# Properties

 Label 405.2.a.i.1.1 Level $405$ Weight $2$ Character 405.1 Self dual yes Analytic conductor $3.234$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.51414$$ of defining polynomial Character $$\chi$$ $$=$$ 405.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} -0.514137 q^{7} -5.83502 q^{8} +O(q^{10})$$ $$q-2.51414 q^{2} +4.32088 q^{4} -1.00000 q^{5} -0.514137 q^{7} -5.83502 q^{8} +2.51414 q^{10} +3.32088 q^{11} -1.32088 q^{13} +1.29261 q^{14} +6.02827 q^{16} +3.32088 q^{17} -1.32088 q^{19} -4.32088 q^{20} -8.34916 q^{22} -4.12763 q^{23} +1.00000 q^{25} +3.32088 q^{26} -2.22153 q^{28} +1.38650 q^{29} +8.73566 q^{31} -3.48586 q^{32} -8.34916 q^{34} +0.514137 q^{35} +0.292611 q^{37} +3.32088 q^{38} +5.83502 q^{40} +11.3492 q^{41} +10.3492 q^{43} +14.3492 q^{44} +10.3774 q^{46} +4.86330 q^{47} -6.73566 q^{49} -2.51414 q^{50} -5.70739 q^{52} +5.02827 q^{53} -3.32088 q^{55} +3.00000 q^{56} -3.48586 q^{58} +5.02827 q^{59} +7.34916 q^{61} -21.9627 q^{62} -3.29261 q^{64} +1.32088 q^{65} +9.44852 q^{67} +14.3492 q^{68} -1.29261 q^{70} -8.99093 q^{71} +6.05655 q^{73} -0.735663 q^{74} -5.70739 q^{76} -1.70739 q^{77} -8.05655 q^{79} -6.02827 q^{80} -28.5333 q^{82} -1.54241 q^{83} -3.32088 q^{85} -26.0192 q^{86} -19.3774 q^{88} +3.00000 q^{89} +0.679116 q^{91} -17.8350 q^{92} -12.2270 q^{94} +1.32088 q^{95} -12.2553 q^{97} +16.9344 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 $$3 q - q^{2} + 5 q^{4} - 3 q^{5} + 5 q^{7} - 3 q^{8} + q^{10} + 2 q^{11} + 4 q^{13} + 9 q^{14} + 5 q^{16} + 2 q^{17} + 4 q^{19} - 5 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 2 q^{26} + 5 q^{28} + 7 q^{29} + 8 q^{31} - 17 q^{32} - 4 q^{34} - 5 q^{35} + 6 q^{37} + 2 q^{38} + 3 q^{40} + 13 q^{41} + 10 q^{43} + 22 q^{44} - 3 q^{46} - 13 q^{47} - 2 q^{49} - q^{50} - 12 q^{52} + 2 q^{53} - 2 q^{55} + 9 q^{56} - 17 q^{58} + 2 q^{59} + q^{61} - 42 q^{62} - 15 q^{64} - 4 q^{65} + 11 q^{67} + 22 q^{68} - 9 q^{70} + 10 q^{71} - 8 q^{73} + 16 q^{74} - 12 q^{76} + 2 q^{79} - 5 q^{80} - 29 q^{82} + 15 q^{83} - 2 q^{85} - 28 q^{86} - 24 q^{88} + 9 q^{89} + 10 q^{91} - 39 q^{92} - 31 q^{94} - 4 q^{95} - 18 q^{97} + 40 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 + 5 * q^7 - 3 * q^8 + q^10 + 2 * q^11 + 4 * q^13 + 9 * q^14 + 5 * q^16 + 2 * q^17 + 4 * q^19 - 5 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 2 * q^26 + 5 * q^28 + 7 * q^29 + 8 * q^31 - 17 * q^32 - 4 * q^34 - 5 * q^35 + 6 * q^37 + 2 * q^38 + 3 * q^40 + 13 * q^41 + 10 * q^43 + 22 * q^44 - 3 * q^46 - 13 * q^47 - 2 * q^49 - q^50 - 12 * q^52 + 2 * q^53 - 2 * q^55 + 9 * q^56 - 17 * q^58 + 2 * q^59 + q^61 - 42 * q^62 - 15 * q^64 - 4 * q^65 + 11 * q^67 + 22 * q^68 - 9 * q^70 + 10 * q^71 - 8 * q^73 + 16 * q^74 - 12 * q^76 + 2 * q^79 - 5 * q^80 - 29 * q^82 + 15 * q^83 - 2 * q^85 - 28 * q^86 - 24 * q^88 + 9 * q^89 + 10 * q^91 - 39 * q^92 - 31 * q^94 - 4 * q^95 - 18 * q^97 + 40 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.51414 −1.77776 −0.888882 0.458137i $$-0.848517\pi$$
−0.888882 + 0.458137i $$0.848517\pi$$
$$3$$ 0 0
$$4$$ 4.32088 2.16044
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.514137 −0.194325 −0.0971627 0.995269i $$-0.530977\pi$$
−0.0971627 + 0.995269i $$0.530977\pi$$
$$8$$ −5.83502 −2.06299
$$9$$ 0 0
$$10$$ 2.51414 0.795040
$$11$$ 3.32088 1.00128 0.500642 0.865654i $$-0.333097\pi$$
0.500642 + 0.865654i $$0.333097\pi$$
$$12$$ 0 0
$$13$$ −1.32088 −0.366347 −0.183174 0.983081i $$-0.558637\pi$$
−0.183174 + 0.983081i $$0.558637\pi$$
$$14$$ 1.29261 0.345465
$$15$$ 0 0
$$16$$ 6.02827 1.50707
$$17$$ 3.32088 0.805433 0.402716 0.915325i $$-0.368066\pi$$
0.402716 + 0.915325i $$0.368066\pi$$
$$18$$ 0 0
$$19$$ −1.32088 −0.303032 −0.151516 0.988455i $$-0.548415\pi$$
−0.151516 + 0.988455i $$0.548415\pi$$
$$20$$ −4.32088 −0.966179
$$21$$ 0 0
$$22$$ −8.34916 −1.78005
$$23$$ −4.12763 −0.860671 −0.430335 0.902669i $$-0.641605\pi$$
−0.430335 + 0.902669i $$0.641605\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 3.32088 0.651279
$$27$$ 0 0
$$28$$ −2.22153 −0.419829
$$29$$ 1.38650 0.257467 0.128734 0.991679i $$-0.458909\pi$$
0.128734 + 0.991679i $$0.458909\pi$$
$$30$$ 0 0
$$31$$ 8.73566 1.56897 0.784486 0.620147i $$-0.212927\pi$$
0.784486 + 0.620147i $$0.212927\pi$$
$$32$$ −3.48586 −0.616219
$$33$$ 0 0
$$34$$ −8.34916 −1.43187
$$35$$ 0.514137 0.0869050
$$36$$ 0 0
$$37$$ 0.292611 0.0481049 0.0240524 0.999711i $$-0.492343\pi$$
0.0240524 + 0.999711i $$0.492343\pi$$
$$38$$ 3.32088 0.538719
$$39$$ 0 0
$$40$$ 5.83502 0.922598
$$41$$ 11.3492 1.77244 0.886220 0.463264i $$-0.153322\pi$$
0.886220 + 0.463264i $$0.153322\pi$$
$$42$$ 0 0
$$43$$ 10.3492 1.57823 0.789116 0.614244i $$-0.210539\pi$$
0.789116 + 0.614244i $$0.210539\pi$$
$$44$$ 14.3492 2.16322
$$45$$ 0 0
$$46$$ 10.3774 1.53007
