# Properties

 Label 289.4.a.g.1.11 Level $289$ Weight $4$ Character 289.1 Self dual yes Analytic conductor $17.052$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(1,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.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: $$17.0515519917$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156$$ x^12 - 4*x^11 - 58*x^10 + 204*x^9 + 1191*x^8 - 3456*x^7 - 10364*x^6 + 21448*x^5 + 38476*x^4 - 32336*x^3 - 57024*x^2 - 15776*x + 1156 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.11 Root $$4.91828$$ of defining polynomial Character $$\chi$$ $$=$$ 289.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.15292 q^{2} -1.99138 q^{3} +1.94089 q^{4} +5.00761 q^{5} -6.27866 q^{6} -2.78516 q^{7} -19.1039 q^{8} -23.0344 q^{9} +O(q^{10})$$ $$q+3.15292 q^{2} -1.99138 q^{3} +1.94089 q^{4} +5.00761 q^{5} -6.27866 q^{6} -2.78516 q^{7} -19.1039 q^{8} -23.0344 q^{9} +15.7886 q^{10} +27.2453 q^{11} -3.86505 q^{12} -59.7352 q^{13} -8.78140 q^{14} -9.97206 q^{15} -75.7601 q^{16} -72.6256 q^{18} +33.2605 q^{19} +9.71922 q^{20} +5.54633 q^{21} +85.9023 q^{22} -210.884 q^{23} +38.0431 q^{24} -99.9239 q^{25} -188.340 q^{26} +99.6376 q^{27} -5.40570 q^{28} +20.0521 q^{29} -31.4411 q^{30} -133.659 q^{31} -86.0343 q^{32} -54.2559 q^{33} -13.9470 q^{35} -44.7072 q^{36} +152.772 q^{37} +104.868 q^{38} +118.956 q^{39} -95.6647 q^{40} +261.840 q^{41} +17.4871 q^{42} -316.820 q^{43} +52.8802 q^{44} -115.347 q^{45} -664.899 q^{46} +329.443 q^{47} +150.867 q^{48} -335.243 q^{49} -315.052 q^{50} -115.939 q^{52} -310.540 q^{53} +314.149 q^{54} +136.434 q^{55} +53.2074 q^{56} -66.2344 q^{57} +63.2227 q^{58} -54.7388 q^{59} -19.3547 q^{60} -818.751 q^{61} -421.417 q^{62} +64.1546 q^{63} +334.822 q^{64} -299.130 q^{65} -171.064 q^{66} +731.181 q^{67} +419.950 q^{69} -43.9738 q^{70} +629.179 q^{71} +440.046 q^{72} +496.481 q^{73} +481.678 q^{74} +198.987 q^{75} +64.5550 q^{76} -75.8828 q^{77} +375.057 q^{78} -90.0262 q^{79} -379.377 q^{80} +423.512 q^{81} +825.559 q^{82} +364.025 q^{83} +10.7648 q^{84} -998.907 q^{86} -39.9315 q^{87} -520.492 q^{88} -192.079 q^{89} -363.680 q^{90} +166.372 q^{91} -409.302 q^{92} +266.167 q^{93} +1038.71 q^{94} +166.556 q^{95} +171.327 q^{96} +1350.20 q^{97} -1056.99 q^{98} -627.580 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 $$12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100})$$ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.15292 1.11472 0.557362 0.830269i $$-0.311813\pi$$
0.557362 + 0.830269i $$0.311813\pi$$
$$3$$ −1.99138 −0.383242 −0.191621 0.981469i $$-0.561374\pi$$
−0.191621 + 0.981469i $$0.561374\pi$$
$$4$$ 1.94089 0.242611
$$5$$ 5.00761 0.447894 0.223947 0.974601i $$-0.428106\pi$$
0.223947 + 0.974601i $$0.428106\pi$$
$$6$$ −6.27866 −0.427209
$$7$$ −2.78516 −0.150385 −0.0751924 0.997169i $$-0.523957\pi$$
−0.0751924 + 0.997169i $$0.523957\pi$$
$$8$$ −19.1039 −0.844280
$$9$$ −23.0344 −0.853126
$$10$$ 15.7886 0.499279
$$11$$ 27.2453 0.746798 0.373399 0.927671i $$-0.378192\pi$$
0.373399 + 0.927671i $$0.378192\pi$$
$$12$$ −3.86505 −0.0929787
$$13$$ −59.7352 −1.27443 −0.637214 0.770687i $$-0.719913\pi$$
−0.637214 + 0.770687i $$0.719913\pi$$
$$14$$ −8.78140 −0.167638
$$15$$ −9.97206 −0.171652
$$16$$ −75.7601 −1.18375
$$17$$ 0 0
$$18$$ −72.6256 −0.951000
$$19$$ 33.2605 0.401604 0.200802 0.979632i $$-0.435645\pi$$
0.200802 + 0.979632i $$0.435645\pi$$
$$20$$ 9.71922 0.108664
$$21$$ 5.54633 0.0576337
$$22$$ 85.9023 0.832475
$$23$$ −210.884 −1.91184 −0.955919 0.293630i $$-0.905137\pi$$
−0.955919 + 0.293630i $$0.905137\pi$$
$$24$$ 38.0431 0.323563
$$25$$ −99.9239 −0.799391
$$26$$ −188.340 −1.42064
$$27$$ 99.6376 0.710195
$$28$$ −5.40570 −0.0364850
$$29$$ 20.0521 0.128400 0.0641998 0.997937i $$-0.479551\pi$$
0.0641998 + 0.997937i $$0.479551\pi$$
$$30$$ −31.4411 −0.191344
$$31$$ −133.659 −0.774385 −0.387192 0.921999i $$-0.626555\pi$$
−0.387192 + 0.921999i $$0.626555\pi$$
$$32$$ −86.0343 −0.475277
$$33$$ −54.2559 −0.286204
$$34$$ 0 0
$$35$$ −13.9470 −0.0673565
$$36$$ −44.7072 −0.206978
$$37$$ 152.772 0.678800 0.339400 0.940642i $$-0.389776\pi$$
0.339400 + 0.940642i $$0.389776\pi$$
$$38$$ 104.868 0.447678
$$39$$ 118.956 0.488414
$$40$$ −95.6647 −0.378148
$$41$$ 261.840 0.997377 0.498689 0.866781i $$-0.333815\pi$$
0.498689 + 0.866781i $$0.333815\pi$$
$$42$$ 17.4871 0.0642457
$$43$$ −316.820 −1.12359 −0.561797 0.827275i $$-0.689890\pi$$
−0.561797 + 0.827275i $$0.689890\pi$$
$$44$$ 52.8802 0.181182
$$45$$ −115.347 −0.382110
$$46$$ −664.899 −2.13117
$$47$$ 329.443 1.02243 0.511215 0.859453i $$-0.329196\pi$$
0.511215 + 0.859453i $$0.329196\pi$$
$$48$$ 150.867 0.453663
$$49$$ −335.243 −0.977384
$$50$$ −315.052 −0.891101
$$51$$ 0 0
$$52$$ −115.939 −0.309190
$$53$$ −310.540 −0.804829 −0.402415 0.915458i $$-0.631829\pi$$
−0.402415 + 0.915458i $$0.631829\pi$$
$$54$$ 314.149 0.791672
$$55$$ 136.434 0.334487
$$56$$ 53.2074 0.126967
$$57$$ −66.2344 −0.153911
$$58$$ 63.2227 0.143130
$$59$$ −54.7388 −0.120786 −0.0603931 0.998175i $$-0.519235\pi$$
−0.0603931 + 0.998175i $$0.519235\pi$$
$$60$$ −19.3547 −0.0416446
