# Properties

 Label 6400.2.a.bg.1.1 Level $6400$ Weight $2$ Character 6400.1 Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6400 = 2^{8} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.1042572936$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.64575$$ of defining polynomial Character $$\chi$$ $$=$$ 6400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} +O(q^{10})$$ $$q-2.64575 q^{3} -4.00000 q^{7} +4.00000 q^{9} -2.64575 q^{11} -3.00000 q^{17} +2.64575 q^{19} +10.5830 q^{21} -4.00000 q^{23} -2.64575 q^{27} +4.00000 q^{31} +7.00000 q^{33} +10.5830 q^{37} +5.00000 q^{41} +5.29150 q^{43} -8.00000 q^{47} +9.00000 q^{49} +7.93725 q^{51} -10.5830 q^{53} -7.00000 q^{57} -5.29150 q^{59} +10.5830 q^{61} -16.0000 q^{63} +7.93725 q^{67} +10.5830 q^{69} -8.00000 q^{71} +7.00000 q^{73} +10.5830 q^{77} +4.00000 q^{79} -5.00000 q^{81} +7.93725 q^{83} +1.00000 q^{89} -10.5830 q^{93} -2.00000 q^{97} -10.5830 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} + 8 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 + 8 * q^9 $$2 q - 8 q^{7} + 8 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{31} + 14 q^{33} + 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} + 8 q^{79} - 10 q^{81} + 2 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 + 8 * q^9 - 6 * q^17 - 8 * q^23 + 8 * q^31 + 14 * q^33 + 10 * q^41 - 16 * q^47 + 18 * q^49 - 14 * q^57 - 32 * q^63 - 16 * q^71 + 14 * q^73 + 8 * q^79 - 10 * q^81 + 2 * q^89 - 4 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.64575 −1.52753 −0.763763 0.645497i $$-0.776650\pi$$
−0.763763 + 0.645497i $$0.776650\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ 4.00000 1.33333
$$10$$ 0 0
$$11$$ −2.64575 −0.797724 −0.398862 0.917011i $$-0.630595\pi$$
−0.398862 + 0.917011i $$0.630595\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 2.64575 0.606977 0.303488 0.952835i $$-0.401849\pi$$
0.303488 + 0.952835i $$0.401849\pi$$
$$20$$ 0 0
$$21$$ 10.5830 2.30940
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2.64575 −0.509175
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 7.00000 1.21854
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.5830 1.73984 0.869918 0.493197i $$-0.164172\pi$$
0.869918 + 0.493197i $$0.164172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 5.29150 0.806947 0.403473 0.914991i $$-0.367803\pi$$
0.403473 + 0.914991i $$0.367803\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 7.93725 1.11144
$$52$$ 0 0
$$53$$ −10.5830 −1.45369 −0.726844 0.686803i $$-0.759014\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −7.00000 −0.927173
$$58$$ 0 0
$$59$$ −5.29150 −0.688895 −0.344447 0.938806i $$-0.611934\pi$$
−0.344447 + 0.938806i $$0.611934\pi$$
$$60$$ 0 0
$$61$$ 10.5830 1.35501 0.677507 0.735516i $$-0.263060\pi$$
0.677507 + 0.735516i $$0.263060\pi$$
$$62$$ 0 0
$$63$$ −16.0000 −2.01581
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.93725 0.969690 0.484845 0.874600i $$-0.338876\pi$$
0.484845 + 0.874600i $$0.338876\pi$$
$$68$$ 0 0
$$69$$ 10.5830 1.27404
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 10.5830 1.20605
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ 7.93725 0.871227 0.435613 0.900134i $$-0.356531\pi$$
