2148 −1.36969
$$625$$ −25.0000 −1.00000
$$626$$ −69.0132 −2.75832
$$627$$ −4.23607 −0.169172
$$628$$ 16.5836 0.661757
$$629$$ −3.16718 −0.126284
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 18.4721 0.734782
$$633$$ −18.9443 −0.752967
$$634$$ −70.5410 −2.80154
$$635$$ −35.7771 −1.41977
$$636$$ 62.8328 2.49148
$$637$$ 0 0
$$638$$ −26.7984 −1.06096
$$639$$ 4.47214 0.176915
$$640$$ 2.43769 0.0963583
$$641$$ 31.5279 1.24528 0.622638 0.782510i $$-0.286061\pi$$
0.622638 + 0.782510i $$0.286061\pi$$
$$642$$ 52.2148 2.06075
$$643$$ 5.41641 0.213602 0.106801 0.994280i $$-0.465939\pi$$
0.106801 + 0.994280i $$0.465939\pi$$
$$644$$ 0 0
$$645$$ −1.05573 −0.0415693
$$646$$ 66.5410 2.61802
$$647$$ −4.41641 −0.173627 −0.0868135 0.996225i $$-0.527668\pi$$
−0.0868135 + 0.996225i $$0.527668\pi$$
$$648$$ −7.47214 −0.293533
$$649$$ 11.9443 0.468854
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −87.9787 −3.44551
$$653$$ 5.88854 0.230437 0.115218 0.993340i $$-0.463243\pi$$
0.115218 + 0.993340i $$0.463243\pi$$
$$654$$ 15.7082 0.614239
$$655$$ 10.0000 0.390732
$$656$$ 59.1246 2.30843
$$657$$ −1.00000 −0.0390137
$$658$$ 0 0
$$659$$ 32.8885 1.28116 0.640578 0.767893i $$-0.278695\pi$$
0.640578 + 0.767893i $$0.278695\pi$$
$$660$$ −10.8541 −0.422495
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 41.8885 1.62804
$$663$$ −20.8328 −0.809080
$$664$$ −93.1935 −3.61661
$$665$$ 0 0
$$666$$ 1.38197 0.0535501
$$667$$ 25.3050 0.979812
$$668$$ 89.6656 3.46927
$$669$$ −18.0000 −0.695920
$$670$$ 4.14590 0.160170
$$671$$ 6.94427 0.268081
$$672$$ 0 0
$$673$$ −18.5836 −0.716345 −0.358172 0.933655i $$-0.616600\pi$$
−0.358172 + 0.933655i $$0.616600\pi$$
$$674$$ −29.8885 −1.15126
$$675$$ 0 0
$$676$$ −4.58359 −0.176292
$$677$$ 43.4164 1.66863 0.834314 0.551289i $$-0.185864\pi$$
0.834314 + 0.551289i $$0.185864\pi$$
$$678$$ 16.9443 0.650741
$$679$$ 0 0
$$680$$ 100.249 3.84438
$$681$$ −26.8328 −1.02824
$$682$$ 22.1803 0.849329
$$683$$ 9.52786 0.364574 0.182287 0.983245i $$-0.441650\pi$$
0.182287 + 0.983245i $$0.441650\pi$$
$$684$$ −20.5623 −0.786219
$$685$$ −13.4164 −0.512615
$$686$$ 0 0
$$687$$ −15.4164 −0.588173
$$688$$ −4.65248 −0.177374
$$689$$ −44.9443 −1.71224
$$690$$ 14.4721 0.550945
$$691$$ 6.58359 0.250452 0.125226 0.992128i $$-0.460034\pi$$
0.125226 + 0.992128i $$0.460034\pi$$
$$692$$ 26.8328 1.02003
$$693$$ 0 0
$$694$$ −2.47214 −0.0938410
$$695$$ −17.8885 −0.678551
$$696$$ −76.4853 −2.89917
$$697$$ 36.0000 1.36360
$$698$$ 6.32624 0.239452
$$699$$ −13.4164 −0.507455
$$700$$ 0 0
$$701$$ −13.4164 −0.506731 −0.253365 0.967371i $$-0.581537\pi$$
−0.253365 + 0.967371i $$0.581537\pi$$
