# Properties

 Label 1840.2.e.c.369.1 Level $1840$ Weight $2$ Character 1840.369 Analytic conductor $14.692$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.e (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 369.1 Root $$-1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.369 Dual form 1840.2.e.c.369.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +O(q^{10})$$ $$q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +5.61803 q^{11} +2.61803i q^{13} +3.61803i q^{15} -0.854102i q^{17} -0.145898 q^{19} -3.00000 q^{21} -1.00000i q^{23} +5.00000 q^{25} -5.47214i q^{27} +9.70820 q^{29} +2.14590 q^{31} -9.09017i q^{33} +4.14590i q^{35} +9.70820i q^{37} +4.23607 q^{39} -5.61803 q^{41} -11.2361i q^{43} -0.854102 q^{45} +1.70820i q^{47} +3.56231 q^{49} -1.38197 q^{51} -2.00000i q^{53} -12.5623 q^{55} +0.236068i q^{57} -6.00000 q^{59} +2.85410 q^{61} -0.708204i q^{63} -5.85410i q^{65} +5.23607i q^{67} -1.61803 q^{69} -0.381966 q^{71} -16.4721i q^{73} -8.09017i q^{75} -10.4164i q^{77} -7.70820 q^{79} -7.70820 q^{81} -7.70820i q^{83} +1.90983i q^{85} -15.7082i q^{87} -3.70820 q^{89} +4.85410 q^{91} -3.47214i q^{93} +0.326238 q^{95} -13.0344i q^{97} +2.14590 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{9}+O(q^{10})$$ 4 * q + 6 * q^9 $$4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71} - 4 q^{79} - 4 q^{81} + 12 q^{89} + 6 q^{91} - 30 q^{95} + 22 q^{99}+O(q^{100})$$ 4 * q + 6 * q^9 + 18 * q^11 - 14 * q^19 - 12 * q^21 + 20 * q^25 + 12 * q^29 + 22 * q^31 + 8 * q^39 - 18 * q^41 + 10 * q^45 - 26 * q^49 - 10 * q^51 - 10 * q^55 - 24 * q^59 - 2 * q^61 - 2 * q^69 - 6 * q^71 - 4 * q^79 - 4 * q^81 + 12 * q^89 + 6 * q^91 - 30 * q^95 + 22 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 0
$$3$$ − 1.61803i − 0.934172i −0.884212 0.467086i $$-0.845304\pi$$
0.884212 0.467086i $$-0.154696\pi$$
$$4$$ 0 0
$$5$$ −2.23607 −1.00000
$$6$$ 0 0
$$7$$ − 1.85410i − 0.700785i −0.936603 0.350392i $$-0.886048\pi$$
0.936603 0.350392i $$-0.113952\pi$$
$$8$$ 0 0
$$9$$ 0.381966 0.127322
$$10$$ 0 0
$$11$$ 5.61803 1.69390 0.846950 0.531672i $$-0.178436\pi$$
0.846950 + 0.531672i $$0.178436\pi$$
$$12$$ 0 0
$$13$$ 2.61803i 0.726112i 0.931767 + 0.363056i $$0.118267\pi$$
−0.931767 + 0.363056i $$0.881733\pi$$
$$14$$ 0 0
$$15$$ 3.61803i 0.934172i
$$16$$ 0 0
$$17$$ − 0.854102i − 0.207150i −0.994622 0.103575i $$-0.966972\pi$$
0.994622 0.103575i $$-0.0330282\pi$$
$$18$$ 0 0
$$19$$ −0.145898 −0.0334713 −0.0167357 0.999860i $$-0.505327\pi$$
−0.0167357 + 0.999860i $$0.505327\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 5.47214i − 1.05311i
$$28$$ 0 0
$$29$$ 9.70820 1.80277 0.901384 0.433020i $$-0.142552\pi$$
0.901384 + 0.433020i $$0.142552\pi$$
$$30$$ 0 0
$$31$$ 2.14590 0.385415 0.192707 0.981256i $$-0.438273\pi$$
0.192707 + 0.981256i $$0.438273\pi$$
$$32$$ 0 0
$$33$$ − 9.09017i − 1.58240i
$$34$$ 0 0
$$35$$ 4.14590i 0.700785i
$$36$$ 0 0
$$37$$ 9.70820i 1.59602i 0.602645 + 0.798009i $$0.294114\pi$$
−0.602645 + 0.798009i $$0.705886\pi$$
$$38$$ 0 0
$$39$$ 4.23607 0.678314
$$40$$ 0 0
$$41$$ −5.61803 −0.877390 −0.438695 0.898636i $$-0.644559\pi$$
−0.438695 + 0.898636i $$0.644559\pi$$
$$42$$ 0 0
$$43$$ − 11.2361i − 1.71348i −0.515745 0.856742i $$-0.672485\pi$$
0.515745 0.856742i $$-0.327515\pi$$
$$44$$ 0 0
$$45$$ −0.854102 −0.127322
$$46$$ 0 0
$$47$$ 1.70820i 0.249167i 0.992209 + 0.124584i $$0.0397595\pi$$
−0.992209 + 0.124584i $$0.960241\pi$$
$$48$$ 0 0
$$49$$ 3.56231 0.508901
$$50$$ 0 0
$$51$$ −1.38197 −0.193514
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ −12.5623 −1.69390
$$56$$ 0 0
$$57$$ 0.236068i 0.0312680i
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 2.85410 0.365430 0.182715 0.983166i $$-0.441511\pi$$
0.182715 + 0.983166i $$0.441511\pi$$
$$62$$ 0 0
$$63$$ − 0.708204i − 0.0892253i
$$64$$ 0 0
$$65$$ − 5.85410i − 0.726112i
$$66$$ 0 0
$$67$$ 5.23607i 0.639688i 0.947470 + 0.319844i $$0.103630\pi$$
−0.947470 + 0.319844i $$0.896370\pi$$
$$68$$ 0 0
$$69$$ −1.61803 −0.194788
$$70$$ 0 0
$$71$$ −0.381966 −0.0453310 −0.0226655 0.999743i $$-0.507215\pi$$
−0.0226655 + 0.999743i $$0.507215\pi$$
$$72$$ 0 0
$$73$$ − 16.4721i − 1.92792i −0.266051 0.963959i $$-0.585719\pi$$
0.266051 0.963959i $$-0.414281\pi$$
$$74$$ 0 0
