# Properties

 Label 49.6.a.g.1.4 Level $49$ Weight $6$ Character 49.1 Self dual yes Analytic conductor $7.859$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.85880717084$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$7.22929$$ of defining polynomial Character $$\chi$$ $$=$$ 49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.81507 q^{2} +6.54802 q^{3} -24.0754 q^{4} -45.9910 q^{5} +18.4331 q^{6} -157.856 q^{8} -200.123 q^{9} +O(q^{10})$$ $$q+2.81507 q^{2} +6.54802 q^{3} -24.0754 q^{4} -45.9910 q^{5} +18.4331 q^{6} -157.856 q^{8} -200.123 q^{9} -129.468 q^{10} -551.781 q^{11} -157.646 q^{12} +1094.10 q^{13} -301.150 q^{15} +326.035 q^{16} -1180.71 q^{17} -563.362 q^{18} -1166.13 q^{19} +1107.25 q^{20} -1553.30 q^{22} +44.3851 q^{23} -1033.65 q^{24} -1009.82 q^{25} +3079.97 q^{26} -2901.58 q^{27} +3329.02 q^{29} -847.759 q^{30} +8784.01 q^{31} +5969.21 q^{32} -3613.07 q^{33} -3323.80 q^{34} +4818.05 q^{36} -2557.12 q^{37} -3282.73 q^{38} +7164.17 q^{39} +7259.97 q^{40} -12761.3 q^{41} -96.7714 q^{43} +13284.3 q^{44} +9203.89 q^{45} +124.947 q^{46} +7679.15 q^{47} +2134.88 q^{48} -2842.73 q^{50} -7731.33 q^{51} -26340.8 q^{52} -11953.3 q^{53} -8168.16 q^{54} +25377.0 q^{55} -7635.81 q^{57} +9371.43 q^{58} +9857.24 q^{59} +7250.30 q^{60} -38517.9 q^{61} +24727.6 q^{62} +6370.65 q^{64} -50318.7 q^{65} -10171.1 q^{66} -67548.9 q^{67} +28426.1 q^{68} +290.634 q^{69} -61374.6 q^{71} +31590.7 q^{72} +1850.40 q^{73} -7198.49 q^{74} -6612.34 q^{75} +28074.9 q^{76} +20167.7 q^{78} -8.52913 q^{79} -14994.7 q^{80} +29630.4 q^{81} -35923.9 q^{82} -95039.3 q^{83} +54302.3 q^{85} -272.419 q^{86} +21798.5 q^{87} +87102.1 q^{88} +53605.6 q^{89} +25909.6 q^{90} -1068.59 q^{92} +57517.9 q^{93} +21617.4 q^{94} +53631.4 q^{95} +39086.5 q^{96} +3110.79 q^{97} +110424. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9}+O(q^{10})$$ 4 * q - 10 * q^2 + 10 * q^4 - 270 * q^8 + 220 * q^9 $$4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36} - 9208 q^{37} + 2464 q^{39} + 20448 q^{43} + 1900 q^{44} + 56712 q^{46} - 43070 q^{50} - 67408 q^{51} - 102920 q^{53} - 15576 q^{57} + 96972 q^{58} - 87080 q^{60} - 40318 q^{64} - 63168 q^{65} - 22896 q^{67} - 153824 q^{71} + 77358 q^{72} + 17596 q^{74} + 133056 q^{78} - 90688 q^{79} - 17204 q^{81} + 272656 q^{85} - 161860 q^{86} + 154812 q^{88} - 212200 q^{92} + 247760 q^{93} + 108224 q^{95} - 42272 q^{99}+O(q^{100})$$ 4 * q - 10 * q^2 + 10 * q^4 - 270 * q^8 + 220 * q^9 - 1952 * q^11 - 4096 * q^15 - 1566 * q^16 - 5974 * q^18 + 3524 * q^22 - 7136 * q^23 + 2764 * q^25 - 3352 * q^29 + 25608 * q^30 + 27810 * q^32 + 27670 * q^36 - 9208 * q^37 + 2464 * q^39 + 20448 * q^43 + 1900 * q^44 + 56712 * q^46 - 43070 * q^50 - 67408 * q^51 - 102920 * q^53 - 15576 * q^57 + 96972 * q^58 - 87080 * q^60 - 40318 * q^64 - 63168 * q^65 - 22896 * q^67 - 153824 * q^71 + 77358 * q^72 + 17596 * q^74 + 133056 * q^78 - 90688 * q^79 - 17204 * q^81 + 272656 * q^85 - 161860 * q^86 + 154812 * q^88 - 212200 * q^92 + 247760 * q^93 + 108224 * q^95 - 42272 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.81507 0.497639 0.248820 0.968550i $$-0.419957\pi$$
0.248820 + 0.968550i $$0.419957\pi$$
$$3$$ 6.54802 0.420055 0.210028 0.977695i $$-0.432645\pi$$
0.210028 + 0.977695i $$0.432645\pi$$
$$4$$ −24.0754 −0.752355
$$5$$ −45.9910 −0.822713 −0.411356 0.911475i $$-0.634945\pi$$
−0.411356 + 0.911475i $$0.634945\pi$$
$$6$$ 18.4331 0.209036
$$7$$ 0 0
$$8$$ −157.856 −0.872041
$$9$$ −200.123 −0.823553
$$10$$ −129.468 −0.409414
$$11$$ −551.781 −1.37494 −0.687472 0.726211i $$-0.741280\pi$$
−0.687472 + 0.726211i $$0.741280\pi$$
$$12$$ −157.646 −0.316031
$$13$$ 1094.10 1.79555 0.897776 0.440453i $$-0.145182\pi$$
0.897776 + 0.440453i $$0.145182\pi$$
$$14$$ 0 0
$$15$$ −301.150 −0.345585
$$16$$ 326.035 0.318393
$$17$$ −1180.71 −0.990883 −0.495442 0.868641i $$-0.664994\pi$$
−0.495442 + 0.868641i $$0.664994\pi$$
$$18$$ −563.362 −0.409833
$$19$$ −1166.13 −0.741074 −0.370537 0.928818i $$-0.620826\pi$$
−0.370537 + 0.928818i $$0.620826\pi$$
$$20$$ 1107.25 0.618972
$$21$$ 0 0
$$22$$ −1553.30 −0.684226
$$23$$ 44.3851 0.0174951 0.00874757 0.999962i $$-0.497216\pi$$
0.00874757 + 0.999962i $$0.497216\pi$$
$$24$$ −1033.65 −0.366305
$$25$$ −1009.82 −0.323143
$$26$$ 3079.97 0.893537
$$27$$ −2901.58 −0.765993
$$28$$ 0 0
$$29$$ 3329.02 0.735057 0.367529 0.930012i $$-0.380204\pi$$
0.367529 + 0.930012i $$0.380204\pi$$
$$30$$ −847.759 −0.171977
$$31$$ 8784.01 1.64168 0.820841 0.571157i $$-0.193505\pi$$
0.820841 + 0.571157i $$0.193505\pi$$
$$32$$ 5969.21 1.03049
$$33$$ −3613.07 −0.577552
$$34$$ −3323.80 −0.493102
$$35$$ 0 0
$$36$$ 4818.05 0.619605
$$37$$ −2557.12 −0.307077 −0.153539 0.988143i $$-0.549067\pi$$
−0.153539 + 0.988143i $$0.549067\pi$$
$$38$$ −3282.73 −0.368787
$$39$$ 7164.17 0.754231
$$40$$ 7259.97 0.717439
$$41$$ −12761.3 −1.18559 −0.592794 0.805354i $$-0.701975\pi$$
−0.592794 + 0.805354i $$0.701975\pi$$
$$42$$ 0 0
$$43$$ −96.7714 −0.00798135 −0.00399067 0.999992i $$-0.501270\pi$$
−0.00399067 + 0.999992i $$0.501270\pi$$
$$44$$ 13284.3 1.03445
$$45$$ 9203.89 0.677548
$$46$$ 124.947 0.00870627
$$47$$ 7679.15 0.507071 0.253535 0.967326i $$-0.418407\pi$$
