# Properties

 Label 7728.2.a.bd.1.2 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ 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] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.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: $$61.7083906820$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.47214 q^{13} -2.00000 q^{15} -4.47214 q^{17} +6.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.47214 q^{31} -2.00000 q^{35} -10.9443 q^{37} -4.47214 q^{39} -6.00000 q^{41} -12.9443 q^{43} +2.00000 q^{45} -6.47214 q^{47} +1.00000 q^{49} +4.47214 q^{51} +6.94427 q^{53} -6.47214 q^{57} -4.00000 q^{59} -6.00000 q^{61} -1.00000 q^{63} +8.94427 q^{65} +12.9443 q^{67} -1.00000 q^{69} +12.9443 q^{71} -2.94427 q^{73} +1.00000 q^{75} -12.9443 q^{79} +1.00000 q^{81} +6.47214 q^{83} -8.94427 q^{85} +2.00000 q^{87} -17.4164 q^{89} -4.47214 q^{91} +6.47214 q^{93} +12.9443 q^{95} +8.47214 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 - 2 * q^25 - 2 * q^27 - 4 * q^29 - 4 * q^31 - 4 * q^35 - 4 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 - 4 * q^47 + 2 * q^49 - 4 * q^53 - 4 * q^57 - 8 * q^59 - 12 * q^61 - 2 * q^63 + 8 * q^67 - 2 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 8 * q^95 + 8 * 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.47214 1.24035 0.620174 0.784465i $$-0.287062\pi$$
0.620174 + 0.784465i $$0.287062\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 0 0
$$19$$ 6.47214 1.48481 0.742405 0.669951i $$-0.233685\pi$$
0.742405 + 0.669951i $$0.233685\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −6.47214 −1.16243 −0.581215 0.813750i $$-0.697422\pi$$
−0.581215 + 0.813750i $$0.697422\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −10.9443 −1.79923 −0.899614 0.436687i $$-0.856152\pi$$
−0.899614 + 0.436687i $$0.856152\pi$$
$$38$$ 0 0
$$39$$ −4.47214 −0.716115
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −12.9443 −1.97398 −0.986991 0.160773i $$-0.948601\pi$$
−0.986991 + 0.160773i $$0.948601\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −6.47214 −0.944058 −0.472029 0.881583i $$-0.656478\pi$$
−0.472029 + 0.881583i $$0.656478\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.47214 0.626224
$$52$$ 0 0
$$53$$ 6.94427 0.953869 0.476935 0.878939i $$-0.341748\pi$$
0.476935 + 0.878939i $$0.341748\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.47214 −0.857255
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 8.94427 1.10940
$$66$$ 0 0
$$67$$ 12.9443 1.58139 0.790697 0.612207i $$-0.209718\pi$$
0.790697 + 0.612207i $$0.209718\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 12.9443 1.53620 0.768101 0.640328i $$-0.221202\pi$$
0.768101 + 0.640328i $$0.221202\pi$$
$$72$$ 0 0
$$73$$ −2.94427 −0.344601 −0.172300 0.985044i $$-0.555120\pi$$
−0.172300 + 0.985044i $$0.555120\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.9443 −1.45634 −0.728172 0.685394i $$-0.759630\pi$$
−0.728172 + 0.685394i $$0.759630\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.47214 0.710409 0.355205 0.934789i $$-0.384411\pi$$
0.355205 + 0.934789i $$0.384411\pi$$
$$84$$ 0 0
$$85$$ −8.94427 −0.970143
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −17.4164 −1.84614 −0.923068 0.384637i $$-0.874327\pi$$
−0.923068 + 0.384637i $$0.874327\pi$$
$$90$$ 0 0
$$91$$ −4.47214 −0.468807
$$92$$ 0 0
$$93$$ 6.47214 0.671129
$$94$$ 0 0
$$95$$ 12.9443 1.32805
$$96$$ 0 0
$$97$$ 8.47214 0.860215 0.430108 0.902778i $$-0.358476\pi$$
