# Properties

 Label 1150.2.a.j.1.1 Level $1150$ Weight $2$ Character 1150.1 Self dual yes Analytic conductor $9.183$ Analytic rank $1$ 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] = [1150,2,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.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: $$9.18279623245$$ Analytic rank: $$1$$ 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: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +1.61803 q^{6} +0.618034 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +1.61803 q^{6} +0.618034 q^{7} -1.00000 q^{8} -0.381966 q^{9} -2.85410 q^{11} -1.61803 q^{12} +7.09017 q^{13} -0.618034 q^{14} +1.00000 q^{16} -6.09017 q^{17} +0.381966 q^{18} +1.85410 q^{19} -1.00000 q^{21} +2.85410 q^{22} -1.00000 q^{23} +1.61803 q^{24} -7.09017 q^{26} +5.47214 q^{27} +0.618034 q^{28} -9.23607 q^{29} +9.09017 q^{31} -1.00000 q^{32} +4.61803 q^{33} +6.09017 q^{34} -0.381966 q^{36} -6.47214 q^{37} -1.85410 q^{38} -11.4721 q^{39} +3.32624 q^{41} +1.00000 q^{42} -2.85410 q^{44} +1.00000 q^{46} +3.70820 q^{47} -1.61803 q^{48} -6.61803 q^{49} +9.85410 q^{51} +7.09017 q^{52} -0.472136 q^{53} -5.47214 q^{54} -0.618034 q^{56} -3.00000 q^{57} +9.23607 q^{58} +1.70820 q^{59} -9.32624 q^{61} -9.09017 q^{62} -0.236068 q^{63} +1.00000 q^{64} -4.61803 q^{66} -14.4721 q^{67} -6.09017 q^{68} +1.61803 q^{69} -4.09017 q^{71} +0.381966 q^{72} -3.23607 q^{73} +6.47214 q^{74} +1.85410 q^{76} -1.76393 q^{77} +11.4721 q^{78} +1.52786 q^{79} -7.70820 q^{81} -3.32624 q^{82} +6.94427 q^{83} -1.00000 q^{84} +14.9443 q^{87} +2.85410 q^{88} -10.4721 q^{89} +4.38197 q^{91} -1.00000 q^{92} -14.7082 q^{93} -3.70820 q^{94} +1.61803 q^{96} -12.3820 q^{97} +6.61803 q^{98} +1.09017 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{6} - q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 + q^6 - q^7 - 2 * q^8 - 3 * q^9 $$2 q - 2 q^{2} - q^{3} + 2 q^{4} + q^{6} - q^{7} - 2 q^{8} - 3 q^{9} + q^{11} - q^{12} + 3 q^{13} + q^{14} + 2 q^{16} - q^{17} + 3 q^{18} - 3 q^{19} - 2 q^{21} - q^{22} - 2 q^{23} + q^{24} - 3 q^{26} + 2 q^{27} - q^{28} - 14 q^{29} + 7 q^{31} - 2 q^{32} + 7 q^{33} + q^{34} - 3 q^{36} - 4 q^{37} + 3 q^{38} - 14 q^{39} - 9 q^{41} + 2 q^{42} + q^{44} + 2 q^{46} - 6 q^{47} - q^{48} - 11 q^{49} + 13 q^{51} + 3 q^{52} + 8 q^{53} - 2 q^{54} + q^{56} - 6 q^{57} + 14 q^{58} - 10 q^{59} - 3 q^{61} - 7 q^{62} + 4 q^{63} + 2 q^{64} - 7 q^{66} - 20 q^{67} - q^{68} + q^{69} + 3 q^{71} + 3 q^{72} - 2 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} + 14 q^{78} + 12 q^{79} - 2 q^{81} + 9 q^{82} - 4 q^{83} - 2 q^{84} + 12 q^{87} - q^{88} - 12 q^{89} + 11 q^{91} - 2 q^{92} - 16 q^{93} + 6 q^{94} + q^{96} - 27 q^{97} + 11 q^{98} - 9 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 + q^6 - q^7 - 2 * q^8 - 3 * q^9 + q^11 - q^12 + 3 * q^13 + q^14 + 2 * q^16 - q^17 + 3 * q^18 - 3 * q^19 - 2 * q^21 - q^22 - 2 * q^23 + q^24 - 3 * q^26 + 2 * q^27 - q^28 - 14 * q^29 + 7 * q^31 - 2 * q^32 + 7 * q^33 + q^34 - 3 * q^36 - 4 * q^37 + 3 * q^38 - 14 * q^39 - 9 * q^41 + 2 * q^42 + q^44 + 2 * q^46 - 6 * q^47 - q^48 - 11 * q^49 + 13 * q^51 + 3 * q^52 + 8 * q^53 - 2 * q^54 + q^56 - 6 * q^57 + 14 * q^58 - 10 * q^59 - 3 * q^61 - 7 * q^62 + 4 * q^63 + 2 * q^64 - 7 * q^66 - 20 * q^67 - q^68 + q^69 + 3 * q^71 + 3 * q^72 - 2 * q^73 + 4 * q^74 - 3 * q^76 - 8 * q^77 + 14 * q^78 + 12 * q^79 - 2 * q^81 + 9 * q^82 - 4 * q^83 - 2 * q^84 + 12 * q^87 - q^88 - 12 * q^89 + 11 * q^91 - 2 * q^92 - 16 * q^93 + 6 * q^94 + q^96 - 27 * q^97 + 11 * q^98 - 9 * 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
$$3$$ −1.61803 −0.934172 −0.467086 0.884212i $$-0.654696\pi$$
−0.467086 + 0.884212i $$0.654696\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.61803 0.660560
$$7$$ 0.618034 0.233595 0.116797 0.993156i $$-0.462737\pi$$
0.116797 + 0.993156i $$0.462737\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −0.381966 −0.127322
$$10$$ 0 0
$$11$$ −2.85410 −0.860544 −0.430272 0.902699i $$-0.641582\pi$$
−0.430272 + 0.902699i $$0.641582\pi$$
$$12$$ −1.61803 −0.467086
$$13$$ 7.09017 1.96646 0.983230 0.182372i $$-0.0583774\pi$$
0.983230 + 0.182372i $$0.0583774\pi$$
$$14$$ −0.618034 −0.165177
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.09017 −1.47708 −0.738542 0.674208i $$-0.764485\pi$$
−0.738542 + 0.674208i $$0.764485\pi$$
$$18$$ 0.381966 0.0900303
$$19$$ 1.85410 0.425360 0.212680 0.977122i $$-0.431781\pi$$
0.212680 + 0.977122i $$0.431781\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 2.85410 0.608497
$$23$$ −1.00000 −0.208514
$$24$$ 1.61803 0.330280
$$25$$ 0 0
$$26$$ −7.09017 −1.39050
$$27$$ 5.47214 1.05311
$$28$$ 0.618034 0.116797
$$29$$ −9.23607 −1.71509 −0.857547 0.514405i $$-0.828013\pi$$
−0.857547 + 0.514405i $$0.828013\pi$$
$$30$$ 0 0
$$31$$ 9.09017 1.63264 0.816321 0.577598i $$-0.196010\pi$$
0.816321 + 0.577598i $$0.196010\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.61803 0.803897
