# Properties

 Label 6384.2.a.cb.1.2 Level $6384$ Weight $2$ Character 6384.1 Self dual yes Analytic conductor $50.976$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.9764966504$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.9248.1 Defining polynomial: $$x^{4} - 5x^{2} + 2$$ x^4 - 5*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.662153$$ of defining polynomial Character $$\chi$$ $$=$$ 6384.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.69614 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.69614 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.32431 q^{11} -1.69614 q^{15} +6.14355 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.91779 q^{23} -2.12311 q^{25} +1.00000 q^{27} -1.42696 q^{29} +3.32431 q^{31} +1.32431 q^{33} -1.69614 q^{35} +9.83969 q^{37} -8.04090 q^{41} -7.16400 q^{43} -1.69614 q^{45} +7.66906 q^{47} +1.00000 q^{49} +6.14355 q^{51} +8.81925 q^{53} -2.24621 q^{55} +1.00000 q^{57} -9.43318 q^{59} +4.64861 q^{61} +1.00000 q^{63} +11.5705 q^{67} -2.91779 q^{69} +3.22165 q^{71} -8.04090 q^{73} -2.12311 q^{75} +1.32431 q^{77} +10.3101 q^{79} +1.00000 q^{81} +4.77835 q^{83} -10.4203 q^{85} -1.42696 q^{87} -11.6385 q^{89} +3.32431 q^{93} -1.69614 q^{95} -6.71659 q^{97} +1.32431 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9 $$4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 4 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 4 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{37} - 8 q^{41} + 12 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{57} + 8 q^{61} + 4 q^{63} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 8 q^{73} + 8 q^{75} + 20 q^{79} + 4 q^{81} + 20 q^{83} - 4 q^{85} + 4 q^{87} + 8 q^{93} - 8 q^{97}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9 - 4 * q^17 + 4 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 + 4 * q^27 + 4 * q^29 + 8 * q^31 + 4 * q^37 - 8 * q^41 + 12 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^51 + 12 * q^53 + 24 * q^55 + 4 * q^57 + 8 * q^61 + 4 * q^63 + 8 * q^67 - 4 * q^69 + 12 * q^71 - 8 * q^73 + 8 * q^75 + 20 * q^79 + 4 * q^81 + 20 * q^83 - 4 * q^85 + 4 * q^87 + 8 * q^93 - 8 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.69614 −0.758537 −0.379269 0.925287i $$-0.623824\pi$$
−0.379269 + 0.925287i $$0.623824\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.32431 0.399294 0.199647 0.979868i $$-0.436021\pi$$
0.199647 + 0.979868i $$0.436021\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −1.69614 −0.437942
$$16$$ 0 0
$$17$$ 6.14355 1.49003 0.745015 0.667047i $$-0.232442\pi$$
0.745015 + 0.667047i $$0.232442\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −2.91779 −0.608401 −0.304201 0.952608i $$-0.598389\pi$$
−0.304201 + 0.952608i $$0.598389\pi$$
$$24$$ 0 0
$$25$$ −2.12311 −0.424621
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.42696 −0.264981 −0.132490 0.991184i $$-0.542297\pi$$
−0.132490 + 0.991184i $$0.542297\pi$$
$$30$$ 0 0
$$31$$ 3.32431 0.597063 0.298532 0.954400i $$-0.403503\pi$$
0.298532 + 0.954400i $$0.403503\pi$$
$$32$$ 0 0
$$33$$ 1.32431 0.230532
$$34$$ 0 0
$$35$$ −1.69614 −0.286700
$$36$$ 0 0
$$37$$ 9.83969 1.61764 0.808818 0.588059i $$-0.200108\pi$$
0.808818 + 0.588059i $$0.200108\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.04090 −1.25578 −0.627888 0.778303i $$-0.716081\pi$$
−0.627888 + 0.778303i $$0.716081\pi$$
$$42$$ 0 0
$$43$$ −7.16400 −1.09250 −0.546250 0.837622i $$-0.683945\pi$$
−0.546250 + 0.837622i $$0.683945\pi$$
$$44$$ 0 0
$$45$$ −1.69614 −0.252846
$$46$$ 0 0
$$47$$ 7.66906 1.11865 0.559324 0.828949i $$-0.311061\pi$$
0.559324 + 0.828949i $$0.311061\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.14355 0.860270
$$52$$ 0 0
$$53$$ 8.81925 1.21142 0.605708 0.795687i $$-0.292890\pi$$
0.605708 + 0.795687i $$0.292890\pi$$
$$54$$ 0 0
$$55$$ −2.24621 −0.302879
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −9.43318 −1.22810 −0.614048 0.789269i $$-0.710460\pi$$
−0.614048 + 0.789269i $$0.710460\pi$$
$$60$$ 0 0
$$61$$ 4.64861 0.595194 0.297597 0.954692i $$-0.403815\pi$$
0.297597 + 0.954692i $$0.403815\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.5705 1.41356 0.706782 0.707432i $$-0.250146\pi$$
