# Properties

 Label 1575.2.a.p.1.2 Level $1575$ Weight $2$ Character 1575.1 Self dual yes Analytic conductor $12.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} +O(q^{10})$$ $$q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} -2.56155 q^{11} -4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} -4.56155 q^{17} +1.12311 q^{19} -4.00000 q^{22} -5.12311 q^{23} -7.12311 q^{26} +0.438447 q^{28} +5.68466 q^{29} -2.43845 q^{32} -7.12311 q^{34} -6.00000 q^{37} +1.75379 q^{38} +3.12311 q^{41} -9.12311 q^{43} -1.12311 q^{44} -8.00000 q^{46} +3.68466 q^{47} +1.00000 q^{49} -2.00000 q^{52} +3.12311 q^{53} -2.43845 q^{56} +8.87689 q^{58} +4.00000 q^{59} -9.36932 q^{61} +5.56155 q^{64} +6.24621 q^{67} -2.00000 q^{68} -8.00000 q^{71} -4.24621 q^{73} -9.36932 q^{74} +0.492423 q^{76} -2.56155 q^{77} -6.56155 q^{79} +4.87689 q^{82} +4.00000 q^{83} -14.2462 q^{86} +6.24621 q^{88} -7.12311 q^{89} -4.56155 q^{91} -2.24621 q^{92} +5.75379 q^{94} +14.8078 q^{97} +1.56155 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 5q^{4} + 2q^{7} - 9q^{8} + O(q^{10})$$ $$2q - q^{2} + 5q^{4} + 2q^{7} - 9q^{8} - q^{11} - 5q^{13} - q^{14} + 3q^{16} - 5q^{17} - 6q^{19} - 8q^{22} - 2q^{23} - 6q^{26} + 5q^{28} - q^{29} - 9q^{32} - 6q^{34} - 12q^{37} + 20q^{38} - 2q^{41} - 10q^{43} + 6q^{44} - 16q^{46} - 5q^{47} + 2q^{49} - 4q^{52} - 2q^{53} - 9q^{56} + 26q^{58} + 8q^{59} + 6q^{61} + 7q^{64} - 4q^{67} - 4q^{68} - 16q^{71} + 8q^{73} + 6q^{74} - 32q^{76} - q^{77} - 9q^{79} + 18q^{82} + 8q^{83} - 12q^{86} - 4q^{88} - 6q^{89} - 5q^{91} + 12q^{92} + 28q^{94} + 9q^{97} - q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.56155 1.10418 0.552092 0.833783i $$-0.313830\pi$$
0.552092 + 0.833783i $$0.313830\pi$$
$$3$$ 0 0
$$4$$ 0.438447 0.219224
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ −2.43845 −0.862121
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.56155 −0.772337 −0.386169 0.922428i $$-0.626202\pi$$
−0.386169 + 0.922428i $$0.626202\pi$$
$$12$$ 0 0
$$13$$ −4.56155 −1.26515 −0.632574 0.774500i $$-0.718001\pi$$
−0.632574 + 0.774500i $$0.718001\pi$$
$$14$$ 1.56155 0.417343
$$15$$ 0 0
$$16$$ −4.68466 −1.17116
$$17$$ −4.56155 −1.10634 −0.553170 0.833069i $$-0.686582\pi$$
−0.553170 + 0.833069i $$0.686582\pi$$
$$18$$ 0 0
$$19$$ 1.12311 0.257658 0.128829 0.991667i $$-0.458878\pi$$
0.128829 + 0.991667i $$0.458878\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ −5.12311 −1.06824 −0.534121 0.845408i $$-0.679357\pi$$
−0.534121 + 0.845408i $$0.679357\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −7.12311 −1.39696
$$27$$ 0 0
$$28$$ 0.438447 0.0828587
$$29$$ 5.68466 1.05561 0.527807 0.849364i $$-0.323014\pi$$
0.527807 + 0.849364i $$0.323014\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −2.43845 −0.431061
$$33$$ 0 0
$$34$$ −7.12311 −1.22160
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 1.75379 0.284502
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.12311 0.487747 0.243874 0.969807i $$-0.421582\pi$$
0.243874 + 0.969807i $$0.421582\pi$$
$$42$$ 0 0
$$43$$ −9.12311 −1.39126 −0.695630 0.718400i $$-0.744875\pi$$
−0.695630 + 0.718400i $$0.744875\pi$$
$$44$$ −1.12311 −0.169315
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 3.68466 0.537463 0.268731 0.963215i $$-0.413396\pi$$
0.268731 + 0.963215i $$0.413396\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 3.12311 0.428992 0.214496 0.976725i $$-0.431189\pi$$
0.214496 + 0.976725i $$0.431189\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.43845 −0.325851
$$57$$ 0 0
$$58$$ 8.87689 1.16559
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −9.36932 −1.19962 −0.599809 0.800143i $$-0.704757\pi$$
−0.599809 + 0.800143i $$0.704757\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 5.56155 0.695194
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.24621 0.763096 0.381548 0.924349i $$-0.375391\pi$$
0.381548 + 0.924349i $$0.375391\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −4.24621 −0.496981 −0.248491 0.968634i $$-0.579935\pi$$
−0.248491 + 0.968634i $$0.579935\pi$$
$$74$$ −9.36932 −1.08916
$$75$$ 0 0
$$76$$ 0.492423 0.0564847
$$77$$ −2.56155 −0.291916
$$78$$ 0 0
$$79$$ −6.56155 −0.738232 −0.369116 0.929383i $$-0.620340\pi$$
−0.369116 + 0.929383i $$0.620340\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 4.87689 0.538563
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −14.2462 −1.53621
$$87$$ 0 0
$$88$$ 6.24621 0.665848
