# Properties

 Label 1850.2.a.bb.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ 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] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 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 $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} -3.28726 q^{7} +1.00000 q^{8} +4.15633 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.67513 q^{3} +1.00000 q^{4} -2.67513 q^{6} -3.28726 q^{7} +1.00000 q^{8} +4.15633 q^{9} +4.83146 q^{11} -2.67513 q^{12} -5.67513 q^{13} -3.28726 q^{14} +1.00000 q^{16} +5.63752 q^{17} +4.15633 q^{18} +4.76845 q^{19} +8.79384 q^{21} +4.83146 q^{22} -2.38787 q^{23} -2.67513 q^{24} -5.67513 q^{26} -3.09332 q^{27} -3.28726 q^{28} -7.92478 q^{29} -7.44358 q^{31} +1.00000 q^{32} -12.9248 q^{33} +5.63752 q^{34} +4.15633 q^{36} -1.00000 q^{37} +4.76845 q^{38} +15.1817 q^{39} -10.3503 q^{41} +8.79384 q^{42} +11.4314 q^{43} +4.83146 q^{44} -2.38787 q^{46} +5.92478 q^{47} -2.67513 q^{48} +3.80606 q^{49} -15.0811 q^{51} -5.67513 q^{52} -9.53690 q^{53} -3.09332 q^{54} -3.28726 q^{56} -12.7562 q^{57} -7.92478 q^{58} -11.4314 q^{59} -4.89938 q^{61} -7.44358 q^{62} -13.6629 q^{63} +1.00000 q^{64} -12.9248 q^{66} -2.10062 q^{67} +5.63752 q^{68} +6.38787 q^{69} -4.89938 q^{71} +4.15633 q^{72} -10.1939 q^{73} -1.00000 q^{74} +4.76845 q^{76} -15.8822 q^{77} +15.1817 q^{78} +7.61213 q^{79} -4.19394 q^{81} -10.3503 q^{82} -13.9199 q^{83} +8.79384 q^{84} +11.4314 q^{86} +21.1998 q^{87} +4.83146 q^{88} +4.44358 q^{89} +18.6556 q^{91} -2.38787 q^{92} +19.9126 q^{93} +5.92478 q^{94} -2.67513 q^{96} -5.35026 q^{97} +3.80606 q^{98} +20.0811 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9} - q^{11} - 3 q^{12} - 12 q^{13} - 4 q^{14} + 3 q^{16} + q^{17} + 2 q^{18} + 3 q^{19} - q^{22} - 8 q^{23} - 3 q^{24} - 12 q^{26} - 3 q^{27} - 4 q^{28} - 2 q^{29} - 6 q^{31} + 3 q^{32} - 17 q^{33} + q^{34} + 2 q^{36} - 3 q^{37} + 3 q^{38} + 20 q^{39} - 21 q^{41} - 8 q^{43} - q^{44} - 8 q^{46} - 4 q^{47} - 3 q^{48} + 11 q^{49} - 13 q^{51} - 12 q^{52} - 6 q^{53} - 3 q^{54} - 4 q^{56} - q^{57} - 2 q^{58} + 8 q^{59} - 8 q^{61} - 6 q^{62} - 10 q^{63} + 3 q^{64} - 17 q^{66} - 13 q^{67} + q^{68} + 20 q^{69} - 8 q^{71} + 2 q^{72} - 31 q^{73} - 3 q^{74} + 3 q^{76} - 2 q^{77} + 20 q^{78} + 22 q^{79} - 13 q^{81} - 21 q^{82} - 7 q^{83} - 8 q^{86} + 10 q^{87} - q^{88} - 3 q^{89} + 12 q^{91} - 8 q^{92} + 12 q^{93} - 4 q^{94} - 3 q^{96} - 6 q^{97} + 11 q^{98} + 28 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 - q^11 - 3 * q^12 - 12 * q^13 - 4 * q^14 + 3 * q^16 + q^17 + 2 * q^18 + 3 * q^19 - q^22 - 8 * q^23 - 3 * q^24 - 12 * q^26 - 3 * q^27 - 4 * q^28 - 2 * q^29 - 6 * q^31 + 3 * q^32 - 17 * q^33 + q^34 + 2 * q^36 - 3 * q^37 + 3 * q^38 + 20 * q^39 - 21 * q^41 - 8 * q^43 - q^44 - 8 * q^46 - 4 * q^47 - 3 * q^48 + 11 * q^49 - 13 * q^51 - 12 * q^52 - 6 * q^53 - 3 * q^54 - 4 * q^56 - q^57 - 2 * q^58 + 8 * q^59 - 8 * q^61 - 6 * q^62 - 10 * q^63 + 3 * q^64 - 17 * q^66 - 13 * q^67 + q^68 + 20 * q^69 - 8 * q^71 + 2 * q^72 - 31 * q^73 - 3 * q^74 + 3 * q^76 - 2 * q^77 + 20 * q^78 + 22 * q^79 - 13 * q^81 - 21 * q^82 - 7 * q^83 - 8 * q^86 + 10 * q^87 - q^88 - 3 * q^89 + 12 * q^91 - 8 * q^92 + 12 * q^93 - 4 * q^94 - 3 * q^96 - 6 * q^97 + 11 * q^98 + 28 * 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
$$3$$ −2.67513 −1.54449 −0.772244 0.635326i $$-0.780866\pi$$
−0.772244 + 0.635326i $$0.780866\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −2.67513 −1.09212
$$7$$ −3.28726 −1.24247 −0.621233 0.783626i $$-0.713368\pi$$
−0.621233 + 0.783626i $$0.713368\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.15633 1.38544
$$10$$ 0 0
$$11$$ 4.83146 1.45674 0.728369 0.685185i $$-0.240279\pi$$
0.728369 + 0.685185i $$0.240279\pi$$
$$12$$ −2.67513 −0.772244
$$13$$ −5.67513 −1.57400 −0.786999 0.616954i $$-0.788366\pi$$
−0.786999 + 0.616954i $$0.788366\pi$$
$$14$$ −3.28726 −0.878557
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.63752 1.36730 0.683650 0.729810i $$-0.260392\pi$$
0.683650 + 0.729810i $$0.260392\pi$$
$$18$$ 4.15633 0.979655
$$19$$ 4.76845 1.09396 0.546979 0.837146i $$-0.315778\pi$$
0.546979 + 0.837146i $$0.315778\pi$$
$$20$$ 0 0
$$21$$ 8.79384 1.91897
$$22$$ 4.83146 1.03007
$$23$$ −2.38787 −0.497906 −0.248953 0.968516i $$-0.580086\pi$$
−0.248953 + 0.968516i $$0.580086\pi$$
$$24$$ −2.67513 −0.546059
$$25$$ 0 0
$$26$$ −5.67513 −1.11298
$$27$$ −3.09332 −0.595310
$$28$$ −3.28726 −0.621233
$$29$$ −7.92478 −1.47159 −0.735797 0.677202i $$-0.763192\pi$$
−0.735797 + 0.677202i $$0.763192\pi$$
$$30$$ 0 0
$$31$$ −7.44358 −1.33691 −0.668453 0.743754i $$-0.733043\pi$$
−0.668453 + 0.743754i $$0.733043\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −12.9248 −2.24991
$$34$$ 5.63752 0.966827
$$35$$ 0 0
$$36$$ 4.15633 0.692721
$$37$$ −1.00000 −0.164399
$$38$$ 4.76845 0.773545
$$39$$ 15.1817 2.43102
$$40$$ 0 0
