# Properties

 Label 387.6.a.c.1.7 Level $387$ Weight $6$ Character 387.1 Self dual yes Analytic conductor $62.069$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 387.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.0685382676$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984$$ x^8 - 4*x^7 - 173*x^6 + 462*x^5 + 9118*x^4 - 14192*x^3 - 167688*x^2 + 106368*x + 681984 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$-6.09504$$ of defining polynomial Character $$\chi$$ $$=$$ 387.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +O(q^{10})$$ $$q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +513.028 q^{10} +631.897 q^{11} +28.5724 q^{13} +1809.15 q^{14} -972.709 q^{16} +1743.07 q^{17} -2027.92 q^{19} +2124.97 q^{20} +5115.23 q^{22} -2980.86 q^{23} +891.469 q^{25} +231.295 q^{26} +7493.51 q^{28} -766.139 q^{29} -8355.33 q^{31} -8270.38 q^{32} +14110.3 q^{34} +14163.7 q^{35} +14892.6 q^{37} -16416.1 q^{38} +784.783 q^{40} +5342.20 q^{41} -1849.00 q^{43} +21187.3 q^{44} -24130.2 q^{46} +6282.09 q^{47} +33140.1 q^{49} +7216.48 q^{50} +958.024 q^{52} +915.172 q^{53} +40046.8 q^{55} +2767.47 q^{56} -6201.92 q^{58} +14644.5 q^{59} -21324.9 q^{61} -67636.7 q^{62} -35822.4 q^{64} +1810.79 q^{65} -12868.9 q^{67} +58444.8 q^{68} +114656. q^{70} -56454.6 q^{71} -25591.3 q^{73} +120556. q^{74} -67995.4 q^{76} +141222. q^{77} +5795.13 q^{79} -61646.1 q^{80} +43245.4 q^{82} +7857.24 q^{83} +110468. q^{85} -14967.7 q^{86} +7824.80 q^{88} +7560.11 q^{89} +6385.60 q^{91} -99947.5 q^{92} +50853.8 q^{94} -128520. q^{95} +111712. q^{97} +268271. q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10})$$ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 $$8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100})$$ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.09504 1.43101 0.715507 0.698605i $$-0.246196\pi$$
0.715507 + 0.698605i $$0.246196\pi$$
$$3$$ 0 0
$$4$$ 33.5297 1.04780
$$5$$ 63.3756 1.13370 0.566849 0.823822i $$-0.308162\pi$$
0.566849 + 0.823822i $$0.308162\pi$$
$$6$$ 0 0
$$7$$ 223.489 1.72389 0.861946 0.507000i $$-0.169245\pi$$
0.861946 + 0.507000i $$0.169245\pi$$
$$8$$ 12.3830 0.0684073
$$9$$ 0 0
$$10$$ 513.028 1.62234
$$11$$ 631.897 1.57458 0.787289 0.616584i $$-0.211484\pi$$
0.787289 + 0.616584i $$0.211484\pi$$
$$12$$ 0 0
$$13$$ 28.5724 0.0468909 0.0234454 0.999725i $$-0.492536\pi$$
0.0234454 + 0.999725i $$0.492536\pi$$
$$14$$ 1809.15 2.46692
$$15$$ 0 0
$$16$$ −972.709 −0.949911
$$17$$ 1743.07 1.46283 0.731414 0.681933i $$-0.238861\pi$$
0.731414 + 0.681933i $$0.238861\pi$$
$$18$$ 0 0
$$19$$ −2027.92 −1.28874 −0.644371 0.764713i $$-0.722881\pi$$
−0.644371 + 0.764713i $$0.722881\pi$$
$$20$$ 2124.97 1.18789
$$21$$ 0 0
$$22$$ 5115.23 2.25324
$$23$$ −2980.86 −1.17496 −0.587479 0.809239i $$-0.699880\pi$$
−0.587479 + 0.809239i $$0.699880\pi$$
$$24$$ 0 0
$$25$$ 891.469 0.285270
$$26$$ 231.295 0.0671015
$$27$$ 0 0
$$28$$ 7493.51 1.80630
$$29$$ −766.139 −0.169166 −0.0845829 0.996416i $$-0.526956\pi$$
−0.0845829 + 0.996416i $$0.526956\pi$$
$$30$$ 0 0
$$31$$ −8355.33 −1.56156 −0.780781 0.624805i $$-0.785178\pi$$
−0.780781 + 0.624805i $$0.785178\pi$$
$$32$$ −8270.38 −1.42774
$$33$$ 0 0
$$34$$ 14110.3 2.09333
$$35$$ 14163.7 1.95437
$$36$$ 0 0
$$37$$ 14892.6 1.78841 0.894204 0.447661i $$-0.147743\pi$$
0.894204 + 0.447661i $$0.147743\pi$$
$$38$$ −16416.1 −1.84421
$$39$$ 0 0
$$40$$ 784.783 0.0775532
$$41$$ 5342.20 0.496319 0.248159 0.968719i $$-0.420174\pi$$
0.248159 + 0.968719i $$0.420174\pi$$
$$42$$ 0 0
$$43$$ −1849.00 −0.152499
$$44$$ 21187.3 1.64985
$$45$$ 0 0
$$46$$ −24130.2 −1.68138
$$47$$ 6282.09 0.414820 0.207410 0.978254i $$-0.433497\pi$$
0.207410 + 0.978254i $$0.433497\pi$$
$$48$$ 0 0
$$49$$ 33140.1 1.97181
$$50$$ 7216.48 0.408226
$$51$$ 0 0
$$52$$ 958.024 0.0491324
$$53$$ 915.172 0.0447521 0.0223760 0.999750i $$-0.492877\pi$$
0.0223760 + 0.999750i $$0.492877\pi$$
$$54$$ 0 0
$$55$$ 40046.8 1.78510
$$56$$ 2767.47 0.117927
$$57$$ 0 0
$$58$$ −6201.92 −0.242079
$$59$$ 14644.5 0.547704 0.273852 0.961772i $$-0.411702\pi$$
0.273852 + 0.961772i $$0.411702\pi$$
$$60$$ 0 0
$$61$$ −21324.9 −0.733775 −0.366887 0.930265i $$-0.619577\pi$$
−0.366887 + 0.930265i $$0.619577\pi$$
$$62$$ −67636.7 −2.23462
$$63$$ 0 0
$$64$$ −35822.4 −1.09321
$$65$$ 1810.79 0.0531601
$$66$$ 0 0
$$67$$ −12868.9 −0.350232 −0.175116 0.984548i $$-0.556030\pi$$
−0.175116 + 0.984548i $$0.556030\pi$$
$$68$$ 58444.8 1.53276
$$69$$ 0 0
$$70$$ 114656. 2.79674
$$71$$ −56454.6 −1.32909 −0.664544 0.747249i $$-0.731374\pi$$
−0.664544 + 0.747249i $$0.731374\pi$$
$$72$$ 0 0
