# Properties

 Label 4851.2.a.bh.1.1 Level $4851$ Weight $2$ Character 4851.1 Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.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: $$38.7354300205$$ Analytic rank: $$0$$ 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: no (minimal twist has level 1617) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 4851.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} +O(q^{10})$$ $$q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} -0.854102 q^{10} +1.00000 q^{11} -5.47214 q^{13} +3.14590 q^{16} -6.00000 q^{17} +0.236068 q^{19} +4.14590 q^{20} +0.381966 q^{22} -6.47214 q^{23} -2.09017 q^{26} +5.76393 q^{29} -0.472136 q^{31} +4.14590 q^{32} -2.29180 q^{34} -9.47214 q^{37} +0.0901699 q^{38} +3.29180 q^{40} -6.00000 q^{41} +8.47214 q^{43} -1.85410 q^{44} -2.47214 q^{46} -2.52786 q^{47} +10.1459 q^{52} -4.94427 q^{53} -2.23607 q^{55} +2.20163 q^{58} -5.94427 q^{59} +10.9443 q^{61} -0.180340 q^{62} -4.70820 q^{64} +12.2361 q^{65} -12.7082 q^{67} +11.1246 q^{68} +4.47214 q^{71} -1.00000 q^{73} -3.61803 q^{74} -0.437694 q^{76} +6.47214 q^{79} -7.03444 q^{80} -2.29180 q^{82} -3.52786 q^{83} +13.4164 q^{85} +3.23607 q^{86} -1.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} -0.965558 q^{94} -0.527864 q^{95} +10.9443 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 5 q^{10} + 2 q^{11} - 2 q^{13} + 13 q^{16} - 12 q^{17} - 4 q^{19} + 15 q^{20} + 3 q^{22} - 4 q^{23} + 7 q^{26} + 16 q^{29} + 8 q^{31} + 15 q^{32} - 18 q^{34} - 10 q^{37} - 11 q^{38} + 20 q^{40} - 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} + 27 q^{52} + 8 q^{53} + 29 q^{58} + 6 q^{59} + 4 q^{61} + 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} - 18 q^{68} - 2 q^{73} - 5 q^{74} - 21 q^{76} + 4 q^{79} + 15 q^{80} - 18 q^{82} - 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} - 31 q^{94} - 10 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 + 5 * q^10 + 2 * q^11 - 2 * q^13 + 13 * q^16 - 12 * q^17 - 4 * q^19 + 15 * q^20 + 3 * q^22 - 4 * q^23 + 7 * q^26 + 16 * q^29 + 8 * q^31 + 15 * q^32 - 18 * q^34 - 10 * q^37 - 11 * q^38 + 20 * q^40 - 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 + 27 * q^52 + 8 * q^53 + 29 * q^58 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 - 18 * q^68 - 2 * q^73 - 5 * q^74 - 21 * q^76 + 4 * q^79 + 15 * q^80 - 18 * q^82 - 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 - 31 * q^94 - 10 * q^95 + 4 * q^97

## 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 $$-0.456882\pi$$
0.135045 + 0.990839i $$0.456882\pi$$
$$3$$ 0 0
$$4$$ −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 0
$$7$$ 0 0
$$8$$ −1.47214 −0.520479
$$9$$ 0 0
$$10$$ −0.854102 −0.270091
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −5.47214 −1.51770 −0.758849 0.651267i $$-0.774238\pi$$
−0.758849 + 0.651267i $$0.774238\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 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 0
$$19$$ 0.236068 0.0541577 0.0270789 0.999633i $$-0.491379\pi$$
0.0270789 + 0.999633i $$0.491379\pi$$
$$20$$ 4.14590 0.927051
$$21$$ 0 0
$$22$$ 0.381966 0.0814354
$$23$$ −6.47214 −1.34953 −0.674767 0.738031i $$-0.735756\pi$$
−0.674767 + 0.738031i $$0.735756\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.09017 −0.409916
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5.76393 1.07034 0.535168 0.844746i $$-0.320248\pi$$
0.535168 + 0.844746i $$0.320248\pi$$
$$30$$ 0 0
$$31$$ −0.472136 −0.0847981 −0.0423991 0.999101i $$-0.513500\pi$$
−0.0423991 + 0.999101i $$0.513500\pi$$
$$32$$ 4.14590 0.732898
$$33$$ 0 0
$$34$$ −2.29180 −0.393040
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.47214 −1.55721 −0.778605 0.627515i $$-0.784072\pi$$
−0.778605 + 0.627515i $$0.784072\pi$$
$$38$$ 0.0901699 0.0146275
$$39$$ 0 0
$$40$$ 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
