# Properties

 Label 33.4.a.c.1.1 Level $33$ Weight $4$ Character 33.1 Self dual yes Analytic conductor $1.947$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.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: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.42443$$ of defining polynomial Character $$\chi$$ $$=$$ 33.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} +13.2733 q^{6} +31.6977 q^{7} -15.8199 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} +13.2733 q^{6} +31.6977 q^{7} -15.8199 q^{8} +9.00000 q^{9} -12.6046 q^{10} -11.0000 q^{11} -34.7267 q^{12} +5.15114 q^{13} -140.244 q^{14} -8.54657 q^{15} -22.6107 q^{16} +121.942 q^{17} -39.8199 q^{18} +34.8489 q^{19} +32.9772 q^{20} -95.0931 q^{21} +48.6687 q^{22} +116.244 q^{23} +47.4596 q^{24} -116.884 q^{25} -22.7909 q^{26} -27.0000 q^{27} +366.919 q^{28} -69.4534 q^{29} +37.8137 q^{30} +140.605 q^{31} +226.598 q^{32} +33.0000 q^{33} -539.524 q^{34} +90.3023 q^{35} +104.180 q^{36} -420.070 q^{37} -154.186 q^{38} -15.4534 q^{39} -45.0685 q^{40} -322.058 q^{41} +420.733 q^{42} +321.035 q^{43} -127.331 q^{44} +25.6397 q^{45} -514.315 q^{46} -231.408 q^{47} +67.8322 q^{48} +661.745 q^{49} +517.145 q^{50} -365.826 q^{51} +59.6274 q^{52} +4.91916 q^{53} +119.460 q^{54} -31.3374 q^{55} -501.453 q^{56} -104.547 q^{57} +307.292 q^{58} +406.443 q^{59} -98.9315 q^{60} -556.431 q^{61} -622.095 q^{62} +285.279 q^{63} -821.683 q^{64} +14.6749 q^{65} -146.006 q^{66} +84.7452 q^{67} +1411.55 q^{68} -348.733 q^{69} -399.536 q^{70} +49.0808 q^{71} -142.379 q^{72} +785.884 q^{73} +1858.57 q^{74} +350.652 q^{75} +403.395 q^{76} -348.675 q^{77} +68.3726 q^{78} -383.118 q^{79} -64.4147 q^{80} +81.0000 q^{81} +1424.92 q^{82} -930.211 q^{83} -1100.76 q^{84} +347.395 q^{85} -1420.40 q^{86} +208.360 q^{87} +174.018 q^{88} -732.559 q^{89} -113.441 q^{90} +163.279 q^{91} +1345.59 q^{92} -421.814 q^{93} +1023.85 q^{94} +99.2794 q^{95} -679.795 q^{96} -1171.49 q^{97} -2927.84 q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9 $$2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19} - 328 q^{20} - 72 q^{21} - 11 q^{22} + 134 q^{23} - 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} + 202 q^{28} - 198 q^{29} + 312 q^{30} + 360 q^{31} + 857 q^{32} + 66 q^{33} - 626 q^{34} + 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} - 90 q^{39} - 1272 q^{40} - 782 q^{41} + 546 q^{42} + 386 q^{43} - 363 q^{44} - 126 q^{45} - 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} + 1379 q^{50} - 318 q^{51} + 592 q^{52} - 522 q^{53} - 27 q^{54} + 154 q^{55} - 1062 q^{56} - 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} - 778 q^{61} + 568 q^{62} + 216 q^{63} + 809 q^{64} - 404 q^{65} + 33 q^{66} - 776 q^{67} + 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} + 513 q^{72} + 1296 q^{73} + 2358 q^{74} - 126 q^{75} + 728 q^{76} - 264 q^{77} - 336 q^{78} + 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} - 324 q^{83} - 606 q^{84} + 616 q^{85} - 1068 q^{86} + 594 q^{87} - 627 q^{88} - 756 q^{89} - 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} + 3722 q^{94} - 156 q^{95} - 2571 q^{96} - 452 q^{97} - 4467 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 - 6 * q^3 + 33 * q^4 - 14 * q^5 - 3 * q^6 + 24 * q^7 + 57 * q^8 + 18 * q^9 - 104 * q^10 - 22 * q^11 - 99 * q^12 + 30 * q^13 - 182 * q^14 + 42 * q^15 + 201 * q^16 + 106 * q^17 + 9 * q^18 + 50 * q^19 - 328 * q^20 - 72 * q^21 - 11 * q^22 + 134 * q^23 - 171 * q^24 + 42 * q^25 + 112 * q^26 - 54 * q^27 + 202 * q^28 - 198 * q^29 + 312 * q^30 + 360 * q^31 + 857 * q^32 + 66 * q^33 - 626 * q^34 + 220 * q^35 + 297 * q^36 - 328 * q^37 - 72 * q^38 - 90 * q^39 - 1272 * q^40 - 782 * q^41 + 546 * q^42 + 386 * q^43 - 363 * q^44 - 126 * q^45 - 418 * q^46 + 266 * q^47 - 603 * q^48 + 378 * q^49 + 1379 * q^50 - 318 * q^51 + 592 * q^52 - 522 * q^53 - 27 * q^54 + 154 * q^55 - 1062 * q^56 - 150 * q^57 - 390 * q^58 - 172 * q^59 + 984 * q^60 - 778 * q^61 + 568 * q^62 + 216 * q^63 + 809 * q^64 - 404 * q^65 + 33 * q^66 - 776 * q^67 + 1070 * q^68 - 402 * q^69 + 304 * q^70 + 630 * q^71 + 513 * q^72 + 1296 * q^73 + 2358 * q^74 - 126 * q^75 + 728 * q^76 - 264 * q^77 - 336 * q^78 + 652 * q^79 - 3832 * q^80 + 162 * q^81 - 1070 * q^82 - 324 * q^83 - 606 * q^84 + 616 * q^85 - 1068 * q^86 + 594 * q^87 - 627 * q^88 - 756 * q^89 - 936 * q^90 - 28 * q^91 + 1726 * q^92 - 1080 * q^93 + 3722 * q^94 - 156 * q^95 - 2571 * q^96 - 452 * q^97 - 4467 * q^98 - 198 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.42443 −1.56427 −0.782136 0.623108i $$-0.785870\pi$$
−0.782136 + 0.623108i $$0.785870\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 11.5756 1.44695
$$5$$ 2.84886 0.254810 0.127405 0.991851i $$-0.459335\pi$$
0.127405 + 0.991851i $$0.459335\pi$$
$$6$$ 13.2733 0.903133
$$7$$ 31.6977 1.71152 0.855758 0.517377i $$-0.173091\pi$$
0.855758 + 0.517377i $$0.173091\pi$$
$$8$$ −15.8199 −0.699146
$$9$$ 9.00000 0.333333
$$10$$ −12.6046 −0.398591
$$11$$ −11.0000 −0.301511
$$12$$ −34.7267 −0.835395
$$13$$ 5.15114 0.109898 0.0549488 0.998489i $$-0.482500\pi$$
0.0549488 + 0.998489i $$0.482500\pi$$
$$14$$ −140.244 −2.67728
$$15$$ −8.54657 −0.147114
$$16$$ −22.6107 −0.353293
$$17$$ 121.942 1.73972 0.869861 0.493297i $$-0.164208\pi$$
0.869861 + 0.493297i $$0.164208\pi$$
$$18$$ −39.8199 −0.521424
$$19$$ 34.8489 0.420783 0.210391 0.977617i $$-0.432526\pi$$
0.210391 + 0.977617i $$0.432526\pi$$
$$20$$ 32.9772 0.368696
$$21$$ −95.0931 −0.988144
