# Properties

 Label 825.4.a.p.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1957.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 10$$ x^3 - x^2 - 9*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.12946$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.59486 q^{2} +3.00000 q^{3} +13.1127 q^{4} -13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.59486 q^{2} +3.00000 q^{3} +13.1127 q^{4} -13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +11.0000 q^{11} +39.3381 q^{12} +15.6584 q^{13} +94.8302 q^{14} +3.04132 q^{16} -72.9507 q^{17} -41.3537 q^{18} +61.0513 q^{19} -61.9150 q^{21} -50.5434 q^{22} +13.6605 q^{23} -70.4764 q^{24} -71.9483 q^{26} +27.0000 q^{27} -270.624 q^{28} -31.4663 q^{29} -243.008 q^{31} +173.963 q^{32} +33.0000 q^{33} +335.198 q^{34} +118.014 q^{36} +65.4018 q^{37} -280.522 q^{38} +46.9753 q^{39} -109.087 q^{41} +284.491 q^{42} +121.750 q^{43} +144.240 q^{44} -62.7678 q^{46} +519.530 q^{47} +9.12396 q^{48} +82.9413 q^{49} -218.852 q^{51} +205.324 q^{52} +542.673 q^{53} -124.061 q^{54} +484.839 q^{56} +183.154 q^{57} +144.583 q^{58} +109.478 q^{59} -89.6156 q^{61} +1116.59 q^{62} -185.745 q^{63} -823.664 q^{64} -151.630 q^{66} -488.446 q^{67} -956.581 q^{68} +40.9814 q^{69} +837.423 q^{71} -211.429 q^{72} -351.216 q^{73} -300.512 q^{74} +800.547 q^{76} -227.022 q^{77} -215.845 q^{78} -831.205 q^{79} +81.0000 q^{81} +501.238 q^{82} -1389.13 q^{83} -811.873 q^{84} -559.423 q^{86} -94.3988 q^{87} -258.413 q^{88} +1523.70 q^{89} -323.164 q^{91} +179.125 q^{92} -729.025 q^{93} -2387.17 q^{94} +521.888 q^{96} +426.612 q^{97} -381.103 q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 9 q^{3} + 17 q^{4} - 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - q^2 + 9 * q^3 + 17 * q^4 - 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 $$3 q - q^{2} + 9 q^{3} + 17 q^{4} - 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} + 33 q^{11} + 51 q^{12} + 20 q^{13} + 144 q^{14} + 25 q^{16} - 32 q^{17} - 9 q^{18} + 116 q^{19} - 18 q^{21} - 11 q^{22} - 240 q^{23} + 9 q^{24} + 302 q^{26} + 81 q^{27} - 160 q^{28} + 238 q^{29} + 92 q^{31} - 197 q^{32} + 99 q^{33} + 354 q^{34} + 153 q^{36} + 90 q^{37} - 324 q^{38} + 60 q^{39} - 46 q^{41} + 432 q^{42} + 134 q^{43} + 187 q^{44} - 240 q^{46} + 220 q^{47} + 75 q^{48} - 457 q^{49} - 96 q^{51} + 1530 q^{52} + 798 q^{53} - 27 q^{54} + 688 q^{56} + 348 q^{57} + 978 q^{58} + 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 54 q^{63} - 1919 q^{64} - 33 q^{66} - 764 q^{67} - 1074 q^{68} - 720 q^{69} + 1816 q^{71} + 27 q^{72} - 100 q^{73} - 1874 q^{74} + 396 q^{76} - 66 q^{77} + 906 q^{78} - 96 q^{79} + 243 q^{81} + 910 q^{82} - 858 q^{83} - 480 q^{84} + 188 q^{86} + 714 q^{87} + 33 q^{88} + 838 q^{89} + 332 q^{91} + 688 q^{92} + 276 q^{93} - 3112 q^{94} - 591 q^{96} + 1322 q^{97} - 1017 q^{98} + 297 q^{99}+O(q^{100})$$ 3 * q - q^2 + 9 * q^3 + 17 * q^4 - 3 * q^6 - 6 * q^7 + 3 * q^8 + 27 * q^9 + 33 * q^11 + 51 * q^12 + 20 * q^13 + 144 * q^14 + 25 * q^16 - 32 * q^17 - 9 * q^18 + 116 * q^19 - 18 * q^21 - 11 * q^22 - 240 * q^23 + 9 * q^24 + 302 * q^26 + 81 * q^27 - 160 * q^28 + 238 * q^29 + 92 * q^31 - 197 * q^32 + 99 * q^33 + 354 * q^34 + 153 * q^36 + 90 * q^37 - 324 * q^38 + 60 * q^39 - 46 * q^41 + 432 * q^42 + 134 * q^43 + 187 * q^44 - 240 * q^46 + 220 * q^47 + 75 * q^48 - 457 * q^49 - 96 * q^51 + 1530 * q^52 + 798 * q^53 - 27 * q^54 + 688 * q^56 + 348 * q^57 + 978 * q^58 + 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 54 * q^63 - 1919 * q^64 - 33 * q^66 - 764 * q^67 - 1074 * q^68 - 720 * q^69 + 1816 * q^71 + 27 * q^72 - 100 * q^73 - 1874 * q^74 + 396 * q^76 - 66 * q^77 + 906 * q^78 - 96 * q^79 + 243 * q^81 + 910 * q^82 - 858 * q^83 - 480 * q^84 + 188 * q^86 + 714 * q^87 + 33 * q^88 + 838 * q^89 + 332 * q^91 + 688 * q^92 + 276 * q^93 - 3112 * q^94 - 591 * q^96 + 1322 * q^97 - 1017 * q^98 + 297 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.59486 −1.62453 −0.812263 0.583291i $$-0.801765\pi$$
−0.812263 + 0.583291i $$0.801765\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 13.1127 1.63909
$$5$$ 0 0
$$6$$ −13.7846 −0.937921
$$7$$ −20.6383 −1.11437 −0.557183 0.830390i $$-0.688118\pi$$
−0.557183 + 0.830390i $$0.688118\pi$$
$$8$$ −23.4921 −1.03822
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 39.3381 0.946328
$$13$$ 15.6584 0.334067 0.167033 0.985951i $$-0.446581\pi$$
0.167033 + 0.985951i $$0.446581\pi$$
$$14$$ 94.8302 1.81032
$$15$$ 0 0
$$16$$ 3.04132 0.0475206
$$17$$ −72.9507 −1.04077 −0.520387 0.853931i $$-0.674212\pi$$
−0.520387 + 0.853931i $$0.674212\pi$$
$$18$$ −41.3537 −0.541509
$$19$$ 61.0513 0.737165 0.368582 0.929595i $$-0.379843\pi$$
0.368582 + 0.929595i $$0.379843\pi$$
$$20$$ 0 0
$$21$$ −61.9150 −0.643379
$$22$$ −50.5434 −0.489813
$$23$$ 13.6605 0.123844 0.0619218 0.998081i $$-0.480277\pi$$
0.0619218 + 0.998081i $$0.480277\pi$$
$$24$$ −70.4764 −0.599414
$$25$$ 0 0
$$26$$ −71.9483 −0.542701
$$27$$ 27.0000 0.192450
$$28$$ −270.624 −1.82654