0.430108 + 0.902778i $$0.358476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 17.4164 1.73300 0.866499 0.499179i $$-0.166365\pi$$
0.866499 + 0.499179i $$0.166365\pi$$
$$102$$ 0 0
$$103$$ −12.9443 −1.27544 −0.637719 0.770270i $$-0.720122\pi$$
−0.637719 + 0.770270i $$0.720122\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 14.9443 1.43140 0.715701 0.698407i $$-0.246107\pi$$
0.715701 + 0.698407i $$0.246107\pi$$
$$110$$ 0 0
$$111$$ 10.9443 1.03878
$$112$$ 0 0
$$113$$ −1.05573 −0.0993145 −0.0496573 0.998766i $$-0.515813\pi$$
−0.0496573 + 0.998766i $$0.515813\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 0 0
$$117$$ 4.47214 0.413449
$$118$$ 0 0
$$119$$ 4.47214 0.409960
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −3.05573 −0.271152 −0.135576 0.990767i $$-0.543288\pi$$
−0.135576 + 0.990767i $$0.543288\pi$$
$$128$$ 0 0
$$129$$ 12.9443 1.13968
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −6.47214 −0.561205
$$134$$ 0 0
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ −14.0000 −1.19610 −0.598050 0.801459i $$-0.704058\pi$$
−0.598050 + 0.801459i $$0.704058\pi$$
$$138$$ 0 0
$$139$$ 8.94427 0.758643 0.379322 0.925265i $$-0.376157\pi$$
0.379322 + 0.925265i $$0.376157\pi$$
$$140$$ 0 0
$$141$$ 6.47214 0.545052
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −18.9443 −1.55198 −0.775988 0.630748i $$-0.782748\pi$$
−0.775988 + 0.630748i $$0.782748\pi$$
$$150$$ 0 0
$$151$$ 12.9443 1.05339 0.526695 0.850054i $$-0.323431\pi$$
0.526695 + 0.850054i $$0.323431\pi$$
$$152$$ 0 0
$$153$$ −4.47214 −0.361551
$$154$$ 0 0
$$155$$ −12.9443 −1.03971
$$156$$ 0 0
$$157$$ 14.9443 1.19268 0.596341 0.802731i $$-0.296620\pi$$
0.596341 + 0.802731i $$0.296620\pi$$
$$158$$ 0 0
$$159$$ −6.94427 −0.550717
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ −8.94427 −0.700569 −0.350285 0.936643i $$-0.613915\pi$$
−0.350285 + 0.936643i $$0.613915\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −11.4164 −0.883428 −0.441714 0.897156i $$-0.645629\pi$$
−0.441714 + 0.897156i $$0.645629\pi$$
$$168$$ 0 0
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ 6.47214 0.494937
$$172$$ 0 0
$$173$$ −8.47214 −0.644125 −0.322062 0.946718i $$-0.604376\pi$$
−0.322062 + 0.946718i $$0.604376\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ −21.8885 −1.60928
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 20.9443 1.51547 0.757737 0.652560i $$-0.226305\pi$$
0.757737 + 0.652560i $$0.226305\pi$$
$$192$$ 0 0
$$193$$ −23.8885 −1.71954 −0.859768 0.510686i $$-0.829392\pi$$
−0.859768 + 0.510686i $$0.829392\pi$$
$$194$$ 0 0
$$195$$ −8.94427 −0.640513
$$196$$ 0 0
$$197$$ 2.94427 0.209771 0.104885 0.994484i $$-0.466552\pi$$
0.104885 + 0.994484i $$0.466552\pi$$
$$198$$ 0 0
$$199$$ −20.9443 −1.48470 −0.742350 0.670012i $$-0.766289\pi$$
−0.742350 + 0.670012i $$0.766289\pi$$
$$200$$ 0 0
$$201$$ −12.9443 −0.913019
$$202$$ 0 0
$$203$$ 2.00000 0.140372
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.9443 −1.16649 −0.583246 0.812296i $$-0.698218\pi$$
−0.583246 + 0.812296i $$0.698218\pi$$
$$212$$ 0 0
$$213$$ −12.9443 −0.886927
$$214$$ 0 0
$$215$$ −25.8885 −1.76558
$$216$$ 0 0
$$217$$ 6.47214 0.439357
$$218$$ 0 0
$$219$$ 2.94427 0.198955
$$220$$ 0 0
$$221$$ −20.0000 −1.34535
$$222$$ 0 0
$$223$$ −9.52786 −0.638033 −0.319016 0.947749i $$-0.603353\pi$$
−0.319016 + 0.947749i $$0.603353\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ −14.4721 −0.960549 −0.480275 0.877118i $$-0.659463\pi$$
−0.480275 + 0.877118i $$0.659463\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ −12.9443 −0.844391
$$236$$ 0 0
