# Properties

 Label 209.2.a.d.1.6 Level $209$ Weight $2$ Character 209.1 Self dual yes Analytic conductor $1.669$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [209,2,Mod(1,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, 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("209.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.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: $$1.66887340224$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 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 1.6 Root $$-2.03821$$ of defining polynomial Character $$\chi$$ $$=$$ 209.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.03821 q^{2} +1.87275 q^{3} +2.15429 q^{4} -3.24760 q^{5} +3.81704 q^{6} +1.92338 q^{7} +0.314472 q^{8} +0.507178 q^{9} +O(q^{10})$$ $$q+2.03821 q^{2} +1.87275 q^{3} +2.15429 q^{4} -3.24760 q^{5} +3.81704 q^{6} +1.92338 q^{7} +0.314472 q^{8} +0.507178 q^{9} -6.61928 q^{10} -1.00000 q^{11} +4.03444 q^{12} +2.85122 q^{13} +3.92024 q^{14} -6.08193 q^{15} -3.66762 q^{16} -2.33033 q^{17} +1.03373 q^{18} +1.00000 q^{19} -6.99626 q^{20} +3.60199 q^{21} -2.03821 q^{22} -2.74653 q^{23} +0.588926 q^{24} +5.54689 q^{25} +5.81138 q^{26} -4.66842 q^{27} +4.14350 q^{28} -0.972965 q^{29} -12.3962 q^{30} -0.00551178 q^{31} -8.10431 q^{32} -1.87275 q^{33} -4.74970 q^{34} -6.24635 q^{35} +1.09261 q^{36} +9.67124 q^{37} +2.03821 q^{38} +5.33962 q^{39} -1.02128 q^{40} +6.65137 q^{41} +7.34161 q^{42} +7.99413 q^{43} -2.15429 q^{44} -1.64711 q^{45} -5.59800 q^{46} +3.46982 q^{47} -6.86852 q^{48} -3.30063 q^{49} +11.3057 q^{50} -4.36412 q^{51} +6.14236 q^{52} +10.5493 q^{53} -9.51521 q^{54} +3.24760 q^{55} +0.604847 q^{56} +1.87275 q^{57} -1.98311 q^{58} -13.7814 q^{59} -13.1022 q^{60} +3.74608 q^{61} -0.0112342 q^{62} +0.975494 q^{63} -9.18303 q^{64} -9.25963 q^{65} -3.81704 q^{66} -3.97172 q^{67} -5.02021 q^{68} -5.14356 q^{69} -12.7314 q^{70} +14.2688 q^{71} +0.159493 q^{72} -13.2263 q^{73} +19.7120 q^{74} +10.3879 q^{75} +2.15429 q^{76} -1.92338 q^{77} +10.8832 q^{78} -1.87656 q^{79} +11.9109 q^{80} -10.2643 q^{81} +13.5569 q^{82} -10.9619 q^{83} +7.75973 q^{84} +7.56799 q^{85} +16.2937 q^{86} -1.82212 q^{87} -0.314472 q^{88} +15.0195 q^{89} -3.35715 q^{90} +5.48397 q^{91} -5.91682 q^{92} -0.0103222 q^{93} +7.07220 q^{94} -3.24760 q^{95} -15.1773 q^{96} -7.57248 q^{97} -6.72736 q^{98} -0.507178 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 9 q^{8} + 11 q^{9}+O(q^{10})$$ 7 * q - q^2 + 2 * q^3 + 15 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 - 9 * q^8 + 11 * q^9 $$7 q - q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 9 q^{8} + 11 q^{9} - 6 q^{10} - 7 q^{11} - 16 q^{12} - 4 q^{13} + 6 q^{14} + 12 q^{15} + 27 q^{16} + 2 q^{17} + 9 q^{18} + 7 q^{19} - 4 q^{20} - 14 q^{21} + q^{22} + 10 q^{23} - 2 q^{24} + 9 q^{25} - 8 q^{26} - 4 q^{27} + 26 q^{28} - 18 q^{29} - 42 q^{30} + 24 q^{31} - 49 q^{32} - 2 q^{33} - 6 q^{34} + 8 q^{35} + 29 q^{36} - q^{38} + 24 q^{39} - 2 q^{40} - 12 q^{41} - 44 q^{42} + 2 q^{43} - 15 q^{44} - 4 q^{45} - 4 q^{46} + 8 q^{47} - 72 q^{48} + 17 q^{49} - 33 q^{50} - 24 q^{51} - 60 q^{52} + 2 q^{53} - 52 q^{54} - 2 q^{55} + 26 q^{56} + 2 q^{57} - 8 q^{58} - 10 q^{59} + 42 q^{60} + 14 q^{61} + 14 q^{62} + 55 q^{64} - 14 q^{65} + 2 q^{66} + 8 q^{67} - 18 q^{68} - 6 q^{69} - 66 q^{70} + 10 q^{71} + 53 q^{72} - 6 q^{73} + 26 q^{74} + 26 q^{75} + 15 q^{76} - 10 q^{77} + 22 q^{78} + 52 q^{79} - 12 q^{80} - q^{81} + 24 q^{82} - 10 q^{83} - 6 q^{84} - 12 q^{85} + 8 q^{86} + 6 q^{87} + 9 q^{88} + 20 q^{90} + 12 q^{91} + 2 q^{93} + 24 q^{94} + 2 q^{95} + 6 q^{96} - 24 q^{97} + 19 q^{98} - 11 q^{99}+O(q^{100})$$ 7 * q - q^2 + 2 * q^3 + 15 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 - 9 * q^8 + 11 * q^9 - 6 * q^10 - 7 * q^11 - 16 * q^12 - 4 * q^13 + 6 * q^14 + 12 * q^15 + 27 * q^16 + 2 * q^17 + 9 * q^18 + 7 * q^19 - 4 * q^20 - 14 * q^21 + q^22 + 10 * q^23 - 2 * q^24 + 9 * q^25 - 8 * q^26 - 4 * q^27 + 26 * q^28 - 18 * q^29 - 42 * q^30 + 24 * q^31 - 49 * q^32 - 2 * q^33 - 6 * q^34 + 8 * q^35 + 29 * q^36 - q^38 + 24 * q^39 - 2 * q^40 - 12 * q^41 - 44 * q^42 + 2 * q^43 - 15 * q^44 - 4 * q^45 - 4 * q^46 + 8 * q^47 - 72 * q^48 + 17 * q^49 - 33 * q^50 - 24 * q^51 - 60 * q^52 + 2 * q^53 - 52 * q^54 - 2 * q^55 + 26 * q^56 + 2 * q^57 - 8 * q^58 - 10 * q^59 + 42 * q^60 + 14 * q^61 + 14 * q^62 + 55 * q^64 - 14 * q^65 + 2 * q^66 + 8 * q^67 - 18 * q^68 - 6 * q^69 - 66 * q^70 + 10 * q^71 + 53 * q^72 - 6 * q^73 + 26 * q^74 + 26 * q^75 + 15 * q^76 - 10 * q^77 + 22 * q^78 + 52 * q^79 - 12 * q^80 - q^81 + 24 * q^82 - 10 * q^83 - 6 * q^84 - 12 * q^85 + 8 * q^86 + 6 * q^87 + 9 * q^88 + 20 * q^90 + 12 * q^91 + 2 * q^93 + 24 * q^94 + 2 * q^95 + 6 * q^96 - 24 * q^97 + 19 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.03821 1.44123 0.720615 0.693335i $$-0.243860\pi$$
0.720615 + 0.693335i $$0.243860\pi$$
$$3$$ 1.87275 1.08123 0.540615 0.841270i $$-0.318191\pi$$
0.540615 + 0.841270i $$0.318191\pi$$
$$4$$ 2.15429 1.07714
$$5$$ −3.24760 −1.45237 −0.726185 0.687499i $$-0.758708\pi$$
−0.726185 + 0.687499i $$0.758708\pi$$
$$6$$ 3.81704 1.55830
$$7$$ 1.92338 0.726967 0.363484 0.931601i $$-0.381587\pi$$
0.363484 + 0.931601i $$0.381587\pi$$
$$8$$ 0.314472 0.111183
$$9$$ 0.507178 0.169059
$$10$$ −6.61928 −2.09320
$$11$$ −1.00000 −0.301511
$$12$$ 4.03444 1.16464
$$13$$ 2.85122 0.790787 0.395394 0.918512i $$-0.370608\pi$$
0.395394 + 0.918512i $$0.370608\pi$$
$$14$$ 3.92024 1.04773
