# Properties

 Label 2816.2.g.a.1407.3 Level $2816$ Weight $2$ Character 2816.1407 Analytic conductor $22.486$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2816 = 2^{8} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2816.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: $$22.4858732092$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 176) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 1407.3 Root $$1.65831 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2816.1407 Dual form 2816.2.g.a.1407.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.31662 q^{3} -3.00000i q^{5} +8.00000 q^{9} +O(q^{10})$$ $$q+3.31662 q^{3} -3.00000i q^{5} +8.00000 q^{9} +3.31662 q^{11} -9.94987i q^{15} +3.31662i q^{23} -4.00000 q^{25} +16.5831 q^{27} +9.94987i q^{31} +11.0000 q^{33} -7.00000i q^{37} -24.0000i q^{45} -6.63325i q^{47} -7.00000 q^{49} -6.00000i q^{53} -9.94987i q^{55} -3.31662 q^{59} -9.94987 q^{67} +11.0000i q^{69} +16.5831i q^{71} -13.2665 q^{75} +31.0000 q^{81} +9.00000 q^{89} +33.0000i q^{93} -17.0000 q^{97} +26.5330 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{9}+O(q^{10})$$ 4 * q + 32 * q^9 $$4 q + 32 q^{9} - 16 q^{25} + 44 q^{33} - 28 q^{49} + 124 q^{81} + 36 q^{89} - 68 q^{97}+O(q^{100})$$ 4 * q + 32 * q^9 - 16 * q^25 + 44 * q^33 - 28 * q^49 + 124 * q^81 + 36 * q^89 - 68 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$1541$$ $$2047$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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
$$3$$ 3.31662 1.91485 0.957427 0.288675i $$-0.0932147\pi$$
0.957427 + 0.288675i $$0.0932147\pi$$
$$4$$ 0 0
$$5$$ − 3.00000i − 1.34164i −0.741620 0.670820i $$-0.765942\pi$$
0.741620 0.670820i $$-0.234058\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 8.00000 2.66667
$$10$$ 0 0
$$11$$ 3.31662 1.00000
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ − 9.94987i − 2.56905i
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.31662i 0.691564i 0.938315 + 0.345782i $$0.112386\pi$$
−0.938315 + 0.345782i $$0.887614\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 16.5831 3.19142
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 9.94987i 1.78705i 0.449013 + 0.893525i $$0.351776\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ 11.0000 1.91485
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 7.00000i − 1.15079i −0.817875 0.575396i $$-0.804848\pi$$
0.817875 0.575396i $$-0.195152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ − 24.0000i − 3.57771i
$$46$$ 0 0
$$47$$ − 6.63325i − 0.967559i −0.875190 0.483779i $$-0.839264\pi$$
0.875190 0.483779i $$-0.160736\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ − 9.94987i − 1.34164i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −3.31662 −0.431788 −0.215894 0.976417i $$-0.569267\pi$$
−0.215894 + 0.976417i $$0.569267\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.94987 −1.21557 −0.607785 0.794101i $$-0.707942\pi$$
−0.607785 + 0.794101i $$0.707942\pi$$
$$68$$ 0 0
$$69$$ 11.0000i 1.32424i
$$70$$ 0 0
$$71$$ 16.5831i 1.96805i 0.178017 + 0.984027i $$0.443032\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ −13.2665 −1.53188
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 31.0000 3.44444
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.00000 0.953998 0.476999 0.878904i $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 33.0000i 3.42194i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.0000 −1.72609 −0.863044 0.505128i $$-0.831445\pi$$
−0.863044 + 0.505128i $$0.831445\pi$$
$$98$$ 0 0
$$99$$ 26.5330 2.66667
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 19.8997i − 1.96078i −0.197066 0.980390i $$-0.563141\pi$$
0.197066 0.980390i $$-0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ − 23.2164i − 2.20360i
$$112$$ 0 0
$$113$$ −21.0000 −1.97551 −0.987757 0.156001i $$-0.950140\pi$$
−0.987757 + 0.156001i $$0.950140\pi$$
$$114$$ 0 0
$$115$$ 9.94987 0.927831
