# Properties

 Label 5610.2.a.bu.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: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.23607 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.23607 q^{13} +3.23607 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -7.70820 q^{19} -1.00000 q^{20} -3.23607 q^{21} +1.00000 q^{22} -5.23607 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -8.47214 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} -3.23607 q^{35} +1.00000 q^{36} -6.00000 q^{37} -7.70820 q^{38} -1.23607 q^{39} -1.00000 q^{40} -8.94427 q^{41} -3.23607 q^{42} -3.23607 q^{43} +1.00000 q^{44} -1.00000 q^{45} -5.23607 q^{46} +2.47214 q^{47} -1.00000 q^{48} +3.47214 q^{49} +1.00000 q^{50} +1.00000 q^{51} +1.23607 q^{52} +3.23607 q^{53} -1.00000 q^{54} -1.00000 q^{55} +3.23607 q^{56} +7.70820 q^{57} -8.47214 q^{58} -3.23607 q^{59} +1.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -1.23607 q^{65} -1.00000 q^{66} +6.00000 q^{67} -1.00000 q^{68} +5.23607 q^{69} -3.23607 q^{70} -0.763932 q^{71} +1.00000 q^{72} +1.23607 q^{73} -6.00000 q^{74} -1.00000 q^{75} -7.70820 q^{76} +3.23607 q^{77} -1.23607 q^{78} +3.70820 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +4.94427 q^{83} -3.23607 q^{84} +1.00000 q^{85} -3.23607 q^{86} +8.47214 q^{87} +1.00000 q^{88} -6.94427 q^{89} -1.00000 q^{90} +4.00000 q^{91} -5.23607 q^{92} +4.00000 q^{93} +2.47214 q^{94} +7.70820 q^{95} -1.00000 q^{96} +3.47214 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} + 2 q^{28} - 8 q^{29} + 2 q^{30} - 8 q^{31} + 2 q^{32} - 2 q^{33} - 2 q^{34} - 2 q^{35} + 2 q^{36} - 12 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} - 2 q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 2 q^{57} - 8 q^{58} - 2 q^{59} + 2 q^{60} + 12 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 2 q^{66} + 12 q^{67} - 2 q^{68} + 6 q^{69} - 2 q^{70} - 6 q^{71} + 2 q^{72} - 2 q^{73} - 12 q^{74} - 2 q^{75} - 2 q^{76} + 2 q^{77} + 2 q^{78} - 6 q^{79} - 2 q^{80} + 2 q^{81} - 8 q^{83} - 2 q^{84} + 2 q^{85} - 2 q^{86} + 8 q^{87} + 2 q^{88} + 4 q^{89} - 2 q^{90} + 8 q^{91} - 6 q^{92} + 8 q^{93} - 4 q^{94} + 2 q^{95} - 2 q^{96} - 2 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 - 2 * q^12 - 2 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^21 + 2 * q^22 - 6 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^26 - 2 * q^27 + 2 * q^28 - 8 * q^29 + 2 * q^30 - 8 * q^31 + 2 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 2 * q^36 - 12 * q^37 - 2 * q^38 + 2 * q^39 - 2 * q^40 - 2 * q^42 - 2 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 - 4 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 + 2 * q^51 - 2 * q^52 + 2 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 + 2 * q^57 - 8 * q^58 - 2 * q^59 + 2 * q^60 + 12 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 + 2 * q^65 - 2 * q^66 + 12 * q^67 - 2 * q^68 + 6 * q^69 - 2 * q^70 - 6 * q^71 + 2 * q^72 - 2 * q^73 - 12 * q^74 - 2 * q^75 - 2 * q^76 + 2 * q^77 + 2 * q^78 - 6 * q^79 - 2 * q^80 + 2 * q^81 - 8 * q^83 - 2 * q^84 + 2 * q^85 - 2 * q^86 + 8 * q^87 + 2 * q^88 + 4 * q^89 - 2 * q^90 + 8 * q^91 - 6 * q^92 + 8 * q^93 - 4 * q^94 + 2 * q^95 - 2 * q^96 - 2 * 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
$$7$$ 3.23607 1.22312 0.611559 0.791199i $$-0.290543\pi$$
0.611559 + 0.791199i $$0.290543\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 1.00000 0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 1.23607 0.342824 0.171412 0.985199i $$-0.445167\pi$$
0.171412 + 0.985199i $$0.445167\pi$$
$$14$$ 3.23607 0.864876
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ 1.00000 0.235702
$$19$$ −7.70820 −1.76838 −0.884192 0.467124i $$-0.845290\pi$$
−0.884192 + 0.467124i $$0.845290\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −3.23607 −0.706168
$$22$$ 1.00000 0.213201
$$23$$ −5.23607 −1.09180 −0.545898 0.837852i $$-0.683811\pi$$
−0.545898 + 0.837852i $$0.683811\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 1.23607 0.242413
$$27$$ −1.00000 −0.192450
$$28$$ 3.23607 0.611559
$$29$$ −8.47214 −1.57324 −0.786618 0.617440i $$-0.788170\pi$$
−0.786618 + 0.617440i $$0.788170\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ −3.23607 −0.546995
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −7.70820 −1.25044
$$39$$ −1.23607 −0.197929
$$40$$ −1.00000 −0.158114
$$41$$ −8.94427 −1.39686 −0.698430 0.715678i $$-0.746118\pi$$
−0.698430 + 0.715678i $$0.746118\pi$$
$$42$$ −3.23607 −0.499336
$$43$$ −3.23607 −0.493496 −0.246748 0.969080i $$-0.579362\pi$$
−0.246748 + 0.969080i $$0.579362\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −5.23607 −0.772016
$$47$$ 2.47214 0.360598 0.180299 0.983612i $$-0.442293\pi$$
0.180299 + 0.983612i $$0.442293\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.47214 0.496019
$$50$$ 1.00000 0.141421
$$51$$ 1.00000 0.140028
$$52$$ 1.23607 0.171412
$$53$$ 3.23607 0.444508 0.222254 0.974989i $$-0.428659\pi$$
0.222254 + 0.974989i $$0.428659\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 3.23607 0.432438
