Properties

 Label 75.4.a.f.1.1 Level $75$ Weight $4$ Character 75.1 Self dual yes Analytic conductor $4.425$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 75.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.70156 q^{2} +3.00000 q^{3} -5.10469 q^{4} -5.10469 q^{6} +22.2094 q^{7} +22.2984 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.70156 q^{2} +3.00000 q^{3} -5.10469 q^{4} -5.10469 q^{6} +22.2094 q^{7} +22.2984 q^{8} +9.00000 q^{9} -1.79063 q^{11} -15.3141 q^{12} +58.2094 q^{13} -37.7906 q^{14} +2.89531 q^{16} +18.9844 q^{17} -15.3141 q^{18} +104.837 q^{19} +66.6281 q^{21} +3.04686 q^{22} -49.6125 q^{23} +66.8953 q^{24} -99.0469 q^{26} +27.0000 q^{27} -113.372 q^{28} -293.466 q^{29} +64.4187 q^{31} -183.314 q^{32} -5.37188 q^{33} -32.3031 q^{34} -45.9422 q^{36} -19.8844 q^{37} -178.388 q^{38} +174.628 q^{39} -165.581 q^{41} -113.372 q^{42} -247.350 q^{43} +9.14059 q^{44} +84.4187 q^{46} +384.544 q^{47} +8.68594 q^{48} +150.256 q^{49} +56.9531 q^{51} -297.141 q^{52} +463.528 q^{53} -45.9422 q^{54} +495.234 q^{56} +314.512 q^{57} +499.350 q^{58} -73.7906 q^{59} -137.350 q^{61} -109.612 q^{62} +199.884 q^{63} +288.758 q^{64} +9.14059 q^{66} +173.906 q^{67} -96.9093 q^{68} -148.837 q^{69} -594.281 q^{71} +200.686 q^{72} +320.231 q^{73} +33.8345 q^{74} -535.163 q^{76} -39.7687 q^{77} -297.141 q^{78} -770.469 q^{79} +81.0000 q^{81} +281.747 q^{82} +173.925 q^{83} -340.116 q^{84} +420.881 q^{86} -880.397 q^{87} -39.9282 q^{88} +1019.02 q^{89} +1292.79 q^{91} +253.256 q^{92} +193.256 q^{93} -654.325 q^{94} -549.942 q^{96} +384.375 q^{97} -255.670 q^{98} -16.1156 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 + 9 * q^6 + 6 * q^7 + 51 * q^8 + 18 * q^9 $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} + 9 q^{6} + 6 q^{7} + 51 q^{8} + 18 q^{9} - 42 q^{11} + 27 q^{12} + 78 q^{13} - 114 q^{14} + 25 q^{16} + 102 q^{17} + 27 q^{18} + 56 q^{19} + 18 q^{21} - 186 q^{22} - 48 q^{23} + 153 q^{24} - 6 q^{26} + 54 q^{27} - 342 q^{28} - 318 q^{29} + 52 q^{31} - 309 q^{32} - 126 q^{33} + 358 q^{34} + 81 q^{36} + 306 q^{37} - 408 q^{38} + 234 q^{39} - 408 q^{41} - 342 q^{42} + 120 q^{43} - 558 q^{44} + 92 q^{46} + 180 q^{47} + 75 q^{48} + 70 q^{49} + 306 q^{51} - 18 q^{52} + 402 q^{53} + 81 q^{54} + 30 q^{56} + 168 q^{57} + 384 q^{58} - 186 q^{59} + 340 q^{61} - 168 q^{62} + 54 q^{63} - 479 q^{64} - 558 q^{66} + 732 q^{67} + 1074 q^{68} - 144 q^{69} - 36 q^{71} + 459 q^{72} + 1332 q^{73} + 1566 q^{74} - 1224 q^{76} + 612 q^{77} - 18 q^{78} + 380 q^{79} + 162 q^{81} - 858 q^{82} - 984 q^{83} - 1026 q^{84} + 2148 q^{86} - 954 q^{87} - 1194 q^{88} + 1116 q^{89} + 972 q^{91} + 276 q^{92} + 156 q^{93} - 1616 q^{94} - 927 q^{96} - 768 q^{97} - 633 q^{98} - 378 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 + 9 * q^6 + 6 * q^7 + 51 * q^8 + 18 * q^9 - 42 * q^11 + 27 * q^12 + 78 * q^13 - 114 * q^14 + 25 * q^16 + 102 * q^17 + 27 * q^18 + 56 * q^19 + 18 * q^21 - 186 * q^22 - 48 * q^23 + 153 * q^24 - 6 * q^26 + 54 * q^27 - 342 * q^28 - 318 * q^29 + 52 * q^31 - 309 * q^32 - 126 * q^33 + 358 * q^34 + 81 * q^36 + 306 * q^37 - 408 * q^38 + 234 * q^39 - 408 * q^41 - 342 * q^42 + 120 * q^43 - 558 * q^44 + 92 * q^46 + 180 * q^47 + 75 * q^48 + 70 * q^49 + 306 * q^51 - 18 * q^52 + 402 * q^53 + 81 * q^54 + 30 * q^56 + 168 * q^57 + 384 * q^58 - 186 * q^59 + 340 * q^61 - 168 * q^62 + 54 * q^63 - 479 * q^64 - 558 * q^66 + 732 * q^67 + 1074 * q^68 - 144 * q^69 - 36 * q^71 + 459 * q^72 + 1332 * q^73 + 1566 * q^74 - 1224 * q^76 + 612 * q^77 - 18 * q^78 + 380 * q^79 + 162 * q^81 - 858 * q^82 - 984 * q^83 - 1026 * q^84 + 2148 * q^86 - 954 * q^87 - 1194 * q^88 + 1116 * q^89 + 972 * q^91 + 276 * q^92 + 156 * q^93 - 1616 * q^94 - 927 * q^96 - 768 * q^97 - 633 * q^98 - 378 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.70156 −0.601593 −0.300797 0.953688i $$-0.597253\pi$$
−0.300797 + 0.953688i $$0.597253\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −5.10469 −0.638086
$$5$$ 0 0
$$6$$ −5.10469 −0.347330
$$7$$ 22.2094 1.19919 0.599597 0.800302i $$-0.295328\pi$$
0.599597 + 0.800302i $$0.295328\pi$$
$$8$$ 22.2984 0.985461
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −1.79063 −0.0490813 −0.0245407 0.999699i $$-0.507812\pi$$
−0.0245407 + 0.999699i $$0.507812\pi$$
$$12$$ −15.3141 −0.368399
$$13$$ 58.2094 1.24188 0.620938 0.783860i $$-0.286752\pi$$
0.620938 + 0.783860i $$0.286752\pi$$
$$14$$ −37.7906 −0.721426
$$15$$ 0 0
$$16$$ 2.89531 0.0452393
$$17$$ 18.9844 0.270846 0.135423 0.990788i $$-0.456761\pi$$
0.135423 + 0.990788i $$0.456761\pi$$
$$18$$ −15.3141 −0.200531
$$19$$ 104.837 1.26586 0.632931 0.774208i $$-0.281852\pi$$
0.632931 + 0.774208i $$0.281852\pi$$
$$20$$ 0 0
$$21$$ 66.6281 0.692355
$$22$$ 3.04686 0.0295270
$$23$$ −49.6125 −0.449779 −0.224890 0.974384i $$-0.572202\pi$$
−0.224890 + 0.974384i $$0.572202\pi$$
$$24$$ 66.8953 0.568956
$$25$$ 0 0
$$26$$ −99.0469 −0.747103
$$27$$ 27.0000 0.192450
$$28$$ −113.372 −0.765188
$$29$$ −293.466 −1.87914 −0.939572 0.342350i $$-0.888777\pi$$
−0.939572 + 0.342350i $$0.888777\pi$$
$$30$$ 0 0
