# Properties

 Label 9025.2.a.n.1.1 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $2$ 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] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$2$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 361) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} +O(q^{10})$$ $$q-1.61803 q^{2} -0.381966 q^{3} +0.618034 q^{4} +0.618034 q^{6} -3.00000 q^{7} +2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} -0.236068 q^{12} +1.00000 q^{13} +4.85410 q^{14} -4.85410 q^{16} -0.763932 q^{17} +4.61803 q^{18} +1.14590 q^{21} +2.61803 q^{22} -5.38197 q^{23} -0.854102 q^{24} -1.61803 q^{26} +2.23607 q^{27} -1.85410 q^{28} -3.61803 q^{29} -8.85410 q^{31} +3.38197 q^{32} +0.618034 q^{33} +1.23607 q^{34} -1.76393 q^{36} -8.85410 q^{37} -0.381966 q^{39} -3.00000 q^{41} -1.85410 q^{42} +0.145898 q^{43} -1.00000 q^{44} +8.70820 q^{46} -3.00000 q^{47} +1.85410 q^{48} +2.00000 q^{49} +0.291796 q^{51} +0.618034 q^{52} +6.32624 q^{53} -3.61803 q^{54} -6.70820 q^{56} +5.85410 q^{58} +0.326238 q^{59} -10.2361 q^{61} +14.3262 q^{62} +8.56231 q^{63} +4.23607 q^{64} -1.00000 q^{66} +7.00000 q^{67} -0.472136 q^{68} +2.05573 q^{69} -7.47214 q^{71} -6.38197 q^{72} +2.70820 q^{73} +14.3262 q^{74} +4.85410 q^{77} +0.618034 q^{78} +13.4164 q^{79} +7.70820 q^{81} +4.85410 q^{82} -8.47214 q^{83} +0.708204 q^{84} -0.236068 q^{86} +1.38197 q^{87} -3.61803 q^{88} -7.76393 q^{89} -3.00000 q^{91} -3.32624 q^{92} +3.38197 q^{93} +4.85410 q^{94} -1.29180 q^{96} -13.8541 q^{97} -3.23607 q^{98} +4.61803 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - q^{11} + 4 q^{12} + 2 q^{13} + 3 q^{14} - 3 q^{16} - 6 q^{17} + 7 q^{18} + 9 q^{21} + 3 q^{22} - 13 q^{23} + 5 q^{24} - q^{26} + 3 q^{28} - 5 q^{29} - 11 q^{31} + 9 q^{32} - q^{33} - 2 q^{34} - 8 q^{36} - 11 q^{37} - 3 q^{39} - 6 q^{41} + 3 q^{42} + 7 q^{43} - 2 q^{44} + 4 q^{46} - 6 q^{47} - 3 q^{48} + 4 q^{49} + 14 q^{51} - q^{52} - 3 q^{53} - 5 q^{54} + 5 q^{58} - 15 q^{59} - 16 q^{61} + 13 q^{62} - 3 q^{63} + 4 q^{64} - 2 q^{66} + 14 q^{67} + 8 q^{68} + 22 q^{69} - 6 q^{71} - 15 q^{72} - 8 q^{73} + 13 q^{74} + 3 q^{77} - q^{78} + 2 q^{81} + 3 q^{82} - 8 q^{83} - 12 q^{84} + 4 q^{86} + 5 q^{87} - 5 q^{88} - 20 q^{89} - 6 q^{91} + 9 q^{92} + 9 q^{93} + 3 q^{94} - 16 q^{96} - 21 q^{97} - 2 q^{98} + 7 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 - q^11 + 4 * q^12 + 2 * q^13 + 3 * q^14 - 3 * q^16 - 6 * q^17 + 7 * q^18 + 9 * q^21 + 3 * q^22 - 13 * q^23 + 5 * q^24 - q^26 + 3 * q^28 - 5 * q^29 - 11 * q^31 + 9 * q^32 - q^33 - 2 * q^34 - 8 * q^36 - 11 * q^37 - 3 * q^39 - 6 * q^41 + 3 * q^42 + 7 * q^43 - 2 * q^44 + 4 * q^46 - 6 * q^47 - 3 * q^48 + 4 * q^49 + 14 * q^51 - q^52 - 3 * q^53 - 5 * q^54 + 5 * q^58 - 15 * q^59 - 16 * q^61 + 13 * q^62 - 3 * q^63 + 4 * q^64 - 2 * q^66 + 14 * q^67 + 8 * q^68 + 22 * q^69 - 6 * q^71 - 15 * q^72 - 8 * q^73 + 13 * q^74 + 3 * q^77 - q^78 + 2 * q^81 + 3 * q^82 - 8 * q^83 - 12 * q^84 + 4 * q^86 + 5 * q^87 - 5 * q^88 - 20 * q^89 - 6 * q^91 + 9 * q^92 + 9 * q^93 + 3 * q^94 - 16 * q^96 - 21 * q^97 - 2 * q^98 + 7 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61803 −1.14412 −0.572061 0.820211i $$-0.693856\pi$$
−0.572061 + 0.820211i $$0.693856\pi$$
$$3$$ −0.381966 −0.220528 −0.110264 0.993902i $$-0.535170\pi$$
−0.110264 + 0.993902i $$0.535170\pi$$
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 0.618034 0.252311
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 2.23607 0.790569
$$9$$ −2.85410 −0.951367
$$10$$ 0 0
$$11$$ −1.61803 −0.487856 −0.243928 0.969793i $$-0.578436\pi$$
−0.243928 + 0.969793i $$0.578436\pi$$
$$12$$ −0.236068 −0.0681470
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 4.85410 1.29731
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −0.763932 −0.185281 −0.0926404 0.995700i $$-0.529531\pi$$
−0.0926404 + 0.995700i $$0.529531\pi$$
$$18$$ 4.61803 1.08848
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 1.14590 0.250055
$$22$$ 2.61803 0.558167
$$23$$ −5.38197 −1.12222 −0.561109 0.827742i $$-0.689625\pi$$
−0.561109 + 0.827742i $$0.689625\pi$$
$$24$$ −0.854102 −0.174343
$$25$$ 0 0
$$26$$ −1.61803 −0.317323
$$27$$ 2.23607 0.430331
$$28$$ −1.85410 −0.350392
$$29$$ −3.61803 −0.671852 −0.335926 0.941888i $$-0.609049\pi$$
−0.335926 + 0.941888i $$0.609049\pi$$
$$30$$ 0 0
$$31$$ −8.85410 −1.59024 −0.795122 0.606450i $$-0.792593\pi$$
−0.795122 + 0.606450i $$0.792593\pi$$
$$32$$ 3.38197 0.597853
$$33$$ 0.618034 0.107586
$$34$$ 1.23607 0.211984
$$35$$ 0 0
$$36$$ −1.76393 −0.293989
$$37$$ −8.85410 −1.45561 −0.727803 0.685787i $$-0.759458\pi$$
−0.727803 + 0.685787i $$0.759458\pi$$
$$38$$ 0 0
$$39$$ −0.381966 −0.0611635
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ −1.85410 −0.286094
$$43$$ 0.145898 0.0222492 0.0111246 0.999938i $$-0.496459\pi$$
0.0111246 + 0.999938i $$0.496459\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 8.70820 1.28395
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 1.85410 0.267617
