# Properties

 Label 4275.2.a.br.1.6 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.1360468641$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$1.30397$$ of defining polynomial Character $$\chi$$ $$=$$ 4275.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41987 q^{2} +3.85577 q^{4} -3.18676 q^{7} +4.49073 q^{8} +O(q^{10})$$ $$q+2.41987 q^{2} +3.85577 q^{4} -3.18676 q^{7} +4.49073 q^{8} -4.15544 q^{11} +2.07086 q^{13} -7.71155 q^{14} +3.15544 q^{16} -5.79470 q^{17} -1.00000 q^{19} -10.0556 q^{22} +2.60794 q^{23} +5.01121 q^{26} -12.2874 q^{28} -6.00000 q^{29} +2.59933 q^{31} -1.34571 q^{32} -14.0224 q^{34} +4.30266 q^{37} -2.41987 q^{38} +0.599328 q^{41} +3.18676 q^{43} -16.0224 q^{44} +6.31087 q^{46} -11.7086 q^{47} +3.15544 q^{49} +7.98476 q^{52} -11.7503 q^{53} -14.3109 q^{56} -14.5192 q^{58} +1.71155 q^{59} -8.75476 q^{61} +6.29004 q^{62} -9.56732 q^{64} +4.76228 q^{67} -22.3430 q^{68} -13.7115 q^{71} -2.72714 q^{73} +10.4119 q^{74} -3.85577 q^{76} +13.2424 q^{77} -1.40067 q^{79} +1.45030 q^{82} +7.07154 q^{83} +7.71155 q^{86} -18.6609 q^{88} -16.5353 q^{89} -6.59933 q^{91} +10.0556 q^{92} -28.3333 q^{94} +2.07086 q^{97} +7.63575 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4}+O(q^{10})$$ 6 * q + 8 * q^4 $$6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100})$$ 6 * q + 8 * q^4 - 2 * q^11 - 16 * q^14 - 4 * q^16 - 6 * q^19 - 8 * q^26 - 36 * q^29 - 8 * q^34 - 12 * q^41 - 20 * q^44 - 8 * q^46 - 4 * q^49 - 40 * q^56 - 20 * q^59 - 14 * q^61 - 12 * q^64 - 52 * q^71 + 40 * q^74 - 8 * q^76 - 24 * q^79 + 16 * q^86 - 24 * q^89 - 24 * q^91 - 48 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41987 1.71111 0.855553 0.517715i $$-0.173217\pi$$
0.855553 + 0.517715i $$0.173217\pi$$
$$3$$ 0 0
$$4$$ 3.85577 1.92789
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.18676 −1.20448 −0.602241 0.798314i $$-0.705725\pi$$
−0.602241 + 0.798314i $$0.705725\pi$$
$$8$$ 4.49073 1.58771
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.15544 −1.25291 −0.626456 0.779457i $$-0.715495\pi$$
−0.626456 + 0.779457i $$0.715495\pi$$
$$12$$ 0 0
$$13$$ 2.07086 0.574353 0.287176 0.957878i $$-0.407283\pi$$
0.287176 + 0.957878i $$0.407283\pi$$
$$14$$ −7.71155 −2.06100
$$15$$ 0 0
$$16$$ 3.15544 0.788859
$$17$$ −5.79470 −1.40542 −0.702710 0.711476i $$-0.748027\pi$$
−0.702710 + 0.711476i $$0.748027\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −10.0556 −2.14386
$$23$$ 2.60794 0.543793 0.271896 0.962327i $$-0.412349\pi$$
0.271896 + 0.962327i $$0.412349\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.01121 0.982779
$$27$$ 0 0
$$28$$ −12.2874 −2.32210
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 2.59933 0.466853 0.233427 0.972374i $$-0.425006\pi$$
0.233427 + 0.972374i $$0.425006\pi$$
$$32$$ −1.34571 −0.237890
$$33$$ 0 0
$$34$$ −14.0224 −2.40482
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.30266 0.707353 0.353677 0.935368i $$-0.384931\pi$$
0.353677 + 0.935368i $$0.384931\pi$$
$$38$$ −2.41987 −0.392555
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.599328 0.0935993 0.0467997 0.998904i $$-0.485098\pi$$
0.0467997 + 0.998904i $$0.485098\pi$$
$$42$$ 0 0
$$43$$ 3.18676 0.485976 0.242988 0.970029i $$-0.421872\pi$$
0.242988 + 0.970029i $$0.421872\pi$$
$$44$$ −16.0224 −2.41547
$$45$$ 0 0
$$46$$ 6.31087 0.930487
$$47$$ −11.7086 −1.70787 −0.853937 0.520376i $$-0.825792\pi$$
−0.853937 + 0.520376i $$0.825792\pi$$
$$48$$ 0 0
$$49$$ 3.15544 0.450777
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 7.98476 1.10729
$$53$$ −11.7503 −1.61403 −0.807017 0.590529i $$-0.798919\pi$$
−0.807017 + 0.590529i $$0.798919\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −14.3109 −1.91237
$$57$$ 0 0
$$58$$ −14.5192 −1.90647
$$59$$ 1.71155 0.222824 0.111412 0.993774i $$-0.464463\pi$$
0.111412 + 0.993774i $$0.464463\pi$$
$$60$$ 0 0
$$61$$ −8.75476 −1.12093 −0.560466 0.828177i $$-0.689378\pi$$
−0.560466 + 0.828177i $$0.689378\pi$$
$$62$$ 6.29004 0.798836
$$63$$ 0 0
$$64$$ −9.56732 −1.19591
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.76228 0.581805 0.290902 0.956753i $$-0.406044\pi$$
0.290902 + 0.956753i $$0.406044\pi$$
$$68$$ −22.3430 −2.70949
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.7115 −1.62726 −0.813631 0.581382i $$-0.802512\pi$$
−0.813631 + 0.581382i $$0.802512\pi$$
$$72$$ 0 0
$$73$$ −2.72714 −0.319188 −0.159594 0.987183i $$-0.551018\pi$$
−0.159594 + 0.987183i $$0.551018\pi$$
$$74$$ 10.4119 1.21036
$$75$$ 0 0
$$76$$ −3.85577 −0.442287
$$77$$ 13.2424 1.50911
$$78$$ 0 0
$$79$$ −1.40067 −0.157588 −0.0787939 0.996891i $$-0.525107\pi$$
