# Properties

 Label 7448.2.a.be.1.1 Level $7448$ Weight $2$ Character 7448.1 Self dual yes Analytic conductor $59.473$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7448 = 2^{3} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4725794254$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 7448.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -5.27492 q^{11} -2.00000 q^{13} +2.00000 q^{15} +0.274917 q^{17} +1.00000 q^{19} +7.54983 q^{23} -4.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} +4.00000 q^{31} -10.5498 q^{33} +2.00000 q^{37} -4.00000 q^{39} -10.5498 q^{41} -1.00000 q^{43} +1.00000 q^{45} -5.27492 q^{47} +0.549834 q^{51} +8.54983 q^{53} -5.27492 q^{55} +2.00000 q^{57} +4.00000 q^{59} -4.72508 q^{61} -2.00000 q^{65} +15.0997 q^{69} -4.54983 q^{71} -11.2749 q^{73} -8.00000 q^{75} -14.5498 q^{79} -11.0000 q^{81} +3.54983 q^{83} +0.274917 q^{85} -8.00000 q^{87} -1.45017 q^{89} +8.00000 q^{93} +1.00000 q^{95} +6.54983 q^{97} -5.27492 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} + O(q^{10})$$ $$2 q + 4 q^{3} + 2 q^{5} + 2 q^{9} - 3 q^{11} - 4 q^{13} + 4 q^{15} - 7 q^{17} + 2 q^{19} - 8 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{31} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 6 q^{41} - 2 q^{43} + 2 q^{45} - 3 q^{47} - 14 q^{51} + 2 q^{53} - 3 q^{55} + 4 q^{57} + 8 q^{59} - 17 q^{61} - 4 q^{65} + 6 q^{71} - 15 q^{73} - 16 q^{75} - 14 q^{79} - 22 q^{81} - 8 q^{83} - 7 q^{85} - 16 q^{87} - 18 q^{89} + 16 q^{93} + 2 q^{95} - 2 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.27492 −1.59045 −0.795224 0.606316i $$-0.792647\pi$$
−0.795224 + 0.606316i $$0.792647\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 0.274917 0.0666772 0.0333386 0.999444i $$-0.489386\pi$$
0.0333386 + 0.999444i $$0.489386\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.54983 1.57425 0.787125 0.616794i $$-0.211569\pi$$
0.787125 + 0.616794i $$0.211569\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −10.5498 −1.83649
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −10.5498 −1.64761 −0.823804 0.566875i $$-0.808152\pi$$
−0.823804 + 0.566875i $$0.808152\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −5.27492 −0.769426 −0.384713 0.923036i $$-0.625700\pi$$
−0.384713 + 0.923036i $$0.625700\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.549834 0.0769922
$$52$$ 0 0
$$53$$ 8.54983 1.17441 0.587205 0.809438i $$-0.300228\pi$$
0.587205 + 0.809438i $$0.300228\pi$$
$$54$$ 0 0
$$55$$ −5.27492 −0.711270
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −4.72508 −0.604985 −0.302492 0.953152i $$-0.597819\pi$$
−0.302492 + 0.953152i $$0.597819\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 15.0997 1.81779
$$70$$ 0 0
$$71$$ −4.54983 −0.539966 −0.269983 0.962865i $$-0.587018\pi$$
−0.269983 + 0.962865i $$0.587018\pi$$
$$72$$ 0 0
$$73$$ −11.2749 −1.31963 −0.659815 0.751428i $$-0.729365\pi$$
−0.659815 + 0.751428i $$0.729365\pi$$
$$74$$ 0 0
$$75$$ −8.00000 −0.923760
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.5498 −1.63698 −0.818492 0.574518i $$-0.805190\pi$$
−0.818492 + 0.574518i $$0.805190\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 3.54983 0.389645 0.194822 0.980839i $$-0.437587\pi$$
0.194822 + 0.980839i $$0.437587\pi$$
$$84$$ 0 0
$$85$$ 0.274917 0.0298190
$$86$$ 0 0
$$87$$ −8.00000 −0.857690
$$88$$ 0 0
$$89$$ −1.45017 −0.153717 −0.0768586 0.997042i $$-0.524489\pi$$
−0.0768586 + 0.997042i $$0.524489\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 6.54983 0.665035 0.332517 0.943097i $$-0.392102\pi$$
0.332517 + 0.943097i $$0.392102\pi$$
$$98$$ 0 0
$$99$$ −5.27492 −0.530149