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0.291796 0.0408596
$$52$$ 0.618034 0.0857059
$$53$$ 6.32624 0.868976 0.434488 0.900678i $$-0.356929\pi$$
0.434488 + 0.900678i $$0.356929\pi$$
$$54$$ −3.61803 −0.492352
$$55$$ 0 0
$$56$$ −6.70820 −0.896421
$$57$$ 0 0
$$58$$ 5.85410 0.768681
$$59$$ 0.326238 0.0424726 0.0212363 0.999774i $$-0.493240\pi$$
0.0212363 + 0.999774i $$0.493240\pi$$
$$60$$ 0 0
$$61$$ −10.2361 −1.31059 −0.655297 0.755371i $$-0.727457\pi$$
−0.655297 + 0.755371i $$0.727457\pi$$
$$62$$ 14.3262 1.81943
$$63$$ 8.56231 1.07875
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ −0.472136 −0.0572549
$$69$$ 2.05573 0.247481
$$70$$ 0 0
$$71$$ −7.47214 −0.886779 −0.443390 0.896329i $$-0.646224\pi$$
−0.443390 + 0.896329i $$0.646224\pi$$
$$72$$ −6.38197 −0.752122
$$73$$ 2.70820 0.316971 0.158486 0.987361i $$-0.449339\pi$$
0.158486 + 0.987361i $$0.449339\pi$$
$$74$$ 14.3262 1.66539
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.85410 0.553176
$$78$$ 0.618034 0.0699786
$$79$$ 13.4164 1.50946 0.754732 0.656033i $$-0.227767\pi$$
0.754732 + 0.656033i $$0.227767\pi$$
$$80$$ 0 0
$$81$$ 7.70820 0.856467
$$82$$ 4.85410 0.536046
$$83$$ −8.47214 −0.929938 −0.464969 0.885327i $$-0.653934\pi$$
−0.464969 + 0.885327i $$0.653934\pi$$
$$84$$ 0.708204 0.0772714
$$85$$ 0 0
$$86$$ −0.236068 −0.0254559
$$87$$ 1.38197 0.148162
$$88$$ −3.61803 −0.385684
$$89$$ −7.76393 −0.822975 −0.411488 0.911415i $$-0.634991\pi$$
−0.411488 + 0.911415i $$0.634991\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ −3.32624 −0.346784
$$93$$ 3.38197 0.350694
$$94$$ 4.85410 0.500662
$$95$$ 0 0
$$96$$ −1.29180 −0.131843
$$97$$ −13.8541 −1.40667 −0.703335 0.710858i $$-0.748307\pi$$
−0.703335 + 0.710858i $$0.748307\pi$$
$$98$$ −3.23607 −0.326892
$$99$$ 4.61803 0.464130
$$100$$ 0 0
$$101$$ −9.18034 −0.913478 −0.456739 0.889601i $$-0.650983\pi$$
−0.456739 + 0.889601i $$0.650983\pi$$
$$102$$ −0.472136 −0.0467484
$$103$$ −14.3262 −1.41161 −0.705803 0.708408i $$-0.749414\pi$$
−0.705803 + 0.708408i $$0.749414\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −10.2361 −0.994215
$$107$$ −16.4164 −1.58703 −0.793517 0.608548i $$-0.791752\pi$$
−0.793517 + 0.608548i $$0.791752\pi$$
$$108$$ 1.38197 0.132980
$$109$$ −3.29180 −0.315297 −0.157648 0.987495i $$-0.550391\pi$$
−0.157648 + 0.987495i $$0.550391\pi$$
$$110$$ 0 0
$$111$$ 3.38197 0.321002
$$112$$ 14.5623 1.37601
$$113$$ −6.76393 −0.636297 −0.318149 0.948041i $$-0.603061\pi$$
−0.318149 + 0.948041i $$0.603061\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ −2.85410 −0.263862
$$118$$ −0.527864 −0.0485938
$$119$$ 2.29180 0.210089
$$120$$ 0 0
$$121$$ −8.38197 −0.761997
$$122$$ 16.5623 1.49948
$$123$$ 1.14590 0.103322
$$124$$ −5.47214 −0.491412
$$125$$ 0 0
$$126$$ −13.8541 −1.23422
$$127$$ −5.76393 −0.511466 −0.255733 0.966747i $$-0.582317\pi$$
−0.255733 + 0.966747i $$0.582317\pi$$
$$128$$ −13.6180 −1.20368
$$129$$ −0.0557281 −0.00490658
$$130$$ 0 0
$$131$$ 15.0902 1.31843 0.659217 0.751953i $$-0.270888\pi$$
0.659217 + 0.751953i $$0.270888\pi$$
$$132$$ 0.381966 0.0332459
$$133$$ 0 0
$$134$$ −11.3262 −0.978438
$$135$$ 0 0
$$136$$ −1.70820 −0.146477
$$137$$ −7.47214 −0.638388 −0.319194 0.947689i $$-0.603412\pi$$
−0.319194 + 0.947689i $$0.603412\pi$$
$$138$$ −3.32624 −0.283148
$$139$$ 14.7984 1.25518 0.627591 0.778543i $$-0.284041\pi$$
0.627591 + 0.778543i $$0.284041\pi$$
$$140$$ 0 0
$$141$$ 1.14590 0.0965020
$$142$$ 12.0902 1.01458
$$143$$ −1.61803 −0.135307
$$144$$ 13.8541 1.15451
$$145$$ 0 0
$$146$$ −4.38197 −0.362654
$$147$$ −0.763932 −0.0630081
$$148$$ −5.47214 −0.449807
$$149$$ −1.90983 −0.156459 −0.0782297 0.996935i $$-0.524927\pi$$
−0.0782297 + 0.996935i $$0.524927\pi$$
$$150$$ 0 0
$$151$$ −21.0902 −1.71629 −0.858147 0.513404i $$-0.828384\pi$$
−0.858147 + 0.513404i $$0.828384\pi$$
$$152$$ 0 0
$$153$$ 2.18034 0.176270
$$154$$ −7.85410 −0.632902
$$155$$ 0 0
$$156$$ −0.236068 −0.0189006
$$157$$ 17.8541 1.42491 0.712456 0.701717i $$-0.247583\pi$$
0.712456 + 0.701717i $$0.247583\pi$$
$$158$$ −21.7082 −1.72701
$$159$$ −2.41641 −0.191634
$$160$$ 0 0
$$161$$ 16.1459 1.27248
$$162$$ −12.4721 −0.979904
$$163$$ −1.76393 −0.138162 −0.0690809 0.997611i $$-0.522007\pi$$
−0.0690809 + 0.997611i $$0.522007\pi$$
$$164$$ −1.85410 −0.144781
$$165$$ 0 0
$$166$$ 13.7082 1.06396
$$167$$ −20.2361 −1.56591 −0.782957 0.622076i $$-0.786290\pi$$
−0.782957 + 0.622076i $$0.786290\pi$$
$$168$$ 2.56231 0.197686
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0.0901699 0.00687539
$$173$$ 0.472136 0.0358958 0.0179479 0.999839i $$-0.494287\pi$$
0.0179479 + 0.999839i $$0.494287\pi$$
$$174$$ −2.23607 −0.169516
$$175$$ 0 0
$$176$$ 7.85410 0.592025
$$177$$ −0.124612 −0.00936640
$$178$$ 12.5623 0.941585
$$179$$ 12.2361 0.914567 0.457283 0.889321i $$-0.348823\pi$$
0.457283 + 0.889321i $$0.348823\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 4.85410 0.359810
$$183$$ 3.90983 0.289023
$$184$$ −12.0344 −0.887191
$$185$$ 0 0
