# Properties

 Label 4410.2.a.r.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{16} +4.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} -9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +7.00000 q^{41} -5.00000 q^{43} +2.00000 q^{44} +3.00000 q^{46} -8.00000 q^{47} -1.00000 q^{50} +2.00000 q^{53} +2.00000 q^{55} +9.00000 q^{58} -10.0000 q^{59} +1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -9.00000 q^{67} +4.00000 q^{68} -2.00000 q^{71} -4.00000 q^{73} +4.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} -7.00000 q^{82} +7.00000 q^{83} +4.00000 q^{85} +5.00000 q^{86} -2.00000 q^{88} -1.00000 q^{89} -3.00000 q^{92} +8.00000 q^{94} -6.00000 q^{95} +14.0000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 9.00000 1.18176
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.00000 −1.09952 −0.549762 0.835321i $$-0.685282\pi$$
−0.549762 + 0.835321i $$0.685282\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −7.00000 −0.773021
$$83$$ 7.00000 0.768350 0.384175 0.923260i $$-0.374486\pi$$
0.384175 + 0.923260i $$0.374486\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 5.00000 0.539164
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ −1.00000 −0.106000 −0.0529999 0.998595i $$-0.516878\pi$$
−0.0529999 + 0.998595i $$0.516878\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ −9.00000 −0.835629
$$117$$ 0 0
$$118$$ 10.0000 0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −1.00000 −0.0905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 9.00000 0.777482
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −9.00000 −0.747409
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ −3.00000 −0.245770 −0.122885 0.992421i $$-0.539215\pi$$
−0.122885 + 0.992421i $$0.539215\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ −7.00000 −0.543305
$$167$$ 21.0000 1.62503 0.812514 0.582941i $$-0.198098\pi$$
0.812514 + 0.582941i $$0.198098\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ −5.00000 −0.381246
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 1.00000 0.0749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 3.00000 0.211079
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 7.00000 0.488901
$$206$$ −1.00000 −0.0696733
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ −26.0000 −1.78991 −0.894957 0.446153i $$-0.852794\pi$$
−0.894957 + 0.446153i $$0.852794\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 3.00000 0.205076
$$215$$ −5.00000 −0.340997
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 9.00000 0.609557
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 28.0000 1.87502 0.937509 0.347960i $$-0.113126\pi$$
0.937509 + 0.347960i $$0.113126\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 0 0
$$232$$ 9.00000 0.590879
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ 1.00000 0.0640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 8.00000 0.494242
$$263$$ −5.00000 −0.308313 −0.154157 0.988046i $$-0.549266\pi$$
−0.154157 + 0.988046i $$0.549266\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −9.00000 −0.549762
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ −6.00000 −0.364474 −0.182237 0.983255i $$-0.558334\pi$$
−0.182237 + 0.983255i $$0.558334\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ 12.0000 0.721010 0.360505 0.932757i $$-0.382604\pi$$
0.360505 + 0.932757i $$0.382604\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 9.00000 0.528498
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ −28.0000 −1.63578 −0.817889 0.575376i $$-0.804856\pi$$
−0.817889 + 0.575376i $$0.804856\pi$$
$$294$$ 0 0
$$295$$ −10.0000 −0.582223
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ 3.00000 0.173785
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 1.00000 0.0572598
$$306$$ 0 0
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 32.0000 1.79730 0.898650 0.438667i $$-0.144549\pi$$
0.898650 + 0.438667i $$0.144549\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −7.00000 −0.386510
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 7.00000 0.384175
$$333$$ 0 0
$$334$$ −21.0000 −1.14907
$$335$$ −9.00000 −0.491723
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 5.00000 0.269582
$$345$$ 0 0
$$346$$ 8.00000 0.430083
$$347$$ −19.0000 −1.01997 −0.509987 0.860182i $$-0.670350\pi$$
−0.509987 + 0.860182i $$0.670350\pi$$
$$348$$ 0 0
$$349$$ −35.0000 −1.87351 −0.936754 0.349990i $$-0.886185\pi$$
−0.936754 + 0.349990i $$0.886185\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −2.00000 −0.106149
$$356$$ −1.00000 −0.0529999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −7.00000 −0.367912
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.00000 −0.209370
