# Properties

 Label 1150.2.b.i.599.2 Level $1150$ Weight $2$ Character 1150.599 Analytic conductor $9.183$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,2,Mod(599,1150)]

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([1, 0]))

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

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

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 599.2 Root $$1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 1150.599 Dual form 1150.2.b.i.599.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} +0.618034i q^{7} +1.00000i q^{8} +0.381966 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} +0.618034i q^{7} +1.00000i q^{8} +0.381966 q^{9} -2.85410 q^{11} -1.61803i q^{12} -7.09017i q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017i q^{17} -0.381966i q^{18} -1.85410 q^{19} -1.00000 q^{21} +2.85410i q^{22} +1.00000i q^{23} -1.61803 q^{24} -7.09017 q^{26} +5.47214i q^{27} -0.618034i q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000i q^{32} -4.61803i q^{33} -6.09017 q^{34} -0.381966 q^{36} -6.47214i q^{37} +1.85410i q^{38} +11.4721 q^{39} +3.32624 q^{41} +1.00000i q^{42} +2.85410 q^{44} +1.00000 q^{46} +3.70820i q^{47} +1.61803i q^{48} +6.61803 q^{49} +9.85410 q^{51} +7.09017i q^{52} +0.472136i q^{53} +5.47214 q^{54} -0.618034 q^{56} -3.00000i q^{57} -9.23607i q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017i q^{62} +0.236068i q^{63} -1.00000 q^{64} -4.61803 q^{66} -14.4721i q^{67} +6.09017i q^{68} -1.61803 q^{69} -4.09017 q^{71} +0.381966i q^{72} +3.23607i q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393i q^{77} -11.4721i q^{78} -1.52786 q^{79} -7.70820 q^{81} -3.32624i q^{82} -6.94427i q^{83} +1.00000 q^{84} +14.9443i q^{87} -2.85410i q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000i q^{92} +14.7082i q^{93} +3.70820 q^{94} +1.61803 q^{96} -12.3820i q^{97} -6.61803i q^{98} -1.09017 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 2 q^{6} + 6 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 2 * q^6 + 6 * q^9 $$4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46} + 22 q^{49} + 26 q^{51} + 4 q^{54} + 2 q^{56} + 20 q^{59} - 6 q^{61} - 4 q^{64} - 14 q^{66} - 2 q^{69} + 6 q^{71} - 8 q^{74} - 6 q^{76} - 24 q^{79} - 4 q^{81} + 4 q^{84} + 24 q^{89} + 22 q^{91} - 12 q^{94} + 2 q^{96} + 18 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 2 * q^6 + 6 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * q^19 - 4 * q^21 - 2 * q^24 - 6 * q^26 + 28 * q^29 + 14 * q^31 - 2 * q^34 - 6 * q^36 + 28 * q^39 - 18 * q^41 - 2 * q^44 + 4 * q^46 + 22 * q^49 + 26 * q^51 + 4 * q^54 + 2 * q^56 + 20 * q^59 - 6 * q^61 - 4 * q^64 - 14 * q^66 - 2 * q^69 + 6 * q^71 - 8 * q^74 - 6 * q^76 - 24 * q^79 - 4 * q^81 + 4 * q^84 + 24 * q^89 + 22 * q^91 - 12 * q^94 + 2 * q^96 + 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.61803i 0.934172i 0.884212 + 0.467086i $$0.154696\pi$$
−0.884212 + 0.467086i $$0.845304\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.61803 0.660560
$$7$$ 0.618034i 0.233595i 0.993156 + 0.116797i $$0.0372628\pi$$
−0.993156 + 0.116797i $$0.962737\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 0.381966 0.127322
$$10$$ 0 0
$$11$$ −2.85410 −0.860544 −0.430272 0.902699i $$-0.641582\pi$$
−0.430272 + 0.902699i $$0.641582\pi$$
$$12$$ − 1.61803i − 0.467086i
$$13$$ − 7.09017i − 1.96646i −0.182372 0.983230i $$-0.558377\pi$$
0.182372 0.983230i $$-0.441623\pi$$
$$14$$ 0.618034 0.165177
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.09017i − 1.47708i −0.674208 0.738542i $$-0.735515\pi$$
0.674208 0.738542i $$-0.264485\pi$$
$$18$$ − 0.381966i − 0.0900303i
$$19$$ −1.85410 −0.425360 −0.212680 0.977122i $$-0.568219\pi$$
−0.212680 + 0.977122i $$0.568219\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 2.85410i 0.608497i
$$23$$ 1.00000i 0.208514i
$$24$$ −1.61803 −0.330280
$$25$$ 0 0
$$26$$ −7.09017 −1.39050
$$27$$ 5.47214i 1.05311i
$$28$$ − 0.618034i − 0.116797i
$$29$$ 9.23607 1.71509 0.857547 0.514405i $$-0.171987\pi$$
0.857547 + 0.514405i $$0.171987\pi$$
$$30$$ 0 0
$$31$$ 9.09017 1.63264 0.816321 0.577598i $$-0.196010\pi$$
0.816321 + 0.577598i $$0.196010\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.61803i − 0.803897i
$$34$$ −6.09017 −1.04446
$$35$$ 0 0
$$36$$ −0.381966 −0.0636610
$$37$$ − 6.47214i − 1.06401i −0.846740 0.532006i $$-0.821438\pi$$
0.846740 0.532006i $$-0.178562\pi$$
$$38$$ 1.85410i 0.300775i
$$39$$ 11.4721 1.83701
$$40$$ 0 0
$$41$$ 3.32624 0.519471 0.259736 0.965680i $$-0.416365\pi$$
0.259736 + 0.965680i $$0.416365\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 2.85410 0.430272
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ 3.70820i 0.540897i 0.962734 + 0.270449i $$0.0871720\pi$$
−0.962734 + 0.270449i $$0.912828\pi$$
$$48$$ 1.61803i 0.233543i
$$49$$ 6.61803 0.945433
$$50$$ 0 0
$$51$$ 9.85410 1.37985
$$52$$ 7.09017i 0.983230i
$$53$$ 0.472136i 0.0648529i 0.999474 + 0.0324264i $$0.0103235\pi$$
