# Properties

 Label 5625.2.a.bb.1.1 Level $5625$ Weight $2$ Character 5625.1 Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45$$ x^8 - 15*x^6 + 70*x^4 - 105*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.66202$$ of defining polynomial Character $$\chi$$ $$=$$ 5625.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.66202 q^{2} +5.08634 q^{4} +3.46831 q^{7} -8.21589 q^{8} +O(q^{10})$$ $$q-2.66202 q^{2} +5.08634 q^{4} +3.46831 q^{7} -8.21589 q^{8} -3.43248 q^{11} +5.70437 q^{13} -9.23269 q^{14} +11.6982 q^{16} +1.89155 q^{17} +2.46831 q^{19} +9.13733 q^{22} +8.84431 q^{23} -15.1851 q^{26} +17.6410 q^{28} +8.98636 q^{29} -1.90746 q^{31} -14.7090 q^{32} -5.03535 q^{34} +5.09254 q^{37} -6.57068 q^{38} +6.35103 q^{41} -3.43296 q^{43} -17.4588 q^{44} -23.5437 q^{46} +8.61447 q^{47} +5.02915 q^{49} +29.0144 q^{52} +2.03360 q^{53} -28.4952 q^{56} -23.9219 q^{58} -8.47242 q^{59} -7.90746 q^{61} +5.07770 q^{62} +15.7592 q^{64} +6.95846 q^{67} +9.62108 q^{68} +2.91855 q^{71} -11.9280 q^{73} -13.5564 q^{74} +12.5546 q^{76} -11.9049 q^{77} -6.60564 q^{79} -16.9066 q^{82} +12.9993 q^{83} +9.13860 q^{86} +28.2009 q^{88} -11.5873 q^{89} +19.7845 q^{91} +44.9852 q^{92} -22.9319 q^{94} -2.17888 q^{97} -13.3877 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 14 q^{4} + 10 q^{7}+O(q^{10})$$ 8 * q + 14 * q^4 + 10 * q^7 $$8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100})$$ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.66202 −1.88233 −0.941166 0.337946i $$-0.890268\pi$$
−0.941166 + 0.337946i $$0.890268\pi$$
$$3$$ 0 0
$$4$$ 5.08634 2.54317
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.46831 1.31090 0.655448 0.755240i $$-0.272480\pi$$
0.655448 + 0.755240i $$0.272480\pi$$
$$8$$ −8.21589 −2.90476
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.43248 −1.03493 −0.517466 0.855703i $$-0.673125\pi$$
−0.517466 + 0.855703i $$0.673125\pi$$
$$12$$ 0 0
$$13$$ 5.70437 1.58211 0.791054 0.611746i $$-0.209532\pi$$
0.791054 + 0.611746i $$0.209532\pi$$
$$14$$ −9.23269 −2.46754
$$15$$ 0 0
$$16$$ 11.6982 2.92454
$$17$$ 1.89155 0.458769 0.229384 0.973336i $$-0.426329\pi$$
0.229384 + 0.973336i $$0.426329\pi$$
$$18$$ 0 0
$$19$$ 2.46831 0.566268 0.283134 0.959080i $$-0.408626\pi$$
0.283134 + 0.959080i $$0.408626\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 9.13733 1.94809
$$23$$ 8.84431 1.84417 0.922083 0.386992i $$-0.126486\pi$$
0.922083 + 0.386992i $$0.126486\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −15.1851 −2.97805
$$27$$ 0 0
$$28$$ 17.6410 3.33383
$$29$$ 8.98636 1.66873 0.834363 0.551216i $$-0.185836\pi$$
0.834363 + 0.551216i $$0.185836\pi$$
$$30$$ 0 0
$$31$$ −1.90746 −0.342591 −0.171295 0.985220i $$-0.554795\pi$$
−0.171295 + 0.985220i $$0.554795\pi$$
$$32$$ −14.7090 −2.60020
$$33$$ 0 0
$$34$$ −5.03535 −0.863555
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.09254 0.837208 0.418604 0.908169i $$-0.362520\pi$$
0.418604 + 0.908169i $$0.362520\pi$$
$$38$$ −6.57068 −1.06590
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.35103 0.991864 0.495932 0.868361i $$-0.334826\pi$$
0.495932 + 0.868361i $$0.334826\pi$$
$$42$$ 0 0
$$43$$ −3.43296 −0.523522 −0.261761 0.965133i $$-0.584303\pi$$
−0.261761 + 0.965133i $$0.584303\pi$$
$$44$$ −17.4588 −2.63201
$$45$$ 0 0
$$46$$ −23.5437 −3.47133
$$47$$ 8.61447 1.25655 0.628275 0.777991i $$-0.283761\pi$$
0.628275 + 0.777991i $$0.283761\pi$$
$$48$$ 0 0
$$49$$ 5.02915 0.718450
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 29.0144 4.02357
$$53$$ 2.03360 0.279337 0.139668 0.990198i $$-0.455396\pi$$
0.139668 + 0.990198i $$0.455396\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −28.4952 −3.80784
$$57$$ 0 0
$$58$$ −23.9219 −3.14109
$$59$$ −8.47242 −1.10302 −0.551508 0.834170i $$-0.685947\pi$$
−0.551508 + 0.834170i $$0.685947\pi$$
$$60$$ 0 0
$$61$$ −7.90746 −1.01245 −0.506223 0.862402i $$-0.668959\pi$$
−0.506223 + 0.862402i $$0.668959\pi$$
$$62$$ 5.07770 0.644869
$$63$$ 0 0
$$64$$ 15.7592 1.96990
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.95846 0.850111 0.425055 0.905167i $$-0.360255\pi$$
0.425055 + 0.905167i $$0.360255\pi$$
$$68$$ 9.62108 1.16673
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.91855 0.346368 0.173184 0.984890i $$-0.444595\pi$$
0.173184 + 0.984890i $$0.444595\pi$$
$$72$$ 0 0
$$73$$ −11.9280 −1.39607 −0.698036 0.716062i $$-0.745943\pi$$
−0.698036 + 0.716062i $$0.745943\pi$$
$$74$$ −13.5564 −1.57590
$$75$$ 0 0
$$76$$ 12.5546 1.44012
$$77$$ −11.9049 −1.35669
$$78$$ 0 0