$$73$$ −25591.3 −0.562064 −0.281032 0.959698i $$-0.590677\pi$$
−0.281032 + 0.959698i $$0.590677\pi$$
$$74$$ 120556. 2.55924
$$75$$ 0 0
$$76$$ −67995.4 −1.35035
$$77$$ 141222. 2.71440
$$78$$ 0 0
$$79$$ 5795.13 0.104471 0.0522355 0.998635i $$-0.483365\pi$$
0.0522355 + 0.998635i $$0.483365\pi$$
$$80$$ −61646.1 −1.07691
$$81$$ 0 0
$$82$$ 43245.4 0.710240
$$83$$ 7857.24 0.125192 0.0625958 0.998039i $$-0.480062\pi$$
0.0625958 + 0.998039i $$0.480062\pi$$
$$84$$ 0 0
$$85$$ 110468. 1.65841
$$86$$ −14967.7 −0.218228
$$87$$ 0 0
$$88$$ 7824.80 0.107713
$$89$$ 7560.11 0.101170 0.0505852 0.998720i $$-0.483891\pi$$
0.0505852 + 0.998720i $$0.483891\pi$$
$$90$$ 0 0
$$91$$ 6385.60 0.0808348
$$92$$ −99947.5 −1.23113
$$93$$ 0 0
$$94$$ 50853.8 0.593613
$$95$$ −128520. −1.46104
$$96$$ 0 0
$$97$$ 111712. 1.20551 0.602754 0.797927i $$-0.294070\pi$$
0.602754 + 0.797927i $$0.294070\pi$$
$$98$$ 268271. 2.82168
$$99$$ 0 0
$$100$$ 29890.7 0.298907
$$101$$ −10492.3 −0.102345 −0.0511724 0.998690i $$-0.516296\pi$$
−0.0511724 + 0.998690i $$0.516296\pi$$
$$102$$ 0 0
$$103$$ −11286.6 −0.104826 −0.0524130 0.998625i $$-0.516691\pi$$
−0.0524130 + 0.998625i $$0.516691\pi$$
$$104$$ 353.813 0.00320768
$$105$$ 0 0
$$106$$ 7408.36 0.0640409
$$107$$ −39045.4 −0.329693 −0.164847 0.986319i $$-0.552713\pi$$
−0.164847 + 0.986319i $$0.552713\pi$$
$$108$$ 0 0
$$109$$ −14622.8 −0.117887 −0.0589433 0.998261i $$-0.518773\pi$$
−0.0589433 + 0.998261i $$0.518773\pi$$
$$110$$ 324181. 2.55450
$$111$$ 0 0
$$112$$ −217389. −1.63755
$$113$$ 859.121 0.00632934 0.00316467 0.999995i $$-0.498993\pi$$
0.00316467 + 0.999995i $$0.498993\pi$$
$$114$$ 0 0
$$115$$ −188914. −1.33205
$$116$$ −25688.4 −0.177252
$$117$$ 0 0
$$118$$ 118548. 0.783772
$$119$$ 389557. 2.52176
$$120$$ 0 0
$$121$$ 238242. 1.47930
$$122$$ −172626. −1.05004
$$123$$ 0 0
$$124$$ −280152. −1.63621
$$125$$ −141551. −0.810288
$$126$$ 0 0
$$127$$ 193396. 1.06399 0.531997 0.846746i $$-0.321442\pi$$
0.531997 + 0.846746i $$0.321442\pi$$
$$128$$ −25331.5 −0.136658
$$129$$ 0 0
$$130$$ 14658.4 0.0760728
$$131$$ 269599. 1.37259 0.686293 0.727325i $$-0.259237\pi$$
0.686293 + 0.727325i $$0.259237\pi$$
$$132$$ 0 0
$$133$$ −453216. −2.22165
$$134$$ −104175. −0.501187
$$135$$ 0 0
$$136$$ 21584.6 0.100068
$$137$$ −148987. −0.678183 −0.339091 0.940753i $$-0.610120\pi$$
−0.339091 + 0.940753i $$0.610120\pi$$
$$138$$ 0 0
$$139$$ −16355.6 −0.0718008 −0.0359004 0.999355i $$-0.511430\pi$$
−0.0359004 + 0.999355i $$0.511430\pi$$
$$140$$ 474906. 2.04780
$$141$$ 0 0
$$142$$ −457003. −1.90194
$$143$$ 18054.8 0.0738333
$$144$$ 0 0
$$145$$ −48554.5 −0.191783
$$146$$ −207163. −0.804322
$$147$$ 0 0
$$148$$ 499345. 1.87390
$$149$$ 150536. 0.555489 0.277744 0.960655i $$-0.410413\pi$$
0.277744 + 0.960655i $$0.410413\pi$$
$$150$$ 0 0
$$151$$ 428141. 1.52807 0.764037 0.645172i $$-0.223214\pi$$
0.764037 + 0.645172i $$0.223214\pi$$
$$152$$ −25111.8 −0.0881594
$$153$$ 0 0
$$154$$ 1.14320e6 3.88435
$$155$$ −529524. −1.77034
$$156$$ 0 0
$$157$$ −434190. −1.40582 −0.702912 0.711277i $$-0.748117\pi$$
−0.702912 + 0.711277i $$0.748117\pi$$
$$158$$ 46911.8 0.149499
$$159$$ 0 0
$$160$$ −524140. −1.61863
$$161$$ −666189. −2.02550
$$162$$ 0 0
$$163$$ −571830. −1.68577 −0.842884 0.538095i $$-0.819144\pi$$
−0.842884 + 0.538095i $$0.819144\pi$$
$$164$$ 179122. 0.520044
$$165$$ 0 0
$$166$$ 63604.7 0.179151
$$167$$ −605744. −1.68073 −0.840365 0.542020i $$-0.817660\pi$$
−0.840365 + 0.542020i $$0.817660\pi$$
$$168$$ 0 0
$$169$$ −370477. −0.997801
$$170$$ 894246. 2.37320
$$171$$ 0 0
$$172$$ −61996.4 −0.159789
$$173$$ 85238.8 0.216532 0.108266 0.994122i $$-0.465470\pi$$
0.108266 + 0.994122i $$0.465470\pi$$
$$174$$ 0 0
$$175$$ 199233. 0.491775
$$176$$ −614652. −1.49571
$$177$$ 0 0
$$178$$ 61199.4 0.144776
$$179$$ −77941.8 −0.181818 −0.0909092 0.995859i $$-0.528977\pi$$
−0.0909092 + 0.995859i $$0.528977\pi$$
$$180$$ 0 0
$$181$$ −347661. −0.788787 −0.394393 0.918942i $$-0.629045\pi$$
−0.394393 + 0.918942i $$0.629045\pi$$
$$182$$ 51691.7 0.115676
$$183$$ 0 0
$$184$$ −36912.2 −0.0803757
$$185$$ 943828. 2.02751
$$186$$ 0 0
$$187$$ 1.10144e6 2.30334
$$188$$ 210637. 0.434650
$$189$$ 0 0
$$190$$ −1.04038e6 −2.09078
$$191$$ 192163. 0.381142 0.190571 0.981673i $$-0.438966\pi$$
0.190571 + 0.981673i $$0.438966\pi$$
$$192$$ 0 0
$$193$$ −271724. −0.525091 −0.262546 0.964920i $$-0.584562\pi$$
−0.262546 + 0.964920i $$0.584562\pi$$
$$194$$ 904313. 1.72510
$$195$$ 0 0
$$196$$ 1.11118e6 2.06606
$$197$$ −76788.1 −0.140971 −0.0704853 0.997513i $$-0.522455\pi$$
−0.0704853 + 0.997513i $$0.522455\pi$$
$$198$$ 0 0
