# Properties

 Label 1045.2.a.h.1.1 Level $1045$ Weight $2$ Character 1045.1 Self dual yes Analytic conductor $8.344$ 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] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ 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: $$x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11$$ x^7 - x^6 - 10*x^5 + 8*x^4 + 27*x^3 - 16*x^2 - 18*x + 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.40300$$ of defining polynomial Character $$\chi$$ $$=$$ 1045.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.40300 q^{2} -1.17935 q^{3} +3.77440 q^{4} -1.00000 q^{5} +2.83397 q^{6} -2.15598 q^{7} -4.26389 q^{8} -1.60914 q^{9} +O(q^{10})$$ $$q-2.40300 q^{2} -1.17935 q^{3} +3.77440 q^{4} -1.00000 q^{5} +2.83397 q^{6} -2.15598 q^{7} -4.26389 q^{8} -1.60914 q^{9} +2.40300 q^{10} +1.00000 q^{11} -4.45134 q^{12} -3.54733 q^{13} +5.18083 q^{14} +1.17935 q^{15} +2.69732 q^{16} -0.622194 q^{17} +3.86675 q^{18} -1.00000 q^{19} -3.77440 q^{20} +2.54266 q^{21} -2.40300 q^{22} -5.67863 q^{23} +5.02862 q^{24} +1.00000 q^{25} +8.52423 q^{26} +5.43578 q^{27} -8.13756 q^{28} +7.32397 q^{29} -2.83397 q^{30} -7.89640 q^{31} +2.04612 q^{32} -1.17935 q^{33} +1.49513 q^{34} +2.15598 q^{35} -6.07353 q^{36} -9.19792 q^{37} +2.40300 q^{38} +4.18354 q^{39} +4.26389 q^{40} +2.29509 q^{41} -6.11000 q^{42} +0.194427 q^{43} +3.77440 q^{44} +1.60914 q^{45} +13.6457 q^{46} +0.340863 q^{47} -3.18108 q^{48} -2.35173 q^{49} -2.40300 q^{50} +0.733784 q^{51} -13.3891 q^{52} -6.94545 q^{53} -13.0622 q^{54} -1.00000 q^{55} +9.19289 q^{56} +1.17935 q^{57} -17.5995 q^{58} +8.99709 q^{59} +4.45134 q^{60} -13.7381 q^{61} +18.9750 q^{62} +3.46927 q^{63} -10.3115 q^{64} +3.54733 q^{65} +2.83397 q^{66} -0.226658 q^{67} -2.34841 q^{68} +6.69709 q^{69} -5.18083 q^{70} +0.343554 q^{71} +6.86119 q^{72} +9.18704 q^{73} +22.1026 q^{74} -1.17935 q^{75} -3.77440 q^{76} -2.15598 q^{77} -10.0530 q^{78} +14.4980 q^{79} -2.69732 q^{80} -1.58327 q^{81} -5.51510 q^{82} +3.15514 q^{83} +9.59702 q^{84} +0.622194 q^{85} -0.467207 q^{86} -8.63752 q^{87} -4.26389 q^{88} +5.44854 q^{89} -3.86675 q^{90} +7.64799 q^{91} -21.4335 q^{92} +9.31261 q^{93} -0.819094 q^{94} +1.00000 q^{95} -2.41309 q^{96} +13.7041 q^{97} +5.65121 q^{98} -1.60914 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10})$$ 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 $$7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100})$$ 7 * q + q^2 + 3 * q^3 + 7 * q^4 - 7 * q^5 + 8 * q^6 - q^7 + 3 * q^8 + 2 * q^9 - q^10 + 7 * q^11 + 13 * q^12 + q^13 + 12 * q^14 - 3 * q^15 + 3 * q^16 + q^17 + 7 * q^18 - 7 * q^19 - 7 * q^20 + 5 * q^21 + q^22 - 8 * q^23 + 25 * q^24 + 7 * q^25 + 12 * q^27 + 4 * q^28 + 11 * q^29 - 8 * q^30 + 7 * q^31 + 12 * q^32 + 3 * q^33 - 14 * q^34 + q^35 + 7 * q^36 - 17 * q^37 - q^38 + 30 * q^39 - 3 * q^40 + 17 * q^41 + 33 * q^42 - 3 * q^43 + 7 * q^44 - 2 * q^45 + 18 * q^46 + 14 * q^47 - 12 * q^48 + 6 * q^49 + q^50 + 8 * q^51 - 17 * q^52 + 7 * q^53 - 27 * q^54 - 7 * q^55 + 36 * q^56 - 3 * q^57 - 15 * q^58 + 35 * q^59 - 13 * q^60 + 17 * q^61 + 46 * q^62 - 22 * q^63 + 5 * q^64 - q^65 + 8 * q^66 + 4 * q^67 - 35 * q^68 - 4 * q^69 - 12 * q^70 + 10 * q^71 + 12 * q^72 + 22 * q^73 - 11 * q^74 + 3 * q^75 - 7 * q^76 - q^77 - 41 * q^78 + 11 * q^79 - 3 * q^80 - 21 * q^81 - 14 * q^82 + 39 * q^83 + 21 * q^84 - q^85 - 24 * q^86 - 2 * q^87 + 3 * q^88 + 18 * q^89 - 7 * q^90 - 22 * q^91 - 51 * q^92 + 10 * q^93 + 14 * q^94 + 7 * q^95 - 11 * q^96 - 4 * q^97 - 26 * q^98 + 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.40300 −1.69918 −0.849589 0.527446i $$-0.823150\pi$$
−0.849589 + 0.527446i $$0.823150\pi$$
$$3$$ −1.17935 −0.680897 −0.340449 0.940263i $$-0.610579\pi$$
−0.340449 + 0.940263i $$0.610579\pi$$
$$4$$ 3.77440 1.88720
$$5$$ −1.00000 −0.447214
$$6$$ 2.83397 1.15697
$$7$$ −2.15598 −0.814885 −0.407443 0.913231i $$-0.633579\pi$$
−0.407443 + 0.913231i $$0.633579\pi$$
$$8$$ −4.26389 −1.50751
$$9$$ −1.60914 −0.536379
$$10$$ 2.40300 0.759895
$$11$$ 1.00000 0.301511
$$12$$ −4.45134 −1.28499
$$13$$ −3.54733 −0.983852 −0.491926 0.870637i $$-0.663707\pi$$
−0.491926 + 0.870637i $$0.663707\pi$$
$$14$$ 5.18083 1.38463
$$15$$ 1.17935 0.304507
$$16$$ 2.69732 0.674331
$$17$$ −0.622194 −0.150904 −0.0754521 0.997149i $$-0.524040\pi$$
−0.0754521 + 0.997149i $$0.524040\pi$$
$$18$$ 3.86675 0.911403
$$19$$ −1.00000 −0.229416
$$20$$ −3.77440 −0.843983
$$21$$ 2.54266 0.554853
$$22$$ −2.40300 −0.512321
$$23$$ −5.67863 −1.18408 −0.592038 0.805910i $$-0.701677\pi$$
−0.592038 + 0.805910i $$0.701677\pi$$
$$24$$ 5.02862 1.02646
$$25$$ 1.00000 0.200000
$$26$$ 8.52423 1.67174
$$27$$ 5.43578 1.04612
$$28$$ −8.13756 −1.53785
$$29$$ 7.32397 1.36003 0.680014 0.733199i $$-0.261974\pi$$
0.680014 + 0.733199i $$0.261974\pi$$
$$30$$ −2.83397 −0.517410
$$31$$ −7.89640 −1.41824 −0.709118 0.705090i $$-0.750907\pi$$
−0.709118 + 0.705090i $$0.750907\pi$$
$$32$$ 2.04612 0.361707
$$33$$ −1.17935 −0.205298
$$34$$ 1.49513 0.256413
$$35$$ 2.15598 0.364428
$$36$$ −6.07353 −1.01226
$$37$$ −9.19792 −1.51213 −0.756065 0.654497i $$-0.772881\pi$$
−0.756065 + 0.654497i $$0.772881\pi$$
$$38$$ 2.40300 0.389818
$$39$$ 4.18354 0.669902
$$40$$ 4.26389 0.674181
$$41$$ 2.29509 0.358433 0.179216 0.983810i $$-0.442644\pi$$
