# Properties

 Label 75.4.a.d.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: $$\Q(\sqrt{19})$$ Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.35890$$ of defining polynomial Character $$\chi$$ $$=$$ 75.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.35890 q^{2} +3.00000 q^{3} +20.7178 q^{4} -16.0767 q^{6} -4.43560 q^{7} -68.1534 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.35890 q^{2} +3.00000 q^{3} +20.7178 q^{4} -16.0767 q^{6} -4.43560 q^{7} -68.1534 q^{8} +9.00000 q^{9} -3.43560 q^{11} +62.1534 q^{12} +78.7424 q^{13} +23.7699 q^{14} +199.485 q^{16} +53.1780 q^{17} -48.2301 q^{18} +20.4356 q^{19} -13.3068 q^{21} +18.4110 q^{22} +118.307 q^{23} -204.460 q^{24} -421.972 q^{26} +27.0000 q^{27} -91.8958 q^{28} +168.049 q^{29} -61.0492 q^{31} -523.792 q^{32} -10.3068 q^{33} -284.975 q^{34} +186.460 q^{36} -246.614 q^{37} -109.512 q^{38} +236.227 q^{39} +422.663 q^{41} +71.3097 q^{42} +362.436 q^{43} -71.1780 q^{44} -633.994 q^{46} -170.515 q^{47} +598.454 q^{48} -323.325 q^{49} +159.534 q^{51} +1631.37 q^{52} -546.049 q^{53} -144.690 q^{54} +302.301 q^{56} +61.3068 q^{57} -900.559 q^{58} -216.970 q^{59} +130.902 q^{61} +327.156 q^{62} -39.9204 q^{63} +1211.07 q^{64} +55.2330 q^{66} -614.890 q^{67} +1101.73 q^{68} +354.920 q^{69} +324.822 q^{71} -613.381 q^{72} -88.8712 q^{73} +1321.58 q^{74} +423.381 q^{76} +15.2389 q^{77} -1265.92 q^{78} -1137.42 q^{79} +81.0000 q^{81} -2265.01 q^{82} -758.909 q^{83} -275.687 q^{84} -1942.26 q^{86} +504.148 q^{87} +234.148 q^{88} +195.681 q^{89} -349.269 q^{91} +2451.06 q^{92} -183.148 q^{93} +913.774 q^{94} -1571.37 q^{96} +521.000 q^{97} +1732.67 q^{98} -30.9204 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 6 * q^3 + 24 * q^4 - 6 * q^6 + 26 * q^7 - 84 * q^8 + 18 * q^9 $$2 q - 2 q^{2} + 6 q^{3} + 24 q^{4} - 6 q^{6} + 26 q^{7} - 84 q^{8} + 18 q^{9} + 28 q^{11} + 72 q^{12} + 18 q^{13} + 126 q^{14} + 120 q^{16} - 68 q^{17} - 18 q^{18} + 6 q^{19} + 78 q^{21} + 124 q^{22} + 132 q^{23} - 252 q^{24} - 626 q^{26} + 54 q^{27} + 8 q^{28} + 92 q^{29} + 122 q^{31} - 664 q^{32} + 84 q^{33} - 692 q^{34} + 216 q^{36} - 284 q^{37} - 158 q^{38} + 54 q^{39} + 392 q^{41} + 378 q^{42} + 690 q^{43} + 32 q^{44} - 588 q^{46} - 620 q^{47} + 360 q^{48} + 260 q^{49} - 204 q^{51} + 1432 q^{52} - 848 q^{53} - 54 q^{54} - 180 q^{56} + 18 q^{57} - 1156 q^{58} + 124 q^{59} + 750 q^{61} + 942 q^{62} + 234 q^{63} + 1376 q^{64} + 372 q^{66} - 358 q^{67} + 704 q^{68} + 396 q^{69} + 824 q^{71} - 756 q^{72} - 108 q^{73} + 1196 q^{74} + 376 q^{76} + 972 q^{77} - 1878 q^{78} - 880 q^{79} + 162 q^{81} - 2368 q^{82} + 156 q^{83} + 24 q^{84} - 842 q^{86} + 276 q^{87} - 264 q^{88} - 864 q^{89} - 2198 q^{91} + 2496 q^{92} + 366 q^{93} - 596 q^{94} - 1992 q^{96} + 1042 q^{97} + 3692 q^{98} + 252 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 6 * q^3 + 24 * q^4 - 6 * q^6 + 26 * q^7 - 84 * q^8 + 18 * q^9 + 28 * q^11 + 72 * q^12 + 18 * q^13 + 126 * q^14 + 120 * q^16 - 68 * q^17 - 18 * q^18 + 6 * q^19 + 78 * q^21 + 124 * q^22 + 132 * q^23 - 252 * q^24 - 626 * q^26 + 54 * q^27 + 8 * q^28 + 92 * q^29 + 122 * q^31 - 664 * q^32 + 84 * q^33 - 692 * q^34 + 216 * q^36 - 284 * q^37 - 158 * q^38 + 54 * q^39 + 392 * q^41 + 378 * q^42 + 690 * q^43 + 32 * q^44 - 588 * q^46 - 620 * q^47 + 360 * q^48 + 260 * q^49 - 204 * q^51 + 1432 * q^52 - 848 * q^53 - 54 * q^54 - 180 * q^56 + 18 * q^57 - 1156 * q^58 + 124 * q^59 + 750 * q^61 + 942 * q^62 + 234 * q^63 + 1376 * q^64 + 372 * q^66 - 358 * q^67 + 704 * q^68 + 396 * q^69 + 824 * q^71 - 756 * q^72 - 108 * q^73 + 1196 * q^74 + 376 * q^76 + 972 * q^77 - 1878 * q^78 - 880 * q^79 + 162 * q^81 - 2368 * q^82 + 156 * q^83 + 24 * q^84 - 842 * q^86 + 276 * q^87 - 264 * q^88 - 864 * q^89 - 2198 * q^91 + 2496 * q^92 + 366 * q^93 - 596 * q^94 - 1992 * q^96 + 1042 * q^97 + 3692 * q^98 + 252 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.35890 −1.89466 −0.947328 0.320264i $$-0.896228\pi$$
−0.947328 + 0.320264i $$0.896228\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 20.7178 2.58972
$$5$$ 0 0
$$6$$ −16.0767 −1.09388
$$7$$ −4.43560 −0.239500 −0.119750 0.992804i $$-0.538209\pi$$
−0.119750 + 0.992804i $$0.538209\pi$$
$$8$$ −68.1534 −3.01198
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −3.43560 −0.0941701 −0.0470851 0.998891i $$-0.514993\pi$$
−0.0470851 + 0.998891i $$0.514993\pi$$
$$12$$ 62.1534 1.49518
$$13$$ 78.7424 1.67994 0.839970 0.542634i $$-0.182573\pi$$
0.839970 + 0.542634i $$0.182573\pi$$
$$14$$ 23.7699 0.453770
$$15$$ 0 0
$$16$$ 199.485 3.11695
$$17$$ 53.1780 0.758680 0.379340 0.925257i $$-0.376151\pi$$
0.379340 + 0.925257i $$0.376151\pi$$
$$18$$ −48.2301 −0.631552
$$19$$ 20.4356 0.246750 0.123375 0.992360i $$-0.460628\pi$$
0.123375 + 0.992360i $$0.460628\pi$$
$$20$$ 0 0
$$21$$ −13.3068 −0.138275
$$22$$ 18.4110 0.178420
$$23$$ 118.307 1.07255 0.536275 0.844043i $$-0.319831\pi$$
0.536275 + 0.844043i $$0.319831\pi$$
$$24$$ −204.460 −1.73897
$$25$$ 0 0
$$26$$ −421.972 −3.18291
$$27$$ 27.0000 0.192450