$$186$$ −5.47214 −0.401236
$$187$$ 1.23607 0.0903902
$$188$$ −1.85410 −0.135224
$$189$$ −6.70820 −0.487950
$$190$$ 0 0
$$191$$ 14.2361 1.03009 0.515043 0.857164i $$-0.327776\pi$$
0.515043 + 0.857164i $$0.327776\pi$$
$$192$$ −1.61803 −0.116772
$$193$$ −5.05573 −0.363919 −0.181960 0.983306i $$-0.558244\pi$$
−0.181960 + 0.983306i $$0.558244\pi$$
$$194$$ 22.4164 1.60940
$$195$$ 0 0
$$196$$ 1.23607 0.0882906
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ −7.47214 −0.531022
$$199$$ −13.4164 −0.951064 −0.475532 0.879698i $$-0.657744\pi$$
−0.475532 + 0.879698i $$0.657744\pi$$
$$200$$ 0 0
$$201$$ −2.67376 −0.188593
$$202$$ 14.8541 1.04513
$$203$$ 10.8541 0.761809
$$204$$ 0.180340 0.0126263
$$205$$ 0 0
$$206$$ 23.1803 1.61505
$$207$$ 15.3607 1.06764
$$208$$ −4.85410 −0.336571
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −3.85410 −0.265327 −0.132664 0.991161i $$-0.542353\pi$$
−0.132664 + 0.991161i $$0.542353\pi$$
$$212$$ 3.90983 0.268528
$$213$$ 2.85410 0.195560
$$214$$ 26.5623 1.81576
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 26.5623 1.80317
$$218$$ 5.32624 0.360738
$$219$$ −1.03444 −0.0699011
$$220$$ 0 0
$$221$$ −0.763932 −0.0513876
$$222$$ −5.47214 −0.367266
$$223$$ 11.6525 0.780307 0.390154 0.920750i $$-0.372422\pi$$
0.390154 + 0.920750i $$0.372422\pi$$
$$224$$ −10.1459 −0.677901
$$225$$ 0 0
$$226$$ 10.9443 0.728002
$$227$$ −16.4164 −1.08960 −0.544798 0.838568i $$-0.683394\pi$$
−0.544798 + 0.838568i $$0.683394\pi$$
$$228$$ 0 0
$$229$$ −13.6180 −0.899905 −0.449953 0.893052i $$-0.648559\pi$$
−0.449953 + 0.893052i $$0.648559\pi$$
$$230$$ 0 0
$$231$$ −1.85410 −0.121991
$$232$$ −8.09017 −0.531146
$$233$$ −4.52786 −0.296630 −0.148315 0.988940i $$-0.547385\pi$$
−0.148315 + 0.988940i $$0.547385\pi$$
$$234$$ 4.61803 0.301890
$$235$$ 0 0
$$236$$ 0.201626 0.0131247
$$237$$ −5.12461 −0.332879
$$238$$ −3.70820 −0.240367
$$239$$ −0.326238 −0.0211026 −0.0105513 0.999944i $$-0.503359\pi$$
−0.0105513 + 0.999944i $$0.503359\pi$$
$$240$$ 0 0
$$241$$ 3.18034 0.204864 0.102432 0.994740i $$-0.467338\pi$$
0.102432 + 0.994740i $$0.467338\pi$$
$$242$$ 13.5623 0.871818
$$243$$ −9.65248 −0.619207
$$244$$ −6.32624 −0.404996
$$245$$ 0 0
$$246$$ −1.85410 −0.118213
$$247$$ 0 0
$$248$$ −19.7984 −1.25720
$$249$$ 3.23607 0.205077
$$250$$ 0 0
$$251$$ −25.3607 −1.60075 −0.800376 0.599498i $$-0.795367\pi$$
−0.800376 + 0.599498i $$0.795367\pi$$
$$252$$ 5.29180 0.333352
$$253$$ 8.70820 0.547480
$$254$$ 9.32624 0.585180
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −20.3607 −1.27006 −0.635032 0.772486i $$-0.719013\pi$$
−0.635032 + 0.772486i $$0.719013\pi$$
$$258$$ 0.0901699 0.00561374
$$259$$ 26.5623 1.65050
$$260$$ 0 0
$$261$$ 10.3262 0.639178
$$262$$ −24.4164 −1.50845
$$263$$ 14.9443 0.921503 0.460752 0.887529i $$-0.347580\pi$$
0.460752 + 0.887529i $$0.347580\pi$$
$$264$$ 1.38197 0.0850541
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.96556 0.181489
$$268$$ 4.32624 0.264267
$$269$$ 30.3262 1.84902 0.924512 0.381154i $$-0.124473\pi$$
0.924512 + 0.381154i $$0.124473\pi$$
$$270$$ 0 0
$$271$$ 1.14590 0.0696083 0.0348042 0.999394i $$-0.488919\pi$$
0.0348042 + 0.999394i $$0.488919\pi$$
$$272$$ 3.70820 0.224843
$$273$$ 1.14590 0.0693529
$$274$$ 12.0902 0.730394
$$275$$ 0 0
$$276$$ 1.27051 0.0764757
$$277$$ −11.4164 −0.685945 −0.342973 0.939345i $$-0.611434\pi$$
−0.342973 + 0.939345i $$0.611434\pi$$
$$278$$ −23.9443 −1.43608
$$279$$ 25.2705 1.51291
$$280$$ 0 0
$$281$$ −29.5066 −1.76021 −0.880107 0.474775i $$-0.842530\pi$$
−0.880107 + 0.474775i $$0.842530\pi$$
$$282$$ −1.85410 −0.110410
$$283$$ −26.0344 −1.54759 −0.773793 0.633438i $$-0.781643\pi$$
−0.773793 + 0.633438i $$0.781643\pi$$
$$284$$ −4.61803 −0.274030
$$285$$ 0 0
$$286$$ 2.61803 0.154808
$$287$$ 9.00000 0.531253
$$288$$ −9.65248 −0.568778
$$289$$ −16.4164 −0.965671
$$290$$ 0 0
$$291$$ 5.29180 0.310211
$$292$$ 1.67376 0.0979495
$$293$$ 10.1459 0.592730 0.296365 0.955075i $$-0.404226\pi$$
0.296365 + 0.955075i $$0.404226\pi$$
$$294$$ 1.23607 0.0720889
$$295$$ 0 0
$$296$$ −19.7984 −1.15076
$$297$$ −3.61803 −0.209940
$$298$$ 3.09017 0.179009
$$299$$ −5.38197 −0.311247
$$300$$ 0 0
$$301$$ −0.437694 −0.0252283
$$302$$ 34.1246 1.96365
$$303$$ 3.50658 0.201448
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −3.52786 −0.201675
$$307$$ 17.3262 0.988861 0.494430 0.869217i $$-0.335377\pi$$
0.494430 + 0.869217i $$0.335377\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 5.47214 0.311299
$$310$$ 0 0
$$311$$ 12.6525 0.717456 0.358728 0.933442i $$-0.383211\pi$$
0.358728 + 0.933442i $$0.383211\pi$$
$$312$$ −0.854102 −0.0483540
$$313$$ 11.3262 0.640197 0.320098 0.947384i $$-0.396284\pi$$
0.320098 + 0.947384i $$0.396284\pi$$
$$314$$ −28.8885 −1.63027
$$315$$ 0 0
$$316$$ 8.29180 0.466450
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 3.90983 0.219252
$$319$$ 5.85410 0.327767
$$320$$ 0 0
$$321$$ 6.27051 0.349986
$$322$$ −26.1246 −1.45587
$$323$$ 0 0
$$324$$ 4.76393 0.264663
$$325$$ 0 0
$$326$$ 2.85410 0.158074