$$79$$ −6.60564 −0.743193 −0.371596 0.928394i $$-0.621189\pi$$
−0.371596 + 0.928394i $$0.621189\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −16.9066 −1.86702
$$83$$ 12.9993 1.42686 0.713429 0.700727i $$-0.247141\pi$$
0.713429 + 0.700727i $$0.247141\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 9.13860 0.985441
$$87$$ 0 0
$$88$$ 28.2009 3.00623
$$89$$ −11.5873 −1.22825 −0.614124 0.789209i $$-0.710491\pi$$
−0.614124 + 0.789209i $$0.710491\pi$$
$$90$$ 0 0
$$91$$ 19.7845 2.07398
$$92$$ 44.9852 4.69003
$$93$$ 0 0
$$94$$ −22.9319 −2.36524
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.17888 −0.221231 −0.110616 0.993863i $$-0.535282\pi$$
−0.110616 + 0.993863i $$0.535282\pi$$
$$98$$ −13.3877 −1.35236
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.30399 0.925782 0.462891 0.886415i $$-0.346812\pi$$
0.462891 + 0.886415i $$0.346812\pi$$
$$102$$ 0 0
$$103$$ −0.176510 −0.0173921 −0.00869604 0.999962i $$-0.502768\pi$$
−0.00869604 + 0.999962i $$0.502768\pi$$
$$104$$ −46.8665 −4.59564
$$105$$ 0 0
$$106$$ −5.41348 −0.525804
$$107$$ −1.37761 −0.133179 −0.0665895 0.997780i $$-0.521212\pi$$
−0.0665895 + 0.997780i $$0.521212\pi$$
$$108$$ 0 0
$$109$$ 1.89744 0.181742 0.0908708 0.995863i $$-0.471035\pi$$
0.0908708 + 0.995863i $$0.471035\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 40.5729 3.83378
$$113$$ 2.35123 0.221185 0.110593 0.993866i $$-0.464725\pi$$
0.110593 + 0.993866i $$0.464725\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 45.7077 4.24385
$$117$$ 0 0
$$118$$ 22.5537 2.07624
$$119$$ 6.56048 0.601398
$$120$$ 0 0
$$121$$ 0.781947 0.0710861
$$122$$ 21.0498 1.90576
$$123$$ 0 0
$$124$$ −9.70201 −0.871266
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −3.78069 −0.335482 −0.167741 0.985831i $$-0.553647\pi$$
−0.167741 + 0.985831i $$0.553647\pi$$
$$128$$ −12.5333 −1.10780
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.797155 −0.0696477 −0.0348239 0.999393i $$-0.511087\pi$$
−0.0348239 + 0.999393i $$0.511087\pi$$
$$132$$ 0 0
$$133$$ 8.56084 0.742319
$$134$$ −18.5235 −1.60019
$$135$$ 0 0
$$136$$ −15.5408 −1.33261
$$137$$ −12.4188 −1.06101 −0.530507 0.847681i $$-0.677998\pi$$
−0.530507 + 0.847681i $$0.677998\pi$$
$$138$$ 0 0
$$139$$ 12.2399 1.03817 0.519087 0.854721i $$-0.326272\pi$$
0.519087 + 0.854721i $$0.326272\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.76922 −0.651979
$$143$$ −19.5802 −1.63738
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 31.7527 2.62787
$$147$$ 0 0
$$148$$ 25.9024 2.12916
$$149$$ −1.31109 −0.107409 −0.0537044 0.998557i $$-0.517103\pi$$
−0.0537044 + 0.998557i $$0.517103\pi$$
$$150$$ 0 0
$$151$$ 2.57266 0.209360 0.104680 0.994506i $$-0.466618\pi$$
0.104680 + 0.994506i $$0.466618\pi$$
$$152$$ −20.2793 −1.64487
$$153$$ 0 0
$$154$$ 31.6911 2.55374
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.7038 1.25330 0.626650 0.779301i $$-0.284426\pi$$
0.626650 + 0.779301i $$0.284426\pi$$
$$158$$ 17.5843 1.39893
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 30.6748 2.41751
$$162$$ 0 0
$$163$$ −7.42913 −0.581894 −0.290947 0.956739i $$-0.593970\pi$$
−0.290947 + 0.956739i $$0.593970\pi$$
$$164$$ 32.3035 2.52248
$$165$$ 0 0
$$166$$ −34.6044 −2.68582
$$167$$ −4.12200 −0.318970 −0.159485 0.987200i $$-0.550983\pi$$
−0.159485 + 0.987200i $$0.550983\pi$$
$$168$$ 0 0
$$169$$ 19.5399 1.50307
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −17.4612 −1.33140
$$173$$ −19.7765 −1.50358 −0.751789 0.659404i $$-0.770809\pi$$
−0.751789 + 0.659404i $$0.770809\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −40.1538 −3.02671
$$177$$ 0 0
$$178$$ 30.8455 2.31197
$$179$$ −7.18260 −0.536853 −0.268426 0.963300i $$-0.586504\pi$$
−0.268426 + 0.963300i $$0.586504\pi$$
$$180$$ 0 0
$$181$$ 15.9100 1.18258 0.591292 0.806458i $$-0.298618\pi$$
0.591292 + 0.806458i $$0.298618\pi$$
$$182$$ −52.6667 −3.90392
$$183$$ 0 0
$$184$$ −72.6639 −5.35686
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.49272 −0.474795
$$188$$ 43.8161 3.19562
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 14.5190 1.05056 0.525278 0.850931i $$-0.323961\pi$$
0.525278 + 0.850931i $$0.323961\pi$$
$$192$$ 0 0
$$193$$ −13.0763 −0.941254 −0.470627 0.882332i $$-0.655972\pi$$
−0.470627 + 0.882332i $$0.655972\pi$$
$$194$$ 5.80021 0.416431
$$195$$ 0 0
$$196$$ 25.5800 1.82714
$$197$$ −17.8849 −1.27425 −0.637124 0.770761i $$-0.719876\pi$$
−0.637124 + 0.770761i $$0.719876\pi$$
$$198$$ 0 0
$$199$$ −21.1565 −1.49974 −0.749871 0.661584i $$-0.769884\pi$$