$$702$$ 9.09017 0.343086
$$703$$ 2.23607 0.0843349
$$704$$ −8.70820 −0.328203
$$705$$ 25.6525 0.966128
$$706$$ 11.0902 0.417384
$$707$$ 0 0
$$708$$ 57.9787 2.17897
$$709$$ 22.4164 0.841866 0.420933 0.907092i $$-0.361703\pi$$
0.420933 + 0.907092i $$0.361703\pi$$
$$710$$ 26.1803 0.982531
$$711$$ −2.47214 −0.0927123
$$712$$ 100.249 3.75700
$$713$$ −20.9443 −0.784369
$$714$$ 0 0
$$715$$ 7.76393 0.290355
$$716$$ −26.8328 −1.00279
$$717$$ −17.3607 −0.648346
$$718$$ 54.8328 2.04634
$$719$$ −4.63932 −0.173017 −0.0865087 0.996251i $$-0.527571\pi$$
−0.0865087 + 0.996251i $$0.527571\pi$$
$$720$$ −22.0344 −0.821175
$$721$$ 0 0
$$722$$ 2.76393 0.102863
$$723$$ 19.0000 0.706618
$$724$$ 2.29180 0.0851739
$$725$$ 0 0
$$726$$ 2.61803 0.0971644
$$727$$ 4.58359 0.169996 0.0849980 0.996381i $$-0.472912\pi$$
0.0849980 + 0.996381i $$0.472912\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −5.85410 −0.216670
$$731$$ −2.83282 −0.104775
$$732$$ 33.7082 1.24589
$$733$$ 29.0557 1.07320 0.536599 0.843837i $$-0.319709\pi$$
0.536599 + 0.843837i $$0.319709\pi$$
$$734$$ 66.5410 2.45607
$$735$$ 0 0
$$736$$ 26.8328 0.989071
$$737$$ −0.708204 −0.0260870
$$738$$ −15.7082 −0.578227
$$739$$ 43.4164 1.59710 0.798549 0.601930i $$-0.205601\pi$$
0.798549 + 0.601930i $$0.205601\pi$$
$$740$$ 5.72949 0.210620
$$741$$ 14.7082 0.540319
$$742$$ 0 0
$$743$$ 12.3050 0.451425 0.225712 0.974194i $$-0.427529\pi$$
0.225712 + 0.974194i $$0.427529\pi$$
$$744$$ 63.3050 2.32087
$$745$$ −35.0000 −1.28230
$$746$$ 44.3607 1.62416
$$747$$ 12.4721 0.456332
$$748$$ −29.1246 −1.06490
$$749$$ 0 0
$$750$$ 29.2705 1.06881
$$751$$ 12.1246 0.442433 0.221217 0.975225i $$-0.428997\pi$$
0.221217 + 0.975225i $$0.428997\pi$$
$$752$$ 113.048 4.12242
$$753$$ −3.94427 −0.143737
$$754$$ 93.0476 3.38859
$$755$$ 15.5279 0.565117
$$756$$ 0 0
$$757$$ −11.5836 −0.421013 −0.210506 0.977592i $$-0.567511\pi$$
−0.210506 + 0.977592i $$0.567511\pi$$
$$758$$ 67.1591 2.43933
$$759$$ −2.47214 −0.0897329
$$760$$ −70.7771 −2.56735
$$761$$ 13.5279 0.490385 0.245192 0.969474i $$-0.421149\pi$$
0.245192 + 0.969474i $$0.421149\pi$$
$$762$$ 41.8885 1.51746
$$763$$ 0 0
$$764$$ 58.2492 2.10738
$$765$$ −13.4164 −0.485071
$$766$$ −23.4164 −0.846069
$$767$$ −41.4721 −1.49747
$$768$$ 14.5623 0.525472
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ 7.76393 0.279611
$$772$$ 125.666 4.52281
$$773$$ −20.3475 −0.731850 −0.365925 0.930644i $$-0.619247\pi$$
−0.365925 + 0.930644i $$0.619247\pi$$
$$774$$ 1.23607 0.0444295
$$775$$ 0 0
$$776$$ 51.8885 1.86269
$$777$$ 0 0
$$778$$ −54.5410 −1.95539
$$779$$ −25.4164 −0.910637
$$780$$ 37.6869 1.34941