$$100$$ 0 0
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.5498 −1.79328 −0.896640 0.442760i $$-0.853999\pi$$
−0.896640 + 0.442760i $$0.853999\pi$$
$$108$$ 0 0
$$109$$ 4.54983 0.435795 0.217898 0.975972i $$-0.430080\pi$$
0.217898 + 0.975972i $$0.430080\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 3.45017 0.324564 0.162282 0.986744i $$-0.448115\pi$$
0.162282 + 0.986744i $$0.448115\pi$$
$$114$$ 0 0
$$115$$ 7.54983 0.704026
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 16.8248 1.52952
$$122$$ 0 0
$$123$$ −21.0997 −1.90249
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 15.0997 1.33988 0.669939 0.742416i $$-0.266320\pi$$
0.669939 + 0.742416i $$0.266320\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −6.82475 −0.596281 −0.298141 0.954522i $$-0.596366\pi$$
−0.298141 + 0.954522i $$0.596366\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ 11.2749 0.963281 0.481641 0.876369i $$-0.340041\pi$$
0.481641 + 0.876369i $$0.340041\pi$$
$$138$$ 0 0
$$139$$ −11.8248 −1.00296 −0.501481 0.865169i $$-0.667211\pi$$
−0.501481 + 0.865169i $$0.667211\pi$$
$$140$$ 0 0
$$141$$ −10.5498 −0.888456
$$142$$ 0 0
$$143$$ 10.5498 0.882221
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −16.0997 −1.31894 −0.659468 0.751733i $$-0.729218\pi$$
−0.659468 + 0.751733i $$0.729218\pi$$
$$150$$ 0 0
$$151$$ 6.54983 0.533018 0.266509 0.963832i $$-0.414130\pi$$
0.266509 + 0.963832i $$0.414130\pi$$
$$152$$ 0 0
$$153$$ 0.274917 0.0222257
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −14.0997 −1.12528 −0.562638 0.826703i $$-0.690214\pi$$
−0.562638 + 0.826703i $$0.690214\pi$$
$$158$$ 0 0
$$159$$ 17.0997 1.35609
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −21.5498 −1.68791 −0.843957 0.536411i $$-0.819780\pi$$
−0.843957 + 0.536411i $$0.819780\pi$$
$$164$$ 0 0
$$165$$ −10.5498 −0.821303
$$166$$ 0 0
$$167$$ −13.0997 −1.01368 −0.506841 0.862039i $$-0.669187\pi$$
−0.506841 + 0.862039i $$0.669187\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −20.5498 −1.56237 −0.781187 0.624296i $$-0.785386\pi$$
−0.781187 + 0.624296i $$0.785386\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 8.00000 0.601317
$$178$$ 0 0
$$179$$ −15.0997 −1.12860 −0.564301 0.825569i $$-0.690854\pi$$
−0.564301 + 0.825569i $$0.690854\pi$$
$$180$$ 0 0
$$181$$ −0.549834 −0.0408689 −0.0204344 0.999791i $$-0.506505\pi$$
−0.0204344 + 0.999791i $$0.506505\pi$$
$$182$$ 0 0
$$183$$ −9.45017 −0.698576
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −1.45017 −0.106047
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −21.0000 −1.51951 −0.759753 0.650211i $$-0.774680\pi$$
−0.759753 + 0.650211i $$0.774680\pi$$
$$192$$ 0 0
$$193$$ −3.09967 −0.223119 −0.111560 0.993758i $$-0.535585\pi$$
−0.111560 + 0.993758i $$0.535585\pi$$
$$194$$ 0 0
$$195$$ −4.00000 −0.286446
$$196$$ 0 0
$$197$$ 12.0997 0.862066 0.431033 0.902336i $$-0.358149\pi$$
0.431033 + 0.902336i $$0.358149\pi$$
$$198$$ 0 0
$$199$$ −3.00000 −0.212664 −0.106332 0.994331i $$-0.533911\pi$$
−0.106332 + 0.994331i $$0.533911\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.5498 −0.736832
$$206$$ 0 0
$$207$$ 7.54983 0.524750
$$208$$ 0 0
$$209$$ −5.27492 −0.364874
$$210$$ 0 0
$$211$$ −25.6495 −1.76578 −0.882892 0.469576i $$-0.844407\pi$$
−0.882892 + 0.469576i $$0.844407\pi$$
$$212$$ 0 0
$$213$$ −9.09967 −0.623499
$$214$$ 0 0
$$215$$ −1.00000 −0.0681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −22.5498 −1.52378
$$220$$ 0 0
$$221$$ −0.549834 −0.0369859
$$222$$ 0 0
$$223$$ 17.0997 1.14508 0.572539 0.819877i $$-0.305958\pi$$