−0.999474 + 0.0324264i $$0.989677\pi$$
$$54$$ 5.47214 0.744663
$$55$$ 0 0
$$56$$ −0.618034 −0.0825883
$$57$$ − 3.00000i − 0.397360i
$$58$$ − 9.23607i − 1.21276i
$$59$$ −1.70820 −0.222389 −0.111195 0.993799i $$-0.535468\pi$$
−0.111195 + 0.993799i $$0.535468\pi$$
$$60$$ 0 0
$$61$$ −9.32624 −1.19410 −0.597051 0.802203i $$-0.703661\pi$$
−0.597051 + 0.802203i $$0.703661\pi$$
$$62$$ − 9.09017i − 1.15445i
$$63$$ 0.236068i 0.0297418i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.61803 −0.568441
$$67$$ − 14.4721i − 1.76805i −0.467437 0.884026i $$-0.654823\pi$$
0.467437 0.884026i $$-0.345177\pi$$
$$68$$ 6.09017i 0.738542i
$$69$$ −1.61803 −0.194788
$$70$$ 0 0
$$71$$ −4.09017 −0.485414 −0.242707 0.970100i $$-0.578035\pi$$
−0.242707 + 0.970100i $$0.578035\pi$$
$$72$$ 0.381966i 0.0450151i
$$73$$ 3.23607i 0.378753i 0.981905 + 0.189377i $$0.0606467\pi$$
−0.981905 + 0.189377i $$0.939353\pi$$
$$74$$ −6.47214 −0.752371
$$75$$ 0 0
$$76$$ 1.85410 0.212680
$$77$$ − 1.76393i − 0.201019i
$$78$$ − 11.4721i − 1.29896i
$$79$$ −1.52786 −0.171898 −0.0859491 0.996300i $$-0.527392\pi$$
−0.0859491 + 0.996300i $$0.527392\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ − 3.32624i − 0.367322i
$$83$$ − 6.94427i − 0.762233i −0.924527 0.381116i $$-0.875540\pi$$
0.924527 0.381116i $$-0.124460\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 14.9443i 1.60219i
$$88$$ − 2.85410i − 0.304248i
$$89$$ 10.4721 1.11004 0.555022 0.831836i $$-0.312710\pi$$
0.555022 + 0.831836i $$0.312710\pi$$
$$90$$ 0 0
$$91$$ 4.38197 0.459355
$$92$$ − 1.00000i − 0.104257i
$$93$$ 14.7082i 1.52517i
$$94$$ 3.70820 0.382472
$$95$$ 0 0
$$96$$ 1.61803 0.165140
$$97$$ − 12.3820i − 1.25720i −0.777730 0.628599i $$-0.783629\pi$$
0.777730 0.628599i $$-0.216371\pi$$
$$98$$ − 6.61803i − 0.668522i
$$99$$ −1.09017 −0.109566
$$100$$ 0 0
$$101$$ −0.291796 −0.0290348 −0.0145174 0.999895i $$-0.504621\pi$$
−0.0145174 + 0.999895i $$0.504621\pi$$
$$102$$ − 9.85410i − 0.975701i
$$103$$ 16.5623i 1.63193i 0.578100 + 0.815966i $$0.303795\pi$$
−0.578100 + 0.815966i $$0.696205\pi$$
$$104$$ 7.09017 0.695248
$$105$$ 0 0
$$106$$ 0.472136 0.0458579
$$107$$ − 18.1803i − 1.75756i −0.477227 0.878780i $$-0.658358\pi$$
0.477227 0.878780i $$-0.341642\pi$$
$$108$$ − 5.47214i − 0.526557i
$$109$$ 11.5623 1.10747 0.553734 0.832694i $$-0.313202\pi$$
0.553734 + 0.832694i $$0.313202\pi$$
$$110$$ 0 0
$$111$$ 10.4721 0.993971
$$112$$ 0.618034i 0.0583987i
$$113$$ 1.05573i 0.0993145i 0.998766 + 0.0496573i $$0.0158129\pi$$
−0.998766 + 0.0496573i $$0.984187\pi$$
$$114$$ −3.00000 −0.280976
$$115$$ 0 0
$$116$$ −9.23607 −0.857547
$$117$$ − 2.70820i − 0.250374i
$$118$$ 1.70820i 0.157253i
$$119$$ 3.76393 0.345039
$$120$$ 0 0
$$121$$ −2.85410 −0.259464
$$122$$ 9.32624i 0.844358i
$$123$$ 5.38197i 0.485276i
$$124$$ −9.09017 −0.816321
$$125$$ 0 0
$$126$$ 0.236068 0.0210306
$$127$$ − 16.1803i − 1.43577i −0.696160 0.717886i $$-0.745110\pi$$
0.696160 0.717886i $$-0.254890\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.94427 0.257242 0.128621 0.991694i $$-0.458945\pi$$
0.128621 + 0.991694i $$0.458945\pi$$
$$132$$ 4.61803i 0.401948i
$$133$$ − 1.14590i − 0.0993620i
$$134$$ −14.4721 −1.25020
$$135$$ 0 0
$$136$$ 6.09017 0.522228
$$137$$ 10.3262i 0.882230i 0.897451 + 0.441115i $$0.145417\pi$$
−0.897451 + 0.441115i $$0.854583\pi$$
$$138$$ 1.61803i 0.137736i
$$139$$ −12.7639 −1.08262 −0.541311 0.840822i $$-0.682072\pi$$
−0.541311 + 0.840822i $$0.682072\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 4.09017i 0.343239i
$$143$$ 20.2361i 1.69223i
$$144$$ 0.381966 0.0318305
$$145$$ 0 0
$$146$$ 3.23607 0.267819
$$147$$ 10.7082i 0.883198i
$$148$$ 6.47214i 0.532006i
$$149$$ 7.85410 0.643433 0.321717 0.946836i $$-0.395740\pi$$
0.321717 + 0.946836i $$0.395740\pi$$
$$150$$ 0 0
$$151$$ −2.56231 −0.208517 −0.104259 0.994550i $$-0.533247\pi$$
−0.104259 + 0.994550i $$0.533247\pi$$
$$152$$ − 1.85410i − 0.150388i
$$153$$ − 2.32624i − 0.188065i
$$154$$ −1.76393 −0.142142
$$155$$ 0 0
$$156$$ −11.4721 −0.918506
$$157$$ 3.70820i 0.295947i 0.988991 + 0.147973i $$0.0472750\pi$$
−0.988991 + 0.147973i $$0.952725\pi$$
$$158$$ 1.52786i 0.121550i
$$159$$ −0.763932 −0.0605838
$$160$$ 0 0
$$161$$ −0.618034 −0.0487079
$$162$$ 7.70820i 0.605614i
$$163$$ 1.38197i 0.108244i 0.998534 + 0.0541220i $$0.0172360\pi$$
−0.998534 + 0.0541220i $$0.982764\pi$$
$$164$$ −3.32624 −0.259736
$$165$$ 0 0
$$166$$ −6.94427 −0.538980
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ − 1.00000i − 0.0771517i
$$169$$ −37.2705 −2.86696
$$170$$ 0 0
$$171$$ −0.708204 −0.0541577
$$172$$ 0 0
$$173$$ − 1.43769i − 0.109306i −0.998505 0.0546529i $$-0.982595\pi$$
0.998505 0.0546529i $$-0.0174052\pi$$
$$174$$ 14.9443 1.13292
$$175$$ 0 0
$$176$$ −2.85410 −0.215136
$$177$$ − 2.76393i − 0.207750i
$$178$$ − 10.4721i − 0.784920i
$$179$$ −2.18034 −0.162966 −0.0814831 0.996675i $$-0.525966\pi$$
−0.0814831 + 0.996675i $$0.525966\pi$$
$$180$$ 0 0
$$181$$ −12.1459 −0.902797 −0.451399 0.892322i $$-0.649075\pi$$
