# Properties

 Label 1200.4.a.bt.1.2 Level $1200$ Weight $4$ Character 1200.1 Self dual yes Analytic conductor $70.802$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +22.2094 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +22.2094 q^{7} +9.00000 q^{9} +1.79063 q^{11} -58.2094 q^{13} -18.9844 q^{17} -104.837 q^{19} +66.6281 q^{21} -49.6125 q^{23} +27.0000 q^{27} -293.466 q^{29} -64.4187 q^{31} +5.37188 q^{33} +19.8844 q^{37} -174.628 q^{39} -165.581 q^{41} -247.350 q^{43} +384.544 q^{47} +150.256 q^{49} -56.9531 q^{51} -463.528 q^{53} -314.512 q^{57} +73.7906 q^{59} -137.350 q^{61} +199.884 q^{63} +173.906 q^{67} -148.837 q^{69} +594.281 q^{71} -320.231 q^{73} +39.7687 q^{77} +770.469 q^{79} +81.0000 q^{81} +173.925 q^{83} -880.397 q^{87} +1019.02 q^{89} -1292.79 q^{91} -193.256 q^{93} -384.375 q^{97} +16.1156 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + 42 q^{11} - 78 q^{13} - 102 q^{17} - 56 q^{19} + 18 q^{21} - 48 q^{23} + 54 q^{27} - 318 q^{29} - 52 q^{31} + 126 q^{33} - 306 q^{37} - 234 q^{39} - 408 q^{41} + 120 q^{43} + 180 q^{47} + 70 q^{49} - 306 q^{51} - 402 q^{53} - 168 q^{57} + 186 q^{59} + 340 q^{61} + 54 q^{63} + 732 q^{67} - 144 q^{69} + 36 q^{71} - 1332 q^{73} - 612 q^{77} - 380 q^{79} + 162 q^{81} - 984 q^{83} - 954 q^{87} + 1116 q^{89} - 972 q^{91} - 156 q^{93} + 768 q^{97} + 378 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 + 42 * q^11 - 78 * q^13 - 102 * q^17 - 56 * q^19 + 18 * q^21 - 48 * q^23 + 54 * q^27 - 318 * q^29 - 52 * q^31 + 126 * q^33 - 306 * q^37 - 234 * q^39 - 408 * q^41 + 120 * q^43 + 180 * q^47 + 70 * q^49 - 306 * q^51 - 402 * q^53 - 168 * q^57 + 186 * q^59 + 340 * q^61 + 54 * q^63 + 732 * q^67 - 144 * q^69 + 36 * q^71 - 1332 * q^73 - 612 * q^77 - 380 * q^79 + 162 * q^81 - 984 * q^83 - 954 * q^87 + 1116 * q^89 - 972 * q^91 - 156 * q^93 + 768 * q^97 + 378 * 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 22.2094 1.19919 0.599597 0.800302i $$-0.295328\pi$$
0.599597 + 0.800302i $$0.295328\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 1.79063 0.0490813 0.0245407 0.999699i $$-0.492188\pi$$
0.0245407 + 0.999699i $$0.492188\pi$$
$$12$$ 0 0
$$13$$ −58.2094 −1.24188 −0.620938 0.783860i $$-0.713248\pi$$
−0.620938 + 0.783860i $$0.713248\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −18.9844 −0.270846 −0.135423 0.990788i $$-0.543239\pi$$
−0.135423 + 0.990788i $$0.543239\pi$$
$$18$$ 0 0
$$19$$ −104.837 −1.26586 −0.632931 0.774208i $$-0.718148\pi$$
−0.632931 + 0.774208i $$0.718148\pi$$
$$20$$ 0 0
$$21$$ 66.6281 0.692355
$$22$$ 0 0
$$23$$ −49.6125 −0.449779 −0.224890 0.974384i $$-0.572202\pi$$
−0.224890 + 0.974384i $$0.572202\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −293.466 −1.87914 −0.939572 0.342350i $$-0.888777\pi$$
−0.939572 + 0.342350i $$0.888777\pi$$
$$30$$ 0 0
$$31$$ −64.4187 −0.373224 −0.186612 0.982434i $$-0.559751\pi$$
−0.186612 + 0.982434i $$0.559751\pi$$
$$32$$ 0 0
$$33$$ 5.37188 0.0283371
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 19.8844 0.0883505 0.0441752 0.999024i $$-0.485934\pi$$
0.0441752 + 0.999024i $$0.485934\pi$$
$$38$$ 0 0
$$39$$ −174.628 −0.716997
$$40$$ 0 0
$$41$$ −165.581 −0.630718 −0.315359 0.948972i $$-0.602125\pi$$
−0.315359 + 0.948972i $$0.602125\pi$$
$$42$$ 0 0
$$43$$ −247.350 −0.877221 −0.438611 0.898677i $$-0.644529\pi$$
−0.438611 + 0.898677i $$0.644529\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 384.544 1.19344 0.596718 0.802451i $$-0.296471\pi$$
0.596718 + 0.802451i $$0.296471\pi$$
$$48$$ 0 0
$$49$$ 150.256 0.438065
$$50$$ 0 0
$$51$$ −56.9531 −0.156373
$$52$$ 0 0
$$53$$ −463.528 −1.20133 −0.600665 0.799501i $$-0.705097\pi$$
−0.600665 + 0.799501i $$0.705097\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −314.512 −0.730846
$$58$$ 0 0
$$59$$ 73.7906 0.162826 0.0814129 0.996680i $$-0.474057\pi$$
0.0814129 + 0.996680i $$0.474057\pi$$
$$60$$ 0 0
$$61$$ −137.350 −0.288293 −0.144146 0.989556i $$-0.546044\pi$$
−0.144146 + 0.989556i $$0.546044\pi$$
$$62$$ 0 0
$$63$$ 199.884 0.399731
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 173.906 0.317105 0.158552 0.987351i $$-0.449317\pi$$
0.158552 + 0.987351i $$0.449317\pi$$
$$68$$ 0 0
$$69$$ −148.837 −0.259680
