# Properties

 Label 1521.4.a.o.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +O(q^{10})$$ $$q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +3.00000 q^{10} +13.8564 q^{11} +24.0000 q^{14} +1.00000 q^{16} +117.000 q^{17} +114.315 q^{19} +8.66025 q^{20} -24.0000 q^{22} -78.0000 q^{23} -122.000 q^{25} +69.2820 q^{28} +141.000 q^{29} -155.885 q^{31} -181.865 q^{32} -202.650 q^{34} +24.0000 q^{35} +143.760 q^{37} -198.000 q^{38} -39.0000 q^{40} -271.932 q^{41} -104.000 q^{43} -69.2820 q^{44} +135.100 q^{46} -301.377 q^{47} -151.000 q^{49} +211.310 q^{50} -93.0000 q^{53} -24.0000 q^{55} -312.000 q^{56} -244.219 q^{58} +284.056 q^{59} +145.000 q^{61} +270.000 q^{62} +307.000 q^{64} -786.351 q^{67} -585.000 q^{68} -41.5692 q^{70} +1056.55 q^{71} +458.993 q^{73} -249.000 q^{74} -571.577 q^{76} -192.000 q^{77} +1276.00 q^{79} -1.73205 q^{80} +471.000 q^{82} -789.815 q^{83} -202.650 q^{85} +180.133 q^{86} +312.000 q^{88} -976.877 q^{89} +390.000 q^{92} +522.000 q^{94} -198.000 q^{95} -200.918 q^{97} +261.540 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{4}+O(q^{10})$$ 2 * q - 10 * q^4 $$2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100})$$ 2 * q - 10 * q^4 + 6 * q^10 + 48 * q^14 + 2 * q^16 + 234 * q^17 - 48 * q^22 - 156 * q^23 - 244 * q^25 + 282 * q^29 + 48 * q^35 - 396 * q^38 - 78 * q^40 - 208 * q^43 - 302 * q^49 - 186 * q^53 - 48 * q^55 - 624 * q^56 + 290 * q^61 + 540 * q^62 + 614 * q^64 - 1170 * q^68 - 498 * q^74 - 384 * q^77 + 2552 * q^79 + 942 * q^82 + 624 * q^88 + 780 * q^92 + 1044 * q^94 - 396 * q^95

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −0.612372 −0.306186 0.951972i $$-0.599053\pi$$
−0.306186 + 0.951972i $$0.599053\pi$$
$$3$$ 0 0
$$4$$ −5.00000 −0.625000
$$5$$ −1.73205 −0.154919 −0.0774597 0.996995i $$-0.524681\pi$$
−0.0774597 + 0.996995i $$0.524681\pi$$
$$6$$ 0 0
$$7$$ −13.8564 −0.748176 −0.374088 0.927393i $$-0.622044\pi$$
−0.374088 + 0.927393i $$0.622044\pi$$
$$8$$ 22.5167 0.995105
$$9$$ 0 0
$$10$$ 3.00000 0.0948683
$$11$$ 13.8564 0.379806 0.189903 0.981803i $$-0.439183\pi$$
0.189903 + 0.981803i $$0.439183\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 24.0000 0.458162
$$15$$ 0 0
$$16$$ 1.00000 0.0156250
$$17$$ 117.000 1.66922 0.834608 0.550845i $$-0.185694\pi$$
0.834608 + 0.550845i $$0.185694\pi$$
$$18$$ 0 0
$$19$$ 114.315 1.38030 0.690151 0.723665i $$-0.257544\pi$$
0.690151 + 0.723665i $$0.257544\pi$$
$$20$$ 8.66025 0.0968246
$$21$$ 0 0
$$22$$ −24.0000 −0.232583
$$23$$ −78.0000 −0.707136 −0.353568 0.935409i $$-0.615032\pi$$
−0.353568 + 0.935409i $$0.615032\pi$$
$$24$$ 0 0
$$25$$ −122.000 −0.976000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 69.2820 0.467610
$$29$$ 141.000 0.902864 0.451432 0.892306i $$-0.350913\pi$$
0.451432 + 0.892306i $$0.350913\pi$$
$$30$$ 0 0
$$31$$ −155.885 −0.903151 −0.451576 0.892233i $$-0.649138\pi$$
−0.451576 + 0.892233i $$0.649138\pi$$
$$32$$ −181.865 −1.00467
$$33$$ 0 0
$$34$$ −202.650 −1.02218
$$35$$ 24.0000 0.115907
$$36$$ 0 0
$$37$$ 143.760 0.638758 0.319379 0.947627i $$-0.396526\pi$$
0.319379 + 0.947627i $$0.396526\pi$$
$$38$$ −198.000 −0.845259
$$39$$ 0 0
$$40$$ −39.0000 −0.154161
$$41$$ −271.932 −1.03582 −0.517910 0.855435i $$-0.673290\pi$$
−0.517910 + 0.855435i $$0.673290\pi$$
$$42$$ 0 0
$$43$$ −104.000 −0.368834 −0.184417 0.982848i $$-0.559040\pi$$
−0.184417 + 0.982848i $$0.559040\pi$$
$$44$$ −69.2820 −0.237379
$$45$$ 0 0
$$46$$ 135.100 0.433030
$$47$$ −301.377 −0.935326 −0.467663 0.883907i $$-0.654904\pi$$
−0.467663 + 0.883907i $$0.654904\pi$$
$$48$$ 0 0
$$49$$ −151.000 −0.440233
$$50$$ 211.310 0.597675
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −93.0000 −0.241029 −0.120514 0.992712i $$-0.538454\pi$$
−0.120514 + 0.992712i $$0.538454\pi$$
$$54$$ 0 0
$$55$$ −24.0000 −0.0588393
$$56$$ −312.000 −0.744513
$$57$$ 0 0
$$58$$ −244.219 −0.552889
$$59$$ 284.056 0.626796 0.313398 0.949622i $$-0.398533\pi$$
0.313398 + 0.949622i $$0.398533\pi$$
$$60$$ 0 0
$$61$$ 145.000 0.304350 0.152175 0.988354i $$-0.451372\pi$$
0.152175 + 0.988354i $$0.451372\pi$$
$$62$$ 270.000 0.553065
$$63$$ 0 0
$$64$$ 307.000 0.599609
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −786.351 −1.43385 −0.716926 0.697149i $$-0.754451\pi$$
−0.716926 + 0.697149i $$0.754451\pi$$
$$68$$ −585.000 −1.04326
$$69$$ 0 0
$$70$$ −41.5692 −0.0709782
$$71$$ 1056.55 1.76605 0.883025 0.469326i $$-0.155503\pi$$
0.883025 + 0.469326i $$0.155503\pi$$
$$72$$ 0 0
$$73$$ 458.993 0.735906 0.367953 0.929844i $$-0.380059\pi$$
0.367953 + 0.929844i $$0.380059\pi$$
$$74$$ −249.000 −0.391158
$$75$$ 0 0
$$76$$ −571.577 −0.862689
$$77$$ −192.000 −0.284161