$$47$$ 4.86330 0.709385 0.354692 0.934983i $$-0.384586\pi$$
0.354692 + 0.934983i $$0.384586\pi$$
$$48$$ 0 0
$$49$$ −6.73566 −0.962238
$$50$$ −2.51414 −0.355553
$$51$$ 0 0
$$52$$ −5.70739 −0.791472
$$53$$ 5.02827 0.690687 0.345343 0.938476i $$-0.387762\pi$$
0.345343 + 0.938476i $$0.387762\pi$$
$$54$$ 0 0
$$55$$ −3.32088 −0.447788
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −3.48586 −0.457716
$$59$$ 5.02827 0.654625 0.327313 0.944916i $$-0.393857\pi$$
0.327313 + 0.944916i $$0.393857\pi$$
$$60$$ 0 0
$$61$$ 7.34916 0.940963 0.470482 0.882410i $$-0.344080\pi$$
0.470482 + 0.882410i $$0.344080\pi$$
$$62$$ −21.9627 −2.78926
$$63$$ 0 0
$$64$$ −3.29261 −0.411576
$$65$$ 1.32088 0.163836
$$66$$ 0 0
$$67$$ 9.44852 1.15432 0.577160 0.816631i $$-0.304161\pi$$
0.577160 + 0.816631i $$0.304161\pi$$
$$68$$ 14.3492 1.74009
$$69$$ 0 0
$$70$$ −1.29261 −0.154497
$$71$$ −8.99093 −1.06703 −0.533513 0.845792i $$-0.679129\pi$$
−0.533513 + 0.845792i $$0.679129\pi$$
$$72$$ 0 0
$$73$$ 6.05655 0.708865 0.354433 0.935082i $$-0.384674\pi$$
0.354433 + 0.935082i $$0.384674\pi$$
$$74$$ −0.735663 −0.0855191
$$75$$ 0 0
$$76$$ −5.70739 −0.654682
$$77$$ −1.70739 −0.194575
$$78$$ 0 0
$$79$$ −8.05655 −0.906432 −0.453216 0.891401i $$-0.649723\pi$$
−0.453216 + 0.891401i $$0.649723\pi$$
$$80$$ −6.02827 −0.673982
$$81$$ 0 0
$$82$$ −28.5333 −3.15098
$$83$$ −1.54241 −0.169302 −0.0846508 0.996411i $$-0.526977\pi$$
−0.0846508 + 0.996411i $$0.526977\pi$$
$$84$$ 0 0
$$85$$ −3.32088 −0.360200
$$86$$ −26.0192 −2.80572
$$87$$ 0 0
$$88$$ −19.3774 −2.06564
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 0.679116 0.0711906
$$92$$ −17.8350 −1.85943
$$93$$ 0 0
$$94$$ −12.2270 −1.26112
$$95$$ 1.32088 0.135520
$$96$$ 0 0
$$97$$ −12.2553 −1.24433 −0.622167 0.782885i $$-0.713747\pi$$
−0.622167 + 0.782885i $$0.713747\pi$$
$$98$$ 16.9344 1.71063
$$99$$ 0 0
$$100$$ 4.32088 0.432088
$$101$$ −11.6700 −1.16121 −0.580606 0.814184i $$-0.697184\pi$$
−0.580606 + 0.814184i $$0.697184\pi$$
$$102$$ 0 0
$$103$$ 0.292611 0.0288318 0.0144159 0.999896i $$-0.495411\pi$$
0.0144159 + 0.999896i $$0.495411\pi$$
$$104$$ 7.70739 0.755772
$$105$$ 0 0
$$106$$ −12.6418 −1.22788
$$107$$ −1.87237 −0.181009 −0.0905043 0.995896i $$-0.528848\pi$$
−0.0905043 + 0.995896i $$0.528848\pi$$
$$108$$ 0 0
$$109$$ 5.54787 0.531390 0.265695 0.964057i $$-0.414399\pi$$
0.265695 + 0.964057i $$0.414399\pi$$
$$110$$ 8.34916 0.796061
$$111$$ 0 0
$$112$$ −3.09936 −0.292862
$$113$$ −7.80128 −0.733883 −0.366942 0.930244i $$-0.619595\pi$$
−0.366942 + 0.930244i $$0.619595\pi$$
$$114$$ 0 0
$$115$$ 4.12763 0.384904
$$116$$ 5.99093 0.556244
$$117$$ 0 0
$$118$$ −12.6418 −1.16377
$$119$$ −1.70739 −0.156516
$$120$$ 0 0
$$121$$ 0.0282739 0.00257035
$$122$$ −18.4768 −1.67281
$$123$$ 0 0
$$124$$ 37.7458 3.38967
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 17.8916 1.58762 0.793810 0.608166i $$-0.208094\pi$$
0.793810 + 0.608166i $$0.208094\pi$$
$$128$$ 15.2498 1.34790
$$129$$ 0 0
$$130$$ −3.32088 −0.291261
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 0.679116 0.0588868
$$134$$ −23.7549 −2.05211
$$135$$ 0 0
$$136$$ −19.3774 −1.66160
$$137$$ −5.67004 −0.484424 −0.242212 0.970223i $$-0.577873\pi$$
−0.242212 + 0.970223i $$0.577873\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 2.22153 0.187753
$$141$$ 0 0
$$142$$ 22.6044 1.89692
$$143$$ −4.38650 −0.366818
$$144$$ 0 0
$$145$$ −1.38650 −0.115143
$$146$$ −15.2270 −1.26019
$$147$$ 0 0
$$148$$ 1.26434 0.103928
$$149$$ −17.6610 −1.44684 −0.723422 0.690407i $$-0.757432\pi$$
−0.723422 + 0.690407i $$0.757432\pi$$
$$150$$ 0 0
$$151$$ 1.26434 0.102890 0.0514451 0.998676i $$-0.483617\pi$$
0.0514451 + 0.998676i $$0.483617\pi$$
$$152$$ 7.70739 0.625152
$$153$$ 0 0
$$154$$ 4.29261 0.345908
$$155$$ −8.73566 −0.701665
$$156$$ 0 0
$$157$$ −15.6700 −1.25061 −0.625303 0.780382i $$-0.715025\pi$$
−0.625303 + 0.780382i $$0.715025\pi$$
$$158$$ 20.2553 1.61142
$$159$$ 0 0
$$160$$ 3.48586 0.275582
$$161$$ 2.12217 0.167250
$$162$$ 0 0
$$163$$ 15.7074 1.23030 0.615149 0.788411i $$-0.289096\pi$$
0.615149 + 0.788411i $$0.289096\pi$$
$$164$$ 49.0384 3.82926
$$165$$ 0 0
$$166$$ 3.87783 0.300978
$$167$$ −6.16498 −0.477060 −0.238530 0.971135i $$-0.576666\pi$$
−0.238530 + 0.971135i $$0.576666\pi$$
$$168$$ 0 0
$$169$$ −11.2553 −0.865790
$$170$$ 8.34916 0.640351
$$171$$ 0 0
$$172$$ 44.7175 3.40968
$$173$$ −8.58522 −0.652722 −0.326361 0.945245i $$-0.605823\pi$$
−0.326361 + 0.945245i $$0.605823\pi$$
$$174$$ 0 0
$$175$$ −0.514137 −0.0388651
$$176$$ 20.0192 1.50900
$$177$$ 0 0
$$178$$ −7.54241 −0.565328
$$179$$ 1.06562 0.0796482 0.0398241 0.999207i $$-0.487320\pi$$
0.0398241 + 0.999207i $$0.487320\pi$$
$$180$$ 0 0
$$181$$ −12.6700 −0.941757 −0.470878 0.882198i $$-0.656063\pi$$
−0.470878 + 0.882198i $$0.656063\pi$$
$$182$$ −1.70739 −0.126560