$$366$$ 0 0
$$367$$ −11.0000 −0.574195 −0.287098 0.957901i $$-0.592690\pi$$
−0.287098 + 0.957901i $$0.592690\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 30.0000 1.54100 0.770498 0.637442i $$-0.220007\pi$$
0.770498 + 0.637442i $$0.220007\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 0 0
$$382$$ −18.0000 −0.920960
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.0000 −1.32337
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 10.0000 0.503155
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −31.0000 −1.54807 −0.774033 0.633145i $$-0.781764\pi$$
−0.774033 + 0.633145i $$0.781764\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −3.00000 −0.149256
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 3.00000 0.148340 0.0741702 0.997246i $$-0.476369\pi$$
0.0741702 + 0.997246i $$0.476369\pi$$
$$410$$ −7.00000 −0.345705
$$411$$ 0 0
$$412$$ 1.00000 0.0492665
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 7.00000 0.343616
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 12.0000 0.586939
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 26.0000 1.26566
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.00000 −0.145010
$$429$$ 0 0
$$430$$ 5.00000 0.241121
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −9.00000 −0.431022
$$437$$ 18.0000 0.861057
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −31.0000 −1.47285 −0.736427 0.676517i $$-0.763489\pi$$
−0.736427 + 0.676517i $$0.763489\pi$$
$$444$$ 0 0
$$445$$ −1.00000 −0.0474045
$$446$$ −28.0000 −1.32584
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 33.0000 1.55737 0.778683 0.627417i $$-0.215888\pi$$
0.778683 + 0.627417i $$0.215888\pi$$
$$450$$ 0 0
$$451$$ 14.0000 0.659234
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −32.0000 −1.49690 −0.748448 0.663193i $$-0.769201\pi$$
−0.748448 + 0.663193i $$0.769201\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 0 0
$$460$$ −3.00000 −0.139876
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ −19.0000 −0.883005 −0.441502 0.897260i $$-0.645554\pi$$
−0.441502 + 0.897260i $$0.645554\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 13.0000 0.601568 0.300784 0.953692i $$-0.402752\pi$$
0.300784 + 0.953692i $$0.402752\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 10.0000 0.460287
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 14.0000 0.635707
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −1.00000 −0.0452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −18.0000 −0.805791 −0.402895 0.915246i $$-0.631996\pi$$
−0.402895 + 0.915246i $$0.631996\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ −3.00000 −0.133498
$$506$$ 6.00000 0.266733
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ −1.00000 −0.0443242 −0.0221621 0.999754i $$-0.507055\pi$$
−0.0221621 + 0.999754i $$0.507055\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 8.00000 0.352865
$$515$$ 1.00000 0.0440653
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 5.00000 0.218010
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −3.00000 −0.129701
$$536$$ 9.00000 0.388741
$$537$$ 0 0
$$538$$ 3.00000 0.129339
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ 6.00000 0.257722
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −9.00000 −0.385518
$$546$$ 0 0
$$547$$ −33.0000 −1.41098 −0.705489 0.708721i $$-0.749273\pi$$
−0.705489 + 0.708721i $$0.749273\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ 54.0000 2.30048
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −12.0000 −0.509831
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000 0.0843649
$$563$$ −17.0000 −0.716465 −0.358232 0.933632i $$-0.616620\pi$$
−0.358232 + 0.933632i $$0.616620\pi$$
$$564$$ 0 0
$$565$$ −2.00000 −0.0841406
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 2.00000 0.0839181
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ −30.0000 −1.25546 −0.627730 0.778431i $$-0.716016\pi$$
−0.627730 + 0.778431i $$0.716016\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −9.00000 −0.373705
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 28.0000 1.15667
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 10.0000 0.411693
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −3.00000 −0.122885
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 1.00000 0.0405887 0.0202944 0.999794i $$-0.493540\pi$$
0.0202944 + 0.999794i $$0.493540\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ −1.00000 −0.0404888
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12.0000 0.484675 0.242338 0.970192i $$-0.422086\pi$$
0.242338 + 0.970192i $$0.422086\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −44.0000 −1.77137 −0.885687 0.464283i $$-0.846312\pi$$
