# Properties

 Label 350.2.a.h.1.1 Level $350$ Weight $2$ Character 350.1 Self dual yes Analytic conductor $2.795$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.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: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 350.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +0.449490 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +6.44949 q^{19} -2.44949 q^{21} +4.89898 q^{22} +6.89898 q^{23} -2.44949 q^{24} +0.449490 q^{26} +1.00000 q^{28} -2.89898 q^{29} -0.898979 q^{31} +1.00000 q^{32} -12.0000 q^{33} -2.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} +6.44949 q^{38} -1.10102 q^{39} -10.8990 q^{41} -2.44949 q^{42} -8.89898 q^{43} +4.89898 q^{44} +6.89898 q^{46} +0.898979 q^{47} -2.44949 q^{48} +1.00000 q^{49} +4.89898 q^{51} +0.449490 q^{52} +1.10102 q^{53} +1.00000 q^{56} -15.7980 q^{57} -2.89898 q^{58} -6.44949 q^{59} +8.44949 q^{61} -0.898979 q^{62} +3.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} +8.00000 q^{67} -2.00000 q^{68} -16.8990 q^{69} -10.8990 q^{71} +3.00000 q^{72} +6.89898 q^{73} -2.00000 q^{74} +6.44949 q^{76} +4.89898 q^{77} -1.10102 q^{78} -2.89898 q^{79} -9.00000 q^{81} -10.8990 q^{82} -2.44949 q^{83} -2.44949 q^{84} -8.89898 q^{86} +7.10102 q^{87} +4.89898 q^{88} -10.0000 q^{89} +0.449490 q^{91} +6.89898 q^{92} +2.20204 q^{93} +0.898979 q^{94} -2.44949 q^{96} +3.79796 q^{97} +1.00000 q^{98} +14.6969 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{9} - 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 8 q^{19} + 4 q^{23} - 4 q^{26} + 2 q^{28} + 4 q^{29} + 8 q^{31} + 2 q^{32} - 24 q^{33} - 4 q^{34} + 6 q^{36} - 4 q^{37} + 8 q^{38} - 12 q^{39} - 12 q^{41} - 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} - 4 q^{52} + 12 q^{53} + 2 q^{56} - 12 q^{57} + 4 q^{58} - 8 q^{59} + 12 q^{61} + 8 q^{62} + 6 q^{63} + 2 q^{64} - 24 q^{66} + 16 q^{67} - 4 q^{68} - 24 q^{69} - 12 q^{71} + 6 q^{72} + 4 q^{73} - 4 q^{74} + 8 q^{76} - 12 q^{78} + 4 q^{79} - 18 q^{81} - 12 q^{82} - 8 q^{86} + 24 q^{87} - 20 q^{89} - 4 q^{91} + 4 q^{92} + 24 q^{93} - 8 q^{94} - 12 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^9 - 4 * q^13 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 6 * q^18 + 8 * q^19 + 4 * q^23 - 4 * q^26 + 2 * q^28 + 4 * q^29 + 8 * q^31 + 2 * q^32 - 24 * q^33 - 4 * q^34 + 6 * q^36 - 4 * q^37 + 8 * q^38 - 12 * q^39 - 12 * q^41 - 8 * q^43 + 4 * q^46 - 8 * q^47 + 2 * q^49 - 4 * q^52 + 12 * q^53 + 2 * q^56 - 12 * q^57 + 4 * q^58 - 8 * q^59 + 12 * q^61 + 8 * q^62 + 6 * q^63 + 2 * q^64 - 24 * q^66 + 16 * q^67 - 4 * q^68 - 24 * q^69 - 12 * q^71 + 6 * q^72 + 4 * q^73 - 4 * q^74 + 8 * q^76 - 12 * q^78 + 4 * q^79 - 18 * q^81 - 12 * q^82 - 8 * q^86 + 24 * q^87 - 20 * q^89 - 4 * q^91 + 4 * q^92 + 24 * q^93 - 8 * q^94 - 12 * q^97 + 2 * q^98

## 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
$$3$$ −2.44949 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −2.44949 −1.00000
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 4.89898 1.47710 0.738549 0.674200i $$-0.235511\pi$$
0.738549 + 0.674200i $$0.235511\pi$$
$$12$$ −2.44949 −0.707107
$$13$$ 0.449490 0.124666 0.0623330 0.998055i $$-0.480146\pi$$
0.0623330 + 0.998055i $$0.480146\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 6.44949 1.47961 0.739807 0.672819i $$-0.234917\pi$$
0.739807 + 0.672819i $$0.234917\pi$$
$$20$$ 0 0
$$21$$ −2.44949 −0.534522
$$22$$ 4.89898 1.04447
$$23$$ 6.89898 1.43854 0.719268 0.694732i $$-0.244477\pi$$
0.719268 + 0.694732i $$0.244477\pi$$
$$24$$ −2.44949 −0.500000
$$25$$ 0 0
$$26$$ 0.449490 0.0881522
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −2.89898 −0.538327 −0.269163 0.963095i $$-0.586747\pi$$
−0.269163 + 0.963095i $$0.586747\pi$$
$$30$$ 0 0
$$31$$ −0.898979 −0.161461 −0.0807307 0.996736i $$-0.525725\pi$$
−0.0807307 + 0.996736i $$0.525725\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −12.0000 −2.08893
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 6.44949 1.04625
$$39$$ −1.10102 −0.176304
$$40$$ 0 0
$$41$$ −10.8990 −1.70213 −0.851067 0.525057i $$-0.824044\pi$$
−0.851067 + 0.525057i $$0.824044\pi$$
$$42$$ −2.44949 −0.377964
$$43$$ −8.89898 −1.35708 −0.678541 0.734563i $$-0.737387\pi$$
−0.678541 + 0.734563i $$0.737387\pi$$
$$44$$ 4.89898 0.738549
$$45$$ 0 0
$$46$$ 6.89898 1.01720
$$47$$ 0.898979 0.131130 0.0655648 0.997848i $$-0.479115\pi$$
0.0655648 + 0.997848i $$0.479115\pi$$
$$48$$ −2.44949 −0.353553
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.89898 0.685994
$$52$$ 0.449490 0.0623330
$$53$$ 1.10102 0.151237 0.0756184 0.997137i $$-0.475907\pi$$
0.0756184 + 0.997137i $$0.475907\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −15.7980 −2.09249
$$58$$ −2.89898 −0.380655
$$59$$ −6.44949 −0.839652 −0.419826 0.907605i $$-0.637909\pi$$
−0.419826 + 0.907605i $$0.637909\pi$$
$$60$$ 0 0
$$61$$ 8.44949 1.08185 0.540923 0.841072i $$-0.318075\pi$$