$$61$$ −818.751 −1.71853 −0.859265 0.511531i $$-0.829079\pi$$
−0.859265 + 0.511531i $$0.829079\pi$$
$$62$$ −421.417 −0.863226
$$63$$ 64.1546 0.128297
$$64$$ 334.822 0.653948
$$65$$ −299.130 −0.570808
$$66$$ −171.064 −0.319039
$$67$$ 731.181 1.33325 0.666627 0.745392i $$-0.267738\pi$$
0.666627 + 0.745392i $$0.267738\pi$$
$$68$$ 0 0
$$69$$ 419.950 0.732696
$$70$$ −43.9738 −0.0750839
$$71$$ 629.179 1.05169 0.525844 0.850581i $$-0.323750\pi$$
0.525844 + 0.850581i $$0.323750\pi$$
$$72$$ 440.046 0.720277
$$73$$ 496.481 0.796010 0.398005 0.917383i $$-0.369703\pi$$
0.398005 + 0.917383i $$0.369703\pi$$
$$74$$ 481.678 0.756675
$$75$$ 198.987 0.306360
$$76$$ 64.5550 0.0974337
$$77$$ −75.8828 −0.112307
$$78$$ 375.057 0.544447
$$79$$ −90.0262 −0.128212 −0.0641059 0.997943i $$-0.520420\pi$$
−0.0641059 + 0.997943i $$0.520420\pi$$
$$80$$ −379.377 −0.530195
$$81$$ 423.512 0.580950
$$82$$ 825.559 1.11180
$$83$$ 364.025 0.481409 0.240704 0.970599i $$-0.422622\pi$$
0.240704 + 0.970599i $$0.422622\pi$$
$$84$$ 10.7648 0.0139826
$$85$$ 0 0
$$86$$ −998.907 −1.25250
$$87$$ −39.9315 −0.0492081
$$88$$ −520.492 −0.630507
$$89$$ −192.079 −0.228767 −0.114384 0.993437i $$-0.536489\pi$$
−0.114384 + 0.993437i $$0.536489\pi$$
$$90$$ −363.680 −0.425948
$$91$$ 166.372 0.191654
$$92$$ −409.302 −0.463833
$$93$$ 266.167 0.296776
$$94$$ 1038.71 1.13973
$$95$$ 166.556 0.179876
$$96$$ 171.327 0.182146
$$97$$ 1350.20 1.41332 0.706659 0.707554i $$-0.250202\pi$$
0.706659 + 0.707554i $$0.250202\pi$$
$$98$$ −1056.99 −1.08951
$$99$$ −627.580 −0.637113
$$100$$ −193.941 −0.193941
$$101$$ −304.020 −0.299516 −0.149758 0.988723i $$-0.547850\pi$$
−0.149758 + 0.988723i $$0.547850\pi$$
$$102$$ 0 0
$$103$$ 988.515 0.945643 0.472822 0.881158i $$-0.343236\pi$$
0.472822 + 0.881158i $$0.343236\pi$$
$$104$$ 1141.17 1.07597
$$105$$ 27.7738 0.0258138
$$106$$ −979.107 −0.897163
$$107$$ −1175.01 −1.06162 −0.530808 0.847492i $$-0.678111\pi$$
−0.530808 + 0.847492i $$0.678111\pi$$
$$108$$ 193.386 0.172301
$$109$$ −838.865 −0.737144 −0.368572 0.929599i $$-0.620153\pi$$
−0.368572 + 0.929599i $$0.620153\pi$$
$$110$$ 430.165 0.372860
$$111$$ −304.228 −0.260144
$$112$$ 211.004 0.178018
$$113$$ 2026.81 1.68731 0.843655 0.536885i $$-0.180399\pi$$
0.843655 + 0.536885i $$0.180399\pi$$
$$114$$ −208.831 −0.171569
$$115$$ −1056.02 −0.856301
$$116$$ 38.9190 0.0311512
$$117$$ 1375.96 1.08725
$$118$$ −172.587 −0.134643
$$119$$ 0 0
$$120$$ 190.505 0.144922
$$121$$ −588.691 −0.442292
$$122$$ −2581.45 −1.91569
$$123$$ −521.423 −0.382237
$$124$$ −259.418 −0.187874
$$125$$ −1126.33 −0.805937
$$126$$ 202.274 0.143016
$$127$$ −1600.30 −1.11814 −0.559070 0.829120i $$-0.688842\pi$$
−0.559070 + 0.829120i $$0.688842\pi$$
$$128$$ 1743.94 1.20425
$$129$$ 630.909 0.430608
$$130$$ −943.133 −0.636294
$$131$$ 2615.07 1.74412 0.872061 0.489397i $$-0.162783\pi$$
0.872061 + 0.489397i $$0.162783\pi$$
$$132$$ −105.305 −0.0694364
$$133$$ −92.6360 −0.0603952
$$134$$ 2305.35 1.48621
$$135$$ 498.946 0.318092
$$136$$ 0 0
$$137$$ 745.711 0.465039 0.232520 0.972592i $$-0.425303\pi$$
0.232520 + 0.972592i $$0.425303\pi$$
$$138$$ 1324.07 0.816754
$$139$$ −2532.45 −1.54532 −0.772661 0.634818i $$-0.781075\pi$$
−0.772661 + 0.634818i $$0.781075\pi$$
$$140$$ −27.0696 −0.0163414
$$141$$ −656.047 −0.391837
$$142$$ 1983.75 1.17234
$$143$$ −1627.51 −0.951740
$$144$$ 1745.09 1.00989
$$145$$ 100.413 0.0575094
$$146$$ 1565.36 0.887332
$$147$$ 667.597 0.374574
$$148$$ 296.514 0.164684
$$149$$ −1816.70 −0.998858 −0.499429 0.866355i $$-0.666457\pi$$
−0.499429 + 0.866355i $$0.666457\pi$$
$$150$$ 627.388 0.341507
$$151$$ −2120.50 −1.14280 −0.571402 0.820670i $$-0.693600\pi$$
−0.571402 + 0.820670i $$0.693600\pi$$
$$152$$ −635.404 −0.339066
$$153$$ 0 0
$$154$$ −239.252 −0.125191
$$155$$ −669.314 −0.346842
$$156$$ 230.880 0.118495
$$157$$ −1607.82 −0.817314 −0.408657 0.912688i $$-0.634003\pi$$
−0.408657 + 0.912688i $$0.634003\pi$$
$$158$$ −283.845 −0.142921
$$159$$ 618.404 0.308444
$$160$$ −430.826 −0.212874
$$161$$ 587.346 0.287511
$$162$$ 1335.30 0.647599
$$163$$ −2278.00 −1.09464 −0.547322 0.836922i $$-0.684353\pi$$
−0.547322 + 0.836922i $$0.684353\pi$$
$$164$$ 508.202 0.241975
$$165$$ −271.692 −0.128189
$$166$$ 1147.74 0.536638
$$167$$ −1652.12 −0.765538 −0.382769 0.923844i $$-0.625029\pi$$
−0.382769 + 0.923844i $$0.625029\pi$$
$$168$$ −105.956 −0.0486590
$$169$$ 1371.29 0.624165
$$170$$ 0 0
$$171$$ −766.135 −0.342619
$$172$$ −614.912 −0.272597
$$173$$ 2210.82 0.971595 0.485797 0.874071i $$-0.338529\pi$$
0.485797 + 0.874071i $$0.338529\pi$$
$$174$$ −125.901 −0.0548534
$$175$$ 278.304 0.120216
$$176$$ −2064.11 −0.884023
$$177$$ 109.006 0.0462903
$$178$$ −605.608 −0.255013
$$179$$ 20.7548 0.00866640 0.00433320 0.999991i $$-0.498621\pi$$
0.00433320 + 0.999991i $$0.498621\pi$$
$$180$$ −223.876 −0.0927042
$$181$$ −436.498 −0.179252 −0.0896262 0.995975i $$-0.528567\pi$$
−0.0896262 + 0.995975i $$0.528567\pi$$
$$182$$ 524.558 0.213642
$$183$$ 1630.45 0.658612
$$184$$ 4028.69 1.61413
$$185$$ 765.023 0.304030
$$186$$ 839.202 0.330824
$$187$$ 0 0
$$188$$ 639.412 0.248053
$$189$$ −277.507 −0.106803
$$190$$ 525.136 0.200512
