# Properties

 Label 1075.6.a.a.1.7 Level $1075$ Weight $6$ Character 1075.1 Self dual yes Analytic conductor $172.413$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1075 = 5^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$172.412606299$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\chi$$ $$=$$ 1075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+8.09504 q^{2} -11.1803 q^{3} +33.5297 q^{4} -90.5052 q^{6} -223.489 q^{7} +12.3830 q^{8} -118.000 q^{9} +O(q^{10})$$ $$q+8.09504 q^{2} -11.1803 q^{3} +33.5297 q^{4} -90.5052 q^{6} -223.489 q^{7} +12.3830 q^{8} -118.000 q^{9} -631.897 q^{11} -374.873 q^{12} -28.5724 q^{13} -1809.15 q^{14} -972.709 q^{16} +1743.07 q^{17} -955.217 q^{18} -2027.92 q^{19} +2498.68 q^{21} -5115.23 q^{22} -2980.86 q^{23} -138.447 q^{24} -231.295 q^{26} +4036.10 q^{27} -7493.51 q^{28} +766.139 q^{29} -8355.33 q^{31} -8270.38 q^{32} +7064.81 q^{33} +14110.3 q^{34} -3956.51 q^{36} -14892.6 q^{37} -16416.1 q^{38} +319.449 q^{39} -5342.20 q^{41} +20226.9 q^{42} +1849.00 q^{43} -21187.3 q^{44} -24130.2 q^{46} +6282.09 q^{47} +10875.2 q^{48} +33140.1 q^{49} -19488.1 q^{51} -958.024 q^{52} +915.172 q^{53} +32672.4 q^{54} -2767.47 q^{56} +22672.8 q^{57} +6201.92 q^{58} -14644.5 q^{59} -21324.9 q^{61} -67636.7 q^{62} +26371.7 q^{63} -35822.4 q^{64} +57189.9 q^{66} +12868.9 q^{67} +58444.8 q^{68} +33327.0 q^{69} +56454.6 q^{71} -1461.20 q^{72} +25591.3 q^{73} -120556. q^{74} -67995.4 q^{76} +141222. q^{77} +2585.95 q^{78} +5795.13 q^{79} -16450.9 q^{81} -43245.4 q^{82} +7857.24 q^{83} +83779.9 q^{84} +14967.7 q^{86} -8565.68 q^{87} -7824.80 q^{88} -7560.11 q^{89} +6385.60 q^{91} -99947.5 q^{92} +93415.3 q^{93} +50853.8 q^{94} +92465.6 q^{96} -111712. q^{97} +268271. q^{98} +74563.9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10})$$ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 $$8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100})$$ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.09504 1.43101 0.715507 0.698605i $$-0.246196\pi$$
0.715507 + 0.698605i $$0.246196\pi$$
$$3$$ −11.1803 −0.717218 −0.358609 0.933488i $$-0.616749\pi$$
−0.358609 + 0.933488i $$0.616749\pi$$
$$4$$ 33.5297 1.04780
$$5$$ 0 0
$$6$$ −90.5052 −1.02635
$$7$$ −223.489 −1.72389 −0.861946 0.507000i $$-0.830755\pi$$
−0.861946 + 0.507000i $$0.830755\pi$$
$$8$$ 12.3830 0.0684073
$$9$$ −118.000 −0.485598
$$10$$ 0 0
$$11$$ −631.897 −1.57458 −0.787289 0.616584i $$-0.788516\pi$$
−0.787289 + 0.616584i $$0.788516\pi$$
$$12$$ −374.873 −0.751504
$$13$$ −28.5724 −0.0468909 −0.0234454 0.999725i $$-0.507464\pi$$
−0.0234454 + 0.999725i $$0.507464\pi$$
$$14$$ −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 $$0.238861\pi$$
$$18$$ −955.217 −0.694897
$$19$$ −2027.92 −1.28874 −0.644371 0.764713i $$-0.722881\pi$$
−0.644371 + 0.764713i $$0.722881\pi$$
$$20$$ 0 0
$$21$$ 2498.68 1.23641
$$22$$ −5115.23 −2.25324
$$23$$ −2980.86 −1.17496 −0.587479 0.809239i $$-0.699880\pi$$
−0.587479 + 0.809239i $$0.699880\pi$$
$$24$$ −138.447 −0.0490630
$$25$$ 0 0
$$26$$ −231.295 −0.0671015
$$27$$ 4036.10 1.06550
$$28$$ −7493.51 −1.80630
$$29$$ 766.139 0.169166 0.0845829 0.996416i $$-0.473044\pi$$
0.0845829 + 0.996416i $$0.473044\pi$$
$$30$$ 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
$$33$$ 7064.81 1.12932
$$34$$ 14110.3 2.09333
$$35$$ 0 0
$$36$$ −3956.51 −0.508811
$$37$$ −14892.6 −1.78841 −0.894204 0.447661i $$-0.852257\pi$$
−0.894204 + 0.447661i $$0.852257\pi$$
$$38$$ −16416.1 −1.84421
$$39$$ 319.449 0.0336310
$$40$$ 0 0
$$41$$ −5342.20 −0.496319 −0.248159 0.968719i $$-0.579826\pi$$
−0.248159 + 0.968719i $$0.579826\pi$$
$$42$$ 20226.9 1.76932
$$43$$ 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 $$0.433497\pi$$
$$48$$ 10875.2 0.681294
$$49$$ 33140.1 1.97181
$$50$$ 0 0
$$51$$ −19488.1 −1.04917
$$52$$ −958.024 −0.0491324
$$53$$ 915.172 0.0447521 0.0223760 0.999750i $$-0.492877\pi$$
0.0223760 + 0.999750i $$0.492877\pi$$
$$54$$ 32672.4 1.52474
$$55$$ 0 0
$$56$$ −2767.47 −0.117927
$$57$$ 22672.8 0.924310
$$58$$ 6201.92 0.242079
$$59$$ −14644.5 −0.547704 −0.273852 0.961772i $$-0.588298\pi$$
−0.273852 + 0.961772i $$0.588298\pi$$
$$60$$ 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
$$63$$ 26371.7 0.837118
$$64$$ −35822.4 −1.09321
$$65$$ 0 0
$$66$$ 57189.9 1.61607
$$67$$ 12868.9 0.350232 0.175116 0.984548i $$-0.443970\pi$$
