# Properties

 Label 2366.2.d.b.337.1 Level $2366$ Weight $2$ Character 2366.337 Analytic conductor $18.893$ 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] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.d (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: $$18.8926051182$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2366.337 Dual form 2366.2.d.b.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} +2.00000i q^{21} -2.00000i q^{24} +5.00000 q^{25} +4.00000 q^{27} +1.00000i q^{28} -6.00000 q^{29} -4.00000i q^{31} -1.00000i q^{32} +6.00000i q^{34} -1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{38} +6.00000i q^{41} +2.00000 q^{42} -8.00000 q^{43} +12.0000i q^{47} -2.00000 q^{48} -1.00000 q^{49} -5.00000i q^{50} +12.0000 q^{51} +6.00000 q^{53} -4.00000i q^{54} +1.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +6.00000i q^{59} +8.00000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{68} +1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} -10.0000 q^{75} -2.00000i q^{76} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} +8.00000i q^{86} +12.0000 q^{87} +6.00000i q^{89} +8.00000i q^{93} +12.0000 q^{94} +2.00000i q^{96} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 4 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{12} - 2 q^{14} + 2 q^{16} - 12 q^{17} + 10 q^{25} + 8 q^{27} - 12 q^{29} - 2 q^{36} + 4 q^{38} + 4 q^{42} - 16 q^{43} - 4 q^{48} - 2 q^{49} + 24 q^{51} + 12 q^{53} + 2 q^{56} + 16 q^{61} - 8 q^{62} - 2 q^{64} + 12 q^{68} - 4 q^{74} - 20 q^{75} + 16 q^{79} - 22 q^{81} + 12 q^{82} + 24 q^{87} + 24 q^{94}+O(q^{100})$$ 2 * q - 4 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^12 - 2 * q^14 + 2 * q^16 - 12 * q^17 + 10 * q^25 + 8 * q^27 - 12 * q^29 - 2 * q^36 + 4 * q^38 + 4 * q^42 - 16 * q^43 - 4 * q^48 - 2 * q^49 + 24 * q^51 + 12 * q^53 + 2 * q^56 + 16 * q^61 - 8 * q^62 - 2 * q^64 + 12 * q^68 - 4 * q^74 - 20 * q^75 + 16 * q^79 - 22 * q^81 + 12 * q^82 + 24 * q^87 + 24 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$2199$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 2.00000i 0.816497i
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ − 2.00000i − 0.408248i
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 1.00000i 0.188982i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ − 4.00000i − 0.718421i −0.933257 0.359211i $$-0.883046\pi$$
0.933257 0.359211i $$-0.116954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 6.00000i 1.02899i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000i 0.937043i 0.883452 + 0.468521i $$0.155213\pi$$
−0.883452 + 0.468521i $$0.844787\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ −1.00000 −0.142857
$$50$$ − 5.00000i − 0.707107i
$$51$$ 12.0000 1.68034
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ − 4.00000i − 0.544331i
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ − 4.00000i − 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ 6.00000i 0.781133i 0.920575 + 0.390567i $$0.127721\pi$$
−0.920575 + 0.390567i $$0.872279\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ − 1.00000i − 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −10.0000 −1.15470
$$76$$ − 2.00000i − 0.229416i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 6.00000 0.662589
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 0 0
$$86$$ 8.00000i 0.862662i
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ 6.00000i 0.635999i 0.948091 + 0.317999i $$0.103011\pi$$
−0.948091 + 0.317999i $$0.896989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 2.00000i 0.204124i
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ − 12.0000i − 1.18818i
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ − 6.00000i − 0.582772i