$$34$$ 6.09017 1.04446
$$35$$ 0 0
$$36$$ −0.381966 −0.0636610
$$37$$ −6.47214 −1.06401 −0.532006 0.846740i $$-0.678562\pi$$
−0.532006 + 0.846740i $$0.678562\pi$$
$$38$$ −1.85410 −0.300775
$$39$$ −11.4721 −1.83701
$$40$$ 0 0
$$41$$ 3.32624 0.519471 0.259736 0.965680i $$-0.416365\pi$$
0.259736 + 0.965680i $$0.416365\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −2.85410 −0.430272
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 3.70820 0.540897 0.270449 0.962734i $$-0.412828\pi$$
0.270449 + 0.962734i $$0.412828\pi$$
$$48$$ −1.61803 −0.233543
$$49$$ −6.61803 −0.945433
$$50$$ 0 0
$$51$$ 9.85410 1.37985
$$52$$ 7.09017 0.983230
$$53$$ −0.472136 −0.0648529 −0.0324264 0.999474i $$-0.510323\pi$$
−0.0324264 + 0.999474i $$0.510323\pi$$
$$54$$ −5.47214 −0.744663
$$55$$ 0 0
$$56$$ −0.618034 −0.0825883
$$57$$ −3.00000 −0.397360
$$58$$ 9.23607 1.21276
$$59$$ 1.70820 0.222389 0.111195 0.993799i $$-0.464532\pi$$
0.111195 + 0.993799i $$0.464532\pi$$
$$60$$ 0 0
$$61$$ −9.32624 −1.19410 −0.597051 0.802203i $$-0.703661\pi$$
−0.597051 + 0.802203i $$0.703661\pi$$
$$62$$ −9.09017 −1.15445
$$63$$ −0.236068 −0.0297418
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.61803 −0.568441
$$67$$ −14.4721 −1.76805 −0.884026 0.467437i $$-0.845177\pi$$
−0.884026 + 0.467437i $$0.845177\pi$$
$$68$$ −6.09017 −0.738542
$$69$$ 1.61803 0.194788
$$70$$ 0 0
$$71$$ −4.09017 −0.485414 −0.242707 0.970100i $$-0.578035\pi$$
−0.242707 + 0.970100i $$0.578035\pi$$
$$72$$ 0.381966 0.0450151
$$73$$ −3.23607 −0.378753 −0.189377 0.981905i $$-0.560647\pi$$
−0.189377 + 0.981905i $$0.560647\pi$$
$$74$$ 6.47214 0.752371
$$75$$ 0 0
$$76$$ 1.85410 0.212680
$$77$$ −1.76393 −0.201019
$$78$$ 11.4721 1.29896
$$79$$ 1.52786 0.171898 0.0859491 0.996300i $$-0.472608\pi$$
0.0859491 + 0.996300i $$0.472608\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ −3.32624 −0.367322
$$83$$ 6.94427 0.762233 0.381116 0.924527i $$-0.375540\pi$$
0.381116 + 0.924527i $$0.375540\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 14.9443 1.60219
$$88$$ 2.85410 0.304248
$$89$$ −10.4721 −1.11004 −0.555022 0.831836i $$-0.687290\pi$$
−0.555022 + 0.831836i $$0.687290\pi$$
$$90$$ 0 0
$$91$$ 4.38197 0.459355
$$92$$ −1.00000 −0.104257
$$93$$ −14.7082 −1.52517
$$94$$ −3.70820 −0.382472
$$95$$ 0 0
$$96$$ 1.61803 0.165140
$$97$$ −12.3820 −1.25720 −0.628599 0.777730i $$-0.716371\pi$$
−0.628599 + 0.777730i $$0.716371\pi$$
$$98$$ 6.61803 0.668522
$$99$$ 1.09017 0.109566
$$100$$ 0 0
$$101$$ −0.291796 −0.0290348 −0.0145174 0.999895i $$-0.504621\pi$$
−0.0145174 + 0.999895i $$0.504621\pi$$
$$102$$ −9.85410 −0.975701
$$103$$ −16.5623 −1.63193 −0.815966 0.578100i $$-0.803795\pi$$
−0.815966 + 0.578100i $$0.803795\pi$$
$$104$$ −7.09017 −0.695248
$$105$$ 0 0
$$106$$ 0.472136 0.0458579
$$107$$ −18.1803 −1.75756 −0.878780 0.477227i $$-0.841642\pi$$
−0.878780 + 0.477227i $$0.841642\pi$$
$$108$$ 5.47214 0.526557
$$109$$ −11.5623 −1.10747 −0.553734 0.832694i $$-0.686798\pi$$
−0.553734 + 0.832694i $$0.686798\pi$$
$$110$$ 0 0
$$111$$ 10.4721 0.993971
$$112$$ 0.618034 0.0583987
$$113$$ −1.05573 −0.0993145 −0.0496573 0.998766i $$-0.515813\pi$$
−0.0496573 + 0.998766i $$0.515813\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ −9.23607 −0.857547
$$117$$ −2.70820 −0.250374
$$118$$ −1.70820 −0.157253
$$119$$ −3.76393 −0.345039
$$120$$ 0 0
$$121$$ −2.85410 −0.259464
$$122$$ 9.32624 0.844358
$$123$$ −5.38197 −0.485276
$$124$$ 9.09017 0.816321
$$125$$ 0 0
$$126$$ 0.236068 0.0210306
$$127$$ −16.1803 −1.43577 −0.717886 0.696160i $$-0.754890\pi$$
−0.717886 + 0.696160i $$0.754890\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.94427 0.257242 0.128621 0.991694i $$-0.458945\pi$$
0.128621 + 0.991694i $$0.458945\pi$$
$$132$$ 4.61803 0.401948
$$133$$ 1.14590 0.0993620
$$134$$ 14.4721 1.25020
$$135$$ 0 0
$$136$$ 6.09017 0.522228
$$137$$ 10.3262 0.882230 0.441115 0.897451i $$-0.354583\pi$$
0.441115 + 0.897451i $$0.354583\pi$$
$$138$$ −1.61803 −0.137736
$$139$$ 12.7639 1.08262 0.541311 0.840822i $$-0.317928\pi$$
0.541311 + 0.840822i $$0.317928\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 4.09017 0.343239
$$143$$ −20.2361 −1.69223
$$144$$ −0.381966 −0.0318305
$$145$$ 0 0
$$146$$ 3.23607 0.267819
$$147$$ 10.7082 0.883198
$$148$$ −6.47214 −0.532006
$$149$$ −7.85410 −0.643433 −0.321717 0.946836i $$-0.604260\pi$$
−0.321717 + 0.946836i $$0.604260\pi$$
$$150$$ 0 0
$$151$$ −2.56231 −0.208517 −0.104259 0.994550i $$-0.533247\pi$$
−0.104259 + 0.994550i $$0.533247\pi$$
$$152$$ −1.85410 −0.150388
$$153$$ 2.32624 0.188065
$$154$$ 1.76393 0.142142
$$155$$ 0 0
$$156$$ −11.4721 −0.918506
$$157$$ 3.70820 0.295947 0.147973 0.988991i $$-0.452725\pi$$
0.147973 + 0.988991i $$0.452725\pi$$
$$158$$ −1.52786 −0.121550
$$159$$ 0.763932 0.0605838
$$160$$ 0 0
$$161$$ −0.618034 −0.0487079
$$162$$ 7.70820 0.605614
$$163$$ −1.38197 −0.108244 −0.0541220 0.998534i $$-0.517236\pi$$
