# Properties

 Label 950.2.a.k.1.1 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.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: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 3$$ x^3 - x^2 - 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.713538$$ of defining polynomial Character $$\chi$$ $$=$$ 950.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.77733 q^{3} +1.00000 q^{4} +2.77733 q^{6} +4.69527 q^{7} -1.00000 q^{8} +4.71354 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.77733 q^{3} +1.00000 q^{4} +2.77733 q^{6} +4.69527 q^{7} -1.00000 q^{8} +4.71354 q^{9} +6.40880 q^{11} -2.77733 q^{12} -1.06379 q^{13} -4.69527 q^{14} +1.00000 q^{16} -1.91794 q^{17} -4.71354 q^{18} -1.00000 q^{19} -13.0403 q^{21} -6.40880 q^{22} -1.79560 q^{23} +2.77733 q^{24} +1.06379 q^{26} -4.75905 q^{27} +4.69527 q^{28} +2.93621 q^{29} -5.55465 q^{31} -1.00000 q^{32} -17.7993 q^{33} +1.91794 q^{34} +4.71354 q^{36} +11.4088 q^{37} +1.00000 q^{38} +2.95449 q^{39} -1.14585 q^{41} +13.0403 q^{42} +3.55465 q^{43} +6.40880 q^{44} +1.79560 q^{46} -10.8359 q^{47} -2.77733 q^{48} +15.0455 q^{49} +5.32674 q^{51} -1.06379 q^{52} +8.69527 q^{53} +4.75905 q^{54} -4.69527 q^{56} +2.77733 q^{57} -2.93621 q^{58} -5.63148 q^{59} -3.39053 q^{61} +5.55465 q^{62} +22.1313 q^{63} +1.00000 q^{64} +17.7993 q^{66} +8.82284 q^{67} -1.91794 q^{68} +4.98696 q^{69} -1.42708 q^{71} -4.71354 q^{72} +12.6132 q^{73} -11.4088 q^{74} -1.00000 q^{76} +30.0910 q^{77} -2.95449 q^{78} -1.96345 q^{79} -0.923174 q^{81} +1.14585 q^{82} -16.2447 q^{83} -13.0403 q^{84} -3.55465 q^{86} -8.15482 q^{87} -6.40880 q^{88} -10.0000 q^{89} -4.99477 q^{91} -1.79560 q^{92} +15.4271 q^{93} +10.8359 q^{94} +2.77733 q^{96} +14.9452 q^{97} -15.0455 q^{98} +30.2081 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} + 13 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 + 13 * q^9 $$3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} + 13 q^{9} + 2 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 4 q^{17} - 13 q^{18} - 3 q^{19} - 11 q^{21} - 2 q^{22} - 14 q^{23} + 2 q^{24} - 2 q^{26} + 7 q^{27} - 2 q^{28} + 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{33} - 4 q^{34} + 13 q^{36} + 17 q^{37} + 3 q^{38} + 29 q^{39} - 8 q^{41} + 11 q^{42} - 2 q^{43} + 2 q^{44} + 14 q^{46} - 13 q^{47} - 2 q^{48} + 25 q^{49} - 11 q^{51} + 2 q^{52} + 10 q^{53} - 7 q^{54} + 2 q^{56} + 2 q^{57} - 14 q^{58} - 6 q^{59} + 22 q^{61} + 4 q^{62} - 2 q^{63} + 3 q^{64} + 4 q^{66} + 4 q^{68} + 8 q^{69} - 2 q^{71} - 13 q^{72} + 12 q^{73} - 17 q^{74} - 3 q^{76} + 50 q^{77} - 29 q^{78} + 24 q^{79} - q^{81} + 8 q^{82} - 12 q^{83} - 11 q^{84} + 2 q^{86} + 21 q^{87} - 2 q^{88} - 30 q^{89} - 7 q^{91} - 14 q^{92} + 44 q^{93} + 13 q^{94} + 2 q^{96} - 25 q^{98} + 24 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 + 13 * q^9 + 2 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^14 + 3 * q^16 + 4 * q^17 - 13 * q^18 - 3 * q^19 - 11 * q^21 - 2 * q^22 - 14 * q^23 + 2 * q^24 - 2 * q^26 + 7 * q^27 - 2 * q^28 + 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^33 - 4 * q^34 + 13 * q^36 + 17 * q^37 + 3 * q^38 + 29 * q^39 - 8 * q^41 + 11 * q^42 - 2 * q^43 + 2 * q^44 + 14 * q^46 - 13 * q^47 - 2 * q^48 + 25 * q^49 - 11 * q^51 + 2 * q^52 + 10 * q^53 - 7 * q^54 + 2 * q^56 + 2 * q^57 - 14 * q^58 - 6 * q^59 + 22 * q^61 + 4 * q^62 - 2 * q^63 + 3 * q^64 + 4 * q^66 + 4 * q^68 + 8 * q^69 - 2 * q^71 - 13 * q^72 + 12 * q^73 - 17 * q^74 - 3 * q^76 + 50 * q^77 - 29 * q^78 + 24 * q^79 - q^81 + 8 * q^82 - 12 * q^83 - 11 * q^84 + 2 * q^86 + 21 * q^87 - 2 * q^88 - 30 * q^89 - 7 * q^91 - 14 * q^92 + 44 * q^93 + 13 * q^94 + 2 * q^96 - 25 * q^98 + 24 * 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
$$3$$ −2.77733 −1.60349 −0.801745 0.597666i $$-0.796095\pi$$
−0.801745 + 0.597666i $$0.796095\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.77733 1.13384
$$7$$ 4.69527 1.77464 0.887322 0.461151i $$-0.152563\pi$$
0.887322 + 0.461151i $$0.152563\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 4.71354 1.57118
$$10$$ 0 0
$$11$$ 6.40880 1.93233 0.966163 0.257931i $$-0.0830406\pi$$
0.966163 + 0.257931i $$0.0830406\pi$$
$$12$$ −2.77733 −0.801745
$$13$$ −1.06379 −0.295042 −0.147521 0.989059i $$-0.547129\pi$$
−0.147521 + 0.989059i $$0.547129\pi$$
$$14$$ −4.69527 −1.25486
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.91794 −0.465169 −0.232584 0.972576i $$-0.574718\pi$$
−0.232584 + 0.972576i $$0.574718\pi$$
$$18$$ −4.71354 −1.11099
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −13.0403 −2.84562
$$22$$ −6.40880 −1.36636
$$23$$ −1.79560 −0.374408 −0.187204 0.982321i $$-0.559943\pi$$
−0.187204 + 0.982321i $$0.559943\pi$$
$$24$$ 2.77733 0.566919
$$25$$ 0 0
$$26$$ 1.06379 0.208626
$$27$$ −4.75905 −0.915880
$$28$$ 4.69527 0.887322
$$29$$ 2.93621 0.545241 0.272620 0.962122i $$-0.412110\pi$$
0.272620 + 0.962122i $$0.412110\pi$$
$$30$$ 0 0
$$31$$ −5.55465 −0.997645 −0.498822 0.866704i $$-0.666234\pi$$
