# Properties

 Label 1183.2.c.d.337.4 Level $1183$ Weight $2$ Character 1183.337 Analytic conductor $9.446$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.c (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: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.4 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.337 Dual form 1183.2.c.d.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421i q^{2} +1.41421 q^{3} -1.58579i q^{5} +2.00000i q^{6} +1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.41421i q^{2} +1.41421 q^{3} -1.58579i q^{5} +2.00000i q^{6} +1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +2.24264 q^{10} +4.24264i q^{11} -1.41421 q^{14} -2.24264i q^{15} -4.00000 q^{16} -1.41421 q^{17} -1.41421i q^{18} +7.24264i q^{19} +1.41421i q^{21} -6.00000 q^{22} +5.82843 q^{23} +4.00000i q^{24} +2.48528 q^{25} -5.65685 q^{27} +0.171573 q^{29} +3.17157 q^{30} -3.24264i q^{31} +6.00000i q^{33} -2.00000i q^{34} +1.58579 q^{35} +2.24264i q^{37} -10.2426 q^{38} +4.48528 q^{40} -8.82843i q^{41} -2.00000 q^{42} +5.00000 q^{43} +1.58579i q^{45} +8.24264i q^{46} +1.58579i q^{47} -5.65685 q^{48} -1.00000 q^{49} +3.51472i q^{50} -2.00000 q^{51} -0.171573 q^{53} -8.00000i q^{54} +6.72792 q^{55} -2.82843 q^{56} +10.2426i q^{57} +0.242641i q^{58} +0.343146i q^{59} +6.00000 q^{61} +4.58579 q^{62} -1.00000i q^{63} -8.00000 q^{64} -8.48528 q^{66} +14.4853i q^{67} +8.24264 q^{69} +2.24264i q^{70} +13.0711i q^{71} -2.82843i q^{72} -9.24264i q^{73} -3.17157 q^{74} +3.51472 q^{75} -4.24264 q^{77} +15.4853 q^{79} +6.34315i q^{80} -5.00000 q^{81} +12.4853 q^{82} -13.2426i q^{83} +2.24264i q^{85} +7.07107i q^{86} +0.242641 q^{87} -12.0000 q^{88} +1.58579i q^{89} -2.24264 q^{90} -4.58579i q^{93} -2.24264 q^{94} +11.4853 q^{95} -11.7279i q^{97} -1.41421i q^{98} -4.24264i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^9 - 8 * q^10 - 16 * q^16 - 24 * q^22 + 12 * q^23 - 24 * q^25 + 12 * q^29 + 24 * q^30 + 12 * q^35 - 24 * q^38 - 16 * q^40 - 8 * q^42 + 20 * q^43 - 4 * q^49 - 8 * q^51 - 12 * q^53 - 24 * q^55 + 24 * q^61 + 24 * q^62 - 32 * q^64 + 16 * q^69 - 24 * q^74 + 48 * q^75 + 28 * q^79 - 20 * q^81 + 16 * q^82 - 16 * q^87 - 48 * q^88 + 8 * q^90 + 8 * q^94 + 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.41421i 1.00000i 0.866025 + 0.500000i $$0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$3$$ 1.41421 0.816497 0.408248 0.912871i $$-0.366140\pi$$
0.408248 + 0.912871i $$0.366140\pi$$
$$4$$ 0 0
$$5$$ − 1.58579i − 0.709185i −0.935021 0.354593i $$-0.884620\pi$$
0.935021 0.354593i $$-0.115380\pi$$
$$6$$ 2.00000i 0.816497i
$$7$$ 1.00000i 0.377964i
$$8$$ 2.82843i 1.00000i
$$9$$ −1.00000 −0.333333
$$10$$ 2.24264 0.709185
$$11$$ 4.24264i 1.27920i 0.768706 + 0.639602i $$0.220901\pi$$
−0.768706 + 0.639602i $$0.779099\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −1.41421 −0.377964
$$15$$ − 2.24264i − 0.579047i
$$16$$ −4.00000 −1.00000
$$17$$ −1.41421 −0.342997 −0.171499 0.985184i $$-0.554861\pi$$
−0.171499 + 0.985184i $$0.554861\pi$$
$$18$$ − 1.41421i − 0.333333i
$$19$$ 7.24264i 1.66158i 0.556589 + 0.830788i $$0.312110\pi$$
−0.556589 + 0.830788i $$0.687890\pi$$
$$20$$ 0 0
$$21$$ 1.41421i 0.308607i
$$22$$ −6.00000 −1.27920
$$23$$ 5.82843 1.21531 0.607656 0.794201i $$-0.292110\pi$$
0.607656 + 0.794201i $$0.292110\pi$$
$$24$$ 4.00000i 0.816497i
$$25$$ 2.48528 0.497056
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ 0.171573 0.0318603 0.0159301 0.999873i $$-0.494929\pi$$
0.0159301 + 0.999873i $$0.494929\pi$$
$$30$$ 3.17157 0.579047
$$31$$ − 3.24264i − 0.582395i −0.956663 0.291198i $$-0.905946\pi$$
0.956663 0.291198i $$-0.0940538\pi$$
$$32$$ 0 0
$$33$$ 6.00000i 1.04447i
$$34$$ − 2.00000i − 0.342997i
$$35$$ 1.58579 0.268047
$$36$$ 0 0
$$37$$ 2.24264i 0.368688i 0.982862 + 0.184344i $$0.0590160\pi$$
−0.982862 + 0.184344i $$0.940984\pi$$
$$38$$ −10.2426 −1.66158
$$39$$ 0 0
$$40$$ 4.48528 0.709185
$$41$$ − 8.82843i − 1.37877i −0.724396 0.689384i $$-0.757881\pi$$
0.724396 0.689384i $$-0.242119\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 0 0
$$45$$ 1.58579i 0.236395i
$$46$$ 8.24264i 1.21531i
$$47$$ 1.58579i 0.231311i 0.993289 + 0.115655i $$0.0368968\pi$$
−0.993289 + 0.115655i $$0.963103\pi$$
$$48$$ −5.65685 −0.816497
$$49$$ −1.00000 −0.142857
$$50$$ 3.51472i 0.497056i
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ −0.171573 −0.0235673 −0.0117837 0.999931i $$-0.503751\pi$$
−0.0117837 + 0.999931i $$0.503751\pi$$
$$54$$ − 8.00000i − 1.08866i
$$55$$ 6.72792 0.907193
$$56$$ −2.82843 −0.377964
$$57$$ 10.2426i 1.35667i
$$58$$ 0.242641i 0.0318603i
$$59$$ 0.343146i 0.0446738i 0.999751 + 0.0223369i $$0.00711064\pi$$
−0.999751 + 0.0223369i $$0.992889\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.58579 0.582395
$$63$$ − 1.00000i − 0.125988i
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ −8.48528 −1.04447
$$67$$ 14.4853i 1.76966i 0.465915 + 0.884829i $$0.345725\pi$$
−0.465915 + 0.884829i $$0.654275\pi$$
$$68$$ 0 0
$$69$$ 8.24264 0.992297