$$191$$ 787.808 0.298449 0.149225 0.988803i $$-0.452322\pi$$
0.149225 + 0.988803i $$0.452322\pi$$
$$192$$ −666.758 −0.250620
$$193$$ 3726.86 1.38998 0.694988 0.719021i $$-0.255410\pi$$
0.694988 + 0.719021i $$0.255410\pi$$
$$194$$ 4257.06 1.57546
$$195$$ 595.683 0.218758
$$196$$ −650.669 −0.237124
$$197$$ 451.885 0.163429 0.0817144 0.996656i $$-0.473960\pi$$
0.0817144 + 0.996656i $$0.473960\pi$$
$$198$$ −1978.71 −0.710206
$$199$$ −3480.16 −1.23971 −0.619855 0.784716i $$-0.712809\pi$$
−0.619855 + 0.784716i $$0.712809\pi$$
$$200$$ 1908.93 0.674910
$$201$$ −1456.06 −0.510958
$$202$$ −958.551 −0.333878
$$203$$ −55.8485 −0.0193093
$$204$$ 0 0
$$205$$ 1311.19 0.446719
$$206$$ 3116.71 1.05413
$$207$$ 4857.58 1.63104
$$208$$ 4525.54 1.50860
$$209$$ 906.194 0.299917
$$210$$ 87.5686 0.0287753
$$211$$ −1818.51 −0.593323 −0.296661 0.954983i $$-0.595873\pi$$
−0.296661 + 0.954983i $$0.595873\pi$$
$$212$$ −602.724 −0.195261
$$213$$ −1252.94 −0.403051
$$214$$ −3704.72 −1.18341
$$215$$ −1586.51 −0.503251
$$216$$ −1903.46 −0.599603
$$217$$ 372.263 0.116456
$$218$$ −2644.87 −0.821712
$$219$$ −988.684 −0.305064
$$220$$ 264.803 0.0811502
$$221$$ 0 0
$$222$$ −959.205 −0.289989
$$223$$ −6507.83 −1.95425 −0.977123 0.212676i $$-0.931782\pi$$
−0.977123 + 0.212676i $$0.931782\pi$$
$$224$$ 239.620 0.0714744
$$225$$ 2301.69 0.681981
$$226$$ 6390.36 1.88089
$$227$$ −2427.16 −0.709676 −0.354838 0.934928i $$-0.615464\pi$$
−0.354838 + 0.934928i $$0.615464\pi$$
$$228$$ −128.554 −0.0373407
$$229$$ 655.706 0.189215 0.0946076 0.995515i $$-0.469840\pi$$
0.0946076 + 0.995515i $$0.469840\pi$$
$$230$$ −3329.55 −0.954540
$$231$$ 151.112 0.0430408
$$232$$ −383.073 −0.108405
$$233$$ −1171.70 −0.329445 −0.164723 0.986340i $$-0.552673\pi$$
−0.164723 + 0.986340i $$0.552673\pi$$
$$234$$ 4338.30 1.21198
$$235$$ 1649.72 0.457940
$$236$$ −106.242 −0.0293041
$$237$$ 179.277 0.0491361
$$238$$ 0 0
$$239$$ −5281.06 −1.42930 −0.714651 0.699481i $$-0.753415\pi$$
−0.714651 + 0.699481i $$0.753415\pi$$
$$240$$ 755.484 0.203193
$$241$$ 1327.84 0.354912 0.177456 0.984129i $$-0.443213\pi$$
0.177456 + 0.984129i $$0.443213\pi$$
$$242$$ −1856.09 −0.493034
$$243$$ −3533.59 −0.932839
$$244$$ −1589.11 −0.416935
$$245$$ −1678.77 −0.437765
$$246$$ −1644.00 −0.426089
$$247$$ −1986.82 −0.511815
$$248$$ 2553.41 0.653797
$$249$$ −724.912 −0.184496
$$250$$ −3551.23 −0.898397
$$251$$ −4280.39 −1.07640 −0.538198 0.842818i $$-0.680895\pi$$
−0.538198 + 0.842818i $$0.680895\pi$$
$$252$$ 124.517 0.0311263
$$253$$ −5745.60 −1.42776
$$254$$ −5045.62 −1.24642
$$255$$ 0 0
$$256$$ 2819.92 0.688458
$$257$$ 242.890 0.0589536 0.0294768 0.999565i $$-0.490616\pi$$
0.0294768 + 0.999565i $$0.490616\pi$$
$$258$$ 1989.20 0.480010
$$259$$ −425.496 −0.102081
$$260$$ −580.579 −0.138485
$$261$$ −461.889 −0.109541
$$262$$ 8245.11 1.94422
$$263$$ −3153.49 −0.739364 −0.369682 0.929158i $$-0.620533\pi$$
−0.369682 + 0.929158i $$0.620533\pi$$
$$264$$ 1036.50 0.241636
$$265$$ −1555.06 −0.360478
$$266$$ −292.074 −0.0673240
$$267$$ 382.502 0.0876732
$$268$$ 1419.14 0.323462
$$269$$ 210.898 0.0478018 0.0239009 0.999714i $$-0.492391\pi$$
0.0239009 + 0.999714i $$0.492391\pi$$
$$270$$ 1573.14 0.354585
$$271$$ −1627.36 −0.364780 −0.182390 0.983226i $$-0.558383\pi$$
−0.182390 + 0.983226i $$0.558383\pi$$
$$272$$ 0 0
$$273$$ −331.311 −0.0734500
$$274$$ 2351.16 0.518391
$$275$$ −2722.46 −0.596984
$$276$$ 815.076 0.177760
$$277$$ 4626.89 1.00362 0.501810 0.864978i $$-0.332668\pi$$
0.501810 + 0.864978i $$0.332668\pi$$
$$278$$ −7984.62 −1.72261
$$279$$ 3078.76 0.660648
$$280$$ 266.442 0.0568677
$$281$$ −5076.28 −1.07767 −0.538836 0.842411i $$-0.681136\pi$$
−0.538836 + 0.842411i $$0.681136\pi$$
$$282$$ −2068.46 −0.436791
$$283$$ 2437.04 0.511898 0.255949 0.966690i $$-0.417612\pi$$
0.255949 + 0.966690i $$0.417612\pi$$
$$284$$ 1221.17 0.255151
$$285$$ −331.676 −0.0689360
$$286$$ −5131.39 −1.06093
$$287$$ −729.266 −0.149990
$$288$$ 1981.75 0.405471
$$289$$ 0 0
$$290$$ 316.595 0.0641072
$$291$$ −2688.76 −0.541642
$$292$$ 963.616 0.193121
$$293$$ −3300.30 −0.658041 −0.329020 0.944323i $$-0.606718\pi$$
−0.329020 + 0.944323i $$0.606718\pi$$
$$294$$ 2104.88 0.417547
$$295$$ −274.111 −0.0540994
$$296$$ −2918.54 −0.573097
$$297$$ 2714.66 0.530372
$$298$$ −5727.91 −1.11345
$$299$$ 12597.2 2.43650
$$300$$ 386.211 0.0743263
$$301$$ 882.395 0.168971
$$302$$ −6685.75 −1.27391
$$303$$ 605.420 0.114787
$$304$$ −2519.82 −0.475399
$$305$$ −4099.98 −0.769719
$$306$$ 0 0
$$307$$ −1186.40 −0.220558 −0.110279 0.993901i $$-0.535174\pi$$
−0.110279 + 0.993901i $$0.535174\pi$$
$$308$$ −147.280 −0.0272470
$$309$$ −1968.51 −0.362410
$$310$$ −2110.29 −0.386634
$$311$$ −657.473 −0.119877 −0.0599387 0.998202i $$-0.519091\pi$$
−0.0599387 + 0.998202i $$0.519091\pi$$
$$312$$ −2272.51 −0.412358
$$313$$ −6073.66 −1.09682 −0.548408 0.836211i $$-0.684766\pi$$
−0.548408 + 0.836211i $$0.684766\pi$$
$$314$$ −5069.34 −0.911080
$$315$$ 321.261 0.0574635
$$316$$ −174.731 −0.0311056
$$317$$ 5753.26 1.01935 0.509677 0.860366i $$-0.329765\pi$$
0.509677 + 0.860366i $$0.329765\pi$$
$$318$$ 1949.78 0.343830
$$319$$ 546.327 0.0958886
