# Properties

 Label 495.4.a.i.1.1 Level $495$ Weight $4$ Character 495.1 Self dual yes Analytic conductor $29.206$ Analytic rank $1$ 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] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 495.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: $$29.2059454528$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 495.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +O(q^{10})$$ $$q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +22.9743 q^{10} -11.0000 q^{11} -15.6584 q^{13} -94.8302 q^{14} +3.04132 q^{16} -72.9507 q^{17} +61.0513 q^{19} -65.5635 q^{20} +50.5434 q^{22} +13.6605 q^{23} +25.0000 q^{25} +71.9483 q^{26} +270.624 q^{28} +31.4663 q^{29} -243.008 q^{31} +173.963 q^{32} +335.198 q^{34} -103.192 q^{35} -65.4018 q^{37} -280.522 q^{38} +117.461 q^{40} +109.087 q^{41} -121.750 q^{43} -144.240 q^{44} -62.7678 q^{46} +519.530 q^{47} +82.9413 q^{49} -114.871 q^{50} -205.324 q^{52} +542.673 q^{53} +55.0000 q^{55} -484.839 q^{56} -144.583 q^{58} -109.478 q^{59} -89.6156 q^{61} +1116.59 q^{62} -823.664 q^{64} +78.2922 q^{65} +488.446 q^{67} -956.581 q^{68} +474.151 q^{70} -837.423 q^{71} +351.216 q^{73} +300.512 q^{74} +800.547 q^{76} -227.022 q^{77} -831.205 q^{79} -15.2066 q^{80} -501.238 q^{82} -1389.13 q^{83} +364.754 q^{85} +559.423 q^{86} +258.413 q^{88} -1523.70 q^{89} -323.164 q^{91} +179.125 q^{92} -2387.17 q^{94} -305.256 q^{95} -426.612 q^{97} -381.103 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 $$3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8} + 5 q^{10} - 33 q^{11} - 20 q^{13} - 144 q^{14} + 25 q^{16} - 32 q^{17} + 116 q^{19} - 85 q^{20} + 11 q^{22} - 240 q^{23} + 75 q^{25} - 302 q^{26} + 160 q^{28} - 238 q^{29} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} - 90 q^{37} - 324 q^{38} - 15 q^{40} + 46 q^{41} - 134 q^{43} - 187 q^{44} - 240 q^{46} + 220 q^{47} - 457 q^{49} - 25 q^{50} - 1530 q^{52} + 798 q^{53} + 165 q^{55} - 688 q^{56} - 978 q^{58} - 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} + 720 q^{70} - 1816 q^{71} + 100 q^{73} + 1874 q^{74} + 396 q^{76} - 66 q^{77} - 96 q^{79} - 125 q^{80} - 910 q^{82} - 858 q^{83} + 160 q^{85} - 188 q^{86} - 33 q^{88} - 838 q^{89} + 332 q^{91} + 688 q^{92} - 3112 q^{94} - 580 q^{95} - 1322 q^{97} - 1017 q^{98}+O(q^{100})$$ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 + 5 * q^10 - 33 * q^11 - 20 * q^13 - 144 * q^14 + 25 * q^16 - 32 * q^17 + 116 * q^19 - 85 * q^20 + 11 * q^22 - 240 * q^23 + 75 * q^25 - 302 * q^26 + 160 * q^28 - 238 * q^29 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 - 90 * q^37 - 324 * q^38 - 15 * q^40 + 46 * q^41 - 134 * q^43 - 187 * q^44 - 240 * q^46 + 220 * q^47 - 457 * q^49 - 25 * q^50 - 1530 * q^52 + 798 * q^53 + 165 * q^55 - 688 * q^56 - 978 * q^58 - 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 + 720 * q^70 - 1816 * q^71 + 100 * q^73 + 1874 * q^74 + 396 * q^76 - 66 * q^77 - 96 * q^79 - 125 * q^80 - 910 * q^82 - 858 * q^83 + 160 * q^85 - 188 * q^86 - 33 * q^88 - 838 * q^89 + 332 * q^91 + 688 * q^92 - 3112 * q^94 - 580 * q^95 - 1322 * q^97 - 1017 * q^98

## 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 $$0.801765\pi$$
$$3$$ 0 0
$$4$$ 13.1127 1.63909
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 20.6383 1.11437 0.557183 0.830390i $$-0.311882\pi$$
0.557183 + 0.830390i $$0.311882\pi$$
$$8$$ −23.4921 −1.03822
$$9$$ 0 0
$$10$$ 22.9743 0.726511
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −15.6584 −0.334067 −0.167033 0.985951i $$-0.553419\pi$$
−0.167033 + 0.985951i $$0.553419\pi$$
$$14$$ −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 $$0.674212\pi$$
$$18$$ 0 0
$$19$$ 61.0513 0.737165 0.368582 0.929595i $$-0.379843\pi$$
0.368582 + 0.929595i $$0.379843\pi$$
$$20$$ −65.5635 −0.733022
$$21$$ 0 0
$$22$$ 50.5434 0.489813
$$23$$ 13.6605 0.123844 0.0619218 0.998081i $$-0.480277\pi$$
0.0619218 + 0.998081i $$0.480277\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 71.9483 0.542701
$$27$$ 0 0
$$28$$ 270.624 1.82654
$$29$$ 31.4663 0.201487 0.100744 0.994912i $$-0.467878\pi$$