−0.749871 + 0.661584i $$0.769884\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −24.7674 −1.74263
$$203$$ 31.1675 2.18753
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0.469874 0.0327377
$$207$$ 0 0
$$208$$ 66.7308 4.62695
$$209$$ −8.47242 −0.586050
$$210$$ 0 0
$$211$$ 13.4002 0.922507 0.461253 0.887268i $$-0.347400\pi$$
0.461253 + 0.887268i $$0.347400\pi$$
$$212$$ 10.3436 0.710400
$$213$$ 0 0
$$214$$ 3.66724 0.250687
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.61567 −0.449101
$$218$$ −5.05101 −0.342098
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10.7901 0.725822
$$222$$ 0 0
$$223$$ 25.2555 1.69124 0.845618 0.533788i $$-0.179232\pi$$
0.845618 + 0.533788i $$0.179232\pi$$
$$224$$ −51.0152 −3.40860
$$225$$ 0 0
$$226$$ −6.25902 −0.416344
$$227$$ −1.89155 −0.125547 −0.0627734 0.998028i $$-0.519995\pi$$
−0.0627734 + 0.998028i $$0.519995\pi$$
$$228$$ 0 0
$$229$$ −4.78069 −0.315917 −0.157958 0.987446i $$-0.550491\pi$$
−0.157958 + 0.987446i $$0.550491\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −73.8310 −4.84724
$$233$$ −3.44563 −0.225731 −0.112865 0.993610i $$-0.536003\pi$$
−0.112865 + 0.993610i $$0.536003\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −43.0936 −2.80516
$$237$$ 0 0
$$238$$ −17.4641 −1.13203
$$239$$ −24.8164 −1.60524 −0.802620 0.596490i $$-0.796561\pi$$
−0.802620 + 0.596490i $$0.796561\pi$$
$$240$$ 0 0
$$241$$ −20.5045 −1.32081 −0.660407 0.750908i $$-0.729616\pi$$
−0.660407 + 0.750908i $$0.729616\pi$$
$$242$$ −2.08156 −0.133808
$$243$$ 0 0
$$244$$ −40.2201 −2.57482
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 14.0801 0.895898
$$248$$ 15.6715 0.995142
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.10825 0.133071 0.0665357 0.997784i $$-0.478805\pi$$
0.0665357 + 0.997784i $$0.478805\pi$$
$$252$$ 0 0
$$253$$ −30.3580 −1.90859
$$254$$ 10.0643 0.631488
$$255$$ 0 0
$$256$$ 1.84554 0.115346
$$257$$ −1.45314 −0.0906445 −0.0453222 0.998972i $$-0.514431\pi$$
−0.0453222 + 0.998972i $$0.514431\pi$$
$$258$$ 0 0
$$259$$ 17.6625 1.09749
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.12204 0.131100
$$263$$ −25.7688 −1.58897 −0.794485 0.607284i $$-0.792259\pi$$
−0.794485 + 0.607284i $$0.792259\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −22.7891 −1.39729
$$267$$ 0 0
$$268$$ 35.3931 2.16198
$$269$$ −31.9778 −1.94972 −0.974859 0.222823i $$-0.928473\pi$$
−0.974859 + 0.222823i $$0.928473\pi$$
$$270$$ 0 0
$$271$$ 25.1821 1.52971 0.764853 0.644205i $$-0.222812\pi$$
0.764853 + 0.644205i $$0.222812\pi$$
$$272$$ 22.1277 1.34169
$$273$$ 0 0
$$274$$ 33.0592 1.99718
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 16.8847 1.01450 0.507252 0.861798i $$-0.330661\pi$$
0.507252 + 0.861798i $$0.330661\pi$$
$$278$$ −32.5828 −1.95419
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7.35764 0.438920 0.219460 0.975622i $$-0.429570\pi$$
0.219460 + 0.975622i $$0.429570\pi$$
$$282$$ 0 0
$$283$$ 12.1220 0.720581 0.360290 0.932840i $$-0.382678\pi$$
0.360290 + 0.932840i $$0.382678\pi$$
$$284$$ 14.8447 0.880872
$$285$$ 0 0
$$286$$ 52.1228 3.08208
$$287$$ 22.0273 1.30023
$$288$$ 0 0
$$289$$ −13.4220 −0.789531
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −60.6701 −3.55045
$$293$$ −24.6612 −1.44072 −0.720362 0.693598i $$-0.756024\pi$$
−0.720362 + 0.693598i $$0.756024\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −41.8397 −2.43189
$$297$$ 0 0
$$298$$ 3.49015 0.202179
$$299$$ 50.4513 2.91767
$$300$$ 0 0
$$301$$ −11.9066 −0.686283
$$302$$ −6.84847 −0.394085
$$303$$ 0 0
$$304$$ 28.8747 1.65608
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −19.9964 −1.14125 −0.570627 0.821210i $$-0.693300\pi$$
−0.570627 + 0.821210i $$0.693300\pi$$
$$308$$ −60.5524 −3.45029
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −3.11485 −0.176627 −0.0883136 0.996093i $$-0.528148\pi$$
−0.0883136 + 0.996093i $$0.528148\pi$$
$$312$$ 0 0
$$313$$ −10.1492 −0.573664 −0.286832 0.957981i $$-0.592602\pi$$
−0.286832 + 0.957981i $$0.592602\pi$$
$$314$$ −41.8038 −2.35913
$$315$$ 0 0
$$316$$ −33.5985 −1.89007
$$317$$ 8.18921 0.459952 0.229976 0.973196i $$-0.426135\pi$$
0.229976 + 0.973196i $$0.426135\pi$$
$$318$$ 0 0
$$319$$ −30.8455 −1.72702
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −81.6568 −4.55056
$$323$$ 4.66893 0.259786
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.7765 1.09532
$$327$$ 0 0
$$328$$ −52.1794 −2.88113
$$329$$ 29.8776 1.64721
$$330$$ 0 0
$$331$$ 19.3339 1.06269 0.531344 0.847156i $$-0.321687\pi$$
0.531344 + 0.847156i $$0.321687\pi$$
$$332$$ 66.1189 3.62875
$$333$$ 0 0