$$28$$ −91.8958 −0.620238
$$29$$ 168.049 1.07607 0.538034 0.842923i $$-0.319167\pi$$
0.538034 + 0.842923i $$0.319167\pi$$
$$30$$ 0 0
$$31$$ −61.0492 −0.353702 −0.176851 0.984238i $$-0.556591\pi$$
−0.176851 + 0.984238i $$0.556591\pi$$
$$32$$ −523.792 −2.89357
$$33$$ −10.3068 −0.0543691
$$34$$ −284.975 −1.43744
$$35$$ 0 0
$$36$$ 186.460 0.863242
$$37$$ −246.614 −1.09576 −0.547879 0.836558i $$-0.684564\pi$$
−0.547879 + 0.836558i $$0.684564\pi$$
$$38$$ −109.512 −0.467506
$$39$$ 236.227 0.969913
$$40$$ 0 0
$$41$$ 422.663 1.60997 0.804986 0.593294i $$-0.202173\pi$$
0.804986 + 0.593294i $$0.202173\pi$$
$$42$$ 71.3097 0.261984
$$43$$ 362.436 1.28537 0.642685 0.766131i $$-0.277820\pi$$
0.642685 + 0.766131i $$0.277820\pi$$
$$44$$ −71.1780 −0.243875
$$45$$ 0 0
$$46$$ −633.994 −2.03212
$$47$$ −170.515 −0.529196 −0.264598 0.964359i $$-0.585239\pi$$
−0.264598 + 0.964359i $$0.585239\pi$$
$$48$$ 598.454 1.79957
$$49$$ −323.325 −0.942640
$$50$$ 0 0
$$51$$ 159.534 0.438024
$$52$$ 1631.37 4.35058
$$53$$ −546.049 −1.41520 −0.707600 0.706613i $$-0.750222\pi$$
−0.707600 + 0.706613i $$0.750222\pi$$
$$54$$ −144.690 −0.364627
$$55$$ 0 0
$$56$$ 302.301 0.721369
$$57$$ 61.3068 0.142461
$$58$$ −900.559 −2.03878
$$59$$ −216.970 −0.478763 −0.239382 0.970926i $$-0.576945\pi$$
−0.239382 + 0.970926i $$0.576945\pi$$
$$60$$ 0 0
$$61$$ 130.902 0.274758 0.137379 0.990519i $$-0.456132\pi$$
0.137379 + 0.990519i $$0.456132\pi$$
$$62$$ 327.156 0.670143
$$63$$ −39.9204 −0.0798332
$$64$$ 1211.07 2.36537
$$65$$ 0 0
$$66$$ 55.2330 0.103011
$$67$$ −614.890 −1.12121 −0.560603 0.828085i $$-0.689430\pi$$
−0.560603 + 0.828085i $$0.689430\pi$$
$$68$$ 1101.73 1.96477
$$69$$ 354.920 0.619238
$$70$$ 0 0
$$71$$ 324.822 0.542948 0.271474 0.962446i $$-0.412489\pi$$
0.271474 + 0.962446i $$0.412489\pi$$
$$72$$ −613.381 −1.00399
$$73$$ −88.8712 −0.142487 −0.0712437 0.997459i $$-0.522697\pi$$
−0.0712437 + 0.997459i $$0.522697\pi$$
$$74$$ 1321.58 2.07608
$$75$$ 0 0
$$76$$ 423.381 0.639014
$$77$$ 15.2389 0.0225537
$$78$$ −1265.92 −1.83765
$$79$$ −1137.42 −1.61988 −0.809938 0.586516i $$-0.800499\pi$$
−0.809938 + 0.586516i $$0.800499\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −2265.01 −3.05034
$$83$$ −758.909 −1.00363 −0.501813 0.864976i $$-0.667334\pi$$
−0.501813 + 0.864976i $$0.667334\pi$$
$$84$$ −275.687 −0.358095
$$85$$ 0 0
$$86$$ −1942.26 −2.43534
$$87$$ 504.148 0.621268
$$88$$ 234.148 0.283639
$$89$$ 195.681 0.233058 0.116529 0.993187i $$-0.462823\pi$$
0.116529 + 0.993187i $$0.462823\pi$$
$$90$$ 0 0
$$91$$ −349.269 −0.402345
$$92$$ 2451.06 2.77761
$$93$$ −183.148 −0.204210
$$94$$ 913.774 1.00264
$$95$$ 0 0
$$96$$ −1571.37 −1.67060
$$97$$ 521.000 0.545356 0.272678 0.962105i $$-0.412091\pi$$
0.272678 + 0.962105i $$0.412091\pi$$
$$98$$ 1732.67 1.78598
$$99$$ −30.9204 −0.0313900
$$100$$ 0 0
$$101$$ 660.920 0.651129 0.325565 0.945520i $$-0.394446\pi$$
0.325565 + 0.945520i $$0.394446\pi$$
$$102$$ −854.926 −0.829905
$$103$$ 1530.75 1.46436 0.732181 0.681110i $$-0.238503\pi$$
0.732181 + 0.681110i $$0.238503\pi$$
$$104$$ −5366.56 −5.05995
$$105$$ 0 0
$$106$$ 2926.22 2.68132
$$107$$ −264.625 −0.239087 −0.119543 0.992829i $$-0.538143\pi$$
−0.119543 + 0.992829i $$0.538143\pi$$
$$108$$ 559.381 0.498393
$$109$$ 1117.61 0.982091 0.491046 0.871134i $$-0.336615\pi$$
0.491046 + 0.871134i $$0.336615\pi$$
$$110$$ 0 0
$$111$$ −739.841 −0.632636
$$112$$ −884.834 −0.746508
$$113$$ −934.061 −0.777602 −0.388801 0.921322i $$-0.627111\pi$$
−0.388801 + 0.921322i $$0.627111\pi$$
$$114$$ −328.537 −0.269915
$$115$$ 0 0
$$116$$ 3481.61 2.78672
$$117$$ 708.681 0.559980
$$118$$ 1162.72 0.907092
$$119$$ −235.876 −0.181704
$$120$$ 0 0
$$121$$ −1319.20 −0.991132
$$122$$ −701.489 −0.520572
$$123$$ 1267.99 0.929517
$$124$$ −1264.80 −0.915990
$$125$$ 0 0
$$126$$ 213.929 0.151257
$$127$$ 630.356 0.440433 0.220217 0.975451i $$-0.429324\pi$$
0.220217 + 0.975451i $$0.429324\pi$$
$$128$$ −2299.66 −1.58799
$$129$$ 1087.31 0.742109
$$130$$ 0 0
$$131$$ −2163.06 −1.44265 −0.721325 0.692597i $$-0.756466\pi$$
−0.721325 + 0.692597i $$0.756466\pi$$
$$132$$ −213.534 −0.140801
$$133$$ −90.6440 −0.0590965
$$134$$ 3295.13 2.12430
$$135$$ 0 0
$$136$$ −3624.26 −2.28513
$$137$$ 1118.61 0.697588 0.348794 0.937199i $$-0.386591\pi$$
0.348794 + 0.937199i $$0.386591\pi$$
$$138$$ −1901.98 −1.17324
$$139$$ −166.478 −0.101586 −0.0507930 0.998709i $$-0.516175\pi$$
−0.0507930 + 0.998709i $$0.516175\pi$$
$$140$$ 0 0
$$141$$ −511.546 −0.305531
$$142$$ −1740.69 −1.02870
$$143$$ −270.527 −0.158200
$$144$$ 1795.36 1.03898
$$145$$ 0 0
$$146$$ 476.252 0.269965
$$147$$ −969.976 −0.544233
$$148$$ −5109.29 −2.83771
$$149$$ −653.143 −0.359111 −0.179555 0.983748i $$-0.557466\pi$$
−0.179555 + 0.983748i $$0.557466\pi$$
$$150$$ 0 0
$$151$$ −1929.38 −1.03981 −0.519903 0.854225i $$-0.674032\pi$$
−0.519903 + 0.854225i $$0.674032\pi$$
$$152$$ −1392.76 −0.743206
$$153$$ 478.602 0.252893
$$154$$ −81.6638 −0.0427315
$$155$$ 0 0
$$156$$ 4894.11 2.51181
$$157$$ 2169.75 1.10296 0.551480 0.834188i $$-0.314063\pi$$
0.551480 + 0.834188i $$0.314063\pi$$
