# Properties

 Label 1850.2.a.be.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.1791440.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 9x^{3} + 13x - 4$$ x^5 - 9*x^3 + 13*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.72987$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} -4.14336 q^{7} +1.00000 q^{8} +4.45216 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.72987 q^{3} +1.00000 q^{4} -2.72987 q^{6} -4.14336 q^{7} +1.00000 q^{8} +4.45216 q^{9} -4.76853 q^{11} -2.72987 q^{12} -3.91744 q^{13} -4.14336 q^{14} +1.00000 q^{16} +3.31637 q^{17} +4.45216 q^{18} -1.85109 q^{19} +11.3108 q^{21} -4.76853 q^{22} -1.54229 q^{23} -2.72987 q^{24} -3.91744 q^{26} -3.96421 q^{27} -4.14336 q^{28} +8.87323 q^{29} +9.75286 q^{31} +1.00000 q^{32} +13.0174 q^{33} +3.31637 q^{34} +4.45216 q^{36} +1.00000 q^{37} -1.85109 q^{38} +10.6941 q^{39} -5.06977 q^{41} +11.3108 q^{42} -9.99446 q^{43} -4.76853 q^{44} -1.54229 q^{46} +4.82700 q^{47} -2.72987 q^{48} +10.1675 q^{49} -9.05324 q^{51} -3.91744 q^{52} -5.13611 q^{53} -3.96421 q^{54} -4.14336 q^{56} +5.05324 q^{57} +8.87323 q^{58} -1.05324 q^{59} -4.14422 q^{61} +9.75286 q^{62} -18.4469 q^{63} +1.00000 q^{64} +13.0174 q^{66} +1.00811 q^{67} +3.31637 q^{68} +4.21025 q^{69} +6.45248 q^{71} +4.45216 q^{72} +10.7714 q^{73} +1.00000 q^{74} -1.85109 q^{76} +19.7578 q^{77} +10.6941 q^{78} -1.19856 q^{79} -2.53473 q^{81} -5.06977 q^{82} +10.6932 q^{83} +11.3108 q^{84} -9.99446 q^{86} -24.2227 q^{87} -4.76853 q^{88} +7.29569 q^{89} +16.2314 q^{91} -1.54229 q^{92} -26.6240 q^{93} +4.82700 q^{94} -2.72987 q^{96} +14.8988 q^{97} +10.1675 q^{98} -21.2303 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 $$5 q + 5 q^{2} + 5 q^{4} - q^{7} + 5 q^{8} + 3 q^{9} + 3 q^{11} - 6 q^{13} - q^{14} + 5 q^{16} + 9 q^{17} + 3 q^{18} + 4 q^{19} + 16 q^{21} + 3 q^{22} + 6 q^{23} - 6 q^{26} - q^{28} + 11 q^{29} + 23 q^{31} + 5 q^{32} + 20 q^{33} + 9 q^{34} + 3 q^{36} + 5 q^{37} + 4 q^{38} + 20 q^{39} - 7 q^{41} + 16 q^{42} - 17 q^{43} + 3 q^{44} + 6 q^{46} + 12 q^{47} + 30 q^{49} - 20 q^{51} - 6 q^{52} - 7 q^{53} - q^{56} + 11 q^{58} + 20 q^{59} - 9 q^{61} + 23 q^{62} - 33 q^{63} + 5 q^{64} + 20 q^{66} + 12 q^{67} + 9 q^{68} + 16 q^{69} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 5 q^{74} + 4 q^{76} - q^{77} + 20 q^{78} + 20 q^{79} - 7 q^{81} - 7 q^{82} + 12 q^{83} + 16 q^{84} - 17 q^{86} - 34 q^{87} + 3 q^{88} + 12 q^{89} + 16 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 3 q^{97} + 30 q^{98} - 11 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 + 5 * q^4 - q^7 + 5 * q^8 + 3 * q^9 + 3 * q^11 - 6 * q^13 - q^14 + 5 * q^16 + 9 * q^17 + 3 * q^18 + 4 * q^19 + 16 * q^21 + 3 * q^22 + 6 * q^23 - 6 * q^26 - q^28 + 11 * q^29 + 23 * q^31 + 5 * q^32 + 20 * q^33 + 9 * q^34 + 3 * q^36 + 5 * q^37 + 4 * q^38 + 20 * q^39 - 7 * q^41 + 16 * q^42 - 17 * q^43 + 3 * q^44 + 6 * q^46 + 12 * q^47 + 30 * q^49 - 20 * q^51 - 6 * q^52 - 7 * q^53 - q^56 + 11 * q^58 + 20 * q^59 - 9 * q^61 + 23 * q^62 - 33 * q^63 + 5 * q^64 + 20 * q^66 + 12 * q^67 + 9 * q^68 + 16 * q^69 + 6 * q^71 + 3 * q^72 - 6 * q^73 + 5 * q^74 + 4 * q^76 - q^77 + 20 * q^78 + 20 * q^79 - 7 * q^81 - 7 * q^82 + 12 * q^83 + 16 * q^84 - 17 * q^86 - 34 * q^87 + 3 * q^88 + 12 * q^89 + 16 * q^91 + 6 * q^92 - 4 * q^93 + 12 * q^94 + 3 * q^97 + 30 * q^98 - 11 * 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
$$3$$ −2.72987 −1.57609 −0.788044 0.615619i $$-0.788906\pi$$
−0.788044 + 0.615619i $$0.788906\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −2.72987 −1.11446
$$7$$ −4.14336 −1.56604 −0.783022 0.621994i $$-0.786323\pi$$
−0.783022 + 0.621994i $$0.786323\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 4.45216 1.48405
$$10$$ 0 0
$$11$$ −4.76853 −1.43777 −0.718883 0.695131i $$-0.755346\pi$$
−0.718883 + 0.695131i $$0.755346\pi$$
$$12$$ −2.72987 −0.788044
$$13$$ −3.91744 −1.08650 −0.543251 0.839570i $$-0.682807\pi$$
−0.543251 + 0.839570i $$0.682807\pi$$
$$14$$ −4.14336 −1.10736
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.31637 0.804337 0.402169 0.915566i $$-0.368257\pi$$
0.402169 + 0.915566i $$0.368257\pi$$
$$18$$ 4.45216 1.04939
$$19$$ −1.85109 −0.424670 −0.212335 0.977197i $$-0.568107\pi$$
−0.212335 + 0.977197i $$0.568107\pi$$
$$20$$ 0 0
$$21$$ 11.3108 2.46822
$$22$$ −4.76853 −1.01665
$$23$$ −1.54229 −0.321590 −0.160795 0.986988i $$-0.551406\pi$$
−0.160795 + 0.986988i $$0.551406\pi$$
$$24$$ −2.72987 −0.557231
$$25$$ 0 0
$$26$$ −3.91744 −0.768273
$$27$$ −3.96421 −0.762913
$$28$$ −4.14336 −0.783022
$$29$$ 8.87323 1.64772 0.823859 0.566795i $$-0.191817\pi$$
0.823859 + 0.566795i $$0.191817\pi$$
$$30$$ 0 0
$$31$$ 9.75286 1.75166 0.875832 0.482615i $$-0.160313\pi$$
0.875832 + 0.482615i $$0.160313\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 13.0174 2.26605
$$34$$ 3.31637 0.568752
$$35$$ 0 0
$$36$$ 4.45216 0.742027
$$37$$ 1.00000 0.164399
$$38$$ −1.85109 −0.300287