$$70$$ 0 0
$$71$$ 594.281 0.993355 0.496677 0.867935i $$-0.334553\pi$$
0.496677 + 0.867935i $$0.334553\pi$$
$$72$$ 0 0
$$73$$ −320.231 −0.513428 −0.256714 0.966487i $$-0.582640\pi$$
−0.256714 + 0.966487i $$0.582640\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 39.7687 0.0588580
$$78$$ 0 0
$$79$$ 770.469 1.09727 0.548636 0.836061i $$-0.315147\pi$$
0.548636 + 0.836061i $$0.315147\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 173.925 0.230009 0.115004 0.993365i $$-0.463312\pi$$
0.115004 + 0.993365i $$0.463312\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −880.397 −1.08492
$$88$$ 0 0
$$89$$ 1019.02 1.21367 0.606834 0.794829i $$-0.292439\pi$$
0.606834 + 0.794829i $$0.292439\pi$$
$$90$$ 0 0
$$91$$ −1292.79 −1.48925
$$92$$ 0 0
$$93$$ −193.256 −0.215481
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −384.375 −0.402344 −0.201172 0.979556i $$-0.564475\pi$$
−0.201172 + 0.979556i $$0.564475\pi$$
$$98$$ 0 0
$$99$$ 16.1156 0.0163604
$$100$$ 0 0
$$101$$ 34.4906 0.0339796 0.0169898 0.999856i $$-0.494592\pi$$
0.0169898 + 0.999856i $$0.494592\pi$$
$$102$$ 0 0
$$103$$ −1756.30 −1.68013 −0.840066 0.542484i $$-0.817484\pi$$
−0.840066 + 0.542484i $$0.817484\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1361.74 −1.23032 −0.615159 0.788403i $$-0.710908\pi$$
−0.615159 + 0.788403i $$0.710908\pi$$
$$108$$ 0 0
$$109$$ 321.119 0.282180 0.141090 0.989997i $$-0.454939\pi$$
0.141090 + 0.989997i $$0.454939\pi$$
$$110$$ 0 0
$$111$$ 59.6531 0.0510092
$$112$$ 0 0
$$113$$ 1582.25 1.31721 0.658607 0.752487i $$-0.271146\pi$$
0.658607 + 0.752487i $$0.271146\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −523.884 −0.413958
$$118$$ 0 0
$$119$$ −421.631 −0.324797
$$120$$ 0 0
$$121$$ −1327.79 −0.997591
$$122$$ 0 0
$$123$$ −496.744 −0.364145
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1197.14 −0.836449 −0.418225 0.908344i $$-0.637348\pi$$
−0.418225 + 0.908344i $$0.637348\pi$$
$$128$$ 0 0
$$129$$ −742.050 −0.506464
$$130$$ 0 0
$$131$$ 321.647 0.214522 0.107261 0.994231i $$-0.465792\pi$$
0.107261 + 0.994231i $$0.465792\pi$$
$$132$$ 0 0
$$133$$ −2328.37 −1.51801
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −354.291 −0.220942 −0.110471 0.993879i $$-0.535236\pi$$
−0.110471 + 0.993879i $$0.535236\pi$$
$$138$$ 0 0
$$139$$ −77.2562 −0.0471424 −0.0235712 0.999722i $$-0.507504\pi$$
−0.0235712 + 0.999722i $$0.507504\pi$$
$$140$$ 0 0
$$141$$ 1153.63 0.689030
$$142$$ 0 0
$$143$$ −104.231 −0.0609529
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 450.769 0.252917
$$148$$ 0 0
$$149$$ −1705.38 −0.937651 −0.468826 0.883291i $$-0.655323\pi$$
−0.468826 + 0.883291i $$0.655323\pi$$
$$150$$ 0 0
$$151$$ −758.281 −0.408663 −0.204331 0.978902i $$-0.565502\pi$$
−0.204331 + 0.978902i $$0.565502\pi$$
$$152$$ 0 0
$$153$$ −170.859 −0.0902821
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1769.05 0.899273 0.449636 0.893212i $$-0.351554\pi$$
0.449636 + 0.893212i $$0.351554\pi$$
$$158$$ 0 0
$$159$$ −1390.58 −0.693588
$$160$$ 0 0
$$161$$ −1101.86 −0.539372
$$162$$ 0 0
$$163$$ −881.719 −0.423690 −0.211845 0.977303i $$-0.567947\pi$$
−0.211845 + 0.977303i $$0.567947\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −216.900 −0.100504 −0.0502522 0.998737i $$-0.516003\pi$$
−0.0502522 + 0.998737i $$0.516003\pi$$
$$168$$ 0 0
$$169$$ 1191.33 0.542254
$$170$$ 0 0
$$171$$ −943.537 −0.421954
$$172$$ 0 0
$$173$$ −4125.91 −1.81322 −0.906610 0.421970i $$-0.861339\pi$$
−0.906610 + 0.421970i $$0.861339\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 221.372 0.0940075
$$178$$ 0 0
$$179$$ 3213.14 1.34168 0.670842 0.741600i $$-0.265933\pi$$
0.670842 + 0.741600i $$0.265933\pi$$
$$180$$ 0 0
$$181$$ 3394.42 1.39395 0.696976 0.717095i $$-0.254529\pi$$
0.696976 + 0.717095i $$0.254529\pi$$
$$182$$ 0 0
$$183$$ −412.050 −0.166446
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −33.9939 −0.0132935
$$188$$ 0 0
$$189$$ 599.653 0.230785
$$190$$ 0 0
$$191$$ 3467.49 1.31361 0.656804 0.754062i $$-0.271908\pi$$
0.656804 + 0.754062i $$0.271908\pi$$
$$192$$ 0 0
$$193$$ −1792.14 −0.668401 −0.334200 0.942502i $$-0.608466\pi$$
−0.334200 + 0.942502i $$0.608466\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1678.19 −0.606935 −0.303467 0.952842i $$-0.598144\pi$$