−0.498822 + 0.866704i $$0.666234\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −17.7993 −3.09847
$$34$$ 1.91794 0.328924
$$35$$ 0 0
$$36$$ 4.71354 0.785590
$$37$$ 11.4088 1.87560 0.937798 0.347182i $$-0.112861\pi$$
0.937798 + 0.347182i $$0.112861\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 2.95449 0.473096
$$40$$ 0 0
$$41$$ −1.14585 −0.178951 −0.0894757 0.995989i $$-0.528519\pi$$
−0.0894757 + 0.995989i $$0.528519\pi$$
$$42$$ 13.0403 2.01216
$$43$$ 3.55465 0.542079 0.271040 0.962568i $$-0.412633\pi$$
0.271040 + 0.962568i $$0.412633\pi$$
$$44$$ 6.40880 0.966163
$$45$$ 0 0
$$46$$ 1.79560 0.264747
$$47$$ −10.8359 −1.58058 −0.790288 0.612736i $$-0.790069\pi$$
−0.790288 + 0.612736i $$0.790069\pi$$
$$48$$ −2.77733 −0.400872
$$49$$ 15.0455 2.14936
$$50$$ 0 0
$$51$$ 5.32674 0.745893
$$52$$ −1.06379 −0.147521
$$53$$ 8.69527 1.19439 0.597193 0.802097i $$-0.296283\pi$$
0.597193 + 0.802097i $$0.296283\pi$$
$$54$$ 4.75905 0.647625
$$55$$ 0 0
$$56$$ −4.69527 −0.627431
$$57$$ 2.77733 0.367866
$$58$$ −2.93621 −0.385544
$$59$$ −5.63148 −0.733156 −0.366578 0.930387i $$-0.619471\pi$$
−0.366578 + 0.930387i $$0.619471\pi$$
$$60$$ 0 0
$$61$$ −3.39053 −0.434113 −0.217056 0.976159i $$-0.569646\pi$$
−0.217056 + 0.976159i $$0.569646\pi$$
$$62$$ 5.55465 0.705441
$$63$$ 22.1313 2.78828
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 17.7993 2.19095
$$67$$ 8.82284 1.07788 0.538941 0.842344i $$-0.318825\pi$$
0.538941 + 0.842344i $$0.318825\pi$$
$$68$$ −1.91794 −0.232584
$$69$$ 4.98696 0.600360
$$70$$ 0 0
$$71$$ −1.42708 −0.169363 −0.0846814 0.996408i $$-0.526987\pi$$
−0.0846814 + 0.996408i $$0.526987\pi$$
$$72$$ −4.71354 −0.555496
$$73$$ 12.6132 1.47626 0.738132 0.674656i $$-0.235708\pi$$
0.738132 + 0.674656i $$0.235708\pi$$
$$74$$ −11.4088 −1.32625
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 30.0910 3.42919
$$78$$ −2.95449 −0.334530
$$79$$ −1.96345 −0.220906 −0.110453 0.993881i $$-0.535230\pi$$
−0.110453 + 0.993881i $$0.535230\pi$$
$$80$$ 0 0
$$81$$ −0.923174 −0.102575
$$82$$ 1.14585 0.126538
$$83$$ −16.2447 −1.78309 −0.891543 0.452937i $$-0.850376\pi$$
−0.891543 + 0.452937i $$0.850376\pi$$
$$84$$ −13.0403 −1.42281
$$85$$ 0 0
$$86$$ −3.55465 −0.383308
$$87$$ −8.15482 −0.874288
$$88$$ −6.40880 −0.683181
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −4.99477 −0.523594
$$92$$ −1.79560 −0.187204
$$93$$ 15.4271 1.59971
$$94$$ 10.8359 1.11764
$$95$$ 0 0
$$96$$ 2.77733 0.283460
$$97$$ 14.9452 1.51745 0.758727 0.651409i $$-0.225822\pi$$
0.758727 + 0.651409i $$0.225822\pi$$
$$98$$ −15.0455 −1.51983
$$99$$ 30.2081 3.03603
$$100$$ 0 0
$$101$$ 10.5364 1.04841 0.524204 0.851592i $$-0.324363\pi$$
0.524204 + 0.851592i $$0.324363\pi$$
$$102$$ −5.32674 −0.527426
$$103$$ 16.9817 1.67326 0.836630 0.547769i $$-0.184523\pi$$
0.836630 + 0.547769i $$0.184523\pi$$
$$104$$ 1.06379 0.104313
$$105$$ 0 0
$$106$$ −8.69527 −0.844559
$$107$$ 1.79036 0.173081 0.0865405 0.996248i $$-0.472419\pi$$
0.0865405 + 0.996248i $$0.472419\pi$$
$$108$$ −4.75905 −0.457940
$$109$$ 2.41404 0.231223 0.115611 0.993295i $$-0.463117\pi$$
0.115611 + 0.993295i $$0.463117\pi$$
$$110$$ 0 0
$$111$$ −31.6860 −3.00750
$$112$$ 4.69527 0.443661
$$113$$ −7.14585 −0.672225 −0.336112 0.941822i $$-0.609112\pi$$
−0.336112 + 0.941822i $$0.609112\pi$$
$$114$$ −2.77733 −0.260120
$$115$$ 0 0
$$116$$ 2.93621 0.272620
$$117$$ −5.01420 −0.463563
$$118$$ 5.63148 0.518420
$$119$$ −9.00523 −0.825508
$$120$$ 0 0
$$121$$ 30.0728 2.73389
$$122$$ 3.39053 0.306964
$$123$$ 3.18239 0.286947
$$124$$ −5.55465 −0.498822
$$125$$ 0 0
$$126$$ −22.1313 −1.97161
$$127$$ −9.26295 −0.821954 −0.410977 0.911646i $$-0.634812\pi$$
−0.410977 + 0.911646i $$0.634812\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −9.87242 −0.869219
$$130$$ 0 0
$$131$$ 11.5181 1.00634 0.503171 0.864187i $$-0.332167\pi$$
0.503171 + 0.864187i $$0.332167\pi$$
$$132$$ −17.7993 −1.54923
$$133$$ −4.69527 −0.407131
$$134$$ −8.82284 −0.762177
$$135$$ 0 0
$$136$$ 1.91794 0.164462
$$137$$ −15.1041 −1.29043 −0.645214 0.764002i $$-0.723232\pi$$
−0.645214 + 0.764002i $$0.723232\pi$$
$$138$$ −4.98696 −0.424518
$$139$$ −0.700500 −0.0594156 −0.0297078 0.999559i $$-0.509458\pi$$
−0.0297078 + 0.999559i $$0.509458\pi$$
$$140$$ 0 0
$$141$$ 30.0948 2.53444
$$142$$ 1.42708 0.119758
$$143$$ −6.81761 −0.570117
$$144$$ 4.71354 0.392795
$$145$$ 0 0
$$146$$ −12.6132 −1.04388
$$147$$ −41.7863 −3.44648
$$148$$ 11.4088 0.937798
$$149$$ 12.9452 1.06051 0.530255 0.847838i $$-0.322096\pi$$
0.530255 + 0.847838i $$0.322096\pi$$
$$150$$ 0 0
$$151$$ 5.70830 0.464535 0.232268 0.972652i $$-0.425385\pi$$
0.232268 + 0.972652i $$0.425385\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −9.04028 −0.730863
$$154$$ −30.0910 −2.42480
$$155$$ 0 0
$$156$$ 2.95449 0.236548
$$157$$ −9.26295 −0.739264 −0.369632 0.929178i $$-0.620516\pi$$
−0.369632 + 0.929178i $$0.620516\pi$$
$$158$$ 1.96345 0.156204
$$159$$ −24.1496 −1.91519
$$160$$ 0 0
$$161$$ −8.43081 −0.664441
$$162$$ 0.923174 0.0725314
