# Properties

 Label 825.6.a.c.1.2 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} -121.000 q^{11} +169.557 q^{12} +277.549 q^{13} +910.853 q^{14} -66.1983 q^{16} -704.489 q^{17} -293.845 q^{18} -2861.18 q^{19} +2259.73 q^{21} +438.954 q^{22} +1066.85 q^{23} -1659.89 q^{24} -1006.87 q^{26} -729.000 q^{27} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} -5661.67 q^{32} +1089.00 q^{33} +2555.69 q^{34} -1526.01 q^{36} +9042.34 q^{37} +10379.6 q^{38} -2497.94 q^{39} +18219.0 q^{41} -8197.68 q^{42} +4054.54 q^{43} +2279.60 q^{44} -3870.24 q^{46} -20750.8 q^{47} +595.785 q^{48} +46234.9 q^{49} +6340.40 q^{51} -5228.92 q^{52} +26485.9 q^{53} +2644.61 q^{54} -46307.4 q^{56} +25750.7 q^{57} +14283.9 q^{58} +4293.12 q^{59} -6831.76 q^{61} +2337.52 q^{62} -20337.6 q^{63} +22657.3 q^{64} -3950.59 q^{66} +56749.5 q^{67} +13272.3 q^{68} -9601.68 q^{69} +3187.09 q^{71} +14939.0 q^{72} +7397.14 q^{73} -32803.1 q^{74} +53903.7 q^{76} +30380.9 q^{77} +9061.82 q^{78} +24393.7 q^{79} +6561.00 q^{81} -66093.5 q^{82} -102795. q^{83} -42572.6 q^{84} -14708.7 q^{86} +35437.0 q^{87} -22316.3 q^{88} +49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} +5799.15 q^{93} +75278.1 q^{94} +50955.1 q^{96} +92279.5 q^{97} -167727. q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 $$2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100})$$ 2 * q - 13 * q^2 - 18 * q^3 + 37 * q^4 + 117 * q^6 - 146 * q^7 - 39 * q^8 + 162 * q^9 - 242 * q^11 - 333 * q^12 + 130 * q^13 - 74 * q^14 + 241 * q^16 + 728 * q^17 - 1053 * q^18 - 828 * q^19 + 1314 * q^21 + 1573 * q^22 + 238 * q^23 + 351 * q^24 + 376 * q^26 - 1458 * q^27 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 - 1391 * q^32 + 2178 * q^33 - 10870 * q^34 + 2997 * q^36 + 1908 * q^37 - 8676 * q^38 - 1170 * q^39 + 36484 * q^41 + 666 * q^42 - 9768 * q^43 - 4477 * q^44 + 3898 * q^46 - 43742 * q^47 - 2169 * q^48 + 40470 * q^49 - 6552 * q^51 - 13468 * q^52 + 12174 * q^53 + 9477 * q^54 - 69786 * q^56 + 7452 * q^57 - 29142 * q^58 - 2788 * q^59 - 25302 * q^61 + 94520 * q^62 - 11826 * q^63 - 27199 * q^64 - 14157 * q^66 + 40520 * q^67 + 93262 * q^68 - 2142 * q^69 + 31386 * q^71 - 3159 * q^72 + 46780 * q^73 + 34062 * q^74 + 167436 * q^76 + 17666 * q^77 - 3384 * q^78 - 16850 * q^79 + 13122 * q^81 - 237278 * q^82 - 79440 * q^83 - 95382 * q^84 + 114840 * q^86 - 6264 * q^87 + 4719 * q^88 - 54204 * q^89 - 85192 * q^91 - 66382 * q^92 + 94320 * q^93 + 290758 * q^94 + 12519 * q^96 + 241568 * q^97 - 113697 * q^98 - 19602 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.62772 −0.641296 −0.320648 0.947198i $$-0.603901\pi$$
−0.320648 + 0.947198i $$0.603901\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −18.8397 −0.588739
$$5$$ 0 0
$$6$$ 32.6495 0.370252
$$7$$ −251.081 −1.93673 −0.968366 0.249534i $$-0.919722\pi$$
−0.968366 + 0.249534i $$0.919722\pi$$
$$8$$ 184.432 1.01885
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 169.557 0.339909
$$13$$ 277.549 0.455492 0.227746 0.973721i $$-0.426864\pi$$
0.227746 + 0.973721i $$0.426864\pi$$
$$14$$ 910.853 1.24202
$$15$$ 0 0
$$16$$ −66.1983 −0.0646468
$$17$$ −704.489 −0.591224 −0.295612 0.955308i $$-0.595523\pi$$
−0.295612 + 0.955308i $$0.595523\pi$$
$$18$$ −293.845 −0.213765
$$19$$ −2861.18 −1.81828 −0.909142 0.416486i $$-0.863261\pi$$
−0.909142 + 0.416486i $$0.863261\pi$$
$$20$$ 0 0
$$21$$ 2259.73 1.11817
$$22$$ 438.954 0.193358
$$23$$ 1066.85 0.420518 0.210259 0.977646i $$-0.432569\pi$$
0.210259 + 0.977646i $$0.432569\pi$$
$$24$$ −1659.89 −0.588235
$$25$$ 0 0
$$26$$ −1006.87 −0.292105
$$27$$ −729.000 −0.192450
$$28$$ 4730.29 1.14023
$$29$$ −3937.44 −0.869399 −0.434700 0.900575i $$-0.643145\pi$$
−0.434700 + 0.900575i $$0.643145\pi$$
$$30$$ 0 0
$$31$$ −644.350 −0.120425 −0.0602126 0.998186i $$-0.519178\pi$$
−0.0602126 + 0.998186i $$0.519178\pi$$
$$32$$ −5661.67 −0.977395
$$33$$ 1089.00 0.174078
$$34$$ 2555.69 0.379149
$$35$$ 0 0
$$36$$ −1526.01 −0.196246
$$37$$ 9042.34 1.08587 0.542934 0.839776i $$-0.317314\pi$$
0.542934 + 0.839776i $$0.317314\pi$$
$$38$$ 10379.6 1.16606
$$39$$ −2497.94 −0.262979
$$40$$ 0 0
$$41$$ 18219.0 1.69264 0.846322 0.532672i $$-0.178812\pi$$
0.846322 + 0.532672i $$0.178812\pi$$
$$42$$ −8197.68 −0.717080
$$43$$ 4054.54 0.334403 0.167202 0.985923i $$-0.446527\pi$$
0.167202 + 0.985923i $$0.446527\pi$$
$$44$$ 2279.60 0.177512
$$45$$ 0 0
$$46$$ −3870.24 −0.269677
$$47$$ −20750.8 −1.37022 −0.685110 0.728439i $$-0.740246\pi$$
−0.685110 + 0.728439i $$0.740246\pi$$
$$48$$ 595.785 0.0373238
$$49$$ 46234.9 2.75093
$$50$$ 0 0
$$51$$ 6340.40 0.341343
$$52$$ −5228.92 −0.268166
$$53$$ 26485.9 1.29517 0.647583 0.761995i $$-0.275780\pi$$
0.647583 + 0.761995i $$0.275780\pi$$
$$54$$ 2644.61 0.123417
$$55$$ 0 0
$$56$$ −46307.4 −1.97324
$$57$$ 25750.7 1.04979
$$58$$ 14283.9 0.557542
$$59$$ 4293.12 0.160562 0.0802810 0.996772i $$-0.474418\pi$$
0.0802810 + 0.996772i $$0.474418\pi$$
$$60$$ 0 0
$$61$$ −6831.76 −0.235076 −0.117538 0.993068i $$-0.537500\pi$$
