# Properties

 Label 1183.2.c.d.337.3 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 337.3 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.337 Dual form 1183.2.c.d.337.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.41421i q^{2} -1.41421 q^{3} +4.41421i q^{5} -2.00000i q^{6} -1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.41421i q^{2} -1.41421 q^{3} +4.41421i q^{5} -2.00000i q^{6} -1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} -6.24264 q^{10} +4.24264i q^{11} +1.41421 q^{14} -6.24264i q^{15} -4.00000 q^{16} +1.41421 q^{17} -1.41421i q^{18} +1.24264i q^{19} +1.41421i q^{21} -6.00000 q^{22} +0.171573 q^{23} -4.00000i q^{24} -14.4853 q^{25} +5.65685 q^{27} +5.82843 q^{29} +8.82843 q^{30} -5.24264i q^{31} -6.00000i q^{33} +2.00000i q^{34} +4.41421 q^{35} +6.24264i q^{37} -1.75736 q^{38} -12.4853 q^{40} +3.17157i q^{41} -2.00000 q^{42} +5.00000 q^{43} -4.41421i q^{45} +0.242641i q^{46} -4.41421i q^{47} +5.65685 q^{48} -1.00000 q^{49} -20.4853i q^{50} -2.00000 q^{51} -5.82843 q^{53} +8.00000i q^{54} -18.7279 q^{55} +2.82843 q^{56} -1.75736i q^{57} +8.24264i q^{58} -11.6569i q^{59} +6.00000 q^{61} +7.41421 q^{62} +1.00000i q^{63} -8.00000 q^{64} +8.48528 q^{66} +2.48528i q^{67} -0.242641 q^{69} +6.24264i q^{70} +1.07107i q^{71} -2.82843i q^{72} +0.757359i q^{73} -8.82843 q^{74} +20.4853 q^{75} +4.24264 q^{77} -1.48528 q^{79} -17.6569i q^{80} -5.00000 q^{81} -4.48528 q^{82} +4.75736i q^{83} +6.24264i q^{85} +7.07107i q^{86} -8.24264 q^{87} -12.0000 q^{88} -4.41421i q^{89} +6.24264 q^{90} +7.41421i q^{93} +6.24264 q^{94} -5.48528 q^{95} -13.7279i q^{97} -1.41421i q^{98} -4.24264i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 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 $$-0.633860\pi$$
−0.408248 + 0.912871i $$0.633860\pi$$
$$4$$ 0 0
$$5$$ 4.41421i 1.97410i 0.160424 + 0.987048i $$0.448714\pi$$
−0.160424 + 0.987048i $$0.551286\pi$$
$$6$$ − 2.00000i − 0.816497i
$$7$$ − 1.00000i − 0.377964i
$$8$$ 2.82843i 1.00000i
$$9$$ −1.00000 −0.333333
$$10$$ −6.24264 −1.97410
$$11$$ 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
$$15$$ − 6.24264i − 1.61184i
$$16$$ −4.00000 −1.00000
$$17$$ 1.41421 0.342997 0.171499 0.985184i $$-0.445139\pi$$
0.171499 + 0.985184i $$0.445139\pi$$
$$18$$ − 1.41421i − 0.333333i
$$19$$ 1.24264i 0.285081i 0.989789 + 0.142541i $$0.0455272\pi$$
−0.989789 + 0.142541i $$0.954473\pi$$
$$20$$ 0 0
$$21$$ 1.41421i 0.308607i
$$22$$ −6.00000 −1.27920
$$23$$ 0.171573 0.0357754 0.0178877 0.999840i $$-0.494306\pi$$
0.0178877 + 0.999840i $$0.494306\pi$$
$$24$$ − 4.00000i − 0.816497i
$$25$$ −14.4853 −2.89706
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ 5.82843 1.08231 0.541156 0.840922i $$-0.317987\pi$$
0.541156 + 0.840922i $$0.317987\pi$$
$$30$$ 8.82843 1.61184
$$31$$ − 5.24264i − 0.941606i −0.882238 0.470803i $$-0.843964\pi$$
0.882238 0.470803i $$-0.156036\pi$$
$$32$$ 0 0
$$33$$ − 6.00000i − 1.04447i
$$34$$ 2.00000i 0.342997i
$$35$$ 4.41421 0.746138
$$36$$ 0 0
$$37$$ 6.24264i 1.02628i 0.858304 + 0.513142i $$0.171519\pi$$
−0.858304 + 0.513142i $$0.828481\pi$$
$$38$$ −1.75736 −0.285081
$$39$$ 0 0
$$40$$ −12.4853 −1.97410
$$41$$ 3.17157i 0.495316i 0.968847 + 0.247658i $$0.0796610\pi$$
−0.968847 + 0.247658i $$0.920339\pi$$
$$42$$ −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
$$45$$ − 4.41421i − 0.658032i
$$46$$ 0.242641i 0.0357754i
$$47$$ − 4.41421i − 0.643879i −0.946760 0.321940i $$-0.895665\pi$$
0.946760 0.321940i $$-0.104335\pi$$
$$48$$ 5.65685 0.816497
$$49$$ −1.00000 −0.142857
$$50$$ − 20.4853i − 2.89706i
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ −5.82843 −0.800596 −0.400298 0.916385i $$-0.631093\pi$$
−0.400298 + 0.916385i $$0.631093\pi$$
$$54$$ 8.00000i 1.08866i
$$55$$ −18.7279 −2.52527
$$56$$ 2.82843 0.377964
$$57$$ − 1.75736i − 0.232768i
$$58$$ 8.24264i 1.08231i
$$59$$ − 11.6569i − 1.51759i −0.651328 0.758797i $$-0.725788\pi$$
0.651328 0.758797i $$-0.274212\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 7.41421 0.941606
$$63$$ 1.00000i 0.125988i
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 8.48528 1.04447
$$67$$ 2.48528i 0.303625i 0.988409 + 0.151813i $$0.0485111\pi$$
−0.988409 + 0.151813i $$0.951489\pi$$
$$68$$ 0 0
$$69$$ −0.242641 −0.0292105
$$70$$ 6.24264i 0.746138i
$$71$$ 1.07107i 0.127112i 0.997978 + 0.0635562i $$0.0202442\pi$$
−0.997978 + 0.0635562i $$0.979756\pi$$
$$72$$ − 2.82843i − 0.333333i
$$73$$ 0.757359i 0.0886422i 0.999017 + 0.0443211i $$0.0141125\pi$$
−0.999017 + 0.0443211i $$0.985888\pi$$
$$74$$ −8.82843 −1.02628
$$75$$ 20.4853 2.36544
$$76$$ 0 0
$$77$$ 4.24264 0.483494
$$78$$ 0 0
$$79$$ −1.48528 −0.167107 −0.0835536 0.996503i $$-0.526627\pi$$
−0.0835536 + 0.996503i $$0.526627\pi$$
$$80$$ − 17.6569i − 1.97410i
$$81$$ −5.00000 −0.555556
$$82$$ −4.48528 −0.495316
$$83$$ 4.75736i 0.522188i 0.965313 + 0.261094i $$0.0840833\pi$$