$$15$$ −6.08193 −1.57035
$$16$$ −3.66762 −0.916905
$$17$$ −2.33033 −0.565189 −0.282594 0.959239i $$-0.591195\pi$$
−0.282594 + 0.959239i $$0.591195\pi$$
$$18$$ 1.03373 0.243653
$$19$$ 1.00000 0.229416
$$20$$ −6.99626 −1.56441
$$21$$ 3.60199 0.786019
$$22$$ −2.03821 −0.434547
$$23$$ −2.74653 −0.572691 −0.286346 0.958126i $$-0.592441\pi$$
−0.286346 + 0.958126i $$0.592441\pi$$
$$24$$ 0.588926 0.120214
$$25$$ 5.54689 1.10938
$$26$$ 5.81138 1.13971
$$27$$ −4.66842 −0.898438
$$28$$ 4.14350 0.783049
$$29$$ −0.972965 −0.180675 −0.0903376 0.995911i $$-0.528795\pi$$
−0.0903376 + 0.995911i $$0.528795\pi$$
$$30$$ −12.3962 −2.26323
$$31$$ −0.00551178 −0.000989945 0 −0.000494973 1.00000i $$-0.500158\pi$$
−0.000494973 1.00000i $$0.500158\pi$$
$$32$$ −8.10431 −1.43265
$$33$$ −1.87275 −0.326003
$$34$$ −4.74970 −0.814567
$$35$$ −6.24635 −1.05583
$$36$$ 1.09261 0.182101
$$37$$ 9.67124 1.58994 0.794971 0.606647i $$-0.207486\pi$$
0.794971 + 0.606647i $$0.207486\pi$$
$$38$$ 2.03821 0.330641
$$39$$ 5.33962 0.855023
$$40$$ −1.02128 −0.161478
$$41$$ 6.65137 1.03877 0.519385 0.854540i $$-0.326161\pi$$
0.519385 + 0.854540i $$0.326161\pi$$
$$42$$ 7.34161 1.13283
$$43$$ 7.99413 1.21909 0.609547 0.792750i $$-0.291352\pi$$
0.609547 + 0.792750i $$0.291352\pi$$
$$44$$ −2.15429 −0.324771
$$45$$ −1.64711 −0.245537
$$46$$ −5.59800 −0.825380
$$47$$ 3.46982 0.506125 0.253062 0.967450i $$-0.418562\pi$$
0.253062 + 0.967450i $$0.418562\pi$$
$$48$$ −6.86852 −0.991385
$$49$$ −3.30063 −0.471518
$$50$$ 11.3057 1.59887
$$51$$ −4.36412 −0.611100
$$52$$ 6.14236 0.851792
$$53$$ 10.5493 1.44905 0.724526 0.689247i $$-0.242058\pi$$
0.724526 + 0.689247i $$0.242058\pi$$
$$54$$ −9.51521 −1.29486
$$55$$ 3.24760 0.437906
$$56$$ 0.604847 0.0808261
$$57$$ 1.87275 0.248051
$$58$$ −1.98311 −0.260394
$$59$$ −13.7814 −1.79419 −0.897096 0.441836i $$-0.854327\pi$$
−0.897096 + 0.441836i $$0.854327\pi$$
$$60$$ −13.1022 −1.69149
$$61$$ 3.74608 0.479636 0.239818 0.970818i $$-0.422912\pi$$
0.239818 + 0.970818i $$0.422912\pi$$
$$62$$ −0.0112342 −0.00142674
$$63$$ 0.975494 0.122901
$$64$$ −9.18303 −1.14788
$$65$$ −9.25963 −1.14852
$$66$$ −3.81704 −0.469846
$$67$$ −3.97172 −0.485223 −0.242612 0.970124i $$-0.578004\pi$$
−0.242612 + 0.970124i $$0.578004\pi$$
$$68$$ −5.02021 −0.608790
$$69$$ −5.14356 −0.619211
$$70$$ −12.7314 −1.52169
$$71$$ 14.2688 1.69339 0.846695 0.532078i $$-0.178589\pi$$
0.846695 + 0.532078i $$0.178589\pi$$
$$72$$ 0.159493 0.0187965
$$73$$ −13.2263 −1.54803 −0.774013 0.633170i $$-0.781753\pi$$
−0.774013 + 0.633170i $$0.781753\pi$$
$$74$$ 19.7120 2.29147
$$75$$ 10.3879 1.19949
$$76$$ 2.15429 0.247114
$$77$$ −1.92338 −0.219189
$$78$$ 10.8832 1.23229
$$79$$ −1.87656 −0.211130 −0.105565 0.994412i $$-0.533665\pi$$
−0.105565 + 0.994412i $$0.533665\pi$$
$$80$$ 11.9109 1.33168
$$81$$ −10.2643 −1.14048
$$82$$ 13.5569 1.49711
$$83$$ −10.9619 −1.20322 −0.601612 0.798789i $$-0.705474\pi$$
−0.601612 + 0.798789i $$0.705474\pi$$
$$84$$ 7.75973 0.846656
$$85$$ 7.56799 0.820863
$$86$$ 16.2937 1.75699
$$87$$ −1.82212 −0.195351
$$88$$ −0.314472 −0.0335228
$$89$$ 15.0195 1.59207 0.796034 0.605253i $$-0.206928\pi$$
0.796034 + 0.605253i $$0.206928\pi$$
$$90$$ −3.35715 −0.353875
$$91$$ 5.48397 0.574876
$$92$$ −5.91682 −0.616871
$$93$$ −0.0103222 −0.00107036
$$94$$ 7.07220 0.729442
$$95$$ −3.24760 −0.333196
$$96$$ −15.1773 −1.54903
$$97$$ −7.57248 −0.768869 −0.384434 0.923152i $$-0.625604\pi$$
−0.384434 + 0.923152i $$0.625604\pi$$
$$98$$ −6.72736 −0.679566
$$99$$ −0.507178 −0.0509733
$$100$$ 11.9496 1.19496
$$101$$ −15.0513 −1.49766 −0.748831 0.662761i $$-0.769385\pi$$
−0.748831 + 0.662761i $$0.769385\pi$$
$$102$$ −8.89499 −0.880735
$$103$$ −0.543451 −0.0535478 −0.0267739 0.999642i $$-0.508523\pi$$
−0.0267739 + 0.999642i $$0.508523\pi$$
$$104$$ 0.896629 0.0879218
$$105$$ −11.6978 −1.14159
$$106$$ 21.5016 2.08842
$$107$$ 14.7371 1.42469 0.712344 0.701831i $$-0.247634\pi$$
0.712344 + 0.701831i $$0.247634\pi$$
$$108$$ −10.0571 −0.967748
$$109$$ −17.3711 −1.66385 −0.831925 0.554888i $$-0.812761\pi$$
−0.831925 + 0.554888i $$0.812761\pi$$
$$110$$ 6.61928 0.631123
$$111$$ 18.1118 1.71909
$$112$$ −7.05421 −0.666560
$$113$$ 12.8865 1.21226 0.606132 0.795364i $$-0.292720\pi$$
0.606132 + 0.795364i $$0.292720\pi$$
$$114$$ 3.81704 0.357499
$$115$$ 8.91963 0.831760
$$116$$ −2.09605 −0.194613
$$117$$ 1.44608 0.133690
$$118$$ −28.0894 −2.58584
$$119$$ −4.48211 −0.410874
$$120$$ −1.91259 −0.174595
$$121$$ 1.00000 0.0909091
$$122$$ 7.63529 0.691266
$$123$$ 12.4563 1.12315
$$124$$ −0.0118740 −0.00106631
$$125$$ −1.77608 −0.158858
$$126$$ 1.98826 0.177128
$$127$$ 15.4342 1.36957 0.684784 0.728746i $$-0.259897\pi$$
0.684784 + 0.728746i $$0.259897\pi$$
$$128$$ −2.50829 −0.221704
$$129$$ 14.9710 1.31812
$$130$$ −18.8730 −1.65527
$$131$$ −12.0655 −1.05417 −0.527083 0.849814i $$-0.676714\pi$$
−0.527083 + 0.849814i $$0.676714\pi$$
$$132$$ −4.03444 −0.351153
$$133$$ 1.92338 0.166778
$$134$$ −8.09519 −0.699318
$$135$$ 15.1612 1.30486
$$136$$ −0.732824 −0.0628392
$$137$$ −5.53253 −0.472676 −0.236338 0.971671i $$-0.575947\pi$$
−0.236338 + 0.971671i $$0.575947\pi$$
$$138$$ −10.4836 −0.892426
$$139$$ −8.66764 −0.735180 −0.367590 0.929988i $$-0.619817\pi$$
−0.367590 + 0.929988i $$0.619817\pi$$
$$140$$ −13.4564 −1.13728
$$141$$ 6.49808 0.547237
$$142$$ 29.0827 2.44057
$$143$$ −2.85122 −0.238431
$$144$$ −1.86014 −0.155011
$$145$$ 3.15980 0.262407
$$146$$ −26.9580 −2.23106
$$147$$ −6.18124 −0.509820
$$148$$ 20.8346 1.71260
