# Properties

 Label 1617.2.a.l.1.2 Level $1617$ Weight $2$ Character 1617.1 Self dual yes Analytic conductor $12.912$ 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] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.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: $$12.9118100068$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1617.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +0.854102 q^{10} -1.00000 q^{11} -1.85410 q^{12} +5.47214 q^{13} -2.23607 q^{15} +3.14590 q^{16} -6.00000 q^{17} -0.381966 q^{18} -0.236068 q^{19} +4.14590 q^{20} +0.381966 q^{22} +6.47214 q^{23} +1.47214 q^{24} -2.09017 q^{26} +1.00000 q^{27} -5.76393 q^{29} +0.854102 q^{30} +0.472136 q^{31} -4.14590 q^{32} -1.00000 q^{33} +2.29180 q^{34} -1.85410 q^{36} -9.47214 q^{37} +0.0901699 q^{38} +5.47214 q^{39} -3.29180 q^{40} -6.00000 q^{41} +8.47214 q^{43} +1.85410 q^{44} -2.23607 q^{45} -2.47214 q^{46} -2.52786 q^{47} +3.14590 q^{48} -6.00000 q^{51} -10.1459 q^{52} +4.94427 q^{53} -0.381966 q^{54} +2.23607 q^{55} -0.236068 q^{57} +2.20163 q^{58} -5.94427 q^{59} +4.14590 q^{60} -10.9443 q^{61} -0.180340 q^{62} -4.70820 q^{64} -12.2361 q^{65} +0.381966 q^{66} -12.7082 q^{67} +11.1246 q^{68} +6.47214 q^{69} -4.47214 q^{71} +1.47214 q^{72} +1.00000 q^{73} +3.61803 q^{74} +0.437694 q^{76} -2.09017 q^{78} +6.47214 q^{79} -7.03444 q^{80} +1.00000 q^{81} +2.29180 q^{82} -3.52786 q^{83} +13.4164 q^{85} -3.23607 q^{86} -5.76393 q^{87} -1.47214 q^{88} -13.4164 q^{89} +0.854102 q^{90} -12.0000 q^{92} +0.472136 q^{93} +0.965558 q^{94} +0.527864 q^{95} -4.14590 q^{96} -10.9443 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 5 q^{10} - 2 q^{11} + 3 q^{12} + 2 q^{13} + 13 q^{16} - 12 q^{17} - 3 q^{18} + 4 q^{19} + 15 q^{20} + 3 q^{22} + 4 q^{23} - 6 q^{24} + 7 q^{26} + 2 q^{27} - 16 q^{29} - 5 q^{30} - 8 q^{31} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 10 q^{37} - 11 q^{38} + 2 q^{39} - 20 q^{40} - 12 q^{41} + 8 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} + 13 q^{48} - 12 q^{51} - 27 q^{52} - 8 q^{53} - 3 q^{54} + 4 q^{57} + 29 q^{58} + 6 q^{59} + 15 q^{60} - 4 q^{61} + 22 q^{62} + 4 q^{64} - 20 q^{65} + 3 q^{66} - 12 q^{67} - 18 q^{68} + 4 q^{69} - 6 q^{72} + 2 q^{73} + 5 q^{74} + 21 q^{76} + 7 q^{78} + 4 q^{79} + 15 q^{80} + 2 q^{81} + 18 q^{82} - 16 q^{83} - 2 q^{86} - 16 q^{87} + 6 q^{88} - 5 q^{90} - 24 q^{92} - 8 q^{93} + 31 q^{94} + 10 q^{95} - 15 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 5 * q^10 - 2 * q^11 + 3 * q^12 + 2 * q^13 + 13 * q^16 - 12 * q^17 - 3 * q^18 + 4 * q^19 + 15 * q^20 + 3 * q^22 + 4 * q^23 - 6 * q^24 + 7 * q^26 + 2 * q^27 - 16 * q^29 - 5 * q^30 - 8 * q^31 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 10 * q^37 - 11 * q^38 + 2 * q^39 - 20 * q^40 - 12 * q^41 + 8 * q^43 - 3 * q^44 + 4 * q^46 - 14 * q^47 + 13 * q^48 - 12 * q^51 - 27 * q^52 - 8 * q^53 - 3 * q^54 + 4 * q^57 + 29 * q^58 + 6 * q^59 + 15 * q^60 - 4 * q^61 + 22 * q^62 + 4 * q^64 - 20 * q^65 + 3 * q^66 - 12 * q^67 - 18 * q^68 + 4 * q^69 - 6 * q^72 + 2 * q^73 + 5 * q^74 + 21 * q^76 + 7 * q^78 + 4 * q^79 + 15 * q^80 + 2 * q^81 + 18 * q^82 - 16 * q^83 - 2 * q^86 - 16 * q^87 + 6 * q^88 - 5 * q^90 - 24 * q^92 - 8 * q^93 + 31 * q^94 + 10 * q^95 - 15 * q^96 - 4 * q^97 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.381966 −0.270091 −0.135045 0.990839i $$-0.543118\pi$$
−0.135045 + 0.990839i $$0.543118\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.85410 −0.927051
$$5$$ −2.23607 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$6$$ −0.381966 −0.155937
$$7$$ 0 0
$$8$$ 1.47214 0.520479
$$9$$ 1.00000 0.333333
$$10$$ 0.854102 0.270091
$$11$$ −1.00000 −0.301511
$$12$$ −1.85410 −0.535233
$$13$$ 5.47214 1.51770 0.758849 0.651267i $$-0.225762\pi$$
0.758849 + 0.651267i $$0.225762\pi$$
$$14$$ 0 0
$$15$$ −2.23607 −0.577350
$$16$$ 3.14590 0.786475
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −0.381966 −0.0900303
$$19$$ −0.236068 −0.0541577 −0.0270789 0.999633i $$-0.508621\pi$$
−0.0270789 + 0.999633i $$0.508621\pi$$
$$20$$ 4.14590 0.927051
$$21$$ 0 0
$$22$$ 0.381966 0.0814354
$$23$$ 6.47214 1.34953 0.674767 0.738031i $$-0.264244\pi$$
0.674767 + 0.738031i $$0.264244\pi$$
$$24$$ 1.47214 0.300498
$$25$$ 0 0
$$26$$ −2.09017 −0.409916
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −5.76393 −1.07034 −0.535168 0.844746i $$-0.679752\pi$$
−0.535168 + 0.844746i $$0.679752\pi$$
$$30$$ 0.854102 0.155937
$$31$$ 0.472136 0.0847981 0.0423991 0.999101i $$-0.486500\pi$$
0.0423991 + 0.999101i $$0.486500\pi$$
$$32$$ −4.14590 −0.732898
$$33$$ −1.00000 −0.174078
$$34$$ 2.29180 0.393040
$$35$$ 0 0
$$36$$ −1.85410 −0.309017
$$37$$ −9.47214 −1.55721 −0.778605 0.627515i $$-0.784072\pi$$
−0.778605 + 0.627515i $$0.784072\pi$$
$$38$$ 0.0901699 0.0146275
$$39$$ 5.47214 0.876243
$$40$$ −3.29180 −0.520479
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.47214 1.29199 0.645994 0.763342i $$-0.276443\pi$$
