# Properties

 Label 3420.2.a.g.1.1 Level $3420$ Weight $2$ Character 3420.1 Self dual yes Analytic conductor $27.309$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3420.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$27.3088374913$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3420.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -4.82843 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -4.82843 q^{7} +2.00000 q^{11} +2.24264 q^{13} +4.82843 q^{17} -1.00000 q^{19} +6.00000 q^{23} +1.00000 q^{25} -10.4853 q^{29} -1.17157 q^{31} +4.82843 q^{35} -10.2426 q^{37} +7.65685 q^{41} -0.828427 q^{43} -0.828427 q^{47} +16.3137 q^{49} +5.07107 q^{53} -2.00000 q^{55} -2.34315 q^{59} -2.34315 q^{61} -2.24264 q^{65} -0.585786 q^{67} -10.8284 q^{71} +14.4853 q^{73} -9.65685 q^{77} -9.65685 q^{79} -9.31371 q^{83} -4.82843 q^{85} -10.4853 q^{89} -10.8284 q^{91} +1.00000 q^{95} -1.75736 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 4 * q^7 $$2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −4.82843 −1.82497 −0.912487 0.409106i $$-0.865841\pi$$
−0.912487 + 0.409106i $$0.865841\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 2.24264 0.621997 0.310998 0.950410i $$-0.399337\pi$$
0.310998 + 0.950410i $$0.399337\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.82843 1.17107 0.585533 0.810649i $$-0.300885\pi$$
0.585533 + 0.810649i $$0.300885\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.4853 −1.94707 −0.973534 0.228543i $$-0.926604\pi$$
−0.973534 + 0.228543i $$0.926604\pi$$
$$30$$ 0 0
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.82843 0.816153
$$36$$ 0 0
$$37$$ −10.2426 −1.68388 −0.841940 0.539571i $$-0.818586\pi$$
−0.841940 + 0.539571i $$0.818586\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 7.65685 1.19580 0.597900 0.801571i $$-0.296002\pi$$
0.597900 + 0.801571i $$0.296002\pi$$
$$42$$ 0 0
$$43$$ −0.828427 −0.126334 −0.0631670 0.998003i $$-0.520120\pi$$
−0.0631670 + 0.998003i $$0.520120\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.828427 −0.120839 −0.0604193 0.998173i $$-0.519244\pi$$
−0.0604193 + 0.998173i $$0.519244\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 5.07107 0.696565 0.348282 0.937390i $$-0.386765\pi$$
0.348282 + 0.937390i $$0.386765\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −2.34315 −0.305052 −0.152526 0.988299i $$-0.548741\pi$$
−0.152526 + 0.988299i $$0.548741\pi$$
$$60$$ 0 0
$$61$$ −2.34315 −0.300009 −0.150005 0.988685i $$-0.547929\pi$$
−0.150005 + 0.988685i $$0.547929\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.24264 −0.278165
$$66$$ 0 0
$$67$$ −0.585786 −0.0715652 −0.0357826 0.999360i $$-0.511392\pi$$
−0.0357826 + 0.999360i $$0.511392\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.8284 −1.28510 −0.642549 0.766245i $$-0.722123\pi$$
−0.642549 + 0.766245i $$0.722123\pi$$
$$72$$ 0 0
$$73$$ 14.4853 1.69537 0.847687 0.530497i $$-0.177995\pi$$
0.847687 + 0.530497i $$0.177995\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −9.65685 −1.10050
$$78$$ 0 0
$$79$$ −9.65685 −1.08648 −0.543240 0.839577i $$-0.682803\pi$$
−0.543240 + 0.839577i $$0.682803\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.31371 −1.02231 −0.511156 0.859488i $$-0.670783\pi$$
−0.511156 + 0.859488i $$0.670783\pi$$
$$84$$ 0 0
$$85$$ −4.82843 −0.523716
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.4853 −1.11144 −0.555719 0.831370i $$-0.687557\pi$$
−0.555719 + 0.831370i $$0.687557\pi$$
$$90$$ 0 0
$$91$$ −10.8284 −1.13513
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −1.75736 −0.178433 −0.0892164 0.996012i $$-0.528436\pi$$
