# Properties

 Label 245.6.a.i.1.6 Level $245$ Weight $6$ Character 245.1 Self dual yes Analytic conductor $39.294$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560$$ x^6 - x^5 - 109*x^4 + 41*x^3 + 2208*x^2 - 3204*x + 560 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$9.72556$$ of defining polynomial Character $$\chi$$ $$=$$ 245.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+8.72556 q^{2} +4.73965 q^{3} +44.1354 q^{4} -25.0000 q^{5} +41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +O(q^{10})$$ $$q+8.72556 q^{2} +4.73965 q^{3} +44.1354 q^{4} -25.0000 q^{5} +41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} -218.139 q^{10} -679.156 q^{11} +209.186 q^{12} +58.8790 q^{13} -118.491 q^{15} -488.399 q^{16} +1427.56 q^{17} -1924.30 q^{18} -2197.34 q^{19} -1103.39 q^{20} -5926.02 q^{22} -2223.33 q^{23} +501.873 q^{24} +625.000 q^{25} +513.753 q^{26} -2197.00 q^{27} +2079.59 q^{29} -1033.90 q^{30} +3599.58 q^{31} -7649.98 q^{32} -3218.96 q^{33} +12456.2 q^{34} -9733.43 q^{36} +9181.95 q^{37} -19173.0 q^{38} +279.066 q^{39} -2647.21 q^{40} +10061.1 q^{41} -13956.6 q^{43} -29974.8 q^{44} +5513.39 q^{45} -19399.8 q^{46} -16272.5 q^{47} -2314.84 q^{48} +5453.48 q^{50} +6766.12 q^{51} +2598.65 q^{52} +1267.14 q^{53} -19170.0 q^{54} +16978.9 q^{55} -10414.6 q^{57} +18145.6 q^{58} -13465.7 q^{59} -5229.66 q^{60} +35031.5 q^{61} +31408.4 q^{62} -51121.6 q^{64} -1471.98 q^{65} -28087.2 q^{66} -36919.0 q^{67} +63005.9 q^{68} -10537.8 q^{69} +4118.92 q^{71} -23352.1 q^{72} -3517.37 q^{73} +80117.7 q^{74} +2962.28 q^{75} -96980.4 q^{76} +2435.01 q^{78} -72440.8 q^{79} +12210.0 q^{80} +43177.2 q^{81} +87789.0 q^{82} -42252.6 q^{83} -35688.9 q^{85} -121779. q^{86} +9856.51 q^{87} -71914.7 q^{88} +94517.0 q^{89} +48107.5 q^{90} -98127.7 q^{92} +17060.8 q^{93} -141987. q^{94} +54933.4 q^{95} -36258.2 q^{96} +85271.1 q^{97} +149778. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10})$$ 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 $$6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100})$$ 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 + 125 * q^10 - 924 * q^11 + 370 * q^12 - 150 * q^13 - 500 * q^15 + 435 * q^16 + 1540 * q^17 + 195 * q^18 + 92 * q^19 - 775 * q^20 - 6855 * q^22 - 3920 * q^23 - 7200 * q^24 + 3750 * q^25 + 2635 * q^26 + 2060 * q^27 + 1264 * q^29 - 2400 * q^30 - 7160 * q^31 + 9105 * q^32 + 4460 * q^33 + 2166 * q^34 - 26375 * q^36 - 14170 * q^37 - 46215 * q^38 - 15376 * q^39 + 3375 * q^40 + 4098 * q^41 - 24460 * q^43 - 27873 * q^44 - 9450 * q^45 + 6815 * q^46 + 42940 * q^47 + 11610 * q^48 - 3125 * q^50 - 42008 * q^51 - 36115 * q^52 - 2450 * q^53 + 19566 * q^54 + 23100 * q^55 - 97100 * q^57 - 36110 * q^58 - 64600 * q^59 - 9250 * q^60 + 73620 * q^61 + 111440 * q^62 - 157997 * q^64 + 3750 * q^65 - 139138 * q^66 - 142620 * q^67 + 124330 * q^68 + 17344 * q^69 - 154256 * q^71 - 117495 * q^72 - 5120 * q^73 + 2785 * q^74 + 12500 * q^75 + 7775 * q^76 - 214090 * q^78 - 222504 * q^79 - 10875 * q^80 - 43986 * q^81 + 31665 * q^82 + 179580 * q^83 - 38500 * q^85 - 207160 * q^86 - 209300 * q^87 - 45145 * q^88 + 41648 * q^89 - 4875 * q^90 - 292185 * q^92 - 198520 * q^93 - 333699 * q^94 - 2300 * q^95 + 61824 * q^96 - 73980 * q^97 - 190772 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.72556 1.54248 0.771238 0.636547i $$-0.219638\pi$$
0.771238 + 0.636547i $$0.219638\pi$$
$$3$$ 4.73965 0.304048 0.152024 0.988377i $$-0.451421\pi$$
0.152024 + 0.988377i $$0.451421\pi$$
$$4$$ 44.1354 1.37923
$$5$$ −25.0000 −0.447214
$$6$$ 41.3561 0.468987
$$7$$ 0 0
$$8$$ 105.888 0.584955
$$9$$ −220.536 −0.907555
$$10$$ −218.139 −0.689816
$$11$$ −679.156 −1.69234 −0.846171 0.532912i $$-0.821098\pi$$
−0.846171 + 0.532912i $$0.821098\pi$$
$$12$$ 209.186 0.419353
$$13$$ 58.8790 0.0966279 0.0483139 0.998832i $$-0.484615\pi$$
0.0483139 + 0.998832i $$0.484615\pi$$
$$14$$ 0 0
$$15$$ −118.491 −0.135975
$$16$$ −488.399 −0.476952
$$17$$ 1427.56 1.19804 0.599020 0.800734i $$-0.295557\pi$$
0.599020 + 0.800734i $$0.295557\pi$$
$$18$$ −1924.30 −1.39988
$$19$$ −2197.34 −1.39641 −0.698205 0.715898i $$-0.746017\pi$$
−0.698205 + 0.715898i $$0.746017\pi$$
$$20$$ −1103.39 −0.616811
$$21$$ 0 0
$$22$$ −5926.02 −2.61040
$$23$$ −2223.33 −0.876365 −0.438182 0.898886i $$-0.644378\pi$$
−0.438182 + 0.898886i $$0.644378\pi$$
$$24$$ 501.873 0.177855
$$25$$ 625.000 0.200000
$$26$$ 513.753 0.149046
$$27$$ −2197.00 −0.579989
$$28$$ 0 0
$$29$$ 2079.59 0.459179 0.229590 0.973288i $$-0.426262\pi$$
0.229590 + 0.973288i $$0.426262\pi$$
$$30$$ −1033.90 −0.209738
$$31$$ 3599.58 0.672741 0.336371 0.941730i $$-0.390801\pi$$
0.336371 + 0.941730i $$0.390801\pi$$
$$32$$ −7649.98 −1.32064
$$33$$ −3218.96 −0.514554
$$34$$ 12456.2 1.84795
$$35$$ 0 0
$$36$$ −9733.43 −1.25173
$$37$$ 9181.95 1.10263 0.551316 0.834296i $$-0.314126\pi$$
0.551316 + 0.834296i $$0.314126\pi$$
$$38$$ −19173.0 −2.15393
$$39$$ 279.066 0.0293796
$$40$$ −2647.21 −0.261600
$$41$$ 10061.1 0.934732 0.467366 0.884064i $$-0.345203\pi$$
0.467366 + 0.884064i $$0.345203\pi$$
$$42$$ 0 0
$$43$$ −13956.6 −1.15108 −0.575542 0.817772i $$-0.695209\pi$$
−0.575542 + 0.817772i $$0.695209\pi$$