0.179216 + 0.983810i $$0.442644\pi$$
$$42$$ −6.11000 −0.942794
$$43$$ 0.194427 0.0296498 0.0148249 0.999890i $$-0.495281\pi$$
0.0148249 + 0.999890i $$0.495281\pi$$
$$44$$ 3.77440 0.569013
$$45$$ 1.60914 0.239876
$$46$$ 13.6457 2.01196
$$47$$ 0.340863 0.0497200 0.0248600 0.999691i $$-0.492086\pi$$
0.0248600 + 0.999691i $$0.492086\pi$$
$$48$$ −3.18108 −0.459150
$$49$$ −2.35173 −0.335962
$$50$$ −2.40300 −0.339835
$$51$$ 0.733784 0.102750
$$52$$ −13.3891 −1.85673
$$53$$ −6.94545 −0.954031 −0.477015 0.878895i $$-0.658281\pi$$
−0.477015 + 0.878895i $$0.658281\pi$$
$$54$$ −13.0622 −1.77754
$$55$$ −1.00000 −0.134840
$$56$$ 9.19289 1.22845
$$57$$ 1.17935 0.156209
$$58$$ −17.5995 −2.31093
$$59$$ 8.99709 1.17132 0.585660 0.810557i $$-0.300835\pi$$
0.585660 + 0.810557i $$0.300835\pi$$
$$60$$ 4.45134 0.574665
$$61$$ −13.7381 −1.75899 −0.879494 0.475910i $$-0.842119\pi$$
−0.879494 + 0.475910i $$0.842119\pi$$
$$62$$ 18.9750 2.40983
$$63$$ 3.46927 0.437087
$$64$$ −10.3115 −1.28893
$$65$$ 3.54733 0.439992
$$66$$ 2.83397 0.348838
$$67$$ −0.226658 −0.0276907 −0.0138453 0.999904i $$-0.504407\pi$$
−0.0138453 + 0.999904i $$0.504407\pi$$
$$68$$ −2.34841 −0.284787
$$69$$ 6.69709 0.806234
$$70$$ −5.18083 −0.619227
$$71$$ 0.343554 0.0407724 0.0203862 0.999792i $$-0.493510\pi$$
0.0203862 + 0.999792i $$0.493510\pi$$
$$72$$ 6.86119 0.808599
$$73$$ 9.18704 1.07526 0.537631 0.843180i $$-0.319319\pi$$
0.537631 + 0.843180i $$0.319319\pi$$
$$74$$ 22.1026 2.56938
$$75$$ −1.17935 −0.136179
$$76$$ −3.77440 −0.432954
$$77$$ −2.15598 −0.245697
$$78$$ −10.0530 −1.13828
$$79$$ 14.4980 1.63115 0.815574 0.578653i $$-0.196422\pi$$
0.815574 + 0.578653i $$0.196422\pi$$
$$80$$ −2.69732 −0.301570
$$81$$ −1.58327 −0.175919
$$82$$ −5.51510 −0.609041
$$83$$ 3.15514 0.346321 0.173161 0.984894i $$-0.444602\pi$$
0.173161 + 0.984894i $$0.444602\pi$$
$$84$$ 9.59702 1.04712
$$85$$ 0.622194 0.0674864
$$86$$ −0.467207 −0.0503802
$$87$$ −8.63752 −0.926039
$$88$$ −4.26389 −0.454533
$$89$$ 5.44854 0.577544 0.288772 0.957398i $$-0.406753\pi$$
0.288772 + 0.957398i $$0.406753\pi$$
$$90$$ −3.86675 −0.407592
$$91$$ 7.64799 0.801727
$$92$$ −21.4335 −2.23459
$$93$$ 9.31261 0.965673
$$94$$ −0.819094 −0.0844831
$$95$$ 1.00000 0.102598
$$96$$ −2.41309 −0.246285
$$97$$ 13.7041 1.39144 0.695722 0.718311i $$-0.255084\pi$$
0.695722 + 0.718311i $$0.255084\pi$$
$$98$$ 5.65121 0.570858
$$99$$ −1.60914 −0.161724
$$100$$ 3.77440 0.377440
$$101$$ −4.70463 −0.468128 −0.234064 0.972221i $$-0.575203\pi$$
−0.234064 + 0.972221i $$0.575203\pi$$
$$102$$ −1.76328 −0.174591
$$103$$ 13.3333 1.31377 0.656885 0.753991i $$-0.271874\pi$$
0.656885 + 0.753991i $$0.271874\pi$$
$$104$$ 15.1254 1.48317
$$105$$ −2.54266 −0.248138
$$106$$ 16.6899 1.62107
$$107$$ −7.07306 −0.683779 −0.341889 0.939740i $$-0.611067\pi$$
−0.341889 + 0.939740i $$0.611067\pi$$
$$108$$ 20.5168 1.97423
$$109$$ 8.93287 0.855614 0.427807 0.903870i $$-0.359286\pi$$
0.427807 + 0.903870i $$0.359286\pi$$
$$110$$ 2.40300 0.229117
$$111$$ 10.8476 1.02960
$$112$$ −5.81539 −0.549502
$$113$$ 8.90054 0.837293 0.418646 0.908149i $$-0.362505\pi$$
0.418646 + 0.908149i $$0.362505\pi$$
$$114$$ −2.83397 −0.265426
$$115$$ 5.67863 0.529535
$$116$$ 27.6436 2.56665
$$117$$ 5.70814 0.527718
$$118$$ −21.6200 −1.99028
$$119$$ 1.34144 0.122970
$$120$$ −5.02862 −0.459048
$$121$$ 1.00000 0.0909091
$$122$$ 33.0127 2.98883
$$123$$ −2.70671 −0.244056
$$124$$ −29.8042 −2.67650
$$125$$ −1.00000 −0.0894427
$$126$$ −8.33666 −0.742689
$$127$$ 14.9809 1.32934 0.664672 0.747135i $$-0.268571\pi$$
0.664672 + 0.747135i $$0.268571\pi$$
$$128$$ 20.6862 1.82842
$$129$$ −0.229297 −0.0201885
$$130$$ −8.52423 −0.747625
$$131$$ 7.81438 0.682746 0.341373 0.939928i $$-0.389108\pi$$
0.341373 + 0.939928i $$0.389108\pi$$
$$132$$ −4.45134 −0.387439
$$133$$ 2.15598 0.186948
$$134$$ 0.544659 0.0470513
$$135$$ −5.43578 −0.467837
$$136$$ 2.65297 0.227490
$$137$$ 14.1390 1.20798 0.603988 0.796994i $$-0.293578\pi$$
0.603988 + 0.796994i $$0.293578\pi$$
$$138$$ −16.0931 −1.36994
$$139$$ 12.5953 1.06832 0.534158 0.845385i $$-0.320629\pi$$
0.534158 + 0.845385i $$0.320629\pi$$
$$140$$ 8.13756 0.687749
$$141$$ −0.401997 −0.0338542
$$142$$ −0.825561 −0.0692795
$$143$$ −3.54733 −0.296643
$$144$$ −4.34036 −0.361697
$$145$$ −7.32397 −0.608223
$$146$$ −22.0764 −1.82706
$$147$$ 2.77351 0.228755
$$148$$ −34.7167 −2.85369
$$149$$ 4.77915 0.391523 0.195761 0.980652i $$-0.437282\pi$$
0.195761 + 0.980652i $$0.437282\pi$$
$$150$$ 2.83397 0.231393
$$151$$ −24.0449 −1.95674 −0.978371 0.206856i $$-0.933677\pi$$
−0.978371 + 0.206856i $$0.933677\pi$$
$$152$$ 4.26389 0.345847
$$153$$ 1.00119 0.0809418
$$154$$ 5.18083 0.417483
$$155$$ 7.89640 0.634254
$$156$$ 15.7904 1.26424
$$157$$ −12.9931 −1.03696 −0.518481 0.855089i $$-0.673502\pi$$
−0.518481 + 0.855089i $$0.673502\pi$$
$$158$$ −34.8386 −2.77161
$$159$$ 8.19110 0.649597
$$160$$ −2.04612 −0.161760
$$161$$ 12.2430 0.964887
$$162$$ 3.80459 0.298917
$$163$$ −19.7251 −1.54499 −0.772495 0.635021i $$-0.780992\pi$$
−0.772495 + 0.635021i $$0.780992\pi$$
$$164$$ 8.66260 0.676435
$$165$$ 1.17935 0.0918122
$$166$$ −7.58180 −0.588461
$$167$$ 9.91934 0.767582 0.383791 0.923420i $$-0.374618\pi$$
0.383791 + 0.923420i $$0.374618\pi$$
$$168$$ −10.8416 −0.836449
$$169$$ −0.416450 −0.0320346
$$170$$ −1.49513 −0.114671