$$75$$ − 8.09017i − 0.934172i
$$76$$ 0 0
$$77$$ − 10.4164i − 1.18706i
$$78$$ 0 0
$$79$$ −7.70820 −0.867241 −0.433620 0.901096i $$-0.642764\pi$$
−0.433620 + 0.901096i $$0.642764\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ 0 0
$$83$$ − 7.70820i − 0.846085i −0.906110 0.423043i $$-0.860962\pi$$
0.906110 0.423043i $$-0.139038\pi$$
$$84$$ 0 0
$$85$$ 1.90983i 0.207150i
$$86$$ 0 0
$$87$$ − 15.7082i − 1.68410i
$$88$$ 0 0
$$89$$ −3.70820 −0.393069 −0.196534 0.980497i $$-0.562969\pi$$
−0.196534 + 0.980497i $$0.562969\pi$$
$$90$$ 0 0
$$91$$ 4.85410 0.508848
$$92$$ 0 0
$$93$$ − 3.47214i − 0.360044i
$$94$$ 0 0
$$95$$ 0.326238 0.0334713
$$96$$ 0 0
$$97$$ − 13.0344i − 1.32345i −0.749748 0.661724i $$-0.769825\pi$$
0.749748 0.661724i $$-0.230175\pi$$
$$98$$ 0 0
$$99$$ 2.14590 0.215671
$$100$$ 0 0
$$101$$ −1.52786 −0.152028 −0.0760141 0.997107i $$-0.524219\pi$$
−0.0760141 + 0.997107i $$0.524219\pi$$
$$102$$ 0 0
$$103$$ − 10.8541i − 1.06949i −0.845015 0.534743i $$-0.820408\pi$$
0.845015 0.534743i $$-0.179592\pi$$
$$104$$ 0 0
$$105$$ 6.70820 0.654654
$$106$$ 0 0
$$107$$ 11.7082i 1.13187i 0.824448 + 0.565937i $$0.191486\pi$$
−0.824448 + 0.565937i $$0.808514\pi$$
$$108$$ 0 0
$$109$$ −7.56231 −0.724338 −0.362169 0.932113i $$-0.617964\pi$$
−0.362169 + 0.932113i $$0.617964\pi$$
$$110$$ 0 0
$$111$$ 15.7082 1.49096
$$112$$ 0 0
$$113$$ − 13.4164i − 1.26211i −0.775738 0.631055i $$-0.782622\pi$$
0.775738 0.631055i $$-0.217378\pi$$
$$114$$ 0 0
$$115$$ 2.23607i 0.208514i
$$116$$ 0 0
$$117$$ 1.00000i 0.0924500i
$$118$$ 0 0
$$119$$ −1.58359 −0.145168
$$120$$ 0 0
$$121$$ 20.5623 1.86930
$$122$$ 0 0
$$123$$ 9.09017i 0.819633i
$$124$$ 0 0
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ − 9.70820i − 0.861464i −0.902480 0.430732i $$-0.858255\pi$$
0.902480 0.430732i $$-0.141745\pi$$
$$128$$ 0 0
$$129$$ −18.1803 −1.60069
$$130$$ 0 0
$$131$$ 14.1803 1.23894 0.619471 0.785020i $$-0.287347\pi$$
0.619471 + 0.785020i $$0.287347\pi$$
$$132$$ 0 0
$$133$$ 0.270510i 0.0234562i
$$134$$ 0 0
$$135$$ 12.2361i 1.05311i
$$136$$ 0 0
$$137$$ 13.8541i 1.18364i 0.806072 + 0.591818i $$0.201590\pi$$
−0.806072 + 0.591818i $$0.798410\pi$$
$$138$$ 0 0
$$139$$ 4.29180 0.364025 0.182013 0.983296i $$-0.441739\pi$$
0.182013 + 0.983296i $$0.441739\pi$$
$$140$$ 0 0
$$141$$ 2.76393 0.232765
$$142$$ 0 0
$$143$$ 14.7082i 1.22996i
$$144$$ 0 0
$$145$$ −21.7082 −1.80277
$$146$$ 0 0
$$147$$ − 5.76393i − 0.475401i
$$148$$ 0 0
$$149$$ 2.61803 0.214478 0.107239 0.994233i $$-0.465799\pi$$
0.107239 + 0.994233i $$0.465799\pi$$
$$150$$ 0 0
$$151$$ −14.2705 −1.16132 −0.580659 0.814147i $$-0.697205\pi$$
−0.580659 + 0.814147i $$0.697205\pi$$
$$152$$ 0 0
$$153$$ − 0.326238i − 0.0263748i
$$154$$ 0 0
$$155$$ −4.79837 −0.385415
$$156$$ 0 0
$$157$$ − 2.29180i − 0.182905i −0.995809 0.0914526i $$-0.970849\pi$$
0.995809 0.0914526i $$-0.0291510\pi$$
$$158$$ 0 0
$$159$$ −3.23607 −0.256637
$$160$$ 0 0
$$161$$ −1.85410 −0.146124
$$162$$ 0 0
$$163$$ 22.0344i 1.72587i 0.505314 + 0.862935i $$0.331377\pi$$
−0.505314 + 0.862935i $$0.668623\pi$$
$$164$$ 0 0
$$165$$ 20.3262i 1.58240i
$$166$$ 0 0
$$167$$ − 9.70820i − 0.751243i −0.926773 0.375622i $$-0.877429\pi$$
0.926773 0.375622i $$-0.122571\pi$$
$$168$$ 0 0
$$169$$ 6.14590 0.472761
$$170$$ 0 0
$$171$$ −0.0557281 −0.00426163
$$172$$ 0 0
$$173$$ 16.5623i 1.25921i 0.776916 + 0.629604i $$0.216783\pi$$
−0.776916 + 0.629604i $$0.783217\pi$$
$$174$$ 0 0
$$175$$ − 9.27051i − 0.700785i
$$176$$ 0 0
$$177$$ 9.70820i 0.729713i
$$178$$ 0 0
$$179$$ 7.52786 0.562659 0.281329 0.959611i $$-0.409225\pi$$
0.281329 + 0.959611i $$0.409225\pi$$
$$180$$ 0 0
$$181$$ 15.5623 1.15674 0.578369 0.815776i $$-0.303690\pi$$
0.578369 + 0.815776i $$0.303690\pi$$
$$182$$ 0 0
$$183$$ − 4.61803i − 0.341375i
$$184$$ 0 0
$$185$$ − 21.7082i − 1.59602i
$$186$$ 0 0
$$187$$ − 4.79837i − 0.350892i
$$188$$ 0 0
$$189$$ −10.1459 −0.738005
$$190$$ 0 0
$$191$$ −25.4164 −1.83907 −0.919533 0.393012i $$-0.871433\pi$$
−0.919533 + 0.393012i $$0.871433\pi$$
$$192$$ 0 0
$$193$$ − 15.7082i − 1.13070i −0.824851 0.565351i $$-0.808741\pi$$
0.824851 0.565351i $$-0.191259\pi$$
$$194$$ 0 0
$$195$$ −9.47214 −0.678314
$$196$$ 0 0
$$197$$ 20.5623i 1.46500i 0.680765 + 0.732502i $$0.261647\pi$$
−0.680765 + 0.732502i $$0.738353\pi$$