$$22$$ 48.6687 0.471646
$$23$$ 116.244 1.05385 0.526926 0.849911i $$-0.323344\pi$$
0.526926 + 0.849911i $$0.323344\pi$$
$$24$$ 47.4596 0.403652
$$25$$ −116.884 −0.935072
$$26$$ −22.7909 −0.171910
$$27$$ −27.0000 −0.192450
$$28$$ 366.919 2.47647
$$29$$ −69.4534 −0.444730 −0.222365 0.974963i $$-0.571378\pi$$
−0.222365 + 0.974963i $$0.571378\pi$$
$$30$$ 37.8137 0.230127
$$31$$ 140.605 0.814623 0.407312 0.913289i $$-0.366466\pi$$
0.407312 + 0.913289i $$0.366466\pi$$
$$32$$ 226.598 1.25179
$$33$$ 33.0000 0.174078
$$34$$ −539.524 −2.72140
$$35$$ 90.3023 0.436111
$$36$$ 104.180 0.482315
$$37$$ −420.070 −1.86646 −0.933232 0.359276i $$-0.883024\pi$$
−0.933232 + 0.359276i $$0.883024\pi$$
$$38$$ −154.186 −0.658219
$$39$$ −15.4534 −0.0634495
$$40$$ −45.0685 −0.178149
$$41$$ −322.058 −1.22676 −0.613378 0.789789i $$-0.710190\pi$$
−0.613378 + 0.789789i $$0.710190\pi$$
$$42$$ 420.733 1.54573
$$43$$ 321.035 1.13854 0.569272 0.822149i $$-0.307225\pi$$
0.569272 + 0.822149i $$0.307225\pi$$
$$44$$ −127.331 −0.436271
$$45$$ 25.6397 0.0849365
$$46$$ −514.315 −1.64851
$$47$$ −231.408 −0.718176 −0.359088 0.933304i $$-0.616912\pi$$
−0.359088 + 0.933304i $$0.616912\pi$$
$$48$$ 67.8322 0.203974
$$49$$ 661.745 1.92929
$$50$$ 517.145 1.46271
$$51$$ −365.826 −1.00443
$$52$$ 59.6274 0.159016
$$53$$ 4.91916 0.0127490 0.00637452 0.999980i $$-0.497971\pi$$
0.00637452 + 0.999980i $$0.497971\pi$$
$$54$$ 119.460 0.301044
$$55$$ −31.3374 −0.0768280
$$56$$ −501.453 −1.19660
$$57$$ −104.547 −0.242939
$$58$$ 307.292 0.695679
$$59$$ 406.443 0.896854 0.448427 0.893820i $$-0.351984\pi$$
0.448427 + 0.893820i $$0.351984\pi$$
$$60$$ −98.9315 −0.212867
$$61$$ −556.431 −1.16793 −0.583964 0.811779i $$-0.698499\pi$$
−0.583964 + 0.811779i $$0.698499\pi$$
$$62$$ −622.095 −1.27429
$$63$$ 285.279 0.570505
$$64$$ −821.683 −1.60485
$$65$$ 14.6749 0.0280030
$$66$$ −146.006 −0.272305
$$67$$ 84.7452 0.154526 0.0772632 0.997011i $$-0.475382\pi$$
0.0772632 + 0.997011i $$0.475382\pi$$
$$68$$ 1411.55 2.51728
$$69$$ −348.733 −0.608442
$$70$$ −399.536 −0.682196
$$71$$ 49.0808 0.0820398 0.0410199 0.999158i $$-0.486939\pi$$
0.0410199 + 0.999158i $$0.486939\pi$$
$$72$$ −142.379 −0.233049
$$73$$ 785.884 1.26001 0.630005 0.776591i $$-0.283053\pi$$
0.630005 + 0.776591i $$0.283053\pi$$
$$74$$ 1858.57 2.91966
$$75$$ 350.652 0.539864
$$76$$ 403.395 0.608850
$$77$$ −348.675 −0.516041
$$78$$ 68.3726 0.0992522
$$79$$ −383.118 −0.545622 −0.272811 0.962068i $$-0.587953\pi$$
−0.272811 + 0.962068i $$0.587953\pi$$
$$80$$ −64.4147 −0.0900223
$$81$$ 81.0000 0.111111
$$82$$ 1424.92 1.91898
$$83$$ −930.211 −1.23017 −0.615084 0.788462i $$-0.710878\pi$$
−0.615084 + 0.788462i $$0.710878\pi$$
$$84$$ −1100.76 −1.42979
$$85$$ 347.395 0.443298
$$86$$ −1420.40 −1.78099
$$87$$ 208.360 0.256765
$$88$$ 174.018 0.210800
$$89$$ −732.559 −0.872484 −0.436242 0.899829i $$-0.643691\pi$$
−0.436242 + 0.899829i $$0.643691\pi$$
$$90$$ −113.441 −0.132864
$$91$$ 163.279 0.188092
$$92$$ 1345.59 1.52487
$$93$$ −421.814 −0.470323
$$94$$ 1023.85 1.12342
$$95$$ 99.2794 0.107220
$$96$$ −679.795 −0.722722
$$97$$ −1171.49 −1.22626 −0.613128 0.789984i $$-0.710089\pi$$
−0.613128 + 0.789984i $$0.710089\pi$$
$$98$$ −2927.84 −3.01793
$$99$$ −99.0000 −0.100504
$$100$$ −1353.00 −1.35300
$$101$$ −1221.27 −1.20318 −0.601589 0.798806i $$-0.705465\pi$$
−0.601589 + 0.798806i $$0.705465\pi$$
$$102$$ 1618.57 1.57120
$$103$$ 516.745 0.494334 0.247167 0.968973i $$-0.420500\pi$$
0.247167 + 0.968973i $$0.420500\pi$$
$$104$$ −81.4903 −0.0768345
$$105$$ −270.907 −0.251789
$$106$$ −21.7645 −0.0199430
$$107$$ −152.025 −0.137353 −0.0686765 0.997639i $$-0.521878\pi$$
−0.0686765 + 0.997639i $$0.521878\pi$$
$$108$$ −312.540 −0.278465
$$109$$ 2170.32 1.90714 0.953572 0.301164i $$-0.0973752\pi$$
0.953572 + 0.301164i $$0.0973752\pi$$
$$110$$ 138.650 0.120180
$$111$$ 1260.21 1.07760
$$112$$ −716.708 −0.604666
$$113$$ −646.397 −0.538123 −0.269062 0.963123i $$-0.586714\pi$$
−0.269062 + 0.963123i $$0.586714\pi$$
$$114$$ 462.559 0.380023
$$115$$ 331.163 0.268532
$$116$$ −803.963 −0.643501
$$117$$ 46.3603 0.0366326
$$118$$ −1798.28 −1.40292
$$119$$ 3865.28 2.97756
$$120$$ 135.206 0.102854
$$121$$ 121.000 0.0909091
$$122$$ 2461.89 1.82696
$$123$$ 966.174 0.708268
$$124$$ 1627.58 1.17872
$$125$$ −689.093 −0.493075
$$126$$ −1262.20 −0.892425
$$127$$ −993.304 −0.694027 −0.347014 0.937860i $$-0.612804\pi$$
−0.347014 + 0.937860i $$0.612804\pi$$
$$128$$ 1822.69 1.25863
$$129$$ −963.105 −0.657339
$$130$$ −64.9279 −0.0438043
$$131$$ 385.814 0.257318 0.128659 0.991689i $$-0.458933\pi$$
0.128659 + 0.991689i $$0.458933\pi$$
$$132$$ 381.994 0.251881
$$133$$ 1104.63 0.720177
$$134$$ −374.949 −0.241721
$$135$$ −76.9192 −0.0490381
$$136$$ −1929.11 −1.21632
$$137$$ 884.840 0.551803 0.275901 0.961186i $$-0.411024\pi$$
0.275901 + 0.961186i $$0.411024\pi$$
$$138$$ 1542.94 0.951769
$$139$$ −1091.94 −0.666312 −0.333156 0.942872i $$-0.608114\pi$$
−0.333156 + 0.942872i $$0.608114\pi$$
$$140$$ 1045.30 0.631029
$$141$$ 694.223 0.414639
$$142$$ −217.155 −0.128333
$$143$$ −56.6626 −0.0331354
$$144$$ −203.497 −0.117764
$$145$$ −197.863 −0.113322
$$146$$ −3477.09 −1.97100
$$147$$ −1985.24 −1.11387
$$148$$ −4862.55 −2.70067
$$149$$ 297.014 0.163304 0.0816522 0.996661i $$-0.473980\pi$$
0.0816522 + 0.996661i $$0.473980\pi$$
$$150$$ −1551.43 −0.844494
$$151$$ −1887.86 −1.01743 −0.508716 0.860935i $$-0.669880\pi$$
−0.508716 + 0.860935i $$0.669880\pi$$
$$152$$ −551.304 −0.294189
$$153$$ 1097.48 0.579907