$$163$$ −4.28123 −0.335332 −0.167666 0.985844i $$-0.553623\pi$$
−0.167666 + 0.985844i $$0.553623\pi$$
$$164$$ −1.14585 −0.0894757
$$165$$ 0 0
$$166$$ 16.2447 1.26083
$$167$$ 15.2264 1.17825 0.589127 0.808040i $$-0.299472\pi$$
0.589127 + 0.808040i $$0.299472\pi$$
$$168$$ 13.0403 1.00608
$$169$$ −11.8684 −0.912950
$$170$$ 0 0
$$171$$ −4.71354 −0.360453
$$172$$ 3.55465 0.271040
$$173$$ 12.2630 0.932335 0.466168 0.884696i $$-0.345634\pi$$
0.466168 + 0.884696i $$0.345634\pi$$
$$174$$ 8.15482 0.618215
$$175$$ 0 0
$$176$$ 6.40880 0.483082
$$177$$ 15.6404 1.17561
$$178$$ 10.0000 0.749532
$$179$$ −1.57292 −0.117566 −0.0587829 0.998271i $$-0.518722\pi$$
−0.0587829 + 0.998271i $$0.518722\pi$$
$$180$$ 0 0
$$181$$ 11.4088 0.848010 0.424005 0.905660i $$-0.360624\pi$$
0.424005 + 0.905660i $$0.360624\pi$$
$$182$$ 4.99477 0.370237
$$183$$ 9.41661 0.696096
$$184$$ 1.79560 0.132373
$$185$$ 0 0
$$186$$ −15.4271 −1.13117
$$187$$ −12.2917 −0.898858
$$188$$ −10.8359 −0.790288
$$189$$ −22.3450 −1.62536
$$190$$ 0 0
$$191$$ −6.35805 −0.460053 −0.230026 0.973184i $$-0.573881\pi$$
−0.230026 + 0.973184i $$0.573881\pi$$
$$192$$ −2.77733 −0.200436
$$193$$ −14.4998 −1.04372 −0.521860 0.853031i $$-0.674762\pi$$
−0.521860 + 0.853031i $$0.674762\pi$$
$$194$$ −14.9452 −1.07300
$$195$$ 0 0
$$196$$ 15.0455 1.07468
$$197$$ 5.14585 0.366627 0.183313 0.983055i $$-0.441318\pi$$
0.183313 + 0.983055i $$0.441318\pi$$
$$198$$ −30.2081 −2.14680
$$199$$ 3.87766 0.274880 0.137440 0.990510i $$-0.456113\pi$$
0.137440 + 0.990510i $$0.456113\pi$$
$$200$$ 0 0
$$201$$ −24.5039 −1.72837
$$202$$ −10.5364 −0.741337
$$203$$ 13.7863 0.967608
$$204$$ 5.32674 0.372947
$$205$$ 0 0
$$206$$ −16.9817 −1.18317
$$207$$ −8.46362 −0.588262
$$208$$ −1.06379 −0.0737604
$$209$$ −6.40880 −0.443306
$$210$$ 0 0
$$211$$ 17.1496 1.18063 0.590313 0.807174i $$-0.299004\pi$$
0.590313 + 0.807174i $$0.299004\pi$$
$$212$$ 8.69527 0.597193
$$213$$ 3.96345 0.271571
$$214$$ −1.79036 −0.122387
$$215$$ 0 0
$$216$$ 4.75905 0.323813
$$217$$ −26.0806 −1.77046
$$218$$ −2.41404 −0.163499
$$219$$ −35.0310 −2.36717
$$220$$ 0 0
$$221$$ 2.04028 0.137244
$$222$$ 31.6860 2.12662
$$223$$ −7.84635 −0.525430 −0.262715 0.964873i $$-0.584618\pi$$
−0.262715 + 0.964873i $$0.584618\pi$$
$$224$$ −4.69527 −0.313716
$$225$$ 0 0
$$226$$ 7.14585 0.475335
$$227$$ −6.28646 −0.417247 −0.208624 0.977996i $$-0.566898\pi$$
−0.208624 + 0.977996i $$0.566898\pi$$
$$228$$ 2.77733 0.183933
$$229$$ −3.14585 −0.207884 −0.103942 0.994583i $$-0.533146\pi$$
−0.103942 + 0.994583i $$0.533146\pi$$
$$230$$ 0 0
$$231$$ −83.5726 −5.49867
$$232$$ −2.93621 −0.192772
$$233$$ −0.182394 −0.0119490 −0.00597451 0.999982i $$-0.501902\pi$$
−0.00597451 + 0.999982i $$0.501902\pi$$
$$234$$ 5.01420 0.327789
$$235$$ 0 0
$$236$$ −5.63148 −0.366578
$$237$$ 5.45315 0.354220
$$238$$ 9.00523 0.583723
$$239$$ −11.5039 −0.744126 −0.372063 0.928208i $$-0.621349\pi$$
−0.372063 + 0.928208i $$0.621349\pi$$
$$240$$ 0 0
$$241$$ −0.445349 −0.0286874 −0.0143437 0.999897i $$-0.504566\pi$$
−0.0143437 + 0.999897i $$0.504566\pi$$
$$242$$ −30.0728 −1.93315
$$243$$ 16.8411 1.08036
$$244$$ −3.39053 −0.217056
$$245$$ 0 0
$$246$$ −3.18239 −0.202902
$$247$$ 1.06379 0.0676872
$$248$$ 5.55465 0.352721
$$249$$ 45.1168 2.85916
$$250$$ 0 0
$$251$$ 13.2630 0.837150 0.418575 0.908182i $$-0.362530\pi$$
0.418575 + 0.908182i $$0.362530\pi$$
$$252$$ 22.1313 1.39414
$$253$$ −11.5076 −0.723479
$$254$$ 9.26295 0.581209
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.4816 1.46474 0.732370 0.680907i $$-0.238414\pi$$
0.732370 + 0.680907i $$0.238414\pi$$
$$258$$ 9.87242 0.614630
$$259$$ 53.5674 3.32851
$$260$$ 0 0
$$261$$ 13.8399 0.856671
$$262$$ −11.5181 −0.711591
$$263$$ −27.9269 −1.72205 −0.861023 0.508565i $$-0.830176\pi$$
−0.861023 + 0.508565i $$0.830176\pi$$
$$264$$ 17.7993 1.09547
$$265$$ 0 0
$$266$$ 4.69527 0.287885
$$267$$ 27.7733 1.69970
$$268$$ 8.82284 0.538941
$$269$$ 16.4816 1.00490 0.502449 0.864607i $$-0.332432\pi$$
0.502449 + 0.864607i $$0.332432\pi$$
$$270$$ 0 0
$$271$$ −1.46736 −0.0891356 −0.0445678 0.999006i $$-0.514191\pi$$
−0.0445678 + 0.999006i $$0.514191\pi$$
$$272$$ −1.91794 −0.116292
$$273$$ 13.8721 0.839577
$$274$$ 15.1041 0.912470
$$275$$ 0 0
$$276$$ 4.98696 0.300180
$$277$$ −15.8359 −0.951486 −0.475743 0.879584i $$-0.657821\pi$$
−0.475743 + 0.879584i $$0.657821\pi$$
$$278$$ 0.700500 0.0420132
$$279$$ −26.1821 −1.56748
$$280$$ 0 0
$$281$$ −6.25515 −0.373151 −0.186576 0.982441i $$-0.559739\pi$$
−0.186576 + 0.982441i $$0.559739\pi$$
$$282$$ −30.0948 −1.79212
$$283$$ 16.7005 0.992742 0.496371 0.868111i $$-0.334666\pi$$
0.496371 + 0.868111i $$0.334666\pi$$
$$284$$ −1.42708 −0.0846814
$$285$$ 0 0
$$286$$ 6.81761 0.403134
$$287$$ −5.38006 −0.317575
$$288$$ −4.71354 −0.277748
$$289$$ −13.3215 −0.783618
$$290$$ 0 0
$$291$$ −41.5076 −2.43322
$$292$$ 12.6132 0.738132
$$293$$ −0.644516 −0.0376530 −0.0188265 0.999823i $$-0.505993\pi$$
−0.0188265 + 0.999823i $$0.505993\pi$$
$$294$$ 41.7863 2.43703
$$295$$ 0 0
$$296$$ −11.4088 −0.663123