$$78$$ 0 0
$$79$$ 1276.00 1.81723 0.908615 0.417634i $$-0.137141\pi$$
0.908615 + 0.417634i $$0.137141\pi$$
$$80$$ −1.73205 −0.00242061
$$81$$ 0 0
$$82$$ 471.000 0.634308
$$83$$ −789.815 −1.04450 −0.522250 0.852793i $$-0.674907\pi$$
−0.522250 + 0.852793i $$0.674907\pi$$
$$84$$ 0 0
$$85$$ −202.650 −0.258594
$$86$$ 180.133 0.225864
$$87$$ 0 0
$$88$$ 312.000 0.377947
$$89$$ −976.877 −1.16347 −0.581734 0.813379i $$-0.697626\pi$$
−0.581734 + 0.813379i $$0.697626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 390.000 0.441960
$$93$$ 0 0
$$94$$ 522.000 0.572768
$$95$$ −198.000 −0.213835
$$96$$ 0 0
$$97$$ −200.918 −0.210311 −0.105155 0.994456i $$-0.533534\pi$$
−0.105155 + 0.994456i $$0.533534\pi$$
$$98$$ 261.540 0.269587
$$99$$ 0 0
$$100$$ 610.000 0.610000
$$101$$ 429.000 0.422645 0.211322 0.977416i $$-0.432223\pi$$
0.211322 + 0.977416i $$0.432223\pi$$
$$102$$ 0 0
$$103$$ −182.000 −0.174107 −0.0870534 0.996204i $$-0.527745\pi$$
−0.0870534 + 0.996204i $$0.527745\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 161.081 0.147599
$$107$$ 1506.00 1.36066 0.680330 0.732906i $$-0.261837\pi$$
0.680330 + 0.732906i $$0.261837\pi$$
$$108$$ 0 0
$$109$$ −1551.92 −1.36373 −0.681866 0.731477i $$-0.738831\pi$$
−0.681866 + 0.731477i $$0.738831\pi$$
$$110$$ 41.5692 0.0360315
$$111$$ 0 0
$$112$$ −13.8564 −0.0116902
$$113$$ 687.000 0.571925 0.285962 0.958241i $$-0.407687\pi$$
0.285962 + 0.958241i $$0.407687\pi$$
$$114$$ 0 0
$$115$$ 135.100 0.109549
$$116$$ −705.000 −0.564290
$$117$$ 0 0
$$118$$ −492.000 −0.383833
$$119$$ −1621.20 −1.24887
$$120$$ 0 0
$$121$$ −1139.00 −0.855748
$$122$$ −251.147 −0.186376
$$123$$ 0 0
$$124$$ 779.423 0.564470
$$125$$ 427.817 0.306121
$$126$$ 0 0
$$127$$ −286.000 −0.199830 −0.0999149 0.994996i $$-0.531857\pi$$
−0.0999149 + 0.994996i $$0.531857\pi$$
$$128$$ 923.183 0.637489
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1974.00 1.31656 0.658279 0.752774i $$-0.271285\pi$$
0.658279 + 0.752774i $$0.271285\pi$$
$$132$$ 0 0
$$133$$ −1584.00 −1.03271
$$134$$ 1362.00 0.878051
$$135$$ 0 0
$$136$$ 2634.45 1.66105
$$137$$ 846.973 0.528188 0.264094 0.964497i $$-0.414927\pi$$
0.264094 + 0.964497i $$0.414927\pi$$
$$138$$ 0 0
$$139$$ 236.000 0.144009 0.0720045 0.997404i $$-0.477060\pi$$
0.0720045 + 0.997404i $$0.477060\pi$$
$$140$$ −120.000 −0.0724418
$$141$$ 0 0
$$142$$ −1830.00 −1.08148
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −244.219 −0.139871
$$146$$ −795.000 −0.450648
$$147$$ 0 0
$$148$$ −718.801 −0.399224
$$149$$ −46.7654 −0.0257125 −0.0128563 0.999917i $$-0.504092\pi$$
−0.0128563 + 0.999917i $$0.504092\pi$$
$$150$$ 0 0
$$151$$ −1770.16 −0.953995 −0.476998 0.878905i $$-0.658275\pi$$
−0.476998 + 0.878905i $$0.658275\pi$$
$$152$$ 2574.00 1.37355
$$153$$ 0 0
$$154$$ 332.554 0.174013
$$155$$ 270.000 0.139916
$$156$$ 0 0
$$157$$ 1211.00 0.615594 0.307797 0.951452i $$-0.400408\pi$$
0.307797 + 0.951452i $$0.400408\pi$$
$$158$$ −2210.10 −1.11282
$$159$$ 0 0
$$160$$ 315.000 0.155643
$$161$$ 1080.80 0.529062
$$162$$ 0 0
$$163$$ −1004.59 −0.482733 −0.241367 0.970434i $$-0.577596\pi$$
−0.241367 + 0.970434i $$0.577596\pi$$
$$164$$ 1359.66 0.647388
$$165$$ 0 0
$$166$$ 1368.00 0.639623
$$167$$ 914.523 0.423760 0.211880 0.977296i $$-0.432041\pi$$
0.211880 + 0.977296i $$0.432041\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 351.000 0.158356
$$171$$ 0 0
$$172$$ 520.000 0.230521
$$173$$ 2574.00 1.13120 0.565600 0.824680i $$-0.308645\pi$$
0.565600 + 0.824680i $$0.308645\pi$$
$$174$$ 0 0
$$175$$ 1690.48 0.730219
$$176$$ 13.8564 0.00593447
$$177$$ 0 0
$$178$$ 1692.00 0.712476
$$179$$ 3744.00 1.56335 0.781675 0.623686i $$-0.214366\pi$$
0.781675 + 0.623686i $$0.214366\pi$$
$$180$$ 0 0
$$181$$ 637.000 0.261590 0.130795 0.991409i $$-0.458247\pi$$
0.130795 + 0.991409i $$0.458247\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1756.30 −0.703675
$$185$$ −249.000 −0.0989559
$$186$$ 0 0
$$187$$ 1621.20 0.633978
$$188$$ 1506.88 0.584579
$$189$$ 0 0
$$190$$ 342.946 0.130947
$$191$$ 2598.00 0.984213 0.492106 0.870535i $$-0.336227\pi$$
0.492106 + 0.870535i $$0.336227\pi$$
$$192$$ 0 0
$$193$$ −1117.17 −0.416662 −0.208331 0.978058i $$-0.566803\pi$$
−0.208331 + 0.978058i $$0.566803\pi$$
$$194$$ 348.000 0.128788
$$195$$ 0 0
$$196$$ 755.000 0.275146
$$197$$ −2050.75 −0.741674 −0.370837 0.928698i $$-0.620929\pi$$
−0.370837 + 0.928698i $$0.620929\pi$$
$$198$$ 0 0
$$199$$ 2522.00 0.898391 0.449196 0.893433i $$-0.351711\pi$$
0.449196 + 0.893433i $$0.351711\pi$$
$$200$$ −2747.03 −0.971223
$$201$$ 0 0
$$202$$ −743.050 −0.258816
$$203$$ −1953.75 −0.675500
$$204$$ 0 0
$$205$$ 471.000 0.160469
