# Properties

 Label 825.4.a.r.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.6765757547$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.47528.1 Defining polynomial: $$x^{3} - x^{2} - 26x - 22$$ x^3 - x^2 - 26*x - 22 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.97123$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.97123 q^{2} -3.00000 q^{3} +16.7131 q^{4} +14.9137 q^{6} -5.48376 q^{7} -43.3148 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.97123 q^{2} -3.00000 q^{3} +16.7131 q^{4} +14.9137 q^{6} -5.48376 q^{7} -43.3148 q^{8} +9.00000 q^{9} +11.0000 q^{11} -50.1393 q^{12} -24.5738 q^{13} +27.2610 q^{14} +81.6231 q^{16} +59.3687 q^{17} -44.7411 q^{18} +5.89234 q^{19} +16.4513 q^{21} -54.6835 q^{22} +68.4570 q^{23} +129.945 q^{24} +122.162 q^{26} -27.0000 q^{27} -91.6506 q^{28} -265.022 q^{29} -196.554 q^{31} -59.2482 q^{32} -33.0000 q^{33} -295.135 q^{34} +150.418 q^{36} -166.575 q^{37} -29.2922 q^{38} +73.7214 q^{39} +424.101 q^{41} -81.7830 q^{42} +177.351 q^{43} +183.844 q^{44} -340.316 q^{46} -141.148 q^{47} -244.869 q^{48} -312.928 q^{49} -178.106 q^{51} -410.704 q^{52} +339.828 q^{53} +134.223 q^{54} +237.528 q^{56} -17.6770 q^{57} +1317.48 q^{58} +416.784 q^{59} +662.146 q^{61} +977.115 q^{62} -49.3538 q^{63} -358.448 q^{64} +164.051 q^{66} -313.713 q^{67} +992.235 q^{68} -205.371 q^{69} -153.693 q^{71} -389.834 q^{72} -153.866 q^{73} +828.084 q^{74} +98.4793 q^{76} -60.3213 q^{77} -366.486 q^{78} +403.000 q^{79} +81.0000 q^{81} -2108.30 q^{82} +652.313 q^{83} +274.952 q^{84} -881.650 q^{86} +795.065 q^{87} -476.463 q^{88} +1226.77 q^{89} +134.757 q^{91} +1144.13 q^{92} +589.662 q^{93} +701.677 q^{94} +177.745 q^{96} -959.746 q^{97} +1555.64 q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 9 q^{3} + 30 q^{4} - 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 - 9 * q^3 + 30 * q^4 - 6 * q^6 - 10 * q^7 - 18 * q^8 + 27 * q^9 $$3 q + 2 q^{2} - 9 q^{3} + 30 q^{4} - 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} + 33 q^{11} - 90 q^{12} - 114 q^{13} - 68 q^{14} + 178 q^{16} + 104 q^{17} + 18 q^{18} - 58 q^{19} + 30 q^{21} + 22 q^{22} - 120 q^{23} + 54 q^{24} - 120 q^{26} - 81 q^{27} - 676 q^{28} - 220 q^{29} + 248 q^{31} + 258 q^{32} - 99 q^{33} - 80 q^{34} + 270 q^{36} - 838 q^{37} - 600 q^{38} + 342 q^{39} + 156 q^{41} + 204 q^{42} - 122 q^{43} + 330 q^{44} - 1256 q^{46} - 504 q^{47} - 534 q^{48} + 279 q^{49} - 312 q^{51} - 520 q^{52} - 282 q^{53} - 54 q^{54} - 1644 q^{56} + 174 q^{57} + 1644 q^{58} + 548 q^{59} + 414 q^{61} + 2448 q^{62} - 90 q^{63} - 58 q^{64} - 66 q^{66} + 428 q^{67} + 1704 q^{68} + 360 q^{69} - 912 q^{71} - 162 q^{72} - 618 q^{73} - 1612 q^{74} - 2752 q^{76} - 110 q^{77} + 360 q^{78} - 542 q^{79} + 243 q^{81} - 3372 q^{82} + 2028 q^{84} - 1548 q^{86} + 660 q^{87} - 198 q^{88} + 790 q^{89} - 772 q^{91} - 1912 q^{92} - 744 q^{93} - 424 q^{94} - 774 q^{96} - 2074 q^{97} + 3978 q^{98} + 297 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 - 9 * q^3 + 30 * q^4 - 6 * q^6 - 10 * q^7 - 18 * q^8 + 27 * q^9 + 33 * q^11 - 90 * q^12 - 114 * q^13 - 68 * q^14 + 178 * q^16 + 104 * q^17 + 18 * q^18 - 58 * q^19 + 30 * q^21 + 22 * q^22 - 120 * q^23 + 54 * q^24 - 120 * q^26 - 81 * q^27 - 676 * q^28 - 220 * q^29 + 248 * q^31 + 258 * q^32 - 99 * q^33 - 80 * q^34 + 270 * q^36 - 838 * q^37 - 600 * q^38 + 342 * q^39 + 156 * q^41 + 204 * q^42 - 122 * q^43 + 330 * q^44 - 1256 * q^46 - 504 * q^47 - 534 * q^48 + 279 * q^49 - 312 * q^51 - 520 * q^52 - 282 * q^53 - 54 * q^54 - 1644 * q^56 + 174 * q^57 + 1644 * q^58 + 548 * q^59 + 414 * q^61 + 2448 * q^62 - 90 * q^63 - 58 * q^64 - 66 * q^66 + 428 * q^67 + 1704 * q^68 + 360 * q^69 - 912 * q^71 - 162 * q^72 - 618 * q^73 - 1612 * q^74 - 2752 * q^76 - 110 * q^77 + 360 * q^78 - 542 * q^79 + 243 * q^81 - 3372 * q^82 + 2028 * q^84 - 1548 * q^86 + 660 * q^87 - 198 * q^88 + 790 * q^89 - 772 * q^91 - 1912 * q^92 - 744 * q^93 - 424 * q^94 - 774 * q^96 - 2074 * q^97 + 3978 * q^98 + 297 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.97123 −1.75759 −0.878797 0.477195i $$-0.841653\pi$$
−0.878797 + 0.477195i $$0.841653\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 16.7131 2.08914
$$5$$ 0 0
$$6$$ 14.9137 1.01475
$$7$$ −5.48376 −0.296095 −0.148048 0.988980i $$-0.547299\pi$$
−0.148048 + 0.988980i $$0.547299\pi$$
$$8$$ −43.3148 −1.91426
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ −50.1393 −1.20616
$$13$$ −24.5738 −0.524272 −0.262136 0.965031i $$-0.584427\pi$$
−0.262136 + 0.965031i $$0.584427\pi$$
$$14$$ 27.2610 0.520415
$$15$$ 0 0
$$16$$ 81.6231 1.27536
$$17$$ 59.3687 0.847001 0.423501 0.905896i $$-0.360801\pi$$
0.423501 + 0.905896i $$0.360801\pi$$
$$18$$ −44.7411 −0.585865
$$19$$ 5.89234 0.0711471 0.0355736 0.999367i $$-0.488674\pi$$
0.0355736 + 0.999367i $$0.488674\pi$$
$$20$$ 0 0
$$21$$ 16.4513 0.170951
$$22$$ −54.6835 −0.529935
$$23$$ 68.4570 0.620621 0.310310 0.950635i $$-0.399567\pi$$
0.310310 + 0.950635i $$0.399567\pi$$
$$24$$ 129.945 1.10520
$$25$$ 0 0