$$149$$ −19.3027 −1.58134 −0.790671 0.612241i $$-0.790268\pi$$
−0.790671 + 0.612241i $$0.790268\pi$$
$$150$$ 21.1727 1.72875
$$151$$ 8.71384 0.709122 0.354561 0.935033i $$-0.384630\pi$$
0.354561 + 0.935033i $$0.384630\pi$$
$$152$$ 0.314472 0.0255070
$$153$$ −1.18189 −0.0955505
$$154$$ −3.92024 −0.315902
$$155$$ 0.0179000 0.00143777
$$156$$ 11.5031 0.920983
$$157$$ −5.86640 −0.468189 −0.234095 0.972214i $$-0.575213\pi$$
−0.234095 + 0.972214i $$0.575213\pi$$
$$158$$ −3.82483 −0.304287
$$159$$ 19.7561 1.56676
$$160$$ 26.3195 2.08074
$$161$$ −5.28261 −0.416328
$$162$$ −20.9208 −1.64369
$$163$$ 14.8802 1.16551 0.582753 0.812649i $$-0.301975\pi$$
0.582753 + 0.812649i $$0.301975\pi$$
$$164$$ 14.3290 1.11891
$$165$$ 6.08193 0.473477
$$166$$ −22.3426 −1.73412
$$167$$ 4.18971 0.324209 0.162105 0.986774i $$-0.448172\pi$$
0.162105 + 0.986774i $$0.448172\pi$$
$$168$$ 1.13273 0.0873917
$$169$$ −4.87053 −0.374656
$$170$$ 15.4251 1.18305
$$171$$ 0.507178 0.0387849
$$172$$ 17.2217 1.31314
$$173$$ 0.707136 0.0537626 0.0268813 0.999639i $$-0.491442\pi$$
0.0268813 + 0.999639i $$0.491442\pi$$
$$174$$ −3.71385 −0.281546
$$175$$ 10.6688 0.806482
$$176$$ 3.66762 0.276457
$$177$$ −25.8091 −1.93993
$$178$$ 30.6129 2.29454
$$179$$ −21.7962 −1.62913 −0.814563 0.580076i $$-0.803023\pi$$
−0.814563 + 0.580076i $$0.803023\pi$$
$$180$$ −3.54835 −0.264478
$$181$$ −2.93416 −0.218094 −0.109047 0.994037i $$-0.534780\pi$$
−0.109047 + 0.994037i $$0.534780\pi$$
$$182$$ 11.1775 0.828529
$$183$$ 7.01546 0.518598
$$184$$ −0.863707 −0.0636733
$$185$$ −31.4083 −2.30918
$$186$$ −0.0210387 −0.00154263
$$187$$ 2.33033 0.170411
$$188$$ 7.47498 0.545169
$$189$$ −8.97913 −0.653135
$$190$$ −6.61928 −0.480213
$$191$$ 22.4018 1.62094 0.810468 0.585783i $$-0.199213\pi$$
0.810468 + 0.585783i $$0.199213\pi$$
$$192$$ −17.1975 −1.24112
$$193$$ 2.55447 0.183875 0.0919375 0.995765i $$-0.470694\pi$$
0.0919375 + 0.995765i $$0.470694\pi$$
$$194$$ −15.4343 −1.10812
$$195$$ −17.3409 −1.24181
$$196$$ −7.11051 −0.507893
$$197$$ −12.6968 −0.904607 −0.452303 0.891864i $$-0.649398\pi$$
−0.452303 + 0.891864i $$0.649398\pi$$
$$198$$ −1.03373 −0.0734643
$$199$$ 10.1553 0.719892 0.359946 0.932973i $$-0.382795\pi$$
0.359946 + 0.932973i $$0.382795\pi$$
$$200$$ 1.74434 0.123344
$$201$$ −7.43803 −0.524638
$$202$$ −30.6777 −2.15848
$$203$$ −1.87138 −0.131345
$$204$$ −9.40158 −0.658242
$$205$$ −21.6010 −1.50868
$$206$$ −1.10767 −0.0771747
$$207$$ −1.39298 −0.0968188
$$208$$ −10.4572 −0.725076
$$209$$ −1.00000 −0.0691714
$$210$$ −23.8426 −1.64530
$$211$$ −8.25858 −0.568544 −0.284272 0.958744i $$-0.591752\pi$$
−0.284272 + 0.958744i $$0.591752\pi$$
$$212$$ 22.7262 1.56084
$$213$$ 26.7218 1.83095
$$214$$ 30.0372 2.05330
$$215$$ −25.9617 −1.77057
$$216$$ −1.46809 −0.0998907
$$217$$ −0.0106012 −0.000719658 0
$$218$$ −35.4059 −2.39799
$$219$$ −24.7696 −1.67377
$$220$$ 6.99626 0.471688
$$221$$ −6.64430 −0.446944
$$222$$ 36.9156 2.47761
$$223$$ −21.2289 −1.42159 −0.710797 0.703398i $$-0.751665\pi$$
−0.710797 + 0.703398i $$0.751665\pi$$
$$224$$ −15.5876 −1.04149
$$225$$ 2.81326 0.187551
$$226$$ 26.2655 1.74715
$$227$$ −9.06652 −0.601766 −0.300883 0.953661i $$-0.597281\pi$$
−0.300883 + 0.953661i $$0.597281\pi$$
$$228$$ 4.03444 0.267187
$$229$$ −5.25556 −0.347297 −0.173649 0.984808i $$-0.555556\pi$$
−0.173649 + 0.984808i $$0.555556\pi$$
$$230$$ 18.1800 1.19876
$$231$$ −3.60199 −0.236994
$$232$$ −0.305970 −0.0200879
$$233$$ −5.68870 −0.372679 −0.186340 0.982485i $$-0.559662\pi$$
−0.186340 + 0.982485i $$0.559662\pi$$
$$234$$ 2.94741 0.192678
$$235$$ −11.2686 −0.735080
$$236$$ −29.6892 −1.93260
$$237$$ −3.51433 −0.228280
$$238$$ −9.13546 −0.592164
$$239$$ −20.3787 −1.31819 −0.659094 0.752060i $$-0.729060\pi$$
−0.659094 + 0.752060i $$0.729060\pi$$
$$240$$ 22.3062 1.43986
$$241$$ 17.6930 1.13971 0.569854 0.821746i $$-0.307000\pi$$
0.569854 + 0.821746i $$0.307000\pi$$
$$242$$ 2.03821 0.131021
$$243$$ −5.21717 −0.334682
$$244$$ 8.07014 0.516638
$$245$$ 10.7191 0.684819
$$246$$ 25.3886 1.61872
$$247$$ 2.85122 0.181419
$$248$$ −0.00173330 −0.000110065 0
$$249$$ −20.5288 −1.30096
$$250$$ −3.62002 −0.228950
$$251$$ −0.776543 −0.0490149 −0.0245075 0.999700i $$-0.507802\pi$$
−0.0245075 + 0.999700i $$0.507802\pi$$
$$252$$ 2.10149 0.132382
$$253$$ 2.74653 0.172673
$$254$$ 31.4582 1.97386
$$255$$ 14.1729 0.887543
$$256$$ 13.2536 0.828352
$$257$$ 29.2762 1.82620 0.913100 0.407736i $$-0.133682\pi$$
0.913100 + 0.407736i $$0.133682\pi$$
$$258$$ 30.5139 1.89972
$$259$$ 18.6014 1.15584
$$260$$ −19.9479 −1.23712
$$261$$ −0.493467 −0.0305448
$$262$$ −24.5919 −1.51929
$$263$$ −5.90041 −0.363835 −0.181918 0.983314i $$-0.558230\pi$$
−0.181918 + 0.983314i $$0.558230\pi$$
$$264$$ −0.588926 −0.0362459
$$265$$ −34.2598 −2.10456
$$266$$ 3.92024 0.240365
$$267$$ 28.1278 1.72139
$$268$$ −8.55623 −0.522655
$$269$$ −10.7278 −0.654087 −0.327044 0.945009i $$-0.606052\pi$$
−0.327044 + 0.945009i $$0.606052\pi$$
$$270$$ 30.9016 1.88061
$$271$$ 16.2707 0.988375 0.494188 0.869355i $$-0.335466\pi$$
0.494188 + 0.869355i $$0.335466\pi$$
$$272$$ 8.54677 0.518224
$$273$$ 10.2701 0.621574
$$274$$ −11.2765 −0.681235
$$275$$ −5.54689 −0.334490
$$276$$ −11.0807 −0.666980
$$277$$ 6.77040 0.406794 0.203397 0.979096i $$-0.434802\pi$$
0.203397 + 0.979096i $$0.434802\pi$$
$$278$$ −17.6664 −1.05956
$$279$$ −0.00279545 −0.000167359 0
$$280$$ −1.96430 −0.117389
$$281$$ −17.1455 −1.02281 −0.511407 0.859339i $$-0.670876\pi$$
−0.511407 + 0.859339i $$0.670876\pi$$
$$282$$ 13.2444 0.788695
