# Properties

 Label 6240.2.a.bw.1.1 Level $6240$ Weight $2$ Character 6240.1 Self dual yes Analytic conductor $49.827$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.8266508613$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.86081$$ of defining polynomial Character $$\chi$$ $$=$$ 6240.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -4.32340 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -4.32340 q^{7} +1.00000 q^{9} +1.39821 q^{11} +1.00000 q^{13} +1.00000 q^{15} +5.11982 q^{17} -2.92520 q^{19} +4.32340 q^{21} -8.97021 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.36842 q^{29} -4.00000 q^{31} -1.39821 q^{33} +4.32340 q^{35} +3.39821 q^{37} -1.00000 q^{39} -0.601793 q^{41} +2.79641 q^{43} -1.00000 q^{45} +9.29362 q^{47} +11.6918 q^{49} -5.11982 q^{51} -2.19462 q^{53} -1.39821 q^{55} +2.92520 q^{57} -7.44322 q^{59} -5.24860 q^{61} -4.32340 q^{63} -1.00000 q^{65} +8.97021 q^{69} -10.0450 q^{71} -13.0152 q^{73} -1.00000 q^{75} -6.04502 q^{77} -1.95498 q^{79} +1.00000 q^{81} +17.2936 q^{83} -5.11982 q^{85} +8.36842 q^{87} +4.04502 q^{89} -4.32340 q^{91} +4.00000 q^{93} +2.92520 q^{95} +10.3234 q^{97} +1.39821 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 12 * q^31 - q^33 + 5 * q^35 + 7 * q^37 - 3 * q^39 - 5 * q^41 + 2 * q^43 - 3 * q^45 - 4 * q^47 - q^51 + 3 * q^53 - q^55 + 4 * q^57 - 3 * q^61 - 5 * q^63 - 3 * q^65 + 3 * q^69 - 11 * q^71 + 4 * q^73 - 3 * q^75 + q^77 - 25 * q^79 + 3 * q^81 + 20 * q^83 - q^85 - 2 * q^87 - 7 * q^89 - 5 * q^91 + 12 * q^93 + 4 * q^95 + 23 * q^97 + q^99

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.32340 −1.63409 −0.817047 0.576572i $$-0.804390\pi$$
−0.817047 + 0.576572i $$0.804390\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.39821 0.421575 0.210788 0.977532i $$-0.432397\pi$$
0.210788 + 0.977532i $$0.432397\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 5.11982 1.24174 0.620869 0.783914i $$-0.286780\pi$$
0.620869 + 0.783914i $$0.286780\pi$$
$$18$$ 0 0
$$19$$ −2.92520 −0.671086 −0.335543 0.942025i $$-0.608920\pi$$
−0.335543 + 0.942025i $$0.608920\pi$$
$$20$$ 0 0
$$21$$ 4.32340 0.943444
$$22$$ 0 0
$$23$$ −8.97021 −1.87042 −0.935209 0.354095i $$-0.884789\pi$$
−0.935209 + 0.354095i $$0.884789\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −8.36842 −1.55398 −0.776988 0.629515i $$-0.783254\pi$$
−0.776988 + 0.629515i $$0.783254\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −1.39821 −0.243397
$$34$$ 0 0
$$35$$ 4.32340 0.730789
$$36$$ 0 0
$$37$$ 3.39821 0.558662 0.279331 0.960195i $$-0.409887\pi$$
0.279331 + 0.960195i $$0.409887\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −0.601793 −0.0939842 −0.0469921 0.998895i $$-0.514964\pi$$
−0.0469921 + 0.998895i $$0.514964\pi$$
$$42$$ 0 0
$$43$$ 2.79641 0.426449 0.213225 0.977003i $$-0.431603\pi$$
0.213225 + 0.977003i $$0.431603\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 9.29362 1.35561 0.677807 0.735240i $$-0.262931\pi$$
0.677807 + 0.735240i $$0.262931\pi$$
$$48$$ 0 0
$$49$$ 11.6918 1.67026
$$50$$ 0 0
$$51$$ −5.11982 −0.716918
$$52$$ 0 0
$$53$$ −2.19462 −0.301455 −0.150727 0.988575i $$-0.548162\pi$$
−0.150727 + 0.988575i $$0.548162\pi$$
$$54$$ 0 0
$$55$$ −1.39821 −0.188534
$$56$$ 0 0
$$57$$ 2.92520 0.387452
$$58$$ 0 0
$$59$$ −7.44322 −0.969025 −0.484513 0.874784i $$-0.661003\pi$$
−0.484513 + 0.874784i $$0.661003\pi$$
$$60$$ 0 0
$$61$$ −5.24860 −0.672015 −0.336007 0.941859i $$-0.609077\pi$$
−0.336007 + 0.941859i $$0.609077\pi$$
$$62$$ 0 0
$$63$$ −4.32340 −0.544698
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 8.97021 1.07989
$$70$$ 0 0
$$71$$ −10.0450 −1.19212 −0.596062 0.802938i $$-0.703269\pi$$
−0.596062 + 0.802938i $$0.703269\pi$$
$$72$$ 0 0
$$73$$ −13.0152 −1.52332 −0.761659 0.647978i $$-0.775615\pi$$
−0.761659 + 0.647978i $$0.775615\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −6.04502 −0.688894
$$78$$ 0 0
$$79$$ −1.95498 −0.219953 −0.109976 0.993934i $$-0.535078\pi$$
−0.109976 + 0.993934i $$0.535078\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 17.2936 1.89822 0.949111 0.314943i $$-0.101985\pi$$