0.100744 + 0.994912i $$0.467878\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ 335.198 1.69076
$$35$$ −103.192 −0.498360
$$36$$ 0 0
$$37$$ −65.4018 −0.290594 −0.145297 0.989388i $$-0.546414\pi$$
−0.145297 + 0.989388i $$0.546414\pi$$
$$38$$ −280.522 −1.19754
$$39$$ 0 0
$$40$$ 117.461 0.464304
$$41$$ 109.087 0.415524 0.207762 0.978179i $$-0.433382\pi$$
0.207762 + 0.978179i $$0.433382\pi$$
$$42$$ 0 0
$$43$$ −121.750 −0.431783 −0.215891 0.976417i $$-0.569266\pi$$
−0.215891 + 0.976417i $$0.569266\pi$$
$$44$$ −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 $$0.201529\pi$$
$$48$$ 0 0
$$49$$ 82.9413 0.241811
$$50$$ −114.871 −0.324905
$$51$$ 0 0
$$52$$ −205.324 −0.547565
$$53$$ 542.673 1.40645 0.703226 0.710967i $$-0.251742\pi$$
0.703226 + 0.710967i $$0.251742\pi$$
$$54$$ 0 0
$$55$$ 55.0000 0.134840
$$56$$ −484.839 −1.15695
$$57$$ 0 0
$$58$$ −144.583 −0.327322
$$59$$ −109.478 −0.241574 −0.120787 0.992678i $$-0.538542\pi$$
−0.120787 + 0.992678i $$0.538542\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ −823.664 −1.60872
$$65$$ 78.2922 0.149399
$$66$$ 0 0
$$67$$ 488.446 0.890644 0.445322 0.895371i $$-0.353089\pi$$
0.445322 + 0.895371i $$0.353089\pi$$
$$68$$ −956.581 −1.70592
$$69$$ 0 0
$$70$$ 474.151 0.809599
$$71$$ −837.423 −1.39977 −0.699887 0.714254i $$-0.746766\pi$$
−0.699887 + 0.714254i $$0.746766\pi$$
$$72$$ 0 0
$$73$$ 351.216 0.563105 0.281553 0.959546i $$-0.409151\pi$$
0.281553 + 0.959546i $$0.409151\pi$$
$$74$$ 300.512 0.472078
$$75$$ 0 0
$$76$$ 800.547 1.20828
$$77$$ −227.022 −0.335994
$$78$$ 0 0
$$79$$ −831.205 −1.18377 −0.591885 0.806022i $$-0.701616\pi$$
−0.591885 + 0.806022i $$0.701616\pi$$
$$80$$ −15.2066 −0.0212519
$$81$$ 0 0
$$82$$ −501.238 −0.675031
$$83$$ −1389.13 −1.83707 −0.918537 0.395335i $$-0.870629\pi$$
−0.918537 + 0.395335i $$0.870629\pi$$
$$84$$ 0 0
$$85$$ 364.754 0.465448
$$86$$ 559.423 0.701443
$$87$$ 0 0
$$88$$ 258.413 0.313034
$$89$$ −1523.70 −1.81474 −0.907369 0.420335i $$-0.861912\pi$$
−0.907369 + 0.420335i $$0.861912\pi$$
$$90$$ 0 0
$$91$$ −323.164 −0.372273
$$92$$ 179.125 0.202990
$$93$$ 0 0
$$94$$ −2387.17 −2.61933
$$95$$ −305.256 −0.329670
$$96$$ 0 0
$$97$$ −426.612 −0.446555 −0.223278 0.974755i $$-0.571676\pi$$
−0.223278 + 0.974755i $$0.571676\pi$$
$$98$$ −381.103 −0.392829
$$99$$ 0 0
$$100$$ 327.818 0.327818
$$101$$ −74.1387 −0.0730403 −0.0365202 0.999333i $$-0.511627\pi$$
−0.0365202 + 0.999333i $$0.511627\pi$$
$$102$$ 0 0
$$103$$ −69.3916 −0.0663821 −0.0331911 0.999449i $$-0.510567\pi$$
−0.0331911 + 0.999449i $$0.510567\pi$$
$$104$$ 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 $$0.672491\pi$$
$$108$$ 0 0
$$109$$ −2226.85 −1.95682 −0.978409 0.206680i $$-0.933734\pi$$
−0.978409 + 0.206680i $$0.933734\pi$$
$$110$$ −252.717 −0.219051
$$111$$ 0 0
$$112$$ 62.7678 0.0529554
$$113$$ 1719.76 1.43169 0.715847 0.698257i $$-0.246041\pi$$
0.715847 + 0.698257i $$0.246041\pi$$
$$114$$ 0 0
$$115$$ −68.3023 −0.0553845
$$116$$ 412.608 0.330256
$$117$$ 0 0
$$118$$ 503.038 0.392444
$$119$$ −1505.58 −1.15980
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 411.771 0.305574
$$123$$ 0 0
$$124$$ −3186.49 −2.30771
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1601.63 −1.11907 −0.559534 0.828807i $$-0.689020\pi$$
−0.559534 + 0.828807i $$0.689020\pi$$
$$128$$ 2392.91 1.65239
$$129$$ 0 0
$$130$$ −359.741 −0.242703
$$131$$ −2004.13 −1.33665 −0.668327 0.743868i $$-0.732989\pi$$
−0.668327 + 0.743868i $$0.732989\pi$$
$$132$$ 0 0
$$133$$ 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 $$0.325331\pi$$
$$138$$ 0 0
$$139$$ 2540.38 1.55016 0.775080 0.631863i $$-0.217709\pi$$
0.775080 + 0.631863i $$0.217709\pi$$
$$140$$ −1353.12 −0.816855
$$141$$ 0 0
$$142$$ 3847.84 2.27397
$$143$$ 172.243 0.100725
$$144$$ 0 0
$$145$$ −157.331 −0.0901079
$$146$$ −1613.79 −0.914780
$$147$$ 0 0
$$148$$ −857.594 −0.476310
$$149$$ −3090.68 −1.69932 −0.849658 0.527334i $$-0.823192\pi$$
−0.849658 + 0.527334i $$0.823192\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 1043.13 0.545831
$$155$$ 1215.04 0.629642
$$156$$ 0 0
$$157$$ −1011.95 −0.514411 −0.257205 0.966357i $$-0.582802\pi$$
