# Properties

 Label 2205.4.a.bz.1.6 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.1163891200.1 Defining polynomial: $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ x^6 - 2*x^5 - 23*x^4 + 12*x^3 + 154*x^2 + 152*x + 28 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2\cdot 7$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$4.10376$$ of defining polynomial Character $$\chi$$ $$=$$ 2205.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.51797 q^{2} +22.4480 q^{4} -5.00000 q^{5} +79.7239 q^{8} +O(q^{10})$$ $$q+5.51797 q^{2} +22.4480 q^{4} -5.00000 q^{5} +79.7239 q^{8} -27.5899 q^{10} -34.5211 q^{11} -68.8935 q^{13} +260.330 q^{16} -91.4346 q^{17} -11.8278 q^{19} -112.240 q^{20} -190.486 q^{22} +0.104165 q^{23} +25.0000 q^{25} -380.152 q^{26} -190.863 q^{29} -159.802 q^{31} +798.703 q^{32} -504.534 q^{34} -177.908 q^{37} -65.2657 q^{38} -398.619 q^{40} +145.247 q^{41} +8.25729 q^{43} -774.930 q^{44} +0.574782 q^{46} +260.529 q^{47} +137.949 q^{50} -1546.52 q^{52} -353.107 q^{53} +172.605 q^{55} -1053.18 q^{58} +240.495 q^{59} -778.188 q^{61} -881.783 q^{62} +2324.58 q^{64} +344.467 q^{65} +151.945 q^{67} -2052.53 q^{68} +311.449 q^{71} -639.888 q^{73} -981.690 q^{74} -265.512 q^{76} +391.186 q^{79} -1301.65 q^{80} +801.472 q^{82} +493.205 q^{83} +457.173 q^{85} +45.5635 q^{86} -2752.15 q^{88} +473.850 q^{89} +2.33831 q^{92} +1437.59 q^{94} +59.1392 q^{95} -839.005 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8 $$6 q + 2 q^{2} + 14 q^{4} - 30 q^{5} + 66 q^{8} - 10 q^{10} + 16 q^{11} - 168 q^{13} + 298 q^{16} - 4 q^{17} - 308 q^{19} - 70 q^{20} - 236 q^{22} + 336 q^{23} + 150 q^{25} + 56 q^{26} - 176 q^{29} - 392 q^{31} + 770 q^{32} - 812 q^{34} - 140 q^{37} + 20 q^{38} - 330 q^{40} + 656 q^{41} - 388 q^{43} + 160 q^{44} - 388 q^{46} + 628 q^{47} + 50 q^{50} - 1520 q^{52} + 676 q^{53} - 80 q^{55} - 2012 q^{58} + 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 840 q^{65} + 1768 q^{67} - 2940 q^{68} + 224 q^{71} - 2640 q^{73} - 928 q^{74} + 1340 q^{76} + 1636 q^{79} - 1490 q^{80} + 1756 q^{82} + 140 q^{83} + 20 q^{85} - 1180 q^{86} - 5652 q^{88} - 1904 q^{89} + 1952 q^{92} + 3332 q^{94} + 1540 q^{95} - 516 q^{97}+O(q^{100})$$ 6 * q + 2 * q^2 + 14 * q^4 - 30 * q^5 + 66 * q^8 - 10 * q^10 + 16 * q^11 - 168 * q^13 + 298 * q^16 - 4 * q^17 - 308 * q^19 - 70 * q^20 - 236 * q^22 + 336 * q^23 + 150 * q^25 + 56 * q^26 - 176 * q^29 - 392 * q^31 + 770 * q^32 - 812 * q^34 - 140 * q^37 + 20 * q^38 - 330 * q^40 + 656 * q^41 - 388 * q^43 + 160 * q^44 - 388 * q^46 + 628 * q^47 + 50 * q^50 - 1520 * q^52 + 676 * q^53 - 80 * q^55 - 2012 * q^58 + 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 840 * q^65 + 1768 * q^67 - 2940 * q^68 + 224 * q^71 - 2640 * q^73 - 928 * q^74 + 1340 * q^76 + 1636 * q^79 - 1490 * q^80 + 1756 * q^82 + 140 * q^83 + 20 * q^85 - 1180 * q^86 - 5652 * q^88 - 1904 * q^89 + 1952 * q^92 + 3332 * q^94 + 1540 * q^95 - 516 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.51797 1.95090 0.975449 0.220225i $$-0.0706790\pi$$
0.975449 + 0.220225i $$0.0706790\pi$$
$$3$$ 0 0
$$4$$ 22.4480 2.80600
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 79.7239 3.52333
$$9$$ 0 0
$$10$$ −27.5899 −0.872468
$$11$$ −34.5211 −0.946227 −0.473113 0.881002i $$-0.656870\pi$$
−0.473113 + 0.881002i $$0.656870\pi$$
$$12$$ 0 0
$$13$$ −68.8935 −1.46982 −0.734908 0.678167i $$-0.762775\pi$$
−0.734908 + 0.678167i $$0.762775\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 260.330 4.06766
$$17$$ −91.4346 −1.30448 −0.652239 0.758013i $$-0.726170\pi$$
−0.652239 + 0.758013i $$0.726170\pi$$
$$18$$ 0 0
$$19$$ −11.8278 −0.142815 −0.0714077 0.997447i $$-0.522749\pi$$
−0.0714077 + 0.997447i $$0.522749\pi$$
$$20$$ −112.240 −1.25488
$$21$$ 0 0
$$22$$ −190.486 −1.84599
$$23$$ 0.104165 0.000944348 0 0.000472174 1.00000i $$-0.499850\pi$$
0.000472174 1.00000i $$0.499850\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −380.152 −2.86746
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −190.863 −1.22215 −0.611076 0.791572i $$-0.709263\pi$$
−0.611076 + 0.791572i $$0.709263\pi$$
$$30$$ 0 0
$$31$$ −159.802 −0.925847 −0.462924 0.886398i $$-0.653200\pi$$
−0.462924 + 0.886398i $$0.653200\pi$$
$$32$$ 798.703 4.41225
$$33$$ 0 0
$$34$$ −504.534 −2.54491
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −177.908 −0.790482 −0.395241 0.918577i $$-0.629339\pi$$
−0.395241 + 0.918577i $$0.629339\pi$$
$$38$$ −65.2657 −0.278618
$$39$$ 0 0
$$40$$ −398.619 −1.57568
$$41$$ 145.247 0.553264 0.276632 0.960976i $$-0.410782\pi$$
0.276632 + 0.960976i $$0.410782\pi$$
$$42$$ 0 0
$$43$$ 8.25729 0.0292843 0.0146421 0.999893i $$-0.495339\pi$$
0.0146421 + 0.999893i $$0.495339\pi$$
$$44$$ −774.930 −2.65512
