# Properties

 Label 2800.2.g.m.449.2 Level $2800$ Weight $2$ Character 2800.449 Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (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: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.449 Dual form 2800.2.g.m.449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -1.00000i q^{7} +2.00000 q^{9} +5.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} -6.00000 q^{19} +1.00000 q^{21} +6.00000i q^{23} +5.00000i q^{27} +9.00000 q^{29} +5.00000i q^{33} -6.00000i q^{37} -1.00000 q^{39} +8.00000 q^{41} -6.00000i q^{43} +3.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} -12.0000i q^{53} -6.00000i q^{57} +8.00000 q^{59} -4.00000 q^{61} -2.00000i q^{63} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000i q^{73} -5.00000i q^{77} -3.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +9.00000i q^{87} +16.0000 q^{89} +1.00000 q^{91} -7.00000i q^{97} +10.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 10 q^{11} - 12 q^{19} + 2 q^{21} + 18 q^{29} - 2 q^{39} + 16 q^{41} - 2 q^{49} + 6 q^{51} + 16 q^{59} - 8 q^{61} - 12 q^{69} - 16 q^{71} - 6 q^{79} + 2 q^{81} + 32 q^{89} + 2 q^{91} + 20 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 10 * q^11 - 12 * q^19 + 2 * q^21 + 18 * q^29 - 2 * q^39 + 16 * q^41 - 2 * q^49 + 6 * q^51 + 16 * q^59 - 8 * q^61 - 12 * q^69 - 16 * q^71 - 6 * q^79 + 2 * q^81 + 32 * q^89 + 2 * q^91 + 20 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 5.00000i 0.870388i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 6.00000i − 0.794719i
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 5.00000i − 0.569803i
$$78$$ 0 0
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.00000i 0.964901i
$$88$$ 0 0
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.00000i − 0.710742i −0.934725 0.355371i $$-0.884354\pi$$
0.934725 0.355371i $$-0.115646\pi$$
$$98$$ 0 0
$$99$$ 10.0000 1.00504
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 9.00000i 0.886796i 0.896325 + 0.443398i $$0.146227\pi$$
−0.896325 + 0.443398i $$0.853773\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 8.00000i 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 6.00000i 0.520266i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 16.0000i 1.36697i 0.729964 + 0.683486i $$0.239537\pi$$
−0.729964 + 0.683486i $$0.760463\pi$$
$$138$$ 0 0
$$139$$ 18.0000 1.52674 0.763370 0.645961i $$-0.223543\pi$$
0.763370 + 0.645961i $$0.223543\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ 0 0
$$143$$ 5.00000i 0.418121i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 1.00000i − 0.0824786i
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 9.00000i − 0.696441i −0.937413 0.348220i $$-0.886786\pi$$
0.937413 0.348220i $$-0.113214\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ −12.0000 −0.917663
$$172$$ 0 0
$$173$$ 19.0000i 1.44454i 0.691609 + 0.722272i $$0.256902\pi$$
−0.691609 + 0.722272i $$0.743098\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 8.00000i 0.601317i
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 15.0000i − 1.09691i
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ −11.0000 −0.795932 −0.397966 0.917400i $$-0.630284\pi$$
−0.397966 + 0.917400i $$0.630284\pi$$
$$192$$ 0 0
$$193$$ 8.00000i 0.575853i 0.957653 + 0.287926i $$0.0929658\pi$$
−0.957653 + 0.287926i $$0.907034\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ − 9.00000i − 0.631676i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.0000i 0.834058i
$$208$$ 0 0
$$209$$ −30.0000 −2.07514
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ 0 0