0.435613 + 0.900134i $$0.356531\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1.00000 0.106000 0.0529999 0.998595i $$-0.483122\pi$$
0.0529999 + 0.998595i $$0.483122\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −10.5830 −1.09741
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ −10.5830 −1.06363
$$100$$ 0 0
$$101$$ 10.5830 1.05305 0.526524 0.850160i $$-0.323495\pi$$
0.526524 + 0.850160i $$0.323495\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.64575 0.255774 0.127887 0.991789i $$-0.459180\pi$$
0.127887 + 0.991789i $$0.459180\pi$$
$$108$$ 0 0
$$109$$ 10.5830 1.01367 0.506834 0.862044i $$-0.330816\pi$$
0.506834 + 0.862044i $$0.330816\pi$$
$$110$$ 0 0
$$111$$ −28.0000 −2.65764
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −4.00000 −0.363636
$$122$$ 0 0
$$123$$ −13.2288 −1.19280
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 0 0
$$129$$ −14.0000 −1.23263
$$130$$ 0 0
$$131$$ −15.8745 −1.38696 −0.693481 0.720475i $$-0.743924\pi$$
−0.693481 + 0.720475i $$0.743924\pi$$
$$132$$ 0 0
$$133$$ −10.5830 −0.917663
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.0000 1.62328 0.811640 0.584158i $$-0.198575\pi$$
0.811640 + 0.584158i $$0.198575\pi$$
$$138$$ 0 0
$$139$$ 18.5203 1.57087 0.785434 0.618945i $$-0.212440\pi$$
0.785434 + 0.618945i $$0.212440\pi$$
$$140$$ 0 0
$$141$$ 21.1660 1.78250
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −23.8118 −1.96396
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.5830 0.844616 0.422308 0.906452i $$-0.361220\pi$$
0.422308 + 0.906452i $$0.361220\pi$$
$$158$$ 0 0
$$159$$ 28.0000 2.22054
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ −13.2288 −1.03616 −0.518078 0.855333i $$-0.673352\pi$$
−0.518078 + 0.855333i $$0.673352\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 10.5830 0.809303
$$172$$ 0 0
$$173$$ −21.1660 −1.60922 −0.804611 0.593802i $$-0.797626\pi$$
−0.804611 + 0.593802i $$0.797626\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.0000 1.05230
$$178$$ 0 0
$$179$$ 23.8118 1.77977 0.889887 0.456180i $$-0.150783\pi$$
0.889887 + 0.456180i $$0.150783\pi$$
$$180$$ 0 0
$$181$$ −10.5830 −0.786629 −0.393314 0.919404i $$-0.628672\pi$$
−0.393314 + 0.919404i $$0.628672\pi$$
$$182$$ 0 0
$$183$$ −28.0000 −2.06982
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 7.93725 0.580429
$$188$$ 0 0
$$189$$ 10.5830 0.769800
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 5.00000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.5830 −0.754008 −0.377004 0.926212i $$-0.623046\pi$$
−0.377004 + 0.926212i $$0.623046\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ −21.0000 −1.48123
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −16.0000 −1.11208
$$208$$ 0 0
$$209$$ −7.00000 −0.484200
$$210$$ 0 0
$$211$$ −7.93725 −0.546423 −0.273212 0.961954i $$-0.588086\pi$$
−0.273212 + 0.961954i $$0.588086\pi$$
$$212$$ 0 0
$$213$$ 21.1660 1.45027
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ −18.5203 −1.25148
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 15.8745 1.05363 0.526814 0.849981i $$-0.323386\pi$$
0.526814 + 0.849981i $$0.323386\pi$$
$$228$$ 0 0
$$229$$ −21.1660 −1.39869 −0.699345 0.714785i $$-0.746525\pi$$
−0.699345 + 0.714785i $$0.746525\pi$$
$$230$$ 0 0
$$231$$ −28.0000 −1.84226
