# Properties

 Label 5610.2.a.bw.1.2 Level $5610$ Weight $2$ Character 5610.1 Self dual yes Analytic conductor $44.796$ Analytic rank $1$ 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] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 5610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +1.12311 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -5.12311 q^{19} -1.00000 q^{20} -1.00000 q^{22} -9.12311 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.12311 q^{26} +1.00000 q^{27} -9.12311 q^{29} -1.00000 q^{30} -7.12311 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +1.12311 q^{37} -5.12311 q^{38} +1.12311 q^{39} -1.00000 q^{40} +3.12311 q^{41} -1.00000 q^{44} -1.00000 q^{45} -9.12311 q^{46} +7.12311 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} +1.12311 q^{52} +4.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -5.12311 q^{57} -9.12311 q^{58} +10.2462 q^{59} -1.00000 q^{60} -8.24621 q^{61} -7.12311 q^{62} +1.00000 q^{64} -1.12311 q^{65} -1.00000 q^{66} -6.87689 q^{67} -1.00000 q^{68} -9.12311 q^{69} -8.00000 q^{71} +1.00000 q^{72} -16.2462 q^{73} +1.12311 q^{74} +1.00000 q^{75} -5.12311 q^{76} +1.12311 q^{78} -5.12311 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.12311 q^{82} +6.24621 q^{83} +1.00000 q^{85} -9.12311 q^{87} -1.00000 q^{88} +12.2462 q^{89} -1.00000 q^{90} -9.12311 q^{92} -7.12311 q^{93} +7.12311 q^{94} +5.12311 q^{95} +1.00000 q^{96} +11.1231 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{24} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 10 q^{29} - 2 q^{30} - 6 q^{31} + 2 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{36} - 6 q^{37} - 2 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 2 q^{44} - 2 q^{45} - 10 q^{46} + 6 q^{47} + 2 q^{48} - 14 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} + 8 q^{53} + 2 q^{54} + 2 q^{55} - 2 q^{57} - 10 q^{58} + 4 q^{59} - 2 q^{60} - 6 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{66} - 22 q^{67} - 2 q^{68} - 10 q^{69} - 16 q^{71} + 2 q^{72} - 16 q^{73} - 6 q^{74} + 2 q^{75} - 2 q^{76} - 6 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} - 2 q^{82} - 4 q^{83} + 2 q^{85} - 10 q^{87} - 2 q^{88} + 8 q^{89} - 2 q^{90} - 10 q^{92} - 6 q^{93} + 6 q^{94} + 2 q^{95} + 2 q^{96} + 14 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 - 6 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 10 * q^23 + 2 * q^24 + 2 * q^25 - 6 * q^26 + 2 * q^27 - 10 * q^29 - 2 * q^30 - 6 * q^31 + 2 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^36 - 6 * q^37 - 2 * q^38 - 6 * q^39 - 2 * q^40 - 2 * q^41 - 2 * q^44 - 2 * q^45 - 10 * q^46 + 6 * q^47 + 2 * q^48 - 14 * q^49 + 2 * q^50 - 2 * q^51 - 6 * q^52 + 8 * q^53 + 2 * q^54 + 2 * q^55 - 2 * q^57 - 10 * q^58 + 4 * q^59 - 2 * q^60 - 6 * q^62 + 2 * q^64 + 6 * q^65 - 2 * q^66 - 22 * q^67 - 2 * q^68 - 10 * q^69 - 16 * q^71 + 2 * q^72 - 16 * q^73 - 6 * q^74 + 2 * q^75 - 2 * q^76 - 6 * q^78 - 2 * q^79 - 2 * q^80 + 2 * q^81 - 2 * q^82 - 4 * q^83 + 2 * q^85 - 10 * q^87 - 2 * q^88 + 8 * q^89 - 2 * q^90 - 10 * q^92 - 6 * q^93 + 6 * q^94 + 2 * q^95 + 2 * q^96 + 14 * q^97 - 14 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ 1.12311 0.311493 0.155747 0.987797i $$-0.450222\pi$$
0.155747 + 0.987797i $$0.450222\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ 1.00000 0.235702
$$19$$ −5.12311 −1.17532 −0.587661 0.809108i $$-0.699951\pi$$
−0.587661 + 0.809108i $$0.699951\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −9.12311 −1.90230 −0.951150 0.308731i $$-0.900096\pi$$
−0.951150 + 0.308731i $$0.900096\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 1.12311 0.220259
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −9.12311 −1.69412 −0.847059 0.531499i $$-0.821629\pi$$
−0.847059 + 0.531499i $$0.821629\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −7.12311 −1.27935 −0.639674 0.768647i $$-0.720931\pi$$
−0.639674 + 0.768647i $$0.720931\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.12311 0.184637 0.0923187 0.995730i $$-0.470572\pi$$
0.0923187 + 0.995730i $$0.470572\pi$$
$$38$$ −5.12311 −0.831077
$$39$$ 1.12311 0.179841
$$40$$ −1.00000 −0.158114
$$41$$ 3.12311 0.487747 0.243874 0.969807i $$-0.421582\pi$$
0.243874 + 0.969807i $$0.421582\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −9.12311 −1.34513
$$47$$ 7.12311 1.03901 0.519506 0.854467i $$-0.326116\pi$$
0.519506 + 0.854467i $$0.326116\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ −1.00000 −0.140028
$$52$$ 1.12311 0.155747
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ −5.12311 −0.678572
$$58$$ −9.12311 −1.19792
$$59$$ 10.2462 1.33394 0.666972 0.745083i $$-0.267590\pi$$