−0.0892164 + 0.996012i $$0.528436\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ −15.8995 −1.56662 −0.783312 0.621629i $$-0.786471\pi$$
−0.783312 + 0.621629i $$0.786471\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.58579 −0.443325 −0.221662 0.975123i $$-0.571148\pi$$
−0.221662 + 0.975123i $$0.571148\pi$$
$$108$$ 0 0
$$109$$ −8.82843 −0.845610 −0.422805 0.906221i $$-0.638954\pi$$
−0.422805 + 0.906221i $$0.638954\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −5.07107 −0.477046 −0.238523 0.971137i $$-0.576663\pi$$
−0.238523 + 0.971137i $$0.576663\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −23.3137 −2.13716
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −21.5563 −1.91282 −0.956408 0.292033i $$-0.905668\pi$$
−0.956408 + 0.292033i $$0.905668\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.65685 0.494242 0.247121 0.968985i $$-0.420516\pi$$
0.247121 + 0.968985i $$0.420516\pi$$
$$132$$ 0 0
$$133$$ 4.82843 0.418678
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 20.1421 1.72086 0.860429 0.509570i $$-0.170195\pi$$
0.860429 + 0.509570i $$0.170195\pi$$
$$138$$ 0 0
$$139$$ −17.3137 −1.46853 −0.734265 0.678863i $$-0.762473\pi$$
−0.734265 + 0.678863i $$0.762473\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.48528 0.375078
$$144$$ 0 0
$$145$$ 10.4853 0.870755
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ 18.1421 1.47639 0.738193 0.674590i $$-0.235679\pi$$
0.738193 + 0.674590i $$0.235679\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1.17157 0.0941030
$$156$$ 0 0
$$157$$ 3.17157 0.253119 0.126560 0.991959i $$-0.459607\pi$$
0.126560 + 0.991959i $$0.459607\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −28.9706 −2.28320
$$162$$ 0 0
$$163$$ −2.48528 −0.194662 −0.0973311 0.995252i $$-0.531031\pi$$
−0.0973311 + 0.995252i $$0.531031\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.7279 0.830152 0.415076 0.909787i $$-0.363755\pi$$
0.415076 + 0.909787i $$0.363755\pi$$
$$168$$ 0 0
$$169$$ −7.97056 −0.613120
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 3.89949 0.296473 0.148237 0.988952i $$-0.452640\pi$$
0.148237 + 0.988952i $$0.452640\pi$$
$$174$$ 0 0
$$175$$ −4.82843 −0.364995
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −21.6569 −1.61871 −0.809355 0.587320i $$-0.800183\pi$$
−0.809355 + 0.587320i $$0.800183\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.2426 0.753054
$$186$$ 0 0
$$187$$ 9.65685 0.706179
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ 0.585786 0.0421658 0.0210829 0.999778i $$-0.493289\pi$$
0.0210829 + 0.999778i $$0.493289\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.31371 0.378586 0.189293 0.981921i $$-0.439380\pi$$
0.189293 + 0.981921i $$0.439380\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 50.6274 3.55335
$$204$$ 0 0
$$205$$ −7.65685 −0.534778
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −11.5147 −0.792706 −0.396353 0.918098i $$-0.629724\pi$$
−0.396353 + 0.918098i $$0.629724\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0.828427 0.0564983
$$216$$ 0 0
$$217$$ 5.65685 0.384012
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10.8284 0.728399
$$222$$ 0 0
$$223$$ 19.2132 1.28661 0.643306 0.765609i $$-0.277562\pi$$
0.643306 + 0.765609i $$0.277562\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −20.5858 −1.36633 −0.683163 0.730266i $$-0.739396\pi$$
−0.683163 + 0.730266i $$0.739396\pi$$
$$228$$ 0 0
$$229$$ −21.6569 −1.43113 −0.715563 0.698549i $$-0.753830\pi$$