$$70$$ 2.24264i 0.268047i
$$71$$ 13.0711i 1.55125i 0.631194 + 0.775625i $$0.282565\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$72$$ − 2.82843i − 0.333333i
$$73$$ − 9.24264i − 1.08177i −0.841097 0.540885i $$-0.818090\pi$$
0.841097 0.540885i $$-0.181910\pi$$
$$74$$ −3.17157 −0.368688
$$75$$ 3.51472 0.405845
$$76$$ 0 0
$$77$$ −4.24264 −0.483494
$$78$$ 0 0
$$79$$ 15.4853 1.74223 0.871115 0.491079i $$-0.163397\pi$$
0.871115 + 0.491079i $$0.163397\pi$$
$$80$$ 6.34315i 0.709185i
$$81$$ −5.00000 −0.555556
$$82$$ 12.4853 1.37877
$$83$$ − 13.2426i − 1.45357i −0.686866 0.726784i $$-0.741014\pi$$
0.686866 0.726784i $$-0.258986\pi$$
$$84$$ 0 0
$$85$$ 2.24264i 0.243249i
$$86$$ 7.07107i 0.762493i
$$87$$ 0.242641 0.0260138
$$88$$ −12.0000 −1.27920
$$89$$ 1.58579i 0.168093i 0.996462 + 0.0840465i $$0.0267844\pi$$
−0.996462 + 0.0840465i $$0.973216\pi$$
$$90$$ −2.24264 −0.236395
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 4.58579i − 0.475524i
$$94$$ −2.24264 −0.231311
$$95$$ 11.4853 1.17837
$$96$$ 0 0
$$97$$ − 11.7279i − 1.19079i −0.803433 0.595395i $$-0.796996\pi$$
0.803433 0.595395i $$-0.203004\pi$$
$$98$$ − 1.41421i − 0.142857i
$$99$$ − 4.24264i − 0.426401i
$$100$$ 0 0
$$101$$ −10.2426 −1.01918 −0.509590 0.860417i $$-0.670203\pi$$
−0.509590 + 0.860417i $$0.670203\pi$$
$$102$$ − 2.82843i − 0.280056i
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 2.24264 0.218859
$$106$$ − 0.242641i − 0.0235673i
$$107$$ −20.1421 −1.94721 −0.973607 0.228232i $$-0.926706\pi$$
−0.973607 + 0.228232i $$0.926706\pi$$
$$108$$ 0 0
$$109$$ − 16.7279i − 1.60224i −0.598501 0.801122i $$-0.704237\pi$$
0.598501 0.801122i $$-0.295763\pi$$
$$110$$ 9.51472i 0.907193i
$$111$$ 3.17157i 0.301032i
$$112$$ − 4.00000i − 0.377964i
$$113$$ 2.31371 0.217655 0.108828 0.994061i $$-0.465290\pi$$
0.108828 + 0.994061i $$0.465290\pi$$
$$114$$ −14.4853 −1.35667
$$115$$ − 9.24264i − 0.861881i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ −0.485281 −0.0446738
$$119$$ − 1.41421i − 0.129641i
$$120$$ 6.34315 0.579047
$$121$$ −7.00000 −0.636364
$$122$$ 8.48528i 0.768221i
$$123$$ − 12.4853i − 1.12576i
$$124$$ 0 0
$$125$$ − 11.8701i − 1.06169i
$$126$$ 1.41421 0.125988
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ − 11.3137i − 1.00000i
$$129$$ 7.07107 0.622573
$$130$$ 0 0
$$131$$ −2.82843 −0.247121 −0.123560 0.992337i $$-0.539431\pi$$
−0.123560 + 0.992337i $$0.539431\pi$$
$$132$$ 0 0
$$133$$ −7.24264 −0.628017
$$134$$ −20.4853 −1.76966
$$135$$ 8.97056i 0.772063i
$$136$$ − 4.00000i − 0.342997i
$$137$$ − 4.58579i − 0.391790i −0.980625 0.195895i $$-0.937239\pi$$
0.980625 0.195895i $$-0.0627612\pi$$
$$138$$ 11.6569i 0.992297i
$$139$$ 6.24264 0.529494 0.264747 0.964318i $$-0.414712\pi$$
0.264747 + 0.964318i $$0.414712\pi$$
$$140$$ 0 0
$$141$$ 2.24264i 0.188864i
$$142$$ −18.4853 −1.55125
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ − 0.272078i − 0.0225948i
$$146$$ 13.0711 1.08177
$$147$$ −1.41421 −0.116642
$$148$$ 0 0
$$149$$ − 16.2426i − 1.33065i −0.746554 0.665324i $$-0.768293\pi$$
0.746554 0.665324i $$-0.231707\pi$$
$$150$$ 4.97056i 0.405845i
$$151$$ − 9.75736i − 0.794043i −0.917809 0.397021i $$-0.870044\pi$$
0.917809 0.397021i $$-0.129956\pi$$
$$152$$ −20.4853 −1.66158
$$153$$ 1.41421 0.114332
$$154$$ − 6.00000i − 0.483494i
$$155$$ −5.14214 −0.413026
$$156$$ 0 0
$$157$$ −3.75736 −0.299870 −0.149935 0.988696i $$-0.547906\pi$$
−0.149935 + 0.988696i $$0.547906\pi$$
$$158$$ 21.8995i 1.74223i
$$159$$ −0.242641 −0.0192427
$$160$$ 0 0
$$161$$ 5.82843i 0.459344i
$$162$$ − 7.07107i − 0.555556i
$$163$$ 8.48528i 0.664619i 0.943170 + 0.332309i $$0.107828\pi$$
−0.943170 + 0.332309i $$0.892172\pi$$
$$164$$ 0 0
$$165$$ 9.51472 0.740720
$$166$$ 18.7279 1.45357
$$167$$ 15.3848i 1.19051i 0.803537 + 0.595255i $$0.202949\pi$$
−0.803537 + 0.595255i $$0.797051\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 0 0
$$170$$ −3.17157 −0.243249
$$171$$ − 7.24264i − 0.553859i
$$172$$ 0 0
$$173$$ −24.7279 −1.88003 −0.940015 0.341134i $$-0.889189\pi$$
−0.940015 + 0.341134i $$0.889189\pi$$
$$174$$ 0.343146i 0.0260138i
$$175$$ 2.48528i 0.187870i
$$176$$ − 16.9706i − 1.27920i
$$177$$ 0.485281i 0.0364760i
$$178$$ −2.24264 −0.168093
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ 18.7279 1.39204 0.696018 0.718025i $$-0.254953\pi$$
0.696018 + 0.718025i $$0.254953\pi$$
$$182$$ 0 0
$$183$$ 8.48528 0.627250
$$184$$ 16.4853i 1.21531i
$$185$$ 3.55635 0.261468
$$186$$ 6.48528 0.475524
$$187$$ − 6.00000i − 0.438763i
$$188$$ 0 0
$$189$$ − 5.65685i − 0.411476i
$$190$$ 16.2426i 1.17837i
$$191$$ 15.1716 1.09778 0.548888 0.835896i $$-0.315051\pi$$
0.548888 + 0.835896i $$0.315051\pi$$
$$192$$ −11.3137 −0.816497
$$193$$ − 14.4853i − 1.04267i −0.853351 0.521337i $$-0.825434\pi$$
0.853351 0.521337i $$-0.174566\pi$$
$$194$$ 16.5858 1.19079
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 12.3431i − 0.879413i −0.898142 0.439706i $$-0.855083\pi$$
0.898142 0.439706i $$-0.144917\pi$$
$$198$$ 6.00000 0.426401
$$199$$ 3.75736 0.266352 0.133176 0.991092i $$-0.457482\pi$$