$$41$$ −10.3503 −1.61644 −0.808220 0.588881i $$-0.799569\pi$$
−0.808220 + 0.588881i $$0.799569\pi$$
$$42$$ 8.79384 1.35692
$$43$$ 11.4314 1.74327 0.871633 0.490158i $$-0.163061\pi$$
0.871633 + 0.490158i $$0.163061\pi$$
$$44$$ 4.83146 0.728369
$$45$$ 0 0
$$46$$ −2.38787 −0.352073
$$47$$ 5.92478 0.864218 0.432109 0.901821i $$-0.357770\pi$$
0.432109 + 0.901821i $$0.357770\pi$$
$$48$$ −2.67513 −0.386122
$$49$$ 3.80606 0.543723
$$50$$ 0 0
$$51$$ −15.0811 −2.11178
$$52$$ −5.67513 −0.786999
$$53$$ −9.53690 −1.30999 −0.654997 0.755631i $$-0.727330\pi$$
−0.654997 + 0.755631i $$0.727330\pi$$
$$54$$ −3.09332 −0.420948
$$55$$ 0 0
$$56$$ −3.28726 −0.439278
$$57$$ −12.7562 −1.68960
$$58$$ −7.92478 −1.04057
$$59$$ −11.4314 −1.48824 −0.744118 0.668048i $$-0.767130\pi$$
−0.744118 + 0.668048i $$0.767130\pi$$
$$60$$ 0 0
$$61$$ −4.89938 −0.627302 −0.313651 0.949538i $$-0.601552\pi$$
−0.313651 + 0.949538i $$0.601552\pi$$
$$62$$ −7.44358 −0.945336
$$63$$ −13.6629 −1.72137
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −12.9248 −1.59093
$$67$$ −2.10062 −0.256631 −0.128316 0.991733i $$-0.540957\pi$$
−0.128316 + 0.991733i $$0.540957\pi$$
$$68$$ 5.63752 0.683650
$$69$$ 6.38787 0.769010
$$70$$ 0 0
$$71$$ −4.89938 −0.581450 −0.290725 0.956807i $$-0.593896\pi$$
−0.290725 + 0.956807i $$0.593896\pi$$
$$72$$ 4.15633 0.489828
$$73$$ −10.1939 −1.19311 −0.596555 0.802572i $$-0.703464\pi$$
−0.596555 + 0.802572i $$0.703464\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 4.76845 0.546979
$$77$$ −15.8822 −1.80995
$$78$$ 15.1817 1.71899
$$79$$ 7.61213 0.856431 0.428216 0.903677i $$-0.359142\pi$$
0.428216 + 0.903677i $$0.359142\pi$$
$$80$$ 0 0
$$81$$ −4.19394 −0.465993
$$82$$ −10.3503 −1.14300
$$83$$ −13.9199 −1.52790 −0.763951 0.645274i $$-0.776743\pi$$
−0.763951 + 0.645274i $$0.776743\pi$$
$$84$$ 8.79384 0.959487
$$85$$ 0 0
$$86$$ 11.4314 1.23268
$$87$$ 21.1998 2.27286
$$88$$ 4.83146 0.515035
$$89$$ 4.44358 0.471019 0.235509 0.971872i $$-0.424324\pi$$
0.235509 + 0.971872i $$0.424324\pi$$
$$90$$ 0 0
$$91$$ 18.6556 1.95564
$$92$$ −2.38787 −0.248953
$$93$$ 19.9126 2.06484
$$94$$ 5.92478 0.611094
$$95$$ 0 0
$$96$$ −2.67513 −0.273029
$$97$$ −5.35026 −0.543237 −0.271618 0.962405i $$-0.587559\pi$$
−0.271618 + 0.962405i $$0.587559\pi$$
$$98$$ 3.80606 0.384470
$$99$$ 20.0811 2.01823
$$100$$ 0 0
$$101$$ −8.12601 −0.808568 −0.404284 0.914634i $$-0.632479\pi$$
−0.404284 + 0.914634i $$0.632479\pi$$
$$102$$ −15.0811 −1.49325
$$103$$ 1.90668 0.187871 0.0939353 0.995578i $$-0.470055\pi$$
0.0939353 + 0.995578i $$0.470055\pi$$
$$104$$ −5.67513 −0.556492
$$105$$ 0 0
$$106$$ −9.53690 −0.926306
$$107$$ 2.41327 0.233299 0.116650 0.993173i $$-0.462785\pi$$
0.116650 + 0.993173i $$0.462785\pi$$
$$108$$ −3.09332 −0.297655
$$109$$ −1.73813 −0.166483 −0.0832416 0.996529i $$-0.526527\pi$$
−0.0832416 + 0.996529i $$0.526527\pi$$
$$110$$ 0 0
$$111$$ 2.67513 0.253912
$$112$$ −3.28726 −0.310617
$$113$$ 5.09332 0.479139 0.239570 0.970879i $$-0.422994\pi$$
0.239570 + 0.970879i $$0.422994\pi$$
$$114$$ −12.7562 −1.19473
$$115$$ 0 0
$$116$$ −7.92478 −0.735797
$$117$$ −23.5877 −2.18068
$$118$$ −11.4314 −1.05234
$$119$$ −18.5320 −1.69882
$$120$$ 0 0
$$121$$ 12.3430 1.12209
$$122$$ −4.89938 −0.443569
$$123$$ 27.6883 2.49657
$$124$$ −7.44358 −0.668453
$$125$$ 0 0
$$126$$ −13.6629 −1.21719
$$127$$ 7.59991 0.674383 0.337191 0.941436i $$-0.390523\pi$$
0.337191 + 0.941436i $$0.390523\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −30.5804 −2.69245
$$130$$ 0 0
$$131$$ 13.3806 1.16907 0.584533 0.811370i $$-0.301278\pi$$
0.584533 + 0.811370i $$0.301278\pi$$
$$132$$ −12.9248 −1.12496
$$133$$ −15.6751 −1.35921
$$134$$ −2.10062 −0.181466
$$135$$ 0 0
$$136$$ 5.63752 0.483413
$$137$$ −11.1866 −0.955739 −0.477870 0.878431i $$-0.658591\pi$$
−0.477870 + 0.878431i $$0.658591\pi$$
$$138$$ 6.38787 0.543772
$$139$$ −17.0689 −1.44776 −0.723882 0.689924i $$-0.757644\pi$$
−0.723882 + 0.689924i $$0.757644\pi$$
$$140$$ 0 0
$$141$$ −15.8496 −1.33477
$$142$$ −4.89938 −0.411147
$$143$$ −27.4191 −2.29290
$$144$$ 4.15633 0.346360
$$145$$ 0 0
$$146$$ −10.1939 −0.843656
$$147$$ −10.1817 −0.839774
$$148$$ −1.00000 −0.0821995
$$149$$ −7.56959 −0.620125 −0.310063 0.950716i $$-0.600350\pi$$
−0.310063 + 0.950716i $$0.600350\pi$$
$$150$$ 0 0
$$151$$ 0.649738 0.0528749 0.0264375 0.999650i $$-0.491584\pi$$
0.0264375 + 0.999650i $$0.491584\pi$$
$$152$$ 4.76845 0.386773
$$153$$ 23.4314 1.89431
$$154$$ −15.8822 −1.27983
$$155$$ 0 0
$$156$$ 15.1817 1.21551
$$157$$ 3.36836 0.268824 0.134412 0.990926i $$-0.457085\pi$$
0.134412 + 0.990926i $$0.457085\pi$$
$$158$$ 7.61213 0.605588
$$159$$ 25.5125 2.02327
$$160$$ 0 0
$$161$$ 7.84955 0.618632
$$162$$ −4.19394 −0.329507
$$163$$ 4.31994 0.338364 0.169182 0.985585i $$-0.445887\pi$$
0.169182 + 0.985585i $$0.445887\pi$$
$$164$$ −10.3503 −0.808220
$$165$$ 0 0
$$166$$ −13.9199 −1.08039
$$167$$ −14.6678 −1.13503 −0.567516 0.823363i $$-0.692095\pi$$