−0.715563 + 0.698549i $$0.753830\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0.828427 0.0540406
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −10.3431 −0.669042 −0.334521 0.942388i $$-0.608575\pi$$
−0.334521 + 0.942388i $$0.608575\pi$$
$$240$$ 0 0
$$241$$ −30.2843 −1.95078 −0.975391 0.220484i $$-0.929236\pi$$
−0.975391 + 0.220484i $$0.929236\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −16.3137 −1.04224
$$246$$ 0 0
$$247$$ −2.24264 −0.142696
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 13.6569 0.862013 0.431006 0.902349i $$-0.358159\pi$$
0.431006 + 0.902349i $$0.358159\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.24264 0.139892 0.0699460 0.997551i $$-0.477717\pi$$
0.0699460 + 0.997551i $$0.477717\pi$$
$$258$$ 0 0
$$259$$ 49.4558 3.07304
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3.65685 −0.225491 −0.112746 0.993624i $$-0.535965\pi$$
−0.112746 + 0.993624i $$0.535965\pi$$
$$264$$ 0 0
$$265$$ −5.07107 −0.311513
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 22.4853 1.37095 0.685476 0.728095i $$-0.259594\pi$$
0.685476 + 0.728095i $$0.259594\pi$$
$$270$$ 0 0
$$271$$ 23.6569 1.43705 0.718526 0.695500i $$-0.244817\pi$$
0.718526 + 0.695500i $$0.244817\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ 16.8284 1.01112 0.505561 0.862791i $$-0.331286\pi$$
0.505561 + 0.862791i $$0.331286\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.9706 −1.13169 −0.565844 0.824512i $$-0.691450\pi$$
−0.565844 + 0.824512i $$0.691450\pi$$
$$282$$ 0 0
$$283$$ −2.68629 −0.159683 −0.0798417 0.996808i $$-0.525441\pi$$
−0.0798417 + 0.996808i $$0.525441\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −36.9706 −2.18230
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 23.4142 1.36787 0.683936 0.729542i $$-0.260267\pi$$
0.683936 + 0.729542i $$0.260267\pi$$
$$294$$ 0 0
$$295$$ 2.34315 0.136423
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 13.4558 0.778172
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 2.34315 0.134168
$$306$$ 0 0
$$307$$ 7.89949 0.450848 0.225424 0.974261i $$-0.427623\pi$$
0.225424 + 0.974261i $$0.427623\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.75736 0.323366 0.161683 0.986843i $$-0.448308\pi$$
0.161683 + 0.986843i $$0.448308\pi$$
$$318$$ 0 0
$$319$$ −20.9706 −1.17413
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.82843 −0.268661
$$324$$ 0 0
$$325$$ 2.24264 0.124399
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 1.17157 0.0643955 0.0321977 0.999482i $$-0.489749\pi$$
0.0321977 + 0.999482i $$0.489749\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0.585786 0.0320049
$$336$$ 0 0
$$337$$ 17.7574 0.967305 0.483652 0.875260i $$-0.339310\pi$$
0.483652 + 0.875260i $$0.339310\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.34315 −0.126888
$$342$$ 0 0
$$343$$ −44.9706 −2.42818
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −22.9706 −1.23312 −0.616562 0.787306i $$-0.711475\pi$$
−0.616562 + 0.787306i $$0.711475\pi$$
$$348$$ 0 0
$$349$$ −0.343146 −0.0183682 −0.00918409 0.999958i $$-0.502923\pi$$
−0.00918409 + 0.999958i $$0.502923\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 3.85786 0.205333 0.102667 0.994716i $$-0.467262\pi$$
0.102667 + 0.994716i $$0.467262\pi$$
$$354$$ 0 0
$$355$$ 10.8284 0.574713
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 21.3137 1.12489 0.562447 0.826833i $$-0.309860\pi$$
0.562447 + 0.826833i $$0.309860\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −14.4853 −0.758194
$$366$$ 0 0
$$367$$ −28.1421 −1.46901 −0.734504 0.678605i $$-0.762585\pi$$