−0.303467 + 0.952842i $$0.598144\pi$$
$$198$$ 0 0
$$199$$ −3108.23 −1.10722 −0.553610 0.832776i $$-0.686750\pi$$
−0.553610 + 0.832776i $$0.686750\pi$$
$$200$$ 0 0
$$201$$ 521.719 0.183081
$$202$$ 0 0
$$203$$ −6517.69 −2.25346
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −446.512 −0.149926
$$208$$ 0 0
$$209$$ −187.725 −0.0621301
$$210$$ 0 0
$$211$$ 4473.27 1.45949 0.729745 0.683719i $$-0.239639\pi$$
0.729745 + 0.683719i $$0.239639\pi$$
$$212$$ 0 0
$$213$$ 1782.84 0.573514
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1430.70 −0.447568
$$218$$ 0 0
$$219$$ −960.694 −0.296428
$$220$$ 0 0
$$221$$ 1105.07 0.336357
$$222$$ 0 0
$$223$$ −1753.42 −0.526535 −0.263268 0.964723i $$-0.584800\pi$$
−0.263268 + 0.964723i $$0.584800\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −936.900 −0.273939 −0.136970 0.990575i $$-0.543736\pi$$
−0.136970 + 0.990575i $$0.543736\pi$$
$$228$$ 0 0
$$229$$ −2582.06 −0.745096 −0.372548 0.928013i $$-0.621516\pi$$
−0.372548 + 0.928013i $$0.621516\pi$$
$$230$$ 0 0
$$231$$ 119.306 0.0339817
$$232$$ 0 0
$$233$$ −2295.01 −0.645284 −0.322642 0.946521i $$-0.604571\pi$$
−0.322642 + 0.946521i $$0.604571\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2311.41 0.633510
$$238$$ 0 0
$$239$$ 2294.01 0.620866 0.310433 0.950595i $$-0.399526\pi$$
0.310433 + 0.950595i $$0.399526\pi$$
$$240$$ 0 0
$$241$$ 382.287 0.102180 0.0510898 0.998694i $$-0.483731\pi$$
0.0510898 + 0.998694i $$0.483731\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6102.52 1.57204
$$248$$ 0 0
$$249$$ 521.775 0.132796
$$250$$ 0 0
$$251$$ 2259.98 0.568322 0.284161 0.958777i $$-0.408285\pi$$
0.284161 + 0.958777i $$0.408285\pi$$
$$252$$ 0 0
$$253$$ −88.8375 −0.0220758
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 92.7843 0.0225203 0.0112602 0.999937i $$-0.496416\pi$$
0.0112602 + 0.999937i $$0.496416\pi$$
$$258$$ 0 0
$$259$$ 441.619 0.105949
$$260$$ 0 0
$$261$$ −2641.19 −0.626382
$$262$$ 0 0
$$263$$ −568.312 −0.133246 −0.0666229 0.997778i $$-0.521222\pi$$
−0.0666229 + 0.997778i $$0.521222\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 3057.07 0.700711
$$268$$ 0 0
$$269$$ 7582.41 1.71862 0.859309 0.511458i $$-0.170894\pi$$
0.859309 + 0.511458i $$0.170894\pi$$
$$270$$ 0 0
$$271$$ −7943.69 −1.78061 −0.890304 0.455366i $$-0.849508\pi$$
−0.890304 + 0.455366i $$0.849508\pi$$
$$272$$ 0 0
$$273$$ −3878.38 −0.859818
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6823.00 −1.47998 −0.739990 0.672618i $$-0.765170\pi$$
−0.739990 + 0.672618i $$0.765170\pi$$
$$278$$ 0 0
$$279$$ −579.769 −0.124408
$$280$$ 0 0
$$281$$ 3315.86 0.703942 0.351971 0.936011i $$-0.385512\pi$$
0.351971 + 0.936011i $$0.385512\pi$$
$$282$$ 0 0
$$283$$ 6602.76 1.38690 0.693451 0.720504i $$-0.256090\pi$$
0.693451 + 0.720504i $$0.256090\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3677.46 −0.756353
$$288$$ 0 0
$$289$$ −4552.59 −0.926642
$$290$$ 0 0
$$291$$ −1153.12 −0.232293
$$292$$ 0 0
$$293$$ −5814.14 −1.15927 −0.579634 0.814877i $$-0.696805\pi$$
−0.579634 + 0.814877i $$0.696805\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 48.3469 0.00944570
$$298$$ 0 0
$$299$$ 2887.91 0.558570
$$300$$ 0 0
$$301$$ −5493.49 −1.05196
$$302$$ 0 0
$$303$$ 103.472 0.0196181
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −8124.86 −1.51046 −0.755229 0.655462i $$-0.772474\pi$$
−0.755229 + 0.655462i $$0.772474\pi$$
$$308$$ 0 0
$$309$$ −5268.91 −0.970025
$$310$$ 0 0
$$311$$ −7336.26 −1.33762 −0.668812 0.743432i $$-0.733197\pi$$
−0.668812 + 0.743432i $$0.733197\pi$$
$$312$$ 0 0
$$313$$ 2202.66 0.397768 0.198884 0.980023i $$-0.436268\pi$$
0.198884 + 0.980023i $$0.436268\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10008.9 −1.77336 −0.886679 0.462386i $$-0.846993\pi$$
−0.886679 + 0.462386i $$0.846993\pi$$
$$318$$ 0 0
$$319$$ −525.488 −0.0922309
$$320$$ 0 0
$$321$$ −4085.21 −0.710325
$$322$$ 0 0
$$323$$ 1990.27 0.342854
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 963.356 0.162917
$$328$$ 0 0
$$329$$ 8540.47 1.43116
$$330$$ 0 0
$$331$$ 8695.94 1.44402 0.722012 0.691881i $$-0.243218\pi$$
0.722012 + 0.691881i $$0.243218\pi$$
$$332$$ 0 0
$$333$$ 178.959 0.0294502