$$213$$ − 8.00000i − 0.548151i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 25.0000i 1.67412i 0.547108 + 0.837062i $$0.315729\pi$$
−0.547108 + 0.837062i $$0.684271\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 13.0000i − 0.862840i −0.902151 0.431420i $$-0.858013\pi$$
0.902151 0.431420i $$-0.141987\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 0 0
$$233$$ 8.00000i 0.524097i 0.965055 + 0.262049i $$0.0843981\pi$$
−0.965055 + 0.262049i $$0.915602\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 3.00000i − 0.194871i
$$238$$ 0 0
$$239$$ −7.00000 −0.452792 −0.226396 0.974035i $$-0.572694\pi$$
−0.226396 + 0.974035i $$0.572694\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ 16.0000i 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 6.00000i − 0.381771i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 0 0
$$253$$ 30.0000i 1.88608i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ 0 0
$$263$$ − 18.0000i − 1.10993i −0.831875 0.554964i $$-0.812732\pi$$
0.831875 0.554964i $$-0.187268\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000i 0.979184i
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ 0 0
$$273$$ 1.00000i 0.0605228i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 14.0000i − 0.841178i −0.907251 0.420589i $$-0.861823\pi$$
0.907251 0.420589i $$-0.138177\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.0000 −0.775515 −0.387757 0.921761i $$-0.626750\pi$$
−0.387757 + 0.921761i $$0.626750\pi$$
$$282$$ 0 0
$$283$$ − 29.0000i − 1.72387i −0.507018 0.861936i $$-0.669252\pi$$
0.507018 0.861936i $$-0.330748\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ − 1.00000i − 0.0584206i −0.999573 0.0292103i $$-0.990701\pi$$
0.999573 0.0292103i $$-0.00929925\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 25.0000i 1.45065i
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −6.00000 −0.345834
$$302$$ 0 0
$$303$$ − 14.0000i − 0.804279i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 27.0000i − 1.54097i −0.637457 0.770486i $$-0.720014\pi$$
0.637457 0.770486i $$-0.279986\pi$$
$$308$$ 0 0
$$309$$ −9.00000 −0.511992
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 29.0000i 1.63918i 0.572953 + 0.819588i $$0.305798\pi$$
−0.572953 + 0.819588i $$0.694202\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ 0 0
$$319$$ 45.0000 2.51952
$$320$$ 0 0
$$321$$ −2.00000 −0.111629
$$322$$ 0 0
$$323$$ 18.0000i 1.00155i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 11.0000i 0.608301i
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ − 12.0000i − 0.657596i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.0000i 1.41631i 0.706057 + 0.708155i $$0.250472\pi$$
−0.706057 + 0.708155i $$0.749528\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 10.0000i − 0.536828i −0.963304 0.268414i $$-0.913500\pi$$
0.963304 0.268414i $$-0.0864995\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ − 33.0000i − 1.75641i −0.478282 0.878206i $$-0.658740\pi$$
0.478282 0.878206i $$-0.341260\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 3.00000i − 0.158777i
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 14.0000i 0.734809i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 29.0000i − 1.51379i −0.653538 0.756894i $$-0.726716\pi$$
0.653538 0.756894i $$-0.273284\pi$$
$$368$$ 0 0
$$369$$ 16.0000 0.832927
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.00000i 0.463524i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 12.0000i − 0.609994i
$$388$$ 0 0
$$389$$ −25.0000 −1.26755 −0.633775 0.773517i $$-0.718496\pi$$