$$283$$ −2.94787 −0.175232 −0.0876162 0.996154i $$-0.527925\pi$$
−0.0876162 + 0.996154i $$0.527925\pi$$
$$284$$ 30.7390 1.82403
$$285$$ −6.08193 −0.360262
$$286$$ −5.81138 −0.343634
$$287$$ 12.7931 0.755152
$$288$$ −4.11033 −0.242203
$$289$$ −11.5695 −0.680561
$$290$$ 6.44033 0.378189
$$291$$ −14.1813 −0.831325
$$292$$ −28.4933 −1.66745
$$293$$ 2.57851 0.150638 0.0753192 0.997159i $$-0.476002\pi$$
0.0753192 + 0.997159i $$0.476002\pi$$
$$294$$ −12.5986 −0.734768
$$295$$ 44.7566 2.60583
$$296$$ 3.04133 0.176774
$$297$$ 4.66842 0.270889
$$298$$ −39.3430 −2.27908
$$299$$ −7.83097 −0.452877
$$300$$ 22.3786 1.29203
$$301$$ 15.3757 0.886241
$$302$$ 17.7606 1.02201
$$303$$ −28.1873 −1.61932
$$304$$ −3.66762 −0.210352
$$305$$ −12.1658 −0.696610
$$306$$ −2.40895 −0.137710
$$307$$ −19.7888 −1.12941 −0.564704 0.825293i $$-0.691010\pi$$
−0.564704 + 0.825293i $$0.691010\pi$$
$$308$$ −4.14350 −0.236098
$$309$$ −1.01775 −0.0578975
$$310$$ 0.0364840 0.00207215
$$311$$ 19.7979 1.12264 0.561319 0.827599i $$-0.310294\pi$$
0.561319 + 0.827599i $$0.310294\pi$$
$$312$$ 1.67916 0.0950637
$$313$$ 10.9847 0.620890 0.310445 0.950591i $$-0.399522\pi$$
0.310445 + 0.950591i $$0.399522\pi$$
$$314$$ −11.9569 −0.674769
$$315$$ −3.16801 −0.178497
$$316$$ −4.04266 −0.227417
$$317$$ 9.88351 0.555113 0.277557 0.960709i $$-0.410475\pi$$
0.277557 + 0.960709i $$0.410475\pi$$
$$318$$ 40.2670 2.25806
$$319$$ 0.972965 0.0544756
$$320$$ 29.8228 1.66714
$$321$$ 27.5988 1.54042
$$322$$ −10.7671 −0.600024
$$323$$ −2.33033 −0.129663
$$324$$ −22.1123 −1.22846
$$325$$ 15.8154 0.877282
$$326$$ 30.3289 1.67976
$$327$$ −32.5317 −1.79901
$$328$$ 2.09167 0.115493
$$329$$ 6.67376 0.367936
$$330$$ 12.3962 0.682390
$$331$$ 25.7597 1.41588 0.707942 0.706271i $$-0.249624\pi$$
0.707942 + 0.706271i $$0.249624\pi$$
$$332$$ −23.6151 −1.29604
$$333$$ 4.90504 0.268795
$$334$$ 8.53949 0.467260
$$335$$ 12.8986 0.704723
$$336$$ −13.2107 −0.720705
$$337$$ 21.8924 1.19256 0.596278 0.802778i $$-0.296646\pi$$
0.596278 + 0.802778i $$0.296646\pi$$
$$338$$ −9.92714 −0.539965
$$339$$ 24.1332 1.31074
$$340$$ 16.3036 0.884188
$$341$$ 0.00551178 0.000298480 0
$$342$$ 1.03373 0.0558979
$$343$$ −19.8120 −1.06975
$$344$$ 2.51393 0.135542
$$345$$ 16.7042 0.899324
$$346$$ 1.44129 0.0774842
$$347$$ 19.2857 1.03531 0.517656 0.855589i $$-0.326805\pi$$
0.517656 + 0.855589i $$0.326805\pi$$
$$348$$ −3.92537 −0.210422
$$349$$ 15.5220 0.830872 0.415436 0.909622i $$-0.363629\pi$$
0.415436 + 0.909622i $$0.363629\pi$$
$$350$$ 21.7451 1.16233
$$351$$ −13.3107 −0.710473
$$352$$ 8.10431 0.431961
$$353$$ −8.92859 −0.475221 −0.237610 0.971361i $$-0.576364\pi$$
−0.237610 + 0.971361i $$0.576364\pi$$
$$354$$ −52.6044 −2.79589
$$355$$ −46.3392 −2.45943
$$356$$ 32.3564 1.71489
$$357$$ −8.39385 −0.444249
$$358$$ −44.4252 −2.34794
$$359$$ 26.2672 1.38633 0.693165 0.720779i $$-0.256216\pi$$
0.693165 + 0.720779i $$0.256216\pi$$
$$360$$ −0.517970 −0.0272994
$$361$$ 1.00000 0.0526316
$$362$$ −5.98042 −0.314324
$$363$$ 1.87275 0.0982937
$$364$$ 11.8141 0.619225
$$365$$ 42.9538 2.24830
$$366$$ 14.2990 0.747418
$$367$$ 10.2560 0.535358 0.267679 0.963508i $$-0.413743\pi$$
0.267679 + 0.963508i $$0.413743\pi$$
$$368$$ 10.0732 0.525103
$$369$$ 3.37343 0.175614
$$370$$ −64.0166 −3.32807
$$371$$ 20.2902 1.05341
$$372$$ −0.0222369 −0.00115293
$$373$$ −20.2242 −1.04717 −0.523584 0.851974i $$-0.675405\pi$$
−0.523584 + 0.851974i $$0.675405\pi$$
$$374$$ 4.74970 0.245601
$$375$$ −3.32615 −0.171762
$$376$$ 1.09116 0.0562722
$$377$$ −2.77414 −0.142876
$$378$$ −18.3013 −0.941318
$$379$$ 14.1534 0.727011 0.363505 0.931592i $$-0.381580\pi$$
0.363505 + 0.931592i $$0.381580\pi$$
$$380$$ −6.99626 −0.358901
$$381$$ 28.9044 1.48082
$$382$$ 45.6595 2.33614
$$383$$ 12.3217 0.629611 0.314806 0.949156i $$-0.398061\pi$$
0.314806 + 0.949156i $$0.398061\pi$$
$$384$$ −4.69739 −0.239713
$$385$$ 6.24635 0.318343
$$386$$ 5.20655 0.265006
$$387$$ 4.05445 0.206099
$$388$$ −16.3133 −0.828183
$$389$$ 33.9248 1.72006 0.860029 0.510245i $$-0.170445\pi$$
0.860029 + 0.510245i $$0.170445\pi$$
$$390$$ −35.3444 −1.78973
$$391$$ 6.40033 0.323679
$$392$$ −1.03795 −0.0524246
$$393$$ −22.5956 −1.13980
$$394$$ −25.8786 −1.30375
$$395$$ 6.09432 0.306639
$$396$$ −1.09261 −0.0549056
$$397$$ −20.3357 −1.02062 −0.510309 0.859991i $$-0.670469\pi$$
−0.510309 + 0.859991i $$0.670469\pi$$
$$398$$ 20.6987 1.03753
$$399$$ 3.60199 0.180325
$$400$$ −20.3439 −1.01719
$$401$$ −30.6815 −1.53216 −0.766080 0.642746i $$-0.777795\pi$$
−0.766080 + 0.642746i $$0.777795\pi$$
$$402$$ −15.1602 −0.756124
$$403$$ −0.0157153 −0.000782836 0
$$404$$ −32.4249 −1.61320
$$405$$ 33.3343 1.65640
$$406$$ −3.81425 −0.189298
$$407$$ −9.67124 −0.479386
$$408$$ −1.37239 −0.0679436
$$409$$ −34.8086 −1.72117 −0.860586 0.509305i $$-0.829903\pi$$
−0.860586 + 0.509305i $$0.829903\pi$$
$$410$$ −44.0273 −2.17435
$$411$$ −10.3610 −0.511072
$$412$$ −1.17075 −0.0576787
$$413$$ −26.5069 −1.30432
$$414$$ −2.83918 −0.139538
$$415$$ 35.5998 1.74752
$$416$$ −23.1072 −1.13292
$$417$$ −16.2323 −0.794899
$$418$$ −2.03821 −0.0996920
$$419$$ 9.04478 0.441866 0.220933 0.975289i $$-0.429090\pi$$
0.220933 + 0.975289i $$0.429090\pi$$
$$420$$ −25.2005 −1.22966
$$421$$ 29.9089 1.45767 0.728836 0.684688i $$-0.240062\pi$$
0.728836 + 0.684688i $$0.240062\pi$$
$$422$$ −16.8327 −0.819402
$$423$$ 1.75981 0.0855651
$$424$$ 3.31745 0.161109
$$425$$ −12.9261 −0.627008
$$426$$ 54.4645 2.63881
$$427$$ 7.20512 0.348680
$$428$$ 31.7479 1.53459