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −10.5830 −0.687440
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −21.0000 −1.35273 −0.676364 0.736567i $$-0.736446\pi$$
−0.676364 + 0.736567i $$0.736446\pi$$
$$242$$ 0 0
$$243$$ 21.1660 1.35780
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −21.0000 −1.33082
$$250$$ 0 0
$$251$$ 7.93725 0.500995 0.250498 0.968117i $$-0.419406\pi$$
0.250498 + 0.968117i $$0.419406\pi$$
$$252$$ 0 0
$$253$$ 10.5830 0.665348
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ −42.3320 −2.63038
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.64575 −0.161917
$$268$$ 0 0
$$269$$ −21.1660 −1.29051 −0.645257 0.763965i $$-0.723250\pi$$
−0.645257 + 0.763965i $$0.723250\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.1660 1.27174 0.635871 0.771795i $$-0.280641\pi$$
0.635871 + 0.771795i $$0.280641\pi$$
$$278$$ 0 0
$$279$$ 16.0000 0.957895
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ 13.2288 0.786368 0.393184 0.919460i $$-0.371374\pi$$
0.393184 + 0.919460i $$0.371374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 5.29150 0.310193
$$292$$ 0 0
$$293$$ 10.5830 0.618266 0.309133 0.951019i $$-0.399961\pi$$
0.309133 + 0.951019i $$0.399961\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 7.00000 0.406181
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −21.1660 −1.21999
$$302$$ 0 0
$$303$$ −28.0000 −1.60856
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.64575 −0.151001 −0.0755005 0.997146i $$-0.524055\pi$$
−0.0755005 + 0.997146i $$0.524055\pi$$
$$308$$ 0 0
$$309$$ 21.1660 1.20409
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −7.00000 −0.390702
$$322$$ 0 0
$$323$$ −7.93725 −0.441641
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −28.0000 −1.54840
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ −2.64575 −0.145424 −0.0727118 0.997353i $$-0.523165\pi$$
−0.0727118 + 0.997353i $$0.523165\pi$$
$$332$$ 0 0
$$333$$ 42.3320 2.31978
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −15.0000 −0.817102 −0.408551 0.912735i $$-0.633966\pi$$
−0.408551 + 0.912735i $$0.633966\pi$$
$$338$$ 0 0
$$339$$ 39.6863 2.15546
$$340$$ 0 0
$$341$$ −10.5830 −0.573102
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.64575 0.142031 0.0710157 0.997475i $$-0.477376\pi$$
0.0710157 + 0.997475i $$0.477376\pi$$
$$348$$ 0 0
$$349$$ −10.5830 −0.566495 −0.283248 0.959047i $$-0.591412\pi$$
−0.283248 + 0.959047i $$0.591412\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −31.7490 −1.68034
$$358$$ 0 0
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ −12.0000 −0.631579
$$362$$ 0 0
$$363$$ 10.5830 0.555464
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 20.0000 1.04116
$$370$$ 0 0
$$371$$ 42.3320 2.19777
$$372$$ 0 0
$$373$$ −10.5830 −0.547967 −0.273984 0.961734i $$-0.588341\pi$$
−0.273984 + 0.961734i $$0.588341\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 7.93725 0.407709 0.203855 0.979001i $$-0.434653\pi$$
0.203855 + 0.979001i $$0.434653\pi$$
$$380$$ 0 0
$$381$$ −31.7490 −1.62655
$$382$$ 0 0
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 21.1660 1.07593
$$388$$ 0 0
$$389$$ −10.5830 −0.536580 −0.268290 0.963338i $$-0.586458\pi$$