−0.567516 + 0.823363i $$0.692095\pi$$
$$168$$ 8.79384 0.678460
$$169$$ 19.2071 1.47747
$$170$$ 0 0
$$171$$ 19.8192 1.51561
$$172$$ 11.4314 0.871633
$$173$$ 8.05079 0.612090 0.306045 0.952017i $$-0.400994\pi$$
0.306045 + 0.952017i $$0.400994\pi$$
$$174$$ 21.1998 1.60715
$$175$$ 0 0
$$176$$ 4.83146 0.364185
$$177$$ 30.5804 2.29856
$$178$$ 4.44358 0.333061
$$179$$ −5.05079 −0.377513 −0.188757 0.982024i $$-0.560446\pi$$
−0.188757 + 0.982024i $$0.560446\pi$$
$$180$$ 0 0
$$181$$ 3.73813 0.277853 0.138927 0.990303i $$-0.455635\pi$$
0.138927 + 0.990303i $$0.455635\pi$$
$$182$$ 18.6556 1.38285
$$183$$ 13.1065 0.968860
$$184$$ −2.38787 −0.176036
$$185$$ 0 0
$$186$$ 19.9126 1.46006
$$187$$ 27.2374 1.99180
$$188$$ 5.92478 0.432109
$$189$$ 10.1685 0.739653
$$190$$ 0 0
$$191$$ −12.5501 −0.908092 −0.454046 0.890978i $$-0.650020\pi$$
−0.454046 + 0.890978i $$0.650020\pi$$
$$192$$ −2.67513 −0.193061
$$193$$ −20.8822 −1.50314 −0.751568 0.659655i $$-0.770702\pi$$
−0.751568 + 0.659655i $$0.770702\pi$$
$$194$$ −5.35026 −0.384126
$$195$$ 0 0
$$196$$ 3.80606 0.271862
$$197$$ 1.07030 0.0762556 0.0381278 0.999273i $$-0.487861\pi$$
0.0381278 + 0.999273i $$0.487861\pi$$
$$198$$ 20.0811 1.42710
$$199$$ −22.1016 −1.56674 −0.783369 0.621556i $$-0.786501\pi$$
−0.783369 + 0.621556i $$0.786501\pi$$
$$200$$ 0 0
$$201$$ 5.61942 0.396363
$$202$$ −8.12601 −0.571744
$$203$$ 26.0508 1.82841
$$204$$ −15.0811 −1.05589
$$205$$ 0 0
$$206$$ 1.90668 0.132845
$$207$$ −9.92478 −0.689820
$$208$$ −5.67513 −0.393500
$$209$$ 23.0386 1.59361
$$210$$ 0 0
$$211$$ −16.3987 −1.12893 −0.564466 0.825456i $$-0.690918\pi$$
−0.564466 + 0.825456i $$0.690918\pi$$
$$212$$ −9.53690 −0.654997
$$213$$ 13.1065 0.898042
$$214$$ 2.41327 0.164967
$$215$$ 0 0
$$216$$ −3.09332 −0.210474
$$217$$ 24.4690 1.66106
$$218$$ −1.73813 −0.117721
$$219$$ 27.2701 1.84274
$$220$$ 0 0
$$221$$ −31.9937 −2.15213
$$222$$ 2.67513 0.179543
$$223$$ −18.4387 −1.23474 −0.617372 0.786671i $$-0.711803\pi$$
−0.617372 + 0.786671i $$0.711803\pi$$
$$224$$ −3.28726 −0.219639
$$225$$ 0 0
$$226$$ 5.09332 0.338803
$$227$$ 11.4558 0.760348 0.380174 0.924915i $$-0.375864\pi$$
0.380174 + 0.924915i $$0.375864\pi$$
$$228$$ −12.7562 −0.844802
$$229$$ 25.6385 1.69424 0.847119 0.531403i $$-0.178335\pi$$
0.847119 + 0.531403i $$0.178335\pi$$
$$230$$ 0 0
$$231$$ 42.4871 2.79544
$$232$$ −7.92478 −0.520287
$$233$$ −9.76845 −0.639953 −0.319976 0.947426i $$-0.603675\pi$$
−0.319976 + 0.947426i $$0.603675\pi$$
$$234$$ −23.5877 −1.54198
$$235$$ 0 0
$$236$$ −11.4314 −0.744118
$$237$$ −20.3634 −1.32275
$$238$$ −18.5320 −1.20125
$$239$$ 16.2882 1.05360 0.526798 0.849990i $$-0.323392\pi$$
0.526798 + 0.849990i $$0.323392\pi$$
$$240$$ 0 0
$$241$$ 13.1744 0.848639 0.424320 0.905512i $$-0.360513\pi$$
0.424320 + 0.905512i $$0.360513\pi$$
$$242$$ 12.3430 0.793436
$$243$$ 20.4993 1.31503
$$244$$ −4.89938 −0.313651
$$245$$ 0 0
$$246$$ 27.6883 1.76534
$$247$$ −27.0616 −1.72189
$$248$$ −7.44358 −0.472668
$$249$$ 37.2374 2.35983
$$250$$ 0 0
$$251$$ 13.0508 0.823758 0.411879 0.911238i $$-0.364873\pi$$
0.411879 + 0.911238i $$0.364873\pi$$
$$252$$ −13.6629 −0.860683
$$253$$ −11.5369 −0.725319
$$254$$ 7.59991 0.476861
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −20.7005 −1.29126 −0.645632 0.763649i $$-0.723406\pi$$
−0.645632 + 0.763649i $$0.723406\pi$$
$$258$$ −30.5804 −1.90385
$$259$$ 3.28726 0.204260
$$260$$ 0 0
$$261$$ −32.9380 −2.03881
$$262$$ 13.3806 0.826655
$$263$$ 20.4387 1.26030 0.630151 0.776473i $$-0.282993\pi$$
0.630151 + 0.776473i $$0.282993\pi$$
$$264$$ −12.9248 −0.795465
$$265$$ 0 0
$$266$$ −15.6751 −0.961104
$$267$$ −11.8872 −0.727483
$$268$$ −2.10062 −0.128316
$$269$$ 6.25202 0.381192 0.190596 0.981669i $$-0.438958\pi$$
0.190596 + 0.981669i $$0.438958\pi$$
$$270$$ 0 0
$$271$$ −0.824162 −0.0500643 −0.0250321 0.999687i $$-0.507969\pi$$
−0.0250321 + 0.999687i $$0.507969\pi$$
$$272$$ 5.63752 0.341825
$$273$$ −49.9062 −3.02046
$$274$$ −11.1866 −0.675810
$$275$$ 0 0
$$276$$ 6.38787 0.384505
$$277$$ 18.1866 1.09273 0.546365 0.837547i $$-0.316011\pi$$
0.546365 + 0.837547i $$0.316011\pi$$
$$278$$ −17.0689 −1.02372
$$279$$ −30.9380 −1.85221
$$280$$ 0 0
$$281$$ 25.0435 1.49397 0.746985 0.664841i $$-0.231501\pi$$
0.746985 + 0.664841i $$0.231501\pi$$
$$282$$ −15.8496 −0.943827
$$283$$ −13.1055 −0.779043 −0.389522 0.921017i $$-0.627360\pi$$
−0.389522 + 0.921017i $$0.627360\pi$$
$$284$$ −4.89938 −0.290725
$$285$$ 0 0
$$286$$ −27.4191 −1.62133
$$287$$ 34.0240 2.00837
$$288$$ 4.15633 0.244914
$$289$$ 14.7816 0.869507
$$290$$ 0 0
$$291$$ 14.3127 0.839022
$$292$$ −10.1939 −0.596555
$$293$$ −9.68101 −0.565571 −0.282785 0.959183i $$-0.591258\pi$$
−0.282785 + 0.959183i $$0.591258\pi$$
$$294$$ −10.1817 −0.593810
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −14.9452 −0.867211
$$298$$ −7.56959 −0.438495
$$299$$ 13.5515 0.783703
$$300$$ 0 0
$$301$$ −37.5778 −2.16595