0.949111 + 0.314943i $$0.101985\pi$$
$$84$$ 0 0
$$85$$ −5.11982 −0.555322
$$86$$ 0 0
$$87$$ 8.36842 0.897189
$$88$$ 0 0
$$89$$ 4.04502 0.428771 0.214385 0.976749i $$-0.431225\pi$$
0.214385 + 0.976749i $$0.431225\pi$$
$$90$$ 0 0
$$91$$ −4.32340 −0.453216
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 2.92520 0.300119
$$96$$ 0 0
$$97$$ 10.3234 1.04818 0.524091 0.851662i $$-0.324405\pi$$
0.524091 + 0.851662i $$0.324405\pi$$
$$98$$ 0 0
$$99$$ 1.39821 0.140525
$$100$$ 0 0
$$101$$ −4.92520 −0.490075 −0.245038 0.969514i $$-0.578800\pi$$
−0.245038 + 0.969514i $$0.578800\pi$$
$$102$$ 0 0
$$103$$ 1.29362 0.127464 0.0637319 0.997967i $$-0.479700\pi$$
0.0637319 + 0.997967i $$0.479700\pi$$
$$104$$ 0 0
$$105$$ −4.32340 −0.421921
$$106$$ 0 0
$$107$$ 0.751399 0.0726405 0.0363202 0.999340i $$-0.488436\pi$$
0.0363202 + 0.999340i $$0.488436\pi$$
$$108$$ 0 0
$$109$$ 14.5180 1.39057 0.695287 0.718732i $$-0.255277\pi$$
0.695287 + 0.718732i $$0.255277\pi$$
$$110$$ 0 0
$$111$$ −3.39821 −0.322544
$$112$$ 0 0
$$113$$ −7.16484 −0.674011 −0.337005 0.941503i $$-0.609414\pi$$
−0.337005 + 0.941503i $$0.609414\pi$$
$$114$$ 0 0
$$115$$ 8.97021 0.836477
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −22.1350 −2.02912
$$120$$ 0 0
$$121$$ −9.04502 −0.822274
$$122$$ 0 0
$$123$$ 0.601793 0.0542618
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −11.7008 −1.03828 −0.519138 0.854690i $$-0.673747\pi$$
−0.519138 + 0.854690i $$0.673747\pi$$
$$128$$ 0 0
$$129$$ −2.79641 −0.246211
$$130$$ 0 0
$$131$$ −19.0152 −1.66137 −0.830684 0.556744i $$-0.812050\pi$$
−0.830684 + 0.556744i $$0.812050\pi$$
$$132$$ 0 0
$$133$$ 12.6468 1.09662
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −14.7368 −1.25905 −0.629527 0.776979i $$-0.716751\pi$$
−0.629527 + 0.776979i $$0.716751\pi$$
$$138$$ 0 0
$$139$$ 5.09899 0.432491 0.216246 0.976339i $$-0.430619\pi$$
0.216246 + 0.976339i $$0.430619\pi$$
$$140$$ 0 0
$$141$$ −9.29362 −0.782664
$$142$$ 0 0
$$143$$ 1.39821 0.116924
$$144$$ 0 0
$$145$$ 8.36842 0.694959
$$146$$ 0 0
$$147$$ −11.6918 −0.964325
$$148$$ 0 0
$$149$$ −4.69182 −0.384369 −0.192185 0.981359i $$-0.561557\pi$$
−0.192185 + 0.981359i $$0.561557\pi$$
$$150$$ 0 0
$$151$$ −18.8864 −1.53696 −0.768479 0.639875i $$-0.778986\pi$$
−0.768479 + 0.639875i $$0.778986\pi$$
$$152$$ 0 0
$$153$$ 5.11982 0.413913
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ 17.1440 1.36824 0.684121 0.729369i $$-0.260186\pi$$
0.684121 + 0.729369i $$0.260186\pi$$
$$158$$ 0 0
$$159$$ 2.19462 0.174045
$$160$$ 0 0
$$161$$ 38.7819 3.05644
$$162$$ 0 0
$$163$$ −0.194622 −0.0152440 −0.00762200 0.999971i $$-0.502426\pi$$
−0.00762200 + 0.999971i $$0.502426\pi$$
$$164$$ 0 0
$$165$$ 1.39821 0.108850
$$166$$ 0 0
$$167$$ 11.4432 0.885503 0.442752 0.896644i $$-0.354002\pi$$
0.442752 + 0.896644i $$0.354002\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −2.92520 −0.223695
$$172$$ 0 0
$$173$$ 9.44322 0.717955 0.358977 0.933346i $$-0.383125\pi$$
0.358977 + 0.933346i $$0.383125\pi$$
$$174$$ 0 0
$$175$$ −4.32340 −0.326819
$$176$$ 0 0
$$177$$ 7.44322 0.559467
$$178$$ 0 0
$$179$$ 6.92520 0.517614 0.258807 0.965929i $$-0.416671\pi$$
0.258807 + 0.965929i $$0.416671\pi$$
$$180$$ 0 0
$$181$$ 7.48824 0.556596 0.278298 0.960495i $$-0.410230\pi$$
0.278298 + 0.960495i $$0.410230\pi$$
$$182$$ 0 0
$$183$$ 5.24860 0.387988
$$184$$ 0 0
$$185$$ −3.39821 −0.249841
$$186$$ 0 0
$$187$$ 7.15857 0.523486
$$188$$ 0 0
$$189$$ 4.32340 0.314481
$$190$$ 0 0
$$191$$ 12.0900 0.874804 0.437402 0.899266i $$-0.355899\pi$$
0.437402 + 0.899266i $$0.355899\pi$$
$$192$$ 0 0
$$193$$ 24.5630 1.76809 0.884043 0.467405i $$-0.154811\pi$$
0.884043 + 0.467405i $$0.154811\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 17.4432 1.24278 0.621389 0.783502i $$-0.286569\pi$$
0.621389 + 0.783502i $$0.286569\pi$$
$$198$$ 0 0
$$199$$ 14.8864 1.05527 0.527636 0.849470i $$-0.323078\pi$$
0.527636 + 0.849470i $$0.323078\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 36.1801 2.53934