−0.117538 + 0.993068i $$0.537500\pi$$
$$62$$ 2337.52 0.0772282
$$63$$ −20337.6 −0.645577
$$64$$ 22657.3 0.691446
$$65$$ 0 0
$$66$$ −3950.59 −0.111635
$$67$$ 56749.5 1.54445 0.772227 0.635347i $$-0.219143\pi$$
0.772227 + 0.635347i $$0.219143\pi$$
$$68$$ 13272.3 0.348077
$$69$$ −9601.68 −0.242786
$$70$$ 0 0
$$71$$ 3187.09 0.0750323 0.0375161 0.999296i $$-0.488055\pi$$
0.0375161 + 0.999296i $$0.488055\pi$$
$$72$$ 14939.0 0.339617
$$73$$ 7397.14 0.162464 0.0812319 0.996695i $$-0.474115\pi$$
0.0812319 + 0.996695i $$0.474115\pi$$
$$74$$ −32803.1 −0.696362
$$75$$ 0 0
$$76$$ 53903.7 1.07050
$$77$$ 30380.9 0.583947
$$78$$ 9061.82 0.168647
$$79$$ 24393.7 0.439754 0.219877 0.975528i $$-0.429434\pi$$
0.219877 + 0.975528i $$0.429434\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −66093.5 −1.08549
$$83$$ −102795. −1.63786 −0.818932 0.573890i $$-0.805434\pi$$
−0.818932 + 0.573890i $$0.805434\pi$$
$$84$$ −42572.6 −0.658312
$$85$$ 0 0
$$86$$ −14708.7 −0.214451
$$87$$ 35437.0 0.501948
$$88$$ −22316.3 −0.307196
$$89$$ 49599.4 0.663745 0.331873 0.943324i $$-0.392320\pi$$
0.331873 + 0.943324i $$0.392320\pi$$
$$90$$ 0 0
$$91$$ −69687.4 −0.882166
$$92$$ −20099.1 −0.247576
$$93$$ 5799.15 0.0695275
$$94$$ 75278.1 0.878717
$$95$$ 0 0
$$96$$ 50955.1 0.564299
$$97$$ 92279.5 0.995808 0.497904 0.867232i $$-0.334103\pi$$
0.497904 + 0.867232i $$0.334103\pi$$
$$98$$ −167727. −1.76416
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ −35546.1 −0.346728 −0.173364 0.984858i $$-0.555464\pi$$
−0.173364 + 0.984858i $$0.555464\pi$$
$$102$$ −23001.2 −0.218902
$$103$$ 59876.8 0.556116 0.278058 0.960564i $$-0.410309\pi$$
0.278058 + 0.960564i $$0.410309\pi$$
$$104$$ 51188.9 0.464079
$$105$$ 0 0
$$106$$ −96083.5 −0.830586
$$107$$ −89253.8 −0.753646 −0.376823 0.926285i $$-0.622983\pi$$
−0.376823 + 0.926285i $$0.622983\pi$$
$$108$$ 13734.1 0.113303
$$109$$ 22796.0 0.183777 0.0918887 0.995769i $$-0.470710\pi$$
0.0918887 + 0.995769i $$0.470710\pi$$
$$110$$ 0 0
$$111$$ −81381.1 −0.626926
$$112$$ 16621.2 0.125203
$$113$$ 166064. 1.22343 0.611717 0.791077i $$-0.290479\pi$$
0.611717 + 0.791077i $$0.290479\pi$$
$$114$$ −93416.1 −0.673224
$$115$$ 0 0
$$116$$ 74180.1 0.511850
$$117$$ 22481.5 0.151831
$$118$$ −15574.2 −0.102968
$$119$$ 176884. 1.14504
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 24783.7 0.150753
$$123$$ −163971. −0.977248
$$124$$ 12139.3 0.0708991
$$125$$ 0 0
$$126$$ 73779.1 0.414006
$$127$$ −159190. −0.875802 −0.437901 0.899023i $$-0.644278\pi$$
−0.437901 + 0.899023i $$0.644278\pi$$
$$128$$ 98979.2 0.533973
$$129$$ −36490.9 −0.193068
$$130$$ 0 0
$$131$$ 232174. 1.18205 0.591025 0.806654i $$-0.298724\pi$$
0.591025 + 0.806654i $$0.298724\pi$$
$$132$$ −20516.4 −0.102486
$$133$$ 718390. 3.52153
$$134$$ −205871. −0.990452
$$135$$ 0 0
$$136$$ −129930. −0.602369
$$137$$ −68205.4 −0.310468 −0.155234 0.987878i $$-0.549613\pi$$
−0.155234 + 0.987878i $$0.549613\pi$$
$$138$$ 34832.2 0.155698
$$139$$ 298162. 1.30893 0.654464 0.756093i $$-0.272894\pi$$
0.654464 + 0.756093i $$0.272894\pi$$
$$140$$ 0 0
$$141$$ 186757. 0.791097
$$142$$ −11561.9 −0.0481179
$$143$$ −33583.4 −0.137336
$$144$$ −5362.06 −0.0215489
$$145$$ 0 0
$$146$$ −26834.7 −0.104187
$$147$$ −416114. −1.58825
$$148$$ −170355. −0.639293
$$149$$ 83131.5 0.306761 0.153380 0.988167i $$-0.450984\pi$$
0.153380 + 0.988167i $$0.450984\pi$$
$$150$$ 0 0
$$151$$ 167598. 0.598171 0.299085 0.954226i $$-0.403318\pi$$
0.299085 + 0.954226i $$0.403318\pi$$
$$152$$ −527694. −1.85256
$$153$$ −57063.6 −0.197075
$$154$$ −110213. −0.374483
$$155$$ 0 0
$$156$$ 47060.3 0.154826
$$157$$ 63259.7 0.204823 0.102411 0.994742i $$-0.467344\pi$$
0.102411 + 0.994742i $$0.467344\pi$$
$$158$$ −88493.4 −0.282012
$$159$$ −238373. −0.747765
$$160$$ 0 0
$$161$$ −267867. −0.814431
$$162$$ −23801.5 −0.0712551
$$163$$ −237727. −0.700825 −0.350412 0.936596i $$-0.613959\pi$$
−0.350412 + 0.936596i $$0.613959\pi$$
$$164$$ −343240. −0.996526
$$165$$ 0 0
$$166$$ 372912. 1.05036
$$167$$ −487880. −1.35370 −0.676849 0.736122i $$-0.736655\pi$$
−0.676849 + 0.736122i $$0.736655\pi$$
$$168$$ 416767. 1.13925
$$169$$ −294260. −0.792527
$$170$$ 0 0
$$171$$ −231756. −0.606095
$$172$$ −76386.1 −0.196876
$$173$$ −325942. −0.827991 −0.413996 0.910279i $$-0.635867\pi$$
−0.413996 + 0.910279i $$0.635867\pi$$
$$174$$ −128555. −0.321897
$$175$$ 0 0
$$176$$ 8009.99 0.0194917
$$177$$ −38638.1 −0.0927005
$$178$$ −179933. −0.425657
$$179$$ 462906. 1.07984 0.539921 0.841716i $$-0.318454\pi$$
0.539921 + 0.841716i $$0.318454\pi$$
$$180$$ 0 0
$$181$$ 23901.3 0.0542281 0.0271141 0.999632i $$-0.491368\pi$$
0.0271141 + 0.999632i $$0.491368\pi$$
$$182$$ 252806. 0.565730
$$183$$ 61485.8 0.135721
$$184$$ 196762. 0.428446
$$185$$ 0 0
$$186$$ −21037.7 −0.0445877
$$187$$ 85243.1 0.178261
$$188$$ 390938. 0.806703
$$189$$ 183038. 0.372724
$$190$$ 0 0
$$191$$ −565986. −1.12259 −0.561297 0.827615i $$-0.689697\pi$$
−0.561297 + 0.827615i $$0.689697\pi$$