−0.0541220 + 0.998534i $$0.517236\pi$$
$$164$$ 3.32624 0.259736
$$165$$ 0 0
$$166$$ −6.94427 −0.538980
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 37.2705 2.86696
$$170$$ 0 0
$$171$$ −0.708204 −0.0541577
$$172$$ 0 0
$$173$$ 1.43769 0.109306 0.0546529 0.998505i $$-0.482595\pi$$
0.0546529 + 0.998505i $$0.482595\pi$$
$$174$$ −14.9443 −1.13292
$$175$$ 0 0
$$176$$ −2.85410 −0.215136
$$177$$ −2.76393 −0.207750
$$178$$ 10.4721 0.784920
$$179$$ 2.18034 0.162966 0.0814831 0.996675i $$-0.474034\pi$$
0.0814831 + 0.996675i $$0.474034\pi$$
$$180$$ 0 0
$$181$$ −12.1459 −0.902797 −0.451399 0.892322i $$-0.649075\pi$$
−0.451399 + 0.892322i $$0.649075\pi$$
$$182$$ −4.38197 −0.324813
$$183$$ 15.0902 1.11550
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 14.7082 1.07846
$$187$$ 17.3820 1.27110
$$188$$ 3.70820 0.270449
$$189$$ 3.38197 0.246002
$$190$$ 0 0
$$191$$ −13.7082 −0.991891 −0.495945 0.868354i $$-0.665178\pi$$
−0.495945 + 0.868354i $$0.665178\pi$$
$$192$$ −1.61803 −0.116772
$$193$$ −0.763932 −0.0549890 −0.0274945 0.999622i $$-0.508753\pi$$
−0.0274945 + 0.999622i $$0.508753\pi$$
$$194$$ 12.3820 0.888973
$$195$$ 0 0
$$196$$ −6.61803 −0.472717
$$197$$ 22.5623 1.60750 0.803749 0.594969i $$-0.202836\pi$$
0.803749 + 0.594969i $$0.202836\pi$$
$$198$$ −1.09017 −0.0774750
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 23.4164 1.65167
$$202$$ 0.291796 0.0205307
$$203$$ −5.70820 −0.400637
$$204$$ 9.85410 0.689925
$$205$$ 0 0
$$206$$ 16.5623 1.15395
$$207$$ 0.381966 0.0265485
$$208$$ 7.09017 0.491615
$$209$$ −5.29180 −0.366041
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ −0.472136 −0.0324264
$$213$$ 6.61803 0.453460
$$214$$ 18.1803 1.24278
$$215$$ 0 0
$$216$$ −5.47214 −0.372332
$$217$$ 5.61803 0.381377
$$218$$ 11.5623 0.783098
$$219$$ 5.23607 0.353821
$$220$$ 0 0
$$221$$ −43.1803 −2.90462
$$222$$ −10.4721 −0.702844
$$223$$ 20.9443 1.40253 0.701266 0.712900i $$-0.252618\pi$$
0.701266 + 0.712900i $$0.252618\pi$$
$$224$$ −0.618034 −0.0412941
$$225$$ 0 0
$$226$$ 1.05573 0.0702260
$$227$$ 18.7639 1.24541 0.622703 0.782458i $$-0.286035\pi$$
0.622703 + 0.782458i $$0.286035\pi$$
$$228$$ −3.00000 −0.198680
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 2.85410 0.187786
$$232$$ 9.23607 0.606378
$$233$$ −6.29180 −0.412189 −0.206095 0.978532i $$-0.566075\pi$$
−0.206095 + 0.978532i $$0.566075\pi$$
$$234$$ 2.70820 0.177041
$$235$$ 0 0
$$236$$ 1.70820 0.111195
$$237$$ −2.47214 −0.160582
$$238$$ 3.76393 0.243979
$$239$$ −20.3607 −1.31702 −0.658511 0.752571i $$-0.728814\pi$$
−0.658511 + 0.752571i $$0.728814\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 2.85410 0.183469
$$243$$ −3.94427 −0.253025
$$244$$ −9.32624 −0.597051
$$245$$ 0 0
$$246$$ 5.38197 0.343142
$$247$$ 13.1459 0.836453
$$248$$ −9.09017 −0.577226
$$249$$ −11.2361 −0.712057
$$250$$ 0 0
$$251$$ 6.14590 0.387926 0.193963 0.981009i $$-0.437866\pi$$
0.193963 + 0.981009i $$0.437866\pi$$
$$252$$ −0.236068 −0.0148709
$$253$$ 2.85410 0.179436
$$254$$ 16.1803 1.01524
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 7.81966 0.487777 0.243888 0.969803i $$-0.421577\pi$$
0.243888 + 0.969803i $$0.421577\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 3.52786 0.218369
$$262$$ −2.94427 −0.181898
$$263$$ −20.7426 −1.27905 −0.639523 0.768772i $$-0.720868\pi$$
−0.639523 + 0.768772i $$0.720868\pi$$
$$264$$ −4.61803 −0.284220
$$265$$ 0 0
$$266$$ −1.14590 −0.0702595
$$267$$ 16.9443 1.03697
$$268$$ −14.4721 −0.884026
$$269$$ −14.1803 −0.864591 −0.432295 0.901732i $$-0.642296\pi$$
−0.432295 + 0.901732i $$0.642296\pi$$
$$270$$ 0 0
$$271$$ −30.3262 −1.84219 −0.921094 0.389341i $$-0.872703\pi$$
−0.921094 + 0.389341i $$0.872703\pi$$
$$272$$ −6.09017 −0.369271
$$273$$ −7.09017 −0.429117
$$274$$ −10.3262 −0.623831
$$275$$ 0 0
$$276$$ 1.61803 0.0973942
$$277$$ −29.4164 −1.76746 −0.883730 0.467996i $$-0.844976\pi$$
−0.883730 + 0.467996i $$0.844976\pi$$
$$278$$ −12.7639 −0.765530
$$279$$ −3.47214 −0.207871
$$280$$ 0 0
$$281$$ 22.7639 1.35798 0.678991 0.734146i $$-0.262417\pi$$
0.678991 + 0.734146i $$0.262417\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 26.9443 1.60167 0.800835 0.598885i $$-0.204389\pi$$
0.800835 + 0.598885i $$0.204389\pi$$
$$284$$ −4.09017 −0.242707
$$285$$ 0 0
$$286$$ 20.2361 1.19658
$$287$$ 2.05573 0.121346
$$288$$ 0.381966 0.0225076
$$289$$ 20.0902 1.18177
$$290$$ 0 0
$$291$$ 20.0344 1.17444
$$292$$ −3.23607 −0.189377
$$293$$ 19.8885 1.16190 0.580951 0.813939i $$-0.302681\pi$$
0.580951 + 0.813939i $$0.302681\pi$$
$$294$$ −10.7082 −0.624515
$$295$$ 0 0
$$296$$ 6.47214 0.376185
$$297$$ −15.6180 −0.906250
$$298$$ 7.85410 0.454976
$$299$$ −7.09017 −0.410035
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2.56231 0.147444
$$303$$ 0.472136 0.0271235
$$304$$ 1.85410 0.106340
$$305$$ 0 0
$$306$$ −2.32624 −0.132982
$$307$$ 28.4508 1.62378 0.811888 0.583813i $$-0.198440\pi$$