$$171$$ 1.60914 0.123054
$$172$$ 0.733845 0.0559552
$$173$$ 10.7461 0.817009 0.408504 0.912756i $$-0.366050\pi$$
0.408504 + 0.912756i $$0.366050\pi$$
$$174$$ 20.7559 1.57350
$$175$$ −2.15598 −0.162977
$$176$$ 2.69732 0.203318
$$177$$ −10.6107 −0.797549
$$178$$ −13.0928 −0.981350
$$179$$ 9.08355 0.678936 0.339468 0.940618i $$-0.389753\pi$$
0.339468 + 0.940618i $$0.389753\pi$$
$$180$$ 6.07353 0.452694
$$181$$ −6.00048 −0.446012 −0.223006 0.974817i $$-0.571587\pi$$
−0.223006 + 0.974817i $$0.571587\pi$$
$$182$$ −18.3781 −1.36228
$$183$$ 16.2020 1.19769
$$184$$ 24.2131 1.78501
$$185$$ 9.19792 0.676245
$$186$$ −22.3782 −1.64085
$$187$$ −0.622194 −0.0454993
$$188$$ 1.28656 0.0938318
$$189$$ −11.7195 −0.852465
$$190$$ −2.40300 −0.174332
$$191$$ −12.6793 −0.917443 −0.458721 0.888580i $$-0.651692\pi$$
−0.458721 + 0.888580i $$0.651692\pi$$
$$192$$ 12.1608 0.877632
$$193$$ 14.5221 1.04533 0.522664 0.852539i $$-0.324938\pi$$
0.522664 + 0.852539i $$0.324938\pi$$
$$194$$ −32.9310 −2.36431
$$195$$ −4.18354 −0.299589
$$196$$ −8.87639 −0.634028
$$197$$ −1.83384 −0.130656 −0.0653278 0.997864i $$-0.520809\pi$$
−0.0653278 + 0.997864i $$0.520809\pi$$
$$198$$ 3.86675 0.274798
$$199$$ −12.9881 −0.920701 −0.460350 0.887737i $$-0.652276\pi$$
−0.460350 + 0.887737i $$0.652276\pi$$
$$200$$ −4.26389 −0.301503
$$201$$ 0.267309 0.0188545
$$202$$ 11.3052 0.795433
$$203$$ −15.7904 −1.10827
$$204$$ 2.76960 0.193911
$$205$$ −2.29509 −0.160296
$$206$$ −32.0399 −2.23233
$$207$$ 9.13769 0.635114
$$208$$ −9.56830 −0.663442
$$209$$ −1.00000 −0.0691714
$$210$$ 6.11000 0.421630
$$211$$ 9.51881 0.655302 0.327651 0.944799i $$-0.393743\pi$$
0.327651 + 0.944799i $$0.393743\pi$$
$$212$$ −26.2149 −1.80045
$$213$$ −0.405170 −0.0277618
$$214$$ 16.9966 1.16186
$$215$$ −0.194427 −0.0132598
$$216$$ −23.1776 −1.57703
$$217$$ 17.0245 1.15570
$$218$$ −21.4657 −1.45384
$$219$$ −10.8347 −0.732143
$$220$$ −3.77440 −0.254470
$$221$$ 2.20713 0.148467
$$222$$ −26.0667 −1.74948
$$223$$ 0.494807 0.0331348 0.0165674 0.999863i $$-0.494726\pi$$
0.0165674 + 0.999863i $$0.494726\pi$$
$$224$$ −4.41141 −0.294750
$$225$$ −1.60914 −0.107276
$$226$$ −21.3880 −1.42271
$$227$$ 27.7697 1.84314 0.921571 0.388210i $$-0.126906\pi$$
0.921571 + 0.388210i $$0.126906\pi$$
$$228$$ 4.45134 0.294797
$$229$$ −2.65545 −0.175477 −0.0877386 0.996144i $$-0.527964\pi$$
−0.0877386 + 0.996144i $$0.527964\pi$$
$$230$$ −13.6457 −0.899774
$$231$$ 2.54266 0.167295
$$232$$ −31.2286 −2.05026
$$233$$ −19.8348 −1.29942 −0.649711 0.760182i $$-0.725110\pi$$
−0.649711 + 0.760182i $$0.725110\pi$$
$$234$$ −13.7167 −0.896686
$$235$$ −0.340863 −0.0222355
$$236$$ 33.9586 2.21052
$$237$$ −17.0981 −1.11064
$$238$$ −3.22348 −0.208947
$$239$$ 9.34316 0.604359 0.302179 0.953251i $$-0.402286\pi$$
0.302179 + 0.953251i $$0.402286\pi$$
$$240$$ 3.18108 0.205338
$$241$$ −14.6621 −0.944470 −0.472235 0.881473i $$-0.656553\pi$$
−0.472235 + 0.881473i $$0.656553\pi$$
$$242$$ −2.40300 −0.154471
$$243$$ −14.4401 −0.926334
$$244$$ −51.8533 −3.31957
$$245$$ 2.35173 0.150247
$$246$$ 6.50422 0.414694
$$247$$ 3.54733 0.225711
$$248$$ 33.6694 2.13801
$$249$$ −3.72101 −0.235809
$$250$$ 2.40300 0.151979
$$251$$ 18.0162 1.13717 0.568586 0.822624i $$-0.307491\pi$$
0.568586 + 0.822624i $$0.307491\pi$$
$$252$$ 13.0944 0.824872
$$253$$ −5.67863 −0.357013
$$254$$ −35.9992 −2.25879
$$255$$ −0.733784 −0.0459513
$$256$$ −29.0860 −1.81788
$$257$$ −17.0567 −1.06397 −0.531984 0.846754i $$-0.678554\pi$$
−0.531984 + 0.846754i $$0.678554\pi$$
$$258$$ 0.551000 0.0343038
$$259$$ 19.8306 1.23221
$$260$$ 13.3891 0.830354
$$261$$ −11.7853 −0.729490
$$262$$ −18.7780 −1.16011
$$263$$ 2.21160 0.136373 0.0681866 0.997673i $$-0.478279\pi$$
0.0681866 + 0.997673i $$0.478279\pi$$
$$264$$ 5.02862 0.309490
$$265$$ 6.94545 0.426655
$$266$$ −5.18083 −0.317657
$$267$$ −6.42573 −0.393248
$$268$$ −0.855498 −0.0522579
$$269$$ −27.1402 −1.65477 −0.827384 0.561637i $$-0.810172\pi$$
−0.827384 + 0.561637i $$0.810172\pi$$
$$270$$ 13.0622 0.794939
$$271$$ 21.4161 1.30094 0.650468 0.759534i $$-0.274573\pi$$
0.650468 + 0.759534i $$0.274573\pi$$
$$272$$ −1.67826 −0.101759
$$273$$ −9.01964 −0.545894
$$274$$ −33.9760 −2.05256
$$275$$ 1.00000 0.0603023
$$276$$ 25.2775 1.52153
$$277$$ 11.4909 0.690424 0.345212 0.938525i $$-0.387807\pi$$
0.345212 + 0.938525i $$0.387807\pi$$
$$278$$ −30.2664 −1.81526
$$279$$ 12.7064 0.760712
$$280$$ −9.19289 −0.549380
$$281$$ −27.5824 −1.64543 −0.822715 0.568454i $$-0.807542\pi$$
−0.822715 + 0.568454i $$0.807542\pi$$
$$282$$ 0.965998 0.0575243
$$283$$ −14.5401 −0.864318 −0.432159 0.901797i $$-0.642248\pi$$
−0.432159 + 0.901797i $$0.642248\pi$$
$$284$$ 1.29671 0.0769457
$$285$$ −1.17935 −0.0698586
$$286$$ 8.52423 0.504048
$$287$$ −4.94818 −0.292082
$$288$$ −3.29249 −0.194012
$$289$$ −16.6129 −0.977228
$$290$$ 17.5995 1.03348
$$291$$ −16.1620 −0.947431
$$292$$ 34.6756 2.02924
$$293$$ 15.1038 0.882373 0.441186 0.897416i $$-0.354558\pi$$
0.441186 + 0.897416i $$0.354558\pi$$
$$294$$ −6.66475 −0.388696
$$295$$ −8.99709 −0.523831
$$296$$ 39.2190 2.27956
$$297$$ 5.43578 0.315416
$$298$$ −11.4843 −0.665267
$$299$$ 20.1440 1.16496
$$300$$ −4.45134 −0.256998
$$301$$ −0.419181 −0.0241612
$$302$$ 57.7798 3.32485
$$303$$ 5.54840 0.318747
$$304$$ −2.69732 −0.154702
$$305$$ 13.7381 0.786643