0.645994 + 0.763342i $$0.276443\pi$$
$$44$$ 1.85410 0.279516
$$45$$ −2.23607 −0.333333
$$46$$ −2.47214 −0.364497
$$47$$ −2.52786 −0.368727 −0.184363 0.982858i $$-0.559022\pi$$
−0.184363 + 0.982858i $$0.559022\pi$$
$$48$$ 3.14590 0.454071
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ −10.1459 −1.40698
$$53$$ 4.94427 0.679148 0.339574 0.940579i $$-0.389717\pi$$
0.339574 + 0.940579i $$0.389717\pi$$
$$54$$ −0.381966 −0.0519790
$$55$$ 2.23607 0.301511
$$56$$ 0 0
$$57$$ −0.236068 −0.0312680
$$58$$ 2.20163 0.289088
$$59$$ −5.94427 −0.773878 −0.386939 0.922105i $$-0.626468\pi$$
−0.386939 + 0.922105i $$0.626468\pi$$
$$60$$ 4.14590 0.535233
$$61$$ −10.9443 −1.40127 −0.700635 0.713520i $$-0.747100\pi$$
−0.700635 + 0.713520i $$0.747100\pi$$
$$62$$ −0.180340 −0.0229032
$$63$$ 0 0
$$64$$ −4.70820 −0.588525
$$65$$ −12.2361 −1.51770
$$66$$ 0.381966 0.0470168
$$67$$ −12.7082 −1.55255 −0.776277 0.630392i $$-0.782894\pi$$
−0.776277 + 0.630392i $$0.782894\pi$$
$$68$$ 11.1246 1.34906
$$69$$ 6.47214 0.779154
$$70$$ 0 0
$$71$$ −4.47214 −0.530745 −0.265372 0.964146i $$-0.585495\pi$$
−0.265372 + 0.964146i $$0.585495\pi$$
$$72$$ 1.47214 0.173493
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 3.61803 0.420588
$$75$$ 0 0
$$76$$ 0.437694 0.0502070
$$77$$ 0 0
$$78$$ −2.09017 −0.236665
$$79$$ 6.47214 0.728172 0.364086 0.931365i $$-0.381381\pi$$
0.364086 + 0.931365i $$0.381381\pi$$
$$80$$ −7.03444 −0.786475
$$81$$ 1.00000 0.111111
$$82$$ 2.29180 0.253087
$$83$$ −3.52786 −0.387233 −0.193617 0.981077i $$-0.562022\pi$$
−0.193617 + 0.981077i $$0.562022\pi$$
$$84$$ 0 0
$$85$$ 13.4164 1.45521
$$86$$ −3.23607 −0.348954
$$87$$ −5.76393 −0.617958
$$88$$ −1.47214 −0.156930
$$89$$ −13.4164 −1.42214 −0.711068 0.703123i $$-0.751788\pi$$
−0.711068 + 0.703123i $$0.751788\pi$$
$$90$$ 0.854102 0.0900303
$$91$$ 0 0
$$92$$ −12.0000 −1.25109
$$93$$ 0.472136 0.0489582
$$94$$ 0.965558 0.0995897
$$95$$ 0.527864 0.0541577
$$96$$ −4.14590 −0.423139
$$97$$ −10.9443 −1.11122 −0.555611 0.831442i $$-0.687516\pi$$
−0.555611 + 0.831442i $$0.687516\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −12.4721 −1.24102 −0.620512 0.784197i $$-0.713075\pi$$
−0.620512 + 0.784197i $$0.713075\pi$$
$$102$$ 2.29180 0.226922
$$103$$ −19.4164 −1.91316 −0.956578 0.291477i $$-0.905853\pi$$
−0.956578 + 0.291477i $$0.905853\pi$$
$$104$$ 8.05573 0.789929
$$105$$ 0 0
$$106$$ −1.88854 −0.183432
$$107$$ 2.05573 0.198735 0.0993674 0.995051i $$-0.468318\pi$$
0.0993674 + 0.995051i $$0.468318\pi$$
$$108$$ −1.85410 −0.178411
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −0.854102 −0.0814354
$$111$$ −9.47214 −0.899055
$$112$$ 0 0
$$113$$ −2.47214 −0.232559 −0.116279 0.993217i $$-0.537097\pi$$
−0.116279 + 0.993217i $$0.537097\pi$$
$$114$$ 0.0901699 0.00844519
$$115$$ −14.4721 −1.34953
$$116$$ 10.6869 0.992255
$$117$$ 5.47214 0.505899
$$118$$ 2.27051 0.209017
$$119$$ 0 0
$$120$$ −3.29180 −0.300498
$$121$$ 1.00000 0.0909091
$$122$$ 4.18034 0.378470
$$123$$ −6.00000 −0.541002
$$124$$ −0.875388 −0.0786122
$$125$$ 11.1803 1.00000
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 10.0902 0.891853
$$129$$ 8.47214 0.745930
$$130$$ 4.67376 0.409916
$$131$$ −4.47214 −0.390732 −0.195366 0.980730i $$-0.562590\pi$$
−0.195366 + 0.980730i $$0.562590\pi$$
$$132$$ 1.85410 0.161379
$$133$$ 0 0
$$134$$ 4.85410 0.419331
$$135$$ −2.23607 −0.192450
$$136$$ −8.83282 −0.757408
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ −2.47214 −0.210442
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −2.52786 −0.212885
$$142$$ 1.70820 0.143349
$$143$$ −5.47214 −0.457603
$$144$$ 3.14590 0.262158
$$145$$ 12.8885 1.07034
$$146$$ −0.381966 −0.0316117
$$147$$ 0 0
$$148$$ 17.5623 1.44361
$$149$$ −15.6525 −1.28230 −0.641150 0.767415i $$-0.721543\pi$$
−0.641150 + 0.767415i $$0.721543\pi$$
$$150$$ 0 0
$$151$$ 10.9443 0.890632 0.445316 0.895373i $$-0.353091\pi$$
0.445316 + 0.895373i $$0.353091\pi$$
$$152$$ −0.347524 −0.0281879
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ −1.05573 −0.0847981
$$156$$ −10.1459 −0.812322
$$157$$ 23.4164 1.86883 0.934416 0.356183i $$-0.115922\pi$$
0.934416 + 0.356183i $$0.115922\pi$$
$$158$$ −2.47214 −0.196673
$$159$$ 4.94427 0.392106
$$160$$ 9.27051 0.732898
$$161$$ 0 0
$$162$$ −0.381966 −0.0300101
$$163$$ 22.1246 1.73293 0.866467 0.499235i $$-0.166386\pi$$
0.866467 + 0.499235i $$0.166386\pi$$
$$164$$ 11.1246 0.868686
$$165$$ 2.23607 0.174078
$$166$$ 1.34752 0.104588
$$167$$ −9.52786 −0.737288 −0.368644 0.929571i $$-0.620178\pi$$
−0.368644 + 0.929571i $$0.620178\pi$$
$$168$$ 0 0
$$169$$ 16.9443 1.30341
$$170$$ −5.12461 −0.393040
$$171$$ −0.236068 −0.0180526
$$172$$ −15.7082 −1.19774
$$173$$ −14.4721 −1.10030 −0.550148 0.835067i $$-0.685429\pi$$
−0.550148 + 0.835067i $$0.685429\pi$$
$$174$$ 2.20163 0.166905
$$175$$ 0 0
$$176$$ −3.14590 −0.237131
$$177$$ −5.94427 −0.446799