$$302$$ 0.649738 0.0373882
$$303$$ 21.7381 1.24882
$$304$$ 4.76845 0.273489
$$305$$ 0 0
$$306$$ 23.4314 1.33948
$$307$$ 20.4337 1.16621 0.583107 0.812395i $$-0.301837\pi$$
0.583107 + 0.812395i $$0.301837\pi$$
$$308$$ −15.8822 −0.904975
$$309$$ −5.10062 −0.290164
$$310$$ 0 0
$$311$$ 10.5320 0.597214 0.298607 0.954376i $$-0.403478\pi$$
0.298607 + 0.954376i $$0.403478\pi$$
$$312$$ 15.1817 0.859496
$$313$$ −9.96968 −0.563520 −0.281760 0.959485i $$-0.590918\pi$$
−0.281760 + 0.959485i $$0.590918\pi$$
$$314$$ 3.36836 0.190088
$$315$$ 0 0
$$316$$ 7.61213 0.428216
$$317$$ 24.3961 1.37022 0.685111 0.728438i $$-0.259754\pi$$
0.685111 + 0.728438i $$0.259754\pi$$
$$318$$ 25.5125 1.43067
$$319$$ −38.2882 −2.14373
$$320$$ 0 0
$$321$$ −6.45580 −0.360328
$$322$$ 7.84955 0.437439
$$323$$ 26.8822 1.49577
$$324$$ −4.19394 −0.232996
$$325$$ 0 0
$$326$$ 4.31994 0.239260
$$327$$ 4.64974 0.257131
$$328$$ −10.3503 −0.571498
$$329$$ −19.4763 −1.07376
$$330$$ 0 0
$$331$$ 34.3561 1.88838 0.944192 0.329395i $$-0.106845\pi$$
0.944192 + 0.329395i $$0.106845\pi$$
$$332$$ −13.9199 −0.763951
$$333$$ −4.15633 −0.227765
$$334$$ −14.6678 −0.802588
$$335$$ 0 0
$$336$$ 8.79384 0.479744
$$337$$ −20.6385 −1.12425 −0.562125 0.827053i $$-0.690016\pi$$
−0.562125 + 0.827053i $$0.690016\pi$$
$$338$$ 19.2071 1.04473
$$339$$ −13.6253 −0.740025
$$340$$ 0 0
$$341$$ −35.9633 −1.94752
$$342$$ 19.8192 1.07170
$$343$$ 10.4993 0.566909
$$344$$ 11.4314 0.616338
$$345$$ 0 0
$$346$$ 8.05079 0.432813
$$347$$ −4.48612 −0.240827 −0.120414 0.992724i $$-0.538422\pi$$
−0.120414 + 0.992724i $$0.538422\pi$$
$$348$$ 21.1998 1.13643
$$349$$ −20.1949 −1.08101 −0.540504 0.841342i $$-0.681766\pi$$
−0.540504 + 0.841342i $$0.681766\pi$$
$$350$$ 0 0
$$351$$ 17.5550 0.937017
$$352$$ 4.83146 0.257517
$$353$$ 25.9102 1.37906 0.689530 0.724257i $$-0.257817\pi$$
0.689530 + 0.724257i $$0.257817\pi$$
$$354$$ 30.5804 1.62533
$$355$$ 0 0
$$356$$ 4.44358 0.235509
$$357$$ 49.5755 2.62381
$$358$$ −5.05079 −0.266942
$$359$$ 4.43866 0.234263 0.117132 0.993116i $$-0.462630\pi$$
0.117132 + 0.993116i $$0.462630\pi$$
$$360$$ 0 0
$$361$$ 3.73813 0.196744
$$362$$ 3.73813 0.196472
$$363$$ −33.0191 −1.73305
$$364$$ 18.6556 0.977820
$$365$$ 0 0
$$366$$ 13.1065 0.685087
$$367$$ −28.2858 −1.47651 −0.738254 0.674522i $$-0.764350\pi$$
−0.738254 + 0.674522i $$0.764350\pi$$
$$368$$ −2.38787 −0.124476
$$369$$ −43.0191 −2.23948
$$370$$ 0 0
$$371$$ 31.3503 1.62762
$$372$$ 19.9126 1.03242
$$373$$ 2.71862 0.140765 0.0703825 0.997520i $$-0.477578\pi$$
0.0703825 + 0.997520i $$0.477578\pi$$
$$374$$ 27.2374 1.40841
$$375$$ 0 0
$$376$$ 5.92478 0.305547
$$377$$ 44.9741 2.31629
$$378$$ 10.1685 0.523013
$$379$$ 18.1514 0.932375 0.466187 0.884686i $$-0.345627\pi$$
0.466187 + 0.884686i $$0.345627\pi$$
$$380$$ 0 0
$$381$$ −20.3307 −1.04158
$$382$$ −12.5501 −0.642118
$$383$$ 3.90668 0.199622 0.0998110 0.995006i $$-0.468176\pi$$
0.0998110 + 0.995006i $$0.468176\pi$$
$$384$$ −2.67513 −0.136515
$$385$$ 0 0
$$386$$ −20.8822 −1.06288
$$387$$ 47.5125 2.41519
$$388$$ −5.35026 −0.271618
$$389$$ 5.28726 0.268075 0.134037 0.990976i $$-0.457206\pi$$
0.134037 + 0.990976i $$0.457206\pi$$
$$390$$ 0 0
$$391$$ −13.4617 −0.680786
$$392$$ 3.80606 0.192235
$$393$$ −35.7948 −1.80561
$$394$$ 1.07030 0.0539208
$$395$$ 0 0
$$396$$ 20.0811 1.00911
$$397$$ −4.15396 −0.208481 −0.104241 0.994552i $$-0.533241\pi$$
−0.104241 + 0.994552i $$0.533241\pi$$
$$398$$ −22.1016 −1.10785
$$399$$ 41.9330 2.09928
$$400$$ 0 0
$$401$$ −22.1817 −1.10770 −0.553851 0.832616i $$-0.686842\pi$$
−0.553851 + 0.832616i $$0.686842\pi$$
$$402$$ 5.61942 0.280271
$$403$$ 42.2433 2.10429
$$404$$ −8.12601 −0.404284
$$405$$ 0 0
$$406$$ 26.0508 1.29288
$$407$$ −4.83146 −0.239486
$$408$$ −15.0811 −0.746626
$$409$$ 0.650693 0.0321747 0.0160874 0.999871i $$-0.494879\pi$$
0.0160874 + 0.999871i $$0.494879\pi$$
$$410$$ 0 0
$$411$$ 29.9257 1.47613
$$412$$ 1.90668 0.0939353
$$413$$ 37.5778 1.84908
$$414$$ −9.92478 −0.487776
$$415$$ 0 0
$$416$$ −5.67513 −0.278246
$$417$$ 45.6615 2.23605
$$418$$ 23.0386 1.12685
$$419$$ 10.8169 0.528439 0.264219 0.964463i $$-0.414886\pi$$
0.264219 + 0.964463i $$0.414886\pi$$
$$420$$ 0 0
$$421$$ 2.05079 0.0999492 0.0499746 0.998750i $$-0.484086\pi$$
0.0499746 + 0.998750i $$0.484086\pi$$
$$422$$ −16.3987 −0.798275
$$423$$ 24.6253 1.19732
$$424$$ −9.53690 −0.463153
$$425$$ 0 0
$$426$$ 13.1065 0.635012
$$427$$ 16.1055 0.779402
$$428$$ 2.41327 0.116650
$$429$$ 73.3498 3.54136
$$430$$ 0 0
$$431$$ 15.4861 0.745940 0.372970 0.927843i $$-0.378339\pi$$
0.372970 + 0.927843i $$0.378339\pi$$
$$432$$ −3.09332 −0.148827
$$433$$ −35.8773 −1.72415 −0.862077 0.506777i $$-0.830837\pi$$
−0.862077 + 0.506777i $$0.830837\pi$$
$$434$$ 24.4690 1.17455
$$435$$ 0 0
$$436$$ −1.73813 −0.0832416
$$437$$ −11.3865 −0.544688
$$438$$ 27.2701 1.30302
$$439$$ −26.7005 −1.27435 −0.637173 0.770721i $$-0.719896\pi$$
−0.637173 + 0.770721i $$0.719896\pi$$