$$89$$ −7.12311 −0.755048 −0.377524 0.926000i $$-0.623224\pi$$
−0.377524 + 0.926000i $$0.623224\pi$$
$$90$$ 0 0
$$91$$ −4.56155 −0.478181
$$92$$ −2.24621 −0.234184
$$93$$ 0 0
$$94$$ 5.75379 0.593458
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.8078 1.50350 0.751750 0.659448i $$-0.229210\pi$$
0.751750 + 0.659448i $$0.229210\pi$$
$$98$$ 1.56155 0.157741
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ −1.43845 −0.141734 −0.0708672 0.997486i $$-0.522577\pi$$
−0.0708672 + 0.997486i $$0.522577\pi$$
$$104$$ 11.1231 1.09071
$$105$$ 0 0
$$106$$ 4.87689 0.473686
$$107$$ −11.3693 −1.09911 −0.549557 0.835456i $$-0.685203\pi$$
−0.549557 + 0.835456i $$0.685203\pi$$
$$108$$ 0 0
$$109$$ 17.6847 1.69388 0.846942 0.531686i $$-0.178441\pi$$
0.846942 + 0.531686i $$0.178441\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.68466 −0.442659
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.49242 0.231416
$$117$$ 0 0
$$118$$ 6.24621 0.575010
$$119$$ −4.56155 −0.418157
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ −14.6307 −1.32460
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.2462 −0.909204 −0.454602 0.890695i $$-0.650219\pi$$
−0.454602 + 0.890695i $$0.650219\pi$$
$$128$$ 13.5616 1.19868
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.12311 0.797089 0.398545 0.917149i $$-0.369515\pi$$
0.398545 + 0.917149i $$0.369515\pi$$
$$132$$ 0 0
$$133$$ 1.12311 0.0973856
$$134$$ 9.75379 0.842599
$$135$$ 0 0
$$136$$ 11.1231 0.953798
$$137$$ −8.87689 −0.758404 −0.379202 0.925314i $$-0.623801\pi$$
−0.379202 + 0.925314i $$0.623801\pi$$
$$138$$ 0 0
$$139$$ −6.87689 −0.583291 −0.291645 0.956527i $$-0.594203\pi$$
−0.291645 + 0.956527i $$0.594203\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.4924 −1.04834
$$143$$ 11.6847 0.977120
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −6.63068 −0.548759
$$147$$ 0 0
$$148$$ −2.63068 −0.216241
$$149$$ 4.24621 0.347863 0.173932 0.984758i $$-0.444353\pi$$
0.173932 + 0.984758i $$0.444353\pi$$
$$150$$ 0 0
$$151$$ 21.9309 1.78471 0.892354 0.451335i $$-0.149052\pi$$
0.892354 + 0.451335i $$0.149052\pi$$
$$152$$ −2.73863 −0.222133
$$153$$ 0 0
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.75379 −0.299585 −0.149792 0.988717i $$-0.547861\pi$$
−0.149792 + 0.988717i $$0.547861\pi$$
$$158$$ −10.2462 −0.815145
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.12311 −0.403757
$$162$$ 0 0
$$163$$ −1.12311 −0.0879684 −0.0439842 0.999032i $$-0.514005\pi$$
−0.0439842 + 0.999032i $$0.514005\pi$$
$$164$$ 1.36932 0.106926
$$165$$ 0 0
$$166$$ 6.24621 0.484800
$$167$$ 21.9309 1.69706 0.848531 0.529146i $$-0.177488\pi$$
0.848531 + 0.529146i $$0.177488\pi$$
$$168$$ 0 0
$$169$$ 7.80776 0.600597
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −8.56155 −0.650923 −0.325461 0.945555i $$-0.605520\pi$$
−0.325461 + 0.945555i $$0.605520\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ −11.1231 −0.833712
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 23.6155 1.75533 0.877664 0.479276i $$-0.159101\pi$$
0.877664 + 0.479276i $$0.159101\pi$$
$$182$$ −7.12311 −0.528000
$$183$$ 0 0
$$184$$ 12.4924 0.920954
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 11.6847 0.854467
$$188$$ 1.61553 0.117824
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 9.43845 0.682942 0.341471 0.939892i $$-0.389075\pi$$
0.341471 + 0.939892i $$0.389075\pi$$
$$192$$ 0 0
$$193$$ 5.36932 0.386492 0.193246 0.981150i $$-0.438098\pi$$
0.193246 + 0.981150i $$0.438098\pi$$
$$194$$ 23.1231 1.66014
$$195$$ 0 0
$$196$$ 0.438447 0.0313177
$$197$$ −7.12311 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$198$$ 0 0
$$199$$ −18.2462 −1.29344 −0.646720 0.762728i $$-0.723860\pi$$
−0.646720 + 0.762728i $$0.723860\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −0.384472 −0.0270513
$$203$$ 5.68466 0.398985
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.24621 −0.156501
$$207$$ 0 0
$$208$$ 21.3693 1.48170
$$209$$ −2.87689 −0.198999
$$210$$ 0 0
$$211$$ −23.0540 −1.58710 −0.793551 0.608504i $$-0.791770\pi$$
−0.793551 + 0.608504i $$0.791770\pi$$
$$212$$ 1.36932 0.0940451
$$213$$ 0 0
$$214$$ −17.7538 −1.21362
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 27.6155 1.87036
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.8078 1.39968
$$222$$ 0 0
$$223$$ 6.56155 0.439394 0.219697 0.975568i $$-0.429493\pi$$
0.219697 + 0.975568i $$0.429493\pi$$
$$224$$ −2.43845 −0.162926
$$225$$ 0 0
$$226$$ −21.8617 −1.45422