−0.965313 + 0.261094i $$0.915917\pi$$
$$84$$ 0 0
$$85$$ 6.24264i 0.677109i
$$86$$ 7.07107i 0.762493i
$$87$$ −8.24264 −0.883704
$$88$$ −12.0000 −1.27920
$$89$$ − 4.41421i − 0.467906i −0.972248 0.233953i $$-0.924834\pi$$
0.972248 0.233953i $$-0.0751661\pi$$
$$90$$ 6.24264 0.658032
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 7.41421i 0.768818i
$$94$$ 6.24264 0.643879
$$95$$ −5.48528 −0.562778
$$96$$ 0 0
$$97$$ − 13.7279i − 1.39386i −0.717139 0.696930i $$-0.754549\pi$$
0.717139 0.696930i $$-0.245451\pi$$
$$98$$ − 1.41421i − 0.142857i
$$99$$ − 4.24264i − 0.426401i
$$100$$ 0 0
$$101$$ −1.75736 −0.174864 −0.0874319 0.996170i $$-0.527866\pi$$
−0.0874319 + 0.996170i $$0.527866\pi$$
$$102$$ − 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
$$105$$ −6.24264 −0.609219
$$106$$ − 8.24264i − 0.800596i
$$107$$ 8.14214 0.787130 0.393565 0.919297i $$-0.371242\pi$$
0.393565 + 0.919297i $$0.371242\pi$$
$$108$$ 0 0
$$109$$ − 8.72792i − 0.835983i −0.908451 0.417992i $$-0.862734\pi$$
0.908451 0.417992i $$-0.137266\pi$$
$$110$$ − 26.4853i − 2.52527i
$$111$$ − 8.82843i − 0.837957i
$$112$$ 4.00000i 0.377964i
$$113$$ −20.3137 −1.91095 −0.955476 0.295067i $$-0.904658\pi$$
−0.955476 + 0.295067i $$0.904658\pi$$
$$114$$ 2.48528 0.232768
$$115$$ 0.757359i 0.0706241i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 16.4853 1.51759
$$119$$ − 1.41421i − 0.129641i
$$120$$ 17.6569 1.61184
$$121$$ −7.00000 −0.636364
$$122$$ 8.48528i 0.768221i
$$123$$ − 4.48528i − 0.404424i
$$124$$ 0 0
$$125$$ − 41.8701i − 3.74497i
$$126$$ −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 $$-0.460569\pi$$
0.123560 + 0.992337i $$0.460569\pi$$
$$132$$ 0 0
$$133$$ 1.24264 0.107751
$$134$$ −3.51472 −0.303625
$$135$$ 24.9706i 2.14912i
$$136$$ 4.00000i 0.342997i
$$137$$ 7.41421i 0.633439i 0.948519 + 0.316720i $$0.102581\pi$$
−0.948519 + 0.316720i $$0.897419\pi$$
$$138$$ − 0.343146i − 0.0292105i
$$139$$ −2.24264 −0.190218 −0.0951092 0.995467i $$-0.530320\pi$$
−0.0951092 + 0.995467i $$0.530320\pi$$
$$140$$ 0 0
$$141$$ 6.24264i 0.525725i
$$142$$ −1.51472 −0.127112
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ 25.7279i 2.13659i
$$146$$ −1.07107 −0.0886422
$$147$$ 1.41421 0.116642
$$148$$ 0 0
$$149$$ 7.75736i 0.635508i 0.948173 + 0.317754i $$0.102929\pi$$
−0.948173 + 0.317754i $$0.897071\pi$$
$$150$$ 28.9706i 2.36544i
$$151$$ 18.2426i 1.48457i 0.670087 + 0.742283i $$0.266257\pi$$
−0.670087 + 0.742283i $$0.733743\pi$$
$$152$$ −3.51472 −0.285081
$$153$$ −1.41421 −0.114332
$$154$$ 6.00000i 0.483494i
$$155$$ 23.1421 1.85882
$$156$$ 0 0
$$157$$ −12.2426 −0.977069 −0.488535 0.872545i $$-0.662468\pi$$
−0.488535 + 0.872545i $$0.662468\pi$$
$$158$$ − 2.10051i − 0.167107i
$$159$$ 8.24264 0.653684
$$160$$ 0 0
$$161$$ − 0.171573i − 0.0135218i
$$162$$ − 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
$$165$$ 26.4853 2.06188
$$166$$ −6.72792 −0.522188
$$167$$ 21.3848i 1.65480i 0.561610 + 0.827402i $$0.310182\pi$$
−0.561610 + 0.827402i $$0.689818\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 0 0
$$170$$ −8.82843 −0.677109
$$171$$ − 1.24264i − 0.0950271i
$$172$$ 0 0
$$173$$ 0.727922 0.0553429 0.0276714 0.999617i $$-0.491191\pi$$
0.0276714 + 0.999617i $$0.491191\pi$$
$$174$$ − 11.6569i − 0.883704i
$$175$$ 14.4853i 1.09498i
$$176$$ − 16.9706i − 1.27920i
$$177$$ 16.4853i 1.23911i
$$178$$ 6.24264 0.467906
$$179$$ 9.00000 0.672692 0.336346 0.941739i $$-0.390809\pi$$
0.336346 + 0.941739i $$0.390809\pi$$
$$180$$ 0 0
$$181$$ −6.72792 −0.500083 −0.250041 0.968235i $$-0.580444\pi$$
−0.250041 + 0.968235i $$0.580444\pi$$
$$182$$ 0 0
$$183$$ −8.48528 −0.627250
$$184$$ 0.485281i 0.0357754i
$$185$$ −27.5563 −2.02598
$$186$$ −10.4853 −0.768818
$$187$$ 6.00000i 0.438763i
$$188$$ 0 0
$$189$$ − 5.65685i − 0.411476i
$$190$$ − 7.75736i − 0.562778i
$$191$$ 20.8284 1.50709 0.753546 0.657395i $$-0.228342\pi$$
0.753546 + 0.657395i $$0.228342\pi$$
$$192$$ 11.3137 0.816497
$$193$$ − 2.48528i − 0.178894i −0.995992 0.0894472i $$-0.971490\pi$$
0.995992 0.0894472i $$-0.0285100\pi$$
$$194$$ 19.4142 1.39386
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.6569i 1.68548i 0.538320 + 0.842741i $$0.319059\pi$$
−0.538320 + 0.842741i $$0.680941\pi$$
$$198$$ 6.00000 0.426401
$$199$$ 12.2426 0.867858 0.433929 0.900947i $$-0.357127\pi$$
0.433929 + 0.900947i $$0.357127\pi$$
$$200$$ − 40.9706i − 2.89706i
$$201$$ − 3.51472i − 0.247909i
$$202$$ − 2.48528i − 0.174864i
$$203$$ − 5.82843i − 0.409075i
$$204$$ 0 0
$$205$$ −14.0000 −0.977802
$$206$$ 11.3137i 0.788263i
$$207$$ −0.171573 −0.0119251
$$208$$ 0 0
$$209$$ −5.27208 −0.364677
$$210$$ − 8.82843i − 0.609219i
$$211$$ 17.9706 1.23714 0.618572 0.785728i $$-0.287711\pi$$
0.618572 + 0.785728i $$0.287711\pi$$
$$212$$ 0 0
$$213$$ − 1.51472i − 0.103787i
$$214$$ 11.5147i 0.787130i
$$215$$ 22.0711i 1.50523i
$$216$$ 16.0000i 1.08866i