$$237$$ 12.9443 0.840821
$$238$$ 0 0
$$239$$ −3.05573 −0.197659 −0.0988293 0.995104i $$-0.531510\pi$$
−0.0988293 + 0.995104i $$0.531510\pi$$
$$240$$ 0 0
$$241$$ 11.5279 0.742575 0.371288 0.928518i $$-0.378916\pi$$
0.371288 + 0.928518i $$0.378916\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ 28.9443 1.84168
$$248$$ 0 0
$$249$$ −6.47214 −0.410155
$$250$$ 0 0
$$251$$ 14.4721 0.913473 0.456737 0.889602i $$-0.349018\pi$$
0.456737 + 0.889602i $$0.349018\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 8.94427 0.560112
$$256$$ 0 0
$$257$$ 14.9443 0.932198 0.466099 0.884733i $$-0.345659\pi$$
0.466099 + 0.884733i $$0.345659\pi$$
$$258$$ 0 0
$$259$$ 10.9443 0.680044
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 13.8885 0.853166
$$266$$ 0 0
$$267$$ 17.4164 1.06587
$$268$$ 0 0
$$269$$ 7.52786 0.458982 0.229491 0.973311i $$-0.426294\pi$$
0.229491 + 0.973311i $$0.426294\pi$$
$$270$$ 0 0
$$271$$ 11.4164 0.693497 0.346749 0.937958i $$-0.387286\pi$$
0.346749 + 0.937958i $$0.387286\pi$$
$$272$$ 0 0
$$273$$ 4.47214 0.270666
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 23.8885 1.43532 0.717662 0.696392i $$-0.245212\pi$$
0.717662 + 0.696392i $$0.245212\pi$$
$$278$$ 0 0
$$279$$ −6.47214 −0.387477
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −9.52786 −0.566373 −0.283186 0.959065i $$-0.591391\pi$$
−0.283186 + 0.959065i $$0.591391\pi$$
$$284$$ 0 0
$$285$$ −12.9443 −0.766752
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ −8.47214 −0.496645
$$292$$ 0 0
$$293$$ −7.88854 −0.460854 −0.230427 0.973090i $$-0.574012\pi$$
−0.230427 + 0.973090i $$0.574012\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.47214 0.258630
$$300$$ 0 0
$$301$$ 12.9443 0.746095
$$302$$ 0 0
$$303$$ −17.4164 −1.00055
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ −7.05573 −0.402692 −0.201346 0.979520i $$-0.564532\pi$$
−0.201346 + 0.979520i $$0.564532\pi$$
$$308$$ 0 0
$$309$$ 12.9443 0.736374
$$310$$ 0 0
$$311$$ −6.47214 −0.367001 −0.183501 0.983020i $$-0.558743\pi$$
−0.183501 + 0.983020i $$0.558743\pi$$
$$312$$ 0 0
$$313$$ 13.4164 0.758340 0.379170 0.925327i $$-0.376210\pi$$
0.379170 + 0.925327i $$0.376210\pi$$
$$314$$ 0 0
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ −30.9443 −1.73800 −0.869002 0.494809i $$-0.835238\pi$$
−0.869002 + 0.494809i $$0.835238\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ −28.9443 −1.61050
$$324$$ 0 0
$$325$$ −4.47214 −0.248069
$$326$$ 0 0
$$327$$ −14.9443 −0.826420
$$328$$ 0 0
$$329$$ 6.47214 0.356820
$$330$$ 0 0
$$331$$ −21.8885 −1.20310 −0.601552 0.798834i $$-0.705451\pi$$
−0.601552 + 0.798834i $$0.705451\pi$$
$$332$$ 0 0
$$333$$ −10.9443 −0.599742
$$334$$ 0 0
$$335$$ 25.8885 1.41444
$$336$$ 0 0
$$337$$ 14.9443 0.814066 0.407033 0.913413i $$-0.366563\pi$$
0.407033 + 0.913413i $$0.366563\pi$$
$$338$$ 0 0
$$339$$ 1.05573 0.0573393
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −2.00000 −0.107676
$$346$$ 0 0
$$347$$ −16.9443 −0.909616 −0.454808 0.890589i $$-0.650292\pi$$
−0.454808 + 0.890589i $$0.650292\pi$$
$$348$$ 0 0
$$349$$ 15.5279 0.831188 0.415594 0.909550i $$-0.363574\pi$$
0.415594 + 0.909550i $$0.363574\pi$$
$$350$$ 0 0
$$351$$ −4.47214 −0.238705
$$352$$ 0 0
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 0 0
$$355$$ 25.8885 1.37402
$$356$$ 0 0
$$357$$ −4.47214 −0.236691
$$358$$ 0 0
$$359$$ −22.8328 −1.20507 −0.602535 0.798092i $$-0.705843\pi$$
−0.602535 + 0.798092i $$0.705843\pi$$
$$360$$ 0 0
$$361$$ 22.8885 1.20466
$$362$$ 0 0
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ −5.88854 −0.308220