0.706782 + 0.707432i $$0.250146\pi$$
$$68$$ 0 0
$$69$$ −2.91779 −0.351261
$$70$$ 0 0
$$71$$ 3.22165 0.382339 0.191170 0.981557i $$-0.438772\pi$$
0.191170 + 0.981557i $$0.438772\pi$$
$$72$$ 0 0
$$73$$ −8.04090 −0.941116 −0.470558 0.882369i $$-0.655947\pi$$
−0.470558 + 0.882369i $$0.655947\pi$$
$$74$$ 0 0
$$75$$ −2.12311 −0.245155
$$76$$ 0 0
$$77$$ 1.32431 0.150919
$$78$$ 0 0
$$79$$ 10.3101 1.15997 0.579987 0.814626i $$-0.303058\pi$$
0.579987 + 0.814626i $$0.303058\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.77835 0.524492 0.262246 0.965001i $$-0.415537\pi$$
0.262246 + 0.965001i $$0.415537\pi$$
$$84$$ 0 0
$$85$$ −10.4203 −1.13024
$$86$$ 0 0
$$87$$ −1.42696 −0.152987
$$88$$ 0 0
$$89$$ −11.6385 −1.23368 −0.616839 0.787089i $$-0.711587\pi$$
−0.616839 + 0.787089i $$0.711587\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.32431 0.344715
$$94$$ 0 0
$$95$$ −1.69614 −0.174020
$$96$$ 0 0
$$97$$ −6.71659 −0.681966 −0.340983 0.940069i $$-0.610760\pi$$
−0.340983 + 0.940069i $$0.610760\pi$$
$$98$$ 0 0
$$99$$ 1.32431 0.133098
$$100$$ 0 0
$$101$$ 13.7779 1.37096 0.685478 0.728094i $$-0.259593\pi$$
0.685478 + 0.728094i $$0.259593\pi$$
$$102$$ 0 0
$$103$$ 13.1911 1.29976 0.649878 0.760039i $$-0.274820\pi$$
0.649878 + 0.760039i $$0.274820\pi$$
$$104$$ 0 0
$$105$$ −1.69614 −0.165526
$$106$$ 0 0
$$107$$ −5.46786 −0.528598 −0.264299 0.964441i $$-0.585141\pi$$
−0.264299 + 0.964441i $$0.585141\pi$$
$$108$$ 0 0
$$109$$ −14.5292 −1.39165 −0.695823 0.718214i $$-0.744960\pi$$
−0.695823 + 0.718214i $$0.744960\pi$$
$$110$$ 0 0
$$111$$ 9.83969 0.933942
$$112$$ 0 0
$$113$$ −2.77835 −0.261365 −0.130683 0.991424i $$-0.541717\pi$$
−0.130683 + 0.991424i $$0.541717\pi$$
$$114$$ 0 0
$$115$$ 4.94898 0.461495
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.14355 0.563179
$$120$$ 0 0
$$121$$ −9.24621 −0.840565
$$122$$ 0 0
$$123$$ −8.04090 −0.725023
$$124$$ 0 0
$$125$$ 12.0818 1.08063
$$126$$ 0 0
$$127$$ −1.16400 −0.103288 −0.0516442 0.998666i $$-0.516446\pi$$
−0.0516442 + 0.998666i $$0.516446\pi$$
$$128$$ 0 0
$$129$$ −7.16400 −0.630755
$$130$$ 0 0
$$131$$ 0.573035 0.0500663 0.0250332 0.999687i $$-0.492031\pi$$
0.0250332 + 0.999687i $$0.492031\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ −1.69614 −0.145981
$$136$$ 0 0
$$137$$ −3.83558 −0.327696 −0.163848 0.986486i $$-0.552391\pi$$
−0.163848 + 0.986486i $$0.552391\pi$$
$$138$$ 0 0
$$139$$ 20.8255 1.76639 0.883196 0.469004i $$-0.155387\pi$$
0.883196 + 0.469004i $$0.155387\pi$$
$$140$$ 0 0
$$141$$ 7.66906 0.645852
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 2.42033 0.200998
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −1.63438 −0.133894 −0.0669468 0.997757i $$-0.521326\pi$$
−0.0669468 + 0.997757i $$0.521326\pi$$
$$150$$ 0 0
$$151$$ 5.08221 0.413584 0.206792 0.978385i $$-0.433698\pi$$
0.206792 + 0.978385i $$0.433698\pi$$
$$152$$ 0 0
$$153$$ 6.14355 0.496677
$$154$$ 0 0
$$155$$ −5.63849 −0.452895
$$156$$ 0 0
$$157$$ 4.24621 0.338885 0.169442 0.985540i $$-0.445803\pi$$
0.169442 + 0.985540i $$0.445803\pi$$
$$158$$ 0 0
$$159$$ 8.81925 0.699412
$$160$$ 0 0
$$161$$ −2.91779 −0.229954
$$162$$ 0 0
$$163$$ 21.9486 1.71914 0.859572 0.511014i $$-0.170730\pi$$
0.859572 + 0.511014i $$0.170730\pi$$
$$164$$ 0 0
$$165$$ −2.24621 −0.174867
$$166$$ 0 0
$$167$$ 8.20532 0.634946 0.317473 0.948267i $$-0.397166\pi$$
0.317473 + 0.948267i $$0.397166\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 1.39228 0.105853 0.0529266 0.998598i $$-0.483145\pi$$
0.0529266 + 0.998598i $$0.483145\pi$$
$$174$$ 0 0
$$175$$ −2.12311 −0.160492
$$176$$ 0 0
$$177$$ −9.43318 −0.709041
$$178$$ 0 0
$$179$$ −19.1064 −1.42808 −0.714038 0.700107i $$-0.753136\pi$$
−0.714038 + 0.700107i $$0.753136\pi$$
$$180$$ 0 0
$$181$$ 19.9895 1.48580 0.742902 0.669400i $$-0.233449\pi$$
0.742902 + 0.669400i $$0.233449\pi$$
$$182$$ 0 0
$$183$$ 4.64861 0.343635
$$184$$ 0 0
$$185$$ −16.6895 −1.22704
$$186$$ 0 0
$$187$$ 8.13595 0.594960
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 5.70235 0.412608 0.206304 0.978488i $$-0.433856\pi$$
0.206304 + 0.978488i $$0.433856\pi$$