0.666972 + 0.745083i $$0.267590\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −8.24621 −1.05582 −0.527910 0.849301i $$-0.677024\pi$$
−0.527910 + 0.849301i $$0.677024\pi$$
$$62$$ −7.12311 −0.904635
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.12311 −0.139304
$$66$$ −1.00000 −0.123091
$$67$$ −6.87689 −0.840146 −0.420073 0.907490i $$-0.637996\pi$$
−0.420073 + 0.907490i $$0.637996\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −9.12311 −1.09829
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −16.2462 −1.90148 −0.950738 0.309997i $$-0.899672\pi$$
−0.950738 + 0.309997i $$0.899672\pi$$
$$74$$ 1.12311 0.130558
$$75$$ 1.00000 0.115470
$$76$$ −5.12311 −0.587661
$$77$$ 0 0
$$78$$ 1.12311 0.127167
$$79$$ −5.12311 −0.576394 −0.288197 0.957571i $$-0.593056\pi$$
−0.288197 + 0.957571i $$0.593056\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 3.12311 0.344889
$$83$$ 6.24621 0.685611 0.342805 0.939406i $$-0.388623\pi$$
0.342805 + 0.939406i $$0.388623\pi$$
$$84$$ 0 0
$$85$$ 1.00000 0.108465
$$86$$ 0 0
$$87$$ −9.12311 −0.978100
$$88$$ −1.00000 −0.106600
$$89$$ 12.2462 1.29810 0.649048 0.760748i $$-0.275167\pi$$
0.649048 + 0.760748i $$0.275167\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −9.12311 −0.951150
$$93$$ −7.12311 −0.738632
$$94$$ 7.12311 0.734692
$$95$$ 5.12311 0.525620
$$96$$ 1.00000 0.102062
$$97$$ 11.1231 1.12938 0.564690 0.825303i $$-0.308996\pi$$
0.564690 + 0.825303i $$0.308996\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ −9.36932 −0.932282 −0.466141 0.884710i $$-0.654356\pi$$
−0.466141 + 0.884710i $$0.654356\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ 2.24621 0.221326 0.110663 0.993858i $$-0.464703\pi$$
0.110663 + 0.993858i $$0.464703\pi$$
$$104$$ 1.12311 0.110130
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 1.00000 0.0953463
$$111$$ 1.12311 0.106600
$$112$$ 0 0
$$113$$ 9.36932 0.881391 0.440696 0.897657i $$-0.354732\pi$$
0.440696 + 0.897657i $$0.354732\pi$$
$$114$$ −5.12311 −0.479823
$$115$$ 9.12311 0.850734
$$116$$ −9.12311 −0.847059
$$117$$ 1.12311 0.103831
$$118$$ 10.2462 0.943240
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ −8.24621 −0.746577
$$123$$ 3.12311 0.281601
$$124$$ −7.12311 −0.639674
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 9.36932 0.831392 0.415696 0.909504i $$-0.363538\pi$$
0.415696 + 0.909504i $$0.363538\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −1.12311 −0.0985029
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ −6.87689 −0.594073
$$135$$ −1.00000 −0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ −4.24621 −0.362778 −0.181389 0.983411i $$-0.558059\pi$$
−0.181389 + 0.983411i $$0.558059\pi$$
$$138$$ −9.12311 −0.776610
$$139$$ 14.2462 1.20835 0.604174 0.796852i $$-0.293503\pi$$
0.604174 + 0.796852i $$0.293503\pi$$
$$140$$ 0 0
$$141$$ 7.12311 0.599874
$$142$$ −8.00000 −0.671345
$$143$$ −1.12311 −0.0939188
$$144$$ 1.00000 0.0833333
$$145$$ 9.12311 0.757633
$$146$$ −16.2462 −1.34455
$$147$$ −7.00000 −0.577350
$$148$$ 1.12311 0.0923187
$$149$$ 5.36932 0.439872 0.219936 0.975514i $$-0.429415\pi$$
0.219936 + 0.975514i $$0.429415\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 9.36932 0.762464 0.381232 0.924479i $$-0.375500\pi$$
0.381232 + 0.924479i $$0.375500\pi$$
$$152$$ −5.12311 −0.415539
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 7.12311 0.572142
$$156$$ 1.12311 0.0899204
$$157$$ −23.6155 −1.88472 −0.942362 0.334595i $$-0.891401\pi$$
−0.942362 + 0.334595i $$0.891401\pi$$
$$158$$ −5.12311 −0.407572
$$159$$ 4.00000 0.317221
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 8.49242 0.665178 0.332589 0.943072i $$-0.392078\pi$$
0.332589 + 0.943072i $$0.392078\pi$$
$$164$$ 3.12311 0.243874
$$165$$ 1.00000 0.0778499
$$166$$ 6.24621 0.484800
$$167$$ −23.1231 −1.78932 −0.894660 0.446748i $$-0.852582\pi$$
−0.894660 + 0.446748i $$0.852582\pi$$
$$168$$ 0 0
$$169$$ −11.7386 −0.902972
$$170$$ 1.00000 0.0766965
$$171$$ −5.12311 −0.391774
$$172$$ 0 0
$$173$$ −14.8769 −1.13107 −0.565535 0.824725i $$-0.691330\pi$$
−0.565535 + 0.824725i $$0.691330\pi$$
$$174$$ −9.12311 −0.691621
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 10.2462 0.770152
$$178$$ 12.2462 0.917892
$$179$$ −16.4924 −1.23270 −0.616351 0.787472i $$-0.711390\pi$$
−0.616351 + 0.787472i $$0.711390\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −17.6155 −1.30935 −0.654676 0.755910i $$-0.727195\pi$$
−0.654676 + 0.755910i $$0.727195\pi$$
$$182$$ 0 0
$$183$$ −8.24621 −0.609577
$$184$$ −9.12311 −0.672564
$$185$$ −1.12311 −0.0825724
$$186$$ −7.12311 −0.522291
$$187$$ 1.00000 0.0731272
$$188$$ 7.12311 0.519506
$$189$$ 0 0
$$190$$ 5.12311 0.371669
$$191$$ 12.8769 0.931739 0.465870 0.884853i $$-0.345742\pi$$
0.465870 + 0.884853i $$0.345742\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −22.4924 −1.61904 −0.809520 0.587092i $$-0.800273\pi$$
