# Properties

 Label 147.4.a.k.1.1 Level $147$ Weight $4$ Character 147.1 Self dual yes Analytic conductor $8.673$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ 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 $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} +3.00000 q^{3} -7.82843 q^{4} -0.100505 q^{5} -1.24264 q^{6} +6.55635 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} +3.00000 q^{3} -7.82843 q^{4} -0.100505 q^{5} -1.24264 q^{6} +6.55635 q^{8} +9.00000 q^{9} +0.0416306 q^{10} -43.9411 q^{11} -23.4853 q^{12} -16.6447 q^{13} -0.301515 q^{15} +59.9117 q^{16} -121.640 q^{17} -3.72792 q^{18} -127.113 q^{19} +0.786797 q^{20} +18.2010 q^{22} +53.5980 q^{23} +19.6690 q^{24} -124.990 q^{25} +6.89444 q^{26} +27.0000 q^{27} +235.681 q^{29} +0.124892 q^{30} -18.7107 q^{31} -77.2670 q^{32} -131.823 q^{33} +50.3848 q^{34} -70.4558 q^{36} -191.882 q^{37} +52.6518 q^{38} -49.9340 q^{39} -0.658946 q^{40} -319.713 q^{41} -218.579 q^{43} +343.990 q^{44} -0.904546 q^{45} -22.2010 q^{46} +401.553 q^{47} +179.735 q^{48} +51.7725 q^{50} -364.919 q^{51} +130.302 q^{52} +643.117 q^{53} -11.1838 q^{54} +4.41631 q^{55} -381.338 q^{57} -97.6224 q^{58} -11.6123 q^{59} +2.36039 q^{60} -12.2426 q^{61} +7.75022 q^{62} -447.288 q^{64} +1.67287 q^{65} +54.6030 q^{66} +669.048 q^{67} +952.247 q^{68} +160.794 q^{69} +822.098 q^{71} +59.0071 q^{72} +515.100 q^{73} +79.4802 q^{74} -374.970 q^{75} +995.092 q^{76} +20.6833 q^{78} -805.754 q^{79} -6.02143 q^{80} +81.0000 q^{81} +132.429 q^{82} -394.863 q^{83} +12.2254 q^{85} +90.5382 q^{86} +707.044 q^{87} -288.093 q^{88} +673.418 q^{89} +0.374675 q^{90} -419.588 q^{92} -56.1320 q^{93} -166.329 q^{94} +12.7755 q^{95} -231.801 q^{96} -1091.11 q^{97} -395.470 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 6q^{3} - 10q^{4} - 20q^{5} + 6q^{6} - 18q^{8} + 18q^{9} - 48q^{10} - 20q^{11} - 30q^{12} - 104q^{13} - 60q^{15} + 18q^{16} - 116q^{17} + 18q^{18} - 192q^{19} + 44q^{20} + 76q^{22} + 28q^{23} - 54q^{24} + 146q^{25} - 204q^{26} + 54q^{27} + 296q^{29} - 144q^{30} + 104q^{31} + 18q^{32} - 60q^{33} + 64q^{34} - 90q^{36} - 248q^{37} - 104q^{38} - 312q^{39} + 488q^{40} - 20q^{41} - 720q^{43} + 292q^{44} - 180q^{45} - 84q^{46} + 96q^{47} + 54q^{48} + 706q^{50} - 348q^{51} + 320q^{52} + 268q^{53} + 54q^{54} - 472q^{55} - 576q^{57} + 48q^{58} + 616q^{59} + 132q^{60} - 16q^{61} + 304q^{62} + 118q^{64} + 1740q^{65} + 228q^{66} - 144q^{67} + 940q^{68} + 84q^{69} + 988q^{71} - 162q^{72} - 104q^{73} - 56q^{74} + 438q^{75} + 1136q^{76} - 612q^{78} - 944q^{79} + 828q^{80} + 162q^{81} + 856q^{82} - 1016q^{83} - 100q^{85} - 1120q^{86} + 888q^{87} - 876q^{88} + 388q^{89} - 432q^{90} - 364q^{92} + 312q^{93} - 904q^{94} + 1304q^{95} + 54q^{96} - 488q^{97} - 180q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.146447 −0.0732233 0.997316i $$-0.523329\pi$$
−0.0732233 + 0.997316i $$0.523329\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −7.82843 −0.978553
$$5$$ −0.100505 −0.00898945 −0.00449472 0.999990i $$-0.501431\pi$$
−0.00449472 + 0.999990i $$0.501431\pi$$
$$6$$ −1.24264 −0.0845510
$$7$$ 0 0
$$8$$ 6.55635 0.289752
$$9$$ 9.00000 0.333333
$$10$$ 0.0416306 0.00131647
$$11$$ −43.9411 −1.20443 −0.602216 0.798333i $$-0.705715\pi$$
−0.602216 + 0.798333i $$0.705715\pi$$
$$12$$ −23.4853 −0.564968
$$13$$ −16.6447 −0.355108 −0.177554 0.984111i $$-0.556818\pi$$
−0.177554 + 0.984111i $$0.556818\pi$$
$$14$$ 0 0
$$15$$ −0.301515 −0.00519006
$$16$$ 59.9117 0.936120
$$17$$ −121.640 −1.73541 −0.867704 0.497081i $$-0.834405\pi$$
−0.867704 + 0.497081i $$0.834405\pi$$
$$18$$ −3.72792 −0.0488155
$$19$$ −127.113 −1.53482 −0.767412 0.641154i $$-0.778456\pi$$
−0.767412 + 0.641154i $$0.778456\pi$$
$$20$$ 0.786797 0.00879665
$$21$$ 0 0
$$22$$ 18.2010 0.176385
$$23$$ 53.5980 0.485911 0.242955 0.970037i $$-0.421883\pi$$
0.242955 + 0.970037i $$0.421883\pi$$
$$24$$ 19.6690 0.167289
$$25$$ −124.990 −0.999919
$$26$$ 6.89444 0.0520043
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 235.681 1.50913 0.754567 0.656223i $$-0.227847\pi$$
0.754567 + 0.656223i $$0.227847\pi$$
$$30$$ 0.124892 0.000760067 0
$$31$$ −18.7107 −0.108404 −0.0542022 0.998530i $$-0.517262\pi$$
−0.0542022 + 0.998530i $$0.517262\pi$$
$$32$$ −77.2670 −0.426844
$$33$$ −131.823 −0.695379
$$34$$ 50.3848 0.254145
$$35$$ 0 0
$$36$$ −70.4558 −0.326184
$$37$$ −191.882 −0.852574 −0.426287 0.904588i $$-0.640179\pi$$
−0.426287 + 0.904588i $$0.640179\pi$$
$$38$$ 52.6518 0.224770
$$39$$ −49.9340 −0.205021
$$40$$ −0.658946 −0.00260471
$$41$$ −319.713 −1.21782 −0.608912 0.793238i $$-0.708394\pi$$
−0.608912 + 0.793238i $$0.708394\pi$$
$$42$$ 0 0
$$43$$ −218.579 −0.775184 −0.387592 0.921831i $$-0.626693\pi$$
−0.387592 + 0.921831i $$0.626693\pi$$
$$44$$ 343.990 1.17860
$$45$$ −0.904546 −0.00299648
$$46$$ −22.2010 −0.0711600
$$47$$ 401.553 1.24623 0.623113 0.782132i $$-0.285868\pi$$
0.623113 + 0.782132i $$0.285868\pi$$
$$48$$ 179.735 0.540469
$$49$$ 0 0
$$50$$ 51.7725 0.146435
$$51$$ −364.919 −1.00194
$$52$$ 130.302 0.347492
$$53$$ 643.117 1.66677 0.833386 0.552692i $$-0.186399\pi$$