$$192$$ 0 0
$$193$$ 22.9766 1.65389 0.826947 0.562281i $$-0.190076\pi$$
0.826947 + 0.562281i $$0.190076\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.24210 −0.444731 −0.222366 0.974963i $$-0.571378\pi$$
−0.222366 + 0.974963i $$0.571378\pi$$
$$198$$ 0 0
$$199$$ −17.7203 −1.25616 −0.628079 0.778150i $$-0.716158\pi$$
−0.628079 + 0.778150i $$0.716158\pi$$
$$200$$ 0 0
$$201$$ 11.5705 0.816121
$$202$$ 0 0
$$203$$ −1.42696 −0.100153
$$204$$ 0 0
$$205$$ 13.6385 0.952554
$$206$$ 0 0
$$207$$ −2.91779 −0.202800
$$208$$ 0 0
$$209$$ 1.32431 0.0916042
$$210$$ 0 0
$$211$$ 14.8268 1.02072 0.510361 0.859960i $$-0.329512\pi$$
0.510361 + 0.859960i $$0.329512\pi$$
$$212$$ 0 0
$$213$$ 3.22165 0.220744
$$214$$ 0 0
$$215$$ 12.1512 0.828702
$$216$$ 0 0
$$217$$ 3.32431 0.225669
$$218$$ 0 0
$$219$$ −8.04090 −0.543353
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 18.9219 1.26710 0.633552 0.773700i $$-0.281596\pi$$
0.633552 + 0.773700i $$0.281596\pi$$
$$224$$ 0 0
$$225$$ −2.12311 −0.141540
$$226$$ 0 0
$$227$$ 11.3381 0.752538 0.376269 0.926511i $$-0.377207\pi$$
0.376269 + 0.926511i $$0.377207\pi$$
$$228$$ 0 0
$$229$$ −15.9867 −1.05643 −0.528217 0.849110i $$-0.677139\pi$$
−0.528217 + 0.849110i $$0.677139\pi$$
$$230$$ 0 0
$$231$$ 1.32431 0.0871330
$$232$$ 0 0
$$233$$ 9.87648 0.647029 0.323515 0.946223i $$-0.395135\pi$$
0.323515 + 0.946223i $$0.395135\pi$$
$$234$$ 0 0
$$235$$ −13.0078 −0.848536
$$236$$ 0 0
$$237$$ 10.3101 0.669711
$$238$$ 0 0
$$239$$ 11.6155 0.751346 0.375673 0.926752i $$-0.377412\pi$$
0.375673 + 0.926752i $$0.377412\pi$$
$$240$$ 0 0
$$241$$ −14.3142 −0.922058 −0.461029 0.887385i $$-0.652520\pi$$
−0.461029 + 0.887385i $$0.652520\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −1.69614 −0.108362
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.77835 0.302816
$$250$$ 0 0
$$251$$ 25.0779 1.58290 0.791451 0.611233i $$-0.209326\pi$$
0.791451 + 0.611233i $$0.209326\pi$$
$$252$$ 0 0
$$253$$ −3.86405 −0.242931
$$254$$ 0 0
$$255$$ −10.4203 −0.652547
$$256$$ 0 0
$$257$$ −28.7304 −1.79215 −0.896077 0.443899i $$-0.853595\pi$$
−0.896077 + 0.443899i $$0.853595\pi$$
$$258$$ 0 0
$$259$$ 9.83969 0.611409
$$260$$ 0 0
$$261$$ −1.42696 −0.0883269
$$262$$ 0 0
$$263$$ 5.09464 0.314149 0.157074 0.987587i $$-0.449794\pi$$
0.157074 + 0.987587i $$0.449794\pi$$
$$264$$ 0 0
$$265$$ −14.9587 −0.918905
$$266$$ 0 0
$$267$$ −11.6385 −0.712264
$$268$$ 0 0
$$269$$ −13.1410 −0.801223 −0.400612 0.916248i $$-0.631202\pi$$
−0.400612 + 0.916248i $$0.631202\pi$$
$$270$$ 0 0
$$271$$ 13.2972 0.807749 0.403875 0.914814i $$-0.367663\pi$$
0.403875 + 0.914814i $$0.367663\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.81164 −0.169548
$$276$$ 0 0
$$277$$ 23.2049 1.39425 0.697124 0.716951i $$-0.254463\pi$$
0.697124 + 0.716951i $$0.254463\pi$$
$$278$$ 0 0
$$279$$ 3.32431 0.199021
$$280$$ 0 0
$$281$$ 26.4036 1.57511 0.787553 0.616247i $$-0.211348\pi$$
0.787553 + 0.616247i $$0.211348\pi$$
$$282$$ 0 0
$$283$$ 28.7795 1.71077 0.855383 0.517997i $$-0.173322\pi$$
0.855383 + 0.517997i $$0.173322\pi$$
$$284$$ 0 0
$$285$$ −1.69614 −0.100471
$$286$$ 0 0
$$287$$ −8.04090 −0.474639
$$288$$ 0 0
$$289$$ 20.7433 1.22019
$$290$$ 0 0
$$291$$ −6.71659 −0.393733
$$292$$ 0 0
$$293$$ −26.3689 −1.54049 −0.770244 0.637750i $$-0.779865\pi$$
−0.770244 + 0.637750i $$0.779865\pi$$
$$294$$ 0 0
$$295$$ 16.0000 0.931556
$$296$$ 0 0
$$297$$ 1.32431 0.0768441
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −7.16400 −0.412926
$$302$$ 0 0
$$303$$ 13.7779 0.791522
$$304$$ 0 0
$$305$$ −7.88470 −0.451477
$$306$$ 0 0
$$307$$ 9.73082 0.555367 0.277684 0.960673i $$-0.410433\pi$$
0.277684 + 0.960673i $$0.410433\pi$$
$$308$$ 0 0
$$309$$ 13.1911 0.750414
$$310$$ 0 0
$$311$$ −27.4844 −1.55850 −0.779249 0.626715i $$-0.784399\pi$$
−0.779249 + 0.626715i $$0.784399\pi$$
$$312$$ 0 0
$$313$$ 22.9766 1.29872 0.649358 0.760483i $$-0.275038\pi$$
0.649358 + 0.760483i $$0.275038\pi$$
$$314$$ 0 0
$$315$$ −1.69614 −0.0955667
$$316$$ 0 0
$$317$$ 30.1983 1.69610 0.848052 0.529913i $$-0.177776\pi$$
0.848052 + 0.529913i $$0.177776\pi$$
$$318$$ 0 0
$$319$$ −1.88974 −0.105805
$$320$$ 0 0
$$321$$ −5.46786 −0.305186