$$198$$ 0 0
$$199$$ 11.4164 0.809288 0.404644 0.914474i $$-0.367395\pi$$
0.404644 + 0.914474i $$0.367395\pi$$
$$200$$ 0 0
$$201$$ 8.47214 0.597578
$$202$$ 0 0
$$203$$ − 18.0000i − 1.26335i
$$204$$ 0 0
$$205$$ 12.5623 0.877390
$$206$$ 0 0
$$207$$ − 0.381966i − 0.0265485i
$$208$$ 0 0
$$209$$ −0.819660 −0.0566971
$$210$$ 0 0
$$211$$ 7.70820 0.530655 0.265327 0.964158i $$-0.414520\pi$$
0.265327 + 0.964158i $$0.414520\pi$$
$$212$$ 0 0
$$213$$ 0.618034i 0.0423470i
$$214$$ 0 0
$$215$$ 25.1246i 1.71348i
$$216$$ 0 0
$$217$$ − 3.97871i − 0.270093i
$$218$$ 0 0
$$219$$ −26.6525 −1.80101
$$220$$ 0 0
$$221$$ 2.23607 0.150414
$$222$$ 0 0
$$223$$ − 22.3607i − 1.49738i −0.662919 0.748691i $$-0.730683\pi$$
0.662919 0.748691i $$-0.269317\pi$$
$$224$$ 0 0
$$225$$ 1.90983 0.127322
$$226$$ 0 0
$$227$$ 10.2918i 0.683090i 0.939865 + 0.341545i $$0.110950\pi$$
−0.939865 + 0.341545i $$0.889050\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −16.8541 −1.10892
$$232$$ 0 0
$$233$$ 21.1246i 1.38392i 0.721936 + 0.691960i $$0.243252\pi$$
−0.721936 + 0.691960i $$0.756748\pi$$
$$234$$ 0 0
$$235$$ − 3.81966i − 0.249167i
$$236$$ 0 0
$$237$$ 12.4721i 0.810152i
$$238$$ 0 0
$$239$$ −10.4721 −0.677386 −0.338693 0.940897i $$-0.609985\pi$$
−0.338693 + 0.940897i $$0.609985\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ − 3.94427i − 0.253025i
$$244$$ 0 0
$$245$$ −7.96556 −0.508901
$$246$$ 0 0
$$247$$ − 0.381966i − 0.0243039i
$$248$$ 0 0
$$249$$ −12.4721 −0.790390
$$250$$ 0 0
$$251$$ 16.7984 1.06030 0.530152 0.847903i $$-0.322135\pi$$
0.530152 + 0.847903i $$0.322135\pi$$
$$252$$ 0 0
$$253$$ − 5.61803i − 0.353203i
$$254$$ 0 0
$$255$$ 3.09017 0.193514
$$256$$ 0 0
$$257$$ 11.4164i 0.712136i 0.934460 + 0.356068i $$0.115883\pi$$
−0.934460 + 0.356068i $$0.884117\pi$$
$$258$$ 0 0
$$259$$ 18.0000 1.11847
$$260$$ 0 0
$$261$$ 3.70820 0.229532
$$262$$ 0 0
$$263$$ 18.2705i 1.12661i 0.826250 + 0.563304i $$0.190470\pi$$
−0.826250 + 0.563304i $$0.809530\pi$$
$$264$$ 0 0
$$265$$ 4.47214i 0.274721i
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ 0 0
$$269$$ 1.52786 0.0931555 0.0465778 0.998915i $$-0.485168\pi$$
0.0465778 + 0.998915i $$0.485168\pi$$
$$270$$ 0 0
$$271$$ −25.8541 −1.57052 −0.785262 0.619163i $$-0.787472\pi$$
−0.785262 + 0.619163i $$0.787472\pi$$
$$272$$ 0 0
$$273$$ − 7.85410i − 0.475352i
$$274$$ 0 0
$$275$$ 28.0902 1.69390
$$276$$ 0 0
$$277$$ − 7.52786i − 0.452306i −0.974092 0.226153i $$-0.927385\pi$$
0.974092 0.226153i $$-0.0726149\pi$$
$$278$$ 0 0
$$279$$ 0.819660 0.0490718
$$280$$ 0 0
$$281$$ 30.6525 1.82857 0.914287 0.405068i $$-0.132752\pi$$
0.914287 + 0.405068i $$0.132752\pi$$
$$282$$ 0 0
$$283$$ − 20.9443i − 1.24501i −0.782617 0.622504i $$-0.786115\pi$$
0.782617 0.622504i $$-0.213885\pi$$
$$284$$ 0 0
$$285$$ − 0.527864i − 0.0312680i
$$286$$ 0 0
$$287$$ 10.4164i 0.614861i
$$288$$ 0 0
$$289$$ 16.2705 0.957089
$$290$$ 0 0
$$291$$ −21.0902 −1.23633
$$292$$ 0 0
$$293$$ − 14.2918i − 0.834936i −0.908692 0.417468i $$-0.862918\pi$$
0.908692 0.417468i $$-0.137082\pi$$
$$294$$ 0 0
$$295$$ 13.4164 0.781133
$$296$$ 0 0
$$297$$ − 30.7426i − 1.78387i
$$298$$ 0 0
$$299$$ 2.61803 0.151405
$$300$$ 0 0
$$301$$ −20.8328 −1.20078
$$302$$ 0 0
$$303$$ 2.47214i 0.142020i
$$304$$ 0 0
$$305$$ −6.38197 −0.365430
$$306$$ 0 0
$$307$$ 16.8541i 0.961914i 0.876744 + 0.480957i $$0.159711\pi$$
−0.876744 + 0.480957i $$0.840289\pi$$
$$308$$ 0 0
$$309$$ −17.5623 −0.999085
$$310$$ 0 0
$$311$$ 22.4721 1.27428 0.637139 0.770749i $$-0.280118\pi$$
0.637139 + 0.770749i $$0.280118\pi$$
$$312$$ 0 0
$$313$$ 25.8541i 1.46136i 0.682720 + 0.730680i $$0.260797\pi$$
−0.682720 + 0.730680i $$0.739203\pi$$
$$314$$ 0 0
$$315$$ 1.58359i 0.0892253i
$$316$$ 0 0
$$317$$ 7.27051i 0.408353i 0.978934 + 0.204176i $$0.0654516\pi$$
−0.978934 + 0.204176i $$0.934548\pi$$
$$318$$ 0 0
$$319$$ 54.5410 3.05371
$$320$$ 0 0
$$321$$ 18.9443 1.05737
$$322$$ 0 0
$$323$$ 0.124612i 0.00693359i
$$324$$ 0 0
$$325$$ 13.0902i 0.726112i
$$326$$ 0 0
$$327$$ 12.2361i 0.676656i
$$328$$ 0 0
$$329$$ 3.16718 0.174613
$$330$$ 0 0
$$331$$ 25.1246 1.38097 0.690487 0.723345i $$-0.257396\pi$$
0.690487 + 0.723345i $$0.257396\pi$$
$$332$$ 0 0
$$333$$ 3.70820i 0.203208i
$$334$$ 0 0
$$335$$ − 11.7082i − 0.639688i
$$336$$ 0 0
