# Properties

 Label 4205.2.a.g.1.4 Level $4205$ Weight $2$ Character 4205.1 Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.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: $$33.5770940499$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.93185$$ of defining polynomial Character $$\chi$$ $$=$$ 4205.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.93185 q^{2} +1.41421 q^{3} +1.73205 q^{4} +1.00000 q^{5} +2.73205 q^{6} -2.73205 q^{7} -0.517638 q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.93185 q^{2} +1.41421 q^{3} +1.73205 q^{4} +1.00000 q^{5} +2.73205 q^{6} -2.73205 q^{7} -0.517638 q^{8} -1.00000 q^{9} +1.93185 q^{10} +0.378937 q^{11} +2.44949 q^{12} -5.46410 q^{13} -5.27792 q^{14} +1.41421 q^{15} -4.46410 q^{16} +3.48477 q^{17} -1.93185 q^{18} -4.24264 q^{19} +1.73205 q^{20} -3.86370 q^{21} +0.732051 q^{22} +2.19615 q^{23} -0.732051 q^{24} +1.00000 q^{25} -10.5558 q^{26} -5.65685 q^{27} -4.73205 q^{28} +2.73205 q^{30} -4.24264 q^{31} -7.58871 q^{32} +0.535898 q^{33} +6.73205 q^{34} -2.73205 q^{35} -1.73205 q^{36} -4.24264 q^{37} -8.19615 q^{38} -7.72741 q^{39} -0.517638 q^{40} -5.93426 q^{41} -7.46410 q^{42} -4.24264 q^{43} +0.656339 q^{44} -1.00000 q^{45} +4.24264 q^{46} +11.2122 q^{47} -6.31319 q^{48} +0.464102 q^{49} +1.93185 q^{50} +4.92820 q^{51} -9.46410 q^{52} -10.9282 q^{54} +0.378937 q^{55} +1.41421 q^{56} -6.00000 q^{57} +6.00000 q^{59} +2.44949 q^{60} -11.5911 q^{61} -8.19615 q^{62} +2.73205 q^{63} -5.73205 q^{64} -5.46410 q^{65} +1.03528 q^{66} +13.1244 q^{67} +6.03579 q^{68} +3.10583 q^{69} -5.27792 q^{70} -6.00000 q^{71} +0.517638 q^{72} +15.8338 q^{73} -8.19615 q^{74} +1.41421 q^{75} -7.34847 q^{76} -1.03528 q^{77} -14.9282 q^{78} +1.13681 q^{79} -4.46410 q^{80} -5.00000 q^{81} -11.4641 q^{82} -8.19615 q^{83} -6.69213 q^{84} +3.48477 q^{85} -8.19615 q^{86} -0.196152 q^{88} +7.72741 q^{89} -1.93185 q^{90} +14.9282 q^{91} +3.80385 q^{92} -6.00000 q^{93} +21.6603 q^{94} -4.24264 q^{95} -10.7321 q^{96} -7.34847 q^{97} +0.896575 q^{98} -0.378937 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^9 $$4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 12 q^{28} + 4 q^{30} + 16 q^{33} + 20 q^{34} - 4 q^{35} - 12 q^{38} - 16 q^{42} - 4 q^{45} - 12 q^{49} - 8 q^{51} - 24 q^{52} - 16 q^{54} - 24 q^{57} + 24 q^{59} - 12 q^{62} + 4 q^{63} - 16 q^{64} - 8 q^{65} + 4 q^{67} - 24 q^{71} - 12 q^{74} - 32 q^{78} - 4 q^{80} - 20 q^{81} - 32 q^{82} - 12 q^{83} - 12 q^{86} + 20 q^{88} + 32 q^{91} + 36 q^{92} - 24 q^{93} + 52 q^{94} - 36 q^{96}+O(q^{100})$$ 4 * q + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^9 - 8 * q^13 - 4 * q^16 - 4 * q^22 - 12 * q^23 + 4 * q^24 + 4 * q^25 - 12 * q^28 + 4 * q^30 + 16 * q^33 + 20 * q^34 - 4 * q^35 - 12 * q^38 - 16 * q^42 - 4 * q^45 - 12 * q^49 - 8 * q^51 - 24 * q^52 - 16 * q^54 - 24 * q^57 + 24 * q^59 - 12 * q^62 + 4 * q^63 - 16 * q^64 - 8 * q^65 + 4 * q^67 - 24 * q^71 - 12 * q^74 - 32 * q^78 - 4 * q^80 - 20 * q^81 - 32 * q^82 - 12 * q^83 - 12 * q^86 + 20 * q^88 + 32 * q^91 + 36 * q^92 - 24 * q^93 + 52 * q^94 - 36 * q^96

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.93185 1.36603 0.683013 0.730406i $$-0.260669\pi$$
0.683013 + 0.730406i $$0.260669\pi$$
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 1.73205 0.866025
$$5$$ 1.00000 0.447214
$$6$$ 2.73205 1.11536
$$7$$ −2.73205 −1.03262 −0.516309 0.856402i $$-0.672694\pi$$
−0.516309 + 0.856402i $$0.672694\pi$$
$$8$$ −0.517638 −0.183013
$$9$$ −1.00000 −0.333333
$$10$$ 1.93185 0.610905
$$11$$ 0.378937 0.114254 0.0571270 0.998367i $$-0.481806\pi$$
0.0571270 + 0.998367i $$0.481806\pi$$
$$12$$ 2.44949 0.707107
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ −5.27792 −1.41058
$$15$$ 1.41421 0.365148
$$16$$ −4.46410 −1.11603
$$17$$ 3.48477 0.845180 0.422590 0.906321i $$-0.361121\pi$$
0.422590 + 0.906321i $$0.361121\pi$$
$$18$$ −1.93185 −0.455342
$$19$$ −4.24264 −0.973329 −0.486664 0.873589i $$-0.661786\pi$$
−0.486664 + 0.873589i $$0.661786\pi$$
$$20$$ 1.73205 0.387298
$$21$$ −3.86370 −0.843129
$$22$$ 0.732051 0.156074
$$23$$ 2.19615 0.457929 0.228965 0.973435i $$-0.426466\pi$$
0.228965 + 0.973435i $$0.426466\pi$$
$$24$$ −0.732051 −0.149429
$$25$$ 1.00000 0.200000
$$26$$ −10.5558 −2.07017
$$27$$ −5.65685 −1.08866
$$28$$ −4.73205 −0.894274
$$29$$ 0 0
$$30$$ 2.73205 0.498802
$$31$$ −4.24264 −0.762001 −0.381000 0.924575i $$-0.624420\pi$$
−0.381000 + 0.924575i $$0.624420\pi$$
$$32$$ −7.58871 −1.34151
$$33$$ 0.535898 0.0932879
$$34$$ 6.73205 1.15454
$$35$$ −2.73205 −0.461801
$$36$$ −1.73205 −0.288675
$$37$$ −4.24264 −0.697486 −0.348743 0.937218i $$-0.613391\pi$$
−0.348743 + 0.937218i $$0.613391\pi$$
$$38$$ −8.19615 −1.32959
$$39$$ −7.72741 −1.23738
$$40$$ −0.517638 −0.0818458
$$41$$ −5.93426 −0.926775 −0.463388 0.886156i $$-0.653366\pi$$
−0.463388 + 0.886156i $$0.653366\pi$$
$$42$$ −7.46410 −1.15174
$$43$$ −4.24264 −0.646997 −0.323498 0.946229i $$-0.604859\pi$$
−0.323498 + 0.946229i $$0.604859\pi$$
$$44$$ 0.656339 0.0989468
$$45$$ −1.00000 −0.149071
$$46$$ 4.24264 0.625543
$$47$$ 11.2122 1.63546 0.817732 0.575600i $$-0.195231\pi$$
0.817732 + 0.575600i $$0.195231\pi$$
$$48$$ −6.31319 −0.911231