$$57$$ 7.70820 1.02098
$$58$$ −8.47214 −1.11245
$$59$$ −3.23607 −0.421300 −0.210650 0.977562i $$-0.567558\pi$$
−0.210650 + 0.977562i $$0.567558\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 3.23607 0.407706
$$64$$ 1.00000 0.125000
$$65$$ −1.23607 −0.153315
$$66$$ −1.00000 −0.123091
$$67$$ 6.00000 0.733017 0.366508 0.930415i $$-0.380553\pi$$
0.366508 + 0.930415i $$0.380553\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 5.23607 0.630349
$$70$$ −3.23607 −0.386784
$$71$$ −0.763932 −0.0906621 −0.0453310 0.998972i $$-0.514434\pi$$
−0.0453310 + 0.998972i $$0.514434\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 1.23607 0.144671 0.0723354 0.997380i $$-0.476955\pi$$
0.0723354 + 0.997380i $$0.476955\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ −7.70820 −0.884192
$$77$$ 3.23607 0.368784
$$78$$ −1.23607 −0.139957
$$79$$ 3.70820 0.417206 0.208603 0.978000i $$-0.433108\pi$$
0.208603 + 0.978000i $$0.433108\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −8.94427 −0.987730
$$83$$ 4.94427 0.542704 0.271352 0.962480i $$-0.412529\pi$$
0.271352 + 0.962480i $$0.412529\pi$$
$$84$$ −3.23607 −0.353084
$$85$$ 1.00000 0.108465
$$86$$ −3.23607 −0.348954
$$87$$ 8.47214 0.908308
$$88$$ 1.00000 0.106600
$$89$$ −6.94427 −0.736091 −0.368046 0.929808i $$-0.619973\pi$$
−0.368046 + 0.929808i $$0.619973\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 4.00000 0.419314
$$92$$ −5.23607 −0.545898
$$93$$ 4.00000 0.414781
$$94$$ 2.47214 0.254981
$$95$$ 7.70820 0.790845
$$96$$ −1.00000 −0.102062
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 3.47214 0.350739
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ 8.94427 0.889988 0.444994 0.895533i $$-0.353206\pi$$
0.444994 + 0.895533i $$0.353206\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 1.23607 0.121206
$$105$$ 3.23607 0.315808
$$106$$ 3.23607 0.314315
$$107$$ 0.944272 0.0912862 0.0456431 0.998958i $$-0.485466\pi$$
0.0456431 + 0.998958i $$0.485466\pi$$
$$108$$ −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
$$111$$ 6.00000 0.569495
$$112$$ 3.23607 0.305780
$$113$$ −15.2361 −1.43329 −0.716644 0.697439i $$-0.754323\pi$$
−0.716644 + 0.697439i $$0.754323\pi$$
$$114$$ 7.70820 0.721939
$$115$$ 5.23607 0.488266
$$116$$ −8.47214 −0.786618
$$117$$ 1.23607 0.114275
$$118$$ −3.23607 −0.297904
$$119$$ −3.23607 −0.296650
$$120$$ 1.00000 0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ 6.00000 0.543214
$$123$$ 8.94427 0.806478
$$124$$ −4.00000 −0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 3.23607 0.288292
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 3.23607 0.284920
$$130$$ −1.23607 −0.108410
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −24.9443 −2.16294
$$134$$ 6.00000 0.518321
$$135$$ 1.00000 0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ −15.8885 −1.35745 −0.678725 0.734393i $$-0.737467\pi$$
−0.678725 + 0.734393i $$0.737467\pi$$
$$138$$ 5.23607 0.445724
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −3.23607 −0.273498
$$141$$ −2.47214 −0.208191
$$142$$ −0.763932 −0.0641078
$$143$$ 1.23607 0.103365
$$144$$ 1.00000 0.0833333
$$145$$ 8.47214 0.703573
$$146$$ 1.23607 0.102298
$$147$$ −3.47214 −0.286377
$$148$$ −6.00000 −0.493197
$$149$$ −9.52786 −0.780553 −0.390277 0.920698i $$-0.627621\pi$$
−0.390277 + 0.920698i $$0.627621\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −14.4721 −1.17773 −0.588863 0.808233i $$-0.700424\pi$$
−0.588863 + 0.808233i $$0.700424\pi$$
$$152$$ −7.70820 −0.625218
$$153$$ −1.00000 −0.0808452
$$154$$ 3.23607 0.260770
$$155$$ 4.00000 0.321288
$$156$$ −1.23607 −0.0989646
$$157$$ 19.4164 1.54960 0.774799 0.632208i $$-0.217851\pi$$
0.774799 + 0.632208i $$0.217851\pi$$
$$158$$ 3.70820 0.295009
$$159$$ −3.23607 −0.256637
$$160$$ −1.00000 −0.0790569
$$161$$ −16.9443 −1.33540
$$162$$ 1.00000 0.0785674
$$163$$ −6.47214 −0.506937 −0.253468 0.967344i $$-0.581571\pi$$
−0.253468 + 0.967344i $$0.581571\pi$$
$$164$$ −8.94427 −0.698430
$$165$$ 1.00000 0.0778499
$$166$$ 4.94427 0.383750
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −3.23607 −0.249668
$$169$$ −11.4721 −0.882472
$$170$$ 1.00000 0.0766965
$$171$$ −7.70820 −0.589461
$$172$$ −3.23607 −0.246748
$$173$$ 13.4164 1.02003 0.510015 0.860165i $$-0.329640\pi$$
0.510015 + 0.860165i $$0.329640\pi$$
$$174$$ 8.47214 0.642271
$$175$$ 3.23607 0.244624
$$176$$ 1.00000 0.0753778
$$177$$ 3.23607 0.243238
$$178$$ −6.94427 −0.520495
$$179$$ −12.7639 −0.954021 −0.477011 0.878898i $$-0.658280\pi$$
−0.477011 + 0.878898i $$0.658280\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 8.47214 0.629729 0.314864 0.949137i $$-0.398041\pi$$
0.314864 + 0.949137i $$0.398041\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −6.00000 −0.443533
$$184$$ −5.23607 −0.386008
$$185$$ 6.00000 0.441129
$$186$$ 4.00000 0.293294
$$187$$ −1.00000 −0.0731272
$$188$$ 2.47214 0.180299
$$189$$ −3.23607 −0.235389
$$190$$ 7.70820 0.559212
$$191$$ 1.52786 0.110552 0.0552762 0.998471i $$-0.482396\pi$$