−0.257205 + 0.966357i $$0.582802\pi$$
$$158$$ 3819.27 1.92307
$$159$$ 0 0
$$160$$ −869.813 −0.429780
$$161$$ 281.929 0.138007
$$162$$ 0 0
$$163$$ −2816.37 −1.35334 −0.676672 0.736285i $$-0.736578\pi$$
−0.676672 + 0.736285i $$0.736578\pi$$
$$164$$ 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 $$0.794665\pi$$
$$168$$ 0 0
$$169$$ −1951.81 −0.888399
$$170$$ −1675.99 −0.756133
$$171$$ 0 0
$$172$$ −1596.47 −0.707730
$$173$$ 2287.85 1.00545 0.502723 0.864448i $$-0.332332\pi$$
0.502723 + 0.864448i $$0.332332\pi$$
$$174$$ 0 0
$$175$$ 515.959 0.222873
$$176$$ −33.4545 −0.0143280
$$177$$ 0 0
$$178$$ 7001.17 2.94809
$$179$$ −3249.06 −1.35668 −0.678340 0.734748i $$-0.737300\pi$$
−0.678340 + 0.734748i $$0.737300\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ −320.913 −0.128576
$$185$$ 327.009 0.129958
$$186$$ 0 0
$$187$$ 802.458 0.313805
$$188$$ 6812.44 2.64281
$$189$$ 0 0
$$190$$ 1402.61 0.535558
$$191$$ 2760.35 1.04572 0.522859 0.852419i $$-0.324866\pi$$
0.522859 + 0.852419i $$0.324866\pi$$
$$192$$ 0 0
$$193$$ −1250.61 −0.466430 −0.233215 0.972425i $$-0.574925\pi$$
−0.233215 + 0.972425i $$0.574925\pi$$
$$194$$ 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 $$0.491711\pi$$
$$198$$ 0 0
$$199$$ 761.249 0.271174 0.135587 0.990765i $$-0.456708\pi$$
0.135587 + 0.990765i $$0.456708\pi$$
$$200$$ −587.303 −0.207643
$$201$$ 0 0
$$202$$ 340.657 0.118656
$$203$$ 649.411 0.224531
$$204$$ 0 0
$$205$$ −545.434 −0.185828
$$206$$ 318.844 0.107840
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ 5245.97 1.67573
$$215$$ 608.749 0.193099
$$216$$ 0 0
$$217$$ −5015.29 −1.56894
$$218$$ 10232.0 3.17890
$$219$$ 0 0
$$220$$ 721.199 0.221015
$$221$$ 1142.29 0.347688
$$222$$ 0 0
$$223$$ 908.084 0.272690 0.136345 0.990661i $$-0.456464\pi$$
0.136345 + 0.990661i $$0.456464\pi$$
$$224$$ 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 $$0.402521\pi$$
$$228$$ 0 0
$$229$$ 4077.47 1.17662 0.588312 0.808634i $$-0.299793\pi$$
0.588312 + 0.808634i $$0.299793\pi$$
$$230$$ 313.839 0.0899737
$$231$$ 0 0
$$232$$ −739.209 −0.209187
$$233$$ −1682.76 −0.473138 −0.236569 0.971615i $$-0.576023\pi$$
−0.236569 + 0.971615i $$0.576023\pi$$
$$234$$ 0 0
$$235$$ −2597.65 −0.721073
$$236$$ −1435.56 −0.395961
$$237$$ 0 0
$$238$$ 6917.93 1.88413
$$239$$ −4024.96 −1.08934 −0.544672 0.838649i $$-0.683346\pi$$
−0.544672 + 0.838649i $$0.683346\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ −1175.10 −0.308313
$$245$$ −414.707 −0.108141
$$246$$ 0 0
$$247$$ −955.968 −0.246262
$$248$$ 5708.78 1.46173
$$249$$ 0 0
$$250$$ 574.357 0.145302
$$251$$ 1827.60 0.459591 0.229796 0.973239i $$-0.426194\pi$$
0.229796 + 0.973239i $$0.426194\pi$$
$$252$$ 0 0
$$253$$ −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 $$0.522624\pi$$
$$258$$ 0 0
$$259$$ −1349.78 −0.323828
$$260$$ 1026.62 0.244878
$$261$$ 0 0
$$262$$ 9208.69 2.17143
$$263$$ 238.098 0.0558241 0.0279120 0.999610i $$-0.491114\pi$$
0.0279120 + 0.999610i $$0.491114\pi$$
$$264$$ 0 0
$$265$$ −2713.37 −0.628984
$$266$$ −5789.51 −1.33450
$$267$$ 0 0
$$268$$ 6404.84 1.45984
$$269$$ −4618.46 −1.04681 −0.523406 0.852083i $$-0.675339\pi$$
−0.523406 + 0.852083i $$0.675339\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ −7686.51 −1.69474
$$275$$ −275.000 −0.0603023
$$276$$ 0 0
$$277$$ −8602.51 −1.86597 −0.932987 0.359911i $$-0.882807\pi$$
−0.932987 + 0.359911i $$0.882807\pi$$
$$278$$ −11672.7 −2.51828
$$279$$ 0 0
$$280$$ 2424.19 0.517404
$$281$$ 2992.81 0.635360 0.317680 0.948198i $$-0.397096\pi$$
0.317680 + 0.948198i $$0.397096\pi$$
$$282$$ 0 0
$$283$$ −6858.89 −1.44070 −0.720351 0.693610i $$-0.756019\pi$$
−0.720351 + 0.693610i $$0.756019\pi$$
$$284$$ −10980.9 −2.29435
$$285$$ 0 0
$$286$$ −791.431 −0.163630
$$287$$ 2251.37 0.463046
$$288$$ 0 0
$$289$$ 408.809 0.0832096
$$290$$ 722.915 0.146383
$$291$$ 0 0
$$292$$ 4605.39 0.922979
$$293$$ −4049.70 −0.807461 −0.403731 0.914878i $$-0.632287\pi$$
−0.403731 + 0.914878i $$0.632287\pi$$
$$294$$ 0 0
$$295$$ 547.392 0.108035
$$296$$ 1536.43 0.301699
$$297$$ 0 0
$$298$$ 14201.2 2.76059
$$299$$ −213.901 −0.0413720
$$300$$ 0 0
$$301$$ −2512.71 −0.481164