$$44$$ −29974.8 −2.33413
$$45$$ 5513.39 0.405871
$$46$$ −19399.8 −1.35177
$$47$$ −16272.5 −1.07451 −0.537254 0.843420i $$-0.680538\pi$$
−0.537254 + 0.843420i $$0.680538\pi$$
$$48$$ −2314.84 −0.145017
$$49$$ 0 0
$$50$$ 5453.48 0.308495
$$51$$ 6766.12 0.364262
$$52$$ 2598.65 0.133272
$$53$$ 1267.14 0.0619634 0.0309817 0.999520i $$-0.490137\pi$$
0.0309817 + 0.999520i $$0.490137\pi$$
$$54$$ −19170.0 −0.894619
$$55$$ 16978.9 0.756838
$$56$$ 0 0
$$57$$ −10414.6 −0.424576
$$58$$ 18145.6 0.708273
$$59$$ −13465.7 −0.503615 −0.251807 0.967777i $$-0.581025\pi$$
−0.251807 + 0.967777i $$0.581025\pi$$
$$60$$ −5229.66 −0.187540
$$61$$ 35031.5 1.20541 0.602703 0.797965i $$-0.294090\pi$$
0.602703 + 0.797965i $$0.294090\pi$$
$$62$$ 31408.4 1.03769
$$63$$ 0 0
$$64$$ −51121.6 −1.56011
$$65$$ −1471.98 −0.0432133
$$66$$ −28087.2 −0.793687
$$67$$ −36919.0 −1.00476 −0.502380 0.864647i $$-0.667542\pi$$
−0.502380 + 0.864647i $$0.667542\pi$$
$$68$$ 63005.9 1.65237
$$69$$ −10537.8 −0.266457
$$70$$ 0 0
$$71$$ 4118.92 0.0969700 0.0484850 0.998824i $$-0.484561\pi$$
0.0484850 + 0.998824i $$0.484561\pi$$
$$72$$ −23352.1 −0.530879
$$73$$ −3517.37 −0.0772521 −0.0386261 0.999254i $$-0.512298\pi$$
−0.0386261 + 0.999254i $$0.512298\pi$$
$$74$$ 80117.7 1.70078
$$75$$ 2962.28 0.0608097
$$76$$ −96980.4 −1.92597
$$77$$ 0 0
$$78$$ 2435.01 0.0453173
$$79$$ −72440.8 −1.30592 −0.652959 0.757394i $$-0.726472\pi$$
−0.652959 + 0.757394i $$0.726472\pi$$
$$80$$ 12210.0 0.213299
$$81$$ 43177.2 0.731210
$$82$$ 87789.0 1.44180
$$83$$ −42252.6 −0.673223 −0.336611 0.941644i $$-0.609281\pi$$
−0.336611 + 0.941644i $$0.609281\pi$$
$$84$$ 0 0
$$85$$ −35688.9 −0.535780
$$86$$ −121779. −1.77552
$$87$$ 9856.51 0.139613
$$88$$ −71914.7 −0.989944
$$89$$ 94517.0 1.26484 0.632419 0.774627i $$-0.282062\pi$$
0.632419 + 0.774627i $$0.282062\pi$$
$$90$$ 48107.5 0.626046
$$91$$ 0 0
$$92$$ −98127.7 −1.20871
$$93$$ 17060.8 0.204546
$$94$$ −141987. −1.65740
$$95$$ 54933.4 0.624493
$$96$$ −36258.2 −0.401539
$$97$$ 85271.1 0.920180 0.460090 0.887872i $$-0.347817\pi$$
0.460090 + 0.887872i $$0.347817\pi$$
$$98$$ 0 0
$$99$$ 149778. 1.53589
$$100$$ 27584.6 0.275846
$$101$$ −180306. −1.75876 −0.879380 0.476121i $$-0.842042\pi$$
−0.879380 + 0.476121i $$0.842042\pi$$
$$102$$ 59038.2 0.561866
$$103$$ −173368. −1.61018 −0.805091 0.593152i $$-0.797883\pi$$
−0.805091 + 0.593152i $$0.797883\pi$$
$$104$$ 6234.60 0.0565230
$$105$$ 0 0
$$106$$ 11056.5 0.0955770
$$107$$ 155442. 1.31253 0.656264 0.754532i $$-0.272136\pi$$
0.656264 + 0.754532i $$0.272136\pi$$
$$108$$ −96965.3 −0.799939
$$109$$ 19088.6 0.153889 0.0769444 0.997035i $$-0.475484\pi$$
0.0769444 + 0.997035i $$0.475484\pi$$
$$110$$ 148151. 1.16740
$$111$$ 43519.2 0.335254
$$112$$ 0 0
$$113$$ 101150. 0.745197 0.372599 0.927993i $$-0.378467\pi$$
0.372599 + 0.927993i $$0.378467\pi$$
$$114$$ −90873.3 −0.654899
$$115$$ 55583.3 0.391922
$$116$$ 91783.5 0.633315
$$117$$ −12984.9 −0.0876951
$$118$$ −117496. −0.776814
$$119$$ 0 0
$$120$$ −12546.8 −0.0795391
$$121$$ 300202. 1.86402
$$122$$ 305669. 1.85931
$$123$$ 47686.2 0.284204
$$124$$ 158869. 0.927866
$$125$$ −15625.0 −0.0894427
$$126$$ 0 0
$$127$$ −112788. −0.620514 −0.310257 0.950653i $$-0.600415\pi$$
−0.310257 + 0.950653i $$0.600415\pi$$
$$128$$ −201265. −1.08578
$$129$$ −66149.1 −0.349985
$$130$$ −12843.8 −0.0666555
$$131$$ −183577. −0.934632 −0.467316 0.884090i $$-0.654779\pi$$
−0.467316 + 0.884090i $$0.654779\pi$$
$$132$$ −142070. −0.709689
$$133$$ 0 0
$$134$$ −322139. −1.54982
$$135$$ 54924.9 0.259379
$$136$$ 151162. 0.700800
$$137$$ −92212.1 −0.419746 −0.209873 0.977729i $$-0.567305\pi$$
−0.209873 + 0.977729i $$0.567305\pi$$
$$138$$ −91948.3 −0.411004
$$139$$ 424349. 1.86288 0.931442 0.363889i $$-0.118551\pi$$
0.931442 + 0.363889i $$0.118551\pi$$
$$140$$ 0 0
$$141$$ −77125.9 −0.326703
$$142$$ 35939.9 0.149574
$$143$$ −39988.1 −0.163527
$$144$$ 107709. 0.432860
$$145$$ −51989.7 −0.205351
$$146$$ −30691.0 −0.119160
$$147$$ 0 0
$$148$$ 405249. 1.52079
$$149$$ 137894. 0.508838 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$150$$ 25847.5 0.0937975
$$151$$ −308590. −1.10139 −0.550693 0.834708i $$-0.685636\pi$$
−0.550693 + 0.834708i $$0.685636\pi$$
$$152$$ −232672. −0.816837
$$153$$ −314828. −1.08729
$$154$$ 0 0
$$155$$ −89989.6 −0.300859
$$156$$ 12316.7 0.0405212
$$157$$ −209003. −0.676710 −0.338355 0.941018i $$-0.609871\pi$$
−0.338355 + 0.941018i $$0.609871\pi$$
$$158$$ −632087. −2.01435
$$159$$ 6005.80 0.0188399
$$160$$ 191249. 0.590609
$$161$$ 0 0
$$162$$ 376745. 1.12787
$$163$$ −249015. −0.734103 −0.367051 0.930201i $$-0.619633\pi$$
−0.367051 + 0.930201i $$0.619633\pi$$
$$164$$ 444052. 1.28921
$$165$$ 80474.0 0.230116
$$166$$ −368678. −1.03843
$$167$$ 550497. 1.52744 0.763720 0.645548i $$-0.223371\pi$$
0.763720 + 0.645548i $$0.223371\pi$$
$$168$$ 0 0
$$169$$ −367826. −0.990663
$$170$$ −311406. −0.826428
$$171$$ 484592. 1.26732
$$172$$ −615978. −1.58761
$$173$$ −103026. −0.261717 −0.130859 0.991401i $$-0.541773\pi$$
−0.130859 + 0.991401i $$0.541773\pi$$
$$174$$ 86003.6 0.215349
$$175$$ 0 0
$$176$$ 331699. 0.807166
$$177$$ −63822.6 −0.153123
$$178$$ 824714. 1.95098
