# Properties

 Label 3640.2.a.l.1.1 Level $3640$ Weight $2$ Character 3640.1 Self dual yes Analytic conductor $29.066$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.0655463357$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3640.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})$$ $$q-2.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.30278 q^{9} +2.60555 q^{11} -1.00000 q^{13} -2.30278 q^{15} +2.69722 q^{17} -2.69722 q^{19} -2.30278 q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.60555 q^{27} -8.90833 q^{29} -4.30278 q^{31} -6.00000 q^{33} +1.00000 q^{35} +6.90833 q^{37} +2.30278 q^{39} -0.697224 q^{41} -2.00000 q^{43} +2.30278 q^{45} +2.60555 q^{47} +1.00000 q^{49} -6.21110 q^{51} +4.60555 q^{53} +2.60555 q^{55} +6.21110 q^{57} -14.1194 q^{59} +13.2111 q^{61} +2.30278 q^{63} -1.00000 q^{65} -1.90833 q^{67} +18.4222 q^{69} +4.60555 q^{71} -1.39445 q^{73} -2.30278 q^{75} +2.60555 q^{77} -14.5139 q^{79} -10.6056 q^{81} +9.81665 q^{83} +2.69722 q^{85} +20.5139 q^{87} -3.69722 q^{89} -1.00000 q^{91} +9.90833 q^{93} -2.69722 q^{95} +1.21110 q^{97} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9 $$2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 9 q^{17} - 9 q^{19} - q^{21} - 16 q^{23} + 2 q^{25} - 4 q^{27} - 7 q^{29} - 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 5 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{55} - 2 q^{57} - 3 q^{59} + 12 q^{61} + q^{63} - 2 q^{65} + 7 q^{67} + 8 q^{69} + 2 q^{71} - 10 q^{73} - q^{75} - 2 q^{77} - 11 q^{79} - 14 q^{81} - 2 q^{83} + 9 q^{85} + 23 q^{87} - 11 q^{89} - 2 q^{91} + 9 q^{93} - 9 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9 - 2 * q^11 - 2 * q^13 - q^15 + 9 * q^17 - 9 * q^19 - q^21 - 16 * q^23 + 2 * q^25 - 4 * q^27 - 7 * q^29 - 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 5 * q^41 - 4 * q^43 + q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 + 2 * q^53 - 2 * q^55 - 2 * q^57 - 3 * q^59 + 12 * q^61 + q^63 - 2 * q^65 + 7 * q^67 + 8 * q^69 + 2 * q^71 - 10 * q^73 - q^75 - 2 * q^77 - 11 * q^79 - 14 * q^81 - 2 * q^83 + 9 * q^85 + 23 * q^87 - 11 * q^89 - 2 * q^91 + 9 * q^93 - 9 * q^95 - 12 * q^97 + 12 * 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
$$3$$ −2.30278 −1.32951 −0.664754 0.747062i $$-0.731464\pi$$
−0.664754 + 0.747062i $$0.731464\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ 2.60555 0.785603 0.392802 0.919623i $$-0.371506\pi$$
0.392802 + 0.919623i $$0.371506\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −2.30278 −0.594574
$$16$$ 0 0
$$17$$ 2.69722 0.654173 0.327086 0.944994i $$-0.393933\pi$$
0.327086 + 0.944994i $$0.393933\pi$$
$$18$$ 0 0
$$19$$ −2.69722 −0.618786 −0.309393 0.950934i $$-0.600126\pi$$
−0.309393 + 0.950934i $$0.600126\pi$$
$$20$$ 0 0
$$21$$ −2.30278 −0.502507
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.60555 0.308988
$$28$$ 0 0
$$29$$ −8.90833 −1.65423 −0.827117 0.562029i $$-0.810021\pi$$
−0.827117 + 0.562029i $$0.810021\pi$$
$$30$$ 0 0
$$31$$ −4.30278 −0.772801 −0.386401 0.922331i $$-0.626282\pi$$
−0.386401 + 0.922331i $$0.626282\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 1.00000 0.169031
$$36$$ 0 0
$$37$$ 6.90833 1.13572 0.567861 0.823124i $$-0.307771\pi$$
0.567861 + 0.823124i $$0.307771\pi$$
$$38$$ 0 0
$$39$$ 2.30278 0.368739
$$40$$ 0 0
$$41$$ −0.697224 −0.108888 −0.0544441 0.998517i $$-0.517339\pi$$
−0.0544441 + 0.998517i $$0.517339\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ 2.30278 0.343278
$$46$$ 0 0
$$47$$ 2.60555 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.21110 −0.869728
$$52$$ 0 0
$$53$$ 4.60555 0.632621 0.316311 0.948656i $$-0.397556\pi$$
0.316311 + 0.948656i $$0.397556\pi$$
$$54$$ 0 0
$$55$$ 2.60555 0.351332
$$56$$ 0 0
$$57$$ 6.21110 0.822681
$$58$$ 0 0
$$59$$ −14.1194 −1.83819 −0.919097 0.394032i $$-0.871080\pi$$
−0.919097 + 0.394032i $$0.871080\pi$$
$$60$$ 0 0
$$61$$ 13.2111 1.69151 0.845754 0.533573i $$-0.179151\pi$$
0.845754 + 0.533573i $$0.179151\pi$$
$$62$$ 0 0
$$63$$ 2.30278 0.290122