$$192$$ −203916. −0.399207
$$193$$ −91762.3 −0.177325 −0.0886627 0.996062i $$-0.528259\pi$$
−0.0886627 + 0.996062i $$0.528259\pi$$
$$194$$ −334764. −0.638608
$$195$$ 0 0
$$196$$ −871049. −1.61958
$$197$$ −485247. −0.890836 −0.445418 0.895323i $$-0.646945\pi$$
−0.445418 + 0.895323i $$0.646945\pi$$
$$198$$ 35555.3 0.0644527
$$199$$ 692632. 1.23985 0.619926 0.784660i $$-0.287162\pi$$
0.619926 + 0.784660i $$0.287162\pi$$
$$200$$ 0 0
$$201$$ −510745. −0.891690
$$202$$ 128951. 0.222355
$$203$$ 988619. 1.68379
$$204$$ −119451. −0.200962
$$205$$ 0 0
$$206$$ −217216. −0.356635
$$207$$ 86415.1 0.140173
$$208$$ −18373.3 −0.0294461
$$209$$ 346203. 0.548233
$$210$$ 0 0
$$211$$ −441020. −0.681949 −0.340975 0.940073i $$-0.610757\pi$$
−0.340975 + 0.940073i $$0.610757\pi$$
$$212$$ −498986. −0.762516
$$213$$ −28683.8 −0.0433199
$$214$$ 323788. 0.483310
$$215$$ 0 0
$$216$$ −134451. −0.196078
$$217$$ 161784. 0.233231
$$218$$ −82697.4 −0.117856
$$219$$ −66574.2 −0.0937985
$$220$$ 0 0
$$221$$ −195530. −0.269298
$$222$$ 295228. 0.402045
$$223$$ −1.13133e6 −1.52345 −0.761726 0.647899i $$-0.775648\pi$$
−0.761726 + 0.647899i $$0.775648\pi$$
$$224$$ 1.42154e6 1.89295
$$225$$ 0 0
$$226$$ −602435. −0.784583
$$227$$ 820354. 1.05666 0.528332 0.849038i $$-0.322818\pi$$
0.528332 + 0.849038i $$0.322818\pi$$
$$228$$ −485133. −0.618051
$$229$$ −1.00301e6 −1.26392 −0.631958 0.775003i $$-0.717748\pi$$
−0.631958 + 0.775003i $$0.717748\pi$$
$$230$$ 0 0
$$231$$ −273428. −0.337142
$$232$$ −726191. −0.885790
$$233$$ −734.569 −0.000886427 0 −0.000443214 1.00000i $$-0.500141\pi$$
−0.000443214 1.00000i $$0.500141\pi$$
$$234$$ −81556.4 −0.0973685
$$235$$ 0 0
$$236$$ −80880.9 −0.0945291
$$237$$ −219543. −0.253892
$$238$$ −641685. −0.734311
$$239$$ −1.06403e6 −1.20492 −0.602460 0.798149i $$-0.705813\pi$$
−0.602460 + 0.798149i $$0.705813\pi$$
$$240$$ 0 0
$$241$$ −661636. −0.733798 −0.366899 0.930261i $$-0.619581\pi$$
−0.366899 + 0.930261i $$0.619581\pi$$
$$242$$ −53113.4 −0.0582996
$$243$$ −59049.0 −0.0641500
$$244$$ 128708. 0.138398
$$245$$ 0 0
$$246$$ 594841. 0.626705
$$247$$ −794118. −0.828214
$$248$$ −118839. −0.122696
$$249$$ 925158. 0.945622
$$250$$ 0 0
$$251$$ 1.34864e6 1.35118 0.675590 0.737277i $$-0.263889\pi$$
0.675590 + 0.737277i $$0.263889\pi$$
$$252$$ 383153. 0.380077
$$253$$ −129089. −0.126791
$$254$$ 577496. 0.561648
$$255$$ 0 0
$$256$$ −1.08410e6 −1.03388
$$257$$ 1.65015e6 1.55844 0.779220 0.626751i $$-0.215616\pi$$
0.779220 + 0.626751i $$0.215616\pi$$
$$258$$ 132379. 0.123814
$$259$$ −2.27036e6 −2.10303
$$260$$ 0 0
$$261$$ −318933. −0.289800
$$262$$ −842262. −0.758044
$$263$$ 868163. 0.773948 0.386974 0.922091i $$-0.373520\pi$$
0.386974 + 0.922091i $$0.373520\pi$$
$$264$$ 200846. 0.177359
$$265$$ 0 0
$$266$$ −2.60612e6 −2.25834
$$267$$ −446395. −0.383213
$$268$$ −1.06914e6 −0.909280
$$269$$ 271547. 0.228804 0.114402 0.993435i $$-0.463505\pi$$
0.114402 + 0.993435i $$0.463505\pi$$
$$270$$ 0 0
$$271$$ −752770. −0.622643 −0.311322 0.950305i $$-0.600772\pi$$
−0.311322 + 0.950305i $$0.600772\pi$$
$$272$$ 46635.9 0.0382207
$$273$$ 627186. 0.509319
$$274$$ 247430. 0.199102
$$275$$ 0 0
$$276$$ 180892. 0.142938
$$277$$ −811445. −0.635418 −0.317709 0.948188i $$-0.602914\pi$$
−0.317709 + 0.948188i $$0.602914\pi$$
$$278$$ −1.08165e6 −0.839411
$$279$$ −52192.3 −0.0401417
$$280$$ 0 0
$$281$$ −1.72395e6 −1.30244 −0.651221 0.758888i $$-0.725743\pi$$
−0.651221 + 0.758888i $$0.725743\pi$$
$$282$$ −677503. −0.507327
$$283$$ −272979. −0.202611 −0.101305 0.994855i $$-0.532302\pi$$
−0.101305 + 0.994855i $$0.532302\pi$$
$$284$$ −60043.6 −0.0441744
$$285$$ 0 0
$$286$$ 121831. 0.0880731
$$287$$ −4.57446e6 −3.27820
$$288$$ −458596. −0.325798
$$289$$ −923553. −0.650455
$$290$$ 0 0
$$291$$ −830515. −0.574930
$$292$$ −139360. −0.0956488
$$293$$ 713441. 0.485500 0.242750 0.970089i $$-0.421951\pi$$
0.242750 + 0.970089i $$0.421951\pi$$
$$294$$ 1.50954e6 1.01854
$$295$$ 0 0
$$296$$ 1.66770e6 1.10634
$$297$$ 88209.0 0.0580259
$$298$$ −301578. −0.196725
$$299$$ 296104. 0.191543
$$300$$ 0 0
$$301$$ −1.01802e6 −0.647649
$$302$$ −607997. −0.383605
$$303$$ 319915. 0.200184
$$304$$ 189405. 0.117546
$$305$$ 0 0
$$306$$ 207011. 0.126383
$$307$$ 2.67734e6 1.62128 0.810640 0.585545i $$-0.199119\pi$$
0.810640 + 0.585545i $$0.199119\pi$$
$$308$$ −572365. −0.343792
$$309$$ −538891. −0.321074
$$310$$ 0 0
$$311$$ −1.28078e6 −0.750886 −0.375443 0.926846i $$-0.622509\pi$$
−0.375443 + 0.926846i $$0.622509\pi$$
$$312$$ −460700. −0.267936
$$313$$ −1.36808e6 −0.789313 −0.394656 0.918829i $$-0.629136\pi$$
−0.394656 + 0.918829i $$0.629136\pi$$
$$314$$ −229488. −0.131352
$$315$$ 0 0
$$316$$ −459569. −0.258900
$$317$$ 7686.28 0.00429604 0.00214802 0.999998i $$-0.499316\pi$$
0.00214802 + 0.999998i $$0.499316\pi$$
$$318$$ 864752. 0.479539
$$319$$ 476431. 0.262134
$$320$$ 0 0
$$321$$ 803284. 0.435117
$$322$$ 971746. 0.522292
$$323$$ 2.01567e6 1.07501
$$324$$ −123607. −0.0654155
$$325$$ 0 0