$$179$$ 470596. 1.09778 0.548891 0.835894i $$-0.315050\pi$$
0.548891 + 0.835894i $$0.315050\pi$$
$$180$$ 243336. 0.559790
$$181$$ −529125. −1.20050 −0.600250 0.799812i $$-0.704932\pi$$
−0.600250 + 0.799812i $$0.704932\pi$$
$$182$$ 0 0
$$183$$ 166037. 0.366502
$$184$$ −235425. −0.512634
$$185$$ −229549. −0.493112
$$186$$ 148865. 0.315507
$$187$$ −969535. −2.02749
$$188$$ −718194. −1.48200
$$189$$ 0 0
$$190$$ 479325. 0.963266
$$191$$ 180604. 0.358215 0.179108 0.983829i $$-0.442679\pi$$
0.179108 + 0.983829i $$0.442679\pi$$
$$192$$ −242298. −0.474348
$$193$$ −512865. −0.991083 −0.495541 0.868584i $$-0.665030\pi$$
−0.495541 + 0.868584i $$0.665030\pi$$
$$194$$ 744038. 1.41935
$$195$$ −6976.65 −0.0131389
$$196$$ 0 0
$$197$$ −28199.0 −0.0517687 −0.0258844 0.999665i $$-0.508240\pi$$
−0.0258844 + 0.999665i $$0.508240\pi$$
$$198$$ 1.30690e6 2.36908
$$199$$ 934000. 1.67192 0.835958 0.548794i $$-0.184913\pi$$
0.835958 + 0.548794i $$0.184913\pi$$
$$200$$ 66180.1 0.116991
$$201$$ −174983. −0.305496
$$202$$ −1.57327e6 −2.71284
$$203$$ 0 0
$$204$$ 298625. 0.502402
$$205$$ −251528. −0.418025
$$206$$ −1.51273e6 −2.48367
$$207$$ 490324. 0.795349
$$208$$ −28756.5 −0.0460869
$$209$$ 1.49234e6 2.36320
$$210$$ 0 0
$$211$$ −206876. −0.319893 −0.159946 0.987126i $$-0.551132\pi$$
−0.159946 + 0.987126i $$0.551132\pi$$
$$212$$ 55925.8 0.0854619
$$213$$ 19522.2 0.0294836
$$214$$ 1.35632e6 2.02454
$$215$$ 348914. 0.514780
$$216$$ −232636. −0.339268
$$217$$ 0 0
$$218$$ 166559. 0.237370
$$219$$ −16671.1 −0.0234884
$$220$$ 749371. 1.04386
$$221$$ 84053.3 0.115764
$$222$$ 379730. 0.517121
$$223$$ 347934. 0.468527 0.234263 0.972173i $$-0.424732\pi$$
0.234263 + 0.972173i $$0.424732\pi$$
$$224$$ 0 0
$$225$$ −137835. −0.181511
$$226$$ 882594. 1.14945
$$227$$ −792997. −1.02143 −0.510713 0.859751i $$-0.670619\pi$$
−0.510713 + 0.859751i $$0.670619\pi$$
$$228$$ −459653. −0.585589
$$229$$ −275845. −0.347598 −0.173799 0.984781i $$-0.555604\pi$$
−0.173799 + 0.984781i $$0.555604\pi$$
$$230$$ 484996. 0.604531
$$231$$ 0 0
$$232$$ 220204. 0.268599
$$233$$ 1.43590e6 1.73275 0.866373 0.499398i $$-0.166446\pi$$
0.866373 + 0.499398i $$0.166446\pi$$
$$234$$ −113301. −0.135267
$$235$$ 406813. 0.480535
$$236$$ −594314. −0.694602
$$237$$ −343344. −0.397062
$$238$$ 0 0
$$239$$ 127902. 0.144838 0.0724191 0.997374i $$-0.476928\pi$$
0.0724191 + 0.997374i $$0.476928\pi$$
$$240$$ 57871.0 0.0648534
$$241$$ 602677. 0.668409 0.334204 0.942501i $$-0.391532\pi$$
0.334204 + 0.942501i $$0.391532\pi$$
$$242$$ 2.61943e6 2.87521
$$243$$ 738515. 0.802312
$$244$$ 1.54613e6 1.66253
$$245$$ 0 0
$$246$$ 416089. 0.438377
$$247$$ −129377. −0.134932
$$248$$ 381154. 0.393524
$$249$$ −200263. −0.204692
$$250$$ −136337. −0.137963
$$251$$ −756978. −0.758401 −0.379201 0.925314i $$-0.623801\pi$$
−0.379201 + 0.925314i $$0.623801\pi$$
$$252$$ 0 0
$$253$$ 1.50999e6 1.48311
$$254$$ −984135. −0.957128
$$255$$ −169153. −0.162903
$$256$$ −120261. −0.114689
$$257$$ 606610. 0.572898 0.286449 0.958096i $$-0.407525\pi$$
0.286449 + 0.958096i $$0.407525\pi$$
$$258$$ −577188. −0.539844
$$259$$ 0 0
$$260$$ −64966.3 −0.0596011
$$261$$ −458624. −0.416730
$$262$$ −1.60181e6 −1.44165
$$263$$ −1.41962e6 −1.26556 −0.632780 0.774331i $$-0.718086\pi$$
−0.632780 + 0.774331i $$0.718086\pi$$
$$264$$ −340850. −0.300991
$$265$$ −31678.5 −0.0277109
$$266$$ 0 0
$$267$$ 447977. 0.384572
$$268$$ −1.62943e6 −1.38580
$$269$$ −496984. −0.418756 −0.209378 0.977835i $$-0.567144\pi$$
−0.209378 + 0.977835i $$0.567144\pi$$
$$270$$ 479250. 0.400086
$$271$$ 155568. 0.128676 0.0643378 0.997928i $$-0.479506\pi$$
0.0643378 + 0.997928i $$0.479506\pi$$
$$272$$ −697218. −0.571408
$$273$$ 0 0
$$274$$ −804602. −0.647448
$$275$$ −424473. −0.338468
$$276$$ −465091. −0.367506
$$277$$ −1.77069e6 −1.38657 −0.693285 0.720663i $$-0.743837\pi$$
−0.693285 + 0.720663i $$0.743837\pi$$
$$278$$ 3.70268e6 2.87345
$$279$$ −793837. −0.610549
$$280$$ 0 0
$$281$$ −2.38775e6 −1.80394 −0.901972 0.431794i $$-0.857881\pi$$
−0.901972 + 0.431794i $$0.857881\pi$$
$$282$$ −672967. −0.503931
$$283$$ −193515. −0.143631 −0.0718156 0.997418i $$-0.522879\pi$$
−0.0718156 + 0.997418i $$0.522879\pi$$
$$284$$ 181790. 0.133744
$$285$$ 260365. 0.189876
$$286$$ −348918. −0.252237
$$287$$ 0 0
$$288$$ 1.68709e6 1.19855
$$289$$ 618065. 0.435301
$$290$$ −453639. −0.316749
$$291$$ 404155. 0.279779
$$292$$ −155240. −0.106549
$$293$$ −66529.2 −0.0452734 −0.0226367 0.999744i $$-0.507206\pi$$
−0.0226367 + 0.999744i $$0.507206\pi$$
$$294$$ 0 0
$$295$$ 336642. 0.225223
$$296$$ 972261. 0.644991
$$297$$ 1.49210e6 0.981540
$$298$$ 1.20320e6 0.784870
$$299$$ −130908. −0.0846813
$$300$$ 130741. 0.0838706
$$301$$ 0 0
$$302$$ −2.69262e6 −1.69886
$$303$$ −854586. −0.534748
$$304$$ 1.07318e6 0.666020
$$305$$ −875786. −0.539074
$$306$$ −2.74705e6 −1.67711
$$307$$ 1.08295e6 0.655787 0.327893 0.944715i $$-0.393661\pi$$
0.327893 + 0.944715i $$0.393661\pi$$
$$308$$ 0 0
$$309$$ −821701. −0.489573
$$310$$ −785210. −0.464068
$$311$$ −1.48395e6 −0.869999 −0.435000 0.900431i $$-0.643251\pi$$
−0.435000 + 0.900431i $$0.643251\pi$$
$$312$$ 29549.8 0.0171857
$$313$$ −2.59381e6 −1.49650 −0.748251 0.663416i $$-0.769106\pi$$