0.253535 + 0.967326i $$0.418407\pi$$
$$48$$ 2134.88 0.133743
$$49$$ 0 0
$$50$$ −2842.73 −0.160809
$$51$$ −7731.33 −0.416226
$$52$$ −26340.8 −1.35089
$$53$$ −11953.3 −0.584520 −0.292260 0.956339i $$-0.594407\pi$$
−0.292260 + 0.956339i $$0.594407\pi$$
$$54$$ −8168.16 −0.381188
$$55$$ 25377.0 1.13118
$$56$$ 0 0
$$57$$ −7635.81 −0.311292
$$58$$ 9371.43 0.365793
$$59$$ 9857.24 0.368659 0.184330 0.982864i $$-0.440989\pi$$
0.184330 + 0.982864i $$0.440989\pi$$
$$60$$ 7250.30 0.260003
$$61$$ −38517.9 −1.32537 −0.662686 0.748897i $$-0.730584\pi$$
−0.662686 + 0.748897i $$0.730584\pi$$
$$62$$ 24727.6 0.816965
$$63$$ 0 0
$$64$$ 6370.65 0.194417
$$65$$ −50318.7 −1.47722
$$66$$ −10171.1 −0.287413
$$67$$ −67548.9 −1.83836 −0.919182 0.393833i $$-0.871149\pi$$
−0.919182 + 0.393833i $$0.871149\pi$$
$$68$$ 28426.1 0.745496
$$69$$ 290.634 0.00734893
$$70$$ 0 0
$$71$$ −61374.6 −1.44492 −0.722458 0.691415i $$-0.756988\pi$$
−0.722458 + 0.691415i $$0.756988\pi$$
$$72$$ 31590.7 0.718172
$$73$$ 1850.40 0.0406404 0.0203202 0.999794i $$-0.493531\pi$$
0.0203202 + 0.999794i $$0.493531\pi$$
$$74$$ −7198.49 −0.152814
$$75$$ −6612.34 −0.135738
$$76$$ 28074.9 0.557551
$$77$$ 0 0
$$78$$ 20167.7 0.375335
$$79$$ −8.52913 −0.000153758 0 −7.68788e−5 1.00000i $$-0.500024\pi$$
−7.68788e−5 1.00000i $$0.500024\pi$$
$$80$$ −14994.7 −0.261946
$$81$$ 29630.4 0.501794
$$82$$ −35923.9 −0.589996
$$83$$ −95039.3 −1.51429 −0.757143 0.653249i $$-0.773405\pi$$
−0.757143 + 0.653249i $$0.773405\pi$$
$$84$$ 0 0
$$85$$ 54302.3 0.815212
$$86$$ −272.419 −0.00397183
$$87$$ 21798.5 0.308765
$$88$$ 87102.1 1.19901
$$89$$ 53605.6 0.717357 0.358678 0.933461i $$-0.383227\pi$$
0.358678 + 0.933461i $$0.383227\pi$$
$$90$$ 25909.6 0.337175
$$91$$ 0 0
$$92$$ −1068.59 −0.0131626
$$93$$ 57517.9 0.689597
$$94$$ 21617.4 0.252338
$$95$$ 53631.4 0.609691
$$96$$ 39086.5 0.432861
$$97$$ 3110.79 0.0335693 0.0167846 0.999859i $$-0.494657\pi$$
0.0167846 + 0.999859i $$0.494657\pi$$
$$98$$ 0 0
$$99$$ 110424. 1.13234
$$100$$ 24311.9 0.243119
$$101$$ 21835.9 0.212994 0.106497 0.994313i $$-0.466037\pi$$
0.106497 + 0.994313i $$0.466037\pi$$
$$102$$ −21764.3 −0.207130
$$103$$ 65341.4 0.606870 0.303435 0.952852i $$-0.401866\pi$$
0.303435 + 0.952852i $$0.401866\pi$$
$$104$$ −172710. −1.56579
$$105$$ 0 0
$$106$$ −33649.5 −0.290880
$$107$$ −108957. −0.920020 −0.460010 0.887914i $$-0.652154\pi$$
−0.460010 + 0.887914i $$0.652154\pi$$
$$108$$ 69856.6 0.576299
$$109$$ 86728.7 0.699192 0.349596 0.936901i $$-0.386319\pi$$
0.349596 + 0.936901i $$0.386319\pi$$
$$110$$ 71438.1 0.562922
$$111$$ −16744.1 −0.128989
$$112$$ 0 0
$$113$$ −101496. −0.747746 −0.373873 0.927480i $$-0.621970\pi$$
−0.373873 + 0.927480i $$0.621970\pi$$
$$114$$ −21495.4 −0.154911
$$115$$ −2041.32 −0.0143935
$$116$$ −80147.3 −0.553024
$$117$$ −218955. −1.47873
$$118$$ 27748.9 0.183459
$$119$$ 0 0
$$120$$ 47538.4 0.301364
$$121$$ 143411. 0.890470
$$122$$ −108431. −0.659557
$$123$$ −83560.9 −0.498013
$$124$$ −211478. −1.23513
$$125$$ 190165. 1.08857
$$126$$ 0 0
$$127$$ −3094.61 −0.0170253 −0.00851267 0.999964i $$-0.502710\pi$$
−0.00851267 + 0.999964i $$0.502710\pi$$
$$128$$ −173081. −0.933736
$$129$$ −633.661 −0.00335261
$$130$$ −141651. −0.735124
$$131$$ 253431. 1.29027 0.645136 0.764067i $$-0.276801\pi$$
0.645136 + 0.764067i $$0.276801\pi$$
$$132$$ 86986.0 0.434525
$$133$$ 0 0
$$134$$ −190155. −0.914842
$$135$$ 133447. 0.630193
$$136$$ 186383. 0.864091
$$137$$ −97152.9 −0.442236 −0.221118 0.975247i $$-0.570971\pi$$
−0.221118 + 0.975247i $$0.570971\pi$$
$$138$$ 818.156 0.00365711
$$139$$ 210308. 0.923249 0.461624 0.887076i $$-0.347267\pi$$
0.461624 + 0.887076i $$0.347267\pi$$
$$140$$ 0 0
$$141$$ 50283.2 0.212998
$$142$$ −172774. −0.719047
$$143$$ −603702. −2.46878
$$144$$ −65247.2 −0.262214
$$145$$ −153105. −0.604741
$$146$$ 5209.01 0.0202243
$$147$$ 0 0
$$148$$ 61563.7 0.231031
$$149$$ 140406. 0.518109 0.259055 0.965863i $$-0.416589\pi$$
0.259055 + 0.965863i $$0.416589\pi$$
$$150$$ −18614.2 −0.0675486
$$151$$ 119696. 0.427205 0.213603 0.976921i $$-0.431480\pi$$
0.213603 + 0.976921i $$0.431480\pi$$
$$152$$ 184080. 0.646247
$$153$$ 236289. 0.816045
$$154$$ 0 0
$$155$$ −403986. −1.35063
$$156$$ −172480. −0.567450
$$157$$ 97616.9 0.316065 0.158032 0.987434i $$-0.449485\pi$$
0.158032 + 0.987434i $$0.449485\pi$$
$$158$$ −24.0101 −7.65159e−5 0
$$159$$ −78270.6 −0.245531
$$160$$ −274530. −0.847794
$$161$$ 0 0
$$162$$ 83411.8 0.249712
$$163$$ 182678. 0.538539 0.269270 0.963065i $$-0.413218\pi$$
0.269270 + 0.963065i $$0.413218\pi$$
$$164$$ 307232. 0.891984
$$165$$ 166169. 0.475160
$$166$$ −267542. −0.753568
$$167$$ −451674. −1.25324 −0.626619 0.779326i $$-0.715562\pi$$
−0.626619 + 0.779326i $$0.715562\pi$$
$$168$$ 0 0
$$169$$ 825757. 2.22400
$$170$$ 152865. 0.405682
$$171$$ 233369. 0.610314
$$172$$ 2329.81 0.00600481
$$173$$ 371647. 0.944095 0.472047 0.881573i $$-0.343515\pi$$
0.472047 + 0.881573i $$0.343515\pi$$
$$174$$ 61364.2 0.153653
$$175$$ 0 0
$$176$$ −179900. −0.437773
$$177$$ 64545.4 0.154857
$$178$$ 150904. 0.356985
$$179$$ 85003.4 0.198291 0.0991457 0.995073i $$-0.468389\pi$$
0.0991457 + 0.995073i $$0.468389\pi$$
$$180$$ −221587. −0.509757