$$206$$ 315.233 0.106618
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1584.00 0.524247
$$210$$ 0 0
$$211$$ 1042.00 0.339973 0.169986 0.985446i $$-0.445628\pi$$
0.169986 + 0.985446i $$0.445628\pi$$
$$212$$ 465.000 0.150643
$$213$$ 0 0
$$214$$ −2608.47 −0.833230
$$215$$ 180.133 0.0571395
$$216$$ 0 0
$$217$$ 2160.00 0.675716
$$218$$ 2688.00 0.835112
$$219$$ 0 0
$$220$$ 120.000 0.0367745
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2407.55 −0.722966 −0.361483 0.932379i $$-0.617730\pi$$
−0.361483 + 0.932379i $$0.617730\pi$$
$$224$$ 2520.00 0.751672
$$225$$ 0 0
$$226$$ −1189.92 −0.350231
$$227$$ 2407.55 0.703942 0.351971 0.936011i $$-0.385512\pi$$
0.351971 + 0.936011i $$0.385512\pi$$
$$228$$ 0 0
$$229$$ 2508.01 0.723729 0.361864 0.932231i $$-0.382140\pi$$
0.361864 + 0.932231i $$0.382140\pi$$
$$230$$ −234.000 −0.0670848
$$231$$ 0 0
$$232$$ 3174.85 0.898444
$$233$$ −5850.00 −1.64483 −0.822417 0.568885i $$-0.807375\pi$$
−0.822417 + 0.568885i $$0.807375\pi$$
$$234$$ 0 0
$$235$$ 522.000 0.144900
$$236$$ −1420.28 −0.391748
$$237$$ 0 0
$$238$$ 2808.00 0.764771
$$239$$ −5383.21 −1.45695 −0.728475 0.685072i $$-0.759771\pi$$
−0.728475 + 0.685072i $$0.759771\pi$$
$$240$$ 0 0
$$241$$ 4917.29 1.31432 0.657159 0.753752i $$-0.271758\pi$$
0.657159 + 0.753752i $$0.271758\pi$$
$$242$$ 1972.81 0.524036
$$243$$ 0 0
$$244$$ −725.000 −0.190219
$$245$$ 261.540 0.0682006
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −3510.00 −0.898731
$$249$$ 0 0
$$250$$ −741.000 −0.187460
$$251$$ −3978.00 −1.00036 −0.500178 0.865923i $$-0.666732\pi$$
−0.500178 + 0.865923i $$0.666732\pi$$
$$252$$ 0 0
$$253$$ −1080.80 −0.268574
$$254$$ 495.367 0.122370
$$255$$ 0 0
$$256$$ −4055.00 −0.989990
$$257$$ 2067.00 0.501696 0.250848 0.968026i $$-0.419291\pi$$
0.250848 + 0.968026i $$0.419291\pi$$
$$258$$ 0 0
$$259$$ −1992.00 −0.477903
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −3419.07 −0.806224
$$263$$ 2052.00 0.481109 0.240555 0.970636i $$-0.422671\pi$$
0.240555 + 0.970636i $$0.422671\pi$$
$$264$$ 0 0
$$265$$ 161.081 0.0373400
$$266$$ 2743.57 0.632402
$$267$$ 0 0
$$268$$ 3931.76 0.896157
$$269$$ −3330.00 −0.754772 −0.377386 0.926056i $$-0.623177\pi$$
−0.377386 + 0.926056i $$0.623177\pi$$
$$270$$ 0 0
$$271$$ −2805.92 −0.628958 −0.314479 0.949264i $$-0.601830\pi$$
−0.314479 + 0.949264i $$0.601830\pi$$
$$272$$ 117.000 0.0260815
$$273$$ 0 0
$$274$$ −1467.00 −0.323448
$$275$$ −1690.48 −0.370690
$$276$$ 0 0
$$277$$ −377.000 −0.0817752 −0.0408876 0.999164i $$-0.513019\pi$$
−0.0408876 + 0.999164i $$0.513019\pi$$
$$278$$ −408.764 −0.0881872
$$279$$ 0 0
$$280$$ 540.400 0.115340
$$281$$ 36.3731 0.00772183 0.00386092 0.999993i $$-0.498771\pi$$
0.00386092 + 0.999993i $$0.498771\pi$$
$$282$$ 0 0
$$283$$ 7124.00 1.49639 0.748194 0.663480i $$-0.230921\pi$$
0.748194 + 0.663480i $$0.230921\pi$$
$$284$$ −5282.75 −1.10378
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3768.00 0.774976
$$288$$ 0 0
$$289$$ 8776.00 1.78628
$$290$$ 423.000 0.0856532
$$291$$ 0 0
$$292$$ −2294.97 −0.459941
$$293$$ 8322.50 1.65941 0.829703 0.558205i $$-0.188510\pi$$
0.829703 + 0.558205i $$0.188510\pi$$
$$294$$ 0 0
$$295$$ −492.000 −0.0971029
$$296$$ 3237.00 0.635631
$$297$$ 0 0
$$298$$ 81.0000 0.0157457
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1441.07 0.275952
$$302$$ 3066.00 0.584200
$$303$$ 0 0
$$304$$ 114.315 0.0215672
$$305$$ −251.147 −0.0471497
$$306$$ 0 0
$$307$$ −2220.49 −0.412801 −0.206401 0.978468i $$-0.566175\pi$$
−0.206401 + 0.978468i $$0.566175\pi$$
$$308$$ 960.000 0.177601
$$309$$ 0 0
$$310$$ −467.654 −0.0856805
$$311$$ 4914.00 0.895972 0.447986 0.894041i $$-0.352141\pi$$
0.447986 + 0.894041i $$0.352141\pi$$
$$312$$ 0 0
$$313$$ −518.000 −0.0935434 −0.0467717 0.998906i $$-0.514893\pi$$
−0.0467717 + 0.998906i $$0.514893\pi$$
$$314$$ −2097.51 −0.376973
$$315$$ 0 0
$$316$$ −6380.00 −1.13577
$$317$$ 3916.17 0.693861 0.346930 0.937891i $$-0.387224\pi$$
0.346930 + 0.937891i $$0.387224\pi$$
$$318$$ 0 0
$$319$$ 1953.75 0.342913
$$320$$ −531.740 −0.0928911
$$321$$ 0 0
$$322$$ −1872.00 −0.323983
$$323$$ 13374.9 2.30402
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 1740.00 0.295613
$$327$$ 0 0
$$328$$ −6123.00 −1.03075
$$329$$ 4176.00 0.699788
$$330$$ 0 0
$$331$$ 7454.75 1.23792 0.618958 0.785424i $$-0.287555\pi$$
0.618958 + 0.785424i $$0.287555\pi$$
$$332$$ 3949.08 0.652812
$$333$$ 0 0
$$334$$ −1584.00 −0.259499
$$335$$ 1362.00 0.222131
$$336$$ 0 0
$$337$$ 3575.00 0.577871 0.288936 0.957349i $$-0.406699\pi$$
0.288936 + 0.957349i $$0.406699\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 1013.25 0.161621
$$341$$ −2160.00 −0.343022
