# Properties

 Label 315.2.a.b.1.1 Level $315$ Weight $2$ Character 315.1 Self dual yes Analytic conductor $2.515$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 315.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 315.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q-2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{11} +5.00000 q^{13} +4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} -2.00000 q^{20} +6.00000 q^{23} +1.00000 q^{25} -2.00000 q^{28} -3.00000 q^{29} -4.00000 q^{31} +1.00000 q^{35} +2.00000 q^{37} +12.0000 q^{41} -10.0000 q^{43} -6.00000 q^{44} -9.00000 q^{47} +1.00000 q^{49} -10.0000 q^{52} -12.0000 q^{53} +3.00000 q^{55} +8.00000 q^{61} -8.00000 q^{64} +5.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +2.00000 q^{73} -4.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} +4.00000 q^{80} -12.0000 q^{83} -3.00000 q^{85} +12.0000 q^{89} +5.00000 q^{91} -12.0000 q^{92} +2.00000 q^{95} -1.00000 q^{97} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.00000 −1.31278 −0.656392 0.754420i $$-0.727918\pi$$
−0.656392 + 0.754420i $$0.727918\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −10.0000 −1.38675
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 5.00000 0.620174
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ −12.0000 −1.25109
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 5.00000 0.492665 0.246332 0.969185i $$-0.420775\pi$$
0.246332 + 0.969185i $$0.420775\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 15.0000 1.25436
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −24.0000 −1.87409
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 20.0000 1.52499
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −9.00000 −0.658145
$$188$$ 18.0000 1.31278
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 20.0000 1.38675
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 24.0000 1.64833
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ 0 0
$$220$$ −6.00000 −0.404520
$$221$$ −15.0000 −1.00901
$$222$$ 0 0
$$223$$ −19.0000 −1.27233 −0.636167 0.771551i $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ 0 0
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −16.0000 −1.02430
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 10.0000 0.636285
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ −10.0000 −0.620174
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3.00000 0.180907
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.00000 −0.178965 −0.0894825 0.995988i $$-0.528521\pi$$
−0.0894825 + 0.995988i $$0.528521\pi$$
$$282$$ 0 0
$$283$$ −13.0000 −0.772770 −0.386385 0.922338i $$-0.626276\pi$$
−0.386385 + 0.922338i $$0.626276\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 21.0000 1.22683 0.613417 0.789760i $$-0.289795\pi$$
0.613417 + 0.789760i $$0.289795\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 30.0000 1.73494
$$300$$ 0 0
$$301$$ −10.0000 −0.576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ 11.0000 0.627803 0.313902 0.949456i $$-0.398364\pi$$
0.313902 + 0.949456i $$0.398364\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ −8.00000 −0.447214
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 24.0000 1.31717
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 6.00000 0.325396
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −15.0000 −0.798369 −0.399185 0.916871i $$-0.630707\pi$$
−0.399185 + 0.916871i $$0.630707\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −24.0000 −1.27200
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ −10.0000 −0.524142
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 24.0000 1.25109
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −15.0000 −0.772539
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 2.00000 0.101535
$$389$$ 3.00000 0.152106 0.0760530 0.997104i $$-0.475768\pi$$
0.0760530 + 0.997104i $$0.475768\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1.00000 −0.0503155
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 15.0000 0.749064 0.374532 0.927214i $$-0.377803\pi$$
0.374532 + 0.927214i $$0.377803\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −10.0000 −0.492665
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.0000 −1.01153 −0.505767 0.862670i $$-0.668791\pi$$
−0.505767 + 0.862670i $$0.668791\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −8.00000 −0.377964
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 12.0000 0.564433
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 5.00000 0.234404
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ −12.0000 −0.559503
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −12.0000 −0.557086