$$302$$ −6243.20 −1.18959
$$303$$ 0 0
$$304$$ 185.677 0.0350305
$$305$$ 448.078 0.0841209
$$306$$ 0 0
$$307$$ −9572.69 −1.77962 −0.889808 0.456335i $$-0.849162\pi$$
−0.889808 + 0.456335i $$0.849162\pi$$
$$308$$ −2976.87 −0.550724
$$309$$ 0 0
$$310$$ −5582.94 −1.02287
$$311$$ −5396.42 −0.983932 −0.491966 0.870614i $$-0.663722\pi$$
−0.491966 + 0.870614i $$0.663722\pi$$
$$312$$ 0 0
$$313$$ 9755.04 1.76162 0.880811 0.473469i $$-0.156998\pi$$
0.880811 + 0.473469i $$0.156998\pi$$
$$314$$ 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 $$0.373962\pi$$
$$318$$ 0 0
$$319$$ −346.129 −0.0607508
$$320$$ 4118.32 0.719440
$$321$$ 0 0
$$322$$ −1295.42 −0.224196
$$323$$ −4453.74 −0.767221
$$324$$ 0 0
$$325$$ −391.461 −0.0668134
$$326$$ 12940.8 2.19854
$$327$$ 0 0
$$328$$ −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
$$333$$ 0 0
$$334$$ 15847.2 2.59616
$$335$$ −2442.23 −0.398308
$$336$$ 0 0
$$337$$ −4500.27 −0.727434 −0.363717 0.931509i $$-0.618492\pi$$
−0.363717 + 0.931509i $$0.618492\pi$$
$$338$$ 8968.30 1.44323
$$339$$ 0 0
$$340$$ 4782.91 0.762910
$$341$$ 2673.09 0.424504
$$342$$ 0 0
$$343$$ −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 $$0.651030\pi$$
$$348$$ 0 0
$$349$$ 3636.26 0.557721 0.278860 0.960332i $$-0.410043\pi$$
0.278860 + 0.960332i $$0.410043\pi$$
$$350$$ −2370.76 −0.362063
$$351$$ 0 0
$$352$$ −1913.59 −0.289757
$$353$$ −210.408 −0.0317248 −0.0158624 0.999874i $$-0.505049\pi$$
−0.0158624 + 0.999874i $$0.505049\pi$$
$$354$$ 0 0
$$355$$ 4187.12 0.625998
$$356$$ −19979.8 −2.97451
$$357$$ 0 0
$$358$$ 14928.9 2.20396
$$359$$ −2499.68 −0.367488 −0.183744 0.982974i $$-0.558822\pi$$
−0.183744 + 0.982974i $$0.558822\pi$$
$$360$$ 0 0
$$361$$ −3131.74 −0.456588
$$362$$ −5378.03 −0.780837
$$363$$ 0 0
$$364$$ −4237.56 −0.610188
$$365$$ −1756.08 −0.251828
$$366$$ 0 0
$$367$$ 5748.70 0.817656 0.408828 0.912612i $$-0.365938\pi$$
0.408828 + 0.912612i $$0.365938\pi$$
$$368$$ 41.5458 0.00588512
$$369$$ 0 0
$$370$$ −1502.56 −0.211120
$$371$$ 11199.9 1.56730
$$372$$ 0 0
$$373$$ 4467.78 0.620196 0.310098 0.950705i $$-0.399638\pi$$
0.310098 + 0.950705i $$0.399638\pi$$
$$374$$ −3687.18 −0.509785
$$375$$ 0 0
$$376$$ −12204.9 −1.67398
$$377$$ −492.712 −0.0673103
$$378$$ 0 0
$$379$$ 7804.08 1.05770 0.528851 0.848715i $$-0.322623\pi$$
0.528851 + 0.848715i $$0.322623\pi$$
$$380$$ −4002.74 −0.540358
$$381$$ 0 0
$$382$$ −12683.4 −1.69880
$$383$$ 11161.1 1.48904 0.744522 0.667597i $$-0.232677\pi$$
0.744522 + 0.667597i $$0.232677\pi$$
$$384$$ 0 0
$$385$$ 1135.11 0.150261
$$386$$ 5746.38 0.757728
$$387$$ 0 0
$$388$$ −5594.04 −0.731944
$$389$$ −8490.24 −1.10661 −0.553306 0.832978i $$-0.686634\pi$$
−0.553306 + 0.832978i $$0.686634\pi$$
$$390$$ 0 0
$$391$$ −996.540 −0.128893
$$392$$ −1948.47 −0.251052
$$393$$ 0 0
$$394$$ −661.616 −0.0845983
$$395$$ 4156.03 0.529398
$$396$$ 0 0
$$397$$ 6019.74 0.761013 0.380507 0.924778i $$-0.375750\pi$$
0.380507 + 0.924778i $$0.375750\pi$$
$$398$$ −3497.83 −0.440529
$$399$$ 0 0
$$400$$ 76.0330 0.00950413
$$401$$ 10398.8 1.29499 0.647495 0.762069i $$-0.275816\pi$$
0.647495 + 0.762069i $$0.275816\pi$$
$$402$$ 0 0
$$403$$ 3805.13 0.470340
$$404$$ −972.158 −0.119720
$$405$$ 0 0
$$406$$ −2983.95 −0.364756
$$407$$ 719.420 0.0876175
$$408$$ 0 0
$$409$$ −4733.68 −0.572287 −0.286144 0.958187i $$-0.592373\pi$$
−0.286144 + 0.958187i $$0.592373\pi$$
$$410$$ 2506.19 0.301883
$$411$$ 0 0
$$412$$ −909.911 −0.108806
$$413$$ −2259.45 −0.269202
$$414$$ 0 0
$$415$$ 6945.67 0.821565
$$416$$ −2723.98 −0.321044
$$417$$ 0 0
$$418$$ 3085.74 0.361073
$$419$$ 8117.57 0.946466 0.473233 0.880937i $$-0.343087\pi$$
0.473233 + 0.880937i $$0.343087\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ −12748.5 −1.46020
$$425$$ −1823.77 −0.208155
$$426$$ 0 0
$$427$$ −1849.52 −0.209612
$$428$$ −14970.8 −1.69075
$$429$$ 0 0
$$430$$ −2797.11 −0.313695
$$431$$ −9335.16 −1.04329 −0.521646 0.853162i $$-0.674682\pi$$
−0.521646 + 0.853162i $$0.674682\pi$$
$$432$$ 0 0
$$433$$ −2983.02 −0.331074 −0.165537 0.986204i $$-0.552936\pi$$
−0.165537 + 0.986204i $$0.552936\pi$$
$$434$$ 23044.5 2.54878
$$435$$ 0 0