$$227$$ 23.6847 1.57201 0.786003 0.618223i $$-0.212147\pi$$
0.786003 + 0.618223i $$0.212147\pi$$
$$228$$ 0 0
$$229$$ 19.1231 1.26369 0.631845 0.775095i $$-0.282298\pi$$
0.631845 + 0.775095i $$0.282298\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −13.8617 −0.910068
$$233$$ −3.12311 −0.204601 −0.102301 0.994754i $$-0.532620\pi$$
−0.102301 + 0.994754i $$0.532620\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1.75379 0.114162
$$237$$ 0 0
$$238$$ −7.12311 −0.461722
$$239$$ 0.807764 0.0522499 0.0261250 0.999659i $$-0.491683\pi$$
0.0261250 + 0.999659i $$0.491683\pi$$
$$240$$ 0 0
$$241$$ 12.2462 0.788848 0.394424 0.918929i $$-0.370944\pi$$
0.394424 + 0.918929i $$0.370944\pi$$
$$242$$ −6.93087 −0.445533
$$243$$ 0 0
$$244$$ −4.10795 −0.262985
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.12311 −0.325975
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 17.1231 1.08080 0.540400 0.841408i $$-0.318273\pi$$
0.540400 + 0.841408i $$0.318273\pi$$
$$252$$ 0 0
$$253$$ 13.1231 0.825043
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 10.0540 0.628373
$$257$$ 22.4924 1.40304 0.701519 0.712650i $$-0.252505\pi$$
0.701519 + 0.712650i $$0.252505\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 14.2462 0.880134
$$263$$ −21.1231 −1.30251 −0.651253 0.758860i $$-0.725756\pi$$
−0.651253 + 0.758860i $$0.725756\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1.75379 0.107532
$$267$$ 0 0
$$268$$ 2.73863 0.167289
$$269$$ −28.7386 −1.75223 −0.876113 0.482106i $$-0.839872\pi$$
−0.876113 + 0.482106i $$0.839872\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 21.3693 1.29571
$$273$$ 0 0
$$274$$ −13.8617 −0.837418
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −16.2462 −0.976140 −0.488070 0.872804i $$-0.662299\pi$$
−0.488070 + 0.872804i $$0.662299\pi$$
$$278$$ −10.7386 −0.644060
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −16.5616 −0.987979 −0.493990 0.869468i $$-0.664462\pi$$
−0.493990 + 0.869468i $$0.664462\pi$$
$$282$$ 0 0
$$283$$ 23.6847 1.40791 0.703953 0.710246i $$-0.251416\pi$$
0.703953 + 0.710246i $$0.251416\pi$$
$$284$$ −3.50758 −0.208136
$$285$$ 0 0
$$286$$ 18.2462 1.07892
$$287$$ 3.12311 0.184351
$$288$$ 0 0
$$289$$ 3.80776 0.223986
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1.86174 −0.108950
$$293$$ 9.68466 0.565784 0.282892 0.959152i $$-0.408706\pi$$
0.282892 + 0.959152i $$0.408706\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 14.6307 0.850391
$$297$$ 0 0
$$298$$ 6.63068 0.384105
$$299$$ 23.3693 1.35148
$$300$$ 0 0
$$301$$ −9.12311 −0.525847
$$302$$ 34.2462 1.97065
$$303$$ 0 0
$$304$$ −5.26137 −0.301760
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 31.6847 1.80834 0.904169 0.427174i $$-0.140491\pi$$
0.904169 + 0.427174i $$0.140491\pi$$
$$308$$ −1.12311 −0.0639949
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9.61553 0.545247 0.272623 0.962121i $$-0.412109\pi$$
0.272623 + 0.962121i $$0.412109\pi$$
$$312$$ 0 0
$$313$$ −31.3002 −1.76919 −0.884596 0.466359i $$-0.845566\pi$$
−0.884596 + 0.466359i $$0.845566\pi$$
$$314$$ −5.86174 −0.330797
$$315$$ 0 0
$$316$$ −2.87689 −0.161838
$$317$$ −22.4924 −1.26330 −0.631650 0.775254i $$-0.717622\pi$$
−0.631650 + 0.775254i $$0.717622\pi$$
$$318$$ 0 0
$$319$$ −14.5616 −0.815290
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ −5.12311 −0.285057
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −1.75379 −0.0971334
$$327$$ 0 0
$$328$$ −7.61553 −0.420497
$$329$$ 3.68466 0.203142
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 1.75379 0.0962517
$$333$$ 0 0
$$334$$ 34.2462 1.87387
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 34.4924 1.87892 0.939461 0.342656i $$-0.111326\pi$$
0.939461 + 0.342656i $$0.111326\pi$$
$$338$$ 12.1922 0.663170
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 22.2462 1.19944
$$345$$ 0 0
$$346$$ −13.3693 −0.718739
$$347$$ −1.12311 −0.0602915 −0.0301457 0.999546i $$-0.509597\pi$$
−0.0301457 + 0.999546i $$0.509597\pi$$
$$348$$ 0 0
$$349$$ −22.4924 −1.20399 −0.601996 0.798499i $$-0.705628\pi$$
−0.601996 + 0.798499i $$0.705628\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6.24621 0.332924
$$353$$ −14.8078 −0.788138 −0.394069 0.919081i $$-0.628933\pi$$
−0.394069 + 0.919081i $$0.628933\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3.12311 −0.165524
$$357$$ 0 0
$$358$$ −31.2311 −1.65061
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −17.7386 −0.933612
$$362$$ 36.8769 1.93821
