# Properties

 Label 154.2.a.d.1.1 Level $154$ Weight $2$ Character 154.1 Self dual yes Analytic conductor $1.230$ 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] = [154,2,Mod(1,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, 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("154.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 154.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +3.23607 q^{5} -3.23607 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.47214 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +3.23607 q^{5} -3.23607 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.47214 q^{9} +3.23607 q^{10} +1.00000 q^{11} -3.23607 q^{12} +1.23607 q^{13} +1.00000 q^{14} -10.4721 q^{15} +1.00000 q^{16} -6.47214 q^{17} +7.47214 q^{18} -2.76393 q^{19} +3.23607 q^{20} -3.23607 q^{21} +1.00000 q^{22} +4.00000 q^{23} -3.23607 q^{24} +5.47214 q^{25} +1.23607 q^{26} -14.4721 q^{27} +1.00000 q^{28} -4.47214 q^{29} -10.4721 q^{30} +2.00000 q^{31} +1.00000 q^{32} -3.23607 q^{33} -6.47214 q^{34} +3.23607 q^{35} +7.47214 q^{36} -10.9443 q^{37} -2.76393 q^{38} -4.00000 q^{39} +3.23607 q^{40} +6.47214 q^{41} -3.23607 q^{42} -1.52786 q^{43} +1.00000 q^{44} +24.1803 q^{45} +4.00000 q^{46} -2.00000 q^{47} -3.23607 q^{48} +1.00000 q^{49} +5.47214 q^{50} +20.9443 q^{51} +1.23607 q^{52} -0.472136 q^{53} -14.4721 q^{54} +3.23607 q^{55} +1.00000 q^{56} +8.94427 q^{57} -4.47214 q^{58} +7.23607 q^{59} -10.4721 q^{60} -5.23607 q^{61} +2.00000 q^{62} +7.47214 q^{63} +1.00000 q^{64} +4.00000 q^{65} -3.23607 q^{66} -15.4164 q^{67} -6.47214 q^{68} -12.9443 q^{69} +3.23607 q^{70} -2.47214 q^{71} +7.47214 q^{72} -4.94427 q^{73} -10.9443 q^{74} -17.7082 q^{75} -2.76393 q^{76} +1.00000 q^{77} -4.00000 q^{78} +3.23607 q^{80} +24.4164 q^{81} +6.47214 q^{82} +10.1803 q^{83} -3.23607 q^{84} -20.9443 q^{85} -1.52786 q^{86} +14.4721 q^{87} +1.00000 q^{88} +10.0000 q^{89} +24.1803 q^{90} +1.23607 q^{91} +4.00000 q^{92} -6.47214 q^{93} -2.00000 q^{94} -8.94427 q^{95} -3.23607 q^{96} +3.52786 q^{97} +1.00000 q^{98} +7.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 6 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} - 12 q^{15} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 10 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 20 q^{27} + 2 q^{28} - 12 q^{30} + 4 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} + 2 q^{35} + 6 q^{36} - 4 q^{37} - 10 q^{38} - 8 q^{39} + 2 q^{40} + 4 q^{41} - 2 q^{42} - 12 q^{43} + 2 q^{44} + 26 q^{45} + 8 q^{46} - 4 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 24 q^{51} - 2 q^{52} + 8 q^{53} - 20 q^{54} + 2 q^{55} + 2 q^{56} + 10 q^{59} - 12 q^{60} - 6 q^{61} + 4 q^{62} + 6 q^{63} + 2 q^{64} + 8 q^{65} - 2 q^{66} - 4 q^{67} - 4 q^{68} - 8 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 8 q^{73} - 4 q^{74} - 22 q^{75} - 10 q^{76} + 2 q^{77} - 8 q^{78} + 2 q^{80} + 22 q^{81} + 4 q^{82} - 2 q^{83} - 2 q^{84} - 24 q^{85} - 12 q^{86} + 20 q^{87} + 2 q^{88} + 20 q^{89} + 26 q^{90} - 2 q^{91} + 8 q^{92} - 4 q^{93} - 4 q^{94} - 2 q^{96} + 16 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 6 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^14 - 12 * q^15 + 2 * q^16 - 4 * q^17 + 6 * q^18 - 10 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 - 20 * q^27 + 2 * q^28 - 12 * q^30 + 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^35 + 6 * q^36 - 4 * q^37 - 10 * q^38 - 8 * q^39 + 2 * q^40 + 4 * q^41 - 2 * q^42 - 12 * q^43 + 2 * q^44 + 26 * q^45 + 8 * q^46 - 4 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 + 24 * q^51 - 2 * q^52 + 8 * q^53 - 20 * q^54 + 2 * q^55 + 2 * q^56 + 10 * q^59 - 12 * q^60 - 6 * q^61 + 4 * q^62 + 6 * q^63 + 2 * q^64 + 8 * q^65 - 2 * q^66 - 4 * q^67 - 4 * q^68 - 8 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 8 * q^73 - 4 * q^74 - 22 * q^75 - 10 * q^76 + 2 * q^77 - 8 * q^78 + 2 * q^80 + 22 * q^81 + 4 * q^82 - 2 * q^83 - 2 * q^84 - 24 * q^85 - 12 * q^86 + 20 * q^87 + 2 * q^88 + 20 * q^89 + 26 * q^90 - 2 * q^91 + 8 * q^92 - 4 * q^93 - 4 * q^94 - 2 * q^96 + 16 * q^97 + 2 * q^98 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −3.23607 −1.86834 −0.934172 0.356822i $$-0.883860\pi$$
−0.934172 + 0.356822i $$0.883860\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 3.23607 1.44721 0.723607 0.690212i $$-0.242483\pi$$
0.723607 + 0.690212i $$0.242483\pi$$
$$6$$ −3.23607 −1.32112
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 7.47214 2.49071
$$10$$ 3.23607 1.02333
$$11$$ 1.00000 0.301511
$$12$$ −3.23607 −0.934172
$$13$$ 1.23607 0.342824 0.171412 0.985199i $$-0.445167\pi$$
0.171412 + 0.985199i $$0.445167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −10.4721 −2.70389
$$16$$ 1.00000 0.250000
$$17$$ −6.47214 −1.56972 −0.784862 0.619671i $$-0.787266\pi$$
−0.784862 + 0.619671i $$0.787266\pi$$
$$18$$ 7.47214 1.76120
$$19$$ −2.76393 −0.634089 −0.317045 0.948411i $$-0.602691\pi$$
−0.317045 + 0.948411i $$0.602691\pi$$
$$20$$ 3.23607 0.723607
$$21$$ −3.23607 −0.706168
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −3.23607 −0.660560
$$25$$ 5.47214 1.09443
$$26$$ 1.23607 0.242413
$$27$$ −14.4721 −2.78516
$$28$$ 1.00000 0.188982
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ −10.4721 −1.91194
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −3.23607 −0.563327