0.572539 + 0.819877i $$0.305958\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ 9.45017 0.627230 0.313615 0.949550i $$-0.398460\pi$$
0.313615 + 0.949550i $$0.398460\pi$$
$$228$$ 0 0
$$229$$ 16.2749 1.07548 0.537738 0.843112i $$-0.319279\pi$$
0.537738 + 0.843112i $$0.319279\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 4.27492 0.280059 0.140030 0.990147i $$-0.455280\pi$$
0.140030 + 0.990147i $$0.455280\pi$$
$$234$$ 0 0
$$235$$ −5.27492 −0.344098
$$236$$ 0 0
$$237$$ −29.0997 −1.89023
$$238$$ 0 0
$$239$$ 23.3746 1.51198 0.755988 0.654585i $$-0.227157\pi$$
0.755988 + 0.654585i $$0.227157\pi$$
$$240$$ 0 0
$$241$$ −10.5498 −0.679575 −0.339787 0.940502i $$-0.610355\pi$$
−0.339787 + 0.940502i $$0.610355\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 7.09967 0.449923
$$250$$ 0 0
$$251$$ 28.0997 1.77364 0.886818 0.462119i $$-0.152911\pi$$
0.886818 + 0.462119i $$0.152911\pi$$
$$252$$ 0 0
$$253$$ −39.8248 −2.50376
$$254$$ 0 0
$$255$$ 0.549834 0.0344320
$$256$$ 0 0
$$257$$ 22.5498 1.40662 0.703310 0.710883i $$-0.251705\pi$$
0.703310 + 0.710883i $$0.251705\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ 22.8248 1.40743 0.703717 0.710480i $$-0.251522\pi$$
0.703717 + 0.710480i $$0.251522\pi$$
$$264$$ 0 0
$$265$$ 8.54983 0.525212
$$266$$ 0 0
$$267$$ −2.90033 −0.177497
$$268$$ 0 0
$$269$$ 24.5498 1.49683 0.748415 0.663231i $$-0.230815\pi$$
0.748415 + 0.663231i $$0.230815\pi$$
$$270$$ 0 0
$$271$$ −17.5498 −1.06608 −0.533038 0.846091i $$-0.678950\pi$$
−0.533038 + 0.846091i $$0.678950\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 21.0997 1.27236
$$276$$ 0 0
$$277$$ −0.725083 −0.0435660 −0.0217830 0.999763i $$-0.506934\pi$$
−0.0217830 + 0.999763i $$0.506934\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ 0.549834 0.0328004 0.0164002 0.999866i $$-0.494779\pi$$
0.0164002 + 0.999866i $$0.494779\pi$$
$$282$$ 0 0
$$283$$ 17.0000 1.01055 0.505273 0.862960i $$-0.331392\pi$$
0.505273 + 0.862960i $$0.331392\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.9244 −0.995554
$$290$$ 0 0
$$291$$ 13.0997 0.767916
$$292$$ 0 0
$$293$$ −27.6495 −1.61530 −0.807651 0.589661i $$-0.799261\pi$$
−0.807651 + 0.589661i $$0.799261\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ 21.0997 1.22433
$$298$$ 0 0
$$299$$ −15.0997 −0.873236
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ −4.72508 −0.270557
$$306$$ 0 0
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ −20.0000 −1.13776
$$310$$ 0 0
$$311$$ 10.2749 0.582637 0.291319 0.956626i $$-0.405906\pi$$
0.291319 + 0.956626i $$0.405906\pi$$
$$312$$ 0 0
$$313$$ 31.1993 1.76349 0.881745 0.471726i $$-0.156369\pi$$
0.881745 + 0.471726i $$0.156369\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.09967 0.511088 0.255544 0.966797i $$-0.417745\pi$$
0.255544 + 0.966797i $$0.417745\pi$$
$$318$$ 0 0
$$319$$ 21.0997 1.18135
$$320$$ 0 0
$$321$$ −37.0997 −2.07070
$$322$$ 0 0
$$323$$ 0.274917 0.0152968
$$324$$ 0 0
$$325$$ 8.00000 0.443760
$$326$$ 0 0
$$327$$ 9.09967 0.503213
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −14.0000 −0.769510 −0.384755 0.923019i $$-0.625714\pi$$
−0.384755 + 0.923019i $$0.625714\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 9.09967 0.495691 0.247845 0.968800i $$-0.420278\pi$$
0.247845 + 0.968800i $$0.420278\pi$$
$$338$$ 0 0
$$339$$ 6.90033 0.374775
$$340$$ 0 0
$$341$$ −21.0997 −1.14261
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 15.0997 0.812939
$$346$$ 0 0
$$347$$ 27.1993 1.46014 0.730068 0.683374i $$-0.239488\pi$$
0.730068 + 0.683374i $$0.239488\pi$$
$$348$$ 0 0
$$349$$ 9.37459 0.501810 0.250905 0.968012i $$-0.419272\pi$$