$$326$$ 862406. 0.449436
$$327$$ −205164. −0.106104
$$328$$ 3.36017e6 1.72455
$$329$$ 5.21014e6 2.65375
$$330$$ 0 0
$$331$$ 958347. 0.480787 0.240393 0.970676i $$-0.422724\pi$$
0.240393 + 0.970676i $$0.422724\pi$$
$$332$$ 1.93663e6 0.964275
$$333$$ 732430. 0.361956
$$334$$ 1.76989e6 0.868121
$$335$$ 0 0
$$336$$ −149590. −0.0722862
$$337$$ −4.08768e6 −1.96066 −0.980330 0.197365i $$-0.936762\pi$$
−0.980330 + 0.197365i $$0.936762\pi$$
$$338$$ 1.06749e6 0.508244
$$339$$ −1.49458e6 −0.706350
$$340$$ 0 0
$$341$$ 77966.3 0.0363096
$$342$$ 840745. 0.388686
$$343$$ −7.38880e6 −3.39108
$$344$$ 747787. 0.340707
$$345$$ 0 0
$$346$$ 1.18243e6 0.530988
$$347$$ −84250.2 −0.0375619 −0.0187809 0.999824i $$-0.505979\pi$$
−0.0187809 + 0.999824i $$0.505979\pi$$
$$348$$ −667621. −0.295517
$$349$$ 1.11859e6 0.491597 0.245798 0.969321i $$-0.420950\pi$$
0.245798 + 0.969321i $$0.420950\pi$$
$$350$$ 0 0
$$351$$ −202333. −0.0876595
$$352$$ 685063. 0.294696
$$353$$ 3.73570e6 1.59564 0.797820 0.602896i $$-0.205987\pi$$
0.797820 + 0.602896i $$0.205987\pi$$
$$354$$ 140168. 0.0594485
$$355$$ 0 0
$$356$$ −934436. −0.390773
$$357$$ −1.59196e6 −0.661090
$$358$$ −1.67929e6 −0.692499
$$359$$ 1.45342e6 0.595189 0.297595 0.954692i $$-0.403816\pi$$
0.297595 + 0.954692i $$0.403816\pi$$
$$360$$ 0 0
$$361$$ 5.71027e6 2.30616
$$362$$ −86707.1 −0.0347763
$$363$$ −131769. −0.0524864
$$364$$ 1.31289e6 0.519366
$$365$$ 0 0
$$366$$ −223053. −0.0870374
$$367$$ −25363.4 −0.00982975 −0.00491487 0.999988i $$-0.501564\pi$$
−0.00491487 + 0.999988i $$0.501564\pi$$
$$368$$ −70623.8 −0.0271851
$$369$$ 1.47574e6 0.564214
$$370$$ 0 0
$$371$$ −6.65013e6 −2.50839
$$372$$ −109254. −0.0409336
$$373$$ 1.72695e6 0.642699 0.321350 0.946961i $$-0.395864\pi$$
0.321350 + 0.946961i $$0.395864\pi$$
$$374$$ −309238. −0.114318
$$375$$ 0 0
$$376$$ −3.82711e6 −1.39605
$$377$$ −1.09283e6 −0.396005
$$378$$ −664012. −0.239027
$$379$$ −4.29401e6 −1.53555 −0.767777 0.640718i $$-0.778637\pi$$
−0.767777 + 0.640718i $$0.778637\pi$$
$$380$$ 0 0
$$381$$ 1.43271e6 0.505644
$$382$$ 2.05324e6 0.719915
$$383$$ −1.29297e6 −0.450392 −0.225196 0.974313i $$-0.572302\pi$$
−0.225196 + 0.974313i $$0.572302\pi$$
$$384$$ −890813. −0.308289
$$385$$ 0 0
$$386$$ 332888. 0.113718
$$387$$ 328418. 0.111468
$$388$$ −1.73851e6 −0.586272
$$389$$ 1.55257e6 0.520209 0.260104 0.965581i $$-0.416243\pi$$
0.260104 + 0.965581i $$0.416243\pi$$
$$390$$ 0 0
$$391$$ −751586. −0.248620
$$392$$ 8.52719e6 2.80279
$$393$$ −2.08957e6 −0.682456
$$394$$ 1.76034e6 0.571290
$$395$$ 0 0
$$396$$ 184647. 0.0591705
$$397$$ −819569. −0.260981 −0.130491 0.991450i $$-0.541655\pi$$
−0.130491 + 0.991450i $$0.541655\pi$$
$$398$$ −2.51268e6 −0.795113
$$399$$ −6.46551e6 −2.03316
$$400$$ 0 0
$$401$$ 1.38956e6 0.431536 0.215768 0.976445i $$-0.430774\pi$$
0.215768 + 0.976445i $$0.430774\pi$$
$$402$$ 1.85284e6 0.571838
$$403$$ −178839. −0.0548528
$$404$$ 669677. 0.204132
$$405$$ 0 0
$$406$$ −3.58643e6 −1.07981
$$407$$ −1.09412e6 −0.327401
$$408$$ 1.16937e6 0.347778
$$409$$ −3.53091e6 −1.04371 −0.521854 0.853035i $$-0.674759\pi$$
−0.521854 + 0.853035i $$0.674759\pi$$
$$410$$ 0 0
$$411$$ 613849. 0.179249
$$412$$ −1.12806e6 −0.327407
$$413$$ −1.07792e6 −0.310966
$$414$$ −313490. −0.0898923
$$415$$ 0 0
$$416$$ −1.57139e6 −0.445196
$$417$$ −2.68346e6 −0.755710
$$418$$ −1.25593e6 −0.351580
$$419$$ −6.69977e6 −1.86434 −0.932170 0.362021i $$-0.882087\pi$$
−0.932170 + 0.362021i $$0.882087\pi$$
$$420$$ 0 0
$$421$$ −5.01124e6 −1.37797 −0.688986 0.724775i $$-0.741944\pi$$
−0.688986 + 0.724775i $$0.741944\pi$$
$$422$$ 1.59990e6 0.437331
$$423$$ −1.68082e6 −0.456740
$$424$$ 4.88485e6 1.31958
$$425$$ 0 0
$$426$$ 104057. 0.0277809
$$427$$ 1.71533e6 0.455279
$$428$$ 1.68151e6 0.443701
$$429$$ 302251. 0.0792910
$$430$$ 0 0
$$431$$ −1.14741e6 −0.297526 −0.148763 0.988873i $$-0.547529\pi$$
−0.148763 + 0.988873i $$0.547529\pi$$
$$432$$ 48258.6 0.0124413
$$433$$ 1.13393e6 0.290649 0.145324 0.989384i $$-0.453577\pi$$
0.145324 + 0.989384i $$0.453577\pi$$
$$434$$ −586908. −0.149570
$$435$$ 0 0
$$436$$ −429469. −0.108197
$$437$$ −3.05246e6 −0.764622
$$438$$ 241513. 0.0601526
$$439$$ 7.44692e6 1.84423 0.922115 0.386915i $$-0.126459\pi$$
0.922115 + 0.386915i $$0.126459\pi$$
$$440$$ 0 0
$$441$$ 3.74503e6 0.916977
$$442$$ 709328. 0.172700
$$443$$ −6.72521e6 −1.62816 −0.814079 0.580754i $$-0.802758\pi$$
−0.814079 + 0.580754i $$0.802758\pi$$
$$444$$ 1.53319e6 0.369096
$$445$$ 0 0
$$446$$ 4.10416e6 0.976984
$$447$$ −748183. −0.177108
$$448$$ −5.68883e6 −1.33915
$$449$$ 2.17793e6 0.509834 0.254917 0.966963i $$-0.417952\pi$$
0.254917 + 0.966963i $$0.417952\pi$$
$$450$$ 0 0
$$451$$ −2.20450e6 −0.510351
$$452$$ −3.12860e6 −0.720284
$$453$$ −1.50838e6 −0.345354
$$454$$ −2.97601e6 −0.677634
$$455$$ 0 0
$$456$$ 4.74924e6 1.06958
$$457$$ 3.72380e6 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$458$$ 3.63865e6 0.810544
$$459$$ 513572. 0.113781
$$460$$ 0 0