−0.809520 + 0.587092i $$0.800273\pi$$
$$194$$ 11.1231 0.798592
$$195$$ −1.12311 −0.0804273
$$196$$ −7.00000 −0.500000
$$197$$ 17.6155 1.25505 0.627527 0.778595i $$-0.284067\pi$$
0.627527 + 0.778595i $$0.284067\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −15.1231 −1.07205 −0.536024 0.844203i $$-0.680074\pi$$
−0.536024 + 0.844203i $$0.680074\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −6.87689 −0.485059
$$202$$ −9.36932 −0.659223
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ −3.12311 −0.218127
$$206$$ 2.24621 0.156501
$$207$$ −9.12311 −0.634100
$$208$$ 1.12311 0.0778734
$$209$$ 5.12311 0.354373
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 4.00000 0.274721
$$213$$ −8.00000 −0.548151
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ −16.2462 −1.09782
$$220$$ 1.00000 0.0674200
$$221$$ −1.12311 −0.0755483
$$222$$ 1.12311 0.0753779
$$223$$ 9.75379 0.653162 0.326581 0.945169i $$-0.394103\pi$$
0.326581 + 0.945169i $$0.394103\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 9.36932 0.623238
$$227$$ −18.2462 −1.21104 −0.605522 0.795829i $$-0.707036\pi$$
−0.605522 + 0.795829i $$0.707036\pi$$
$$228$$ −5.12311 −0.339286
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 9.12311 0.601560
$$231$$ 0 0
$$232$$ −9.12311 −0.598961
$$233$$ −10.4924 −0.687381 −0.343691 0.939083i $$-0.611677\pi$$
−0.343691 + 0.939083i $$0.611677\pi$$
$$234$$ 1.12311 0.0734197
$$235$$ −7.12311 −0.464660
$$236$$ 10.2462 0.666972
$$237$$ −5.12311 −0.332781
$$238$$ 0 0
$$239$$ 26.2462 1.69773 0.848863 0.528613i $$-0.177288\pi$$
0.848863 + 0.528613i $$0.177288\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 15.6155 1.00588 0.502942 0.864320i $$-0.332251\pi$$
0.502942 + 0.864320i $$0.332251\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ −8.24621 −0.527910
$$245$$ 7.00000 0.447214
$$246$$ 3.12311 0.199122
$$247$$ −5.75379 −0.366105
$$248$$ −7.12311 −0.452318
$$249$$ 6.24621 0.395838
$$250$$ −1.00000 −0.0632456
$$251$$ −26.7386 −1.68773 −0.843864 0.536557i $$-0.819724\pi$$
−0.843864 + 0.536557i $$0.819724\pi$$
$$252$$ 0 0
$$253$$ 9.12311 0.573565
$$254$$ 9.36932 0.587883
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −10.4924 −0.654499 −0.327250 0.944938i $$-0.606122\pi$$
−0.327250 + 0.944938i $$0.606122\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1.12311 −0.0696521
$$261$$ −9.12311 −0.564706
$$262$$ 0 0
$$263$$ 30.2462 1.86506 0.932531 0.361091i $$-0.117596\pi$$
0.932531 + 0.361091i $$0.117596\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −4.00000 −0.245718
$$266$$ 0 0
$$267$$ 12.2462 0.749456
$$268$$ −6.87689 −0.420073
$$269$$ 7.75379 0.472757 0.236378 0.971661i $$-0.424040\pi$$
0.236378 + 0.971661i $$0.424040\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −2.63068 −0.159803 −0.0799013 0.996803i $$-0.525461\pi$$
−0.0799013 + 0.996803i $$0.525461\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ −4.24621 −0.256523
$$275$$ −1.00000 −0.0603023
$$276$$ −9.12311 −0.549146
$$277$$ −12.2462 −0.735804 −0.367902 0.929865i $$-0.619924\pi$$
−0.367902 + 0.929865i $$0.619924\pi$$
$$278$$ 14.2462 0.854431
$$279$$ −7.12311 −0.426449
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 7.12311 0.424175
$$283$$ 8.49242 0.504822 0.252411 0.967620i $$-0.418776\pi$$
0.252411 + 0.967620i $$0.418776\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 5.12311 0.303467
$$286$$ −1.12311 −0.0664106
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 9.12311 0.535727
$$291$$ 11.1231 0.652048
$$292$$ −16.2462 −0.950738
$$293$$ −14.4924 −0.846656 −0.423328 0.905976i $$-0.639138\pi$$
−0.423328 + 0.905976i $$0.639138\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ −10.2462 −0.596557
$$296$$ 1.12311 0.0652792
$$297$$ −1.00000 −0.0580259
$$298$$ 5.36932 0.311036
$$299$$ −10.2462 −0.592554
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 9.36932 0.539144
$$303$$ −9.36932 −0.538253
$$304$$ −5.12311 −0.293830
$$305$$ 8.24621 0.472177
$$306$$ −1.00000 −0.0571662
$$307$$ −16.4924 −0.941272 −0.470636 0.882327i $$-0.655976\pi$$
−0.470636 + 0.882327i $$0.655976\pi$$
$$308$$ 0 0
$$309$$ 2.24621 0.127782
$$310$$ 7.12311 0.404565
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 1.12311 0.0635833
$$313$$ 3.12311 0.176528 0.0882642 0.996097i $$-0.471868\pi$$
0.0882642 + 0.996097i $$0.471868\pi$$
$$314$$ −23.6155 −1.33270
$$315$$ 0 0
$$316$$ −5.12311 −0.288197
$$317$$ 28.2462 1.58647 0.793233 0.608919i $$-0.208396\pi$$
0.793233 + 0.608919i $$0.208396\pi$$
$$318$$ 4.00000 0.224309
$$319$$ 9.12311 0.510796
$$320$$ −1.00000 −0.0559017
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 5.12311 0.285057
$$324$$ 1.00000 0.0555556