0.175116 + 0.984548i $$0.443970\pi$$
$$68$$ 58444.8 1.53276
$$69$$ 33327.0 0.842702
$$70$$ 0 0
$$71$$ 56454.6 1.32909 0.664544 0.747249i $$-0.268626\pi$$
0.664544 + 0.747249i $$0.268626\pi$$
$$72$$ −1461.20 −0.0332184
$$73$$ 25591.3 0.562064 0.281032 0.959698i $$-0.409323\pi$$
0.281032 + 0.959698i $$0.409323\pi$$
$$74$$ −120556. −2.55924
$$75$$ 0 0
$$76$$ −67995.4 −1.35035
$$77$$ 141222. 2.71440
$$78$$ 2585.95 0.0481265
$$79$$ 5795.13 0.104471 0.0522355 0.998635i $$-0.483365\pi$$
0.0522355 + 0.998635i $$0.483365\pi$$
$$80$$ 0 0
$$81$$ −16450.9 −0.278597
$$82$$ −43245.4 −0.710240
$$83$$ 7857.24 0.125192 0.0625958 0.998039i $$-0.480062\pi$$
0.0625958 + 0.998039i $$0.480062\pi$$
$$84$$ 83779.9 1.29551
$$85$$ 0 0
$$86$$ 14967.7 0.218228
$$87$$ −8565.68 −0.121329
$$88$$ −7824.80 −0.107713
$$89$$ −7560.11 −0.101170 −0.0505852 0.998720i $$-0.516109\pi$$
−0.0505852 + 0.998720i $$0.516109\pi$$
$$90$$ 0 0
$$91$$ 6385.60 0.0808348
$$92$$ −99947.5 −1.23113
$$93$$ 93415.3 1.11998
$$94$$ 50853.8 0.593613
$$95$$ 0 0
$$96$$ 92465.6 1.02400
$$97$$ −111712. −1.20551 −0.602754 0.797927i $$-0.705930\pi$$
−0.602754 + 0.797927i $$0.705930\pi$$
$$98$$ 268271. 2.82168
$$99$$ 74563.9 0.764611
$$100$$ 0 0
$$101$$ 10492.3 0.102345 0.0511724 0.998690i $$-0.483704\pi$$
0.0511724 + 0.998690i $$0.483704\pi$$
$$102$$ −157757. −1.50137
$$103$$ 11286.6 0.104826 0.0524130 0.998625i $$-0.483309\pi$$
0.0524130 + 0.998625i $$0.483309\pi$$
$$104$$ −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 $$0.552713\pi$$
$$108$$ 135329. 1.11643
$$109$$ −14622.8 −0.117887 −0.0589433 0.998261i $$-0.518773\pi$$
−0.0589433 + 0.998261i $$0.518773\pi$$
$$110$$ 0 0
$$111$$ 166504. 1.28268
$$112$$ 217389. 1.63755
$$113$$ 859.121 0.00632934 0.00316467 0.999995i $$-0.498993\pi$$
0.00316467 + 0.999995i $$0.498993\pi$$
$$114$$ 183537. 1.32270
$$115$$ 0 0
$$116$$ 25688.4 0.177252
$$117$$ 3371.55 0.0227701
$$118$$ −118548. −0.783772
$$119$$ −389557. −2.52176
$$120$$ 0 0
$$121$$ 238242. 1.47930
$$122$$ −172626. −1.05004
$$123$$ 59727.6 0.355969
$$124$$ −280152. −1.63621
$$125$$ 0 0
$$126$$ 213480. 1.19793
$$127$$ −193396. −1.06399 −0.531997 0.846746i $$-0.678558\pi$$
−0.531997 + 0.846746i $$0.678558\pi$$
$$128$$ −25331.5 −0.136658
$$129$$ −20672.4 −0.109375
$$130$$ 0 0
$$131$$ −269599. −1.37259 −0.686293 0.727325i $$-0.740763\pi$$
−0.686293 + 0.727325i $$0.740763\pi$$
$$132$$ 236881. 1.18330
$$133$$ 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 $$0.610120\pi$$
$$138$$ 269784. 1.20592
$$139$$ −16355.6 −0.0718008 −0.0359004 0.999355i $$-0.511430\pi$$
−0.0359004 + 0.999355i $$0.511430\pi$$
$$140$$ 0 0
$$141$$ −70235.8 −0.297516
$$142$$ 457003. 1.90194
$$143$$ 18054.8 0.0738333
$$144$$ 114780. 0.461275
$$145$$ 0 0
$$146$$ 207163. 0.804322
$$147$$ −370518. −1.41422
$$148$$ −499345. −1.87390
$$149$$ −150536. −0.555489 −0.277744 0.960655i $$-0.589587\pi$$
−0.277744 + 0.960655i $$0.589587\pi$$
$$150$$ 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
$$153$$ −205683. −0.710346
$$154$$ 1.14320e6 3.88435
$$155$$ 0 0
$$156$$ 10711.0 0.0352387
$$157$$ 434190. 1.40582 0.702912 0.711277i $$-0.251883\pi$$
0.702912 + 0.711277i $$0.251883\pi$$
$$158$$ 46911.8 0.149499
$$159$$ −10231.9 −0.0320970
$$160$$ 0 0
$$161$$ 666189. 2.02550
$$162$$ −133171. −0.398677
$$163$$ 571830. 1.68577 0.842884 0.538095i $$-0.180856\pi$$
0.842884 + 0.538095i $$0.180856\pi$$
$$164$$ −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 $$0.817660\pi$$
$$168$$ 30941.2 0.0845793
$$169$$ −370477. −0.997801
$$170$$ 0 0
$$171$$ 239295. 0.625810
$$172$$ 61996.4 0.159789
$$173$$ 85238.8 0.216532 0.108266 0.994122i $$-0.465470\pi$$
0.108266 + 0.994122i $$0.465470\pi$$
$$174$$ −69339.6 −0.173623
$$175$$ 0 0
$$176$$ 614652. 1.49571
$$177$$ 163731. 0.392823
$$178$$ −61199.4 −0.144776
$$179$$ 77941.8 0.181818 0.0909092 0.995859i $$-0.471023\pi$$
0.0909092 + 0.995859i $$0.471023\pi$$
$$180$$ 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
$$183$$ 238420. 0.526277
$$184$$ −36912.2 −0.0803757
$$185$$ 0 0
$$186$$ 756201. 1.60271
$$187$$ −1.10144e6 −2.30334
$$188$$ 210637. 0.434650
$$189$$ −902023. −1.83680
$$190$$ 0 0
$$191$$ −192163. −0.381142 −0.190571 0.981673i $$-0.561034\pi$$
−0.190571 + 0.981673i $$0.561034\pi$$
$$192$$ 400506. 0.784072
$$193$$ 271724. 0.525091 0.262546 0.964920i $$-0.415438\pi$$
0.262546 + 0.964920i $$0.415438\pi$$