$$34$$ −6.47214 −1.10996
$$35$$ 3.23607 0.546995
$$36$$ 7.47214 1.24536
$$37$$ −10.9443 −1.79923 −0.899614 0.436687i $$-0.856152\pi$$
−0.899614 + 0.436687i $$0.856152\pi$$
$$38$$ −2.76393 −0.448369
$$39$$ −4.00000 −0.640513
$$40$$ 3.23607 0.511667
$$41$$ 6.47214 1.01078 0.505389 0.862892i $$-0.331349\pi$$
0.505389 + 0.862892i $$0.331349\pi$$
$$42$$ −3.23607 −0.499336
$$43$$ −1.52786 −0.232997 −0.116499 0.993191i $$-0.537167\pi$$
−0.116499 + 0.993191i $$0.537167\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 24.1803 3.60459
$$46$$ 4.00000 0.589768
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ −3.23607 −0.467086
$$49$$ 1.00000 0.142857
$$50$$ 5.47214 0.773877
$$51$$ 20.9443 2.93278
$$52$$ 1.23607 0.171412
$$53$$ −0.472136 −0.0648529 −0.0324264 0.999474i $$-0.510323\pi$$
−0.0324264 + 0.999474i $$0.510323\pi$$
$$54$$ −14.4721 −1.96941
$$55$$ 3.23607 0.436351
$$56$$ 1.00000 0.133631
$$57$$ 8.94427 1.18470
$$58$$ −4.47214 −0.587220
$$59$$ 7.23607 0.942056 0.471028 0.882118i $$-0.343883\pi$$
0.471028 + 0.882118i $$0.343883\pi$$
$$60$$ −10.4721 −1.35195
$$61$$ −5.23607 −0.670410 −0.335205 0.942145i $$-0.608806\pi$$
−0.335205 + 0.942145i $$0.608806\pi$$
$$62$$ 2.00000 0.254000
$$63$$ 7.47214 0.941401
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ −3.23607 −0.398332
$$67$$ −15.4164 −1.88341 −0.941707 0.336434i $$-0.890779\pi$$
−0.941707 + 0.336434i $$0.890779\pi$$
$$68$$ −6.47214 −0.784862
$$69$$ −12.9443 −1.55831
$$70$$ 3.23607 0.386784
$$71$$ −2.47214 −0.293389 −0.146694 0.989182i $$-0.546863\pi$$
−0.146694 + 0.989182i $$0.546863\pi$$
$$72$$ 7.47214 0.880600
$$73$$ −4.94427 −0.578683 −0.289342 0.957226i $$-0.593436\pi$$
−0.289342 + 0.957226i $$0.593436\pi$$
$$74$$ −10.9443 −1.27225
$$75$$ −17.7082 −2.04477
$$76$$ −2.76393 −0.317045
$$77$$ 1.00000 0.113961
$$78$$ −4.00000 −0.452911
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 3.23607 0.361803
$$81$$ 24.4164 2.71293
$$82$$ 6.47214 0.714728
$$83$$ 10.1803 1.11744 0.558719 0.829357i $$-0.311293\pi$$
0.558719 + 0.829357i $$0.311293\pi$$
$$84$$ −3.23607 −0.353084
$$85$$ −20.9443 −2.27173
$$86$$ −1.52786 −0.164754
$$87$$ 14.4721 1.55158
$$88$$ 1.00000 0.106600
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 24.1803 2.54883
$$91$$ 1.23607 0.129575
$$92$$ 4.00000 0.417029
$$93$$ −6.47214 −0.671129
$$94$$ −2.00000 −0.206284
$$95$$ −8.94427 −0.917663
$$96$$ −3.23607 −0.330280
$$97$$ 3.52786 0.358200 0.179100 0.983831i $$-0.442681\pi$$
0.179100 + 0.983831i $$0.442681\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 7.47214 0.750978
$$100$$ 5.47214 0.547214
$$101$$ −14.1803 −1.41100 −0.705498 0.708712i $$-0.749277\pi$$
−0.705498 + 0.708712i $$0.749277\pi$$
$$102$$ 20.9443 2.07379
$$103$$ 2.94427 0.290108 0.145054 0.989424i $$-0.453664\pi$$
0.145054 + 0.989424i $$0.453664\pi$$
$$104$$ 1.23607 0.121206
$$105$$ −10.4721 −1.02198
$$106$$ −0.472136 −0.0458579
$$107$$ −6.47214 −0.625685 −0.312842 0.949805i $$-0.601281\pi$$
−0.312842 + 0.949805i $$0.601281\pi$$
$$108$$ −14.4721 −1.39258
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 3.23607 0.308547
$$111$$ 35.4164 3.36158
$$112$$ 1.00000 0.0944911
$$113$$ 8.47214 0.796992 0.398496 0.917170i $$-0.369532\pi$$
0.398496 + 0.917170i $$0.369532\pi$$
$$114$$ 8.94427 0.837708
$$115$$ 12.9443 1.20706
$$116$$ −4.47214 −0.415227
$$117$$ 9.23607 0.853875
$$118$$ 7.23607 0.666134
$$119$$ −6.47214 −0.593300
$$120$$ −10.4721 −0.955971
$$121$$ 1.00000 0.0909091
$$122$$ −5.23607 −0.474051
$$123$$ −20.9443 −1.88848
$$124$$ 2.00000 0.179605
$$125$$ 1.52786 0.136656
$$126$$ 7.47214 0.665671
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.94427 0.435319
$$130$$ 4.00000 0.350823
$$131$$ 9.23607 0.806959 0.403480 0.914989i $$-0.367801\pi$$
0.403480 + 0.914989i $$0.367801\pi$$
$$132$$ −3.23607 −0.281664
$$133$$ −2.76393 −0.239663
$$134$$ −15.4164 −1.33177
$$135$$ −46.8328 −4.03073
$$136$$ −6.47214 −0.554981
$$137$$ 15.8885 1.35745 0.678725 0.734393i $$-0.262533\pi$$
0.678725 + 0.734393i $$0.262533\pi$$
$$138$$ −12.9443 −1.10189
$$139$$ −8.29180 −0.703301 −0.351650 0.936131i $$-0.614379\pi$$
−0.351650 + 0.936131i $$0.614379\pi$$
$$140$$ 3.23607 0.273498
$$141$$ 6.47214 0.545052
$$142$$ −2.47214 −0.207457
$$143$$ 1.23607 0.103365
$$144$$ 7.47214 0.622678
$$145$$ −14.4721 −1.20185
$$146$$ −4.94427 −0.409191
$$147$$ −3.23607 −0.266906
$$148$$ −10.9443 −0.899614
$$149$$ 22.3607 1.83186 0.915929 0.401340i $$-0.131455\pi$$
0.915929 + 0.401340i $$0.131455\pi$$
$$150$$ −17.7082 −1.44587
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ −2.76393 −0.224184
$$153$$ −48.3607 −3.90973
$$154$$ 1.00000 0.0805823
$$155$$ 6.47214 0.519854
$$156$$ −4.00000 −0.320256
$$157$$ 18.6525 1.48863 0.744315 0.667829i $$-0.232776\pi$$
0.744315 + 0.667829i $$0.232776\pi$$
$$158$$ 0 0
$$159$$ 1.52786 0.121168
$$160$$ 3.23607 0.255834
$$161$$ 4.00000 0.315244
$$162$$ 24.4164 1.91833
$$163$$ 7.41641 0.580898 0.290449 0.956890i $$-0.406195\pi$$
0.290449 + 0.956890i $$0.406195\pi$$
$$164$$ 6.47214 0.505389
$$165$$ −10.4721 −0.815255
$$166$$ 10.1803 0.790148
$$167$$ −15.4164 −1.19296 −0.596479 0.802629i $$-0.703434\pi$$
−0.596479 + 0.802629i $$0.703434\pi$$
$$168$$ −3.23607 −0.249668
$$169$$ −11.4721 −0.882472
$$170$$ −20.9443 −1.60635
$$171$$ −20.6525 −1.57933