$$178$$ 5.12461 0.384106
$$179$$ −14.4721 −1.08170 −0.540849 0.841120i $$-0.681897\pi$$
−0.540849 + 0.841120i $$0.681897\pi$$
$$180$$ 4.14590 0.309017
$$181$$ 8.47214 0.629729 0.314864 0.949137i $$-0.398041\pi$$
0.314864 + 0.949137i $$0.398041\pi$$
$$182$$ 0 0
$$183$$ −10.9443 −0.809024
$$184$$ 9.52786 0.702403
$$185$$ 21.1803 1.55721
$$186$$ −0.180340 −0.0132232
$$187$$ 6.00000 0.438763
$$188$$ 4.68692 0.341829
$$189$$ 0 0
$$190$$ −0.201626 −0.0146275
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −4.70820 −0.339785
$$193$$ −9.88854 −0.711793 −0.355896 0.934525i $$-0.615824\pi$$
−0.355896 + 0.934525i $$0.615824\pi$$
$$194$$ 4.18034 0.300131
$$195$$ −12.2361 −0.876243
$$196$$ 0 0
$$197$$ 0.472136 0.0336383 0.0168191 0.999859i $$-0.494646\pi$$
0.0168191 + 0.999859i $$0.494646\pi$$
$$198$$ 0.381966 0.0271451
$$199$$ −15.5279 −1.10074 −0.550371 0.834921i $$-0.685514\pi$$
−0.550371 + 0.834921i $$0.685514\pi$$
$$200$$ 0 0
$$201$$ −12.7082 −0.896368
$$202$$ 4.76393 0.335189
$$203$$ 0 0
$$204$$ 11.1246 0.778879
$$205$$ 13.4164 0.937043
$$206$$ 7.41641 0.516726
$$207$$ 6.47214 0.449845
$$208$$ 17.2148 1.19363
$$209$$ 0.236068 0.0163292
$$210$$ 0 0
$$211$$ 1.05573 0.0726793 0.0363397 0.999339i $$-0.488430\pi$$
0.0363397 + 0.999339i $$0.488430\pi$$
$$212$$ −9.16718 −0.629605
$$213$$ −4.47214 −0.306426
$$214$$ −0.785218 −0.0536764
$$215$$ −18.9443 −1.29199
$$216$$ 1.47214 0.100166
$$217$$ 0 0
$$218$$ −2.29180 −0.155220
$$219$$ 1.00000 0.0675737
$$220$$ −4.14590 −0.279516
$$221$$ −32.8328 −2.20857
$$222$$ 3.61803 0.242827
$$223$$ −18.0000 −1.20537 −0.602685 0.797980i $$-0.705902\pi$$
−0.602685 + 0.797980i $$0.705902\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0.944272 0.0628120
$$227$$ 26.8328 1.78096 0.890478 0.455026i $$-0.150370\pi$$
0.890478 + 0.455026i $$0.150370\pi$$
$$228$$ 0.437694 0.0289870
$$229$$ 11.4164 0.754417 0.377209 0.926128i $$-0.376884\pi$$
0.377209 + 0.926128i $$0.376884\pi$$
$$230$$ 5.52786 0.364497
$$231$$ 0 0
$$232$$ −8.48529 −0.557087
$$233$$ −13.4164 −0.878938 −0.439469 0.898258i $$-0.644833\pi$$
−0.439469 + 0.898258i $$0.644833\pi$$
$$234$$ −2.09017 −0.136639
$$235$$ 5.65248 0.368727
$$236$$ 11.0213 0.717425
$$237$$ 6.47214 0.420410
$$238$$ 0 0
$$239$$ −27.3607 −1.76982 −0.884908 0.465767i $$-0.845779\pi$$
−0.884908 + 0.465767i $$0.845779\pi$$
$$240$$ −7.03444 −0.454071
$$241$$ 19.0000 1.22390 0.611949 0.790897i $$-0.290386\pi$$
0.611949 + 0.790897i $$0.290386\pi$$
$$242$$ −0.381966 −0.0245537
$$243$$ 1.00000 0.0641500
$$244$$ 20.2918 1.29905
$$245$$ 0 0
$$246$$ 2.29180 0.146120
$$247$$ −1.29180 −0.0821950
$$248$$ 0.695048 0.0441356
$$249$$ −3.52786 −0.223569
$$250$$ −4.27051 −0.270091
$$251$$ 13.9443 0.880155 0.440077 0.897960i $$-0.354951\pi$$
0.440077 + 0.897960i $$0.354951\pi$$
$$252$$ 0 0
$$253$$ −6.47214 −0.406900
$$254$$ −6.11146 −0.383467
$$255$$ 13.4164 0.840168
$$256$$ 5.56231 0.347644
$$257$$ 12.2361 0.763265 0.381632 0.924314i $$-0.375362\pi$$
0.381632 + 0.924314i $$0.375362\pi$$
$$258$$ −3.23607 −0.201469
$$259$$ 0 0
$$260$$ 22.6869 1.40698
$$261$$ −5.76393 −0.356778
$$262$$ 1.70820 0.105533
$$263$$ 22.4164 1.38225 0.691127 0.722733i $$-0.257114\pi$$
0.691127 + 0.722733i $$0.257114\pi$$
$$264$$ −1.47214 −0.0906037
$$265$$ −11.0557 −0.679148
$$266$$ 0 0
$$267$$ −13.4164 −0.821071
$$268$$ 23.5623 1.43930
$$269$$ 7.52786 0.458982 0.229491 0.973311i $$-0.426294\pi$$
0.229491 + 0.973311i $$0.426294\pi$$
$$270$$ 0.854102 0.0519790
$$271$$ −18.7082 −1.13644 −0.568221 0.822876i $$-0.692368\pi$$
−0.568221 + 0.822876i $$0.692368\pi$$
$$272$$ −18.8754 −1.14449
$$273$$ 0 0
$$274$$ −2.29180 −0.138452
$$275$$ 0 0
$$276$$ −12.0000 −0.722315
$$277$$ −1.05573 −0.0634326 −0.0317163 0.999497i $$-0.510097\pi$$
−0.0317163 + 0.999497i $$0.510097\pi$$
$$278$$ 3.05573 0.183270
$$279$$ 0.472136 0.0282660
$$280$$ 0 0
$$281$$ −17.1803 −1.02489 −0.512447 0.858719i $$-0.671261\pi$$
−0.512447 + 0.858719i $$0.671261\pi$$
$$282$$ 0.965558 0.0574982
$$283$$ 8.70820 0.517649 0.258824 0.965924i $$-0.416665\pi$$
0.258824 + 0.965924i $$0.416665\pi$$
$$284$$ 8.29180 0.492028
$$285$$ 0.527864 0.0312680
$$286$$ 2.09017 0.123594
$$287$$ 0 0
$$288$$ −4.14590 −0.244299
$$289$$ 19.0000 1.11765
$$290$$ −4.92299 −0.289088
$$291$$ −10.9443 −0.641565
$$292$$ −1.85410 −0.108503
$$293$$ −25.8885 −1.51242 −0.756212 0.654326i $$-0.772952\pi$$
−0.756212 + 0.654326i $$0.772952\pi$$
$$294$$ 0 0
$$295$$ 13.2918 0.773878
$$296$$ −13.9443 −0.810494
$$297$$ −1.00000 −0.0580259
$$298$$ 5.97871 0.346338
$$299$$ 35.4164 2.04818
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4.18034 −0.240552
$$303$$ −12.4721 −0.716505
$$304$$ −0.742646 −0.0425937
$$305$$ 24.4721 1.40127
$$306$$ 2.29180 0.131013
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −19.4164 −1.10456
$$310$$ 0.403252 0.0229032
$$311$$ −4.94427 −0.280364 −0.140182 0.990126i $$-0.544769\pi$$