−0.748251 + 0.663416i $$0.769106\pi$$
$$314$$ −1.82367e6 −1.04381
$$315$$ 0 0
$$316$$ −3.19721e6 −1.80116
$$317$$ −2.77640e6 −1.55180 −0.775898 0.630858i $$-0.782703\pi$$
−0.775898 + 0.630858i $$0.782703\pi$$
$$318$$ 52404.0 0.0290601
$$319$$ −1.41237e6 −0.777088
$$320$$ 1.27804e6 0.697701
$$321$$ 736739. 0.399072
$$322$$ 0 0
$$323$$ −3.13683e6 −1.67295
$$324$$ 1.90564e6 1.00851
$$325$$ 36799.4 0.0193256
$$326$$ −2.17280e6 −1.13234
$$327$$ 90473.1 0.0467897
$$328$$ 1.06535e6 0.546776
$$329$$ 0 0
$$330$$ 702181. 0.354948
$$331$$ −1.99398e6 −1.00035 −0.500174 0.865925i $$-0.666731\pi$$
−0.500174 + 0.865925i $$0.666731\pi$$
$$332$$ −1.86484e6 −0.928530
$$333$$ −2.02495e6 −1.00070
$$334$$ 4.80340e6 2.35604
$$335$$ 922974. 0.449343
$$336$$ 0 0
$$337$$ −19232.2 −0.00922473 −0.00461236 0.999989i $$-0.501468\pi$$
−0.00461236 + 0.999989i $$0.501468\pi$$
$$338$$ −3.20949e6 −1.52807
$$339$$ 479417. 0.226576
$$340$$ −1.57515e6 −0.738965
$$341$$ −2.44468e6 −1.13851
$$342$$ 4.22833e6 1.95481
$$343$$ 0 0
$$344$$ −1.47783e6 −0.673333
$$345$$ 263445. 0.119163
$$346$$ −898961. −0.403692
$$347$$ −2.23061e6 −0.994489 −0.497245 0.867610i $$-0.665655\pi$$
−0.497245 + 0.867610i $$0.665655\pi$$
$$348$$ 435021. 0.192558
$$349$$ −313791. −0.137904 −0.0689520 0.997620i $$-0.521966\pi$$
−0.0689520 + 0.997620i $$0.521966\pi$$
$$350$$ 0 0
$$351$$ −129357. −0.0560431
$$352$$ 5.19553e6 2.23498
$$353$$ 1.47348e6 0.629370 0.314685 0.949196i $$-0.398101\pi$$
0.314685 + 0.949196i $$0.398101\pi$$
$$354$$ −556888. −0.236189
$$355$$ −102973. −0.0433663
$$356$$ 4.17155e6 1.74450
$$357$$ 0 0
$$358$$ 4.10622e6 1.69330
$$359$$ −1.21115e6 −0.495977 −0.247988 0.968763i $$-0.579769\pi$$
−0.247988 + 0.968763i $$0.579769\pi$$
$$360$$ 583803. 0.237416
$$361$$ 2.35219e6 0.949960
$$362$$ −4.61692e6 −1.85174
$$363$$ 1.42285e6 0.566753
$$364$$ 0 0
$$365$$ 87934.1 0.0345482
$$366$$ 1.44876e6 0.565321
$$367$$ 2.43854e6 0.945073 0.472536 0.881311i $$-0.343339\pi$$
0.472536 + 0.881311i $$0.343339\pi$$
$$368$$ 1.08587e6 0.417984
$$369$$ −2.21884e6 −0.848320
$$370$$ −2.00294e6 −0.760614
$$371$$ 0 0
$$372$$ 752983. 0.282116
$$373$$ −3.48803e6 −1.29810 −0.649050 0.760746i $$-0.724833\pi$$
−0.649050 + 0.760746i $$0.724833\pi$$
$$374$$ −8.45974e6 −3.12736
$$375$$ −74057.0 −0.0271949
$$376$$ −1.72307e6 −0.628539
$$377$$ 122444. 0.0443695
$$378$$ 0 0
$$379$$ −964503. −0.344910 −0.172455 0.985017i $$-0.555170\pi$$
−0.172455 + 0.985017i $$0.555170\pi$$
$$380$$ 2.42451e6 0.861321
$$381$$ −534573. −0.188666
$$382$$ 1.57587e6 0.552539
$$383$$ 3.36433e6 1.17193 0.585965 0.810336i $$-0.300716\pi$$
0.585965 + 0.810336i $$0.300716\pi$$
$$384$$ −953926. −0.330131
$$385$$ 0 0
$$386$$ −4.47504e6 −1.52872
$$387$$ 3.07792e6 1.04467
$$388$$ 3.76348e6 1.26914
$$389$$ −1.18786e6 −0.398007 −0.199003 0.979999i $$-0.563770\pi$$
−0.199003 + 0.979999i $$0.563770\pi$$
$$390$$ −60875.2 −0.0202665
$$391$$ −3.17394e6 −1.04992
$$392$$ 0 0
$$393$$ −870091. −0.284173
$$394$$ −246052. −0.0798520
$$395$$ 1.81102e6 0.584024
$$396$$ 6.61052e6 2.11835
$$397$$ 1.63665e6 0.521171 0.260585 0.965451i $$-0.416084\pi$$
0.260585 + 0.965451i $$0.416084\pi$$
$$398$$ 8.14967e6 2.57889
$$399$$ 0 0
$$400$$ −305249. −0.0953904
$$401$$ −892182. −0.277072 −0.138536 0.990357i $$-0.544240\pi$$
−0.138536 + 0.990357i $$0.544240\pi$$
$$402$$ −1.52682e6 −0.471220
$$403$$ 211940. 0.0650056
$$404$$ −7.95787e6 −2.42574
$$405$$ −1.07943e6 −0.327007
$$406$$ 0 0
$$407$$ −6.23598e6 −1.86603
$$408$$ 716453. 0.213077
$$409$$ −3.38800e6 −1.00146 −0.500732 0.865603i $$-0.666936\pi$$
−0.500732 + 0.865603i $$0.666936\pi$$
$$410$$ −2.19472e6 −0.644793
$$411$$ −437053. −0.127623
$$412$$ −7.65165e6 −2.22081
$$413$$ 0 0
$$414$$ 4.27836e6 1.22681
$$415$$ 1.05632e6 0.301074
$$416$$ −450423. −0.127611
$$417$$ 2.01126e6 0.566407
$$418$$ 1.30215e7 3.64518
$$419$$ −5.29618e6 −1.47376 −0.736882 0.676022i $$-0.763703\pi$$
−0.736882 + 0.676022i $$0.763703\pi$$
$$420$$ 0 0
$$421$$ 5.38110e6 1.47967 0.739836 0.672787i $$-0.234903\pi$$
0.739836 + 0.672787i $$0.234903\pi$$
$$422$$ −1.80511e6 −0.493427
$$423$$ 3.58867e6 0.975175
$$424$$ 134175. 0.0362458
$$425$$ 892224. 0.239608
$$426$$ 170342. 0.0454777
$$427$$ 0 0
$$428$$ 6.86049e6 1.81028
$$429$$ −189529. −0.0497203
$$430$$ 3.04447e6 0.794036
$$431$$ −6.29797e6 −1.63308 −0.816540 0.577290i $$-0.804110\pi$$
−0.816540 + 0.577290i $$0.804110\pi$$
$$432$$ 1.07301e6 0.276627
$$433$$ −4.54572e6 −1.16515 −0.582577 0.812775i $$-0.697956\pi$$
−0.582577 + 0.812775i $$0.697956\pi$$
$$434$$ 0 0
$$435$$ −246413. −0.0624367
$$436$$ 842482. 0.212248
$$437$$ 4.88541e6 1.22376
$$438$$ −145464. −0.0362303
$$439$$ −2.69494e6 −0.667402 −0.333701 0.942679i $$-0.608298\pi$$
−0.333701 + 0.942679i $$0.608298\pi$$
$$440$$ 1.79787e6 0.442717
$$441$$ 0 0
$$442$$ 733412. 0.178563
$$443$$ −3.00924e6 −0.728530 −0.364265 0.931295i $$-0.618680\pi$$
−0.364265 + 0.931295i $$0.618680\pi$$
$$444$$ 1.92074e6 0.462393
$$445$$ −2.36292e6 −0.565653
$$446$$ 3.03592e6 0.722691
$$447$$ 653569. 0.154711
$$448$$ 0 0
$$449$$ 3.52092e6 0.824214 0.412107 0.911135i $$-0.364793\pi$$
0.412107 + 0.911135i $$0.364793\pi$$