$$172$$ −1.52786 −0.116499
$$173$$ 1.23607 0.0939765 0.0469883 0.998895i $$-0.485038\pi$$
0.0469883 + 0.998895i $$0.485038\pi$$
$$174$$ 14.4721 1.09713
$$175$$ 5.47214 0.413655
$$176$$ 1.00000 0.0753778
$$177$$ −23.4164 −1.76008
$$178$$ 10.0000 0.749532
$$179$$ 8.94427 0.668526 0.334263 0.942480i $$-0.391513\pi$$
0.334263 + 0.942480i $$0.391513\pi$$
$$180$$ 24.1803 1.80230
$$181$$ 4.76393 0.354100 0.177050 0.984202i $$-0.443345\pi$$
0.177050 + 0.984202i $$0.443345\pi$$
$$182$$ 1.23607 0.0916235
$$183$$ 16.9443 1.25256
$$184$$ 4.00000 0.294884
$$185$$ −35.4164 −2.60387
$$186$$ −6.47214 −0.474560
$$187$$ −6.47214 −0.473289
$$188$$ −2.00000 −0.145865
$$189$$ −14.4721 −1.05269
$$190$$ −8.94427 −0.648886
$$191$$ 6.47214 0.468307 0.234154 0.972200i $$-0.424768\pi$$
0.234154 + 0.972200i $$0.424768\pi$$
$$192$$ −3.23607 −0.233543
$$193$$ 2.94427 0.211933 0.105967 0.994370i $$-0.466206\pi$$
0.105967 + 0.994370i $$0.466206\pi$$
$$194$$ 3.52786 0.253286
$$195$$ −12.9443 −0.926959
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 7.47214 0.531022
$$199$$ 1.05573 0.0748386 0.0374193 0.999300i $$-0.488086\pi$$
0.0374193 + 0.999300i $$0.488086\pi$$
$$200$$ 5.47214 0.386938
$$201$$ 49.8885 3.51887
$$202$$ −14.1803 −0.997725
$$203$$ −4.47214 −0.313882
$$204$$ 20.9443 1.46639
$$205$$ 20.9443 1.46281
$$206$$ 2.94427 0.205137
$$207$$ 29.8885 2.07740
$$208$$ 1.23607 0.0857059
$$209$$ −2.76393 −0.191185
$$210$$ −10.4721 −0.722646
$$211$$ −22.4721 −1.54705 −0.773523 0.633768i $$-0.781507\pi$$
−0.773523 + 0.633768i $$0.781507\pi$$
$$212$$ −0.472136 −0.0324264
$$213$$ 8.00000 0.548151
$$214$$ −6.47214 −0.442426
$$215$$ −4.94427 −0.337197
$$216$$ −14.4721 −0.984704
$$217$$ 2.00000 0.135769
$$218$$ −10.0000 −0.677285
$$219$$ 16.0000 1.08118
$$220$$ 3.23607 0.218176
$$221$$ −8.00000 −0.538138
$$222$$ 35.4164 2.37699
$$223$$ 8.47214 0.567336 0.283668 0.958923i $$-0.408449\pi$$
0.283668 + 0.958923i $$0.408449\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 40.8885 2.72590
$$226$$ 8.47214 0.563558
$$227$$ −14.7639 −0.979917 −0.489958 0.871746i $$-0.662988\pi$$
−0.489958 + 0.871746i $$0.662988\pi$$
$$228$$ 8.94427 0.592349
$$229$$ 12.7639 0.843464 0.421732 0.906720i $$-0.361422\pi$$
0.421732 + 0.906720i $$0.361422\pi$$
$$230$$ 12.9443 0.853520
$$231$$ −3.23607 −0.212918
$$232$$ −4.47214 −0.293610
$$233$$ 2.94427 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$234$$ 9.23607 0.603781
$$235$$ −6.47214 −0.422196
$$236$$ 7.23607 0.471028
$$237$$ 0 0
$$238$$ −6.47214 −0.419526
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ −10.4721 −0.675973
$$241$$ −11.4164 −0.735395 −0.367698 0.929945i $$-0.619854\pi$$
−0.367698 + 0.929945i $$0.619854\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −35.5967 −2.28353
$$244$$ −5.23607 −0.335205
$$245$$ 3.23607 0.206745
$$246$$ −20.9443 −1.33536
$$247$$ −3.41641 −0.217381
$$248$$ 2.00000 0.127000
$$249$$ −32.9443 −2.08776
$$250$$ 1.52786 0.0966306
$$251$$ 24.7639 1.56309 0.781543 0.623852i $$-0.214433\pi$$
0.781543 + 0.623852i $$0.214433\pi$$
$$252$$ 7.47214 0.470700
$$253$$ 4.00000 0.251478
$$254$$ −12.0000 −0.752947
$$255$$ 67.7771 4.24437
$$256$$ 1.00000 0.0625000
$$257$$ −10.9443 −0.682685 −0.341342 0.939939i $$-0.610882\pi$$
−0.341342 + 0.939939i $$0.610882\pi$$
$$258$$ 4.94427 0.307817
$$259$$ −10.9443 −0.680044
$$260$$ 4.00000 0.248069
$$261$$ −33.4164 −2.06842
$$262$$ 9.23607 0.570606
$$263$$ 12.9443 0.798178 0.399089 0.916912i $$-0.369326\pi$$
0.399089 + 0.916912i $$0.369326\pi$$
$$264$$ −3.23607 −0.199166
$$265$$ −1.52786 −0.0938559
$$266$$ −2.76393 −0.169468
$$267$$ −32.3607 −1.98044
$$268$$ −15.4164 −0.941707
$$269$$ −27.2361 −1.66061 −0.830306 0.557307i $$-0.811834\pi$$
−0.830306 + 0.557307i $$0.811834\pi$$
$$270$$ −46.8328 −2.85015
$$271$$ −16.9443 −1.02929 −0.514646 0.857403i $$-0.672076\pi$$
−0.514646 + 0.857403i $$0.672076\pi$$
$$272$$ −6.47214 −0.392431
$$273$$ −4.00000 −0.242091
$$274$$ 15.8885 0.959862
$$275$$ 5.47214 0.329982
$$276$$ −12.9443 −0.779154
$$277$$ 12.4721 0.749378 0.374689 0.927151i $$-0.377749\pi$$
0.374689 + 0.927151i $$0.377749\pi$$
$$278$$ −8.29180 −0.497309
$$279$$ 14.9443 0.894690
$$280$$ 3.23607 0.193392
$$281$$ −24.8328 −1.48140 −0.740701 0.671835i $$-0.765506\pi$$
−0.740701 + 0.671835i $$0.765506\pi$$
$$282$$ 6.47214 0.385410
$$283$$ −16.6525 −0.989887 −0.494943 0.868925i $$-0.664811\pi$$
−0.494943 + 0.868925i $$0.664811\pi$$
$$284$$ −2.47214 −0.146694
$$285$$ 28.9443 1.71451
$$286$$ 1.23607 0.0730902
$$287$$ 6.47214 0.382038
$$288$$ 7.47214 0.440300
$$289$$ 24.8885 1.46403
$$290$$ −14.4721 −0.849833
$$291$$ −11.4164 −0.669242
$$292$$ −4.94427 −0.289342
$$293$$ 4.65248 0.271801 0.135900 0.990723i $$-0.456607\pi$$
0.135900 + 0.990723i $$0.456607\pi$$
$$294$$ −3.23607 −0.188731
$$295$$ 23.4164 1.36336
$$296$$ −10.9443 −0.636123
$$297$$ −14.4721 −0.839759
$$298$$ 22.3607 1.29532
$$299$$ 4.94427 0.285935
$$300$$ −17.7082 −1.02238
$$301$$ −1.52786 −0.0880646
$$302$$ 12.0000 0.690522
$$303$$ 45.8885 2.63623
$$304$$ −2.76393 −0.158522
$$305$$ −16.9443 −0.970226
$$306$$ −48.3607 −2.76460
$$307$$ 32.0689 1.83027 0.915134 0.403150i $$-0.132085\pi$$
0.915134 + 0.403150i $$0.132085\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ −9.52786 −0.542021
$$310$$ 6.47214 0.367593