$$26$$ 122.162 0.921458
$$27$$ −27.0000 −0.192450
$$28$$ −91.6506 −0.618584
$$29$$ −265.022 −1.69701 −0.848505 0.529187i $$-0.822497\pi$$
−0.848505 + 0.529187i $$0.822497\pi$$
$$30$$ 0 0
$$31$$ −196.554 −1.13878 −0.569389 0.822068i $$-0.692820\pi$$
−0.569389 + 0.822068i $$0.692820\pi$$
$$32$$ −59.2482 −0.327303
$$33$$ −33.0000 −0.174078
$$34$$ −295.135 −1.48868
$$35$$ 0 0
$$36$$ 150.418 0.696379
$$37$$ −166.575 −0.740131 −0.370065 0.929006i $$-0.620665\pi$$
−0.370065 + 0.929006i $$0.620665\pi$$
$$38$$ −29.2922 −0.125048
$$39$$ 73.7214 0.302689
$$40$$ 0 0
$$41$$ 424.101 1.61545 0.807725 0.589559i $$-0.200699\pi$$
0.807725 + 0.589559i $$0.200699\pi$$
$$42$$ −81.7830 −0.300462
$$43$$ 177.351 0.628970 0.314485 0.949262i $$-0.398168\pi$$
0.314485 + 0.949262i $$0.398168\pi$$
$$44$$ 183.844 0.629899
$$45$$ 0 0
$$46$$ −340.316 −1.09080
$$47$$ −141.148 −0.438053 −0.219026 0.975719i $$-0.570288\pi$$
−0.219026 + 0.975719i $$0.570288\pi$$
$$48$$ −244.869 −0.736330
$$49$$ −312.928 −0.912328
$$50$$ 0 0
$$51$$ −178.106 −0.489016
$$52$$ −410.704 −1.09528
$$53$$ 339.828 0.880736 0.440368 0.897817i $$-0.354848\pi$$
0.440368 + 0.897817i $$0.354848\pi$$
$$54$$ 134.223 0.338249
$$55$$ 0 0
$$56$$ 237.528 0.566804
$$57$$ −17.6770 −0.0410768
$$58$$ 1317.48 2.98266
$$59$$ 416.784 0.919672 0.459836 0.888004i $$-0.347908\pi$$
0.459836 + 0.888004i $$0.347908\pi$$
$$60$$ 0 0
$$61$$ 662.146 1.38982 0.694911 0.719096i $$-0.255444\pi$$
0.694911 + 0.719096i $$0.255444\pi$$
$$62$$ 977.115 2.00151
$$63$$ −49.3538 −0.0986984
$$64$$ −358.448 −0.700094
$$65$$ 0 0
$$66$$ 164.051 0.305958
$$67$$ −313.713 −0.572032 −0.286016 0.958225i $$-0.592331\pi$$
−0.286016 + 0.958225i $$0.592331\pi$$
$$68$$ 992.235 1.76950
$$69$$ −205.371 −0.358316
$$70$$ 0 0
$$71$$ −153.693 −0.256902 −0.128451 0.991716i $$-0.541000\pi$$
−0.128451 + 0.991716i $$0.541000\pi$$
$$72$$ −389.834 −0.638088
$$73$$ −153.866 −0.246694 −0.123347 0.992364i $$-0.539363\pi$$
−0.123347 + 0.992364i $$0.539363\pi$$
$$74$$ 828.084 1.30085
$$75$$ 0 0
$$76$$ 98.4793 0.148636
$$77$$ −60.3213 −0.0892760
$$78$$ −366.486 −0.532004
$$79$$ 403.000 0.573938 0.286969 0.957940i $$-0.407352\pi$$
0.286969 + 0.957940i $$0.407352\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −2108.30 −2.83931
$$83$$ 652.313 0.862659 0.431329 0.902195i $$-0.358045\pi$$
0.431329 + 0.902195i $$0.358045\pi$$
$$84$$ 274.952 0.357139
$$85$$ 0 0
$$86$$ −881.650 −1.10547
$$87$$ 795.065 0.979769
$$88$$ −476.463 −0.577172
$$89$$ 1226.77 1.46109 0.730545 0.682864i $$-0.239266\pi$$
0.730545 + 0.682864i $$0.239266\pi$$
$$90$$ 0 0
$$91$$ 134.757 0.155234
$$92$$ 1144.13 1.29656
$$93$$ 589.662 0.657474
$$94$$ 701.677 0.769919
$$95$$ 0 0
$$96$$ 177.745 0.188969
$$97$$ −959.746 −1.00461 −0.502306 0.864690i $$-0.667515\pi$$
−0.502306 + 0.864690i $$0.667515\pi$$
$$98$$ 1555.64 1.60350
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ −1258.52 −1.23988 −0.619939 0.784650i $$-0.712843\pi$$
−0.619939 + 0.784650i $$0.712843\pi$$
$$102$$ 885.406 0.859492
$$103$$ 493.175 0.471786 0.235893 0.971779i $$-0.424198\pi$$
0.235893 + 0.971779i $$0.424198\pi$$
$$104$$ 1064.41 1.00360
$$105$$ 0 0
$$106$$ −1689.36 −1.54798
$$107$$ 1099.05 0.992983 0.496492 0.868042i $$-0.334621\pi$$
0.496492 + 0.868042i $$0.334621\pi$$
$$108$$ −451.254 −0.402055
$$109$$ 1275.04 1.12043 0.560216 0.828347i $$-0.310718\pi$$
0.560216 + 0.828347i $$0.310718\pi$$
$$110$$ 0 0
$$111$$ 499.726 0.427315
$$112$$ −447.601 −0.377628
$$113$$ 946.433 0.787902 0.393951 0.919131i $$-0.371108\pi$$
0.393951 + 0.919131i $$0.371108\pi$$
$$114$$ 87.8765 0.0721964
$$115$$ 0 0
$$116$$ −4429.34 −3.54529
$$117$$ −221.164 −0.174757
$$118$$ −2071.93 −1.61641
$$119$$ −325.563 −0.250793
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −3291.68 −2.44274
$$123$$ −1272.30 −0.932681
$$124$$ −3285.03 −2.37907
$$125$$ 0 0
$$126$$ 245.349 0.173472
$$127$$ −2260.77 −1.57962 −0.789808 0.613354i $$-0.789820\pi$$
−0.789808 + 0.613354i $$0.789820\pi$$
$$128$$ 2255.91 1.55778
$$129$$ −532.052 −0.363136
$$130$$ 0 0
$$131$$ −57.1277 −0.0381013 −0.0190507 0.999819i $$-0.506064\pi$$
−0.0190507 + 0.999819i $$0.506064\pi$$
$$132$$ −551.533 −0.363672
$$133$$ −32.3122 −0.0210663
$$134$$ 1559.54 1.00540
$$135$$ 0 0
$$136$$ −2571.54 −1.62138
$$137$$ −1919.27 −1.19689 −0.598447 0.801162i $$-0.704215\pi$$
−0.598447 + 0.801162i $$0.704215\pi$$
$$138$$ 1020.95 0.629774
$$139$$ −1690.46 −1.03153 −0.515766 0.856729i $$-0.672493\pi$$
−0.515766 + 0.856729i $$0.672493\pi$$
$$140$$ 0 0
$$141$$ 423.443 0.252910
$$142$$ 764.044 0.451529
$$143$$ −270.312 −0.158074
$$144$$ 734.608 0.425120
$$145$$ 0 0
$$146$$ 764.904 0.433588
$$147$$ 938.785 0.526733
$$148$$ −2783.99 −1.54624
$$149$$ −2077.88 −1.14246 −0.571231 0.820789i $$-0.693534\pi$$
−0.571231 + 0.820789i $$0.693534\pi$$
$$150$$ 0 0
$$151$$ 2711.16 1.46113 0.730566 0.682842i $$-0.239256\pi$$
0.730566 + 0.682842i $$0.239256\pi$$
$$152$$ −255.226 −0.136194
$$153$$ 534.318 0.282334
$$154$$ 299.871 0.156911
$$155$$ 0 0
$$156$$ 1232.11 0.632359