$$204$$ 0 0
$$205$$ 0.601793 0.0420310
$$206$$ 0 0
$$207$$ −8.97021 −0.623473
$$208$$ 0 0
$$209$$ −4.09003 −0.282913
$$210$$ 0 0
$$211$$ 5.29362 0.364428 0.182214 0.983259i $$-0.441674\pi$$
0.182214 + 0.983259i $$0.441674\pi$$
$$212$$ 0 0
$$213$$ 10.0450 0.688273
$$214$$ 0 0
$$215$$ −2.79641 −0.190714
$$216$$ 0 0
$$217$$ 17.2936 1.17397
$$218$$ 0 0
$$219$$ 13.0152 0.879488
$$220$$ 0 0
$$221$$ 5.11982 0.344396
$$222$$ 0 0
$$223$$ −0.427995 −0.0286606 −0.0143303 0.999897i $$-0.504562\pi$$
−0.0143303 + 0.999897i $$0.504562\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 21.6829 1.43914 0.719571 0.694419i $$-0.244338\pi$$
0.719571 + 0.694419i $$0.244338\pi$$
$$228$$ 0 0
$$229$$ 5.22441 0.345239 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$230$$ 0 0
$$231$$ 6.04502 0.397733
$$232$$ 0 0
$$233$$ 11.6170 0.761056 0.380528 0.924769i $$-0.375742\pi$$
0.380528 + 0.924769i $$0.375742\pi$$
$$234$$ 0 0
$$235$$ −9.29362 −0.606249
$$236$$ 0 0
$$237$$ 1.95498 0.126990
$$238$$ 0 0
$$239$$ 11.2486 0.727612 0.363806 0.931475i $$-0.381477\pi$$
0.363806 + 0.931475i $$0.381477\pi$$
$$240$$ 0 0
$$241$$ −6.38924 −0.411567 −0.205784 0.978597i $$-0.565974\pi$$
−0.205784 + 0.978597i $$0.565974\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −11.6918 −0.746963
$$246$$ 0 0
$$247$$ −2.92520 −0.186126
$$248$$ 0 0
$$249$$ −17.2936 −1.09594
$$250$$ 0 0
$$251$$ 25.7216 1.62353 0.811767 0.583982i $$-0.198506\pi$$
0.811767 + 0.583982i $$0.198506\pi$$
$$252$$ 0 0
$$253$$ −12.5422 −0.788523
$$254$$ 0 0
$$255$$ 5.11982 0.320616
$$256$$ 0 0
$$257$$ −13.2728 −0.827934 −0.413967 0.910292i $$-0.635857\pi$$
−0.413967 + 0.910292i $$0.635857\pi$$
$$258$$ 0 0
$$259$$ −14.6918 −0.912906
$$260$$ 0 0
$$261$$ −8.36842 −0.517992
$$262$$ 0 0
$$263$$ 28.6081 1.76405 0.882024 0.471204i $$-0.156180\pi$$
0.882024 + 0.471204i $$0.156180\pi$$
$$264$$ 0 0
$$265$$ 2.19462 0.134815
$$266$$ 0 0
$$267$$ −4.04502 −0.247551
$$268$$ 0 0
$$269$$ 8.66763 0.528475 0.264237 0.964458i $$-0.414880\pi$$
0.264237 + 0.964458i $$0.414880\pi$$
$$270$$ 0 0
$$271$$ −29.6829 −1.80311 −0.901553 0.432669i $$-0.857572\pi$$
−0.901553 + 0.432669i $$0.857572\pi$$
$$272$$ 0 0
$$273$$ 4.32340 0.261664
$$274$$ 0 0
$$275$$ 1.39821 0.0843151
$$276$$ 0 0
$$277$$ 13.0540 0.784338 0.392169 0.919893i $$-0.371725\pi$$
0.392169 + 0.919893i $$0.371725\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 9.70079 0.578700 0.289350 0.957223i $$-0.406561\pi$$
0.289350 + 0.957223i $$0.406561\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ −2.92520 −0.173274
$$286$$ 0 0
$$287$$ 2.60179 0.153579
$$288$$ 0 0
$$289$$ 9.21255 0.541915
$$290$$ 0 0
$$291$$ −10.3234 −0.605169
$$292$$ 0 0
$$293$$ 22.6468 1.32304 0.661520 0.749927i $$-0.269912\pi$$
0.661520 + 0.749927i $$0.269912\pi$$
$$294$$ 0 0
$$295$$ 7.44322 0.433361
$$296$$ 0 0
$$297$$ −1.39821 −0.0811322
$$298$$ 0 0
$$299$$ −8.97021 −0.518761
$$300$$ 0 0
$$301$$ −12.0900 −0.696858
$$302$$ 0 0
$$303$$ 4.92520 0.282945
$$304$$ 0 0
$$305$$ 5.24860 0.300534
$$306$$ 0 0
$$307$$ 14.9494 0.853207 0.426603 0.904439i $$-0.359710\pi$$
0.426603 + 0.904439i $$0.359710\pi$$
$$308$$ 0 0
$$309$$ −1.29362 −0.0735913
$$310$$ 0 0
$$311$$ 6.49720 0.368423 0.184211 0.982887i $$-0.441027\pi$$
0.184211 + 0.982887i $$0.441027\pi$$
$$312$$ 0 0
$$313$$ 19.0361 1.07598 0.537991 0.842951i $$-0.319184\pi$$
0.537991 + 0.842951i $$0.319184\pi$$
$$314$$ 0 0
$$315$$ 4.32340 0.243596
$$316$$ 0 0
$$317$$ 12.4072 0.696856 0.348428 0.937336i $$-0.386716\pi$$
0.348428 + 0.937336i $$0.386716\pi$$
$$318$$ 0 0
$$319$$ −11.7008 −0.655118
$$320$$ 0 0
$$321$$ −0.751399 −0.0419390
$$322$$ 0 0
$$323$$ −14.9765 −0.833314
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ −14.5180 −0.802849
$$328$$ 0 0
$$329$$ −40.1801 −2.21520
$$330$$ 0 0
$$331$$ 25.5541 1.40458 0.702290 0.711891i $$-0.252161\pi$$
0.702290 + 0.711891i $$0.252161\pi$$
$$332$$ 0 0
$$333$$ 3.39821 0.186221
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −20.0305 −1.09113 −0.545564 0.838069i $$-0.683685\pi$$