$$320$$ 1676.66 0.292900
$$321$$ 2339.90 0.406855
$$322$$ 1851.85 0.320496
$$323$$ 0 0
$$324$$ 821.991 0.140945
$$325$$ 5968.97 1.01877
$$326$$ −7182.35 −1.22023
$$327$$ 1670.50 0.282504
$$328$$ −5002.15 −0.842066
$$329$$ −917.553 −0.153758
$$330$$ −856.623 −0.142896
$$331$$ −89.8427 −0.0149190 −0.00745952 0.999972i $$-0.502374\pi$$
−0.00745952 + 0.999972i $$0.502374\pi$$
$$332$$ 706.532 0.116795
$$333$$ −3519.01 −0.579101
$$334$$ −5209.00 −0.853364
$$335$$ 3661.47 0.597156
$$336$$ −420.190 −0.0682240
$$337$$ −847.308 −0.136961 −0.0684804 0.997652i $$-0.521815\pi$$
−0.0684804 + 0.997652i $$0.521815\pi$$
$$338$$ 4323.56 0.695772
$$339$$ −4036.15 −0.646648
$$340$$ 0 0
$$341$$ −3641.60 −0.578309
$$342$$ −2415.56 −0.381926
$$343$$ 1889.02 0.297369
$$344$$ 6052.48 0.948628
$$345$$ 2102.94 0.328170
$$346$$ 6970.55 1.08306
$$347$$ 590.322 0.0913260 0.0456630 0.998957i $$-0.485460\pi$$
0.0456630 + 0.998957i $$0.485460\pi$$
$$348$$ −77.5026 −0.0119384
$$349$$ 9387.68 1.43986 0.719930 0.694047i $$-0.244174\pi$$
0.719930 + 0.694047i $$0.244174\pi$$
$$350$$ 877.471 0.134008
$$351$$ −5951.87 −0.905092
$$352$$ −2344.03 −0.354936
$$353$$ −6176.09 −0.931218 −0.465609 0.884990i $$-0.654165\pi$$
−0.465609 + 0.884990i $$0.654165\pi$$
$$354$$ 343.687 0.0516009
$$355$$ 3150.68 0.471045
$$356$$ −372.804 −0.0555015
$$357$$ 0 0
$$358$$ 65.4382 0.00966066
$$359$$ 7151.14 1.05132 0.525658 0.850696i $$-0.323819\pi$$
0.525658 + 0.850696i $$0.323819\pi$$
$$360$$ 2203.58 0.322608
$$361$$ −5752.74 −0.838714
$$362$$ −1376.24 −0.199817
$$363$$ 1172.31 0.169505
$$364$$ 322.910 0.0464975
$$365$$ 2486.18 0.356528
$$366$$ 5140.66 0.734171
$$367$$ −3358.05 −0.477626 −0.238813 0.971066i $$-0.576758\pi$$
−0.238813 + 0.971066i $$0.576758\pi$$
$$368$$ 15976.6 2.26314
$$369$$ −6031.32 −0.850888
$$370$$ 2412.05 0.338910
$$371$$ 864.905 0.121034
$$372$$ 516.600 0.0720013
$$373$$ 8379.14 1.16315 0.581576 0.813492i $$-0.302436\pi$$
0.581576 + 0.813492i $$0.302436\pi$$
$$374$$ 0 0
$$375$$ 2242.95 0.308868
$$376$$ −6293.64 −0.863217
$$377$$ −1197.82 −0.163636
$$378$$ −874.957 −0.119055
$$379$$ −6395.71 −0.866821 −0.433411 0.901196i $$-0.642690\pi$$
−0.433411 + 0.901196i $$0.642690\pi$$
$$380$$ 323.266 0.0436400
$$381$$ 3186.81 0.428518
$$382$$ 2483.89 0.332689
$$383$$ −204.687 −0.0273081 −0.0136541 0.999907i $$-0.504346\pi$$
−0.0136541 + 0.999907i $$0.504346\pi$$
$$384$$ −3472.85 −0.461518
$$385$$ −379.991 −0.0503017
$$386$$ 11750.5 1.54944
$$387$$ 7297.75 0.958567
$$388$$ 2620.59 0.342887
$$389$$ 5770.56 0.752131 0.376066 0.926593i $$-0.377277\pi$$
0.376066 + 0.926593i $$0.377277\pi$$
$$390$$ 1878.14 0.243854
$$391$$ 0 0
$$392$$ 6404.44 0.825186
$$393$$ −5207.61 −0.668420
$$394$$ 1424.76 0.182178
$$395$$ −450.816 −0.0574253
$$396$$ −1218.06 −0.154571
$$397$$ 7144.37 0.903188 0.451594 0.892223i $$-0.350855\pi$$
0.451594 + 0.892223i $$0.350855\pi$$
$$398$$ −10972.7 −1.38194
$$399$$ 184.474 0.0231459
$$400$$ 7570.24 0.946280
$$401$$ −6364.52 −0.792591 −0.396296 0.918123i $$-0.629704\pi$$
−0.396296 + 0.918123i $$0.629704\pi$$
$$402$$ −4590.84 −0.569578
$$403$$ 7984.16 0.986897
$$404$$ −590.070 −0.0726660
$$405$$ 2120.78 0.260204
$$406$$ −176.086 −0.0215246
$$407$$ 4162.33 0.506926
$$408$$ 0 0
$$409$$ 2997.87 0.362433 0.181217 0.983443i $$-0.441997\pi$$
0.181217 + 0.983443i $$0.441997\pi$$
$$410$$ 4134.07 0.497969
$$411$$ −1485.00 −0.178222
$$412$$ 1918.60 0.229424
$$413$$ 152.457 0.0181644
$$414$$ 15315.5 1.81816
$$415$$ 1822.89 0.215620
$$416$$ 5139.27 0.605705
$$417$$ 5043.08 0.592232
$$418$$ 2857.15 0.334325
$$419$$ −11747.3 −1.36967 −0.684835 0.728698i $$-0.740126\pi$$
−0.684835 + 0.728698i $$0.740126\pi$$
$$420$$ 53.9060 0.00626272
$$421$$ 8842.15 1.02361 0.511805 0.859102i $$-0.328977\pi$$
0.511805 + 0.859102i $$0.328977\pi$$
$$422$$ −5733.60 −0.661391
$$423$$ −7588.52 −0.872261
$$424$$ 5932.52 0.679501
$$425$$ 0 0
$$426$$ −3950.41 −0.449291
$$427$$ 2280.36 0.258441
$$428$$ −2280.57 −0.257560
$$429$$ 3240.98 0.364746
$$430$$ −5002.13 −0.560987
$$431$$ 3099.38 0.346385 0.173192 0.984888i $$-0.444592\pi$$
0.173192 + 0.984888i $$0.444592\pi$$
$$432$$ −7548.55 −0.840694
$$433$$ 10072.8 1.11794 0.558970 0.829188i $$-0.311197\pi$$
0.558970 + 0.829188i $$0.311197\pi$$
$$434$$ 1173.72 0.129816
$$435$$ −199.961 −0.0220400
$$436$$ −1628.14 −0.178839
$$437$$ −7014.09 −0.767802
$$438$$ −3117.24 −0.340063
$$439$$ 17357.9 1.88712 0.943561 0.331199i $$-0.107453\pi$$
0.943561 + 0.331199i $$0.107453\pi$$
$$440$$ −2606.42 −0.282400
$$441$$ 7722.12 0.833832
$$442$$ 0 0
$$443$$ 6979.90 0.748589 0.374295 0.927310i $$-0.377885\pi$$
0.374295 + 0.927310i $$0.377885\pi$$
$$444$$ −590.472 −0.0631139
$$445$$ −961.855 −0.102464
$$446$$ −20518.7 −2.17845
$$447$$ 3617.74 0.382804
$$448$$ −932.533 −0.0983439
$$449$$ 9230.50 0.970188 0.485094 0.874462i $$-0.338785\pi$$
0.485094 + 0.874462i $$0.338785\pi$$
$$450$$ 7257.03 0.760221
$$451$$ 7133.91 0.744840
$$452$$ 3933.81 0.409361
$$453$$ 4222.72 0.437970
$$454$$ −7652.64 −0.791093
$$455$$ 833.127 0.0858409
$$456$$ 1265.33 0.129944
$$457$$ 12250.6 1.25396 0.626979 0.779036i $$-0.284291\pi$$