$$31$$ 64.4187 0.373224 0.186612 0.982434i $$-0.440249\pi$$
0.186612 + 0.982434i $$0.440249\pi$$
$$32$$ −183.314 −1.01268
$$33$$ −5.37188 −0.0283371
$$34$$ −32.3031 −0.162939
$$35$$ 0 0
$$36$$ −45.9422 −0.212695
$$37$$ −19.8844 −0.0883505 −0.0441752 0.999024i $$-0.514066\pi$$
−0.0441752 + 0.999024i $$0.514066\pi$$
$$38$$ −178.388 −0.761534
$$39$$ 174.628 0.716997
$$40$$ 0 0
$$41$$ −165.581 −0.630718 −0.315359 0.948972i $$-0.602125\pi$$
−0.315359 + 0.948972i $$0.602125\pi$$
$$42$$ −113.372 −0.416516
$$43$$ −247.350 −0.877221 −0.438611 0.898677i $$-0.644529\pi$$
−0.438611 + 0.898677i $$0.644529\pi$$
$$44$$ 9.14059 0.0313181
$$45$$ 0 0
$$46$$ 84.4187 0.270584
$$47$$ 384.544 1.19344 0.596718 0.802451i $$-0.296471\pi$$
0.596718 + 0.802451i $$0.296471\pi$$
$$48$$ 8.68594 0.0261189
$$49$$ 150.256 0.438065
$$50$$ 0 0
$$51$$ 56.9531 0.156373
$$52$$ −297.141 −0.792423
$$53$$ 463.528 1.20133 0.600665 0.799501i $$-0.294903\pi$$
0.600665 + 0.799501i $$0.294903\pi$$
$$54$$ −45.9422 −0.115777
$$55$$ 0 0
$$56$$ 495.234 1.18176
$$57$$ 314.512 0.730846
$$58$$ 499.350 1.13048
$$59$$ −73.7906 −0.162826 −0.0814129 0.996680i $$-0.525943\pi$$
−0.0814129 + 0.996680i $$0.525943\pi$$
$$60$$ 0 0
$$61$$ −137.350 −0.288293 −0.144146 0.989556i $$-0.546044\pi$$
−0.144146 + 0.989556i $$0.546044\pi$$
$$62$$ −109.612 −0.224529
$$63$$ 199.884 0.399731
$$64$$ 288.758 0.563980
$$65$$ 0 0
$$66$$ 9.14059 0.0170474
$$67$$ 173.906 0.317105 0.158552 0.987351i $$-0.449317\pi$$
0.158552 + 0.987351i $$0.449317\pi$$
$$68$$ −96.9093 −0.172823
$$69$$ −148.837 −0.259680
$$70$$ 0 0
$$71$$ −594.281 −0.993355 −0.496677 0.867935i $$-0.665447\pi$$
−0.496677 + 0.867935i $$0.665447\pi$$
$$72$$ 200.686 0.328487
$$73$$ 320.231 0.513428 0.256714 0.966487i $$-0.417360\pi$$
0.256714 + 0.966487i $$0.417360\pi$$
$$74$$ 33.8345 0.0531510
$$75$$ 0 0
$$76$$ −535.163 −0.807728
$$77$$ −39.7687 −0.0588580
$$78$$ −297.141 −0.431340
$$79$$ −770.469 −1.09727 −0.548636 0.836061i $$-0.684853\pi$$
−0.548636 + 0.836061i $$0.684853\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 281.747 0.379436
$$83$$ 173.925 0.230009 0.115004 0.993365i $$-0.463312\pi$$
0.115004 + 0.993365i $$0.463312\pi$$
$$84$$ −340.116 −0.441782
$$85$$ 0 0
$$86$$ 420.881 0.527730
$$87$$ −880.397 −1.08492
$$88$$ −39.9282 −0.0483677
$$89$$ 1019.02 1.21367 0.606834 0.794829i $$-0.292439\pi$$
0.606834 + 0.794829i $$0.292439\pi$$
$$90$$ 0 0
$$91$$ 1292.79 1.48925
$$92$$ 253.256 0.286998
$$93$$ 193.256 0.215481
$$94$$ −654.325 −0.717962
$$95$$ 0 0
$$96$$ −549.942 −0.584669
$$97$$ 384.375 0.402344 0.201172 0.979556i $$-0.435525\pi$$
0.201172 + 0.979556i $$0.435525\pi$$
$$98$$ −255.670 −0.263537
$$99$$ −16.1156 −0.0163604
$$100$$ 0 0
$$101$$ 34.4906 0.0339796 0.0169898 0.999856i $$-0.494592\pi$$
0.0169898 + 0.999856i $$0.494592\pi$$
$$102$$ −96.9093 −0.0940730
$$103$$ −1756.30 −1.68013 −0.840066 0.542484i $$-0.817484\pi$$
−0.840066 + 0.542484i $$0.817484\pi$$
$$104$$ 1297.98 1.22382
$$105$$ 0 0
$$106$$ −788.722 −0.722712
$$107$$ −1361.74 −1.23032 −0.615159 0.788403i $$-0.710908\pi$$
−0.615159 + 0.788403i $$0.710908\pi$$
$$108$$ −137.827 −0.122800
$$109$$ 321.119 0.282180 0.141090 0.989997i $$-0.454939\pi$$
0.141090 + 0.989997i $$0.454939\pi$$
$$110$$ 0 0
$$111$$ −59.6531 −0.0510092
$$112$$ 64.3031 0.0542506
$$113$$ −1582.25 −1.31721 −0.658607 0.752487i $$-0.728854\pi$$
−0.658607 + 0.752487i $$0.728854\pi$$
$$114$$ −535.163 −0.439672
$$115$$ 0 0
$$116$$ 1498.05 1.19906
$$117$$ 523.884 0.413958
$$118$$ 125.559 0.0979549
$$119$$ 421.631 0.324797
$$120$$ 0 0
$$121$$ −1327.79 −0.997591
$$122$$ 233.709 0.173435
$$123$$ −496.744 −0.364145
$$124$$ −328.837 −0.238149
$$125$$ 0 0
$$126$$ −340.116 −0.240475
$$127$$ −1197.14 −0.836449 −0.418225 0.908344i $$-0.637348\pi$$
−0.418225 + 0.908344i $$0.637348\pi$$
$$128$$ 975.173 0.673390
$$129$$ −742.050 −0.506464
$$130$$ 0 0
$$131$$ −321.647 −0.214522 −0.107261 0.994231i $$-0.534208\pi$$
−0.107261 + 0.994231i $$0.534208\pi$$
$$132$$ 27.4218 0.0180815
$$133$$ 2328.37 1.51801
$$134$$ −295.912 −0.190768
$$135$$ 0 0
$$136$$ 423.322 0.266909
$$137$$ 354.291 0.220942 0.110471 0.993879i $$-0.464764\pi$$
0.110471 + 0.993879i $$0.464764\pi$$
$$138$$ 253.256 0.156222
$$139$$ 77.2562 0.0471424 0.0235712 0.999722i $$-0.492496\pi$$
0.0235712 + 0.999722i $$0.492496\pi$$
$$140$$ 0 0
$$141$$ 1153.63 0.689030
$$142$$ 1011.21 0.597595
$$143$$ −104.231 −0.0609529
$$144$$ 26.0578 0.0150798
$$145$$ 0 0
$$146$$ −544.893 −0.308875
$$147$$ 450.769 0.252917
$$148$$ 101.503 0.0563752
$$149$$ −1705.38 −0.937651 −0.468826 0.883291i $$-0.655323\pi$$
−0.468826 + 0.883291i $$0.655323\pi$$
$$150$$ 0 0
$$151$$ 758.281 0.408663 0.204331 0.978902i $$-0.434498\pi$$
0.204331 + 0.978902i $$0.434498\pi$$
$$152$$ 2337.71 1.24746
$$153$$ 170.859 0.0902821
$$154$$ 67.6689 0.0354086
$$155$$ 0 0
$$156$$ −891.422 −0.457506
$$157$$ −1769.05 −0.899273 −0.449636 0.893212i $$-0.648446\pi$$
−0.449636 + 0.893212i $$0.648446\pi$$
$$158$$ 1311.00 0.660111
$$159$$ 1390.58 0.693588
$$160$$ 0 0
$$161$$ −1101.86 −0.539372
$$162$$ −137.827 −0.0668437
$$163$$ −881.719 −0.423690 −0.211845 0.977303i $$-0.567947\pi$$