0.133176 + 0.991092i $$0.457482\pi$$
$$200$$ 7.02944i 0.497056i
$$201$$ 20.4853i 1.44492i
$$202$$ − 14.4853i − 1.01918i
$$203$$ 0.171573i 0.0120421i
$$204$$ 0 0
$$205$$ −14.0000 −0.977802
$$206$$ 11.3137i 0.788263i
$$207$$ −5.82843 −0.405104
$$208$$ 0 0
$$209$$ −30.7279 −2.12549
$$210$$ 3.17157i 0.218859i
$$211$$ −15.9706 −1.09946 −0.549729 0.835343i $$-0.685269\pi$$
−0.549729 + 0.835343i $$0.685269\pi$$
$$212$$ 0 0
$$213$$ 18.4853i 1.26659i
$$214$$ − 28.4853i − 1.94721i
$$215$$ − 7.92893i − 0.540749i
$$216$$ − 16.0000i − 1.08866i
$$217$$ 3.24264 0.220125
$$218$$ 23.6569 1.60224
$$219$$ − 13.0711i − 0.883261i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −4.48528 −0.301032
$$223$$ − 0.757359i − 0.0507165i −0.999678 0.0253583i $$-0.991927\pi$$
0.999678 0.0253583i $$-0.00807265\pi$$
$$224$$ 0 0
$$225$$ −2.48528 −0.165685
$$226$$ 3.27208i 0.217655i
$$227$$ − 26.8284i − 1.78067i −0.455311 0.890333i $$-0.650472\pi$$
0.455311 0.890333i $$-0.349528\pi$$
$$228$$ 0 0
$$229$$ 29.4558i 1.94650i 0.229755 + 0.973248i $$0.426207\pi$$
−0.229755 + 0.973248i $$0.573793\pi$$
$$230$$ 13.0711 0.861881
$$231$$ −6.00000 −0.394771
$$232$$ 0.485281i 0.0318603i
$$233$$ 14.6569 0.960202 0.480101 0.877213i $$-0.340600\pi$$
0.480101 + 0.877213i $$0.340600\pi$$
$$234$$ 0 0
$$235$$ 2.51472 0.164042
$$236$$ 0 0
$$237$$ 21.8995 1.42253
$$238$$ 2.00000 0.129641
$$239$$ − 3.51472i − 0.227348i −0.993518 0.113674i $$-0.963738\pi$$
0.993518 0.113674i $$-0.0362620\pi$$
$$240$$ 8.97056i 0.579047i
$$241$$ − 20.2132i − 1.30205i −0.759058 0.651023i $$-0.774340\pi$$
0.759058 0.651023i $$-0.225660\pi$$
$$242$$ − 9.89949i − 0.636364i
$$243$$ 9.89949 0.635053
$$244$$ 0 0
$$245$$ 1.58579i 0.101312i
$$246$$ 17.6569 1.12576
$$247$$ 0 0
$$248$$ 9.17157 0.582395
$$249$$ − 18.7279i − 1.18683i
$$250$$ 16.7868 1.06169
$$251$$ −16.5858 −1.04689 −0.523443 0.852061i $$-0.675353\pi$$
−0.523443 + 0.852061i $$0.675353\pi$$
$$252$$ 0 0
$$253$$ 24.7279i 1.55463i
$$254$$ − 2.82843i − 0.177471i
$$255$$ 3.17157i 0.198612i
$$256$$ 0 0
$$257$$ 19.4142 1.21103 0.605513 0.795836i $$-0.292968\pi$$
0.605513 + 0.795836i $$0.292968\pi$$
$$258$$ 10.0000i 0.622573i
$$259$$ −2.24264 −0.139351
$$260$$ 0 0
$$261$$ −0.171573 −0.0106201
$$262$$ − 4.00000i − 0.247121i
$$263$$ 1.97056 0.121510 0.0607551 0.998153i $$-0.480649\pi$$
0.0607551 + 0.998153i $$0.480649\pi$$
$$264$$ −16.9706 −1.04447
$$265$$ 0.272078i 0.0167136i
$$266$$ − 10.2426i − 0.628017i
$$267$$ 2.24264i 0.137247i
$$268$$ 0 0
$$269$$ 9.17157 0.559201 0.279600 0.960116i $$-0.409798\pi$$
0.279600 + 0.960116i $$0.409798\pi$$
$$270$$ −12.6863 −0.772063
$$271$$ 20.0000i 1.21491i 0.794353 + 0.607457i $$0.207810\pi$$
−0.794353 + 0.607457i $$0.792190\pi$$
$$272$$ 5.65685 0.342997
$$273$$ 0 0
$$274$$ 6.48528 0.391790
$$275$$ 10.5442i 0.635837i
$$276$$ 0 0
$$277$$ 7.48528 0.449747 0.224873 0.974388i $$-0.427803\pi$$
0.224873 + 0.974388i $$0.427803\pi$$
$$278$$ 8.82843i 0.529494i
$$279$$ 3.24264i 0.194132i
$$280$$ 4.48528i 0.268047i
$$281$$ 15.5563i 0.928014i 0.885832 + 0.464007i $$0.153589\pi$$
−0.885832 + 0.464007i $$0.846411\pi$$
$$282$$ −3.17157 −0.188864
$$283$$ 8.48528 0.504398 0.252199 0.967675i $$-0.418846\pi$$
0.252199 + 0.967675i $$0.418846\pi$$
$$284$$ 0 0
$$285$$ 16.2426 0.962131
$$286$$ 0 0
$$287$$ 8.82843 0.521126
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 0.384776 0.0225948
$$291$$ − 16.5858i − 0.972276i
$$292$$ 0 0
$$293$$ 15.3848i 0.898788i 0.893334 + 0.449394i $$0.148360\pi$$
−0.893334 + 0.449394i $$0.851640\pi$$
$$294$$ − 2.00000i − 0.116642i
$$295$$ 0.544156 0.0316820
$$296$$ −6.34315 −0.368688
$$297$$ − 24.0000i − 1.39262i
$$298$$ 22.9706 1.33065
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 5.00000i 0.288195i
$$302$$ 13.7990 0.794043
$$303$$ −14.4853 −0.832158
$$304$$ − 28.9706i − 1.66158i
$$305$$ − 9.51472i − 0.544811i
$$306$$ 2.00000i 0.114332i
$$307$$ 13.2426i 0.755797i 0.925847 + 0.377899i $$0.123353\pi$$
−0.925847 + 0.377899i $$0.876647\pi$$
$$308$$ 0 0
$$309$$ 11.3137 0.643614
$$310$$ − 7.27208i − 0.413026i
$$311$$ 7.41421 0.420421 0.210211 0.977656i $$-0.432585\pi$$
0.210211 + 0.977656i $$0.432585\pi$$
$$312$$ 0 0
$$313$$ 23.2132 1.31209 0.656044 0.754723i $$-0.272229\pi$$
0.656044 + 0.754723i $$0.272229\pi$$
$$314$$ − 5.31371i − 0.299870i
$$315$$ −1.58579 −0.0893489
$$316$$ 0 0
$$317$$ 11.3137i 0.635441i 0.948184 + 0.317721i $$0.102917\pi$$
−0.948184 + 0.317721i $$0.897083\pi$$
$$318$$ − 0.343146i − 0.0192427i
$$319$$ 0.727922i 0.0407558i
$$320$$ 12.6863i 0.709185i
$$321$$ −28.4853 −1.58989
$$322$$ −8.24264 −0.459344
$$323$$ − 10.2426i − 0.569916i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ − 23.6569i − 1.30823i
$$328$$ 24.9706 1.37877
$$329$$ −1.58579 −0.0874272
$$330$$ 13.4558i 0.740720i
$$331$$ 18.0000i 0.989369i 0.869072 + 0.494685i $$0.164716\pi$$
−0.869072 + 0.494685i $$0.835284\pi$$
$$332$$ 0 0
$$333$$ − 2.24264i − 0.122896i
$$334$$ −21.7574 −1.19051
$$335$$ 22.9706 1.25502
$$336$$ − 5.65685i − 0.308607i