0.811888 + 0.583813i $$0.198440\pi$$
$$308$$ −1.76393 −0.100509
$$309$$ 26.7984 1.52451
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 11.4721 0.649482
$$313$$ −12.7984 −0.723407 −0.361703 0.932293i $$-0.617805\pi$$
−0.361703 + 0.932293i $$0.617805\pi$$
$$314$$ −3.70820 −0.209266
$$315$$ 0 0
$$316$$ 1.52786 0.0859491
$$317$$ 11.0902 0.622886 0.311443 0.950265i $$-0.399188\pi$$
0.311443 + 0.950265i $$0.399188\pi$$
$$318$$ −0.763932 −0.0428392
$$319$$ 26.3607 1.47591
$$320$$ 0 0
$$321$$ 29.4164 1.64186
$$322$$ 0.618034 0.0344417
$$323$$ −11.2918 −0.628292
$$324$$ −7.70820 −0.428234
$$325$$ 0 0
$$326$$ 1.38197 0.0765400
$$327$$ 18.7082 1.03457
$$328$$ −3.32624 −0.183661
$$329$$ 2.29180 0.126351
$$330$$ 0 0
$$331$$ 19.2361 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$332$$ 6.94427 0.381116
$$333$$ 2.47214 0.135472
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −13.6738 −0.744857 −0.372429 0.928061i $$-0.621475\pi$$
−0.372429 + 0.928061i $$0.621475\pi$$
$$338$$ −37.2705 −2.02725
$$339$$ 1.70820 0.0927769
$$340$$ 0 0
$$341$$ −25.9443 −1.40496
$$342$$ 0.708204 0.0382953
$$343$$ −8.41641 −0.454443
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −1.43769 −0.0772909
$$347$$ 6.38197 0.342602 0.171301 0.985219i $$-0.445203\pi$$
0.171301 + 0.985219i $$0.445203\pi$$
$$348$$ 14.9443 0.801097
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 38.7984 2.07090
$$352$$ 2.85410 0.152124
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 2.76393 0.146901
$$355$$ 0 0
$$356$$ −10.4721 −0.555022
$$357$$ 6.09017 0.322326
$$358$$ −2.18034 −0.115235
$$359$$ 26.3607 1.39126 0.695632 0.718399i $$-0.255125\pi$$
0.695632 + 0.718399i $$0.255125\pi$$
$$360$$ 0 0
$$361$$ −15.5623 −0.819069
$$362$$ 12.1459 0.638374
$$363$$ 4.61803 0.242384
$$364$$ 4.38197 0.229677
$$365$$ 0 0
$$366$$ −15.0902 −0.788776
$$367$$ −6.47214 −0.337843 −0.168921 0.985630i $$-0.554028\pi$$
−0.168921 + 0.985630i $$0.554028\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −1.27051 −0.0661401
$$370$$ 0 0
$$371$$ −0.291796 −0.0151493
$$372$$ −14.7082 −0.762585
$$373$$ −20.1803 −1.04490 −0.522449 0.852670i $$-0.674982\pi$$
−0.522449 + 0.852670i $$0.674982\pi$$
$$374$$ −17.3820 −0.898800
$$375$$ 0 0
$$376$$ −3.70820 −0.191236
$$377$$ −65.4853 −3.37266
$$378$$ −3.38197 −0.173950
$$379$$ −22.4508 −1.15322 −0.576611 0.817019i $$-0.695625\pi$$
−0.576611 + 0.817019i $$0.695625\pi$$
$$380$$ 0 0
$$381$$ 26.1803 1.34126
$$382$$ 13.7082 0.701373
$$383$$ 17.8885 0.914062 0.457031 0.889451i $$-0.348913\pi$$
0.457031 + 0.889451i $$0.348913\pi$$
$$384$$ 1.61803 0.0825700
$$385$$ 0 0
$$386$$ 0.763932 0.0388831
$$387$$ 0 0
$$388$$ −12.3820 −0.628599
$$389$$ 21.3262 1.08128 0.540642 0.841253i $$-0.318182\pi$$
0.540642 + 0.841253i $$0.318182\pi$$
$$390$$ 0 0
$$391$$ 6.09017 0.307993
$$392$$ 6.61803 0.334261
$$393$$ −4.76393 −0.240309
$$394$$ −22.5623 −1.13667
$$395$$ 0 0
$$396$$ 1.09017 0.0547831
$$397$$ −7.32624 −0.367693 −0.183847 0.982955i $$-0.558855\pi$$
−0.183847 + 0.982955i $$0.558855\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ −1.85410 −0.0928212
$$400$$ 0 0
$$401$$ −1.70820 −0.0853036 −0.0426518 0.999090i $$-0.513581\pi$$
−0.0426518 + 0.999090i $$0.513581\pi$$
$$402$$ −23.4164 −1.16790
$$403$$ 64.4508 3.21053
$$404$$ −0.291796 −0.0145174
$$405$$ 0 0
$$406$$ 5.70820 0.283293
$$407$$ 18.4721 0.915630
$$408$$ −9.85410 −0.487851
$$409$$ −30.2148 −1.49402 −0.747012 0.664810i $$-0.768512\pi$$
−0.747012 + 0.664810i $$0.768512\pi$$
$$410$$ 0 0
$$411$$ −16.7082 −0.824155
$$412$$ −16.5623 −0.815966
$$413$$ 1.05573 0.0519490
$$414$$ −0.381966 −0.0187726
$$415$$ 0 0
$$416$$ −7.09017 −0.347624
$$417$$ −20.6525 −1.01136
$$418$$ 5.29180 0.258830
$$419$$ 14.4721 0.707010 0.353505 0.935433i $$-0.384990\pi$$
0.353505 + 0.935433i $$0.384990\pi$$
$$420$$ 0 0
$$421$$ 13.7426 0.669776 0.334888 0.942258i $$-0.391302\pi$$
0.334888 + 0.942258i $$0.391302\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ −1.41641 −0.0688681
$$424$$ 0.472136 0.0229289
$$425$$ 0 0
$$426$$ −6.61803 −0.320645
$$427$$ −5.76393 −0.278936
$$428$$ −18.1803 −0.878780
$$429$$ 32.7426 1.58083
$$430$$ 0 0
$$431$$ 3.34752 0.161245 0.0806223 0.996745i $$-0.474309\pi$$
0.0806223 + 0.996745i $$0.474309\pi$$
$$432$$ 5.47214 0.263278
$$433$$ −8.50658 −0.408800 −0.204400 0.978887i $$-0.565524\pi$$
−0.204400 + 0.978887i $$0.565524\pi$$
$$434$$ −5.61803 −0.269674
$$435$$ 0 0
$$436$$ −11.5623 −0.553734
$$437$$ −1.85410 −0.0886937
$$438$$ −5.23607 −0.250189
$$439$$ −13.3820 −0.638686 −0.319343 0.947639i $$-0.603462\pi$$
−0.319343 + 0.947639i $$0.603462\pi$$
$$440$$ 0 0
$$441$$ 2.52786 0.120374
$$442$$ 43.1803 2.05388
$$443$$ −25.0902 −1.19207 −0.596035 0.802958i $$-0.703258\pi$$
−0.596035 + 0.802958i $$0.703258\pi$$
$$444$$ 10.4721 0.496986
$$445$$ 0 0
$$446$$ −20.9443 −0.991740
$$447$$ 12.7082 0.601077
$$448$$ 0.618034 0.0291994
