# Properties

 Label 825.6.a.v.1.2 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $13$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$1$$ Dimension: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400$$ x^13 - 306*x^11 - 206*x^10 + 34574*x^9 + 39928*x^8 - 1788312*x^7 - 2591628*x^6 + 42852537*x^5 + 63733360*x^4 - 448113518*x^3 - 549984598*x^2 + 1518551280*x + 522579400 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 5^{7}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-9.04603$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-10.0460 q^{2} -9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} +29.0524 q^{7} -370.927 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-10.0460 q^{2} -9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} +29.0524 q^{7} -370.927 q^{8} +81.0000 q^{9} +121.000 q^{11} -620.305 q^{12} -1023.51 q^{13} -291.862 q^{14} +1520.82 q^{16} +1509.69 q^{17} -813.728 q^{18} +1643.04 q^{19} -261.472 q^{21} -1215.57 q^{22} -1478.11 q^{23} +3338.34 q^{24} +10282.2 q^{26} -729.000 q^{27} +2002.37 q^{28} -4572.33 q^{29} +7531.62 q^{31} -3408.50 q^{32} -1089.00 q^{33} -15166.4 q^{34} +5582.74 q^{36} -4408.40 q^{37} -16506.0 q^{38} +9211.62 q^{39} +5629.62 q^{41} +2626.75 q^{42} +2283.11 q^{43} +8339.65 q^{44} +14849.1 q^{46} -5980.48 q^{47} -13687.3 q^{48} -15963.0 q^{49} -13587.2 q^{51} -70543.3 q^{52} -28498.0 q^{53} +7323.56 q^{54} -10776.3 q^{56} -14787.3 q^{57} +45933.7 q^{58} +25942.8 q^{59} -51360.7 q^{61} -75662.8 q^{62} +2353.25 q^{63} -14424.2 q^{64} +10940.1 q^{66} +39180.2 q^{67} +104052. q^{68} +13303.0 q^{69} +37560.3 q^{71} -30045.1 q^{72} -58687.6 q^{73} +44286.9 q^{74} +113242. q^{76} +3515.34 q^{77} -92540.2 q^{78} -8274.06 q^{79} +6561.00 q^{81} -56555.3 q^{82} +21355.4 q^{83} -18021.4 q^{84} -22936.2 q^{86} +41150.9 q^{87} -44882.1 q^{88} +78385.8 q^{89} -29735.5 q^{91} -101875. q^{92} -67784.5 q^{93} +60080.1 q^{94} +30676.5 q^{96} -64574.7 q^{97} +160364. q^{98} +9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10})$$ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 $$13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100})$$ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.0460 −1.77590 −0.887952 0.459936i $$-0.847872\pi$$
−0.887952 + 0.459936i $$0.847872\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 68.9227 2.15384
$$5$$ 0 0
$$6$$ 90.4143 1.02532
$$7$$ 29.0524 0.224098 0.112049 0.993703i $$-0.464259\pi$$
0.112049 + 0.993703i $$0.464259\pi$$
$$8$$ −370.927 −2.04910
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ −620.305 −1.24352
$$13$$ −1023.51 −1.67971 −0.839856 0.542809i $$-0.817361\pi$$
−0.839856 + 0.542809i $$0.817361\pi$$
$$14$$ −291.862 −0.397976
$$15$$ 0 0
$$16$$ 1520.82 1.48517
$$17$$ 1509.69 1.26696 0.633482 0.773757i $$-0.281625\pi$$
0.633482 + 0.773757i $$0.281625\pi$$
$$18$$ −813.728 −0.591968
$$19$$ 1643.04 1.04415 0.522075 0.852900i $$-0.325158\pi$$
0.522075 + 0.852900i $$0.325158\pi$$
$$20$$ 0 0
$$21$$ −261.472 −0.129383
$$22$$ −1215.57 −0.535455
$$23$$ −1478.11 −0.582622 −0.291311 0.956628i $$-0.594092\pi$$
−0.291311 + 0.956628i $$0.594092\pi$$
$$24$$ 3338.34 1.18305
$$25$$ 0 0
$$26$$ 10282.2 2.98301
$$27$$ −729.000 −0.192450
$$28$$ 2002.37 0.482669
$$29$$ −4572.33 −1.00958 −0.504792 0.863241i $$-0.668431\pi$$
−0.504792 + 0.863241i $$0.668431\pi$$
$$30$$ 0 0
$$31$$ 7531.62 1.40762 0.703808 0.710391i $$-0.251482\pi$$
0.703808 + 0.710391i $$0.251482\pi$$
$$32$$ −3408.50 −0.588421
$$33$$ −1089.00 −0.174078
$$34$$ −15166.4 −2.25001
$$35$$ 0 0
$$36$$ 5582.74 0.717945
$$37$$ −4408.40 −0.529391 −0.264695 0.964332i $$-0.585271\pi$$
−0.264695 + 0.964332i $$0.585271\pi$$
$$38$$ −16506.0 −1.85431
$$39$$ 9211.62 0.969783
$$40$$ 0 0
$$41$$ 5629.62 0.523021 0.261511 0.965201i $$-0.415779\pi$$
0.261511 + 0.965201i $$0.415779\pi$$
$$42$$ 2626.75 0.229771
$$43$$ 2283.11 0.188302 0.0941510 0.995558i $$-0.469986\pi$$
0.0941510 + 0.995558i $$0.469986\pi$$
$$44$$ 8339.65 0.649406
$$45$$ 0 0
$$46$$ 14849.1 1.03468
$$47$$ −5980.48 −0.394904 −0.197452 0.980313i $$-0.563267\pi$$
−0.197452 + 0.980313i $$0.563267\pi$$
$$48$$ −13687.3 −0.857464
$$49$$ −15963.0 −0.949780
$$50$$ 0 0
$$51$$ −13587.2 −0.731482
$$52$$ −70543.3 −3.61782
$$53$$ −28498.0 −1.39356 −0.696778 0.717287i $$-0.745384\pi$$
−0.696778 + 0.717287i $$0.745384\pi$$
$$54$$ 7323.56 0.341773
$$55$$ 0 0
$$56$$ −10776.3 −0.459199
$$57$$ −14787.3 −0.602840
$$58$$ 45933.7 1.79292
$$59$$ 25942.8 0.970258 0.485129 0.874443i $$-0.338773\pi$$
0.485129 + 0.874443i $$0.338773\pi$$
$$60$$ 0 0
$$61$$ −51360.7 −1.76728 −0.883642 0.468163i $$-0.844916\pi$$
−0.883642 + 0.468163i $$0.844916\pi$$
$$62$$ −75662.8 −2.49979
$$63$$ 2353.25 0.0746992
$$64$$ −14424.2 −0.440193
$$65$$ 0 0
$$66$$ 10940.1 0.309145
$$67$$ 39180.2 1.06630 0.533151 0.846020i $$-0.321008\pi$$
0.533151 + 0.846020i $$0.321008\pi$$
$$68$$ 104052. 2.72883