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 3.00000i − 0.268328i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ − 49.7494i − 4.28174i
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ − 22.0000i − 1.85273i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −23.2164 −1.91485
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 29.8496 2.39758
$$156$$ 0 0
$$157$$ 23.0000i 1.83560i 0.397043 + 0.917800i $$0.370036\pi$$
−0.397043 + 0.917800i $$0.629964\pi$$
$$158$$ 0 0
$$159$$ − 19.8997i − 1.57815i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 19.8997 1.55867 0.779334 0.626608i $$-0.215557\pi$$
0.779334 + 0.626608i $$0.215557\pi$$
$$164$$ 0 0
$$165$$ − 33.0000i − 2.56905i
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −11.0000 −0.826811
$$178$$ 0 0
$$179$$ 16.5831 1.23948 0.619740 0.784807i $$-0.287238\pi$$
0.619740 + 0.784807i $$0.287238\pi$$
$$180$$ 0 0
$$181$$ 25.0000i 1.85824i 0.369784 + 0.929118i $$0.379432\pi$$
−0.369784 + 0.929118i $$0.620568\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −21.0000 −1.54395
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 23.2164i 1.67988i 0.542681 + 0.839939i $$0.317409\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ − 19.8997i − 1.41066i −0.708881 0.705328i $$-0.750800\pi$$
0.708881 0.705328i $$-0.249200\pi$$
$$200$$ 0 0
$$201$$ −33.0000 −2.32764
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 26.5330i 1.84417i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 55.0000i 3.76854i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 29.8496i − 1.99888i −0.0334825 0.999439i $$-0.510660\pi$$
0.0334825 0.999439i $$-0.489340\pi$$
$$224$$ 0 0
$$225$$ −32.0000 −2.13333
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 5.00000i 0.330409i 0.986259 + 0.165205i $$0.0528285\pi$$
−0.986259 + 0.165205i $$0.947172\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ −19.8997 −1.29812
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 53.0660 3.40419
$$244$$ 0 0
$$245$$ 21.0000i 1.34164i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.5831 −1.04672 −0.523359 0.852112i $$-0.675321\pi$$
−0.523359 + 0.852112i $$0.675321\pi$$
$$252$$ 0 0
$$253$$ 11.0000i 0.691564i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ 0 0
$$267$$ 29.8496 1.82677
$$268$$ 0 0
$$269$$ 30.0000i 1.82913i 0.404436 + 0.914566i $$0.367468\pi$$
−0.404436 + 0.914566i $$0.632532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −13.2665 −0.800000
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 79.5990i 4.76547i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −56.3826 −3.30521
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 9.94987i 0.579304i
$$296$$ 0 0
$$297$$ 55.0000 3.19142
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ − 66.0000i − 3.75461i
$$310$$ 0 0
$$311$$ 33.1662i 1.88069i 0.340229 + 0.940343i $$0.389495\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.0000i 1.51647i 0.651981 + 0.758236i $$0.273938\pi$$
−0.651981 + 0.758236i $$0.726062\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 9.94987 0.546895 0.273447 0.961887i $$-0.411836\pi$$
0.273447 + 0.961887i $$0.411836\pi$$
$$332$$ 0 0
$$333$$ − 56.0000i − 3.06878i
$$334$$ 0 0
$$335$$ 29.8496i 1.63086i
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ −69.6491 −3.78282
$$340$$ 0 0
$$341$$ 33.0000i 1.78705i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 33.0000 1.77666
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −9.00000 −0.479022 −0.239511 0.970894i $$-0.576987\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$354$$ 0 0
$$355$$ 49.7494 2.64042
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 36.4829 1.91485
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.94987i 0.519379i 0.965692 + 0.259690i $$0.0836203\pi$$
−0.965692 + 0.259690i $$0.916380\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ − 9.94987i − 0.513809i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −29.8496 −1.53327 −0.766636 0.642082i $$-0.778071\pi$$