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 2.00000i 0.191565i 0.995402 + 0.0957826i $$0.0305354\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ 0 0
$$111$$ 4.00000i 0.379663i
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 6.00000i 0.550019i
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ − 8.00000i − 0.724286i
$$123$$ − 12.0000i − 1.08200i
$$124$$ 4.00000i 0.359211i
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 16.0000 1.40872
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ − 6.00000i − 0.514496i
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ − 24.0000i − 2.02116i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 2.00000 0.164957
$$148$$ 2.00000i 0.164399i
$$149$$ − 18.0000i − 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 10.0000i 0.816497i
$$151$$ − 8.00000i − 0.651031i −0.945537 0.325515i $$-0.894462\pi$$
0.945537 0.325515i $$-0.105538\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 11.0000i 0.864242i
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ − 6.00000i − 0.468521i
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 2.00000i 0.152944i
$$172$$ 8.00000 0.609994
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ − 12.0000i − 0.909718i
$$175$$ − 5.00000i − 0.377964i
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ 6.00000 0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ −16.0000 −1.18275
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ − 12.0000i − 0.875190i
$$189$$ − 4.00000i − 0.290957i
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 2.00000 0.144338
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 5.00000i 0.353553i
$$201$$ 8.00000i 0.564276i
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ − 4.00000i − 0.278693i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ − 12.0000i − 0.820303i
$$215$$ 0 0
$$216$$ 4.00000i 0.272166i
$$217$$ −4.00000 −0.271538
$$218$$ 2.00000 0.135457
$$219$$ 4.00000i 0.270295i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000 0.268462
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 5.00000 0.333333
$$226$$ − 6.00000i − 0.399114i
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 4.00000i 0.264327i 0.991228 + 0.132164i $$0.0421925\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ − 6.00000i − 0.390567i
$$237$$ −16.0000 −1.03931
$$238$$ 6.00000 0.388922
$$239$$ 24.0000i 1.55243i 0.630468 + 0.776215i $$0.282863\pi$$
−0.630468 + 0.776215i $$0.717137\pi$$
$$240$$ 0 0
$$241$$ 10.0000i 0.644157i 0.946713 + 0.322078i $$0.104381\pi$$
−0.946713 + 0.322078i $$0.895619\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 10.0000 0.641500
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 12.0000i 0.760469i
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ 0 0
$$254$$ − 16.0000i − 1.00393i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ − 16.0000i − 0.996116i
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ − 18.0000i − 1.11204i
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ − 2.00000i − 0.122628i
$$267$$ − 12.0000i − 0.734388i
$$268$$ 4.00000i 0.244339i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 16.0000i 0.971931i 0.873978 + 0.485965i $$0.161532\pi$$
−0.873978 + 0.485965i $$0.838468\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ − 14.0000i − 0.839664i
$$279$$ − 4.00000i − 0.239474i
$$280$$ 0 0
$$281$$ 6.00000i 0.357930i 0.983855 + 0.178965i $$0.0572749\pi$$
−0.983855 + 0.178965i $$0.942725\pi$$
$$282$$ −24.0000 −1.42918
$$283$$ 22.0000 1.30776 0.653882 0.756596i $$-0.273139\pi$$
0.653882 + 0.756596i $$0.273139\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 20.0000i 1.17242i