$$49$$ 0.464102 0.0663002
$$50$$ 1.93185 0.273205
$$51$$ 4.92820 0.690086
$$52$$ −9.46410 −1.31243
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ −10.9282 −1.48714
$$55$$ 0.378937 0.0510959
$$56$$ 1.41421 0.188982
$$57$$ −6.00000 −0.794719
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 2.44949 0.316228
$$61$$ −11.5911 −1.48409 −0.742045 0.670350i $$-0.766144\pi$$
−0.742045 + 0.670350i $$0.766144\pi$$
$$62$$ −8.19615 −1.04091
$$63$$ 2.73205 0.344206
$$64$$ −5.73205 −0.716506
$$65$$ −5.46410 −0.677738
$$66$$ 1.03528 0.127434
$$67$$ 13.1244 1.60340 0.801698 0.597730i $$-0.203930\pi$$
0.801698 + 0.597730i $$0.203930\pi$$
$$68$$ 6.03579 0.731947
$$69$$ 3.10583 0.373898
$$70$$ −5.27792 −0.630832
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0.517638 0.0610042
$$73$$ 15.8338 1.85320 0.926600 0.376048i $$-0.122717\pi$$
0.926600 + 0.376048i $$0.122717\pi$$
$$74$$ −8.19615 −0.952783
$$75$$ 1.41421 0.163299
$$76$$ −7.34847 −0.842927
$$77$$ −1.03528 −0.117981
$$78$$ −14.9282 −1.69029
$$79$$ 1.13681 0.127901 0.0639507 0.997953i $$-0.479630\pi$$
0.0639507 + 0.997953i $$0.479630\pi$$
$$80$$ −4.46410 −0.499102
$$81$$ −5.00000 −0.555556
$$82$$ −11.4641 −1.26600
$$83$$ −8.19615 −0.899645 −0.449822 0.893118i $$-0.648513\pi$$
−0.449822 + 0.893118i $$0.648513\pi$$
$$84$$ −6.69213 −0.730171
$$85$$ 3.48477 0.377976
$$86$$ −8.19615 −0.883814
$$87$$ 0 0
$$88$$ −0.196152 −0.0209099
$$89$$ 7.72741 0.819103 0.409552 0.912287i $$-0.365685\pi$$
0.409552 + 0.912287i $$0.365685\pi$$
$$90$$ −1.93185 −0.203635
$$91$$ 14.9282 1.56490
$$92$$ 3.80385 0.396579
$$93$$ −6.00000 −0.622171
$$94$$ 21.6603 2.23408
$$95$$ −4.24264 −0.435286
$$96$$ −10.7321 −1.09534
$$97$$ −7.34847 −0.746124 −0.373062 0.927806i $$-0.621692\pi$$
−0.373062 + 0.927806i $$0.621692\pi$$
$$98$$ 0.896575 0.0905678
$$99$$ −0.378937 −0.0380846
$$100$$ 1.73205 0.173205
$$101$$ −15.4548 −1.53781 −0.768906 0.639362i $$-0.779198\pi$$
−0.768906 + 0.639362i $$0.779198\pi$$
$$102$$ 9.52056 0.942676
$$103$$ 10.1962 1.00466 0.502328 0.864677i $$-0.332477\pi$$
0.502328 + 0.864677i $$0.332477\pi$$
$$104$$ 2.82843 0.277350
$$105$$ −3.86370 −0.377059
$$106$$ 0 0
$$107$$ −8.19615 −0.792352 −0.396176 0.918175i $$-0.629663\pi$$
−0.396176 + 0.918175i $$0.629663\pi$$
$$108$$ −9.79796 −0.942809
$$109$$ −5.46410 −0.523366 −0.261683 0.965154i $$-0.584277\pi$$
−0.261683 + 0.965154i $$0.584277\pi$$
$$110$$ 0.732051 0.0697983
$$111$$ −6.00000 −0.569495
$$112$$ 12.1962 1.15243
$$113$$ −8.86422 −0.833876 −0.416938 0.908935i $$-0.636897\pi$$
−0.416938 + 0.908935i $$0.636897\pi$$
$$114$$ −11.5911 −1.08561
$$115$$ 2.19615 0.204792
$$116$$ 0 0
$$117$$ 5.46410 0.505156
$$118$$ 11.5911 1.06705
$$119$$ −9.52056 −0.872748
$$120$$ −0.732051 −0.0668268
$$121$$ −10.8564 −0.986946
$$122$$ −22.3923 −2.02730
$$123$$ −8.39230 −0.756709
$$124$$ −7.34847 −0.659912
$$125$$ 1.00000 0.0894427
$$126$$ 5.27792 0.470194
$$127$$ 12.7279 1.12942 0.564710 0.825289i $$-0.308988\pi$$
0.564710 + 0.825289i $$0.308988\pi$$
$$128$$ 4.10394 0.362740
$$129$$ −6.00000 −0.528271
$$130$$ −10.5558 −0.925808
$$131$$ −11.9700 −1.04583 −0.522914 0.852385i $$-0.675155\pi$$
−0.522914 + 0.852385i $$0.675155\pi$$
$$132$$ 0.928203 0.0807897
$$133$$ 11.5911 1.00508
$$134$$ 25.3543 2.19028
$$135$$ −5.65685 −0.486864
$$136$$ −1.80385 −0.154679
$$137$$ −9.89949 −0.845771 −0.422885 0.906183i $$-0.638983\pi$$
−0.422885 + 0.906183i $$0.638983\pi$$
$$138$$ 6.00000 0.510754
$$139$$ −6.53590 −0.554368 −0.277184 0.960817i $$-0.589401\pi$$
−0.277184 + 0.960817i $$0.589401\pi$$
$$140$$ −4.73205 −0.399931
$$141$$ 15.8564 1.33535
$$142$$ −11.5911 −0.972704
$$143$$ −2.07055 −0.173148
$$144$$ 4.46410 0.372008
$$145$$ 0 0
$$146$$ 30.5885 2.53152
$$147$$ 0.656339 0.0541339
$$148$$ −7.34847 −0.604040
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 2.73205 0.223071
$$151$$ 2.39230 0.194683 0.0973415 0.995251i $$-0.468966\pi$$
0.0973415 + 0.995251i $$0.468966\pi$$
$$152$$ 2.19615 0.178131
$$153$$ −3.48477 −0.281727
$$154$$ −2.00000 −0.161165
$$155$$ −4.24264 −0.340777
$$156$$ −13.3843 −1.07160
$$157$$ 7.34847 0.586472 0.293236 0.956040i $$-0.405268\pi$$
0.293236 + 0.956040i $$0.405268\pi$$
$$158$$ 2.19615 0.174717
$$159$$ 0 0
$$160$$ −7.58871 −0.599940
$$161$$ −6.00000 −0.472866
$$162$$ −9.65926 −0.758903
$$163$$ −4.24264 −0.332309 −0.166155 0.986100i $$-0.553135\pi$$
−0.166155 + 0.986100i $$0.553135\pi$$
$$164$$ −10.2784 −0.802611
$$165$$ 0.535898 0.0417196
$$166$$ −15.8338 −1.22894
$$167$$ −14.1962 −1.09853 −0.549266 0.835648i $$-0.685092\pi$$
−0.549266 + 0.835648i $$0.685092\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 16.8564 1.29665
$$170$$ 6.73205 0.516325
$$171$$ 4.24264 0.324443
$$172$$ −7.34847 −0.560316
$$173$$ 20.7846 1.58022 0.790112 0.612962i $$-0.210022\pi$$
0.790112 + 0.612962i $$0.210022\pi$$
$$174$$ 0 0
$$175$$ −2.73205 −0.206524
$$176$$ −1.69161 −0.127510
$$177$$ 8.48528 0.637793
$$178$$ 14.9282 1.11892
$$179$$ 4.39230 0.328296 0.164148 0.986436i $$-0.447512\pi$$
0.164148 + 0.986436i $$0.447512\pi$$
$$180$$ −1.73205 −0.129099
$$181$$ 21.8564 1.62457 0.812287 0.583258i $$-0.198222\pi$$
0.812287 + 0.583258i $$0.198222\pi$$