$$436$$ −29200.0 −3.20739
$$437$$ 833.988 0.0912931
$$438$$ 0 0
$$439$$ −5232.32 −0.568850 −0.284425 0.958698i $$-0.591803\pi$$
−0.284425 + 0.958698i $$0.591803\pi$$
$$440$$ −1292.07 −0.139993
$$441$$ 0 0
$$442$$ −5248.68 −0.564828
$$443$$ −7517.71 −0.806269 −0.403135 0.915141i $$-0.632079\pi$$
−0.403135 + 0.915141i $$0.632079\pi$$
$$444$$ 0 0
$$445$$ 7618.49 0.811575
$$446$$ −4172.52 −0.442992
$$447$$ 0 0
$$448$$ −16999.1 −1.79270
$$449$$ −16070.9 −1.68916 −0.844581 0.535428i $$-0.820150\pi$$
−0.844581 + 0.535428i $$0.820150\pi$$
$$450$$ 0 0
$$451$$ −1199.96 −0.125285
$$452$$ 22550.7 2.34667
$$453$$ 0 0
$$454$$ −9475.29 −0.979510
$$455$$ 1615.82 0.166485
$$456$$ 0 0
$$457$$ 9718.51 0.994776 0.497388 0.867528i $$-0.334293\pi$$
0.497388 + 0.867528i $$0.334293\pi$$
$$458$$ −18735.4 −1.91146
$$459$$ 0 0
$$460$$ −895.627 −0.0907801
$$461$$ 14538.0 1.46877 0.734385 0.678733i $$-0.237471\pi$$
0.734385 + 0.678733i $$0.237471\pi$$
$$462$$ 0 0
$$463$$ −9978.17 −1.00157 −0.500783 0.865573i $$-0.666955\pi$$
−0.500783 + 0.865573i $$0.666955\pi$$
$$464$$ 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 $$0.771147\pi$$
$$468$$ 0 0
$$469$$ 10080.7 0.992503
$$470$$ 11935.8 1.17140
$$471$$ 0 0
$$472$$ 2571.88 0.250806
$$473$$ 1339.25 0.130187
$$474$$ 0 0
$$475$$ 1526.28 0.147433
$$476$$ −19742.3 −1.90102
$$477$$ 0 0
$$478$$ 18494.1 1.76967
$$479$$ −11330.8 −1.08083 −0.540415 0.841399i $$-0.681733\pi$$
−0.540415 + 0.841399i $$0.681733\pi$$
$$480$$ 0 0
$$481$$ 1024.09 0.0970779
$$482$$ 12793.3 1.20896
$$483$$ 0 0
$$484$$ 1586.64 0.149008
$$485$$ 2133.06 0.199706
$$486$$ 0 0
$$487$$ 19086.9 1.77599 0.887997 0.459850i $$-0.152097\pi$$
0.887997 + 0.459850i $$0.152097\pi$$
$$488$$ 2105.26 0.195288
$$489$$ 0 0
$$490$$ 1905.52 0.175679
$$491$$ 8112.85 0.745677 0.372839 0.927896i $$-0.378384\pi$$
0.372839 + 0.927896i $$0.378384\pi$$
$$492$$ 0 0
$$493$$ −2295.49 −0.209703
$$494$$ 4392.53 0.400060
$$495$$ 0 0
$$496$$ −739.066 −0.0669053
$$497$$ −17283.0 −1.55986
$$498$$ 0 0
$$499$$ 18329.1 1.64433 0.822167 0.569246i $$-0.192765\pi$$
0.822167 + 0.569246i $$0.192765\pi$$
$$500$$ −1639.09 −0.146604
$$501$$ 0 0
$$502$$ −8397.58 −0.746618
$$503$$ −7739.57 −0.686064 −0.343032 0.939324i $$-0.611454\pi$$
−0.343032 + 0.939324i $$0.611454\pi$$
$$504$$ 0 0
$$505$$ 370.693 0.0326646
$$506$$ 690.446 0.0606602
$$507$$ 0 0
$$508$$ −21001.7 −1.83425
$$509$$ −15914.9 −1.38589 −0.692943 0.720993i $$-0.743686\pi$$
−0.692943 + 0.720993i $$0.743686\pi$$
$$510$$ 0 0
$$511$$ 7248.51 0.627505
$$512$$ 1100.65 0.0950048
$$513$$ 0 0
$$514$$ 2688.78 0.230733
$$515$$ 346.958 0.0296870
$$516$$ 0 0
$$517$$ −5714.83 −0.486147
$$518$$ 6202.07 0.526068
$$519$$ 0 0
$$520$$ −1839.25 −0.155109
$$521$$ −2274.50 −0.191262 −0.0956312 0.995417i $$-0.530487\pi$$
−0.0956312 + 0.995417i $$0.530487\pi$$
$$522$$ 0 0
$$523$$ 10971.1 0.917274 0.458637 0.888624i $$-0.348338\pi$$
0.458637 + 0.888624i $$0.348338\pi$$
$$524$$ −26279.6 −2.19089
$$525$$ 0 0
$$526$$ −1094.02 −0.0906877
$$527$$ 17727.6 1.46533
$$528$$ 0 0
$$529$$ −11980.4 −0.984663
$$530$$ 12467.5 1.02180
$$531$$ 0 0
$$532$$ 16522.0 1.34646
$$533$$ −1708.13 −0.138813
$$534$$ 0 0
$$535$$ 5708.53 0.461310
$$536$$ −11474.6 −0.924680
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ −12690.7 −1.00020
$$545$$ 11134.2 0.875115
$$546$$ 0 0
$$547$$ 20685.1 1.61688 0.808439 0.588581i $$-0.200313\pi$$
0.808439 + 0.588581i $$0.200313\pi$$
$$548$$ 21935.6 1.70993
$$549$$ 0 0
$$550$$ 1263.59 0.0979627
$$551$$ 1921.06 0.148529
$$552$$ 0 0
$$553$$ −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 $$0.364540\pi$$
$$558$$ 0 0
$$559$$ 1906.41 0.144244
$$560$$ −313.839 −0.0236824
$$561$$ 0 0
$$562$$ −13751.5 −1.03216
$$563$$ 15381.2 1.15141 0.575704 0.817658i $$-0.304728\pi$$
0.575704 + 0.817658i $$0.304728\pi$$
$$564$$ 0 0
$$565$$ −8598.80 −0.640273
$$566$$ 31515.6 2.34046
$$567$$ 0 0
$$568$$ 19672.9 1.45327
$$569$$ 1348.88 0.0993814 0.0496907 0.998765i $$-0.484176\pi$$
0.0496907 + 0.998765i $$0.484176\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −10344.7 −0.752231
$$575$$ 341.511 0.0247687
$$576$$ 0 0