$$292$$ 2.00000i 0.117041i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ − 2.00000i − 0.116642i
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 10.0000 0.577350
$$301$$ 8.00000i 0.461112i
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 2.00000i 0.114708i
$$305$$ 0 0
$$306$$ 6.00000i 0.342997i
$$307$$ − 2.00000i − 0.114146i −0.998370 0.0570730i $$-0.981823\pi$$
0.998370 0.0570730i $$-0.0181768\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 4.00000i 0.225733i
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 12.0000i 0.672927i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ − 12.0000i − 0.667698i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ − 4.00000i − 0.221201i
$$328$$ −6.00000 −0.331295
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 8.00000i 0.439720i 0.975531 + 0.219860i $$0.0705600\pi$$
−0.975531 + 0.219860i $$0.929440\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 2.00000i 0.109109i
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000 0.108148
$$343$$ 1.00000i 0.0539949i
$$344$$ − 8.00000i − 0.431331i
$$345$$ 0 0
$$346$$ − 12.0000i − 0.645124i
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 28.0000i 1.49881i 0.662114 + 0.749403i $$0.269659\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ − 6.00000i − 0.317999i
$$357$$ − 12.0000i − 0.635107i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 20.0000i 1.05118i
$$363$$ −22.0000 −1.15470
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 16.0000i 0.836333i
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 6.00000i 0.312348i
$$370$$ 0 0
$$371$$ − 6.00000i − 0.311504i
$$372$$ − 8.00000i − 0.414781i
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ −4.00000 −0.205738
$$379$$ − 16.0000i − 0.821865i −0.911666 0.410932i $$-0.865203\pi$$
0.911666 0.410932i $$-0.134797\pi$$
$$380$$ 0 0
$$381$$ −32.0000 −1.63941
$$382$$ − 24.0000i − 1.22795i
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ − 2.00000i − 0.102062i
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −8.00000 −0.406663
$$388$$ 10.0000i 0.507673i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 1.00000i − 0.0505076i
$$393$$ −36.0000 −1.81596
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 20.0000i − 1.00377i −0.864934 0.501886i $$-0.832640\pi$$
0.864934 0.501886i $$-0.167360\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ −4.00000 −0.200250
$$400$$ 5.00000 0.250000
$$401$$ 18.0000i 0.898877i 0.893311 + 0.449439i $$0.148376\pi$$
−0.893311 + 0.449439i $$0.851624\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 0 0
$$408$$ 12.0000i 0.594089i
$$409$$ 14.0000i 0.692255i 0.938187 + 0.346128i $$0.112504\pi$$
−0.938187 + 0.346128i $$0.887496\pi$$
$$410$$ 0 0
$$411$$ 36.0000i 1.77575i
$$412$$ −4.00000 −0.197066
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −28.0000 −1.37117
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ − 10.0000i − 0.487370i −0.969854 0.243685i $$-0.921644\pi$$
0.969854 0.243685i $$-0.0783563\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000i 0.291386i
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000i 1.15604i 0.816023 + 0.578020i $$0.196174\pi$$
−0.816023 + 0.578020i $$0.803826\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 4.00000i 0.192006i
$$435$$ 0 0
$$436$$ − 2.00000i − 0.0957826i
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ − 4.00000i − 0.189832i
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 36.0000i 1.70274i
$$448$$ 1.00000i 0.0472456i
$$449$$ − 18.0000i − 0.849473i −0.905317 0.424736i $$-0.860367\pi$$
0.905317 0.424736i $$-0.139633\pi$$
$$450$$ − 5.00000i − 0.235702i
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 16.0000i 0.751746i
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 4.00000 0.186908