$$322$$ 0 0
$$323$$ 6.14355 0.341836
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −14.5292 −0.803467
$$328$$ 0 0
$$329$$ 7.66906 0.422809
$$330$$ 0 0
$$331$$ 24.3266 1.33711 0.668556 0.743662i $$-0.266913\pi$$
0.668556 + 0.743662i $$0.266913\pi$$
$$332$$ 0 0
$$333$$ 9.83969 0.539212
$$334$$ 0 0
$$335$$ −19.6252 −1.07224
$$336$$ 0 0
$$337$$ 14.3689 0.782724 0.391362 0.920237i $$-0.372004\pi$$
0.391362 + 0.920237i $$0.372004\pi$$
$$338$$ 0 0
$$339$$ −2.77835 −0.150899
$$340$$ 0 0
$$341$$ 4.40240 0.238403
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 4.94898 0.266444
$$346$$ 0 0
$$347$$ −18.0138 −0.967032 −0.483516 0.875335i $$-0.660641\pi$$
−0.483516 + 0.875335i $$0.660641\pi$$
$$348$$ 0 0
$$349$$ −10.4924 −0.561646 −0.280823 0.959760i $$-0.590607\pi$$
−0.280823 + 0.959760i $$0.590607\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −12.6562 −0.673622 −0.336811 0.941572i $$-0.609348\pi$$
−0.336811 + 0.941572i $$0.609348\pi$$
$$354$$ 0 0
$$355$$ −5.46437 −0.290019
$$356$$ 0 0
$$357$$ 6.14355 0.325151
$$358$$ 0 0
$$359$$ −36.3050 −1.91611 −0.958053 0.286590i $$-0.907478\pi$$
−0.958053 + 0.286590i $$0.907478\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −9.24621 −0.485300
$$364$$ 0 0
$$365$$ 13.6385 0.713871
$$366$$ 0 0
$$367$$ 23.0634 1.20390 0.601951 0.798533i $$-0.294390\pi$$
0.601951 + 0.798533i $$0.294390\pi$$
$$368$$ 0 0
$$369$$ −8.04090 −0.418592
$$370$$ 0 0
$$371$$ 8.81925 0.457872
$$372$$ 0 0
$$373$$ −15.1369 −0.783760 −0.391880 0.920016i $$-0.628175\pi$$
−0.391880 + 0.920016i $$0.628175\pi$$
$$374$$ 0 0
$$375$$ 12.0818 0.623901
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 4.42948 0.227527 0.113764 0.993508i $$-0.463709\pi$$
0.113764 + 0.993508i $$0.463709\pi$$
$$380$$ 0 0
$$381$$ −1.16400 −0.0596336
$$382$$ 0 0
$$383$$ −5.98674 −0.305908 −0.152954 0.988233i $$-0.548879\pi$$
−0.152954 + 0.988233i $$0.548879\pi$$
$$384$$ 0 0
$$385$$ −2.24621 −0.114478
$$386$$ 0 0
$$387$$ −7.16400 −0.364167
$$388$$ 0 0
$$389$$ 26.9858 1.36823 0.684116 0.729373i $$-0.260188\pi$$
0.684116 + 0.729373i $$0.260188\pi$$
$$390$$ 0 0
$$391$$ −17.9256 −0.906537
$$392$$ 0 0
$$393$$ 0.573035 0.0289058
$$394$$ 0 0
$$395$$ −17.4873 −0.879883
$$396$$ 0 0
$$397$$ −14.1360 −0.709463 −0.354732 0.934968i $$-0.615428\pi$$
−0.354732 + 0.934968i $$0.615428\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 4.33505 0.216482 0.108241 0.994125i $$-0.465478\pi$$
0.108241 + 0.994125i $$0.465478\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.69614 −0.0842819
$$406$$ 0 0
$$407$$ 13.0308 0.645912
$$408$$ 0 0
$$409$$ 19.5191 0.965157 0.482578 0.875853i $$-0.339700\pi$$
0.482578 + 0.875853i $$0.339700\pi$$
$$410$$ 0 0
$$411$$ −3.83558 −0.189195
$$412$$ 0 0
$$413$$ −9.43318 −0.464176
$$414$$ 0 0
$$415$$ −8.10476 −0.397847
$$416$$ 0 0
$$417$$ 20.8255 1.01983
$$418$$ 0 0
$$419$$ 10.9552 0.535196 0.267598 0.963531i $$-0.413770\pi$$
0.267598 + 0.963531i $$0.413770\pi$$
$$420$$ 0 0
$$421$$ −1.49843 −0.0730290 −0.0365145 0.999333i $$-0.511626\pi$$
−0.0365145 + 0.999333i $$0.511626\pi$$
$$422$$ 0 0
$$423$$ 7.66906 0.372883
$$424$$ 0 0
$$425$$ −13.0434 −0.632698
$$426$$ 0 0
$$427$$ 4.64861 0.224962
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 7.64471 0.368233 0.184116 0.982904i $$-0.441058\pi$$
0.184116 + 0.982904i $$0.441058\pi$$
$$432$$ 0 0
$$433$$ −26.3418 −1.26591 −0.632954 0.774190i $$-0.718158\pi$$
−0.632954 + 0.774190i $$0.718158\pi$$
$$434$$ 0 0
$$435$$ 2.42033 0.116046
$$436$$ 0 0
$$437$$ −2.91779 −0.139577
$$438$$ 0 0
$$439$$ −5.29584 −0.252757 −0.126378 0.991982i $$-0.540335\pi$$
−0.126378 + 0.991982i $$0.540335\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 11.9118 0.565946 0.282973 0.959128i $$-0.408679\pi$$
0.282973 + 0.959128i $$0.408679\pi$$
$$444$$ 0 0
$$445$$ 19.7405 0.935791
$$446$$ 0 0
$$447$$ −1.63438 −0.0773035
$$448$$ 0 0
$$449$$ −19.8091 −0.934850 −0.467425 0.884033i $$-0.654818\pi$$
−0.467425 + 0.884033i $$0.654818\pi$$
$$450$$ 0 0
$$451$$ −10.6486 −0.501424
$$452$$ 0 0
$$453$$ 5.08221 0.238783
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.9077 0.837685 0.418843 0.908059i $$-0.362436\pi$$
