# Properties

 Label 1176.2.a.d.1.1 Level $1176$ Weight $2$ Character 1176.1 Self dual yes Analytic conductor $9.390$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.39040727770$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} -6.00000 q^{11} -3.00000 q^{13} -2.00000 q^{15} +4.00000 q^{17} -5.00000 q^{19} -4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -4.00000 q^{29} +7.00000 q^{31} +6.00000 q^{33} -9.00000 q^{37} +3.00000 q^{39} -2.00000 q^{41} -1.00000 q^{43} +2.00000 q^{45} +2.00000 q^{47} -4.00000 q^{51} +8.00000 q^{53} -12.0000 q^{55} +5.00000 q^{57} +10.0000 q^{61} -6.00000 q^{65} -15.0000 q^{67} +4.00000 q^{69} -6.00000 q^{71} -11.0000 q^{73} +1.00000 q^{75} +1.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +8.00000 q^{85} +4.00000 q^{87} -8.00000 q^{89} -7.00000 q^{93} -10.0000 q^{95} -14.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ −3.00000 −0.832050 −0.416025 0.909353i $$-0.636577\pi$$
−0.416025 + 0.909353i $$0.636577\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.00000 −1.47959 −0.739795 0.672832i $$-0.765078\pi$$
−0.739795 + 0.672832i $$0.765078\pi$$
$$38$$ 0 0
$$39$$ 3.00000 0.480384
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ 0 0
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 5.00000 0.662266
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −15.0000 −1.83254 −0.916271 0.400559i $$-0.868816\pi$$
−0.916271 + 0.400559i $$0.868816\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ −8.00000 −0.847998 −0.423999 0.905663i $$-0.639374\pi$$
−0.423999 + 0.905663i $$0.639374\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −7.00000 −0.725866
$$94$$ 0 0
$$95$$ −10.0000 −1.02598
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −9.00000 −0.886796 −0.443398 0.896325i $$-0.646227\pi$$
−0.443398 + 0.896325i $$0.646227\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 9.00000 0.854242
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ −3.00000 −0.277350
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −1.00000 −0.0887357 −0.0443678 0.999015i $$-0.514127\pi$$
−0.0443678 + 0.999015i $$0.514127\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ 20.0000 1.70872 0.854358 0.519685i $$-0.173951\pi$$
0.854358 + 0.519685i $$0.173951\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 18.0000 1.50524
$$144$$ 0 0
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ 14.0000 1.12451
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ 0 0
$$173$$ 20.0000 1.52057 0.760286 0.649589i $$-0.225059\pi$$
0.760286 + 0.649589i $$0.225059\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 26.0000 1.94333 0.971666 0.236360i $$-0.0759544\pi$$
0.971666 + 0.236360i $$0.0759544\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −18.0000 −1.32339
$$186$$ 0 0
$$187$$ −24.0000 −1.75505
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ 3.00000 0.215945 0.107972 0.994154i $$-0.465564\pi$$
0.107972 + 0.994154i $$0.465564\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ 0 0
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 15.0000 1.05802
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ 30.0000 2.07514
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ 14.0000 0.929213 0.464606 0.885517i $$-0.346196\pi$$
0.464606 + 0.885517i $$0.346196\pi$$
$$228$$ 0 0
$$229$$ −7.00000 −0.462573 −0.231287 0.972886i $$-0.574293\pi$$
−0.231287 + 0.972886i $$0.574293\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ −1.00000 −0.0649570
$$238$$ 0 0
$$239$$ 2.00000 0.129369 0.0646846 0.997906i $$-0.479396\pi$$
0.0646846 + 0.997906i $$0.479396\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.0000 0.954427
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 24.0000 1.50887
$$254$$ 0 0
$$255$$ −8.00000 −0.500979
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 16.0000 0.982872
$$266$$ 0 0
$$267$$ 8.00000 0.489592
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ 0 0