$$342$$ 0 0
$$343$$ 6845.06 1.07755
$$344$$ −2341.73 −0.367028
$$345$$ 0 0
$$346$$ −4458.30 −0.692716
$$347$$ 6966.00 1.07768 0.538839 0.842409i $$-0.318863\pi$$
0.538839 + 0.842409i $$0.318863\pi$$
$$348$$ 0 0
$$349$$ 6651.08 1.02013 0.510063 0.860137i $$-0.329622\pi$$
0.510063 + 0.860137i $$0.329622\pi$$
$$350$$ −2928.00 −0.447166
$$351$$ 0 0
$$352$$ −2520.00 −0.381581
$$353$$ −5630.90 −0.849015 −0.424508 0.905424i $$-0.639553\pi$$
−0.424508 + 0.905424i $$0.639553\pi$$
$$354$$ 0 0
$$355$$ −1830.00 −0.273595
$$356$$ 4884.38 0.727168
$$357$$ 0 0
$$358$$ −6484.80 −0.957353
$$359$$ 7129.12 1.04808 0.524040 0.851694i $$-0.324424\pi$$
0.524040 + 0.851694i $$0.324424\pi$$
$$360$$ 0 0
$$361$$ 6209.00 0.905234
$$362$$ −1103.32 −0.160191
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −795.000 −0.114006
$$366$$ 0 0
$$367$$ 2.00000 0.000284466 0 0.000142233 1.00000i $$-0.499955\pi$$
0.000142233 1.00000i $$0.499955\pi$$
$$368$$ −78.0000 −0.0110490
$$369$$ 0 0
$$370$$ 431.281 0.0605979
$$371$$ 1288.65 0.180332
$$372$$ 0 0
$$373$$ 3499.00 0.485714 0.242857 0.970062i $$-0.421915\pi$$
0.242857 + 0.970062i $$0.421915\pi$$
$$374$$ −2808.00 −0.388231
$$375$$ 0 0
$$376$$ −6786.00 −0.930748
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 5518.31 0.747907 0.373953 0.927447i $$-0.378002\pi$$
0.373953 + 0.927447i $$0.378002\pi$$
$$380$$ 990.000 0.133647
$$381$$ 0 0
$$382$$ −4499.87 −0.602705
$$383$$ −7364.68 −0.982552 −0.491276 0.871004i $$-0.663469\pi$$
−0.491276 + 0.871004i $$0.663469\pi$$
$$384$$ 0 0
$$385$$ 332.554 0.0440221
$$386$$ 1935.00 0.255153
$$387$$ 0 0
$$388$$ 1004.59 0.131444
$$389$$ −1209.00 −0.157580 −0.0787901 0.996891i $$-0.525106\pi$$
−0.0787901 + 0.996891i $$0.525106\pi$$
$$390$$ 0 0
$$391$$ −9126.00 −1.18036
$$392$$ −3400.02 −0.438078
$$393$$ 0 0
$$394$$ 3552.00 0.454181
$$395$$ −2210.10 −0.281524
$$396$$ 0 0
$$397$$ 11694.8 1.47845 0.739226 0.673457i $$-0.235191\pi$$
0.739226 + 0.673457i $$0.235191\pi$$
$$398$$ −4368.23 −0.550150
$$399$$ 0 0
$$400$$ −122.000 −0.0152500
$$401$$ −2980.86 −0.371215 −0.185607 0.982624i $$-0.559425\pi$$
−0.185607 + 0.982624i $$0.559425\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −2145.00 −0.264153
$$405$$ 0 0
$$406$$ 3384.00 0.413658
$$407$$ 1992.00 0.242604
$$408$$ 0 0
$$409$$ −43.3013 −0.00523499 −0.00261749 0.999997i $$-0.500833\pi$$
−0.00261749 + 0.999997i $$0.500833\pi$$
$$410$$ −815.796 −0.0982666
$$411$$ 0 0
$$412$$ 910.000 0.108817
$$413$$ −3936.00 −0.468954
$$414$$ 0 0
$$415$$ 1368.00 0.161813
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −2743.57 −0.321034
$$419$$ 9462.00 1.10322 0.551610 0.834102i $$-0.314014\pi$$
0.551610 + 0.834102i $$0.314014\pi$$
$$420$$ 0 0
$$421$$ −7068.50 −0.818284 −0.409142 0.912471i $$-0.634172\pi$$
−0.409142 + 0.912471i $$0.634172\pi$$
$$422$$ −1804.80 −0.208190
$$423$$ 0 0
$$424$$ −2094.05 −0.239849
$$425$$ −14274.0 −1.62915
$$426$$ 0 0
$$427$$ −2009.18 −0.227707
$$428$$ −7530.00 −0.850412
$$429$$ 0 0
$$430$$ −312.000 −0.0349906
$$431$$ 9928.12 1.10956 0.554780 0.831997i $$-0.312802\pi$$
0.554780 + 0.831997i $$0.312802\pi$$
$$432$$ 0 0
$$433$$ 6617.00 0.734394 0.367197 0.930143i $$-0.380317\pi$$
0.367197 + 0.930143i $$0.380317\pi$$
$$434$$ −3741.23 −0.413790
$$435$$ 0 0
$$436$$ 7759.59 0.852332
$$437$$ −8916.60 −0.976061
$$438$$ 0 0
$$439$$ −13988.0 −1.52075 −0.760377 0.649482i $$-0.774986\pi$$
−0.760377 + 0.649482i $$0.774986\pi$$
$$440$$ −540.400 −0.0585513
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −2004.00 −0.214928 −0.107464 0.994209i $$-0.534273\pi$$
−0.107464 + 0.994209i $$0.534273\pi$$
$$444$$ 0 0
$$445$$ 1692.00 0.180244
$$446$$ 4170.00 0.442725
$$447$$ 0 0
$$448$$ −4253.92 −0.448613
$$449$$ −9082.87 −0.954671 −0.477336 0.878721i $$-0.658397\pi$$
−0.477336 + 0.878721i $$0.658397\pi$$
$$450$$ 0 0
$$451$$ −3768.00 −0.393411
$$452$$ −3435.00 −0.357453
$$453$$ 0 0
$$454$$ −4170.00 −0.431074
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2523.60 0.258313 0.129156 0.991624i $$-0.458773\pi$$
0.129156 + 0.991624i $$0.458773\pi$$
$$458$$ −4344.00 −0.443192
$$459$$ 0 0
$$460$$ −675.500 −0.0684681
$$461$$ 19587.8 1.97894 0.989472 0.144725i $$-0.0462299\pi$$
0.989472 + 0.144725i $$0.0462299\pi$$
$$462$$ 0 0
$$463$$ 8632.54 0.866497 0.433249 0.901274i $$-0.357367\pi$$
0.433249 + 0.901274i $$0.357367\pi$$
$$464$$ 141.000 0.0141072
$$465$$ 0 0
$$466$$ 10132.5 1.00725
$$467$$ 5460.00 0.541025 0.270512 0.962716i $$-0.412807\pi$$
0.270512 + 0.962716i $$0.412807\pi$$
$$468$$ 0 0
$$469$$ 10896.0 1.07277
$$470$$ −904.131 −0.0887328
$$471$$ 0 0
$$472$$ 6396.00 0.623728
$$473$$ −1441.07 −0.140085