$$337$$ 3.38197i 0.184227i 0.995748 + 0.0921137i $$0.0293623\pi$$
−0.995748 + 0.0921137i $$0.970638\pi$$
$$338$$ 0 0
$$339$$ −21.7082 −1.17903
$$340$$ 0 0
$$341$$ 12.0557 0.652854
$$342$$ 0 0
$$343$$ − 19.5836i − 1.05741i
$$344$$ 0 0
$$345$$ 3.61803 0.194788
$$346$$ 0 0
$$347$$ 14.5623i 0.781746i 0.920445 + 0.390873i $$0.127827\pi$$
−0.920445 + 0.390873i $$0.872173\pi$$
$$348$$ 0 0
$$349$$ −27.7082 −1.48319 −0.741593 0.670850i $$-0.765929\pi$$
−0.741593 + 0.670850i $$0.765929\pi$$
$$350$$ 0 0
$$351$$ 14.3262 0.764678
$$352$$ 0 0
$$353$$ 8.00000i 0.425797i 0.977074 + 0.212899i $$0.0682904\pi$$
−0.977074 + 0.212899i $$0.931710\pi$$
$$354$$ 0 0
$$355$$ 0.854102 0.0453310
$$356$$ 0 0
$$357$$ 2.56231i 0.135612i
$$358$$ 0 0
$$359$$ −13.4164 −0.708091 −0.354045 0.935228i $$-0.615194\pi$$
−0.354045 + 0.935228i $$0.615194\pi$$
$$360$$ 0 0
$$361$$ −18.9787 −0.998880
$$362$$ 0 0
$$363$$ − 33.2705i − 1.74625i
$$364$$ 0 0
$$365$$ 36.8328i 1.92792i
$$366$$ 0 0
$$367$$ 12.0000i 0.626395i 0.949688 + 0.313197i $$0.101400\pi$$
−0.949688 + 0.313197i $$0.898600\pi$$
$$368$$ 0 0
$$369$$ −2.14590 −0.111711
$$370$$ 0 0
$$371$$ −3.70820 −0.192520
$$372$$ 0 0
$$373$$ − 20.9443i − 1.08445i −0.840232 0.542227i $$-0.817581\pi$$
0.840232 0.542227i $$-0.182419\pi$$
$$374$$ 0 0
$$375$$ 18.0902i 0.934172i
$$376$$ 0 0
$$377$$ 25.4164i 1.30901i
$$378$$ 0 0
$$379$$ −24.2705 −1.24669 −0.623346 0.781946i $$-0.714227\pi$$
−0.623346 + 0.781946i $$0.714227\pi$$
$$380$$ 0 0
$$381$$ −15.7082 −0.804756
$$382$$ 0 0
$$383$$ − 0.583592i − 0.0298202i −0.999889 0.0149101i $$-0.995254\pi$$
0.999889 0.0149101i $$-0.00474620\pi$$
$$384$$ 0 0
$$385$$ 23.2918i 1.18706i
$$386$$ 0 0
$$387$$ − 4.29180i − 0.218164i
$$388$$ 0 0
$$389$$ 21.3262 1.08128 0.540642 0.841253i $$-0.318182\pi$$
0.540642 + 0.841253i $$0.318182\pi$$
$$390$$ 0 0
$$391$$ −0.854102 −0.0431938
$$392$$ 0 0
$$393$$ − 22.9443i − 1.15739i
$$394$$ 0 0
$$395$$ 17.2361 0.867241
$$396$$ 0 0
$$397$$ − 15.4377i − 0.774796i −0.921913 0.387398i $$-0.873374\pi$$
0.921913 0.387398i $$-0.126626\pi$$
$$398$$ 0 0
$$399$$ 0.437694 0.0219121
$$400$$ 0 0
$$401$$ −25.5279 −1.27480 −0.637400 0.770533i $$-0.719990\pi$$
−0.637400 + 0.770533i $$0.719990\pi$$
$$402$$ 0 0
$$403$$ 5.61803i 0.279854i
$$404$$ 0 0
$$405$$ 17.2361 0.856467
$$406$$ 0 0
$$407$$ 54.5410i 2.70350i
$$408$$ 0 0
$$409$$ −13.5623 −0.670613 −0.335306 0.942109i $$-0.608840\pi$$
−0.335306 + 0.942109i $$0.608840\pi$$
$$410$$ 0 0
$$411$$ 22.4164 1.10572
$$412$$ 0 0
$$413$$ 11.1246i 0.547406i
$$414$$ 0 0
$$415$$ 17.2361i 0.846085i
$$416$$ 0 0
$$417$$ − 6.94427i − 0.340062i
$$418$$ 0 0
$$419$$ −29.8885 −1.46015 −0.730075 0.683367i $$-0.760515\pi$$
−0.730075 + 0.683367i $$0.760515\pi$$
$$420$$ 0 0
$$421$$ 0.145898 0.00711064 0.00355532 0.999994i $$-0.498868\pi$$
0.00355532 + 0.999994i $$0.498868\pi$$
$$422$$ 0 0
$$423$$ 0.652476i 0.0317245i
$$424$$ 0 0
$$425$$ − 4.27051i − 0.207150i
$$426$$ 0 0
$$427$$ − 5.29180i − 0.256088i
$$428$$ 0 0
$$429$$ 23.7984 1.14900
$$430$$ 0 0
$$431$$ 11.2361 0.541222 0.270611 0.962689i $$-0.412774\pi$$
0.270611 + 0.962689i $$0.412774\pi$$
$$432$$ 0 0
$$433$$ − 27.3820i − 1.31589i −0.753065 0.657947i $$-0.771425\pi$$
0.753065 0.657947i $$-0.228575\pi$$
$$434$$ 0 0
$$435$$ 35.1246i 1.68410i
$$436$$ 0 0
$$437$$ 0.145898i 0.00697925i
$$438$$ 0 0
$$439$$ 33.2705 1.58791 0.793957 0.607973i $$-0.208017\pi$$
0.793957 + 0.607973i $$0.208017\pi$$
$$440$$ 0 0
$$441$$ 1.36068 0.0647943
$$442$$ 0 0
$$443$$ − 0.145898i − 0.00693182i −0.999994 0.00346591i $$-0.998897\pi$$
0.999994 0.00346591i $$-0.00110324\pi$$
$$444$$ 0 0
$$445$$ 8.29180 0.393069
$$446$$ 0 0
$$447$$ − 4.23607i − 0.200359i
$$448$$ 0 0
$$449$$ 17.5623 0.828816 0.414408 0.910091i $$-0.363989\pi$$
0.414408 + 0.910091i $$0.363989\pi$$
$$450$$ 0 0
$$451$$ −31.5623 −1.48621
$$452$$ 0 0
$$453$$ 23.0902i 1.08487i
$$454$$ 0 0
$$455$$ −10.8541 −0.508848
$$456$$ 0 0
$$457$$ − 1.41641i − 0.0662568i −0.999451 0.0331284i $$-0.989453\pi$$
0.999451 0.0331284i $$-0.0105470\pi$$
$$458$$ 0 0
$$459$$ −4.67376 −0.218153
$$460$$ 0 0
$$461$$ 37.3050 1.73746 0.868732 0.495282i $$-0.164935\pi$$
0.868732 + 0.495282i $$0.164935\pi$$
$$462$$ 0 0
$$463$$ 15.7082i 0.730022i 0.931003 + 0.365011i $$0.118935\pi$$
−0.931003 + 0.365011i $$0.881065\pi$$