$$45$$ 0 0
$$46$$ −2.47214 −0.364497
$$47$$ −2.52786 −0.368727 −0.184363 0.982858i $$-0.559022\pi$$
−0.184363 + 0.982858i $$0.559022\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 10.1459 1.40698
$$53$$ −4.94427 −0.679148 −0.339574 0.940579i $$-0.610283\pi$$
−0.339574 + 0.940579i $$0.610283\pi$$
$$54$$ 0 0
$$55$$ −2.23607 −0.301511
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.20163 0.289088
$$59$$ −5.94427 −0.773878 −0.386939 0.922105i $$-0.626468\pi$$
−0.386939 + 0.922105i $$0.626468\pi$$
$$60$$ 0 0
$$61$$ 10.9443 1.40127 0.700635 0.713520i $$-0.252900\pi$$
0.700635 + 0.713520i $$0.252900\pi$$
$$62$$ −0.180340 −0.0229032
$$63$$ 0 0
$$64$$ −4.70820 −0.588525
$$65$$ 12.2361 1.51770
$$66$$ 0 0
$$67$$ −12.7082 −1.55255 −0.776277 0.630392i $$-0.782894\pi$$
−0.776277 + 0.630392i $$0.782894\pi$$
$$68$$ 11.1246 1.34906
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.47214 0.530745 0.265372 0.964146i $$-0.414505\pi$$
0.265372 + 0.964146i $$0.414505\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ −3.61803 −0.420588
$$75$$ 0 0
$$76$$ −0.437694 −0.0502070
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.47214 0.728172 0.364086 0.931365i $$-0.381381\pi$$
0.364086 + 0.931365i $$0.381381\pi$$
$$80$$ −7.03444 −0.786475
$$81$$ 0 0
$$82$$ −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
$$87$$ 0 0
$$88$$ −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 0
$$91$$ 0 0
$$92$$ 12.0000 1.25109
$$93$$ 0 0
$$94$$ −0.965558 −0.0995897
$$95$$ −0.527864 −0.0541577
$$96$$ 0 0
$$97$$ 10.9443 1.11122 0.555611 0.831442i $$-0.312484\pi$$
0.555611 + 0.831442i $$0.312484\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.4721 −1.24102 −0.620512 0.784197i $$-0.713075\pi$$
−0.620512 + 0.784197i $$0.713075\pi$$
$$102$$ 0 0
$$103$$ 19.4164 1.91316 0.956578 0.291477i $$-0.0941468\pi$$
0.956578 + 0.291477i $$0.0941468\pi$$
$$104$$ 8.05573 0.789929
$$105$$ 0 0
$$106$$ −1.88854 −0.183432
$$107$$ −2.05573 −0.198735 −0.0993674 0.995051i $$-0.531682\pi$$
−0.0993674 + 0.995051i $$0.531682\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −0.854102 −0.0814354
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.47214 0.232559 0.116279 0.993217i $$-0.462903\pi$$
0.116279 + 0.993217i $$0.462903\pi$$
$$114$$ 0 0
$$115$$ 14.4721 1.34953
$$116$$ −10.6869 −0.992255
$$117$$ 0 0
$$118$$ −2.27051 −0.209017
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.18034 0.378470
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 4.67376 0.409916
$$131$$ −4.47214 −0.390732 −0.195366 0.980730i $$-0.562590\pi$$
−0.195366 + 0.980730i $$0.562590\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.85410 −0.419331
$$135$$ 0 0
$$136$$ 8.83282 0.757408
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1.70820 0.143349
$$143$$ −5.47214 −0.457603
$$144$$ 0 0
$$145$$ −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 $$-0.278457\pi$$
0.641150 + 0.767415i $$0.278457\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.05573 0.0847981
$$156$$ 0 0
$$157$$ −23.4164 −1.86883 −0.934416 0.356183i $$-0.884078\pi$$
−0.934416 + 0.356183i $$0.884078\pi$$
$$158$$ 2.47214 0.196673
$$159$$ 0 0
$$160$$ −9.27051 −0.732898
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 22.1246 1.73293 0.866467 0.499235i $$-0.166386\pi$$
0.866467 + 0.499235i $$0.166386\pi$$
$$164$$ 11.1246 0.868686
$$165$$ 0 0
$$166$$ −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 0
$$172$$ −15.7082 −1.19774
$$173$$ −14.4721 −1.10030 −0.550148 0.835067i $$-0.685429\pi$$
−0.550148 + 0.835067i $$0.685429\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.14590 0.237131
$$177$$ 0 0
$$178$$ −5.12461 −0.384106
$$179$$ 14.4721 1.08170 0.540849 0.841120i $$-0.318103\pi$$
0.540849 + 0.841120i $$0.318103\pi$$
$$180$$ 0 0
$$181$$ −8.47214 −0.629729 −0.314864 0.949137i $$-0.601959\pi$$