−0.0787939 + 0.996891i $$0.525107\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1.45030 0.160158
$$83$$ 7.07154 0.776203 0.388101 0.921617i $$-0.373131\pi$$
0.388101 + 0.921617i $$0.373131\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 7.71155 0.831557
$$87$$ 0 0
$$88$$ −18.6609 −1.98926
$$89$$ −16.5353 −1.75274 −0.876370 0.481639i $$-0.840042\pi$$
−0.876370 + 0.481639i $$0.840042\pi$$
$$90$$ 0 0
$$91$$ −6.59933 −0.691798
$$92$$ 10.0556 1.04837
$$93$$ 0 0
$$94$$ −28.3333 −2.92236
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.07086 0.210264 0.105132 0.994458i $$-0.466474\pi$$
0.105132 + 0.994458i $$0.466474\pi$$
$$98$$ 7.63575 0.771327
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.71155 0.170305 0.0851525 0.996368i $$-0.472862\pi$$
0.0851525 + 0.996368i $$0.472862\pi$$
$$102$$ 0 0
$$103$$ 5.75296 0.566856 0.283428 0.958994i $$-0.408528\pi$$
0.283428 + 0.958994i $$0.408528\pi$$
$$104$$ 9.29966 0.911907
$$105$$ 0 0
$$106$$ −28.4343 −2.76178
$$107$$ 15.4324 1.49191 0.745955 0.665996i $$-0.231993\pi$$
0.745955 + 0.665996i $$0.231993\pi$$
$$108$$ 0 0
$$109$$ 11.7115 1.12176 0.560881 0.827896i $$-0.310462\pi$$
0.560881 + 0.827896i $$0.310462\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −10.0556 −0.950167
$$113$$ 10.5927 0.996477 0.498239 0.867040i $$-0.333980\pi$$
0.498239 + 0.867040i $$0.333980\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −23.1346 −2.14800
$$117$$ 0 0
$$118$$ 4.14172 0.381276
$$119$$ 18.4663 1.69280
$$120$$ 0 0
$$121$$ 6.26765 0.569787
$$122$$ −21.1854 −1.91804
$$123$$ 0 0
$$124$$ 10.0224 0.900040
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.07484 −0.539055 −0.269528 0.962993i $$-0.586868\pi$$
−0.269528 + 0.962993i $$0.586868\pi$$
$$128$$ −20.4602 −1.80845
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.5785 1.18636 0.593181 0.805069i $$-0.297872\pi$$
0.593181 + 0.805069i $$0.297872\pi$$
$$132$$ 0 0
$$133$$ 3.18676 0.276327
$$134$$ 11.5241 0.995530
$$135$$ 0 0
$$136$$ −26.0224 −2.23140
$$137$$ 7.94302 0.678618 0.339309 0.940675i $$-0.389807\pi$$
0.339309 + 0.940675i $$0.389807\pi$$
$$138$$ 0 0
$$139$$ 3.26765 0.277159 0.138579 0.990351i $$-0.455746\pi$$
0.138579 + 0.990351i $$0.455746\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −33.1802 −2.78442
$$143$$ −8.60532 −0.719613
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −6.59933 −0.546164
$$147$$ 0 0
$$148$$ 16.5901 1.36370
$$149$$ −8.44389 −0.691751 −0.345875 0.938280i $$-0.612418\pi$$
−0.345875 + 0.938280i $$0.612418\pi$$
$$150$$ 0 0
$$151$$ 0.887783 0.0722468 0.0361234 0.999347i $$-0.488499\pi$$
0.0361234 + 0.999347i $$0.488499\pi$$
$$152$$ −4.49073 −0.364246
$$153$$ 0 0
$$154$$ 32.0448 2.58225
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.14172 0.330545 0.165273 0.986248i $$-0.447150\pi$$
0.165273 + 0.986248i $$0.447150\pi$$
$$158$$ −3.38944 −0.269650
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.31087 −0.654989
$$162$$ 0 0
$$163$$ −24.7126 −1.93564 −0.967819 0.251647i $$-0.919028\pi$$
−0.967819 + 0.251647i $$0.919028\pi$$
$$164$$ 2.31087 0.180449
$$165$$ 0 0
$$166$$ 17.1122 1.32817
$$167$$ 3.60464 0.278935 0.139468 0.990227i $$-0.455461\pi$$
0.139468 + 0.990227i $$0.455461\pi$$
$$168$$ 0 0
$$169$$ −8.71155 −0.670119
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.2874 0.936907
$$173$$ 22.4205 1.70460 0.852300 0.523054i $$-0.175207\pi$$
0.852300 + 0.523054i $$0.175207\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −13.1122 −0.988371
$$177$$ 0 0
$$178$$ −40.0133 −2.99912
$$179$$ 5.13464 0.383781 0.191890 0.981416i $$-0.438538\pi$$
0.191890 + 0.981416i $$0.438538\pi$$
$$180$$ 0 0
$$181$$ 20.8462 1.54948 0.774742 0.632277i $$-0.217880\pi$$
0.774742 + 0.632277i $$0.217880\pi$$
$$182$$ −15.9695 −1.18374
$$183$$ 0 0
$$184$$ 11.7115 0.863387
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 24.0795 1.76087
$$188$$ −45.1457 −3.29259
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 5.26765 0.381154 0.190577 0.981672i $$-0.438964\pi$$
0.190577 + 0.981672i $$0.438964\pi$$
$$192$$ 0 0
$$193$$ 2.07086 0.149064 0.0745318 0.997219i $$-0.476254\pi$$
0.0745318 + 0.997219i $$0.476254\pi$$
$$194$$ 5.01121 0.359784
$$195$$ 0 0
$$196$$ 12.1666 0.869046
$$197$$ −10.4318 −0.743232 −0.371616 0.928387i $$-0.621196\pi$$
−0.371616 + 0.928387i $$0.621196\pi$$
$$198$$ 0 0
$$199$$ −2.73235 −0.193691 −0.0968455 0.995299i $$-0.530875\pi$$
−0.0968455 + 0.995299i $$0.530875\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 4.14172 0.291410
$$203$$ 19.1206 1.34200
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.9214 0.969951