−0.268290 + 0.963338i $$0.586458\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 42.0000 2.11862
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −21.1660 −1.06229 −0.531146 0.847280i $$-0.678238\pi$$
−0.531146 + 0.847280i $$0.678238\pi$$
$$398$$ 0 0
$$399$$ 28.0000 1.40175
$$400$$ 0 0
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −28.0000 −1.38791
$$408$$ 0 0
$$409$$ −3.00000 −0.148340 −0.0741702 0.997246i $$-0.523631\pi$$
−0.0741702 + 0.997246i $$0.523631\pi$$
$$410$$ 0 0
$$411$$ −50.2693 −2.47960
$$412$$ 0 0
$$413$$ 21.1660 1.04151
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −49.0000 −2.39954
$$418$$ 0 0
$$419$$ −18.5203 −0.904774 −0.452387 0.891822i $$-0.649427\pi$$
−0.452387 + 0.891822i $$0.649427\pi$$
$$420$$ 0 0
$$421$$ −21.1660 −1.03157 −0.515784 0.856719i $$-0.672499\pi$$
−0.515784 + 0.856719i $$0.672499\pi$$
$$422$$ 0 0
$$423$$ −32.0000 −1.55589
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −42.3320 −2.04859
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ 37.0000 1.77811 0.889053 0.457804i $$-0.151364\pi$$
0.889053 + 0.457804i $$0.151364\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −10.5830 −0.506254
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 36.0000 1.71429
$$442$$ 0 0
$$443$$ −29.1033 −1.38274 −0.691369 0.722502i $$-0.742992\pi$$
−0.691369 + 0.722502i $$0.742992\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 27.0000 1.27421 0.637104 0.770778i $$-0.280132\pi$$
0.637104 + 0.770778i $$0.280132\pi$$
$$450$$ 0 0
$$451$$ −13.2288 −0.622918
$$452$$ 0 0
$$453$$ 10.5830 0.497233
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 27.0000 1.26301 0.631503 0.775373i $$-0.282438\pi$$
0.631503 + 0.775373i $$0.282438\pi$$
$$458$$ 0 0
$$459$$ 7.93725 0.370479
$$460$$ 0 0
$$461$$ −42.3320 −1.97160 −0.985799 0.167927i $$-0.946293\pi$$
−0.985799 + 0.167927i $$0.946293\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −26.4575 −1.22431 −0.612154 0.790739i $$-0.709697\pi$$
−0.612154 + 0.790739i $$0.709697\pi$$
$$468$$ 0 0
$$469$$ −31.7490 −1.46603
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ −14.0000 −0.643721
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −42.3320 −1.93825
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −42.3320 −1.92617
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ 35.0000 1.58275
$$490$$ 0 0
$$491$$ −5.29150 −0.238802 −0.119401 0.992846i $$-0.538097\pi$$
−0.119401 + 0.992846i $$0.538097\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000 1.43540
$$498$$ 0 0
$$499$$ 26.4575 1.18440 0.592200 0.805791i $$-0.298259\pi$$
0.592200 + 0.805791i $$0.298259\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 34.3948 1.52753
$$508$$ 0 0
$$509$$ −31.7490 −1.40725 −0.703625 0.710571i $$-0.748437\pi$$
−0.703625 + 0.710571i $$0.748437\pi$$
$$510$$ 0 0
$$511$$ −28.0000 −1.23865
$$512$$ 0 0
$$513$$ −7.00000 −0.309058
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 21.1660 0.930880
$$518$$ 0 0
$$519$$ 56.0000 2.45813
$$520$$ 0 0
$$521$$ −3.00000 −0.131432 −0.0657162 0.997838i $$-0.520933\pi$$
−0.0657162 + 0.997838i $$0.520933\pi$$
$$522$$ 0 0
$$523$$ 2.64575 0.115691 0.0578453 0.998326i $$-0.481577\pi$$
0.0578453 + 0.998326i $$0.481577\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −21.1660 −0.918527