$$366$$ 0 0
$$367$$ 22.8328 1.19186 0.595932 0.803035i $$-0.296783\pi$$
0.595932 + 0.803035i $$0.296783\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −6.94427 −0.360529
$$372$$ 0 0
$$373$$ −10.9443 −0.566673 −0.283336 0.959021i $$-0.591441\pi$$
−0.283336 + 0.959021i $$0.591441\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ −8.94427 −0.460653
$$378$$ 0 0
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ 0 0
$$381$$ 3.05573 0.156550
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −12.9443 −0.657994
$$388$$ 0 0
$$389$$ −1.05573 −0.0535275 −0.0267638 0.999642i $$-0.508520\pi$$
−0.0267638 + 0.999642i $$0.508520\pi$$
$$390$$ 0 0
$$391$$ −4.47214 −0.226166
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ −25.8885 −1.30259
$$396$$ 0 0
$$397$$ −27.5279 −1.38158 −0.690792 0.723054i $$-0.742738\pi$$
−0.690792 + 0.723054i $$0.742738\pi$$
$$398$$ 0 0
$$399$$ 6.47214 0.324012
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ −28.9443 −1.44182
$$404$$ 0 0
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.8885 0.983425 0.491713 0.870758i $$-0.336371\pi$$
0.491713 + 0.870758i $$0.336371\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ 0 0
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ 12.9443 0.635409
$$416$$ 0 0
$$417$$ −8.94427 −0.438003
$$418$$ 0 0
$$419$$ −19.4164 −0.948554 −0.474277 0.880376i $$-0.657290\pi$$
−0.474277 + 0.880376i $$0.657290\pi$$
$$420$$ 0 0
$$421$$ −20.8328 −1.01533 −0.507665 0.861555i $$-0.669491\pi$$
−0.507665 + 0.861555i $$0.669491\pi$$
$$422$$ 0 0
$$423$$ −6.47214 −0.314686
$$424$$ 0 0
$$425$$ 4.47214 0.216930
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 17.8885 0.861661 0.430830 0.902433i $$-0.358221\pi$$
0.430830 + 0.902433i $$0.358221\pi$$
$$432$$ 0 0
$$433$$ −1.41641 −0.0680682 −0.0340341 0.999421i $$-0.510835\pi$$
−0.0340341 + 0.999421i $$0.510835\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ 0 0
$$437$$ 6.47214 0.309604
$$438$$ 0 0
$$439$$ 16.3607 0.780853 0.390426 0.920634i $$-0.372328\pi$$
0.390426 + 0.920634i $$0.372328\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 18.8328 0.894774 0.447387 0.894340i $$-0.352355\pi$$
0.447387 + 0.894340i $$0.352355\pi$$
$$444$$ 0 0
$$445$$ −34.8328 −1.65123
$$446$$ 0 0
$$447$$ 18.9443 0.896033
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −12.9443 −0.608175
$$454$$ 0 0
$$455$$ −8.94427 −0.419314
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 0 0
$$459$$ 4.47214 0.208741
$$460$$ 0 0
$$461$$ 33.4164 1.55636 0.778179 0.628043i $$-0.216144\pi$$
0.778179 + 0.628043i $$0.216144\pi$$
$$462$$ 0 0
$$463$$ −41.8885 −1.94673 −0.973363 0.229270i $$-0.926366\pi$$
−0.973363 + 0.229270i $$0.926366\pi$$
$$464$$ 0 0
$$465$$ 12.9443 0.600276
$$466$$ 0 0
$$467$$ 29.3050 1.35607 0.678036 0.735029i $$-0.262831\pi$$
0.678036 + 0.735029i $$0.262831\pi$$
$$468$$ 0 0
$$469$$ −12.9443 −0.597711
$$470$$ 0 0
$$471$$ −14.9443 −0.688596
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.47214 −0.296962
$$476$$ 0 0
$$477$$ 6.94427 0.317956
$$478$$ 0 0
$$479$$ −12.9443 −0.591439 −0.295719 0.955275i $$-0.595559\pi$$
−0.295719 + 0.955275i $$0.595559\pi$$
$$480$$ 0 0
$$481$$ −48.9443 −2.23167
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ 16.9443 0.769400
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ 0 0
$$489$$ 8.94427 0.404474
$$490$$ 0 0
$$491$$ −23.0557 −1.04049 −0.520245 0.854017i $$-0.674159\pi$$
−0.520245 + 0.854017i $$0.674159\pi$$
$$492$$ 0 0
$$493$$ 8.94427 0.402830
