# Properties

 Label 176.2.e.a.175.2 Level $176$ Weight $2$ Character 176.175 Analytic conductor $1.405$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("176.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.e (of order $$2$$, degree $$1$$, 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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 175.2 Root $$0.500000 - 1.65831i$$ of defining polynomial Character $$\chi$$ $$=$$ 176.175 Dual form 176.2.e.a.175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.31662i q^{3} +3.00000 q^{5} -8.00000 q^{9} +O(q^{10})$$ $$q+3.31662i q^{3} +3.00000 q^{5} -8.00000 q^{9} -3.31662i q^{11} +9.94987i q^{15} +3.31662i q^{23} +4.00000 q^{25} -16.5831i q^{27} -9.94987i q^{31} +11.0000 q^{33} +7.00000 q^{37} -24.0000 q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000 q^{53} -9.94987i q^{55} +3.31662i q^{59} -9.94987i q^{67} -11.0000 q^{69} +16.5831i q^{71} +13.2665i q^{75} +31.0000 q^{81} -9.00000 q^{89} +33.0000 q^{93} -17.0000 q^{97} +26.5330i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} - 16 q^{9}+O(q^{10})$$ 2 * q + 6 * q^5 - 16 * q^9 $$2 q + 6 q^{5} - 16 q^{9} + 8 q^{25} + 22 q^{33} + 14 q^{37} - 48 q^{45} - 14 q^{49} + 12 q^{53} - 22 q^{69} + 62 q^{81} - 18 q^{89} + 66 q^{93} - 34 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 - 16 * q^9 + 8 * q^25 + 22 * q^33 + 14 * q^37 - 48 * q^45 - 14 * q^49 + 12 * q^53 - 22 * q^69 + 62 * q^81 - 18 * q^89 + 66 * q^93 - 34 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$133$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 3.31662i 1.91485i 0.288675 + 0.957427i $$0.406785\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 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
$$11$$ − 3.31662i − 1.00000i
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 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
$$27$$ − 16.5831i − 3.19142i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ − 9.94987i − 1.78705i −0.449013 0.893525i $$-0.648224\pi$$
0.449013 0.893525i $$-0.351776\pi$$
$$32$$ 0 0
$$33$$ 11.0000 1.91485
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −24.0000 −3.57771
$$46$$ 0 0
$$47$$ 6.63325i 0.967559i 0.875190 + 0.483779i $$0.160736\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ − 9.94987i − 1.34164i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.31662i 0.431788i 0.976417 + 0.215894i $$0.0692665\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 9.94987i − 1.21557i −0.794101 0.607785i $$-0.792058\pi$$
0.794101 0.607785i $$-0.207942\pi$$
$$68$$ 0 0
$$69$$ −11.0000 −1.32424
$$70$$ 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
$$75$$ 13.2665i 1.53188i
$$76$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 33.0000 3.42194
$$94$$ 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
$$99$$ 26.5330i 2.66667i
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 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
$$115$$ 9.94987i 0.927831i
$$116$$ 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
$$125$$ −3.00000 −0.268328
$$126$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 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 $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −22.0000 −1.85273
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 23.2164i − 1.91485i
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 29.8496i − 2.39758i
$$156$$ 0 0
$$157$$ 23.0000 1.83560 0.917800 0.397043i $$-0.129964\pi$$
0.917800 + 0.397043i $$0.129964\pi$$
$$158$$ 0 0
$$159$$ 19.8997i 1.57815i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 19.8997i 1.55867i 0.626608 + 0.779334i $$0.284443\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 33.0000 2.56905
$$166$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −11.0000 −0.826811
$$178$$ 0 0
$$179$$ 16.5831i 1.23948i 0.784807 + 0.619740i $$0.212762\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ −25.0000 −1.85824 −0.929118 0.369784i $$-0.879432\pi$$
−0.929118 + 0.369784i $$0.879432\pi$$
$$182$$ 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 $$-0.682591\pi$$
0.542681 0.839939i $$-0.317409\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ −55.0000 −3.76854
$$214$$ 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 $$0.489340\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ 0 0
$$225$$ −32.0000 −2.13333
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 19.8997i 1.29812i
$$236$$ 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
$$243$$ 53.0660i 3.40419i
$$244$$ 0 0
$$245$$ −21.0000 −1.34164
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.5831i 1.04672i 0.852112 + 0.523359i $$0.175321\pi$$
−0.852112 + 0.523359i $$0.824679\pi$$
$$252$$ 0 0
$$253$$ 11.0000 0.691564
$$254$$ 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
$$267$$ − 29.8496i − 1.82677i
$$268$$ 0 0
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 13.2665i − 0.800000i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ − 56.3826i − 3.30521i
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 66.0000 3.75461
