# Properties

 Label 2450.2.a.bb.1.1 Level $2450$ Weight $2$ Character 2450.1 Self dual yes Analytic conductor $19.563$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +3.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +5.00000 q^{19} +3.00000 q^{22} -7.00000 q^{23} +5.00000 q^{26} -4.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} +5.00000 q^{38} -3.00000 q^{41} +2.00000 q^{43} +3.00000 q^{44} -7.00000 q^{46} +7.00000 q^{47} +5.00000 q^{52} +9.00000 q^{53} -4.00000 q^{58} +4.00000 q^{59} -6.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{67} +2.00000 q^{68} -6.00000 q^{71} -3.00000 q^{72} +16.0000 q^{73} +1.00000 q^{74} +5.00000 q^{76} +14.0000 q^{79} +9.00000 q^{81} -3.00000 q^{82} +6.00000 q^{83} +2.00000 q^{86} +3.00000 q^{88} -2.00000 q^{89} -7.00000 q^{92} +7.00000 q^{94} +12.0000 q^{97} -9.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000 0.639602
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.00000 0.980581
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −7.00000 −1.03209
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.00000 0.693375
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 16.0000 1.87266 0.936329 0.351123i $$-0.114200\pi$$
0.936329 + 0.351123i $$0.114200\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 5.00000 0.573539
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ −3.00000 −0.331295
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −7.00000 −0.729800
$$93$$ 0 0
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 0 0
$$99$$ −9.00000 −0.904534
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 5.00000 0.490290
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ −15.0000 −1.38675
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −6.00000 −0.543214
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −6.00000 −0.503509
$$143$$ 15.0000 1.25436
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 16.0000 1.32417
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −6.00000 −0.488273 −0.244137 0.969741i $$-0.578505\pi$$
−0.244137 + 0.969741i $$0.578505\pi$$
$$152$$ 5.00000 0.405554
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.00000 0.718278 0.359139 0.933284i $$-0.383070\pi$$
0.359139 + 0.933284i $$0.383070\pi$$
$$158$$ 14.0000 1.11378
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −15.0000 −1.16073 −0.580367 0.814355i $$-0.697091\pi$$
−0.580367 + 0.814355i $$0.697091\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −15.0000 −1.14708
$$172$$ 2.00000 0.152499
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ 13.0000 0.971666 0.485833 0.874052i $$-0.338516\pi$$
0.485833 + 0.874052i $$0.338516\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −7.00000 −0.516047
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ 7.00000 0.510527
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −5.00000 −0.356235 −0.178118 0.984009i $$-0.557001\pi$$
−0.178118 + 0.984009i $$0.557001\pi$$
$$198$$ −9.00000 −0.639602
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 21.0000 1.45960
$$208$$ 5.00000 0.346688
$$209$$ 15.0000 1.03757
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 9.00000 0.618123
$$213$$ 0 0
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10.0000 0.672673
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ −15.0000 −0.980581
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 9.00000 0.579741 0.289870 0.957066i $$-0.406388\pi$$
0.289870 + 0.957066i $$0.406388\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 25.0000 1.59071
$$248$$ 2.00000 0.127000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5.00000 −0.315597 −0.157799 0.987471i $$-0.550440\pi$$
−0.157799 + 0.987471i $$0.550440\pi$$
$$252$$ 0 0
$$253$$ −21.0000 −1.32026
$$254$$ 7.00000 0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 1.00000 0.0617802
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 9.00000 0.536895 0.268447 0.963294i $$-0.413489\pi$$
0.268447 + 0.963294i $$0.413489\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 15.0000 0.886969
$$287$$ 0 0
$$288$$ −3.00000 −0.176777
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 16.0000 0.936329