$$194$$ −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 $$0.522455\pi$$
$$198$$ 603598. 1.09417
$$199$$ −694776. −1.24369 −0.621845 0.783140i $$-0.713617\pi$$
−0.621845 + 0.783140i $$0.713617\pi$$
$$200$$ 0 0
$$201$$ −143879. −0.251193
$$202$$ 84935.3 0.146457
$$203$$ −171223. −0.291624
$$204$$ −653432. −1.09932
$$205$$ 0 0
$$206$$ 91365.3 0.150008
$$207$$ 351743. 0.570557
$$208$$ 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
$$213$$ −631181. −0.953246
$$214$$ −316074. −0.471796
$$215$$ 0 0
$$216$$ 49979.2 0.0728878
$$217$$ 1.86732e6 2.69196
$$218$$ −118372. −0.168698
$$219$$ −286119. −0.403123
$$220$$ 0 0
$$221$$ −49803.8 −0.0685933
$$222$$ 1.34786e6 1.83553
$$223$$ −1.04204e6 −1.40320 −0.701602 0.712569i $$-0.747531\pi$$
−0.701602 + 0.712569i $$0.747531\pi$$
$$224$$ 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 $$0.703128\pi$$
$$228$$ 760211. 0.968495
$$229$$ −137140. −0.172812 −0.0864062 0.996260i $$-0.527538\pi$$
−0.0864062 + 0.996260i $$0.527538\pi$$
$$230$$ 0 0
$$231$$ −1.57890e6 −1.94682
$$232$$ 9487.13 0.0115722
$$233$$ −428602. −0.517208 −0.258604 0.965983i $$-0.583262\pi$$
−0.258604 + 0.965983i $$0.583262\pi$$
$$234$$ 27292.8 0.0325843
$$235$$ 0 0
$$236$$ −491027. −0.573886
$$237$$ −64791.5 −0.0749285
$$238$$ −3.15348e6 −3.60868
$$239$$ −1.17134e6 −1.32645 −0.663223 0.748422i $$-0.730812\pi$$
−0.663223 + 0.748422i $$0.730812\pi$$
$$240$$ 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
$$243$$ −796846. −0.865683
$$244$$ −715018. −0.768852
$$245$$ 0 0
$$246$$ 483497. 0.509397
$$247$$ 57942.4 0.0604302
$$248$$ −103464. −0.106822
$$249$$ −87846.6 −0.0897897
$$250$$ 0 0
$$251$$ −141272. −0.141538 −0.0707690 0.997493i $$-0.522545\pi$$
−0.0707690 + 0.997493i $$0.522545\pi$$
$$252$$ 884235. 0.877135
$$253$$ 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 $$0.470384\pi$$
$$258$$ −167344. −0.156517
$$259$$ 3.32833e6 3.08302
$$260$$ 0 0
$$261$$ −90404.5 −0.0821465
$$262$$ −2.18241e6 −1.96419
$$263$$ 1.96879e6 1.75513 0.877565 0.479457i $$-0.159166\pi$$
0.877565 + 0.479457i $$0.159166\pi$$
$$264$$ 87483.9 0.0772535
$$265$$ 0 0
$$266$$ 3.66880e6 3.17922
$$267$$ 84524.6 0.0725612
$$268$$ 431492. 0.366974
$$269$$ −531007. −0.447424 −0.223712 0.974655i $$-0.571818\pi$$
−0.223712 + 0.974655i $$0.571818\pi$$
$$270$$ 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
$$273$$ −71393.2 −0.0579762
$$274$$ −1.20606e6 −0.970489
$$275$$ 0 0
$$276$$ 1.11745e6 0.882986
$$277$$ −13630.1 −0.0106733 −0.00533665 0.999986i $$-0.501699\pi$$
−0.00533665 + 0.999986i $$0.501699\pi$$
$$278$$ −132399. −0.102748
$$279$$ 985930. 0.758291
$$280$$ 0 0
$$281$$ 1.12070e6 0.846692 0.423346 0.905968i $$-0.360855\pi$$
0.423346 + 0.905968i $$0.360855\pi$$
$$282$$ −568562. −0.425750
$$283$$ 1.52947e6 1.13521 0.567605 0.823301i $$-0.307870\pi$$
0.567605 + 0.823301i $$0.307870\pi$$
$$284$$ 1.89291e6 1.39262
$$285$$ 0 0
$$286$$ 146154. 0.105657
$$287$$ 1.19392e6 0.855600
$$288$$ 975907. 0.693309
$$289$$ 1.61845e6 1.13987
$$290$$ 0 0
$$291$$ 1.24898e6 0.864613
$$292$$ 858070. 0.588932
$$293$$ 1.58484e6 1.07849 0.539246 0.842148i $$-0.318709\pi$$
0.539246 + 0.842148i $$0.318709\pi$$
$$294$$ −2.99936e6 −2.02376
$$295$$ 0 0
$$296$$ −184416. −0.122340
$$297$$ −2.55040e6 −1.67771
$$298$$ −1.21860e6 −0.794912
$$299$$ 85170.4 0.0550948
$$300$$ 0 0
$$301$$ −413230. −0.262891
$$302$$ 3.46582e6 2.18670
$$303$$ −117307. −0.0734035
$$304$$ 1.97257e6 1.22419
$$305$$ 0 0
$$306$$ −1.66501e6 −1.01652
$$307$$ −695751. −0.421316 −0.210658 0.977560i $$-0.567561\pi$$
−0.210658 + 0.977560i $$0.567561\pi$$
$$308$$ 4.73512e6 2.84416
$$309$$ −126188. −0.0751831
$$310$$ 0 0
$$311$$ −988763. −0.579684 −0.289842 0.957075i $$-0.593603\pi$$
−0.289842 + 0.957075i $$0.593603\pi$$
$$312$$ 3955.75 0.00230061
$$313$$ 3.07551e6 1.77442 0.887208 0.461369i $$-0.152642\pi$$
0.887208 + 0.461369i $$0.152642\pi$$
$$314$$ 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 $$0.581345\pi$$
$$318$$ −82827.9 −0.0459313
$$319$$ −484120. −0.266365
$$320$$ 0 0
$$321$$ 436540. 0.236462
$$322$$ 5.39283e6 2.89852
$$323$$ −3.53481e6 −1.88521
$$324$$ −551594. −0.291915
$$325$$ 0 0
$$326$$ 4.62899e6 2.41236
$$327$$ 163488. 0.0845505
$$328$$ −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
$$333$$ 1.75733e6 0.868446
$$334$$ −4.90353e6 −2.40515
$$335$$ 0 0
$$336$$ −2.43049e6 −1.17448