0.418843 + 0.908059i $$0.362436\pi$$
$$458$$ 0 0
$$459$$ 6.14355 0.286757
$$460$$ 0 0
$$461$$ −1.49083 −0.0694347 −0.0347173 0.999397i $$-0.511053\pi$$
−0.0347173 + 0.999397i $$0.511053\pi$$
$$462$$ 0 0
$$463$$ 18.7028 0.869192 0.434596 0.900626i $$-0.356891\pi$$
0.434596 + 0.900626i $$0.356891\pi$$
$$464$$ 0 0
$$465$$ −5.63849 −0.261479
$$466$$ 0 0
$$467$$ −22.1574 −1.02532 −0.512660 0.858591i $$-0.671340\pi$$
−0.512660 + 0.858591i $$0.671340\pi$$
$$468$$ 0 0
$$469$$ 11.5705 0.534277
$$470$$ 0 0
$$471$$ 4.24621 0.195655
$$472$$ 0 0
$$473$$ −9.48734 −0.436228
$$474$$ 0 0
$$475$$ −2.12311 −0.0974148
$$476$$ 0 0
$$477$$ 8.81925 0.403806
$$478$$ 0 0
$$479$$ −16.9387 −0.773947 −0.386973 0.922091i $$-0.626480\pi$$
−0.386973 + 0.922091i $$0.626480\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −2.91779 −0.132764
$$484$$ 0 0
$$485$$ 11.3923 0.517297
$$486$$ 0 0
$$487$$ −10.4460 −0.473354 −0.236677 0.971588i $$-0.576058\pi$$
−0.236677 + 0.971588i $$0.576058\pi$$
$$488$$ 0 0
$$489$$ 21.9486 0.992548
$$490$$ 0 0
$$491$$ 5.66557 0.255684 0.127842 0.991795i $$-0.459195\pi$$
0.127842 + 0.991795i $$0.459195\pi$$
$$492$$ 0 0
$$493$$ −8.76663 −0.394829
$$494$$ 0 0
$$495$$ −2.24621 −0.100960
$$496$$ 0 0
$$497$$ 3.22165 0.144511
$$498$$ 0 0
$$499$$ −29.8438 −1.33599 −0.667996 0.744165i $$-0.732848\pi$$
−0.667996 + 0.744165i $$0.732848\pi$$
$$500$$ 0 0
$$501$$ 8.20532 0.366586
$$502$$ 0 0
$$503$$ −9.02868 −0.402569 −0.201284 0.979533i $$-0.564512\pi$$
−0.201284 + 0.979533i $$0.564512\pi$$
$$504$$ 0 0
$$505$$ −23.3693 −1.03992
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 0 0
$$509$$ 23.9256 1.06048 0.530242 0.847846i $$-0.322101\pi$$
0.530242 + 0.847846i $$0.322101\pi$$
$$510$$ 0 0
$$511$$ −8.04090 −0.355708
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ −22.3739 −0.985913
$$516$$ 0 0
$$517$$ 10.1562 0.446669
$$518$$ 0 0
$$519$$ 1.39228 0.0611144
$$520$$ 0 0
$$521$$ 26.5742 1.16424 0.582119 0.813104i $$-0.302224\pi$$
0.582119 + 0.813104i $$0.302224\pi$$
$$522$$ 0 0
$$523$$ 9.73082 0.425499 0.212750 0.977107i $$-0.431758\pi$$
0.212750 + 0.977107i $$0.431758\pi$$
$$524$$ 0 0
$$525$$ −2.12311 −0.0926599
$$526$$ 0 0
$$527$$ 20.4231 0.889642
$$528$$ 0 0
$$529$$ −14.4865 −0.629848
$$530$$ 0 0
$$531$$ −9.43318 −0.409365
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 9.27426 0.400961
$$536$$ 0 0
$$537$$ −19.1064 −0.824500
$$538$$ 0 0
$$539$$ 1.32431 0.0570419
$$540$$ 0 0
$$541$$ −2.49242 −0.107158 −0.0535788 0.998564i $$-0.517063\pi$$
−0.0535788 + 0.998564i $$0.517063\pi$$
$$542$$ 0 0
$$543$$ 19.9895 0.857830
$$544$$ 0 0
$$545$$ 24.6436 1.05561
$$546$$ 0 0
$$547$$ 5.12722 0.219224 0.109612 0.993974i $$-0.465039\pi$$
0.109612 + 0.993974i $$0.465039\pi$$
$$548$$ 0 0
$$549$$ 4.64861 0.198398
$$550$$ 0 0
$$551$$ −1.42696 −0.0607907
$$552$$ 0 0
$$553$$ 10.3101 0.438429
$$554$$ 0 0
$$555$$ −16.6895 −0.708430
$$556$$ 0 0
$$557$$ −7.96238 −0.337377 −0.168688 0.985669i $$-0.553953\pi$$
−0.168688 + 0.985669i $$0.553953\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.13595 0.343500
$$562$$ 0 0
$$563$$ −18.9899 −0.800328 −0.400164 0.916444i $$-0.631047\pi$$
−0.400164 + 0.916444i $$0.631047\pi$$
$$564$$ 0 0
$$565$$ 4.71247 0.198255
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −9.71407 −0.407235 −0.203618 0.979051i $$-0.565270\pi$$
−0.203618 + 0.979051i $$0.565270\pi$$
$$570$$ 0 0
$$571$$ −17.7079 −0.741051 −0.370525 0.928822i $$-0.620822\pi$$
−0.370525 + 0.928822i $$0.620822\pi$$
$$572$$ 0 0
$$573$$ 5.70235 0.238219
$$574$$ 0 0
$$575$$ 6.19478 0.258340
$$576$$ 0 0
$$577$$ −38.1636 −1.58877 −0.794385 0.607414i $$-0.792207\pi$$
−0.794385 + 0.607414i $$0.792207\pi$$
$$578$$ 0 0
$$579$$ 22.9766 0.954876
$$580$$ 0 0
$$581$$ 4.77835 0.198239
$$582$$ 0 0
$$583$$ 11.6794 0.483711
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 41.9186 1.73016 0.865082 0.501631i $$-0.167266\pi$$
0.865082 + 0.501631i $$0.167266\pi$$
$$588$$ 0 0
$$589$$ 3.32431 0.136976
$$590$$ 0 0
$$591$$ −6.24210 −0.256766
$$592$$ 0 0
$$593$$ −22.0200 −0.904254 −0.452127 0.891954i $$-0.649335\pi$$
−0.452127 + 0.891954i $$0.649335\pi$$
$$594$$ 0 0
$$595$$ −10.4203 −0.427192