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ −35.0000 −2.02410
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −6.00000 −0.345261
$$303$$ 0 0
$$304$$ 5.00000 0.286770
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 9.00000 0.507899
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 10.0000 0.556415
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 0 0
$$328$$ −3.00000 −0.165647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5.00000 0.274825 0.137412 0.990514i $$-0.456121\pi$$
0.137412 + 0.990514i $$0.456121\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −3.00000 −0.164399
$$334$$ −15.0000 −0.820763
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ −15.0000 −0.811107
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 12.0000 0.642345 0.321173 0.947021i $$-0.395923\pi$$
0.321173 + 0.947021i $$0.395923\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 13.0000 0.687071
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −26.0000 −1.36653
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −13.0000 −0.678594 −0.339297 0.940679i $$-0.610189\pi$$
−0.339297 + 0.940679i $$0.610189\pi$$
$$368$$ −7.00000 −0.364900
$$369$$ 9.00000 0.468521
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ −20.0000 −1.03005
$$378$$ 0 0
$$379$$ −29.0000 −1.48963 −0.744815 0.667271i $$-0.767462\pi$$
−0.744815 + 0.667271i $$0.767462\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −20.0000 −1.02329
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −6.00000 −0.304997
$$388$$ 12.0000 0.609208
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −14.0000 −0.708010
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −5.00000 −0.251896
$$395$$ 0 0
$$396$$ −9.00000 −0.452267
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −18.0000 −0.902258
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.0000 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$402$$ 0 0
$$403$$ 10.0000 0.498135
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.00000 0.148704
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 21.0000 1.03209
$$415$$ 0 0
$$416$$ 5.00000 0.245145
$$417$$ 0 0
$$418$$ 15.0000 0.733674
$$419$$ −35.0000 −1.70986 −0.854931 0.518742i $$-0.826401\pi$$
−0.854931 + 0.518742i $$0.826401\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ −9.00000 −0.438113
$$423$$ −21.0000 −1.02105
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −16.0000 −0.773389
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.00000 0.0963366 0.0481683 0.998839i $$-0.484662\pi$$
0.0481683 + 0.998839i $$0.484662\pi$$
$$432$$ 0 0
$$433$$ −28.0000 −1.34559 −0.672797 0.739827i $$-0.734907\pi$$
−0.672797 + 0.739827i $$0.734907\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −35.0000 −1.67428
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 10.0000 0.475651
$$443$$ 30.0000 1.42534 0.712672 0.701498i $$-0.247485\pi$$
0.712672 + 0.701498i $$0.247485\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.00000 0.235965 0.117982 0.993016i $$-0.462357\pi$$
0.117982 + 0.993016i $$0.462357\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 32.0000 1.49039 0.745194 0.666847i $$-0.232357\pi$$
0.745194 + 0.666847i $$0.232357\pi$$
$$462$$ 0 0
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −8.00000 −0.370593
$$467$$ −34.0000 −1.57333 −0.786666 0.617379i $$-0.788195\pi$$
−0.786666 + 0.617379i $$0.788195\pi$$
$$468$$ −15.0000 −0.693375
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000 0.184115
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −27.0000 −1.23625
$$478$$ 20.0000 0.914779
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ 9.00000 0.409939
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ −8.00000 −0.360302
$$494$$ 25.0000 1.12480
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −5.00000 −0.223161
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −21.0000 −0.933564
$$507$$ 0 0
$$508$$ 7.00000 0.310575
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 21.0000 0.923579
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 27.0000 1.18289 0.591446 0.806345i $$-0.298557\pi$$
0.591446 + 0.806345i $$0.298557\pi$$