$$157$$ −2341.23 −1.19013 −0.595066 0.803677i $$-0.702874\pi$$
−0.595066 + 0.803677i $$0.702874\pi$$
$$158$$ −2003.41 −1.00875
$$159$$ −1019.48 −0.508493
$$160$$ 0 0
$$161$$ −375.402 −0.183763
$$162$$ −402.669 −0.195288
$$163$$ −3523.01 −1.69291 −0.846453 0.532464i $$-0.821266\pi$$
−0.846453 + 0.532464i $$0.821266\pi$$
$$164$$ 7088.05 3.37490
$$165$$ 0 0
$$166$$ −3242.80 −1.51620
$$167$$ −4011.90 −1.85898 −0.929492 0.368842i $$-0.879754\pi$$
−0.929492 + 0.368842i $$0.879754\pi$$
$$168$$ −712.584 −0.327244
$$169$$ −1593.13 −0.725138
$$170$$ 0 0
$$171$$ 53.0310 0.0237157
$$172$$ 2964.08 1.31400
$$173$$ −3325.30 −1.46137 −0.730687 0.682713i $$-0.760800\pi$$
−0.730687 + 0.682713i $$0.760800\pi$$
$$174$$ −3952.45 −1.72204
$$175$$ 0 0
$$176$$ 897.854 0.384536
$$177$$ −1250.35 −0.530973
$$178$$ −6098.54 −2.56800
$$179$$ −26.6350 −0.0111217 −0.00556087 0.999985i $$-0.501770\pi$$
−0.00556087 + 0.999985i $$0.501770\pi$$
$$180$$ 0 0
$$181$$ 1934.25 0.794320 0.397160 0.917749i $$-0.369996\pi$$
0.397160 + 0.917749i $$0.369996\pi$$
$$182$$ −669.906 −0.272839
$$183$$ −1986.44 −0.802414
$$184$$ −2965.21 −1.18803
$$185$$ 0 0
$$186$$ −2931.34 −1.15557
$$187$$ 653.055 0.255380
$$188$$ −2359.01 −0.915153
$$189$$ 148.061 0.0569835
$$190$$ 0 0
$$191$$ −3341.59 −1.26591 −0.632956 0.774188i $$-0.718158\pi$$
−0.632956 + 0.774188i $$0.718158\pi$$
$$192$$ 1075.34 0.404200
$$193$$ 1293.97 0.482600 0.241300 0.970451i $$-0.422426\pi$$
0.241300 + 0.970451i $$0.422426\pi$$
$$194$$ 4771.12 1.76570
$$195$$ 0 0
$$196$$ −5230.01 −1.90598
$$197$$ 2301.17 0.832240 0.416120 0.909310i $$-0.363390\pi$$
0.416120 + 0.909310i $$0.363390\pi$$
$$198$$ −492.152 −0.176645
$$199$$ −3039.64 −1.08279 −0.541393 0.840770i $$-0.682103\pi$$
−0.541393 + 0.840770i $$0.682103\pi$$
$$200$$ 0 0
$$201$$ 941.139 0.330263
$$202$$ 6256.40 2.17920
$$203$$ 1453.31 0.502476
$$204$$ −2976.70 −1.02162
$$205$$ 0 0
$$206$$ −2451.69 −0.829209
$$207$$ 616.113 0.206874
$$208$$ −2005.79 −0.668636
$$209$$ 64.8157 0.0214517
$$210$$ 0 0
$$211$$ 2807.86 0.916119 0.458060 0.888921i $$-0.348545\pi$$
0.458060 + 0.888921i $$0.348545\pi$$
$$212$$ 5679.58 1.83998
$$213$$ 461.080 0.148322
$$214$$ −5463.63 −1.74526
$$215$$ 0 0
$$216$$ 1169.50 0.368400
$$217$$ 1077.85 0.337187
$$218$$ −6338.53 −1.96926
$$219$$ 461.598 0.142429
$$220$$ 0 0
$$221$$ −1458.91 −0.444059
$$222$$ −2484.25 −0.751046
$$223$$ −1443.09 −0.433348 −0.216674 0.976244i $$-0.569521\pi$$
−0.216674 + 0.976244i $$0.569521\pi$$
$$224$$ 324.903 0.0969129
$$225$$ 0 0
$$226$$ −4704.94 −1.38481
$$227$$ −3658.71 −1.06977 −0.534884 0.844926i $$-0.679645\pi$$
−0.534884 + 0.844926i $$0.679645\pi$$
$$228$$ −295.438 −0.0858151
$$229$$ −1695.50 −0.489265 −0.244632 0.969616i $$-0.578667\pi$$
−0.244632 + 0.969616i $$0.578667\pi$$
$$230$$ 0 0
$$231$$ 180.964 0.0515435
$$232$$ 11479.4 3.24853
$$233$$ 5782.24 1.62578 0.812890 0.582417i $$-0.197893\pi$$
0.812890 + 0.582417i $$0.197893\pi$$
$$234$$ 1099.46 0.307153
$$235$$ 0 0
$$236$$ 6965.76 1.92132
$$237$$ −1209.00 −0.331363
$$238$$ 1618.45 0.440792
$$239$$ −5836.16 −1.57954 −0.789770 0.613404i $$-0.789800\pi$$
−0.789770 + 0.613404i $$0.789800\pi$$
$$240$$ 0 0
$$241$$ −982.308 −0.262556 −0.131278 0.991346i $$-0.541908\pi$$
−0.131278 + 0.991346i $$0.541908\pi$$
$$242$$ −601.519 −0.159781
$$243$$ −243.000 −0.0641500
$$244$$ 11066.5 2.90353
$$245$$ 0 0
$$246$$ 6324.91 1.63927
$$247$$ −144.797 −0.0373005
$$248$$ 8513.70 2.17992
$$249$$ −1956.94 −0.498056
$$250$$ 0 0
$$251$$ 4320.23 1.08642 0.543209 0.839598i $$-0.317209\pi$$
0.543209 + 0.839598i $$0.317209\pi$$
$$252$$ −824.856 −0.206195
$$253$$ 753.027 0.187124
$$254$$ 11238.8 2.77633
$$255$$ 0 0
$$256$$ −8347.07 −2.03786
$$257$$ 6224.29 1.51074 0.755371 0.655298i $$-0.227457\pi$$
0.755371 + 0.655298i $$0.227457\pi$$
$$258$$ 2644.95 0.638246
$$259$$ 913.459 0.219149
$$260$$ 0 0
$$261$$ −2385.20 −0.565670
$$262$$ 283.995 0.0669666
$$263$$ 2588.95 0.607002 0.303501 0.952831i $$-0.401844\pi$$
0.303501 + 0.952831i $$0.401844\pi$$
$$264$$ 1429.39 0.333230
$$265$$ 0 0
$$266$$ 160.631 0.0370260
$$267$$ −3680.30 −0.843561
$$268$$ −5243.12 −1.19505
$$269$$ −5871.53 −1.33083 −0.665415 0.746473i $$-0.731746\pi$$
−0.665415 + 0.746473i $$0.731746\pi$$
$$270$$ 0 0
$$271$$ −3306.93 −0.741261 −0.370630 0.928780i $$-0.620858\pi$$
−0.370630 + 0.928780i $$0.620858\pi$$
$$272$$ 4845.85 1.08023
$$273$$ −404.270 −0.0896247
$$274$$ 9541.14 2.10366
$$275$$ 0 0
$$276$$ −3432.39 −0.748571
$$277$$ 2861.37 0.620660 0.310330 0.950629i $$-0.399560\pi$$
0.310330 + 0.950629i $$0.399560\pi$$
$$278$$ 8403.66 1.81302
$$279$$ −1768.99 −0.379593
$$280$$ 0 0
$$281$$ −2988.05 −0.634349 −0.317174 0.948367i $$-0.602734\pi$$
−0.317174 + 0.948367i $$0.602734\pi$$
$$282$$ −2105.03 −0.444513
$$283$$ −1454.60 −0.305536 −0.152768 0.988262i $$-0.548819\pi$$
−0.152768 + 0.988262i $$0.548819\pi$$
$$284$$ −2568.69 −0.536704
$$285$$ 0 0
$$286$$ 1343.78 0.277830
$$287$$ −2325.67 −0.478327
$$288$$ −533.234 −0.109101
$$289$$ −1388.36 −0.282589