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.68466 −0.192338 −0.0961688 0.995365i $$-0.530659\pi$$
−0.0961688 + 0.995365i $$0.530659\pi$$
$$368$$ 24.0000 1.25109
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.12311 0.162144
$$372$$ 0 0
$$373$$ −29.3693 −1.52069 −0.760343 0.649522i $$-0.774969\pi$$
−0.760343 + 0.649522i $$0.774969\pi$$
$$374$$ 18.2462 0.943489
$$375$$ 0 0
$$376$$ −8.98485 −0.463358
$$377$$ −25.9309 −1.33551
$$378$$ 0 0
$$379$$ 16.4924 0.847159 0.423579 0.905859i $$-0.360773\pi$$
0.423579 + 0.905859i $$0.360773\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 14.7386 0.754094
$$383$$ −10.2462 −0.523557 −0.261778 0.965128i $$-0.584309\pi$$
−0.261778 + 0.965128i $$0.584309\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 8.38447 0.426758
$$387$$ 0 0
$$388$$ 6.49242 0.329603
$$389$$ −3.93087 −0.199303 −0.0996515 0.995022i $$-0.531773\pi$$
−0.0996515 + 0.995022i $$0.531773\pi$$
$$390$$ 0 0
$$391$$ 23.3693 1.18184
$$392$$ −2.43845 −0.123160
$$393$$ 0 0
$$394$$ −11.1231 −0.560374
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −23.4384 −1.17634 −0.588171 0.808737i $$-0.700152\pi$$
−0.588171 + 0.808737i $$0.700152\pi$$
$$398$$ −28.4924 −1.42820
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −27.4384 −1.37021 −0.685105 0.728444i $$-0.740244\pi$$
−0.685105 + 0.728444i $$0.740244\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −0.107951 −0.00537074
$$405$$ 0 0
$$406$$ 8.87689 0.440553
$$407$$ 15.3693 0.761829
$$408$$ 0 0
$$409$$ −26.4924 −1.30997 −0.654983 0.755644i $$-0.727324\pi$$
−0.654983 + 0.755644i $$0.727324\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −0.630683 −0.0310715
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 11.1231 0.545355
$$417$$ 0 0
$$418$$ −4.49242 −0.219732
$$419$$ −9.75379 −0.476504 −0.238252 0.971203i $$-0.576574\pi$$
−0.238252 + 0.971203i $$0.576574\pi$$
$$420$$ 0 0
$$421$$ 9.68466 0.472001 0.236001 0.971753i $$-0.424163\pi$$
0.236001 + 0.971753i $$0.424163\pi$$
$$422$$ −36.0000 −1.75245
$$423$$ 0 0
$$424$$ −7.61553 −0.369843
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −9.36932 −0.453413
$$428$$ −4.98485 −0.240952
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −0.807764 −0.0389086 −0.0194543 0.999811i $$-0.506193\pi$$
−0.0194543 + 0.999811i $$0.506193\pi$$
$$432$$ 0 0
$$433$$ 8.24621 0.396288 0.198144 0.980173i $$-0.436509\pi$$
0.198144 + 0.980173i $$0.436509\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7.75379 0.371339
$$437$$ −5.75379 −0.275241
$$438$$ 0 0
$$439$$ −15.3693 −0.733537 −0.366769 0.930312i $$-0.619536\pi$$
−0.366769 + 0.930312i $$0.619536\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 32.4924 1.54551
$$443$$ −27.3693 −1.30036 −0.650178 0.759782i $$-0.725306\pi$$
−0.650178 + 0.759782i $$0.725306\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 10.2462 0.485172
$$447$$ 0 0
$$448$$ 5.56155 0.262759
$$449$$ −18.8078 −0.887593 −0.443797 0.896128i $$-0.646369\pi$$
−0.443797 + 0.896128i $$0.646369\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ −6.13826 −0.288719
$$453$$ 0 0
$$454$$ 36.9848 1.73578
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.87689 0.415244 0.207622 0.978209i $$-0.433428\pi$$
0.207622 + 0.978209i $$0.433428\pi$$
$$458$$ 29.8617 1.39535
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 4.87689 0.227140 0.113570 0.993530i $$-0.463771\pi$$
0.113570 + 0.993530i $$0.463771\pi$$
$$462$$ 0 0
$$463$$ 20.4924 0.952364 0.476182 0.879347i $$-0.342020\pi$$
0.476182 + 0.879347i $$0.342020\pi$$
$$464$$ −26.6307 −1.23630
$$465$$ 0 0
$$466$$ −4.87689 −0.225918
$$467$$ 26.5616 1.22912 0.614561 0.788869i $$-0.289333\pi$$
0.614561 + 0.788869i $$0.289333\pi$$
$$468$$ 0 0
$$469$$ 6.24621 0.288423
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −9.75379 −0.448955
$$473$$ 23.3693 1.07452
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 0 0
$$478$$ 1.26137 0.0576935
$$479$$ −13.1231 −0.599610 −0.299805 0.954001i $$-0.596922\pi$$
−0.299805 + 0.954001i $$0.596922\pi$$
$$480$$ 0 0
$$481$$ 27.3693 1.24793
$$482$$ 19.1231 0.871034
$$483$$ 0 0
$$484$$ −1.94602 −0.0884557
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5.12311 −0.232150 −0.116075 0.993240i $$-0.537031\pi$$
−0.116075 + 0.993240i $$0.537031\pi$$
$$488$$ 22.8466 1.03422
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.17708 −0.188509 −0.0942545 0.995548i $$-0.530047\pi$$
−0.0942545 + 0.995548i $$0.530047\pi$$