$$464$$ 0 0
$$465$$ 7.76393i 0.360044i
$$466$$ 0 0
$$467$$ 23.1246i 1.07008i 0.844827 + 0.535040i $$0.179703\pi$$
−0.844827 + 0.535040i $$0.820297\pi$$
$$468$$ 0 0
$$469$$ 9.70820 0.448283
$$470$$ 0 0
$$471$$ −3.70820 −0.170865
$$472$$ 0 0
$$473$$ − 63.1246i − 2.90247i
$$474$$ 0 0
$$475$$ −0.729490 −0.0334713
$$476$$ 0 0
$$477$$ − 0.763932i − 0.0349780i
$$478$$ 0 0
$$479$$ −28.4721 −1.30093 −0.650463 0.759538i $$-0.725425\pi$$
−0.650463 + 0.759538i $$0.725425\pi$$
$$480$$ 0 0
$$481$$ −25.4164 −1.15889
$$482$$ 0 0
$$483$$ 3.00000i 0.136505i
$$484$$ 0 0
$$485$$ 29.1459i 1.32345i
$$486$$ 0 0
$$487$$ − 8.29180i − 0.375737i −0.982194 0.187869i $$-0.939842\pi$$
0.982194 0.187869i $$-0.0601579\pi$$
$$488$$ 0 0
$$489$$ 35.6525 1.61226
$$490$$ 0 0
$$491$$ −26.8328 −1.21095 −0.605474 0.795865i $$-0.707016\pi$$
−0.605474 + 0.795865i $$0.707016\pi$$
$$492$$ 0 0
$$493$$ − 8.29180i − 0.373444i
$$494$$ 0 0
$$495$$ −4.79837 −0.215671
$$496$$ 0 0
$$497$$ 0.708204i 0.0317673i
$$498$$ 0 0
$$499$$ −15.4164 −0.690133 −0.345067 0.938578i $$-0.612144\pi$$
−0.345067 + 0.938578i $$0.612144\pi$$
$$500$$ 0 0
$$501$$ −15.7082 −0.701791
$$502$$ 0 0
$$503$$ − 10.1459i − 0.452383i −0.974083 0.226192i $$-0.927372\pi$$
0.974083 0.226192i $$-0.0726276\pi$$
$$504$$ 0 0
$$505$$ 3.41641 0.152028
$$506$$ 0 0
$$507$$ − 9.94427i − 0.441641i
$$508$$ 0 0
$$509$$ −6.65248 −0.294866 −0.147433 0.989072i $$-0.547101\pi$$
−0.147433 + 0.989072i $$0.547101\pi$$
$$510$$ 0 0
$$511$$ −30.5410 −1.35106
$$512$$ 0 0
$$513$$ 0.798374i 0.0352491i
$$514$$ 0 0
$$515$$ 24.2705i 1.06949i
$$516$$ 0 0
$$517$$ 9.59675i 0.422064i
$$518$$ 0 0
$$519$$ 26.7984 1.17632
$$520$$ 0 0
$$521$$ −38.0689 −1.66783 −0.833914 0.551894i $$-0.813905\pi$$
−0.833914 + 0.551894i $$0.813905\pi$$
$$522$$ 0 0
$$523$$ − 4.36068i − 0.190679i −0.995445 0.0953396i $$-0.969606\pi$$
0.995445 0.0953396i $$-0.0303937\pi$$
$$524$$ 0 0
$$525$$ −15.0000 −0.654654
$$526$$ 0 0
$$527$$ − 1.83282i − 0.0798387i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −2.29180 −0.0994555
$$532$$ 0 0
$$533$$ − 14.7082i − 0.637083i
$$534$$ 0 0
$$535$$ − 26.1803i − 1.13187i
$$536$$ 0 0
$$537$$ − 12.1803i − 0.525620i
$$538$$ 0 0
$$539$$ 20.0132 0.862028
$$540$$ 0 0
$$541$$ −0.583592 −0.0250906 −0.0125453 0.999921i $$-0.503993\pi$$
−0.0125453 + 0.999921i $$0.503993\pi$$
$$542$$ 0 0
$$543$$ − 25.1803i − 1.08059i
$$544$$ 0 0
$$545$$ 16.9098 0.724338
$$546$$ 0 0
$$547$$ − 7.85410i − 0.335817i −0.985803 0.167909i $$-0.946299\pi$$
0.985803 0.167909i $$-0.0537013\pi$$
$$548$$ 0 0
$$549$$ 1.09017 0.0465273
$$550$$ 0 0
$$551$$ −1.41641 −0.0603410
$$552$$ 0 0
$$553$$ 14.2918i 0.607749i
$$554$$ 0 0
$$555$$ −35.1246 −1.49096
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ 29.4164 1.24418
$$560$$ 0 0
$$561$$ −7.76393 −0.327793
$$562$$ 0 0
$$563$$ 15.7082i 0.662022i 0.943627 + 0.331011i $$0.107390\pi$$
−0.943627 + 0.331011i $$0.892610\pi$$
$$564$$ 0 0
$$565$$ 30.0000i 1.26211i
$$566$$ 0 0
$$567$$ 14.2918i 0.600199i
$$568$$ 0 0
$$569$$ 16.3607 0.685875 0.342938 0.939358i $$-0.388578\pi$$
0.342938 + 0.939358i $$0.388578\pi$$
$$570$$ 0 0
$$571$$ −8.27051 −0.346110 −0.173055 0.984912i $$-0.555364\pi$$
−0.173055 + 0.984912i $$0.555364\pi$$
$$572$$ 0 0
$$573$$ 41.1246i 1.71801i
$$574$$ 0 0
$$575$$ − 5.00000i − 0.208514i
$$576$$ 0 0
$$577$$ 39.7082i 1.65307i 0.562882 + 0.826537i $$0.309692\pi$$
−0.562882 + 0.826537i $$0.690308\pi$$
$$578$$ 0 0
$$579$$ −25.4164 −1.05627
$$580$$ 0 0
$$581$$ −14.2918 −0.592924
$$582$$ 0 0
$$583$$ − 11.2361i − 0.465350i
$$584$$ 0 0
$$585$$ − 2.23607i − 0.0924500i
$$586$$ 0 0
$$587$$ − 8.85410i − 0.365448i −0.983164 0.182724i $$-0.941509\pi$$
0.983164 0.182724i $$-0.0584915\pi$$
$$588$$ 0 0
$$589$$ −0.313082 −0.0129003
$$590$$ 0 0
$$591$$ 33.2705 1.36857
$$592$$ 0 0
$$593$$ 6.58359i 0.270356i 0.990821 + 0.135178i $$0.0431606\pi$$
−0.990821 + 0.135178i $$0.956839\pi$$
$$594$$ 0 0
$$595$$ 3.54102 0.145168
$$596$$ 0 0
$$597$$ − 18.4721i − 0.756014i
$$598$$ 0 0
$$599$$ 23.6180 0.965007 0.482503 0.875894i $$-0.339728\pi$$
0.482503 + 0.875894i $$0.339728\pi$$
$$600$$ 0 0
$$601$$ −9.85410 −0.401957 −0.200979 0.979596i $$-0.564412\pi$$
−0.200979 + 0.979596i $$0.564412\pi$$
$$602$$ 0 0
$$603$$ 2.00000i 0.0814463i