$$182$$ 28.8391 2.13769
$$183$$ −16.3923 −1.21175
$$184$$ −1.13681 −0.0838069
$$185$$ −4.24264 −0.311925
$$186$$ −11.5911 −0.849901
$$187$$ 1.32051 0.0965651
$$188$$ 19.4201 1.41635
$$189$$ 15.4548 1.12417
$$190$$ −8.19615 −0.594611
$$191$$ 24.8738 1.79981 0.899904 0.436089i $$-0.143637\pi$$
0.899904 + 0.436089i $$0.143637\pi$$
$$192$$ −8.10634 −0.585025
$$193$$ 12.7279 0.916176 0.458088 0.888907i $$-0.348534\pi$$
0.458088 + 0.888907i $$0.348534\pi$$
$$194$$ −14.1962 −1.01922
$$195$$ −7.72741 −0.553371
$$196$$ 0.803848 0.0574177
$$197$$ −22.3923 −1.59539 −0.797693 0.603064i $$-0.793946\pi$$
−0.797693 + 0.603064i $$0.793946\pi$$
$$198$$ −0.732051 −0.0520246
$$199$$ −12.5359 −0.888646 −0.444323 0.895867i $$-0.646556\pi$$
−0.444323 + 0.895867i $$0.646556\pi$$
$$200$$ −0.517638 −0.0366025
$$201$$ 18.5606 1.30917
$$202$$ −29.8564 −2.10069
$$203$$ 0 0
$$204$$ 8.53590 0.597632
$$205$$ −5.93426 −0.414466
$$206$$ 19.6975 1.37239
$$207$$ −2.19615 −0.152643
$$208$$ 24.3923 1.69130
$$209$$ −1.60770 −0.111207
$$210$$ −7.46410 −0.515072
$$211$$ −22.0454 −1.51767 −0.758834 0.651284i $$-0.774231\pi$$
−0.758834 + 0.651284i $$0.774231\pi$$
$$212$$ 0 0
$$213$$ −8.48528 −0.581402
$$214$$ −15.8338 −1.08237
$$215$$ −4.24264 −0.289346
$$216$$ 2.92820 0.199239
$$217$$ 11.5911 0.786856
$$218$$ −10.5558 −0.714931
$$219$$ 22.3923 1.51313
$$220$$ 0.656339 0.0442504
$$221$$ −19.0411 −1.28084
$$222$$ −11.5911 −0.777944
$$223$$ 7.66025 0.512969 0.256484 0.966548i $$-0.417436\pi$$
0.256484 + 0.966548i $$0.417436\pi$$
$$224$$ 20.7327 1.38526
$$225$$ −1.00000 −0.0666667
$$226$$ −17.1244 −1.13910
$$227$$ −26.1962 −1.73870 −0.869350 0.494197i $$-0.835462\pi$$
−0.869350 + 0.494197i $$0.835462\pi$$
$$228$$ −10.3923 −0.688247
$$229$$ 25.4558 1.68217 0.841085 0.540903i $$-0.181918\pi$$
0.841085 + 0.540903i $$0.181918\pi$$
$$230$$ 4.24264 0.279751
$$231$$ −1.46410 −0.0963308
$$232$$ 0 0
$$233$$ 1.60770 0.105324 0.0526618 0.998612i $$-0.483229\pi$$
0.0526618 + 0.998612i $$0.483229\pi$$
$$234$$ 10.5558 0.690056
$$235$$ 11.2122 0.731401
$$236$$ 10.3923 0.676481
$$237$$ 1.60770 0.104431
$$238$$ −18.3923 −1.19220
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ −6.31319 −0.407515
$$241$$ 4.92820 0.317453 0.158727 0.987323i $$-0.449261\pi$$
0.158727 + 0.987323i $$0.449261\pi$$
$$242$$ −20.9730 −1.34819
$$243$$ 9.89949 0.635053
$$244$$ −20.0764 −1.28526
$$245$$ 0.464102 0.0296504
$$246$$ −16.2127 −1.03368
$$247$$ 23.1822 1.47505
$$248$$ 2.19615 0.139456
$$249$$ −11.5911 −0.734557
$$250$$ 1.93185 0.122181
$$251$$ 8.86422 0.559505 0.279752 0.960072i $$-0.409748\pi$$
0.279752 + 0.960072i $$0.409748\pi$$
$$252$$ 4.73205 0.298091
$$253$$ 0.832204 0.0523202
$$254$$ 24.5885 1.54282
$$255$$ 4.92820 0.308616
$$256$$ 19.3923 1.21202
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ −11.5911 −0.721631
$$259$$ 11.5911 0.720237
$$260$$ −9.46410 −0.586939
$$261$$ 0 0
$$262$$ −23.1244 −1.42863
$$263$$ −24.5964 −1.51668 −0.758341 0.651859i $$-0.773990\pi$$
−0.758341 + 0.651859i $$0.773990\pi$$
$$264$$ −0.277401 −0.0170729
$$265$$ 0 0
$$266$$ 22.3923 1.37296
$$267$$ 10.9282 0.668795
$$268$$ 22.7321 1.38858
$$269$$ 1.59008 0.0969488 0.0484744 0.998824i $$-0.484564\pi$$
0.0484744 + 0.998824i $$0.484564\pi$$
$$270$$ −10.9282 −0.665069
$$271$$ 24.3190 1.47728 0.738638 0.674102i $$-0.235469\pi$$
0.738638 + 0.674102i $$0.235469\pi$$
$$272$$ −15.5563 −0.943242
$$273$$ 21.1117 1.27774
$$274$$ −19.1244 −1.15534
$$275$$ 0.378937 0.0228508
$$276$$ 5.37945 0.323805
$$277$$ −10.9282 −0.656612 −0.328306 0.944571i $$-0.606478\pi$$
−0.328306 + 0.944571i $$0.606478\pi$$
$$278$$ −12.6264 −0.757280
$$279$$ 4.24264 0.254000
$$280$$ 1.41421 0.0845154
$$281$$ 16.3923 0.977883 0.488941 0.872317i $$-0.337383\pi$$
0.488941 + 0.872317i $$0.337383\pi$$
$$282$$ 30.6322 1.82412
$$283$$ −30.9808 −1.84162 −0.920808 0.390017i $$-0.872469\pi$$
−0.920808 + 0.390017i $$0.872469\pi$$
$$284$$ −10.3923 −0.616670
$$285$$ −6.00000 −0.355409
$$286$$ −4.00000 −0.236525
$$287$$ 16.2127 0.957005
$$288$$ 7.58871 0.447169
$$289$$ −4.85641 −0.285671
$$290$$ 0 0
$$291$$ −10.3923 −0.609208
$$292$$ 27.4249 1.60492
$$293$$ −16.5916 −0.969293 −0.484647 0.874710i $$-0.661052\pi$$
−0.484647 + 0.874710i $$0.661052\pi$$
$$294$$ 1.26795 0.0739483
$$295$$ 6.00000 0.349334
$$296$$ 2.19615 0.127649
$$297$$ −2.14359 −0.124384
$$298$$ 34.7733 2.01436
$$299$$ −12.0000 −0.693978
$$300$$ 2.44949 0.141421
$$301$$ 11.5911 0.668100
$$302$$ 4.62158 0.265942
$$303$$ −21.8564 −1.25562
$$304$$ 18.9396 1.08626
$$305$$ −11.5911 −0.663705
$$306$$ −6.73205 −0.384846
$$307$$ −33.6365 −1.91974 −0.959869 0.280450i $$-0.909516\pi$$
−0.959869 + 0.280450i $$0.909516\pi$$
$$308$$ −1.79315 −0.102174
$$309$$ 14.4195 0.820299
$$310$$ −8.19615 −0.465510
$$311$$ −19.6975 −1.11694 −0.558470 0.829525i $$-0.688611\pi$$
−0.558470 + 0.829525i $$0.688611\pi$$
$$312$$ 4.00000 0.226455
$$313$$ −18.5359 −1.04771 −0.523855 0.851807i $$-0.675507\pi$$
−0.523855 + 0.851807i $$0.675507\pi$$
$$314$$ 14.1962 0.801135
$$315$$ 2.73205 0.153934
$$316$$ 1.96902 0.110766
$$317$$ −18.1817 −1.02119 −0.510593 0.859822i $$-0.670574\pi$$