$$522$$ 12.0000 0.525226
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 1.00000 0.0436852
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −15.0000 −0.649722
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ −24.0000 −1.03089
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −26.0000 −1.11168 −0.555840 0.831289i $$-0.687603\pi$$
−0.555840 + 0.831289i $$0.687603\pi$$
$$548$$ −8.00000 −0.341743
$$549$$ 18.0000 0.768221
$$550$$ 0 0
$$551$$ −20.0000 −0.852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 23.0000 0.974541 0.487271 0.873251i $$-0.337993\pi$$
0.487271 + 0.873251i $$0.337993\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 10.0000 0.422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 9.00000 0.379642
$$563$$ −2.00000 −0.0842900 −0.0421450 0.999112i $$-0.513419\pi$$
−0.0421450 + 0.999112i $$0.513419\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 15.0000 0.627182
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 27.0000 1.11823
$$584$$ 16.0000 0.662085
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ −34.0000 −1.40333 −0.701665 0.712507i $$-0.747560\pi$$
−0.701665 + 0.712507i $$0.747560\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000 0.0410997
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ −35.0000 −1.43126
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ −6.00000 −0.244339
$$604$$ −6.00000 −0.244137
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 13.0000 0.527654 0.263827 0.964570i $$-0.415015\pi$$
0.263827 + 0.964570i $$0.415015\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 35.0000 1.41595
$$612$$ −6.00000 −0.242536
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ −19.0000 −0.763674 −0.381837 0.924230i $$-0.624709\pi$$
−0.381837 + 0.924230i $$0.624709\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −6.00000 −0.240578
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 22.0000 0.879297
$$627$$ 0 0
$$628$$ 9.00000 0.359139
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ −18.0000 −0.716569 −0.358284 0.933613i $$-0.616638\pi$$
−0.358284 + 0.933613i $$0.616638\pi$$
$$632$$ 14.0000 0.556890
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −12.0000 −0.475085
$$639$$ 18.0000 0.712069
$$640$$ 0 0
$$641$$ −5.00000 −0.197488 −0.0987441 0.995113i $$-0.531483\pi$$
−0.0987441 + 0.995113i $$0.531483\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 10.0000 0.393445
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 3.00000 0.117399 0.0586995 0.998276i $$-0.481305\pi$$
0.0586995 + 0.998276i $$0.481305\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ −48.0000 −1.87266
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 5.00000 0.194331
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ 28.0000 1.08416
$$668$$ −15.0000 −0.580367
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −18.0000 −0.694882
$$672$$ 0 0
$$673$$ 32.0000 1.23351 0.616755 0.787155i $$-0.288447\pi$$
0.616755 + 0.787155i $$0.288447\pi$$
$$674$$ 10.0000 0.385186
$$675$$ 0 0
$$676$$ 12.0000 0.461538
$$677$$ 17.0000 0.653363 0.326682 0.945134i $$-0.394070\pi$$
0.326682 + 0.945134i $$0.394070\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 6.00000 0.229752
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ −15.0000 −0.573539
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ 45.0000 1.71436
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ −9.00000 −0.342129
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ 12.0000 0.454207
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.0000 0.982006 0.491003 0.871158i $$-0.336630\pi$$
0.491003 + 0.871158i $$0.336630\pi$$
$$702$$ 0 0
$$703$$ 5.00000 0.188579
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 12.0000 0.450669 0.225335 0.974281i $$-0.427652\pi$$
0.225335 + 0.974281i $$0.427652\pi$$
$$710$$ 0 0
$$711$$ −42.0000 −1.57512
$$712$$ −2.00000 −0.0749532
$$713$$ −14.0000 −0.524304
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 13.0000 0.485833
$$717$$ 0 0
$$718$$ 16.0000 0.597115
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 6.00000 0.223297
$$723$$ 0 0
$$724$$ −26.0000 −0.966282