$$45$$ 0 0
$$46$$ 0.574782 0.00184233
$$47$$ 260.529 0.808553 0.404277 0.914637i $$-0.367523\pi$$
0.404277 + 0.914637i $$0.367523\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 137.949 0.390180
$$51$$ 0 0
$$52$$ −1546.52 −4.12431
$$53$$ −353.107 −0.915151 −0.457576 0.889171i $$-0.651282\pi$$
−0.457576 + 0.889171i $$0.651282\pi$$
$$54$$ 0 0
$$55$$ 172.605 0.423165
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −1053.18 −2.38429
$$59$$ 240.495 0.530673 0.265337 0.964156i $$-0.414517\pi$$
0.265337 + 0.964156i $$0.414517\pi$$
$$60$$ 0 0
$$61$$ −778.188 −1.63339 −0.816695 0.577069i $$-0.804196\pi$$
−0.816695 + 0.577069i $$0.804196\pi$$
$$62$$ −881.783 −1.80623
$$63$$ 0 0
$$64$$ 2324.58 4.54020
$$65$$ 344.467 0.657322
$$66$$ 0 0
$$67$$ 151.945 0.277059 0.138530 0.990358i $$-0.455762\pi$$
0.138530 + 0.990358i $$0.455762\pi$$
$$68$$ −2052.53 −3.66037
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 311.449 0.520594 0.260297 0.965529i $$-0.416180\pi$$
0.260297 + 0.965529i $$0.416180\pi$$
$$72$$ 0 0
$$73$$ −639.888 −1.02593 −0.512967 0.858408i $$-0.671454\pi$$
−0.512967 + 0.858408i $$0.671454\pi$$
$$74$$ −981.690 −1.54215
$$75$$ 0 0
$$76$$ −265.512 −0.400740
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 391.186 0.557112 0.278556 0.960420i $$-0.410144\pi$$
0.278556 + 0.960420i $$0.410144\pi$$
$$80$$ −1301.65 −1.81911
$$81$$ 0 0
$$82$$ 801.472 1.07936
$$83$$ 493.205 0.652245 0.326122 0.945328i $$-0.394258\pi$$
0.326122 + 0.945328i $$0.394258\pi$$
$$84$$ 0 0
$$85$$ 457.173 0.583381
$$86$$ 45.5635 0.0571307
$$87$$ 0 0
$$88$$ −2752.15 −3.33387
$$89$$ 473.850 0.564359 0.282180 0.959362i $$-0.408943\pi$$
0.282180 + 0.959362i $$0.408943\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.33831 0.00264984
$$93$$ 0 0
$$94$$ 1437.59 1.57741
$$95$$ 59.1392 0.0638690
$$96$$ 0 0
$$97$$ −839.005 −0.878227 −0.439114 0.898431i $$-0.644707\pi$$
−0.439114 + 0.898431i $$0.644707\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 561.201 0.561201
$$101$$ 887.440 0.874293 0.437146 0.899390i $$-0.355989\pi$$
0.437146 + 0.899390i $$0.355989\pi$$
$$102$$ 0 0
$$103$$ −619.087 −0.592238 −0.296119 0.955151i $$-0.595692\pi$$
−0.296119 + 0.955151i $$0.595692\pi$$
$$104$$ −5492.46 −5.17865
$$105$$ 0 0
$$106$$ −1948.44 −1.78537
$$107$$ 2151.30 1.94368 0.971841 0.235639i $$-0.0757183\pi$$
0.971841 + 0.235639i $$0.0757183\pi$$
$$108$$ 0 0
$$109$$ −407.076 −0.357714 −0.178857 0.983875i $$-0.557240\pi$$
−0.178857 + 0.983875i $$0.557240\pi$$
$$110$$ 952.432 0.825553
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 349.581 0.291025 0.145513 0.989356i $$-0.453517\pi$$
0.145513 + 0.989356i $$0.453517\pi$$
$$114$$ 0 0
$$115$$ −0.520827 −0.000422325 0
$$116$$ −4284.50 −3.42936
$$117$$ 0 0
$$118$$ 1327.04 1.03529
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −139.296 −0.104655
$$122$$ −4294.02 −3.18658
$$123$$ 0 0
$$124$$ −3587.24 −2.59793
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1183.78 0.827114 0.413557 0.910478i $$-0.364286\pi$$
0.413557 + 0.910478i $$0.364286\pi$$
$$128$$ 6437.36 4.44522
$$129$$ 0 0
$$130$$ 1900.76 1.28237
$$131$$ 223.357 0.148968 0.0744840 0.997222i $$-0.476269\pi$$
0.0744840 + 0.997222i $$0.476269\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 838.426 0.540515
$$135$$ 0 0
$$136$$ −7289.52 −4.59611
$$137$$ −2036.66 −1.27010 −0.635050 0.772471i $$-0.719021\pi$$
−0.635050 + 0.772471i $$0.719021\pi$$
$$138$$ 0 0
$$139$$ −2687.00 −1.63963 −0.819815 0.572629i $$-0.805924\pi$$
−0.819815 + 0.572629i $$0.805924\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1718.57 1.01563
$$143$$ 2378.28 1.39078
$$144$$ 0 0
$$145$$ 954.316 0.546563
$$146$$ −3530.88 −2.00149
$$147$$ 0 0
$$148$$ −3993.68 −2.21810
$$149$$ −673.500 −0.370304 −0.185152 0.982710i $$-0.559278\pi$$
−0.185152 + 0.982710i $$0.559278\pi$$
$$150$$ 0 0
$$151$$ −2125.18 −1.14533 −0.572664 0.819790i $$-0.694090\pi$$
−0.572664 + 0.819790i $$0.694090\pi$$
$$152$$ −942.961 −0.503186
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 799.010 0.414052
$$156$$ 0 0
$$157$$ 2813.03 1.42997 0.714983 0.699142i $$-0.246435\pi$$
0.714983 + 0.699142i $$0.246435\pi$$
$$158$$ 2158.55 1.08687
$$159$$ 0 0
$$160$$ −3993.52 −1.97322
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1344.42 −0.646032 −0.323016 0.946394i $$-0.604697\pi$$
−0.323016 + 0.946394i $$0.604697\pi$$
$$164$$ 3260.52 1.55246
$$165$$ 0 0
$$166$$ 2721.49 1.27246
$$167$$ 1451.24 0.672456 0.336228 0.941781i $$-0.390849\pi$$
0.336228 + 0.941781i $$0.390849\pi$$
$$168$$ 0 0
$$169$$ 2549.31 1.16036
$$170$$ 2522.67 1.13812
$$171$$ 0 0
$$172$$ 185.360 0.0821718
$$173$$ −1979.16 −0.869784 −0.434892 0.900483i $$-0.643213\pi$$