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −15.0000 −0.694117 −0.347059 0.937843i $$-0.612820\pi$$
−0.347059 + 0.937843i $$0.612820\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −30.0000 −1.37940
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 6.00000 0.275010
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 4.00000 0.181818
$$485$$ −1.00000 −0.0454077
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15.0000 −0.676941 −0.338470 0.940977i $$-0.609909\pi$$
−0.338470 + 0.940977i $$0.609909\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −16.0000 −0.718421
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −31.0000 −1.38775 −0.693875 0.720095i $$-0.744098\pi$$
−0.693875 + 0.720095i $$0.744098\pi$$
$$500$$ −2.00000 −0.0894427
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −27.0000 −1.20387 −0.601935 0.798545i $$-0.705603\pi$$
−0.601935 + 0.798545i $$0.705603\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 32.0000 1.41977
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 5.00000 0.220326
$$516$$ 0 0
$$517$$ −27.0000 −1.18746
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 60.0000 2.59889
$$534$$ 0 0
$$535$$ −6.00000 −0.259403
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −7.00000 −0.299847
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −24.0000 −1.02523
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ −1.00000 −0.0425243
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −28.0000 −1.18746
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ 0 0
$$559$$ −50.0000 −2.11477
$$560$$ 4.00000 0.169031
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −30.0000 −1.25436
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000 0.328798
$$593$$ 39.0000 1.60154 0.800769 0.598973i $$-0.204424\pi$$
0.800769 + 0.598973i $$0.204424\pi$$
$$594$$ 0 0
$$595$$ −3.00000 −0.122988
$$596$$ −12.0000 −0.491539
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −45.0000 −1.83865 −0.919325 0.393499i $$-0.871265\pi$$
−0.919325 + 0.393499i $$0.871265\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 2.00000 0.0813788
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ −13.0000 −0.527654 −0.263827 0.964570i $$-0.584985\pi$$
−0.263827 + 0.964570i $$0.584985\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −45.0000 −1.82051
$$612$$ 0 0
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −28.0000 −1.11732
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ 5.00000 0.198107
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 41.0000 1.61688 0.808441 0.588577i $$-0.200312\pi$$
0.808441 + 0.588577i $$0.200312\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 48.0000 1.87409
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ −6.00000 −0.232147
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −28.0000 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ −45.0000 −1.72949 −0.864745 0.502211i $$-0.832520\pi$$
−0.864745 + 0.502211i $$0.832520\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −40.0000 −1.52499
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 14.0000 0.531050
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −2.00000 −0.0755929
$$701$$ 9.00000 0.339925 0.169963 0.985451i $$-0.445635\pi$$
0.169963 + 0.985451i $$0.445635\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ −24.0000 −0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 15.0000 0.560968
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ 5.00000 0.186210
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −40.0000 −1.48659
$$725$$ −3.00000 −0.111417
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 30.0000 1.10959
$$732$$ 0 0
$$733$$ −31.0000 −1.14501 −0.572506 0.819901i $$-0.694029\pi$$
−0.572506 + 0.819901i $$0.694029\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ −43.0000 −1.58178 −0.790890 0.611958i $$-0.790382\pi$$
−0.790890 + 0.611958i $$0.790382\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 18.0000 0.658145
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ −36.0000 −1.31278
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.00000 −0.0363937
$$756$$ 0 0
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ −7.00000 −0.253417
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ −21.0000 −0.755318 −0.377659 0.925945i $$-0.623271\pi$$
−0.377659 + 0.925945i $$0.623271\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 4.00000 0.142857
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ 5.00000 0.178231 0.0891154 0.996021i $$-0.471596\pi$$
0.0891154 + 0.996021i $$0.471596\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 40.0000 1.42044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 32.0000 1.13421