$$183$$ 0 0
$$184$$ 24.0848 1.77556
$$185$$ −0.292611 −0.0215132
$$186$$ 0 0
$$187$$ 11.0283 0.806467
$$188$$ 21.0137 1.53258
$$189$$ 0 0
$$190$$ −3.32088 −0.240922
$$191$$ 16.9344 1.22533 0.612664 0.790343i $$-0.290098\pi$$
0.612664 + 0.790343i $$0.290098\pi$$
$$192$$ 0 0
$$193$$ 26.7175 1.92317 0.961585 0.274509i $$-0.0885153\pi$$
0.961585 + 0.274509i $$0.0885153\pi$$
$$194$$ 30.8114 2.21213
$$195$$ 0 0
$$196$$ −29.1040 −2.07886
$$197$$ 14.2553 1.01565 0.507823 0.861462i $$-0.330450\pi$$
0.507823 + 0.861462i $$0.330450\pi$$
$$198$$ 0 0
$$199$$ −24.6610 −1.74817 −0.874085 0.485773i $$-0.838538\pi$$
−0.874085 + 0.485773i $$0.838538\pi$$
$$200$$ −5.83502 −0.412598
$$201$$ 0 0
$$202$$ 29.3401 2.06436
$$203$$ −0.712853 −0.0500325
$$204$$ 0 0
$$205$$ −11.3492 −0.792660
$$206$$ −0.735663 −0.0512561
$$207$$ 0 0
$$208$$ −7.96265 −0.552111
$$209$$ −4.38650 −0.303421
$$210$$ 0 0
$$211$$ −5.37743 −0.370198 −0.185099 0.982720i $$-0.559261\pi$$
−0.185099 + 0.982720i $$0.559261\pi$$
$$212$$ 21.7266 1.49219
$$213$$ 0 0
$$214$$ 4.70739 0.321791
$$215$$ −10.3492 −0.705807
$$216$$ 0 0
$$217$$ −4.49133 −0.304891
$$218$$ −13.9481 −0.944686
$$219$$ 0 0
$$220$$ −14.3492 −0.967420
$$221$$ −4.38650 −0.295068
$$222$$ 0 0
$$223$$ 8.66458 0.580223 0.290112 0.956993i $$-0.406308\pi$$
0.290112 + 0.956993i $$0.406308\pi$$
$$224$$ 1.79221 0.119747
$$225$$ 0 0
$$226$$ 19.6135 1.30467
$$227$$ −3.32088 −0.220415 −0.110207 0.993909i $$-0.535152\pi$$
−0.110207 + 0.993909i $$0.535152\pi$$
$$228$$ 0 0
$$229$$ −25.3118 −1.67265 −0.836326 0.548233i $$-0.815301\pi$$
−0.836326 + 0.548233i $$0.815301\pi$$
$$230$$ −10.3774 −0.684268
$$231$$ 0 0
$$232$$ −8.09029 −0.531153
$$233$$ −27.6327 −1.81028 −0.905139 0.425116i $$-0.860233\pi$$
−0.905139 + 0.425116i $$0.860233\pi$$
$$234$$ 0 0
$$235$$ −4.86330 −0.317246
$$236$$ 21.7266 1.41428
$$237$$ 0 0
$$238$$ 4.29261 0.278249
$$239$$ 4.19872 0.271592 0.135796 0.990737i $$-0.456641\pi$$
0.135796 + 0.990737i $$0.456641\pi$$
$$240$$ 0 0
$$241$$ 3.60442 0.232181 0.116091 0.993239i $$-0.462964\pi$$
0.116091 + 0.993239i $$0.462964\pi$$
$$242$$ −0.0710844 −0.00456948
$$243$$ 0 0
$$244$$ 31.7549 2.03290
$$245$$ 6.73566 0.430326
$$246$$ 0 0
$$247$$ 1.74474 0.111015
$$248$$ −50.9728 −3.23677
$$249$$ 0 0
$$250$$ 2.51414 0.159008
$$251$$ 6.87783 0.434125 0.217062 0.976158i $$-0.430352\pi$$
0.217062 + 0.976158i $$0.430352\pi$$
$$252$$ 0 0
$$253$$ −13.7074 −0.861776
$$254$$ −44.9819 −2.82241
$$255$$ 0 0
$$256$$ −31.7549 −1.98468
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −0.150442 −0.00934801
$$260$$ 5.70739 0.353957
$$261$$ 0 0
$$262$$ −15.0848 −0.931943
$$263$$ −6.23606 −0.384532 −0.192266 0.981343i $$-0.561584\pi$$
−0.192266 + 0.981343i $$0.561584\pi$$
$$264$$ 0 0
$$265$$ −5.02827 −0.308884
$$266$$ −1.70739 −0.104687
$$267$$ 0 0
$$268$$ 40.8259 2.49384
$$269$$ −9.92345 −0.605044 −0.302522 0.953142i $$-0.597828\pi$$
−0.302522 + 0.953142i $$0.597828\pi$$
$$270$$ 0 0
$$271$$ 6.60442 0.401190 0.200595 0.979674i $$-0.435712\pi$$
0.200595 + 0.979674i $$0.435712\pi$$
$$272$$ 20.0192 1.21384
$$273$$ 0 0
$$274$$ 14.2553 0.861192
$$275$$ 3.32088 0.200257
$$276$$ 0 0
$$277$$ 22.6610 1.36157 0.680783 0.732485i $$-0.261640\pi$$
0.680783 + 0.732485i $$0.261640\pi$$
$$278$$ −20.1131 −1.20630
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ 15.5479 0.927508 0.463754 0.885964i $$-0.346502\pi$$
0.463754 + 0.885964i $$0.346502\pi$$
$$282$$ 0 0
$$283$$ 0.645378 0.0383637 0.0191819 0.999816i $$-0.493894\pi$$
0.0191819 + 0.999816i $$0.493894\pi$$
$$284$$ −38.8488 −2.30525
$$285$$ 0 0
$$286$$ 11.0283 0.652116
$$287$$ −5.83502 −0.344430
$$288$$ 0 0
$$289$$ −5.97173 −0.351278
$$290$$ 3.48586 0.204697
$$291$$ 0 0
$$292$$ 26.1696 1.53146
$$293$$ 1.37743 0.0804704 0.0402352 0.999190i $$-0.487189\pi$$
0.0402352 + 0.999190i $$0.487189\pi$$
$$294$$ 0 0
$$295$$ −5.02827 −0.292757
$$296$$ −1.70739 −0.0992400
$$297$$ 0 0
$$298$$ 44.4021 2.57214
$$299$$ 5.45213 0.315305
$$300$$ 0 0
$$301$$ −5.32088 −0.306691
$$302$$ −3.17872 −0.182915
$$303$$ 0 0
$$304$$ −7.96265 −0.456689
$$305$$ −7.34916 −0.420812
$$306$$ 0 0
$$307$$ −7.98546 −0.455754 −0.227877 0.973690i $$-0.573178\pi$$
−0.227877 + 0.973690i $$0.573178\pi$$
$$308$$ −7.37743 −0.420368
$$309$$ 0 0
$$310$$ 21.9627 1.24739
$$311$$ −9.63270 −0.546220 −0.273110 0.961983i $$-0.588052\pi$$
−0.273110 + 0.961983i $$0.588052\pi$$
$$312$$ 0 0
$$313$$ −24.5369 −1.38691 −0.693455 0.720500i $$-0.743912\pi$$
−0.693455 + 0.720500i $$0.743912\pi$$
$$314$$ 39.3966 2.22328
$$315$$ 0 0
$$316$$ −34.8114 −1.95829
$$317$$ 20.3492 1.14292 0.571461 0.820629i $$-0.306377\pi$$
0.571461 + 0.820629i $$0.306377\pi$$
$$318$$ 0 0
$$319$$ 4.60442 0.257798
$$320$$ 3.29261 0.184063
$$321$$ 0 0
$$322$$ −5.33542 −0.297331
$$323$$ −4.38650 −0.244072
$$324$$ 0 0