−0.510593 + 0.859822i $$0.670574\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −5.73205 −0.320431
$$321$$ −11.5911 −0.646953
$$322$$ −11.5911 −0.645947
$$323$$ −14.7846 −0.822638
$$324$$ −8.66025 −0.481125
$$325$$ −5.46410 −0.303094
$$326$$ −8.19615 −0.453943
$$327$$ −7.72741 −0.427327
$$328$$ 3.07180 0.169612
$$329$$ −30.6322 −1.68881
$$330$$ 1.03528 0.0569901
$$331$$ 10.4543 0.574620 0.287310 0.957838i $$-0.407239\pi$$
0.287310 + 0.957838i $$0.407239\pi$$
$$332$$ −14.1962 −0.779115
$$333$$ 4.24264 0.232495
$$334$$ −27.4249 −1.50062
$$335$$ 13.1244 0.717060
$$336$$ 17.2480 0.940954
$$337$$ 32.8043 1.78696 0.893482 0.449098i $$-0.148255\pi$$
0.893482 + 0.449098i $$0.148255\pi$$
$$338$$ 32.5641 1.77125
$$339$$ −12.5359 −0.680857
$$340$$ 6.03579 0.327337
$$341$$ −1.60770 −0.0870616
$$342$$ 8.19615 0.443197
$$343$$ 17.8564 0.964155
$$344$$ 2.19615 0.118409
$$345$$ 3.10583 0.167212
$$346$$ 40.1528 2.15863
$$347$$ −15.8038 −0.848395 −0.424197 0.905570i $$-0.639444\pi$$
−0.424197 + 0.905570i $$0.639444\pi$$
$$348$$ 0 0
$$349$$ −2.39230 −0.128057 −0.0640286 0.997948i $$-0.520395\pi$$
−0.0640286 + 0.997948i $$0.520395\pi$$
$$350$$ −5.27792 −0.282117
$$351$$ 30.9096 1.64983
$$352$$ −2.87564 −0.153272
$$353$$ 8.78461 0.467558 0.233779 0.972290i $$-0.424891\pi$$
0.233779 + 0.972290i $$0.424891\pi$$
$$354$$ 16.3923 0.871241
$$355$$ −6.00000 −0.318447
$$356$$ 13.3843 0.709364
$$357$$ −13.4641 −0.712596
$$358$$ 8.48528 0.448461
$$359$$ 7.07107 0.373197 0.186598 0.982436i $$-0.440254\pi$$
0.186598 + 0.982436i $$0.440254\pi$$
$$360$$ 0.517638 0.0272819
$$361$$ −1.00000 −0.0526316
$$362$$ 42.2233 2.21921
$$363$$ −15.3533 −0.805838
$$364$$ 25.8564 1.35524
$$365$$ 15.8338 0.828776
$$366$$ −31.6675 −1.65529
$$367$$ 27.4249 1.43157 0.715783 0.698323i $$-0.246070\pi$$
0.715783 + 0.698323i $$0.246070\pi$$
$$368$$ −9.80385 −0.511061
$$369$$ 5.93426 0.308925
$$370$$ −8.19615 −0.426098
$$371$$ 0 0
$$372$$ −10.3923 −0.538816
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 2.55103 0.131910
$$375$$ 1.41421 0.0730297
$$376$$ −5.80385 −0.299311
$$377$$ 0 0
$$378$$ 29.8564 1.53565
$$379$$ 1.13681 0.0583941 0.0291971 0.999574i $$-0.490705\pi$$
0.0291971 + 0.999574i $$0.490705\pi$$
$$380$$ −7.34847 −0.376969
$$381$$ 18.0000 0.922168
$$382$$ 48.0526 2.45858
$$383$$ −8.19615 −0.418804 −0.209402 0.977830i $$-0.567152\pi$$
−0.209402 + 0.977830i $$0.567152\pi$$
$$384$$ 5.80385 0.296176
$$385$$ −1.03528 −0.0527626
$$386$$ 24.5885 1.25152
$$387$$ 4.24264 0.215666
$$388$$ −12.7279 −0.646162
$$389$$ 1.59008 0.0806202 0.0403101 0.999187i $$-0.487165\pi$$
0.0403101 + 0.999187i $$0.487165\pi$$
$$390$$ −14.9282 −0.755919
$$391$$ 7.65308 0.387033
$$392$$ −0.240237 −0.0121338
$$393$$ −16.9282 −0.853915
$$394$$ −43.2586 −2.17934
$$395$$ 1.13681 0.0571992
$$396$$ −0.656339 −0.0329823
$$397$$ 9.60770 0.482196 0.241098 0.970501i $$-0.422492\pi$$
0.241098 + 0.970501i $$0.422492\pi$$
$$398$$ −24.2175 −1.21391
$$399$$ 16.3923 0.820642
$$400$$ −4.46410 −0.223205
$$401$$ −32.7846 −1.63719 −0.818593 0.574375i $$-0.805245\pi$$
−0.818593 + 0.574375i $$0.805245\pi$$
$$402$$ 35.8564 1.78836
$$403$$ 23.1822 1.15479
$$404$$ −26.7685 −1.33178
$$405$$ −5.00000 −0.248452
$$406$$ 0 0
$$407$$ −1.60770 −0.0796905
$$408$$ −2.55103 −0.126295
$$409$$ 26.2880 1.29986 0.649930 0.759994i $$-0.274798\pi$$
0.649930 + 0.759994i $$0.274798\pi$$
$$410$$ −11.4641 −0.566172
$$411$$ −14.0000 −0.690569
$$412$$ 17.6603 0.870058
$$413$$ −16.3923 −0.806613
$$414$$ −4.24264 −0.208514
$$415$$ −8.19615 −0.402333
$$416$$ 41.4655 2.03301
$$417$$ −9.24316 −0.452639
$$418$$ −3.10583 −0.151911
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ −6.69213 −0.326543
$$421$$ −26.2880 −1.28120 −0.640601 0.767874i $$-0.721315\pi$$
−0.640601 + 0.767874i $$0.721315\pi$$
$$422$$ −42.5885 −2.07317
$$423$$ −11.2122 −0.545154
$$424$$ 0 0
$$425$$ 3.48477 0.169036
$$426$$ −16.3923 −0.794210
$$427$$ 31.6675 1.53250
$$428$$ −14.1962 −0.686197
$$429$$ −2.92820 −0.141375
$$430$$ −8.19615 −0.395254
$$431$$ −16.3923 −0.789590 −0.394795 0.918769i $$-0.629184\pi$$
−0.394795 + 0.918769i $$0.629184\pi$$
$$432$$ 25.2528 1.21497
$$433$$ −12.7279 −0.611665 −0.305832 0.952085i $$-0.598935\pi$$
−0.305832 + 0.952085i $$0.598935\pi$$
$$434$$ 22.3923 1.07487
$$435$$ 0 0
$$436$$ −9.46410 −0.453248
$$437$$ −9.31749 −0.445716
$$438$$ 43.2586 2.06698
$$439$$ 4.92820 0.235210 0.117605 0.993060i $$-0.462478\pi$$
0.117605 + 0.993060i $$0.462478\pi$$
$$440$$ −0.196152 −0.00935120
$$441$$ −0.464102 −0.0221001
$$442$$ −36.7846 −1.74967
$$443$$ 5.00052 0.237582 0.118791 0.992919i $$-0.462098\pi$$
0.118791 + 0.992919i $$0.462098\pi$$
$$444$$ −10.3923 −0.493197
$$445$$ 7.72741 0.366314
$$446$$ 14.7985 0.700728
$$447$$ 25.4558 1.20402
$$448$$ 15.6603 0.739877
$$449$$ −6.13733 −0.289638 −0.144819 0.989458i $$-0.546260\pi$$
−0.144819 + 0.989458i $$0.546260\pi$$
$$450$$ −1.93185 −0.0910684
$$451$$ −2.24871 −0.105888
$$452$$ −15.3533 −0.722157
$$453$$ 3.38323 0.158958
$$454$$ −50.6071 −2.37511
$$455$$ 14.9282 0.699845
$$456$$ 3.10583 0.145444
$$457$$ −13.0718 −0.611473 −0.305736 0.952116i $$-0.598903\pi$$
−0.305736 + 0.952116i $$0.598903\pi$$
$$458$$ 49.1769 2.29789
$$459$$ −19.7128 −0.920115