$$325$$ 1.12311 0.0622987
$$326$$ 8.49242 0.470352
$$327$$ −6.00000 −0.331801
$$328$$ 3.12311 0.172445
$$329$$ 0 0
$$330$$ 1.00000 0.0550482
$$331$$ −30.2462 −1.66248 −0.831241 0.555912i $$-0.812369\pi$$
−0.831241 + 0.555912i $$0.812369\pi$$
$$332$$ 6.24621 0.342805
$$333$$ 1.12311 0.0615458
$$334$$ −23.1231 −1.26524
$$335$$ 6.87689 0.375725
$$336$$ 0 0
$$337$$ 10.4924 0.571559 0.285779 0.958295i $$-0.407748\pi$$
0.285779 + 0.958295i $$0.407748\pi$$
$$338$$ −11.7386 −0.638498
$$339$$ 9.36932 0.508871
$$340$$ 1.00000 0.0542326
$$341$$ 7.12311 0.385738
$$342$$ −5.12311 −0.277026
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 9.12311 0.491171
$$346$$ −14.8769 −0.799787
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ −9.12311 −0.489050
$$349$$ 4.00000 0.214115 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$350$$ 0 0
$$351$$ 1.12311 0.0599469
$$352$$ −1.00000 −0.0533002
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 10.2462 0.544580
$$355$$ 8.00000 0.424596
$$356$$ 12.2462 0.649048
$$357$$ 0 0
$$358$$ −16.4924 −0.871652
$$359$$ −22.2462 −1.17411 −0.587055 0.809547i $$-0.699713\pi$$
−0.587055 + 0.809547i $$0.699713\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 7.24621 0.381380
$$362$$ −17.6155 −0.925852
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 16.2462 0.850366
$$366$$ −8.24621 −0.431036
$$367$$ −14.4924 −0.756498 −0.378249 0.925704i $$-0.623474\pi$$
−0.378249 + 0.925704i $$0.623474\pi$$
$$368$$ −9.12311 −0.475575
$$369$$ 3.12311 0.162582
$$370$$ −1.12311 −0.0583875
$$371$$ 0 0
$$372$$ −7.12311 −0.369316
$$373$$ −37.6155 −1.94766 −0.973829 0.227281i $$-0.927016\pi$$
−0.973829 + 0.227281i $$0.927016\pi$$
$$374$$ 1.00000 0.0517088
$$375$$ −1.00000 −0.0516398
$$376$$ 7.12311 0.367346
$$377$$ −10.2462 −0.527707
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 5.12311 0.262810
$$381$$ 9.36932 0.480005
$$382$$ 12.8769 0.658839
$$383$$ 5.36932 0.274359 0.137180 0.990546i $$-0.456196\pi$$
0.137180 + 0.990546i $$0.456196\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −22.4924 −1.14483
$$387$$ 0 0
$$388$$ 11.1231 0.564690
$$389$$ 33.6155 1.70437 0.852187 0.523237i $$-0.175276\pi$$
0.852187 + 0.523237i $$0.175276\pi$$
$$390$$ −1.12311 −0.0568707
$$391$$ 9.12311 0.461375
$$392$$ −7.00000 −0.353553
$$393$$ 0 0
$$394$$ 17.6155 0.887457
$$395$$ 5.12311 0.257771
$$396$$ −1.00000 −0.0502519
$$397$$ 35.8617 1.79985 0.899925 0.436046i $$-0.143621\pi$$
0.899925 + 0.436046i $$0.143621\pi$$
$$398$$ −15.1231 −0.758053
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ −6.87689 −0.342988
$$403$$ −8.00000 −0.398508
$$404$$ −9.36932 −0.466141
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −1.12311 −0.0556703
$$408$$ −1.00000 −0.0495074
$$409$$ −22.4924 −1.11218 −0.556089 0.831123i $$-0.687699\pi$$
−0.556089 + 0.831123i $$0.687699\pi$$
$$410$$ −3.12311 −0.154239
$$411$$ −4.24621 −0.209450
$$412$$ 2.24621 0.110663
$$413$$ 0 0
$$414$$ −9.12311 −0.448376
$$415$$ −6.24621 −0.306614
$$416$$ 1.12311 0.0550648
$$417$$ 14.2462 0.697640
$$418$$ 5.12311 0.250579
$$419$$ −16.4924 −0.805708 −0.402854 0.915264i $$-0.631982\pi$$
−0.402854 + 0.915264i $$0.631982\pi$$
$$420$$ 0 0
$$421$$ 8.73863 0.425895 0.212947 0.977064i $$-0.431694\pi$$
0.212947 + 0.977064i $$0.431694\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 7.12311 0.346337
$$424$$ 4.00000 0.194257
$$425$$ −1.00000 −0.0485071
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −1.12311 −0.0542241
$$430$$ 0 0
$$431$$ 34.9848 1.68516 0.842580 0.538571i $$-0.181036\pi$$
0.842580 + 0.538571i $$0.181036\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 2.49242 0.119778 0.0598891 0.998205i $$-0.480925\pi$$
0.0598891 + 0.998205i $$0.480925\pi$$
$$434$$ 0 0
$$435$$ 9.12311 0.437419
$$436$$ −6.00000 −0.287348
$$437$$ 46.7386 2.23581
$$438$$ −16.2462 −0.776274
$$439$$ 37.6155 1.79529 0.897646 0.440718i $$-0.145276\pi$$
0.897646 + 0.440718i $$0.145276\pi$$
$$440$$ 1.00000 0.0476731
$$441$$ −7.00000 −0.333333
$$442$$ −1.12311 −0.0534207
$$443$$ −17.1231 −0.813543 −0.406772 0.913530i $$-0.633346\pi$$
−0.406772 + 0.913530i $$0.633346\pi$$
$$444$$ 1.12311 0.0533002
$$445$$ −12.2462 −0.580526
$$446$$ 9.75379 0.461855
$$447$$ 5.36932 0.253960
$$448$$ 0 0
$$449$$ −32.7386 −1.54503 −0.772516 0.634996i $$-0.781002\pi$$
−0.772516 + 0.634996i $$0.781002\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −3.12311 −0.147061
$$452$$ 9.36932 0.440696
$$453$$ 9.36932 0.440209
$$454$$ −18.2462 −0.856337
$$455$$ 0 0
$$456$$ −5.12311 −0.239911
$$457$$ 18.4924 0.865039 0.432520 0.901625i $$-0.357625\pi$$
0.432520 + 0.901625i $$0.357625\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −1.00000 −0.0466760
$$460$$ 9.12311 0.425367