−0.451399 + 0.892322i $$0.649075\pi$$
$$182$$ − 4.38197i − 0.324813i
$$183$$ − 15.0902i − 1.11550i
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 14.7082 1.07846
$$187$$ 17.3820i 1.27110i
$$188$$ − 3.70820i − 0.270449i
$$189$$ −3.38197 −0.246002
$$190$$ 0 0
$$191$$ −13.7082 −0.991891 −0.495945 0.868354i $$-0.665178\pi$$
−0.495945 + 0.868354i $$0.665178\pi$$
$$192$$ − 1.61803i − 0.116772i
$$193$$ 0.763932i 0.0549890i 0.999622 + 0.0274945i $$0.00875288\pi$$
−0.999622 + 0.0274945i $$0.991247\pi$$
$$194$$ −12.3820 −0.888973
$$195$$ 0 0
$$196$$ −6.61803 −0.472717
$$197$$ 22.5623i 1.60750i 0.594969 + 0.803749i $$0.297164\pi$$
−0.594969 + 0.803749i $$0.702836\pi$$
$$198$$ 1.09017i 0.0774750i
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ 23.4164 1.65167
$$202$$ 0.291796i 0.0205307i
$$203$$ 5.70820i 0.400637i
$$204$$ −9.85410 −0.689925
$$205$$ 0 0
$$206$$ 16.5623 1.15395
$$207$$ 0.381966i 0.0265485i
$$208$$ − 7.09017i − 0.491615i
$$209$$ 5.29180 0.366041
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ − 0.472136i − 0.0324264i
$$213$$ − 6.61803i − 0.453460i
$$214$$ −18.1803 −1.24278
$$215$$ 0 0
$$216$$ −5.47214 −0.372332
$$217$$ 5.61803i 0.381377i
$$218$$ − 11.5623i − 0.783098i
$$219$$ −5.23607 −0.353821
$$220$$ 0 0
$$221$$ −43.1803 −2.90462
$$222$$ − 10.4721i − 0.702844i
$$223$$ − 20.9443i − 1.40253i −0.712900 0.701266i $$-0.752618\pi$$
0.712900 0.701266i $$-0.247382\pi$$
$$224$$ 0.618034 0.0412941
$$225$$ 0 0
$$226$$ 1.05573 0.0702260
$$227$$ 18.7639i 1.24541i 0.782458 + 0.622703i $$0.213965\pi$$
−0.782458 + 0.622703i $$0.786035\pi$$
$$228$$ 3.00000i 0.198680i
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 2.85410 0.187786
$$232$$ 9.23607i 0.606378i
$$233$$ 6.29180i 0.412189i 0.978532 + 0.206095i $$0.0660755\pi$$
−0.978532 + 0.206095i $$0.933925\pi$$
$$234$$ −2.70820 −0.177041
$$235$$ 0 0
$$236$$ 1.70820 0.111195
$$237$$ − 2.47214i − 0.160582i
$$238$$ − 3.76393i − 0.243979i
$$239$$ 20.3607 1.31702 0.658511 0.752571i $$-0.271186\pi$$
0.658511 + 0.752571i $$0.271186\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 2.85410i 0.183469i
$$243$$ 3.94427i 0.253025i
$$244$$ 9.32624 0.597051
$$245$$ 0 0
$$246$$ 5.38197 0.343142
$$247$$ 13.1459i 0.836453i
$$248$$ 9.09017i 0.577226i
$$249$$ 11.2361 0.712057
$$250$$ 0 0
$$251$$ 6.14590 0.387926 0.193963 0.981009i $$-0.437866\pi$$
0.193963 + 0.981009i $$0.437866\pi$$
$$252$$ − 0.236068i − 0.0148709i
$$253$$ − 2.85410i − 0.179436i
$$254$$ −16.1803 −1.01524
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 7.81966i 0.487777i 0.969803 + 0.243888i $$0.0784231\pi$$
−0.969803 + 0.243888i $$0.921577\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 3.52786 0.218369
$$262$$ − 2.94427i − 0.181898i
$$263$$ 20.7426i 1.27905i 0.768772 + 0.639523i $$0.220868\pi$$
−0.768772 + 0.639523i $$0.779132\pi$$
$$264$$ 4.61803 0.284220
$$265$$ 0 0
$$266$$ −1.14590 −0.0702595
$$267$$ 16.9443i 1.03697i
$$268$$ 14.4721i 0.884026i
$$269$$ 14.1803 0.864591 0.432295 0.901732i $$-0.357704\pi$$
0.432295 + 0.901732i $$0.357704\pi$$
$$270$$ 0 0
$$271$$ −30.3262 −1.84219 −0.921094 0.389341i $$-0.872703\pi$$
−0.921094 + 0.389341i $$0.872703\pi$$
$$272$$ − 6.09017i − 0.369271i
$$273$$ 7.09017i 0.429117i
$$274$$ 10.3262 0.623831
$$275$$ 0 0
$$276$$ 1.61803 0.0973942
$$277$$ − 29.4164i − 1.76746i −0.467996 0.883730i $$-0.655024\pi$$
0.467996 0.883730i $$-0.344976\pi$$
$$278$$ 12.7639i 0.765530i
$$279$$ 3.47214 0.207871
$$280$$ 0 0
$$281$$ 22.7639 1.35798 0.678991 0.734146i $$-0.262417\pi$$
0.678991 + 0.734146i $$0.262417\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ − 26.9443i − 1.60167i −0.598885 0.800835i $$-0.704389\pi$$
0.598885 0.800835i $$-0.295611\pi$$
$$284$$ 4.09017 0.242707
$$285$$ 0 0
$$286$$ 20.2361 1.19658
$$287$$ 2.05573i 0.121346i
$$288$$ − 0.381966i − 0.0225076i
$$289$$ −20.0902 −1.18177
$$290$$ 0 0
$$291$$ 20.0344 1.17444
$$292$$ − 3.23607i − 0.189377i
$$293$$ − 19.8885i − 1.16190i −0.813939 0.580951i $$-0.802681\pi$$
0.813939 0.580951i $$-0.197319\pi$$
$$294$$ 10.7082 0.624515
$$295$$ 0 0
$$296$$ 6.47214 0.376185
$$297$$ − 15.6180i − 0.906250i
$$298$$ − 7.85410i − 0.454976i
$$299$$ 7.09017 0.410035
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2.56231i 0.147444i
$$303$$ − 0.472136i − 0.0271235i
$$304$$ −1.85410 −0.106340
$$305$$ 0 0
$$306$$ −2.32624 −0.132982
$$307$$ 28.4508i 1.62378i 0.583813 + 0.811888i $$0.301560\pi$$
−0.583813 + 0.811888i $$0.698440\pi$$
$$308$$ 1.76393i 0.100509i
$$309$$ −26.7984 −1.52451
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 11.4721i 0.649482i
$$313$$ 12.7984i 0.723407i 0.932293 + 0.361703i $$0.117805\pi$$
−0.932293 + 0.361703i $$0.882195\pi$$
$$314$$ 3.70820 0.209266
$$315$$ 0 0
$$316$$ 1.52786 0.0859491
$$317$$ 11.0902i 0.622886i 0.950265 + 0.311443i $$0.100812\pi$$
−0.950265 + 0.311443i $$0.899188\pi$$
$$318$$ 0.763932i 0.0428392i
$$319$$ −26.3607 −1.47591
$$320$$ 0 0
$$321$$ 29.4164 1.64186