$$154$$ 1542.69 0.807229
$$155$$ 400.562 0.207574
$$156$$ −178.882 −0.0918080
$$157$$ −56.5343 −0.0287384 −0.0143692 0.999897i $$-0.504574\pi$$
−0.0143692 + 0.999897i $$0.504574\pi$$
$$158$$ 1695.08 0.853501
$$159$$ −14.7575 −0.00736066
$$160$$ 645.547 0.318968
$$161$$ 3684.68 1.80369
$$162$$ −358.379 −0.173808
$$163$$ −49.2338 −0.0236582 −0.0118291 0.999930i $$-0.503765\pi$$
−0.0118291 + 0.999930i $$0.503765\pi$$
$$164$$ −3728.01 −1.77505
$$165$$ 94.0123 0.0443567
$$166$$ 4115.65 1.92432
$$167$$ 2068.75 0.958589 0.479294 0.877654i $$-0.340893\pi$$
0.479294 + 0.877654i $$0.340893\pi$$
$$168$$ 1504.36 0.690857
$$169$$ −2170.47 −0.987923
$$170$$ −1537.03 −0.693438
$$171$$ 313.640 0.140261
$$172$$ 3716.17 1.64741
$$173$$ −604.012 −0.265446 −0.132723 0.991153i $$-0.542372\pi$$
−0.132723 + 0.991153i $$0.542372\pi$$
$$174$$ −921.875 −0.401650
$$175$$ −3704.96 −1.60039
$$176$$ 248.718 0.106522
$$177$$ −1219.33 −0.517799
$$178$$ 3241.15 1.36480
$$179$$ −2132.02 −0.890251 −0.445126 0.895468i $$-0.646841\pi$$
−0.445126 + 0.895468i $$0.646841\pi$$
$$180$$ 296.794 0.122899
$$181$$ −589.371 −0.242031 −0.121015 0.992651i $$-0.538615\pi$$
−0.121015 + 0.992651i $$0.538615\pi$$
$$182$$ −722.418 −0.294226
$$183$$ 1669.29 0.674304
$$184$$ −1838.97 −0.736796
$$185$$ −1196.72 −0.475593
$$186$$ 1866.28 0.735713
$$187$$ −1341.36 −0.524546
$$188$$ −2678.68 −1.03916
$$189$$ −855.838 −0.329381
$$190$$ −439.255 −0.167720
$$191$$ −2160.90 −0.818624 −0.409312 0.912395i $$-0.634231\pi$$
−0.409312 + 0.912395i $$0.634231\pi$$
$$192$$ 2465.05 0.926560
$$193$$ −1490.91 −0.556052 −0.278026 0.960574i $$-0.589680\pi$$
−0.278026 + 0.960574i $$0.589680\pi$$
$$194$$ 5183.18 1.91820
$$195$$ −44.0246 −0.0161675
$$196$$ 7660.08 2.79157
$$197$$ −230.529 −0.0833732 −0.0416866 0.999131i $$-0.513273\pi$$
−0.0416866 + 0.999131i $$0.513273\pi$$
$$198$$ 438.018 0.157215
$$199$$ 22.4007 0.00797963 0.00398982 0.999992i $$-0.498730\pi$$
0.00398982 + 0.999992i $$0.498730\pi$$
$$200$$ 1849.09 0.653752
$$201$$ −254.236 −0.0892159
$$202$$ 5403.43 1.88210
$$203$$ −2201.51 −0.761163
$$204$$ −4234.65 −1.45336
$$205$$ −917.497 −0.312589
$$206$$ −2286.30 −0.773273
$$207$$ 1046.20 0.351284
$$208$$ −116.471 −0.0388260
$$209$$ −383.337 −0.126871
$$210$$ 1198.61 0.393866
$$211$$ −1051.64 −0.343117 −0.171558 0.985174i $$-0.554880\pi$$
−0.171558 + 0.985174i $$0.554880\pi$$
$$212$$ 56.9421 0.0184472
$$213$$ −147.243 −0.0473657
$$214$$ 672.622 0.214857
$$215$$ 914.583 0.290112
$$216$$ 427.136 0.134551
$$217$$ 4456.84 1.39424
$$218$$ −9602.42 −2.98329
$$219$$ −2357.65 −0.727467
$$220$$ −362.749 −0.111166
$$221$$ 628.141 0.191191
$$222$$ −5575.71 −1.68566
$$223$$ 3861.80 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$224$$ 7182.65 2.14246
$$225$$ −1051.96 −0.311691
$$226$$ 2859.94 0.841771
$$227$$ −872.721 −0.255174 −0.127587 0.991827i $$-0.540723\pi$$
−0.127587 + 0.991827i $$0.540723\pi$$
$$228$$ −1210.19 −0.351520
$$229$$ 1841.72 0.531459 0.265730 0.964048i $$-0.414387\pi$$
0.265730 + 0.964048i $$0.414387\pi$$
$$230$$ −1465.21 −0.420057
$$231$$ 1046.02 0.297937
$$232$$ 1098.74 0.310931
$$233$$ 3932.14 1.10559 0.552796 0.833317i $$-0.313561\pi$$
0.552796 + 0.833317i $$0.313561\pi$$
$$234$$ −205.118 −0.0573033
$$235$$ −659.248 −0.182998
$$236$$ 4704.81 1.29770
$$237$$ 1149.35 0.315015
$$238$$ −17101.7 −4.65772
$$239$$ 4772.10 1.29155 0.645777 0.763526i $$-0.276534\pi$$
0.645777 + 0.763526i $$0.276534\pi$$
$$240$$ 193.244 0.0519744
$$241$$ 3988.84 1.06616 0.533078 0.846066i $$-0.321035\pi$$
0.533078 + 0.846066i $$0.321035\pi$$
$$242$$ −535.356 −0.142207
$$243$$ −243.000 −0.0641500
$$244$$ −6441.00 −1.68993
$$245$$ 1885.22 0.491601
$$246$$ −4274.77 −1.10792
$$247$$ 179.511 0.0462431
$$248$$ −2224.34 −0.569540
$$249$$ 2790.63 0.710238
$$250$$ 3048.84 0.771303
$$251$$ −5474.22 −1.37661 −0.688306 0.725421i $$-0.741645\pi$$
−0.688306 + 0.725421i $$0.741645\pi$$
$$252$$ 3302.27 0.825491
$$253$$ −1278.69 −0.317749
$$254$$ 4394.80 1.08565
$$255$$ −1042.19 −0.255938
$$256$$ −1490.90 −0.363989
$$257$$ −6434.01 −1.56164 −0.780822 0.624754i $$-0.785199\pi$$
−0.780822 + 0.624754i $$0.785199\pi$$
$$258$$ 4261.19 1.02826
$$259$$ −13315.3 −3.19448
$$260$$ 169.870 0.0405188
$$261$$ −625.081 −0.148243
$$262$$ −1707.01 −0.402516
$$263$$ 7589.00 1.77931 0.889654 0.456636i $$-0.150946\pi$$
0.889654 + 0.456636i $$0.150946\pi$$
$$264$$ −522.055 −0.121706
$$265$$ 14.0140 0.00324858
$$266$$ −4887.35 −1.12655
$$267$$ 2197.68 0.503729
$$268$$ 980.974 0.223591
$$269$$ 478.178 0.108383 0.0541914 0.998531i $$-0.482742\pi$$
0.0541914 + 0.998531i $$0.482742\pi$$
$$270$$ 340.323 0.0767090
$$271$$ −122.323 −0.0274192 −0.0137096 0.999906i $$-0.504364\pi$$
−0.0137096 + 0.999906i $$0.504364\pi$$
$$272$$ −2757.20 −0.614631
$$273$$ −489.838 −0.108595
$$274$$ −3914.91 −0.863170
$$275$$ 1285.72 0.281935
$$276$$ −4036.78 −0.880383
$$277$$ 8199.41 1.77854 0.889269 0.457385i $$-0.151214\pi$$
0.889269 + 0.457385i $$0.151214\pi$$
$$278$$ 4831.22 1.04229
$$279$$ 1265.44 0.271541
$$280$$ −1428.57 −0.304905
$$281$$ 6943.79 1.47413 0.737067 0.675820i $$-0.236210\pi$$
0.737067 + 0.675820i $$0.236210\pi$$
$$282$$ −3071.54 −0.648609
$$283$$ 1035.14 0.217429 0.108715 0.994073i $$-0.465327\pi$$
0.108715 + 0.994073i $$0.465327\pi$$
$$284$$ 568.139 0.118707
$$285$$ −297.838 −0.0619032
$$286$$ 250.699 0.0518328
$$287$$ −10208.5 −2.09961
$$288$$ 2039.39 0.417264
$$289$$ 9956.85 2.02663
$$290$$ 875.430 0.177266
$$291$$ 3514.47 0.707979
$$292$$ 9097.06 1.82317