$$158$$ 6095.34 3.06911
$$159$$ −1638.15 −0.817066
$$160$$ 0 0
$$161$$ −524.761 −0.256876
$$162$$ −434.071 −0.210517
$$163$$ −763.738 −0.366997 −0.183499 0.983020i $$-0.558742\pi$$
−0.183499 + 0.983020i $$0.558742\pi$$
$$164$$ 8756.64 4.16938
$$165$$ 0 0
$$166$$ 4066.91 1.90153
$$167$$ −2564.28 −1.18820 −0.594102 0.804389i $$-0.702493\pi$$
−0.594102 + 0.804389i $$0.702493\pi$$
$$168$$ 906.903 0.416483
$$169$$ 4003.36 1.82220
$$170$$ 0 0
$$171$$ 183.920 0.0822500
$$172$$ 7508.87 3.32875
$$173$$ 51.8290 0.0227774 0.0113887 0.999935i $$-0.496375\pi$$
0.0113887 + 0.999935i $$0.496375\pi$$
$$174$$ −2701.68 −1.17709
$$175$$ 0 0
$$176$$ −685.349 −0.293523
$$177$$ −650.909 −0.276414
$$178$$ −1048.64 −0.441566
$$179$$ −3956.63 −1.65214 −0.826068 0.563571i $$-0.809427\pi$$
−0.826068 + 0.563571i $$0.809427\pi$$
$$180$$ 0 0
$$181$$ 1804.04 0.740848 0.370424 0.928863i $$-0.379212\pi$$
0.370424 + 0.928863i $$0.379212\pi$$
$$182$$ 1871.70 0.762305
$$183$$ 392.705 0.158632
$$184$$ −8063.01 −3.23050
$$185$$ 0 0
$$186$$ 981.469 0.386908
$$187$$ −182.698 −0.0714449
$$188$$ −3532.70 −1.37047
$$189$$ −119.761 −0.0460917
$$190$$ 0 0
$$191$$ 3666.75 1.38909 0.694547 0.719448i $$-0.255605\pi$$
0.694547 + 0.719448i $$0.255605\pi$$
$$192$$ 3633.20 1.36565
$$193$$ −2716.98 −1.01333 −0.506664 0.862144i $$-0.669121\pi$$
−0.506664 + 0.862144i $$0.669121\pi$$
$$194$$ −2791.99 −1.03326
$$195$$ 0 0
$$196$$ −6698.59 −2.44118
$$197$$ 2034.30 0.735723 0.367862 0.929881i $$-0.380090\pi$$
0.367862 + 0.929881i $$0.380090\pi$$
$$198$$ 165.699 0.0594733
$$199$$ −1551.27 −0.552596 −0.276298 0.961072i $$-0.589108\pi$$
−0.276298 + 0.961072i $$0.589108\pi$$
$$200$$ 0 0
$$201$$ −1844.67 −0.647328
$$202$$ −3541.81 −1.23367
$$203$$ −745.398 −0.257718
$$204$$ 3305.19 1.13436
$$205$$ 0 0
$$206$$ −8203.13 −2.77446
$$207$$ 1064.76 0.357517
$$208$$ 15707.9 5.23629
$$209$$ −70.2084 −0.0232365
$$210$$ 0 0
$$211$$ 3192.51 1.04162 0.520809 0.853673i $$-0.325630\pi$$
0.520809 + 0.853673i $$0.325630\pi$$
$$212$$ −11312.9 −3.66498
$$213$$ 974.466 0.313471
$$214$$ 1418.10 0.452988
$$215$$ 0 0
$$216$$ −1840.14 −0.579656
$$217$$ 270.789 0.0847115
$$218$$ −5989.18 −1.86073
$$219$$ −266.614 −0.0822652
$$220$$ 0 0
$$221$$ 4187.36 1.27454
$$222$$ 3964.73 1.19863
$$223$$ −1555.55 −0.467120 −0.233560 0.972342i $$-0.575037\pi$$
−0.233560 + 0.972342i $$0.575037\pi$$
$$224$$ 2323.33 0.693008
$$225$$ 0 0
$$226$$ 5005.54 1.47329
$$227$$ −6206.86 −1.81482 −0.907409 0.420248i $$-0.861943\pi$$
−0.907409 + 0.420248i $$0.861943\pi$$
$$228$$ 1270.14 0.368935
$$229$$ 4679.51 1.35035 0.675176 0.737657i $$-0.264068\pi$$
0.675176 + 0.737657i $$0.264068\pi$$
$$230$$ 0 0
$$231$$ 45.7167 0.0130214
$$232$$ −11453.1 −3.24110
$$233$$ −3244.53 −0.912259 −0.456129 0.889913i $$-0.650765\pi$$
−0.456129 + 0.889913i $$0.650765\pi$$
$$234$$ −3797.75 −1.06097
$$235$$ 0 0
$$236$$ −4495.13 −1.23986
$$237$$ −3412.27 −0.935236
$$238$$ 1264.04 0.344266
$$239$$ −3658.62 −0.990193 −0.495097 0.868838i $$-0.664867\pi$$
−0.495097 + 0.868838i $$0.664867\pi$$
$$240$$ 0 0
$$241$$ 1931.01 0.516131 0.258065 0.966127i $$-0.416915\pi$$
0.258065 + 0.966127i $$0.416915\pi$$
$$242$$ 7069.44 1.87786
$$243$$ 243.000 0.0641500
$$244$$ 2711.99 0.711548
$$245$$ 0 0
$$246$$ −6795.02 −1.76112
$$247$$ 1609.15 0.414525
$$248$$ 4160.71 1.06534
$$249$$ −2276.73 −0.579444
$$250$$ 0 0
$$251$$ −5843.34 −1.46944 −0.734718 0.678373i $$-0.762685\pi$$
−0.734718 + 0.678373i $$0.762685\pi$$
$$252$$ −827.062 −0.206746
$$253$$ −406.454 −0.101002
$$254$$ −3378.01 −0.834470
$$255$$ 0 0
$$256$$ 2635.09 0.643333
$$257$$ 4506.11 1.09371 0.546855 0.837227i $$-0.315825\pi$$
0.546855 + 0.837227i $$0.315825\pi$$
$$258$$ −5826.77 −1.40604
$$259$$ 1093.88 0.262434
$$260$$ 0 0
$$261$$ 1512.44 0.358689
$$262$$ 11591.6 2.73333
$$263$$ 5340.16 1.25205 0.626024 0.779804i $$-0.284681\pi$$
0.626024 + 0.779804i $$0.284681\pi$$
$$264$$ 702.443 0.163759
$$265$$ 0 0
$$266$$ 485.752 0.111968
$$267$$ 587.044 0.134556
$$268$$ −12739.2 −2.90361
$$269$$ 2809.79 0.636863 0.318431 0.947946i $$-0.396844\pi$$
0.318431 + 0.947946i $$0.396844\pi$$
$$270$$ 0 0
$$271$$ 3102.95 0.695537 0.347769 0.937580i $$-0.386940\pi$$
0.347769 + 0.937580i $$0.386940\pi$$
$$272$$ 10608.2 2.36477
$$273$$ −1047.81 −0.232294
$$274$$ −5994.54 −1.32169
$$275$$ 0 0
$$276$$ 7353.17 1.60365
$$277$$ −4598.93 −0.997555 −0.498777 0.866730i $$-0.666217\pi$$
−0.498777 + 0.866730i $$0.666217\pi$$
$$278$$ 892.138 0.192471
$$279$$ −549.443 −0.117901
$$280$$ 0 0
$$281$$ 2571.83 0.545987 0.272994 0.962016i $$-0.411986\pi$$
0.272994 + 0.962016i $$0.411986\pi$$
$$282$$ 2741.32 0.578877
$$283$$ 5575.31 1.17109 0.585544 0.810641i $$-0.300881\pi$$
0.585544 + 0.810641i $$0.300881\pi$$
$$284$$ 6729.60 1.40608
$$285$$ 0 0
$$286$$ 1449.73 0.299735
$$287$$ −1874.76 −0.385588
$$288$$ −4714.12 −0.964522
$$289$$ −2085.10 −0.424405
$$290$$ 0 0
$$291$$ 1563.00 0.314861
$$292$$ −1841.22 −0.369003
$$293$$ 5794.27 1.15531 0.577654 0.816282i $$-0.303968\pi$$
0.577654 + 0.816282i $$0.303968\pi$$