$$322$$ 0.618034i 0.0344417i
$$323$$ 11.2918i 0.628292i
$$324$$ 7.70820 0.428234
$$325$$ 0 0
$$326$$ 1.38197 0.0765400
$$327$$ 18.7082i 1.03457i
$$328$$ 3.32624i 0.183661i
$$329$$ −2.29180 −0.126351
$$330$$ 0 0
$$331$$ 19.2361 1.05731 0.528655 0.848837i $$-0.322697\pi$$
0.528655 + 0.848837i $$0.322697\pi$$
$$332$$ 6.94427i 0.381116i
$$333$$ − 2.47214i − 0.135472i
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ − 13.6738i − 0.744857i −0.928061 0.372429i $$-0.878525\pi$$
0.928061 0.372429i $$-0.121475\pi$$
$$338$$ 37.2705i 2.02725i
$$339$$ −1.70820 −0.0927769
$$340$$ 0 0
$$341$$ −25.9443 −1.40496
$$342$$ 0.708204i 0.0382953i
$$343$$ 8.41641i 0.454443i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −1.43769 −0.0772909
$$347$$ 6.38197i 0.342602i 0.985219 + 0.171301i $$0.0547970\pi$$
−0.985219 + 0.171301i $$0.945203\pi$$
$$348$$ − 14.9443i − 0.801097i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 38.7984 2.07090
$$352$$ 2.85410i 0.152124i
$$353$$ 24.0000i 1.27739i 0.769460 + 0.638696i $$0.220526\pi$$
−0.769460 + 0.638696i $$0.779474\pi$$
$$354$$ −2.76393 −0.146901
$$355$$ 0 0
$$356$$ −10.4721 −0.555022
$$357$$ 6.09017i 0.322326i
$$358$$ 2.18034i 0.115235i
$$359$$ −26.3607 −1.39126 −0.695632 0.718399i $$-0.744875\pi$$
−0.695632 + 0.718399i $$0.744875\pi$$
$$360$$ 0 0
$$361$$ −15.5623 −0.819069
$$362$$ 12.1459i 0.638374i
$$363$$ − 4.61803i − 0.242384i
$$364$$ −4.38197 −0.229677
$$365$$ 0 0
$$366$$ −15.0902 −0.788776
$$367$$ − 6.47214i − 0.337843i −0.985630 0.168921i $$-0.945972\pi$$
0.985630 0.168921i $$-0.0540284\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 1.27051 0.0661401
$$370$$ 0 0
$$371$$ −0.291796 −0.0151493
$$372$$ − 14.7082i − 0.762585i
$$373$$ 20.1803i 1.04490i 0.852670 + 0.522449i $$0.174982\pi$$
−0.852670 + 0.522449i $$0.825018\pi$$
$$374$$ 17.3820 0.898800
$$375$$ 0 0
$$376$$ −3.70820 −0.191236
$$377$$ − 65.4853i − 3.37266i
$$378$$ 3.38197i 0.173950i
$$379$$ 22.4508 1.15322 0.576611 0.817019i $$-0.304375\pi$$
0.576611 + 0.817019i $$0.304375\pi$$
$$380$$ 0 0
$$381$$ 26.1803 1.34126
$$382$$ 13.7082i 0.701373i
$$383$$ − 17.8885i − 0.914062i −0.889451 0.457031i $$-0.848913\pi$$
0.889451 0.457031i $$-0.151087\pi$$
$$384$$ −1.61803 −0.0825700
$$385$$ 0 0
$$386$$ 0.763932 0.0388831
$$387$$ 0 0
$$388$$ 12.3820i 0.628599i
$$389$$ −21.3262 −1.08128 −0.540642 0.841253i $$-0.681818\pi$$
−0.540642 + 0.841253i $$0.681818\pi$$
$$390$$ 0 0
$$391$$ 6.09017 0.307993
$$392$$ 6.61803i 0.334261i
$$393$$ 4.76393i 0.240309i
$$394$$ 22.5623 1.13667
$$395$$ 0 0
$$396$$ 1.09017 0.0547831
$$397$$ − 7.32624i − 0.367693i −0.982955 0.183847i $$-0.941145\pi$$
0.982955 0.183847i $$-0.0588550\pi$$
$$398$$ 2.00000i 0.100251i
$$399$$ 1.85410 0.0928212
$$400$$ 0 0
$$401$$ −1.70820 −0.0853036 −0.0426518 0.999090i $$-0.513581\pi$$
−0.0426518 + 0.999090i $$0.513581\pi$$
$$402$$ − 23.4164i − 1.16790i
$$403$$ − 64.4508i − 3.21053i
$$404$$ 0.291796 0.0145174
$$405$$ 0 0
$$406$$ 5.70820 0.283293
$$407$$ 18.4721i 0.915630i
$$408$$ 9.85410i 0.487851i
$$409$$ 30.2148 1.49402 0.747012 0.664810i $$-0.231488\pi$$
0.747012 + 0.664810i $$0.231488\pi$$
$$410$$ 0 0
$$411$$ −16.7082 −0.824155
$$412$$ − 16.5623i − 0.815966i
$$413$$ − 1.05573i − 0.0519490i
$$414$$ 0.381966 0.0187726
$$415$$ 0 0
$$416$$ −7.09017 −0.347624
$$417$$ − 20.6525i − 1.01136i
$$418$$ − 5.29180i − 0.258830i
$$419$$ −14.4721 −0.707010 −0.353505 0.935433i $$-0.615010\pi$$
−0.353505 + 0.935433i $$0.615010\pi$$
$$420$$ 0 0
$$421$$ 13.7426 0.669776 0.334888 0.942258i $$-0.391302\pi$$
0.334888 + 0.942258i $$0.391302\pi$$
$$422$$ − 14.0000i − 0.681509i
$$423$$ 1.41641i 0.0688681i
$$424$$ −0.472136 −0.0229289
$$425$$ 0 0
$$426$$ −6.61803 −0.320645
$$427$$ − 5.76393i − 0.278936i
$$428$$ 18.1803i 0.878780i
$$429$$ −32.7426 −1.58083
$$430$$ 0 0
$$431$$ 3.34752 0.161245 0.0806223 0.996745i $$-0.474309\pi$$
0.0806223 + 0.996745i $$0.474309\pi$$
$$432$$ 5.47214i 0.263278i
$$433$$ 8.50658i 0.408800i 0.978887 + 0.204400i $$0.0655243\pi$$
−0.978887 + 0.204400i $$0.934476\pi$$
$$434$$ 5.61803 0.269674
$$435$$ 0 0
$$436$$ −11.5623 −0.553734
$$437$$ − 1.85410i − 0.0886937i
$$438$$ 5.23607i 0.250189i
$$439$$ 13.3820 0.638686 0.319343 0.947639i $$-0.396538\pi$$
0.319343 + 0.947639i $$0.396538\pi$$
$$440$$ 0 0
$$441$$ 2.52786 0.120374
$$442$$ 43.1803i 2.05388i
$$443$$ 25.0902i 1.19207i 0.802958 + 0.596035i $$0.203258\pi$$
−0.802958 + 0.596035i $$0.796742\pi$$
$$444$$ −10.4721 −0.496986
$$445$$ 0 0
$$446$$ −20.9443 −0.991740
$$447$$ 12.7082i 0.601077i
$$448$$ − 0.618034i − 0.0291994i
$$449$$ 1.56231 0.0737298 0.0368649 0.999320i $$-0.488263\pi$$
0.0368649 + 0.999320i $$0.488263\pi$$
$$450$$ 0 0
$$451$$ −9.49342 −0.447028
$$452$$ − 1.05573i − 0.0496573i
$$453$$ − 4.14590i − 0.194791i
$$454$$ 18.7639 0.880635
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ 37.7771i 1.76714i 0.468301 + 0.883569i $$0.344866\pi$$
−0.468301 + 0.883569i $$0.655134\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 33.3262 1.55554