−0.140182 + 0.990126i $$0.544769\pi$$
$$312$$ 8.05573 0.456066
$$313$$ 18.3607 1.03781 0.518903 0.854833i $$-0.326340\pi$$
0.518903 + 0.854833i $$0.326340\pi$$
$$314$$ −8.94427 −0.504754
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 9.05573 0.508620 0.254310 0.967123i $$-0.418152\pi$$
0.254310 + 0.967123i $$0.418152\pi$$
$$318$$ −1.88854 −0.105904
$$319$$ 5.76393 0.322718
$$320$$ 10.5279 0.588525
$$321$$ 2.05573 0.114740
$$322$$ 0 0
$$323$$ 1.41641 0.0788110
$$324$$ −1.85410 −0.103006
$$325$$ 0 0
$$326$$ −8.45085 −0.468049
$$327$$ 6.00000 0.331801
$$328$$ −8.83282 −0.487711
$$329$$ 0 0
$$330$$ −0.854102 −0.0470168
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 6.54102 0.358985
$$333$$ −9.47214 −0.519070
$$334$$ 3.63932 0.199135
$$335$$ 28.4164 1.55255
$$336$$ 0 0
$$337$$ −15.4164 −0.839785 −0.419893 0.907574i $$-0.637932\pi$$
−0.419893 + 0.907574i $$0.637932\pi$$
$$338$$ −6.47214 −0.352038
$$339$$ −2.47214 −0.134268
$$340$$ −24.8754 −1.34906
$$341$$ −0.472136 −0.0255676
$$342$$ 0.0901699 0.00487583
$$343$$ 0 0
$$344$$ 12.4721 0.672453
$$345$$ −14.4721 −0.779154
$$346$$ 5.52786 0.297180
$$347$$ −16.9443 −0.909616 −0.454808 0.890589i $$-0.650292\pi$$
−0.454808 + 0.890589i $$0.650292\pi$$
$$348$$ 10.6869 0.572879
$$349$$ −24.4164 −1.30698 −0.653490 0.756935i $$-0.726696\pi$$
−0.653490 + 0.756935i $$0.726696\pi$$
$$350$$ 0 0
$$351$$ 5.47214 0.292081
$$352$$ 4.14590 0.220977
$$353$$ −0.236068 −0.0125646 −0.00628232 0.999980i $$-0.502000\pi$$
−0.00628232 + 0.999980i $$0.502000\pi$$
$$354$$ 2.27051 0.120676
$$355$$ 10.0000 0.530745
$$356$$ 24.8754 1.31839
$$357$$ 0 0
$$358$$ 5.52786 0.292157
$$359$$ −3.05573 −0.161275 −0.0806376 0.996743i $$-0.525696\pi$$
−0.0806376 + 0.996743i $$0.525696\pi$$
$$360$$ −3.29180 −0.173493
$$361$$ −18.9443 −0.997067
$$362$$ −3.23607 −0.170084
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ −2.23607 −0.117041
$$366$$ 4.18034 0.218510
$$367$$ −1.41641 −0.0739359 −0.0369679 0.999316i $$-0.511770\pi$$
−0.0369679 + 0.999316i $$0.511770\pi$$
$$368$$ 20.3607 1.06137
$$369$$ −6.00000 −0.312348
$$370$$ −8.09017 −0.420588
$$371$$ 0 0
$$372$$ −0.875388 −0.0453868
$$373$$ 0.944272 0.0488925 0.0244463 0.999701i $$-0.492218\pi$$
0.0244463 + 0.999701i $$0.492218\pi$$
$$374$$ −2.29180 −0.118506
$$375$$ 11.1803 0.577350
$$376$$ −3.72136 −0.191914
$$377$$ −31.5410 −1.62445
$$378$$ 0 0
$$379$$ 5.65248 0.290348 0.145174 0.989406i $$-0.453626\pi$$
0.145174 + 0.989406i $$0.453626\pi$$
$$380$$ −0.978714 −0.0502070
$$381$$ 16.0000 0.819705
$$382$$ −4.58359 −0.234517
$$383$$ 8.94427 0.457031 0.228515 0.973540i $$-0.426613\pi$$
0.228515 + 0.973540i $$0.426613\pi$$
$$384$$ 10.0902 0.514912
$$385$$ 0 0
$$386$$ 3.77709 0.192249
$$387$$ 8.47214 0.430663
$$388$$ 20.2918 1.03016
$$389$$ −32.8328 −1.66469 −0.832345 0.554258i $$-0.813002\pi$$
−0.832345 + 0.554258i $$0.813002\pi$$
$$390$$ 4.67376 0.236665
$$391$$ −38.8328 −1.96386
$$392$$ 0 0
$$393$$ −4.47214 −0.225589
$$394$$ −0.180340 −0.00908539
$$395$$ −14.4721 −0.728172
$$396$$ 1.85410 0.0931721
$$397$$ 17.4164 0.874104 0.437052 0.899436i $$-0.356022\pi$$
0.437052 + 0.899436i $$0.356022\pi$$
$$398$$ 5.93112 0.297300
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0.472136 0.0235773 0.0117887 0.999931i $$-0.496247\pi$$
0.0117887 + 0.999931i $$0.496247\pi$$
$$402$$ 4.85410 0.242101
$$403$$ 2.58359 0.128698
$$404$$ 23.1246 1.15049
$$405$$ −2.23607 −0.111111
$$406$$ 0 0
$$407$$ 9.47214 0.469516
$$408$$ −8.83282 −0.437290
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ −5.12461 −0.253087
$$411$$ 6.00000 0.295958
$$412$$ 36.0000 1.77359
$$413$$ 0 0
$$414$$ −2.47214 −0.121499
$$415$$ 7.88854 0.387233
$$416$$ −22.6869 −1.11232
$$417$$ −8.00000 −0.391762
$$418$$ −0.0901699 −0.00441036
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ 20.5279 1.00047 0.500233 0.865891i $$-0.333248\pi$$
0.500233 + 0.865891i $$0.333248\pi$$
$$422$$ −0.403252 −0.0196300
$$423$$ −2.52786 −0.122909
$$424$$ 7.27864 0.353482
$$425$$ 0 0
$$426$$ 1.70820 0.0827628
$$427$$ 0 0
$$428$$ −3.81153 −0.184237
$$429$$ −5.47214 −0.264197
$$430$$ 7.23607 0.348954
$$431$$ 27.3607 1.31792 0.658959 0.752179i $$-0.270997\pi$$
0.658959 + 0.752179i $$0.270997\pi$$
$$432$$ 3.14590 0.151357
$$433$$ 39.4164 1.89423 0.947116 0.320892i $$-0.103983\pi$$
0.947116 + 0.320892i $$0.103983\pi$$
$$434$$ 0 0
$$435$$ 12.8885 0.617958
$$436$$ −11.1246 −0.532772
$$437$$ −1.52786 −0.0730876
$$438$$ −0.381966 −0.0182510
$$439$$ 32.1246 1.53322 0.766612 0.642111i $$-0.221941\pi$$
0.766612 + 0.642111i $$0.221941\pi$$
$$440$$ 3.29180 0.156930
$$441$$ 0 0
$$442$$ 12.5410 0.596515
$$443$$ 26.9443 1.28016 0.640080 0.768308i $$-0.278901\pi$$
0.640080 + 0.768308i $$0.278901\pi$$
$$444$$ 17.5623 0.833470
$$445$$ 30.0000 1.42214
$$446$$ 6.87539 0.325559
$$447$$ −15.6525 −0.740337
$$448$$ 0 0
$$449$$ 1.41641 0.0668444 0.0334222 0.999441i $$-0.489359\pi$$