$$39$$ 10.6941 1.71242
$$40$$ 0 0
$$41$$ −5.06977 −0.791764 −0.395882 0.918301i $$-0.629561\pi$$
−0.395882 + 0.918301i $$0.629561\pi$$
$$42$$ 11.3108 1.74530
$$43$$ −9.99446 −1.52414 −0.762070 0.647494i $$-0.775817\pi$$
−0.762070 + 0.647494i $$0.775817\pi$$
$$44$$ −4.76853 −0.718883
$$45$$ 0 0
$$46$$ −1.54229 −0.227399
$$47$$ 4.82700 0.704090 0.352045 0.935983i $$-0.385486\pi$$
0.352045 + 0.935983i $$0.385486\pi$$
$$48$$ −2.72987 −0.394022
$$49$$ 10.1675 1.45249
$$50$$ 0 0
$$51$$ −9.05324 −1.26771
$$52$$ −3.91744 −0.543251
$$53$$ −5.13611 −0.705499 −0.352750 0.935718i $$-0.614753\pi$$
−0.352750 + 0.935718i $$0.614753\pi$$
$$54$$ −3.96421 −0.539461
$$55$$ 0 0
$$56$$ −4.14336 −0.553680
$$57$$ 5.05324 0.669317
$$58$$ 8.87323 1.16511
$$59$$ −1.05324 −0.137120 −0.0685598 0.997647i $$-0.521840\pi$$
−0.0685598 + 0.997647i $$0.521840\pi$$
$$60$$ 0 0
$$61$$ −4.14422 −0.530613 −0.265306 0.964164i $$-0.585473\pi$$
−0.265306 + 0.964164i $$0.585473\pi$$
$$62$$ 9.75286 1.23861
$$63$$ −18.4469 −2.32410
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 13.0174 1.60234
$$67$$ 1.00811 0.123160 0.0615800 0.998102i $$-0.480386\pi$$
0.0615800 + 0.998102i $$0.480386\pi$$
$$68$$ 3.31637 0.402169
$$69$$ 4.21025 0.506855
$$70$$ 0 0
$$71$$ 6.45248 0.765768 0.382884 0.923796i $$-0.374931\pi$$
0.382884 + 0.923796i $$0.374931\pi$$
$$72$$ 4.45216 0.524693
$$73$$ 10.7714 1.26070 0.630349 0.776312i $$-0.282912\pi$$
0.630349 + 0.776312i $$0.282912\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −1.85109 −0.212335
$$77$$ 19.7578 2.25161
$$78$$ 10.6941 1.21087
$$79$$ −1.19856 −0.134849 −0.0674243 0.997724i $$-0.521478\pi$$
−0.0674243 + 0.997724i $$0.521478\pi$$
$$80$$ 0 0
$$81$$ −2.53473 −0.281636
$$82$$ −5.06977 −0.559862
$$83$$ 10.6932 1.17373 0.586867 0.809683i $$-0.300361\pi$$
0.586867 + 0.809683i $$0.300361\pi$$
$$84$$ 11.3108 1.23411
$$85$$ 0 0
$$86$$ −9.99446 −1.07773
$$87$$ −24.2227 −2.59695
$$88$$ −4.76853 −0.508327
$$89$$ 7.29569 0.773342 0.386671 0.922218i $$-0.373625\pi$$
0.386671 + 0.922218i $$0.373625\pi$$
$$90$$ 0 0
$$91$$ 16.2314 1.70151
$$92$$ −1.54229 −0.160795
$$93$$ −26.6240 −2.76078
$$94$$ 4.82700 0.497867
$$95$$ 0 0
$$96$$ −2.72987 −0.278616
$$97$$ 14.8988 1.51274 0.756371 0.654143i $$-0.226970\pi$$
0.756371 + 0.654143i $$0.226970\pi$$
$$98$$ 10.1675 1.02707
$$99$$ −21.2303 −2.13372
$$100$$ 0 0
$$101$$ 18.5903 1.84980 0.924902 0.380206i $$-0.124147\pi$$
0.924902 + 0.380206i $$0.124147\pi$$
$$102$$ −9.05324 −0.896404
$$103$$ −13.3033 −1.31081 −0.655404 0.755278i $$-0.727502\pi$$
−0.655404 + 0.755278i $$0.727502\pi$$
$$104$$ −3.91744 −0.384136
$$105$$ 0 0
$$106$$ −5.13611 −0.498863
$$107$$ 12.4763 1.20613 0.603066 0.797691i $$-0.293946\pi$$
0.603066 + 0.797691i $$0.293946\pi$$
$$108$$ −3.96421 −0.381457
$$109$$ −4.35447 −0.417083 −0.208541 0.978014i $$-0.566872\pi$$
−0.208541 + 0.978014i $$0.566872\pi$$
$$110$$ 0 0
$$111$$ −2.72987 −0.259107
$$112$$ −4.14336 −0.391511
$$113$$ 9.54261 0.897693 0.448846 0.893609i $$-0.351835\pi$$
0.448846 + 0.893609i $$0.351835\pi$$
$$114$$ 5.05324 0.473279
$$115$$ 0 0
$$116$$ 8.87323 0.823859
$$117$$ −17.4411 −1.61243
$$118$$ −1.05324 −0.0969582
$$119$$ −13.7409 −1.25963
$$120$$ 0 0
$$121$$ 11.7389 1.06717
$$122$$ −4.14422 −0.375200
$$123$$ 13.8398 1.24789
$$124$$ 9.75286 0.875832
$$125$$ 0 0
$$126$$ −18.4469 −1.64338
$$127$$ 8.33492 0.739605 0.369802 0.929110i $$-0.379425\pi$$
0.369802 + 0.929110i $$0.379425\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 27.2835 2.40218
$$130$$ 0 0
$$131$$ 3.68597 0.322045 0.161022 0.986951i $$-0.448521\pi$$
0.161022 + 0.986951i $$0.448521\pi$$
$$132$$ 13.0174 1.13302
$$133$$ 7.66975 0.665052
$$134$$ 1.00811 0.0870873
$$135$$ 0 0
$$136$$ 3.31637 0.284376
$$137$$ −12.3027 −1.05109 −0.525546 0.850765i $$-0.676139\pi$$
−0.525546 + 0.850765i $$0.676139\pi$$
$$138$$ 4.21025 0.358400
$$139$$ −0.842442 −0.0714550 −0.0357275 0.999362i $$-0.511375\pi$$
−0.0357275 + 0.999362i $$0.511375\pi$$
$$140$$ 0 0
$$141$$ −13.1770 −1.10971
$$142$$ 6.45248 0.541480
$$143$$ 18.6804 1.56214
$$144$$ 4.45216 0.371014
$$145$$ 0 0
$$146$$ 10.7714 0.891448
$$147$$ −27.7558 −2.28926
$$148$$ 1.00000 0.0821995
$$149$$ −16.7723 −1.37404 −0.687019 0.726640i $$-0.741081\pi$$
−0.687019 + 0.726640i $$0.741081\pi$$
$$150$$ 0 0
$$151$$ −7.59755 −0.618280 −0.309140 0.951017i $$-0.600041\pi$$
−0.309140 + 0.951017i $$0.600041\pi$$
$$152$$ −1.85109 −0.150143
$$153$$ 14.7650 1.19368
$$154$$ 19.7578 1.59213
$$155$$ 0 0
$$156$$ 10.6941 0.856211
$$157$$ 4.45631 0.355652 0.177826 0.984062i $$-0.443094\pi$$
0.177826 + 0.984062i $$0.443094\pi$$
$$158$$ −1.19856 −0.0953523
$$159$$ 14.0209 1.11193
$$160$$ 0 0
$$161$$ 6.39028 0.503624
$$162$$ −2.53473 −0.199147
$$163$$ 19.4599 1.52422 0.762110 0.647447i $$-0.224163\pi$$
0.762110 + 0.647447i $$0.224163\pi$$
$$164$$ −5.06977 −0.395882
$$165$$ 0 0
$$166$$ 10.6932 0.829955
$$167$$ 7.13813 0.552365 0.276183 0.961105i $$-0.410931\pi$$