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ 12.0000i 0.558896i 0.960161 + 0.279448i $$0.0901514\pi$$
−0.960161 + 0.279448i $$0.909849\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ − 6.00000i − 0.277945i
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ −6.00000 −0.276172
$$473$$ 0 0
$$474$$ 16.0000i 0.734904i
$$475$$ 10.0000i 0.458831i
$$476$$ − 6.00000i − 0.275010i
$$477$$ 6.00000 0.274721
$$478$$ 24.0000 1.09773
$$479$$ 36.0000i 1.64488i 0.568850 + 0.822441i $$0.307388\pi$$
−0.568850 + 0.822441i $$0.692612\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ − 10.0000i − 0.453609i
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ − 32.0000i − 1.44709i
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 12.0000i 0.541002i
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 4.00000i − 0.179605i
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ − 24.0000i − 1.07224i
$$502$$ − 18.0000i − 0.803379i
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ 36.0000i 1.59567i 0.602875 + 0.797836i $$0.294022\pi$$
−0.602875 + 0.797836i $$0.705978\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 8.00000i 0.353209i
$$514$$ 18.0000i 0.793946i
$$515$$ 0 0
$$516$$ −16.0000 −0.704361
$$517$$ 0 0
$$518$$ 2.00000i 0.0878750i
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ −18.0000 −0.786334
$$525$$ 10.0000i 0.436436i
$$526$$ 0 0
$$527$$ 24.0000i 1.04546i
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 6.00000i 0.260378i
$$532$$ −2.00000 −0.0867110
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −24.0000 −1.03568
$$538$$ 12.0000i 0.517357i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 38.0000i − 1.63375i −0.576816 0.816874i $$-0.695705\pi$$
0.576816 0.816874i $$-0.304295\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 40.0000 1.71656
$$544$$ 6.00000i 0.257248i
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ − 12.0000i − 0.511217i
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ − 10.0000i − 0.424859i
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −30.0000 −1.26435 −0.632175 0.774826i $$-0.717837\pi$$
−0.632175 + 0.774826i $$0.717837\pi$$
$$564$$ 24.0000i 1.01058i
$$565$$ 0 0
$$566$$ − 22.0000i − 0.924729i
$$567$$ 11.0000i 0.461957i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −2.00523
$$574$$ − 6.00000i − 0.250435i
$$575$$ 0 0
$$576$$ −1.00000 −0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 28.0000i 1.16364i
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ 20.0000 0.829027
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ − 42.0000i − 1.73353i −0.498721 0.866763i $$-0.666197\pi$$
0.498721 0.866763i $$-0.333803\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 36.0000i 1.48084i
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000i 0.737309i
$$597$$ 40.0000 1.63709
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ − 10.0000i − 0.408248i
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 8.00000 0.326056
$$603$$ − 4.00000i − 0.162893i
$$604$$ 8.00000i 0.325515i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ − 12.0000i − 0.486265i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ − 26.0000i − 1.04503i −0.852631 0.522514i $$-0.824994\pi$$
0.852631 0.522514i $$-0.175006\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 10.0000i 0.399680i
$$627$$ 0 0
$$628$$ 4.00000 0.159617
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ 16.0000i 0.636950i 0.947931 + 0.318475i $$0.103171\pi$$
−0.947931 + 0.318475i $$0.896829\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 8.00000 0.317971
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ − 11.0000i − 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ − 16.0000i − 0.626608i