$$311$$ 5.41641 0.307136 0.153568 0.988138i $$-0.450924\pi$$
0.153568 + 0.988138i $$0.450924\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 28.4721 1.60934 0.804670 0.593722i $$-0.202342\pi$$
0.804670 + 0.593722i $$0.202342\pi$$
$$314$$ 18.6525 1.05262
$$315$$ 24.1803 1.36241
$$316$$ 0 0
$$317$$ −13.0557 −0.733283 −0.366641 0.930362i $$-0.619492\pi$$
−0.366641 + 0.930362i $$0.619492\pi$$
$$318$$ 1.52786 0.0856784
$$319$$ −4.47214 −0.250392
$$320$$ 3.23607 0.180902
$$321$$ 20.9443 1.16900
$$322$$ 4.00000 0.222911
$$323$$ 17.8885 0.995345
$$324$$ 24.4164 1.35647
$$325$$ 6.76393 0.375195
$$326$$ 7.41641 0.410757
$$327$$ 32.3607 1.78955
$$328$$ 6.47214 0.357364
$$329$$ −2.00000 −0.110264
$$330$$ −10.4721 −0.576472
$$331$$ 0.944272 0.0519019 0.0259509 0.999663i $$-0.491739\pi$$
0.0259509 + 0.999663i $$0.491739\pi$$
$$332$$ 10.1803 0.558719
$$333$$ −81.7771 −4.48136
$$334$$ −15.4164 −0.843548
$$335$$ −49.8885 −2.72570
$$336$$ −3.23607 −0.176542
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −11.4721 −0.624002
$$339$$ −27.4164 −1.48905
$$340$$ −20.9443 −1.13586
$$341$$ 2.00000 0.108306
$$342$$ −20.6525 −1.11676
$$343$$ 1.00000 0.0539949
$$344$$ −1.52786 −0.0823769
$$345$$ −41.8885 −2.25520
$$346$$ 1.23607 0.0664514
$$347$$ −6.47214 −0.347442 −0.173721 0.984795i $$-0.555579\pi$$
−0.173721 + 0.984795i $$0.555579\pi$$
$$348$$ 14.4721 0.775788
$$349$$ 8.29180 0.443850 0.221925 0.975064i $$-0.428766\pi$$
0.221925 + 0.975064i $$0.428766\pi$$
$$350$$ 5.47214 0.292498
$$351$$ −17.8885 −0.954820
$$352$$ 1.00000 0.0533002
$$353$$ −34.9443 −1.85990 −0.929948 0.367691i $$-0.880148\pi$$
−0.929948 + 0.367691i $$0.880148\pi$$
$$354$$ −23.4164 −1.24457
$$355$$ −8.00000 −0.424596
$$356$$ 10.0000 0.529999
$$357$$ 20.9443 1.10849
$$358$$ 8.94427 0.472719
$$359$$ −26.8328 −1.41618 −0.708091 0.706121i $$-0.750443\pi$$
−0.708091 + 0.706121i $$0.750443\pi$$
$$360$$ 24.1803 1.27442
$$361$$ −11.3607 −0.597931
$$362$$ 4.76393 0.250387
$$363$$ −3.23607 −0.169850
$$364$$ 1.23607 0.0647876
$$365$$ −16.0000 −0.837478
$$366$$ 16.9443 0.885691
$$367$$ 21.4164 1.11793 0.558964 0.829192i $$-0.311199\pi$$
0.558964 + 0.829192i $$0.311199\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 48.3607 2.51756
$$370$$ −35.4164 −1.84121
$$371$$ −0.472136 −0.0245121
$$372$$ −6.47214 −0.335565
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ −6.47214 −0.334666
$$375$$ −4.94427 −0.255321
$$376$$ −2.00000 −0.103142
$$377$$ −5.52786 −0.284699
$$378$$ −14.4721 −0.744366
$$379$$ 5.52786 0.283947 0.141974 0.989870i $$-0.454655\pi$$
0.141974 + 0.989870i $$0.454655\pi$$
$$380$$ −8.94427 −0.458831
$$381$$ 38.8328 1.98947
$$382$$ 6.47214 0.331143
$$383$$ 11.8885 0.607476 0.303738 0.952756i $$-0.401765\pi$$
0.303738 + 0.952756i $$0.401765\pi$$
$$384$$ −3.23607 −0.165140
$$385$$ 3.23607 0.164925
$$386$$ 2.94427 0.149859
$$387$$ −11.4164 −0.580329
$$388$$ 3.52786 0.179100
$$389$$ 6.58359 0.333801 0.166901 0.985974i $$-0.446624\pi$$
0.166901 + 0.985974i $$0.446624\pi$$
$$390$$ −12.9443 −0.655459
$$391$$ −25.8885 −1.30924
$$392$$ 1.00000 0.0505076
$$393$$ −29.8885 −1.50768
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 7.47214 0.375489
$$397$$ −10.2918 −0.516530 −0.258265 0.966074i $$-0.583151\pi$$
−0.258265 + 0.966074i $$0.583151\pi$$
$$398$$ 1.05573 0.0529189
$$399$$ 8.94427 0.447774
$$400$$ 5.47214 0.273607
$$401$$ −30.3607 −1.51614 −0.758070 0.652173i $$-0.773857\pi$$
−0.758070 + 0.652173i $$0.773857\pi$$
$$402$$ 49.8885 2.48821
$$403$$ 2.47214 0.123146
$$404$$ −14.1803 −0.705498
$$405$$ 79.0132 3.92620
$$406$$ −4.47214 −0.221948
$$407$$ −10.9443 −0.542487
$$408$$ 20.9443 1.03690
$$409$$ −23.4164 −1.15787 −0.578933 0.815375i $$-0.696531\pi$$
−0.578933 + 0.815375i $$0.696531\pi$$
$$410$$ 20.9443 1.03436
$$411$$ −51.4164 −2.53618
$$412$$ 2.94427 0.145054
$$413$$ 7.23607 0.356064
$$414$$ 29.8885 1.46894
$$415$$ 32.9443 1.61717
$$416$$ 1.23607 0.0606032
$$417$$ 26.8328 1.31401
$$418$$ −2.76393 −0.135188
$$419$$ 12.7639 0.623559 0.311779 0.950155i $$-0.399075\pi$$
0.311779 + 0.950155i $$0.399075\pi$$
$$420$$ −10.4721 −0.510988
$$421$$ 7.52786 0.366886 0.183443 0.983030i $$-0.441276\pi$$
0.183443 + 0.983030i $$0.441276\pi$$
$$422$$ −22.4721 −1.09393
$$423$$ −14.9443 −0.726615
$$424$$ −0.472136 −0.0229289
$$425$$ −35.4164 −1.71795
$$426$$ 8.00000 0.387601
$$427$$ −5.23607 −0.253391
$$428$$ −6.47214 −0.312842
$$429$$ −4.00000 −0.193122
$$430$$ −4.94427 −0.238434
$$431$$ 40.9443 1.97222 0.986108 0.166105i $$-0.0531190\pi$$
0.986108 + 0.166105i $$0.0531190\pi$$
$$432$$ −14.4721 −0.696291
$$433$$ 19.5279 0.938449 0.469225 0.883079i $$-0.344533\pi$$
0.469225 + 0.883079i $$0.344533\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 46.8328 2.24546
$$436$$ −10.0000 −0.478913
$$437$$ −11.0557 −0.528867
$$438$$ 16.0000 0.764510
$$439$$ −8.94427 −0.426887 −0.213443 0.976955i $$-0.568468\pi$$
−0.213443 + 0.976955i $$0.568468\pi$$
$$440$$ 3.23607 0.154273
$$441$$ 7.47214 0.355816
$$442$$ −8.00000 −0.380521
$$443$$ −7.05573 −0.335228 −0.167614 0.985853i $$-0.553606\pi$$
−0.167614 + 0.985853i $$0.553606\pi$$
$$444$$ 35.4164 1.68079
$$445$$ 32.3607 1.53404
$$446$$ 8.47214 0.401167
$$447$$ −72.3607 −3.42254
$$448$$ 1.00000 0.0472456
$$449$$ 1.05573 0.0498229 0.0249114 0.999690i $$-0.492070\pi$$
0.0249114 + 0.999690i $$0.492070\pi$$