−0.885687 + 0.464283i $$0.846312\pi$$
$$618$$ 0 0
$$619$$ −46.0000 −1.84890 −0.924448 0.381308i $$-0.875474\pi$$
−0.924448 + 0.381308i $$0.875474\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 0 0
$$634$$ −32.0000 −1.27088
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −5.00000 −0.197488 −0.0987441 0.995113i $$-0.531483\pi$$
−0.0987441 + 0.995113i $$0.531483\pi$$
$$642$$ 0 0
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 11.0000 0.432455 0.216227 0.976343i $$-0.430625\pi$$
0.216227 + 0.976343i $$0.430625\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 4.00000 0.156532 0.0782660 0.996933i $$-0.475062\pi$$
0.0782660 + 0.996933i $$0.475062\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 7.00000 0.273304
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 0 0
$$661$$ −11.0000 −0.427850 −0.213925 0.976850i $$-0.568625\pi$$
−0.213925 + 0.976850i $$0.568625\pi$$
$$662$$ 32.0000 1.24372
$$663$$ 0 0
$$664$$ −7.00000 −0.271653
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 27.0000 1.04544
$$668$$ 21.0000 0.812514
$$669$$ 0 0
$$670$$ 9.00000 0.347700
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ 8.00000 0.306336
$$683$$ 37.0000 1.41577 0.707883 0.706330i $$-0.249650\pi$$
0.707883 + 0.706330i $$0.249650\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −5.00000 −0.190623
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 22.0000 0.836919 0.418460 0.908235i $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ −8.00000 −0.304114
$$693$$ 0 0
$$694$$ 19.0000 0.721230
$$695$$ −14.0000 −0.531050
$$696$$ 0 0
$$697$$ 28.0000 1.06058
$$698$$ 35.0000 1.32477
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 47.0000 1.77517 0.887583 0.460648i $$-0.152383\pi$$
0.887583 + 0.460648i $$0.152383\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −11.0000 −0.413114 −0.206557 0.978435i $$-0.566226\pi$$
−0.206557 + 0.978435i $$0.566226\pi$$
$$710$$ 2.00000 0.0750587
$$711$$ 0 0
$$712$$ 1.00000 0.0374766
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −4.00000 −0.149279
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 7.00000 0.260153
$$725$$ −9.00000 −0.334252
$$726$$ 0 0
$$727$$ −21.0000 −0.778847 −0.389423 0.921059i $$-0.627326\pi$$
−0.389423 + 0.921059i $$0.627326\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 4.00000 0.148047
$$731$$ −20.0000 −0.739727
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 11.0000 0.406017
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ −18.0000 −0.663039
$$738$$ 0 0
$$739$$ −2.00000 −0.0735712 −0.0367856 0.999323i $$-0.511712\pi$$
−0.0367856 + 0.999323i $$0.511712\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −9.00000 −0.330178 −0.165089 0.986279i $$-0.552791\pi$$
−0.165089 + 0.986279i $$0.552791\pi$$
$$744$$ 0 0
$$745$$ −3.00000 −0.109911
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ 8.00000 0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ 16.0000 0.581530 0.290765 0.956795i $$-0.406090\pi$$
0.290765 + 0.956795i $$0.406090\pi$$
$$758$$ −30.0000 −1.08965
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 15.0000 0.541972
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000 0.935760
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ −42.0000 −1.50481
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 12.0000 0.429119
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 31.0000 1.10503 0.552515 0.833503i $$-0.313668\pi$$
0.552515 + 0.833503i $$0.313668\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ −10.0000 −0.355784
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 31.0000 1.09465
$$803$$ −8.00000 −0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 3.00000 0.105540
$$809$$ 51.0000 1.79306 0.896532 0.442978i $$-0.146078\pi$$
0.896532 + 0.442978i $$0.146078\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 30.0000 1.04957
$$818$$ −3.00000 −0.104893
$$819$$ 0 0
$$820$$ 7.00000 0.244451
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 19.0000 0.662298 0.331149 0.943578i $$-0.392564\pi$$
0.331149 + 0.943578i $$0.392564\pi$$
$$824$$ −1.00000 −0.0348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 19.0000 0.660695 0.330347 0.943859i $$-0.392834\pi$$
0.330347 + 0.943859i $$0.392834\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ −7.00000 −0.242974
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 21.0000 0.726735
$$836$$ −12.0000 −0.415029
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −14.0000 −0.483334 −0.241667 0.970359i $$-0.577694\pi$$
−0.241667 + 0.970359i $$0.577694\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 19.0000 0.654783