$$297$$ −30.4998 −1.76978
$$298$$ −12.9452 −0.749894
$$299$$ 1.91014 0.110466
$$300$$ 0 0
$$301$$ 16.6900 0.961997
$$302$$ −5.70830 −0.328476
$$303$$ −29.2630 −1.68111
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 9.04028 0.516798
$$307$$ −26.6457 −1.52075 −0.760375 0.649485i $$-0.774985\pi$$
−0.760375 + 0.649485i $$0.774985\pi$$
$$308$$ 30.0910 1.71460
$$309$$ −47.1638 −2.68305
$$310$$ 0 0
$$311$$ 11.5494 0.654907 0.327454 0.944867i $$-0.393809\pi$$
0.327454 + 0.944867i $$0.393809\pi$$
$$312$$ −2.95449 −0.167265
$$313$$ 23.0638 1.30364 0.651821 0.758373i $$-0.274005\pi$$
0.651821 + 0.758373i $$0.274005\pi$$
$$314$$ 9.26295 0.522739
$$315$$ 0 0
$$316$$ −1.96345 −0.110453
$$317$$ 13.9232 0.782003 0.391002 0.920390i $$-0.372129\pi$$
0.391002 + 0.920390i $$0.372129\pi$$
$$318$$ 24.1496 1.35424
$$319$$ 18.8176 1.05358
$$320$$ 0 0
$$321$$ −4.97242 −0.277534
$$322$$ 8.43081 0.469831
$$323$$ 1.91794 0.106717
$$324$$ −0.923174 −0.0512874
$$325$$ 0 0
$$326$$ 4.28123 0.237115
$$327$$ −6.70457 −0.370763
$$328$$ 1.14585 0.0632689
$$329$$ −50.8773 −2.80496
$$330$$ 0 0
$$331$$ −0.735546 −0.0404292 −0.0202146 0.999796i $$-0.506435\pi$$
−0.0202146 + 0.999796i $$0.506435\pi$$
$$332$$ −16.2447 −0.891543
$$333$$ 53.7758 2.94690
$$334$$ −15.2264 −0.833152
$$335$$ 0 0
$$336$$ −13.0403 −0.711406
$$337$$ 2.28123 0.124266 0.0621332 0.998068i $$-0.480210\pi$$
0.0621332 + 0.998068i $$0.480210\pi$$
$$338$$ 11.8684 0.645553
$$339$$ 19.8463 1.07791
$$340$$ 0 0
$$341$$ −35.5987 −1.92778
$$342$$ 4.71354 0.254879
$$343$$ 37.7758 2.03970
$$344$$ −3.55465 −0.191654
$$345$$ 0 0
$$346$$ −12.2630 −0.659261
$$347$$ 2.56246 0.137560 0.0687799 0.997632i $$-0.478089\pi$$
0.0687799 + 0.997632i $$0.478089\pi$$
$$348$$ −8.15482 −0.437144
$$349$$ −5.67176 −0.303602 −0.151801 0.988411i $$-0.548507\pi$$
−0.151801 + 0.988411i $$0.548507\pi$$
$$350$$ 0 0
$$351$$ 5.06262 0.270223
$$352$$ −6.40880 −0.341590
$$353$$ 23.6665 1.25964 0.629821 0.776740i $$-0.283128\pi$$
0.629821 + 0.776740i $$0.283128\pi$$
$$354$$ −15.6404 −0.831280
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 25.0105 1.32369
$$358$$ 1.57292 0.0831316
$$359$$ 31.9724 1.68744 0.843720 0.536784i $$-0.180361\pi$$
0.843720 + 0.536784i $$0.180361\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −11.4088 −0.599633
$$363$$ −83.5218 −4.38376
$$364$$ −4.99477 −0.261797
$$365$$ 0 0
$$366$$ −9.41661 −0.492214
$$367$$ −1.14585 −0.0598128 −0.0299064 0.999553i $$-0.509521\pi$$
−0.0299064 + 0.999553i $$0.509521\pi$$
$$368$$ −1.79560 −0.0936020
$$369$$ −5.40100 −0.281165
$$370$$ 0 0
$$371$$ 40.8266 2.11961
$$372$$ 15.4271 0.799857
$$373$$ −32.8579 −1.70132 −0.850658 0.525719i $$-0.823796\pi$$
−0.850658 + 0.525719i $$0.823796\pi$$
$$374$$ 12.2917 0.635588
$$375$$ 0 0
$$376$$ 10.8359 0.558818
$$377$$ −3.12351 −0.160869
$$378$$ 22.3450 1.14930
$$379$$ −18.2865 −0.939312 −0.469656 0.882849i $$-0.655622\pi$$
−0.469656 + 0.882849i $$0.655622\pi$$
$$380$$ 0 0
$$381$$ 25.7262 1.31800
$$382$$ 6.35805 0.325306
$$383$$ −20.8542 −1.06560 −0.532799 0.846242i $$-0.678860\pi$$
−0.532799 + 0.846242i $$0.678860\pi$$
$$384$$ 2.77733 0.141730
$$385$$ 0 0
$$386$$ 14.4998 0.738022
$$387$$ 16.7550 0.851704
$$388$$ 14.9452 0.758727
$$389$$ −24.9086 −1.26292 −0.631459 0.775409i $$-0.717544\pi$$
−0.631459 + 0.775409i $$0.717544\pi$$
$$390$$ 0 0
$$391$$ 3.44385 0.174163
$$392$$ −15.0455 −0.759913
$$393$$ −31.9895 −1.61366
$$394$$ −5.14585 −0.259244
$$395$$ 0 0
$$396$$ 30.2081 1.51802
$$397$$ 24.7445 1.24189 0.620946 0.783853i $$-0.286749\pi$$
0.620946 + 0.783853i $$0.286749\pi$$
$$398$$ −3.87766 −0.194369
$$399$$ 13.0403 0.652831
$$400$$ 0 0
$$401$$ 15.6457 0.781308 0.390654 0.920538i $$-0.372249\pi$$
0.390654 + 0.920538i $$0.372249\pi$$
$$402$$ 24.5039 1.22214
$$403$$ 5.90897 0.294347
$$404$$ 10.5364 0.524204
$$405$$ 0 0
$$406$$ −13.7863 −0.684202
$$407$$ 73.1168 3.62426
$$408$$ −5.32674 −0.263713
$$409$$ 30.5804 1.51210 0.756052 0.654512i $$-0.227126\pi$$
0.756052 + 0.654512i $$0.227126\pi$$
$$410$$ 0 0
$$411$$ 41.9489 2.06919
$$412$$ 16.9817 0.836630
$$413$$ −26.4413 −1.30109
$$414$$ 8.46362 0.415964
$$415$$ 0 0
$$416$$ 1.06379 0.0521565
$$417$$ 1.94552 0.0952723
$$418$$ 6.40880 0.313465
$$419$$ 1.96345 0.0959210 0.0479605 0.998849i $$-0.484728\pi$$
0.0479605 + 0.998849i $$0.484728\pi$$
$$420$$ 0 0
$$421$$ −25.5949 −1.24742 −0.623710 0.781656i $$-0.714376\pi$$
−0.623710 + 0.781656i $$0.714376\pi$$
$$422$$ −17.1496 −0.834829
$$423$$ −51.0753 −2.48337
$$424$$ −8.69527 −0.422279
$$425$$ 0 0
$$426$$ −3.96345 −0.192030
$$427$$ −15.9194 −0.770396
$$428$$ 1.79036 0.0865405
$$429$$ 18.9347 0.914177
$$430$$ 0 0
$$431$$ −15.3540 −0.739575 −0.369788 0.929116i $$-0.620570\pi$$
−0.369788 + 0.929116i $$0.620570\pi$$
$$432$$ −4.75905 −0.228970
$$433$$ −16.6640 −0.800819 −0.400409 0.916336i $$-0.631132\pi$$
−0.400409 + 0.916336i $$0.631132\pi$$
$$434$$ 26.0806 1.25191
$$435$$ 0 0
$$436$$ 2.41404 0.115611
$$437$$ 1.79560 0.0858951
$$438$$ 35.0310 1.67384