$$429$$ −5.33962 −0.257799
$$430$$ −52.9154 −2.55180
$$431$$ −13.7402 −0.661840 −0.330920 0.943659i $$-0.607359\pi$$
−0.330920 + 0.943659i $$0.607359\pi$$
$$432$$ 17.1220 0.823782
$$433$$ 3.28875 0.158047 0.0790236 0.996873i $$-0.474820\pi$$
0.0790236 + 0.996873i $$0.474820\pi$$
$$434$$ −0.0216075 −0.00103719
$$435$$ 5.91750 0.283723
$$436$$ −37.4224 −1.79221
$$437$$ −2.74653 −0.131384
$$438$$ −50.4855 −2.41229
$$439$$ 13.1729 0.628707 0.314353 0.949306i $$-0.398212\pi$$
0.314353 + 0.949306i $$0.398212\pi$$
$$440$$ 1.02128 0.0486875
$$441$$ −1.67401 −0.0797146
$$442$$ −13.5425 −0.644149
$$443$$ −24.9484 −1.18533 −0.592666 0.805448i $$-0.701925\pi$$
−0.592666 + 0.805448i $$0.701925\pi$$
$$444$$ 39.0180 1.85171
$$445$$ −48.7774 −2.31227
$$446$$ −43.2689 −2.04884
$$447$$ −36.1491 −1.70980
$$448$$ −17.6624 −0.834470
$$449$$ 15.5530 0.733993 0.366996 0.930222i $$-0.380386\pi$$
0.366996 + 0.930222i $$0.380386\pi$$
$$450$$ 5.73401 0.270304
$$451$$ −6.65137 −0.313201
$$452$$ 27.7613 1.30578
$$453$$ 16.3188 0.766724
$$454$$ −18.4794 −0.867283
$$455$$ −17.8097 −0.834933
$$456$$ 0.588926 0.0275790
$$457$$ −19.1451 −0.895571 −0.447786 0.894141i $$-0.647787\pi$$
−0.447786 + 0.894141i $$0.647787\pi$$
$$458$$ −10.7119 −0.500535
$$459$$ 10.8790 0.507787
$$460$$ 19.2155 0.895925
$$461$$ 33.5045 1.56046 0.780229 0.625493i $$-0.215102\pi$$
0.780229 + 0.625493i $$0.215102\pi$$
$$462$$ −7.34161 −0.341563
$$463$$ 2.83101 0.131568 0.0657842 0.997834i $$-0.479045\pi$$
0.0657842 + 0.997834i $$0.479045\pi$$
$$464$$ 3.56847 0.165662
$$465$$ 0.0335222 0.00155456
$$466$$ −11.5948 −0.537117
$$467$$ −20.3717 −0.942689 −0.471344 0.881949i $$-0.656231\pi$$
−0.471344 + 0.881949i $$0.656231\pi$$
$$468$$ 3.11527 0.144003
$$469$$ −7.63911 −0.352741
$$470$$ −22.9677 −1.05942
$$471$$ −10.9863 −0.506221
$$472$$ −4.33388 −0.199483
$$473$$ −7.99413 −0.367570
$$474$$ −7.16293 −0.329004
$$475$$ 5.54689 0.254509
$$476$$ −9.65575 −0.442571
$$477$$ 5.35036 0.244976
$$478$$ −41.5360 −1.89981
$$479$$ −7.81572 −0.357109 −0.178555 0.983930i $$-0.557142\pi$$
−0.178555 + 0.983930i $$0.557142\pi$$
$$480$$ 49.2898 2.24976
$$481$$ 27.5749 1.25731
$$482$$ 36.0621 1.64258
$$483$$ −9.89299 −0.450146
$$484$$ 2.15429 0.0979222
$$485$$ 24.5924 1.11668
$$486$$ −10.6337 −0.482353
$$487$$ −9.10523 −0.412597 −0.206299 0.978489i $$-0.566142\pi$$
−0.206299 + 0.978489i $$0.566142\pi$$
$$488$$ 1.17804 0.0533272
$$489$$ 27.8668 1.26018
$$490$$ 21.8478 0.986982
$$491$$ 34.4175 1.55324 0.776619 0.629970i $$-0.216933\pi$$
0.776619 + 0.629970i $$0.216933\pi$$
$$492$$ 26.8345 1.20979
$$493$$ 2.26733 0.102116
$$494$$ 5.81138 0.261467
$$495$$ 1.64711 0.0740321
$$496$$ 0.0202151 0.000907685 0
$$497$$ 27.4442 1.23104
$$498$$ −41.8420 −1.87498
$$499$$ 7.80798 0.349533 0.174767 0.984610i $$-0.444083\pi$$
0.174767 + 0.984610i $$0.444083\pi$$
$$500$$ −3.82619 −0.171113
$$501$$ 7.84625 0.350545
$$502$$ −1.58275 −0.0706418
$$503$$ 34.9580 1.55870 0.779350 0.626589i $$-0.215550\pi$$
0.779350 + 0.626589i $$0.215550\pi$$
$$504$$ 0.306765 0.0136644
$$505$$ 48.8806 2.17516
$$506$$ 5.59800 0.248861
$$507$$ −9.12126 −0.405089
$$508$$ 33.2498 1.47522
$$509$$ −11.3952 −0.505082 −0.252541 0.967586i $$-0.581266\pi$$
−0.252541 + 0.967586i $$0.581266\pi$$
$$510$$ 28.8873 1.27915
$$511$$ −25.4392 −1.12536
$$512$$ 32.0302 1.41555
$$513$$ −4.66842 −0.206116
$$514$$ 59.6710 2.63197
$$515$$ 1.76491 0.0777712
$$516$$ 32.2518 1.41981
$$517$$ −3.46982 −0.152602
$$518$$ 37.9136 1.66583
$$519$$ 1.32429 0.0581297
$$520$$ −2.91189 −0.127695
$$521$$ −28.0648 −1.22954 −0.614770 0.788707i $$-0.710751\pi$$
−0.614770 + 0.788707i $$0.710751\pi$$
$$522$$ −1.00579 −0.0440221
$$523$$ 20.5683 0.899389 0.449694 0.893182i $$-0.351533\pi$$
0.449694 + 0.893182i $$0.351533\pi$$
$$524$$ −25.9925 −1.13549
$$525$$ 19.9799 0.871993
$$526$$ −12.0263 −0.524370
$$527$$ 0.0128443 0.000559506 0
$$528$$ 6.86852 0.298914
$$529$$ −15.4566 −0.672025
$$530$$ −69.8285 −3.03316
$$531$$ −6.98965 −0.303325
$$532$$ 4.14350 0.179644
$$533$$ 18.9645 0.821446
$$534$$ 57.3302 2.48092
$$535$$ −47.8601 −2.06917
$$536$$ −1.24899 −0.0539484
$$537$$ −40.8188 −1.76146
$$538$$ −21.8655 −0.942690
$$539$$ 3.30063 0.142168
$$540$$ 32.6615 1.40553
$$541$$ −4.13908 −0.177953 −0.0889766 0.996034i $$-0.528360\pi$$
−0.0889766 + 0.996034i $$0.528360\pi$$
$$542$$ 33.1631 1.42448
$$543$$ −5.49493 −0.235810
$$544$$ 18.8857 0.809720
$$545$$ 56.4144 2.41653
$$546$$ 20.9326 0.895831
$$547$$ −30.7624 −1.31530 −0.657652 0.753322i $$-0.728450\pi$$
−0.657652 + 0.753322i $$0.728450\pi$$
$$548$$ −11.9187 −0.509141
$$549$$ 1.89993 0.0810870
$$550$$ −11.3057 −0.482077
$$551$$ −0.972965 −0.0414497
$$552$$ −1.61750 −0.0688455
$$553$$ −3.60934 −0.153485
$$554$$ 13.7995 0.586284
$$555$$ −58.8198 −2.49676
$$556$$ −18.6726 −0.791895
$$557$$ 8.84004 0.374565 0.187282 0.982306i $$-0.440032\pi$$
0.187282 + 0.982306i $$0.440032\pi$$
$$558$$ −0.00569772 −0.000241204 0
$$559$$ 22.7930 0.964043
$$560$$ 22.9092 0.968091
$$561$$ 4.36412 0.184253
$$562$$ −34.9461 −1.47411
$$563$$ −3.15807 −0.133097 −0.0665484 0.997783i $$-0.521199\pi$$
−0.0665484 + 0.997783i $$0.521199\pi$$
$$564$$ 13.9987 0.589454
$$565$$ −41.8503 −1.76066
$$566$$ −6.00836 −0.252550
$$567$$ −19.7421 −0.829091
$$568$$ 4.48712 0.188276
$$569$$ 18.7192 0.784749 0.392375 0.919806i $$-0.371654\pi$$
0.392375 + 0.919806i $$0.371654\pi$$
$$570$$ −12.3962 −0.519221
$$571$$ 37.6834 1.57700 0.788500 0.615034i $$-0.210858\pi$$
0.788500 + 0.615034i $$0.210858\pi$$
