# Properties

 Label 1856.4.a.z.1.1 Level $1856$ Weight $4$ Character 1856.1 Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198$$ x^5 - x^4 - 34*x^3 + 74*x^2 + 94*x - 198 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 232) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.30242$$ of defining polynomial Character $$\chi$$ $$=$$ 1856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-7.82921 q^{3} +7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +O(q^{10})$$ $$q-7.82921 q^{3} +7.04208 q^{5} +14.1473 q^{7} +34.2965 q^{9} +16.9982 q^{11} -62.0351 q^{13} -55.1339 q^{15} +104.842 q^{17} +9.38323 q^{19} -110.762 q^{21} +173.620 q^{23} -75.4091 q^{25} -57.1255 q^{27} -29.0000 q^{29} -28.2215 q^{31} -133.082 q^{33} +99.6261 q^{35} +171.527 q^{37} +485.685 q^{39} +92.0027 q^{41} +519.014 q^{43} +241.518 q^{45} +109.612 q^{47} -142.855 q^{49} -820.830 q^{51} -140.553 q^{53} +119.703 q^{55} -73.4632 q^{57} -353.254 q^{59} +185.355 q^{61} +485.201 q^{63} -436.856 q^{65} -574.840 q^{67} -1359.30 q^{69} +535.054 q^{71} -508.723 q^{73} +590.393 q^{75} +240.478 q^{77} -61.0156 q^{79} -478.757 q^{81} +826.753 q^{83} +738.306 q^{85} +227.047 q^{87} -938.445 q^{89} -877.626 q^{91} +220.952 q^{93} +66.0775 q^{95} -888.766 q^{97} +582.978 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10})$$ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 $$5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100})$$ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * 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
$$3$$ −7.82921 −1.50673 −0.753366 0.657602i $$-0.771571\pi$$
−0.753366 + 0.657602i $$0.771571\pi$$
$$4$$ 0 0
$$5$$ 7.04208 0.629863 0.314931 0.949114i $$-0.398018\pi$$
0.314931 + 0.949114i $$0.398018\pi$$
$$6$$ 0 0
$$7$$ 14.1473 0.763880 0.381940 0.924187i $$-0.375256\pi$$
0.381940 + 0.924187i $$0.375256\pi$$
$$8$$ 0 0
$$9$$ 34.2965 1.27024
$$10$$ 0 0
$$11$$ 16.9982 0.465922 0.232961 0.972486i $$-0.425158\pi$$
0.232961 + 0.972486i $$0.425158\pi$$
$$12$$ 0 0
$$13$$ −62.0351 −1.32349 −0.661747 0.749727i $$-0.730185\pi$$
−0.661747 + 0.749727i $$0.730185\pi$$
$$14$$ 0 0
$$15$$ −55.1339 −0.949034
$$16$$ 0 0
$$17$$ 104.842 1.49576 0.747881 0.663833i $$-0.231071\pi$$
0.747881 + 0.663833i $$0.231071\pi$$
$$18$$ 0 0
$$19$$ 9.38323 0.113298 0.0566490 0.998394i $$-0.481958\pi$$
0.0566490 + 0.998394i $$0.481958\pi$$
$$20$$ 0 0
$$21$$ −110.762 −1.15096
$$22$$ 0 0
$$23$$ 173.620 1.57401 0.787005 0.616947i $$-0.211631\pi$$
0.787005 + 0.616947i $$0.211631\pi$$
$$24$$ 0 0
$$25$$ −75.4091 −0.603273
$$26$$ 0 0
$$27$$ −57.1255 −0.407178
$$28$$ 0 0
$$29$$ −29.0000 −0.185695
$$30$$ 0 0
$$31$$ −28.2215 −0.163507 −0.0817536 0.996653i $$-0.526052\pi$$
−0.0817536 + 0.996653i $$0.526052\pi$$
$$32$$ 0 0
$$33$$ −133.082 −0.702020
$$34$$ 0 0
$$35$$ 99.6261 0.481140
$$36$$ 0 0
$$37$$ 171.527 0.762130 0.381065 0.924548i $$-0.375557\pi$$
0.381065 + 0.924548i $$0.375557\pi$$
$$38$$ 0 0
$$39$$ 485.685 1.99415
$$40$$ 0 0
$$41$$ 92.0027 0.350449 0.175225 0.984529i $$-0.443935\pi$$
0.175225 + 0.984529i $$0.443935\pi$$
$$42$$ 0 0
$$43$$ 519.014 1.84067 0.920335 0.391131i $$-0.127916\pi$$
0.920335 + 0.391131i $$0.127916\pi$$
$$44$$ 0 0
$$45$$ 241.518 0.800077
$$46$$ 0 0
$$47$$ 109.612 0.340182 0.170091 0.985428i $$-0.445594\pi$$
0.170091 + 0.985428i $$0.445594\pi$$
$$48$$ 0 0
$$49$$ −142.855 −0.416487
$$50$$ 0 0
$$51$$ −820.830 −2.25371
$$52$$ 0 0
$$53$$ −140.553 −0.364273 −0.182136 0.983273i $$-0.558301\pi$$
−0.182136 + 0.983273i $$0.558301\pi$$
$$54$$ 0 0
$$55$$ 119.703 0.293467
$$56$$ 0 0
$$57$$ −73.4632 −0.170710
$$58$$ 0 0
$$59$$ −353.254 −0.779487 −0.389744 0.920923i $$-0.627436\pi$$
−0.389744 + 0.920923i $$0.627436\pi$$
$$60$$ 0 0
$$61$$ 185.355 0.389053 0.194526 0.980897i $$-0.437683\pi$$
0.194526 + 0.980897i $$0.437683\pi$$
$$62$$ 0 0
$$63$$ 485.201 0.970311
$$64$$ 0 0
$$65$$ −436.856 −0.833620
$$66$$ 0 0
$$67$$ −574.840 −1.04818 −0.524088 0.851664i $$-0.675594\pi$$
−0.524088 + 0.851664i $$0.675594\pi$$
$$68$$ 0 0
$$69$$ −1359.30 −2.37161
$$70$$ 0 0
$$71$$ 535.054 0.894355 0.447177 0.894445i $$-0.352429\pi$$
0.447177 + 0.894445i $$0.352429\pi$$
$$72$$ 0 0