$$290$$ 0 0
$$291$$ 2879.24 0.580014
$$292$$ −2571.58 −0.515378
$$293$$ 8586.52 1.71205 0.856023 0.516937i $$-0.172928\pi$$
0.856023 + 0.516937i $$0.172928\pi$$
$$294$$ −4666.92 −0.925782
$$295$$ 0 0
$$296$$ 7215.19 1.41681
$$297$$ −297.000 −0.0580259
$$298$$ 10329.6 2.00798
$$299$$ −1682.25 −0.325374
$$300$$ 0 0
$$301$$ −972.547 −0.186235
$$302$$ −13477.8 −2.56808
$$303$$ 3775.57 0.715844
$$304$$ 480.951 0.0907382
$$305$$ 0 0
$$306$$ −2656.22 −0.496228
$$307$$ −529.029 −0.0983495 −0.0491748 0.998790i $$-0.515659\pi$$
−0.0491748 + 0.998790i $$0.515659\pi$$
$$308$$ −1008.16 −0.186510
$$309$$ −1479.53 −0.272386
$$310$$ 0 0
$$311$$ −5008.00 −0.913111 −0.456555 0.889695i $$-0.650917\pi$$
−0.456555 + 0.889695i $$0.650917\pi$$
$$312$$ −3193.23 −0.579426
$$313$$ −7858.53 −1.41914 −0.709570 0.704635i $$-0.751111\pi$$
−0.709570 + 0.704635i $$0.751111\pi$$
$$314$$ 11638.8 2.09177
$$315$$ 0 0
$$316$$ 6735.39 1.19904
$$317$$ −7747.04 −1.37261 −0.686304 0.727315i $$-0.740768\pi$$
−0.686304 + 0.727315i $$0.740768\pi$$
$$318$$ 5068.09 0.893724
$$319$$ −2915.24 −0.511668
$$320$$ 0 0
$$321$$ −3297.15 −0.573299
$$322$$ 1866.21 0.322980
$$323$$ 349.820 0.0602617
$$324$$ 1353.76 0.232126
$$325$$ 0 0
$$326$$ 17513.7 2.97544
$$327$$ −3825.13 −0.646882
$$328$$ −18369.9 −3.09240
$$329$$ 774.019 0.129705
$$330$$ 0 0
$$331$$ 1355.08 0.225022 0.112511 0.993650i $$-0.464111\pi$$
0.112511 + 0.993650i $$0.464111\pi$$
$$332$$ 10902.2 1.80221
$$333$$ −1499.18 −0.246710
$$334$$ 19944.1 3.26734
$$335$$ 0 0
$$336$$ 1342.80 0.218024
$$337$$ −2596.97 −0.419780 −0.209890 0.977725i $$-0.567311\pi$$
−0.209890 + 0.977725i $$0.567311\pi$$
$$338$$ 7919.81 1.27450
$$339$$ −2839.30 −0.454896
$$340$$ 0 0
$$341$$ −2162.09 −0.343355
$$342$$ −263.629 −0.0416826
$$343$$ 3596.95 0.566231
$$344$$ −7681.91 −1.20401
$$345$$ 0 0
$$346$$ 16530.8 2.56850
$$347$$ −266.820 −0.0412785 −0.0206392 0.999787i $$-0.506570\pi$$
−0.0206392 + 0.999787i $$0.506570\pi$$
$$348$$ 13288.0 2.04687
$$349$$ −10549.4 −1.61804 −0.809021 0.587780i $$-0.800002\pi$$
−0.809021 + 0.587780i $$0.800002\pi$$
$$350$$ 0 0
$$351$$ 663.492 0.100896
$$352$$ −651.730 −0.0986856
$$353$$ −7152.63 −1.07846 −0.539229 0.842159i $$-0.681284\pi$$
−0.539229 + 0.842159i $$0.681284\pi$$
$$354$$ 6215.79 0.933235
$$355$$ 0 0
$$356$$ 20503.1 3.05242
$$357$$ 976.690 0.144795
$$358$$ 132.409 0.0195475
$$359$$ 359.182 0.0528047 0.0264024 0.999651i $$-0.491595\pi$$
0.0264024 + 0.999651i $$0.491595\pi$$
$$360$$ 0 0
$$361$$ −6824.28 −0.994938
$$362$$ −9615.62 −1.39609
$$363$$ −363.000 −0.0524864
$$364$$ 2252.20 0.324306
$$365$$ 0 0
$$366$$ 9875.04 1.41032
$$367$$ −9042.47 −1.28614 −0.643070 0.765808i $$-0.722339\pi$$
−0.643070 + 0.765808i $$0.722339\pi$$
$$368$$ 5587.67 0.791515
$$369$$ 3816.91 0.538483
$$370$$ 0 0
$$371$$ −1863.54 −0.260781
$$372$$ 9855.08 1.37355
$$373$$ 11929.0 1.65592 0.827960 0.560787i $$-0.189501\pi$$
0.827960 + 0.560787i $$0.189501\pi$$
$$374$$ −3246.49 −0.448855
$$375$$ 0 0
$$376$$ 6113.78 0.838549
$$377$$ 6512.59 0.889696
$$378$$ −736.047 −0.100154
$$379$$ 6556.08 0.888557 0.444279 0.895889i $$-0.353460\pi$$
0.444279 + 0.895889i $$0.353460\pi$$
$$380$$ 0 0
$$381$$ 6782.32 0.911992
$$382$$ 16611.8 2.22496
$$383$$ −10703.1 −1.42795 −0.713975 0.700172i $$-0.753107\pi$$
−0.713975 + 0.700172i $$0.753107\pi$$
$$384$$ −6767.74 −0.899388
$$385$$ 0 0
$$386$$ −6432.60 −0.848214
$$387$$ 1596.15 0.209657
$$388$$ −16040.3 −2.09878
$$389$$ −6450.30 −0.840728 −0.420364 0.907355i $$-0.638098\pi$$
−0.420364 + 0.907355i $$0.638098\pi$$
$$390$$ 0 0
$$391$$ 4064.20 0.525667
$$392$$ 13554.4 1.74644
$$393$$ 171.383 0.0219978
$$394$$ −11439.6 −1.46274
$$395$$ 0 0
$$396$$ 1654.60 0.209966
$$397$$ 12526.0 1.58353 0.791764 0.610828i $$-0.209163\pi$$
0.791764 + 0.610828i $$0.209163\pi$$
$$398$$ 15110.7 1.90310
$$399$$ 96.9365 0.0121626
$$400$$ 0 0
$$401$$ 9119.52 1.13568 0.567839 0.823139i $$-0.307779\pi$$
0.567839 + 0.823139i $$0.307779\pi$$
$$402$$ −4678.62 −0.580468
$$403$$ 4830.08 0.597030
$$404$$ −21033.8 −2.59028
$$405$$ 0 0
$$406$$ −7224.76 −0.883150
$$407$$ −1832.33 −0.223158
$$408$$ 7714.63 0.936106
$$409$$ −596.921 −0.0721658 −0.0360829 0.999349i $$-0.511488\pi$$
−0.0360829 + 0.999349i $$0.511488\pi$$
$$410$$ 0 0
$$411$$ 5757.82 0.691027
$$412$$ 8242.49 0.985627
$$413$$ −2285.54 −0.272310
$$414$$ −3062.84 −0.363600
$$415$$ 0 0
$$416$$ 1455.95 0.171596
$$417$$ 5071.38 0.595555
$$418$$ −322.214 −0.0377033
$$419$$ 11560.5 1.34789 0.673944 0.738782i $$-0.264599\pi$$
0.673944 + 0.738782i $$0.264599\pi$$
$$420$$ 0 0
$$421$$ 15182.6 1.75761 0.878805 0.477182i $$-0.158342\pi$$
0.878805 + 0.477182i $$0.158342\pi$$
$$422$$ −13958.5 −1.61017
$$423$$ −1270.33 −0.146018
$$424$$ −14719.6 −1.68596
$$425$$ 0 0
$$426$$ −2292.13 −0.260691
$$427$$ −3631.05 −0.411519
$$428$$ 18368.5 2.07448
$$429$$ 810.935 0.0912641
$$430$$ 0 0
$$431$$ −14902.7 −1.66551 −0.832757 0.553639i $$-0.813239\pi$$
−0.832757 + 0.553639i $$0.813239\pi$$
$$432$$ −2203.82 −0.245443
$$433$$ 2269.85 0.251922 0.125961 0.992035i $$-0.459799\pi$$