−0.545564 + 0.838069i $$0.683685\pi$$
$$338$$ 0 0
$$339$$ 7.16484 0.389140
$$340$$ 0 0
$$341$$ −5.59283 −0.302869
$$342$$ 0 0
$$343$$ −20.2847 −1.09527
$$344$$ 0 0
$$345$$ −8.97021 −0.482940
$$346$$ 0 0
$$347$$ −26.8719 −1.44256 −0.721279 0.692644i $$-0.756446\pi$$
−0.721279 + 0.692644i $$0.756446\pi$$
$$348$$ 0 0
$$349$$ 23.4224 1.25377 0.626886 0.779111i $$-0.284329\pi$$
0.626886 + 0.779111i $$0.284329\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −17.5333 −0.933201 −0.466601 0.884468i $$-0.654521\pi$$
−0.466601 + 0.884468i $$0.654521\pi$$
$$354$$ 0 0
$$355$$ 10.0450 0.533134
$$356$$ 0 0
$$357$$ 22.1350 1.17151
$$358$$ 0 0
$$359$$ 13.2936 0.701610 0.350805 0.936448i $$-0.385908\pi$$
0.350805 + 0.936448i $$0.385908\pi$$
$$360$$ 0 0
$$361$$ −10.4432 −0.549643
$$362$$ 0 0
$$363$$ 9.04502 0.474740
$$364$$ 0 0
$$365$$ 13.0152 0.681248
$$366$$ 0 0
$$367$$ −12.0900 −0.631095 −0.315547 0.948910i $$-0.602188\pi$$
−0.315547 + 0.948910i $$0.602188\pi$$
$$368$$ 0 0
$$369$$ −0.601793 −0.0313281
$$370$$ 0 0
$$371$$ 9.48824 0.492605
$$372$$ 0 0
$$373$$ 3.55118 0.183873 0.0919366 0.995765i $$-0.470694\pi$$
0.0919366 + 0.995765i $$0.470694\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −8.36842 −0.430996
$$378$$ 0 0
$$379$$ 12.6081 0.647632 0.323816 0.946120i $$-0.395034\pi$$
0.323816 + 0.946120i $$0.395034\pi$$
$$380$$ 0 0
$$381$$ 11.7008 0.599449
$$382$$ 0 0
$$383$$ −4.94602 −0.252730 −0.126365 0.991984i $$-0.540331\pi$$
−0.126365 + 0.991984i $$0.540331\pi$$
$$384$$ 0 0
$$385$$ 6.04502 0.308083
$$386$$ 0 0
$$387$$ 2.79641 0.142150
$$388$$ 0 0
$$389$$ 33.3628 1.69156 0.845781 0.533530i $$-0.179135\pi$$
0.845781 + 0.533530i $$0.179135\pi$$
$$390$$ 0 0
$$391$$ −45.9259 −2.32257
$$392$$ 0 0
$$393$$ 19.0152 0.959191
$$394$$ 0 0
$$395$$ 1.95498 0.0983659
$$396$$ 0 0
$$397$$ −7.18903 −0.360807 −0.180403 0.983593i $$-0.557740\pi$$
−0.180403 + 0.983593i $$0.557740\pi$$
$$398$$ 0 0
$$399$$ −12.6468 −0.633132
$$400$$ 0 0
$$401$$ 19.0361 0.950615 0.475308 0.879820i $$-0.342337\pi$$
0.475308 + 0.879820i $$0.342337\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 4.75140 0.235518
$$408$$ 0 0
$$409$$ −21.5333 −1.06475 −0.532375 0.846508i $$-0.678701\pi$$
−0.532375 + 0.846508i $$0.678701\pi$$
$$410$$ 0 0
$$411$$ 14.7368 0.726915
$$412$$ 0 0
$$413$$ 32.1801 1.58348
$$414$$ 0 0
$$415$$ −17.2936 −0.848910
$$416$$ 0 0
$$417$$ −5.09899 −0.249699
$$418$$ 0 0
$$419$$ −35.9196 −1.75479 −0.877394 0.479771i $$-0.840720\pi$$
−0.877394 + 0.479771i $$0.840720\pi$$
$$420$$ 0 0
$$421$$ −23.5124 −1.14593 −0.572963 0.819581i $$-0.694206\pi$$
−0.572963 + 0.819581i $$0.694206\pi$$
$$422$$ 0 0
$$423$$ 9.29362 0.451871
$$424$$ 0 0
$$425$$ 5.11982 0.248348
$$426$$ 0 0
$$427$$ 22.6918 1.09813
$$428$$ 0 0
$$429$$ −1.39821 −0.0675061
$$430$$ 0 0
$$431$$ −10.5872 −0.509969 −0.254985 0.966945i $$-0.582070\pi$$
−0.254985 + 0.966945i $$0.582070\pi$$
$$432$$ 0 0
$$433$$ −38.4376 −1.84719 −0.923597 0.383364i $$-0.874765\pi$$
−0.923597 + 0.383364i $$0.874765\pi$$
$$434$$ 0 0
$$435$$ −8.36842 −0.401235
$$436$$ 0 0
$$437$$ 26.2396 1.25521
$$438$$ 0 0
$$439$$ −15.6378 −0.746354 −0.373177 0.927760i $$-0.621732\pi$$
−0.373177 + 0.927760i $$0.621732\pi$$
$$440$$ 0 0
$$441$$ 11.6918 0.556754
$$442$$ 0 0
$$443$$ 17.7875 0.845107 0.422554 0.906338i $$-0.361134\pi$$
0.422554 + 0.906338i $$0.361134\pi$$
$$444$$ 0 0
$$445$$ −4.04502 −0.191752
$$446$$ 0 0
$$447$$ 4.69182 0.221916
$$448$$ 0 0
$$449$$ 29.7279 1.40295 0.701473 0.712696i $$-0.252526\pi$$
0.701473 + 0.712696i $$0.252526\pi$$
$$450$$ 0 0
$$451$$ −0.841431 −0.0396214
$$452$$ 0 0
$$453$$ 18.8864 0.887363
$$454$$ 0 0
$$455$$ 4.32340 0.202684
$$456$$ 0 0
$$457$$ 3.61702 0.169197 0.0845986 0.996415i $$-0.473039\pi$$
0.0845986 + 0.996415i $$0.473039\pi$$
$$458$$ 0 0
$$459$$ −5.11982 −0.238973
$$460$$ 0 0
$$461$$ 12.1350 0.565186 0.282593 0.959240i $$-0.408805\pi$$
0.282593 + 0.959240i $$0.408805\pi$$
$$462$$ 0 0
$$463$$ 7.11982 0.330886 0.165443 0.986219i $$-0.447095\pi$$