$$781$$ −4.47214 −0.160026
$$782$$ 38.8328 1.38866
$$783$$ 10.2361 0.365807
$$784$$ 0 0
$$785$$ −7.63932 −0.272659
$$786$$ −11.7082 −0.417618
$$787$$ 38.1246 1.35900 0.679498 0.733678i $$-0.262198\pi$$
0.679498 + 0.733678i $$0.262198\pi$$
$$788$$ −41.1246 −1.46500
$$789$$ 4.41641 0.157228
$$790$$ −14.4721 −0.514895
$$791$$ 0 0
$$792$$ 7.47214 0.265511
$$793$$ −24.1115 −0.856223
$$794$$ −24.6525 −0.874884
$$795$$ −28.9443 −1.02655
$$796$$ 118.790 4.21041
$$797$$ −45.7639 −1.62104 −0.810521 0.585710i $$-0.800816\pi$$
−0.810521 + 0.585710i $$0.800816\pi$$
$$798$$ 0 0
$$799$$ 68.8328 2.43513
$$800$$ 0 0
$$801$$ −13.4164 −0.474045
$$802$$ 22.1803 0.783215
$$803$$ 1.00000 0.0352892
$$804$$ −3.43769 −0.121238
$$805$$ 0 0
$$806$$ −77.0132 −2.71267
$$807$$ 16.4721 0.579847
$$808$$ −26.3607 −0.927365
$$809$$ −14.8197 −0.521032 −0.260516 0.965470i $$-0.583893\pi$$
−0.260516 + 0.965470i $$0.583893\pi$$
$$810$$ 5.85410 0.205692
$$811$$ −27.5410 −0.967096 −0.483548 0.875318i $$-0.660652\pi$$
−0.483548 + 0.875318i $$0.660652\pi$$
$$812$$ 0 0
$$813$$ −5.29180 −0.185591
$$814$$ −1.38197 −0.0484379
$$815$$ 40.5279 1.41963
$$816$$ −59.1246 −2.06978
$$817$$ 2.00000 0.0699711
$$818$$ −36.6525 −1.28152
$$819$$ 0 0
$$820$$ −65.1246 −2.27425
$$821$$ −33.5410 −1.17059 −0.585295 0.810821i $$-0.699021\pi$$
−0.585295 + 0.810821i $$0.699021\pi$$
$$822$$ 15.7082 0.547887
$$823$$ −45.5410 −1.58746 −0.793730 0.608270i $$-0.791864\pi$$
−0.793730 + 0.608270i $$0.791864\pi$$
$$824$$ 55.4164 1.93052
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0.167184 0.00581357 0.00290678 0.999996i $$-0.499075\pi$$
0.00290678 + 0.999996i $$0.499075\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −34.8328 −1.20979 −0.604897 0.796304i $$-0.706786\pi$$
−0.604897 + 0.796304i $$0.706786\pi$$
$$830$$ 73.0132 2.53432
$$831$$ 18.9443 0.657170
$$832$$ 30.2361 1.04825
$$833$$ 0 0
$$834$$ 20.9443 0.725241
$$835$$ −41.3050 −1.42942
$$836$$ 20.5623 0.711162
$$837$$ −8.47214 −0.292840
$$838$$ −7.85410 −0.271315
$$839$$ −20.5279 −0.708701 −0.354350 0.935113i $$-0.615298\pi$$
−0.354350 + 0.935113i $$0.615298\pi$$
$$840$$ 0 0
$$841$$ 75.7771 2.61300
$$842$$ −77.1591 −2.65908
$$843$$ −5.18034 −0.178420
$$844$$ 91.9574 3.16531
$$845$$ 2.11146 0.0726363
$$846$$ −30.0344 −1.03261
$$847$$ 0 0
$$848$$ −127.554 −4.38023
$$849$$ −4.70820 −0.161585
$$850$$ 0 0
$$851$$ 1.30495 0.0447332
$$852$$ −21.7082 −0.743711
$$853$$ −54.7214 −1.87362 −0.936812 0.349834i $$-0.886238\pi$$
−0.936812 + 0.349834i $$0.886238\pi$$
$$854$$ 0 0
$$855$$ 9.47214 0.323940
$$856$$ −149.026 −5.09361