$$29$$ −31.4663 −0.201487 −0.100744 0.994912i $$-0.532122\pi$$
−0.100744 + 0.994912i $$0.532122\pi$$
$$30$$ 0 0
$$31$$ −243.008 −1.40792 −0.703961 0.710239i $$-0.748587\pi$$
−0.703961 + 0.710239i $$0.748587\pi$$
$$32$$ 173.963 0.961016
$$33$$ 33.0000 0.174078
$$34$$ 335.198 1.69076
$$35$$ 0 0
$$36$$ 118.014 0.546363
$$37$$ 65.4018 0.290594 0.145297 0.989388i $$-0.453586\pi$$
0.145297 + 0.989388i $$0.453586\pi$$
$$38$$ −280.522 −1.19754
$$39$$ 46.9753 0.192874
$$40$$ 0 0
$$41$$ −109.087 −0.415524 −0.207762 0.978179i $$-0.566618\pi$$
−0.207762 + 0.978179i $$0.566618\pi$$
$$42$$ 284.491 1.04519
$$43$$ 121.750 0.431783 0.215891 0.976417i $$-0.430734\pi$$
0.215891 + 0.976417i $$0.430734\pi$$
$$44$$ 144.240 0.494204
$$45$$ 0 0
$$46$$ −62.7678 −0.201187
$$47$$ 519.530 1.61237 0.806184 0.591665i $$-0.201529\pi$$
0.806184 + 0.591665i $$0.201529\pi$$
$$48$$ 9.12396 0.0274361
$$49$$ 82.9413 0.241811
$$50$$ 0 0
$$51$$ −218.852 −0.600891
$$52$$ 205.324 0.547565
$$53$$ 542.673 1.40645 0.703226 0.710967i $$-0.251742\pi$$
0.703226 + 0.710967i $$0.251742\pi$$
$$54$$ −124.061 −0.312640
$$55$$ 0 0
$$56$$ 484.839 1.15695
$$57$$ 183.154 0.425602
$$58$$ 144.583 0.327322
$$59$$ 109.478 0.241574 0.120787 0.992678i $$-0.461458\pi$$
0.120787 + 0.992678i $$0.461458\pi$$
$$60$$ 0 0
$$61$$ −89.6156 −0.188100 −0.0940501 0.995567i $$-0.529981\pi$$
−0.0940501 + 0.995567i $$0.529981\pi$$
$$62$$ 1116.59 2.28721
$$63$$ −185.745 −0.371455
$$64$$ −823.664 −1.60872
$$65$$ 0 0
$$66$$ −151.630 −0.282794
$$67$$ −488.446 −0.890644 −0.445322 0.895371i $$-0.646911\pi$$
−0.445322 + 0.895371i $$0.646911\pi$$
$$68$$ −956.581 −1.70592
$$69$$ 40.9814 0.0715011
$$70$$ 0 0
$$71$$ 837.423 1.39977 0.699887 0.714254i $$-0.253234\pi$$
0.699887 + 0.714254i $$0.253234\pi$$
$$72$$ −211.429 −0.346072
$$73$$ −351.216 −0.563105 −0.281553 0.959546i $$-0.590849\pi$$
−0.281553 + 0.959546i $$0.590849\pi$$
$$74$$ −300.512 −0.472078
$$75$$ 0 0
$$76$$ 800.547 1.20828
$$77$$ −227.022 −0.335994
$$78$$ −215.845 −0.313328
$$79$$ −831.205 −1.18377 −0.591885 0.806022i $$-0.701616\pi$$
−0.591885 + 0.806022i $$0.701616\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 501.238 0.675031
$$83$$ −1389.13 −1.83707 −0.918537 0.395335i $$-0.870629\pi$$
−0.918537 + 0.395335i $$0.870629\pi$$
$$84$$ −811.873 −1.05456
$$85$$ 0 0
$$86$$ −559.423 −0.701443
$$87$$ −94.3988 −0.116329
$$88$$ −258.413 −0.313034
$$89$$ 1523.70 1.81474 0.907369 0.420335i $$-0.138088\pi$$
0.907369 + 0.420335i $$0.138088\pi$$
$$90$$ 0 0
$$91$$ −323.164 −0.372273
$$92$$ 179.125 0.202990
$$93$$ −729.025 −0.812864
$$94$$ −2387.17 −2.61933
$$95$$ 0 0
$$96$$ 521.888 0.554843
$$97$$ 426.612 0.446555 0.223278 0.974755i $$-0.428324\pi$$
0.223278 + 0.974755i $$0.428324\pi$$
$$98$$ −381.103 −0.392829
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 74.1387 0.0730403 0.0365202 0.999333i $$-0.488373\pi$$
0.0365202 + 0.999333i $$0.488373\pi$$
$$102$$ 1005.59 0.976163
$$103$$ 69.3916 0.0663821 0.0331911 0.999449i $$-0.489433\pi$$
0.0331911 + 0.999449i $$0.489433\pi$$
$$104$$ −367.850 −0.346833
$$105$$ 0 0
$$106$$ −2493.51 −2.28482
$$107$$ −1141.71 −1.03152 −0.515761 0.856733i $$-0.672491\pi$$
−0.515761 + 0.856733i $$0.672491\pi$$
$$108$$ 354.043 0.315443
$$109$$ −2226.85 −1.95682 −0.978409 0.206680i $$-0.933734\pi$$
−0.978409 + 0.206680i $$0.933734\pi$$
$$110$$ 0 0
$$111$$ 196.205 0.167775
$$112$$ −62.7678 −0.0529554
$$113$$ 1719.76 1.43169 0.715847 0.698257i $$-0.246041\pi$$
0.715847 + 0.698257i $$0.246041\pi$$
$$114$$ −841.566 −0.691402
$$115$$ 0 0
$$116$$ −412.608 −0.330256
$$117$$ 140.926 0.111356
$$118$$ −503.038 −0.392444
$$119$$ 1505.58 1.15980
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 411.771 0.305574
$$123$$ −327.260 −0.239903
$$124$$ −3186.49 −2.30771
$$125$$ 0 0
$$126$$ 853.472 0.603439
$$127$$ 1601.63 1.11907 0.559534 0.828807i $$-0.310980\pi$$
0.559534 + 0.828807i $$0.310980\pi$$
$$128$$ 2392.91 1.65239
$$129$$ 365.249 0.249290
$$130$$ 0 0
$$131$$ 2004.13 1.33665 0.668327 0.743868i $$-0.267011\pi$$
0.668327 + 0.743868i $$0.267011\pi$$
$$132$$ 432.719 0.285329
$$133$$ −1260.00 −0.821471
$$134$$ 2244.34 1.44687
$$135$$ 0 0
$$136$$ 1713.77 1.08055
$$137$$ 1672.85 1.04322 0.521610 0.853184i $$-0.325331\pi$$
0.521610 + 0.853184i $$0.325331\pi$$
$$138$$ −188.303 −0.116155
$$139$$ 2540.38 1.55016 0.775080 0.631863i $$-0.217709\pi$$
0.775080 + 0.631863i $$0.217709\pi$$
$$140$$ 0 0
$$141$$ 1558.59 0.930901
$$142$$ −3847.84 −2.27397
$$143$$ 172.243 0.100725
$$144$$ 27.3719 0.0158402
$$145$$ 0 0
$$146$$ 1613.79 0.914780
$$147$$ 248.824 0.139610
$$148$$ 857.594 0.476310
$$149$$ 3090.68 1.69932 0.849658 0.527334i $$-0.176808\pi$$
0.849658 + 0.527334i $$0.176808\pi$$
$$150$$ 0 0
$$151$$ 1358.74 0.732267 0.366134 0.930562i $$-0.380681\pi$$
0.366134 + 0.930562i $$0.380681\pi$$
$$152$$ −1434.22 −0.765335
$$153$$ −656.557 −0.346925
$$154$$ 1043.13 0.545831
$$155$$ 0 0
$$156$$ 615.973 0.316137
$$157$$ 1011.95 0.514411 0.257205 0.966357i $$-0.417198\pi$$