$$460$$ 0 0
$$461$$ −39.2361 −1.82741 −0.913703 0.406383i $$-0.866790\pi$$
−0.913703 + 0.406383i $$0.866790\pi$$
$$462$$ − 2.85410i − 0.132785i
$$463$$ 2.00000i 0.0929479i 0.998920 + 0.0464739i $$0.0147984\pi$$
−0.998920 + 0.0464739i $$0.985202\pi$$
$$464$$ 9.23607 0.428774
$$465$$ 0 0
$$466$$ 6.29180 0.291462
$$467$$ 17.1246i 0.792433i 0.918157 + 0.396216i $$0.129677\pi$$
−0.918157 + 0.396216i $$0.870323\pi$$
$$468$$ 2.70820i 0.125187i
$$469$$ 8.94427 0.413008
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ − 1.70820i − 0.0786265i
$$473$$ 0 0
$$474$$ −2.47214 −0.113549
$$475$$ 0 0
$$476$$ −3.76393 −0.172520
$$477$$ 0.180340i 0.00825720i
$$478$$ − 20.3607i − 0.931276i
$$479$$ 31.8885 1.45702 0.728512 0.685033i $$-0.240212\pi$$
0.728512 + 0.685033i $$0.240212\pi$$
$$480$$ 0 0
$$481$$ −45.8885 −2.09234
$$482$$ 0 0
$$483$$ − 1.00000i − 0.0455016i
$$484$$ 2.85410 0.129732
$$485$$ 0 0
$$486$$ 3.94427 0.178916
$$487$$ 19.8197i 0.898115i 0.893503 + 0.449057i $$0.148240\pi$$
−0.893503 + 0.449057i $$0.851760\pi$$
$$488$$ − 9.32624i − 0.422179i
$$489$$ −2.23607 −0.101118
$$490$$ 0 0
$$491$$ 6.18034 0.278915 0.139457 0.990228i $$-0.455464\pi$$
0.139457 + 0.990228i $$0.455464\pi$$
$$492$$ − 5.38197i − 0.242638i
$$493$$ − 56.2492i − 2.53334i
$$494$$ 13.1459 0.591462
$$495$$ 0 0
$$496$$ 9.09017 0.408161
$$497$$ − 2.52786i − 0.113390i
$$498$$ − 11.2361i − 0.503500i
$$499$$ −12.3607 −0.553340 −0.276670 0.960965i $$-0.589231\pi$$
−0.276670 + 0.960965i $$0.589231\pi$$
$$500$$ 0 0
$$501$$ −12.9443 −0.578307
$$502$$ − 6.14590i − 0.274305i
$$503$$ 36.3262i 1.61971i 0.586632 + 0.809853i $$0.300453\pi$$
−0.586632 + 0.809853i $$0.699547\pi$$
$$504$$ −0.236068 −0.0105153
$$505$$ 0 0
$$506$$ −2.85410 −0.126880
$$507$$ − 60.3050i − 2.67824i
$$508$$ 16.1803i 0.717886i
$$509$$ −36.6525 −1.62459 −0.812296 0.583245i $$-0.801783\pi$$
−0.812296 + 0.583245i $$0.801783\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 10.1459i − 0.447952i
$$514$$ 7.81966 0.344910
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 10.5836i − 0.465466i
$$518$$ − 4.00000i − 0.175750i
$$519$$ 2.32624 0.102111
$$520$$ 0 0
$$521$$ −15.5279 −0.680288 −0.340144 0.940373i $$-0.610476\pi$$
−0.340144 + 0.940373i $$0.610476\pi$$
$$522$$ − 3.52786i − 0.154410i
$$523$$ − 26.0000i − 1.13690i −0.822718 0.568450i $$-0.807543\pi$$
0.822718 0.568450i $$-0.192457\pi$$
$$524$$ −2.94427 −0.128621
$$525$$ 0 0
$$526$$ 20.7426 0.904422
$$527$$ − 55.3607i − 2.41155i
$$528$$ − 4.61803i − 0.200974i
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −0.652476 −0.0283150
$$532$$ 1.14590i 0.0496810i
$$533$$ − 23.5836i − 1.02152i
$$534$$ 16.9443 0.733250
$$535$$ 0 0
$$536$$ 14.4721 0.625101
$$537$$ − 3.52786i − 0.152239i
$$538$$ − 14.1803i − 0.611358i
$$539$$ −18.8885 −0.813587
$$540$$ 0 0
$$541$$ 22.8328 0.981659 0.490830 0.871256i $$-0.336694\pi$$
0.490830 + 0.871256i $$0.336694\pi$$
$$542$$ 30.3262i 1.30262i
$$543$$ − 19.6525i − 0.843368i
$$544$$ −6.09017 −0.261114
$$545$$ 0 0
$$546$$ 7.09017 0.303431
$$547$$ − 27.9230i − 1.19390i −0.802278 0.596950i $$-0.796379\pi$$
0.802278 0.596950i $$-0.203621\pi$$
$$548$$ − 10.3262i − 0.441115i
$$549$$ −3.56231 −0.152036
$$550$$ 0 0
$$551$$ −17.1246 −0.729533
$$552$$ − 1.61803i − 0.0688681i
$$553$$ − 0.944272i − 0.0401545i
$$554$$ −29.4164 −1.24978
$$555$$ 0 0
$$556$$ 12.7639 0.541311
$$557$$ 22.8328i 0.967457i 0.875218 + 0.483729i $$0.160718\pi$$
−0.875218 + 0.483729i $$0.839282\pi$$
$$558$$ − 3.47214i − 0.146987i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −28.1246 −1.18742
$$562$$ − 22.7639i − 0.960239i
$$563$$ − 13.8885i − 0.585332i −0.956215 0.292666i $$-0.905458\pi$$
0.956215 0.292666i $$-0.0945425\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ −26.9443 −1.13255
$$567$$ − 4.76393i − 0.200066i
$$568$$ − 4.09017i − 0.171620i
$$569$$ 2.00000 0.0838444 0.0419222 0.999121i $$-0.486652\pi$$
0.0419222 + 0.999121i $$0.486652\pi$$
$$570$$ 0 0
$$571$$ 15.9787 0.668688 0.334344 0.942451i $$-0.391485\pi$$
0.334344 + 0.942451i $$0.391485\pi$$
$$572$$ − 20.2361i − 0.846113i
$$573$$ − 22.1803i − 0.926597i
$$574$$ 2.05573 0.0858044
$$575$$ 0 0
$$576$$ −0.381966 −0.0159153
$$577$$ 3.52786i 0.146867i 0.997300 + 0.0734335i $$0.0233957\pi$$
−0.997300 + 0.0734335i $$0.976604\pi$$
$$578$$ 20.0902i 0.835641i
$$579$$ −1.23607 −0.0513692
$$580$$ 0 0
$$581$$ 4.29180 0.178054
$$582$$ − 20.0344i − 0.830454i
$$583$$ − 1.34752i − 0.0558087i
$$584$$ −3.23607 −0.133909
$$585$$ 0 0
$$586$$ −19.8885 −0.821588
$$587$$ − 13.6180i − 0.562076i −0.959697 0.281038i $$-0.909321\pi$$
0.959697 0.281038i $$-0.0906788\pi$$
$$588$$ − 10.7082i − 0.441599i
$$589$$ −16.8541 −0.694461
$$590$$ 0 0
$$591$$ −36.5066 −1.50168
$$592$$ − 6.47214i − 0.266003i
$$593$$ 39.2361i 1.61123i 0.592438 + 0.805616i $$0.298166\pi$$
−0.592438 + 0.805616i $$0.701834\pi$$
$$594$$ −15.6180 −0.640816
$$595$$ 0 0
$$596$$ −7.85410 −0.321717
$$597$$ − 3.23607i − 0.132443i
$$598$$ − 7.09017i − 0.289939i