$$337$$ 33.0000 1.79762 0.898812 0.438334i $$-0.144431\pi$$
0.898812 + 0.438334i $$0.144431\pi$$
$$338$$ 0 0
$$339$$ 3.27208 0.177715
$$340$$ 0 0
$$341$$ 13.7574 0.745003
$$342$$ 10.2426 0.553859
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 14.1421i 0.762493i
$$345$$ − 13.0711i − 0.703723i
$$346$$ − 34.9706i − 1.88003i
$$347$$ −5.65685 −0.303676 −0.151838 0.988405i $$-0.548519\pi$$
−0.151838 + 0.988405i $$0.548519\pi$$
$$348$$ 0 0
$$349$$ − 25.7279i − 1.37718i −0.725149 0.688592i $$-0.758229\pi$$
0.725149 0.688592i $$-0.241771\pi$$
$$350$$ −3.51472 −0.187870
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8.48528i 0.451626i 0.974171 + 0.225813i $$0.0725038\pi$$
−0.974171 + 0.225813i $$0.927496\pi$$
$$354$$ −0.686292 −0.0364760
$$355$$ 20.7279 1.10012
$$356$$ 0 0
$$357$$ − 2.00000i − 0.105851i
$$358$$ 12.7279i 0.672692i
$$359$$ − 27.8995i − 1.47248i −0.676721 0.736240i $$-0.736600\pi$$
0.676721 0.736240i $$-0.263400\pi$$
$$360$$ −4.48528 −0.236395
$$361$$ −33.4558 −1.76083
$$362$$ 26.4853i 1.39204i
$$363$$ −9.89949 −0.519589
$$364$$ 0 0
$$365$$ −14.6569 −0.767175
$$366$$ 12.0000i 0.627250i
$$367$$ 10.2426 0.534661 0.267331 0.963605i $$-0.413858\pi$$
0.267331 + 0.963605i $$0.413858\pi$$
$$368$$ −23.3137 −1.21531
$$369$$ 8.82843i 0.459590i
$$370$$ 5.02944i 0.261468i
$$371$$ − 0.171573i − 0.00890762i
$$372$$ 0 0
$$373$$ 8.48528 0.439351 0.219676 0.975573i $$-0.429500\pi$$
0.219676 + 0.975573i $$0.429500\pi$$
$$374$$ 8.48528 0.438763
$$375$$ − 16.7868i − 0.866866i
$$376$$ −4.48528 −0.231311
$$377$$ 0 0
$$378$$ 8.00000 0.411476
$$379$$ − 23.7574i − 1.22033i −0.792273 0.610167i $$-0.791102\pi$$
0.792273 0.610167i $$-0.208898\pi$$
$$380$$ 0 0
$$381$$ −2.82843 −0.144905
$$382$$ 21.4558i 1.09778i
$$383$$ 20.4853i 1.04675i 0.852103 + 0.523374i $$0.175327\pi$$
−0.852103 + 0.523374i $$0.824673\pi$$
$$384$$ − 16.0000i − 0.816497i
$$385$$ 6.72792i 0.342887i
$$386$$ 20.4853 1.04267
$$387$$ −5.00000 −0.254164
$$388$$ 0 0
$$389$$ −29.6569 −1.50366 −0.751831 0.659356i $$-0.770829\pi$$
−0.751831 + 0.659356i $$0.770829\pi$$
$$390$$ 0 0
$$391$$ −8.24264 −0.416848
$$392$$ − 2.82843i − 0.142857i
$$393$$ −4.00000 −0.201773
$$394$$ 17.4558 0.879413
$$395$$ − 24.5563i − 1.23556i
$$396$$ 0 0
$$397$$ − 18.2132i − 0.914094i −0.889442 0.457047i $$-0.848907\pi$$
0.889442 0.457047i $$-0.151093\pi$$
$$398$$ 5.31371i 0.266352i
$$399$$ −10.2426 −0.512773
$$400$$ −9.94113 −0.497056
$$401$$ 6.34315i 0.316762i 0.987378 + 0.158381i $$0.0506274\pi$$
−0.987378 + 0.158381i $$0.949373\pi$$
$$402$$ −28.9706 −1.44492
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 7.92893i 0.393992i
$$406$$ −0.242641 −0.0120421
$$407$$ −9.51472 −0.471627
$$408$$ − 5.65685i − 0.280056i
$$409$$ − 3.24264i − 0.160338i −0.996781 0.0801691i $$-0.974454\pi$$
0.996781 0.0801691i $$-0.0255460\pi$$
$$410$$ − 19.7990i − 0.977802i
$$411$$ − 6.48528i − 0.319895i
$$412$$ 0 0
$$413$$ −0.343146 −0.0168851
$$414$$ − 8.24264i − 0.405104i
$$415$$ −21.0000 −1.03085
$$416$$ 0 0
$$417$$ 8.82843 0.432330
$$418$$ − 43.4558i − 2.12549i
$$419$$ −20.8701 −1.01957 −0.509785 0.860302i $$-0.670275\pi$$
−0.509785 + 0.860302i $$0.670275\pi$$
$$420$$ 0 0
$$421$$ − 6.72792i − 0.327899i −0.986469 0.163949i $$-0.947577\pi$$
0.986469 0.163949i $$-0.0524234\pi$$
$$422$$ − 22.5858i − 1.09946i
$$423$$ − 1.58579i − 0.0771036i
$$424$$ − 0.485281i − 0.0235673i
$$425$$ −3.51472 −0.170489
$$426$$ −26.1421 −1.26659
$$427$$ 6.00000i 0.290360i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 11.2132 0.540749
$$431$$ 12.3431i 0.594548i 0.954792 + 0.297274i $$0.0960775\pi$$
−0.954792 + 0.297274i $$0.903922\pi$$
$$432$$ 22.6274 1.08866
$$433$$ 24.9706 1.20001 0.600004 0.799997i $$-0.295166\pi$$
0.600004 + 0.799997i $$0.295166\pi$$
$$434$$ 4.58579i 0.220125i
$$435$$ − 0.384776i − 0.0184486i
$$436$$ 0 0
$$437$$ 42.2132i 2.01933i
$$438$$ 18.4853 0.883261
$$439$$ 34.4853 1.64589 0.822946 0.568119i $$-0.192329\pi$$
0.822946 + 0.568119i $$0.192329\pi$$
$$440$$ 19.0294i 0.907193i
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −3.68629 −0.175141 −0.0875705 0.996158i $$-0.527910\pi$$
−0.0875705 + 0.996158i $$0.527910\pi$$
$$444$$ 0 0
$$445$$ 2.51472 0.119209
$$446$$ 1.07107 0.0507165
$$447$$ − 22.9706i − 1.08647i
$$448$$ − 8.00000i − 0.377964i
$$449$$ 21.1716i 0.999148i 0.866271 + 0.499574i $$0.166510\pi$$
−0.866271 + 0.499574i $$0.833490\pi$$
$$450$$ − 3.51472i − 0.165685i
$$451$$ 37.4558 1.76373
$$452$$ 0 0
$$453$$ − 13.7990i − 0.648333i
$$454$$ 37.9411 1.78067
$$455$$ 0 0
$$456$$ −28.9706 −1.35667
$$457$$ 7.21320i 0.337419i 0.985666 + 0.168710i $$0.0539600\pi$$
−0.985666 + 0.168710i $$0.946040\pi$$
$$458$$ −41.6569 −1.94650
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ − 8.82843i − 0.411181i −0.978638 0.205590i $$-0.934089\pi$$
0.978638 0.205590i $$-0.0659115\pi$$
$$462$$ − 8.48528i − 0.394771i
$$463$$ − 4.24264i − 0.197172i −0.995129 0.0985861i $$-0.968568\pi$$
0.995129 0.0985861i $$-0.0314320\pi$$
$$464$$ −0.686292 −0.0318603