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 29.0000 1.07555 0.537775 0.843088i $$-0.319265\pi$$
0.537775 + 0.843088i $$0.319265\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −41.0000 −1.51437 −0.757185 0.653201i $$-0.773426\pi$$
−0.757185 + 0.653201i $$0.773426\pi$$
$$734$$ −13.0000 −0.479839
$$735$$ 0 0
$$736$$ −7.00000 −0.258023
$$737$$ 6.00000 0.221013
$$738$$ 9.00000 0.331295
$$739$$ −29.0000 −1.06678 −0.533391 0.845869i $$-0.679083\pi$$
−0.533391 + 0.845869i $$0.679083\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ −18.0000 −0.658586
$$748$$ 6.00000 0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.0000 1.02173 0.510867 0.859660i $$-0.329324\pi$$
0.510867 + 0.859660i $$0.329324\pi$$
$$752$$ 7.00000 0.255264
$$753$$ 0 0
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000 1.52652 0.763258 0.646094i $$-0.223599\pi$$
0.763258 + 0.646094i $$0.223599\pi$$
$$758$$ −29.0000 −1.05333
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.00000 −0.0362500 −0.0181250 0.999836i $$-0.505770\pi$$
−0.0181250 + 0.999836i $$0.505770\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −20.0000 −0.723575
$$765$$ 0 0
$$766$$ −21.0000 −0.758761
$$767$$ 20.0000 0.722158
$$768$$ 0 0
$$769$$ 29.0000 1.04577 0.522883 0.852404i $$-0.324856\pi$$
0.522883 + 0.852404i $$0.324856\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ −45.0000 −1.61854 −0.809269 0.587439i $$-0.800136\pi$$
−0.809269 + 0.587439i $$0.800136\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ 12.0000 0.430775
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ −15.0000 −0.537431
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ −14.0000 −0.500639
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.0000 −0.641631 −0.320815 0.947142i $$-0.603957\pi$$
−0.320815 + 0.947142i $$0.603957\pi$$
$$788$$ −5.00000 −0.178118
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −9.00000 −0.319801
$$793$$ −30.0000 −1.06533
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −18.0000 −0.637993
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 14.0000 0.495284
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ −15.0000 −0.529668
$$803$$ 48.0000 1.69388
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 10.0000 0.352235
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −5.00000 −0.175791 −0.0878953 0.996130i $$-0.528014\pi$$
−0.0878953 + 0.996130i $$0.528014\pi$$
$$810$$ 0 0
$$811$$ 33.0000 1.15879 0.579393 0.815048i $$-0.303290\pi$$
0.579393 + 0.815048i $$0.303290\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 3.00000 0.105150
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ 14.0000 0.489499
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ −24.0000 −0.836587 −0.418294 0.908312i $$-0.637372\pi$$
−0.418294 + 0.908312i $$0.637372\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −22.0000 −0.765015 −0.382507 0.923952i $$-0.624939\pi$$
−0.382507 + 0.923952i $$0.624939\pi$$
$$828$$ 21.0000 0.729800
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 5.00000 0.173344
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 15.0000 0.518786
$$837$$ 0 0
$$838$$ −35.0000 −1.20905
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −20.0000 −0.689246
$$843$$ 0 0
$$844$$ −9.00000 −0.309793
$$845$$ 0 0
$$846$$ −21.0000 −0.721995
$$847$$ 0 0
$$848$$ 9.00000 0.309061
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −7.00000 −0.239957
$$852$$ 0 0
$$853$$ 43.0000 1.47229 0.736146 0.676823i $$-0.236644\pi$$
0.736146 + 0.676823i $$0.236644\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −16.0000 −0.546869
$$857$$ −8.00000 −0.273275 −0.136637 0.990621i $$-0.543630\pi$$
−0.136637 + 0.990621i $$0.543630\pi$$
$$858$$ 0 0
$$859$$ 12.0000 0.409435 0.204717 0.978821i $$-0.434372\pi$$
0.204717 + 0.978821i $$0.434372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 2.00000 0.0681203
$$863$$ −11.0000 −0.374444 −0.187222 0.982318i $$-0.559948\pi$$
−0.187222 + 0.982318i $$0.559948\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −28.0000 −0.951479
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 42.0000 1.42475
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 2.00000 0.0677285