$$449$$ −1.56231 −0.0737298 −0.0368649 0.999320i $$-0.511737\pi$$
−0.0368649 + 0.999320i $$0.511737\pi$$
$$450$$ 0 0
$$451$$ −9.49342 −0.447028
$$452$$ −1.05573 −0.0496573
$$453$$ 4.14590 0.194791
$$454$$ −18.7639 −0.880635
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ 37.7771 1.76714 0.883569 0.468301i $$-0.155134\pi$$
0.883569 + 0.468301i $$0.155134\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −33.3262 −1.55554
$$460$$ 0 0
$$461$$ −39.2361 −1.82741 −0.913703 0.406383i $$-0.866790\pi$$
−0.913703 + 0.406383i $$0.866790\pi$$
$$462$$ −2.85410 −0.132785
$$463$$ −2.00000 −0.0929479 −0.0464739 0.998920i $$-0.514798\pi$$
−0.0464739 + 0.998920i $$0.514798\pi$$
$$464$$ −9.23607 −0.428774
$$465$$ 0 0
$$466$$ 6.29180 0.291462
$$467$$ 17.1246 0.792433 0.396216 0.918157i $$-0.370323\pi$$
0.396216 + 0.918157i $$0.370323\pi$$
$$468$$ −2.70820 −0.125187
$$469$$ −8.94427 −0.413008
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ −1.70820 −0.0786265
$$473$$ 0 0
$$474$$ 2.47214 0.113549
$$475$$ 0 0
$$476$$ −3.76393 −0.172520
$$477$$ 0.180340 0.00825720
$$478$$ 20.3607 0.931276
$$479$$ −31.8885 −1.45702 −0.728512 0.685033i $$-0.759788\pi$$
−0.728512 + 0.685033i $$0.759788\pi$$
$$480$$ 0 0
$$481$$ −45.8885 −2.09234
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ −2.85410 −0.129732
$$485$$ 0 0
$$486$$ 3.94427 0.178916
$$487$$ 19.8197 0.898115 0.449057 0.893503i $$-0.351760\pi$$
0.449057 + 0.893503i $$0.351760\pi$$
$$488$$ 9.32624 0.422179
$$489$$ 2.23607 0.101118
$$490$$ 0 0
$$491$$ 6.18034 0.278915 0.139457 0.990228i $$-0.455464\pi$$
0.139457 + 0.990228i $$0.455464\pi$$
$$492$$ −5.38197 −0.242638
$$493$$ 56.2492 2.53334
$$494$$ −13.1459 −0.591462
$$495$$ 0 0
$$496$$ 9.09017 0.408161
$$497$$ −2.52786 −0.113390
$$498$$ 11.2361 0.503500
$$499$$ 12.3607 0.553340 0.276670 0.960965i $$-0.410769\pi$$
0.276670 + 0.960965i $$0.410769\pi$$
$$500$$ 0 0
$$501$$ −12.9443 −0.578307
$$502$$ −6.14590 −0.274305
$$503$$ −36.3262 −1.61971 −0.809853 0.586632i $$-0.800453\pi$$
−0.809853 + 0.586632i $$0.800453\pi$$
$$504$$ 0.236068 0.0105153
$$505$$ 0 0
$$506$$ −2.85410 −0.126880
$$507$$ −60.3050 −2.67824
$$508$$ −16.1803 −0.717886
$$509$$ 36.6525 1.62459 0.812296 0.583245i $$-0.198217\pi$$
0.812296 + 0.583245i $$0.198217\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 10.1459 0.447952
$$514$$ −7.81966 −0.344910
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −10.5836 −0.465466
$$518$$ 4.00000 0.175750
$$519$$ −2.32624 −0.102111
$$520$$ 0 0
$$521$$ −15.5279 −0.680288 −0.340144 0.940373i $$-0.610476\pi$$
−0.340144 + 0.940373i $$0.610476\pi$$
$$522$$ −3.52786 −0.154410
$$523$$ 26.0000 1.13690 0.568450 0.822718i $$-0.307543\pi$$
0.568450 + 0.822718i $$0.307543\pi$$
$$524$$ 2.94427 0.128621
$$525$$ 0 0
$$526$$ 20.7426 0.904422
$$527$$ −55.3607 −2.41155
$$528$$ 4.61803 0.200974
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −0.652476 −0.0283150
$$532$$ 1.14590 0.0496810
$$533$$ 23.5836 1.02152
$$534$$ −16.9443 −0.733250
$$535$$ 0 0
$$536$$ 14.4721 0.625101
$$537$$ −3.52786 −0.152239
$$538$$ 14.1803 0.611358
$$539$$ 18.8885 0.813587
$$540$$ 0 0
$$541$$ 22.8328 0.981659 0.490830 0.871256i $$-0.336694\pi$$
0.490830 + 0.871256i $$0.336694\pi$$
$$542$$ 30.3262 1.30262
$$543$$ 19.6525 0.843368
$$544$$ 6.09017 0.261114
$$545$$ 0 0
$$546$$ 7.09017 0.303431
$$547$$ −27.9230 −1.19390 −0.596950 0.802278i $$-0.703621\pi$$
−0.596950 + 0.802278i $$0.703621\pi$$
$$548$$ 10.3262 0.441115
$$549$$ 3.56231 0.152036
$$550$$ 0 0
$$551$$ −17.1246 −0.729533
$$552$$ −1.61803 −0.0688681
$$553$$ 0.944272 0.0401545
$$554$$ 29.4164 1.24978
$$555$$ 0 0
$$556$$ 12.7639 0.541311
$$557$$ 22.8328 0.967457 0.483729 0.875218i $$-0.339282\pi$$
0.483729 + 0.875218i $$0.339282\pi$$
$$558$$ 3.47214 0.146987
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −28.1246 −1.18742
$$562$$ −22.7639 −0.960239
$$563$$ 13.8885 0.585332 0.292666 0.956215i $$-0.405458\pi$$
0.292666 + 0.956215i $$0.405458\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −26.9443 −1.13255
$$567$$ −4.76393 −0.200066
$$568$$ 4.09017 0.171620
$$569$$ −2.00000 −0.0838444 −0.0419222 0.999121i $$-0.513348\pi$$
−0.0419222 + 0.999121i $$0.513348\pi$$
$$570$$ 0 0
$$571$$ 15.9787 0.668688 0.334344 0.942451i $$-0.391485\pi$$
0.334344 + 0.942451i $$0.391485\pi$$
$$572$$ −20.2361 −0.846113
$$573$$ 22.1803 0.926597
$$574$$ −2.05573 −0.0858044
$$575$$ 0 0
$$576$$ −0.381966 −0.0159153
$$577$$ 3.52786 0.146867 0.0734335 0.997300i $$-0.476604\pi$$
0.0734335 + 0.997300i $$0.476604\pi$$
$$578$$ −20.0902 −0.835641
$$579$$ 1.23607 0.0513692
$$580$$ 0 0
$$581$$ 4.29180 0.178054
$$582$$ −20.0344 −0.830454
$$583$$ 1.34752 0.0558087
$$584$$ 3.23607 0.133909
$$585$$ 0 0
$$586$$ −19.8885 −0.821588
$$587$$ −13.6180 −0.562076 −0.281038 0.959697i $$-0.590679\pi$$
−0.281038 + 0.959697i $$0.590679\pi$$
$$588$$ 10.7082 0.441599
$$589$$ 16.8541 0.694461
$$590$$ 0 0
$$591$$ −36.5066 −1.50168