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7400.61 1.19625 0.598126 0.801402i $$-0.295912\pi$$
0.598126 + 0.801402i $$0.295912\pi$$
$$338$$ 0 0
$$339$$ 4746.74 0.760494
$$340$$ 0 0
$$341$$ −115.350 −0.0183183
$$342$$ 0 0
$$343$$ −4280.72 −0.673869
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7841.44 1.21311 0.606557 0.795040i $$-0.292550\pi$$
0.606557 + 0.795040i $$0.292550\pi$$
$$348$$ 0 0
$$349$$ −4961.26 −0.760946 −0.380473 0.924792i $$-0.624239\pi$$
−0.380473 + 0.924792i $$0.624239\pi$$
$$350$$ 0 0
$$351$$ −1571.65 −0.238999
$$352$$ 0 0
$$353$$ 12163.0 1.83392 0.916959 0.398981i $$-0.130636\pi$$
0.916959 + 0.398981i $$0.130636\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1264.89 −0.187522
$$358$$ 0 0
$$359$$ −5193.79 −0.763559 −0.381779 0.924253i $$-0.624689\pi$$
−0.381779 + 0.924253i $$0.624689\pi$$
$$360$$ 0 0
$$361$$ 4131.90 0.602406
$$362$$ 0 0
$$363$$ −3983.38 −0.575959
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −6086.09 −0.865644 −0.432822 0.901479i $$-0.642482\pi$$
−0.432822 + 0.901479i $$0.642482\pi$$
$$368$$ 0 0
$$369$$ −1490.23 −0.210239
$$370$$ 0 0
$$371$$ −10294.7 −1.44063
$$372$$ 0 0
$$373$$ 10581.9 1.46893 0.734466 0.678646i $$-0.237433\pi$$
0.734466 + 0.678646i $$0.237433\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 17082.4 2.33366
$$378$$ 0 0
$$379$$ −11655.2 −1.57964 −0.789822 0.613336i $$-0.789827\pi$$
−0.789822 + 0.613336i $$0.789827\pi$$
$$380$$ 0 0
$$381$$ −3591.42 −0.482924
$$382$$ 0 0
$$383$$ 6364.97 0.849177 0.424588 0.905387i $$-0.360419\pi$$
0.424588 + 0.905387i $$0.360419\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2226.15 −0.292407
$$388$$ 0 0
$$389$$ 6134.33 0.799545 0.399773 0.916614i $$-0.369089\pi$$
0.399773 + 0.916614i $$0.369089\pi$$
$$390$$ 0 0
$$391$$ 941.862 0.121821
$$392$$ 0 0
$$393$$ 964.941 0.123855
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9746.46 −1.23214 −0.616072 0.787690i $$-0.711277\pi$$
−0.616072 + 0.787690i $$0.711277\pi$$
$$398$$ 0 0
$$399$$ −6985.12 −0.876425
$$400$$ 0 0
$$401$$ −1306.44 −0.162695 −0.0813474 0.996686i $$-0.525922\pi$$
−0.0813474 + 0.996686i $$0.525922\pi$$
$$402$$ 0 0
$$403$$ 3749.77 0.463498
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 35.6055 0.00433636
$$408$$ 0 0
$$409$$ −3876.93 −0.468709 −0.234354 0.972151i $$-0.575298\pi$$
−0.234354 + 0.972151i $$0.575298\pi$$
$$410$$ 0 0
$$411$$ −1062.87 −0.127561
$$412$$ 0 0
$$413$$ 1638.84 0.195260
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −231.769 −0.0272177
$$418$$ 0 0
$$419$$ 16022.5 1.86814 0.934071 0.357088i $$-0.116230\pi$$
0.934071 + 0.357088i $$0.116230\pi$$
$$420$$ 0 0
$$421$$ −8119.73 −0.939980 −0.469990 0.882672i $$-0.655742\pi$$
−0.469990 + 0.882672i $$0.655742\pi$$
$$422$$ 0 0
$$423$$ 3460.89 0.397812
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3050.46 −0.345719
$$428$$ 0 0
$$429$$ −312.694 −0.0351911
$$430$$ 0 0
$$431$$ 5713.99 0.638592 0.319296 0.947655i $$-0.396554\pi$$
0.319296 + 0.947655i $$0.396554\pi$$
$$432$$ 0 0
$$433$$ −6251.34 −0.693811 −0.346906 0.937900i $$-0.612768\pi$$
−0.346906 + 0.937900i $$0.612768\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5201.25 0.569358
$$438$$ 0 0
$$439$$ 4230.97 0.459984 0.229992 0.973192i $$-0.426130\pi$$
0.229992 + 0.973192i $$0.426130\pi$$
$$440$$ 0 0
$$441$$ 1352.31 0.146022
$$442$$ 0 0
$$443$$ 6314.29 0.677203 0.338601 0.940930i $$-0.390046\pi$$
0.338601 + 0.940930i $$0.390046\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −5116.13 −0.541353
$$448$$ 0 0
$$449$$ −9349.71 −0.982717 −0.491358 0.870957i $$-0.663499\pi$$
−0.491358 + 0.870957i $$0.663499\pi$$
$$450$$ 0 0
$$451$$ −296.494 −0.0309565
$$452$$ 0 0
$$453$$ −2274.84 −0.235941
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9547.46 0.977268 0.488634 0.872489i $$-0.337495\pi$$
0.488634 + 0.872489i $$0.337495\pi$$
$$458$$ 0 0
$$459$$ −512.578 −0.0521244
$$460$$ 0 0
$$461$$ 6237.23 0.630145 0.315073 0.949068i $$-0.397971\pi$$
0.315073 + 0.949068i $$0.397971\pi$$
$$462$$ 0 0
$$463$$ 6469.98 0.649428 0.324714 0.945812i $$-0.394732\pi$$
0.324714 + 0.945812i $$0.394732\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 7206.64 0.714097 0.357049 0.934086i $$-0.383783\pi$$