$$334$$ 10.9728 0.600408
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −0.902375 −0.0491555 −0.0245778 0.999698i $$-0.507824\pi$$
−0.0245778 + 0.999698i $$0.507824\pi$$
$$338$$ −52.0155 −2.82927
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.54734 0.354558
$$342$$ 0 0
$$343$$ −6.83551 −0.369083
$$344$$ 28.2048 1.52070
$$345$$ 0 0
$$346$$ 52.6454 2.83023
$$347$$ 0.676375 0.0363097 0.0181548 0.999835i $$-0.494221\pi$$
0.0181548 + 0.999835i $$0.494221\pi$$
$$348$$ 0 0
$$349$$ 10.0604 0.538523 0.269262 0.963067i $$-0.413220\pi$$
0.269262 + 0.963067i $$0.413220\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 50.4883 2.69104
$$353$$ 7.29023 0.388020 0.194010 0.981000i $$-0.437851\pi$$
0.194010 + 0.981000i $$0.437851\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −58.9368 −3.12365
$$357$$ 0 0
$$358$$ 19.1202 1.01053
$$359$$ 15.8301 0.835479 0.417739 0.908567i $$-0.362823\pi$$
0.417739 + 0.908567i $$0.362823\pi$$
$$360$$ 0 0
$$361$$ −12.9075 −0.679340
$$362$$ −42.3528 −2.22601
$$363$$ 0 0
$$364$$ 100.631 5.27449
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 23.5116 1.22730 0.613649 0.789579i $$-0.289701\pi$$
0.613649 + 0.789579i $$0.289701\pi$$
$$368$$ 103.462 5.39335
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 7.05315 0.366181
$$372$$ 0 0
$$373$$ −8.27467 −0.428446 −0.214223 0.976785i $$-0.568722\pi$$
−0.214223 + 0.976785i $$0.568722\pi$$
$$374$$ 17.2837 0.893721
$$375$$ 0 0
$$376$$ −70.7756 −3.64997
$$377$$ 51.2616 2.64010
$$378$$ 0 0
$$379$$ −27.3648 −1.40564 −0.702819 0.711369i $$-0.748075\pi$$
−0.702819 + 0.711369i $$0.748075\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −38.6498 −1.97749
$$383$$ 15.6880 0.801620 0.400810 0.916161i $$-0.368729\pi$$
0.400810 + 0.916161i $$0.368729\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 34.8094 1.77175
$$387$$ 0 0
$$388$$ −11.0825 −0.562629
$$389$$ −20.6950 −1.04928 −0.524638 0.851325i $$-0.675799\pi$$
−0.524638 + 0.851325i $$0.675799\pi$$
$$390$$ 0 0
$$391$$ 16.7295 0.846046
$$392$$ −41.3190 −2.08692
$$393$$ 0 0
$$394$$ 47.6100 2.39856
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 10.3616 0.520033 0.260016 0.965604i $$-0.416272\pi$$
0.260016 + 0.965604i $$0.416272\pi$$
$$398$$ 56.3189 2.82301
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0.600848 0.0300049 0.0150025 0.999887i $$-0.495224\pi$$
0.0150025 + 0.999887i $$0.495224\pi$$
$$402$$ 0 0
$$403$$ −10.8809 −0.542015
$$404$$ 47.3233 2.35442
$$405$$ 0 0
$$406$$ −82.9683 −4.11765
$$407$$ −17.4801 −0.866454
$$408$$ 0 0
$$409$$ −20.0064 −0.989253 −0.494626 0.869106i $$-0.664695\pi$$
−0.494626 + 0.869106i $$0.664695\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −0.897792 −0.0442310
$$413$$ −29.3850 −1.44594
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −83.9055 −4.11381
$$417$$ 0 0
$$418$$ 22.5537 1.10314
$$419$$ −10.8245 −0.528813 −0.264407 0.964411i $$-0.585176\pi$$
−0.264407 + 0.964411i $$0.585176\pi$$
$$420$$ 0 0
$$421$$ −31.3321 −1.52703 −0.763516 0.645789i $$-0.776528\pi$$
−0.763516 + 0.645789i $$0.776528\pi$$
$$422$$ −35.6715 −1.73646
$$423$$ 0 0
$$424$$ −16.7078 −0.811405
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −27.4255 −1.32721
$$428$$ −7.00702 −0.338697
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 31.6470 1.52438 0.762191 0.647353i $$-0.224124\pi$$
0.762191 + 0.647353i $$0.224124\pi$$
$$432$$ 0 0
$$433$$ 19.9075 0.956692 0.478346 0.878172i $$-0.341237\pi$$
0.478346 + 0.878172i $$0.341237\pi$$
$$434$$ 17.6110 0.845356
$$435$$ 0 0
$$436$$ 9.65101 0.462200
$$437$$ 21.8305 1.04429
$$438$$ 0 0
$$439$$ −23.7707 −1.13451 −0.567256 0.823542i $$-0.691995\pi$$
−0.567256 + 0.823542i $$0.691995\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −28.7235 −1.36624
$$443$$ −8.36336 −0.397355 −0.198678 0.980065i $$-0.563665\pi$$
−0.198678 + 0.980065i $$0.563665\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −67.2307 −3.18347
$$447$$ 0 0
$$448$$ 54.6577 2.58234
$$449$$ 35.2889 1.66539 0.832693 0.553734i $$-0.186798\pi$$
0.832693 + 0.553734i $$0.186798\pi$$
$$450$$ 0 0
$$451$$ −21.7998 −1.02651
$$452$$ 11.9592 0.562512
$$453$$ 0 0
$$454$$ 5.03535 0.236320
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.9534 0.605933 0.302967 0.953001i $$-0.402023\pi$$
0.302967 + 0.953001i $$0.402023\pi$$
$$458$$ 12.7263 0.594660
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −29.8908 −1.39215 −0.696076 0.717968i $$-0.745072\pi$$
−0.696076 + 0.717968i $$0.745072\pi$$
$$462$$ 0 0