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −1.90833 −0.233139 −0.116570 0.993183i $$-0.537190\pi$$
−0.116570 + 0.993183i $$0.537190\pi$$
$$68$$ 0 0
$$69$$ 18.4222 2.21777
$$70$$ 0 0
$$71$$ 4.60555 0.546578 0.273289 0.961932i $$-0.411888\pi$$
0.273289 + 0.961932i $$0.411888\pi$$
$$72$$ 0 0
$$73$$ −1.39445 −0.163208 −0.0816039 0.996665i $$-0.526004\pi$$
−0.0816039 + 0.996665i $$0.526004\pi$$
$$74$$ 0 0
$$75$$ −2.30278 −0.265902
$$76$$ 0 0
$$77$$ 2.60555 0.296930
$$78$$ 0 0
$$79$$ −14.5139 −1.63294 −0.816469 0.577389i $$-0.804072\pi$$
−0.816469 + 0.577389i $$0.804072\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 0 0
$$83$$ 9.81665 1.07752 0.538759 0.842460i $$-0.318893\pi$$
0.538759 + 0.842460i $$0.318893\pi$$
$$84$$ 0 0
$$85$$ 2.69722 0.292555
$$86$$ 0 0
$$87$$ 20.5139 2.19932
$$88$$ 0 0
$$89$$ −3.69722 −0.391905 −0.195952 0.980613i $$-0.562780\pi$$
−0.195952 + 0.980613i $$0.562780\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ 9.90833 1.02745
$$94$$ 0 0
$$95$$ −2.69722 −0.276729
$$96$$ 0 0
$$97$$ 1.21110 0.122969 0.0614844 0.998108i $$-0.480417\pi$$
0.0614844 + 0.998108i $$0.480417\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 1.90833 0.188033 0.0940165 0.995571i $$-0.470029\pi$$
0.0940165 + 0.995571i $$0.470029\pi$$
$$104$$ 0 0
$$105$$ −2.30278 −0.224728
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ −15.9083 −1.50995
$$112$$ 0 0
$$113$$ −13.8167 −1.29976 −0.649881 0.760036i $$-0.725181\pi$$
−0.649881 + 0.760036i $$0.725181\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ −2.30278 −0.212892
$$118$$ 0 0
$$119$$ 2.69722 0.247254
$$120$$ 0 0
$$121$$ −4.21110 −0.382828
$$122$$ 0 0
$$123$$ 1.60555 0.144768
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −1.39445 −0.123737 −0.0618687 0.998084i $$-0.519706\pi$$
−0.0618687 + 0.998084i $$0.519706\pi$$
$$128$$ 0 0
$$129$$ 4.60555 0.405496
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −2.69722 −0.233879
$$134$$ 0 0
$$135$$ 1.60555 0.138184
$$136$$ 0 0
$$137$$ 6.30278 0.538482 0.269241 0.963073i $$-0.413227\pi$$
0.269241 + 0.963073i $$0.413227\pi$$
$$138$$ 0 0
$$139$$ −12.4222 −1.05364 −0.526819 0.849978i $$-0.676615\pi$$
−0.526819 + 0.849978i $$0.676615\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −2.60555 −0.217887
$$144$$ 0 0
$$145$$ −8.90833 −0.739796
$$146$$ 0 0
$$147$$ −2.30278 −0.189930
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ 6.21110 0.502138
$$154$$ 0 0
$$155$$ −4.30278 −0.345607
$$156$$ 0 0
$$157$$ −7.51388 −0.599673 −0.299836 0.953991i $$-0.596932\pi$$
−0.299836 + 0.953991i $$0.596932\pi$$
$$158$$ 0 0
$$159$$ −10.6056 −0.841075
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −0.119429 −0.00935444 −0.00467722 0.999989i $$-0.501489\pi$$
−0.00467722 + 0.999989i $$0.501489\pi$$
$$164$$ 0 0
$$165$$ −6.00000 −0.467099
$$166$$ 0 0
$$167$$ −8.78890 −0.680105 −0.340053 0.940406i $$-0.610445\pi$$
−0.340053 + 0.940406i $$0.610445\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −6.21110 −0.474975
$$172$$ 0 0
$$173$$ 22.3028 1.69565 0.847824 0.530277i $$-0.177912\pi$$
0.847824 + 0.530277i $$0.177912\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 32.5139 2.44389
$$178$$ 0 0
$$179$$ −19.9083 −1.48802 −0.744009 0.668169i $$-0.767078\pi$$
−0.744009 + 0.668169i $$0.767078\pi$$
$$180$$ 0 0
$$181$$ −11.8167 −0.878325 −0.439162 0.898408i $$-0.644725\pi$$
−0.439162 + 0.898408i $$0.644725\pi$$
$$182$$ 0 0
$$183$$ −30.4222 −2.24887
$$184$$ 0 0
$$185$$ 6.90833 0.507910
$$186$$ 0 0
$$187$$ 7.02776 0.513920
$$188$$ 0 0
$$189$$ 1.60555 0.116787
$$190$$ 0 0
$$191$$ 1.51388 0.109540 0.0547702 0.998499i $$-0.482557\pi$$
0.0547702 + 0.998499i $$0.482557\pi$$
$$192$$ 0 0
$$193$$ 0.0916731 0.00659877 0.00329939 0.999995i $$-0.498950\pi$$
0.00329939 + 0.999995i $$0.498950\pi$$
$$194$$ 0 0
$$195$$ 2.30278 0.164905
$$196$$ 0 0
$$197$$ −12.1194 −0.863474 −0.431737 0.902000i $$-0.642099\pi$$
−0.431737 + 0.902000i $$0.642099\pi$$
$$198$$ 0 0
$$199$$ −13.2111 −0.936510 −0.468255 0.883593i $$-0.655117\pi$$