$$797$$ −15.0000 −0.531327 −0.265664 0.964066i $$-0.585591\pi$$
−0.265664 + 0.964066i $$0.585591\pi$$
$$798$$ 0 0
$$799$$ 27.0000 0.955191
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 6.00000 0.210559
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 2.00000 0.0700569
$$816$$ 0 0
$$817$$ −20.0000 −0.699711
$$818$$ 0 0
$$819$$ 0 0
$$820$$ −24.0000 −0.838116
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −54.0000 −1.87776 −0.938882 0.344239i $$-0.888137\pi$$
−0.938882 + 0.344239i $$0.888137\pi$$
$$828$$ 0 0
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −40.0000 −1.38675
$$833$$ −3.00000 −0.103944
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 42.0000 1.45000 0.725001 0.688748i $$-0.241839\pi$$
0.725001 + 0.688748i $$0.241839\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 26.0000 0.894957
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ −48.0000 −1.64833
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 20.0000 0.681994
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 9.00000 0.306009
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 8.00000 0.271538
$$869$$ −3.00000 −0.101768
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 12.0000 0.404520
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 30.0000 1.00901
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 38.0000 1.27233
$$893$$ −18.0000 −0.602347
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 20.0000 0.664822
$$906$$ 0 0
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ −6.00000 −0.199117
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 8.00000 0.264327
$$917$$ 6.00000 0.198137
$$918$$ 0 0
$$919$$ 11.0000 0.362857 0.181428 0.983404i $$-0.441928\pi$$
0.181428 + 0.983404i $$0.441928\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 48.0000 1.57229
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −9.00000 −0.294331
$$936$$ 0 0
$$937$$ 47.0000 1.53542 0.767712 0.640796i $$-0.221395\pi$$
0.767712 + 0.640796i $$0.221395\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 18.0000 0.587095
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 72.0000 2.34464
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ −9.00000 −0.291233
$$956$$ −42.0000 −1.35838
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 20.0000 0.644157
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ 14.0000 0.448819
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 32.0000 1.02430
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ −2.00000 −0.0638877
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 21.0000 0.669796 0.334898 0.942254i $$-0.391298\pi$$
0.334898 + 0.942254i $$0.391298\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −20.0000 −0.636285
$$989$$ −60.0000 −1.90789
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ −37.0000 −1.17180 −0.585901 0.810383i $$-0.699259\pi$$
−0.585901 + 0.810383i $$0.699259\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 315.2.a.b.1.1 1
3.2 odd 2 35.2.a.a.1.1 1
4.3 odd 2 5040.2.a.v.1.1 1
5.2 odd 4 1575.2.d.c.1324.2 2
5.3 odd 4 1575.2.d.c.1324.1 2
5.4 even 2 1575.2.a.f.1.1 1
7.6 odd 2 2205.2.a.e.1.1 1
12.11 even 2 560.2.a.b.1.1 1
15.2 even 4 175.2.b.a.99.1 2
15.8 even 4 175.2.b.a.99.2 2
15.14 odd 2 175.2.a.b.1.1 1
21.2 odd 6 245.2.e.a.116.1 2
21.5 even 6 245.2.e.b.116.1 2
21.11 odd 6 245.2.e.a.226.1 2
21.17 even 6 245.2.e.b.226.1 2
21.20 even 2 245.2.a.c.1.1 1
24.5 odd 2 2240.2.a.k.1.1 1
24.11 even 2 2240.2.a.u.1.1 1
33.32 even 2 4235.2.a.c.1.1 1
39.38 odd 2 5915.2.a.f.1.1 1
60.23 odd 4 2800.2.g.l.449.1 2
60.47 odd 4 2800.2.g.l.449.2 2
60.59 even 2 2800.2.a.z.1.1 1
84.83 odd 2 3920.2.a.ba.1.1 1
105.62 odd 4 1225.2.b.d.99.2 2
105.83 odd 4 1225.2.b.d.99.1 2
105.104 even 2 1225.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 1 3.2 odd 2
175.2.a.b.1.1 1 15.14 odd 2
175.2.b.a.99.1 2 15.2 even 4
175.2.b.a.99.2 2 15.8 even 4
245.2.a.c.1.1 1 21.20 even 2
245.2.e.a.116.1 2 21.2 odd 6
245.2.e.a.226.1 2 21.11 odd 6
245.2.e.b.116.1 2 21.5 even 6
245.2.e.b.226.1 2 21.17 even 6
315.2.a.b.1.1 1 1.1 even 1 trivial
560.2.a.b.1.1 1 12.11 even 2
1225.2.a.e.1.1 1 105.104 even 2
1225.2.b.d.99.1 2 105.83 odd 4
1225.2.b.d.99.2 2 105.62 odd 4
1575.2.a.f.1.1 1 5.4 even 2
1575.2.d.c.1324.1 2 5.3 odd 4
1575.2.d.c.1324.2 2 5.2 odd 4
2205.2.a.e.1.1 1 7.6 odd 2
2240.2.a.k.1.1 1 24.5 odd 2
2240.2.a.u.1.1 1 24.11 even 2
2800.2.a.z.1.1 1 60.59 even 2
2800.2.g.l.449.1 2 60.23 odd 4
2800.2.g.l.449.2 2 60.47 odd 4
3920.2.a.ba.1.1 1 84.83 odd 2
4235.2.a.c.1.1 1 33.32 even 2
5040.2.a.v.1.1 1 4.3 odd 2
5915.2.a.f.1.1 1 39.38 odd 2