$$69$$ 13303.0 0.336377
$$70$$ 0 0
$$71$$ 37560.3 0.884267 0.442133 0.896949i $$-0.354222\pi$$
0.442133 + 0.896949i $$0.354222\pi$$
$$72$$ −30045.1 −0.683034
$$73$$ −58687.6 −1.28896 −0.644480 0.764621i $$-0.722926\pi$$
−0.644480 + 0.764621i $$0.722926\pi$$
$$74$$ 44286.9 0.940147
$$75$$ 0 0
$$76$$ 113242. 2.24893
$$77$$ 3515.34 0.0675680
$$78$$ −92540.2 −1.72224
$$79$$ −8274.06 −0.149160 −0.0745798 0.997215i $$-0.523762\pi$$
−0.0745798 + 0.997215i $$0.523762\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −56555.3 −0.928836
$$83$$ 21355.4 0.340261 0.170130 0.985422i $$-0.445581\pi$$
0.170130 + 0.985422i $$0.445581\pi$$
$$84$$ −18021.4 −0.278669
$$85$$ 0 0
$$86$$ −22936.2 −0.334406
$$87$$ 41150.9 0.582883
$$88$$ −44882.1 −0.617827
$$89$$ 78385.8 1.04897 0.524484 0.851420i $$-0.324258\pi$$
0.524484 + 0.851420i $$0.324258\pi$$
$$90$$ 0 0
$$91$$ −29735.5 −0.376420
$$92$$ −101875. −1.25487
$$93$$ −67784.5 −0.812687
$$94$$ 60080.1 0.701312
$$95$$ 0 0
$$96$$ 30676.5 0.339725
$$97$$ −64574.7 −0.696840 −0.348420 0.937339i $$-0.613282\pi$$
−0.348420 + 0.937339i $$0.613282\pi$$
$$98$$ 160364. 1.68672
$$99$$ 9801.00 0.100504
$$100$$ 0 0
$$101$$ 48364.4 0.471762 0.235881 0.971782i $$-0.424203\pi$$
0.235881 + 0.971782i $$0.424203\pi$$
$$102$$ 136497. 1.29904
$$103$$ −121923. −1.13238 −0.566192 0.824273i $$-0.691584\pi$$
−0.566192 + 0.824273i $$0.691584\pi$$
$$104$$ 379648. 3.44190
$$105$$ 0 0
$$106$$ 286291. 2.47482
$$107$$ 191883. 1.62023 0.810117 0.586269i $$-0.199404\pi$$
0.810117 + 0.586269i $$0.199404\pi$$
$$108$$ −50244.7 −0.414506
$$109$$ −15660.0 −0.126248 −0.0631240 0.998006i $$-0.520106\pi$$
−0.0631240 + 0.998006i $$0.520106\pi$$
$$110$$ 0 0
$$111$$ 39675.6 0.305644
$$112$$ 44183.4 0.332823
$$113$$ 15119.6 0.111389 0.0556947 0.998448i $$-0.482263\pi$$
0.0556947 + 0.998448i $$0.482263\pi$$
$$114$$ 148554. 1.07059
$$115$$ 0 0
$$116$$ −315137. −2.17448
$$117$$ −82904.6 −0.559904
$$118$$ −260622. −1.72309
$$119$$ 43860.0 0.283924
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 515971. 3.13853
$$123$$ −50666.6 −0.301966
$$124$$ 519099. 3.03177
$$125$$ 0 0
$$126$$ −23640.8 −0.132659
$$127$$ 223751. 1.23099 0.615496 0.788140i $$-0.288956\pi$$
0.615496 + 0.788140i $$0.288956\pi$$
$$128$$ 253978. 1.37016
$$129$$ −20548.0 −0.108716
$$130$$ 0 0
$$131$$ 269255. 1.37084 0.685418 0.728149i $$-0.259619\pi$$
0.685418 + 0.728149i $$0.259619\pi$$
$$132$$ −75056.9 −0.374935
$$133$$ 47734.1 0.233991
$$134$$ −393606. −1.89365
$$135$$ 0 0
$$136$$ −559983. −2.59614
$$137$$ −172187. −0.783790 −0.391895 0.920010i $$-0.628180\pi$$
−0.391895 + 0.920010i $$0.628180\pi$$
$$138$$ −133642. −0.597374
$$139$$ −350209. −1.53741 −0.768707 0.639601i $$-0.779099\pi$$
−0.768707 + 0.639601i $$0.779099\pi$$
$$140$$ 0 0
$$141$$ 53824.4 0.227998
$$142$$ −377332. −1.57037
$$143$$ −123845. −0.506452
$$144$$ 123186. 0.495057
$$145$$ 0 0
$$146$$ 589578. 2.28907
$$147$$ 143667. 0.548356
$$148$$ −303839. −1.14022
$$149$$ 519184. 1.91582 0.957912 0.287062i $$-0.0926785\pi$$
0.957912 + 0.287062i $$0.0926785\pi$$
$$150$$ 0 0
$$151$$ −230203. −0.821617 −0.410808 0.911722i $$-0.634753\pi$$
−0.410808 + 0.911722i $$0.634753\pi$$
$$152$$ −609446. −2.13957
$$153$$ 122285. 0.422321
$$154$$ −35315.2 −0.119994
$$155$$ 0 0
$$156$$ 634890. 2.08875
$$157$$ 328864. 1.06480 0.532399 0.846494i $$-0.321291\pi$$
0.532399 + 0.846494i $$0.321291\pi$$
$$158$$ 83121.5 0.264893
$$159$$ 256482. 0.804569
$$160$$ 0 0
$$161$$ −42942.7 −0.130564
$$162$$ −65912.0 −0.197323
$$163$$ 246382. 0.726341 0.363171 0.931723i $$-0.381694\pi$$
0.363171 + 0.931723i $$0.381694\pi$$
$$164$$ 388009. 1.12650
$$165$$ 0 0
$$166$$ −214537. −0.604270
$$167$$ 581413. 1.61322 0.806610 0.591084i $$-0.201300\pi$$
0.806610 + 0.591084i $$0.201300\pi$$
$$168$$ 96986.9 0.265118
$$169$$ 676286. 1.82143
$$170$$ 0 0
$$171$$ 133086. 0.348050
$$172$$ 157358. 0.405572
$$173$$ 264650. 0.672290 0.336145 0.941810i $$-0.390877\pi$$
0.336145 + 0.941810i $$0.390877\pi$$
$$174$$ −413404. −1.03515
$$175$$ 0 0
$$176$$ 184019. 0.447796
$$177$$ −233485. −0.560179
$$178$$ −787466. −1.86287
$$179$$ −13886.5 −0.0323937 −0.0161969 0.999869i $$-0.505156\pi$$
−0.0161969 + 0.999869i $$0.505156\pi$$
$$180$$ 0 0
$$181$$ −752593. −1.70751 −0.853756 0.520673i $$-0.825681\pi$$
−0.853756 + 0.520673i $$0.825681\pi$$
$$182$$ 298724. 0.668485
$$183$$ 462246. 1.02034
$$184$$ 548271. 1.19385
$$185$$ 0 0
$$186$$ 680966. 1.44325
$$187$$ 182672. 0.382004
$$188$$ −412191. −0.850558
$$189$$ −21179.2 −0.0431276
$$190$$ 0 0
$$191$$ −25416.1 −0.0504110 −0.0252055 0.999682i $$-0.508024\pi$$
−0.0252055 + 0.999682i $$0.508024\pi$$
$$192$$ 129818. 0.254145
$$193$$ 489822. 0.946553 0.473277 0.880914i $$-0.343071\pi$$
0.473277 + 0.880914i $$0.343071\pi$$
$$194$$ 648719. 1.23752
$$195$$ 0 0
$$196$$ −1.10021e6 −2.04567
$$197$$ 324827. 0.596329 0.298164 0.954515i $$-0.403626\pi$$