−0.734504 + 0.678605i $$0.762585\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.4853 −1.27121
$$372$$ 0 0
$$373$$ 9.55635 0.494809 0.247405 0.968912i $$-0.420422\pi$$
0.247405 + 0.968912i $$0.420422\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −23.5147 −1.21107
$$378$$ 0 0
$$379$$ 22.3431 1.14769 0.573845 0.818964i $$-0.305451\pi$$
0.573845 + 0.818964i $$0.305451\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.92893 −0.149661 −0.0748307 0.997196i $$-0.523842\pi$$
−0.0748307 + 0.997196i $$0.523842\pi$$
$$384$$ 0 0
$$385$$ 9.65685 0.492159
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 28.6274 1.45147 0.725734 0.687976i $$-0.241500\pi$$
0.725734 + 0.687976i $$0.241500\pi$$
$$390$$ 0 0
$$391$$ 28.9706 1.46510
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 9.65685 0.485889
$$396$$ 0 0
$$397$$ −32.1421 −1.61317 −0.806584 0.591120i $$-0.798686\pi$$
−0.806584 + 0.591120i $$0.798686\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.68629 0.333897 0.166949 0.985966i $$-0.446609\pi$$
0.166949 + 0.985966i $$0.446609\pi$$
$$402$$ 0 0
$$403$$ −2.62742 −0.130881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.4853 −1.01542
$$408$$ 0 0
$$409$$ 9.51472 0.470473 0.235236 0.971938i $$-0.424414\pi$$
0.235236 + 0.971938i $$0.424414\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 11.3137 0.556711
$$414$$ 0 0
$$415$$ 9.31371 0.457192
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −9.65685 −0.471768 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$420$$ 0 0
$$421$$ −8.34315 −0.406620 −0.203310 0.979114i $$-0.565170\pi$$
−0.203310 + 0.979114i $$0.565170\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4.82843 0.234213
$$426$$ 0 0
$$427$$ 11.3137 0.547509
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.4558 −1.61151 −0.805756 0.592248i $$-0.798241\pi$$
−0.805756 + 0.592248i $$0.798241\pi$$
$$432$$ 0 0
$$433$$ −15.2132 −0.731100 −0.365550 0.930792i $$-0.619119\pi$$
−0.365550 + 0.930792i $$0.619119\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −29.6569 −1.41544 −0.707722 0.706491i $$-0.750277\pi$$
−0.707722 + 0.706491i $$0.750277\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −34.2843 −1.62889 −0.814447 0.580237i $$-0.802960\pi$$
−0.814447 + 0.580237i $$0.802960\pi$$
$$444$$ 0 0
$$445$$ 10.4853 0.497050
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13.5147 0.637799 0.318900 0.947789i $$-0.396687\pi$$
0.318900 + 0.947789i $$0.396687\pi$$
$$450$$ 0 0
$$451$$ 15.3137 0.721094
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 10.8284 0.507644
$$456$$ 0 0
$$457$$ 18.9706 0.887405 0.443703 0.896174i $$-0.353665\pi$$
0.443703 + 0.896174i $$0.353665\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5.31371 0.247484 0.123742 0.992314i $$-0.460510\pi$$
0.123742 + 0.992314i $$0.460510\pi$$
$$462$$ 0 0
$$463$$ −9.31371 −0.432845 −0.216422 0.976300i $$-0.569439\pi$$
−0.216422 + 0.976300i $$0.569439\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9.31371 −0.430987 −0.215494 0.976505i $$-0.569136\pi$$
−0.215494 + 0.976505i $$0.569136\pi$$
$$468$$ 0 0
$$469$$ 2.82843 0.130605
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −1.65685 −0.0761822
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ −22.9706 −1.04737
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.75736 0.0797976
$$486$$ 0 0
$$487$$ 24.3848 1.10498 0.552490 0.833520i $$-0.313678\pi$$
0.552490 + 0.833520i $$0.313678\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −35.3137 −1.59369 −0.796843 0.604187i $$-0.793498\pi$$
−0.796843 + 0.604187i $$0.793498\pi$$
$$492$$ 0 0