$$293$$ −6144.81 −1.22520 −0.612600 0.790393i $$-0.709876\pi$$
−0.612600 + 0.790393i $$0.709876\pi$$
$$294$$ 8783.53 1.74240
$$295$$ 1157.90 0.228527
$$296$$ 6645.45 1.30493
$$297$$ 297.000 0.0580259
$$298$$ −1314.12 −0.255452
$$299$$ 598.791 0.115816
$$300$$ 4059.00 0.781154
$$301$$ 10176.1 1.94864
$$302$$ 8352.72 1.59154
$$303$$ 3663.81 0.694655
$$304$$ −787.958 −0.148659
$$305$$ −1585.19 −0.297599
$$306$$ −4855.71 −0.907133
$$307$$ −2186.09 −0.406406 −0.203203 0.979137i $$-0.565135\pi$$
−0.203203 + 0.979137i $$0.565135\pi$$
$$308$$ −4036.11 −0.746684
$$309$$ −1550.24 −0.285404
$$310$$ −1772.26 −0.324702
$$311$$ −7484.83 −1.36471 −0.682357 0.731019i $$-0.739045\pi$$
−0.682357 + 0.731019i $$0.739045\pi$$
$$312$$ 244.471 0.0443604
$$313$$ −6833.33 −1.23400 −0.617001 0.786962i $$-0.711653\pi$$
−0.617001 + 0.786962i $$0.711653\pi$$
$$314$$ 250.132 0.0449546
$$315$$ 812.721 0.145370
$$316$$ −4434.81 −0.789485
$$317$$ 924.265 0.163760 0.0818800 0.996642i $$-0.473908\pi$$
0.0818800 + 0.996642i $$0.473908\pi$$
$$318$$ 65.2934 0.0115141
$$319$$ 763.988 0.134091
$$320$$ −2340.86 −0.408931
$$321$$ 456.074 0.0793008
$$322$$ −16302.6 −2.82145
$$323$$ 4249.54 0.732046
$$324$$ 937.621 0.160772
$$325$$ −602.086 −0.102762
$$326$$ 217.831 0.0370078
$$327$$ −6510.95 −1.10109
$$328$$ 5094.91 0.857681
$$329$$ −7335.10 −1.22917
$$330$$ −415.951 −0.0693859
$$331$$ −9820.46 −1.63076 −0.815380 0.578927i $$-0.803472\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$332$$ −10767.7 −1.77999
$$333$$ −3780.63 −0.622154
$$334$$ −9153.02 −1.49949
$$335$$ 241.427 0.0393748
$$336$$ 2150.12 0.349104
$$337$$ 600.808 0.0971161 0.0485580 0.998820i $$-0.484537\pi$$
0.0485580 + 0.998820i $$0.484537\pi$$
$$338$$ 9603.07 1.54538
$$339$$ 1939.19 0.310686
$$340$$ 4021.30 0.641428
$$341$$ −1546.65 −0.245618
$$342$$ −1387.68 −0.219406
$$343$$ 10103.5 1.59049
$$344$$ −5078.73 −0.796008
$$345$$ −993.490 −0.155037
$$346$$ 2672.41 0.415230
$$347$$ −3143.41 −0.486303 −0.243152 0.969988i $$-0.578181\pi$$
−0.243152 + 0.969988i $$0.578181\pi$$
$$348$$ 2411.89 0.371525
$$349$$ 720.663 0.110533 0.0552667 0.998472i $$-0.482399\pi$$
0.0552667 + 0.998472i $$0.482399\pi$$
$$350$$ 16392.3 2.50345
$$351$$ −139.081 −0.0211498
$$352$$ −2492.58 −0.377429
$$353$$ 1207.12 0.182007 0.0910034 0.995851i $$-0.470993\pi$$
0.0910034 + 0.995851i $$0.470993\pi$$
$$354$$ 5394.83 0.809978
$$355$$ 139.824 0.0209045
$$356$$ −8479.79 −1.26244
$$357$$ −11595.8 −1.71910
$$358$$ 9432.99 1.39260
$$359$$ 8748.31 1.28612 0.643062 0.765814i $$-0.277664\pi$$
0.643062 + 0.765814i $$0.277664\pi$$
$$360$$ −405.617 −0.0593830
$$361$$ −5644.56 −0.822942
$$362$$ 2607.63 0.378602
$$363$$ −363.000 −0.0524864
$$364$$ 1890.05 0.272158
$$365$$ 2238.87 0.321063
$$366$$ −7385.66 −1.05479
$$367$$ −6730.45 −0.957293 −0.478647 0.878008i $$-0.658872\pi$$
−0.478647 + 0.878008i $$0.658872\pi$$
$$368$$ −2628.37 −0.372318
$$369$$ −2898.52 −0.408919
$$370$$ 5294.81 0.743956
$$371$$ 155.926 0.0218202
$$372$$ −4882.73 −0.680532
$$373$$ −227.394 −0.0315657 −0.0157828 0.999875i $$-0.505024\pi$$
−0.0157828 + 0.999875i $$0.505024\pi$$
$$374$$ 5934.76 0.820533
$$375$$ 2067.28 0.284677
$$376$$ 3660.84 0.502110
$$377$$ −357.764 −0.0488748
$$378$$ 3786.60 0.515242
$$379$$ 11356.2 1.53913 0.769565 0.638568i $$-0.220473\pi$$
0.769565 + 0.638568i $$0.220473\pi$$
$$380$$ 1149.22 0.155141
$$381$$ 2979.91 0.400697
$$382$$ 9560.74 1.28055
$$383$$ 10753.6 1.43468 0.717338 0.696725i $$-0.245360\pi$$
0.717338 + 0.696725i $$0.245360\pi$$
$$384$$ −5468.07 −0.726670
$$385$$ −993.325 −0.131492
$$386$$ 6596.43 0.869817
$$387$$ 2889.32 0.379515
$$388$$ −13560.7 −1.77433
$$389$$ −11727.1 −1.52850 −0.764252 0.644918i $$-0.776891\pi$$
−0.764252 + 0.644918i $$0.776891\pi$$
$$390$$ 194.784 0.0252904
$$391$$ 14175.1 1.83341
$$392$$ −10468.7 −1.34885
$$393$$ −1157.44 −0.148563
$$394$$ 1019.96 0.130418
$$395$$ −1091.45 −0.139030
$$396$$ −1145.98 −0.145424
$$397$$ −359.905 −0.0454990 −0.0227495 0.999741i $$-0.507242\pi$$
−0.0227495 + 0.999741i $$0.507242\pi$$
$$398$$ −99.1105 −0.0124823
$$399$$ −3313.89 −0.415794
$$400$$ 2642.83 0.330354
$$401$$ −4066.71 −0.506438 −0.253219 0.967409i $$-0.581489\pi$$
−0.253219 + 0.967409i $$0.581489\pi$$
$$402$$ 1124.85 0.139558
$$403$$ 724.274 0.0895252
$$404$$ −14136.9 −1.74093
$$405$$ 230.757 0.0283122
$$406$$ 9740.45 1.19067
$$407$$ 4620.77 0.562760
$$408$$ 5787.32 0.702242
$$409$$ −13488.8 −1.63076 −0.815379 0.578927i $$-0.803472\pi$$
−0.815379 + 0.578927i $$0.803472\pi$$
$$410$$ 4059.40 0.488975
$$411$$ −2654.52 −0.318584
$$412$$ 5981.62 0.715275
$$413$$ 12883.3 1.53498
$$414$$ −4628.83 −0.549504
$$415$$ −2650.04 −0.313459
$$416$$ 1167.24 0.137569
$$417$$ 3275.83 0.384695
$$418$$ 1696.05 0.198460
$$419$$ −7040.12 −0.820841 −0.410420 0.911896i $$-0.634618\pi$$
−0.410420 + 0.911896i $$0.634618\pi$$
$$420$$ −3135.90 −0.364325
$$421$$ 9171.74 1.06177 0.530883 0.847445i $$-0.321860\pi$$
0.530883 + 0.847445i $$0.321860\pi$$
$$422$$ 4652.89 0.536728
$$423$$ −2082.67 −0.239392
$$424$$ −77.8204 −0.00891343
$$425$$ −14253.1 −1.62677
$$426$$ 651.464 0.0740928
$$427$$ −17637.6 −1.99893
$$428$$ −1759.77 −0.198742
$$429$$ 169.988 0.0191307
$$430$$ −4046.51 −0.453814
$$431$$ 992.995 0.110976 0.0554882 0.998459i $$-0.482328\pi$$
0.0554882 + 0.998459i $$0.482328\pi$$
$$432$$ 610.490 0.0679912
$$433$$ 3790.21 0.420660 0.210330 0.977630i $$-0.432546\pi$$
0.210330 + 0.977630i $$0.432546\pi$$
$$434$$ −19719.0 −2.18097