0.540923 + 0.841072i $$0.318075\pi$$
$$62$$ −0.898979 −0.114171
$$63$$ 3.00000 0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −12.0000 −1.47710
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −16.8990 −2.03440
$$70$$ 0 0
$$71$$ −10.8990 −1.29347 −0.646735 0.762714i $$-0.723866\pi$$
−0.646735 + 0.762714i $$0.723866\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 6.89898 0.807464 0.403732 0.914877i $$-0.367713\pi$$
0.403732 + 0.914877i $$0.367713\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 6.44949 0.739807
$$77$$ 4.89898 0.558291
$$78$$ −1.10102 −0.124666
$$79$$ −2.89898 −0.326161 −0.163080 0.986613i $$-0.552143\pi$$
−0.163080 + 0.986613i $$0.552143\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ −10.8990 −1.20359
$$83$$ −2.44949 −0.268866 −0.134433 0.990923i $$-0.542921\pi$$
−0.134433 + 0.990923i $$0.542921\pi$$
$$84$$ −2.44949 −0.267261
$$85$$ 0 0
$$86$$ −8.89898 −0.959602
$$87$$ 7.10102 0.761309
$$88$$ 4.89898 0.522233
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0.449490 0.0471193
$$92$$ 6.89898 0.719268
$$93$$ 2.20204 0.228341
$$94$$ 0.898979 0.0927227
$$95$$ 0 0
$$96$$ −2.44949 −0.250000
$$97$$ 3.79796 0.385624 0.192812 0.981236i $$-0.438239\pi$$
0.192812 + 0.981236i $$0.438239\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 14.6969 1.47710
$$100$$ 0 0
$$101$$ 8.44949 0.840756 0.420378 0.907349i $$-0.361898\pi$$
0.420378 + 0.907349i $$0.361898\pi$$
$$102$$ 4.89898 0.485071
$$103$$ −3.10102 −0.305553 −0.152776 0.988261i $$-0.548821\pi$$
−0.152776 + 0.988261i $$0.548821\pi$$
$$104$$ 0.449490 0.0440761
$$105$$ 0 0
$$106$$ 1.10102 0.106941
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ −2.89898 −0.277672 −0.138836 0.990315i $$-0.544336\pi$$
−0.138836 + 0.990315i $$0.544336\pi$$
$$110$$ 0 0
$$111$$ 4.89898 0.464991
$$112$$ 1.00000 0.0944911
$$113$$ −0.202041 −0.0190064 −0.00950321 0.999955i $$-0.503025\pi$$
−0.00950321 + 0.999955i $$0.503025\pi$$
$$114$$ −15.7980 −1.47961
$$115$$ 0 0
$$116$$ −2.89898 −0.269163
$$117$$ 1.34847 0.124666
$$118$$ −6.44949 −0.593724
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ 8.44949 0.764981
$$123$$ 26.6969 2.40718
$$124$$ −0.898979 −0.0807307
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ 5.10102 0.452642 0.226321 0.974053i $$-0.427330\pi$$
0.226321 + 0.974053i $$0.427330\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 21.7980 1.91920
$$130$$ 0 0
$$131$$ −1.55051 −0.135469 −0.0677344 0.997703i $$-0.521577\pi$$
−0.0677344 + 0.997703i $$0.521577\pi$$
$$132$$ −12.0000 −1.04447
$$133$$ 6.44949 0.559242
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −17.7980 −1.52058 −0.760291 0.649582i $$-0.774944\pi$$
−0.760291 + 0.649582i $$0.774944\pi$$
$$138$$ −16.8990 −1.43854
$$139$$ 6.44949 0.547039 0.273519 0.961867i $$-0.411812\pi$$
0.273519 + 0.961867i $$0.411812\pi$$
$$140$$ 0 0
$$141$$ −2.20204 −0.185445
$$142$$ −10.8990 −0.914622
$$143$$ 2.20204 0.184144
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 6.89898 0.570964
$$147$$ −2.44949 −0.202031
$$148$$ −2.00000 −0.164399
$$149$$ 15.7980 1.29422 0.647110 0.762397i $$-0.275978\pi$$
0.647110 + 0.762397i $$0.275978\pi$$
$$150$$ 0 0
$$151$$ −19.5959 −1.59469 −0.797347 0.603522i $$-0.793764\pi$$
−0.797347 + 0.603522i $$0.793764\pi$$
$$152$$ 6.44949 0.523123
$$153$$ −6.00000 −0.485071
$$154$$ 4.89898 0.394771
$$155$$ 0 0
$$156$$ −1.10102 −0.0881522
$$157$$ −8.44949 −0.674343 −0.337171 0.941443i $$-0.609470\pi$$
−0.337171 + 0.941443i $$0.609470\pi$$
$$158$$ −2.89898 −0.230630
$$159$$ −2.69694 −0.213881
$$160$$ 0 0
$$161$$ 6.89898 0.543716
$$162$$ −9.00000 −0.707107
$$163$$ 16.8990 1.32363 0.661815 0.749667i $$-0.269786\pi$$
0.661815 + 0.749667i $$0.269786\pi$$
$$164$$ −10.8990 −0.851067
$$165$$ 0 0
$$166$$ −2.44949 −0.190117
$$167$$ −4.89898 −0.379094 −0.189547 0.981872i $$-0.560702\pi$$
−0.189547 + 0.981872i $$0.560702\pi$$
$$168$$ −2.44949 −0.188982
$$169$$ −12.7980 −0.984458
$$170$$ 0 0
$$171$$ 19.3485 1.47961
$$172$$ −8.89898 −0.678541
$$173$$ −18.2474 −1.38733 −0.693664 0.720299i $$-0.744005\pi$$
−0.693664 + 0.720299i $$0.744005\pi$$
$$174$$ 7.10102 0.538327
$$175$$ 0 0
$$176$$ 4.89898 0.369274
$$177$$ 15.7980 1.18745
$$178$$ −10.0000 −0.749532
$$179$$ −5.79796 −0.433360 −0.216680 0.976243i $$-0.569523\pi$$
−0.216680 + 0.976243i $$0.569523\pi$$
$$180$$ 0 0
$$181$$ 14.2474 1.05900 0.529502 0.848309i $$-0.322379\pi$$
0.529502 + 0.848309i $$0.322379\pi$$
$$182$$ 0.449490 0.0333184
$$183$$ −20.6969 −1.52996
$$184$$ 6.89898 0.508600
$$185$$ 0 0
$$186$$ 2.20204 0.161461
$$187$$ −9.79796 −0.716498
$$188$$ 0.898979 0.0655648
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.6969 −1.20815 −0.604074 0.796928i $$-0.706457\pi$$
−0.604074 + 0.796928i $$0.706457\pi$$
$$192$$ −2.44949 −0.176777
$$193$$ −17.5959 −1.26658 −0.633291 0.773914i $$-0.718296\pi$$
−0.633291 + 0.773914i $$0.718296\pi$$
$$194$$ 3.79796 0.272678