−0.211845 + 0.977303i $$0.567947\pi$$
$$164$$ 845.240 0.402452
$$165$$ 0 0
$$166$$ −295.944 −0.138372
$$167$$ −216.900 −0.100504 −0.0502522 0.998737i $$-0.516003\pi$$
−0.0502522 + 0.998737i $$0.516003\pi$$
$$168$$ 1485.70 0.682289
$$169$$ 1191.33 0.542254
$$170$$ 0 0
$$171$$ 943.537 0.421954
$$172$$ 1262.64 0.559742
$$173$$ 4125.91 1.81322 0.906610 0.421970i $$-0.138661\pi$$
0.906610 + 0.421970i $$0.138661\pi$$
$$174$$ 1498.05 0.652683
$$175$$ 0 0
$$176$$ −5.18443 −0.00222040
$$177$$ −221.372 −0.0940075
$$178$$ −1733.93 −0.730134
$$179$$ −3213.14 −1.34168 −0.670842 0.741600i $$-0.734067\pi$$
−0.670842 + 0.741600i $$0.734067\pi$$
$$180$$ 0 0
$$181$$ 3394.42 1.39395 0.696976 0.717095i $$-0.254529\pi$$
0.696976 + 0.717095i $$0.254529\pi$$
$$182$$ −2199.77 −0.895921
$$183$$ −412.050 −0.166446
$$184$$ −1106.28 −0.443240
$$185$$ 0 0
$$186$$ −328.837 −0.129632
$$187$$ −33.9939 −0.0132935
$$188$$ −1962.98 −0.761514
$$189$$ 599.653 0.230785
$$190$$ 0 0
$$191$$ −3467.49 −1.31361 −0.656804 0.754062i $$-0.728092\pi$$
−0.656804 + 0.754062i $$0.728092\pi$$
$$192$$ 866.273 0.325614
$$193$$ 1792.14 0.668401 0.334200 0.942502i $$-0.391534\pi$$
0.334200 + 0.942502i $$0.391534\pi$$
$$194$$ −654.038 −0.242047
$$195$$ 0 0
$$196$$ −767.011 −0.279523
$$197$$ 1678.19 0.606935 0.303467 0.952842i $$-0.401856\pi$$
0.303467 + 0.952842i $$0.401856\pi$$
$$198$$ 27.4218 0.00984233
$$199$$ 3108.23 1.10722 0.553610 0.832776i $$-0.313250\pi$$
0.553610 + 0.832776i $$0.313250\pi$$
$$200$$ 0 0
$$201$$ 521.719 0.183081
$$202$$ −58.6878 −0.0204419
$$203$$ −6517.69 −2.25346
$$204$$ −290.728 −0.0997795
$$205$$ 0 0
$$206$$ 2988.46 1.01076
$$207$$ −446.512 −0.149926
$$208$$ 168.534 0.0561815
$$209$$ −187.725 −0.0621301
$$210$$ 0 0
$$211$$ −4473.27 −1.45949 −0.729745 0.683719i $$-0.760361\pi$$
−0.729745 + 0.683719i $$0.760361\pi$$
$$212$$ −2366.17 −0.766551
$$213$$ −1782.84 −0.573514
$$214$$ 2317.08 0.740151
$$215$$ 0 0
$$216$$ 602.058 0.189652
$$217$$ 1430.70 0.447568
$$218$$ −546.403 −0.169757
$$219$$ 960.694 0.296428
$$220$$ 0 0
$$221$$ 1105.07 0.336357
$$222$$ 101.503 0.0306868
$$223$$ −1753.42 −0.526535 −0.263268 0.964723i $$-0.584800\pi$$
−0.263268 + 0.964723i $$0.584800\pi$$
$$224$$ −4071.29 −1.21440
$$225$$ 0 0
$$226$$ 2692.29 0.792427
$$227$$ −936.900 −0.273939 −0.136970 0.990575i $$-0.543736\pi$$
−0.136970 + 0.990575i $$0.543736\pi$$
$$228$$ −1605.49 −0.466342
$$229$$ −2582.06 −0.745096 −0.372548 0.928013i $$-0.621516\pi$$
−0.372548 + 0.928013i $$0.621516\pi$$
$$230$$ 0 0
$$231$$ −119.306 −0.0339817
$$232$$ −6543.82 −1.85182
$$233$$ 2295.01 0.645284 0.322642 0.946521i $$-0.395429\pi$$
0.322642 + 0.946521i $$0.395429\pi$$
$$234$$ −891.422 −0.249034
$$235$$ 0 0
$$236$$ 376.678 0.103897
$$237$$ −2311.41 −0.633510
$$238$$ −717.432 −0.195396
$$239$$ −2294.01 −0.620866 −0.310433 0.950595i $$-0.600474\pi$$
−0.310433 + 0.950595i $$0.600474\pi$$
$$240$$ 0 0
$$241$$ 382.287 0.102180 0.0510898 0.998694i $$-0.483731\pi$$
0.0510898 + 0.998694i $$0.483731\pi$$
$$242$$ 2259.32 0.600144
$$243$$ 243.000 0.0641500
$$244$$ 701.128 0.183956
$$245$$ 0 0
$$246$$ 845.240 0.219067
$$247$$ 6102.52 1.57204
$$248$$ 1436.44 0.367798
$$249$$ 521.775 0.132796
$$250$$ 0 0
$$251$$ −2259.98 −0.568322 −0.284161 0.958777i $$-0.591715\pi$$
−0.284161 + 0.958777i $$0.591715\pi$$
$$252$$ −1020.35 −0.255063
$$253$$ 88.8375 0.0220758
$$254$$ 2037.01 0.503202
$$255$$ 0 0
$$256$$ −3969.38 −0.969087
$$257$$ −92.7843 −0.0225203 −0.0112602 0.999937i $$-0.503584\pi$$
−0.0112602 + 0.999937i $$0.503584\pi$$
$$258$$ 1262.64 0.304685
$$259$$ −441.619 −0.105949
$$260$$ 0 0
$$261$$ −2641.19 −0.626382
$$262$$ 547.302 0.129055
$$263$$ −568.312 −0.133246 −0.0666229 0.997778i $$-0.521222\pi$$
−0.0666229 + 0.997778i $$0.521222\pi$$
$$264$$ −119.785 −0.0279251
$$265$$ 0 0
$$266$$ −3961.87 −0.913226
$$267$$ 3057.07 0.700711
$$268$$ −887.737 −0.202340
$$269$$ 7582.41 1.71862 0.859309 0.511458i $$-0.170894\pi$$
0.859309 + 0.511458i $$0.170894\pi$$
$$270$$ 0 0
$$271$$ 7943.69 1.78061 0.890304 0.455366i $$-0.150492\pi$$
0.890304 + 0.455366i $$0.150492\pi$$
$$272$$ 54.9657 0.0122529
$$273$$ 3878.38 0.859818
$$274$$ −602.847 −0.132917
$$275$$ 0 0
$$276$$ 759.769 0.165698
$$277$$ 6823.00 1.47998 0.739990 0.672618i $$-0.234830\pi$$
0.739990 + 0.672618i $$0.234830\pi$$
$$278$$ −131.456 −0.0283605
$$279$$ 579.769 0.124408
$$280$$ 0 0
$$281$$ 3315.86 0.703942 0.351971 0.936011i $$-0.385512\pi$$
0.351971 + 0.936011i $$0.385512\pi$$
$$282$$ −1962.98 −0.414516
$$283$$ 6602.76 1.38690 0.693451 0.720504i $$-0.256090\pi$$
0.693451 + 0.720504i $$0.256090\pi$$
$$284$$ 3033.62 0.633846
$$285$$ 0 0
$$286$$ 177.356 0.0366688
$$287$$ −3677.46 −0.756353
$$288$$ −1649.83 −0.337559
$$289$$ −4552.59 −0.926642
$$290$$ 0 0
$$291$$ 1153.12 0.232293
$$292$$ −1634.68 −0.327611
$$293$$ 5814.14 1.15927 0.579634 0.814877i $$-0.303195\pi$$
0.579634 + 0.814877i $$0.303195\pi$$
$$294$$ −767.011 −0.152153
$$295$$ 0 0
$$296$$ −443.390 −0.0870660
$$297$$ −48.3469 −0.00944570
$$298$$ 2901.81 0.564084
$$299$$ −2887.91 −0.558570
$$300$$ 0 0
$$301$$ −5493.49 −1.05196