$$439$$ −19.5987 −0.935393 −0.467697 0.883889i $$-0.654916\pi$$
−0.467697 + 0.883889i $$0.654916\pi$$
$$440$$ 0 0
$$441$$ 70.9176 3.37703
$$442$$ −2.04028 −0.0970462
$$443$$ −13.0183 −0.618517 −0.309258 0.950978i $$-0.600081\pi$$
−0.309258 + 0.950978i $$0.600081\pi$$
$$444$$ −31.6860 −1.50375
$$445$$ 0 0
$$446$$ 7.84635 0.371535
$$447$$ −35.9530 −1.70052
$$448$$ 4.69527 0.221830
$$449$$ 6.77359 0.319666 0.159833 0.987144i $$-0.448905\pi$$
0.159833 + 0.987144i $$0.448905\pi$$
$$450$$ 0 0
$$451$$ −7.34352 −0.345793
$$452$$ −7.14585 −0.336112
$$453$$ −15.8538 −0.744877
$$454$$ 6.28646 0.295038
$$455$$ 0 0
$$456$$ −2.77733 −0.130060
$$457$$ 27.3189 1.27793 0.638963 0.769237i $$-0.279364\pi$$
0.638963 + 0.769237i $$0.279364\pi$$
$$458$$ 3.14585 0.146996
$$459$$ 9.12758 0.426039
$$460$$ 0 0
$$461$$ −27.7993 −1.29474 −0.647372 0.762174i $$-0.724132\pi$$
−0.647372 + 0.762174i $$0.724132\pi$$
$$462$$ 83.5726 3.88815
$$463$$ −38.2369 −1.77702 −0.888509 0.458859i $$-0.848258\pi$$
−0.888509 + 0.458859i $$0.848258\pi$$
$$464$$ 2.93621 0.136310
$$465$$ 0 0
$$466$$ 0.182394 0.00844923
$$467$$ −23.4711 −1.08611 −0.543056 0.839696i $$-0.682733\pi$$
−0.543056 + 0.839696i $$0.682733\pi$$
$$468$$ −5.01420 −0.231782
$$469$$ 41.4256 1.91286
$$470$$ 0 0
$$471$$ 25.7262 1.18540
$$472$$ 5.63148 0.259210
$$473$$ 22.7811 1.04747
$$474$$ −5.45315 −0.250472
$$475$$ 0 0
$$476$$ −9.00523 −0.412754
$$477$$ 40.9855 1.87660
$$478$$ 11.5039 0.526176
$$479$$ −19.6900 −0.899660 −0.449830 0.893114i $$-0.648516\pi$$
−0.449830 + 0.893114i $$0.648516\pi$$
$$480$$ 0 0
$$481$$ −12.1365 −0.553379
$$482$$ 0.445349 0.0202851
$$483$$ 23.4151 1.06542
$$484$$ 30.0728 1.36694
$$485$$ 0 0
$$486$$ −16.8411 −0.763928
$$487$$ −16.3357 −0.740242 −0.370121 0.928984i $$-0.620684\pi$$
−0.370121 + 0.928984i $$0.620684\pi$$
$$488$$ 3.39053 0.153482
$$489$$ 11.8904 0.537701
$$490$$ 0 0
$$491$$ 27.1093 1.22343 0.611713 0.791080i $$-0.290481\pi$$
0.611713 + 0.791080i $$0.290481\pi$$
$$492$$ 3.18239 0.143473
$$493$$ −5.63148 −0.253629
$$494$$ −1.06379 −0.0478621
$$495$$ 0 0
$$496$$ −5.55465 −0.249411
$$497$$ −6.70050 −0.300558
$$498$$ −45.1168 −2.02173
$$499$$ −4.69003 −0.209955 −0.104977 0.994475i $$-0.533477\pi$$
−0.104977 + 0.994475i $$0.533477\pi$$
$$500$$ 0 0
$$501$$ −42.2887 −1.88932
$$502$$ −13.2630 −0.591955
$$503$$ −5.19136 −0.231471 −0.115736 0.993280i $$-0.536923\pi$$
−0.115736 + 0.993280i $$0.536923\pi$$
$$504$$ −22.1313 −0.985807
$$505$$ 0 0
$$506$$ 11.5076 0.511577
$$507$$ 32.9623 1.46391
$$508$$ −9.26295 −0.410977
$$509$$ −4.51811 −0.200262 −0.100131 0.994974i $$-0.531926\pi$$
−0.100131 + 0.994974i $$0.531926\pi$$
$$510$$ 0 0
$$511$$ 59.2223 2.61984
$$512$$ −1.00000 −0.0441942
$$513$$ 4.75905 0.210117
$$514$$ −23.4816 −1.03573
$$515$$ 0 0
$$516$$ −9.87242 −0.434609
$$517$$ −69.4450 −3.05419
$$518$$ −53.5674 −2.35361
$$519$$ −34.0582 −1.49499
$$520$$ 0 0
$$521$$ 14.4453 0.632862 0.316431 0.948615i $$-0.397515\pi$$
0.316431 + 0.948615i $$0.397515\pi$$
$$522$$ −13.8399 −0.605758
$$523$$ 18.2134 0.796415 0.398208 0.917295i $$-0.369632\pi$$
0.398208 + 0.917295i $$0.369632\pi$$
$$524$$ 11.5181 0.503171
$$525$$ 0 0
$$526$$ 27.9269 1.21767
$$527$$ 10.6535 0.464073
$$528$$ −17.7993 −0.774617
$$529$$ −19.7758 −0.859819
$$530$$ 0 0
$$531$$ −26.5442 −1.15192
$$532$$ −4.69527 −0.203566
$$533$$ 1.21894 0.0527981
$$534$$ −27.7733 −1.20187
$$535$$ 0 0
$$536$$ −8.82284 −0.381089
$$537$$ 4.36852 0.188516
$$538$$ −16.4816 −0.710571
$$539$$ 96.4237 4.15326
$$540$$ 0 0
$$541$$ 37.3905 1.60754 0.803772 0.594937i $$-0.202823\pi$$
0.803772 + 0.594937i $$0.202823\pi$$
$$542$$ 1.46736 0.0630284
$$543$$ −31.6860 −1.35977
$$544$$ 1.91794 0.0822310
$$545$$ 0 0
$$546$$ −13.8721 −0.593671
$$547$$ −29.5621 −1.26399 −0.631993 0.774974i $$-0.717763\pi$$
−0.631993 + 0.774974i $$0.717763\pi$$
$$548$$ −15.1041 −0.645214
$$549$$ −15.9814 −0.682069
$$550$$ 0 0
$$551$$ −2.93621 −0.125087
$$552$$ −4.98696 −0.212259
$$553$$ −9.21894 −0.392029
$$554$$ 15.8359 0.672802
$$555$$ 0 0
$$556$$ −0.700500 −0.0297078
$$557$$ −14.3723 −0.608972 −0.304486 0.952517i $$-0.598485\pi$$
−0.304486 + 0.952517i $$0.598485\pi$$
$$558$$ 26.1821 1.10837
$$559$$ −3.78139 −0.159936
$$560$$ 0 0
$$561$$ 34.1380 1.44131
$$562$$ 6.25515 0.263858
$$563$$ 6.39053 0.269329 0.134664 0.990891i $$-0.457004\pi$$
0.134664 + 0.990891i $$0.457004\pi$$
$$564$$ 30.0948 1.26722
$$565$$ 0 0
$$566$$ −16.7005 −0.701974
$$567$$ −4.33455 −0.182034
$$568$$ 1.42708 0.0598788
$$569$$ −44.9086 −1.88267 −0.941334 0.337476i $$-0.890427\pi$$
−0.941334 + 0.337476i $$0.890427\pi$$
$$570$$ 0 0
$$571$$ 12.4193 0.519730 0.259865 0.965645i $$-0.416322\pi$$
0.259865 + 0.965645i $$0.416322\pi$$
$$572$$ −6.81761 −0.285058
$$573$$ 17.6584 0.737690
$$574$$ 5.38006 0.224559
$$575$$ 0 0
$$576$$ 4.71354 0.196397
$$577$$ 4.20440 0.175032 0.0875158 0.996163i $$-0.472107\pi$$
0.0875158 + 0.996163i $$0.472107\pi$$
$$578$$ 13.3215 0.554102
$$579$$ 40.2708 1.67360
$$580$$ 0 0
$$581$$ −76.2731 −3.16434
$$582$$ 41.5076 1.72055
$$583$$ 55.7262 2.30795
