# Properties

 Label 3150.2.g.t.2899.1 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.1 Root $$-1.00000i$$ Character $$\chi$$ = 3150.2899 Dual form 3150.2.g.t.2899.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} -4.00000i q^{22} +8.00000i q^{23} -2.00000 q^{26} -1.00000i q^{28} +10.0000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} -2.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} +4.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{56} -10.0000i q^{58} +4.00000 q^{59} -6.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +6.00000i q^{68} +12.0000 q^{71} -6.00000i q^{73} -2.00000 q^{74} +4.00000i q^{77} +8.00000 q^{79} -2.00000i q^{82} +4.00000i q^{83} +8.00000 q^{86} +4.00000i q^{88} +14.0000 q^{89} +2.00000 q^{91} -8.00000i q^{92} +4.00000 q^{94} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 8q^{11} + 2q^{14} + 2q^{16} - 4q^{26} + 20q^{29} - 16q^{31} - 12q^{34} + 4q^{41} - 8q^{44} + 16q^{46} - 2q^{49} - 2q^{56} + 8q^{59} - 12q^{61} - 2q^{64} + 24q^{71} - 4q^{74} + 16q^{79} + 16q^{86} + 28q^{89} + 4q^{91} + 8q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ − 10.0000i − 1.31306i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.00000i 0.455842i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 2.00000i − 0.220863i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 4.00000i 0.426401i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 8.00000i − 0.834058i
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ 0 0
$$118$$ − 4.00000i − 0.368230i
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 6.00000i 0.543214i
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 12.0000i − 1.00702i
$$143$$ − 8.00000i − 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ − 14.0000i − 1.04934i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ 0 0
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 22.0000i − 1.56744i −0.621117 0.783718i $$-0.713321\pi$$
0.621117 0.783718i $$-0.286679\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 6.00000i − 0.422159i
$$203$$ 10.0000i 0.701862i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.00000i − 0.543075i
$$218$$ − 18.0000i − 1.21911i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ − 24.0000i − 1.60716i −0.595198 0.803579i $$-0.702926\pi$$
0.595198 0.803579i $$-0.297074\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 10.0000i 0.656532i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ − 6.00000i − 0.388922i
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4.00000i 0.247121i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ − 16.0000i − 0.959616i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 2.00000i 0.118056i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6.00000i 0.351123i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ − 4.00000i − 0.227921i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ 0 0
$$319$$ 40.0000 2.23957
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00000i 0.445823i
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 2.00000i 0.110432i
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.00000i 0.326841i 0.986557 + 0.163420i $$0.0522527\pi$$
−0.986557 + 0.163420i $$0.947747\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 20.0000i 1.07366i 0.843692 + 0.536828i $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 4.00000i 0.211407i
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 18.0000i − 0.946059i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 10.0000 0.519174
$$372$$ 0 0
$$373$$ − 30.0000i − 1.55334i −0.629907 0.776671i $$-0.716907\pi$$
0.629907 0.776671i $$-0.283093\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ − 20.0000i − 1.03005i
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12.0000i 0.613973i
$$383$$ 28.0000i 1.43073i 0.698749 + 0.715367i $$0.253740\pi$$
−0.698749 + 0.715367i $$0.746260\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ 2.00000i 0.101535i
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 24.0000i 1.20301i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 10.0000 0.496292
$$407$$ − 8.00000i − 0.396545i
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 16.0000i 0.788263i
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 6.00000i − 0.290360i
$$428$$ 4.00000i 0.193347i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 12.0000i 0.570782i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 6.00000i 0.282216i
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000i 0.280668i 0.990104 + 0.140334i $$0.0448177\pi$$
−0.990104 + 0.140334i $$0.955182\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000i 0.184115i
$$473$$ 32.0000i 1.47136i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ 0 0
$$478$$ 12.0000i 0.548867i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 14.0000i 0.637683i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ − 60.0000i − 2.70226i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 20.0000i − 0.892644i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 16.0000i 0.703679i
$$518$$ − 2.00000i − 0.0878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ − 28.0000i − 1.22435i −0.790721 0.612177i $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 48.0000i 2.09091i
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ − 2.00000i − 0.0862261i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 22.0000i − 0.928014i
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 0 0
$$568$$ 12.0000i 0.503509i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 0 0
$$574$$ 2.00000 0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.0000i 0.582828i 0.956597 + 0.291414i $$0.0941257\pi$$
−0.956597 + 0.291414i $$0.905874\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ − 40.0000i − 1.65663i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 38.0000i 1.56047i 0.625485 + 0.780236i $$0.284901\pi$$
−0.625485 + 0.780236i $$0.715099\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ − 16.0000i − 0.654289i
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 14.0000i 0.560898i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ − 10.0000i − 0.399043i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 0 0
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ − 40.0000i − 1.58362i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 16.0000i − 0.626608i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 4.00000i 0.155936i
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 80.0000i 3.09761i