$$337$$ 647484. 0.310566 0.155283 0.987870i $$-0.450371\pi$$
0.155283 + 0.987870i $$0.450371\pi$$
$$338$$ −2.99902e6 −1.42787
$$339$$ −9605.25 −0.00453952
$$340$$ 0 0
$$341$$ 5.27970e6 2.45880
$$342$$ 1.93710e6 0.895544
$$343$$ −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 $$0.467158\pi$$
$$348$$ −287205. −0.127129
$$349$$ 2.85075e6 1.25284 0.626420 0.779486i $$-0.284520\pi$$
0.626420 + 0.779486i $$0.284520\pi$$
$$350$$ 0 0
$$351$$ −115321. −0.0499621
$$352$$ 5.22602e6 2.24810
$$353$$ 809444. 0.345740 0.172870 0.984945i $$-0.444696\pi$$
0.172870 + 0.984945i $$0.444696\pi$$
$$354$$ 1.32541e6 0.562136
$$355$$ 0 0
$$356$$ −253488. −0.106007
$$357$$ 4.35538e6 1.80865
$$358$$ 630942. 0.260185
$$359$$ 3.46118e6 1.41739 0.708694 0.705516i $$-0.249285\pi$$
0.708694 + 0.705516i $$0.249285\pi$$
$$360$$ 0 0
$$361$$ 1.63635e6 0.660856
$$362$$ −2.81433e6 −1.12877
$$363$$ −2.66363e6 −1.06098
$$364$$ 214107. 0.0846990
$$365$$ 0 0
$$366$$ 1.93002e6 0.753110
$$367$$ −2.13602e6 −0.827827 −0.413914 0.910316i $$-0.635838\pi$$
−0.413914 + 0.910316i $$0.635838\pi$$
$$368$$ 2.89951e6 1.11611
$$369$$ 630381. 0.241011
$$370$$ 0 0
$$371$$ −204531. −0.0771478
$$372$$ 3.13219e6 1.17352
$$373$$ −2.37193e6 −0.882734 −0.441367 0.897327i $$-0.645506\pi$$
−0.441367 + 0.897327i $$0.645506\pi$$
$$374$$ −8.91622e6 −3.29611
$$375$$ 0 0
$$376$$ 77791.4 0.0283767
$$377$$ −21890.4 −0.00793233
$$378$$ −7.30191e6 −2.62849
$$379$$ −5.47562e6 −1.95810 −0.979051 0.203616i $$-0.934731\pi$$
−0.979051 + 0.203616i $$0.934731\pi$$
$$380$$ 0 0
$$381$$ 2.16224e6 0.763116
$$382$$ −1.55557e6 −0.545419
$$383$$ −3.16966e6 −1.10412 −0.552059 0.833805i $$-0.686158\pi$$
−0.552059 + 0.833805i $$0.686158\pi$$
$$384$$ 283214. 0.0980138
$$385$$ 0 0
$$386$$ 2.19962e6 0.751413
$$387$$ −218182. −0.0740529
$$388$$ −3.74567e6 −1.26314
$$389$$ −311235. −0.104283 −0.0521416 0.998640i $$-0.516605\pi$$
−0.0521416 + 0.998640i $$0.516605\pi$$
$$390$$ 0 0
$$391$$ −5.19587e6 −1.71876
$$392$$ 410376. 0.134886
$$393$$ 3.01420e6 0.984444
$$394$$ −621603. −0.201731
$$395$$ 0 0
$$396$$ 2.50011e6 0.801162
$$397$$ 496314. 0.158045 0.0790224 0.996873i $$-0.474820\pi$$
0.0790224 + 0.996873i $$0.474820\pi$$
$$398$$ −5.62424e6 −1.77974
$$399$$ −5.06711e6 −1.59341
$$400$$ 0 0
$$401$$ −5.63670e6 −1.75051 −0.875253 0.483665i $$-0.839305\pi$$
−0.875253 + 0.483665i $$0.839305\pi$$
$$402$$ −1.16471e6 −0.359461
$$403$$ 238732. 0.0732230
$$404$$ 351802. 0.107237
$$405$$ 0 0
$$406$$ −1.38606e6 −0.417318
$$407$$ 9.41059e6 2.81599
$$408$$ −241323. −0.0717707
$$409$$ −720515. −0.212978 −0.106489 0.994314i $$-0.533961\pi$$
−0.106489 + 0.994314i $$0.533961\pi$$
$$410$$ 0 0
$$411$$ 1.66572e6 0.486405
$$412$$ 378435. 0.109837
$$413$$ 3.27289e6 0.944182
$$414$$ 2.84737e6 0.816476
$$415$$ 0 0
$$416$$ 236305. 0.0669482
$$417$$ 182861. 0.0514969
$$418$$ 1.03733e7 2.90385
$$419$$ −3.94151e6 −1.09680 −0.548400 0.836216i $$-0.684763\pi$$
−0.548400 + 0.836216i $$0.684763\pi$$
$$420$$ 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
$$423$$ −741288. −0.201436
$$424$$ 11332.6 0.00306137
$$425$$ 0 0
$$426$$ −5.10944e6 −1.36411
$$427$$ 4.76588e6 1.26495
$$428$$ −1.30918e6 −0.345454
$$429$$ −201859. −0.0529546
$$430$$ 0 0
$$431$$ 1.24913e6 0.323903 0.161951 0.986799i $$-0.448221\pi$$
0.161951 + 0.986799i $$0.448221\pi$$
$$432$$ −3.92595e6 −1.01213
$$433$$ −971902. −0.249117 −0.124558 0.992212i $$-0.539751\pi$$
−0.124558 + 0.992212i $$0.539751\pi$$
$$434$$ 1.51160e7 3.85224
$$435$$ 0 0
$$436$$ −490299. −0.123522
$$437$$ 6.04494e6 1.51422
$$438$$ −2.31615e6 −0.576874
$$439$$ −1.32453e6 −0.328021 −0.164010 0.986459i $$-0.552443\pi$$
−0.164010 + 0.986459i $$0.552443\pi$$
$$440$$ 0 0
$$441$$ −3.91054e6 −0.957504
$$442$$ −403164. −0.0981581
$$443$$ −4.76446e6 −1.15346 −0.576732 0.816933i $$-0.695672\pi$$
−0.576732 + 0.816933i $$0.695672\pi$$
$$444$$ 5.58284e6 1.34399
$$445$$ 0 0
$$446$$ −8.43533e6 −2.00801
$$447$$ 1.68304e6 0.398407
$$448$$ 8.00589e6 1.88458
$$449$$ −5.40007e6 −1.26411 −0.632053 0.774926i $$-0.717787\pi$$
−0.632053 + 0.774926i $$0.717787\pi$$
$$450$$ 0 0
$$451$$ 3.37572e6 0.781493
$$452$$ 28806.1 0.00663190
$$453$$ −4.78676e6 −1.09596
$$454$$ −7.48767e6 −1.70493
$$455$$ 0 0
$$456$$ 280758. 0.0632295
$$457$$ −2.12752e6 −0.476523 −0.238261 0.971201i $$-0.576577\pi$$
−0.238261 + 0.971201i $$0.576577\pi$$
$$458$$ −1.11015e6 −0.247297