$$325$$ −1.32088 −0.0732695
$$326$$ −39.4905 −2.18718
$$327$$ 0 0
$$328$$ −66.2226 −3.65653
$$329$$ −2.50040 −0.137851
$$330$$ 0 0
$$331$$ 16.4431 0.903792 0.451896 0.892071i $$-0.350748\pi$$
0.451896 + 0.892071i $$0.350748\pi$$
$$332$$ −6.66458 −0.365766
$$333$$ 0 0
$$334$$ 15.4996 0.848100
$$335$$ −9.44852 −0.516228
$$336$$ 0 0
$$337$$ 4.89703 0.266758 0.133379 0.991065i $$-0.457417\pi$$
0.133379 + 0.991065i $$0.457417\pi$$
$$338$$ 28.2973 1.53917
$$339$$ 0 0
$$340$$ −14.3492 −0.778192
$$341$$ 29.0101 1.57099
$$342$$ 0 0
$$343$$ 7.06201 0.381313
$$344$$ −60.3876 −3.25588
$$345$$ 0 0
$$346$$ 21.5844 1.16039
$$347$$ −22.2745 −1.19576 −0.597878 0.801587i $$-0.703989\pi$$
−0.597878 + 0.801587i $$0.703989\pi$$
$$348$$ 0 0
$$349$$ −2.94345 −0.157559 −0.0787797 0.996892i $$-0.525102\pi$$
−0.0787797 + 0.996892i $$0.525102\pi$$
$$350$$ 1.29261 0.0690929
$$351$$ 0 0
$$352$$ −11.5761 −0.617011
$$353$$ −18.8296 −1.00220 −0.501098 0.865390i $$-0.667070\pi$$
−0.501098 + 0.865390i $$0.667070\pi$$
$$354$$ 0 0
$$355$$ 8.99093 0.477189
$$356$$ 12.9627 0.687019
$$357$$ 0 0
$$358$$ −2.67912 −0.141596
$$359$$ 31.8770 1.68241 0.841203 0.540720i $$-0.181848\pi$$
0.841203 + 0.540720i $$0.181848\pi$$
$$360$$ 0 0
$$361$$ −17.2553 −0.908172
$$362$$ 31.8542 1.67422
$$363$$ 0 0
$$364$$ 2.93438 0.153803
$$365$$ −6.05655 −0.317014
$$366$$ 0 0
$$367$$ −18.3492 −0.957818 −0.478909 0.877864i $$-0.658968\pi$$
−0.478909 + 0.877864i $$0.658968\pi$$
$$368$$ −24.8825 −1.29709
$$369$$ 0 0
$$370$$ 0.735663 0.0382453
$$371$$ −2.58522 −0.134218
$$372$$ 0 0
$$373$$ −2.19872 −0.113845 −0.0569226 0.998379i $$-0.518129\pi$$
−0.0569226 + 0.998379i $$0.518129\pi$$
$$374$$ −27.7266 −1.43371
$$375$$ 0 0
$$376$$ −28.3774 −1.46345
$$377$$ −1.83141 −0.0943226
$$378$$ 0 0
$$379$$ 15.4713 0.794709 0.397354 0.917665i $$-0.369928\pi$$
0.397354 + 0.917665i $$0.369928\pi$$
$$380$$ 5.70739 0.292783
$$381$$ 0 0
$$382$$ −42.5753 −2.17834
$$383$$ −7.70739 −0.393829 −0.196915 0.980421i $$-0.563092\pi$$
−0.196915 + 0.980421i $$0.563092\pi$$
$$384$$ 0 0
$$385$$ 1.70739 0.0870166
$$386$$ −67.1715 −3.41894
$$387$$ 0 0
$$388$$ −52.9536 −2.68831
$$389$$ 24.6327 1.24893 0.624464 0.781054i $$-0.285318\pi$$
0.624464 + 0.781054i $$0.285318\pi$$
$$390$$ 0 0
$$391$$ −13.7074 −0.693212
$$392$$ 39.3027 1.98509
$$393$$ 0 0
$$394$$ −35.8397 −1.80558
$$395$$ 8.05655 0.405369
$$396$$ 0 0
$$397$$ −6.77301 −0.339928 −0.169964 0.985450i $$-0.554365\pi$$
−0.169964 + 0.985450i $$0.554365\pi$$
$$398$$ 62.0011 3.10783
$$399$$ 0 0
$$400$$ 6.02827 0.301414
$$401$$ 18.4996 0.923826 0.461913 0.886925i $$-0.347163\pi$$
0.461913 + 0.886925i $$0.347163\pi$$
$$402$$ 0 0
$$403$$ −11.5388 −0.574789
$$404$$ −50.4249 −2.50873
$$405$$ 0 0
$$406$$ 1.79221 0.0889459
$$407$$ 0.971726 0.0481667
$$408$$ 0 0
$$409$$ −13.4148 −0.663318 −0.331659 0.943399i $$-0.607608\pi$$
−0.331659 + 0.943399i $$0.607608\pi$$
$$410$$ 28.5333 1.40916
$$411$$ 0 0
$$412$$ 1.26434 0.0622894
$$413$$ −2.58522 −0.127210
$$414$$ 0 0
$$415$$ 1.54241 0.0757140
$$416$$ 4.60442 0.225750
$$417$$ 0 0
$$418$$ 11.0283 0.539411
$$419$$ 33.1150 1.61777 0.808886 0.587966i $$-0.200071\pi$$
0.808886 + 0.587966i $$0.200071\pi$$
$$420$$ 0 0
$$421$$ −14.6983 −0.716352 −0.358176 0.933654i $$-0.616601\pi$$
−0.358176 + 0.933654i $$0.616601\pi$$
$$422$$ 13.5196 0.658124
$$423$$ 0 0
$$424$$ −29.3401 −1.42488
$$425$$ 3.32088 0.161087
$$426$$ 0 0
$$427$$ −3.77847 −0.182853
$$428$$ −8.09029 −0.391059
$$429$$ 0 0
$$430$$ 26.0192 1.25476
$$431$$ 32.7549 1.57775 0.788873 0.614556i $$-0.210665\pi$$
0.788873 + 0.614556i $$0.210665\pi$$
$$432$$ 0 0
$$433$$ −11.8314 −0.568581 −0.284291 0.958738i $$-0.591758\pi$$
−0.284291 + 0.958738i $$0.591758\pi$$
$$434$$ 11.2918 0.542024
$$435$$ 0 0
$$436$$ 23.9717 1.14804
$$437$$ 5.45213 0.260811
$$438$$ 0 0
$$439$$ 8.31181 0.396701 0.198351 0.980131i $$-0.436442\pi$$
0.198351 + 0.980131i $$0.436442\pi$$
$$440$$ 19.3774 0.923783
$$441$$ 0 0
$$442$$ 11.0283 0.524561
$$443$$ 29.1751 1.38615 0.693076 0.720865i $$-0.256255\pi$$
0.693076 + 0.720865i $$0.256255\pi$$
$$444$$ 0 0
$$445$$ −3.00000 −0.142214
$$446$$ −21.7839 −1.03150
$$447$$ 0 0
$$448$$ 1.69285 0.0799798
$$449$$ −18.9717 −0.895331 −0.447666 0.894201i $$-0.647744\pi$$
−0.447666 + 0.894201i $$0.647744\pi$$
$$450$$ 0 0
$$451$$ 37.6892 1.77472
$$452$$ −33.7084 −1.58551
$$453$$ 0 0
$$454$$ 8.34916 0.391845
$$455$$ −0.679116 −0.0318374
$$456$$ 0 0
$$457$$ −23.2353 −1.08690 −0.543450 0.839442i $$-0.682882\pi$$
−0.543450 + 0.839442i $$0.682882\pi$$
$$458$$ 63.6374 2.97358
$$459$$ 0 0
$$460$$ 17.8350 0.831562
$$461$$ −4.42571 −0.206126 −0.103063 0.994675i $$-0.532864\pi$$
−0.103063 + 0.994675i $$0.532864\pi$$
$$462$$ 0 0
$$463$$ −19.5087 −0.906645 −0.453322 0.891347i $$-0.649761\pi$$
−0.453322 + 0.891347i $$0.649761\pi$$