$$302$$ −1290.26 −0.245849
$$303$$ 103.472 0.0196181
$$304$$ 303.537 0.0572667
$$305$$ 0 0
$$306$$ −290.728 −0.0543131
$$307$$ −8124.86 −1.51046 −0.755229 0.655462i $$-0.772474\pi$$
−0.755229 + 0.655462i $$0.772474\pi$$
$$308$$ 203.007 0.0375564
$$309$$ −5268.91 −0.970025
$$310$$ 0 0
$$311$$ 7336.26 1.33762 0.668812 0.743432i $$-0.266803\pi$$
0.668812 + 0.743432i $$0.266803\pi$$
$$312$$ 3893.93 0.706572
$$313$$ −2202.66 −0.397768 −0.198884 0.980023i $$-0.563732\pi$$
−0.198884 + 0.980023i $$0.563732\pi$$
$$314$$ 3010.15 0.540996
$$315$$ 0 0
$$316$$ 3933.00 0.700154
$$317$$ 10008.9 1.77336 0.886679 0.462386i $$-0.153007\pi$$
0.886679 + 0.462386i $$0.153007\pi$$
$$318$$ −2366.17 −0.417258
$$319$$ 525.488 0.0922309
$$320$$ 0 0
$$321$$ −4085.21 −0.710325
$$322$$ 1874.89 0.324483
$$323$$ 1990.27 0.342854
$$324$$ −413.480 −0.0708984
$$325$$ 0 0
$$326$$ 1500.30 0.254889
$$327$$ 963.356 0.162917
$$328$$ −3692.20 −0.621548
$$329$$ 8540.47 1.43116
$$330$$ 0 0
$$331$$ −8695.94 −1.44402 −0.722012 0.691881i $$-0.756782\pi$$
−0.722012 + 0.691881i $$0.756782\pi$$
$$332$$ −887.832 −0.146765
$$333$$ −178.959 −0.0294502
$$334$$ 369.069 0.0604627
$$335$$ 0 0
$$336$$ 192.909 0.0313216
$$337$$ −7400.61 −1.19625 −0.598126 0.801402i $$-0.704088\pi$$
−0.598126 + 0.801402i $$0.704088\pi$$
$$338$$ −2027.12 −0.326216
$$339$$ −4746.74 −0.760494
$$340$$ 0 0
$$341$$ −115.350 −0.0183183
$$342$$ −1605.49 −0.253845
$$343$$ −4280.72 −0.673869
$$344$$ −5515.52 −0.864467
$$345$$ 0 0
$$346$$ −7020.49 −1.09082
$$347$$ 7841.44 1.21311 0.606557 0.795040i $$-0.292550\pi$$
0.606557 + 0.795040i $$0.292550\pi$$
$$348$$ 4494.15 0.692275
$$349$$ −4961.26 −0.760946 −0.380473 0.924792i $$-0.624239\pi$$
−0.380473 + 0.924792i $$0.624239\pi$$
$$350$$ 0 0
$$351$$ 1571.65 0.238999
$$352$$ 328.247 0.0497035
$$353$$ −12163.0 −1.83392 −0.916959 0.398981i $$-0.869364\pi$$
−0.916959 + 0.398981i $$0.869364\pi$$
$$354$$ 376.678 0.0565543
$$355$$ 0 0
$$356$$ −5201.80 −0.774424
$$357$$ 1264.89 0.187522
$$358$$ 5467.36 0.807148
$$359$$ 5193.79 0.763559 0.381779 0.924253i $$-0.375311\pi$$
0.381779 + 0.924253i $$0.375311\pi$$
$$360$$ 0 0
$$361$$ 4131.90 0.602406
$$362$$ −5775.81 −0.838591
$$363$$ −3983.38 −0.575959
$$364$$ −6599.31 −0.950268
$$365$$ 0 0
$$366$$ 701.128 0.100133
$$367$$ −6086.09 −0.865644 −0.432822 0.901479i $$-0.642482\pi$$
−0.432822 + 0.901479i $$0.642482\pi$$
$$368$$ −143.644 −0.0203477
$$369$$ −1490.23 −0.210239
$$370$$ 0 0
$$371$$ 10294.7 1.44063
$$372$$ −986.512 −0.137495
$$373$$ −10581.9 −1.46893 −0.734466 0.678646i $$-0.762567\pi$$
−0.734466 + 0.678646i $$0.762567\pi$$
$$374$$ 57.8428 0.00799727
$$375$$ 0 0
$$376$$ 8574.72 1.17608
$$377$$ −17082.4 −2.33366
$$378$$ −1020.35 −0.138839
$$379$$ 11655.2 1.57964 0.789822 0.613336i $$-0.210173\pi$$
0.789822 + 0.613336i $$0.210173\pi$$
$$380$$ 0 0
$$381$$ −3591.42 −0.482924
$$382$$ 5900.16 0.790257
$$383$$ 6364.97 0.849177 0.424588 0.905387i $$-0.360419\pi$$
0.424588 + 0.905387i $$0.360419\pi$$
$$384$$ 2925.52 0.388782
$$385$$ 0 0
$$386$$ −3049.44 −0.402105
$$387$$ −2226.15 −0.292407
$$388$$ −1962.11 −0.256730
$$389$$ 6134.33 0.799545 0.399773 0.916614i $$-0.369089\pi$$
0.399773 + 0.916614i $$0.369089\pi$$
$$390$$ 0 0
$$391$$ −941.862 −0.121821
$$392$$ 3350.48 0.431696
$$393$$ −964.941 −0.123855
$$394$$ −2855.55 −0.365128
$$395$$ 0 0
$$396$$ 82.2653 0.0104394
$$397$$ 9746.46 1.23214 0.616072 0.787690i $$-0.288723\pi$$
0.616072 + 0.787690i $$0.288723\pi$$
$$398$$ −5288.85 −0.666096
$$399$$ 6985.12 0.876425
$$400$$ 0 0
$$401$$ −1306.44 −0.162695 −0.0813474 0.996686i $$-0.525922\pi$$
−0.0813474 + 0.996686i $$0.525922\pi$$
$$402$$ −887.737 −0.110140
$$403$$ 3749.77 0.463498
$$404$$ −176.063 −0.0216819
$$405$$ 0 0
$$406$$ 11090.2 1.35566
$$407$$ 35.6055 0.00433636
$$408$$ 1269.97 0.154100
$$409$$ −3876.93 −0.468709 −0.234354 0.972151i $$-0.575298\pi$$
−0.234354 + 0.972151i $$0.575298\pi$$
$$410$$ 0 0
$$411$$ 1062.87 0.127561
$$412$$ 8965.38 1.07207
$$413$$ −1638.84 −0.195260
$$414$$ 759.769 0.0901947
$$415$$ 0 0
$$416$$ −10670.6 −1.25762
$$417$$ 231.769 0.0272177
$$418$$ 319.426 0.0373771
$$419$$ −16022.5 −1.86814 −0.934071 0.357088i $$-0.883770\pi$$
−0.934071 + 0.357088i $$0.883770\pi$$
$$420$$ 0 0
$$421$$ −8119.73 −0.939980 −0.469990 0.882672i $$-0.655742\pi$$
−0.469990 + 0.882672i $$0.655742\pi$$
$$422$$ 7611.54 0.878019
$$423$$ 3460.89 0.397812
$$424$$ 10336.0 1.18386
$$425$$ 0 0
$$426$$ 3033.62 0.345022
$$427$$ −3050.46 −0.345719
$$428$$ 6951.24 0.785049
$$429$$ −312.694 −0.0351911
$$430$$ 0 0
$$431$$ −5713.99 −0.638592 −0.319296 0.947655i $$-0.603446\pi$$
−0.319296 + 0.947655i $$0.603446\pi$$
$$432$$ 78.1735 0.00870630
$$433$$ 6251.34 0.693811 0.346906 0.937900i $$-0.387232\pi$$
0.346906 + 0.937900i $$0.387232\pi$$
$$434$$ −2434.42 −0.269254
$$435$$ 0 0
$$436$$ −1639.21 −0.180055
$$437$$ −5201.25 −0.569358
$$438$$ −1634.68 −0.178329
$$439$$ −4230.97 −0.459984 −0.229992 0.973192i $$-0.573870\pi$$
−0.229992 + 0.973192i $$0.573870\pi$$
$$440$$ 0 0
$$441$$ 1352.31 0.146022
$$442$$ −1880.34 −0.202350
$$443$$ 6314.29 0.677203 0.338601 0.940930i $$-0.390046\pi$$