$$459$$ 7.03522e6 1.55864
$$460$$ 0 0
$$461$$ 2.80905e6 0.615612 0.307806 0.951449i $$-0.400405\pi$$
0.307806 + 0.951449i $$0.400405\pi$$
$$462$$ −1.27813e7 −2.78593
$$463$$ 2.85342e6 0.618605 0.309302 0.950964i $$-0.399904\pi$$
0.309302 + 0.950964i $$0.399904\pi$$
$$464$$ −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 $$0.481096\pi$$
$$468$$ 113047. 0.0238586
$$469$$ −2.87606e6 −0.603762
$$470$$ 0 0
$$471$$ −4.85439e6 −1.00828
$$472$$ −181344. −0.0374669
$$473$$ −1.16838e6 −0.240121
$$474$$ −524490. −0.107224
$$475$$ 0 0
$$476$$ −1.30617e7 −2.64231
$$477$$ −107991. −0.0217315
$$478$$ −9.48208e6 −1.89816
$$479$$ 5.46747e6 1.08880 0.544399 0.838826i $$-0.316758\pi$$
0.544399 + 0.838826i $$0.316758\pi$$
$$480$$ 0 0
$$481$$ 425517. 0.0838600
$$482$$ 965349. 0.189263
$$483$$ −7.44821e6 −1.45273
$$484$$ 7.98819e6 1.55001
$$485$$ 0 0
$$486$$ −6.45050e6 −1.23881
$$487$$ −9.34759e6 −1.78598 −0.892991 0.450074i $$-0.851397\pi$$
−0.892991 + 0.450074i $$0.851397\pi$$
$$488$$ −264067. −0.0501955
$$489$$ −6.39325e6 −1.20906
$$490$$ 0 0
$$491$$ −1.04964e7 −1.96488 −0.982438 0.186589i $$-0.940257\pi$$
−0.982438 + 0.186589i $$0.940257\pi$$
$$492$$ 2.00265e6 0.372986
$$493$$ 1.33544e6 0.247460
$$494$$ 469046. 0.0864766
$$495$$ 0 0
$$496$$ 8.12730e6 1.48335
$$497$$ −1.26170e7 −2.29120
$$498$$ −711122. −0.128490
$$499$$ 5.74842e6 1.03347 0.516734 0.856146i $$-0.327148\pi$$
0.516734 + 0.856146i $$0.327148\pi$$
$$500$$ 0 0
$$501$$ 6.77242e6 1.20545
$$502$$ −1.14361e6 −0.202543
$$503$$ −3.10341e6 −0.546915 −0.273457 0.961884i $$-0.588167\pi$$
−0.273457 + 0.961884i $$0.588167\pi$$
$$504$$ 326562. 0.0572650
$$505$$ 0 0
$$506$$ 1.52478e7 2.64747
$$507$$ 4.14205e6 0.715642
$$508$$ −6.48453e6 −1.11486
$$509$$ −7.92714e6 −1.35620 −0.678098 0.734972i $$-0.737195\pi$$
−0.678098 + 0.734972i $$0.737195\pi$$
$$510$$ 0 0
$$511$$ −5.71937e6 −0.968938
$$512$$ 8.43012e6 1.42121
$$513$$ −8.18488e6 −1.37315
$$514$$ 1.59269e6 0.265904
$$515$$ 0 0
$$516$$ −693141. −0.114603
$$517$$ −3.96963e6 −0.653166
$$518$$ 2.69429e7 4.41185
$$519$$ −952998. −0.155301
$$520$$ 0 0
$$521$$ −4.37136e6 −0.705541 −0.352771 0.935710i $$-0.614760\pi$$
−0.352771 + 0.935710i $$0.614760\pi$$
$$522$$ −731828. −0.117553
$$523$$ −1.11680e6 −0.178534 −0.0892671 0.996008i $$-0.528452\pi$$
−0.0892671 + 0.996008i $$0.528452\pi$$
$$524$$ −9.03957e6 −1.43820
$$525$$ 0 0
$$526$$ 1.59374e7 2.51162
$$527$$ −1.45640e7 −2.28430
$$528$$ −6.87201e6 −1.07275
$$529$$ 2.44920e6 0.380527
$$530$$ 0 0
$$531$$ 1.72806e6 0.265964
$$532$$ 1.51962e7 2.32786
$$533$$ 152640. 0.0232728
$$534$$ 684230. 0.103836
$$535$$ 0 0
$$536$$ 159357. 0.0239584
$$537$$ −871415. −0.130403
$$538$$ −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
$$543$$ 3.88696e6 0.565732
$$544$$ −1.44159e7 −2.08855
$$545$$ 0 0
$$546$$ −577931. −0.0829648
$$547$$ 9.82839e6 1.40448 0.702238 0.711942i $$-0.252184\pi$$
0.702238 + 0.711942i $$0.252184\pi$$
$$548$$ −4.99549e6 −0.710602
$$549$$ 2.51635e6 0.356319
$$550$$ 0 0
$$551$$ −1.55367e6 −0.218011
$$552$$ 412690. 0.0576470
$$553$$ −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 $$0.359153\pi$$
$$558$$ 7.98115e6 1.08513
$$559$$ −52830.4 −0.00715079
$$560$$ 0 0
$$561$$ 1.23145e7 1.65200
$$562$$ 9.07215e6 1.21163
$$563$$ 6.71129e6 0.892350 0.446175 0.894946i $$-0.352786\pi$$
0.446175 + 0.894946i $$0.352786\pi$$
$$564$$ −2.35499e6 −0.311739
$$565$$ 0 0
$$566$$ 1.23812e7 1.62450
$$567$$ 3.67659e6 0.480272
$$568$$ 699080. 0.0909193
$$569$$ −7.01697e6 −0.908592 −0.454296 0.890851i $$-0.650109\pi$$
−0.454296 + 0.890851i $$0.650109\pi$$
$$570$$ 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
$$573$$ 2.14845e6 0.273362
$$574$$ 9.66484e6 1.22438
$$575$$ 0 0
$$576$$ 4.22705e6 0.530861
$$577$$ −1.08712e7 −1.35937 −0.679686 0.733503i $$-0.737884\pi$$
−0.679686 + 0.733503i $$0.737884\pi$$
$$578$$ 1.31014e7 1.63117
$$579$$ −3.03796e6 −0.376605
$$580$$ 0 0
$$581$$ −1.75600e6 −0.215817
$$582$$ 1.01105e7 1.23727
$$583$$ −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 $$0.673422\pi$$
$$588$$ −1.24233e7 −1.48182
$$589$$ 1.69439e7 2.01245
$$590$$ 0 0
$$591$$ 858517. 0.101107
$$592$$ 1.44862e7 1.69883
$$593$$ −5.13574e6 −0.599745 −0.299872 0.953979i $$-0.596944\pi$$
−0.299872 + 0.953979i $$0.596944\pi$$
$$594$$ −2.06456e7 −2.40083
$$595$$ 0 0
$$596$$ −5.04743e6 −0.582043
$$597$$ 7.76782e6 0.891997