$$492$$ 0 0
$$493$$ −25.9309 −1.16787
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −4.17708 −0.186992 −0.0934959 0.995620i $$-0.529804\pi$$
−0.0934959 + 0.995620i $$0.529804\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 26.7386 1.19340
$$503$$ 10.0691 0.448960 0.224480 0.974479i $$-0.427932\pi$$
0.224480 + 0.974479i $$0.427932\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 20.4924 0.910999
$$507$$ 0 0
$$508$$ −4.49242 −0.199319
$$509$$ 28.2462 1.25199 0.625996 0.779827i $$-0.284693\pi$$
0.625996 + 0.779827i $$0.284693\pi$$
$$510$$ 0 0
$$511$$ −4.24621 −0.187841
$$512$$ −11.4233 −0.504843
$$513$$ 0 0
$$514$$ 35.1231 1.54921
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −9.43845 −0.415102
$$518$$ −9.36932 −0.411664
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ −7.50758 −0.328283 −0.164142 0.986437i $$-0.552485\pi$$
−0.164142 + 0.986437i $$0.552485\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −32.9848 −1.43821
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 3.24621 0.141140
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.492423 0.0213492
$$533$$ −14.2462 −0.617072
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −15.2311 −0.657881
$$537$$ 0 0
$$538$$ −44.8769 −1.93478
$$539$$ −2.56155 −0.110334
$$540$$ 0 0
$$541$$ −17.1922 −0.739152 −0.369576 0.929201i $$-0.620497\pi$$
−0.369576 + 0.929201i $$0.620497\pi$$
$$542$$ −24.9848 −1.07319
$$543$$ 0 0
$$544$$ 11.1231 0.476899
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −14.2462 −0.609124 −0.304562 0.952493i $$-0.598510\pi$$
−0.304562 + 0.952493i $$0.598510\pi$$
$$548$$ −3.89205 −0.166260
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 6.38447 0.271988
$$552$$ 0 0
$$553$$ −6.56155 −0.279026
$$554$$ −25.3693 −1.07784
$$555$$ 0 0
$$556$$ −3.01515 −0.127871
$$557$$ −4.87689 −0.206641 −0.103320 0.994648i $$-0.532947\pi$$
−0.103320 + 0.994648i $$0.532947\pi$$
$$558$$ 0 0
$$559$$ 41.6155 1.76015
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −25.8617 −1.09091
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 36.9848 1.55459
$$567$$ 0 0
$$568$$ 19.5076 0.818520
$$569$$ −34.9848 −1.46664 −0.733320 0.679883i $$-0.762031\pi$$
−0.733320 + 0.679883i $$0.762031\pi$$
$$570$$ 0 0
$$571$$ 7.50758 0.314182 0.157091 0.987584i $$-0.449788\pi$$
0.157091 + 0.987584i $$0.449788\pi$$
$$572$$ 5.12311 0.214208
$$573$$ 0 0
$$574$$ 4.87689 0.203558
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −13.0540 −0.543444 −0.271722 0.962376i $$-0.587593\pi$$
−0.271722 + 0.962376i $$0.587593\pi$$
$$578$$ 5.94602 0.247322
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 10.3542 0.428458
$$585$$ 0 0
$$586$$ 15.1231 0.624730
$$587$$ −9.75379 −0.402582 −0.201291 0.979531i $$-0.564514\pi$$
−0.201291 + 0.979531i $$0.564514\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 28.1080 1.15523
$$593$$ −23.4384 −0.962502 −0.481251 0.876583i $$-0.659817\pi$$
−0.481251 + 0.876583i $$0.659817\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.86174 0.0762598
$$597$$ 0 0
$$598$$ 36.4924 1.49229
$$599$$ −8.80776 −0.359875 −0.179938 0.983678i $$-0.557590\pi$$
−0.179938 + 0.983678i $$0.557590\pi$$
$$600$$ 0 0
$$601$$ −26.4924 −1.08065 −0.540324 0.841457i $$-0.681698\pi$$
−0.540324 + 0.841457i $$0.681698\pi$$
$$602$$ −14.2462 −0.580632
$$603$$ 0 0
$$604$$ 9.61553 0.391250
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4.94602 0.200753 0.100376 0.994950i $$-0.467995\pi$$
0.100376 + 0.994950i $$0.467995\pi$$
$$608$$ −2.73863 −0.111066
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.8078 −0.679969
$$612$$ 0 0
$$613$$ 8.73863 0.352950 0.176475 0.984305i $$-0.443531\pi$$
0.176475 + 0.984305i $$0.443531\pi$$
$$614$$ 49.4773 1.99674
$$615$$ 0 0
$$616$$ 6.24621 0.251667
$$617$$ 15.7538 0.634224 0.317112 0.948388i $$-0.397287\pi$$
0.317112 + 0.948388i $$0.397287\pi$$
$$618$$ 0 0
$$619$$ −42.1080 −1.69246 −0.846231 0.532817i $$-0.821134\pi$$
−0.846231 + 0.532817i $$0.821134\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 15.0152 0.602053
$$623$$ −7.12311 −0.285381
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −48.8769 −1.95351
$$627$$ 0 0
$$628$$ −1.64584 −0.0656761
$$629$$ 27.3693 1.09129
$$630$$ 0 0
$$631$$ 8.80776 0.350632 0.175316 0.984512i $$-0.443905\pi$$
0.175316 + 0.984512i $$0.443905\pi$$
$$632$$ 16.0000 0.636446
$$633$$ 0 0
$$634$$ −35.1231 −1.39492
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4.56155 −0.180735