0.257205 + 0.966357i $$0.417198\pi$$
$$158$$ 3819.27 1.92307
$$159$$ 1628.02 0.812015
$$160$$ 0 0
$$161$$ −281.929 −0.138007
$$162$$ −372.183 −0.180503
$$163$$ 2816.37 1.35334 0.676672 0.736285i $$-0.263422\pi$$
0.676672 + 0.736285i $$0.263422\pi$$
$$164$$ −1430.42 −0.681081
$$165$$ 0 0
$$166$$ 6382.87 2.98438
$$167$$ −3448.89 −1.59810 −0.799052 0.601262i $$-0.794665\pi$$
−0.799052 + 0.601262i $$0.794665\pi$$
$$168$$ 1454.52 0.667966
$$169$$ −1951.81 −0.888399
$$170$$ 0 0
$$171$$ 549.462 0.245722
$$172$$ 1596.47 0.707730
$$173$$ 2287.85 1.00545 0.502723 0.864448i $$-0.332332\pi$$
0.502723 + 0.864448i $$0.332332\pi$$
$$174$$ 433.749 0.188979
$$175$$ 0 0
$$176$$ 33.4545 0.0143280
$$177$$ 328.435 0.139473
$$178$$ −7001.17 −2.94809
$$179$$ 3249.06 1.35668 0.678340 0.734748i $$-0.262700\pi$$
0.678340 + 0.734748i $$0.262700\pi$$
$$180$$ 0 0
$$181$$ 1170.45 0.480655 0.240328 0.970692i $$-0.422745\pi$$
0.240328 + 0.970692i $$0.422745\pi$$
$$182$$ 1484.89 0.604767
$$183$$ −268.847 −0.108600
$$184$$ −320.913 −0.128576
$$185$$ 0 0
$$186$$ 3349.76 1.32052
$$187$$ −802.458 −0.313805
$$188$$ 6812.44 2.64281
$$189$$ −557.235 −0.214460
$$190$$ 0 0
$$191$$ −2760.35 −1.04572 −0.522859 0.852419i $$-0.675134\pi$$
−0.522859 + 0.852419i $$0.675134\pi$$
$$192$$ −2470.99 −0.928794
$$193$$ 1250.61 0.466430 0.233215 0.972425i $$-0.425075\pi$$
0.233215 + 0.972425i $$0.425075\pi$$
$$194$$ −1960.22 −0.725441
$$195$$ 0 0
$$196$$ 1087.58 0.396350
$$197$$ 143.991 0.0520756 0.0260378 0.999661i $$-0.491711\pi$$
0.0260378 + 0.999661i $$0.491711\pi$$
$$198$$ −454.891 −0.163271
$$199$$ 761.249 0.271174 0.135587 0.990765i $$-0.456708\pi$$
0.135587 + 0.990765i $$0.456708\pi$$
$$200$$ 0 0
$$201$$ −1465.34 −0.514213
$$202$$ −340.657 −0.118656
$$203$$ 649.411 0.224531
$$204$$ −2869.74 −0.984913
$$205$$ 0 0
$$206$$ −318.844 −0.107840
$$207$$ 122.944 0.0412812
$$208$$ 47.6223 0.0158751
$$209$$ 671.564 0.222263
$$210$$ 0 0
$$211$$ 3976.58 1.29743 0.648717 0.761029i $$-0.275306\pi$$
0.648717 + 0.761029i $$0.275306\pi$$
$$212$$ 7115.91 2.30530
$$213$$ 2512.27 0.808159
$$214$$ 5245.97 1.67573
$$215$$ 0 0
$$216$$ −634.287 −0.199805
$$217$$ 5015.29 1.56894
$$218$$ 10232.0 3.17890
$$219$$ −1053.65 −0.325109
$$220$$ 0 0
$$221$$ −1142.29 −0.347688
$$222$$ −901.535 −0.272555
$$223$$ −908.084 −0.272690 −0.136345 0.990661i $$-0.543536\pi$$
−0.136345 + 0.990661i $$0.543536\pi$$
$$224$$ −3590.30 −1.07092
$$225$$ 0 0
$$226$$ −7902.05 −2.32583
$$227$$ 2062.15 0.602951 0.301475 0.953474i $$-0.402521\pi$$
0.301475 + 0.953474i $$0.402521\pi$$
$$228$$ 2401.64 0.697599
$$229$$ 4077.47 1.17662 0.588312 0.808634i $$-0.299793\pi$$
0.588312 + 0.808634i $$0.299793\pi$$
$$230$$ 0 0
$$231$$ −681.065 −0.193986
$$232$$ 739.209 0.209187
$$233$$ −1682.76 −0.473138 −0.236569 0.971615i $$-0.576023\pi$$
−0.236569 + 0.971615i $$0.576023\pi$$
$$234$$ −647.534 −0.180900
$$235$$ 0 0
$$236$$ 1435.56 0.395961
$$237$$ −2493.62 −0.683450
$$238$$ −6917.93 −1.88413
$$239$$ 4024.96 1.08934 0.544672 0.838649i $$-0.316654\pi$$
0.544672 + 0.838649i $$0.316654\pi$$
$$240$$ 0 0
$$241$$ −2784.27 −0.744194 −0.372097 0.928194i $$-0.621361\pi$$
−0.372097 + 0.928194i $$0.621361\pi$$
$$242$$ −555.978 −0.147684
$$243$$ 243.000 0.0641500
$$244$$ −1175.10 −0.308313
$$245$$ 0 0
$$246$$ 1503.71 0.389729
$$247$$ 955.968 0.246262
$$248$$ 5708.78 1.46173
$$249$$ −4167.40 −1.06064
$$250$$ 0 0
$$251$$ −1827.60 −0.459591 −0.229796 0.973239i $$-0.573806\pi$$
−0.229796 + 0.973239i $$0.573806\pi$$
$$252$$ −2435.62 −0.608848
$$253$$ 150.265 0.0373402
$$254$$ −7359.26 −1.81796
$$255$$ 0 0
$$256$$ −4405.79 −1.07563
$$257$$ −585.171 −0.142031 −0.0710155 0.997475i $$-0.522624\pi$$
−0.0710155 + 0.997475i $$0.522624\pi$$
$$258$$ −1678.27 −0.404978
$$259$$ −1349.78 −0.323828
$$260$$ 0 0
$$261$$ −283.196 −0.0671625
$$262$$ −9208.69 −2.17143
$$263$$ 238.098 0.0558241 0.0279120 0.999610i $$-0.491114\pi$$
0.0279120 + 0.999610i $$0.491114\pi$$
$$264$$ −775.240 −0.180730
$$265$$ 0 0
$$266$$ 5789.51 1.33450
$$267$$ 4571.09 1.04774
$$268$$ −6404.84 −1.45984
$$269$$ 4618.46 1.04681 0.523406 0.852083i $$-0.324661\pi$$
0.523406 + 0.852083i $$0.324661\pi$$
$$270$$ 0 0
$$271$$ −143.439 −0.0321525 −0.0160762 0.999871i $$-0.505117\pi$$
−0.0160762 + 0.999871i $$0.505117\pi$$
$$272$$ −221.867 −0.0494582
$$273$$ −969.493 −0.214932
$$274$$ −7686.51 −1.69474
$$275$$ 0 0
$$276$$ 537.376 0.117197
$$277$$ 8602.51 1.86597 0.932987 0.359911i $$-0.117193\pi$$
0.932987 + 0.359911i $$0.117193\pi$$
$$278$$ −11672.7 −2.51828
$$279$$ −2187.07 −0.469307
$$280$$ 0 0
$$281$$ −2992.81 −0.635360 −0.317680 0.948198i $$-0.602904\pi$$
−0.317680 + 0.948198i $$0.602904\pi$$
$$282$$ −7161.50 −1.51227
$$283$$ 6858.89 1.44070 0.720351 0.693610i $$-0.243981\pi$$
0.720351 + 0.693610i $$0.243981\pi$$
$$284$$ 10980.9 2.29435
$$285$$ 0 0
$$286$$ −791.431 −0.163630
$$287$$ 2251.37 0.463046
$$288$$ 1565.66 0.320339
$$289$$ 408.809 0.0832096
$$290$$ 0 0
$$291$$ 1279.84 0.257819
$$292$$ −4605.39 −0.922979