0.357049 + 0.934086i $$0.383783\pi$$
$$468$$ 0 0
$$469$$ 3862.35 0.380270
$$470$$ 0 0
$$471$$ 5307.16 0.519195
$$472$$ 0 0
$$473$$ −442.912 −0.0430552
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −4171.75 −0.400443
$$478$$ 0 0
$$479$$ 10851.8 1.03514 0.517571 0.855640i $$-0.326836\pi$$
0.517571 + 0.855640i $$0.326836\pi$$
$$480$$ 0 0
$$481$$ −1157.46 −0.109720
$$482$$ 0 0
$$483$$ −3305.59 −0.311407
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12757.1 1.18702 0.593510 0.804827i $$-0.297742\pi$$
0.593510 + 0.804827i $$0.297742\pi$$
$$488$$ 0 0
$$489$$ −2645.16 −0.244618
$$490$$ 0 0
$$491$$ 7016.52 0.644911 0.322455 0.946585i $$-0.395492\pi$$
0.322455 + 0.946585i $$0.395492\pi$$
$$492$$ 0 0
$$493$$ 5571.26 0.508960
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 13198.6 1.19122
$$498$$ 0 0
$$499$$ −11372.3 −1.02023 −0.510113 0.860107i $$-0.670396\pi$$
−0.510113 + 0.860107i $$0.670396\pi$$
$$500$$ 0 0
$$501$$ −650.700 −0.0580262
$$502$$ 0 0
$$503$$ −5587.37 −0.495285 −0.247643 0.968851i $$-0.579656\pi$$
−0.247643 + 0.968851i $$0.579656\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3573.99 0.313070
$$508$$ 0 0
$$509$$ 16256.7 1.41565 0.707825 0.706388i $$-0.249676\pi$$
0.707825 + 0.706388i $$0.249676\pi$$
$$510$$ 0 0
$$511$$ −7112.14 −0.615699
$$512$$ 0 0
$$513$$ −2830.61 −0.243615
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 688.574 0.0585754
$$518$$ 0 0
$$519$$ −12377.7 −1.04686
$$520$$ 0 0
$$521$$ 19748.4 1.66064 0.830320 0.557286i $$-0.188157\pi$$
0.830320 + 0.557286i $$0.188157\pi$$
$$522$$ 0 0
$$523$$ −7843.44 −0.655774 −0.327887 0.944717i $$-0.606337\pi$$
−0.327887 + 0.944717i $$0.606337\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1222.95 0.101086
$$528$$ 0 0
$$529$$ −9705.60 −0.797699
$$530$$ 0 0
$$531$$ 664.116 0.0542753
$$532$$ 0 0
$$533$$ 9638.38 0.783273
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 9639.42 0.774622
$$538$$ 0 0
$$539$$ 269.053 0.0215008
$$540$$ 0 0
$$541$$ 7383.29 0.586751 0.293376 0.955997i $$-0.405221\pi$$
0.293376 + 0.955997i $$0.405221\pi$$
$$542$$ 0 0
$$543$$ 10183.3 0.804798
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3354.90 −0.262240 −0.131120 0.991367i $$-0.541857\pi$$
−0.131120 + 0.991367i $$0.541857\pi$$
$$548$$ 0 0
$$549$$ −1236.15 −0.0960976
$$550$$ 0 0
$$551$$ 30766.2 2.37874
$$552$$ 0 0
$$553$$ 17111.6 1.31584
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −20771.8 −1.58012 −0.790061 0.613028i $$-0.789951\pi$$
−0.790061 + 0.613028i $$0.789951\pi$$
$$558$$ 0 0
$$559$$ 14398.1 1.08940
$$560$$ 0 0
$$561$$ −101.982 −0.00767500
$$562$$ 0 0
$$563$$ 7194.86 0.538592 0.269296 0.963057i $$-0.413209\pi$$
0.269296 + 0.963057i $$0.413209\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1798.96 0.133244
$$568$$ 0 0
$$569$$ 11549.5 0.850931 0.425466 0.904975i $$-0.360110\pi$$
0.425466 + 0.904975i $$0.360110\pi$$
$$570$$ 0 0
$$571$$ −1482.54 −0.108655 −0.0543277 0.998523i $$-0.517302\pi$$
−0.0543277 + 0.998523i $$0.517302\pi$$
$$572$$ 0 0
$$573$$ 10402.5 0.758412
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15264.0 1.10130 0.550649 0.834737i $$-0.314380\pi$$
0.550649 + 0.834737i $$0.314380\pi$$
$$578$$ 0 0
$$579$$ −5376.43 −0.385901
$$580$$ 0 0
$$581$$ 3862.76 0.275825
$$582$$ 0 0
$$583$$ −830.006 −0.0589628
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1736.89 0.122128 0.0610639 0.998134i $$-0.480551\pi$$
0.0610639 + 0.998134i $$0.480551\pi$$
$$588$$ 0 0
$$589$$ 6753.50 0.472450
$$590$$ 0 0
$$591$$ −5034.57 −0.350414
$$592$$ 0 0
$$593$$ 11764.8 0.814707 0.407353 0.913271i $$-0.366452\pi$$
0.407353 + 0.913271i $$0.366452\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9324.69 −0.639253
$$598$$ 0 0
$$599$$ 9451.99 0.644737 0.322369 0.946614i $$-0.395521\pi$$
0.322369 + 0.946614i $$0.395521\pi$$
$$600$$ 0 0
$$601$$ −3131.93 −0.212569 −0.106285 0.994336i $$-0.533895\pi$$
−0.106285 + 0.994336i $$0.533895\pi$$
$$602$$ 0 0
$$603$$ 1565.16 0.105702
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 22700.8 1.51795 0.758975 0.651120i $$-0.225700\pi$$
0.758975 + 0.651120i $$0.225700\pi$$
$$608$$ 0 0
$$609$$ −19553.1 −1.30103
$$610$$ 0 0