0.250905 + 0.968012i $$0.419272\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ 7.09967 0.377877 0.188939 0.981989i $$-0.439495\pi$$
0.188939 + 0.981989i $$0.439495\pi$$
$$354$$ 0 0
$$355$$ −4.54983 −0.241480
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −25.2749 −1.33396 −0.666980 0.745076i $$-0.732413\pi$$
−0.666980 + 0.745076i $$0.732413\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 33.6495 1.76614
$$364$$ 0 0
$$365$$ −11.2749 −0.590156
$$366$$ 0 0
$$367$$ −34.1993 −1.78519 −0.892595 0.450858i $$-0.851118\pi$$
−0.892595 + 0.450858i $$0.851118\pi$$
$$368$$ 0 0
$$369$$ −10.5498 −0.549202
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 8.54983 0.442694 0.221347 0.975195i $$-0.428955\pi$$
0.221347 + 0.975195i $$0.428955\pi$$
$$374$$ 0 0
$$375$$ −18.0000 −0.929516
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ −27.0997 −1.39202 −0.696008 0.718034i $$-0.745042\pi$$
−0.696008 + 0.718034i $$0.745042\pi$$
$$380$$ 0 0
$$381$$ 30.1993 1.54716
$$382$$ 0 0
$$383$$ −22.0000 −1.12415 −0.562074 0.827087i $$-0.689996\pi$$
−0.562074 + 0.827087i $$0.689996\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 0 0
$$389$$ 15.1752 0.769416 0.384708 0.923038i $$-0.374302\pi$$
0.384708 + 0.923038i $$0.374302\pi$$
$$390$$ 0 0
$$391$$ 2.07558 0.104967
$$392$$ 0 0
$$393$$ −13.6495 −0.688526
$$394$$ 0 0
$$395$$ −14.5498 −0.732082
$$396$$ 0 0
$$397$$ 28.2749 1.41908 0.709539 0.704666i $$-0.248903\pi$$
0.709539 + 0.704666i $$0.248903\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9.09967 −0.454416 −0.227208 0.973846i $$-0.572960\pi$$
−0.227208 + 0.973846i $$0.572960\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ 0 0
$$407$$ −10.5498 −0.522936
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 22.5498 1.11230
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 3.54983 0.174255
$$416$$ 0 0
$$417$$ −23.6495 −1.15812
$$418$$ 0 0
$$419$$ −4.45017 −0.217405 −0.108702 0.994074i $$-0.534670\pi$$
−0.108702 + 0.994074i $$0.534670\pi$$
$$420$$ 0 0
$$421$$ 3.45017 0.168151 0.0840754 0.996459i $$-0.473206\pi$$
0.0840754 + 0.996459i $$0.473206\pi$$
$$422$$ 0 0
$$423$$ −5.27492 −0.256475
$$424$$ 0 0
$$425$$ −1.09967 −0.0533418
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 21.0997 1.01870
$$430$$ 0 0
$$431$$ −21.0997 −1.01634 −0.508168 0.861258i $$-0.669677\pi$$
−0.508168 + 0.861258i $$0.669677\pi$$
$$432$$ 0 0
$$433$$ 5.09967 0.245074 0.122537 0.992464i $$-0.460897\pi$$
0.122537 + 0.992464i $$0.460897\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ 0 0
$$437$$ 7.54983 0.361158
$$438$$ 0 0
$$439$$ 26.1993 1.25043 0.625213 0.780454i $$-0.285012\pi$$
0.625213 + 0.780454i $$0.285012\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 22.8248 1.08444 0.542218 0.840238i $$-0.317585\pi$$
0.542218 + 0.840238i $$0.317585\pi$$
$$444$$ 0 0
$$445$$ −1.45017 −0.0687444
$$446$$ 0 0
$$447$$ −32.1993 −1.52298
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 55.6495 2.62043
$$452$$ 0 0
$$453$$ 13.0997 0.615476
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.72508 0.221030 0.110515 0.993874i $$-0.464750\pi$$
0.110515 + 0.993874i $$0.464750\pi$$
$$458$$ 0 0
$$459$$ −1.09967 −0.0513281
$$460$$ 0 0
$$461$$ −23.2749 −1.08402 −0.542010 0.840372i $$-0.682337\pi$$
−0.542010 + 0.840372i $$0.682337\pi$$
$$462$$ 0 0
$$463$$ −19.8248 −0.921334 −0.460667 0.887573i $$-0.652390\pi$$
−0.460667 + 0.887573i $$0.652390\pi$$
$$464$$ 0 0
$$465$$ 8.00000 0.370991
$$466$$ 0 0
$$467$$ 10.7251 0.496298 0.248149 0.968722i $$-0.420178\pi$$
0.248149 + 0.968722i $$0.420178\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −28.1993 −1.29936
$$472$$ 0 0
$$473$$ 5.27492 0.242541