−0.468255 + 0.883593i $$0.655117\pi$$
$$200$$ 0 0
$$201$$ 4.39445 0.309961
$$202$$ 0 0
$$203$$ −8.90833 −0.625242
$$204$$ 0 0
$$205$$ −0.697224 −0.0486963
$$206$$ 0 0
$$207$$ −18.4222 −1.28043
$$208$$ 0 0
$$209$$ −7.02776 −0.486120
$$210$$ 0 0
$$211$$ −6.51388 −0.448434 −0.224217 0.974539i $$-0.571982\pi$$
−0.224217 + 0.974539i $$0.571982\pi$$
$$212$$ 0 0
$$213$$ −10.6056 −0.726680
$$214$$ 0 0
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ −4.30278 −0.292091
$$218$$ 0 0
$$219$$ 3.21110 0.216986
$$220$$ 0 0
$$221$$ −2.69722 −0.181435
$$222$$ 0 0
$$223$$ −1.21110 −0.0811014 −0.0405507 0.999177i $$-0.512911\pi$$
−0.0405507 + 0.999177i $$0.512911\pi$$
$$224$$ 0 0
$$225$$ 2.30278 0.153518
$$226$$ 0 0
$$227$$ 1.81665 0.120576 0.0602878 0.998181i $$-0.480798\pi$$
0.0602878 + 0.998181i $$0.480798\pi$$
$$228$$ 0 0
$$229$$ 3.30278 0.218254 0.109127 0.994028i $$-0.465195\pi$$
0.109127 + 0.994028i $$0.465195\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 19.8167 1.29823 0.649116 0.760689i $$-0.275139\pi$$
0.649116 + 0.760689i $$0.275139\pi$$
$$234$$ 0 0
$$235$$ 2.60555 0.169967
$$236$$ 0 0
$$237$$ 33.4222 2.17101
$$238$$ 0 0
$$239$$ −21.6333 −1.39934 −0.699671 0.714465i $$-0.746670\pi$$
−0.699671 + 0.714465i $$0.746670\pi$$
$$240$$ 0 0
$$241$$ −6.09167 −0.392399 −0.196200 0.980564i $$-0.562860\pi$$
−0.196200 + 0.980564i $$0.562860\pi$$
$$242$$ 0 0
$$243$$ 19.6056 1.25770
$$244$$ 0 0
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ 2.69722 0.171620
$$248$$ 0 0
$$249$$ −22.6056 −1.43257
$$250$$ 0 0
$$251$$ 15.6333 0.986766 0.493383 0.869812i $$-0.335760\pi$$
0.493383 + 0.869812i $$0.335760\pi$$
$$252$$ 0 0
$$253$$ −20.8444 −1.31048
$$254$$ 0 0
$$255$$ −6.21110 −0.388954
$$256$$ 0 0
$$257$$ −21.6333 −1.34945 −0.674724 0.738070i $$-0.735737\pi$$
−0.674724 + 0.738070i $$0.735737\pi$$
$$258$$ 0 0
$$259$$ 6.90833 0.429263
$$260$$ 0 0
$$261$$ −20.5139 −1.26978
$$262$$ 0 0
$$263$$ −27.6333 −1.70394 −0.851971 0.523588i $$-0.824593\pi$$
−0.851971 + 0.523588i $$0.824593\pi$$
$$264$$ 0 0
$$265$$ 4.60555 0.282917
$$266$$ 0 0
$$267$$ 8.51388 0.521041
$$268$$ 0 0
$$269$$ 3.81665 0.232705 0.116353 0.993208i $$-0.462880\pi$$
0.116353 + 0.993208i $$0.462880\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 2.30278 0.139370
$$274$$ 0 0
$$275$$ 2.60555 0.157121
$$276$$ 0 0
$$277$$ −13.2111 −0.793778 −0.396889 0.917867i $$-0.629910\pi$$
−0.396889 + 0.917867i $$0.629910\pi$$
$$278$$ 0 0
$$279$$ −9.90833 −0.593196
$$280$$ 0 0
$$281$$ −32.0000 −1.90896 −0.954480 0.298275i $$-0.903589\pi$$
−0.954480 + 0.298275i $$0.903589\pi$$
$$282$$ 0 0
$$283$$ −25.9083 −1.54009 −0.770045 0.637989i $$-0.779766\pi$$
−0.770045 + 0.637989i $$0.779766\pi$$
$$284$$ 0 0
$$285$$ 6.21110 0.367914
$$286$$ 0 0
$$287$$ −0.697224 −0.0411559
$$288$$ 0 0
$$289$$ −9.72498 −0.572058
$$290$$ 0 0
$$291$$ −2.78890 −0.163488
$$292$$ 0 0
$$293$$ −26.4222 −1.54360 −0.771801 0.635864i $$-0.780644\pi$$
−0.771801 + 0.635864i $$0.780644\pi$$
$$294$$ 0 0
$$295$$ −14.1194 −0.822065
$$296$$ 0 0
$$297$$ 4.18335 0.242742
$$298$$ 0 0
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 0 0
$$303$$ −13.8167 −0.793746
$$304$$ 0 0
$$305$$ 13.2111 0.756466
$$306$$ 0 0
$$307$$ 14.6056 0.833583 0.416791 0.909002i $$-0.363155\pi$$
0.416791 + 0.909002i $$0.363155\pi$$
$$308$$ 0 0
$$309$$ −4.39445 −0.249991
$$310$$ 0 0
$$311$$ 13.0278 0.738736 0.369368 0.929283i $$-0.379574\pi$$
0.369368 + 0.929283i $$0.379574\pi$$
$$312$$ 0 0
$$313$$ 9.33053 0.527393 0.263696 0.964606i $$-0.415058\pi$$
0.263696 + 0.964606i $$0.415058\pi$$
$$314$$ 0 0
$$315$$ 2.30278 0.129747
$$316$$ 0 0
$$317$$ 12.4222 0.697701 0.348850 0.937178i $$-0.386572\pi$$
0.348850 + 0.937178i $$0.386572\pi$$
$$318$$ 0 0
$$319$$ −23.2111 −1.29957
$$320$$ 0 0
$$321$$ 13.8167 0.771170
$$322$$ 0 0
$$323$$ −7.27502 −0.404793
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ 9.21110 0.509375
$$328$$ 0 0
$$329$$ 2.60555 0.143649
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 0 0
$$333$$ 15.9083 0.871771
