# Properties

 Label 3626.2.a.a.1.2 Level $3626$ Weight $2$ Character 3626.1 Self dual yes Analytic conductor $28.954$ 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] = [3626,2,Mod(1,3626)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3626, 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("3626.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.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: $$28.9537557729$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3626.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} -1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{10} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -0.394449 q^{15} +1.00000 q^{16} +6.00000 q^{17} +2.90833 q^{18} -2.00000 q^{19} -1.30278 q^{20} -1.30278 q^{22} -6.90833 q^{23} -0.302776 q^{24} -3.30278 q^{25} -2.30278 q^{26} -1.78890 q^{27} +6.90833 q^{29} +0.394449 q^{30} -3.30278 q^{31} -1.00000 q^{32} +0.394449 q^{33} -6.00000 q^{34} -2.90833 q^{36} +1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} +1.30278 q^{40} +0.908327 q^{41} -6.60555 q^{43} +1.30278 q^{44} +3.78890 q^{45} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +3.30278 q^{50} +1.81665 q^{51} +2.30278 q^{52} -6.00000 q^{53} +1.78890 q^{54} -1.69722 q^{55} -0.605551 q^{57} -6.90833 q^{58} -3.39445 q^{59} -0.394449 q^{60} +10.5139 q^{61} +3.30278 q^{62} +1.00000 q^{64} -3.00000 q^{65} -0.394449 q^{66} +14.5139 q^{67} +6.00000 q^{68} -2.09167 q^{69} +6.00000 q^{71} +2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -0.697224 q^{78} -16.1194 q^{79} -1.30278 q^{80} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -7.81665 q^{85} +6.60555 q^{86} +2.09167 q^{87} -1.30278 q^{88} -5.21110 q^{89} -3.78890 q^{90} -6.90833 q^{92} -1.00000 q^{93} -2.60555 q^{94} +2.60555 q^{95} -0.302776 q^{96} -12.4222 q^{97} -3.78890 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9} - q^{10} - q^{11} - 3 q^{12} + q^{13} - 8 q^{15} + 2 q^{16} + 12 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + q^{22} - 3 q^{23} + 3 q^{24} - 3 q^{25} - q^{26} - 18 q^{27} + 3 q^{29} + 8 q^{30} - 3 q^{31} - 2 q^{32} + 8 q^{33} - 12 q^{34} + 5 q^{36} + 2 q^{37} + 4 q^{38} + 5 q^{39} - q^{40} - 9 q^{41} - 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 3 q^{50} - 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 6 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 3 q^{61} + 3 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} + 12 q^{71} - 5 q^{72} + 21 q^{73} - 2 q^{74} - 2 q^{75} - 4 q^{76} - 5 q^{78} - 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} - 20 q^{83} + 6 q^{85} + 6 q^{86} + 15 q^{87} + q^{88} + 4 q^{89} - 22 q^{90} - 3 q^{92} - 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} - 22 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 - 2 * q^8 + 5 * q^9 - q^10 - q^11 - 3 * q^12 + q^13 - 8 * q^15 + 2 * q^16 + 12 * q^17 - 5 * q^18 - 4 * q^19 + q^20 + q^22 - 3 * q^23 + 3 * q^24 - 3 * q^25 - q^26 - 18 * q^27 + 3 * q^29 + 8 * q^30 - 3 * q^31 - 2 * q^32 + 8 * q^33 - 12 * q^34 + 5 * q^36 + 2 * q^37 + 4 * q^38 + 5 * q^39 - q^40 - 9 * q^41 - 6 * q^43 - q^44 + 22 * q^45 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 3 * q^50 - 18 * q^51 + q^52 - 12 * q^53 + 18 * q^54 - 7 * q^55 + 6 * q^57 - 3 * q^58 - 14 * q^59 - 8 * q^60 + 3 * q^61 + 3 * q^62 + 2 * q^64 - 6 * q^65 - 8 * q^66 + 11 * q^67 + 12 * q^68 - 15 * q^69 + 12 * q^71 - 5 * q^72 + 21 * q^73 - 2 * q^74 - 2 * q^75 - 4 * q^76 - 5 * q^78 - 7 * q^79 + q^80 + 38 * q^81 + 9 * q^82 - 20 * q^83 + 6 * q^85 + 6 * q^86 + 15 * q^87 + q^88 + 4 * q^89 - 22 * q^90 - 3 * q^92 - 2 * q^93 + 2 * q^94 - 2 * q^95 + 3 * q^96 + 4 * q^97 - 22 * 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
$$3$$ 0.302776 0.174808 0.0874038 0.996173i $$-0.472143\pi$$
0.0874038 + 0.996173i $$0.472143\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.30278 −0.582619 −0.291309 0.956629i $$-0.594091\pi$$
−0.291309 + 0.956629i $$0.594091\pi$$
$$6$$ −0.302776 −0.123608
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ −2.90833 −0.969442
$$10$$ 1.30278 0.411974
$$11$$ 1.30278 0.392802 0.196401 0.980524i $$-0.437075\pi$$
0.196401 + 0.980524i $$0.437075\pi$$
$$12$$ 0.302776 0.0874038
$$13$$ 2.30278 0.638675 0.319338 0.947641i $$-0.396540\pi$$
0.319338 + 0.947641i $$0.396540\pi$$
$$14$$ 0 0
$$15$$ −0.394449 −0.101846
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 2.90833 0.685499
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −1.30278 −0.291309
$$21$$ 0 0
$$22$$ −1.30278 −0.277753
$$23$$ −6.90833 −1.44049 −0.720243 0.693722i $$-0.755970\pi$$
−0.720243 + 0.693722i $$0.755970\pi$$
$$24$$ −0.302776 −0.0618038
$$25$$ −3.30278 −0.660555
$$26$$ −2.30278 −0.451611
$$27$$ −1.78890 −0.344273
$$28$$ 0 0
$$29$$ 6.90833 1.28284 0.641422 0.767188i $$-0.278345\pi$$
0.641422 + 0.767188i $$0.278345\pi$$
$$30$$ 0.394449 0.0720162
$$31$$ −3.30278 −0.593196 −0.296598 0.955002i $$-0.595852\pi$$
−0.296598 + 0.955002i $$0.595852\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0.394449 0.0686647