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 8.54983 0.391470
$$478$$ 0 0
$$479$$ 14.4502 0.660245 0.330122 0.943938i $$-0.392910\pi$$
0.330122 + 0.943938i $$0.392910\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6.54983 0.297413
$$486$$ 0 0
$$487$$ 7.09967 0.321717 0.160858 0.986978i $$-0.448574\pi$$
0.160858 + 0.986978i $$0.448574\pi$$
$$488$$ 0 0
$$489$$ −43.0997 −1.94903
$$490$$ 0 0
$$491$$ 5.54983 0.250461 0.125230 0.992128i $$-0.460033\pi$$
0.125230 + 0.992128i $$0.460033\pi$$
$$492$$ 0 0
$$493$$ −1.09967 −0.0495266
$$494$$ 0 0
$$495$$ −5.27492 −0.237090
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.1993 1.39667 0.698337 0.715769i $$-0.253924\pi$$
0.698337 + 0.715769i $$0.253924\pi$$
$$500$$ 0 0
$$501$$ −26.1993 −1.17050
$$502$$ 0 0
$$503$$ 10.6495 0.474838 0.237419 0.971407i $$-0.423699\pi$$
0.237419 + 0.971407i $$0.423699\pi$$
$$504$$ 0 0
$$505$$ 3.00000 0.133498
$$506$$ 0 0
$$507$$ −18.0000 −0.799408
$$508$$ 0 0
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ −10.0000 −0.440653
$$516$$ 0 0
$$517$$ 27.8248 1.22373
$$518$$ 0 0
$$519$$ −41.0997 −1.80408
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ −3.64950 −0.159582 −0.0797908 0.996812i $$-0.525425\pi$$
−0.0797908 + 0.996812i $$0.525425\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.09967 0.0479023
$$528$$ 0 0
$$529$$ 34.0000 1.47826
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 21.0997 0.913928
$$534$$ 0 0
$$535$$ −18.5498 −0.801979
$$536$$ 0 0
$$537$$ −30.1993 −1.30320
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.0000 −0.730887 −0.365444 0.930834i $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ −1.09967 −0.0471913
$$544$$ 0 0
$$545$$ 4.54983 0.194893
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 0 0
$$549$$ −4.72508 −0.201662
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 4.00000 0.169791
$$556$$ 0 0
$$557$$ −28.0997 −1.19062 −0.595311 0.803496i $$-0.702971\pi$$
−0.595311 + 0.803496i $$0.702971\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ −2.90033 −0.122452
$$562$$ 0 0
$$563$$ 26.5498 1.11894 0.559471 0.828850i $$-0.311004\pi$$
0.559471 + 0.828850i $$0.311004\pi$$
$$564$$ 0 0
$$565$$ 3.45017 0.145150
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 44.1993 1.85293 0.926466 0.376378i $$-0.122830\pi$$
0.926466 + 0.376378i $$0.122830\pi$$
$$570$$ 0 0
$$571$$ −35.5498 −1.48771 −0.743857 0.668339i $$-0.767006\pi$$
−0.743857 + 0.668339i $$0.767006\pi$$
$$572$$ 0 0
$$573$$ −42.0000 −1.75458
$$574$$ 0 0
$$575$$ −30.1993 −1.25940
$$576$$ 0 0
$$577$$ 24.3746 1.01473 0.507364 0.861732i $$-0.330620\pi$$
0.507364 + 0.861732i $$0.330620\pi$$
$$578$$ 0 0
$$579$$ −6.19934 −0.257636
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −45.0997 −1.86784
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 10.2749 0.424091 0.212046 0.977260i $$-0.431987\pi$$
0.212046 + 0.977260i $$0.431987\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 24.1993 0.995428
$$592$$ 0 0
$$593$$ −3.00000 −0.123195 −0.0615976 0.998101i $$-0.519620\pi$$
−0.0615976 + 0.998101i $$0.519620\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.00000 −0.245564
$$598$$ 0 0
$$599$$ −29.6495 −1.21145 −0.605723 0.795676i $$-0.707116\pi$$
−0.605723 + 0.795676i $$0.707116\pi$$
$$600$$ 0 0
$$601$$ 26.5498 1.08299 0.541495 0.840704i $$-0.317858\pi$$
0.541495 + 0.840704i $$0.317858\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 16.8248 0.684023
$$606$$ 0 0
$$607$$ 19.6495 0.797549 0.398774 0.917049i $$-0.369436\pi$$
0.398774 + 0.917049i $$0.369436\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.5498 0.426801
$$612$$ 0 0