$$181$$ −379442. −0.860892 −0.430446 0.902616i $$-0.641644\pi$$
−0.430446 + 0.902616i $$0.641644\pi$$
$$182$$ 0 0
$$183$$ −252216. −0.556730
$$184$$ −7006.46 −0.0152565
$$185$$ 117605. 0.252636
$$186$$ 161917. 0.343171
$$187$$ 651496. 1.36241
$$188$$ −184878. −0.381497
$$189$$ 0 0
$$190$$ 150976. 0.303406
$$191$$ −922196. −1.82911 −0.914555 0.404462i $$-0.867459\pi$$
−0.914555 + 0.404462i $$0.867459\pi$$
$$192$$ 41715.1 0.0816658
$$193$$ 505107. 0.976090 0.488045 0.872818i $$-0.337710\pi$$
0.488045 + 0.872818i $$0.337710\pi$$
$$194$$ 8757.11 0.0167054
$$195$$ −329488. −0.620516
$$196$$ 0 0
$$197$$ 251505. 0.461723 0.230861 0.972987i $$-0.425846\pi$$
0.230861 + 0.972987i $$0.425846\pi$$
$$198$$ 310853. 0.563497
$$199$$ −208033. −0.372392 −0.186196 0.982513i $$-0.559616\pi$$
−0.186196 + 0.982513i $$0.559616\pi$$
$$200$$ 159407. 0.281794
$$201$$ −442311. −0.772215
$$202$$ 61469.7 0.105994
$$203$$ 0 0
$$204$$ 186135. 0.313150
$$205$$ 586904. 0.975399
$$206$$ 183941. 0.302002
$$207$$ −8882.50 −0.0144082
$$208$$ 356714. 0.571692
$$209$$ 643446. 1.01893
$$210$$ 0 0
$$211$$ −640577. −0.990525 −0.495262 0.868744i $$-0.664928\pi$$
−0.495262 + 0.868744i $$0.664928\pi$$
$$212$$ 287781. 0.439767
$$213$$ −401882. −0.606945
$$214$$ −306723. −0.457838
$$215$$ 4450.62 0.00656636
$$216$$ 458032. 0.667977
$$217$$ 0 0
$$218$$ 244148. 0.347945
$$219$$ 12116.4 0.0170712
$$220$$ −610960. −0.851052
$$221$$ −1.29182e6 −1.77918
$$222$$ −47135.8 −0.0641902
$$223$$ 390135. 0.525354 0.262677 0.964884i $$-0.415395\pi$$
0.262677 + 0.964884i $$0.415395\pi$$
$$224$$ 0 0
$$225$$ 202089. 0.266126
$$226$$ −285720. −0.372108
$$227$$ 291353. 0.375279 0.187639 0.982238i $$-0.439916\pi$$
0.187639 + 0.982238i $$0.439916\pi$$
$$228$$ 183835. 0.234202
$$229$$ −1.23040e6 −1.55045 −0.775227 0.631682i $$-0.782365\pi$$
−0.775227 + 0.631682i $$0.782365\pi$$
$$230$$ −5746.45 −0.00716276
$$231$$ 0 0
$$232$$ −525506. −0.641000
$$233$$ 114279. 0.137903 0.0689517 0.997620i $$-0.478035\pi$$
0.0689517 + 0.997620i $$0.478035\pi$$
$$234$$ −616373. −0.735875
$$235$$ −353172. −0.417174
$$236$$ −237317. −0.277363
$$237$$ −55.8489 −6.45867e−5 0
$$238$$ 0 0
$$239$$ −1.14782e6 −1.29981 −0.649906 0.760014i $$-0.725192\pi$$
−0.649906 + 0.760014i $$0.725192\pi$$
$$240$$ −98185.4 −0.110032
$$241$$ 812708. 0.901346 0.450673 0.892689i $$-0.351184\pi$$
0.450673 + 0.892689i $$0.351184\pi$$
$$242$$ 403713. 0.443133
$$243$$ 899104. 0.976775
$$244$$ 927332. 0.997150
$$245$$ 0 0
$$246$$ −235230. −0.247831
$$247$$ −1.27586e6 −1.33064
$$248$$ −1.38661e6 −1.43161
$$249$$ −622318. −0.636084
$$250$$ 535328. 0.541714
$$251$$ −406772. −0.407537 −0.203768 0.979019i $$-0.565319\pi$$
−0.203768 + 0.979019i $$0.565319\pi$$
$$252$$ 0 0
$$253$$ −24490.8 −0.0240548
$$254$$ −8711.54 −0.00847248
$$255$$ 355572. 0.342434
$$256$$ −691096. −0.659081
$$257$$ −1.69712e6 −1.60281 −0.801403 0.598125i $$-0.795913\pi$$
−0.801403 + 0.598125i $$0.795913\pi$$
$$258$$ −1783.80 −0.00166839
$$259$$ 0 0
$$260$$ 1.21144e6 1.11140
$$261$$ −666215. −0.605359
$$262$$ 713427. 0.642090
$$263$$ 205694. 0.183372 0.0916859 0.995788i $$-0.470774\pi$$
0.0916859 + 0.995788i $$0.470774\pi$$
$$264$$ 570346. 0.503649
$$265$$ 549746. 0.480892
$$266$$ 0 0
$$267$$ 351010. 0.301330
$$268$$ 1.62627e6 1.38310
$$269$$ 1.73425e6 1.46127 0.730635 0.682769i $$-0.239224\pi$$
0.730635 + 0.682769i $$0.239224\pi$$
$$270$$ 375662. 0.313609
$$271$$ 369042. 0.305248 0.152624 0.988284i $$-0.451228\pi$$
0.152624 + 0.988284i $$0.451228\pi$$
$$272$$ −384954. −0.315491
$$273$$ 0 0
$$274$$ −273492. −0.220074
$$275$$ 557201. 0.444304
$$276$$ −6997.12 −0.00552900
$$277$$ 1.22767e6 0.961351 0.480676 0.876899i $$-0.340391\pi$$
0.480676 + 0.876899i $$0.340391\pi$$
$$278$$ 592032. 0.459445
$$279$$ −1.75789e6 −1.35201
$$280$$ 0 0
$$281$$ 2.00671e6 1.51607 0.758035 0.652214i $$-0.226159\pi$$
0.758035 + 0.652214i $$0.226159\pi$$
$$282$$ 141551. 0.105996
$$283$$ −1.78581e6 −1.32547 −0.662734 0.748855i $$-0.730604\pi$$
−0.662734 + 0.748855i $$0.730604\pi$$
$$284$$ 1.47762e6 1.08709
$$285$$ 351179. 0.256104
$$286$$ −1.69947e6 −1.22856
$$287$$ 0 0
$$288$$ −1.19458e6 −0.848660
$$289$$ −25770.8 −0.0181503
$$290$$ −431002. −0.300943
$$291$$ 20369.5 0.0141009
$$292$$ −44549.0 −0.0305760
$$293$$ 853248. 0.580639 0.290319 0.956930i $$-0.406238\pi$$
0.290319 + 0.956930i $$0.406238\pi$$
$$294$$ 0 0
$$295$$ −453345. −0.303301
$$296$$ 403658. 0.267784
$$297$$ 1.60104e6 1.05320
$$298$$ 395254. 0.257831
$$299$$ 48561.6 0.0314134
$$300$$ 159194. 0.102123
$$301$$ 0 0
$$302$$ 336952. 0.212594
$$303$$ 142982. 0.0894693
$$304$$ −380198. −0.235953
$$305$$ 1.77148e6 1.09040
$$306$$ 665170. 0.406096
$$307$$ −1.96068e6 −1.18730 −0.593652 0.804722i $$-0.702314\pi$$
−0.593652 + 0.804722i $$0.702314\pi$$
$$308$$ 0 0
$$309$$ 427857. 0.254919
$$310$$ −1.13725e6 −0.672128
$$311$$ 863604. 0.506307 0.253153 0.967426i $$-0.418532\pi$$
0.253153 + 0.967426i $$0.418532\pi$$
$$312$$ −1.13091e6 −0.657720
$$313$$ 1.10047e6 0.634918 0.317459 0.948272i $$-0.397170\pi$$
0.317459 + 0.948272i $$0.397170\pi$$
$$314$$ 274799. 0.157286
$$315$$ 0 0
$$316$$ 205.342 0.000115680 0
$$317$$ 1.49591e6 0.836097 0.418048 0.908425i $$-0.362714\pi$$