−0.314864 + 0.949137i $$0.601959\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 9.52786 0.702403
$$185$$ 21.1803 1.55721
$$186$$ 0 0
$$187$$ −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 $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −9.88854 −0.711793 −0.355896 0.934525i $$-0.615824\pi$$
−0.355896 + 0.934525i $$0.615824\pi$$
$$194$$ 4.18034 0.300131
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −0.472136 −0.0336383 −0.0168191 0.999859i $$-0.505354\pi$$
−0.0168191 + 0.999859i $$0.505354\pi$$
$$198$$ 0 0
$$199$$ 15.5279 1.10074 0.550371 0.834921i $$-0.314486\pi$$
0.550371 + 0.834921i $$0.314486\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −4.76393 −0.335189
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 13.4164 0.937043
$$206$$ 7.41641 0.516726
$$207$$ 0 0
$$208$$ −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
$$213$$ 0 0
$$214$$ −0.785218 −0.0536764
$$215$$ −18.9443 −1.29199
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.29180 0.155220
$$219$$ 0 0
$$220$$ 4.14590 0.279516
$$221$$ 32.8328 2.20857
$$222$$ 0 0
$$223$$ 18.0000 1.20537 0.602685 0.797980i $$-0.294098\pi$$
0.602685 + 0.797980i $$0.294098\pi$$
$$224$$ 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 0
$$229$$ −11.4164 −0.754417 −0.377209 0.926128i $$-0.623116\pi$$
−0.377209 + 0.926128i $$0.623116\pi$$
$$230$$ 5.52786 0.364497
$$231$$ 0 0
$$232$$ −8.48529 −0.557087
$$233$$ 13.4164 0.878938 0.439469 0.898258i $$-0.355167\pi$$
0.439469 + 0.898258i $$0.355167\pi$$
$$234$$ 0 0
$$235$$ 5.65248 0.368727
$$236$$ 11.0213 0.717425
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 27.3607 1.76982 0.884908 0.465767i $$-0.154221\pi$$
0.884908 + 0.465767i $$0.154221\pi$$
$$240$$ 0 0
$$241$$ −19.0000 −1.22390 −0.611949 0.790897i $$-0.709614\pi$$
−0.611949 + 0.790897i $$0.709614\pi$$
$$242$$ 0.381966 0.0245537
$$243$$ 0 0
$$244$$ −20.2918 −1.29905
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.29180 −0.0821950
$$248$$ 0.695048 0.0441356
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 5.56231 0.347644
$$257$$ 12.2361 0.763265 0.381632 0.924314i $$-0.375362\pi$$
0.381632 + 0.924314i $$0.375362\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −22.6869 −1.40698
$$261$$ 0 0
$$262$$ −1.70820 −0.105533
$$263$$ −22.4164 −1.38225 −0.691127 0.722733i $$-0.742886\pi$$
−0.691127 + 0.722733i $$0.742886\pi$$
$$264$$ 0 0
$$265$$ 11.0557 0.679148
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 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 0
$$271$$ 18.7082 1.13644 0.568221 0.822876i $$-0.307632\pi$$
0.568221 + 0.822876i $$0.307632\pi$$
$$272$$ −18.8754 −1.14449
$$273$$ 0 0
$$274$$ −2.29180 −0.138452
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −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 0
$$280$$ 0 0
$$281$$ 17.1803 1.02489 0.512447 0.858719i $$-0.328739\pi$$
0.512447 + 0.858719i $$0.328739\pi$$
$$282$$ 0 0
$$283$$ −8.70820 −0.517649 −0.258824 0.965924i $$-0.583335\pi$$
−0.258824 + 0.965924i $$0.583335\pi$$
$$284$$ −8.29180 −0.492028
$$285$$ 0 0
$$286$$ −2.09017 −0.123594
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −4.92299 −0.289088
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ 5.97871 0.346338
$$299$$ 35.4164 2.04818
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 4.18034 0.240552
$$303$$ 0 0
$$304$$ 0.742646 0.0425937
$$305$$ −24.4721 −1.40127
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0.403252 0.0229032
$$311$$ −4.94427 −0.280364 −0.140182 0.990126i $$-0.544769\pi$$
−0.140182 + 0.990126i $$0.544769\pi$$
$$312$$ 0 0
$$313$$ −18.3607 −1.03781 −0.518903 0.854833i $$-0.673660\pi$$
−0.518903 + 0.854833i $$0.673660\pi$$
$$314$$ −8.94427 −0.504754
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −9.05573 −0.508620 −0.254310 0.967123i $$-0.581848\pi$$