0.338601 + 0.940930i $$0.390046\pi$$
$$444$$ 304.510 0.0325482
$$445$$ 0 0
$$446$$ 2983.55 0.316760
$$447$$ −5116.13 −0.541353
$$448$$ 6413.13 0.676321
$$449$$ −9349.71 −0.982717 −0.491358 0.870957i $$-0.663499\pi$$
−0.491358 + 0.870957i $$0.663499\pi$$
$$450$$ 0 0
$$451$$ 296.494 0.0309565
$$452$$ 8076.87 0.840496
$$453$$ 2274.84 0.235941
$$454$$ 1594.19 0.164800
$$455$$ 0 0
$$456$$ 7013.14 0.720220
$$457$$ −9547.46 −0.977268 −0.488634 0.872489i $$-0.662505\pi$$
−0.488634 + 0.872489i $$0.662505\pi$$
$$458$$ 4393.53 0.448245
$$459$$ 512.578 0.0521244
$$460$$ 0 0
$$461$$ 6237.23 0.630145 0.315073 0.949068i $$-0.397971\pi$$
0.315073 + 0.949068i $$0.397971\pi$$
$$462$$ 203.007 0.0204431
$$463$$ 6469.98 0.649428 0.324714 0.945812i $$-0.394732\pi$$
0.324714 + 0.945812i $$0.394732\pi$$
$$464$$ −849.675 −0.0850111
$$465$$ 0 0
$$466$$ −3905.10 −0.388198
$$467$$ 7206.64 0.714097 0.357049 0.934086i $$-0.383783\pi$$
0.357049 + 0.934086i $$0.383783\pi$$
$$468$$ −2674.27 −0.264141
$$469$$ 3862.35 0.380270
$$470$$ 0 0
$$471$$ −5307.16 −0.519195
$$472$$ −1645.42 −0.160458
$$473$$ 442.912 0.0430552
$$474$$ 3933.00 0.381115
$$475$$ 0 0
$$476$$ −2152.29 −0.207248
$$477$$ 4171.75 0.400443
$$478$$ 3903.39 0.373509
$$479$$ −10851.8 −1.03514 −0.517571 0.855640i $$-0.673164\pi$$
−0.517571 + 0.855640i $$0.673164\pi$$
$$480$$ 0 0
$$481$$ −1157.46 −0.109720
$$482$$ −650.485 −0.0614705
$$483$$ −3305.59 −0.311407
$$484$$ 6777.97 0.636549
$$485$$ 0 0
$$486$$ −413.480 −0.0385922
$$487$$ 12757.1 1.18702 0.593510 0.804827i $$-0.297742\pi$$
0.593510 + 0.804827i $$0.297742\pi$$
$$488$$ −3062.69 −0.284101
$$489$$ −2645.16 −0.244618
$$490$$ 0 0
$$491$$ −7016.52 −0.644911 −0.322455 0.946585i $$-0.604508\pi$$
−0.322455 + 0.946585i $$0.604508\pi$$
$$492$$ 2535.72 0.232356
$$493$$ −5571.26 −0.508960
$$494$$ −10383.8 −0.945729
$$495$$ 0 0
$$496$$ 186.512 0.0168844
$$497$$ −13198.6 −1.19122
$$498$$ −887.832 −0.0798890
$$499$$ 11372.3 1.02023 0.510113 0.860107i $$-0.329604\pi$$
0.510113 + 0.860107i $$0.329604\pi$$
$$500$$ 0 0
$$501$$ −650.700 −0.0580262
$$502$$ 3845.50 0.341899
$$503$$ −5587.37 −0.495285 −0.247643 0.968851i $$-0.579656\pi$$
−0.247643 + 0.968851i $$0.579656\pi$$
$$504$$ 4457.11 0.393919
$$505$$ 0 0
$$506$$ −151.163 −0.0132806
$$507$$ 3573.99 0.313070
$$508$$ 6111.03 0.533726
$$509$$ 16256.7 1.41565 0.707825 0.706388i $$-0.249676\pi$$
0.707825 + 0.706388i $$0.249676\pi$$
$$510$$ 0 0
$$511$$ 7112.14 0.615699
$$512$$ −1047.24 −0.0903943
$$513$$ 2830.61 0.243615
$$514$$ 157.878 0.0135481
$$515$$ 0 0
$$516$$ 3787.93 0.323167
$$517$$ −688.574 −0.0585754
$$518$$ 751.442 0.0637384
$$519$$ 12377.7 1.04686
$$520$$ 0 0
$$521$$ 19748.4 1.66064 0.830320 0.557286i $$-0.188157\pi$$
0.830320 + 0.557286i $$0.188157\pi$$
$$522$$ 4494.15 0.376827
$$523$$ −7843.44 −0.655774 −0.327887 0.944717i $$-0.606337\pi$$
−0.327887 + 0.944717i $$0.606337\pi$$
$$524$$ 1641.91 0.136884
$$525$$ 0 0
$$526$$ 967.019 0.0801597
$$527$$ 1222.95 0.101086
$$528$$ −15.5533 −0.00128195
$$529$$ −9705.60 −0.797699
$$530$$ 0 0
$$531$$ −664.116 −0.0542753
$$532$$ −11885.6 −0.968622
$$533$$ −9638.38 −0.783273
$$534$$ −5201.80 −0.421543
$$535$$ 0 0
$$536$$ 3877.84 0.312495
$$537$$ −9639.42 −0.774622
$$538$$ −12902.0 −1.03391
$$539$$ −269.053 −0.0215008
$$540$$ 0 0
$$541$$ 7383.29 0.586751 0.293376 0.955997i $$-0.405221\pi$$
0.293376 + 0.955997i $$0.405221\pi$$
$$542$$ −13516.7 −1.07120
$$543$$ 10183.3 0.804798
$$544$$ −3480.10 −0.274280
$$545$$ 0 0
$$546$$ −6599.31 −0.517260
$$547$$ −3354.90 −0.262240 −0.131120 0.991367i $$-0.541857\pi$$
−0.131120 + 0.991367i $$0.541857\pi$$
$$548$$ −1808.54 −0.140980
$$549$$ −1236.15 −0.0960976
$$550$$ 0 0
$$551$$ −30766.2 −2.37874
$$552$$ −3318.84 −0.255905
$$553$$ −17111.6 −1.31584
$$554$$ −11609.8 −0.890346
$$555$$ 0 0
$$556$$ −394.369 −0.0300809
$$557$$ 20771.8 1.58012 0.790061 0.613028i $$-0.210049\pi$$
0.790061 + 0.613028i $$0.210049\pi$$
$$558$$ −986.512 −0.0748430
$$559$$ −14398.1 −1.08940
$$560$$ 0 0
$$561$$ −101.982 −0.00767500
$$562$$ −5642.15 −0.423487
$$563$$ 7194.86 0.538592 0.269296 0.963057i $$-0.413209\pi$$
0.269296 + 0.963057i $$0.413209\pi$$
$$564$$ −5888.93 −0.439660
$$565$$ 0 0
$$566$$ −11235.0 −0.834350
$$567$$ 1798.96 0.133244
$$568$$ −13251.5 −0.978913
$$569$$ 11549.5 0.850931 0.425466 0.904975i $$-0.360110\pi$$
0.425466 + 0.904975i $$0.360110\pi$$
$$570$$ 0 0
$$571$$ 1482.54 0.108655 0.0543277 0.998523i $$-0.482698\pi$$
0.0543277 + 0.998523i $$0.482698\pi$$
$$572$$ 532.068 0.0388932
$$573$$ −10402.5 −0.758412
$$574$$ 6257.42 0.455017
$$575$$ 0 0
$$576$$ 2598.82 0.187993
$$577$$ −15264.0 −1.10130 −0.550649 0.834737i $$-0.685620\pi$$
−0.550649 + 0.834737i $$0.685620\pi$$
$$578$$ 7746.52 0.557462
$$579$$ 5376.43 0.385901
$$580$$ 0 0
$$581$$ 3862.76 0.275825
$$582$$ −1962.11 −0.139746
$$583$$ −830.006 −0.0589628
$$584$$ 7140.66 0.505963
$$585$$ 0 0
$$586$$ −9893.12 −0.697408
$$587$$ 1736.89 0.122128 0.0610639 0.998134i $$-0.480551\pi$$
0.0610639 + 0.998134i $$0.480551\pi$$
$$588$$ −2301.03 −0.161383
$$589$$ 6753.50 0.472450
$$590$$ 0 0
$$591$$ 5034.57 0.350414
$$592$$ −57.5714 −0.00399691
