# Properties

 Label 3360.2.a.ba.1.1 Level $3360$ Weight $2$ Character 3360.1 Self dual yes Analytic conductor $26.830$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.8297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 3360.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.12311 q^{11} -3.12311 q^{13} +1.00000 q^{15} +2.00000 q^{17} +1.12311 q^{19} +1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -5.12311 q^{31} +5.12311 q^{33} +1.00000 q^{35} -2.00000 q^{37} +3.12311 q^{39} +2.00000 q^{41} -10.2462 q^{43} -1.00000 q^{45} +1.00000 q^{49} -2.00000 q^{51} +13.3693 q^{53} +5.12311 q^{55} -1.12311 q^{57} +4.00000 q^{59} -8.24621 q^{61} -1.00000 q^{63} +3.12311 q^{65} +2.24621 q^{67} +5.12311 q^{71} +15.1231 q^{73} -1.00000 q^{75} +5.12311 q^{77} +2.24621 q^{79} +1.00000 q^{81} +4.00000 q^{83} -2.00000 q^{85} +2.00000 q^{87} -0.246211 q^{89} +3.12311 q^{91} +5.12311 q^{93} -1.12311 q^{95} +4.87689 q^{97} -5.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} + 2 q^{35} - 4 q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} - 4 q^{51} + 2 q^{53} + 2 q^{55} + 6 q^{57} + 8 q^{59} - 2 q^{63} - 2 q^{65} - 12 q^{67} + 2 q^{71} + 22 q^{73} - 2 q^{75} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} + 4 q^{87} + 16 q^{89} - 2 q^{91} + 2 q^{93} + 6 q^{95} + 18 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 2 * q^31 + 2 * q^33 + 2 * q^35 - 4 * q^37 - 2 * q^39 + 4 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^49 - 4 * q^51 + 2 * q^53 + 2 * q^55 + 6 * q^57 + 8 * q^59 - 2 * q^63 - 2 * q^65 - 12 * q^67 + 2 * q^71 + 22 * q^73 - 2 * q^75 + 2 * q^77 - 12 * q^79 + 2 * q^81 + 8 * q^83 - 4 * q^85 + 4 * q^87 + 16 * q^89 - 2 * q^91 + 2 * q^93 + 6 * q^95 + 18 * q^97 - 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.12311 −1.54467 −0.772337 0.635213i $$-0.780912\pi$$
−0.772337 + 0.635213i $$0.780912\pi$$
$$12$$ 0 0
$$13$$ −3.12311 −0.866194 −0.433097 0.901347i $$-0.642579\pi$$
−0.433097 + 0.901347i $$0.642579\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.12311 0.257658 0.128829 0.991667i $$-0.458878\pi$$
0.128829 + 0.991667i $$0.458878\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −5.12311 −0.920137 −0.460068 0.887883i $$-0.652175\pi$$
−0.460068 + 0.887883i $$0.652175\pi$$
$$32$$ 0 0
$$33$$ 5.12311 0.891818
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 3.12311 0.500097
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −10.2462 −1.56253 −0.781266 0.624198i $$-0.785426\pi$$
−0.781266 + 0.624198i $$0.785426\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 13.3693 1.83642 0.918208 0.396098i $$-0.129636\pi$$
0.918208 + 0.396098i $$0.129636\pi$$
$$54$$ 0 0
$$55$$ 5.12311 0.690799
$$56$$ 0 0
$$57$$ −1.12311 −0.148759
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −8.24621 −1.05582 −0.527910 0.849301i $$-0.677024\pi$$
−0.527910 + 0.849301i $$0.677024\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 3.12311 0.387374
$$66$$ 0 0
$$67$$ 2.24621 0.274418 0.137209 0.990542i $$-0.456187\pi$$
0.137209 + 0.990542i $$0.456187\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.12311 0.608001 0.304000 0.952672i $$-0.401678\pi$$
0.304000 + 0.952672i $$0.401678\pi$$
$$72$$ 0 0
$$73$$ 15.1231 1.77003 0.885013 0.465567i $$-0.154149\pi$$
0.885013 + 0.465567i $$0.154149\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 5.12311 0.583832
$$78$$ 0 0
$$79$$ 2.24621 0.252719 0.126359 0.991985i $$-0.459671\pi$$
0.126359 + 0.991985i $$0.459671\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −0.246211 −0.0260983 −0.0130492 0.999915i $$-0.504154\pi$$
−0.0130492 + 0.999915i $$0.504154\pi$$
$$90$$ 0 0