$$596$$ 0 0
$$597$$ −17.7203 −0.725243
$$598$$ 0 0
$$599$$ 14.0880 0.575620 0.287810 0.957687i $$-0.407073\pi$$
0.287810 + 0.957687i $$0.407073\pi$$
$$600$$ 0 0
$$601$$ −5.01696 −0.204646 −0.102323 0.994751i $$-0.532628\pi$$
−0.102323 + 0.994751i $$0.532628\pi$$
$$602$$ 0 0
$$603$$ 11.5705 0.471188
$$604$$ 0 0
$$605$$ 15.6829 0.637600
$$606$$ 0 0
$$607$$ 35.9215 1.45801 0.729004 0.684509i $$-0.239984\pi$$
0.729004 + 0.684509i $$0.239984\pi$$
$$608$$ 0 0
$$609$$ −1.42696 −0.0578235
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 29.5053 1.19171 0.595853 0.803093i $$-0.296814\pi$$
0.595853 + 0.803093i $$0.296814\pi$$
$$614$$ 0 0
$$615$$ 13.6385 0.549957
$$616$$ 0 0
$$617$$ −12.3822 −0.498487 −0.249244 0.968441i $$-0.580182\pi$$
−0.249244 + 0.968441i $$0.580182\pi$$
$$618$$ 0 0
$$619$$ −26.5333 −1.06646 −0.533232 0.845969i $$-0.679023\pi$$
−0.533232 + 0.845969i $$0.679023\pi$$
$$620$$ 0 0
$$621$$ −2.91779 −0.117087
$$622$$ 0 0
$$623$$ −11.6385 −0.466286
$$624$$ 0 0
$$625$$ −9.87689 −0.395076
$$626$$ 0 0
$$627$$ 1.32431 0.0528877
$$628$$ 0 0
$$629$$ 60.4507 2.41033
$$630$$ 0 0
$$631$$ −20.0588 −0.798529 −0.399265 0.916836i $$-0.630735\pi$$
−0.399265 + 0.916836i $$0.630735\pi$$
$$632$$ 0 0
$$633$$ 14.8268 0.589314
$$634$$ 0 0
$$635$$ 1.97431 0.0783481
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3.22165 0.127446
$$640$$ 0 0
$$641$$ 43.0237 1.69934 0.849668 0.527319i $$-0.176803\pi$$
0.849668 + 0.527319i $$0.176803\pi$$
$$642$$ 0 0
$$643$$ −31.9665 −1.26064 −0.630318 0.776337i $$-0.717075\pi$$
−0.630318 + 0.776337i $$0.717075\pi$$
$$644$$ 0 0
$$645$$ 12.1512 0.478451
$$646$$ 0 0
$$647$$ −15.3820 −0.604727 −0.302364 0.953193i $$-0.597776\pi$$
−0.302364 + 0.953193i $$0.597776\pi$$
$$648$$ 0 0
$$649$$ −12.4924 −0.490370
$$650$$ 0 0
$$651$$ 3.32431 0.130290
$$652$$ 0 0
$$653$$ −17.3013 −0.677054 −0.338527 0.940957i $$-0.609929\pi$$
−0.338527 + 0.940957i $$0.609929\pi$$
$$654$$ 0 0
$$655$$ −0.971949 −0.0379772
$$656$$ 0 0
$$657$$ −8.04090 −0.313705
$$658$$ 0 0
$$659$$ −46.0421 −1.79354 −0.896772 0.442492i $$-0.854094\pi$$
−0.896772 + 0.442492i $$0.854094\pi$$
$$660$$ 0 0
$$661$$ 5.69963 0.221690 0.110845 0.993838i $$-0.464644\pi$$
0.110845 + 0.993838i $$0.464644\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.69614 −0.0657735
$$666$$ 0 0
$$667$$ 4.16358 0.161215
$$668$$ 0 0
$$669$$ 18.9219 0.731563
$$670$$ 0 0
$$671$$ 6.15619 0.237657
$$672$$ 0 0
$$673$$ −18.2095 −0.701925 −0.350963 0.936389i $$-0.614146\pi$$
−0.350963 + 0.936389i $$0.614146\pi$$
$$674$$ 0 0
$$675$$ −2.12311 −0.0817184
$$676$$ 0 0
$$677$$ −43.1126 −1.65695 −0.828475 0.560026i $$-0.810791\pi$$
−0.828475 + 0.560026i $$0.810791\pi$$
$$678$$ 0 0
$$679$$ −6.71659 −0.257759
$$680$$ 0 0
$$681$$ 11.3381 0.434478
$$682$$ 0 0
$$683$$ 1.60381 0.0613681 0.0306841 0.999529i $$-0.490231\pi$$
0.0306841 + 0.999529i $$0.490231\pi$$
$$684$$ 0 0
$$685$$ 6.50569 0.248569
$$686$$ 0 0
$$687$$ −15.9867 −0.609932
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 18.3739 0.698977 0.349489 0.936941i $$-0.386355\pi$$
0.349489 + 0.936941i $$0.386355\pi$$
$$692$$ 0 0
$$693$$ 1.32431 0.0503063
$$694$$ 0 0
$$695$$ −35.3229 −1.33987
$$696$$ 0 0
$$697$$ −49.3997 −1.87115
$$698$$ 0 0
$$699$$ 9.87648 0.373563
$$700$$ 0 0
$$701$$ −30.5958 −1.15559 −0.577794 0.816183i $$-0.696086\pi$$
−0.577794 + 0.816183i $$0.696086\pi$$
$$702$$ 0 0
$$703$$ 9.83969 0.371111
$$704$$ 0 0
$$705$$ −13.0078 −0.489902
$$706$$ 0 0
$$707$$ 13.7779 0.518172
$$708$$ 0 0
$$709$$ 25.2334 0.947659 0.473829 0.880617i $$-0.342871\pi$$
0.473829 + 0.880617i $$0.342871\pi$$
$$710$$ 0 0
$$711$$ 10.3101 0.386658
$$712$$ 0 0
$$713$$ −9.69963 −0.363254
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 11.6155 0.433790
$$718$$ 0 0
$$719$$ −21.8459 −0.814715 −0.407357 0.913269i $$-0.633550\pi$$
−0.407357 + 0.913269i $$0.633550\pi$$
$$720$$ 0 0
$$721$$ 13.1911 0.491262
$$722$$ 0 0
$$723$$ −14.3142 −0.532350
$$724$$ 0 0
$$725$$ 3.02960 0.112516
$$726$$ 0 0
$$727$$ −39.8889 −1.47940 −0.739699 0.672938i $$-0.765032\pi$$
−0.739699 + 0.672938i $$0.765032\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −44.0124 −1.62786