$$435$$ 593.589 0.0654262
$$436$$ 25122.7 2.75954
$$437$$ 4050.98 0.443443
$$438$$ 10431.3 1.13796
$$439$$ −5136.97 −0.558483 −0.279242 0.960221i $$-0.590083\pi$$
−0.279242 + 0.960221i $$0.590083\pi$$
$$440$$ 495.754 0.0537139
$$441$$ 5955.71 0.643095
$$442$$ −2779.16 −0.299075
$$443$$ 10676.8 1.14508 0.572541 0.819876i $$-0.305958\pi$$
0.572541 + 0.819876i $$0.305958\pi$$
$$444$$ 14587.7 1.55923
$$445$$ −2086.96 −0.222317
$$446$$ −17086.2 −1.81403
$$447$$ −891.042 −0.0942838
$$448$$ −26045.5 −2.74672
$$449$$ 10529.9 1.10676 0.553379 0.832929i $$-0.313338\pi$$
0.553379 + 0.832929i $$0.313338\pi$$
$$450$$ 4654.30 0.487569
$$451$$ 3542.64 0.369881
$$452$$ −7482.42 −0.778636
$$453$$ 5663.59 0.587414
$$454$$ 3861.29 0.399162
$$455$$ 465.160 0.0479275
$$456$$ 1653.91 0.169850
$$457$$ −14072.5 −1.44045 −0.720225 0.693741i $$-0.755961\pi$$
−0.720225 + 0.693741i $$0.755961\pi$$
$$458$$ −8148.55 −0.831347
$$459$$ −3292.43 −0.334810
$$460$$ 3833.41 0.388551
$$461$$ −30.8173 −0.00311346 −0.00155673 0.999999i $$-0.500496\pi$$
−0.00155673 + 0.999999i $$0.500496\pi$$
$$462$$ −4628.06 −0.466054
$$463$$ 17591.3 1.76573 0.882867 0.469622i $$-0.155610\pi$$
0.882867 + 0.469622i $$0.155610\pi$$
$$464$$ 1570.39 0.157120
$$465$$ −1201.69 −0.119843
$$466$$ −17397.5 −1.72945
$$467$$ 13273.1 1.31522 0.657609 0.753360i $$-0.271568\pi$$
0.657609 + 0.753360i $$0.271568\pi$$
$$468$$ 536.647 0.0530053
$$469$$ 2686.23 0.264474
$$470$$ 2916.79 0.286259
$$471$$ 169.603 0.0165921
$$472$$ −6429.87 −0.627031
$$473$$ −3531.39 −0.343284
$$474$$ −5085.23 −0.492769
$$475$$ −4073.27 −0.393462
$$476$$ 44742.9 4.30837
$$477$$ 44.2724 0.00424968
$$478$$ −21113.8 −2.02034
$$479$$ 2496.68 0.238155 0.119077 0.992885i $$-0.462006\pi$$
0.119077 + 0.992885i $$0.462006\pi$$
$$480$$ −1936.64 −0.184156
$$481$$ −2163.84 −0.205120
$$482$$ −17648.3 −1.66776
$$483$$ −11054.0 −1.04136
$$484$$ 1400.64 0.131541
$$485$$ −3337.41 −0.312462
$$486$$ 1075.14 0.100348
$$487$$ −3464.42 −0.322357 −0.161178 0.986925i $$-0.551529\pi$$
−0.161178 + 0.986925i $$0.551529\pi$$
$$488$$ 8802.65 0.816552
$$489$$ 147.701 0.0136591
$$490$$ −8341.01 −0.768997
$$491$$ −16224.6 −1.49125 −0.745625 0.666366i $$-0.767849\pi$$
−0.745625 + 0.666366i $$0.767849\pi$$
$$492$$ 11184.0 1.02483
$$493$$ −8469.29 −0.773707
$$494$$ −794.236 −0.0723367
$$495$$ −282.037 −0.0256093
$$496$$ −3179.17 −0.287800
$$497$$ 1555.75 0.140412
$$498$$ −12347.0 −1.11100
$$499$$ 9993.81 0.896562 0.448281 0.893893i $$-0.352036\pi$$
0.448281 + 0.893893i $$0.352036\pi$$
$$500$$ −7976.65 −0.713453
$$501$$ −6206.24 −0.553441
$$502$$ 24220.3 2.15340
$$503$$ −15334.8 −1.35933 −0.679667 0.733520i $$-0.737876\pi$$
−0.679667 + 0.733520i $$0.737876\pi$$
$$504$$ −4513.08 −0.398866
$$505$$ −3479.23 −0.306581
$$506$$ 5657.46 0.497045
$$507$$ 6511.40 0.570377
$$508$$ −11498.1 −1.00422
$$509$$ −7291.23 −0.634927 −0.317464 0.948270i $$-0.602831\pi$$
−0.317464 + 0.948270i $$0.602831\pi$$
$$510$$ 4611.08 0.400357
$$511$$ 24910.7 2.15653
$$512$$ −7985.14 −0.689251
$$513$$ −940.919 −0.0809797
$$514$$ 28466.8 2.44283
$$515$$ 1472.13 0.125961
$$516$$ −11148.5 −0.951134
$$517$$ 2545.49 0.216538
$$518$$ 58912.5 4.99704
$$519$$ 1812.04 0.153255
$$520$$ −232.154 −0.0195782
$$521$$ 16794.3 1.41223 0.706114 0.708098i $$-0.250447\pi$$
0.706114 + 0.708098i $$0.250447\pi$$
$$522$$ 2765.63 0.231893
$$523$$ −21009.4 −1.75655 −0.878275 0.478157i $$-0.841305\pi$$
−0.878275 + 0.478157i $$0.841305\pi$$
$$524$$ 4466.01 0.372326
$$525$$ 11114.9 0.923986
$$526$$ −33577.0 −2.78332
$$527$$ 17145.6 1.41722
$$528$$ −746.154 −0.0615003
$$529$$ 1345.73 0.110605
$$530$$ −62.0039 −0.00508166
$$531$$ 3657.99 0.298951
$$532$$ 12786.7 1.04206
$$533$$ −1658.97 −0.134818
$$534$$ −9723.46 −0.787969
$$535$$ −433.097 −0.0349989
$$536$$ −1340.66 −0.108036
$$537$$ 6396.07 0.513987
$$538$$ −2115.66 −0.169540
$$539$$ −7279.20 −0.581702
$$540$$ −890.383 −0.0709555
$$541$$ −16802.8 −1.33532 −0.667662 0.744464i $$-0.732705\pi$$
−0.667662 + 0.744464i $$0.732705\pi$$
$$542$$ 541.211 0.0428911
$$543$$ 1768.11 0.139737
$$544$$ 27631.9 2.17777
$$545$$ 6182.93 0.485959
$$546$$ 2167.25 0.169872
$$547$$ 16784.5 1.31198 0.655990 0.754770i $$-0.272251\pi$$
0.655990 + 0.754770i $$0.272251\pi$$
$$548$$ 10242.5 0.798429
$$549$$ −5007.88 −0.389309
$$550$$ −5688.59 −0.441023
$$551$$ −2420.37 −0.187135
$$552$$ 5516.91 0.425390
$$553$$ −12144.0 −0.933840
$$554$$ −36277.7 −2.78212
$$555$$ 3590.16 0.274584
$$556$$ −12639.9 −0.964117
$$557$$ 18127.0 1.37893 0.689467 0.724317i $$-0.257845\pi$$
0.689467 + 0.724317i $$0.257845\pi$$
$$558$$ −5598.85 −0.424764
$$559$$ 1653.70 0.125123
$$560$$ −2041.80 −0.154075
$$561$$ 4024.09 0.302847
$$562$$ −30722.3 −2.30595
$$563$$ 2090.88 0.156518 0.0782592 0.996933i $$-0.475064\pi$$
0.0782592 + 0.996933i $$0.475064\pi$$
$$564$$ 8036.03 0.599961
$$565$$ −1841.49 −0.137119
$$566$$ −4579.89 −0.340119
$$567$$ 2567.51 0.190168
$$568$$ −776.452 −0.0573578
$$569$$ 6249.23 0.460424 0.230212 0.973140i $$-0.426058\pi$$
0.230212 + 0.973140i $$0.426058\pi$$
$$570$$ 1317.76 0.0968335
$$571$$ 6048.79 0.443317 0.221659 0.975124i $$-0.428853\pi$$
0.221659 + 0.975124i $$0.428853\pi$$
$$572$$ −655.902 −0.0479451
$$573$$ 6482.69 0.472633
$$574$$ 45166.8 3.28437
$$575$$ −13587.1 −0.985428
$$576$$ −7395.15 −0.534950
$$577$$ −15729.1 −1.13486 −0.567429 0.823423i $$-0.692062\pi$$
−0.567429 + 0.823423i $$0.692062\pi$$
$$578$$ −44053.4 −3.17021
$$579$$ 4472.73 0.321037
$$580$$ −2290.38 −0.163970
$$581$$ −29485.6 −2.10545