$$493$$ −50.6274 −2.28014
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 52.2843 2.34527
$$498$$ 0 0
$$499$$ −26.9706 −1.20737 −0.603684 0.797224i $$-0.706301\pi$$
−0.603684 + 0.797224i $$0.706301\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −2.97056 −0.132451 −0.0662254 0.997805i $$-0.521096\pi$$
−0.0662254 + 0.997805i $$0.521096\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 33.7990 1.49811 0.749057 0.662506i $$-0.230507\pi$$
0.749057 + 0.662506i $$0.230507\pi$$
$$510$$ 0 0
$$511$$ −69.9411 −3.09401
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 15.8995 0.700615
$$516$$ 0 0
$$517$$ −1.65685 −0.0728684
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −28.3431 −1.24174 −0.620868 0.783915i $$-0.713220\pi$$
−0.620868 + 0.783915i $$0.713220\pi$$
$$522$$ 0 0
$$523$$ −6.72792 −0.294191 −0.147096 0.989122i $$-0.546993\pi$$
−0.147096 + 0.989122i $$0.546993\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.65685 −0.246416
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 17.1716 0.743783
$$534$$ 0 0
$$535$$ 4.58579 0.198261
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 32.6274 1.40536
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 8.82843 0.378168
$$546$$ 0 0
$$547$$ 36.3848 1.55570 0.777850 0.628450i $$-0.216310\pi$$
0.777850 + 0.628450i $$0.216310\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 10.4853 0.446688
$$552$$ 0 0
$$553$$ 46.6274 1.98280
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −41.7990 −1.77108 −0.885540 0.464563i $$-0.846211\pi$$
−0.885540 + 0.464563i $$0.846211\pi$$
$$558$$ 0 0
$$559$$ −1.85786 −0.0785793
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 11.4142 0.481052 0.240526 0.970643i $$-0.422680\pi$$
0.240526 + 0.970643i $$0.422680\pi$$
$$564$$ 0 0
$$565$$ 5.07107 0.213341
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −0.828427 −0.0347295 −0.0173647 0.999849i $$-0.505528\pi$$
−0.0173647 + 0.999849i $$0.505528\pi$$
$$570$$ 0 0
$$571$$ −14.6863 −0.614602 −0.307301 0.951612i $$-0.599426\pi$$
−0.307301 + 0.951612i $$0.599426\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 44.9706 1.86569
$$582$$ 0 0
$$583$$ 10.1421 0.420044
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.9706 0.617901 0.308951 0.951078i $$-0.400022\pi$$
0.308951 + 0.951078i $$0.400022\pi$$
$$588$$ 0 0
$$589$$ 1.17157 0.0482738
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −8.62742 −0.354286 −0.177143 0.984185i $$-0.556685\pi$$
−0.177143 + 0.984185i $$0.556685\pi$$
$$594$$ 0 0
$$595$$ 23.3137 0.955769
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 4.68629 0.191477 0.0957383 0.995407i $$-0.469479\pi$$
0.0957383 + 0.995407i $$0.469479\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 5.07107 0.205828 0.102914 0.994690i $$-0.467183\pi$$
0.102914 + 0.994690i $$0.467183\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.85786 −0.0751611
$$612$$ 0 0
$$613$$ 17.5147 0.707413 0.353706 0.935356i $$-0.384921\pi$$
0.353706 + 0.935356i $$0.384921\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.20101 0.249643 0.124822 0.992179i $$-0.460164\pi$$
0.124822 + 0.992179i $$0.460164\pi$$
$$618$$ 0 0
$$619$$ 18.0000 0.723481 0.361741 0.932279i $$-0.382183\pi$$
0.361741 + 0.932279i $$0.382183\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 50.6274 2.02834
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −49.4558 −1.97193
$$630$$ 0 0
$$631$$ −3.65685 −0.145577 −0.0727885 0.997347i $$-0.523190\pi$$
−0.0727885 + 0.997347i $$0.523190\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 21.5563 0.855438
