# Properties

 Label 405.2.a.j.1.3 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 1.3 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 $$-0.151483\pi$$
0.888882 + 0.458137i $$0.151483\pi$$
$$3$$ 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 $$-0.666903\pi$$
−0.500642 + 0.865654i $$0.666903\pi$$
$$12$$ 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 $$-0.631934\pi$$
−0.402716 + 0.915325i $$0.631934\pi$$
$$18$$ 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 $$-0.358395\pi$$
0.430335 + 0.902669i $$0.358395\pi$$
$$24$$ 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 $$-0.541091\pi$$
−0.128734 + 0.991679i $$0.541091\pi$$
$$30$$ 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 $$-0.846678\pi$$
−0.886220 + 0.463264i $$0.846678\pi$$
$$42$$ 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 $$-0.615414\pi$$
−0.354692 + 0.934983i $$0.615414\pi$$
$$48$$ 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 $$-0.612238\pi$$
−0.345343 + 0.938476i $$0.612238\pi$$
$$54$$ 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 $$-0.606143\pi$$
−0.327313 + 0.944916i $$0.606143\pi$$
$$60$$ 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 $$-0.320871\pi$$
0.533513 + 0.845792i $$0.320871\pi$$
$$72$$ 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 $$-0.473023\pi$$
0.0846508 + 0.996411i $$0.473023\pi$$
$$84$$ 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 $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 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 $$-0.302816\pi$$
0.580606 + 0.814184i $$0.302816\pi$$
$$102$$ 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 $$-0.471152\pi$$
0.0905043 + 0.995896i $$0.471152\pi$$
$$108$$ 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 $$-0.380405\pi$$
0.366942 + 0.930244i $$0.380405\pi$$
$$114$$ 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 $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 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 $$-0.422127\pi$$
0.242212 + 0.970223i $$0.422127\pi$$
$$138$$ 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 $$-0.242568\pi$$
0.723422 + 0.690407i $$0.242568\pi$$
$$150$$ 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 $$-0.423334\pi$$
0.238530 + 0.971135i $$0.423334\pi$$
$$168$$ 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 $$-0.394177\pi$$
0.326361 + 0.945245i $$0.394177\pi$$
$$174$$ 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 $$-0.512680\pi$$
−0.0398241 + 0.999207i $$0.512680\pi$$
$$180$$ 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 $$-0.709902\pi$$
−0.612664 + 0.790343i $$0.709902\pi$$
$$192$$ 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 $$-0.669550\pi$$
−0.507823 + 0.861462i $$0.669550\pi$$
$$198$$ 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 $$-0.464848\pi$$
0.110207 + 0.993909i $$0.464848\pi$$
$$228$$ 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 $$-0.139767\pi$$
0.905139 + 0.425116i $$0.139767\pi$$
$$234$$ 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 $$-0.543359\pi$$
−0.135796 + 0.990737i $$0.543359\pi$$
$$240$$ 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 $$-0.569648\pi$$
−0.217062 + 0.976158i $$0.569648\pi$$
$$252$$ 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 $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 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 $$-0.438416\pi$$
0.192266 + 0.981343i $$0.438416\pi$$
$$264$$ 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 $$-0.402172\pi$$
0.302522 + 0.953142i $$0.402172\pi$$
$$270$$ 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 $$-0.653498\pi$$
−0.463754 + 0.885964i $$0.653498\pi$$
$$282$$ 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 $$-0.512811\pi$$
−0.0402352 + 0.999190i $$0.512811\pi$$
$$294$$ 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 $$-0.411948\pi$$
0.273110 + 0.961983i $$0.411948\pi$$
$$312$$ 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 $$-0.693623\pi$$
−0.571461 + 0.820629i $$0.693623\pi$$
$$318$$ 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 $$-0.296011\pi$$
0.597878 + 0.801587i $$0.296011\pi$$
$$348$$ 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 $$-0.332930\pi$$
0.501098 + 0.865390i $$0.332930\pi$$
$$354$$ 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 $$-0.818152\pi$$
−0.841203 + 0.540720i $$0.818152\pi$$
$$360$$ 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 $$-0.436908\pi$$
0.196915 + 0.980421i $$0.436908\pi$$
$$384$$ 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 $$-0.714682\pi$$
−0.624464 + 0.781054i $$0.714682\pi$$
$$390$$ 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 $$-0.652837\pi$$
−0.461913 + 0.886925i $$0.652837\pi$$
$$402$$ 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 $$-0.799929\pi$$
−0.808886 + 0.587966i $$0.799929\pi$$
$$420$$ 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 $$-0.789335\pi$$
−0.788873 + 0.614556i $$0.789335\pi$$
$$432$$ 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 $$-0.743745\pi$$
−0.693076 + 0.720865i $$0.743745\pi$$
$$444$$ 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 $$-0.352256\pi$$
0.447666 + 0.894201i $$0.352256\pi$$
$$450$$ 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 $$-0.467136\pi$$
0.103063 + 0.994675i $$0.467136\pi$$
$$462$$ 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 $$-0.692679\pi$$
−0.569026 + 0.822320i $$0.692679\pi$$
$$468$$ 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 $$-0.230868\pi$$
0.748304 + 0.663356i $$0.230868\pi$$