$$91$$ 3.12311 0.327390
$$92$$ 0 0
$$93$$ 5.12311 0.531241
$$94$$ 0 0
$$95$$ −1.12311 −0.115228
$$96$$ 0 0
$$97$$ 4.87689 0.495174 0.247587 0.968866i $$-0.420362\pi$$
0.247587 + 0.968866i $$0.420362\pi$$
$$98$$ 0 0
$$99$$ −5.12311 −0.514891
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 8.24621 0.789844 0.394922 0.918715i $$-0.370772\pi$$
0.394922 + 0.918715i $$0.370772\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −5.36932 −0.505103 −0.252551 0.967583i $$-0.581270\pi$$
−0.252551 + 0.967583i $$0.581270\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −3.12311 −0.288731
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 15.2462 1.38602
$$122$$ 0 0
$$123$$ −2.00000 −0.180334
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −20.4924 −1.81841 −0.909204 0.416350i $$-0.863309\pi$$
−0.909204 + 0.416350i $$0.863309\pi$$
$$128$$ 0 0
$$129$$ 10.2462 0.902129
$$130$$ 0 0
$$131$$ −1.75379 −0.153229 −0.0766146 0.997061i $$-0.524411\pi$$
−0.0766146 + 0.997061i $$0.524411\pi$$
$$132$$ 0 0
$$133$$ −1.12311 −0.0973856
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 12.8769 1.10015 0.550074 0.835116i $$-0.314600\pi$$
0.550074 + 0.835116i $$0.314600\pi$$
$$138$$ 0 0
$$139$$ 9.12311 0.773812 0.386906 0.922119i $$-0.373544\pi$$
0.386906 + 0.922119i $$0.373544\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 16.0000 1.33799
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −12.2462 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$150$$ 0 0
$$151$$ −20.4924 −1.66765 −0.833825 0.552029i $$-0.813854\pi$$
−0.833825 + 0.552029i $$0.813854\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 5.12311 0.411498
$$156$$ 0 0
$$157$$ 7.12311 0.568486 0.284243 0.958752i $$-0.408258\pi$$
0.284243 + 0.958752i $$0.408258\pi$$
$$158$$ 0 0
$$159$$ −13.3693 −1.06026
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.2462 −0.802545 −0.401273 0.915959i $$-0.631432\pi$$
−0.401273 + 0.915959i $$0.631432\pi$$
$$164$$ 0 0
$$165$$ −5.12311 −0.398833
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −3.24621 −0.249709
$$170$$ 0 0
$$171$$ 1.12311 0.0858860
$$172$$ 0 0
$$173$$ −0.246211 −0.0187191 −0.00935955 0.999956i $$-0.502979\pi$$
−0.00935955 + 0.999956i $$0.502979\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ 2.87689 0.215029 0.107515 0.994204i $$-0.465711\pi$$
0.107515 + 0.994204i $$0.465711\pi$$
$$180$$ 0 0
$$181$$ 22.4924 1.67185 0.835924 0.548845i $$-0.184932\pi$$
0.835924 + 0.548845i $$0.184932\pi$$
$$182$$ 0 0
$$183$$ 8.24621 0.609577
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −10.2462 −0.749277
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −5.12311 −0.370695 −0.185347 0.982673i $$-0.559341\pi$$
−0.185347 + 0.982673i $$0.559341\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 0 0
$$195$$ −3.12311 −0.223650
$$196$$ 0 0
$$197$$ 23.6155 1.68254 0.841268 0.540618i $$-0.181809\pi$$
0.841268 + 0.540618i $$0.181809\pi$$
$$198$$ 0 0
$$199$$ −5.12311 −0.363167 −0.181584 0.983375i $$-0.558122\pi$$
−0.181584 + 0.983375i $$0.558122\pi$$
$$200$$ 0 0
$$201$$ −2.24621 −0.158436
$$202$$ 0 0
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −5.75379 −0.397998
$$210$$ 0 0
$$211$$ −5.75379 −0.396107 −0.198054 0.980191i $$-0.563462\pi$$
−0.198054 + 0.980191i $$0.563462\pi$$
$$212$$ 0 0
$$213$$ −5.12311 −0.351029
$$214$$ 0 0
$$215$$ 10.2462 0.698786
$$216$$ 0 0
$$217$$ 5.12311 0.347779
$$218$$ 0 0
$$219$$ −15.1231 −1.02192
$$220$$ 0 0
$$221$$ −6.24621 −0.420166
$$222$$ 0 0
$$223$$ 18.2462 1.22186 0.610928 0.791686i $$-0.290796\pi$$
0.610928 + 0.791686i $$0.290796\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −3.75379 −0.248057 −0.124029 0.992279i $$-0.539581\pi$$