0.165443 + 0.986219i $$0.447095\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ −0.542218 −0.0250909 −0.0125454 0.999921i $$-0.503993\pi$$
−0.0125454 + 0.999921i $$0.503993\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −17.1440 −0.789954
$$472$$ 0 0
$$473$$ 3.90997 0.179781
$$474$$ 0 0
$$475$$ −2.92520 −0.134217
$$476$$ 0 0
$$477$$ −2.19462 −0.100485
$$478$$ 0 0
$$479$$ 17.7458 0.810826 0.405413 0.914134i $$-0.367128\pi$$
0.405413 + 0.914134i $$0.367128\pi$$
$$480$$ 0 0
$$481$$ 3.39821 0.154945
$$482$$ 0 0
$$483$$ −38.7819 −1.76464
$$484$$ 0 0
$$485$$ −10.3234 −0.468762
$$486$$ 0 0
$$487$$ −17.5270 −0.794224 −0.397112 0.917770i $$-0.629988\pi$$
−0.397112 + 0.917770i $$0.629988\pi$$
$$488$$ 0 0
$$489$$ 0.194622 0.00880112
$$490$$ 0 0
$$491$$ −19.4045 −0.875712 −0.437856 0.899045i $$-0.644262\pi$$
−0.437856 + 0.899045i $$0.644262\pi$$
$$492$$ 0 0
$$493$$ −42.8448 −1.92963
$$494$$ 0 0
$$495$$ −1.39821 −0.0628448
$$496$$ 0 0
$$497$$ 43.4287 1.94804
$$498$$ 0 0
$$499$$ −15.6620 −0.701129 −0.350565 0.936539i $$-0.614010\pi$$
−0.350565 + 0.936539i $$0.614010\pi$$
$$500$$ 0 0
$$501$$ −11.4432 −0.511246
$$502$$ 0 0
$$503$$ 8.90727 0.397156 0.198578 0.980085i $$-0.436368\pi$$
0.198578 + 0.980085i $$0.436368\pi$$
$$504$$ 0 0
$$505$$ 4.92520 0.219168
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 40.2251 1.78295 0.891473 0.453074i $$-0.149673\pi$$
0.891473 + 0.453074i $$0.149673\pi$$
$$510$$ 0 0
$$511$$ 56.2701 2.48924
$$512$$ 0 0
$$513$$ 2.92520 0.129151
$$514$$ 0 0
$$515$$ −1.29362 −0.0570036
$$516$$ 0 0
$$517$$ 12.9944 0.571493
$$518$$ 0 0
$$519$$ −9.44322 −0.414512
$$520$$ 0 0
$$521$$ −36.0305 −1.57852 −0.789262 0.614057i $$-0.789536\pi$$
−0.789262 + 0.614057i $$0.789536\pi$$
$$522$$ 0 0
$$523$$ −41.3836 −1.80958 −0.904790 0.425857i $$-0.859973\pi$$
−0.904790 + 0.425857i $$0.859973\pi$$
$$524$$ 0 0
$$525$$ 4.32340 0.188689
$$526$$ 0 0
$$527$$ −20.4793 −0.892091
$$528$$ 0 0
$$529$$ 57.4647 2.49847
$$530$$ 0 0
$$531$$ −7.44322 −0.323008
$$532$$ 0 0
$$533$$ −0.601793 −0.0260665
$$534$$ 0 0
$$535$$ −0.751399 −0.0324858
$$536$$ 0 0
$$537$$ −6.92520 −0.298844
$$538$$ 0 0
$$539$$ 16.3476 0.704141
$$540$$ 0 0
$$541$$ −29.3144 −1.26033 −0.630163 0.776463i $$-0.717012\pi$$
−0.630163 + 0.776463i $$0.717012\pi$$
$$542$$ 0 0
$$543$$ −7.48824 −0.321351
$$544$$ 0 0
$$545$$ −14.5180 −0.621884
$$546$$ 0 0
$$547$$ 30.5872 1.30782 0.653908 0.756574i $$-0.273128\pi$$
0.653908 + 0.756574i $$0.273128\pi$$
$$548$$ 0 0
$$549$$ −5.24860 −0.224005
$$550$$ 0 0
$$551$$ 24.4793 1.04285
$$552$$ 0 0
$$553$$ 8.45219 0.359424
$$554$$ 0 0
$$555$$ 3.39821 0.144246
$$556$$ 0 0
$$557$$ 13.9100 0.589384 0.294692 0.955592i $$-0.404783\pi$$
0.294692 + 0.955592i $$0.404783\pi$$
$$558$$ 0 0
$$559$$ 2.79641 0.118276
$$560$$ 0 0
$$561$$ −7.15857 −0.302235
$$562$$ 0 0
$$563$$ 35.5962 1.50020 0.750100 0.661324i $$-0.230005\pi$$
0.750100 + 0.661324i $$0.230005\pi$$
$$564$$ 0 0
$$565$$ 7.16484 0.301427
$$566$$ 0 0
$$567$$ −4.32340 −0.181566
$$568$$ 0 0
$$569$$ −9.44322 −0.395881 −0.197940 0.980214i $$-0.563425\pi$$
−0.197940 + 0.980214i $$0.563425\pi$$
$$570$$ 0 0
$$571$$ 18.3026 0.765939 0.382970 0.923761i $$-0.374901\pi$$
0.382970 + 0.923761i $$0.374901\pi$$
$$572$$ 0 0
$$573$$ −12.0900 −0.505068
$$574$$ 0 0
$$575$$ −8.97021 −0.374084
$$576$$ 0 0
$$577$$ 4.29295 0.178718 0.0893589 0.995999i $$-0.471518\pi$$
0.0893589 + 0.995999i $$0.471518\pi$$
$$578$$ 0 0
$$579$$ −24.5630 −1.02081
$$580$$ 0 0
$$581$$ −74.7673 −3.10187
$$582$$ 0 0
$$583$$ −3.06854 −0.127086
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ 30.1980 1.24640 0.623202 0.782061i $$-0.285831\pi$$
0.623202 + 0.782061i $$0.285831\pi$$
$$588$$ 0 0
$$589$$ 11.7008 0.482123
$$590$$ 0 0
$$591$$ −17.4432 −0.717518
$$592$$ 0 0
$$593$$ 21.7908 0.894842 0.447421 0.894324i $$-0.352343\pi$$
0.447421 + 0.894324i $$0.352343\pi$$
$$594$$ 0 0
$$595$$ 22.1350 0.907448
$$596$$ 0 0
$$597$$ −14.8864 −0.609262
$$598$$ 0 0
$$599$$ 4.81434 0.196709 0.0983543 0.995151i $$-0.468642\pi$$
0.0983543 + 0.995151i $$0.468642\pi$$
$$600$$ 0 0
$$601$$ 0.991037 0.0404252 0.0202126 0.999796i $$-0.493566\pi$$