$$217$$ −5.24264 −0.355894
$$218$$ 12.3431 0.835983
$$219$$ − 1.07107i − 0.0723761i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 12.4853 0.837957
$$223$$ 9.24264i 0.618933i 0.950910 + 0.309466i $$0.100150\pi$$
−0.950910 + 0.309466i $$0.899850\pi$$
$$224$$ 0 0
$$225$$ 14.4853 0.965685
$$226$$ − 28.7279i − 1.91095i
$$227$$ 21.1716i 1.40521i 0.711582 + 0.702603i $$0.247979\pi$$
−0.711582 + 0.702603i $$0.752021\pi$$
$$228$$ 0 0
$$229$$ 21.4558i 1.41784i 0.705288 + 0.708921i $$0.250818\pi$$
−0.705288 + 0.708921i $$0.749182\pi$$
$$230$$ −1.07107 −0.0706241
$$231$$ −6.00000 −0.394771
$$232$$ 16.4853i 1.08231i
$$233$$ 3.34315 0.219017 0.109508 0.993986i $$-0.465072\pi$$
0.109508 + 0.993986i $$0.465072\pi$$
$$234$$ 0 0
$$235$$ 19.4853 1.27108
$$236$$ 0 0
$$237$$ 2.10051 0.136442
$$238$$ 2.00000 0.129641
$$239$$ 20.4853i 1.32508i 0.749025 + 0.662541i $$0.230522\pi$$
−0.749025 + 0.662541i $$0.769478\pi$$
$$240$$ 24.9706i 1.61184i
$$241$$ − 22.2132i − 1.43088i −0.698675 0.715439i $$-0.746227\pi$$
0.698675 0.715439i $$-0.253773\pi$$
$$242$$ − 9.89949i − 0.636364i
$$243$$ −9.89949 −0.635053
$$244$$ 0 0
$$245$$ − 4.41421i − 0.282014i
$$246$$ 6.34315 0.404424
$$247$$ 0 0
$$248$$ 14.8284 0.941606
$$249$$ − 6.72792i − 0.426365i
$$250$$ 59.2132 3.74497
$$251$$ −19.4142 −1.22541 −0.612707 0.790310i $$-0.709919\pi$$
−0.612707 + 0.790310i $$0.709919\pi$$
$$252$$ 0 0
$$253$$ 0.727922i 0.0457641i
$$254$$ − 2.82843i − 0.177471i
$$255$$ − 8.82843i − 0.552858i
$$256$$ 0 0
$$257$$ 16.5858 1.03459 0.517296 0.855806i $$-0.326938\pi$$
0.517296 + 0.855806i $$0.326938\pi$$
$$258$$ − 10.0000i − 0.622573i
$$259$$ 6.24264 0.387899
$$260$$ 0 0
$$261$$ −5.82843 −0.360771
$$262$$ 4.00000i 0.247121i
$$263$$ −31.9706 −1.97139 −0.985695 0.168541i $$-0.946095\pi$$
−0.985695 + 0.168541i $$0.946095\pi$$
$$264$$ 16.9706 1.04447
$$265$$ − 25.7279i − 1.58045i
$$266$$ 1.75736i 0.107751i
$$267$$ 6.24264i 0.382043i
$$268$$ 0 0
$$269$$ 14.8284 0.904105 0.452053 0.891991i $$-0.350692\pi$$
0.452053 + 0.891991i $$0.350692\pi$$
$$270$$ −35.3137 −2.14912
$$271$$ − 20.0000i − 1.21491i −0.794353 0.607457i $$-0.792190\pi$$
0.794353 0.607457i $$-0.207810\pi$$
$$272$$ −5.65685 −0.342997
$$273$$ 0 0
$$274$$ −10.4853 −0.633439
$$275$$ − 61.4558i − 3.70593i
$$276$$ 0 0
$$277$$ −9.48528 −0.569915 −0.284958 0.958540i $$-0.591980\pi$$
−0.284958 + 0.958540i $$0.591980\pi$$
$$278$$ − 3.17157i − 0.190218i
$$279$$ 5.24264i 0.313869i
$$280$$ 12.4853i 0.746138i
$$281$$ 15.5563i 0.928014i 0.885832 + 0.464007i $$0.153589\pi$$
−0.885832 + 0.464007i $$0.846411\pi$$
$$282$$ −8.82843 −0.525725
$$283$$ −8.48528 −0.504398 −0.252199 0.967675i $$-0.581154\pi$$
−0.252199 + 0.967675i $$0.581154\pi$$
$$284$$ 0 0
$$285$$ 7.75736 0.459506
$$286$$ 0 0
$$287$$ 3.17157 0.187212
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ −36.3848 −2.13659
$$291$$ 19.4142i 1.13808i
$$292$$ 0 0
$$293$$ 21.3848i 1.24931i 0.780900 + 0.624656i $$0.214761\pi$$
−0.780900 + 0.624656i $$0.785239\pi$$
$$294$$ 2.00000i 0.116642i
$$295$$ 51.4558 2.99588
$$296$$ −17.6569 −1.02628
$$297$$ 24.0000i 1.39262i
$$298$$ −10.9706 −0.635508
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 5.00000i − 0.288195i
$$302$$ −25.7990 −1.48457
$$303$$ 2.48528 0.142776
$$304$$ − 4.97056i − 0.285081i
$$305$$ 26.4853i 1.51654i
$$306$$ − 2.00000i − 0.114332i
$$307$$ − 4.75736i − 0.271517i −0.990742 0.135758i $$-0.956653\pi$$
0.990742 0.135758i $$-0.0433471\pi$$
$$308$$ 0 0
$$309$$ −11.3137 −0.643614
$$310$$ 32.7279i 1.85882i
$$311$$ 4.58579 0.260036 0.130018 0.991512i $$-0.458496\pi$$
0.130018 + 0.991512i $$0.458496\pi$$
$$312$$ 0 0
$$313$$ −19.2132 −1.08599 −0.542997 0.839734i $$-0.682711\pi$$
−0.542997 + 0.839734i $$0.682711\pi$$
$$314$$ − 17.3137i − 0.977069i
$$315$$ −4.41421 −0.248713
$$316$$ 0 0
$$317$$ 11.3137i 0.635441i 0.948184 + 0.317721i $$0.102917\pi$$
−0.948184 + 0.317721i $$0.897083\pi$$
$$318$$ 11.6569i 0.653684i
$$319$$ 24.7279i 1.38450i
$$320$$ − 35.3137i − 1.97410i
$$321$$ −11.5147 −0.642689
$$322$$ 0.242641 0.0135218
$$323$$ 1.75736i 0.0977821i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 12.3431i 0.682578i
$$328$$ −8.97056 −0.495316
$$329$$ −4.41421 −0.243363
$$330$$ 37.4558i 2.06188i
$$331$$ − 18.0000i − 0.989369i −0.869072 0.494685i $$-0.835284\pi$$
0.869072 0.494685i $$-0.164716\pi$$
$$332$$ 0 0
$$333$$ − 6.24264i − 0.342095i
$$334$$ −30.2426 −1.65480
$$335$$ −10.9706 −0.599386
$$336$$ − 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
$$339$$ 28.7279 1.56029
$$340$$ 0 0
$$341$$ 22.2426 1.20451
$$342$$ 1.75736 0.0950271
$$343$$ 1.00000i 0.0539949i
$$344$$ 14.1421i 0.762493i
$$345$$ − 1.07107i − 0.0576644i
$$346$$ 1.02944i 0.0553429i
$$347$$ 5.65685 0.303676 0.151838 0.988405i $$-0.451481\pi$$
0.151838 + 0.988405i $$0.451481\pi$$
$$348$$ 0 0
$$349$$ 0.272078i 0.0145640i 0.999973 + 0.00728200i $$0.00231795\pi$$