0.0334222 + 0.999441i $$0.489359\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 4.58359 0.215594
$$453$$ 10.9443 0.514207
$$454$$ −10.2492 −0.481020
$$455$$ 0 0
$$456$$ −0.347524 −0.0162743
$$457$$ 15.4164 0.721149 0.360575 0.932730i $$-0.382581\pi$$
0.360575 + 0.932730i $$0.382581\pi$$
$$458$$ −4.36068 −0.203761
$$459$$ −6.00000 −0.280056
$$460$$ 26.8328 1.25109
$$461$$ −7.52786 −0.350608 −0.175304 0.984514i $$-0.556091\pi$$
−0.175304 + 0.984514i $$0.556091\pi$$
$$462$$ 0 0
$$463$$ 10.7082 0.497652 0.248826 0.968548i $$-0.419955\pi$$
0.248826 + 0.968548i $$0.419955\pi$$
$$464$$ −18.1327 −0.841791
$$465$$ −1.05573 −0.0489582
$$466$$ 5.12461 0.237393
$$467$$ 29.8328 1.38050 0.690249 0.723572i $$-0.257501\pi$$
0.690249 + 0.723572i $$0.257501\pi$$
$$468$$ −10.1459 −0.468994
$$469$$ 0 0
$$470$$ −2.15905 −0.0995897
$$471$$ 23.4164 1.07897
$$472$$ −8.75078 −0.402787
$$473$$ −8.47214 −0.389549
$$474$$ −2.47214 −0.113549
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4.94427 0.226383
$$478$$ 10.4508 0.478011
$$479$$ 13.8885 0.634584 0.317292 0.948328i $$-0.397227\pi$$
0.317292 + 0.948328i $$0.397227\pi$$
$$480$$ 9.27051 0.423139
$$481$$ −51.8328 −2.36337
$$482$$ −7.25735 −0.330563
$$483$$ 0 0
$$484$$ −1.85410 −0.0842774
$$485$$ 24.4721 1.11122
$$486$$ −0.381966 −0.0173263
$$487$$ 13.8885 0.629350 0.314675 0.949199i $$-0.398104\pi$$
0.314675 + 0.949199i $$0.398104\pi$$
$$488$$ −16.1115 −0.729331
$$489$$ 22.1246 1.00051
$$490$$ 0 0
$$491$$ 24.8885 1.12320 0.561602 0.827407i $$-0.310185\pi$$
0.561602 + 0.827407i $$0.310185\pi$$
$$492$$ 11.1246 0.501536
$$493$$ 34.5836 1.55757
$$494$$ 0.493422 0.0222001
$$495$$ 2.23607 0.100504
$$496$$ 1.48529 0.0666916
$$497$$ 0 0
$$498$$ 1.34752 0.0603840
$$499$$ −18.1246 −0.811369 −0.405685 0.914013i $$-0.632967\pi$$
−0.405685 + 0.914013i $$0.632967\pi$$
$$500$$ −20.7295 −0.927051
$$501$$ −9.52786 −0.425674
$$502$$ −5.32624 −0.237722
$$503$$ −37.4164 −1.66832 −0.834158 0.551526i $$-0.814046\pi$$
−0.834158 + 0.551526i $$0.814046\pi$$
$$504$$ 0 0
$$505$$ 27.8885 1.24102
$$506$$ 2.47214 0.109900
$$507$$ 16.9443 0.752522
$$508$$ −29.6656 −1.31620
$$509$$ −44.4721 −1.97119 −0.985596 0.169115i $$-0.945909\pi$$
−0.985596 + 0.169115i $$0.945909\pi$$
$$510$$ −5.12461 −0.226922
$$511$$ 0 0
$$512$$ −22.3050 −0.985749
$$513$$ −0.236068 −0.0104227
$$514$$ −4.67376 −0.206151
$$515$$ 43.4164 1.91316
$$516$$ −15.7082 −0.691515
$$517$$ 2.52786 0.111175
$$518$$ 0 0
$$519$$ −14.4721 −0.635256
$$520$$ −18.0132 −0.789929
$$521$$ 28.2361 1.23704 0.618522 0.785767i $$-0.287732\pi$$
0.618522 + 0.785767i $$0.287732\pi$$
$$522$$ 2.20163 0.0963626
$$523$$ −16.7082 −0.730599 −0.365299 0.930890i $$-0.619033\pi$$
−0.365299 + 0.930890i $$0.619033\pi$$
$$524$$ 8.29180 0.362229
$$525$$ 0 0
$$526$$ −8.56231 −0.373334
$$527$$ −2.83282 −0.123399
$$528$$ −3.14590 −0.136908
$$529$$ 18.8885 0.821241
$$530$$ 4.22291 0.183432
$$531$$ −5.94427 −0.257959
$$532$$ 0 0
$$533$$ −32.8328 −1.42215
$$534$$ 5.12461 0.221764
$$535$$ −4.59675 −0.198735
$$536$$ −18.7082 −0.808071
$$537$$ −14.4721 −0.624519
$$538$$ −2.87539 −0.123967
$$539$$ 0 0
$$540$$ 4.14590 0.178411
$$541$$ −5.41641 −0.232870 −0.116435 0.993198i $$-0.537147\pi$$
−0.116435 + 0.993198i $$0.537147\pi$$
$$542$$ 7.14590 0.306943
$$543$$ 8.47214 0.363574
$$544$$ 24.8754 1.06652
$$545$$ −13.4164 −0.574696
$$546$$ 0 0
$$547$$ 13.4164 0.573644 0.286822 0.957984i $$-0.407401\pi$$
0.286822 + 0.957984i $$0.407401\pi$$
$$548$$ −11.1246 −0.475220
$$549$$ −10.9443 −0.467090
$$550$$ 0 0
$$551$$ 1.36068 0.0579669
$$552$$ 9.52786 0.405533
$$553$$ 0 0
$$554$$ 0.403252 0.0171325
$$555$$ 21.1803 0.899055
$$556$$ 14.8328 0.629052
$$557$$ 32.1246 1.36116 0.680582 0.732672i $$-0.261727\pi$$
0.680582 + 0.732672i $$0.261727\pi$$
$$558$$ −0.180340 −0.00763440
$$559$$ 46.3607 1.96085
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 6.56231 0.276814
$$563$$ −4.47214 −0.188478 −0.0942390 0.995550i $$-0.530042\pi$$
−0.0942390 + 0.995550i $$0.530042\pi$$
$$564$$ 4.68692 0.197355
$$565$$ 5.52786 0.232559
$$566$$ −3.32624 −0.139812
$$567$$ 0 0
$$568$$ −6.58359 −0.276241
$$569$$ −3.52786 −0.147896 −0.0739479 0.997262i $$-0.523560\pi$$
−0.0739479 + 0.997262i $$0.523560\pi$$
$$570$$ −0.201626 −0.00844519
$$571$$ −37.4164 −1.56583 −0.782914 0.622130i $$-0.786268\pi$$
−0.782914 + 0.622130i $$0.786268\pi$$
$$572$$ 10.1459 0.424221
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −4.70820 −0.196175
$$577$$ −17.0557 −0.710039 −0.355020 0.934859i $$-0.615526\pi$$
−0.355020 + 0.934859i $$0.615526\pi$$
$$578$$ −7.25735 −0.301866
$$579$$ −9.88854 −0.410954
$$580$$ −23.8967 −0.992255
$$581$$ 0 0
$$582$$ 4.18034 0.173281
$$583$$ −4.94427 −0.204771
$$584$$ 1.47214 0.0609174
$$585$$ −12.2361 −0.505899
$$586$$ 9.88854 0.408492
$$587$$ −31.9443 −1.31848 −0.659241 0.751932i $$-0.729122\pi$$
−0.659241 + 0.751932i $$0.729122\pi$$
$$588$$ 0 0