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −2.90833 −0.484721
$$37$$ 1.00000 0.164399
$$38$$ 2.00000 0.324443
$$39$$ 0.697224 0.111645
$$40$$ 1.30278 0.205987
$$41$$ 0.908327 0.141857 0.0709284 0.997481i $$-0.477404\pi$$
0.0709284 + 0.997481i $$0.477404\pi$$
$$42$$ 0 0
$$43$$ −6.60555 −1.00734 −0.503669 0.863897i $$-0.668017\pi$$
−0.503669 + 0.863897i $$0.668017\pi$$
$$44$$ 1.30278 0.196401
$$45$$ 3.78890 0.564815
$$46$$ 6.90833 1.01858
$$47$$ 2.60555 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$48$$ 0.302776 0.0437019
$$49$$ 0 0
$$50$$ 3.30278 0.467083
$$51$$ 1.81665 0.254382
$$52$$ 2.30278 0.319338
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.78890 0.243438
$$55$$ −1.69722 −0.228854
$$56$$ 0 0
$$57$$ −0.605551 −0.0802072
$$58$$ −6.90833 −0.907108
$$59$$ −3.39445 −0.441920 −0.220960 0.975283i $$-0.570919\pi$$
−0.220960 + 0.975283i $$0.570919\pi$$
$$60$$ −0.394449 −0.0509231
$$61$$ 10.5139 1.34616 0.673082 0.739568i $$-0.264970\pi$$
0.673082 + 0.739568i $$0.264970\pi$$
$$62$$ 3.30278 0.419453
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ −0.394449 −0.0485533
$$67$$ 14.5139 1.77315 0.886576 0.462583i $$-0.153077\pi$$
0.886576 + 0.462583i $$0.153077\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −2.09167 −0.251808
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 2.90833 0.342750
$$73$$ 8.69722 1.01793 0.508967 0.860786i $$-0.330028\pi$$
0.508967 + 0.860786i $$0.330028\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ −1.00000 −0.115470
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ −0.697224 −0.0789451
$$79$$ −16.1194 −1.81358 −0.906789 0.421585i $$-0.861474\pi$$
−0.906789 + 0.421585i $$0.861474\pi$$
$$80$$ −1.30278 −0.145655
$$81$$ 8.18335 0.909261
$$82$$ −0.908327 −0.100308
$$83$$ −17.2111 −1.88916 −0.944582 0.328276i $$-0.893533\pi$$
−0.944582 + 0.328276i $$0.893533\pi$$
$$84$$ 0 0
$$85$$ −7.81665 −0.847835
$$86$$ 6.60555 0.712295
$$87$$ 2.09167 0.224251
$$88$$ −1.30278 −0.138876
$$89$$ −5.21110 −0.552376 −0.276188 0.961104i $$-0.589071\pi$$
−0.276188 + 0.961104i $$0.589071\pi$$
$$90$$ −3.78890 −0.399385
$$91$$ 0 0
$$92$$ −6.90833 −0.720243
$$93$$ −1.00000 −0.103695
$$94$$ −2.60555 −0.268742
$$95$$ 2.60555 0.267324
$$96$$ −0.302776 −0.0309019
$$97$$ −12.4222 −1.26128 −0.630642 0.776074i $$-0.717208\pi$$
−0.630642 + 0.776074i $$0.717208\pi$$
$$98$$ 0 0
$$99$$ −3.78890 −0.380799
$$100$$ −3.30278 −0.330278
$$101$$ −16.4222 −1.63407 −0.817035 0.576588i $$-0.804384\pi$$
−0.817035 + 0.576588i $$0.804384\pi$$
$$102$$ −1.81665 −0.179876
$$103$$ −3.30278 −0.325432 −0.162716 0.986673i $$-0.552025\pi$$
−0.162716 + 0.986673i $$0.552025\pi$$
$$104$$ −2.30278 −0.225806
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 4.30278 0.415965 0.207983 0.978133i $$-0.433310\pi$$
0.207983 + 0.978133i $$0.433310\pi$$
$$108$$ −1.78890 −0.172137
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 1.69722 0.161824
$$111$$ 0.302776 0.0287382
$$112$$ 0 0
$$113$$ 11.2111 1.05465 0.527326 0.849663i $$-0.323195\pi$$
0.527326 + 0.849663i $$0.323195\pi$$
$$114$$ 0.605551 0.0567151
$$115$$ 9.00000 0.839254
$$116$$ 6.90833 0.641422
$$117$$ −6.69722 −0.619159
$$118$$ 3.39445 0.312484
$$119$$ 0 0
$$120$$ 0.394449 0.0360081
$$121$$ −9.30278 −0.845707
$$122$$ −10.5139 −0.951882
$$123$$ 0.275019 0.0247977
$$124$$ −3.30278 −0.296598
$$125$$ 10.8167 0.967471
$$126$$ 0 0
$$127$$ −4.78890 −0.424946 −0.212473 0.977167i $$-0.568152\pi$$
−0.212473 + 0.977167i $$0.568152\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ 3.00000 0.263117
$$131$$ −3.39445 −0.296574 −0.148287 0.988944i $$-0.547376\pi$$
−0.148287 + 0.988944i $$0.547376\pi$$
$$132$$ 0.394449 0.0343324
$$133$$ 0 0
$$134$$ −14.5139 −1.25381
$$135$$ 2.33053 0.200580
$$136$$ −6.00000 −0.514496
$$137$$ −9.90833 −0.846525 −0.423263 0.906007i $$-0.639115\pi$$
−0.423263 + 0.906007i $$0.639115\pi$$
$$138$$ 2.09167 0.178055
$$139$$ −8.90833 −0.755594 −0.377797 0.925888i $$-0.623318\pi$$
−0.377797 + 0.925888i $$0.623318\pi$$
$$140$$ 0 0
$$141$$ 0.788897 0.0664372
$$142$$ −6.00000 −0.503509
$$143$$ 3.00000 0.250873
$$144$$ −2.90833 −0.242361
$$145$$ −9.00000 −0.747409
$$146$$ −8.69722 −0.719787
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ −1.81665 −0.148826 −0.0744130 0.997228i $$-0.523708\pi$$
−0.0744130 + 0.997228i $$0.523708\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −13.3944 −1.09002 −0.545012 0.838428i $$-0.683475\pi$$
−0.545012 + 0.838428i $$0.683475\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −17.4500 −1.41075
$$154$$ 0 0
$$155$$ 4.30278 0.345607
$$156$$ 0.697224 0.0558226
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ 16.1194 1.28239
$$159$$ −1.81665 −0.144070
$$160$$ 1.30278 0.102993
$$161$$ 0 0
$$162$$ −8.18335 −0.642944
$$163$$ −20.4222 −1.59959 −0.799795 0.600273i $$-0.795059\pi$$
−0.799795 + 0.600273i $$0.795059\pi$$
$$164$$ 0.908327 0.0709284
$$165$$ −0.513878 −0.0400054
$$166$$ 17.2111 1.33584
$$167$$ −12.5139 −0.968353 −0.484176 0.874970i $$-0.660881\pi$$