0.125961 + 0.992035i $$0.459799\pi$$
$$434$$ −5358.26 −0.592637
$$435$$ 0 0
$$436$$ 21309.9 2.34074
$$437$$ 403.372 0.0441554
$$438$$ −2294.71 −0.250332
$$439$$ −13418.7 −1.45886 −0.729431 0.684054i $$-0.760215\pi$$
−0.729431 + 0.684054i $$0.760215\pi$$
$$440$$ 0 0
$$441$$ −2816.36 −0.304109
$$442$$ 7252.59 0.780476
$$443$$ −11507.5 −1.23417 −0.617086 0.786896i $$-0.711687\pi$$
−0.617086 + 0.786896i $$0.711687\pi$$
$$444$$ 8351.98 0.892719
$$445$$ 0 0
$$446$$ 7173.94 0.761650
$$447$$ 6233.65 0.659601
$$448$$ 1965.64 0.207294
$$449$$ −16203.8 −1.70313 −0.851564 0.524250i $$-0.824346\pi$$
−0.851564 + 0.524250i $$0.824346\pi$$
$$450$$ 0 0
$$451$$ 4665.11 0.487077
$$452$$ 15817.8 1.64604
$$453$$ −8133.48 −0.843585
$$454$$ 18188.3 1.88022
$$455$$ 0 0
$$456$$ 765.677 0.0786318
$$457$$ −4912.79 −0.502868 −0.251434 0.967874i $$-0.580902\pi$$
−0.251434 + 0.967874i $$0.580902\pi$$
$$458$$ 8428.70 0.859929
$$459$$ −1602.95 −0.163005
$$460$$ 0 0
$$461$$ −14270.8 −1.44178 −0.720888 0.693051i $$-0.756266\pi$$
−0.720888 + 0.693051i $$0.756266\pi$$
$$462$$ −899.613 −0.0905926
$$463$$ −6656.46 −0.668146 −0.334073 0.942547i $$-0.608423\pi$$
−0.334073 + 0.942547i $$0.608423\pi$$
$$464$$ −21631.9 −2.16430
$$465$$ 0 0
$$466$$ −28744.8 −2.85746
$$467$$ −19347.7 −1.91714 −0.958571 0.284853i $$-0.908055\pi$$
−0.958571 + 0.284853i $$0.908055\pi$$
$$468$$ −3696.34 −0.365093
$$469$$ 1720.33 0.169376
$$470$$ 0 0
$$471$$ 7023.70 0.687123
$$472$$ −18052.9 −1.76050
$$473$$ 1950.86 0.189642
$$474$$ 6010.22 0.582402
$$475$$ 0 0
$$476$$ −5441.18 −0.523941
$$477$$ 3058.45 0.293579
$$478$$ 29012.9 2.77619
$$479$$ 9826.69 0.937355 0.468678 0.883369i $$-0.344731\pi$$
0.468678 + 0.883369i $$0.344731\pi$$
$$480$$ 0 0
$$481$$ 4093.39 0.388030
$$482$$ 4883.28 0.461467
$$483$$ 1126.21 0.106095
$$484$$ 2022.29 0.189922
$$485$$ 0 0
$$486$$ 1208.01 0.112750
$$487$$ −10278.1 −0.956353 −0.478176 0.878264i $$-0.658702\pi$$
−0.478176 + 0.878264i $$0.658702\pi$$
$$488$$ −28680.7 −2.66048
$$489$$ 10569.0 0.977400
$$490$$ 0 0
$$491$$ 11397.1 1.04755 0.523773 0.851858i $$-0.324524\pi$$
0.523773 + 0.851858i $$0.324524\pi$$
$$492$$ −21264.1 −1.94850
$$493$$ −15734.0 −1.43737
$$494$$ 719.819 0.0655591
$$495$$ 0 0
$$496$$ −16043.3 −1.45235
$$497$$ 842.817 0.0760674
$$498$$ 9728.39 0.875381
$$499$$ −9179.17 −0.823479 −0.411740 0.911302i $$-0.635079\pi$$
−0.411740 + 0.911302i $$0.635079\pi$$
$$500$$ 0 0
$$501$$ 12035.7 1.07329
$$502$$ −21476.9 −1.90948
$$503$$ −6782.80 −0.601253 −0.300626 0.953742i $$-0.597196\pi$$
−0.300626 + 0.953742i $$0.597196\pi$$
$$504$$ 2137.75 0.188935
$$505$$ 0 0
$$506$$ −3743.47 −0.328889
$$507$$ 4779.39 0.418659
$$508$$ −37784.6 −3.30004
$$509$$ 6814.24 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$510$$ 0 0
$$511$$ 843.765 0.0730449
$$512$$ 23447.9 2.02395
$$513$$ −159.093 −0.0136923
$$514$$ −30942.4 −2.65527
$$515$$ 0 0
$$516$$ −8892.23 −0.758641
$$517$$ −1552.62 −0.132078
$$518$$ −4541.01 −0.385175
$$519$$ 9975.89 0.843724
$$520$$ 0 0
$$521$$ 206.712 0.0173824 0.00869118 0.999962i $$-0.497233\pi$$
0.00869118 + 0.999962i $$0.497233\pi$$
$$522$$ 11857.4 0.994219
$$523$$ 11375.9 0.951118 0.475559 0.879684i $$-0.342246\pi$$
0.475559 + 0.879684i $$0.342246\pi$$
$$524$$ −954.781 −0.0795989
$$525$$ 0 0
$$526$$ −12870.3 −1.06686
$$527$$ −11669.1 −0.964547
$$528$$ −2693.56 −0.222012
$$529$$ −7480.63 −0.614830
$$530$$ 0 0
$$531$$ 3751.06 0.306557
$$532$$ −540.036 −0.0440104
$$533$$ −10421.8 −0.846936
$$534$$ 18295.6 1.48264
$$535$$ 0 0
$$536$$ 13588.4 1.09502
$$537$$ 79.9049 0.00642114
$$538$$ 29188.7 2.33906
$$539$$ −3442.21 −0.275077
$$540$$ 0 0
$$541$$ −10228.1 −0.812831 −0.406415 0.913688i $$-0.633221\pi$$
−0.406415 + 0.913688i $$0.633221\pi$$
$$542$$ 16439.5 1.30284
$$543$$ −5802.76 −0.458601
$$544$$ −3517.49 −0.277226
$$545$$ 0 0
$$546$$ 2009.72 0.157524
$$547$$ 15538.9 1.21461 0.607307 0.794467i $$-0.292250\pi$$
0.607307 + 0.794467i $$0.292250\pi$$
$$548$$ −32077.0 −2.50048
$$549$$ 5959.31 0.463274
$$550$$ 0 0
$$551$$ −1561.60 −0.120737
$$552$$ 8895.62 0.685911
$$553$$ −2209.96 −0.169940
$$554$$ −14224.5 −1.09087
$$555$$ 0 0
$$556$$ −28252.8 −2.15501
$$557$$ 5191.17 0.394896 0.197448 0.980313i $$-0.436735\pi$$
0.197448 + 0.980313i $$0.436735\pi$$
$$558$$ 8794.03 0.667170
$$559$$ −4358.17 −0.329752
$$560$$ 0 0
$$561$$ −1959.17 −0.147444
$$562$$ 14854.3 1.11493
$$563$$ −13722.9 −1.02727 −0.513635 0.858009i $$-0.671701\pi$$
−0.513635 + 0.858009i $$0.671701\pi$$
$$564$$ 7077.04 0.528364
$$565$$ 0 0
$$566$$ 7231.13 0.537009
$$567$$ −444.184 −0.0328995
$$568$$ 6657.20 0.491778
$$569$$ 23209.0 1.70997 0.854986 0.518652i $$-0.173566\pi$$
0.854986 + 0.518652i $$0.173566\pi$$
$$570$$ 0 0
$$571$$ −13130.5 −0.962337 −0.481168 0.876628i $$-0.659787\pi$$
−0.481168 + 0.876628i $$0.659787\pi$$
$$572$$ −4517.75 −0.330239
$$573$$ 10024.8 0.730874
$$574$$ 11561.4 0.840705
$$575$$ 0 0
$$576$$ −3226.03 −0.233365
$$577$$ 12261.0 0.884633 0.442317 0.896859i $$-0.354157\pi$$