$$857$$ −50.7214 −1.73261 −0.866304 0.499517i $$-0.833511\pi$$
−0.866304 + 0.499517i $$0.833511\pi$$
$$858$$ −9.09017 −0.310333
$$859$$ 15.4164 0.526001 0.263001 0.964796i $$-0.415288\pi$$
0.263001 + 0.964796i $$0.415288\pi$$
$$860$$ 5.12461 0.174748
$$861$$ 0 0
$$862$$ 45.4508 1.54806
$$863$$ 35.3050 1.20179 0.600897 0.799326i $$-0.294810\pi$$
0.600897 + 0.799326i $$0.294810\pi$$
$$864$$ 10.8541 0.369264
$$865$$ −12.3607 −0.420276
$$866$$ 32.9443 1.11949
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 2.47214 0.0838615
$$870$$ 59.9230 2.03158
$$871$$ 2.45898 0.0833194
$$872$$ −44.8328 −1.51823
$$873$$ −6.94427 −0.235028
$$874$$ −27.4164 −0.927374
$$875$$ 0 0
$$876$$ 4.85410 0.164005
$$877$$ −21.8885 −0.739124 −0.369562 0.929206i $$-0.620492\pi$$
−0.369562 + 0.929206i $$0.620492\pi$$
$$878$$ −21.2705 −0.717845
$$879$$ 9.88854 0.333532
$$880$$ 22.0344 0.742781
$$881$$ −22.5967 −0.761304 −0.380652 0.924718i $$-0.624300\pi$$
−0.380652 + 0.924718i $$0.624300\pi$$
$$882$$ 0 0
$$883$$ 4.23607 0.142555 0.0712775 0.997457i $$-0.477292\pi$$
0.0712775 + 0.997457i $$0.477292\pi$$
$$884$$ 101.125 3.40119
$$885$$ −26.7082 −0.897786
$$886$$ −23.7082 −0.796493
$$887$$ −28.3607 −0.952258 −0.476129 0.879375i $$-0.657961\pi$$
−0.476129 + 0.879375i $$0.657961\pi$$
$$888$$ −3.94427 −0.132361
$$889$$ 0 0
$$890$$ −78.5410 −2.63270
$$891$$ −1.00000 −0.0335013
$$892$$ 87.3738 2.92549
$$893$$ −48.5967 −1.62623
$$894$$ 40.9787 1.37053
$$895$$ 12.3607 0.413172
$$896$$ 0 0
$$897$$ 8.58359 0.286598
$$898$$ 66.5410 2.22050
$$899$$ −86.7214 −2.89232
$$900$$ 0 0
$$901$$ −77.6656 −2.58742
$$902$$ 15.7082 0.523026
$$903$$ 0 0
$$904$$ −48.3607 −1.60845
$$905$$ −1.05573 −0.0350936
$$906$$ −18.1803 −0.604001
$$907$$ −14.1115 −0.468563 −0.234282 0.972169i $$-0.575274\pi$$
−0.234282 + 0.972169i $$0.575274\pi$$
$$908$$ 130.249 4.32247
$$909$$ 3.52786 0.117012
$$910$$ 0 0
$$911$$ −31.4164 −1.04087 −0.520436 0.853901i $$-0.674231\pi$$
−0.520436 + 0.853901i $$0.674231\pi$$
$$912$$ 41.7426 1.38224
$$913$$ −12.4721 −0.412767
$$914$$ 29.8885 0.988625
$$915$$ −15.5279 −0.513335
$$916$$ 74.8328 2.47255
$$917$$ 0 0
$$918$$ 15.7082 0.518448
$$919$$ 1.88854 0.0622973 0.0311487 0.999515i $$-0.490083\pi$$
0.0311487 + 0.999515i $$0.490083\pi$$
$$920$$ −41.3050 −1.36178
$$921$$ −16.0000 −0.527218
$$922$$ −43.1246 −1.42023
$$923$$ 15.5279 0.511106
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 7.09017 0.232997
$$927$$ −7.41641 −0.243587
$$928$$ 111.103 3.64715
$$929$$ 52.4853 1.72199 0.860993 0.508616i $$-0.169843\pi$$
0.860993 + 0.508616i $$0.169843\pi$$
$$930$$ −49.5967 −1.62634
$$931$$ 0 0
$$932$$ 65.1246 2.13323