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 0 0
$$656$$ 6.00000i 0.234261i
$$657$$ − 2.00000i − 0.0780274i
$$658$$ − 12.0000i − 0.467809i
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 40.0000i 1.55582i 0.628376 + 0.777910i $$0.283720\pi$$
−0.628376 + 0.777910i $$0.716280\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ − 12.0000i − 0.464294i
$$669$$ − 16.0000i − 0.618596i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000 0.0771517
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 14.0000i 0.539260i
$$675$$ 20.0000 0.769800
$$676$$ 0 0
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ − 36.0000i − 1.37952i
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ − 2.00000i − 0.0764719i
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ − 8.00000i − 0.305219i
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 46.0000i − 1.74992i −0.484193 0.874961i $$-0.660887\pi$$
0.484193 0.874961i $$-0.339113\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 24.0000i 0.911028i
$$695$$ 0 0
$$696$$ 12.0000i 0.454859i
$$697$$ − 36.0000i − 1.36360i
$$698$$ 28.0000 1.05982
$$699$$ −12.0000 −0.453882
$$700$$ 5.00000i 0.188982i
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ 46.0000i 1.72757i 0.503864 + 0.863783i $$0.331911\pi$$
−0.503864 + 0.863783i $$0.668089\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 48.0000i − 1.79259i
$$718$$ 24.0000 0.895672
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ − 4.00000i − 0.148968i
$$722$$ − 15.0000i − 0.558242i
$$723$$ − 20.0000i − 0.743808i
$$724$$ 20.0000 0.743294
$$725$$ −30.0000 −1.11417
$$726$$ 22.0000i 0.816497i
$$727$$ −44.0000 −1.63187 −0.815935 0.578144i $$-0.803777\pi$$
−0.815935 + 0.578144i $$0.803777\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 16.0000 0.591377
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ − 8.00000i − 0.295285i
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ 16.0000i 0.588570i 0.955718 + 0.294285i $$0.0950814\pi$$
−0.955718 + 0.294285i $$0.904919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −6.00000 −0.220267
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ − 14.0000i − 0.512576i
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ − 12.0000i − 0.438470i
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 4.00000i 0.145479i
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000i 0.652499i 0.945284 + 0.326250i $$0.105785\pi$$
−0.945284 + 0.326250i $$0.894215\pi$$
$$762$$ 32.0000i 1.15924i
$$763$$ 2.00000 0.0724049
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ 0 0
$$768$$ −2.00000 −0.0721688
$$769$$ 14.0000i 0.504853i 0.967616 + 0.252426i $$0.0812286\pi$$
−0.967616 + 0.252426i $$0.918771\pi$$
$$770$$ 0 0
$$771$$ 36.0000 1.29651
$$772$$ 14.0000i 0.503871i
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 8.00000i 0.287554i
$$775$$ − 20.0000i − 0.718421i
$$776$$ 10.0000 0.358979
$$777$$ 4.00000 0.143499
$$778$$ 18.0000i 0.645331i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 36.0000i 1.28408i
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 6.00000i − 0.213335i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ − 72.0000i − 2.54718i
$$800$$ − 5.00000i − 0.176777i
$$801$$ 6.00000i 0.212000i
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ − 8.00000i − 0.282138i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 24.0000 0.844840
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 2.00000i 0.0702295i 0.999383 + 0.0351147i $$0.0111797\pi$$
−0.999383 + 0.0351147i $$0.988820\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ − 16.0000i − 0.559769i
$$818$$ 14.0000 0.489499
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ 36.0000 1.25564