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −1.00000 −0.0594438 −0.0297219 0.999558i $$-0.509462\pi$$
−0.0297219 + 0.999558i $$0.509462\pi$$
$$284$$ 0 0
$$285$$ 10.0000 0.592349
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 6.00000 0.348155
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ −11.0000 −0.627803 −0.313902 0.949456i $$-0.601636\pi$$
−0.313902 + 0.949456i $$0.601636\pi$$
$$308$$ 0 0
$$309$$ 9.00000 0.511992
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −20.0000 −1.11283
$$324$$ 0 0
$$325$$ 3.00000 0.166410
$$326$$ 0 0
$$327$$ 11.0000 0.608301
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5.00000 0.274825 0.137412 0.990514i $$-0.456121\pi$$
0.137412 + 0.990514i $$0.456121\pi$$
$$332$$ 0 0
$$333$$ −9.00000 −0.493197
$$334$$ 0 0
$$335$$ −30.0000 −1.63908
$$336$$ 0 0
$$337$$ 29.0000 1.57973 0.789865 0.613280i $$-0.210150\pi$$
0.789865 + 0.613280i $$0.210150\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −42.0000 −2.27443
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 8.00000 0.430706
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ −25.0000 −1.31216
$$364$$ 0 0
$$365$$ −22.0000 −1.15153
$$366$$ 0 0
$$367$$ −7.00000 −0.365397 −0.182699 0.983169i $$-0.558483\pi$$
−0.182699 + 0.983169i $$0.558483\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −13.0000 −0.673114 −0.336557 0.941663i $$-0.609263\pi$$
−0.336557 + 0.941663i $$0.609263\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ −15.0000 −0.770498 −0.385249 0.922813i $$-0.625884\pi$$
−0.385249 + 0.922813i $$0.625884\pi$$
$$380$$ 0 0
$$381$$ 1.00000 0.0512316
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 0 0
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ −14.0000 −0.706207
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ −5.00000 −0.250943 −0.125471 0.992097i $$-0.540044\pi$$
−0.125471 + 0.992097i $$0.540044\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ −21.0000 −1.04608
$$404$$ 0 0
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 54.0000 2.67668
$$408$$ 0 0
$$409$$ −3.00000 −0.148340 −0.0741702 0.997246i $$-0.523631\pi$$
−0.0741702 + 0.997246i $$0.523631\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 9.00000 0.440732
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ −35.0000 −1.70580 −0.852898 0.522078i $$-0.825157\pi$$
−0.852898 + 0.522078i $$0.825157\pi$$
$$422$$ 0 0
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −18.0000 −0.869048
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ 31.0000 1.48976 0.744882 0.667196i $$-0.232506\pi$$
0.744882 + 0.667196i $$0.232506\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ 20.0000 0.956730
$$438$$ 0 0
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ −16.0000 −0.758473
$$446$$ 0 0
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ −38.0000 −1.79333 −0.896665 0.442709i $$-0.854018\pi$$
−0.896665 + 0.442709i $$0.854018\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 0 0
$$453$$ 8.00000 0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.0000 0.608114 0.304057 0.952654i $$-0.401659\pi$$
0.304057 + 0.952654i $$0.401659\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 17.0000 0.790057 0.395029 0.918669i $$-0.370735\pi$$
0.395029 + 0.918669i $$0.370735\pi$$
$$464$$ 0 0
$$465$$ −14.0000 −0.649234
$$466$$ 0 0
$$467$$ 30.0000 1.38823 0.694117 0.719862i $$-0.255795\pi$$
0.694117 + 0.719862i $$0.255795\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 5.00000 0.229416
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 27.0000 1.23109
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −28.0000 −1.27141
$$486$$ 0 0
$$487$$ 25.0000 1.13286 0.566429 0.824110i $$-0.308325\pi$$
0.566429 + 0.824110i $$0.308325\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −17.0000 −0.761025 −0.380512 0.924776i $$-0.624252\pi$$
−0.380512 + 0.924776i $$0.624252\pi$$
$$500$$ 0 0
$$501$$ −18.0000 −0.804181
$$502$$ 0 0
$$503$$ −14.0000 −0.624229 −0.312115 0.950044i $$-0.601037\pi$$