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.9443 −0.580630
$$498$$ 0 0
$$499$$ −15.0557 −0.673987 −0.336993 0.941507i $$-0.609410\pi$$
−0.336993 + 0.941507i $$0.609410\pi$$
$$500$$ 0 0
$$501$$ 11.4164 0.510047
$$502$$ 0 0
$$503$$ −25.8885 −1.15431 −0.577157 0.816634i $$-0.695838\pi$$
−0.577157 + 0.816634i $$0.695838\pi$$
$$504$$ 0 0
$$505$$ 34.8328 1.55004
$$506$$ 0 0
$$507$$ −7.00000 −0.310881
$$508$$ 0 0
$$509$$ 33.4164 1.48116 0.740578 0.671970i $$-0.234552\pi$$
0.740578 + 0.671970i $$0.234552\pi$$
$$510$$ 0 0
$$511$$ 2.94427 0.130247
$$512$$ 0 0
$$513$$ −6.47214 −0.285752
$$514$$ 0 0
$$515$$ −25.8885 −1.14079
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 8.47214 0.371885
$$520$$ 0 0
$$521$$ 6.58359 0.288432 0.144216 0.989546i $$-0.453934\pi$$
0.144216 + 0.989546i $$0.453934\pi$$
$$522$$ 0 0
$$523$$ −37.3050 −1.63123 −0.815616 0.578594i $$-0.803602\pi$$
−0.815616 + 0.578594i $$0.803602\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −0.0436436
$$526$$ 0 0
$$527$$ 28.9443 1.26083
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ −26.8328 −1.16226
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ 0 0
$$537$$ −20.0000 −0.863064
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ −2.00000 −0.0858282
$$544$$ 0 0
$$545$$ 29.8885 1.28028
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −12.9443 −0.551445
$$552$$ 0 0
$$553$$ 12.9443 0.550446
$$554$$ 0 0
$$555$$ 21.8885 0.929117
$$556$$ 0 0
$$557$$ −34.9443 −1.48064 −0.740318 0.672257i $$-0.765325\pi$$
−0.740318 + 0.672257i $$0.765325\pi$$
$$558$$ 0 0
$$559$$ −57.8885 −2.44842
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −17.5279 −0.738711 −0.369356 0.929288i $$-0.620422\pi$$
−0.369356 + 0.929288i $$0.620422\pi$$
$$564$$ 0 0
$$565$$ −2.11146 −0.0888296
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 27.8885 1.16915 0.584574 0.811340i $$-0.301262\pi$$
0.584574 + 0.811340i $$0.301262\pi$$
$$570$$ 0 0
$$571$$ −4.94427 −0.206911 −0.103456 0.994634i $$-0.532990\pi$$
−0.103456 + 0.994634i $$0.532990\pi$$
$$572$$ 0 0
$$573$$ −20.9443 −0.874960
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ 14.9443 0.622138 0.311069 0.950387i $$-0.399313\pi$$
0.311069 + 0.950387i $$0.399313\pi$$
$$578$$ 0 0
$$579$$ 23.8885 0.992774
$$580$$ 0 0
$$581$$ −6.47214 −0.268509
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 8.94427 0.369800
$$586$$ 0 0
$$587$$ −16.9443 −0.699365 −0.349682 0.936868i $$-0.613711\pi$$
−0.349682 + 0.936868i $$0.613711\pi$$
$$588$$ 0 0
$$589$$ −41.8885 −1.72599
$$590$$ 0 0
$$591$$ −2.94427 −0.121111
$$592$$ 0 0
$$593$$ 24.8328 1.01976 0.509881 0.860245i $$-0.329690\pi$$
0.509881 + 0.860245i $$0.329690\pi$$
$$594$$ 0 0
$$595$$ 8.94427 0.366679
$$596$$ 0 0
$$597$$ 20.9443 0.857192
$$598$$ 0 0
$$599$$ 1.88854 0.0771638 0.0385819 0.999255i $$-0.487716\pi$$
0.0385819 + 0.999255i $$0.487716\pi$$
$$600$$ 0 0
$$601$$ −12.8328 −0.523461 −0.261731 0.965141i $$-0.584293\pi$$
−0.261731 + 0.965141i $$0.584293\pi$$
$$602$$ 0 0
$$603$$ 12.9443 0.527132
$$604$$ 0 0
$$605$$ −22.0000 −0.894427
$$606$$ 0 0
$$607$$ −16.3607 −0.664060 −0.332030 0.943269i $$-0.607733\pi$$
−0.332030 + 0.943269i $$0.607733\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ −28.9443 −1.17096
$$612$$ 0 0
$$613$$ −12.8328 −0.518313 −0.259156 0.965835i $$-0.583444\pi$$
−0.259156 + 0.965835i $$0.583444\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 0 0
$$619$$ 25.5279 1.02605 0.513026 0.858373i $$-0.328525\pi$$
0.513026 + 0.858373i $$0.328525\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ 17.4164 0.697774