$$450$$ −1.20269e6 −0.279976
$$451$$ −6.83308e6 −1.58189
$$452$$ 4.46431e6 1.02780
$$453$$ −1.46261e6 −0.334875
$$454$$ −6.91935e6 −1.57553
$$455$$ 0 0
$$456$$ −1.10278e6 −0.248358
$$457$$ 2.51620e6 0.563578 0.281789 0.959476i $$-0.409072\pi$$
0.281789 + 0.959476i $$0.409072\pi$$
$$458$$ −2.40691e6 −0.536161
$$459$$ −3.13634e6 −0.694850
$$460$$ 2.45319e6 0.540552
$$461$$ −200800. −0.0440060 −0.0220030 0.999758i $$-0.507004\pi$$
−0.0220030 + 0.999758i $$0.507004\pi$$
$$462$$ 0 0
$$463$$ 192190. 0.0416656 0.0208328 0.999783i $$-0.493368\pi$$
0.0208328 + 0.999783i $$0.493368\pi$$
$$464$$ −1.01567e6 −0.219007
$$465$$ −426519. −0.0914757
$$466$$ 1.25290e7 2.67272
$$467$$ 1.49891e6 0.318040 0.159020 0.987275i $$-0.449167\pi$$
0.159020 + 0.987275i $$0.449167\pi$$
$$468$$ −573095. −0.120952
$$469$$ 0 0
$$470$$ 3.54967e6 0.741213
$$471$$ −990599. −0.205753
$$472$$ −1.42586e6 −0.294592
$$473$$ 9.47868e6 1.94803
$$474$$ −2.99587e6 −0.612459
$$475$$ −1.37334e6 −0.279282
$$476$$ 0 0
$$477$$ −279450. −0.0562352
$$478$$ 1.11602e6 0.223409
$$479$$ 7.48400e6 1.49037 0.745186 0.666857i $$-0.232361\pi$$
0.745186 + 0.666857i $$0.232361\pi$$
$$480$$ 906455. 0.179574
$$481$$ 540625. 0.106545
$$482$$ 5.25870e6 1.03100
$$483$$ 0 0
$$484$$ 1.32496e7 2.57092
$$485$$ −2.13178e6 −0.411517
$$486$$ 6.44395e6 1.23755
$$487$$ −2.19160e6 −0.418734 −0.209367 0.977837i $$-0.567140\pi$$
−0.209367 + 0.977837i $$0.567140\pi$$
$$488$$ 3.70942e6 0.705109
$$489$$ −1.18024e6 −0.223203
$$490$$ 0 0
$$491$$ 3.71570e6 0.695564 0.347782 0.937575i $$-0.386935\pi$$
0.347782 + 0.937575i $$0.386935\pi$$
$$492$$ 2.10465e6 0.391983
$$493$$ 2.96873e6 0.550115
$$494$$ −1.12889e6 −0.208129
$$495$$ −3.74446e6 −0.686872
$$496$$ −1.75803e6 −0.320865
$$497$$ 0 0
$$498$$ −1.74740e6 −0.315733
$$499$$ 5.74147e6 1.03222 0.516109 0.856523i $$-0.327380\pi$$
0.516109 + 0.856523i $$0.327380\pi$$
$$500$$ −689616. −0.123362
$$501$$ 2.60916e6 0.464416
$$502$$ −6.60506e6 −1.16982
$$503$$ 8.85276e6 1.56012 0.780061 0.625703i $$-0.215188\pi$$
0.780061 + 0.625703i $$0.215188\pi$$
$$504$$ 0 0
$$505$$ 4.50765e6 0.786541
$$506$$ 1.31755e7 2.28766
$$507$$ −1.74337e6 −0.301210
$$508$$ −4.97792e6 −0.855833
$$509$$ 478571. 0.0818751 0.0409376 0.999162i $$-0.486966\pi$$
0.0409376 + 0.999162i $$0.486966\pi$$
$$510$$ −1.47595e6 −0.251274
$$511$$ 0 0
$$512$$ 5.39114e6 0.908879
$$513$$ 4.82754e6 0.809902
$$514$$ 5.29302e6 0.883681
$$515$$ 4.33419e6 0.720095
$$516$$ −2.91952e6 −0.482711
$$517$$ 1.10516e7 1.81844
$$518$$ 0 0
$$519$$ −488307. −0.0795747
$$520$$ −155865. −0.0252778
$$521$$ −8.96169e6 −1.44642 −0.723212 0.690626i $$-0.757335\pi$$
−0.723212 + 0.690626i $$0.757335\pi$$
$$522$$ −4.00175e6 −0.642796
$$523$$ 1.08551e7 1.73531 0.867656 0.497164i $$-0.165625\pi$$
0.867656 + 0.497164i $$0.165625\pi$$
$$524$$ −8.10226e6 −1.28907
$$525$$ 0 0
$$526$$ −1.23870e7 −1.95210
$$527$$ 5.13861e6 0.805971
$$528$$ 1.57214e6 0.245418
$$529$$ −1.49313e6 −0.231984
$$530$$ −276413. −0.0427434
$$531$$ 2.96967e6 0.457058
$$532$$ 0 0
$$533$$ 592389. 0.0903211
$$534$$ 3.90885e6 0.593193
$$535$$ −3.88604e6 −0.586980
$$536$$ −3.90929e6 −0.587740
$$537$$ 2.23046e6 0.333779
$$538$$ −4.33646e6 −0.645922
$$539$$ 0 0
$$540$$ 2.42413e6 0.357744
$$541$$ −1.15845e7 −1.70170 −0.850852 0.525405i $$-0.823914\pi$$
−0.850852 + 0.525405i $$0.823914\pi$$
$$542$$ 1.35742e6 0.198479
$$543$$ −2.50787e6 −0.365010
$$544$$ −1.09208e7 −1.58218
$$545$$ −477214. −0.0688212
$$546$$ 0 0
$$547$$ 594715. 0.0849846 0.0424923 0.999097i $$-0.486470\pi$$
0.0424923 + 0.999097i $$0.486470\pi$$
$$548$$ −4.06982e6 −0.578927
$$549$$ −7.72569e6 −1.09397
$$550$$ −3.70376e6 −0.522079
$$551$$ −4.56956e6 −0.641202
$$552$$ −1.11583e6 −0.155866
$$553$$ 0 0
$$554$$ −1.54502e7 −2.13875
$$555$$ −1.08798e6 −0.149930
$$556$$ 1.87288e7 2.56935
$$557$$ 4.57366e6 0.624634 0.312317 0.949978i $$-0.398895\pi$$
0.312317 + 0.949978i $$0.398895\pi$$
$$558$$ −6.92667e6 −0.941758
$$559$$ −821748. −0.111227
$$560$$ 0 0
$$561$$ −4.59525e6 −0.616456
$$562$$ −2.08345e7 −2.78254
$$563$$ 2.88071e6 0.383027 0.191513 0.981490i $$-0.438660\pi$$
0.191513 + 0.981490i $$0.438660\pi$$
$$564$$ −3.40398e6 −0.450599
$$565$$ −2.52876e6 −0.333262
$$566$$ −1.68853e6 −0.221548
$$567$$ 0 0
$$568$$ 436145. 0.0567231
$$569$$ 1.18609e7 1.53581 0.767903 0.640566i $$-0.221300\pi$$
0.767903 + 0.640566i $$0.221300\pi$$
$$570$$ 2.27183e6 0.292880
$$571$$ −3.94838e6 −0.506791 −0.253396 0.967363i $$-0.581547\pi$$
−0.253396 + 0.967363i $$0.581547\pi$$
$$572$$ −1.76489e6 −0.225542
$$573$$ 856000. 0.108915
$$574$$ 0 0
$$575$$ −1.38958e6 −0.175273
$$576$$ 1.12741e7 1.41588
$$577$$ 1.33953e6 0.167500 0.0837498 0.996487i $$-0.473310\pi$$
0.0837498 + 0.996487i $$0.473310\pi$$
$$578$$ 5.39296e6 0.671441
$$579$$ −2.43080e6 −0.301337
$$580$$ −2.29459e6 −0.283227
$$581$$ 0 0
$$582$$ 3.52648e6 0.431553
$$583$$ −860587. −0.104863
$$584$$ −372448. −0.0451890
$$585$$ 324623. 0.0392184
$$586$$ −580504. −0.0698331
$$587$$ −1.53600e7 −1.83990 −0.919951 0.392034i $$-0.871772\pi$$
−0.919951 + 0.392034i $$0.871772\pi$$
$$588$$ 0 0
$$589$$ −7.90950e6 −0.939422
$$590$$ 2.93739e6 0.347402