−0.434892 + 0.900483i $$0.643213\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −8986.87 −3.84893
$$177$$ 0 0
$$178$$ 2614.69 1.10101
$$179$$ 4358.66 1.82001 0.910005 0.414598i $$-0.136078\pi$$
0.910005 + 0.414598i $$0.136078\pi$$
$$180$$ 0 0
$$181$$ 377.923 0.155198 0.0775988 0.996985i $$-0.475275\pi$$
0.0775988 + 0.996985i $$0.475275\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 8.30447 0.00332725
$$185$$ 889.538 0.353514
$$186$$ 0 0
$$187$$ 3156.42 1.23433
$$188$$ 5848.36 2.26880
$$189$$ 0 0
$$190$$ 326.328 0.124602
$$191$$ 2425.95 0.919033 0.459517 0.888169i $$-0.348023\pi$$
0.459517 + 0.888169i $$0.348023\pi$$
$$192$$ 0 0
$$193$$ −622.923 −0.232326 −0.116163 0.993230i $$-0.537060\pi$$
−0.116163 + 0.993230i $$0.537060\pi$$
$$194$$ −4629.61 −1.71333
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2842.29 −1.02794 −0.513971 0.857807i $$-0.671826\pi$$
−0.513971 + 0.857807i $$0.671826\pi$$
$$198$$ 0 0
$$199$$ 867.364 0.308974 0.154487 0.987995i $$-0.450628\pi$$
0.154487 + 0.987995i $$0.450628\pi$$
$$200$$ 1993.10 0.704666
$$201$$ 0 0
$$202$$ 4896.87 1.70566
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −726.237 −0.247427
$$206$$ −3416.11 −1.15540
$$207$$ 0 0
$$208$$ −17935.0 −5.97871
$$209$$ 408.309 0.135136
$$210$$ 0 0
$$211$$ −5975.92 −1.94976 −0.974880 0.222730i $$-0.928503\pi$$
−0.974880 + 0.222730i $$0.928503\pi$$
$$212$$ −7926.57 −2.56792
$$213$$ 0 0
$$214$$ 11870.8 3.79192
$$215$$ −41.2864 −0.0130963
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −2246.24 −0.697864
$$219$$ 0 0
$$220$$ 3874.65 1.18740
$$221$$ 6299.25 1.91734
$$222$$ 0 0
$$223$$ 5181.58 1.55598 0.777992 0.628274i $$-0.216238\pi$$
0.777992 + 0.628274i $$0.216238\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 1928.98 0.567761
$$227$$ 3753.06 1.09735 0.548677 0.836035i $$-0.315132\pi$$
0.548677 + 0.836035i $$0.315132\pi$$
$$228$$ 0 0
$$229$$ 6258.14 1.80589 0.902947 0.429752i $$-0.141399\pi$$
0.902947 + 0.429752i $$0.141399\pi$$
$$230$$ −2.87391 −0.000823913 0
$$231$$ 0 0
$$232$$ −15216.4 −4.30605
$$233$$ −1779.96 −0.500469 −0.250234 0.968185i $$-0.580508\pi$$
−0.250234 + 0.968185i $$0.580508\pi$$
$$234$$ 0 0
$$235$$ −1302.64 −0.361596
$$236$$ 5398.63 1.48907
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −3519.46 −0.952532 −0.476266 0.879301i $$-0.658010\pi$$
−0.476266 + 0.879301i $$0.658010\pi$$
$$240$$ 0 0
$$241$$ −362.930 −0.0970058 −0.0485029 0.998823i $$-0.515445\pi$$
−0.0485029 + 0.998823i $$0.515445\pi$$
$$242$$ −768.629 −0.204171
$$243$$ 0 0
$$244$$ −17468.8 −4.58330
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 814.861 0.209912
$$248$$ −12740.0 −3.26207
$$249$$ 0 0
$$250$$ −689.747 −0.174494
$$251$$ −5333.85 −1.34131 −0.670656 0.741768i $$-0.733988\pi$$
−0.670656 + 0.741768i $$0.733988\pi$$
$$252$$ 0 0
$$253$$ −3.59590 −0.000893567 0
$$254$$ 6532.06 1.61361
$$255$$ 0 0
$$256$$ 16924.5 4.13197
$$257$$ −2438.78 −0.591933 −0.295966 0.955198i $$-0.595642\pi$$
−0.295966 + 0.955198i $$0.595642\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 7732.62 1.84445
$$261$$ 0 0
$$262$$ 1232.48 0.290621
$$263$$ −1526.25 −0.357843 −0.178921 0.983863i $$-0.557261\pi$$
−0.178921 + 0.983863i $$0.557261\pi$$
$$264$$ 0 0
$$265$$ 1765.54 0.409268
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 3410.86 0.777430
$$269$$ −7564.12 −1.71447 −0.857235 0.514925i $$-0.827820\pi$$
−0.857235 + 0.514925i $$0.827820\pi$$
$$270$$ 0 0
$$271$$ −4282.68 −0.959980 −0.479990 0.877274i $$-0.659360\pi$$
−0.479990 + 0.877274i $$0.659360\pi$$
$$272$$ −23803.2 −5.30617
$$273$$ 0 0
$$274$$ −11238.2 −2.47784
$$275$$ −863.027 −0.189245
$$276$$ 0 0
$$277$$ −4008.41 −0.869465 −0.434732 0.900560i $$-0.643157\pi$$
−0.434732 + 0.900560i $$0.643157\pi$$
$$278$$ −14826.8 −3.19875
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6935.44 −1.47236 −0.736181 0.676785i $$-0.763373\pi$$
−0.736181 + 0.676785i $$0.763373\pi$$
$$282$$ 0 0
$$283$$ 2666.64 0.560124 0.280062 0.959982i $$-0.409645\pi$$
0.280062 + 0.959982i $$0.409645\pi$$
$$284$$ 6991.41 1.46079
$$285$$ 0 0
$$286$$ 13123.3 2.71327
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 3447.28 0.701665
$$290$$ 5265.89 1.06629
$$291$$ 0 0
$$292$$ −14364.2 −2.87878
$$293$$ 5939.10 1.18418 0.592092 0.805870i $$-0.298302\pi$$
0.592092 + 0.805870i $$0.298302\pi$$
$$294$$ 0 0
$$295$$ −1202.47 −0.237324
$$296$$ −14183.5 −2.78513
$$297$$ 0 0
$$298$$ −3716.36 −0.722425
$$299$$ −7.17632 −0.00138802
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −11726.7 −2.23442
$$303$$ 0 0
$$304$$ −3079.14 −0.580924
$$305$$ 3890.94 0.730474
$$306$$ 0 0
$$307$$ −10381.5 −1.92998 −0.964992 0.262278i $$-0.915526\pi$$
−0.964992 + 0.262278i $$0.915526\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4408.91 0.807772