$$199$$ −694776. −1.24369 −0.621845 0.783140i $$-0.713617\pi$$
−0.621845 + 0.783140i $$0.713617\pi$$
$$200$$ 11039.1 0.0195146
$$201$$ 0 0
$$202$$ −84935.3 −0.146457
$$203$$ −171223. −0.291624
$$204$$ 0 0
$$205$$ 338565. 0.562675
$$206$$ −91365.3 −0.150008
$$207$$ 0 0
$$208$$ −27792.6 −0.0445422
$$209$$ −1.28143e6 −2.02923
$$210$$ 0 0
$$211$$ −433533. −0.670372 −0.335186 0.942152i $$-0.608799\pi$$
−0.335186 + 0.942152i $$0.608799\pi$$
$$212$$ 30685.5 0.0468914
$$213$$ 0 0
$$214$$ −316074. −0.471796
$$215$$ −117182. −0.172887
$$216$$ 0 0
$$217$$ −1.86732e6 −2.69196
$$218$$ −118372. −0.168698
$$219$$ 0 0
$$220$$ 1.34276e6 1.87043
$$221$$ 49803.8 0.0685933
$$222$$ 0 0
$$223$$ 1.04204e6 1.40320 0.701602 0.712569i $$-0.252469\pi$$
0.701602 + 0.712569i $$0.252469\pi$$
$$224$$ −1.84834e6 −2.46128
$$225$$ 0 0
$$226$$ 6954.62 0.00905737
$$227$$ −924970. −1.19141 −0.595707 0.803202i $$-0.703128\pi$$
−0.595707 + 0.803202i $$0.703128\pi$$
$$228$$ 0 0
$$229$$ −137140. −0.172812 −0.0864062 0.996260i $$-0.527538\pi$$
−0.0864062 + 0.996260i $$0.527538\pi$$
$$230$$ −1.52927e6 −1.90618
$$231$$ 0 0
$$232$$ −9487.13 −0.0115722
$$233$$ −428602. −0.517208 −0.258604 0.965983i $$-0.583262\pi$$
−0.258604 + 0.965983i $$0.583262\pi$$
$$234$$ 0 0
$$235$$ 398131. 0.470280
$$236$$ 491027. 0.573886
$$237$$ 0 0
$$238$$ 3.15348e6 3.60868
$$239$$ 1.17134e6 1.32645 0.663223 0.748422i $$-0.269188\pi$$
0.663223 + 0.748422i $$0.269188\pi$$
$$240$$ 0 0
$$241$$ 119252. 0.132258 0.0661291 0.997811i $$-0.478935\pi$$
0.0661291 + 0.997811i $$0.478935\pi$$
$$242$$ 1.92858e6 2.11690
$$243$$ 0 0
$$244$$ −715018. −0.768852
$$245$$ 2.10028e6 2.23543
$$246$$ 0 0
$$247$$ −57942.4 −0.0604302
$$248$$ −103464. −0.106822
$$249$$ 0 0
$$250$$ −1.14586e6 −1.15953
$$251$$ 141272. 0.141538 0.0707690 0.997493i $$-0.477455\pi$$
0.0707690 + 0.997493i $$0.477455\pi$$
$$252$$ 0 0
$$253$$ −1.88360e6 −1.85006
$$254$$ 1.56555e6 1.52259
$$255$$ 0 0
$$256$$ 941257. 0.897652
$$257$$ 196749. 0.185815 0.0929074 0.995675i $$-0.470384\pi$$
0.0929074 + 0.995675i $$0.470384\pi$$
$$258$$ 0 0
$$259$$ 3.32833e6 3.08302
$$260$$ 60715.4 0.0557013
$$261$$ 0 0
$$262$$ 2.18241e6 1.96419
$$263$$ 1.96879e6 1.75513 0.877565 0.479457i $$-0.159166\pi$$
0.877565 + 0.479457i $$0.159166\pi$$
$$264$$ 0 0
$$265$$ 57999.6 0.0507353
$$266$$ −3.66880e6 −3.17922
$$267$$ 0 0
$$268$$ −431492. −0.366974
$$269$$ 531007. 0.447424 0.223712 0.974655i $$-0.428182\pi$$
0.223712 + 0.974655i $$0.428182\pi$$
$$270$$ 0 0
$$271$$ −1.68769e6 −1.39595 −0.697975 0.716122i $$-0.745916\pi$$
−0.697975 + 0.716122i $$0.745916\pi$$
$$272$$ −1.69550e6 −1.38956
$$273$$ 0 0
$$274$$ −1.20606e6 −0.970489
$$275$$ 563316. 0.449180
$$276$$ 0 0
$$277$$ 13630.1 0.0106733 0.00533665 0.999986i $$-0.498301\pi$$
0.00533665 + 0.999986i $$0.498301\pi$$
$$278$$ −132399. −0.102748
$$279$$ 0 0
$$280$$ 175390. 0.133693
$$281$$ −1.12070e6 −0.846692 −0.423346 0.905968i $$-0.639145\pi$$
−0.423346 + 0.905968i $$0.639145\pi$$
$$282$$ 0 0
$$283$$ −1.52947e6 −1.13521 −0.567605 0.823301i $$-0.692130\pi$$
−0.567605 + 0.823301i $$0.692130\pi$$
$$284$$ −1.89291e6 −1.39262
$$285$$ 0 0
$$286$$ 146154. 0.105657
$$287$$ 1.19392e6 0.855600
$$288$$ 0 0
$$289$$ 1.61845e6 1.13987
$$290$$ −393051. −0.274444
$$291$$ 0 0
$$292$$ −858070. −0.588932
$$293$$ 1.58484e6 1.07849 0.539246 0.842148i $$-0.318709\pi$$
0.539246 + 0.842148i $$0.318709\pi$$
$$294$$ 0 0
$$295$$ 928107. 0.620930
$$296$$ 184416. 0.122340
$$297$$ 0 0
$$298$$ 1.21860e6 0.794912
$$299$$ −85170.4 −0.0550948
$$300$$ 0 0
$$301$$ −413230. −0.262891
$$302$$ 3.46582e6 2.18670
$$303$$ 0 0
$$304$$ 1.97257e6 1.22419
$$305$$ −1.35148e6 −0.831879
$$306$$ 0 0
$$307$$ 695751. 0.421316 0.210658 0.977560i $$-0.432439\pi$$
0.210658 + 0.977560i $$0.432439\pi$$
$$308$$ 4.73512e6 2.84416
$$309$$ 0 0
$$310$$ −4.28652e6 −2.53338
$$311$$ 988763. 0.579684 0.289842 0.957075i $$-0.406397\pi$$
0.289842 + 0.957075i $$0.406397\pi$$
$$312$$ 0 0
$$313$$ −3.07551e6 −1.77442 −0.887208 0.461369i $$-0.847358\pi$$
−0.887208 + 0.461369i $$0.847358\pi$$
$$314$$ −3.51479e6 −2.01176
$$315$$ 0 0
$$316$$ 194309. 0.109465
$$317$$ −904527. −0.505561 −0.252780 0.967524i $$-0.581345\pi$$
−0.252780 + 0.967524i $$0.581345\pi$$
$$318$$ 0 0
$$319$$ −484120. −0.266365
$$320$$ −2.27027e6 −1.23937
$$321$$ 0 0
$$322$$ −5.39283e6 −2.89852
$$323$$ −3.53481e6 −1.88521
$$324$$ 0 0
$$325$$ 25471.4 0.0133766
$$326$$ −4.62899e6 −2.41236
$$327$$ 0 0
$$328$$ 66152.7 0.0339518
$$329$$ 1.40398e6 0.715105
$$330$$ 0 0
$$331$$ −2.09309e6 −1.05007 −0.525034 0.851081i $$-0.675947\pi$$
−0.525034 + 0.851081i $$0.675947\pi$$