−0.124029 + 0.992279i $$0.539581\pi$$
$$230$$ 0 0
$$231$$ −5.12311 −0.337076
$$232$$ 0 0
$$233$$ 23.1231 1.51485 0.757423 0.652925i $$-0.226458\pi$$
0.757423 + 0.652925i $$0.226458\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.24621 −0.145907
$$238$$ 0 0
$$239$$ 13.1231 0.848863 0.424432 0.905460i $$-0.360474\pi$$
0.424432 + 0.905460i $$0.360474\pi$$
$$240$$ 0 0
$$241$$ 20.2462 1.30417 0.652087 0.758145i $$-0.273894\pi$$
0.652087 + 0.758145i $$0.273894\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −3.50758 −0.223182
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 2.00000 0.125245
$$256$$ 0 0
$$257$$ 20.2462 1.26292 0.631462 0.775407i $$-0.282455\pi$$
0.631462 + 0.775407i $$0.282455\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 18.2462 1.12511 0.562555 0.826760i $$-0.309819\pi$$
0.562555 + 0.826760i $$0.309819\pi$$
$$264$$ 0 0
$$265$$ −13.3693 −0.821271
$$266$$ 0 0
$$267$$ 0.246211 0.0150679
$$268$$ 0 0
$$269$$ −16.2462 −0.990549 −0.495274 0.868737i $$-0.664933\pi$$
−0.495274 + 0.868737i $$0.664933\pi$$
$$270$$ 0 0
$$271$$ 17.6155 1.07007 0.535034 0.844831i $$-0.320299\pi$$
0.535034 + 0.844831i $$0.320299\pi$$
$$272$$ 0 0
$$273$$ −3.12311 −0.189019
$$274$$ 0 0
$$275$$ −5.12311 −0.308935
$$276$$ 0 0
$$277$$ −14.4924 −0.870765 −0.435383 0.900245i $$-0.643387\pi$$
−0.435383 + 0.900245i $$0.643387\pi$$
$$278$$ 0 0
$$279$$ −5.12311 −0.306712
$$280$$ 0 0
$$281$$ 24.7386 1.47578 0.737892 0.674919i $$-0.235822\pi$$
0.737892 + 0.674919i $$0.235822\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ 0 0
$$285$$ 1.12311 0.0665270
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −4.87689 −0.285889
$$292$$ 0 0
$$293$$ −16.2462 −0.949114 −0.474557 0.880225i $$-0.657392\pi$$
−0.474557 + 0.880225i $$0.657392\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 5.12311 0.297273
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 10.2462 0.590582
$$302$$ 0 0
$$303$$ 0.246211 0.0141445
$$304$$ 0 0
$$305$$ 8.24621 0.472177
$$306$$ 0 0
$$307$$ 9.75379 0.556678 0.278339 0.960483i $$-0.410216\pi$$
0.278339 + 0.960483i $$0.410216\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ 33.3693 1.88615 0.943073 0.332587i $$-0.107921\pi$$
0.943073 + 0.332587i $$0.107921\pi$$
$$314$$ 0 0
$$315$$ 1.00000 0.0563436
$$316$$ 0 0
$$317$$ −15.1231 −0.849398 −0.424699 0.905335i $$-0.639620\pi$$
−0.424699 + 0.905335i $$0.639620\pi$$
$$318$$ 0 0
$$319$$ 10.2462 0.573678
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ 2.24621 0.124983
$$324$$ 0 0
$$325$$ −3.12311 −0.173239
$$326$$ 0 0
$$327$$ −8.24621 −0.456017
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.24621 −0.123463 −0.0617315 0.998093i $$-0.519662\pi$$
−0.0617315 + 0.998093i $$0.519662\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ −2.24621 −0.122724
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 5.36932 0.291621
$$340$$ 0 0
$$341$$ 26.2462 1.42131
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 26.2462 1.40897 0.704485 0.709719i $$-0.251178\pi$$
0.704485 + 0.709719i $$0.251178\pi$$
$$348$$ 0 0
$$349$$ −11.7538 −0.629166 −0.314583 0.949230i $$-0.601865\pi$$
−0.314583 + 0.949230i $$0.601865\pi$$
$$350$$ 0 0
$$351$$ 3.12311 0.166699
$$352$$ 0 0
$$353$$ −10.4924 −0.558455 −0.279228 0.960225i $$-0.590078\pi$$
−0.279228 + 0.960225i $$0.590078\pi$$
$$354$$ 0 0
$$355$$ −5.12311 −0.271906
$$356$$ 0 0
$$357$$ 2.00000 0.105851
$$358$$ 0 0
$$359$$ −33.6155 −1.77416 −0.887080 0.461616i $$-0.847270\pi$$
−0.887080 + 0.461616i $$0.847270\pi$$
$$360$$ 0 0
$$361$$ −17.7386 −0.933612
$$362$$ 0 0
$$363$$ −15.2462 −0.800219