$$598$$ 689458. 0.0788415
$$599$$ 7.56853e6 0.861875 0.430937 0.902382i $$-0.358183\pi$$
0.430937 + 0.902382i $$0.358183\pi$$
$$600$$ 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
$$603$$ −1.51854e6 −0.170072
$$604$$ 1.43554e7 1.60112
$$605$$ 0 0
$$606$$ −949604. −0.105042
$$607$$ −2.62172e6 −0.288812 −0.144406 0.989519i $$-0.546127\pi$$
−0.144406 + 0.989519i $$0.546127\pi$$
$$608$$ 1.67716e7 1.83999
$$609$$ 1.91433e6 0.209158
$$610$$ 0 0
$$611$$ −179494. −0.0194513
$$612$$ −6.89650e6 −0.744303
$$613$$ −537167. −0.0577376 −0.0288688 0.999583i $$-0.509190\pi$$
−0.0288688 + 0.999583i $$0.509190\pi$$
$$614$$ −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 $$0.709164\pi$$
$$618$$ −1.02149e6 −0.107588
$$619$$ 2.20752e6 0.231568 0.115784 0.993274i $$-0.463062\pi$$
0.115784 + 0.993274i $$0.463062\pi$$
$$620$$ 0 0
$$621$$ −1.20311e7 −1.25192
$$622$$ −8.00408e6 −0.829536
$$623$$ 1.68960e6 0.174407
$$624$$ −310731. −0.0319465
$$625$$ 0 0
$$626$$ 2.48963e7 2.53922
$$627$$ −1.43268e7 −1.45540
$$628$$ 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
$$633$$ 4.84704e6 0.480803
$$634$$ −7.32219e6 −0.723465
$$635$$ 0 0
$$636$$ −343074. −0.0336314
$$637$$ −946893. −0.0924597
$$638$$ −3.91897e6 −0.381172
$$639$$ −6.66166e6 −0.645402
$$640$$ 0 0
$$641$$ 4.90224e6 0.471248 0.235624 0.971844i $$-0.424287\pi$$
0.235624 + 0.971844i $$0.424287\pi$$
$$642$$ 3.53381e6 0.338381
$$643$$ 1.80253e7 1.71931 0.859656 0.510874i $$-0.170678\pi$$
0.859656 + 0.510874i $$0.170678\pi$$
$$644$$ 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 $$0.301716\pi$$
$$648$$ −203712. −0.0190581
$$649$$ 9.25383e6 0.862402
$$650$$ 0 0
$$651$$ −2.08772e7 −1.93073
$$652$$ 1.91733e7 1.76635
$$653$$ 1.28553e7 1.17977 0.589887 0.807486i $$-0.299172\pi$$
0.589887 + 0.807486i $$0.299172\pi$$
$$654$$ 1.32344e6 0.120993
$$655$$ 0 0
$$656$$ 5.19641e6 0.471459
$$657$$ −3.01978e6 −0.272937
$$658$$ −1.13652e7 −1.02333
$$659$$ −5.18218e6 −0.464835 −0.232417 0.972616i $$-0.574664\pi$$
−0.232417 + 0.972616i $$0.574664\pi$$
$$660$$ 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
$$663$$ 556823. 0.0491964
$$664$$ 97296.6 0.00856402
$$665$$ 0 0
$$666$$ 1.42257e7 1.24276
$$667$$ −2.28375e6 −0.198763
$$668$$ −2.03104e7 −1.76108
$$669$$ 1.16503e7 1.00640
$$670$$ 0 0
$$671$$ 1.34751e7 1.15539
$$672$$ −2.06650e7 −1.76527
$$673$$ −6.21171e6 −0.528657 −0.264328 0.964433i $$-0.585150\pi$$
−0.264328 + 0.964433i $$0.585150\pi$$
$$674$$ 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 $$0.782747\pi$$
$$678$$ −77754.9 −0.00649611
$$679$$ 2.49663e7 2.07817
$$680$$ 0 0
$$681$$ 1.03415e7 0.854504
$$682$$ 4.27394e7 3.51858
$$683$$ 4.91590e6 0.403228 0.201614 0.979465i $$-0.435381\pi$$
0.201614 + 0.979465i $$0.435381\pi$$
$$684$$ 8.02348e6 0.655726
$$685$$ 0 0
$$686$$ −2.95491e7 −2.39736
$$687$$ 1.53327e6 0.123944
$$688$$ −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
$$693$$ −1.66642e7 −1.31811
$$694$$ 3.74005e6 0.294767
$$695$$ 0 0
$$696$$ −106069. −0.00829977
$$697$$ −9.31185e6 −0.726029
$$698$$ 2.30769e7 1.79283
$$699$$ 4.79192e6 0.370951
$$700$$ 0 0
$$701$$ 9.02819e6 0.693914 0.346957 0.937881i $$-0.387215\pi$$
0.346957 + 0.937881i $$0.387215\pi$$
$$702$$ −933529. −0.0714966
$$703$$ 3.02010e7 2.30480
$$704$$ 2.26360e7 1.72135
$$705$$ 0 0
$$706$$ 6.55248e6 0.494760
$$707$$ −2.34490e6 −0.176431
$$708$$ 5.48984e6 0.411601
$$709$$ 4.91059e6 0.366875 0.183438 0.983031i $$-0.441277\pi$$
0.183438 + 0.983031i $$0.441277\pi$$
$$710$$ 0 0
$$711$$ −683827. −0.0507308
$$712$$ −93617.2 −0.00692079
$$713$$ 2.49061e7 1.83477
$$714$$ 3.52570e7 2.58821
$$715$$ 0 0
$$716$$ 2.61336e6 0.190510
$$717$$ 1.30960e7 0.951352
$$718$$ 2.80184e7 2.02830
$$719$$ 1.44751e7 1.04424 0.522119 0.852873i $$-0.325142\pi$$
0.522119 + 0.852873i $$0.325142\pi$$
$$720$$ 0 0
$$721$$ −2.52242e6 −0.180709
$$722$$ 1.32463e7 0.945695
$$723$$ −1.33328e6 −0.0948580
$$724$$ −1.16570e7 −0.826493
$$725$$ 0 0
$$726$$ −2.15622e7 −1.51828
$$727$$ −1.55189e7 −1.08899 −0.544495 0.838764i $$-0.683279\pi$$
−0.544495 + 0.838764i $$0.683279\pi$$
$$728$$ 79073.2 0.00552969
$$729$$ 1.29066e7 0.899481
$$730$$ 0 0
$$731$$ 3.22294e6 0.223079
$$732$$ 7.99414e6 0.551435
$$733$$ −1.03870e6 −0.0714050 −0.0357025 0.999362i $$-0.511367\pi$$
−0.0357025 + 0.999362i $$0.511367\pi$$
$$734$$ −1.72912e7 −1.18463
$$735$$ 0 0
$$736$$ 2.46529e7 1.67754