$$294$$ 5198.01 1.03114
$$295$$ 0 0
$$296$$ 16807.6 3.30040
$$297$$ −92.7611 −0.0181230
$$298$$ 3500.13 0.680392
$$299$$ 9315.76 1.80182
$$300$$ 0 0
$$301$$ −1607.62 −0.307846
$$302$$ 10339.4 1.97008
$$303$$ 1982.76 0.375930
$$304$$ 4076.59 0.769107
$$305$$ 0 0
$$306$$ −2564.78 −0.479146
$$307$$ −1404.47 −0.261099 −0.130550 0.991442i $$-0.541674\pi$$
−0.130550 + 0.991442i $$0.541674\pi$$
$$308$$ 315.717 0.0584079
$$309$$ 4592.25 0.845449
$$310$$ 0 0
$$311$$ −4096.75 −0.746963 −0.373481 0.927638i $$-0.621836\pi$$
−0.373481 + 0.927638i $$0.621836\pi$$
$$312$$ −16099.7 −2.92136
$$313$$ 974.611 0.176001 0.0880004 0.996120i $$-0.471952\pi$$
0.0880004 + 0.996120i $$0.471952\pi$$
$$314$$ −11627.5 −2.08973
$$315$$ 0 0
$$316$$ −23564.9 −4.19503
$$317$$ −2071.69 −0.367058 −0.183529 0.983014i $$-0.558752\pi$$
−0.183529 + 0.983014i $$0.558752\pi$$
$$318$$ 8778.67 1.54806
$$319$$ −577.349 −0.101333
$$320$$ 0 0
$$321$$ −793.876 −0.138037
$$322$$ 2812.14 0.486691
$$323$$ 1086.72 0.187204
$$324$$ 1678.14 0.287747
$$325$$ 0 0
$$326$$ 4092.79 0.695334
$$327$$ 3352.84 0.567011
$$328$$ −28805.9 −4.84921
$$329$$ 756.337 0.126742
$$330$$ 0 0
$$331$$ −6159.17 −1.02278 −0.511388 0.859350i $$-0.670868\pi$$
−0.511388 + 0.859350i $$0.670868\pi$$
$$332$$ −15722.9 −2.59912
$$333$$ −2219.52 −0.365252
$$334$$ 13741.7 2.25124
$$335$$ 0 0
$$336$$ −2654.50 −0.430997
$$337$$ 2791.26 0.451186 0.225593 0.974222i $$-0.427568\pi$$
0.225593 + 0.974222i $$0.427568\pi$$
$$338$$ −21453.6 −3.45243
$$339$$ −2802.18 −0.448949
$$340$$ 0 0
$$341$$ 209.740 0.0333081
$$342$$ −985.611 −0.155835
$$343$$ 2955.55 0.465262
$$344$$ −24701.2 −3.87151
$$345$$ 0 0
$$346$$ −277.746 −0.0431553
$$347$$ 940.848 0.145554 0.0727772 0.997348i $$-0.476814\pi$$
0.0727772 + 0.997348i $$0.476814\pi$$
$$348$$ 10444.8 1.60891
$$349$$ −3519.62 −0.539831 −0.269915 0.962884i $$-0.586996\pi$$
−0.269915 + 0.962884i $$0.586996\pi$$
$$350$$ 0 0
$$351$$ 2126.04 0.323304
$$352$$ 1799.54 0.272487
$$353$$ 5021.60 0.757147 0.378573 0.925571i $$-0.376415\pi$$
0.378573 + 0.925571i $$0.376415\pi$$
$$354$$ 3488.15 0.523710
$$355$$ 0 0
$$356$$ 4054.09 0.603557
$$357$$ −707.628 −0.104907
$$358$$ 21203.2 3.13023
$$359$$ 6811.99 1.00146 0.500728 0.865604i $$-0.333066\pi$$
0.500728 + 0.865604i $$0.333066\pi$$
$$360$$ 0 0
$$361$$ −6441.39 −0.939115
$$362$$ −9667.69 −1.40365
$$363$$ −3957.59 −0.572230
$$364$$ −7236.09 −1.04196
$$365$$ 0 0
$$366$$ −2104.47 −0.300553
$$367$$ 3748.07 0.533099 0.266550 0.963821i $$-0.414116\pi$$
0.266550 + 0.963821i $$0.414116\pi$$
$$368$$ 23600.4 3.34309
$$369$$ 3803.96 0.536657
$$370$$ 0 0
$$371$$ 2422.05 0.338940
$$372$$ −3794.41 −0.528847
$$373$$ −898.302 −0.124698 −0.0623489 0.998054i $$-0.519859\pi$$
−0.0623489 + 0.998054i $$0.519859\pi$$
$$374$$ 979.060 0.135364
$$375$$ 0 0
$$376$$ 11621.2 1.59393
$$377$$ 13232.6 1.80773
$$378$$ 641.788 0.0873280
$$379$$ −9378.99 −1.27115 −0.635576 0.772038i $$-0.719237\pi$$
−0.635576 + 0.772038i $$0.719237\pi$$
$$380$$ 0 0
$$381$$ 1891.07 0.254284
$$382$$ −19649.8 −2.63186
$$383$$ −9446.29 −1.26027 −0.630134 0.776486i $$-0.717000\pi$$
−0.630134 + 0.776486i $$0.717000\pi$$
$$384$$ −6898.97 −0.916828
$$385$$ 0 0
$$386$$ 14560.0 1.91991
$$387$$ 3261.92 0.428457
$$388$$ 10794.0 1.41232
$$389$$ −7643.23 −0.996214 −0.498107 0.867116i $$-0.665971\pi$$
−0.498107 + 0.867116i $$0.665971\pi$$
$$390$$ 0 0
$$391$$ 6291.32 0.813723
$$392$$ 22035.7 2.83922
$$393$$ −6489.17 −0.832914
$$394$$ −10901.6 −1.39394
$$395$$ 0 0
$$396$$ −640.602 −0.0812915
$$397$$ −12013.6 −1.51876 −0.759378 0.650650i $$-0.774497\pi$$
−0.759378 + 0.650650i $$0.774497\pi$$
$$398$$ 8313.10 1.04698
$$399$$ −271.932 −0.0341194
$$400$$ 0 0
$$401$$ −8538.51 −1.06332 −0.531662 0.846957i $$-0.678432\pi$$
−0.531662 + 0.846957i $$0.678432\pi$$
$$402$$ 9885.40 1.22646
$$403$$ −4807.16 −0.594197
$$404$$ 13692.8 1.68625
$$405$$ 0 0
$$406$$ 3994.51 0.488287
$$407$$ 847.265 0.103188
$$408$$ −10872.8 −1.31932
$$409$$ −12267.6 −1.48312 −0.741558 0.670889i $$-0.765913\pi$$
−0.741558 + 0.670889i $$0.765913\pi$$
$$410$$ 0 0
$$411$$ 3355.84 0.402753
$$412$$ 31713.8 3.79229
$$413$$ 962.389 0.114664
$$414$$ −5705.95 −0.677372
$$415$$ 0 0
$$416$$ −41244.6 −4.86102
$$417$$ −499.433 −0.0586508
$$418$$ 376.240 0.0440251
$$419$$ 15493.0 1.80641 0.903204 0.429212i $$-0.141209\pi$$
0.903204 + 0.429212i $$0.141209\pi$$
$$420$$ 0 0
$$421$$ 7510.67 0.869472 0.434736 0.900558i $$-0.356842\pi$$
0.434736 + 0.900558i $$0.356842\pi$$
$$422$$ −17108.3 −1.97351
$$423$$ −1534.64 −0.176399
$$424$$ 37215.1 4.26256
$$425$$ 0 0
$$426$$ −5222.07 −0.593920
$$427$$ −580.627 −0.0658045
$$428$$ −5482.45 −0.619169
$$429$$ −811.581 −0.0913368
$$430$$ 0 0
$$431$$ −15675.3 −1.75186 −0.875932 0.482434i $$-0.839753\pi$$
−0.875932 + 0.482434i $$0.839753\pi$$
$$432$$ 5386.09 0.599857
$$433$$ −9604.78 −1.06600 −0.532998 0.846117i $$-0.678935\pi$$
−0.532998 + 0.846117i $$0.678935\pi$$
$$434$$ −1451.13 −0.160499
$$435$$ 0 0
$$436$$ 23154.5 2.54335
$$437$$ 2417.67 0.264652
$$438$$ 1428.76 0.155864