−0.999973 + 0.00728200i $$0.997682\pi$$
$$350$$ −20.4853 −1.09498
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8.48528i 0.451626i 0.974171 + 0.225813i $$0.0725038\pi$$
−0.974171 + 0.225813i $$0.927496\pi$$
$$354$$ −23.3137 −1.23911
$$355$$ −4.72792 −0.250932
$$356$$ 0 0
$$357$$ 2.00000i 0.105851i
$$358$$ 12.7279i 0.672692i
$$359$$ 8.10051i 0.427528i 0.976885 + 0.213764i $$0.0685724\pi$$
−0.976885 + 0.213764i $$0.931428\pi$$
$$360$$ 12.4853 0.658032
$$361$$ 17.4558 0.918729
$$362$$ − 9.51472i − 0.500083i
$$363$$ 9.89949 0.519589
$$364$$ 0 0
$$365$$ −3.34315 −0.174988
$$366$$ − 12.0000i − 0.627250i
$$367$$ 1.75736 0.0917334 0.0458667 0.998948i $$-0.485395\pi$$
0.0458667 + 0.998948i $$0.485395\pi$$
$$368$$ −0.686292 −0.0357754
$$369$$ − 3.17157i − 0.165105i
$$370$$ − 38.9706i − 2.02598i
$$371$$ 5.82843i 0.302597i
$$372$$ 0 0
$$373$$ −8.48528 −0.439351 −0.219676 0.975573i $$-0.570500\pi$$
−0.219676 + 0.975573i $$0.570500\pi$$
$$374$$ −8.48528 −0.438763
$$375$$ 59.2132i 3.05776i
$$376$$ 12.4853 0.643879
$$377$$ 0 0
$$378$$ 8.00000 0.411476
$$379$$ 32.2426i 1.65619i 0.560585 + 0.828097i $$0.310576\pi$$
−0.560585 + 0.828097i $$0.689424\pi$$
$$380$$ 0 0
$$381$$ 2.82843 0.144905
$$382$$ 29.4558i 1.50709i
$$383$$ − 3.51472i − 0.179594i −0.995960 0.0897969i $$-0.971378\pi$$
0.995960 0.0897969i $$-0.0286218\pi$$
$$384$$ 16.0000i 0.816497i
$$385$$ 18.7279i 0.954463i
$$386$$ 3.51472 0.178894
$$387$$ −5.00000 −0.254164
$$388$$ 0 0
$$389$$ −18.3431 −0.930034 −0.465017 0.885302i $$-0.653952\pi$$
−0.465017 + 0.885302i $$0.653952\pi$$
$$390$$ 0 0
$$391$$ 0.242641 0.0122709
$$392$$ − 2.82843i − 0.142857i
$$393$$ −4.00000 −0.201773
$$394$$ −33.4558 −1.68548
$$395$$ − 6.55635i − 0.329886i
$$396$$ 0 0
$$397$$ − 24.2132i − 1.21523i −0.794233 0.607613i $$-0.792127\pi$$
0.794233 0.607613i $$-0.207873\pi$$
$$398$$ 17.3137i 0.867858i
$$399$$ −1.75736 −0.0879780
$$400$$ 57.9411 2.89706
$$401$$ − 17.6569i − 0.881741i −0.897571 0.440871i $$-0.854670\pi$$
0.897571 0.440871i $$-0.145330\pi$$
$$402$$ 4.97056 0.247909
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 22.0711i − 1.09672i
$$406$$ 8.24264 0.409075
$$407$$ −26.4853 −1.31283
$$408$$ − 5.65685i − 0.280056i
$$409$$ − 5.24264i − 0.259232i −0.991564 0.129616i $$-0.958626\pi$$
0.991564 0.129616i $$-0.0413744\pi$$
$$410$$ − 19.7990i − 0.977802i
$$411$$ − 10.4853i − 0.517201i
$$412$$ 0 0
$$413$$ −11.6569 −0.573596
$$414$$ − 0.242641i − 0.0119251i
$$415$$ −21.0000 −1.03085
$$416$$ 0 0
$$417$$ 3.17157 0.155313
$$418$$ − 7.45584i − 0.364677i
$$419$$ 32.8701 1.60581 0.802904 0.596109i $$-0.203287\pi$$
0.802904 + 0.596109i $$0.203287\pi$$
$$420$$ 0 0
$$421$$ − 18.7279i − 0.912743i −0.889789 0.456372i $$-0.849149\pi$$
0.889789 0.456372i $$-0.150851\pi$$
$$422$$ 25.4142i 1.23714i
$$423$$ 4.41421i 0.214626i
$$424$$ − 16.4853i − 0.800596i
$$425$$ −20.4853 −0.993682
$$426$$ 2.14214 0.103787
$$427$$ − 6.00000i − 0.290360i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −31.2132 −1.50523
$$431$$ − 23.6569i − 1.13951i −0.821814 0.569755i $$-0.807038\pi$$
0.821814 0.569755i $$-0.192962\pi$$
$$432$$ −22.6274 −1.08866
$$433$$ −8.97056 −0.431098 −0.215549 0.976493i $$-0.569154\pi$$
−0.215549 + 0.976493i $$0.569154\pi$$
$$434$$ − 7.41421i − 0.355894i
$$435$$ − 36.3848i − 1.74452i
$$436$$ 0 0
$$437$$ 0.213203i 0.0101989i
$$438$$ 1.51472 0.0723761
$$439$$ 17.5147 0.835932 0.417966 0.908463i $$-0.362743\pi$$
0.417966 + 0.908463i $$0.362743\pi$$
$$440$$ − 52.9706i − 2.52527i
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −26.3137 −1.25020 −0.625101 0.780544i $$-0.714942\pi$$
−0.625101 + 0.780544i $$0.714942\pi$$
$$444$$ 0 0
$$445$$ 19.4853 0.923691
$$446$$ −13.0711 −0.618933
$$447$$ − 10.9706i − 0.518890i
$$448$$ 8.00000i 0.377964i
$$449$$ − 26.8284i − 1.26611i −0.774106 0.633056i $$-0.781800\pi$$
0.774106 0.633056i $$-0.218200\pi$$
$$450$$ 20.4853i 0.965685i
$$451$$ −13.4558 −0.633611
$$452$$ 0 0
$$453$$ − 25.7990i − 1.21214i
$$454$$ −29.9411 −1.40521
$$455$$ 0 0
$$456$$ 4.97056 0.232768
$$457$$ 35.2132i 1.64720i 0.567168 + 0.823602i $$0.308039\pi$$
−0.567168 + 0.823602i $$0.691961\pi$$
$$458$$ −30.3431 −1.41784
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ 3.17157i 0.147715i 0.997269 + 0.0738574i $$0.0235310\pi$$
−0.997269 + 0.0738574i $$0.976469\pi$$
$$462$$ − 8.48528i − 0.394771i
$$463$$ − 4.24264i − 0.197172i −0.995129 0.0985861i $$-0.968568\pi$$
0.995129 0.0985861i $$-0.0314320\pi$$
$$464$$ −23.3137 −1.08231
$$465$$ −32.7279 −1.51772
$$466$$ 4.72792i 0.219017i
$$467$$ −15.8995 −0.735741 −0.367870 0.929877i $$-0.619913\pi$$
−0.367870 + 0.929877i $$0.619913\pi$$
$$468$$ 0 0
$$469$$ 2.48528 0.114760
$$470$$ 27.5563i 1.27108i
$$471$$ 17.3137 0.797774
$$472$$ 32.9706 1.51759
$$473$$ 21.2132i 0.975384i
$$474$$ 2.97056i 0.136442i
$$475$$ − 18.0000i − 0.825897i
$$476$$ 0 0
$$477$$ 5.82843 0.266865