$$293$$ −4049.70 −0.807461 −0.403731 0.914878i $$-0.632287\pi$$
−0.403731 + 0.914878i $$0.632287\pi$$
$$294$$ −1143.31 −0.226800
$$295$$ 0 0
$$296$$ −1536.43 −0.301699
$$297$$ 297.000 0.0580259
$$298$$ −14201.2 −2.76059
$$299$$ 213.901 0.0413720
$$300$$ 0 0
$$301$$ −2512.71 −0.481164
$$302$$ −6243.20 −1.18959
$$303$$ 222.416 0.0421699
$$304$$ 185.677 0.0350305
$$305$$ 0 0
$$306$$ 3016.78 0.563588
$$307$$ 9572.69 1.77962 0.889808 0.456335i $$-0.150838\pi$$
0.889808 + 0.456335i $$0.150838\pi$$
$$308$$ −2976.87 −0.550724
$$309$$ 208.175 0.0383257
$$310$$ 0 0
$$311$$ 5396.42 0.983932 0.491966 0.870614i $$-0.336278\pi$$
0.491966 + 0.870614i $$0.336278\pi$$
$$312$$ −1103.55 −0.200244
$$313$$ −9755.04 −1.76162 −0.880811 0.473469i $$-0.843002\pi$$
−0.880811 + 0.473469i $$0.843002\pi$$
$$314$$ −4649.77 −0.835674
$$315$$ 0 0
$$316$$ −10899.3 −1.94030
$$317$$ 4353.75 0.771391 0.385695 0.922626i $$-0.373962\pi$$
0.385695 + 0.922626i $$0.373962\pi$$
$$318$$ −7480.52 −1.31914
$$319$$ −346.129 −0.0607508
$$320$$ 0 0
$$321$$ −3425.12 −0.595549
$$322$$ 1295.42 0.224196
$$323$$ −4453.74 −0.767221
$$324$$ 1062.13 0.182121
$$325$$ 0 0
$$326$$ −12940.8 −2.19854
$$327$$ −6680.54 −1.12977
$$328$$ 2562.68 0.431404
$$329$$ −10722.2 −1.79677
$$330$$ 0 0
$$331$$ 5387.64 0.894656 0.447328 0.894370i $$-0.352376\pi$$
0.447328 + 0.894370i $$0.352376\pi$$
$$332$$ −18215.3 −3.01113
$$333$$ 588.616 0.0968648
$$334$$ 15847.2 2.59616
$$335$$ 0 0
$$336$$ −188.303 −0.0305738
$$337$$ 4500.27 0.727434 0.363717 0.931509i $$-0.381508\pi$$
0.363717 + 0.931509i $$0.381508\pi$$
$$338$$ 8968.30 1.44323
$$339$$ 5159.28 0.826589
$$340$$ 0 0
$$341$$ −2673.09 −0.424504
$$342$$ −2524.70 −0.399181
$$343$$ 5367.18 0.844899
$$344$$ −2860.16 −0.448283
$$345$$ 0 0
$$346$$ −10512.4 −1.63337
$$347$$ −5906.32 −0.913740 −0.456870 0.889533i $$-0.651030\pi$$
−0.456870 + 0.889533i $$0.651030\pi$$
$$348$$ −1237.82 −0.190673
$$349$$ 3636.26 0.557721 0.278860 0.960332i $$-0.410043\pi$$
0.278860 + 0.960332i $$0.410043\pi$$
$$350$$ 0 0
$$351$$ 422.778 0.0642912
$$352$$ 1913.59 0.289757
$$353$$ −210.408 −0.0317248 −0.0158624 0.999874i $$-0.505049\pi$$
−0.0158624 + 0.999874i $$0.505049\pi$$
$$354$$ −1509.11 −0.226577
$$355$$ 0 0
$$356$$ 19979.8 2.97451
$$357$$ 4516.75 0.669612
$$358$$ −14928.9 −2.20396
$$359$$ 2499.68 0.367488 0.183744 0.982974i $$-0.441178\pi$$
0.183744 + 0.982974i $$0.441178\pi$$
$$360$$ 0 0
$$361$$ −3131.74 −0.456588
$$362$$ −5378.03 −0.780837
$$363$$ 363.000 0.0524864
$$364$$ −4237.56 −0.610188
$$365$$ 0 0
$$366$$ 1235.31 0.176423
$$367$$ −5748.70 −0.817656 −0.408828 0.912612i $$-0.634062\pi$$
−0.408828 + 0.912612i $$0.634062\pi$$
$$368$$ 41.5458 0.00588512
$$369$$ −981.781 −0.138508
$$370$$ 0 0
$$371$$ −11199.9 −1.56730
$$372$$ −9559.48 −1.33236
$$373$$ −4467.78 −0.620196 −0.310098 0.950705i $$-0.600362\pi$$
−0.310098 + 0.950705i $$0.600362\pi$$
$$374$$ 3687.18 0.509785
$$375$$ 0 0
$$376$$ −12204.9 −1.67398
$$377$$ −492.712 −0.0673103
$$378$$ 2560.42 0.348396
$$379$$ 7804.08 1.05770 0.528851 0.848715i $$-0.322623\pi$$
0.528851 + 0.848715i $$0.322623\pi$$
$$380$$ 0 0
$$381$$ 4804.89 0.646094
$$382$$ 12683.4 1.69880
$$383$$ 11161.1 1.48904 0.744522 0.667597i $$-0.232677\pi$$
0.744522 + 0.667597i $$0.232677\pi$$
$$384$$ 7178.74 0.954007
$$385$$ 0 0
$$386$$ −5746.38 −0.757728
$$387$$ 1095.75 0.143928
$$388$$ 5594.04 0.731944
$$389$$ 8490.24 1.10661 0.553306 0.832978i $$-0.313366\pi$$
0.553306 + 0.832978i $$0.313366\pi$$
$$390$$ 0 0
$$391$$ −996.540 −0.128893
$$392$$ −1948.47 −0.251052
$$393$$ 6012.39 0.771717
$$394$$ −661.616 −0.0845983
$$395$$ 0 0
$$396$$ 1298.16 0.164735
$$397$$ −6019.74 −0.761013 −0.380507 0.924778i $$-0.624250\pi$$
−0.380507 + 0.924778i $$0.624250\pi$$
$$398$$ −3497.83 −0.440529
$$399$$ −3779.99 −0.474277
$$400$$ 0 0
$$401$$ −10398.8 −1.29499 −0.647495 0.762069i $$-0.724184\pi$$
−0.647495 + 0.762069i $$0.724184\pi$$
$$402$$ 6733.01 0.835354
$$403$$ −3805.13 −0.470340
$$404$$ 972.158 0.119720
$$405$$ 0 0
$$406$$ −2983.95 −0.364756
$$407$$ 719.420 0.0876175
$$408$$ 5141.30 0.623854
$$409$$ −4733.68 −0.572287 −0.286144 0.958187i $$-0.592373\pi$$
−0.286144 + 0.958187i $$0.592373\pi$$
$$410$$ 0 0
$$411$$ 5018.55 0.602304
$$412$$ 909.911 0.108806
$$413$$ −2259.45 −0.269202
$$414$$ −564.910 −0.0670624
$$415$$ 0 0
$$416$$ 2723.98 0.321044
$$417$$ 7621.14 0.894985
$$418$$ −3085.74 −0.361073
$$419$$ −8117.57 −0.946466 −0.473233 0.880937i $$-0.656913\pi$$
−0.473233 + 0.880937i $$0.656913\pi$$
$$420$$ 0 0
$$421$$ −9484.27 −1.09795 −0.548973 0.835840i $$-0.684981\pi$$
−0.548973 + 0.835840i $$0.684981\pi$$
$$422$$ −18271.8 −2.10772
$$423$$ 4675.77 0.537456
$$424$$ −12748.5 −1.46020
$$425$$ 0 0
$$426$$ −11543.5 −1.31288
$$427$$ 1849.52 0.209612
$$428$$ −14970.8 −1.69075
$$429$$ 516.728 0.0581536
$$430$$ 0 0
$$431$$ 9335.16 1.04329 0.521646 0.853162i $$-0.325318\pi$$
0.521646 + 0.853162i $$0.325318\pi$$
$$432$$ 82.1157 0.00914535
$$433$$ 2983.02 0.331074 0.165537 0.986204i $$-0.447064\pi$$
0.165537 + 0.986204i $$0.447064\pi$$