−0.633775 + 0.773517i $$0.718496\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 6.00000i 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 29.0000i 1.45547i 0.685859 + 0.727734i $$0.259427\pi$$
−0.685859 + 0.727734i $$0.740573\pi$$
$$398$$ 0 0
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ 9.00000 0.449439 0.224719 0.974424i $$-0.427853\pi$$
0.224719 + 0.974424i $$0.427853\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 30.0000i − 1.48704i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −16.0000 −0.789222
$$412$$ 0 0
$$413$$ − 8.00000i − 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 18.0000i 0.881464i
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −19.0000 −0.926003 −0.463002 0.886357i $$-0.653228\pi$$
−0.463002 + 0.886357i $$0.653228\pi$$
$$422$$ 0 0
$$423$$ 6.00000i 0.291730i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ −5.00000 −0.241402
$$430$$ 0 0
$$431$$ −23.0000 −1.10787 −0.553936 0.832560i $$-0.686875\pi$$
−0.553936 + 0.832560i $$0.686875\pi$$
$$432$$ 0 0
$$433$$ − 30.0000i − 1.44171i −0.693087 0.720854i $$-0.743750\pi$$
0.693087 0.720854i $$-0.256250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 36.0000i − 1.72211i
$$438$$ 0 0
$$439$$ −34.0000 −1.62273 −0.811366 0.584539i $$-0.801275\pi$$
−0.811366 + 0.584539i $$0.801275\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ − 30.0000i − 1.42534i −0.701498 0.712672i $$-0.747485\pi$$
0.701498 0.712672i $$-0.252515\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.0000i 0.662177i
$$448$$ 0 0
$$449$$ 33.0000 1.55737 0.778683 0.627417i $$-0.215888\pi$$
0.778683 + 0.627417i $$0.215888\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ 0 0
$$453$$ 19.0000i 0.892698i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.0000i 1.30978i 0.755722 + 0.654892i $$0.227286\pi$$
−0.755722 + 0.654892i $$0.772714\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 0 0
$$463$$ − 20.0000i − 0.929479i −0.885448 0.464739i $$-0.846148\pi$$
0.885448 0.464739i $$-0.153852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 33.0000i 1.52706i 0.645774 + 0.763529i $$0.276535\pi$$
−0.645774 + 0.763529i $$0.723465\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ − 30.0000i − 1.37940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 24.0000i − 1.09888i
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 0 0
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ −33.0000 −1.48927 −0.744635 0.667472i $$-0.767376\pi$$
−0.744635 + 0.667472i $$0.767376\pi$$
$$492$$ 0 0
$$493$$ − 27.0000i − 1.21602i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ 0 0
$$503$$ 31.0000i 1.38222i 0.722749 + 0.691111i $$0.242878\pi$$
−0.722749 + 0.691111i $$0.757122\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ − 30.0000i − 1.32453i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 15.0000i 0.659699i
$$518$$ 0 0
$$519$$ −19.0000 −0.834007
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 16.0000 0.694341
$$532$$ 0 0
$$533$$ 8.00000i 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.00000i 0.172613i
$$538$$ 0 0
$$539$$ −5.00000 −0.215365
$$540$$ 0 0
$$541$$ −9.00000 −0.386940 −0.193470 0.981106i $$-0.561974\pi$$
−0.193470 + 0.981106i $$0.561974\pi$$
$$542$$ 0 0
$$543$$ − 20.0000i − 0.858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 0 0
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ −54.0000 −2.30048
$$552$$ 0 0
$$553$$ 3.00000i 0.127573i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 24.0000i − 1.01691i −0.861088 0.508456i $$-0.830216\pi$$
0.861088 0.508456i $$-0.169784\pi$$
$$558$$ 0 0