$$478$$ −28.9706 −1.32508
$$479$$ − 36.2132i − 1.65462i −0.561743 0.827312i $$-0.689869\pi$$
0.561743 0.827312i $$-0.310131\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 31.4142 1.43088
$$483$$ 0.242641i 0.0110405i
$$484$$ 0 0
$$485$$ 60.5980 2.75161
$$486$$ − 14.0000i − 0.635053i
$$487$$ 39.4558i 1.78791i 0.448152 + 0.893957i $$0.352082\pi$$
−0.448152 + 0.893957i $$0.647918\pi$$
$$488$$ 16.9706i 0.768221i
$$489$$ − 12.0000i − 0.542659i
$$490$$ 6.24264 0.282014
$$491$$ 28.6274 1.29194 0.645969 0.763364i $$-0.276454\pi$$
0.645969 + 0.763364i $$0.276454\pi$$
$$492$$ 0 0
$$493$$ 8.24264 0.371230
$$494$$ 0 0
$$495$$ 18.7279 0.841757
$$496$$ 20.9706i 0.941606i
$$497$$ 1.07107 0.0480440
$$498$$ 9.51472 0.426365
$$499$$ − 38.7279i − 1.73370i −0.498569 0.866850i $$-0.666141\pi$$
0.498569 0.866850i $$-0.333859\pi$$
$$500$$ 0 0
$$501$$ − 30.2426i − 1.35114i
$$502$$ − 27.4558i − 1.22541i
$$503$$ 16.6274 0.741380 0.370690 0.928757i $$-0.379121\pi$$
0.370690 + 0.928757i $$0.379121\pi$$
$$504$$ −2.82843 −0.125988
$$505$$ − 7.75736i − 0.345198i
$$506$$ −1.02944 −0.0457641
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 24.8995i − 1.10365i −0.833960 0.551825i $$-0.813931\pi$$
0.833960 0.551825i $$-0.186069\pi$$
$$510$$ 12.4853 0.552858
$$511$$ 0.757359 0.0335036
$$512$$ − 22.6274i − 1.00000i
$$513$$ 7.02944i 0.310357i
$$514$$ 23.4558i 1.03459i
$$515$$ 35.3137i 1.55611i
$$516$$ 0 0
$$517$$ 18.7279 0.823653
$$518$$ 8.82843i 0.387899i
$$519$$ −1.02944 −0.0451873
$$520$$ 0 0
$$521$$ −17.6569 −0.773561 −0.386780 0.922172i $$-0.626413\pi$$
−0.386780 + 0.922172i $$0.626413\pi$$
$$522$$ − 8.24264i − 0.360771i
$$523$$ −2.97056 −0.129894 −0.0649468 0.997889i $$-0.520688\pi$$
−0.0649468 + 0.997889i $$0.520688\pi$$
$$524$$ 0 0
$$525$$ − 20.4853i − 0.894051i
$$526$$ − 45.2132i − 1.97139i
$$527$$ − 7.41421i − 0.322968i
$$528$$ 24.0000i 1.04447i
$$529$$ −22.9706 −0.998720
$$530$$ 36.3848 1.58045
$$531$$ 11.6569i 0.505864i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −8.82843 −0.382043
$$535$$ 35.9411i 1.55387i
$$536$$ −7.02944 −0.303625
$$537$$ −12.7279 −0.549250
$$538$$ 20.9706i 0.904105i
$$539$$ − 4.24264i − 0.182743i
$$540$$ 0 0
$$541$$ − 35.2132i − 1.51393i −0.653453 0.756967i $$-0.726680\pi$$
0.653453 0.756967i $$-0.273320\pi$$
$$542$$ 28.2843 1.21491
$$543$$ 9.51472 0.408316
$$544$$ 0 0
$$545$$ 38.5269 1.65031
$$546$$ 0 0
$$547$$ 18.5147 0.791632 0.395816 0.918330i $$-0.370462\pi$$
0.395816 + 0.918330i $$0.370462\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 86.9117 3.70593
$$551$$ 7.24264i 0.308547i
$$552$$ − 0.686292i − 0.0292105i
$$553$$ 1.48528i 0.0631606i
$$554$$ − 13.4142i − 0.569915i
$$555$$ 38.9706 1.65421
$$556$$ 0 0
$$557$$ − 12.0000i − 0.508456i −0.967144 0.254228i $$-0.918179\pi$$
0.967144 0.254228i $$-0.0818214\pi$$
$$558$$ −7.41421 −0.313869
$$559$$ 0 0
$$560$$ −17.6569 −0.746138
$$561$$ − 8.48528i − 0.358249i
$$562$$ −22.0000 −0.928014
$$563$$ 17.6569 0.744148 0.372074 0.928203i $$-0.378647\pi$$
0.372074 + 0.928203i $$0.378647\pi$$
$$564$$ 0 0
$$565$$ − 89.6690i − 3.77241i
$$566$$ − 12.0000i − 0.504398i
$$567$$ 5.00000i 0.209980i
$$568$$ −3.02944 −0.127112
$$569$$ 23.1421 0.970169 0.485084 0.874467i $$-0.338789\pi$$
0.485084 + 0.874467i $$0.338789\pi$$
$$570$$ 10.9706i 0.459506i
$$571$$ 8.45584 0.353866 0.176933 0.984223i $$-0.443382\pi$$
0.176933 + 0.984223i $$0.443382\pi$$
$$572$$ 0 0
$$573$$ −29.4558 −1.23054
$$574$$ 4.48528i 0.187212i
$$575$$ −2.48528 −0.103643
$$576$$ 8.00000 0.333333
$$577$$ 26.9706i 1.12280i 0.827545 + 0.561400i $$0.189737\pi$$
−0.827545 + 0.561400i $$0.810263\pi$$
$$578$$ − 21.2132i − 0.882353i
$$579$$ 3.51472i 0.146067i
$$580$$ 0 0
$$581$$ 4.75736 0.197369
$$582$$ −27.4558 −1.13808
$$583$$ − 24.7279i − 1.02413i
$$584$$ −2.14214 −0.0886422
$$585$$ 0 0
$$586$$ −30.2426 −1.24931
$$587$$ 24.5563i 1.01355i 0.862079 + 0.506775i $$0.169162\pi$$
−0.862079 + 0.506775i $$0.830838\pi$$
$$588$$ 0 0
$$589$$ 6.51472 0.268434
$$590$$ 72.7696i 2.99588i
$$591$$ − 33.4558i − 1.37619i
$$592$$ − 24.9706i − 1.02628i
$$593$$ − 30.5563i − 1.25480i −0.778698 0.627399i $$-0.784119\pi$$
0.778698 0.627399i $$-0.215881\pi$$
$$594$$ −33.9411 −1.39262
$$595$$ 6.24264 0.255923
$$596$$ 0 0
$$597$$ −17.3137 −0.708603
$$598$$ 0 0
$$599$$ 10.7990 0.441235 0.220617 0.975360i $$-0.429193\pi$$
0.220617 + 0.975360i $$0.429193\pi$$
$$600$$ 57.9411i 2.36544i
$$601$$ −5.02944 −0.205155 −0.102578 0.994725i $$-0.532709\pi$$
−0.102578 + 0.994725i $$0.532709\pi$$
$$602$$ 7.07107 0.288195
$$603$$ − 2.48528i − 0.101208i
$$604$$ 0 0
$$605$$ − 30.8995i − 1.25624i
$$606$$ 3.51472i 0.142776i
$$607$$ −1.27208 −0.0516321 −0.0258160 0.999667i $$-0.508218\pi$$
−0.0258160 + 0.999667i $$0.508218\pi$$
$$608$$ 0 0
$$609$$ 8.24264i 0.334009i
$$610$$ −37.4558 −1.51654
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 16.0000i − 0.646234i −0.946359 0.323117i $$-0.895269\pi$$
