# Properties

 Label 7448.2.a.bn.1.6 Level $7448$ Weight $2$ Character 7448.1 Self dual yes Analytic conductor $59.473$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

## 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.98211824.1 Defining polynomial: $$x^{6} - 3 x^{5} - 6 x^{4} + 15 x^{3} + 8 x^{2} - 9 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-2.04436$$ of defining polynomial Character $$\chi$$ $$=$$ 7448.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.04436 q^{3} -2.60221 q^{5} +6.26815 q^{9} +O(q^{10})$$ $$q+3.04436 q^{3} -2.60221 q^{5} +6.26815 q^{9} +3.68293 q^{11} +6.17208 q^{13} -7.92207 q^{15} -6.92207 q^{17} -1.00000 q^{19} -2.61920 q^{23} +1.77149 q^{25} +9.94945 q^{27} +3.12082 q^{29} +6.97542 q^{31} +11.2122 q^{33} +0.292486 q^{37} +18.7901 q^{39} +5.29265 q^{41} -1.96364 q^{43} -16.3111 q^{45} +7.99528 q^{47} -21.0733 q^{51} +1.79173 q^{53} -9.58376 q^{55} -3.04436 q^{57} +7.12345 q^{59} -2.14820 q^{61} -16.0610 q^{65} -7.63547 q^{67} -7.97378 q^{69} +3.29274 q^{71} +8.82925 q^{73} +5.39308 q^{75} -6.39094 q^{79} +11.4853 q^{81} +7.35997 q^{83} +18.0127 q^{85} +9.50092 q^{87} -13.5002 q^{89} +21.2357 q^{93} +2.60221 q^{95} -8.90409 q^{97} +23.0852 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} + q^{5} + 3 q^{9} + O(q^{10})$$ $$6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99} + O(q^{100})$$

## 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
$$3$$ 3.04436 1.75766 0.878832 0.477131i $$-0.158323\pi$$
0.878832 + 0.477131i $$0.158323\pi$$
$$4$$ 0 0
$$5$$ −2.60221 −1.16374 −0.581872 0.813281i $$-0.697679\pi$$
−0.581872 + 0.813281i $$0.697679\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 6.26815 2.08938
$$10$$ 0 0
$$11$$ 3.68293 1.11045 0.555223 0.831702i $$-0.312633\pi$$
0.555223 + 0.831702i $$0.312633\pi$$
$$12$$ 0 0
$$13$$ 6.17208 1.71183 0.855913 0.517119i $$-0.172996\pi$$
0.855913 + 0.517119i $$0.172996\pi$$
$$14$$ 0 0
$$15$$ −7.92207 −2.04547
$$16$$ 0 0
$$17$$ −6.92207 −1.67885 −0.839425 0.543476i $$-0.817108\pi$$
−0.839425 + 0.543476i $$0.817108\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.61920 −0.546140 −0.273070 0.961994i $$-0.588039\pi$$
−0.273070 + 0.961994i $$0.588039\pi$$
$$24$$ 0 0
$$25$$ 1.77149 0.354299
$$26$$ 0 0
$$27$$ 9.94945 1.91477
$$28$$ 0 0
$$29$$ 3.12082 0.579522 0.289761 0.957099i $$-0.406424\pi$$
0.289761 + 0.957099i $$0.406424\pi$$
$$30$$ 0 0
$$31$$ 6.97542 1.25282 0.626411 0.779493i $$-0.284523\pi$$
0.626411 + 0.779493i $$0.284523\pi$$
$$32$$ 0 0
$$33$$ 11.2122 1.95179
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.292486 0.0480843 0.0240422 0.999711i $$-0.492346\pi$$
0.0240422 + 0.999711i $$0.492346\pi$$
$$38$$ 0 0
$$39$$ 18.7901 3.00882
$$40$$ 0 0
$$41$$ 5.29265 0.826573 0.413287 0.910601i $$-0.364381\pi$$
0.413287 + 0.910601i $$0.364381\pi$$
$$42$$ 0 0
$$43$$ −1.96364 −0.299453 −0.149726 0.988727i $$-0.547839\pi$$
−0.149726 + 0.988727i $$0.547839\pi$$
$$44$$ 0 0
$$45$$ −16.3111 −2.43151
$$46$$ 0 0
$$47$$ 7.99528 1.16623 0.583116 0.812389i $$-0.301833\pi$$
0.583116 + 0.812389i $$0.301833\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −21.0733 −2.95085
$$52$$ 0 0
$$53$$ 1.79173 0.246113 0.123056 0.992400i $$-0.460730\pi$$
0.123056 + 0.992400i $$0.460730\pi$$
$$54$$ 0 0
$$55$$ −9.58376 −1.29227
$$56$$ 0 0
$$57$$ −3.04436 −0.403236
$$58$$ 0 0
$$59$$ 7.12345 0.927395 0.463697 0.885994i $$-0.346523\pi$$
0.463697 + 0.885994i $$0.346523\pi$$
$$60$$ 0 0
$$61$$ −2.14820 −0.275049 −0.137525 0.990498i $$-0.543915\pi$$
−0.137525 + 0.990498i $$0.543915\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −16.0610 −1.99213
$$66$$ 0 0
$$67$$ −7.63547 −0.932821 −0.466410 0.884568i $$-0.654453\pi$$
−0.466410 + 0.884568i $$0.654453\pi$$
$$68$$ 0 0
$$69$$ −7.97378 −0.959931
$$70$$ 0 0
$$71$$ 3.29274 0.390776 0.195388 0.980726i $$-0.437403\pi$$
0.195388 + 0.980726i $$0.437403\pi$$
$$72$$ 0 0
$$73$$ 8.82925 1.03339 0.516693 0.856171i $$-0.327163\pi$$
0.516693 + 0.856171i $$0.327163\pi$$
$$74$$ 0 0
$$75$$ 5.39308 0.622739