0.276183 + 0.961105i $$0.410931\pi$$
$$168$$ 11.3108 0.872649
$$169$$ 2.34632 0.180486
$$170$$ 0 0
$$171$$ −8.24137 −0.630233
$$172$$ −9.99446 −0.762070
$$173$$ −11.6199 −0.883448 −0.441724 0.897151i $$-0.645633\pi$$
−0.441724 + 0.897151i $$0.645633\pi$$
$$174$$ −24.2227 −1.83632
$$175$$ 0 0
$$176$$ −4.76853 −0.359442
$$177$$ 2.87519 0.216113
$$178$$ 7.29569 0.546835
$$179$$ −17.2224 −1.28726 −0.643631 0.765336i $$-0.722573\pi$$
−0.643631 + 0.765336i $$0.722573\pi$$
$$180$$ 0 0
$$181$$ −23.5098 −1.74747 −0.873733 0.486405i $$-0.838308\pi$$
−0.873733 + 0.486405i $$0.838308\pi$$
$$182$$ 16.2314 1.20315
$$183$$ 11.3132 0.836293
$$184$$ −1.54229 −0.113699
$$185$$ 0 0
$$186$$ −26.6240 −1.95217
$$187$$ −15.8142 −1.15645
$$188$$ 4.82700 0.352045
$$189$$ 16.4252 1.19476
$$190$$ 0 0
$$191$$ 1.16660 0.0844125 0.0422063 0.999109i $$-0.486561\pi$$
0.0422063 + 0.999109i $$0.486561\pi$$
$$192$$ −2.72987 −0.197011
$$193$$ −9.36727 −0.674271 −0.337135 0.941456i $$-0.609458\pi$$
−0.337135 + 0.941456i $$0.609458\pi$$
$$194$$ 14.8988 1.06967
$$195$$ 0 0
$$196$$ 10.1675 0.726247
$$197$$ 3.21836 0.229299 0.114649 0.993406i $$-0.463426\pi$$
0.114649 + 0.993406i $$0.463426\pi$$
$$198$$ −21.2303 −1.50877
$$199$$ −2.99104 −0.212029 −0.106014 0.994365i $$-0.533809\pi$$
−0.106014 + 0.994365i $$0.533809\pi$$
$$200$$ 0 0
$$201$$ −2.75200 −0.194111
$$202$$ 18.5903 1.30801
$$203$$ −36.7650 −2.58040
$$204$$ −9.05324 −0.633853
$$205$$ 0 0
$$206$$ −13.3033 −0.926882
$$207$$ −6.86654 −0.477257
$$208$$ −3.91744 −0.271625
$$209$$ 8.82700 0.610576
$$210$$ 0 0
$$211$$ 4.75340 0.327237 0.163619 0.986524i $$-0.447683\pi$$
0.163619 + 0.986524i $$0.447683\pi$$
$$212$$ −5.13611 −0.352750
$$213$$ −17.6144 −1.20692
$$214$$ 12.4763 0.852864
$$215$$ 0 0
$$216$$ −3.96421 −0.269731
$$217$$ −40.4096 −2.74318
$$218$$ −4.35447 −0.294922
$$219$$ −29.4045 −1.98697
$$220$$ 0 0
$$221$$ −12.9917 −0.873914
$$222$$ −2.72987 −0.183217
$$223$$ −6.95635 −0.465832 −0.232916 0.972497i $$-0.574827\pi$$
−0.232916 + 0.972497i $$0.574827\pi$$
$$224$$ −4.14336 −0.276840
$$225$$ 0 0
$$226$$ 9.54261 0.634765
$$227$$ −22.2980 −1.47997 −0.739986 0.672622i $$-0.765168\pi$$
−0.739986 + 0.672622i $$0.765168\pi$$
$$228$$ 5.05324 0.334659
$$229$$ 12.9648 0.856739 0.428370 0.903604i $$-0.359088\pi$$
0.428370 + 0.903604i $$0.359088\pi$$
$$230$$ 0 0
$$231$$ −53.9360 −3.54873
$$232$$ 8.87323 0.582556
$$233$$ −10.4327 −0.683468 −0.341734 0.939797i $$-0.611014\pi$$
−0.341734 + 0.939797i $$0.611014\pi$$
$$234$$ −17.4411 −1.14016
$$235$$ 0 0
$$236$$ −1.05324 −0.0685598
$$237$$ 3.27191 0.212533
$$238$$ −13.7409 −0.890691
$$239$$ 1.58711 0.102661 0.0513307 0.998682i $$-0.483654\pi$$
0.0513307 + 0.998682i $$0.483654\pi$$
$$240$$ 0 0
$$241$$ 16.2576 1.04724 0.523622 0.851951i $$-0.324581\pi$$
0.523622 + 0.851951i $$0.324581\pi$$
$$242$$ 11.7389 0.754604
$$243$$ 18.8121 1.20680
$$244$$ −4.14422 −0.265306
$$245$$ 0 0
$$246$$ 13.8398 0.882392
$$247$$ 7.25154 0.461405
$$248$$ 9.75286 0.619307
$$249$$ −29.1911 −1.84991
$$250$$ 0 0
$$251$$ 14.9486 0.943547 0.471774 0.881720i $$-0.343614\pi$$
0.471774 + 0.881720i $$0.343614\pi$$
$$252$$ −18.4469 −1.16205
$$253$$ 7.35447 0.462372
$$254$$ 8.33492 0.522979
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 14.9968 0.935474 0.467737 0.883868i $$-0.345069\pi$$
0.467737 + 0.883868i $$0.345069\pi$$
$$258$$ 27.2835 1.69860
$$259$$ −4.14336 −0.257456
$$260$$ 0 0
$$261$$ 39.5051 2.44530
$$262$$ 3.68597 0.227720
$$263$$ 22.9610 1.41583 0.707917 0.706295i $$-0.249635\pi$$
0.707917 + 0.706295i $$0.249635\pi$$
$$264$$ 13.0174 0.801169
$$265$$ 0 0
$$266$$ 7.66975 0.470263
$$267$$ −19.9163 −1.21885
$$268$$ 1.00811 0.0615800
$$269$$ 10.0896 0.615175 0.307588 0.951520i $$-0.400478\pi$$
0.307588 + 0.951520i $$0.400478\pi$$
$$270$$ 0 0
$$271$$ 3.30678 0.200872 0.100436 0.994943i $$-0.467976\pi$$
0.100436 + 0.994943i $$0.467976\pi$$
$$272$$ 3.31637 0.201084
$$273$$ −44.3095 −2.68173
$$274$$ −12.3027 −0.743234
$$275$$ 0 0
$$276$$ 4.21025 0.253427
$$277$$ −15.9046 −0.955617 −0.477809 0.878464i $$-0.658569\pi$$
−0.477809 + 0.878464i $$0.658569\pi$$
$$278$$ −0.842442 −0.0505263
$$279$$ 43.4213 2.59957
$$280$$ 0 0
$$281$$ 11.3609 0.677732 0.338866 0.940835i $$-0.389957\pi$$
0.338866 + 0.940835i $$0.389957\pi$$
$$282$$ −13.1770 −0.784682
$$283$$ 20.1530 1.19797 0.598984 0.800761i $$-0.295571\pi$$
0.598984 + 0.800761i $$0.295571\pi$$
$$284$$ 6.45248 0.382884
$$285$$ 0 0
$$286$$ 18.6804 1.10460
$$287$$ 21.0059 1.23994
$$288$$ 4.45216 0.262346
$$289$$ −6.00171 −0.353042
$$290$$ 0 0
$$291$$ −40.6717 −2.38422
$$292$$ 10.7714 0.630349
$$293$$ −21.0517 −1.22986 −0.614928 0.788583i $$-0.710815\pi$$
−0.614928 + 0.788583i $$0.710815\pi$$
$$294$$ −27.7558 −1.61875
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 18.9035 1.09689
$$298$$ −16.7723 −0.971591
$$299$$ 6.04184 0.349408
$$300$$ 0 0
$$301$$ 41.4107 2.38687