0.946359 0.323117i $$-0.104731\pi$$
$$614$$ 6.72792 0.271517
$$615$$ 19.7990 0.798372
$$616$$ 12.0000i 0.483494i
$$617$$ 23.6569i 0.952389i 0.879340 + 0.476195i $$0.157984\pi$$
−0.879340 + 0.476195i $$0.842016\pi$$
$$618$$ − 16.0000i − 0.643614i
$$619$$ 32.9706i 1.32520i 0.748974 + 0.662599i $$0.230547\pi$$
−0.748974 + 0.662599i $$0.769453\pi$$
$$620$$ 0 0
$$621$$ 0.970563 0.0389473
$$622$$ 6.48528i 0.260036i
$$623$$ −4.41421 −0.176852
$$624$$ 0 0
$$625$$ 112.397 4.49588
$$626$$ − 27.1716i − 1.08599i
$$627$$ 7.45584 0.297758
$$628$$ 0 0
$$629$$ 8.82843i 0.352012i
$$630$$ − 6.24264i − 0.248713i
$$631$$ − 2.00000i − 0.0796187i −0.999207 0.0398094i $$-0.987325\pi$$
0.999207 0.0398094i $$-0.0126751\pi$$
$$632$$ − 4.20101i − 0.167107i
$$633$$ −25.4142 −1.01012
$$634$$ −16.0000 −0.635441
$$635$$ − 8.82843i − 0.350345i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −34.9706 −1.38450
$$639$$ − 1.07107i − 0.0423708i
$$640$$ 49.9411 1.97410
$$641$$ −26.6569 −1.05288 −0.526441 0.850212i $$-0.676474\pi$$
−0.526441 + 0.850212i $$0.676474\pi$$
$$642$$ − 16.2843i − 0.642689i
$$643$$ 4.48528i 0.176882i 0.996081 + 0.0884411i $$0.0281885\pi$$
−0.996081 + 0.0884411i $$0.971811\pi$$
$$644$$ 0 0
$$645$$ − 31.2132i − 1.22902i
$$646$$ −2.48528 −0.0977821
$$647$$ 50.5269 1.98642 0.993209 0.116344i $$-0.0371176\pi$$
0.993209 + 0.116344i $$0.0371176\pi$$
$$648$$ − 14.1421i − 0.555556i
$$649$$ 49.4558 1.94131
$$650$$ 0 0
$$651$$ 7.41421 0.290586
$$652$$ 0 0
$$653$$ −5.31371 −0.207941 −0.103971 0.994580i $$-0.533155\pi$$
−0.103971 + 0.994580i $$0.533155\pi$$
$$654$$ −17.4558 −0.682578
$$655$$ 12.4853i 0.487840i
$$656$$ − 12.6863i − 0.495316i
$$657$$ − 0.757359i − 0.0295474i
$$658$$ − 6.24264i − 0.243363i
$$659$$ −26.6569 −1.03840 −0.519202 0.854652i $$-0.673771\pi$$
−0.519202 + 0.854652i $$0.673771\pi$$
$$660$$ 0 0
$$661$$ 19.2426i 0.748452i 0.927338 + 0.374226i $$0.122092\pi$$
−0.927338 + 0.374226i $$0.877908\pi$$
$$662$$ 25.4558 0.989369
$$663$$ 0 0
$$664$$ −13.4558 −0.522188
$$665$$ 5.48528i 0.212710i
$$666$$ 8.82843 0.342095
$$667$$ 1.00000 0.0387202
$$668$$ 0 0
$$669$$ − 13.0711i − 0.505357i
$$670$$ − 15.5147i − 0.599386i
$$671$$ 25.4558i 0.982712i
$$672$$ 0 0
$$673$$ −40.9411 −1.57816 −0.789082 0.614288i $$-0.789443\pi$$
−0.789082 + 0.614288i $$0.789443\pi$$
$$674$$ 46.6690i 1.79762i
$$675$$ −81.9411 −3.15392
$$676$$ 0 0
$$677$$ 29.3553 1.12822 0.564109 0.825701i $$-0.309220\pi$$
0.564109 + 0.825701i $$0.309220\pi$$
$$678$$ 40.6274i 1.56029i
$$679$$ −13.7279 −0.526829
$$680$$ −17.6569 −0.677109
$$681$$ − 29.9411i − 1.14735i
$$682$$ 31.4558i 1.20451i
$$683$$ 26.8284i 1.02656i 0.858221 + 0.513281i $$0.171570\pi$$
−0.858221 + 0.513281i $$0.828430\pi$$
$$684$$ 0 0
$$685$$ −32.7279 −1.25047
$$686$$ −1.41421 −0.0539949
$$687$$ − 30.3431i − 1.15766i
$$688$$ −20.0000 −0.762493
$$689$$ 0 0
$$690$$ 1.51472 0.0576644
$$691$$ − 30.6985i − 1.16783i −0.811816 0.583913i $$-0.801521\pi$$
0.811816 0.583913i $$-0.198479\pi$$
$$692$$ 0 0
$$693$$ −4.24264 −0.161165
$$694$$ 8.00000i 0.303676i
$$695$$ − 9.89949i − 0.375509i
$$696$$ − 23.3137i − 0.883704i
$$697$$ 4.48528i 0.169892i
$$698$$ −0.384776 −0.0145640
$$699$$ −4.72792 −0.178826
$$700$$ 0 0
$$701$$ −28.7990 −1.08772 −0.543861 0.839175i $$-0.683038\pi$$
−0.543861 + 0.839175i $$0.683038\pi$$
$$702$$ 0 0
$$703$$ −7.75736 −0.292574
$$704$$ − 33.9411i − 1.27920i
$$705$$ −27.5563 −1.03783
$$706$$ −12.0000 −0.451626
$$707$$ 1.75736i 0.0660923i
$$708$$ 0 0
$$709$$ 24.7279i 0.928677i 0.885658 + 0.464338i $$0.153708\pi$$
−0.885658 + 0.464338i $$0.846292\pi$$
$$710$$ − 6.68629i − 0.250932i
$$711$$ 1.48528 0.0557024
$$712$$ 12.4853 0.467906
$$713$$ − 0.899495i − 0.0336864i
$$714$$ −2.82843 −0.105851
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 28.9706i − 1.08193i
$$718$$ −11.4558 −0.427528
$$719$$ 10.2426 0.381986 0.190993 0.981591i $$-0.438829\pi$$
0.190993 + 0.981591i $$0.438829\pi$$
$$720$$ 17.6569i 0.658032i
$$721$$ − 8.00000i − 0.297936i
$$722$$ 24.6863i 0.918729i
$$723$$ 31.4142i 1.16831i
$$724$$ 0 0
$$725$$ −84.4264 −3.13552
$$726$$ 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
$$730$$ − 4.72792i − 0.174988i
$$731$$ 7.07107 0.261533
$$732$$ 0 0
$$733$$ 42.6985i 1.57710i 0.614968 + 0.788552i $$0.289169\pi$$
−0.614968 + 0.788552i $$0.710831\pi$$
$$734$$ 2.48528i 0.0917334i
$$735$$ 6.24264i 0.230263i
$$736$$ 0 0
$$737$$ −10.5442 −0.388399
$$738$$ 4.48528 0.165105
$$739$$ − 41.6985i − 1.53390i −0.641705 0.766952i $$-0.721773\pi$$
0.641705 0.766952i $$-0.278227\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −8.24264 −0.302597
$$743$$ − 18.3431i − 0.672945i −0.941693 0.336472i $$-0.890766\pi$$
0.941693 0.336472i $$-0.109234\pi$$
$$744$$ −20.9706 −0.768818
$$745$$ −34.2426 −1.25455
$$746$$ − 12.0000i − 0.439351i
$$747$$ − 4.75736i − 0.174063i