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −63.0000 −2.71865
$$538$$ 0 0
$$539$$ −23.8118 −1.02565
$$540$$ 0 0
$$541$$ −21.1660 −0.909998 −0.454999 0.890492i $$-0.650360\pi$$
−0.454999 + 0.890492i $$0.650360\pi$$
$$542$$ 0 0
$$543$$ 28.0000 1.20160
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18.5203 0.791869 0.395935 0.918279i $$-0.370421\pi$$
0.395935 + 0.918279i $$0.370421\pi$$
$$548$$ 0 0
$$549$$ 42.3320 1.80669
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −16.0000 −0.680389
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −21.0000 −0.886621
$$562$$ 0 0
$$563$$ 15.8745 0.669031 0.334515 0.942390i $$-0.391427\pi$$
0.334515 + 0.942390i $$0.391427\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 20.0000 0.839921
$$568$$ 0 0
$$569$$ −11.0000 −0.461144 −0.230572 0.973055i $$-0.574060\pi$$
−0.230572 + 0.973055i $$0.574060\pi$$
$$570$$ 0 0
$$571$$ 37.0405 1.55010 0.775049 0.631901i $$-0.217725\pi$$
0.775049 + 0.631901i $$0.217725\pi$$
$$572$$ 0 0
$$573$$ −10.5830 −0.442111
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ 0 0
$$579$$ −13.2288 −0.549768
$$580$$ 0 0
$$581$$ −31.7490 −1.31717
$$582$$ 0 0
$$583$$ 28.0000 1.15964
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7.93725 −0.327606 −0.163803 0.986493i $$-0.552376\pi$$
−0.163803 + 0.986493i $$0.552376\pi$$
$$588$$ 0 0
$$589$$ 10.5830 0.436065
$$590$$ 0 0
$$591$$ 28.0000 1.15177
$$592$$ 0 0
$$593$$ 41.0000 1.68367 0.841834 0.539736i $$-0.181476\pi$$
0.841834 + 0.539736i $$0.181476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −63.4980 −2.59880
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 0 0
$$603$$ 31.7490 1.29292
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 10.5830 0.427444 0.213722 0.976895i $$-0.431441\pi$$
0.213722 + 0.976895i $$0.431441\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −5.29150 −0.212683 −0.106342 0.994330i $$-0.533914\pi$$
−0.106342 + 0.994330i $$0.533914\pi$$
$$620$$ 0 0
$$621$$ 10.5830 0.424681
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 18.5203 0.739628
$$628$$ 0 0
$$629$$ −31.7490 −1.26592
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ 0 0
$$633$$ 21.0000 0.834675
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −32.0000 −1.26590
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 15.8745 0.626029 0.313015 0.949748i $$-0.398661\pi$$
0.313015 + 0.949748i $$0.398661\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 14.0000 0.549548
$$650$$ 0 0
$$651$$ 42.3320 1.65912
$$652$$ 0 0
$$653$$ −31.7490 −1.24243 −0.621217 0.783638i $$-0.713362\pi$$
−0.621217 + 0.783638i $$0.713362\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 28.0000 1.09238
$$658$$ 0 0
$$659$$ −7.93725 −0.309192 −0.154596 0.987978i $$-0.549408\pi$$
−0.154596 + 0.987978i $$0.549408\pi$$
$$660$$ 0 0
$$661$$ −21.1660 −0.823262 −0.411631 0.911351i $$-0.635041\pi$$
−0.411631 + 0.911351i $$0.635041\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 42.3320 1.63665
$$670$$ 0 0
$$671$$ −28.0000 −1.08093
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 42.3320 1.62695 0.813476 0.581599i $$-0.197573\pi$$
0.813476 + 0.581599i $$0.197573\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −42.0000 −1.60944