$$434$$ −23044.5 −2.54878
$$435$$ 0 0
$$436$$ −29200.0 −3.20739
$$437$$ 833.988 0.0912931
$$438$$ 4841.36 0.528148
$$439$$ −5232.32 −0.568850 −0.284425 0.958698i $$-0.591803\pi$$
−0.284425 + 0.958698i $$0.591803\pi$$
$$440$$ 0 0
$$441$$ 746.472 0.0806038
$$442$$ 5248.68 0.564828
$$443$$ −7517.71 −0.806269 −0.403135 0.915141i $$-0.632079\pi$$
−0.403135 + 0.915141i $$0.632079\pi$$
$$444$$ 2572.78 0.274997
$$445$$ 0 0
$$446$$ 4172.52 0.442992
$$447$$ 9272.03 0.981101
$$448$$ 16999.1 1.79270
$$449$$ 16070.9 1.68916 0.844581 0.535428i $$-0.179850\pi$$
0.844581 + 0.535428i $$0.179850\pi$$
$$450$$ 0 0
$$451$$ −1199.96 −0.125285
$$452$$ 22550.7 2.34667
$$453$$ 4076.21 0.422775
$$454$$ −9475.29 −0.979510
$$455$$ 0 0
$$456$$ −4302.67 −0.441867
$$457$$ −9718.51 −0.994776 −0.497388 0.867528i $$-0.665707\pi$$
−0.497388 + 0.867528i $$0.665707\pi$$
$$458$$ −18735.4 −1.91146
$$459$$ −1969.67 −0.200297
$$460$$ 0 0
$$461$$ −14538.0 −1.46877 −0.734385 0.678733i $$-0.762529\pi$$
−0.734385 + 0.678733i $$0.762529\pi$$
$$462$$ 3129.40 0.315136
$$463$$ 9978.17 1.00157 0.500783 0.865573i $$-0.333045\pi$$
0.500783 + 0.865573i $$0.333045\pi$$
$$464$$ −95.6990 −0.00957481
$$465$$ 0 0
$$466$$ 7732.03 0.768625
$$467$$ −15188.2 −1.50498 −0.752489 0.658605i $$-0.771147\pi$$
−0.752489 + 0.658605i $$0.771147\pi$$
$$468$$ 1847.92 0.182522
$$469$$ 10080.7 0.992503
$$470$$ 0 0
$$471$$ 3035.85 0.296995
$$472$$ −2571.88 −0.250806
$$473$$ 1339.25 0.130187
$$474$$ 11457.8 1.11028
$$475$$ 0 0
$$476$$ 19742.3 1.90102
$$477$$ 4884.06 0.468817
$$478$$ −18494.1 −1.76967
$$479$$ 11330.8 1.08083 0.540415 0.841399i $$-0.318267\pi$$
0.540415 + 0.841399i $$0.318267\pi$$
$$480$$ 0 0
$$481$$ 1024.09 0.0970779
$$482$$ 12793.3 1.20896
$$483$$ −845.788 −0.0796784
$$484$$ 1586.64 0.149008
$$485$$ 0 0
$$486$$ −1116.55 −0.104213
$$487$$ −19086.9 −1.77599 −0.887997 0.459850i $$-0.847903\pi$$
−0.887997 + 0.459850i $$0.847903\pi$$
$$488$$ 2105.26 0.195288
$$489$$ 8449.11 0.781353
$$490$$ 0 0
$$491$$ −8112.85 −0.745677 −0.372839 0.927896i $$-0.621616\pi$$
−0.372839 + 0.927896i $$0.621616\pi$$
$$492$$ −4291.27 −0.393222
$$493$$ 2295.49 0.209703
$$494$$ −4392.53 −0.400060
$$495$$ 0 0
$$496$$ −739.066 −0.0669053
$$497$$ −17283.0 −1.55986
$$498$$ 19148.6 1.72303
$$499$$ 18329.1 1.64433 0.822167 0.569246i $$-0.192765\pi$$
0.822167 + 0.569246i $$0.192765\pi$$
$$500$$ 0 0
$$501$$ −10346.7 −0.922666
$$502$$ 8397.58 0.746618
$$503$$ −7739.57 −0.686064 −0.343032 0.939324i $$-0.611454\pi$$
−0.343032 + 0.939324i $$0.611454\pi$$
$$504$$ 4363.55 0.385651
$$505$$ 0 0
$$506$$ −690.446 −0.0606602
$$507$$ −5855.44 −0.512918
$$508$$ 21001.7 1.83425
$$509$$ 15914.9 1.38589 0.692943 0.720993i $$-0.256314\pi$$
0.692943 + 0.720993i $$0.256314\pi$$
$$510$$ 0 0
$$511$$ 7248.51 0.627505
$$512$$ 1100.65 0.0950048
$$513$$ 1648.38 0.141867
$$514$$ 2688.78 0.230733
$$515$$ 0 0
$$516$$ 4789.40 0.408608
$$517$$ 5714.83 0.486147
$$518$$ 6202.07 0.526068
$$519$$ 6863.56 0.580495
$$520$$ 0 0
$$521$$ 2274.50 0.191262 0.0956312 0.995417i $$-0.469513\pi$$
0.0956312 + 0.995417i $$0.469513\pi$$
$$522$$ 1301.25 0.109107
$$523$$ −10971.1 −0.917274 −0.458637 0.888624i $$-0.651662\pi$$
−0.458637 + 0.888624i $$0.651662\pi$$
$$524$$ 26279.6 2.19089
$$525$$ 0 0
$$526$$ −1094.02 −0.0906877
$$527$$ 17727.6 1.46533
$$528$$ 100.364 0.00827228
$$529$$ −11980.4 −0.984663
$$530$$ 0 0
$$531$$ 985.306 0.0805247
$$532$$ −16522.0 −1.34646
$$533$$ −1708.13 −0.138813
$$534$$ −21003.5 −1.70208
$$535$$ 0 0
$$536$$ 11474.6 0.924680
$$537$$ 9747.17 0.783280
$$538$$ −21221.2 −1.70058
$$539$$ 912.355 0.0729089
$$540$$ 0 0
$$541$$ 5313.05 0.422229 0.211115 0.977461i $$-0.432291\pi$$
0.211115 + 0.977461i $$0.432291\pi$$
$$542$$ 659.084 0.0522326
$$543$$ 3511.34 0.277506
$$544$$ −12690.7 −1.00020
$$545$$ 0 0
$$546$$ 4454.68 0.349162
$$547$$ −20685.1 −1.61688 −0.808439 0.588581i $$-0.799687\pi$$
−0.808439 + 0.588581i $$0.799687\pi$$
$$548$$ 21935.6 1.70993
$$549$$ −806.541 −0.0627000
$$550$$ 0 0
$$551$$ −1921.06 −0.148529
$$552$$ −962.739 −0.0742335
$$553$$ 17154.7 1.31915
$$554$$ −39527.3 −3.03132
$$555$$ 0 0
$$556$$ 33311.2 2.54085
$$557$$ 10853.8 0.825659 0.412830 0.910808i $$-0.364540\pi$$
0.412830 + 0.910808i $$0.364540\pi$$
$$558$$ 10049.3 0.762402
$$559$$ 1906.41 0.144244
$$560$$ 0 0
$$561$$ −2407.37 −0.181175
$$562$$ 13751.5 1.03216
$$563$$ 15381.2 1.15141 0.575704 0.817658i $$-0.304728\pi$$
0.575704 + 0.817658i $$0.304728\pi$$
$$564$$ 20437.3 1.52583
$$565$$ 0 0
$$566$$ −31515.6 −2.34046
$$567$$ −1671.71 −0.123818
$$568$$ −19672.9 −1.45327
$$569$$ −1348.88 −0.0993814 −0.0496907 0.998765i $$-0.515824\pi$$
−0.0496907 + 0.998765i $$0.515824\pi$$
$$570$$ 0 0
$$571$$ 3463.51 0.253841 0.126920 0.991913i $$-0.459491\pi$$
0.126920 + 0.991913i $$0.459491\pi$$
$$572$$ 2258.57 0.165097
$$573$$ −8281.05 −0.603745
$$574$$ −10344.7 −0.752231
$$575$$ 0 0
$$576$$ −7412.97 −0.536239
$$577$$ −12052.6 −0.869598 −0.434799 0.900528i $$-0.643181\pi$$
−0.434799 + 0.900528i $$0.643181\pi$$