$$737$$ −8.13184e6 −0.551467
$$738$$ 5.10296e6 0.344891
$$739$$ 1.33918e6 0.0902046 0.0451023 0.998982i $$-0.485639\pi$$
0.0451023 + 0.998982i $$0.485639\pi$$
$$740$$ 0 0
$$741$$ −647816. −0.0433417
$$742$$ −1.65568e6 −0.110400
$$743$$ −1.56518e7 −1.04014 −0.520070 0.854124i $$-0.674094\pi$$
−0.520070 + 0.854124i $$0.674094\pi$$
$$744$$ 1.15677e6 0.0766149
$$745$$ 0 0
$$746$$ −1.92009e7 −1.26321
$$747$$ −927157. −0.0607927
$$748$$ −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
$$753$$ 1.57947e6 0.101514
$$754$$ −177204. −0.0113513
$$755$$ 0 0
$$756$$ −3.02446e7 −1.92461
$$757$$ 2.73627e7 1.73548 0.867739 0.497021i $$-0.165573\pi$$
0.867739 + 0.497021i $$0.165573\pi$$
$$758$$ −4.43254e7 −2.80207
$$759$$ −2.10592e7 −1.32690
$$760$$ 0 0
$$761$$ 5.86301e6 0.366994 0.183497 0.983020i $$-0.441258\pi$$
0.183497 + 0.983020i $$0.441258\pi$$
$$762$$ 1.75034e7 1.09203
$$763$$ 3.26803e6 0.203224
$$764$$ −6.44317e6 −0.399361
$$765$$ 0 0
$$766$$ −2.56585e7 −1.58001
$$767$$ 418430. 0.0256823
$$768$$ −1.05236e7 −0.643813
$$769$$ −1.36162e7 −0.830308 −0.415154 0.909751i $$-0.636272\pi$$
−0.415154 + 0.909751i $$0.636272\pi$$
$$770$$ 0 0
$$771$$ −2.19972e6 −0.133270
$$772$$ 9.11083e6 0.550192
$$773$$ 9.44787e6 0.568703 0.284351 0.958720i $$-0.408222\pi$$
0.284351 + 0.958720i $$0.408222\pi$$
$$774$$ −1.76620e6 −0.105971
$$775$$ 0 0
$$776$$ −1.38333e6 −0.0824655
$$777$$ −3.72118e7 −2.21120
$$778$$ −2.51946e6 −0.149231
$$779$$ 1.08335e7 0.639627
$$780$$ 0 0
$$781$$ −3.56735e7 −2.09275
$$782$$ −4.20608e7 −2.45958
$$783$$ 3.09221e6 0.180246
$$784$$ −3.22357e7 −1.87304
$$785$$ 0 0
$$786$$ 2.44001e7 1.40875
$$787$$ −2.97678e7 −1.71320 −0.856602 0.515977i $$-0.827429\pi$$
−0.856602 + 0.515977i $$0.827429\pi$$
$$788$$ −2.57468e6 −0.147709
$$789$$ −2.20117e7 −1.25881
$$790$$ 0 0
$$791$$ −192004. −0.0109111
$$792$$ 923328. 0.0523050
$$793$$ 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 $$0.331694\pi$$
$$798$$ −4.10184e7 −2.28019
$$799$$ 1.09501e7 0.606810
$$800$$ 0 0
$$801$$ 892095. 0.0491281
$$802$$ −4.56293e7 −2.50500
$$803$$ −1.61711e7 −0.885013
$$804$$ −4.82422e6 −0.263201
$$805$$ 0 0
$$806$$ 1.93254e6 0.104783
$$807$$ 5.93683e6 0.320901
$$808$$ 129926. 0.00700113
$$809$$ −1.57182e7 −0.844366 −0.422183 0.906511i $$-0.638736\pi$$
−0.422183 + 0.906511i $$0.638736\pi$$
$$810$$ 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
$$813$$ 1.88690e7 1.00120
$$814$$ 7.61791e7 4.02972
$$815$$ 0 0
$$816$$ 1.89563e7 0.996617
$$817$$ −3.74962e6 −0.196531
$$818$$ −5.83260e6 −0.304775
$$819$$ −753503. −0.0392532
$$820$$ 0 0
$$821$$ 2.61911e7 1.35611 0.678056 0.735010i $$-0.262823\pi$$
0.678056 + 0.735010i $$0.262823\pi$$
$$822$$ 1.34841e7 0.696053
$$823$$ 3.66910e7 1.88825 0.944126 0.329585i $$-0.106909\pi$$
0.944126 + 0.329585i $$0.106909\pi$$
$$824$$ 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 $$0.614445\pi$$
$$828$$ 1.17938e7 0.597832
$$829$$ −2.83219e7 −1.43132 −0.715660 0.698449i $$-0.753874\pi$$
−0.715660 + 0.698449i $$0.753874\pi$$
$$830$$ 0 0
$$831$$ 152389. 0.00765509
$$832$$ 1.02353e6 0.0512617
$$833$$ 5.77657e7 2.88441
$$834$$ 1.48027e6 0.0736928
$$835$$ 0 0
$$836$$ 4.29661e7 2.12623
$$837$$ −3.37229e7 −1.66384
$$838$$ −3.19067e7 −1.56954
$$839$$ 2.80231e7 1.37439 0.687197 0.726471i $$-0.258841\pi$$
0.687197 + 0.726471i $$0.258841\pi$$
$$840$$ 0 0
$$841$$ −1.99242e7 −0.971383
$$842$$ 3.55782e7 1.72943
$$843$$ −1.25298e7 −0.607263
$$844$$ −1.45362e7 −0.702418
$$845$$ 0 0
$$846$$ −6.00076e6 −0.288257
$$847$$ −5.32444e7 −2.55015
$$848$$ −890197. −0.0425105
$$849$$ −1.71000e7 −0.814193
$$850$$ 0 0
$$851$$ 4.43928e7 2.10130
$$852$$ −2.11633e7 −0.998815
$$853$$ 2.38300e7 1.12138 0.560689 0.828027i $$-0.310537\pi$$
0.560689 + 0.828027i $$0.310537\pi$$
$$854$$ 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 $$0.150025\pi$$
$$858$$ −1.63405e6 −0.0757789
$$859$$ 2.68650e7 1.24224 0.621118 0.783717i $$-0.286679\pi$$
0.621118 + 0.783717i $$0.286679\pi$$
$$860$$ 0 0
$$861$$ −1.33484e7 −0.613652
$$862$$ 1.01118e7 0.463510
$$863$$ 1.63558e7 0.747558 0.373779 0.927518i $$-0.378062\pi$$
0.373779 + 0.927518i $$0.378062\pi$$
$$864$$ −3.33801e7 −1.52126
$$865$$ 0 0
$$866$$ −7.86758e6 −0.356489
$$867$$ −1.80948e7 −0.817535
$$868$$ 6.26107e7 2.82065
$$869$$ −3.66192e6 −0.164498
$$870$$ 0 0
$$871$$ −367696. −0.0164227
$$872$$ −181075. −0.00806431