$$464$$ 8.35823 0.388021
$$465$$ 0 0
$$466$$ 69.4724 3.21825
$$467$$ 24.5935 1.13805 0.569026 0.822320i $$-0.307321\pi$$
0.569026 + 0.822320i $$0.307321\pi$$
$$468$$ 0 0
$$469$$ −4.85783 −0.224314
$$470$$ 12.2270 0.563989
$$471$$ 0 0
$$472$$ −29.3401 −1.35049
$$473$$ 34.3684 1.58026
$$474$$ 0 0
$$475$$ −1.32088 −0.0606063
$$476$$ −7.37743 −0.338144
$$477$$ 0 0
$$478$$ −10.5561 −0.482827
$$479$$ −32.7549 −1.49661 −0.748304 0.663356i $$-0.769132\pi$$
−0.748304 + 0.663356i $$0.769132\pi$$
$$480$$ 0 0
$$481$$ −0.386505 −0.0176231
$$482$$ −9.06201 −0.412763
$$483$$ 0 0
$$484$$ 0.122168 0.00555309
$$485$$ 12.2553 0.556483
$$486$$ 0 0
$$487$$ −6.03735 −0.273578 −0.136789 0.990600i $$-0.543678\pi$$
−0.136789 + 0.990600i $$0.543678\pi$$
$$488$$ −42.8825 −1.94120
$$489$$ 0 0
$$490$$ −16.9344 −0.765017
$$491$$ −14.4431 −0.651806 −0.325903 0.945403i $$-0.605668\pi$$
−0.325903 + 0.945403i $$0.605668\pi$$
$$492$$ 0 0
$$493$$ 4.60442 0.207373
$$494$$ −4.38650 −0.197358
$$495$$ 0 0
$$496$$ 52.6610 2.36455
$$497$$ 4.62257 0.207351
$$498$$ 0 0
$$499$$ 20.9717 0.938823 0.469412 0.882979i $$-0.344466\pi$$
0.469412 + 0.882979i $$0.344466\pi$$
$$500$$ −4.32088 −0.193236
$$501$$ 0 0
$$502$$ −17.2918 −0.771771
$$503$$ 5.31728 0.237086 0.118543 0.992949i $$-0.462178\pi$$
0.118543 + 0.992949i $$0.462178\pi$$
$$504$$ 0 0
$$505$$ 11.6700 0.519310
$$506$$ 34.4623 1.53203
$$507$$ 0 0
$$508$$ 77.3074 3.42996
$$509$$ −18.2270 −0.807897 −0.403949 0.914782i $$-0.632363\pi$$
−0.403949 + 0.914782i $$0.632363\pi$$
$$510$$ 0 0
$$511$$ −3.11389 −0.137751
$$512$$ 49.3365 2.18038
$$513$$ 0 0
$$514$$ −45.2545 −1.99609
$$515$$ −0.292611 −0.0128940
$$516$$ 0 0
$$517$$ 16.1504 0.710296
$$518$$ 0.378232 0.0166185
$$519$$ 0 0
$$520$$ −7.70739 −0.337991
$$521$$ −40.1232 −1.75783 −0.878915 0.476978i $$-0.841732\pi$$
−0.878915 + 0.476978i $$0.841732\pi$$
$$522$$ 0 0
$$523$$ 18.9873 0.830257 0.415129 0.909763i $$-0.363737\pi$$
0.415129 + 0.909763i $$0.363737\pi$$
$$524$$ 25.9253 1.13255
$$525$$ 0 0
$$526$$ 15.6783 0.683607
$$527$$ 29.0101 1.26370
$$528$$ 0 0
$$529$$ −5.96265 −0.259246
$$530$$ 12.6418 0.549123
$$531$$ 0 0
$$532$$ 2.93438 0.127221
$$533$$ −14.9909 −0.649329
$$534$$ 0 0
$$535$$ 1.87237 0.0809495
$$536$$ −55.1323 −2.38135
$$537$$ 0 0
$$538$$ 24.9489 1.07562
$$539$$ −22.3684 −0.963473
$$540$$ 0 0
$$541$$ 16.5279 0.710589 0.355294 0.934754i $$-0.384381\pi$$
0.355294 + 0.934754i $$0.384381\pi$$
$$542$$ −16.6044 −0.713221
$$543$$ 0 0
$$544$$ −11.5761 −0.496323
$$545$$ −5.54787 −0.237645
$$546$$ 0 0
$$547$$ 17.6737 0.755671 0.377835 0.925873i $$-0.376669\pi$$
0.377835 + 0.925873i $$0.376669\pi$$
$$548$$ −24.4996 −1.04657
$$549$$ 0 0
$$550$$ −8.34916 −0.356009
$$551$$ −1.83141 −0.0780208
$$552$$ 0 0
$$553$$ 4.14217 0.176143
$$554$$ −56.9728 −2.42054
$$555$$ 0 0
$$556$$ 34.5671 1.46597
$$557$$ −17.3401 −0.734723 −0.367362 0.930078i $$-0.619739\pi$$
−0.367362 + 0.930078i $$0.619739\pi$$
$$558$$ 0 0
$$559$$ −13.6700 −0.578181
$$560$$ 3.09936 0.130972
$$561$$ 0 0
$$562$$ −39.0895 −1.64889
$$563$$ −12.9945 −0.547654 −0.273827 0.961779i $$-0.588290\pi$$
−0.273827 + 0.961779i $$0.588290\pi$$
$$564$$ 0 0
$$565$$ 7.80128 0.328202
$$566$$ −1.62257 −0.0682016
$$567$$ 0 0
$$568$$ 52.4623 2.20127
$$569$$ 16.6802 0.699269 0.349635 0.936886i $$-0.386306\pi$$
0.349635 + 0.936886i $$0.386306\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −18.9536 −0.792489
$$573$$ 0 0
$$574$$ 14.6700 0.612316
$$575$$ −4.12763 −0.172134
$$576$$ 0 0
$$577$$ −23.5953 −0.982287 −0.491144 0.871079i $$-0.663421\pi$$
−0.491144 + 0.871079i $$0.663421\pi$$
$$578$$ 15.0137 0.624489
$$579$$ 0 0
$$580$$ −5.99093 −0.248760
$$581$$ 0.793010 0.0328996
$$582$$ 0 0
$$583$$ 16.6983 0.691574
$$584$$ −35.3401 −1.46238
$$585$$ 0 0
$$586$$ −3.46305 −0.143057
$$587$$ −28.1276 −1.16095 −0.580476 0.814277i $$-0.697133\pi$$
−0.580476 + 0.814277i $$0.697133\pi$$
$$588$$ 0 0
$$589$$ −11.5388 −0.475448
$$590$$ 12.6418 0.520453
$$591$$ 0 0
$$592$$ 1.76394 0.0724974
$$593$$ 9.17872 0.376925 0.188462 0.982080i $$-0.439650\pi$$
0.188462 + 0.982080i $$0.439650\pi$$
$$594$$ 0 0
$$595$$ 1.70739 0.0699961
$$596$$ −76.3110 −3.12582
$$597$$ 0 0
$$598$$ −13.7074 −0.560537
$$599$$ 31.4713 1.28588 0.642942 0.765915i $$-0.277714\pi$$
0.642942 + 0.765915i $$0.277714\pi$$
$$600$$ 0 0
$$601$$ −29.2654 −1.19376 −0.596880 0.802330i $$-0.703593\pi$$
−0.596880 + 0.802330i $$0.703593\pi$$
$$602$$ 13.3774 0.545223
$$603$$ 0 0
$$604$$ 5.46305 0.222288
$$605$$ −0.0282739 −0.00114950
$$606$$ 0 0
$$607$$ −44.2034 −1.79416 −0.897080 0.441868i $$-0.854316\pi$$
−0.897080 + 0.441868i $$0.854316\pi$$
$$608$$ 4.60442 0.186734
$$609$$ 0 0
$$610$$ 18.4768 0.748103
$$611$$ −6.42385 −0.259881
$$612$$ 0 0
$$613$$ −35.1715 −1.42056 −0.710282 0.703918i $$-0.751432\pi$$