0.298164 + 0.954515i $$0.403626\pi$$
$$198$$ −98461.1 −0.178485
$$199$$ −398516. −0.713368 −0.356684 0.934225i $$-0.616093\pi$$
−0.356684 + 0.934225i $$0.616093\pi$$
$$200$$ 0 0
$$201$$ −352622. −0.615629
$$202$$ −485870. −0.837803
$$203$$ −132837. −0.226245
$$204$$ −936465. −1.57549
$$205$$ 0 0
$$206$$ 1.22485e6 2.01101
$$207$$ −119727. −0.194207
$$208$$ −1.55657e6 −2.49466
$$209$$ 198807. 0.314823
$$210$$ 0 0
$$211$$ −868645. −1.34319 −0.671593 0.740920i $$-0.734390\pi$$
−0.671593 + 0.740920i $$0.734390\pi$$
$$212$$ −1.96416e6 −3.00149
$$213$$ −338043. −0.510532
$$214$$ −1.92766e6 −2.87738
$$215$$ 0 0
$$216$$ 270406. 0.394350
$$217$$ 218812. 0.315443
$$218$$ 157320. 0.224204
$$219$$ 528189. 0.744181
$$220$$ 0 0
$$221$$ −1.54518e6 −2.12814
$$222$$ −398582. −0.542794
$$223$$ 48571.9 0.0654068 0.0327034 0.999465i $$-0.489588\pi$$
0.0327034 + 0.999465i $$0.489588\pi$$
$$224$$ −99025.1 −0.131864
$$225$$ 0 0
$$226$$ −151892. −0.197817
$$227$$ −78669.6 −0.101331 −0.0506655 0.998716i $$-0.516134\pi$$
−0.0506655 + 0.998716i $$0.516134\pi$$
$$228$$ −1.01918e6 −1.29842
$$229$$ −324090. −0.408391 −0.204196 0.978930i $$-0.565458\pi$$
−0.204196 + 0.978930i $$0.565458\pi$$
$$230$$ 0 0
$$231$$ −31638.1 −0.0390104
$$232$$ 1.69600e6 2.06874
$$233$$ −1.26828e6 −1.53048 −0.765238 0.643747i $$-0.777379\pi$$
−0.765238 + 0.643747i $$0.777379\pi$$
$$234$$ 832862. 0.994336
$$235$$ 0 0
$$236$$ 1.78805e6 2.08978
$$237$$ 74466.6 0.0861173
$$238$$ −440619. −0.504221
$$239$$ −1.35381e6 −1.53307 −0.766535 0.642202i $$-0.778021\pi$$
−0.766535 + 0.642202i $$0.778021\pi$$
$$240$$ 0 0
$$241$$ 1.14198e6 1.26653 0.633264 0.773936i $$-0.281715\pi$$
0.633264 + 0.773936i $$0.281715\pi$$
$$242$$ −147084. −0.161446
$$243$$ −59049.0 −0.0641500
$$244$$ −3.53992e6 −3.80644
$$245$$ 0 0
$$246$$ 508998. 0.536264
$$247$$ −1.68167e6 −1.75387
$$248$$ −2.79368e6 −2.88434
$$249$$ −192198. −0.196449
$$250$$ 0 0
$$251$$ −321578. −0.322183 −0.161091 0.986940i $$-0.551501\pi$$
−0.161091 + 0.986940i $$0.551501\pi$$
$$252$$ 162192. 0.160890
$$253$$ −178851. −0.175667
$$254$$ −2.24781e6 −2.18612
$$255$$ 0 0
$$256$$ −2.08990e6 −1.99308
$$257$$ −1.44275e6 −1.36257 −0.681286 0.732018i $$-0.738579\pi$$
−0.681286 + 0.732018i $$0.738579\pi$$
$$258$$ 206425. 0.193070
$$259$$ −128075. −0.118635
$$260$$ 0 0
$$261$$ −370359. −0.336528
$$262$$ −2.70495e6 −2.43447
$$263$$ −838618. −0.747610 −0.373805 0.927507i $$-0.621947\pi$$
−0.373805 + 0.927507i $$0.621947\pi$$
$$264$$ 403939. 0.356703
$$265$$ 0 0
$$266$$ −479539. −0.415546
$$267$$ −705472. −0.605622
$$268$$ 2.70041e6 2.29664
$$269$$ −415262. −0.349898 −0.174949 0.984577i $$-0.555976\pi$$
−0.174949 + 0.984577i $$0.555976\pi$$
$$270$$ 0 0
$$271$$ 647209. 0.535330 0.267665 0.963512i $$-0.413748\pi$$
0.267665 + 0.963512i $$0.413748\pi$$
$$272$$ 2.29595e6 1.88166
$$273$$ 267620. 0.217326
$$274$$ 1.72980e6 1.39193
$$275$$ 0 0
$$276$$ 916878. 0.724501
$$277$$ −679011. −0.531713 −0.265856 0.964013i $$-0.585655\pi$$
−0.265856 + 0.964013i $$0.585655\pi$$
$$278$$ 3.51821e6 2.73030
$$279$$ 610061. 0.469205
$$280$$ 0 0
$$281$$ −1.78685e6 −1.34997 −0.674983 0.737834i $$-0.735849\pi$$
−0.674983 + 0.737834i $$0.735849\pi$$
$$282$$ −540721. −0.404903
$$283$$ 872555. 0.647630 0.323815 0.946120i $$-0.395034\pi$$
0.323815 + 0.946120i $$0.395034\pi$$
$$284$$ 2.58876e6 1.90457
$$285$$ 0 0
$$286$$ 1.24415e6 0.899411
$$287$$ 163554. 0.117208
$$288$$ −276088. −0.196140
$$289$$ 859295. 0.605198
$$290$$ 0 0
$$291$$ 581172. 0.402321
$$292$$ −4.04491e6 −2.77621
$$293$$ −653130. −0.444458 −0.222229 0.974995i $$-0.571333\pi$$
−0.222229 + 0.974995i $$0.571333\pi$$
$$294$$ −1.44328e6 −0.973827
$$295$$ 0 0
$$296$$ 1.63519e6 1.08477
$$297$$ −88209.0 −0.0580259
$$298$$ −5.21574e6 −3.40232
$$299$$ 1.51286e6 0.978638
$$300$$ 0 0
$$301$$ 66329.8 0.0421980
$$302$$ 2.31263e6 1.45911
$$303$$ −435280. −0.272372
$$304$$ 2.49875e6 1.55074
$$305$$ 0 0
$$306$$ −1.22847e6 −0.750002
$$307$$ 1.87392e6 1.13476 0.567381 0.823456i $$-0.307957\pi$$
0.567381 + 0.823456i $$0.307957\pi$$
$$308$$ 242287. 0.145530
$$309$$ 1.09731e6 0.653783
$$310$$ 0 0
$$311$$ −1.60626e6 −0.941705 −0.470853 0.882212i $$-0.656054\pi$$
−0.470853 + 0.882212i $$0.656054\pi$$
$$312$$ −3.41684e6 −1.98718
$$313$$ 2.58202e6 1.48970 0.744849 0.667234i $$-0.232522\pi$$
0.744849 + 0.667234i $$0.232522\pi$$
$$314$$ −3.30377e6 −1.89098
$$315$$ 0 0
$$316$$ −570271. −0.321265
$$317$$ 1.34588e6 0.752242 0.376121 0.926571i $$-0.377258\pi$$
0.376121 + 0.926571i $$0.377258\pi$$
$$318$$ −2.57662e6 −1.42884
$$319$$ −553252. −0.304401
$$320$$ 0 0
$$321$$ −1.72695e6 −0.935442
$$322$$ 431403. 0.231870
$$323$$ 2.48047e6 1.32290
$$324$$ 452202. 0.239315
$$325$$ 0 0
$$326$$ −2.47516e6 −1.28991
$$327$$ 140940. 0.0728893
$$328$$ −2.08818e6 −1.07172
$$329$$ −173748. −0.0884971
$$330$$ 0 0
$$331$$ 2.74357e6 1.37641 0.688203 0.725518i $$-0.258400\pi$$
0.688203 + 0.725518i $$0.258400\pi$$