$$364$$ 0 0
$$365$$ −15.1231 −0.791580
$$366$$ 0 0
$$367$$ −13.7538 −0.717942 −0.358971 0.933349i $$-0.616872\pi$$
−0.358971 + 0.933349i $$0.616872\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ −13.3693 −0.694100
$$372$$ 0 0
$$373$$ −20.2462 −1.04831 −0.524155 0.851623i $$-0.675619\pi$$
−0.524155 + 0.851623i $$0.675619\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 6.24621 0.321696
$$378$$ 0 0
$$379$$ −12.4924 −0.641693 −0.320846 0.947131i $$-0.603967\pi$$
−0.320846 + 0.947131i $$0.603967\pi$$
$$380$$ 0 0
$$381$$ 20.4924 1.04986
$$382$$ 0 0
$$383$$ −4.49242 −0.229552 −0.114776 0.993391i $$-0.536615\pi$$
−0.114776 + 0.993391i $$0.536615\pi$$
$$384$$ 0 0
$$385$$ −5.12311 −0.261098
$$386$$ 0 0
$$387$$ −10.2462 −0.520844
$$388$$ 0 0
$$389$$ −20.2462 −1.02652 −0.513262 0.858232i $$-0.671563\pi$$
−0.513262 + 0.858232i $$0.671563\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 1.75379 0.0884669
$$394$$ 0 0
$$395$$ −2.24621 −0.113019
$$396$$ 0 0
$$397$$ −19.1231 −0.959761 −0.479881 0.877334i $$-0.659320\pi$$
−0.479881 + 0.877334i $$0.659320\pi$$
$$398$$ 0 0
$$399$$ 1.12311 0.0562256
$$400$$ 0 0
$$401$$ −38.9848 −1.94681 −0.973405 0.229090i $$-0.926425\pi$$
−0.973405 + 0.229090i $$0.926425\pi$$
$$402$$ 0 0
$$403$$ 16.0000 0.797017
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 10.2462 0.507886
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −12.8769 −0.635170
$$412$$ 0 0
$$413$$ −4.00000 −0.196827
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ −9.12311 −0.446760
$$418$$ 0 0
$$419$$ 17.7538 0.867329 0.433665 0.901074i $$-0.357220\pi$$
0.433665 + 0.901074i $$0.357220\pi$$
$$420$$ 0 0
$$421$$ −6.49242 −0.316421 −0.158211 0.987405i $$-0.550573\pi$$
−0.158211 + 0.987405i $$0.550573\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 8.24621 0.399062
$$428$$ 0 0
$$429$$ −16.0000 −0.772487
$$430$$ 0 0
$$431$$ 33.6155 1.61920 0.809602 0.586980i $$-0.199683\pi$$
0.809602 + 0.586980i $$0.199683\pi$$
$$432$$ 0 0
$$433$$ 4.87689 0.234369 0.117184 0.993110i $$-0.462613\pi$$
0.117184 + 0.993110i $$0.462613\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −33.6155 −1.60438 −0.802191 0.597068i $$-0.796332\pi$$
−0.802191 + 0.597068i $$0.796332\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 22.7386 1.08035 0.540173 0.841554i $$-0.318359\pi$$
0.540173 + 0.841554i $$0.318359\pi$$
$$444$$ 0 0
$$445$$ 0.246211 0.0116715
$$446$$ 0 0
$$447$$ 12.2462 0.579226
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ −10.2462 −0.482475
$$452$$ 0 0
$$453$$ 20.4924 0.962818
$$454$$ 0 0
$$455$$ −3.12311 −0.146413
$$456$$ 0 0
$$457$$ 4.24621 0.198629 0.0993147 0.995056i $$-0.468335\pi$$
0.0993147 + 0.995056i $$0.468335\pi$$
$$458$$ 0 0
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 14.4924 0.674979 0.337490 0.941329i $$-0.390422\pi$$
0.337490 + 0.941329i $$0.390422\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ −5.12311 −0.237578
$$466$$ 0 0
$$467$$ −36.9848 −1.71145 −0.855727 0.517427i $$-0.826890\pi$$
−0.855727 + 0.517427i $$0.826890\pi$$
$$468$$ 0 0
$$469$$ −2.24621 −0.103720
$$470$$ 0 0
$$471$$ −7.12311 −0.328215
$$472$$ 0 0
$$473$$ 52.4924 2.41360
$$474$$ 0 0
$$475$$ 1.12311 0.0515316
$$476$$ 0 0
$$477$$ 13.3693 0.612139
$$478$$ 0 0
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 6.24621 0.284803
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.87689 −0.221448
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ 10.2462 0.463350
$$490$$ 0 0
$$491$$ −13.1231 −0.592237 −0.296119 0.955151i $$-0.595692\pi$$
−0.296119 + 0.955151i $$0.595692\pi$$
$$492$$ 0 0
$$493$$ −4.00000 −0.180151
$$494$$ 0 0