$$613$$ −15.1752 −0.612923 −0.306461 0.951883i $$-0.599145\pi$$
−0.306461 + 0.951883i $$0.599145\pi$$
$$614$$ 0 0
$$615$$ −21.0997 −0.850821
$$616$$ 0 0
$$617$$ 6.09967 0.245563 0.122782 0.992434i $$-0.460818\pi$$
0.122782 + 0.992434i $$0.460818\pi$$
$$618$$ 0 0
$$619$$ 3.54983 0.142680 0.0713399 0.997452i $$-0.477272\pi$$
0.0713399 + 0.997452i $$0.477272\pi$$
$$620$$ 0 0
$$621$$ −30.1993 −1.21186
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −10.5498 −0.421320
$$628$$ 0 0
$$629$$ 0.549834 0.0219233
$$630$$ 0 0
$$631$$ −17.0000 −0.676759 −0.338380 0.941010i $$-0.609879\pi$$
−0.338380 + 0.941010i $$0.609879\pi$$
$$632$$ 0 0
$$633$$ −51.2990 −2.03895
$$634$$ 0 0
$$635$$ 15.0997 0.599212
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −4.54983 −0.179989
$$640$$ 0 0
$$641$$ 31.4502 1.24221 0.621103 0.783729i $$-0.286685\pi$$
0.621103 + 0.783729i $$0.286685\pi$$
$$642$$ 0 0
$$643$$ 22.8248 0.900120 0.450060 0.892998i $$-0.351403\pi$$
0.450060 + 0.892998i $$0.351403\pi$$
$$644$$ 0 0
$$645$$ −2.00000 −0.0787499
$$646$$ 0 0
$$647$$ −16.0997 −0.632943 −0.316472 0.948602i $$-0.602498\pi$$
−0.316472 + 0.948602i $$0.602498\pi$$
$$648$$ 0 0
$$649$$ −21.0997 −0.828234
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8.27492 −0.323823 −0.161911 0.986805i $$-0.551766\pi$$
−0.161911 + 0.986805i $$0.551766\pi$$
$$654$$ 0 0
$$655$$ −6.82475 −0.266665
$$656$$ 0 0
$$657$$ −11.2749 −0.439876
$$658$$ 0 0
$$659$$ −40.5498 −1.57960 −0.789799 0.613366i $$-0.789815\pi$$
−0.789799 + 0.613366i $$0.789815\pi$$
$$660$$ 0 0
$$661$$ 29.4502 1.14548 0.572739 0.819738i $$-0.305881\pi$$
0.572739 + 0.819738i $$0.305881\pi$$
$$662$$ 0 0
$$663$$ −1.09967 −0.0427076
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −30.1993 −1.16932
$$668$$ 0 0
$$669$$ 34.1993 1.32222
$$670$$ 0 0
$$671$$ 24.9244 0.962197
$$672$$ 0 0
$$673$$ −4.54983 −0.175383 −0.0876916 0.996148i $$-0.527949\pi$$
−0.0876916 + 0.996148i $$0.527949\pi$$
$$674$$ 0 0
$$675$$ 16.0000 0.615840
$$676$$ 0 0
$$677$$ −6.90033 −0.265201 −0.132601 0.991170i $$-0.542333\pi$$
−0.132601 + 0.991170i $$0.542333\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.9003 0.724262
$$682$$ 0 0
$$683$$ 27.6495 1.05798 0.528989 0.848628i $$-0.322571\pi$$
0.528989 + 0.848628i $$0.322571\pi$$
$$684$$ 0 0
$$685$$ 11.2749 0.430792
$$686$$ 0 0
$$687$$ 32.5498 1.24185
$$688$$ 0 0
$$689$$ −17.0997 −0.651446
$$690$$ 0 0
$$691$$ −15.9244 −0.605794 −0.302897 0.953023i $$-0.597954\pi$$
−0.302897 + 0.953023i $$0.597954\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −11.8248 −0.448538
$$696$$ 0 0
$$697$$ −2.90033 −0.109858
$$698$$ 0 0
$$699$$ 8.54983 0.323384
$$700$$ 0 0
$$701$$ −15.0000 −0.566542 −0.283271 0.959040i $$-0.591420\pi$$
−0.283271 + 0.959040i $$0.591420\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ −10.5498 −0.397330
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −15.0000 −0.563337 −0.281668 0.959512i $$-0.590888\pi$$
−0.281668 + 0.959512i $$0.590888\pi$$
$$710$$ 0 0
$$711$$ −14.5498 −0.545661
$$712$$ 0 0
$$713$$ 30.1993 1.13097
$$714$$ 0 0
$$715$$ 10.5498 0.394541
$$716$$ 0 0
$$717$$ 46.7492 1.74588
$$718$$ 0 0
$$719$$ −34.2749 −1.27824 −0.639119 0.769108i $$-0.720701\pi$$
−0.639119 + 0.769108i $$0.720701\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −21.0997 −0.784705
$$724$$ 0 0
$$725$$ 16.0000 0.594225
$$726$$ 0 0
$$727$$ 44.0997 1.63557 0.817783 0.575527i $$-0.195203\pi$$
0.817783 + 0.575527i $$0.195203\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −0.274917 −0.0101682
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −26.8248 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 0 0