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 48.9443 1.95154
$$630$$ 0 0
$$631$$ 28.9443 1.15225 0.576127 0.817360i $$-0.304564\pi$$
0.576127 + 0.817360i $$0.304564\pi$$
$$632$$ 0 0
$$633$$ 16.9443 0.673474
$$634$$ 0 0
$$635$$ −6.11146 −0.242526
$$636$$ 0 0
$$637$$ 4.47214 0.177192
$$638$$ 0 0
$$639$$ 12.9443 0.512067
$$640$$ 0 0
$$641$$ 10.0000 0.394976 0.197488 0.980305i $$-0.436722\pi$$
0.197488 + 0.980305i $$0.436722\pi$$
$$642$$ 0 0
$$643$$ 1.52786 0.0602531 0.0301265 0.999546i $$-0.490409\pi$$
0.0301265 + 0.999546i $$0.490409\pi$$
$$644$$ 0 0
$$645$$ 25.8885 1.01936
$$646$$ 0 0
$$647$$ 32.3607 1.27223 0.636115 0.771594i $$-0.280540\pi$$
0.636115 + 0.771594i $$0.280540\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −6.47214 −0.253663
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ 0 0
$$657$$ −2.94427 −0.114867
$$658$$ 0 0
$$659$$ 6.83282 0.266169 0.133084 0.991105i $$-0.457512\pi$$
0.133084 + 0.991105i $$0.457512\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 0 0
$$663$$ 20.0000 0.776736
$$664$$ 0 0
$$665$$ −12.9443 −0.501957
$$666$$ 0 0
$$667$$ −2.00000 −0.0774403
$$668$$ 0 0
$$669$$ 9.52786 0.368369
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −39.8885 −1.53304 −0.766521 0.642220i $$-0.778014\pi$$
−0.766521 + 0.642220i $$0.778014\pi$$
$$678$$ 0 0
$$679$$ −8.47214 −0.325131
$$680$$ 0 0
$$681$$ 14.4721 0.554573
$$682$$ 0 0
$$683$$ 0.944272 0.0361316 0.0180658 0.999837i $$-0.494249\pi$$
0.0180658 + 0.999837i $$0.494249\pi$$
$$684$$ 0 0
$$685$$ −28.0000 −1.06983
$$686$$ 0 0
$$687$$ 14.0000 0.534133
$$688$$ 0 0
$$689$$ 31.0557 1.18313
$$690$$ 0 0
$$691$$ 32.9443 1.25326 0.626630 0.779317i $$-0.284434\pi$$
0.626630 + 0.779317i $$0.284434\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.8885 0.678551
$$696$$ 0 0
$$697$$ 26.8328 1.01637
$$698$$ 0 0
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 32.8328 1.24008 0.620039 0.784571i $$-0.287117\pi$$
0.620039 + 0.784571i $$0.287117\pi$$
$$702$$ 0 0
$$703$$ −70.8328 −2.67151
$$704$$ 0 0
$$705$$ 12.9443 0.487509
$$706$$ 0 0
$$707$$ −17.4164 −0.655011
$$708$$ 0 0
$$709$$ −10.9443 −0.411021 −0.205510 0.978655i $$-0.565885\pi$$
−0.205510 + 0.978655i $$0.565885\pi$$
$$710$$ 0 0
$$711$$ −12.9443 −0.485448
$$712$$ 0 0
$$713$$ −6.47214 −0.242383
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.05573 0.114118
$$718$$ 0 0
$$719$$ 14.4721 0.539720 0.269860 0.962900i $$-0.413023\pi$$
0.269860 + 0.962900i $$0.413023\pi$$
$$720$$ 0 0
$$721$$ 12.9443 0.482070
$$722$$ 0 0
$$723$$ −11.5279 −0.428726
$$724$$ 0 0
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ −20.9443 −0.776780 −0.388390 0.921495i $$-0.626969\pi$$
−0.388390 + 0.921495i $$0.626969\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 57.8885 2.14109
$$732$$ 0 0
$$733$$ −33.0557 −1.22094 −0.610471 0.792039i $$-0.709020\pi$$
−0.610471 + 0.792039i $$0.709020\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −8.94427 −0.329020 −0.164510 0.986375i $$-0.552604\pi$$
−0.164510 + 0.986375i $$0.552604\pi$$
$$740$$ 0 0
$$741$$ −28.9443 −1.06329
$$742$$ 0 0
$$743$$ −30.8328 −1.13115 −0.565573 0.824698i $$-0.691345\pi$$
−0.565573 + 0.824698i $$0.691345\pi$$
$$744$$ 0 0
$$745$$ −37.8885 −1.38813
$$746$$ 0 0
$$747$$ 6.47214 0.236803
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 17.8885 0.652762 0.326381 0.945238i $$-0.394171\pi$$
0.326381 + 0.945238i $$0.394171\pi$$
$$752$$ 0 0
$$753$$ −14.4721 −0.527394
$$754$$ 0 0
$$755$$ 25.8885 0.942181
$$756$$ 0 0