−0.484176 + 0.874970i $$0.660881\pi$$
$$168$$ 0 0
$$169$$ −7.69722 −0.592094
$$170$$ 7.81665 0.599510
$$171$$ 5.81665 0.444811
$$172$$ −6.60555 −0.503669
$$173$$ 23.2111 1.76471 0.882354 0.470587i $$-0.155958\pi$$
0.882354 + 0.470587i $$0.155958\pi$$
$$174$$ −2.09167 −0.158569
$$175$$ 0 0
$$176$$ 1.30278 0.0982004
$$177$$ −1.02776 −0.0772509
$$178$$ 5.21110 0.390589
$$179$$ 7.81665 0.584244 0.292122 0.956381i $$-0.405639\pi$$
0.292122 + 0.956381i $$0.405639\pi$$
$$180$$ 3.78890 0.282408
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 3.18335 0.235320
$$184$$ 6.90833 0.509289
$$185$$ −1.30278 −0.0957820
$$186$$ 1.00000 0.0733236
$$187$$ 7.81665 0.571610
$$188$$ 2.60555 0.190029
$$189$$ 0 0
$$190$$ −2.60555 −0.189027
$$191$$ 12.5139 0.905472 0.452736 0.891644i $$-0.350448\pi$$
0.452736 + 0.891644i $$0.350448\pi$$
$$192$$ 0.302776 0.0218509
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 12.4222 0.891862
$$195$$ −0.908327 −0.0650466
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 3.78890 0.269265
$$199$$ 2.42221 0.171706 0.0858528 0.996308i $$-0.472639\pi$$
0.0858528 + 0.996308i $$0.472639\pi$$
$$200$$ 3.30278 0.233542
$$201$$ 4.39445 0.309961
$$202$$ 16.4222 1.15546
$$203$$ 0 0
$$204$$ 1.81665 0.127191
$$205$$ −1.18335 −0.0826485
$$206$$ 3.30278 0.230115
$$207$$ 20.0917 1.39647
$$208$$ 2.30278 0.159669
$$209$$ −2.60555 −0.180230
$$210$$ 0 0
$$211$$ 6.69722 0.461056 0.230528 0.973066i $$-0.425955\pi$$
0.230528 + 0.973066i $$0.425955\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 1.81665 0.124475
$$214$$ −4.30278 −0.294132
$$215$$ 8.60555 0.586894
$$216$$ 1.78890 0.121719
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ 2.63331 0.177942
$$220$$ −1.69722 −0.114427
$$221$$ 13.8167 0.929409
$$222$$ −0.302776 −0.0203210
$$223$$ −15.8167 −1.05916 −0.529581 0.848260i $$-0.677651\pi$$
−0.529581 + 0.848260i $$0.677651\pi$$
$$224$$ 0 0
$$225$$ 9.60555 0.640370
$$226$$ −11.2111 −0.745751
$$227$$ 7.81665 0.518810 0.259405 0.965769i $$-0.416474\pi$$
0.259405 + 0.965769i $$0.416474\pi$$
$$228$$ −0.605551 −0.0401036
$$229$$ −17.3944 −1.14946 −0.574729 0.818344i $$-0.694892\pi$$
−0.574729 + 0.818344i $$0.694892\pi$$
$$230$$ −9.00000 −0.593442
$$231$$ 0 0
$$232$$ −6.90833 −0.453554
$$233$$ −9.51388 −0.623275 −0.311637 0.950201i $$-0.600877\pi$$
−0.311637 + 0.950201i $$0.600877\pi$$
$$234$$ 6.69722 0.437811
$$235$$ −3.39445 −0.221429
$$236$$ −3.39445 −0.220960
$$237$$ −4.88057 −0.317027
$$238$$ 0 0
$$239$$ 0.513878 0.0332400 0.0166200 0.999862i $$-0.494709\pi$$
0.0166200 + 0.999862i $$0.494709\pi$$
$$240$$ −0.394449 −0.0254616
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 9.30278 0.598005
$$243$$ 7.84441 0.503219
$$244$$ 10.5139 0.673082
$$245$$ 0 0
$$246$$ −0.275019 −0.0175346
$$247$$ −4.60555 −0.293044
$$248$$ 3.30278 0.209726
$$249$$ −5.21110 −0.330240
$$250$$ −10.8167 −0.684105
$$251$$ 6.78890 0.428511 0.214256 0.976778i $$-0.431267\pi$$
0.214256 + 0.976778i $$0.431267\pi$$
$$252$$ 0 0
$$253$$ −9.00000 −0.565825
$$254$$ 4.78890 0.300482
$$255$$ −2.36669 −0.148208
$$256$$ 1.00000 0.0625000
$$257$$ 11.2111 0.699329 0.349665 0.936875i $$-0.386296\pi$$
0.349665 + 0.936875i $$0.386296\pi$$
$$258$$ 2.00000 0.124515
$$259$$ 0 0
$$260$$ −3.00000 −0.186052
$$261$$ −20.0917 −1.24364
$$262$$ 3.39445 0.209710
$$263$$ −7.81665 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$264$$ −0.394449 −0.0242766
$$265$$ 7.81665 0.480173
$$266$$ 0 0
$$267$$ −1.57779 −0.0965595
$$268$$ 14.5139 0.886576
$$269$$ 6.78890 0.413926 0.206963 0.978349i $$-0.433642\pi$$
0.206963 + 0.978349i $$0.433642\pi$$
$$270$$ −2.33053 −0.141832
$$271$$ −6.42221 −0.390121 −0.195061 0.980791i $$-0.562490\pi$$
−0.195061 + 0.980791i $$0.562490\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 9.90833 0.598584
$$275$$ −4.30278 −0.259467
$$276$$ −2.09167 −0.125904
$$277$$ −25.1194 −1.50928 −0.754640 0.656139i $$-0.772189\pi$$
−0.754640 + 0.656139i $$0.772189\pi$$
$$278$$ 8.90833 0.534286
$$279$$ 9.60555 0.575069
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ −0.788897 −0.0469782
$$283$$ −17.3944 −1.03399 −0.516996 0.855988i $$-0.672950\pi$$
−0.516996 + 0.855988i $$0.672950\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0.788897 0.0467303
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ 2.90833 0.171375
$$289$$ 19.0000 1.11765
$$290$$ 9.00000 0.528498
$$291$$ −3.76114 −0.220482
$$292$$ 8.69722 0.508967
$$293$$ 25.0278 1.46214 0.731069 0.682304i $$-0.239022\pi$$
0.731069 + 0.682304i $$0.239022\pi$$
$$294$$ 0 0
$$295$$ 4.42221 0.257471
$$296$$ −1.00000 −0.0581238
$$297$$ −2.33053 −0.135231
$$298$$ 1.81665 0.105236
$$299$$ −15.9083 −0.920002
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 13.3944 0.770764
$$303$$ −4.97224 −0.285648
$$304$$ −2.00000 −0.114708
$$305$$ −13.6972 −0.784301
$$306$$ 17.4500 0.997548
$$307$$ −7.09167 −0.404743 −0.202372 0.979309i $$-0.564865\pi$$
−0.202372 + 0.979309i $$0.564865\pi$$
$$308$$ 0 0
$$309$$ −1.00000 −0.0568880