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −6.39094 −0.719037 −0.359519 0.933138i $$-0.617059\pi$$
−0.359519 + 0.933138i $$0.617059\pi$$
$$80$$ 0 0
$$81$$ 11.4853 1.27614
$$82$$ 0 0
$$83$$ 7.35997 0.807861 0.403931 0.914790i $$-0.367644\pi$$
0.403931 + 0.914790i $$0.367644\pi$$
$$84$$ 0 0
$$85$$ 18.0127 1.95375
$$86$$ 0 0
$$87$$ 9.50092 1.01861
$$88$$ 0 0
$$89$$ −13.5002 −1.43102 −0.715509 0.698603i $$-0.753805\pi$$
−0.715509 + 0.698603i $$0.753805\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 21.2357 2.20204
$$94$$ 0 0
$$95$$ 2.60221 0.266981
$$96$$ 0 0
$$97$$ −8.90409 −0.904073 −0.452037 0.891999i $$-0.649302\pi$$
−0.452037 + 0.891999i $$0.649302\pi$$
$$98$$ 0 0
$$99$$ 23.0852 2.32015
$$100$$ 0 0
$$101$$ 14.8128 1.47392 0.736962 0.675934i $$-0.236259\pi$$
0.736962 + 0.675934i $$0.236259\pi$$
$$102$$ 0 0
$$103$$ 16.8373 1.65902 0.829512 0.558489i $$-0.188619\pi$$
0.829512 + 0.558489i $$0.188619\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −13.1650 −1.27271 −0.636353 0.771398i $$-0.719558\pi$$
−0.636353 + 0.771398i $$0.719558\pi$$
$$108$$ 0 0
$$109$$ 10.0004 0.957866 0.478933 0.877852i $$-0.341024\pi$$
0.478933 + 0.877852i $$0.341024\pi$$
$$110$$ 0 0
$$111$$ 0.890433 0.0845162
$$112$$ 0 0
$$113$$ −17.6013 −1.65579 −0.827895 0.560883i $$-0.810462\pi$$
−0.827895 + 0.560883i $$0.810462\pi$$
$$114$$ 0 0
$$115$$ 6.81569 0.635567
$$116$$ 0 0
$$117$$ 38.6876 3.57667
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2.56398 0.233089
$$122$$ 0 0
$$123$$ 16.1128 1.45284
$$124$$ 0 0
$$125$$ 8.40125 0.751430
$$126$$ 0 0
$$127$$ 13.8533 1.22928 0.614640 0.788807i $$-0.289301\pi$$
0.614640 + 0.788807i $$0.289301\pi$$
$$128$$ 0 0
$$129$$ −5.97805 −0.526338
$$130$$ 0 0
$$131$$ 5.67842 0.496126 0.248063 0.968744i $$-0.420206\pi$$
0.248063 + 0.968744i $$0.420206\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −25.8906 −2.22830
$$136$$ 0 0
$$137$$ 20.8928 1.78499 0.892495 0.451057i $$-0.148953\pi$$
0.892495 + 0.451057i $$0.148953\pi$$
$$138$$ 0 0
$$139$$ −18.3740 −1.55847 −0.779233 0.626735i $$-0.784391\pi$$
−0.779233 + 0.626735i $$0.784391\pi$$
$$140$$ 0 0
$$141$$ 24.3406 2.04984
$$142$$ 0 0
$$143$$ 22.7313 1.90089
$$144$$ 0 0
$$145$$ −8.12104 −0.674415
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3.86236 −0.316417 −0.158208 0.987406i $$-0.550572\pi$$
−0.158208 + 0.987406i $$0.550572\pi$$
$$150$$ 0 0
$$151$$ 21.4781 1.74786 0.873932 0.486047i $$-0.161562\pi$$
0.873932 + 0.486047i $$0.161562\pi$$
$$152$$ 0 0
$$153$$ −43.3886 −3.50776
$$154$$ 0 0
$$155$$ −18.1515 −1.45796
$$156$$ 0 0
$$157$$ −1.22729 −0.0979485 −0.0489742 0.998800i $$-0.515595\pi$$
−0.0489742 + 0.998800i $$0.515595\pi$$
$$158$$ 0 0
$$159$$ 5.45467 0.432584
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −13.6255 −1.06723 −0.533615 0.845727i $$-0.679167\pi$$
−0.533615 + 0.845727i $$0.679167\pi$$
$$164$$ 0 0
$$165$$ −29.1764 −2.27138
$$166$$ 0 0
$$167$$ −23.8703 −1.84714 −0.923570 0.383430i $$-0.874743\pi$$
−0.923570 + 0.383430i $$0.874743\pi$$
$$168$$ 0 0
$$169$$ 25.0946 1.93035
$$170$$ 0 0
$$171$$ −6.26815 −0.479338
$$172$$ 0 0
$$173$$ 2.40276 0.182678 0.0913392 0.995820i $$-0.470885\pi$$
0.0913392 + 0.995820i $$0.470885\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 21.6864 1.63005
$$178$$ 0 0
$$179$$ −4.56753 −0.341393 −0.170697 0.985324i $$-0.554602\pi$$
−0.170697 + 0.985324i $$0.554602\pi$$
$$180$$ 0 0
$$181$$ 18.6789 1.38839 0.694196 0.719786i $$-0.255760\pi$$
0.694196 + 0.719786i $$0.255760\pi$$
$$182$$ 0 0
$$183$$ −6.53991 −0.483444
$$184$$ 0 0
$$185$$ −0.761109 −0.0559578
$$186$$ 0 0
$$187$$ −25.4935 −1.86427
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.89439 −0.137074 −0.0685368 0.997649i $$-0.521833\pi$$
−0.0685368 + 0.997649i $$0.521833\pi$$
$$192$$ 0 0
$$193$$ 26.7270 1.92385 0.961927 0.273306i $$-0.0881173\pi$$
0.961927 + 0.273306i $$0.0881173\pi$$
$$194$$ 0 0
$$195$$ −48.8957 −3.50149
$$196$$ 0 0
$$197$$ −13.5172 −0.963060 −0.481530 0.876430i $$-0.659919\pi$$
−0.481530 + 0.876430i $$0.659919\pi$$
$$198$$ 0 0