$$439$$ −6362.06 −0.691673 −0.345837 0.938295i $$-0.612405\pi$$
−0.345837 + 0.938295i $$0.612405\pi$$
$$440$$ 0 0
$$441$$ −2909.93 −0.314213
$$442$$ −22439.6 −2.41481
$$443$$ 931.658 0.0999196 0.0499598 0.998751i $$-0.484091\pi$$
0.0499598 + 0.998751i $$0.484091\pi$$
$$444$$ −15327.9 −1.63835
$$445$$ 0 0
$$446$$ 8336.06 0.885031
$$447$$ −1959.43 −0.207333
$$448$$ −5371.81 −0.566505
$$449$$ −18684.1 −1.96383 −0.981914 0.189329i $$-0.939369\pi$$
−0.981914 + 0.189329i $$0.939369\pi$$
$$450$$ 0 0
$$451$$ −1452.10 −0.151611
$$452$$ −19351.7 −2.01378
$$453$$ −5788.14 −0.600333
$$454$$ 33261.9 3.43846
$$455$$ 0 0
$$456$$ −4178.27 −0.429090
$$457$$ −11565.9 −1.18387 −0.591936 0.805985i $$-0.701636\pi$$
−0.591936 + 0.805985i $$0.701636\pi$$
$$458$$ −25077.0 −2.55845
$$459$$ 1435.81 0.146008
$$460$$ 0 0
$$461$$ 19401.0 1.96008 0.980039 0.198806i $$-0.0637062\pi$$
0.980039 + 0.198806i $$0.0637062\pi$$
$$462$$ −244.991 −0.0246711
$$463$$ 1576.28 0.158220 0.0791099 0.996866i $$-0.474792\pi$$
0.0791099 + 0.996866i $$0.474792\pi$$
$$464$$ 33523.2 3.35405
$$465$$ 0 0
$$466$$ 17387.1 1.72842
$$467$$ 3256.55 0.322687 0.161344 0.986898i $$-0.448417\pi$$
0.161344 + 0.986898i $$0.448417\pi$$
$$468$$ 14682.3 1.45019
$$469$$ 2727.40 0.268528
$$470$$ 0 0
$$471$$ 6509.25 0.636795
$$472$$ 14787.2 1.44203
$$473$$ −1245.18 −0.121043
$$474$$ 18286.0 1.77195
$$475$$ 0 0
$$476$$ −4886.83 −0.470562
$$477$$ −4914.44 −0.471733
$$478$$ 19606.2 1.87608
$$479$$ 8291.59 0.790924 0.395462 0.918482i $$-0.370585\pi$$
0.395462 + 0.918482i $$0.370585\pi$$
$$480$$ 0 0
$$481$$ −19418.9 −1.84081
$$482$$ −10348.1 −0.977891
$$483$$ −1574.28 −0.148307
$$484$$ −27330.9 −2.56676
$$485$$ 0 0
$$486$$ −1302.21 −0.121542
$$487$$ 4758.55 0.442773 0.221387 0.975186i $$-0.428942\pi$$
0.221387 + 0.975186i $$0.428942\pi$$
$$488$$ −8921.39 −0.827567
$$489$$ −2291.21 −0.211886
$$490$$ 0 0
$$491$$ 3906.46 0.359055 0.179528 0.983753i $$-0.442543\pi$$
0.179528 + 0.983753i $$0.442543\pi$$
$$492$$ 26269.9 2.40719
$$493$$ 8936.52 0.816390
$$494$$ −8623.26 −0.785382
$$495$$ 0 0
$$496$$ −12178.4 −1.10247
$$497$$ −1440.78 −0.130036
$$498$$ 12200.7 1.09785
$$499$$ −3093.31 −0.277506 −0.138753 0.990327i $$-0.544309\pi$$
−0.138753 + 0.990327i $$0.544309\pi$$
$$500$$ 0 0
$$501$$ −7692.85 −0.686010
$$502$$ 31313.9 2.78408
$$503$$ −18153.9 −1.60923 −0.804616 0.593796i $$-0.797629\pi$$
−0.804616 + 0.593796i $$0.797629\pi$$
$$504$$ 2720.71 0.240456
$$505$$ 0 0
$$506$$ 2178.15 0.191365
$$507$$ 12010.1 1.05204
$$508$$ 13059.6 1.14060
$$509$$ 2281.32 0.198660 0.0993298 0.995055i $$-0.468330\pi$$
0.0993298 + 0.995055i $$0.468330\pi$$
$$510$$ 0 0
$$511$$ 394.197 0.0341257
$$512$$ 4276.07 0.369097
$$513$$ 551.761 0.0474870
$$514$$ −24147.8 −2.07221
$$515$$ 0 0
$$516$$ 22526.6 1.92186
$$517$$ 585.821 0.0498344
$$518$$ −5861.98 −0.497221
$$519$$ 155.487 0.0131505
$$520$$ 0 0
$$521$$ −16691.9 −1.40362 −0.701809 0.712366i $$-0.747624\pi$$
−0.701809 + 0.712366i $$0.747624\pi$$
$$522$$ −8105.03 −0.679593
$$523$$ −17090.4 −1.42889 −0.714446 0.699690i $$-0.753321\pi$$
−0.714446 + 0.699690i $$0.753321\pi$$
$$524$$ −44813.8 −3.73607
$$525$$ 0 0
$$526$$ −28617.4 −2.37220
$$527$$ −3246.47 −0.268346
$$528$$ −2056.05 −0.169466
$$529$$ 1829.50 0.150365
$$530$$ 0 0
$$531$$ −1952.73 −0.159588
$$532$$ −1877.94 −0.153044
$$533$$ 33281.5 2.70465
$$534$$ −3145.91 −0.254938
$$535$$ 0 0
$$536$$ 41906.8 3.37705
$$537$$ −11869.9 −0.953861
$$538$$ −15057.4 −1.20664
$$539$$ 1110.82 0.0887685
$$540$$ 0 0
$$541$$ 5271.40 0.418919 0.209459 0.977817i $$-0.432830\pi$$
0.209459 + 0.977817i $$0.432830\pi$$
$$542$$ −16628.4 −1.31780
$$543$$ 5412.13 0.427729
$$544$$ −27854.2 −2.19529
$$545$$ 0 0
$$546$$ 5615.10 0.440117
$$547$$ 15182.2 1.18673 0.593366 0.804933i $$-0.297799\pi$$
0.593366 + 0.804933i $$0.297799\pi$$
$$548$$ 23175.2 1.80656
$$549$$ 1178.11 0.0915860
$$550$$ 0 0
$$551$$ 3434.18 0.265519
$$552$$ −24189.0 −1.86513
$$553$$ 5045.15 0.387960
$$554$$ 24645.2 1.89002
$$555$$ 0 0
$$556$$ −3449.05 −0.263080
$$557$$ 12241.2 0.931198 0.465599 0.884996i $$-0.345839\pi$$
0.465599 + 0.884996i $$0.345839\pi$$
$$558$$ 2944.41 0.223381
$$559$$ 28539.0 2.15934
$$560$$ 0 0
$$561$$ −548.094 −0.0412488
$$562$$ −13782.2 −1.03446
$$563$$ 14196.4 1.06271 0.531355 0.847149i $$-0.321683\pi$$
0.531355 + 0.847149i $$0.321683\pi$$
$$564$$ −10598.1 −0.791242
$$565$$ 0 0
$$566$$ −29877.5 −2.21881
$$567$$ −359.283 −0.0266111
$$568$$ −22137.7 −1.63535
$$569$$ 9150.05 0.674148 0.337074 0.941478i $$-0.390563\pi$$
0.337074 + 0.941478i $$0.390563\pi$$
$$570$$ 0 0
$$571$$ 23582.1 1.72833 0.864167 0.503206i $$-0.167846\pi$$
0.864167 + 0.503206i $$0.167846\pi$$
$$572$$ −5604.72 −0.409695
$$573$$ 11000.3 0.801993
$$574$$ 10046.7 0.730556
$$575$$ 0 0
$$576$$ 10899.6 0.788456
$$577$$ 3906.22 0.281834 0.140917 0.990021i $$-0.454995\pi$$
0.140917 + 0.990021i $$0.454995\pi$$
$$578$$ 11173.9 0.804102
$$579$$ −8150.93 −0.585045
$$580$$ 0 0
$$581$$ 3366.21 0.240368
$$582$$ −8375.96 −0.596554
$$583$$ 1876.00 0.133270
$$584$$ 6056.87 0.429170