0.0552762 + 0.998471i $$0.482396\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 7.70820 0.554849 0.277424 0.960747i $$-0.410519\pi$$
0.277424 + 0.960747i $$0.410519\pi$$
$$194$$ 0 0
$$195$$ 1.23607 0.0885167
$$196$$ 3.47214 0.248010
$$197$$ 14.9443 1.06474 0.532368 0.846513i $$-0.321302\pi$$
0.532368 + 0.846513i $$0.321302\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −6.00000 −0.423207
$$202$$ 8.94427 0.629317
$$203$$ −27.4164 −1.92425
$$204$$ 1.00000 0.0700140
$$205$$ 8.94427 0.624695
$$206$$ 4.00000 0.278693
$$207$$ −5.23607 −0.363932
$$208$$ 1.23607 0.0857059
$$209$$ −7.70820 −0.533188
$$210$$ 3.23607 0.223310
$$211$$ 20.3607 1.40169 0.700844 0.713315i $$-0.252807\pi$$
0.700844 + 0.713315i $$0.252807\pi$$
$$212$$ 3.23607 0.222254
$$213$$ 0.763932 0.0523438
$$214$$ 0.944272 0.0645491
$$215$$ 3.23607 0.220698
$$216$$ −1.00000 −0.0680414
$$217$$ −12.9443 −0.878714
$$218$$ −6.00000 −0.406371
$$219$$ −1.23607 −0.0835257
$$220$$ −1.00000 −0.0674200
$$221$$ −1.23607 −0.0831469
$$222$$ 6.00000 0.402694
$$223$$ −23.4164 −1.56808 −0.784039 0.620711i $$-0.786844\pi$$
−0.784039 + 0.620711i $$0.786844\pi$$
$$224$$ 3.23607 0.216219
$$225$$ 1.00000 0.0666667
$$226$$ −15.2361 −1.01349
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 7.70820 0.510488
$$229$$ 29.4164 1.94389 0.971945 0.235206i $$-0.0755766\pi$$
0.971945 + 0.235206i $$0.0755766\pi$$
$$230$$ 5.23607 0.345256
$$231$$ −3.23607 −0.212918
$$232$$ −8.47214 −0.556223
$$233$$ −7.52786 −0.493167 −0.246583 0.969122i $$-0.579308\pi$$
−0.246583 + 0.969122i $$0.579308\pi$$
$$234$$ 1.23607 0.0808043
$$235$$ −2.47214 −0.161264
$$236$$ −3.23607 −0.210650
$$237$$ −3.70820 −0.240874
$$238$$ −3.23607 −0.209763
$$239$$ −16.3607 −1.05828 −0.529142 0.848533i $$-0.677486\pi$$
−0.529142 + 0.848533i $$0.677486\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 4.18034 0.269279 0.134640 0.990895i $$-0.457012\pi$$
0.134640 + 0.990895i $$0.457012\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ −3.47214 −0.221827
$$246$$ 8.94427 0.570266
$$247$$ −9.52786 −0.606243
$$248$$ −4.00000 −0.254000
$$249$$ −4.94427 −0.313331
$$250$$ −1.00000 −0.0632456
$$251$$ 23.2361 1.46665 0.733324 0.679880i $$-0.237968\pi$$
0.733324 + 0.679880i $$0.237968\pi$$
$$252$$ 3.23607 0.203853
$$253$$ −5.23607 −0.329189
$$254$$ −12.0000 −0.752947
$$255$$ −1.00000 −0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −4.47214 −0.278964 −0.139482 0.990225i $$-0.544544\pi$$
−0.139482 + 0.990225i $$0.544544\pi$$
$$258$$ 3.23607 0.201469
$$259$$ −19.4164 −1.20648
$$260$$ −1.23607 −0.0766577
$$261$$ −8.47214 −0.524412
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ −3.23607 −0.198790
$$266$$ −24.9443 −1.52943
$$267$$ 6.94427 0.424983
$$268$$ 6.00000 0.366508
$$269$$ −24.4721 −1.49209 −0.746046 0.665894i $$-0.768050\pi$$
−0.746046 + 0.665894i $$0.768050\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ −4.00000 −0.242091
$$274$$ −15.8885 −0.959862
$$275$$ 1.00000 0.0603023
$$276$$ 5.23607 0.315174
$$277$$ 14.0000 0.841178 0.420589 0.907251i $$-0.361823\pi$$
0.420589 + 0.907251i $$0.361823\pi$$
$$278$$ 8.00000 0.479808
$$279$$ −4.00000 −0.239474
$$280$$ −3.23607 −0.193392
$$281$$ −16.4721 −0.982645 −0.491323 0.870978i $$-0.663486\pi$$
−0.491323 + 0.870978i $$0.663486\pi$$
$$282$$ −2.47214 −0.147214
$$283$$ −3.41641 −0.203084 −0.101542 0.994831i $$-0.532378\pi$$
−0.101542 + 0.994831i $$0.532378\pi$$
$$284$$ −0.763932 −0.0453310
$$285$$ −7.70820 −0.456595
$$286$$ 1.23607 0.0730902
$$287$$ −28.9443 −1.70853
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 8.47214 0.497501
$$291$$ 0 0
$$292$$ 1.23607 0.0723354
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ −3.47214 −0.202499
$$295$$ 3.23607 0.188411
$$296$$ −6.00000 −0.348743
$$297$$ −1.00000 −0.0580259
$$298$$ −9.52786 −0.551934
$$299$$ −6.47214 −0.374293
$$300$$ −1.00000 −0.0577350
$$301$$ −10.4721 −0.603604
$$302$$ −14.4721 −0.832778
$$303$$ −8.94427 −0.513835
$$304$$ −7.70820 −0.442096
$$305$$ −6.00000 −0.343559
$$306$$ −1.00000 −0.0571662
$$307$$ 10.2918 0.587384 0.293692 0.955900i $$-0.405116\pi$$
0.293692 + 0.955900i $$0.405116\pi$$
$$308$$ 3.23607 0.184392
$$309$$ −4.00000 −0.227552
$$310$$ 4.00000 0.227185
$$311$$ 11.2361 0.637139 0.318569 0.947900i $$-0.396798\pi$$
0.318569 + 0.947900i $$0.396798\pi$$
$$312$$ −1.23607 −0.0699786
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 19.4164 1.09573
$$315$$ −3.23607 −0.182332
$$316$$ 3.70820 0.208603
$$317$$ −13.0557 −0.733283 −0.366641 0.930362i $$-0.619492\pi$$
−0.366641 + 0.930362i $$0.619492\pi$$
$$318$$ −3.23607 −0.181470
$$319$$ −8.47214 −0.474349
$$320$$ −1.00000 −0.0559017
$$321$$ −0.944272 −0.0527041
$$322$$ −16.9443 −0.944267
$$323$$ 7.70820 0.428896
$$324$$ 1.00000 0.0555556
$$325$$ 1.23607 0.0685647
$$326$$ −6.47214 −0.358458
$$327$$ 6.00000 0.331801
$$328$$ −8.94427 −0.493865
$$329$$ 8.00000 0.441054
$$330$$ 1.00000 0.0550482
$$331$$ 33.8885 1.86268 0.931341 0.364147i $$-0.118639\pi$$
0.931341 + 0.364147i $$0.118639\pi$$
$$332$$ 4.94427 0.271352
$$333$$ −6.00000 −0.328798
$$334$$ −12.0000 −0.656611
$$335$$ −6.00000 −0.327815