$$332$$ 263451. 0.131176
$$333$$ 0 0
$$334$$ −4.90353e6 −2.40515
$$335$$ −815577. −0.397057
$$336$$ 0 0
$$337$$ −647484. −0.310566 −0.155283 0.987870i $$-0.549629\pi$$
−0.155283 + 0.987870i $$0.549629\pi$$
$$338$$ −2.99902e6 −1.42787
$$339$$ 0 0
$$340$$ 3.70397e6 1.73768
$$341$$ −5.27970e6 −2.45880
$$342$$ 0 0
$$343$$ 3.65027e6 1.67529
$$344$$ −22896.2 −0.0104320
$$345$$ 0 0
$$346$$ 690012. 0.309860
$$347$$ 462018. 0.205985 0.102992 0.994682i $$-0.467158\pi$$
0.102992 + 0.994682i $$0.467158\pi$$
$$348$$ 0 0
$$349$$ 2.85075e6 1.25284 0.626420 0.779486i $$-0.284520\pi$$
0.626420 + 0.779486i $$0.284520\pi$$
$$350$$ 1.61280e6 0.703737
$$351$$ 0 0
$$352$$ −5.22602e6 −2.24810
$$353$$ 809444. 0.345740 0.172870 0.984945i $$-0.444696\pi$$
0.172870 + 0.984945i $$0.444696\pi$$
$$354$$ 0 0
$$355$$ −3.57785e6 −1.50678
$$356$$ 253488. 0.106007
$$357$$ 0 0
$$358$$ −630942. −0.260185
$$359$$ −3.46118e6 −1.41739 −0.708694 0.705516i $$-0.750715\pi$$
−0.708694 + 0.705516i $$0.750715\pi$$
$$360$$ 0 0
$$361$$ 1.63635e6 0.660856
$$362$$ −2.81433e6 −1.12877
$$363$$ 0 0
$$364$$ 214107. 0.0846990
$$365$$ −1.62187e6 −0.637210
$$366$$ 0 0
$$367$$ 2.13602e6 0.827827 0.413914 0.910316i $$-0.364162\pi$$
0.413914 + 0.910316i $$0.364162\pi$$
$$368$$ 2.89951e6 1.11611
$$369$$ 0 0
$$370$$ 7.64033e6 2.90140
$$371$$ 204531. 0.0771478
$$372$$ 0 0
$$373$$ 2.37193e6 0.882734 0.441367 0.897327i $$-0.354494\pi$$
0.441367 + 0.897327i $$0.354494\pi$$
$$374$$ 8.91622e6 3.29611
$$375$$ 0 0
$$376$$ 77791.4 0.0283767
$$377$$ −21890.4 −0.00793233
$$378$$ 0 0
$$379$$ −5.47562e6 −1.95810 −0.979051 0.203616i $$-0.934731\pi$$
−0.979051 + 0.203616i $$0.934731\pi$$
$$380$$ −4.30925e6 −1.53089
$$381$$ 0 0
$$382$$ 1.55557e6 0.545419
$$383$$ −3.16966e6 −1.10412 −0.552059 0.833805i $$-0.686158\pi$$
−0.552059 + 0.833805i $$0.686158\pi$$
$$384$$ 0 0
$$385$$ 8.95001e6 3.07731
$$386$$ −2.19962e6 −0.751413
$$387$$ 0 0
$$388$$ 3.74567e6 1.26314
$$389$$ 311235. 0.104283 0.0521416 0.998640i $$-0.483395\pi$$
0.0521416 + 0.998640i $$0.483395\pi$$
$$390$$ 0 0
$$391$$ −5.19587e6 −1.71876
$$392$$ 410376. 0.134886
$$393$$ 0 0
$$394$$ −621603. −0.201731
$$395$$ 367270. 0.118438
$$396$$ 0 0
$$397$$ −496314. −0.158045 −0.0790224 0.996873i $$-0.525180\pi$$
−0.0790224 + 0.996873i $$0.525180\pi$$
$$398$$ −5.62424e6 −1.77974
$$399$$ 0 0
$$400$$ −867140. −0.270981
$$401$$ 5.63670e6 1.75051 0.875253 0.483665i $$-0.160695\pi$$
0.875253 + 0.483665i $$0.160695\pi$$
$$402$$ 0 0
$$403$$ −238732. −0.0732230
$$404$$ −351802. −0.107237
$$405$$ 0 0
$$406$$ −1.38606e6 −0.417318
$$407$$ 9.41059e6 2.81599
$$408$$ 0 0
$$409$$ −720515. −0.212978 −0.106489 0.994314i $$-0.533961\pi$$
−0.106489 + 0.994314i $$0.533961\pi$$
$$410$$ 2.74070e6 0.805197
$$411$$ 0 0
$$412$$ −378435. −0.109837
$$413$$ 3.27289e6 0.944182
$$414$$ 0 0
$$415$$ 497958. 0.141929
$$416$$ −236305. −0.0669482
$$417$$ 0 0
$$418$$ −1.03733e7 −2.90385
$$419$$ 3.94151e6 1.09680 0.548400 0.836216i $$-0.315237\pi$$
0.548400 + 0.836216i $$0.315237\pi$$
$$420$$ 0 0
$$421$$ 4.39506e6 1.20854 0.604268 0.796781i $$-0.293466\pi$$
0.604268 + 0.796781i $$0.293466\pi$$
$$422$$ −3.50947e6 −0.959312
$$423$$ 0 0
$$424$$ 11332.6 0.00306137
$$425$$ 1.55390e6 0.417301
$$426$$ 0 0
$$427$$ −4.76588e6 −1.26495
$$428$$ −1.30918e6 −0.345454
$$429$$ 0 0
$$430$$ −948589. −0.247404
$$431$$ −1.24913e6 −0.323903 −0.161951 0.986799i $$-0.551779\pi$$
−0.161951 + 0.986799i $$0.551779\pi$$
$$432$$ 0 0
$$433$$ 971902. 0.249117 0.124558 0.992212i $$-0.460249\pi$$
0.124558 + 0.992212i $$0.460249\pi$$
$$434$$ −1.51160e7 −3.85224
$$435$$ 0 0
$$436$$ −490299. −0.123522
$$437$$ 6.04494e6 1.51422
$$438$$ 0 0
$$439$$ −1.32453e6 −0.328021 −0.164010 0.986459i $$-0.552443\pi$$
−0.164010 + 0.986459i $$0.552443\pi$$
$$440$$ 495902. 0.122114
$$441$$ 0 0
$$442$$ 403164. 0.0981581
$$443$$ −4.76446e6 −1.15346 −0.576732 0.816933i $$-0.695672\pi$$
−0.576732 + 0.816933i $$0.695672\pi$$
$$444$$ 0 0
$$445$$ 479127. 0.114697
$$446$$ 8.43533e6 2.00801
$$447$$ 0 0
$$448$$ −8.00589e6 −1.88458
$$449$$ 5.40007e6 1.26411 0.632053 0.774926i $$-0.282213\pi$$
0.632053 + 0.774926i $$0.282213\pi$$
$$450$$ 0 0
$$451$$ 3.37572e6 0.781493
$$452$$ 28806.1 0.00663190
$$453$$ 0 0
$$454$$ −7.48767e6 −1.70493
$$455$$ 404692. 0.0916422
$$456$$ 0 0
$$457$$ 2.12752e6 0.476523 0.238261 0.971201i $$-0.423423\pi$$
0.238261 + 0.971201i $$0.423423\pi$$
$$458$$ −1.11015e6 −0.247297
$$459$$ 0 0
$$460$$ −6.33423e6 −1.39572
$$461$$ −2.80905e6 −0.615612 −0.307806 0.951449i $$-0.599595\pi$$
−0.307806 + 0.951449i $$0.599595\pi$$
$$462$$ 0 0
