# Properties

 Label 289.4.a.g.1.12 Level $289$ Weight $4$ Character 289.1 Self dual yes Analytic conductor $17.052$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Learn more

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 1.12 Root $$3.38755$$ 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 $$-0.438626\pi$$
0.191621 + 0.981469i $$0.438626\pi$$
$$4$$ 1.94089 0.242611
$$5$$ −5.00761 −0.447894 −0.223947 0.974601i $$-0.571894\pi$$
−0.223947 + 0.974601i $$0.571894\pi$$
$$6$$ 6.27866 0.427209
$$7$$ 2.78516 0.150385 0.0751924 0.997169i $$-0.476043\pi$$
0.0751924 + 0.997169i $$0.476043\pi$$
$$8$$ −19.1039 −0.844280
$$9$$ −23.0344 −0.853126
$$10$$ −15.7886 −0.499279
$$11$$ −27.2453 −0.746798 −0.373399 0.927671i $$-0.621808\pi$$
−0.373399 + 0.927671i $$0.621808\pi$$
$$12$$ 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 $$-0.0948635\pi$$
0.955919 + 0.293630i $$0.0948635\pi$$
$$24$$ −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 $$-0.520449\pi$$
−0.0641998 + 0.997937i $$0.520449\pi$$
$$30$$ −31.4411 −0.191344
$$31$$ 133.659 0.774385 0.387192 0.921999i $$-0.373445\pi$$
0.387192 + 0.921999i $$0.373445\pi$$
$$32$$ −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 $$-0.610224\pi$$
−0.339400 + 0.940642i $$0.610224\pi$$
$$38$$ 104.868 0.447678
$$39$$ −118.956 −0.488414
$$40$$ 95.6647 0.378148
$$41$$ −261.840 −0.997377 −0.498689 0.866781i $$-0.666185\pi$$
−0.498689 + 0.866781i $$0.666185\pi$$
$$42$$ 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 $$-0.170921\pi$$
0.859265 + 0.511531i $$0.170921\pi$$
$$62$$ 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 $$-0.676250\pi$$
−0.525844 + 0.850581i $$0.676250\pi$$
$$72$$ 440.046 0.720277
$$73$$ −496.481 −0.796010 −0.398005 0.917383i $$-0.630297\pi$$
−0.398005 + 0.917383i $$0.630297\pi$$
$$74$$ −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 $$-0.479580\pi$$
0.0641059 + 0.997943i $$0.479580\pi$$
$$80$$ 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 $$-0.749798\pi$$
−0.706659 + 0.707554i $$0.749798\pi$$
$$98$$ −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 $$-0.321889\pi$$
0.530808 + 0.847492i $$0.321889\pi$$
$$108$$ −193.386 −0.172301
$$109$$ 838.865 0.737144 0.368572 0.929599i $$-0.379847\pi$$
0.368572 + 0.929599i $$0.379847\pi$$
$$110$$ 430.165 0.372860
$$111$$ −304.228 −0.260144
$$112$$ −211.004 −0.178018
$$113$$ −2026.81 −1.68731 −0.843655 0.536885i $$-0.819601\pi$$
−0.843655 + 0.536885i $$0.819601\pi$$
$$114$$ 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 $$-0.837217\pi$$
−0.872061 + 0.489397i $$0.837217\pi$$
$$132$$ −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 $$-0.218925\pi$$
0.772661 + 0.634818i $$0.218925\pi$$
$$140$$ −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 $$-0.315647\pi$$
0.547322 + 0.836922i $$0.315647\pi$$
$$164$$ −508.202 −0.241975
$$165$$ 271.692 0.128189
$$166$$ 1147.74 0.536638
$$167$$ 1652.12 0.765538 0.382769 0.923844i $$-0.374971\pi$$
0.382769 + 0.923844i $$0.374971\pi$$
$$168$$ −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 $$-0.661471\pi$$
−0.485797 + 0.874071i $$0.661471\pi$$
$$174$$ −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 $$-0.471433\pi$$
0.0896262 + 0.995975i $$0.471433\pi$$
$$182$$ −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 $$-0.744590\pi$$
−0.694988 + 0.719021i $$0.744590\pi$$
$$194$$ −4257.06 −1.57546
$$195$$ 595.683 0.218758
$$196$$ −650.669 −0.237124
$$197$$ −451.885 −0.163429 −0.0817144 0.996656i $$-0.526040\pi$$
−0.0817144 + 0.996656i $$0.526040\pi$$
$$198$$ 1978.71 0.710206
$$199$$ 3480.16 1.23971 0.619855 0.784716i $$-0.287191\pi$$
0.619855 + 0.784716i $$0.287191\pi$$
$$200$$ 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 $$-0.404127\pi$$
0.296661 + 0.954983i $$0.404127\pi$$
$$212$$ −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 $$-0.384536\pi$$
0.354838 + 0.934928i $$0.384536\pi$$
$$228$$ 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 $$-0.447327\pi$$
0.164723 + 0.986340i $$0.447327\pi$$
$$234$$ 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 $$-0.556787\pi$$