0.626979 + 0.779036i $$0.284291\pi$$
$$458$$ 2067.39 0.210923
$$459$$ 0 0
$$460$$ −2049.62 −0.207748
$$461$$ 6259.50 0.632395 0.316198 0.948693i $$-0.397594\pi$$
0.316198 + 0.948693i $$0.397594\pi$$
$$462$$ 476.442 0.0479786
$$463$$ 10195.0 1.02333 0.511665 0.859185i $$-0.329029\pi$$
0.511665 + 0.859185i $$0.329029\pi$$
$$464$$ −1519.15 −0.151993
$$465$$ 1332.86 0.132924
$$466$$ −3694.28 −0.367241
$$467$$ −14783.0 −1.46483 −0.732417 0.680856i $$-0.761608\pi$$
−0.732417 + 0.680856i $$0.761608\pi$$
$$468$$ 2670.59 0.263778
$$469$$ −2036.46 −0.200501
$$470$$ 5201.43 0.510477
$$471$$ 3201.79 0.313229
$$472$$ 1045.72 0.101977
$$473$$ −8631.86 −0.839098
$$474$$ 565.244 0.0547732
$$475$$ −3323.52 −0.321039
$$476$$ 0 0
$$477$$ 7153.10 0.686620
$$478$$ −16650.7 −1.59328
$$479$$ −16370.0 −1.56152 −0.780758 0.624834i $$-0.785167\pi$$
−0.780758 + 0.624834i $$0.785167\pi$$
$$480$$ 857.939 0.0815820
$$481$$ −9125.87 −0.865081
$$482$$ 4186.57 0.395629
$$483$$ −1169.63 −0.110186
$$484$$ −1142.58 −0.107305
$$485$$ 6761.26 0.633017
$$486$$ −11141.1 −1.03986
$$487$$ −4106.31 −0.382084 −0.191042 0.981582i $$-0.561187\pi$$
−0.191042 + 0.981582i $$0.561187\pi$$
$$488$$ 15641.3 1.45092
$$489$$ 4536.37 0.419513
$$490$$ −5293.01 −0.487987
$$491$$ −878.001 −0.0806999 −0.0403499 0.999186i $$-0.512847\pi$$
−0.0403499 + 0.999186i $$0.512847\pi$$
$$492$$ −1012.02 −0.0927349
$$493$$ 0 0
$$494$$ −6264.28 −0.570533
$$495$$ −3142.68 −0.285359
$$496$$ 10126.0 0.916679
$$497$$ −1752.37 −0.158158
$$498$$ −2285.59 −0.205662
$$499$$ −13819.6 −1.23978 −0.619892 0.784687i $$-0.712824\pi$$
−0.619892 + 0.784687i $$0.712824\pi$$
$$500$$ −2186.08 −0.195529
$$501$$ 3290.00 0.293386
$$502$$ −13495.7 −1.19989
$$503$$ 4721.43 0.418525 0.209263 0.977859i $$-0.432894\pi$$
0.209263 + 0.977859i $$0.432894\pi$$
$$504$$ −1225.60 −0.108319
$$505$$ −1522.41 −0.134152
$$506$$ −18115.4 −1.59156
$$507$$ −2730.76 −0.239206
$$508$$ −3106.01 −0.271273
$$509$$ −16554.3 −1.44156 −0.720782 0.693161i $$-0.756217\pi$$
−0.720782 + 0.693161i $$0.756217\pi$$
$$510$$ 0 0
$$511$$ −1382.78 −0.119708
$$512$$ −5060.53 −0.436808
$$513$$ 3314.00 0.285217
$$514$$ 765.813 0.0657171
$$515$$ 4950.09 0.423548
$$516$$ 1224.53 0.104470
$$517$$ 8975.79 0.763548
$$518$$ −1341.55 −0.113792
$$519$$ −4402.60 −0.372356
$$520$$ 5714.55 0.481922
$$521$$ 14755.5 1.24079 0.620394 0.784290i $$-0.286973\pi$$
0.620394 + 0.784290i $$0.286973\pi$$
$$522$$ −1456.30 −0.122108
$$523$$ 7800.86 0.652214 0.326107 0.945333i $$-0.394263\pi$$
0.326107 + 0.945333i $$0.394263\pi$$
$$524$$ 5075.57 0.423144
$$525$$ −554.210 −0.0460719
$$526$$ −9942.71 −0.824188
$$527$$ 0 0
$$528$$ 4110.43 0.338795
$$529$$ 32304.9 2.65512
$$530$$ −4902.98 −0.401834
$$531$$ 1260.88 0.103046
$$532$$ −179.796 −0.0146525
$$533$$ −15641.0 −1.27108
$$534$$ 1206.00 0.0977315
$$535$$ −5884.01 −0.475491
$$536$$ −13968.4 −1.12564
$$537$$ −41.3307 −0.00332133
$$538$$ 664.944 0.0532859
$$539$$ −9133.81 −0.729909
$$540$$ 968.399 0.0771727
$$541$$ −23028.5 −1.83007 −0.915037 0.403369i $$-0.867839\pi$$
−0.915037 + 0.403369i $$0.867839\pi$$
$$542$$ −5130.95 −0.406629
$$543$$ 869.235 0.0686970
$$544$$ 0 0
$$545$$ −4200.71 −0.330162
$$546$$ −1044.60 −0.0818765
$$547$$ −2641.77 −0.206497 −0.103249 0.994656i $$-0.532924\pi$$
−0.103249 + 0.994656i $$0.532924\pi$$
$$548$$ 1447.34 0.112824
$$549$$ 18859.4 1.46612
$$550$$ −8583.69 −0.665473
$$551$$ 666.944 0.0515658
$$552$$ −8022.67 −0.618601
$$553$$ 250.738 0.0192811
$$554$$ 14588.2 1.11876
$$555$$ −1523.45 −0.116517
$$556$$ −4915.21 −0.374913
$$557$$ −19800.8 −1.50626 −0.753129 0.657873i $$-0.771456\pi$$
−0.753129 + 0.657873i $$0.771456\pi$$
$$558$$ 9707.08 0.736440
$$559$$ 18925.3 1.43194
$$560$$ 1056.63 0.0797333
$$561$$ 0 0
$$562$$ −16005.1 −1.20131
$$563$$ −9733.93 −0.728661 −0.364331 0.931270i $$-0.618702\pi$$
−0.364331 + 0.931270i $$0.618702\pi$$
$$564$$ −1273.31 −0.0950642
$$565$$ 10149.5 0.755736
$$566$$ 7683.80 0.570626
$$567$$ −1179.55 −0.0873660
$$568$$ −12019.8 −0.887919
$$569$$ 6340.67 0.467161 0.233581 0.972337i $$-0.424956\pi$$
0.233581 + 0.972337i $$0.424956\pi$$
$$570$$ −1045.75 −0.0768447
$$571$$ 12377.7 0.907166 0.453583 0.891214i $$-0.350146\pi$$
0.453583 + 0.891214i $$0.350146\pi$$
$$572$$ −3158.81 −0.230903
$$573$$ −1568.83 −0.114378
$$574$$ −2299.32 −0.167198
$$575$$ 21072.3 1.52831
$$576$$ −7712.41 −0.557900
$$577$$ −36.6040 −0.00264098 −0.00132049 0.999999i $$-0.500420\pi$$
−0.00132049 + 0.999999i $$0.500420\pi$$
$$578$$ 0 0
$$579$$ −7421.61 −0.532697
$$580$$ 194.891 0.0139524
$$581$$ −1013.87 −0.0723965
$$582$$ −8477.44 −0.603782
$$583$$ −8460.77 −0.601045
$$584$$ −9484.72 −0.672055
$$585$$ 6890.29 0.486971
$$586$$ −10405.6 −0.733534
$$587$$ −20350.4 −1.43092 −0.715462 0.698651i $$-0.753784\pi$$
−0.715462 + 0.698651i $$0.753784\pi$$
$$588$$ 1295.73 0.0908760
$$589$$ −4445.58 −0.310996
$$590$$ −864.248 −0.0603060
$$591$$ −899.876 −0.0626328
$$592$$ −11574.0 −0.803530
$$593$$ −20387.4 −1.41182 −0.705910 0.708301i $$-0.749462\pi$$
−0.705910 + 0.708301i $$0.749462\pi$$
$$594$$ 8559.10 0.591219
$$595$$ 0 0
$$596$$ −3526.02 −0.242334
$$597$$ 6930.33 0.475108