0.833386 + 0.552692i $$0.186399\pi$$
$$54$$ −11.1838 −0.0281837
$$55$$ 4.41631 0.0108272
$$56$$ 0 0
$$57$$ −381.338 −0.886131
$$58$$ −97.6224 −0.221008
$$59$$ −11.6123 −0.0256235 −0.0128118 0.999918i $$-0.504078\pi$$
−0.0128118 + 0.999918i $$0.504078\pi$$
$$60$$ 2.36039 0.00507875
$$61$$ −12.2426 −0.0256969 −0.0128484 0.999917i $$-0.504090\pi$$
−0.0128484 + 0.999917i $$0.504090\pi$$
$$62$$ 7.75022 0.0158755
$$63$$ 0 0
$$64$$ −447.288 −0.873610
$$65$$ 1.67287 0.00319222
$$66$$ 54.6030 0.101836
$$67$$ 669.048 1.21996 0.609979 0.792417i $$-0.291178\pi$$
0.609979 + 0.792417i $$0.291178\pi$$
$$68$$ 952.247 1.69819
$$69$$ 160.794 0.280541
$$70$$ 0 0
$$71$$ 822.098 1.37416 0.687078 0.726584i $$-0.258893\pi$$
0.687078 + 0.726584i $$0.258893\pi$$
$$72$$ 59.0071 0.0965841
$$73$$ 515.100 0.825861 0.412930 0.910763i $$-0.364505\pi$$
0.412930 + 0.910763i $$0.364505\pi$$
$$74$$ 79.4802 0.124857
$$75$$ −374.970 −0.577304
$$76$$ 995.092 1.50191
$$77$$ 0 0
$$78$$ 20.6833 0.0300247
$$79$$ −805.754 −1.14752 −0.573762 0.819022i $$-0.694517\pi$$
−0.573762 + 0.819022i $$0.694517\pi$$
$$80$$ −6.02143 −0.00841520
$$81$$ 81.0000 0.111111
$$82$$ 132.429 0.178346
$$83$$ −394.863 −0.522191 −0.261095 0.965313i $$-0.584084\pi$$
−0.261095 + 0.965313i $$0.584084\pi$$
$$84$$ 0 0
$$85$$ 12.2254 0.0156004
$$86$$ 90.5382 0.113523
$$87$$ 707.044 0.871299
$$88$$ −288.093 −0.348987
$$89$$ 673.418 0.802047 0.401024 0.916068i $$-0.368655\pi$$
0.401024 + 0.916068i $$0.368655\pi$$
$$90$$ 0.374675 0.000438825 0
$$91$$ 0 0
$$92$$ −419.588 −0.475490
$$93$$ −56.1320 −0.0625873
$$94$$ −166.329 −0.182505
$$95$$ 12.7755 0.0137972
$$96$$ −231.801 −0.246439
$$97$$ −1091.11 −1.14212 −0.571061 0.820908i $$-0.693468\pi$$
−0.571061 + 0.820908i $$0.693468\pi$$
$$98$$ 0 0
$$99$$ −395.470 −0.401477
$$100$$ 978.474 0.978474
$$101$$ −1370.79 −1.35048 −0.675242 0.737597i $$-0.735961\pi$$
−0.675242 + 0.737597i $$0.735961\pi$$
$$102$$ 151.154 0.146730
$$103$$ −1413.96 −1.35263 −0.676316 0.736611i $$-0.736425\pi$$
−0.676316 + 0.736611i $$0.736425\pi$$
$$104$$ −109.128 −0.102893
$$105$$ 0 0
$$106$$ −266.388 −0.244093
$$107$$ −343.539 −0.310385 −0.155192 0.987884i $$-0.549600\pi$$
−0.155192 + 0.987884i $$0.549600\pi$$
$$108$$ −211.368 −0.188323
$$109$$ −317.657 −0.279138 −0.139569 0.990212i $$-0.544572\pi$$
−0.139569 + 0.990212i $$0.544572\pi$$
$$110$$ −1.82929 −0.00158560
$$111$$ −575.647 −0.492234
$$112$$ 0 0
$$113$$ 798.373 0.664643 0.332321 0.943166i $$-0.392168\pi$$
0.332321 + 0.943166i $$0.392168\pi$$
$$114$$ 157.955 0.129771
$$115$$ −5.38687 −0.00436807
$$116$$ −1845.01 −1.47677
$$117$$ −149.802 −0.118369
$$118$$ 4.80996 0.00375248
$$119$$ 0 0
$$120$$ −1.97684 −0.00150383
$$121$$ 599.823 0.450656
$$122$$ 5.07107 0.00376322
$$123$$ −959.138 −0.703110
$$124$$ 146.475 0.106080
$$125$$ 25.1253 0.0179782
$$126$$ 0 0
$$127$$ 1071.40 0.748593 0.374297 0.927309i $$-0.377884\pi$$
0.374297 + 0.927309i $$0.377884\pi$$
$$128$$ 803.409 0.554781
$$129$$ −655.736 −0.447553
$$130$$ −0.692927 −0.000467490 0
$$131$$ −2515.02 −1.67739 −0.838695 0.544601i $$-0.816681\pi$$
−0.838695 + 0.544601i $$0.816681\pi$$
$$132$$ 1031.97 0.680465
$$133$$ 0 0
$$134$$ −277.129 −0.178659
$$135$$ −2.71364 −0.00173002
$$136$$ −797.512 −0.502839
$$137$$ 251.064 0.156568 0.0782841 0.996931i $$-0.475056\pi$$
0.0782841 + 0.996931i $$0.475056\pi$$
$$138$$ −66.6030 −0.0410842
$$139$$ 886.067 0.540685 0.270343 0.962764i $$-0.412863\pi$$
0.270343 + 0.962764i $$0.412863\pi$$
$$140$$ 0 0
$$141$$ 1204.66 0.719508
$$142$$ −340.524 −0.201240
$$143$$ 731.385 0.427703
$$144$$ 539.205 0.312040
$$145$$ −23.6872 −0.0135663
$$146$$ −213.361 −0.120945
$$147$$ 0 0
$$148$$ 1502.14 0.834289
$$149$$ 582.626 0.320339 0.160170 0.987090i $$-0.448796\pi$$
0.160170 + 0.987090i $$0.448796\pi$$
$$150$$ 155.318 0.0845442
$$151$$ −2811.76 −1.51535 −0.757676 0.652631i $$-0.773665\pi$$
−0.757676 + 0.652631i $$0.773665\pi$$
$$152$$ −833.395 −0.444719
$$153$$ −1094.76 −0.578469
$$154$$ 0 0
$$155$$ 1.88052 0.000974496 0
$$156$$ 390.905 0.200624
$$157$$ −1691.34 −0.859770 −0.429885 0.902884i $$-0.641446\pi$$
−0.429885 + 0.902884i $$0.641446\pi$$
$$158$$ 333.754 0.168051
$$159$$ 1929.35 0.962311
$$160$$ 7.76573 0.00383709
$$161$$ 0 0
$$162$$ −33.5513 −0.0162718
$$163$$ −40.7232 −0.0195686 −0.00978432 0.999952i $$-0.503114\pi$$
−0.00978432 + 0.999952i $$0.503114\pi$$
$$164$$ 2502.85 1.19170
$$165$$ 13.2489 0.00625107
$$166$$ 163.558 0.0764731
$$167$$ 2900.47 1.34398 0.671990 0.740560i $$-0.265440\pi$$
0.671990 + 0.740560i $$0.265440\pi$$
$$168$$ 0 0
$$169$$ −1919.96 −0.873899
$$170$$ −5.06393 −0.00228462
$$171$$ −1144.01 −0.511608
$$172$$ 1711.13 0.758559
$$173$$ −2146.15 −0.943171 −0.471585 0.881820i $$-0.656318\pi$$
−0.471585 + 0.881820i $$0.656318\pi$$
$$174$$ −292.867 −0.127599
$$175$$ 0 0
$$176$$ −2632.59 −1.12749
$$177$$ −34.8368 −0.0147938
$$178$$ −278.939 −0.117457
$$179$$ 1203.54 0.502552 0.251276 0.967916i $$-0.419150\pi$$
0.251276 + 0.967916i $$0.419150\pi$$
$$180$$ 7.08117 0.00293222
$$181$$ −2990.47 −1.22807 −0.614033 0.789280i $$-0.710454\pi$$
−0.614033 + 0.789280i $$0.710454\pi$$