$$327$$ 1.25735 0.0695318
$$328$$ −6.70820 −0.370399
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 10.9443 0.601552 0.300776 0.953695i $$-0.402754\pi$$
0.300776 + 0.953695i $$0.402754\pi$$
$$332$$ −5.23607 −0.287367
$$333$$ 25.2705 1.38482
$$334$$ 32.7426 1.79160
$$335$$ 0 0
$$336$$ −5.56231 −0.303449
$$337$$ 17.1246 0.932837 0.466419 0.884564i $$-0.345544\pi$$
0.466419 + 0.884564i $$0.345544\pi$$
$$338$$ 19.4164 1.05611
$$339$$ 2.58359 0.140321
$$340$$ 0 0
$$341$$ 14.3262 0.775809
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0.326238 0.0175896
$$345$$ 0 0
$$346$$ −0.763932 −0.0410692
$$347$$ 25.4164 1.36442 0.682212 0.731154i $$-0.261018\pi$$
0.682212 + 0.731154i $$0.261018\pi$$
$$348$$ 0.854102 0.0457847
$$349$$ −20.9787 −1.12296 −0.561482 0.827489i $$-0.689769\pi$$
−0.561482 + 0.827489i $$0.689769\pi$$
$$350$$ 0 0
$$351$$ 2.23607 0.119352
$$352$$ −5.47214 −0.291666
$$353$$ −24.4508 −1.30139 −0.650694 0.759340i $$-0.725522\pi$$
−0.650694 + 0.759340i $$0.725522\pi$$
$$354$$ 0.201626 0.0107163
$$355$$ 0 0
$$356$$ −4.79837 −0.254313
$$357$$ −0.875388 −0.0463305
$$358$$ −19.7984 −1.04638
$$359$$ 7.03444 0.371264 0.185632 0.982619i $$-0.440567\pi$$
0.185632 + 0.982619i $$0.440567\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ −19.4164 −1.02050
$$363$$ 3.20163 0.168042
$$364$$ −1.85410 −0.0971813
$$365$$ 0 0
$$366$$ −6.32624 −0.330678
$$367$$ 15.9443 0.832284 0.416142 0.909300i $$-0.363382\pi$$
0.416142 + 0.909300i $$0.363382\pi$$
$$368$$ 26.1246 1.36184
$$369$$ 8.56231 0.445736
$$370$$ 0 0
$$371$$ −18.9787 −0.985326
$$372$$ 2.09017 0.108370
$$373$$ 5.47214 0.283336 0.141668 0.989914i $$-0.454753\pi$$
0.141668 + 0.989914i $$0.454753\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ −6.70820 −0.345949
$$377$$ −3.61803 −0.186338
$$378$$ 10.8541 0.558275
$$379$$ −15.1246 −0.776899 −0.388450 0.921470i $$-0.626989\pi$$
−0.388450 + 0.921470i $$0.626989\pi$$
$$380$$ 0 0
$$381$$ 2.20163 0.112793
$$382$$ −23.0344 −1.17854
$$383$$ −0.381966 −0.0195176 −0.00975878 0.999952i $$-0.503106\pi$$
−0.00975878 + 0.999952i $$0.503106\pi$$
$$384$$ 5.20163 0.265444
$$385$$ 0 0
$$386$$ 8.18034 0.416368
$$387$$ −0.416408 −0.0211672
$$388$$ −8.56231 −0.434685
$$389$$ 9.27051 0.470034 0.235017 0.971991i $$-0.424485\pi$$
0.235017 + 0.971991i $$0.424485\pi$$
$$390$$ 0 0
$$391$$ 4.11146 0.207925
$$392$$ 4.47214 0.225877
$$393$$ −5.76393 −0.290752
$$394$$ 4.85410 0.244546
$$395$$ 0 0
$$396$$ 2.85410 0.143424
$$397$$ 11.4721 0.575770 0.287885 0.957665i $$-0.407048\pi$$
0.287885 + 0.957665i $$0.407048\pi$$
$$398$$ 21.7082 1.08813
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −35.8885 −1.79219 −0.896094 0.443864i $$-0.853607\pi$$
−0.896094 + 0.443864i $$0.853607\pi$$
$$402$$ 4.32624 0.215773
$$403$$ −8.85410 −0.441054
$$404$$ −5.67376 −0.282280
$$405$$ 0 0
$$406$$ −17.5623 −0.871603
$$407$$ 14.3262 0.710125
$$408$$ 0.652476 0.0323024
$$409$$ −8.29180 −0.410003 −0.205001 0.978762i $$-0.565720\pi$$
−0.205001 + 0.978762i $$0.565720\pi$$
$$410$$ 0 0
$$411$$ 2.85410 0.140782
$$412$$ −8.85410 −0.436210
$$413$$ −0.978714 −0.0481594
$$414$$ −24.8541 −1.22151
$$415$$ 0 0
$$416$$ 3.38197 0.165815
$$417$$ −5.65248 −0.276803
$$418$$ 0 0
$$419$$ −8.94427 −0.436956 −0.218478 0.975842i $$-0.570109\pi$$
−0.218478 + 0.975842i $$0.570109\pi$$
$$420$$ 0 0
$$421$$ −27.4721 −1.33891 −0.669455 0.742853i $$-0.733472\pi$$
−0.669455 + 0.742853i $$0.733472\pi$$
$$422$$ 6.23607 0.303567
$$423$$ 8.56231 0.416314
$$424$$ 14.1459 0.686986
$$425$$ 0 0
$$426$$ −4.61803 −0.223744
$$427$$ 30.7082 1.48607
$$428$$ −10.1459 −0.490420
$$429$$ 0.618034 0.0298390
$$430$$ 0 0
$$431$$ 27.6525 1.33197 0.665986 0.745964i $$-0.268011\pi$$
0.665986 + 0.745964i $$0.268011\pi$$
$$432$$ −10.8541 −0.522218
$$433$$ 3.43769 0.165205 0.0826025 0.996583i $$-0.473677\pi$$
0.0826025 + 0.996583i $$0.473677\pi$$
$$434$$ −42.9787 −2.06304
$$435$$ 0 0
$$436$$ −2.03444 −0.0974321
$$437$$ 0 0
$$438$$ 1.67376 0.0799754
$$439$$ −34.5967 −1.65121 −0.825606 0.564247i $$-0.809167\pi$$
−0.825606 + 0.564247i $$0.809167\pi$$
$$440$$ 0 0
$$441$$ −5.70820 −0.271819
$$442$$ 1.23607 0.0587938
$$443$$ 7.58359 0.360307 0.180154 0.983638i $$-0.442340\pi$$
0.180154 + 0.983638i $$0.442340\pi$$
$$444$$ 2.09017 0.0991951
$$445$$ 0 0
$$446$$ −18.8541 −0.892768
$$447$$ 0.729490 0.0345037
$$448$$ −12.7082 −0.600406
$$449$$ 2.88854 0.136319 0.0681594 0.997674i $$-0.478287\pi$$
0.0681594 + 0.997674i $$0.478287\pi$$
$$450$$ 0 0
$$451$$ 4.85410 0.228571
$$452$$ −4.18034 −0.196627
$$453$$ 8.05573 0.378491
$$454$$ 26.5623 1.24663
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −19.7082 −0.921911 −0.460955 0.887423i $$-0.652493\pi$$
−0.460955 + 0.887423i $$0.652493\pi$$
$$458$$ 22.0344 1.02960
$$459$$ −1.70820 −0.0797321
$$460$$ 0 0
$$461$$ 20.9443 0.975472 0.487736 0.872991i $$-0.337823\pi$$
0.487736 + 0.872991i $$0.337823\pi$$
$$462$$ 3.00000 0.139573
$$463$$ −28.2705 −1.31384 −0.656921 0.753959i $$-0.728142\pi$$