$$474$$ 0 0
$$475$$ −13946.5 −1.34717
$$476$$ 8106.00 0.780542
$$477$$ 0 0
$$478$$ 9324.00 0.892196
$$479$$ −2553.04 −0.243531 −0.121766 0.992559i $$-0.538856\pi$$
−0.121766 + 0.992559i $$0.538856\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −8517.00 −0.804852
$$483$$ 0 0
$$484$$ 5695.00 0.534842
$$485$$ 348.000 0.0325812
$$486$$ 0 0
$$487$$ −10828.8 −1.00760 −0.503798 0.863822i $$-0.668064\pi$$
−0.503798 + 0.863822i $$0.668064\pi$$
$$488$$ 3264.92 0.302860
$$489$$ 0 0
$$490$$ −453.000 −0.0417642
$$491$$ 11388.0 1.04671 0.523354 0.852116i $$-0.324681\pi$$
0.523354 + 0.852116i $$0.324681\pi$$
$$492$$ 0 0
$$493$$ 16497.0 1.50707
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −155.885 −0.0141117
$$497$$ −14640.0 −1.32132
$$498$$ 0 0
$$499$$ 17677.3 1.58586 0.792931 0.609311i $$-0.208554\pi$$
0.792931 + 0.609311i $$0.208554\pi$$
$$500$$ −2139.08 −0.191325
$$501$$ 0 0
$$502$$ 6890.10 0.612590
$$503$$ −3876.00 −0.343583 −0.171792 0.985133i $$-0.554956\pi$$
−0.171792 + 0.985133i $$0.554956\pi$$
$$504$$ 0 0
$$505$$ −743.050 −0.0654758
$$506$$ 1872.00 0.164467
$$507$$ 0 0
$$508$$ 1430.00 0.124894
$$509$$ 17065.9 1.48612 0.743058 0.669228i $$-0.233375\pi$$
0.743058 + 0.669228i $$0.233375\pi$$
$$510$$ 0 0
$$511$$ −6360.00 −0.550587
$$512$$ −361.999 −0.0312465
$$513$$ 0 0
$$514$$ −3580.15 −0.307225
$$515$$ 315.233 0.0269725
$$516$$ 0 0
$$517$$ −4176.00 −0.355242
$$518$$ 3450.25 0.292655
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2121.00 −0.178355 −0.0891773 0.996016i $$-0.528424\pi$$
−0.0891773 + 0.996016i $$0.528424\pi$$
$$522$$ 0 0
$$523$$ −11464.0 −0.958481 −0.479241 0.877684i $$-0.659088\pi$$
−0.479241 + 0.877684i $$0.659088\pi$$
$$524$$ −9870.00 −0.822849
$$525$$ 0 0
$$526$$ −3554.17 −0.294618
$$527$$ −18238.5 −1.50755
$$528$$ 0 0
$$529$$ −6083.00 −0.499959
$$530$$ −279.000 −0.0228660
$$531$$ 0 0
$$532$$ 7920.00 0.645443
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −2608.47 −0.210792
$$536$$ −17706.0 −1.42683
$$537$$ 0 0
$$538$$ 5767.73 0.462202
$$539$$ −2092.32 −0.167203
$$540$$ 0 0
$$541$$ −4764.87 −0.378665 −0.189333 0.981913i $$-0.560632\pi$$
−0.189333 + 0.981913i $$0.560632\pi$$
$$542$$ 4860.00 0.385157
$$543$$ 0 0
$$544$$ −21278.2 −1.67702
$$545$$ 2688.00 0.211268
$$546$$ 0 0
$$547$$ 6554.00 0.512301 0.256151 0.966637i $$-0.417546\pi$$
0.256151 + 0.966637i $$0.417546\pi$$
$$548$$ −4234.86 −0.330118
$$549$$ 0 0
$$550$$ 2928.00 0.227001
$$551$$ 16118.5 1.24622
$$552$$ 0 0
$$553$$ −17680.8 −1.35961
$$554$$ 652.983 0.0500769
$$555$$ 0 0
$$556$$ −1180.00 −0.0900057
$$557$$ −18112.1 −1.37780 −0.688898 0.724858i $$-0.741905\pi$$
−0.688898 + 0.724858i $$0.741905\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 24.0000 0.00181104
$$561$$ 0 0
$$562$$ −63.0000 −0.00472864
$$563$$ −12168.0 −0.910870 −0.455435 0.890269i $$-0.650516\pi$$
−0.455435 + 0.890269i $$0.650516\pi$$
$$564$$ 0 0
$$565$$ −1189.92 −0.0886022
$$566$$ −12339.1 −0.916347
$$567$$ 0 0
$$568$$ 23790.0 1.75741
$$569$$ −7722.00 −0.568933 −0.284467 0.958686i $$-0.591817\pi$$
−0.284467 + 0.958686i $$0.591817\pi$$
$$570$$ 0 0
$$571$$ −11440.0 −0.838440 −0.419220 0.907885i $$-0.637696\pi$$
−0.419220 + 0.907885i $$0.637696\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −6526.37 −0.474574
$$575$$ 9516.00 0.690165
$$576$$ 0 0
$$577$$ −15444.7 −1.11433 −0.557167 0.830400i $$-0.688112\pi$$
−0.557167 + 0.830400i $$0.688112\pi$$
$$578$$ −15200.5 −1.09387
$$579$$ 0 0
$$580$$ 1221.10 0.0874194
$$581$$ 10944.0 0.781469
$$582$$ 0 0
$$583$$ −1288.65 −0.0915442
$$584$$ 10335.0 0.732304
$$585$$ 0 0
$$586$$ −14415.0 −1.01617
$$587$$ 14071.2 0.989403 0.494702 0.869063i $$-0.335277\pi$$
0.494702 + 0.869063i $$0.335277\pi$$
$$588$$ 0 0
$$589$$ −17820.0 −1.24662
$$590$$ 852.169 0.0594631
$$591$$ 0 0
$$592$$ 143.760 0.00998059
$$593$$ −26938.6 −1.86549 −0.932745 0.360538i $$-0.882593\pi$$
−0.932745 + 0.360538i $$0.882593\pi$$
$$594$$ 0 0
$$595$$ 2808.00 0.193474
$$596$$ 233.827 0.0160703
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 10554.0 0.719908 0.359954 0.932970i $$-0.382792\pi$$
0.359954 + 0.932970i $$0.382792\pi$$
$$600$$ 0 0
$$601$$ −14831.0 −1.00660 −0.503302 0.864111i $$-0.667882\pi$$
−0.503302 + 0.864111i $$0.667882\pi$$
$$602$$ −2496.00 −0.168986
$$603$$ 0 0
$$604$$ 8850.78 0.596247
$$605$$ 1972.81 0.132572
$$606$$ 0 0
$$607$$ −7954.00 −0.531866 −0.265933 0.963991i $$-0.585680\pi$$
−0.265933 + 0.963991i $$0.585680\pi$$
$$608$$ −20790.0 −1.38675
$$609$$ 0 0
$$610$$ 435.000 0.0288732
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 25220.4 1.66173 0.830866 0.556472i $$-0.187845\pi$$
0.830866 + 0.556472i $$0.187845\pi$$