$$757$$ −25.0557 −0.910666 −0.455333 0.890321i $$-0.650480\pi$$
−0.455333 + 0.890321i $$0.650480\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.9443 1.41173 0.705864 0.708347i $$-0.250559\pi$$
0.705864 + 0.708347i $$0.250559\pi$$
$$762$$ 0 0
$$763$$ −14.9443 −0.541019
$$764$$ 0 0
$$765$$ −8.94427 −0.323381
$$766$$ 0 0
$$767$$ −17.8885 −0.645918
$$768$$ 0 0
$$769$$ −14.3607 −0.517859 −0.258930 0.965896i $$-0.583370\pi$$
−0.258930 + 0.965896i $$0.583370\pi$$
$$770$$ 0 0
$$771$$ −14.9443 −0.538205
$$772$$ 0 0
$$773$$ 22.9443 0.825248 0.412624 0.910901i $$-0.364612\pi$$
0.412624 + 0.910901i $$0.364612\pi$$
$$774$$ 0 0
$$775$$ 6.47214 0.232486
$$776$$ 0 0
$$777$$ −10.9443 −0.392624
$$778$$ 0 0
$$779$$ −38.8328 −1.39133
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 29.8885 1.06677
$$786$$ 0 0
$$787$$ 27.4164 0.977289 0.488645 0.872483i $$-0.337491\pi$$
0.488645 + 0.872483i $$0.337491\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 1.05573 0.0375374
$$792$$ 0 0
$$793$$ −26.8328 −0.952861
$$794$$ 0 0
$$795$$ −13.8885 −0.492576
$$796$$ 0 0
$$797$$ 24.8328 0.879623 0.439812 0.898090i $$-0.355045\pi$$
0.439812 + 0.898090i $$0.355045\pi$$
$$798$$ 0 0
$$799$$ 28.9443 1.02397
$$800$$ 0 0
$$801$$ −17.4164 −0.615379
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −2.00000 −0.0704907
$$806$$ 0 0
$$807$$ −7.52786 −0.264993
$$808$$ 0 0
$$809$$ 0.111456 0.00391859 0.00195930 0.999998i $$-0.499376\pi$$
0.00195930 + 0.999998i $$0.499376\pi$$
$$810$$ 0 0
$$811$$ 2.11146 0.0741433 0.0370716 0.999313i $$-0.488197\pi$$
0.0370716 + 0.999313i $$0.488197\pi$$
$$812$$ 0 0
$$813$$ −11.4164 −0.400391
$$814$$ 0 0
$$815$$ −17.8885 −0.626608
$$816$$ 0 0
$$817$$ −83.7771 −2.93099
$$818$$ 0 0
$$819$$ −4.47214 −0.156269
$$820$$ 0 0
$$821$$ 36.8328 1.28547 0.642737 0.766087i $$-0.277799\pi$$
0.642737 + 0.766087i $$0.277799\pi$$
$$822$$ 0 0
$$823$$ −3.05573 −0.106516 −0.0532580 0.998581i $$-0.516961\pi$$
−0.0532580 + 0.998581i $$0.516961\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.88854 0.0656711 0.0328356 0.999461i $$-0.489546\pi$$
0.0328356 + 0.999461i $$0.489546\pi$$
$$828$$ 0 0
$$829$$ −13.4164 −0.465971 −0.232986 0.972480i $$-0.574849\pi$$
−0.232986 + 0.972480i $$0.574849\pi$$
$$830$$ 0 0
$$831$$ −23.8885 −0.828684
$$832$$ 0 0
$$833$$ −4.47214 −0.154950
$$834$$ 0 0
$$835$$ −22.8328 −0.790162
$$836$$ 0 0
$$837$$ 6.47214 0.223710
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −10.0000 −0.344418
$$844$$ 0 0
$$845$$ 14.0000 0.481615
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ 9.52786 0.326995
$$850$$ 0 0
$$851$$ −10.9443 −0.375165
$$852$$ 0 0
$$853$$ 38.3607 1.31344 0.656722 0.754132i $$-0.271942\pi$$
0.656722 + 0.754132i $$0.271942\pi$$
$$854$$ 0 0
$$855$$ 12.9443 0.442685
$$856$$ 0 0
$$857$$ −12.1115 −0.413719 −0.206860 0.978371i $$-0.566324\pi$$
−0.206860 + 0.978371i $$0.566324\pi$$
$$858$$ 0 0
$$859$$ −15.0557 −0.513695 −0.256847 0.966452i $$-0.582684\pi$$
−0.256847 + 0.966452i $$0.582684\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ −3.05573 −0.104018 −0.0520091 0.998647i $$-0.516562\pi$$
−0.0520091 + 0.998647i $$0.516562\pi$$
$$864$$ 0 0
$$865$$ −16.9443 −0.576123
$$866$$ 0 0
$$867$$ −3.00000 −0.101885
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 57.8885 1.96148
$$872$$ 0 0
$$873$$ 8.47214 0.286738
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 0 0
$$877$$ 54.7214 1.84781 0.923905 0.382623i $$-0.124979\pi$$
0.923905 + 0.382623i $$0.124979\pi$$
$$878$$ 0 0
$$879$$ 7.88854 0.266074
$$880$$ 0 0