$$592$$ −6.47214 −0.266003
$$593$$ −39.2361 −1.61123 −0.805616 0.592438i $$-0.798166\pi$$
−0.805616 + 0.592438i $$0.798166\pi$$
$$594$$ 15.6180 0.640816
$$595$$ 0 0
$$596$$ −7.85410 −0.321717
$$597$$ −3.23607 −0.132443
$$598$$ 7.09017 0.289939
$$599$$ −18.3820 −0.751067 −0.375533 0.926809i $$-0.622540\pi$$
−0.375533 + 0.926809i $$0.622540\pi$$
$$600$$ 0 0
$$601$$ −33.2705 −1.35713 −0.678566 0.734539i $$-0.737398\pi$$
−0.678566 + 0.734539i $$0.737398\pi$$
$$602$$ 0 0
$$603$$ 5.52786 0.225112
$$604$$ −2.56231 −0.104259
$$605$$ 0 0
$$606$$ −0.472136 −0.0191792
$$607$$ −26.4721 −1.07447 −0.537235 0.843432i $$-0.680531\pi$$
−0.537235 + 0.843432i $$0.680531\pi$$
$$608$$ −1.85410 −0.0751938
$$609$$ 9.23607 0.374264
$$610$$ 0 0
$$611$$ 26.2918 1.06365
$$612$$ 2.32624 0.0940326
$$613$$ −19.3050 −0.779720 −0.389860 0.920874i $$-0.627477\pi$$
−0.389860 + 0.920874i $$0.627477\pi$$
$$614$$ −28.4508 −1.14818
$$615$$ 0 0
$$616$$ 1.76393 0.0710708
$$617$$ −34.0902 −1.37242 −0.686209 0.727404i $$-0.740727\pi$$
−0.686209 + 0.727404i $$0.740727\pi$$
$$618$$ −26.7984 −1.07799
$$619$$ −2.79837 −0.112476 −0.0562381 0.998417i $$-0.517911\pi$$
−0.0562381 + 0.998417i $$0.517911\pi$$
$$620$$ 0 0
$$621$$ −5.47214 −0.219589
$$622$$ −4.00000 −0.160385
$$623$$ −6.47214 −0.259301
$$624$$ −11.4721 −0.459253
$$625$$ 0 0
$$626$$ 12.7984 0.511526
$$627$$ 8.56231 0.341946
$$628$$ 3.70820 0.147973
$$629$$ 39.4164 1.57164
$$630$$ 0 0
$$631$$ 42.0689 1.67474 0.837368 0.546640i $$-0.184093\pi$$
0.837368 + 0.546640i $$0.184093\pi$$
$$632$$ −1.52786 −0.0607752
$$633$$ −22.6525 −0.900355
$$634$$ −11.0902 −0.440447
$$635$$ 0 0
$$636$$ 0.763932 0.0302919
$$637$$ −46.9230 −1.85916
$$638$$ −26.3607 −1.04363
$$639$$ 1.56231 0.0618039
$$640$$ 0 0
$$641$$ 0.360680 0.0142460 0.00712300 0.999975i $$-0.497733\pi$$
0.00712300 + 0.999975i $$0.497733\pi$$
$$642$$ −29.4164 −1.16097
$$643$$ 8.29180 0.326997 0.163498 0.986544i $$-0.447722\pi$$
0.163498 + 0.986544i $$0.447722\pi$$
$$644$$ −0.618034 −0.0243540
$$645$$ 0 0
$$646$$ 11.2918 0.444270
$$647$$ 36.2492 1.42510 0.712552 0.701619i $$-0.247539\pi$$
0.712552 + 0.701619i $$0.247539\pi$$
$$648$$ 7.70820 0.302807
$$649$$ −4.87539 −0.191376
$$650$$ 0 0
$$651$$ −9.09017 −0.356272
$$652$$ −1.38197 −0.0541220
$$653$$ 8.03444 0.314412 0.157206 0.987566i $$-0.449751\pi$$
0.157206 + 0.987566i $$0.449751\pi$$
$$654$$ −18.7082 −0.731549
$$655$$ 0 0
$$656$$ 3.32624 0.129868
$$657$$ 1.23607 0.0482236
$$658$$ −2.29180 −0.0893435
$$659$$ −46.2492 −1.80161 −0.900807 0.434220i $$-0.857024\pi$$
−0.900807 + 0.434220i $$0.857024\pi$$
$$660$$ 0 0
$$661$$ 18.6738 0.726325 0.363163 0.931726i $$-0.381697\pi$$
0.363163 + 0.931726i $$0.381697\pi$$
$$662$$ −19.2361 −0.747631
$$663$$ 69.8673 2.71342
$$664$$ −6.94427 −0.269490
$$665$$ 0 0
$$666$$ −2.47214 −0.0957933
$$667$$ 9.23607 0.357622
$$668$$ 8.00000 0.309529
$$669$$ −33.8885 −1.31021
$$670$$ 0 0
$$671$$ 26.6180 1.02758
$$672$$ 1.00000 0.0385758
$$673$$ 10.9443 0.421871 0.210935 0.977500i $$-0.432349\pi$$
0.210935 + 0.977500i $$0.432349\pi$$
$$674$$ 13.6738 0.526694
$$675$$ 0 0
$$676$$ 37.2705 1.43348
$$677$$ 50.9443 1.95795 0.978974 0.203986i $$-0.0653899\pi$$
0.978974 + 0.203986i $$0.0653899\pi$$
$$678$$ −1.70820 −0.0656032
$$679$$ −7.65248 −0.293675
$$680$$ 0 0
$$681$$ −30.3607 −1.16342
$$682$$ 25.9443 0.993458
$$683$$ 31.5623 1.20770 0.603849 0.797099i $$-0.293633\pi$$
0.603849 + 0.797099i $$0.293633\pi$$
$$684$$ −0.708204 −0.0270789
$$685$$ 0 0
$$686$$ 8.41641 0.321340
$$687$$ −16.1803 −0.617318
$$688$$ 0 0
$$689$$ −3.34752 −0.127531
$$690$$ 0 0
$$691$$ −29.2361 −1.11219 −0.556096 0.831118i $$-0.687701\pi$$
−0.556096 + 0.831118i $$0.687701\pi$$
$$692$$ 1.43769 0.0546529
$$693$$ 0.673762 0.0255941
$$694$$ −6.38197 −0.242256
$$695$$ 0 0
$$696$$ −14.9443 −0.566461
$$697$$ −20.2574 −0.767302
$$698$$ 2.00000 0.0757011
$$699$$ 10.1803 0.385056
$$700$$ 0 0
$$701$$ 43.3394 1.63691 0.818453 0.574573i $$-0.194832\pi$$
0.818453 + 0.574573i $$0.194832\pi$$
$$702$$ −38.7984 −1.46435
$$703$$ −12.0000 −0.452589
$$704$$ −2.85410 −0.107568
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ −0.180340 −0.00678238
$$708$$ −2.76393 −0.103875
$$709$$ −26.0902 −0.979837 −0.489918 0.871768i $$-0.662973\pi$$
−0.489918 + 0.871768i $$0.662973\pi$$
$$710$$ 0 0
$$711$$ −0.583592 −0.0218864
$$712$$ 10.4721 0.392460
$$713$$ −9.09017 −0.340430
$$714$$ −6.09017 −0.227919
$$715$$ 0 0
$$716$$ 2.18034 0.0814831
$$717$$ 32.9443 1.23033
$$718$$ −26.3607 −0.983772
$$719$$ 35.2705 1.31537 0.657684 0.753294i $$-0.271536\pi$$
0.657684 + 0.753294i $$0.271536\pi$$
$$720$$ 0 0
$$721$$ −10.2361 −0.381211
$$722$$ 15.5623 0.579169
$$723$$ 0 0
$$724$$ −12.1459 −0.451399
$$725$$ 0 0
$$726$$ −4.61803 −0.171391
$$727$$ −28.2016 −1.04594 −0.522970 0.852351i $$-0.675176\pi$$
−0.522970 + 0.852351i $$0.675176\pi$$
$$728$$ −4.38197 −0.162406
$$729$$ 29.5066 1.09284