$$182$$ 0 0
$$183$$ −36.7279 −0.0148361
$$184$$ 351.407 0.140794
$$185$$ 19.2851 0.00766417
$$186$$ 23.2506 0.00916570
$$187$$ 5344.98 2.09018
$$188$$ −3143.53 −1.21950
$$189$$ 0 0
$$190$$ −5.29177 −0.00202056
$$191$$ −2807.41 −1.06354 −0.531772 0.846887i $$-0.678474\pi$$
−0.531772 + 0.846887i $$0.678474\pi$$
$$192$$ −1341.87 −0.504379
$$193$$ 3336.37 1.24434 0.622169 0.782883i $$-0.286252\pi$$
0.622169 + 0.782883i $$0.286252\pi$$
$$194$$ 451.954 0.167260
$$195$$ 5.01862 0.00184303
$$196$$ 0 0
$$197$$ −4226.65 −1.52861 −0.764305 0.644855i $$-0.776918\pi$$
−0.764305 + 0.644855i $$0.776918\pi$$
$$198$$ 163.809 0.0587950
$$199$$ 4385.69 1.56228 0.781140 0.624356i $$-0.214639\pi$$
0.781140 + 0.624356i $$0.214639\pi$$
$$200$$ −819.477 −0.289729
$$201$$ 2007.14 0.704343
$$202$$ 567.800 0.197774
$$203$$ 0 0
$$204$$ 2856.74 0.980450
$$205$$ 32.1328 0.0109476
$$206$$ 585.680 0.198088
$$207$$ 482.382 0.161970
$$208$$ −997.210 −0.332423
$$209$$ 5585.48 1.84859
$$210$$ 0 0
$$211$$ 2291.56 0.747665 0.373833 0.927496i $$-0.378043\pi$$
0.373833 + 0.927496i $$0.378043\pi$$
$$212$$ −5034.59 −1.63103
$$213$$ 2466.29 0.793369
$$214$$ 142.299 0.0454548
$$215$$ 21.9683 0.00696848
$$216$$ 177.021 0.0557629
$$217$$ 0 0
$$218$$ 131.578 0.0408788
$$219$$ 1545.30 0.476811
$$220$$ −34.5727 −0.0105950
$$221$$ 2024.65 0.616257
$$222$$ 238.441 0.0720860
$$223$$ −217.970 −0.0654544 −0.0327272 0.999464i $$-0.510419\pi$$
−0.0327272 + 0.999464i $$0.510419\pi$$
$$224$$ 0 0
$$225$$ −1124.91 −0.333306
$$226$$ −330.697 −0.0973347
$$227$$ 1835.36 0.536639 0.268320 0.963330i $$-0.413532\pi$$
0.268320 + 0.963330i $$0.413532\pi$$
$$228$$ 2985.28 0.867126
$$229$$ 2774.01 0.800488 0.400244 0.916409i $$-0.368925\pi$$
0.400244 + 0.916409i $$0.368925\pi$$
$$230$$ 2.23131 0.000639689 0
$$231$$ 0 0
$$232$$ 1545.21 0.437275
$$233$$ −988.712 −0.277994 −0.138997 0.990293i $$-0.544388\pi$$
−0.138997 + 0.990293i $$0.544388\pi$$
$$234$$ 62.0500 0.0173348
$$235$$ −40.3581 −0.0112029
$$236$$ 90.9058 0.0250740
$$237$$ −2417.26 −0.662524
$$238$$ 0 0
$$239$$ −837.928 −0.226783 −0.113391 0.993550i $$-0.536171\pi$$
−0.113391 + 0.993550i $$0.536171\pi$$
$$240$$ −18.0643 −0.00485852
$$241$$ −3454.99 −0.923466 −0.461733 0.887019i $$-0.652772\pi$$
−0.461733 + 0.887019i $$0.652772\pi$$
$$242$$ −248.455 −0.0659970
$$243$$ 243.000 0.0641500
$$244$$ 95.8406 0.0251458
$$245$$ 0 0
$$246$$ 397.288 0.102968
$$247$$ 2115.75 0.545028
$$248$$ −122.674 −0.0314104
$$249$$ −1184.59 −0.301487
$$250$$ −10.4072 −0.00263284
$$251$$ 5635.01 1.41705 0.708523 0.705688i $$-0.249362\pi$$
0.708523 + 0.705688i $$0.249362\pi$$
$$252$$ 0 0
$$253$$ −2355.16 −0.585246
$$254$$ −443.788 −0.109629
$$255$$ 36.6762 0.00900687
$$256$$ 3245.52 0.792364
$$257$$ −2271.16 −0.551248 −0.275624 0.961265i $$-0.588885\pi$$
−0.275624 + 0.961265i $$0.588885\pi$$
$$258$$ 271.615 0.0655426
$$259$$ 0 0
$$260$$ −13.0960 −0.00312376
$$261$$ 2121.13 0.503045
$$262$$ 1041.75 0.245648
$$263$$ 163.867 0.0384201 0.0192101 0.999815i $$-0.493885\pi$$
0.0192101 + 0.999815i $$0.493885\pi$$
$$264$$ −864.280 −0.201488
$$265$$ −64.6365 −0.0149834
$$266$$ 0 0
$$267$$ 2020.26 0.463062
$$268$$ −5237.59 −1.19379
$$269$$ −5167.10 −1.17116 −0.585582 0.810613i $$-0.699134\pi$$
−0.585582 + 0.810613i $$0.699134\pi$$
$$270$$ 1.12403 0.000253356 0
$$271$$ −1622.27 −0.363638 −0.181819 0.983332i $$-0.558199\pi$$
−0.181819 + 0.983332i $$0.558199\pi$$
$$272$$ −7287.63 −1.62455
$$273$$ 0 0
$$274$$ −103.994 −0.0229289
$$275$$ 5492.20 1.20433
$$276$$ −1258.76 −0.274524
$$277$$ −4612.37 −1.00047 −0.500235 0.865890i $$-0.666753\pi$$
−0.500235 + 0.865890i $$0.666753\pi$$
$$278$$ −367.021 −0.0791815
$$279$$ −168.396 −0.0361348
$$280$$ 0 0
$$281$$ −2125.22 −0.451174 −0.225587 0.974223i $$-0.572430\pi$$
−0.225587 + 0.974223i $$0.572430\pi$$
$$282$$ −498.987 −0.105370
$$283$$ 2571.54 0.540149 0.270075 0.962839i $$-0.412952\pi$$
0.270075 + 0.962839i $$0.412952\pi$$
$$284$$ −6435.73 −1.34468
$$285$$ 38.3264 0.00796583
$$286$$ −302.950 −0.0626356
$$287$$ 0 0
$$288$$ −695.403 −0.142281
$$289$$ 9883.19 2.01164
$$290$$ 9.81154 0.00198674
$$291$$ −3273.34 −0.659404
$$292$$ −4032.42 −0.808149
$$293$$ −3324.96 −0.662957 −0.331478 0.943463i $$-0.607547\pi$$
−0.331478 + 0.943463i $$0.607547\pi$$
$$294$$ 0 0
$$295$$ 1.16709 0.000230341 0
$$296$$ −1258.05 −0.247035
$$297$$ −1186.41 −0.231793
$$298$$ −241.331 −0.0469126
$$299$$ −892.120 −0.172551
$$300$$ 2935.42 0.564922
$$301$$ 0 0
$$302$$ 1164.67 0.221918
$$303$$ −4112.37 −0.779702
$$304$$ −7615.54 −1.43678
$$305$$ 1.23045 0.000231001 0
$$306$$ 453.463 0.0847149
$$307$$ 887.096 0.164916 0.0824580 0.996595i $$-0.473723\pi$$
0.0824580 + 0.996595i $$0.473723\pi$$
$$308$$ 0 0
$$309$$ −4241.87 −0.780943
$$310$$ −0.778936 −0.000142712 0
$$311$$ 4510.82 0.822460 0.411230 0.911532i $$-0.365099\pi$$
0.411230 + 0.911532i $$0.365099\pi$$
$$312$$ −327.385 −0.0594055
$$313$$ −3715.78 −0.671018 −0.335509 0.942037i $$-0.608908\pi$$
−0.335509 + 0.942037i $$0.608908\pi$$
$$314$$ 700.577 0.125910
$$315$$ 0 0
$$316$$ 6307.79 1.12291