−0.656921 + 0.753959i $$0.728142\pi$$
$$464$$ 17.5623 0.815310
$$465$$ 0 0
$$466$$ 7.32624 0.339381
$$467$$ 15.9443 0.737813 0.368906 0.929467i $$-0.379732\pi$$
0.368906 + 0.929467i $$0.379732\pi$$
$$468$$ −1.76393 −0.0815378
$$469$$ −21.0000 −0.969690
$$470$$ 0 0
$$471$$ −6.81966 −0.314233
$$472$$ 0.729490 0.0335775
$$473$$ −0.236068 −0.0108544
$$474$$ 8.29180 0.380855
$$475$$ 0 0
$$476$$ 1.41641 0.0649209
$$477$$ −18.0557 −0.826715
$$478$$ 0.527864 0.0241439
$$479$$ 23.0902 1.05502 0.527508 0.849550i $$-0.323126\pi$$
0.527508 + 0.849550i $$0.323126\pi$$
$$480$$ 0 0
$$481$$ −8.85410 −0.403712
$$482$$ −5.14590 −0.234389
$$483$$ −6.16718 −0.280617
$$484$$ −5.18034 −0.235470
$$485$$ 0 0
$$486$$ 15.6180 0.708448
$$487$$ −4.18034 −0.189429 −0.0947146 0.995504i $$-0.530194\pi$$
−0.0947146 + 0.995504i $$0.530194\pi$$
$$488$$ −22.8885 −1.03612
$$489$$ 0.673762 0.0304686
$$490$$ 0 0
$$491$$ −21.2148 −0.957410 −0.478705 0.877976i $$-0.658894\pi$$
−0.478705 + 0.877976i $$0.658894\pi$$
$$492$$ 0.708204 0.0319283
$$493$$ 2.76393 0.124481
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 42.9787 1.92980
$$497$$ 22.4164 1.00551
$$498$$ −5.23607 −0.234634
$$499$$ 25.1246 1.12473 0.562366 0.826888i $$-0.309891\pi$$
0.562366 + 0.826888i $$0.309891\pi$$
$$500$$ 0 0
$$501$$ 7.72949 0.345328
$$502$$ 41.0344 1.83146
$$503$$ −35.8328 −1.59771 −0.798853 0.601526i $$-0.794560\pi$$
−0.798853 + 0.601526i $$0.794560\pi$$
$$504$$ 19.1459 0.852826
$$505$$ 0 0
$$506$$ −14.0902 −0.626384
$$507$$ 4.58359 0.203564
$$508$$ −3.56231 −0.158052
$$509$$ 2.03444 0.0901750 0.0450875 0.998983i $$-0.485643\pi$$
0.0450875 + 0.998983i $$0.485643\pi$$
$$510$$ 0 0
$$511$$ −8.12461 −0.359412
$$512$$ 5.29180 0.233867
$$513$$ 0 0
$$514$$ 32.9443 1.45311
$$515$$ 0 0
$$516$$ −0.0344419 −0.00151622
$$517$$ 4.85410 0.213483
$$518$$ −42.9787 −1.88838
$$519$$ −0.180340 −0.00791604
$$520$$ 0 0
$$521$$ 6.27051 0.274716 0.137358 0.990521i $$-0.456139\pi$$
0.137358 + 0.990521i $$0.456139\pi$$
$$522$$ −16.7082 −0.731298
$$523$$ 4.41641 0.193116 0.0965580 0.995327i $$-0.469217\pi$$
0.0965580 + 0.995327i $$0.469217\pi$$
$$524$$ 9.32624 0.407419
$$525$$ 0 0
$$526$$ −24.1803 −1.05431
$$527$$ 6.76393 0.294642
$$528$$ −3.00000 −0.130558
$$529$$ 5.96556 0.259372
$$530$$ 0 0
$$531$$ −0.931116 −0.0404070
$$532$$ 0 0
$$533$$ −3.00000 −0.129944
$$534$$ −4.79837 −0.207646
$$535$$ 0 0
$$536$$ 15.6525 0.676084
$$537$$ −4.67376 −0.201688
$$538$$ −49.0689 −2.11551
$$539$$ −3.23607 −0.139387
$$540$$ 0 0
$$541$$ −29.8328 −1.28261 −0.641306 0.767285i $$-0.721607\pi$$
−0.641306 + 0.767285i $$0.721607\pi$$
$$542$$ −1.85410 −0.0796405
$$543$$ −4.58359 −0.196701
$$544$$ −2.58359 −0.110771
$$545$$ 0 0
$$546$$ −1.85410 −0.0793482
$$547$$ 26.9230 1.15114 0.575572 0.817751i $$-0.304779\pi$$
0.575572 + 0.817751i $$0.304779\pi$$
$$548$$ −4.61803 −0.197273
$$549$$ 29.2148 1.24686
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.59675 0.195651
$$553$$ −40.2492 −1.71157
$$554$$ 18.4721 0.784806
$$555$$ 0 0
$$556$$ 9.14590 0.387872
$$557$$ 0.819660 0.0347301 0.0173651 0.999849i $$-0.494472\pi$$
0.0173651 + 0.999849i $$0.494472\pi$$
$$558$$ −40.8885 −1.73095
$$559$$ 0.145898 0.00617083
$$560$$ 0 0
$$561$$ −0.472136 −0.0199336
$$562$$ 47.7426 2.01390
$$563$$ 32.8328 1.38374 0.691869 0.722023i $$-0.256788\pi$$
0.691869 + 0.722023i $$0.256788\pi$$
$$564$$ 0.708204 0.0298208
$$565$$ 0 0
$$566$$ 42.1246 1.77063
$$567$$ −23.1246 −0.971142
$$568$$ −16.7082 −0.701061
$$569$$ 16.9098 0.708897 0.354448 0.935076i $$-0.384669\pi$$
0.354448 + 0.935076i $$0.384669\pi$$
$$570$$ 0 0
$$571$$ 6.67376 0.279288 0.139644 0.990202i $$-0.455404\pi$$
0.139644 + 0.990202i $$0.455404\pi$$
$$572$$ −1.00000 −0.0418121
$$573$$ −5.43769 −0.227163
$$574$$ −14.5623 −0.607819
$$575$$ 0 0
$$576$$ −12.0902 −0.503757
$$577$$ 12.1246 0.504754 0.252377 0.967629i $$-0.418788\pi$$
0.252377 + 0.967629i $$0.418788\pi$$
$$578$$ 26.5623 1.10485
$$579$$ 1.93112 0.0802545
$$580$$ 0 0
$$581$$ 25.4164 1.05445
$$582$$ −8.56231 −0.354919
$$583$$ −10.2361 −0.423935
$$584$$ 6.05573 0.250588
$$585$$ 0 0
$$586$$ −16.4164 −0.678156
$$587$$ 2.12461 0.0876921 0.0438461 0.999038i $$-0.486039\pi$$
0.0438461 + 0.999038i $$0.486039\pi$$
$$588$$ −0.472136 −0.0194706
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 1.14590 0.0471359
$$592$$ 42.9787 1.76641
$$593$$ −0.708204 −0.0290824 −0.0145412 0.999894i $$-0.504629\pi$$
−0.0145412 + 0.999894i $$0.504629\pi$$
$$594$$ 5.85410 0.240197
$$595$$ 0 0
$$596$$ −1.18034 −0.0483486
$$597$$ 5.12461 0.209736
$$598$$ 8.70820 0.356105
$$599$$ 28.4164 1.16106 0.580531 0.814238i $$-0.302845\pi$$
0.580531 + 0.814238i $$0.302845\pi$$
$$600$$ 0 0
$$601$$ 20.2918 0.827720 0.413860 0.910341i $$-0.364180\pi$$
0.413860 + 0.910341i $$0.364180\pi$$
$$602$$ 0.708204 0.0288642
$$603$$ −19.9787 −0.813596
$$604$$ −13.0344 −0.530364
$$605$$ 0 0
$$606$$ −5.67376 −0.230481
$$607$$ 6.27051 0.254512 0.127256 0.991870i $$-0.459383\pi$$
0.127256 + 0.991870i $$0.459383\pi$$