$$461$$ −3.12311 −0.145458 −0.0727288 0.997352i $$-0.523171\pi$$
−0.0727288 + 0.997352i $$0.523171\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −9.12311 −0.423530
$$465$$ 7.12311 0.330326
$$466$$ −10.4924 −0.486052
$$467$$ 39.3693 1.82179 0.910897 0.412633i $$-0.135391\pi$$
0.910897 + 0.412633i $$0.135391\pi$$
$$468$$ 1.12311 0.0519156
$$469$$ 0 0
$$470$$ −7.12311 −0.328564
$$471$$ −23.6155 −1.08815
$$472$$ 10.2462 0.471620
$$473$$ 0 0
$$474$$ −5.12311 −0.235312
$$475$$ −5.12311 −0.235064
$$476$$ 0 0
$$477$$ 4.00000 0.183147
$$478$$ 26.2462 1.20047
$$479$$ −12.2462 −0.559544 −0.279772 0.960067i $$-0.590259\pi$$
−0.279772 + 0.960067i $$0.590259\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 1.26137 0.0575134
$$482$$ 15.6155 0.711268
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −11.1231 −0.505074
$$486$$ 1.00000 0.0453609
$$487$$ −42.4924 −1.92552 −0.962758 0.270366i $$-0.912855\pi$$
−0.962758 + 0.270366i $$0.912855\pi$$
$$488$$ −8.24621 −0.373288
$$489$$ 8.49242 0.384041
$$490$$ 7.00000 0.316228
$$491$$ −29.6155 −1.33653 −0.668265 0.743923i $$-0.732963\pi$$
−0.668265 + 0.743923i $$0.732963\pi$$
$$492$$ 3.12311 0.140800
$$493$$ 9.12311 0.410884
$$494$$ −5.75379 −0.258875
$$495$$ 1.00000 0.0449467
$$496$$ −7.12311 −0.319837
$$497$$ 0 0
$$498$$ 6.24621 0.279899
$$499$$ 34.2462 1.53307 0.766535 0.642202i $$-0.221979\pi$$
0.766535 + 0.642202i $$0.221979\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −23.1231 −1.03306
$$502$$ −26.7386 −1.19340
$$503$$ −39.1231 −1.74441 −0.872207 0.489138i $$-0.837312\pi$$
−0.872207 + 0.489138i $$0.837312\pi$$
$$504$$ 0 0
$$505$$ 9.36932 0.416929
$$506$$ 9.12311 0.405572
$$507$$ −11.7386 −0.521331
$$508$$ 9.36932 0.415696
$$509$$ 38.1080 1.68911 0.844553 0.535473i $$-0.179866\pi$$
0.844553 + 0.535473i $$0.179866\pi$$
$$510$$ 1.00000 0.0442807
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −5.12311 −0.226191
$$514$$ −10.4924 −0.462801
$$515$$ −2.24621 −0.0989799
$$516$$ 0 0
$$517$$ −7.12311 −0.313274
$$518$$ 0 0
$$519$$ −14.8769 −0.653023
$$520$$ −1.12311 −0.0492514
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ −9.12311 −0.399307
$$523$$ −18.7386 −0.819383 −0.409692 0.912224i $$-0.634364\pi$$
−0.409692 + 0.912224i $$0.634364\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 30.2462 1.31880
$$527$$ 7.12311 0.310287
$$528$$ −1.00000 −0.0435194
$$529$$ 60.2311 2.61874
$$530$$ −4.00000 −0.173749
$$531$$ 10.2462 0.444648
$$532$$ 0 0
$$533$$ 3.50758 0.151930
$$534$$ 12.2462 0.529945
$$535$$ −12.0000 −0.518805
$$536$$ −6.87689 −0.297037
$$537$$ −16.4924 −0.711701
$$538$$ 7.75379 0.334290
$$539$$ 7.00000 0.301511
$$540$$ −1.00000 −0.0430331
$$541$$ 18.4924 0.795051 0.397526 0.917591i $$-0.369869\pi$$
0.397526 + 0.917591i $$0.369869\pi$$
$$542$$ −2.63068 −0.112998
$$543$$ −17.6155 −0.755955
$$544$$ −1.00000 −0.0428746
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −40.4924 −1.73133 −0.865665 0.500623i $$-0.833104\pi$$
−0.865665 + 0.500623i $$0.833104\pi$$
$$548$$ −4.24621 −0.181389
$$549$$ −8.24621 −0.351940
$$550$$ −1.00000 −0.0426401
$$551$$ 46.7386 1.99113
$$552$$ −9.12311 −0.388305
$$553$$ 0 0
$$554$$ −12.2462 −0.520292
$$555$$ −1.12311 −0.0476732
$$556$$ 14.2462 0.604174
$$557$$ −34.4924 −1.46149 −0.730745 0.682650i $$-0.760827\pi$$
−0.730745 + 0.682650i $$0.760827\pi$$
$$558$$ −7.12311 −0.301545
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ −18.0000 −0.759284
$$563$$ 26.7386 1.12690 0.563450 0.826150i $$-0.309474\pi$$
0.563450 + 0.826150i $$0.309474\pi$$
$$564$$ 7.12311 0.299937
$$565$$ −9.36932 −0.394170
$$566$$ 8.49242 0.356963
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 5.12311 0.214583
$$571$$ 6.24621 0.261396 0.130698 0.991422i $$-0.458278\pi$$
0.130698 + 0.991422i $$0.458278\pi$$
$$572$$ −1.12311 −0.0469594
$$573$$ 12.8769 0.537940
$$574$$ 0 0
$$575$$ −9.12311 −0.380460
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −22.4924 −0.934753
$$580$$ 9.12311 0.378816
$$581$$ 0 0
$$582$$ 11.1231 0.461068
$$583$$ −4.00000 −0.165663
$$584$$ −16.2462 −0.672273
$$585$$ −1.12311 −0.0464347
$$586$$ −14.4924 −0.598676
$$587$$ 35.3693 1.45985 0.729924 0.683528i $$-0.239555\pi$$
0.729924 + 0.683528i $$0.239555\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 36.4924 1.50364
$$590$$ −10.2462 −0.421830
$$591$$ 17.6155 0.724606
$$592$$ 1.12311 0.0461594
$$593$$ 7.75379 0.318410 0.159205 0.987246i $$-0.449107\pi$$
0.159205 + 0.987246i $$0.449107\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 5.36932 0.219936
$$597$$ −15.1231 −0.618948
$$598$$ −10.2462 −0.418999
$$599$$ 33.3693 1.36343 0.681717 0.731616i $$-0.261234\pi$$
0.681717 + 0.731616i $$0.261234\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −19.1231 −0.780048 −0.390024 0.920805i $$-0.627533\pi$$