$$465$$ −7.27208 −0.337235
$$466$$ 20.7279i 0.960202i
$$467$$ 3.89949 0.180447 0.0902236 0.995922i $$-0.471242\pi$$
0.0902236 + 0.995922i $$0.471242\pi$$
$$468$$ 0 0
$$469$$ −14.4853 −0.668868
$$470$$ 3.55635i 0.164042i
$$471$$ −5.31371 −0.244843
$$472$$ −0.970563 −0.0446738
$$473$$ 21.2132i 0.975384i
$$474$$ 30.9706i 1.42253i
$$475$$ 18.0000i 0.825897i
$$476$$ 0 0
$$477$$ 0.171573 0.00785578
$$478$$ 4.97056 0.227348
$$479$$ − 6.21320i − 0.283889i −0.989875 0.141944i $$-0.954665\pi$$
0.989875 0.141944i $$-0.0453354\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 28.5858 1.30205
$$483$$ 8.24264i 0.375053i
$$484$$ 0 0
$$485$$ −18.5980 −0.844491
$$486$$ 14.0000i 0.635053i
$$487$$ 11.4558i 0.519114i 0.965728 + 0.259557i $$0.0835765\pi$$
−0.965728 + 0.259557i $$0.916423\pi$$
$$488$$ 16.9706i 0.768221i
$$489$$ 12.0000i 0.542659i
$$490$$ −2.24264 −0.101312
$$491$$ −16.6274 −0.750385 −0.375192 0.926947i $$-0.622423\pi$$
−0.375192 + 0.926947i $$0.622423\pi$$
$$492$$ 0 0
$$493$$ −0.242641 −0.0109280
$$494$$ 0 0
$$495$$ −6.72792 −0.302398
$$496$$ 12.9706i 0.582395i
$$497$$ −13.0711 −0.586318
$$498$$ 26.4853 1.18683
$$499$$ 13.2721i 0.594140i 0.954856 + 0.297070i $$0.0960094\pi$$
−0.954856 + 0.297070i $$0.903991\pi$$
$$500$$ 0 0
$$501$$ 21.7574i 0.972047i
$$502$$ − 23.4558i − 1.04689i
$$503$$ −28.6274 −1.27643 −0.638217 0.769857i $$-0.720328\pi$$
−0.638217 + 0.769857i $$0.720328\pi$$
$$504$$ 2.82843 0.125988
$$505$$ 16.2426i 0.722788i
$$506$$ −34.9706 −1.55463
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 5.10051i 0.226076i 0.993591 + 0.113038i $$0.0360582\pi$$
−0.993591 + 0.113038i $$0.963942\pi$$
$$510$$ −4.48528 −0.198612
$$511$$ 9.24264 0.408870
$$512$$ − 22.6274i − 1.00000i
$$513$$ − 40.9706i − 1.80889i
$$514$$ 27.4558i 1.21103i
$$515$$ − 12.6863i − 0.559025i
$$516$$ 0 0
$$517$$ −6.72792 −0.295894
$$518$$ − 3.17157i − 0.139351i
$$519$$ −34.9706 −1.53504
$$520$$ 0 0
$$521$$ −6.34315 −0.277898 −0.138949 0.990300i $$-0.544372\pi$$
−0.138949 + 0.990300i $$0.544372\pi$$
$$522$$ − 0.242641i − 0.0106201i
$$523$$ 30.9706 1.35425 0.677124 0.735869i $$-0.263226\pi$$
0.677124 + 0.735869i $$0.263226\pi$$
$$524$$ 0 0
$$525$$ 3.51472i 0.153395i
$$526$$ 2.78680i 0.121510i
$$527$$ 4.58579i 0.199760i
$$528$$ − 24.0000i − 1.04447i
$$529$$ 10.9706 0.476981
$$530$$ −0.384776 −0.0167136
$$531$$ − 0.343146i − 0.0148913i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −3.17157 −0.137247
$$535$$ 31.9411i 1.38094i
$$536$$ −40.9706 −1.76966
$$537$$ 12.7279 0.549250
$$538$$ 12.9706i 0.559201i
$$539$$ − 4.24264i − 0.182743i
$$540$$ 0 0
$$541$$ − 7.21320i − 0.310120i −0.987905 0.155060i $$-0.950443\pi$$
0.987905 0.155060i $$-0.0495571\pi$$
$$542$$ −28.2843 −1.21491
$$543$$ 26.4853 1.13659
$$544$$ 0 0
$$545$$ −26.5269 −1.13629
$$546$$ 0 0
$$547$$ 35.4853 1.51724 0.758621 0.651533i $$-0.225874\pi$$
0.758621 + 0.651533i $$0.225874\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ −14.9117 −0.635837
$$551$$ 1.24264i 0.0529383i
$$552$$ 23.3137i 0.992297i
$$553$$ 15.4853i 0.658501i
$$554$$ 10.5858i 0.449747i
$$555$$ 5.02944 0.213488
$$556$$ 0 0
$$557$$ 12.0000i 0.508456i 0.967144 + 0.254228i $$0.0818214\pi$$
−0.967144 + 0.254228i $$0.918179\pi$$
$$558$$ −4.58579 −0.194132
$$559$$ 0 0
$$560$$ −6.34315 −0.268047
$$561$$ − 8.48528i − 0.358249i
$$562$$ −22.0000 −0.928014
$$563$$ 6.34315 0.267332 0.133666 0.991026i $$-0.457325\pi$$
0.133666 + 0.991026i $$0.457325\pi$$
$$564$$ 0 0
$$565$$ − 3.66905i − 0.154358i
$$566$$ 12.0000i 0.504398i
$$567$$ − 5.00000i − 0.209980i
$$568$$ −36.9706 −1.55125
$$569$$ −5.14214 −0.215570 −0.107785 0.994174i $$-0.534376\pi$$
−0.107785 + 0.994174i $$0.534376\pi$$
$$570$$ 22.9706i 0.962131i
$$571$$ −42.4558 −1.77672 −0.888361 0.459146i $$-0.848156\pi$$
−0.888361 + 0.459146i $$0.848156\pi$$
$$572$$ 0 0
$$573$$ 21.4558 0.896331
$$574$$ 12.4853i 0.521126i
$$575$$ 14.4853 0.604078
$$576$$ 8.00000 0.333333
$$577$$ 6.97056i 0.290188i 0.989418 + 0.145094i $$0.0463485\pi$$
−0.989418 + 0.145094i $$0.953651\pi$$
$$578$$ − 21.2132i − 0.882353i
$$579$$ − 20.4853i − 0.851339i
$$580$$ 0 0
$$581$$ 13.2426 0.549397
$$582$$ 23.4558 0.972276
$$583$$ − 0.727922i − 0.0301475i
$$584$$ 26.1421 1.08177
$$585$$ 0 0
$$586$$ −21.7574 −0.898788
$$587$$ 6.55635i 0.270609i 0.990804 + 0.135305i $$0.0432013\pi$$
−0.990804 + 0.135305i $$0.956799\pi$$
$$588$$ 0 0
$$589$$ 23.4853 0.967694
$$590$$ 0.769553i 0.0316820i
$$591$$ − 17.4558i − 0.718037i
$$592$$ − 8.97056i − 0.368688i
$$593$$ − 0.556349i − 0.0228465i −0.999935 0.0114233i $$-0.996364\pi$$
0.999935 0.0114233i $$-0.00363622\pi$$
$$594$$ 33.9411 1.39262
$$595$$ −2.24264 −0.0919393
$$596$$ 0 0
$$597$$ 5.31371 0.217476
$$598$$ 0 0
$$599$$ −28.7990 −1.17669 −0.588347 0.808608i $$-0.700221\pi$$
−0.588347 + 0.808608i $$0.700221\pi$$
$$600$$ 9.94113i 0.405845i
$$601$$ −38.9706 −1.58964 −0.794821 0.606844i $$-0.792435\pi$$
−0.794821 + 0.606844i $$0.792435\pi$$
$$602$$ −7.07107 −0.288195
$$603$$ − 14.4853i − 0.589886i
$$604$$ 0 0
$$605$$ 11.1005i 0.451300i