$$450$$ 40.8885 1.92750
$$451$$ 6.47214 0.304761
$$452$$ 8.47214 0.398496
$$453$$ −38.8328 −1.82452
$$454$$ −14.7639 −0.692906
$$455$$ 4.00000 0.187523
$$456$$ 8.94427 0.418854
$$457$$ 9.05573 0.423609 0.211805 0.977312i $$-0.432066\pi$$
0.211805 + 0.977312i $$0.432066\pi$$
$$458$$ 12.7639 0.596419
$$459$$ 93.6656 4.37194
$$460$$ 12.9443 0.603530
$$461$$ 29.2361 1.36166 0.680830 0.732442i $$-0.261619\pi$$
0.680830 + 0.732442i $$0.261619\pi$$
$$462$$ −3.23607 −0.150556
$$463$$ −21.5279 −1.00048 −0.500242 0.865885i $$-0.666756\pi$$
−0.500242 + 0.865885i $$0.666756\pi$$
$$464$$ −4.47214 −0.207614
$$465$$ −20.9443 −0.971267
$$466$$ 2.94427 0.136391
$$467$$ 13.1246 0.607335 0.303667 0.952778i $$-0.401789\pi$$
0.303667 + 0.952778i $$0.401789\pi$$
$$468$$ 9.23607 0.426937
$$469$$ −15.4164 −0.711864
$$470$$ −6.47214 −0.298537
$$471$$ −60.3607 −2.78127
$$472$$ 7.23607 0.333067
$$473$$ −1.52786 −0.0702513
$$474$$ 0 0
$$475$$ −15.1246 −0.693965
$$476$$ −6.47214 −0.296650
$$477$$ −3.52786 −0.161530
$$478$$ 20.0000 0.914779
$$479$$ −32.3607 −1.47860 −0.739299 0.673378i $$-0.764843\pi$$
−0.739299 + 0.673378i $$0.764843\pi$$
$$480$$ −10.4721 −0.477985
$$481$$ −13.5279 −0.616818
$$482$$ −11.4164 −0.520003
$$483$$ −12.9443 −0.588985
$$484$$ 1.00000 0.0454545
$$485$$ 11.4164 0.518392
$$486$$ −35.5967 −1.61470
$$487$$ −0.944272 −0.0427890 −0.0213945 0.999771i $$-0.506811\pi$$
−0.0213945 + 0.999771i $$0.506811\pi$$
$$488$$ −5.23607 −0.237026
$$489$$ −24.0000 −1.08532
$$490$$ 3.23607 0.146191
$$491$$ 0.944272 0.0426144 0.0213072 0.999773i $$-0.493217\pi$$
0.0213072 + 0.999773i $$0.493217\pi$$
$$492$$ −20.9443 −0.944241
$$493$$ 28.9443 1.30358
$$494$$ −3.41641 −0.153711
$$495$$ 24.1803 1.08683
$$496$$ 2.00000 0.0898027
$$497$$ −2.47214 −0.110890
$$498$$ −32.9443 −1.47627
$$499$$ 12.3607 0.553340 0.276670 0.960965i $$-0.410769\pi$$
0.276670 + 0.960965i $$0.410769\pi$$
$$500$$ 1.52786 0.0683282
$$501$$ 49.8885 2.22886
$$502$$ 24.7639 1.10527
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ 7.47214 0.332835
$$505$$ −45.8885 −2.04201
$$506$$ 4.00000 0.177822
$$507$$ 37.1246 1.64876
$$508$$ −12.0000 −0.532414
$$509$$ −34.0689 −1.51008 −0.755038 0.655681i $$-0.772382\pi$$
−0.755038 + 0.655681i $$0.772382\pi$$
$$510$$ 67.7771 3.00122
$$511$$ −4.94427 −0.218722
$$512$$ 1.00000 0.0441942
$$513$$ 40.0000 1.76604
$$514$$ −10.9443 −0.482731
$$515$$ 9.52786 0.419848
$$516$$ 4.94427 0.217659
$$517$$ −2.00000 −0.0879599
$$518$$ −10.9443 −0.480864
$$519$$ −4.00000 −0.175581
$$520$$ 4.00000 0.175412
$$521$$ 34.3607 1.50537 0.752684 0.658382i $$-0.228759\pi$$
0.752684 + 0.658382i $$0.228759\pi$$
$$522$$ −33.4164 −1.46260
$$523$$ −27.7082 −1.21160 −0.605798 0.795619i $$-0.707146\pi$$
−0.605798 + 0.795619i $$0.707146\pi$$
$$524$$ 9.23607 0.403480
$$525$$ −17.7082 −0.772849
$$526$$ 12.9443 0.564397
$$527$$ −12.9443 −0.563861
$$528$$ −3.23607 −0.140832
$$529$$ −7.00000 −0.304348
$$530$$ −1.52786 −0.0663662
$$531$$ 54.0689 2.34639
$$532$$ −2.76393 −0.119832
$$533$$ 8.00000 0.346518
$$534$$ −32.3607 −1.40038
$$535$$ −20.9443 −0.905500
$$536$$ −15.4164 −0.665887
$$537$$ −28.9443 −1.24904
$$538$$ −27.2361 −1.17423
$$539$$ 1.00000 0.0430730
$$540$$ −46.8328 −2.01536
$$541$$ −9.05573 −0.389336 −0.194668 0.980869i $$-0.562363\pi$$
−0.194668 + 0.980869i $$0.562363\pi$$
$$542$$ −16.9443 −0.727819
$$543$$ −15.4164 −0.661581
$$544$$ −6.47214 −0.277491
$$545$$ −32.3607 −1.38618
$$546$$ −4.00000 −0.171184
$$547$$ 16.9443 0.724485 0.362242 0.932084i $$-0.382011\pi$$
0.362242 + 0.932084i $$0.382011\pi$$
$$548$$ 15.8885 0.678725
$$549$$ −39.1246 −1.66980
$$550$$ 5.47214 0.233333
$$551$$ 12.3607 0.526583
$$552$$ −12.9443 −0.550945
$$553$$ 0 0
$$554$$ 12.4721 0.529890
$$555$$ 114.610 4.86492
$$556$$ −8.29180 −0.351650
$$557$$ −28.8328 −1.22169 −0.610843 0.791752i $$-0.709169\pi$$
−0.610843 + 0.791752i $$0.709169\pi$$
$$558$$ 14.9443 0.632641
$$559$$ −1.88854 −0.0798769
$$560$$ 3.23607 0.136749
$$561$$ 20.9443 0.884268
$$562$$ −24.8328 −1.04751
$$563$$ 26.7639 1.12797 0.563983 0.825787i $$-0.309268\pi$$
0.563983 + 0.825787i $$0.309268\pi$$
$$564$$ 6.47214 0.272526
$$565$$ 27.4164 1.15342
$$566$$ −16.6525 −0.699956
$$567$$ 24.4164 1.02539
$$568$$ −2.47214 −0.103729
$$569$$ 16.8328 0.705668 0.352834 0.935686i $$-0.385218\pi$$
0.352834 + 0.935686i $$0.385218\pi$$
$$570$$ 28.9443 1.21234
$$571$$ −45.8885 −1.92038 −0.960188 0.279355i $$-0.909879\pi$$
−0.960188 + 0.279355i $$0.909879\pi$$
$$572$$ 1.23607 0.0516826
$$573$$ −20.9443 −0.874960
$$574$$ 6.47214 0.270142
$$575$$ 21.8885 0.912815
$$576$$ 7.47214 0.311339
$$577$$ 9.05573 0.376995 0.188497 0.982074i $$-0.439638\pi$$
0.188497 + 0.982074i $$0.439638\pi$$
$$578$$ 24.8885 1.03523
$$579$$ −9.52786 −0.395965
$$580$$ −14.4721 −0.600923
$$581$$ 10.1803 0.422352
$$582$$ −11.4164 −0.473225
$$583$$ −0.472136 −0.0195539
$$584$$ −4.94427 −0.204595
$$585$$ 29.8885 1.23574
$$586$$ 4.65248 0.192192
$$587$$ −28.1803 −1.16313 −0.581564 0.813501i $$-0.697559\pi$$
−0.581564 + 0.813501i $$0.697559\pi$$
$$588$$ −3.23607 −0.133453
$$589$$ −5.52786 −0.227772
$$590$$ 23.4164 0.964038
$$591$$ −58.2492 −2.39605
$$592$$ −10.9443 −0.449807
$$593$$ 24.0000 0.985562 0.492781 0.870153i $$-0.335980\pi$$
0.492781 + 0.870153i $$0.335980\pi$$
$$594$$ −14.4721 −0.593799