$$495$$ 5.12311 0.230266
$$496$$ 0 0
$$497$$ −5.12311 −0.229803
$$498$$ 0 0
$$499$$ 20.4924 0.917367 0.458683 0.888600i $$-0.348321\pi$$
0.458683 + 0.888600i $$0.348321\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ 3.50758 0.156395 0.0781976 0.996938i $$-0.475084\pi$$
0.0781976 + 0.996938i $$0.475084\pi$$
$$504$$ 0 0
$$505$$ 0.246211 0.0109563
$$506$$ 0 0
$$507$$ 3.24621 0.144169
$$508$$ 0 0
$$509$$ −11.7538 −0.520978 −0.260489 0.965477i $$-0.583884\pi$$
−0.260489 + 0.965477i $$0.583884\pi$$
$$510$$ 0 0
$$511$$ −15.1231 −0.669007
$$512$$ 0 0
$$513$$ −1.12311 −0.0495863
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0.246211 0.0108075
$$520$$ 0 0
$$521$$ 22.4924 0.985411 0.492705 0.870196i $$-0.336008\pi$$
0.492705 + 0.870196i $$0.336008\pi$$
$$522$$ 0 0
$$523$$ 34.7386 1.51901 0.759507 0.650499i $$-0.225440\pi$$
0.759507 + 0.650499i $$0.225440\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ −10.2462 −0.446332
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −6.24621 −0.270553
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ 0 0
$$537$$ −2.87689 −0.124147
$$538$$ 0 0
$$539$$ −5.12311 −0.220668
$$540$$ 0 0
$$541$$ −14.4924 −0.623078 −0.311539 0.950233i $$-0.600844\pi$$
−0.311539 + 0.950233i $$0.600844\pi$$
$$542$$ 0 0
$$543$$ −22.4924 −0.965242
$$544$$ 0 0
$$545$$ −8.24621 −0.353229
$$546$$ 0 0
$$547$$ −21.7538 −0.930125 −0.465062 0.885278i $$-0.653968\pi$$
−0.465062 + 0.885278i $$0.653968\pi$$
$$548$$ 0 0
$$549$$ −8.24621 −0.351940
$$550$$ 0 0
$$551$$ −2.24621 −0.0956918
$$552$$ 0 0
$$553$$ −2.24621 −0.0955186
$$554$$ 0 0
$$555$$ −2.00000 −0.0848953
$$556$$ 0 0
$$557$$ −15.1231 −0.640787 −0.320393 0.947285i $$-0.603815\pi$$
−0.320393 + 0.947285i $$0.603815\pi$$
$$558$$ 0 0
$$559$$ 32.0000 1.35346
$$560$$ 0 0
$$561$$ 10.2462 0.432595
$$562$$ 0 0
$$563$$ 16.4924 0.695073 0.347536 0.937667i $$-0.387018\pi$$
0.347536 + 0.937667i $$0.387018\pi$$
$$564$$ 0 0
$$565$$ 5.36932 0.225889
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 3.50758 0.146788 0.0733938 0.997303i $$-0.476617\pi$$
0.0733938 + 0.997303i $$0.476617\pi$$
$$572$$ 0 0
$$573$$ 5.12311 0.214021
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23.6155 −0.983127 −0.491564 0.870842i $$-0.663574\pi$$
−0.491564 + 0.870842i $$0.663574\pi$$
$$578$$ 0 0
$$579$$ −18.0000 −0.748054
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 0 0
$$583$$ −68.4924 −2.83667
$$584$$ 0 0
$$585$$ 3.12311 0.129125
$$586$$ 0 0
$$587$$ 32.4924 1.34111 0.670553 0.741862i $$-0.266057\pi$$
0.670553 + 0.741862i $$0.266057\pi$$
$$588$$ 0 0
$$589$$ −5.75379 −0.237081
$$590$$ 0 0
$$591$$ −23.6155 −0.971413
$$592$$ 0 0
$$593$$ 30.4924 1.25217 0.626087 0.779753i $$-0.284656\pi$$
0.626087 + 0.779753i $$0.284656\pi$$
$$594$$ 0 0
$$595$$ 2.00000 0.0819920
$$596$$ 0 0
$$597$$ 5.12311 0.209675
$$598$$ 0 0
$$599$$ −5.12311 −0.209324 −0.104662 0.994508i $$-0.533376\pi$$
−0.104662 + 0.994508i $$0.533376\pi$$
$$600$$ 0 0
$$601$$ −16.2462 −0.662697 −0.331348 0.943508i $$-0.607504\pi$$
−0.331348 + 0.943508i $$0.607504\pi$$
$$602$$ 0 0
$$603$$ 2.24621 0.0914728
$$604$$ 0 0
$$605$$ −15.2462 −0.619847
$$606$$ 0 0
$$607$$ −13.7538 −0.558249 −0.279125 0.960255i $$-0.590044\pi$$
−0.279125 + 0.960255i $$0.590044\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 22.9848 0.928349 0.464175 0.885744i $$-0.346351\pi$$
0.464175 + 0.885744i $$0.346351\pi$$
$$614$$ 0 0
$$615$$ 2.00000 0.0806478
$$616$$ 0 0
$$617$$ −13.3693 −0.538228 −0.269114 0.963108i $$-0.586731\pi$$
−0.269114 + 0.963108i $$0.586731\pi$$
$$618$$ 0 0
$$619$$ 17.1231 0.688236 0.344118 0.938926i $$-0.388178\pi$$