$$599$$ 18.3820 0.751067 0.375533 0.926809i $$-0.377460\pi$$
0.375533 + 0.926809i $$0.377460\pi$$
$$600$$ 0 0
$$601$$ −33.2705 −1.35713 −0.678566 0.734539i $$-0.737398\pi$$
−0.678566 + 0.734539i $$0.737398\pi$$
$$602$$ 0 0
$$603$$ − 5.52786i − 0.225112i
$$604$$ 2.56231 0.104259
$$605$$ 0 0
$$606$$ −0.472136 −0.0191792
$$607$$ − 26.4721i − 1.07447i −0.843432 0.537235i $$-0.819469\pi$$
0.843432 0.537235i $$-0.180531\pi$$
$$608$$ 1.85410i 0.0751938i
$$609$$ −9.23607 −0.374264
$$610$$ 0 0
$$611$$ 26.2918 1.06365
$$612$$ 2.32624i 0.0940326i
$$613$$ 19.3050i 0.779720i 0.920874 + 0.389860i $$0.127477\pi$$
−0.920874 + 0.389860i $$0.872523\pi$$
$$614$$ 28.4508 1.14818
$$615$$ 0 0
$$616$$ 1.76393 0.0710708
$$617$$ − 34.0902i − 1.37242i −0.727404 0.686209i $$-0.759273\pi$$
0.727404 0.686209i $$-0.240727\pi$$
$$618$$ 26.7984i 1.07799i
$$619$$ 2.79837 0.112476 0.0562381 0.998417i $$-0.482089\pi$$
0.0562381 + 0.998417i $$0.482089\pi$$
$$620$$ 0 0
$$621$$ −5.47214 −0.219589
$$622$$ − 4.00000i − 0.160385i
$$623$$ 6.47214i 0.259301i
$$624$$ 11.4721 0.459253
$$625$$ 0 0
$$626$$ 12.7984 0.511526
$$627$$ 8.56231i 0.341946i
$$628$$ − 3.70820i − 0.147973i
$$629$$ −39.4164 −1.57164
$$630$$ 0 0
$$631$$ 42.0689 1.67474 0.837368 0.546640i $$-0.184093\pi$$
0.837368 + 0.546640i $$0.184093\pi$$
$$632$$ − 1.52786i − 0.0607752i
$$633$$ 22.6525i 0.900355i
$$634$$ 11.0902 0.440447
$$635$$ 0 0
$$636$$ 0.763932 0.0302919
$$637$$ − 46.9230i − 1.85916i
$$638$$ 26.3607i 1.04363i
$$639$$ −1.56231 −0.0618039
$$640$$ 0 0
$$641$$ 0.360680 0.0142460 0.00712300 0.999975i $$-0.497733\pi$$
0.00712300 + 0.999975i $$0.497733\pi$$
$$642$$ − 29.4164i − 1.16097i
$$643$$ − 8.29180i − 0.326997i −0.986544 0.163498i $$-0.947722\pi$$
0.986544 0.163498i $$-0.0522778\pi$$
$$644$$ 0.618034 0.0243540
$$645$$ 0 0
$$646$$ 11.2918 0.444270
$$647$$ 36.2492i 1.42510i 0.701619 + 0.712552i $$0.252461\pi$$
−0.701619 + 0.712552i $$0.747539\pi$$
$$648$$ − 7.70820i − 0.302807i
$$649$$ 4.87539 0.191376
$$650$$ 0 0
$$651$$ −9.09017 −0.356272
$$652$$ − 1.38197i − 0.0541220i
$$653$$ − 8.03444i − 0.314412i −0.987566 0.157206i $$-0.949751\pi$$
0.987566 0.157206i $$-0.0502487\pi$$
$$654$$ 18.7082 0.731549
$$655$$ 0 0
$$656$$ 3.32624 0.129868
$$657$$ 1.23607i 0.0482236i
$$658$$ 2.29180i 0.0893435i
$$659$$ 46.2492 1.80161 0.900807 0.434220i $$-0.142976\pi$$
0.900807 + 0.434220i $$0.142976\pi$$
$$660$$ 0 0
$$661$$ 18.6738 0.726325 0.363163 0.931726i $$-0.381697\pi$$
0.363163 + 0.931726i $$0.381697\pi$$
$$662$$ − 19.2361i − 0.747631i
$$663$$ − 69.8673i − 2.71342i
$$664$$ 6.94427 0.269490
$$665$$ 0 0
$$666$$ −2.47214 −0.0957933
$$667$$ 9.23607i 0.357622i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 33.8885 1.31021
$$670$$ 0 0
$$671$$ 26.6180 1.02758
$$672$$ 1.00000i 0.0385758i
$$673$$ − 10.9443i − 0.421871i −0.977500 0.210935i $$-0.932349\pi$$
0.977500 0.210935i $$-0.0676510\pi$$
$$674$$ −13.6738 −0.526694
$$675$$ 0 0
$$676$$ 37.2705 1.43348
$$677$$ 50.9443i 1.95795i 0.203986 + 0.978974i $$0.434610\pi$$
−0.203986 + 0.978974i $$0.565390\pi$$
$$678$$ 1.70820i 0.0656032i
$$679$$ 7.65248 0.293675
$$680$$ 0 0
$$681$$ −30.3607 −1.16342
$$682$$ 25.9443i 0.993458i
$$683$$ − 31.5623i − 1.20770i −0.797099 0.603849i $$-0.793633\pi$$
0.797099 0.603849i $$-0.206367\pi$$
$$684$$ 0.708204 0.0270789
$$685$$ 0 0
$$686$$ 8.41641 0.321340
$$687$$ − 16.1803i − 0.617318i
$$688$$ 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 $$0.687701\pi$$
$$692$$ 1.43769i 0.0546529i
$$693$$ − 0.673762i − 0.0255941i
$$694$$ 6.38197 0.242256
$$695$$ 0 0
$$696$$ −14.9443 −0.566461
$$697$$ − 20.2574i − 0.767302i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ −10.1803 −0.385056
$$700$$ 0 0
$$701$$ 43.3394 1.63691 0.818453 0.574573i $$-0.194832\pi$$
0.818453 + 0.574573i $$0.194832\pi$$
$$702$$ − 38.7984i − 1.46435i
$$703$$ 12.0000i 0.452589i
$$704$$ 2.85410 0.107568
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ − 0.180340i − 0.00678238i
$$708$$ 2.76393i 0.103875i
$$709$$ 26.0902 0.979837 0.489918 0.871768i $$-0.337027\pi$$
0.489918 + 0.871768i $$0.337027\pi$$
$$710$$ 0 0
$$711$$ −0.583592 −0.0218864
$$712$$ 10.4721i 0.392460i
$$713$$ 9.09017i 0.340430i
$$714$$ 6.09017 0.227919
$$715$$ 0 0
$$716$$ 2.18034 0.0814831
$$717$$ 32.9443i 1.23033i
$$718$$ 26.3607i 0.983772i
$$719$$ −35.2705 −1.31537 −0.657684 0.753294i $$-0.728464\pi$$
−0.657684 + 0.753294i $$0.728464\pi$$
$$720$$ 0 0
$$721$$ −10.2361 −0.381211
$$722$$ 15.5623i 0.579169i
$$723$$ 0 0
$$724$$ 12.1459 0.451399
$$725$$ 0 0
$$726$$ −4.61803 −0.171391
$$727$$ − 28.2016i − 1.04594i −0.852351 0.522970i $$-0.824824\pi$$
0.852351 0.522970i $$-0.175176\pi$$
$$728$$ 4.38197i 0.162406i
$$729$$ −29.5066 −1.09284
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 15.0902i 0.557749i
$$733$$ − 29.4164i − 1.08652i −0.839565 0.543260i $$-0.817190\pi$$
0.839565 0.543260i $$-0.182810\pi$$
$$734$$ −6.47214 −0.238891
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 41.3050i 1.52149i