−0.312115 + 0.950044i $$0.601037\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 4.00000 0.177646
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 5.00000 0.220755
$$514$$ 0 0
$$515$$ −18.0000 −0.793175
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ −20.0000 −0.877903
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 28.0000 1.21970
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ −24.0000 −1.03761
$$536$$ 0 0
$$537$$ −26.0000 −1.12198
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.00000 0.0429934 0.0214967 0.999769i $$-0.493157\pi$$
0.0214967 + 0.999769i $$0.493157\pi$$
$$542$$ 0 0
$$543$$ 7.00000 0.300399
$$544$$ 0 0
$$545$$ −22.0000 −0.942376
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 20.0000 0.852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 18.0000 0.764057
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ 3.00000 0.126886
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 0 0
$$563$$ −2.00000 −0.0842900 −0.0421450 0.999112i $$-0.513419\pi$$
−0.0421450 + 0.999112i $$0.513419\pi$$
$$564$$ 0 0
$$565$$ 12.0000 0.504844
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 23.0000 0.962520 0.481260 0.876578i $$-0.340179\pi$$
0.481260 + 0.876578i $$0.340179\pi$$
$$572$$ 0 0
$$573$$ −10.0000 −0.417756
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 39.0000 1.62359 0.811796 0.583942i $$-0.198490\pi$$
0.811796 + 0.583942i $$0.198490\pi$$
$$578$$ 0 0
$$579$$ −3.00000 −0.124676
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −48.0000 −1.98796
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ 0 0
$$587$$ 16.0000 0.660391 0.330195 0.943913i $$-0.392885\pi$$
0.330195 + 0.943913i $$0.392885\pi$$
$$588$$ 0 0
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ 31.0000 1.26452 0.632258 0.774758i $$-0.282128\pi$$
0.632258 + 0.774758i $$0.282128\pi$$
$$602$$ 0 0
$$603$$ −15.0000 −0.610847
$$604$$ 0 0
$$605$$ 50.0000 2.03279
$$606$$ 0 0
$$607$$ 1.00000 0.0405887 0.0202944 0.999794i $$-0.493540\pi$$
0.0202944 + 0.999794i $$0.493540\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ 0 0
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 9.00000 0.361741 0.180870 0.983507i $$-0.442109\pi$$
0.180870 + 0.983507i $$0.442109\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ −30.0000 −1.19808
$$628$$ 0 0
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 0 0
$$635$$ −2.00000 −0.0793676
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −20.0000 −0.789953 −0.394976 0.918691i $$-0.629247\pi$$
−0.394976 + 0.918691i $$0.629247\pi$$
$$642$$ 0 0
$$643$$ −17.0000 −0.670415 −0.335207 0.942144i $$-0.608806\pi$$
−0.335207 + 0.942144i $$0.608806\pi$$
$$644$$ 0 0
$$645$$ 2.00000 0.0787499
$$646$$ 0 0
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −22.0000 −0.860927 −0.430463 0.902608i $$-0.641650\pi$$
−0.430463 + 0.902608i $$0.641650\pi$$
$$654$$ 0 0
$$655$$ 28.0000 1.09405
$$656$$ 0 0
$$657$$ −11.0000 −0.429151
$$658$$ 0 0
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ −24.0000 −0.927894
$$670$$ 0 0
$$671$$ −60.0000 −2.31627
$$672$$ 0 0
$$673$$ 7.00000 0.269830 0.134915 0.990857i $$-0.456924\pi$$
0.134915 + 0.990857i $$0.456924\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −14.0000 −0.536481
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 40.0000 1.52832
$$686$$ 0 0
$$687$$ 7.00000 0.267067
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −7.00000 −0.266293 −0.133146 0.991096i $$-0.542508\pi$$
−0.133146 + 0.991096i $$0.542508\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −18.0000 −0.682779
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 0 0
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 28.0000 1.05755 0.528773 0.848763i $$-0.322652\pi$$
0.528773 + 0.848763i $$0.322652\pi$$
$$702$$ 0 0
$$703$$ 45.0000 1.69721
$$704$$ 0 0
$$705$$ −4.00000 −0.150649
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −50.0000 −1.87779 −0.938895 0.344204i $$-0.888149\pi$$
−0.938895 + 0.344204i $$0.888149\pi$$
$$710$$ 0 0
$$711$$ 1.00000 0.0375029