$$593$$ −11764.8 −0.814707 −0.407353 0.913271i $$-0.633548\pi$$
−0.407353 + 0.913271i $$0.633548\pi$$
$$594$$ 82.2653 0.00568247
$$595$$ 0 0
$$596$$ 8705.42 0.598302
$$597$$ 9324.69 0.639253
$$598$$ 4913.96 0.336032
$$599$$ −9451.99 −0.644737 −0.322369 0.946614i $$-0.604479\pi$$
−0.322369 + 0.946614i $$0.604479\pi$$
$$600$$ 0 0
$$601$$ −3131.93 −0.212569 −0.106285 0.994336i $$-0.533895\pi$$
−0.106285 + 0.994336i $$0.533895\pi$$
$$602$$ 9347.51 0.632851
$$603$$ 1565.16 0.105702
$$604$$ −3870.79 −0.260762
$$605$$ 0 0
$$606$$ −176.063 −0.0118021
$$607$$ 22700.8 1.51795 0.758975 0.651120i $$-0.225700\pi$$
0.758975 + 0.651120i $$0.225700\pi$$
$$608$$ −19218.2 −1.28191
$$609$$ −19553.1 −1.30103
$$610$$ 0 0
$$611$$ 22384.0 1.48210
$$612$$ −872.184 −0.0576077
$$613$$ 28911.6 1.90494 0.952471 0.304629i $$-0.0985325\pi$$
0.952471 + 0.304629i $$0.0985325\pi$$
$$614$$ 13825.0 0.908681
$$615$$ 0 0
$$616$$ −886.780 −0.0580023
$$617$$ −5566.87 −0.363231 −0.181616 0.983370i $$-0.558133\pi$$
−0.181616 + 0.983370i $$0.558133\pi$$
$$618$$ 8965.38 0.583560
$$619$$ 4150.32 0.269492 0.134746 0.990880i $$-0.456978\pi$$
0.134746 + 0.990880i $$0.456978\pi$$
$$620$$ 0 0
$$621$$ −1339.54 −0.0865600
$$622$$ −12483.1 −0.804705
$$623$$ 22631.9 1.45542
$$624$$ 505.603 0.0324364
$$625$$ 0 0
$$626$$ 3747.96 0.239295
$$627$$ −563.175 −0.0358709
$$628$$ 9030.46 0.573813
$$629$$ −377.492 −0.0239294
$$630$$ 0 0
$$631$$ −4090.09 −0.258041 −0.129021 0.991642i $$-0.541183\pi$$
−0.129021 + 0.991642i $$0.541183\pi$$
$$632$$ −17180.2 −1.08132
$$633$$ −13419.8 −0.842637
$$634$$ −17030.7 −1.06684
$$635$$ 0 0
$$636$$ −7098.50 −0.442569
$$637$$ 8746.32 0.544022
$$638$$ −894.150 −0.0554855
$$639$$ −5348.53 −0.331118
$$640$$ 0 0
$$641$$ 3909.35 0.240890 0.120445 0.992720i $$-0.461568\pi$$
0.120445 + 0.992720i $$0.461568\pi$$
$$642$$ 6951.24 0.427327
$$643$$ 30539.5 1.87303 0.936516 0.350624i $$-0.114031\pi$$
0.936516 + 0.350624i $$0.114031\pi$$
$$644$$ 5624.66 0.344166
$$645$$ 0 0
$$646$$ −3386.58 −0.206259
$$647$$ −12707.7 −0.772167 −0.386083 0.922464i $$-0.626172\pi$$
−0.386083 + 0.922464i $$0.626172\pi$$
$$648$$ 1806.17 0.109496
$$649$$ 132.132 0.00799170
$$650$$ 0 0
$$651$$ 4292.10 0.258403
$$652$$ 4500.90 0.270351
$$653$$ 12777.6 0.765737 0.382869 0.923803i $$-0.374936\pi$$
0.382869 + 0.923803i $$0.374936\pi$$
$$654$$ −1639.21 −0.0980095
$$655$$ 0 0
$$656$$ −479.410 −0.0285332
$$657$$ 2882.08 0.171143
$$658$$ −14532.1 −0.860976
$$659$$ 23563.5 1.39287 0.696435 0.717620i $$-0.254768\pi$$
0.696435 + 0.717620i $$0.254768\pi$$
$$660$$ 0 0
$$661$$ −4361.31 −0.256634 −0.128317 0.991733i $$-0.540958\pi$$
−0.128317 + 0.991733i $$0.540958\pi$$
$$662$$ 14796.7 0.868715
$$663$$ 3315.21 0.194196
$$664$$ 3878.25 0.226665
$$665$$ 0 0
$$666$$ 304.510 0.0177170
$$667$$ 14559.6 0.845200
$$668$$ 1107.21 0.0641304
$$669$$ −5260.25 −0.303995
$$670$$ 0 0
$$671$$ 245.943 0.0141498
$$672$$ −12213.9 −0.701131
$$673$$ −8203.52 −0.469870 −0.234935 0.972011i $$-0.575488\pi$$
−0.234935 + 0.972011i $$0.575488\pi$$
$$674$$ 12592.6 0.719657
$$675$$ 0 0
$$676$$ −6081.37 −0.346004
$$677$$ 28057.1 1.59279 0.796397 0.604774i $$-0.206737\pi$$
0.796397 + 0.604774i $$0.206737\pi$$
$$678$$ 8076.87 0.457508
$$679$$ 8536.73 0.482488
$$680$$ 0 0
$$681$$ −2810.70 −0.158159
$$682$$ 196.275 0.0110202
$$683$$ −3344.62 −0.187377 −0.0936885 0.995602i $$-0.529866\pi$$
−0.0936885 + 0.995602i $$0.529866\pi$$
$$684$$ −4816.46 −0.269243
$$685$$ 0 0
$$686$$ 7283.91 0.405395
$$687$$ −7746.17 −0.430182
$$688$$ −716.156 −0.0396849
$$689$$ 26981.7 1.49190
$$690$$ 0 0
$$691$$ 12964.8 0.713757 0.356879 0.934151i $$-0.383841\pi$$
0.356879 + 0.934151i $$0.383841\pi$$
$$692$$ −21061.5 −1.15699
$$693$$ −357.918 −0.0196193
$$694$$ −13342.7 −0.729801
$$695$$ 0 0
$$696$$ −19631.5 −1.06915
$$697$$ −3143.46 −0.170828
$$698$$ 8441.90 0.457780
$$699$$ 6885.03 0.372555
$$700$$ 0 0
$$701$$ −16162.1 −0.870806 −0.435403 0.900236i $$-0.643394\pi$$
−0.435403 + 0.900236i $$0.643394\pi$$
$$702$$ −2674.27 −0.143780
$$703$$ −2084.63 −0.111839
$$704$$ −517.058 −0.0276809
$$705$$ 0 0
$$706$$ 20696.2 1.10327
$$707$$ 766.014 0.0407481
$$708$$ 1130.03 0.0599849
$$709$$ −14244.4 −0.754529 −0.377265 0.926105i $$-0.623135\pi$$
−0.377265 + 0.926105i $$0.623135\pi$$
$$710$$ 0 0
$$711$$ −6934.22 −0.365757
$$712$$ 22722.7 1.19602
$$713$$ −3195.97 −0.167868
$$714$$ −2152.29 −0.112812
$$715$$ 0 0
$$716$$ 16402.1 0.856109
$$717$$ −6882.02 −0.358457
$$718$$ −8837.55 −0.459352
$$719$$ −27638.5 −1.43358 −0.716790 0.697289i $$-0.754389\pi$$
−0.716790 + 0.697289i $$0.754389\pi$$
$$720$$ 0 0
$$721$$ −39006.4 −2.01480
$$722$$ −7030.68 −0.362403
$$723$$ 1146.86 0.0589934
$$724$$ −17327.4 −0.889460
$$725$$ 0 0
$$726$$ 6777.97 0.346493
$$727$$ −2525.52 −0.128840 −0.0644199 0.997923i $$-0.520520\pi$$
−0.0644199 + 0.997923i $$0.520520\pi$$
$$728$$ 28827.3 1.46760
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −4695.79 −0.237592
$$732$$ 2103.39 0.106207
$$733$$ −8400.27 −0.423289 −0.211645 0.977347i $$-0.567882\pi$$
−0.211645 + 0.977347i $$0.567882\pi$$
$$734$$ 10355.9 0.520765
$$735$$ 0 0