$$608$$ 0 0
$$609$$ −4.14590 −0.168000
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 1.34752 0.0544704
$$613$$ 19.9443 0.805542 0.402771 0.915301i $$-0.368047\pi$$
0.402771 + 0.915301i $$0.368047\pi$$
$$614$$ −28.0344 −1.13138
$$615$$ 0 0
$$616$$ 10.8541 0.437324
$$617$$ −7.67376 −0.308934 −0.154467 0.987998i $$-0.549366\pi$$
−0.154467 + 0.987998i $$0.549366\pi$$
$$618$$ −8.85410 −0.356164
$$619$$ 10.1246 0.406943 0.203471 0.979081i $$-0.434778\pi$$
0.203471 + 0.979081i $$0.434778\pi$$
$$620$$ 0 0
$$621$$ −12.0344 −0.482926
$$622$$ −20.4721 −0.820858
$$623$$ 23.2918 0.933166
$$624$$ 1.85410 0.0742235
$$625$$ 0 0
$$626$$ −18.3262 −0.732464
$$627$$ 0 0
$$628$$ 11.0344 0.440322
$$629$$ 6.76393 0.269696
$$630$$ 0 0
$$631$$ 29.3607 1.16883 0.584415 0.811455i $$-0.301324\pi$$
0.584415 + 0.811455i $$0.301324\pi$$
$$632$$ 30.0000 1.19334
$$633$$ 1.47214 0.0585122
$$634$$ 29.1246 1.15669
$$635$$ 0 0
$$636$$ −1.49342 −0.0592180
$$637$$ 2.00000 0.0792429
$$638$$ −9.47214 −0.375005
$$639$$ 21.3262 0.843653
$$640$$ 0 0
$$641$$ −39.5066 −1.56042 −0.780208 0.625520i $$-0.784887\pi$$
−0.780208 + 0.625520i $$0.784887\pi$$
$$642$$ −10.1459 −0.400427
$$643$$ 24.2918 0.957975 0.478987 0.877822i $$-0.341004\pi$$
0.478987 + 0.877822i $$0.341004\pi$$
$$644$$ 9.97871 0.393216
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −7.47214 −0.293760 −0.146880 0.989154i $$-0.546923\pi$$
−0.146880 + 0.989154i $$0.546923\pi$$
$$648$$ 17.2361 0.677097
$$649$$ −0.527864 −0.0207205
$$650$$ 0 0
$$651$$ −10.1459 −0.397649
$$652$$ −1.09017 −0.0426944
$$653$$ 23.5623 0.922064 0.461032 0.887383i $$-0.347479\pi$$
0.461032 + 0.887383i $$0.347479\pi$$
$$654$$ −2.03444 −0.0795530
$$655$$ 0 0
$$656$$ 14.5623 0.568563
$$657$$ −7.72949 −0.301556
$$658$$ −14.5623 −0.567698
$$659$$ −25.7771 −1.00413 −0.502066 0.864829i $$-0.667427\pi$$
−0.502066 + 0.864829i $$0.667427\pi$$
$$660$$ 0 0
$$661$$ 5.41641 0.210674 0.105337 0.994437i $$-0.466408\pi$$
0.105337 + 0.994437i $$0.466408\pi$$
$$662$$ −17.7082 −0.688249
$$663$$ 0.291796 0.0113324
$$664$$ −18.9443 −0.735180
$$665$$ 0 0
$$666$$ −40.8885 −1.58440
$$667$$ 19.4721 0.753964
$$668$$ −12.5066 −0.483894
$$669$$ −4.45085 −0.172080
$$670$$ 0 0
$$671$$ 16.5623 0.639381
$$672$$ 3.87539 0.149496
$$673$$ −34.1246 −1.31541 −0.657704 0.753277i $$-0.728472\pi$$
−0.657704 + 0.753277i $$0.728472\pi$$
$$674$$ −27.7082 −1.06728
$$675$$ 0 0
$$676$$ −7.41641 −0.285246
$$677$$ 30.7426 1.18154 0.590768 0.806842i $$-0.298825\pi$$
0.590768 + 0.806842i $$0.298825\pi$$
$$678$$ −4.18034 −0.160545
$$679$$ 41.5623 1.59501
$$680$$ 0 0
$$681$$ 6.27051 0.240286
$$682$$ −23.1803 −0.887621
$$683$$ −9.65248 −0.369342 −0.184671 0.982800i $$-0.559122\pi$$
−0.184671 + 0.982800i $$0.559122\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −24.2705 −0.926652
$$687$$ 5.20163 0.198454
$$688$$ −0.708204 −0.0270000
$$689$$ 6.32624 0.241010
$$690$$ 0 0
$$691$$ −39.1803 −1.49049 −0.745245 0.666791i $$-0.767668\pi$$
−0.745245 + 0.666791i $$0.767668\pi$$
$$692$$ 0.291796 0.0110924
$$693$$ −13.8541 −0.526274
$$694$$ −41.1246 −1.56107
$$695$$ 0 0
$$696$$ 3.09017 0.117133
$$697$$ 2.29180 0.0868080
$$698$$ 33.9443 1.28481
$$699$$ 1.72949 0.0654153
$$700$$ 0 0
$$701$$ −39.6312 −1.49685 −0.748425 0.663220i $$-0.769189\pi$$
−0.748425 + 0.663220i $$0.769189\pi$$
$$702$$ −3.61803 −0.136554
$$703$$ 0 0
$$704$$ −6.85410 −0.258324
$$705$$ 0 0
$$706$$ 39.5623 1.48895
$$707$$ 27.5410 1.03579
$$708$$ −0.0770143 −0.00289438
$$709$$ 16.5836 0.622810 0.311405 0.950277i $$-0.399200\pi$$
0.311405 + 0.950277i $$0.399200\pi$$
$$710$$ 0 0
$$711$$ −38.2918 −1.43605
$$712$$ −17.3607 −0.650619
$$713$$ 47.6525 1.78460
$$714$$ 1.41641 0.0530077
$$715$$ 0 0
$$716$$ 7.56231 0.282617
$$717$$ 0.124612 0.00465371
$$718$$ −11.3820 −0.424771
$$719$$ 47.0344 1.75409 0.877044 0.480409i $$-0.159512\pi$$
0.877044 + 0.480409i $$0.159512\pi$$
$$720$$ 0 0
$$721$$ 42.9787 1.60061
$$722$$ 0 0
$$723$$ −1.21478 −0.0451782
$$724$$ 7.41641 0.275629
$$725$$ 0 0
$$726$$ −5.18034 −0.192260
$$727$$ 16.0689 0.595962 0.297981 0.954572i $$-0.403687\pi$$
0.297981 + 0.954572i $$0.403687\pi$$
$$728$$ −6.70820 −0.248623
$$729$$ −19.4377 −0.719915
$$730$$ 0 0
$$731$$ −0.111456 −0.00412236
$$732$$ 2.41641 0.0893130
$$733$$ 52.9574 1.95603 0.978014 0.208541i $$-0.0668715\pi$$
0.978014 + 0.208541i $$0.0668715\pi$$
$$734$$ −25.7984 −0.952235
$$735$$ 0 0
$$736$$ −18.2016 −0.670921
$$737$$ −11.3262 −0.417207
$$738$$ −13.8541 −0.509977
$$739$$ −25.0000 −0.919640 −0.459820 0.888012i $$-0.652086\pi$$
−0.459820 + 0.888012i $$0.652086\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 30.7082 1.12733
$$743$$ 3.36068 0.123291 0.0616457 0.998098i $$-0.480365\pi$$
0.0616457 + 0.998098i $$0.480365\pi$$
$$744$$ 7.56231 0.277248
$$745$$ 0 0
$$746$$ −8.85410 −0.324172
$$747$$ 24.1803 0.884712
$$748$$ 0.763932 0.0279321
$$749$$ 49.2492 1.79953
$$750$$ 0 0
$$751$$ −17.1459 −0.625663 −0.312831 0.949809i $$-0.601277\pi$$