$$463$$ −14.1155 −0.656002 −0.328001 0.944677i $$-0.606375\pi$$
−0.328001 + 0.944677i $$0.606375\pi$$
$$464$$ 105.124 4.88026
$$465$$ 0 0
$$466$$ 9.17233 0.424900
$$467$$ 4.00068 0.185129 0.0925647 0.995707i $$-0.470494\pi$$
0.0925647 + 0.995707i $$0.470494\pi$$
$$468$$ 0 0
$$469$$ 24.1341 1.11441
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 69.6085 3.20399
$$473$$ 11.7836 0.541810
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 33.3688 1.52946
$$477$$ 0 0
$$478$$ 66.0618 3.02159
$$479$$ 19.2413 0.879156 0.439578 0.898204i $$-0.355128\pi$$
0.439578 + 0.898204i $$0.355128\pi$$
$$480$$ 0 0
$$481$$ 29.0497 1.32455
$$482$$ 54.5835 2.48621
$$483$$ 0 0
$$484$$ 3.97725 0.180784
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −13.9219 −0.630859 −0.315430 0.948949i $$-0.602149\pi$$
−0.315430 + 0.948949i $$0.602149\pi$$
$$488$$ 64.9669 2.94091
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −14.5058 −0.654639 −0.327319 0.944914i $$-0.606145\pi$$
−0.327319 + 0.944914i $$0.606145\pi$$
$$492$$ 0 0
$$493$$ 16.9982 0.765559
$$494$$ −37.4816 −1.68638
$$495$$ 0 0
$$496$$ −22.3138 −1.00192
$$497$$ 10.1224 0.454052
$$498$$ 0 0
$$499$$ 1.26905 0.0568103 0.0284052 0.999596i $$-0.490957\pi$$
0.0284052 + 0.999596i $$0.490957\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −5.61219 −0.250484
$$503$$ 8.50684 0.379301 0.189651 0.981852i $$-0.439264\pi$$
0.189651 + 0.981852i $$0.439264\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 80.8134 3.59260
$$507$$ 0 0
$$508$$ −19.2299 −0.853188
$$509$$ −36.4250 −1.61451 −0.807254 0.590204i $$-0.799047\pi$$
−0.807254 + 0.590204i $$0.799047\pi$$
$$510$$ 0 0
$$511$$ −41.3701 −1.83011
$$512$$ 20.1538 0.890680
$$513$$ 0 0
$$514$$ 3.86829 0.170623
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −29.5690 −1.30044
$$518$$ −47.0178 −2.06585
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 31.9990 1.40190 0.700951 0.713209i $$-0.252759\pi$$
0.700951 + 0.713209i $$0.252759\pi$$
$$522$$ 0 0
$$523$$ 29.0448 1.27004 0.635020 0.772496i $$-0.280992\pi$$
0.635020 + 0.772496i $$0.280992\pi$$
$$524$$ −4.05460 −0.177126
$$525$$ 0 0
$$526$$ 68.5969 2.99097
$$527$$ −3.60807 −0.157170
$$528$$ 0 0
$$529$$ 55.2218 2.40095
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 43.5434 1.88784
$$533$$ 36.2287 1.56924
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −57.1700 −2.46937
$$537$$ 0 0
$$538$$ 85.1254 3.67001
$$539$$ −17.2625 −0.743547
$$540$$ 0 0
$$541$$ −18.2546 −0.784827 −0.392414 0.919789i $$-0.628360\pi$$
−0.392414 + 0.919789i $$0.628360\pi$$
$$542$$ −67.0353 −2.87941
$$543$$ 0 0
$$544$$ −27.8228 −1.19289
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6.92805 0.296222 0.148111 0.988971i $$-0.452681\pi$$
0.148111 + 0.988971i $$0.452681\pi$$
$$548$$ −63.1665 −2.69834
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 22.1811 0.944946
$$552$$ 0 0
$$553$$ −22.9104 −0.974249
$$554$$ −44.9474 −1.90963
$$555$$ 0 0
$$556$$ 62.2563 2.64025
$$557$$ 11.5873 0.490969 0.245484 0.969401i $$-0.421053\pi$$
0.245484 + 0.969401i $$0.421053\pi$$
$$558$$ 0 0
$$559$$ −19.5829 −0.828268
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −19.5862 −0.826192
$$563$$ 4.92004 0.207355 0.103677 0.994611i $$-0.466939\pi$$
0.103677 + 0.994611i $$0.466939\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −32.2691 −1.35637
$$567$$ 0 0
$$568$$ −23.9785 −1.00611
$$569$$ 37.5316 1.57341 0.786704 0.617331i $$-0.211786\pi$$
0.786704 + 0.617331i $$0.211786\pi$$
$$570$$ 0 0
$$571$$ −31.5956 −1.32224 −0.661118 0.750282i $$-0.729918\pi$$
−0.661118 + 0.750282i $$0.729918\pi$$
$$572$$ −99.5914 −4.16413
$$573$$ 0 0
$$574$$ −58.6371 −2.44747
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 36.4055 1.51558 0.757790 0.652499i $$-0.226279\pi$$
0.757790 + 0.652499i $$0.226279\pi$$
$$578$$ 35.7297 1.48616
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 45.0856 1.87046
$$582$$ 0 0
$$583$$ −6.98030 −0.289095
$$584$$ 97.9996 4.05525
$$585$$ 0 0
$$586$$ 65.6486 2.71192
$$587$$ −35.5061 −1.46550 −0.732748 0.680500i $$-0.761763\pi$$
−0.732748 + 0.680500i $$0.761763\pi$$
$$588$$ 0 0
$$589$$ −4.70820 −0.193998
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 59.5734 2.44845
$$593$$ −45.7086 −1.87703 −0.938513 0.345244i $$-0.887796\pi$$
−0.938513 + 0.345244i $$0.887796\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.66866 −0.273159
$$597$$ 0 0
$$598$$ −134.302 −5.49202
$$599$$ 3.58625 0.146530 0.0732652 0.997312i $$-0.476658\pi$$
0.0732652 + 0.997312i $$0.476658\pi$$
$$600$$ 0 0