$$207$$ 0 0
$$208$$ 6.53446 0.453083
$$209$$ 4.15544 0.287438
$$210$$ 0 0
$$211$$ −15.7340 −1.08317 −0.541585 0.840646i $$-0.682176\pi$$
−0.541585 + 0.840646i $$0.682176\pi$$
$$212$$ −45.3066 −3.11167
$$213$$ 0 0
$$214$$ 37.3445 2.55282
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.28343 −0.562316
$$218$$ 28.3404 1.91946
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 18.8219 1.26041 0.630203 0.776430i $$-0.282972\pi$$
0.630203 + 0.776430i $$0.282972\pi$$
$$224$$ 4.28845 0.286534
$$225$$ 0 0
$$226$$ 25.6330 1.70508
$$227$$ 14.4418 0.958533 0.479267 0.877669i $$-0.340903\pi$$
0.479267 + 0.877669i $$0.340903\pi$$
$$228$$ 0 0
$$229$$ −4.17785 −0.276080 −0.138040 0.990427i $$-0.544080\pi$$
−0.138040 + 0.990427i $$0.544080\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −26.9444 −1.76898
$$233$$ −12.0847 −0.791697 −0.395849 0.918316i $$-0.629550\pi$$
−0.395849 + 0.918316i $$0.629550\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.59933 0.429580
$$237$$ 0 0
$$238$$ 44.6861 2.89657
$$239$$ 11.3541 0.734435 0.367218 0.930135i $$-0.380310\pi$$
0.367218 + 0.930135i $$0.380310\pi$$
$$240$$ 0 0
$$241$$ −3.40067 −0.219057 −0.109528 0.993984i $$-0.534934\pi$$
−0.109528 + 0.993984i $$0.534934\pi$$
$$242$$ 15.1669 0.974966
$$243$$ 0 0
$$244$$ −33.7564 −2.16103
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.07086 −0.131766
$$248$$ 11.6729 0.741229
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3.04322 −0.192086 −0.0960432 0.995377i $$-0.530619\pi$$
−0.0960432 + 0.995377i $$0.530619\pi$$
$$252$$ 0 0
$$253$$ −10.8371 −0.681324
$$254$$ −14.7003 −0.922381
$$255$$ 0 0
$$256$$ −30.3765 −1.89853
$$257$$ −17.2881 −1.07840 −0.539201 0.842177i $$-0.681274\pi$$
−0.539201 + 0.842177i $$0.681274\pi$$
$$258$$ 0 0
$$259$$ −13.7115 −0.851994
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 32.8583 2.02999
$$263$$ 1.19336 0.0735859 0.0367930 0.999323i $$-0.488286\pi$$
0.0367930 + 0.999323i $$0.488286\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 7.71155 0.472825
$$267$$ 0 0
$$268$$ 18.3623 1.12165
$$269$$ −22.1089 −1.34800 −0.674000 0.738731i $$-0.735425\pi$$
−0.674000 + 0.738731i $$0.735425\pi$$
$$270$$ 0 0
$$271$$ −4.08644 −0.248234 −0.124117 0.992268i $$-0.539610\pi$$
−0.124117 + 0.992268i $$0.539610\pi$$
$$272$$ −18.2848 −1.10868
$$273$$ 0 0
$$274$$ 19.2211 1.16119
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.0199 −0.602037 −0.301019 0.953618i $$-0.597327\pi$$
−0.301019 + 0.953618i $$0.597327\pi$$
$$278$$ 7.90730 0.474248
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −0.599328 −0.0357529 −0.0178765 0.999840i $$-0.505691\pi$$
−0.0178765 + 0.999840i $$0.505691\pi$$
$$282$$ 0 0
$$283$$ 5.41856 0.322100 0.161050 0.986946i $$-0.448512\pi$$
0.161050 + 0.986946i $$0.448512\pi$$
$$284$$ −52.8686 −3.13717
$$285$$ 0 0
$$286$$ −20.8238 −1.23133
$$287$$ −1.90991 −0.112739
$$288$$ 0 0
$$289$$ 16.5785 0.975207
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −10.5152 −0.615358
$$293$$ −3.46691 −0.202539 −0.101269 0.994859i $$-0.532290\pi$$
−0.101269 + 0.994859i $$0.532290\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 19.3221 1.12307
$$297$$ 0 0
$$298$$ −20.4331 −1.18366
$$299$$ 5.40067 0.312329
$$300$$ 0 0
$$301$$ −10.1554 −0.585350
$$302$$ 2.14832 0.123622
$$303$$ 0 0
$$304$$ −3.15544 −0.180977
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.5901 −0.946846 −0.473423 0.880835i $$-0.656982\pi$$
−0.473423 + 0.880835i $$0.656982\pi$$
$$308$$ 51.0596 2.90939
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4.15544 0.235633 0.117817 0.993035i $$-0.462411\pi$$
0.117817 + 0.993035i $$0.462411\pi$$
$$312$$ 0 0
$$313$$ −0.919237 −0.0519583 −0.0259792 0.999662i $$-0.508270\pi$$
−0.0259792 + 0.999662i $$0.508270\pi$$
$$314$$ 10.0224 0.565598
$$315$$ 0 0
$$316$$ −5.40067 −0.303812
$$317$$ 26.7292 1.50126 0.750630 0.660723i $$-0.229750\pi$$
0.750630 + 0.660723i $$0.229750\pi$$
$$318$$ 0 0
$$319$$ 24.9326 1.39596
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −20.1112 −1.12076
$$323$$ 5.79470 0.322426
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −59.8012 −3.31208
$$327$$ 0 0
$$328$$ 2.69142 0.148609
$$329$$ 37.3125 2.05710
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 27.2663 1.49643
$$333$$ 0 0
$$334$$ 8.72275 0.477288
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.5040 1.22587 0.612935 0.790133i $$-0.289989\pi$$
0.612935 + 0.790133i $$0.289989\pi$$
$$338$$ −21.0808 −1.14664
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.8013 −0.584926
$$342$$ 0 0
$$343$$ 12.2517 0.661530
$$344$$ 14.3109 0.771591
$$345$$ 0 0