$$199$$ −10.8090 −0.766228 −0.383114 0.923701i $$-0.625148\pi$$
−0.383114 + 0.923701i $$0.625148\pi$$
$$200$$ 0 0
$$201$$ −23.2451 −1.63959
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −13.7726 −0.961919
$$206$$ 0 0
$$207$$ −16.4175 −1.14110
$$208$$ 0 0
$$209$$ −3.68293 −0.254754
$$210$$ 0 0
$$211$$ −9.11248 −0.627329 −0.313664 0.949534i $$-0.601557\pi$$
−0.313664 + 0.949534i $$0.601557\pi$$
$$212$$ 0 0
$$213$$ 10.0243 0.686853
$$214$$ 0 0
$$215$$ 5.10981 0.348486
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 26.8794 1.81634
$$220$$ 0 0
$$221$$ −42.7236 −2.87390
$$222$$ 0 0
$$223$$ 26.2526 1.75801 0.879003 0.476817i $$-0.158210\pi$$
0.879003 + 0.476817i $$0.158210\pi$$
$$224$$ 0 0
$$225$$ 11.1040 0.740267
$$226$$ 0 0
$$227$$ −22.4173 −1.48789 −0.743945 0.668241i $$-0.767047\pi$$
−0.743945 + 0.668241i $$0.767047\pi$$
$$228$$ 0 0
$$229$$ 15.0486 0.994441 0.497221 0.867624i $$-0.334354\pi$$
0.497221 + 0.867624i $$0.334354\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4.82983 −0.316413 −0.158206 0.987406i $$-0.550571\pi$$
−0.158206 + 0.987406i $$0.550571\pi$$
$$234$$ 0 0
$$235$$ −20.8054 −1.35720
$$236$$ 0 0
$$237$$ −19.4564 −1.26383
$$238$$ 0 0
$$239$$ −7.09173 −0.458726 −0.229363 0.973341i $$-0.573664\pi$$
−0.229363 + 0.973341i $$0.573664\pi$$
$$240$$ 0 0
$$241$$ 18.4405 1.18786 0.593930 0.804517i $$-0.297576\pi$$
0.593930 + 0.804517i $$0.297576\pi$$
$$242$$ 0 0
$$243$$ 5.11707 0.328260
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.17208 −0.392720
$$248$$ 0 0
$$249$$ 22.4064 1.41995
$$250$$ 0 0
$$251$$ 14.6808 0.926641 0.463321 0.886191i $$-0.346658\pi$$
0.463321 + 0.886191i $$0.346658\pi$$
$$252$$ 0 0
$$253$$ −9.64631 −0.606459
$$254$$ 0 0
$$255$$ 54.8372 3.43404
$$256$$ 0 0
$$257$$ 10.0101 0.624415 0.312208 0.950014i $$-0.398932\pi$$
0.312208 + 0.950014i $$0.398932\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 19.5618 1.21085
$$262$$ 0 0
$$263$$ −11.2916 −0.696268 −0.348134 0.937445i $$-0.613185\pi$$
−0.348134 + 0.937445i $$0.613185\pi$$
$$264$$ 0 0
$$265$$ −4.66245 −0.286412
$$266$$ 0 0
$$267$$ −41.0995 −2.51525
$$268$$ 0 0
$$269$$ −25.9497 −1.58218 −0.791089 0.611701i $$-0.790486\pi$$
−0.791089 + 0.611701i $$0.790486\pi$$
$$270$$ 0 0
$$271$$ −20.2412 −1.22956 −0.614781 0.788698i $$-0.710756\pi$$
−0.614781 + 0.788698i $$0.710756\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.52429 0.393430
$$276$$ 0 0
$$277$$ 13.0873 0.786340 0.393170 0.919466i $$-0.371378\pi$$
0.393170 + 0.919466i $$0.371378\pi$$
$$278$$ 0 0
$$279$$ 43.7230 2.61763
$$280$$ 0 0
$$281$$ 21.1772 1.26333 0.631664 0.775242i $$-0.282372\pi$$
0.631664 + 0.775242i $$0.282372\pi$$
$$282$$ 0 0
$$283$$ 7.32367 0.435347 0.217673 0.976022i $$-0.430153\pi$$
0.217673 + 0.976022i $$0.430153\pi$$
$$284$$ 0 0
$$285$$ 7.92207 0.469263
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 30.9151 1.81854
$$290$$ 0 0
$$291$$ −27.1073 −1.58906
$$292$$ 0 0
$$293$$ 17.5488 1.02521 0.512606 0.858624i $$-0.328680\pi$$
0.512606 + 0.858624i $$0.328680\pi$$
$$294$$ 0 0
$$295$$ −18.5367 −1.07925
$$296$$ 0 0
$$297$$ 36.6431 2.12625
$$298$$ 0 0
$$299$$ −16.1659 −0.934897
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 45.0954 2.59067
$$304$$ 0 0
$$305$$ 5.59007 0.320087
$$306$$ 0 0
$$307$$ 32.5131 1.85562 0.927810 0.373053i $$-0.121689\pi$$
0.927810 + 0.373053i $$0.121689\pi$$
$$308$$ 0 0
$$309$$ 51.2587 2.91601
$$310$$ 0 0
$$311$$ −8.72637 −0.494827 −0.247414 0.968910i $$-0.579581\pi$$
−0.247414 + 0.968910i $$0.579581\pi$$
$$312$$ 0 0
$$313$$ 10.6345 0.601096 0.300548 0.953767i $$-0.402830\pi$$
0.300548 + 0.953767i $$0.402830\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.2316 −0.799326 −0.399663 0.916662i $$-0.630873\pi$$
−0.399663 + 0.916662i $$0.630873\pi$$
$$318$$ 0 0
$$319$$ 11.4938 0.643528
$$320$$ 0 0
$$321$$ −40.0790 −2.23699
$$322$$ 0 0
$$323$$ 6.92207 0.385155
$$324$$ 0 0
$$325$$ 10.9338 0.606498
$$326$$ 0 0
$$327$$ 30.4449 1.68361
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 29.8929 1.64306 0.821532 0.570163i $$-0.193120\pi$$