−0.710282 + 0.703918i $$0.751432\pi$$
$$614$$ 20.0765 0.810224
$$615$$ 0 0
$$616$$ 9.96265 0.401407
$$617$$ 7.42571 0.298948 0.149474 0.988766i $$-0.452242\pi$$
0.149474 + 0.988766i $$0.452242\pi$$
$$618$$ 0 0
$$619$$ 8.54787 0.343568 0.171784 0.985135i $$-0.445047\pi$$
0.171784 + 0.985135i $$0.445047\pi$$
$$620$$ −37.7458 −1.51591
$$621$$ 0 0
$$622$$ 24.2179 0.971050
$$623$$ −1.54241 −0.0617954
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 61.6892 2.46560
$$627$$ 0 0
$$628$$ −67.7084 −2.70186
$$629$$ 0.971726 0.0387453
$$630$$ 0 0
$$631$$ −2.36836 −0.0942829 −0.0471415 0.998888i $$-0.515011\pi$$
−0.0471415 + 0.998888i $$0.515011\pi$$
$$632$$ 47.0101 1.86996
$$633$$ 0 0
$$634$$ −51.1606 −2.03185
$$635$$ −17.8916 −0.710005
$$636$$ 0 0
$$637$$ 8.89703 0.352513
$$638$$ −11.5761 −0.458304
$$639$$ 0 0
$$640$$ −15.2498 −0.602801
$$641$$ 0.133096 0.00525698 0.00262849 0.999997i $$-0.499163\pi$$
0.00262849 + 0.999997i $$0.499163\pi$$
$$642$$ 0 0
$$643$$ −22.6464 −0.893088 −0.446544 0.894762i $$-0.647345\pi$$
−0.446544 + 0.894762i $$0.647345\pi$$
$$644$$ 9.16964 0.361335
$$645$$ 0 0
$$646$$ 11.0283 0.433902
$$647$$ −46.3912 −1.82383 −0.911913 0.410385i $$-0.865394\pi$$
−0.911913 + 0.410385i $$0.865394\pi$$
$$648$$ 0 0
$$649$$ 16.6983 0.655466
$$650$$ 3.32088 0.130256
$$651$$ 0 0
$$652$$ 67.8698 2.65799
$$653$$ −36.4057 −1.42467 −0.712333 0.701842i $$-0.752361\pi$$
−0.712333 + 0.701842i $$0.752361\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 68.4158 2.67119
$$657$$ 0 0
$$658$$ 6.28635 0.245067
$$659$$ −19.1414 −0.745642 −0.372821 0.927903i $$-0.621609\pi$$
−0.372821 + 0.927903i $$0.621609\pi$$
$$660$$ 0 0
$$661$$ 39.9072 1.55221 0.776104 0.630605i $$-0.217193\pi$$
0.776104 + 0.630605i $$0.217193\pi$$
$$662$$ −41.3401 −1.60673
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ −0.679116 −0.0263350
$$666$$ 0 0
$$667$$ −5.72298 −0.221595
$$668$$ −26.6382 −1.03066
$$669$$ 0 0
$$670$$ 23.7549 0.917730
$$671$$ 24.4057 0.942172
$$672$$ 0 0
$$673$$ 23.6508 0.911673 0.455836 0.890064i $$-0.349340\pi$$
0.455836 + 0.890064i $$0.349340\pi$$
$$674$$ −12.3118 −0.474233
$$675$$ 0 0
$$676$$ −48.6327 −1.87049
$$677$$ 14.8031 0.568931 0.284465 0.958686i $$-0.408184\pi$$
0.284465 + 0.958686i $$0.408184\pi$$
$$678$$ 0 0
$$679$$ 6.30088 0.241806
$$680$$ 19.3774 0.743091
$$681$$ 0 0
$$682$$ −72.9354 −2.79284
$$683$$ 4.95252 0.189503 0.0947515 0.995501i $$-0.469794\pi$$
0.0947515 + 0.995501i $$0.469794\pi$$
$$684$$ 0 0
$$685$$ 5.67004 0.216641
$$686$$ −17.7549 −0.677884
$$687$$ 0 0
$$688$$ 62.3876 2.37850
$$689$$ −6.64177 −0.253031
$$690$$ 0 0
$$691$$ −19.2088 −0.730739 −0.365369 0.930863i $$-0.619057\pi$$
−0.365369 + 0.930863i $$0.619057\pi$$
$$692$$ −37.0957 −1.41017
$$693$$ 0 0
$$694$$ 56.0011 2.12577
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ 37.6892 1.42758
$$698$$ 7.40024 0.280103
$$699$$ 0 0
$$700$$ −2.22153 −0.0839658
$$701$$ 29.3492 1.10850 0.554251 0.832349i $$-0.313005\pi$$
0.554251 + 0.832349i $$0.313005\pi$$
$$702$$ 0 0
$$703$$ −0.386505 −0.0145773
$$704$$ −10.9344 −0.412105
$$705$$ 0 0
$$706$$ 47.3401 1.78167
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ 38.7266 1.45441 0.727204 0.686422i $$-0.240820\pi$$
0.727204 + 0.686422i $$0.240820\pi$$
$$710$$ −22.6044 −0.848329
$$711$$ 0 0
$$712$$ −17.5051 −0.656030
$$713$$ −36.0576 −1.35037
$$714$$ 0 0
$$715$$ 4.38650 0.164046
$$716$$ 4.60442 0.172075
$$717$$ 0 0
$$718$$ −80.1432 −2.99092
$$719$$ 15.0848 0.562569 0.281284 0.959624i $$-0.409240\pi$$
0.281284 + 0.959624i $$0.409240\pi$$
$$720$$ 0 0
$$721$$ −0.150442 −0.00560275
$$722$$ 43.3821 1.61451
$$723$$ 0 0
$$724$$ −54.7458 −2.03461
$$725$$ 1.38650 0.0514935
$$726$$ 0 0
$$727$$ 12.3455 0.457871 0.228936 0.973442i $$-0.426475\pi$$
0.228936 + 0.973442i $$0.426475\pi$$
$$728$$ −3.96265 −0.146866
$$729$$ 0 0
$$730$$ 15.2270 0.563576
$$731$$ 34.3684 1.27116
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 46.1323 1.70277
$$735$$ 0 0
$$736$$ 14.3884 0.530362
$$737$$ 31.3774 1.15580
$$738$$ 0 0
$$739$$ 29.7266 1.09351 0.546755 0.837293i $$-0.315863\pi$$
0.546755 + 0.837293i $$0.315863\pi$$
$$740$$ −1.26434 −0.0464779
$$741$$ 0 0
$$742$$ 6.49960 0.238608
$$743$$ −48.3648 −1.77433 −0.887165 0.461452i $$-0.847329\pi$$
−0.887165 + 0.461452i $$0.847329\pi$$
$$744$$ 0 0
$$745$$ 17.6610 0.647048
$$746$$ 5.52787 0.202390
$$747$$ 0 0
$$748$$ 47.6519 1.74233
$$749$$ 0.962653 0.0351746
$$750$$ 0 0
$$751$$ −31.8205 −1.16115 −0.580573 0.814208i $$-0.697171\pi$$
−0.580573 + 0.814208i $$0.697171\pi$$
$$752$$ 29.3173 1.06909
$$753$$ 0 0
$$754$$ 4.60442 0.167683
$$755$$ −1.26434 −0.0460139
$$756$$ 0 0
$$757$$ 4.94531 0.179740 0.0898701 0.995953i $$-0.471355\pi$$
0.0898701 + 0.995953i $$0.471355\pi$$
$$758$$ −38.8970 −1.41280
$$759$$ 0 0
$$760$$ −7.70739 −0.279576