$$336$$ −3.23607 −0.176542
$$337$$ −3.70820 −0.201999 −0.100999 0.994886i $$-0.532204\pi$$
−0.100999 + 0.994886i $$0.532204\pi$$
$$338$$ −11.4721 −0.624002
$$339$$ 15.2361 0.827510
$$340$$ 1.00000 0.0542326
$$341$$ −4.00000 −0.216612
$$342$$ −7.70820 −0.416812
$$343$$ −11.4164 −0.616428
$$344$$ −3.23607 −0.174477
$$345$$ −5.23607 −0.281900
$$346$$ 13.4164 0.721271
$$347$$ 12.9443 0.694885 0.347442 0.937701i $$-0.387050\pi$$
0.347442 + 0.937701i $$0.387050\pi$$
$$348$$ 8.47214 0.454154
$$349$$ 1.70820 0.0914381 0.0457190 0.998954i $$-0.485442\pi$$
0.0457190 + 0.998954i $$0.485442\pi$$
$$350$$ 3.23607 0.172975
$$351$$ −1.23607 −0.0659764
$$352$$ 1.00000 0.0533002
$$353$$ −4.47214 −0.238028 −0.119014 0.992893i $$-0.537973\pi$$
−0.119014 + 0.992893i $$0.537973\pi$$
$$354$$ 3.23607 0.171995
$$355$$ 0.763932 0.0405453
$$356$$ −6.94427 −0.368046
$$357$$ 3.23607 0.171271
$$358$$ −12.7639 −0.674595
$$359$$ −7.41641 −0.391423 −0.195712 0.980662i $$-0.562702\pi$$
−0.195712 + 0.980662i $$0.562702\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 40.4164 2.12718
$$362$$ 8.47214 0.445286
$$363$$ −1.00000 −0.0524864
$$364$$ 4.00000 0.209657
$$365$$ −1.23607 −0.0646988
$$366$$ −6.00000 −0.313625
$$367$$ 28.8328 1.50506 0.752530 0.658558i $$-0.228833\pi$$
0.752530 + 0.658558i $$0.228833\pi$$
$$368$$ −5.23607 −0.272949
$$369$$ −8.94427 −0.465620
$$370$$ 6.00000 0.311925
$$371$$ 10.4721 0.543686
$$372$$ 4.00000 0.207390
$$373$$ −15.7082 −0.813340 −0.406670 0.913575i $$-0.633310\pi$$
−0.406670 + 0.913575i $$0.633310\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ 1.00000 0.0516398
$$376$$ 2.47214 0.127491
$$377$$ −10.4721 −0.539342
$$378$$ −3.23607 −0.166445
$$379$$ −24.9443 −1.28130 −0.640651 0.767833i $$-0.721335\pi$$
−0.640651 + 0.767833i $$0.721335\pi$$
$$380$$ 7.70820 0.395423
$$381$$ 12.0000 0.614779
$$382$$ 1.52786 0.0781723
$$383$$ 10.4721 0.535101 0.267551 0.963544i $$-0.413786\pi$$
0.267551 + 0.963544i $$0.413786\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −3.23607 −0.164925
$$386$$ 7.70820 0.392337
$$387$$ −3.23607 −0.164499
$$388$$ 0 0
$$389$$ 25.5967 1.29781 0.648903 0.760871i $$-0.275228\pi$$
0.648903 + 0.760871i $$0.275228\pi$$
$$390$$ 1.23607 0.0625907
$$391$$ 5.23607 0.264799
$$392$$ 3.47214 0.175369
$$393$$ 0 0
$$394$$ 14.9443 0.752882
$$395$$ −3.70820 −0.186580
$$396$$ 1.00000 0.0502519
$$397$$ −19.8885 −0.998177 −0.499089 0.866551i $$-0.666332\pi$$
−0.499089 + 0.866551i $$0.666332\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 24.9443 1.24878
$$400$$ 1.00000 0.0500000
$$401$$ −20.2918 −1.01332 −0.506662 0.862145i $$-0.669121\pi$$
−0.506662 + 0.862145i $$0.669121\pi$$
$$402$$ −6.00000 −0.299253
$$403$$ −4.94427 −0.246292
$$404$$ 8.94427 0.444994
$$405$$ −1.00000 −0.0496904
$$406$$ −27.4164 −1.36065
$$407$$ −6.00000 −0.297409
$$408$$ 1.00000 0.0495074
$$409$$ 10.9443 0.541159 0.270580 0.962698i $$-0.412785\pi$$
0.270580 + 0.962698i $$0.412785\pi$$
$$410$$ 8.94427 0.441726
$$411$$ 15.8885 0.783724
$$412$$ 4.00000 0.197066
$$413$$ −10.4721 −0.515300
$$414$$ −5.23607 −0.257339
$$415$$ −4.94427 −0.242705
$$416$$ 1.23607 0.0606032
$$417$$ −8.00000 −0.391762
$$418$$ −7.70820 −0.377021
$$419$$ 10.4721 0.511597 0.255799 0.966730i $$-0.417662\pi$$
0.255799 + 0.966730i $$0.417662\pi$$
$$420$$ 3.23607 0.157904
$$421$$ −6.94427 −0.338443 −0.169222 0.985578i $$-0.554125\pi$$
−0.169222 + 0.985578i $$0.554125\pi$$
$$422$$ 20.3607 0.991142
$$423$$ 2.47214 0.120199
$$424$$ 3.23607 0.157157
$$425$$ −1.00000 −0.0485071
$$426$$ 0.763932 0.0370126
$$427$$ 19.4164 0.939626
$$428$$ 0.944272 0.0456431
$$429$$ −1.23607 −0.0596779
$$430$$ 3.23607 0.156057
$$431$$ −5.05573 −0.243526 −0.121763 0.992559i $$-0.538855\pi$$
−0.121763 + 0.992559i $$0.538855\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ −12.9443 −0.621345
$$435$$ −8.47214 −0.406208
$$436$$ −6.00000 −0.287348
$$437$$ 40.3607 1.93071
$$438$$ −1.23607 −0.0590616
$$439$$ −4.29180 −0.204836 −0.102418 0.994741i $$-0.532658\pi$$
−0.102418 + 0.994741i $$0.532658\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ 3.47214 0.165340
$$442$$ −1.23607 −0.0587938
$$443$$ −3.34752 −0.159046 −0.0795228 0.996833i $$-0.525340\pi$$
−0.0795228 + 0.996833i $$0.525340\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 6.94427 0.329190
$$446$$ −23.4164 −1.10880
$$447$$ 9.52786 0.450653
$$448$$ 3.23607 0.152890
$$449$$ 6.18034 0.291668 0.145834 0.989309i $$-0.453413\pi$$
0.145834 + 0.989309i $$0.453413\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −8.94427 −0.421169
$$452$$ −15.2361 −0.716644
$$453$$ 14.4721 0.679960
$$454$$ 0 0
$$455$$ −4.00000 −0.187523
$$456$$ 7.70820 0.360970
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 29.4164 1.37454
$$459$$ 1.00000 0.0466760
$$460$$ 5.23607 0.244133
$$461$$ 7.41641 0.345417 0.172708 0.984973i $$-0.444748\pi$$
0.172708 + 0.984973i $$0.444748\pi$$
$$462$$ −3.23607 −0.150556
$$463$$ 20.3607 0.946241 0.473121 0.880998i $$-0.343128\pi$$
0.473121 + 0.880998i $$0.343128\pi$$
$$464$$ −8.47214 −0.393309