$$589$$ −0.111456 −0.00459247
$$590$$ −5.07701 −0.209017
$$591$$ 0.472136 0.0194211
$$592$$ −29.7984 −1.22471
$$593$$ 10.5836 0.434616 0.217308 0.976103i $$-0.430272\pi$$
0.217308 + 0.976103i $$0.430272\pi$$
$$594$$ 0.381966 0.0156723
$$595$$ 0 0
$$596$$ 29.0213 1.18876
$$597$$ −15.5279 −0.635513
$$598$$ −13.5279 −0.553195
$$599$$ −37.3050 −1.52424 −0.762120 0.647436i $$-0.775841\pi$$
−0.762120 + 0.647436i $$0.775841\pi$$
$$600$$ 0 0
$$601$$ 47.8328 1.95114 0.975571 0.219686i $$-0.0705033\pi$$
0.975571 + 0.219686i $$0.0705033\pi$$
$$602$$ 0 0
$$603$$ −12.7082 −0.517518
$$604$$ −20.2918 −0.825661
$$605$$ −2.23607 −0.0909091
$$606$$ 4.76393 0.193522
$$607$$ 9.29180 0.377142 0.188571 0.982060i $$-0.439614\pi$$
0.188571 + 0.982060i $$0.439614\pi$$
$$608$$ 0.978714 0.0396921
$$609$$ 0 0
$$610$$ −9.34752 −0.378470
$$611$$ −13.8328 −0.559616
$$612$$ 11.1246 0.449686
$$613$$ −7.41641 −0.299546 −0.149773 0.988720i $$-0.547854\pi$$
−0.149773 + 0.988720i $$0.547854\pi$$
$$614$$ 6.11146 0.246638
$$615$$ 13.4164 0.541002
$$616$$ 0 0
$$617$$ 40.2492 1.62037 0.810186 0.586172i $$-0.199366\pi$$
0.810186 + 0.586172i $$0.199366\pi$$
$$618$$ 7.41641 0.298332
$$619$$ −0.583592 −0.0234565 −0.0117283 0.999931i $$-0.503733\pi$$
−0.0117283 + 0.999931i $$0.503733\pi$$
$$620$$ 1.95743 0.0786122
$$621$$ 6.47214 0.259718
$$622$$ 1.88854 0.0757237
$$623$$ 0 0
$$624$$ 17.2148 0.689143
$$625$$ −25.0000 −1.00000
$$626$$ −7.01316 −0.280302
$$627$$ 0.236068 0.00942765
$$628$$ −43.4164 −1.73250
$$629$$ 56.8328 2.26607
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 9.52786 0.378998
$$633$$ 1.05573 0.0419614
$$634$$ −3.45898 −0.137374
$$635$$ −35.7771 −1.41977
$$636$$ −9.16718 −0.363503
$$637$$ 0 0
$$638$$ −2.20163 −0.0871632
$$639$$ −4.47214 −0.176915
$$640$$ −22.5623 −0.891853
$$641$$ 40.4721 1.59855 0.799277 0.600963i $$-0.205216\pi$$
0.799277 + 0.600963i $$0.205216\pi$$
$$642$$ −0.785218 −0.0309901
$$643$$ 21.4164 0.844581 0.422290 0.906461i $$-0.361226\pi$$
0.422290 + 0.906461i $$0.361226\pi$$
$$644$$ 0 0
$$645$$ −18.9443 −0.745930
$$646$$ −0.541020 −0.0212861
$$647$$ −22.4164 −0.881280 −0.440640 0.897684i $$-0.645248\pi$$
−0.440640 + 0.897684i $$0.645248\pi$$
$$648$$ 1.47214 0.0578310
$$649$$ 5.94427 0.233333
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −41.0213 −1.60652
$$653$$ −29.8885 −1.16963 −0.584815 0.811167i $$-0.698833\pi$$
−0.584815 + 0.811167i $$0.698833\pi$$
$$654$$ −2.29180 −0.0896163
$$655$$ 10.0000 0.390732
$$656$$ −18.8754 −0.736960
$$657$$ 1.00000 0.0390137
$$658$$ 0 0
$$659$$ −2.88854 −0.112522 −0.0562608 0.998416i $$-0.517918\pi$$
−0.0562608 + 0.998416i $$0.517918\pi$$
$$660$$ −4.14590 −0.161379
$$661$$ 4.00000 0.155582 0.0777910 0.996970i $$-0.475213\pi$$
0.0777910 + 0.996970i $$0.475213\pi$$
$$662$$ 6.11146 0.237528
$$663$$ −32.8328 −1.27512
$$664$$ −5.19350 −0.201547
$$665$$ 0 0
$$666$$ 3.61803 0.140196
$$667$$ −37.3050 −1.44445
$$668$$ 17.6656 0.683504
$$669$$ −18.0000 −0.695920
$$670$$ −10.8541 −0.419331
$$671$$ 10.9443 0.422499
$$672$$ 0 0
$$673$$ −45.4164 −1.75067 −0.875337 0.483513i $$-0.839360\pi$$
−0.875337 + 0.483513i $$0.839360\pi$$
$$674$$ 5.88854 0.226818
$$675$$ 0 0
$$676$$ −31.4164 −1.20832
$$677$$ −16.5836 −0.637359 −0.318680 0.947862i $$-0.603239\pi$$
−0.318680 + 0.947862i $$0.603239\pi$$
$$678$$ 0.944272 0.0362645
$$679$$ 0 0
$$680$$ 19.7508 0.757408
$$681$$ 26.8328 1.02824
$$682$$ 0.180340 0.00690557
$$683$$ 18.4721 0.706817 0.353408 0.935469i $$-0.385023\pi$$
0.353408 + 0.935469i $$0.385023\pi$$
$$684$$ 0.437694 0.0167357
$$685$$ −13.4164 −0.512615
$$686$$ 0 0
$$687$$ 11.4164 0.435563
$$688$$ 26.6525 1.01612
$$689$$ 27.0557 1.03074
$$690$$ 5.52786 0.210442
$$691$$ −33.4164 −1.27122 −0.635610 0.772010i $$-0.719251\pi$$
−0.635610 + 0.772010i $$0.719251\pi$$
$$692$$ 26.8328 1.02003
$$693$$ 0 0
$$694$$ 6.47214 0.245679
$$695$$ 17.8885 0.678551
$$696$$ −8.48529 −0.321634
$$697$$ 36.0000 1.36360
$$698$$ 9.32624 0.353003
$$699$$ −13.4164 −0.507455
$$700$$ 0 0
$$701$$ 13.4164 0.506731 0.253365 0.967371i $$-0.418463\pi$$
0.253365 + 0.967371i $$0.418463\pi$$
$$702$$ −2.09017 −0.0788884
$$703$$ 2.23607 0.0843349
$$704$$ 4.70820 0.177447
$$705$$ 5.65248 0.212885
$$706$$ 0.0901699 0.00339359
$$707$$ 0 0
$$708$$ 11.0213 0.414205
$$709$$ −4.41641 −0.165862 −0.0829308 0.996555i $$-0.526428\pi$$
−0.0829308 + 0.996555i $$0.526428\pi$$
$$710$$ −3.81966 −0.143349
$$711$$ 6.47214 0.242724
$$712$$ −19.7508 −0.740192
$$713$$ 3.05573 0.114438
$$714$$ 0 0
$$715$$ 12.2361 0.457603
$$716$$ 26.8328 1.00279
$$717$$ −27.3607 −1.02180
$$718$$ 1.16718 0.0435589
$$719$$ 49.3607 1.84084 0.920421 0.390928i $$-0.127846\pi$$
0.920421 + 0.390928i $$0.127846\pi$$
$$720$$ −7.03444 −0.262158
$$721$$ 0 0
$$722$$ 7.23607 0.269299
$$723$$ 19.0000 0.706618
$$724$$ −15.7082 −0.583791
$$725$$ 0 0
$$726$$ −0.381966 −0.0141761
$$727$$ −31.4164 −1.16517 −0.582585 0.812770i $$-0.697959\pi$$