$$317$$ 6954.52 1.23219 0.616096 0.787671i $$-0.288713\pi$$
0.616096 + 0.787671i $$0.288713\pi$$
$$318$$ −799.163 −0.140927
$$319$$ −10356.1 −1.81765
$$320$$ 44.9548 0.00785327
$$321$$ −1030.62 −0.179201
$$322$$ 0 0
$$323$$ 15461.9 2.66355
$$324$$ −634.103 −0.108728
$$325$$ 2080.41 0.355079
$$326$$ 16.8681 0.00286576
$$327$$ −952.971 −0.161160
$$328$$ −2096.15 −0.352867
$$329$$ 0 0
$$330$$ −5.48788 −0.000915448 0
$$331$$ −9863.18 −1.63785 −0.818926 0.573899i $$-0.805430\pi$$
−0.818926 + 0.573899i $$0.805430\pi$$
$$332$$ 3091.16 0.510992
$$333$$ −1726.94 −0.284191
$$334$$ −1201.41 −0.196821
$$335$$ −67.2427 −0.0109668
$$336$$ 0 0
$$337$$ −5945.06 −0.960974 −0.480487 0.877002i $$-0.659540\pi$$
−0.480487 + 0.877002i $$0.659540\pi$$
$$338$$ 795.272 0.127979
$$339$$ 2395.12 0.383732
$$340$$ −95.7056 −0.0152658
$$341$$ 822.168 0.130566
$$342$$ 473.866 0.0749232
$$343$$ 0 0
$$344$$ −1433.08 −0.224612
$$345$$ −16.1606 −0.00252191
$$346$$ 888.963 0.138124
$$347$$ 1169.57 0.180939 0.0904697 0.995899i $$-0.471163\pi$$
0.0904697 + 0.995899i $$0.471163\pi$$
$$348$$ −5535.04 −0.852613
$$349$$ 9176.66 1.40749 0.703747 0.710451i $$-0.251509\pi$$
0.703747 + 0.710451i $$0.251509\pi$$
$$350$$ 0 0
$$351$$ −449.406 −0.0683405
$$352$$ 3395.20 0.514104
$$353$$ −10587.2 −1.59632 −0.798158 0.602448i $$-0.794192\pi$$
−0.798158 + 0.602448i $$0.794192\pi$$
$$354$$ 14.4299 0.00216649
$$355$$ −82.6250 −0.0123529
$$356$$ −5271.81 −0.784846
$$357$$ 0 0
$$358$$ −498.522 −0.0735970
$$359$$ 8615.21 1.26656 0.633278 0.773924i $$-0.281709\pi$$
0.633278 + 0.773924i $$0.281709\pi$$
$$360$$ −5.93052 −0.000868238 0
$$361$$ 9298.64 1.35568
$$362$$ 1238.69 0.179846
$$363$$ 1799.47 0.260186
$$364$$ 0 0
$$365$$ −51.7701 −0.00742403
$$366$$ 15.2132 0.00217270
$$367$$ −8297.79 −1.18022 −0.590110 0.807323i $$-0.700916\pi$$
−0.590110 + 0.807323i $$0.700916\pi$$
$$368$$ 3211.15 0.454871
$$369$$ −2877.41 −0.405941
$$370$$ −7.98817 −0.00112239
$$371$$ 0 0
$$372$$ 439.426 0.0612450
$$373$$ −5123.86 −0.711269 −0.355634 0.934625i $$-0.615735\pi$$
−0.355634 + 0.934625i $$0.615735\pi$$
$$374$$ −2213.96 −0.306100
$$375$$ 75.3758 0.0103797
$$376$$ 2632.72 0.361097
$$377$$ −3922.83 −0.535905
$$378$$ 0 0
$$379$$ −1502.49 −0.203635 −0.101817 0.994803i $$-0.532466\pi$$
−0.101817 + 0.994803i $$0.532466\pi$$
$$380$$ −100.012 −0.0135013
$$381$$ 3214.20 0.432200
$$382$$ 1162.87 0.155753
$$383$$ 10872.9 1.45060 0.725301 0.688431i $$-0.241700\pi$$
0.725301 + 0.688431i $$0.241700\pi$$
$$384$$ 2410.23 0.320303
$$385$$ 0 0
$$386$$ −1381.97 −0.182229
$$387$$ −1967.21 −0.258395
$$388$$ 8541.71 1.11763
$$389$$ 4618.99 0.602036 0.301018 0.953618i $$-0.402674\pi$$
0.301018 + 0.953618i $$0.402674\pi$$
$$390$$ −2.07878 −0.000269905 0
$$391$$ −6519.64 −0.843254
$$392$$ 0 0
$$393$$ −7545.05 −0.968442
$$394$$ 1750.73 0.223860
$$395$$ 80.9824 0.0103156
$$396$$ 3095.91 0.392867
$$397$$ −9606.95 −1.21451 −0.607253 0.794508i $$-0.707729\pi$$
−0.607253 + 0.794508i $$0.707729\pi$$
$$398$$ −1816.61 −0.228791
$$399$$ 0 0
$$400$$ −7488.36 −0.936044
$$401$$ −10501.0 −1.30772 −0.653862 0.756614i $$-0.726852\pi$$
−0.653862 + 0.756614i $$0.726852\pi$$
$$402$$ −831.386 −0.103149
$$403$$ 311.433 0.0384952
$$404$$ 10731.1 1.32152
$$405$$ −8.14091 −0.000998827 0
$$406$$ 0 0
$$407$$ 8431.52 1.02687
$$408$$ −2392.54 −0.290314
$$409$$ 12066.9 1.45885 0.729427 0.684059i $$-0.239787\pi$$
0.729427 + 0.684059i $$0.239787\pi$$
$$410$$ −13.3098 −0.00160323
$$411$$ 753.192 0.0903947
$$412$$ 11069.0 1.32362
$$413$$ 0 0
$$414$$ −199.809 −0.0237200
$$415$$ 39.6857 0.00469421
$$416$$ 1286.08 0.151576
$$417$$ 2658.20 0.312165
$$418$$ −2313.58 −0.270720
$$419$$ 6366.31 0.742278 0.371139 0.928577i $$-0.378967\pi$$
0.371139 + 0.928577i $$0.378967\pi$$
$$420$$ 0 0
$$421$$ −4731.84 −0.547781 −0.273890 0.961761i $$-0.588311\pi$$
−0.273890 + 0.961761i $$0.588311\pi$$
$$422$$ −949.194 −0.109493
$$423$$ 3613.98 0.415408
$$424$$ 4216.50 0.482951
$$425$$ 15203.7 1.73527
$$426$$ −1021.57 −0.116186
$$427$$ 0 0
$$428$$ 2689.37 0.303728
$$429$$ 2194.16 0.246934
$$430$$ −9.09955 −0.00102051
$$431$$ 3752.78 0.419409 0.209704 0.977765i $$-0.432750\pi$$
0.209704 + 0.977765i $$0.432750\pi$$
$$432$$ 1617.62 0.180156
$$433$$ −11709.2 −1.29956 −0.649780 0.760122i $$-0.725139\pi$$
−0.649780 + 0.760122i $$0.725139\pi$$
$$434$$ 0 0
$$435$$ −71.0615 −0.00783250
$$436$$ 2486.75 0.273151
$$437$$ −6812.98 −0.745788
$$438$$ −640.084 −0.0698274
$$439$$ 14924.7 1.62259 0.811296 0.584635i $$-0.198762\pi$$
0.811296 + 0.584635i $$0.198762\pi$$
$$440$$ 28.9548 0.00313720
$$441$$ 0 0
$$442$$ −838.638 −0.0902487
$$443$$ −8517.66 −0.913513 −0.456756 0.889592i $$-0.650989\pi$$
−0.456756 + 0.889592i $$0.650989\pi$$
$$444$$ 4506.41 0.481677
$$445$$ −67.6820 −0.00720996
$$446$$ 90.2860 0.00958557
$$447$$ 1747.88 0.184948
$$448$$ 0 0
$$449$$ −5965.73 −0.627038 −0.313519 0.949582i $$-0.601508\pi$$
−0.313519 + 0.949582i $$0.601508\pi$$
$$450$$ 465.953 0.0488116
$$451$$ 14048.5 1.46678
$$452$$ −6250.01 −0.650389
$$453$$ −8435.29 −0.874889
$$454$$ −760.231 −0.0785890
$$455$$ 0 0
$$456$$ −2500.19 −0.256759