$$595$$ −20.9443 −0.858631
$$596$$ 22.3607 0.915929
$$597$$ −3.41641 −0.139824
$$598$$ 4.94427 0.202186
$$599$$ 12.3607 0.505044 0.252522 0.967591i $$-0.418740\pi$$
0.252522 + 0.967591i $$0.418740\pi$$
$$600$$ −17.7082 −0.722934
$$601$$ 18.8328 0.768207 0.384103 0.923290i $$-0.374511\pi$$
0.384103 + 0.923290i $$0.374511\pi$$
$$602$$ −1.52786 −0.0622711
$$603$$ −115.193 −4.69104
$$604$$ 12.0000 0.488273
$$605$$ 3.23607 0.131565
$$606$$ 45.8885 1.86409
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ −2.76393 −0.112092
$$609$$ 14.4721 0.586441
$$610$$ −16.9443 −0.686054
$$611$$ −2.47214 −0.100012
$$612$$ −48.3607 −1.95486
$$613$$ 19.5279 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$614$$ 32.0689 1.29419
$$615$$ −67.7771 −2.73304
$$616$$ 1.00000 0.0402911
$$617$$ −5.41641 −0.218056 −0.109028 0.994039i $$-0.534774\pi$$
−0.109028 + 0.994039i $$0.534774\pi$$
$$618$$ −9.52786 −0.383267
$$619$$ −48.5410 −1.95103 −0.975514 0.219937i $$-0.929415\pi$$
−0.975514 + 0.219937i $$0.929415\pi$$
$$620$$ 6.47214 0.259927
$$621$$ −57.8885 −2.32299
$$622$$ 5.41641 0.217178
$$623$$ 10.0000 0.400642
$$624$$ −4.00000 −0.160128
$$625$$ −22.4164 −0.896656
$$626$$ 28.4721 1.13798
$$627$$ 8.94427 0.357200
$$628$$ 18.6525 0.744315
$$629$$ 70.8328 2.82429
$$630$$ 24.1803 0.963368
$$631$$ −4.58359 −0.182470 −0.0912350 0.995829i $$-0.529081\pi$$
−0.0912350 + 0.995829i $$0.529081\pi$$
$$632$$ 0 0
$$633$$ 72.7214 2.89041
$$634$$ −13.0557 −0.518509
$$635$$ −38.8328 −1.54103
$$636$$ 1.52786 0.0605838
$$637$$ 1.23607 0.0489748
$$638$$ −4.47214 −0.177054
$$639$$ −18.4721 −0.730746
$$640$$ 3.23607 0.127917
$$641$$ 36.4721 1.44056 0.720281 0.693682i $$-0.244013\pi$$
0.720281 + 0.693682i $$0.244013\pi$$
$$642$$ 20.9443 0.826604
$$643$$ −23.2361 −0.916341 −0.458171 0.888864i $$-0.651495\pi$$
−0.458171 + 0.888864i $$0.651495\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 16.0000 0.629999
$$646$$ 17.8885 0.703815
$$647$$ 24.8328 0.976279 0.488139 0.872766i $$-0.337676\pi$$
0.488139 + 0.872766i $$0.337676\pi$$
$$648$$ 24.4164 0.959167
$$649$$ 7.23607 0.284041
$$650$$ 6.76393 0.265303
$$651$$ −6.47214 −0.253663
$$652$$ 7.41641 0.290449
$$653$$ 1.63932 0.0641516 0.0320758 0.999485i $$-0.489788\pi$$
0.0320758 + 0.999485i $$0.489788\pi$$
$$654$$ 32.3607 1.26540
$$655$$ 29.8885 1.16784
$$656$$ 6.47214 0.252694
$$657$$ −36.9443 −1.44133
$$658$$ −2.00000 −0.0779681
$$659$$ 43.4164 1.69126 0.845632 0.533767i $$-0.179224\pi$$
0.845632 + 0.533767i $$0.179224\pi$$
$$660$$ −10.4721 −0.407627
$$661$$ 37.1246 1.44398 0.721990 0.691903i $$-0.243228\pi$$
0.721990 + 0.691903i $$0.243228\pi$$
$$662$$ 0.944272 0.0367002
$$663$$ 25.8885 1.00543
$$664$$ 10.1803 0.395074
$$665$$ −8.94427 −0.346844
$$666$$ −81.7771 −3.16880
$$667$$ −17.8885 −0.692647
$$668$$ −15.4164 −0.596479
$$669$$ −27.4164 −1.05998
$$670$$ −49.8885 −1.92736
$$671$$ −5.23607 −0.202136
$$672$$ −3.23607 −0.124834
$$673$$ 31.8885 1.22921 0.614607 0.788834i $$-0.289315\pi$$
0.614607 + 0.788834i $$0.289315\pi$$
$$674$$ 18.0000 0.693334
$$675$$ −79.1935 −3.04816
$$676$$ −11.4721 −0.441236
$$677$$ 32.0689 1.23251 0.616254 0.787548i $$-0.288650\pi$$
0.616254 + 0.787548i $$0.288650\pi$$
$$678$$ −27.4164 −1.05292
$$679$$ 3.52786 0.135387
$$680$$ −20.9443 −0.803176
$$681$$ 47.7771 1.83082
$$682$$ 2.00000 0.0765840
$$683$$ 15.0557 0.576091 0.288046 0.957617i $$-0.406994\pi$$
0.288046 + 0.957617i $$0.406994\pi$$
$$684$$ −20.6525 −0.789667
$$685$$ 51.4164 1.96452
$$686$$ 1.00000 0.0381802
$$687$$ −41.3050 −1.57588
$$688$$ −1.52786 −0.0582493
$$689$$ −0.583592 −0.0222331
$$690$$ −41.8885 −1.59467
$$691$$ −18.6525 −0.709574 −0.354787 0.934947i $$-0.615447\pi$$
−0.354787 + 0.934947i $$0.615447\pi$$
$$692$$ 1.23607 0.0469883
$$693$$ 7.47214 0.283843
$$694$$ −6.47214 −0.245679
$$695$$ −26.8328 −1.01783
$$696$$ 14.4721 0.548565
$$697$$ −41.8885 −1.58664
$$698$$ 8.29180 0.313849
$$699$$ −9.52786 −0.360377
$$700$$ 5.47214 0.206827
$$701$$ 46.7214 1.76464 0.882321 0.470649i $$-0.155980\pi$$
0.882321 + 0.470649i $$0.155980\pi$$
$$702$$ −17.8885 −0.675160
$$703$$ 30.2492 1.14087
$$704$$ 1.00000 0.0376889
$$705$$ 20.9443 0.788807
$$706$$ −34.9443 −1.31515
$$707$$ −14.1803 −0.533307
$$708$$ −23.4164 −0.880042
$$709$$ −4.47214 −0.167955 −0.0839773 0.996468i $$-0.526762\pi$$
−0.0839773 + 0.996468i $$0.526762\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 8.00000 0.299602
$$714$$ 20.9443 0.783820
$$715$$ 4.00000 0.149592
$$716$$ 8.94427 0.334263
$$717$$ −64.7214 −2.41706
$$718$$ −26.8328 −1.00139
$$719$$ −36.8328 −1.37363 −0.686816 0.726831i $$-0.740992\pi$$
−0.686816 + 0.726831i $$0.740992\pi$$
$$720$$ 24.1803 0.901148
$$721$$ 2.94427 0.109650
$$722$$ −11.3607 −0.422801
$$723$$ 36.9443 1.37397
$$724$$ 4.76393 0.177050
$$725$$ −24.4721 −0.908872
$$726$$ −3.23607 −0.120102
$$727$$ 18.0000 0.667583 0.333792 0.942647i $$-0.391672\pi$$
0.333792 + 0.942647i $$0.391672\pi$$
$$728$$ 1.23607 0.0458117
$$729$$ 41.9443 1.55349
$$730$$ −16.0000 −0.592187
$$731$$ 9.88854 0.365741
$$732$$ 16.9443 0.626278
$$733$$ 8.87539 0.327820 0.163910 0.986475i $$-0.447589\pi$$
0.163910 + 0.986475i $$0.447589\pi$$
$$734$$ 21.4164 0.790494
$$735$$ −10.4721 −0.386271
$$736$$ 4.00000 0.147442
$$737$$ −15.4164 −0.567871
$$738$$ 48.3607 1.78018
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ −35.4164 −1.30193