−0.312831 + 0.949809i $$0.601277\pi$$
$$752$$ 14.5623 0.531033
$$753$$ 9.68692 0.353011
$$754$$ 5.85410 0.213194
$$755$$ 0 0
$$756$$ −4.14590 −0.150785
$$757$$ −26.7426 −0.971978 −0.485989 0.873965i $$-0.661540\pi$$
−0.485989 + 0.873965i $$0.661540\pi$$
$$758$$ 24.4721 0.888868
$$759$$ −3.32624 −0.120735
$$760$$ 0 0
$$761$$ 4.88854 0.177210 0.0886048 0.996067i $$-0.471759\pi$$
0.0886048 + 0.996067i $$0.471759\pi$$
$$762$$ −3.56231 −0.129049
$$763$$ 9.87539 0.357513
$$764$$ 8.79837 0.318314
$$765$$ 0 0
$$766$$ 0.618034 0.0223305
$$767$$ 0.326238 0.0117798
$$768$$ −5.18034 −0.186929
$$769$$ 36.6312 1.32095 0.660477 0.750846i $$-0.270354\pi$$
0.660477 + 0.750846i $$0.270354\pi$$
$$770$$ 0 0
$$771$$ 7.77709 0.280085
$$772$$ −3.12461 −0.112457
$$773$$ 35.9230 1.29206 0.646030 0.763312i $$-0.276428\pi$$
0.646030 + 0.763312i $$0.276428\pi$$
$$774$$ 0.673762 0.0242179
$$775$$ 0 0
$$776$$ −30.9787 −1.11207
$$777$$ −10.1459 −0.363982
$$778$$ −15.0000 −0.537776
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 12.0902 0.432620
$$782$$ −6.65248 −0.237892
$$783$$ −8.09017 −0.289119
$$784$$ −9.70820 −0.346722
$$785$$ 0 0
$$786$$ 9.32624 0.332656
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ −1.85410 −0.0660496
$$789$$ −5.70820 −0.203217
$$790$$ 0 0
$$791$$ 20.2918 0.721493
$$792$$ 10.3262 0.366927
$$793$$ −10.2361 −0.363493
$$794$$ −18.5623 −0.658752
$$795$$ 0 0
$$796$$ −8.29180 −0.293895
$$797$$ 20.2918 0.718772 0.359386 0.933189i $$-0.382986\pi$$
0.359386 + 0.933189i $$0.382986\pi$$
$$798$$ 0 0
$$799$$ 2.29180 0.0810779
$$800$$ 0 0
$$801$$ 22.1591 0.782952
$$802$$ 58.0689 2.05048
$$803$$ −4.38197 −0.154636
$$804$$ −1.65248 −0.0582783
$$805$$ 0 0
$$806$$ 14.3262 0.504620
$$807$$ −11.5836 −0.407762
$$808$$ −20.5279 −0.722168
$$809$$ −24.7984 −0.871864 −0.435932 0.899980i $$-0.643581\pi$$
−0.435932 + 0.899980i $$0.643581\pi$$
$$810$$ 0 0
$$811$$ 22.9787 0.806892 0.403446 0.915004i $$-0.367812\pi$$
0.403446 + 0.915004i $$0.367812\pi$$
$$812$$ 6.70820 0.235412
$$813$$ −0.437694 −0.0153506
$$814$$ −23.1803 −0.812470
$$815$$ 0 0
$$816$$ −1.41641 −0.0495842
$$817$$ 0 0
$$818$$ 13.4164 0.469094
$$819$$ 8.56231 0.299191
$$820$$ 0 0
$$821$$ 52.0476 1.81647 0.908237 0.418457i $$-0.137429\pi$$
0.908237 + 0.418457i $$0.137429\pi$$
$$822$$ −4.61803 −0.161072
$$823$$ 25.5967 0.892247 0.446123 0.894972i $$-0.352804\pi$$
0.446123 + 0.894972i $$0.352804\pi$$
$$824$$ −32.0344 −1.11597
$$825$$ 0 0
$$826$$ 1.58359 0.0551002
$$827$$ −32.5967 −1.13350 −0.566750 0.823890i $$-0.691799\pi$$
−0.566750 + 0.823890i $$0.691799\pi$$
$$828$$ 9.49342 0.329919
$$829$$ 4.67376 0.162326 0.0811632 0.996701i $$-0.474136\pi$$
0.0811632 + 0.996701i $$0.474136\pi$$
$$830$$ 0 0
$$831$$ 4.36068 0.151270
$$832$$ 4.23607 0.146859
$$833$$ −1.52786 −0.0529374
$$834$$ 9.14590 0.316697
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −19.7984 −0.684332
$$838$$ 14.4721 0.499932
$$839$$ 15.2016 0.524818 0.262409 0.964957i $$-0.415483\pi$$
0.262409 + 0.964957i $$0.415483\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ 44.4508 1.53188
$$843$$ 11.2705 0.388177
$$844$$ −2.38197 −0.0819907
$$845$$ 0 0
$$846$$ −13.8541 −0.476314
$$847$$ 25.1459 0.864023
$$848$$ −30.7082 −1.05452
$$849$$ 9.94427 0.341287
$$850$$ 0 0
$$851$$ 47.6525 1.63351
$$852$$ 1.76393 0.0604313
$$853$$ −30.7082 −1.05143 −0.525714 0.850661i $$-0.676202\pi$$
−0.525714 + 0.850661i $$0.676202\pi$$
$$854$$ −49.6869 −1.70025
$$855$$ 0 0
$$856$$ −36.7082 −1.25466
$$857$$ 10.6180 0.362705 0.181353 0.983418i $$-0.441952\pi$$
0.181353 + 0.983418i $$0.441952\pi$$
$$858$$ −1.00000 −0.0341394
$$859$$ −48.5410 −1.65620 −0.828099 0.560582i $$-0.810578\pi$$
−0.828099 + 0.560582i $$0.810578\pi$$
$$860$$ 0 0
$$861$$ −3.43769 −0.117156
$$862$$ −44.7426 −1.52394
$$863$$ 27.0557 0.920988 0.460494 0.887663i $$-0.347672\pi$$
0.460494 + 0.887663i $$0.347672\pi$$
$$864$$ 7.56231 0.257275
$$865$$ 0 0
$$866$$ −5.56231 −0.189015
$$867$$ 6.27051 0.212958
$$868$$ 16.4164 0.557209
$$869$$ −21.7082 −0.736400
$$870$$ 0 0
$$871$$ 7.00000 0.237186
$$872$$ −7.36068 −0.249264
$$873$$ 39.5410 1.33826
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −0.639320 −0.0216006
$$877$$ −11.8197 −0.399122 −0.199561 0.979885i $$-0.563952\pi$$
−0.199561 + 0.979885i $$0.563952\pi$$
$$878$$ 55.9787 1.88919
$$879$$ −3.87539 −0.130714
$$880$$ 0 0
$$881$$ 32.4508 1.09330 0.546648 0.837362i $$-0.315903\pi$$
0.546648 + 0.837362i $$0.315903\pi$$
$$882$$ 9.23607 0.310995
$$883$$ 50.9230 1.71369 0.856847 0.515570i $$-0.172420\pi$$
0.856847 + 0.515570i $$0.172420\pi$$
$$884$$ −0.472136 −0.0158797
$$885$$ 0 0
$$886$$ −12.2705 −0.412236
$$887$$ −27.3475 −0.918240 −0.459120 0.888374i $$-0.651835\pi$$
−0.459120 + 0.888374i $$0.651835\pi$$
$$888$$ 7.56231 0.253774
$$889$$ 17.2918 0.579948
$$890$$ 0 0
$$891$$ −12.4721 −0.417832
$$892$$ 7.20163 0.241128
$$893$$ 0 0
$$894$$ −1.18034 −0.0394765
$$895$$ 0 0
$$896$$ 40.8541 1.36484
$$897$$ 2.05573 0.0686388
$$898$$ −4.67376 −0.155965
$$899$$ 32.0344 1.06841