$$73$$ −508.723 −0.815637 −0.407818 0.913063i $$-0.633710\pi$$
−0.407818 + 0.913063i $$0.633710\pi$$
$$74$$ 0 0
$$75$$ 590.393 0.908970
$$76$$ 0 0
$$77$$ 240.478 0.355909
$$78$$ 0 0
$$79$$ −61.0156 −0.0868961 −0.0434481 0.999056i $$-0.513834\pi$$
−0.0434481 + 0.999056i $$0.513834\pi$$
$$80$$ 0 0
$$81$$ −478.757 −0.656731
$$82$$ 0 0
$$83$$ 826.753 1.09335 0.546674 0.837345i $$-0.315894\pi$$
0.546674 + 0.837345i $$0.315894\pi$$
$$84$$ 0 0
$$85$$ 738.306 0.942124
$$86$$ 0 0
$$87$$ 227.047 0.279793
$$88$$ 0 0
$$89$$ −938.445 −1.11770 −0.558848 0.829270i $$-0.688757\pi$$
−0.558848 + 0.829270i $$0.688757\pi$$
$$90$$ 0 0
$$91$$ −877.626 −1.01099
$$92$$ 0 0
$$93$$ 220.952 0.246361
$$94$$ 0 0
$$95$$ 66.0775 0.0713621
$$96$$ 0 0
$$97$$ −888.766 −0.930315 −0.465158 0.885228i $$-0.654002\pi$$
−0.465158 + 0.885228i $$0.654002\pi$$
$$98$$ 0 0
$$99$$ 582.978 0.591833
$$100$$ 0 0
$$101$$ 1551.83 1.52884 0.764420 0.644719i $$-0.223026\pi$$
0.764420 + 0.644719i $$0.223026\pi$$
$$102$$ 0 0
$$103$$ −402.438 −0.384984 −0.192492 0.981299i $$-0.561657\pi$$
−0.192492 + 0.981299i $$0.561657\pi$$
$$104$$ 0 0
$$105$$ −779.994 −0.724948
$$106$$ 0 0
$$107$$ 821.553 0.742267 0.371133 0.928580i $$-0.378969\pi$$
0.371133 + 0.928580i $$0.378969\pi$$
$$108$$ 0 0
$$109$$ −803.052 −0.705674 −0.352837 0.935685i $$-0.614783\pi$$
−0.352837 + 0.935685i $$0.614783\pi$$
$$110$$ 0 0
$$111$$ −1342.92 −1.14832
$$112$$ 0 0
$$113$$ −735.506 −0.612306 −0.306153 0.951982i $$-0.599042\pi$$
−0.306153 + 0.951982i $$0.599042\pi$$
$$114$$ 0 0
$$115$$ 1222.64 0.991410
$$116$$ 0 0
$$117$$ −2127.58 −1.68116
$$118$$ 0 0
$$119$$ 1483.23 1.14258
$$120$$ 0 0
$$121$$ −1042.06 −0.782916
$$122$$ 0 0
$$123$$ −720.308 −0.528033
$$124$$ 0 0
$$125$$ −1411.30 −1.00984
$$126$$ 0 0
$$127$$ −1895.16 −1.32416 −0.662078 0.749435i $$-0.730325\pi$$
−0.662078 + 0.749435i $$0.730325\pi$$
$$128$$ 0 0
$$129$$ −4063.46 −2.77340
$$130$$ 0 0
$$131$$ 1145.54 0.764020 0.382010 0.924158i $$-0.375232\pi$$
0.382010 + 0.924158i $$0.375232\pi$$
$$132$$ 0 0
$$133$$ 132.747 0.0865461
$$134$$ 0 0
$$135$$ −402.283 −0.256466
$$136$$ 0 0
$$137$$ 2871.41 1.79067 0.895334 0.445396i $$-0.146937\pi$$
0.895334 + 0.445396i $$0.146937\pi$$
$$138$$ 0 0
$$139$$ −777.734 −0.474579 −0.237290 0.971439i $$-0.576259\pi$$
−0.237290 + 0.971439i $$0.576259\pi$$
$$140$$ 0 0
$$141$$ −858.174 −0.512562
$$142$$ 0 0
$$143$$ −1054.48 −0.616646
$$144$$ 0 0
$$145$$ −204.220 −0.116963
$$146$$ 0 0
$$147$$ 1118.44 0.627534
$$148$$ 0 0
$$149$$ −2373.29 −1.30488 −0.652440 0.757840i $$-0.726255\pi$$
−0.652440 + 0.757840i $$0.726255\pi$$
$$150$$ 0 0
$$151$$ 102.190 0.0550733 0.0275366 0.999621i $$-0.491234\pi$$
0.0275366 + 0.999621i $$0.491234\pi$$
$$152$$ 0 0
$$153$$ 3595.71 1.89998
$$154$$ 0 0
$$155$$ −198.738 −0.102987
$$156$$ 0 0
$$157$$ −585.462 −0.297611 −0.148805 0.988866i $$-0.547543\pi$$
−0.148805 + 0.988866i $$0.547543\pi$$
$$158$$ 0 0
$$159$$ 1100.42 0.548861
$$160$$ 0 0
$$161$$ 2456.24 1.20235
$$162$$ 0 0
$$163$$ −1440.36 −0.692131 −0.346066 0.938210i $$-0.612483\pi$$
−0.346066 + 0.938210i $$0.612483\pi$$
$$164$$ 0 0
$$165$$ −937.176 −0.442176
$$166$$ 0 0
$$167$$ 3338.00 1.54672 0.773359 0.633968i $$-0.218575\pi$$
0.773359 + 0.633968i $$0.218575\pi$$
$$168$$ 0 0
$$169$$ 1651.35 0.751638
$$170$$ 0 0
$$171$$ 321.812 0.143916
$$172$$ 0 0
$$173$$ 3612.07 1.58740 0.793701 0.608308i $$-0.208151\pi$$
0.793701 + 0.608308i $$0.208151\pi$$
$$174$$ 0 0
$$175$$ −1066.83 −0.460828
$$176$$ 0 0
$$177$$ 2765.70 1.17448
$$178$$ 0 0
$$179$$ 2415.54 1.00864 0.504319 0.863517i $$-0.331744\pi$$
0.504319 + 0.863517i $$0.331744\pi$$
$$180$$ 0 0
$$181$$ 1755.89 0.721073 0.360537 0.932745i $$-0.382594\pi$$
0.360537 + 0.932745i $$0.382594\pi$$
$$182$$ 0 0
$$183$$ −1451.18 −0.586198
$$184$$ 0 0
$$185$$ 1207.90 0.480037
$$186$$ 0 0
$$187$$ 1782.13 0.696909
$$188$$ 0 0
$$189$$ −808.170 −0.311035
$$190$$ 0 0
$$191$$ 1156.10 0.437972 0.218986 0.975728i $$-0.429725\pi$$
0.218986 + 0.975728i $$0.429725\pi$$
$$192$$ 0 0
$$193$$ −3214.73 −1.19897 −0.599485 0.800386i $$-0.704628\pi$$
−0.599485 + 0.800386i $$0.704628\pi$$
$$194$$ 0 0
$$195$$ 3420.23 1.25604
$$196$$ 0 0