$$761$$ −35.4249 −1.28415 −0.642076 0.766641i $$-0.721927\pi$$
−0.642076 + 0.766641i $$0.721927\pi$$
$$762$$ 0 0
$$763$$ −2.85237 −0.103263
$$764$$ 73.1715 2.64725
$$765$$ 0 0
$$766$$ 19.3774 0.700135
$$767$$ −6.64177 −0.239820
$$768$$ 0 0
$$769$$ 49.4249 1.78231 0.891154 0.453701i $$-0.149897\pi$$
0.891154 + 0.453701i $$0.149897\pi$$
$$770$$ −4.29261 −0.154695
$$771$$ 0 0
$$772$$ 115.443 4.15490
$$773$$ −12.6599 −0.455345 −0.227673 0.973738i $$-0.573112\pi$$
−0.227673 + 0.973738i $$0.573112\pi$$
$$774$$ 0 0
$$775$$ 8.73566 0.313794
$$776$$ 71.5097 2.56705
$$777$$ 0 0
$$778$$ −61.9300 −2.22030
$$779$$ −14.9909 −0.537106
$$780$$ 0 0
$$781$$ −29.8578 −1.06840
$$782$$ 34.4623 1.23237
$$783$$ 0 0
$$784$$ −40.6044 −1.45016
$$785$$ 15.6700 0.559288
$$786$$ 0 0
$$787$$ 30.9344 1.10269 0.551346 0.834277i $$-0.314115\pi$$
0.551346 + 0.834277i $$0.314115\pi$$
$$788$$ 61.5953 2.19424
$$789$$ 0 0
$$790$$ −20.2553 −0.720650
$$791$$ 4.01093 0.142612
$$792$$ 0 0
$$793$$ −9.70739 −0.344720
$$794$$ 17.0283 0.604311
$$795$$ 0 0
$$796$$ −106.557 −3.77682
$$797$$ 30.5935 1.08368 0.541839 0.840483i $$-0.317728\pi$$
0.541839 + 0.840483i $$0.317728\pi$$
$$798$$ 0 0
$$799$$ 16.1504 0.571362
$$800$$ −3.48586 −0.123244
$$801$$ 0 0
$$802$$ −46.5105 −1.64234
$$803$$ 20.1131 0.709776
$$804$$ 0 0
$$805$$ −2.12217 −0.0747966
$$806$$ 29.0101 1.02184
$$807$$ 0 0
$$808$$ 68.0950 2.39557
$$809$$ 2.89703 0.101854 0.0509271 0.998702i $$-0.483782\pi$$
0.0509271 + 0.998702i $$0.483782\pi$$
$$810$$ 0 0
$$811$$ −14.8861 −0.522722 −0.261361 0.965241i $$-0.584171\pi$$
−0.261361 + 0.965241i $$0.584171\pi$$
$$812$$ −3.08016 −0.108092
$$813$$ 0 0
$$814$$ −2.44305 −0.0856289
$$815$$ −15.7074 −0.550206
$$816$$ 0 0
$$817$$ −13.6700 −0.478254
$$818$$ 33.7266 1.17922
$$819$$ 0 0
$$820$$ −49.0384 −1.71250
$$821$$ 8.95173 0.312417 0.156209 0.987724i $$-0.450073\pi$$
0.156209 + 0.987724i $$0.450073\pi$$
$$822$$ 0 0
$$823$$ 2.99454 0.104383 0.0521915 0.998637i $$-0.483379\pi$$
0.0521915 + 0.998637i $$0.483379\pi$$
$$824$$ −1.70739 −0.0594797
$$825$$ 0 0
$$826$$ 6.49960 0.226150
$$827$$ −31.9663 −1.11158 −0.555788 0.831324i $$-0.687583\pi$$
−0.555788 + 0.831324i $$0.687583\pi$$
$$828$$ 0 0
$$829$$ 22.7458 0.789994 0.394997 0.918682i $$-0.370746\pi$$
0.394997 + 0.918682i $$0.370746\pi$$
$$830$$ −3.87783 −0.134602
$$831$$ 0 0
$$832$$ 4.34916 0.150780
$$833$$ −22.3684 −0.775018
$$834$$ 0 0
$$835$$ 6.16498 0.213348
$$836$$ −18.9536 −0.655523
$$837$$ 0 0
$$838$$ −83.2555 −2.87601
$$839$$ 23.2643 0.803174 0.401587 0.915821i $$-0.368459\pi$$
0.401587 + 0.915821i $$0.368459\pi$$
$$840$$ 0 0
$$841$$ −27.0776 −0.933710
$$842$$ 36.9536 1.27350
$$843$$ 0 0
$$844$$ −23.2353 −0.799791
$$845$$ 11.2553 0.387193
$$846$$ 0 0
$$847$$ −0.0145366 −0.000499485 0
$$848$$ 30.3118 1.04091
$$849$$ 0 0
$$850$$ −8.34916 −0.286374
$$851$$ −1.20779 −0.0414025
$$852$$ 0 0
$$853$$ 10.9909 0.376322 0.188161 0.982138i $$-0.439747\pi$$
0.188161 + 0.982138i $$0.439747\pi$$
$$854$$ 9.49960 0.325070
$$855$$ 0 0
$$856$$ 10.9253 0.373419
$$857$$ 16.1504 0.551689 0.275844 0.961202i $$-0.411043\pi$$
0.275844 + 0.961202i $$0.411043\pi$$
$$858$$ 0 0
$$859$$ 28.5188 0.973049 0.486524 0.873667i $$-0.338264\pi$$
0.486524 + 0.873667i $$0.338264\pi$$
$$860$$ −44.7175 −1.52485
$$861$$ 0 0
$$862$$ −82.3502 −2.80486
$$863$$ −12.2890 −0.418322 −0.209161 0.977881i $$-0.567073\pi$$
−0.209161 + 0.977881i $$0.567073\pi$$
$$864$$ 0 0
$$865$$ 8.58522 0.291906
$$866$$ 29.7458 1.01080
$$867$$ 0 0
$$868$$ −19.4065 −0.658700
$$869$$ −26.7549 −0.907597
$$870$$ 0 0
$$871$$ −12.4804 −0.422882
$$872$$ −32.3720 −1.09625
$$873$$ 0 0
$$874$$ −13.7074 −0.463659
$$875$$ 0.514137 0.0173810
$$876$$ 0 0
$$877$$ −39.7002 −1.34058 −0.670290 0.742099i $$-0.733830\pi$$
−0.670290 + 0.742099i $$0.733830\pi$$
$$878$$ −20.8970 −0.705241
$$879$$ 0 0
$$880$$ −20.0192 −0.674847
$$881$$ −32.1040 −1.08161 −0.540806 0.841147i $$-0.681881\pi$$
−0.540806 + 0.841147i $$0.681881\pi$$
$$882$$ 0 0
$$883$$ 13.5051 0.454482 0.227241 0.973839i $$-0.427030\pi$$
0.227241 + 0.973839i $$0.427030\pi$$
$$884$$ −18.9536 −0.637478
$$885$$ 0 0
$$886$$ −73.3502 −2.46425
$$887$$ −35.1222 −1.17929 −0.589643 0.807664i $$-0.700732\pi$$
−0.589643 + 0.807664i $$0.700732\pi$$
$$888$$ 0 0
$$889$$ −9.19872 −0.308515
$$890$$ 7.54241 0.252822
$$891$$ 0 0
$$892$$ 37.4386 1.25354
$$893$$ −6.42385 −0.214966
$$894$$ 0 0
$$895$$ −1.06562 −0.0356198
$$896$$ −7.84049 −0.261932
$$897$$ 0 0
$$898$$ 47.6975 1.59169
$$899$$ 12.1120 0.403959
$$900$$ 0 0
$$901$$ 16.6983 0.556302
$$902$$ −94.7559 −3.15503
$$903$$ 0 0
$$904$$ 45.5207 1.51399
$$905$$ 12.6700 0.421166
$$906$$ 0 0
$$907$$ 15.1186 0.502004 0.251002 0.967987i $$-0.419240\pi$$
0.251002 + 0.967987i $$0.419240\pi$$
$$908$$ −14.3492 −0.476194
$$909$$ 0 0
$$910$$ 1.70739 0.0565994