$$332$$ 1.47187e6 0.732865
$$333$$ −357080. −0.176464
$$334$$ −5.84089e6 −2.86492
$$335$$ 0 0
$$336$$ −397650. −0.192156
$$337$$ −3.70365e6 −1.77646 −0.888229 0.459400i $$-0.848064\pi$$
−0.888229 + 0.459400i $$0.848064\pi$$
$$338$$ −6.79399e6 −3.23469
$$339$$ −136076. −0.0643107
$$340$$ 0 0
$$341$$ 911325. 0.424412
$$342$$ −1.33698e6 −0.618103
$$343$$ −952047. −0.436941
$$344$$ −846866. −0.385850
$$345$$ 0 0
$$346$$ −2.65868e6 −1.19392
$$347$$ 1.17221e6 0.522616 0.261308 0.965255i $$-0.415846\pi$$
0.261308 + 0.965255i $$0.415846\pi$$
$$348$$ 2.83624e6 1.25543
$$349$$ −360082. −0.158248 −0.0791240 0.996865i $$-0.525212\pi$$
−0.0791240 + 0.996865i $$0.525212\pi$$
$$350$$ 0 0
$$351$$ 746141. 0.323261
$$352$$ −412428. −0.177415
$$353$$ −998723. −0.426588 −0.213294 0.976988i $$-0.568419\pi$$
−0.213294 + 0.976988i $$0.568419\pi$$
$$354$$ 2.34560e6 0.994824
$$355$$ 0 0
$$356$$ 5.40256e6 2.25930
$$357$$ −394740. −0.163923
$$358$$ 139504. 0.0575282
$$359$$ 1.47706e6 0.604871 0.302436 0.953170i $$-0.402200\pi$$
0.302436 + 0.953170i $$0.402200\pi$$
$$360$$ 0 0
$$361$$ 223465. 0.0902489
$$362$$ 7.56057e6 3.03238
$$363$$ −131769. −0.0524864
$$364$$ −2.04945e6 −0.810746
$$365$$ 0 0
$$366$$ −4.64374e6 −1.81203
$$367$$ −3.21905e6 −1.24756 −0.623781 0.781599i $$-0.714404\pi$$
−0.623781 + 0.781599i $$0.714404\pi$$
$$368$$ −2.24793e6 −0.865294
$$369$$ 455999. 0.174340
$$370$$ 0 0
$$371$$ −827935. −0.312292
$$372$$ −4.67190e6 −1.75039
$$373$$ 4.65790e6 1.73348 0.866739 0.498762i $$-0.166212\pi$$
0.866739 + 0.498762i $$0.166212\pi$$
$$374$$ −1.83513e6 −0.678403
$$375$$ 0 0
$$376$$ 2.21832e6 0.809198
$$377$$ 4.67984e6 1.69581
$$378$$ 212767. 0.0765905
$$379$$ −4.04644e6 −1.44702 −0.723510 0.690314i $$-0.757472\pi$$
−0.723510 + 0.690314i $$0.757472\pi$$
$$380$$ 0 0
$$381$$ −2.01376e6 −0.710714
$$382$$ 255331. 0.0895252
$$383$$ −824148. −0.287084 −0.143542 0.989644i $$-0.545849\pi$$
−0.143542 + 0.989644i $$0.545849\pi$$
$$384$$ −2.28580e6 −0.791062
$$385$$ 0 0
$$386$$ −4.92077e6 −1.68099
$$387$$ 184932. 0.0627674
$$388$$ −4.45066e6 −1.50088
$$389$$ −2.54462e6 −0.852607 −0.426303 0.904580i $$-0.640184\pi$$
−0.426303 + 0.904580i $$0.640184\pi$$
$$390$$ 0 0
$$391$$ −2.23148e6 −0.738162
$$392$$ 5.92109e6 1.94620
$$393$$ −2.42330e6 −0.791453
$$394$$ −3.26322e6 −1.05902
$$395$$ 0 0
$$396$$ 675512. 0.216469
$$397$$ 5.17776e6 1.64879 0.824395 0.566015i $$-0.191516\pi$$
0.824395 + 0.566015i $$0.191516\pi$$
$$398$$ 4.00351e6 1.26687
$$399$$ −429607. −0.135095
$$400$$ 0 0
$$401$$ −4.82239e6 −1.49762 −0.748809 0.662785i $$-0.769374\pi$$
−0.748809 + 0.662785i $$0.769374\pi$$
$$402$$ 3.54245e6 1.09330
$$403$$ −7.70871e6 −2.36439
$$404$$ 3.33341e6 1.01610
$$405$$ 0 0
$$406$$ 1.33449e6 0.401790
$$407$$ −533416. −0.159617
$$408$$ 5.03985e6 1.49888
$$409$$ −3.68626e6 −1.08963 −0.544813 0.838558i $$-0.683399\pi$$
−0.544813 + 0.838558i $$0.683399\pi$$
$$410$$ 0 0
$$411$$ 1.54968e6 0.452521
$$412$$ −8.40329e6 −2.43897
$$413$$ 753702. 0.217433
$$414$$ 1.20278e6 0.344894
$$415$$ 0 0
$$416$$ 3.48864e6 0.988377
$$417$$ 3.15188e6 0.887626
$$418$$ −1.99722e6 −0.559095
$$419$$ 1.31904e6 0.367049 0.183525 0.983015i $$-0.441249\pi$$
0.183525 + 0.983015i $$0.441249\pi$$
$$420$$ 0 0
$$421$$ −3.29518e6 −0.906095 −0.453048 0.891486i $$-0.649663\pi$$
−0.453048 + 0.891486i $$0.649663\pi$$
$$422$$ 8.72643e6 2.38537
$$423$$ −484419. −0.131635
$$424$$ 1.05707e7 2.85553
$$425$$ 0 0
$$426$$ 3.39599e6 0.906655
$$427$$ −1.49215e6 −0.396044
$$428$$ 1.32251e7 3.48972
$$429$$ 1.11461e6 0.292400
$$430$$ 0 0
$$431$$ 1.87137e6 0.485251 0.242625 0.970120i $$-0.421991\pi$$
0.242625 + 0.970120i $$0.421991\pi$$
$$432$$ −1.10867e6 −0.285821
$$433$$ −4.31194e6 −1.10523 −0.552615 0.833437i $$-0.686370\pi$$
−0.552615 + 0.833437i $$0.686370\pi$$
$$434$$ −2.19819e6 −0.560197
$$435$$ 0 0
$$436$$ −1.07933e6 −0.271917
$$437$$ −2.42859e6 −0.608345
$$438$$ −5.30620e6 −1.32159
$$439$$ 1.13612e6 0.281360 0.140680 0.990055i $$-0.455071\pi$$
0.140680 + 0.990055i $$0.455071\pi$$
$$440$$ 0 0
$$441$$ −1.29300e6 −0.316593
$$442$$ 1.55230e7 3.77936
$$443$$ 875875. 0.212047 0.106024 0.994364i $$-0.466188\pi$$
0.106024 + 0.994364i $$0.466188\pi$$
$$444$$ 2.73455e6 0.658306
$$445$$ 0 0
$$446$$ −487955. −0.116156
$$447$$ −4.67266e6 −1.10610
$$448$$ −419059. −0.0986461
$$449$$ 4.08806e6 0.956977 0.478489 0.878094i $$-0.341185\pi$$
0.478489 + 0.878094i $$0.341185\pi$$
$$450$$ 0 0
$$451$$ 681184. 0.157697
$$452$$ 1.04208e6 0.239915
$$453$$ 2.07183e6 0.474361
$$454$$ 790317. 0.179954
$$455$$ 0 0
$$456$$ 5.48501e6 1.23528
$$457$$ −2.85279e6 −0.638968 −0.319484 0.947592i $$-0.603510\pi$$
−0.319484 + 0.947592i $$0.603510\pi$$
$$458$$ 3.25581e6 0.725263
$$459$$ −1.10056e6 −0.243827
$$460$$ 0 0
$$461$$ 6.16387e6 1.35083 0.675416 0.737437i $$-0.263964\pi$$
0.675416 + 0.737437i $$0.263964\pi$$
$$462$$ 317837. 0.0692787
$$463$$ 4.52412e6 0.980804 0.490402 0.871496i $$-0.336850\pi$$