$$460$$ 3.80385 0.177355
$$461$$ 22.6274 1.05386 0.526932 0.849907i $$-0.323342\pi$$
0.526932 + 0.849907i $$0.323342\pi$$
$$462$$ −2.82843 −0.131590
$$463$$ 4.33975 0.201685 0.100843 0.994902i $$-0.467846\pi$$
0.100843 + 0.994902i $$0.467846\pi$$
$$464$$ 0 0
$$465$$ −6.00000 −0.278243
$$466$$ 3.10583 0.143875
$$467$$ −7.62587 −0.352883 −0.176442 0.984311i $$-0.556459\pi$$
−0.176442 + 0.984311i $$0.556459\pi$$
$$468$$ 9.46410 0.437478
$$469$$ −35.8564 −1.65570
$$470$$ 21.6603 0.999113
$$471$$ 10.3923 0.478852
$$472$$ −3.10583 −0.142957
$$473$$ −1.60770 −0.0739219
$$474$$ 3.10583 0.142655
$$475$$ −4.24264 −0.194666
$$476$$ −16.4901 −0.755822
$$477$$ 0 0
$$478$$ 11.5911 0.530165
$$479$$ −10.1769 −0.464994 −0.232497 0.972597i $$-0.574690\pi$$
−0.232497 + 0.972597i $$0.574690\pi$$
$$480$$ −10.7321 −0.489849
$$481$$ 23.1822 1.05702
$$482$$ 9.52056 0.433650
$$483$$ −8.48528 −0.386094
$$484$$ −18.8038 −0.854720
$$485$$ −7.34847 −0.333677
$$486$$ 19.1244 0.867498
$$487$$ 8.58846 0.389180 0.194590 0.980885i $$-0.437662\pi$$
0.194590 + 0.980885i $$0.437662\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −6.00000 −0.271329
$$490$$ 0.896575 0.0405032
$$491$$ 1.41421 0.0638226 0.0319113 0.999491i $$-0.489841\pi$$
0.0319113 + 0.999491i $$0.489841\pi$$
$$492$$ −14.5359 −0.655329
$$493$$ 0 0
$$494$$ 44.7846 2.01495
$$495$$ −0.378937 −0.0170320
$$496$$ 18.9396 0.850412
$$497$$ 16.3923 0.735295
$$498$$ −22.3923 −1.00342
$$499$$ −21.8564 −0.978427 −0.489214 0.872164i $$-0.662716\pi$$
−0.489214 + 0.872164i $$0.662716\pi$$
$$500$$ 1.73205 0.0774597
$$501$$ −20.0764 −0.896947
$$502$$ 17.1244 0.764297
$$503$$ 5.00052 0.222962 0.111481 0.993767i $$-0.464441\pi$$
0.111481 + 0.993767i $$0.464441\pi$$
$$504$$ −1.41421 −0.0629941
$$505$$ −15.4548 −0.687730
$$506$$ 1.60770 0.0714708
$$507$$ 23.8386 1.05871
$$508$$ 22.0454 0.978107
$$509$$ −20.7846 −0.921262 −0.460631 0.887592i $$-0.652377\pi$$
−0.460631 + 0.887592i $$0.652377\pi$$
$$510$$ 9.52056 0.421577
$$511$$ −43.2586 −1.91365
$$512$$ 29.2552 1.29291
$$513$$ 24.0000 1.05963
$$514$$ 23.1822 1.02252
$$515$$ 10.1962 0.449296
$$516$$ −10.3923 −0.457496
$$517$$ 4.24871 0.186858
$$518$$ 22.3923 0.983861
$$519$$ 29.3939 1.29025
$$520$$ 2.82843 0.124035
$$521$$ −8.78461 −0.384861 −0.192430 0.981311i $$-0.561637\pi$$
−0.192430 + 0.981311i $$0.561637\pi$$
$$522$$ 0 0
$$523$$ −4.33975 −0.189764 −0.0948819 0.995489i $$-0.530247\pi$$
−0.0948819 + 0.995489i $$0.530247\pi$$
$$524$$ −20.7327 −0.905714
$$525$$ −3.86370 −0.168626
$$526$$ −47.5167 −2.07182
$$527$$ −14.7846 −0.644028
$$528$$ −2.39230 −0.104112
$$529$$ −18.1769 −0.790301
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 20.0764 0.870422
$$533$$ 32.4254 1.40450
$$534$$ 21.1117 0.913591
$$535$$ −8.19615 −0.354351
$$536$$ −6.79367 −0.293442
$$537$$ 6.21166 0.268053
$$538$$ 3.07180 0.132435
$$539$$ 0.175865 0.00757506
$$540$$ −9.79796 −0.421637
$$541$$ −25.4558 −1.09443 −0.547216 0.836991i $$-0.684312\pi$$
−0.547216 + 0.836991i $$0.684312\pi$$
$$542$$ 46.9808 2.01800
$$543$$ 30.9096 1.32646
$$544$$ −26.4449 −1.13381
$$545$$ −5.46410 −0.234056
$$546$$ 40.7846 1.74542
$$547$$ −23.5167 −1.00550 −0.502750 0.864432i $$-0.667678\pi$$
−0.502750 + 0.864432i $$0.667678\pi$$
$$548$$ −17.1464 −0.732459
$$549$$ 11.5911 0.494697
$$550$$ 0.732051 0.0312148
$$551$$ 0 0
$$552$$ −1.60770 −0.0684280
$$553$$ −3.10583 −0.132073
$$554$$ −21.1117 −0.896949
$$555$$ −6.00000 −0.254686
$$556$$ −11.3205 −0.480096
$$557$$ 1.60770 0.0681202 0.0340601 0.999420i $$-0.489156\pi$$
0.0340601 + 0.999420i $$0.489156\pi$$
$$558$$ 8.19615 0.346971
$$559$$ 23.1822 0.980503
$$560$$ 12.1962 0.515382
$$561$$ 1.86748 0.0788451
$$562$$ 31.6675 1.33581
$$563$$ 2.72689 0.114925 0.0574624 0.998348i $$-0.481699\pi$$
0.0574624 + 0.998348i $$0.481699\pi$$
$$564$$ 27.4641 1.15645
$$565$$ −8.86422 −0.372920
$$566$$ −59.8502 −2.51569
$$567$$ 13.6603 0.573677
$$568$$ 3.10583 0.130318
$$569$$ 20.8343 0.873418 0.436709 0.899603i $$-0.356144\pi$$
0.436709 + 0.899603i $$0.356144\pi$$
$$570$$ −11.5911 −0.485498
$$571$$ −32.3923 −1.35558 −0.677788 0.735257i $$-0.737061\pi$$
−0.677788 + 0.735257i $$0.737061\pi$$
$$572$$ −3.58630 −0.149951
$$573$$ 35.1769 1.46954
$$574$$ 31.3205 1.30729
$$575$$ 2.19615 0.0915859
$$576$$ 5.73205 0.238835
$$577$$ −15.0015 −0.624523 −0.312261 0.949996i $$-0.601086\pi$$
−0.312261 + 0.949996i $$0.601086\pi$$
$$578$$ −9.38186 −0.390234
$$579$$ 18.0000 0.748054
$$580$$ 0 0
$$581$$ 22.3923 0.928989
$$582$$ −20.0764 −0.832193
$$583$$ 0 0
$$584$$ −8.19615 −0.339159
$$585$$ 5.46410 0.225913
$$586$$ −32.0526 −1.32408
$$587$$ −33.8038 −1.39523 −0.697617 0.716471i $$-0.745756\pi$$
−0.697617 + 0.716471i $$0.745756\pi$$
$$588$$ 1.13681 0.0468813
$$589$$ 18.0000 0.741677
$$590$$ 11.5911 0.477198
$$591$$ −31.6675 −1.30263
$$592$$ 18.9396 0.778412
$$593$$ 40.3923 1.65871 0.829357 0.558720i $$-0.188707\pi$$
0.829357 + 0.558720i $$0.188707\pi$$
$$594$$ −4.14110 −0.169912
$$595$$ −9.52056 −0.390305
$$596$$ 31.1769 1.27706
$$597$$ −17.7284 −0.725577
$$598$$ −23.1822 −0.947991
$$599$$ 46.7434 1.90988 0.954941 0.296795i $$-0.0959177\pi$$
0.954941 + 0.296795i $$0.0959177\pi$$