$$682$$ 0 0
$$683$$ 23.8118 0.911132 0.455566 0.890202i $$-0.349437\pi$$
0.455566 + 0.890202i $$0.349437\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 56.0000 2.13653
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 23.8118 0.905842 0.452921 0.891551i $$-0.350382\pi$$
0.452921 + 0.891551i $$0.350382\pi$$
$$692$$ 0 0
$$693$$ 42.3320 1.60806
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −15.0000 −0.568166
$$698$$ 0 0
$$699$$ 15.8745 0.600429
$$700$$ 0 0
$$701$$ 21.1660 0.799429 0.399715 0.916640i $$-0.369109\pi$$
0.399715 + 0.916640i $$0.369109\pi$$
$$702$$ 0 0
$$703$$ 28.0000 1.05604
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −42.3320 −1.59206
$$708$$ 0 0
$$709$$ −21.1660 −0.794906 −0.397453 0.917622i $$-0.630106\pi$$
−0.397453 + 0.917622i $$0.630106\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 21.1660 0.790459
$$718$$ 0 0
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ 55.5608 2.06633
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ −41.0000 −1.51852
$$730$$ 0 0
$$731$$ −15.8745 −0.587140
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −21.0000 −0.773545
$$738$$ 0 0
$$739$$ −15.8745 −0.583953 −0.291977 0.956425i $$-0.594313\pi$$
−0.291977 + 0.956425i $$0.594313\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 31.7490 1.16164
$$748$$ 0 0
$$749$$ −10.5830 −0.386695
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ −21.0000 −0.765283
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −21.1660 −0.769292 −0.384646 0.923064i $$-0.625676\pi$$
−0.384646 + 0.923064i $$0.625676\pi$$
$$758$$ 0 0
$$759$$ −28.0000 −1.01634
$$760$$ 0 0
$$761$$ 29.0000 1.05125 0.525625 0.850717i $$-0.323832\pi$$
0.525625 + 0.850717i $$0.323832\pi$$
$$762$$ 0 0
$$763$$ −42.3320 −1.53252
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −21.0000 −0.757279 −0.378640 0.925544i $$-0.623608\pi$$
−0.378640 + 0.925544i $$0.623608\pi$$
$$770$$ 0 0
$$771$$ −37.0405 −1.33398
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 112.000 4.01798
$$778$$ 0 0
$$779$$ 13.2288 0.473969
$$780$$ 0 0
$$781$$ 21.1660 0.757379
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −26.4575 −0.943108 −0.471554 0.881837i $$-0.656307\pi$$
−0.471554 + 0.881837i $$0.656307\pi$$
$$788$$ 0 0
$$789$$ 31.7490 1.13029
$$790$$ 0 0
$$791$$ 60.0000 2.13335
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 31.7490 1.12461 0.562304 0.826931i $$-0.309915\pi$$
0.562304 + 0.826931i $$0.309915\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ −18.5203 −0.653566
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 56.0000 1.97129
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −5.29150 −0.185810 −0.0929049 0.995675i $$-0.529615\pi$$
−0.0929049 + 0.995675i $$0.529615\pi$$
$$812$$ 0 0
$$813$$ 52.9150 1.85581
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 14.0000 0.489798
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.5830 0.369349 0.184675 0.982800i $$-0.440877\pi$$
0.184675 + 0.982800i $$0.440877\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 13.2288 0.460009 0.230004 0.973190i $$-0.426126\pi$$
0.230004 + 0.973190i $$0.426126\pi$$
$$828$$ 0 0
$$829$$ −21.1660 −0.735126 −0.367563 0.929999i $$-0.619808\pi$$