$$306$$ −2.40587 −0.137534
$$307$$ 14.1979 0.810318 0.405159 0.914246i $$-0.367216\pi$$
0.405159 + 0.914246i $$0.367216\pi$$
$$308$$ −8.13756 −0.463680
$$309$$ −15.7246 −0.894543
$$310$$ −18.9750 −1.07771
$$311$$ 11.4136 0.647203 0.323602 0.946193i $$-0.395106\pi$$
0.323602 + 0.946193i $$0.395106\pi$$
$$312$$ −17.8382 −1.00989
$$313$$ −17.5462 −0.991773 −0.495886 0.868387i $$-0.665157\pi$$
−0.495886 + 0.868387i $$0.665157\pi$$
$$314$$ 31.2224 1.76198
$$315$$ −3.46927 −0.195471
$$316$$ 54.7211 3.07830
$$317$$ 9.50403 0.533800 0.266900 0.963724i $$-0.414001\pi$$
0.266900 + 0.963724i $$0.414001\pi$$
$$318$$ −19.6832 −1.10378
$$319$$ 7.32397 0.410064
$$320$$ 10.3115 0.576429
$$321$$ 8.34160 0.465583
$$322$$ −29.4200 −1.63951
$$323$$ 0.622194 0.0346198
$$324$$ −5.97590 −0.331994
$$325$$ −3.54733 −0.196770
$$326$$ 47.3994 2.62521
$$327$$ −10.5350 −0.582585
$$328$$ −9.78602 −0.540343
$$329$$ −0.734896 −0.0405161
$$330$$ −2.83397 −0.156005
$$331$$ 18.8296 1.03497 0.517483 0.855693i $$-0.326869\pi$$
0.517483 + 0.855693i $$0.326869\pi$$
$$332$$ 11.9088 0.653579
$$333$$ 14.8007 0.811074
$$334$$ −23.8362 −1.30426
$$335$$ 0.226658 0.0123836
$$336$$ 6.85837 0.374155
$$337$$ −3.44511 −0.187667 −0.0938336 0.995588i $$-0.529912\pi$$
−0.0938336 + 0.995588i $$0.529912\pi$$
$$338$$ 1.00073 0.0544324
$$339$$ −10.4968 −0.570110
$$340$$ 2.34841 0.127361
$$341$$ −7.89640 −0.427614
$$342$$ −3.86675 −0.209090
$$343$$ 20.1622 1.08866
$$344$$ −0.829015 −0.0446975
$$345$$ −6.69709 −0.360559
$$346$$ −25.8228 −1.38824
$$347$$ 13.2512 0.711361 0.355681 0.934608i $$-0.384249\pi$$
0.355681 + 0.934608i $$0.384249\pi$$
$$348$$ −32.6015 −1.74762
$$349$$ 14.8843 0.796737 0.398369 0.917225i $$-0.369576\pi$$
0.398369 + 0.917225i $$0.369576\pi$$
$$350$$ 5.18083 0.276927
$$351$$ −19.2825 −1.02922
$$352$$ 2.04612 0.109059
$$353$$ −26.2048 −1.39474 −0.697371 0.716710i $$-0.745647\pi$$
−0.697371 + 0.716710i $$0.745647\pi$$
$$354$$ 25.4975 1.35518
$$355$$ −0.343554 −0.0182340
$$356$$ 20.5650 1.08994
$$357$$ −1.58203 −0.0837297
$$358$$ −21.8278 −1.15363
$$359$$ −24.3560 −1.28546 −0.642731 0.766092i $$-0.722199\pi$$
−0.642731 + 0.766092i $$0.722199\pi$$
$$360$$ −6.86119 −0.361616
$$361$$ 1.00000 0.0526316
$$362$$ 14.4191 0.757854
$$363$$ −1.17935 −0.0618998
$$364$$ 28.8666 1.51302
$$365$$ −9.18704 −0.480871
$$366$$ −38.9335 −2.03509
$$367$$ 27.2556 1.42273 0.711365 0.702823i $$-0.248077\pi$$
0.711365 + 0.702823i $$0.248077\pi$$
$$368$$ −15.3171 −0.798459
$$369$$ −3.69311 −0.192256
$$370$$ −22.1026 −1.14906
$$371$$ 14.9743 0.777426
$$372$$ 35.1496 1.82242
$$373$$ −28.9149 −1.49716 −0.748579 0.663045i $$-0.769264\pi$$
−0.748579 + 0.663045i $$0.769264\pi$$
$$374$$ 1.49513 0.0773114
$$375$$ 1.17935 0.0609013
$$376$$ −1.45341 −0.0749537
$$377$$ −25.9805 −1.33807
$$378$$ 28.1618 1.44849
$$379$$ −31.0010 −1.59241 −0.796207 0.605025i $$-0.793163\pi$$
−0.796207 + 0.605025i $$0.793163\pi$$
$$380$$ 3.77440 0.193623
$$381$$ −17.6678 −0.905147
$$382$$ 30.4684 1.55890
$$383$$ 23.3273 1.19197 0.595985 0.802996i $$-0.296762\pi$$
0.595985 + 0.802996i $$0.296762\pi$$
$$384$$ −24.3963 −1.24497
$$385$$ 2.15598 0.109879
$$386$$ −34.8967 −1.77620
$$387$$ −0.312859 −0.0159035
$$388$$ 51.7250 2.62594
$$389$$ 13.0462 0.661468 0.330734 0.943724i $$-0.392704\pi$$
0.330734 + 0.943724i $$0.392704\pi$$
$$390$$ 10.0530 0.509056
$$391$$ 3.53321 0.178682
$$392$$ 10.0275 0.506467
$$393$$ −9.21588 −0.464880
$$394$$ 4.40671 0.222007
$$395$$ −14.4980 −0.729471
$$396$$ −6.07353 −0.305207
$$397$$ 19.3816 0.972733 0.486366 0.873755i $$-0.338322\pi$$
0.486366 + 0.873755i $$0.338322\pi$$
$$398$$ 31.2104 1.56443
$$399$$ −2.54266 −0.127292
$$400$$ 2.69732 0.134866
$$401$$ −17.1713 −0.857496 −0.428748 0.903424i $$-0.641045\pi$$
−0.428748 + 0.903424i $$0.641045\pi$$
$$402$$ −0.642342 −0.0320371
$$403$$ 28.0111 1.39533
$$404$$ −17.7572 −0.883453
$$405$$ 1.58327 0.0786733
$$406$$ 37.9442 1.88314
$$407$$ −9.19792 −0.455924
$$408$$ −3.12877 −0.154897
$$409$$ 32.5084 1.60744 0.803718 0.595011i $$-0.202852\pi$$
0.803718 + 0.595011i $$0.202852\pi$$
$$410$$ 5.51510 0.272371
$$411$$ −16.6748 −0.822507
$$412$$ 50.3253 2.47935
$$413$$ −19.3976 −0.954492
$$414$$ −21.9579 −1.07917
$$415$$ −3.15514 −0.154880
$$416$$ −7.25827 −0.355866
$$417$$ −14.8542 −0.727414
$$418$$ 2.40300 0.117535
$$419$$ 0.917867 0.0448407 0.0224204 0.999749i $$-0.492863\pi$$
0.0224204 + 0.999749i $$0.492863\pi$$
$$420$$ −9.59702 −0.468287
$$421$$ 14.2065 0.692382 0.346191 0.938164i $$-0.387475\pi$$
0.346191 + 0.938164i $$0.387475\pi$$
$$422$$ −22.8737 −1.11347
$$423$$ −0.548496 −0.0266688
$$424$$ 29.6147 1.43821
$$425$$ −0.622194 −0.0301808
$$426$$ 0.973624 0.0471722
$$427$$ 29.6192 1.43337
$$428$$ −26.6966 −1.29043
$$429$$ 4.18354 0.201983
$$430$$ 0.467207 0.0225307
$$431$$ −24.3672 −1.17373 −0.586863 0.809687i $$-0.699637\pi$$
−0.586863 + 0.809687i $$0.699637\pi$$
$$432$$ 14.6621 0.705428
$$433$$ −34.1956 −1.64333 −0.821667 0.569967i $$-0.806956\pi$$
−0.821667 + 0.569967i $$0.806956\pi$$
$$434$$ −40.9099 −1.96374
$$435$$ 8.63752 0.414137
$$436$$ 33.7163 1.61472
$$437$$ 5.67863 0.271646
$$438$$ 26.0358 1.24404
$$439$$ −14.2883 −0.681942 −0.340971 0.940074i $$-0.610756\pi$$
−0.340971 + 0.940074i $$0.610756\pi$$
$$440$$ 4.26389 0.203273
$$441$$ 3.78426 0.180203