0.490402 + 0.871496i $$0.336850\pi$$
$$464$$ −6.95366e6 −1.49940
$$465$$ 0 0
$$466$$ 1.27412e7 2.71798
$$467$$ −2.23084e6 −0.473344 −0.236672 0.971590i $$-0.576057\pi$$
−0.236672 + 0.971590i $$0.576057\pi$$
$$468$$ −5.71401e6 −1.20594
$$469$$ 1.13828e6 0.238956
$$470$$ 0 0
$$471$$ −2.95977e6 −0.614761
$$472$$ −9.62289e6 −1.98816
$$473$$ 276256. 0.0567752
$$474$$ −748093. −0.152936
$$475$$ 0 0
$$476$$ 3.02295e6 0.611525
$$477$$ −2.30834e6 −0.464518
$$478$$ 1.36004e7 2.72259
$$479$$ −6.62215e6 −1.31874 −0.659372 0.751817i $$-0.729178\pi$$
−0.659372 + 0.751817i $$0.729178\pi$$
$$480$$ 0 0
$$481$$ 4.51205e6 0.889224
$$482$$ −1.14723e7 −2.24923
$$483$$ 386484. 0.0753813
$$484$$ 1.00910e6 0.195803
$$485$$ 0 0
$$486$$ 593208. 0.113924
$$487$$ −2.17583e6 −0.415721 −0.207860 0.978158i $$-0.566650\pi$$
−0.207860 + 0.978158i $$0.566650\pi$$
$$488$$ 1.90511e7 3.62134
$$489$$ −2.21744e6 −0.419353
$$490$$ 0 0
$$491$$ 7.67198e6 1.43616 0.718081 0.695959i $$-0.245021\pi$$
0.718081 + 0.695959i $$0.245021\pi$$
$$492$$ −3.49208e6 −0.650386
$$493$$ −6.90278e6 −1.27911
$$494$$ 1.68941e7 3.11471
$$495$$ 0 0
$$496$$ 1.14542e7 2.09055
$$497$$ 1.09122e6 0.198162
$$498$$ 1.93083e6 0.348875
$$499$$ 6.65009e6 1.19557 0.597787 0.801655i $$-0.296047\pi$$
0.597787 + 0.801655i $$0.296047\pi$$
$$500$$ 0 0
$$501$$ −5.23272e6 −0.931393
$$502$$ 3.23058e6 0.572165
$$503$$ −4.18197e6 −0.736989 −0.368495 0.929630i $$-0.620127\pi$$
−0.368495 + 0.929630i $$0.620127\pi$$
$$504$$ −872882. −0.153066
$$505$$ 0 0
$$506$$ 1.79675e6 0.311968
$$507$$ −6.08657e6 −1.05161
$$508$$ 1.54215e7 2.65135
$$509$$ −1.01322e7 −1.73345 −0.866725 0.498787i $$-0.833779\pi$$
−0.866725 + 0.498787i $$0.833779\pi$$
$$510$$ 0 0
$$511$$ −1.70502e6 −0.288853
$$512$$ 1.28679e7 2.16936
$$513$$ −1.19777e6 −0.200947
$$514$$ 1.44939e7 2.41980
$$515$$ 0 0
$$516$$ −1.41622e6 −0.234157
$$517$$ −723639. −0.119068
$$518$$ 1.28664e6 0.210685
$$519$$ −2.38185e6 −0.388147
$$520$$ 0 0
$$521$$ −6.42546e6 −1.03707 −0.518537 0.855055i $$-0.673523\pi$$
−0.518537 + 0.855055i $$0.673523\pi$$
$$522$$ 3.72063e6 0.597641
$$523$$ 1.17398e6 0.187674 0.0938372 0.995588i $$-0.470087\pi$$
0.0938372 + 0.995588i $$0.470087\pi$$
$$524$$ 1.85578e7 2.95256
$$525$$ 0 0
$$526$$ 8.42478e6 1.32768
$$527$$ 1.13704e7 1.78340
$$528$$ −1.65617e6 −0.258535
$$529$$ −4.25153e6 −0.660551
$$530$$ 0 0
$$531$$ 2.10137e6 0.323419
$$532$$ 3.28997e6 0.503979
$$533$$ −5.76199e6 −0.878525
$$534$$ 7.08719e6 1.07553
$$535$$ 0 0
$$536$$ −1.45330e7 −2.18496
$$537$$ 124979. 0.0187025
$$538$$ 4.17174e6 0.621386
$$539$$ −1.93152e6 −0.286370
$$540$$ 0 0
$$541$$ −5.25283e6 −0.771614 −0.385807 0.922580i $$-0.626077\pi$$
−0.385807 + 0.922580i $$0.626077\pi$$
$$542$$ −6.50188e6 −0.950694
$$543$$ 6.77334e6 0.985833
$$544$$ −5.14576e6 −0.745508
$$545$$ 0 0
$$546$$ −2.68852e6 −0.385950
$$547$$ −1.13849e7 −1.62689 −0.813447 0.581640i $$-0.802411\pi$$
−0.813447 + 0.581640i $$0.802411\pi$$
$$548$$ −1.18676e7 −1.68815
$$549$$ −4.16022e6 −0.589095
$$550$$ 0 0
$$551$$ −7.51249e6 −1.05416
$$552$$ −4.93444e6 −0.689271
$$553$$ −240382. −0.0334263
$$554$$ 6.82136e6 0.944271
$$555$$ 0 0
$$556$$ −2.41374e7 −3.31134
$$557$$ −4.54355e6 −0.620522 −0.310261 0.950651i $$-0.600416\pi$$
−0.310261 + 0.950651i $$0.600416\pi$$
$$558$$ −6.12869e6 −0.833263
$$559$$ −2.33679e6 −0.316293
$$560$$ 0 0
$$561$$ −1.64405e6 −0.220550
$$562$$ 1.79508e7 2.39741
$$563$$ 1.22077e7 1.62317 0.811583 0.584237i $$-0.198606\pi$$
0.811583 + 0.584237i $$0.198606\pi$$
$$564$$ 3.70972e6 0.491070
$$565$$ 0 0
$$566$$ −8.76572e6 −1.15013
$$567$$ 190613. 0.0248997
$$568$$ −1.39321e7 −1.81195
$$569$$ −1.58948e6 −0.205814 −0.102907 0.994691i $$-0.532814\pi$$
−0.102907 + 0.994691i $$0.532814\pi$$
$$570$$ 0 0
$$571$$ −692853. −0.0889306 −0.0444653 0.999011i $$-0.514158\pi$$
−0.0444653 + 0.999011i $$0.514158\pi$$
$$572$$ −8.53574e6 −1.09082
$$573$$ 228745. 0.0291048
$$574$$ −1.64307e6 −0.208150
$$575$$ 0 0
$$576$$ −1.16836e6 −0.146731
$$577$$ −1.24815e7 −1.56073 −0.780365 0.625324i $$-0.784967\pi$$
−0.780365 + 0.625324i $$0.784967\pi$$
$$578$$ −8.63250e6 −1.07477
$$579$$ −4.40840e6 −0.546493
$$580$$ 0 0
$$581$$ 620425. 0.0762516
$$582$$ −5.83847e6 −0.714483
$$583$$ −3.44825e6 −0.420173
$$584$$ 2.17688e7 2.64121
$$585$$ 0 0
$$586$$ 6.56136e6 0.789314
$$587$$ 9.13291e6 1.09399 0.546996 0.837135i $$-0.315771\pi$$
0.546996 + 0.837135i $$0.315771\pi$$
$$588$$ 9.90189e6 1.18107
$$589$$ 1.23747e7 1.46976
$$590$$ 0 0
$$591$$ −2.92344e6 −0.344291
$$592$$ −6.70436e6 −0.786236
$$593$$ −1.08272e7 −1.26438 −0.632191 0.774813i $$-0.717844\pi$$
−0.632191 + 0.774813i $$0.717844\pi$$
$$594$$ 886150. 0.103048
$$595$$ 0 0
$$596$$ 3.57836e7 4.12637
$$597$$ 3.58665e6 0.411863
$$598$$ −1.51983e7 −1.73797
$$599$$ 7.47502e6 0.851227 0.425614 0.904905i $$-0.360058\pi$$
0.425614 + 0.904905i $$0.360058\pi$$
$$600$$ 0 0