$$611$$ −22384.0 −1.48210
$$612$$ 0 0
$$613$$ −28911.6 −1.90494 −0.952471 0.304629i $$-0.901468\pi$$
−0.952471 + 0.304629i $$0.901468\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5566.87 0.363231 0.181616 0.983370i $$-0.441867\pi$$
0.181616 + 0.983370i $$0.441867\pi$$
$$618$$ 0 0
$$619$$ −4150.32 −0.269492 −0.134746 0.990880i $$-0.543022\pi$$
−0.134746 + 0.990880i $$0.543022\pi$$
$$620$$ 0 0
$$621$$ −1339.54 −0.0865600
$$622$$ 0 0
$$623$$ 22631.9 1.45542
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −563.175 −0.0358709
$$628$$ 0 0
$$629$$ −377.492 −0.0239294
$$630$$ 0 0
$$631$$ 4090.09 0.258041 0.129021 0.991642i $$-0.458817\pi$$
0.129021 + 0.991642i $$0.458817\pi$$
$$632$$ 0 0
$$633$$ 13419.8 0.842637
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −8746.32 −0.544022
$$638$$ 0 0
$$639$$ 5348.53 0.331118
$$640$$ 0 0
$$641$$ 3909.35 0.240890 0.120445 0.992720i $$-0.461568\pi$$
0.120445 + 0.992720i $$0.461568\pi$$
$$642$$ 0 0
$$643$$ 30539.5 1.87303 0.936516 0.350624i $$-0.114031\pi$$
0.936516 + 0.350624i $$0.114031\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12707.7 −0.772167 −0.386083 0.922464i $$-0.626172\pi$$
−0.386083 + 0.922464i $$0.626172\pi$$
$$648$$ 0 0
$$649$$ 132.132 0.00799170
$$650$$ 0 0
$$651$$ −4292.10 −0.258403
$$652$$ 0 0
$$653$$ −12777.6 −0.765737 −0.382869 0.923803i $$-0.625064\pi$$
−0.382869 + 0.923803i $$0.625064\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −2882.08 −0.171143
$$658$$ 0 0
$$659$$ −23563.5 −1.39287 −0.696435 0.717620i $$-0.745232\pi$$
−0.696435 + 0.717620i $$0.745232\pi$$
$$660$$ 0 0
$$661$$ −4361.31 −0.256634 −0.128317 0.991733i $$-0.540958\pi$$
−0.128317 + 0.991733i $$0.540958\pi$$
$$662$$ 0 0
$$663$$ 3315.21 0.194196
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 14559.6 0.845200
$$668$$ 0 0
$$669$$ −5260.25 −0.303995
$$670$$ 0 0
$$671$$ −245.943 −0.0141498
$$672$$ 0 0
$$673$$ 8203.52 0.469870 0.234935 0.972011i $$-0.424512\pi$$
0.234935 + 0.972011i $$0.424512\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −28057.1 −1.59279 −0.796397 0.604774i $$-0.793263\pi$$
−0.796397 + 0.604774i $$0.793263\pi$$
$$678$$ 0 0
$$679$$ −8536.73 −0.482488
$$680$$ 0 0
$$681$$ −2810.70 −0.158159
$$682$$ 0 0
$$683$$ −3344.62 −0.187377 −0.0936885 0.995602i $$-0.529866\pi$$
−0.0936885 + 0.995602i $$0.529866\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −7746.17 −0.430182
$$688$$ 0 0
$$689$$ 26981.7 1.49190
$$690$$ 0 0
$$691$$ −12964.8 −0.713757 −0.356879 0.934151i $$-0.616159\pi$$
−0.356879 + 0.934151i $$0.616159\pi$$
$$692$$ 0 0
$$693$$ 357.918 0.0196193
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3143.46 0.170828
$$698$$ 0 0
$$699$$ −6885.03 −0.372555
$$700$$ 0 0
$$701$$ −16162.1 −0.870806 −0.435403 0.900236i $$-0.643394\pi$$
−0.435403 + 0.900236i $$0.643394\pi$$
$$702$$ 0 0
$$703$$ −2084.63 −0.111839
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 766.014 0.0407481
$$708$$ 0 0
$$709$$ −14244.4 −0.754529 −0.377265 0.926105i $$-0.623135\pi$$
−0.377265 + 0.926105i $$0.623135\pi$$
$$710$$ 0 0
$$711$$ 6934.22 0.365757
$$712$$ 0 0
$$713$$ 3195.97 0.167868
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6882.02 0.358457
$$718$$ 0 0
$$719$$ 27638.5 1.43358 0.716790 0.697289i $$-0.245611\pi$$
0.716790 + 0.697289i $$0.245611\pi$$
$$720$$ 0 0
$$721$$ −39006.4 −2.01480
$$722$$ 0 0
$$723$$ 1146.86 0.0589934
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2525.52 −0.128840 −0.0644199 0.997923i $$-0.520520\pi$$
−0.0644199 + 0.997923i $$0.520520\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4695.79 0.237592
$$732$$ 0 0
$$733$$ 8400.27 0.423289 0.211645 0.977347i $$-0.432118\pi$$
0.211645 + 0.977347i $$0.432118\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 311.401 0.0155639
$$738$$ 0 0
$$739$$ 19689.1 0.980074 0.490037 0.871702i $$-0.336983\pi$$
0.490037 + 0.871702i $$0.336983\pi$$
$$740$$ 0 0
$$741$$ 18307.6 0.907619
$$742$$ 0 0
$$743$$ −22526.6 −1.11227 −0.556137 0.831091i $$-0.687717\pi$$
−0.556137 + 0.831091i $$0.687717\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1565.32 0.0766696
$$748$$ 0 0
$$749$$ −30243.3 −1.47539
$$750$$ 0 0
$$751$$ −34691.1 −1.68562 −0.842808 0.538215i $$-0.819099\pi$$