0.344118 + 0.938926i $$0.388178\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0.246211 0.00986425
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 5.75379 0.229784
$$628$$ 0 0
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 2.24621 0.0894203 0.0447101 0.999000i $$-0.485764\pi$$
0.0447101 + 0.999000i $$0.485764\pi$$
$$632$$ 0 0
$$633$$ 5.75379 0.228693
$$634$$ 0 0
$$635$$ 20.4924 0.813217
$$636$$ 0 0
$$637$$ −3.12311 −0.123742
$$638$$ 0 0
$$639$$ 5.12311 0.202667
$$640$$ 0 0
$$641$$ −8.24621 −0.325706 −0.162853 0.986650i $$-0.552070\pi$$
−0.162853 + 0.986650i $$0.552070\pi$$
$$642$$ 0 0
$$643$$ 22.2462 0.877305 0.438652 0.898657i $$-0.355456\pi$$
0.438652 + 0.898657i $$0.355456\pi$$
$$644$$ 0 0
$$645$$ −10.2462 −0.403444
$$646$$ 0 0
$$647$$ 40.9848 1.61128 0.805640 0.592405i $$-0.201821\pi$$
0.805640 + 0.592405i $$0.201821\pi$$
$$648$$ 0 0
$$649$$ −20.4924 −0.804398
$$650$$ 0 0
$$651$$ −5.12311 −0.200790
$$652$$ 0 0
$$653$$ 7.61553 0.298019 0.149009 0.988836i $$-0.452392\pi$$
0.149009 + 0.988836i $$0.452392\pi$$
$$654$$ 0 0
$$655$$ 1.75379 0.0685262
$$656$$ 0 0
$$657$$ 15.1231 0.590009
$$658$$ 0 0
$$659$$ 9.61553 0.374568 0.187284 0.982306i $$-0.440032\pi$$
0.187284 + 0.982306i $$0.440032\pi$$
$$660$$ 0 0
$$661$$ 31.7538 1.23508 0.617540 0.786540i $$-0.288130\pi$$
0.617540 + 0.786540i $$0.288130\pi$$
$$662$$ 0 0
$$663$$ 6.24621 0.242583
$$664$$ 0 0
$$665$$ 1.12311 0.0435522
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −18.2462 −0.705439
$$670$$ 0 0
$$671$$ 42.2462 1.63090
$$672$$ 0 0
$$673$$ 37.2311 1.43515 0.717576 0.696480i $$-0.245252\pi$$
0.717576 + 0.696480i $$0.245252\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 7.75379 0.298002 0.149001 0.988837i $$-0.452394\pi$$
0.149001 + 0.988837i $$0.452394\pi$$
$$678$$ 0 0
$$679$$ −4.87689 −0.187158
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 13.7538 0.526274 0.263137 0.964758i $$-0.415243\pi$$
0.263137 + 0.964758i $$0.415243\pi$$
$$684$$ 0 0
$$685$$ −12.8769 −0.492001
$$686$$ 0 0
$$687$$ 3.75379 0.143216
$$688$$ 0 0
$$689$$ −41.7538 −1.59069
$$690$$ 0 0
$$691$$ −20.6307 −0.784828 −0.392414 0.919789i $$-0.628360\pi$$
−0.392414 + 0.919789i $$0.628360\pi$$
$$692$$ 0 0
$$693$$ 5.12311 0.194611
$$694$$ 0 0
$$695$$ −9.12311 −0.346059
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ −23.1231 −0.874596
$$700$$ 0 0
$$701$$ −38.4924 −1.45384 −0.726919 0.686723i $$-0.759049\pi$$
−0.726919 + 0.686723i $$0.759049\pi$$
$$702$$ 0 0
$$703$$ −2.24621 −0.0847175
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0.246211 0.00925973
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 2.24621 0.0842395
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ 0 0
$$717$$ −13.1231 −0.490091
$$718$$ 0 0
$$719$$ 43.2311 1.61225 0.806123 0.591748i $$-0.201562\pi$$
0.806123 + 0.591748i $$0.201562\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −20.2462 −0.752965
$$724$$ 0 0
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −30.7386 −1.14003 −0.570016 0.821633i $$-0.693063\pi$$
−0.570016 + 0.821633i $$0.693063\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −20.4924 −0.757940
$$732$$ 0 0
$$733$$ −47.6155 −1.75872 −0.879360 0.476158i $$-0.842029\pi$$
−0.879360 + 0.476158i $$0.842029\pi$$
$$734$$ 0 0
$$735$$ 1.00000 0.0368856
$$736$$ 0 0
$$737$$ −11.5076 −0.423887
$$738$$ 0 0
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 3.50758 0.128854
$$742$$ 0 0
$$743$$ −14.7386 −0.540708 −0.270354 0.962761i $$-0.587141\pi$$
−0.270354 + 0.962761i $$0.587141\pi$$
$$744$$ 0 0
$$745$$ 12.2462 0.448666
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −10.2462 −0.373890 −0.186945 0.982370i $$-0.559859\pi$$