−0.367563 + 0.929999i $$0.619808\pi$$
$$830$$ 0 0
$$831$$ −56.0000 −1.94262
$$832$$ 0 0
$$833$$ −27.0000 −0.935495
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.5830 −0.365802
$$838$$ 0 0
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ −58.2065 −2.00474
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 16.0000 0.549767
$$848$$ 0 0
$$849$$ −35.0000 −1.20120
$$850$$ 0 0
$$851$$ −42.3320 −1.45112
$$852$$ 0 0
$$853$$ −52.9150 −1.81178 −0.905888 0.423517i $$-0.860795\pi$$
−0.905888 + 0.423517i $$0.860795\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −21.0000 −0.717346 −0.358673 0.933463i $$-0.616771\pi$$
−0.358673 + 0.933463i $$0.616771\pi$$
$$858$$ 0 0
$$859$$ −2.64575 −0.0902719 −0.0451359 0.998981i $$-0.514372\pi$$
−0.0451359 + 0.998981i $$0.514372\pi$$
$$860$$ 0 0
$$861$$ 52.9150 1.80334
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 21.1660 0.718835
$$868$$ 0 0
$$869$$ −10.5830 −0.359004
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 42.3320 1.42945 0.714725 0.699405i $$-0.246552\pi$$
0.714725 + 0.699405i $$0.246552\pi$$
$$878$$ 0 0
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ −44.9778 −1.51362 −0.756811 0.653633i $$-0.773244\pi$$
−0.756811 + 0.653633i $$0.773244\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −56.0000 −1.88030 −0.940148 0.340766i $$-0.889313\pi$$
−0.940148 + 0.340766i $$0.889313\pi$$
$$888$$ 0 0
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ 13.2288 0.443180
$$892$$ 0 0
$$893$$ −21.1660 −0.708294
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 31.7490 1.05771
$$902$$ 0 0
$$903$$ 56.0000 1.86356
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 5.29150 0.175701 0.0878507 0.996134i $$-0.472000\pi$$
0.0878507 + 0.996134i $$0.472000\pi$$
$$908$$ 0 0
$$909$$ 42.3320 1.40406
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ −21.0000 −0.694999
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 63.4980 2.09689
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 7.00000 0.230658
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −32.0000 −1.05102
$$928$$ 0 0
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 23.8118 0.780399
$$932$$ 0 0
$$933$$ −10.5830 −0.346472
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ 15.8745 0.518045
$$940$$ 0 0
$$941$$ 42.3320 1.37998 0.689992 0.723817i $$-0.257614\pi$$
0.689992 + 0.723817i $$0.257614\pi$$
$$942$$ 0 0
$$943$$ −20.0000 −0.651290
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 15.8745 0.515852 0.257926 0.966165i $$-0.416961\pi$$
0.257926 + 0.966165i $$0.416961\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −5.00000 −0.161966 −0.0809829 0.996715i $$-0.525806\pi$$
−0.0809829 + 0.996715i $$0.525806\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −76.0000 −2.45417
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 10.5830 0.341033
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 21.0000 0.674617
$$970$$ 0 0
$$971$$ −23.8118 −0.764156 −0.382078 0.924130i $$-0.624791\pi$$
−0.382078 + 0.924130i $$0.624791\pi$$
$$972$$ 0 0
$$973$$ −74.0810 −2.37493
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37.0000 1.18373 0.591867 0.806035i $$-0.298391\pi$$
0.591867 + 0.806035i $$0.298391\pi$$