$$197$$ 3924.03 1.41916 0.709582 0.704623i $$-0.248884\pi$$
0.709582 + 0.704623i $$0.248884\pi$$
$$198$$ 0 0
$$199$$ 4463.80 1.59010 0.795051 0.606543i $$-0.207444\pi$$
0.795051 + 0.606543i $$0.207444\pi$$
$$200$$ 0 0
$$201$$ 4500.54 1.57932
$$202$$ 0 0
$$203$$ −410.271 −0.141849
$$204$$ 0 0
$$205$$ 647.890 0.220735
$$206$$ 0 0
$$207$$ 5954.54 1.99937
$$208$$ 0 0
$$209$$ 159.498 0.0527881
$$210$$ 0 0
$$211$$ −4881.35 −1.59263 −0.796317 0.604880i $$-0.793221\pi$$
−0.796317 + 0.604880i $$0.793221\pi$$
$$212$$ 0 0
$$213$$ −4189.05 −1.34755
$$214$$ 0 0
$$215$$ 3654.93 1.15937
$$216$$ 0 0
$$217$$ −399.256 −0.124900
$$218$$ 0 0
$$219$$ 3982.90 1.22895
$$220$$ 0 0
$$221$$ −6503.89 −1.97963
$$222$$ 0 0
$$223$$ −1533.22 −0.460414 −0.230207 0.973142i $$-0.573940\pi$$
−0.230207 + 0.973142i $$0.573940\pi$$
$$224$$ 0 0
$$225$$ −2586.27 −0.766301
$$226$$ 0 0
$$227$$ 3110.06 0.909348 0.454674 0.890658i $$-0.349756\pi$$
0.454674 + 0.890658i $$0.349756\pi$$
$$228$$ 0 0
$$229$$ 4686.03 1.35223 0.676117 0.736794i $$-0.263661\pi$$
0.676117 + 0.736794i $$0.263661\pi$$
$$230$$ 0 0
$$231$$ −1882.75 −0.536259
$$232$$ 0 0
$$233$$ −908.748 −0.255511 −0.127756 0.991806i $$-0.540777\pi$$
−0.127756 + 0.991806i $$0.540777\pi$$
$$234$$ 0 0
$$235$$ 771.896 0.214268
$$236$$ 0 0
$$237$$ 477.704 0.130929
$$238$$ 0 0
$$239$$ −18.3950 −0.00497854 −0.00248927 0.999997i $$-0.500792\pi$$
−0.00248927 + 0.999997i $$0.500792\pi$$
$$240$$ 0 0
$$241$$ 6199.96 1.65716 0.828578 0.559874i $$-0.189151\pi$$
0.828578 + 0.559874i $$0.189151\pi$$
$$242$$ 0 0
$$243$$ 5290.68 1.39670
$$244$$ 0 0
$$245$$ −1006.00 −0.262330
$$246$$ 0 0
$$247$$ −582.089 −0.149949
$$248$$ 0 0
$$249$$ −6472.82 −1.64738
$$250$$ 0 0
$$251$$ −3325.41 −0.836248 −0.418124 0.908390i $$-0.637312\pi$$
−0.418124 + 0.908390i $$0.637312\pi$$
$$252$$ 0 0
$$253$$ 2951.22 0.733366
$$254$$ 0 0
$$255$$ −5780.35 −1.41953
$$256$$ 0 0
$$257$$ 2713.20 0.658541 0.329271 0.944236i $$-0.393197\pi$$
0.329271 + 0.944236i $$0.393197\pi$$
$$258$$ 0 0
$$259$$ 2426.63 0.582176
$$260$$ 0 0
$$261$$ −994.597 −0.235878
$$262$$ 0 0
$$263$$ 6234.94 1.46184 0.730918 0.682465i $$-0.239092\pi$$
0.730918 + 0.682465i $$0.239092\pi$$
$$264$$ 0 0
$$265$$ −989.787 −0.229442
$$266$$ 0 0
$$267$$ 7347.28 1.68407
$$268$$ 0 0
$$269$$ 5745.23 1.30220 0.651102 0.758990i $$-0.274307\pi$$
0.651102 + 0.758990i $$0.274307\pi$$
$$270$$ 0 0
$$271$$ 8855.53 1.98500 0.992501 0.122238i $$-0.0390072\pi$$
0.992501 + 0.122238i $$0.0390072\pi$$
$$272$$ 0 0
$$273$$ 6871.12 1.52329
$$274$$ 0 0
$$275$$ −1281.82 −0.281078
$$276$$ 0 0
$$277$$ 2805.71 0.608588 0.304294 0.952578i $$-0.401580\pi$$
0.304294 + 0.952578i $$0.401580\pi$$
$$278$$ 0 0
$$279$$ −967.896 −0.207693
$$280$$ 0 0
$$281$$ 5133.23 1.08976 0.544881 0.838514i $$-0.316575\pi$$
0.544881 + 0.838514i $$0.316575\pi$$
$$282$$ 0 0
$$283$$ −2928.86 −0.615204 −0.307602 0.951515i $$-0.599527\pi$$
−0.307602 + 0.951515i $$0.599527\pi$$
$$284$$ 0 0
$$285$$ −517.334 −0.107524
$$286$$ 0 0
$$287$$ 1301.59 0.267701
$$288$$ 0 0
$$289$$ 6078.87 1.23730
$$290$$ 0 0
$$291$$ 6958.34 1.40173
$$292$$ 0 0
$$293$$ −5685.37 −1.13359 −0.566796 0.823858i $$-0.691817\pi$$
−0.566796 + 0.823858i $$0.691817\pi$$
$$294$$ 0 0
$$295$$ −2487.64 −0.490970
$$296$$ 0 0
$$297$$ −971.031 −0.189713
$$298$$ 0 0
$$299$$ −10770.5 −2.08319
$$300$$ 0 0
$$301$$ 7342.62 1.40605
$$302$$ 0 0
$$303$$ −12149.6 −2.30355
$$304$$ 0 0
$$305$$ 1305.28 0.245050
$$306$$ 0 0
$$307$$ −2879.21 −0.535262 −0.267631 0.963522i $$-0.586241\pi$$
−0.267631 + 0.963522i $$0.586241\pi$$
$$308$$ 0 0
$$309$$ 3150.77 0.580068
$$310$$ 0 0
$$311$$ 513.825 0.0936860 0.0468430 0.998902i $$-0.485084\pi$$
0.0468430 + 0.998902i $$0.485084\pi$$
$$312$$ 0 0
$$313$$ −1508.76 −0.272460 −0.136230 0.990677i $$-0.543499\pi$$
−0.136230 + 0.990677i $$0.543499\pi$$
$$314$$ 0 0
$$315$$ 3416.82 0.611163
$$316$$ 0 0
$$317$$ −5411.89 −0.958872 −0.479436 0.877577i $$-0.659159\pi$$
−0.479436 + 0.877577i $$0.659159\pi$$
$$318$$ 0 0
$$319$$ −492.947 −0.0865196
$$320$$ 0 0
$$321$$ −6432.11 −1.11840
$$322$$ 0 0
$$323$$ 983.758 0.169467
$$324$$ 0 0
$$325$$ 4678.01 0.798429
$$326$$ 0 0
$$327$$ 6287.26 1.06326
$$328$$ 0 0
$$329$$ 1550.71 0.259858