$$346$$ 54.2547 2.91675
$$347$$ 14.4543 0.775946 0.387973 0.921671i $$-0.373175\pi$$
0.387973 + 0.921671i $$0.373175\pi$$
$$348$$ 0 0
$$349$$ −13.3541 −0.714828 −0.357414 0.933946i $$-0.616342\pi$$
−0.357414 + 0.933946i $$0.616342\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.59201 0.298055
$$353$$ −17.6410 −0.938937 −0.469469 0.882949i $$-0.655554\pi$$
−0.469469 + 0.882949i $$0.655554\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −63.7564 −3.37908
$$357$$ 0 0
$$358$$ 12.4252 0.656690
$$359$$ −12.4663 −0.657947 −0.328973 0.944339i $$-0.606703\pi$$
−0.328973 + 0.944339i $$0.606703\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 50.4451 2.65133
$$363$$ 0 0
$$364$$ −25.4455 −1.33371
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −16.4291 −0.857594 −0.428797 0.903401i $$-0.641062\pi$$
−0.428797 + 0.903401i $$0.641062\pi$$
$$368$$ 8.22918 0.428976
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 37.4455 1.94407
$$372$$ 0 0
$$373$$ −5.29334 −0.274079 −0.137039 0.990566i $$-0.543759\pi$$
−0.137039 + 0.990566i $$0.543759\pi$$
$$374$$ 58.2693 3.01303
$$375$$ 0 0
$$376$$ −52.5801 −2.71161
$$377$$ −12.4252 −0.639928
$$378$$ 0 0
$$379$$ 14.5353 0.746629 0.373314 0.927705i $$-0.378221\pi$$
0.373314 + 0.927705i $$0.378221\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12.7470 0.652195
$$383$$ −0.453598 −0.0231778 −0.0115889 0.999933i $$-0.503689\pi$$
−0.0115889 + 0.999933i $$0.503689\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5.01121 0.255064
$$387$$ 0 0
$$388$$ 7.98476 0.405365
$$389$$ −16.1554 −0.819113 −0.409557 0.912285i $$-0.634317\pi$$
−0.409557 + 0.912285i $$0.634317\pi$$
$$390$$ 0 0
$$391$$ −15.1122 −0.764258
$$392$$ 14.1702 0.715704
$$393$$ 0 0
$$394$$ −25.2435 −1.27175
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.7563 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$398$$ −6.61192 −0.331426
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0864 −0.603568 −0.301784 0.953376i $$-0.597582\pi$$
−0.301784 + 0.953376i $$0.597582\pi$$
$$402$$ 0 0
$$403$$ 5.38284 0.268138
$$404$$ 6.59933 0.328329
$$405$$ 0 0
$$406$$ 46.2693 2.29631
$$407$$ −17.8794 −0.886251
$$408$$ 0 0
$$409$$ −19.1346 −0.946147 −0.473073 0.881023i $$-0.656855\pi$$
−0.473073 + 0.881023i $$0.656855\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 22.1821 1.09283
$$413$$ −5.45428 −0.268388
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.78678 −0.136633
$$417$$ 0 0
$$418$$ 10.0556 0.491836
$$419$$ −8.04484 −0.393016 −0.196508 0.980502i $$-0.562960\pi$$
−0.196508 + 0.980502i $$0.562960\pi$$
$$420$$ 0 0
$$421$$ −29.3591 −1.43087 −0.715437 0.698678i $$-0.753772\pi$$
−0.715437 + 0.698678i $$0.753772\pi$$
$$422$$ −38.0742 −1.85342
$$423$$ 0 0
$$424$$ −52.7676 −2.56262
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 27.8993 1.35014
$$428$$ 59.5040 2.87623
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 32.0448 1.54355 0.771773 0.635898i $$-0.219370\pi$$
0.771773 + 0.635898i $$0.219370\pi$$
$$432$$ 0 0
$$433$$ 0.482831 0.0232034 0.0116017 0.999933i $$-0.496307\pi$$
0.0116017 + 0.999933i $$0.496307\pi$$
$$434$$ −20.0448 −0.962183
$$435$$ 0 0
$$436$$ 45.1571 2.16263
$$437$$ −2.60794 −0.124755
$$438$$ 0 0
$$439$$ −27.3591 −1.30578 −0.652889 0.757454i $$-0.726443\pi$$
−0.652889 + 0.757454i $$0.726443\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −29.0384 −1.38122
$$443$$ −23.3815 −1.11089 −0.555444 0.831554i $$-0.687452\pi$$
−0.555444 + 0.831554i $$0.687452\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 45.5465 2.15669
$$447$$ 0 0
$$448$$ 30.4887 1.44046
$$449$$ −23.1346 −1.09179 −0.545895 0.837853i $$-0.683810\pi$$
−0.545895 + 0.837853i $$0.683810\pi$$
$$450$$ 0 0
$$451$$ −2.49047 −0.117272
$$452$$ 40.8430 1.92109
$$453$$ 0 0
$$454$$ 34.9472 1.64015
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 21.2503 0.994049 0.497025 0.867736i $$-0.334426\pi$$
0.497025 + 0.867736i $$0.334426\pi$$
$$458$$ −10.1099 −0.472403
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −31.5785 −1.47076 −0.735379 0.677656i $$-0.762996\pi$$
−0.735379 + 0.677656i $$0.762996\pi$$
$$462$$ 0 0
$$463$$ −15.6119 −0.725547 −0.362774 0.931877i $$-0.618170\pi$$
−0.362774 + 0.931877i $$0.618170\pi$$
$$464$$ −18.9326 −0.878925
$$465$$ 0 0
$$466$$ −29.2435 −1.35468
$$467$$ 29.8264 1.38020 0.690101 0.723713i $$-0.257566\pi$$
0.690101 + 0.723713i $$0.257566\pi$$
$$468$$ 0 0
$$469$$ −15.1762 −0.700774
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 7.68608 0.353781
$$473$$ −13.2424 −0.608885
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 71.2019 3.26353