0.418048 + 0.908425i $$0.362714\pi$$
$$318$$ −220337. −0.122186
$$319$$ −1.83689e6 −1.01066
$$320$$ −292993. −0.159949
$$321$$ −713454. −0.386459
$$322$$ 0 0
$$323$$ 1.37686e6 0.734318
$$324$$ −713363. −0.377527
$$325$$ −1.10485e6 −0.580221
$$326$$ 514252. 0.267998
$$327$$ 567901. 0.293699
$$328$$ 2.01445e6 1.03388
$$329$$ 0 0
$$330$$ 467777. 0.236458
$$331$$ −2.74015e6 −1.37469 −0.687345 0.726331i $$-0.741224\pi$$
−0.687345 + 0.726331i $$0.741224\pi$$
$$332$$ 2.28810e6 1.13928
$$333$$ 511741. 0.252894
$$334$$ −1.27149e6 −0.623661
$$335$$ 3.10665e6 1.51245
$$336$$ 0 0
$$337$$ −2.31353e6 −1.10968 −0.554842 0.831956i $$-0.687221\pi$$
−0.554842 + 0.831956i $$0.687221\pi$$
$$338$$ 2.32457e6 1.10675
$$339$$ −664600. −0.314095
$$340$$ −1.30735e6 −0.613329
$$341$$ −4.84685e6 −2.25722
$$342$$ 656951. 0.303716
$$343$$ 0 0
$$344$$ 15276.0 0.00696006
$$345$$ −13366.6 −0.00604606
$$346$$ 1.04621e6 0.469819
$$347$$ −3.05926e6 −1.36393 −0.681966 0.731384i $$-0.738875\pi$$
−0.681966 + 0.731384i $$0.738875\pi$$
$$348$$ −524806. −0.232301
$$349$$ −210232. −0.0923921 −0.0461961 0.998932i $$-0.514710\pi$$
−0.0461961 + 0.998932i $$0.514710\pi$$
$$350$$ 0 0
$$351$$ −3.17461e6 −1.37538
$$352$$ −3.29370e6 −1.41686
$$353$$ −3.76790e6 −1.60939 −0.804697 0.593686i $$-0.797672\pi$$
−0.804697 + 0.593686i $$0.797672\pi$$
$$354$$ 181700. 0.0770631
$$355$$ 2.82268e6 1.18875
$$356$$ −1.29057e6 −0.539707
$$357$$ 0 0
$$358$$ 239291. 0.0986776
$$359$$ 1.00722e6 0.412465 0.206232 0.978503i $$-0.433880\pi$$
0.206232 + 0.978503i $$0.433880\pi$$
$$360$$ −1.45289e6 −0.590850
$$361$$ −1.11625e6 −0.450809
$$362$$ −1.06816e6 −0.428414
$$363$$ 939058. 0.374047
$$364$$ 0 0
$$365$$ −85101.8 −0.0334354
$$366$$ −710005. −0.277050
$$367$$ −1.52650e6 −0.591603 −0.295802 0.955249i $$-0.595587\pi$$
−0.295802 + 0.955249i $$0.595587\pi$$
$$368$$ 14471.1 0.00557034
$$369$$ 2.55383e6 0.976396
$$370$$ 331066. 0.125722
$$371$$ 0 0
$$372$$ −1.38476e6 −0.518822
$$373$$ 4.86297e6 1.80980 0.904898 0.425629i $$-0.139947\pi$$
0.904898 + 0.425629i $$0.139947\pi$$
$$374$$ 1.83401e6 0.677988
$$375$$ 1.24520e6 0.457258
$$376$$ −1.21220e6 −0.442186
$$377$$ 3.64227e6 1.31983
$$378$$ 0 0
$$379$$ 630878. 0.225604 0.112802 0.993617i $$-0.464017\pi$$
0.112802 + 0.993617i $$0.464017\pi$$
$$380$$ −1.29119e6 −0.458704
$$381$$ −20263.5 −0.00715159
$$382$$ −2.59605e6 −0.910237
$$383$$ 565644. 0.197036 0.0985182 0.995135i $$-0.468590\pi$$
0.0985182 + 0.995135i $$0.468590\pi$$
$$384$$ −1.13334e6 −0.392221
$$385$$ 0 0
$$386$$ 1.42191e6 0.485741
$$387$$ 19366.2 0.00657306
$$388$$ −74893.5 −0.0252560
$$389$$ 592212. 0.198428 0.0992140 0.995066i $$-0.468367\pi$$
0.0992140 + 0.995066i $$0.468367\pi$$
$$390$$ −927532. −0.308793
$$391$$ −52406.1 −0.0173356
$$392$$ 0 0
$$393$$ 1.65947e6 0.541986
$$394$$ 708005. 0.229771
$$395$$ 392.264 0.000126498 0
$$396$$ −2.65851e6 −0.851922
$$397$$ 1.34312e6 0.427698 0.213849 0.976867i $$-0.431400\pi$$
0.213849 + 0.976867i $$0.431400\pi$$
$$398$$ −585629. −0.185317
$$399$$ 0 0
$$400$$ −329238. −0.102887
$$401$$ −3.68716e6 −1.14507 −0.572534 0.819881i $$-0.694040\pi$$
−0.572534 + 0.819881i $$0.694040\pi$$
$$402$$ −1.24514e6 −0.384284
$$403$$ 9.61057e6 2.94772
$$404$$ −525707. −0.160247
$$405$$ −1.36273e6 −0.412832
$$406$$ 0 0
$$407$$ 1.41097e6 0.422214
$$408$$ 1.22044e6 0.362966
$$409$$ 1.45630e6 0.430470 0.215235 0.976562i $$-0.430948\pi$$
0.215235 + 0.976562i $$0.430948\pi$$
$$410$$ 1.65218e6 0.485397
$$411$$ −636159. −0.185764
$$412$$ −1.57312e6 −0.456582
$$413$$ 0 0
$$414$$ −25004.9 −0.00717008
$$415$$ 4.37096e6 1.24582
$$416$$ 6.53090e6 1.85029
$$417$$ 1.37710e6 0.387816
$$418$$ 1.81135e6 0.507062
$$419$$ −2.92192e6 −0.813080 −0.406540 0.913633i $$-0.633265\pi$$
−0.406540 + 0.913633i $$0.633265\pi$$
$$420$$ 0 0
$$421$$ 2.01999e6 0.555450 0.277725 0.960661i $$-0.410420\pi$$
0.277725 + 0.960661i $$0.410420\pi$$
$$422$$ −1.80327e6 −0.492924
$$423$$ −1.53678e6 −0.417600
$$424$$ 1.88691e6 0.509725
$$425$$ 1.19231e6 0.320197
$$426$$ −1.13133e6 −0.302040
$$427$$ 0 0
$$428$$ 2.62319e6 0.692182
$$429$$ −3.95305e6 −1.03703
$$430$$ 12528.8 0.00326768
$$431$$ −5.43800e6 −1.41009 −0.705043 0.709164i $$-0.749072\pi$$
−0.705043 + 0.709164i $$0.749072\pi$$
$$432$$ −946016. −0.243887
$$433$$ −3.77335e6 −0.967179 −0.483590 0.875295i $$-0.660667\pi$$
−0.483590 + 0.875295i $$0.660667\pi$$
$$434$$ 0 0
$$435$$ −1.00253e6 −0.254025
$$436$$ −2.08802e6 −0.526041
$$437$$ −51758.6 −0.0129652
$$438$$ 34108.7 0.00849531
$$439$$ 2.35150e6 0.582350 0.291175 0.956670i $$-0.405954\pi$$
0.291175 + 0.956670i $$0.405954\pi$$
$$440$$ −4.00591e6 −0.986439
$$441$$ 0 0
$$442$$ −3.63656e6 −0.885391
$$443$$ 4.80377e6 1.16298 0.581491 0.813553i $$-0.302470\pi$$
0.581491 + 0.813553i $$0.302470\pi$$
$$444$$ 403120. 0.0970458
$$445$$ −2.46538e6 −0.590179
$$446$$ 1.09826e6 0.261437
$$447$$ 919383. 0.217634
$$448$$ 0 0
$$449$$ −2.76805e6 −0.647975 −0.323987 0.946061i $$-0.605024\pi$$
−0.323987 + 0.946061i $$0.605024\pi$$
$$450$$ 568896. 0.132435
$$451$$ 7.04142e6 1.63012
$$452$$ 2.44356e6 0.562571
$$453$$ 783770. 0.179450
$$454$$ 820179. 0.186754
$$455$$ 0 0
$$456$$ 1.20536e6 0.271459
$$457$$ 241566. 0.0541061 0.0270530 0.999634i $$-0.491388\pi$$