$$442$$ −5.30372 −0.252272
$$443$$ 26.4567 1.25700 0.628498 0.777812i $$-0.283670\pi$$
0.628498 + 0.777812i $$0.283670\pi$$
$$444$$ 40.9431 1.94307
$$445$$ −5.44854 −0.258286
$$446$$ −1.18902 −0.0563018
$$447$$ −5.63628 −0.266587
$$448$$ 22.2314 1.05033
$$449$$ −9.81985 −0.463428 −0.231714 0.972784i $$-0.574433\pi$$
−0.231714 + 0.972784i $$0.574433\pi$$
$$450$$ 3.86675 0.182281
$$451$$ 2.29509 0.108072
$$452$$ 33.5942 1.58014
$$453$$ 28.3573 1.33234
$$454$$ −66.7307 −3.13182
$$455$$ −7.64799 −0.358543
$$456$$ −5.02862 −0.235487
$$457$$ −8.44589 −0.395082 −0.197541 0.980295i $$-0.563296\pi$$
−0.197541 + 0.980295i $$0.563296\pi$$
$$458$$ 6.38105 0.298167
$$459$$ −3.38211 −0.157863
$$460$$ 21.4335 0.999340
$$461$$ −25.0196 −1.16528 −0.582639 0.812731i $$-0.697980\pi$$
−0.582639 + 0.812731i $$0.697980\pi$$
$$462$$ −6.11000 −0.284263
$$463$$ −5.25947 −0.244428 −0.122214 0.992504i $$-0.538999\pi$$
−0.122214 + 0.992504i $$0.538999\pi$$
$$464$$ 19.7551 0.917108
$$465$$ −9.31261 −0.431862
$$466$$ 47.6630 2.20795
$$467$$ 16.4456 0.761013 0.380507 0.924778i $$-0.375750\pi$$
0.380507 + 0.924778i $$0.375750\pi$$
$$468$$ 21.5448 0.995910
$$469$$ 0.488671 0.0225647
$$470$$ 0.819094 0.0377820
$$471$$ 15.3234 0.706065
$$472$$ −38.3626 −1.76578
$$473$$ 0.194427 0.00893975
$$474$$ 41.0868 1.88718
$$475$$ −1.00000 −0.0458831
$$476$$ 5.06314 0.232069
$$477$$ 11.1762 0.511722
$$478$$ −22.4516 −1.02691
$$479$$ −21.0736 −0.962876 −0.481438 0.876480i $$-0.659885\pi$$
−0.481438 + 0.876480i $$0.659885\pi$$
$$480$$ 2.41309 0.110142
$$481$$ 32.6281 1.48771
$$482$$ 35.2330 1.60482
$$483$$ −14.4388 −0.656989
$$484$$ 3.77440 0.171564
$$485$$ −13.7041 −0.622273
$$486$$ 34.6996 1.57400
$$487$$ −19.2996 −0.874547 −0.437274 0.899329i $$-0.644056\pi$$
−0.437274 + 0.899329i $$0.644056\pi$$
$$488$$ 58.5779 2.65170
$$489$$ 23.2628 1.05198
$$490$$ −5.65121 −0.255296
$$491$$ 0.412492 0.0186155 0.00930775 0.999957i $$-0.497037\pi$$
0.00930775 + 0.999957i $$0.497037\pi$$
$$492$$ −10.2162 −0.460583
$$493$$ −4.55693 −0.205234
$$494$$ −8.52423 −0.383523
$$495$$ 1.60914 0.0723253
$$496$$ −21.2991 −0.956360
$$497$$ −0.740698 −0.0332248
$$498$$ 8.94158 0.400682
$$499$$ 35.3500 1.58248 0.791241 0.611504i $$-0.209435\pi$$
0.791241 + 0.611504i $$0.209435\pi$$
$$500$$ −3.77440 −0.168797
$$501$$ −11.6984 −0.522644
$$502$$ −43.2929 −1.93226
$$503$$ 9.66966 0.431149 0.215575 0.976487i $$-0.430838\pi$$
0.215575 + 0.976487i $$0.430838\pi$$
$$504$$ −14.7926 −0.658915
$$505$$ 4.70463 0.209353
$$506$$ 13.6457 0.606627
$$507$$ 0.491139 0.0218123
$$508$$ 56.5442 2.50874
$$509$$ 32.5575 1.44308 0.721542 0.692370i $$-0.243434\pi$$
0.721542 + 0.692370i $$0.243434\pi$$
$$510$$ 1.76328 0.0780794
$$511$$ −19.8071 −0.876215
$$512$$ 28.5213 1.26047
$$513$$ −5.43578 −0.239996
$$514$$ 40.9873 1.80787
$$515$$ −13.3333 −0.587536
$$516$$ −0.865459 −0.0380997
$$517$$ 0.340863 0.0149912
$$518$$ −47.6529 −2.09375
$$519$$ −12.6734 −0.556299
$$520$$ −15.1254 −0.663294
$$521$$ 13.0671 0.572482 0.286241 0.958158i $$-0.407594\pi$$
0.286241 + 0.958158i $$0.407594\pi$$
$$522$$ 28.3200 1.23953
$$523$$ 12.2092 0.533870 0.266935 0.963714i $$-0.413989\pi$$
0.266935 + 0.963714i $$0.413989\pi$$
$$524$$ 29.4946 1.28848
$$525$$ 2.54266 0.110971
$$526$$ −5.31448 −0.231722
$$527$$ 4.91309 0.214018
$$528$$ −3.18108 −0.138439
$$529$$ 9.24686 0.402037
$$530$$ −16.6899 −0.724963
$$531$$ −14.4775 −0.628272
$$532$$ 8.13756 0.352808
$$533$$ −8.14144 −0.352645
$$534$$ 15.4410 0.668198
$$535$$ 7.07306 0.305795
$$536$$ 0.966445 0.0417441
$$537$$ −10.7127 −0.462286
$$538$$ 65.2179 2.81174
$$539$$ −2.35173 −0.101296
$$540$$ −20.5168 −0.882904
$$541$$ 16.5139 0.709990 0.354995 0.934868i $$-0.384483\pi$$
0.354995 + 0.934868i $$0.384483\pi$$
$$542$$ −51.4629 −2.21052
$$543$$ 7.07666 0.303689
$$544$$ −1.27308 −0.0545831
$$545$$ −8.93287 −0.382642
$$546$$ 21.6742 0.927570
$$547$$ 5.86659 0.250837 0.125419 0.992104i $$-0.459973\pi$$
0.125419 + 0.992104i $$0.459973\pi$$
$$548$$ 53.3663 2.27969
$$549$$ 22.1065 0.943484
$$550$$ −2.40300 −0.102464
$$551$$ −7.32397 −0.312012
$$552$$ −28.5557 −1.21541
$$553$$ −31.2574 −1.32920
$$554$$ −27.6127 −1.17315
$$555$$ −10.8476 −0.460453
$$556$$ 47.5396 2.01613
$$557$$ −27.3711 −1.15975 −0.579875 0.814705i $$-0.696899\pi$$
−0.579875 + 0.814705i $$0.696899\pi$$
$$558$$ −30.5334 −1.29258
$$559$$ −0.689696 −0.0291710
$$560$$ 5.81539 0.245745
$$561$$ 0.733784 0.0309804
$$562$$ 66.2806 2.79588
$$563$$ 35.1477 1.48130 0.740649 0.671892i $$-0.234518\pi$$
0.740649 + 0.671892i $$0.234518\pi$$
$$564$$ −1.51730 −0.0638898
$$565$$ −8.90054 −0.374449
$$566$$ 34.9398 1.46863
$$567$$ 3.41350 0.143354
$$568$$ −1.46488 −0.0614649
$$569$$ −32.5565 −1.36484 −0.682420 0.730960i $$-0.739073\pi$$
−0.682420 + 0.730960i $$0.739073\pi$$
$$570$$ 2.83397 0.118702
$$571$$ 27.4850 1.15021 0.575106 0.818079i $$-0.304961\pi$$
0.575106 + 0.818079i $$0.304961\pi$$
$$572$$ −13.3891 −0.559825
$$573$$ 14.9533 0.624684
$$574$$ 11.8905 0.496299
$$575$$ −5.67863 −0.236815
$$576$$ 16.5926 0.691357
$$577$$ −3.42422 −0.142552 −0.0712760 0.997457i $$-0.522707\pi$$
−0.0712760 + 0.997457i $$0.522707\pi$$
$$578$$ 39.9207 1.66048
$$579$$ −17.1267 −0.711760
$$580$$ −27.6436 −1.14784
$$581$$ −6.80243 −0.282212
$$582$$ 38.8372 1.60985
$$583$$ −6.94545 −0.287651