$$310$$ −4.30278 −0.244381
$$311$$ −5.09167 −0.288722 −0.144361 0.989525i $$-0.546113\pi$$
−0.144361 + 0.989525i $$0.546113\pi$$
$$312$$ −0.697224 −0.0394726
$$313$$ −27.0278 −1.52770 −0.763850 0.645394i $$-0.776693\pi$$
−0.763850 + 0.645394i $$0.776693\pi$$
$$314$$ 7.21110 0.406946
$$315$$ 0 0
$$316$$ −16.1194 −0.906789
$$317$$ −5.21110 −0.292685 −0.146342 0.989234i $$-0.546750\pi$$
−0.146342 + 0.989234i $$0.546750\pi$$
$$318$$ 1.81665 0.101873
$$319$$ 9.00000 0.503903
$$320$$ −1.30278 −0.0728274
$$321$$ 1.30278 0.0727138
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 8.18335 0.454630
$$325$$ −7.60555 −0.421880
$$326$$ 20.4222 1.13108
$$327$$ 0.605551 0.0334871
$$328$$ −0.908327 −0.0501540
$$329$$ 0 0
$$330$$ 0.513878 0.0282881
$$331$$ 1.21110 0.0665682 0.0332841 0.999446i $$-0.489403\pi$$
0.0332841 + 0.999446i $$0.489403\pi$$
$$332$$ −17.2111 −0.944582
$$333$$ −2.90833 −0.159375
$$334$$ 12.5139 0.684729
$$335$$ −18.9083 −1.03307
$$336$$ 0 0
$$337$$ −19.1194 −1.04150 −0.520751 0.853709i $$-0.674348\pi$$
−0.520751 + 0.853709i $$0.674348\pi$$
$$338$$ 7.69722 0.418674
$$339$$ 3.39445 0.184361
$$340$$ −7.81665 −0.423918
$$341$$ −4.30278 −0.233008
$$342$$ −5.81665 −0.314529
$$343$$ 0 0
$$344$$ 6.60555 0.356147
$$345$$ 2.72498 0.146708
$$346$$ −23.2111 −1.24784
$$347$$ 31.8167 1.70801 0.854004 0.520267i $$-0.174168\pi$$
0.854004 + 0.520267i $$0.174168\pi$$
$$348$$ 2.09167 0.112125
$$349$$ 22.2389 1.19042 0.595209 0.803571i $$-0.297069\pi$$
0.595209 + 0.803571i $$0.297069\pi$$
$$350$$ 0 0
$$351$$ −4.11943 −0.219879
$$352$$ −1.30278 −0.0694382
$$353$$ −31.8167 −1.69343 −0.846715 0.532047i $$-0.821423\pi$$
−0.846715 + 0.532047i $$0.821423\pi$$
$$354$$ 1.02776 0.0546246
$$355$$ −7.81665 −0.414865
$$356$$ −5.21110 −0.276188
$$357$$ 0 0
$$358$$ −7.81665 −0.413123
$$359$$ −11.2111 −0.591699 −0.295850 0.955235i $$-0.595603\pi$$
−0.295850 + 0.955235i $$0.595603\pi$$
$$360$$ −3.78890 −0.199692
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000 1.05118
$$363$$ −2.81665 −0.147836
$$364$$ 0 0
$$365$$ −11.3305 −0.593067
$$366$$ −3.18335 −0.166396
$$367$$ 17.8167 0.930022 0.465011 0.885305i $$-0.346050\pi$$
0.465011 + 0.885305i $$0.346050\pi$$
$$368$$ −6.90833 −0.360121
$$369$$ −2.64171 −0.137522
$$370$$ 1.30278 0.0677281
$$371$$ 0 0
$$372$$ −1.00000 −0.0518476
$$373$$ 3.81665 0.197619 0.0988094 0.995106i $$-0.468497\pi$$
0.0988094 + 0.995106i $$0.468497\pi$$
$$374$$ −7.81665 −0.404190
$$375$$ 3.27502 0.169121
$$376$$ −2.60555 −0.134371
$$377$$ 15.9083 0.819321
$$378$$ 0 0
$$379$$ −15.3305 −0.787477 −0.393738 0.919223i $$-0.628818\pi$$
−0.393738 + 0.919223i $$0.628818\pi$$
$$380$$ 2.60555 0.133662
$$381$$ −1.44996 −0.0742838
$$382$$ −12.5139 −0.640266
$$383$$ −20.8444 −1.06510 −0.532550 0.846399i $$-0.678766\pi$$
−0.532550 + 0.846399i $$0.678766\pi$$
$$384$$ −0.302776 −0.0154510
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 19.2111 0.976555
$$388$$ −12.4222 −0.630642
$$389$$ −11.8806 −0.602369 −0.301184 0.953566i $$-0.597382\pi$$
−0.301184 + 0.953566i $$0.597382\pi$$
$$390$$ 0.908327 0.0459949
$$391$$ −41.4500 −2.09621
$$392$$ 0 0
$$393$$ −1.02776 −0.0518435
$$394$$ 6.00000 0.302276
$$395$$ 21.0000 1.05662
$$396$$ −3.78890 −0.190399
$$397$$ −27.8167 −1.39608 −0.698039 0.716060i $$-0.745944\pi$$
−0.698039 + 0.716060i $$0.745944\pi$$
$$398$$ −2.42221 −0.121414
$$399$$ 0 0
$$400$$ −3.30278 −0.165139
$$401$$ 13.8167 0.689971 0.344985 0.938608i $$-0.387884\pi$$
0.344985 + 0.938608i $$0.387884\pi$$
$$402$$ −4.39445 −0.219175
$$403$$ −7.60555 −0.378859
$$404$$ −16.4222 −0.817035
$$405$$ −10.6611 −0.529753
$$406$$ 0 0
$$407$$ 1.30278 0.0645762
$$408$$ −1.81665 −0.0899378
$$409$$ 5.02776 0.248607 0.124303 0.992244i $$-0.460330\pi$$
0.124303 + 0.992244i $$0.460330\pi$$
$$410$$ 1.18335 0.0584413
$$411$$ −3.00000 −0.147979
$$412$$ −3.30278 −0.162716
$$413$$ 0 0
$$414$$ −20.0917 −0.987452
$$415$$ 22.4222 1.10066
$$416$$ −2.30278 −0.112903
$$417$$ −2.69722 −0.132084
$$418$$ 2.60555 0.127442
$$419$$ 25.1472 1.22852 0.614260 0.789104i $$-0.289455\pi$$
0.614260 + 0.789104i $$0.289455\pi$$
$$420$$ 0 0
$$421$$ 28.7250 1.39997 0.699985 0.714158i $$-0.253190\pi$$
0.699985 + 0.714158i $$0.253190\pi$$
$$422$$ −6.69722 −0.326016
$$423$$ −7.57779 −0.368445
$$424$$ 6.00000 0.291386
$$425$$ −19.8167 −0.961249
$$426$$ −1.81665 −0.0880172
$$427$$ 0 0
$$428$$ 4.30278 0.207983
$$429$$ 0.908327 0.0438544
$$430$$ −8.60555 −0.414997
$$431$$ −5.21110 −0.251010 −0.125505 0.992093i $$-0.540055\pi$$
−0.125505 + 0.992093i $$0.540055\pi$$
$$432$$ −1.78890 −0.0860684
$$433$$ 11.9361 0.573612 0.286806 0.957989i $$-0.407407\pi$$
0.286806 + 0.957989i $$0.407407\pi$$
$$434$$ 0 0
$$435$$ −2.72498 −0.130653
$$436$$ 2.00000 0.0957826
$$437$$ 13.8167 0.660940
$$438$$ −2.63331 −0.125824
$$439$$ 9.33053 0.445322 0.222661 0.974896i $$-0.428526\pi$$
0.222661 + 0.974896i $$0.428526\pi$$
$$440$$ 1.69722 0.0809120
$$441$$ 0 0
$$442$$ −13.8167 −0.657191
$$443$$ 0.275019 0.0130666 0.00653328 0.999979i $$-0.497920\pi$$