0.0270530 + 0.999634i $$0.491388\pi$$
$$458$$ −3.46368e6 −0.771567
$$459$$ 3.42594e6 0.759010
$$460$$ 49145.4 0.0108290
$$461$$ 990579. 0.217088 0.108544 0.994092i $$-0.465381\pi$$
0.108544 + 0.994092i $$0.465381\pi$$
$$462$$ 0 0
$$463$$ 6.20488e6 1.34518 0.672591 0.740014i $$-0.265181\pi$$
0.672591 + 0.740014i $$0.265181\pi$$
$$464$$ 1.08538e6 0.234037
$$465$$ −2.64531e6 −0.567340
$$466$$ 321702. 0.0686261
$$467$$ −6.88497e6 −1.46086 −0.730432 0.682985i $$-0.760681\pi$$
−0.730432 + 0.682985i $$0.760681\pi$$
$$468$$ 5.27141e6 1.11253
$$469$$ 0 0
$$470$$ −994206. −0.207602
$$471$$ 639197. 0.132765
$$472$$ −1.55603e6 −0.321486
$$473$$ 53396.6 0.0109739
$$474$$ −157.219 −3.21409e−5 0
$$475$$ 1.17758e6 0.239473
$$476$$ 0 0
$$477$$ 2.39214e6 0.481383
$$478$$ −3.23121e6 −0.646838
$$479$$ −5.41288e6 −1.07793 −0.538963 0.842329i $$-0.681184\pi$$
−0.538963 + 0.842329i $$0.681184\pi$$
$$480$$ −1.79763e6 −0.356120
$$481$$ −2.79774e6 −0.551373
$$482$$ 2.28783e6 0.448545
$$483$$ 0 0
$$484$$ −3.45268e6 −0.669950
$$485$$ −143069. −0.0276179
$$486$$ 2.53104e6 0.486081
$$487$$ −3.01043e6 −0.575182 −0.287591 0.957753i $$-0.592854\pi$$
−0.287591 + 0.957753i $$0.592854\pi$$
$$488$$ 6.08029e6 1.15578
$$489$$ 1.19618e6 0.226216
$$490$$ 0 0
$$491$$ −7.24498e6 −1.35623 −0.678115 0.734956i $$-0.737203\pi$$
−0.678115 + 0.734956i $$0.737203\pi$$
$$492$$ 2.01176e6 0.374683
$$493$$ −3.93062e6 −0.728356
$$494$$ −3.59163e6 −0.662177
$$495$$ −5.07853e6 −0.931591
$$496$$ 2.86389e6 0.522700
$$497$$ 0 0
$$498$$ −1.75187e6 −0.316540
$$499$$ −5.12788e6 −0.921906 −0.460953 0.887425i $$-0.652492\pi$$
−0.460953 + 0.887425i $$0.652492\pi$$
$$500$$ −4.57829e6 −0.818989
$$501$$ −2.95757e6 −0.526429
$$502$$ −1.14509e6 −0.202806
$$503$$ 1.05978e7 1.86766 0.933830 0.357718i $$-0.116445\pi$$
0.933830 + 0.357718i $$0.116445\pi$$
$$504$$ 0 0
$$505$$ −1.00426e6 −0.175233
$$506$$ −68943.5 −0.0119706
$$507$$ 5.40707e6 0.934205
$$508$$ 74503.8 0.0128091
$$509$$ 8.78840e6 1.50354 0.751770 0.659425i $$-0.229200\pi$$
0.751770 + 0.659425i $$0.229200\pi$$
$$510$$ 1.00096e6 0.170409
$$511$$ 0 0
$$512$$ 3.59310e6 0.605752
$$513$$ 3.38361e6 0.567658
$$514$$ −4.77753e6 −0.797619
$$515$$ −3.00512e6 −0.499280
$$516$$ 15255.6 0.00252235
$$517$$ −4.23721e6 −0.697194
$$518$$ 0 0
$$519$$ 2.43355e6 0.396572
$$520$$ 7.94312e6 1.28820
$$521$$ 162133. 0.0261684 0.0130842 0.999914i $$-0.495835\pi$$
0.0130842 + 0.999914i $$0.495835\pi$$
$$522$$ −1.87544e6 −0.301250
$$523$$ −7.14844e6 −1.14277 −0.571383 0.820684i $$-0.693593\pi$$
−0.571383 + 0.820684i $$0.693593\pi$$
$$524$$ −6.10144e6 −0.970743
$$525$$ 0 0
$$526$$ 579044. 0.0912530
$$527$$ −1.03714e7 −1.62671
$$528$$ −1.17799e6 −0.183889
$$529$$ −6.43437e6 −0.999694
$$530$$ 1.54758e6 0.239311
$$531$$ −1.97267e6 −0.303611
$$532$$ 0 0
$$533$$ −1.39621e7 −2.12879
$$534$$ 988120. 0.149953
$$535$$ 5.01106e6 0.756912
$$536$$ 1.06630e7 1.60313
$$537$$ 556604. 0.0832934
$$538$$ 4.88203e6 0.727185
$$539$$ 0 0
$$540$$ −3.21278e6 −0.474129
$$541$$ −4.83604e6 −0.710390 −0.355195 0.934792i $$-0.615586\pi$$
−0.355195 + 0.934792i $$0.615586\pi$$
$$542$$ 1.03888e6 0.151903
$$543$$ −2.48459e6 −0.361622
$$544$$ −7.04793e6 −1.02109
$$545$$ −3.98874e6 −0.575234
$$546$$ 0 0
$$547$$ 9.98777e6 1.42725 0.713626 0.700527i $$-0.247052\pi$$
0.713626 + 0.700527i $$0.247052\pi$$
$$548$$ 2.33899e6 0.332719
$$549$$ 7.70833e6 1.09151
$$550$$ 1.56856e6 0.221103
$$551$$ −3.88205e6 −0.544732
$$552$$ −45878.4 −0.00640856
$$553$$ 0 0
$$554$$ 3.45598e6 0.478406
$$555$$ 770078. 0.106121
$$556$$ −5.06324e6 −0.694611
$$557$$ 1.74619e6 0.238481 0.119241 0.992865i $$-0.461954\pi$$
0.119241 + 0.992865i $$0.461954\pi$$
$$558$$ −4.94858e6 −0.672814
$$559$$ −105877. −0.0143309
$$560$$ 0 0
$$561$$ 4.26600e6 0.572287
$$562$$ 5.64904e6 0.754456
$$563$$ −755218. −0.100416 −0.0502078 0.998739i $$-0.515988\pi$$
−0.0502078 + 0.998739i $$0.515988\pi$$
$$564$$ −1.21059e6 −0.160250
$$565$$ 4.66792e6 0.615181
$$566$$ −5.02719e6 −0.659605
$$567$$ 0 0
$$568$$ 9.68836e6 1.26003
$$569$$ −4.39534e6 −0.569131 −0.284565 0.958657i $$-0.591849\pi$$
−0.284565 + 0.958657i $$0.591849\pi$$
$$570$$ 988594. 0.127447
$$571$$ 1.16104e7 1.49024 0.745121 0.666930i $$-0.232392\pi$$
0.745121 + 0.666930i $$0.232392\pi$$
$$572$$ 1.45344e7 1.85740
$$573$$ −6.03855e6 −0.768327
$$574$$ 0 0
$$575$$ −44821.1 −0.00565344
$$576$$ −1.27492e6 −0.160113
$$577$$ 1.06643e7 1.33350 0.666748 0.745283i $$-0.267686\pi$$
0.666748 + 0.745283i $$0.267686\pi$$
$$578$$ −72546.6 −0.00903228
$$579$$ 3.30745e6 0.410012
$$580$$ 3.68606e6 0.454980
$$581$$ 0 0
$$582$$ 57341.7 0.00701719
$$583$$ 6.59562e6 0.803682
$$584$$ −292097. −0.0354401
$$585$$ 1.00700e7 1.21657
$$586$$ 2.40195e6 0.288949
$$587$$ −1.39482e7 −1.67079 −0.835396 0.549648i $$-0.814762\pi$$
−0.835396 + 0.549648i $$0.814762\pi$$
$$588$$ 0 0
$$589$$ −1.02433e7 −1.21661
$$590$$ −1.27620e6 −0.150934
$$591$$ 1.64686e6 0.193949
$$592$$ −833711. −0.0977713
$$593$$ 1.17933e7 1.37720 0.688600 0.725142i $$-0.258226\pi$$
0.688600 + 0.725142i $$0.258226\pi$$
$$594$$ 4.50703e6 0.524113
$$595$$ 0 0
$$596$$ −3.38033e6 −0.389802
$$597$$ −1.36220e6 −0.156425
$$598$$ 136705. 0.0156326