$$638$$ −22.7386 −0.900231
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 2.56155 0.101018 0.0505089 0.998724i $$-0.483916\pi$$
0.0505089 + 0.998724i $$0.483916\pi$$
$$644$$ −2.24621 −0.0885131
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 3.50758 0.137897 0.0689486 0.997620i $$-0.478036\pi$$
0.0689486 + 0.997620i $$0.478036\pi$$
$$648$$ 0 0
$$649$$ −10.2462 −0.402199
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −0.492423 −0.0192848
$$653$$ 49.2311 1.92656 0.963280 0.268499i $$-0.0865275\pi$$
0.963280 + 0.268499i $$0.0865275\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −14.6307 −0.571232
$$657$$ 0 0
$$658$$ 5.75379 0.224306
$$659$$ 36.1771 1.40926 0.704629 0.709575i $$-0.251113\pi$$
0.704629 + 0.709575i $$0.251113\pi$$
$$660$$ 0 0
$$661$$ 3.12311 0.121475 0.0607374 0.998154i $$-0.480655\pi$$
0.0607374 + 0.998154i $$0.480655\pi$$
$$662$$ 18.7386 0.728298
$$663$$ 0 0
$$664$$ −9.75379 −0.378520
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −29.1231 −1.12765
$$668$$ 9.61553 0.372036
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 25.8617 0.996897 0.498448 0.866919i $$-0.333903\pi$$
0.498448 + 0.866919i $$0.333903\pi$$
$$674$$ 53.8617 2.07468
$$675$$ 0 0
$$676$$ 3.42329 0.131665
$$677$$ −23.9309 −0.919738 −0.459869 0.887987i $$-0.652104\pi$$
−0.459869 + 0.887987i $$0.652104\pi$$
$$678$$ 0 0
$$679$$ 14.8078 0.568270
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 42.7386 1.63535 0.817674 0.575681i $$-0.195263\pi$$
0.817674 + 0.575681i $$0.195263\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.56155 0.0596204
$$687$$ 0 0
$$688$$ 42.7386 1.62940
$$689$$ −14.2462 −0.542737
$$690$$ 0 0
$$691$$ 8.49242 0.323067 0.161533 0.986867i $$-0.448356\pi$$
0.161533 + 0.986867i $$0.448356\pi$$
$$692$$ −3.75379 −0.142698
$$693$$ 0 0
$$694$$ −1.75379 −0.0665729
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −14.2462 −0.539614
$$698$$ −35.1231 −1.32943
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −0.0691303 −0.00261102 −0.00130551 0.999999i $$-0.500416\pi$$
−0.00130551 + 0.999999i $$0.500416\pi$$
$$702$$ 0 0
$$703$$ −6.73863 −0.254152
$$704$$ −14.2462 −0.536924
$$705$$ 0 0
$$706$$ −23.1231 −0.870250
$$707$$ −0.246211 −0.00925973
$$708$$ 0 0
$$709$$ −18.1771 −0.682655 −0.341327 0.939945i $$-0.610876\pi$$
−0.341327 + 0.939945i $$0.610876\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 17.3693 0.650943
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −8.76894 −0.327711
$$717$$ 0 0
$$718$$ −12.4924 −0.466213
$$719$$ −49.6155 −1.85035 −0.925173 0.379544i $$-0.876081\pi$$
−0.925173 + 0.379544i $$0.876081\pi$$
$$720$$ 0 0
$$721$$ −1.43845 −0.0535706
$$722$$ −27.6998 −1.03088
$$723$$ 0 0
$$724$$ 10.3542 0.384809
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −19.5076 −0.723496 −0.361748 0.932276i $$-0.617820\pi$$
−0.361748 + 0.932276i $$0.617820\pi$$
$$728$$ 11.1231 0.412250
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 41.6155 1.53921
$$732$$ 0 0
$$733$$ 5.68466 0.209968 0.104984 0.994474i $$-0.466521\pi$$
0.104984 + 0.994474i $$0.466521\pi$$
$$734$$ −5.75379 −0.212376
$$735$$ 0 0
$$736$$ 12.4924 0.460477
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 6.06913 0.223257 0.111628 0.993750i $$-0.464393\pi$$
0.111628 + 0.993750i $$0.464393\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.87689 0.179036
$$743$$ 32.9848 1.21010 0.605048 0.796189i $$-0.293154\pi$$
0.605048 + 0.796189i $$0.293154\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −45.8617 −1.67912
$$747$$ 0 0
$$748$$ 5.12311 0.187319
$$749$$ −11.3693 −0.415426
$$750$$ 0 0
$$751$$ 45.9309 1.67604 0.838021 0.545639i $$-0.183713\pi$$
0.838021 + 0.545639i $$0.183713\pi$$
$$752$$ −17.2614 −0.629457
$$753$$ 0 0
$$754$$ −40.4924 −1.47465
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −14.6307 −0.531761 −0.265881 0.964006i $$-0.585663\pi$$
−0.265881 + 0.964006i $$0.585663\pi$$
$$758$$ 25.7538 0.935420
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −31.7538 −1.15107 −0.575537 0.817776i $$-0.695207\pi$$
−0.575537 + 0.817776i $$0.695207\pi$$
$$762$$ 0 0
$$763$$ 17.6847 0.640228
$$764$$ 4.13826 0.149717
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ −18.2462 −0.658833
$$768$$ 0 0
$$769$$ −9.50758 −0.342852 −0.171426 0.985197i $$-0.554837\pi$$
−0.171426 + 0.985197i $$0.554837\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.35416 0.0847281