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 15.0902 0.557749
$$733$$ 29.4164 1.08652 0.543260 0.839565i $$-0.317190\pi$$
0.543260 + 0.839565i $$0.317190\pi$$
$$734$$ 6.47214 0.238891
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 41.3050 1.52149
$$738$$ 1.27051 0.0467681
$$739$$ −13.8885 −0.510898 −0.255449 0.966822i $$-0.582223\pi$$
−0.255449 + 0.966822i $$0.582223\pi$$
$$740$$ 0 0
$$741$$ −21.2705 −0.781392
$$742$$ 0.291796 0.0107122
$$743$$ −33.6312 −1.23381 −0.616904 0.787038i $$-0.711613\pi$$
−0.616904 + 0.787038i $$0.711613\pi$$
$$744$$ 14.7082 0.539229
$$745$$ 0 0
$$746$$ 20.1803 0.738855
$$747$$ −2.65248 −0.0970490
$$748$$ 17.3820 0.635548
$$749$$ −11.2361 −0.410557
$$750$$ 0 0
$$751$$ −47.0132 −1.71553 −0.857767 0.514038i $$-0.828149\pi$$
−0.857767 + 0.514038i $$0.828149\pi$$
$$752$$ 3.70820 0.135224
$$753$$ −9.94427 −0.362389
$$754$$ 65.4853 2.38483
$$755$$ 0 0
$$756$$ 3.38197 0.123001
$$757$$ 17.8885 0.650170 0.325085 0.945685i $$-0.394607\pi$$
0.325085 + 0.945685i $$0.394607\pi$$
$$758$$ 22.4508 0.815452
$$759$$ −4.61803 −0.167624
$$760$$ 0 0
$$761$$ 46.8673 1.69894 0.849468 0.527640i $$-0.176923\pi$$
0.849468 + 0.527640i $$0.176923\pi$$
$$762$$ −26.1803 −0.948414
$$763$$ −7.14590 −0.258699
$$764$$ −13.7082 −0.495945
$$765$$ 0 0
$$766$$ −17.8885 −0.646339
$$767$$ 12.1115 0.437319
$$768$$ −1.61803 −0.0583858
$$769$$ −6.58359 −0.237410 −0.118705 0.992930i $$-0.537874\pi$$
−0.118705 + 0.992930i $$0.537874\pi$$
$$770$$ 0 0
$$771$$ −12.6525 −0.455668
$$772$$ −0.763932 −0.0274945
$$773$$ −28.9443 −1.04105 −0.520527 0.853845i $$-0.674264\pi$$
−0.520527 + 0.853845i $$0.674264\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12.3820 0.444487
$$777$$ 6.47214 0.232187
$$778$$ −21.3262 −0.764583
$$779$$ 6.16718 0.220962
$$780$$ 0 0
$$781$$ 11.6738 0.417720
$$782$$ −6.09017 −0.217784
$$783$$ −50.5410 −1.80619
$$784$$ −6.61803 −0.236358
$$785$$ 0 0
$$786$$ 4.76393 0.169924
$$787$$ 2.87539 0.102497 0.0512483 0.998686i $$-0.483680\pi$$
0.0512483 + 0.998686i $$0.483680\pi$$
$$788$$ 22.5623 0.803749
$$789$$ 33.5623 1.19485
$$790$$ 0 0
$$791$$ −0.652476 −0.0231994
$$792$$ −1.09017 −0.0387375
$$793$$ −66.1246 −2.34815
$$794$$ 7.32624 0.259998
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ −13.7082 −0.485569 −0.242785 0.970080i $$-0.578061\pi$$
−0.242785 + 0.970080i $$0.578061\pi$$
$$798$$ 1.85410 0.0656345
$$799$$ −22.5836 −0.798950
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 1.70820 0.0603188
$$803$$ 9.23607 0.325934
$$804$$ 23.4164 0.825833
$$805$$ 0 0
$$806$$ −64.4508 −2.27018
$$807$$ 22.9443 0.807677
$$808$$ 0.291796 0.0102653
$$809$$ −46.7426 −1.64338 −0.821692 0.569932i $$-0.806970\pi$$
−0.821692 + 0.569932i $$0.806970\pi$$
$$810$$ 0 0
$$811$$ 21.8197 0.766192 0.383096 0.923709i $$-0.374858\pi$$
0.383096 + 0.923709i $$0.374858\pi$$
$$812$$ −5.70820 −0.200319
$$813$$ 49.0689 1.72092
$$814$$ −18.4721 −0.647448
$$815$$ 0 0
$$816$$ 9.85410 0.344963
$$817$$ 0 0
$$818$$ 30.2148 1.05644
$$819$$ −1.67376 −0.0584860
$$820$$ 0 0
$$821$$ −33.0557 −1.15365 −0.576826 0.816867i $$-0.695709\pi$$
−0.576826 + 0.816867i $$0.695709\pi$$
$$822$$ 16.7082 0.582766
$$823$$ −25.4164 −0.885960 −0.442980 0.896531i $$-0.646079\pi$$
−0.442980 + 0.896531i $$0.646079\pi$$
$$824$$ 16.5623 0.576975
$$825$$ 0 0
$$826$$ −1.05573 −0.0367335
$$827$$ −21.7082 −0.754868 −0.377434 0.926036i $$-0.623194\pi$$
−0.377434 + 0.926036i $$0.623194\pi$$
$$828$$ 0.381966 0.0132742
$$829$$ 18.9443 0.657962 0.328981 0.944337i $$-0.393295\pi$$
0.328981 + 0.944337i $$0.393295\pi$$
$$830$$ 0 0
$$831$$ 47.5967 1.65111
$$832$$ 7.09017 0.245807
$$833$$ 40.3050 1.39648
$$834$$ 20.6525 0.715137
$$835$$ 0 0
$$836$$ −5.29180 −0.183021
$$837$$ 49.7426 1.71936
$$838$$ −14.4721 −0.499932
$$839$$ 33.0132 1.13974 0.569870 0.821735i $$-0.306993\pi$$
0.569870 + 0.821735i $$0.306993\pi$$
$$840$$ 0 0
$$841$$ 56.3050 1.94155
$$842$$ −13.7426 −0.473603
$$843$$ −36.8328 −1.26859
$$844$$ 14.0000 0.481900
$$845$$ 0 0
$$846$$ 1.41641 0.0486971
$$847$$ −1.76393 −0.0606094
$$848$$ −0.472136 −0.0162132
$$849$$ −43.5967 −1.49624
$$850$$ 0 0
$$851$$ 6.47214 0.221862
$$852$$ 6.61803 0.226730
$$853$$ 10.7984 0.369729 0.184865 0.982764i $$-0.440815\pi$$
0.184865 + 0.982764i $$0.440815\pi$$
$$854$$ 5.76393 0.197238
$$855$$ 0 0
$$856$$ 18.1803 0.621391
$$857$$ 6.58359 0.224891 0.112446 0.993658i $$-0.464132\pi$$
0.112446 + 0.993658i $$0.464132\pi$$
$$858$$ −32.7426 −1.11782
$$859$$ −24.0689 −0.821220 −0.410610 0.911811i $$-0.634684\pi$$
−0.410610 + 0.911811i $$0.634684\pi$$
$$860$$ 0 0
$$861$$ −3.32624 −0.113358
$$862$$ −3.34752 −0.114017
$$863$$ −32.7639 −1.11530 −0.557649 0.830077i $$-0.688296\pi$$
−0.557649 + 0.830077i $$0.688296\pi$$
$$864$$ −5.47214 −0.186166
$$865$$ 0 0
$$866$$ 8.50658 0.289065
$$867$$ −32.5066 −1.10398
$$868$$ 5.61803 0.190688
$$869$$ −4.36068 −0.147926
$$870$$ 0 0
$$871$$ −102.610 −3.47680
$$872$$ 11.5623 0.391549