$$604$$ 0 0
$$605$$ −45.9787 −1.86930
$$606$$ 0 0
$$607$$ 21.0557i 0.854626i 0.904104 + 0.427313i $$0.140540\pi$$
−0.904104 + 0.427313i $$0.859460\pi$$
$$608$$ 0 0
$$609$$ −29.1246 −1.18019
$$610$$ 0 0
$$611$$ −4.47214 −0.180923
$$612$$ 0 0
$$613$$ − 0.763932i − 0.0308549i −0.999881 0.0154275i $$-0.995089\pi$$
0.999881 0.0154275i $$-0.00491091\pi$$
$$614$$ 0 0
$$615$$ − 20.3262i − 0.819633i
$$616$$ 0 0
$$617$$ 2.56231i 0.103155i 0.998669 + 0.0515773i $$0.0164248\pi$$
−0.998669 + 0.0515773i $$0.983575\pi$$
$$618$$ 0 0
$$619$$ −27.8541 −1.11955 −0.559775 0.828644i $$-0.689113\pi$$
−0.559775 + 0.828644i $$0.689113\pi$$
$$620$$ 0 0
$$621$$ −5.47214 −0.219589
$$622$$ 0 0
$$623$$ 6.87539i 0.275457i
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 1.32624i 0.0529648i
$$628$$ 0 0
$$629$$ 8.29180 0.330616
$$630$$ 0 0
$$631$$ 23.4164 0.932192 0.466096 0.884734i $$-0.345660\pi$$
0.466096 + 0.884734i $$0.345660\pi$$
$$632$$ 0 0
$$633$$ − 12.4721i − 0.495723i
$$634$$ 0 0
$$635$$ 21.7082i 0.861464i
$$636$$ 0 0
$$637$$ 9.32624i 0.369519i
$$638$$ 0 0
$$639$$ −0.145898 −0.00577164
$$640$$ 0 0
$$641$$ −0.652476 −0.0257712 −0.0128856 0.999917i $$-0.504102\pi$$
−0.0128856 + 0.999917i $$0.504102\pi$$
$$642$$ 0 0
$$643$$ 41.0132i 1.61740i 0.588221 + 0.808700i $$0.299829\pi$$
−0.588221 + 0.808700i $$0.700171\pi$$
$$644$$ 0 0
$$645$$ 40.6525 1.60069
$$646$$ 0 0
$$647$$ − 13.7082i − 0.538925i −0.963011 0.269463i $$-0.913154\pi$$
0.963011 0.269463i $$-0.0868460\pi$$
$$648$$ 0 0
$$649$$ −33.7082 −1.32316
$$650$$ 0 0
$$651$$ −6.43769 −0.252313
$$652$$ 0 0
$$653$$ 34.9787i 1.36882i 0.729096 + 0.684411i $$0.239941\pi$$
−0.729096 + 0.684411i $$0.760059\pi$$
$$654$$ 0 0
$$655$$ −31.7082 −1.23894
$$656$$ 0 0
$$657$$ − 6.29180i − 0.245466i
$$658$$ 0 0
$$659$$ −17.8885 −0.696839 −0.348419 0.937339i $$-0.613281\pi$$
−0.348419 + 0.937339i $$0.613281\pi$$
$$660$$ 0 0
$$661$$ −39.3951 −1.53229 −0.766146 0.642666i $$-0.777828\pi$$
−0.766146 + 0.642666i $$0.777828\pi$$
$$662$$ 0 0
$$663$$ − 3.61803i − 0.140513i
$$664$$ 0 0
$$665$$ − 0.604878i − 0.0234562i
$$666$$ 0 0
$$667$$ − 9.70820i − 0.375903i
$$668$$ 0 0
$$669$$ −36.1803 −1.39881
$$670$$ 0 0
$$671$$ 16.0344 0.619003
$$672$$ 0 0
$$673$$ − 27.7082i − 1.06807i −0.845461 0.534036i $$-0.820675\pi$$
0.845461 0.534036i $$-0.179325\pi$$
$$674$$ 0 0
$$675$$ − 27.3607i − 1.05311i
$$676$$ 0 0
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 0 0
$$679$$ −24.1672 −0.927451
$$680$$ 0 0
$$681$$ 16.6525 0.638124
$$682$$ 0 0
$$683$$ 10.9787i 0.420089i 0.977692 + 0.210044i $$0.0673609\pi$$
−0.977692 + 0.210044i $$0.932639\pi$$
$$684$$ 0 0
$$685$$ − 30.9787i − 1.18364i
$$686$$ 0 0
$$687$$ − 16.1803i − 0.617318i
$$688$$ 0 0
$$689$$ 5.23607 0.199478
$$690$$ 0 0
$$691$$ 21.1246 0.803618 0.401809 0.915723i $$-0.368382\pi$$
0.401809 + 0.915723i $$0.368382\pi$$
$$692$$ 0 0
$$693$$ − 3.97871i − 0.151139i
$$694$$ 0 0
$$695$$ −9.59675 −0.364025
$$696$$ 0 0
$$697$$ 4.79837i 0.181751i
$$698$$ 0 0
$$699$$ 34.1803 1.29282
$$700$$ 0 0
$$701$$ −27.9230 −1.05464 −0.527318 0.849668i $$-0.676802\pi$$
−0.527318 + 0.849668i $$0.676802\pi$$
$$702$$ 0 0
$$703$$ − 1.41641i − 0.0534208i
$$704$$ 0 0
$$705$$ −6.18034 −0.232765
$$706$$ 0 0
$$707$$ 2.83282i 0.106539i
$$708$$ 0 0
$$709$$ −8.56231 −0.321564 −0.160782 0.986990i $$-0.551402\pi$$
−0.160782 + 0.986990i $$0.551402\pi$$
$$710$$ 0 0
$$711$$ −2.94427 −0.110419
$$712$$ 0 0
$$713$$ − 2.14590i − 0.0803645i
$$714$$ 0 0
$$715$$ − 32.8885i − 1.22996i
$$716$$ 0 0
$$717$$ 16.9443i 0.632795i
$$718$$ 0 0
$$719$$ 23.4508 0.874569 0.437285 0.899323i $$-0.355940\pi$$
0.437285 + 0.899323i $$0.355940\pi$$
$$720$$ 0 0
$$721$$ −20.1246 −0.749480
$$722$$ 0 0
$$723$$ 3.23607i 0.120351i
$$724$$ 0 0
$$725$$ 48.5410 1.80277
$$726$$ 0 0
$$727$$ 40.7426i 1.51106i 0.655113 + 0.755531i $$0.272621\pi$$
−0.655113 + 0.755531i $$0.727379\pi$$
$$728$$ 0 0
$$729$$ −29.5066 −1.09284
$$730$$ 0 0
$$731$$ −9.59675 −0.354949
$$732$$ 0 0
$$733$$ 0.111456i 0.00411673i 0.999998 + 0.00205836i $$0.000655198\pi$$
−0.999998 + 0.00205836i $$0.999345\pi$$
$$734$$ 0 0
$$735$$ 12.8885i 0.475401i
$$736$$ 0 0
$$737$$ 29.4164i 1.08357i
$$738$$ 0 0
$$739$$ 34.5410 1.27061 0.635306 0.772261i $$-0.280874\pi$$
0.635306 + 0.772261i $$0.280874\pi$$