$$461$$ 4.74202e6 1.03923 0.519615 0.854401i $$-0.326076\pi$$
0.519615 + 0.854401i $$0.326076\pi$$
$$462$$ 991919. 0.216208
$$463$$ 4.82803e6 1.04669 0.523344 0.852121i $$-0.324684\pi$$
0.523344 + 0.852121i $$0.324684\pi$$
$$464$$ 260652. 0.0562039
$$465$$ 0 0
$$466$$ 2664.81 0.000568462 0
$$467$$ −7.56254e6 −1.60463 −0.802316 0.596900i $$-0.796399\pi$$
−0.802316 + 0.596900i $$0.796399\pi$$
$$468$$ −423543. −0.0893887
$$469$$ −1.42487e7 −2.99119
$$470$$ 0 0
$$471$$ −569337. −0.118254
$$472$$ 791788. 0.163589
$$473$$ −490599. −0.100826
$$474$$ 796441. 0.162820
$$475$$ 0 0
$$476$$ −3.33243e6 −0.674131
$$477$$ 2.14536e6 0.431722
$$478$$ 3.85999e6 0.772710
$$479$$ −1.71897e6 −0.342319 −0.171159 0.985243i $$-0.554751\pi$$
−0.171159 + 0.985243i $$0.554751\pi$$
$$480$$ 0 0
$$481$$ 2.50969e6 0.494604
$$482$$ 2.40023e6 0.470582
$$483$$ 2.41080e6 0.470212
$$484$$ −275831. −0.0535218
$$485$$ 0 0
$$486$$ 214213. 0.0411392
$$487$$ 3.08125e6 0.588715 0.294357 0.955695i $$-0.404894\pi$$
0.294357 + 0.955695i $$0.404894\pi$$
$$488$$ −1.25999e6 −0.239508
$$489$$ 2.13954e6 0.404621
$$490$$ 0 0
$$491$$ −1.05909e6 −0.198258 −0.0991290 0.995075i $$-0.531606\pi$$
−0.0991290 + 0.995075i $$0.531606\pi$$
$$492$$ 3.08916e6 0.575344
$$493$$ 2.77388e6 0.514009
$$494$$ 2.88084e6 0.531131
$$495$$ 0 0
$$496$$ 42654.9 0.00778510
$$497$$ −800218. −0.145317
$$498$$ −3.35621e6 −0.606423
$$499$$ 2.26415e6 0.407056 0.203528 0.979069i $$-0.434759\pi$$
0.203528 + 0.979069i $$0.434759\pi$$
$$500$$ 0 0
$$501$$ 4.39092e6 0.781558
$$502$$ −4.89250e6 −0.866507
$$503$$ 7.65222e6 1.34855 0.674276 0.738480i $$-0.264456\pi$$
0.674276 + 0.738480i $$0.264456\pi$$
$$504$$ −3.75090e6 −0.657748
$$505$$ 0 0
$$506$$ 468299. 0.0813106
$$507$$ 2.64834e6 0.457566
$$508$$ 2.99908e6 0.515619
$$509$$ 489107. 0.0836777 0.0418388 0.999124i $$-0.486678\pi$$
0.0418388 + 0.999124i $$0.486678\pi$$
$$510$$ 0 0
$$511$$ −1.85728e6 −0.314649
$$512$$ 765484. 0.129051
$$513$$ 2.08580e6 0.349929
$$514$$ −5.98627e6 −0.999421
$$515$$ 0 0
$$516$$ 687475. 0.113667
$$517$$ 2.51085e6 0.413137
$$518$$ 8.23624e6 1.34867
$$519$$ 2.93348e6 0.478041
$$520$$ 0 0
$$521$$ 1.36556e6 0.220402 0.110201 0.993909i $$-0.464851\pi$$
0.110201 + 0.993909i $$0.464851\pi$$
$$522$$ 1.15700e6 0.185847
$$523$$ 5.63994e6 0.901613 0.450807 0.892622i $$-0.351136\pi$$
0.450807 + 0.892622i $$0.351136\pi$$
$$524$$ −4.37408e6 −0.695919
$$525$$ 0 0
$$526$$ −3.14945e6 −0.496330
$$527$$ 453937. 0.0711982
$$528$$ −72089.9 −0.0112536
$$529$$ −5.29817e6 −0.823164
$$530$$ 0 0
$$531$$ 347742. 0.0535207
$$532$$ −1.35342e7 −2.07326
$$533$$ 5.05667e6 0.770986
$$534$$ 1.61939e6 0.245753
$$535$$ 0 0
$$536$$ 1.04664e7 1.57357
$$537$$ −4.16615e6 −0.623447
$$538$$ −985095. −0.146731
$$539$$ −5.59442e6 −0.829437
$$540$$ 0 0
$$541$$ 8.24934e6 1.21179 0.605893 0.795546i $$-0.292816\pi$$
0.605893 + 0.795546i $$0.292816\pi$$
$$542$$ 2.73084e6 0.399299
$$543$$ −215111. −0.0313086
$$544$$ 3.98859e6 0.577859
$$545$$ 0 0
$$546$$ −2.27526e6 −0.326624
$$547$$ 4.74537e6 0.678112 0.339056 0.940766i $$-0.389892\pi$$
0.339056 + 0.940766i $$0.389892\pi$$
$$548$$ 1.28497e6 0.182785
$$549$$ −553372. −0.0783586
$$550$$ 0 0
$$551$$ 1.12657e7 1.58082
$$552$$ −1.77086e6 −0.247363
$$553$$ −6.12480e6 −0.851685
$$554$$ 2.94369e6 0.407491
$$555$$ 0 0
$$556$$ −5.61728e6 −0.770617
$$557$$ −2.99690e6 −0.409293 −0.204647 0.978836i $$-0.565605\pi$$
−0.204647 + 0.978836i $$0.565605\pi$$
$$558$$ 189339. 0.0257427
$$559$$ 1.12533e6 0.152318
$$560$$ 0 0
$$561$$ −767188. −0.102919
$$562$$ 6.25400e6 0.835251
$$563$$ 3.89491e6 0.517876 0.258938 0.965894i $$-0.416627\pi$$
0.258938 + 0.965894i $$0.416627\pi$$
$$564$$ −3.51844e6 −0.465750
$$565$$ 0 0
$$566$$ 990290. 0.129934
$$567$$ −1.64735e6 −0.215192
$$568$$ 587801. 0.0764468
$$569$$ 2.08127e6 0.269493 0.134747 0.990880i $$-0.456978\pi$$
0.134747 + 0.990880i $$0.456978\pi$$
$$570$$ 0 0
$$571$$ 1.28958e7 1.65523 0.827615 0.561297i $$-0.189697\pi$$
0.827615 + 0.561297i $$0.189697\pi$$
$$572$$ 632700. 0.0808551
$$573$$ 5.09387e6 0.648129
$$574$$ 1.65948e7 2.10229
$$575$$ 0 0
$$576$$ 1.83524e6 0.230482
$$577$$ −1.26163e6 −0.157759 −0.0788795 0.996884i $$-0.525134\pi$$
−0.0788795 + 0.996884i $$0.525134\pi$$
$$578$$ 3.35039e6 0.417134
$$579$$ 825861. 0.102379
$$580$$ 0 0
$$581$$ 2.58100e7 3.17210
$$582$$ 3.01288e6 0.368701
$$583$$ −3.20480e6 −0.390508
$$584$$ 1.36427e6 0.165527
$$585$$ 0 0
$$586$$ −2.58816e6 −0.311349
$$587$$ 1.25673e7 1.50538 0.752689 0.658376i $$-0.228756\pi$$
0.752689 + 0.658376i $$0.228756\pi$$
$$588$$ 7.83945e6 0.935065
$$589$$ 1.84360e6 0.218967
$$590$$ 0 0
$$591$$ 4.36723e6 0.514324
$$592$$ −598588. −0.0701978
$$593$$ 4.32620e6 0.505207 0.252604 0.967570i $$-0.418713\pi$$
0.252604 + 0.967570i $$0.418713\pi$$
$$594$$ −319997. −0.0372118
$$595$$ 0 0
$$596$$ −1.56617e6 −0.180602
$$597$$ −6.23369e6 −0.715829
$$598$$ −1.07418e6 −0.122836
$$599$$ 1.45262e7 1.65419 0.827097 0.562060i $$-0.189991\pi$$
0.827097 + 0.562060i $$0.189991\pi$$