$$748$$ 0 0
$$749$$ − 8.14214i − 0.297507i
$$750$$ −83.7401 −3.05776
$$751$$ 1.48528 0.0541987 0.0270993 0.999633i $$-0.491373\pi$$
0.0270993 + 0.999633i $$0.491373\pi$$
$$752$$ 17.6569i 0.643879i
$$753$$ 27.4558 1.00055
$$754$$ 0 0
$$755$$ −80.5269 −2.93067
$$756$$ 0 0
$$757$$ 4.51472 0.164090 0.0820451 0.996629i $$-0.473855\pi$$
0.0820451 + 0.996629i $$0.473855\pi$$
$$758$$ −45.5980 −1.65619
$$759$$ − 1.02944i − 0.0373662i
$$760$$ − 15.5147i − 0.562778i
$$761$$ − 43.2426i − 1.56754i −0.621048 0.783772i $$-0.713293\pi$$
0.621048 0.783772i $$-0.286707\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ −8.72792 −0.315972
$$764$$ 0 0
$$765$$ − 6.24264i − 0.225703i
$$766$$ 4.97056 0.179594
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 9.78680i 0.352921i 0.984308 + 0.176460i $$0.0564648\pi$$
−0.984308 + 0.176460i $$0.943535\pi$$
$$770$$ −26.4853 −0.954463
$$771$$ −23.4558 −0.844742
$$772$$ 0 0
$$773$$ − 27.1716i − 0.977294i −0.872482 0.488647i $$-0.837491\pi$$
0.872482 0.488647i $$-0.162509\pi$$
$$774$$ − 7.07107i − 0.254164i
$$775$$ 75.9411i 2.72789i
$$776$$ 38.8284 1.39386
$$777$$ −8.82843 −0.316718
$$778$$ − 25.9411i − 0.930034i
$$779$$ −3.94113 −0.141205
$$780$$ 0 0
$$781$$ −4.54416 −0.162603
$$782$$ 0.343146i 0.0122709i
$$783$$ 32.9706 1.17827
$$784$$ 4.00000 0.142857
$$785$$ − 54.0416i − 1.92883i
$$786$$ − 5.65685i − 0.201773i
$$787$$ 33.2426i 1.18497i 0.805581 + 0.592486i $$0.201853\pi$$
−0.805581 + 0.592486i $$0.798147\pi$$
$$788$$ 0 0
$$789$$ 45.2132 1.60963
$$790$$ 9.27208 0.329886
$$791$$ 20.3137i 0.722272i
$$792$$ 12.0000 0.426401
$$793$$ 0 0
$$794$$ 34.2426 1.21523
$$795$$ 36.3848i 1.29044i
$$796$$ 0 0
$$797$$ 35.6569 1.26303 0.631515 0.775363i $$-0.282433\pi$$
0.631515 + 0.775363i $$0.282433\pi$$
$$798$$ − 2.48528i − 0.0879780i
$$799$$ − 6.24264i − 0.220849i
$$800$$ 0 0
$$801$$ 4.41421i 0.155969i
$$802$$ 24.9706 0.881741
$$803$$ −3.21320 −0.113391
$$804$$ 0 0
$$805$$ 0.757359 0.0266934
$$806$$ 0 0
$$807$$ −20.9706 −0.738199
$$808$$ − 4.97056i − 0.174864i
$$809$$ 0.514719 0.0180965 0.00904827 0.999959i $$-0.497120\pi$$
0.00904827 + 0.999959i $$0.497120\pi$$
$$810$$ 31.2132 1.09672
$$811$$ 45.9411i 1.61321i 0.591090 + 0.806606i $$0.298698\pi$$
−0.591090 + 0.806606i $$0.701302\pi$$
$$812$$ 0 0
$$813$$ 28.2843i 0.991973i
$$814$$ − 37.4558i − 1.31283i
$$815$$ −37.4558 −1.31202
$$816$$ 8.00000 0.280056
$$817$$ 6.21320i 0.217372i
$$818$$ 7.41421 0.259232
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.14214i 0.284162i 0.989855 + 0.142081i $$0.0453794\pi$$
−0.989855 + 0.142081i $$0.954621\pi$$
$$822$$ 14.8284 0.517201
$$823$$ 6.48528 0.226063 0.113031 0.993591i $$-0.463944\pi$$
0.113031 + 0.993591i $$0.463944\pi$$
$$824$$ 22.6274i 0.788263i
$$825$$ 86.9117i 3.02588i
$$826$$ − 16.4853i − 0.573596i
$$827$$ − 1.45584i − 0.0506247i −0.999680 0.0253123i $$-0.991942\pi$$
0.999680 0.0253123i $$-0.00805803\pi$$
$$828$$ 0 0
$$829$$ −25.2721 −0.877736 −0.438868 0.898552i $$-0.644620\pi$$
−0.438868 + 0.898552i $$0.644620\pi$$
$$830$$ − 29.6985i − 1.03085i
$$831$$ 13.4142 0.465334
$$832$$ 0 0
$$833$$ −1.41421 −0.0489996
$$834$$ 4.48528i 0.155313i
$$835$$ −94.3970 −3.26674
$$836$$ 0 0
$$837$$ − 29.6569i − 1.02509i
$$838$$ 46.4853i 1.60581i
$$839$$ 38.8284i 1.34051i 0.742133 + 0.670253i $$0.233814\pi$$
−0.742133 + 0.670253i $$0.766186\pi$$
$$840$$ − 17.6569i − 0.609219i
$$841$$ 4.97056 0.171399
$$842$$ 26.4853 0.912743
$$843$$ − 22.0000i − 0.757720i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ −6.24264 −0.214626
$$847$$ 7.00000i 0.240523i
$$848$$ 23.3137 0.800596
$$849$$ 12.0000 0.411839
$$850$$ − 28.9706i − 0.993682i
$$851$$ 1.07107i 0.0367157i
$$852$$ 0 0
$$853$$ − 14.2721i − 0.488667i −0.969691 0.244333i $$-0.921431\pi$$
0.969691 0.244333i $$-0.0785691\pi$$
$$854$$ 8.48528 0.290360
$$855$$ 5.48528 0.187593
$$856$$ 23.0294i 0.787130i
$$857$$ −18.7696 −0.641156 −0.320578 0.947222i $$-0.603877\pi$$
−0.320578 + 0.947222i $$0.603877\pi$$
$$858$$ 0 0
$$859$$ −2.97056 −0.101354 −0.0506771 0.998715i $$-0.516138\pi$$
−0.0506771 + 0.998715i $$0.516138\pi$$
$$860$$ 0 0
$$861$$ −4.48528 −0.152858
$$862$$ 33.4558 1.13951
$$863$$ − 28.2843i − 0.962808i −0.876499 0.481404i $$-0.840127\pi$$
0.876499 0.481404i $$-0.159873\pi$$
$$864$$ 0 0
$$865$$ 3.21320i 0.109252i
$$866$$ − 12.6863i − 0.431098i
$$867$$ 21.2132 0.720438
$$868$$ 0 0
$$869$$ − 6.30152i − 0.213764i
$$870$$ 51.4558 1.74452
$$871$$ 0 0
$$872$$ 24.6863 0.835983
$$873$$ 13.7279i 0.464620i
$$874$$ −0.301515 −0.0101989
$$875$$ −41.8701 −1.41547
$$876$$ 0 0
$$877$$ − 1.75736i − 0.0593418i −0.999560 0.0296709i $$-0.990554\pi$$
0.999560 0.0296709i $$-0.00944593\pi$$
$$878$$ 24.7696i 0.835932i
$$879$$ − 30.2426i − 1.02006i
$$880$$ 74.9117 2.52527
$$881$$ 49.1127 1.65465 0.827324 0.561724i $$-0.189862\pi$$
0.827324 + 0.561724i $$0.189862\pi$$