$$740$$ 0 0
$$741$$ −0.618034 −0.0227040
$$742$$ 0 0
$$743$$ − 36.9787i − 1.35662i −0.734777 0.678309i $$-0.762713\pi$$
0.734777 0.678309i $$-0.237287\pi$$
$$744$$ 0 0
$$745$$ −5.85410 −0.214478
$$746$$ 0 0
$$747$$ − 2.94427i − 0.107725i
$$748$$ 0 0
$$749$$ 21.7082 0.793201
$$750$$ 0 0
$$751$$ 0.875388 0.0319434 0.0159717 0.999872i $$-0.494916\pi$$
0.0159717 + 0.999872i $$0.494916\pi$$
$$752$$ 0 0
$$753$$ − 27.1803i − 0.990507i
$$754$$ 0 0
$$755$$ 31.9098 1.16132
$$756$$ 0 0
$$757$$ − 29.0132i − 1.05450i −0.849710 0.527251i $$-0.823223\pi$$
0.849710 0.527251i $$-0.176777\pi$$
$$758$$ 0 0
$$759$$ −9.09017 −0.329952
$$760$$ 0 0
$$761$$ 26.5066 0.960863 0.480431 0.877032i $$-0.340480\pi$$
0.480431 + 0.877032i $$0.340480\pi$$
$$762$$ 0 0
$$763$$ 14.0213i 0.507605i
$$764$$ 0 0
$$765$$ 0.729490i 0.0263748i
$$766$$ 0 0
$$767$$ − 15.7082i − 0.567190i
$$768$$ 0 0
$$769$$ 47.7082 1.72040 0.860201 0.509955i $$-0.170338\pi$$
0.860201 + 0.509955i $$0.170338\pi$$
$$770$$ 0 0
$$771$$ 18.4721 0.665258
$$772$$ 0 0
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ 0 0
$$775$$ 10.7295 0.385415
$$776$$ 0 0
$$777$$ − 29.1246i − 1.04484i
$$778$$ 0 0
$$779$$ 0.819660 0.0293674
$$780$$ 0 0
$$781$$ −2.14590 −0.0767863
$$782$$ 0 0
$$783$$ − 53.1246i − 1.89852i
$$784$$ 0 0
$$785$$ 5.12461i 0.182905i
$$786$$ 0 0
$$787$$ 4.58359i 0.163387i 0.996657 + 0.0816937i $$0.0260329\pi$$
−0.996657 + 0.0816937i $$0.973967\pi$$
$$788$$ 0 0
$$789$$ 29.5623 1.05245
$$790$$ 0 0
$$791$$ −24.8754 −0.884467
$$792$$ 0 0
$$793$$ 7.47214i 0.265343i
$$794$$ 0 0
$$795$$ 7.23607 0.256637
$$796$$ 0 0
$$797$$ − 45.4164i − 1.60873i −0.594134 0.804366i $$-0.702505\pi$$
0.594134 0.804366i $$-0.297495\pi$$
$$798$$ 0 0
$$799$$ 1.45898 0.0516150
$$800$$ 0 0
$$801$$ −1.41641 −0.0500463
$$802$$ 0 0
$$803$$ − 92.5410i − 3.26570i
$$804$$ 0 0
$$805$$ 4.14590 0.146124
$$806$$ 0 0
$$807$$ − 2.47214i − 0.0870233i
$$808$$ 0 0
$$809$$ 42.1591 1.48223 0.741117 0.671376i $$-0.234297\pi$$
0.741117 + 0.671376i $$0.234297\pi$$
$$810$$ 0 0
$$811$$ −5.70820 −0.200442 −0.100221 0.994965i $$-0.531955\pi$$
−0.100221 + 0.994965i $$0.531955\pi$$
$$812$$ 0 0
$$813$$ 41.8328i 1.46714i
$$814$$ 0 0
$$815$$ − 49.2705i − 1.72587i
$$816$$ 0 0
$$817$$ 1.63932i 0.0573526i
$$818$$ 0 0
$$819$$ 1.85410 0.0647876
$$820$$ 0 0
$$821$$ 14.9443 0.521559 0.260779 0.965398i $$-0.416020\pi$$
0.260779 + 0.965398i $$0.416020\pi$$
$$822$$ 0 0
$$823$$ 32.8328i 1.14448i 0.820086 + 0.572240i $$0.193925\pi$$
−0.820086 + 0.572240i $$0.806075\pi$$
$$824$$ 0 0
$$825$$ − 45.4508i − 1.58240i
$$826$$ 0 0
$$827$$ − 32.2492i − 1.12142i −0.828014 0.560708i $$-0.810529\pi$$
0.828014 0.560708i $$-0.189471\pi$$
$$828$$ 0 0
$$829$$ 34.5410 1.19966 0.599830 0.800128i $$-0.295235\pi$$
0.599830 + 0.800128i $$0.295235\pi$$
$$830$$ 0 0
$$831$$ −12.1803 −0.422531
$$832$$ 0 0
$$833$$ − 3.04257i − 0.105419i
$$834$$ 0 0
$$835$$ 21.7082i 0.751243i
$$836$$ 0 0
$$837$$ − 11.7426i − 0.405885i
$$838$$ 0 0
$$839$$ −11.2361 −0.387912 −0.193956 0.981010i $$-0.562132\pi$$
−0.193956 + 0.981010i $$0.562132\pi$$
$$840$$ 0 0
$$841$$ 65.2492 2.24997
$$842$$ 0 0
$$843$$ − 49.5967i − 1.70820i
$$844$$ 0 0
$$845$$ −13.7426 −0.472761
$$846$$ 0 0
$$847$$ − 38.1246i − 1.30998i
$$848$$ 0 0
$$849$$ −33.8885 −1.16305
$$850$$ 0 0
$$851$$ 9.70820 0.332793
$$852$$ 0 0
$$853$$ 0.214782i 0.00735399i 0.999993 + 0.00367699i $$0.00117043\pi$$
−0.999993 + 0.00367699i $$0.998830\pi$$
$$854$$ 0 0
$$855$$ 0.124612 0.00426163
$$856$$ 0 0
$$857$$ 36.0000i 1.22974i 0.788630 + 0.614868i $$0.210791\pi$$
−0.788630 + 0.614868i $$0.789209\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 16.8541 0.574386
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 0 0
$$865$$ − 37.0344i − 1.25921i
$$866$$ 0 0
$$867$$ − 26.3262i − 0.894086i
$$868$$ 0 0
$$869$$ −43.3050 −1.46902
$$870$$ 0 0
$$871$$ −13.7082 −0.464485
$$872$$ 0 0
$$873$$ − 4.97871i − 0.168504i
$$874$$ 0 0
$$875$$ 20.7295i 0.700785i
$$876$$ 0 0
$$877$$ − 25.6869i − 0.867386i −0.901061 0.433693i $$-0.857210\pi$$
0.901061 0.433693i $$-0.142790\pi$$
$$878$$ 0 0
$$879$$ −23.1246 −0.779974
$$880$$ 0 0
$$881$$ −51.7082 −1.74209 −0.871047 0.491200i $$-0.836558\pi$$
−0.871047 + 0.491200i $$0.836558\pi$$
$$882$$ 0 0