$$311$$ 4240.54 0.773181 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$312$$ 0 0
$$313$$ 283.903 0.0512688 0.0256344 0.999671i $$-0.491839\pi$$
0.0256344 + 0.999671i $$0.491839\pi$$
$$314$$ 15522.2 2.78972
$$315$$ 0 0
$$316$$ 8781.36 1.56326
$$317$$ −1739.49 −0.308201 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$318$$ 0 0
$$319$$ 6588.80 1.15643
$$320$$ −11622.9 −2.03044
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1081.47 0.186300
$$324$$ 0 0
$$325$$ −1722.34 −0.293963
$$326$$ −7418.48 −1.26034
$$327$$ 0 0
$$328$$ 11579.7 1.94933
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5606.96 0.931077 0.465538 0.885028i $$-0.345861\pi$$
0.465538 + 0.885028i $$0.345861\pi$$
$$332$$ 11071.5 1.83020
$$333$$ 0 0
$$334$$ 8007.89 1.31189
$$335$$ −759.723 −0.123905
$$336$$ 0 0
$$337$$ −9427.44 −1.52387 −0.761937 0.647652i $$-0.775751\pi$$
−0.761937 + 0.647652i $$0.775751\pi$$
$$338$$ 14067.0 2.26375
$$339$$ 0 0
$$340$$ 10262.6 1.63697
$$341$$ 5516.53 0.876062
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 658.303 0.103178
$$345$$ 0 0
$$346$$ −10920.9 −1.69686
$$347$$ −11634.2 −1.79988 −0.899939 0.436017i $$-0.856389\pi$$
−0.899939 + 0.436017i $$0.856389\pi$$
$$348$$ 0 0
$$349$$ 1317.10 0.202013 0.101006 0.994886i $$-0.467794\pi$$
0.101006 + 0.994886i $$0.467794\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −27572.1 −4.17499
$$353$$ 5848.82 0.881874 0.440937 0.897538i $$-0.354646\pi$$
0.440937 + 0.897538i $$0.354646\pi$$
$$354$$ 0 0
$$355$$ −1557.24 −0.232817
$$356$$ 10637.0 1.58359
$$357$$ 0 0
$$358$$ 24051.0 3.55065
$$359$$ 12422.0 1.82621 0.913104 0.407727i $$-0.133679\pi$$
0.913104 + 0.407727i $$0.133679\pi$$
$$360$$ 0 0
$$361$$ −6719.10 −0.979604
$$362$$ 2085.37 0.302775
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3199.44 0.458812
$$366$$ 0 0
$$367$$ 6590.78 0.937427 0.468713 0.883350i $$-0.344718\pi$$
0.468713 + 0.883350i $$0.344718\pi$$
$$368$$ 27.1174 0.00384128
$$369$$ 0 0
$$370$$ 4908.45 0.689670
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 344.278 0.0477910 0.0238955 0.999714i $$-0.492393\pi$$
0.0238955 + 0.999714i $$0.492393\pi$$
$$374$$ 17417.0 2.40806
$$375$$ 0 0
$$376$$ 20770.4 2.84880
$$377$$ 13149.2 1.79634
$$378$$ 0 0
$$379$$ 5241.23 0.710353 0.355177 0.934799i $$-0.384421\pi$$
0.355177 + 0.934799i $$0.384421\pi$$
$$380$$ 1327.56 0.179217
$$381$$ 0 0
$$382$$ 13386.3 1.79294
$$383$$ 7597.37 1.01360 0.506798 0.862065i $$-0.330829\pi$$
0.506798 + 0.862065i $$0.330829\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3437.27 −0.453245
$$387$$ 0 0
$$388$$ −18834.0 −2.46431
$$389$$ −3101.84 −0.404291 −0.202146 0.979355i $$-0.564791\pi$$
−0.202146 + 0.979355i $$0.564791\pi$$
$$390$$ 0 0
$$391$$ −9.52432 −0.00123188
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −15683.7 −2.00541
$$395$$ −1955.93 −0.249148
$$396$$ 0 0
$$397$$ 2932.06 0.370669 0.185335 0.982675i $$-0.440663\pi$$
0.185335 + 0.982675i $$0.440663\pi$$
$$398$$ 4786.09 0.602777
$$399$$ 0 0
$$400$$ 6508.25 0.813531
$$401$$ −89.2375 −0.0111130 −0.00555649 0.999985i $$-0.501769\pi$$
−0.00555649 + 0.999985i $$0.501769\pi$$
$$402$$ 0 0
$$403$$ 11009.3 1.36083
$$404$$ 19921.3 2.45327
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6141.56 0.747975
$$408$$ 0 0
$$409$$ −147.388 −0.0178188 −0.00890940 0.999960i $$-0.502836\pi$$
−0.00890940 + 0.999960i $$0.502836\pi$$
$$410$$ −4007.36 −0.482706
$$411$$ 0 0
$$412$$ −13897.3 −1.66182
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2466.03 −0.291693
$$416$$ −55025.4 −6.48520
$$417$$ 0 0
$$418$$ 2253.04 0.263636
$$419$$ −3781.67 −0.440923 −0.220462 0.975396i $$-0.570756\pi$$
−0.220462 + 0.975396i $$0.570756\pi$$
$$420$$ 0 0
$$421$$ −10899.2 −1.26175 −0.630874 0.775885i $$-0.717304\pi$$
−0.630874 + 0.775885i $$0.717304\pi$$
$$422$$ −32975.0 −3.80378
$$423$$ 0 0
$$424$$ −28151.1 −3.22438
$$425$$ −2285.86 −0.260896
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 48292.4 5.45398
$$429$$ 0 0
$$430$$ −227.817 −0.0255496
$$431$$ 11283.4 1.26102 0.630512 0.776180i $$-0.282845\pi$$
0.630512 + 0.776180i $$0.282845\pi$$
$$432$$ 0 0
$$433$$ 8906.19 0.988462 0.494231 0.869331i $$-0.335450\pi$$
0.494231 + 0.869331i $$0.335450\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −9138.07 −1.00375
$$437$$ −1.23205 −0.000134867 0
$$438$$ 0 0
$$439$$ −8149.49 −0.886000 −0.443000 0.896522i $$-0.646086\pi$$
−0.443000 + 0.896522i $$0.646086\pi$$
$$440$$ 13760.8 1.49095
$$441$$ 0 0
$$442$$ 34759.1 3.74054
$$443$$ 11472.8 1.23045 0.615223 0.788353i $$-0.289066\pi$$
0.615223 + 0.788353i $$0.289066\pi$$
$$444$$ 0 0
$$445$$ −2369.25 −0.252389
$$446$$ 28591.8 3.03557
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1963.80 −0.206409 −0.103204 0.994660i $$-0.532910\pi$$