$$732$$ 0 0
$$733$$ −34.7713 −1.28431 −0.642154 0.766576i $$-0.721959\pi$$
−0.642154 + 0.766576i $$0.721959\pi$$
$$734$$ 0 0
$$735$$ −1.69614 −0.0625631
$$736$$ 0 0
$$737$$ 15.3229 0.564427
$$738$$ 0 0
$$739$$ 3.02805 0.111389 0.0556943 0.998448i $$-0.482263\pi$$
0.0556943 + 0.998448i $$0.482263\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25.0737 −0.919864 −0.459932 0.887954i $$-0.652126\pi$$
−0.459932 + 0.887954i $$0.652126\pi$$
$$744$$ 0 0
$$745$$ 2.77214 0.101563
$$746$$ 0 0
$$747$$ 4.77835 0.174831
$$748$$ 0 0
$$749$$ −5.46786 −0.199791
$$750$$ 0 0
$$751$$ −12.7616 −0.465677 −0.232839 0.972515i $$-0.574801\pi$$
−0.232839 + 0.972515i $$0.574801\pi$$
$$752$$ 0 0
$$753$$ 25.0779 0.913889
$$754$$ 0 0
$$755$$ −8.62014 −0.313719
$$756$$ 0 0
$$757$$ 16.3198 0.593152 0.296576 0.955009i $$-0.404155\pi$$
0.296576 + 0.955009i $$0.404155\pi$$
$$758$$ 0 0
$$759$$ −3.86405 −0.140256
$$760$$ 0 0
$$761$$ −24.6492 −0.893534 −0.446767 0.894650i $$-0.647425\pi$$
−0.446767 + 0.894650i $$0.647425\pi$$
$$762$$ 0 0
$$763$$ −14.5292 −0.525993
$$764$$ 0 0
$$765$$ −10.4203 −0.376748
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 12.1103 0.436707 0.218354 0.975870i $$-0.429931\pi$$
0.218354 + 0.975870i $$0.429931\pi$$
$$770$$ 0 0
$$771$$ −28.7304 −1.03470
$$772$$ 0 0
$$773$$ −35.4456 −1.27489 −0.637445 0.770496i $$-0.720009\pi$$
−0.637445 + 0.770496i $$0.720009\pi$$
$$774$$ 0 0
$$775$$ −7.05785 −0.253526
$$776$$ 0 0
$$777$$ 9.83969 0.352997
$$778$$ 0 0
$$779$$ −8.04090 −0.288095
$$780$$ 0 0
$$781$$ 4.26645 0.152666
$$782$$ 0 0
$$783$$ −1.42696 −0.0509956
$$784$$ 0 0
$$785$$ −7.20217 −0.257057
$$786$$ 0 0
$$787$$ −24.5890 −0.876501 −0.438251 0.898853i $$-0.644402\pi$$
−0.438251 + 0.898853i $$0.644402\pi$$
$$788$$ 0 0
$$789$$ 5.09464 0.181374
$$790$$ 0 0
$$791$$ −2.77835 −0.0987868
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −14.9587 −0.530530
$$796$$ 0 0
$$797$$ −8.85393 −0.313622 −0.156811 0.987629i $$-0.550121\pi$$
−0.156811 + 0.987629i $$0.550121\pi$$
$$798$$ 0 0
$$799$$ 47.1153 1.66682
$$800$$ 0 0
$$801$$ −11.6385 −0.411226
$$802$$ 0 0
$$803$$ −10.6486 −0.375781
$$804$$ 0 0
$$805$$ 4.94898 0.174429
$$806$$ 0 0
$$807$$ −13.1410 −0.462586
$$808$$ 0 0
$$809$$ −5.05924 −0.177874 −0.0889368 0.996037i $$-0.528347\pi$$
−0.0889368 + 0.996037i $$0.528347\pi$$
$$810$$ 0 0
$$811$$ −24.2095 −0.850111 −0.425056 0.905167i $$-0.639745\pi$$
−0.425056 + 0.905167i $$0.639745\pi$$
$$812$$ 0 0
$$813$$ 13.2972 0.466354
$$814$$ 0 0
$$815$$ −37.2279 −1.30404
$$816$$ 0 0
$$817$$ −7.16400 −0.250637
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 32.3372 1.12857 0.564287 0.825579i $$-0.309151\pi$$
0.564287 + 0.825579i $$0.309151\pi$$
$$822$$ 0 0
$$823$$ −43.2734 −1.50842 −0.754208 0.656635i $$-0.771979\pi$$
−0.754208 + 0.656635i $$0.771979\pi$$
$$824$$ 0 0
$$825$$ −2.81164 −0.0978889
$$826$$ 0 0
$$827$$ 0.975438 0.0339193 0.0169597 0.999856i $$-0.494601\pi$$
0.0169597 + 0.999856i $$0.494601\pi$$
$$828$$ 0 0
$$829$$ −20.0436 −0.696144 −0.348072 0.937468i $$-0.613163\pi$$
−0.348072 + 0.937468i $$0.613163\pi$$
$$830$$ 0 0
$$831$$ 23.2049 0.804969
$$832$$ 0 0
$$833$$ 6.14355 0.212862
$$834$$ 0 0
$$835$$ −13.9174 −0.481631
$$836$$ 0 0
$$837$$ 3.32431 0.114905
$$838$$ 0 0
$$839$$ 47.4947 1.63970 0.819850 0.572578i $$-0.194057\pi$$
0.819850 + 0.572578i $$0.194057\pi$$
$$840$$ 0 0
$$841$$ −26.9638 −0.929785
$$842$$ 0 0
$$843$$ 26.4036 0.909388
$$844$$ 0 0
$$845$$ 22.0498 0.758537
$$846$$ 0 0
$$847$$ −9.24621 −0.317704
$$848$$ 0 0
$$849$$ 28.7795 0.987711
$$850$$ 0 0
$$851$$ −28.7102 −0.984172
$$852$$ 0 0
$$853$$ 4.64861 0.159166 0.0795828 0.996828i $$-0.474641\pi$$
0.0795828 + 0.996828i $$0.474641\pi$$
$$854$$ 0 0
$$855$$ −1.69614 −0.0580068
$$856$$ 0 0
$$857$$ −20.3974 −0.696761 −0.348380 0.937353i $$-0.613268\pi$$
−0.348380 + 0.937353i $$0.613268\pi$$
$$858$$ 0 0
$$859$$ −5.07949 −0.173310 −0.0866549 0.996238i $$-0.527618\pi$$
−0.0866549 + 0.996238i $$0.527618\pi$$
$$860$$ 0 0
$$861$$ −8.04090 −0.274033
$$862$$ 0 0
$$863$$ −12.5119 −0.425910 −0.212955 0.977062i $$-0.568309\pi$$
−0.212955 + 0.977062i $$0.568309\pi$$
$$864$$ 0 0
$$865$$ −2.36151 −0.0802936