$$577$$ 12052.6 0.869598 0.434799 0.900528i $$-0.356819\pi$$
0.434799 + 0.900528i $$0.356819\pi$$
$$578$$ −1878.42 −0.135176
$$579$$ 0 0
$$580$$ −2063.04 −0.147695
$$581$$ −28669.4 −2.04717
$$582$$ 0 0
$$583$$ −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 $$0.371989\pi$$
$$588$$ 0 0
$$589$$ −14836.0 −1.03787
$$590$$ −2515.19 −0.175506
$$591$$ 0 0
$$592$$ −198.908 −0.0138092
$$593$$ 7939.69 0.549821 0.274911 0.961470i $$-0.411352\pi$$
0.274911 + 0.961470i $$0.411352\pi$$
$$594$$ 0 0
$$595$$ 7527.91 0.518680
$$596$$ −40527.1 −2.78533
$$597$$ 0 0
$$598$$ 982.846 0.0672100
$$599$$ 19474.7 1.32840 0.664202 0.747553i $$-0.268771\pi$$
0.664202 + 0.747553i $$0.268771\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 17816.7 1.20025
$$605$$ −605.000 −0.0406558
$$606$$ 0 0
$$607$$ 1427.44 0.0954496 0.0477248 0.998861i $$-0.484803\pi$$
0.0477248 + 0.998861i $$0.484803\pi$$
$$608$$ 10620.6 0.708427
$$609$$ 0 0
$$610$$ −2058.85 −0.136657
$$611$$ −8135.03 −0.538638
$$612$$ 0 0
$$613$$ −8029.40 −0.529045 −0.264522 0.964380i $$-0.585214\pi$$
−0.264522 + 0.964380i $$0.585214\pi$$
$$614$$ 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 $$0.737344\pi$$
$$618$$ 0 0
$$619$$ 1677.43 0.108920 0.0544602 0.998516i $$-0.482656\pi$$
0.0544602 + 0.998516i $$0.482656\pi$$
$$620$$ 15932.5 1.03204
$$621$$ 0 0
$$622$$ 24795.8 1.59842
$$623$$ −31446.6 −2.02228
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −44823.0 −2.86180
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ −20004.8 −1.25315
$$635$$ 8008.15 0.500462
$$636$$ 0 0
$$637$$ −1298.73 −0.0807812
$$638$$ 1590.41 0.0986912
$$639$$ 0 0
$$640$$ −11964.6 −0.738971
$$641$$ −15165.3 −0.934468 −0.467234 0.884134i $$-0.654749\pi$$
−0.467234 + 0.884134i $$0.654749\pi$$
$$642$$ 0 0
$$643$$ 27156.1 1.66553 0.832763 0.553630i $$-0.186758\pi$$
0.832763 + 0.553630i $$0.186758\pi$$
$$644$$ 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 $$0.846374\pi$$
$$648$$ 0 0
$$649$$ 1204.26 0.0728374
$$650$$ 1798.71 0.108540
$$651$$ 0 0
$$652$$ −36930.2 −2.21825
$$653$$ 19141.7 1.14713 0.573564 0.819161i $$-0.305560\pi$$
0.573564 + 0.819161i $$0.305560\pi$$
$$654$$ 0 0
$$655$$ 10020.7 0.597770
$$656$$ 331.768 0.0197460
$$657$$ 0 0
$$658$$ −49267.2 −2.91890
$$659$$ 24939.6 1.47422 0.737110 0.675773i $$-0.236190\pi$$
0.737110 + 0.675773i $$0.236190\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ 32633.7 1.90728
$$665$$ −6299.99 −0.367373
$$666$$ 0 0
$$667$$ 429.843 0.0249529
$$668$$ −45224.3 −2.61943
$$669$$ 0 0
$$670$$ 11221.7 0.647062
$$671$$ 985.772 0.0567143
$$672$$ 0 0
$$673$$ −13855.8 −0.793615 −0.396807 0.917902i $$-0.629882\pi$$
−0.396807 + 0.917902i $$0.629882\pi$$
$$674$$ 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 $$0.248957\pi$$
$$678$$ 0 0
$$679$$ −8804.57 −0.497626
$$680$$ −8568.84 −0.483235
$$681$$ 0 0
$$682$$ −12282.5 −0.689619
$$683$$ 14420.5 0.807887 0.403943 0.914784i $$-0.367639\pi$$
0.403943 + 0.914784i $$0.367639\pi$$
$$684$$ 0 0
$$685$$ −8364.25 −0.466543
$$686$$ 24661.4 1.37256
$$687$$ 0 0
$$688$$ −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
$$693$$ 0 0
$$694$$ 27138.7 1.48440
$$695$$ −12701.9 −0.693253
$$696$$ 0 0
$$697$$ −7957.96 −0.432467
$$698$$ −16708.1 −0.906032
$$699$$ 0 0
$$700$$ 6765.61 0.365309
$$701$$ 9151.47 0.493076 0.246538 0.969133i $$-0.420707\pi$$
0.246538 + 0.969133i $$0.420707\pi$$
$$702$$ 0 0
$$703$$ −3992.86 −0.214216
$$704$$ 9060.30 0.485047
$$705$$ 0 0
$$706$$ 966.793 0.0515378
$$707$$ −1530.10 −0.0813937
$$708$$ 0 0
$$709$$ −6261.96 −0.331697 −0.165848 0.986151i $$-0.553036\pi$$
−0.165848 + 0.986151i $$0.553036\pi$$
$$710$$ −19239.2 −1.01695
$$711$$ 0 0
$$712$$ 35794.9 1.88409
$$713$$ −3319.60 −0.174362
$$714$$ 0 0
$$715$$ −861.214 −0.0450456
$$716$$ −42603.9 −2.22372
$$717$$ 0 0
$$718$$ 11485.7 0.596995
$$719$$ 18228.7 0.945500 0.472750 0.881197i $$-0.343261\pi$$
0.472750 + 0.881197i $$0.343261\pi$$
$$720$$ 0 0
$$721$$ −1432.13 −0.0739740
$$722$$ 14389.9 0.741740
$$723$$ 0 0
$$724$$ 15347.7 0.787836
$$725$$ 786.656 0.0402975
$$726$$ 0 0
$$727$$ −7233.66 −0.369026 −0.184513 0.982830i $$-0.559071\pi$$