$$601$$ 10.0366 0.409400 0.204700 0.978825i $$-0.434378\pi$$
0.204700 + 0.978825i $$0.434378\pi$$
$$602$$ 31.6955 1.29181
$$603$$ 0 0
$$604$$ 13.0854 0.532439
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.8691 1.41529 0.707646 0.706567i $$-0.249757\pi$$
0.707646 + 0.706567i $$0.249757\pi$$
$$608$$ −36.3063 −1.47241
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 49.1402 1.98800
$$612$$ 0 0
$$613$$ 23.8674 0.963994 0.481997 0.876173i $$-0.339912\pi$$
0.481997 + 0.876173i $$0.339912\pi$$
$$614$$ 53.2307 2.14822
$$615$$ 0 0
$$616$$ 97.8095 3.94086
$$617$$ 6.81159 0.274224 0.137112 0.990556i $$-0.456218\pi$$
0.137112 + 0.990556i $$0.456218\pi$$
$$618$$ 0 0
$$619$$ −24.7256 −0.993808 −0.496904 0.867806i $$-0.665530\pi$$
−0.496904 + 0.867806i $$0.665530\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 8.29180 0.332471
$$623$$ −40.1882 −1.61011
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 27.0172 1.07983
$$627$$ 0 0
$$628$$ 79.8749 3.18735
$$629$$ 9.63280 0.384085
$$630$$ 0 0
$$631$$ −23.4178 −0.932250 −0.466125 0.884719i $$-0.654350\pi$$
−0.466125 + 0.884719i $$0.654350\pi$$
$$632$$ 54.2713 2.15879
$$633$$ 0 0
$$634$$ −21.7998 −0.865781
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 28.6882 1.13667
$$638$$ 82.1114 3.25082
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.9591 1.06482 0.532410 0.846487i $$-0.321287\pi$$
0.532410 + 0.846487i $$0.321287\pi$$
$$642$$ 0 0
$$643$$ −9.56996 −0.377403 −0.188701 0.982035i $$-0.560428\pi$$
−0.188701 + 0.982035i $$0.560428\pi$$
$$644$$ 156.022 6.14814
$$645$$ 0 0
$$646$$ −12.4288 −0.489004
$$647$$ 20.0941 0.789981 0.394991 0.918685i $$-0.370748\pi$$
0.394991 + 0.918685i $$0.370748\pi$$
$$648$$ 0 0
$$649$$ 29.0815 1.14155
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −37.7871 −1.47986
$$653$$ 40.7211 1.59354 0.796770 0.604282i $$-0.206540\pi$$
0.796770 + 0.604282i $$0.206540\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 74.2955 2.90075
$$657$$ 0 0
$$658$$ −79.5348 −3.10059
$$659$$ 11.4254 0.445070 0.222535 0.974925i $$-0.428567\pi$$
0.222535 + 0.974925i $$0.428567\pi$$
$$660$$ 0 0
$$661$$ −33.5985 −1.30683 −0.653416 0.756999i $$-0.726665\pi$$
−0.653416 + 0.756999i $$0.726665\pi$$
$$662$$ −51.4672 −2.00033
$$663$$ 0 0
$$664$$ −106.801 −4.14468
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 79.4782 3.07741
$$668$$ −20.9659 −0.811196
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 27.1422 1.04781
$$672$$ 0 0
$$673$$ 27.7871 1.07111 0.535557 0.844499i $$-0.320102\pi$$
0.535557 + 0.844499i $$0.320102\pi$$
$$674$$ 2.40214 0.0925270
$$675$$ 0 0
$$676$$ 99.3865 3.82256
$$677$$ −31.3520 −1.20496 −0.602478 0.798135i $$-0.705820\pi$$
−0.602478 + 0.798135i $$0.705820\pi$$
$$678$$ 0 0
$$679$$ −7.55701 −0.290012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −17.4291 −0.667396
$$683$$ −0.655106 −0.0250669 −0.0125335 0.999921i $$-0.503990\pi$$
−0.0125335 + 0.999921i $$0.503990\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 18.1963 0.694736
$$687$$ 0 0
$$688$$ −40.1594 −1.53106
$$689$$ 11.6004 0.441941
$$690$$ 0 0
$$691$$ 28.9381 1.10086 0.550428 0.834883i $$-0.314464\pi$$
0.550428 + 0.834883i $$0.314464\pi$$
$$692$$ −100.590 −3.82385
$$693$$ 0 0
$$694$$ −1.80052 −0.0683469
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0133 0.455036
$$698$$ −26.7811 −1.01368
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 31.0249 1.17179 0.585896 0.810386i $$-0.300743\pi$$
0.585896 + 0.810386i $$0.300743\pi$$
$$702$$ 0 0
$$703$$ 12.5699 0.474084
$$704$$ −54.0932 −2.03871
$$705$$ 0 0
$$706$$ −19.4067 −0.730382
$$707$$ 32.2691 1.21360
$$708$$ 0 0
$$709$$ 41.2676 1.54984 0.774918 0.632062i $$-0.217791\pi$$
0.774918 + 0.632062i $$0.217791\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 95.1998 3.56776
$$713$$ −16.8702 −0.631794
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −36.5331 −1.36531
$$717$$ 0 0
$$718$$ −42.1399 −1.57265
$$719$$ 2.15581 0.0803980 0.0401990 0.999192i $$-0.487201\pi$$
0.0401990 + 0.999192i $$0.487201\pi$$
$$720$$ 0 0
$$721$$ −0.612192 −0.0227992
$$722$$ 34.3599 1.27874
$$723$$ 0 0
$$724$$ 80.9239 3.00751
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 43.8091 1.62479 0.812396 0.583107i $$-0.198163\pi$$
0.812396 + 0.583107i $$0.198163\pi$$
$$728$$ −162.548 −6.02441
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −6.49362 −0.240175
$$732$$ 0 0
$$733$$ −38.6326 −1.42693 −0.713464 0.700692i $$-0.752875\pi$$
−0.713464 + 0.700692i $$0.752875\pi$$
$$734$$ −62.5884 −2.31018