0.821532 + 0.570163i $$0.193120\pi$$
$$332$$ 0 0
$$333$$ 1.83335 0.100467
$$334$$ 0 0
$$335$$ 19.8691 1.08556
$$336$$ 0 0
$$337$$ 17.6020 0.958841 0.479421 0.877585i $$-0.340847\pi$$
0.479421 + 0.877585i $$0.340847\pi$$
$$338$$ 0 0
$$339$$ −53.5847 −2.91032
$$340$$ 0 0
$$341$$ 25.6900 1.39119
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 20.7495 1.11711
$$346$$ 0 0
$$347$$ −1.55959 −0.0837234 −0.0418617 0.999123i $$-0.513329\pi$$
−0.0418617 + 0.999123i $$0.513329\pi$$
$$348$$ 0 0
$$349$$ −27.9765 −1.49755 −0.748773 0.662826i $$-0.769357\pi$$
−0.748773 + 0.662826i $$0.769357\pi$$
$$350$$ 0 0
$$351$$ 61.4088 3.27776
$$352$$ 0 0
$$353$$ −4.89276 −0.260415 −0.130208 0.991487i $$-0.541564\pi$$
−0.130208 + 0.991487i $$0.541564\pi$$
$$354$$ 0 0
$$355$$ −8.56840 −0.454763
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −37.7566 −1.99272 −0.996360 0.0852473i $$-0.972832\pi$$
−0.996360 + 0.0852473i $$0.972832\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 7.80568 0.409692
$$364$$ 0 0
$$365$$ −22.9756 −1.20260
$$366$$ 0 0
$$367$$ 14.1666 0.739493 0.369746 0.929133i $$-0.379445\pi$$
0.369746 + 0.929133i $$0.379445\pi$$
$$368$$ 0 0
$$369$$ 33.1751 1.72703
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.80652 −0.0935380 −0.0467690 0.998906i $$-0.514892\pi$$
−0.0467690 + 0.998906i $$0.514892\pi$$
$$374$$ 0 0
$$375$$ 25.5765 1.32076
$$376$$ 0 0
$$377$$ 19.2620 0.992042
$$378$$ 0 0
$$379$$ −35.3424 −1.81542 −0.907710 0.419599i $$-0.862171\pi$$
−0.907710 + 0.419599i $$0.862171\pi$$
$$380$$ 0 0
$$381$$ 42.1745 2.16066
$$382$$ 0 0
$$383$$ 19.8292 1.01323 0.506614 0.862173i $$-0.330897\pi$$
0.506614 + 0.862173i $$0.330897\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −12.3084 −0.625672
$$388$$ 0 0
$$389$$ −22.2057 −1.12587 −0.562936 0.826501i $$-0.690328\pi$$
−0.562936 + 0.826501i $$0.690328\pi$$
$$390$$ 0 0
$$391$$ 18.1303 0.916887
$$392$$ 0 0
$$393$$ 17.2872 0.872023
$$394$$ 0 0
$$395$$ 16.6306 0.836775
$$396$$ 0 0
$$397$$ −0.875987 −0.0439645 −0.0219823 0.999758i $$-0.506998\pi$$
−0.0219823 + 0.999758i $$0.506998\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −38.6864 −1.93191 −0.965953 0.258717i $$-0.916700\pi$$
−0.965953 + 0.258717i $$0.916700\pi$$
$$402$$ 0 0
$$403$$ 43.0528 2.14461
$$404$$ 0 0
$$405$$ −29.8871 −1.48510
$$406$$ 0 0
$$407$$ 1.07720 0.0533950
$$408$$ 0 0
$$409$$ −11.6808 −0.577580 −0.288790 0.957392i $$-0.593253\pi$$
−0.288790 + 0.957392i $$0.593253\pi$$
$$410$$ 0 0
$$411$$ 63.6052 3.13741
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −19.1522 −0.940143
$$416$$ 0 0
$$417$$ −55.9373 −2.73926
$$418$$ 0 0
$$419$$ −11.2773 −0.550932 −0.275466 0.961311i $$-0.588832\pi$$
−0.275466 + 0.961311i $$0.588832\pi$$
$$420$$ 0 0
$$421$$ −5.72033 −0.278792 −0.139396 0.990237i $$-0.544516\pi$$
−0.139396 + 0.990237i $$0.544516\pi$$
$$422$$ 0 0
$$423$$ 50.1157 2.43671
$$424$$ 0 0
$$425$$ −12.2624 −0.594815
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 69.2025 3.34113
$$430$$ 0 0
$$431$$ 15.3845 0.741045 0.370522 0.928824i $$-0.379179\pi$$
0.370522 + 0.928824i $$0.379179\pi$$
$$432$$ 0 0
$$433$$ 20.1376 0.967749 0.483875 0.875137i $$-0.339229\pi$$
0.483875 + 0.875137i $$0.339229\pi$$
$$434$$ 0 0
$$435$$ −24.7234 −1.18540
$$436$$ 0 0
$$437$$ 2.61920 0.125293
$$438$$ 0 0
$$439$$ 10.7327 0.512245 0.256122 0.966644i $$-0.417555\pi$$
0.256122 + 0.966644i $$0.417555\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 35.1006 1.66768 0.833840 0.552006i $$-0.186137\pi$$
0.833840 + 0.552006i $$0.186137\pi$$
$$444$$ 0 0
$$445$$ 35.1304 1.66534
$$446$$ 0 0
$$447$$ −11.7584 −0.556154
$$448$$ 0 0
$$449$$ −35.3230 −1.66699 −0.833497 0.552524i $$-0.813665\pi$$
−0.833497 + 0.552524i $$0.813665\pi$$
$$450$$ 0 0
$$451$$ 19.4925 0.917864
$$452$$ 0 0
$$453$$ 65.3872 3.07216
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 30.6444 1.43348 0.716741 0.697339i $$-0.245633\pi$$
0.716741 + 0.697339i $$0.245633\pi$$
$$458$$ 0 0
$$459$$ −68.8709 −3.21462
$$460$$ 0 0
$$461$$ 2.47914 0.115465 0.0577325 0.998332i $$-0.481613\pi$$