−0.184513 + 0.982830i $$0.559071\pi$$
$$728$$ 7591.81 0.386499
$$729$$ 0 0
$$730$$ 8068.93 0.409102
$$731$$ 8881.73 0.449388
$$732$$ 0 0
$$733$$ −13444.8 −0.677485 −0.338743 0.940879i $$-0.610002\pi$$
−0.338743 + 0.940879i $$0.610002\pi$$
$$734$$ −26414.4 −1.32830
$$735$$ 0 0
$$736$$ 2376.41 0.119016
$$737$$ −5372.90 −0.268539
$$738$$ 0 0
$$739$$ 18490.9 0.920432 0.460216 0.887807i $$-0.347772\pi$$
0.460216 + 0.887807i $$0.347772\pi$$
$$740$$ 4287.97 0.213012
$$741$$ 0 0
$$742$$ −51461.8 −2.54612
$$743$$ 25160.9 1.24235 0.621173 0.783674i $$-0.286657\pi$$
0.621173 + 0.783674i $$0.286657\pi$$
$$744$$ 0 0
$$745$$ 15453.4 0.759957
$$746$$ −20528.8 −1.00752
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 2263.94 0.109347
$$755$$ −6793.68 −0.327480
$$756$$ 0 0
$$757$$ −7014.90 −0.336804 −0.168402 0.985718i $$-0.553861\pi$$
−0.168402 + 0.985718i $$0.553861\pi$$
$$758$$ −35858.6 −1.71826
$$759$$ 0 0
$$760$$ 7171.12 0.342268
$$761$$ 30156.9 1.43651 0.718256 0.695779i $$-0.244941\pi$$
0.718256 + 0.695779i $$0.244941\pi$$
$$762$$ 0 0
$$763$$ −45958.4 −2.18061
$$764$$ 36195.7 1.71402
$$765$$ 0 0
$$766$$ −51283.5 −2.41899
$$767$$ 1714.26 0.0807019
$$768$$ 0 0
$$769$$ 11292.2 0.529530 0.264765 0.964313i $$-0.414706\pi$$
0.264765 + 0.964313i $$0.414706\pi$$
$$770$$ −5215.66 −0.244103
$$771$$ 0 0
$$772$$ −16398.9 −0.764519
$$773$$ −8524.10 −0.396624 −0.198312 0.980139i $$-0.563546\pi$$
−0.198312 + 0.980139i $$0.563546\pi$$
$$774$$ 0 0
$$775$$ −6075.21 −0.281584
$$776$$ 10022.0 0.463621
$$777$$ 0 0
$$778$$ 39011.4 1.79772
$$779$$ 6659.89 0.306310
$$780$$ 0 0
$$781$$ 9211.66 0.422047
$$782$$ 4578.96 0.209390
$$783$$ 0 0
$$784$$ 252.251 0.0114910
$$785$$ 5059.76 0.230052
$$786$$ 0 0
$$787$$ −14983.9 −0.678676 −0.339338 0.940665i $$-0.610203\pi$$
−0.339338 + 0.940665i $$0.610203\pi$$
$$788$$ 1888.10 0.0853565
$$789$$ 0 0
$$790$$ −19096.3 −0.860022
$$791$$ 35493.0 1.59543
$$792$$ 0 0
$$793$$ 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 $$0.809412\pi$$
$$798$$ 0 0
$$799$$ −37900.1 −1.67811
$$800$$ 4349.06 0.192203
$$801$$ 0 0
$$802$$ −47781.0 −2.10375
$$803$$ −3863.37 −0.169783
$$804$$ 0 0
$$805$$ −1409.65 −0.0617186
$$806$$ −17484.0 −0.764080
$$807$$ 0 0
$$808$$ 1741.68 0.0758316
$$809$$ −23797.1 −1.03419 −0.517096 0.855928i $$-0.672987\pi$$
−0.517096 + 0.855928i $$0.672987\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ −3305.63 −0.142337
$$815$$ 14081.8 0.605234
$$816$$ 0 0
$$817$$ −7432.98 −0.318295
$$818$$ 21750.6 0.929696
$$819$$ 0 0
$$820$$ −7152.11 −0.304589
$$821$$ 25156.8 1.06940 0.534702 0.845041i $$-0.320424\pi$$
0.534702 + 0.845041i $$0.320424\pi$$
$$822$$ 0 0
$$823$$ 1318.51 0.0558447 0.0279224 0.999610i $$-0.491111\pi$$
0.0279224 + 0.999610i $$0.491111\pi$$
$$824$$ 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 $$0.500836\pi$$
$$828$$ 0 0
$$829$$ −8886.80 −0.372318 −0.186159 0.982520i $$-0.559604\pi$$
−0.186159 + 0.982520i $$0.559604\pi$$
$$830$$ −31914.3 −1.33465
$$831$$ 0 0
$$832$$ 12897.3 0.537419
$$833$$ −6050.63 −0.251671
$$834$$ 0 0
$$835$$ 17244.5 0.714694
$$836$$ −8806.02 −0.364309
$$837$$ 0 0
$$838$$ −37299.1 −1.53756
$$839$$ −2995.21 −0.123249 −0.0616247 0.998099i $$-0.519628\pi$$
−0.0616247 + 0.998099i $$0.519628\pi$$
$$840$$ 0 0
$$841$$ −23398.9 −0.959403
$$842$$ 43578.8 1.78364
$$843$$ 0 0
$$844$$ 52143.6 2.12661
$$845$$ 9759.07 0.397304
$$846$$ 0 0
$$847$$ 2497.24 0.101306
$$848$$ 1650.44 0.0668355
$$849$$ 0 0
$$850$$ 8379.95 0.338153
$$851$$ −893.418 −0.0359882
$$852$$ 0 0
$$853$$ 18130.5 0.727757 0.363878 0.931446i $$-0.381452\pi$$
0.363878 + 0.931446i $$0.381452\pi$$
$$854$$ 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 $$0.676318\pi$$
$$858$$ 0 0
$$859$$ −29456.2 −1.17000 −0.585002 0.811032i $$-0.698906\pi$$
−0.585002 + 0.811032i $$0.698906\pi$$
$$860$$ 7982.34 0.316506
$$861$$ 0 0
$$862$$ 42893.7 1.69486
$$863$$ 762.616 0.0300808 0.0150404 0.999887i $$-0.495212\pi$$
0.0150404 + 0.999887i $$0.495212\pi$$
$$864$$ 0 0
$$865$$ −11439.3 −0.449649
$$866$$ 13706.6 0.537838
$$867$$ 0 0
$$868$$ −65764.0 −2.57163
$$869$$ 9143.26 0.356920