$$584$$ −12.6132 −0.521938
$$585$$ 0 0
$$586$$ 0.644516 0.0266247
$$587$$ −24.7915 −1.02326 −0.511628 0.859207i $$-0.670957\pi$$
−0.511628 + 0.859207i $$0.670957\pi$$
$$588$$ −41.7863 −1.72324
$$589$$ 5.55465 0.228875
$$590$$ 0 0
$$591$$ −14.2917 −0.587882
$$592$$ 11.4088 0.468899
$$593$$ −4.67176 −0.191846 −0.0959231 0.995389i $$-0.530580\pi$$
−0.0959231 + 0.995389i $$0.530580\pi$$
$$594$$ 30.4998 1.25142
$$595$$ 0 0
$$596$$ 12.9452 0.530255
$$597$$ −10.7695 −0.440767
$$598$$ −1.91014 −0.0781113
$$599$$ 39.2369 1.60318 0.801588 0.597877i $$-0.203989\pi$$
0.801588 + 0.597877i $$0.203989\pi$$
$$600$$ 0 0
$$601$$ −42.4267 −1.73062 −0.865311 0.501235i $$-0.832879\pi$$
−0.865311 + 0.501235i $$0.832879\pi$$
$$602$$ −16.6900 −0.680235
$$603$$ 41.5868 1.69355
$$604$$ 5.70830 0.232268
$$605$$ 0 0
$$606$$ 29.2630 1.18873
$$607$$ 2.98173 0.121025 0.0605123 0.998167i $$-0.480727\pi$$
0.0605123 + 0.998167i $$0.480727\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −38.2890 −1.55155
$$610$$ 0 0
$$611$$ 11.5271 0.466336
$$612$$ −9.04028 −0.365432
$$613$$ 28.9452 1.16908 0.584542 0.811363i $$-0.301274\pi$$
0.584542 + 0.811363i $$0.301274\pi$$
$$614$$ 26.6457 1.07533
$$615$$ 0 0
$$616$$ −30.0910 −1.21240
$$617$$ 15.0365 0.605349 0.302674 0.953094i $$-0.402121\pi$$
0.302674 + 0.953094i $$0.402121\pi$$
$$618$$ 47.1638 1.89721
$$619$$ 8.25515 0.331803 0.165901 0.986142i $$-0.446947\pi$$
0.165901 + 0.986142i $$0.446947\pi$$
$$620$$ 0 0
$$621$$ 8.54535 0.342913
$$622$$ −11.5494 −0.463089
$$623$$ −46.9527 −1.88112
$$624$$ 2.95449 0.118274
$$625$$ 0 0
$$626$$ −23.0638 −0.921814
$$627$$ 17.7993 0.710837
$$628$$ −9.26295 −0.369632
$$629$$ −21.8814 −0.872468
$$630$$ 0 0
$$631$$ −9.09103 −0.361908 −0.180954 0.983492i $$-0.557919\pi$$
−0.180954 + 0.983492i $$0.557919\pi$$
$$632$$ 1.96345 0.0781020
$$633$$ −47.6300 −1.89312
$$634$$ −13.9232 −0.552960
$$635$$ 0 0
$$636$$ −24.1496 −0.957593
$$637$$ −16.0052 −0.634150
$$638$$ −18.8176 −0.744996
$$639$$ −6.72658 −0.266099
$$640$$ 0 0
$$641$$ 30.1171 1.18955 0.594777 0.803891i $$-0.297240\pi$$
0.594777 + 0.803891i $$0.297240\pi$$
$$642$$ 4.97242 0.196246
$$643$$ 35.3174 1.39278 0.696392 0.717662i $$-0.254788\pi$$
0.696392 + 0.717662i $$0.254788\pi$$
$$644$$ −8.43081 −0.332220
$$645$$ 0 0
$$646$$ −1.91794 −0.0754603
$$647$$ −4.12234 −0.162066 −0.0810330 0.996711i $$-0.525822\pi$$
−0.0810330 + 0.996711i $$0.525822\pi$$
$$648$$ 0.923174 0.0362657
$$649$$ −36.0910 −1.41670
$$650$$ 0 0
$$651$$ 72.4342 2.83892
$$652$$ −4.28123 −0.167666
$$653$$ −8.62741 −0.337617 −0.168808 0.985649i $$-0.553992\pi$$
−0.168808 + 0.985649i $$0.553992\pi$$
$$654$$ 6.70457 0.262169
$$655$$ 0 0
$$656$$ −1.14585 −0.0447379
$$657$$ 59.4528 2.31948
$$658$$ 50.8773 1.98340
$$659$$ −37.3853 −1.45632 −0.728162 0.685405i $$-0.759625\pi$$
−0.728162 + 0.685405i $$0.759625\pi$$
$$660$$ 0 0
$$661$$ −12.8997 −0.501739 −0.250869 0.968021i $$-0.580716\pi$$
−0.250869 + 0.968021i $$0.580716\pi$$
$$662$$ 0.735546 0.0285878
$$663$$ −5.66652 −0.220070
$$664$$ 16.2447 0.630416
$$665$$ 0 0
$$666$$ −53.7758 −2.08377
$$667$$ −5.27226 −0.204143
$$668$$ 15.2264 0.589127
$$669$$ 21.7919 0.842522
$$670$$ 0 0
$$671$$ −21.7292 −0.838848
$$672$$ 13.0403 0.503040
$$673$$ −10.4349 −0.402235 −0.201118 0.979567i $$-0.564457\pi$$
−0.201118 + 0.979567i $$0.564457\pi$$
$$674$$ −2.28123 −0.0878696
$$675$$ 0 0
$$676$$ −11.8684 −0.456475
$$677$$ −12.4401 −0.478112 −0.239056 0.971006i $$-0.576838\pi$$
−0.239056 + 0.971006i $$0.576838\pi$$
$$678$$ −19.8463 −0.762194
$$679$$ 70.1716 2.69294
$$680$$ 0 0
$$681$$ 17.4596 0.669052
$$682$$ 35.5987 1.36314
$$683$$ −17.5364 −0.671011 −0.335505 0.942038i $$-0.608907\pi$$
−0.335505 + 0.942038i $$0.608907\pi$$
$$684$$ −4.71354 −0.180227
$$685$$ 0 0
$$686$$ −37.7758 −1.44229
$$687$$ 8.73705 0.333339
$$688$$ 3.55465 0.135520
$$689$$ −9.24992 −0.352394
$$690$$ 0 0
$$691$$ −25.2734 −0.961446 −0.480723 0.876872i $$-0.659626\pi$$
−0.480723 + 0.876872i $$0.659626\pi$$
$$692$$ 12.2630 0.466168
$$693$$ 141.835 5.38787
$$694$$ −2.56246 −0.0972695
$$695$$ 0 0
$$696$$ 8.15482 0.309108
$$697$$ 2.19767 0.0832426
$$698$$ 5.67176 0.214679
$$699$$ 0.506567 0.0191601
$$700$$ 0 0
$$701$$ −16.0545 −0.606370 −0.303185 0.952932i $$-0.598050\pi$$
−0.303185 + 0.952932i $$0.598050\pi$$
$$702$$ −5.06262 −0.191076
$$703$$ −11.4088 −0.430291
$$704$$ 6.40880 0.241541
$$705$$ 0 0
$$706$$ −23.6665 −0.890701
$$707$$ 49.4711 1.86055
$$708$$ 15.6404 0.587804
$$709$$ 13.5076 0.507290 0.253645 0.967297i $$-0.418371\pi$$
0.253645 + 0.967297i $$0.418371\pi$$
$$710$$ 0 0
$$711$$ −9.25482 −0.347083
$$712$$ 10.0000 0.374766
$$713$$ 9.97392 0.373526
$$714$$ −25.0105 −0.935993
$$715$$ 0 0
$$716$$ −1.57292 −0.0587829
$$717$$ 31.9501 1.19320
$$718$$ −31.9724 −1.19320
$$719$$ 0.803402 0.0299619 0.0149809 0.999888i $$-0.495231\pi$$
0.0149809 + 0.999888i $$0.495231\pi$$
$$720$$ 0 0
$$721$$ 79.7337 2.96944
$$722$$ −1.00000 −0.0372161
$$723$$ 1.23688 0.0460000
$$724$$ 11.4088 0.424005
$$725$$ 0 0