$$463$$ −2.85342e6 −0.618605 −0.309302 0.950964i $$-0.600096\pi$$
−0.309302 + 0.950964i $$0.600096\pi$$
$$464$$ 745230. 0.160692
$$465$$ 0 0
$$466$$ −3.46955e6 −0.740132
$$467$$ 559463. 0.118708 0.0593539 0.998237i $$-0.481096\pi$$
0.0593539 + 0.998237i $$0.481096\pi$$
$$468$$ 0 0
$$469$$ −2.87606e6 −0.603762
$$470$$ 3.22289e6 0.672978
$$471$$ 0 0
$$472$$ 181344. 0.0374669
$$473$$ −1.16838e6 −0.240121
$$474$$ 0 0
$$475$$ −1.80782e6 −0.367640
$$476$$ 1.30617e7 2.64231
$$477$$ 0 0
$$478$$ 9.48208e6 1.89816
$$479$$ −5.46747e6 −1.08880 −0.544399 0.838826i $$-0.683242\pi$$
−0.544399 + 0.838826i $$0.683242\pi$$
$$480$$ 0 0
$$481$$ 425517. 0.0838600
$$482$$ 965349. 0.189263
$$483$$ 0 0
$$484$$ 7.98819e6 1.55001
$$485$$ 7.07981e6 1.36668
$$486$$ 0 0
$$487$$ 9.34759e6 1.78598 0.892991 0.450074i $$-0.148603\pi$$
0.892991 + 0.450074i $$0.148603\pi$$
$$488$$ −264067. −0.0501955
$$489$$ 0 0
$$490$$ 1.70018e7 3.19893
$$491$$ 1.04964e7 1.96488 0.982438 0.186589i $$-0.0597433\pi$$
0.982438 + 0.186589i $$0.0597433\pi$$
$$492$$ 0 0
$$493$$ −1.33544e6 −0.247460
$$494$$ −469046. −0.0864766
$$495$$ 0 0
$$496$$ 8.12730e6 1.48335
$$497$$ −1.26170e7 −2.29120
$$498$$ 0 0
$$499$$ 5.74842e6 1.03347 0.516734 0.856146i $$-0.327148\pi$$
0.516734 + 0.856146i $$0.327148\pi$$
$$500$$ −4.74618e6 −0.849022
$$501$$ 0 0
$$502$$ 1.14361e6 0.202543
$$503$$ −3.10341e6 −0.546915 −0.273457 0.961884i $$-0.588167\pi$$
−0.273457 + 0.961884i $$0.588167\pi$$
$$504$$ 0 0
$$505$$ −664953. −0.116028
$$506$$ −1.52478e7 −2.64747
$$507$$ 0 0
$$508$$ 6.48453e6 1.11486
$$509$$ 7.92714e6 1.35620 0.678098 0.734972i $$-0.262805\pi$$
0.678098 + 0.734972i $$0.262805\pi$$
$$510$$ 0 0
$$511$$ −5.71937e6 −0.968938
$$512$$ 8.43012e6 1.42121
$$513$$ 0 0
$$514$$ 1.59269e6 0.265904
$$515$$ −715293. −0.118841
$$516$$ 0 0
$$517$$ 3.96963e6 0.653166
$$518$$ 2.69429e7 4.41185
$$519$$ 0 0
$$520$$ 22423.1 0.00363654
$$521$$ 4.37136e6 0.705541 0.352771 0.935710i $$-0.385240\pi$$
0.352771 + 0.935710i $$0.385240\pi$$
$$522$$ 0 0
$$523$$ 1.11680e6 0.178534 0.0892671 0.996008i $$-0.471548\pi$$
0.0892671 + 0.996008i $$0.471548\pi$$
$$524$$ 9.03957e6 1.43820
$$525$$ 0 0
$$526$$ 1.59374e7 2.51162
$$527$$ −1.45640e7 −2.28430
$$528$$ 0 0
$$529$$ 2.44920e6 0.380527
$$530$$ 469509. 0.0726030
$$531$$ 0 0
$$532$$ −1.51962e7 −2.32786
$$533$$ 152640. 0.0232728
$$534$$ 0 0
$$535$$ −2.47453e6 −0.373773
$$536$$ −159357. −0.0239584
$$537$$ 0 0
$$538$$ 4.29852e6 0.640271
$$539$$ 2.09411e7 3.10476
$$540$$ 0 0
$$541$$ −6212.12 −0.000912528 0 −0.000456264 1.00000i $$-0.500145\pi$$
−0.000456264 1.00000i $$0.500145\pi$$
$$542$$ −1.36619e7 −1.99763
$$543$$ 0 0
$$544$$ −1.44159e7 −2.08855
$$545$$ −926730. −0.133648
$$546$$ 0 0
$$547$$ −9.82839e6 −1.40448 −0.702238 0.711942i $$-0.747816\pi$$
−0.702238 + 0.711942i $$0.747816\pi$$
$$548$$ −4.99549e6 −0.710602
$$549$$ 0 0
$$550$$ 4.56007e6 0.642783
$$551$$ 1.55367e6 0.218011
$$552$$ 0 0
$$553$$ 1.29515e6 0.180097
$$554$$ 110336. 0.0152737
$$555$$ 0 0
$$556$$ −548398. −0.0752332
$$557$$ 6.27045e6 0.856369 0.428184 0.903691i $$-0.359153\pi$$
0.428184 + 0.903691i $$0.359153\pi$$
$$558$$ 0 0
$$559$$ −52830.4 −0.00715079
$$560$$ −1.37772e7 −1.85648
$$561$$ 0 0
$$562$$ −9.07215e6 −1.21163
$$563$$ 6.71129e6 0.892350 0.446175 0.894946i $$-0.352786\pi$$
0.446175 + 0.894946i $$0.352786\pi$$
$$564$$ 0 0
$$565$$ 54447.3 0.00717555
$$566$$ −1.23812e7 −1.62450
$$567$$ 0 0
$$568$$ −699080. −0.0909193
$$569$$ 7.01697e6 0.908592 0.454296 0.890851i $$-0.349891\pi$$
0.454296 + 0.890851i $$0.349891\pi$$
$$570$$ 0 0
$$571$$ 1.61627e6 0.207455 0.103727 0.994606i $$-0.466923\pi$$
0.103727 + 0.994606i $$0.466923\pi$$
$$572$$ 605372. 0.0773628
$$573$$ 0 0
$$574$$ 9.66484e6 1.22438
$$575$$ −2.65735e6 −0.335181
$$576$$ 0 0
$$577$$ 1.08712e7 1.35937 0.679686 0.733503i $$-0.262116\pi$$
0.679686 + 0.733503i $$0.262116\pi$$
$$578$$ 1.31014e7 1.63117
$$579$$ 0 0
$$580$$ −1.62802e6 −0.200951
$$581$$ 1.75600e6 0.215817
$$582$$ 0 0
$$583$$ 578294. 0.0704656
$$584$$ −316898. −0.0384493
$$585$$ 0 0
$$586$$ 1.28294e7 1.54334
$$587$$ −8.65320e6 −1.03653 −0.518264 0.855221i $$-0.673422\pi$$
−0.518264 + 0.855221i $$0.673422\pi$$
$$588$$ 0 0
$$589$$ 1.69439e7 2.01245
$$590$$ 7.51306e6 0.888560
$$591$$ 0 0
$$592$$ −1.44862e7 −1.69883
$$593$$ −5.13574e6 −0.599745 −0.299872 0.953979i $$-0.596944\pi$$
−0.299872 + 0.953979i $$0.596944\pi$$
$$594$$ 0 0
$$595$$ 2.46884e7 2.85891
$$596$$ 5.04743e6 0.582043
$$597$$ 0 0
$$598$$ −689458. −0.0788415
$$599$$ −7.56853e6 −0.861875 −0.430937 0.902382i $$-0.641817\pi$$
−0.430937 + 0.902382i $$0.641817\pi$$
$$600$$ 0 0