$$736$$ 9094.67 0.455481
$$737$$ −311.401 −0.0155639
$$738$$ 2535.72 0.126479
$$739$$ −19689.1 −0.980074 −0.490037 0.871702i $$-0.663017\pi$$
−0.490037 + 0.871702i $$0.663017\pi$$
$$740$$ 0 0
$$741$$ 18307.6 0.907619
$$742$$ −17517.0 −0.866671
$$743$$ −22526.6 −1.11227 −0.556137 0.831091i $$-0.687717\pi$$
−0.556137 + 0.831091i $$0.687717\pi$$
$$744$$ 4309.31 0.212348
$$745$$ 0 0
$$746$$ 18005.8 0.883699
$$747$$ 1565.32 0.0766696
$$748$$ 173.528 0.00848239
$$749$$ −30243.3 −1.47539
$$750$$ 0 0
$$751$$ 34691.1 1.68562 0.842808 0.538215i $$-0.180901\pi$$
0.842808 + 0.538215i $$0.180901\pi$$
$$752$$ 1113.37 0.0539902
$$753$$ −6779.95 −0.328121
$$754$$ 29066.8 1.40392
$$755$$ 0 0
$$756$$ −3061.04 −0.147261
$$757$$ 6619.98 0.317843 0.158922 0.987291i $$-0.449198\pi$$
0.158922 + 0.987291i $$0.449198\pi$$
$$758$$ −19832.0 −0.950303
$$759$$ 266.512 0.0127454
$$760$$ 0 0
$$761$$ −29368.7 −1.39897 −0.699483 0.714649i $$-0.746586\pi$$
−0.699483 + 0.714649i $$0.746586\pi$$
$$762$$ 6111.03 0.290524
$$763$$ 7131.84 0.338388
$$764$$ 17700.5 0.838194
$$765$$ 0 0
$$766$$ −10830.4 −0.510859
$$767$$ −4295.31 −0.202209
$$768$$ −11908.1 −0.559503
$$769$$ 32677.4 1.53235 0.766174 0.642633i $$-0.222158\pi$$
0.766174 + 0.642633i $$0.222158\pi$$
$$770$$ 0 0
$$771$$ −278.353 −0.0130021
$$772$$ −9148.33 −0.426497
$$773$$ −28047.5 −1.30504 −0.652522 0.757770i $$-0.726289\pi$$
−0.652522 + 0.757770i $$0.726289\pi$$
$$774$$ 3787.93 0.175910
$$775$$ 0 0
$$776$$ 8570.96 0.396494
$$777$$ −1324.86 −0.0611699
$$778$$ −10438.0 −0.481001
$$779$$ −17359.1 −0.798402
$$780$$ 0 0
$$781$$ 1064.14 0.0487552
$$782$$ 1602.64 0.0732867
$$783$$ −7923.57 −0.361642
$$784$$ 435.039 0.0198177
$$785$$ 0 0
$$786$$ 1641.91 0.0745100
$$787$$ −22172.1 −1.00426 −0.502128 0.864793i $$-0.667449\pi$$
−0.502128 + 0.864793i $$0.667449\pi$$
$$788$$ −8566.64 −0.387276
$$789$$ −1704.94 −0.0769295
$$790$$ 0 0
$$791$$ −35140.7 −1.57960
$$792$$ −359.354 −0.0161226
$$793$$ −7995.06 −0.358024
$$794$$ −16584.2 −0.741249
$$795$$ 0 0
$$796$$ −15866.5 −0.706501
$$797$$ −24170.3 −1.07422 −0.537112 0.843511i $$-0.680485\pi$$
−0.537112 + 0.843511i $$0.680485\pi$$
$$798$$ −11885.6 −0.527251
$$799$$ 7300.32 0.323238
$$800$$ 0 0
$$801$$ 9171.22 0.404556
$$802$$ 2222.99 0.0978761
$$803$$ −573.415 −0.0251997
$$804$$ −2663.21 −0.116821
$$805$$ 0 0
$$806$$ −6380.47 −0.278837
$$807$$ 22747.2 0.992244
$$808$$ 769.085 0.0334856
$$809$$ 15304.2 0.665102 0.332551 0.943085i $$-0.392091\pi$$
0.332551 + 0.943085i $$0.392091\pi$$
$$810$$ 0 0
$$811$$ −27002.2 −1.16914 −0.584572 0.811342i $$-0.698738\pi$$
−0.584572 + 0.811342i $$0.698738\pi$$
$$812$$ 33270.7 1.43790
$$813$$ 23831.1 1.02803
$$814$$ −60.5849 −0.00260872
$$815$$ 0 0
$$816$$ 164.897 0.00707421
$$817$$ −25931.5 −1.11044
$$818$$ 6596.84 0.281972
$$819$$ 11635.1 0.496416
$$820$$ 0 0
$$821$$ 25061.4 1.06535 0.532673 0.846321i $$-0.321188\pi$$
0.532673 + 0.846321i $$0.321188\pi$$
$$822$$ −1808.54 −0.0767398
$$823$$ −24896.4 −1.05448 −0.527238 0.849718i $$-0.676772\pi$$
−0.527238 + 0.849718i $$0.676772\pi$$
$$824$$ −39162.8 −1.65571
$$825$$ 0 0
$$826$$ 2788.59 0.117467
$$827$$ −20063.2 −0.843612 −0.421806 0.906686i $$-0.638604\pi$$
−0.421806 + 0.906686i $$0.638604\pi$$
$$828$$ 2279.31 0.0956659
$$829$$ −13884.2 −0.581687 −0.290844 0.956771i $$-0.593936\pi$$
−0.290844 + 0.956771i $$0.593936\pi$$
$$830$$ 0 0
$$831$$ 20469.0 0.854467
$$832$$ 16808.4 0.700393
$$833$$ 2852.52 0.118648
$$834$$ −394.369 −0.0163740
$$835$$ 0 0
$$836$$ 958.277 0.0396444
$$837$$ 1739.31 0.0718270
$$838$$ 27263.3 1.12386
$$839$$ −13678.1 −0.562838 −0.281419 0.959585i $$-0.590805\pi$$
−0.281419 + 0.959585i $$0.590805\pi$$
$$840$$ 0 0
$$841$$ 61733.1 2.53118
$$842$$ 13816.2 0.565485
$$843$$ 9947.59 0.406421
$$844$$ 22834.6 0.931280
$$845$$ 0 0
$$846$$ −5888.93 −0.239321
$$847$$ −29489.5 −1.19630
$$848$$ 1342.06 0.0543473
$$849$$ 19808.3 0.800728
$$850$$ 0 0
$$851$$ 986.512 0.0397382
$$852$$ 9100.86 0.365951
$$853$$ 29802.9 1.19629 0.598143 0.801390i $$-0.295906\pi$$
0.598143 + 0.801390i $$0.295906\pi$$
$$854$$ 5190.54 0.207982
$$855$$ 0 0
$$856$$ −30364.6 −1.21243
$$857$$ 22045.2 0.878706 0.439353 0.898314i $$-0.355208\pi$$
0.439353 + 0.898314i $$0.355208\pi$$
$$858$$ 532.068 0.0211708
$$859$$ 33609.5 1.33497 0.667487 0.744622i $$-0.267370\pi$$
0.667487 + 0.744622i $$0.267370\pi$$
$$860$$ 0 0
$$861$$ −11032.4 −0.436681
$$862$$ 9722.70 0.384172
$$863$$ 33775.6 1.33226 0.666128 0.745838i $$-0.267951\pi$$
0.666128 + 0.745838i $$0.267951\pi$$
$$864$$ −4949.48 −0.194890
$$865$$ 0 0
$$866$$ −10637.0 −0.417392
$$867$$ −13657.8 −0.534997
$$868$$ −7303.27 −0.285587
$$869$$ 1379.62 0.0538556
$$870$$ 0 0
$$871$$ 10123.0 0.393805
$$872$$ 7160.44 0.278077
$$873$$ 3459.37 0.134115
$$874$$ 8850.25 0.342522
$$875$$ 0 0
$$876$$ −4904.04 −0.189146
$$877$$ 12637.0 0.486570 0.243285 0.969955i $$-0.421775\pi$$
0.243285 + 0.969955i $$0.421775\pi$$
$$878$$ 7199.25 0.276723
$$879$$ 17442.4 0.669304
$$880$$ 0 0
$$881$$ −6579.45 −0.251609 −0.125804 0.992055i $$-0.540151\pi$$
−0.125804 + 0.992055i $$0.540151\pi$$
$$882$$ −2301.03 −0.0878456