−0.842808 + 0.538215i $$0.819099\pi$$
$$752$$ 0 0
$$753$$ 6779.95 0.328121
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6619.98 −0.317843 −0.158922 0.987291i $$-0.550802\pi$$
−0.158922 + 0.987291i $$0.550802\pi$$
$$758$$ 0 0
$$759$$ −266.512 −0.0127454
$$760$$ 0 0
$$761$$ −29368.7 −1.39897 −0.699483 0.714649i $$-0.746586\pi$$
−0.699483 + 0.714649i $$0.746586\pi$$
$$762$$ 0 0
$$763$$ 7131.84 0.338388
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −4295.31 −0.202209
$$768$$ 0 0
$$769$$ 32677.4 1.53235 0.766174 0.642633i $$-0.222158\pi$$
0.766174 + 0.642633i $$0.222158\pi$$
$$770$$ 0 0
$$771$$ 278.353 0.0130021
$$772$$ 0 0
$$773$$ 28047.5 1.30504 0.652522 0.757770i $$-0.273711\pi$$
0.652522 + 0.757770i $$0.273711\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1324.86 0.0611699
$$778$$ 0 0
$$779$$ 17359.1 0.798402
$$780$$ 0 0
$$781$$ 1064.14 0.0487552
$$782$$ 0 0
$$783$$ −7923.57 −0.361642
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22172.1 −1.00426 −0.502128 0.864793i $$-0.667449\pi$$
−0.502128 + 0.864793i $$0.667449\pi$$
$$788$$ 0 0
$$789$$ −1704.94 −0.0769295
$$790$$ 0 0
$$791$$ 35140.7 1.57960
$$792$$ 0 0
$$793$$ 7995.06 0.358024
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 24170.3 1.07422 0.537112 0.843511i $$-0.319515\pi$$
0.537112 + 0.843511i $$0.319515\pi$$
$$798$$ 0 0
$$799$$ −7300.32 −0.323238
$$800$$ 0 0
$$801$$ 9171.22 0.404556
$$802$$ 0 0
$$803$$ −573.415 −0.0251997
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 22747.2 0.992244
$$808$$ 0 0
$$809$$ 15304.2 0.665102 0.332551 0.943085i $$-0.392091\pi$$
0.332551 + 0.943085i $$0.392091\pi$$
$$810$$ 0 0
$$811$$ 27002.2 1.16914 0.584572 0.811342i $$-0.301262\pi$$
0.584572 + 0.811342i $$0.301262\pi$$
$$812$$ 0 0
$$813$$ −23831.1 −1.02803
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 25931.5 1.11044
$$818$$ 0 0
$$819$$ −11635.1 −0.496416
$$820$$ 0 0
$$821$$ 25061.4 1.06535 0.532673 0.846321i $$-0.321188\pi$$
0.532673 + 0.846321i $$0.321188\pi$$
$$822$$ 0 0
$$823$$ −24896.4 −1.05448 −0.527238 0.849718i $$-0.676772\pi$$
−0.527238 + 0.849718i $$0.676772\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20063.2 −0.843612 −0.421806 0.906686i $$-0.638604\pi$$
−0.421806 + 0.906686i $$0.638604\pi$$
$$828$$ 0 0
$$829$$ −13884.2 −0.581687 −0.290844 0.956771i $$-0.593936\pi$$
−0.290844 + 0.956771i $$0.593936\pi$$
$$830$$ 0 0
$$831$$ −20469.0 −0.854467
$$832$$ 0 0
$$833$$ −2852.52 −0.118648
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1739.31 −0.0718270
$$838$$ 0 0
$$839$$ 13678.1 0.562838 0.281419 0.959585i $$-0.409195\pi$$
0.281419 + 0.959585i $$0.409195\pi$$
$$840$$ 0 0
$$841$$ 61733.1 2.53118
$$842$$ 0 0
$$843$$ 9947.59 0.406421
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −29489.5 −1.19630
$$848$$ 0 0
$$849$$ 19808.3 0.800728
$$850$$ 0 0
$$851$$ −986.512 −0.0397382
$$852$$ 0 0
$$853$$ −29802.9 −1.19629 −0.598143 0.801390i $$-0.704094\pi$$
−0.598143 + 0.801390i $$0.704094\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −22045.2 −0.878706 −0.439353 0.898314i $$-0.644792\pi$$
−0.439353 + 0.898314i $$0.644792\pi$$
$$858$$ 0 0
$$859$$ −33609.5 −1.33497 −0.667487 0.744622i $$-0.732630\pi$$
−0.667487 + 0.744622i $$0.732630\pi$$
$$860$$ 0 0
$$861$$ −11032.4 −0.436681
$$862$$ 0 0
$$863$$ 33775.6 1.33226 0.666128 0.745838i $$-0.267951\pi$$
0.666128 + 0.745838i $$0.267951\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −13657.8 −0.534997
$$868$$ 0 0
$$869$$ 1379.62 0.0538556
$$870$$ 0 0
$$871$$ −10123.0 −0.393805
$$872$$ 0 0
$$873$$ −3459.37 −0.134115
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12637.0 −0.486570 −0.243285 0.969955i $$-0.578225\pi$$
−0.243285 + 0.969955i $$0.578225\pi$$
$$878$$ 0 0
$$879$$ −17442.4 −0.669304
$$880$$ 0 0
$$881$$ −6579.45 −0.251609 −0.125804 0.992055i $$-0.540151\pi$$
−0.125804 + 0.992055i $$0.540151\pi$$
$$882$$ 0 0
$$883$$ −50442.1 −1.92244 −0.961219 0.275786i $$-0.911062\pi$$
−0.961219 + 0.275786i $$0.911062\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 984.823 0.0372797 0.0186399 0.999826i $$-0.494066\pi$$
0.0186399 + 0.999826i $$0.494066\pi$$
$$888$$ 0 0
$$889$$ −26587.7 −1.00306
$$890$$ 0 0
$$891$$ 145.041 0.00545348