−0.186945 + 0.982370i $$0.559859\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 20.4924 0.745796
$$756$$ 0 0
$$757$$ 35.7538 1.29949 0.649747 0.760151i $$-0.274875\pi$$
0.649747 + 0.760151i $$0.274875\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −28.7386 −1.04177 −0.520887 0.853625i $$-0.674399\pi$$
−0.520887 + 0.853625i $$0.674399\pi$$
$$762$$ 0 0
$$763$$ −8.24621 −0.298533
$$764$$ 0 0
$$765$$ −2.00000 −0.0723102
$$766$$ 0 0
$$767$$ −12.4924 −0.451075
$$768$$ 0 0
$$769$$ −32.2462 −1.16283 −0.581414 0.813608i $$-0.697500\pi$$
−0.581414 + 0.813608i $$0.697500\pi$$
$$770$$ 0 0
$$771$$ −20.2462 −0.729149
$$772$$ 0 0
$$773$$ −16.2462 −0.584336 −0.292168 0.956367i $$-0.594377\pi$$
−0.292168 + 0.956367i $$0.594377\pi$$
$$774$$ 0 0
$$775$$ −5.12311 −0.184027
$$776$$ 0 0
$$777$$ −2.00000 −0.0717496
$$778$$ 0 0
$$779$$ 2.24621 0.0804789
$$780$$ 0 0
$$781$$ −26.2462 −0.939163
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ −7.12311 −0.254235
$$786$$ 0 0
$$787$$ −31.2311 −1.11327 −0.556633 0.830758i $$-0.687907\pi$$
−0.556633 + 0.830758i $$0.687907\pi$$
$$788$$ 0 0
$$789$$ −18.2462 −0.649582
$$790$$ 0 0
$$791$$ 5.36932 0.190911
$$792$$ 0 0
$$793$$ 25.7538 0.914544
$$794$$ 0 0
$$795$$ 13.3693 0.474161
$$796$$ 0 0
$$797$$ −36.7386 −1.30135 −0.650675 0.759357i $$-0.725514\pi$$
−0.650675 + 0.759357i $$0.725514\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −0.246211 −0.00869945
$$802$$ 0 0
$$803$$ −77.4773 −2.73411
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 16.2462 0.571894
$$808$$ 0 0
$$809$$ 20.2462 0.711819 0.355909 0.934520i $$-0.384171\pi$$
0.355909 + 0.934520i $$0.384171\pi$$
$$810$$ 0 0
$$811$$ 53.6155 1.88270 0.941348 0.337438i $$-0.109560\pi$$
0.941348 + 0.337438i $$0.109560\pi$$
$$812$$ 0 0
$$813$$ −17.6155 −0.617804
$$814$$ 0 0
$$815$$ 10.2462 0.358909
$$816$$ 0 0
$$817$$ −11.5076 −0.402599
$$818$$ 0 0
$$819$$ 3.12311 0.109130
$$820$$ 0 0
$$821$$ 54.9848 1.91898 0.959492 0.281735i $$-0.0909100\pi$$
0.959492 + 0.281735i $$0.0909100\pi$$
$$822$$ 0 0
$$823$$ 12.4924 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$824$$ 0 0
$$825$$ 5.12311 0.178364
$$826$$ 0 0
$$827$$ −48.9848 −1.70337 −0.851685 0.524054i $$-0.824419\pi$$
−0.851685 + 0.524054i $$0.824419\pi$$
$$828$$ 0 0
$$829$$ 6.49242 0.225491 0.112746 0.993624i $$-0.464035\pi$$
0.112746 + 0.993624i $$0.464035\pi$$
$$830$$ 0 0
$$831$$ 14.4924 0.502737
$$832$$ 0 0
$$833$$ 2.00000 0.0692959
$$834$$ 0 0
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 5.12311 0.177080
$$838$$ 0 0
$$839$$ 38.7386 1.33741 0.668703 0.743530i $$-0.266850\pi$$
0.668703 + 0.743530i $$0.266850\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −24.7386 −0.852044
$$844$$ 0 0
$$845$$ 3.24621 0.111673
$$846$$ 0 0
$$847$$ −15.2462 −0.523866
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0.384472 0.0131641 0.00658203 0.999978i $$-0.497905\pi$$
0.00658203 + 0.999978i $$0.497905\pi$$
$$854$$ 0 0
$$855$$ −1.12311 −0.0384094
$$856$$ 0 0
$$857$$ −48.2462 −1.64806 −0.824030 0.566547i $$-0.808279\pi$$
−0.824030 + 0.566547i $$0.808279\pi$$
$$858$$ 0 0
$$859$$ −21.6155 −0.737512 −0.368756 0.929526i $$-0.620216\pi$$
−0.368756 + 0.929526i $$0.620216\pi$$
$$860$$ 0 0
$$861$$ 2.00000 0.0681598
$$862$$ 0 0
$$863$$ −40.0000 −1.36162 −0.680808 0.732462i $$-0.738371\pi$$
−0.680808 + 0.732462i $$0.738371\pi$$
$$864$$ 0 0
$$865$$ 0.246211 0.00837143
$$866$$ 0 0
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ −11.5076 −0.390368
$$870$$ 0 0
$$871$$ −7.01515 −0.237699
$$872$$ 0 0
$$873$$ 4.87689 0.165058
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 10.4924 0.354304 0.177152 0.984184i $$-0.443312\pi$$