$$330$$ 0 0
$$331$$ −10354.1 −1.71938 −0.859688 0.510820i $$-0.829342\pi$$
−0.859688 + 0.510820i $$0.829342\pi$$
$$332$$ 0 0
$$333$$ 5882.75 0.968087
$$334$$ 0 0
$$335$$ −4048.07 −0.660208
$$336$$ 0 0
$$337$$ 4707.20 0.760882 0.380441 0.924805i $$-0.375772\pi$$
0.380441 + 0.924805i $$0.375772\pi$$
$$338$$ 0 0
$$339$$ 5758.43 0.922581
$$340$$ 0 0
$$341$$ −479.714 −0.0761817
$$342$$ 0 0
$$343$$ −6873.52 −1.08203
$$344$$ 0 0
$$345$$ −9572.33 −1.49379
$$346$$ 0 0
$$347$$ 6733.27 1.04167 0.520837 0.853656i $$-0.325620\pi$$
0.520837 + 0.853656i $$0.325620\pi$$
$$348$$ 0 0
$$349$$ 9272.50 1.42219 0.711097 0.703094i $$-0.248199\pi$$
0.711097 + 0.703094i $$0.248199\pi$$
$$350$$ 0 0
$$351$$ 3543.79 0.538898
$$352$$ 0 0
$$353$$ 5776.97 0.871041 0.435520 0.900179i $$-0.356564\pi$$
0.435520 + 0.900179i $$0.356564\pi$$
$$354$$ 0 0
$$355$$ 3767.89 0.563321
$$356$$ 0 0
$$357$$ −11612.5 −1.72157
$$358$$ 0 0
$$359$$ 3821.93 0.561877 0.280938 0.959726i $$-0.409354\pi$$
0.280938 + 0.959726i $$0.409354\pi$$
$$360$$ 0 0
$$361$$ −6770.95 −0.987164
$$362$$ 0 0
$$363$$ 8158.51 1.17964
$$364$$ 0 0
$$365$$ −3582.47 −0.513739
$$366$$ 0 0
$$367$$ −9481.16 −1.34854 −0.674268 0.738487i $$-0.735541\pi$$
−0.674268 + 0.738487i $$0.735541\pi$$
$$368$$ 0 0
$$369$$ 3155.37 0.445154
$$370$$ 0 0
$$371$$ −1988.44 −0.278261
$$372$$ 0 0
$$373$$ 11849.6 1.64490 0.822450 0.568837i $$-0.192606\pi$$
0.822450 + 0.568837i $$0.192606\pi$$
$$374$$ 0 0
$$375$$ 11049.3 1.52156
$$376$$ 0 0
$$377$$ 1799.02 0.245767
$$378$$ 0 0
$$379$$ 14650.4 1.98560 0.992799 0.119794i $$-0.0382235\pi$$
0.992799 + 0.119794i $$0.0382235\pi$$
$$380$$ 0 0
$$381$$ 14837.6 1.99515
$$382$$ 0 0
$$383$$ −2202.45 −0.293837 −0.146919 0.989149i $$-0.546936\pi$$
−0.146919 + 0.989149i $$0.546936\pi$$
$$384$$ 0 0
$$385$$ 1693.46 0.224174
$$386$$ 0 0
$$387$$ 17800.3 2.33809
$$388$$ 0 0
$$389$$ −5614.61 −0.731804 −0.365902 0.930653i $$-0.619240\pi$$
−0.365902 + 0.930653i $$0.619240\pi$$
$$390$$ 0 0
$$391$$ 18202.7 2.35434
$$392$$ 0 0
$$393$$ −8968.70 −1.15117
$$394$$ 0 0
$$395$$ −429.677 −0.0547326
$$396$$ 0 0
$$397$$ −1672.32 −0.211414 −0.105707 0.994397i $$-0.533711\pi$$
−0.105707 + 0.994397i $$0.533711\pi$$
$$398$$ 0 0
$$399$$ −1039.30 −0.130402
$$400$$ 0 0
$$401$$ −1183.01 −0.147324 −0.0736619 0.997283i $$-0.523469\pi$$
−0.0736619 + 0.997283i $$0.523469\pi$$
$$402$$ 0 0
$$403$$ 1750.72 0.216401
$$404$$ 0 0
$$405$$ −3371.45 −0.413651
$$406$$ 0 0
$$407$$ 2915.64 0.355093
$$408$$ 0 0
$$409$$ −6264.74 −0.757388 −0.378694 0.925522i $$-0.623627\pi$$
−0.378694 + 0.925522i $$0.623627\pi$$
$$410$$ 0 0
$$411$$ −22480.9 −2.69805
$$412$$ 0 0
$$413$$ −4997.58 −0.595435
$$414$$ 0 0
$$415$$ 5822.06 0.688659
$$416$$ 0 0
$$417$$ 6089.04 0.715063
$$418$$ 0 0
$$419$$ 4224.75 0.492584 0.246292 0.969196i $$-0.420788\pi$$
0.246292 + 0.969196i $$0.420788\pi$$
$$420$$ 0 0
$$421$$ −10801.7 −1.25046 −0.625231 0.780440i $$-0.714995\pi$$
−0.625231 + 0.780440i $$0.714995\pi$$
$$422$$ 0 0
$$423$$ 3759.30 0.432112
$$424$$ 0 0
$$425$$ −7906.05 −0.902352
$$426$$ 0 0
$$427$$ 2622.26 0.297190
$$428$$ 0 0
$$429$$ 8255.77 0.929120
$$430$$ 0 0
$$431$$ 13127.7 1.46715 0.733574 0.679610i $$-0.237851\pi$$
0.733574 + 0.679610i $$0.237851\pi$$
$$432$$ 0 0
$$433$$ −11403.2 −1.26559 −0.632797 0.774318i $$-0.718093\pi$$
−0.632797 + 0.774318i $$0.718093\pi$$
$$434$$ 0 0
$$435$$ 1598.88 0.176231
$$436$$ 0 0
$$437$$ 1629.11 0.178332
$$438$$ 0 0
$$439$$ 14656.2 1.59339 0.796697 0.604378i $$-0.206578\pi$$
0.796697 + 0.604378i $$0.206578\pi$$
$$440$$ 0 0
$$441$$ −4899.42 −0.529038
$$442$$ 0 0
$$443$$ −6136.95 −0.658184 −0.329092 0.944298i $$-0.606743\pi$$
−0.329092 + 0.944298i $$0.606743\pi$$
$$444$$ 0 0
$$445$$ −6608.61 −0.703995
$$446$$ 0 0
$$447$$ 18580.9 1.96610
$$448$$ 0 0
$$449$$ −18012.1 −1.89319 −0.946594 0.322428i $$-0.895501\pi$$
−0.946594 + 0.322428i $$0.895501\pi$$
$$450$$ 0 0
$$451$$ 1563.88 0.163282
$$452$$ 0 0
$$453$$ −800.063 −0.0829807
$$454$$ 0 0
$$455$$ −6180.31 −0.636786
$$456$$ 0 0
$$457$$ 715.440 0.0732316 0.0366158 0.999329i $$-0.488342\pi$$
0.0366158 + 0.999329i $$0.488342\pi$$
$$458$$ 0 0
$$459$$ −5989.16 −0.609042
$$460$$ 0 0