$$636$$ 0 0
$$637$$ 36.5858 1.44958
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21.3137 0.841841 0.420920 0.907098i $$-0.361707\pi$$
0.420920 + 0.907098i $$0.361707\pi$$
$$642$$ 0 0
$$643$$ 24.6274 0.971211 0.485605 0.874178i $$-0.338599\pi$$
0.485605 + 0.874178i $$0.338599\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −6.68629 −0.262865 −0.131433 0.991325i $$-0.541958\pi$$
−0.131433 + 0.991325i $$0.541958\pi$$
$$648$$ 0 0
$$649$$ −4.68629 −0.183953
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13.7990 −0.539996 −0.269998 0.962861i $$-0.587023\pi$$
−0.269998 + 0.962861i $$0.587023\pi$$
$$654$$ 0 0
$$655$$ −5.65685 −0.221032
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 34.6274 1.34889 0.674446 0.738324i $$-0.264382\pi$$
0.674446 + 0.738324i $$0.264382\pi$$
$$660$$ 0 0
$$661$$ −12.6274 −0.491150 −0.245575 0.969378i $$-0.578977\pi$$
−0.245575 + 0.969378i $$0.578977\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.82843 −0.187238
$$666$$ 0 0
$$667$$ −62.9117 −2.43595
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.68629 −0.180912
$$672$$ 0 0
$$673$$ 10.2426 0.394825 0.197412 0.980321i $$-0.436746\pi$$
0.197412 + 0.980321i $$0.436746\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −25.5563 −0.982210 −0.491105 0.871100i $$-0.663407\pi$$
−0.491105 + 0.871100i $$0.663407\pi$$
$$678$$ 0 0
$$679$$ 8.48528 0.325635
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 38.7279 1.48188 0.740941 0.671570i $$-0.234380\pi$$
0.740941 + 0.671570i $$0.234380\pi$$
$$684$$ 0 0
$$685$$ −20.1421 −0.769591
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 11.3726 0.433261
$$690$$ 0 0
$$691$$ −15.6569 −0.595615 −0.297807 0.954626i $$-0.596255\pi$$
−0.297807 + 0.954626i $$0.596255\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.3137 0.656746
$$696$$ 0 0
$$697$$ 36.9706 1.40036
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −49.6569 −1.87551 −0.937757 0.347293i $$-0.887101\pi$$
−0.937757 + 0.347293i $$0.887101\pi$$
$$702$$ 0 0
$$703$$ 10.2426 0.386309
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 19.3137 0.726367
$$708$$ 0 0
$$709$$ 18.9706 0.712454 0.356227 0.934399i $$-0.384063\pi$$
0.356227 + 0.934399i $$0.384063\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −7.02944 −0.263254
$$714$$ 0 0
$$715$$ −4.48528 −0.167740
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −38.2843 −1.42776 −0.713881 0.700267i $$-0.753064\pi$$
−0.713881 + 0.700267i $$0.753064\pi$$
$$720$$ 0 0
$$721$$ 76.7696 2.85905
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −10.4853 −0.389414
$$726$$ 0 0
$$727$$ −29.5147 −1.09464 −0.547320 0.836923i $$-0.684352\pi$$
−0.547320 + 0.836923i $$0.684352\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 39.6569 1.46476 0.732380 0.680896i $$-0.238410\pi$$
0.732380 + 0.680896i $$0.238410\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.17157 −0.0431554
$$738$$ 0 0
$$739$$ 34.6274 1.27379 0.636895 0.770951i $$-0.280218\pi$$
0.636895 + 0.770951i $$0.280218\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −14.0416 −0.515137 −0.257569 0.966260i $$-0.582921\pi$$
−0.257569 + 0.966260i $$0.582921\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 22.1421 0.809056
$$750$$ 0 0
$$751$$ 14.1421 0.516054 0.258027 0.966138i $$-0.416928\pi$$
0.258027 + 0.966138i $$0.416928\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −18.1421 −0.660260
$$756$$ 0 0
$$757$$ 19.6569 0.714441 0.357220 0.934020i $$-0.383725\pi$$
0.357220 + 0.934020i $$0.383725\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.6274 −1.11024 −0.555121 0.831769i $$-0.687328\pi$$