−0.103204 + 0.994660i $$0.532910\pi$$
$$450$$ 0 0
$$451$$ −5014.10 −0.523514
$$452$$ 7847.42 0.816618
$$453$$ 0 0
$$454$$ 20709.3 2.14083
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5589.85 0.572171 0.286086 0.958204i $$-0.407646\pi$$
0.286086 + 0.958204i $$0.407646\pi$$
$$458$$ 34532.3 3.52311
$$459$$ 0 0
$$460$$ −11.6915 −0.00118505
$$461$$ −18790.9 −1.89844 −0.949219 0.314618i $$-0.898124\pi$$
−0.949219 + 0.314618i $$0.898124\pi$$
$$462$$ 0 0
$$463$$ −7892.22 −0.792187 −0.396094 0.918210i $$-0.629634\pi$$
−0.396094 + 0.918210i $$0.629634\pi$$
$$464$$ −49687.4 −4.97130
$$465$$ 0 0
$$466$$ −9821.78 −0.976363
$$467$$ 385.511 0.0381998 0.0190999 0.999818i $$-0.493920\pi$$
0.0190999 + 0.999818i $$0.493920\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −7187.95 −0.705437
$$471$$ 0 0
$$472$$ 19173.2 1.86974
$$473$$ −285.050 −0.0277096
$$474$$ 0 0
$$475$$ −295.696 −0.0285631
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −19420.3 −1.85829
$$479$$ 1694.46 0.161632 0.0808160 0.996729i $$-0.474247\pi$$
0.0808160 + 0.996729i $$0.474247\pi$$
$$480$$ 0 0
$$481$$ 12256.7 1.16186
$$482$$ −2002.64 −0.189248
$$483$$ 0 0
$$484$$ −3126.91 −0.293662
$$485$$ 4195.03 0.392755
$$486$$ 0 0
$$487$$ 15711.2 1.46189 0.730945 0.682436i $$-0.239079\pi$$
0.730945 + 0.682436i $$0.239079\pi$$
$$488$$ −62040.2 −5.75498
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2716.34 0.249667 0.124834 0.992178i $$-0.460160\pi$$
0.124834 + 0.992178i $$0.460160\pi$$
$$492$$ 0 0
$$493$$ 17451.5 1.59427
$$494$$ 4496.38 0.409518
$$495$$ 0 0
$$496$$ −41601.2 −3.76603
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4295.34 0.385342 0.192671 0.981263i $$-0.438285\pi$$
0.192671 + 0.981263i $$0.438285\pi$$
$$500$$ −2806.00 −0.250977
$$501$$ 0 0
$$502$$ −29432.0 −2.61677
$$503$$ 6515.26 0.577537 0.288769 0.957399i $$-0.406754\pi$$
0.288769 + 0.957399i $$0.406754\pi$$
$$504$$ 0 0
$$505$$ −4437.20 −0.390996
$$506$$ −19.8421 −0.00174326
$$507$$ 0 0
$$508$$ 26573.5 2.32088
$$509$$ 14391.8 1.25325 0.626624 0.779321i $$-0.284436\pi$$
0.626624 + 0.779321i $$0.284436\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 41890.2 3.61583
$$513$$ 0 0
$$514$$ −13457.1 −1.15480
$$515$$ 3095.44 0.264857
$$516$$ 0 0
$$517$$ −8993.73 −0.765075
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 27462.3 2.31596
$$521$$ 2913.36 0.244984 0.122492 0.992469i $$-0.460911\pi$$
0.122492 + 0.992469i $$0.460911\pi$$
$$522$$ 0 0
$$523$$ −16870.2 −1.41048 −0.705242 0.708967i $$-0.749162\pi$$
−0.705242 + 0.708967i $$0.749162\pi$$
$$524$$ 5013.93 0.418005
$$525$$ 0 0
$$526$$ −8421.82 −0.698115
$$527$$ 14611.4 1.20775
$$528$$ 0 0
$$529$$ −12167.0 −0.999999
$$530$$ 9742.18 0.798441
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10006.6 −0.813197
$$534$$ 0 0
$$535$$ −10756.5 −0.869241
$$536$$ 12113.6 0.976172
$$537$$ 0 0
$$538$$ −41738.6 −3.34476
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2229.59 −0.177186 −0.0885929 0.996068i $$-0.528237\pi$$
−0.0885929 + 0.996068i $$0.528237\pi$$
$$542$$ −23631.7 −1.87282
$$543$$ 0 0
$$544$$ −73029.1 −5.75569
$$545$$ 2035.38 0.159975
$$546$$ 0 0
$$547$$ −1218.73 −0.0952639 −0.0476319 0.998865i $$-0.515167\pi$$
−0.0476319 + 0.998865i $$0.515167\pi$$
$$548$$ −45719.1 −3.56391
$$549$$ 0 0
$$550$$ −4762.16 −0.369198
$$551$$ 2257.50 0.174542
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −22118.3 −1.69624
$$555$$ 0 0
$$556$$ −60317.9 −4.60081
$$557$$ 22734.5 1.72943 0.864714 0.502264i $$-0.167499\pi$$
0.864714 + 0.502264i $$0.167499\pi$$
$$558$$ 0 0
$$559$$ −568.873 −0.0430425
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −38269.6 −2.87243
$$563$$ 4302.11 0.322047 0.161023 0.986951i $$-0.448521\pi$$
0.161023 + 0.986951i $$0.448521\pi$$
$$564$$ 0 0
$$565$$ −1747.91 −0.130150
$$566$$ 14714.4 1.09275
$$567$$ 0 0
$$568$$ 24829.9 1.83422
$$569$$ −14866.9 −1.09535 −0.547675 0.836691i $$-0.684487\pi$$
−0.547675 + 0.836691i $$0.684487\pi$$
$$570$$ 0 0
$$571$$ 16514.5 1.21035 0.605174 0.796093i $$-0.293103\pi$$
0.605174 + 0.796093i $$0.293103\pi$$
$$572$$ 53387.6 3.90253
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.60414 0.000188870 0
$$576$$ 0 0
$$577$$ −1867.97 −0.134774 −0.0673870 0.997727i $$-0.521466\pi$$
−0.0673870 + 0.997727i $$0.521466\pi$$
$$578$$ 19022.0 1.36888
$$579$$ 0 0
$$580$$ 21422.5 1.53366
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 12189.6 0.865941
$$584$$ −51014.3 −3.61471
$$585$$ 0 0
$$586$$ 32771.8 2.31022
$$587$$ 3926.15 0.276064 0.138032 0.990428i $$-0.455922\pi$$
0.138032 + 0.990428i $$0.455922\pi$$
$$588$$ 0 0
$$589$$ 1890.11 0.132225
$$590$$ −6635.21 −0.462996
$$591$$ 0 0
$$592$$ −46314.7 −3.21541