$$743$$ 28.7492 1.05470 0.527352 0.849647i $$-0.323185\pi$$
0.527352 + 0.849647i $$0.323185\pi$$
$$744$$ 0 0
$$745$$ −16.0997 −0.589846
$$746$$ 0 0
$$747$$ 3.54983 0.129882
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 56.1993 2.04802
$$754$$ 0 0
$$755$$ 6.54983 0.238373
$$756$$ 0 0
$$757$$ −12.3746 −0.449762 −0.224881 0.974386i $$-0.572199\pi$$
−0.224881 + 0.974386i $$0.572199\pi$$
$$758$$ 0 0
$$759$$ −79.6495 −2.89109
$$760$$ 0 0
$$761$$ 43.0000 1.55875 0.779374 0.626559i $$-0.215537\pi$$
0.779374 + 0.626559i $$0.215537\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0.274917 0.00993965
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −17.1993 −0.620224 −0.310112 0.950700i $$-0.600367\pi$$
−0.310112 + 0.950700i $$0.600367\pi$$
$$770$$ 0 0
$$771$$ 45.0997 1.62422
$$772$$ 0 0
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −10.5498 −0.377987
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ 16.0000 0.571793
$$784$$ 0 0
$$785$$ −14.0997 −0.503239
$$786$$ 0 0
$$787$$ −14.0000 −0.499046 −0.249523 0.968369i $$-0.580274\pi$$
−0.249523 + 0.968369i $$0.580274\pi$$
$$788$$ 0 0
$$789$$ 45.6495 1.62517
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 9.45017 0.335585
$$794$$ 0 0
$$795$$ 17.0997 0.606463
$$796$$ 0 0
$$797$$ 1.45017 0.0513675 0.0256837 0.999670i $$-0.491824\pi$$
0.0256837 + 0.999670i $$0.491824\pi$$
$$798$$ 0 0
$$799$$ −1.45017 −0.0513032
$$800$$ 0 0
$$801$$ −1.45017 −0.0512391
$$802$$ 0 0
$$803$$ 59.4743 2.09880
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 49.0997 1.72839
$$808$$ 0 0
$$809$$ 12.0997 0.425402 0.212701 0.977117i $$-0.431774\pi$$
0.212701 + 0.977117i $$0.431774\pi$$
$$810$$ 0 0
$$811$$ −52.7492 −1.85227 −0.926137 0.377187i $$-0.876891\pi$$
−0.926137 + 0.377187i $$0.876891\pi$$
$$812$$ 0 0
$$813$$ −35.0997 −1.23100
$$814$$ 0 0
$$815$$ −21.5498 −0.754858
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26.9244 −0.939669 −0.469834 0.882755i $$-0.655686\pi$$
−0.469834 + 0.882755i $$0.655686\pi$$
$$822$$ 0 0
$$823$$ 4.92442 0.171655 0.0858273 0.996310i $$-0.472647\pi$$
0.0858273 + 0.996310i $$0.472647\pi$$
$$824$$ 0 0
$$825$$ 42.1993 1.46919
$$826$$ 0 0
$$827$$ −41.6495 −1.44830 −0.724148 0.689645i $$-0.757767\pi$$
−0.724148 + 0.689645i $$0.757767\pi$$
$$828$$ 0 0
$$829$$ 14.9003 0.517510 0.258755 0.965943i $$-0.416688\pi$$
0.258755 + 0.965943i $$0.416688\pi$$
$$830$$ 0 0
$$831$$ −1.45017 −0.0503057
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −13.0997 −0.453333
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ 0 0
$$839$$ −37.6495 −1.29981 −0.649903 0.760018i $$-0.725190\pi$$
−0.649903 + 0.760018i $$0.725190\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 1.09967 0.0378746
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 34.0000 1.16688
$$850$$ 0 0
$$851$$ 15.0997 0.517610
$$852$$ 0 0
$$853$$ 42.0997 1.44147 0.720733 0.693213i $$-0.243806\pi$$
0.720733 + 0.693213i $$0.243806\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −41.6495 −1.42272 −0.711360 0.702828i $$-0.751920\pi$$
−0.711360 + 0.702828i $$0.751920\pi$$
$$858$$ 0 0
$$859$$ 10.0997 0.344596 0.172298 0.985045i $$-0.444881\pi$$
0.172298 + 0.985045i $$0.444881\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −2.90033 −0.0987284 −0.0493642 0.998781i $$-0.515720\pi$$
−0.0493642 + 0.998781i $$0.515720\pi$$
$$864$$ 0 0
$$865$$ −20.5498 −0.698715
$$866$$ 0 0
$$867$$ −33.8488 −1.14957
$$868$$ 0 0
$$869$$ 76.7492 2.60354
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 6.54983 0.221678
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.7492 1.10586 0.552930 0.833227i $$-0.313510\pi$$