$$866$$ 0 0
$$867$$ 20.7433 0.704478
$$868$$ 0 0
$$869$$ 13.6537 0.463170
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −6.71659 −0.227322
$$874$$ 0 0
$$875$$ 12.0818 0.408439
$$876$$ 0 0
$$877$$ 52.6601 1.77821 0.889103 0.457707i $$-0.151329\pi$$
0.889103 + 0.457707i $$0.151329\pi$$
$$878$$ 0 0
$$879$$ −26.3689 −0.889401
$$880$$ 0 0
$$881$$ 23.5892 0.794739 0.397369 0.917659i $$-0.369923\pi$$
0.397369 + 0.917659i $$0.369923\pi$$
$$882$$ 0 0
$$883$$ 41.9256 1.41091 0.705454 0.708755i $$-0.250743\pi$$
0.705454 + 0.708755i $$0.250743\pi$$
$$884$$ 0 0
$$885$$ 16.0000 0.537834
$$886$$ 0 0
$$887$$ −2.31280 −0.0776561 −0.0388281 0.999246i $$-0.512362\pi$$
−0.0388281 + 0.999246i $$0.512362\pi$$
$$888$$ 0 0
$$889$$ −1.16400 −0.0390394
$$890$$ 0 0
$$891$$ 1.32431 0.0443660
$$892$$ 0 0
$$893$$ 7.66906 0.256635
$$894$$ 0 0
$$895$$ 32.4071 1.08325
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −4.74367 −0.158210
$$900$$ 0 0
$$901$$ 54.1815 1.80505
$$902$$ 0 0
$$903$$ −7.16400 −0.238403
$$904$$ 0 0
$$905$$ −33.9049 −1.12704
$$906$$ 0 0
$$907$$ 14.4980 0.481399 0.240699 0.970600i $$-0.422623\pi$$
0.240699 + 0.970600i $$0.422623\pi$$
$$908$$ 0 0
$$909$$ 13.7779 0.456985
$$910$$ 0 0
$$911$$ 25.1320 0.832662 0.416331 0.909213i $$-0.363316\pi$$
0.416331 + 0.909213i $$0.363316\pi$$
$$912$$ 0 0
$$913$$ 6.32800 0.209426
$$914$$ 0 0
$$915$$ −7.88470 −0.260660
$$916$$ 0 0
$$917$$ 0.573035 0.0189233
$$918$$ 0 0
$$919$$ −7.17874 −0.236805 −0.118402 0.992966i $$-0.537777\pi$$
−0.118402 + 0.992966i $$0.537777\pi$$
$$920$$ 0 0
$$921$$ 9.73082 0.320642
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −20.8907 −0.686882
$$926$$ 0 0
$$927$$ 13.1911 0.433252
$$928$$ 0 0
$$929$$ −45.1334 −1.48078 −0.740390 0.672178i $$-0.765359\pi$$
−0.740390 + 0.672178i $$0.765359\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ −27.4844 −0.899799
$$934$$ 0 0
$$935$$ −13.7997 −0.451299
$$936$$ 0 0
$$937$$ 52.4916 1.71483 0.857413 0.514629i $$-0.172071\pi$$
0.857413 + 0.514629i $$0.172071\pi$$
$$938$$ 0 0
$$939$$ 22.9766 0.749814
$$940$$ 0 0
$$941$$ −35.3055 −1.15092 −0.575462 0.817828i $$-0.695178\pi$$
−0.575462 + 0.817828i $$0.695178\pi$$
$$942$$ 0 0
$$943$$ 23.4616 0.764016
$$944$$ 0 0
$$945$$ −1.69614 −0.0551755
$$946$$ 0 0
$$947$$ 23.7800 0.772747 0.386374 0.922342i $$-0.373728\pi$$
0.386374 + 0.922342i $$0.373728\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 30.1983 0.979246
$$952$$ 0 0
$$953$$ −16.6957 −0.540828 −0.270414 0.962744i $$-0.587160\pi$$
−0.270414 + 0.962744i $$0.587160\pi$$
$$954$$ 0 0
$$955$$ −9.67200 −0.312978
$$956$$ 0 0
$$957$$ −1.88974 −0.0610866
$$958$$ 0 0
$$959$$ −3.83558 −0.123857
$$960$$ 0 0
$$961$$ −19.9490 −0.643516
$$962$$ 0 0
$$963$$ −5.46786 −0.176199
$$964$$ 0 0
$$965$$ −38.9716 −1.25454
$$966$$ 0 0
$$967$$ 47.7300 1.53489 0.767446 0.641113i $$-0.221527\pi$$
0.767446 + 0.641113i $$0.221527\pi$$
$$968$$ 0 0
$$969$$ 6.14355 0.197359
$$970$$ 0 0
$$971$$ −18.6486 −0.598462 −0.299231 0.954181i $$-0.596730\pi$$
−0.299231 + 0.954181i $$0.596730\pi$$
$$972$$ 0 0
$$973$$ 20.8255 0.667634
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −25.3728 −0.811748 −0.405874 0.913929i $$-0.633033\pi$$
−0.405874 + 0.913929i $$0.633033\pi$$
$$978$$ 0 0
$$979$$ −15.4129 −0.492600
$$980$$ 0 0
$$981$$ −14.5292 −0.463882
$$982$$ 0 0
$$983$$ −44.3841 −1.41563 −0.707817 0.706396i $$-0.750320\pi$$
−0.707817 + 0.706396i $$0.750320\pi$$
$$984$$ 0 0
$$985$$ 10.5875 0.337345
$$986$$ 0 0
$$987$$ 7.66906 0.244109
$$988$$ 0 0
$$989$$ 20.9031 0.664678
$$990$$ 0 0
$$991$$ 34.7483 1.10382 0.551909 0.833905i $$-0.313900\pi$$
0.551909 + 0.833905i $$0.313900\pi$$
$$992$$ 0 0
$$993$$ 24.3266 0.771982
$$994$$ 0 0
$$995$$ 30.0561 0.952843
$$996$$ 0 0
$$997$$ −1.32292 −0.0418972 −0.0209486 0.999781i $$-0.506669\pi$$
−0.0209486 + 0.999781i $$0.506669\pi$$
$$998$$ 0 0
$$999$$ 9.83969 0.311314
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 6384.2.a.cb.1.2 4
4.3 odd 2 3192.2.a.x.1.2 4
12.11 even 2 9576.2.a.cj.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.x.1.2 4 4.3 odd 2
6384.2.a.cb.1.2 4 1.1 even 1 trivial
9576.2.a.cj.1.3 4 12.11 even 2