$$480$$ 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 $$-0.394332\pi$$
0.325903 + 0.945403i $$0.394332\pi$$
$$492$$ 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 $$-0.537822\pi$$
−0.118543 + 0.992949i $$0.537822\pi$$
$$504$$ 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 $$-0.367637\pi$$
0.403949 + 0.914782i $$0.367637\pi$$
$$510$$ 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 $$-0.158268\pi$$
0.878915 + 0.476978i $$0.158268\pi$$
$$522$$ 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 $$-0.380261\pi$$
0.367362 + 0.930078i $$0.380261\pi$$
$$558$$ 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 $$-0.411710\pi$$
0.273827 + 0.961779i $$0.411710\pi$$
$$564$$ 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 $$-0.613694\pi$$
−0.349635 + 0.936886i $$0.613694\pi$$
$$570$$ 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 $$-0.302867\pi$$
0.580476 + 0.814277i $$0.302867\pi$$
$$588$$ 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 $$-0.560350\pi$$
−0.188462 + 0.982080i $$0.560350\pi$$
$$594$$ 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 $$-0.722286\pi$$
−0.642942 + 0.765915i $$0.722286\pi$$
$$600$$ 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 $$-0.547758\pi$$
−0.149474 + 0.988766i $$0.547758\pi$$
$$618$$ 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 $$-0.500837\pi$$
−0.00262849 + 0.999997i $$0.500837\pi$$
$$642$$ 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 $$-0.134606\pi$$
0.911913 + 0.410385i $$0.134606\pi$$
$$648$$ 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 $$-0.247639\pi$$
0.712333 + 0.701842i $$0.247639\pi$$
$$654$$ 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 $$-0.378391\pi$$
0.372821 + 0.927903i $$0.378391\pi$$
$$660$$ 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 $$-0.591816\pi$$
−0.284465 + 0.958686i $$0.591816\pi$$
$$678$$ 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 $$-0.530206\pi$$
−0.0947515 + 0.995501i $$0.530206\pi$$
$$684$$ 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 $$-0.686995\pi$$
−0.554251 + 0.832349i $$0.686995\pi$$
$$702$$ 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 $$-0.590760\pi$$
−0.281284 + 0.959624i $$0.590760\pi$$
$$720$$ 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 $$-0.152671\pi$$
0.887165 + 0.461452i $$0.152671\pi$$
$$744$$ 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 $$-0.278073\pi$$
0.642076 + 0.766641i $$0.278073\pi$$
$$762$$ 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 $$-0.426888\pi$$
0.227673 + 0.973738i $$0.426888\pi$$
$$774$$ 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 $$-0.682272\pi$$
−0.541839 + 0.840483i $$0.682272\pi$$
$$798$$ 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 $$-0.516218\pi$$
−0.0509271 + 0.998702i $$0.516218\pi$$
$$810$$ 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 $$-0.549927\pi$$
−0.156209 + 0.987724i $$0.549927\pi$$
$$822$$ 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 $$-0.312417\pi$$
0.555788 + 0.831324i $$0.312417\pi$$
$$828$$ 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 $$-0.631541\pi$$
−0.401587 + 0.915821i $$0.631541\pi$$
$$840$$ 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 $$-0.588957\pi$$
−0.275844 + 0.961202i $$0.588957\pi$$
$$858$$ 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 $$-0.432927\pi$$
0.209161 + 0.977881i $$0.432927\pi$$
$$864$$ 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 $$-0.318119\pi$$
0.540806 + 0.841147i $$0.318119\pi$$
$$882$$ 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 $$-0.299268\pi$$
0.589643 + 0.807664i $$0.299268\pi$$
$$888$$ 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 $$-0.163711\pi$$
0.870631 + 0.491936i $$0.163711\pi$$
$$912$$ 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 $$-0.391496\pi$$
0.334311 + 0.942463i $$0.391496\pi$$
$$930$$ 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 $$-0.623647\pi$$
−0.378754 + 0.925497i $$0.623647\pi$$
$$942$$ 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 $$-0.293638\pi$$
0.603837 + 0.797108i $$0.293638\pi$$
$$948$$ 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 $$-0.624714\pi$$
−0.381853 + 0.924223i $$0.624714\pi$$
$$954$$ 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 $$-0.432025\pi$$
0.211929 + 0.977285i $$0.432025\pi$$
$$972$$ 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 $$-0.573626\pi$$
−0.229245 + 0.973369i $$0.573626\pi$$
$$978$$ 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 $$-0.672296\pi$$
−0.515236 + 0.857048i $$0.672296\pi$$
$$984$$ 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.j.1.3 3
3.2 odd 2 405.2.a.i.1.1 3
4.3 odd 2 6480.2.a.bv.1.3 3
5.2 odd 4 2025.2.b.l.649.6 6
5.3 odd 4 2025.2.b.l.649.1 6
5.4 even 2 2025.2.a.n.1.1 3
9.2 odd 6 135.2.e.b.91.3 6
9.4 even 3 45.2.e.b.16.1 6
9.5 odd 6 135.2.e.b.46.3 6
9.7 even 3 45.2.e.b.31.1 yes 6
12.11 even 2 6480.2.a.bs.1.3 3
15.2 even 4 2025.2.b.m.649.1 6
15.8 even 4 2025.2.b.m.649.6 6
15.14 odd 2 2025.2.a.o.1.3 3
36.7 odd 6 720.2.q.i.481.1 6
36.11 even 6 2160.2.q.k.1441.1 6
36.23 even 6 2160.2.q.k.721.1 6
36.31 odd 6 720.2.q.i.241.1 6
45.2 even 12 675.2.k.b.199.1 12
45.4 even 6 225.2.e.b.151.3 6
45.7 odd 12 225.2.k.b.49.6 12
45.13 odd 12 225.2.k.b.124.6 12
45.14 odd 6 675.2.e.b.451.1 6
45.22 odd 12 225.2.k.b.124.1 12
45.23 even 12 675.2.k.b.424.1 12
45.29 odd 6 675.2.e.b.226.1 6
45.32 even 12 675.2.k.b.424.6 12
45.34 even 6 225.2.e.b.76.3 6
45.38 even 12 675.2.k.b.199.6 12
45.43 odd 12 225.2.k.b.49.1 12

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