$$900$$ 0 0
$$901$$ −4.83282 −0.161004
$$902$$ −7.85410 −0.261513
$$903$$ 0.167184 0.00556354
$$904$$ −15.1246 −0.503037
$$905$$ 0 0
$$906$$ −13.0344 −0.433040
$$907$$ 26.4721 0.878993 0.439496 0.898244i $$-0.355157\pi$$
0.439496 + 0.898244i $$0.355157\pi$$
$$908$$ −10.1459 −0.336703
$$909$$ 26.2016 0.869053
$$910$$ 0 0
$$911$$ 3.38197 0.112050 0.0560248 0.998429i $$-0.482157\pi$$
0.0560248 + 0.998429i $$0.482157\pi$$
$$912$$ 0 0
$$913$$ 13.7082 0.453675
$$914$$ 31.8885 1.05478
$$915$$ 0 0
$$916$$ −8.41641 −0.278086
$$917$$ −45.2705 −1.49496
$$918$$ 2.76393 0.0912234
$$919$$ 23.2918 0.768325 0.384163 0.923265i $$-0.374490\pi$$
0.384163 + 0.923265i $$0.374490\pi$$
$$920$$ 0 0
$$921$$ −6.61803 −0.218072
$$922$$ −33.8885 −1.11606
$$923$$ −7.47214 −0.245948
$$924$$ −1.14590 −0.0376973
$$925$$ 0 0
$$926$$ 45.7426 1.50320
$$927$$ 40.8885 1.34296
$$928$$ −12.2361 −0.401669
$$929$$ 16.3820 0.537475 0.268737 0.963213i $$-0.413394\pi$$
0.268737 + 0.963213i $$0.413394\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2.79837 −0.0916638
$$933$$ −4.83282 −0.158219
$$934$$ −25.7984 −0.844149
$$935$$ 0 0
$$936$$ −6.38197 −0.208601
$$937$$ −40.5623 −1.32511 −0.662556 0.749012i $$-0.730529\pi$$
−0.662556 + 0.749012i $$0.730529\pi$$
$$938$$ 33.9787 1.10944
$$939$$ −4.32624 −0.141181
$$940$$ 0 0
$$941$$ −10.6869 −0.348384 −0.174192 0.984712i $$-0.555731\pi$$
−0.174192 + 0.984712i $$0.555731\pi$$
$$942$$ 11.0344 0.359522
$$943$$ 16.1459 0.525783
$$944$$ −1.58359 −0.0515415
$$945$$ 0 0
$$946$$ 0.381966 0.0124188
$$947$$ 32.6525 1.06106 0.530531 0.847665i $$-0.321992\pi$$
0.530531 + 0.847665i $$0.321992\pi$$
$$948$$ −3.16718 −0.102865
$$949$$ 2.70820 0.0879120
$$950$$ 0 0
$$951$$ 6.87539 0.222950
$$952$$ 5.12461 0.166090
$$953$$ −17.2918 −0.560136 −0.280068 0.959980i $$-0.590357\pi$$
−0.280068 + 0.959980i $$0.590357\pi$$
$$954$$ 29.2148 0.945863
$$955$$ 0 0
$$956$$ −0.201626 −0.00652105
$$957$$ −2.23607 −0.0722818
$$958$$ −37.3607 −1.20707
$$959$$ 22.4164 0.723864
$$960$$ 0 0
$$961$$ 47.3951 1.52887
$$962$$ 14.3262 0.461896
$$963$$ 46.8541 1.50985
$$964$$ 1.96556 0.0633064
$$965$$ 0 0
$$966$$ 9.97871 0.321060
$$967$$ −6.54102 −0.210345 −0.105173 0.994454i $$-0.533539\pi$$
−0.105173 + 0.994454i $$0.533539\pi$$
$$968$$ −18.7426 −0.602411
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 23.5066 0.754362 0.377181 0.926140i $$-0.376893\pi$$
0.377181 + 0.926140i $$0.376893\pi$$
$$972$$ −5.96556 −0.191345
$$973$$ −44.3951 −1.42324
$$974$$ 6.76393 0.216730
$$975$$ 0 0
$$976$$ 49.6869 1.59044
$$977$$ −10.6393 −0.340382 −0.170191 0.985411i $$-0.554438\pi$$
−0.170191 + 0.985411i $$0.554438\pi$$
$$978$$ −1.09017 −0.0348598
$$979$$ 12.5623 0.401493
$$980$$ 0 0
$$981$$ 9.39512 0.299963
$$982$$ 34.3262 1.09539
$$983$$ −32.6180 −1.04035 −0.520177 0.854059i $$-0.674134\pi$$
−0.520177 + 0.854059i $$0.674134\pi$$
$$984$$ 2.56231 0.0816833
$$985$$ 0 0
$$986$$ −4.47214 −0.142422
$$987$$ −3.43769 −0.109423
$$988$$ 0 0
$$989$$ −0.785218 −0.0249685
$$990$$ 0 0
$$991$$ 7.45085 0.236684 0.118342 0.992973i $$-0.462242\pi$$
0.118342 + 0.992973i $$0.462242\pi$$
$$992$$ −29.9443 −0.950732
$$993$$ −4.18034 −0.132659
$$994$$ −36.2705 −1.15043
$$995$$ 0 0
$$996$$ 2.00000 0.0633724
$$997$$ −39.7082 −1.25757 −0.628786 0.777579i $$-0.716448\pi$$
−0.628786 + 0.777579i $$0.716448\pi$$
$$998$$ −40.6525 −1.28683
$$999$$ −19.7984 −0.626393
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 9025.2.a.n.1.1 2
5.4 even 2 361.2.a.f.1.2 yes 2
15.14 odd 2 3249.2.a.i.1.1 2
19.18 odd 2 9025.2.a.s.1.2 2
20.19 odd 2 5776.2.a.s.1.2 2
95.4 even 18 361.2.e.j.54.2 12
95.9 even 18 361.2.e.j.62.1 12
95.14 odd 18 361.2.e.i.234.1 12
95.24 even 18 361.2.e.j.234.2 12
95.29 odd 18 361.2.e.i.62.2 12
95.34 odd 18 361.2.e.i.54.1 12
95.44 even 18 361.2.e.j.245.2 12
95.49 even 6 361.2.c.d.292.1 4
95.54 even 18 361.2.e.j.28.2 12
95.59 odd 18 361.2.e.i.99.2 12
95.64 even 6 361.2.c.d.68.1 4
95.69 odd 6 361.2.c.g.68.2 4
95.74 even 18 361.2.e.j.99.1 12
95.79 odd 18 361.2.e.i.28.1 12
95.84 odd 6 361.2.c.g.292.2 4
95.89 odd 18 361.2.e.i.245.1 12
95.94 odd 2 361.2.a.c.1.1 2
285.284 even 2 3249.2.a.o.1.2 2
380.379 even 2 5776.2.a.bg.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
361.2.a.c.1.1 2 95.94 odd 2
361.2.a.f.1.2 yes 2 5.4 even 2
361.2.c.d.68.1 4 95.64 even 6
361.2.c.d.292.1 4 95.49 even 6
361.2.c.g.68.2 4 95.69 odd 6
361.2.c.g.292.2 4 95.84 odd 6
361.2.e.i.28.1 12 95.79 odd 18
361.2.e.i.54.1 12 95.34 odd 18
361.2.e.i.62.2 12 95.29 odd 18
361.2.e.i.99.2 12 95.59 odd 18
361.2.e.i.234.1 12 95.14 odd 18
361.2.e.i.245.1 12 95.89 odd 18
361.2.e.j.28.2 12 95.54 even 18
361.2.e.j.54.2 12 95.4 even 18
361.2.e.j.62.1 12 95.9 even 18
361.2.e.j.99.1 12 95.74 even 18
361.2.e.j.234.2 12 95.24 even 18
361.2.e.j.245.2 12 95.44 even 18
3249.2.a.i.1.1 2 15.14 odd 2
3249.2.a.o.1.2 2 285.284 even 2
5776.2.a.s.1.2 2 20.19 odd 2
5776.2.a.bg.1.1 2 380.379 even 2
9025.2.a.n.1.1 2 1.1 even 1 trivial
9025.2.a.s.1.2 2 19.18 odd 2