$$334$$ 0 0
$$335$$ −1.90833 −0.104263
$$336$$ 0 0
$$337$$ 4.42221 0.240893 0.120446 0.992720i $$-0.461567\pi$$
0.120446 + 0.992720i $$0.461567\pi$$
$$338$$ 0 0
$$339$$ 31.8167 1.72804
$$340$$ 0 0
$$341$$ −11.2111 −0.607115
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 18.4222 0.991818
$$346$$ 0 0
$$347$$ −8.18335 −0.439305 −0.219653 0.975578i $$-0.570492\pi$$
−0.219653 + 0.975578i $$0.570492\pi$$
$$348$$ 0 0
$$349$$ 23.9361 1.28127 0.640635 0.767846i $$-0.278671\pi$$
0.640635 + 0.767846i $$0.278671\pi$$
$$350$$ 0 0
$$351$$ −1.60555 −0.0856980
$$352$$ 0 0
$$353$$ −25.6333 −1.36432 −0.682162 0.731201i $$-0.738960\pi$$
−0.682162 + 0.731201i $$0.738960\pi$$
$$354$$ 0 0
$$355$$ 4.60555 0.244437
$$356$$ 0 0
$$357$$ −6.21110 −0.328726
$$358$$ 0 0
$$359$$ −11.0278 −0.582023 −0.291011 0.956720i $$-0.593992\pi$$
−0.291011 + 0.956720i $$0.593992\pi$$
$$360$$ 0 0
$$361$$ −11.7250 −0.617104
$$362$$ 0 0
$$363$$ 9.69722 0.508972
$$364$$ 0 0
$$365$$ −1.39445 −0.0729888
$$366$$ 0 0
$$367$$ 1.57779 0.0823602 0.0411801 0.999152i $$-0.486888\pi$$
0.0411801 + 0.999152i $$0.486888\pi$$
$$368$$ 0 0
$$369$$ −1.60555 −0.0835817
$$370$$ 0 0
$$371$$ 4.60555 0.239108
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ −2.30278 −0.118915
$$376$$ 0 0
$$377$$ 8.90833 0.458802
$$378$$ 0 0
$$379$$ 13.6333 0.700296 0.350148 0.936694i $$-0.386131\pi$$
0.350148 + 0.936694i $$0.386131\pi$$
$$380$$ 0 0
$$381$$ 3.21110 0.164510
$$382$$ 0 0
$$383$$ 26.2389 1.34074 0.670372 0.742026i $$-0.266135\pi$$
0.670372 + 0.742026i $$0.266135\pi$$
$$384$$ 0 0
$$385$$ 2.60555 0.132791
$$386$$ 0 0
$$387$$ −4.60555 −0.234113
$$388$$ 0 0
$$389$$ −8.51388 −0.431671 −0.215835 0.976430i $$-0.569247\pi$$
−0.215835 + 0.976430i $$0.569247\pi$$
$$390$$ 0 0
$$391$$ −21.5778 −1.09124
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −14.5139 −0.730272
$$396$$ 0 0
$$397$$ 8.18335 0.410710 0.205355 0.978688i $$-0.434165\pi$$
0.205355 + 0.978688i $$0.434165\pi$$
$$398$$ 0 0
$$399$$ 6.21110 0.310944
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ 4.30278 0.214337
$$404$$ 0 0
$$405$$ −10.6056 −0.526994
$$406$$ 0 0
$$407$$ 18.0000 0.892227
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ −14.5139 −0.715917
$$412$$ 0 0
$$413$$ −14.1194 −0.694772
$$414$$ 0 0
$$415$$ 9.81665 0.481881
$$416$$ 0 0
$$417$$ 28.6056 1.40082
$$418$$ 0 0
$$419$$ −15.6333 −0.763737 −0.381869 0.924217i $$-0.624719\pi$$
−0.381869 + 0.924217i $$0.624719\pi$$
$$420$$ 0 0
$$421$$ −12.4222 −0.605421 −0.302711 0.953083i $$-0.597892\pi$$
−0.302711 + 0.953083i $$0.597892\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 2.69722 0.130835
$$426$$ 0 0
$$427$$ 13.2111 0.639330
$$428$$ 0 0
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ 0 0
$$433$$ −24.9361 −1.19835 −0.599176 0.800617i $$-0.704505\pi$$
−0.599176 + 0.800617i $$0.704505\pi$$
$$434$$ 0 0
$$435$$ 20.5139 0.983565
$$436$$ 0 0
$$437$$ 21.5778 1.03221
$$438$$ 0 0
$$439$$ 19.2111 0.916896 0.458448 0.888721i $$-0.348406\pi$$
0.458448 + 0.888721i $$0.348406\pi$$
$$440$$ 0 0
$$441$$ 2.30278 0.109656
$$442$$ 0 0
$$443$$ 16.6056 0.788954 0.394477 0.918906i $$-0.370926\pi$$
0.394477 + 0.918906i $$0.370926\pi$$
$$444$$ 0 0
$$445$$ −3.69722 −0.175265
$$446$$ 0 0
$$447$$ −18.4222 −0.871340
$$448$$ 0 0
$$449$$ −35.0278 −1.65306 −0.826531 0.562891i $$-0.809689\pi$$
−0.826531 + 0.562891i $$0.809689\pi$$
$$450$$ 0 0
$$451$$ −1.81665 −0.0855429
$$452$$ 0 0
$$453$$ 23.0278 1.08194
$$454$$ 0 0
$$455$$ −1.00000 −0.0468807
$$456$$ 0 0
$$457$$ 27.5139 1.28704 0.643522 0.765427i $$-0.277472\pi$$
0.643522 + 0.765427i $$0.277472\pi$$
$$458$$ 0 0
$$459$$ 4.33053 0.202132
$$460$$ 0 0
$$461$$ 24.9083 1.16010 0.580048 0.814582i $$-0.303034\pi$$
0.580048 + 0.814582i $$0.303034\pi$$
$$462$$ 0 0
$$463$$ 5.72498 0.266062 0.133031 0.991112i $$-0.457529\pi$$
0.133031 + 0.991112i $$0.457529\pi$$
$$464$$ 0 0
$$465$$ 9.90833 0.459488
$$466$$ 0 0
$$467$$ −19.4861 −0.901710 −0.450855 0.892597i $$-0.648881\pi$$
−0.450855 + 0.892597i $$0.648881\pi$$