$$465$$ −4.00000 −0.185496
$$466$$ −7.52786 −0.348722
$$467$$ 14.7639 0.683193 0.341597 0.939847i $$-0.389032\pi$$
0.341597 + 0.939847i $$0.389032\pi$$
$$468$$ 1.23607 0.0571373
$$469$$ 19.4164 0.896566
$$470$$ −2.47214 −0.114031
$$471$$ −19.4164 −0.894661
$$472$$ −3.23607 −0.148952
$$473$$ −3.23607 −0.148795
$$474$$ −3.70820 −0.170323
$$475$$ −7.70820 −0.353677
$$476$$ −3.23607 −0.148325
$$477$$ 3.23607 0.148169
$$478$$ −16.3607 −0.748320
$$479$$ −27.5279 −1.25778 −0.628890 0.777494i $$-0.716490\pi$$
−0.628890 + 0.777494i $$0.716490\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −7.41641 −0.338159
$$482$$ 4.18034 0.190409
$$483$$ 16.9443 0.770991
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −20.8328 −0.944025 −0.472012 0.881592i $$-0.656472\pi$$
−0.472012 + 0.881592i $$0.656472\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 6.47214 0.292680
$$490$$ −3.47214 −0.156855
$$491$$ 1.05573 0.0476443 0.0238222 0.999716i $$-0.492416\pi$$
0.0238222 + 0.999716i $$0.492416\pi$$
$$492$$ 8.94427 0.403239
$$493$$ 8.47214 0.381566
$$494$$ −9.52786 −0.428679
$$495$$ −1.00000 −0.0449467
$$496$$ −4.00000 −0.179605
$$497$$ −2.47214 −0.110890
$$498$$ −4.94427 −0.221558
$$499$$ −22.8328 −1.02214 −0.511069 0.859540i $$-0.670750\pi$$
−0.511069 + 0.859540i $$0.670750\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 12.0000 0.536120
$$502$$ 23.2361 1.03708
$$503$$ 18.4721 0.823632 0.411816 0.911267i $$-0.364895\pi$$
0.411816 + 0.911267i $$0.364895\pi$$
$$504$$ 3.23607 0.144146
$$505$$ −8.94427 −0.398015
$$506$$ −5.23607 −0.232772
$$507$$ 11.4721 0.509495
$$508$$ −12.0000 −0.532414
$$509$$ 34.7639 1.54088 0.770442 0.637510i $$-0.220036\pi$$
0.770442 + 0.637510i $$0.220036\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ 4.00000 0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 7.70820 0.340326
$$514$$ −4.47214 −0.197257
$$515$$ −4.00000 −0.176261
$$516$$ 3.23607 0.142460
$$517$$ 2.47214 0.108724
$$518$$ −19.4164 −0.853108
$$519$$ −13.4164 −0.588915
$$520$$ −1.23607 −0.0542052
$$521$$ −35.1246 −1.53884 −0.769419 0.638745i $$-0.779454\pi$$
−0.769419 + 0.638745i $$0.779454\pi$$
$$522$$ −8.47214 −0.370815
$$523$$ 21.1246 0.923715 0.461857 0.886954i $$-0.347183\pi$$
0.461857 + 0.886954i $$0.347183\pi$$
$$524$$ 0 0
$$525$$ −3.23607 −0.141234
$$526$$ −12.0000 −0.523225
$$527$$ 4.00000 0.174243
$$528$$ −1.00000 −0.0435194
$$529$$ 4.41641 0.192018
$$530$$ −3.23607 −0.140566
$$531$$ −3.23607 −0.140433
$$532$$ −24.9443 −1.08147
$$533$$ −11.0557 −0.478877
$$534$$ 6.94427 0.300508
$$535$$ −0.944272 −0.0408244
$$536$$ 6.00000 0.259161
$$537$$ 12.7639 0.550804
$$538$$ −24.4721 −1.05507
$$539$$ 3.47214 0.149555
$$540$$ 1.00000 0.0430331
$$541$$ −45.7771 −1.96811 −0.984055 0.177862i $$-0.943082\pi$$
−0.984055 + 0.177862i $$0.943082\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −8.47214 −0.363574
$$544$$ −1.00000 −0.0428746
$$545$$ 6.00000 0.257012
$$546$$ −4.00000 −0.171184
$$547$$ 4.58359 0.195980 0.0979901 0.995187i $$-0.468759\pi$$
0.0979901 + 0.995187i $$0.468759\pi$$
$$548$$ −15.8885 −0.678725
$$549$$ 6.00000 0.256074
$$550$$ 1.00000 0.0426401
$$551$$ 65.3050 2.78208
$$552$$ 5.23607 0.222862
$$553$$ 12.0000 0.510292
$$554$$ 14.0000 0.594803
$$555$$ −6.00000 −0.254686
$$556$$ 8.00000 0.339276
$$557$$ −20.8328 −0.882715 −0.441357 0.897331i $$-0.645503\pi$$
−0.441357 + 0.897331i $$0.645503\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ −4.00000 −0.169182
$$560$$ −3.23607 −0.136749
$$561$$ 1.00000 0.0422200
$$562$$ −16.4721 −0.694835
$$563$$ −35.7771 −1.50782 −0.753912 0.656975i $$-0.771836\pi$$
−0.753912 + 0.656975i $$0.771836\pi$$
$$564$$ −2.47214 −0.104096
$$565$$ 15.2361 0.640986
$$566$$ −3.41641 −0.143602
$$567$$ 3.23607 0.135902
$$568$$ −0.763932 −0.0320539
$$569$$ 19.3050 0.809306 0.404653 0.914470i $$-0.367392\pi$$
0.404653 + 0.914470i $$0.367392\pi$$
$$570$$ −7.70820 −0.322861
$$571$$ −25.8885 −1.08340 −0.541701 0.840571i $$-0.682219\pi$$
−0.541701 + 0.840571i $$0.682219\pi$$
$$572$$ 1.23607 0.0516826
$$573$$ −1.52786 −0.0638274
$$574$$ −28.9443 −1.20811
$$575$$ −5.23607 −0.218359
$$576$$ 1.00000 0.0416667
$$577$$ −15.8885 −0.661449 −0.330724 0.943727i $$-0.607293\pi$$
−0.330724 + 0.943727i $$0.607293\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −7.70820 −0.320342
$$580$$ 8.47214 0.351786
$$581$$ 16.0000 0.663792
$$582$$ 0 0
$$583$$ 3.23607 0.134024
$$584$$ 1.23607 0.0511489
$$585$$ −1.23607 −0.0511051
$$586$$ −18.0000 −0.743573
$$587$$ −2.18034 −0.0899923 −0.0449961 0.998987i $$-0.514328\pi$$
−0.0449961 + 0.998987i $$0.514328\pi$$
$$588$$ −3.47214 −0.143188
$$589$$ 30.8328 1.27044
$$590$$ 3.23607 0.133227
$$591$$ −14.9443 −0.614725
$$592$$ −6.00000 −0.246598
$$593$$ 13.4164 0.550946 0.275473 0.961309i $$-0.411166\pi$$
0.275473 + 0.961309i $$0.411166\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 3.23607 0.132666
$$596$$ −9.52786 −0.390277
$$597$$ 12.0000 0.491127
$$598$$ −6.47214 −0.264665
$$599$$ 4.58359 0.187280 0.0936402 0.995606i $$-0.470150\pi$$