$$601$$ −1.02516e6 −0.115772 −0.0578861 0.998323i $$-0.518436\pi$$
−0.0578861 + 0.998323i $$0.518436\pi$$
$$602$$ −666351. −0.0749397
$$603$$ 3.17360e6 0.355434
$$604$$ −1.58662e7 −1.76963
$$605$$ 0 0
$$606$$ 4.37283e6 0.483706
$$607$$ −1.33264e7 −1.46805 −0.734024 0.679123i $$-0.762360\pi$$
−0.734024 + 0.679123i $$0.762360\pi$$
$$608$$ −5.60028e6 −0.614399
$$609$$ 1.19553e6 0.130623
$$610$$ 0 0
$$611$$ 6.12110e6 0.663325
$$612$$ 8.42819e6 0.909611
$$613$$ 1.57435e7 1.69220 0.846098 0.533027i $$-0.178946\pi$$
0.846098 + 0.533027i $$0.178946\pi$$
$$614$$ −1.88254e7 −2.01523
$$615$$ 0 0
$$616$$ −1.30394e6 −0.138454
$$617$$ 5.87960e6 0.621777 0.310888 0.950446i $$-0.399373\pi$$
0.310888 + 0.950446i $$0.399373\pi$$
$$618$$ −1.10236e7 −1.16106
$$619$$ −1.81138e6 −0.190013 −0.0950063 0.995477i $$-0.530287\pi$$
−0.0950063 + 0.995477i $$0.530287\pi$$
$$620$$ 0 0
$$621$$ 1.07754e6 0.112126
$$622$$ 1.61365e7 1.67238
$$623$$ 2.27730e6 0.235071
$$624$$ 1.40092e7 1.44029
$$625$$ 0 0
$$626$$ −2.59390e7 −2.64556
$$627$$ −1.78927e6 −0.181763
$$628$$ 2.26662e7 2.29340
$$629$$ −6.65529e6 −0.670719
$$630$$ 0 0
$$631$$ −1.43835e7 −1.43811 −0.719054 0.694954i $$-0.755425\pi$$
−0.719054 + 0.694954i $$0.755425\pi$$
$$632$$ 3.06907e6 0.305643
$$633$$ 7.81780e6 0.775489
$$634$$ −1.35207e7 −1.33591
$$635$$ 0 0
$$636$$ 1.76774e7 1.73291
$$637$$ 1.63383e7 1.59536
$$638$$ 5.55798e6 0.540587
$$639$$ 3.04239e6 0.294756
$$640$$ 0 0
$$641$$ −3.80867e6 −0.366124 −0.183062 0.983101i $$-0.558601\pi$$
−0.183062 + 0.983101i $$0.558601\pi$$
$$642$$ 1.73490e7 1.66126
$$643$$ −483920. −0.0461579 −0.0230789 0.999734i $$-0.507347\pi$$
−0.0230789 + 0.999734i $$0.507347\pi$$
$$644$$ −2.95973e6 −0.281214
$$645$$ 0 0
$$646$$ −2.49188e7 −2.34934
$$647$$ −1.80652e7 −1.69661 −0.848306 0.529506i $$-0.822377\pi$$
−0.848306 + 0.529506i $$0.822377\pi$$
$$648$$ −2.43365e6 −0.227678
$$649$$ 3.13908e6 0.292544
$$650$$ 0 0
$$651$$ −1.96931e6 −0.182121
$$652$$ 1.69813e7 1.56442
$$653$$ −2.03744e7 −1.86983 −0.934915 0.354871i $$-0.884525\pi$$
−0.934915 + 0.354871i $$0.884525\pi$$
$$654$$ −1.41588e6 −0.129444
$$655$$ 0 0
$$656$$ 8.56161e6 0.776776
$$657$$ −4.75370e6 −0.429653
$$658$$ 1.74547e6 0.157162
$$659$$ −1.37586e7 −1.23413 −0.617066 0.786912i $$-0.711679\pi$$
−0.617066 + 0.786912i $$0.711679\pi$$
$$660$$ 0 0
$$661$$ −1.34803e7 −1.20004 −0.600019 0.799985i $$-0.704840\pi$$
−0.600019 + 0.799985i $$0.704840\pi$$
$$662$$ −2.75620e7 −2.44437
$$663$$ 1.39066e7 1.22868
$$664$$ −7.92127e6 −0.697228
$$665$$ 0 0
$$666$$ 3.58724e6 0.313382
$$667$$ 6.75840e6 0.588206
$$668$$ 4.00726e7 3.47461
$$669$$ −437147. −0.0377627
$$670$$ 0 0
$$671$$ −6.21465e6 −0.532856
$$672$$ 891226. 0.0761315
$$673$$ 680925. 0.0579511 0.0289755 0.999580i $$-0.490776\pi$$
0.0289755 + 0.999580i $$0.490776\pi$$
$$674$$ 3.72070e7 3.15482
$$675$$ 0 0
$$676$$ 4.66115e7 3.92307
$$677$$ −1.36705e7 −1.14634 −0.573171 0.819436i $$-0.694287\pi$$
−0.573171 + 0.819436i $$0.694287\pi$$
$$678$$ 1.36703e6 0.114210
$$679$$ −1.87605e6 −0.156160
$$680$$ 0 0
$$681$$ 708026. 0.0585034
$$682$$ −9.15520e6 −0.753715
$$683$$ −6.79648e6 −0.557484 −0.278742 0.960366i $$-0.589917\pi$$
−0.278742 + 0.960366i $$0.589917\pi$$
$$684$$ 9.17264e6 0.749642
$$685$$ 0 0
$$686$$ 9.56429e6 0.775966
$$687$$ 2.91681e6 0.235785
$$688$$ 3.47218e6 0.279661
$$689$$ 2.91680e7 2.34077
$$690$$ 0 0
$$691$$ 9.11491e6 0.726202 0.363101 0.931750i $$-0.381718\pi$$
0.363101 + 0.931750i $$0.381718\pi$$
$$692$$ 1.82404e7 1.44800
$$693$$ 284743. 0.0225227
$$694$$ −1.17761e7 −0.928116
$$695$$ 0 0
$$696$$ −1.52640e7 −1.19439
$$697$$ 8.49896e6 0.662649
$$698$$ 3.61740e6 0.281033
$$699$$ 1.14146e7 0.883621
$$700$$ 0 0
$$701$$ −1.03729e7 −0.797269 −0.398634 0.917110i $$-0.630516\pi$$
−0.398634 + 0.917110i $$0.630516\pi$$
$$702$$ −7.49575e6 −0.574080
$$703$$ −7.24315e6 −0.552763
$$704$$ −1.74533e6 −0.132723
$$705$$ 0 0
$$706$$ 1.00332e7 0.757579
$$707$$ 1.40510e6 0.105721
$$708$$ −1.60925e7 −1.20653
$$709$$ 1.78828e7 1.33604 0.668022 0.744142i $$-0.267141\pi$$
0.668022 + 0.744142i $$0.267141\pi$$
$$710$$ 0 0
$$711$$ −670199. −0.0497199
$$712$$ −2.90754e7 −2.14944
$$713$$ −1.11326e7 −0.820108
$$714$$ 3.96557e6 0.291112
$$715$$ 0 0
$$716$$ −957097. −0.0697708
$$717$$ 1.21843e7 0.885119
$$718$$ −1.48386e7 −1.07419
$$719$$ −2.62904e7 −1.89660 −0.948298 0.317382i $$-0.897196\pi$$
−0.948298 + 0.317382i $$0.897196\pi$$
$$720$$ 0 0
$$721$$ −3.54217e6 −0.253765
$$722$$ −2.24494e6 −0.160273
$$723$$ −1.02778e7 −0.731231
$$724$$ −5.18708e7 −3.67770
$$725$$ 0 0
$$726$$ 1.32376e6 0.0932108
$$727$$ −1.40763e7 −0.987763 −0.493882 0.869529i $$-0.664422\pi$$
−0.493882 + 0.869529i $$0.664422\pi$$
$$728$$ 1.10297e7 0.771322
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 3.44677e6 0.238572
$$732$$ 3.18593e7 2.19765
$$733$$ 5.60463e6 0.385290 0.192645 0.981269i $$-0.438293\pi$$
0.192645 + 0.981269i $$0.438293\pi$$
$$734$$ 3.23387e7 2.21555