0.0577325 + 0.998332i $$0.481613\pi$$
$$462$$ 0 0
$$463$$ −6.46751 −0.300571 −0.150285 0.988643i $$-0.548019\pi$$
−0.150285 + 0.988643i $$0.548019\pi$$
$$464$$ 0 0
$$465$$ −55.2598 −2.56261
$$466$$ 0 0
$$467$$ 6.61582 0.306144 0.153072 0.988215i $$-0.451083\pi$$
0.153072 + 0.988215i $$0.451083\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −3.73632 −0.172161
$$472$$ 0 0
$$473$$ −7.23196 −0.332526
$$474$$ 0 0
$$475$$ −1.77149 −0.0812817
$$476$$ 0 0
$$477$$ 11.2308 0.514224
$$478$$ 0 0
$$479$$ −10.5627 −0.482621 −0.241310 0.970448i $$-0.577577\pi$$
−0.241310 + 0.970448i $$0.577577\pi$$
$$480$$ 0 0
$$481$$ 1.80524 0.0823121
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 23.1703 1.05211
$$486$$ 0 0
$$487$$ −25.1801 −1.14102 −0.570510 0.821291i $$-0.693254\pi$$
−0.570510 + 0.821291i $$0.693254\pi$$
$$488$$ 0 0
$$489$$ −41.4810 −1.87583
$$490$$ 0 0
$$491$$ −10.3251 −0.465966 −0.232983 0.972481i $$-0.574849\pi$$
−0.232983 + 0.972481i $$0.574849\pi$$
$$492$$ 0 0
$$493$$ −21.6026 −0.972931
$$494$$ 0 0
$$495$$ −60.0725 −2.70006
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −23.0441 −1.03160 −0.515798 0.856710i $$-0.672504\pi$$
−0.515798 + 0.856710i $$0.672504\pi$$
$$500$$ 0 0
$$501$$ −72.6699 −3.24665
$$502$$ 0 0
$$503$$ 5.09453 0.227154 0.113577 0.993529i $$-0.463769\pi$$
0.113577 + 0.993529i $$0.463769\pi$$
$$504$$ 0 0
$$505$$ −38.5459 −1.71527
$$506$$ 0 0
$$507$$ 76.3970 3.39291
$$508$$ 0 0
$$509$$ 39.2110 1.73800 0.868999 0.494815i $$-0.164764\pi$$
0.868999 + 0.494815i $$0.164764\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −9.94945 −0.439279
$$514$$ 0 0
$$515$$ −43.8141 −1.93068
$$516$$ 0 0
$$517$$ 29.4461 1.29504
$$518$$ 0 0
$$519$$ 7.31487 0.321087
$$520$$ 0 0
$$521$$ −8.68475 −0.380486 −0.190243 0.981737i $$-0.560928\pi$$
−0.190243 + 0.981737i $$0.560928\pi$$
$$522$$ 0 0
$$523$$ −33.5727 −1.46803 −0.734017 0.679131i $$-0.762357\pi$$
−0.734017 + 0.679131i $$0.762357\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −48.2843 −2.10330
$$528$$ 0 0
$$529$$ −16.1398 −0.701731
$$530$$ 0 0
$$531$$ 44.6509 1.93768
$$532$$ 0 0
$$533$$ 32.6667 1.41495
$$534$$ 0 0
$$535$$ 34.2580 1.48110
$$536$$ 0 0
$$537$$ −13.9052 −0.600055
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −40.5514 −1.74344 −0.871720 0.490004i $$-0.836995\pi$$
−0.871720 + 0.490004i $$0.836995\pi$$
$$542$$ 0 0
$$543$$ 56.8654 2.44033
$$544$$ 0 0
$$545$$ −26.0232 −1.11471
$$546$$ 0 0
$$547$$ −12.7094 −0.543414 −0.271707 0.962380i $$-0.587588\pi$$
−0.271707 + 0.962380i $$0.587588\pi$$
$$548$$ 0 0
$$549$$ −13.4653 −0.574683
$$550$$ 0 0
$$551$$ −3.12082 −0.132952
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −2.31709 −0.0983551
$$556$$ 0 0
$$557$$ −16.8513 −0.714011 −0.357005 0.934102i $$-0.616202\pi$$
−0.357005 + 0.934102i $$0.616202\pi$$
$$558$$ 0 0
$$559$$ −12.1198 −0.512611
$$560$$ 0 0
$$561$$ −77.6116 −3.27676
$$562$$ 0 0
$$563$$ 5.30235 0.223467 0.111734 0.993738i $$-0.464360\pi$$
0.111734 + 0.993738i $$0.464360\pi$$
$$564$$ 0 0
$$565$$ 45.8022 1.92691
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 12.4621 0.522438 0.261219 0.965280i $$-0.415876\pi$$
0.261219 + 0.965280i $$0.415876\pi$$
$$570$$ 0 0
$$571$$ 15.5137 0.649230 0.324615 0.945846i $$-0.394765\pi$$
0.324615 + 0.945846i $$0.394765\pi$$
$$572$$ 0 0
$$573$$ −5.76723 −0.240929
$$574$$ 0 0
$$575$$ −4.63989 −0.193497
$$576$$ 0 0
$$577$$ 11.8485 0.493260 0.246630 0.969110i $$-0.420677\pi$$
0.246630 + 0.969110i $$0.420677\pi$$
$$578$$ 0 0
$$579$$ 81.3668 3.38149
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.59881 0.273295
$$584$$ 0 0
$$585$$ −100.673 −4.16232
$$586$$ 0 0
$$587$$ 21.7714 0.898603 0.449302 0.893380i $$-0.351673\pi$$
0.449302 + 0.893380i $$0.351673\pi$$
$$588$$ 0 0
$$589$$ −6.97542 −0.287417
$$590$$ 0 0
$$591$$ −41.1513 −1.69274
$$592$$ 0 0
$$593$$ 28.0822 1.15320 0.576599 0.817028i $$-0.304380\pi$$
0.576599 + 0.817028i $$0.304380\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −32.9065 −1.34677
$$598$$ 0 0
$$599$$ 29.0002 1.18492 0.592459 0.805601i $$-0.298157\pi$$