$$578$$ −1878.42 −0.135176
$$579$$ 3751.83 0.269293
$$580$$ 0 0
$$581$$ 28669.4 2.04717
$$582$$ −5880.66 −0.418834
$$583$$ 5969.41 0.424061
$$584$$ 8250.80 0.584624
$$585$$ 0 0
$$586$$ 18607.8 1.31174
$$587$$ 11133.1 0.782813 0.391407 0.920218i $$-0.371989\pi$$
0.391407 + 0.920218i $$0.371989\pi$$
$$588$$ 3262.75 0.228833
$$589$$ −14836.0 −1.03787
$$590$$ 0 0
$$591$$ 431.972 0.0300659
$$592$$ 198.908 0.0138092
$$593$$ 7939.69 0.549821 0.274911 0.961470i $$-0.411352\pi$$
0.274911 + 0.961470i $$0.411352\pi$$
$$594$$ −1364.67 −0.0942646
$$595$$ 0 0
$$596$$ 40527.1 2.78533
$$597$$ 2283.75 0.156562
$$598$$ −982.846 −0.0672100
$$599$$ −19474.7 −1.32840 −0.664202 0.747553i $$-0.731229\pi$$
−0.664202 + 0.747553i $$0.731229\pi$$
$$600$$ 0 0
$$601$$ −19946.1 −1.35377 −0.676887 0.736087i $$-0.736671\pi$$
−0.676887 + 0.736087i $$0.736671\pi$$
$$602$$ 11545.6 0.781664
$$603$$ −4396.01 −0.296881
$$604$$ 17816.7 1.20025
$$605$$ 0 0
$$606$$ −1021.97 −0.0685061
$$607$$ −1427.44 −0.0954496 −0.0477248 0.998861i $$-0.515197\pi$$
−0.0477248 + 0.998861i $$0.515197\pi$$
$$608$$ 10620.6 0.708427
$$609$$ 1948.23 0.129633
$$610$$ 0 0
$$611$$ 8135.03 0.538638
$$612$$ −8609.23 −0.568640
$$613$$ 8029.40 0.529045 0.264522 0.964380i $$-0.414786\pi$$
0.264522 + 0.964380i $$0.414786\pi$$
$$614$$ −43985.1 −2.89103
$$615$$ 0 0
$$616$$ 5333.22 0.348834
$$617$$ −20795.5 −1.35688 −0.678440 0.734655i $$-0.737344\pi$$
−0.678440 + 0.734655i $$0.737344\pi$$
$$618$$ −956.533 −0.0622612
$$619$$ 1677.43 0.108920 0.0544602 0.998516i $$-0.482656\pi$$
0.0544602 + 0.998516i $$0.482656\pi$$
$$620$$ 0 0
$$621$$ 368.832 0.0238337
$$622$$ −24795.8 −1.59842
$$623$$ −31446.6 −2.02228
$$624$$ 142.867 0.00916547
$$625$$ 0 0
$$626$$ 44823.0 2.86180
$$627$$ 2014.69 0.128324
$$628$$ 13269.4 0.843165
$$629$$ −4771.11 −0.302443
$$630$$ 0 0
$$631$$ −25225.2 −1.59144 −0.795719 0.605666i $$-0.792907\pi$$
−0.795719 + 0.605666i $$0.792907\pi$$
$$632$$ 19526.8 1.22901
$$633$$ 11929.7 0.749074
$$634$$ −20004.8 −1.25315
$$635$$ 0 0
$$636$$ 21347.7 1.33096
$$637$$ 1298.73 0.0807812
$$638$$ 1590.41 0.0986912
$$639$$ 7536.81 0.466591
$$640$$ 0 0
$$641$$ 15165.3 0.934468 0.467234 0.884134i $$-0.345251\pi$$
0.467234 + 0.884134i $$0.345251\pi$$
$$642$$ 15737.9 0.967486
$$643$$ −27156.1 −1.66553 −0.832763 0.553630i $$-0.813242\pi$$
−0.832763 + 0.553630i $$0.813242\pi$$
$$644$$ −3696.85 −0.226206
$$645$$ 0 0
$$646$$ 20464.3 1.24637
$$647$$ −29154.9 −1.77156 −0.885778 0.464110i $$-0.846374\pi$$
−0.885778 + 0.464110i $$0.846374\pi$$
$$648$$ −1902.86 −0.115357
$$649$$ 1204.26 0.0728374
$$650$$ 0 0
$$651$$ 15045.9 0.905828
$$652$$ 36930.2 2.21825
$$653$$ 19141.7 1.14713 0.573564 0.819161i $$-0.305560\pi$$
0.573564 + 0.819161i $$0.305560\pi$$
$$654$$ 30696.1 1.83534
$$655$$ 0 0
$$656$$ −331.768 −0.0197460
$$657$$ −3160.94 −0.187702
$$658$$ 49267.2 2.91890
$$659$$ −24939.6 −1.47422 −0.737110 0.675773i $$-0.763810\pi$$
−0.737110 + 0.675773i $$0.763810\pi$$
$$660$$ 0 0
$$661$$ 22617.7 1.33090 0.665452 0.746440i $$-0.268239\pi$$
0.665452 + 0.746440i $$0.268239\pi$$
$$662$$ −24755.4 −1.45339
$$663$$ −3426.88 −0.200738
$$664$$ 32633.7 1.90728
$$665$$ 0 0
$$666$$ −2704.61 −0.157359
$$667$$ −429.843 −0.0249529
$$668$$ −45224.3 −2.61943
$$669$$ −2724.25 −0.157438
$$670$$ 0 0
$$671$$ −985.772 −0.0567143
$$672$$ −10770.9 −0.618298
$$673$$ 13855.8 0.793615 0.396807 0.917902i $$-0.370118\pi$$
0.396807 + 0.917902i $$0.370118\pi$$
$$674$$ −20678.1 −1.18174
$$675$$ 0 0
$$676$$ −25593.5 −1.45616
$$677$$ 24992.8 1.41884 0.709419 0.704787i $$-0.248957\pi$$
0.709419 + 0.704787i $$0.248957\pi$$
$$678$$ −23706.1 −1.34282
$$679$$ −8804.57 −0.497626
$$680$$ 0 0
$$681$$ 6186.46 0.348114
$$682$$ 12282.5 0.689619
$$683$$ 14420.5 0.807887 0.403943 0.914784i $$-0.367639\pi$$
0.403943 + 0.914784i $$0.367639\pi$$
$$684$$ 7204.93 0.402759
$$685$$ 0 0
$$686$$ −24661.4 −1.37256
$$687$$ 12232.4 0.679324
$$688$$ 370.280 0.0205186
$$689$$ 8497.42 0.469849
$$690$$ 0 0
$$691$$ 30552.4 1.68201 0.841005 0.541027i $$-0.181964\pi$$
0.841005 + 0.541027i $$0.181964\pi$$
$$692$$ 29999.9 1.64801
$$693$$ −2043.20 −0.111998
$$694$$ 27138.7 1.48440
$$695$$ 0 0
$$696$$ 2217.63 0.120774
$$697$$ 7957.96 0.432467
$$698$$ −16708.1 −0.906032
$$699$$ −5048.27 −0.273166
$$700$$ 0 0
$$701$$ −9151.47 −0.493076 −0.246538 0.969133i $$-0.579293\pi$$
−0.246538 + 0.969133i $$0.579293\pi$$
$$702$$ −1942.60 −0.104443
$$703$$ 3992.86 0.214216
$$704$$ −9060.30 −0.485047
$$705$$ 0 0
$$706$$ 966.793 0.0515378
$$707$$ −1530.10 −0.0813937
$$708$$ 4306.67 0.228608
$$709$$ −6261.96 −0.331697 −0.165848 0.986151i $$-0.553036\pi$$
−0.165848 + 0.986151i $$0.553036\pi$$
$$710$$ 0 0
$$711$$ −7480.85 −0.394590
$$712$$ −35794.9 −1.88409
$$713$$ −3319.60 −0.174362
$$714$$ −20753.8 −1.08780
$$715$$ 0 0
$$716$$ 42603.9 2.22372
$$717$$ 12074.9 0.628933
$$718$$ −11485.7 −0.596995
$$719$$ −18228.7 −0.945500 −0.472750 0.881197i $$-0.656739\pi$$
−0.472750 + 0.881197i $$0.656739\pi$$
$$720$$ 0 0
$$721$$ −1432.13 −0.0739740