$$881$$ −22.3607 −0.753350 −0.376675 0.926345i $$-0.622933\pi$$
−0.376675 + 0.926345i $$0.622933\pi$$
$$882$$ 0 0
$$883$$ 50.8328 1.71066 0.855330 0.518083i $$-0.173354\pi$$
0.855330 + 0.518083i $$0.173354\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ 0 0
$$887$$ 14.4721 0.485927 0.242963 0.970035i $$-0.421881\pi$$
0.242963 + 0.970035i $$0.421881\pi$$
$$888$$ 0 0
$$889$$ 3.05573 0.102486
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −41.8885 −1.40175
$$894$$ 0 0
$$895$$ 40.0000 1.33705
$$896$$ 0 0
$$897$$ −4.47214 −0.149320
$$898$$ 0 0
$$899$$ 12.9443 0.431716
$$900$$ 0 0
$$901$$ −31.0557 −1.03462
$$902$$ 0 0
$$903$$ −12.9443 −0.430758
$$904$$ 0 0
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ 6.11146 0.202928 0.101464 0.994839i $$-0.467647\pi$$
0.101464 + 0.994839i $$0.467647\pi$$
$$908$$ 0 0
$$909$$ 17.4164 0.577666
$$910$$ 0 0
$$911$$ −40.7214 −1.34916 −0.674579 0.738202i $$-0.735675\pi$$
−0.674579 + 0.738202i $$0.735675\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 12.0000 0.396708
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ −52.9443 −1.74647 −0.873235 0.487299i $$-0.837982\pi$$
−0.873235 + 0.487299i $$0.837982\pi$$
$$920$$ 0 0
$$921$$ 7.05573 0.232494
$$922$$ 0 0
$$923$$ 57.8885 1.90542
$$924$$ 0 0
$$925$$ 10.9443 0.359845
$$926$$ 0 0
$$927$$ −12.9443 −0.425146
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 6.47214 0.212116
$$932$$ 0 0
$$933$$ 6.47214 0.211888
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −40.2492 −1.31488 −0.657442 0.753505i $$-0.728362\pi$$
−0.657442 + 0.753505i $$0.728362\pi$$
$$938$$ 0 0
$$939$$ −13.4164 −0.437828
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 0 0
$$943$$ −6.00000 −0.195387
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 16.9443 0.550615 0.275307 0.961356i $$-0.411220\pi$$
0.275307 + 0.961356i $$0.411220\pi$$
$$948$$ 0 0
$$949$$ −13.1672 −0.427425
$$950$$ 0 0
$$951$$ 30.9443 1.00344
$$952$$ 0 0
$$953$$ −26.9443 −0.872811 −0.436405 0.899750i $$-0.643749\pi$$
−0.436405 + 0.899750i $$0.643749\pi$$
$$954$$ 0 0
$$955$$ 41.8885 1.35548
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 14.0000 0.452084
$$960$$ 0 0
$$961$$ 10.8885 0.351243
$$962$$ 0 0
$$963$$ 8.00000 0.257796
$$964$$ 0 0
$$965$$ −47.7771 −1.53800
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 0 0
$$969$$ 28.9443 0.929824
$$970$$ 0 0
$$971$$ −3.41641 −0.109638 −0.0548189 0.998496i $$-0.517458\pi$$
−0.0548189 + 0.998496i $$0.517458\pi$$
$$972$$ 0 0
$$973$$ −8.94427 −0.286740
$$974$$ 0 0
$$975$$ 4.47214 0.143223
$$976$$ 0 0
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 14.9443 0.477134
$$982$$ 0 0
$$983$$ 57.8885 1.84636 0.923179 0.384371i $$-0.125581\pi$$
0.923179 + 0.384371i $$0.125581\pi$$
$$984$$ 0 0
$$985$$ 5.88854 0.187625
$$986$$ 0 0
$$987$$ −6.47214 −0.206010
$$988$$ 0 0
$$989$$ −12.9443 −0.411604
$$990$$ 0 0
$$991$$ −43.0557 −1.36771 −0.683855 0.729618i $$-0.739698\pi$$
−0.683855 + 0.729618i $$0.739698\pi$$
$$992$$ 0 0
$$993$$ 21.8885 0.694612
$$994$$ 0 0
$$995$$ −41.8885 −1.32796
$$996$$ 0 0
$$997$$ −1.63932 −0.0519178 −0.0259589 0.999663i $$-0.508264\pi$$
−0.0259589 + 0.999663i $$0.508264\pi$$
$$998$$ 0 0
$$999$$ 10.9443 0.346261
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 7728.2.a.bd.1.2 2
4.3 odd 2 966.2.a.p.1.2 2
12.11 even 2 2898.2.a.v.1.2 2
28.27 even 2 6762.2.a.bz.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.2 2 4.3 odd 2
2898.2.a.v.1.2 2 12.11 even 2
6762.2.a.bz.1.1 2 28.27 even 2
7728.2.a.bd.1.2 2 1.1 even 1 trivial