$$582$$ −15549.5 −1.10747
$$583$$ −54.1108 −0.00384398
$$584$$ −12432.6 −0.880931
$$585$$ 132.074 0.00933433
$$586$$ 27187.3 1.91655
$$587$$ 15620.5 1.09835 0.549173 0.835709i $$-0.314943\pi$$
0.549173 + 0.835709i $$0.314943\pi$$
$$588$$ −22980.2 −1.61172
$$589$$ 4899.91 0.342780
$$590$$ −5123.04 −0.357478
$$591$$ 691.587 0.0481355
$$592$$ 9498.09 0.659407
$$593$$ −493.541 −0.0341776 −0.0170888 0.999854i $$-0.505440\pi$$
−0.0170888 + 0.999854i $$0.505440\pi$$
$$594$$ −1314.06 −0.0907683
$$595$$ 11011.6 0.758711
$$596$$ 3438.11 0.236293
$$597$$ −67.2022 −0.00460704
$$598$$ −2649.31 −0.181168
$$599$$ −12455.1 −0.849585 −0.424793 0.905291i $$-0.639653\pi$$
−0.424793 + 0.905291i $$0.639653\pi$$
$$600$$ −5547.27 −0.377444
$$601$$ 12454.8 0.845329 0.422664 0.906286i $$-0.361095\pi$$
0.422664 + 0.906286i $$0.361095\pi$$
$$602$$ −45023.3 −3.04820
$$603$$ 762.707 0.0515088
$$604$$ −21853.1 −1.47217
$$605$$ 344.712 0.0231645
$$606$$ −16210.3 −1.08663
$$607$$ −4243.19 −0.283733 −0.141867 0.989886i $$-0.545310\pi$$
−0.141867 + 0.989886i $$0.545310\pi$$
$$608$$ 7896.70 0.526732
$$609$$ 6604.54 0.439458
$$610$$ 7013.57 0.465526
$$611$$ −1192.01 −0.0789259
$$612$$ 12703.9 0.839095
$$613$$ 5733.14 0.377748 0.188874 0.982001i $$-0.439516\pi$$
0.188874 + 0.982001i $$0.439516\pi$$
$$614$$ 9672.18 0.635729
$$615$$ 2752.49 0.180473
$$616$$ 5515.99 0.360788
$$617$$ 15642.1 1.02063 0.510314 0.859988i $$-0.329529\pi$$
0.510314 + 0.859988i $$0.329529\pi$$
$$618$$ 6858.91 0.446449
$$619$$ −7467.40 −0.484879 −0.242440 0.970167i $$-0.577948\pi$$
−0.242440 + 0.970167i $$0.577948\pi$$
$$620$$ 4636.74 0.300348
$$621$$ −3138.60 −0.202814
$$622$$ 33116.1 2.13478
$$623$$ −23220.4 −1.49327
$$624$$ 349.413 0.0224162
$$625$$ 12647.4 0.809432
$$626$$ 30233.6 1.93031
$$627$$ 1150.01 0.0732489
$$628$$ −654.416 −0.0415829
$$629$$ −51224.2 −3.24713
$$630$$ −3595.82 −0.227399
$$631$$ −1486.38 −0.0937745 −0.0468872 0.998900i $$-0.514930\pi$$
−0.0468872 + 0.998900i $$0.514930\pi$$
$$632$$ 6060.87 0.381469
$$633$$ 3154.91 0.198099
$$634$$ −4089.35 −0.256165
$$635$$ −2829.78 −0.176845
$$636$$ −170.826 −0.0106505
$$637$$ 3408.74 0.212024
$$638$$ −3380.21 −0.209755
$$639$$ 441.728 0.0273466
$$640$$ 5192.58 0.320711
$$641$$ 12386.0 0.763211 0.381606 0.924325i $$-0.375371\pi$$
0.381606 + 0.924325i $$0.375371\pi$$
$$642$$ −2017.87 −0.124048
$$643$$ −14458.1 −0.886737 −0.443369 0.896339i $$-0.646217\pi$$
−0.443369 + 0.896339i $$0.646217\pi$$
$$644$$ 42652.3 2.60984
$$645$$ −2743.75 −0.167496
$$646$$ −18801.8 −1.14512
$$647$$ 15792.8 0.959625 0.479813 0.877371i $$-0.340705\pi$$
0.479813 + 0.877371i $$0.340705\pi$$
$$648$$ −1281.41 −0.0776828
$$649$$ −4470.87 −0.270412
$$650$$ 2663.89 0.160748
$$651$$ −13370.5 −0.804965
$$652$$ −569.909 −0.0342321
$$653$$ −3179.93 −0.190567 −0.0952837 0.995450i $$-0.530376\pi$$
−0.0952837 + 0.995450i $$0.530376\pi$$
$$654$$ 28807.3 1.72240
$$655$$ 1099.13 0.0655672
$$656$$ 7281.96 0.433404
$$657$$ 7072.96 0.420003
$$658$$ 32453.6 1.92276
$$659$$ 11593.5 0.685308 0.342654 0.939462i $$-0.388674\pi$$
0.342654 + 0.939462i $$0.388674\pi$$
$$660$$ 1088.25 0.0641817
$$661$$ 3233.88 0.190293 0.0951464 0.995463i $$-0.469668\pi$$
0.0951464 + 0.995463i $$0.469668\pi$$
$$662$$ 43449.9 2.55095
$$663$$ −1884.42 −0.110384
$$664$$ 14715.8 0.860066
$$665$$ 3146.93 0.183508
$$666$$ 16727.1 0.973219
$$667$$ −8073.56 −0.468680
$$668$$ 23946.9 1.38703
$$669$$ −11585.4 −0.669532
$$670$$ −1068.18 −0.0615929
$$671$$ 6120.74 0.352144
$$672$$ −21548.0 −1.23695
$$673$$ −5495.72 −0.314776 −0.157388 0.987537i $$-0.550307\pi$$
−0.157388 + 0.987537i $$0.550307\pi$$
$$674$$ −2658.23 −0.151916
$$675$$ 3155.87 0.179955
$$676$$ −25124.4 −1.42947
$$677$$ 33836.7 1.92090 0.960451 0.278448i $$-0.0898200\pi$$
0.960451 + 0.278448i $$0.0898200\pi$$
$$678$$ −8579.82 −0.485997
$$679$$ −37133.6 −2.09876
$$680$$ −5495.75 −0.309930
$$681$$ 2618.16 0.147325
$$682$$ 6843.04 0.384214
$$683$$ −21080.3 −1.18099 −0.590493 0.807043i $$-0.701067\pi$$
−0.590493 + 0.807043i $$0.701067\pi$$
$$684$$ 3630.56 0.202950
$$685$$ 2520.78 0.140605
$$686$$ −44702.2 −2.48796
$$687$$ −5525.16 −0.306838
$$688$$ −7258.84 −0.402239
$$689$$ 25.3393 0.00140109
$$690$$ 4395.63 0.242520
$$691$$ 11811.3 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$692$$ −6991.79 −0.384087
$$693$$ −3138.07 −0.172014
$$694$$ 13907.8 0.760711
$$695$$ −3110.79 −0.169783
$$696$$ −3296.23 −0.179516
$$697$$ −39272.4 −2.13422
$$698$$ −3188.52 −0.172904
$$699$$ −11796.4 −0.638313
$$700$$ −42887.0 −2.31568
$$701$$ 4244.99 0.228718 0.114359 0.993440i $$-0.463519\pi$$
0.114359 + 0.993440i $$0.463519\pi$$
$$702$$ 615.353 0.0330841
$$703$$ −14639.0 −0.785376
$$704$$ 9038.51 0.483880
$$705$$ 1977.74 0.105654
$$706$$ −5340.81 −0.284708
$$707$$ −38711.5 −2.05926
$$708$$ −14114.4 −0.749227
$$709$$ −898.822 −0.0476107 −0.0238053 0.999717i $$-0.507578\pi$$
−0.0238053 + 0.999717i $$0.507578\pi$$
$$710$$ −618.643 −0.0327004
$$711$$ −3448.06 −0.181874
$$712$$ 11589.0 0.609993
$$713$$ 16344.5 0.858493
$$714$$ 51305.0 2.68913
$$715$$ −161.424 −0.00844322
$$716$$ −24679.4 −1.28815
$$717$$ −14316.3 −0.745679
$$718$$ −38706.3 −2.01185
$$719$$ 10741.8 0.557165 0.278582 0.960412i $$-0.410135\pi$$
0.278582 + 0.960412i $$0.410135\pi$$
$$720$$ −579.733 −0.0300074
$$721$$ 16379.6 0.846061
$$722$$ 24973.9 1.28730
$$723$$ −11966.5 −0.615546
$$724$$ −6822.30 −0.350206
$$725$$ 8117.99 0.415855
$$726$$ 1606.07 0.0821030
$$727$$ 16794.2 0.856758 0.428379 0.903599i $$-0.359085\pi$$