$$310$$ 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 $$-0.319568\pi$$
0.536972 + 0.843600i $$0.319568\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.0000 1.51647 0.758236 0.651981i $$-0.226062\pi$$
0.758236 + 0.651981i $$0.226062\pi$$
$$318$$ 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
$$331$$ − 9.94987i − 0.546895i −0.961887 0.273447i $$-0.911836\pi$$
0.961887 0.273447i $$-0.0881639\pi$$
$$332$$ 0 0
$$333$$ −56.0000 −3.06878
$$334$$ 0 0
$$335$$ − 29.8496i − 1.63086i
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ − 69.6491i − 3.78282i
$$340$$ 0 0
$$341$$ −33.0000 −1.78705
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −33.0000 −1.77666
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 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
$$355$$ 49.7494i 2.64042i
$$356$$ 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
$$363$$ − 36.4829i − 1.91485i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 9.94987i − 0.519379i −0.965692 0.259690i $$-0.916380\pi$$
0.965692 0.259690i $$-0.0836203\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ − 9.94987i − 0.513809i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 29.8496i 1.53327i 0.642082 + 0.766636i $$0.278071\pi$$
−0.642082 + 0.766636i $$0.721929\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.31662i 0.169472i 0.996403 + 0.0847358i $$0.0270046\pi$$
−0.996403 + 0.0847358i $$0.972995\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 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
$$405$$ 93.0000 4.62121
$$406$$ 0 0
$$407$$ − 23.2164i − 1.15079i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 9.94987i 0.490791i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 33.1662i − 1.62028i −0.586238 0.810139i $$-0.699392\pi$$
0.586238 0.810139i $$-0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 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
$$443$$ − 36.4829i − 1.73335i −0.498870 0.866677i $$-0.666252\pi$$
0.498870 0.866677i $$-0.333748\pi$$
$$444$$ 0 0
$$445$$ −27.0000 −1.27992
$$446$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 29.8496i 1.38723i 0.720346 + 0.693615i $$0.243983\pi$$
−0.720346 + 0.693615i $$0.756017\pi$$
$$464$$ 0 0
$$465$$ 99.0000 4.59102
$$466$$ 0 0
$$467$$ 43.1161i 1.99518i 0.0694117 + 0.997588i $$0.477888\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 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
$$477$$ −48.0000 −2.19777
$$478$$ 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
$$485$$ −51.0000 −2.31579
$$486$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 79.5990i 3.57771i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 19.8997i 0.890835i 0.895323 + 0.445418i $$0.146945\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 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
$$507$$ 43.1161i 1.91485i
$$508$$ 0 0
$$509$$ −45.0000 −1.99459 −0.997295 0.0735034i $$-0.976582\pi$$
−0.997295 + 0.0735034i $$0.976582\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 59.6992i − 2.63066i
$$516$$ 0 0
$$517$$ 22.0000 0.967559
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 15.0000 0.657162 0.328581 0.944476i $$-0.393430\pi$$
0.328581 + 0.944476i $$0.393430\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 12.0000 0.521739
$$530$$ 0 0
$$531$$ − 26.5330i − 1.15143i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −55.0000 −2.37343
$$538$$ 0 0
$$539$$ 23.2164i 1.00000i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ − 82.9156i − 3.55825i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 69.6491i 2.95644i
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ −63.0000 −2.65043
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 77.0000 3.21672
$$574$$ 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
$$587$$ − 6.63325i − 0.273784i −0.990586 0.136892i $$-0.956289\pi$$
0.990586 0.136892i $$-0.0437113\pi$$
$$588$$ 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
$$597$$ 66.0000 2.70120
$$598$$ 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
$$603$$ 79.5990i 3.24152i
$$604$$ 0 0
$$605$$ −33.0000 −1.34164
$$606$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000 1.69086 0.845428 0.534089i $$-0.179345\pi$$
0.845428 + 0.534089i $$0.179345\pi$$
$$618$$ 0 0
$$619$$ − 49.7494i − 1.99960i −0.0200967 0.999798i $$-0.506397\pi$$
0.0200967 0.999798i $$-0.493603\pi$$
$$620$$ 0 0
$$621$$ 55.0000 2.20707
$$622$$ 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
$$643$$ 29.8496i 1.17715i 0.808441 + 0.588577i $$0.200312\pi$$
−0.808441 + 0.588577i $$0.799688\pi$$
$$644$$ 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
$$653$$ 51.0000 1.99578 0.997892 0.0648948i $$-0.0206712\pi$$
0.997892 + 0.0648948i $$0.0206712\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −99.0000 −3.82756
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ − 66.3325i − 2.55314i
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 46.4327i 1.77670i 0.459167 + 0.888350i $$0.348148\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ 9.00000 0.343872
$$686$$ 0 0