$$477$$ 0 0
$$478$$ 27.4754 1.25670
$$479$$ −4.53531 −0.207223 −0.103612 0.994618i $$-0.533040\pi$$
−0.103612 + 0.994618i $$0.533040\pi$$
$$480$$ 0 0
$$481$$ 8.91020 0.406270
$$482$$ −8.22918 −0.374829
$$483$$ 0 0
$$484$$ 24.1666 1.09848
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 39.2550 1.77881 0.889407 0.457116i $$-0.151118\pi$$
0.889407 + 0.457116i $$0.151118\pi$$
$$488$$ −39.3153 −1.77972
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −38.8910 −1.75513 −0.877564 0.479460i $$-0.840832\pi$$
−0.877564 + 0.479460i $$0.840832\pi$$
$$492$$ 0 0
$$493$$ 34.7682 1.56588
$$494$$ −5.01121 −0.225465
$$495$$ 0 0
$$496$$ 8.20202 0.368281
$$497$$ 43.6954 1.96001
$$498$$ 0 0
$$499$$ 4.73235 0.211849 0.105924 0.994374i $$-0.466220\pi$$
0.105924 + 0.994374i $$0.466220\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −7.36420 −0.328680
$$503$$ −1.85567 −0.0827400 −0.0413700 0.999144i $$-0.513172\pi$$
−0.0413700 + 0.999144i $$0.513172\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −26.2244 −1.16582
$$507$$ 0 0
$$508$$ −23.4232 −1.03924
$$509$$ −22.8878 −1.01448 −0.507242 0.861804i $$-0.669335\pi$$
−0.507242 + 0.861804i $$0.669335\pi$$
$$510$$ 0 0
$$511$$ 8.69074 0.384456
$$512$$ −32.5867 −1.44014
$$513$$ 0 0
$$514$$ −41.8350 −1.84526
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 48.6543 2.13982
$$518$$ −33.1802 −1.45785
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 29.7340 1.30267 0.651334 0.758791i $$-0.274210\pi$$
0.651334 + 0.758791i $$0.274210\pi$$
$$522$$ 0 0
$$523$$ 30.6497 1.34022 0.670109 0.742263i $$-0.266248\pi$$
0.670109 + 0.742263i $$0.266248\pi$$
$$524$$ 52.3557 2.28717
$$525$$ 0 0
$$526$$ 2.88778 0.125913
$$527$$ −15.0623 −0.656125
$$528$$ 0 0
$$529$$ −16.1987 −0.704289
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 12.2874 0.532727
$$533$$ 1.24112 0.0537590
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 21.3861 0.923739
$$537$$ 0 0
$$538$$ −53.5006 −2.30657
$$539$$ −13.1122 −0.564783
$$540$$ 0 0
$$541$$ 2.21946 0.0954220 0.0477110 0.998861i $$-0.484807\pi$$
0.0477110 + 0.998861i $$0.484807\pi$$
$$542$$ −9.88865 −0.424754
$$543$$ 0 0
$$544$$ 7.79798 0.334336
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −14.4297 −0.616970 −0.308485 0.951229i $$-0.599822\pi$$
−0.308485 + 0.951229i $$0.599822\pi$$
$$548$$ 30.6265 1.30830
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 4.46360 0.189812
$$554$$ −24.2469 −1.03015
$$555$$ 0 0
$$556$$ 12.5993 0.534331
$$557$$ −19.4610 −0.824588 −0.412294 0.911051i $$-0.635272\pi$$
−0.412294 + 0.911051i $$0.635272\pi$$
$$558$$ 0 0
$$559$$ 6.59933 0.279122
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.45030 −0.0611771
$$563$$ 12.3649 0.521118 0.260559 0.965458i $$-0.416093\pi$$
0.260559 + 0.965458i $$0.416093\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 13.1122 0.551148
$$567$$ 0 0
$$568$$ −61.5748 −2.58362
$$569$$ −15.0898 −0.632597 −0.316299 0.948660i $$-0.602440\pi$$
−0.316299 + 0.948660i $$0.602440\pi$$
$$570$$ 0 0
$$571$$ 17.1571 0.718000 0.359000 0.933337i $$-0.383118\pi$$
0.359000 + 0.933337i $$0.383118\pi$$
$$572$$ −33.1802 −1.38733
$$573$$ 0 0
$$574$$ −4.62175 −0.192908
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −22.5165 −0.937374 −0.468687 0.883364i $$-0.655273\pi$$
−0.468687 + 0.883364i $$0.655273\pi$$
$$578$$ 40.1179 1.66868
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −22.5353 −0.934922
$$582$$ 0 0
$$583$$ 48.8278 2.02224
$$584$$ −12.2469 −0.506778
$$585$$ 0 0
$$586$$ −8.38946 −0.346566
$$587$$ −7.49544 −0.309370 −0.154685 0.987964i $$-0.549436\pi$$
−0.154685 + 0.987964i $$0.549436\pi$$
$$588$$ 0 0
$$589$$ −2.59933 −0.107103
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 13.5768 0.558002
$$593$$ −27.8094 −1.14199 −0.570997 0.820952i $$-0.693443\pi$$
−0.570997 + 0.820952i $$0.693443\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −32.5577 −1.33362
$$597$$ 0 0
$$598$$ 13.0689 0.534428
$$599$$ 45.4903 1.85869 0.929343 0.369219i $$-0.120375\pi$$
0.929343 + 0.369219i $$0.120375\pi$$
$$600$$ 0 0
$$601$$ −16.5993 −0.677101 −0.338550 0.940948i $$-0.609937\pi$$
−0.338550 + 0.940948i $$0.609937\pi$$
$$602$$ −24.5748 −1.00160
$$603$$ 0 0
$$604$$ 3.42309 0.139284
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −5.08417 −0.206360 −0.103180 0.994663i $$-0.532902\pi$$
−0.103180 + 0.994663i $$0.532902\pi$$
$$608$$ 1.34571 0.0545758
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.2469 −0.980923
$$612$$ 0 0
$$613$$ 4.63706 0.187289 0.0936445 0.995606i $$-0.470148\pi$$
0.0936445 + 0.995606i $$0.470148\pi$$