−0.390024 + 0.920805i $$0.627533\pi$$
$$602$$ 0 0
$$603$$ −6.87689 −0.280049
$$604$$ 9.36932 0.381232
$$605$$ −1.00000 −0.0406558
$$606$$ −9.36932 −0.380602
$$607$$ 36.4924 1.48118 0.740591 0.671956i $$-0.234546\pi$$
0.740591 + 0.671956i $$0.234546\pi$$
$$608$$ −5.12311 −0.207769
$$609$$ 0 0
$$610$$ 8.24621 0.333879
$$611$$ 8.00000 0.323645
$$612$$ −1.00000 −0.0404226
$$613$$ −21.6155 −0.873043 −0.436521 0.899694i $$-0.643790\pi$$
−0.436521 + 0.899694i $$0.643790\pi$$
$$614$$ −16.4924 −0.665580
$$615$$ −3.12311 −0.125936
$$616$$ 0 0
$$617$$ 6.63068 0.266941 0.133471 0.991053i $$-0.457388\pi$$
0.133471 + 0.991053i $$0.457388\pi$$
$$618$$ 2.24621 0.0903559
$$619$$ 26.2462 1.05492 0.527462 0.849579i $$-0.323144\pi$$
0.527462 + 0.849579i $$0.323144\pi$$
$$620$$ 7.12311 0.286071
$$621$$ −9.12311 −0.366098
$$622$$ −4.00000 −0.160385
$$623$$ 0 0
$$624$$ 1.12311 0.0449602
$$625$$ 1.00000 0.0400000
$$626$$ 3.12311 0.124824
$$627$$ 5.12311 0.204597
$$628$$ −23.6155 −0.942362
$$629$$ −1.12311 −0.0447812
$$630$$ 0 0
$$631$$ −17.7538 −0.706767 −0.353384 0.935479i $$-0.614969\pi$$
−0.353384 + 0.935479i $$0.614969\pi$$
$$632$$ −5.12311 −0.203786
$$633$$ 20.0000 0.794929
$$634$$ 28.2462 1.12180
$$635$$ −9.36932 −0.371810
$$636$$ 4.00000 0.158610
$$637$$ −7.86174 −0.311493
$$638$$ 9.12311 0.361187
$$639$$ −8.00000 −0.316475
$$640$$ −1.00000 −0.0395285
$$641$$ −12.7386 −0.503146 −0.251573 0.967838i $$-0.580948\pi$$
−0.251573 + 0.967838i $$0.580948\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5.12311 0.201566
$$647$$ 6.63068 0.260679 0.130340 0.991469i $$-0.458393\pi$$
0.130340 + 0.991469i $$0.458393\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −10.2462 −0.402199
$$650$$ 1.12311 0.0440518
$$651$$ 0 0
$$652$$ 8.49242 0.332589
$$653$$ −28.2462 −1.10536 −0.552680 0.833394i $$-0.686395\pi$$
−0.552680 + 0.833394i $$0.686395\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ 3.12311 0.121937
$$657$$ −16.2462 −0.633825
$$658$$ 0 0
$$659$$ −42.1080 −1.64029 −0.820146 0.572154i $$-0.806108\pi$$
−0.820146 + 0.572154i $$0.806108\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −34.9848 −1.36075 −0.680376 0.732863i $$-0.738184\pi$$
−0.680376 + 0.732863i $$0.738184\pi$$
$$662$$ −30.2462 −1.17555
$$663$$ −1.12311 −0.0436178
$$664$$ 6.24621 0.242400
$$665$$ 0 0
$$666$$ 1.12311 0.0435195
$$667$$ 83.2311 3.22272
$$668$$ −23.1231 −0.894660
$$669$$ 9.75379 0.377103
$$670$$ 6.87689 0.265678
$$671$$ 8.24621 0.318341
$$672$$ 0 0
$$673$$ −42.9848 −1.65694 −0.828472 0.560030i $$-0.810789\pi$$
−0.828472 + 0.560030i $$0.810789\pi$$
$$674$$ 10.4924 0.404153
$$675$$ 1.00000 0.0384900
$$676$$ −11.7386 −0.451486
$$677$$ 38.1080 1.46461 0.732304 0.680978i $$-0.238445\pi$$
0.732304 + 0.680978i $$0.238445\pi$$
$$678$$ 9.36932 0.359826
$$679$$ 0 0
$$680$$ 1.00000 0.0383482
$$681$$ −18.2462 −0.699196
$$682$$ 7.12311 0.272758
$$683$$ −32.4924 −1.24329 −0.621644 0.783300i $$-0.713535\pi$$
−0.621644 + 0.783300i $$0.713535\pi$$
$$684$$ −5.12311 −0.195887
$$685$$ 4.24621 0.162239
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ 0 0
$$689$$ 4.49242 0.171148
$$690$$ 9.12311 0.347311
$$691$$ −16.4924 −0.627401 −0.313701 0.949522i $$-0.601569\pi$$
−0.313701 + 0.949522i $$0.601569\pi$$
$$692$$ −14.8769 −0.565535
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ −14.2462 −0.540390
$$696$$ −9.12311 −0.345810
$$697$$ −3.12311 −0.118296
$$698$$ 4.00000 0.151402
$$699$$ −10.4924 −0.396860
$$700$$ 0 0
$$701$$ 1.36932 0.0517184 0.0258592 0.999666i $$-0.491768\pi$$
0.0258592 + 0.999666i $$0.491768\pi$$
$$702$$ 1.12311 0.0423889
$$703$$ −5.75379 −0.217008
$$704$$ −1.00000 −0.0376889
$$705$$ −7.12311 −0.268272
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ 10.2462 0.385076
$$709$$ −19.3693 −0.727430 −0.363715 0.931510i $$-0.618492\pi$$
−0.363715 + 0.931510i $$0.618492\pi$$
$$710$$ 8.00000 0.300235
$$711$$ −5.12311 −0.192131
$$712$$ 12.2462 0.458946
$$713$$ 64.9848 2.43370
$$714$$ 0 0
$$715$$ 1.12311 0.0420018
$$716$$ −16.4924 −0.616351
$$717$$ 26.2462 0.980183
$$718$$ −22.2462 −0.830221
$$719$$ 30.7386 1.14636 0.573179 0.819430i $$-0.305710\pi$$
0.573179 + 0.819430i $$0.305710\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 7.24621 0.269676
$$723$$ 15.6155 0.580748
$$724$$ −17.6155 −0.654676
$$725$$ −9.12311 −0.338824
$$726$$ 1.00000 0.0371135
$$727$$ −40.4924 −1.50178 −0.750890 0.660427i $$-0.770375\pi$$
−0.750890 + 0.660427i $$0.770375\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 16.2462 0.601299
$$731$$ 0 0
$$732$$ −8.24621 −0.304789
$$733$$ −33.1231 −1.22343 −0.611715 0.791078i $$-0.709520\pi$$
−0.611715 + 0.791078i $$0.709520\pi$$
$$734$$ −14.4924 −0.534925