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −9.10102 −0.648421 −0.324210 0.945985i $$-0.605099\pi$$
−0.324210 + 0.945985i $$0.605099\pi$$
$$198$$ 14.6969 1.04447
$$199$$ 7.10102 0.503378 0.251689 0.967808i $$-0.419014\pi$$
0.251689 + 0.967808i $$0.419014\pi$$
$$200$$ 0 0
$$201$$ −19.5959 −1.38219
$$202$$ 8.44949 0.594504
$$203$$ −2.89898 −0.203468
$$204$$ 4.89898 0.342997
$$205$$ 0 0
$$206$$ −3.10102 −0.216058
$$207$$ 20.6969 1.43854
$$208$$ 0.449490 0.0311665
$$209$$ 31.5959 2.18554
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 1.10102 0.0756184
$$213$$ 26.6969 1.82924
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.898979 −0.0610267
$$218$$ −2.89898 −0.196344
$$219$$ −16.8990 −1.14193
$$220$$ 0 0
$$221$$ −0.898979 −0.0604719
$$222$$ 4.89898 0.328798
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −0.202041 −0.0134396
$$227$$ 7.34847 0.487735 0.243868 0.969809i $$-0.421584\pi$$
0.243868 + 0.969809i $$0.421584\pi$$
$$228$$ −15.7980 −1.04625
$$229$$ 15.1464 1.00090 0.500452 0.865764i $$-0.333167\pi$$
0.500452 + 0.865764i $$0.333167\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ −2.89898 −0.190327
$$233$$ −10.2020 −0.668358 −0.334179 0.942510i $$-0.608459\pi$$
−0.334179 + 0.942510i $$0.608459\pi$$
$$234$$ 1.34847 0.0881522
$$235$$ 0 0
$$236$$ −6.44949 −0.419826
$$237$$ 7.10102 0.461261
$$238$$ −2.00000 −0.129641
$$239$$ −25.7980 −1.66873 −0.834366 0.551211i $$-0.814166\pi$$
−0.834366 + 0.551211i $$0.814166\pi$$
$$240$$ 0 0
$$241$$ 20.6969 1.33321 0.666604 0.745412i $$-0.267747\pi$$
0.666604 + 0.745412i $$0.267747\pi$$
$$242$$ 13.0000 0.835672
$$243$$ 22.0454 1.41421
$$244$$ 8.44949 0.540923
$$245$$ 0 0
$$246$$ 26.6969 1.70213
$$247$$ 2.89898 0.184458
$$248$$ −0.898979 −0.0570853
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ −1.55051 −0.0978673 −0.0489337 0.998802i $$-0.515582\pi$$
−0.0489337 + 0.998802i $$0.515582\pi$$
$$252$$ 3.00000 0.188982
$$253$$ 33.7980 2.12486
$$254$$ 5.10102 0.320066
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −20.6969 −1.29104 −0.645520 0.763744i $$-0.723359\pi$$
−0.645520 + 0.763744i $$0.723359\pi$$
$$258$$ 21.7980 1.35708
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −8.69694 −0.538327
$$262$$ −1.55051 −0.0957908
$$263$$ 9.79796 0.604168 0.302084 0.953281i $$-0.402318\pi$$
0.302084 + 0.953281i $$0.402318\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ 6.44949 0.395444
$$267$$ 24.4949 1.49906
$$268$$ 8.00000 0.488678
$$269$$ −15.1464 −0.923494 −0.461747 0.887012i $$-0.652777\pi$$
−0.461747 + 0.887012i $$0.652777\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −1.10102 −0.0666368
$$274$$ −17.7980 −1.07521
$$275$$ 0 0
$$276$$ −16.8990 −1.01720
$$277$$ 5.10102 0.306491 0.153245 0.988188i $$-0.451028\pi$$
0.153245 + 0.988188i $$0.451028\pi$$
$$278$$ 6.44949 0.386815
$$279$$ −2.69694 −0.161461
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −2.20204 −0.131130
$$283$$ −28.2474 −1.67914 −0.839568 0.543254i $$-0.817192\pi$$
−0.839568 + 0.543254i $$0.817192\pi$$
$$284$$ −10.8990 −0.646735
$$285$$ 0 0
$$286$$ 2.20204 0.130209
$$287$$ −10.8990 −0.643346
$$288$$ 3.00000 0.176777
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −9.30306 −0.545355
$$292$$ 6.89898 0.403732
$$293$$ 6.24745 0.364980 0.182490 0.983208i $$-0.441584\pi$$
0.182490 + 0.983208i $$0.441584\pi$$
$$294$$ −2.44949 −0.142857
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 15.7980 0.915151
$$299$$ 3.10102 0.179337
$$300$$ 0 0
$$301$$ −8.89898 −0.512929
$$302$$ −19.5959 −1.12762
$$303$$ −20.6969 −1.18901
$$304$$ 6.44949 0.369904
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −4.24745 −0.242415 −0.121207 0.992627i $$-0.538677\pi$$
−0.121207 + 0.992627i $$0.538677\pi$$
$$308$$ 4.89898 0.279145
$$309$$ 7.59592 0.432117
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ −1.10102 −0.0623330
$$313$$ −17.5959 −0.994580 −0.497290 0.867584i $$-0.665672\pi$$
−0.497290 + 0.867584i $$0.665672\pi$$
$$314$$ −8.44949 −0.476832
$$315$$ 0 0
$$316$$ −2.89898 −0.163080
$$317$$ −26.4949 −1.48810 −0.744051 0.668123i $$-0.767098\pi$$
−0.744051 + 0.668123i $$0.767098\pi$$
$$318$$ −2.69694 −0.151237
$$319$$ −14.2020 −0.795162
$$320$$ 0 0
$$321$$ −19.5959 −1.09374
$$322$$ 6.89898 0.384465
$$323$$ −12.8990 −0.717718
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ 16.8990 0.935948
$$327$$ 7.10102 0.392687
$$328$$ −10.8990 −0.601795
$$329$$ 0.898979 0.0495623
$$330$$ 0 0
$$331$$ 10.6969 0.587957 0.293978 0.955812i $$-0.405021\pi$$
0.293978 + 0.955812i $$0.405021\pi$$
$$332$$ −2.44949 −0.134433
$$333$$ −6.00000 −0.328798
$$334$$ −4.89898 −0.268060
$$335$$ 0 0
$$336$$ −2.44949 −0.133631
$$337$$ 29.5959 1.61219 0.806096 0.591785i $$-0.201576\pi$$
0.806096 + 0.591785i $$0.201576\pi$$
$$338$$ −12.7980 −0.696117
$$339$$ 0.494897 0.0268791
$$340$$ 0 0
$$341$$ −4.40408 −0.238494
$$342$$ 19.3485 1.04625
$$343$$ 1.00000 0.0539949
$$344$$ −8.89898 −0.479801