$$584$$ −39.1725 −1.62097
$$585$$ −5.70814 −0.236002
$$586$$ −36.2944 −1.49931
$$587$$ −18.7672 −0.774604 −0.387302 0.921953i $$-0.626593\pi$$
−0.387302 + 0.921953i $$0.626593\pi$$
$$588$$ 10.4684 0.431708
$$589$$ 7.89640 0.325366
$$590$$ 21.6200 0.890081
$$591$$ 2.16273 0.0889630
$$592$$ −24.8098 −1.01968
$$593$$ 10.8009 0.443538 0.221769 0.975099i $$-0.428817\pi$$
0.221769 + 0.975099i $$0.428817\pi$$
$$594$$ −13.0622 −0.535947
$$595$$ −1.34144 −0.0549937
$$596$$ 18.0384 0.738883
$$597$$ 15.3175 0.626903
$$598$$ −48.4060 −1.97947
$$599$$ 6.00417 0.245324 0.122662 0.992449i $$-0.460857\pi$$
0.122662 + 0.992449i $$0.460857\pi$$
$$600$$ 5.02862 0.205292
$$601$$ −34.2605 −1.39751 −0.698757 0.715359i $$-0.746263\pi$$
−0.698757 + 0.715359i $$0.746263\pi$$
$$602$$ 1.00729 0.0410541
$$603$$ 0.364723 0.0148527
$$604$$ −90.7550 −3.69277
$$605$$ −1.00000 −0.0406558
$$606$$ −13.3328 −0.541608
$$607$$ −26.8024 −1.08788 −0.543938 0.839126i $$-0.683067\pi$$
−0.543938 + 0.839126i $$0.683067\pi$$
$$608$$ −2.04612 −0.0829812
$$609$$ 18.6224 0.754616
$$610$$ −33.0127 −1.33665
$$611$$ −1.20915 −0.0489172
$$612$$ 3.77892 0.152754
$$613$$ −20.7648 −0.838682 −0.419341 0.907829i $$-0.637739\pi$$
−0.419341 + 0.907829i $$0.637739\pi$$
$$614$$ −34.1176 −1.37687
$$615$$ 2.70671 0.109145
$$616$$ 9.19289 0.370392
$$617$$ −1.40456 −0.0565455 −0.0282728 0.999600i $$-0.509001\pi$$
−0.0282728 + 0.999600i $$0.509001\pi$$
$$618$$ 37.7863 1.51999
$$619$$ 47.5216 1.91005 0.955027 0.296519i $$-0.0958258\pi$$
0.955027 + 0.296519i $$0.0958258\pi$$
$$620$$ 29.8042 1.19697
$$621$$ −30.8678 −1.23868
$$622$$ −27.4268 −1.09971
$$623$$ −11.7470 −0.470632
$$624$$ 11.2844 0.451736
$$625$$ 1.00000 0.0400000
$$626$$ 42.1636 1.68520
$$627$$ 1.17935 0.0470987
$$628$$ −49.0412 −1.95696
$$629$$ 5.72289 0.228187
$$630$$ 8.33666 0.332141
$$631$$ −25.1060 −0.999454 −0.499727 0.866183i $$-0.666566\pi$$
−0.499727 + 0.866183i $$0.666566\pi$$
$$632$$ −61.8177 −2.45898
$$633$$ −11.2260 −0.446193
$$634$$ −22.8382 −0.907020
$$635$$ −14.9809 −0.594501
$$636$$ 30.9165 1.22592
$$637$$ 8.34237 0.330537
$$638$$ −17.5995 −0.696771
$$639$$ −0.552826 −0.0218694
$$640$$ −20.6862 −0.817695
$$641$$ 21.3481 0.843199 0.421600 0.906782i $$-0.361469\pi$$
0.421600 + 0.906782i $$0.361469\pi$$
$$642$$ −20.0449 −0.791108
$$643$$ −14.5139 −0.572371 −0.286185 0.958174i $$-0.592387\pi$$
−0.286185 + 0.958174i $$0.592387\pi$$
$$644$$ 46.2102 1.82094
$$645$$ 0.229297 0.00902855
$$646$$ −1.49513 −0.0588252
$$647$$ −23.8976 −0.939511 −0.469755 0.882797i $$-0.655658\pi$$
−0.469755 + 0.882797i $$0.655658\pi$$
$$648$$ 6.75089 0.265200
$$649$$ 8.99709 0.353167
$$650$$ 8.52423 0.334348
$$651$$ −20.0778 −0.786913
$$652$$ −74.4506 −2.91571
$$653$$ 16.3847 0.641181 0.320591 0.947218i $$-0.396119\pi$$
0.320591 + 0.947218i $$0.396119\pi$$
$$654$$ 25.3155 0.989916
$$655$$ −7.81438 −0.305333
$$656$$ 6.19060 0.241702
$$657$$ −14.7832 −0.576747
$$658$$ 1.76595 0.0688441
$$659$$ 3.86889 0.150711 0.0753554 0.997157i $$-0.475991\pi$$
0.0753554 + 0.997157i $$0.475991\pi$$
$$660$$ 4.45134 0.173268
$$661$$ 4.57508 0.177950 0.0889749 0.996034i $$-0.471641\pi$$
0.0889749 + 0.996034i $$0.471641\pi$$
$$662$$ −45.2474 −1.75859
$$663$$ −2.60297 −0.101091
$$664$$ −13.4532 −0.522084
$$665$$ −2.15598 −0.0836055
$$666$$ −35.5661 −1.37816
$$667$$ −41.5901 −1.61038
$$668$$ 37.4396 1.44858
$$669$$ −0.583550 −0.0225614
$$670$$ −0.544659 −0.0210420
$$671$$ −13.7381 −0.530355
$$672$$ 5.20259 0.200694
$$673$$ 22.2879 0.859135 0.429568 0.903035i $$-0.358666\pi$$
0.429568 + 0.903035i $$0.358666\pi$$
$$674$$ 8.27860 0.318880
$$675$$ 5.43578 0.209223
$$676$$ −1.57185 −0.0604557
$$677$$ −22.4962 −0.864600 −0.432300 0.901730i $$-0.642298\pi$$
−0.432300 + 0.901730i $$0.642298\pi$$
$$678$$ 25.2239 0.968718
$$679$$ −29.5459 −1.13387
$$680$$ −2.65297 −0.101737
$$681$$ −32.7502 −1.25499
$$682$$ 18.9750 0.726592
$$683$$ 12.8458 0.491532 0.245766 0.969329i $$-0.420961\pi$$
0.245766 + 0.969329i $$0.420961\pi$$
$$684$$ 6.07353 0.232227
$$685$$ −14.1390 −0.540223
$$686$$ −48.4497 −1.84982
$$687$$ 3.13170 0.119482
$$688$$ 0.524432 0.0199938
$$689$$ 24.6378 0.938625
$$690$$ 16.0931 0.612654
$$691$$ −24.2131 −0.921110 −0.460555 0.887631i $$-0.652350\pi$$
−0.460555 + 0.887631i $$0.652350\pi$$
$$692$$ 40.5600 1.54186
$$693$$ 3.46927 0.131787
$$694$$ −31.8426 −1.20873
$$695$$ −12.5953 −0.477766
$$696$$ 36.8295 1.39602
$$697$$ −1.42799 −0.0540890
$$698$$ −35.7669 −1.35380
$$699$$ 23.3922 0.884772
$$700$$ −8.13756 −0.307571
$$701$$ −36.6327 −1.38360 −0.691798 0.722091i $$-0.743181\pi$$
−0.691798 + 0.722091i $$0.743181\pi$$
$$702$$ 46.3358 1.74883
$$703$$ 9.19792 0.346906
$$704$$ −10.3115 −0.388628
$$705$$ 0.401997 0.0151401
$$706$$ 62.9702 2.36991
$$707$$ 10.1431 0.381471
$$708$$ −40.0491 −1.50514
$$709$$ 37.1353 1.39464 0.697322 0.716758i $$-0.254375\pi$$
0.697322 + 0.716758i $$0.254375\pi$$
$$710$$ 0.825561 0.0309827
$$711$$ −23.3292 −0.874913
$$712$$ −23.2320 −0.870656
$$713$$ 44.8408 1.67930
$$714$$ 3.80161 0.142272
$$715$$ 3.54733 0.132663
$$716$$ 34.2850 1.28129
$$717$$ −11.0188 −0.411506
$$718$$ 58.5275 2.18423
$$719$$ 5.84694 0.218054 0.109027 0.994039i $$-0.465226\pi$$
0.109027 + 0.994039i $$0.465226\pi$$
$$720$$ 4.34036 0.161756
$$721$$ −28.7464 −1.07057