$$598$$ 39717.8 2.71603
$$599$$ 316.417 0.0215834 0.0107917 0.999942i $$-0.496565\pi$$
0.0107917 + 0.999942i $$0.496565\pi$$
$$600$$ −3801.41 −0.258653
$$601$$ 1331.85 0.0903950 0.0451975 0.998978i $$-0.485608\pi$$
0.0451975 + 0.998978i $$0.485608\pi$$
$$602$$ 2782.12 0.188357
$$603$$ −16842.3 −1.13743
$$604$$ −4115.65 −0.277257
$$605$$ −2947.93 −0.198100
$$606$$ 1908.84 0.127956
$$607$$ 13873.2 0.927668 0.463834 0.885922i $$-0.346473\pi$$
0.463834 + 0.885922i $$0.346473\pi$$
$$608$$ −2861.54 −0.190873
$$609$$ 111.216 0.00740014
$$610$$ −12926.9 −0.858025
$$611$$ −19679.3 −1.30301
$$612$$ 0 0
$$613$$ 15297.0 1.00790 0.503948 0.863734i $$-0.331880\pi$$
0.503948 + 0.863734i $$0.331880\pi$$
$$614$$ −3740.61 −0.245861
$$615$$ −2611.08 −0.171201
$$616$$ 1449.66 0.0948186
$$617$$ 15116.8 0.986354 0.493177 0.869929i $$-0.335835\pi$$
0.493177 + 0.869929i $$0.335835\pi$$
$$618$$ −6206.55 −0.403987
$$619$$ 22412.1 1.45528 0.727639 0.685960i $$-0.240618\pi$$
0.727639 + 0.685960i $$0.240618\pi$$
$$620$$ −1299.06 −0.0841479
$$621$$ −21011.9 −1.35778
$$622$$ −2072.96 −0.133630
$$623$$ 534.971 0.0344031
$$624$$ −9012.08 −0.578160
$$625$$ 6850.26 0.438417
$$626$$ −19149.7 −1.22265
$$627$$ −1804.58 −0.114941
$$628$$ −3120.61 −0.198290
$$629$$ 0 0
$$630$$ 1012.91 0.0640560
$$631$$ −4830.94 −0.304781 −0.152390 0.988320i $$-0.548697\pi$$
−0.152390 + 0.988320i $$0.548697\pi$$
$$632$$ 1719.85 0.108247
$$633$$ 3621.34 0.227386
$$634$$ 18139.6 1.13630
$$635$$ −8013.68 −0.500808
$$636$$ 1200.25 0.0748320
$$637$$ 20025.8 1.24561
$$638$$ 1722.53 0.106889
$$639$$ −14492.8 −0.897222
$$640$$ 8732.96 0.539376
$$641$$ −23007.3 −1.41768 −0.708839 0.705370i $$-0.750781\pi$$
−0.708839 + 0.705370i $$0.750781\pi$$
$$642$$ 7377.52 0.453532
$$643$$ 5689.90 0.348970 0.174485 0.984660i $$-0.444174\pi$$
0.174485 + 0.984660i $$0.444174\pi$$
$$644$$ 1139.97 0.0697535
$$645$$ 3159.35 0.192867
$$646$$ 0 0
$$647$$ −15949.0 −0.969122 −0.484561 0.874758i $$-0.661021\pi$$
−0.484561 + 0.874758i $$0.661021\pi$$
$$648$$ −8090.72 −0.490484
$$649$$ −1491.38 −0.0902029
$$650$$ 18819.7 1.13564
$$651$$ −741.318 −0.0446307
$$652$$ −4421.35 −0.265573
$$653$$ −10764.6 −0.645101 −0.322550 0.946552i $$-0.604540\pi$$
−0.322550 + 0.946552i $$0.604540\pi$$
$$654$$ 5266.95 0.314914
$$655$$ 13095.3 0.781182
$$656$$ −19837.0 −1.18065
$$657$$ −11436.1 −0.679097
$$658$$ −2892.97 −0.171398
$$659$$ 25208.2 1.49010 0.745048 0.667011i $$-0.232427\pi$$
0.745048 + 0.667011i $$0.232427\pi$$
$$660$$ −527.325 −0.0311001
$$661$$ 14419.5 0.848492 0.424246 0.905547i $$-0.360539\pi$$
0.424246 + 0.905547i $$0.360539\pi$$
$$662$$ −283.267 −0.0166306
$$663$$ 0 0
$$664$$ −6954.28 −0.406444
$$665$$ −463.885 −0.0270506
$$666$$ −11095.2 −0.645539
$$667$$ −4228.67 −0.245479
$$668$$ −3206.58 −0.185728
$$669$$ 12959.6 0.748948
$$670$$ 11544.3 0.665665
$$671$$ −22307.2 −1.28339
$$672$$ −477.174 −0.0273920
$$673$$ −2110.64 −0.120890 −0.0604451 0.998172i $$-0.519252\pi$$
−0.0604451 + 0.998172i $$0.519252\pi$$
$$674$$ −2671.49 −0.152674
$$675$$ −9956.17 −0.567723
$$676$$ 2661.52 0.151429
$$677$$ 11944.5 0.678087 0.339043 0.940771i $$-0.389897\pi$$
0.339043 + 0.940771i $$0.389897\pi$$
$$678$$ −12725.6 −0.720834
$$679$$ −3760.52 −0.212541
$$680$$ 0 0
$$681$$ 4833.41 0.271977
$$682$$ −11481.6 −0.644656
$$683$$ 19085.8 1.06925 0.534626 0.845089i $$-0.320453\pi$$
0.534626 + 0.845089i $$0.320453\pi$$
$$684$$ −1486.98 −0.0831232
$$685$$ 3734.23 0.208288
$$686$$ 5955.92 0.331484
$$687$$ −1305.76 −0.0725152
$$688$$ 24002.3 1.33006
$$689$$ 18550.2 1.02570
$$690$$ 6630.41 0.365819
$$691$$ −28316.8 −1.55893 −0.779465 0.626445i $$-0.784509\pi$$
−0.779465 + 0.626445i $$0.784509\pi$$
$$692$$ 4290.97 0.235720
$$693$$ 1747.91 0.0958121
$$694$$ 1861.24 0.101803
$$695$$ −12681.5 −0.692141
$$696$$ 762.846 0.0415454
$$697$$ 0 0
$$698$$ 29598.6 1.60505
$$699$$ 2333.31 0.126257
$$700$$ 540.158 0.0291658
$$701$$ −5916.75 −0.318791 −0.159396 0.987215i $$-0.550955\pi$$
−0.159396 + 0.987215i $$0.550955\pi$$
$$702$$ −18765.8 −1.00893
$$703$$ 5081.28 0.272609
$$704$$ 9122.33 0.488368
$$705$$ −3285.22 −0.175502
$$706$$ −19472.7 −1.03805
$$707$$ 846.747 0.0450427
$$708$$ 211.568 0.0112305
$$709$$ −18499.8 −0.979936 −0.489968 0.871740i $$-0.662992\pi$$
−0.489968 + 0.871740i $$0.662992\pi$$
$$710$$ 9933.85 0.525085
$$711$$ 2073.70 0.109381
$$712$$ 3669.45 0.193144
$$713$$ 28186.6 1.48050
$$714$$ 0 0
$$715$$ −8149.91 −0.426279
$$716$$ 40.2828 0.00210257
$$717$$ 10516.6 0.547768
$$718$$ 22546.9 1.17193
$$719$$ −22206.5 −1.15182 −0.575912 0.817511i $$-0.695353\pi$$
−0.575912 + 0.817511i $$0.695353\pi$$
$$720$$ 8738.72 0.452323
$$721$$ −2753.18 −0.142210
$$722$$ −18137.9 −0.934935
$$723$$ −2644.24 −0.136017
$$724$$ −847.195 −0.0434886
$$725$$ −2003.69 −0.102641
$$726$$ 3696.19 0.188951
$$727$$ 3777.02 0.192685 0.0963424 0.995348i $$-0.469286\pi$$
0.0963424 + 0.995348i $$0.469286\pi$$
$$728$$ −3178.36 −0.161810
$$729$$ −4398.10 −0.223447
$$730$$ 7838.73 0.397431
$$731$$ 0 0
$$732$$ 3164.52 0.159787
$$733$$ −19956.4 −1.00560 −0.502801 0.864402i $$-0.667697\pi$$
−0.502801 + 0.864402i $$0.667697\pi$$