$$345$$ 0 0
$$346$$ −18.2474 −0.980989
$$347$$ −19.1010 −1.02540 −0.512698 0.858569i $$-0.671354\pi$$
−0.512698 + 0.858569i $$0.671354\pi$$
$$348$$ 7.10102 0.380655
$$349$$ −3.55051 −0.190054 −0.0950272 0.995475i $$-0.530294\pi$$
−0.0950272 + 0.995475i $$0.530294\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.89898 0.261116
$$353$$ −13.1010 −0.697297 −0.348648 0.937254i $$-0.613359\pi$$
−0.348648 + 0.937254i $$0.613359\pi$$
$$354$$ 15.7980 0.839652
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 4.89898 0.259281
$$358$$ −5.79796 −0.306432
$$359$$ 11.5959 0.612009 0.306005 0.952030i $$-0.401008\pi$$
0.306005 + 0.952030i $$0.401008\pi$$
$$360$$ 0 0
$$361$$ 22.5959 1.18926
$$362$$ 14.2474 0.748829
$$363$$ −31.8434 −1.67134
$$364$$ 0.449490 0.0235597
$$365$$ 0 0
$$366$$ −20.6969 −1.08185
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 6.89898 0.359634
$$369$$ −32.6969 −1.70213
$$370$$ 0 0
$$371$$ 1.10102 0.0571621
$$372$$ 2.20204 0.114171
$$373$$ −24.6969 −1.27876 −0.639380 0.768891i $$-0.720809\pi$$
−0.639380 + 0.768891i $$0.720809\pi$$
$$374$$ −9.79796 −0.506640
$$375$$ 0 0
$$376$$ 0.898979 0.0463613
$$377$$ −1.30306 −0.0671111
$$378$$ 0 0
$$379$$ 1.30306 0.0669338 0.0334669 0.999440i $$-0.489345\pi$$
0.0334669 + 0.999440i $$0.489345\pi$$
$$380$$ 0 0
$$381$$ −12.4949 −0.640133
$$382$$ −16.6969 −0.854290
$$383$$ 16.8990 0.863498 0.431749 0.901994i $$-0.357897\pi$$
0.431749 + 0.901994i $$0.357897\pi$$
$$384$$ −2.44949 −0.125000
$$385$$ 0 0
$$386$$ −17.5959 −0.895609
$$387$$ −26.6969 −1.35708
$$388$$ 3.79796 0.192812
$$389$$ 22.8990 1.16102 0.580512 0.814252i $$-0.302852\pi$$
0.580512 + 0.814252i $$0.302852\pi$$
$$390$$ 0 0
$$391$$ −13.7980 −0.697793
$$392$$ 1.00000 0.0505076
$$393$$ 3.79796 0.191582
$$394$$ −9.10102 −0.458503
$$395$$ 0 0
$$396$$ 14.6969 0.738549
$$397$$ 17.3485 0.870695 0.435347 0.900263i $$-0.356626\pi$$
0.435347 + 0.900263i $$0.356626\pi$$
$$398$$ 7.10102 0.355942
$$399$$ −15.7980 −0.790887
$$400$$ 0 0
$$401$$ 29.3939 1.46786 0.733930 0.679225i $$-0.237684\pi$$
0.733930 + 0.679225i $$0.237684\pi$$
$$402$$ −19.5959 −0.977356
$$403$$ −0.404082 −0.0201288
$$404$$ 8.44949 0.420378
$$405$$ 0 0
$$406$$ −2.89898 −0.143874
$$407$$ −9.79796 −0.485667
$$408$$ 4.89898 0.242536
$$409$$ −14.4949 −0.716727 −0.358363 0.933582i $$-0.616665\pi$$
−0.358363 + 0.933582i $$0.616665\pi$$
$$410$$ 0 0
$$411$$ 43.5959 2.15043
$$412$$ −3.10102 −0.152776
$$413$$ −6.44949 −0.317359
$$414$$ 20.6969 1.01720
$$415$$ 0 0
$$416$$ 0.449490 0.0220380
$$417$$ −15.7980 −0.773629
$$418$$ 31.5959 1.54541
$$419$$ −6.44949 −0.315078 −0.157539 0.987513i $$-0.550356\pi$$
−0.157539 + 0.987513i $$0.550356\pi$$
$$420$$ 0 0
$$421$$ −23.7980 −1.15984 −0.579921 0.814673i $$-0.696917\pi$$
−0.579921 + 0.814673i $$0.696917\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 2.69694 0.131130
$$424$$ 1.10102 0.0534703
$$425$$ 0 0
$$426$$ 26.6969 1.29347
$$427$$ 8.44949 0.408899
$$428$$ 8.00000 0.386695
$$429$$ −5.39388 −0.260419
$$430$$ 0 0
$$431$$ 17.7980 0.857298 0.428649 0.903471i $$-0.358990\pi$$
0.428649 + 0.903471i $$0.358990\pi$$
$$432$$ 0 0
$$433$$ 19.7980 0.951429 0.475715 0.879600i $$-0.342190\pi$$
0.475715 + 0.879600i $$0.342190\pi$$
$$434$$ −0.898979 −0.0431524
$$435$$ 0 0
$$436$$ −2.89898 −0.138836
$$437$$ 44.4949 2.12848
$$438$$ −16.8990 −0.807464
$$439$$ 37.3939 1.78471 0.892356 0.451332i $$-0.149051\pi$$
0.892356 + 0.451332i $$0.149051\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ −0.898979 −0.0427601
$$443$$ 9.79796 0.465515 0.232758 0.972535i $$-0.425225\pi$$
0.232758 + 0.972535i $$0.425225\pi$$
$$444$$ 4.89898 0.232495
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ −38.6969 −1.83030
$$448$$ 1.00000 0.0472456
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −53.3939 −2.51422
$$452$$ −0.202041 −0.00950321
$$453$$ 48.0000 2.25524
$$454$$ 7.34847 0.344881
$$455$$ 0 0
$$456$$ −15.7980 −0.739807
$$457$$ 9.59592 0.448878 0.224439 0.974488i $$-0.427945\pi$$
0.224439 + 0.974488i $$0.427945\pi$$
$$458$$ 15.1464 0.707746
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2.65153 0.123494 0.0617470 0.998092i $$-0.480333\pi$$
0.0617470 + 0.998092i $$0.480333\pi$$
$$462$$ −12.0000 −0.558291
$$463$$ 35.5959 1.65428 0.827141 0.561994i $$-0.189966\pi$$
0.827141 + 0.561994i $$0.189966\pi$$
$$464$$ −2.89898 −0.134582
$$465$$ 0 0
$$466$$ −10.2020 −0.472600
$$467$$ −5.55051 −0.256847 −0.128423 0.991719i $$-0.540992\pi$$
−0.128423 + 0.991719i $$0.540992\pi$$
$$468$$ 1.34847 0.0623330
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 20.6969 0.953665
$$472$$ −6.44949 −0.296862
$$473$$ −43.5959 −2.00454
$$474$$ 7.10102 0.326161
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 3.30306 0.151237
$$478$$ −25.7980 −1.17997
$$479$$ 38.6969 1.76811 0.884054 0.467385i $$-0.154804\pi$$
0.884054 + 0.467385i $$0.154804\pi$$
$$480$$ 0 0
$$481$$ −0.898979 −0.0409899
$$482$$ 20.6969 0.942720