$$722$$ −2.40300 −0.0894304
$$723$$ 17.2917 0.643087
$$724$$ −22.6482 −0.841715
$$725$$ 7.32397 0.272006
$$726$$ 2.83397 0.105179
$$727$$ 52.5801 1.95009 0.975044 0.222013i $$-0.0712627\pi$$
0.975044 + 0.222013i $$0.0712627\pi$$
$$728$$ −32.6102 −1.20861
$$729$$ 21.7797 0.806657
$$730$$ 22.0764 0.817086
$$731$$ −0.120971 −0.00447428
$$732$$ 61.1531 2.26028
$$733$$ 31.6350 1.16847 0.584233 0.811586i $$-0.301395\pi$$
0.584233 + 0.811586i $$0.301395\pi$$
$$734$$ −65.4951 −2.41747
$$735$$ −2.77351 −0.102303
$$736$$ −11.6192 −0.428288
$$737$$ −0.226658 −0.00834905
$$738$$ 8.87455 0.326677
$$739$$ 25.8243 0.949963 0.474982 0.879996i $$-0.342455\pi$$
0.474982 + 0.879996i $$0.342455\pi$$
$$740$$ 34.7167 1.27621
$$741$$ −4.18354 −0.153686
$$742$$ −35.9832 −1.32098
$$743$$ −22.0227 −0.807933 −0.403967 0.914774i $$-0.632369\pi$$
−0.403967 + 0.914774i $$0.632369\pi$$
$$744$$ −39.7080 −1.45577
$$745$$ −4.77915 −0.175094
$$746$$ 69.4826 2.54394
$$747$$ −5.07705 −0.185760
$$748$$ −2.34841 −0.0858664
$$749$$ 15.2494 0.557201
$$750$$ −2.83397 −0.103482
$$751$$ 25.1467 0.917618 0.458809 0.888535i $$-0.348276\pi$$
0.458809 + 0.888535i $$0.348276\pi$$
$$752$$ 0.919419 0.0335277
$$753$$ −21.2474 −0.774297
$$754$$ 62.4312 2.27361
$$755$$ 24.0449 0.875082
$$756$$ −44.2340 −1.60877
$$757$$ −24.6908 −0.897403 −0.448702 0.893682i $$-0.648113\pi$$
−0.448702 + 0.893682i $$0.648113\pi$$
$$758$$ 74.4953 2.70579
$$759$$ 6.69709 0.243089
$$760$$ −4.26389 −0.154668
$$761$$ 22.1274 0.802119 0.401059 0.916052i $$-0.368642\pi$$
0.401059 + 0.916052i $$0.368642\pi$$
$$762$$ 42.4556 1.53800
$$763$$ −19.2591 −0.697227
$$764$$ −47.8568 −1.73140
$$765$$ −1.00119 −0.0361983
$$766$$ −56.0555 −2.02537
$$767$$ −31.9156 −1.15241
$$768$$ 34.3026 1.23779
$$769$$ 0.0310222 0.00111869 0.000559344 1.00000i $$-0.499822\pi$$
0.000559344 1.00000i $$0.499822\pi$$
$$770$$ −5.18083 −0.186704
$$771$$ 20.1158 0.724454
$$772$$ 54.8125 1.97274
$$773$$ 42.3683 1.52388 0.761940 0.647647i $$-0.224247\pi$$
0.761940 + 0.647647i $$0.224247\pi$$
$$774$$ 0.751800 0.0270229
$$775$$ −7.89640 −0.283647
$$776$$ −58.4330 −2.09762
$$777$$ −23.3872 −0.839010
$$778$$ −31.3500 −1.12395
$$779$$ −2.29509 −0.0822301
$$780$$ −15.7904 −0.565386
$$781$$ 0.343554 0.0122933
$$782$$ −8.49030 −0.303612
$$783$$ 39.8115 1.42275
$$784$$ −6.34338 −0.226549
$$785$$ 12.9931 0.463744
$$786$$ 22.1458 0.789913
$$787$$ −33.4522 −1.19244 −0.596222 0.802820i $$-0.703332\pi$$
−0.596222 + 0.802820i $$0.703332\pi$$
$$788$$ −6.92165 −0.246573
$$789$$ −2.60825 −0.0928562
$$790$$ 34.8386 1.23950
$$791$$ −19.1894 −0.682298
$$792$$ 6.86119 0.243802
$$793$$ 48.7337 1.73058
$$794$$ −46.5739 −1.65285
$$795$$ −8.19110 −0.290509
$$796$$ −49.0223 −1.73755
$$797$$ −12.2471 −0.433816 −0.216908 0.976192i $$-0.569597\pi$$
−0.216908 + 0.976192i $$0.569597\pi$$
$$798$$ 6.11000 0.216292
$$799$$ −0.212083 −0.00750296
$$800$$ 2.04612 0.0723413
$$801$$ −8.76745 −0.309783
$$802$$ 41.2627 1.45704
$$803$$ 9.18704 0.324203
$$804$$ 1.00893 0.0355822
$$805$$ −12.2430 −0.431510
$$806$$ −67.3108 −2.37092
$$807$$ 32.0078 1.12673
$$808$$ 20.0601 0.705710
$$809$$ −25.7316 −0.904676 −0.452338 0.891846i $$-0.649410\pi$$
−0.452338 + 0.891846i $$0.649410\pi$$
$$810$$ −3.80459 −0.133680
$$811$$ −17.4658 −0.613309 −0.306654 0.951821i $$-0.599210\pi$$
−0.306654 + 0.951821i $$0.599210\pi$$
$$812$$ −59.5993 −2.09152
$$813$$ −25.2570 −0.885803
$$814$$ 22.1026 0.774696
$$815$$ 19.7251 0.690941
$$816$$ 1.97925 0.0692877
$$817$$ −0.194427 −0.00680213
$$818$$ −78.1176 −2.73132
$$819$$ −12.3067 −0.430029
$$820$$ −8.66260 −0.302511
$$821$$ 18.5642 0.647896 0.323948 0.946075i $$-0.394990\pi$$
0.323948 + 0.946075i $$0.394990\pi$$
$$822$$ 40.0695 1.39758
$$823$$ −19.1102 −0.666139 −0.333069 0.942902i $$-0.608084\pi$$
−0.333069 + 0.942902i $$0.608084\pi$$
$$824$$ −56.8518 −1.98053
$$825$$ −1.17935 −0.0410597
$$826$$ 46.6124 1.62185
$$827$$ 38.2855 1.33132 0.665658 0.746257i $$-0.268151\pi$$
0.665658 + 0.746257i $$0.268151\pi$$
$$828$$ 34.4894 1.19859
$$829$$ 8.64603 0.300289 0.150144 0.988664i $$-0.452026\pi$$
0.150144 + 0.988664i $$0.452026\pi$$
$$830$$ 7.58180 0.263168
$$831$$ −13.5518 −0.470108
$$832$$ 36.5782 1.26812
$$833$$ 1.46323 0.0506980
$$834$$ 35.6946 1.23600
$$835$$ −9.91934 −0.343273
$$836$$ −3.77440 −0.130541
$$837$$ −42.9231 −1.48364
$$838$$ −2.20563 −0.0761924
$$839$$ 24.7123 0.853165 0.426582 0.904449i $$-0.359717\pi$$
0.426582 + 0.904449i $$0.359717\pi$$
$$840$$ 10.8416 0.374071
$$841$$ 24.6406 0.849675
$$842$$ −34.1382 −1.17648
$$843$$ 32.5293 1.12037
$$844$$ 35.9279 1.23669
$$845$$ 0.416450 0.0143263
$$846$$ 1.31803 0.0453150
$$847$$ −2.15598 −0.0740805
$$848$$ −18.7341 −0.643332
$$849$$ 17.1478 0.588512
$$850$$ 1.49513 0.0512826
$$851$$ 52.2316 1.79048
$$852$$ −1.52928 −0.0523921
$$853$$ 7.26768 0.248841 0.124420 0.992230i $$-0.460293\pi$$
0.124420 + 0.992230i $$0.460293\pi$$
$$854$$ −71.1749 −2.43556
$$855$$ −1.60914 −0.0550313
$$856$$ 30.1588 1.03081
$$857$$ −52.0192 −1.77694 −0.888471 0.458932i $$-0.848232\pi$$
−0.888471 + 0.458932i $$0.848232\pi$$
$$858$$ −10.0530 −0.343205
$$859$$ −33.3223 −1.13694 −0.568472 0.822703i $$-0.692465\pi$$
−0.568472 + 0.822703i $$0.692465\pi$$
$$860$$ −0.733845 −0.0250239
$$861$$ 5.83563 0.198878
$$862$$ 58.5543 1.99437