0.442317 + 0.896859i $$0.354157\pi$$
$$578$$ 6901.86 0.496677
$$579$$ −3881.90 −0.278629
$$580$$ 0 0
$$581$$ −3577.13 −0.255429
$$582$$ −14313.3 −1.01943
$$583$$ 3738.11 0.265552
$$584$$ 6664.69 0.472238
$$585$$ 0 0
$$586$$ −42685.5 −3.00908
$$587$$ 3553.42 0.249856 0.124928 0.992166i $$-0.460130\pi$$
0.124928 + 0.992166i $$0.460130\pi$$
$$588$$ 15690.0 1.10042
$$589$$ −1158.16 −0.0810208
$$590$$ 0 0
$$591$$ −6903.50 −0.480494
$$592$$ −13596.4 −0.943933
$$593$$ −1942.83 −0.134540 −0.0672701 0.997735i $$-0.521429\pi$$
−0.0672701 + 0.997735i $$0.521429\pi$$
$$594$$ 1476.45 0.101986
$$595$$ 0 0
$$596$$ −34727.9 −2.38676
$$597$$ 9118.92 0.625147
$$598$$ 8362.84 0.571876
$$599$$ 18585.4 1.26774 0.633871 0.773439i $$-0.281465\pi$$
0.633871 + 0.773439i $$0.281465\pi$$
$$600$$ 0 0
$$601$$ 8010.10 0.543658 0.271829 0.962346i $$-0.412371\pi$$
0.271829 + 0.962346i $$0.412371\pi$$
$$602$$ 4834.75 0.327325
$$603$$ −2823.42 −0.190677
$$604$$ 45311.9 3.05251
$$605$$ 0 0
$$606$$ −18769.2 −1.25816
$$607$$ 1537.69 0.102822 0.0514111 0.998678i $$-0.483628\pi$$
0.0514111 + 0.998678i $$0.483628\pi$$
$$608$$ −349.110 −0.0232867
$$609$$ −4359.94 −0.290105
$$610$$ 0 0
$$611$$ 3468.53 0.229659
$$612$$ 8930.11 0.589834
$$613$$ −2298.75 −0.151461 −0.0757304 0.997128i $$-0.524129\pi$$
−0.0757304 + 0.997128i $$0.524129\pi$$
$$614$$ 2629.93 0.172859
$$615$$ 0 0
$$616$$ 2612.81 0.170898
$$617$$ −6089.51 −0.397333 −0.198666 0.980067i $$-0.563661\pi$$
−0.198666 + 0.980067i $$0.563661\pi$$
$$618$$ 7355.06 0.478744
$$619$$ 12255.4 0.795775 0.397887 0.917434i $$-0.369743\pi$$
0.397887 + 0.917434i $$0.369743\pi$$
$$620$$ 0 0
$$621$$ −1848.34 −0.119439
$$622$$ 24895.9 1.60488
$$623$$ −6727.29 −0.432622
$$624$$ 6017.36 0.386037
$$625$$ 0 0
$$626$$ 39066.6 2.49427
$$627$$ −194.447 −0.0123851
$$628$$ −39129.3 −2.48635
$$629$$ −9889.36 −0.626891
$$630$$ 0 0
$$631$$ −25746.5 −1.62433 −0.812165 0.583428i $$-0.801711\pi$$
−0.812165 + 0.583428i $$0.801711\pi$$
$$632$$ −17455.9 −1.09867
$$633$$ −8423.58 −0.528922
$$634$$ 38512.3 2.41249
$$635$$ 0 0
$$636$$ −17038.8 −1.06231
$$637$$ 7689.84 0.478308
$$638$$ 14492.3 0.899305
$$639$$ −1383.24 −0.0856340
$$640$$ 0 0
$$641$$ −598.022 −0.0368494 −0.0184247 0.999830i $$-0.505865\pi$$
−0.0184247 + 0.999830i $$0.505865\pi$$
$$642$$ 16390.9 1.00763
$$643$$ −14610.9 −0.896109 −0.448054 0.894006i $$-0.647883\pi$$
−0.448054 + 0.894006i $$0.647883\pi$$
$$644$$ −6274.13 −0.383906
$$645$$ 0 0
$$646$$ −1739.04 −0.105916
$$647$$ 17252.3 1.04831 0.524156 0.851622i $$-0.324381\pi$$
0.524156 + 0.851622i $$0.324381\pi$$
$$648$$ −3508.50 −0.212696
$$649$$ 4584.63 0.277292
$$650$$ 0 0
$$651$$ −3233.56 −0.194675
$$652$$ −58880.5 −3.53671
$$653$$ 17768.6 1.06484 0.532418 0.846481i $$-0.321283\pi$$
0.532418 + 0.846481i $$0.321283\pi$$
$$654$$ 19015.6 1.13696
$$655$$ 0 0
$$656$$ 34616.4 2.06028
$$657$$ −1384.80 −0.0822314
$$658$$ −3847.82 −0.227969
$$659$$ −17594.9 −1.04006 −0.520031 0.854147i $$-0.674080\pi$$
−0.520031 + 0.854147i $$0.674080\pi$$
$$660$$ 0 0
$$661$$ 23355.9 1.37434 0.687170 0.726497i $$-0.258853\pi$$
0.687170 + 0.726497i $$0.258853\pi$$
$$662$$ −6736.44 −0.395497
$$663$$ 4376.74 0.256378
$$664$$ −28254.8 −1.65136
$$665$$ 0 0
$$666$$ 7452.76 0.433617
$$667$$ −18142.6 −1.05320
$$668$$ −67051.4 −3.88368
$$669$$ 4329.28 0.250194
$$670$$ 0 0
$$671$$ 7283.61 0.419047
$$672$$ −974.708 −0.0559527
$$673$$ 2340.47 0.134054 0.0670271 0.997751i $$-0.478649\pi$$
0.0670271 + 0.997751i $$0.478649\pi$$
$$674$$ 12910.1 0.737803
$$675$$ 0 0
$$676$$ −26626.1 −1.51491
$$677$$ 16562.2 0.940234 0.470117 0.882604i $$-0.344212\pi$$
0.470117 + 0.882604i $$0.344212\pi$$
$$678$$ 14114.8 0.799522
$$679$$ 5263.01 0.297461
$$680$$ 0 0
$$681$$ 10976.1 0.617630
$$682$$ 10748.3 0.603478
$$683$$ −22303.4 −1.24951 −0.624755 0.780821i $$-0.714801\pi$$
−0.624755 + 0.780821i $$0.714801\pi$$
$$684$$ 886.313 0.0495454
$$685$$ 0 0
$$686$$ −17881.3 −0.995204
$$687$$ 5086.49 0.282477
$$688$$ 14475.9 0.802163
$$689$$ −8350.87 −0.461745
$$690$$ 0 0
$$691$$ 6244.91 0.343802 0.171901 0.985114i $$-0.445009\pi$$
0.171901 + 0.985114i $$0.445009\pi$$
$$692$$ −55576.0 −3.05301
$$693$$ −542.892 −0.0297587
$$694$$ 1326.42 0.0725508
$$695$$ 0 0
$$696$$ −34438.1 −1.87554
$$697$$ 25178.3 1.36829
$$698$$ 52443.5 2.84386
$$699$$ −17346.7 −0.938645
$$700$$ 0 0
$$701$$ −15277.7 −0.823154 −0.411577 0.911375i $$-0.635022\pi$$
−0.411577 + 0.911375i $$0.635022\pi$$
$$702$$ −3298.37 −0.177335
$$703$$ −981.519 −0.0526582
$$704$$ −3942.93 −0.211086
$$705$$ 0 0
$$706$$ 35557.3 1.89549
$$707$$ 6901.43 0.367122
$$708$$ −20897.3 −1.10928
$$709$$ −11732.1 −0.621449 −0.310725 0.950500i $$-0.600572\pi$$
−0.310725 + 0.950500i $$0.600572\pi$$
$$710$$ 0 0
$$711$$ 3627.00 0.191313
$$712$$ −53137.2 −2.79691
$$713$$ −13455.5 −0.706750
$$714$$ −4855.35 −0.254491
$$715$$ 0 0
$$716$$ −445.153 −0.0232349
$$717$$ 17508.5 0.911947
$$718$$ −1785.57 −0.0928093
$$719$$ −19006.2 −0.985828 −0.492914 0.870078i $$-0.664068\pi$$
−0.492914 + 0.870078i $$0.664068\pi$$