0.592459 + 0.805601i $$0.298157\pi$$
$$600$$ 0 0
$$601$$ −5.11828 −0.208779 −0.104390 0.994536i $$-0.533289\pi$$
−0.104390 + 0.994536i $$0.533289\pi$$
$$602$$ 0 0
$$603$$ −47.8603 −1.94902
$$604$$ 0 0
$$605$$ −6.67200 −0.271256
$$606$$ 0 0
$$607$$ −1.89786 −0.0770317 −0.0385159 0.999258i $$-0.512263\pi$$
−0.0385159 + 0.999258i $$0.512263\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 49.3475 1.99639
$$612$$ 0 0
$$613$$ −30.0880 −1.21524 −0.607621 0.794227i $$-0.707876\pi$$
−0.607621 + 0.794227i $$0.707876\pi$$
$$614$$ 0 0
$$615$$ −41.9288 −1.69073
$$616$$ 0 0
$$617$$ −14.3309 −0.576941 −0.288470 0.957489i $$-0.593147\pi$$
−0.288470 + 0.957489i $$0.593147\pi$$
$$618$$ 0 0
$$619$$ 16.3218 0.656027 0.328013 0.944673i $$-0.393621\pi$$
0.328013 + 0.944673i $$0.393621\pi$$
$$620$$ 0 0
$$621$$ −26.0596 −1.04573
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −30.7193 −1.22877
$$626$$ 0 0
$$627$$ −11.2122 −0.447771
$$628$$ 0 0
$$629$$ −2.02461 −0.0807264
$$630$$ 0 0
$$631$$ 12.0842 0.481065 0.240533 0.970641i $$-0.422678\pi$$
0.240533 + 0.970641i $$0.422678\pi$$
$$632$$ 0 0
$$633$$ −27.7417 −1.10263
$$634$$ 0 0
$$635$$ −36.0492 −1.43057
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 20.6394 0.816482
$$640$$ 0 0
$$641$$ −8.04908 −0.317919 −0.158960 0.987285i $$-0.550814\pi$$
−0.158960 + 0.987285i $$0.550814\pi$$
$$642$$ 0 0
$$643$$ 4.82685 0.190352 0.0951762 0.995460i $$-0.469659\pi$$
0.0951762 + 0.995460i $$0.469659\pi$$
$$644$$ 0 0
$$645$$ 15.5561 0.612522
$$646$$ 0 0
$$647$$ −26.1071 −1.02637 −0.513187 0.858277i $$-0.671535\pi$$
−0.513187 + 0.858277i $$0.671535\pi$$
$$648$$ 0 0
$$649$$ 26.2352 1.02982
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −39.5954 −1.54949 −0.774743 0.632276i $$-0.782121\pi$$
−0.774743 + 0.632276i $$0.782121\pi$$
$$654$$ 0 0
$$655$$ −14.7764 −0.577364
$$656$$ 0 0
$$657$$ 55.3431 2.15914
$$658$$ 0 0
$$659$$ 34.5877 1.34735 0.673674 0.739029i $$-0.264715\pi$$
0.673674 + 0.739029i $$0.264715\pi$$
$$660$$ 0 0
$$661$$ −16.4839 −0.641149 −0.320574 0.947223i $$-0.603876\pi$$
−0.320574 + 0.947223i $$0.603876\pi$$
$$662$$ 0 0
$$663$$ −130.066 −5.05135
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.17404 −0.316500
$$668$$ 0 0
$$669$$ 79.9225 3.08998
$$670$$ 0 0
$$671$$ −7.91168 −0.305427
$$672$$ 0 0
$$673$$ −23.4114 −0.902442 −0.451221 0.892412i $$-0.649011\pi$$
−0.451221 + 0.892412i $$0.649011\pi$$
$$674$$ 0 0
$$675$$ 17.6254 0.678402
$$676$$ 0 0
$$677$$ 35.5410 1.36595 0.682977 0.730440i $$-0.260685\pi$$
0.682977 + 0.730440i $$0.260685\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −68.2465 −2.61521
$$682$$ 0 0
$$683$$ −16.7886 −0.642397 −0.321198 0.947012i $$-0.604086\pi$$
−0.321198 + 0.947012i $$0.604086\pi$$
$$684$$ 0 0
$$685$$ −54.3674 −2.07727
$$686$$ 0 0
$$687$$ 45.8135 1.74789
$$688$$ 0 0
$$689$$ 11.0587 0.421302
$$690$$ 0 0
$$691$$ −40.5064 −1.54094 −0.770468 0.637478i $$-0.779978\pi$$
−0.770468 + 0.637478i $$0.779978\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 47.8131 1.81365
$$696$$ 0 0
$$697$$ −36.6361 −1.38769
$$698$$ 0 0
$$699$$ −14.7038 −0.556148
$$700$$ 0 0
$$701$$ −1.84233 −0.0695838 −0.0347919 0.999395i $$-0.511077\pi$$
−0.0347919 + 0.999395i $$0.511077\pi$$
$$702$$ 0 0
$$703$$ −0.292486 −0.0110313
$$704$$ 0 0
$$705$$ −63.3392 −2.38549
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3.60227 0.135286 0.0676430 0.997710i $$-0.478452\pi$$
0.0676430 + 0.997710i $$0.478452\pi$$
$$710$$ 0 0
$$711$$ −40.0594 −1.50235
$$712$$ 0 0
$$713$$ −18.2700 −0.684216
$$714$$ 0 0
$$715$$ −59.1517 −2.21215
$$716$$ 0 0
$$717$$ −21.5898 −0.806286
$$718$$ 0 0
$$719$$ 36.7155 1.36926 0.684629 0.728892i $$-0.259964\pi$$
0.684629 + 0.728892i $$0.259964\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 56.1397 2.08786
$$724$$ 0 0
$$725$$ 5.52852 0.205324
$$726$$ 0 0
$$727$$ −26.0217 −0.965090 −0.482545 0.875871i $$-0.660287\pi$$
−0.482545 + 0.875871i $$0.660287\pi$$
$$728$$ 0 0
$$729$$ −18.8777 −0.699173
$$730$$ 0 0
$$731$$ 13.5925 0.502736
$$732$$ 0 0