0.0936402 + 0.995606i $$0.470150\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −1.12461 −0.0458739 −0.0229369 0.999737i $$-0.507302\pi$$
−0.0229369 + 0.999737i $$0.507302\pi$$
$$602$$ −10.4721 −0.426812
$$603$$ 6.00000 0.244339
$$604$$ −14.4721 −0.588863
$$605$$ −1.00000 −0.0406558
$$606$$ −8.94427 −0.363336
$$607$$ 29.1246 1.18213 0.591066 0.806623i $$-0.298707\pi$$
0.591066 + 0.806623i $$0.298707\pi$$
$$608$$ −7.70820 −0.312609
$$609$$ 27.4164 1.11097
$$610$$ −6.00000 −0.242933
$$611$$ 3.05573 0.123622
$$612$$ −1.00000 −0.0404226
$$613$$ 7.12461 0.287760 0.143880 0.989595i $$-0.454042\pi$$
0.143880 + 0.989595i $$0.454042\pi$$
$$614$$ 10.2918 0.415343
$$615$$ −8.94427 −0.360668
$$616$$ 3.23607 0.130385
$$617$$ −0.763932 −0.0307547 −0.0153774 0.999882i $$-0.504895\pi$$
−0.0153774 + 0.999882i $$0.504895\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ −9.88854 −0.397454 −0.198727 0.980055i $$-0.563681\pi$$
−0.198727 + 0.980055i $$0.563681\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 5.23607 0.210116
$$622$$ 11.2361 0.450525
$$623$$ −22.4721 −0.900327
$$624$$ −1.23607 −0.0494823
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 7.70820 0.307836
$$628$$ 19.4164 0.774799
$$629$$ 6.00000 0.239236
$$630$$ −3.23607 −0.128928
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 3.70820 0.147504
$$633$$ −20.3607 −0.809264
$$634$$ −13.0557 −0.518509
$$635$$ 12.0000 0.476205
$$636$$ −3.23607 −0.128318
$$637$$ 4.29180 0.170047
$$638$$ −8.47214 −0.335415
$$639$$ −0.763932 −0.0302207
$$640$$ −1.00000 −0.0395285
$$641$$ 8.65248 0.341752 0.170876 0.985293i $$-0.445340\pi$$
0.170876 + 0.985293i $$0.445340\pi$$
$$642$$ −0.944272 −0.0372674
$$643$$ −36.9443 −1.45694 −0.728470 0.685078i $$-0.759768\pi$$
−0.728470 + 0.685078i $$0.759768\pi$$
$$644$$ −16.9443 −0.667698
$$645$$ −3.23607 −0.127420
$$646$$ 7.70820 0.303275
$$647$$ −14.8328 −0.583138 −0.291569 0.956550i $$-0.594177\pi$$
−0.291569 + 0.956550i $$0.594177\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −3.23607 −0.127027
$$650$$ 1.23607 0.0484826
$$651$$ 12.9443 0.507326
$$652$$ −6.47214 −0.253468
$$653$$ 11.8885 0.465235 0.232617 0.972568i $$-0.425271\pi$$
0.232617 + 0.972568i $$0.425271\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 0 0
$$656$$ −8.94427 −0.349215
$$657$$ 1.23607 0.0482236
$$658$$ 8.00000 0.311872
$$659$$ 44.8328 1.74644 0.873219 0.487328i $$-0.162028\pi$$
0.873219 + 0.487328i $$0.162028\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ 1.41641 0.0550919 0.0275459 0.999621i $$-0.491231\pi$$
0.0275459 + 0.999621i $$0.491231\pi$$
$$662$$ 33.8885 1.31712
$$663$$ 1.23607 0.0480049
$$664$$ 4.94427 0.191875
$$665$$ 24.9443 0.967297
$$666$$ −6.00000 −0.232495
$$667$$ 44.3607 1.71765
$$668$$ −12.0000 −0.464294
$$669$$ 23.4164 0.905331
$$670$$ −6.00000 −0.231800
$$671$$ 6.00000 0.231627
$$672$$ −3.23607 −0.124834
$$673$$ −45.2361 −1.74372 −0.871861 0.489753i $$-0.837087\pi$$
−0.871861 + 0.489753i $$0.837087\pi$$
$$674$$ −3.70820 −0.142835
$$675$$ −1.00000 −0.0384900
$$676$$ −11.4721 −0.441236
$$677$$ −39.8885 −1.53304 −0.766521 0.642220i $$-0.778014\pi$$
−0.766521 + 0.642220i $$0.778014\pi$$
$$678$$ 15.2361 0.585138
$$679$$ 0 0
$$680$$ 1.00000 0.0383482
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ −23.7771 −0.909805 −0.454902 0.890541i $$-0.650326\pi$$
−0.454902 + 0.890541i $$0.650326\pi$$
$$684$$ −7.70820 −0.294731
$$685$$ 15.8885 0.607070
$$686$$ −11.4164 −0.435880
$$687$$ −29.4164 −1.12231
$$688$$ −3.23607 −0.123374
$$689$$ 4.00000 0.152388
$$690$$ −5.23607 −0.199334
$$691$$ 42.8328 1.62944 0.814719 0.579857i $$-0.196891\pi$$
0.814719 + 0.579857i $$0.196891\pi$$
$$692$$ 13.4164 0.510015
$$693$$ 3.23607 0.122928
$$694$$ 12.9443 0.491358
$$695$$ −8.00000 −0.303457
$$696$$ 8.47214 0.321135
$$697$$ 8.94427 0.338788
$$698$$ 1.70820 0.0646565
$$699$$ 7.52786 0.284730
$$700$$ 3.23607 0.122312
$$701$$ 26.4721 0.999839 0.499919 0.866072i $$-0.333363\pi$$
0.499919 + 0.866072i $$0.333363\pi$$
$$702$$ −1.23607 −0.0466524
$$703$$ 46.2492 1.74432
$$704$$ 1.00000 0.0376889
$$705$$ 2.47214 0.0931060
$$706$$ −4.47214 −0.168311
$$707$$ 28.9443 1.08856
$$708$$ 3.23607 0.121619
$$709$$ −24.8328 −0.932616 −0.466308 0.884622i $$-0.654416\pi$$
−0.466308 + 0.884622i $$0.654416\pi$$
$$710$$ 0.763932 0.0286699
$$711$$ 3.70820 0.139069
$$712$$ −6.94427 −0.260248
$$713$$ 20.9443 0.784369
$$714$$ 3.23607 0.121107
$$715$$ −1.23607 −0.0462263
$$716$$ −12.7639 −0.477011
$$717$$ 16.3607 0.611001
$$718$$ −7.41641 −0.276778
$$719$$ 48.5410 1.81027 0.905137 0.425119i $$-0.139768\pi$$
0.905137 + 0.425119i $$0.139768\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 12.9443 0.482070
$$722$$ 40.4164 1.50414
$$723$$ −4.18034 −0.155469
$$724$$ 8.47214 0.314864
$$725$$ −8.47214 −0.314647
$$726$$ −1.00000 −0.0371135
$$727$$ −18.8328 −0.698470 −0.349235 0.937035i $$-0.613559\pi$$
−0.349235 + 0.937035i $$0.613559\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 1.00000 0.0370370
$$730$$ −1.23607 −0.0457489
$$731$$ 3.23607 0.119690
$$732$$ −6.00000 −0.221766