$$726$$ 83.5218 3.09979
$$727$$ −51.6860 −1.91693 −0.958463 0.285217i $$-0.907934\pi$$
−0.958463 + 0.285217i $$0.907934\pi$$
$$728$$ 4.99477 0.185118
$$729$$ −44.0037 −1.62977
$$730$$ 0 0
$$731$$ −6.81761 −0.252158
$$732$$ 9.41661 0.348048
$$733$$ 22.3723 0.826338 0.413169 0.910654i $$-0.364422\pi$$
0.413169 + 0.910654i $$0.364422\pi$$
$$734$$ 1.14585 0.0422940
$$735$$ 0 0
$$736$$ 1.79560 0.0661866
$$737$$ 56.5438 2.08282
$$738$$ 5.40100 0.198814
$$739$$ 33.4271 1.22963 0.614817 0.788669i $$-0.289230\pi$$
0.614817 + 0.788669i $$0.289230\pi$$
$$740$$ 0 0
$$741$$ −2.95449 −0.108536
$$742$$ −40.8266 −1.49879
$$743$$ −14.4998 −0.531947 −0.265974 0.963980i $$-0.585693\pi$$
−0.265974 + 0.963980i $$0.585693\pi$$
$$744$$ −15.4271 −0.565584
$$745$$ 0 0
$$746$$ 32.8579 1.20301
$$747$$ −76.5699 −2.80155
$$748$$ −12.2917 −0.449429
$$749$$ 8.40623 0.307157
$$750$$ 0 0
$$751$$ −26.6169 −0.971266 −0.485633 0.874163i $$-0.661411\pi$$
−0.485633 + 0.874163i $$0.661411\pi$$
$$752$$ −10.8359 −0.395144
$$753$$ −36.8355 −1.34236
$$754$$ 3.12351 0.113751
$$755$$ 0 0
$$756$$ −22.3450 −0.812680
$$757$$ 31.0362 1.12803 0.564015 0.825764i $$-0.309256\pi$$
0.564015 + 0.825764i $$0.309256\pi$$
$$758$$ 18.2865 0.664194
$$759$$ 31.9605 1.16009
$$760$$ 0 0
$$761$$ −27.8083 −1.00805 −0.504025 0.863689i $$-0.668148\pi$$
−0.504025 + 0.863689i $$0.668148\pi$$
$$762$$ −25.7262 −0.931963
$$763$$ 11.3345 0.410338
$$764$$ −6.35805 −0.230026
$$765$$ 0 0
$$766$$ 20.8542 0.753491
$$767$$ 5.99070 0.216312
$$768$$ −2.77733 −0.100218
$$769$$ −19.2279 −0.693376 −0.346688 0.937980i $$-0.612694\pi$$
−0.346688 + 0.937980i $$0.612694\pi$$
$$770$$ 0 0
$$771$$ −65.2159 −2.34869
$$772$$ −14.4998 −0.521860
$$773$$ −6.00373 −0.215939 −0.107970 0.994154i $$-0.534435\pi$$
−0.107970 + 0.994154i $$0.534435\pi$$
$$774$$ −16.7550 −0.602245
$$775$$ 0 0
$$776$$ −14.9452 −0.536501
$$777$$ −148.774 −5.33724
$$778$$ 24.9086 0.893018
$$779$$ 1.14585 0.0410543
$$780$$ 0 0
$$781$$ −9.14585 −0.327264
$$782$$ −3.44385 −0.123152
$$783$$ −13.9736 −0.499375
$$784$$ 15.0455 0.537340
$$785$$ 0 0
$$786$$ 31.9895 1.14103
$$787$$ −22.2797 −0.794187 −0.397093 0.917778i $$-0.629981\pi$$
−0.397093 + 0.917778i $$0.629981\pi$$
$$788$$ 5.14585 0.183313
$$789$$ 77.5621 2.76128
$$790$$ 0 0
$$791$$ −33.5517 −1.19296
$$792$$ −30.2081 −1.07340
$$793$$ 3.60680 0.128081
$$794$$ −24.7445 −0.878150
$$795$$ 0 0
$$796$$ 3.87766 0.137440
$$797$$ −26.6259 −0.943138 −0.471569 0.881829i $$-0.656312\pi$$
−0.471569 + 0.881829i $$0.656312\pi$$
$$798$$ −13.0403 −0.461621
$$799$$ 20.7826 0.735234
$$800$$ 0 0
$$801$$ −47.1354 −1.66545
$$802$$ −15.6457 −0.552468
$$803$$ 80.8355 2.85262
$$804$$ −24.5039 −0.864186
$$805$$ 0 0
$$806$$ −5.90897 −0.208135
$$807$$ −45.7747 −1.61134
$$808$$ −10.5364 −0.370669
$$809$$ 0.651250 0.0228967 0.0114484 0.999934i $$-0.496356\pi$$
0.0114484 + 0.999934i $$0.496356\pi$$
$$810$$ 0 0
$$811$$ −13.0780 −0.459230 −0.229615 0.973281i $$-0.573747\pi$$
−0.229615 + 0.973281i $$0.573747\pi$$
$$812$$ 13.7863 0.483804
$$813$$ 4.07533 0.142928
$$814$$ −73.1168 −2.56274
$$815$$ 0 0
$$816$$ 5.32674 0.186473
$$817$$ −3.55465 −0.124362
$$818$$ −30.5804 −1.06922
$$819$$ −23.5430 −0.822660
$$820$$ 0 0
$$821$$ 14.6640 0.511776 0.255888 0.966706i $$-0.417632\pi$$
0.255888 + 0.966706i $$0.417632\pi$$
$$822$$ −41.9489 −1.46314
$$823$$ 18.2850 0.637374 0.318687 0.947860i $$-0.396758\pi$$
0.318687 + 0.947860i $$0.396758\pi$$
$$824$$ −16.9817 −0.591587
$$825$$ 0 0
$$826$$ 26.4413 0.920010
$$827$$ −40.1910 −1.39758 −0.698790 0.715327i $$-0.746278\pi$$
−0.698790 + 0.715327i $$0.746278\pi$$
$$828$$ −8.46362 −0.294131
$$829$$ 28.3760 0.985539 0.492769 0.870160i $$-0.335985\pi$$
0.492769 + 0.870160i $$0.335985\pi$$
$$830$$ 0 0
$$831$$ 43.9814 1.52570
$$832$$ −1.06379 −0.0368802
$$833$$ −28.8564 −0.999815
$$834$$ −1.94552 −0.0673677
$$835$$ 0 0
$$836$$ −6.40880 −0.221653
$$837$$ 26.4349 0.913723
$$838$$ −1.96345 −0.0678264
$$839$$ 6.93471 0.239413 0.119706 0.992809i $$-0.461805\pi$$
0.119706 + 0.992809i $$0.461805\pi$$
$$840$$ 0 0
$$841$$ −20.3787 −0.702712
$$842$$ 25.5949 0.882060
$$843$$ 17.3726 0.598344
$$844$$ 17.1496 0.590313
$$845$$ 0 0
$$846$$ 51.0753 1.75601
$$847$$ 141.200 4.85167
$$848$$ 8.69527 0.298597
$$849$$ −46.3827 −1.59185
$$850$$ 0 0
$$851$$ −20.4856 −0.702238
$$852$$ 3.96345 0.135786
$$853$$ −21.9739 −0.752373 −0.376186 0.926544i $$-0.622765\pi$$
−0.376186 + 0.926544i $$0.622765\pi$$
$$854$$ 15.9194 0.544752
$$855$$ 0 0
$$856$$ −1.79036 −0.0611934
$$857$$ −13.6091 −0.464879 −0.232440 0.972611i $$-0.574671\pi$$
−0.232440 + 0.972611i $$0.574671\pi$$
$$858$$ −18.9347 −0.646420
$$859$$ 5.17192 0.176464 0.0882319 0.996100i $$-0.471878\pi$$
0.0882319 + 0.996100i $$0.471878\pi$$
$$860$$ 0 0
$$861$$ 14.9422 0.509228
$$862$$ 15.3540 0.522959
$$863$$ −15.9635 −0.543402 −0.271701 0.962382i $$-0.587586\pi$$
−0.271701 + 0.962382i $$0.587586\pi$$
$$864$$ 4.75905 0.161906
$$865$$ 0 0
$$866$$ 16.6640 0.566264
$$867$$ 36.9982 1.25652
$$868$$ −26.0806 −0.885232