$$457$$ −13860.6 −1.41875 −0.709376 0.704830i $$-0.751023\pi$$
−0.709376 + 0.704830i $$0.751023\pi$$
$$458$$ −1149.03 −0.117229
$$459$$ −3284.27 −0.333979
$$460$$ 42.1707 0.00427439
$$461$$ 149.312 0.0150850 0.00754249 0.999972i $$-0.497599\pi$$
0.00754249 + 0.999972i $$0.497599\pi$$
$$462$$ 0 0
$$463$$ 5403.95 0.542425 0.271213 0.962519i $$-0.412575\pi$$
0.271213 + 0.962519i $$0.412575\pi$$
$$464$$ 14120.1 1.41273
$$465$$ 5.64155 0.000562625 0
$$466$$ 409.538 0.0407113
$$467$$ −3704.66 −0.367090 −0.183545 0.983011i $$-0.558757\pi$$
−0.183545 + 0.983011i $$0.558757\pi$$
$$468$$ 1172.71 0.115831
$$469$$ 0 0
$$470$$ 16.7169 0.00164062
$$471$$ −5074.03 −0.496389
$$472$$ −76.1341 −0.00742448
$$473$$ 9604.59 0.933657
$$474$$ 1001.26 0.0970244
$$475$$ 15887.8 1.53470
$$476$$ 0 0
$$477$$ 5788.05 0.555591
$$478$$ 347.081 0.0332115
$$479$$ 10671.8 1.01796 0.508982 0.860777i $$-0.330022\pi$$
0.508982 + 0.860777i $$0.330022\pi$$
$$480$$ 23.2972 0.00221535
$$481$$ 3193.82 0.302756
$$482$$ 1431.10 0.135238
$$483$$ 0 0
$$484$$ −4695.67 −0.440990
$$485$$ 109.662 0.0102670
$$486$$ −100.654 −0.00939455
$$487$$ 5853.92 0.544695 0.272348 0.962199i $$-0.412200\pi$$
0.272348 + 0.962199i $$0.412200\pi$$
$$488$$ −80.2670 −0.00744573
$$489$$ −122.170 −0.0112980
$$490$$ 0 0
$$491$$ 4065.31 0.373656 0.186828 0.982393i $$-0.440179\pi$$
0.186828 + 0.982393i $$0.440179\pi$$
$$492$$ 7508.54 0.688031
$$493$$ −28668.2 −2.61896
$$494$$ −876.371 −0.0798174
$$495$$ 39.7468 0.00360906
$$496$$ −1120.99 −0.101480
$$497$$ 0 0
$$498$$ 490.673 0.0441517
$$499$$ 4811.19 0.431620 0.215810 0.976435i $$-0.430761\pi$$
0.215810 + 0.976435i $$0.430761\pi$$
$$500$$ −196.691 −0.0175926
$$501$$ 8701.40 0.775947
$$502$$ −2334.10 −0.207522
$$503$$ −17001.2 −1.50705 −0.753526 0.657418i $$-0.771649\pi$$
−0.753526 + 0.657418i $$0.771649\pi$$
$$504$$ 0 0
$$505$$ 137.771 0.0121401
$$506$$ 975.537 0.0857074
$$507$$ −5759.87 −0.504546
$$508$$ −8387.37 −0.732538
$$509$$ −13797.2 −1.20148 −0.600738 0.799446i $$-0.705126\pi$$
−0.600738 + 0.799446i $$0.705126\pi$$
$$510$$ −15.1918 −0.00131903
$$511$$ 0 0
$$512$$ −7771.61 −0.670820
$$513$$ −3432.04 −0.295377
$$514$$ 940.744 0.0807284
$$515$$ 142.110 0.0121594
$$516$$ 5133.38 0.437954
$$517$$ −17644.7 −1.50099
$$518$$ 0 0
$$519$$ −6438.44 −0.544540
$$520$$ 10.9679 0.000924954 0
$$521$$ −3936.61 −0.331029 −0.165515 0.986207i $$-0.552928\pi$$
−0.165515 + 0.986207i $$0.552928\pi$$
$$522$$ −878.601 −0.0736692
$$523$$ −17459.3 −1.45973 −0.729866 0.683590i $$-0.760418\pi$$
−0.729866 + 0.683590i $$0.760418\pi$$
$$524$$ 19688.6 1.64142
$$525$$ 0 0
$$526$$ −67.8760 −0.00562649
$$527$$ 2275.96 0.188126
$$528$$ −7897.76 −0.650958
$$529$$ −9294.26 −0.763891
$$530$$ 26.7733 0.00219426
$$531$$ −104.510 −0.00854118
$$532$$ 0 0
$$533$$ 5321.51 0.432458
$$534$$ −836.817 −0.0678139
$$535$$ 34.5274 0.00279019
$$536$$ 4386.51 0.353486
$$537$$ 3610.62 0.290148
$$538$$ 2140.28 0.171513
$$539$$ 0 0
$$540$$ 21.2435 0.00169292
$$541$$ 19101.3 1.51798 0.758992 0.651100i $$-0.225692\pi$$
0.758992 + 0.651100i $$0.225692\pi$$
$$542$$ 671.967 0.0532536
$$543$$ −8971.41 −0.709024
$$544$$ 9398.73 0.740749
$$545$$ 31.9261 0.00250929
$$546$$ 0 0
$$547$$ 15413.5 1.20481 0.602407 0.798189i $$-0.294208\pi$$
0.602407 + 0.798189i $$0.294208\pi$$
$$548$$ −1965.44 −0.153210
$$549$$ −110.184 −0.00856563
$$550$$ −2274.94 −0.176371
$$551$$ −29958.1 −2.31626
$$552$$ 1054.22 0.0812874
$$553$$ 0 0
$$554$$ 1910.51 0.146516
$$555$$ 57.8554 0.00442491
$$556$$ −6936.51 −0.529089
$$557$$ −20492.9 −1.55891 −0.779453 0.626460i $$-0.784503\pi$$
−0.779453 + 0.626460i $$0.784503\pi$$
$$558$$ 69.7519 0.00529182
$$559$$ 3638.17 0.275274
$$560$$ 0 0
$$561$$ 16034.9 1.20677
$$562$$ 880.294 0.0660729
$$563$$ 7142.49 0.534671 0.267336 0.963603i $$-0.413857\pi$$
0.267336 + 0.963603i $$0.413857\pi$$
$$564$$ −9430.59 −0.704077
$$565$$ −80.2406 −0.00597477
$$566$$ −1065.17 −0.0791030
$$567$$ 0 0
$$568$$ 5389.96 0.398165
$$569$$ 4097.91 0.301922 0.150961 0.988540i $$-0.451763\pi$$
0.150961 + 0.988540i $$0.451763\pi$$
$$570$$ −15.8753 −0.00116657
$$571$$ −2838.48 −0.208033 −0.104016 0.994576i $$-0.533169\pi$$
−0.104016 + 0.994576i $$0.533169\pi$$
$$572$$ −5725.60 −0.418530
$$573$$ −8422.23 −0.614038
$$574$$ 0 0
$$575$$ −6699.21 −0.485872
$$576$$ −4025.60 −0.291203
$$577$$ 15464.5 1.11576 0.557881 0.829921i $$-0.311615\pi$$
0.557881 + 0.829921i $$0.311615\pi$$
$$578$$ −4093.75 −0.294598
$$579$$ 10009.1 0.718418
$$580$$ 185.433 0.0132753
$$581$$ 0 0
$$582$$ 1355.86 0.0965675
$$583$$ −28259.3 −2.00751
$$584$$ 3377.17 0.239295
$$585$$ 15.0559 0.00106407
$$586$$ 1377.24 0.0970878
$$587$$ −14003.6 −0.984652 −0.492326 0.870411i $$-0.663853\pi$$
−0.492326 + 0.870411i $$0.663853\pi$$
$$588$$ 0 0
$$589$$ 2378.36 0.166382
$$590$$ −0.483425 −3.37327e−5 0
$$591$$ −12679.9 −0.882543
$$592$$ −11496.0 −0.798112
$$593$$ −6504.50 −0.450435 −0.225217 0.974309i $$-0.572309\pi$$
−0.225217 + 0.974309i $$0.572309\pi$$
$$594$$ 491.427 0.0339453
$$595$$ 0 0
$$596$$ −4561.04 −0.313469
$$597$$ 13157.1 0.901982
$$598$$ 369.528 0.0252695
$$599$$ −12616.1 −0.860567 −0.430284 0.902694i $$-0.641586\pi$$