$$593$$ −7554.18 −0.523125 −0.261562 0.965187i $$-0.584238\pi$$
−0.261562 + 0.965187i $$0.584238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −15118.8 −1.03907
$$597$$ 0 0
$$598$$ −39.5988 −0.00270788
$$599$$ −28210.9 −1.92432 −0.962160 0.272487i $$-0.912154\pi$$
−0.962160 + 0.272487i $$0.912154\pi$$
$$600$$ 0 0
$$601$$ 23181.4 1.57336 0.786679 0.617363i $$-0.211799\pi$$
0.786679 + 0.617363i $$0.211799\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −47706.1 −3.21380
$$605$$ 696.478 0.0468031
$$606$$ 0 0
$$607$$ 24863.6 1.66257 0.831287 0.555844i $$-0.187605\pi$$
0.831287 + 0.555844i $$0.187605\pi$$
$$608$$ −9446.93 −0.630137
$$609$$ 0 0
$$610$$ 21470.1 1.42508
$$611$$ −17948.7 −1.18843
$$612$$ 0 0
$$613$$ −12792.7 −0.842893 −0.421446 0.906853i $$-0.638477\pi$$
−0.421446 + 0.906853i $$0.638477\pi$$
$$614$$ −57285.0 −3.76520
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −19793.3 −1.29149 −0.645744 0.763554i $$-0.723453\pi$$
−0.645744 + 0.763554i $$0.723453\pi$$
$$618$$ 0 0
$$619$$ 20951.7 1.36045 0.680226 0.733002i $$-0.261882\pi$$
0.680226 + 0.733002i $$0.261882\pi$$
$$620$$ 17936.2 1.16183
$$621$$ 0 0
$$622$$ 23399.2 1.50840
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 1566.57 0.100020
$$627$$ 0 0
$$628$$ 63147.1 4.01249
$$629$$ 16266.9 1.03117
$$630$$ 0 0
$$631$$ 23273.5 1.46831 0.734156 0.678981i $$-0.237578\pi$$
0.734156 + 0.678981i $$0.237578\pi$$
$$632$$ 31186.9 1.96289
$$633$$ 0 0
$$634$$ −9598.47 −0.601268
$$635$$ −5918.90 −0.369896
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 36356.8 2.25608
$$639$$ 0 0
$$640$$ −32186.8 −1.98796
$$641$$ 22117.7 1.36287 0.681434 0.731880i $$-0.261357\pi$$
0.681434 + 0.731880i $$0.261357\pi$$
$$642$$ 0 0
$$643$$ −20269.9 −1.24318 −0.621591 0.783342i $$-0.713513\pi$$
−0.621591 + 0.783342i $$0.713513\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5967.54 0.363451
$$647$$ −3145.25 −0.191117 −0.0955583 0.995424i $$-0.530464\pi$$
−0.0955583 + 0.995424i $$0.530464\pi$$
$$648$$ 0 0
$$649$$ −8302.13 −0.502137
$$650$$ −9503.81 −0.573493
$$651$$ 0 0
$$652$$ −30179.6 −1.81277
$$653$$ 5953.35 0.356773 0.178386 0.983961i $$-0.442912\pi$$
0.178386 + 0.983961i $$0.442912\pi$$
$$654$$ 0 0
$$655$$ −1116.79 −0.0666205
$$656$$ 37812.3 2.25049
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −26277.5 −1.55330 −0.776652 0.629930i $$-0.783084\pi$$
−0.776652 + 0.629930i $$0.783084\pi$$
$$660$$ 0 0
$$661$$ 24004.3 1.41250 0.706248 0.707964i $$-0.250386\pi$$
0.706248 + 0.707964i $$0.250386\pi$$
$$662$$ 30939.1 1.81644
$$663$$ 0 0
$$664$$ 39320.3 2.29808
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −19.8814 −0.00115414
$$668$$ 32577.4 1.88691
$$669$$ 0 0
$$670$$ −4192.13 −0.241725
$$671$$ 26863.9 1.54556
$$672$$ 0 0
$$673$$ 9205.36 0.527252 0.263626 0.964625i $$-0.415082\pi$$
0.263626 + 0.964625i $$0.415082\pi$$
$$674$$ −52020.4 −2.97292
$$675$$ 0 0
$$676$$ 57227.1 3.25598
$$677$$ −18773.1 −1.06575 −0.532873 0.846195i $$-0.678888\pi$$
−0.532873 + 0.846195i $$0.678888\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 36447.6 2.05544
$$681$$ 0 0
$$682$$ 30440.1 1.70911
$$683$$ 10222.2 0.572684 0.286342 0.958127i $$-0.407561\pi$$
0.286342 + 0.958127i $$0.407561\pi$$
$$684$$ 0 0
$$685$$ 10183.3 0.568006
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2149.62 0.119118
$$689$$ 24326.8 1.34510
$$690$$ 0 0
$$691$$ −22355.1 −1.23072 −0.615361 0.788246i $$-0.710990\pi$$
−0.615361 + 0.788246i $$0.710990\pi$$
$$692$$ −44428.2 −2.44062
$$693$$ 0 0
$$694$$ −64197.3 −3.51138
$$695$$ 13435.0 0.733264
$$696$$ 0 0
$$697$$ −13280.6 −0.721722
$$698$$ 7267.70 0.394107
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −16217.3 −0.873779 −0.436890 0.899515i $$-0.643920\pi$$
−0.436890 + 0.899515i $$0.643920\pi$$
$$702$$ 0 0
$$703$$ 2104.26 0.112893
$$704$$ −80247.1 −4.29606
$$705$$ 0 0
$$706$$ 32273.7 1.72045
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3922.92 0.207798 0.103899 0.994588i $$-0.466868\pi$$
0.103899 + 0.994588i $$0.466868\pi$$
$$710$$ −8592.83 −0.454202
$$711$$ 0 0
$$712$$ 37777.1 1.98842
$$713$$ −16.6458 −0.000874322 0
$$714$$ 0 0
$$715$$ −11891.4 −0.621976
$$716$$ 97843.4 5.10695
$$717$$ 0 0
$$718$$ 68544.3 3.56275
$$719$$ 23908.6 1.24011 0.620055 0.784558i $$-0.287110\pi$$
0.620055 + 0.784558i $$0.287110\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −37075.8 −1.91111
$$723$$ 0 0
$$724$$ 8483.62 0.435485
$$725$$ −4771.58 −0.244430
$$726$$ 0 0
$$727$$ 26906.4 1.37263 0.686315 0.727305i $$-0.259227\pi$$
0.686315 + 0.727305i $$0.259227\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 17654.4 0.895095
$$731$$ −755.001 −0.0382007
$$732$$ 0 0
$$733$$ −27637.5 −1.39265 −0.696325 0.717726i $$-0.745183\pi$$