0.0202126 + 0.999796i $$0.493566\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 9.04502 0.367732
$$606$$ 0 0
$$607$$ −28.0900 −1.14014 −0.570070 0.821596i $$-0.693084\pi$$
−0.570070 + 0.821596i $$0.693084\pi$$
$$608$$ 0 0
$$609$$ −36.1801 −1.46609
$$610$$ 0 0
$$611$$ 9.29362 0.375980
$$612$$ 0 0
$$613$$ −23.3982 −0.945045 −0.472522 0.881319i $$-0.656656\pi$$
−0.472522 + 0.881319i $$0.656656\pi$$
$$614$$ 0 0
$$615$$ −0.601793 −0.0242666
$$616$$ 0 0
$$617$$ 28.0305 1.12846 0.564232 0.825616i $$-0.309172\pi$$
0.564232 + 0.825616i $$0.309172\pi$$
$$618$$ 0 0
$$619$$ −27.7521 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$620$$ 0 0
$$621$$ 8.97021 0.359962
$$622$$ 0 0
$$623$$ −17.4882 −0.700652
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.09003 0.163340
$$628$$ 0 0
$$629$$ 17.3982 0.693712
$$630$$ 0 0
$$631$$ 50.2105 1.99885 0.999425 0.0339171i $$-0.0107982\pi$$
0.999425 + 0.0339171i $$0.0107982\pi$$
$$632$$ 0 0
$$633$$ −5.29362 −0.210402
$$634$$ 0 0
$$635$$ 11.7008 0.464332
$$636$$ 0 0
$$637$$ 11.6918 0.463247
$$638$$ 0 0
$$639$$ −10.0450 −0.397375
$$640$$ 0 0
$$641$$ −23.2936 −0.920043 −0.460021 0.887908i $$-0.652158\pi$$
−0.460021 + 0.887908i $$0.652158\pi$$
$$642$$ 0 0
$$643$$ 4.54222 0.179128 0.0895638 0.995981i $$-0.471453\pi$$
0.0895638 + 0.995981i $$0.471453\pi$$
$$644$$ 0 0
$$645$$ 2.79641 0.110109
$$646$$ 0 0
$$647$$ −2.73057 −0.107350 −0.0536750 0.998558i $$-0.517093\pi$$
−0.0536750 + 0.998558i $$0.517093\pi$$
$$648$$ 0 0
$$649$$ −10.4072 −0.408517
$$650$$ 0 0
$$651$$ −17.2936 −0.677790
$$652$$ 0 0
$$653$$ −49.5816 −1.94028 −0.970140 0.242547i $$-0.922017\pi$$
−0.970140 + 0.242547i $$0.922017\pi$$
$$654$$ 0 0
$$655$$ 19.0152 0.742986
$$656$$ 0 0
$$657$$ −13.0152 −0.507772
$$658$$ 0 0
$$659$$ 20.0513 0.781087 0.390544 0.920584i $$-0.372287\pi$$
0.390544 + 0.920584i $$0.372287\pi$$
$$660$$ 0 0
$$661$$ −15.5415 −0.604496 −0.302248 0.953229i $$-0.597737\pi$$
−0.302248 + 0.953229i $$0.597737\pi$$
$$662$$ 0 0
$$663$$ −5.11982 −0.198837
$$664$$ 0 0
$$665$$ −12.6468 −0.490422
$$666$$ 0 0
$$667$$ 75.0665 2.90659
$$668$$ 0 0
$$669$$ 0.427995 0.0165472
$$670$$ 0 0
$$671$$ −7.33863 −0.283305
$$672$$ 0 0
$$673$$ 15.2036 0.586055 0.293028 0.956104i $$-0.405337\pi$$
0.293028 + 0.956104i $$0.405337\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 26.5422 1.02010 0.510050 0.860145i $$-0.329627\pi$$
0.510050 + 0.860145i $$0.329627\pi$$
$$678$$ 0 0
$$679$$ −44.6323 −1.71283
$$680$$ 0 0
$$681$$ −21.6829 −0.830889
$$682$$ 0 0
$$683$$ 34.1980 1.30855 0.654275 0.756257i $$-0.272974\pi$$
0.654275 + 0.756257i $$0.272974\pi$$
$$684$$ 0 0
$$685$$ 14.7368 0.563066
$$686$$ 0 0
$$687$$ −5.22441 −0.199324
$$688$$ 0 0
$$689$$ −2.19462 −0.0836085
$$690$$ 0 0
$$691$$ −35.6204 −1.35506 −0.677532 0.735494i $$-0.736950\pi$$
−0.677532 + 0.735494i $$0.736950\pi$$
$$692$$ 0 0
$$693$$ −6.04502 −0.229631
$$694$$ 0 0
$$695$$ −5.09899 −0.193416
$$696$$ 0 0
$$697$$ −3.08107 −0.116704
$$698$$ 0 0
$$699$$ −11.6170 −0.439396
$$700$$ 0 0
$$701$$ −32.6260 −1.23227 −0.616133 0.787642i $$-0.711302\pi$$
−0.616133 + 0.787642i $$0.711302\pi$$
$$702$$ 0 0
$$703$$ −9.94043 −0.374910
$$704$$ 0 0
$$705$$ 9.29362 0.350018
$$706$$ 0 0
$$707$$ 21.2936 0.800829
$$708$$ 0 0
$$709$$ 42.3989 1.59232 0.796162 0.605084i $$-0.206860\pi$$
0.796162 + 0.605084i $$0.206860\pi$$
$$710$$ 0 0
$$711$$ −1.95498 −0.0733176
$$712$$ 0 0
$$713$$ 35.8809 1.34375
$$714$$ 0 0
$$715$$ −1.39821 −0.0522900
$$716$$ 0 0
$$717$$ −11.2486 −0.420087
$$718$$ 0 0
$$719$$ 0.855989 0.0319230 0.0159615 0.999873i $$-0.494919\pi$$
0.0159615 + 0.999873i $$0.494919\pi$$
$$720$$ 0 0
$$721$$ −5.59283 −0.208288
$$722$$ 0 0
$$723$$ 6.38924 0.237619
$$724$$ 0 0
$$725$$ −8.36842 −0.310795
$$726$$ 0 0
$$727$$ 12.3476 0.457947 0.228973 0.973433i $$-0.426463\pi$$
0.228973 + 0.973433i $$0.426463\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 14.3171 0.529538
$$732$$ 0 0
$$733$$ −27.4882 −1.01530 −0.507651 0.861563i $$-0.669486\pi$$
−0.507651 + 0.861563i $$0.669486\pi$$