$$440$$ 0 0
$$441$$ 15.8192 0.753297
$$442$$ −31.9937 −1.52178
$$443$$ −5.75765 −0.273554 −0.136777 0.990602i $$-0.543674\pi$$
−0.136777 + 0.990602i $$0.543674\pi$$
$$444$$ 2.67513 0.126956
$$445$$ 0 0
$$446$$ −18.4387 −0.873096
$$447$$ 20.2496 0.957775
$$448$$ −3.28726 −0.155308
$$449$$ −31.1803 −1.47149 −0.735745 0.677259i $$-0.763168\pi$$
−0.735745 + 0.677259i $$0.763168\pi$$
$$450$$ 0 0
$$451$$ −50.0068 −2.35473
$$452$$ 5.09332 0.239570
$$453$$ −1.73813 −0.0816647
$$454$$ 11.4558 0.537647
$$455$$ 0 0
$$456$$ −12.7562 −0.597365
$$457$$ −7.27011 −0.340082 −0.170041 0.985437i $$-0.554390\pi$$
−0.170041 + 0.985437i $$0.554390\pi$$
$$458$$ 25.6385 1.19801
$$459$$ −17.4387 −0.813967
$$460$$ 0 0
$$461$$ −18.6253 −0.867467 −0.433733 0.901041i $$-0.642804\pi$$
−0.433733 + 0.901041i $$0.642804\pi$$
$$462$$ 42.4871 1.97668
$$463$$ −1.89209 −0.0879329 −0.0439664 0.999033i $$-0.513999\pi$$
−0.0439664 + 0.999033i $$0.513999\pi$$
$$464$$ −7.92478 −0.367899
$$465$$ 0 0
$$466$$ −9.76845 −0.452515
$$467$$ −1.21440 −0.0561959 −0.0280980 0.999605i $$-0.508945\pi$$
−0.0280980 + 0.999605i $$0.508945\pi$$
$$468$$ −23.5877 −1.09034
$$469$$ 6.90526 0.318855
$$470$$ 0 0
$$471$$ −9.01080 −0.415196
$$472$$ −11.4314 −0.526171
$$473$$ 55.2301 2.53948
$$474$$ −20.3634 −0.935324
$$475$$ 0 0
$$476$$ −18.5320 −0.849412
$$477$$ −39.6385 −1.81492
$$478$$ 16.2882 0.745006
$$479$$ −40.7694 −1.86280 −0.931401 0.363995i $$-0.881412\pi$$
−0.931401 + 0.363995i $$0.881412\pi$$
$$480$$ 0 0
$$481$$ 5.67513 0.258764
$$482$$ 13.1744 0.600079
$$483$$ −20.9986 −0.955469
$$484$$ 12.3430 0.561044
$$485$$ 0 0
$$486$$ 20.4993 0.929867
$$487$$ 23.8578 1.08110 0.540550 0.841312i $$-0.318216\pi$$
0.540550 + 0.841312i $$0.318216\pi$$
$$488$$ −4.89938 −0.221785
$$489$$ −11.5564 −0.522599
$$490$$ 0 0
$$491$$ 36.0362 1.62629 0.813145 0.582061i $$-0.197753\pi$$
0.813145 + 0.582061i $$0.197753\pi$$
$$492$$ 27.6883 1.24829
$$493$$ −44.6761 −2.01211
$$494$$ −27.0616 −1.21756
$$495$$ 0 0
$$496$$ −7.44358 −0.334227
$$497$$ 16.1055 0.722432
$$498$$ 37.2374 1.66865
$$499$$ −31.4558 −1.40816 −0.704078 0.710123i $$-0.748639\pi$$
−0.704078 + 0.710123i $$0.748639\pi$$
$$500$$ 0 0
$$501$$ 39.2384 1.75304
$$502$$ 13.0508 0.582485
$$503$$ −3.07381 −0.137054 −0.0685272 0.997649i $$-0.521830\pi$$
−0.0685272 + 0.997649i $$0.521830\pi$$
$$504$$ −13.6629 −0.608594
$$505$$ 0 0
$$506$$ −11.5369 −0.512878
$$507$$ −51.3815 −2.28193
$$508$$ 7.59991 0.337191
$$509$$ −3.01951 −0.133838 −0.0669188 0.997758i $$-0.521317\pi$$
−0.0669188 + 0.997758i $$0.521317\pi$$
$$510$$ 0 0
$$511$$ 33.5101 1.48240
$$512$$ 1.00000 0.0441942
$$513$$ −14.7504 −0.651244
$$514$$ −20.7005 −0.913061
$$515$$ 0 0
$$516$$ −30.5804 −1.34623
$$517$$ 28.6253 1.25894
$$518$$ 3.28726 0.144434
$$519$$ −21.5369 −0.945365
$$520$$ 0 0
$$521$$ 28.8700 1.26482 0.632409 0.774634i $$-0.282066\pi$$
0.632409 + 0.774634i $$0.282066\pi$$
$$522$$ −32.9380 −1.44165
$$523$$ −3.37328 −0.147503 −0.0737517 0.997277i $$-0.523497\pi$$
−0.0737517 + 0.997277i $$0.523497\pi$$
$$524$$ 13.3806 0.584533
$$525$$ 0 0
$$526$$ 20.4387 0.891168
$$527$$ −41.9633 −1.82795
$$528$$ −12.9248 −0.562479
$$529$$ −17.2981 −0.752090
$$530$$ 0 0
$$531$$ −47.5125 −2.06187
$$532$$ −15.6751 −0.679603
$$533$$ 58.7391 2.54427
$$534$$ −11.8872 −0.514408
$$535$$ 0 0
$$536$$ −2.10062 −0.0907328
$$537$$ 13.5115 0.583065
$$538$$ 6.25202 0.269544
$$539$$ 18.3888 0.792063
$$540$$ 0 0
$$541$$ 10.3004 0.442850 0.221425 0.975177i $$-0.428929\pi$$
0.221425 + 0.975177i $$0.428929\pi$$
$$542$$ −0.824162 −0.0354008
$$543$$ −10.0000 −0.429141
$$544$$ 5.63752 0.241707
$$545$$ 0 0
$$546$$ −49.9062 −2.13579
$$547$$ −15.3576 −0.656642 −0.328321 0.944566i $$-0.606483\pi$$
−0.328321 + 0.944566i $$0.606483\pi$$
$$548$$ −11.1866 −0.477870
$$549$$ −20.3634 −0.869090
$$550$$ 0 0
$$551$$ −37.7889 −1.60986
$$552$$ 6.38787 0.271886
$$553$$ −25.0230 −1.06409
$$554$$ 18.1866 0.772676
$$555$$ 0 0
$$556$$ −17.0689 −0.723882
$$557$$ −14.8119 −0.627602 −0.313801 0.949489i $$-0.601603\pi$$
−0.313801 + 0.949489i $$0.601603\pi$$
$$558$$ −30.9380 −1.30971
$$559$$ −64.8745 −2.74390
$$560$$ 0 0
$$561$$ −72.8637 −3.07631
$$562$$ 25.0435 1.05640
$$563$$ −3.24472 −0.136749 −0.0683744 0.997660i $$-0.521781\pi$$
−0.0683744 + 0.997660i $$0.521781\pi$$
$$564$$ −15.8496 −0.667387
$$565$$ 0 0
$$566$$ −13.1055 −0.550867
$$567$$ 13.7866 0.578981
$$568$$ −4.89938 −0.205574
$$569$$ −17.1587 −0.719330 −0.359665 0.933082i $$-0.617109\pi$$
−0.359665 + 0.933082i $$0.617109\pi$$
$$570$$ 0 0
$$571$$ 0.0244376 0.00102268 0.000511340 1.00000i $$-0.499837\pi$$
0.000511340 1.00000i $$0.499837\pi$$
$$572$$ −27.4191 −1.14645
$$573$$ 33.5731 1.40254
$$574$$ 34.0240 1.42013
$$575$$ 0 0
$$576$$ 4.15633 0.173180
$$577$$ 34.5002 1.43626 0.718132 0.695907i $$-0.244997\pi$$
0.718132 + 0.695907i $$0.244997\pi$$
$$578$$ 14.7816 0.614835
$$579$$ 55.8627 2.32158
$$580$$ 0 0
$$581$$ 45.7581 1.89837