0.00653328 + 0.999979i $$0.497920\pi$$
$$444$$ 0.302776 0.0143691
$$445$$ 6.78890 0.321825
$$446$$ 15.8167 0.748940
$$447$$ −0.550039 −0.0260159
$$448$$ 0 0
$$449$$ −0.788897 −0.0372304 −0.0186152 0.999827i $$-0.505926\pi$$
−0.0186152 + 0.999827i $$0.505926\pi$$
$$450$$ −9.60555 −0.452810
$$451$$ 1.18335 0.0557216
$$452$$ 11.2111 0.527326
$$453$$ −4.05551 −0.190545
$$454$$ −7.81665 −0.366854
$$455$$ 0 0
$$456$$ 0.605551 0.0283575
$$457$$ 4.60555 0.215439 0.107719 0.994181i $$-0.465645\pi$$
0.107719 + 0.994181i $$0.465645\pi$$
$$458$$ 17.3944 0.812789
$$459$$ −10.7334 −0.500991
$$460$$ 9.00000 0.419627
$$461$$ 16.4222 0.764858 0.382429 0.923985i $$-0.375088\pi$$
0.382429 + 0.923985i $$0.375088\pi$$
$$462$$ 0 0
$$463$$ 30.3028 1.40829 0.704145 0.710056i $$-0.251331\pi$$
0.704145 + 0.710056i $$0.251331\pi$$
$$464$$ 6.90833 0.320711
$$465$$ 1.30278 0.0604148
$$466$$ 9.51388 0.440722
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ −6.69722 −0.309579
$$469$$ 0 0
$$470$$ 3.39445 0.156574
$$471$$ −2.18335 −0.100603
$$472$$ 3.39445 0.156242
$$473$$ −8.60555 −0.395684
$$474$$ 4.88057 0.224172
$$475$$ 6.60555 0.303083
$$476$$ 0 0
$$477$$ 17.4500 0.798979
$$478$$ −0.513878 −0.0235042
$$479$$ −12.1194 −0.553751 −0.276875 0.960906i $$-0.589299\pi$$
−0.276875 + 0.960906i $$0.589299\pi$$
$$480$$ 0.394449 0.0180040
$$481$$ 2.30278 0.104998
$$482$$ 8.00000 0.364390
$$483$$ 0 0
$$484$$ −9.30278 −0.422853
$$485$$ 16.1833 0.734848
$$486$$ −7.84441 −0.355830
$$487$$ −22.7889 −1.03266 −0.516332 0.856389i $$-0.672703\pi$$
−0.516332 + 0.856389i $$0.672703\pi$$
$$488$$ −10.5139 −0.475941
$$489$$ −6.18335 −0.279621
$$490$$ 0 0
$$491$$ −14.7250 −0.664529 −0.332265 0.943186i $$-0.607813\pi$$
−0.332265 + 0.943186i $$0.607813\pi$$
$$492$$ 0.275019 0.0123988
$$493$$ 41.4500 1.86681
$$494$$ 4.60555 0.207214
$$495$$ 4.93608 0.221860
$$496$$ −3.30278 −0.148299
$$497$$ 0 0
$$498$$ 5.21110 0.233515
$$499$$ 8.23886 0.368822 0.184411 0.982849i $$-0.440962\pi$$
0.184411 + 0.982849i $$0.440962\pi$$
$$500$$ 10.8167 0.483735
$$501$$ −3.78890 −0.169275
$$502$$ −6.78890 −0.303003
$$503$$ 24.5139 1.09302 0.546510 0.837453i $$-0.315956\pi$$
0.546510 + 0.837453i $$0.315956\pi$$
$$504$$ 0 0
$$505$$ 21.3944 0.952040
$$506$$ 9.00000 0.400099
$$507$$ −2.33053 −0.103503
$$508$$ −4.78890 −0.212473
$$509$$ 25.8167 1.14430 0.572152 0.820148i $$-0.306109\pi$$
0.572152 + 0.820148i $$0.306109\pi$$
$$510$$ 2.36669 0.104799
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 3.57779 0.157964
$$514$$ −11.2111 −0.494501
$$515$$ 4.30278 0.189603
$$516$$ −2.00000 −0.0880451
$$517$$ 3.39445 0.149288
$$518$$ 0 0
$$519$$ 7.02776 0.308484
$$520$$ 3.00000 0.131559
$$521$$ 9.63331 0.422043 0.211021 0.977481i $$-0.432321\pi$$
0.211021 + 0.977481i $$0.432321\pi$$
$$522$$ 20.0917 0.879389
$$523$$ −32.2389 −1.40971 −0.704853 0.709353i $$-0.748987\pi$$
−0.704853 + 0.709353i $$0.748987\pi$$
$$524$$ −3.39445 −0.148287
$$525$$ 0 0
$$526$$ 7.81665 0.340822
$$527$$ −19.8167 −0.863227
$$528$$ 0.394449 0.0171662
$$529$$ 24.7250 1.07500
$$530$$ −7.81665 −0.339534
$$531$$ 9.87217 0.428416
$$532$$ 0 0
$$533$$ 2.09167 0.0906004
$$534$$ 1.57779 0.0682779
$$535$$ −5.60555 −0.242349
$$536$$ −14.5139 −0.626904
$$537$$ 2.36669 0.102130
$$538$$ −6.78890 −0.292690
$$539$$ 0 0
$$540$$ 2.33053 0.100290
$$541$$ −20.9361 −0.900113 −0.450056 0.893000i $$-0.648596\pi$$
−0.450056 + 0.893000i $$0.648596\pi$$
$$542$$ 6.42221 0.275857
$$543$$ −6.05551 −0.259867
$$544$$ −6.00000 −0.257248
$$545$$ −2.60555 −0.111610
$$546$$ 0 0
$$547$$ −13.3944 −0.572705 −0.286353 0.958124i $$-0.592443\pi$$
−0.286353 + 0.958124i $$0.592443\pi$$
$$548$$ −9.90833 −0.423263
$$549$$ −30.5778 −1.30503
$$550$$ 4.30278 0.183471
$$551$$ −13.8167 −0.588609
$$552$$ 2.09167 0.0890275
$$553$$ 0 0
$$554$$ 25.1194 1.06722
$$555$$ −0.394449 −0.0167434
$$556$$ −8.90833 −0.377797
$$557$$ −6.51388 −0.276002 −0.138001 0.990432i $$-0.544068\pi$$
−0.138001 + 0.990432i $$0.544068\pi$$
$$558$$ −9.60555 −0.406635
$$559$$ −15.2111 −0.643361
$$560$$ 0 0
$$561$$ 2.36669 0.0999218
$$562$$ 12.0000 0.506189
$$563$$ −44.0555 −1.85672 −0.928359 0.371684i $$-0.878780\pi$$
−0.928359 + 0.371684i $$0.878780\pi$$
$$564$$ 0.788897 0.0332186
$$565$$ −14.6056 −0.614460
$$566$$ 17.3944 0.731143
$$567$$ 0 0
$$568$$ −6.00000 −0.251754
$$569$$ −10.4222 −0.436922 −0.218461 0.975846i $$-0.570104\pi$$
−0.218461 + 0.975846i $$0.570104\pi$$
$$570$$ −0.788897 −0.0330433
$$571$$ −20.3028 −0.849645 −0.424822 0.905277i $$-0.639663\pi$$
−0.424822 + 0.905277i $$0.639663\pi$$
$$572$$ 3.00000 0.125436
$$573$$ 3.78890 0.158283
$$574$$ 0 0
$$575$$ 22.8167 0.951520
$$576$$ −2.90833 −0.121180
$$577$$ 28.2389 1.17560 0.587800 0.809007i $$-0.299994\pi$$
0.587800 + 0.809007i $$0.299994\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −1.21110 −0.0503317
$$580$$ −9.00000 −0.373705
$$581$$ 0 0
$$582$$ 3.76114 0.155904
$$583$$ −7.81665 −0.323733
$$584$$ −8.69722 −0.359894
$$585$$ 8.72498 0.360734
$$586$$ −25.0278 −1.03389
$$587$$ 2.36669 0.0976838 0.0488419 0.998807i $$-0.484447\pi$$