$$591$$ −133653. −0.0157402
$$592$$ −4.48446e6 −0.525903
$$593$$ −963371. −0.112501 −0.0562506 0.998417i $$-0.517915\pi$$
−0.0562506 + 0.998417i $$0.517915\pi$$
$$594$$ 1.30194e7 1.51400
$$595$$ 0 0
$$596$$ 6.08600e6 0.701805
$$597$$ 4.42683e6 0.508343
$$598$$ −1.14224e6 −0.130619
$$599$$ −1.39812e6 −0.159212 −0.0796062 0.996826i $$-0.525366\pi$$
−0.0796062 + 0.996826i $$0.525366\pi$$
$$600$$ 313670. 0.0355710
$$601$$ 9.79835e6 1.10654 0.553270 0.833002i $$-0.313380\pi$$
0.553270 + 0.833002i $$0.313380\pi$$
$$602$$ 0 0
$$603$$ 8.14195e6 0.911875
$$604$$ −1.36197e7 −1.51907
$$605$$ −7.50506e6 −0.833616
$$606$$ −7.45674e6 −0.824836
$$607$$ −1.05240e7 −1.15933 −0.579666 0.814854i $$-0.696817\pi$$
−0.579666 + 0.814854i $$0.696817\pi$$
$$608$$ 1.68096e7 1.84416
$$609$$ 0 0
$$610$$ −7.64173e6 −0.831509
$$611$$ −958110. −0.103827
$$612$$ −1.38950e7 −1.49962
$$613$$ −1.17388e7 −1.26175 −0.630874 0.775886i $$-0.717303\pi$$
−0.630874 + 0.775886i $$0.717303\pi$$
$$614$$ 9.44935e6 1.01154
$$615$$ −1.19215e6 −0.127100
$$616$$ 0 0
$$617$$ −6.83389e6 −0.722695 −0.361348 0.932431i $$-0.617683\pi$$
−0.361348 + 0.932431i $$0.617683\pi$$
$$618$$ −7.16980e6 −0.755155
$$619$$ 1.06954e7 1.12194 0.560969 0.827837i $$-0.310429\pi$$
0.560969 + 0.827837i $$0.310429\pi$$
$$620$$ −3.97173e6 −0.414954
$$621$$ 4.88465e6 0.508282
$$622$$ −1.29483e7 −1.34195
$$623$$ 0 0
$$624$$ −136295. −0.0140126
$$625$$ 390625. 0.0400000
$$626$$ −2.26324e7 −2.30832
$$627$$ 7.07315e6 0.718528
$$628$$ −9.22442e6 −0.933340
$$629$$ 1.31078e7 1.32100
$$630$$ 0 0
$$631$$ 5.17826e6 0.517739 0.258869 0.965912i $$-0.416650\pi$$
0.258869 + 0.965912i $$0.416650\pi$$
$$632$$ −7.67063e6 −0.763903
$$633$$ −980521. −0.0972630
$$634$$ −2.42257e7 −2.39361
$$635$$ 2.81969e6 0.277502
$$636$$ 265068. 0.0259846
$$637$$ 0 0
$$638$$ −1.23237e7 −1.19864
$$639$$ −908369. −0.0880055
$$640$$ 5.03163e6 0.485578
$$641$$ 2.00548e7 1.92785 0.963927 0.266165i $$-0.0857568\pi$$
0.963927 + 0.266165i $$0.0857568\pi$$
$$642$$ 6.42846e6 0.615559
$$643$$ −1.00964e6 −0.0963032 −0.0481516 0.998840i $$-0.515333\pi$$
−0.0481516 + 0.998840i $$0.515333\pi$$
$$644$$ 0 0
$$645$$ 1.65373e6 0.156518
$$646$$ −2.73706e7 −2.58049
$$647$$ 2.66309e6 0.250107 0.125053 0.992150i $$-0.460090\pi$$
0.125053 + 0.992150i $$0.460090\pi$$
$$648$$ 4.57196e6 0.427725
$$649$$ 9.14531e6 0.852289
$$650$$ 321095. 0.0298092
$$651$$ 0 0
$$652$$ −1.09904e7 −1.01250
$$653$$ 6.98872e6 0.641379 0.320689 0.947184i $$-0.396085\pi$$
0.320689 + 0.947184i $$0.396085\pi$$
$$654$$ 789429. 0.0721720
$$655$$ 4.58943e6 0.417980
$$656$$ −4.91384e6 −0.445822
$$657$$ 775705. 0.0701105
$$658$$ 0 0
$$659$$ 1.22406e7 1.09797 0.548985 0.835833i $$-0.315015\pi$$
0.548985 + 0.835833i $$0.315015\pi$$
$$660$$ 3.55175e6 0.317383
$$661$$ 8.94566e6 0.796359 0.398180 0.917307i $$-0.369642\pi$$
0.398180 + 0.917307i $$0.369642\pi$$
$$662$$ −1.73986e7 −1.54301
$$663$$ 398383. 0.0351979
$$664$$ −4.47406e6 −0.393805
$$665$$ 0 0
$$666$$ −1.76688e7 −1.54355
$$667$$ −4.62362e6 −0.402409
$$668$$ 2.42964e7 2.10669
$$669$$ 1.64908e6 0.142455
$$670$$ 8.05347e6 0.693100
$$671$$ −2.37918e7 −2.03996
$$672$$ 0 0
$$673$$ −8.40438e6 −0.715267 −0.357633 0.933862i $$-0.616416\pi$$
−0.357633 + 0.933862i $$0.616416\pi$$
$$674$$ −167811. −0.0142289
$$675$$ −1.37312e6 −0.115998
$$676$$ −1.62342e7 −1.36635
$$677$$ 2.67443e6 0.224264 0.112132 0.993693i $$-0.464232\pi$$
0.112132 + 0.993693i $$0.464232\pi$$
$$678$$ 4.18318e6 0.349488
$$679$$ 0 0
$$680$$ −3.77904e6 −0.313407
$$681$$ −3.75853e6 −0.310563
$$682$$ −2.13312e7 −1.75612
$$683$$ 8.95973e6 0.734925 0.367463 0.930038i $$-0.380227\pi$$
0.367463 + 0.930038i $$0.380227\pi$$
$$684$$ 2.13876e7 1.74792
$$685$$ 2.30530e6 0.187716
$$686$$ 0 0
$$687$$ −1.30741e6 −0.105687
$$688$$ 6.81636e6 0.549012
$$689$$ 74608.0 0.00598739
$$690$$ 2.29871e6 0.183807
$$691$$ −22616.0 −0.00180186 −0.000900928 1.00000i $$-0.500287\pi$$
−0.000900928 1.00000i $$0.500287\pi$$
$$692$$ −4.54710e6 −0.360968
$$693$$ 0 0
$$694$$ −1.94633e7 −1.53398
$$695$$ −1.06087e7 −0.833107
$$696$$ 1.04369e6 0.0816672
$$697$$ 1.43628e7 1.11985
$$698$$ −2.73800e6 −0.212714
$$699$$ 6.80566e6 0.526839
$$700$$ 0 0
$$701$$ 1.80427e7 1.38677 0.693386 0.720566i $$-0.256118\pi$$
0.693386 + 0.720566i $$0.256118\pi$$
$$702$$ −1.12871e6 −0.0864451
$$703$$ −2.01759e7 −1.53973
$$704$$ 3.47195e7 2.64023
$$705$$ 1.92815e6 0.146106
$$706$$ 1.28569e7 0.970789
$$707$$ 0 0
$$708$$ −2.81684e6 −0.211193
$$709$$ 1.52807e7 1.14164 0.570819 0.821076i $$-0.306626\pi$$
0.570819 + 0.821076i $$0.306626\pi$$
$$710$$ −898497. −0.0668915
$$711$$ 1.59758e7 1.18519
$$712$$ 1.00082e7 0.739874
$$713$$ −8.00308e6 −0.589567
$$714$$ 0 0
$$715$$ 999702. 0.0731317
$$716$$ 2.07700e7 1.51409
$$717$$ 606211. 0.0440378
$$718$$ −1.05679e7 −0.765032
$$719$$ 1.45192e7 1.04742 0.523710 0.851897i $$-0.324548\pi$$
0.523710 + 0.851897i $$0.324548\pi$$
$$720$$ −2.69274e6 −0.193581
$$721$$ 0 0
$$722$$ 2.05242e7 1.46529
$$723$$ 2.85648e6 0.203229
$$724$$ −2.33532e7 −1.65577
$$725$$ 1.29974e6 0.0918359
$$726$$ 1.24152e7 0.874202
$$727$$ 1.98706e7 1.39436 0.697179 0.716897i $$-0.254438\pi$$
0.697179 + 0.716897i $$0.254438\pi$$