0.428379 + 0.903599i $$0.359085\pi$$
$$728$$ −2583.06 −0.131503
$$729$$ 729.000 0.0370370
$$730$$ −9905.73 −0.502229
$$731$$ 39147.7 1.98075
$$732$$ 19323.0 0.975681
$$733$$ 8659.40 0.436347 0.218173 0.975910i $$-0.429990\pi$$
0.218173 + 0.975910i $$0.429990\pi$$
$$734$$ 29778.4 1.49747
$$735$$ −5655.65 −0.283826
$$736$$ 26340.8 1.31920
$$737$$ −932.197 −0.0465915
$$738$$ 12824.3 0.639660
$$739$$ 16705.7 0.831567 0.415783 0.909464i $$-0.363507\pi$$
0.415783 + 0.909464i $$0.363507\pi$$
$$740$$ −13852.7 −0.688157
$$741$$ −538.534 −0.0266984
$$742$$ −689.884 −0.0341327
$$743$$ 1292.12 0.0637996 0.0318998 0.999491i $$-0.489844\pi$$
0.0318998 + 0.999491i $$0.489844\pi$$
$$744$$ 6673.03 0.328824
$$745$$ 846.151 0.0416115
$$746$$ 1006.09 0.0493773
$$747$$ −8371.90 −0.410056
$$748$$ −15527.0 −0.758990
$$749$$ −4818.83 −0.235082
$$750$$ −9146.53 −0.445312
$$751$$ −14980.4 −0.727886 −0.363943 0.931421i $$-0.618570\pi$$
−0.363943 + 0.931421i $$0.618570\pi$$
$$752$$ 5232.30 0.253726
$$753$$ 16422.7 0.794787
$$754$$ 1582.90 0.0764535
$$755$$ −5378.25 −0.259251
$$756$$ −9906.82 −0.476597
$$757$$ 3003.41 0.144202 0.0721010 0.997397i $$-0.477030\pi$$
0.0721010 + 0.997397i $$0.477030\pi$$
$$758$$ −50244.8 −2.40762
$$759$$ 3836.06 0.183452
$$760$$ −1570.59 −0.0749621
$$761$$ −20375.0 −0.970555 −0.485277 0.874360i $$-0.661281\pi$$
−0.485277 + 0.874360i $$0.661281\pi$$
$$762$$ −13184.4 −0.626799
$$763$$ 68794.1 3.26411
$$764$$ −25013.6 −1.18450
$$765$$ 3126.56 0.147766
$$766$$ −47578.4 −2.24422
$$767$$ 2093.65 0.0985621
$$768$$ 4472.70 0.210149
$$769$$ −12372.4 −0.580184 −0.290092 0.956999i $$-0.593686\pi$$
−0.290092 + 0.956999i $$0.593686\pi$$
$$770$$ 4394.90 0.205690
$$771$$ 19302.0 0.901615
$$772$$ −17258.1 −0.804578
$$773$$ 21023.6 0.978225 0.489113 0.872221i $$-0.337321\pi$$
0.489113 + 0.872221i $$0.337321\pi$$
$$774$$ −12783.6 −0.593664
$$775$$ −16434.4 −0.761732
$$776$$ 18532.8 0.857331
$$777$$ 39945.8 1.84433
$$778$$ 51885.7 2.39099
$$779$$ −11223.4 −0.516198
$$780$$ −509.610 −0.0233935
$$781$$ −539.889 −0.0247359
$$782$$ −62716.6 −2.86795
$$783$$ 1875.24 0.0855884
$$784$$ −14962.5 −0.681602
$$785$$ −161.058 −0.00732281
$$786$$ 5121.02 0.232393
$$787$$ 30286.2 1.37177 0.685886 0.727709i $$-0.259415\pi$$
0.685886 + 0.727709i $$0.259415\pi$$
$$788$$ −2668.51 −0.120637
$$789$$ −22767.0 −1.02728
$$790$$ 4829.03 0.217480
$$791$$ −20489.3 −0.921007
$$792$$ 1566.17 0.0702668
$$793$$ −2866.25 −0.128353
$$794$$ 1592.37 0.0711729
$$795$$ −42.0420 −0.00187557
$$796$$ 259.301 0.0115461
$$797$$ 32337.8 1.43722 0.718610 0.695413i $$-0.244779\pi$$
0.718610 + 0.695413i $$0.244779\pi$$
$$798$$ 14662.1 0.650415
$$799$$ −28218.3 −1.24943
$$800$$ −26485.7 −1.17052
$$801$$ −6593.03 −0.290828
$$802$$ 17992.9 0.792207
$$803$$ −8644.72 −0.379907
$$804$$ −2942.92 −0.129091
$$805$$ 10497.1 0.459596
$$806$$ −3204.50 −0.140042
$$807$$ −1434.53 −0.0625749
$$808$$ 19320.3 0.841197
$$809$$ −891.707 −0.0387525 −0.0193762 0.999812i $$-0.506168\pi$$
−0.0193762 + 0.999812i $$0.506168\pi$$
$$810$$ −1020.97 −0.0442879
$$811$$ −10114.9 −0.437957 −0.218978 0.975730i $$-0.570272\pi$$
−0.218978 + 0.975730i $$0.570272\pi$$
$$812$$ −25483.8 −1.10136
$$813$$ 366.970 0.0158305
$$814$$ −20444.3 −0.880309
$$815$$ −140.260 −0.00602833
$$816$$ 8271.59 0.354857
$$817$$ 11187.7 0.479080
$$818$$ 59680.4 2.55095
$$819$$ 1469.51 0.0626972
$$820$$ −10620.6 −0.452300
$$821$$ 10833.5 0.460525 0.230262 0.973129i $$-0.426042\pi$$
0.230262 + 0.973129i $$0.426042\pi$$
$$822$$ 11744.7 0.498351
$$823$$ 31958.5 1.35359 0.676794 0.736173i $$-0.263369\pi$$
0.676794 + 0.736173i $$0.263369\pi$$
$$824$$ −8174.84 −0.345612
$$825$$ −3857.17 −0.162775
$$826$$ −57001.3 −2.40112
$$827$$ 34847.3 1.46525 0.732624 0.680634i $$-0.238296\pi$$
0.732624 + 0.680634i $$0.238296\pi$$
$$828$$ 12110.3 0.508289
$$829$$ 6537.91 0.273910 0.136955 0.990577i $$-0.456268\pi$$
0.136955 + 0.990577i $$0.456268\pi$$
$$830$$ 11724.9 0.490334
$$831$$ −24598.2 −1.02684
$$832$$ −4232.60 −0.176369
$$833$$ 80694.5 3.35642
$$834$$ −14493.7 −0.601768
$$835$$ 5893.56 0.244258
$$836$$ −4437.35 −0.183575
$$837$$ −3796.32 −0.156774
$$838$$ 31148.5 1.28402
$$839$$ 2710.34 0.111527 0.0557635 0.998444i $$-0.482241\pi$$
0.0557635 + 0.998444i $$0.482241\pi$$
$$840$$ 4285.71 0.176037
$$841$$ −19565.2 −0.802215
$$842$$ −40579.7 −1.66089
$$843$$ −20831.4 −0.851092
$$844$$ −12173.3 −0.496471
$$845$$ −6183.35 −0.251732
$$846$$ 9214.62 0.374474
$$847$$ 3835.42 0.155592
$$848$$ −111.226 −0.00450414
$$849$$ −3105.41 −0.125533
$$850$$ 63061.7 2.54470
$$851$$ −48830.8 −1.96698
$$852$$ −1704.42 −0.0685356
$$853$$ 9759.32 0.391738 0.195869 0.980630i $$-0.437247\pi$$
0.195869 + 0.980630i $$0.437247\pi$$
$$854$$ 78036.2 3.12687
$$855$$ 893.515 0.0357398
$$856$$ 2405.01 0.0960298
$$857$$ −13649.8 −0.544072 −0.272036 0.962287i $$-0.587697\pi$$
−0.272036 + 0.962287i $$0.587697\pi$$
$$858$$ −752.098 −0.0299257
$$859$$ 7796.42 0.309674 0.154837 0.987940i $$-0.450515\pi$$
0.154837 + 0.987940i $$0.450515\pi$$
$$860$$ 10586.8 0.419776
$$861$$ 30625.5 1.21221
$$862$$ −4393.43 −0.173597
$$863$$ 7183.57 0.283350 0.141675 0.989913i $$-0.454751\pi$$
0.141675 + 0.989913i $$0.454751\pi$$
$$864$$ −6118.16 −0.240907
$$865$$ −1720.75 −0.0676383
$$866$$ −16769.5 −0.658026
$$867$$ −29870.6 −1.17008
$$868$$ 51590.5 2.01739
$$869$$ 4214.30 0.164511
$$870$$ −2626.29 −0.102344
$$871$$ 436.534 0.0169821
$$872$$ −34334.1 −1.33337
$$873$$ −10543.4 −0.408752