$$733$$ −16.8036 −0.620656 −0.310328 0.950630i $$-0.600439\pi$$
−0.310328 + 0.950630i $$0.600439\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −28.1209 −1.03585
$$738$$ 0 0
$$739$$ 0.614452 0.0226030 0.0113015 0.999936i $$-0.496403\pi$$
0.0113015 + 0.999936i $$0.496403\pi$$
$$740$$ 0 0
$$741$$ −18.7901 −0.690270
$$742$$ 0 0
$$743$$ 4.75699 0.174517 0.0872585 0.996186i $$-0.472189\pi$$
0.0872585 + 0.996186i $$0.472189\pi$$
$$744$$ 0 0
$$745$$ 10.0507 0.368228
$$746$$ 0 0
$$747$$ 46.1334 1.68793
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 25.0603 0.914463 0.457232 0.889348i $$-0.348841\pi$$
0.457232 + 0.889348i $$0.348841\pi$$
$$752$$ 0 0
$$753$$ 44.6936 1.62872
$$754$$ 0 0
$$755$$ −55.8906 −2.03407
$$756$$ 0 0
$$757$$ −13.3559 −0.485429 −0.242715 0.970098i $$-0.578038\pi$$
−0.242715 + 0.970098i $$0.578038\pi$$
$$758$$ 0 0
$$759$$ −29.3669 −1.06595
$$760$$ 0 0
$$761$$ −45.7947 −1.66005 −0.830027 0.557723i $$-0.811675\pi$$
−0.830027 + 0.557723i $$0.811675\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 112.906 4.08214
$$766$$ 0 0
$$767$$ 43.9665 1.58754
$$768$$ 0 0
$$769$$ −25.8371 −0.931708 −0.465854 0.884862i $$-0.654253\pi$$
−0.465854 + 0.884862i $$0.654253\pi$$
$$770$$ 0 0
$$771$$ 30.4745 1.09751
$$772$$ 0 0
$$773$$ −15.7479 −0.566412 −0.283206 0.959059i $$-0.591398\pi$$
−0.283206 + 0.959059i $$0.591398\pi$$
$$774$$ 0 0
$$775$$ 12.3569 0.443873
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5.29265 −0.189629
$$780$$ 0 0
$$781$$ 12.1269 0.433936
$$782$$ 0 0
$$783$$ 31.0505 1.10965
$$784$$ 0 0
$$785$$ 3.19367 0.113987
$$786$$ 0 0
$$787$$ −1.22949 −0.0438267 −0.0219133 0.999760i $$-0.506976\pi$$
−0.0219133 + 0.999760i $$0.506976\pi$$
$$788$$ 0 0
$$789$$ −34.3756 −1.22380
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −13.2589 −0.470836
$$794$$ 0 0
$$795$$ −14.1942 −0.503416
$$796$$ 0 0
$$797$$ −47.3799 −1.67828 −0.839141 0.543913i $$-0.816942\pi$$
−0.839141 + 0.543913i $$0.816942\pi$$
$$798$$ 0 0
$$799$$ −55.3440 −1.95793
$$800$$ 0 0
$$801$$ −84.6214 −2.98995
$$802$$ 0 0
$$803$$ 32.5175 1.14752
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −79.0002 −2.78094
$$808$$ 0 0
$$809$$ −19.8717 −0.698653 −0.349326 0.937001i $$-0.613590\pi$$
−0.349326 + 0.937001i $$0.613590\pi$$
$$810$$ 0 0
$$811$$ −23.5779 −0.827931 −0.413966 0.910293i $$-0.635857\pi$$
−0.413966 + 0.910293i $$0.635857\pi$$
$$812$$ 0 0
$$813$$ −61.6214 −2.16116
$$814$$ 0 0
$$815$$ 35.4564 1.24198
$$816$$ 0 0
$$817$$ 1.96364 0.0686992
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −28.4995 −0.994638 −0.497319 0.867568i $$-0.665682\pi$$
−0.497319 + 0.867568i $$0.665682\pi$$
$$822$$ 0 0
$$823$$ −21.0443 −0.733558 −0.366779 0.930308i $$-0.619540\pi$$
−0.366779 + 0.930308i $$0.619540\pi$$
$$824$$ 0 0
$$825$$ 19.8623 0.691517
$$826$$ 0 0
$$827$$ −0.283949 −0.00987387 −0.00493693 0.999988i $$-0.501571\pi$$
−0.00493693 + 0.999988i $$0.501571\pi$$
$$828$$ 0 0
$$829$$ 47.7341 1.65787 0.828937 0.559342i $$-0.188946\pi$$
0.828937 + 0.559342i $$0.188946\pi$$
$$830$$ 0 0
$$831$$ 39.8425 1.38212
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 62.1156 2.14960
$$836$$ 0 0
$$837$$ 69.4016 2.39887
$$838$$ 0 0
$$839$$ −24.9296 −0.860664 −0.430332 0.902671i $$-0.641603\pi$$
−0.430332 + 0.902671i $$0.641603\pi$$
$$840$$ 0 0
$$841$$ −19.2605 −0.664154
$$842$$ 0 0
$$843$$ 64.4712 2.22051
$$844$$ 0 0
$$845$$ −65.3013 −2.24643
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 22.2959 0.765194
$$850$$ 0 0
$$851$$ −0.766077 −0.0262608
$$852$$ 0 0
$$853$$ 30.1668 1.03289 0.516447 0.856319i $$-0.327254\pi$$
0.516447 + 0.856319i $$0.327254\pi$$
$$854$$ 0 0
$$855$$ 16.3111 0.557826
$$856$$ 0 0
$$857$$ −30.7276 −1.04963 −0.524817 0.851215i $$-0.675866\pi$$
−0.524817 + 0.851215i $$0.675866\pi$$
$$858$$ 0 0
$$859$$ −29.1043 −0.993025 −0.496513 0.868029i $$-0.665386\pi$$
−0.496513 + 0.868029i $$0.665386\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 25.8047 0.878401 0.439201 0.898389i $$-0.355262\pi$$
0.439201 + 0.898389i $$0.355262\pi$$
$$864$$ 0 0
$$865$$ −6.25248 −0.212591