$$773$$ 8.06913 0.290226 0.145113 0.989415i $$-0.453645\pi$$
0.145113 + 0.989415i $$0.453645\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −36.1080 −1.29620
$$777$$ 0 0
$$778$$ −6.13826 −0.220067
$$779$$ 3.50758 0.125672
$$780$$ 0 0
$$781$$ 20.4924 0.733277
$$782$$ 36.4924 1.30497
$$783$$ 0 0
$$784$$ −4.68466 −0.167309
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.82292 0.136272 0.0681362 0.997676i $$-0.478295\pi$$
0.0681362 + 0.997676i $$0.478295\pi$$
$$788$$ −3.12311 −0.111256
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 42.7386 1.51769
$$794$$ −36.6004 −1.29890
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ −13.0540 −0.462396 −0.231198 0.972907i $$-0.574264\pi$$
−0.231198 + 0.972907i $$0.574264\pi$$
$$798$$ 0 0
$$799$$ −16.8078 −0.594616
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −42.8466 −1.51297
$$803$$ 10.8769 0.383837
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0.600373 0.0211211
$$809$$ 53.5464 1.88259 0.941296 0.337584i $$-0.109610\pi$$
0.941296 + 0.337584i $$0.109610\pi$$
$$810$$ 0 0
$$811$$ −21.6155 −0.759024 −0.379512 0.925187i $$-0.623908\pi$$
−0.379512 + 0.925187i $$0.623908\pi$$
$$812$$ 2.49242 0.0874669
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10.2462 −0.358470
$$818$$ −41.3693 −1.44644
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −40.4233 −1.41078 −0.705391 0.708818i $$-0.749229\pi$$
−0.705391 + 0.708818i $$0.749229\pi$$
$$822$$ 0 0
$$823$$ 3.50758 0.122266 0.0611332 0.998130i $$-0.480529\pi$$
0.0611332 + 0.998130i $$0.480529\pi$$
$$824$$ 3.50758 0.122192
$$825$$ 0 0
$$826$$ 6.24621 0.217333
$$827$$ 19.3693 0.673537 0.336769 0.941587i $$-0.390666\pi$$
0.336769 + 0.941587i $$0.390666\pi$$
$$828$$ 0 0
$$829$$ 43.1231 1.49773 0.748864 0.662724i $$-0.230600\pi$$
0.748864 + 0.662724i $$0.230600\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −25.3693 −0.879523
$$833$$ −4.56155 −0.158048
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1.26137 −0.0436253
$$837$$ 0 0
$$838$$ −15.2311 −0.526148
$$839$$ 37.1231 1.28163 0.640816 0.767695i $$-0.278596\pi$$
0.640816 + 0.767695i $$0.278596\pi$$
$$840$$ 0 0
$$841$$ 3.31534 0.114322
$$842$$ 15.1231 0.521177
$$843$$ 0 0
$$844$$ −10.1080 −0.347930
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4.43845 −0.152507
$$848$$ −14.6307 −0.502420
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 30.7386 1.05371
$$852$$ 0 0
$$853$$ 56.7386 1.94269 0.971347 0.237666i $$-0.0763824\pi$$
0.971347 + 0.237666i $$0.0763824\pi$$
$$854$$ −14.6307 −0.500652
$$855$$ 0 0
$$856$$ 27.7235 0.947569
$$857$$ −32.2462 −1.10151 −0.550755 0.834667i $$-0.685660\pi$$
−0.550755 + 0.834667i $$0.685660\pi$$
$$858$$ 0 0
$$859$$ 16.4924 0.562714 0.281357 0.959603i $$-0.409215\pi$$
0.281357 + 0.959603i $$0.409215\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −1.26137 −0.0429623
$$863$$ −42.2462 −1.43808 −0.719039 0.694970i $$-0.755418\pi$$
−0.719039 + 0.694970i $$0.755418\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 12.8769 0.437575
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.8078 0.570164
$$870$$ 0 0
$$871$$ −28.4924 −0.965429
$$872$$ −43.1231 −1.46033
$$873$$ 0 0
$$874$$ −8.98485 −0.303917
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.7538 0.802108 0.401054 0.916054i $$-0.368644\pi$$
0.401054 + 0.916054i $$0.368644\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −45.8617 −1.54512 −0.772561 0.634941i $$-0.781024\pi$$
−0.772561 + 0.634941i $$0.781024\pi$$
$$882$$ 0 0
$$883$$ −24.4924 −0.824236 −0.412118 0.911131i $$-0.635211\pi$$
−0.412118 + 0.911131i $$0.635211\pi$$
$$884$$ 9.12311 0.306843
$$885$$ 0 0
$$886$$ −42.7386 −1.43583
$$887$$ −12.4924 −0.419454 −0.209727 0.977760i $$-0.567258\pi$$
−0.209727 + 0.977760i $$0.567258\pi$$
$$888$$ 0 0
$$889$$ −10.2462 −0.343647
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.87689 0.0963255
$$893$$ 4.13826 0.138482
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 13.5616 0.453060
$$897$$ 0 0
$$898$$ −29.3693 −0.980067
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −14.2462 −0.474610
$$902$$ −12.4924 −0.415952
$$903$$ 0 0
$$904$$ 34.1383 1.13542
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −50.1080 −1.66381 −0.831904 0.554920i $$-0.812749\pi$$
−0.831904 + 0.554920i $$0.812749\pi$$
$$908$$ 10.3845 0.344621
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −4.49242 −0.148841 −0.0744203 0.997227i $$-0.523711\pi$$
−0.0744203 + 0.997227i $$0.523711\pi$$