−0.254310 + 0.967123i $$0.581848\pi$$
$$318$$ 0 0
$$319$$ 5.76393 0.322718
$$320$$ 10.5279 0.588525
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1.41641 −0.0788110
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.45085 0.468049
$$327$$ 0 0
$$328$$ 8.83282 0.487711
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 6.54102 0.358985
$$333$$ 0 0
$$334$$ −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
$$339$$ 0 0
$$340$$ −24.8754 −1.34906
$$341$$ −0.472136 −0.0255676
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −12.4721 −0.672453
$$345$$ 0 0
$$346$$ −5.52786 −0.297180
$$347$$ 16.9443 0.909616 0.454808 0.890589i $$-0.349708\pi$$
0.454808 + 0.890589i $$0.349708\pi$$
$$348$$ 0 0
$$349$$ 24.4164 1.30698 0.653490 0.756935i $$-0.273304\pi$$
0.653490 + 0.756935i $$0.273304\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.14590 0.220977
$$353$$ −0.236068 −0.0125646 −0.00628232 0.999980i $$-0.502000\pi$$
−0.00628232 + 0.999980i $$0.502000\pi$$
$$354$$ 0 0
$$355$$ −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 $$-0.474304\pi$$
0.0806376 + 0.996743i $$0.474304\pi$$
$$360$$ 0 0
$$361$$ −18.9443 −0.997067
$$362$$ −3.23607 −0.170084
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.23607 0.117041
$$366$$ 0 0
$$367$$ 1.41641 0.0739359 0.0369679 0.999316i $$-0.488230\pi$$
0.0369679 + 0.999316i $$0.488230\pi$$
$$368$$ −20.3607 −1.06137
$$369$$ 0 0
$$370$$ 8.09017 0.420588
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0.944272 0.0488925 0.0244463 0.999701i $$-0.492218\pi$$
0.0244463 + 0.999701i $$0.492218\pi$$
$$374$$ −2.29180 −0.118506
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ −4.58359 −0.234517
$$383$$ 8.94427 0.457031 0.228515 0.973540i $$-0.426613\pi$$
0.228515 + 0.973540i $$0.426613\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.77709 −0.192249
$$387$$ 0 0
$$388$$ −20.2918 −1.03016
$$389$$ 32.8328 1.66469 0.832345 0.554258i $$-0.186998\pi$$
0.832345 + 0.554258i $$0.186998\pi$$
$$390$$ 0 0
$$391$$ 38.8328 1.96386
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −0.180340 −0.00908539
$$395$$ −14.4721 −0.728172
$$396$$ 0 0
$$397$$ −17.4164 −0.874104 −0.437052 0.899436i $$-0.643978\pi$$
−0.437052 + 0.899436i $$0.643978\pi$$
$$398$$ 5.93112 0.297300
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −0.472136 −0.0235773 −0.0117887 0.999931i $$-0.503753\pi$$
−0.0117887 + 0.999931i $$0.503753\pi$$
$$402$$ 0 0
$$403$$ 2.58359 0.128698
$$404$$ 23.1246 1.15049
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −9.47214 −0.469516
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 5.12461 0.253087
$$411$$ 0 0
$$412$$ −36.0000 −1.77359
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 7.88854 0.387233
$$416$$ −22.6869 −1.11232
$$417$$ 0 0
$$418$$ 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
$$423$$ 0 0
$$424$$ 7.27864 0.353482
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 3.81153 0.184237
$$429$$ 0 0
$$430$$ −7.23607 −0.348954
$$431$$ −27.3607 −1.31792 −0.658959 0.752179i $$-0.729003\pi$$
−0.658959 + 0.752179i $$0.729003\pi$$
$$432$$ 0 0
$$433$$ −39.4164 −1.89423 −0.947116 0.320892i $$-0.896017\pi$$
−0.947116 + 0.320892i $$0.896017\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −11.1246 −0.532772
$$437$$ −1.52786 −0.0730876
$$438$$ 0 0
$$439$$ −32.1246 −1.53322 −0.766612 0.642111i $$-0.778059\pi$$
−0.766612 + 0.642111i $$0.778059\pi$$
$$440$$ 3.29180 0.156930
$$441$$ 0 0
$$442$$ 12.5410 0.596515
$$443$$ −26.9443 −1.28016 −0.640080 0.768308i $$-0.721099\pi$$
−0.640080 + 0.768308i $$0.721099\pi$$
$$444$$ 0 0
$$445$$ 30.0000 1.42214
$$446$$ 6.87539 0.325559
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.41641 −0.0668444 −0.0334222 0.999441i $$-0.510641\pi$$
−0.0334222 + 0.999441i $$0.510641\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ −4.58359 −0.215594