$$606$$ − 20.4853i − 0.832158i
$$607$$ −26.7279 −1.08485 −0.542426 0.840103i $$-0.682494\pi$$
−0.542426 + 0.840103i $$0.682494\pi$$
$$608$$ 0 0
$$609$$ 0.242641i 0.00983230i
$$610$$ 13.4558 0.544811
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ −18.7279 −0.755797
$$615$$ −19.7990 −0.798372
$$616$$ − 12.0000i − 0.483494i
$$617$$ − 12.3431i − 0.496916i −0.968643 0.248458i $$-0.920076\pi$$
0.968643 0.248458i $$-0.0799239\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ 0.970563i 0.0390102i 0.999810 + 0.0195051i $$0.00620906\pi$$
−0.999810 + 0.0195051i $$0.993791\pi$$
$$620$$ 0 0
$$621$$ −32.9706 −1.32306
$$622$$ 10.4853i 0.420421i
$$623$$ −1.58579 −0.0635332
$$624$$ 0 0
$$625$$ −6.39697 −0.255879
$$626$$ 32.8284i 1.31209i
$$627$$ −43.4558 −1.73546
$$628$$ 0 0
$$629$$ − 3.17157i − 0.126459i
$$630$$ − 2.24264i − 0.0893489i
$$631$$ 2.00000i 0.0796187i 0.999207 + 0.0398094i $$0.0126751\pi$$
−0.999207 + 0.0398094i $$0.987325\pi$$
$$632$$ 43.7990i 1.74223i
$$633$$ −22.5858 −0.897704
$$634$$ −16.0000 −0.635441
$$635$$ 3.17157i 0.125860i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.02944 −0.0407558
$$639$$ − 13.0711i − 0.517083i
$$640$$ −17.9411 −0.709185
$$641$$ −15.3431 −0.606018 −0.303009 0.952988i $$-0.597991\pi$$
−0.303009 + 0.952988i $$0.597991\pi$$
$$642$$ − 40.2843i − 1.58989i
$$643$$ 12.4853i 0.492371i 0.969223 + 0.246186i $$0.0791773\pi$$
−0.969223 + 0.246186i $$0.920823\pi$$
$$644$$ 0 0
$$645$$ − 11.2132i − 0.441519i
$$646$$ 14.4853 0.569916
$$647$$ −14.5269 −0.571112 −0.285556 0.958362i $$-0.592178\pi$$
−0.285556 + 0.958362i $$0.592178\pi$$
$$648$$ − 14.1421i − 0.555556i
$$649$$ −1.45584 −0.0571469
$$650$$ 0 0
$$651$$ 4.58579 0.179731
$$652$$ 0 0
$$653$$ 17.3137 0.677538 0.338769 0.940870i $$-0.389990\pi$$
0.338769 + 0.940870i $$0.389990\pi$$
$$654$$ 33.4558 1.30823
$$655$$ 4.48528i 0.175254i
$$656$$ 35.3137i 1.37877i
$$657$$ 9.24264i 0.360590i
$$658$$ − 2.24264i − 0.0874272i
$$659$$ −15.3431 −0.597684 −0.298842 0.954303i $$-0.596600\pi$$
−0.298842 + 0.954303i $$0.596600\pi$$
$$660$$ 0 0
$$661$$ − 10.7574i − 0.418413i −0.977872 0.209206i $$-0.932912\pi$$
0.977872 0.209206i $$-0.0670880\pi$$
$$662$$ −25.4558 −0.989369
$$663$$ 0 0
$$664$$ 37.4558 1.45357
$$665$$ 11.4853i 0.445380i
$$666$$ 3.17157 0.122896
$$667$$ 1.00000 0.0387202
$$668$$ 0 0
$$669$$ − 1.07107i − 0.0414099i
$$670$$ 32.4853i 1.25502i
$$671$$ 25.4558i 0.982712i
$$672$$ 0 0
$$673$$ 26.9411 1.03850 0.519252 0.854621i $$-0.326211\pi$$
0.519252 + 0.854621i $$0.326211\pi$$
$$674$$ 46.6690i 1.79762i
$$675$$ −14.0589 −0.541126
$$676$$ 0 0
$$677$$ −41.3553 −1.58941 −0.794707 0.606993i $$-0.792376\pi$$
−0.794707 + 0.606993i $$0.792376\pi$$
$$678$$ 4.62742i 0.177715i
$$679$$ 11.7279 0.450076
$$680$$ −6.34315 −0.243249
$$681$$ − 37.9411i − 1.45391i
$$682$$ 19.4558i 0.745003i
$$683$$ − 21.1716i − 0.810108i −0.914293 0.405054i $$-0.867253\pi$$
0.914293 0.405054i $$-0.132747\pi$$
$$684$$ 0 0
$$685$$ −7.27208 −0.277852
$$686$$ 1.41421 0.0539949
$$687$$ 41.6569i 1.58931i
$$688$$ −20.0000 −0.762493
$$689$$ 0 0
$$690$$ 18.4853 0.703723
$$691$$ − 28.6985i − 1.09174i −0.837869 0.545871i $$-0.816199\pi$$
0.837869 0.545871i $$-0.183801\pi$$
$$692$$ 0 0
$$693$$ 4.24264 0.161165
$$694$$ − 8.00000i − 0.303676i
$$695$$ − 9.89949i − 0.375509i
$$696$$ 0.686292i 0.0260138i
$$697$$ 12.4853i 0.472914i
$$698$$ 36.3848 1.37718
$$699$$ 20.7279 0.784002
$$700$$ 0 0
$$701$$ 10.7990 0.407872 0.203936 0.978984i $$-0.434627\pi$$
0.203936 + 0.978984i $$0.434627\pi$$
$$702$$ 0 0
$$703$$ −16.2426 −0.612603
$$704$$ − 33.9411i − 1.27920i
$$705$$ 3.55635 0.133940
$$706$$ −12.0000 −0.451626
$$707$$ − 10.2426i − 0.385214i
$$708$$ 0 0
$$709$$ 0.727922i 0.0273377i 0.999907 + 0.0136688i $$0.00435106\pi$$
−0.999907 + 0.0136688i $$0.995649\pi$$
$$710$$ 29.3137i 1.10012i
$$711$$ −15.4853 −0.580743
$$712$$ −4.48528 −0.168093
$$713$$ − 18.8995i − 0.707792i
$$714$$ 2.82843 0.105851
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 4.97056i − 0.185629i
$$718$$ 39.4558 1.47248
$$719$$ 1.75736 0.0655384 0.0327692 0.999463i $$-0.489567\pi$$
0.0327692 + 0.999463i $$0.489567\pi$$
$$720$$ − 6.34315i − 0.236395i
$$721$$ 8.00000i 0.297936i
$$722$$ − 47.3137i − 1.76083i
$$723$$ − 28.5858i − 1.06312i
$$724$$ 0 0
$$725$$ 0.426407 0.0158364
$$726$$ − 14.0000i − 0.519589i
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ − 20.7279i − 0.767175i
$$731$$ −7.07107 −0.261533
$$732$$ 0 0
$$733$$ 16.6985i 0.616773i 0.951261 + 0.308386i $$0.0997889\pi$$
−0.951261 + 0.308386i $$0.900211\pi$$
$$734$$ 14.4853i 0.534661i
$$735$$ 2.24264i 0.0827210i
$$736$$ 0 0
$$737$$ −61.4558 −2.26376
$$738$$ −12.4853 −0.459590
$$739$$ − 17.6985i − 0.651049i −0.945534 0.325525i $$-0.894459\pi$$
0.945534 0.325525i $$-0.105541\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.242641 0.00890762
$$743$$ 29.6569i 1.08800i 0.839084 + 0.544002i $$0.183092\pi$$
−0.839084 + 0.544002i $$0.816908\pi$$
$$744$$ 12.9706 0.475524
$$745$$ −25.7574 −0.943677
$$746$$ 12.0000i 0.439351i