$$735$$ 0 0
$$736$$ −130.091 −4.79521
$$737$$ −23.8848 −0.879808
$$738$$ 0 0
$$739$$ 20.4743 0.753159 0.376579 0.926384i $$-0.377100\pi$$
0.376579 + 0.926384i $$0.377100\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −18.7756 −0.689275
$$743$$ 4.86490 0.178476 0.0892379 0.996010i $$-0.471557\pi$$
0.0892379 + 0.996010i $$0.471557\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.0273 0.806478
$$747$$ 0 0
$$748$$ −33.0242 −1.20748
$$749$$ −4.77799 −0.174584
$$750$$ 0 0
$$751$$ −36.7353 −1.34049 −0.670245 0.742140i $$-0.733811\pi$$
−0.670245 + 0.742140i $$0.733811\pi$$
$$752$$ 100.774 3.67484
$$753$$ 0 0
$$754$$ −136.459 −4.96955
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 13.1609 0.478340 0.239170 0.970978i $$-0.423125\pi$$
0.239170 + 0.970978i $$0.423125\pi$$
$$758$$ 72.8457 2.64587
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 17.9859 0.651987 0.325994 0.945372i $$-0.394301\pi$$
0.325994 + 0.945372i $$0.394301\pi$$
$$762$$ 0 0
$$763$$ 6.58089 0.238244
$$764$$ 73.8484 2.67174
$$765$$ 0 0
$$766$$ −41.7618 −1.50891
$$767$$ −48.3299 −1.74509
$$768$$ 0 0
$$769$$ 3.28562 0.118483 0.0592413 0.998244i $$-0.481132\pi$$
0.0592413 + 0.998244i $$0.481132\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −66.5106 −2.39377
$$773$$ 10.0265 0.360628 0.180314 0.983609i $$-0.442289\pi$$
0.180314 + 0.983609i $$0.442289\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 17.9014 0.642624
$$777$$ 0 0
$$778$$ 55.0904 1.97509
$$779$$ 15.6763 0.561661
$$780$$ 0 0
$$781$$ −10.0179 −0.358467
$$782$$ −44.5342 −1.59254
$$783$$ 0 0
$$784$$ 58.8319 2.10114
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43.1508 1.53816 0.769080 0.639153i $$-0.220715\pi$$
0.769080 + 0.639153i $$0.220715\pi$$
$$788$$ −90.9688 −3.24063
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8.15479 0.289951
$$792$$ 0 0
$$793$$ −45.1071 −1.60180
$$794$$ −27.5827 −0.978874
$$795$$ 0 0
$$796$$ −107.609 −3.81410
$$797$$ −23.2836 −0.824748 −0.412374 0.911015i $$-0.635300\pi$$
−0.412374 + 0.911015i $$0.635300\pi$$
$$798$$ 0 0
$$799$$ 16.2947 0.576466
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −1.59947 −0.0564792
$$803$$ 40.9428 1.44484
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 28.9651 1.02025
$$807$$ 0 0
$$808$$ −76.4406 −2.68917
$$809$$ −6.18914 −0.217598 −0.108799 0.994064i $$-0.534701\pi$$
−0.108799 + 0.994064i $$0.534701\pi$$
$$810$$ 0 0
$$811$$ −2.53809 −0.0891245 −0.0445623 0.999007i $$-0.514189\pi$$
−0.0445623 + 0.999007i $$0.514189\pi$$
$$812$$ 158.528 5.56325
$$813$$ 0 0
$$814$$ 46.5322 1.63095
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.47360 −0.296454
$$818$$ 53.2574 1.86210
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −44.4179 −1.55019 −0.775097 0.631842i $$-0.782299\pi$$
−0.775097 + 0.631842i $$0.782299\pi$$
$$822$$ 0 0
$$823$$ 19.6434 0.684724 0.342362 0.939568i $$-0.388773\pi$$
0.342362 + 0.939568i $$0.388773\pi$$
$$824$$ 1.45019 0.0505198
$$825$$ 0 0
$$826$$ 78.2233 2.72174
$$827$$ −31.1557 −1.08339 −0.541695 0.840575i $$-0.682217\pi$$
−0.541695 + 0.840575i $$0.682217\pi$$
$$828$$ 0 0
$$829$$ 2.46831 0.0857278 0.0428639 0.999081i $$-0.486352\pi$$
0.0428639 + 0.999081i $$0.486352\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 89.8964 3.11660
$$833$$ 9.51290 0.329602
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −43.0936 −1.49042
$$837$$ 0 0
$$838$$ 28.8151 0.995401
$$839$$ −46.0334 −1.58925 −0.794625 0.607100i $$-0.792333\pi$$
−0.794625 + 0.607100i $$0.792333\pi$$
$$840$$ 0 0
$$841$$ 51.7547 1.78464
$$842$$ 83.4065 2.87438
$$843$$ 0 0
$$844$$ 68.1579 2.34609
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.71203 0.0931866
$$848$$ 23.7894 0.816932
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 45.0400 1.54395
$$852$$ 0 0
$$853$$ −9.24700 −0.316611 −0.158306 0.987390i $$-0.550603\pi$$
−0.158306 + 0.987390i $$0.550603\pi$$
$$854$$ 73.0072 2.49825
$$855$$ 0 0
$$856$$ 11.3183 0.386853
$$857$$ 32.3505 1.10507 0.552536 0.833489i $$-0.313660\pi$$
0.552536 + 0.833489i $$0.313660\pi$$
$$858$$ 0 0
$$859$$ −14.0883 −0.480688 −0.240344 0.970688i $$-0.577260\pi$$
−0.240344 + 0.970688i $$0.577260\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −84.2448 −2.86939
$$863$$ 52.5301 1.78815 0.894073 0.447921i $$-0.147836\pi$$
0.894073 + 0.447921i $$0.147836\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −52.9940 −1.80081
$$867$$ 0 0
$$868$$ −33.6495 −1.14214
$$869$$ 22.6738 0.769155
$$870$$ 0 0
$$871$$ 39.6936 1.34497
$$872$$ −15.5891 −0.527915