$$728$$ 0 0
$$729$$ −6.99176e6 −0.487268
$$730$$ 767275. 0.0532898
$$731$$ −1.99238e7 −1.37904
$$732$$ 7.32810e6 0.505491
$$733$$ −1.58619e7 −1.09042 −0.545211 0.838299i $$-0.683550\pi$$
−0.545211 + 0.838299i $$0.683550\pi$$
$$734$$ 2.12777e7 1.45775
$$735$$ 0 0
$$736$$ 1.70085e7 1.15736
$$737$$ 2.50738e7 1.70040
$$738$$ −1.93606e7 −1.30851
$$739$$ 2.51524e7 1.69421 0.847107 0.531422i $$-0.178342\pi$$
0.847107 + 0.531422i $$0.178342\pi$$
$$740$$ −1.01312e7 −0.680116
$$741$$ −613202. −0.0410259
$$742$$ 0 0
$$743$$ 7.30561e6 0.485495 0.242747 0.970090i $$-0.421951\pi$$
0.242747 + 0.970090i $$0.421951\pi$$
$$744$$ 1.80653e6 0.119650
$$745$$ −3.44735e6 −0.227559
$$746$$ −3.04350e7 −2.00229
$$747$$ 9.31822e6 0.610986
$$748$$ −4.27908e7 −2.79638
$$749$$ 0 0
$$750$$ −646189. −0.0419475
$$751$$ −826829. −0.0534953 −0.0267477 0.999642i $$-0.508515\pi$$
−0.0267477 + 0.999642i $$0.508515\pi$$
$$752$$ 7.94747e6 0.512489
$$753$$ −3.58781e6 −0.230591
$$754$$ 1.06839e6 0.0684389
$$755$$ 7.71475e6 0.492554
$$756$$ 0 0
$$757$$ 4.70431e6 0.298371 0.149185 0.988809i $$-0.452335\pi$$
0.149185 + 0.988809i $$0.452335\pi$$
$$758$$ −8.41583e6 −0.532015
$$759$$ 7.15683e6 0.450937
$$760$$ 5.81680e6 0.365301
$$761$$ −1.42681e6 −0.0893107 −0.0446553 0.999002i $$-0.514219\pi$$
−0.0446553 + 0.999002i $$0.514219\pi$$
$$762$$ −4.66445e6 −0.291013
$$763$$ 0 0
$$764$$ 7.97104e6 0.494062
$$765$$ 7.87069e6 0.486250
$$766$$ 2.93557e7 1.80767
$$767$$ −792847. −0.0486632
$$768$$ −569993. −0.0348712
$$769$$ −9.32444e6 −0.568600 −0.284300 0.958735i $$-0.591761\pi$$
−0.284300 + 0.958735i $$0.591761\pi$$
$$770$$ 0 0
$$771$$ 2.87512e6 0.174189
$$772$$ −2.26355e7 −1.36693
$$773$$ −1.52132e7 −0.915741 −0.457870 0.889019i $$-0.651388\pi$$
−0.457870 + 0.889019i $$0.651388\pi$$
$$774$$ 2.68566e7 1.61138
$$775$$ 2.24974e6 0.134548
$$776$$ 9.02921e6 0.538264
$$777$$ 0 0
$$778$$ −1.03647e7 −0.613916
$$779$$ −2.21077e7 −1.30527
$$780$$ −307917. −0.0181216
$$781$$ −2.79739e6 −0.164106
$$782$$ −2.76944e7 −1.61948
$$783$$ −4.56885e6 −0.266319
$$784$$ 0 0
$$785$$ 5.22507e6 0.302634
$$786$$ −7.59203e6 −0.438331
$$787$$ 2.20016e7 1.26624 0.633122 0.774052i $$-0.281773\pi$$
0.633122 + 0.774052i $$0.281773\pi$$
$$788$$ −1.24457e6 −0.0714011
$$789$$ −6.72850e6 −0.384792
$$790$$ 1.58022e7 0.900843
$$791$$ 0 0
$$792$$ 1.58598e7 0.898428
$$793$$ 2.06262e6 0.116476
$$794$$ 1.42807e7 0.803893
$$795$$ −150145. −0.00842545
$$796$$ 4.12225e7 2.30596
$$797$$ 3.02905e7 1.68912 0.844559 0.535462i $$-0.179862\pi$$
0.844559 + 0.535462i $$0.179862\pi$$
$$798$$ 0 0
$$799$$ −2.32299e7 −1.28730
$$800$$ −4.78124e6 −0.264128
$$801$$ −2.08444e7 −1.14791
$$802$$ −7.78479e6 −0.427377
$$803$$ 2.38884e6 0.130737
$$804$$ −7.72294e6 −0.421350
$$805$$ 0 0
$$806$$ 1.84930e6 0.100269
$$807$$ −2.35553e6 −0.127322
$$808$$ −1.90923e7 −1.02880
$$809$$ −1.29572e7 −0.696050 −0.348025 0.937485i $$-0.613148\pi$$
−0.348025 + 0.937485i $$0.613148\pi$$
$$810$$ −9.41863e6 −0.504400
$$811$$ −1.87248e7 −0.999688 −0.499844 0.866115i $$-0.666609\pi$$
−0.499844 + 0.866115i $$0.666609\pi$$
$$812$$ 0 0
$$813$$ 737336. 0.0391236
$$814$$ −5.44124e7 −2.87831
$$815$$ 6.22538e6 0.328301
$$816$$ −3.30457e6 −0.173736
$$817$$ 3.06673e7 1.60738
$$818$$ −2.95622e7 −1.54473
$$819$$ 0 0
$$820$$ −1.11013e7 −0.576553
$$821$$ 2.07863e7 1.07626 0.538132 0.842861i $$-0.319130\pi$$
0.538132 + 0.842861i $$0.319130\pi$$
$$822$$ −3.81353e6 −0.196856
$$823$$ −2.40815e7 −1.23932 −0.619660 0.784870i $$-0.712730\pi$$
−0.619660 + 0.784870i $$0.712730\pi$$
$$824$$ −1.83576e7 −0.941884
$$825$$ −2.01185e6 −0.102911
$$826$$ 0 0
$$827$$ −1.53428e6 −0.0780081 −0.0390041 0.999239i $$-0.512419\pi$$
−0.0390041 + 0.999239i $$0.512419\pi$$
$$828$$ 2.16407e7 1.09697
$$829$$ −1.47802e7 −0.746952 −0.373476 0.927640i $$-0.621834\pi$$
−0.373476 + 0.927640i $$0.621834\pi$$
$$830$$ 9.21695e6 0.464400
$$831$$ −8.39242e6 −0.421585
$$832$$ −3.00999e6 −0.150750
$$833$$ 0 0
$$834$$ 1.75494e7 0.873669
$$835$$ −1.37624e7 −0.683092
$$836$$ 6.58649e7 3.25940
$$837$$ −7.90827e6 −0.390183
$$838$$ −4.62121e7 −2.27324
$$839$$ 2.41688e6 0.118536 0.0592681 0.998242i $$-0.481123\pi$$
0.0592681 + 0.998242i $$0.481123\pi$$
$$840$$ 0 0
$$841$$ −1.61865e7 −0.789154
$$842$$ 4.69531e7 2.28236
$$843$$ −1.13171e7 −0.548486
$$844$$ −9.13057e6 −0.441206
$$845$$ 9.19566e6 0.443038
$$846$$ 3.13132e7 1.50418
$$847$$ 0 0
$$848$$ −618870. −0.0295536
$$849$$ −917194. −0.0436709
$$850$$ 7.78515e6 0.369590
$$851$$ −2.04145e7 −0.966309
$$852$$ 861621. 0.0406647
$$853$$ −3.98657e7 −1.87598 −0.937988 0.346669i $$-0.887313\pi$$
−0.937988 + 0.346669i $$0.887313\pi$$
$$854$$ 0 0
$$855$$ −1.21148e7 −0.566762
$$856$$ 1.64595e7 0.767770
$$857$$ 3.36474e7 1.56495 0.782473 0.622684i $$-0.213958\pi$$
0.782473 + 0.622684i $$0.213958\pi$$
$$858$$ −1.65375e6 −0.0766923
$$859$$ 8.27879e6 0.382811 0.191405 0.981511i $$-0.438695\pi$$
0.191405 + 0.981511i $$0.438695\pi$$
$$860$$ 1.53995e7 0.710001
$$861$$ 0 0
$$862$$ −5.49533e7 −2.51898
$$863$$ 1.06111e7 0.484990 0.242495 0.970153i $$-0.422034\pi$$
0.242495 + 0.970153i $$0.422034\pi$$
$$864$$ 1.68070e7 0.765958
$$865$$ 2.57565e6 0.117043