$$614$$ 3846.00 0.252788
$$615$$ 0 0
$$616$$ −4323.20 −0.282771
$$617$$ −17384.6 −1.13432 −0.567162 0.823607i $$-0.691959\pi$$
−0.567162 + 0.823607i $$0.691959\pi$$
$$618$$ 0 0
$$619$$ −8209.92 −0.533093 −0.266547 0.963822i $$-0.585883\pi$$
−0.266547 + 0.963822i $$0.585883\pi$$
$$620$$ −1350.00 −0.0874473
$$621$$ 0 0
$$622$$ −8511.30 −0.548669
$$623$$ 13536.0 0.870479
$$624$$ 0 0
$$625$$ 14509.0 0.928576
$$626$$ 897.202 0.0572834
$$627$$ 0 0
$$628$$ −6055.00 −0.384747
$$629$$ 16819.9 1.06622
$$630$$ 0 0
$$631$$ 12865.7 0.811687 0.405843 0.913943i $$-0.366978\pi$$
0.405843 + 0.913943i $$0.366978\pi$$
$$632$$ 28731.3 1.80834
$$633$$ 0 0
$$634$$ −6783.00 −0.424901
$$635$$ 495.367 0.0309575
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −3384.00 −0.209990
$$639$$ 0 0
$$640$$ −1599.00 −0.0987594
$$641$$ 6201.00 0.382098 0.191049 0.981581i $$-0.438811\pi$$
0.191049 + 0.981581i $$0.438811\pi$$
$$642$$ 0 0
$$643$$ 16821.7 1.03170 0.515849 0.856679i $$-0.327476\pi$$
0.515849 + 0.856679i $$0.327476\pi$$
$$644$$ −5404.00 −0.330664
$$645$$ 0 0
$$646$$ −23166.0 −1.41092
$$647$$ −13494.0 −0.819944 −0.409972 0.912098i $$-0.634462\pi$$
−0.409972 + 0.912098i $$0.634462\pi$$
$$648$$ 0 0
$$649$$ 3936.00 0.238061
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5022.95 0.301708
$$653$$ 11334.0 0.679225 0.339612 0.940566i $$-0.389704\pi$$
0.339612 + 0.940566i $$0.389704\pi$$
$$654$$ 0 0
$$655$$ −3419.07 −0.203960
$$656$$ −271.932 −0.0161847
$$657$$ 0 0
$$658$$ −7233.04 −0.428531
$$659$$ −13236.0 −0.782400 −0.391200 0.920306i $$-0.627940\pi$$
−0.391200 + 0.920306i $$0.627940\pi$$
$$660$$ 0 0
$$661$$ 11852.4 0.697437 0.348718 0.937228i $$-0.386617\pi$$
0.348718 + 0.937228i $$0.386617\pi$$
$$662$$ −12912.0 −0.758065
$$663$$ 0 0
$$664$$ −17784.0 −1.03939
$$665$$ 2743.57 0.159986
$$666$$ 0 0
$$667$$ −10998.0 −0.638447
$$668$$ −4572.61 −0.264850
$$669$$ 0 0
$$670$$ −2359.05 −0.136027
$$671$$ 2009.18 0.115594
$$672$$ 0 0
$$673$$ −8021.00 −0.459416 −0.229708 0.973260i $$-0.573777\pi$$
−0.229708 + 0.973260i $$0.573777\pi$$
$$674$$ −6192.08 −0.353873
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 21630.0 1.22793 0.613965 0.789333i $$-0.289574\pi$$
0.613965 + 0.789333i $$0.289574\pi$$
$$678$$ 0 0
$$679$$ 2784.00 0.157349
$$680$$ −4563.00 −0.257328
$$681$$ 0 0
$$682$$ 3741.23 0.210057
$$683$$ 26538.5 1.48677 0.743387 0.668861i $$-0.233218\pi$$
0.743387 + 0.668861i $$0.233218\pi$$
$$684$$ 0 0
$$685$$ −1467.00 −0.0818266
$$686$$ −11856.0 −0.659860
$$687$$ 0 0
$$688$$ −104.000 −0.00576303
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −831.384 −0.0457704 −0.0228852 0.999738i $$-0.507285\pi$$
−0.0228852 + 0.999738i $$0.507285\pi$$
$$692$$ −12870.0 −0.707000
$$693$$ 0 0
$$694$$ −12065.5 −0.659941
$$695$$ −408.764 −0.0223098
$$696$$ 0 0
$$697$$ −31816.0 −1.72901
$$698$$ −11520.0 −0.624697
$$699$$ 0 0
$$700$$ −8452.41 −0.456387
$$701$$ 30186.0 1.62640 0.813202 0.581981i $$-0.197722\pi$$
0.813202 + 0.581981i $$0.197722\pi$$
$$702$$ 0 0
$$703$$ 16434.0 0.881679
$$704$$ 4253.92 0.227735
$$705$$ 0 0
$$706$$ 9753.00 0.519914
$$707$$ −5944.40 −0.316212
$$708$$ 0 0
$$709$$ −11880.1 −0.629292 −0.314646 0.949209i $$-0.601886\pi$$
−0.314646 + 0.949209i $$0.601886\pi$$
$$710$$ 3169.65 0.167542
$$711$$ 0 0
$$712$$ −21996.0 −1.15777
$$713$$ 12159.0 0.638651
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −18720.0 −0.977094
$$717$$ 0 0
$$718$$ −12348.0 −0.641815
$$719$$ 18408.0 0.954802 0.477401 0.878686i $$-0.341579\pi$$
0.477401 + 0.878686i $$0.341579\pi$$
$$720$$ 0 0
$$721$$ 2521.87 0.130262
$$722$$ −10754.3 −0.554340
$$723$$ 0 0
$$724$$ −3185.00 −0.163494
$$725$$ −17202.0 −0.881195
$$726$$ 0 0
$$727$$ −21112.0 −1.07703 −0.538515 0.842616i $$-0.681014\pi$$
−0.538515 + 0.842616i $$0.681014\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1376.98 0.0698142
$$731$$ −12168.0 −0.615663
$$732$$ 0 0
$$733$$ 23959.5 1.20732 0.603658 0.797243i $$-0.293709\pi$$
0.603658 + 0.797243i $$0.293709\pi$$
$$734$$ −3.46410 −0.000174199 0
$$735$$ 0 0
$$736$$ 14185.5 0.710441
$$737$$ −10896.0 −0.544585
$$738$$ 0 0
$$739$$ −3166.19 −0.157605 −0.0788025 0.996890i $$-0.525110\pi$$
−0.0788025 + 0.996890i $$0.525110\pi$$
$$740$$ 1245.00 0.0618474
$$741$$ 0 0
$$742$$ −2232.00 −0.110430
$$743$$ 30103.0 1.48637 0.743185 0.669086i $$-0.233314\pi$$
0.743185 + 0.669086i $$0.233314\pi$$
$$744$$ 0 0
$$745$$ 81.0000 0.00398337
$$746$$ −6060.45 −0.297438
$$747$$ 0 0
$$748$$ −8106.00 −0.396236
$$749$$ −20867.7 −1.01801
$$750$$ 0 0
$$751$$ −28496.0 −1.38460 −0.692299 0.721610i $$-0.743402\pi$$
−0.692299 + 0.721610i $$0.743402\pi$$