$$873$$ 0 0
$$874$$ −58.1131 −1.96571
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6.58359 −0.222312 −0.111156 0.993803i $$-0.535455\pi$$
−0.111156 + 0.993803i $$0.535455\pi$$
$$878$$ 63.2779 2.13553
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −36.4594 −1.22835 −0.614174 0.789171i $$-0.710511\pi$$
−0.614174 + 0.789171i $$0.710511\pi$$
$$882$$ 0 0
$$883$$ 17.5190 0.589560 0.294780 0.955565i $$-0.404754\pi$$
0.294780 + 0.955565i $$0.404754\pi$$
$$884$$ 54.8822 1.84589
$$885$$ 0 0
$$886$$ 22.2634 0.747954
$$887$$ 24.6277 0.826917 0.413459 0.910523i $$-0.364321\pi$$
0.413459 + 0.910523i $$0.364321\pi$$
$$888$$ 0 0
$$889$$ −13.1126 −0.439782
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 128.458 4.30110
$$893$$ 21.2632 0.711544
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −43.4694 −1.45221
$$897$$ 0 0
$$898$$ −93.9397 −3.13481
$$899$$ −17.1412 −0.571689
$$900$$ 0 0
$$901$$ 3.84666 0.128151
$$902$$ 58.0315 1.93224
$$903$$ 0 0
$$904$$ −19.3175 −0.642490
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.47955 −0.148741 −0.0743705 0.997231i $$-0.523695\pi$$
−0.0743705 + 0.997231i $$0.523695\pi$$
$$908$$ −9.62108 −0.319287
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 41.9576 1.39012 0.695058 0.718953i $$-0.255379\pi$$
0.695058 + 0.718953i $$0.255379\pi$$
$$912$$ 0 0
$$913$$ −44.6199 −1.47670
$$914$$ −34.4821 −1.14057
$$915$$ 0 0
$$916$$ −24.3162 −0.803430
$$917$$ −2.76478 −0.0913010
$$918$$ 0 0
$$919$$ −37.6692 −1.24259 −0.621297 0.783575i $$-0.713394\pi$$
−0.621297 + 0.783575i $$0.713394\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 79.5698 2.62049
$$923$$ 16.6485 0.547992
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 37.5757 1.23481
$$927$$ 0 0
$$928$$ −132.180 −4.33903
$$929$$ 17.1493 0.562649 0.281325 0.959613i $$-0.409226\pi$$
0.281325 + 0.959613i $$0.409226\pi$$
$$930$$ 0 0
$$931$$ 12.4135 0.406835
$$932$$ −17.5256 −0.574071
$$933$$ 0 0
$$934$$ −10.6499 −0.348475
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 37.1944 1.21509 0.607544 0.794286i $$-0.292155\pi$$
0.607544 + 0.794286i $$0.292155\pi$$
$$938$$ −64.2453 −2.09768
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −29.5387 −0.962935 −0.481468 0.876464i $$-0.659896\pi$$
−0.481468 + 0.876464i $$0.659896\pi$$
$$942$$ 0 0
$$943$$ 56.1705 1.82916
$$944$$ −99.1119 −3.22582
$$945$$ 0 0
$$946$$ −31.3681 −1.01987
$$947$$ −34.0869 −1.10767 −0.553837 0.832625i $$-0.686837\pi$$
−0.553837 + 0.832625i $$0.686837\pi$$
$$948$$ 0 0
$$949$$ −68.0421 −2.20874
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −53.9002 −1.74692
$$953$$ 9.67588 0.313432 0.156716 0.987644i $$-0.449909\pi$$
0.156716 + 0.987644i $$0.449909\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −126.225 −4.08240
$$957$$ 0 0
$$958$$ −51.2206 −1.65486
$$959$$ −43.0724 −1.39088
$$960$$ 0 0
$$961$$ −27.3616 −0.882632
$$962$$ −77.3309 −2.49325
$$963$$ 0 0
$$964$$ −104.293 −3.35905
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −17.2537 −0.554842 −0.277421 0.960748i $$-0.589480\pi$$
−0.277421 + 0.960748i $$0.589480\pi$$
$$968$$ −6.42440 −0.206488
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1.92509 0.0617789 0.0308895 0.999523i $$-0.490166\pi$$
0.0308895 + 0.999523i $$0.490166\pi$$
$$972$$ 0 0
$$973$$ 42.4517 1.36094
$$974$$ 37.0602 1.18749
$$975$$ 0 0
$$976$$ −92.5029 −2.96095
$$977$$ −26.1185 −0.835605 −0.417803 0.908538i $$-0.637200\pi$$
−0.417803 + 0.908538i $$0.637200\pi$$
$$978$$ 0 0
$$979$$ 39.7731 1.27116
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 38.6148 1.23225
$$983$$ −9.28360 −0.296101 −0.148050 0.988980i $$-0.547300\pi$$
−0.148050 + 0.988980i $$0.547300\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −45.2494 −1.44104
$$987$$ 0 0
$$988$$ 71.6164 2.27842
$$989$$ −30.3622 −0.965461
$$990$$ 0 0
$$991$$ 41.1337 1.30666 0.653328 0.757075i $$-0.273372\pi$$
0.653328 + 0.757075i $$0.273372\pi$$
$$992$$ 28.0568 0.890805
$$993$$ 0 0
$$994$$ −26.9461 −0.854677
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 4.60496 0.145840 0.0729202 0.997338i $$-0.476768\pi$$
0.0729202 + 0.997338i $$0.476768\pi$$
$$998$$ −3.37823 −0.106936
$$999$$ 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 5625.2.a.bb.1.1 yes 8
3.2 odd 2 inner 5625.2.a.bb.1.8 yes 8
5.4 even 2 5625.2.a.z.1.8 yes 8
15.14 odd 2 5625.2.a.z.1.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.1 8 15.14 odd 2
5625.2.a.z.1.8 yes 8 5.4 even 2
5625.2.a.bb.1.1 yes 8 1.1 even 1 trivial
5625.2.a.bb.1.8 yes 8 3.2 odd 2 inner