$$582$$ 14.3127 0.593278
$$583$$ −46.0771 −1.90832
$$584$$ −10.1939 −0.421828
$$585$$ 0 0
$$586$$ −9.68101 −0.399919
$$587$$ 4.84367 0.199920 0.0999599 0.994991i $$-0.468129\pi$$
0.0999599 + 0.994991i $$0.468129\pi$$
$$588$$ −10.1817 −0.419887
$$589$$ −35.4944 −1.46252
$$590$$ 0 0
$$591$$ −2.86319 −0.117776
$$592$$ −1.00000 −0.0410997
$$593$$ −31.0263 −1.27410 −0.637050 0.770823i $$-0.719845\pi$$
−0.637050 + 0.770823i $$0.719845\pi$$
$$594$$ −14.9452 −0.613211
$$595$$ 0 0
$$596$$ −7.56959 −0.310063
$$597$$ 59.1246 2.41981
$$598$$ 13.5515 0.554162
$$599$$ −32.0240 −1.30846 −0.654232 0.756294i $$-0.727008\pi$$
−0.654232 + 0.756294i $$0.727008\pi$$
$$600$$ 0 0
$$601$$ −9.14174 −0.372899 −0.186450 0.982465i $$-0.559698\pi$$
−0.186450 + 0.982465i $$0.559698\pi$$
$$602$$ −37.5778 −1.53156
$$603$$ −8.73084 −0.355547
$$604$$ 0.649738 0.0264375
$$605$$ 0 0
$$606$$ 21.7381 0.883051
$$607$$ −36.8691 −1.49647 −0.748235 0.663434i $$-0.769098\pi$$
−0.748235 + 0.663434i $$0.769098\pi$$
$$608$$ 4.76845 0.193386
$$609$$ −69.6893 −2.82395
$$610$$ 0 0
$$611$$ −33.6239 −1.36028
$$612$$ 23.4314 0.947157
$$613$$ 27.9902 1.13051 0.565256 0.824916i $$-0.308777\pi$$
0.565256 + 0.824916i $$0.308777\pi$$
$$614$$ 20.4337 0.824638
$$615$$ 0 0
$$616$$ −15.8822 −0.639914
$$617$$ 22.5804 0.909052 0.454526 0.890733i $$-0.349809\pi$$
0.454526 + 0.890733i $$0.349809\pi$$
$$618$$ −5.10062 −0.205177
$$619$$ 40.0362 1.60919 0.804595 0.593824i $$-0.202382\pi$$
0.804595 + 0.593824i $$0.202382\pi$$
$$620$$ 0 0
$$621$$ 7.38646 0.296408
$$622$$ 10.5320 0.422294
$$623$$ −14.6072 −0.585225
$$624$$ 15.1817 0.607755
$$625$$ 0 0
$$626$$ −9.96968 −0.398469
$$627$$ −61.6312 −2.46131
$$628$$ 3.36836 0.134412
$$629$$ −5.63752 −0.224783
$$630$$ 0 0
$$631$$ 26.1768 1.04208 0.521041 0.853532i $$-0.325544\pi$$
0.521041 + 0.853532i $$0.325544\pi$$
$$632$$ 7.61213 0.302794
$$633$$ 43.8686 1.74362
$$634$$ 24.3961 0.968894
$$635$$ 0 0
$$636$$ 25.5125 1.01164
$$637$$ −21.5999 −0.855820
$$638$$ −38.2882 −1.51584
$$639$$ −20.3634 −0.805565
$$640$$ 0 0
$$641$$ 20.9525 0.827576 0.413788 0.910373i $$-0.364206\pi$$
0.413788 + 0.910373i $$0.364206\pi$$
$$642$$ −6.45580 −0.254790
$$643$$ −17.7586 −0.700331 −0.350165 0.936688i $$-0.613875\pi$$
−0.350165 + 0.936688i $$0.613875\pi$$
$$644$$ 7.84955 0.309316
$$645$$ 0 0
$$646$$ 26.8822 1.05767
$$647$$ 11.5550 0.454274 0.227137 0.973863i $$-0.427063\pi$$
0.227137 + 0.973863i $$0.427063\pi$$
$$648$$ −4.19394 −0.164753
$$649$$ −55.2301 −2.16797
$$650$$ 0 0
$$651$$ −65.4577 −2.56549
$$652$$ 4.31994 0.169182
$$653$$ −7.78655 −0.304711 −0.152356 0.988326i $$-0.548686\pi$$
−0.152356 + 0.988326i $$0.548686\pi$$
$$654$$ 4.64974 0.181819
$$655$$ 0 0
$$656$$ −10.3503 −0.404110
$$657$$ −42.3693 −1.65298
$$658$$ −19.4763 −0.759264
$$659$$ −13.7029 −0.533789 −0.266894 0.963726i $$-0.585998\pi$$
−0.266894 + 0.963726i $$0.585998\pi$$
$$660$$ 0 0
$$661$$ 42.0263 1.63464 0.817318 0.576187i $$-0.195460\pi$$
0.817318 + 0.576187i $$0.195460\pi$$
$$662$$ 34.3561 1.33529
$$663$$ 85.5872 3.32393
$$664$$ −13.9199 −0.540195
$$665$$ 0 0
$$666$$ −4.15633 −0.161054
$$667$$ 18.9234 0.732716
$$668$$ −14.6678 −0.567516
$$669$$ 49.3258 1.90705
$$670$$ 0 0
$$671$$ −23.6712 −0.913815
$$672$$ 8.79384 0.339230
$$673$$ 29.0943 1.12150 0.560751 0.827985i $$-0.310513\pi$$
0.560751 + 0.827985i $$0.310513\pi$$
$$674$$ −20.6385 −0.794964
$$675$$ 0 0
$$676$$ 19.2071 0.738735
$$677$$ −25.3112 −0.972790 −0.486395 0.873739i $$-0.661688\pi$$
−0.486395 + 0.873739i $$0.661688\pi$$
$$678$$ −13.6253 −0.523277
$$679$$ 17.5877 0.674954
$$680$$ 0 0
$$681$$ −30.6458 −1.17435
$$682$$ −35.9633 −1.37711
$$683$$ −4.04349 −0.154720 −0.0773599 0.997003i $$-0.524649\pi$$
−0.0773599 + 0.997003i $$0.524649\pi$$
$$684$$ 19.8192 0.757807
$$685$$ 0 0
$$686$$ 10.4993 0.400865
$$687$$ −68.5863 −2.61673
$$688$$ 11.4314 0.435817
$$689$$ 54.1232 2.06193
$$690$$ 0 0
$$691$$ 7.92970 0.301660 0.150830 0.988560i $$-0.451805\pi$$
0.150830 + 0.988560i $$0.451805\pi$$
$$692$$ 8.05079 0.306045
$$693$$ −66.0118 −2.50758
$$694$$ −4.48612 −0.170291
$$695$$ 0 0
$$696$$ 21.1998 0.803577
$$697$$ −58.3498 −2.21016
$$698$$ −20.1949 −0.764388
$$699$$ 26.1319 0.988399
$$700$$ 0 0
$$701$$ −48.3611 −1.82657 −0.913286 0.407319i $$-0.866464\pi$$
−0.913286 + 0.407319i $$0.866464\pi$$
$$702$$ 17.5550 0.662571
$$703$$ −4.76845 −0.179846
$$704$$ 4.83146 0.182092
$$705$$ 0 0
$$706$$ 25.9102 0.975143
$$707$$ 26.7123 1.00462
$$708$$ 30.5804 1.14928
$$709$$ 7.45183 0.279859 0.139930 0.990161i $$-0.455312\pi$$
0.139930 + 0.990161i $$0.455312\pi$$
$$710$$ 0 0
$$711$$ 31.6385 1.18654
$$712$$ 4.44358 0.166530
$$713$$ 17.7743 0.665654
$$714$$ 49.5755 1.85532
$$715$$ 0 0
$$716$$ −5.05079 −0.188757
$$717$$ −43.5731 −1.62727
$$718$$ 4.43866 0.165649
$$719$$ −6.45088 −0.240577 −0.120289 0.992739i $$-0.538382\pi$$
−0.120289 + 0.992739i $$0.538382\pi$$
$$720$$ 0 0
$$721$$ −6.26774 −0.233423