0.177152 + 0.984184i $$0.443312\pi$$
$$878$$ 0 0
$$879$$ 16.2462 0.547971
$$880$$ 0 0
$$881$$ −40.2462 −1.35593 −0.677965 0.735095i $$-0.737138\pi$$
−0.677965 + 0.735095i $$0.737138\pi$$
$$882$$ 0 0
$$883$$ −18.2462 −0.614034 −0.307017 0.951704i $$-0.599331\pi$$
−0.307017 + 0.951704i $$0.599331\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 0 0
$$889$$ 20.4924 0.687294
$$890$$ 0 0
$$891$$ −5.12311 −0.171630
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −2.87689 −0.0961640
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 10.2462 0.341730
$$900$$ 0 0
$$901$$ 26.7386 0.890793
$$902$$ 0 0
$$903$$ −10.2462 −0.340973
$$904$$ 0 0
$$905$$ −22.4924 −0.747673
$$906$$ 0 0
$$907$$ 38.7386 1.28630 0.643148 0.765742i $$-0.277628\pi$$
0.643148 + 0.765742i $$0.277628\pi$$
$$908$$ 0 0
$$909$$ −0.246211 −0.00816631
$$910$$ 0 0
$$911$$ −1.61553 −0.0535248 −0.0267624 0.999642i $$-0.508520\pi$$
−0.0267624 + 0.999642i $$0.508520\pi$$
$$912$$ 0 0
$$913$$ −20.4924 −0.678200
$$914$$ 0 0
$$915$$ −8.24621 −0.272611
$$916$$ 0 0
$$917$$ 1.75379 0.0579152
$$918$$ 0 0
$$919$$ 18.2462 0.601887 0.300943 0.953642i $$-0.402698\pi$$
0.300943 + 0.953642i $$0.402698\pi$$
$$920$$ 0 0
$$921$$ −9.75379 −0.321398
$$922$$ 0 0
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −11.7538 −0.385629 −0.192815 0.981235i $$-0.561762\pi$$
−0.192815 + 0.981235i $$0.561762\pi$$
$$930$$ 0 0
$$931$$ 1.12311 0.0368083
$$932$$ 0 0
$$933$$ −16.0000 −0.523816
$$934$$ 0 0
$$935$$ 10.2462 0.335087
$$936$$ 0 0
$$937$$ 40.1080 1.31027 0.655135 0.755512i $$-0.272612\pi$$
0.655135 + 0.755512i $$0.272612\pi$$
$$938$$ 0 0
$$939$$ −33.3693 −1.08897
$$940$$ 0 0
$$941$$ 42.9848 1.40127 0.700633 0.713522i $$-0.252901\pi$$
0.700633 + 0.713522i $$0.252901\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −1.00000 −0.0325300
$$946$$ 0 0
$$947$$ 51.2311 1.66479 0.832393 0.554186i $$-0.186970\pi$$
0.832393 + 0.554186i $$0.186970\pi$$
$$948$$ 0 0
$$949$$ −47.2311 −1.53318
$$950$$ 0 0
$$951$$ 15.1231 0.490400
$$952$$ 0 0
$$953$$ 2.63068 0.0852162 0.0426081 0.999092i $$-0.486433\pi$$
0.0426081 + 0.999092i $$0.486433\pi$$
$$954$$ 0 0
$$955$$ 5.12311 0.165780
$$956$$ 0 0
$$957$$ −10.2462 −0.331213
$$958$$ 0 0
$$959$$ −12.8769 −0.415817
$$960$$ 0 0
$$961$$ −4.75379 −0.153348
$$962$$ 0 0
$$963$$ 8.00000 0.257796
$$964$$ 0 0
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 0 0
$$969$$ −2.24621 −0.0721587
$$970$$ 0 0
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ 0 0
$$973$$ −9.12311 −0.292473
$$974$$ 0 0
$$975$$ 3.12311 0.100019
$$976$$ 0 0
$$977$$ 10.6307 0.340106 0.170053 0.985435i $$-0.445606\pi$$
0.170053 + 0.985435i $$0.445606\pi$$
$$978$$ 0 0
$$979$$ 1.26137 0.0403134
$$980$$ 0 0
$$981$$ 8.24621 0.263281
$$982$$ 0 0
$$983$$ 11.5076 0.367035 0.183517 0.983016i $$-0.441252\pi$$
0.183517 + 0.983016i $$0.441252\pi$$
$$984$$ 0 0
$$985$$ −23.6155 −0.752453
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 9.26137 0.294197 0.147098 0.989122i $$-0.453007\pi$$
0.147098 + 0.989122i $$0.453007\pi$$
$$992$$ 0 0
$$993$$ 2.24621 0.0712814
$$994$$ 0 0
$$995$$ 5.12311 0.162413
$$996$$ 0 0
$$997$$ 37.8617 1.19909 0.599547 0.800340i $$-0.295348\pi$$
0.599547 + 0.800340i $$0.295348\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
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 3360.2.a.ba.1.1 2
4.3 odd 2 3360.2.a.bg.1.2 yes 2
8.3 odd 2 6720.2.a.ct.1.1 2
8.5 even 2 6720.2.a.cw.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.ba.1.1 2 1.1 even 1 trivial
3360.2.a.bg.1.2 yes 2 4.3 odd 2
6720.2.a.ct.1.1 2 8.3 odd 2
6720.2.a.cw.1.2 2 8.5 even 2