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ 15.0000 0.633300
$$562$$ 0 0
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 0 0
$$573$$ − 11.0000i − 0.459532i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 17.0000i − 0.707719i −0.935299 0.353860i $$-0.884869\pi$$
0.935299 0.353860i $$-0.115131\pi$$
$$578$$ 0 0
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ − 60.0000i − 2.48495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ − 7.00000i − 0.287456i −0.989617 0.143728i $$-0.954091\pi$$
0.989617 0.143728i $$-0.0459090\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −33.0000 −1.34834 −0.674172 0.738575i $$-0.735499\pi$$
−0.674172 + 0.738575i $$0.735499\pi$$
$$600$$ 0 0
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.0000i 0.690009i 0.938601 + 0.345004i $$0.112123\pi$$
−0.938601 + 0.345004i $$0.887877\pi$$
$$608$$ 0 0
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ − 22.0000i − 0.888572i −0.895885 0.444286i $$-0.853457\pi$$
0.895885 0.444286i $$-0.146543\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 10.0000i − 0.402585i −0.979531 0.201292i $$-0.935486\pi$$
0.979531 0.201292i $$-0.0645141\pi$$
$$618$$ 0 0
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 0 0
$$623$$ − 16.0000i − 0.641026i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 30.0000i − 1.19808i
$$628$$ 0 0
$$629$$ −18.0000 −0.717707
$$630$$ 0 0
$$631$$ 9.00000 0.358284 0.179142 0.983823i $$-0.442668\pi$$
0.179142 + 0.983823i $$0.442668\pi$$
$$632$$ 0 0
$$633$$ − 13.0000i − 0.516704i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 1.00000i − 0.0396214i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 0 0
$$643$$ − 47.0000i − 1.85350i −0.375680 0.926750i $$-0.622591\pi$$
0.375680 0.926750i $$-0.377409\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 40.0000 1.57014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 20.0000i 0.780274i
$$658$$ 0 0
$$659$$ −25.0000 −0.973862 −0.486931 0.873441i $$-0.661884\pi$$
−0.486931 + 0.873441i $$0.661884\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 0 0
$$663$$ 3.00000i 0.116510i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 54.0000i 2.09089i
$$668$$ 0 0
$$669$$ −25.0000 −0.966556
$$670$$ 0 0
$$671$$ −20.0000 −0.772091
$$672$$ 0 0
$$673$$ 32.0000i 1.23351i 0.787155 + 0.616755i $$0.211553\pi$$
−0.787155 + 0.616755i $$0.788447\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 33.0000i − 1.26829i −0.773213 0.634147i $$-0.781352\pi$$
0.773213 0.634147i $$-0.218648\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 13.0000 0.498161
$$682$$ 0 0
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 16.0000i − 0.610438i
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 0 0
$$693$$ − 10.0000i − 0.379869i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 24.0000i − 0.909065i
$$698$$ 0 0
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ −21.0000 −0.793159 −0.396580 0.918000i $$-0.629803\pi$$
−0.396580 + 0.918000i $$0.629803\pi$$
$$702$$ 0 0
$$703$$ 36.0000i 1.35777i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 14.0000i 0.526524i
$$708$$ 0 0
$$709$$ 41.0000 1.53979 0.769894 0.638172i $$-0.220309\pi$$
0.769894 + 0.638172i $$0.220309\pi$$
$$710$$ 0 0
$$711$$ −6.00000 −0.225018
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 7.00000i − 0.261420i
$$718$$ 0 0
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ 9.00000 0.335178
$$722$$ 0 0
$$723$$ 18.0000i 0.669427i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ 0 0
$$733$$ 5.00000i 0.184679i 0.995728 + 0.0923396i $$0.0294345\pi$$