$$741$$ 11.0557 0.406142
$$742$$ −0.472136 −0.0173327
$$743$$ −13.8885 −0.509521 −0.254761 0.967004i $$-0.581997\pi$$
−0.254761 + 0.967004i $$0.581997\pi$$
$$744$$ −6.47214 −0.237280
$$745$$ 72.3607 2.65109
$$746$$ −6.00000 −0.219676
$$747$$ 76.0689 2.78321
$$748$$ −6.47214 −0.236645
$$749$$ −6.47214 −0.236487
$$750$$ −4.94427 −0.180539
$$751$$ 0.944272 0.0344570 0.0172285 0.999852i $$-0.494516\pi$$
0.0172285 + 0.999852i $$0.494516\pi$$
$$752$$ −2.00000 −0.0729325
$$753$$ −80.1378 −2.92038
$$754$$ −5.52786 −0.201313
$$755$$ 38.8328 1.41327
$$756$$ −14.4721 −0.526346
$$757$$ 39.3050 1.42856 0.714281 0.699859i $$-0.246754\pi$$
0.714281 + 0.699859i $$0.246754\pi$$
$$758$$ 5.52786 0.200781
$$759$$ −12.9443 −0.469847
$$760$$ −8.94427 −0.324443
$$761$$ 15.4164 0.558844 0.279422 0.960168i $$-0.409857\pi$$
0.279422 + 0.960168i $$0.409857\pi$$
$$762$$ 38.8328 1.40676
$$763$$ −10.0000 −0.362024
$$764$$ 6.47214 0.234154
$$765$$ −156.498 −5.65821
$$766$$ 11.8885 0.429551
$$767$$ 8.94427 0.322959
$$768$$ −3.23607 −0.116772
$$769$$ −16.5836 −0.598020 −0.299010 0.954250i $$-0.596656\pi$$
−0.299010 + 0.954250i $$0.596656\pi$$
$$770$$ 3.23607 0.116620
$$771$$ 35.4164 1.27549
$$772$$ 2.94427 0.105967
$$773$$ 2.29180 0.0824302 0.0412151 0.999150i $$-0.486877\pi$$
0.0412151 + 0.999150i $$0.486877\pi$$
$$774$$ −11.4164 −0.410354
$$775$$ 10.9443 0.393130
$$776$$ 3.52786 0.126643
$$777$$ 35.4164 1.27056
$$778$$ 6.58359 0.236033
$$779$$ −17.8885 −0.640924
$$780$$ −12.9443 −0.463479
$$781$$ −2.47214 −0.0884600
$$782$$ −25.8885 −0.925772
$$783$$ 64.7214 2.31295
$$784$$ 1.00000 0.0357143
$$785$$ 60.3607 2.15437
$$786$$ −29.8885 −1.06609
$$787$$ −5.81966 −0.207448 −0.103724 0.994606i $$-0.533076\pi$$
−0.103724 + 0.994606i $$0.533076\pi$$
$$788$$ 18.0000 0.641223
$$789$$ −41.8885 −1.49127
$$790$$ 0 0
$$791$$ 8.47214 0.301234
$$792$$ 7.47214 0.265511
$$793$$ −6.47214 −0.229832
$$794$$ −10.2918 −0.365242
$$795$$ 4.94427 0.175355
$$796$$ 1.05573 0.0374193
$$797$$ 7.59675 0.269091 0.134545 0.990907i $$-0.457043\pi$$
0.134545 + 0.990907i $$0.457043\pi$$
$$798$$ 8.94427 0.316624
$$799$$ 12.9443 0.457935
$$800$$ 5.47214 0.193469
$$801$$ 74.7214 2.64015
$$802$$ −30.3607 −1.07207
$$803$$ −4.94427 −0.174480
$$804$$ 49.8885 1.75943
$$805$$ 12.9443 0.456226
$$806$$ 2.47214 0.0870773
$$807$$ 88.1378 3.10260
$$808$$ −14.1803 −0.498863
$$809$$ −38.9443 −1.36921 −0.684604 0.728915i $$-0.740025\pi$$
−0.684604 + 0.728915i $$0.740025\pi$$
$$810$$ 79.0132 2.77624
$$811$$ 9.23607 0.324322 0.162161 0.986764i $$-0.448154\pi$$
0.162161 + 0.986764i $$0.448154\pi$$
$$812$$ −4.47214 −0.156941
$$813$$ 54.8328 1.92307
$$814$$ −10.9443 −0.383597
$$815$$ 24.0000 0.840683
$$816$$ 20.9443 0.733196
$$817$$ 4.22291 0.147741
$$818$$ −23.4164 −0.818736
$$819$$ 9.23607 0.322734
$$820$$ 20.9443 0.731406
$$821$$ 25.4164 0.887039 0.443519 0.896265i $$-0.353730\pi$$
0.443519 + 0.896265i $$0.353730\pi$$
$$822$$ −51.4164 −1.79335
$$823$$ 34.2492 1.19385 0.596926 0.802296i $$-0.296388\pi$$
0.596926 + 0.802296i $$0.296388\pi$$
$$824$$ 2.94427 0.102569
$$825$$ −17.7082 −0.616521
$$826$$ 7.23607 0.251775
$$827$$ −0.944272 −0.0328356 −0.0164178 0.999865i $$-0.505226\pi$$
−0.0164178 + 0.999865i $$0.505226\pi$$
$$828$$ 29.8885 1.03870
$$829$$ −1.70820 −0.0593284 −0.0296642 0.999560i $$-0.509444\pi$$
−0.0296642 + 0.999560i $$0.509444\pi$$
$$830$$ 32.9443 1.14351
$$831$$ −40.3607 −1.40010
$$832$$ 1.23607 0.0428529
$$833$$ −6.47214 −0.224246
$$834$$ 26.8328 0.929144
$$835$$ −49.8885 −1.72646
$$836$$ −2.76393 −0.0955926
$$837$$ −28.9443 −1.00046
$$838$$ 12.7639 0.440923
$$839$$ −36.8328 −1.27161 −0.635805 0.771850i $$-0.719332\pi$$
−0.635805 + 0.771850i $$0.719332\pi$$
$$840$$ −10.4721 −0.361323
$$841$$ −9.00000 −0.310345
$$842$$ 7.52786 0.259427
$$843$$ 80.3607 2.76777
$$844$$ −22.4721 −0.773523
$$845$$ −37.1246 −1.27713
$$846$$ −14.9443 −0.513795
$$847$$ 1.00000 0.0343604
$$848$$ −0.472136 −0.0162132
$$849$$ 53.8885 1.84945
$$850$$ −35.4164 −1.21477
$$851$$ −43.7771 −1.50066
$$852$$ 8.00000 0.274075
$$853$$ −34.5410 −1.18266 −0.591331 0.806429i $$-0.701397\pi$$
−0.591331 + 0.806429i $$0.701397\pi$$
$$854$$ −5.23607 −0.179175
$$855$$ −66.8328 −2.28563
$$856$$ −6.47214 −0.221213
$$857$$ −37.5279 −1.28193 −0.640964 0.767571i $$-0.721465\pi$$
−0.640964 + 0.767571i $$0.721465\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ −25.1246 −0.857241 −0.428620 0.903485i $$-0.641000\pi$$
−0.428620 + 0.903485i $$0.641000\pi$$
$$860$$ −4.94427 −0.168598
$$861$$ −20.9443 −0.713779
$$862$$ 40.9443 1.39457
$$863$$ 27.4164 0.933265 0.466633 0.884451i $$-0.345467\pi$$
0.466633 + 0.884451i $$0.345467\pi$$
$$864$$ −14.4721 −0.492352
$$865$$ 4.00000 0.136004
$$866$$ 19.5279 0.663584
$$867$$ −80.5410 −2.73532
$$868$$ 2.00000 0.0678844
$$869$$ 0 0
$$870$$ 46.8328 1.58778
$$871$$ −19.0557 −0.645679
$$872$$ −10.0000 −0.338643
$$873$$ 26.3607 0.892174
$$874$$ −11.0557 −0.373966
$$875$$ 1.52786 0.0516512
$$876$$ 16.0000 0.540590
$$877$$ 26.9443 0.909843 0.454922 0.890531i $$-0.349667\pi$$
0.454922 + 0.890531i $$0.349667\pi$$
$$878$$ −8.94427 −0.301855
$$879$$ −15.0557 −0.507817
$$880$$ 3.23607 0.109088
$$881$$ −24.8328 −0.836639 −0.418319 0.908300i $$-0.637381\pi$$
−0.418319 + 0.908300i $$0.637381\pi$$
$$882$$ 7.47214 0.251600