$$734$$ −10587.7 −0.532422
$$735$$ 3343.06 0.167770
$$736$$ 18143.2 0.908652
$$737$$ 19921.3 0.995671
$$738$$ −19016.2 −0.948506
$$739$$ 23268.4 1.15824 0.579121 0.815241i $$-0.303396\pi$$
0.579121 + 0.815241i $$0.303396\pi$$
$$740$$ 1484.83 0.0737612
$$741$$ 3956.52 0.196149
$$742$$ 2726.97 0.134920
$$743$$ 12587.1 0.621502 0.310751 0.950491i $$-0.399420\pi$$
0.310751 + 0.950491i $$0.399420\pi$$
$$744$$ −5084.82 −0.250562
$$745$$ −9097.32 −0.447383
$$746$$ 26418.7 1.29659
$$747$$ −8385.09 −0.410702
$$748$$ 0 0
$$749$$ 3272.61 0.159651
$$750$$ 7071.85 0.344303
$$751$$ 2662.79 0.129383 0.0646914 0.997905i $$-0.479394\pi$$
0.0646914 + 0.997905i $$0.479394\pi$$
$$752$$ −24958.6 −1.21030
$$753$$ 8523.89 0.412520
$$754$$ −3776.62 −0.182409
$$755$$ −10618.6 −0.511855
$$756$$ −538.611 −0.0259115
$$757$$ 26430.2 1.26899 0.634493 0.772929i $$-0.281209\pi$$
0.634493 + 0.772929i $$0.281209\pi$$
$$758$$ −20165.1 −0.966267
$$759$$ 11441.7 0.547176
$$760$$ −3181.86 −0.151866
$$761$$ −8469.75 −0.403454 −0.201727 0.979442i $$-0.564655\pi$$
−0.201727 + 0.979442i $$0.564655\pi$$
$$762$$ 10047.8 0.477680
$$763$$ 2336.38 0.110855
$$764$$ 1529.05 0.0724071
$$765$$ 0 0
$$766$$ −645.361 −0.0304410
$$767$$ 3269.83 0.153933
$$768$$ −5615.54 −0.263846
$$769$$ −8452.54 −0.396367 −0.198184 0.980165i $$-0.563504\pi$$
−0.198184 + 0.980165i $$0.563504\pi$$
$$770$$ −1198.08 −0.0560725
$$771$$ −483.687 −0.0225935
$$772$$ 7233.43 0.337224
$$773$$ 33936.2 1.57904 0.789521 0.613724i $$-0.210329\pi$$
0.789521 + 0.613724i $$0.210329\pi$$
$$774$$ 23009.2 1.06854
$$775$$ 13355.8 0.619036
$$776$$ −25794.0 −1.19324
$$777$$ 847.324 0.0391217
$$778$$ 18194.1 0.838419
$$779$$ 8708.91 0.400551
$$780$$ 1156.15 0.0530730
$$781$$ 17142.2 0.785399
$$782$$ 0 0
$$783$$ 1997.95 0.0911887
$$784$$ 25398.0 1.15698
$$785$$ −8051.35 −0.366070
$$786$$ −16419.2 −0.745105
$$787$$ −34379.1 −1.55716 −0.778578 0.627547i $$-0.784059\pi$$
−0.778578 + 0.627547i $$0.784059\pi$$
$$788$$ 877.059 0.0396497
$$789$$ 6279.81 0.283355
$$790$$ −1421.39 −0.0640134
$$791$$ −5645.00 −0.253746
$$792$$ 11989.2 0.537902
$$793$$ 48908.2 2.19014
$$794$$ 22525.6 1.00681
$$795$$ 3096.72 0.138150
$$796$$ −6754.61 −0.300768
$$797$$ −3329.26 −0.147965 −0.0739826 0.997260i $$-0.523571\pi$$
−0.0739826 + 0.997260i $$0.523571\pi$$
$$798$$ 581.630 0.0258014
$$799$$ 0 0
$$800$$ 8596.87 0.379932
$$801$$ 4424.42 0.195167
$$802$$ −20066.8 −0.883521
$$803$$ 13526.8 0.594459
$$804$$ −2826.05 −0.123964
$$805$$ 2941.20 0.128775
$$806$$ 25173.4 1.10012
$$807$$ −419.979 −0.0183196
$$808$$ 5807.97 0.252876
$$809$$ −7217.64 −0.313670 −0.156835 0.987625i $$-0.550129\pi$$
−0.156835 + 0.987625i $$0.550129\pi$$
$$810$$ 6686.66 0.290056
$$811$$ 36564.1 1.58316 0.791579 0.611067i $$-0.209259\pi$$
0.791579 + 0.611067i $$0.209259\pi$$
$$812$$ −108.396 −0.00468466
$$813$$ 3240.70 0.139799
$$814$$ 13123.5 0.565083
$$815$$ −11407.3 −0.490284
$$816$$ 0 0
$$817$$ −10537.6 −0.451240
$$818$$ 9452.04 0.404013
$$819$$ −3832.29 −0.163505
$$820$$ 2544.88 0.108379
$$821$$ 23140.5 0.983691 0.491845 0.870683i $$-0.336323\pi$$
0.491845 + 0.870683i $$0.336323\pi$$
$$822$$ −4682.07 −0.198669
$$823$$ −41287.9 −1.74873 −0.874365 0.485269i $$-0.838722\pi$$
−0.874365 + 0.485269i $$0.838722\pi$$
$$824$$ −18884.5 −0.798388
$$825$$ 5421.46 0.228789
$$826$$ 480.683 0.0202483
$$827$$ −10008.0 −0.420811 −0.210405 0.977614i $$-0.567478\pi$$
−0.210405 + 0.977614i $$0.567478\pi$$
$$828$$ 9428.02 0.395708
$$829$$ −44643.6 −1.87037 −0.935185 0.354159i $$-0.884767\pi$$
−0.935185 + 0.354159i $$0.884767\pi$$
$$830$$ 5747.43 0.240357
$$831$$ −9213.90 −0.384629
$$832$$ −20000.6 −0.833410
$$833$$ 0 0
$$834$$ 15900.4 0.660176
$$835$$ −8273.17 −0.342880
$$836$$ 1758.82 0.0727633
$$837$$ −13317.5 −0.549964
$$838$$ −37038.2 −1.52681
$$839$$ 14235.3 0.585765 0.292882 0.956148i $$-0.405386\pi$$
0.292882 + 0.956148i $$0.405386\pi$$
$$840$$ −530.588 −0.0217941
$$841$$ −23986.9 −0.983514
$$842$$ 27878.6 1.14104
$$843$$ 10108.8 0.413008
$$844$$ −3529.52 −0.143947
$$845$$ 6866.88 0.279560
$$846$$ −23926.0 −0.972331
$$847$$ 1639.60 0.0665140
$$848$$ 23526.5 0.952717
$$849$$ −4853.08 −0.196181
$$850$$ 0 0
$$851$$ −32217.1 −1.29775
$$852$$ −2431.81 −0.0977846
$$853$$ −27729.0 −1.11304 −0.556520 0.830834i $$-0.687864\pi$$
−0.556520 + 0.830834i $$0.687864\pi$$
$$854$$ 7189.78 0.288090
$$855$$ −3836.51 −0.153457
$$856$$ 22447.3 0.896301
$$857$$ −31280.7 −1.24682 −0.623411 0.781894i $$-0.714254\pi$$
−0.623411 + 0.781894i $$0.714254\pi$$
$$858$$ 10218.6 0.406592
$$859$$ −55.2890 −0.00219608 −0.00109804 0.999999i $$-0.500350\pi$$
−0.00109804 + 0.999999i $$0.500350\pi$$
$$860$$ −3079.24 −0.122094
$$861$$ 1452.25 0.0574826
$$862$$ 9772.09 0.386124
$$863$$ −22900.4 −0.903291 −0.451646 0.892197i $$-0.649163\pi$$
−0.451646 + 0.892197i $$0.649163\pi$$
$$864$$ −8572.25 −0.337539
$$865$$ 11070.9 0.435172
$$866$$ 31758.7 1.24620
$$867$$ 0 0
$$868$$ 722.522 0.0282535
$$869$$ −2452.79 −0.0957484
$$870$$ −630.461 −0.0245685
$$871$$ −43677.2 −1.69913
$$872$$ 16025.6 0.622356
$$873$$ −31101.0 −1.20574
$$874$$ −22114.9 −0.855888
$$875$$ 3137.02 0.121201