$$722$$ 14389.9 0.741740
$$723$$ −8352.81 −0.429660
$$724$$ 15347.7 0.787836
$$725$$ 0 0
$$726$$ −1667.93 −0.0852655
$$727$$ 7233.66 0.369026 0.184513 0.982830i $$-0.440929\pi$$
0.184513 + 0.982830i $$0.440929\pi$$
$$728$$ 7591.81 0.386499
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −8881.73 −0.449388
$$732$$ −3525.31 −0.178004
$$733$$ 13444.8 0.677485 0.338743 0.940879i $$-0.389998\pi$$
0.338743 + 0.940879i $$0.389998\pi$$
$$734$$ 26414.4 1.32830
$$735$$ 0 0
$$736$$ 2376.41 0.119016
$$737$$ −5372.90 −0.268539
$$738$$ 4511.14 0.225010
$$739$$ 18490.9 0.920432 0.460216 0.887807i $$-0.347772\pi$$
0.460216 + 0.887807i $$0.347772\pi$$
$$740$$ 0 0
$$741$$ 2867.90 0.142180
$$742$$ 51461.8 2.54612
$$743$$ 25160.9 1.24235 0.621173 0.783674i $$-0.286657\pi$$
0.621173 + 0.783674i $$0.286657\pi$$
$$744$$ 17126.3 0.843927
$$745$$ 0 0
$$746$$ 20528.8 1.00752
$$747$$ −12502.2 −0.612358
$$748$$ −10522.4 −0.514354
$$749$$ 23562.9 1.14949
$$750$$ 0 0
$$751$$ −13419.5 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$752$$ 1580.06 0.0766207
$$753$$ −5482.81 −0.265345
$$754$$ 2263.94 0.109347
$$755$$ 0 0
$$756$$ −7306.86 −0.351518
$$757$$ 7014.90 0.336804 0.168402 0.985718i $$-0.446139\pi$$
0.168402 + 0.985718i $$0.446139\pi$$
$$758$$ −35858.6 −1.71826
$$759$$ 450.795 0.0215584
$$760$$ 0 0
$$761$$ −30156.9 −1.43651 −0.718256 0.695779i $$-0.755059\pi$$
−0.718256 + 0.695779i $$0.755059\pi$$
$$762$$ −22077.8 −1.04960
$$763$$ 45958.4 2.18061
$$764$$ −36195.7 −1.71402
$$765$$ 0 0
$$766$$ −51283.5 −2.41899
$$767$$ 1714.26 0.0807019
$$768$$ −13217.4 −0.621017
$$769$$ 11292.2 0.529530 0.264765 0.964313i $$-0.414706\pi$$
0.264765 + 0.964313i $$0.414706\pi$$
$$770$$ 0 0
$$771$$ −1755.51 −0.0820016
$$772$$ 16398.9 0.764519
$$773$$ −8524.10 −0.396624 −0.198312 0.980139i $$-0.563546\pi$$
−0.198312 + 0.980139i $$0.563546\pi$$
$$774$$ −5034.80 −0.233814
$$775$$ 0 0
$$776$$ −10022.0 −0.463621
$$777$$ −4049.35 −0.186962
$$778$$ −39011.4 −1.79772
$$779$$ −6659.89 −0.306310
$$780$$ 0 0
$$781$$ 9211.66 0.422047
$$782$$ 4578.96 0.209390
$$783$$ −849.589 −0.0387763
$$784$$ 252.251 0.0114910
$$785$$ 0 0
$$786$$ −27626.1 −1.25368
$$787$$ 14983.9 0.678676 0.339338 0.940665i $$-0.389797\pi$$
0.339338 + 0.940665i $$0.389797\pi$$
$$788$$ 1888.10 0.0853565
$$789$$ 714.293 0.0322300
$$790$$ 0 0
$$791$$ −35493.0 −1.59543
$$792$$ −2325.72 −0.104345
$$793$$ −1403.24 −0.0628380
$$794$$ 27659.9 1.23629
$$795$$ 0 0
$$796$$ 9982.04 0.444477
$$797$$ −37172.3 −1.65208 −0.826041 0.563610i $$-0.809412\pi$$
−0.826041 + 0.563610i $$0.809412\pi$$
$$798$$ 17368.5 0.770475
$$799$$ −37900.1 −1.67811
$$800$$ 0 0
$$801$$ 13713.3 0.604913
$$802$$ 47781.0 2.10375
$$803$$ −3863.37 −0.169783
$$804$$ −19214.5 −0.842841
$$805$$ 0 0
$$806$$ 17484.0 0.764080
$$807$$ 13855.4 0.604378
$$808$$ −1741.68 −0.0758316
$$809$$ 23797.1 1.03419 0.517096 0.855928i $$-0.327013\pi$$
0.517096 + 0.855928i $$0.327013\pi$$
$$810$$ 0 0
$$811$$ 8988.35 0.389178 0.194589 0.980885i $$-0.437663\pi$$
0.194589 + 0.980885i $$0.437663\pi$$
$$812$$ 8515.54 0.368026
$$813$$ −430.318 −0.0185633
$$814$$ −3305.63 −0.142337
$$815$$ 0 0
$$816$$ −665.600 −0.0285547
$$817$$ 7432.98 0.318295
$$818$$ 21750.6 0.929696
$$819$$ −2908.48 −0.124091
$$820$$ 0 0
$$821$$ −25156.8 −1.06940 −0.534702 0.845041i $$-0.679576\pi$$
−0.534702 + 0.845041i $$0.679576\pi$$
$$822$$ −23059.5 −0.978459
$$823$$ −1318.51 −0.0558447 −0.0279224 0.999610i $$-0.508889\pi$$
−0.0279224 + 0.999610i $$0.508889\pi$$
$$824$$ −1630.16 −0.0689189
$$825$$ 0 0
$$826$$ 10381.9 0.437326
$$827$$ −124.982 −0.00525519 −0.00262760 0.999997i $$-0.500836\pi$$
−0.00262760 + 0.999997i $$0.500836\pi$$
$$828$$ 1612.13 0.0676635
$$829$$ −8886.80 −0.372318 −0.186159 0.982520i $$-0.559604\pi$$
−0.186159 + 0.982520i $$0.559604\pi$$
$$830$$ 0 0
$$831$$ 25807.5 1.07732
$$832$$ −12897.3 −0.537419
$$833$$ −6050.63 −0.251671
$$834$$ −35018.0 −1.45393
$$835$$ 0 0
$$836$$ 8806.02 0.364309
$$837$$ −6561.22 −0.270955
$$838$$ 37299.1 1.53756
$$839$$ 2995.21 0.123249 0.0616247 0.998099i $$-0.480372\pi$$
0.0616247 + 0.998099i $$0.480372\pi$$
$$840$$ 0 0
$$841$$ −23398.9 −0.959403
$$842$$ 43578.8 1.78364
$$843$$ −8978.43 −0.366825
$$844$$ 52143.6 2.12661
$$845$$ 0 0
$$846$$ −21484.5 −0.873111
$$847$$ −2497.24 −0.101306
$$848$$ 1650.44 0.0668355
$$849$$ 20576.7 0.831789
$$850$$ 0 0
$$851$$ 893.418 0.0359882
$$852$$ 32942.7 1.32464
$$853$$ −18130.5 −0.727757 −0.363878 0.931446i $$-0.618548\pi$$
−0.363878 + 0.931446i $$0.618548\pi$$
$$854$$ −8498.27 −0.340521
$$855$$ 0 0
$$856$$ 26821.1 1.07094
$$857$$ −26394.1 −1.05205 −0.526024 0.850470i $$-0.676318\pi$$
−0.526024 + 0.850470i $$0.676318\pi$$
$$858$$ −2374.29 −0.0944720
$$859$$ −29456.2 −1.17000 −0.585002 0.811032i $$-0.698906\pi$$
−0.585002 + 0.811032i $$0.698906\pi$$
$$860$$ 0 0
$$861$$ 6754.12 0.267340
$$862$$ −42893.7 −1.69486
$$863$$ 762.616 0.0300808 0.0150404 0.999887i $$-0.495212\pi$$
0.0150404 + 0.999887i $$0.495212\pi$$
$$864$$ 4696.99 0.184948
$$865$$ 0 0