0.552930 + 0.833227i $$0.313510\pi$$
$$878$$ 0 0
$$879$$ −55.2990 −1.86519
$$880$$ 0 0
$$881$$ −49.9244 −1.68200 −0.840998 0.541038i $$-0.818032\pi$$
−0.840998 + 0.541038i $$0.818032\pi$$
$$882$$ 0 0
$$883$$ −41.0241 −1.38057 −0.690285 0.723537i $$-0.742515\pi$$
−0.690285 + 0.723537i $$0.742515\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ 0 0
$$887$$ 52.1993 1.75268 0.876341 0.481691i $$-0.159977\pi$$
0.876341 + 0.481691i $$0.159977\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 58.0241 1.94388
$$892$$ 0 0
$$893$$ −5.27492 −0.176518
$$894$$ 0 0
$$895$$ −15.0997 −0.504726
$$896$$ 0 0
$$897$$ −30.1993 −1.00833
$$898$$ 0 0
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 2.35050 0.0783064
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −0.549834 −0.0182771
$$906$$ 0 0
$$907$$ 6.90033 0.229122 0.114561 0.993416i $$-0.463454\pi$$
0.114561 + 0.993416i $$0.463454\pi$$
$$908$$ 0 0
$$909$$ 3.00000 0.0995037
$$910$$ 0 0
$$911$$ 11.4502 0.379361 0.189680 0.981846i $$-0.439255\pi$$
0.189680 + 0.981846i $$0.439255\pi$$
$$912$$ 0 0
$$913$$ −18.7251 −0.619710
$$914$$ 0 0
$$915$$ −9.45017 −0.312413
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −39.5498 −1.30463 −0.652314 0.757949i $$-0.726202\pi$$
−0.652314 + 0.757949i $$0.726202\pi$$
$$920$$ 0 0
$$921$$ 52.0000 1.71346
$$922$$ 0 0
$$923$$ 9.09967 0.299519
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 0 0
$$927$$ −10.0000 −0.328443
$$928$$ 0 0
$$929$$ 33.0000 1.08269 0.541347 0.840799i $$-0.317914\pi$$
0.541347 + 0.840799i $$0.317914\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 20.5498 0.672771
$$934$$ 0 0
$$935$$ −1.45017 −0.0474255
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 62.3987 2.03630
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ 0 0
$$943$$ −79.6495 −2.59374
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 44.0000 1.42981 0.714904 0.699223i $$-0.246470\pi$$
0.714904 + 0.699223i $$0.246470\pi$$
$$948$$ 0 0
$$949$$ 22.5498 0.731999
$$950$$ 0 0
$$951$$ 18.1993 0.590154
$$952$$ 0 0
$$953$$ 43.0997 1.39614 0.698068 0.716032i $$-0.254043\pi$$
0.698068 + 0.716032i $$0.254043\pi$$
$$954$$ 0 0
$$955$$ −21.0000 −0.679544
$$956$$ 0 0
$$957$$ 42.1993 1.36411
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −18.5498 −0.597760
$$964$$ 0 0
$$965$$ −3.09967 −0.0997819
$$966$$ 0 0
$$967$$ 42.1993 1.35704 0.678520 0.734582i $$-0.262622\pi$$
0.678520 + 0.734582i $$0.262622\pi$$
$$968$$ 0 0
$$969$$ 0.549834 0.0176632
$$970$$ 0 0
$$971$$ −21.4502 −0.688369 −0.344184 0.938902i $$-0.611844\pi$$
−0.344184 + 0.938902i $$0.611844\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 16.0000 0.512410
$$976$$ 0 0
$$977$$ −35.2990 −1.12932 −0.564658 0.825325i $$-0.690992\pi$$
−0.564658 + 0.825325i $$0.690992\pi$$
$$978$$ 0 0
$$979$$ 7.64950 0.244479
$$980$$ 0 0
$$981$$ 4.54983 0.145265
$$982$$ 0 0
$$983$$ 52.1993 1.66490 0.832450 0.554100i $$-0.186937\pi$$
0.832450 + 0.554100i $$0.186937\pi$$
$$984$$ 0 0
$$985$$ 12.0997 0.385528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7.54983 −0.240071
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ −28.0000 −0.888553
$$994$$ 0 0
$$995$$ −3.00000 −0.0951064
$$996$$ 0 0
$$997$$ 7.17525 0.227242 0.113621 0.993524i $$-0.463755\pi$$
0.113621 + 0.993524i $$0.463755\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 7448.2.a.be.1.1 2
7.3 odd 6 1064.2.q.l.457.1 yes 4
7.5 odd 6 1064.2.q.l.305.1 4
7.6 odd 2 7448.2.a.w.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.l.305.1 4 7.5 odd 6
1064.2.q.l.457.1 yes 4 7.3 odd 6
7448.2.a.w.1.1 2 7.6 odd 2
7448.2.a.be.1.1 2 1.1 even 1 trivial