$$600$$ −0.732051 −0.0298858
$$601$$ 5.37945 0.219432 0.109716 0.993963i $$-0.465006\pi$$
0.109716 + 0.993963i $$0.465006\pi$$
$$602$$ 22.3923 0.912642
$$603$$ −13.1244 −0.534465
$$604$$ 4.14359 0.168600
$$605$$ −10.8564 −0.441376
$$606$$ −42.2233 −1.71521
$$607$$ 12.7279 0.516610 0.258305 0.966063i $$-0.416836\pi$$
0.258305 + 0.966063i $$0.416836\pi$$
$$608$$ 32.1962 1.30573
$$609$$ 0 0
$$610$$ −22.3923 −0.906638
$$611$$ −61.2645 −2.47849
$$612$$ −6.03579 −0.243982
$$613$$ −19.0718 −0.770303 −0.385151 0.922853i $$-0.625851\pi$$
−0.385151 + 0.922853i $$0.625851\pi$$
$$614$$ −64.9808 −2.62241
$$615$$ −8.39230 −0.338410
$$616$$ 0.535898 0.0215920
$$617$$ 21.4906 0.865179 0.432590 0.901591i $$-0.357600\pi$$
0.432590 + 0.901591i $$0.357600\pi$$
$$618$$ 27.8564 1.12055
$$619$$ −30.5307 −1.22713 −0.613566 0.789643i $$-0.710266\pi$$
−0.613566 + 0.789643i $$0.710266\pi$$
$$620$$ −7.34847 −0.295122
$$621$$ −12.4233 −0.498530
$$622$$ −38.0526 −1.52577
$$623$$ −21.1117 −0.845821
$$624$$ 34.4959 1.38094
$$625$$ 1.00000 0.0400000
$$626$$ −35.8086 −1.43120
$$627$$ −2.27362 −0.0907998
$$628$$ 12.7279 0.507899
$$629$$ −14.7846 −0.589501
$$630$$ 5.27792 0.210277
$$631$$ −9.85641 −0.392377 −0.196189 0.980566i $$-0.562857\pi$$
−0.196189 + 0.980566i $$0.562857\pi$$
$$632$$ −0.588457 −0.0234076
$$633$$ −31.1769 −1.23917
$$634$$ −35.1244 −1.39497
$$635$$ 12.7279 0.505092
$$636$$ 0 0
$$637$$ −2.53590 −0.100476
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 4.10394 0.162222
$$641$$ −20.8343 −0.822904 −0.411452 0.911431i $$-0.634978\pi$$
−0.411452 + 0.911431i $$0.634978\pi$$
$$642$$ −22.3923 −0.883754
$$643$$ −48.0526 −1.89501 −0.947504 0.319744i $$-0.896403\pi$$
−0.947504 + 0.319744i $$0.896403\pi$$
$$644$$ −10.3923 −0.409514
$$645$$ −6.00000 −0.236250
$$646$$ −28.5617 −1.12374
$$647$$ −45.3731 −1.78380 −0.891900 0.452233i $$-0.850627\pi$$
−0.891900 + 0.452233i $$0.850627\pi$$
$$648$$ 2.58819 0.101674
$$649$$ 2.27362 0.0892476
$$650$$ −10.5558 −0.414034
$$651$$ 16.3923 0.642465
$$652$$ −7.34847 −0.287788
$$653$$ 1.89469 0.0741448 0.0370724 0.999313i $$-0.488197\pi$$
0.0370724 + 0.999313i $$0.488197\pi$$
$$654$$ −14.9282 −0.583739
$$655$$ −11.9700 −0.467708
$$656$$ 26.4911 1.03430
$$657$$ −15.8338 −0.617733
$$658$$ −59.1769 −2.30696
$$659$$ 20.4553 0.796826 0.398413 0.917206i $$-0.369561\pi$$
0.398413 + 0.917206i $$0.369561\pi$$
$$660$$ 0.928203 0.0361303
$$661$$ 21.8564 0.850116 0.425058 0.905166i $$-0.360254\pi$$
0.425058 + 0.905166i $$0.360254\pi$$
$$662$$ 20.1962 0.784946
$$663$$ −26.9282 −1.04580
$$664$$ 4.24264 0.164646
$$665$$ 11.5911 0.449484
$$666$$ 8.19615 0.317594
$$667$$ 0 0
$$668$$ −24.5885 −0.951356
$$669$$ 10.8332 0.418837
$$670$$ 25.3543 0.979522
$$671$$ −4.39230 −0.169563
$$672$$ 29.3205 1.13106
$$673$$ 9.32051 0.359279 0.179640 0.983732i $$-0.442507\pi$$
0.179640 + 0.983732i $$0.442507\pi$$
$$674$$ 63.3731 2.44104
$$675$$ −5.65685 −0.217732
$$676$$ 29.1962 1.12293
$$677$$ −44.4698 −1.70911 −0.854556 0.519360i $$-0.826170\pi$$
−0.854556 + 0.519360i $$0.826170\pi$$
$$678$$ −24.2175 −0.930067
$$679$$ 20.0764 0.770461
$$680$$ −1.80385 −0.0691744
$$681$$ −37.0470 −1.41964
$$682$$ −3.10583 −0.118928
$$683$$ 21.8038 0.834301 0.417151 0.908837i $$-0.363029\pi$$
0.417151 + 0.908837i $$0.363029\pi$$
$$684$$ 7.34847 0.280976
$$685$$ −9.89949 −0.378240
$$686$$ 34.4959 1.31706
$$687$$ 36.0000 1.37349
$$688$$ 18.9396 0.722065
$$689$$ 0 0
$$690$$ 6.00000 0.228416
$$691$$ −16.9282 −0.643979 −0.321990 0.946743i $$-0.604352\pi$$
−0.321990 + 0.946743i $$0.604352\pi$$
$$692$$ 36.0000 1.36851
$$693$$ 1.03528 0.0393269
$$694$$ −30.5307 −1.15893
$$695$$ −6.53590 −0.247921
$$696$$ 0 0
$$697$$ −20.6795 −0.783292
$$698$$ −4.62158 −0.174929
$$699$$ 2.27362 0.0859964
$$700$$ −4.73205 −0.178855
$$701$$ 8.78461 0.331790 0.165895 0.986143i $$-0.446949\pi$$
0.165895 + 0.986143i $$0.446949\pi$$
$$702$$ 59.7128 2.25371
$$703$$ 18.0000 0.678883
$$704$$ −2.17209 −0.0818637
$$705$$ 15.8564 0.597187
$$706$$ 16.9706 0.638696
$$707$$ 42.2233 1.58797
$$708$$ 14.6969 0.552345
$$709$$ 48.7846 1.83214 0.916072 0.401013i $$-0.131342\pi$$
0.916072 + 0.401013i $$0.131342\pi$$
$$710$$ −11.5911 −0.435007
$$711$$ −1.13681 −0.0426338
$$712$$ −4.00000 −0.149906
$$713$$ −9.31749 −0.348943
$$714$$ −26.0106 −0.973424
$$715$$ −2.07055 −0.0774343
$$716$$ 7.60770 0.284313
$$717$$ 8.48528 0.316889
$$718$$ 13.6603 0.509796
$$719$$ 10.3923 0.387568 0.193784 0.981044i $$-0.437924\pi$$
0.193784 + 0.981044i $$0.437924\pi$$
$$720$$ 4.46410 0.166367
$$721$$ −27.8564 −1.03743
$$722$$ −1.93185 −0.0718961
$$723$$ 6.96953 0.259200
$$724$$ 37.8564 1.40692
$$725$$ 0 0
$$726$$ −29.6603 −1.10080
$$727$$ 8.18067 0.303404 0.151702 0.988426i $$-0.451525\pi$$
0.151702 + 0.988426i $$0.451525\pi$$
$$728$$ −7.72741 −0.286397
$$729$$ 29.0000 1.07407
$$730$$ 30.5885 1.13213
$$731$$ −14.7846 −0.546829
$$732$$ −28.3923 −1.04941
$$733$$ −44.3954 −1.63978 −0.819891 0.572519i $$-0.805966\pi$$
−0.819891 + 0.572519i $$0.805966\pi$$
$$734$$ 52.9808 1.95556
$$735$$ 0.656339 0.0242094
$$736$$ −16.6660 −0.614315
$$737$$ 4.97331 0.183194
$$738$$ 11.4641 0.421999
$$739$$ 42.1218 1.54948 0.774738 0.632283i $$-0.217882\pi$$