−0.766636 + 0.642082i $$0.778071\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 3.31662i − 0.169472i −0.996403 0.0847358i $$-0.972995\pi$$
0.996403 0.0847358i $$-0.0270046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 15.0000i − 0.760530i −0.924878 0.380265i $$-0.875833\pi$$
0.924878 0.380265i $$-0.124167\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 93.0000i − 4.62121i
$$406$$ 0 0
$$407$$ − 23.2164i − 1.15079i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −9.94987 −0.490791
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −33.1662 −1.62028 −0.810139 0.586238i $$-0.800608\pi$$
−0.810139 + 0.586238i $$0.800608\pi$$
$$420$$ 0 0
$$421$$ 10.0000i 0.487370i 0.969854 + 0.243685i $$0.0783563\pi$$
−0.969854 + 0.243685i $$0.921644\pi$$
$$422$$ 0 0
$$423$$ − 53.0660i − 2.58016i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −29.0000 −1.39365 −0.696826 0.717241i $$-0.745405\pi$$
−0.696826 + 0.717241i $$0.745405\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ −56.0000 −2.66667
$$442$$ 0 0
$$443$$ 36.4829 1.73335 0.866677 0.498870i $$-0.166252\pi$$
0.866677 + 0.498870i $$0.166252\pi$$
$$444$$ 0 0
$$445$$ − 27.0000i − 1.27992i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 39.0000 1.84052 0.920262 0.391303i $$-0.127976\pi$$
0.920262 + 0.391303i $$0.127976\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ − 29.8496i − 1.38723i −0.720346 0.693615i $$-0.756017\pi$$
0.720346 0.693615i $$-0.243983\pi$$
$$464$$ 0 0
$$465$$ 99.0000 4.59102
$$466$$ 0 0
$$467$$ 43.1161 1.99518 0.997588 0.0694117i $$-0.0221122\pi$$
0.997588 + 0.0694117i $$0.0221122\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 76.2824i 3.51491i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 48.0000i − 2.19777i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 51.0000i 2.31579i
$$486$$ 0 0
$$487$$ − 9.94987i − 0.450872i −0.974258 0.225436i $$-0.927619\pi$$
0.974258 0.225436i $$-0.0723806\pi$$
$$488$$ 0 0
$$489$$ 66.0000 2.98462
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ − 79.5990i − 3.57771i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 19.8997 0.890835 0.445418 0.895323i $$-0.353055\pi$$
0.445418 + 0.895323i $$0.353055\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −43.1161 −1.91485
$$508$$ 0 0
$$509$$ − 45.0000i − 1.99459i −0.0735034 0.997295i $$-0.523418\pi$$
0.0735034 0.997295i $$-0.476582\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −59.6992 −2.63066
$$516$$ 0 0
$$517$$ − 22.0000i − 0.967559i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 12.0000 0.521739
$$530$$ 0 0
$$531$$ −26.5330 −1.15143
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 55.0000 2.37343
$$538$$ 0 0
$$539$$ −23.2164 −1.00000
$$540$$ 0 0
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ 82.9156i 3.55825i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −69.6491 −2.95644
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 63.0000i 2.65043i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 77.0000i 3.21672i
$$574$$ 0 0
$$575$$ − 13.2665i − 0.553251i
$$576$$ 0 0
$$577$$ 47.0000 1.95664 0.978318 0.207109i $$-0.0664056\pi$$
0.978318 + 0.207109i $$0.0664056\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 19.8997i − 0.824163i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.63325 0.273784 0.136892 0.990586i $$-0.456289\pi$$
0.136892 + 0.990586i $$0.456289\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 66.0000i − 2.70120i
$$598$$ 0 0
$$599$$ 33.1662i 1.35514i 0.735460 + 0.677568i $$0.236966\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −79.5990 −3.24152
$$604$$ 0 0
$$605$$ − 33.0000i − 1.34164i
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 49.7494 1.99960 0.999798 0.0200967i $$-0.00639741\pi$$
0.999798 + 0.0200967i $$0.00639741\pi$$
$$620$$ 0 0
$$621$$ 55.0000i 2.20707i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ − 49.7494i − 1.98049i −0.139333 0.990246i $$-0.544496\pi$$