$$874$$ −17923.3 −0.693666
$$875$$ −21842.7 −0.843905
$$876$$ −27291.2 −1.05261
$$877$$ 17063.1 0.656991 0.328495 0.944506i $$-0.393458\pi$$
0.328495 + 0.944506i $$0.393458\pi$$
$$878$$ 22728.2 0.873620
$$879$$ 18434.4 0.707369
$$880$$ 708.562 0.0271428
$$881$$ −32174.9 −1.23042 −0.615210 0.788363i $$-0.710929\pi$$
−0.615210 + 0.788363i $$0.710929\pi$$
$$882$$ −26350.6 −1.00598
$$883$$ 2843.68 0.108378 0.0541889 0.998531i $$-0.482743\pi$$
0.0541889 + 0.998531i $$0.482743\pi$$
$$884$$ 7271.09 0.276644
$$885$$ −3473.69 −0.131940
$$886$$ −47238.9 −1.79122
$$887$$ −31417.8 −1.18930 −0.594649 0.803985i $$-0.702709\pi$$
−0.594649 + 0.803985i $$0.702709\pi$$
$$888$$ −19936.4 −0.753401
$$889$$ −31485.5 −1.18784
$$890$$ 9233.59 0.347765
$$891$$ −891.000 −0.0335013
$$892$$ 44702.5 1.67797
$$893$$ −8064.30 −0.302196
$$894$$ 3942.35 0.147485
$$895$$ −6073.83 −0.226845
$$896$$ 57775.1 2.15416
$$897$$ −1796.37 −0.0668664
$$898$$ −46588.6 −1.73127
$$899$$ −9765.47 −0.362288
$$900$$ −12177.0 −0.451000
$$901$$ 599.852 0.0221798
$$902$$ −15674.1 −0.578594
$$903$$ −30528.2 −1.12505
$$904$$ 10225.9 0.376227
$$905$$ −1679.03 −0.0616718
$$906$$ −25058.1 −0.918875
$$907$$ 12253.1 0.448573 0.224287 0.974523i $$-0.427995\pi$$
0.224287 + 0.974523i $$0.427995\pi$$
$$908$$ −10102.2 −0.369223
$$909$$ −10991.4 −0.401059
$$910$$ −2058.07 −0.0749717
$$911$$ −48422.4 −1.76104 −0.880518 0.474012i $$-0.842805\pi$$
−0.880518 + 0.474012i $$0.842805\pi$$
$$912$$ 2363.87 0.0858286
$$913$$ 10232.3 0.370909
$$914$$ 62262.9 2.25326
$$915$$ 4755.57 0.171819
$$916$$ 21318.9 0.768993
$$917$$ 12229.4 0.440404
$$918$$ 14567.1 0.523733
$$919$$ 5546.18 0.199077 0.0995385 0.995034i $$-0.468263\pi$$
0.0995385 + 0.995034i $$0.468263\pi$$
$$920$$ −5238.96 −0.187743
$$921$$ 6558.26 0.234638
$$922$$ 136.349 0.00487030
$$923$$ 252.822 0.00901598
$$924$$ 12108.3 0.431098
$$925$$ 49099.5 1.74528
$$926$$ −77831.3 −2.76209
$$927$$ 4650.71 0.164778
$$928$$ −15738.0 −0.556709
$$929$$ −35684.5 −1.26025 −0.630125 0.776494i $$-0.716996\pi$$
−0.630125 + 0.776494i $$0.716996\pi$$
$$930$$ 5316.78 0.187467
$$931$$ 23061.1 0.811811
$$932$$ 45516.7 1.59973
$$933$$ 22454.5 0.787918
$$934$$ −58726.0 −2.05736
$$935$$ −3821.35 −0.133659
$$936$$ −733.413 −0.0256115
$$937$$ −48903.6 −1.70503 −0.852514 0.522705i $$-0.824923\pi$$
−0.852514 + 0.522705i $$0.824923\pi$$
$$938$$ −11885.0 −0.413710
$$939$$ 20500.0 0.712451
$$940$$ −7631.17 −0.264789
$$941$$ −23741.9 −0.822490 −0.411245 0.911525i $$-0.634906\pi$$
−0.411245 + 0.911525i $$0.634906\pi$$
$$942$$ −750.396 −0.0259546
$$943$$ −37437.4 −1.29282
$$944$$ −9189.97 −0.316852
$$945$$ −2438.16 −0.0839295
$$946$$ 15624.4 0.536989
$$947$$ 37612.4 1.29064 0.645321 0.763911i $$-0.276724\pi$$
0.645321 + 0.763911i $$0.276724\pi$$
$$948$$ 13304.4 0.455810
$$949$$ 4048.20 0.138472
$$950$$ 18021.9 0.615482
$$951$$ −2772.80 −0.0945469
$$952$$ −61148.2 −2.08175
$$953$$ −48294.3 −1.64156 −0.820779 0.571246i $$-0.806460\pi$$
−0.820779 + 0.571246i $$0.806460\pi$$
$$954$$ −195.880 −0.00664765
$$955$$ −6156.09 −0.208593
$$956$$ 55239.8 1.86881
$$957$$ −2291.96 −0.0774176
$$958$$ −11046.4 −0.372539
$$959$$ 28047.4 0.944419
$$960$$ 7022.57 0.236096
$$961$$ −10021.4 −0.336389
$$962$$ 9573.76 0.320863
$$963$$ −1368.22 −0.0457843
$$964$$ 46173.1 1.54267
$$965$$ −4247.39 −0.141687
$$966$$ 48907.8 1.62897
$$967$$ 1840.92 0.0612204 0.0306102 0.999531i $$-0.490255\pi$$
0.0306102 + 0.999531i $$0.490255\pi$$
$$968$$ −1914.20 −0.0635587
$$969$$ −12748.6 −0.422647
$$970$$ 14766.1 0.488775
$$971$$ 31461.8 1.03981 0.519906 0.854223i $$-0.325967\pi$$
0.519906 + 0.854223i $$0.325967\pi$$
$$972$$ −2812.86 −0.0928217
$$973$$ −34612.1 −1.14040
$$974$$ 15328.1 0.504254
$$975$$ 1806.26 0.0593298
$$976$$ 12581.3 0.412620
$$977$$ −7040.11 −0.230535 −0.115268 0.993334i $$-0.536773\pi$$
−0.115268 + 0.993334i $$0.536773\pi$$
$$978$$ −653.494 −0.0213665
$$979$$ 8058.15 0.263064
$$980$$ 21822.5 0.711320
$$981$$ 19532.9 0.635715
$$982$$ 71784.4 2.33272
$$983$$ −24610.9 −0.798541 −0.399270 0.916833i $$-0.630737\pi$$
−0.399270 + 0.916833i $$0.630737\pi$$
$$984$$ −15284.7 −0.495183
$$985$$ −656.744 −0.0212443
$$986$$ 37471.8 1.21029
$$987$$ 22005.3 0.709662
$$988$$ 2077.95 0.0669112
$$989$$ 37318.5 1.19986
$$990$$ 1247.85 0.0400599
$$991$$ −40003.3 −1.28229 −0.641144 0.767421i $$-0.721540\pi$$
−0.641144 + 0.767421i $$0.721540\pi$$
$$992$$ 31860.8 1.01974
$$993$$ 29461.4 0.941519
$$994$$ −6883.31 −0.219643
$$995$$ 63.8165 0.00203329
$$996$$ 32303.2 1.02768
$$997$$ −7342.61 −0.233242 −0.116621 0.993176i $$-0.537206\pi$$
−0.116621 + 0.993176i $$0.537206\pi$$
$$998$$ −44216.9 −1.40247
$$999$$ 11341.9 0.359201
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 33.4.a.c.1.1 2
3.2 odd 2 99.4.a.f.1.2 2
4.3 odd 2 528.4.a.p.1.2 2
5.2 odd 4 825.4.c.h.199.2 4
5.3 odd 4 825.4.c.h.199.3 4
5.4 even 2 825.4.a.l.1.2 2
7.6 odd 2 1617.4.a.k.1.1 2
8.3 odd 2 2112.4.a.bg.1.1 2
8.5 even 2 2112.4.a.bn.1.1 2
11.10 odd 2 363.4.a.i.1.2 2
12.11 even 2 1584.4.a.bj.1.1 2
15.14 odd 2 2475.4.a.p.1.1 2
33.32 even 2 1089.4.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 1.1 even 1 trivial
99.4.a.f.1.2 2 3.2 odd 2
363.4.a.i.1.2 2 11.10 odd 2
528.4.a.p.1.2 2 4.3 odd 2
825.4.a.l.1.2 2 5.4 even 2
825.4.c.h.199.2 4 5.2 odd 4
825.4.c.h.199.3 4 5.3 odd 4
1089.4.a.u.1.1 2 33.32 even 2
1584.4.a.bj.1.1 2 12.11 even 2
1617.4.a.k.1.1 2 7.6 odd 2
2112.4.a.bg.1.1 2 8.3 odd 2
2112.4.a.bn.1.1 2 8.5 even 2
2475.4.a.p.1.1 2 15.14 odd 2