−0.555121 + 0.831769i $$0.687328\pi$$
$$762$$ 0 0
$$763$$ 42.6274 1.54322
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.25483 −0.189741
$$768$$ 0 0
$$769$$ 38.6274 1.39294 0.696470 0.717586i $$-0.254753\pi$$
0.696470 + 0.717586i $$0.254753\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0.100505 0.00361492 0.00180746 0.999998i $$-0.499425\pi$$
0.00180746 + 0.999998i $$0.499425\pi$$
$$774$$ 0 0
$$775$$ −1.17157 −0.0420841
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −7.65685 −0.274335
$$780$$ 0 0
$$781$$ −21.6569 −0.774943
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −3.17157 −0.113198
$$786$$ 0 0
$$787$$ −41.8406 −1.49146 −0.745729 0.666250i $$-0.767898\pi$$
−0.745729 + 0.666250i $$0.767898\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 24.4853 0.870596
$$792$$ 0 0
$$793$$ −5.25483 −0.186605
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −11.8995 −0.421502 −0.210751 0.977540i $$-0.567591\pi$$
−0.210751 + 0.977540i $$0.567591\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 28.9706 1.02235
$$804$$ 0 0
$$805$$ 28.9706 1.02108
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 39.2548 1.38013 0.690063 0.723749i $$-0.257583\pi$$
0.690063 + 0.723749i $$0.257583\pi$$
$$810$$ 0 0
$$811$$ 19.7990 0.695237 0.347618 0.937636i $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 2.48528 0.0870556
$$816$$ 0 0
$$817$$ 0.828427 0.0289830
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −15.1716 −0.528848 −0.264424 0.964407i $$-0.585182\pi$$
−0.264424 + 0.964407i $$0.585182\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 29.0711 1.01090 0.505450 0.862856i $$-0.331326\pi$$
0.505450 + 0.862856i $$0.331326\pi$$
$$828$$ 0 0
$$829$$ −40.4264 −1.40407 −0.702034 0.712144i $$-0.747724\pi$$
−0.702034 + 0.712144i $$0.747724\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 78.7696 2.72920
$$834$$ 0 0
$$835$$ −10.7279 −0.371255
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −43.5980 −1.50517 −0.752585 0.658495i $$-0.771193\pi$$
−0.752585 + 0.658495i $$0.771193\pi$$
$$840$$ 0 0
$$841$$ 80.9411 2.79107
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 7.97056 0.274196
$$846$$ 0 0
$$847$$ 33.7990 1.16135
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −61.4558 −2.10668
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.2132 0.929585 0.464793 0.885420i $$-0.346129\pi$$
0.464793 + 0.885420i $$0.346129\pi$$
$$858$$ 0 0
$$859$$ −24.2843 −0.828569 −0.414284 0.910148i $$-0.635968\pi$$
−0.414284 + 0.910148i $$0.635968\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 3.89949 0.132740 0.0663702 0.997795i $$-0.478858\pi$$
0.0663702 + 0.997795i $$0.478858\pi$$
$$864$$ 0 0
$$865$$ −3.89949 −0.132587
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −19.3137 −0.655173
$$870$$ 0 0
$$871$$ −1.31371 −0.0445133
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 4.82843 0.163231
$$876$$ 0 0
$$877$$ −37.0711 −1.25180 −0.625901 0.779903i $$-0.715268\pi$$
−0.625901 + 0.779903i $$0.715268\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −44.2843 −1.49198 −0.745988 0.665960i $$-0.768022\pi$$
−0.745988 + 0.665960i $$0.768022\pi$$
$$882$$ 0 0
$$883$$ −39.1716 −1.31823 −0.659114 0.752043i $$-0.729069\pi$$
−0.659114 + 0.752043i $$0.729069\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 49.0711 1.64765 0.823823 0.566848i $$-0.191837\pi$$
0.823823 + 0.566848i $$0.191837\pi$$
$$888$$ 0 0
$$889$$ 104.083 3.49084
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0.828427 0.0277223