$$302$$ −7.59755 −0.437190
$$303$$ −50.7490 −2.91545
$$304$$ −1.85109 −0.106167
$$305$$ 0 0
$$306$$ 14.7650 0.844059
$$307$$ 17.0946 0.975638 0.487819 0.872945i $$-0.337793\pi$$
0.487819 + 0.872945i $$0.337793\pi$$
$$308$$ 19.7578 1.12580
$$309$$ 36.3161 2.06595
$$310$$ 0 0
$$311$$ 20.0503 1.13695 0.568473 0.822702i $$-0.307534\pi$$
0.568473 + 0.822702i $$0.307534\pi$$
$$312$$ 10.6941 0.605433
$$313$$ 13.3400 0.754019 0.377010 0.926209i $$-0.376952\pi$$
0.377010 + 0.926209i $$0.376952\pi$$
$$314$$ 4.45631 0.251484
$$315$$ 0 0
$$316$$ −1.19856 −0.0674243
$$317$$ 12.8528 0.721885 0.360943 0.932588i $$-0.382455\pi$$
0.360943 + 0.932588i $$0.382455\pi$$
$$318$$ 14.0209 0.786252
$$319$$ −42.3123 −2.36903
$$320$$ 0 0
$$321$$ −34.0587 −1.90097
$$322$$ 6.39028 0.356116
$$323$$ −6.13890 −0.341578
$$324$$ −2.53473 −0.140818
$$325$$ 0 0
$$326$$ 19.4599 1.07779
$$327$$ 11.8871 0.657359
$$328$$ −5.06977 −0.279931
$$329$$ −20.0000 −1.10264
$$330$$ 0 0
$$331$$ −23.5835 −1.29627 −0.648134 0.761526i $$-0.724451\pi$$
−0.648134 + 0.761526i $$0.724451\pi$$
$$332$$ 10.6932 0.586867
$$333$$ 4.45216 0.243977
$$334$$ 7.13813 0.390581
$$335$$ 0 0
$$336$$ 11.3108 0.617056
$$337$$ 26.6074 1.44940 0.724698 0.689067i $$-0.241979\pi$$
0.724698 + 0.689067i $$0.241979\pi$$
$$338$$ 2.34632 0.127623
$$339$$ −26.0500 −1.41484
$$340$$ 0 0
$$341$$ −46.5068 −2.51848
$$342$$ −8.24137 −0.445642
$$343$$ −13.1239 −0.708626
$$344$$ −9.99446 −0.538865
$$345$$ 0 0
$$346$$ −11.6199 −0.624692
$$347$$ −17.2626 −0.926707 −0.463353 0.886174i $$-0.653354\pi$$
−0.463353 + 0.886174i $$0.653354\pi$$
$$348$$ −24.2227 −1.29847
$$349$$ 11.1619 0.597484 0.298742 0.954334i $$-0.403433\pi$$
0.298742 + 0.954334i $$0.403433\pi$$
$$350$$ 0 0
$$351$$ 15.5296 0.828906
$$352$$ −4.76853 −0.254164
$$353$$ −12.6563 −0.673628 −0.336814 0.941571i $$-0.609349\pi$$
−0.336814 + 0.941571i $$0.609349\pi$$
$$354$$ 2.87519 0.152815
$$355$$ 0 0
$$356$$ 7.29569 0.386671
$$357$$ 37.5108 1.98528
$$358$$ −17.2224 −0.910232
$$359$$ −23.2778 −1.22855 −0.614277 0.789091i $$-0.710552\pi$$
−0.614277 + 0.789091i $$0.710552\pi$$
$$360$$ 0 0
$$361$$ −15.5735 −0.819655
$$362$$ −23.5098 −1.23565
$$363$$ −32.0456 −1.68196
$$364$$ 16.2314 0.850755
$$365$$ 0 0
$$366$$ 11.3132 0.591348
$$367$$ −9.09417 −0.474712 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$368$$ −1.54229 −0.0803976
$$369$$ −22.5714 −1.17502
$$370$$ 0 0
$$371$$ 21.2808 1.10484
$$372$$ −26.6240 −1.38039
$$373$$ 5.75976 0.298229 0.149114 0.988820i $$-0.452358\pi$$
0.149114 + 0.988820i $$0.452358\pi$$
$$374$$ −15.8142 −0.817733
$$375$$ 0 0
$$376$$ 4.82700 0.248933
$$377$$ −34.7603 −1.79025
$$378$$ 16.4252 0.844820
$$379$$ 18.0902 0.929229 0.464614 0.885513i $$-0.346193\pi$$
0.464614 + 0.885513i $$0.346193\pi$$
$$380$$ 0 0
$$381$$ −22.7532 −1.16568
$$382$$ 1.16660 0.0596887
$$383$$ 3.54752 0.181270 0.0906350 0.995884i $$-0.471110\pi$$
0.0906350 + 0.995884i $$0.471110\pi$$
$$384$$ −2.72987 −0.139308
$$385$$ 0 0
$$386$$ −9.36727 −0.476781
$$387$$ −44.4970 −2.26191
$$388$$ 14.8988 0.756371
$$389$$ 25.6053 1.29824 0.649119 0.760687i $$-0.275138\pi$$
0.649119 + 0.760687i $$0.275138\pi$$
$$390$$ 0 0
$$391$$ −5.11481 −0.258667
$$392$$ 10.1675 0.513534
$$393$$ −10.0622 −0.507571
$$394$$ 3.21836 0.162139
$$395$$ 0 0
$$396$$ −21.2303 −1.06686
$$397$$ 31.4597 1.57892 0.789459 0.613803i $$-0.210361\pi$$
0.789459 + 0.613803i $$0.210361\pi$$
$$398$$ −2.99104 −0.149927
$$399$$ −20.9374 −1.04818
$$400$$ 0 0
$$401$$ −22.9940 −1.14826 −0.574132 0.818763i $$-0.694660\pi$$
−0.574132 + 0.818763i $$0.694660\pi$$
$$402$$ −2.75200 −0.137257
$$403$$ −38.2062 −1.90319
$$404$$ 18.5903 0.924902
$$405$$ 0 0
$$406$$ −36.7650 −1.82462
$$407$$ −4.76853 −0.236367
$$408$$ −9.05324 −0.448202
$$409$$ −33.3828 −1.65067 −0.825337 0.564640i $$-0.809015\pi$$
−0.825337 + 0.564640i $$0.809015\pi$$
$$410$$ 0 0
$$411$$ 33.5848 1.65661
$$412$$ −13.3033 −0.655404
$$413$$ 4.36394 0.214735
$$414$$ −6.86654 −0.337472
$$415$$ 0 0
$$416$$ −3.91744 −0.192068
$$417$$ 2.29975 0.112619
$$418$$ 8.82700 0.431743
$$419$$ 8.32565 0.406734 0.203367 0.979103i $$-0.434811\pi$$
0.203367 + 0.979103i $$0.434811\pi$$
$$420$$ 0 0
$$421$$ −8.41521 −0.410132 −0.205066 0.978748i $$-0.565741\pi$$
−0.205066 + 0.978748i $$0.565741\pi$$
$$422$$ 4.75340 0.231392
$$423$$ 21.4906 1.04491
$$424$$ −5.13611 −0.249432
$$425$$ 0 0
$$426$$ −17.6144 −0.853420
$$427$$ 17.1710 0.830963
$$428$$ 12.4763 0.603066
$$429$$ −50.9950 −2.46206
$$430$$ 0 0
$$431$$ 3.04769 0.146802 0.0734011 0.997303i $$-0.476615\pi$$
0.0734011 + 0.997303i $$0.476615\pi$$
$$432$$ −3.96421 −0.190728
$$433$$ 22.9439 1.10262 0.551308 0.834302i $$-0.314129\pi$$
0.551308 + 0.834302i $$0.314129\pi$$
$$434$$ −40.4096 −1.93972
$$435$$ 0 0
$$436$$ −4.35447 −0.208541
$$437$$ 2.85493 0.136570
$$438$$ −29.4045 −1.40500
$$439$$ 10.3033 0.491751 0.245875 0.969301i $$-0.420925\pi$$
0.245875 + 0.969301i $$0.420925\pi$$