$$876$$ −1918.93 −0.0740120
$$877$$ −1782.50 −0.0686326 −0.0343163 0.999411i $$-0.510925\pi$$
−0.0343163 + 0.999411i $$0.510925\pi$$
$$878$$ 54728.0 2.10362
$$879$$ 6572.17 0.252189
$$880$$ −10336.3 −0.395949
$$881$$ −17232.6 −0.659001 −0.329500 0.944155i $$-0.606880\pi$$
−0.329500 + 0.944155i $$0.606880\pi$$
$$882$$ 24347.2 0.929493
$$883$$ 10188.2 0.388291 0.194145 0.980973i $$-0.437807\pi$$
0.194145 + 0.980973i $$0.437807\pi$$
$$884$$ 0 0
$$885$$ 545.859 0.0207332
$$886$$ 22007.1 0.834471
$$887$$ −15671.6 −0.593238 −0.296619 0.954996i $$-0.595859\pi$$
−0.296619 + 0.954996i $$0.595859\pi$$
$$888$$ 5811.93 0.219635
$$889$$ 4457.10 0.168151
$$890$$ −3032.65 −0.114219
$$891$$ 11538.7 0.433852
$$892$$ −12631.0 −0.474122
$$893$$ 10957.4 0.410612
$$894$$ 11406.4 0.426721
$$895$$ 103.932 0.00388163
$$896$$ −4857.16 −0.181101
$$897$$ −25085.8 −0.933768
$$898$$ 29103.0 1.08149
$$899$$ −2680.15 −0.0994307
$$900$$ 4467.32 0.165456
$$901$$ 0 0
$$902$$ 22492.6 0.830291
$$903$$ −1757.19 −0.0647569
$$904$$ −38719.9 −1.42456
$$905$$ −2185.81 −0.0802861
$$906$$ 13313.9 0.488216
$$907$$ 16908.3 0.618999 0.309500 0.950900i $$-0.399838\pi$$
0.309500 + 0.950900i $$0.399838\pi$$
$$908$$ −4710.86 −0.172175
$$909$$ 7002.92 0.255525
$$910$$ 2626.78 0.0956890
$$911$$ 36100.6 1.31292 0.656458 0.754362i $$-0.272054\pi$$
0.656458 + 0.754362i $$0.272054\pi$$
$$912$$ 5017.92 0.182193
$$913$$ 9917.98 0.359515
$$914$$ 38625.1 1.39782
$$915$$ 8164.63 0.294988
$$916$$ 1272.65 0.0459057
$$917$$ −7283.41 −0.262290
$$918$$ 0 0
$$919$$ 20016.2 0.718470 0.359235 0.933247i $$-0.383038\pi$$
0.359235 + 0.933247i $$0.383038\pi$$
$$920$$ 20174.1 0.722958
$$921$$ 2362.57 0.0845269
$$922$$ 19735.7 0.704947
$$923$$ −37584.1 −1.34030
$$924$$ 293.291 0.0104422
$$925$$ −15265.6 −0.542626
$$926$$ 32144.0 1.14073
$$927$$ −22769.8 −0.806753
$$928$$ −1725.17 −0.0610253
$$929$$ −30630.3 −1.08175 −0.540875 0.841103i $$-0.681907\pi$$
−0.540875 + 0.841103i $$0.681907\pi$$
$$930$$ 4202.40 0.148174
$$931$$ −11150.3 −0.392522
$$932$$ −2274.15 −0.0799272
$$933$$ 1309.28 0.0459420
$$934$$ −46609.7 −1.63289
$$935$$ 0 0
$$936$$ −26286.2 −0.917941
$$937$$ 3883.68 0.135405 0.0677025 0.997706i $$-0.478433\pi$$
0.0677025 + 0.997706i $$0.478433\pi$$
$$938$$ −6420.79 −0.223503
$$939$$ 12095.0 0.420346
$$940$$ 3201.93 0.111101
$$941$$ 21696.0 0.751613 0.375807 0.926698i $$-0.377366\pi$$
0.375807 + 0.926698i $$0.377366\pi$$
$$942$$ 10095.0 0.349164
$$943$$ −55217.7 −1.90682
$$944$$ 4147.02 0.142981
$$945$$ −1389.65 −0.0478362
$$946$$ −27215.6 −0.935364
$$947$$ −15904.8 −0.545763 −0.272882 0.962048i $$-0.587977\pi$$
−0.272882 + 0.962048i $$0.587977\pi$$
$$948$$ 347.956 0.0119210
$$949$$ −29657.4 −1.01446
$$950$$ −10478.8 −0.357870
$$951$$ −11456.9 −0.390659
$$952$$ 0 0
$$953$$ −81.8493 −0.00278212 −0.00139106 0.999999i $$-0.500443\pi$$
−0.00139106 + 0.999999i $$0.500443\pi$$
$$954$$ 22553.1 0.765393
$$955$$ 3945.03 0.133674
$$956$$ −10250.0 −0.346765
$$957$$ −1087.95 −0.0367485
$$958$$ −51613.4 −1.74066
$$959$$ −2076.93 −0.0699348
$$960$$ −3338.86 −0.112251
$$961$$ −11926.2 −0.400328
$$962$$ −28773.1 −0.964327
$$963$$ 27065.7 0.905692
$$964$$ 2577.19 0.0861056
$$965$$ 18662.7 0.622562
$$966$$ −3687.75 −0.122827
$$967$$ −19279.7 −0.641151 −0.320576 0.947223i $$-0.603876\pi$$
−0.320576 + 0.947223i $$0.603876\pi$$
$$968$$ 11246.3 0.373419
$$969$$ 0 0
$$970$$ 21317.7 0.705639
$$971$$ −53776.3 −1.77731 −0.888653 0.458581i $$-0.848358\pi$$
−0.888653 + 0.458581i $$0.848358\pi$$
$$972$$ −6858.31 −0.226317
$$973$$ 7053.30 0.232393
$$974$$ −12946.9 −0.425918
$$975$$ −11886.5 −0.390433
$$976$$ 62028.6 2.03431
$$977$$ 54738.2 1.79246 0.896228 0.443594i $$-0.146297\pi$$
0.896228 + 0.443594i $$0.146297\pi$$
$$978$$ 14302.8 0.467641
$$979$$ −5233.25 −0.170843
$$980$$ −3258.30 −0.106207
$$981$$ 19322.7 0.628876
$$982$$ −2768.27 −0.0899581
$$983$$ 3405.32 0.110491 0.0552456 0.998473i $$-0.482406\pi$$
0.0552456 + 0.998473i $$0.482406\pi$$
$$984$$ 9961.19 0.322715
$$985$$ 2262.86 0.0731988
$$986$$ 0 0
$$987$$ 1827.20 0.0589264
$$988$$ −3856.20 −0.124172
$$989$$ 66812.1 2.14813
$$990$$ −9908.60 −0.318097
$$991$$ 29925.7 0.959256 0.479628 0.877472i $$-0.340772\pi$$
0.479628 + 0.877472i $$0.340772\pi$$
$$992$$ 11499.3 0.368047
$$993$$ 178.911 0.00571759
$$994$$ −5525.07 −0.176302
$$995$$ −17427.3 −0.555259
$$996$$ −1406.98 −0.0447608
$$997$$ 32714.5 1.03920 0.519598 0.854411i $$-0.326082\pi$$
0.519598 + 0.854411i $$0.326082\pi$$
$$998$$ −43572.2 −1.38202
$$999$$ 15221.8 0.482080
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 289.4.a.g.1.11 12
17.3 odd 16 17.4.d.a.9.3 yes 12
17.4 even 4 289.4.b.e.288.2 12
17.6 odd 16 17.4.d.a.2.3 12
17.13 even 4 289.4.b.e.288.1 12
17.16 even 2 inner 289.4.a.g.1.12 12
51.20 even 16 153.4.l.a.145.1 12
51.23 even 16 153.4.l.a.19.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.3 12 17.6 odd 16
17.4.d.a.9.3 yes 12 17.3 odd 16
153.4.l.a.19.1 12 51.23 even 16
153.4.l.a.145.1 12 51.20 even 16
289.4.a.g.1.11 12 1.1 even 1 trivial
289.4.a.g.1.12 12 17.16 even 2 inner
289.4.b.e.288.1 12 17.13 even 4
289.4.b.e.288.2 12 17.4 even 4