$$453$$ 0 0
$$454$$ 10.2492 0.481020
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15.4164 0.721149 0.360575 0.932730i $$-0.382581\pi$$
0.360575 + 0.932730i $$0.382581\pi$$
$$458$$ −4.36068 −0.203761
$$459$$ 0 0
$$460$$ −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
$$465$$ 0 0
$$466$$ 5.12461 0.237393
$$467$$ 29.8328 1.38050 0.690249 0.723572i $$-0.257501\pi$$
0.690249 + 0.723572i $$0.257501\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 2.15905 0.0995897
$$471$$ 0 0
$$472$$ 8.75078 0.402787
$$473$$ 8.47214 0.389549
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 10.4508 0.478011
$$479$$ 13.8885 0.634584 0.317292 0.948328i $$-0.397227\pi$$
0.317292 + 0.948328i $$0.397227\pi$$
$$480$$ 0 0
$$481$$ 51.8328 2.36337
$$482$$ −7.25735 −0.330563
$$483$$ 0 0
$$484$$ −1.85410 −0.0842774
$$485$$ −24.4721 −1.11122
$$486$$ 0 0
$$487$$ 13.8885 0.629350 0.314675 0.949199i $$-0.398104\pi$$
0.314675 + 0.949199i $$0.398104\pi$$
$$488$$ −16.1115 −0.729331
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −24.8885 −1.12320 −0.561602 0.827407i $$-0.689815\pi$$
−0.561602 + 0.827407i $$0.689815\pi$$
$$492$$ 0 0
$$493$$ −34.5836 −1.55757
$$494$$ −0.493422 −0.0222001
$$495$$ 0 0
$$496$$ −1.48529 −0.0666916
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −18.1246 −0.811369 −0.405685 0.914013i $$-0.632967\pi$$
−0.405685 + 0.914013i $$0.632967\pi$$
$$500$$ −20.7295 −0.927051
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ −29.6656 −1.31620
$$509$$ −44.4721 −1.97119 −0.985596 0.169115i $$-0.945909\pi$$
−0.985596 + 0.169115i $$0.945909\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 22.3050 0.985749
$$513$$ 0 0
$$514$$ 4.67376 0.206151
$$515$$ −43.4164 −1.91316
$$516$$ 0 0
$$517$$ −2.52786 −0.111175
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −18.0132 −0.789929
$$521$$ 28.2361 1.23704 0.618522 0.785767i $$-0.287732\pi$$
0.618522 + 0.785767i $$0.287732\pi$$
$$522$$ 0 0
$$523$$ 16.7082 0.730599 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$524$$ 8.29180 0.362229
$$525$$ 0 0
$$526$$ −8.56231 −0.373334
$$527$$ 2.83282 0.123399
$$528$$ 0 0
$$529$$ 18.8885 0.821241
$$530$$ 4.22291 0.183432
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 32.8328 1.42215
$$534$$ 0 0
$$535$$ 4.59675 0.198735
$$536$$ 18.7082 0.808071
$$537$$ 0 0
$$538$$ 2.87539 0.123967
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.41641 −0.232870 −0.116435 0.993198i $$-0.537147\pi$$
−0.116435 + 0.993198i $$0.537147\pi$$
$$542$$ 7.14590 0.306943
$$543$$ 0 0
$$544$$ −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
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.36068 0.0579669
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −0.403252 −0.0171325
$$555$$ 0 0
$$556$$ −14.8328 −0.629052
$$557$$ −32.1246 −1.36116 −0.680582 0.732672i $$-0.738273\pi$$
−0.680582 + 0.732672i $$0.738273\pi$$
$$558$$ 0 0
$$559$$ −46.3607 −1.96085
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.56231 0.276814
$$563$$ −4.47214 −0.188478 −0.0942390 0.995550i $$-0.530042\pi$$
−0.0942390 + 0.995550i $$0.530042\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.476440\pi$$
0.0739479 + 0.997262i $$0.476440\pi$$
$$570$$ 0 0
$$571$$ −37.4164 −1.56583 −0.782914 0.622130i $$-0.786268\pi$$
−0.782914 + 0.622130i $$0.786268\pi$$
$$572$$ 10.1459 0.424221
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 17.0557 0.710039 0.355020 0.934859i $$-0.384474\pi$$
0.355020 + 0.934859i $$0.384474\pi$$
$$578$$ 7.25735 0.301866
$$579$$ 0 0
$$580$$ 23.8967 0.992255
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.94427 −0.204771
$$584$$ 1.47214 0.0609174
$$585$$ 0 0
$$586$$ −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 0
$$592$$ −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 0
$$595$$ 0 0
$$596$$ −29.0213 −1.18876