$$614$$ −40.1458 −1.62015
$$615$$ 0 0
$$616$$ 59.4679 2.39603
$$617$$ −40.2874 −1.62191 −0.810955 0.585108i $$-0.801052\pi$$
−0.810955 + 0.585108i $$0.801052\pi$$
$$618$$ 0 0
$$619$$ −43.3815 −1.74365 −0.871825 0.489818i $$-0.837063\pi$$
−0.871825 + 0.489818i $$0.837063\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 10.0556 0.403194
$$623$$ 52.6940 2.11114
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2.22443 −0.0889062
$$627$$ 0 0
$$628$$ 15.9695 0.637253
$$629$$ −24.9326 −0.994129
$$630$$ 0 0
$$631$$ 7.53369 0.299911 0.149956 0.988693i $$-0.452087\pi$$
0.149956 + 0.988693i $$0.452087\pi$$
$$632$$ −6.29004 −0.250204
$$633$$ 0 0
$$634$$ 64.6812 2.56882
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.53446 0.258905
$$638$$ 60.3337 2.38863
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23.3075 0.920591 0.460296 0.887766i $$-0.347743\pi$$
0.460296 + 0.887766i $$0.347743\pi$$
$$642$$ 0 0
$$643$$ 1.87419 0.0739110 0.0369555 0.999317i $$-0.488234\pi$$
0.0369555 + 0.999317i $$0.488234\pi$$
$$644$$ −32.0448 −1.26274
$$645$$ 0 0
$$646$$ 14.0224 0.551705
$$647$$ 47.0371 1.84922 0.924609 0.380917i $$-0.124392\pi$$
0.924609 + 0.380917i $$0.124392\pi$$
$$648$$ 0 0
$$649$$ −7.11222 −0.279179
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −95.2861 −3.73169
$$653$$ −24.1630 −0.945571 −0.472785 0.881178i $$-0.656751\pi$$
−0.472785 + 0.881178i $$0.656751\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 1.89114 0.0738367
$$657$$ 0 0
$$658$$ 90.2914 3.51992
$$659$$ −10.2885 −0.400781 −0.200391 0.979716i $$-0.564221\pi$$
−0.200391 + 0.979716i $$0.564221\pi$$
$$660$$ 0 0
$$661$$ 5.93598 0.230883 0.115441 0.993314i $$-0.463172\pi$$
0.115441 + 0.993314i $$0.463172\pi$$
$$662$$ 19.3590 0.752407
$$663$$ 0 0
$$664$$ 31.7564 1.23239
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −15.6476 −0.605879
$$668$$ 13.8987 0.537755
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.3799 1.40443
$$672$$ 0 0
$$673$$ −40.7053 −1.56907 −0.784537 0.620082i $$-0.787099\pi$$
−0.784537 + 0.620082i $$0.787099\pi$$
$$674$$ 54.4567 2.09759
$$675$$ 0 0
$$676$$ −33.5897 −1.29191
$$677$$ −18.8761 −0.725469 −0.362734 0.931893i $$-0.618157\pi$$
−0.362734 + 0.931893i $$0.618157\pi$$
$$678$$ 0 0
$$679$$ −6.59933 −0.253259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −26.1379 −1.00087
$$683$$ 19.6576 0.752179 0.376089 0.926583i $$-0.377269\pi$$
0.376089 + 0.926583i $$0.377269\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 29.6475 1.13195
$$687$$ 0 0
$$688$$ 10.0556 0.383367
$$689$$ −24.3333 −0.927025
$$690$$ 0 0
$$691$$ −48.2451 −1.83533 −0.917665 0.397354i $$-0.869928\pi$$
−0.917665 + 0.397354i $$0.869928\pi$$
$$692$$ 86.4483 3.28627
$$693$$ 0 0
$$694$$ 34.9775 1.32773
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.47293 −0.131546
$$698$$ −32.3152 −1.22315
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −0.512889 −0.0193715 −0.00968577 0.999953i $$-0.503083\pi$$
−0.00968577 + 0.999953i $$0.503083\pi$$
$$702$$ 0 0
$$703$$ −4.30266 −0.162278
$$704$$ 39.7564 1.49838
$$705$$ 0 0
$$706$$ −42.6890 −1.60662
$$707$$ −5.45428 −0.205129
$$708$$ 0 0
$$709$$ −8.84618 −0.332225 −0.166113 0.986107i $$-0.553122\pi$$
−0.166113 + 0.986107i $$0.553122\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −74.2556 −2.78285
$$713$$ 6.77889 0.253871
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 19.7980 0.739885
$$717$$ 0 0
$$718$$ −30.1669 −1.12582
$$719$$ −7.84456 −0.292553 −0.146276 0.989244i $$-0.546729\pi$$
−0.146276 + 0.989244i $$0.546729\pi$$
$$720$$ 0 0
$$721$$ −18.3333 −0.682767
$$722$$ 2.41987 0.0900582
$$723$$ 0 0
$$724$$ 80.3781 2.98723
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2.91130 −0.107974 −0.0539870 0.998542i $$-0.517193\pi$$
−0.0539870 + 0.998542i $$0.517193\pi$$
$$728$$ −29.6358 −1.09838
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −18.4663 −0.683001
$$732$$ 0 0
$$733$$ −33.7775 −1.24760 −0.623800 0.781584i $$-0.714412\pi$$
−0.623800 + 0.781584i $$0.714412\pi$$
$$734$$ −39.7564 −1.46743
$$735$$ 0 0
$$736$$ −3.50953 −0.129363
$$737$$ −19.7893 −0.728950
$$738$$ 0 0
$$739$$ −35.8030 −1.31703 −0.658517 0.752566i $$-0.728816\pi$$
−0.658517 + 0.752566i $$0.728816\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 90.6133 3.32652
$$743$$ 5.66948 0.207993 0.103996 0.994578i $$-0.466837\pi$$
0.103996 + 0.994578i $$0.466837\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −12.8092 −0.468978
$$747$$ 0 0
$$748$$ 92.8451 3.39475
$$749$$ −49.1795 −1.79698
$$750$$ 0 0
$$751$$ −27.4679 −1.00232 −0.501159 0.865355i $$-0.667093\pi$$