$$600$$ 0 0
$$601$$ −380688. −0.0429915 −0.0214958 0.999769i $$-0.506843\pi$$
−0.0214958 + 0.999769i $$0.506843\pi$$
$$602$$ 3.69309e6 0.415335
$$603$$ 4.59671e6 0.514818
$$604$$ −3.15748e6 −0.352167
$$605$$ 0 0
$$606$$ −1.16056e6 −0.128377
$$607$$ 4.99233e6 0.549961 0.274980 0.961450i $$-0.411329\pi$$
0.274980 + 0.961450i $$0.411329\pi$$
$$608$$ 1.61991e7 1.77718
$$609$$ −8.89757e6 −0.972139
$$610$$ 0 0
$$611$$ −5.75936e6 −0.624125
$$612$$ 1.07506e6 0.116026
$$613$$ −2.99353e6 −0.321760 −0.160880 0.986974i $$-0.551433\pi$$
−0.160880 + 0.986974i $$0.551433\pi$$
$$614$$ −9.71265e6 −1.03972
$$615$$ 0 0
$$616$$ 5.60320e6 0.594955
$$617$$ −6.27781e6 −0.663889 −0.331945 0.943299i $$-0.607705\pi$$
−0.331945 + 0.943299i $$0.607705\pi$$
$$618$$ 1.95494e6 0.205903
$$619$$ 3.96477e6 0.415902 0.207951 0.978139i $$-0.433320\pi$$
0.207951 + 0.978139i $$0.433320\pi$$
$$620$$ 0 0
$$621$$ −777736. −0.0809288
$$622$$ 4.64631e6 0.481540
$$623$$ −1.24535e7 −1.28550
$$624$$ 165359. 0.0170007
$$625$$ 0 0
$$626$$ 4.96299e6 0.506183
$$627$$ −3.11583e6 −0.316523
$$628$$ −1.19179e6 −0.120587
$$629$$ −6.37023e6 −0.641990
$$630$$ 0 0
$$631$$ −9.62587e6 −0.962425 −0.481212 0.876604i $$-0.659803\pi$$
−0.481212 + 0.876604i $$0.659803\pi$$
$$632$$ 4.49898e6 0.448044
$$633$$ 3.96918e6 0.393724
$$634$$ −27883.7 −0.00275503
$$635$$ 0 0
$$636$$ 4.49087e6 0.440239
$$637$$ 1.28324e7 1.25303
$$638$$ −1.72836e6 −0.168105
$$639$$ 258154. 0.0250108
$$640$$ 0 0
$$641$$ −2.20090e6 −0.211571 −0.105785 0.994389i $$-0.533736\pi$$
−0.105785 + 0.994389i $$0.533736\pi$$
$$642$$ −2.91409e6 −0.279039
$$643$$ −2.00555e7 −1.91296 −0.956482 0.291791i $$-0.905749\pi$$
−0.956482 + 0.291791i $$0.905749\pi$$
$$644$$ 5.04652e6 0.479488
$$645$$ 0 0
$$646$$ −7.31229e6 −0.689401
$$647$$ −128900. −0.0121057 −0.00605287 0.999982i $$-0.501927\pi$$
−0.00605287 + 0.999982i $$0.501927\pi$$
$$648$$ 1.21006e6 0.113206
$$649$$ −519467. −0.0484113
$$650$$ 0 0
$$651$$ −1.45606e6 −0.134656
$$652$$ 4.47869e6 0.412603
$$653$$ −3.03250e6 −0.278303 −0.139151 0.990271i $$-0.544437\pi$$
−0.139151 + 0.990271i $$0.544437\pi$$
$$654$$ 744277. 0.0680441
$$655$$ 0 0
$$656$$ −1.20607e6 −0.109424
$$657$$ 599168. 0.0541546
$$658$$ −1.89009e7 −1.70184
$$659$$ −1.59132e7 −1.42740 −0.713698 0.700454i $$-0.752981\pi$$
−0.713698 + 0.700454i $$0.752981\pi$$
$$660$$ 0 0
$$661$$ 6.50892e6 0.579436 0.289718 0.957112i $$-0.406439\pi$$
0.289718 + 0.957112i $$0.406439\pi$$
$$662$$ −3.47661e6 −0.308327
$$663$$ 1.75977e6 0.155479
$$664$$ −1.89587e7 −1.66874
$$665$$ 0 0
$$666$$ −2.65705e6 −0.232121
$$667$$ −4.20067e6 −0.365598
$$668$$ 9.19149e6 0.796975
$$669$$ 1.01820e7 0.879565
$$670$$ 0 0
$$671$$ 826643. 0.0708780
$$672$$ −1.27939e7 −1.09290
$$673$$ 5.66035e6 0.481732 0.240866 0.970558i $$-0.422569\pi$$
0.240866 + 0.970558i $$0.422569\pi$$
$$674$$ 1.48290e7 1.25736
$$675$$ 0 0
$$676$$ 5.54375e6 0.466592
$$677$$ 1.53601e7 1.28802 0.644012 0.765016i $$-0.277269\pi$$
0.644012 + 0.765016i $$0.277269\pi$$
$$678$$ 5.42192e6 0.452979
$$679$$ −2.31697e7 −1.92861
$$680$$ 0 0
$$681$$ −7.38319e6 −0.610065
$$682$$ −282840. −0.0232852
$$683$$ −1.47069e6 −0.120634 −0.0603170 0.998179i $$-0.519211\pi$$
−0.0603170 + 0.998179i $$0.519211\pi$$
$$684$$ 4.36620e6 0.356832
$$685$$ 0 0
$$686$$ 2.68045e7 2.17469
$$687$$ 9.02712e6 0.729722
$$688$$ −268404. −0.0216181
$$689$$ 7.35114e6 0.589939
$$690$$ 0 0
$$691$$ −3.65617e6 −0.291294 −0.145647 0.989337i $$-0.546526\pi$$
−0.145647 + 0.989337i $$0.546526\pi$$
$$692$$ 6.14064e6 0.487471
$$693$$ 2.46085e6 0.194649
$$694$$ 305636. 0.0240883
$$695$$ 0 0
$$696$$ 6.53571e6 0.511411
$$697$$ −1.28351e7 −1.00073
$$698$$ −4.05794e6 −0.315259
$$699$$ 6611.12 0.000511779 0
$$700$$ 0 0
$$701$$ −5.21232e6 −0.400623 −0.200312 0.979732i $$-0.564195\pi$$
−0.200312 + 0.979732i $$0.564195\pi$$
$$702$$ 734008. 0.0562157
$$703$$ −2.58718e7 −1.97442
$$704$$ −2.74153e6 −0.208479
$$705$$ 0 0
$$706$$ −1.35521e7 −1.02328
$$707$$ 8.92498e6 0.671519
$$708$$ 727928. 0.0545764
$$709$$ −1.48388e7 −1.10862 −0.554311 0.832310i $$-0.687018\pi$$
−0.554311 + 0.832310i $$0.687018\pi$$
$$710$$ 0 0
$$711$$ 1.97589e6 0.146585
$$712$$ 9.14772e6 0.676258
$$713$$ −687427. −0.0506410
$$714$$ 5.77517e6 0.423954
$$715$$ 0 0
$$716$$ −8.72099e6 −0.635745
$$717$$ 9.57625e6 0.695661
$$718$$ −5.27260e6 −0.381693
$$719$$ 1.37497e7 0.991906 0.495953 0.868349i $$-0.334819\pi$$
0.495953 + 0.868349i $$0.334819\pi$$
$$720$$ 0 0
$$721$$ −1.50339e7 −1.07705
$$722$$ −2.07153e7 −1.47893
$$723$$ 5.95473e6 0.423659
$$724$$ −450292. −0.0319262
$$725$$ 0 0
$$726$$ 478021. 0.0336593
$$727$$ 8.20549e6 0.575796 0.287898 0.957661i $$-0.407044\pi$$
0.287898 + 0.957661i $$0.407044\pi$$
$$728$$ −1.28526e7 −0.898797
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −2.85638e6 −0.197707
$$732$$ −1.15837e6 −0.0799043
$$733$$ −1.20636e7 −0.829312 −0.414656 0.909978i $$-0.636098\pi$$
−0.414656 + 0.909978i $$0.636098\pi$$