$$734$$ 0 0
$$735$$ 11.6918 0.431259
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 36.3989 1.33895 0.669477 0.742833i $$-0.266518\pi$$
0.669477 + 0.742833i $$0.266518\pi$$
$$740$$ 0 0
$$741$$ 2.92520 0.107460
$$742$$ 0 0
$$743$$ −35.2340 −1.29261 −0.646306 0.763078i $$-0.723687\pi$$
−0.646306 + 0.763078i $$0.723687\pi$$
$$744$$ 0 0
$$745$$ 4.69182 0.171895
$$746$$ 0 0
$$747$$ 17.2936 0.632740
$$748$$ 0 0
$$749$$ −3.24860 −0.118701
$$750$$ 0 0
$$751$$ 3.54781 0.129462 0.0647308 0.997903i $$-0.479381\pi$$
0.0647308 + 0.997903i $$0.479381\pi$$
$$752$$ 0 0
$$753$$ −25.7216 −0.937348
$$754$$ 0 0
$$755$$ 18.8864 0.687348
$$756$$ 0 0
$$757$$ −22.6052 −0.821599 −0.410799 0.911726i $$-0.634750\pi$$
−0.410799 + 0.911726i $$0.634750\pi$$
$$758$$ 0 0
$$759$$ 12.5422 0.455254
$$760$$ 0 0
$$761$$ 29.6233 1.07384 0.536922 0.843632i $$-0.319587\pi$$
0.536922 + 0.843632i $$0.319587\pi$$
$$762$$ 0 0
$$763$$ −62.7673 −2.27233
$$764$$ 0 0
$$765$$ −5.11982 −0.185107
$$766$$ 0 0
$$767$$ −7.44322 −0.268759
$$768$$ 0 0
$$769$$ 3.89204 0.140351 0.0701753 0.997535i $$-0.477644\pi$$
0.0701753 + 0.997535i $$0.477644\pi$$
$$770$$ 0 0
$$771$$ 13.2728 0.478008
$$772$$ 0 0
$$773$$ −15.1261 −0.544047 −0.272024 0.962291i $$-0.587693\pi$$
−0.272024 + 0.962291i $$0.587693\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 14.6918 0.527066
$$778$$ 0 0
$$779$$ 1.76036 0.0630715
$$780$$ 0 0
$$781$$ −14.0450 −0.502570
$$782$$ 0 0
$$783$$ 8.36842 0.299063
$$784$$ 0 0
$$785$$ −17.1440 −0.611896
$$786$$ 0 0
$$787$$ −7.14401 −0.254656 −0.127328 0.991861i $$-0.540640\pi$$
−0.127328 + 0.991861i $$0.540640\pi$$
$$788$$ 0 0
$$789$$ −28.6081 −1.01847
$$790$$ 0 0
$$791$$ 30.9765 1.10140
$$792$$ 0 0
$$793$$ −5.24860 −0.186383
$$794$$ 0 0
$$795$$ −2.19462 −0.0778352
$$796$$ 0 0
$$797$$ −24.6918 −0.874629 −0.437315 0.899309i $$-0.644070\pi$$
−0.437315 + 0.899309i $$0.644070\pi$$
$$798$$ 0 0
$$799$$ 47.5816 1.68332
$$800$$ 0 0
$$801$$ 4.04502 0.142924
$$802$$ 0 0
$$803$$ −18.1980 −0.642193
$$804$$ 0 0
$$805$$ −38.7819 −1.36688
$$806$$ 0 0
$$807$$ −8.66763 −0.305115
$$808$$ 0 0
$$809$$ −16.5872 −0.583176 −0.291588 0.956544i $$-0.594184\pi$$
−0.291588 + 0.956544i $$0.594184\pi$$
$$810$$ 0 0
$$811$$ 30.3684 1.06638 0.533190 0.845996i $$-0.320993\pi$$
0.533190 + 0.845996i $$0.320993\pi$$
$$812$$ 0 0
$$813$$ 29.6829 1.04102
$$814$$ 0 0
$$815$$ 0.194622 0.00681732
$$816$$ 0 0
$$817$$ −8.18006 −0.286184
$$818$$ 0 0
$$819$$ −4.32340 −0.151072
$$820$$ 0 0
$$821$$ −12.9010 −0.450248 −0.225124 0.974330i $$-0.572279\pi$$
−0.225124 + 0.974330i $$0.572279\pi$$
$$822$$ 0 0
$$823$$ −14.7064 −0.512632 −0.256316 0.966593i $$-0.582509\pi$$
−0.256316 + 0.966593i $$0.582509\pi$$
$$824$$ 0 0
$$825$$ −1.39821 −0.0486793
$$826$$ 0 0
$$827$$ 37.5928 1.30723 0.653615 0.756827i $$-0.273251\pi$$
0.653615 + 0.756827i $$0.273251\pi$$
$$828$$ 0 0
$$829$$ −29.3657 −1.01991 −0.509957 0.860200i $$-0.670339\pi$$
−0.509957 + 0.860200i $$0.670339\pi$$
$$830$$ 0 0
$$831$$ −13.0540 −0.452838
$$832$$ 0 0
$$833$$ 59.8600 2.07403
$$834$$ 0 0
$$835$$ −11.4432 −0.396009
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 0 0
$$839$$ −18.8594 −0.651097 −0.325549 0.945525i $$-0.605549\pi$$
−0.325549 + 0.945525i $$0.605549\pi$$
$$840$$ 0 0
$$841$$ 41.0305 1.41484
$$842$$ 0 0
$$843$$ −9.70079 −0.334113
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 39.1053 1.34367
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −30.4826 −1.04493
$$852$$ 0 0
$$853$$ −39.5783 −1.35513 −0.677567 0.735461i $$-0.736966\pi$$
−0.677567 + 0.735461i $$0.736966\pi$$
$$854$$ 0 0
$$855$$ 2.92520 0.100040
$$856$$ 0 0
$$857$$ 36.0546 1.23160 0.615802 0.787901i $$-0.288832\pi$$
0.615802 + 0.787901i $$0.288832\pi$$
$$858$$ 0 0
$$859$$ −8.75140 −0.298594 −0.149297 0.988792i $$-0.547701\pi$$
−0.149297 + 0.988792i $$0.547701\pi$$
$$860$$ 0 0
$$861$$ −2.60179 −0.0886689
$$862$$ 0 0
$$863$$ −35.4141 −1.20551 −0.602755 0.797926i $$-0.705930\pi$$
−0.602755 + 0.797926i $$0.705930\pi$$
$$864$$ 0 0
$$865$$ −9.44322 −0.321079