$$733$$ 3.34752 0.123644 0.0618218 0.998087i $$-0.480309\pi$$
0.0618218 + 0.998087i $$0.480309\pi$$
$$734$$ 28.8328 1.06424
$$735$$ 3.47214 0.128072
$$736$$ −5.23607 −0.193004
$$737$$ 6.00000 0.221013
$$738$$ −8.94427 −0.329243
$$739$$ 28.2918 1.04073 0.520365 0.853944i $$-0.325796\pi$$
0.520365 + 0.853944i $$0.325796\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 9.52786 0.350015
$$742$$ 10.4721 0.384444
$$743$$ −22.4721 −0.824423 −0.412211 0.911088i $$-0.635243\pi$$
−0.412211 + 0.911088i $$0.635243\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 9.52786 0.349074
$$746$$ −15.7082 −0.575118
$$747$$ 4.94427 0.180901
$$748$$ −1.00000 −0.0365636
$$749$$ 3.05573 0.111654
$$750$$ 1.00000 0.0365148
$$751$$ 7.41641 0.270629 0.135314 0.990803i $$-0.456796\pi$$
0.135314 + 0.990803i $$0.456796\pi$$
$$752$$ 2.47214 0.0901495
$$753$$ −23.2361 −0.846769
$$754$$ −10.4721 −0.381373
$$755$$ 14.4721 0.526695
$$756$$ −3.23607 −0.117695
$$757$$ 21.8885 0.795553 0.397776 0.917482i $$-0.369782\pi$$
0.397776 + 0.917482i $$0.369782\pi$$
$$758$$ −24.9443 −0.906017
$$759$$ 5.23607 0.190057
$$760$$ 7.70820 0.279606
$$761$$ −36.4721 −1.32211 −0.661057 0.750336i $$-0.729892\pi$$
−0.661057 + 0.750336i $$0.729892\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −19.4164 −0.702921
$$764$$ 1.52786 0.0552762
$$765$$ 1.00000 0.0361551
$$766$$ 10.4721 0.378374
$$767$$ −4.00000 −0.144432
$$768$$ −1.00000 −0.0360844
$$769$$ 24.4721 0.882488 0.441244 0.897387i $$-0.354537\pi$$
0.441244 + 0.897387i $$0.354537\pi$$
$$770$$ −3.23607 −0.116620
$$771$$ 4.47214 0.161060
$$772$$ 7.70820 0.277424
$$773$$ −31.2361 −1.12348 −0.561742 0.827313i $$-0.689868\pi$$
−0.561742 + 0.827313i $$0.689868\pi$$
$$774$$ −3.23607 −0.116318
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 19.4164 0.696560
$$778$$ 25.5967 0.917688
$$779$$ 68.9443 2.47018
$$780$$ 1.23607 0.0442583
$$781$$ −0.763932 −0.0273356
$$782$$ 5.23607 0.187241
$$783$$ 8.47214 0.302769
$$784$$ 3.47214 0.124005
$$785$$ −19.4164 −0.693001
$$786$$ 0 0
$$787$$ 19.0557 0.679263 0.339632 0.940559i $$-0.389698\pi$$
0.339632 + 0.940559i $$0.389698\pi$$
$$788$$ 14.9443 0.532368
$$789$$ 12.0000 0.427211
$$790$$ −3.70820 −0.131932
$$791$$ −49.3050 −1.75308
$$792$$ 1.00000 0.0355335
$$793$$ 7.41641 0.263364
$$794$$ −19.8885 −0.705818
$$795$$ 3.23607 0.114772
$$796$$ −12.0000 −0.425329
$$797$$ 35.0132 1.24023 0.620115 0.784511i $$-0.287086\pi$$
0.620115 + 0.784511i $$0.287086\pi$$
$$798$$ 24.9443 0.883018
$$799$$ −2.47214 −0.0874579
$$800$$ 1.00000 0.0353553
$$801$$ −6.94427 −0.245364
$$802$$ −20.2918 −0.716528
$$803$$ 1.23607 0.0436199
$$804$$ −6.00000 −0.211604
$$805$$ 16.9443 0.597207
$$806$$ −4.94427 −0.174155
$$807$$ 24.4721 0.861460
$$808$$ 8.94427 0.314658
$$809$$ −16.3607 −0.575211 −0.287605 0.957749i $$-0.592859\pi$$
−0.287605 + 0.957749i $$0.592859\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −30.2492 −1.06219 −0.531097 0.847311i $$-0.678220\pi$$
−0.531097 + 0.847311i $$0.678220\pi$$
$$812$$ −27.4164 −0.962127
$$813$$ −20.0000 −0.701431
$$814$$ −6.00000 −0.210300
$$815$$ 6.47214 0.226709
$$816$$ 1.00000 0.0350070
$$817$$ 24.9443 0.872690
$$818$$ 10.9443 0.382657
$$819$$ 4.00000 0.139771
$$820$$ 8.94427 0.312348
$$821$$ −10.3607 −0.361590 −0.180795 0.983521i $$-0.557867\pi$$
−0.180795 + 0.983521i $$0.557867\pi$$
$$822$$ 15.8885 0.554177
$$823$$ 37.4164 1.30425 0.652127 0.758110i $$-0.273877\pi$$
0.652127 + 0.758110i $$0.273877\pi$$
$$824$$ 4.00000 0.139347
$$825$$ −1.00000 −0.0348155
$$826$$ −10.4721 −0.364372
$$827$$ −32.9443 −1.14558 −0.572792 0.819701i $$-0.694140\pi$$
−0.572792 + 0.819701i $$0.694140\pi$$
$$828$$ −5.23607 −0.181966
$$829$$ −49.7771 −1.72883 −0.864415 0.502779i $$-0.832311\pi$$
−0.864415 + 0.502779i $$0.832311\pi$$
$$830$$ −4.94427 −0.171618
$$831$$ −14.0000 −0.485655
$$832$$ 1.23607 0.0428529
$$833$$ −3.47214 −0.120302
$$834$$ −8.00000 −0.277017
$$835$$ 12.0000 0.415277
$$836$$ −7.70820 −0.266594
$$837$$ 4.00000 0.138260
$$838$$ 10.4721 0.361754
$$839$$ 14.2918 0.493408 0.246704 0.969091i $$-0.420653\pi$$
0.246704 + 0.969091i $$0.420653\pi$$
$$840$$ 3.23607 0.111655
$$841$$ 42.7771 1.47507
$$842$$ −6.94427 −0.239315
$$843$$ 16.4721 0.567330
$$844$$ 20.3607 0.700844
$$845$$ 11.4721 0.394653
$$846$$ 2.47214 0.0849938
$$847$$ 3.23607 0.111193
$$848$$ 3.23607 0.111127
$$849$$ 3.41641 0.117251
$$850$$ −1.00000 −0.0342997
$$851$$ 31.4164 1.07694
$$852$$ 0.763932 0.0261719
$$853$$ −14.3607 −0.491700 −0.245850 0.969308i $$-0.579067\pi$$
−0.245850 + 0.969308i $$0.579067\pi$$
$$854$$ 19.4164 0.664416
$$855$$ 7.70820 0.263615
$$856$$ 0.944272 0.0322745
$$857$$ −19.8885 −0.679380 −0.339690 0.940538i $$-0.610322\pi$$
−0.339690 + 0.940538i $$0.610322\pi$$
$$858$$ −1.23607 −0.0421987
$$859$$ 12.9443 0.441653 0.220826 0.975313i $$-0.429125\pi$$
0.220826 + 0.975313i $$0.429125\pi$$
$$860$$ 3.23607 0.110349
$$861$$ 28.9443 0.986418
$$862$$ −5.05573 −0.172199
$$863$$ −52.9443 −1.80224 −0.901122 0.433566i $$-0.857255\pi$$
−0.901122 + 0.433566i $$0.857255\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −13.4164 −0.456172