$$468$$ 0 0
$$469$$ −1.90833 −0.0881183
$$470$$ 0 0
$$471$$ 17.3028 0.797270
$$472$$ 0 0
$$473$$ −5.21110 −0.239607
$$474$$ 0 0
$$475$$ −2.69722 −0.123757
$$476$$ 0 0
$$477$$ 10.6056 0.485595
$$478$$ 0 0
$$479$$ 40.9083 1.86915 0.934575 0.355767i $$-0.115780\pi$$
0.934575 + 0.355767i $$0.115780\pi$$
$$480$$ 0 0
$$481$$ −6.90833 −0.314993
$$482$$ 0 0
$$483$$ 18.4222 0.838239
$$484$$ 0 0
$$485$$ 1.21110 0.0549933
$$486$$ 0 0
$$487$$ 30.3028 1.37315 0.686575 0.727059i $$-0.259113\pi$$
0.686575 + 0.727059i $$0.259113\pi$$
$$488$$ 0 0
$$489$$ 0.275019 0.0124368
$$490$$ 0 0
$$491$$ −30.4222 −1.37293 −0.686467 0.727161i $$-0.740840\pi$$
−0.686467 + 0.727161i $$0.740840\pi$$
$$492$$ 0 0
$$493$$ −24.0278 −1.08216
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ 4.60555 0.206587
$$498$$ 0 0
$$499$$ 26.0000 1.16392 0.581960 0.813217i $$-0.302286\pi$$
0.581960 + 0.813217i $$0.302286\pi$$
$$500$$ 0 0
$$501$$ 20.2389 0.904206
$$502$$ 0 0
$$503$$ −27.6333 −1.23211 −0.616054 0.787704i $$-0.711270\pi$$
−0.616054 + 0.787704i $$0.711270\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ −2.30278 −0.102270
$$508$$ 0 0
$$509$$ −24.3305 −1.07843 −0.539216 0.842168i $$-0.681279\pi$$
−0.539216 + 0.842168i $$0.681279\pi$$
$$510$$ 0 0
$$511$$ −1.39445 −0.0616868
$$512$$ 0 0
$$513$$ −4.33053 −0.191198
$$514$$ 0 0
$$515$$ 1.90833 0.0840909
$$516$$ 0 0
$$517$$ 6.78890 0.298575
$$518$$ 0 0
$$519$$ −51.3583 −2.25438
$$520$$ 0 0
$$521$$ −22.2389 −0.974302 −0.487151 0.873318i $$-0.661964\pi$$
−0.487151 + 0.873318i $$0.661964\pi$$
$$522$$ 0 0
$$523$$ 34.4222 1.50518 0.752589 0.658491i $$-0.228805\pi$$
0.752589 + 0.658491i $$0.228805\pi$$
$$524$$ 0 0
$$525$$ −2.30278 −0.100501
$$526$$ 0 0
$$527$$ −11.6056 −0.505546
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ −32.5139 −1.41098
$$532$$ 0 0
$$533$$ 0.697224 0.0302001
$$534$$ 0 0
$$535$$ −6.00000 −0.259403
$$536$$ 0 0
$$537$$ 45.8444 1.97833
$$538$$ 0 0
$$539$$ 2.60555 0.112229
$$540$$ 0 0
$$541$$ 38.0555 1.63613 0.818067 0.575123i $$-0.195046\pi$$
0.818067 + 0.575123i $$0.195046\pi$$
$$542$$ 0 0
$$543$$ 27.2111 1.16774
$$544$$ 0 0
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ −37.6333 −1.60908 −0.804542 0.593896i $$-0.797589\pi$$
−0.804542 + 0.593896i $$0.797589\pi$$
$$548$$ 0 0
$$549$$ 30.4222 1.29839
$$550$$ 0 0
$$551$$ 24.0278 1.02362
$$552$$ 0 0
$$553$$ −14.5139 −0.617193
$$554$$ 0 0
$$555$$ −15.9083 −0.675271
$$556$$ 0 0
$$557$$ 4.09167 0.173370 0.0866849 0.996236i $$-0.472373\pi$$
0.0866849 + 0.996236i $$0.472373\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ −16.1833 −0.683261
$$562$$ 0 0
$$563$$ 41.5416 1.75077 0.875386 0.483425i $$-0.160608\pi$$
0.875386 + 0.483425i $$0.160608\pi$$
$$564$$ 0 0
$$565$$ −13.8167 −0.581271
$$566$$ 0 0
$$567$$ −10.6056 −0.445391
$$568$$ 0 0
$$569$$ 45.7527 1.91805 0.959027 0.283314i $$-0.0914338\pi$$
0.959027 + 0.283314i $$0.0914338\pi$$
$$570$$ 0 0
$$571$$ −10.7250 −0.448826 −0.224413 0.974494i $$-0.572047\pi$$
−0.224413 + 0.974494i $$0.572047\pi$$
$$572$$ 0 0
$$573$$ −3.48612 −0.145635
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ −14.6056 −0.608037 −0.304019 0.952666i $$-0.598328\pi$$
−0.304019 + 0.952666i $$0.598328\pi$$
$$578$$ 0 0
$$579$$ −0.211103 −0.00877312
$$580$$ 0 0
$$581$$ 9.81665 0.407263
$$582$$ 0 0
$$583$$ 12.0000 0.496989
$$584$$ 0 0
$$585$$ −2.30278 −0.0952081
$$586$$ 0 0
$$587$$ 1.81665 0.0749813 0.0374907 0.999297i $$-0.488064\pi$$
0.0374907 + 0.999297i $$0.488064\pi$$
$$588$$ 0 0
$$589$$ 11.6056 0.478198
$$590$$ 0 0
$$591$$ 27.9083 1.14800
$$592$$ 0 0
$$593$$ −22.1833 −0.910961 −0.455480 0.890246i $$-0.650532\pi$$
−0.455480 + 0.890246i $$0.650532\pi$$
$$594$$ 0 0
$$595$$ 2.69722 0.110575
$$596$$ 0 0
$$597$$ 30.4222 1.24510
$$598$$ 0 0
$$599$$ 18.4222 0.752711 0.376355 0.926475i $$-0.377177\pi$$
0.376355 + 0.926475i $$0.377177\pi$$
$$600$$ 0 0
$$601$$ 37.2111 1.51787 0.758936 0.651165i $$-0.225719\pi$$
0.758936 + 0.651165i $$0.225719\pi$$
$$602$$ 0 0
$$603$$ −4.39445 −0.178956
$$604$$ 0 0