$$978$$ 0 0
$$979$$ −2.64575 −0.0845586
$$980$$ 0 0
$$981$$ 42.3320 1.35156
$$982$$ 0 0
$$983$$ 28.0000 0.893061 0.446531 0.894768i $$-0.352659\pi$$
0.446531 + 0.894768i $$0.352659\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −84.6640 −2.69489
$$988$$ 0 0
$$989$$ −21.1660 −0.673040
$$990$$ 0 0
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 0 0
$$993$$ 7.00000 0.222138
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −42.3320 −1.34067 −0.670334 0.742059i $$-0.733849\pi$$
−0.670334 + 0.742059i $$0.733849\pi$$
$$998$$ 0 0
$$999$$ −28.0000 −0.885881
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 6400.2.a.bg.1.1 2
4.3 odd 2 6400.2.a.cb.1.2 2
5.4 even 2 6400.2.a.cc.1.2 2
8.3 odd 2 6400.2.a.cb.1.1 2
8.5 even 2 inner 6400.2.a.bg.1.2 2
16.3 odd 4 800.2.d.a.401.2 2
16.5 even 4 200.2.d.c.101.2 yes 2
16.11 odd 4 800.2.d.a.401.1 2
16.13 even 4 200.2.d.c.101.1 yes 2
20.19 odd 2 6400.2.a.bh.1.1 2
40.19 odd 2 6400.2.a.bh.1.2 2
40.29 even 2 6400.2.a.cc.1.1 2
48.5 odd 4 1800.2.k.d.901.1 2
48.11 even 4 7200.2.k.b.3601.1 2
48.29 odd 4 1800.2.k.d.901.2 2
48.35 even 4 7200.2.k.b.3601.2 2
80.3 even 4 800.2.f.d.49.2 4
80.13 odd 4 200.2.f.d.149.1 4
80.19 odd 4 800.2.d.d.401.1 2
80.27 even 4 800.2.f.d.49.1 4
80.29 even 4 200.2.d.b.101.2 yes 2
80.37 odd 4 200.2.f.d.149.2 4
80.43 even 4 800.2.f.d.49.4 4
80.53 odd 4 200.2.f.d.149.3 4
80.59 odd 4 800.2.d.d.401.2 2
80.67 even 4 800.2.f.d.49.3 4
80.69 even 4 200.2.d.b.101.1 2
80.77 odd 4 200.2.f.d.149.4 4
240.29 odd 4 1800.2.k.f.901.1 2
240.53 even 4 1800.2.d.m.1549.2 4
240.59 even 4 7200.2.k.i.3601.1 2
240.77 even 4 1800.2.d.m.1549.1 4
240.83 odd 4 7200.2.d.m.2449.4 4
240.107 odd 4 7200.2.d.m.2449.1 4
240.149 odd 4 1800.2.k.f.901.2 2
240.173 even 4 1800.2.d.m.1549.4 4
240.179 even 4 7200.2.k.i.3601.2 2
240.197 even 4 1800.2.d.m.1549.3 4
240.203 odd 4 7200.2.d.m.2449.3 4
240.227 odd 4 7200.2.d.m.2449.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.d.b.101.1 2 80.69 even 4
200.2.d.b.101.2 yes 2 80.29 even 4
200.2.d.c.101.1 yes 2 16.13 even 4
200.2.d.c.101.2 yes 2 16.5 even 4
200.2.f.d.149.1 4 80.13 odd 4
200.2.f.d.149.2 4 80.37 odd 4
200.2.f.d.149.3 4 80.53 odd 4
200.2.f.d.149.4 4 80.77 odd 4
800.2.d.a.401.1 2 16.11 odd 4
800.2.d.a.401.2 2 16.3 odd 4
800.2.d.d.401.1 2 80.19 odd 4
800.2.d.d.401.2 2 80.59 odd 4
800.2.f.d.49.1 4 80.27 even 4
800.2.f.d.49.2 4 80.3 even 4
800.2.f.d.49.3 4 80.67 even 4
800.2.f.d.49.4 4 80.43 even 4
1800.2.d.m.1549.1 4 240.77 even 4
1800.2.d.m.1549.2 4 240.53 even 4
1800.2.d.m.1549.3 4 240.197 even 4
1800.2.d.m.1549.4 4 240.173 even 4
1800.2.k.d.901.1 2 48.5 odd 4
1800.2.k.d.901.2 2 48.29 odd 4
1800.2.k.f.901.1 2 240.29 odd 4
1800.2.k.f.901.2 2 240.149 odd 4
6400.2.a.bg.1.1 2 1.1 even 1 trivial
6400.2.a.bg.1.2 2 8.5 even 2 inner
6400.2.a.bh.1.1 2 20.19 odd 2
6400.2.a.bh.1.2 2 40.19 odd 2
6400.2.a.cb.1.1 2 8.3 odd 2
6400.2.a.cb.1.2 2 4.3 odd 2
6400.2.a.cc.1.1 2 40.29 even 2
6400.2.a.cc.1.2 2 5.4 even 2
7200.2.d.m.2449.1 4 240.107 odd 4
7200.2.d.m.2449.2 4 240.227 odd 4
7200.2.d.m.2449.3 4 240.203 odd 4
7200.2.d.m.2449.4 4 240.83 odd 4
7200.2.k.b.3601.1 2 48.11 even 4
7200.2.k.b.3601.2 2 48.35 even 4
7200.2.k.i.3601.1 2 240.59 even 4
7200.2.k.i.3601.2 2 240.179 even 4