$$461$$ 16471.2 1.66408 0.832038 0.554719i $$-0.187174\pi$$
0.832038 + 0.554719i $$0.187174\pi$$
$$462$$ 0 0
$$463$$ 3844.07 0.385851 0.192925 0.981213i $$-0.438202\pi$$
0.192925 + 0.981213i $$0.438202\pi$$
$$464$$ 0 0
$$465$$ 1555.96 0.155174
$$466$$ 0 0
$$467$$ −15072.5 −1.49352 −0.746759 0.665094i $$-0.768391\pi$$
−0.746759 + 0.665094i $$0.768391\pi$$
$$468$$ 0 0
$$469$$ −8132.41 −0.800682
$$470$$ 0 0
$$471$$ 4583.70 0.448420
$$472$$ 0 0
$$473$$ 8822.29 0.857610
$$474$$ 0 0
$$475$$ −707.581 −0.0683496
$$476$$ 0 0
$$477$$ −4820.48 −0.462714
$$478$$ 0 0
$$479$$ 8856.14 0.844776 0.422388 0.906415i $$-0.361192\pi$$
0.422388 + 0.906415i $$0.361192\pi$$
$$480$$ 0 0
$$481$$ −10640.7 −1.00867
$$482$$ 0 0
$$483$$ −19230.4 −1.81163
$$484$$ 0 0
$$485$$ −6258.76 −0.585971
$$486$$ 0 0
$$487$$ 6998.94 0.651237 0.325618 0.945501i $$-0.394428\pi$$
0.325618 + 0.945501i $$0.394428\pi$$
$$488$$ 0 0
$$489$$ 11276.8 1.04286
$$490$$ 0 0
$$491$$ 2169.60 0.199415 0.0997076 0.995017i $$-0.468209\pi$$
0.0997076 + 0.995017i $$0.468209\pi$$
$$492$$ 0 0
$$493$$ −3040.42 −0.277756
$$494$$ 0 0
$$495$$ 4105.38 0.372774
$$496$$ 0 0
$$497$$ 7569.55 0.683180
$$498$$ 0 0
$$499$$ 17541.6 1.57368 0.786842 0.617155i $$-0.211715\pi$$
0.786842 + 0.617155i $$0.211715\pi$$
$$500$$ 0 0
$$501$$ −26133.9 −2.33049
$$502$$ 0 0
$$503$$ −16190.4 −1.43517 −0.717587 0.696469i $$-0.754753\pi$$
−0.717587 + 0.696469i $$0.754753\pi$$
$$504$$ 0 0
$$505$$ 10928.1 0.962959
$$506$$ 0 0
$$507$$ −12928.8 −1.13252
$$508$$ 0 0
$$509$$ −1493.11 −0.130021 −0.0650106 0.997885i $$-0.520708\pi$$
−0.0650106 + 0.997885i $$0.520708\pi$$
$$510$$ 0 0
$$511$$ −7197.03 −0.623049
$$512$$ 0 0
$$513$$ −536.022 −0.0461325
$$514$$ 0 0
$$515$$ −2834.00 −0.242487
$$516$$ 0 0
$$517$$ 1863.20 0.158498
$$518$$ 0 0
$$519$$ −28279.6 −2.39179
$$520$$ 0 0
$$521$$ −7026.84 −0.590886 −0.295443 0.955360i $$-0.595467\pi$$
−0.295443 + 0.955360i $$0.595467\pi$$
$$522$$ 0 0
$$523$$ −21924.1 −1.83303 −0.916514 0.400003i $$-0.869009\pi$$
−0.916514 + 0.400003i $$0.869009\pi$$
$$524$$ 0 0
$$525$$ 8352.45 0.694345
$$526$$ 0 0
$$527$$ −2958.80 −0.244568
$$528$$ 0 0
$$529$$ 17976.8 1.47750
$$530$$ 0 0
$$531$$ −12115.4 −0.990135
$$532$$ 0 0
$$533$$ −5707.39 −0.463817
$$534$$ 0 0
$$535$$ 5785.44 0.467526
$$536$$ 0 0
$$537$$ −18911.8 −1.51975
$$538$$ 0 0
$$539$$ −2428.28 −0.194051
$$540$$ 0 0
$$541$$ 983.325 0.0781450 0.0390725 0.999236i $$-0.487560\pi$$
0.0390725 + 0.999236i $$0.487560\pi$$
$$542$$ 0 0
$$543$$ −13747.2 −1.08646
$$544$$ 0 0
$$545$$ −5655.16 −0.444478
$$546$$ 0 0
$$547$$ −6100.54 −0.476856 −0.238428 0.971160i $$-0.576632\pi$$
−0.238428 + 0.971160i $$0.576632\pi$$
$$548$$ 0 0
$$549$$ 6357.01 0.494190
$$550$$ 0 0
$$551$$ −272.114 −0.0210389
$$552$$ 0 0
$$553$$ −863.204 −0.0663782
$$554$$ 0 0
$$555$$ −9456.93 −0.723287
$$556$$ 0 0
$$557$$ −7531.50 −0.572926 −0.286463 0.958091i $$-0.592480\pi$$
−0.286463 + 0.958091i $$0.592480\pi$$
$$558$$ 0 0
$$559$$ −32197.0 −2.43612
$$560$$ 0 0
$$561$$ −13952.6 −1.05005
$$562$$ 0 0
$$563$$ 19553.5 1.46373 0.731866 0.681449i $$-0.238650\pi$$
0.731866 + 0.681449i $$0.238650\pi$$
$$564$$ 0 0
$$565$$ −5179.49 −0.385669
$$566$$ 0 0
$$567$$ −6773.10 −0.501664
$$568$$ 0 0
$$569$$ 23656.9 1.74297 0.871483 0.490426i $$-0.163159\pi$$
0.871483 + 0.490426i $$0.163159\pi$$
$$570$$ 0 0
$$571$$ 5466.08 0.400610 0.200305 0.979734i $$-0.435807\pi$$
0.200305 + 0.979734i $$0.435807\pi$$
$$572$$ 0 0
$$573$$ −9051.37 −0.659907
$$574$$ 0 0
$$575$$ −13092.5 −0.949557
$$576$$ 0 0
$$577$$ −653.065 −0.0471187 −0.0235593 0.999722i $$-0.507500\pi$$
−0.0235593 + 0.999722i $$0.507500\pi$$
$$578$$ 0 0
$$579$$ 25168.8 1.80653
$$580$$ 0 0
$$581$$ 11696.3 0.835187
$$582$$ 0 0
$$583$$ −2389.15 −0.169723
$$584$$ 0 0
$$585$$ −14982.6 −1.05890
$$586$$ 0 0
$$587$$ 19414.4 1.36511 0.682554 0.730835i $$-0.260869\pi$$
0.682554 + 0.730835i $$0.260869\pi$$
$$588$$ 0 0
$$589$$ −264.809 −0.0185250
$$590$$ 0 0
$$591$$ −30722.0 −2.13830
$$592$$ 0 0
$$593$$ 18910.6 1.30955 0.654777 0.755822i $$-0.272763\pi$$
0.654777 + 0.755822i $$0.272763\pi$$
$$594$$ 0 0
$$595$$ 10445.0 0.719670
$$596$$ 0 0
$$597$$ −34948.0 −2.39586
$$598$$ 0 0
$$599$$ −12128.7 −0.827322 −0.413661 0.910431i $$-0.635750\pi$$