$$873$$ −36.0000 −1.21842
$$874$$ −35.0000 −1.18389
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31.0000 1.04680 0.523398 0.852088i $$-0.324664\pi$$
0.523398 + 0.852088i $$0.324664\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 15.0000 0.505363 0.252681 0.967550i $$-0.418688\pi$$
0.252681 + 0.967550i $$0.418688\pi$$
$$882$$ 0 0
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 10.0000 0.336336
$$885$$ 0 0
$$886$$ 30.0000 1.00787
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 27.0000 0.904534
$$892$$ 8.00000 0.267860
$$893$$ 35.0000 1.17123
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 5.00000 0.166852
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ −9.00000 −0.299667
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ 6.00000 0.199117
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.00000 0.0662630 0.0331315 0.999451i $$-0.489452\pi$$
0.0331315 + 0.999451i $$0.489452\pi$$
$$912$$ 0 0
$$913$$ 18.0000 0.595713
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 32.0000 1.05386
$$923$$ −30.0000 −0.987462
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −17.0000 −0.558655
$$927$$ −24.0000 −0.788263
$$928$$ −4.00000 −0.131306
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −8.00000 −0.262049
$$933$$ 0 0
$$934$$ −34.0000 −1.11251
$$935$$ 0 0
$$936$$ −15.0000 −0.490290
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 28.0000 0.912774 0.456387 0.889781i $$-0.349143\pi$$
0.456387 + 0.889781i $$0.349143\pi$$
$$942$$ 0 0
$$943$$ 21.0000 0.683854
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 6.00000 0.195077
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ 80.0000 2.59691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ −27.0000 −0.874157
$$955$$ 0 0
$$956$$ 20.0000 0.646846
$$957$$ 0 0
$$958$$ −36.0000 −1.16311
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 5.00000 0.161206
$$963$$ 48.0000 1.54678
$$964$$ 9.00000 0.289870
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 33.0000 1.05902 0.529510 0.848304i $$-0.322376\pi$$
0.529510 + 0.848304i $$0.322376\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 24.0000 0.765871
$$983$$ 29.0000 0.924956 0.462478 0.886631i $$-0.346960\pi$$
0.462478 + 0.886631i $$0.346960\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 25.0000 0.795356
$$989$$ −14.0000 −0.445174
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 2450.2.a.bb.1.1 1
5.2 odd 4 490.2.c.d.99.2 2
5.3 odd 4 490.2.c.d.99.1 2
5.4 even 2 2450.2.a.j.1.1 1
7.3 odd 6 350.2.e.c.51.1 2
7.5 odd 6 350.2.e.c.151.1 2
7.6 odd 2 2450.2.a.ba.1.1 1
35.2 odd 12 490.2.i.a.459.2 4
35.3 even 12 70.2.i.b.9.2 yes 4
35.12 even 12 70.2.i.b.39.2 yes 4
35.13 even 4 490.2.c.a.99.1 2
35.17 even 12 70.2.i.b.9.1 4
35.18 odd 12 490.2.i.a.79.2 4
35.19 odd 6 350.2.e.j.151.1 2
35.23 odd 12 490.2.i.a.459.1 4
35.24 odd 6 350.2.e.j.51.1 2
35.27 even 4 490.2.c.a.99.2 2
35.32 odd 12 490.2.i.a.79.1 4
35.33 even 12 70.2.i.b.39.1 yes 4
35.34 odd 2 2450.2.a.k.1.1 1
105.17 odd 12 630.2.u.a.289.2 4
105.38 odd 12 630.2.u.a.289.1 4
105.47 odd 12 630.2.u.a.109.1 4
105.68 odd 12 630.2.u.a.109.2 4
140.3 odd 12 560.2.bw.d.289.1 4
140.47 odd 12 560.2.bw.d.529.1 4
140.87 odd 12 560.2.bw.d.289.2 4
140.103 odd 12 560.2.bw.d.529.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.i.b.9.1 4 35.17 even 12
70.2.i.b.9.2 yes 4 35.3 even 12
70.2.i.b.39.1 yes 4 35.33 even 12
70.2.i.b.39.2 yes 4 35.12 even 12
350.2.e.c.51.1 2 7.3 odd 6
350.2.e.c.151.1 2 7.5 odd 6
350.2.e.j.51.1 2 35.24 odd 6
350.2.e.j.151.1 2 35.19 odd 6
490.2.c.a.99.1 2 35.13 even 4
490.2.c.a.99.2 2 35.27 even 4
490.2.c.d.99.1 2 5.3 odd 4
490.2.c.d.99.2 2 5.2 odd 4
490.2.i.a.79.1 4 35.32 odd 12
490.2.i.a.79.2 4 35.18 odd 12
490.2.i.a.459.1 4 35.23 odd 12
490.2.i.a.459.2 4 35.2 odd 12
560.2.bw.d.289.1 4 140.3 odd 12
560.2.bw.d.289.2 4 140.87 odd 12
560.2.bw.d.529.1 4 140.47 odd 12
560.2.bw.d.529.2 4 140.103 odd 12
630.2.u.a.109.1 4 105.47 odd 12
630.2.u.a.109.2 4 105.68 odd 12
630.2.u.a.289.1 4 105.38 odd 12
630.2.u.a.289.2 4 105.17 odd 12
2450.2.a.j.1.1 1 5.4 even 2
2450.2.a.k.1.1 1 35.34 odd 2
2450.2.a.ba.1.1 1 7.6 odd 2
2450.2.a.bb.1.1 1 1.1 even 1 trivial