$$843$$ 0 0
$$844$$ −26.0000 −0.894957
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ −5.00000 −0.170499
$$861$$ 0 0
$$862$$ −30.0000 −1.02180
$$863$$ 11.0000 0.374444 0.187222 0.982318i $$-0.440052\pi$$
0.187222 + 0.982318i $$0.440052\pi$$
$$864$$ 0 0
$$865$$ −8.00000 −0.272008
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 9.00000 0.304778
$$873$$ 0 0
$$874$$ −18.0000 −0.608859
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ 20.0000 0.674967
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 7.00000 0.235836 0.117918 0.993023i $$-0.462378\pi$$
0.117918 + 0.993023i $$0.462378\pi$$
$$882$$ 0 0
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 31.0000 1.04147
$$887$$ −29.0000 −0.973725 −0.486862 0.873479i $$-0.661859\pi$$
−0.486862 + 0.873479i $$0.661859\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 1.00000 0.0335201
$$891$$ 0 0
$$892$$ 28.0000 0.937509
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −33.0000 −1.10122
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ 8.00000 0.266519
$$902$$ −14.0000 −0.466149
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 7.00000 0.232688
$$906$$ 0 0
$$907$$ 5.00000 0.166022 0.0830111 0.996549i $$-0.473546\pi$$
0.0830111 + 0.996549i $$0.473546\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 14.0000 0.463332
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 3.00000 0.0989071
$$921$$ 0 0
$$922$$ −14.0000 −0.461065
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 19.0000 0.624379
$$927$$ 0 0
$$928$$ 9.00000 0.295439
$$929$$ −43.0000 −1.41078 −0.705392 0.708817i $$-0.749229\pi$$
−0.705392 + 0.708817i $$0.749229\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ −13.0000 −0.425373
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −28.0000 −0.914720 −0.457360 0.889282i $$-0.651205\pi$$
−0.457360 + 0.889282i $$0.651205\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 0 0
$$943$$ −21.0000 −0.683854
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ 10.0000 0.325128
$$947$$ 25.0000 0.812391 0.406195 0.913786i $$-0.366855\pi$$
0.406195 + 0.913786i $$0.366855\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 12.0000 0.388718 0.194359 0.980930i $$-0.437737\pi$$
0.194359 + 0.980930i $$0.437737\pi$$
$$954$$ 0 0
$$955$$ 18.0000 0.582466
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 26.0000 0.836970
$$966$$ 0 0
$$967$$ 37.0000 1.18984 0.594920 0.803785i $$-0.297184\pi$$
0.594920 + 0.803785i $$0.297184\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ −14.0000 −0.449513
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 1.00000 0.0320092
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −2.00000 −0.0639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ −17.0000 −0.542216 −0.271108 0.962549i $$-0.587390\pi$$
−0.271108 + 0.962549i $$0.587390\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 15.0000 0.476972
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ 18.0000 0.569780
$$999$$ 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 4410.2.a.r.1.1 1
3.2 odd 2 490.2.a.k.1.1 1
7.2 even 3 630.2.k.f.361.1 2
7.4 even 3 630.2.k.f.541.1 2
7.6 odd 2 4410.2.a.h.1.1 1
12.11 even 2 3920.2.a.b.1.1 1
15.2 even 4 2450.2.c.s.99.2 2
15.8 even 4 2450.2.c.s.99.1 2
15.14 odd 2 2450.2.a.b.1.1 1
21.2 odd 6 70.2.e.a.11.1 2
21.5 even 6 490.2.e.f.361.1 2
21.11 odd 6 70.2.e.a.51.1 yes 2
21.17 even 6 490.2.e.f.471.1 2
21.20 even 2 490.2.a.e.1.1 1
84.11 even 6 560.2.q.i.401.1 2
84.23 even 6 560.2.q.i.81.1 2
84.83 odd 2 3920.2.a.bk.1.1 1
105.2 even 12 350.2.j.f.249.2 4
105.23 even 12 350.2.j.f.249.1 4
105.32 even 12 350.2.j.f.149.1 4
105.44 odd 6 350.2.e.l.151.1 2
105.53 even 12 350.2.j.f.149.2 4
105.62 odd 4 2450.2.c.a.99.2 2
105.74 odd 6 350.2.e.l.51.1 2
105.83 odd 4 2450.2.c.a.99.1 2
105.104 even 2 2450.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.a.11.1 2 21.2 odd 6
70.2.e.a.51.1 yes 2 21.11 odd 6
350.2.e.l.51.1 2 105.74 odd 6
350.2.e.l.151.1 2 105.44 odd 6
350.2.j.f.149.1 4 105.32 even 12
350.2.j.f.149.2 4 105.53 even 12
350.2.j.f.249.1 4 105.23 even 12
350.2.j.f.249.2 4 105.2 even 12
490.2.a.e.1.1 1 21.20 even 2
490.2.a.k.1.1 1 3.2 odd 2
490.2.e.f.361.1 2 21.5 even 6
490.2.e.f.471.1 2 21.17 even 6
560.2.q.i.81.1 2 84.23 even 6
560.2.q.i.401.1 2 84.11 even 6
630.2.k.f.361.1 2 7.2 even 3
630.2.k.f.541.1 2 7.4 even 3
2450.2.a.b.1.1 1 15.14 odd 2
2450.2.a.q.1.1 1 105.104 even 2
2450.2.c.a.99.1 2 105.83 odd 4
2450.2.c.a.99.2 2 105.62 odd 4
2450.2.c.s.99.1 2 15.8 even 4
2450.2.c.s.99.2 2 15.2 even 4
3920.2.a.b.1.1 1 12.11 even 2
3920.2.a.bk.1.1 1 84.83 odd 2
4410.2.a.h.1.1 1 7.6 odd 2
4410.2.a.r.1.1 1 1.1 even 1 trivial