$$866$$ −13706.6 −0.537838
$$867$$ 1226.43 0.0480411
$$868$$ 65764.0 2.57163
$$869$$ −9143.26 −0.356920
$$870$$ 0 0
$$871$$ −7648.29 −0.297535
$$872$$ 52313.3 2.03160
$$873$$ 3839.51 0.148852
$$874$$ −3832.06 −0.148308
$$875$$ 0 0
$$876$$ −13816.2 −0.532882
$$877$$ −44767.2 −1.72369 −0.861847 0.507168i $$-0.830692\pi$$
−0.861847 + 0.507168i $$0.830692\pi$$
$$878$$ 24041.8 0.924112
$$879$$ −12149.1 −0.466188
$$880$$ 0 0
$$881$$ −32057.9 −1.22595 −0.612973 0.790104i $$-0.710027\pi$$
−0.612973 + 0.790104i $$0.710027\pi$$
$$882$$ −3429.93 −0.130943
$$883$$ −7078.95 −0.269791 −0.134896 0.990860i $$-0.543070\pi$$
−0.134896 + 0.990860i $$0.543070\pi$$
$$884$$ −14978.6 −0.569891
$$885$$ 0 0
$$886$$ 34542.8 1.30981
$$887$$ 25148.1 0.951964 0.475982 0.879455i $$-0.342093\pi$$
0.475982 + 0.879455i $$0.342093\pi$$
$$888$$ −4609.28 −0.174186
$$889$$ −33055.0 −1.24705
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ −11907.4 −0.446963
$$893$$ 31718.0 1.18858
$$894$$ −42603.6 −1.59382
$$895$$ 0 0
$$896$$ −49385.8 −1.84137
$$897$$ 641.704 0.0238862
$$898$$ −73843.6 −2.74409
$$899$$ 7646.56 0.283679
$$900$$ 0 0
$$901$$ −39588.4 −1.46380
$$902$$ 5513.62 0.203529
$$903$$ −7538.14 −0.277800
$$904$$ −40400.8 −1.48641
$$905$$ 0 0
$$906$$ −18729.6 −0.686809
$$907$$ −1269.76 −0.0464848 −0.0232424 0.999730i $$-0.507399\pi$$
−0.0232424 + 0.999730i $$0.507399\pi$$
$$908$$ 27040.4 0.988289
$$909$$ 667.248 0.0243468
$$910$$ 0 0
$$911$$ −33783.1 −1.22863 −0.614316 0.789060i $$-0.710568\pi$$
−0.614316 + 0.789060i $$0.710568\pi$$
$$912$$ 557.030 0.0202249
$$913$$ −15280.5 −0.553899
$$914$$ 44655.1 1.61604
$$915$$ 0 0
$$916$$ 53466.6 1.92859
$$917$$ −41361.9 −1.48952
$$918$$ 9050.35 0.325388
$$919$$ 39262.5 1.40930 0.704652 0.709553i $$-0.251103\pi$$
0.704652 + 0.709553i $$0.251103\pi$$
$$920$$ 0 0
$$921$$ 28718.1 1.02746
$$922$$ 66800.1 2.38606
$$923$$ 13112.7 0.467618
$$924$$ −8930.61 −0.317960
$$925$$ 0 0
$$926$$ −45848.3 −1.62707
$$927$$ 624.525 0.0221274
$$928$$ −5473.95 −0.193633
$$929$$ −21175.0 −0.747825 −0.373913 0.927464i $$-0.621984\pi$$
−0.373913 + 0.927464i $$0.621984\pi$$
$$930$$ 0 0
$$931$$ 5063.68 0.178255
$$932$$ −22065.5 −0.775515
$$933$$ 16189.3 0.568073
$$934$$ 69787.5 2.44488
$$935$$ 0 0
$$936$$ −3310.65 −0.115611
$$937$$ 5135.11 0.179036 0.0895180 0.995985i $$-0.471467\pi$$
0.0895180 + 0.995985i $$0.471467\pi$$
$$938$$ −46319.4 −1.61235
$$939$$ −29265.1 −1.01707
$$940$$ 0 0
$$941$$ −9702.77 −0.336133 −0.168067 0.985776i $$-0.553752\pi$$
−0.168067 + 0.985776i $$0.553752\pi$$
$$942$$ −13949.3 −0.482477
$$943$$ −1490.18 −0.0514600
$$944$$ 332.959 0.0114798
$$945$$ 0 0
$$946$$ −6153.65 −0.211493
$$947$$ 699.579 0.0240055 0.0120028 0.999928i $$-0.496179\pi$$
0.0120028 + 0.999928i $$0.496179\pi$$
$$948$$ −32698.0 −1.12024
$$949$$ −5499.49 −0.188115
$$950$$ 0 0
$$951$$ 13061.2 0.445363
$$952$$ −35369.3 −1.20412
$$953$$ −42039.3 −1.42895 −0.714473 0.699663i $$-0.753333\pi$$
−0.714473 + 0.699663i $$0.753333\pi$$
$$954$$ −22441.6 −0.761606
$$955$$ 0 0
$$956$$ 52778.1 1.78553
$$957$$ −1038.39 −0.0350745
$$958$$ −52063.4 −1.75584
$$959$$ −34524.9 −1.16253
$$960$$ 0 0
$$961$$ 29262.0 0.982243
$$962$$ −4705.55 −0.157706
$$963$$ −10275.3 −0.343841
$$964$$ −36509.3 −1.21980
$$965$$ 0 0
$$966$$ 3886.27 0.129440
$$967$$ 32794.8 1.09060 0.545299 0.838242i $$-0.316416\pi$$
0.545299 + 0.838242i $$0.316416\pi$$
$$968$$ −2842.55 −0.0943832
$$969$$ −13361.2 −0.442955
$$970$$ 0 0
$$971$$ −3322.53 −0.109810 −0.0549048 0.998492i $$-0.517486\pi$$
−0.0549048 + 0.998492i $$0.517486\pi$$
$$972$$ 3186.39 0.105148
$$973$$ −52429.3 −1.72745
$$974$$ 87701.4 2.88515
$$975$$ 0 0
$$976$$ −272.550 −0.00893864
$$977$$ −22192.5 −0.726716 −0.363358 0.931650i $$-0.618370\pi$$
−0.363358 + 0.931650i $$0.618370\pi$$
$$978$$ −38822.4 −1.26933
$$979$$ 16760.7 0.547164
$$980$$ 0 0
$$981$$ −20041.6 −0.652272
$$982$$ 37277.4 1.21137
$$983$$ −7383.09 −0.239556 −0.119778 0.992801i $$-0.538218\pi$$
−0.119778 + 0.992801i $$0.538218\pi$$
$$984$$ 7688.04 0.249071
$$985$$ 0 0
$$986$$ −10547.4 −0.340668
$$987$$ −32166.7 −1.03736
$$988$$ 12535.3 0.403645
$$989$$ 1663.16 0.0534735
$$990$$ 0 0
$$991$$ 46260.8 1.48287 0.741434 0.671026i $$-0.234146\pi$$
0.741434 + 0.671026i $$0.234146\pi$$
$$992$$ −42274.3 −1.35304
$$993$$ 16162.9 0.516530
$$994$$ 79413.1 2.53403
$$995$$ 0 0
$$996$$ −54645.9 −1.73847
$$997$$ −41196.8 −1.30864 −0.654320 0.756217i $$-0.727045\pi$$
−0.654320 + 0.756217i $$0.727045\pi$$
$$998$$ −84219.5 −2.67127
$$999$$ 1765.85 0.0559249
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 825.4.a.p.1.1 3
3.2 odd 2 2475.4.a.z.1.3 3
5.2 odd 4 825.4.c.m.199.1 6
5.3 odd 4 825.4.c.m.199.6 6
5.4 even 2 165.4.a.g.1.3 3
15.14 odd 2 495.4.a.i.1.1 3
55.54 odd 2 1815.4.a.q.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 5.4 even 2
495.4.a.i.1.1 3 15.14 odd 2
825.4.a.p.1.1 3 1.1 even 1 trivial
825.4.c.m.199.1 6 5.2 odd 4
825.4.c.m.199.6 6 5.3 odd 4
1815.4.a.q.1.1 3 55.54 odd 2
2475.4.a.z.1.3 3 3.2 odd 2