$$605$$ −4.21110 −0.171206
$$606$$ 0 0
$$607$$ −11.5139 −0.467334 −0.233667 0.972317i $$-0.575073\pi$$
−0.233667 + 0.972317i $$0.575073\pi$$
$$608$$ 0 0
$$609$$ 20.5139 0.831264
$$610$$ 0 0
$$611$$ −2.60555 −0.105409
$$612$$ 0 0
$$613$$ −18.8444 −0.761119 −0.380559 0.924757i $$-0.624269\pi$$
−0.380559 + 0.924757i $$0.624269\pi$$
$$614$$ 0 0
$$615$$ 1.60555 0.0647421
$$616$$ 0 0
$$617$$ 3.90833 0.157343 0.0786717 0.996901i $$-0.474932\pi$$
0.0786717 + 0.996901i $$0.474932\pi$$
$$618$$ 0 0
$$619$$ 10.7250 0.431073 0.215537 0.976496i $$-0.430850\pi$$
0.215537 + 0.976496i $$0.430850\pi$$
$$620$$ 0 0
$$621$$ −12.8444 −0.515428
$$622$$ 0 0
$$623$$ −3.69722 −0.148126
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 16.1833 0.646301
$$628$$ 0 0
$$629$$ 18.6333 0.742959
$$630$$ 0 0
$$631$$ 5.81665 0.231557 0.115779 0.993275i $$-0.463064\pi$$
0.115779 + 0.993275i $$0.463064\pi$$
$$632$$ 0 0
$$633$$ 15.0000 0.596196
$$634$$ 0 0
$$635$$ −1.39445 −0.0553370
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 10.6056 0.419549
$$640$$ 0 0
$$641$$ −38.7250 −1.52954 −0.764772 0.644301i $$-0.777149\pi$$
−0.764772 + 0.644301i $$0.777149\pi$$
$$642$$ 0 0
$$643$$ 42.8444 1.68962 0.844809 0.535068i $$-0.179714\pi$$
0.844809 + 0.535068i $$0.179714\pi$$
$$644$$ 0 0
$$645$$ 4.60555 0.181343
$$646$$ 0 0
$$647$$ 0.486122 0.0191114 0.00955571 0.999954i $$-0.496958\pi$$
0.00955571 + 0.999954i $$0.496958\pi$$
$$648$$ 0 0
$$649$$ −36.7889 −1.44409
$$650$$ 0 0
$$651$$ 9.90833 0.388338
$$652$$ 0 0
$$653$$ 2.42221 0.0947882 0.0473941 0.998876i $$-0.484908\pi$$
0.0473941 + 0.998876i $$0.484908\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −3.21110 −0.125277
$$658$$ 0 0
$$659$$ −30.1472 −1.17437 −0.587184 0.809454i $$-0.699763\pi$$
−0.587184 + 0.809454i $$0.699763\pi$$
$$660$$ 0 0
$$661$$ 15.4861 0.602340 0.301170 0.953570i $$-0.402623\pi$$
0.301170 + 0.953570i $$0.402623\pi$$
$$662$$ 0 0
$$663$$ 6.21110 0.241219
$$664$$ 0 0
$$665$$ −2.69722 −0.104594
$$666$$ 0 0
$$667$$ 71.2666 2.75945
$$668$$ 0 0
$$669$$ 2.78890 0.107825
$$670$$ 0 0
$$671$$ 34.4222 1.32885
$$672$$ 0 0
$$673$$ 7.81665 0.301310 0.150655 0.988586i $$-0.451862\pi$$
0.150655 + 0.988586i $$0.451862\pi$$
$$674$$ 0 0
$$675$$ 1.60555 0.0617977
$$676$$ 0 0
$$677$$ −8.42221 −0.323692 −0.161846 0.986816i $$-0.551745\pi$$
−0.161846 + 0.986816i $$0.551745\pi$$
$$678$$ 0 0
$$679$$ 1.21110 0.0464779
$$680$$ 0 0
$$681$$ −4.18335 −0.160306
$$682$$ 0 0
$$683$$ 2.93608 0.112346 0.0561731 0.998421i $$-0.482110\pi$$
0.0561731 + 0.998421i $$0.482110\pi$$
$$684$$ 0 0
$$685$$ 6.30278 0.240817
$$686$$ 0 0
$$687$$ −7.60555 −0.290170
$$688$$ 0 0
$$689$$ −4.60555 −0.175458
$$690$$ 0 0
$$691$$ −3.09167 −0.117613 −0.0588064 0.998269i $$-0.518729\pi$$
−0.0588064 + 0.998269i $$0.518729\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ −12.4222 −0.471201
$$696$$ 0 0
$$697$$ −1.88057 −0.0712317
$$698$$ 0 0
$$699$$ −45.6333 −1.72601
$$700$$ 0 0
$$701$$ −35.1194 −1.32644 −0.663221 0.748423i $$-0.730811\pi$$
−0.663221 + 0.748423i $$0.730811\pi$$
$$702$$ 0 0
$$703$$ −18.6333 −0.702769
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ 0 0
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ 29.2111 1.09705 0.548523 0.836135i $$-0.315190\pi$$
0.548523 + 0.836135i $$0.315190\pi$$
$$710$$ 0 0
$$711$$ −33.4222 −1.25343
$$712$$ 0 0
$$713$$ 34.4222 1.28912
$$714$$ 0 0
$$715$$ −2.60555 −0.0974421
$$716$$ 0 0
$$717$$ 49.8167 1.86044
$$718$$ 0 0
$$719$$ −2.42221 −0.0903330 −0.0451665 0.998979i $$-0.514382\pi$$
−0.0451665 + 0.998979i $$0.514382\pi$$
$$720$$ 0 0
$$721$$ 1.90833 0.0710698
$$722$$ 0 0
$$723$$ 14.0278 0.521698
$$724$$ 0 0
$$725$$ −8.90833 −0.330847
$$726$$ 0 0
$$727$$ −25.6972 −0.953057 −0.476529 0.879159i $$-0.658105\pi$$
−0.476529 + 0.879159i $$0.658105\pi$$
$$728$$ 0 0
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ −5.39445 −0.199521
$$732$$ 0 0
$$733$$ 12.0000 0.443230 0.221615 0.975134i $$-0.428867\pi$$
0.221615 + 0.975134i $$0.428867\pi$$
$$734$$ 0 0
$$735$$ −2.30278 −0.0849392