$$892$$ 0 0
$$893$$ −40314.6 −1.51072
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8663.74 0.322490
$$898$$ 0 0
$$899$$ 18904.7 0.701342
$$900$$ 0 0
$$901$$ 8799.79 0.325376
$$902$$ 0 0
$$903$$ −16480.5 −0.607348
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 43679.9 1.59908 0.799541 0.600612i $$-0.205076\pi$$
0.799541 + 0.600612i $$0.205076\pi$$
$$908$$ 0 0
$$909$$ 310.415 0.0113265
$$910$$ 0 0
$$911$$ 10364.3 0.376930 0.188465 0.982080i $$-0.439649\pi$$
0.188465 + 0.982080i $$0.439649\pi$$
$$912$$ 0 0
$$913$$ 311.435 0.0112891
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 7143.58 0.257254
$$918$$ 0 0
$$919$$ −11451.9 −0.411059 −0.205530 0.978651i $$-0.565892\pi$$
−0.205530 + 0.978651i $$0.565892\pi$$
$$920$$ 0 0
$$921$$ −24374.6 −0.872063
$$922$$ 0 0
$$923$$ −34592.7 −1.23362
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −15806.7 −0.560044
$$928$$ 0 0
$$929$$ −27701.8 −0.978326 −0.489163 0.872192i $$-0.662698\pi$$
−0.489163 + 0.872192i $$0.662698\pi$$
$$930$$ 0 0
$$931$$ −15752.5 −0.554529
$$932$$ 0 0
$$933$$ −22008.8 −0.772277
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5878.01 0.204937 0.102469 0.994736i $$-0.467326\pi$$
0.102469 + 0.994736i $$0.467326\pi$$
$$938$$ 0 0
$$939$$ 6607.97 0.229652
$$940$$ 0 0
$$941$$ −28786.0 −0.997234 −0.498617 0.866823i $$-0.666159\pi$$
−0.498617 + 0.866823i $$0.666159\pi$$
$$942$$ 0 0
$$943$$ 8214.90 0.283684
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1695.04 −0.0581641 −0.0290821 0.999577i $$-0.509258\pi$$
−0.0290821 + 0.999577i $$0.509258\pi$$
$$948$$ 0 0
$$949$$ 18640.5 0.637613
$$950$$ 0 0
$$951$$ −30026.6 −1.02385
$$952$$ 0 0
$$953$$ −31929.4 −1.08530 −0.542651 0.839958i $$-0.682580\pi$$
−0.542651 + 0.839958i $$0.682580\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −1576.46 −0.0532495
$$958$$ 0 0
$$959$$ −7868.57 −0.264952
$$960$$ 0 0
$$961$$ −25641.2 −0.860704
$$962$$ 0 0
$$963$$ −12255.6 −0.410106
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −10897.1 −0.362385 −0.181193 0.983448i $$-0.557996\pi$$
−0.181193 + 0.983448i $$0.557996\pi$$
$$968$$ 0 0
$$969$$ 5970.82 0.197947
$$970$$ 0 0
$$971$$ −7041.97 −0.232737 −0.116368 0.993206i $$-0.537125\pi$$
−0.116368 + 0.993206i $$0.537125\pi$$
$$972$$ 0 0
$$973$$ −1715.81 −0.0565328
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37607.6 1.23150 0.615749 0.787943i $$-0.288854\pi$$
0.615749 + 0.787943i $$0.288854\pi$$
$$978$$ 0 0
$$979$$ 1824.69 0.0595684
$$980$$ 0 0
$$981$$ 2890.07 0.0940599
$$982$$ 0 0
$$983$$ −25297.7 −0.820826 −0.410413 0.911900i $$-0.634615\pi$$
−0.410413 + 0.911900i $$0.634615\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 25621.4 0.826281
$$988$$ 0 0
$$989$$ 12271.6 0.394556
$$990$$ 0 0
$$991$$ 41686.5 1.33624 0.668120 0.744053i $$-0.267099\pi$$
0.668120 + 0.744053i $$0.267099\pi$$
$$992$$ 0 0
$$993$$ 26087.8 0.833708
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −25465.9 −0.808939 −0.404470 0.914551i $$-0.632544\pi$$
−0.404470 + 0.914551i $$0.632544\pi$$
$$998$$ 0 0
$$999$$ 536.878 0.0170031
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 1200.4.a.bt.1.2 2
4.3 odd 2 75.4.a.c.1.2 2
5.2 odd 4 240.4.f.f.49.1 4
5.3 odd 4 240.4.f.f.49.3 4
5.4 even 2 1200.4.a.bn.1.1 2
12.11 even 2 225.4.a.o.1.1 2
15.2 even 4 720.4.f.j.289.4 4
15.8 even 4 720.4.f.j.289.3 4
20.3 even 4 15.4.b.a.4.2 4
20.7 even 4 15.4.b.a.4.3 yes 4
20.19 odd 2 75.4.a.f.1.1 2
40.3 even 4 960.4.f.q.769.4 4
40.13 odd 4 960.4.f.p.769.2 4
40.27 even 4 960.4.f.q.769.2 4
40.37 odd 4 960.4.f.p.769.4 4
60.23 odd 4 45.4.b.b.19.3 4
60.47 odd 4 45.4.b.b.19.2 4
60.59 even 2 225.4.a.i.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.2 4 20.3 even 4
15.4.b.a.4.3 yes 4 20.7 even 4
45.4.b.b.19.2 4 60.47 odd 4
45.4.b.b.19.3 4 60.23 odd 4
75.4.a.c.1.2 2 4.3 odd 2
75.4.a.f.1.1 2 20.19 odd 2
225.4.a.i.1.2 2 60.59 even 2
225.4.a.o.1.1 2 12.11 even 2
240.4.f.f.49.1 4 5.2 odd 4
240.4.f.f.49.3 4 5.3 odd 4
720.4.f.j.289.3 4 15.8 even 4
720.4.f.j.289.4 4 15.2 even 4
960.4.f.p.769.2 4 40.13 odd 4
960.4.f.p.769.4 4 40.37 odd 4
960.4.f.q.769.2 4 40.27 even 4
960.4.f.q.769.4 4 40.3 even 4
1200.4.a.bn.1.1 2 5.4 even 2
1200.4.a.bt.1.2 2 1.1 even 1 trivial