−0.413661 + 0.910431i $$0.635750\pi$$
$$600$$ 0 0
$$601$$ −25803.3 −1.75131 −0.875656 0.482936i $$-0.839570\pi$$
−0.875656 + 0.482936i $$0.839570\pi$$
$$602$$ 0 0
$$603$$ −19715.0 −1.33144
$$604$$ 0 0
$$605$$ −7338.28 −0.493130
$$606$$ 0 0
$$607$$ 12183.6 0.814690 0.407345 0.913274i $$-0.366455\pi$$
0.407345 + 0.913274i $$0.366455\pi$$
$$608$$ 0 0
$$609$$ 3212.09 0.213728
$$610$$ 0 0
$$611$$ −6799.78 −0.450229
$$612$$ 0 0
$$613$$ 16519.2 1.08843 0.544213 0.838947i $$-0.316829\pi$$
0.544213 + 0.838947i $$0.316829\pi$$
$$614$$ 0 0
$$615$$ −5072.47 −0.332588
$$616$$ 0 0
$$617$$ −6518.09 −0.425297 −0.212649 0.977129i $$-0.568209\pi$$
−0.212649 + 0.977129i $$0.568209\pi$$
$$618$$ 0 0
$$619$$ 4344.67 0.282111 0.141056 0.990002i $$-0.454950\pi$$
0.141056 + 0.990002i $$0.454950\pi$$
$$620$$ 0 0
$$621$$ −9918.12 −0.640902
$$622$$ 0 0
$$623$$ −13276.4 −0.853787
$$624$$ 0 0
$$625$$ −512.326 −0.0327889
$$626$$ 0 0
$$627$$ −1248.74 −0.0795374
$$628$$ 0 0
$$629$$ 17983.2 1.13996
$$630$$ 0 0
$$631$$ 2989.38 0.188598 0.0942989 0.995544i $$-0.469939\pi$$
0.0942989 + 0.995544i $$0.469939\pi$$
$$632$$ 0 0
$$633$$ 38217.1 2.39967
$$634$$ 0 0
$$635$$ −13345.8 −0.834037
$$636$$ 0 0
$$637$$ 8862.02 0.551218
$$638$$ 0 0
$$639$$ 18350.5 1.13604
$$640$$ 0 0
$$641$$ 15375.4 0.947411 0.473706 0.880683i $$-0.342916\pi$$
0.473706 + 0.880683i $$0.342916\pi$$
$$642$$ 0 0
$$643$$ 16966.5 1.04058 0.520291 0.853989i $$-0.325823\pi$$
0.520291 + 0.853989i $$0.325823\pi$$
$$644$$ 0 0
$$645$$ −28615.2 −1.74686
$$646$$ 0 0
$$647$$ −12379.7 −0.752234 −0.376117 0.926572i $$-0.622741\pi$$
−0.376117 + 0.926572i $$0.622741\pi$$
$$648$$ 0 0
$$649$$ −6004.68 −0.363181
$$650$$ 0 0
$$651$$ 3125.86 0.188191
$$652$$ 0 0
$$653$$ −30418.3 −1.82291 −0.911453 0.411403i $$-0.865039\pi$$
−0.911453 + 0.411403i $$0.865039\pi$$
$$654$$ 0 0
$$655$$ 8067.01 0.481228
$$656$$ 0 0
$$657$$ −17447.4 −1.03605
$$658$$ 0 0
$$659$$ 6069.11 0.358754 0.179377 0.983780i $$-0.442592\pi$$
0.179377 + 0.983780i $$0.442592\pi$$
$$660$$ 0 0
$$661$$ 24894.6 1.46489 0.732443 0.680829i $$-0.238380\pi$$
0.732443 + 0.680829i $$0.238380\pi$$
$$662$$ 0 0
$$663$$ 50920.3 2.98277
$$664$$ 0 0
$$665$$ 934.815 0.0545121
$$666$$ 0 0
$$667$$ −5034.97 −0.292286
$$668$$ 0 0
$$669$$ 12003.9 0.693720
$$670$$ 0 0
$$671$$ 3150.69 0.181268
$$672$$ 0 0
$$673$$ 21677.5 1.24162 0.620808 0.783963i $$-0.286805\pi$$
0.620808 + 0.783963i $$0.286805\pi$$
$$674$$ 0 0
$$675$$ 4307.79 0.245640
$$676$$ 0 0
$$677$$ 21322.2 1.21046 0.605228 0.796052i $$-0.293082\pi$$
0.605228 + 0.796052i $$0.293082\pi$$
$$678$$ 0 0
$$679$$ −12573.6 −0.710649
$$680$$ 0 0
$$681$$ −24349.3 −1.37014
$$682$$ 0 0
$$683$$ −38.7459 −0.00217068 −0.00108534 0.999999i $$-0.500345\pi$$
−0.00108534 + 0.999999i $$0.500345\pi$$
$$684$$ 0 0
$$685$$ 20220.7 1.12787
$$686$$ 0 0
$$687$$ −36687.9 −2.03745
$$688$$ 0 0
$$689$$ 8719.23 0.482113
$$690$$ 0 0
$$691$$ −27812.0 −1.53114 −0.765571 0.643351i $$-0.777544\pi$$
−0.765571 + 0.643351i $$0.777544\pi$$
$$692$$ 0 0
$$693$$ 8247.54 0.452090
$$694$$ 0 0
$$695$$ −5476.86 −0.298920
$$696$$ 0 0
$$697$$ 9645.76 0.524188
$$698$$ 0 0
$$699$$ 7114.78 0.384987
$$700$$ 0 0
$$701$$ −23404.8 −1.26104 −0.630519 0.776174i $$-0.717158\pi$$
−0.630519 + 0.776174i $$0.717158\pi$$
$$702$$ 0 0
$$703$$ 1609.47 0.0863477
$$704$$ 0 0
$$705$$ −6043.33 −0.322844
$$706$$ 0 0
$$707$$ 21954.1 1.16785
$$708$$ 0 0
$$709$$ −8525.03 −0.451572 −0.225786 0.974177i $$-0.572495\pi$$
−0.225786 + 0.974177i $$0.572495\pi$$
$$710$$ 0 0
$$711$$ −2092.62 −0.110379
$$712$$ 0 0
$$713$$ −4899.80 −0.257362
$$714$$ 0 0
$$715$$ −7425.76 −0.388402
$$716$$ 0 0
$$717$$ 144.018 0.00750132
$$718$$ 0 0
$$719$$ −13242.3 −0.686862 −0.343431 0.939178i $$-0.611589\pi$$
−0.343431 + 0.939178i $$0.611589\pi$$
$$720$$ 0 0
$$721$$ −5693.39 −0.294082
$$722$$ 0 0
$$723$$ −48540.8 −2.49689
$$724$$ 0 0
$$725$$ 2186.86 0.112025
$$726$$ 0 0
$$727$$ 5299.22 0.270340 0.135170 0.990822i $$-0.456842\pi$$
0.135170 + 0.990822i $$0.456842\pi$$
$$728$$ 0 0
$$729$$ −28495.4 −1.44771
$$730$$ 0 0
$$731$$ 54414.5 2.75320
$$732$$ 0 0
$$733$$ −28433.7 −1.43278 −0.716388 0.697702i $$-0.754206\pi$$