$$882$$ 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
$$885$$ −72.7696 −2.44612
$$886$$ − 37.2132i − 1.25020i
$$887$$ −43.1127 −1.44758 −0.723791 0.690019i $$-0.757602\pi$$
−0.723791 + 0.690019i $$0.757602\pi$$
$$888$$ 24.9706 0.837957
$$889$$ 2.00000i 0.0670778i
$$890$$ 27.5563i 0.923691i
$$891$$ − 21.2132i − 0.710669i
$$892$$ 0 0
$$893$$ 5.48528 0.183558
$$894$$ 15.5147 0.518890
$$895$$ 39.7279i 1.32796i
$$896$$ −11.3137 −0.377964
$$897$$ 0 0
$$898$$ 37.9411 1.26611
$$899$$ − 30.5563i − 1.01911i
$$900$$ 0 0
$$901$$ −8.24264 −0.274602
$$902$$ − 19.0294i − 0.633611i
$$903$$ 7.07107i 0.235310i
$$904$$ − 57.4558i − 1.91095i
$$905$$ − 29.6985i − 0.987211i
$$906$$ 36.4853 1.21214
$$907$$ 31.9706 1.06157 0.530783 0.847508i $$-0.321898\pi$$
0.530783 + 0.847508i $$0.321898\pi$$
$$908$$ 0 0
$$909$$ 1.75736 0.0582879
$$910$$ 0 0
$$911$$ −10.0294 −0.332290 −0.166145 0.986101i $$-0.553132\pi$$
−0.166145 + 0.986101i $$0.553132\pi$$
$$912$$ 7.02944i 0.232768i
$$913$$ −20.1838 −0.667985
$$914$$ −49.7990 −1.64720
$$915$$ − 37.4558i − 1.23825i
$$916$$ 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 $$0.370452\pi$$
$$920$$ −2.14214 −0.0706241
$$921$$ 6.72792i 0.221693i
$$922$$ −4.48528 −0.147715
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 90.4264i − 2.97320i
$$926$$ 6.00000 0.197172
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 17.1005i 0.561049i 0.959847 + 0.280525i $$0.0905085\pi$$
−0.959847 + 0.280525i $$0.909492\pi$$
$$930$$ − 46.2843i − 1.51772i
$$931$$ − 1.24264i − 0.0407259i
$$932$$ 0 0
$$933$$ −6.48528 −0.212319
$$934$$ − 22.4853i − 0.735741i
$$935$$ −26.4853 −0.866161
$$936$$ 0 0
$$937$$ 2.78680 0.0910407 0.0455203 0.998963i $$-0.485505\pi$$
0.0455203 + 0.998963i $$0.485505\pi$$
$$938$$ 3.51472i 0.114760i
$$939$$ 27.1716 0.886711
$$940$$ 0 0
$$941$$ 25.9289i 0.845259i 0.906303 + 0.422630i $$0.138893\pi$$
−0.906303 + 0.422630i $$0.861107\pi$$
$$942$$ 24.4853i 0.797774i
$$943$$ 0.544156i 0.0177202i
$$944$$ 46.6274i 1.51759i
$$945$$ 24.9706 0.812292
$$946$$ −30.0000 −0.975384
$$947$$ − 20.8284i − 0.676833i −0.940996 0.338416i $$-0.890109\pi$$
0.940996 0.338416i $$-0.109891\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 25.4558 0.825897
$$951$$ − 16.0000i − 0.518836i
$$952$$ 4.00000 0.129641
$$953$$ 17.1421 0.555288 0.277644 0.960684i $$-0.410446\pi$$
0.277644 + 0.960684i $$0.410446\pi$$
$$954$$ 8.24264i 0.266865i
$$955$$ 91.9411i 2.97514i
$$956$$ 0 0
$$957$$ − 34.9706i − 1.13044i
$$958$$ 51.2132 1.65462
$$959$$ 7.41421 0.239417
$$960$$ 49.9411i 1.61184i
$$961$$ 3.51472 0.113378
$$962$$ 0 0
$$963$$ −8.14214 −0.262377
$$964$$ 0 0
$$965$$ 10.9706 0.353155
$$966$$ −0.343146 −0.0110405
$$967$$ − 41.6985i − 1.34093i −0.741940 0.670466i $$-0.766094\pi$$
0.741940 0.670466i $$-0.233906\pi$$
$$968$$ − 19.7990i − 0.636364i
$$969$$ − 2.48528i − 0.0798387i
$$970$$ 85.6985i 2.75161i
$$971$$ −7.45584 −0.239269 −0.119635 0.992818i $$-0.538172\pi$$
−0.119635 + 0.992818i $$0.538172\pi$$
$$972$$ 0 0
$$973$$ 2.24264i 0.0718958i
$$974$$ −55.7990 −1.78791
$$975$$ 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
$$979$$ 18.7279 0.598547
$$980$$ 0 0
$$981$$ 8.72792i 0.278661i
$$982$$ 40.4853i 1.29194i
$$983$$ − 33.0416i − 1.05386i −0.849907 0.526932i $$-0.823342\pi$$
0.849907 0.526932i $$-0.176658\pi$$
$$984$$ 12.6863 0.404424
$$985$$ −104.426 −3.32730
$$986$$ 11.6569i 0.371230i
$$987$$ 6.24264 0.198705
$$988$$ 0 0
$$989$$ 0.857864 0.0272785
$$990$$ 26.4853i 0.841757i
$$991$$ 34.9706 1.11088 0.555438 0.831558i $$-0.312551\pi$$
0.555438 + 0.831558i $$0.312551\pi$$
$$992$$ 0 0
$$993$$ 25.4558i 0.807817i
$$994$$ 1.51472i 0.0480440i
$$995$$ 54.0416i 1.71323i
$$996$$ 0 0
$$997$$ −28.4853 −0.902138 −0.451069 0.892489i $$-0.648957\pi$$
−0.451069 + 0.892489i $$0.648957\pi$$
$$998$$ 54.7696 1.73370
$$999$$ 35.3137i 1.11728i
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.337.3 4
13.5 odd 4 91.2.a.c.1.2 2
13.8 odd 4 1183.2.a.d.1.1 2
13.12 even 2 inner 1183.2.c.d.337.1 4
39.5 even 4 819.2.a.h.1.1 2
52.31 even 4 1456.2.a.q.1.2 2
65.44 odd 4 2275.2.a.j.1.1 2
91.5 even 12 637.2.e.g.508.1 4
91.18 odd 12 637.2.e.f.79.1 4
91.31 even 12 637.2.e.g.79.1 4
91.34 even 4 8281.2.a.v.1.1 2
91.44 odd 12 637.2.e.f.508.1 4
91.83 even 4 637.2.a.g.1.2 2
104.5 odd 4 5824.2.a.bl.1.2 2
104.83 even 4 5824.2.a.bk.1.1 2
273.83 odd 4 5733.2.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.2 2 13.5 odd 4
637.2.a.g.1.2 2 91.83 even 4
637.2.e.f.79.1 4 91.18 odd 12
637.2.e.f.508.1 4 91.44 odd 12
637.2.e.g.79.1 4 91.31 even 12
637.2.e.g.508.1 4 91.5 even 12
819.2.a.h.1.1 2 39.5 even 4
1183.2.a.d.1.1 2 13.8 odd 4
1183.2.c.d.337.1 4 13.12 even 2 inner
1183.2.c.d.337.3 4 1.1 even 1 trivial
1456.2.a.q.1.2 2 52.31 even 4
2275.2.a.j.1.1 2 65.44 odd 4
5733.2.a.s.1.1 2 273.83 odd 4
5824.2.a.bk.1.1 2 104.83 even 4
5824.2.a.bl.1.2 2 104.5 odd 4
8281.2.a.v.1.1 2 91.34 even 4