0.139333 0.990246i $$-0.455504\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 132.665i 5.24815i
$$640$$ 0 0
$$641$$ −45.0000 −1.77739 −0.888697 0.458496i $$-0.848388\pi$$
−0.888697 + 0.458496i $$0.848388\pi$$
$$642$$ 0 0
$$643$$ 29.8496 1.17715 0.588577 0.808441i $$-0.299688\pi$$
0.588577 + 0.808441i $$0.299688\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 43.1161i 1.69507i 0.530740 + 0.847535i $$0.321914\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ −11.0000 −0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 51.0000i 1.99578i 0.0648948 + 0.997892i $$0.479329\pi$$
−0.0648948 + 0.997892i $$0.520671\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 13.0000i 0.505641i 0.967513 + 0.252821i $$0.0813583\pi$$
−0.967513 + 0.252821i $$0.918642\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ − 99.0000i − 3.82756i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −66.3325 −2.55314
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −46.4327 −1.77670 −0.888350 0.459167i $$-0.848148\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 9.00000i 0.343872i
$$686$$ 0 0
$$687$$ 16.5831i 0.632686i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −49.7494 −1.89256 −0.946278 0.323355i $$-0.895189\pi$$
−0.946278 + 0.323355i $$0.895189\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −66.0000 −2.48570
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 19.0000i − 0.713560i −0.934188 0.356780i $$-0.883875\pi$$
0.934188 0.356780i $$-0.116125\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −33.0000 −1.23586
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 16.5831i − 0.618446i −0.950990 0.309223i $$-0.899931\pi$$
0.950990 0.309223i $$-0.100069\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 9.94987i − 0.369020i −0.982831 0.184510i $$-0.940930\pi$$
0.982831 0.184510i $$-0.0590699\pi$$
$$728$$ 0 0
$$729$$ 83.0000 3.07407
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 69.6491i 2.56905i
$$736$$ 0 0
$$737$$ −33.0000 −1.21557
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 49.7494i 1.81538i 0.419641 + 0.907690i $$0.362156\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 0 0
$$753$$ −55.0000 −2.00431
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 38.0000i − 1.38113i −0.723269 0.690567i $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ 0 0
$$759$$ 36.4829i 1.32424i
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 59.6992 2.15002
$$772$$ 0 0
$$773$$ − 54.0000i − 1.94225i −0.238581 0.971123i $$-0.576682\pi$$
0.238581 0.971123i $$-0.423318\pi$$
$$774$$ 0 0
$$775$$ − 39.7995i − 1.42964i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 55.0000i 1.96805i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 69.0000 2.46272
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −59.6992 −2.11731
$$796$$ 0 0
$$797$$ 3.00000i 0.106265i 0.998587 + 0.0531327i $$0.0169206\pi$$
−0.998587 + 0.0531327i $$0.983079\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 72.0000 2.54399
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 99.4987i 3.50252i
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 59.6992i − 2.09117i
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 29.8496i 1.04049i 0.854016 + 0.520246i $$0.174160\pi$$
−0.854016 + 0.520246i $$0.825840\pi$$
$$824$$ 0 0
$$825$$ −44.0000 −1.53188
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ − 29.0000i − 1.00721i −0.863934 0.503606i $$-0.832006\pi$$
0.863934 0.503606i $$-0.167994\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 165.000i 5.70323i
$$838$$ 0 0
$$839$$ − 36.4829i − 1.25953i −0.776786 0.629764i $$-0.783151\pi$$
0.776786 0.629764i $$-0.216849\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 39.0000i 1.34164i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 23.2164 0.795847
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 49.7494 1.69743 0.848713 0.528853i $$-0.177378\pi$$