$$572$$ −6.14236 −0.256825
$$573$$ 41.9529 1.75261
$$574$$ 26.0750 1.08835
$$575$$ −15.2347 −0.635331
$$576$$ −4.65743 −0.194060
$$577$$ 1.77272 0.0737993 0.0368997 0.999319i $$-0.488252\pi$$
0.0368997 + 0.999319i $$0.488252\pi$$
$$578$$ −23.5811 −0.980846
$$579$$ 4.78388 0.198811
$$580$$ 6.80712 0.282650
$$581$$ −21.0838 −0.874704
$$582$$ −28.9045 −1.19813
$$583$$ −10.5493 −0.436906
$$584$$ −4.15931 −0.172113
$$585$$ −4.69628 −0.194167
$$586$$ 5.25554 0.217105
$$587$$ −32.1703 −1.32781 −0.663906 0.747816i $$-0.731103\pi$$
−0.663906 + 0.747816i $$0.731103\pi$$
$$588$$ −13.3162 −0.549150
$$589$$ −0.00551178 −0.000227109 0
$$590$$ 91.2232 3.75560
$$591$$ −23.7778 −0.978088
$$592$$ −35.4704 −1.45783
$$593$$ 12.2719 0.503946 0.251973 0.967734i $$-0.418921\pi$$
0.251973 + 0.967734i $$0.418921\pi$$
$$594$$ 9.51521 0.390414
$$595$$ 14.5561 0.596741
$$596$$ −41.5837 −1.70333
$$597$$ 19.0183 0.778369
$$598$$ −15.9611 −0.652700
$$599$$ 16.7005 0.682366 0.341183 0.939997i $$-0.389172\pi$$
0.341183 + 0.939997i $$0.389172\pi$$
$$600$$ 3.26671 0.133363
$$601$$ 14.3030 0.583432 0.291716 0.956505i $$-0.405774\pi$$
0.291716 + 0.956505i $$0.405774\pi$$
$$602$$ 31.3389 1.27728
$$603$$ −2.01437 −0.0820315
$$604$$ 18.7721 0.763827
$$605$$ −3.24760 −0.132034
$$606$$ −57.4516 −2.33381
$$607$$ −9.04370 −0.367073 −0.183536 0.983013i $$-0.558754\pi$$
−0.183536 + 0.983013i $$0.558754\pi$$
$$608$$ −8.10431 −0.328673
$$609$$ −3.50461 −0.142014
$$610$$ −24.7963 −1.00397
$$611$$ 9.89322 0.400237
$$612$$ −2.54614 −0.102922
$$613$$ −24.0096 −0.969738 −0.484869 0.874587i $$-0.661133\pi$$
−0.484869 + 0.874587i $$0.661133\pi$$
$$614$$ −40.3337 −1.62774
$$615$$ −40.4532 −1.63123
$$616$$ −0.604847 −0.0243700
$$617$$ 1.30852 0.0526791 0.0263395 0.999653i $$-0.491615\pi$$
0.0263395 + 0.999653i $$0.491615\pi$$
$$618$$ −2.07438 −0.0834436
$$619$$ 32.2747 1.29723 0.648616 0.761116i $$-0.275348\pi$$
0.648616 + 0.761116i $$0.275348\pi$$
$$620$$ 0.0385619 0.00154868
$$621$$ 12.8220 0.514528
$$622$$ 40.3523 1.61798
$$623$$ 28.8882 1.15738
$$624$$ −19.5837 −0.783975
$$625$$ −21.9665 −0.878658
$$626$$ 22.3890 0.894845
$$627$$ −1.87275 −0.0747903
$$628$$ −12.6379 −0.504308
$$629$$ −22.5372 −0.898618
$$630$$ −6.45706 −0.257256
$$631$$ 29.6689 1.18110 0.590551 0.807000i $$-0.298911\pi$$
0.590551 + 0.807000i $$0.298911\pi$$
$$632$$ −0.590127 −0.0234740
$$633$$ −15.4662 −0.614727
$$634$$ 20.1446 0.800046
$$635$$ −50.1242 −1.98912
$$636$$ 42.5603 1.68763
$$637$$ −9.41083 −0.372871
$$638$$ 1.98311 0.0785119
$$639$$ 7.23680 0.286283
$$640$$ 8.14591 0.321995
$$641$$ 22.2716 0.879675 0.439838 0.898077i $$-0.355036\pi$$
0.439838 + 0.898077i $$0.355036\pi$$
$$642$$ 56.2521 2.22009
$$643$$ −16.6461 −0.656456 −0.328228 0.944599i $$-0.606451\pi$$
−0.328228 + 0.944599i $$0.606451\pi$$
$$644$$ −11.3803 −0.448445
$$645$$ −48.6197 −1.91440
$$646$$ −4.74970 −0.186875
$$647$$ −15.9531 −0.627180 −0.313590 0.949559i $$-0.601532\pi$$
−0.313590 + 0.949559i $$0.601532\pi$$
$$648$$ −3.22783 −0.126801
$$649$$ 13.7814 0.540969
$$650$$ 32.2351 1.26437
$$651$$ −0.0198534 −0.000778116 0
$$652$$ 32.0562 1.25542
$$653$$ 14.7613 0.577655 0.288828 0.957381i $$-0.406735\pi$$
0.288828 + 0.957381i $$0.406735\pi$$
$$654$$ −66.3063 −2.59278
$$655$$ 39.1838 1.53104
$$656$$ −24.3947 −0.952453
$$657$$ −6.70811 −0.261708
$$658$$ 13.6025 0.530281
$$659$$ −19.6143 −0.764065 −0.382032 0.924149i $$-0.624776\pi$$
−0.382032 + 0.924149i $$0.624776\pi$$
$$660$$ 13.1022 0.510003
$$661$$ −31.5523 −1.22724 −0.613621 0.789601i $$-0.710288\pi$$
−0.613621 + 0.789601i $$0.710288\pi$$
$$662$$ 52.5037 2.04061
$$663$$ −12.4431 −0.483250
$$664$$ −3.44720 −0.133777
$$665$$ −6.24635 −0.242223
$$666$$ 9.99749 0.387395
$$667$$ 2.67228 0.103471
$$668$$ 9.02583 0.349220
$$669$$ −39.7564 −1.53707
$$670$$ 26.2899 1.01567
$$671$$ −3.74608 −0.144616
$$672$$ −29.1917 −1.12609
$$673$$ 41.3876 1.59537 0.797687 0.603071i $$-0.206057\pi$$
0.797687 + 0.603071i $$0.206057\pi$$
$$674$$ 44.6213 1.71875
$$675$$ −25.8952 −0.996708
$$676$$ −10.4925 −0.403558
$$677$$ −20.6981 −0.795491 −0.397746 0.917496i $$-0.630207\pi$$
−0.397746 + 0.917496i $$0.630207\pi$$
$$678$$ 49.1885 1.88907
$$679$$ −14.5647 −0.558943
$$680$$ 2.37992 0.0912657
$$681$$ −16.9793 −0.650647
$$682$$ 0.0112342 0.000430178 0
$$683$$ −0.658543 −0.0251985 −0.0125992 0.999921i $$-0.504011\pi$$
−0.0125992 + 0.999921i $$0.504011\pi$$
$$684$$ 1.09261 0.0417769
$$685$$ 17.9674 0.686501
$$686$$ −40.3809 −1.54175
$$687$$ −9.84233 −0.375508
$$688$$ −29.3194 −1.11779
$$689$$ 30.0783 1.14589
$$690$$ 34.0466 1.29613
$$691$$ 34.2462 1.30279 0.651393 0.758741i $$-0.274185\pi$$
0.651393 + 0.758741i $$0.274185\pi$$
$$692$$ 1.52338 0.0579100
$$693$$ −0.975494 −0.0370559
$$694$$ 39.3083 1.49212
$$695$$ 28.1490 1.06775
$$696$$ −0.573005 −0.0217197
$$697$$ −15.4999 −0.587101
$$698$$ 31.6370 1.19748
$$699$$ −10.6535 −0.402952
$$700$$ 22.9836 0.868697
$$701$$ 22.4638 0.848445 0.424223 0.905558i $$-0.360547\pi$$
0.424223 + 0.905558i $$0.360547\pi$$
$$702$$ −27.1300 −1.02396
$$703$$ 9.67124 0.364758
$$704$$ 9.18303 0.346098
$$705$$ −21.1032 −0.794791
$$706$$ −18.1983 −0.684902
$$707$$ −28.9493 −1.08875
$$708$$ −55.6003 −2.08959
$$709$$ −0.410520 −0.0154174 −0.00770870 0.999970i $$-0.502454\pi$$
−0.00770870 + 0.999970i $$0.502454\pi$$
$$710$$ −94.4489 −3.54460
$$711$$ −0.951752 −0.0356935
$$712$$ 4.72322 0.177010
$$713$$ 0.0151383 0.000566933 0
$$714$$ −17.1084 −0.640266
$$715$$ 9.25963 0.346290