$$873$$ 1.31820e7 0.585392
$$874$$ 4.89341e7 2.16687
$$875$$ 0 0
$$876$$ −9.59350e6 −0.422393
$$877$$ 2.30604e7 1.01243 0.506217 0.862406i $$-0.331043\pi$$
0.506217 + 0.862406i $$0.331043\pi$$
$$878$$ −1.07221e7 −0.469402
$$879$$ −1.77191e7 −0.773515
$$880$$ 0 0
$$881$$ −3.15095e7 −1.36774 −0.683868 0.729606i $$-0.739704\pi$$
−0.683868 + 0.729606i $$0.739704\pi$$
$$882$$ −3.16560e7 −1.37020
$$883$$ −2.86348e7 −1.23593 −0.617963 0.786208i $$-0.712042\pi$$
−0.617963 + 0.786208i $$0.712042\pi$$
$$884$$ −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 $$0.324357\pi$$
$$888$$ 2.06183e6 0.0877446
$$889$$ 4.32219e7 1.83421
$$890$$ 0 0
$$891$$ 1.03953e7 0.438673
$$892$$ −3.49392e7 −1.47028
$$893$$ −1.27396e7 −0.534596
$$894$$ 1.36243e7 0.570126
$$895$$ 0 0
$$896$$ 5.66130e6 0.235584
$$897$$ −952233. −0.0395150
$$898$$ −4.37138e7 −1.80895
$$899$$ −6.40134e6 −0.264163
$$900$$ 0 0
$$901$$ 1.59521e6 0.0654646
$$902$$ 2.73266e7 1.11833
$$903$$ 4.62005e6 0.188550
$$904$$ 10638.5 0.000432973 0
$$905$$ 0 0
$$906$$ −3.87490e7 −1.56834
$$907$$ −1.90832e7 −0.770254 −0.385127 0.922864i $$-0.625842\pi$$
−0.385127 + 0.922864i $$0.625842\pi$$
$$908$$ −3.10140e7 −1.24837
$$909$$ −1.23809e6 −0.0496984
$$910$$ 0 0
$$911$$ −3.04616e7 −1.21606 −0.608032 0.793912i $$-0.708041\pi$$
−0.608032 + 0.793912i $$0.708041\pi$$
$$912$$ −2.20540e7 −0.878012
$$913$$ −4.96497e6 −0.197124
$$914$$ −1.72224e7 −0.681911
$$915$$ 0 0
$$916$$ −4.59826e6 −0.181073
$$917$$ 6.02522e7 2.36619
$$918$$ 5.69504e7 2.23044
$$919$$ 2.73866e7 1.06967 0.534834 0.844957i $$-0.320374\pi$$
0.534834 + 0.844957i $$0.320374\pi$$
$$920$$ 0 0
$$921$$ 7.77872e6 0.302175
$$922$$ 2.27394e7 0.880950
$$923$$ −1.61304e6 −0.0623221
$$924$$ −5.29402e7 −2.03989
$$925$$ 0 0
$$926$$ 2.30986e7 0.885232
$$927$$ −1.33182e6 −0.0509033
$$928$$ −6.33626e6 −0.241525
$$929$$ −2.65451e7 −1.00913 −0.504563 0.863375i $$-0.668346\pi$$
−0.504563 + 0.863375i $$0.668346\pi$$
$$930$$ 0 0
$$931$$ −6.72054e7 −2.54115
$$932$$ −1.43709e7 −0.541932
$$933$$ 1.10547e7 0.415760
$$934$$ 4.52888e6 0.169873
$$935$$ 0 0
$$936$$ 41750.0 0.00155764
$$937$$ 2.34660e7 0.873151 0.436576 0.899668i $$-0.356191\pi$$
0.436576 + 0.899668i $$0.356191\pi$$
$$938$$ −2.32818e7 −0.863993
$$939$$ −3.43852e7 −1.27264
$$940$$ 0 0
$$941$$ −2.18303e7 −0.803686 −0.401843 0.915709i $$-0.631630\pi$$
−0.401843 + 0.915709i $$0.631630\pi$$
$$942$$ −3.92965e7 −1.44287
$$943$$ 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 $$0.610140\pi$$
$$948$$ −2.17244e6 −0.0785103
$$949$$ −731206. −0.0263557
$$950$$ 0 0
$$951$$ 1.01129e7 0.362598
$$952$$ −4.82390e6 −0.172507
$$953$$ −5.37413e7 −1.91680 −0.958399 0.285432i $$-0.907863\pi$$
−0.958399 + 0.285432i $$0.907863\pi$$
$$954$$ −874188. −0.0310981
$$955$$ 0 0
$$956$$ −3.92748e7 −1.38986
$$957$$ 5.41262e6 0.191042
$$958$$ 4.42594e7 1.55809
$$959$$ 3.32969e7 1.16911
$$960$$ 0 0
$$961$$ 4.11823e7 1.43847
$$962$$ 3.44458e6 0.120005
$$963$$ 4.60736e6 0.160098
$$964$$ 3.99848e6 0.138581
$$965$$ 0 0
$$966$$ −6.02936e7 −2.07887
$$967$$ −1.78136e7 −0.612611 −0.306306 0.951933i $$-0.599093\pi$$
−0.306306 + 0.951933i $$0.599093\pi$$
$$968$$ 2.95016e6 0.101195
$$969$$ 3.95203e7 1.35211
$$970$$ 0 0
$$971$$ 3.06231e7 1.04232 0.521160 0.853459i $$-0.325500\pi$$
0.521160 + 0.853459i $$0.325500\pi$$
$$972$$ −2.67180e7 −0.907065
$$973$$ 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 $$0.394634\pi$$
$$978$$ −5.17536e7 −1.73019
$$979$$ 4.77721e6 0.159301
$$980$$ 0 0
$$981$$ 1.72550e6 0.0572455
$$982$$ −8.49685e7 −2.81177
$$983$$ −3.41089e7 −1.12586 −0.562929 0.826505i $$-0.690325\pi$$
−0.562929 + 0.826505i $$0.690325\pi$$
$$984$$ 739609. 0.0243509
$$985$$ 0 0
$$986$$ 1.08104e7 0.354120
$$987$$ 1.56969e7 0.512886
$$988$$ 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
$$993$$ 2.34014e7 0.753128
$$994$$ −1.02135e8 −3.27875
$$995$$ 0 0
$$996$$ −2.94547e6 −0.0940819
$$997$$ −3.45745e7 −1.10159 −0.550793 0.834642i $$-0.685675\pi$$
−0.550793 + 0.834642i $$0.685675\pi$$
$$998$$ 4.65337e7 1.47891
$$999$$ −6.01081e7 −1.90554
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 1075.6.a.a.1.7 8
5.4 even 2 43.6.a.a.1.2 8
15.14 odd 2 387.6.a.c.1.7 8
20.19 odd 2 688.6.a.e.1.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.2 8 5.4 even 2
387.6.a.c.1.7 8 15.14 odd 2
688.6.a.e.1.2 8 20.19 odd 2
1075.6.a.a.1.7 8 1.1 even 1 trivial