$$599$$ −4.38057e6 −0.498843 −0.249421 0.968395i $$-0.580240\pi$$
−0.249421 + 0.968395i $$0.580240\pi$$
$$600$$ 1.04380e6 0.118369
$$601$$ 688570. 0.0777610 0.0388805 0.999244i $$-0.487621\pi$$
0.0388805 + 0.999244i $$0.487621\pi$$
$$602$$ 0 0
$$603$$ 1.35181e7 1.51399
$$604$$ −2.88172e6 −0.321410
$$605$$ −6.59563e6 −0.732601
$$606$$ 402504. 0.0445235
$$607$$ −9.37319e6 −1.03256 −0.516281 0.856420i $$-0.672684\pi$$
−0.516281 + 0.856420i $$0.672684\pi$$
$$608$$ −6.96085e6 −0.763666
$$609$$ 0 0
$$610$$ 4.98684e6 0.542626
$$611$$ 8.40174e6 0.910472
$$612$$ −5.68874e6 −0.613956
$$613$$ −2.16685e6 −0.232904 −0.116452 0.993196i $$-0.537152\pi$$
−0.116452 + 0.993196i $$0.537152\pi$$
$$614$$ −5.51947e6 −0.590849
$$615$$ 3.84306e6 0.409722
$$616$$ 0 0
$$617$$ −5.07951e6 −0.537166 −0.268583 0.963256i $$-0.586555\pi$$
−0.268583 + 0.963256i $$0.586555\pi$$
$$618$$ 1.20445e6 0.126858
$$619$$ −2.19034e6 −0.229766 −0.114883 0.993379i $$-0.536649\pi$$
−0.114883 + 0.993379i $$0.536649\pi$$
$$620$$ 9.72611e6 1.01616
$$621$$ −128787. −0.0134012
$$622$$ 2.43111e6 0.251958
$$623$$ 0 0
$$624$$ 2.33577e6 0.240142
$$625$$ −5.59018e6 −0.572435
$$626$$ 3.09790e6 0.315960
$$627$$ 4.21329e6 0.428009
$$628$$ −2.35016e6 −0.237793
$$629$$ 3.01923e6 0.304278
$$630$$ 0 0
$$631$$ −7.18693e6 −0.718572 −0.359286 0.933228i $$-0.616980\pi$$
−0.359286 + 0.933228i $$0.616980\pi$$
$$632$$ 1346.38 0.000134083 0
$$633$$ −4.19451e6 −0.416075
$$634$$ 4.21109e6 0.416075
$$635$$ 142324. 0.0140070
$$636$$ 1.88439e6 0.184726
$$637$$ 0 0
$$638$$ −5.17097e6 −0.502945
$$639$$ 1.22825e7 1.18997
$$640$$ 7.96017e6 0.768197
$$641$$ −1.76500e7 −1.69668 −0.848340 0.529452i $$-0.822398\pi$$
−0.848340 + 0.529452i $$0.822398\pi$$
$$642$$ −2.00843e6 −0.192317
$$643$$ 898309. 0.0856837 0.0428419 0.999082i $$-0.486359\pi$$
0.0428419 + 0.999082i $$0.486359\pi$$
$$644$$ 0 0
$$645$$ 29142.7 0.00275823
$$646$$ 3.87597e6 0.365425
$$647$$ −1.38642e6 −0.130207 −0.0651035 0.997879i $$-0.520738\pi$$
−0.0651035 + 0.997879i $$0.520738\pi$$
$$648$$ −4.67735e6 −0.437585
$$649$$ −5.43904e6 −0.506886
$$650$$ −3.11022e6 −0.288741
$$651$$ 0 0
$$652$$ −4.39804e6 −0.405173
$$653$$ −1.75425e7 −1.60994 −0.804968 0.593318i $$-0.797818\pi$$
−0.804968 + 0.593318i $$0.797818\pi$$
$$654$$ 1.59868e6 0.146156
$$655$$ −1.16556e7 −1.06152
$$656$$ −4.16062e6 −0.377484
$$657$$ −370308. −0.0334696
$$658$$ 0 0
$$659$$ 9.87522e6 0.885795 0.442898 0.896572i $$-0.353950\pi$$
0.442898 + 0.896572i $$0.353950\pi$$
$$660$$ −4.00058e6 −0.357489
$$661$$ −8.06792e6 −0.718221 −0.359110 0.933295i $$-0.616920\pi$$
−0.359110 + 0.933295i $$0.616920\pi$$
$$662$$ −7.71373e6 −0.684100
$$663$$ −8.45884e6 −0.747355
$$664$$ 1.50025e7 1.32052
$$665$$ 0 0
$$666$$ 1.44059e6 0.125850
$$667$$ 147759. 0.0128599
$$668$$ 1.08742e7 0.942880
$$669$$ 2.55461e6 0.220678
$$670$$ 8.74544e6 0.752652
$$671$$ 2.12534e7 1.82231
$$672$$ 0 0
$$673$$ −1.12772e7 −0.959762 −0.479881 0.877334i $$-0.659320\pi$$
−0.479881 + 0.877334i $$0.659320\pi$$
$$674$$ −6.51274e6 −0.552223
$$675$$ 2.93008e6 0.247526
$$676$$ −1.98804e7 −1.67324
$$677$$ 5.20372e6 0.436357 0.218179 0.975909i $$-0.429988\pi$$
0.218179 + 0.975909i $$0.429988\pi$$
$$678$$ −1.87090e6 −0.156306
$$679$$ 0 0
$$680$$ −8.57196e6 −0.710899
$$681$$ 1.90778e6 0.157638
$$682$$ −1.36442e7 −1.12328
$$683$$ 6.05915e6 0.497004 0.248502 0.968631i $$-0.420062\pi$$
0.248502 + 0.968631i $$0.420062\pi$$
$$684$$ −5.61845e6 −0.459173
$$685$$ 4.46816e6 0.363833
$$686$$ 0 0
$$687$$ −8.05671e6 −0.651277
$$688$$ −31550.9 −0.00254121
$$689$$ −1.30781e7 −1.04954
$$690$$ −37627.9 −0.00300876
$$691$$ 7.36498e6 0.586781 0.293391 0.955993i $$-0.405216\pi$$
0.293391 + 0.955993i $$0.405216\pi$$
$$692$$ −8.94754e6 −0.710295
$$693$$ 0 0
$$694$$ −8.61204e6 −0.678746
$$695$$ −9.67228e6 −0.759569
$$696$$ −3.44102e6 −0.269255
$$697$$ 1.50674e7 1.17478
$$698$$ −591818. −0.0459779
$$699$$ 748298. 0.0579271
$$700$$ 0 0
$$701$$ 7.80919e6 0.600221 0.300110 0.953904i $$-0.402977\pi$$
0.300110 + 0.953904i $$0.402977\pi$$
$$702$$ −8.93676e6 −0.684443
$$703$$ 2.98193e6 0.227567
$$704$$ −3.51520e6 −0.267312
$$705$$ −2.31258e6 −0.175236
$$706$$ −1.06069e7 −0.800898
$$707$$ 0 0
$$708$$ −1.55395e6 −0.116508
$$709$$ 1.75650e7 1.31230 0.656150 0.754631i $$-0.272184\pi$$
0.656150 + 0.754631i $$0.272184\pi$$
$$710$$ 7.94606e6 0.591569
$$711$$ 1706.88 0.000126628 0
$$712$$ −8.46198e6 −0.625564
$$713$$ 389879. 0.0287214
$$714$$ 0 0
$$715$$ 2.77649e7 2.03110
$$716$$ −2.04649e6 −0.149186
$$717$$ −7.51597e6 −0.545993
$$718$$ 2.83539e6 0.205259
$$719$$ −8.09220e6 −0.583773 −0.291887 0.956453i $$-0.594283\pi$$
−0.291887 + 0.956453i $$0.594283\pi$$
$$720$$ 3.00079e6 0.215727
$$721$$ 0 0
$$722$$ −3.14232e6 −0.224341
$$723$$ 5.32162e6 0.378615
$$724$$ 9.13520e6 0.647697
$$725$$ −3.36172e6 −0.237529
$$726$$ 2.64352e6 0.186140
$$727$$ 1.51986e7 1.06652 0.533258 0.845952i $$-0.320967\pi$$
0.533258 + 0.845952i $$0.320967\pi$$
$$728$$ 0 0
$$729$$ −1.31285e6 −0.0914944
$$730$$ −239568. −0.0166388
$$731$$ 114259. 0.00790858
$$732$$ 6.07218e6 0.418858
$$733$$ 5.83402e6 0.401059 0.200530 0.979688i $$-0.435734\pi$$
0.200530 + 0.979688i $$0.435734\pi$$
$$734$$ −4.29720e6 −0.294405