$$734$$ 92011.3 0.00630378
$$735$$ 0 0
$$736$$ −6.04017e6 −0.411012
$$737$$ −6.86668e6 −0.465670
$$738$$ −5.35357e6 −0.361828
$$739$$ −1.02319e7 −0.689203 −0.344602 0.938749i $$-0.611986\pi$$
−0.344602 + 0.938749i $$0.611986\pi$$
$$740$$ 0 0
$$741$$ 7.14706e6 0.478170
$$742$$ 2.41248e7 1.60862
$$743$$ −2.41980e7 −1.60808 −0.804039 0.594577i $$-0.797320\pi$$
−0.804039 + 0.594577i $$0.797320\pi$$
$$744$$ 1.06955e6 0.0708383
$$745$$ 0 0
$$746$$ −6.26489e6 −0.412161
$$747$$ −8.32642e6 −0.545955
$$748$$ −1.60595e6 −0.104949
$$749$$ 2.24100e7 1.45961
$$750$$ 0 0
$$751$$ 1.10121e7 0.712476 0.356238 0.934395i $$-0.384059\pi$$
0.356238 + 0.934395i $$0.384059\pi$$
$$752$$ 1.37367e6 0.0885803
$$753$$ −1.21378e7 −0.780104
$$754$$ 3.96449e6 0.253956
$$755$$ 0 0
$$756$$ −3.44838e6 −0.219437
$$757$$ 2.08747e7 1.32398 0.661989 0.749513i $$-0.269712\pi$$
0.661989 + 0.749513i $$0.269712\pi$$
$$758$$ 1.55775e7 0.984744
$$759$$ 1.16180e6 0.0732028
$$760$$ 0 0
$$761$$ 2.85723e7 1.78848 0.894240 0.447587i $$-0.147716\pi$$
0.894240 + 0.447587i $$0.147716\pi$$
$$762$$ −5.19746e6 −0.324268
$$763$$ −5.72365e6 −0.355928
$$764$$ 1.06630e7 0.660915
$$765$$ 0 0
$$766$$ 4.69052e6 0.288835
$$767$$ 1.19155e6 0.0731347
$$768$$ 9.75692e6 0.596911
$$769$$ −3.07246e6 −0.187357 −0.0936787 0.995602i $$-0.529863\pi$$
−0.0936787 + 0.995602i $$0.529863\pi$$
$$770$$ 0 0
$$771$$ −1.48513e7 −0.899765
$$772$$ 1.72877e6 0.104398
$$773$$ 3.02076e7 1.81831 0.909153 0.416461i $$-0.136730\pi$$
0.909153 + 0.416461i $$0.136730\pi$$
$$774$$ −1.19141e6 −0.0714838
$$775$$ 0 0
$$776$$ 1.70193e7 1.01458
$$777$$ 2.04333e7 1.21419
$$778$$ −5.63229e6 −0.333608
$$779$$ −5.21280e7 −3.07771
$$780$$ 0 0
$$781$$ −385638. −0.0226231
$$782$$ 2.72654e6 0.159439
$$783$$ 2.87040e6 0.167316
$$784$$ −3.06067e6 −0.177839
$$785$$ 0 0
$$786$$ 7.58036e6 0.437657
$$787$$ 2.07854e7 1.19625 0.598126 0.801402i $$-0.295912\pi$$
0.598126 + 0.801402i $$0.295912\pi$$
$$788$$ 9.14190e6 0.524470
$$789$$ −7.81346e6 −0.446839
$$790$$ 0 0
$$791$$ −4.16957e7 −2.36946
$$792$$ −1.80762e6 −0.102399
$$793$$ −1.89615e6 −0.107075
$$794$$ 2.97317e6 0.167366
$$795$$ 0 0
$$796$$ −1.30490e7 −0.729950
$$797$$ 7.95535e6 0.443622 0.221811 0.975090i $$-0.428803\pi$$
0.221811 + 0.975090i $$0.428803\pi$$
$$798$$ 2.34551e7 1.30385
$$799$$ 1.46187e7 0.810106
$$800$$ 0 0
$$801$$ 4.01755e6 0.221248
$$802$$ −5.04094e6 −0.276742
$$803$$ −895054. −0.0489847
$$804$$ 9.62226e6 0.524973
$$805$$ 0 0
$$806$$ 648776. 0.0351769
$$807$$ −2.44392e6 −0.132100
$$808$$ −6.55584e6 −0.353265
$$809$$ −3.04660e7 −1.63661 −0.818303 0.574787i $$-0.805085\pi$$
−0.818303 + 0.574787i $$0.805085\pi$$
$$810$$ 0 0
$$811$$ 1.28041e7 0.683594 0.341797 0.939774i $$-0.388964\pi$$
0.341797 + 0.939774i $$0.388964\pi$$
$$812$$ −1.86252e7 −0.991316
$$813$$ 6.77493e6 0.359483
$$814$$ 3.96917e6 0.209961
$$815$$ 0 0
$$816$$ −419723. −0.0220667
$$817$$ −1.16008e7 −0.608040
$$818$$ 1.28092e7 0.669325
$$819$$ −5.64468e6 −0.294055
$$820$$ 0 0
$$821$$ 1.13635e7 0.588373 0.294186 0.955748i $$-0.404951\pi$$
0.294186 + 0.955748i $$0.404951\pi$$
$$822$$ −2.22687e6 −0.114952
$$823$$ −8.23741e6 −0.423927 −0.211964 0.977278i $$-0.567986\pi$$
−0.211964 + 0.977278i $$0.567986\pi$$
$$824$$ 1.10432e7 0.566600
$$825$$ 0 0
$$826$$ 3.91040e6 0.199421
$$827$$ −2.21035e7 −1.12382 −0.561909 0.827199i $$-0.689933\pi$$
−0.561909 + 0.827199i $$0.689933\pi$$
$$828$$ −1.62803e6 −0.0825252
$$829$$ −5.57875e6 −0.281936 −0.140968 0.990014i $$-0.545021\pi$$
−0.140968 + 0.990014i $$0.545021\pi$$
$$830$$ 0 0
$$831$$ 7.30300e6 0.366859
$$832$$ 6.28851e6 0.314948
$$833$$ −3.25720e7 −1.62641
$$834$$ 9.73484e6 0.484634
$$835$$ 0 0
$$836$$ −6.52235e6 −0.322766
$$837$$ 469731. 0.0231758
$$838$$ 2.43049e7 1.19559
$$839$$ −2.33576e7 −1.14558 −0.572788 0.819704i $$-0.694138\pi$$
−0.572788 + 0.819704i $$0.694138\pi$$
$$840$$ 0 0
$$841$$ −5.00769e6 −0.244145
$$842$$ 1.81794e7 0.883688
$$843$$ 1.55155e7 0.751965
$$844$$ 8.30866e6 0.401490
$$845$$ 0 0
$$846$$ 6.09753e6 0.292906
$$847$$ −3.67608e6 −0.176067
$$848$$ −1.75332e6 −0.0837284
$$849$$ 2.45681e6 0.116977
$$850$$ 0 0
$$851$$ 9.64685e6 0.456627
$$852$$ 540393. 0.0255041
$$853$$ −8.94855e6 −0.421095 −0.210548 0.977584i $$-0.567525\pi$$
−0.210548 + 0.977584i $$0.567525\pi$$
$$854$$ −6.22273e6 −0.291969
$$855$$ 0 0
$$856$$ −1.64612e7 −0.767853
$$857$$ 2.03190e6 0.0945039 0.0472520 0.998883i $$-0.484954\pi$$
0.0472520 + 0.998883i $$0.484954\pi$$
$$858$$ −1.09648e6 −0.0508490
$$859$$ −1.92359e7 −0.889467 −0.444734 0.895663i $$-0.646702\pi$$
−0.444734 + 0.895663i $$0.646702\pi$$
$$860$$ 0 0
$$861$$ 4.11701e7 1.89267
$$862$$ 4.16247e6 0.190802
$$863$$ −2.00380e7 −0.915856 −0.457928 0.888989i $$-0.651408\pi$$
−0.457928 + 0.888989i $$0.651408\pi$$
$$864$$ 4.12736e6 0.188100
$$865$$ 0 0
$$866$$ −4.11360e6 −0.186392
$$867$$ 8.31197e6 0.375540
$$868$$ −3.04796e6 −0.137312
$$869$$ −2.95164e6 −0.132591
$$870$$ 0 0
$$871$$ 1.57507e7 0.703486
$$872$$ 4.20431e6 0.187242