−0.177456 + 0.984129i $$0.556787\pi$$
$$242$$ −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 $$-0.507609\pi$$
−0.0239009 + 0.999714i $$0.507609\pi$$
$$270$$ 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 $$-0.667332\pi$$
−0.501810 + 0.864978i $$0.667332\pi$$
$$278$$ 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 $$-0.582388\pi$$
−0.255949 + 0.966690i $$0.582388\pi$$
$$284$$ −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 $$-0.480909\pi$$
0.0599387 + 0.998202i $$0.480909\pi$$
$$312$$ 2272.51 0.412358
$$313$$ 6073.66 1.09682 0.548408 0.836211i $$-0.315234\pi$$
0.548408 + 0.836211i $$0.315234\pi$$
$$314$$ −5069.34 −0.911080
$$315$$ 321.261 0.0574635
$$316$$ 174.731 0.0311056
$$317$$ −5753.26 −1.01935 −0.509677 0.860366i $$-0.670235\pi$$
−0.509677 + 0.860366i $$0.670235\pi$$
$$318$$ −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 $$-0.478185\pi$$
0.0684804 + 0.997652i $$0.478185\pi$$
$$338$$ 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 $$-0.514540\pi$$
−0.0456630 + 0.998957i $$0.514540\pi$$
$$348$$ −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 $$-0.423242\pi$$
0.238813 + 0.971066i $$0.423242\pi$$
$$368$$ −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 $$-0.357310\pi$$
0.433411 + 0.901196i $$0.357310\pi$$
$$380$$ −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 $$-0.649145\pi$$
−0.451594 + 0.892223i $$0.649145\pi$$
$$398$$ 10972.7 1.38194
$$399$$ 184.474 0.0231459
$$400$$ 7570.24 0.946280
$$401$$ 6364.52 0.792591 0.396296 0.918123i $$-0.370296\pi$$
0.396296 + 0.918123i $$0.370296\pi$$
$$402$$ 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 $$-0.259874\pi$$
0.684835 + 0.728698i $$0.259874\pi$$
$$420$$ −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 $$-0.555408\pi$$
−0.173192 + 0.984888i $$0.555408\pi$$
$$432$$ 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 $$-0.892547\pi$$
−0.943561 + 0.331199i $$0.892547\pi$$
$$440$$ −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 $$-0.661215\pi$$
−0.485094 + 0.874462i $$0.661215\pi$$
$$450$$ 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 $$-0.214833\pi$$
0.780758 + 0.624834i $$0.214833\pi$$
$$480$$ 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 $$-0.438813\pi$$
0.191042 + 0.981582i $$0.438813\pi$$
$$488$$ −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 $$-0.287176\pi$$
0.619892 + 0.784687i $$0.287176\pi$$
$$500$$ 2186.08 0.195529
$$501$$ 3290.00 0.293386
$$502$$ −13495.7 −1.19989
$$503$$ −4721.43 −0.418525 −0.209263 0.977859i $$-0.567106\pi$$
−0.209263 + 0.977859i $$0.567106\pi$$
$$504$$ 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 $$-0.713027\pi$$
−0.620394 + 0.784290i $$0.713027\pi$$
$$522$$ 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 $$-0.132161\pi$$
0.915037 + 0.403369i $$0.132161\pi$$
$$542$$ −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 $$-0.467076\pi$$
0.103249 + 0.994656i $$0.467076\pi$$
$$548$$ 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 $$-0.649854\pi$$
−0.453583 + 0.891214i $$0.649854\pi$$
$$572$$ 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 $$-0.514392\pi$$
−0.0451975 + 0.998978i $$0.514392\pi$$
$$602$$ −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 $$-0.653527\pi$$
−0.463834 + 0.885922i $$0.653527\pi$$
$$608$$ −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 $$-0.664165\pi$$
−0.493177 + 0.869929i $$0.664165\pi$$
$$618$$ 6206.55 0.403987
$$619$$ −22412.1 −1.45528 −0.727639 0.685960i $$-0.759382\pi$$
−0.727639 + 0.685960i $$0.759382\pi$$
$$620$$ −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 $$-0.249219\pi$$
0.708839 + 0.705370i $$0.249219\pi$$
$$642$$ 7377.52 0.453532
$$643$$ −5689.90 −0.348970 −0.174485 0.984660i $$-0.555826\pi$$
−0.174485 + 0.984660i $$0.555826\pi$$
$$644$$ 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 $$-0.395460\pi$$
0.322550 + 0.946552i $$0.395460\pi$$
$$654$$ 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 $$-0.480748\pi$$
0.0604451 + 0.998172i $$0.480748\pi$$
$$674$$ 2671.49 0.152674