−0.501159 + 0.865355i $$0.667093\pi$$
$$752$$ −36.9457 −1.34727
$$753$$ 0 0
$$754$$ −30.0673 −1.09498
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −41.6370 −1.51332 −0.756662 0.653806i $$-0.773171\pi$$
−0.756662 + 0.653806i $$0.773171\pi$$
$$758$$ 35.1736 1.27756
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9.46967 0.343275 0.171638 0.985160i $$-0.445094\pi$$
0.171638 + 0.985160i $$0.445094\pi$$
$$762$$ 0 0
$$763$$ −37.3219 −1.35114
$$764$$ 20.3109 0.734822
$$765$$ 0 0
$$766$$ −1.09765 −0.0396597
$$767$$ 3.54437 0.127980
$$768$$ 0 0
$$769$$ 1.90858 0.0688253 0.0344127 0.999408i $$-0.489044\pi$$
0.0344127 + 0.999408i $$0.489044\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7.98476 0.287378
$$773$$ 28.4007 1.02150 0.510751 0.859729i $$-0.329367\pi$$
0.510751 + 0.859729i $$0.329367\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 9.29966 0.333838
$$777$$ 0 0
$$778$$ −39.0941 −1.40159
$$779$$ −0.599328 −0.0214732
$$780$$ 0 0
$$781$$ 56.9775 2.03881
$$782$$ −36.5696 −1.30773
$$783$$ 0 0
$$784$$ 9.95678 0.355599
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 15.6708 0.558605 0.279303 0.960203i $$-0.409897\pi$$
0.279303 + 0.960203i $$0.409897\pi$$
$$788$$ −40.2225 −1.43287
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −33.7564 −1.20024
$$792$$ 0 0
$$793$$ −18.1299 −0.643811
$$794$$ 79.2659 2.81304
$$795$$ 0 0
$$796$$ −10.5353 −0.373414
$$797$$ 36.9225 1.30786 0.653932 0.756554i $$-0.273118\pi$$
0.653932 + 0.756554i $$0.273118\pi$$
$$798$$ 0 0
$$799$$ 67.8478 2.40028
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −29.2476 −1.03277
$$803$$ 11.3325 0.399914
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.0258 0.458813
$$807$$ 0 0
$$808$$ 7.68608 0.270396
$$809$$ −43.0465 −1.51343 −0.756716 0.653743i $$-0.773198\pi$$
−0.756716 + 0.653743i $$0.773198\pi$$
$$810$$ 0 0
$$811$$ 32.2469 1.13234 0.566170 0.824288i $$-0.308425\pi$$
0.566170 + 0.824288i $$0.308425\pi$$
$$812$$ 73.7245 2.58722
$$813$$ 0 0
$$814$$ −43.2659 −1.51647
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3.18676 −0.111491
$$818$$ −46.3033 −1.61896
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −5.18121 −0.180826 −0.0904128 0.995904i $$-0.528819\pi$$
−0.0904128 + 0.995904i $$0.528819\pi$$
$$822$$ 0 0
$$823$$ −25.8517 −0.901133 −0.450567 0.892743i $$-0.648778\pi$$
−0.450567 + 0.892743i $$0.648778\pi$$
$$824$$ 25.8350 0.900004
$$825$$ 0 0
$$826$$ −13.1987 −0.459240
$$827$$ 9.97816 0.346974 0.173487 0.984836i $$-0.444496\pi$$
0.173487 + 0.984836i $$0.444496\pi$$
$$828$$ 0 0
$$829$$ 21.9808 0.763425 0.381713 0.924281i $$-0.375334\pi$$
0.381713 + 0.924281i $$0.375334\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −19.8126 −0.686877
$$833$$ −18.2848 −0.633531
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0224 0.554147
$$837$$ 0 0
$$838$$ −19.4675 −0.672492
$$839$$ 16.1089 0.556140 0.278070 0.960561i $$-0.410305\pi$$
0.278070 + 0.960561i $$0.410305\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −71.0451 −2.44838
$$843$$ 0 0
$$844$$ −60.6666 −2.08823
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −19.9735 −0.686298
$$848$$ −37.0775 −1.27324
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 11.2211 0.384653
$$852$$ 0 0
$$853$$ 44.9615 1.53945 0.769727 0.638373i $$-0.220392\pi$$
0.769727 + 0.638373i $$0.220392\pi$$
$$854$$ 67.5128 2.31024
$$855$$ 0 0
$$856$$ 69.3029 2.36872
$$857$$ 46.2431 1.57963 0.789817 0.613343i $$-0.210176\pi$$
0.789817 + 0.613343i $$0.210176\pi$$
$$858$$ 0 0
$$859$$ −10.7772 −0.367713 −0.183856 0.982953i $$-0.558858\pi$$
−0.183856 + 0.982953i $$0.558858\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 77.5443 2.64117
$$863$$ −22.7966 −0.776007 −0.388003 0.921658i $$-0.626835\pi$$
−0.388003 + 0.921658i $$0.626835\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 1.16839 0.0397034
$$867$$ 0 0
$$868$$ −31.9390 −1.08408
$$869$$ 5.82040 0.197444
$$870$$ 0 0
$$871$$ 9.86201 0.334161
$$872$$ 52.5934 1.78104
$$873$$ 0 0
$$874$$ −6.31087 −0.213468
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −4.94644 −0.167029 −0.0835146 0.996507i $$-0.526615\pi$$
−0.0835146 + 0.996507i $$0.526615\pi$$
$$878$$ −66.2054 −2.23432
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.53033 −0.0852490 −0.0426245 0.999091i $$-0.513572\pi$$
−0.0426245 + 0.999091i $$0.513572\pi$$
$$882$$ 0 0
$$883$$ 29.7430 1.00093 0.500465 0.865757i $$-0.333162\pi$$
0.500465 + 0.865757i $$0.333162\pi$$
$$884$$ −46.2693 −1.55620
$$885$$ 0 0
$$886$$ −56.5801 −1.90085
$$887$$ −45.5450 −1.52925 −0.764626 0.644474i $$-0.777076\pi$$
−0.764626 + 0.644474i $$0.777076\pi$$