$$747$$ 13.2426i 0.484523i
$$748$$ 0 0
$$749$$ − 20.1421i − 0.735978i
$$750$$ 23.7401 0.866866
$$751$$ −15.4853 −0.565066 −0.282533 0.959258i $$-0.591175\pi$$
−0.282533 + 0.959258i $$0.591175\pi$$
$$752$$ − 6.34315i − 0.231311i
$$753$$ −23.4558 −0.854778
$$754$$ 0 0
$$755$$ −15.4731 −0.563123
$$756$$ 0 0
$$757$$ 21.4853 0.780896 0.390448 0.920625i $$-0.372320\pi$$
0.390448 + 0.920625i $$0.372320\pi$$
$$758$$ 33.5980 1.22033
$$759$$ 34.9706i 1.26935i
$$760$$ 32.4853i 1.17837i
$$761$$ 34.7574i 1.25995i 0.776614 + 0.629977i $$0.216936\pi$$
−0.776614 + 0.629977i $$0.783064\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 16.7279 0.605591
$$764$$ 0 0
$$765$$ − 2.24264i − 0.0810828i
$$766$$ −28.9706 −1.04675
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 52.2132i − 1.88286i −0.337214 0.941428i $$-0.609484\pi$$
0.337214 0.941428i $$-0.390516\pi$$
$$770$$ −9.51472 −0.342887
$$771$$ 27.4558 0.988798
$$772$$ 0 0
$$773$$ 32.8284i 1.18076i 0.807127 + 0.590378i $$0.201021\pi$$
−0.807127 + 0.590378i $$0.798979\pi$$
$$774$$ − 7.07107i − 0.254164i
$$775$$ − 8.05887i − 0.289483i
$$776$$ 33.1716 1.19079
$$777$$ −3.17157 −0.113780
$$778$$ − 41.9411i − 1.50366i
$$779$$ 63.9411 2.29093
$$780$$ 0 0
$$781$$ −55.4558 −1.98437
$$782$$ − 11.6569i − 0.416848i
$$783$$ −0.970563 −0.0346851
$$784$$ 4.00000 0.142857
$$785$$ 5.95837i 0.212663i
$$786$$ − 5.65685i − 0.201773i
$$787$$ − 24.7574i − 0.882505i −0.897383 0.441252i $$-0.854534\pi$$
0.897383 0.441252i $$-0.145466\pi$$
$$788$$ 0 0
$$789$$ 2.78680 0.0992126
$$790$$ 34.7279 1.23556
$$791$$ 2.31371i 0.0822660i
$$792$$ 12.0000 0.426401
$$793$$ 0 0
$$794$$ 25.7574 0.914094
$$795$$ 0.384776i 0.0136466i
$$796$$ 0 0
$$797$$ 24.3431 0.862278 0.431139 0.902285i $$-0.358112\pi$$
0.431139 + 0.902285i $$0.358112\pi$$
$$798$$ − 14.4853i − 0.512773i
$$799$$ − 2.24264i − 0.0793389i
$$800$$ 0 0
$$801$$ − 1.58579i − 0.0560310i
$$802$$ −8.97056 −0.316762
$$803$$ 39.2132 1.38380
$$804$$ 0 0
$$805$$ 9.24264 0.325760
$$806$$ 0 0
$$807$$ 12.9706 0.456585
$$808$$ − 28.9706i − 1.01918i
$$809$$ 17.4853 0.614750 0.307375 0.951589i $$-0.400549\pi$$
0.307375 + 0.951589i $$0.400549\pi$$
$$810$$ −11.2132 −0.393992
$$811$$ 21.9411i 0.770457i 0.922821 + 0.385229i $$0.125877\pi$$
−0.922821 + 0.385229i $$0.874123\pi$$
$$812$$ 0 0
$$813$$ 28.2843i 0.991973i
$$814$$ − 13.4558i − 0.471627i
$$815$$ 13.4558 0.471338
$$816$$ 8.00000 0.280056
$$817$$ 36.2132i 1.26694i
$$818$$ 4.58579 0.160338
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 20.1421i 0.702965i 0.936194 + 0.351483i $$0.114322\pi$$
−0.936194 + 0.351483i $$0.885678\pi$$
$$822$$ 9.17157 0.319895
$$823$$ −10.4853 −0.365494 −0.182747 0.983160i $$-0.558499\pi$$
−0.182747 + 0.983160i $$0.558499\pi$$
$$824$$ 22.6274i 0.788263i
$$825$$ 14.9117i 0.519158i
$$826$$ − 0.485281i − 0.0168851i
$$827$$ − 49.4558i − 1.71975i −0.510506 0.859874i $$-0.670542\pi$$
0.510506 0.859874i $$-0.329458\pi$$
$$828$$ 0 0
$$829$$ −50.7279 −1.76185 −0.880927 0.473253i $$-0.843080\pi$$
−0.880927 + 0.473253i $$0.843080\pi$$
$$830$$ − 29.6985i − 1.03085i
$$831$$ 10.5858 0.367217
$$832$$ 0 0
$$833$$ 1.41421 0.0489996
$$834$$ 12.4853i 0.432330i
$$835$$ 24.3970 0.844292
$$836$$ 0 0
$$837$$ 18.3431i 0.634032i
$$838$$ − 29.5147i − 1.01957i
$$839$$ − 33.1716i − 1.14521i −0.819831 0.572605i $$-0.805933\pi$$
0.819831 0.572605i $$-0.194067\pi$$
$$840$$ 6.34315i 0.218859i
$$841$$ −28.9706 −0.998985
$$842$$ 9.51472 0.327899
$$843$$ 22.0000i 0.757720i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 2.24264 0.0771036
$$847$$ − 7.00000i − 0.240523i
$$848$$ 0.686292 0.0235673
$$849$$ 12.0000 0.411839
$$850$$ − 4.97056i − 0.170489i
$$851$$ 13.0711i 0.448070i
$$852$$ 0 0
$$853$$ 39.7279i 1.36026i 0.733092 + 0.680129i $$0.238076\pi$$
−0.733092 + 0.680129i $$0.761924\pi$$
$$854$$ −8.48528 −0.290360
$$855$$ −11.4853 −0.392788
$$856$$ − 56.9706i − 1.94721i
$$857$$ 54.7696 1.87089 0.935446 0.353469i $$-0.114998\pi$$
0.935446 + 0.353469i $$0.114998\pi$$
$$858$$ 0 0
$$859$$ 30.9706 1.05670 0.528351 0.849026i $$-0.322811\pi$$
0.528351 + 0.849026i $$0.322811\pi$$
$$860$$ 0 0
$$861$$ 12.4853 0.425497
$$862$$ −17.4558 −0.594548
$$863$$ − 28.2843i − 0.962808i −0.876499 0.481404i $$-0.840127\pi$$
0.876499 0.481404i $$-0.159873\pi$$
$$864$$ 0 0
$$865$$ 39.2132i 1.33329i
$$866$$ 35.3137i 1.20001i
$$867$$ −21.2132 −0.720438
$$868$$ 0 0
$$869$$ 65.6985i 2.22867i
$$870$$ 0.544156 0.0184486
$$871$$ 0 0
$$872$$ 47.3137 1.60224
$$873$$ 11.7279i 0.396930i
$$874$$ −59.6985 −2.01933
$$875$$ 11.8701 0.401281
$$876$$ 0 0
$$877$$ 10.2426i 0.345869i 0.984933 + 0.172935i $$0.0553250\pi$$
−0.984933 + 0.172935i $$0.944675\pi$$
$$878$$ 48.7696i 1.64589i
$$879$$ 21.7574i 0.733858i
$$880$$ −26.9117 −0.907193
$$881$$ −13.1127 −0.441778 −0.220889 0.975299i $$-0.570896\pi$$
−0.220889 + 0.975299i $$0.570896\pi$$
$$882$$ 1.41421i 0.0476190i
$$883$$ −2.00000 −0.0673054 −0.0336527 0.999434i $$-0.510714\pi$$
−0.0336527 + 0.999434i $$0.510714\pi$$
$$884$$ 0 0
$$885$$ 0.769553 0.0258682