−0.430284 + 0.902694i $$0.641586\pi$$
$$600$$ −2458.43 −0.167275
$$601$$ −8270.87 −0.561358 −0.280679 0.959802i $$-0.590560\pi$$
−0.280679 + 0.959802i $$0.590560\pi$$
$$602$$ 0 0
$$603$$ 6021.43 0.406653
$$604$$ 22011.7 1.48285
$$605$$ −60.2852 −0.00405114
$$606$$ 1703.40 0.114185
$$607$$ −3811.84 −0.254889 −0.127445 0.991846i $$-0.540677\pi$$
−0.127445 + 0.991846i $$0.540677\pi$$
$$608$$ 9821.62 0.655130
$$609$$ 0 0
$$610$$ −0.509668 −3.38293e−5 0
$$611$$ −6683.72 −0.442544
$$612$$ 8570.22 0.566063
$$613$$ 11359.3 0.748445 0.374222 0.927339i $$-0.377910\pi$$
0.374222 + 0.927339i $$0.377910\pi$$
$$614$$ −367.447 −0.0241514
$$615$$ 96.3983 0.00632057
$$616$$ 0 0
$$617$$ −18272.2 −1.19224 −0.596118 0.802896i $$-0.703291\pi$$
−0.596118 + 0.802896i $$0.703291\pi$$
$$618$$ 1757.04 0.114366
$$619$$ −29600.2 −1.92203 −0.961013 0.276503i $$-0.910824\pi$$
−0.961013 + 0.276503i $$0.910824\pi$$
$$620$$ −14.7215 −0.000953596 0
$$621$$ 1447.15 0.0935136
$$622$$ −1868.44 −0.120447
$$623$$ 0 0
$$624$$ −2991.63 −0.191925
$$625$$ 15621.2 0.999758
$$626$$ 1539.13 0.0982682
$$627$$ 16756.4 1.06728
$$628$$ 13240.6 0.841331
$$629$$ 23340.5 1.47956
$$630$$ 0 0
$$631$$ 7185.41 0.453322 0.226661 0.973974i $$-0.427219\pi$$
0.226661 + 0.973974i $$0.427219\pi$$
$$632$$ −5282.81 −0.332498
$$633$$ 6874.67 0.431665
$$634$$ −2880.66 −0.180450
$$635$$ −107.681 −0.00672944
$$636$$ −15103.8 −0.941673
$$637$$ 0 0
$$638$$ 4289.64 0.266189
$$639$$ 7398.88 0.458052
$$640$$ −80.7467 −0.00498718
$$641$$ −232.982 −0.0143561 −0.00717803 0.999974i $$-0.502285\pi$$
−0.00717803 + 0.999974i $$0.502285\pi$$
$$642$$ 426.896 0.0262433
$$643$$ 1837.96 0.112725 0.0563624 0.998410i $$-0.482050\pi$$
0.0563624 + 0.998410i $$0.482050\pi$$
$$644$$ 0 0
$$645$$ 65.9048 0.00402325
$$646$$ −6404.54 −0.390067
$$647$$ 18594.7 1.12988 0.564941 0.825131i $$-0.308899\pi$$
0.564941 + 0.825131i $$0.308899\pi$$
$$648$$ 531.064 0.0321947
$$649$$ 510.256 0.0308618
$$650$$ −861.736 −0.0520001
$$651$$ 0 0
$$652$$ 318.799 0.0191490
$$653$$ −28864.7 −1.72980 −0.864902 0.501940i $$-0.832620\pi$$
−0.864902 + 0.501940i $$0.832620\pi$$
$$654$$ 394.733 0.0236014
$$655$$ 252.772 0.0150788
$$656$$ −19154.5 −1.14003
$$657$$ 4635.90 0.275287
$$658$$ 0 0
$$659$$ −29066.3 −1.71815 −0.859076 0.511847i $$-0.828961\pi$$
−0.859076 + 0.511847i $$0.828961\pi$$
$$660$$ −103.718 −0.00611701
$$661$$ 3979.51 0.234168 0.117084 0.993122i $$-0.462645\pi$$
0.117084 + 0.993122i $$0.462645\pi$$
$$662$$ 4085.46 0.239858
$$663$$ 6073.95 0.355796
$$664$$ −2588.86 −0.151306
$$665$$ 0 0
$$666$$ 715.322 0.0416189
$$667$$ 12632.0 0.733305
$$668$$ −22706.1 −1.31516
$$669$$ −653.909 −0.0377901
$$670$$ 27.8528 0.00160604
$$671$$ 537.955 0.0309501
$$672$$ 0 0
$$673$$ −184.229 −0.0105520 −0.00527601 0.999986i $$-0.501679\pi$$
−0.00527601 + 0.999986i $$0.501679\pi$$
$$674$$ 2462.52 0.140731
$$675$$ −3374.73 −0.192435
$$676$$ 15030.2 0.855156
$$677$$ 16683.5 0.947116 0.473558 0.880763i $$-0.342969\pi$$
0.473558 + 0.880763i $$0.342969\pi$$
$$678$$ −992.091 −0.0561962
$$679$$ 0 0
$$680$$ 80.1540 0.00452024
$$681$$ 5506.08 0.309829
$$682$$ −340.553 −0.0191209
$$683$$ 17808.2 0.997676 0.498838 0.866695i $$-0.333760\pi$$
0.498838 + 0.866695i $$0.333760\pi$$
$$684$$ 8955.83 0.500636
$$685$$ −25.2332 −0.00140746
$$686$$ 0 0
$$687$$ 8322.03 0.462162
$$688$$ −13095.4 −0.725666
$$689$$ −10704.5 −0.591883
$$690$$ 6.69394 0.000369325 0
$$691$$ 20145.7 1.10908 0.554542 0.832156i $$-0.312894\pi$$
0.554542 + 0.832156i $$0.312894\pi$$
$$692$$ 16801.0 0.922943
$$693$$ 0 0
$$694$$ −484.453 −0.0264980
$$695$$ −89.0542 −0.00486046
$$696$$ 4635.63 0.252461
$$697$$ 38889.7 2.11342
$$698$$ −3801.10 −0.206123
$$699$$ −2966.14 −0.160500
$$700$$ 0 0
$$701$$ −2719.67 −0.146534 −0.0732672 0.997312i $$-0.523343\pi$$
−0.0732672 + 0.997312i $$0.523343\pi$$
$$702$$ 186.150 0.0100082
$$703$$ 24390.7 1.30855
$$704$$ 19654.4 1.05220
$$705$$ −121.074 −0.00646798
$$706$$ 4385.36 0.233775
$$707$$ 0 0
$$708$$ 272.717 0.0144765
$$709$$ −625.708 −0.0331438 −0.0165719 0.999863i $$-0.505275\pi$$
−0.0165719 + 0.999863i $$0.505275\pi$$
$$710$$ 34.2244 0.00180904
$$711$$ −7251.79 −0.382508
$$712$$ 4415.17 0.232395
$$713$$ −1002.85 −0.0526749
$$714$$ 0 0
$$715$$ −73.5079 −0.00384481
$$716$$ −9421.82 −0.491774
$$717$$ −2513.78 −0.130933
$$718$$ −3568.54 −0.185483
$$719$$ 9577.54 0.496776 0.248388 0.968661i $$-0.420099\pi$$
0.248388 + 0.968661i $$0.420099\pi$$
$$720$$ −54.1929 −0.00280507
$$721$$ 0 0
$$722$$ −3851.62 −0.198535
$$723$$ −10365.0 −0.533163
$$724$$ 23410.7 1.20173
$$725$$ −29457.8 −1.50901
$$726$$ −745.364 −0.0381034
$$727$$ 16741.2 0.854053 0.427027 0.904239i $$-0.359561\pi$$
0.427027 + 0.904239i $$0.359561\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 21.4439 0.00108722
$$731$$ 26587.8 1.34526
$$732$$ 287.522 0.0145179
$$733$$ 6496.04 0.327335 0.163668 0.986516i $$-0.447668\pi$$
0.163668 + 0.986516i $$0.447668\pi$$
$$734$$ 3437.06 0.172839
$$735$$ 0 0
$$736$$ −4141.36 −0.207408
$$737$$ −29398.7 −1.46936
$$738$$ 1191.86 0.0594487