−0.582585 + 0.812770i $$0.697959\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0.854102 0.0316117
$$731$$ −50.8328 −1.88012
$$732$$ 20.2918 0.750006
$$733$$ −46.9443 −1.73393 −0.866963 0.498372i $$-0.833931\pi$$
−0.866963 + 0.498372i $$0.833931\pi$$
$$734$$ 0.541020 0.0199694
$$735$$ 0 0
$$736$$ −26.8328 −0.989071
$$737$$ 12.7082 0.468113
$$738$$ 2.29180 0.0843622
$$739$$ 16.5836 0.610037 0.305019 0.952346i $$-0.401337\pi$$
0.305019 + 0.952346i $$0.401337\pi$$
$$740$$ −39.2705 −1.44361
$$741$$ −1.29180 −0.0474553
$$742$$ 0 0
$$743$$ −50.3050 −1.84551 −0.922755 0.385387i $$-0.874068\pi$$
−0.922755 + 0.385387i $$0.874068\pi$$
$$744$$ 0.695048 0.0254817
$$745$$ 35.0000 1.28230
$$746$$ −0.360680 −0.0132054
$$747$$ −3.52786 −0.129078
$$748$$ −11.1246 −0.406756
$$749$$ 0 0
$$750$$ −4.27051 −0.155937
$$751$$ −28.1246 −1.02628 −0.513141 0.858304i $$-0.671518\pi$$
−0.513141 + 0.858304i $$0.671518\pi$$
$$752$$ −7.95240 −0.289994
$$753$$ 13.9443 0.508158
$$754$$ 12.0476 0.438748
$$755$$ −24.4721 −0.890632
$$756$$ 0 0
$$757$$ −38.4164 −1.39627 −0.698134 0.715967i $$-0.745986\pi$$
−0.698134 + 0.715967i $$0.745986\pi$$
$$758$$ −2.15905 −0.0784204
$$759$$ −6.47214 −0.234924
$$760$$ 0.777088 0.0281879
$$761$$ −22.4721 −0.814614 −0.407307 0.913291i $$-0.633532\pi$$
−0.407307 + 0.913291i $$0.633532\pi$$
$$762$$ −6.11146 −0.221395
$$763$$ 0 0
$$764$$ −22.2492 −0.804949
$$765$$ 13.4164 0.485071
$$766$$ −3.41641 −0.123440
$$767$$ −32.5279 −1.17451
$$768$$ 5.56231 0.200712
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 12.2361 0.440671
$$772$$ 18.3344 0.659868
$$773$$ 51.6525 1.85781 0.928905 0.370318i $$-0.120751\pi$$
0.928905 + 0.370318i $$0.120751\pi$$
$$774$$ −3.23607 −0.116318
$$775$$ 0 0
$$776$$ −16.1115 −0.578368
$$777$$ 0 0
$$778$$ 12.5410 0.449617
$$779$$ 1.41641 0.0507481
$$780$$ 22.6869 0.812322
$$781$$ 4.47214 0.160026
$$782$$ 14.8328 0.530420
$$783$$ −5.76393 −0.205986
$$784$$ 0 0
$$785$$ −52.3607 −1.86883
$$786$$ 1.70820 0.0609296
$$787$$ 2.12461 0.0757342 0.0378671 0.999283i $$-0.487944\pi$$
0.0378671 + 0.999283i $$0.487944\pi$$
$$788$$ −0.875388 −0.0311844
$$789$$ 22.4164 0.798045
$$790$$ 5.52786 0.196673
$$791$$ 0 0
$$792$$ −1.47214 −0.0523101
$$793$$ −59.8885 −2.12670
$$794$$ −6.65248 −0.236088
$$795$$ −11.0557 −0.392106
$$796$$ 28.7902 1.02044
$$797$$ 50.2361 1.77945 0.889726 0.456494i $$-0.150895\pi$$
0.889726 + 0.456494i $$0.150895\pi$$
$$798$$ 0 0
$$799$$ 15.1672 0.536576
$$800$$ 0 0
$$801$$ −13.4164 −0.474045
$$802$$ −0.180340 −0.00636802
$$803$$ −1.00000 −0.0352892
$$804$$ 23.5623 0.830978
$$805$$ 0 0
$$806$$ −0.986844 −0.0347601
$$807$$ 7.52786 0.264993
$$808$$ −18.3607 −0.645926
$$809$$ −37.1803 −1.30719 −0.653596 0.756844i $$-0.726740\pi$$
−0.653596 + 0.756844i $$0.726740\pi$$
$$810$$ 0.854102 0.0300101
$$811$$ −39.5410 −1.38847 −0.694236 0.719747i $$-0.744258\pi$$
−0.694236 + 0.719747i $$0.744258\pi$$
$$812$$ 0 0
$$813$$ −18.7082 −0.656125
$$814$$ −3.61803 −0.126812
$$815$$ −49.4721 −1.73293
$$816$$ −18.8754 −0.660771
$$817$$ −2.00000 −0.0699711
$$818$$ 5.34752 0.186972
$$819$$ 0 0
$$820$$ −24.8754 −0.868686
$$821$$ 33.5410 1.17059 0.585295 0.810821i $$-0.300979\pi$$
0.585295 + 0.810821i $$0.300979\pi$$
$$822$$ −2.29180 −0.0799356
$$823$$ 21.5410 0.750873 0.375436 0.926848i $$-0.377493\pi$$
0.375436 + 0.926848i $$0.377493\pi$$
$$824$$ −28.5836 −0.995757
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 53.8328 1.87195 0.935975 0.352066i $$-0.114521\pi$$
0.935975 + 0.352066i $$0.114521\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −18.8328 −0.654091 −0.327045 0.945009i $$-0.606053\pi$$
−0.327045 + 0.945009i $$0.606053\pi$$
$$830$$ −3.01316 −0.104588
$$831$$ −1.05573 −0.0366228
$$832$$ −25.7639 −0.893204
$$833$$ 0 0
$$834$$ 3.05573 0.105811
$$835$$ 21.3050 0.737288
$$836$$ −0.437694 −0.0151380
$$837$$ 0.472136 0.0163194
$$838$$ 1.14590 0.0395844
$$839$$ 29.4721 1.01749 0.508746 0.860917i $$-0.330109\pi$$
0.508746 + 0.860917i $$0.330109\pi$$
$$840$$ 0 0
$$841$$ 4.22291 0.145618
$$842$$ −7.84095 −0.270217
$$843$$ −17.1803 −0.591722
$$844$$ −1.95743 −0.0673774
$$845$$ −37.8885 −1.30341
$$846$$ 0.965558 0.0331966
$$847$$ 0 0
$$848$$ 15.5542 0.534133
$$849$$ 8.70820 0.298865
$$850$$ 0 0
$$851$$ −61.3050 −2.10151
$$852$$ 8.29180 0.284072
$$853$$ −34.7214 −1.18884 −0.594418 0.804156i $$-0.702618\pi$$
−0.594418 + 0.804156i $$0.702618\pi$$
$$854$$ 0 0
$$855$$ 0.527864 0.0180526
$$856$$ 3.02631 0.103437
$$857$$ −38.7214 −1.32270 −0.661348 0.750079i $$-0.730015\pi$$
−0.661348 + 0.750079i $$0.730015\pi$$
$$858$$ 2.09017 0.0713572
$$859$$ 11.4164 0.389523 0.194761 0.980851i $$-0.437607\pi$$
0.194761 + 0.980851i $$0.437607\pi$$
$$860$$ 35.1246 1.19774
$$861$$ 0 0
$$862$$ −10.4508 −0.355957
$$863$$ −27.3050 −0.929471 −0.464736 0.885449i $$-0.653851\pi$$
−0.464736 + 0.885449i $$0.653851\pi$$
$$864$$ −4.14590 −0.141046
$$865$$ 32.3607 1.10030