$$888$$ 0 0
$$889$$ 19.3591 0.649282
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 72.5729 2.42992
$$893$$ 11.7086 0.391813
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 65.2019 2.17824
$$897$$ 0 0
$$898$$ −55.9828 −1.86817
$$899$$ −15.5960 −0.520155
$$900$$ 0 0
$$901$$ 68.0897 2.26840
$$902$$ −6.02662 −0.200664
$$903$$ 0 0
$$904$$ 47.5689 1.58212
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 33.2034 1.10250 0.551250 0.834340i $$-0.314151\pi$$
0.551250 + 0.834340i $$0.314151\pi$$
$$908$$ 55.6841 1.84794
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −55.7788 −1.84803 −0.924017 0.382351i $$-0.875114\pi$$
−0.924017 + 0.382351i $$0.875114\pi$$
$$912$$ 0 0
$$913$$ −29.3853 −0.972513
$$914$$ 51.4231 1.70092
$$915$$ 0 0
$$916$$ −16.1089 −0.532252
$$917$$ −43.2715 −1.42895
$$918$$ 0 0
$$919$$ −28.5769 −0.942665 −0.471333 0.881956i $$-0.656227\pi$$
−0.471333 + 0.881956i $$0.656227\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −76.4159 −2.51662
$$923$$ −28.3947 −0.934622
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −37.7788 −1.24149
$$927$$ 0 0
$$928$$ 8.07426 0.265051
$$929$$ 54.4937 1.78788 0.893940 0.448186i $$-0.147930\pi$$
0.893940 + 0.448186i $$0.147930\pi$$
$$930$$ 0 0
$$931$$ −3.15544 −0.103415
$$932$$ −46.5960 −1.52630
$$933$$ 0 0
$$934$$ 72.1761 2.36167
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −37.1484 −1.21359 −0.606793 0.794860i $$-0.707544\pi$$
−0.606793 + 0.794860i $$0.707544\pi$$
$$938$$ −36.7245 −1.19910
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −32.3973 −1.05612 −0.528061 0.849206i $$-0.677081\pi$$
−0.528061 + 0.849206i $$0.677081\pi$$
$$942$$ 0 0
$$943$$ 1.56301 0.0508986
$$944$$ 5.40067 0.175777
$$945$$ 0 0
$$946$$ −32.0448 −1.04187
$$947$$ −35.4662 −1.15250 −0.576249 0.817275i $$-0.695484\pi$$
−0.576249 + 0.817275i $$0.695484\pi$$
$$948$$ 0 0
$$949$$ −5.64752 −0.183326
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 82.9272 2.68769
$$953$$ 14.3411 0.464553 0.232277 0.972650i $$-0.425383\pi$$
0.232277 + 0.972650i $$0.425383\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 43.7788 1.41591
$$957$$ 0 0
$$958$$ −10.9749 −0.354581
$$959$$ −25.3125 −0.817383
$$960$$ 0 0
$$961$$ −24.2435 −0.782048
$$962$$ 21.5615 0.695172
$$963$$ 0 0
$$964$$ −13.1122 −0.422316
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −27.9351 −0.898331 −0.449165 0.893449i $$-0.648279\pi$$
−0.449165 + 0.893449i $$0.648279\pi$$
$$968$$ 28.1463 0.904657
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 42.5993 1.36708 0.683539 0.729914i $$-0.260440\pi$$
0.683539 + 0.729914i $$0.260440\pi$$
$$972$$ 0 0
$$973$$ −10.4132 −0.333833
$$974$$ 94.9920 3.04374
$$975$$ 0 0
$$976$$ −27.6251 −0.884258
$$977$$ −39.5597 −1.26563 −0.632813 0.774304i $$-0.718100\pi$$
−0.632813 + 0.774304i $$0.718100\pi$$
$$978$$ 0 0
$$979$$ 68.7114 2.19603
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −94.1112 −3.00321
$$983$$ 39.7689 1.26843 0.634215 0.773157i $$-0.281323\pi$$
0.634215 + 0.773157i $$0.281323\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 84.1345 2.67939
$$987$$ 0 0
$$988$$ −7.98476 −0.254029
$$989$$ 8.31087 0.264270
$$990$$ 0 0
$$991$$ 55.2019 1.75355 0.876773 0.480905i $$-0.159692\pi$$
0.876773 + 0.480905i $$0.159692\pi$$
$$992$$ −3.49794 −0.111060
$$993$$ 0 0
$$994$$ 105.737 3.35378
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −11.6543 −0.369097 −0.184548 0.982823i $$-0.559082\pi$$
−0.184548 + 0.982823i $$0.559082\pi$$
$$998$$ 11.4517 0.362496
$$999$$ 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 4275.2.a.br.1.6 6
3.2 odd 2 475.2.a.j.1.1 6
5.2 odd 4 855.2.c.d.514.6 6
5.3 odd 4 855.2.c.d.514.1 6
5.4 even 2 inner 4275.2.a.br.1.1 6
12.11 even 2 7600.2.a.ck.1.3 6
15.2 even 4 95.2.b.b.39.1 6
15.8 even 4 95.2.b.b.39.6 yes 6
15.14 odd 2 475.2.a.j.1.6 6
57.56 even 2 9025.2.a.bx.1.6 6
60.23 odd 4 1520.2.d.h.609.3 6
60.47 odd 4 1520.2.d.h.609.4 6
60.59 even 2 7600.2.a.ck.1.4 6
285.113 odd 4 1805.2.b.e.1084.1 6
285.227 odd 4 1805.2.b.e.1084.6 6
285.284 even 2 9025.2.a.bx.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.1 6 15.2 even 4
95.2.b.b.39.6 yes 6 15.8 even 4
475.2.a.j.1.1 6 3.2 odd 2
475.2.a.j.1.6 6 15.14 odd 2
855.2.c.d.514.1 6 5.3 odd 4
855.2.c.d.514.6 6 5.2 odd 4
1520.2.d.h.609.3 6 60.23 odd 4
1520.2.d.h.609.4 6 60.47 odd 4
1805.2.b.e.1084.1 6 285.113 odd 4
1805.2.b.e.1084.6 6 285.227 odd 4
4275.2.a.br.1.1 6 5.4 even 2 inner
4275.2.a.br.1.6 6 1.1 even 1 trivial
7600.2.a.ck.1.3 6 12.11 even 2
7600.2.a.ck.1.4 6 60.59 even 2
9025.2.a.bx.1.1 6 285.284 even 2
9025.2.a.bx.1.6 6 57.56 even 2