$$720$$ 0 0
$$721$$ −2704.45 −0.139694
$$722$$ 33925.1 1.74870
$$723$$ 2946.92 0.151587
$$724$$ 32327.4 1.65945
$$725$$ 0 0
$$726$$ 1804.56 0.0922498
$$727$$ −2650.77 −0.135229 −0.0676146 0.997712i $$-0.521539\pi$$
−0.0676146 + 0.997712i $$0.521539\pi$$
$$728$$ −5836.96 −0.297160
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 10529.1 0.532738
$$732$$ −33199.5 −1.67635
$$733$$ 8854.22 0.446164 0.223082 0.974800i $$-0.428388\pi$$
0.223082 + 0.974800i $$0.428388\pi$$
$$734$$ 44952.2 2.26051
$$735$$ 0 0
$$736$$ −4055.96 −0.203131
$$737$$ −3450.84 −0.172474
$$738$$ −18974.7 −0.946435
$$739$$ 17174.8 0.854921 0.427460 0.904034i $$-0.359408\pi$$
0.427460 + 0.904034i $$0.359408\pi$$
$$740$$ 0 0
$$741$$ 434.391 0.0215354
$$742$$ 9264.06 0.458348
$$743$$ 9800.30 0.483901 0.241950 0.970289i $$-0.422213\pi$$
0.241950 + 0.970289i $$0.422213\pi$$
$$744$$ −25541.1 −1.25858
$$745$$ 0 0
$$746$$ −59301.6 −2.91044
$$747$$ 5870.82 0.287553
$$748$$ 10914.6 0.533525
$$749$$ −6026.92 −0.294017
$$750$$ 0 0
$$751$$ 2781.25 0.135139 0.0675695 0.997715i $$-0.478476\pi$$
0.0675695 + 0.997715i $$0.478476\pi$$
$$752$$ −11520.9 −0.558675
$$753$$ −12960.7 −0.627243
$$754$$ −32375.6 −1.56372
$$755$$ 0 0
$$756$$ 2474.57 0.119046
$$757$$ −3010.87 −0.144560 −0.0722800 0.997384i $$-0.523028\pi$$
−0.0722800 + 0.997384i $$0.523028\pi$$
$$758$$ −32591.8 −1.56172
$$759$$ −2259.08 −0.108036
$$760$$ 0 0
$$761$$ −2709.54 −0.129068 −0.0645340 0.997916i $$-0.520556\pi$$
−0.0645340 + 0.997916i $$0.520556\pi$$
$$762$$ −33716.5 −1.60291
$$763$$ −6992.03 −0.331754
$$764$$ −55848.4 −2.64466
$$765$$ 0 0
$$766$$ 53207.7 2.50976
$$767$$ −10242.0 −0.482159
$$768$$ 25041.2 1.17656
$$769$$ −28121.1 −1.31869 −0.659345 0.751841i $$-0.729166\pi$$
−0.659345 + 0.751841i $$0.729166\pi$$
$$770$$ 0 0
$$771$$ −18672.9 −0.872227
$$772$$ 21626.2 1.00822
$$773$$ −4274.61 −0.198897 −0.0994483 0.995043i $$-0.531708\pi$$
−0.0994483 + 0.995043i $$0.531708\pi$$
$$774$$ −7934.85 −0.368491
$$775$$ 0 0
$$776$$ 41571.2 1.92309
$$777$$ −2740.38 −0.126526
$$778$$ 32065.9 1.47766
$$779$$ 2498.95 0.114935
$$780$$ 0 0
$$781$$ −1690.63 −0.0774589
$$782$$ −20204.1 −0.923909
$$783$$ 7155.59 0.326590
$$784$$ −25542.2 −1.16355
$$785$$ 0 0
$$786$$ −851.985 −0.0386632
$$787$$ −19107.9 −0.865469 −0.432734 0.901522i $$-0.642451\pi$$
−0.432734 + 0.901522i $$0.642451\pi$$
$$788$$ 38459.6 1.73866
$$789$$ −7766.85 −0.350453
$$790$$ 0 0
$$791$$ −5190.01 −0.233294
$$792$$ −4288.17 −0.192391
$$793$$ −16271.4 −0.728645
$$794$$ −62269.4 −2.78320
$$795$$ 0 0
$$796$$ −50801.9 −2.26209
$$797$$ −27518.4 −1.22303 −0.611513 0.791234i $$-0.709439\pi$$
−0.611513 + 0.791234i $$0.709439\pi$$
$$798$$ −481.893 −0.0213770
$$799$$ −8379.74 −0.371031
$$800$$ 0 0
$$801$$ 11040.9 0.487030
$$802$$ −45335.2 −1.99606
$$803$$ −1692.53 −0.0743811
$$804$$ 15729.4 0.689965
$$805$$ 0 0
$$806$$ −24011.4 −1.04934
$$807$$ 17614.6 0.768356
$$808$$ 54512.7 2.37345
$$809$$ 21986.6 0.955511 0.477755 0.878493i $$-0.341450\pi$$
0.477755 + 0.878493i $$0.341450\pi$$
$$810$$ 0 0
$$811$$ 11835.1 0.512437 0.256218 0.966619i $$-0.417523\pi$$
0.256218 + 0.966619i $$0.417523\pi$$
$$812$$ 24289.4 1.04974
$$813$$ 9920.79 0.427967
$$814$$ 9108.93 0.392221
$$815$$ 0 0
$$816$$ −14537.6 −0.623672
$$817$$ 1045.01 0.0447494
$$818$$ 2967.43 0.126838
$$819$$ 1212.81 0.0517448
$$820$$ 0 0
$$821$$ −7890.58 −0.335424 −0.167712 0.985836i $$-0.553638\pi$$
−0.167712 + 0.985836i $$0.553638\pi$$
$$822$$ −28623.4 −1.21455
$$823$$ −1000.74 −0.0423861 −0.0211930 0.999775i $$-0.506746\pi$$
−0.0211930 + 0.999775i $$0.506746\pi$$
$$824$$ −21361.8 −0.903123
$$825$$ 0 0
$$826$$ 11362.0 0.478611
$$827$$ −22764.2 −0.957181 −0.478590 0.878038i $$-0.658852\pi$$
−0.478590 + 0.878038i $$0.658852\pi$$
$$828$$ 10297.2 0.432188
$$829$$ 39835.8 1.66894 0.834471 0.551051i $$-0.185773\pi$$
0.834471 + 0.551051i $$0.185773\pi$$
$$830$$ 0 0
$$831$$ −8584.10 −0.358338
$$832$$ 8808.43 0.367040
$$833$$ −18578.1 −0.772742
$$834$$ −25211.0 −1.04674
$$835$$ 0 0
$$836$$ 1083.27 0.0448155
$$837$$ 5306.96 0.219158
$$838$$ −57469.7 −2.36904
$$839$$ 23330.9 0.960037 0.480019 0.877258i $$-0.340630\pi$$
0.480019 + 0.877258i $$0.340630\pi$$
$$840$$ 0 0
$$841$$ 45847.5 1.87984
$$842$$ −75476.0 −3.08916
$$843$$ 8964.14 0.366241
$$844$$ 46928.1 1.91390
$$845$$ 0 0
$$846$$ 6315.09 0.256640
$$847$$ −663.535 −0.0269177
$$848$$ 27737.8 1.12326
$$849$$ 4363.79 0.176402
$$850$$ 0 0
$$851$$ −11403.3 −0.459341
$$852$$ 7706.08 0.309866
$$853$$ −22937.3 −0.920701 −0.460351 0.887737i $$-0.652276\pi$$
−0.460351 + 0.887737i $$0.652276\pi$$
$$854$$ 18050.8 0.723284
$$855$$ 0 0
$$856$$ −47605.2 −1.90083
$$857$$ 38472.3 1.53348 0.766738 0.641961i $$-0.221879\pi$$
0.766738 + 0.641961i $$0.221879\pi$$
$$858$$ −4031.34 −0.160405
$$859$$ −22970.3 −0.912384 −0.456192 0.889881i $$-0.650787\pi$$
−0.456192 + 0.889881i $$0.650787\pi$$
$$860$$ 0 0
$$861$$ 6977.00 0.276162
$$862$$ 74084.5 2.92730
$$863$$ 33587.6 1.32484 0.662420 0.749133i $$-0.269530\pi$$
0.662420 + 0.749133i $$0.269530\pi$$