$$440$$ 0 0
$$441$$ 45.2672 2.15558
$$442$$ −12.9917 −0.617950
$$443$$ −4.84236 −0.230067 −0.115034 0.993362i $$-0.536698\pi$$
−0.115034 + 0.993362i $$0.536698\pi$$
$$444$$ −2.72987 −0.129554
$$445$$ 0 0
$$446$$ −6.95635 −0.329393
$$447$$ 45.7860 2.16560
$$448$$ −4.14336 −0.195756
$$449$$ 25.3149 1.19468 0.597341 0.801987i $$-0.296224\pi$$
0.597341 + 0.801987i $$0.296224\pi$$
$$450$$ 0 0
$$451$$ 24.1753 1.13837
$$452$$ 9.54261 0.448846
$$453$$ 20.7403 0.974464
$$454$$ −22.2980 −1.04650
$$455$$ 0 0
$$456$$ 5.05324 0.236639
$$457$$ 11.9945 0.561077 0.280539 0.959843i $$-0.409487\pi$$
0.280539 + 0.959843i $$0.409487\pi$$
$$458$$ 12.9648 0.605806
$$459$$ −13.1468 −0.613639
$$460$$ 0 0
$$461$$ −23.6570 −1.10182 −0.550909 0.834565i $$-0.685719\pi$$
−0.550909 + 0.834565i $$0.685719\pi$$
$$462$$ −53.9360 −2.50933
$$463$$ −17.3667 −0.807099 −0.403550 0.914958i $$-0.632224\pi$$
−0.403550 + 0.914958i $$0.632224\pi$$
$$464$$ 8.87323 0.411929
$$465$$ 0 0
$$466$$ −10.4327 −0.483285
$$467$$ −20.4476 −0.946200 −0.473100 0.881009i $$-0.656865\pi$$
−0.473100 + 0.881009i $$0.656865\pi$$
$$468$$ −17.4411 −0.806214
$$469$$ −4.17696 −0.192874
$$470$$ 0 0
$$471$$ −12.1651 −0.560539
$$472$$ −1.05324 −0.0484791
$$473$$ 47.6589 2.19136
$$474$$ 3.27191 0.150284
$$475$$ 0 0
$$476$$ −13.7409 −0.629814
$$477$$ −22.8668 −1.04700
$$478$$ 1.58711 0.0725925
$$479$$ −34.1325 −1.55955 −0.779777 0.626057i $$-0.784668\pi$$
−0.779777 + 0.626057i $$0.784668\pi$$
$$480$$ 0 0
$$481$$ −3.91744 −0.178620
$$482$$ 16.2576 0.740513
$$483$$ −17.4446 −0.793757
$$484$$ 11.7389 0.533586
$$485$$ 0 0
$$486$$ 18.8121 0.853334
$$487$$ 6.16917 0.279552 0.139776 0.990183i $$-0.455362\pi$$
0.139776 + 0.990183i $$0.455362\pi$$
$$488$$ −4.14422 −0.187600
$$489$$ −53.1230 −2.40231
$$490$$ 0 0
$$491$$ −11.8290 −0.533836 −0.266918 0.963719i $$-0.586005\pi$$
−0.266918 + 0.963719i $$0.586005\pi$$
$$492$$ 13.8398 0.623945
$$493$$ 29.4269 1.32532
$$494$$ 7.25154 0.326262
$$495$$ 0 0
$$496$$ 9.75286 0.437916
$$497$$ −26.7350 −1.19923
$$498$$ −29.1911 −1.30808
$$499$$ 31.4004 1.40568 0.702839 0.711349i $$-0.251916\pi$$
0.702839 + 0.711349i $$0.251916\pi$$
$$500$$ 0 0
$$501$$ −19.4861 −0.870577
$$502$$ 14.9486 0.667189
$$503$$ 39.7743 1.77345 0.886724 0.462299i $$-0.152975\pi$$
0.886724 + 0.462299i $$0.152975\pi$$
$$504$$ −18.4469 −0.821692
$$505$$ 0 0
$$506$$ 7.35447 0.326946
$$507$$ −6.40514 −0.284462
$$508$$ 8.33492 0.369802
$$509$$ −6.82812 −0.302651 −0.151326 0.988484i $$-0.548354\pi$$
−0.151326 + 0.988484i $$0.548354\pi$$
$$510$$ 0 0
$$511$$ −44.6299 −1.97431
$$512$$ 1.00000 0.0441942
$$513$$ 7.33813 0.323986
$$514$$ 14.9968 0.661480
$$515$$ 0 0
$$516$$ 27.2835 1.20109
$$517$$ −23.0177 −1.01232
$$518$$ −4.14336 −0.182049
$$519$$ 31.7209 1.39239
$$520$$ 0 0
$$521$$ −30.6638 −1.34341 −0.671703 0.740821i $$-0.734437\pi$$
−0.671703 + 0.740821i $$0.734437\pi$$
$$522$$ 39.5051 1.72909
$$523$$ 12.1618 0.531800 0.265900 0.964001i $$-0.414331\pi$$
0.265900 + 0.964001i $$0.414331\pi$$
$$524$$ 3.68597 0.161022
$$525$$ 0 0
$$526$$ 22.9610 1.00115
$$527$$ 32.3441 1.40893
$$528$$ 13.0174 0.566512
$$529$$ −20.6213 −0.896580
$$530$$ 0 0
$$531$$ −4.68918 −0.203493
$$532$$ 7.66975 0.332526
$$533$$ 19.8605 0.860253
$$534$$ −19.9163 −0.861861
$$535$$ 0 0
$$536$$ 1.00811 0.0435437
$$537$$ 47.0148 2.02884
$$538$$ 10.0896 0.434995
$$539$$ −48.4839 −2.08835
$$540$$ 0 0
$$541$$ 10.5469 0.453446 0.226723 0.973959i $$-0.427199\pi$$
0.226723 + 0.973959i $$0.427199\pi$$
$$542$$ 3.30678 0.142038
$$543$$ 64.1785 2.75416
$$544$$ 3.31637 0.142188
$$545$$ 0 0
$$546$$ −44.3095 −1.89627
$$547$$ 4.09517 0.175097 0.0875484 0.996160i $$-0.472097\pi$$
0.0875484 + 0.996160i $$0.472097\pi$$
$$548$$ −12.3027 −0.525546
$$549$$ −18.4507 −0.787459
$$550$$ 0 0
$$551$$ −16.4252 −0.699736
$$552$$ 4.21025 0.179200
$$553$$ 4.96607 0.211179
$$554$$ −15.9046 −0.675723
$$555$$ 0 0
$$556$$ −0.842442 −0.0357275
$$557$$ −2.50711 −0.106230 −0.0531149 0.998588i $$-0.516915\pi$$
−0.0531149 + 0.998588i $$0.516915\pi$$
$$558$$ 43.4213 1.83817
$$559$$ 39.1527 1.65598
$$560$$ 0 0
$$561$$ 43.1706 1.82267
$$562$$ 11.3609 0.479229
$$563$$ 9.87532 0.416195 0.208097 0.978108i $$-0.433273\pi$$
0.208097 + 0.978108i $$0.433273\pi$$
$$564$$ −13.1770 −0.554854
$$565$$ 0 0
$$566$$ 20.1530 0.847092
$$567$$ 10.5023 0.441055
$$568$$ 6.45248 0.270740
$$569$$ 12.5198 0.524858 0.262429 0.964951i $$-0.415477\pi$$
0.262429 + 0.964951i $$0.415477\pi$$
$$570$$ 0 0
$$571$$ 20.3156 0.850179 0.425090 0.905151i $$-0.360243\pi$$
0.425090 + 0.905151i $$0.360243\pi$$
$$572$$ 18.6804 0.781068
$$573$$ −3.18467 −0.133042
$$574$$ 21.0059 0.876769
$$575$$ 0 0
$$576$$ 4.45216 0.185507
$$577$$ 20.0756 0.835759 0.417880 0.908502i $$-0.362773\pi$$
0.417880 + 0.908502i $$0.362773\pi$$
$$578$$ −6.00171 −0.249638
$$579$$ 25.5714 1.06271
$$580$$ 0 0
$$581$$ −44.3059 −1.83812
$$582$$ −40.6717 −1.68590
$$583$$ 24.4917 1.01434
$$584$$ 10.7714 0.445724