$$601$$ 2.56891e6 0.290110 0.145055 0.989424i $$-0.453664\pi$$
0.145055 + 0.989424i $$0.453664\pi$$
$$602$$ −3.34512e6 −0.376201
$$603$$ 0 0
$$604$$ 1.43554e7 1.60112
$$605$$ 1.50987e7 1.67707
$$606$$ 0 0
$$607$$ 2.62172e6 0.288812 0.144406 0.989519i $$-0.453873\pi$$
0.144406 + 0.989519i $$0.453873\pi$$
$$608$$ 1.67716e7 1.83999
$$609$$ 0 0
$$610$$ −1.09403e7 −1.19043
$$611$$ 179494. 0.0194513
$$612$$ 0 0
$$613$$ 537167. 0.0577376 0.0288688 0.999583i $$-0.490810\pi$$
0.0288688 + 0.999583i $$0.490810\pi$$
$$614$$ 5.63213e6 0.602909
$$615$$ 0 0
$$616$$ 1.74875e6 0.185685
$$617$$ −1.15522e7 −1.22166 −0.610830 0.791762i $$-0.709164\pi$$
−0.610830 + 0.791762i $$0.709164\pi$$
$$618$$ 0 0
$$619$$ 2.20752e6 0.231568 0.115784 0.993274i $$-0.463062\pi$$
0.115784 + 0.993274i $$0.463062\pi$$
$$620$$ −1.77548e7 −1.85497
$$621$$ 0 0
$$622$$ 8.00408e6 0.829536
$$623$$ 1.68960e6 0.174407
$$624$$ 0 0
$$625$$ −1.17567e7 −1.20389
$$626$$ −2.48963e7 −2.53922
$$627$$ 0 0
$$628$$ −1.45583e7 −1.47303
$$629$$ 2.59589e7 2.61613
$$630$$ 0 0
$$631$$ −1.43943e7 −1.43919 −0.719594 0.694395i $$-0.755672\pi$$
−0.719594 + 0.694395i $$0.755672\pi$$
$$632$$ 71761.3 0.00714657
$$633$$ 0 0
$$634$$ −7.32219e6 −0.723465
$$635$$ 1.22566e7 1.20625
$$636$$ 0 0
$$637$$ 946893. 0.0924597
$$638$$ −3.91897e6 −0.381172
$$639$$ 0 0
$$640$$ −1.60540e6 −0.154929
$$641$$ −4.90224e6 −0.471248 −0.235624 0.971844i $$-0.575713\pi$$
−0.235624 + 0.971844i $$0.575713\pi$$
$$642$$ 0 0
$$643$$ −1.80253e7 −1.71931 −0.859656 0.510874i $$-0.829322\pi$$
−0.859656 + 0.510874i $$0.829322\pi$$
$$644$$ −2.23371e7 −2.12233
$$645$$ 0 0
$$646$$ −2.86144e7 −2.69776
$$647$$ 1.24242e7 1.16683 0.583415 0.812174i $$-0.301716\pi$$
0.583415 + 0.812174i $$0.301716\pi$$
$$648$$ 0 0
$$649$$ 9.25383e6 0.862402
$$650$$ 206192. 0.0191421
$$651$$ 0 0
$$652$$ −1.91733e7 −1.76635
$$653$$ 1.28553e7 1.17977 0.589887 0.807486i $$-0.299172\pi$$
0.589887 + 0.807486i $$0.299172\pi$$
$$654$$ 0 0
$$655$$ 1.70860e7 1.55610
$$656$$ −5.19641e6 −0.471459
$$657$$ 0 0
$$658$$ 1.13652e7 1.02333
$$659$$ 5.18218e6 0.464835 0.232417 0.972616i $$-0.425336\pi$$
0.232417 + 0.972616i $$0.425336\pi$$
$$660$$ 0 0
$$661$$ 6.47130e6 0.576087 0.288043 0.957617i $$-0.406995\pi$$
0.288043 + 0.957617i $$0.406995\pi$$
$$662$$ −1.69436e7 −1.50266
$$663$$ 0 0
$$664$$ 97296.6 0.00856402
$$665$$ −2.87229e7 −2.51868
$$666$$ 0 0
$$667$$ 2.28375e6 0.198763
$$668$$ −2.03104e7 −1.76108
$$669$$ 0 0
$$670$$ −6.60213e6 −0.568194
$$671$$ −1.34751e7 −1.15539
$$672$$ 0 0
$$673$$ 6.21171e6 0.528657 0.264328 0.964433i $$-0.414850\pi$$
0.264328 + 0.964433i $$0.414850\pi$$
$$674$$ −5.24141e6 −0.444425
$$675$$ 0 0
$$676$$ −1.24220e7 −1.04550
$$677$$ −1.85078e7 −1.55197 −0.775984 0.630752i $$-0.782747\pi$$
−0.775984 + 0.630752i $$0.782747\pi$$
$$678$$ 0 0
$$679$$ 2.49663e7 2.07817
$$680$$ 1.36793e6 0.113447
$$681$$ 0 0
$$682$$ −4.27394e7 −3.51858
$$683$$ 4.91590e6 0.403228 0.201614 0.979465i $$-0.435381\pi$$
0.201614 + 0.979465i $$0.435381\pi$$
$$684$$ 0 0
$$685$$ −9.44214e6 −0.768854
$$686$$ 2.95491e7 2.39736
$$687$$ 0 0
$$688$$ 1.79854e6 0.144860
$$689$$ 26148.7 0.00209846
$$690$$ 0 0
$$691$$ −1.07645e7 −0.857630 −0.428815 0.903392i $$-0.641069\pi$$
−0.428815 + 0.903392i $$0.641069\pi$$
$$692$$ 2.85803e6 0.226883
$$693$$ 0 0
$$694$$ 3.74005e6 0.294767
$$695$$ −1.03655e6 −0.0814004
$$696$$ 0 0
$$697$$ 9.31185e6 0.726029
$$698$$ 2.30769e7 1.79283
$$699$$ 0 0
$$700$$ 6.68023e6 0.515284
$$701$$ −9.02819e6 −0.693914 −0.346957 0.937881i $$-0.612785\pi$$
−0.346957 + 0.937881i $$0.612785\pi$$
$$702$$ 0 0
$$703$$ −3.02010e7 −2.30480
$$704$$ −2.26360e7 −1.72135
$$705$$ 0 0
$$706$$ 6.55248e6 0.494760
$$707$$ −2.34490e6 −0.176431
$$708$$ 0 0
$$709$$ 4.91059e6 0.366875 0.183438 0.983031i $$-0.441277\pi$$
0.183438 + 0.983031i $$0.441277\pi$$
$$710$$ −2.89628e7 −2.15623
$$711$$ 0 0
$$712$$ 93617.2 0.00692079
$$713$$ 2.49061e7 1.83477
$$714$$ 0 0
$$715$$ 1.14423e6 0.0837047
$$716$$ −2.61336e6 −0.190510
$$717$$ 0 0
$$718$$ −2.80184e7 −2.02830
$$719$$ −1.44751e7 −1.04424 −0.522119 0.852873i $$-0.674858\pi$$
−0.522119 + 0.852873i $$0.674858\pi$$
$$720$$ 0 0
$$721$$ −2.52242e6 −0.180709
$$722$$ 1.32463e7 0.945695
$$723$$ 0 0
$$724$$ −1.16570e7 −0.826493
$$725$$ −682989. −0.0482579
$$726$$ 0 0
$$727$$ 1.55189e7 1.08899 0.544495 0.838764i $$-0.316721\pi$$
0.544495 + 0.838764i $$0.316721\pi$$
$$728$$ 79073.2 0.00552969
$$729$$ 0 0
$$730$$ −1.31291e7 −0.911857
$$731$$ −3.22294e6 −0.223079
$$732$$ 0 0
$$733$$ 1.03870e6 0.0714050 0.0357025 0.999362i $$-0.488633\pi$$
0.0357025 + 0.999362i $$0.488633\pi$$
$$734$$ 1.72912e7 1.18463