$$933$$ 12.9443 0.423776
$$934$$ −62.3951 −2.04163
$$935$$ 13.4164 0.438763
$$936$$ −25.9443 −0.848015
$$937$$ 42.7214 1.39565 0.697823 0.716270i $$-0.254152\pi$$
0.697823 + 0.716270i $$0.254152\pi$$
$$938$$ 0 0
$$939$$ −26.3607 −0.860248
$$940$$ −124.520 −4.06139
$$941$$ 51.1935 1.66886 0.834430 0.551114i $$-0.185797\pi$$
0.834430 + 0.551114i $$0.185797\pi$$
$$942$$ 8.94427 0.291420
$$943$$ −14.8328 −0.483023
$$944$$ −117.700 −3.83081
$$945$$ 0 0
$$946$$ −1.23607 −0.0401880
$$947$$ 0.111456 0.00362184 0.00181092 0.999998i $$-0.499424\pi$$
0.00181092 + 0.999998i $$0.499424\pi$$
$$948$$ 12.0000 0.389742
$$949$$ −3.47214 −0.112710
$$950$$ 0 0
$$951$$ −26.9443 −0.873728
$$952$$ 0 0
$$953$$ −28.4853 −0.922729 −0.461365 0.887211i $$-0.652640\pi$$
−0.461365 + 0.887211i $$0.652640\pi$$
$$954$$ 33.8885 1.09718
$$955$$ −26.8328 −0.868290
$$956$$ 84.2705 2.72550
$$957$$ −10.2361 −0.330885
$$958$$ −57.3050 −1.85144
$$959$$ 0 0
$$960$$ 19.4721 0.628460
$$961$$ 40.7771 1.31539
$$962$$ 4.79837 0.154706
$$963$$ 19.9443 0.642695
$$964$$ −92.2279 −2.97046
$$965$$ −57.8885 −1.86350
$$966$$ 0 0
$$967$$ 46.2492 1.48727 0.743637 0.668583i $$-0.233099\pi$$
0.743637 + 0.668583i $$0.233099\pi$$
$$968$$ −7.47214 −0.240164
$$969$$ 25.4164 0.816493
$$970$$ −40.6525 −1.30527
$$971$$ 24.8885 0.798711 0.399356 0.916796i $$-0.369234\pi$$
0.399356 + 0.916796i $$0.369234\pi$$
$$972$$ −4.85410 −0.155695
$$973$$ 0 0
$$974$$ 57.3050 1.83617
$$975$$ 0 0
$$976$$ −68.4296 −2.19038
$$977$$ −13.5279 −0.432795 −0.216397 0.976305i $$-0.569431\pi$$
−0.216397 + 0.976305i $$0.569431\pi$$
$$978$$ −47.4508 −1.51731
$$979$$ 13.4164 0.428790
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ 28.5066 0.909681
$$983$$ 38.8328 1.23857 0.619287 0.785165i $$-0.287422\pi$$
0.619287 + 0.785165i $$0.287422\pi$$
$$984$$ 44.8328 1.42922
$$985$$ 18.9443 0.603615
$$986$$ 160.790 5.12060
$$987$$ 0 0
$$988$$ −71.3951 −2.27138
$$989$$ 1.16718 0.0371143
$$990$$ −5.85410 −0.186056
$$991$$ 13.2918 0.422228 0.211114 0.977461i $$-0.432291\pi$$
0.211114 + 0.977461i $$0.432291\pi$$
$$992$$ −91.9574 −2.91965
$$993$$ 16.0000 0.507745
$$994$$ 0 0
$$995$$ −54.7214 −1.73478
$$996$$ −60.5410 −1.91832
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ −57.9230 −1.83352
$$999$$ 0.527864 0.0167009
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.k.1.1 2
3.2 odd 2 4851.2.a.bh.1.2 2
7.6 odd 2 1617.2.a.l.1.1 yes 2
21.20 even 2 4851.2.a.bg.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.k.1.1 2 1.1 even 1 trivial
1617.2.a.l.1.1 yes 2 7.6 odd 2
4851.2.a.bg.1.2 2 21.20 even 2
4851.2.a.bh.1.2 2 3.2 odd 2