$$687$$ − 16.5831i − 0.632686i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 49.7494i − 1.89256i −0.323355 0.946278i $$-0.604811\pi$$
0.323355 0.946278i $$-0.395189\pi$$
$$692$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −66.0000 −2.48570
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 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 $$0.100069\pi$$
−0.950990 + 0.309223i $$0.899931\pi$$
$$720$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ − 69.6491i − 2.56905i
$$736$$ 0 0
$$737$$ −33.0000 −1.21557
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 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 $$-0.637844\pi$$
0.419641 0.907690i $$-0.362156\pi$$
$$752$$ 0 0
$$753$$ −55.0000 −2.00431
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 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
$$771$$ 59.6992i 2.15002i
$$772$$ 0 0
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ − 39.7995i − 1.42964i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 55.0000 1.96805
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 69.0000 2.46272
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 59.6992i 2.11731i
$$796$$ 0 0
$$797$$ 3.00000 0.106265 0.0531327 0.998587i $$-0.483079\pi$$
0.0531327 + 0.998587i $$0.483079\pi$$
$$798$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −29.0000 −1.00721 −0.503606 0.863934i $$-0.667994\pi$$
−0.503606 + 0.863934i $$0.667994\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −165.000 −5.70323
$$838$$ 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
$$845$$ 39.0000 1.34164
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 23.2164i 0.795847i
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ − 49.7494i − 1.69743i −0.528853 0.848713i $$-0.677378\pi$$
0.528853 0.848713i $$-0.322622\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 46.4327i − 1.58059i −0.612727 0.790295i $$-0.709928\pi$$
0.612727 0.790295i $$-0.290072\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 56.3826i 1.91485i
$$868$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 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
$$883$$ 19.8997i 0.669680i 0.942275 + 0.334840i $$0.108682\pi$$
−0.942275 + 0.334840i $$0.891318\pi$$
$$884$$ 0 0
$$885$$ −33.0000 −1.10928
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ − 102.815i − 3.44444i
$$892$$ 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
$$907$$ − 59.6992i − 1.98228i −0.132818 0.991140i $$-0.542403\pi$$
0.132818 0.991140i $$-0.457597\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.63325i 0.219769i 0.993944 + 0.109885i $$0.0350482\pi$$
−0.993944 + 0.109885i $$0.964952\pi$$
$$912$$ 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
$$925$$ 28.0000 0.920634
$$926$$ 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
$$933$$ −110.000 −3.60124
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 63.0159i 2.05645i
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 23.2164i − 0.754431i −0.926126 0.377215i $$-0.876882\pi$$
0.926126 0.377215i $$-0.123118\pi$$
$$948$$ 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
$$955$$ − 69.6491i − 2.25379i
$$956$$ 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
$$971$$ 43.1161i 1.38366i 0.722059 + 0.691831i $$0.243196\pi$$
−0.722059 + 0.691831i $$0.756804\pi$$
$$972$$ 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
$$979$$ 29.8496i 0.953998i
$$980$$ 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 $$0.397103\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ 0 0
$$993$$ 33.0000 1.04722
$$994$$ 0 0
$$995$$ − 59.6992i − 1.89259i
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 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 176.2.e.a.175.2 yes 2
3.2 odd 2 1584.2.o.a.703.2 2
4.3 odd 2 inner 176.2.e.a.175.1 2
8.3 odd 2 704.2.e.a.703.2 2
8.5 even 2 704.2.e.a.703.1 2
11.10 odd 2 CM 176.2.e.a.175.2 yes 2
12.11 even 2 1584.2.o.a.703.1 2
16.3 odd 4 2816.2.g.a.1407.3 4
16.5 even 4 2816.2.g.a.1407.4 4
16.11 odd 4 2816.2.g.a.1407.2 4
16.13 even 4 2816.2.g.a.1407.1 4
33.32 even 2 1584.2.o.a.703.2 2
44.43 even 2 inner 176.2.e.a.175.1 2
88.21 odd 2 704.2.e.a.703.1 2
88.43 even 2 704.2.e.a.703.2 2
132.131 odd 2 1584.2.o.a.703.1 2
176.21 odd 4 2816.2.g.a.1407.4 4
176.43 even 4 2816.2.g.a.1407.2 4
176.109 odd 4 2816.2.g.a.1407.1 4
176.131 even 4 2816.2.g.a.1407.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.a.175.1 2 4.3 odd 2 inner
176.2.e.a.175.1 2 44.43 even 2 inner
176.2.e.a.175.2 yes 2 1.1 even 1 trivial
176.2.e.a.175.2 yes 2 11.10 odd 2 CM
704.2.e.a.703.1 2 8.5 even 2
704.2.e.a.703.1 2 88.21 odd 2
704.2.e.a.703.2 2 8.3 odd 2
704.2.e.a.703.2 2 88.43 even 2
1584.2.o.a.703.1 2 12.11 even 2
1584.2.o.a.703.1 2 132.131 odd 2
1584.2.o.a.703.2 2 3.2 odd 2
1584.2.o.a.703.2 2 33.32 even 2
2816.2.g.a.1407.1 4 16.13 even 4
2816.2.g.a.1407.1 4 176.109 odd 4
2816.2.g.a.1407.2 4 16.11 odd 4
2816.2.g.a.1407.2 4 176.43 even 4
2816.2.g.a.1407.3 4 16.3 odd 4
2816.2.g.a.1407.3 4 176.131 even 4
2816.2.g.a.1407.4 4 16.5 even 4
2816.2.g.a.1407.4 4 176.21 odd 4