$$722$$ 3.73813 0.139119
$$723$$ −35.2433 −1.31071
$$724$$ 3.73813 0.138927
$$725$$ 0 0
$$726$$ −33.0191 −1.22545
$$727$$ −37.1754 −1.37876 −0.689379 0.724401i $$-0.742117\pi$$
−0.689379 + 0.724401i $$0.742117\pi$$
$$728$$ 18.6556 0.691423
$$729$$ −42.2565 −1.56505
$$730$$ 0 0
$$731$$ 64.4445 2.38357
$$732$$ 13.1065 0.484430
$$733$$ 37.1754 1.37310 0.686552 0.727081i $$-0.259123\pi$$
0.686552 + 0.727081i $$0.259123\pi$$
$$734$$ −28.2858 −1.04405
$$735$$ 0 0
$$736$$ −2.38787 −0.0880182
$$737$$ −10.1490 −0.373844
$$738$$ −43.0191 −1.58355
$$739$$ 6.74940 0.248281 0.124140 0.992265i $$-0.460383\pi$$
0.124140 + 0.992265i $$0.460383\pi$$
$$740$$ 0 0
$$741$$ 72.3933 2.65943
$$742$$ 31.3503 1.15090
$$743$$ 23.9248 0.877715 0.438857 0.898557i $$-0.355383\pi$$
0.438857 + 0.898557i $$0.355383\pi$$
$$744$$ 19.9126 0.730030
$$745$$ 0 0
$$746$$ 2.71862 0.0995358
$$747$$ −57.8554 −2.11682
$$748$$ 27.2374 0.995899
$$749$$ −7.93303 −0.289866
$$750$$ 0 0
$$751$$ −5.27504 −0.192489 −0.0962445 0.995358i $$-0.530683\pi$$
−0.0962445 + 0.995358i $$0.530683\pi$$
$$752$$ 5.92478 0.216054
$$753$$ −34.9126 −1.27228
$$754$$ 44.9741 1.63786
$$755$$ 0 0
$$756$$ 10.1685 0.369826
$$757$$ −3.62672 −0.131815 −0.0659076 0.997826i $$-0.520994\pi$$
−0.0659076 + 0.997826i $$0.520994\pi$$
$$758$$ 18.1514 0.659289
$$759$$ 30.8627 1.12025
$$760$$ 0 0
$$761$$ −4.03173 −0.146150 −0.0730751 0.997326i $$-0.523281\pi$$
−0.0730751 + 0.997326i $$0.523281\pi$$
$$762$$ −20.3307 −0.736505
$$763$$ 5.71370 0.206850
$$764$$ −12.5501 −0.454046
$$765$$ 0 0
$$766$$ 3.90668 0.141154
$$767$$ 64.8745 2.34248
$$768$$ −2.67513 −0.0965305
$$769$$ −27.6072 −0.995541 −0.497771 0.867309i $$-0.665848\pi$$
−0.497771 + 0.867309i $$0.665848\pi$$
$$770$$ 0 0
$$771$$ 55.3766 1.99434
$$772$$ −20.8822 −0.751568
$$773$$ −19.4798 −0.700639 −0.350319 0.936630i $$-0.613927\pi$$
−0.350319 + 0.936630i $$0.613927\pi$$
$$774$$ 47.5125 1.70780
$$775$$ 0 0
$$776$$ −5.35026 −0.192063
$$777$$ −8.79384 −0.315477
$$778$$ 5.28726 0.189557
$$779$$ −49.3547 −1.76832
$$780$$ 0 0
$$781$$ −23.6712 −0.847021
$$782$$ −13.4617 −0.481389
$$783$$ 24.5139 0.876055
$$784$$ 3.80606 0.135931
$$785$$ 0 0
$$786$$ −35.7948 −1.27676
$$787$$ 14.1768 0.505348 0.252674 0.967551i $$-0.418690\pi$$
0.252674 + 0.967551i $$0.418690\pi$$
$$788$$ 1.07030 0.0381278
$$789$$ −54.6761 −1.94652
$$790$$ 0 0
$$791$$ −16.7431 −0.595315
$$792$$ 20.0811 0.713551
$$793$$ 27.8046 0.987372
$$794$$ −4.15396 −0.147418
$$795$$ 0 0
$$796$$ −22.1016 −0.783369
$$797$$ −21.8764 −0.774900 −0.387450 0.921891i $$-0.626644\pi$$
−0.387450 + 0.921891i $$0.626644\pi$$
$$798$$ 41.9330 1.48441
$$799$$ 33.4010 1.18164
$$800$$ 0 0
$$801$$ 18.4690 0.652569
$$802$$ −22.1817 −0.783264
$$803$$ −49.2516 −1.73805
$$804$$ 5.61942 0.198182
$$805$$ 0 0
$$806$$ 42.2433 1.48796
$$807$$ −16.7250 −0.588747
$$808$$ −8.12601 −0.285872
$$809$$ −12.2130 −0.429386 −0.214693 0.976682i $$-0.568875\pi$$
−0.214693 + 0.976682i $$0.568875\pi$$
$$810$$ 0 0
$$811$$ 8.64974 0.303733 0.151867 0.988401i $$-0.451472\pi$$
0.151867 + 0.988401i $$0.451472\pi$$
$$812$$ 26.0508 0.914203
$$813$$ 2.20474 0.0773236
$$814$$ −4.83146 −0.169342
$$815$$ 0 0
$$816$$ −15.0811 −0.527944
$$817$$ 54.5099 1.90706
$$818$$ 0.650693 0.0227510
$$819$$ 77.5388 2.70943
$$820$$ 0 0
$$821$$ −41.7972 −1.45873 −0.729366 0.684124i $$-0.760185\pi$$
−0.729366 + 0.684124i $$0.760185\pi$$
$$822$$ 29.9257 1.04378
$$823$$ −41.9633 −1.46275 −0.731375 0.681975i $$-0.761121\pi$$
−0.731375 + 0.681975i $$0.761121\pi$$
$$824$$ 1.90668 0.0664223
$$825$$ 0 0
$$826$$ 37.5778 1.30750
$$827$$ −23.5950 −0.820478 −0.410239 0.911978i $$-0.634555\pi$$
−0.410239 + 0.911978i $$0.634555\pi$$
$$828$$ −9.92478 −0.344910
$$829$$ 39.4396 1.36979 0.684897 0.728640i $$-0.259847\pi$$
0.684897 + 0.728640i $$0.259847\pi$$
$$830$$ 0 0
$$831$$ −48.6516 −1.68771
$$832$$ −5.67513 −0.196750
$$833$$ 21.4568 0.743433
$$834$$ 45.6615 1.58113
$$835$$ 0 0
$$836$$ 23.0386 0.796805
$$837$$ 23.0254 0.795874
$$838$$ 10.8169 0.373662
$$839$$ 33.4495 1.15480 0.577402 0.816460i $$-0.304067\pi$$
0.577402 + 0.816460i $$0.304067\pi$$
$$840$$ 0 0
$$841$$ 33.8021 1.16559
$$842$$ 2.05079 0.0706747
$$843$$ −66.9946 −2.30742
$$844$$ −16.3987 −0.564466
$$845$$ 0 0
$$846$$ 24.6253 0.846635
$$847$$ −40.5745 −1.39416
$$848$$ −9.53690 −0.327499
$$849$$ 35.0590 1.20322
$$850$$ 0 0
$$851$$ 2.38787 0.0818552
$$852$$ 13.1065 0.449021
$$853$$ 29.5101 1.01041 0.505203 0.863000i $$-0.331418\pi$$
0.505203 + 0.863000i $$0.331418\pi$$
$$854$$ 16.1055 0.551120
$$855$$ 0 0
$$856$$ 2.41327 0.0824837
$$857$$ −36.4133 −1.24385 −0.621927 0.783075i $$-0.713650\pi$$
−0.621927 + 0.783075i $$0.713650\pi$$
$$858$$ 73.3498 2.50412
$$859$$ −50.7685 −1.73220 −0.866099 0.499873i $$-0.833380\pi$$
−0.866099 + 0.499873i $$0.833380\pi$$
$$860$$ 0 0
$$861$$ −91.0186 −3.10191
$$862$$ 15.4861 0.527459
$$863$$ −44.6032 −1.51831 −0.759156 0.650909i $$-0.774388\pi$$