$$866$$ −3.96640e7 −1.79722
$$867$$ 2.92941e6 0.132353
$$868$$ 0 0
$$869$$ 4.91987e7 2.21006
$$870$$ −2.15009e6 −0.0963072
$$871$$ −2.17375e6 −0.0970879
$$872$$ 2.02126e6 0.0900181
$$873$$ −1.88053e7 −0.835113
$$874$$ 4.26280e7 1.88763
$$875$$ 0 0
$$876$$ −735784. −0.0323959
$$877$$ −1.80051e6 −0.0790492 −0.0395246 0.999219i $$-0.512584\pi$$
−0.0395246 + 0.999219i $$0.512584\pi$$
$$878$$ −2.35148e7 −1.02945
$$879$$ −315325. −0.0137653
$$880$$ −8.29248e6 −0.360976
$$881$$ −3.86161e7 −1.67621 −0.838105 0.545508i $$-0.816337\pi$$
−0.838105 + 0.545508i $$0.816337\pi$$
$$882$$ 0 0
$$883$$ 8.80155e6 0.379890 0.189945 0.981795i $$-0.439169\pi$$
0.189945 + 0.981795i $$0.439169\pi$$
$$884$$ 3.70972e6 0.159665
$$885$$ 1.59557e6 0.0684789
$$886$$ −2.62573e7 −1.12374
$$887$$ 2.76706e7 1.18089 0.590444 0.807078i $$-0.298953\pi$$
0.590444 + 0.807078i $$0.298953\pi$$
$$888$$ 4.60817e6 0.196108
$$889$$ 0 0
$$890$$ −2.06178e7 −0.872505
$$891$$ −2.93241e7 −1.23746
$$892$$ 1.53562e7 0.646207
$$893$$ 3.57562e7 1.50045
$$894$$ 5.70275e6 0.238639
$$895$$ −1.17649e7 −0.490943
$$896$$ 0 0
$$897$$ −620457. −0.0257472
$$898$$ 3.07220e7 1.27133
$$899$$ 7.48565e6 0.308909
$$900$$ −6.08340e6 −0.250346
$$901$$ 1.80892e6 0.0742347
$$902$$ −5.96224e7 −2.44002
$$903$$ 0 0
$$904$$ 1.07106e7 0.435907
$$905$$ 1.32281e7 0.536880
$$906$$ −1.27621e7 −0.516536
$$907$$ −3.47961e7 −1.40447 −0.702235 0.711945i $$-0.747815\pi$$
−0.702235 + 0.711945i $$0.747815\pi$$
$$908$$ −3.49993e7 −1.40878
$$909$$ 3.97639e7 1.59617
$$910$$ 0 0
$$911$$ 2.48752e7 0.993049 0.496524 0.868023i $$-0.334609\pi$$
0.496524 + 0.868023i $$0.334609\pi$$
$$912$$ 5.08648e6 0.202502
$$913$$ 2.86962e7 1.13932
$$914$$ 2.19552e7 0.869306
$$915$$ −4.15092e6 −0.163905
$$916$$ −1.21745e7 −0.479418
$$917$$ 0 0
$$918$$ −2.73663e7 −1.07179
$$919$$ −1.88971e7 −0.738086 −0.369043 0.929412i $$-0.620314\pi$$
−0.369043 + 0.929412i $$0.620314\pi$$
$$920$$ 5.88562e6 0.229257
$$921$$ 5.13280e6 0.199391
$$922$$ −1.75209e6 −0.0678782
$$923$$ 242518. 0.00937000
$$924$$ 0 0
$$925$$ 5.73872e6 0.220527
$$926$$ 1.67696e6 0.0642681
$$927$$ 3.82337e7 1.46133
$$928$$ −1.59088e7 −0.606412
$$929$$ −2.29202e7 −0.871324 −0.435662 0.900110i $$-0.643486\pi$$
−0.435662 + 0.900110i $$0.643486\pi$$
$$930$$ −3.72162e6 −0.141099
$$931$$ 0 0
$$932$$ 6.33741e7 2.38986
$$933$$ −7.03341e6 −0.264522
$$934$$ 1.30788e7 0.490570
$$935$$ 2.42384e7 0.906723
$$936$$ −1.37495e6 −0.0512977
$$937$$ 3.70307e7 1.37789 0.688943 0.724815i $$-0.258075\pi$$
0.688943 + 0.724815i $$0.258075\pi$$
$$938$$ 0 0
$$939$$ −1.22937e7 −0.455009
$$940$$ 1.79548e7 0.662769
$$941$$ −1.25128e7 −0.460660 −0.230330 0.973113i $$-0.573981\pi$$
−0.230330 + 0.973113i $$0.573981\pi$$
$$942$$ −8.64353e6 −0.317369
$$943$$ −2.23692e7 −0.819166
$$944$$ 6.57663e6 0.240200
$$945$$ 0 0
$$946$$ 8.27068e7 3.00479
$$947$$ −1.22987e7 −0.445640 −0.222820 0.974860i $$-0.571526\pi$$
−0.222820 + 0.974860i $$0.571526\pi$$
$$948$$ −1.51536e7 −0.547641
$$949$$ −207099. −0.00746471
$$950$$ −1.19831e7 −0.430786
$$951$$ −1.31592e7 −0.471821
$$952$$ 0 0
$$953$$ 3.53421e6 0.126055 0.0630275 0.998012i $$-0.479924\pi$$
0.0630275 + 0.998012i $$0.479924\pi$$
$$954$$ −2.43836e6 −0.0867414
$$955$$ −4.51510e6 −0.160199
$$956$$ 5.64501e6 0.199765
$$957$$ −6.69411e6 −0.236273
$$958$$ 6.53021e7 2.29886
$$959$$ 0 0
$$960$$ 6.05746e6 0.212135
$$961$$ −1.56721e7 −0.547419
$$962$$ 4.71725e6 0.164343
$$963$$ −3.42805e7 −1.19119
$$964$$ 2.65994e7 0.921890
$$965$$ 1.28216e7 0.443226
$$966$$ 0 0
$$967$$ −9.47189e6 −0.325740 −0.162870 0.986648i $$-0.552075\pi$$
−0.162870 + 0.986648i $$0.552075\pi$$
$$968$$ 3.17879e7 1.09037
$$969$$ −1.48675e7 −0.508659
$$970$$ −1.86010e7 −0.634755
$$971$$ 1.18431e7 0.403106 0.201553 0.979478i $$-0.435401\pi$$
0.201553 + 0.979478i $$0.435401\pi$$
$$972$$ 3.25946e7 1.10657
$$973$$ 0 0
$$974$$ −1.91229e7 −0.645887
$$975$$ 174416. 0.00587591
$$976$$ −1.71093e7 −0.574921
$$977$$ −7.17342e6 −0.240431 −0.120215 0.992748i $$-0.538358\pi$$
−0.120215 + 0.992748i $$0.538358\pi$$
$$978$$ −1.02983e7 −0.344285
$$979$$ −6.41918e7 −2.14054
$$980$$ 0 0
$$981$$ −4.20971e6 −0.139663
$$982$$ 3.24216e7 1.07289
$$983$$ 3.00611e7 0.992251 0.496126 0.868251i $$-0.334755\pi$$
0.496126 + 0.868251i $$0.334755\pi$$
$$984$$ 5.04940e6 0.166246
$$985$$ 704974. 0.0231517
$$986$$ 2.59039e7 0.848540
$$987$$ 0 0
$$988$$ −5.71011e6 −0.186103
$$989$$ 3.10301e7 1.00877
$$990$$ −3.26725e7 −1.05948
$$991$$ −2.55378e7 −0.826035 −0.413018 0.910723i $$-0.635525\pi$$
−0.413018 + 0.910723i $$0.635525\pi$$
$$992$$ −2.75367e7 −0.888451
$$993$$ −9.45077e6 −0.304154
$$994$$ 0 0
$$995$$ −2.33500e7 −0.747703
$$996$$ −8.83867e6 −0.282318
$$997$$ −2.42489e7 −0.772598 −0.386299 0.922374i $$-0.626247\pi$$
−0.386299 + 0.922374i $$0.626247\pi$$
$$998$$ 5.00975e7 1.59217
$$999$$ −2.01727e7 −0.639515
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 245.6.a.i.1.6 6
7.2 even 3 35.6.e.a.11.1 12
7.4 even 3 35.6.e.a.16.1 yes 12
7.6 odd 2 245.6.a.h.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.e.a.11.1 12 7.2 even 3
35.6.e.a.16.1 yes 12 7.4 even 3
245.6.a.h.1.6 6 7.6 odd 2
245.6.a.i.1.6 6 1.1 even 1 trivial