$$739$$ 499.280 0.0248529 0.0124265 0.999923i $$-0.496044\pi$$
0.0124265 + 0.999923i $$0.496044\pi$$
$$740$$ −150.972 −0.00749980
$$741$$ 6347.24 0.314672
$$742$$ 0 0
$$743$$ −6367.30 −0.314393 −0.157196 0.987567i $$-0.550246\pi$$
−0.157196 + 0.987567i $$0.550246\pi$$
$$744$$ −368.021 −0.0181348
$$745$$ −58.5568 −0.00287967
$$746$$ 2122.37 0.104163
$$747$$ −3553.77 −0.174064
$$748$$ −41842.8 −2.04535
$$749$$ 0 0
$$750$$ −31.2217 −0.00152007
$$751$$ 496.098 0.0241050 0.0120525 0.999927i $$-0.496163\pi$$
0.0120525 + 0.999927i $$0.496163\pi$$
$$752$$ 24057.7 1.16662
$$753$$ 16905.0 0.818132
$$754$$ 1624.89 0.0784815
$$755$$ 282.597 0.0136222
$$756$$ 0 0
$$757$$ 13025.9 0.625408 0.312704 0.949851i $$-0.398765\pi$$
0.312704 + 0.949851i $$0.398765\pi$$
$$758$$ 622.350 0.0298216
$$759$$ −7065.47 −0.337892
$$760$$ 83.7604 0.00399778
$$761$$ 25474.6 1.21347 0.606736 0.794904i $$-0.292479\pi$$
0.606736 + 0.794904i $$0.292479\pi$$
$$762$$ −1331.36 −0.0632943
$$763$$ 0 0
$$764$$ 21977.6 1.04074
$$765$$ 110.029 0.00520012
$$766$$ −4503.72 −0.212436
$$767$$ 193.282 0.00909911
$$768$$ 9736.57 0.457472
$$769$$ 29054.0 1.36244 0.681218 0.732080i $$-0.261450\pi$$
0.681218 + 0.732080i $$0.261450\pi$$
$$770$$ 0 0
$$771$$ −6813.47 −0.318263
$$772$$ −26118.5 −1.21765
$$773$$ −1897.35 −0.0882834 −0.0441417 0.999025i $$-0.514055\pi$$
−0.0441417 + 0.999025i $$0.514055\pi$$
$$774$$ 814.844 0.0378410
$$775$$ 2338.65 0.108396
$$776$$ −7153.72 −0.330933
$$777$$ 0 0
$$778$$ −1913.25 −0.0881662
$$779$$ 40639.6 1.86914
$$780$$ −39.2879 −0.00180350
$$781$$ −36123.9 −1.65508
$$782$$ 2700.52 0.123492
$$783$$ 6363.39 0.290433
$$784$$ 0 0
$$785$$ 169.989 0.00772886
$$786$$ 3125.26 0.141825
$$787$$ −42650.3 −1.93179 −0.965895 0.258936i $$-0.916628\pi$$
−0.965895 + 0.258936i $$0.916628\pi$$
$$788$$ 33088.0 1.49583
$$789$$ 491.602 0.0221819
$$790$$ −33.5440 −0.00151069
$$791$$ 0 0
$$792$$ −2592.84 −0.116329
$$793$$ 203.775 0.00912516
$$794$$ 3979.33 0.177860
$$795$$ −193.910 −0.00865064
$$796$$ −34333.1 −1.52877
$$797$$ −36822.8 −1.63655 −0.818275 0.574828i $$-0.805069\pi$$
−0.818275 + 0.574828i $$0.805069\pi$$
$$798$$ 0 0
$$799$$ −48844.8 −2.16271
$$800$$ 9657.60 0.426810
$$801$$ 6060.77 0.267349
$$802$$ 4349.68 0.191512
$$803$$ −22634.1 −0.994693
$$804$$ −15712.8 −0.689238
$$805$$ 0 0
$$806$$ −129.000 −0.00563750
$$807$$ −15501.3 −0.676172
$$808$$ −8987.38 −0.391306
$$809$$ 5081.94 0.220855 0.110427 0.993884i $$-0.464778\pi$$
0.110427 + 0.993884i $$0.464778\pi$$
$$810$$ 3.37208 0.000146275 0
$$811$$ −11873.4 −0.514097 −0.257048 0.966399i $$-0.582750\pi$$
−0.257048 + 0.966399i $$0.582750\pi$$
$$812$$ 0 0
$$813$$ −4866.82 −0.209947
$$814$$ −3492.45 −0.150381
$$815$$ 4.09289 0.000175911 0
$$816$$ −21862.9 −0.937935
$$817$$ 27784.1 1.18977
$$818$$ −4998.29 −0.213644
$$819$$ 0 0
$$820$$ −251.549 −0.0107128
$$821$$ 16969.4 0.721361 0.360681 0.932689i $$-0.382544\pi$$
0.360681 + 0.932689i $$0.382544\pi$$
$$822$$ −311.982 −0.0132380
$$823$$ 3995.94 0.169246 0.0846231 0.996413i $$-0.473031\pi$$
0.0846231 + 0.996413i $$0.473031\pi$$
$$824$$ −9270.39 −0.391929
$$825$$ 16476.6 0.695323
$$826$$ 0 0
$$827$$ −13589.7 −0.571417 −0.285708 0.958317i $$-0.592229\pi$$
−0.285708 + 0.958317i $$0.592229\pi$$
$$828$$ −3776.29 −0.158497
$$829$$ 30646.0 1.28393 0.641966 0.766733i $$-0.278119\pi$$
0.641966 + 0.766733i $$0.278119\pi$$
$$830$$ −16.4384 −0.000687451 0
$$831$$ −13837.1 −0.577622
$$832$$ 7444.96 0.310226
$$833$$ 0 0
$$834$$ −1101.06 −0.0457155
$$835$$ −291.511 −0.0120816
$$836$$ −43725.5 −1.80894
$$837$$ −505.188 −0.0208624
$$838$$ −2637.01 −0.108704
$$839$$ 7497.57 0.308516 0.154258 0.988031i $$-0.450701\pi$$
0.154258 + 0.988031i $$0.450701\pi$$
$$840$$ 0 0
$$841$$ 31156.6 1.27749
$$842$$ 1959.99 0.0802207
$$843$$ −6375.65 −0.260485
$$844$$ −17939.3 −0.731630
$$845$$ 192.965 0.00785586
$$846$$ −1496.96 −0.0608351
$$847$$ 0 0
$$848$$ 38530.2 1.56030
$$849$$ 7714.62 0.311855
$$850$$ −6297.59 −0.254124
$$851$$ −10284.5 −0.414275
$$852$$ −19307.2 −0.776354
$$853$$ 10347.6 0.415352 0.207676 0.978198i $$-0.433410\pi$$
0.207676 + 0.978198i $$0.433410\pi$$
$$854$$ 0 0
$$855$$ 114.979 0.00459907
$$856$$ −2252.36 −0.0899348
$$857$$ −1550.00 −0.0617817 −0.0308909 0.999523i $$-0.509834\pi$$
−0.0308909 + 0.999523i $$0.509834\pi$$
$$858$$ −908.849 −0.0361627
$$859$$ −15187.5 −0.603250 −0.301625 0.953427i $$-0.597529\pi$$
−0.301625 + 0.953427i $$0.597529\pi$$
$$860$$ −171.977 −0.00681903
$$861$$ 0 0
$$862$$ −1554.45 −0.0614210
$$863$$ 41550.2 1.63892 0.819459 0.573138i $$-0.194274\pi$$
0.819459 + 0.573138i $$0.194274\pi$$
$$864$$ −2086.21 −0.0821462
$$865$$ 215.699 0.00847858
$$866$$ 4850.12 0.190316
$$867$$ 29649.6 1.16142
$$868$$ 0 0
$$869$$ 35405.8 1.38212
$$870$$ 29.4346 0.00114704
$$871$$ −11136.1 −0.433216
$$872$$ −2082.67 −0.0808808
$$873$$ −9820.03 −0.380707
$$874$$ 2822.03 0.109218
$$875$$ 0 0
$$876$$ −12097.3 −0.466585
$$877$$ 27837.4 1.07184 0.535919 0.844269i $$-0.319965\pi$$
0.535919 + 0.844269i $$0.319965\pi$$
$$878$$ −6182.02 −0.237623
$$879$$ −9974.89 −0.382758
$$880$$ 264.588 0.0101355
$$881$$ 2587.85 0.0989635 0.0494817 0.998775i $$-0.484243\pi$$