−0.995728 + 0.0923396i $$0.970565\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 20.0000i − 0.736709i
$$738$$ 0 0
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ 0 0
$$743$$ 36.0000i 1.32071i 0.750953 + 0.660356i $$0.229595\pi$$
−0.750953 + 0.660356i $$0.770405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.0000i 0.878114i
$$748$$ 0 0
$$749$$ 2.00000 0.0730784
$$750$$ 0 0
$$751$$ 35.0000 1.27717 0.638584 0.769552i $$-0.279520\pi$$
0.638584 + 0.769552i $$0.279520\pi$$
$$752$$ 0 0
$$753$$ − 14.0000i − 0.510188i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 8.00000i − 0.290765i −0.989376 0.145382i $$-0.953559\pi$$
0.989376 0.145382i $$-0.0464413\pi$$
$$758$$ 0 0
$$759$$ −30.0000 −1.08893
$$760$$ 0 0
$$761$$ −46.0000 −1.66750 −0.833749 0.552143i $$-0.813810\pi$$
−0.833749 + 0.552143i $$0.813810\pi$$
$$762$$ 0 0
$$763$$ − 11.0000i − 0.398227i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 0 0
$$773$$ 1.00000i 0.0359675i 0.999838 + 0.0179838i $$0.00572471\pi$$
−0.999838 + 0.0179838i $$0.994275\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 6.00000i − 0.215249i
$$778$$ 0 0
$$779$$ −48.0000 −1.71978
$$780$$ 0 0
$$781$$ −40.0000 −1.43131
$$782$$ 0 0
$$783$$ 45.0000i 1.60817i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 11.0000i 0.392108i 0.980593 + 0.196054i $$0.0628127\pi$$
−0.980593 + 0.196054i $$0.937187\pi$$
$$788$$ 0 0
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ − 4.00000i − 0.142044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 35.0000i − 1.23976i −0.784695 0.619882i $$-0.787181\pi$$
0.784695 0.619882i $$-0.212819\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ 32.0000 1.13066
$$802$$ 0 0
$$803$$ 50.0000i 1.76446i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 10.0000i − 0.352017i
$$808$$ 0 0
$$809$$ 23.0000 0.808637 0.404318 0.914618i $$-0.367509\pi$$
0.404318 + 0.914618i $$0.367509\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 0 0
$$813$$ − 4.00000i − 0.140286i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 36.0000i 1.25948i
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ −7.00000 −0.244302 −0.122151 0.992512i $$-0.538979\pi$$
−0.122151 + 0.992512i $$0.538979\pi$$
$$822$$ 0 0
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.00000i 0.0695468i 0.999395 + 0.0347734i $$0.0110710\pi$$
−0.999395 + 0.0347734i $$0.988929\pi$$
$$828$$ 0 0
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ 3.00000i 0.103944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −26.0000 −0.897620 −0.448810 0.893627i $$-0.648152\pi$$
−0.448810 + 0.893627i $$0.648152\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ − 13.0000i − 0.447744i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 14.0000i − 0.481046i
$$848$$ 0 0
$$849$$ 29.0000 0.995277
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 14.0000i 0.478231i 0.970991 + 0.239115i $$0.0768574\pi$$
−0.970991 + 0.239115i $$0.923143\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ 0 0
$$863$$ − 20.0000i − 0.680808i −0.940279 0.340404i $$-0.889436\pi$$
0.940279 0.340404i $$-0.110564\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 8.00000i 0.271694i
$$868$$ 0 0
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ − 14.0000i − 0.473828i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.0000i 0.337676i 0.985644 + 0.168838i $$0.0540015\pi$$
−0.985644 + 0.168838i $$0.945999\pi$$
$$878$$ 0 0
$$879$$ 1.00000 0.0337292
$$880$$ 0 0
$$881$$ −16.0000 −0.539054 −0.269527 0.962993i $$-0.586867\pi$$
−0.269527 + 0.962993i $$0.586867\pi$$
$$882$$ 0 0
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 0 0