$$735$$ 0 0
$$736$$ 264944. 0.0180285
$$737$$ 3.72722e7 2.52765
$$738$$ 7.18921e6 0.485893
$$739$$ 6.47719e6 0.436290 0.218145 0.975916i $$-0.429999\pi$$
0.218145 + 0.975916i $$0.429999\pi$$
$$740$$ −2.83138e6 −0.190072
$$741$$ −8.35433e6 −0.558941
$$742$$ 0 0
$$743$$ −1.50899e7 −1.00280 −0.501401 0.865215i $$-0.667182\pi$$
−0.501401 + 0.865215i $$0.667182\pi$$
$$744$$ −9.07955e6 −0.601357
$$745$$ −6.45744e6 −0.426255
$$746$$ 1.36896e7 0.900626
$$747$$ 1.90196e7 1.24710
$$748$$ −1.56850e7 −1.02502
$$749$$ 0 0
$$750$$ 3.50534e6 0.227550
$$751$$ −2.13997e6 −0.138455 −0.0692273 0.997601i $$-0.522053\pi$$
−0.0692273 + 0.997601i $$0.522053\pi$$
$$752$$ 2.50367e6 0.161448
$$753$$ −2.66355e6 −0.171188
$$754$$ 1.02533e7 0.656801
$$755$$ −5.50494e6 −0.351467
$$756$$ 0 0
$$757$$ 2.10943e7 1.33791 0.668954 0.743304i $$-0.266742\pi$$
0.668954 + 0.743304i $$0.266742\pi$$
$$758$$ 1.77597e6 0.112270
$$759$$ −160366. −0.0101044
$$760$$ −8.46605e6 −0.531675
$$761$$ 9.79958e6 0.613403 0.306701 0.951806i $$-0.400775\pi$$
0.306701 + 0.951806i $$0.400775\pi$$
$$762$$ −57043.3 −0.00355891
$$763$$ 0 0
$$764$$ 2.22022e7 1.37614
$$765$$ −1.08672e7 −0.671371
$$766$$ 1.59233e6 0.0980530
$$767$$ 1.07848e7 0.661947
$$768$$ −4.52531e6 −0.276850
$$769$$ −3.23493e7 −1.97265 −0.986323 0.164825i $$-0.947294\pi$$
−0.986323 + 0.164825i $$0.947294\pi$$
$$770$$ 0 0
$$771$$ −1.11128e7 −0.673267
$$772$$ −1.21606e7 −0.734366
$$773$$ 9.19713e6 0.553609 0.276805 0.960926i $$-0.410724\pi$$
0.276805 + 0.960926i $$0.410724\pi$$
$$774$$ 54517.4 0.00327102
$$775$$ −8.87030e6 −0.530499
$$776$$ −491058. −0.0292738
$$777$$ 0 0
$$778$$ 1.66712e6 0.0987456
$$779$$ 1.48812e7 0.878609
$$780$$ 7.93254e6 0.466848
$$781$$ 3.38653e7 1.98668
$$782$$ −147527. −0.00862690
$$783$$ −9.65941e6 −0.563049
$$784$$ 0 0
$$785$$ −4.48950e6 −0.260030
$$786$$ 4.67153e6 0.269713
$$787$$ 1.46950e7 0.845731 0.422866 0.906192i $$-0.361024\pi$$
0.422866 + 0.906192i $$0.361024\pi$$
$$788$$ −6.05508e6 −0.347379
$$789$$ 1.34689e6 0.0770263
$$790$$ 1104.25 6.29506e−5 0
$$791$$ 0 0
$$792$$ −1.74312e7 −0.987446
$$793$$ −4.21423e7 −2.37977
$$794$$ 3.78097e6 0.212839
$$795$$ 3.59975e6 0.202001
$$796$$ 5.00847e6 0.280171
$$797$$ 2.33344e7 1.30122 0.650610 0.759412i $$-0.274513\pi$$
0.650610 + 0.759412i $$0.274513\pi$$
$$798$$ 0 0
$$799$$ −9.06688e6 −0.502448
$$800$$ −6.02785e6 −0.332995
$$801$$ −1.07277e7 −0.590782
$$802$$ −1.03796e7 −0.569831
$$803$$ −1.02101e6 −0.0558783
$$804$$ 1.06488e7 0.580980
$$805$$ 0 0
$$806$$ 2.70545e7 1.46690
$$807$$ 1.13559e7 0.613814
$$808$$ −3.44693e6 −0.185740
$$809$$ 1.69301e6 0.0909468 0.0454734 0.998966i $$-0.485520\pi$$
0.0454734 + 0.998966i $$0.485520\pi$$
$$810$$ −3.83620e6 −0.205442
$$811$$ −2.12400e7 −1.13397 −0.566987 0.823727i $$-0.691891\pi$$
−0.566987 + 0.823727i $$0.691891\pi$$
$$812$$ 0 0
$$813$$ 2.41649e6 0.128221
$$814$$ 3.97199e6 0.210110
$$815$$ −8.40155e6 −0.443063
$$816$$ −2.52068e6 −0.132524
$$817$$ 112848. 0.00591477
$$818$$ 4.09960e6 0.214219
$$819$$ 0 0
$$820$$ −1.41299e7 −0.733847
$$821$$ 8.73550e6 0.452304 0.226152 0.974092i $$-0.427385\pi$$
0.226152 + 0.974092i $$0.427385\pi$$
$$822$$ −1.79083e6 −0.0924433
$$823$$ −3.27964e7 −1.68782 −0.843910 0.536485i $$-0.819752\pi$$
−0.843910 + 0.536485i $$0.819752\pi$$
$$824$$ −1.03146e7 −0.529215
$$825$$ 3.64856e6 0.186632
$$826$$ 0 0
$$827$$ 1.31248e7 0.667311 0.333656 0.942695i $$-0.391718\pi$$
0.333656 + 0.942695i $$0.391718\pi$$
$$828$$ 213849. 0.0108401
$$829$$ 2.05402e7 1.03805 0.519026 0.854759i $$-0.326295\pi$$
0.519026 + 0.854759i $$0.326295\pi$$
$$830$$ 1.23046e7 0.619970
$$831$$ 8.03880e6 0.403821
$$832$$ 6.97012e6 0.349085
$$833$$ 0 0
$$834$$ 3.87664e6 0.192992
$$835$$ 2.07729e7 1.03106
$$836$$ −1.54912e7 −0.766601
$$837$$ −2.54875e7 −1.25752
$$838$$ −8.22542e6 −0.404621
$$839$$ 2.83736e7 1.39159 0.695793 0.718243i $$-0.255053\pi$$
0.695793 + 0.718243i $$0.255053\pi$$
$$840$$ 0 0
$$841$$ −9.42879e6 −0.459691
$$842$$ 5.68643e6 0.276414
$$843$$ 1.31400e7 0.636834
$$844$$ 1.54221e7 0.745226
$$845$$ −3.79774e7 −1.82972
$$846$$ −4.32614e6 −0.207814
$$847$$ 0 0
$$848$$ −3.89720e6 −0.186107
$$849$$ −1.16935e7 −0.556770
$$850$$ 3.35645e6 0.159343
$$851$$ −113498. −0.00537236
$$852$$ 9.67545e6 0.456638
$$853$$ 1.02093e7 0.480424 0.240212 0.970720i $$-0.422783\pi$$
0.240212 + 0.970720i $$0.422783\pi$$
$$854$$ 0 0
$$855$$ −1.07329e7 −0.502113
$$856$$ 1.71996e7 0.802295
$$857$$ −8.33206e6 −0.387525 −0.193763 0.981048i $$-0.562069\pi$$
−0.193763 + 0.981048i $$0.562069\pi$$
$$858$$ −1.11281e7 −0.516064
$$859$$ 3.12766e7 1.44623 0.723113 0.690729i $$-0.242710\pi$$
0.723113 + 0.690729i $$0.242710\pi$$
$$860$$ −107150. −0.00494023
$$861$$ 0 0
$$862$$ −1.53084e7 −0.701714
$$863$$ −3.73573e7 −1.70745 −0.853726 0.520722i $$-0.825663\pi$$
−0.853726 + 0.520722i $$0.825663\pi$$
$$864$$ −1.73201e7 −0.789345
$$865$$ −1.70924e7 −0.776719
$$866$$ −1.06222e7 −0.481306
$$867$$ −168747. −0.00762411
$$868$$ 0 0
$$869$$ 4706.21 0.000211408 0
$$870$$ −2.82221e6 −0.126413
$$871$$ −7.39051e7 −3.30088
$$872$$ −1.36907e7 −0.609724
$$873$$ −622543. −0.0276461
$$874$$ −145704. −0.00645199
$$875$$ 0 0
$$876$$ −291708. −0.0128436