0.774738 + 0.632283i $$0.217882\pi$$
$$740$$ −7.34847 −0.270135
$$741$$ 32.7846 1.20437
$$742$$ 0 0
$$743$$ 19.6975 0.722629 0.361315 0.932444i $$-0.382328\pi$$
0.361315 + 0.932444i $$0.382328\pi$$
$$744$$ 3.10583 0.113865
$$745$$ 18.0000 0.659469
$$746$$ −42.5007 −1.55606
$$747$$ 8.19615 0.299882
$$748$$ 2.28719 0.0836278
$$749$$ 22.3923 0.818197
$$750$$ 2.73205 0.0997604
$$751$$ −10.4543 −0.381483 −0.190741 0.981640i $$-0.561089\pi$$
−0.190741 + 0.981640i $$0.561089\pi$$
$$752$$ −50.0523 −1.82522
$$753$$ 12.5359 0.456834
$$754$$ 0 0
$$755$$ 2.39230 0.0870649
$$756$$ 26.7685 0.973562
$$757$$ 13.5601 0.492851 0.246426 0.969162i $$-0.420744\pi$$
0.246426 + 0.969162i $$0.420744\pi$$
$$758$$ 2.19615 0.0797678
$$759$$ 1.17691 0.0427193
$$760$$ 2.19615 0.0796628
$$761$$ 10.3923 0.376721 0.188360 0.982100i $$-0.439683\pi$$
0.188360 + 0.982100i $$0.439683\pi$$
$$762$$ 34.7733 1.25970
$$763$$ 14.9282 0.540437
$$764$$ 43.0827 1.55868
$$765$$ −3.48477 −0.125992
$$766$$ −15.8338 −0.572097
$$767$$ −32.7846 −1.18378
$$768$$ 27.4249 0.989609
$$769$$ 14.6969 0.529985 0.264993 0.964250i $$-0.414630\pi$$
0.264993 + 0.964250i $$0.414630\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 16.9706 0.611180
$$772$$ 22.0454 0.793432
$$773$$ −18.3848 −0.661254 −0.330627 0.943761i $$-0.607260\pi$$
−0.330627 + 0.943761i $$0.607260\pi$$
$$774$$ 8.19615 0.294605
$$775$$ −4.24264 −0.152400
$$776$$ 3.80385 0.136550
$$777$$ 16.3923 0.588071
$$778$$ 3.07180 0.110129
$$779$$ 25.1769 0.902057
$$780$$ −13.3843 −0.479233
$$781$$ −2.27362 −0.0813567
$$782$$ 14.7846 0.528697
$$783$$ 0 0
$$784$$ −2.07180 −0.0739927
$$785$$ 7.34847 0.262278
$$786$$ −32.7028 −1.16647
$$787$$ −7.41154 −0.264193 −0.132096 0.991237i $$-0.542171\pi$$
−0.132096 + 0.991237i $$0.542171\pi$$
$$788$$ −38.7846 −1.38164
$$789$$ −34.7846 −1.23836
$$790$$ 2.19615 0.0781356
$$791$$ 24.2175 0.861075
$$792$$ 0.196152 0.00696997
$$793$$ 63.3350 2.24909
$$794$$ 18.5606 0.658693
$$795$$ 0 0
$$796$$ −21.7128 −0.769590
$$797$$ −21.4906 −0.761236 −0.380618 0.924732i $$-0.624289\pi$$
−0.380618 + 0.924732i $$0.624289\pi$$
$$798$$ 31.6675 1.12102
$$799$$ 39.0718 1.38226
$$800$$ −7.58871 −0.268301
$$801$$ −7.72741 −0.273034
$$802$$ −63.3350 −2.23644
$$803$$ 6.00000 0.211735
$$804$$ 32.1480 1.13377
$$805$$ −6.00000 −0.211472
$$806$$ 44.7846 1.57747
$$807$$ 2.24871 0.0791584
$$808$$ 8.00000 0.281439
$$809$$ −41.6685 −1.46499 −0.732494 0.680774i $$-0.761644\pi$$
−0.732494 + 0.680774i $$0.761644\pi$$
$$810$$ −9.65926 −0.339392
$$811$$ 39.3205 1.38073 0.690365 0.723461i $$-0.257450\pi$$
0.690365 + 0.723461i $$0.257450\pi$$
$$812$$ 0 0
$$813$$ 34.3923 1.20619
$$814$$ −3.10583 −0.108859
$$815$$ −4.24264 −0.148613
$$816$$ −22.0000 −0.770154
$$817$$ 18.0000 0.629740
$$818$$ 50.7846 1.77564
$$819$$ −14.9282 −0.521634
$$820$$ −10.2784 −0.358938
$$821$$ −14.7846 −0.515986 −0.257993 0.966147i $$-0.583061\pi$$
−0.257993 + 0.966147i $$0.583061\pi$$
$$822$$ −27.0459 −0.943335
$$823$$ 8.18067 0.285160 0.142580 0.989783i $$-0.454460\pi$$
0.142580 + 0.989783i $$0.454460\pi$$
$$824$$ −5.27792 −0.183865
$$825$$ 0.535898 0.0186576
$$826$$ −31.6675 −1.10185
$$827$$ 3.68784 0.128239 0.0641193 0.997942i $$-0.479576\pi$$
0.0641193 + 0.997942i $$0.479576\pi$$
$$828$$ −3.80385 −0.132193
$$829$$ 30.8353 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$830$$ −15.8338 −0.549598
$$831$$ −15.4548 −0.536122
$$832$$ 31.3205 1.08584
$$833$$ 1.61729 0.0560356
$$834$$ −17.8564 −0.618317
$$835$$ −14.1962 −0.491278
$$836$$ −2.78461 −0.0963077
$$837$$ 24.0000 0.829561
$$838$$ 34.7733 1.20122
$$839$$ −8.86422 −0.306027 −0.153013 0.988224i $$-0.548898\pi$$
−0.153013 + 0.988224i $$0.548898\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 0 0
$$842$$ −50.7846 −1.75015
$$843$$ 23.1822 0.798438
$$844$$ −38.1838 −1.31434
$$845$$ 16.8564 0.579878
$$846$$ −21.6603 −0.744695
$$847$$ 29.6603 1.01914
$$848$$ 0 0
$$849$$ −43.8134 −1.50367
$$850$$ 6.73205 0.230907
$$851$$ −9.31749 −0.319399
$$852$$ −14.6969 −0.503509
$$853$$ 6.51626 0.223113 0.111556 0.993758i $$-0.464416\pi$$
0.111556 + 0.993758i $$0.464416\pi$$
$$854$$ 61.1769 2.09343
$$855$$ 4.24264 0.145095
$$856$$ 4.24264 0.145010
$$857$$ −3.21539 −0.109836 −0.0549178 0.998491i $$-0.517490\pi$$
−0.0549178 + 0.998491i $$0.517490\pi$$
$$858$$ −5.65685 −0.193122
$$859$$ 39.0160 1.33121 0.665604 0.746305i $$-0.268174\pi$$
0.665604 + 0.746305i $$0.268174\pi$$
$$860$$ −7.34847 −0.250581
$$861$$ 22.9282 0.781391
$$862$$ −31.6675 −1.07860
$$863$$ 27.8038 0.946454 0.473227 0.880941i $$-0.343089\pi$$
0.473227 + 0.880941i $$0.343089\pi$$
$$864$$ 42.9282 1.46045
$$865$$ 20.7846 0.706698
$$866$$ −24.5885 −0.835550
$$867$$ −6.86800 −0.233249
$$868$$ 20.0764 0.681437
$$869$$ 0.430781 0.0146132
$$870$$ 0 0
$$871$$ −71.7128 −2.42990
$$872$$ 2.82843 0.0957826
$$873$$ 7.34847 0.248708
$$874$$ −18.0000 −0.608859
$$875$$ −2.73205 −0.0923602
$$876$$ 38.7846 1.31041
$$877$$ 16.7846 0.566776 0.283388 0.959005i $$-0.408542\pi$$
0.283388 + 0.959005i $$0.408542\pi$$
$$878$$ 9.52056 0.321303
$$879$$ −23.4641 −0.791425
$$880$$ −1.69161 −0.0570243
$$881$$ −6.13733 −0.206772 −0.103386 0.994641i $$-0.532968\pi$$