$$863$$ 12.4776 0.424743 0.212371 0.977189i $$-0.431881\pi$$
0.212371 + 0.977189i $$0.431881\pi$$
$$864$$ 11.1223 0.378387
$$865$$ −10.7461 −0.365377
$$866$$ 82.1720 2.79232
$$867$$ 19.5924 0.665392
$$868$$ 64.2574 2.18104
$$869$$ 14.4980 0.491809
$$870$$ −20.7559 −0.703693
$$871$$ 0.804030 0.0272435
$$872$$ −38.0888 −1.28985
$$873$$ −22.0518 −0.746342
$$874$$ −13.6457 −0.461574
$$875$$ 2.15598 0.0728856
$$876$$ −40.8946 −1.38170
$$877$$ 1.06682 0.0360238 0.0180119 0.999838i $$-0.494266\pi$$
0.0180119 + 0.999838i $$0.494266\pi$$
$$878$$ 34.3347 1.15874
$$879$$ −17.8126 −0.600805
$$880$$ −2.69732 −0.0909267
$$881$$ −34.9901 −1.17885 −0.589423 0.807825i $$-0.700645\pi$$
−0.589423 + 0.807825i $$0.700645\pi$$
$$882$$ −9.09357 −0.306196
$$883$$ −48.3044 −1.62557 −0.812786 0.582562i $$-0.802050\pi$$
−0.812786 + 0.582562i $$0.802050\pi$$
$$884$$ 8.33059 0.280188
$$885$$ 10.6107 0.356675
$$886$$ −63.5754 −2.13586
$$887$$ 8.74597 0.293661 0.146830 0.989162i $$-0.453093\pi$$
0.146830 + 0.989162i $$0.453093\pi$$
$$888$$ −46.2528 −1.55214
$$889$$ −32.2987 −1.08326
$$890$$ 13.0928 0.438873
$$891$$ −1.58327 −0.0530415
$$892$$ 1.86760 0.0625320
$$893$$ −0.340863 −0.0114066
$$894$$ 13.5440 0.452978
$$895$$ −9.08355 −0.303629
$$896$$ −44.5992 −1.48995
$$897$$ −23.7568 −0.793216
$$898$$ 23.5971 0.787445
$$899$$ −57.8330 −1.92884
$$900$$ −6.07353 −0.202451
$$901$$ 4.32141 0.143967
$$902$$ −5.51510 −0.183633
$$903$$ 0.494360 0.0164513
$$904$$ −37.9510 −1.26223
$$905$$ 6.00048 0.199463
$$906$$ −68.1425 −2.26388
$$907$$ −25.8242 −0.857479 −0.428739 0.903428i $$-0.641042\pi$$
−0.428739 + 0.903428i $$0.641042\pi$$
$$908$$ 104.814 3.47838
$$909$$ 7.57040 0.251094
$$910$$ 18.3781 0.609228
$$911$$ −11.3062 −0.374591 −0.187296 0.982304i $$-0.559972\pi$$
−0.187296 + 0.982304i $$0.559972\pi$$
$$912$$ 3.18108 0.105336
$$913$$ 3.15514 0.104420
$$914$$ 20.2955 0.671315
$$915$$ −16.2020 −0.535623
$$916$$ −10.0228 −0.331161
$$917$$ −16.8477 −0.556360
$$918$$ 8.12720 0.268238
$$919$$ −1.21432 −0.0400567 −0.0200284 0.999799i $$-0.506376\pi$$
−0.0200284 + 0.999799i $$0.506376\pi$$
$$920$$ −24.2131 −0.798282
$$921$$ −16.7443 −0.551743
$$922$$ 60.1220 1.98001
$$923$$ −1.21870 −0.0401140
$$924$$ 9.59702 0.315719
$$925$$ −9.19792 −0.302426
$$926$$ 12.6385 0.415327
$$927$$ −21.4551 −0.704679
$$928$$ 14.9857 0.491931
$$929$$ −45.7356 −1.50054 −0.750269 0.661133i $$-0.770076\pi$$
−0.750269 + 0.661133i $$0.770076\pi$$
$$930$$ 22.3782 0.733810
$$931$$ 2.35173 0.0770749
$$932$$ −74.8646 −2.45227
$$933$$ −13.4606 −0.440679
$$934$$ −39.5188 −1.29310
$$935$$ 0.622194 0.0203479
$$936$$ −24.3389 −0.795542
$$937$$ −30.7781 −1.00548 −0.502739 0.864438i $$-0.667674\pi$$
−0.502739 + 0.864438i $$0.667674\pi$$
$$938$$ −1.17428 −0.0383414
$$939$$ 20.6931 0.675295
$$940$$ −1.28656 −0.0419628
$$941$$ 31.6570 1.03199 0.515995 0.856592i $$-0.327422\pi$$
0.515995 + 0.856592i $$0.327422\pi$$
$$942$$ −36.8221 −1.19973
$$943$$ −13.0330 −0.424412
$$944$$ 24.2680 0.789858
$$945$$ 11.7195 0.381234
$$946$$ −0.467207 −0.0151902
$$947$$ 50.8102 1.65111 0.825555 0.564321i $$-0.190862\pi$$
0.825555 + 0.564321i $$0.190862\pi$$
$$948$$ −64.5353 −2.09601
$$949$$ −32.5895 −1.05790
$$950$$ 2.40300 0.0779636
$$951$$ −11.2086 −0.363463
$$952$$ −5.71976 −0.185378
$$953$$ 26.6043 0.861799 0.430899 0.902400i $$-0.358196\pi$$
0.430899 + 0.902400i $$0.358196\pi$$
$$954$$ −26.8563 −0.869506
$$955$$ 12.6793 0.410293
$$956$$ 35.2649 1.14055
$$957$$ −8.63752 −0.279211
$$958$$ 50.6398 1.63610
$$959$$ −30.4834 −0.984361
$$960$$ −12.1608 −0.392489
$$961$$ 31.3532 1.01139
$$962$$ −78.4052 −2.52789
$$963$$ 11.3815 0.366764
$$964$$ −55.3407 −1.78241
$$965$$ −14.5221 −0.467485
$$966$$ 34.6965 1.11634
$$967$$ −24.6428 −0.792458 −0.396229 0.918152i $$-0.629681\pi$$
−0.396229 + 0.918152i $$0.629681\pi$$
$$968$$ −4.26389 −0.137047
$$969$$ −0.733784 −0.0235725
$$970$$ 32.9310 1.05735
$$971$$ 9.87139 0.316788 0.158394 0.987376i $$-0.449368\pi$$
0.158394 + 0.987376i $$0.449368\pi$$
$$972$$ −54.5028 −1.74818
$$973$$ −27.1552 −0.870555
$$974$$ 46.3769 1.48601
$$975$$ 4.18354 0.133980
$$976$$ −37.0562 −1.18614
$$977$$ 15.0169 0.480434 0.240217 0.970719i $$-0.422781\pi$$
0.240217 + 0.970719i $$0.422781\pi$$
$$978$$ −55.9005 −1.78750
$$979$$ 5.44854 0.174136
$$980$$ 8.87639 0.283546
$$981$$ −14.3742 −0.458933
$$982$$ −0.991218 −0.0316310
$$983$$ −7.52621 −0.240049 −0.120024 0.992771i $$-0.538297\pi$$
−0.120024 + 0.992771i $$0.538297\pi$$
$$984$$ 11.5411 0.367918
$$985$$ 1.83384 0.0584309
$$986$$ 10.9503 0.348729
$$987$$ 0.866699 0.0275873
$$988$$ 13.3891 0.425963
$$989$$ −1.10408 −0.0351076
$$990$$ −3.86675 −0.122894
$$991$$ 49.1175 1.56027 0.780135 0.625611i $$-0.215150\pi$$
0.780135 + 0.625611i $$0.215150\pi$$
$$992$$ −16.1570 −0.512985
$$993$$ −22.2066 −0.704706
$$994$$ 1.77990 0.0564549
$$995$$ 12.9881 0.411750
$$996$$ −14.0446 −0.445020
$$997$$ 26.7842 0.848263 0.424132 0.905601i $$-0.360579\pi$$
0.424132 + 0.905601i $$0.360579\pi$$
$$998$$ −84.9460 −2.68892
$$999$$ −49.9979 −1.58186
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 1045.2.a.h.1.1 7
3.2 odd 2 9405.2.a.bd.1.7 7
5.4 even 2 5225.2.a.m.1.7 7

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.1 7 1.1 even 1 trivial
5225.2.a.m.1.7 7 5.4 even 2
9405.2.a.bd.1.7 7 3.2 odd 2