$$894$$ 0 0
$$895$$ 21.6569 0.723909
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 12.2843 0.409703
$$900$$ 0 0
$$901$$ 24.4853 0.815723
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ −20.3848 −0.676865 −0.338433 0.940991i $$-0.609897\pi$$
−0.338433 + 0.940991i $$0.609897\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −4.20101 −0.139186 −0.0695928 0.997575i $$-0.522170\pi$$
−0.0695928 + 0.997575i $$0.522170\pi$$
$$912$$ 0 0
$$913$$ −18.6274 −0.616478
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −27.3137 −0.901978
$$918$$ 0 0
$$919$$ 35.3137 1.16489 0.582446 0.812869i $$-0.302096\pi$$
0.582446 + 0.812869i $$0.302096\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −24.2843 −0.799327
$$924$$ 0 0
$$925$$ −10.2426 −0.336776
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −6.68629 −0.219370 −0.109685 0.993966i $$-0.534984\pi$$
−0.109685 + 0.993966i $$0.534984\pi$$
$$930$$ 0 0
$$931$$ −16.3137 −0.534660
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −9.65685 −0.315813
$$936$$ 0 0
$$937$$ −40.1421 −1.31139 −0.655693 0.755027i $$-0.727624\pi$$
−0.655693 + 0.755027i $$0.727624\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 30.2843 0.987239 0.493620 0.869678i $$-0.335674\pi$$
0.493620 + 0.869678i $$0.335674\pi$$
$$942$$ 0 0
$$943$$ 45.9411 1.49605
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.34315 0.271116 0.135558 0.990769i $$-0.456717\pi$$
0.135558 + 0.990769i $$0.456717\pi$$
$$948$$ 0 0
$$949$$ 32.4853 1.05452
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −5.75736 −0.186499 −0.0932496 0.995643i $$-0.529725\pi$$
−0.0932496 + 0.995643i $$0.529725\pi$$
$$954$$ 0 0
$$955$$ 4.00000 0.129437
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −97.2548 −3.14052
$$960$$ 0 0
$$961$$ −29.6274 −0.955723
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −0.585786 −0.0188571
$$966$$ 0 0
$$967$$ 20.6274 0.663333 0.331667 0.943397i $$-0.392389\pi$$
0.331667 + 0.943397i $$0.392389\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.1127 1.12682 0.563410 0.826177i $$-0.309489\pi$$
0.563410 + 0.826177i $$0.309489\pi$$
$$972$$ 0 0
$$973$$ 83.5980 2.68003
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −9.27208 −0.296640 −0.148320 0.988939i $$-0.547387\pi$$
−0.148320 + 0.988939i $$0.547387\pi$$
$$978$$ 0 0
$$979$$ −20.9706 −0.670222
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 23.4142 0.746797 0.373399 0.927671i $$-0.378192\pi$$
0.373399 + 0.927671i $$0.378192\pi$$
$$984$$ 0 0
$$985$$ −5.31371 −0.169309
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.97056 −0.158055
$$990$$ 0 0
$$991$$ −51.1127 −1.62365 −0.811824 0.583902i $$-0.801525\pi$$
−0.811824 + 0.583902i $$0.801525\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 10.3431 0.327900
$$996$$ 0 0
$$997$$ −13.5147 −0.428015 −0.214008 0.976832i $$-0.568652\pi$$
−0.214008 + 0.976832i $$0.568652\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3420.2.a.g.1.1 2
3.2 odd 2 380.2.a.c.1.2 2
12.11 even 2 1520.2.a.o.1.1 2
15.2 even 4 1900.2.c.d.1749.3 4
15.8 even 4 1900.2.c.d.1749.2 4
15.14 odd 2 1900.2.a.e.1.1 2
24.5 odd 2 6080.2.a.bl.1.1 2
24.11 even 2 6080.2.a.y.1.2 2
57.56 even 2 7220.2.a.m.1.1 2
60.59 even 2 7600.2.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.2 2 3.2 odd 2
1520.2.a.o.1.1 2 12.11 even 2
1900.2.a.e.1.1 2 15.14 odd 2
1900.2.c.d.1749.2 4 15.8 even 4
1900.2.c.d.1749.3 4 15.2 even 4
3420.2.a.g.1.1 2 1.1 even 1 trivial
6080.2.a.y.1.2 2 24.11 even 2
6080.2.a.bl.1.1 2 24.5 odd 2
7220.2.a.m.1.1 2 57.56 even 2
7600.2.a.u.1.2 2 60.59 even 2