$$738$$ − 1.27051i − 0.0467681i
$$739$$ 13.8885 0.510898 0.255449 0.966822i $$-0.417777\pi$$
0.255449 + 0.966822i $$0.417777\pi$$
$$740$$ 0 0
$$741$$ −21.2705 −0.781392
$$742$$ 0.291796i 0.0107122i
$$743$$ 33.6312i 1.23381i 0.787038 + 0.616904i $$0.211613\pi$$
−0.787038 + 0.616904i $$0.788387\pi$$
$$744$$ −14.7082 −0.539229
$$745$$ 0 0
$$746$$ 20.1803 0.738855
$$747$$ − 2.65248i − 0.0970490i
$$748$$ − 17.3820i − 0.635548i
$$749$$ 11.2361 0.410557
$$750$$ 0 0
$$751$$ −47.0132 −1.71553 −0.857767 0.514038i $$-0.828149\pi$$
−0.857767 + 0.514038i $$0.828149\pi$$
$$752$$ 3.70820i 0.135224i
$$753$$ 9.94427i 0.362389i
$$754$$ −65.4853 −2.38483
$$755$$ 0 0
$$756$$ 3.38197 0.123001
$$757$$ 17.8885i 0.650170i 0.945685 + 0.325085i $$0.105393\pi$$
−0.945685 + 0.325085i $$0.894607\pi$$
$$758$$ − 22.4508i − 0.815452i
$$759$$ 4.61803 0.167624
$$760$$ 0 0
$$761$$ 46.8673 1.69894 0.849468 0.527640i $$-0.176923\pi$$
0.849468 + 0.527640i $$0.176923\pi$$
$$762$$ − 26.1803i − 0.948414i
$$763$$ 7.14590i 0.258699i
$$764$$ 13.7082 0.495945
$$765$$ 0 0
$$766$$ −17.8885 −0.646339
$$767$$ 12.1115i 0.437319i
$$768$$ 1.61803i 0.0583858i
$$769$$ 6.58359 0.237410 0.118705 0.992930i $$-0.462126\pi$$
0.118705 + 0.992930i $$0.462126\pi$$
$$770$$ 0 0
$$771$$ −12.6525 −0.455668
$$772$$ − 0.763932i − 0.0274945i
$$773$$ 28.9443i 1.04105i 0.853845 + 0.520527i $$0.174264\pi$$
−0.853845 + 0.520527i $$0.825736\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12.3820 0.444487
$$777$$ 6.47214i 0.232187i
$$778$$ 21.3262i 0.764583i
$$779$$ −6.16718 −0.220962
$$780$$ 0 0
$$781$$ 11.6738 0.417720
$$782$$ − 6.09017i − 0.217784i
$$783$$ 50.5410i 1.80619i
$$784$$ 6.61803 0.236358
$$785$$ 0 0
$$786$$ 4.76393 0.169924
$$787$$ 2.87539i 0.102497i 0.998686 + 0.0512483i $$0.0163200\pi$$
−0.998686 + 0.0512483i $$0.983680\pi$$
$$788$$ − 22.5623i − 0.803749i
$$789$$ −33.5623 −1.19485
$$790$$ 0 0
$$791$$ −0.652476 −0.0231994
$$792$$ − 1.09017i − 0.0387375i
$$793$$ 66.1246i 2.34815i
$$794$$ −7.32624 −0.259998
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ − 13.7082i − 0.485569i −0.970080 0.242785i $$-0.921939\pi$$
0.970080 0.242785i $$-0.0780609\pi$$
$$798$$ − 1.85410i − 0.0656345i
$$799$$ 22.5836 0.798950
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 1.70820i 0.0603188i
$$803$$ − 9.23607i − 0.325934i
$$804$$ −23.4164 −0.825833
$$805$$ 0 0
$$806$$ −64.4508 −2.27018
$$807$$ 22.9443i 0.807677i
$$808$$ − 0.291796i − 0.0102653i
$$809$$ 46.7426 1.64338 0.821692 0.569932i $$-0.193030\pi$$
0.821692 + 0.569932i $$0.193030\pi$$
$$810$$ 0 0
$$811$$ 21.8197 0.766192 0.383096 0.923709i $$-0.374858\pi$$
0.383096 + 0.923709i $$0.374858\pi$$
$$812$$ − 5.70820i − 0.200319i
$$813$$ − 49.0689i − 1.72092i
$$814$$ 18.4721 0.647448
$$815$$ 0 0
$$816$$ 9.85410 0.344963
$$817$$ 0 0
$$818$$ − 30.2148i − 1.05644i
$$819$$ 1.67376 0.0584860
$$820$$ 0 0
$$821$$ −33.0557 −1.15365 −0.576826 0.816867i $$-0.695709\pi$$
−0.576826 + 0.816867i $$0.695709\pi$$
$$822$$ 16.7082i 0.582766i
$$823$$ 25.4164i 0.885960i 0.896531 + 0.442980i $$0.146079\pi$$
−0.896531 + 0.442980i $$0.853921\pi$$
$$824$$ −16.5623 −0.576975
$$825$$ 0 0
$$826$$ −1.05573 −0.0367335
$$827$$ − 21.7082i − 0.754868i −0.926036 0.377434i $$-0.876806\pi$$
0.926036 0.377434i $$-0.123194\pi$$
$$828$$ − 0.381966i − 0.0132742i
$$829$$ −18.9443 −0.657962 −0.328981 0.944337i $$-0.606705\pi$$
−0.328981 + 0.944337i $$0.606705\pi$$
$$830$$ 0 0
$$831$$ 47.5967 1.65111
$$832$$ 7.09017i 0.245807i
$$833$$ − 40.3050i − 1.39648i
$$834$$ −20.6525 −0.715137
$$835$$ 0 0
$$836$$ −5.29180 −0.183021
$$837$$ 49.7426i 1.71936i
$$838$$ 14.4721i 0.499932i
$$839$$ −33.0132 −1.13974 −0.569870 0.821735i $$-0.693007\pi$$
−0.569870 + 0.821735i $$0.693007\pi$$
$$840$$ 0 0
$$841$$ 56.3050 1.94155
$$842$$ − 13.7426i − 0.473603i
$$843$$ 36.8328i 1.26859i
$$844$$ −14.0000 −0.481900
$$845$$ 0 0
$$846$$ 1.41641 0.0486971
$$847$$ − 1.76393i − 0.0606094i
$$848$$ 0.472136i 0.0162132i
$$849$$ 43.5967 1.49624
$$850$$ 0 0
$$851$$ 6.47214 0.221862
$$852$$ 6.61803i 0.226730i
$$853$$ − 10.7984i − 0.369729i −0.982764 0.184865i $$-0.940815\pi$$
0.982764 0.184865i $$-0.0591847\pi$$
$$854$$ −5.76393 −0.197238
$$855$$ 0 0
$$856$$ 18.1803 0.621391
$$857$$ 6.58359i 0.224891i 0.993658 + 0.112446i $$0.0358684\pi$$
−0.993658 + 0.112446i $$0.964132\pi$$
$$858$$ 32.7426i 1.11782i
$$859$$ 24.0689 0.821220 0.410610 0.911811i $$-0.365316\pi$$
0.410610 + 0.911811i $$0.365316\pi$$
$$860$$ 0 0
$$861$$ −3.32624 −0.113358
$$862$$ − 3.34752i − 0.114017i
$$863$$ 32.7639i 1.11530i 0.830077 + 0.557649i $$0.188296\pi$$
−0.830077 + 0.557649i $$0.811704\pi$$
$$864$$ 5.47214 0.186166
$$865$$ 0 0
$$866$$ 8.50658 0.289065
$$867$$ − 32.5066i − 1.10398i
$$868$$ − 5.61803i − 0.190688i
$$869$$ 4.36068 0.147926
$$870$$ 0 0
$$871$$ −102.610 −3.47680
$$872$$ 11.5623i 0.391549i
$$873$$ − 4.72949i − 0.160069i
$$874$$ −1.85410 −0.0627159
$$875$$ 0 0
$$876$$ 5.23607 0.176910