$$585$$ 0 0
$$586$$ −21.0517 −0.869639
$$587$$ 3.71178 0.153201 0.0766007 0.997062i $$-0.475593\pi$$
0.0766007 + 0.997062i $$0.475593\pi$$
$$588$$ −27.7558 −1.14463
$$589$$ −18.0534 −0.743879
$$590$$ 0 0
$$591$$ −8.78569 −0.361395
$$592$$ 1.00000 0.0410997
$$593$$ 36.2899 1.49025 0.745124 0.666926i $$-0.232390\pi$$
0.745124 + 0.666926i $$0.232390\pi$$
$$594$$ 18.9035 0.775619
$$595$$ 0 0
$$596$$ −16.7723 −0.687019
$$597$$ 8.16512 0.334176
$$598$$ 6.04184 0.247069
$$599$$ 23.7528 0.970515 0.485257 0.874371i $$-0.338726\pi$$
0.485257 + 0.874371i $$0.338726\pi$$
$$600$$ 0 0
$$601$$ 15.9718 0.651502 0.325751 0.945456i $$-0.394383\pi$$
0.325751 + 0.945456i $$0.394383\pi$$
$$602$$ 41.4107 1.68777
$$603$$ 4.48827 0.182776
$$604$$ −7.59755 −0.309140
$$605$$ 0 0
$$606$$ −50.7490 −2.06154
$$607$$ 38.0139 1.54294 0.771469 0.636267i $$-0.219522\pi$$
0.771469 + 0.636267i $$0.219522\pi$$
$$608$$ −1.85109 −0.0750717
$$609$$ 100.364 4.06694
$$610$$ 0 0
$$611$$ −18.9095 −0.764995
$$612$$ 14.7650 0.596840
$$613$$ −8.40204 −0.339355 −0.169678 0.985500i $$-0.554273\pi$$
−0.169678 + 0.985500i $$0.554273\pi$$
$$614$$ 17.0946 0.689880
$$615$$ 0 0
$$616$$ 19.7578 0.796063
$$617$$ −6.72707 −0.270822 −0.135411 0.990790i $$-0.543235\pi$$
−0.135411 + 0.990790i $$0.543235\pi$$
$$618$$ 36.3161 1.46085
$$619$$ 24.3689 0.979470 0.489735 0.871871i $$-0.337094\pi$$
0.489735 + 0.871871i $$0.337094\pi$$
$$620$$ 0 0
$$621$$ 6.11398 0.245345
$$622$$ 20.0503 0.803943
$$623$$ −30.2287 −1.21109
$$624$$ 10.6941 0.428106
$$625$$ 0 0
$$626$$ 13.3400 0.533172
$$627$$ −24.0965 −0.962322
$$628$$ 4.45631 0.177826
$$629$$ 3.31637 0.132232
$$630$$ 0 0
$$631$$ 34.7737 1.38432 0.692160 0.721744i $$-0.256659\pi$$
0.692160 + 0.721744i $$0.256659\pi$$
$$632$$ −1.19856 −0.0476762
$$633$$ −12.9761 −0.515755
$$634$$ 12.8528 0.510450
$$635$$ 0 0
$$636$$ 14.0209 0.555964
$$637$$ −39.8304 −1.57814
$$638$$ −42.3123 −1.67516
$$639$$ 28.7275 1.13644
$$640$$ 0 0
$$641$$ −27.8544 −1.10018 −0.550091 0.835105i $$-0.685407\pi$$
−0.550091 + 0.835105i $$0.685407\pi$$
$$642$$ −34.0587 −1.34419
$$643$$ −35.4013 −1.39609 −0.698045 0.716054i $$-0.745947\pi$$
−0.698045 + 0.716054i $$0.745947\pi$$
$$644$$ 6.39028 0.251812
$$645$$ 0 0
$$646$$ −6.13890 −0.241532
$$647$$ 16.1343 0.634305 0.317152 0.948375i $$-0.397273\pi$$
0.317152 + 0.948375i $$0.397273\pi$$
$$648$$ −2.53473 −0.0995734
$$649$$ 5.02239 0.197146
$$650$$ 0 0
$$651$$ 110.313 4.32350
$$652$$ 19.4599 0.762110
$$653$$ 11.9146 0.466256 0.233128 0.972446i $$-0.425104\pi$$
0.233128 + 0.972446i $$0.425104\pi$$
$$654$$ 11.8871 0.464823
$$655$$ 0 0
$$656$$ −5.06977 −0.197941
$$657$$ 47.9561 1.87095
$$658$$ −20.0000 −0.779681
$$659$$ −30.5124 −1.18859 −0.594297 0.804246i $$-0.702569\pi$$
−0.594297 + 0.804246i $$0.702569\pi$$
$$660$$ 0 0
$$661$$ 42.0973 1.63740 0.818698 0.574224i $$-0.194696\pi$$
0.818698 + 0.574224i $$0.194696\pi$$
$$662$$ −23.5835 −0.916600
$$663$$ 35.4655 1.37737
$$664$$ 10.6932 0.414977
$$665$$ 0 0
$$666$$ 4.45216 0.172518
$$667$$ −13.6851 −0.529890
$$668$$ 7.13813 0.276183
$$669$$ 18.9899 0.734192
$$670$$ 0 0
$$671$$ 19.7618 0.762897
$$672$$ 11.3108 0.436324
$$673$$ −41.1540 −1.58637 −0.793184 0.608982i $$-0.791578\pi$$
−0.793184 + 0.608982i $$0.791578\pi$$
$$674$$ 26.6074 1.02488
$$675$$ 0 0
$$676$$ 2.34632 0.0902431
$$677$$ −10.3914 −0.399373 −0.199686 0.979860i $$-0.563992\pi$$
−0.199686 + 0.979860i $$0.563992\pi$$
$$678$$ −26.0500 −1.00045
$$679$$ −61.7311 −2.36902
$$680$$ 0 0
$$681$$ 60.8706 2.33257
$$682$$ −46.5068 −1.78084
$$683$$ 5.05345 0.193365 0.0966824 0.995315i $$-0.469177\pi$$
0.0966824 + 0.995315i $$0.469177\pi$$
$$684$$ −8.24137 −0.315117
$$685$$ 0 0
$$686$$ −13.1239 −0.501074
$$687$$ −35.3922 −1.35030
$$688$$ −9.99446 −0.381035
$$689$$ 20.1204 0.766526
$$690$$ 0 0
$$691$$ 13.7857 0.524432 0.262216 0.965009i $$-0.415547\pi$$
0.262216 + 0.965009i $$0.415547\pi$$
$$692$$ −11.6199 −0.441724
$$693$$ 87.9648 3.34151
$$694$$ −17.2626 −0.655280
$$695$$ 0 0
$$696$$ −24.2227 −0.918160
$$697$$ −16.8132 −0.636846
$$698$$ 11.1619 0.422485
$$699$$ 28.4798 1.07721
$$700$$ 0 0
$$701$$ −2.20512 −0.0832862 −0.0416431 0.999133i $$-0.513259\pi$$
−0.0416431 + 0.999133i $$0.513259\pi$$
$$702$$ 15.5296 0.586125
$$703$$ −1.85109 −0.0698153
$$704$$ −4.76853 −0.179721
$$705$$ 0 0
$$706$$ −12.6563 −0.476327
$$707$$ −77.0264 −2.89687
$$708$$ 2.87519 0.108056
$$709$$ −12.7194 −0.477687 −0.238843 0.971058i $$-0.576768\pi$$
−0.238843 + 0.971058i $$0.576768\pi$$
$$710$$ 0 0
$$711$$ −5.33619 −0.200123
$$712$$ 7.29569 0.273418
$$713$$ −15.0418 −0.563318
$$714$$ 37.5108 1.40381
$$715$$ 0 0
$$716$$ −17.2224 −0.643631
$$717$$ −4.33258 −0.161803
$$718$$ −23.2778 −0.868718
$$719$$ 34.9654 1.30399 0.651996 0.758223i $$-0.273932\pi$$
0.651996 + 0.758223i $$0.273932\pi$$
$$720$$ 0 0
$$721$$ 55.1202 2.05278
$$722$$ −15.5735 −0.579584
$$723$$ −44.3810 −1.65055
$$724$$ −23.5098 −0.873733
$$725$$ 0 0