$$736$$ 0 0
$$737$$ −4.97224 −0.183155
$$738$$ 0 0
$$739$$ 46.0555 1.69418 0.847090 0.531450i $$-0.178353\pi$$
0.847090 + 0.531450i $$0.178353\pi$$
$$740$$ 0 0
$$741$$ −6.21110 −0.228171
$$742$$ 0 0
$$743$$ 26.0917 0.957211 0.478605 0.878030i $$-0.341142\pi$$
0.478605 + 0.878030i $$0.341142\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ 22.6056 0.827094
$$748$$ 0 0
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ −5.51388 −0.201204 −0.100602 0.994927i $$-0.532077\pi$$
−0.100602 + 0.994927i $$0.532077\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ −10.0000 −0.363937
$$756$$ 0 0
$$757$$ 24.1833 0.878959 0.439479 0.898253i $$-0.355163\pi$$
0.439479 + 0.898253i $$0.355163\pi$$
$$758$$ 0 0
$$759$$ 48.0000 1.74229
$$760$$ 0 0
$$761$$ −6.48612 −0.235122 −0.117561 0.993066i $$-0.537508\pi$$
−0.117561 + 0.993066i $$0.537508\pi$$
$$762$$ 0 0
$$763$$ −4.00000 −0.144810
$$764$$ 0 0
$$765$$ 6.21110 0.224563
$$766$$ 0 0
$$767$$ 14.1194 0.509823
$$768$$ 0 0
$$769$$ −44.4222 −1.60191 −0.800953 0.598727i $$-0.795673\pi$$
−0.800953 + 0.598727i $$0.795673\pi$$
$$770$$ 0 0
$$771$$ 49.8167 1.79410
$$772$$ 0 0
$$773$$ 37.8167 1.36017 0.680085 0.733133i $$-0.261943\pi$$
0.680085 + 0.733133i $$0.261943\pi$$
$$774$$ 0 0
$$775$$ −4.30278 −0.154560
$$776$$ 0 0
$$777$$ −15.9083 −0.570708
$$778$$ 0 0
$$779$$ 1.88057 0.0673784
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ −14.3028 −0.511140
$$784$$ 0 0
$$785$$ −7.51388 −0.268182
$$786$$ 0 0
$$787$$ 14.4222 0.514096 0.257048 0.966399i $$-0.417250\pi$$
0.257048 + 0.966399i $$0.417250\pi$$
$$788$$ 0 0
$$789$$ 63.6333 2.26541
$$790$$ 0 0
$$791$$ −13.8167 −0.491264
$$792$$ 0 0
$$793$$ −13.2111 −0.469140
$$794$$ 0 0
$$795$$ −10.6056 −0.376140
$$796$$ 0 0
$$797$$ 10.5416 0.373404 0.186702 0.982417i $$-0.440220\pi$$
0.186702 + 0.982417i $$0.440220\pi$$
$$798$$ 0 0
$$799$$ 7.02776 0.248624
$$800$$ 0 0
$$801$$ −8.51388 −0.300823
$$802$$ 0 0
$$803$$ −3.63331 −0.128217
$$804$$ 0 0
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ −8.78890 −0.309384
$$808$$ 0 0
$$809$$ −3.48612 −0.122566 −0.0612828 0.998120i $$-0.519519\pi$$
−0.0612828 + 0.998120i $$0.519519\pi$$
$$810$$ 0 0
$$811$$ 11.6333 0.408501 0.204250 0.978919i $$-0.434524\pi$$
0.204250 + 0.978919i $$0.434524\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −0.119429 −0.00418343
$$816$$ 0 0
$$817$$ 5.39445 0.188728
$$818$$ 0 0
$$819$$ −2.30278 −0.0804655
$$820$$ 0 0
$$821$$ −54.4777 −1.90129 −0.950643 0.310288i $$-0.899575\pi$$
−0.950643 + 0.310288i $$0.899575\pi$$
$$822$$ 0 0
$$823$$ 33.0278 1.15128 0.575638 0.817705i $$-0.304754\pi$$
0.575638 + 0.817705i $$0.304754\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ 39.6333 1.37819 0.689093 0.724673i $$-0.258009\pi$$
0.689093 + 0.724673i $$0.258009\pi$$
$$828$$ 0 0
$$829$$ −5.21110 −0.180989 −0.0904945 0.995897i $$-0.528845\pi$$
−0.0904945 + 0.995897i $$0.528845\pi$$
$$830$$ 0 0
$$831$$ 30.4222 1.05533
$$832$$ 0 0
$$833$$ 2.69722 0.0934533
$$834$$ 0 0
$$835$$ −8.78890 −0.304152
$$836$$ 0 0
$$837$$ −6.90833 −0.238787
$$838$$ 0 0
$$839$$ 20.8444 0.719629 0.359814 0.933024i $$-0.382840\pi$$
0.359814 + 0.933024i $$0.382840\pi$$
$$840$$ 0 0
$$841$$ 50.3583 1.73649
$$842$$ 0 0
$$843$$ 73.6888 2.53798
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ −4.21110 −0.144695
$$848$$ 0 0
$$849$$ 59.6611 2.04756
$$850$$ 0 0
$$851$$ −55.2666 −1.89452
$$852$$ 0 0
$$853$$ 45.8167 1.56873 0.784366 0.620298i $$-0.212988\pi$$
0.784366 + 0.620298i $$0.212988\pi$$
$$854$$ 0 0
$$855$$ −6.21110 −0.212415
$$856$$ 0 0
$$857$$ 45.6972 1.56099 0.780494 0.625164i $$-0.214968\pi$$
0.780494 + 0.625164i $$0.214968\pi$$
$$858$$ 0 0
$$859$$ −10.4222 −0.355601 −0.177801 0.984067i $$-0.556898\pi$$
−0.177801 + 0.984067i $$0.556898\pi$$
$$860$$ 0 0
$$861$$ 1.60555 0.0547170
$$862$$ 0 0
$$863$$ 18.2750 0.622089 0.311044 0.950395i $$-0.399321\pi$$
0.311044 + 0.950395i $$0.399321\pi$$
$$864$$ 0 0
$$865$$ 22.3028 0.758317
$$866$$ 0 0
$$867$$ 22.3944 0.760555
$$868$$ 0 0
$$869$$ −37.8167 −1.28284