−0.696325 + 0.717726i $$0.745183\pi$$
$$734$$ 36367.7 1.82882
$$735$$ 0 0
$$736$$ 83.1973 0.00416670
$$737$$ −5245.29 −0.262161
$$738$$ 0 0
$$739$$ −14038.9 −0.698821 −0.349410 0.936970i $$-0.613618\pi$$
−0.349410 + 0.936970i $$0.613618\pi$$
$$740$$ 19968.4 0.991963
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −17698.7 −0.873893 −0.436946 0.899488i $$-0.643940\pi$$
−0.436946 + 0.899488i $$0.643940\pi$$
$$744$$ 0 0
$$745$$ 3367.50 0.165605
$$746$$ 1899.72 0.0932354
$$747$$ 0 0
$$748$$ 70855.4 3.46354
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 17199.7 0.835719 0.417859 0.908512i $$-0.362780\pi$$
0.417859 + 0.908512i $$0.362780\pi$$
$$752$$ 67823.4 3.28892
$$753$$ 0 0
$$754$$ 72557.1 3.50448
$$755$$ 10625.9 0.512206
$$756$$ 0 0
$$757$$ −19200.7 −0.921879 −0.460939 0.887432i $$-0.652487\pi$$
−0.460939 + 0.887432i $$0.652487\pi$$
$$758$$ 28921.0 1.38583
$$759$$ 0 0
$$760$$ 4714.80 0.225031
$$761$$ −27918.7 −1.32990 −0.664949 0.746889i $$-0.731547\pi$$
−0.664949 + 0.746889i $$0.731547\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 54457.7 2.57881
$$765$$ 0 0
$$766$$ 41922.1 1.97742
$$767$$ −16568.5 −0.779992
$$768$$ 0 0
$$769$$ 7436.29 0.348712 0.174356 0.984683i $$-0.444216\pi$$
0.174356 + 0.984683i $$0.444216\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13983.4 −0.651908
$$773$$ 4989.84 0.232176 0.116088 0.993239i $$-0.462965\pi$$
0.116088 + 0.993239i $$0.462965\pi$$
$$774$$ 0 0
$$775$$ −3995.05 −0.185169
$$776$$ −66888.7 −3.09429
$$777$$ 0 0
$$778$$ −17115.9 −0.788731
$$779$$ −1717.96 −0.0790146
$$780$$ 0 0
$$781$$ −10751.5 −0.492600
$$782$$ −52.5550 −0.00240328
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −14065.2 −0.639500
$$786$$ 0 0
$$787$$ 2870.69 0.130024 0.0650122 0.997884i $$-0.479291\pi$$
0.0650122 + 0.997884i $$0.479291\pi$$
$$788$$ −63803.8 −2.88441
$$789$$ 0 0
$$790$$ −10792.8 −0.486063
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 53612.1 2.40078
$$794$$ 16179.0 0.723138
$$795$$ 0 0
$$796$$ 19470.6 0.866982
$$797$$ 4676.61 0.207847 0.103923 0.994585i $$-0.466860\pi$$
0.103923 + 0.994585i $$0.466860\pi$$
$$798$$ 0 0
$$799$$ −23821.3 −1.05474
$$800$$ 19967.6 0.882451
$$801$$ 0 0
$$802$$ −492.410 −0.0216803
$$803$$ 22089.6 0.970766
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 60749.1 2.65483
$$807$$ 0 0
$$808$$ 70750.2 3.08042
$$809$$ 15376.9 0.668261 0.334131 0.942527i $$-0.391557\pi$$
0.334131 + 0.942527i $$0.391557\pi$$
$$810$$ 0 0
$$811$$ −36422.9 −1.57704 −0.788522 0.615007i $$-0.789153\pi$$
−0.788522 + 0.615007i $$0.789153\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 33889.0 1.45922
$$815$$ 6722.10 0.288914
$$816$$ 0 0
$$817$$ −97.6658 −0.00418225
$$818$$ −813.285 −0.0347627
$$819$$ 0 0
$$820$$ −16302.6 −0.694282
$$821$$ 25479.0 1.08310 0.541550 0.840669i $$-0.317838\pi$$
0.541550 + 0.840669i $$0.317838\pi$$
$$822$$ 0 0
$$823$$ −17933.3 −0.759557 −0.379779 0.925077i $$-0.624000\pi$$
−0.379779 + 0.925077i $$0.624000\pi$$
$$824$$ −49356.1 −2.08665
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −13021.6 −0.547529 −0.273764 0.961797i $$-0.588269\pi$$
−0.273764 + 0.961797i $$0.588269\pi$$
$$828$$ 0 0
$$829$$ 11397.6 0.477509 0.238754 0.971080i $$-0.423261\pi$$
0.238754 + 0.971080i $$0.423261\pi$$
$$830$$ −13607.5 −0.569063
$$831$$ 0 0
$$832$$ −160149. −6.67326
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −7256.19 −0.300731
$$836$$ 9165.75 0.379191
$$837$$ 0 0
$$838$$ −20867.2 −0.860196
$$839$$ −37681.2 −1.55053 −0.775267 0.631633i $$-0.782385\pi$$
−0.775267 + 0.631633i $$0.782385\pi$$
$$840$$ 0 0
$$841$$ 12039.8 0.493656
$$842$$ −60141.7 −2.46154
$$843$$ 0 0
$$844$$ −134148. −5.47104
$$845$$ −12746.6 −0.518929
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −91924.4 −3.72252
$$849$$ 0 0
$$850$$ −12613.3 −0.508981
$$851$$ −18.5318 −0.000746490 0
$$852$$ 0 0
$$853$$ −21771.6 −0.873911 −0.436956 0.899483i $$-0.643943\pi$$
−0.436956 + 0.899483i $$0.643943\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 171510. 6.84823
$$857$$ 29860.2 1.19021 0.595103 0.803649i $$-0.297111\pi$$
0.595103 + 0.803649i $$0.297111\pi$$
$$858$$ 0 0
$$859$$ −18530.6 −0.736039 −0.368019 0.929818i $$-0.619964\pi$$
−0.368019 + 0.929818i $$0.619964\pi$$
$$860$$ −926.799 −0.0367484
$$861$$ 0 0
$$862$$ 62261.4 2.46013
$$863$$ 21296.5 0.840024 0.420012 0.907519i $$-0.362026\pi$$
0.420012 + 0.907519i $$0.362026\pi$$
$$864$$ 0 0
$$865$$ 9895.79 0.388979
$$866$$ 49144.1 1.92839
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −13504.2 −0.527155
$$870$$ 0 0
$$871$$ −10468.0 −0.407226
$$872$$ −32453.7 −1.26035
$$873$$ 0 0
$$874$$ −6.79843 −0.000263112 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −18793.4 −0.723614 −0.361807 0.932253i $$-0.617840\pi$$