$$726$$ −32.0456 −1.18932
$$727$$ −9.90635 −0.367406 −0.183703 0.982982i $$-0.558809\pi$$
−0.183703 + 0.982982i $$0.558809\pi$$
$$728$$ 16.2314 0.601575
$$729$$ −43.7503 −1.62038
$$730$$ 0 0
$$731$$ −33.1453 −1.22592
$$732$$ 11.3132 0.418146
$$733$$ 37.3826 1.38076 0.690379 0.723448i $$-0.257444\pi$$
0.690379 + 0.723448i $$0.257444\pi$$
$$734$$ −9.09417 −0.335672
$$735$$ 0 0
$$736$$ −1.54229 −0.0568497
$$737$$ −4.80720 −0.177075
$$738$$ −22.5714 −0.830866
$$739$$ 14.4487 0.531506 0.265753 0.964041i $$-0.414380\pi$$
0.265753 + 0.964041i $$0.414380\pi$$
$$740$$ 0 0
$$741$$ −19.7957 −0.727215
$$742$$ 21.2808 0.781242
$$743$$ 45.1826 1.65759 0.828794 0.559553i $$-0.189027\pi$$
0.828794 + 0.559553i $$0.189027\pi$$
$$744$$ −26.6240 −0.976083
$$745$$ 0 0
$$746$$ 5.75976 0.210880
$$747$$ 47.6080 1.74188
$$748$$ −15.8142 −0.578224
$$749$$ −51.6939 −1.88886
$$750$$ 0 0
$$751$$ 49.3127 1.79945 0.899723 0.436461i $$-0.143768\pi$$
0.899723 + 0.436461i $$0.143768\pi$$
$$752$$ 4.82700 0.176022
$$753$$ −40.8077 −1.48711
$$754$$ −34.7603 −1.26590
$$755$$ 0 0
$$756$$ 16.4252 0.597378
$$757$$ 21.5911 0.784741 0.392370 0.919807i $$-0.371655\pi$$
0.392370 + 0.919807i $$0.371655\pi$$
$$758$$ 18.0902 0.657064
$$759$$ −20.0767 −0.728738
$$760$$ 0 0
$$761$$ 18.4253 0.667917 0.333959 0.942588i $$-0.391615\pi$$
0.333959 + 0.942588i $$0.391615\pi$$
$$762$$ −22.7532 −0.824262
$$763$$ 18.0422 0.653170
$$764$$ 1.16660 0.0422063
$$765$$ 0 0
$$766$$ 3.54752 0.128177
$$767$$ 4.12598 0.148981
$$768$$ −2.72987 −0.0985055
$$769$$ 6.50829 0.234695 0.117348 0.993091i $$-0.462561\pi$$
0.117348 + 0.993091i $$0.462561\pi$$
$$770$$ 0 0
$$771$$ −40.9392 −1.47439
$$772$$ −9.36727 −0.337135
$$773$$ −40.5853 −1.45975 −0.729875 0.683581i $$-0.760422\pi$$
−0.729875 + 0.683581i $$0.760422\pi$$
$$774$$ −44.4970 −1.59941
$$775$$ 0 0
$$776$$ 14.8988 0.534835
$$777$$ 11.3108 0.405774
$$778$$ 25.6053 0.917993
$$779$$ 9.38461 0.336239
$$780$$ 0 0
$$781$$ −30.7688 −1.10100
$$782$$ −5.11481 −0.182905
$$783$$ −35.1754 −1.25707
$$784$$ 10.1675 0.363124
$$785$$ 0 0
$$786$$ −10.0622 −0.358907
$$787$$ −37.5389 −1.33812 −0.669059 0.743210i $$-0.733302\pi$$
−0.669059 + 0.743210i $$0.733302\pi$$
$$788$$ 3.21836 0.114649
$$789$$ −62.6804 −2.23148
$$790$$ 0 0
$$791$$ −39.5385 −1.40583
$$792$$ −21.2303 −0.754385
$$793$$ 16.2347 0.576512
$$794$$ 31.4597 1.11646
$$795$$ 0 0
$$796$$ −2.99104 −0.106014
$$797$$ 28.8746 1.02279 0.511396 0.859345i $$-0.329129\pi$$
0.511396 + 0.859345i $$0.329129\pi$$
$$798$$ −20.9374 −0.741176
$$799$$ 16.0081 0.566326
$$800$$ 0 0
$$801$$ 32.4816 1.14768
$$802$$ −22.9940 −0.811945
$$803$$ −51.3638 −1.81259
$$804$$ −2.75200 −0.0970556
$$805$$ 0 0
$$806$$ −38.2062 −1.34576
$$807$$ −27.5433 −0.969571
$$808$$ 18.5903 0.654004
$$809$$ −8.98633 −0.315943 −0.157971 0.987444i $$-0.550495\pi$$
−0.157971 + 0.987444i $$0.550495\pi$$
$$810$$ 0 0
$$811$$ −24.4949 −0.860134 −0.430067 0.902797i $$-0.641510\pi$$
−0.430067 + 0.902797i $$0.641510\pi$$
$$812$$ −36.7650 −1.29020
$$813$$ −9.02706 −0.316593
$$814$$ −4.76853 −0.167137
$$815$$ 0 0
$$816$$ −9.05324 −0.316927
$$817$$ 18.5007 0.647257
$$818$$ −33.3828 −1.16720
$$819$$ 72.2647 2.52513
$$820$$ 0 0
$$821$$ 27.1346 0.947005 0.473502 0.880793i $$-0.342990\pi$$
0.473502 + 0.880793i $$0.342990\pi$$
$$822$$ 33.5848 1.17140
$$823$$ −10.7460 −0.374584 −0.187292 0.982304i $$-0.559971\pi$$
−0.187292 + 0.982304i $$0.559971\pi$$
$$824$$ −13.3033 −0.463441
$$825$$ 0 0
$$826$$ 4.36394 0.151841
$$827$$ 19.1035 0.664293 0.332147 0.943228i $$-0.392227\pi$$
0.332147 + 0.943228i $$0.392227\pi$$
$$828$$ −6.86654 −0.238629
$$829$$ 51.1250 1.77565 0.887823 0.460185i $$-0.152217\pi$$
0.887823 + 0.460185i $$0.152217\pi$$
$$830$$ 0 0
$$831$$ 43.4175 1.50614
$$832$$ −3.91744 −0.135813
$$833$$ 33.7190 1.16830
$$834$$ 2.29975 0.0796340
$$835$$ 0 0
$$836$$ 8.82700 0.305288
$$837$$ −38.6624 −1.33637
$$838$$ 8.32565 0.287605
$$839$$ −42.1947 −1.45672 −0.728361 0.685193i $$-0.759718\pi$$
−0.728361 + 0.685193i $$0.759718\pi$$
$$840$$ 0 0
$$841$$ 49.7342 1.71497
$$842$$ −8.41521 −0.290007
$$843$$ −31.0136 −1.06816
$$844$$ 4.75340 0.163619
$$845$$ 0 0
$$846$$ 21.4906 0.738861
$$847$$ −48.6385 −1.67124
$$848$$ −5.13611 −0.176375
$$849$$ −55.0148 −1.88810
$$850$$ 0 0
$$851$$ −1.54229 −0.0528691
$$852$$ −17.6144 −0.603459
$$853$$ 42.5489 1.45685 0.728423 0.685128i $$-0.240254\pi$$
0.728423 + 0.685128i $$0.240254\pi$$
$$854$$ 17.1710 0.587580
$$855$$ 0 0
$$856$$ 12.4763 0.426432
$$857$$ 29.7588 1.01654 0.508271 0.861197i $$-0.330285\pi$$
0.508271 + 0.861197i $$0.330285\pi$$
$$858$$ −50.9950 −1.74094
$$859$$ 8.40182 0.286666 0.143333 0.989674i $$-0.454218\pi$$
0.143333 + 0.989674i $$0.454218\pi$$
$$860$$ 0 0
$$861$$ −57.3432 −1.95425
$$862$$ 3.04769 0.103805
$$863$$ −36.3794 −1.23837 −0.619185 0.785245i $$-0.712537\pi$$
−0.619185 + 0.785245i $$0.712537\pi$$
$$864$$ −3.96421 −0.134865
$$865$$ 0 0
$$866$$ 22.9439 0.779667