$$873$$ 4.72949 0.160069
$$874$$ 1.85410 0.0627159
$$875$$ 0 0
$$876$$ 5.23607 0.176910
$$877$$ −18.7426 −0.632894 −0.316447 0.948610i $$-0.602490\pi$$
−0.316447 + 0.948610i $$0.602490\pi$$
$$878$$ 13.3820 0.451619
$$879$$ −32.1803 −1.08542
$$880$$ 0 0
$$881$$ 8.58359 0.289189 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$882$$ −2.52786 −0.0851176
$$883$$ −15.5623 −0.523713 −0.261857 0.965107i $$-0.584335\pi$$
−0.261857 + 0.965107i $$0.584335\pi$$
$$884$$ −43.1803 −1.45231
$$885$$ 0 0
$$886$$ 25.0902 0.842921
$$887$$ 5.16718 0.173497 0.0867485 0.996230i $$-0.472352\pi$$
0.0867485 + 0.996230i $$0.472352\pi$$
$$888$$ −10.4721 −0.351422
$$889$$ −10.0000 −0.335389
$$890$$ 0 0
$$891$$ 22.0000 0.737028
$$892$$ 20.9443 0.701266
$$893$$ 6.87539 0.230076
$$894$$ −12.7082 −0.425026
$$895$$ 0 0
$$896$$ −0.618034 −0.0206471
$$897$$ 11.4721 0.383043
$$898$$ 1.56231 0.0521348
$$899$$ −83.9574 −2.80014
$$900$$ 0 0
$$901$$ 2.87539 0.0957931
$$902$$ 9.49342 0.316096
$$903$$ 0 0
$$904$$ 1.05573 0.0351130
$$905$$ 0 0
$$906$$ −4.14590 −0.137738
$$907$$ −7.12461 −0.236569 −0.118284 0.992980i $$-0.537739\pi$$
−0.118284 + 0.992980i $$0.537739\pi$$
$$908$$ 18.7639 0.622703
$$909$$ 0.111456 0.00369677
$$910$$ 0 0
$$911$$ 36.0689 1.19502 0.597508 0.801863i $$-0.296158\pi$$
0.597508 + 0.801863i $$0.296158\pi$$
$$912$$ −3.00000 −0.0993399
$$913$$ −19.8197 −0.655935
$$914$$ −37.7771 −1.24955
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 1.81966 0.0600905
$$918$$ 33.3262 1.09993
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −46.0344 −1.51689
$$922$$ 39.2361 1.29217
$$923$$ −29.0000 −0.954547
$$924$$ 2.85410 0.0938931
$$925$$ 0 0
$$926$$ 2.00000 0.0657241
$$927$$ 6.32624 0.207781
$$928$$ 9.23607 0.303189
$$929$$ −3.52786 −0.115745 −0.0578727 0.998324i $$-0.518432\pi$$
−0.0578727 + 0.998324i $$0.518432\pi$$
$$930$$ 0 0
$$931$$ −12.2705 −0.402150
$$932$$ −6.29180 −0.206095
$$933$$ −6.47214 −0.211888
$$934$$ −17.1246 −0.560334
$$935$$ 0 0
$$936$$ 2.70820 0.0885204
$$937$$ 36.7984 1.20215 0.601075 0.799192i $$-0.294739\pi$$
0.601075 + 0.799192i $$0.294739\pi$$
$$938$$ 8.94427 0.292041
$$939$$ 20.7082 0.675787
$$940$$ 0 0
$$941$$ −22.4934 −0.733265 −0.366632 0.930366i $$-0.619489\pi$$
−0.366632 + 0.930366i $$0.619489\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −3.32624 −0.108317
$$944$$ 1.70820 0.0555973
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 54.6869 1.77709 0.888543 0.458793i $$-0.151718\pi$$
0.888543 + 0.458793i $$0.151718\pi$$
$$948$$ −2.47214 −0.0802912
$$949$$ −22.9443 −0.744803
$$950$$ 0 0
$$951$$ −17.9443 −0.581883
$$952$$ 3.76393 0.121990
$$953$$ −3.79837 −0.123041 −0.0615207 0.998106i $$-0.519595\pi$$
−0.0615207 + 0.998106i $$0.519595\pi$$
$$954$$ −0.180340 −0.00583872
$$955$$ 0 0
$$956$$ −20.3607 −0.658511
$$957$$ −42.6525 −1.37876
$$958$$ 31.8885 1.03027
$$959$$ 6.38197 0.206084
$$960$$ 0 0
$$961$$ 51.6312 1.66552
$$962$$ 45.8885 1.47951
$$963$$ 6.94427 0.223776
$$964$$ 0 0
$$965$$ 0 0
$$966$$ −1.00000 −0.0321745
$$967$$ 16.5410 0.531923 0.265962 0.963984i $$-0.414311\pi$$
0.265962 + 0.963984i $$0.414311\pi$$
$$968$$ 2.85410 0.0917343
$$969$$ 18.2705 0.586933
$$970$$ 0 0
$$971$$ 34.2705 1.09979 0.549896 0.835233i $$-0.314667\pi$$
0.549896 + 0.835233i $$0.314667\pi$$
$$972$$ −3.94427 −0.126513
$$973$$ 7.88854 0.252895
$$974$$ −19.8197 −0.635063
$$975$$ 0 0
$$976$$ −9.32624 −0.298526
$$977$$ 23.5623 0.753825 0.376912 0.926249i $$-0.376986\pi$$
0.376912 + 0.926249i $$0.376986\pi$$
$$978$$ −2.23607 −0.0715016
$$979$$ 29.8885 0.955242
$$980$$ 0 0
$$981$$ 4.41641 0.141005
$$982$$ −6.18034 −0.197223
$$983$$ 14.2705 0.455159 0.227579 0.973760i $$-0.426919\pi$$
0.227579 + 0.973760i $$0.426919\pi$$
$$984$$ 5.38197 0.171571
$$985$$ 0 0
$$986$$ −56.2492 −1.79134
$$987$$ −3.70820 −0.118033
$$988$$ 13.1459 0.418227
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 27.5066 0.873775 0.436888 0.899516i $$-0.356081\pi$$
0.436888 + 0.899516i $$0.356081\pi$$
$$992$$ −9.09017 −0.288613
$$993$$ −31.1246 −0.987710
$$994$$ 2.52786 0.0801790
$$995$$ 0 0
$$996$$ −11.2361 −0.356028
$$997$$ −57.1935 −1.81134 −0.905668 0.423987i $$-0.860630\pi$$
−0.905668 + 0.423987i $$0.860630\pi$$
$$998$$ −12.3607 −0.391270
$$999$$ −35.4164 −1.12053
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 1150.2.a.j.1.1 2
4.3 odd 2 9200.2.a.bu.1.2 2
5.2 odd 4 1150.2.b.i.599.2 4
5.3 odd 4 1150.2.b.i.599.3 4
5.4 even 2 230.2.a.c.1.2 2
15.14 odd 2 2070.2.a.u.1.1 2
20.19 odd 2 1840.2.a.l.1.1 2
40.19 odd 2 7360.2.a.bn.1.2 2
40.29 even 2 7360.2.a.bh.1.1 2
115.114 odd 2 5290.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 5.4 even 2
1150.2.a.j.1.1 2 1.1 even 1 trivial
1150.2.b.i.599.2 4 5.2 odd 4
1150.2.b.i.599.3 4 5.3 odd 4
1840.2.a.l.1.1 2 20.19 odd 2
2070.2.a.u.1.1 2 15.14 odd 2
5290.2.a.o.1.2 2 115.114 odd 2
7360.2.a.bh.1.1 2 40.29 even 2
7360.2.a.bn.1.2 2 40.19 odd 2
9200.2.a.bu.1.2 2 4.3 odd 2