$$873$$ 7.47464e6 0.331936
$$874$$ 1.10735e7 0.490349
$$875$$ 0 0
$$876$$ 1.25424e6 0.0552229
$$877$$ −1.98487e7 −0.871430 −0.435715 0.900085i $$-0.643504\pi$$
−0.435715 + 0.900085i $$0.643504\pi$$
$$878$$ −2.70153e7 −1.18270
$$879$$ −6.42097e6 −0.280304
$$880$$ 0 0
$$881$$ −2.93546e7 −1.27420 −0.637099 0.770782i $$-0.719866\pi$$
−0.637099 + 0.770782i $$0.719866\pi$$
$$882$$ −1.35859e7 −0.588054
$$883$$ −3.85199e7 −1.66258 −0.831291 0.555838i $$-0.812398\pi$$
−0.831291 + 0.555838i $$0.812398\pi$$
$$884$$ 3.68372e6 0.158546
$$885$$ 0 0
$$886$$ 2.43972e7 1.04413
$$887$$ 2.68797e6 0.114714 0.0573569 0.998354i $$-0.481733\pi$$
0.0573569 + 0.998354i $$0.481733\pi$$
$$888$$ −1.50093e7 −0.638745
$$889$$ 3.99696e7 1.69619
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ 2.13139e7 0.896916
$$893$$ 5.93719e7 2.49145
$$894$$ 2.71420e6 0.113579
$$895$$ 0 0
$$896$$ −2.48519e7 −1.03416
$$897$$ −2.66493e6 −0.110587
$$898$$ −7.90093e6 −0.326955
$$899$$ 2.53709e6 0.104698
$$900$$ 0 0
$$901$$ −1.86590e7 −0.765733
$$902$$ 7.99731e6 0.327286
$$903$$ 9.16218e6 0.373921
$$904$$ 3.06276e7 1.24650
$$905$$ 0 0
$$906$$ 5.47197e6 0.221474
$$907$$ 7.78340e6 0.314160 0.157080 0.987586i $$-0.449792\pi$$
0.157080 + 0.987586i $$0.449792\pi$$
$$908$$ −1.54552e7 −0.622099
$$909$$ −2.87924e6 −0.115576
$$910$$ 0 0
$$911$$ 3.15361e7 1.25896 0.629481 0.777016i $$-0.283268\pi$$
0.629481 + 0.777016i $$0.283268\pi$$
$$912$$ −1.70465e6 −0.0678653
$$913$$ 1.24382e7 0.493835
$$914$$ −1.35089e7 −0.534878
$$915$$ 0 0
$$916$$ 1.88964e7 0.744117
$$917$$ −5.82946e7 −2.28931
$$918$$ −1.86310e6 −0.0729673
$$919$$ 4.21184e7 1.64507 0.822533 0.568717i $$-0.192560\pi$$
0.822533 + 0.568717i $$0.192560\pi$$
$$920$$ 0 0
$$921$$ −2.40961e7 −0.936047
$$922$$ −1.72027e7 −0.666454
$$923$$ 884572. 0.0341766
$$924$$ 5.15128e6 0.198489
$$925$$ 0 0
$$926$$ −1.75147e7 −0.671237
$$927$$ 4.85002e6 0.185372
$$928$$ 2.22925e7 0.849746
$$929$$ 9.43595e6 0.358712 0.179356 0.983784i $$-0.442599\pi$$
0.179356 + 0.983784i $$0.442599\pi$$
$$930$$ 0 0
$$931$$ −1.32287e8 −5.00197
$$932$$ 13839.0 0.000521874 0
$$933$$ 1.15270e7 0.433524
$$934$$ 2.74348e7 1.02904
$$935$$ 0 0
$$936$$ 4.14630e6 0.154693
$$937$$ −2.17869e7 −0.810674 −0.405337 0.914167i $$-0.632846\pi$$
−0.405337 + 0.914167i $$0.632846\pi$$
$$938$$ 5.16904e7 1.91824
$$939$$ 1.23127e7 0.455710
$$940$$ 0 0
$$941$$ 2.18678e7 0.805065 0.402532 0.915406i $$-0.368130\pi$$
0.402532 + 0.915406i $$0.368130\pi$$
$$942$$ 2.06540e6 0.0758361
$$943$$ 1.94370e7 0.711787
$$944$$ −284197. −0.0103798
$$945$$ 0 0
$$946$$ 1.77976e6 0.0646595
$$947$$ −2.42335e7 −0.878094 −0.439047 0.898464i $$-0.644684\pi$$
−0.439047 + 0.898464i $$0.644684\pi$$
$$948$$ 4.13612e6 0.149476
$$949$$ 2.05307e6 0.0740010
$$950$$ 0 0
$$951$$ −69176.5 −0.00248032
$$952$$ 3.26231e7 1.16663
$$953$$ −5.56092e7 −1.98342 −0.991710 0.128499i $$-0.958984\pi$$
−0.991710 + 0.128499i $$0.958984\pi$$
$$954$$ −7.78277e6 −0.276862
$$955$$ 0 0
$$956$$ 2.00459e7 0.709383
$$957$$ −4.28788e6 −0.151343
$$958$$ 6.23595e6 0.219528
$$959$$ 1.71251e7 0.601294
$$960$$ 0 0
$$961$$ −2.82140e7 −0.985498
$$962$$ −9.10446e6 −0.317188
$$963$$ −7.22956e6 −0.251215
$$964$$ 1.24650e7 0.432016
$$965$$ 0 0
$$966$$ −8.74571e6 −0.301545
$$967$$ 2.09122e7 0.719173 0.359586 0.933112i $$-0.382918\pi$$
0.359586 + 0.933112i $$0.382918\pi$$
$$968$$ 2.70027e6 0.0926229
$$969$$ −1.81410e7 −0.620659
$$970$$ 0 0
$$971$$ 1.68184e7 0.572448 0.286224 0.958163i $$-0.407600\pi$$
0.286224 + 0.958163i $$0.407600\pi$$
$$972$$ 1.11246e6 0.0377676
$$973$$ −7.48630e7 −2.53504
$$974$$ −1.11779e7 −0.377541
$$975$$ 0 0
$$976$$ 452251. 0.0151969
$$977$$ 3.45472e7 1.15792 0.578958 0.815358i $$-0.303460\pi$$
0.578958 + 0.815358i $$0.303460\pi$$
$$978$$ −7.76166e6 −0.259482
$$979$$ −6.00153e6 −0.200127
$$980$$ 0 0
$$981$$ 1.84648e6 0.0612592
$$982$$ 3.84210e6 0.127142
$$983$$ 3.40232e7 1.12303 0.561516 0.827466i $$-0.310218\pi$$
0.561516 + 0.827466i $$0.310218\pi$$
$$984$$ −3.02415e7 −0.995671
$$985$$ 0 0
$$986$$ −1.00629e7 −0.329632
$$987$$ −4.68913e7 −1.53214
$$988$$ 1.49609e7 0.487602
$$989$$ 4.32560e6 0.140623
$$990$$ 0 0
$$991$$ −4.47458e7 −1.44733 −0.723665 0.690151i $$-0.757544\pi$$
−0.723665 + 0.690151i $$0.757544\pi$$
$$992$$ 3.64810e6 0.117703
$$993$$ −8.62512e6 −0.277583
$$994$$ 2.90297e6 0.0931915
$$995$$ 0 0
$$996$$ −1.74297e7 −0.556725
$$997$$ 5.86494e6 0.186864 0.0934320 0.995626i $$-0.470216\pi$$
0.0934320 + 0.995626i $$0.470216\pi$$
$$998$$ −8.21371e6 −0.261044
$$999$$ −6.59187e6 −0.208975
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.c.1.2 2
5.4 even 2 33.6.a.e.1.1 2
15.14 odd 2 99.6.a.d.1.2 2
20.19 odd 2 528.6.a.o.1.2 2
55.54 odd 2 363.6.a.f.1.2 2
165.164 even 2 1089.6.a.p.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.e.1.1 2 5.4 even 2
99.6.a.d.1.2 2 15.14 odd 2
363.6.a.f.1.2 2 55.54 odd 2
528.6.a.o.1.2 2 20.19 odd 2
825.6.a.c.1.2 2 1.1 even 1 trivial
1089.6.a.p.1.1 2 165.164 even 2