$$735$$ 7.00000 0.258199
$$736$$ −9.12311 −0.336282
$$737$$ 6.87689 0.253314
$$738$$ 3.12311 0.114963
$$739$$ 38.8769 1.43011 0.715055 0.699068i $$-0.246402\pi$$
0.715055 + 0.699068i $$0.246402\pi$$
$$740$$ −1.12311 −0.0412862
$$741$$ −5.75379 −0.211371
$$742$$ 0 0
$$743$$ −0.384472 −0.0141049 −0.00705245 0.999975i $$-0.502245\pi$$
−0.00705245 + 0.999975i $$0.502245\pi$$
$$744$$ −7.12311 −0.261146
$$745$$ −5.36932 −0.196717
$$746$$ −37.6155 −1.37720
$$747$$ 6.24621 0.228537
$$748$$ 1.00000 0.0365636
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ 7.12311 0.259926 0.129963 0.991519i $$-0.458514\pi$$
0.129963 + 0.991519i $$0.458514\pi$$
$$752$$ 7.12311 0.259753
$$753$$ −26.7386 −0.974410
$$754$$ −10.2462 −0.373145
$$755$$ −9.36932 −0.340984
$$756$$ 0 0
$$757$$ −7.12311 −0.258894 −0.129447 0.991586i $$-0.541320\pi$$
−0.129447 + 0.991586i $$0.541320\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 9.12311 0.331148
$$760$$ 5.12311 0.185835
$$761$$ 13.5076 0.489649 0.244825 0.969567i $$-0.421270\pi$$
0.244825 + 0.969567i $$0.421270\pi$$
$$762$$ 9.36932 0.339415
$$763$$ 0 0
$$764$$ 12.8769 0.465870
$$765$$ 1.00000 0.0361551
$$766$$ 5.36932 0.194001
$$767$$ 11.5076 0.415515
$$768$$ 1.00000 0.0360844
$$769$$ −28.2462 −1.01858 −0.509292 0.860594i $$-0.670093\pi$$
−0.509292 + 0.860594i $$0.670093\pi$$
$$770$$ 0 0
$$771$$ −10.4924 −0.377875
$$772$$ −22.4924 −0.809520
$$773$$ 20.4924 0.737061 0.368531 0.929616i $$-0.379861\pi$$
0.368531 + 0.929616i $$0.379861\pi$$
$$774$$ 0 0
$$775$$ −7.12311 −0.255870
$$776$$ 11.1231 0.399296
$$777$$ 0 0
$$778$$ 33.6155 1.20518
$$779$$ −16.0000 −0.573259
$$780$$ −1.12311 −0.0402136
$$781$$ 8.00000 0.286263
$$782$$ 9.12311 0.326242
$$783$$ −9.12311 −0.326033
$$784$$ −7.00000 −0.250000
$$785$$ 23.6155 0.842874
$$786$$ 0 0
$$787$$ −16.4924 −0.587891 −0.293946 0.955822i $$-0.594968\pi$$
−0.293946 + 0.955822i $$0.594968\pi$$
$$788$$ 17.6155 0.627527
$$789$$ 30.2462 1.07679
$$790$$ 5.12311 0.182272
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ −9.26137 −0.328881
$$794$$ 35.8617 1.27269
$$795$$ −4.00000 −0.141865
$$796$$ −15.1231 −0.536024
$$797$$ 25.7538 0.912246 0.456123 0.889917i $$-0.349238\pi$$
0.456123 + 0.889917i $$0.349238\pi$$
$$798$$ 0 0
$$799$$ −7.12311 −0.251997
$$800$$ 1.00000 0.0353553
$$801$$ 12.2462 0.432699
$$802$$ −18.0000 −0.635602
$$803$$ 16.2462 0.573316
$$804$$ −6.87689 −0.242529
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 7.75379 0.272946
$$808$$ −9.36932 −0.329611
$$809$$ −18.6307 −0.655020 −0.327510 0.944848i $$-0.606210\pi$$
−0.327510 + 0.944848i $$0.606210\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 44.9848 1.57963 0.789816 0.613344i $$-0.210176\pi$$
0.789816 + 0.613344i $$0.210176\pi$$
$$812$$ 0 0
$$813$$ −2.63068 −0.0922621
$$814$$ −1.12311 −0.0393648
$$815$$ −8.49242 −0.297477
$$816$$ −1.00000 −0.0350070
$$817$$ 0 0
$$818$$ −22.4924 −0.786429
$$819$$ 0 0
$$820$$ −3.12311 −0.109064
$$821$$ −16.6307 −0.580415 −0.290207 0.956964i $$-0.593724\pi$$
−0.290207 + 0.956964i $$0.593724\pi$$
$$822$$ −4.24621 −0.148104
$$823$$ 33.2311 1.15836 0.579181 0.815199i $$-0.303372\pi$$
0.579181 + 0.815199i $$0.303372\pi$$
$$824$$ 2.24621 0.0782505
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ 50.2462 1.74723 0.873616 0.486616i $$-0.161769\pi$$
0.873616 + 0.486616i $$0.161769\pi$$
$$828$$ −9.12311 −0.317050
$$829$$ −20.2462 −0.703180 −0.351590 0.936154i $$-0.614359\pi$$
−0.351590 + 0.936154i $$0.614359\pi$$
$$830$$ −6.24621 −0.216809
$$831$$ −12.2462 −0.424816
$$832$$ 1.12311 0.0389367
$$833$$ 7.00000 0.242536
$$834$$ 14.2462 0.493306
$$835$$ 23.1231 0.800208
$$836$$ 5.12311 0.177186
$$837$$ −7.12311 −0.246211
$$838$$ −16.4924 −0.569721
$$839$$ −12.9848 −0.448287 −0.224143 0.974556i $$-0.571958\pi$$
−0.224143 + 0.974556i $$0.571958\pi$$
$$840$$ 0 0
$$841$$ 54.2311 1.87004
$$842$$ 8.73863 0.301153
$$843$$ −18.0000 −0.619953
$$844$$ 20.0000 0.688428
$$845$$ 11.7386 0.403821
$$846$$ 7.12311 0.244897
$$847$$ 0 0
$$848$$ 4.00000 0.137361
$$849$$ 8.49242 0.291459
$$850$$ −1.00000 −0.0342997
$$851$$ −10.2462 −0.351236
$$852$$ −8.00000 −0.274075
$$853$$ 10.4924 0.359254 0.179627 0.983735i $$-0.442511\pi$$
0.179627 + 0.983735i $$0.442511\pi$$
$$854$$ 0 0
$$855$$ 5.12311 0.175207
$$856$$ 12.0000 0.410152
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ −1.12311 −0.0383422
$$859$$ −39.2311 −1.33855 −0.669273 0.743016i $$-0.733394\pi$$
−0.669273 + 0.743016i $$0.733394\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 34.9848 1.19159
$$863$$ 19.1231 0.650958 0.325479 0.945549i $$-0.394474\pi$$
0.325479 + 0.945549i $$0.394474\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 14.8769 0.505830
$$866$$ 2.49242 0.0846960