$$864$$ 1599.70 0.0629895
$$865$$ 0 0
$$866$$ −11284.0 −0.442777
$$867$$ 4165.08 0.163153
$$868$$ 18014.3 0.704430
$$869$$ 4433.00 0.173049
$$870$$ 0 0
$$871$$ 7709.12 0.299901
$$872$$ −55228.3 −2.14480
$$873$$ −8637.71 −0.334871
$$874$$ −2005.25 −0.0776073
$$875$$ 0 0
$$876$$ 7714.74 0.297554
$$877$$ 704.130 0.0271115 0.0135558 0.999908i $$-0.495685\pi$$
0.0135558 + 0.999908i $$0.495685\pi$$
$$878$$ 66707.5 2.56409
$$879$$ −25759.5 −0.988450
$$880$$ 0 0
$$881$$ 9746.33 0.372715 0.186358 0.982482i $$-0.440332\pi$$
0.186358 + 0.982482i $$0.440332\pi$$
$$882$$ 14000.7 0.534501
$$883$$ 8774.24 0.334402 0.167201 0.985923i $$-0.446527\pi$$
0.167201 + 0.985923i $$0.446527\pi$$
$$884$$ −24383.0 −0.927701
$$885$$ 0 0
$$886$$ 57206.4 2.16917
$$887$$ 13505.1 0.511224 0.255612 0.966779i $$-0.417723\pi$$
0.255612 + 0.966779i $$0.417723\pi$$
$$888$$ −21645.6 −0.817993
$$889$$ 12397.5 0.467717
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ −24118.6 −0.905324
$$893$$ −831.689 −0.0311662
$$894$$ −30988.9 −1.15931
$$895$$ 0 0
$$896$$ −12370.9 −0.461252
$$897$$ 5046.75 0.187855
$$898$$ 80552.8 2.99341
$$899$$ 52091.1 1.93252
$$900$$ 0 0
$$901$$ 20175.1 0.745984
$$902$$ −23191.3 −0.856083
$$903$$ 2917.64 0.107523
$$904$$ −40994.6 −1.50825
$$905$$ 0 0
$$906$$ 40433.4 1.48268
$$907$$ 2523.23 0.0923732 0.0461866 0.998933i $$-0.485293\pi$$
0.0461866 + 0.998933i $$0.485293\pi$$
$$908$$ −61148.4 −2.23489
$$909$$ −11326.7 −0.413293
$$910$$ 0 0
$$911$$ −32285.3 −1.17416 −0.587080 0.809529i $$-0.699723\pi$$
−0.587080 + 0.809529i $$0.699723\pi$$
$$912$$ −1442.85 −0.0523877
$$913$$ 7175.45 0.260101
$$914$$ 24422.6 0.883838
$$915$$ 0 0
$$916$$ −28337.0 −1.02214
$$917$$ 313.274 0.0112816
$$918$$ 7968.65 0.286497
$$919$$ −40956.5 −1.47011 −0.735055 0.678008i $$-0.762844\pi$$
−0.735055 + 0.678008i $$0.762844\pi$$
$$920$$ 0 0
$$921$$ 1587.09 0.0567821
$$922$$ 70943.6 2.53406
$$923$$ 3776.83 0.134687
$$924$$ 3024.47 0.107682
$$925$$ 0 0
$$926$$ 33090.8 1.17433
$$927$$ 4438.58 0.157262
$$928$$ 15702.1 0.555437
$$929$$ −6728.41 −0.237623 −0.118812 0.992917i $$-0.537908\pi$$
−0.118812 + 0.992917i $$0.537908\pi$$
$$930$$ 0 0
$$931$$ −1843.88 −0.0649095
$$932$$ 96639.1 3.39648
$$933$$ 15024.0 0.527185
$$934$$ 96181.9 3.36956
$$935$$ 0 0
$$936$$ 9579.69 0.334532
$$937$$ 36377.8 1.26831 0.634157 0.773205i $$-0.281347\pi$$
0.634157 + 0.773205i $$0.281347\pi$$
$$938$$ −8552.13 −0.297694
$$939$$ 23575.6 0.819340
$$940$$ 0 0
$$941$$ 25907.0 0.897496 0.448748 0.893658i $$-0.351870\pi$$
0.448748 + 0.893658i $$0.351870\pi$$
$$942$$ −34916.4 −1.20768
$$943$$ 29032.7 1.00258
$$944$$ 34019.2 1.17291
$$945$$ 0 0
$$946$$ −9698.15 −0.333313
$$947$$ −37096.3 −1.27293 −0.636466 0.771304i $$-0.719605\pi$$
−0.636466 + 0.771304i $$0.719605\pi$$
$$948$$ −20206.2 −0.692263
$$949$$ 3781.07 0.129335
$$950$$ 0 0
$$951$$ 23241.1 0.792476
$$952$$ 14101.7 0.480084
$$953$$ 15323.7 0.520865 0.260432 0.965492i $$-0.416135\pi$$
0.260432 + 0.965492i $$0.416135\pi$$
$$954$$ −15204.3 −0.515992
$$955$$ 0 0
$$956$$ −97540.4 −3.29988
$$957$$ 8745.72 0.295412
$$958$$ −48850.7 −1.64749
$$959$$ 10524.8 0.354395
$$960$$ 0 0
$$961$$ 8842.46 0.296817
$$962$$ −20349.2 −0.682000
$$963$$ 9891.45 0.330994
$$964$$ −16417.4 −0.548516
$$965$$ 0 0
$$966$$ −5598.62 −0.186473
$$967$$ 53388.0 1.77543 0.887716 0.460391i $$-0.152291\pi$$
0.887716 + 0.460391i $$0.152291\pi$$
$$968$$ −5241.10 −0.174024
$$969$$ −1049.46 −0.0347921
$$970$$ 0 0
$$971$$ −36325.7 −1.20056 −0.600281 0.799789i $$-0.704945\pi$$
−0.600281 + 0.799789i $$0.704945\pi$$
$$972$$ −4061.28 −0.134018
$$973$$ 9270.07 0.305432
$$974$$ 51094.6 1.68088
$$975$$ 0 0
$$976$$ 54046.4 1.77252
$$977$$ −48608.7 −1.59174 −0.795870 0.605468i $$-0.792986\pi$$
−0.795870 + 0.605468i $$0.792986\pi$$
$$978$$ −52541.1 −1.71787
$$979$$ 13494.4 0.440535
$$980$$ 0 0
$$981$$ 11475.4 0.373477
$$982$$ −56657.7 −1.84116
$$983$$ −31762.6 −1.03059 −0.515295 0.857013i $$-0.672318\pi$$
−0.515295 + 0.857013i $$0.672318\pi$$
$$984$$ 55109.6 1.78540
$$985$$ 0 0
$$986$$ 78217.2 2.52631
$$987$$ −2322.06 −0.0748854
$$988$$ −2420.01 −0.0779258
$$989$$ 12140.9 0.390352
$$990$$ 0 0
$$991$$ −5073.82 −0.162639 −0.0813195 0.996688i $$-0.525913\pi$$
−0.0813195 + 0.996688i $$0.525913\pi$$
$$992$$ 11645.5 0.372726
$$993$$ −4065.25 −0.129916
$$994$$ −4189.83 −0.133696
$$995$$ 0 0
$$996$$ −32706.5 −1.04051
$$997$$ −24607.0 −0.781657 −0.390828 0.920464i $$-0.627811\pi$$
−0.390828 + 0.920464i $$0.627811\pi$$
$$998$$ 45631.8 1.44734
$$999$$ 4497.54 0.142438
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 825.4.a.r.1.1 3
3.2 odd 2 2475.4.a.t.1.3 3
5.2 odd 4 825.4.c.k.199.2 6
5.3 odd 4 825.4.c.k.199.5 6
5.4 even 2 165.4.a.e.1.3 3
15.14 odd 2 495.4.a.k.1.1 3
55.54 odd 2 1815.4.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.e.1.3 3 5.4 even 2
495.4.a.k.1.1 3 15.14 odd 2
825.4.a.r.1.1 3 1.1 even 1 trivial
825.4.c.k.199.2 6 5.2 odd 4
825.4.c.k.199.5 6 5.3 odd 4
1815.4.a.r.1.1 3 55.54 odd 2
2475.4.a.t.1.3 3 3.2 odd 2