$$483$$ −16.8990 −0.768930
$$484$$ 13.0000 0.590909
$$485$$ 0 0
$$486$$ 22.0454 1.00000
$$487$$ 36.6969 1.66290 0.831449 0.555602i $$-0.187512\pi$$
0.831449 + 0.555602i $$0.187512\pi$$
$$488$$ 8.44949 0.382490
$$489$$ −41.3939 −1.87190
$$490$$ 0 0
$$491$$ −19.5959 −0.884351 −0.442176 0.896928i $$-0.645793\pi$$
−0.442176 + 0.896928i $$0.645793\pi$$
$$492$$ 26.6969 1.20359
$$493$$ 5.79796 0.261127
$$494$$ 2.89898 0.130431
$$495$$ 0 0
$$496$$ −0.898979 −0.0403654
$$497$$ −10.8990 −0.488886
$$498$$ 6.00000 0.268866
$$499$$ 25.7980 1.15488 0.577438 0.816435i $$-0.304053\pi$$
0.577438 + 0.816435i $$0.304053\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −1.55051 −0.0692027
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ 33.7980 1.50250
$$507$$ 31.3485 1.39223
$$508$$ 5.10102 0.226321
$$509$$ −36.4495 −1.61560 −0.807798 0.589460i $$-0.799341\pi$$
−0.807798 + 0.589460i $$0.799341\pi$$
$$510$$ 0 0
$$511$$ 6.89898 0.305193
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −20.6969 −0.912903
$$515$$ 0 0
$$516$$ 21.7980 0.959602
$$517$$ 4.40408 0.193691
$$518$$ −2.00000 −0.0878750
$$519$$ 44.6969 1.96198
$$520$$ 0 0
$$521$$ 3.30306 0.144710 0.0723549 0.997379i $$-0.476949\pi$$
0.0723549 + 0.997379i $$0.476949\pi$$
$$522$$ −8.69694 −0.380655
$$523$$ −1.14643 −0.0501298 −0.0250649 0.999686i $$-0.507979\pi$$
−0.0250649 + 0.999686i $$0.507979\pi$$
$$524$$ −1.55051 −0.0677344
$$525$$ 0 0
$$526$$ 9.79796 0.427211
$$527$$ 1.79796 0.0783203
$$528$$ −12.0000 −0.522233
$$529$$ 24.5959 1.06939
$$530$$ 0 0
$$531$$ −19.3485 −0.839652
$$532$$ 6.44949 0.279621
$$533$$ −4.89898 −0.212198
$$534$$ 24.4949 1.06000
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 14.2020 0.612863
$$538$$ −15.1464 −0.653009
$$539$$ 4.89898 0.211014
$$540$$ 0 0
$$541$$ −29.5959 −1.27243 −0.636214 0.771513i $$-0.719500\pi$$
−0.636214 + 0.771513i $$0.719500\pi$$
$$542$$ 12.0000 0.515444
$$543$$ −34.8990 −1.49766
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ −1.10102 −0.0471193
$$547$$ −10.6969 −0.457368 −0.228684 0.973501i $$-0.573442\pi$$
−0.228684 + 0.973501i $$0.573442\pi$$
$$548$$ −17.7980 −0.760291
$$549$$ 25.3485 1.08185
$$550$$ 0 0
$$551$$ −18.6969 −0.796516
$$552$$ −16.8990 −0.719268
$$553$$ −2.89898 −0.123277
$$554$$ 5.10102 0.216722
$$555$$ 0 0
$$556$$ 6.44949 0.273519
$$557$$ 16.6969 0.707472 0.353736 0.935345i $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$558$$ −2.69694 −0.114171
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ −18.0000 −0.759284
$$563$$ −14.0454 −0.591943 −0.295972 0.955197i $$-0.595643\pi$$
−0.295972 + 0.955197i $$0.595643\pi$$
$$564$$ −2.20204 −0.0927227
$$565$$ 0 0
$$566$$ −28.2474 −1.18733
$$567$$ −9.00000 −0.377964
$$568$$ −10.8990 −0.457311
$$569$$ 14.2020 0.595381 0.297690 0.954663i $$-0.403784\pi$$
0.297690 + 0.954663i $$0.403784\pi$$
$$570$$ 0 0
$$571$$ −20.8990 −0.874595 −0.437298 0.899317i $$-0.644064\pi$$
−0.437298 + 0.899317i $$0.644064\pi$$
$$572$$ 2.20204 0.0920720
$$573$$ 40.8990 1.70858
$$574$$ −10.8990 −0.454915
$$575$$ 0 0
$$576$$ 3.00000 0.125000
$$577$$ −46.4949 −1.93561 −0.967804 0.251705i $$-0.919009\pi$$
−0.967804 + 0.251705i $$0.919009\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 43.1010 1.79122
$$580$$ 0 0
$$581$$ −2.44949 −0.101622
$$582$$ −9.30306 −0.385624
$$583$$ 5.39388 0.223392
$$584$$ 6.89898 0.285482
$$585$$ 0 0
$$586$$ 6.24745 0.258080
$$587$$ 33.1464 1.36810 0.684050 0.729435i $$-0.260217\pi$$
0.684050 + 0.729435i $$0.260217\pi$$
$$588$$ −2.44949 −0.101015
$$589$$ −5.79796 −0.238901
$$590$$ 0 0
$$591$$ 22.2929 0.917006
$$592$$ −2.00000 −0.0821995
$$593$$ 1.10102 0.0452135 0.0226067 0.999744i $$-0.492803\pi$$
0.0226067 + 0.999744i $$0.492803\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 15.7980 0.647110
$$597$$ −17.3939 −0.711884
$$598$$ 3.10102 0.126810
$$599$$ −22.8990 −0.935627 −0.467813 0.883827i $$-0.654958\pi$$
−0.467813 + 0.883827i $$0.654958\pi$$
$$600$$ 0 0
$$601$$ 19.3939 0.791093 0.395546 0.918446i $$-0.370555\pi$$
0.395546 + 0.918446i $$0.370555\pi$$
$$602$$ −8.89898 −0.362695
$$603$$ 24.0000 0.977356
$$604$$ −19.5959 −0.797347
$$605$$ 0 0
$$606$$ −20.6969 −0.840756
$$607$$ 25.3939 1.03071 0.515353 0.856978i $$-0.327661\pi$$
0.515353 + 0.856978i $$0.327661\pi$$
$$608$$ 6.44949 0.261561
$$609$$ 7.10102 0.287748
$$610$$ 0 0
$$611$$ 0.404082 0.0163474
$$612$$ −6.00000 −0.242536
$$613$$ 8.20204 0.331277 0.165639 0.986187i $$-0.447031\pi$$
0.165639 + 0.986187i $$0.447031\pi$$
$$614$$ −4.24745 −0.171413
$$615$$ 0 0
$$616$$ 4.89898 0.197386
$$617$$ 9.59592 0.386317 0.193159 0.981168i $$-0.438127\pi$$
0.193159 + 0.981168i $$0.438127\pi$$
$$618$$ 7.59592 0.305553
$$619$$ −46.4495 −1.86696 −0.933481 0.358626i $$-0.883245\pi$$
−0.933481 + 0.358626i $$0.883245\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ −10.0000 −0.400642
$$624$$ −1.10102 −0.0440761
$$625$$ 0 0
$$626$$ −17.5959 −0.703274