$$877$$ 38996.7 0.00171210 0.000856049 1.00000i $$-0.499728\pi$$
0.000856049 1.00000i $$0.499728\pi$$
$$878$$ 6.61965e6 0.289800
$$879$$ 5.58708e6 0.243900
$$880$$ 8.27378e6 0.360162
$$881$$ 3.15554e7 1.36973 0.684864 0.728671i $$-0.259862\pi$$
0.684864 + 0.728671i $$0.259862\pi$$
$$882$$ 0 0
$$883$$ −3.42253e7 −1.47722 −0.738611 0.674132i $$-0.764518\pi$$
−0.738611 + 0.674132i $$0.764518\pi$$
$$884$$ 3.11010e7 1.33858
$$885$$ −2.96851e6 −0.127403
$$886$$ 1.35230e7 0.578745
$$887$$ −2.69886e7 −1.15178 −0.575892 0.817526i $$-0.695345\pi$$
−0.575892 + 0.817526i $$0.695345\pi$$
$$888$$ 2.64316e6 0.112484
$$889$$ 0 0
$$890$$ −6.94022e6 −0.293696
$$891$$ −1.63495e7 −0.689938
$$892$$ −9.39263e6 −0.395253
$$893$$ −8.95486e6 −0.375777
$$894$$ 2.58813e6 0.108303
$$895$$ −3.90940e6 −0.163137
$$896$$ 0 0
$$897$$ 317982. 0.0131954
$$898$$ −7.79226e6 −0.322458
$$899$$ 2.92421e7 1.20673
$$900$$ −4.86538e6 −0.200221
$$901$$ 1.41135e7 0.579191
$$902$$ 1.98221e7 0.811211
$$903$$ 0 0
$$904$$ 1.60218e7 0.652065
$$905$$ 1.74509e7 0.708267
$$906$$ 2.20637e6 0.0893013
$$907$$ 1.92103e7 0.775381 0.387690 0.921790i $$-0.373273\pi$$
0.387690 + 0.921790i $$0.373273\pi$$
$$908$$ −7.01442e6 −0.282343
$$909$$ −4.36988e6 −0.175412
$$910$$ 0 0
$$911$$ 2.86013e7 1.14180 0.570899 0.821020i $$-0.306595\pi$$
0.570899 + 0.821020i $$0.306595\pi$$
$$912$$ −2.48954e6 −0.0991133
$$913$$ 5.24408e7 2.08206
$$914$$ 680027. 0.0269253
$$915$$ 1.15997e7 0.458029
$$916$$ 2.96224e7 1.16649
$$917$$ 0 0
$$918$$ 9.64426e6 0.377713
$$919$$ −4.21754e7 −1.64729 −0.823645 0.567106i $$-0.808063\pi$$
−0.823645 + 0.567106i $$0.808063\pi$$
$$920$$ 322235. 0.0125517
$$921$$ −1.28386e7 −0.498733
$$922$$ 2.78855e6 0.108032
$$923$$ −6.71498e7 −2.59442
$$924$$ 0 0
$$925$$ 2.58224e6 0.0992299
$$926$$ 1.74672e7 0.669416
$$927$$ −1.30764e7 −0.499790
$$928$$ 1.98716e7 0.757466
$$929$$ −3.01886e7 −1.14763 −0.573817 0.818983i $$-0.694538\pi$$
−0.573817 + 0.818983i $$0.694538\pi$$
$$930$$ −7.44673e6 −0.282331
$$931$$ 0 0
$$932$$ −2.75130e6 −0.103752
$$933$$ 5.65489e6 0.212677
$$934$$ −1.93817e7 −0.726984
$$935$$ −2.99630e7 −1.12087
$$936$$ 3.45634e7 1.28952
$$937$$ −3.64068e6 −0.135467 −0.0677335 0.997703i $$-0.521577\pi$$
−0.0677335 + 0.997703i $$0.521577\pi$$
$$938$$ 0 0
$$939$$ 7.20589e6 0.266701
$$940$$ 8.50275e6 0.313863
$$941$$ 1.88601e7 0.694336 0.347168 0.937803i $$-0.387143\pi$$
0.347168 + 0.937803i $$0.387143\pi$$
$$942$$ 1.79939e6 0.0660689
$$943$$ −566410. −0.0207420
$$944$$ 3.21380e6 0.117379
$$945$$ 0 0
$$946$$ 150315. 0.00546104
$$947$$ −1.82172e7 −0.660094 −0.330047 0.943965i $$-0.607065\pi$$
−0.330047 + 0.943965i $$0.607065\pi$$
$$948$$ 1344.58 4.85922e−5 0
$$949$$ 2.02452e6 0.0729720
$$950$$ 3.31498e6 0.119171
$$951$$ 9.79522e6 0.351207
$$952$$ 0 0
$$953$$ 2.76898e7 0.987616 0.493808 0.869571i $$-0.335604\pi$$
0.493808 + 0.869571i $$0.335604\pi$$
$$954$$ 6.73406e6 0.239555
$$955$$ 4.24127e7 1.50483
$$956$$ 2.76343e7 0.977921
$$957$$ −1.20280e7 −0.424534
$$958$$ −1.52376e7 −0.536419
$$959$$ 0 0
$$960$$ −1.91852e6 −0.0671875
$$961$$ 4.85298e7 1.69512
$$962$$ −7.87585e6 −0.274385
$$963$$ 2.18049e7 0.757685
$$964$$ −1.95662e7 −0.678133
$$965$$ −2.32304e7 −0.803042
$$966$$ 0 0
$$967$$ −2.44768e7 −0.841761 −0.420881 0.907116i $$-0.638279\pi$$
−0.420881 + 0.907116i $$0.638279\pi$$
$$968$$ −2.26383e7 −0.776526
$$969$$ 9.01571e6 0.308454
$$970$$ −402749. −0.0137437
$$971$$ 9.50151e6 0.323403 0.161702 0.986840i $$-0.448302\pi$$
0.161702 + 0.986840i $$0.448302\pi$$
$$972$$ −2.16463e7 −0.734881
$$973$$ 0 0
$$974$$ −8.47457e6 −0.286233
$$975$$ −7.23455e6 −0.243725
$$976$$ −1.25582e7 −0.421990
$$977$$ 4.69012e7 1.57198 0.785991 0.618238i $$-0.212153\pi$$
0.785991 + 0.618238i $$0.212153\pi$$
$$978$$ 3.36733e6 0.112574
$$979$$ −2.95785e7 −0.986325
$$980$$ 0 0
$$981$$ −1.73564e7 −0.575822
$$982$$ −2.03951e7 −0.674914
$$983$$ 2.35382e7 0.776945 0.388473 0.921460i $$-0.373003\pi$$
0.388473 + 0.921460i $$0.373003\pi$$
$$984$$ 1.31906e7 0.434288
$$985$$ −1.15670e7 −0.379865
$$986$$ −1.10650e7 −0.362458
$$987$$ 0 0
$$988$$ 3.07167e7 1.00111
$$989$$ −4295.21 −0.000139635 0
$$990$$ −1.42964e7 −0.463596
$$991$$ −2.64104e7 −0.854261 −0.427130 0.904190i $$-0.640475\pi$$
−0.427130 + 0.904190i $$0.640475\pi$$
$$992$$ 5.24336e7 1.69173
$$993$$ −1.79426e7 −0.577446
$$994$$ 0 0
$$995$$ 9.56766e6 0.306371
$$996$$ 1.49825e7 0.478561
$$997$$ −1.95164e7 −0.621815 −0.310907 0.950440i $$-0.600633\pi$$
−0.310907 + 0.950440i $$0.600633\pi$$
$$998$$ −1.44354e7 −0.458777
$$999$$ 7.41970e6 0.235219
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 49.6.a.g.1.4 yes 4
3.2 odd 2 441.6.a.z.1.2 4
4.3 odd 2 784.6.a.bf.1.2 4
7.2 even 3 49.6.c.h.18.1 8
7.3 odd 6 49.6.c.h.30.2 8
7.4 even 3 49.6.c.h.30.1 8
7.5 odd 6 49.6.c.h.18.2 8
7.6 odd 2 inner 49.6.a.g.1.3 4
21.20 even 2 441.6.a.z.1.1 4
28.27 even 2 784.6.a.bf.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 4 7.6 odd 2 inner
49.6.a.g.1.4 yes 4 1.1 even 1 trivial
49.6.c.h.18.1 8 7.2 even 3
49.6.c.h.18.2 8 7.5 odd 6
49.6.c.h.30.1 8 7.4 even 3
49.6.c.h.30.2 8 7.3 odd 6
441.6.a.z.1.1 4 21.20 even 2
441.6.a.z.1.2 4 3.2 odd 2
784.6.a.bf.1.2 4 4.3 odd 2
784.6.a.bf.1.3 4 28.27 even 2