$$883$$ −50442.1 −1.92244 −0.961219 0.275786i $$-0.911062\pi$$
−0.961219 + 0.275786i $$0.911062\pi$$
$$884$$ −5641.03 −0.214625
$$885$$ 0 0
$$886$$ −10744.2 −0.407401
$$887$$ 984.823 0.0372797 0.0186399 0.999826i $$-0.494066\pi$$
0.0186399 + 0.999826i $$0.494066\pi$$
$$888$$ −1330.17 −0.0502676
$$889$$ −26587.7 −1.00306
$$890$$ 0 0
$$891$$ −145.041 −0.00545348
$$892$$ 8950.64 0.335975
$$893$$ 40314.6 1.51072
$$894$$ 8705.42 0.325674
$$895$$ 0 0
$$896$$ 21658.0 0.807525
$$897$$ −8663.74 −0.322490
$$898$$ 15909.1 0.591196
$$899$$ −18904.7 −0.701342
$$900$$ 0 0
$$901$$ 8799.79 0.325376
$$902$$ −504.503 −0.0186232
$$903$$ −16480.5 −0.607348
$$904$$ −35281.6 −1.29806
$$905$$ 0 0
$$906$$ −3870.79 −0.141941
$$907$$ 43679.9 1.59908 0.799541 0.600612i $$-0.205076\pi$$
0.799541 + 0.600612i $$0.205076\pi$$
$$908$$ 4782.58 0.174797
$$909$$ 310.415 0.0113265
$$910$$ 0 0
$$911$$ −10364.3 −0.376930 −0.188465 0.982080i $$-0.560351\pi$$
−0.188465 + 0.982080i $$0.560351\pi$$
$$912$$ 910.612 0.0330629
$$913$$ −311.435 −0.0112891
$$914$$ 16245.6 0.587918
$$915$$ 0 0
$$916$$ 13180.6 0.475435
$$917$$ −7143.58 −0.257254
$$918$$ −872.184 −0.0313577
$$919$$ 11451.9 0.411059 0.205530 0.978651i $$-0.434108\pi$$
0.205530 + 0.978651i $$0.434108\pi$$
$$920$$ 0 0
$$921$$ −24374.6 −0.872063
$$922$$ −10613.0 −0.379091
$$923$$ −34592.7 −1.23362
$$924$$ 609.020 0.0216832
$$925$$ 0 0
$$926$$ −11009.1 −0.390692
$$927$$ −15806.7 −0.560044
$$928$$ 53796.4 1.90297
$$929$$ −27701.8 −0.978326 −0.489163 0.872192i $$-0.662698\pi$$
−0.489163 + 0.872192i $$0.662698\pi$$
$$930$$ 0 0
$$931$$ 15752.5 0.554529
$$932$$ −11715.3 −0.411746
$$933$$ 22008.8 0.772277
$$934$$ −12262.5 −0.429596
$$935$$ 0 0
$$936$$ 11681.8 0.407940
$$937$$ −5878.01 −0.204937 −0.102469 0.994736i $$-0.532674\pi$$
−0.102469 + 0.994736i $$0.532674\pi$$
$$938$$ −6572.03 −0.228768
$$939$$ −6607.97 −0.229652
$$940$$ 0 0
$$941$$ −28786.0 −0.997234 −0.498617 0.866823i $$-0.666159\pi$$
−0.498617 + 0.866823i $$0.666159\pi$$
$$942$$ 9030.46 0.312344
$$943$$ 8214.90 0.283684
$$944$$ −213.647 −0.00736612
$$945$$ 0 0
$$946$$ −753.642 −0.0259017
$$947$$ −1695.04 −0.0581641 −0.0290821 0.999577i $$-0.509258\pi$$
−0.0290821 + 0.999577i $$0.509258\pi$$
$$948$$ 11799.0 0.404234
$$949$$ 18640.5 0.637613
$$950$$ 0 0
$$951$$ 30026.6 1.02385
$$952$$ 9401.72 0.320075
$$953$$ 31929.4 1.08530 0.542651 0.839958i $$-0.317420\pi$$
0.542651 + 0.839958i $$0.317420\pi$$
$$954$$ −7098.50 −0.240904
$$955$$ 0 0
$$956$$ 11710.2 0.396166
$$957$$ 1576.46 0.0532495
$$958$$ 18465.1 0.622735
$$959$$ 7868.57 0.264952
$$960$$ 0 0
$$961$$ −25641.2 −0.860704
$$962$$ 1969.48 0.0660069
$$963$$ −12255.6 −0.410106
$$964$$ −1951.46 −0.0651994
$$965$$ 0 0
$$966$$ 5624.66 0.187340
$$967$$ −10897.1 −0.362385 −0.181193 0.983448i $$-0.557996\pi$$
−0.181193 + 0.983448i $$0.557996\pi$$
$$968$$ −29607.7 −0.983087
$$969$$ 5970.82 0.197947
$$970$$ 0 0
$$971$$ 7041.97 0.232737 0.116368 0.993206i $$-0.462875\pi$$
0.116368 + 0.993206i $$0.462875\pi$$
$$972$$ −1240.44 −0.0409332
$$973$$ 1715.81 0.0565328
$$974$$ −21707.0 −0.714103
$$975$$ 0 0
$$976$$ −397.671 −0.0130422
$$977$$ −37607.6 −1.23150 −0.615749 0.787943i $$-0.711146\pi$$
−0.615749 + 0.787943i $$0.711146\pi$$
$$978$$ 4500.90 0.147160
$$979$$ −1824.69 −0.0595684
$$980$$ 0 0
$$981$$ 2890.07 0.0940599
$$982$$ 11939.0 0.387974
$$983$$ −25297.7 −0.820826 −0.410413 0.911900i $$-0.634615\pi$$
−0.410413 + 0.911900i $$0.634615\pi$$
$$984$$ −11076.6 −0.358851
$$985$$ 0 0
$$986$$ 9479.85 0.306187
$$987$$ 25621.4 0.826281
$$988$$ −31151.5 −1.00310
$$989$$ 12271.6 0.394556
$$990$$ 0 0
$$991$$ −41686.5 −1.33624 −0.668120 0.744053i $$-0.732901\pi$$
−0.668120 + 0.744053i $$0.732901\pi$$
$$992$$ −11808.9 −0.377955
$$993$$ −26087.8 −0.833708
$$994$$ 22458.3 0.716633
$$995$$ 0 0
$$996$$ −2663.50 −0.0847351
$$997$$ 25465.9 0.808939 0.404470 0.914551i $$-0.367456\pi$$
0.404470 + 0.914551i $$0.367456\pi$$
$$998$$ −19350.6 −0.613761
$$999$$ −536.878 −0.0170031
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 75.4.a.f.1.1 2
3.2 odd 2 225.4.a.i.1.2 2
4.3 odd 2 1200.4.a.bn.1.1 2
5.2 odd 4 15.4.b.a.4.2 4
5.3 odd 4 15.4.b.a.4.3 yes 4
5.4 even 2 75.4.a.c.1.2 2
15.2 even 4 45.4.b.b.19.3 4
15.8 even 4 45.4.b.b.19.2 4
15.14 odd 2 225.4.a.o.1.1 2
20.3 even 4 240.4.f.f.49.1 4
20.7 even 4 240.4.f.f.49.3 4
20.19 odd 2 1200.4.a.bt.1.2 2
40.3 even 4 960.4.f.p.769.4 4
40.13 odd 4 960.4.f.q.769.2 4
40.27 even 4 960.4.f.p.769.2 4
40.37 odd 4 960.4.f.q.769.4 4
60.23 odd 4 720.4.f.j.289.4 4
60.47 odd 4 720.4.f.j.289.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.2 4 5.2 odd 4
15.4.b.a.4.3 yes 4 5.3 odd 4
45.4.b.b.19.2 4 15.8 even 4
45.4.b.b.19.3 4 15.2 even 4
75.4.a.c.1.2 2 5.4 even 2
75.4.a.f.1.1 2 1.1 even 1 trivial
225.4.a.i.1.2 2 3.2 odd 2
225.4.a.o.1.1 2 15.14 odd 2
240.4.f.f.49.1 4 20.3 even 4
240.4.f.f.49.3 4 20.7 even 4
720.4.f.j.289.3 4 60.47 odd 4
720.4.f.j.289.4 4 60.23 odd 4
960.4.f.p.769.2 4 40.27 even 4
960.4.f.p.769.4 4 40.3 even 4
960.4.f.q.769.2 4 40.13 odd 4
960.4.f.q.769.4 4 40.37 odd 4
1200.4.a.bn.1.1 2 4.3 odd 2
1200.4.a.bt.1.2 2 20.19 odd 2