$$866$$ 0 0
$$867$$ −9.21255 −0.312875
$$868$$ 0 0
$$869$$ −2.73347 −0.0927267
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 10.3234 0.349394
$$874$$ 0 0
$$875$$ 4.32340 0.146158
$$876$$ 0 0
$$877$$ 53.0665 1.79193 0.895964 0.444126i $$-0.146486\pi$$
0.895964 + 0.444126i $$0.146486\pi$$
$$878$$ 0 0
$$879$$ −22.6468 −0.763858
$$880$$ 0 0
$$881$$ −38.6468 −1.30204 −0.651022 0.759059i $$-0.725659\pi$$
−0.651022 + 0.759059i $$0.725659\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ −7.44322 −0.250201
$$886$$ 0 0
$$887$$ 38.2222 1.28338 0.641688 0.766966i $$-0.278235\pi$$
0.641688 + 0.766966i $$0.278235\pi$$
$$888$$ 0 0
$$889$$ 50.5872 1.69664
$$890$$ 0 0
$$891$$ 1.39821 0.0468417
$$892$$ 0 0
$$893$$ −27.1857 −0.909733
$$894$$ 0 0
$$895$$ −6.92520 −0.231484
$$896$$ 0 0
$$897$$ 8.97021 0.299507
$$898$$ 0 0
$$899$$ 33.4737 1.11641
$$900$$ 0 0
$$901$$ −11.2361 −0.374328
$$902$$ 0 0
$$903$$ 12.0900 0.402331
$$904$$ 0 0
$$905$$ −7.48824 −0.248917
$$906$$ 0 0
$$907$$ −38.8573 −1.29024 −0.645118 0.764083i $$-0.723192\pi$$
−0.645118 + 0.764083i $$0.723192\pi$$
$$908$$ 0 0
$$909$$ −4.92520 −0.163358
$$910$$ 0 0
$$911$$ 32.8269 1.08760 0.543801 0.839214i $$-0.316984\pi$$
0.543801 + 0.839214i $$0.316984\pi$$
$$912$$ 0 0
$$913$$ 24.1801 0.800243
$$914$$ 0 0
$$915$$ −5.24860 −0.173513
$$916$$ 0 0
$$917$$ 82.2105 2.71483
$$918$$ 0 0
$$919$$ −46.3151 −1.52779 −0.763897 0.645338i $$-0.776717\pi$$
−0.763897 + 0.645338i $$0.776717\pi$$
$$920$$ 0 0
$$921$$ −14.9494 −0.492599
$$922$$ 0 0
$$923$$ −10.0450 −0.330636
$$924$$ 0 0
$$925$$ 3.39821 0.111732
$$926$$ 0 0
$$927$$ 1.29362 0.0424880
$$928$$ 0 0
$$929$$ −24.4343 −0.801662 −0.400831 0.916152i $$-0.631279\pi$$
−0.400831 + 0.916152i $$0.631279\pi$$
$$930$$ 0 0
$$931$$ −34.2009 −1.12089
$$932$$ 0 0
$$933$$ −6.49720 −0.212709
$$934$$ 0 0
$$935$$ −7.15857 −0.234110
$$936$$ 0 0
$$937$$ 14.6885 0.479851 0.239925 0.970791i $$-0.422877\pi$$
0.239925 + 0.970791i $$0.422877\pi$$
$$938$$ 0 0
$$939$$ −19.0361 −0.621218
$$940$$ 0 0
$$941$$ 12.0034 0.391299 0.195649 0.980674i $$-0.437319\pi$$
0.195649 + 0.980674i $$0.437319\pi$$
$$942$$ 0 0
$$943$$ 5.39821 0.175790
$$944$$ 0 0
$$945$$ −4.32340 −0.140640
$$946$$ 0 0
$$947$$ 35.7008 1.16012 0.580060 0.814574i $$-0.303029\pi$$
0.580060 + 0.814574i $$0.303029\pi$$
$$948$$ 0 0
$$949$$ −13.0152 −0.422492
$$950$$ 0 0
$$951$$ −12.4072 −0.402330
$$952$$ 0 0
$$953$$ −28.1738 −0.912639 −0.456319 0.889816i $$-0.650833\pi$$
−0.456319 + 0.889816i $$0.650833\pi$$
$$954$$ 0 0
$$955$$ −12.0900 −0.391224
$$956$$ 0 0
$$957$$ 11.7008 0.378233
$$958$$ 0 0
$$959$$ 63.7133 2.05741
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0.751399 0.0242135
$$964$$ 0 0
$$965$$ −24.5630 −0.790712
$$966$$ 0 0
$$967$$ −0.685559 −0.0220461 −0.0110230 0.999939i $$-0.503509\pi$$
−0.0110230 + 0.999939i $$0.503509\pi$$
$$968$$ 0 0
$$969$$ 14.9765 0.481114
$$970$$ 0 0
$$971$$ 33.6925 1.08124 0.540622 0.841266i $$-0.318189\pi$$
0.540622 + 0.841266i $$0.318189\pi$$
$$972$$ 0 0
$$973$$ −22.0450 −0.706731
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ −38.4793 −1.23106 −0.615531 0.788113i $$-0.711058\pi$$
−0.615531 + 0.788113i $$0.711058\pi$$
$$978$$ 0 0
$$979$$ 5.65577 0.180759
$$980$$ 0 0
$$981$$ 14.5180 0.463525
$$982$$ 0 0
$$983$$ −33.7312 −1.07586 −0.537930 0.842990i $$-0.680793\pi$$
−0.537930 + 0.842990i $$0.680793\pi$$
$$984$$ 0 0
$$985$$ −17.4432 −0.555787
$$986$$ 0 0
$$987$$ 40.1801 1.27895
$$988$$ 0 0
$$989$$ −25.0844 −0.797639
$$990$$ 0 0
$$991$$ 33.5366 1.06533 0.532663 0.846327i $$-0.321191\pi$$
0.532663 + 0.846327i $$0.321191\pi$$
$$992$$ 0 0
$$993$$ −25.5541 −0.810934
$$994$$ 0 0
$$995$$ −14.8864 −0.471932
$$996$$ 0 0
$$997$$ −54.2284 −1.71743 −0.858716 0.512452i $$-0.828737\pi$$
−0.858716 + 0.512452i $$0.828737\pi$$
$$998$$ 0 0
$$999$$ −3.39821 −0.107515
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 6240.2.a.bw.1.1 3
4.3 odd 2 6240.2.a.cb.1.3 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bw.1.1 3 1.1 even 1 trivial
6240.2.a.cb.1.3 yes 3 4.3 odd 2