$$735$$ 0 0
$$736$$ 5.03813e6 0.342827
$$737$$ 4.74081e6 0.321502
$$738$$ −4.58098e6 −0.309612
$$739$$ −1.92933e6 −0.129956 −0.0649778 0.997887i $$-0.520698\pi$$
−0.0649778 + 0.997887i $$0.520698\pi$$
$$740$$ 0 0
$$741$$ 1.51350e7 1.01260
$$742$$ 8.31746e6 0.554601
$$743$$ 4.43790e6 0.294921 0.147460 0.989068i $$-0.452890\pi$$
0.147460 + 0.989068i $$0.452890\pi$$
$$744$$ 2.51431e7 1.66528
$$745$$ 0 0
$$746$$ −4.67934e7 −3.07849
$$747$$ 1.72978e6 0.113420
$$748$$ 1.25903e7 0.822774
$$749$$ 5.57467e6 0.363090
$$750$$ 0 0
$$751$$ −1.11577e7 −0.721897 −0.360949 0.932586i $$-0.617547\pi$$
−0.360949 + 0.932586i $$0.617547\pi$$
$$752$$ −9.09521e6 −0.586500
$$753$$ 2.89420e6 0.186012
$$754$$ −4.70138e7 −3.01160
$$755$$ 0 0
$$756$$ −1.45973e6 −0.0928898
$$757$$ −1.16077e7 −0.736216 −0.368108 0.929783i $$-0.619994\pi$$
−0.368108 + 0.929783i $$0.619994\pi$$
$$758$$ 4.06506e7 2.56977
$$759$$ 1.60966e6 0.101422
$$760$$ 0 0
$$761$$ −1.38952e6 −0.0869770 −0.0434885 0.999054i $$-0.513847\pi$$
−0.0434885 + 0.999054i $$0.513847\pi$$
$$762$$ 2.02303e7 1.26216
$$763$$ −454960. −0.0282919
$$764$$ −1.75175e6 −0.108577
$$765$$ 0 0
$$766$$ 8.27942e6 0.509833
$$767$$ −2.65528e7 −1.62975
$$768$$ 1.88091e7 1.15071
$$769$$ −2.19019e7 −1.33557 −0.667784 0.744355i $$-0.732757\pi$$
−0.667784 + 0.744355i $$0.732757\pi$$
$$770$$ 0 0
$$771$$ 1.29848e7 0.786681
$$772$$ 3.37599e7 2.03872
$$773$$ 9.21665e6 0.554784 0.277392 0.960757i $$-0.410530\pi$$
0.277392 + 0.960757i $$0.410530\pi$$
$$774$$ −1.85783e6 −0.111469
$$775$$ 0 0
$$776$$ 2.39525e7 1.42789
$$777$$ 1.15267e6 0.0684941
$$778$$ 2.55633e7 1.51415
$$779$$ 9.24966e6 0.546113
$$780$$ 0 0
$$781$$ 4.54480e6 0.266617
$$782$$ 2.24175e7 1.31090
$$783$$ 3.33323e6 0.194294
$$784$$ −2.42767e7 −1.41059
$$785$$ 0 0
$$786$$ 2.43445e7 1.40554
$$787$$ 2.18581e7 1.25798 0.628992 0.777412i $$-0.283468\pi$$
0.628992 + 0.777412i $$0.283468\pi$$
$$788$$ 2.23879e7 1.28439
$$789$$ 7.54756e6 0.431633
$$790$$ 0 0
$$791$$ 439261. 0.0249621
$$792$$ −3.63545e6 −0.205942
$$793$$ 5.25684e7 2.96853
$$794$$ −5.20159e7 −2.92809
$$795$$ 0 0
$$796$$ −2.74668e7 −1.53648
$$797$$ −2.79333e7 −1.55767 −0.778837 0.627226i $$-0.784190\pi$$
−0.778837 + 0.627226i $$0.784190\pi$$
$$798$$ 4.31585e6 0.239916
$$799$$ −9.02865e6 −0.500329
$$800$$ 0 0
$$801$$ 6.34925e6 0.349656
$$802$$ 4.84459e7 2.65963
$$803$$ −7.10120e6 −0.388636
$$804$$ −2.43037e7 −1.32596
$$805$$ 0 0
$$806$$ 7.74419e7 4.19893
$$807$$ 3.73736e6 0.202014
$$808$$ −1.79397e7 −0.966687
$$809$$ 1.07778e7 0.578974 0.289487 0.957182i $$-0.406515\pi$$
0.289487 + 0.957182i $$0.406515\pi$$
$$810$$ 0 0
$$811$$ −2.71394e7 −1.44893 −0.724467 0.689309i $$-0.757914\pi$$
−0.724467 + 0.689309i $$0.757914\pi$$
$$812$$ −9.15550e6 −0.487295
$$813$$ −5.82488e6 −0.309073
$$814$$ 5.35871e6 0.283465
$$815$$ 0 0
$$816$$ −2.06636e7 −1.08638
$$817$$ 3.75122e6 0.196616
$$818$$ 3.70322e7 1.93507
$$819$$ −2.40858e6 −0.125473
$$820$$ 0 0
$$821$$ 6.84095e6 0.354208 0.177104 0.984192i $$-0.443327\pi$$
0.177104 + 0.984192i $$0.443327\pi$$
$$822$$ −1.55682e7 −0.803634
$$823$$ −1.95930e7 −1.00833 −0.504163 0.863609i $$-0.668199\pi$$
−0.504163 + 0.863609i $$0.668199\pi$$
$$824$$ 4.52247e7 2.32037
$$825$$ 0 0
$$826$$ −7.57171e6 −0.386139
$$827$$ 1.86828e7 0.949902 0.474951 0.880012i $$-0.342466\pi$$
0.474951 + 0.880012i $$0.342466\pi$$
$$828$$ −8.25191e6 −0.418291
$$829$$ −1.58057e7 −0.798779 −0.399390 0.916781i $$-0.630778\pi$$
−0.399390 + 0.916781i $$0.630778\pi$$
$$830$$ 0 0
$$831$$ 6.11109e6 0.306985
$$832$$ 1.47634e7 0.739397
$$833$$ −2.40991e7 −1.20334
$$834$$ −3.16639e7 −1.57634
$$835$$ 0 0
$$836$$ 1.37023e7 0.678077
$$837$$ −5.49055e6 −0.270896
$$838$$ −1.32512e7 −0.651844
$$839$$ −3.69461e7 −1.81202 −0.906011 0.423255i $$-0.860888\pi$$
−0.906011 + 0.423255i $$0.860888\pi$$
$$840$$ 0 0
$$841$$ 395030. 0.0192593
$$842$$ 3.31035e7 1.60914
$$843$$ 1.60817e7 0.779403
$$844$$ −5.98694e7 −2.89300
$$845$$ 0 0
$$846$$ 4.86649e6 0.233771
$$847$$ 425357. 0.0203725
$$848$$ −4.33401e7 −2.06967
$$849$$ −7.85300e6 −0.373909
$$850$$ 0 0
$$851$$ 6.51609e6 0.308435
$$852$$ −2.32988e7 −1.09960
$$853$$ −2.39478e7 −1.12692 −0.563459 0.826144i $$-0.690530\pi$$
−0.563459 + 0.826144i $$0.690530\pi$$
$$854$$ 1.49902e7 0.703337
$$855$$ 0 0
$$856$$ −7.11746e7 −3.32002
$$857$$ 4.09428e6 0.190426 0.0952129 0.995457i $$-0.469647\pi$$
0.0952129 + 0.995457i $$0.469647\pi$$
$$858$$ −1.11974e7 −0.519275
$$859$$ −6.51782e6 −0.301384 −0.150692 0.988581i $$-0.548150\pi$$
−0.150692 + 0.988581i $$0.548150\pi$$
$$860$$ 0 0
$$861$$ −1.47199e6 −0.0676700
$$862$$ −1.87998e7 −0.861759
$$863$$ −8.54609e6 −0.390607 −0.195304 0.980743i $$-0.562569\pi$$
−0.195304 + 0.980743i $$0.562569\pi$$
$$864$$ 2.48479e6 0.113242
$$865$$ 0 0
$$866$$ 4.33178e7 1.96278
$$867$$ −7.73365e6 −0.349411
$$868$$ 1.50811e7 0.679413
$$869$$ −1.00116e6 −0.0449733
$$870$$ 0 0
$$871$$ −4.01015e7 −1.79108