$$869$$ −12.5834 −0.426862
$$870$$ 0 0
$$871$$ −9.38563 −0.318020
$$872$$ −2.41404 −0.0817496
$$873$$ 70.4447 2.38419
$$874$$ −1.79560 −0.0607370
$$875$$ 0 0
$$876$$ −35.0310 −1.18359
$$877$$ 15.8866 0.536453 0.268227 0.963356i $$-0.413562\pi$$
0.268227 + 0.963356i $$0.413562\pi$$
$$878$$ 19.5987 0.661423
$$879$$ 1.79003 0.0603762
$$880$$ 0 0
$$881$$ −16.1458 −0.543967 −0.271984 0.962302i $$-0.587680\pi$$
−0.271984 + 0.962302i $$0.587680\pi$$
$$882$$ −70.9176 −2.38792
$$883$$ −38.2887 −1.28852 −0.644259 0.764808i $$-0.722834\pi$$
−0.644259 + 0.764808i $$0.722834\pi$$
$$884$$ 2.04028 0.0686221
$$885$$ 0 0
$$886$$ 13.0183 0.437357
$$887$$ −19.2809 −0.647389 −0.323695 0.946162i $$-0.604925\pi$$
−0.323695 + 0.946162i $$0.604925\pi$$
$$888$$ 31.6860 1.06331
$$889$$ −43.4920 −1.45868
$$890$$ 0 0
$$891$$ −5.91644 −0.198208
$$892$$ −7.84635 −0.262715
$$893$$ 10.8359 0.362609
$$894$$ 35.9530 1.20245
$$895$$ 0 0
$$896$$ −4.69527 −0.156858
$$897$$ −5.30507 −0.177131
$$898$$ −6.77359 −0.226038
$$899$$ −16.3096 −0.543957
$$900$$ 0 0
$$901$$ −16.6770 −0.555591
$$902$$ 7.34352 0.244512
$$903$$ −46.3537 −1.54255
$$904$$ 7.14585 0.237667
$$905$$ 0 0
$$906$$ 15.8538 0.526708
$$907$$ −43.0205 −1.42847 −0.714236 0.699905i $$-0.753226\pi$$
−0.714236 + 0.699905i $$0.753226\pi$$
$$908$$ −6.28646 −0.208624
$$909$$ 49.6636 1.64724
$$910$$ 0 0
$$911$$ −25.7733 −0.853906 −0.426953 0.904274i $$-0.640413\pi$$
−0.426953 + 0.904274i $$0.640413\pi$$
$$912$$ 2.77733 0.0919664
$$913$$ −104.109 −3.44550
$$914$$ −27.3189 −0.903630
$$915$$ 0 0
$$916$$ −3.14585 −0.103942
$$917$$ 54.0806 1.78590
$$918$$ −9.12758 −0.301255
$$919$$ −29.5897 −0.976074 −0.488037 0.872823i $$-0.662287\pi$$
−0.488037 + 0.872823i $$0.662287\pi$$
$$920$$ 0 0
$$921$$ 74.0037 2.43851
$$922$$ 27.7993 0.915522
$$923$$ 1.51811 0.0499691
$$924$$ −83.5726 −2.74934
$$925$$ 0 0
$$926$$ 38.2369 1.25654
$$927$$ 80.0440 2.62899
$$928$$ −2.93621 −0.0963859
$$929$$ −1.42334 −0.0466983 −0.0233491 0.999727i $$-0.507433\pi$$
−0.0233491 + 0.999727i $$0.507433\pi$$
$$930$$ 0 0
$$931$$ −15.0455 −0.493097
$$932$$ −0.182394 −0.00597451
$$933$$ −32.0765 −1.05014
$$934$$ 23.4711 0.767998
$$935$$ 0 0
$$936$$ 5.01420 0.163894
$$937$$ −28.2276 −0.922155 −0.461077 0.887360i $$-0.652537\pi$$
−0.461077 + 0.887360i $$0.652537\pi$$
$$938$$ −41.4256 −1.35259
$$939$$ −64.0556 −2.09038
$$940$$ 0 0
$$941$$ −15.8672 −0.517256 −0.258628 0.965977i $$-0.583270\pi$$
−0.258628 + 0.965977i $$0.583270\pi$$
$$942$$ −25.7262 −0.838206
$$943$$ 2.05748 0.0670009
$$944$$ −5.63148 −0.183289
$$945$$ 0 0
$$946$$ −22.7811 −0.740676
$$947$$ −43.6718 −1.41914 −0.709571 0.704634i $$-0.751111\pi$$
−0.709571 + 0.704634i $$0.751111\pi$$
$$948$$ 5.45315 0.177110
$$949$$ −13.4178 −0.435559
$$950$$ 0 0
$$951$$ −38.6692 −1.25393
$$952$$ 9.00523 0.291861
$$953$$ −16.0261 −0.519136 −0.259568 0.965725i $$-0.583580\pi$$
−0.259568 + 0.965725i $$0.583580\pi$$
$$954$$ −40.9855 −1.32695
$$955$$ 0 0
$$956$$ −11.5039 −0.372063
$$957$$ −52.2626 −1.68941
$$958$$ 19.6900 0.636156
$$959$$ −70.9176 −2.29005
$$960$$ 0 0
$$961$$ −0.145848 −0.00470478
$$962$$ 12.1365 0.391298
$$963$$ 8.43895 0.271941
$$964$$ −0.445349 −0.0143437
$$965$$ 0 0
$$966$$ −23.4151 −0.753369
$$967$$ 11.5987 0.372988 0.186494 0.982456i $$-0.440288\pi$$
0.186494 + 0.982456i $$0.440288\pi$$
$$968$$ −30.0728 −0.966575
$$969$$ −5.32674 −0.171120
$$970$$ 0 0
$$971$$ −27.0362 −0.867633 −0.433817 0.901001i $$-0.642833\pi$$
−0.433817 + 0.901001i $$0.642833\pi$$
$$972$$ 16.8411 0.540179
$$973$$ −3.28903 −0.105442
$$974$$ 16.3357 0.523430
$$975$$ 0 0
$$976$$ −3.39053 −0.108528
$$977$$ −14.1537 −0.452815 −0.226408 0.974033i $$-0.572698\pi$$
−0.226408 + 0.974033i $$0.572698\pi$$
$$978$$ −11.8904 −0.380212
$$979$$ −64.0880 −2.04826
$$980$$ 0 0
$$981$$ 11.3787 0.363293
$$982$$ −27.1093 −0.865093
$$983$$ 32.8542 1.04788 0.523942 0.851754i $$-0.324461\pi$$
0.523942 + 0.851754i $$0.324461\pi$$
$$984$$ −3.18239 −0.101451
$$985$$ 0 0
$$986$$ 5.63148 0.179343
$$987$$ 141.303 4.49772
$$988$$ 1.06379 0.0338436
$$989$$ −6.38273 −0.202959
$$990$$ 0 0
$$991$$ 47.9709 1.52385 0.761923 0.647667i $$-0.224255\pi$$
0.761923 + 0.647667i $$0.224255\pi$$
$$992$$ 5.55465 0.176360
$$993$$ 2.04285 0.0648279
$$994$$ 6.70050 0.212527
$$995$$ 0 0
$$996$$ 45.1168 1.42958
$$997$$ −44.1716 −1.39893 −0.699464 0.714668i $$-0.746578\pi$$
−0.699464 + 0.714668i $$0.746578\pi$$
$$998$$ 4.69003 0.148460
$$999$$ −54.2951 −1.71782
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 950.2.a.k.1.1 3
3.2 odd 2 8550.2.a.co.1.3 3
4.3 odd 2 7600.2.a.cb.1.3 3
5.2 odd 4 950.2.b.g.799.3 6
5.3 odd 4 950.2.b.g.799.4 6
5.4 even 2 950.2.a.m.1.3 yes 3
15.14 odd 2 8550.2.a.cj.1.1 3
20.19 odd 2 7600.2.a.bm.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 1.1 even 1 trivial
950.2.a.m.1.3 yes 3 5.4 even 2
950.2.b.g.799.3 6 5.2 odd 4
950.2.b.g.799.4 6 5.3 odd 4
7600.2.a.bm.1.1 3 20.19 odd 2
7600.2.a.cb.1.3 3 4.3 odd 2
8550.2.a.cj.1.1 3 15.14 odd 2
8550.2.a.co.1.3 3 3.2 odd 2