$$883$$ 51.2705i 1.72539i 0.505725 + 0.862695i $$0.331225\pi$$
−0.505725 + 0.862695i $$0.668775\pi$$
$$884$$ 0 0
$$885$$ − 21.7082i − 0.729713i
$$886$$ 0 0
$$887$$ 30.5410i 1.02547i 0.858548 + 0.512734i $$0.171367\pi$$
−0.858548 + 0.512734i $$0.828633\pi$$
$$888$$ 0 0
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ −43.3050 −1.45077
$$892$$ 0 0
$$893$$ − 0.249224i − 0.00833995i
$$894$$ 0 0
$$895$$ −16.8328 −0.562659
$$896$$ 0 0
$$897$$ − 4.23607i − 0.141438i
$$898$$ 0 0
$$899$$ 20.8328 0.694813
$$900$$ 0 0
$$901$$ −1.70820 −0.0569085
$$902$$ 0 0
$$903$$ 33.7082i 1.12174i
$$904$$ 0 0
$$905$$ −34.7984 −1.15674
$$906$$ 0 0
$$907$$ − 36.5410i − 1.21332i −0.794960 0.606662i $$-0.792508\pi$$
0.794960 0.606662i $$-0.207492\pi$$
$$908$$ 0 0
$$909$$ −0.583592 −0.0193565
$$910$$ 0 0
$$911$$ 22.3607 0.740842 0.370421 0.928864i $$-0.379213\pi$$
0.370421 + 0.928864i $$0.379213\pi$$
$$912$$ 0 0
$$913$$ − 43.3050i − 1.43318i
$$914$$ 0 0
$$915$$ 10.3262i 0.341375i
$$916$$ 0 0
$$917$$ − 26.2918i − 0.868232i
$$918$$ 0 0
$$919$$ −41.4164 −1.36620 −0.683101 0.730324i $$-0.739369\pi$$
−0.683101 + 0.730324i $$0.739369\pi$$
$$920$$ 0 0
$$921$$ 27.2705 0.898594
$$922$$ 0 0
$$923$$ − 1.00000i − 0.0329154i
$$924$$ 0 0
$$925$$ 48.5410i 1.59602i
$$926$$ 0 0
$$927$$ − 4.14590i − 0.136169i
$$928$$ 0 0
$$929$$ 43.3050 1.42079 0.710395 0.703804i $$-0.248516\pi$$
0.710395 + 0.703804i $$0.248516\pi$$
$$930$$ 0 0
$$931$$ −0.519733 −0.0170336
$$932$$ 0 0
$$933$$ − 36.3607i − 1.19040i
$$934$$ 0 0
$$935$$ 10.7295i 0.350892i
$$936$$ 0 0
$$937$$ 24.3262i 0.794704i 0.917666 + 0.397352i $$0.130071\pi$$
−0.917666 + 0.397352i $$0.869929\pi$$
$$938$$ 0 0
$$939$$ 41.8328 1.36516
$$940$$ 0 0
$$941$$ 27.2148 0.887177 0.443588 0.896231i $$-0.353705\pi$$
0.443588 + 0.896231i $$0.353705\pi$$
$$942$$ 0 0
$$943$$ 5.61803i 0.182948i
$$944$$ 0 0
$$945$$ 22.6869 0.738005
$$946$$ 0 0
$$947$$ 26.3951i 0.857726i 0.903369 + 0.428863i $$0.141086\pi$$
−0.903369 + 0.428863i $$0.858914\pi$$
$$948$$ 0 0
$$949$$ 43.1246 1.39988
$$950$$ 0 0
$$951$$ 11.7639 0.381472
$$952$$ 0 0
$$953$$ 37.3951i 1.21135i 0.795713 + 0.605673i $$0.207096\pi$$
−0.795713 + 0.605673i $$0.792904\pi$$
$$954$$ 0 0
$$955$$ 56.8328 1.83907
$$956$$ 0 0
$$957$$ − 88.2492i − 2.85269i
$$958$$ 0 0
$$959$$ 25.6869 0.829474
$$960$$ 0 0
$$961$$ −26.3951 −0.851456
$$962$$ 0 0
$$963$$ 4.47214i 0.144113i
$$964$$ 0 0
$$965$$ 35.1246i 1.13070i
$$966$$ 0 0
$$967$$ 29.2361i 0.940169i 0.882622 + 0.470084i $$0.155776\pi$$
−0.882622 + 0.470084i $$0.844224\pi$$
$$968$$ 0 0
$$969$$ 0.201626 0.00647716
$$970$$ 0 0
$$971$$ −19.1459 −0.614421 −0.307211 0.951642i $$-0.599396\pi$$
−0.307211 + 0.951642i $$0.599396\pi$$
$$972$$ 0 0
$$973$$ − 7.95743i − 0.255103i
$$974$$ 0 0
$$975$$ 21.1803 0.678314
$$976$$ 0 0
$$977$$ − 1.27051i − 0.0406472i −0.999793 0.0203236i $$-0.993530\pi$$
0.999793 0.0203236i $$-0.00646965\pi$$
$$978$$ 0 0
$$979$$ −20.8328 −0.665820
$$980$$ 0 0
$$981$$ −2.88854 −0.0922241
$$982$$ 0 0
$$983$$ 47.3951i 1.51167i 0.654762 + 0.755835i $$0.272769\pi$$
−0.654762 + 0.755835i $$0.727231\pi$$
$$984$$ 0 0
$$985$$ − 45.9787i − 1.46500i
$$986$$ 0 0
$$987$$ − 5.12461i − 0.163118i
$$988$$ 0 0
$$989$$ −11.2361 −0.357286
$$990$$ 0 0
$$991$$ −17.7295 −0.563196 −0.281598 0.959532i $$-0.590864\pi$$
−0.281598 + 0.959532i $$0.590864\pi$$
$$992$$ 0 0
$$993$$ − 40.6525i − 1.29007i
$$994$$ 0 0
$$995$$ −25.5279 −0.809288
$$996$$ 0 0
$$997$$ 62.7214i 1.98641i 0.116398 + 0.993203i $$0.462865\pi$$
−0.116398 + 0.993203i $$0.537135\pi$$
$$998$$ 0 0
$$999$$ 53.1246 1.68079
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 1840.2.e.c.369.1 4
4.3 odd 2 230.2.b.a.139.4 yes 4
5.2 odd 4 9200.2.a.bo.1.1 2
5.3 odd 4 9200.2.a.by.1.2 2
5.4 even 2 inner 1840.2.e.c.369.4 4
12.11 even 2 2070.2.d.c.829.2 4
20.3 even 4 1150.2.a.n.1.1 2
20.7 even 4 1150.2.a.l.1.2 2
20.19 odd 2 230.2.b.a.139.1 4
60.59 even 2 2070.2.d.c.829.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.1 4 20.19 odd 2
230.2.b.a.139.4 yes 4 4.3 odd 2
1150.2.a.l.1.2 2 20.7 even 4
1150.2.a.n.1.1 2 20.3 even 4
1840.2.e.c.369.1 4 1.1 even 1 trivial
1840.2.e.c.369.4 4 5.4 even 2 inner
2070.2.d.c.829.2 4 12.11 even 2
2070.2.d.c.829.4 4 60.59 even 2
9200.2.a.bo.1.1 2 5.2 odd 4
9200.2.a.by.1.2 2 5.3 odd 4