$$712$$ 0 0
$$713$$ −28.0000 −1.04861
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ −2.00000 −0.0746914
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 0 0
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 5.00000 0.185440 0.0927199 0.995692i $$-0.470444\pi$$
0.0927199 + 0.995692i $$0.470444\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ −11.0000 −0.406294 −0.203147 0.979148i $$-0.565117\pi$$
−0.203147 + 0.979148i $$0.565117\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 90.0000 3.31519
$$738$$ 0 0
$$739$$ −5.00000 −0.183928 −0.0919640 0.995762i $$-0.529314\pi$$
−0.0919640 + 0.995762i $$0.529314\pi$$
$$740$$ 0 0
$$741$$ −15.0000 −0.551039
$$742$$ 0 0
$$743$$ −34.0000 −1.24734 −0.623670 0.781688i $$-0.714359\pi$$
−0.623670 + 0.781688i $$0.714359\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −37.0000 −1.35015 −0.675075 0.737749i $$-0.735889\pi$$
−0.675075 + 0.737749i $$0.735889\pi$$
$$752$$ 0 0
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ −24.0000 −0.871145
$$760$$ 0 0
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 7.00000 0.252426 0.126213 0.992003i $$-0.459718\pi$$
0.126213 + 0.992003i $$0.459718\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ −50.0000 −1.79838 −0.899188 0.437564i $$-0.855842\pi$$
−0.899188 + 0.437564i $$0.855842\pi$$
$$774$$ 0 0
$$775$$ −7.00000 −0.251447
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 4.00000 0.142948
$$784$$ 0 0
$$785$$ 36.0000 1.28490
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 0 0
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −30.0000 −1.06533
$$794$$ 0 0
$$795$$ −16.0000 −0.567462
$$796$$ 0 0
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ 66.0000 2.32909
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000 0.633630
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 5.00000 0.174928
$$818$$ 0 0
$$819$$ 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
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ 18.0000 0.625921 0.312961 0.949766i $$-0.398679\pi$$
0.312961 + 0.949766i $$0.398679\pi$$
$$828$$ 0 0
$$829$$ −43.0000 −1.49345 −0.746726 0.665132i $$-0.768375\pi$$
−0.746726 + 0.665132i $$0.768375\pi$$
$$830$$ 0 0
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 36.0000 1.24583
$$836$$ 0 0
$$837$$ −7.00000 −0.241955
$$838$$ 0 0
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −8.00000 −0.275208
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 1.00000 0.0343199
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ 0 0
$$853$$ −17.0000 −0.582069 −0.291034 0.956713i $$-0.593999\pi$$
−0.291034 + 0.956713i $$0.593999\pi$$
$$854$$ 0 0
$$855$$ −10.0000 −0.341993
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 16.0000 0.545913 0.272956 0.962026i $$-0.411998\pi$$
0.272956 + 0.962026i $$0.411998\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 14.0000 0.476566 0.238283 0.971196i $$-0.423415\pi$$
0.238283 + 0.971196i $$0.423415\pi$$
$$864$$ 0 0
$$865$$ 40.0000 1.36004
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ −6.00000 −0.203536
$$870$$ 0 0
$$871$$ 45.0000 1.52477
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 52.0000 1.75192 0.875962 0.482380i $$-0.160227\pi$$
0.875962 + 0.482380i $$0.160227\pi$$
$$882$$ 0 0
$$883$$ 1.00000 0.0336527 0.0168263 0.999858i $$-0.494644\pi$$
0.0168263 + 0.999858i $$0.494644\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −54.0000 −1.81314 −0.906571 0.422053i $$-0.861310\pi$$
−0.906571 + 0.422053i $$0.861310\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 0 0
$$893$$ −10.0000 −0.334637
$$894$$ 0 0
$$895$$ 52.0000 1.73817
$$896$$ 0 0
$$897$$ −12.0000 −0.400668
$$898$$ 0 0
$$899$$ −28.0000 −0.933852
$$900$$ 0 0
$$901$$ 32.0000 1.06607
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −14.0000 −0.465376
$$906$$ 0 0
$$907$$ 17.0000 0.564476 0.282238 0.959344i $$-0.408923\pi$$
0.282238 + 0.959344i $$0.408923\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 28.0000 0.927681 0.463841 0.885919i $$-0.346471\pi$$