$$886$$ − 5.21320i − 0.175141i
$$887$$ 19.1127 0.641742 0.320871 0.947123i $$-0.396024\pi$$
0.320871 + 0.947123i $$0.396024\pi$$
$$888$$ −8.97056 −0.301032
$$889$$ − 2.00000i − 0.0670778i
$$890$$ 3.55635i 0.119209i
$$891$$ − 21.2132i − 0.710669i
$$892$$ 0 0
$$893$$ −11.4853 −0.384340
$$894$$ 32.4853 1.08647
$$895$$ − 14.2721i − 0.477063i
$$896$$ 11.3137 0.377964
$$897$$ 0 0
$$898$$ −29.9411 −0.999148
$$899$$ − 0.556349i − 0.0185553i
$$900$$ 0 0
$$901$$ 0.242641 0.00808353
$$902$$ 52.9706i 1.76373i
$$903$$ 7.07107i 0.235310i
$$904$$ 6.54416i 0.217655i
$$905$$ − 29.6985i − 0.987211i
$$906$$ 19.5147 0.648333
$$907$$ −1.97056 −0.0654315 −0.0327157 0.999465i $$-0.510416\pi$$
−0.0327157 + 0.999465i $$0.510416\pi$$
$$908$$ 0 0
$$909$$ 10.2426 0.339727
$$910$$ 0 0
$$911$$ −43.9706 −1.45681 −0.728405 0.685147i $$-0.759738\pi$$
−0.728405 + 0.685147i $$0.759738\pi$$
$$912$$ − 40.9706i − 1.35667i
$$913$$ 56.1838 1.85941
$$914$$ −10.2010 −0.337419
$$915$$ − 13.4558i − 0.444836i
$$916$$ 0 0
$$917$$ − 2.82843i − 0.0934029i
$$918$$ 11.3137i 0.373408i
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 26.1421 0.861881
$$921$$ 18.7279i 0.617106i
$$922$$ 12.4853 0.411181
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 5.57359i 0.183259i
$$926$$ 6.00000 0.197172
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ − 36.8995i − 1.21063i −0.795985 0.605317i $$-0.793047\pi$$
0.795985 0.605317i $$-0.206953\pi$$
$$930$$ − 10.2843i − 0.337235i
$$931$$ − 7.24264i − 0.237368i
$$932$$ 0 0
$$933$$ 10.4853 0.343273
$$934$$ 5.51472i 0.180447i
$$935$$ −9.51472 −0.311165
$$936$$ 0 0
$$937$$ 45.2132 1.47705 0.738525 0.674226i $$-0.235522\pi$$
0.738525 + 0.674226i $$0.235522\pi$$
$$938$$ − 20.4853i − 0.668868i
$$939$$ 32.8284 1.07132
$$940$$ 0 0
$$941$$ − 40.0711i − 1.30628i −0.757237 0.653140i $$-0.773451\pi$$
0.757237 0.653140i $$-0.226549\pi$$
$$942$$ − 7.51472i − 0.244843i
$$943$$ − 51.4558i − 1.67563i
$$944$$ − 1.37258i − 0.0446738i
$$945$$ −8.97056 −0.291812
$$946$$ −30.0000 −0.975384
$$947$$ 15.1716i 0.493010i 0.969142 + 0.246505i $$0.0792822\pi$$
−0.969142 + 0.246505i $$0.920718\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −25.4558 −0.825897
$$951$$ 16.0000i 0.518836i
$$952$$ 4.00000 0.129641
$$953$$ −11.1421 −0.360929 −0.180465 0.983581i $$-0.557760\pi$$
−0.180465 + 0.983581i $$0.557760\pi$$
$$954$$ 0.242641i 0.00785578i
$$955$$ − 24.0589i − 0.778527i
$$956$$ 0 0
$$957$$ 1.02944i 0.0332770i
$$958$$ 8.78680 0.283889
$$959$$ 4.58579 0.148083
$$960$$ 17.9411i 0.579047i
$$961$$ 20.4853 0.660816
$$962$$ 0 0
$$963$$ 20.1421 0.649071
$$964$$ 0 0
$$965$$ −22.9706 −0.739449
$$966$$ −11.6569 −0.375053
$$967$$ − 17.6985i − 0.569145i −0.958655 0.284572i $$-0.908148\pi$$
0.958655 0.284572i $$-0.0918516\pi$$
$$968$$ − 19.7990i − 0.636364i
$$969$$ − 14.4853i − 0.465334i
$$970$$ − 26.3015i − 0.844491i
$$971$$ 43.4558 1.39456 0.697282 0.716797i $$-0.254392\pi$$
0.697282 + 0.716797i $$0.254392\pi$$
$$972$$ 0 0
$$973$$ 6.24264i 0.200130i
$$974$$ −16.2010 −0.519114
$$975$$ 0 0
$$976$$ −24.0000 −0.768221
$$977$$ − 24.0416i − 0.769160i −0.923092 0.384580i $$-0.874346\pi$$
0.923092 0.384580i $$-0.125654\pi$$
$$978$$ −16.9706 −0.542659
$$979$$ −6.72792 −0.215025
$$980$$ 0 0
$$981$$ 16.7279i 0.534081i
$$982$$ − 23.5147i − 0.750385i
$$983$$ − 15.0416i − 0.479754i −0.970803 0.239877i $$-0.922893\pi$$
0.970803 0.239877i $$-0.0771070\pi$$
$$984$$ 35.3137 1.12576
$$985$$ −19.5736 −0.623667
$$986$$ − 0.343146i − 0.0109280i
$$987$$ −2.24264 −0.0713840
$$988$$ 0 0
$$989$$ 29.1421 0.926666
$$990$$ − 9.51472i − 0.302398i
$$991$$ 1.02944 0.0327012 0.0163506 0.999866i $$-0.494795\pi$$
0.0163506 + 0.999866i $$0.494795\pi$$
$$992$$ 0 0
$$993$$ 25.4558i 0.807817i
$$994$$ − 18.4853i − 0.586318i
$$995$$ − 5.95837i − 0.188893i
$$996$$ 0 0
$$997$$ −11.5147 −0.364675 −0.182337 0.983236i $$-0.558366\pi$$
−0.182337 + 0.983236i $$0.558366\pi$$
$$998$$ −18.7696 −0.594140
$$999$$ − 12.6863i − 0.401377i
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 1183.2.c.d.337.4 4
13.5 odd 4 1183.2.a.d.1.2 2
13.8 odd 4 91.2.a.c.1.1 2
13.12 even 2 inner 1183.2.c.d.337.2 4
39.8 even 4 819.2.a.h.1.2 2
52.47 even 4 1456.2.a.q.1.1 2
65.34 odd 4 2275.2.a.j.1.2 2
91.34 even 4 637.2.a.g.1.1 2
91.47 even 12 637.2.e.g.508.2 4
91.60 odd 12 637.2.e.f.79.2 4
91.73 even 12 637.2.e.g.79.2 4
91.83 even 4 8281.2.a.v.1.2 2
91.86 odd 12 637.2.e.f.508.2 4
104.21 odd 4 5824.2.a.bl.1.1 2
104.99 even 4 5824.2.a.bk.1.2 2
273.125 odd 4 5733.2.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.1 2 13.8 odd 4
637.2.a.g.1.1 2 91.34 even 4
637.2.e.f.79.2 4 91.60 odd 12
637.2.e.f.508.2 4 91.86 odd 12
637.2.e.g.79.2 4 91.73 even 12
637.2.e.g.508.2 4 91.47 even 12
819.2.a.h.1.2 2 39.8 even 4
1183.2.a.d.1.2 2 13.5 odd 4
1183.2.c.d.337.2 4 13.12 even 2 inner
1183.2.c.d.337.4 4 1.1 even 1 trivial
1456.2.a.q.1.1 2 52.47 even 4
2275.2.a.j.1.2 2 65.34 odd 4
5733.2.a.s.1.2 2 273.125 odd 4
5824.2.a.bk.1.2 2 104.99 even 4
5824.2.a.bl.1.1 2 104.21 odd 4
8281.2.a.v.1.2 2 91.83 even 4