$$675$$ 9956.17 0.567723
$$676$$ 2661.52 0.151429
$$677$$ −11944.5 −0.678087 −0.339043 0.940771i $$-0.610103\pi$$
−0.339043 + 0.940771i $$0.610103\pi$$
$$678$$ −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 $$-0.679547\pi$$
−0.534626 + 0.845089i $$0.679547\pi$$
$$684$$ −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 $$-0.215491\pi$$
0.779465 + 0.626445i $$0.215491\pi$$
$$692$$ −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 $$-0.337008\pi$$
0.489968 + 0.871740i $$0.337008\pi$$
$$710$$ 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 $$-0.304647\pi$$
0.575912 + 0.817511i $$0.304647\pi$$
$$720$$ −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 $$-0.600580\pi$$
−0.310751 + 0.950491i $$0.600580\pi$$
$$744$$ −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 $$-0.520606\pi$$
−0.0646914 + 0.997905i $$0.520606\pi$$
$$752$$ −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 $$-0.215941\pi$$
0.778578 + 0.627547i $$0.215941\pi$$
$$788$$ −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 $$-0.449871\pi$$
0.156835 + 0.987625i $$0.449871\pi$$
$$810$$ −6686.66 −0.290056
$$811$$ −36564.1 −1.58316 −0.791579 0.611067i $$-0.790741\pi$$
−0.791579 + 0.611067i $$0.790741\pi$$
$$812$$ −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 $$-0.663677\pi$$
−0.491845 + 0.870683i $$0.663677\pi$$
$$822$$ 4682.07 0.198669
$$823$$ 41287.9 1.74873 0.874365 0.485269i $$-0.161278\pi$$
0.874365 + 0.485269i $$0.161278\pi$$
$$824$$ −18884.5 −0.798388
$$825$$ 5421.46 0.228789
$$826$$ −480.683 −0.0202483
$$827$$ 10008.0 0.420811 0.210405 0.977614i $$-0.432522\pi$$
0.210405 + 0.977614i $$0.432522\pi$$
$$828$$ −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 $$-0.594614\pi$$
−0.292882 + 0.956148i $$0.594614\pi$$
$$840$$ 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 $$-0.312136\pi$$
0.556520 + 0.830834i $$0.312136\pi$$
$$854$$ 7189.78 0.288090
$$855$$ 3836.51 0.153457
$$856$$ −22447.3 −0.896301
$$857$$ 31280.7 1.24682 0.623411 0.781894i $$-0.285746\pi$$
0.623411 + 0.781894i $$0.285746\pi$$
$$858$$ 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 $$-0.489075\pi$$
0.0343163 + 0.999411i $$0.489075\pi$$
$$878$$ −54728.0 −2.10362
$$879$$ −6572.17 −0.252189
$$880$$ −10336.3 −0.395949
$$881$$ 17232.6 0.659001 0.329500 0.944155i $$-0.393120\pi$$
0.329500 + 0.944155i $$0.393120\pi$$
$$882$$ 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 $$-0.404141\pi$$
0.296619 + 0.954996i $$0.404141\pi$$
$$888$$ 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 $$-0.600162\pi$$
−0.309500 + 0.950900i $$0.600162\pi$$
$$908$$ 4710.86 0.172175
$$909$$ 7002.92 0.255525
$$910$$ 2626.78 0.0956890
$$911$$ −36100.6 −1.31292 −0.656458 0.754362i $$-0.727946\pi$$
−0.656458 + 0.754362i $$0.727946\pi$$
$$912$$ −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 $$-0.318093\pi$$
0.540875 + 0.841103i $$0.318093\pi$$
$$930$$ −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 $$-0.622634\pi$$
−0.375807 + 0.926698i $$0.622634\pi$$
$$942$$ −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 $$-0.412023\pi$$
0.272882 + 0.962048i $$0.412023\pi$$
$$948$$ 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 $$-0.517594\pi$$
−0.0552456 + 0.998473i $$0.517594\pi$$
$$984$$ 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 $$-0.659228\pi$$
−0.479628 + 0.877472i $$0.659228\pi$$
$$992$$ −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 $$-0.673918\pi$$
−0.519598 + 0.854411i $$0.673918\pi$$
$$998$$ 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.1.12 12
17.4 even 4 289.4.b.e.288.1 12
17.11 odd 16 17.4.d.a.2.3 12
17.13 even 4 289.4.b.e.288.2 12
17.14 odd 16 17.4.d.a.9.3 yes 12
17.16 even 2 inner 289.4.a.g.1.11 12
51.11 even 16 153.4.l.a.19.1 12
51.14 even 16 153.4.l.a.145.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.3 12 17.11 odd 16
17.4.d.a.9.3 yes 12 17.14 odd 16
153.4.l.a.19.1 12 51.11 even 16
153.4.l.a.145.1 12 51.14 even 16
289.4.a.g.1.11 12 17.16 even 2 inner
289.4.a.g.1.12 12 1.1 even 1 trivial
289.4.b.e.288.1 12 17.4 even 4
289.4.b.e.288.2 12 17.13 even 4