$$597$$ 0 0
$$598$$ 13.5279 0.553195
$$599$$ 37.3050 1.52424 0.762120 0.647436i $$-0.224159\pi$$
0.762120 + 0.647436i $$0.224159\pi$$
$$600$$ 0 0
$$601$$ −47.8328 −1.95114 −0.975571 0.219686i $$-0.929497\pi$$
−0.975571 + 0.219686i $$0.929497\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −20.2918 −0.825661
$$605$$ −2.23607 −0.0909091
$$606$$ 0 0
$$607$$ −9.29180 −0.377142 −0.188571 0.982060i $$-0.560386\pi$$
−0.188571 + 0.982060i $$0.560386\pi$$
$$608$$ 0.978714 0.0396921
$$609$$ 0 0
$$610$$ −9.34752 −0.378470
$$611$$ 13.8328 0.559616
$$612$$ 0 0
$$613$$ −7.41641 −0.299546 −0.149773 0.988720i $$-0.547854\pi$$
−0.149773 + 0.988720i $$0.547854\pi$$
$$614$$ 6.11146 0.246638
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −40.2492 −1.62037 −0.810186 0.586172i $$-0.800634\pi$$
−0.810186 + 0.586172i $$0.800634\pi$$
$$618$$ 0 0
$$619$$ 0.583592 0.0234565 0.0117283 0.999931i $$-0.496267\pi$$
0.0117283 + 0.999931i $$0.496267\pi$$
$$620$$ −1.95743 −0.0786122
$$621$$ 0 0
$$622$$ −1.88854 −0.0757237
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −25.0000 −1.00000
$$626$$ −7.01316 −0.280302
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ −3.45898 −0.137374
$$635$$ −35.7771 −1.41977
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 2.20163 0.0871632
$$639$$ 0 0
$$640$$ 22.5623 0.891853
$$641$$ −40.4721 −1.59855 −0.799277 0.600963i $$-0.794784\pi$$
−0.799277 + 0.600963i $$0.794784\pi$$
$$642$$ 0 0
$$643$$ −21.4164 −0.844581 −0.422290 0.906461i $$-0.638774\pi$$
−0.422290 + 0.906461i $$0.638774\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −0.541020 −0.0212861
$$647$$ −22.4164 −0.881280 −0.440640 0.897684i $$-0.645248\pi$$
−0.440640 + 0.897684i $$0.645248\pi$$
$$648$$ 0 0
$$649$$ −5.94427 −0.233333
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −41.0213 −1.60652
$$653$$ 29.8885 1.16963 0.584815 0.811167i $$-0.301167\pi$$
0.584815 + 0.811167i $$0.301167\pi$$
$$654$$ 0 0
$$655$$ 10.0000 0.390732
$$656$$ −18.8754 −0.736960
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.88854 0.112522 0.0562608 0.998416i $$-0.482082\pi$$
0.0562608 + 0.998416i $$0.482082\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ −6.11146 −0.237528
$$663$$ 0 0
$$664$$ 5.19350 0.201547
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −37.3050 −1.44445
$$668$$ 17.6656 0.683504
$$669$$ 0 0
$$670$$ 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 0
$$679$$ 0 0
$$680$$ −19.7508 −0.757408
$$681$$ 0 0
$$682$$ −0.180340 −0.00690557
$$683$$ −18.4721 −0.706817 −0.353408 0.935469i $$-0.614977\pi$$
−0.353408 + 0.935469i $$0.614977\pi$$
$$684$$ 0 0
$$685$$ 13.4164 0.512615
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 26.6525 1.01612
$$689$$ 27.0557 1.03074
$$690$$ 0 0
$$691$$ 33.4164 1.27122 0.635610 0.772010i $$-0.280749\pi$$
0.635610 + 0.772010i $$0.280749\pi$$
$$692$$ 26.8328 1.02003
$$693$$ 0 0
$$694$$ 6.47214 0.245679
$$695$$ −17.8885 −0.678551
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 9.32624 0.353003
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13.4164 −0.506731 −0.253365 0.967371i $$-0.581537\pi$$
−0.253365 + 0.967371i $$0.581537\pi$$
$$702$$ 0 0
$$703$$ −2.23607 −0.0843349
$$704$$ −4.70820 −0.177447
$$705$$ 0 0
$$706$$ −0.0901699 −0.00339359
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4.41641 −0.165862 −0.0829308 0.996555i $$-0.526428\pi$$
−0.0829308 + 0.996555i $$0.526428\pi$$
$$710$$ −3.81966 −0.143349
$$711$$ 0 0
$$712$$ 19.7508 0.740192
$$713$$ 3.05573 0.114438
$$714$$ 0 0
$$715$$ 12.2361 0.457603
$$716$$ −26.8328 −1.00279
$$717$$ 0 0
$$718$$ 1.16718 0.0435589
$$719$$ 49.3607 1.84084 0.920421 0.390928i $$-0.127846\pi$$
0.920421 + 0.390928i $$0.127846\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −7.23607 −0.269299
$$723$$ 0 0
$$724$$ 15.7082 0.583791