0.0488419 + 0.998807i $$0.484447\pi$$
$$588$$ 0 0
$$589$$ 6.60555 0.272177
$$590$$ −4.42221 −0.182059
$$591$$ −1.81665 −0.0747272
$$592$$ 1.00000 0.0410997
$$593$$ −36.5139 −1.49945 −0.749723 0.661752i $$-0.769813\pi$$
−0.749723 + 0.661752i $$0.769813\pi$$
$$594$$ 2.33053 0.0956229
$$595$$ 0 0
$$596$$ −1.81665 −0.0744130
$$597$$ 0.733385 0.0300154
$$598$$ 15.9083 0.650540
$$599$$ −35.2111 −1.43869 −0.719343 0.694655i $$-0.755557\pi$$
−0.719343 + 0.694655i $$0.755557\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 20.6972 0.844257 0.422129 0.906536i $$-0.361283\pi$$
0.422129 + 0.906536i $$0.361283\pi$$
$$602$$ 0 0
$$603$$ −42.2111 −1.71897
$$604$$ −13.3944 −0.545012
$$605$$ 12.1194 0.492725
$$606$$ 4.97224 0.201984
$$607$$ 31.5139 1.27911 0.639554 0.768746i $$-0.279119\pi$$
0.639554 + 0.768746i $$0.279119\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 13.6972 0.554584
$$611$$ 6.00000 0.242734
$$612$$ −17.4500 −0.705373
$$613$$ −8.18335 −0.330522 −0.165261 0.986250i $$-0.552847\pi$$
−0.165261 + 0.986250i $$0.552847\pi$$
$$614$$ 7.09167 0.286197
$$615$$ −0.358288 −0.0144476
$$616$$ 0 0
$$617$$ 47.5694 1.91507 0.957536 0.288314i $$-0.0930948\pi$$
0.957536 + 0.288314i $$0.0930948\pi$$
$$618$$ 1.00000 0.0402259
$$619$$ 2.69722 0.108411 0.0542053 0.998530i $$-0.482737\pi$$
0.0542053 + 0.998530i $$0.482737\pi$$
$$620$$ 4.30278 0.172804
$$621$$ 12.3583 0.495921
$$622$$ 5.09167 0.204157
$$623$$ 0 0
$$624$$ 0.697224 0.0279113
$$625$$ 2.42221 0.0968882
$$626$$ 27.0278 1.08025
$$627$$ −0.788897 −0.0315055
$$628$$ −7.21110 −0.287754
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 18.3028 0.728622 0.364311 0.931277i $$-0.381305\pi$$
0.364311 + 0.931277i $$0.381305\pi$$
$$632$$ 16.1194 0.641196
$$633$$ 2.02776 0.0805961
$$634$$ 5.21110 0.206959
$$635$$ 6.23886 0.247582
$$636$$ −1.81665 −0.0720350
$$637$$ 0 0
$$638$$ −9.00000 −0.356313
$$639$$ −17.4500 −0.690310
$$640$$ 1.30278 0.0514967
$$641$$ −2.48612 −0.0981959 −0.0490980 0.998794i $$-0.515635\pi$$
−0.0490980 + 0.998794i $$0.515635\pi$$
$$642$$ −1.30278 −0.0514165
$$643$$ 29.8167 1.17585 0.587927 0.808914i $$-0.299944\pi$$
0.587927 + 0.808914i $$0.299944\pi$$
$$644$$ 0 0
$$645$$ 2.60555 0.102593
$$646$$ 12.0000 0.472134
$$647$$ 25.9361 1.01965 0.509826 0.860277i $$-0.329710\pi$$
0.509826 + 0.860277i $$0.329710\pi$$
$$648$$ −8.18335 −0.321472
$$649$$ −4.42221 −0.173587
$$650$$ 7.60555 0.298314
$$651$$ 0 0
$$652$$ −20.4222 −0.799795
$$653$$ 6.90833 0.270344 0.135172 0.990822i $$-0.456841\pi$$
0.135172 + 0.990822i $$0.456841\pi$$
$$654$$ −0.605551 −0.0236789
$$655$$ 4.42221 0.172790
$$656$$ 0.908327 0.0354642
$$657$$ −25.2944 −0.986827
$$658$$ 0 0
$$659$$ −42.1194 −1.64074 −0.820370 0.571833i $$-0.806233\pi$$
−0.820370 + 0.571833i $$0.806233\pi$$
$$660$$ −0.513878 −0.0200027
$$661$$ 12.4861 0.485654 0.242827 0.970070i $$-0.421925\pi$$
0.242827 + 0.970070i $$0.421925\pi$$
$$662$$ −1.21110 −0.0470708
$$663$$ 4.18335 0.162468
$$664$$ 17.2111 0.667920
$$665$$ 0 0
$$666$$ 2.90833 0.112695
$$667$$ −47.7250 −1.84792
$$668$$ −12.5139 −0.484176
$$669$$ −4.78890 −0.185149
$$670$$ 18.9083 0.730492
$$671$$ 13.6972 0.528775
$$672$$ 0 0
$$673$$ 24.3028 0.936803 0.468402 0.883516i $$-0.344830\pi$$
0.468402 + 0.883516i $$0.344830\pi$$
$$674$$ 19.1194 0.736453
$$675$$ 5.90833 0.227412
$$676$$ −7.69722 −0.296047
$$677$$ 36.2389 1.39277 0.696386 0.717667i $$-0.254790\pi$$
0.696386 + 0.717667i $$0.254790\pi$$
$$678$$ −3.39445 −0.130363
$$679$$ 0 0
$$680$$ 7.81665 0.299755
$$681$$ 2.36669 0.0906918
$$682$$ 4.30278 0.164762
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 5.81665 0.222405
$$685$$ 12.9083 0.493202
$$686$$ 0 0
$$687$$ −5.26662 −0.200934
$$688$$ −6.60555 −0.251834
$$689$$ −13.8167 −0.526373
$$690$$ −2.72498 −0.103738
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 23.2111 0.882354
$$693$$ 0 0
$$694$$ −31.8167 −1.20774
$$695$$ 11.6056 0.440224
$$696$$ −2.09167 −0.0792847
$$697$$ 5.44996 0.206432
$$698$$ −22.2389 −0.841753
$$699$$ −2.88057 −0.108953
$$700$$ 0 0
$$701$$ 14.8806 0.562031 0.281016 0.959703i $$-0.409329\pi$$
0.281016 + 0.959703i $$0.409329\pi$$
$$702$$ 4.11943 0.155478
$$703$$ −2.00000 −0.0754314
$$704$$ 1.30278 0.0491002
$$705$$ −1.02776 −0.0387075
$$706$$ 31.8167 1.19744
$$707$$ 0 0
$$708$$ −1.02776 −0.0386254
$$709$$ −1.66947 −0.0626982 −0.0313491 0.999508i $$-0.509980\pi$$
−0.0313491 + 0.999508i $$0.509980\pi$$
$$710$$ 7.81665 0.293354
$$711$$ 46.8806 1.75816
$$712$$ 5.21110 0.195294
$$713$$ 22.8167 0.854490
$$714$$ 0 0
$$715$$ −3.90833 −0.146163
$$716$$ 7.81665 0.292122
$$717$$ 0.155590 0.00581061
$$718$$ 11.2111 0.418395
$$719$$ 8.36669 0.312025 0.156012 0.987755i $$-0.450136\pi$$
0.156012 + 0.987755i $$0.450136\pi$$
$$720$$ 3.78890 0.141204
$$721$$ 0 0
$$722$$ 15.0000 0.558242
$$723$$ −2.42221 −0.0900828
$$724$$ −20.0000 −0.743294
$$725$$ −22.8167 −0.847389
$$726$$ 2.81665 0.104536
$$727$$ −29.9083 −1.10924 −0.554619 0.832104i $$-0.687136\pi$$
−0.554619 + 0.832104i $$0.687136\pi$$