$$893$$ − 18.0000i − 0.602347i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 6.00000i − 0.199667i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 30.0000i − 0.996134i −0.867139 0.498067i $$-0.834043\pi$$
0.867139 0.498067i $$-0.165957\pi$$
$$908$$ 0 0
$$909$$ −28.0000 −0.928701
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 60.0000i 1.98571i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 6.00000i − 0.198137i
$$918$$ 0 0
$$919$$ 25.0000 0.824674 0.412337 0.911031i $$-0.364713\pi$$
0.412337 + 0.911031i $$0.364713\pi$$
$$920$$ 0 0
$$921$$ 27.0000 0.889680
$$922$$ 0 0
$$923$$ − 8.00000i − 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 18.0000i 0.591198i
$$928$$ 0 0
$$929$$ −36.0000 −1.18112 −0.590561 0.806993i $$-0.701093\pi$$
−0.590561 + 0.806993i $$0.701093\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 0 0
$$933$$ 14.0000i 0.458339i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 7.00000i − 0.228680i −0.993442 0.114340i $$-0.963525\pi$$
0.993442 0.114340i $$-0.0364753\pi$$
$$938$$ 0 0
$$939$$ −29.0000 −0.946379
$$940$$ 0 0
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ 0 0
$$943$$ 48.0000i 1.56310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ 20.0000i 0.647864i 0.946080 + 0.323932i $$0.105005\pi$$
−0.946080 + 0.323932i $$0.894995\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 45.0000i 1.45464i
$$958$$ 0 0
$$959$$ 16.0000 0.516667
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 4.00000i 0.128898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 22.0000i 0.707472i 0.935345 + 0.353736i $$0.115089\pi$$
−0.935345 + 0.353736i $$0.884911\pi$$
$$968$$ 0 0
$$969$$ −18.0000 −0.578243
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ − 18.0000i − 0.577054i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 0 0
$$979$$ 80.0000 2.55681
$$980$$ 0 0
$$981$$ 22.0000 0.702406
$$982$$ 0 0
$$983$$ − 9.00000i − 0.287055i −0.989646 0.143528i $$-0.954155\pi$$
0.989646 0.143528i $$-0.0458446\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 3.00000i 0.0954911i
$$988$$ 0 0
$$989$$ 36.0000 1.14473
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.00000i 0.0316703i 0.999875 + 0.0158352i $$0.00504070\pi$$
−0.999875 + 0.0158352i $$0.994959\pi$$
$$998$$ 0 0
$$999$$ 30.0000 0.949158
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 2800.2.g.m.449.2 2
4.3 odd 2 1400.2.g.e.449.1 2
5.2 odd 4 560.2.a.e.1.1 1
5.3 odd 4 2800.2.a.i.1.1 1
5.4 even 2 inner 2800.2.g.m.449.1 2
15.2 even 4 5040.2.a.be.1.1 1
20.3 even 4 1400.2.a.k.1.1 1
20.7 even 4 280.2.a.b.1.1 1
20.19 odd 2 1400.2.g.e.449.2 2
35.27 even 4 3920.2.a.r.1.1 1
40.27 even 4 2240.2.a.v.1.1 1
40.37 odd 4 2240.2.a.j.1.1 1
60.47 odd 4 2520.2.a.p.1.1 1
140.27 odd 4 1960.2.a.k.1.1 1
140.47 odd 12 1960.2.q.e.361.1 2
140.67 even 12 1960.2.q.m.961.1 2
140.83 odd 4 9800.2.a.n.1.1 1
140.87 odd 12 1960.2.q.e.961.1 2
140.107 even 12 1960.2.q.m.361.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.b.1.1 1 20.7 even 4
560.2.a.e.1.1 1 5.2 odd 4
1400.2.a.k.1.1 1 20.3 even 4
1400.2.g.e.449.1 2 4.3 odd 2
1400.2.g.e.449.2 2 20.19 odd 2
1960.2.a.k.1.1 1 140.27 odd 4
1960.2.q.e.361.1 2 140.47 odd 12
1960.2.q.e.961.1 2 140.87 odd 12
1960.2.q.m.361.1 2 140.107 even 12
1960.2.q.m.961.1 2 140.67 even 12
2240.2.a.j.1.1 1 40.37 odd 4
2240.2.a.v.1.1 1 40.27 even 4
2520.2.a.p.1.1 1 60.47 odd 4
2800.2.a.i.1.1 1 5.3 odd 4
2800.2.g.m.449.1 2 5.4 even 2 inner
2800.2.g.m.449.2 2 1.1 even 1 trivial
3920.2.a.r.1.1 1 35.27 even 4
5040.2.a.be.1.1 1 15.2 even 4
9800.2.a.n.1.1 1 140.83 odd 4