$$735$$ 0 0
$$736$$ 2.46529e7 1.67754
$$737$$ −8.13184e6 −0.551467
$$738$$ 0 0
$$739$$ 1.33918e6 0.0902046 0.0451023 0.998982i $$-0.485639\pi$$
0.0451023 + 0.998982i $$0.485639\pi$$
$$740$$ 3.16463e7 2.12443
$$741$$ 0 0
$$742$$ 1.65568e6 0.110400
$$743$$ −1.56518e7 −1.04014 −0.520070 0.854124i $$-0.674094\pi$$
−0.520070 + 0.854124i $$0.674094\pi$$
$$744$$ 0 0
$$745$$ 9.54032e6 0.629756
$$746$$ 1.92009e7 1.26321
$$747$$ 0 0
$$748$$ 3.69310e7 2.41345
$$749$$ −8.72620e6 −0.568356
$$750$$ 0 0
$$751$$ 1.92274e7 1.24400 0.622002 0.783016i $$-0.286320\pi$$
0.622002 + 0.783016i $$0.286320\pi$$
$$752$$ −6.11065e6 −0.394042
$$753$$ 0 0
$$754$$ −177204. −0.0113513
$$755$$ 2.71337e7 1.73237
$$756$$ 0 0
$$757$$ −2.73627e7 −1.73548 −0.867739 0.497021i $$-0.834427\pi$$
−0.867739 + 0.497021i $$0.834427\pi$$
$$758$$ −4.43254e7 −2.80207
$$759$$ 0 0
$$760$$ −1.59147e6 −0.0999461
$$761$$ −5.86301e6 −0.366994 −0.183497 0.983020i $$-0.558742\pi$$
−0.183497 + 0.983020i $$0.558742\pi$$
$$762$$ 0 0
$$763$$ −3.26803e6 −0.203224
$$764$$ 6.44317e6 0.399361
$$765$$ 0 0
$$766$$ −2.56585e7 −1.58001
$$767$$ 418430. 0.0256823
$$768$$ 0 0
$$769$$ −1.36162e7 −0.830308 −0.415154 0.909751i $$-0.636272\pi$$
−0.415154 + 0.909751i $$0.636272\pi$$
$$770$$ 7.24507e7 4.40368
$$771$$ 0 0
$$772$$ −9.11083e6 −0.550192
$$773$$ 9.44787e6 0.568703 0.284351 0.958720i $$-0.408222\pi$$
0.284351 + 0.958720i $$0.408222\pi$$
$$774$$ 0 0
$$775$$ −7.44851e6 −0.445467
$$776$$ 1.38333e6 0.0824655
$$777$$ 0 0
$$778$$ 2.51946e6 0.149231
$$779$$ −1.08335e7 −0.639627
$$780$$ 0 0
$$781$$ −3.56735e7 −2.09275
$$782$$ −4.20608e7 −2.45958
$$783$$ 0 0
$$784$$ −3.22357e7 −1.87304
$$785$$ −2.75171e7 −1.59378
$$786$$ 0 0
$$787$$ 2.97678e7 1.71320 0.856602 0.515977i $$-0.172571\pi$$
0.856602 + 0.515977i $$0.172571\pi$$
$$788$$ −2.57468e6 −0.147709
$$789$$ 0 0
$$790$$ 2.97307e6 0.169487
$$791$$ 192004. 0.0109111
$$792$$ 0 0
$$793$$ −609304. −0.0344073
$$794$$ −4.01768e6 −0.226164
$$795$$ 0 0
$$796$$ −2.32956e7 −1.30314
$$797$$ 1.80924e7 1.00891 0.504453 0.863439i $$-0.331694\pi$$
0.504453 + 0.863439i $$0.331694\pi$$
$$798$$ 0 0
$$799$$ 1.09501e7 0.606810
$$800$$ −7.37279e6 −0.407293
$$801$$ 0 0
$$802$$ 4.56293e7 2.50500
$$803$$ −1.61711e7 −0.885013
$$804$$ 0 0
$$805$$ −4.22201e7 −2.29631
$$806$$ −1.93254e6 −0.104783
$$807$$ 0 0
$$808$$ −129926. −0.00700113
$$809$$ 1.57182e7 0.844366 0.422183 0.906511i $$-0.361264\pi$$
0.422183 + 0.906511i $$0.361264\pi$$
$$810$$ 0 0
$$811$$ −1.57562e6 −0.0841202 −0.0420601 0.999115i $$-0.513392\pi$$
−0.0420601 + 0.999115i $$0.513392\pi$$
$$812$$ −5.74106e6 −0.305564
$$813$$ 0 0
$$814$$ 7.61791e7 4.02972
$$815$$ −3.62401e7 −1.91115
$$816$$ 0 0
$$817$$ 3.74962e6 0.196531
$$818$$ −5.83260e6 −0.304775
$$819$$ 0 0
$$820$$ 1.13520e7 0.589573
$$821$$ −2.61911e7 −1.35611 −0.678056 0.735010i $$-0.737177\pi$$
−0.678056 + 0.735010i $$0.737177\pi$$
$$822$$ 0 0
$$823$$ −3.66910e7 −1.88825 −0.944126 0.329585i $$-0.893091\pi$$
−0.944126 + 0.329585i $$0.893091\pi$$
$$824$$ −139762. −0.00717086
$$825$$ 0 0
$$826$$ 2.64942e7 1.35114
$$827$$ −1.38402e7 −0.703685 −0.351843 0.936059i $$-0.614445\pi$$
−0.351843 + 0.936059i $$0.614445\pi$$
$$828$$ 0 0
$$829$$ −2.83219e7 −1.43132 −0.715660 0.698449i $$-0.753874\pi$$
−0.715660 + 0.698449i $$0.753874\pi$$
$$830$$ 4.03099e6 0.203103
$$831$$ 0 0
$$832$$ −1.02353e6 −0.0512617
$$833$$ 5.77657e7 2.88441
$$834$$ 0 0
$$835$$ −3.83894e7 −1.90544
$$836$$ −4.29661e7 −2.12623
$$837$$ 0 0
$$838$$ 3.19067e7 1.56954
$$839$$ −2.80231e7 −1.37439 −0.687197 0.726471i $$-0.741159\pi$$
−0.687197 + 0.726471i $$0.741159\pi$$
$$840$$ 0 0
$$841$$ −1.99242e7 −0.971383
$$842$$ 3.55782e7 1.72943
$$843$$ 0 0
$$844$$ −1.45362e7 −0.702418
$$845$$ −2.34792e7 −1.13120
$$846$$ 0 0
$$847$$ 5.32444e7 2.55015
$$848$$ −890197. −0.0425105
$$849$$ 0 0
$$850$$ 1.25789e7 0.597164
$$851$$ −4.43928e7 −2.10130
$$852$$ 0 0
$$853$$ −2.38300e7 −1.12138 −0.560689 0.828027i $$-0.689463\pi$$
−0.560689 + 0.828027i $$0.689463\pi$$
$$854$$ −3.85800e7 −1.81016
$$855$$ 0 0
$$856$$ −483501. −0.0225534
$$857$$ 3.83129e7 1.78194 0.890971 0.454061i $$-0.150025\pi$$
0.890971 + 0.454061i $$0.150025\pi$$
$$858$$ 0 0
$$859$$ 2.68650e7 1.24224 0.621118 0.783717i $$-0.286679\pi$$
0.621118 + 0.783717i $$0.286679\pi$$
$$860$$ −3.92906e6 −0.181152
$$861$$ 0 0
$$862$$ −1.01118e7 −0.463510
$$863$$ 1.63558e7 0.747558 0.373779 0.927518i $$-0.378062\pi$$
0.373779 + 0.927518i $$0.378062\pi$$
$$864$$ 0 0
$$865$$ 5.40206e6 0.245482
$$866$$ 7.86758e6 0.356489
$$867$$ 0 0
$$868$$ −6.26107e7 −2.82065
$$869$$ 3.66192e6 0.164498
$$870$$ 0 0