$$728$$ 0 0
$$729$$ −22.1749 −0.821294
$$730$$ 11.3305 0.419362
$$731$$ −39.6333 −1.46589
$$732$$ 3.18335 0.117660
$$733$$ −29.6333 −1.09453 −0.547266 0.836959i $$-0.684331\pi$$
−0.547266 + 0.836959i $$0.684331\pi$$
$$734$$ −17.8167 −0.657625
$$735$$ 0 0
$$736$$ 6.90833 0.254644
$$737$$ 18.9083 0.696497
$$738$$ 2.64171 0.0972427
$$739$$ −42.3305 −1.55715 −0.778577 0.627549i $$-0.784058\pi$$
−0.778577 + 0.627549i $$0.784058\pi$$
$$740$$ −1.30278 −0.0478910
$$741$$ −1.39445 −0.0512264
$$742$$ 0 0
$$743$$ 35.4500 1.30053 0.650266 0.759706i $$-0.274657\pi$$
0.650266 + 0.759706i $$0.274657\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 2.36669 0.0867089
$$746$$ −3.81665 −0.139738
$$747$$ 50.0555 1.83144
$$748$$ 7.81665 0.285805
$$749$$ 0 0
$$750$$ −3.27502 −0.119587
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 2.60555 0.0950147
$$753$$ 2.05551 0.0749070
$$754$$ −15.9083 −0.579347
$$755$$ 17.4500 0.635069
$$756$$ 0 0
$$757$$ 9.30278 0.338115 0.169058 0.985606i $$-0.445928\pi$$
0.169058 + 0.985606i $$0.445928\pi$$
$$758$$ 15.3305 0.556830
$$759$$ −2.72498 −0.0989105
$$760$$ −2.60555 −0.0945133
$$761$$ −42.1194 −1.52683 −0.763414 0.645909i $$-0.776478\pi$$
−0.763414 + 0.645909i $$0.776478\pi$$
$$762$$ 1.44996 0.0525266
$$763$$ 0 0
$$764$$ 12.5139 0.452736
$$765$$ 22.7334 0.821927
$$766$$ 20.8444 0.753139
$$767$$ −7.81665 −0.282243
$$768$$ 0.302776 0.0109255
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 3.39445 0.122248
$$772$$ −4.00000 −0.143963
$$773$$ 50.0555 1.80037 0.900186 0.435506i $$-0.143431\pi$$
0.900186 + 0.435506i $$0.143431\pi$$
$$774$$ −19.2111 −0.690529
$$775$$ 10.9083 0.391839
$$776$$ 12.4222 0.445931
$$777$$ 0 0
$$778$$ 11.8806 0.425939
$$779$$ −1.81665 −0.0650884
$$780$$ −0.908327 −0.0325233
$$781$$ 7.81665 0.279702
$$782$$ 41.4500 1.48225
$$783$$ −12.3583 −0.441649
$$784$$ 0 0
$$785$$ 9.39445 0.335302
$$786$$ 1.02776 0.0366589
$$787$$ −25.2111 −0.898679 −0.449339 0.893361i $$-0.648341\pi$$
−0.449339 + 0.893361i $$0.648341\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −2.36669 −0.0842565
$$790$$ −21.0000 −0.747146
$$791$$ 0 0
$$792$$ 3.78890 0.134633
$$793$$ 24.2111 0.859761
$$794$$ 27.8167 0.987176
$$795$$ 2.36669 0.0839379
$$796$$ 2.42221 0.0858528
$$797$$ 17.3305 0.613879 0.306939 0.951729i $$-0.400695\pi$$
0.306939 + 0.951729i $$0.400695\pi$$
$$798$$ 0 0
$$799$$ 15.6333 0.553067
$$800$$ 3.30278 0.116771
$$801$$ 15.1556 0.535496
$$802$$ −13.8167 −0.487883
$$803$$ 11.3305 0.399846
$$804$$ 4.39445 0.154980
$$805$$ 0 0
$$806$$ 7.60555 0.267894
$$807$$ 2.05551 0.0723575
$$808$$ 16.4222 0.577731
$$809$$ 29.4500 1.03541 0.517703 0.855561i $$-0.326787\pi$$
0.517703 + 0.855561i $$0.326787\pi$$
$$810$$ 10.6611 0.374592
$$811$$ −54.1472 −1.90136 −0.950682 0.310166i $$-0.899615\pi$$
−0.950682 + 0.310166i $$0.899615\pi$$
$$812$$ 0 0
$$813$$ −1.94449 −0.0681961
$$814$$ −1.30278 −0.0456623
$$815$$ 26.6056 0.931952
$$816$$ 1.81665 0.0635956
$$817$$ 13.2111 0.462198
$$818$$ −5.02776 −0.175791
$$819$$ 0 0
$$820$$ −1.18335 −0.0413242
$$821$$ 11.2111 0.391270 0.195635 0.980677i $$-0.437323\pi$$
0.195635 + 0.980677i $$0.437323\pi$$
$$822$$ 3.00000 0.104637
$$823$$ −12.8444 −0.447728 −0.223864 0.974620i $$-0.571867\pi$$
−0.223864 + 0.974620i $$0.571867\pi$$
$$824$$ 3.30278 0.115058
$$825$$ −1.30278 −0.0453568
$$826$$ 0 0
$$827$$ 27.3944 0.952598 0.476299 0.879283i $$-0.341978\pi$$
0.476299 + 0.879283i $$0.341978\pi$$
$$828$$ 20.0917 0.698234
$$829$$ −4.72498 −0.164105 −0.0820527 0.996628i $$-0.526148\pi$$
−0.0820527 + 0.996628i $$0.526148\pi$$
$$830$$ −22.4222 −0.778286
$$831$$ −7.60555 −0.263834
$$832$$ 2.30278 0.0798344
$$833$$ 0 0
$$834$$ 2.69722 0.0933972
$$835$$ 16.3028 0.564181
$$836$$ −2.60555 −0.0901149
$$837$$ 5.90833 0.204222
$$838$$ −25.1472 −0.868695
$$839$$ 49.0278 1.69263 0.846313 0.532686i $$-0.178817\pi$$
0.846313 + 0.532686i $$0.178817\pi$$
$$840$$ 0 0
$$841$$ 18.7250 0.645689
$$842$$ −28.7250 −0.989928
$$843$$ −3.63331 −0.125138
$$844$$ 6.69722 0.230528
$$845$$ 10.0278 0.344965
$$846$$ 7.57779 0.260530
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −5.26662 −0.180750
$$850$$ 19.8167 0.679706
$$851$$ −6.90833 −0.236814
$$852$$ 1.81665 0.0622375
$$853$$ 11.5416 0.395178 0.197589 0.980285i $$-0.436689\pi$$
0.197589 + 0.980285i $$0.436689\pi$$
$$854$$ 0 0
$$855$$ −7.57779 −0.259155
$$856$$ −4.30278 −0.147066
$$857$$ 14.8444 0.507075 0.253538 0.967326i $$-0.418406\pi$$
0.253538 + 0.967326i $$0.418406\pi$$
$$858$$ −0.908327 −0.0310098
$$859$$ 24.0555 0.820764 0.410382 0.911914i $$-0.365395\pi$$
0.410382 + 0.911914i $$0.365395\pi$$
$$860$$ 8.60555 0.293447
$$861$$ 0 0
$$862$$ 5.21110 0.177491
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.78890 0.0608595
$$865$$ −30.2389 −1.02815
$$866$$ −11.9361 −0.405605
$$867$$ 5.75274 0.195373
$$868$$ 0 0
$$869$$ −21.0000 −0.712376
$$870$$ 2.72498 0.0923855
$$871$$ 33.4222 1.13247
$$872$$ −2.00000 −0.0677285
$$873$$ 36.1278 1.22274
$$874$$ −13.8167 −0.467355
$$875$$ 0 0
$$876$$ 2.63331 0.0889712