−0.103386 + 0.994641i $$0.532968\pi$$
$$882$$ −0.896575 −0.0301893
$$883$$ −8.58846 −0.289025 −0.144512 0.989503i $$-0.546161\pi$$
−0.144512 + 0.989503i $$0.546161\pi$$
$$884$$ −32.9802 −1.10924
$$885$$ 8.48528 0.285230
$$886$$ 9.66025 0.324543
$$887$$ 18.1817 0.610482 0.305241 0.952275i $$-0.401263\pi$$
0.305241 + 0.952275i $$0.401263\pi$$
$$888$$ 3.10583 0.104225
$$889$$ −34.7733 −1.16626
$$890$$ 14.9282 0.500395
$$891$$ −1.89469 −0.0634744
$$892$$ 13.2679 0.444244
$$893$$ −47.5692 −1.59184
$$894$$ 49.1769 1.64472
$$895$$ 4.39230 0.146819
$$896$$ −11.2122 −0.374572
$$897$$ −16.9706 −0.566631
$$898$$ −11.8564 −0.395653
$$899$$ 0 0
$$900$$ −1.73205 −0.0577350
$$901$$ 0 0
$$902$$ −4.34418 −0.144645
$$903$$ 16.3923 0.545502
$$904$$ 4.58846 0.152610
$$905$$ 21.8564 0.726532
$$906$$ 6.53590 0.217141
$$907$$ −6.51626 −0.216369 −0.108185 0.994131i $$-0.534504\pi$$
−0.108185 + 0.994131i $$0.534504\pi$$
$$908$$ −45.3731 −1.50576
$$909$$ 15.4548 0.512604
$$910$$ 28.8391 0.956006
$$911$$ −18.1817 −0.602387 −0.301193 0.953563i $$-0.597385\pi$$
−0.301193 + 0.953563i $$0.597385\pi$$
$$912$$ 26.7846 0.886927
$$913$$ −3.10583 −0.102788
$$914$$ −25.2528 −0.835287
$$915$$ −16.3923 −0.541913
$$916$$ 44.0908 1.45680
$$917$$ 32.7028 1.07994
$$918$$ −38.0822 −1.25690
$$919$$ 3.85641 0.127211 0.0636056 0.997975i $$-0.479740\pi$$
0.0636056 + 0.997975i $$0.479740\pi$$
$$920$$ −1.13681 −0.0374796
$$921$$ −47.5692 −1.56746
$$922$$ 43.7128 1.43960
$$923$$ 32.7846 1.07912
$$924$$ −2.53590 −0.0834249
$$925$$ −4.24264 −0.139497
$$926$$ 8.38375 0.275507
$$927$$ −10.1962 −0.334886
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ −11.5911 −0.380087
$$931$$ −1.96902 −0.0645319
$$932$$ 2.78461 0.0912129
$$933$$ −27.8564 −0.911978
$$934$$ −14.7321 −0.482047
$$935$$ 1.32051 0.0431852
$$936$$ −2.82843 −0.0924500
$$937$$ −2.24871 −0.0734622 −0.0367311 0.999325i $$-0.511695\pi$$
−0.0367311 + 0.999325i $$0.511695\pi$$
$$938$$ −69.2693 −2.26172
$$939$$ −26.2137 −0.855452
$$940$$ 19.4201 0.633412
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 20.0764 0.654124
$$943$$ −13.0325 −0.424398
$$944$$ −26.7846 −0.871765
$$945$$ 15.4548 0.502745
$$946$$ −3.10583 −0.100979
$$947$$ −0.453267 −0.0147292 −0.00736460 0.999973i $$-0.502344\pi$$
−0.00736460 + 0.999973i $$0.502344\pi$$
$$948$$ 2.78461 0.0904399
$$949$$ −86.5172 −2.80847
$$950$$ −8.19615 −0.265918
$$951$$ −25.7128 −0.833795
$$952$$ 4.92820 0.159724
$$953$$ 35.5692 1.15220 0.576100 0.817379i $$-0.304574\pi$$
0.576100 + 0.817379i $$0.304574\pi$$
$$954$$ 0 0
$$955$$ 24.8738 0.804898
$$956$$ 10.3923 0.336111
$$957$$ 0 0
$$958$$ −19.6603 −0.635194
$$959$$ 27.0459 0.873358
$$960$$ −8.10634 −0.261631
$$961$$ −13.0000 −0.419355
$$962$$ 44.7846 1.44391
$$963$$ 8.19615 0.264117
$$964$$ 8.53590 0.274923
$$965$$ 12.7279 0.409726
$$966$$ −16.3923 −0.527414
$$967$$ −0.304608 −0.00979553 −0.00489776 0.999988i $$-0.501559\pi$$
−0.00489776 + 0.999988i $$0.501559\pi$$
$$968$$ 5.61969 0.180624
$$969$$ −20.9086 −0.671681
$$970$$ −14.1962 −0.455811
$$971$$ −22.3228 −0.716373 −0.358187 0.933650i $$-0.616605\pi$$
−0.358187 + 0.933650i $$0.616605\pi$$
$$972$$ 17.1464 0.549972
$$973$$ 17.8564 0.572450
$$974$$ 16.5916 0.531630
$$975$$ −7.72741 −0.247475
$$976$$ 51.7439 1.65628
$$977$$ 48.0000 1.53566 0.767828 0.640656i $$-0.221338\pi$$
0.767828 + 0.640656i $$0.221338\pi$$
$$978$$ −11.5911 −0.370643
$$979$$ 2.92820 0.0935858
$$980$$ 0.803848 0.0256780
$$981$$ 5.46410 0.174455
$$982$$ 2.73205 0.0871832
$$983$$ −30.2533 −0.964930 −0.482465 0.875915i $$-0.660258\pi$$
−0.482465 + 0.875915i $$0.660258\pi$$
$$984$$ 4.34418 0.138487
$$985$$ −22.3923 −0.713478
$$986$$ 0 0
$$987$$ −43.3205 −1.37891
$$988$$ 40.1528 1.27743
$$989$$ −9.31749 −0.296279
$$990$$ −0.732051 −0.0232661
$$991$$ −26.6795 −0.847502 −0.423751 0.905779i $$-0.639287\pi$$
−0.423751 + 0.905779i $$0.639287\pi$$
$$992$$ 32.1962 1.02223
$$993$$ 14.7846 0.469175
$$994$$ 31.6675 1.00443
$$995$$ −12.5359 −0.397415
$$996$$ −20.0764 −0.636145
$$997$$ −13.5601 −0.429454 −0.214727 0.976674i $$-0.568886\pi$$
−0.214727 + 0.976674i $$0.568886\pi$$
$$998$$ −42.2233 −1.33656
$$999$$ 24.0000 0.759326
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 4205.2.a.g.1.4 4
29.12 odd 4 145.2.c.a.86.4 yes 4
29.17 odd 4 145.2.c.a.86.1 4
29.28 even 2 inner 4205.2.a.g.1.1 4
87.17 even 4 1305.2.d.a.811.4 4
87.41 even 4 1305.2.d.a.811.1 4
116.75 even 4 2320.2.g.e.1681.4 4
116.99 even 4 2320.2.g.e.1681.2 4
145.12 even 4 725.2.d.b.724.1 8
145.17 even 4 725.2.d.b.724.7 8
145.99 odd 4 725.2.c.d.376.1 4
145.104 odd 4 725.2.c.d.376.4 4
145.128 even 4 725.2.d.b.724.8 8
145.133 even 4 725.2.d.b.724.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.c.a.86.1 4 29.17 odd 4
145.2.c.a.86.4 yes 4 29.12 odd 4
725.2.c.d.376.1 4 145.99 odd 4
725.2.c.d.376.4 4 145.104 odd 4
725.2.d.b.724.1 8 145.12 even 4
725.2.d.b.724.2 8 145.133 even 4
725.2.d.b.724.7 8 145.17 even 4
725.2.d.b.724.8 8 145.128 even 4
1305.2.d.a.811.1 4 87.41 even 4
1305.2.d.a.811.4 4 87.17 even 4
2320.2.g.e.1681.2 4 116.99 even 4
2320.2.g.e.1681.4 4 116.75 even 4
4205.2.a.g.1.1 4 29.28 even 2 inner
4205.2.a.g.1.4 4 1.1 even 1 trivial