$$883$$ 50.8328 1.71066 0.855330 0.518083i $$-0.173354\pi$$
0.855330 + 0.518083i $$0.173354\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ −75.7771 −2.54722
$$886$$ −7.05573 −0.237042
$$887$$ 0.360680 0.0121104 0.00605522 0.999982i $$-0.498073\pi$$
0.00605522 + 0.999982i $$0.498073\pi$$
$$888$$ 35.4164 1.18850
$$889$$ −12.0000 −0.402467
$$890$$ 32.3607 1.08473
$$891$$ 24.4164 0.817980
$$892$$ 8.47214 0.283668
$$893$$ 5.52786 0.184983
$$894$$ −72.3607 −2.42010
$$895$$ 28.9443 0.967500
$$896$$ 1.00000 0.0334077
$$897$$ −16.0000 −0.534224
$$898$$ 1.05573 0.0352301
$$899$$ −8.94427 −0.298308
$$900$$ 40.8885 1.36295
$$901$$ 3.05573 0.101801
$$902$$ 6.47214 0.215499
$$903$$ 4.94427 0.164535
$$904$$ 8.47214 0.281779
$$905$$ 15.4164 0.512459
$$906$$ −38.8328 −1.29013
$$907$$ 20.3607 0.676065 0.338033 0.941134i $$-0.390239\pi$$
0.338033 + 0.941134i $$0.390239\pi$$
$$908$$ −14.7639 −0.489958
$$909$$ −105.957 −3.51439
$$910$$ 4.00000 0.132599
$$911$$ −28.0000 −0.927681 −0.463841 0.885919i $$-0.653529\pi$$
−0.463841 + 0.885919i $$0.653529\pi$$
$$912$$ 8.94427 0.296174
$$913$$ 10.1803 0.336920
$$914$$ 9.05573 0.299537
$$915$$ 54.8328 1.81272
$$916$$ 12.7639 0.421732
$$917$$ 9.23607 0.305002
$$918$$ 93.6656 3.09143
$$919$$ −57.8885 −1.90957 −0.954783 0.297302i $$-0.903913\pi$$
−0.954783 + 0.297302i $$0.903913\pi$$
$$920$$ 12.9443 0.426760
$$921$$ −103.777 −3.41957
$$922$$ 29.2361 0.962839
$$923$$ −3.05573 −0.100581
$$924$$ −3.23607 −0.106459
$$925$$ −59.8885 −1.96912
$$926$$ −21.5279 −0.707450
$$927$$ 22.0000 0.722575
$$928$$ −4.47214 −0.146805
$$929$$ −40.2492 −1.32053 −0.660267 0.751031i $$-0.729557\pi$$
−0.660267 + 0.751031i $$0.729557\pi$$
$$930$$ −20.9443 −0.686790
$$931$$ −2.76393 −0.0905842
$$932$$ 2.94427 0.0964428
$$933$$ −17.5279 −0.573837
$$934$$ 13.1246 0.429450
$$935$$ −20.9443 −0.684951
$$936$$ 9.23607 0.301890
$$937$$ −20.9443 −0.684220 −0.342110 0.939660i $$-0.611141\pi$$
−0.342110 + 0.939660i $$0.611141\pi$$
$$938$$ −15.4164 −0.503364
$$939$$ −92.1378 −3.00680
$$940$$ −6.47214 −0.211098
$$941$$ −34.1803 −1.11425 −0.557124 0.830430i $$-0.688095\pi$$
−0.557124 + 0.830430i $$0.688095\pi$$
$$942$$ −60.3607 −1.96666
$$943$$ 25.8885 0.843047
$$944$$ 7.23607 0.235514
$$945$$ −46.8328 −1.52347
$$946$$ −1.52786 −0.0496751
$$947$$ −0.944272 −0.0306847 −0.0153424 0.999882i $$-0.504884\pi$$
−0.0153424 + 0.999882i $$0.504884\pi$$
$$948$$ 0 0
$$949$$ −6.11146 −0.198386
$$950$$ −15.1246 −0.490707
$$951$$ 42.2492 1.37002
$$952$$ −6.47214 −0.209763
$$953$$ 5.05573 0.163771 0.0818855 0.996642i $$-0.473906\pi$$
0.0818855 + 0.996642i $$0.473906\pi$$
$$954$$ −3.52786 −0.114219
$$955$$ 20.9443 0.677741
$$956$$ 20.0000 0.646846
$$957$$ 14.4721 0.467818
$$958$$ −32.3607 −1.04553
$$959$$ 15.8885 0.513068
$$960$$ −10.4721 −0.337987
$$961$$ −27.0000 −0.870968
$$962$$ −13.5279 −0.436156
$$963$$ −48.3607 −1.55840
$$964$$ −11.4164 −0.367698
$$965$$ 9.52786 0.306713
$$966$$ −12.9443 −0.416475
$$967$$ 10.1115 0.325163 0.162581 0.986695i $$-0.448018\pi$$
0.162581 + 0.986695i $$0.448018\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −57.8885 −1.85965
$$970$$ 11.4164 0.366559
$$971$$ −16.5410 −0.530827 −0.265413 0.964135i $$-0.585508\pi$$
−0.265413 + 0.964135i $$0.585508\pi$$
$$972$$ −35.5967 −1.14177
$$973$$ −8.29180 −0.265823
$$974$$ −0.944272 −0.0302564
$$975$$ −21.8885 −0.700994
$$976$$ −5.23607 −0.167602
$$977$$ 24.8328 0.794472 0.397236 0.917716i $$-0.369969\pi$$
0.397236 + 0.917716i $$0.369969\pi$$
$$978$$ −24.0000 −0.767435
$$979$$ 10.0000 0.319601
$$980$$ 3.23607 0.103372
$$981$$ −74.7214 −2.38567
$$982$$ 0.944272 0.0301329
$$983$$ 14.0000 0.446531 0.223265 0.974758i $$-0.428328\pi$$
0.223265 + 0.974758i $$0.428328\pi$$
$$984$$ −20.9443 −0.667679
$$985$$ 58.2492 1.85597
$$986$$ 28.9443 0.921773
$$987$$ 6.47214 0.206010
$$988$$ −3.41641 −0.108690
$$989$$ −6.11146 −0.194333
$$990$$ 24.1803 0.768502
$$991$$ 44.3607 1.40916 0.704582 0.709623i $$-0.251135\pi$$
0.704582 + 0.709623i $$0.251135\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ −3.05573 −0.0969706
$$994$$ −2.47214 −0.0784114
$$995$$ 3.41641 0.108307
$$996$$ −32.9443 −1.04388
$$997$$ 1.81966 0.0576292 0.0288146 0.999585i $$-0.490827\pi$$
0.0288146 + 0.999585i $$0.490827\pi$$
$$998$$ 12.3607 0.391270
$$999$$ 158.387 5.01114
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 154.2.a.d.1.1 2
3.2 odd 2 1386.2.a.m.1.1 2
4.3 odd 2 1232.2.a.p.1.2 2
5.2 odd 4 3850.2.c.q.1849.4 4
5.3 odd 4 3850.2.c.q.1849.1 4
5.4 even 2 3850.2.a.bj.1.2 2
7.2 even 3 1078.2.e.q.67.2 4
7.3 odd 6 1078.2.e.n.177.1 4
7.4 even 3 1078.2.e.q.177.2 4
7.5 odd 6 1078.2.e.n.67.1 4
7.6 odd 2 1078.2.a.w.1.2 2
8.3 odd 2 4928.2.a.bk.1.1 2
8.5 even 2 4928.2.a.bt.1.2 2
11.10 odd 2 1694.2.a.l.1.1 2
21.20 even 2 9702.2.a.cu.1.2 2
28.27 even 2 8624.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 2 1.1 even 1 trivial
1078.2.a.w.1.2 2 7.6 odd 2
1078.2.e.n.67.1 4 7.5 odd 6
1078.2.e.n.177.1 4 7.3 odd 6
1078.2.e.q.67.2 4 7.2 even 3
1078.2.e.q.177.2 4 7.4 even 3
1232.2.a.p.1.2 2 4.3 odd 2
1386.2.a.m.1.1 2 3.2 odd 2
1694.2.a.l.1.1 2 11.10 odd 2
3850.2.a.bj.1.2 2 5.4 even 2
3850.2.c.q.1849.1 4 5.3 odd 4
3850.2.c.q.1849.4 4 5.2 odd 4
4928.2.a.bk.1.1 2 8.3 odd 2
4928.2.a.bt.1.2 2 8.5 even 2
8624.2.a.bf.1.1 2 28.27 even 2
9702.2.a.cu.1.2 2 21.20 even 2