$$866$$ 0 0
$$867$$ 94.1169 3.19638
$$868$$ 0 0
$$869$$ −23.5374 −0.798451
$$870$$ 0 0
$$871$$ −47.1267 −1.59683
$$872$$ 0 0
$$873$$ −55.8122 −1.88896
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30.4888 −1.02953 −0.514766 0.857331i $$-0.672121\pi$$
−0.514766 + 0.857331i $$0.672121\pi$$
$$878$$ 0 0
$$879$$ 53.4249 1.80198
$$880$$ 0 0
$$881$$ −12.0111 −0.404663 −0.202331 0.979317i $$-0.564852\pi$$
−0.202331 + 0.979317i $$0.564852\pi$$
$$882$$ 0 0
$$883$$ −29.4398 −0.990729 −0.495365 0.868685i $$-0.664966\pi$$
−0.495365 + 0.868685i $$0.664966\pi$$
$$884$$ 0 0
$$885$$ −56.4325 −1.89696
$$886$$ 0 0
$$887$$ 14.9000 0.500295 0.250147 0.968208i $$-0.419521\pi$$
0.250147 + 0.968208i $$0.419521\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 42.2995 1.41709
$$892$$ 0 0
$$893$$ −7.99528 −0.267552
$$894$$ 0 0
$$895$$ 11.8857 0.397294
$$896$$ 0 0
$$897$$ −49.2148 −1.64324
$$898$$ 0 0
$$899$$ 21.7690 0.726038
$$900$$ 0 0
$$901$$ −12.4025 −0.413186
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −48.6064 −1.61573
$$906$$ 0 0
$$907$$ 15.9921 0.531009 0.265505 0.964110i $$-0.414461\pi$$
0.265505 + 0.964110i $$0.414461\pi$$
$$908$$ 0 0
$$909$$ 92.8487 3.07960
$$910$$ 0 0
$$911$$ 7.54020 0.249818 0.124909 0.992168i $$-0.460136\pi$$
0.124909 + 0.992168i $$0.460136\pi$$
$$912$$ 0 0
$$913$$ 27.1062 0.897086
$$914$$ 0 0
$$915$$ 17.0182 0.562605
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.87076 0.160672 0.0803358 0.996768i $$-0.474401\pi$$
0.0803358 + 0.996768i $$0.474401\pi$$
$$920$$ 0 0
$$921$$ 98.9817 3.26156
$$922$$ 0 0
$$923$$ 20.3230 0.668941
$$924$$ 0 0
$$925$$ 0.518137 0.0170362
$$926$$ 0 0
$$927$$ 105.539 3.46634
$$928$$ 0 0
$$929$$ −49.7616 −1.63263 −0.816313 0.577610i $$-0.803986\pi$$
−0.816313 + 0.577610i $$0.803986\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −26.5663 −0.869740
$$934$$ 0 0
$$935$$ 66.3395 2.16953
$$936$$ 0 0
$$937$$ 19.8753 0.649299 0.324649 0.945834i $$-0.394754\pi$$
0.324649 + 0.945834i $$0.394754\pi$$
$$938$$ 0 0
$$939$$ 32.3752 1.05652
$$940$$ 0 0
$$941$$ 17.1505 0.559090 0.279545 0.960133i $$-0.409816\pi$$
0.279545 + 0.960133i $$0.409816\pi$$
$$942$$ 0 0
$$943$$ −13.8625 −0.451425
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −13.8371 −0.449645 −0.224823 0.974400i $$-0.572180\pi$$
−0.224823 + 0.974400i $$0.572180\pi$$
$$948$$ 0 0
$$949$$ 54.4948 1.76898
$$950$$ 0 0
$$951$$ −43.3262 −1.40495
$$952$$ 0 0
$$953$$ 19.7888 0.641023 0.320511 0.947245i $$-0.396145\pi$$
0.320511 + 0.947245i $$0.396145\pi$$
$$954$$ 0 0
$$955$$ 4.92961 0.159518
$$956$$ 0 0
$$957$$ 34.9912 1.13111
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 17.6564 0.569562
$$962$$ 0 0
$$963$$ −82.5201 −2.65917
$$964$$ 0 0
$$965$$ −69.5493 −2.23887
$$966$$ 0 0
$$967$$ −13.5184 −0.434721 −0.217361 0.976091i $$-0.569745\pi$$
−0.217361 + 0.976091i $$0.569745\pi$$
$$968$$ 0 0
$$969$$ 21.0733 0.676972
$$970$$ 0 0
$$971$$ −26.7127 −0.857250 −0.428625 0.903482i $$-0.641002\pi$$
−0.428625 + 0.903482i $$0.641002\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 33.2865 1.06602
$$976$$ 0 0
$$977$$ 46.5330 1.48872 0.744361 0.667778i $$-0.232754\pi$$
0.744361 + 0.667778i $$0.232754\pi$$
$$978$$ 0 0
$$979$$ −49.7203 −1.58907
$$980$$ 0 0
$$981$$ 62.6841 2.00135
$$982$$ 0 0
$$983$$ −28.4260 −0.906648 −0.453324 0.891346i $$-0.649762\pi$$
−0.453324 + 0.891346i $$0.649762\pi$$
$$984$$ 0 0
$$985$$ 35.1746 1.12076
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 5.14317 0.163543
$$990$$ 0 0
$$991$$ 7.12413 0.226305 0.113153 0.993578i $$-0.463905\pi$$
0.113153 + 0.993578i $$0.463905\pi$$
$$992$$ 0 0
$$993$$ 91.0049 2.88795
$$994$$ 0 0
$$995$$ 28.1272 0.891693
$$996$$ 0 0
$$997$$ 27.2902 0.864290 0.432145 0.901804i $$-0.357757\pi$$
0.432145 + 0.901804i $$0.357757\pi$$
$$998$$ 0 0
$$999$$ 2.91007 0.0920706
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.bn.1.6 yes 6
7.6 odd 2 7448.2.a.bm.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
7448.2.a.bm.1.1 6 7.6 odd 2
7448.2.a.bn.1.6 yes 6 1.1 even 1 trivial