$$585$$ 0 0
$$586$$ −31050.9 −2.18891
$$587$$ −25938.0 −1.82381 −0.911905 0.410401i $$-0.865389\pi$$
−0.911905 + 0.410401i $$0.865389\pi$$
$$588$$ −20095.8 −1.40941
$$589$$ −1247.58 −0.0872759
$$590$$ 0 0
$$591$$ 6102.89 0.424770
$$592$$ −49195.7 −3.41542
$$593$$ 1908.23 0.132145 0.0660723 0.997815i $$-0.478953\pi$$
0.0660723 + 0.997815i $$0.478953\pi$$
$$594$$ 497.097 0.0343370
$$595$$ 0 0
$$596$$ −13531.7 −0.929999
$$597$$ −4653.81 −0.319041
$$598$$ −49922.2 −3.41383
$$599$$ −3495.41 −0.238429 −0.119214 0.992869i $$-0.538038\pi$$
−0.119214 + 0.992869i $$0.538038\pi$$
$$600$$ 0 0
$$601$$ −18267.2 −1.23983 −0.619913 0.784671i $$-0.712832\pi$$
−0.619913 + 0.784671i $$0.712832\pi$$
$$602$$ 8615.06 0.583262
$$603$$ −5534.01 −0.373735
$$604$$ −39972.5 −2.69281
$$605$$ 0 0
$$606$$ −10625.4 −0.712257
$$607$$ 11538.2 0.771534 0.385767 0.922596i $$-0.373937\pi$$
0.385767 + 0.922596i $$0.373937\pi$$
$$608$$ −10704.0 −0.713987
$$609$$ −2236.19 −0.148793
$$610$$ 0 0
$$611$$ −13426.8 −0.889017
$$612$$ 9915.58 0.654924
$$613$$ 21136.9 1.39268 0.696340 0.717713i $$-0.254811\pi$$
0.696340 + 0.717713i $$0.254811\pi$$
$$614$$ 7526.43 0.494694
$$615$$ 0 0
$$616$$ −1038.58 −0.0679314
$$617$$ 15673.2 1.02266 0.511329 0.859385i $$-0.329153\pi$$
0.511329 + 0.859385i $$0.329153\pi$$
$$618$$ −24609.4 −1.60184
$$619$$ 22923.7 1.48850 0.744249 0.667902i $$-0.232807\pi$$
0.744249 + 0.667902i $$0.232807\pi$$
$$620$$ 0 0
$$621$$ 3194.28 0.206413
$$622$$ 21954.1 1.41524
$$623$$ −867.964 −0.0558174
$$624$$ 47123.7 3.02317
$$625$$ 0 0
$$626$$ −5222.84 −0.333461
$$627$$ −210.625 −0.0134156
$$628$$ 44952.4 2.85636
$$629$$ −13114.4 −0.831329
$$630$$ 0 0
$$631$$ 9108.23 0.574632 0.287316 0.957836i $$-0.407237\pi$$
0.287316 + 0.957836i $$0.407237\pi$$
$$632$$ 77519.3 4.87904
$$633$$ 9577.53 0.601379
$$634$$ 11102.0 0.695449
$$635$$ 0 0
$$636$$ −33938.8 −2.11598
$$637$$ −25459.4 −1.58358
$$638$$ 3093.96 0.191992
$$639$$ 2923.40 0.180983
$$640$$ 0 0
$$641$$ 20103.5 1.23875 0.619375 0.785095i $$-0.287386\pi$$
0.619375 + 0.785095i $$0.287386\pi$$
$$642$$ 4254.30 0.261533
$$643$$ 5934.92 0.363997 0.181999 0.983299i $$-0.441743\pi$$
0.181999 + 0.983299i $$0.441743\pi$$
$$644$$ −10871.9 −0.665237
$$645$$ 0 0
$$646$$ −5823.64 −0.354688
$$647$$ −14193.7 −0.862460 −0.431230 0.902242i $$-0.641920\pi$$
−0.431230 + 0.902242i $$0.641920\pi$$
$$648$$ −5520.42 −0.334665
$$649$$ 745.420 0.0450852
$$650$$ 0 0
$$651$$ 812.368 0.0489082
$$652$$ −15823.0 −0.950422
$$653$$ −4795.80 −0.287403 −0.143701 0.989621i $$-0.545900\pi$$
−0.143701 + 0.989621i $$0.545900\pi$$
$$654$$ −17967.5 −1.07429
$$655$$ 0 0
$$656$$ 84314.8 5.01820
$$657$$ −799.841 −0.0474958
$$658$$ −4053.13 −0.240133
$$659$$ 4399.57 0.260065 0.130032 0.991510i $$-0.458492\pi$$
0.130032 + 0.991510i $$0.458492\pi$$
$$660$$ 0 0
$$661$$ 24096.0 1.41789 0.708945 0.705263i $$-0.249171\pi$$
0.708945 + 0.705263i $$0.249171\pi$$
$$662$$ 33006.4 1.93781
$$663$$ 12562.1 0.735853
$$664$$ 51722.2 3.02291
$$665$$ 0 0
$$666$$ 11894.2 0.692028
$$667$$ 19881.4 1.15414
$$668$$ −53126.3 −3.07712
$$669$$ −4666.66 −0.269692
$$670$$ 0 0
$$671$$ −449.725 −0.0258740
$$672$$ 6969.98 0.400109
$$673$$ −27648.3 −1.58360 −0.791800 0.610781i $$-0.790856\pi$$
−0.791800 + 0.610781i $$0.790856\pi$$
$$674$$ −14958.1 −0.854843
$$675$$ 0 0
$$676$$ 82940.9 4.71898
$$677$$ −27605.5 −1.56716 −0.783580 0.621292i $$-0.786608\pi$$
−0.783580 + 0.621292i $$0.786608\pi$$
$$678$$ 15016.6 0.850604
$$679$$ −2310.95 −0.130613
$$680$$ 0 0
$$681$$ −18620.6 −1.04779
$$682$$ −1123.98 −0.0631075
$$683$$ −14949.4 −0.837513 −0.418756 0.908099i $$-0.637534\pi$$
−0.418756 + 0.908099i $$0.637534\pi$$
$$684$$ 3810.42 0.213005
$$685$$ 0 0
$$686$$ −15838.5 −0.881511
$$687$$ 14038.5 0.779626
$$688$$ 72300.4 4.00643
$$689$$ −42997.2 −2.37745
$$690$$ 0 0
$$691$$ 8884.30 0.489110 0.244555 0.969635i $$-0.421358\pi$$
0.244555 + 0.969635i $$0.421358\pi$$
$$692$$ 1073.78 0.0589871
$$693$$ 137.150 0.00751790
$$694$$ −5041.91 −0.275775
$$695$$ 0 0
$$696$$ −34359.4 −1.87125
$$697$$ 22476.4 1.22145
$$698$$ 18861.3 1.02279
$$699$$ −9733.59 −0.526693
$$700$$ 0 0
$$701$$ 10556.9 0.568798 0.284399 0.958706i $$-0.408206\pi$$
0.284399 + 0.958706i $$0.408206\pi$$
$$702$$ −11393.3 −0.612551
$$703$$ −5039.70 −0.270378
$$704$$ −4160.74 −0.222747
$$705$$ 0 0
$$706$$ −26910.2 −1.43453
$$707$$ −2931.58 −0.155945
$$708$$ −13485.4 −0.715836
$$709$$ 25351.9 1.34289 0.671445 0.741055i $$-0.265674\pi$$
0.671445 + 0.741055i $$0.265674\pi$$
$$710$$ 0 0
$$711$$ −10236.8 −0.539959
$$712$$ −13336.4 −0.701968
$$713$$ −7222.53 −0.379363
$$714$$ 3792.11 0.198762
$$715$$ 0 0
$$716$$ −81972.6 −4.27858
$$717$$ −10975.8 −0.571688
$$718$$ −36504.8 −1.89742
$$719$$ 9719.94 0.504162 0.252081 0.967706i $$-0.418885\pi$$
0.252081 + 0.967706i $$0.418885\pi$$
$$720$$ 0 0
$$721$$ −6789.79 −0.350714
$$722$$ 34518.7 1.77930
$$723$$ 5793.04 0.297988
$$724$$ 37375.8 1.91859
$$725$$ 0 0
$$726$$ 21208.3 1.08418
$$727$$ −27509.3 −1.40339 −0.701694 0.712479i $$-0.747572\pi$$