0.0494817 + 0.998775i $$0.484243\pi$$
$$882$$ 0 0
$$883$$ −16382.0 −0.624346 −0.312173 0.950025i $$-0.601057\pi$$
−0.312173 + 0.950025i $$0.601057\pi$$
$$884$$ −15849.8 −0.603040
$$885$$ 3.50127 0.000132988 0
$$886$$ 3528.13 0.133781
$$887$$ 22980.2 0.869896 0.434948 0.900456i $$-0.356767\pi$$
0.434948 + 0.900456i $$0.356767\pi$$
$$888$$ −3774.14 −0.142626
$$889$$ 0 0
$$890$$ 28.0348 0.00105587
$$891$$ −3559.23 −0.133826
$$892$$ 1706.36 0.0640506
$$893$$ −51042.5 −1.91274
$$894$$ −723.994 −0.0270850
$$895$$ −120.962 −0.00451766
$$896$$ 0 0
$$897$$ −2676.36 −0.0996222
$$898$$ 2471.09 0.0918276
$$899$$ −4409.76 −0.163597
$$900$$ 8806.27 0.326158
$$901$$ −78228.5 −2.89253
$$902$$ −5819.10 −0.214806
$$903$$ 0 0
$$904$$ 5234.42 0.192582
$$905$$ 300.557 0.0110396
$$906$$ 3494.01 0.128125
$$907$$ −22715.4 −0.831590 −0.415795 0.909458i $$-0.636497\pi$$
−0.415795 + 0.909458i $$0.636497\pi$$
$$908$$ −14368.0 −0.525130
$$909$$ −12337.1 −0.450161
$$910$$ 0 0
$$911$$ 34922.3 1.27006 0.635031 0.772487i $$-0.280987\pi$$
0.635031 + 0.772487i $$0.280987\pi$$
$$912$$ −22846.6 −0.829525
$$913$$ 17350.7 0.628943
$$914$$ 5741.24 0.207772
$$915$$ 3.69134 0.000133368 0
$$916$$ −21716.1 −0.783320
$$917$$ 0 0
$$918$$ 1360.39 0.0489102
$$919$$ 11702.6 0.420059 0.210030 0.977695i $$-0.432644\pi$$
0.210030 + 0.977695i $$0.432644\pi$$
$$920$$ −35.3182 −0.00126566
$$921$$ 2661.29 0.0952144
$$922$$ −61.8473 −0.00220914
$$923$$ −13683.5 −0.487973
$$924$$ 0 0
$$925$$ 23983.3 0.852505
$$926$$ −2238.39 −0.0794364
$$927$$ −12725.6 −0.450878
$$928$$ −18210.4 −0.644165
$$929$$ 53096.5 1.87518 0.937588 0.347747i $$-0.113053\pi$$
0.937588 + 0.347747i $$0.113053\pi$$
$$930$$ −2.33681 −8.23946e−5 0
$$931$$ 0 0
$$932$$ 7740.06 0.272032
$$933$$ 13532.5 0.474848
$$934$$ 1534.52 0.0537591
$$935$$ −537.198 −0.0187896
$$936$$ −982.154 −0.0342978
$$937$$ 39020.6 1.36046 0.680229 0.733000i $$-0.261880\pi$$
0.680229 + 0.733000i $$0.261880\pi$$
$$938$$ 0 0
$$939$$ −11147.4 −0.387412
$$940$$ 315.941 0.0109626
$$941$$ −30743.8 −1.06506 −0.532528 0.846412i $$-0.678758\pi$$
−0.532528 + 0.846412i $$0.678758\pi$$
$$942$$ 2101.73 0.0726944
$$943$$ −17136.0 −0.591754
$$944$$ −695.710 −0.0239867
$$945$$ 0 0
$$946$$ −3978.35 −0.136731
$$947$$ 16300.3 0.559332 0.279666 0.960097i $$-0.409776\pi$$
0.279666 + 0.960097i $$0.409776\pi$$
$$948$$ 18923.4 0.648315
$$949$$ −8573.66 −0.293269
$$950$$ −6580.94 −0.224752
$$951$$ 20863.6 0.711406
$$952$$ 0 0
$$953$$ −11512.7 −0.391325 −0.195663 0.980671i $$-0.562686\pi$$
−0.195663 + 0.980671i $$0.562686\pi$$
$$954$$ −2397.49 −0.0813644
$$955$$ 282.159 0.00956068
$$956$$ 6559.65 0.221919
$$957$$ −31068.3 −1.04942
$$958$$ −4420.39 −0.149078
$$959$$ 0 0
$$960$$ 134.864 0.00453409
$$961$$ −29440.9 −0.988248
$$962$$ −1322.92 −0.0443375
$$963$$ −3091.85 −0.103462
$$964$$ 27047.1 0.903661
$$965$$ −335.322 −0.0111859
$$966$$ 0 0
$$967$$ −18178.4 −0.604528 −0.302264 0.953224i $$-0.597742\pi$$
−0.302264 + 0.953224i $$0.597742\pi$$
$$968$$ 3932.65 0.130579
$$969$$ 46385.8 1.53780
$$970$$ −45.4237 −0.00150357
$$971$$ 28276.7 0.934543 0.467271 0.884114i $$-0.345237\pi$$
0.467271 + 0.884114i $$0.345237\pi$$
$$972$$ −1902.31 −0.0627742
$$973$$ 0 0
$$974$$ −2424.77 −0.0797688
$$975$$ 6241.24 0.205005
$$976$$ −733.477 −0.0240554
$$977$$ −13947.1 −0.456710 −0.228355 0.973578i $$-0.573335\pi$$
−0.228355 + 0.973578i $$0.573335\pi$$
$$978$$ 50.6043 0.00165455
$$979$$ −29590.8 −0.966011
$$980$$ 0 0
$$981$$ −2858.91 −0.0930459
$$982$$ −1683.91 −0.0547206
$$983$$ −26576.8 −0.862327 −0.431164 0.902274i $$-0.641897\pi$$
−0.431164 + 0.902274i $$0.641897\pi$$
$$984$$ −6288.45 −0.203728
$$985$$ 424.799 0.0137414
$$986$$ 11874.7 0.383539
$$987$$ 0 0
$$988$$ −16563.0 −0.533339
$$989$$ −11715.4 −0.376671
$$990$$ −16.4636 −0.000528534 0
$$991$$ 16249.9 0.520884 0.260442 0.965490i $$-0.416132\pi$$
0.260442 + 0.965490i $$0.416132\pi$$
$$992$$ 1445.72 0.0462718
$$993$$ −29589.5 −0.945615
$$994$$ 0 0
$$995$$ −440.784 −0.0140440
$$996$$ 9273.47 0.295021
$$997$$ 18814.1 0.597642 0.298821 0.954309i $$-0.403407\pi$$
0.298821 + 0.954309i $$0.403407\pi$$
$$998$$ −1992.86 −0.0632092
$$999$$ −5180.82 −0.164078
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 147.4.a.k.1.1 yes 2
3.2 odd 2 441.4.a.o.1.2 2
4.3 odd 2 2352.4.a.bl.1.2 2
7.2 even 3 147.4.e.j.67.2 4
7.3 odd 6 147.4.e.k.79.2 4
7.4 even 3 147.4.e.j.79.2 4
7.5 odd 6 147.4.e.k.67.2 4
7.6 odd 2 147.4.a.j.1.1 2
21.2 odd 6 441.4.e.u.361.1 4
21.5 even 6 441.4.e.v.361.1 4
21.11 odd 6 441.4.e.u.226.1 4
21.17 even 6 441.4.e.v.226.1 4
21.20 even 2 441.4.a.n.1.2 2
28.27 even 2 2352.4.a.cf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.1 2 7.6 odd 2
147.4.a.k.1.1 yes 2 1.1 even 1 trivial
147.4.e.j.67.2 4 7.2 even 3
147.4.e.j.79.2 4 7.4 even 3
147.4.e.k.67.2 4 7.5 odd 6
147.4.e.k.79.2 4 7.3 odd 6
441.4.a.n.1.2 2 21.20 even 2
441.4.a.o.1.2 2 3.2 odd 2
441.4.e.u.226.1 4 21.11 odd 6
441.4.e.u.361.1 4 21.2 odd 6
441.4.e.v.226.1 4 21.17 even 6
441.4.e.v.361.1 4 21.5 even 6
2352.4.a.bl.1.2 2 4.3 odd 2
2352.4.a.cf.1.1 2 28.27 even 2