$$912$$ 0 0
$$913$$ −10.2462 −0.339100
$$914$$ 13.8617 0.458506
$$915$$ 0 0
$$916$$ 8.38447 0.277031
$$917$$ 9.12311 0.301271
$$918$$ 0 0
$$919$$ −13.3002 −0.438733 −0.219366 0.975643i $$-0.570399\pi$$
−0.219366 + 0.975643i $$0.570399\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 7.61553 0.250804
$$923$$ 36.4924 1.20116
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ 0 0
$$928$$ −13.8617 −0.455034
$$929$$ 52.1080 1.70961 0.854803 0.518952i $$-0.173678\pi$$
0.854803 + 0.518952i $$0.173678\pi$$
$$930$$ 0 0
$$931$$ 1.12311 0.0368083
$$932$$ −1.36932 −0.0448535
$$933$$ 0 0
$$934$$ 41.4773 1.35718
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.6695 −0.740580 −0.370290 0.928916i $$-0.620742\pi$$
−0.370290 + 0.928916i $$0.620742\pi$$
$$938$$ 9.75379 0.318472
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 13.8617 0.451880 0.225940 0.974141i $$-0.427455\pi$$
0.225940 + 0.974141i $$0.427455\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ −18.7386 −0.609891
$$945$$ 0 0
$$946$$ 36.4924 1.18647
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ 19.3693 0.628755
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 11.1231 0.360502
$$953$$ −24.8769 −0.805842 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0.354162 0.0114544
$$957$$ 0 0
$$958$$ −20.4924 −0.662080
$$959$$ −8.87689 −0.286650
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 42.7386 1.37795
$$963$$ 0 0
$$964$$ 5.36932 0.172934
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 26.8769 0.864303 0.432151 0.901801i $$-0.357755\pi$$
0.432151 + 0.901801i $$0.357755\pi$$
$$968$$ 10.8229 0.347862
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 49.4773 1.58780 0.793901 0.608048i $$-0.208047\pi$$
0.793901 + 0.608048i $$0.208047\pi$$
$$972$$ 0 0
$$973$$ −6.87689 −0.220463
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 43.8920 1.40495
$$977$$ −49.2311 −1.57504 −0.787521 0.616288i $$-0.788636\pi$$
−0.787521 + 0.616288i $$0.788636\pi$$
$$978$$ 0 0
$$979$$ 18.2462 0.583151
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −6.52273 −0.208149
$$983$$ −10.4233 −0.332451 −0.166226 0.986088i $$-0.553158\pi$$
−0.166226 + 0.986088i $$0.553158\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −40.4924 −1.28954
$$987$$ 0 0
$$988$$ −2.24621 −0.0714615
$$989$$ 46.7386 1.48620
$$990$$ 0 0
$$991$$ −20.4924 −0.650963 −0.325482 0.945548i $$-0.605526\pi$$
−0.325482 + 0.945548i $$0.605526\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −12.4924 −0.396236
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −9.68466 −0.306716 −0.153358 0.988171i $$-0.549009\pi$$
−0.153358 + 0.988171i $$0.549009\pi$$
$$998$$ −6.52273 −0.206473
$$999$$ 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 1575.2.a.p.1.2 2
3.2 odd 2 175.2.a.f.1.1 2
5.2 odd 4 1575.2.d.e.1324.3 4
5.3 odd 4 1575.2.d.e.1324.2 4
5.4 even 2 315.2.a.e.1.1 2
12.11 even 2 2800.2.a.bi.1.1 2
15.2 even 4 175.2.b.b.99.2 4
15.8 even 4 175.2.b.b.99.3 4
15.14 odd 2 35.2.a.b.1.2 2
20.19 odd 2 5040.2.a.bt.1.2 2
21.20 even 2 1225.2.a.s.1.1 2
35.34 odd 2 2205.2.a.x.1.1 2
60.23 odd 4 2800.2.g.t.449.1 4
60.47 odd 4 2800.2.g.t.449.4 4
60.59 even 2 560.2.a.i.1.2 2
105.44 odd 6 245.2.e.i.116.1 4
105.59 even 6 245.2.e.h.226.1 4
105.62 odd 4 1225.2.b.f.99.2 4
105.74 odd 6 245.2.e.i.226.1 4
105.83 odd 4 1225.2.b.f.99.3 4
105.89 even 6 245.2.e.h.116.1 4
105.104 even 2 245.2.a.d.1.2 2
120.29 odd 2 2240.2.a.bh.1.2 2
120.59 even 2 2240.2.a.bd.1.1 2
165.164 even 2 4235.2.a.m.1.1 2
195.194 odd 2 5915.2.a.l.1.1 2
420.419 odd 2 3920.2.a.bs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.b.1.2 2 15.14 odd 2
175.2.a.f.1.1 2 3.2 odd 2
175.2.b.b.99.2 4 15.2 even 4
175.2.b.b.99.3 4 15.8 even 4
245.2.a.d.1.2 2 105.104 even 2
245.2.e.h.116.1 4 105.89 even 6
245.2.e.h.226.1 4 105.59 even 6
245.2.e.i.116.1 4 105.44 odd 6
245.2.e.i.226.1 4 105.74 odd 6
315.2.a.e.1.1 2 5.4 even 2
560.2.a.i.1.2 2 60.59 even 2
1225.2.a.s.1.1 2 21.20 even 2
1225.2.b.f.99.2 4 105.62 odd 4
1225.2.b.f.99.3 4 105.83 odd 4
1575.2.a.p.1.2 2 1.1 even 1 trivial
1575.2.d.e.1324.2 4 5.3 odd 4
1575.2.d.e.1324.3 4 5.2 odd 4
2205.2.a.x.1.1 2 35.34 odd 2
2240.2.a.bd.1.1 2 120.59 even 2
2240.2.a.bh.1.2 2 120.29 odd 2
2800.2.a.bi.1.1 2 12.11 even 2
2800.2.g.t.449.1 4 60.23 odd 4
2800.2.g.t.449.4 4 60.47 odd 4
3920.2.a.bs.1.1 2 420.419 odd 2
4235.2.a.m.1.1 2 165.164 even 2
5040.2.a.bt.1.2 2 20.19 odd 2
5915.2.a.l.1.1 2 195.194 odd 2