−0.701694 + 0.712479i $$0.747572\pi$$
$$728$$ 23803.9 1.21186
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 19273.6 0.975184
$$732$$ 8135.98 0.410812
$$733$$ −7240.49 −0.364848 −0.182424 0.983220i $$-0.558394\pi$$
−0.182424 + 0.983220i $$0.558394\pi$$
$$734$$ −20085.5 −1.01004
$$735$$ 0 0
$$736$$ −61968.1 −3.10350
$$737$$ 2112.51 0.105584
$$738$$ −20385.1 −1.01678
$$739$$ −15875.3 −0.790234 −0.395117 0.918631i $$-0.629296\pi$$
−0.395117 + 0.918631i $$0.629296\pi$$
$$740$$ 0 0
$$741$$ 4827.44 0.239326
$$742$$ −12979.5 −0.642175
$$743$$ −25714.3 −1.26967 −0.634836 0.772647i $$-0.718932\pi$$
−0.634836 + 0.772647i $$0.718932\pi$$
$$744$$ 12482.1 0.615076
$$745$$ 0 0
$$746$$ 4813.91 0.236260
$$747$$ −6830.18 −0.334542
$$748$$ −3785.10 −0.185023
$$749$$ 1173.77 0.0572612
$$750$$ 0 0
$$751$$ −9709.09 −0.471757 −0.235879 0.971783i $$-0.575797\pi$$
−0.235879 + 0.971783i $$0.575797\pi$$
$$752$$ −34015.2 −1.64948
$$753$$ −17530.0 −0.848379
$$754$$ −70912.1 −3.42502
$$755$$ 0 0
$$756$$ −2481.19 −0.119365
$$757$$ −9567.13 −0.459344 −0.229672 0.973268i $$-0.573765\pi$$
−0.229672 + 0.973268i $$0.573765\pi$$
$$758$$ 50261.1 2.40840
$$759$$ −1219.36 −0.0583137
$$760$$ 0 0
$$761$$ −12322.5 −0.586980 −0.293490 0.955962i $$-0.594817\pi$$
−0.293490 + 0.955962i $$0.594817\pi$$
$$762$$ −10134.0 −0.481782
$$763$$ −4957.28 −0.235211
$$764$$ 75967.0 3.59737
$$765$$ 0 0
$$766$$ 50621.7 2.38778
$$767$$ −17084.7 −0.804293
$$768$$ 7905.27 0.371428
$$769$$ 2575.56 0.120776 0.0603881 0.998175i $$-0.480766\pi$$
0.0603881 + 0.998175i $$0.480766\pi$$
$$770$$ 0 0
$$771$$ 13518.3 0.631454
$$772$$ −56289.8 −2.62424
$$773$$ −6606.23 −0.307386 −0.153693 0.988119i $$-0.549117\pi$$
−0.153693 + 0.988119i $$0.549117\pi$$
$$774$$ −17480.3 −0.811778
$$775$$ 0 0
$$776$$ −35507.9 −1.64260
$$777$$ 3281.63 0.151516
$$778$$ 40959.3 1.88748
$$779$$ 8637.37 0.397260
$$780$$ 0 0
$$781$$ −1115.96 −0.0511294
$$782$$ −33714.5 −1.54173
$$783$$ 4537.33 0.207089
$$784$$ −64498.5 −2.93816
$$785$$ 0 0
$$786$$ 34774.8 1.57809
$$787$$ 16417.0 0.743587 0.371793 0.928315i $$-0.378743\pi$$
0.371793 + 0.928315i $$0.378743\pi$$
$$788$$ 42146.1 1.90532
$$789$$ 16020.5 0.722870
$$790$$ 0 0
$$791$$ 4143.12 0.186235
$$792$$ 2107.33 0.0945462
$$793$$ 10307.5 0.461577
$$794$$ 64379.8 2.87752
$$795$$ 0 0
$$796$$ −32138.9 −1.43107
$$797$$ −3944.19 −0.175295 −0.0876477 0.996152i $$-0.527935\pi$$
−0.0876477 + 0.996152i $$0.527935\pi$$
$$798$$ 1457.26 0.0646445
$$799$$ −9067.66 −0.401490
$$800$$ 0 0
$$801$$ 1761.13 0.0776861
$$802$$ 45757.0 2.01463
$$803$$ 305.325 0.0134181
$$804$$ −38217.5 −1.67640
$$805$$ 0 0
$$806$$ 25761.1 1.12580
$$807$$ 8429.37 0.367693
$$808$$ −45044.0 −1.96119
$$809$$ −17960.7 −0.780549 −0.390275 0.920699i $$-0.627620\pi$$
−0.390275 + 0.920699i $$0.627620\pi$$
$$810$$ 0 0
$$811$$ −13162.5 −0.569912 −0.284956 0.958541i $$-0.591979\pi$$
−0.284956 + 0.958541i $$0.591979\pi$$
$$812$$ −15443.0 −0.667418
$$813$$ 9308.84 0.401569
$$814$$ −4540.41 −0.195505
$$815$$ 0 0
$$816$$ 31824.6 1.36530
$$817$$ 7406.59 0.317165
$$818$$ 65740.9 2.81000
$$819$$ −3143.42 −0.134115
$$820$$ 0 0
$$821$$ 26502.4 1.12660 0.563302 0.826251i $$-0.309531\pi$$
0.563302 + 0.826251i $$0.309531\pi$$
$$822$$ −17983.6 −0.763078
$$823$$ 6937.86 0.293850 0.146925 0.989148i $$-0.453062\pi$$
0.146925 + 0.989148i $$0.453062\pi$$
$$824$$ −104326. −4.41063
$$825$$ 0 0
$$826$$ −5157.35 −0.217248
$$827$$ 41197.9 1.73228 0.866138 0.499805i $$-0.166595\pi$$
0.866138 + 0.499805i $$0.166595\pi$$
$$828$$ 22059.5 0.925871
$$829$$ −693.324 −0.0290472 −0.0145236 0.999895i $$-0.504623\pi$$
−0.0145236 + 0.999895i $$0.504623\pi$$
$$830$$ 0 0
$$831$$ −13796.8 −0.575938
$$832$$ 95362.4 3.97367
$$833$$ −17193.8 −0.715162
$$834$$ 2676.41 0.111123
$$835$$ 0 0
$$836$$ −1454.56 −0.0601760
$$837$$ −1648.33 −0.0680699
$$838$$ −83025.7 −3.42252
$$839$$ −6491.28 −0.267108 −0.133554 0.991042i $$-0.542639\pi$$
−0.133554 + 0.991042i $$0.542639\pi$$
$$840$$ 0 0
$$841$$ 3851.52 0.157921
$$842$$ −40248.9 −1.64735
$$843$$ 7715.49 0.315226
$$844$$ 66141.8 2.69750
$$845$$ 0 0
$$846$$ 8223.97 0.334215
$$847$$ 5851.42 0.237376
$$848$$ −108928. −4.41111
$$849$$ 16725.9 0.676128
$$850$$ 0 0
$$851$$ −29176.1 −1.17526
$$852$$ 20188.8 0.811803
$$853$$ −1116.68 −0.0448233 −0.0224117 0.999749i $$-0.507134\pi$$
−0.0224117 + 0.999749i $$0.507134\pi$$
$$854$$ 3111.52 0.124677
$$855$$ 0 0
$$856$$ 18035.1 0.720126
$$857$$ 44383.9 1.76911 0.884554 0.466438i $$-0.154463\pi$$
0.884554 + 0.466438i $$0.154463\pi$$
$$858$$ 4349.18 0.173052
$$859$$ −25579.3 −1.01601 −0.508006 0.861354i $$-0.669617\pi$$
−0.508006 + 0.861354i $$0.669617\pi$$
$$860$$ 0 0
$$861$$ −5624.28 −0.222619
$$862$$ 84002.5 3.31918
$$863$$ −11194.8 −0.441570 −0.220785 0.975323i $$-0.570862\pi$$
−0.220785 + 0.975323i $$0.570862\pi$$
$$864$$ −14142.4 −0.556867
$$865$$ 0 0
$$866$$ 51471.0 2.01970
$$867$$ −6255.31 −0.245030
$$868$$ 5610.16 0.219379
$$869$$ 3907.73 0.152544
$$870$$ 0 0
$$871$$ −48417.9 −1.88356
$$872$$ −76169.2 −2.95804
$$873$$ 4689.00 0.181785
$$874$$ −12956.0 −0.501424