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 0 0
$$826$$ − 6.00000i − 0.208767i
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ −56.0000 −1.94496 −0.972480 0.232986i $$-0.925151\pi$$
−0.972480 + 0.232986i $$0.925151\pi$$
$$830$$ 0 0
$$831$$ −20.0000 −0.693792
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 28.0000i 0.969561i
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ − 6.00000i − 0.207267i
$$839$$ − 12.0000i − 0.414286i −0.978311 0.207143i $$-0.933583\pi$$
0.978311 0.207143i $$-0.0664165\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −10.0000 −0.344623
$$843$$ − 12.0000i − 0.413302i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ − 11.0000i − 0.377964i
$$848$$ 6.00000 0.206041
$$849$$ −44.0000 −1.51008
$$850$$ 30.0000i 1.02899i
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 44.0000i − 1.50653i −0.657716 0.753266i $$-0.728477\pi$$
0.657716 0.753266i $$-0.271523\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 12.0000i 0.410152i
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ 24.0000 0.817443
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ − 4.00000i − 0.136083i
$$865$$ 0 0
$$866$$ − 34.0000i − 1.15537i
$$867$$ −38.0000 −1.29055
$$868$$ 4.00000 0.135769
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −2.00000 −0.0677285
$$873$$ − 10.0000i − 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ − 4.00000i − 0.135147i
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 48.0000i 1.61900i
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000i 0.403148i
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ − 16.0000i − 0.536623i
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 8.00000i − 0.267860i
$$893$$ −24.0000 −0.803129
$$894$$ 36.0000 1.20402
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 24.0000i 0.800445i
$$900$$ −5.00000 −0.166667
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ 6.00000i 0.199557i
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ − 18.0000i − 0.597351i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ − 4.00000i − 0.132164i
$$917$$ − 18.0000i − 0.594412i
$$918$$ 24.0000i 0.792118i
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 4.00000i 0.131804i
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 10.0000i − 0.328798i
$$926$$ −32.0000 −1.05159
$$927$$ 4.00000 0.131377
$$928$$ 6.00000i 0.196960i
$$929$$ 6.00000i 0.196854i 0.995144 + 0.0984268i $$0.0313810\pi$$
−0.995144 + 0.0984268i $$0.968619\pi$$
$$930$$ 0 0
$$931$$ − 2.00000i − 0.0655474i
$$932$$ −6.00000 −0.196537
$$933$$ −48.0000 −1.57145
$$934$$ − 6.00000i − 0.196326i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 4.00000i 0.130605i
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ − 24.0000i − 0.782378i −0.920310 0.391189i $$-0.872064\pi$$
0.920310 0.391189i $$-0.127936\pi$$
$$942$$ − 8.00000i − 0.260654i
$$943$$ 0 0
$$944$$ 6.00000i 0.195283i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 0 0
$$950$$ 10.0000 0.324443
$$951$$ − 12.0000i − 0.389127i
$$952$$ −6.00000 −0.194461
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ 0 0
$$956$$ − 24.0000i − 0.776215i
$$957$$ 0 0
$$958$$ 36.0000 1.16311
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ − 10.0000i − 0.322078i
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 24.0000i 0.770991i
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ − 14.0000i − 0.448819i
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ − 6.00000i − 0.191957i −0.995383 0.0959785i $$-0.969402\pi$$
0.995383 0.0959785i $$-0.0305980\pi$$
$$978$$ −32.0000 −1.02325
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ − 12.0000i − 0.382935i
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ 12.0000 0.382546