$$866$$ −15.0557 −0.511614
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ −6.47214 −0.219552
$$870$$ −4.92299 −0.166905
$$871$$ −69.5410 −2.35631
$$872$$ 8.83282 0.299117
$$873$$ −10.9443 −0.370407
$$874$$ 0.583592 0.0197403
$$875$$ 0 0
$$876$$ −1.85410 −0.0626443
$$877$$ 13.8885 0.468983 0.234491 0.972118i $$-0.424658\pi$$
0.234491 + 0.972118i $$0.424658\pi$$
$$878$$ −12.2705 −0.414110
$$879$$ −25.8885 −0.873199
$$880$$ 7.03444 0.237131
$$881$$ −26.5967 −0.896067 −0.448034 0.894017i $$-0.647876\pi$$
−0.448034 + 0.894017i $$0.647876\pi$$
$$882$$ 0 0
$$883$$ −0.236068 −0.00794432 −0.00397216 0.999992i $$-0.501264\pi$$
−0.00397216 + 0.999992i $$0.501264\pi$$
$$884$$ 60.8754 2.04746
$$885$$ 13.2918 0.446799
$$886$$ −10.2918 −0.345760
$$887$$ −16.3607 −0.549338 −0.274669 0.961539i $$-0.588568\pi$$
−0.274669 + 0.961539i $$0.588568\pi$$
$$888$$ −13.9443 −0.467939
$$889$$ 0 0
$$890$$ −11.4590 −0.384106
$$891$$ −1.00000 −0.0335013
$$892$$ 33.3738 1.11744
$$893$$ 0.596748 0.0199694
$$894$$ 5.97871 0.199958
$$895$$ 32.3607 1.08170
$$896$$ 0 0
$$897$$ 35.4164 1.18252
$$898$$ −0.541020 −0.0180541
$$899$$ −2.72136 −0.0907624
$$900$$ 0 0
$$901$$ −29.6656 −0.988305
$$902$$ −2.29180 −0.0763085
$$903$$ 0 0
$$904$$ −3.63932 −0.121042
$$905$$ −18.9443 −0.629729
$$906$$ −4.18034 −0.138882
$$907$$ −49.8885 −1.65652 −0.828261 0.560343i $$-0.810669\pi$$
−0.828261 + 0.560343i $$0.810669\pi$$
$$908$$ −49.7508 −1.65104
$$909$$ −12.4721 −0.413675
$$910$$ 0 0
$$911$$ −4.58359 −0.151861 −0.0759306 0.997113i $$-0.524193\pi$$
−0.0759306 + 0.997113i $$0.524193\pi$$
$$912$$ −0.742646 −0.0245915
$$913$$ 3.52786 0.116755
$$914$$ −5.88854 −0.194776
$$915$$ 24.4721 0.809024
$$916$$ −21.1672 −0.699383
$$917$$ 0 0
$$918$$ 2.29180 0.0756405
$$919$$ −33.8885 −1.11788 −0.558940 0.829208i $$-0.688792\pi$$
−0.558940 + 0.829208i $$0.688792\pi$$
$$920$$ −21.3050 −0.702403
$$921$$ −16.0000 −0.527218
$$922$$ 2.87539 0.0946959
$$923$$ −24.4721 −0.805510
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.09017 −0.134411
$$927$$ −19.4164 −0.637719
$$928$$ 23.8967 0.784447
$$929$$ 32.4853 1.06581 0.532904 0.846176i $$-0.321101\pi$$
0.532904 + 0.846176i $$0.321101\pi$$
$$930$$ 0.403252 0.0132232
$$931$$ 0 0
$$932$$ 24.8754 0.814820
$$933$$ −4.94427 −0.161868
$$934$$ −11.3951 −0.372860
$$935$$ −13.4164 −0.438763
$$936$$ 8.05573 0.263310
$$937$$ 46.7214 1.52632 0.763160 0.646209i $$-0.223647\pi$$
0.763160 + 0.646209i $$0.223647\pi$$
$$938$$ 0 0
$$939$$ 18.3607 0.599178
$$940$$ −10.4803 −0.341829
$$941$$ 47.1935 1.53846 0.769232 0.638970i $$-0.220639\pi$$
0.769232 + 0.638970i $$0.220639\pi$$
$$942$$ −8.94427 −0.291420
$$943$$ −38.8328 −1.26457
$$944$$ −18.7001 −0.608636
$$945$$ 0 0
$$946$$ 3.23607 0.105214
$$947$$ 35.8885 1.16622 0.583110 0.812393i $$-0.301835\pi$$
0.583110 + 0.812393i $$0.301835\pi$$
$$948$$ −12.0000 −0.389742
$$949$$ 5.47214 0.177633
$$950$$ 0 0
$$951$$ 9.05573 0.293652
$$952$$ 0 0
$$953$$ 56.4853 1.82974 0.914869 0.403751i $$-0.132294\pi$$
0.914869 + 0.403751i $$0.132294\pi$$
$$954$$ −1.88854 −0.0611439
$$955$$ −26.8328 −0.868290
$$956$$ 50.7295 1.64071
$$957$$ 5.76393 0.186321
$$958$$ −5.30495 −0.171395
$$959$$ 0 0
$$960$$ 10.5279 0.339785
$$961$$ −30.7771 −0.992809
$$962$$ 19.7984 0.638325
$$963$$ 2.05573 0.0662449
$$964$$ −35.2279 −1.13462
$$965$$ 22.1115 0.711793
$$966$$ 0 0
$$967$$ −34.2492 −1.10138 −0.550690 0.834710i $$-0.685635\pi$$
−0.550690 + 0.834710i $$0.685635\pi$$
$$968$$ 1.47214 0.0473162
$$969$$ 1.41641 0.0455016
$$970$$ −9.34752 −0.300131
$$971$$ 10.8885 0.349430 0.174715 0.984619i $$-0.444100\pi$$
0.174715 + 0.984619i $$0.444100\pi$$
$$972$$ −1.85410 −0.0594703
$$973$$ 0 0
$$974$$ −5.30495 −0.169982
$$975$$ 0 0
$$976$$ −34.4296 −1.10206
$$977$$ −22.4721 −0.718947 −0.359474 0.933155i $$-0.617044\pi$$
−0.359474 + 0.933155i $$0.617044\pi$$
$$978$$ −8.45085 −0.270228
$$979$$ 13.4164 0.428790
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ −9.50658 −0.303367
$$983$$ 14.8328 0.473093 0.236547 0.971620i $$-0.423984\pi$$
0.236547 + 0.971620i $$0.423984\pi$$
$$984$$ −8.83282 −0.281580
$$985$$ −1.05573 −0.0336383
$$986$$ −13.2098 −0.420684
$$987$$ 0 0
$$988$$ 2.39512 0.0761990
$$989$$ 54.8328 1.74358
$$990$$ −0.854102 −0.0271451
$$991$$ 26.7082 0.848414 0.424207 0.905565i $$-0.360553\pi$$
0.424207 + 0.905565i $$0.360553\pi$$
$$992$$ −1.95743 −0.0621484
$$993$$ −16.0000 −0.507745
$$994$$ 0 0
$$995$$ 34.7214 1.10074
$$996$$ 6.54102 0.207260
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ 6.92299 0.219143
$$999$$ −9.47214 −0.299685
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 1617.2.a.l.1.2 yes 2
3.2 odd 2 4851.2.a.bg.1.1 2
7.6 odd 2 1617.2.a.k.1.2 2
21.20 even 2 4851.2.a.bh.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.k.1.2 2 7.6 odd 2
1617.2.a.l.1.2 yes 2 1.1 even 1 trivial
4851.2.a.bg.1.1 2 3.2 odd 2
4851.2.a.bh.1.1 2 21.20 even 2