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 5.12311 0.173789
$$870$$ 9.12311 0.309302
$$871$$ −7.72348 −0.261700
$$872$$ −6.00000 −0.203186
$$873$$ 11.1231 0.376460
$$874$$ 46.7386 1.58096
$$875$$ 0 0
$$876$$ −16.2462 −0.548909
$$877$$ −12.7386 −0.430153 −0.215077 0.976597i $$-0.569000\pi$$
−0.215077 + 0.976597i $$0.569000\pi$$
$$878$$ 37.6155 1.26946
$$879$$ −14.4924 −0.488817
$$880$$ 1.00000 0.0337100
$$881$$ −4.24621 −0.143058 −0.0715292 0.997439i $$-0.522788\pi$$
−0.0715292 + 0.997439i $$0.522788\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ 1.12311 0.0377955 0.0188978 0.999821i $$-0.493984\pi$$
0.0188978 + 0.999821i $$0.493984\pi$$
$$884$$ −1.12311 −0.0377741
$$885$$ −10.2462 −0.344423
$$886$$ −17.1231 −0.575262
$$887$$ −6.63068 −0.222637 −0.111318 0.993785i $$-0.535507\pi$$
−0.111318 + 0.993785i $$0.535507\pi$$
$$888$$ 1.12311 0.0376890
$$889$$ 0 0
$$890$$ −12.2462 −0.410494
$$891$$ −1.00000 −0.0335013
$$892$$ 9.75379 0.326581
$$893$$ −36.4924 −1.22117
$$894$$ 5.36932 0.179577
$$895$$ 16.4924 0.551281
$$896$$ 0 0
$$897$$ −10.2462 −0.342111
$$898$$ −32.7386 −1.09250
$$899$$ 64.9848 2.16737
$$900$$ 1.00000 0.0333333
$$901$$ −4.00000 −0.133259
$$902$$ −3.12311 −0.103988
$$903$$ 0 0
$$904$$ 9.36932 0.311619
$$905$$ 17.6155 0.585560
$$906$$ 9.36932 0.311275
$$907$$ −5.26137 −0.174701 −0.0873504 0.996178i $$-0.527840\pi$$
−0.0873504 + 0.996178i $$0.527840\pi$$
$$908$$ −18.2462 −0.605522
$$909$$ −9.36932 −0.310761
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ −5.12311 −0.169643
$$913$$ −6.24621 −0.206719
$$914$$ 18.4924 0.611675
$$915$$ 8.24621 0.272611
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ −1.00000 −0.0330049
$$919$$ 36.1080 1.19109 0.595546 0.803321i $$-0.296936\pi$$
0.595546 + 0.803321i $$0.296936\pi$$
$$920$$ 9.12311 0.300780
$$921$$ −16.4924 −0.543444
$$922$$ −3.12311 −0.102854
$$923$$ −8.98485 −0.295740
$$924$$ 0 0
$$925$$ 1.12311 0.0369275
$$926$$ −24.0000 −0.788689
$$927$$ 2.24621 0.0737753
$$928$$ −9.12311 −0.299481
$$929$$ 20.2462 0.664257 0.332128 0.943234i $$-0.392233\pi$$
0.332128 + 0.943234i $$0.392233\pi$$
$$930$$ 7.12311 0.233576
$$931$$ 35.8617 1.17532
$$932$$ −10.4924 −0.343691
$$933$$ −4.00000 −0.130954
$$934$$ 39.3693 1.28820
$$935$$ −1.00000 −0.0327035
$$936$$ 1.12311 0.0367099
$$937$$ 22.4924 0.734795 0.367398 0.930064i $$-0.380249\pi$$
0.367398 + 0.930064i $$0.380249\pi$$
$$938$$ 0 0
$$939$$ 3.12311 0.101919
$$940$$ −7.12311 −0.232330
$$941$$ −49.6155 −1.61742 −0.808710 0.588208i $$-0.799834\pi$$
−0.808710 + 0.588208i $$0.799834\pi$$
$$942$$ −23.6155 −0.769435
$$943$$ −28.4924 −0.927841
$$944$$ 10.2462 0.333486
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 56.4924 1.83576 0.917879 0.396861i $$-0.129901\pi$$
0.917879 + 0.396861i $$0.129901\pi$$
$$948$$ −5.12311 −0.166391
$$949$$ −18.2462 −0.592297
$$950$$ −5.12311 −0.166215
$$951$$ 28.2462 0.915946
$$952$$ 0 0
$$953$$ −25.5076 −0.826271 −0.413136 0.910669i $$-0.635567\pi$$
−0.413136 + 0.910669i $$0.635567\pi$$
$$954$$ 4.00000 0.129505
$$955$$ −12.8769 −0.416687
$$956$$ 26.2462 0.848863
$$957$$ 9.12311 0.294908
$$958$$ −12.2462 −0.395657
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ 19.7386 0.636730
$$962$$ 1.26137 0.0406681
$$963$$ 12.0000 0.386695
$$964$$ 15.6155 0.502942
$$965$$ 22.4924 0.724057
$$966$$ 0 0
$$967$$ −49.3693 −1.58761 −0.793805 0.608172i $$-0.791903\pi$$
−0.793805 + 0.608172i $$0.791903\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 5.12311 0.164578
$$970$$ −11.1231 −0.357141
$$971$$ 45.4773 1.45943 0.729717 0.683749i $$-0.239652\pi$$
0.729717 + 0.683749i $$0.239652\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −42.4924 −1.36155
$$975$$ 1.12311 0.0359682
$$976$$ −8.24621 −0.263955
$$977$$ −21.2311 −0.679242 −0.339621 0.940562i $$-0.610299\pi$$
−0.339621 + 0.940562i $$0.610299\pi$$
$$978$$ 8.49242 0.271558
$$979$$ −12.2462 −0.391391
$$980$$ 7.00000 0.223607
$$981$$ −6.00000 −0.191565
$$982$$ −29.6155 −0.945069
$$983$$ −29.1231 −0.928883 −0.464441 0.885604i $$-0.653745\pi$$
−0.464441 + 0.885604i $$0.653745\pi$$
$$984$$ 3.12311 0.0995610
$$985$$ −17.6155 −0.561277
$$986$$ 9.12311 0.290539
$$987$$ 0 0
$$988$$ −5.75379 −0.183052
$$989$$ 0 0
$$990$$ 1.00000 0.0317821
$$991$$ −11.1231 −0.353337 −0.176669 0.984270i $$-0.556532\pi$$
−0.176669 + 0.984270i $$0.556532\pi$$
$$992$$ −7.12311 −0.226159
$$993$$ −30.2462 −0.959835
$$994$$ 0 0
$$995$$ 15.1231 0.479435
$$996$$ 6.24621 0.197919
$$997$$ 28.7386 0.910162 0.455081 0.890450i $$-0.349610\pi$$
0.455081 + 0.890450i $$0.349610\pi$$
$$998$$ 34.2462 1.08404
$$999$$ 1.12311 0.0355335
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 5610.2.a.bw.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bw.1.2 2 1.1 even 1 trivial