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 31.4164 1.16517 0.582585 0.812770i $$-0.302041\pi$$
0.582585 + 0.812770i $$0.302041\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0.854102 0.0316117
$$731$$ −50.8328 −1.88012
$$732$$ 0 0
$$733$$ 46.9443 1.73393 0.866963 0.498372i $$-0.166069\pi$$
0.866963 + 0.498372i $$0.166069\pi$$
$$734$$ 0.541020 0.0199694
$$735$$ 0 0
$$736$$ −26.8328 −0.989071
$$737$$ −12.7082 −0.468113
$$738$$ 0 0
$$739$$ 16.5836 0.610037 0.305019 0.952346i $$-0.401337\pi$$
0.305019 + 0.952346i $$0.401337\pi$$
$$740$$ −39.2705 −1.44361
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 50.3050 1.84551 0.922755 0.385387i $$-0.125932\pi$$
0.922755 + 0.385387i $$0.125932\pi$$
$$744$$ 0 0
$$745$$ −35.0000 −1.28230
$$746$$ 0.360680 0.0132054
$$747$$ 0 0
$$748$$ 11.1246 0.406756
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −28.1246 −1.02628 −0.513141 0.858304i $$-0.671518\pi$$
−0.513141 + 0.858304i $$0.671518\pi$$
$$752$$ −7.95240 −0.289994
$$753$$ 0 0
$$754$$ −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
$$759$$ 0 0
$$760$$ 0.777088 0.0281879
$$761$$ −22.4721 −0.814614 −0.407307 0.913291i $$-0.633532\pi$$
−0.407307 + 0.913291i $$0.633532\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 22.2492 0.804949
$$765$$ 0 0
$$766$$ 3.41641 0.123440
$$767$$ 32.5279 1.17451
$$768$$ 0 0
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18.3344 0.659868
$$773$$ 51.6525 1.85781 0.928905 0.370318i $$-0.120751\pi$$
0.928905 + 0.370318i $$0.120751\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −16.1115 −0.578368
$$777$$ 0 0
$$778$$ 12.5410 0.449617
$$779$$ −1.41641 −0.0507481
$$780$$ 0 0
$$781$$ 4.47214 0.160026
$$782$$ 14.8328 0.530420
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 52.3607 1.86883
$$786$$ 0 0
$$787$$ −2.12461 −0.0757342 −0.0378671 0.999283i $$-0.512056\pi$$
−0.0378671 + 0.999283i $$0.512056\pi$$
$$788$$ 0.875388 0.0311844
$$789$$ 0 0
$$790$$ −5.52786 −0.196673
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −59.8885 −2.12670
$$794$$ −6.65248 −0.236088
$$795$$ 0 0
$$796$$ −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
$$801$$ 0 0
$$802$$ −0.180340 −0.00636802
$$803$$ −1.00000 −0.0352892
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0.986844 0.0347601
$$807$$ 0 0
$$808$$ 18.3607 0.645926
$$809$$ 37.1803 1.30719 0.653596 0.756844i $$-0.273260\pi$$
0.653596 + 0.756844i $$0.273260\pi$$
$$810$$ 0 0
$$811$$ 39.5410 1.38847 0.694236 0.719747i $$-0.255742\pi$$
0.694236 + 0.719747i $$0.255742\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −3.61803 −0.126812
$$815$$ −49.4721 −1.73293
$$816$$ 0 0
$$817$$ 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 $$-0.699021\pi$$
−0.585295 + 0.810821i $$0.699021\pi$$
$$822$$ 0 0
$$823$$ 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 $$-0.885479\pi$$
−0.935975 + 0.352066i $$0.885479\pi$$
$$828$$ 0 0
$$829$$ 18.8328 0.654091 0.327045 0.945009i $$-0.393947\pi$$
0.327045 + 0.945009i $$0.393947\pi$$
$$830$$ 3.01316 0.104588
$$831$$ 0 0
$$832$$ 25.7639 0.893204
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 21.3050 0.737288
$$836$$ −0.437694 −0.0151380
$$837$$ 0 0
$$838$$ −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
$$843$$ 0 0
$$844$$ −1.95743 −0.0673774
$$845$$ −37.8885 −1.30341
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −15.5542 −0.534133
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 61.3050 2.10151
$$852$$ 0 0
$$853$$ 34.7214 1.18884 0.594418 0.804156i $$-0.297382\pi$$
0.594418 + 0.804156i $$0.297382\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 3.02631 0.103437
$$857$$ −38.7214 −1.32270 −0.661348 0.750079i $$-0.730015\pi$$
−0.661348 + 0.750079i $$0.730015\pi$$
$$858$$ 0 0
$$859$$ −11.4164 −0.389523 −0.194761 0.980851i $$-0.562393\pi$$
−0.194761 + 0.980851i $$0.562393\pi$$
$$860$$ 35.1246 1.19774