$$716$$ −46.9553 −1.75480
$$717$$ −38.1641 −1.42527
$$718$$ 53.5380 1.99802
$$719$$ 36.5145 1.36176 0.680880 0.732395i $$-0.261598\pi$$
0.680880 + 0.732395i $$0.261598\pi$$
$$720$$ 6.04097 0.225134
$$721$$ −1.04526 −0.0389275
$$722$$ 2.03821 0.0758542
$$723$$ 33.1346 1.23229
$$724$$ −6.32102 −0.234919
$$725$$ −5.39693 −0.200437
$$726$$ 3.81704 0.141664
$$727$$ −39.2587 −1.45602 −0.728012 0.685564i $$-0.759556\pi$$
−0.728012 + 0.685564i $$0.759556\pi$$
$$728$$ 1.72455 0.0639163
$$729$$ 21.0225 0.778610
$$730$$ 87.5488 3.24032
$$731$$ −18.6290 −0.689018
$$732$$ 15.1133 0.558604
$$733$$ −34.3259 −1.26786 −0.633928 0.773392i $$-0.718558\pi$$
−0.633928 + 0.773392i $$0.718558\pi$$
$$734$$ 20.9038 0.771575
$$735$$ 20.0742 0.740447
$$736$$ 22.2587 0.820468
$$737$$ 3.97172 0.146300
$$738$$ 6.87575 0.253100
$$739$$ −26.7732 −0.984870 −0.492435 0.870349i $$-0.663893\pi$$
−0.492435 + 0.870349i $$0.663893\pi$$
$$740$$ −67.6625 −2.48732
$$741$$ 5.33962 0.196156
$$742$$ 41.3556 1.51821
$$743$$ −9.38431 −0.344277 −0.172139 0.985073i $$-0.555068\pi$$
−0.172139 + 0.985073i $$0.555068\pi$$
$$744$$ −0.00324603 −0.000119005 0
$$745$$ 62.6876 2.29669
$$746$$ −41.2210 −1.50921
$$747$$ −5.55963 −0.203416
$$748$$ 5.02021 0.183557
$$749$$ 28.3449 1.03570
$$750$$ −6.77939 −0.247548
$$751$$ 7.66846 0.279826 0.139913 0.990164i $$-0.455318\pi$$
0.139913 + 0.990164i $$0.455318\pi$$
$$752$$ −12.7260 −0.464068
$$753$$ −1.45427 −0.0529965
$$754$$ −5.65428 −0.205917
$$755$$ −28.2990 −1.02991
$$756$$ −19.3436 −0.703521
$$757$$ −6.75931 −0.245671 −0.122836 0.992427i $$-0.539199\pi$$
−0.122836 + 0.992427i $$0.539199\pi$$
$$758$$ 28.8475 1.04779
$$759$$ 5.14356 0.186699
$$760$$ −1.02128 −0.0370456
$$761$$ −34.8774 −1.26430 −0.632152 0.774844i $$-0.717828\pi$$
−0.632152 + 0.774844i $$0.717828\pi$$
$$762$$ 58.9132 2.13420
$$763$$ −33.4111 −1.20956
$$764$$ 48.2599 1.74598
$$765$$ 3.83832 0.138775
$$766$$ 25.1142 0.907415
$$767$$ −39.2940 −1.41882
$$768$$ 24.8207 0.895640
$$769$$ −0.00622027 −0.000224309 0 −0.000112154 1.00000i $$-0.500036\pi$$
−0.000112154 1.00000i $$0.500036\pi$$
$$770$$ 12.7314 0.458806
$$771$$ 54.8269 1.97454
$$772$$ 5.50307 0.198060
$$773$$ 10.8548 0.390418 0.195209 0.980762i $$-0.437461\pi$$
0.195209 + 0.980762i $$0.437461\pi$$
$$774$$ 8.26380 0.297036
$$775$$ −0.0305733 −0.00109822
$$776$$ −2.38133 −0.0854848
$$777$$ 34.8357 1.24973
$$778$$ 69.1459 2.47900
$$779$$ 6.65137 0.238310
$$780$$ −37.3574 −1.33761
$$781$$ −14.2688 −0.510576
$$782$$ 13.0452 0.466496
$$783$$ 4.54221 0.162325
$$784$$ 12.1054 0.432337
$$785$$ 19.0517 0.679984
$$786$$ −46.0544 −1.64271
$$787$$ 37.8221 1.34821 0.674106 0.738635i $$-0.264529\pi$$
0.674106 + 0.738635i $$0.264529\pi$$
$$788$$ −27.3525 −0.974392
$$789$$ −11.0500 −0.393390
$$790$$ 12.4215 0.441937
$$791$$ 24.7857 0.881277
$$792$$ −0.159493 −0.00566735
$$793$$ 10.6809 0.379290
$$794$$ −41.4483 −1.47095
$$795$$ −64.1598 −2.27552
$$796$$ 21.8775 0.775428
$$797$$ −49.2853 −1.74577 −0.872887 0.487922i $$-0.837755\pi$$
−0.872887 + 0.487922i $$0.837755\pi$$
$$798$$ 7.34161 0.259890
$$799$$ −8.08583 −0.286056
$$800$$ −44.9537 −1.58935
$$801$$ 7.61758 0.269154
$$802$$ −62.5352 −2.20819
$$803$$ 13.2263 0.466747
$$804$$ −16.0237 −0.565111
$$805$$ 17.1558 0.604662
$$806$$ −0.0320311 −0.00112825
$$807$$ −20.0905 −0.707219
$$808$$ −4.73322 −0.166514
$$809$$ 34.0416 1.19684 0.598420 0.801183i $$-0.295796\pi$$
0.598420 + 0.801183i $$0.295796\pi$$
$$810$$ 67.9423 2.38725
$$811$$ −44.7037 −1.56976 −0.784880 0.619648i $$-0.787275\pi$$
−0.784880 + 0.619648i $$0.787275\pi$$
$$812$$ −4.03149 −0.141477
$$813$$ 30.4709 1.06866
$$814$$ −19.7120 −0.690905
$$815$$ −48.3249 −1.69275
$$816$$ 16.0059 0.560320
$$817$$ 7.99413 0.279679
$$818$$ −70.9470 −2.48061
$$819$$ 2.78135 0.0971882
$$820$$ −46.5348 −1.62506
$$821$$ −48.0778 −1.67793 −0.838963 0.544188i $$-0.816838\pi$$
−0.838963 + 0.544188i $$0.816838\pi$$
$$822$$ −21.1179 −0.736572
$$823$$ 18.6246 0.649211 0.324606 0.945849i $$-0.394768\pi$$
0.324606 + 0.945849i $$0.394768\pi$$
$$824$$ −0.170900 −0.00595358
$$825$$ −10.3879 −0.361661
$$826$$ −54.0265 −1.87982
$$827$$ 6.99744 0.243325 0.121662 0.992572i $$-0.461177\pi$$
0.121662 + 0.992572i $$0.461177\pi$$
$$828$$ −3.00088 −0.104288
$$829$$ 24.9441 0.866344 0.433172 0.901311i $$-0.357394\pi$$
0.433172 + 0.901311i $$0.357394\pi$$
$$830$$ 72.5597 2.51859
$$831$$ 12.6792 0.439838
$$832$$ −26.1829 −0.907727
$$833$$ 7.69157 0.266497
$$834$$ −33.0848 −1.14563
$$835$$ −13.6065 −0.470872
$$836$$ −2.15429 −0.0745076
$$837$$ 0.0257313 0.000889405 0
$$838$$ 18.4351 0.636831
$$839$$ 10.2122 0.352566 0.176283 0.984340i $$-0.443593\pi$$
0.176283 + 0.984340i $$0.443593\pi$$
$$840$$ −3.67864 −0.126925
$$841$$ −28.0533 −0.967356
$$842$$ 60.9606 2.10084
$$843$$ −32.1091 −1.10590
$$844$$ −17.7914 −0.612404
$$845$$ 15.8175 0.544139
$$846$$ 3.58687 0.123319
$$847$$ 1.92338 0.0660879
$$848$$ −38.6907 −1.32864
$$849$$ −5.52060 −0.189467
$$850$$ −26.3461 −0.903663
$$851$$ −26.5624 −0.910546
$$852$$ 57.5664 1.97219
$$853$$ 33.5562 1.14894 0.574472 0.818524i $$-0.305207\pi$$
0.574472 + 0.818524i $$0.305207\pi$$
$$854$$ 14.6855 0.502528
$$855$$ −1.64711 −0.0563300
$$856$$ 4.63440 0.158400
$$857$$ 34.8775 1.19139 0.595696 0.803210i $$-0.296876\pi$$
0.595696 + 0.803210i $$0.296876\pi$$
$$858$$ −10.8832 −0.371548
$$859$$ 13.1320 0.448059 0.224029 0.974582i $$-0.428079\pi$$
0.224029 + 0.974582i $$0.428079\pi$$
$$860$$ −55.9290 −1.90716
$$861$$ 23.9582 0.816493