−0.759156 + 0.650909i $$0.774388\pi$$
$$864$$ −3.09332 −0.105237
$$865$$ 0 0
$$866$$ −35.8773 −1.21916
$$867$$ −39.5428 −1.34294
$$868$$ 24.4690 0.830531
$$869$$ 36.7777 1.24760
$$870$$ 0 0
$$871$$ 11.9213 0.403937
$$872$$ −1.73813 −0.0588607
$$873$$ −22.2374 −0.752623
$$874$$ −11.3865 −0.385153
$$875$$ 0 0
$$876$$ 27.2701 0.921372
$$877$$ 42.4079 1.43201 0.716006 0.698094i $$-0.245968\pi$$
0.716006 + 0.698094i $$0.245968\pi$$
$$878$$ −26.7005 −0.901099
$$879$$ 25.8980 0.873517
$$880$$ 0 0
$$881$$ −11.0435 −0.372065 −0.186032 0.982544i $$-0.559563\pi$$
−0.186032 + 0.982544i $$0.559563\pi$$
$$882$$ 15.8192 0.532661
$$883$$ −3.52847 −0.118742 −0.0593712 0.998236i $$-0.518910\pi$$
−0.0593712 + 0.998236i $$0.518910\pi$$
$$884$$ −31.9937 −1.07606
$$885$$ 0 0
$$886$$ −5.75765 −0.193432
$$887$$ 11.8035 0.396323 0.198162 0.980169i $$-0.436503\pi$$
0.198162 + 0.980169i $$0.436503\pi$$
$$888$$ 2.67513 0.0897715
$$889$$ −24.9829 −0.837898
$$890$$ 0 0
$$891$$ −20.2628 −0.678830
$$892$$ −18.4387 −0.617372
$$893$$ 28.2520 0.945418
$$894$$ 20.2496 0.677249
$$895$$ 0 0
$$896$$ −3.28726 −0.109820
$$897$$ −36.2520 −1.21042
$$898$$ −31.1803 −1.04050
$$899$$ 58.9887 1.96738
$$900$$ 0 0
$$901$$ −53.7645 −1.79115
$$902$$ −50.0068 −1.66505
$$903$$ 100.526 3.34528
$$904$$ 5.09332 0.169401
$$905$$ 0 0
$$906$$ −1.73813 −0.0577457
$$907$$ −34.2579 −1.13751 −0.568757 0.822505i $$-0.692576\pi$$
−0.568757 + 0.822505i $$0.692576\pi$$
$$908$$ 11.4558 0.380174
$$909$$ −33.7743 −1.12022
$$910$$ 0 0
$$911$$ 31.0214 1.02779 0.513893 0.857854i $$-0.328203\pi$$
0.513893 + 0.857854i $$0.328203\pi$$
$$912$$ −12.7562 −0.422401
$$913$$ −67.2532 −2.22575
$$914$$ −7.27011 −0.240474
$$915$$ 0 0
$$916$$ 25.6385 0.847119
$$917$$ −43.9854 −1.45253
$$918$$ −17.4387 −0.575561
$$919$$ 18.9986 0.626706 0.313353 0.949637i $$-0.398548\pi$$
0.313353 + 0.949637i $$0.398548\pi$$
$$920$$ 0 0
$$921$$ −54.6629 −1.80120
$$922$$ −18.6253 −0.613392
$$923$$ 27.8046 0.915201
$$924$$ 42.4871 1.39772
$$925$$ 0 0
$$926$$ −1.89209 −0.0621779
$$927$$ 7.92478 0.260284
$$928$$ −7.92478 −0.260144
$$929$$ 55.4530 1.81935 0.909677 0.415318i $$-0.136330\pi$$
0.909677 + 0.415318i $$0.136330\pi$$
$$930$$ 0 0
$$931$$ 18.1490 0.594810
$$932$$ −9.76845 −0.319976
$$933$$ −28.1744 −0.922389
$$934$$ −1.21440 −0.0397365
$$935$$ 0 0
$$936$$ −23.5877 −0.770988
$$937$$ −27.9683 −0.913683 −0.456842 0.889548i $$-0.651019\pi$$
−0.456842 + 0.889548i $$0.651019\pi$$
$$938$$ 6.90526 0.225465
$$939$$ 26.6702 0.870349
$$940$$ 0 0
$$941$$ 52.3390 1.70620 0.853101 0.521745i $$-0.174719\pi$$
0.853101 + 0.521745i $$0.174719\pi$$
$$942$$ −9.01080 −0.293588
$$943$$ 24.7151 0.804835
$$944$$ −11.4314 −0.372059
$$945$$ 0 0
$$946$$ 55.2301 1.79569
$$947$$ 10.0752 0.327401 0.163700 0.986510i $$-0.447657\pi$$
0.163700 + 0.986510i $$0.447657\pi$$
$$948$$ −20.3634 −0.661374
$$949$$ 57.8519 1.87795
$$950$$ 0 0
$$951$$ −65.2628 −2.11629
$$952$$ −18.5320 −0.600625
$$953$$ −10.2809 −0.333032 −0.166516 0.986039i $$-0.553252\pi$$
−0.166516 + 0.986039i $$0.553252\pi$$
$$954$$ −39.6385 −1.28334
$$955$$ 0 0
$$956$$ 16.2882 0.526798
$$957$$ 102.426 3.31096
$$958$$ −40.7694 −1.31720
$$959$$ 36.7734 1.18747
$$960$$ 0 0
$$961$$ 24.4069 0.787320
$$962$$ 5.67513 0.182974
$$963$$ 10.0303 0.323222
$$964$$ 13.1744 0.424320
$$965$$ 0 0
$$966$$ −20.9986 −0.675618
$$967$$ 30.7612 0.989212 0.494606 0.869117i $$-0.335312\pi$$
0.494606 + 0.869117i $$0.335312\pi$$
$$968$$ 12.3430 0.396718
$$969$$ −71.9135 −2.31019
$$970$$ 0 0
$$971$$ 52.9086 1.69792 0.848959 0.528459i $$-0.177230\pi$$
0.848959 + 0.528459i $$0.177230\pi$$
$$972$$ 20.4993 0.657515
$$973$$ 56.1098 1.79880
$$974$$ 23.8578 0.764453
$$975$$ 0 0
$$976$$ −4.89938 −0.156825
$$977$$ 16.5901 0.530763 0.265382 0.964143i $$-0.414502\pi$$
0.265382 + 0.964143i $$0.414502\pi$$
$$978$$ −11.5564 −0.369533
$$979$$ 21.4690 0.686151
$$980$$ 0 0
$$981$$ −7.22425 −0.230653
$$982$$ 36.0362 1.14996
$$983$$ −36.3146 −1.15825 −0.579127 0.815237i $$-0.696607\pi$$
−0.579127 + 0.815237i $$0.696607\pi$$
$$984$$ 27.6883 0.882671
$$985$$ 0 0
$$986$$ −44.6761 −1.42278
$$987$$ 52.1016 1.65841
$$988$$ −27.0616 −0.860944
$$989$$ −27.2966 −0.867983
$$990$$ 0 0
$$991$$ 3.95272 0.125562 0.0627812 0.998027i $$-0.480003\pi$$
0.0627812 + 0.998027i $$0.480003\pi$$
$$992$$ −7.44358 −0.236334
$$993$$ −91.9072 −2.91659
$$994$$ 16.1055 0.510837
$$995$$ 0 0
$$996$$ 37.2374 1.17991
$$997$$ 41.6625 1.31946 0.659732 0.751501i $$-0.270670\pi$$
0.659732 + 0.751501i $$0.270670\pi$$
$$998$$ −31.4558 −0.995716
$$999$$ 3.09332 0.0978684
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 1850.2.a.bb.1.1 yes 3
5.2 odd 4 1850.2.b.n.149.6 6
5.3 odd 4 1850.2.b.n.149.1 6
5.4 even 2 1850.2.a.ba.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.3 3 5.4 even 2
1850.2.a.bb.1.1 yes 3 1.1 even 1 trivial
1850.2.b.n.149.1 6 5.3 odd 4
1850.2.b.n.149.6 6 5.2 odd 4