0.848713 + 0.528853i $$0.177378\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46.4327i 1.58059i 0.612727 + 0.790295i $$0.290072\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 56.3826 1.91485
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −136.000 −4.60290
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −57.0000 −1.92038 −0.960189 0.279350i $$-0.909881\pi$$
−0.960189 + 0.279350i $$0.909881\pi$$
$$882$$ 0 0
$$883$$ 19.8997 0.669680 0.334840 0.942275i $$-0.391318\pi$$
0.334840 + 0.942275i $$0.391318\pi$$
$$884$$ 0 0
$$885$$ 33.0000i 1.10928i
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 102.815 3.44444
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ − 49.7494i − 1.66294i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 75.0000 2.49308
$$906$$ 0 0
$$907$$ 59.6992 1.98228 0.991140 0.132818i $$-0.0424025\pi$$
0.991140 + 0.132818i $$0.0424025\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 6.63325i − 0.219769i −0.993944 0.109885i $$-0.964952\pi$$
0.993944 0.109885i $$-0.0350482\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 28.0000i 0.920634i
$$926$$ 0 0
$$927$$ − 159.198i − 5.22875i
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 110.000i 3.60124i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ −63.0159 −2.05645
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −23.2164 −0.754431 −0.377215 0.926126i $$-0.623118\pi$$
−0.377215 + 0.926126i $$0.623118\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 89.5489i 2.90382i
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 69.6491 2.25379
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −68.0000 −2.19355
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −43.1161 −1.38366 −0.691831 0.722059i $$-0.743196\pi$$
−0.691831 + 0.722059i $$0.743196\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27.0000 0.863807 0.431903 0.901920i $$-0.357842\pi$$
0.431903 + 0.901920i $$0.357842\pi$$
$$978$$ 0 0
$$979$$ 29.8496 0.953998
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 36.4829i − 1.16362i −0.813324 0.581811i $$-0.802344\pi$$
0.813324 0.581811i $$-0.197656\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 59.6992i − 1.89641i −0.317660 0.948205i $$-0.602897\pi$$
0.317660 0.948205i $$-0.397103\pi$$
$$992$$ 0 0
$$993$$ 33.0000 1.04722
$$994$$ 0 0
$$995$$ −59.6992 −1.89259
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 0 0
$$999$$ − 116.082i − 3.67267i
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 2816.2.g.a.1407.3 4
4.3 odd 2 inner 2816.2.g.a.1407.1 4
8.3 odd 2 inner 2816.2.g.a.1407.4 4
8.5 even 2 inner 2816.2.g.a.1407.2 4
11.10 odd 2 CM 2816.2.g.a.1407.3 4
16.3 odd 4 704.2.e.a.703.1 2
16.5 even 4 176.2.e.a.175.1 2
16.11 odd 4 176.2.e.a.175.2 yes 2
16.13 even 4 704.2.e.a.703.2 2
44.43 even 2 inner 2816.2.g.a.1407.1 4
48.5 odd 4 1584.2.o.a.703.1 2
48.11 even 4 1584.2.o.a.703.2 2
88.21 odd 2 inner 2816.2.g.a.1407.2 4
88.43 even 2 inner 2816.2.g.a.1407.4 4
176.21 odd 4 176.2.e.a.175.1 2
176.43 even 4 176.2.e.a.175.2 yes 2
176.109 odd 4 704.2.e.a.703.2 2
176.131 even 4 704.2.e.a.703.1 2
528.197 even 4 1584.2.o.a.703.1 2
528.395 odd 4 1584.2.o.a.703.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.a.175.1 2 16.5 even 4
176.2.e.a.175.1 2 176.21 odd 4
176.2.e.a.175.2 yes 2 16.11 odd 4
176.2.e.a.175.2 yes 2 176.43 even 4
704.2.e.a.703.1 2 16.3 odd 4
704.2.e.a.703.1 2 176.131 even 4
704.2.e.a.703.2 2 16.13 even 4
704.2.e.a.703.2 2 176.109 odd 4
1584.2.o.a.703.1 2 48.5 odd 4
1584.2.o.a.703.1 2 528.197 even 4
1584.2.o.a.703.2 2 48.11 even 4
1584.2.o.a.703.2 2 528.395 odd 4
2816.2.g.a.1407.1 4 4.3 odd 2 inner
2816.2.g.a.1407.1 4 44.43 even 2 inner
2816.2.g.a.1407.2 4 8.5 even 2 inner
2816.2.g.a.1407.2 4 88.21 odd 2 inner
2816.2.g.a.1407.3 4 1.1 even 1 trivial
2816.2.g.a.1407.3 4 11.10 odd 2 CM
2816.2.g.a.1407.4 4 8.3 odd 2 inner
2816.2.g.a.1407.4 4 88.43 even 2 inner