$$911$$ −52.5561 −1.74126 −0.870631 0.491936i $$-0.836289\pi$$
−0.870631 + 0.491936i $$0.836289\pi$$
$$912$$ 0 0
$$913$$ −5.12217 −0.169519
$$914$$ 58.4166 1.93225
$$915$$ 0 0
$$916$$ −109.369 −3.61367
$$917$$ −3.08482 −0.101870
$$918$$ 0 0
$$919$$ −54.5489 −1.79940 −0.899702 0.436505i $$-0.856216\pi$$
−0.899702 + 0.436505i $$0.856216\pi$$
$$920$$ −24.0848 −0.794053
$$921$$ 0 0
$$922$$ 11.1268 0.366443
$$923$$ 11.8760 0.390903
$$924$$ 0 0
$$925$$ 0.292611 0.00962098
$$926$$ 49.0475 1.61180
$$927$$ 0 0
$$928$$ −4.83317 −0.158656
$$929$$ −20.3793 −0.668623 −0.334311 0.942463i $$-0.608504\pi$$
−0.334311 + 0.942463i $$0.608504\pi$$
$$930$$ 0 0
$$931$$ 8.89703 0.291588
$$932$$ −119.398 −3.91100
$$933$$ 0 0
$$934$$ −61.8314 −2.02319
$$935$$ −11.0283 −0.360663
$$936$$ 0 0
$$937$$ 49.1979 1.60723 0.803613 0.595152i $$-0.202908\pi$$
0.803613 + 0.595152i $$0.202908\pi$$
$$938$$ 12.2133 0.398777
$$939$$ 0 0
$$940$$ −21.0137 −0.685393
$$941$$ 23.2371 0.757508 0.378754 0.925497i $$-0.376353\pi$$
0.378754 + 0.925497i $$0.376353\pi$$
$$942$$ 0 0
$$943$$ −46.8452 −1.52549
$$944$$ 30.3118 0.986565
$$945$$ 0 0
$$946$$ −86.4068 −2.80933
$$947$$ −37.1642 −1.20767 −0.603837 0.797108i $$-0.706362\pi$$
−0.603837 + 0.797108i $$0.706362\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 3.32088 0.107744
$$951$$ 0 0
$$952$$ 9.96265 0.322891
$$953$$ 23.5761 0.763706 0.381853 0.924223i $$-0.375286\pi$$
0.381853 + 0.924223i $$0.375286\pi$$
$$954$$ 0 0
$$955$$ −16.9344 −0.547984
$$956$$ 18.1422 0.586760
$$957$$ 0 0
$$958$$ 82.3502 2.66061
$$959$$ 2.91518 0.0941360
$$960$$ 0 0
$$961$$ 45.3118 1.46167
$$962$$ 0.971726 0.0313297
$$963$$ 0 0
$$964$$ 15.5743 0.501614
$$965$$ −26.7175 −0.860067
$$966$$ 0 0
$$967$$ 8.38290 0.269576 0.134788 0.990874i $$-0.456965\pi$$
0.134788 + 0.990874i $$0.456965\pi$$
$$968$$ −0.164979 −0.00530261
$$969$$ 0 0
$$970$$ −30.8114 −0.989295
$$971$$ −13.2078 −0.423858 −0.211929 0.977285i $$-0.567975\pi$$
−0.211929 + 0.977285i $$0.567975\pi$$
$$972$$ 0 0
$$973$$ −4.11310 −0.131860
$$974$$ 15.1787 0.486357
$$975$$ 0 0
$$976$$ 44.3027 1.41810
$$977$$ 14.3310 0.458490 0.229245 0.973369i $$-0.426374\pi$$
0.229245 + 0.973369i $$0.426374\pi$$
$$978$$ 0 0
$$979$$ 9.96265 0.318408
$$980$$ 29.1040 0.929694
$$981$$ 0 0
$$982$$ 36.3118 1.15876
$$983$$ 32.3082 1.03047 0.515236 0.857048i $$-0.327704\pi$$
0.515236 + 0.857048i $$0.327704\pi$$
$$984$$ 0 0
$$985$$ −14.2553 −0.454210
$$986$$ −11.5761 −0.368660
$$987$$ 0 0
$$988$$ 7.53880 0.239841
$$989$$ −42.7175 −1.35834
$$990$$ 0 0
$$991$$ −39.6700 −1.26016 −0.630080 0.776530i $$-0.716978\pi$$
−0.630080 + 0.776530i $$0.716978\pi$$
$$992$$ −30.4513 −0.966831
$$993$$ 0 0
$$994$$ −11.6218 −0.368620
$$995$$ 24.6610 0.781805
$$996$$ 0 0
$$997$$ 38.6874 1.22524 0.612621 0.790377i $$-0.290115\pi$$
0.612621 + 0.790377i $$0.290115\pi$$
$$998$$ −52.7258 −1.66901
$$999$$ 0 0
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 405.2.a.i.1.1 3
3.2 odd 2 405.2.a.j.1.3 3
4.3 odd 2 6480.2.a.bs.1.3 3
5.2 odd 4 2025.2.b.m.649.1 6
5.3 odd 4 2025.2.b.m.649.6 6
5.4 even 2 2025.2.a.o.1.3 3
9.2 odd 6 45.2.e.b.31.1 yes 6
9.4 even 3 135.2.e.b.46.3 6
9.5 odd 6 45.2.e.b.16.1 6
9.7 even 3 135.2.e.b.91.3 6
12.11 even 2 6480.2.a.bv.1.3 3
15.2 even 4 2025.2.b.l.649.6 6
15.8 even 4 2025.2.b.l.649.1 6
15.14 odd 2 2025.2.a.n.1.1 3
36.7 odd 6 2160.2.q.k.1441.1 6
36.11 even 6 720.2.q.i.481.1 6
36.23 even 6 720.2.q.i.241.1 6
36.31 odd 6 2160.2.q.k.721.1 6
45.2 even 12 225.2.k.b.49.6 12
45.4 even 6 675.2.e.b.451.1 6
45.7 odd 12 675.2.k.b.199.1 12
45.13 odd 12 675.2.k.b.424.1 12
45.14 odd 6 225.2.e.b.151.3 6
45.22 odd 12 675.2.k.b.424.6 12
45.23 even 12 225.2.k.b.124.6 12
45.29 odd 6 225.2.e.b.76.3 6
45.32 even 12 225.2.k.b.124.1 12
45.34 even 6 675.2.e.b.226.1 6
45.38 even 12 225.2.k.b.49.1 12
45.43 odd 12 675.2.k.b.199.6 12

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 9.5 odd 6
45.2.e.b.31.1 yes 6 9.2 odd 6
135.2.e.b.46.3 6 9.4 even 3
135.2.e.b.91.3 6 9.7 even 3
225.2.e.b.76.3 6 45.29 odd 6
225.2.e.b.151.3 6 45.14 odd 6
225.2.k.b.49.1 12 45.38 even 12
225.2.k.b.49.6 12 45.2 even 12
225.2.k.b.124.1 12 45.32 even 12
225.2.k.b.124.6 12 45.23 even 12
405.2.a.i.1.1 3 1.1 even 1 trivial
405.2.a.j.1.3 3 3.2 odd 2
675.2.e.b.226.1 6 45.34 even 6
675.2.e.b.451.1 6 45.4 even 6
675.2.k.b.199.1 12 45.7 odd 12
675.2.k.b.199.6 12 45.43 odd 12
675.2.k.b.424.1 12 45.13 odd 12
675.2.k.b.424.6 12 45.22 odd 12
720.2.q.i.241.1 6 36.23 even 6
720.2.q.i.481.1 6 36.11 even 6
2025.2.a.n.1.1 3 15.14 odd 2
2025.2.a.o.1.3 3 5.4 even 2
2025.2.b.l.649.1 6 15.8 even 4
2025.2.b.l.649.6 6 15.2 even 4
2025.2.b.m.649.1 6 5.2 odd 4
2025.2.b.m.649.6 6 5.3 odd 4
2160.2.q.k.721.1 6 36.31 odd 6
2160.2.q.k.1441.1 6 36.7 odd 6
6480.2.a.bs.1.3 3 4.3 odd 2
6480.2.a.bv.1.3 3 12.11 even 2