$$875$$ 0 0
$$876$$ −5523.65 −0.213044
$$877$$ −5721.75 −0.220308 −0.110154 0.993915i $$-0.535134\pi$$
−0.110154 + 0.993915i $$0.535134\pi$$
$$878$$ 34093.6 1.31048
$$879$$ 17382.8 0.667017
$$880$$ 0 0
$$881$$ −34682.8 −1.32633 −0.663163 0.748475i $$-0.730786\pi$$
−0.663163 + 0.748475i $$0.730786\pi$$
$$882$$ 15594.0 0.595326
$$883$$ 37990.4 1.44788 0.723941 0.689862i $$-0.242329\pi$$
0.723941 + 0.689862i $$0.242329\pi$$
$$884$$ 86752.9 3.30070
$$885$$ 0 0
$$886$$ −4992.66 −0.189313
$$887$$ −28299.0 −1.07124 −0.535618 0.844460i $$-0.679921\pi$$
−0.535618 + 0.844460i $$0.679921\pi$$
$$888$$ 50422.7 1.90549
$$889$$ −2796.00 −0.105484
$$890$$ 0 0
$$891$$ −278.283 −0.0104633
$$892$$ −32227.7 −1.20971
$$893$$ −3484.58 −0.130579
$$894$$ 10500.4 0.392825
$$895$$ 0 0
$$896$$ 10200.4 0.380324
$$897$$ 27947.3 1.04028
$$898$$ 100126. 3.72078
$$899$$ −10259.3 −0.380607
$$900$$ 0 0
$$901$$ −29037.8 −1.07368
$$902$$ 7781.65 0.287251
$$903$$ −4822.85 −0.177735
$$904$$ 63659.4 2.34212
$$905$$ 0 0
$$906$$ 31018.1 1.13742
$$907$$ −17388.0 −0.636559 −0.318280 0.947997i $$-0.603105\pi$$
−0.318280 + 0.947997i $$0.603105\pi$$
$$908$$ −128592. −4.69988
$$909$$ 5948.28 0.217043
$$910$$ 0 0
$$911$$ 23555.3 0.856663 0.428332 0.903622i $$-0.359101\pi$$
0.428332 + 0.903622i $$0.359101\pi$$
$$912$$ 12229.8 0.444044
$$913$$ 2607.30 0.0945117
$$914$$ 61980.4 2.24303
$$915$$ 0 0
$$916$$ 96949.1 3.49704
$$917$$ 9594.44 0.345514
$$918$$ −7694.34 −0.276635
$$919$$ −5983.09 −0.214760 −0.107380 0.994218i $$-0.534246\pi$$
−0.107380 + 0.994218i $$0.534246\pi$$
$$920$$ 0 0
$$921$$ −4213.42 −0.150746
$$922$$ −103968. −3.71368
$$923$$ 25577.3 0.912119
$$924$$ 947.150 0.0337218
$$925$$ 0 0
$$926$$ −8447.10 −0.299772
$$927$$ 13776.7 0.488120
$$928$$ −88022.7 −3.11367
$$929$$ 20576.7 0.726694 0.363347 0.931654i $$-0.381634\pi$$
0.363347 + 0.931654i $$0.381634\pi$$
$$930$$ 0 0
$$931$$ −6607.35 −0.232596
$$932$$ −67219.5 −2.36250
$$933$$ −12290.2 −0.431259
$$934$$ −17451.5 −0.611382
$$935$$ 0 0
$$936$$ −48299.0 −1.68665
$$937$$ −11228.6 −0.391485 −0.195743 0.980655i $$-0.562712\pi$$
−0.195743 + 0.980655i $$0.562712\pi$$
$$938$$ −14615.9 −0.508769
$$939$$ 2923.83 0.101614
$$940$$ 0 0
$$941$$ 38567.6 1.33610 0.668049 0.744118i $$-0.267130\pi$$
0.668049 + 0.744118i $$0.267130\pi$$
$$942$$ −34882.4 −1.20651
$$943$$ 50003.9 1.72678
$$944$$ −43282.1 −1.49228
$$945$$ 0 0
$$946$$ 6672.81 0.229336
$$947$$ 4606.17 0.158057 0.0790287 0.996872i $$-0.474818\pi$$
0.0790287 + 0.996872i $$0.474818\pi$$
$$948$$ −70694.8 −2.42200
$$949$$ −6997.93 −0.239370
$$950$$ 0 0
$$951$$ −6215.06 −0.211921
$$952$$ 16075.8 0.547288
$$953$$ −25559.7 −0.868795 −0.434397 0.900721i $$-0.643039\pi$$
−0.434397 + 0.900721i $$0.643039\pi$$
$$954$$ 26336.0 0.893773
$$955$$ 0 0
$$956$$ −75798.5 −2.56433
$$957$$ −1732.05 −0.0585048
$$958$$ −44433.8 −1.49853
$$959$$ −4961.72 −0.167072
$$960$$ 0 0
$$961$$ −26064.0 −0.874895
$$962$$ 104064. 3.48769
$$963$$ −2381.63 −0.0796956
$$964$$ 40006.4 1.33664
$$965$$ 0 0
$$966$$ 8436.42 0.280991
$$967$$ 37895.8 1.26023 0.630117 0.776500i $$-0.283007\pi$$
0.630117 + 0.776500i $$0.283007\pi$$
$$968$$ 89907.7 2.98527
$$969$$ 3260.17 0.108082
$$970$$ 0 0
$$971$$ −46761.0 −1.54545 −0.772726 0.634740i $$-0.781107\pi$$
−0.772726 + 0.634740i $$0.781107\pi$$
$$972$$ 5034.42 0.166131
$$973$$ 738.428 0.0243298
$$974$$ −25500.6 −0.838903
$$975$$ 0 0
$$976$$ 26112.9 0.856407
$$977$$ 3070.29 0.100540 0.0502698 0.998736i $$-0.483992\pi$$
0.0502698 + 0.998736i $$0.483992\pi$$
$$978$$ 12278.4 0.401451
$$979$$ −672.282 −0.0219471
$$980$$ 0 0
$$981$$ 10058.5 0.327364
$$982$$ −20934.3 −0.680287
$$983$$ 16319.0 0.529498 0.264749 0.964317i $$-0.414711\pi$$
0.264749 + 0.964317i $$0.414711\pi$$
$$984$$ −86417.7 −2.79969
$$985$$ 0 0
$$986$$ −47889.9 −1.54678
$$987$$ 2269.01 0.0731747
$$988$$ 33338.0 1.07350
$$989$$ 42878.6 1.37862
$$990$$ 0 0
$$991$$ 5105.79 0.163664 0.0818319 0.996646i $$-0.473923\pi$$
0.0818319 + 0.996646i $$0.473923\pi$$
$$992$$ 31977.0 1.02346
$$993$$ −18477.5 −0.590500
$$994$$ 7720.99 0.246373
$$995$$ 0 0
$$996$$ −47168.7 −1.50060
$$997$$ 7206.97 0.228934 0.114467 0.993427i $$-0.463484\pi$$
0.114467 + 0.993427i $$0.463484\pi$$
$$998$$ 16576.7 0.525779
$$999$$ −6658.57 −0.210879
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.d.1.1 2
3.2 odd 2 225.4.a.n.1.2 2
4.3 odd 2 1200.4.a.bl.1.2 2
5.2 odd 4 75.4.b.c.49.1 4
5.3 odd 4 75.4.b.c.49.4 4
5.4 even 2 75.4.a.e.1.2 yes 2
15.2 even 4 225.4.b.h.199.4 4
15.8 even 4 225.4.b.h.199.1 4
15.14 odd 2 225.4.a.j.1.1 2
20.3 even 4 1200.4.f.v.49.1 4
20.7 even 4 1200.4.f.v.49.4 4
20.19 odd 2 1200.4.a.bu.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.1 2 1.1 even 1 trivial
75.4.a.e.1.2 yes 2 5.4 even 2
75.4.b.c.49.1 4 5.2 odd 4
75.4.b.c.49.4 4 5.3 odd 4
225.4.a.j.1.1 2 15.14 odd 2
225.4.a.n.1.2 2 3.2 odd 2
225.4.b.h.199.1 4 15.8 even 4
225.4.b.h.199.4 4 15.2 even 4
1200.4.a.bl.1.2 2 4.3 odd 2
1200.4.a.bu.1.1 2 20.19 odd 2
1200.4.f.v.49.1 4 20.3 even 4
1200.4.f.v.49.4 4 20.7 even 4