$$752$$ −301.377 −0.0146145
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 3066.00 0.147792
$$756$$ 0 0
$$757$$ 17422.0 0.836477 0.418239 0.908337i $$-0.362648\pi$$
0.418239 + 0.908337i $$0.362648\pi$$
$$758$$ −9558.00 −0.457998
$$759$$ 0 0
$$760$$ −4458.30 −0.212789
$$761$$ −41326.7 −1.96858 −0.984292 0.176547i $$-0.943507\pi$$
−0.984292 + 0.176547i $$0.943507\pi$$
$$762$$ 0 0
$$763$$ 21504.0 1.02031
$$764$$ −12990.0 −0.615133
$$765$$ 0 0
$$766$$ 12756.0 0.601688
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −14071.2 −0.659844 −0.329922 0.944008i $$-0.607022\pi$$
−0.329922 + 0.944008i $$0.607022\pi$$
$$770$$ −576.000 −0.0269579
$$771$$ 0 0
$$772$$ 5585.86 0.260414
$$773$$ 200.918 0.00934866 0.00467433 0.999989i $$-0.498512\pi$$
0.00467433 + 0.999989i $$0.498512\pi$$
$$774$$ 0 0
$$775$$ 19017.9 0.881476
$$776$$ −4524.00 −0.209281
$$777$$ 0 0
$$778$$ 2094.05 0.0964978
$$779$$ −31086.0 −1.42975
$$780$$ 0 0
$$781$$ 14640.0 0.670756
$$782$$ 15806.7 0.722821
$$783$$ 0 0
$$784$$ −151.000 −0.00687864
$$785$$ −2097.51 −0.0953675
$$786$$ 0 0
$$787$$ −6903.95 −0.312706 −0.156353 0.987701i $$-0.549974\pi$$
−0.156353 + 0.987701i $$0.549974\pi$$
$$788$$ 10253.7 0.463546
$$789$$ 0 0
$$790$$ 3828.00 0.172398
$$791$$ −9519.35 −0.427900
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −20256.0 −0.905363
$$795$$ 0 0
$$796$$ −12610.0 −0.561494
$$797$$ −31278.0 −1.39012 −0.695059 0.718953i $$-0.744622\pi$$
−0.695059 + 0.718953i $$0.744622\pi$$
$$798$$ 0 0
$$799$$ −35261.1 −1.56126
$$800$$ 22187.6 0.980561
$$801$$ 0 0
$$802$$ 5163.00 0.227322
$$803$$ 6360.00 0.279501
$$804$$ 0 0
$$805$$ −1872.00 −0.0819619
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 9659.65 0.420576
$$809$$ −8049.00 −0.349799 −0.174900 0.984586i $$-0.555960\pi$$
−0.174900 + 0.984586i $$0.555960\pi$$
$$810$$ 0 0
$$811$$ −14026.1 −0.607305 −0.303653 0.952783i $$-0.598206\pi$$
−0.303653 + 0.952783i $$0.598206\pi$$
$$812$$ 9768.77 0.422188
$$813$$ 0 0
$$814$$ −3450.25 −0.148564
$$815$$ 1740.00 0.0747847
$$816$$ 0 0
$$817$$ −11888.8 −0.509102
$$818$$ 75.0000 0.00320576
$$819$$ 0 0
$$820$$ −2355.00 −0.100293
$$821$$ 8036.72 0.341636 0.170818 0.985303i $$-0.445359\pi$$
0.170818 + 0.985303i $$0.445359\pi$$
$$822$$ 0 0
$$823$$ 40300.0 1.70689 0.853445 0.521184i $$-0.174509\pi$$
0.853445 + 0.521184i $$0.174509\pi$$
$$824$$ −4098.03 −0.173255
$$825$$ 0 0
$$826$$ 6817.35 0.287174
$$827$$ −39525.4 −1.66195 −0.830975 0.556310i $$-0.812217\pi$$
−0.830975 + 0.556310i $$0.812217\pi$$
$$828$$ 0 0
$$829$$ 12311.0 0.515776 0.257888 0.966175i $$-0.416973\pi$$
0.257888 + 0.966175i $$0.416973\pi$$
$$830$$ −2369.45 −0.0990899
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −17667.0 −0.734844
$$834$$ 0 0
$$835$$ −1584.00 −0.0656486
$$836$$ −7920.00 −0.327654
$$837$$ 0 0
$$838$$ −16388.7 −0.675581
$$839$$ −21467.0 −0.883343 −0.441671 0.897177i $$-0.645614\pi$$
−0.441671 + 0.897177i $$0.645614\pi$$
$$840$$ 0 0
$$841$$ −4508.00 −0.184837
$$842$$ 12243.0 0.501095
$$843$$ 0 0
$$844$$ −5210.00 −0.212483
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15782.4 0.640249
$$848$$ −93.0000 −0.00376608
$$849$$ 0 0
$$850$$ 24723.3 0.997649
$$851$$ −11213.3 −0.451688
$$852$$ 0 0
$$853$$ 774.227 0.0310774 0.0155387 0.999879i $$-0.495054\pi$$
0.0155387 + 0.999879i $$0.495054\pi$$
$$854$$ 3480.00 0.139442
$$855$$ 0 0
$$856$$ 33910.1 1.35400
$$857$$ 13923.0 0.554960 0.277480 0.960731i $$-0.410501\pi$$
0.277480 + 0.960731i $$0.410501\pi$$
$$858$$ 0 0
$$859$$ −22358.0 −0.888062 −0.444031 0.896011i $$-0.646452\pi$$
−0.444031 + 0.896011i $$0.646452\pi$$
$$860$$ −900.666 −0.0357122
$$861$$ 0 0
$$862$$ −17196.0 −0.679464
$$863$$ −2230.88 −0.0879955 −0.0439977 0.999032i $$-0.514009\pi$$
−0.0439977 + 0.999032i $$0.514009\pi$$
$$864$$ 0 0
$$865$$ −4458.30 −0.175245
$$866$$ −11461.0 −0.449723
$$867$$ 0 0
$$868$$ −10800.0 −0.422322
$$869$$ 17680.8 0.690195
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −34944.0 −1.35706
$$873$$ 0 0
$$874$$ 15444.0 0.597713
$$875$$ −5928.00 −0.229032
$$876$$ 0 0
$$877$$ 16754.1 0.645093 0.322547 0.946554i $$-0.395461\pi$$
0.322547 + 0.946554i $$0.395461\pi$$
$$878$$ 24227.9 0.931268
$$879$$ 0 0
$$880$$ −24.0000 −0.000919363 0
$$881$$ 17355.0 0.663683 0.331842 0.943335i $$-0.392330\pi$$
0.331842 + 0.943335i $$0.392330\pi$$
$$882$$ 0 0
$$883$$ 46982.0 1.79057 0.895283 0.445497i $$-0.146973\pi$$
0.895283 + 0.445497i $$0.146973\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 3471.03 0.131616
$$887$$ 8916.00 0.337508 0.168754 0.985658i $$-0.446026\pi$$
0.168754 + 0.985658i $$0.446026\pi$$
$$888$$ 0 0
$$889$$ 3962.93 0.149508