−0.716388 + 0.697702i $$0.754206\pi$$
$$734$$ 0 0
$$735$$ 7876.15 0.395260
$$736$$ 0 0
$$737$$ −9771.23 −0.488369
$$738$$ 0 0
$$739$$ 19450.7 0.968209 0.484105 0.875010i $$-0.339145\pi$$
0.484105 + 0.875010i $$0.339145\pi$$
$$740$$ 0 0
$$741$$ 4557.30 0.225933
$$742$$ 0 0
$$743$$ −15260.1 −0.753484 −0.376742 0.926318i $$-0.622956\pi$$
−0.376742 + 0.926318i $$0.622956\pi$$
$$744$$ 0 0
$$745$$ −16712.9 −0.821896
$$746$$ 0 0
$$747$$ 28354.7 1.38881
$$748$$ 0 0
$$749$$ 11622.7 0.567003
$$750$$ 0 0
$$751$$ 23290.8 1.13168 0.565841 0.824514i $$-0.308551\pi$$
0.565841 + 0.824514i $$0.308551\pi$$
$$752$$ 0 0
$$753$$ 26035.3 1.26000
$$754$$ 0 0
$$755$$ 719.627 0.0346886
$$756$$ 0 0
$$757$$ 22951.5 1.10196 0.550981 0.834517i $$-0.314253\pi$$
0.550981 + 0.834517i $$0.314253\pi$$
$$758$$ 0 0
$$759$$ −23105.7 −1.10499
$$760$$ 0 0
$$761$$ 14975.4 0.713349 0.356675 0.934229i $$-0.383910\pi$$
0.356675 + 0.934229i $$0.383910\pi$$
$$762$$ 0 0
$$763$$ −11361.0 −0.539050
$$764$$ 0 0
$$765$$ 25321.3 1.19672
$$766$$ 0 0
$$767$$ 21914.1 1.03165
$$768$$ 0 0
$$769$$ 30907.7 1.44936 0.724682 0.689084i $$-0.241987\pi$$
0.724682 + 0.689084i $$0.241987\pi$$
$$770$$ 0 0
$$771$$ −21242.2 −0.992244
$$772$$ 0 0
$$773$$ 3728.71 0.173496 0.0867479 0.996230i $$-0.472353\pi$$
0.0867479 + 0.996230i $$0.472353\pi$$
$$774$$ 0 0
$$775$$ 2128.16 0.0986395
$$776$$ 0 0
$$777$$ −18998.6 −0.877182
$$778$$ 0 0
$$779$$ 863.283 0.0397051
$$780$$ 0 0
$$781$$ 9094.94 0.416700
$$782$$ 0 0
$$783$$ 1656.64 0.0756111
$$784$$ 0 0
$$785$$ −4122.87 −0.187454
$$786$$ 0 0
$$787$$ −33291.9 −1.50791 −0.753957 0.656923i $$-0.771858\pi$$
−0.753957 + 0.656923i $$0.771858\pi$$
$$788$$ 0 0
$$789$$ −48814.7 −2.20260
$$790$$ 0 0
$$791$$ −10405.4 −0.467729
$$792$$ 0 0
$$793$$ −11498.5 −0.514909
$$794$$ 0 0
$$795$$ 7749.24 0.345707
$$796$$ 0 0
$$797$$ 36436.3 1.61937 0.809687 0.586862i $$-0.199637\pi$$
0.809687 + 0.586862i $$0.199637\pi$$
$$798$$ 0 0
$$799$$ 11491.9 0.508831
$$800$$ 0 0
$$801$$ −32185.4 −1.41974
$$802$$ 0 0
$$803$$ −8647.37 −0.380024
$$804$$ 0 0
$$805$$ 17297.1 0.757318
$$806$$ 0 0
$$807$$ −44980.6 −1.96207
$$808$$ 0 0
$$809$$ 9776.41 0.424871 0.212435 0.977175i $$-0.431861\pi$$
0.212435 + 0.977175i $$0.431861\pi$$
$$810$$ 0 0
$$811$$ −30799.9 −1.33358 −0.666788 0.745248i $$-0.732331\pi$$
−0.666788 + 0.745248i $$0.732331\pi$$
$$812$$ 0 0
$$813$$ −69331.8 −2.99086
$$814$$ 0 0
$$815$$ −10143.1 −0.435948
$$816$$ 0 0
$$817$$ 4870.02 0.208544
$$818$$ 0 0
$$819$$ −30099.5 −1.28420
$$820$$ 0 0
$$821$$ 3080.42 0.130947 0.0654735 0.997854i $$-0.479144\pi$$
0.0654735 + 0.997854i $$0.479144\pi$$
$$822$$ 0 0
$$823$$ 23048.4 0.976204 0.488102 0.872787i $$-0.337689\pi$$
0.488102 + 0.872787i $$0.337689\pi$$
$$824$$ 0 0
$$825$$ 10035.6 0.423510
$$826$$ 0 0
$$827$$ 13260.3 0.557565 0.278783 0.960354i $$-0.410069\pi$$
0.278783 + 0.960354i $$0.410069\pi$$
$$828$$ 0 0
$$829$$ 34413.4 1.44177 0.720885 0.693055i $$-0.243736\pi$$
0.720885 + 0.693055i $$0.243736\pi$$
$$830$$ 0 0
$$831$$ −21966.5 −0.916978
$$832$$ 0 0
$$833$$ −14977.2 −0.622965
$$834$$ 0 0
$$835$$ 23506.5 0.974221
$$836$$ 0 0
$$837$$ 1612.17 0.0665766
$$838$$ 0 0
$$839$$ 5440.08 0.223853 0.111926 0.993717i $$-0.464298\pi$$
0.111926 + 0.993717i $$0.464298\pi$$
$$840$$ 0 0
$$841$$ 841.000 0.0344828
$$842$$ 0 0
$$843$$ −40189.1 −1.64198
$$844$$ 0 0
$$845$$ 11628.9 0.473429
$$846$$ 0 0
$$847$$ −14742.3 −0.598054
$$848$$ 0 0
$$849$$ 22930.7 0.926947
$$850$$ 0 0
$$851$$ 29780.4 1.19960
$$852$$ 0 0
$$853$$ −22303.5 −0.895261 −0.447631 0.894219i $$-0.647732\pi$$
−0.447631 + 0.894219i $$0.647732\pi$$
$$854$$ 0 0
$$855$$ 2266.22 0.0906470
$$856$$ 0 0
$$857$$ 13859.0 0.552411 0.276205 0.961099i $$-0.410923\pi$$
0.276205 + 0.961099i $$0.410923\pi$$
$$858$$ 0 0
$$859$$ 24629.1 0.978270 0.489135 0.872208i $$-0.337313\pi$$
0.489135 + 0.872208i $$0.337313\pi$$
$$860$$ 0 0
$$861$$ −10190.4 −0.403354
$$862$$ 0 0
$$863$$ −33140.6 −1.30721 −0.653604 0.756837i $$-0.726744\pi$$
−0.653604 + 0.756837i $$0.726744\pi$$
$$864$$ 0 0
$$865$$ 25436.5 0.999846
$$866$$ 0 0
$$867$$ −47592.7 −1.86428
$$868$$ 0 0
$$869$$ −1037.16 −0.0404869
$$870$$ 0 0
$$871$$ 35660.2 1.38726
$$872$$ 0 0