$$872$$ 5.80870e6 0.258695
$$873$$ −5.23055e6 −0.232280
$$874$$ 2.43977e7 1.08036
$$875$$ 0 0
$$876$$ 3.64042e7 1.60284
$$877$$ 3.63188e7 1.59453 0.797266 0.603629i $$-0.206279\pi$$
0.797266 + 0.603629i $$0.206279\pi$$
$$878$$ −1.14135e7 −0.499668
$$879$$ 5.87817e6 0.256608
$$880$$ 0 0
$$881$$ 1.57431e7 0.683364 0.341682 0.939816i $$-0.389004\pi$$
0.341682 + 0.939816i $$0.389004\pi$$
$$882$$ 1.29895e7 0.562240
$$883$$ −1.30059e7 −0.561354 −0.280677 0.959802i $$-0.590559\pi$$
−0.280677 + 0.959802i $$0.590559\pi$$
$$884$$ −1.06498e8 −4.58365
$$885$$ 0 0
$$886$$ −8.79907e6 −0.376576
$$887$$ 3.12186e7 1.33231 0.666154 0.745814i $$-0.267939\pi$$
0.666154 + 0.745814i $$0.267939\pi$$
$$888$$ −1.47167e7 −0.626295
$$889$$ 6.50050e6 0.275862
$$890$$ 0 0
$$891$$ 793881. 0.0335013
$$892$$ 3.34771e6 0.140876
$$893$$ −9.82614e6 −0.412339
$$894$$ 4.69416e7 1.96433
$$895$$ 0 0
$$896$$ 7.37868e6 0.307050
$$897$$ −1.36158e7 −0.565017
$$898$$ −4.10688e7 −1.69950
$$899$$ −3.44370e7 −1.42111
$$900$$ 0 0
$$901$$ −4.30230e7 −1.76558
$$902$$ −6.84319e6 −0.280054
$$903$$ −596968. −0.0243631
$$904$$ −5.60827e6 −0.228248
$$905$$ 0 0
$$906$$ −2.08137e7 −0.842419
$$907$$ −3.59253e7 −1.45005 −0.725023 0.688725i $$-0.758171\pi$$
−0.725023 + 0.688725i $$0.758171\pi$$
$$908$$ −5.42212e6 −0.218250
$$909$$ 3.91752e6 0.157254
$$910$$ 0 0
$$911$$ −1.47634e7 −0.589372 −0.294686 0.955594i $$-0.595215\pi$$
−0.294686 + 0.955594i $$0.595215\pi$$
$$912$$ −2.24888e7 −0.895321
$$913$$ 2.58400e6 0.102592
$$914$$ 2.86592e7 1.13475
$$915$$ 0 0
$$916$$ −2.23371e7 −0.879607
$$917$$ 7.82251e6 0.307201
$$918$$ 1.10563e7 0.433014
$$919$$ −1.26772e7 −0.495146 −0.247573 0.968869i $$-0.579633\pi$$
−0.247573 + 0.968869i $$0.579633\pi$$
$$920$$ 0 0
$$921$$ −1.68653e7 −0.655155
$$922$$ −6.19224e7 −2.39895
$$923$$ −3.84435e7 −1.48531
$$924$$ −2.18058e6 −0.0840220
$$925$$ 0 0
$$926$$ −4.54495e7 −1.74181
$$927$$ −9.87580e6 −0.377462
$$928$$ 1.55848e7 0.594060
$$929$$ 1.20341e7 0.457482 0.228741 0.973487i $$-0.426539\pi$$
0.228741 + 0.973487i $$0.426539\pi$$
$$930$$ 0 0
$$931$$ −2.62277e7 −0.991713
$$932$$ −8.74136e7 −3.29639
$$933$$ 1.44563e7 0.543694
$$934$$ 2.24111e7 0.840614
$$935$$ 0 0
$$936$$ 3.07515e7 1.14730
$$937$$ 4.51344e7 1.67942 0.839708 0.543038i $$-0.182726\pi$$
0.839708 + 0.543038i $$0.182726\pi$$
$$938$$ −1.14352e7 −0.424362
$$939$$ −2.32381e7 −0.860077
$$940$$ 0 0
$$941$$ 8.44095e6 0.310754 0.155377 0.987855i $$-0.450341\pi$$
0.155377 + 0.987855i $$0.450341\pi$$
$$942$$ 2.97340e7 1.09176
$$943$$ −8.32120e6 −0.304724
$$944$$ 3.94543e7 1.44100
$$945$$ 0 0
$$946$$ −2.77528e6 −0.100827
$$947$$ −2.68109e7 −0.971485 −0.485743 0.874102i $$-0.661451\pi$$
−0.485743 + 0.874102i $$0.661451\pi$$
$$948$$ 5.13244e6 0.185483
$$949$$ 6.00675e7 2.16508
$$950$$ 0 0
$$951$$ −1.21129e7 −0.434307
$$952$$ −1.62689e7 −0.581788
$$953$$ −1.42183e7 −0.507127 −0.253563 0.967319i $$-0.581603\pi$$
−0.253563 + 0.967319i $$0.581603\pi$$
$$954$$ 2.31896e7 0.824940
$$955$$ 0 0
$$956$$ −9.33081e7 −3.30198
$$957$$ 4.97926e6 0.175746
$$958$$ 6.65263e7 2.34196
$$959$$ −5.00246e6 −0.175645
$$960$$ 0 0
$$961$$ 2.80961e7 0.981380
$$962$$ −4.53282e7 −1.57918
$$963$$ 1.55425e7 0.540078
$$964$$ 7.87082e7 2.72789
$$965$$ 0 0
$$966$$ −3.88263e6 −0.133870
$$967$$ −277112. −0.00952993 −0.00476496 0.999989i $$-0.501517\pi$$
−0.00476496 + 0.999989i $$0.501517\pi$$
$$968$$ −5.43074e6 −0.186282
$$969$$ −2.23242e7 −0.763777
$$970$$ 0 0
$$971$$ 7.77514e6 0.264643 0.132321 0.991207i $$-0.457757\pi$$
0.132321 + 0.991207i $$0.457757\pi$$
$$972$$ −4.06982e6 −0.138169
$$973$$ −1.01744e7 −0.344531
$$974$$ 2.18584e7 0.738280
$$975$$ 0 0
$$976$$ −7.81102e7 −2.62472
$$977$$ −4.83351e6 −0.162004 −0.0810020 0.996714i $$-0.525812\pi$$
−0.0810020 + 0.996714i $$0.525812\pi$$
$$978$$ 2.22765e7 0.744731
$$979$$ 9.48468e6 0.316276
$$980$$ 0 0
$$981$$ −1.26846e6 −0.0420827
$$982$$ −7.70729e7 −2.55049
$$983$$ 9.75876e6 0.322115 0.161057 0.986945i $$-0.448510\pi$$
0.161057 + 0.986945i $$0.448510\pi$$
$$984$$ 1.87936e7 0.618760
$$985$$ 0 0
$$986$$ 6.93455e7 2.27157
$$987$$ 1.56373e6 0.0510938
$$988$$ −1.15905e8 −3.77755
$$989$$ −3.37468e6 −0.109709
$$990$$ 0 0
$$991$$ 2.72969e7 0.882935 0.441468 0.897277i $$-0.354458\pi$$
0.441468 + 0.897277i $$0.354458\pi$$
$$992$$ −2.56715e7 −0.828270
$$993$$ −2.46922e7 −0.794668
$$994$$ −1.09624e7 −0.351917
$$995$$ 0 0
$$996$$ −1.32468e7 −0.423120
$$997$$ −4.85951e7 −1.54830 −0.774149 0.633004i $$-0.781822\pi$$
−0.774149 + 0.633004i $$0.781822\pi$$
$$998$$ −6.68070e7 −2.12322
$$999$$ 3.21372e6 0.101881
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 825.6.a.v.1.2 13
5.2 odd 4 165.6.c.b.34.3 26
5.3 odd 4 165.6.c.b.34.24 yes 26
5.4 even 2 825.6.a.y.1.12 13

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.3 26 5.2 odd 4
165.6.c.b.34.24 yes 26 5.3 odd 4
825.6.a.v.1.2 13 1.1 even 1 trivial
825.6.a.y.1.12 13 5.4 even 2