$$985$$ 0 0
$$986$$ − 36.0000i − 1.14647i
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ − 16.0000i − 0.507745i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ − 12.0000i − 0.380235i
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ − 8.00000i − 0.253109i
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 2366.2.d.b.337.1 2
13.5 odd 4 14.2.a.a.1.1 1
13.8 odd 4 2366.2.a.j.1.1 1
13.12 even 2 inner 2366.2.d.b.337.2 2
39.5 even 4 126.2.a.b.1.1 1
52.31 even 4 112.2.a.c.1.1 1
65.18 even 4 350.2.c.d.99.2 2
65.44 odd 4 350.2.a.f.1.1 1
65.57 even 4 350.2.c.d.99.1 2
91.5 even 12 98.2.c.a.67.1 2
91.18 odd 12 98.2.c.b.79.1 2
91.31 even 12 98.2.c.a.79.1 2
91.44 odd 12 98.2.c.b.67.1 2
91.83 even 4 98.2.a.a.1.1 1
104.5 odd 4 448.2.a.g.1.1 1
104.83 even 4 448.2.a.a.1.1 1
117.5 even 12 1134.2.f.f.379.1 2
117.31 odd 12 1134.2.f.l.379.1 2
117.70 odd 12 1134.2.f.l.757.1 2
117.83 even 12 1134.2.f.f.757.1 2
143.109 even 4 1694.2.a.e.1.1 1
156.83 odd 4 1008.2.a.h.1.1 1
195.44 even 4 3150.2.a.i.1.1 1
195.83 odd 4 3150.2.g.j.2899.1 2
195.122 odd 4 3150.2.g.j.2899.2 2
208.5 odd 4 1792.2.b.c.897.2 2
208.83 even 4 1792.2.b.g.897.2 2
208.109 odd 4 1792.2.b.c.897.1 2
208.187 even 4 1792.2.b.g.897.1 2
221.135 odd 4 4046.2.a.f.1.1 1
247.18 even 4 5054.2.a.c.1.1 1
260.83 odd 4 2800.2.g.h.449.2 2
260.187 odd 4 2800.2.g.h.449.1 2
260.239 even 4 2800.2.a.g.1.1 1
273.5 odd 12 882.2.g.d.361.1 2
273.44 even 12 882.2.g.c.361.1 2
273.83 odd 4 882.2.a.i.1.1 1
273.122 odd 12 882.2.g.d.667.1 2
273.200 even 12 882.2.g.c.667.1 2
299.252 even 4 7406.2.a.a.1.1 1
312.5 even 4 4032.2.a.w.1.1 1
312.83 odd 4 4032.2.a.r.1.1 1
364.31 odd 12 784.2.i.i.177.1 2
364.83 odd 4 784.2.a.b.1.1 1
364.135 even 12 784.2.i.c.753.1 2
364.187 odd 12 784.2.i.i.753.1 2
364.291 even 12 784.2.i.c.177.1 2
455.83 odd 4 2450.2.c.c.99.2 2
455.174 even 4 2450.2.a.t.1.1 1
455.447 odd 4 2450.2.c.c.99.1 2
728.83 odd 4 3136.2.a.z.1.1 1
728.629 even 4 3136.2.a.e.1.1 1
1092.83 even 4 7056.2.a.bd.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 13.5 odd 4
98.2.a.a.1.1 1 91.83 even 4
98.2.c.a.67.1 2 91.5 even 12
98.2.c.a.79.1 2 91.31 even 12
98.2.c.b.67.1 2 91.44 odd 12
98.2.c.b.79.1 2 91.18 odd 12
112.2.a.c.1.1 1 52.31 even 4
126.2.a.b.1.1 1 39.5 even 4
350.2.a.f.1.1 1 65.44 odd 4
350.2.c.d.99.1 2 65.57 even 4
350.2.c.d.99.2 2 65.18 even 4
448.2.a.a.1.1 1 104.83 even 4
448.2.a.g.1.1 1 104.5 odd 4
784.2.a.b.1.1 1 364.83 odd 4
784.2.i.c.177.1 2 364.291 even 12
784.2.i.c.753.1 2 364.135 even 12
784.2.i.i.177.1 2 364.31 odd 12
784.2.i.i.753.1 2 364.187 odd 12
882.2.a.i.1.1 1 273.83 odd 4
882.2.g.c.361.1 2 273.44 even 12
882.2.g.c.667.1 2 273.200 even 12
882.2.g.d.361.1 2 273.5 odd 12
882.2.g.d.667.1 2 273.122 odd 12
1008.2.a.h.1.1 1 156.83 odd 4
1134.2.f.f.379.1 2 117.5 even 12
1134.2.f.f.757.1 2 117.83 even 12
1134.2.f.l.379.1 2 117.31 odd 12
1134.2.f.l.757.1 2 117.70 odd 12
1694.2.a.e.1.1 1 143.109 even 4
1792.2.b.c.897.1 2 208.109 odd 4
1792.2.b.c.897.2 2 208.5 odd 4
1792.2.b.g.897.1 2 208.187 even 4
1792.2.b.g.897.2 2 208.83 even 4
2366.2.a.j.1.1 1 13.8 odd 4
2366.2.d.b.337.1 2 1.1 even 1 trivial
2366.2.d.b.337.2 2 13.12 even 2 inner
2450.2.a.t.1.1 1 455.174 even 4
2450.2.c.c.99.1 2 455.447 odd 4
2450.2.c.c.99.2 2 455.83 odd 4
2800.2.a.g.1.1 1 260.239 even 4
2800.2.g.h.449.1 2 260.187 odd 4
2800.2.g.h.449.2 2 260.83 odd 4
3136.2.a.e.1.1 1 728.629 even 4
3136.2.a.z.1.1 1 728.83 odd 4
3150.2.a.i.1.1 1 195.44 even 4
3150.2.g.j.2899.1 2 195.83 odd 4
3150.2.g.j.2899.2 2 195.122 odd 4
4032.2.a.r.1.1 1 312.83 odd 4
4032.2.a.w.1.1 1 312.5 even 4
4046.2.a.f.1.1 1 221.135 odd 4
5054.2.a.c.1.1 1 247.18 even 4
7056.2.a.bd.1.1 1 1092.83 even 4
7406.2.a.a.1.1 1 299.252 even 4