−0.361807 + 0.932253i $$0.617840\pi$$
$$878$$ −44968.7 −1.72850
$$879$$ 0 0
$$880$$ 44934.4 1.72129
$$881$$ 1638.62 0.0626634 0.0313317 0.999509i $$-0.490025\pi$$
0.0313317 + 0.999509i $$0.490025\pi$$
$$882$$ 0 0
$$883$$ −35424.1 −1.35008 −0.675038 0.737783i $$-0.735873\pi$$
−0.675038 + 0.737783i $$0.735873\pi$$
$$884$$ 141406. 5.38008
$$885$$ 0 0
$$886$$ 63306.5 2.40048
$$887$$ −5131.41 −0.194246 −0.0971229 0.995272i $$-0.530964\pi$$
−0.0971229 + 0.995272i $$0.530964\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −13073.5 −0.492386
$$891$$ 0 0
$$892$$ 116316. 4.36610
$$893$$ −3081.49 −0.115474
$$894$$ 0 0
$$895$$ −21793.3 −0.813933
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −10836.2 −0.402682
$$899$$ 30500.3 1.13153
$$900$$ 0 0
$$901$$ 32286.2 1.19380
$$902$$ −27667.7 −1.02132
$$903$$ 0 0
$$904$$ 27870.0 1.02538
$$905$$ −1889.61 −0.0694065
$$906$$ 0 0
$$907$$ −19934.6 −0.729787 −0.364893 0.931049i $$-0.618895\pi$$
−0.364893 + 0.931049i $$0.618895\pi$$
$$908$$ 84248.8 3.07918
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −48387.4 −1.75976 −0.879882 0.475193i $$-0.842378\pi$$
−0.879882 + 0.475193i $$0.842378\pi$$
$$912$$ 0 0
$$913$$ −17026.0 −0.617172
$$914$$ 30844.7 1.11625
$$915$$ 0 0
$$916$$ 140483. 5.06735
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −14431.7 −0.518019 −0.259009 0.965875i $$-0.583396\pi$$
−0.259009 + 0.965875i $$0.583396\pi$$
$$920$$ −41.5224 −0.00148799
$$921$$ 0 0
$$922$$ −103688. −3.70366
$$923$$ −21456.8 −0.765177
$$924$$ 0 0
$$925$$ −4447.69 −0.158096
$$926$$ −43549.1 −1.54548
$$927$$ 0 0
$$928$$ −152443. −5.39244
$$929$$ 5549.82 0.196000 0.0979998 0.995186i $$-0.468756\pi$$
0.0979998 + 0.995186i $$0.468756\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −39956.7 −1.40432
$$933$$ 0 0
$$934$$ 2127.24 0.0745239
$$935$$ −15782.1 −0.552010
$$936$$ 0 0
$$937$$ −26231.9 −0.914576 −0.457288 0.889319i $$-0.651179\pi$$
−0.457288 + 0.889319i $$0.651179\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −29241.8 −1.01464
$$941$$ 2836.80 0.0982753 0.0491376 0.998792i $$-0.484353\pi$$
0.0491376 + 0.998792i $$0.484353\pi$$
$$942$$ 0 0
$$943$$ 15.1298 0.000522474 0
$$944$$ 62607.9 2.15860
$$945$$ 0 0
$$946$$ −1572.90 −0.0540586
$$947$$ 40272.9 1.38193 0.690967 0.722886i $$-0.257185\pi$$
0.690967 + 0.722886i $$0.257185\pi$$
$$948$$ 0 0
$$949$$ 44084.1 1.50793
$$950$$ −1631.64 −0.0557236
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −17770.1 −0.604019 −0.302010 0.953305i $$-0.597657\pi$$
−0.302010 + 0.953305i $$0.597657\pi$$
$$954$$ 0 0
$$955$$ −12129.7 −0.411004
$$956$$ −79005.0 −2.67281
$$957$$ 0 0
$$958$$ 9349.97 0.315328
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4254.35 −0.142807
$$962$$ 67632.0 2.26668
$$963$$ 0 0
$$964$$ −8147.07 −0.272199
$$965$$ 3114.61 0.103899
$$966$$ 0 0
$$967$$ −3530.14 −0.117396 −0.0586978 0.998276i $$-0.518695\pi$$
−0.0586978 + 0.998276i $$0.518695\pi$$
$$968$$ −11105.2 −0.368734
$$969$$ 0 0
$$970$$ 23148.0 0.766226
$$971$$ 17650.7 0.583356 0.291678 0.956517i $$-0.405786\pi$$
0.291678 + 0.956517i $$0.405786\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 86693.8 2.85200
$$975$$ 0 0
$$976$$ −202586. −6.64407
$$977$$ 14477.6 0.474085 0.237042 0.971499i $$-0.423822\pi$$
0.237042 + 0.971499i $$0.423822\pi$$
$$978$$ 0 0
$$979$$ −16357.8 −0.534012
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 14988.7 0.487075
$$983$$ −8764.09 −0.284365 −0.142183 0.989840i $$-0.545412\pi$$
−0.142183 + 0.989840i $$0.545412\pi$$
$$984$$ 0 0
$$985$$ 14211.4 0.459710
$$986$$ 96296.9 3.11026
$$987$$ 0 0
$$988$$ 18292.0 0.589015
$$989$$ 0.860124 2.76546e−5 0
$$990$$ 0 0
$$991$$ 33624.1 1.07781 0.538903 0.842368i $$-0.318839\pi$$
0.538903 + 0.842368i $$0.318839\pi$$
$$992$$ −127634. −4.08507
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4336.82 −0.138177
$$996$$ 0 0
$$997$$ −16631.3 −0.528302 −0.264151 0.964481i $$-0.585092\pi$$
−0.264151 + 0.964481i $$0.585092\pi$$
$$998$$ 23701.6 0.751764
$$999$$ 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 2205.4.a.bz.1.6 6
3.2 odd 2 245.4.a.o.1.1 6
7.6 odd 2 2205.4.a.ca.1.6 6
15.14 odd 2 1225.4.a.bj.1.6 6
21.2 odd 6 245.4.e.q.116.6 12
21.5 even 6 245.4.e.p.116.6 12
21.11 odd 6 245.4.e.q.226.6 12
21.17 even 6 245.4.e.p.226.6 12
21.20 even 2 245.4.a.p.1.1 yes 6
105.104 even 2 1225.4.a.bi.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.o.1.1 6 3.2 odd 2
245.4.a.p.1.1 yes 6 21.20 even 2
245.4.e.p.116.6 12 21.5 even 6
245.4.e.p.226.6 12 21.17 even 6
245.4.e.q.116.6 12 21.2 odd 6
245.4.e.q.226.6 12 21.11 odd 6
1225.4.a.bi.1.6 6 105.104 even 2
1225.4.a.bj.1.6 6 15.14 odd 2
2205.4.a.bz.1.6 6 1.1 even 1 trivial
2205.4.a.ca.1.6 6 7.6 odd 2