$$870$$ 0 0
$$871$$ 1.90833 0.0646612
$$872$$ 0 0
$$873$$ 2.78890 0.0943899
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 31.9083 1.07747 0.538734 0.842476i $$-0.318903\pi$$
0.538734 + 0.842476i $$0.318903\pi$$
$$878$$ 0 0
$$879$$ 60.8444 2.05223
$$880$$ 0 0
$$881$$ −35.4500 −1.19434 −0.597170 0.802115i $$-0.703708\pi$$
−0.597170 + 0.802115i $$0.703708\pi$$
$$882$$ 0 0
$$883$$ 21.3944 0.719981 0.359990 0.932956i $$-0.382780\pi$$
0.359990 + 0.932956i $$0.382780\pi$$
$$884$$ 0 0
$$885$$ 32.5139 1.09294
$$886$$ 0 0
$$887$$ −35.1472 −1.18013 −0.590064 0.807357i $$-0.700897\pi$$
−0.590064 + 0.807357i $$0.700897\pi$$
$$888$$ 0 0
$$889$$ −1.39445 −0.0467683
$$890$$ 0 0
$$891$$ −27.6333 −0.925751
$$892$$ 0 0
$$893$$ −7.02776 −0.235175
$$894$$ 0 0
$$895$$ −19.9083 −0.665462
$$896$$ 0 0
$$897$$ −18.4222 −0.615100
$$898$$ 0 0
$$899$$ 38.3305 1.27839
$$900$$ 0 0
$$901$$ 12.4222 0.413844
$$902$$ 0 0
$$903$$ 4.60555 0.153263
$$904$$ 0 0
$$905$$ −11.8167 −0.392799
$$906$$ 0 0
$$907$$ −4.60555 −0.152925 −0.0764624 0.997072i $$-0.524363\pi$$
−0.0764624 + 0.997072i $$0.524363\pi$$
$$908$$ 0 0
$$909$$ 13.8167 0.458269
$$910$$ 0 0
$$911$$ −23.9083 −0.792118 −0.396059 0.918225i $$-0.629622\pi$$
−0.396059 + 0.918225i $$0.629622\pi$$
$$912$$ 0 0
$$913$$ 25.5778 0.846501
$$914$$ 0 0
$$915$$ −30.4222 −1.00573
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −32.5416 −1.07345 −0.536725 0.843757i $$-0.680339\pi$$
−0.536725 + 0.843757i $$0.680339\pi$$
$$920$$ 0 0
$$921$$ −33.6333 −1.10826
$$922$$ 0 0
$$923$$ −4.60555 −0.151594
$$924$$ 0 0
$$925$$ 6.90833 0.227144
$$926$$ 0 0
$$927$$ 4.39445 0.144333
$$928$$ 0 0
$$929$$ −33.7527 −1.10739 −0.553696 0.832719i $$-0.686783\pi$$
−0.553696 + 0.832719i $$0.686783\pi$$
$$930$$ 0 0
$$931$$ −2.69722 −0.0883980
$$932$$ 0 0
$$933$$ −30.0000 −0.982156
$$934$$ 0 0
$$935$$ 7.02776 0.229832
$$936$$ 0 0
$$937$$ −37.3028 −1.21863 −0.609314 0.792929i $$-0.708555\pi$$
−0.609314 + 0.792929i $$0.708555\pi$$
$$938$$ 0 0
$$939$$ −21.4861 −0.701173
$$940$$ 0 0
$$941$$ −27.1194 −0.884068 −0.442034 0.896998i $$-0.645743\pi$$
−0.442034 + 0.896998i $$0.645743\pi$$
$$942$$ 0 0
$$943$$ 5.57779 0.181638
$$944$$ 0 0
$$945$$ 1.60555 0.0522286
$$946$$ 0 0
$$947$$ −56.1472 −1.82454 −0.912269 0.409591i $$-0.865671\pi$$
−0.912269 + 0.409591i $$0.865671\pi$$
$$948$$ 0 0
$$949$$ 1.39445 0.0452657
$$950$$ 0 0
$$951$$ −28.6056 −0.927599
$$952$$ 0 0
$$953$$ 11.0278 0.357224 0.178612 0.983920i $$-0.442839\pi$$
0.178612 + 0.983920i $$0.442839\pi$$
$$954$$ 0 0
$$955$$ 1.51388 0.0489879
$$956$$ 0 0
$$957$$ 53.4500 1.72779
$$958$$ 0 0
$$959$$ 6.30278 0.203527
$$960$$ 0 0
$$961$$ −12.4861 −0.402778
$$962$$ 0 0
$$963$$ −13.8167 −0.445235
$$964$$ 0 0
$$965$$ 0.0916731 0.00295106
$$966$$ 0 0
$$967$$ 12.4861 0.401527 0.200763 0.979640i $$-0.435658\pi$$
0.200763 + 0.979640i $$0.435658\pi$$
$$968$$ 0 0
$$969$$ 16.7527 0.538175
$$970$$ 0 0
$$971$$ 49.0278 1.57338 0.786688 0.617351i $$-0.211794\pi$$
0.786688 + 0.617351i $$0.211794\pi$$
$$972$$ 0 0
$$973$$ −12.4222 −0.398238
$$974$$ 0 0
$$975$$ 2.30278 0.0737478
$$976$$ 0 0
$$977$$ −7.90833 −0.253010 −0.126505 0.991966i $$-0.540376\pi$$
−0.126505 + 0.991966i $$0.540376\pi$$
$$978$$ 0 0
$$979$$ −9.63331 −0.307882
$$980$$ 0 0
$$981$$ −9.21110 −0.294088
$$982$$ 0 0
$$983$$ −53.4500 −1.70479 −0.852395 0.522899i $$-0.824850\pi$$
−0.852395 + 0.522899i $$0.824850\pi$$
$$984$$ 0 0
$$985$$ −12.1194 −0.386157
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −34.1472 −1.08472 −0.542361 0.840146i $$-0.682469\pi$$
−0.542361 + 0.840146i $$0.682469\pi$$
$$992$$ 0 0
$$993$$ −41.4500 −1.31537
$$994$$ 0 0
$$995$$ −13.2111 −0.418820
$$996$$ 0 0
$$997$$ −6.14719 −0.194683 −0.0973417 0.995251i $$-0.531034\pi$$
−0.0973417 + 0.995251i $$0.531034\pi$$
$$998$$ 0 0
$$999$$ 11.0917 0.350925
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 3640.2.a.l.1.1 2
4.3 odd 2 7280.2.a.bh.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.l.1.1 2 1.1 even 1 trivial
7280.2.a.bh.1.2 2 4.3 odd 2