$$890$$ −2930.63 −0.110376
$$891$$ 0 0
$$892$$ 12037.8 0.451854
$$893$$ −34452.0 −1.29103
$$894$$ 0 0
$$895$$ −6484.80 −0.242193
$$896$$ −12792.0 −0.476954
$$897$$ 0 0
$$898$$ 15732.0 0.584614
$$899$$ −21979.7 −0.815423
$$900$$ 0 0
$$901$$ −10881.0 −0.402329
$$902$$ 6526.37 0.240914
$$903$$ 0 0
$$904$$ 15468.9 0.569126
$$905$$ −1103.32 −0.0405254
$$906$$ 0 0
$$907$$ 30836.0 1.12888 0.564439 0.825475i $$-0.309092\pi$$
0.564439 + 0.825475i $$0.309092\pi$$
$$908$$ −12037.8 −0.439964
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 27480.0 0.999400 0.499700 0.866199i $$-0.333444\pi$$
0.499700 + 0.866199i $$0.333444\pi$$
$$912$$ 0 0
$$913$$ −10944.0 −0.396707
$$914$$ −4371.00 −0.158184
$$915$$ 0 0
$$916$$ −12540.0 −0.452331
$$917$$ −27352.5 −0.985017
$$918$$ 0 0
$$919$$ −28442.0 −1.02091 −0.510454 0.859905i $$-0.670523\pi$$
−0.510454 + 0.859905i $$0.670523\pi$$
$$920$$ 3042.00 0.109013
$$921$$ 0 0
$$922$$ −33927.0 −1.21185
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −17538.7 −0.623427
$$926$$ −14952.0 −0.530619
$$927$$ 0 0
$$928$$ −25643.0 −0.907083
$$929$$ 6978.43 0.246453 0.123227 0.992379i $$-0.460676\pi$$
0.123227 + 0.992379i $$0.460676\pi$$
$$930$$ 0 0
$$931$$ −17261.6 −0.607655
$$932$$ 29250.0 1.02802
$$933$$ 0 0
$$934$$ −9457.00 −0.331309
$$935$$ −2808.00 −0.0982154
$$936$$ 0 0
$$937$$ −38465.0 −1.34109 −0.670543 0.741871i $$-0.733939\pi$$
−0.670543 + 0.741871i $$0.733939\pi$$
$$938$$ −18872.4 −0.656937
$$939$$ 0 0
$$940$$ −2610.00 −0.0905626
$$941$$ 4884.38 0.169210 0.0846049 0.996415i $$-0.473037\pi$$
0.0846049 + 0.996415i $$0.473037\pi$$
$$942$$ 0 0
$$943$$ 21210.7 0.732466
$$944$$ 284.056 0.00979369
$$945$$ 0 0
$$946$$ 2496.00 0.0857843
$$947$$ 21765.0 0.746849 0.373424 0.927661i $$-0.378184\pi$$
0.373424 + 0.927661i $$0.378184\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 24156.0 0.824973
$$951$$ 0 0
$$952$$ −36504.0 −1.24275
$$953$$ 6474.00 0.220056 0.110028 0.993928i $$-0.464906\pi$$
0.110028 + 0.993928i $$0.464906\pi$$
$$954$$ 0 0
$$955$$ −4499.87 −0.152474
$$956$$ 26916.1 0.910594
$$957$$ 0 0
$$958$$ 4422.00 0.149132
$$959$$ −11736.0 −0.395177
$$960$$ 0 0
$$961$$ −5491.00 −0.184317
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −24586.5 −0.821449
$$965$$ 1935.00 0.0645491
$$966$$ 0 0
$$967$$ −7541.35 −0.250789 −0.125395 0.992107i $$-0.540020\pi$$
−0.125395 + 0.992107i $$0.540020\pi$$
$$968$$ −25646.5 −0.851559
$$969$$ 0 0
$$970$$ −602.754 −0.0199518
$$971$$ −34998.0 −1.15668 −0.578342 0.815795i $$-0.696300\pi$$
−0.578342 + 0.815795i $$0.696300\pi$$
$$972$$ 0 0
$$973$$ −3270.11 −0.107744
$$974$$ 18756.0 0.617024
$$975$$ 0 0
$$976$$ 145.000 0.00475547
$$977$$ 25216.9 0.825753 0.412877 0.910787i $$-0.364524\pi$$
0.412877 + 0.910787i $$0.364524\pi$$
$$978$$ 0 0
$$979$$ −13536.0 −0.441892
$$980$$ −1307.70 −0.0426254
$$981$$ 0 0
$$982$$ −19724.6 −0.640975
$$983$$ 56440.6 1.83131 0.915654 0.401967i $$-0.131673\pi$$
0.915654 + 0.401967i $$0.131673\pi$$
$$984$$ 0 0
$$985$$ 3552.00 0.114900
$$986$$ −28573.6 −0.922891
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8112.00 0.260816
$$990$$ 0 0
$$991$$ 59282.0 1.90026 0.950129 0.311859i $$-0.100951\pi$$
0.950129 + 0.311859i $$0.100951\pi$$
$$992$$ 28350.0 0.907372
$$993$$ 0 0
$$994$$ 25357.2 0.809137
$$995$$ −4368.23 −0.139178
$$996$$ 0 0
$$997$$ −37711.0 −1.19791 −0.598957 0.800782i $$-0.704418\pi$$
−0.598957 + 0.800782i $$0.704418\pi$$
$$998$$ −30618.0 −0.971138
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.4.a.o.1.1 2
3.2 odd 2 169.4.a.i.1.2 2
13.2 odd 12 117.4.q.a.82.1 2
13.7 odd 12 117.4.q.a.10.1 2
13.12 even 2 inner 1521.4.a.o.1.2 2
39.2 even 12 13.4.e.b.4.1 2
39.5 even 4 169.4.b.d.168.1 2
39.8 even 4 169.4.b.d.168.2 2
39.11 even 12 169.4.e.a.147.1 2
39.17 odd 6 169.4.c.h.146.2 4
39.20 even 12 13.4.e.b.10.1 yes 2
39.23 odd 6 169.4.c.h.22.2 4
39.29 odd 6 169.4.c.h.22.1 4
39.32 even 12 169.4.e.a.23.1 2
39.35 odd 6 169.4.c.h.146.1 4
39.38 odd 2 169.4.a.i.1.1 2
156.59 odd 12 208.4.w.b.49.1 2
156.119 odd 12 208.4.w.b.17.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.b.4.1 2 39.2 even 12
13.4.e.b.10.1 yes 2 39.20 even 12
117.4.q.a.10.1 2 13.7 odd 12
117.4.q.a.82.1 2 13.2 odd 12
169.4.a.i.1.1 2 39.38 odd 2
169.4.a.i.1.2 2 3.2 odd 2
169.4.b.d.168.1 2 39.5 even 4
169.4.b.d.168.2 2 39.8 even 4
169.4.c.h.22.1 4 39.29 odd 6
169.4.c.h.22.2 4 39.23 odd 6
169.4.c.h.146.1 4 39.35 odd 6
169.4.c.h.146.2 4 39.17 odd 6
169.4.e.a.23.1 2 39.32 even 12
169.4.e.a.147.1 2 39.11 even 12
208.4.w.b.17.1 2 156.119 odd 12
208.4.w.b.49.1 2 156.59 odd 12
1521.4.a.o.1.1 2 1.1 even 1 trivial
1521.4.a.o.1.2 2 13.12 even 2 inner