$$867$$ 16.3839 0.556425
$$868$$ −40.4096 −1.37159
$$869$$ 5.71537 0.193881
$$870$$ 0 0
$$871$$ −3.94920 −0.133814
$$872$$ −4.35447 −0.147461
$$873$$ 66.3318 2.24499
$$874$$ 2.85493 0.0965694
$$875$$ 0 0
$$876$$ −29.4045 −0.993486
$$877$$ 41.3380 1.39588 0.697942 0.716154i $$-0.254099\pi$$
0.697942 + 0.716154i $$0.254099\pi$$
$$878$$ 10.3033 0.347720
$$879$$ 57.4684 1.93836
$$880$$ 0 0
$$881$$ 21.1354 0.712071 0.356035 0.934472i $$-0.384128\pi$$
0.356035 + 0.934472i $$0.384128\pi$$
$$882$$ 45.2672 1.52423
$$883$$ −10.7365 −0.361312 −0.180656 0.983546i $$-0.557822\pi$$
−0.180656 + 0.983546i $$0.557822\pi$$
$$884$$ −12.9917 −0.436957
$$885$$ 0 0
$$886$$ −4.84236 −0.162682
$$887$$ −11.1798 −0.375379 −0.187690 0.982228i $$-0.560100\pi$$
−0.187690 + 0.982228i $$0.560100\pi$$
$$888$$ −2.72987 −0.0916083
$$889$$ −34.5346 −1.15825
$$890$$ 0 0
$$891$$ 12.0869 0.404927
$$892$$ −6.95635 −0.232916
$$893$$ −8.93522 −0.299006
$$894$$ 45.7860 1.53131
$$895$$ 0 0
$$896$$ −4.14336 −0.138420
$$897$$ −16.4934 −0.550698
$$898$$ 25.3149 0.844768
$$899$$ 86.5393 2.88625
$$900$$ 0 0
$$901$$ −17.0332 −0.567459
$$902$$ 24.1753 0.804951
$$903$$ −113.046 −3.76192
$$904$$ 9.54261 0.317382
$$905$$ 0 0
$$906$$ 20.7403 0.689050
$$907$$ −20.5343 −0.681831 −0.340915 0.940094i $$-0.610737\pi$$
−0.340915 + 0.940094i $$0.610737\pi$$
$$908$$ −22.2980 −0.739986
$$909$$ 82.7671 2.74521
$$910$$ 0 0
$$911$$ 17.5197 0.580454 0.290227 0.956958i $$-0.406269\pi$$
0.290227 + 0.956958i $$0.406269\pi$$
$$912$$ 5.05324 0.167329
$$913$$ −50.9910 −1.68755
$$914$$ 11.9945 0.396741
$$915$$ 0 0
$$916$$ 12.9648 0.428370
$$917$$ −15.2723 −0.504336
$$918$$ −13.1468 −0.433909
$$919$$ 43.8760 1.44734 0.723668 0.690149i $$-0.242455\pi$$
0.723668 + 0.690149i $$0.242455\pi$$
$$920$$ 0 0
$$921$$ −46.6658 −1.53769
$$922$$ −23.6570 −0.779103
$$923$$ −25.2772 −0.832009
$$924$$ −53.9360 −1.77436
$$925$$ 0 0
$$926$$ −17.3667 −0.570705
$$927$$ −59.2283 −1.94531
$$928$$ 8.87323 0.291278
$$929$$ −21.9800 −0.721142 −0.360571 0.932732i $$-0.617418\pi$$
−0.360571 + 0.932732i $$0.617418\pi$$
$$930$$ 0 0
$$931$$ −18.8209 −0.616831
$$932$$ −10.4327 −0.341734
$$933$$ −54.7346 −1.79193
$$934$$ −20.4476 −0.669065
$$935$$ 0 0
$$936$$ −17.4411 −0.570079
$$937$$ −9.62735 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$938$$ −4.17696 −0.136383
$$939$$ −36.4163 −1.18840
$$940$$ 0 0
$$941$$ −28.3548 −0.924340 −0.462170 0.886791i $$-0.652929\pi$$
−0.462170 + 0.886791i $$0.652929\pi$$
$$942$$ −12.1651 −0.396361
$$943$$ 7.81906 0.254624
$$944$$ −1.05324 −0.0342799
$$945$$ 0 0
$$946$$ 47.6589 1.54952
$$947$$ −12.5179 −0.406778 −0.203389 0.979098i $$-0.565196\pi$$
−0.203389 + 0.979098i $$0.565196\pi$$
$$948$$ 3.27191 0.106267
$$949$$ −42.1963 −1.36975
$$950$$ 0 0
$$951$$ −35.0864 −1.13776
$$952$$ −13.7409 −0.445346
$$953$$ −3.39730 −0.110049 −0.0550247 0.998485i $$-0.517524\pi$$
−0.0550247 + 0.998485i $$0.517524\pi$$
$$954$$ −22.8668 −0.740340
$$955$$ 0 0
$$956$$ 1.58711 0.0513307
$$957$$ 115.507 3.73380
$$958$$ −34.1325 −1.10277
$$959$$ 50.9746 1.64606
$$960$$ 0 0
$$961$$ 64.1182 2.06833
$$962$$ −3.91744 −0.126303
$$963$$ 55.5466 1.78997
$$964$$ 16.2576 0.523622
$$965$$ 0 0
$$966$$ −17.4446 −0.561271
$$967$$ 2.54955 0.0819879 0.0409940 0.999159i $$-0.486948\pi$$
0.0409940 + 0.999159i $$0.486948\pi$$
$$968$$ 11.7389 0.377302
$$969$$ 16.7584 0.538357
$$970$$ 0 0
$$971$$ −35.2223 −1.13034 −0.565168 0.824976i $$-0.691189\pi$$
−0.565168 + 0.824976i $$0.691189\pi$$
$$972$$ 18.8121 0.603398
$$973$$ 3.49054 0.111902
$$974$$ 6.16917 0.197673
$$975$$ 0 0
$$976$$ −4.14422 −0.132653
$$977$$ −5.79786 −0.185490 −0.0927450 0.995690i $$-0.529564\pi$$
−0.0927450 + 0.995690i $$0.529564\pi$$
$$978$$ −53.1230 −1.69869
$$979$$ −34.7897 −1.11188
$$980$$ 0 0
$$981$$ −19.3868 −0.618973
$$982$$ −11.8290 −0.377479
$$983$$ −20.6825 −0.659671 −0.329835 0.944038i $$-0.606993\pi$$
−0.329835 + 0.944038i $$0.606993\pi$$
$$984$$ 13.8398 0.441196
$$985$$ 0 0
$$986$$ 29.4269 0.937143
$$987$$ 54.5973 1.73785
$$988$$ 7.25154 0.230702
$$989$$ 15.4144 0.490149
$$990$$ 0 0
$$991$$ 49.7158 1.57927 0.789637 0.613574i $$-0.210269\pi$$
0.789637 + 0.613574i $$0.210269\pi$$
$$992$$ 9.75286 0.309654
$$993$$ 64.3799 2.04303
$$994$$ −26.7350 −0.847981
$$995$$ 0 0
$$996$$ −29.1911 −0.924954
$$997$$ −53.1550 −1.68344 −0.841718 0.539918i $$-0.818455\pi$$
−0.841718 + 0.539918i $$0.818455\pi$$
$$998$$ 31.4004 0.993964
$$999$$ −3.96421 −0.125422
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.be.1.1 5
5.2 odd 4 370.2.b.d.149.10 yes 10
5.3 odd 4 370.2.b.d.149.1 10
5.4 even 2 1850.2.a.bd.1.5 5
15.2 even 4 3330.2.d.p.1999.5 10
15.8 even 4 3330.2.d.p.1999.10 10

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.1 10 5.3 odd 4
370.2.b.d.149.10 yes 10 5.2 odd 4
1850.2.a.bd.1.5 5 5.4 even 2
1850.2.a.be.1.1 5 1.1 even 1 trivial
3330.2.d.p.1999.5 10 15.2 even 4
3330.2.d.p.1999.10 10 15.8 even 4