$$668$$ 12.0000i 0.464294i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 22.0000i − 0.845529i −0.906240 0.422764i $$-0.861060\pi$$
0.906240 0.422764i $$-0.138940\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 32.0000i 1.22534i
$$683$$ − 28.0000i − 1.07139i −0.844411 0.535695i $$-0.820050\pi$$
0.844411 0.535695i $$-0.179950\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 12.0000i − 0.454532i
$$698$$ 18.0000i 0.681310i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 6.00000i 0.225653i
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 14.0000i 0.524672i
$$713$$ − 64.0000i − 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 20.0000i 0.746393i
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 19.0000i 0.707107i
$$723$$ 0 0
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16.0000i 0.593407i 0.954970 + 0.296704i $$0.0958873\pi$$
−0.954970 + 0.296704i $$0.904113\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 10.0000i − 0.367112i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −30.0000 −1.09838
$$747$$ 0 0
$$748$$ 24.0000i 0.877527i
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 0 0
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 18.0000i 0.651644i
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ − 8.00000i − 0.288863i
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 10.0000i − 0.359908i
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ − 10.0000i − 0.358517i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ − 48.0000i − 1.71648i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 22.0000i 0.783718i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 12.0000i 0.426132i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 34.0000i 1.20058i
$$803$$ − 24.0000i − 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 0 0
$$808$$ 6.00000i 0.211079i
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ − 10.0000i − 0.350931i
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 26.0000i 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000i 0.0693375i
$$833$$ 6.00000i 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 28.0000i − 0.967244i
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 26.0000i 0.896019i
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000i 0.171802i
$$848$$ − 10.0000i − 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 0 0
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ − 38.0000i − 1.29806i −0.760765 0.649028i $$-0.775176\pi$$
0.760765 0.649028i $$-0.224824\pi$$
$$858$$ 0 0
$$859$$ −24.0000 −0.818869 −0.409435 0.912339i $$-0.634274\pi$$
−0.409435 + 0.912339i $$0.634274\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 36.0000i − 1.22616i
$$863$$ 8.00000i 0.272323i 0.990687 + 0.136162i $$0.0434766\pi$$
−0.990687 + 0.136162i $$0.956523\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ 8.00000i 0.271538i
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 18.0000i 0.609557i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 2.00000i − 0.0675352i −0.999430 0.0337676i $$-0.989249\pi$$
0.999430 0.0337676i $$-0.0107506\pi$$
$$878$$ 32.0000i 1.07995i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 48.0000i 1.61533i 0.589643 + 0.807664i $$0.299269\pi$$
−0.589643 + 0.807664i $$0.700731\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 20.0000i 0.671534i 0.941945 + 0.335767i $$0.108996\pi$$
−0.941945 + 0.335767i $$0.891004\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 24.0000i 0.803579i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ − 26.0000i − 0.867631i
$$899$$ −80.0000 −2.66815
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ − 8.00000i − 0.266371i
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 0 0
$$913$$ 16.0000i 0.529523i
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ − 4.00000i − 0.132092i
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 18.0000i 0.592798i
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 0 0
$$928$$ − 10.0000i − 0.328266i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 18.0000i − 0.589610i
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 42.0000i − 1.37208i −0.727564 0.686040i $$-0.759347\pi$$
0.727564 0.686040i $$-0.240653\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 16.0000i 0.521032i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 32.0000 1.04041
$$947$$ 52.0000i 1.68977i 0.534946 + 0.844886i $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.00000i 0.194461i
$$953$$ 50.0000i 1.61966i 0.586665 + 0.809829i $$0.300440\pi$$
−0.586665 + 0.809829i $$0.699560\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 8.00000i 0.258468i
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 4.00000i 0.128965i
$$963$$ 0 0
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 0 0
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 20.0000i 0.638226i
$$983$$ − 28.0000i − 0.893061i −0.894768 0.446531i $$-0.852659\pi$$
0.894768 0.446531i $$-0.147341\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −60.0000 −1.91079
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −64.0000 −2.03508
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 0 0
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 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 3150.2.g.t.2899.1 2
3.2 odd 2 1050.2.g.f.799.2 2
5.2 odd 4 630.2.a.i.1.1 1
5.3 odd 4 3150.2.a.t.1.1 1
5.4 even 2 inner 3150.2.g.t.2899.2 2
15.2 even 4 210.2.a.a.1.1 1
15.8 even 4 1050.2.a.q.1.1 1
15.14 odd 2 1050.2.g.f.799.1 2
20.7 even 4 5040.2.a.bg.1.1 1
35.27 even 4 4410.2.a.bc.1.1 1
60.23 odd 4 8400.2.a.m.1.1 1
60.47 odd 4 1680.2.a.o.1.1 1
105.2 even 12 1470.2.i.t.361.1 2
105.17 odd 12 1470.2.i.n.961.1 2
105.32 even 12 1470.2.i.t.961.1 2
105.47 odd 12 1470.2.i.n.361.1 2
105.62 odd 4 1470.2.a.g.1.1 1
105.83 odd 4 7350.2.a.bo.1.1 1
120.77 even 4 6720.2.a.cg.1.1 1
120.107 odd 4 6720.2.a.z.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.a.1.1 1 15.2 even 4
630.2.a.i.1.1 1 5.2 odd 4
1050.2.a.q.1.1 1 15.8 even 4
1050.2.g.f.799.1 2 15.14 odd 2
1050.2.g.f.799.2 2 3.2 odd 2
1470.2.a.g.1.1 1 105.62 odd 4
1470.2.i.n.361.1 2 105.47 odd 12
1470.2.i.n.961.1 2 105.17 odd 12
1470.2.i.t.361.1 2 105.2 even 12
1470.2.i.t.961.1 2 105.32 even 12
1680.2.a.o.1.1 1 60.47 odd 4
3150.2.a.t.1.1 1 5.3 odd 4
3150.2.g.t.2899.1 2 1.1 even 1 trivial
3150.2.g.t.2899.2 2 5.4 even 2 inner
4410.2.a.bc.1.1 1 35.27 even 4
5040.2.a.bg.1.1 1 20.7 even 4
6720.2.a.z.1.1 1 120.107 odd 4
6720.2.a.cg.1.1 1 120.77 even 4
7350.2.a.bo.1.1 1 105.83 odd 4
8400.2.a.m.1.1 1 60.23 odd 4