$$870$$ 0 0
$$871$$ −7648.29 −0.297535
$$872$$ 52313.3 2.03160
$$873$$ 0 0
$$874$$ −3832.06 −0.148308
$$875$$ −2579.79 −0.0996719
$$876$$ 0 0
$$877$$ 44767.2 1.72369 0.861847 0.507168i $$-0.169308\pi$$
0.861847 + 0.507168i $$0.169308\pi$$
$$878$$ 24041.8 0.924112
$$879$$ 0 0
$$880$$ 167.273 0.00640768
$$881$$ 32057.9 1.22595 0.612973 0.790104i $$-0.289973\pi$$
0.612973 + 0.790104i $$0.289973\pi$$
$$882$$ 0 0
$$883$$ 7078.95 0.269791 0.134896 0.990860i $$-0.456930\pi$$
0.134896 + 0.990860i $$0.456930\pi$$
$$884$$ 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 $$0.342093\pi$$
$$888$$ 0 0
$$889$$ −33055.0 −1.24705
$$890$$ −35005.9 −1.31843
$$891$$ 0 0
$$892$$ 11907.4 0.446963
$$893$$ 31718.0 1.18858
$$894$$ 0 0
$$895$$ 16245.3 0.606726
$$896$$ 49385.8 1.84137
$$897$$ 0 0
$$898$$ 73843.6 2.74409
$$899$$ −7646.56 −0.283679
$$900$$ 0 0
$$901$$ −39588.4 −1.46380
$$902$$ 5513.62 0.203529
$$903$$ 0 0
$$904$$ −40400.8 −1.48641
$$905$$ −5852.23 −0.214955
$$906$$ 0 0
$$907$$ 1269.76 0.0464848 0.0232424 0.999730i $$-0.492601\pi$$
0.0232424 + 0.999730i $$0.492601\pi$$
$$908$$ 27040.4 0.988289
$$909$$ 0 0
$$910$$ −7424.47 −0.270460
$$911$$ 33783.1 1.22863 0.614316 0.789060i $$-0.289432\pi$$
0.614316 + 0.789060i $$0.289432\pi$$
$$912$$ 0 0
$$913$$ 15280.5 0.553899
$$914$$ −44655.1 −1.61604
$$915$$ 0 0
$$916$$ 53466.6 1.92859
$$917$$ −41361.9 −1.48952
$$918$$ 0 0
$$919$$ 39262.5 1.40930 0.704652 0.709553i $$-0.251103\pi$$
0.704652 + 0.709553i $$0.251103\pi$$
$$920$$ 1604.57 0.0575010
$$921$$ 0 0
$$922$$ −66800.1 −2.38606
$$923$$ 13112.7 0.467618
$$924$$ 0 0
$$925$$ −1635.04 −0.0581189
$$926$$ 45848.3 1.62707
$$927$$ 0 0
$$928$$ 5473.95 0.193633
$$929$$ 21175.0 0.747825 0.373913 0.927464i $$-0.378016\pi$$
0.373913 + 0.927464i $$0.378016\pi$$
$$930$$ 0 0
$$931$$ 5063.68 0.178255
$$932$$ −22065.5 −0.775515
$$933$$ 0 0
$$934$$ 69787.5 2.44488
$$935$$ −4012.29 −0.140338
$$936$$ 0 0
$$937$$ −5135.11 −0.179036 −0.0895180 0.995985i $$-0.528533\pi$$
−0.0895180 + 0.995985i $$0.528533\pi$$
$$938$$ −46319.4 −1.61235
$$939$$ 0 0
$$940$$ −34062.2 −1.18190
$$941$$ 9702.77 0.336133 0.168067 0.985776i $$-0.446248\pi$$
0.168067 + 0.985776i $$0.446248\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.496179\pi$$
$$948$$ 0 0
$$949$$ −5499.49 −0.188115
$$950$$ −7013.05 −0.239509
$$951$$ 0 0
$$952$$ 35369.3 1.20412
$$953$$ −42039.3 −1.42895 −0.714473 0.699663i $$-0.753333\pi$$
−0.714473 + 0.699663i $$0.753333\pi$$
$$954$$ 0 0
$$955$$ −13801.8 −0.467659
$$956$$ −52778.1 −1.78553
$$957$$ 0 0
$$958$$ 52063.4 1.75584
$$959$$ 34524.9 1.16253
$$960$$ 0 0
$$961$$ 29262.0 0.982243
$$962$$ −4705.55 −0.157706
$$963$$ 0 0
$$964$$ −36509.3 −1.21980
$$965$$ 6253.05 0.208594
$$966$$ 0 0
$$967$$ −32794.8 −1.09060 −0.545299 0.838242i $$-0.683584\pi$$
−0.545299 + 0.838242i $$0.683584\pi$$
$$968$$ −2842.55 −0.0943832
$$969$$ 0 0
$$970$$ −9801.10 −0.324427
$$971$$ 3322.53 0.109810 0.0549048 0.998492i $$-0.482514\pi$$
0.0549048 + 0.998492i $$0.482514\pi$$
$$972$$ 0 0
$$973$$ 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 $$0.618370\pi$$
$$978$$ 0 0
$$979$$ 16760.7 0.547164
$$980$$ −5437.92 −0.177253
$$981$$ 0 0
$$982$$ −37277.4 −1.21137
$$983$$ −7383.09 −0.239556 −0.119778 0.992801i $$-0.538218\pi$$
−0.119778 + 0.992801i $$0.538218\pi$$
$$984$$ 0 0
$$985$$ −719.953 −0.0232889
$$986$$ 10547.4 0.340668
$$987$$ 0 0
$$988$$ −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
$$993$$ 0 0
$$994$$ 79413.1 2.53403
$$995$$ −3806.25 −0.121273
$$996$$ 0 0
$$997$$ 41196.8 1.30864 0.654320 0.756217i $$-0.272955\pi$$
0.654320 + 0.756217i $$0.272955\pi$$
$$998$$ −84219.5 −2.67127
$$999$$ 0 0
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 495.4.a.i.1.1 3
3.2 odd 2 165.4.a.g.1.3 3
5.4 even 2 2475.4.a.z.1.3 3
15.2 even 4 825.4.c.m.199.6 6
15.8 even 4 825.4.c.m.199.1 6
15.14 odd 2 825.4.a.p.1.1 3
33.32 even 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 3.2 odd 2
495.4.a.i.1.1 3 1.1 even 1 trivial
825.4.a.p.1.1 3 15.14 odd 2
825.4.c.m.199.1 6 15.8 even 4
825.4.c.m.199.6 6 15.2 even 4
1815.4.a.q.1.1 3 33.32 even 2
2475.4.a.z.1.3 3 5.4 even 2