$$862$$ −28.0053 −0.953863
$$863$$ −27.5223 −0.936871 −0.468435 0.883498i $$-0.655182\pi$$
−0.468435 + 0.883498i $$0.655182\pi$$
$$864$$ 37.8343 1.28715
$$865$$ −2.29649 −0.0780831
$$866$$ 6.70315 0.227782
$$867$$ −21.6668 −0.735844
$$868$$ −0.0228381 −0.000775175 0
$$869$$ 1.87656 0.0636581
$$870$$ 12.0611 0.408910
$$871$$ −11.3243 −0.383708
$$872$$ −5.46272 −0.184991
$$873$$ −3.84060 −0.129985
$$874$$ −5.59800 −0.189355
$$875$$ −3.41607 −0.115484
$$876$$ −53.3608 −1.80289
$$877$$ −18.2679 −0.616865 −0.308432 0.951246i $$-0.599804\pi$$
−0.308432 + 0.951246i $$0.599804\pi$$
$$878$$ 26.8490 0.906111
$$879$$ 4.82890 0.162875
$$880$$ −11.9109 −0.401518
$$881$$ −47.0363 −1.58469 −0.792346 0.610072i $$-0.791141\pi$$
−0.792346 + 0.610072i $$0.791141\pi$$
$$882$$ −3.41197 −0.114887
$$883$$ −24.7005 −0.831238 −0.415619 0.909539i $$-0.636435\pi$$
−0.415619 + 0.909539i $$0.636435\pi$$
$$884$$ −14.3137 −0.481423
$$885$$ 83.8177 2.81750
$$886$$ −50.8500 −1.70834
$$887$$ 23.3946 0.785515 0.392758 0.919642i $$-0.371521\pi$$
0.392758 + 0.919642i $$0.371521\pi$$
$$888$$ 5.69564 0.191133
$$889$$ 29.6858 0.995631
$$890$$ −99.4184 −3.33251
$$891$$ 10.2643 0.343867
$$892$$ −45.7332 −1.53126
$$893$$ 3.46982 0.116113
$$894$$ −73.6794 −2.46421
$$895$$ 70.7853 2.36609
$$896$$ −4.82438 −0.161171
$$897$$ −14.6654 −0.489664
$$898$$ 31.7003 1.05785
$$899$$ 0.00536277 0.000178858 0
$$900$$ 6.06058 0.202019
$$901$$ −24.5833 −0.818989
$$902$$ −13.5569 −0.451395
$$903$$ 28.7948 0.958231
$$904$$ 4.05246 0.134783
$$905$$ 9.52897 0.316754
$$906$$ 33.2611 1.10503
$$907$$ 19.2411 0.638892 0.319446 0.947605i $$-0.396503\pi$$
0.319446 + 0.947605i $$0.396503\pi$$
$$908$$ −19.5319 −0.648189
$$909$$ −7.63370 −0.253194
$$910$$ −36.2999 −1.20333
$$911$$ 48.5854 1.60971 0.804853 0.593475i $$-0.202244\pi$$
0.804853 + 0.593475i $$0.202244\pi$$
$$912$$ −6.86852 −0.227439
$$913$$ 10.9619 0.362785
$$914$$ −39.0217 −1.29072
$$915$$ −22.7834 −0.753195
$$916$$ −11.3220 −0.374089
$$917$$ −23.2064 −0.766344
$$918$$ 22.1736 0.731839
$$919$$ 21.2754 0.701812 0.350906 0.936411i $$-0.385874\pi$$
0.350906 + 0.936411i $$0.385874\pi$$
$$920$$ 2.80497 0.0924772
$$921$$ −37.0595 −1.22115
$$922$$ 68.2891 2.24898
$$923$$ 40.6834 1.33911
$$924$$ −7.75973 −0.255276
$$925$$ 53.6453 1.76385
$$926$$ 5.77019 0.189620
$$927$$ −0.275626 −0.00905276
$$928$$ 7.88521 0.258845
$$929$$ 49.9319 1.63821 0.819106 0.573643i $$-0.194470\pi$$
0.819106 + 0.573643i $$0.194470\pi$$
$$930$$ 0.0683253 0.00224047
$$931$$ −3.30063 −0.108174
$$932$$ −12.2551 −0.401429
$$933$$ 37.0765 1.21383
$$934$$ −41.5217 −1.35863
$$935$$ −7.56799 −0.247500
$$936$$ 0.454751 0.0148640
$$937$$ 11.2142 0.366353 0.183176 0.983080i $$-0.441362\pi$$
0.183176 + 0.983080i $$0.441362\pi$$
$$938$$ −15.5701 −0.508381
$$939$$ 20.5715 0.671325
$$940$$ −24.2757 −0.791787
$$941$$ −47.9390 −1.56277 −0.781383 0.624052i $$-0.785485\pi$$
−0.781383 + 0.624052i $$0.785485\pi$$
$$942$$ −22.3923 −0.729580
$$943$$ −18.2682 −0.594894
$$944$$ 50.5451 1.64510
$$945$$ 29.1606 0.948594
$$946$$ −16.2937 −0.529754
$$947$$ 14.4633 0.469993 0.234997 0.971996i $$-0.424492\pi$$
0.234997 + 0.971996i $$0.424492\pi$$
$$948$$ −7.57088 −0.245891
$$949$$ −37.7112 −1.22416
$$950$$ 11.3057 0.366806
$$951$$ 18.5093 0.600206
$$952$$ −1.40950 −0.0456820
$$953$$ 19.3671 0.627363 0.313682 0.949528i $$-0.398438\pi$$
0.313682 + 0.949528i $$0.398438\pi$$
$$954$$ 10.9051 0.353067
$$955$$ −72.7520 −2.35420
$$956$$ −43.9016 −1.41988
$$957$$ 1.82212 0.0589007
$$958$$ −15.9300 −0.514677
$$959$$ −10.6411 −0.343620
$$960$$ 55.8505 1.80257
$$961$$ −31.0000 −0.999999
$$962$$ 56.2033 1.81207
$$963$$ 7.47432 0.240857
$$964$$ 38.1159 1.22763
$$965$$ −8.29591 −0.267055
$$966$$ −20.1640 −0.648765
$$967$$ −29.6452 −0.953326 −0.476663 0.879086i $$-0.658154\pi$$
−0.476663 + 0.879086i $$0.658154\pi$$
$$968$$ 0.314472 0.0101075
$$969$$ −4.36412 −0.140196
$$970$$ 50.1244 1.60940
$$971$$ −27.9572 −0.897190 −0.448595 0.893735i $$-0.648075\pi$$
−0.448595 + 0.893735i $$0.648075\pi$$
$$972$$ −11.2393 −0.360500
$$973$$ −16.6711 −0.534452
$$974$$ −18.5583 −0.594648
$$975$$ 29.6183 0.948544
$$976$$ −13.7392 −0.439781
$$977$$ −25.4860 −0.815369 −0.407684 0.913123i $$-0.633664\pi$$
−0.407684 + 0.913123i $$0.633664\pi$$
$$978$$ 56.7983 1.81621
$$979$$ −15.0195 −0.480026
$$980$$ 23.0921 0.737649
$$981$$ −8.81024 −0.281289
$$982$$ 70.1499 2.23857
$$983$$ −23.7523 −0.757582 −0.378791 0.925482i $$-0.623660\pi$$
−0.378791 + 0.925482i $$0.623660\pi$$
$$984$$ 3.91717 0.124875
$$985$$ 41.2340 1.31382
$$986$$ 4.62130 0.147172
$$987$$ 12.4983 0.397824
$$988$$ 6.14236 0.195414
$$989$$ −21.9561 −0.698164
$$990$$ 3.35715 0.106697
$$991$$ 10.8187 0.343668 0.171834 0.985126i $$-0.445031\pi$$
0.171834 + 0.985126i $$0.445031\pi$$
$$992$$ 0.0446692 0.00141825
$$993$$ 48.2415 1.53090
$$994$$ 55.9369 1.77421
$$995$$ −32.9804 −1.04555
$$996$$ −44.2250 −1.40132
$$997$$ −55.0261 −1.74270 −0.871348 0.490666i $$-0.836753\pi$$
−0.871348 + 0.490666i $$0.836753\pi$$
$$998$$ 15.9143 0.503758
$$999$$ −45.1494 −1.42847
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 209.2.a.d.1.6 7
3.2 odd 2 1881.2.a.p.1.2 7
4.3 odd 2 3344.2.a.ba.1.2 7
5.4 even 2 5225.2.a.n.1.2 7
11.10 odd 2 2299.2.a.q.1.2 7
19.18 odd 2 3971.2.a.i.1.2 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.6 7 1.1 even 1 trivial
1881.2.a.p.1.2 7 3.2 odd 2
2299.2.a.q.1.2 7 11.10 odd 2
3344.2.a.ba.1.2 7 4.3 odd 2
3971.2.a.i.1.2 7 19.18 odd 2
5225.2.a.n.1.2 7 5.4 even 2