$$627$$ −77.3939 −3.09081
$$628$$ −8.44949 −0.337171
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 6.49490 0.258558 0.129279 0.991608i $$-0.458734\pi$$
0.129279 + 0.991608i $$0.458734\pi$$
$$632$$ −2.89898 −0.115315
$$633$$ −29.3939 −1.16830
$$634$$ −26.4949 −1.05225
$$635$$ 0 0
$$636$$ −2.69694 −0.106941
$$637$$ 0.449490 0.0178094
$$638$$ −14.2020 −0.562264
$$639$$ −32.6969 −1.29347
$$640$$ 0 0
$$641$$ 6.20204 0.244966 0.122483 0.992471i $$-0.460914\pi$$
0.122483 + 0.992471i $$0.460914\pi$$
$$642$$ −19.5959 −0.773389
$$643$$ 9.14643 0.360700 0.180350 0.983603i $$-0.442277\pi$$
0.180350 + 0.983603i $$0.442277\pi$$
$$644$$ 6.89898 0.271858
$$645$$ 0 0
$$646$$ −12.8990 −0.507504
$$647$$ −22.2929 −0.876423 −0.438211 0.898872i $$-0.644388\pi$$
−0.438211 + 0.898872i $$0.644388\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −31.5959 −1.24025
$$650$$ 0 0
$$651$$ 2.20204 0.0863048
$$652$$ 16.8990 0.661815
$$653$$ 39.7980 1.55741 0.778707 0.627387i $$-0.215876\pi$$
0.778707 + 0.627387i $$0.215876\pi$$
$$654$$ 7.10102 0.277672
$$655$$ 0 0
$$656$$ −10.8990 −0.425534
$$657$$ 20.6969 0.807464
$$658$$ 0.898979 0.0350459
$$659$$ 7.10102 0.276616 0.138308 0.990389i $$-0.455834\pi$$
0.138308 + 0.990389i $$0.455834\pi$$
$$660$$ 0 0
$$661$$ 12.9444 0.503478 0.251739 0.967795i $$-0.418997\pi$$
0.251739 + 0.967795i $$0.418997\pi$$
$$662$$ 10.6969 0.415748
$$663$$ 2.20204 0.0855202
$$664$$ −2.44949 −0.0950586
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ −20.0000 −0.774403
$$668$$ −4.89898 −0.189547
$$669$$ −9.79796 −0.378811
$$670$$ 0 0
$$671$$ 41.3939 1.59799
$$672$$ −2.44949 −0.0944911
$$673$$ −1.79796 −0.0693062 −0.0346531 0.999399i $$-0.511033\pi$$
−0.0346531 + 0.999399i $$0.511033\pi$$
$$674$$ 29.5959 1.13999
$$675$$ 0 0
$$676$$ −12.7980 −0.492229
$$677$$ 31.5505 1.21258 0.606292 0.795242i $$-0.292656\pi$$
0.606292 + 0.795242i $$0.292656\pi$$
$$678$$ 0.494897 0.0190064
$$679$$ 3.79796 0.145752
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ −4.40408 −0.168641
$$683$$ 35.5959 1.36204 0.681020 0.732265i $$-0.261537\pi$$
0.681020 + 0.732265i $$0.261537\pi$$
$$684$$ 19.3485 0.739807
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −37.1010 −1.41549
$$688$$ −8.89898 −0.339270
$$689$$ 0.494897 0.0188541
$$690$$ 0 0
$$691$$ −13.1464 −0.500114 −0.250057 0.968231i $$-0.580449\pi$$
−0.250057 + 0.968231i $$0.580449\pi$$
$$692$$ −18.2474 −0.693664
$$693$$ 14.6969 0.558291
$$694$$ −19.1010 −0.725065
$$695$$ 0 0
$$696$$ 7.10102 0.269163
$$697$$ 21.7980 0.825657
$$698$$ −3.55051 −0.134389
$$699$$ 24.9898 0.945201
$$700$$ 0 0
$$701$$ 40.6969 1.53710 0.768551 0.639788i $$-0.220978\pi$$
0.768551 + 0.639788i $$0.220978\pi$$
$$702$$ 0 0
$$703$$ −12.8990 −0.486494
$$704$$ 4.89898 0.184637
$$705$$ 0 0
$$706$$ −13.1010 −0.493063
$$707$$ 8.44949 0.317776
$$708$$ 15.7980 0.593724
$$709$$ 40.2929 1.51323 0.756615 0.653861i $$-0.226852\pi$$
0.756615 + 0.653861i $$0.226852\pi$$
$$710$$ 0 0
$$711$$ −8.69694 −0.326161
$$712$$ −10.0000 −0.374766
$$713$$ −6.20204 −0.232268
$$714$$ 4.89898 0.183340
$$715$$ 0 0
$$716$$ −5.79796 −0.216680
$$717$$ 63.1918 2.35994
$$718$$ 11.5959 0.432756
$$719$$ 44.4949 1.65938 0.829690 0.558225i $$-0.188517\pi$$
0.829690 + 0.558225i $$0.188517\pi$$
$$720$$ 0 0
$$721$$ −3.10102 −0.115488
$$722$$ 22.5959 0.840933
$$723$$ −50.6969 −1.88544
$$724$$ 14.2474 0.529502
$$725$$ 0 0
$$726$$ −31.8434 −1.18182
$$727$$ 6.69694 0.248376 0.124188 0.992259i $$-0.460367\pi$$
0.124188 + 0.992259i $$0.460367\pi$$
$$728$$ 0.449490 0.0166592
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 17.7980 0.658281
$$732$$ −20.6969 −0.764981
$$733$$ 43.6413 1.61193 0.805965 0.591964i $$-0.201647\pi$$
0.805965 + 0.591964i $$0.201647\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ 6.89898 0.254300
$$737$$ 39.1918 1.44365
$$738$$ −32.6969 −1.20359
$$739$$ −44.4949 −1.63677 −0.818386 0.574669i $$-0.805131\pi$$
−0.818386 + 0.574669i $$0.805131\pi$$
$$740$$ 0 0
$$741$$ −7.10102 −0.260863
$$742$$ 1.10102 0.0404197
$$743$$ 15.3031 0.561415 0.280707 0.959793i $$-0.409431\pi$$
0.280707 + 0.959793i $$0.409431\pi$$
$$744$$ 2.20204 0.0807307
$$745$$ 0 0
$$746$$ −24.6969 −0.904219
$$747$$ −7.34847 −0.268866
$$748$$ −9.79796 −0.358249
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ −22.2020 −0.810164 −0.405082 0.914280i $$-0.632757\pi$$
−0.405082 + 0.914280i $$0.632757\pi$$
$$752$$ 0.898979 0.0327824
$$753$$ 3.79796 0.138405
$$754$$ −1.30306 −0.0474547
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32.2020 1.17040 0.585202 0.810888i $$-0.301015\pi$$
0.585202 + 0.810888i $$0.301015\pi$$
$$758$$ 1.30306 0.0473293
$$759$$ −82.7878 −3.00501
$$760$$ 0 0
$$761$$ −30.8990 −1.12009 −0.560044 0.828463i $$-0.689216\pi$$
−0.560044 + 0.828463i $$0.689216\pi$$
$$762$$ −12.4949 −0.452642
$$763$$ −2.89898 −0.104950
$$764$$ −16.6969 −0.604074
$$765$$ 0 0
$$766$$ 16.8990 0.610585
$$767$$ −2.89898 −0.104676