$$866$$ −6.00000 −0.203888
$$867$$ −1.00000 −0.0339618
$$868$$ −12.9443 −0.439357
$$869$$ 3.70820 0.125792
$$870$$ −8.47214 −0.287232
$$871$$ 7.41641 0.251295
$$872$$ −6.00000 −0.203186
$$873$$ 0 0
$$874$$ 40.3607 1.36522
$$875$$ −3.23607 −0.109399
$$876$$ −1.23607 −0.0417629
$$877$$ 42.3607 1.43042 0.715209 0.698910i $$-0.246331\pi$$
0.715209 + 0.698910i $$0.246331\pi$$
$$878$$ −4.29180 −0.144841
$$879$$ 18.0000 0.607125
$$880$$ −1.00000 −0.0337100
$$881$$ 14.1803 0.477748 0.238874 0.971051i $$-0.423222\pi$$
0.238874 + 0.971051i $$0.423222\pi$$
$$882$$ 3.47214 0.116913
$$883$$ 33.4164 1.12455 0.562276 0.826950i $$-0.309926\pi$$
0.562276 + 0.826950i $$0.309926\pi$$
$$884$$ −1.23607 −0.0415735
$$885$$ −3.23607 −0.108779
$$886$$ −3.34752 −0.112462
$$887$$ −56.7214 −1.90452 −0.952258 0.305293i $$-0.901246\pi$$
−0.952258 + 0.305293i $$0.901246\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −38.8328 −1.30241
$$890$$ 6.94427 0.232773
$$891$$ 1.00000 0.0335013
$$892$$ −23.4164 −0.784039
$$893$$ −19.0557 −0.637676
$$894$$ 9.52786 0.318659
$$895$$ 12.7639 0.426651
$$896$$ 3.23607 0.108109
$$897$$ 6.47214 0.216098
$$898$$ 6.18034 0.206241
$$899$$ 33.8885 1.13025
$$900$$ 1.00000 0.0333333
$$901$$ −3.23607 −0.107809
$$902$$ −8.94427 −0.297812
$$903$$ 10.4721 0.348491
$$904$$ −15.2361 −0.506744
$$905$$ −8.47214 −0.281623
$$906$$ 14.4721 0.480805
$$907$$ −38.8328 −1.28942 −0.644711 0.764426i $$-0.723022\pi$$
−0.644711 + 0.764426i $$0.723022\pi$$
$$908$$ 0 0
$$909$$ 8.94427 0.296663
$$910$$ −4.00000 −0.132599
$$911$$ 31.2361 1.03490 0.517449 0.855714i $$-0.326882\pi$$
0.517449 + 0.855714i $$0.326882\pi$$
$$912$$ 7.70820 0.255244
$$913$$ 4.94427 0.163632
$$914$$ 18.0000 0.595387
$$915$$ 6.00000 0.198354
$$916$$ 29.4164 0.971945
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ −6.47214 −0.213496 −0.106748 0.994286i $$-0.534044\pi$$
−0.106748 + 0.994286i $$0.534044\pi$$
$$920$$ 5.23607 0.172628
$$921$$ −10.2918 −0.339126
$$922$$ 7.41641 0.244246
$$923$$ −0.944272 −0.0310811
$$924$$ −3.23607 −0.106459
$$925$$ −6.00000 −0.197279
$$926$$ 20.3607 0.669093
$$927$$ 4.00000 0.131377
$$928$$ −8.47214 −0.278111
$$929$$ 33.0132 1.08313 0.541563 0.840660i $$-0.317833\pi$$
0.541563 + 0.840660i $$0.317833\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ −26.7639 −0.877152
$$932$$ −7.52786 −0.246583
$$933$$ −11.2361 −0.367852
$$934$$ 14.7639 0.483091
$$935$$ 1.00000 0.0327035
$$936$$ 1.23607 0.0404021
$$937$$ −60.2492 −1.96826 −0.984128 0.177459i $$-0.943212\pi$$
−0.984128 + 0.177459i $$0.943212\pi$$
$$938$$ 19.4164 0.633968
$$939$$ 0 0
$$940$$ −2.47214 −0.0806322
$$941$$ 12.4721 0.406580 0.203290 0.979119i $$-0.434837\pi$$
0.203290 + 0.979119i $$0.434837\pi$$
$$942$$ −19.4164 −0.632621
$$943$$ 46.8328 1.52509
$$944$$ −3.23607 −0.105325
$$945$$ 3.23607 0.105269
$$946$$ −3.23607 −0.105214
$$947$$ 44.7214 1.45325 0.726624 0.687035i $$-0.241088\pi$$
0.726624 + 0.687035i $$0.241088\pi$$
$$948$$ −3.70820 −0.120437
$$949$$ 1.52786 0.0495966
$$950$$ −7.70820 −0.250087
$$951$$ 13.0557 0.423361
$$952$$ −3.23607 −0.104882
$$953$$ 43.3050 1.40278 0.701392 0.712775i $$-0.252562\pi$$
0.701392 + 0.712775i $$0.252562\pi$$
$$954$$ 3.23607 0.104772
$$955$$ −1.52786 −0.0494405
$$956$$ −16.3607 −0.529142
$$957$$ 8.47214 0.273865
$$958$$ −27.5279 −0.889385
$$959$$ −51.4164 −1.66032
$$960$$ 1.00000 0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ −7.41641 −0.239115
$$963$$ 0.944272 0.0304287
$$964$$ 4.18034 0.134640
$$965$$ −7.70820 −0.248136
$$966$$ 16.9443 0.545173
$$967$$ −7.41641 −0.238496 −0.119248 0.992865i $$-0.538048\pi$$
−0.119248 + 0.992865i $$0.538048\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −7.70820 −0.247623
$$970$$ 0 0
$$971$$ 48.7639 1.56491 0.782455 0.622708i $$-0.213967\pi$$
0.782455 + 0.622708i $$0.213967\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 25.8885 0.829949
$$974$$ −20.8328 −0.667526
$$975$$ −1.23607 −0.0395859
$$976$$ 6.00000 0.192055
$$977$$ −7.52786 −0.240838 −0.120419 0.992723i $$-0.538424\pi$$
−0.120419 + 0.992723i $$0.538424\pi$$
$$978$$ 6.47214 0.206956
$$979$$ −6.94427 −0.221940
$$980$$ −3.47214 −0.110913
$$981$$ −6.00000 −0.191565
$$982$$ 1.05573 0.0336896
$$983$$ 0.652476 0.0208107 0.0104054 0.999946i $$-0.496688\pi$$
0.0104054 + 0.999946i $$0.496688\pi$$
$$984$$ 8.94427 0.285133
$$985$$ −14.9443 −0.476164
$$986$$ 8.47214 0.269808
$$987$$ −8.00000 −0.254643
$$988$$ −9.52786 −0.303122
$$989$$ 16.9443 0.538797
$$990$$ −1.00000 −0.0317821
$$991$$ 34.2492 1.08796 0.543981 0.839097i $$-0.316916\pi$$
0.543981 + 0.839097i $$0.316916\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −33.8885 −1.07542
$$994$$ −2.47214 −0.0784114
$$995$$ 12.0000 0.380426
$$996$$ −4.94427 −0.156665
$$997$$ 5.63932 0.178599 0.0892995 0.996005i $$-0.471537\pi$$
0.0892995 + 0.996005i $$0.471537\pi$$
$$998$$ −22.8328 −0.722760
$$999$$ 6.00000 0.189832
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.bu.1.2 2

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