$$877$$ 7.21110 0.243502 0.121751 0.992561i $$-0.461149\pi$$
0.121751 + 0.992561i $$0.461149\pi$$
$$878$$ −9.33053 −0.314890
$$879$$ 7.57779 0.255593
$$880$$ −1.69722 −0.0572134
$$881$$ −25.5416 −0.860520 −0.430260 0.902705i $$-0.641578\pi$$
−0.430260 + 0.902705i $$0.641578\pi$$
$$882$$ 0 0
$$883$$ −2.42221 −0.0815137 −0.0407568 0.999169i $$-0.512977\pi$$
−0.0407568 + 0.999169i $$0.512977\pi$$
$$884$$ 13.8167 0.464704
$$885$$ 1.33894 0.0450078
$$886$$ −0.275019 −0.00923945
$$887$$ −28.4222 −0.954324 −0.477162 0.878815i $$-0.658335\pi$$
−0.477162 + 0.878815i $$0.658335\pi$$
$$888$$ −0.302776 −0.0101605
$$889$$ 0 0
$$890$$ −6.78890 −0.227564
$$891$$ 10.6611 0.357159
$$892$$ −15.8167 −0.529581
$$893$$ −5.21110 −0.174383
$$894$$ 0.550039 0.0183960
$$895$$ −10.1833 −0.340392
$$896$$ 0 0
$$897$$ −4.81665 −0.160823
$$898$$ 0.788897 0.0263258
$$899$$ −22.8167 −0.760978
$$900$$ 9.60555 0.320185
$$901$$ −36.0000 −1.19933
$$902$$ −1.18335 −0.0394011
$$903$$ 0 0
$$904$$ −11.2111 −0.372876
$$905$$ 26.0555 0.866115
$$906$$ 4.05551 0.134735
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ 7.81665 0.259405
$$909$$ 47.7611 1.58414
$$910$$ 0 0
$$911$$ −46.4222 −1.53804 −0.769018 0.639227i $$-0.779254\pi$$
−0.769018 + 0.639227i $$0.779254\pi$$
$$912$$ −0.605551 −0.0200518
$$913$$ −22.4222 −0.742067
$$914$$ −4.60555 −0.152338
$$915$$ −4.14719 −0.137102
$$916$$ −17.3944 −0.574729
$$917$$ 0 0
$$918$$ 10.7334 0.354254
$$919$$ −38.4222 −1.26743 −0.633716 0.773566i $$-0.718471\pi$$
−0.633716 + 0.773566i $$0.718471\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ −2.14719 −0.0707522
$$922$$ −16.4222 −0.540837
$$923$$ 13.8167 0.454781
$$924$$ 0 0
$$925$$ −3.30278 −0.108595
$$926$$ −30.3028 −0.995811
$$927$$ 9.60555 0.315488
$$928$$ −6.90833 −0.226777
$$929$$ 36.5139 1.19798 0.598991 0.800756i $$-0.295569\pi$$
0.598991 + 0.800756i $$0.295569\pi$$
$$930$$ −1.30278 −0.0427197
$$931$$ 0 0
$$932$$ −9.51388 −0.311637
$$933$$ −1.54163 −0.0504708
$$934$$ 0 0
$$935$$ −10.1833 −0.333031
$$936$$ 6.69722 0.218906
$$937$$ 28.9083 0.944394 0.472197 0.881493i $$-0.343461\pi$$
0.472197 + 0.881493i $$0.343461\pi$$
$$938$$ 0 0
$$939$$ −8.18335 −0.267053
$$940$$ −3.39445 −0.110715
$$941$$ 7.81665 0.254816 0.127408 0.991850i $$-0.459334\pi$$
0.127408 + 0.991850i $$0.459334\pi$$
$$942$$ 2.18335 0.0711373
$$943$$ −6.27502 −0.204343
$$944$$ −3.39445 −0.110480
$$945$$ 0 0
$$946$$ 8.60555 0.279791
$$947$$ 39.6333 1.28791 0.643955 0.765064i $$-0.277293\pi$$
0.643955 + 0.765064i $$0.277293\pi$$
$$948$$ −4.88057 −0.158514
$$949$$ 20.0278 0.650128
$$950$$ −6.60555 −0.214312
$$951$$ −1.57779 −0.0511635
$$952$$ 0 0
$$953$$ −18.7527 −0.607461 −0.303730 0.952758i $$-0.598232\pi$$
−0.303730 + 0.952758i $$0.598232\pi$$
$$954$$ −17.4500 −0.564963
$$955$$ −16.3028 −0.527545
$$956$$ 0.513878 0.0166200
$$957$$ 2.72498 0.0880861
$$958$$ 12.1194 0.391561
$$959$$ 0 0
$$960$$ −0.394449 −0.0127308
$$961$$ −20.0917 −0.648118
$$962$$ −2.30278 −0.0742445
$$963$$ −12.5139 −0.403254
$$964$$ −8.00000 −0.257663
$$965$$ 5.21110 0.167751
$$966$$ 0 0
$$967$$ 25.7250 0.827260 0.413630 0.910445i $$-0.364261\pi$$
0.413630 + 0.910445i $$0.364261\pi$$
$$968$$ 9.30278 0.299003
$$969$$ −3.63331 −0.116719
$$970$$ −16.1833 −0.519616
$$971$$ −31.5416 −1.01222 −0.506110 0.862469i $$-0.668917\pi$$
−0.506110 + 0.862469i $$0.668917\pi$$
$$972$$ 7.84441 0.251610
$$973$$ 0 0
$$974$$ 22.7889 0.730203
$$975$$ −2.30278 −0.0737478
$$976$$ 10.5139 0.336541
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 6.18335 0.197722
$$979$$ −6.78890 −0.216974
$$980$$ 0 0
$$981$$ −5.81665 −0.185711
$$982$$ 14.7250 0.469893
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ −0.275019 −0.00876729
$$985$$ 7.81665 0.249059
$$986$$ −41.4500 −1.32004
$$987$$ 0 0
$$988$$ −4.60555 −0.146522
$$989$$ 45.6333 1.45105
$$990$$ −4.93608 −0.156879
$$991$$ 54.3028 1.72498 0.862492 0.506070i $$-0.168902\pi$$
0.862492 + 0.506070i $$0.168902\pi$$
$$992$$ 3.30278 0.104863
$$993$$ 0.366692 0.0116366
$$994$$ 0 0
$$995$$ −3.15559 −0.100039
$$996$$ −5.21110 −0.165120
$$997$$ 23.5778 0.746716 0.373358 0.927687i $$-0.378206\pi$$
0.373358 + 0.927687i $$0.378206\pi$$
$$998$$ −8.23886 −0.260797
$$999$$ −1.78890 −0.0565982
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 3626.2.a.a.1.2 2
7.6 odd 2 74.2.a.a.1.1 2
21.20 even 2 666.2.a.j.1.1 2
28.27 even 2 592.2.a.f.1.2 2
35.13 even 4 1850.2.b.i.149.3 4
35.27 even 4 1850.2.b.i.149.2 4
35.34 odd 2 1850.2.a.u.1.2 2
56.13 odd 2 2368.2.a.s.1.2 2
56.27 even 2 2368.2.a.ba.1.1 2
77.76 even 2 8954.2.a.p.1.1 2
84.83 odd 2 5328.2.a.bf.1.1 2
259.258 odd 2 2738.2.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 7.6 odd 2
592.2.a.f.1.2 2 28.27 even 2
666.2.a.j.1.1 2 21.20 even 2
1850.2.a.u.1.2 2 35.34 odd 2
1850.2.b.i.149.2 4 35.27 even 4
1850.2.b.i.149.3 4 35.13 even 4
2368.2.a.s.1.2 2 56.13 odd 2
2368.2.a.ba.1.1 2 56.27 even 2
2738.2.a.l.1.1 2 259.258 odd 2
3626.2.a.a.1.2 2 1.1 even 1 trivial
5328.2.a.bf.1.1 2 84.83 odd 2
8954.2.a.p.1.1 2 77.76 even 2