0.463841 + 0.885919i $$0.346471\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ −20.0000 −0.661180
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ 11.0000 0.362462
$$922$$ 0 0
$$923$$ 18.0000 0.592477
$$924$$ 0 0
$$925$$ 9.00000 0.295918
$$926$$ 0 0
$$927$$ −9.00000 −0.295599
$$928$$ 0 0
$$929$$ 58.0000 1.90292 0.951459 0.307775i $$-0.0995844\pi$$
0.951459 + 0.307775i $$0.0995844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 18.0000 0.589294
$$934$$ 0 0
$$935$$ −48.0000 −1.56977
$$936$$ 0 0
$$937$$ −33.0000 −1.07806 −0.539032 0.842286i $$-0.681210\pi$$
−0.539032 + 0.842286i $$0.681210\pi$$
$$938$$ 0 0
$$939$$ 1.00000 0.0326338
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −38.0000 −1.23483 −0.617417 0.786636i $$-0.711821\pi$$
−0.617417 + 0.786636i $$0.711821\pi$$
$$948$$ 0 0
$$949$$ 33.0000 1.07123
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.00000 −0.259145 −0.129573 0.991570i $$-0.541361\pi$$
−0.129573 + 0.991570i $$0.541361\pi$$
$$954$$ 0 0
$$955$$ 20.0000 0.647185
$$956$$ 0 0
$$957$$ −24.0000 −0.775810
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ 6.00000 0.193147
$$966$$ 0 0
$$967$$ −27.0000 −0.868261 −0.434131 0.900850i $$-0.642944\pi$$
−0.434131 + 0.900850i $$0.642944\pi$$
$$968$$ 0 0
$$969$$ 20.0000 0.642493
$$970$$ 0 0
$$971$$ −56.0000 −1.79713 −0.898563 0.438845i $$-0.855388\pi$$
−0.898563 + 0.438845i $$0.855388\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −3.00000 −0.0960769
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 48.0000 1.53409
$$980$$ 0 0
$$981$$ −11.0000 −0.351203
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ −24.0000 −0.764704
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −33.0000 −1.04828 −0.524140 0.851632i $$-0.675613\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ 0 0
$$993$$ −5.00000 −0.158670
$$994$$ 0 0
$$995$$ −32.0000 −1.01447
$$996$$ 0 0
$$997$$ −17.0000 −0.538395 −0.269198 0.963085i $$-0.586759\pi$$
−0.269198 + 0.963085i $$0.586759\pi$$
$$998$$ 0 0
$$999$$ 9.00000 0.284747
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 1176.2.a.d.1.1 1
3.2 odd 2 3528.2.a.f.1.1 1
4.3 odd 2 2352.2.a.x.1.1 1
7.2 even 3 168.2.q.b.25.1 2
7.3 odd 6 1176.2.q.e.961.1 2
7.4 even 3 168.2.q.b.121.1 yes 2
7.5 odd 6 1176.2.q.e.361.1 2
7.6 odd 2 1176.2.a.e.1.1 1
8.3 odd 2 9408.2.a.f.1.1 1
8.5 even 2 9408.2.a.cd.1.1 1
12.11 even 2 7056.2.a.i.1.1 1
21.2 odd 6 504.2.s.g.361.1 2
21.5 even 6 3528.2.s.d.361.1 2
21.11 odd 6 504.2.s.g.289.1 2
21.17 even 6 3528.2.s.d.3313.1 2
21.20 even 2 3528.2.a.y.1.1 1
28.3 even 6 2352.2.q.v.961.1 2
28.11 odd 6 336.2.q.a.289.1 2
28.19 even 6 2352.2.q.v.1537.1 2
28.23 odd 6 336.2.q.a.193.1 2
28.27 even 2 2352.2.a.e.1.1 1
56.11 odd 6 1344.2.q.t.961.1 2
56.13 odd 2 9408.2.a.bk.1.1 1
56.27 even 2 9408.2.a.cs.1.1 1
56.37 even 6 1344.2.q.i.193.1 2
56.51 odd 6 1344.2.q.t.193.1 2
56.53 even 6 1344.2.q.i.961.1 2
84.11 even 6 1008.2.s.m.289.1 2
84.23 even 6 1008.2.s.m.865.1 2
84.83 odd 2 7056.2.a.bn.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.b.25.1 2 7.2 even 3
168.2.q.b.121.1 yes 2 7.4 even 3
336.2.q.a.193.1 2 28.23 odd 6
336.2.q.a.289.1 2 28.11 odd 6
504.2.s.g.289.1 2 21.11 odd 6
504.2.s.g.361.1 2 21.2 odd 6
1008.2.s.m.289.1 2 84.11 even 6
1008.2.s.m.865.1 2 84.23 even 6
1176.2.a.d.1.1 1 1.1 even 1 trivial
1176.2.a.e.1.1 1 7.6 odd 2
1176.2.q.e.361.1 2 7.5 odd 6
1176.2.q.e.961.1 2 7.3 odd 6
1344.2.q.i.193.1 2 56.37 even 6
1344.2.q.i.961.1 2 56.53 even 6
1344.2.q.t.193.1 2 56.51 odd 6
1344.2.q.t.961.1 2 56.11 odd 6
2352.2.a.e.1.1 1 28.27 even 2
2352.2.a.x.1.1 1 4.3 odd 2
2352.2.q.v.961.1 2 28.3 even 6
2352.2.q.v.1537.1 2 28.19 even 6
3528.2.a.f.1.1 1 3.2 odd 2
3528.2.a.y.1.1 1 21.20 even 2
3528.2.s.d.361.1 2 21.5 even 6
3528.2.s.d.3313.1 2 21.17 even 6
7056.2.a.i.1.1 1 12.11 even 2
7056.2.a.bn.1.1 1 84.83 odd 2
9408.2.a.f.1.1 1 8.3 odd 2
9408.2.a.bk.1.1 1 56.13 odd 2
9408.2.a.cd.1.1 1 8.5 even 2
9408.2.a.cs.1.1 1 56.27 even 2