$$768$$ −2.44949 −0.0883883
$$769$$ −11.3031 −0.407599 −0.203799 0.979013i $$-0.565329\pi$$
−0.203799 + 0.979013i $$0.565329\pi$$
$$770$$ 0 0
$$771$$ 50.6969 1.82581
$$772$$ −17.5959 −0.633291
$$773$$ 13.3485 0.480111 0.240056 0.970759i $$-0.422834\pi$$
0.240056 + 0.970759i $$0.422834\pi$$
$$774$$ −26.6969 −0.959602
$$775$$ 0 0
$$776$$ 3.79796 0.136339
$$777$$ 4.89898 0.175750
$$778$$ 22.8990 0.820968
$$779$$ −70.2929 −2.51850
$$780$$ 0 0
$$781$$ −53.3939 −1.91058
$$782$$ −13.7980 −0.493414
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 3.79796 0.135469
$$787$$ −45.5505 −1.62370 −0.811850 0.583866i $$-0.801539\pi$$
−0.811850 + 0.583866i $$0.801539\pi$$
$$788$$ −9.10102 −0.324210
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −0.202041 −0.00718375
$$792$$ 14.6969 0.522233
$$793$$ 3.79796 0.134869
$$794$$ 17.3485 0.615674
$$795$$ 0 0
$$796$$ 7.10102 0.251689
$$797$$ −52.9444 −1.87539 −0.937693 0.347464i $$-0.887043\pi$$
−0.937693 + 0.347464i $$0.887043\pi$$
$$798$$ −15.7980 −0.559242
$$799$$ −1.79796 −0.0636072
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 29.3939 1.03793
$$803$$ 33.7980 1.19270
$$804$$ −19.5959 −0.691095
$$805$$ 0 0
$$806$$ −0.404082 −0.0142332
$$807$$ 37.1010 1.30602
$$808$$ 8.44949 0.297252
$$809$$ 8.40408 0.295472 0.147736 0.989027i $$-0.452801\pi$$
0.147736 + 0.989027i $$0.452801\pi$$
$$810$$ 0 0
$$811$$ −38.9444 −1.36752 −0.683761 0.729706i $$-0.739657\pi$$
−0.683761 + 0.729706i $$0.739657\pi$$
$$812$$ −2.89898 −0.101734
$$813$$ −29.3939 −1.03089
$$814$$ −9.79796 −0.343418
$$815$$ 0 0
$$816$$ 4.89898 0.171499
$$817$$ −57.3939 −2.00796
$$818$$ −14.4949 −0.506802
$$819$$ 1.34847 0.0471193
$$820$$ 0 0
$$821$$ 27.7980 0.970155 0.485078 0.874471i $$-0.338791\pi$$
0.485078 + 0.874471i $$0.338791\pi$$
$$822$$ 43.5959 1.52058
$$823$$ −39.1918 −1.36614 −0.683071 0.730352i $$-0.739356\pi$$
−0.683071 + 0.730352i $$0.739356\pi$$
$$824$$ −3.10102 −0.108029
$$825$$ 0 0
$$826$$ −6.44949 −0.224406
$$827$$ −23.5959 −0.820510 −0.410255 0.911971i $$-0.634560\pi$$
−0.410255 + 0.911971i $$0.634560\pi$$
$$828$$ 20.6969 0.719268
$$829$$ 39.6413 1.37680 0.688400 0.725331i $$-0.258313\pi$$
0.688400 + 0.725331i $$0.258313\pi$$
$$830$$ 0 0
$$831$$ −12.4949 −0.433443
$$832$$ 0.449490 0.0155833
$$833$$ −2.00000 −0.0692959
$$834$$ −15.7980 −0.547039
$$835$$ 0 0
$$836$$ 31.5959 1.09277
$$837$$ 0 0
$$838$$ −6.44949 −0.222794
$$839$$ 27.1010 0.935631 0.467816 0.883826i $$-0.345041\pi$$
0.467816 + 0.883826i $$0.345041\pi$$
$$840$$ 0 0
$$841$$ −20.5959 −0.710204
$$842$$ −23.7980 −0.820132
$$843$$ 44.0908 1.51857
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 2.69694 0.0927227
$$847$$ 13.0000 0.446685
$$848$$ 1.10102 0.0378092
$$849$$ 69.1918 2.37466
$$850$$ 0 0
$$851$$ −13.7980 −0.472988
$$852$$ 26.6969 0.914622
$$853$$ −29.8434 −1.02182 −0.510909 0.859635i $$-0.670691\pi$$
−0.510909 + 0.859635i $$0.670691\pi$$
$$854$$ 8.44949 0.289136
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ −25.1918 −0.860537 −0.430268 0.902701i $$-0.641581\pi$$
−0.430268 + 0.902701i $$0.641581\pi$$
$$858$$ −5.39388 −0.184144
$$859$$ −29.6413 −1.01135 −0.505674 0.862724i $$-0.668756\pi$$
−0.505674 + 0.862724i $$0.668756\pi$$
$$860$$ 0 0
$$861$$ 26.6969 0.909829
$$862$$ 17.7980 0.606201
$$863$$ −13.3939 −0.455933 −0.227966 0.973669i $$-0.573208\pi$$
−0.227966 + 0.973669i $$0.573208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.7980 0.672762
$$867$$ 31.8434 1.08146
$$868$$ −0.898979 −0.0305134
$$869$$ −14.2020 −0.481771
$$870$$ 0 0
$$871$$ 3.59592 0.121843
$$872$$ −2.89898 −0.0981718
$$873$$ 11.3939 0.385624
$$874$$ 44.4949 1.50506
$$875$$ 0 0
$$876$$ −16.8990 −0.570964
$$877$$ −19.3939 −0.654885 −0.327442 0.944871i $$-0.606187\pi$$
−0.327442 + 0.944871i $$0.606187\pi$$
$$878$$ 37.3939 1.26198
$$879$$ −15.3031 −0.516159
$$880$$ 0 0
$$881$$ 27.7980 0.936537 0.468269 0.883586i $$-0.344878\pi$$
0.468269 + 0.883586i $$0.344878\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −41.7980 −1.40661 −0.703307 0.710887i $$-0.748294\pi$$
−0.703307 + 0.710887i $$0.748294\pi$$
$$884$$ −0.898979 −0.0302360
$$885$$ 0 0
$$886$$ 9.79796 0.329169
$$887$$ 26.6969 0.896395 0.448198 0.893934i $$-0.352066\pi$$
0.448198 + 0.893934i $$0.352066\pi$$
$$888$$ 4.89898 0.164399
$$889$$ 5.10102 0.171083
$$890$$ 0 0
$$891$$ −44.0908 −1.47710
$$892$$ 4.00000 0.133930
$$893$$ 5.79796 0.194021
$$894$$ −38.6969 −1.29422
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ −7.59592 −0.253620
$$898$$ −10.0000 −0.333704
$$899$$ 2.60612 0.0869191
$$900$$ 0 0
$$901$$ −2.20204 −0.0733606
$$902$$ −53.3939 −1.77782
$$903$$ 21.7980 0.725391
$$904$$ −0.202041 −0.00671978
$$905$$ 0 0
$$906$$ 48.0000 1.59469
$$907$$ 22.2020 0.737207 0.368603 0.929587i $$-0.379836\pi$$
0.368603 + 0.929587i $$0.379836\pi$$
$$908$$ 7.34847 0.243868
$$909$$ 25.3485 0.840756
$$910$$ 0 0
$$911$$ 3.59592 0.119138 0.0595690 0.998224i $$-0.481027\pi$$