$$871$$ −367696. −0.0164227
$$872$$ −181075. −0.00806431
$$873$$ 0 0
$$874$$ 4.89341e7 2.16687
$$875$$ −3.16351e7 −1.39685
$$876$$ 0 0
$$877$$ −2.30604e7 −1.01243 −0.506217 0.862406i $$-0.668957\pi$$
−0.506217 + 0.862406i $$0.668957\pi$$
$$878$$ −1.07221e7 −0.469402
$$879$$ 0 0
$$880$$ −3.89539e7 −1.69568
$$881$$ 3.15095e7 1.36774 0.683868 0.729606i $$-0.260296\pi$$
0.683868 + 0.729606i $$0.260296\pi$$
$$882$$ 0 0
$$883$$ 2.86348e7 1.23593 0.617963 0.786208i $$-0.287958\pi$$
0.617963 + 0.786208i $$0.287958\pi$$
$$884$$ 1.66991e6 0.0718723
$$885$$ 0 0
$$886$$ −3.85685e7 −1.65063
$$887$$ 2.45670e7 1.04844 0.524220 0.851583i $$-0.324357\pi$$
0.524220 + 0.851583i $$0.324357\pi$$
$$888$$ 0 0
$$889$$ 4.32219e7 1.83421
$$890$$ 3.87855e6 0.164132
$$891$$ 0 0
$$892$$ 3.49392e7 1.47028
$$893$$ −1.27396e7 −0.534596
$$894$$ 0 0
$$895$$ −4.93961e6 −0.206127
$$896$$ −5.66130e6 −0.235584
$$897$$ 0 0
$$898$$ 4.37138e7 1.80895
$$899$$ 6.40134e6 0.264163
$$900$$ 0 0
$$901$$ 1.59521e6 0.0654646
$$902$$ 2.73266e7 1.11833
$$903$$ 0 0
$$904$$ 10638.5 0.000432973 0
$$905$$ −2.20332e7 −0.894245
$$906$$ 0 0
$$907$$ 1.90832e7 0.770254 0.385127 0.922864i $$-0.374158\pi$$
0.385127 + 0.922864i $$0.374158\pi$$
$$908$$ −3.10140e7 −1.24837
$$909$$ 0 0
$$910$$ 3.27600e6 0.131141
$$911$$ 3.04616e7 1.21606 0.608032 0.793912i $$-0.291959\pi$$
0.608032 + 0.793912i $$0.291959\pi$$
$$912$$ 0 0
$$913$$ 4.96497e6 0.197124
$$914$$ 1.72224e7 0.681911
$$915$$ 0 0
$$916$$ −4.59826e6 −0.181073
$$917$$ 6.02522e7 2.36619
$$918$$ 0 0
$$919$$ 2.73866e7 1.06967 0.534834 0.844957i $$-0.320374\pi$$
0.534834 + 0.844957i $$0.320374\pi$$
$$920$$ −2.33933e6 −0.0911218
$$921$$ 0 0
$$922$$ −2.27394e7 −0.880950
$$923$$ −1.61304e6 −0.0623221
$$924$$ 0 0
$$925$$ 1.32763e7 0.510179
$$926$$ −2.30986e7 −0.885232
$$927$$ 0 0
$$928$$ 6.33626e6 0.241525
$$929$$ 2.65451e7 1.00913 0.504563 0.863375i $$-0.331654\pi$$
0.504563 + 0.863375i $$0.331654\pi$$
$$930$$ 0 0
$$931$$ −6.72054e7 −2.54115
$$932$$ −1.43709e7 −0.541932
$$933$$ 0 0
$$934$$ 4.52888e6 0.169873
$$935$$ 6.98046e7 2.61129
$$936$$ 0 0
$$937$$ −2.34660e7 −0.873151 −0.436576 0.899668i $$-0.643809\pi$$
−0.436576 + 0.899668i $$0.643809\pi$$
$$938$$ −2.32818e7 −0.863993
$$939$$ 0 0
$$940$$ 1.33492e7 0.492761
$$941$$ 2.18303e7 0.803686 0.401843 0.915709i $$-0.368370\pi$$
0.401843 + 0.915709i $$0.368370\pi$$
$$942$$ 0 0
$$943$$ −1.59244e7 −0.583154
$$944$$ −1.42449e7 −0.520270
$$945$$ 0 0
$$946$$ −9.45806e6 −0.343617
$$947$$ −1.87197e7 −0.678305 −0.339152 0.940731i $$-0.610140\pi$$
−0.339152 + 0.940731i $$0.610140\pi$$
$$948$$ 0 0
$$949$$ −731206. −0.0263557
$$950$$ −1.46344e7 −0.526098
$$951$$ 0 0
$$952$$ 4.82390e6 0.172507
$$953$$ −5.37413e7 −1.91680 −0.958399 0.285432i $$-0.907863\pi$$
−0.958399 + 0.285432i $$0.907863\pi$$
$$954$$ 0 0
$$955$$ 1.21785e7 0.432099
$$956$$ 3.92748e7 1.38986
$$957$$ 0 0
$$958$$ −4.42594e7 −1.55809
$$959$$ −3.32969e7 −1.16911
$$960$$ 0 0
$$961$$ 4.11823e7 1.43847
$$962$$ 3.44458e6 0.120005
$$963$$ 0 0
$$964$$ 3.99848e6 0.138581
$$965$$ −1.72207e7 −0.595295
$$966$$ 0 0
$$967$$ 1.78136e7 0.612611 0.306306 0.951933i $$-0.400907\pi$$
0.306306 + 0.951933i $$0.400907\pi$$
$$968$$ 2.95016e6 0.101195
$$969$$ 0 0
$$970$$ 5.73114e7 1.95574
$$971$$ −3.06231e7 −1.04232 −0.521160 0.853459i $$-0.674500\pi$$
−0.521160 + 0.853459i $$0.674500\pi$$
$$972$$ 0 0
$$973$$ −3.65529e6 −0.123777
$$974$$ 7.56691e7 2.55577
$$975$$ 0 0
$$976$$ 2.07429e7 0.697021
$$977$$ 1.93935e7 0.650011 0.325005 0.945712i $$-0.394634\pi$$
0.325005 + 0.945712i $$0.394634\pi$$
$$978$$ 0 0
$$979$$ 4.77721e6 0.159301
$$980$$ 7.04217e7 2.34229
$$981$$ 0 0
$$982$$ 8.49685e7 2.81177
$$983$$ −3.41089e7 −1.12586 −0.562929 0.826505i $$-0.690325\pi$$
−0.562929 + 0.826505i $$0.690325\pi$$
$$984$$ 0 0
$$985$$ −4.86650e6 −0.159818
$$986$$ −1.08104e7 −0.354120
$$987$$ 0 0
$$988$$ −1.94279e6 −0.0633190
$$989$$ 5.51162e6 0.179179
$$990$$ 0 0
$$991$$ −1.15321e7 −0.373013 −0.186507 0.982454i $$-0.559717\pi$$
−0.186507 + 0.982454i $$0.559717\pi$$
$$992$$ 6.91017e7 2.22951
$$993$$ 0 0
$$994$$ −1.02135e8 −3.27875
$$995$$ −4.40319e7 −1.40997
$$996$$ 0 0
$$997$$ 3.45745e7 1.10159 0.550793 0.834642i $$-0.314325\pi$$
0.550793 + 0.834642i $$0.314325\pi$$
$$998$$ 4.65337e7 1.47891
$$999$$ 0 0
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 387.6.a.c.1.7 8
3.2 odd 2 43.6.a.a.1.2 8
12.11 even 2 688.6.a.e.1.2 8
15.14 odd 2 1075.6.a.a.1.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.2 8 3.2 odd 2
387.6.a.c.1.7 8 1.1 even 1 trivial
688.6.a.e.1.2 8 12.11 even 2
1075.6.a.a.1.7 8 15.14 odd 2