$$873$$ −30481.5 −1.18172
$$874$$ 0 0
$$875$$ −19966.0 −0.771398
$$876$$ 0 0
$$877$$ −2224.74 −0.0856604 −0.0428302 0.999082i $$-0.513637\pi$$
−0.0428302 + 0.999082i $$0.513637\pi$$
$$878$$ 0 0
$$879$$ 44511.9 1.70802
$$880$$ 0 0
$$881$$ 24242.9 0.927087 0.463544 0.886074i $$-0.346578\pi$$
0.463544 + 0.886074i $$0.346578\pi$$
$$882$$ 0 0
$$883$$ −24060.4 −0.916984 −0.458492 0.888698i $$-0.651610\pi$$
−0.458492 + 0.888698i $$0.651610\pi$$
$$884$$ 0 0
$$885$$ 19476.3 0.739760
$$886$$ 0 0
$$887$$ 9727.38 0.368223 0.184111 0.982905i $$-0.441059\pi$$
0.184111 + 0.982905i $$0.441059\pi$$
$$888$$ 0 0
$$889$$ −26811.3 −1.01150
$$890$$ 0 0
$$891$$ −8138.00 −0.305986
$$892$$ 0 0
$$893$$ 1028.51 0.0385419
$$894$$ 0 0
$$895$$ 17010.5 0.635304
$$896$$ 0 0
$$897$$ 84324.5 3.13881
$$898$$ 0 0
$$899$$ 818.422 0.0303625
$$900$$ 0 0
$$901$$ −14735.9 −0.544865
$$902$$ 0 0
$$903$$ −57486.9 −2.11854
$$904$$ 0 0
$$905$$ 12365.1 0.454177
$$906$$ 0 0
$$907$$ −4523.40 −0.165598 −0.0827988 0.996566i $$-0.526386\pi$$
−0.0827988 + 0.996566i $$0.526386\pi$$
$$908$$ 0 0
$$909$$ 53222.2 1.94199
$$910$$ 0 0
$$911$$ 24396.1 0.887245 0.443623 0.896214i $$-0.353693\pi$$
0.443623 + 0.896214i $$0.353693\pi$$
$$912$$ 0 0
$$913$$ 14053.3 0.509415
$$914$$ 0 0
$$915$$ −10219.3 −0.369224
$$916$$ 0 0
$$917$$ 16206.3 0.583620
$$918$$ 0 0
$$919$$ −51701.0 −1.85578 −0.927888 0.372858i $$-0.878378\pi$$
−0.927888 + 0.372858i $$0.878378\pi$$
$$920$$ 0 0
$$921$$ 22541.9 0.806496
$$922$$ 0 0
$$923$$ −33192.1 −1.18367
$$924$$ 0 0
$$925$$ −12934.7 −0.459772
$$926$$ 0 0
$$927$$ −13802.2 −0.489022
$$928$$ 0 0
$$929$$ −8172.60 −0.288627 −0.144313 0.989532i $$-0.546097\pi$$
−0.144313 + 0.989532i $$0.546097\pi$$
$$930$$ 0 0
$$931$$ −1340.44 −0.0471871
$$932$$ 0 0
$$933$$ −4022.84 −0.141160
$$934$$ 0 0
$$935$$ 12549.9 0.438957
$$936$$ 0 0
$$937$$ 40385.8 1.40806 0.704028 0.710173i $$-0.251383\pi$$
0.704028 + 0.710173i $$0.251383\pi$$
$$938$$ 0 0
$$939$$ 11812.4 0.410525
$$940$$ 0 0
$$941$$ −3540.29 −0.122646 −0.0613232 0.998118i $$-0.519532\pi$$
−0.0613232 + 0.998118i $$0.519532\pi$$
$$942$$ 0 0
$$943$$ 15973.5 0.551610
$$944$$ 0 0
$$945$$ −5691.20 −0.195910
$$946$$ 0 0
$$947$$ −53975.8 −1.85214 −0.926071 0.377349i $$-0.876836\pi$$
−0.926071 + 0.377349i $$0.876836\pi$$
$$948$$ 0 0
$$949$$ 31558.7 1.07949
$$950$$ 0 0
$$951$$ 42370.8 1.44476
$$952$$ 0 0
$$953$$ −19500.7 −0.662843 −0.331421 0.943483i $$-0.607528\pi$$
−0.331421 + 0.943483i $$0.607528\pi$$
$$954$$ 0 0
$$955$$ 8141.37 0.275862
$$956$$ 0 0
$$957$$ 3859.39 0.130362
$$958$$ 0 0
$$959$$ 40622.6 1.36786
$$960$$ 0 0
$$961$$ −28994.5 −0.973265
$$962$$ 0 0
$$963$$ 28176.4 0.942857
$$964$$ 0 0
$$965$$ −22638.4 −0.755186
$$966$$ 0 0
$$967$$ −18203.2 −0.605354 −0.302677 0.953093i $$-0.597880\pi$$
−0.302677 + 0.953093i $$0.597880\pi$$
$$968$$ 0 0
$$969$$ −7702.04 −0.255341
$$970$$ 0 0
$$971$$ −10588.8 −0.349961 −0.174981 0.984572i $$-0.555986\pi$$
−0.174981 + 0.984572i $$0.555986\pi$$
$$972$$ 0 0
$$973$$ −11002.8 −0.362522
$$974$$ 0 0
$$975$$ −36625.1 −1.20302
$$976$$ 0 0
$$977$$ 24805.8 0.812290 0.406145 0.913809i $$-0.366873\pi$$
0.406145 + 0.913809i $$0.366873\pi$$
$$978$$ 0 0
$$979$$ −15951.9 −0.520760
$$980$$ 0 0
$$981$$ −27541.8 −0.896375
$$982$$ 0 0
$$983$$ −2760.27 −0.0895613 −0.0447807 0.998997i $$-0.514259\pi$$
−0.0447807 + 0.998997i $$0.514259\pi$$
$$984$$ 0 0
$$985$$ 27633.3 0.893878
$$986$$ 0 0
$$987$$ −12140.8 −0.391536
$$988$$ 0 0
$$989$$ 90111.0 2.89723
$$990$$ 0 0
$$991$$ −5609.17 −0.179799 −0.0898997 0.995951i $$-0.528655\pi$$
−0.0898997 + 0.995951i $$0.528655\pi$$
$$992$$ 0 0
$$993$$ 81064.5 2.59064
$$994$$ 0 0
$$995$$ 31434.4 1.00155
$$996$$ 0 0
$$997$$ 3629.31 0.115287 0.0576436 0.998337i $$-0.481641\pi$$
0.0576436 + 0.998337i $$0.481641\pi$$
$$998$$ 0 0
$$999$$ −9798.55 −0.310323
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 1856.4.a.z.1.1 5
4.3 odd 2 1856.4.a.ba.1.5 5
8.3 odd 2 464.4.a.m.1.1 5
8.5 even 2 232.4.a.d.1.5 5
24.5 odd 2 2088.4.a.f.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.d.1.5 5 8.5 even 2
464.4.a.m.1.1 5 8.3 odd 2
1856.4.a.z.1.1 5 1.1 even 1 trivial
1856.4.a.ba.1.5 5 4.3 odd 2
2088.4.a.f.1.4 5 24.5 odd 2