0.0595690 + 0.998224i $$0.481027\pi$$
$$912$$ −15.7980 −0.523123
$$913$$ −12.0000 −0.397142
$$914$$ 9.59592 0.317405
$$915$$ 0 0
$$916$$ 15.1464 0.500452
$$917$$ −1.55051 −0.0512024
$$918$$ 0 0
$$919$$ −17.1010 −0.564111 −0.282055 0.959398i $$-0.591016\pi$$
−0.282055 + 0.959398i $$0.591016\pi$$
$$920$$ 0 0
$$921$$ 10.4041 0.342826
$$922$$ 2.65153 0.0873235
$$923$$ −4.89898 −0.161252
$$924$$ −12.0000 −0.394771
$$925$$ 0 0
$$926$$ 35.5959 1.16975
$$927$$ −9.30306 −0.305553
$$928$$ −2.89898 −0.0951637
$$929$$ 40.2929 1.32197 0.660983 0.750401i $$-0.270140\pi$$
0.660983 + 0.750401i $$0.270140\pi$$
$$930$$ 0 0
$$931$$ 6.44949 0.211373
$$932$$ −10.2020 −0.334179
$$933$$ −29.3939 −0.962312
$$934$$ −5.55051 −0.181618
$$935$$ 0 0
$$936$$ 1.34847 0.0440761
$$937$$ 50.8990 1.66280 0.831399 0.555677i $$-0.187541\pi$$
0.831399 + 0.555677i $$0.187541\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 43.1010 1.40655
$$940$$ 0 0
$$941$$ −24.4495 −0.797031 −0.398515 0.917162i $$-0.630474\pi$$
−0.398515 + 0.917162i $$0.630474\pi$$
$$942$$ 20.6969 0.674343
$$943$$ −75.1918 −2.44858
$$944$$ −6.44949 −0.209913
$$945$$ 0 0
$$946$$ −43.5959 −1.41743
$$947$$ 44.0908 1.43276 0.716379 0.697711i $$-0.245798\pi$$
0.716379 + 0.697711i $$0.245798\pi$$
$$948$$ 7.10102 0.230630
$$949$$ 3.10102 0.100663
$$950$$ 0 0
$$951$$ 64.8990 2.10449
$$952$$ −2.00000 −0.0648204
$$953$$ −21.7980 −0.706105 −0.353053 0.935603i $$-0.614856\pi$$
−0.353053 + 0.935603i $$0.614856\pi$$
$$954$$ 3.30306 0.106941
$$955$$ 0 0
$$956$$ −25.7980 −0.834366
$$957$$ 34.7878 1.12453
$$958$$ 38.6969 1.25024
$$959$$ −17.7980 −0.574726
$$960$$ 0 0
$$961$$ −30.1918 −0.973930
$$962$$ −0.898979 −0.0289843
$$963$$ 24.0000 0.773389
$$964$$ 20.6969 0.666604
$$965$$ 0 0
$$966$$ −16.8990 −0.543716
$$967$$ −32.2929 −1.03847 −0.519234 0.854632i $$-0.673783\pi$$
−0.519234 + 0.854632i $$0.673783\pi$$
$$968$$ 13.0000 0.417836
$$969$$ 31.5959 1.01501
$$970$$ 0 0
$$971$$ −14.4495 −0.463706 −0.231853 0.972751i $$-0.574479\pi$$
−0.231853 + 0.972751i $$0.574479\pi$$
$$972$$ 22.0454 0.707107
$$973$$ 6.44949 0.206761
$$974$$ 36.6969 1.17585
$$975$$ 0 0
$$976$$ 8.44949 0.270462
$$977$$ −29.3939 −0.940393 −0.470197 0.882562i $$-0.655817\pi$$
−0.470197 + 0.882562i $$0.655817\pi$$
$$978$$ −41.3939 −1.32363
$$979$$ −48.9898 −1.56572
$$980$$ 0 0
$$981$$ −8.69694 −0.277672
$$982$$ −19.5959 −0.625331
$$983$$ 42.6969 1.36182 0.680910 0.732367i $$-0.261584\pi$$
0.680910 + 0.732367i $$0.261584\pi$$
$$984$$ 26.6969 0.851067
$$985$$ 0 0
$$986$$ 5.79796 0.184645
$$987$$ −2.20204 −0.0700917
$$988$$ 2.89898 0.0922288
$$989$$ −61.3939 −1.95221
$$990$$ 0 0
$$991$$ 60.6969 1.92810 0.964051 0.265718i $$-0.0856089\pi$$
0.964051 + 0.265718i $$0.0856089\pi$$
$$992$$ −0.898979 −0.0285426
$$993$$ −26.2020 −0.831497
$$994$$ −10.8990 −0.345695
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ −42.6515 −1.35079 −0.675394 0.737457i $$-0.736026\pi$$
−0.675394 + 0.737457i $$0.736026\pi$$
$$998$$ 25.7980 0.816620
$$999$$ 0 0
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 350.2.a.h.1.1 2
3.2 odd 2 3150.2.a.bs.1.1 2
4.3 odd 2 2800.2.a.bl.1.2 2
5.2 odd 4 70.2.c.a.29.4 yes 4
5.3 odd 4 70.2.c.a.29.1 4
5.4 even 2 350.2.a.g.1.2 2
7.6 odd 2 2450.2.a.bq.1.2 2
15.2 even 4 630.2.g.g.379.2 4
15.8 even 4 630.2.g.g.379.4 4
15.14 odd 2 3150.2.a.bt.1.1 2
20.3 even 4 560.2.g.e.449.3 4
20.7 even 4 560.2.g.e.449.1 4
20.19 odd 2 2800.2.a.bm.1.1 2
35.2 odd 12 490.2.i.c.459.3 8
35.3 even 12 490.2.i.f.79.4 8
35.12 even 12 490.2.i.f.459.4 8
35.13 even 4 490.2.c.e.99.2 4
35.17 even 12 490.2.i.f.79.1 8
35.18 odd 12 490.2.i.c.79.3 8
35.23 odd 12 490.2.i.c.459.2 8
35.27 even 4 490.2.c.e.99.3 4
35.32 odd 12 490.2.i.c.79.2 8
35.33 even 12 490.2.i.f.459.1 8
35.34 odd 2 2450.2.a.bl.1.1 2
40.3 even 4 2240.2.g.i.449.2 4
40.13 odd 4 2240.2.g.j.449.4 4
40.27 even 4 2240.2.g.i.449.4 4
40.37 odd 4 2240.2.g.j.449.2 4
60.23 odd 4 5040.2.t.t.1009.3 4
60.47 odd 4 5040.2.t.t.1009.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.1 4 5.3 odd 4
70.2.c.a.29.4 yes 4 5.2 odd 4
350.2.a.g.1.2 2 5.4 even 2
350.2.a.h.1.1 2 1.1 even 1 trivial
490.2.c.e.99.2 4 35.13 even 4
490.2.c.e.99.3 4 35.27 even 4
490.2.i.c.79.2 8 35.32 odd 12
490.2.i.c.79.3 8 35.18 odd 12
490.2.i.c.459.2 8 35.23 odd 12
490.2.i.c.459.3 8 35.2 odd 12
490.2.i.f.79.1 8 35.17 even 12
490.2.i.f.79.4 8 35.3 even 12
490.2.i.f.459.1 8 35.33 even 12
490.2.i.f.459.4 8 35.12 even 12
560.2.g.e.449.1 4 20.7 even 4
560.2.g.e.449.3 4 20.3 even 4
630.2.g.g.379.2 4 15.2 even 4
630.2.g.g.379.4 4 15.8 even 4
2240.2.g.i.449.2 4 40.3 even 4
2240.2.g.i.449.4 4 40.27 even 4
2240.2.g.j.449.2 4 40.37 odd 4
2240.2.g.j.449.4 4 40.13 odd 4
2450.2.a.bl.1.1 2 35.34 odd 2
2450.2.a.bq.1.2 2 7.6 odd 2
2800.2.a.bl.1.2 2 4.3 odd 2
2800.2.a.bm.1.1 2 20.19 odd 2
3150.2.a.bs.1.1 2 3.2 odd 2
3150.2.a.bt.1.1 2 15.14 odd 2
5040.2.t.t.1009.3 4 60.23 odd 4
5040.2.t.t.1009.4 4 60.47 odd 4