$$861$$ 0 0
$$862$$ −10.4508 −0.355957
$$863$$ 27.3050 0.929471 0.464736 0.885449i $$-0.346149\pi$$
0.464736 + 0.885449i $$0.346149\pi$$
$$864$$ 0 0
$$865$$ 32.3607 1.10030
$$866$$ −15.0557 −0.511614
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 6.47214 0.219552
$$870$$ 0 0
$$871$$ 69.5410 2.35631
$$872$$ −8.83282 −0.299117
$$873$$ 0 0
$$874$$ −0.583592 −0.0197403
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13.8885 0.468983 0.234491 0.972118i $$-0.424658\pi$$
0.234491 + 0.972118i $$0.424658\pi$$
$$878$$ −12.2705 −0.414110
$$879$$ 0 0
$$880$$ −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
$$885$$ 0 0
$$886$$ −10.2918 −0.345760
$$887$$ −16.3607 −0.549338 −0.274669 0.961539i $$-0.588568\pi$$
−0.274669 + 0.961539i $$0.588568\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 11.4590 0.384106
$$891$$ 0 0
$$892$$ −33.3738 −1.11744
$$893$$ −0.596748 −0.0199694
$$894$$ 0 0
$$895$$ −32.3607 −1.08170
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −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
$$906$$ 0 0
$$907$$ −49.8885 −1.65652 −0.828261 0.560343i $$-0.810669\pi$$
−0.828261 + 0.560343i $$0.810669\pi$$
$$908$$ −49.7508 −1.65104
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 4.58359 0.151861 0.0759306 0.997113i $$-0.475807\pi$$
0.0759306 + 0.997113i $$0.475807\pi$$
$$912$$ 0 0
$$913$$ −3.52786 −0.116755
$$914$$ 5.88854 0.194776
$$915$$ 0 0
$$916$$ 21.1672 0.699383
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −33.8885 −1.11788 −0.558940 0.829208i $$-0.688792\pi$$
−0.558940 + 0.829208i $$0.688792\pi$$
$$920$$ −21.3050 −0.702403
$$921$$ 0 0
$$922$$ −2.87539 −0.0946959
$$923$$ −24.4721 −0.805510
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4.09017 0.134411
$$927$$ 0 0
$$928$$ 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 0
$$931$$ 0 0
$$932$$ −24.8754 −0.814820
$$933$$ 0 0
$$934$$ 11.3951 0.372860
$$935$$ 13.4164 0.438763
$$936$$ 0 0
$$937$$ −46.7214 −1.52632 −0.763160 0.646209i $$-0.776353\pi$$
−0.763160 + 0.646209i $$0.776353\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −10.4803 −0.341829
$$941$$ 47.1935 1.53846 0.769232 0.638970i $$-0.220639\pi$$
0.769232 + 0.638970i $$0.220639\pi$$
$$942$$ 0 0
$$943$$ 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 $$-0.698165\pi$$
−0.583110 + 0.812393i $$0.698165\pi$$
$$948$$ 0 0
$$949$$ 5.47214 0.177633
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −56.4853 −1.82974 −0.914869 0.403751i $$-0.867706\pi$$
−0.914869 + 0.403751i $$0.867706\pi$$
$$954$$ 0 0
$$955$$ 26.8328 0.868290
$$956$$ −50.7295 −1.64071
$$957$$ 0 0
$$958$$ 5.30495 0.171395
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.7771 −0.992809
$$962$$ 19.7984 0.638325
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ −9.34752 −0.300131
$$971$$ 10.8885 0.349430 0.174715 0.984619i $$-0.444100\pi$$
0.174715 + 0.984619i $$0.444100\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 5.30495 0.169982
$$975$$ 0 0
$$976$$ 34.4296 1.10206
$$977$$ 22.4721 0.718947 0.359474 0.933155i $$-0.382956\pi$$
0.359474 + 0.933155i $$0.382956\pi$$
$$978$$ 0 0
$$979$$ −13.4164 −0.428790
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −9.50658 −0.303367
$$983$$ 14.8328 0.473093 0.236547 0.971620i $$-0.423984\pi$$
0.236547 + 0.971620i $$0.423984\pi$$
$$984$$ 0 0
$$985$$ 1.05573 0.0336383
$$986$$ −13.2098 −0.420684
$$987$$ 0 0
$$988$$ 2.39512 0.0761990
$$989$$ −54.8328 −1.74358
$$990$$ 0 0
$$991$$ 26.7082 0.848414 0.424207 0.905565i $$-0.360553\pi$$
0.424207 + 0.905565i $$0.360553\pi$$
$$992$$ −1.95743 −0.0621484
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −34.7214 −1.10074
$$996$$ 0 0
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ −6.92299 −0.219143
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

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