$$877$$ − 18.7426i − 0.632894i −0.948610 0.316447i $$-0.897510\pi$$
0.948610 0.316447i $$-0.102490\pi$$
$$878$$ − 13.3820i − 0.451619i
$$879$$ 32.1803 1.08542
$$880$$ 0 0
$$881$$ 8.58359 0.289189 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$882$$ − 2.52786i − 0.0851176i
$$883$$ 15.5623i 0.523713i 0.965107 + 0.261857i $$0.0843348\pi$$
−0.965107 + 0.261857i $$0.915665\pi$$
$$884$$ 43.1803 1.45231
$$885$$ 0 0
$$886$$ 25.0902 0.842921
$$887$$ 5.16718i 0.173497i 0.996230 + 0.0867485i $$0.0276477\pi$$
−0.996230 + 0.0867485i $$0.972352\pi$$
$$888$$ 10.4721i 0.351422i
$$889$$ 10.0000 0.335389
$$890$$ 0 0
$$891$$ 22.0000 0.737028
$$892$$ 20.9443i 0.701266i
$$893$$ − 6.87539i − 0.230076i
$$894$$ 12.7082 0.425026
$$895$$ 0 0
$$896$$ −0.618034 −0.0206471
$$897$$ 11.4721i 0.383043i
$$898$$ − 1.56231i − 0.0521348i
$$899$$ 83.9574 2.80014
$$900$$ 0 0
$$901$$ 2.87539 0.0957931
$$902$$ 9.49342i 0.316096i
$$903$$ 0 0
$$904$$ −1.05573 −0.0351130
$$905$$ 0 0
$$906$$ −4.14590 −0.137738
$$907$$ − 7.12461i − 0.236569i −0.992980 0.118284i $$-0.962261\pi$$
0.992980 0.118284i $$-0.0377395\pi$$
$$908$$ − 18.7639i − 0.622703i
$$909$$ −0.111456 −0.00369677
$$910$$ 0 0
$$911$$ 36.0689 1.19502 0.597508 0.801863i $$-0.296158\pi$$
0.597508 + 0.801863i $$0.296158\pi$$
$$912$$ − 3.00000i − 0.0993399i
$$913$$ 19.8197i 0.655935i
$$914$$ 37.7771 1.24955
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 1.81966i 0.0600905i
$$918$$ − 33.3262i − 1.09993i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −46.0344 −1.51689
$$922$$ 39.2361i 1.29217i
$$923$$ 29.0000i 0.954547i
$$924$$ −2.85410 −0.0938931
$$925$$ 0 0
$$926$$ 2.00000 0.0657241
$$927$$ 6.32624i 0.207781i
$$928$$ − 9.23607i − 0.303189i
$$929$$ 3.52786 0.115745 0.0578727 0.998324i $$-0.481568\pi$$
0.0578727 + 0.998324i $$0.481568\pi$$
$$930$$ 0 0
$$931$$ −12.2705 −0.402150
$$932$$ − 6.29180i − 0.206095i
$$933$$ 6.47214i 0.211888i
$$934$$ 17.1246 0.560334
$$935$$ 0 0
$$936$$ 2.70820 0.0885204
$$937$$ 36.7984i 1.20215i 0.799192 + 0.601075i $$0.205261\pi$$
−0.799192 + 0.601075i $$0.794739\pi$$
$$938$$ − 8.94427i − 0.292041i
$$939$$ −20.7082 −0.675787
$$940$$ 0 0
$$941$$ −22.4934 −0.733265 −0.366632 0.930366i $$-0.619489\pi$$
−0.366632 + 0.930366i $$0.619489\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ 3.32624i 0.108317i
$$944$$ −1.70820 −0.0555973
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 54.6869i 1.77709i 0.458793 + 0.888543i $$0.348282\pi$$
−0.458793 + 0.888543i $$0.651718\pi$$
$$948$$ 2.47214i 0.0802912i
$$949$$ 22.9443 0.744803
$$950$$ 0 0
$$951$$ −17.9443 −0.581883
$$952$$ 3.76393i 0.121990i
$$953$$ 3.79837i 0.123041i 0.998106 + 0.0615207i $$0.0195950\pi$$
−0.998106 + 0.0615207i $$0.980405\pi$$
$$954$$ 0.180340 0.00583872
$$955$$ 0 0
$$956$$ −20.3607 −0.658511
$$957$$ − 42.6525i − 1.37876i
$$958$$ − 31.8885i − 1.03027i
$$959$$ −6.38197 −0.206084
$$960$$ 0 0
$$961$$ 51.6312 1.66552
$$962$$ 45.8885i 1.47951i
$$963$$ − 6.94427i − 0.223776i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ −1.00000 −0.0321745
$$967$$ 16.5410i 0.531923i 0.963984 + 0.265962i $$0.0856895\pi$$
−0.963984 + 0.265962i $$0.914311\pi$$
$$968$$ − 2.85410i − 0.0917343i
$$969$$ −18.2705 −0.586933
$$970$$ 0 0
$$971$$ 34.2705 1.09979 0.549896 0.835233i $$-0.314667\pi$$
0.549896 + 0.835233i $$0.314667\pi$$
$$972$$ − 3.94427i − 0.126513i
$$973$$ − 7.88854i − 0.252895i
$$974$$ 19.8197 0.635063
$$975$$ 0 0
$$976$$ −9.32624 −0.298526
$$977$$ 23.5623i 0.753825i 0.926249 + 0.376912i $$0.123014\pi$$
−0.926249 + 0.376912i $$0.876986\pi$$
$$978$$ 2.23607i 0.0715016i
$$979$$ −29.8885 −0.955242
$$980$$ 0 0
$$981$$ 4.41641 0.141005
$$982$$ − 6.18034i − 0.197223i
$$983$$ − 14.2705i − 0.455159i −0.973760 0.227579i $$-0.926919\pi$$
0.973760 0.227579i $$-0.0730811\pi$$
$$984$$ −5.38197 −0.171571
$$985$$ 0 0
$$986$$ −56.2492 −1.79134
$$987$$ − 3.70820i − 0.118033i
$$988$$ − 13.1459i − 0.418227i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 27.5066 0.873775 0.436888 0.899516i $$-0.356081\pi$$
0.436888 + 0.899516i $$0.356081\pi$$
$$992$$ − 9.09017i − 0.288613i
$$993$$ 31.1246i 0.987710i
$$994$$ −2.52786 −0.0801790
$$995$$ 0 0
$$996$$ −11.2361 −0.356028
$$997$$ − 57.1935i − 1.81134i −0.423987 0.905668i $$-0.639370\pi$$
0.423987 0.905668i $$-0.360630\pi$$
$$998$$ 12.3607i 0.391270i
$$999$$ 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.b.i.599.2 4
5.2 odd 4 230.2.a.c.1.2 2
5.3 odd 4 1150.2.a.j.1.1 2
5.4 even 2 inner 1150.2.b.i.599.3 4
15.2 even 4 2070.2.a.u.1.1 2
20.3 even 4 9200.2.a.bu.1.2 2
20.7 even 4 1840.2.a.l.1.1 2
40.27 even 4 7360.2.a.bn.1.2 2
40.37 odd 4 7360.2.a.bh.1.1 2
115.22 even 4 5290.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 5.2 odd 4
1150.2.a.j.1.1 2 5.3 odd 4
1150.2.b.i.599.2 4 1.1 even 1 trivial
1150.2.b.i.599.3 4 5.4 even 2 inner
1840.2.a.l.1.1 2 20.7 even 4
2070.2.a.u.1.1 2 15.2 even 4
5290.2.a.o.1.2 2 115.22 even 4
7360.2.a.bh.1.1 2 40.37 odd 4
7360.2.a.bn.1.2 2 40.27 even 4
9200.2.a.bu.1.2 2 20.3 even 4