# Properties

 Label 7056.2.k.b.881.3 Level $7056$ Weight $2$ Character 7056.881 Analytic conductor $56.342$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(881,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7056.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.k (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: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 881.3 Root $$1.22474 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 7056.881 Dual form 7056.2.k.b.881.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.44949 q^{5} +O(q^{10})$$ $$q+2.44949 q^{5} -1.41421i q^{11} -5.19615i q^{13} +4.89898 q^{17} -1.73205i q^{19} -5.65685i q^{23} +1.00000 q^{25} -2.82843i q^{29} +1.73205i q^{31} -1.00000 q^{37} -7.34847 q^{41} +1.00000 q^{43} +12.2474 q^{47} -2.82843i q^{53} -3.46410i q^{55} -4.89898 q^{59} -3.46410i q^{61} -12.7279i q^{65} -11.0000 q^{67} +7.07107i q^{71} -1.73205i q^{73} -5.00000 q^{79} -7.34847 q^{83} +12.0000 q^{85} -4.89898 q^{89} -4.24264i q^{95} -10.3923i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{25} - 4 q^{37} + 4 q^{43} - 44 q^{67} - 20 q^{79} + 48 q^{85}+O(q^{100})$$ 4 * q + 4 * q^25 - 4 * q^37 + 4 * q^43 - 44 * q^67 - 20 * q^79 + 48 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1765$$ $$4609$$ $$6175$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.44949 1.09545 0.547723 0.836660i $$-0.315495\pi$$
0.547723 + 0.836660i $$0.315495\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 1.41421i − 0.426401i −0.977008 0.213201i $$-0.931611\pi$$
0.977008 0.213201i $$-0.0683888\pi$$
$$12$$ 0 0
$$13$$ − 5.19615i − 1.44115i −0.693375 0.720577i $$-0.743877\pi$$
0.693375 0.720577i $$-0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.89898 1.18818 0.594089 0.804400i $$-0.297513\pi$$
0.594089 + 0.804400i $$0.297513\pi$$
$$18$$ 0 0
$$19$$ − 1.73205i − 0.397360i −0.980064 0.198680i $$-0.936335\pi$$
0.980064 0.198680i $$-0.0636654\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 5.65685i − 1.17954i −0.807573 0.589768i $$-0.799219\pi$$
0.807573 0.589768i $$-0.200781\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 2.82843i − 0.525226i −0.964901 0.262613i $$-0.915416\pi$$
0.964901 0.262613i $$-0.0845842\pi$$
$$30$$ 0 0
$$31$$ 1.73205i 0.311086i 0.987829 + 0.155543i $$0.0497126\pi$$
−0.987829 + 0.155543i $$0.950287\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.34847 −1.14764 −0.573819 0.818982i $$-0.694539\pi$$
−0.573819 + 0.818982i $$0.694539\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.2474 1.78647 0.893237 0.449586i $$-0.148429\pi$$
0.893237 + 0.449586i $$0.148429\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 2.82843i − 0.388514i −0.980951 0.194257i $$-0.937770\pi$$
0.980951 0.194257i $$-0.0622296\pi$$
$$54$$ 0 0
$$55$$ − 3.46410i − 0.467099i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ 0 0
$$61$$ − 3.46410i − 0.443533i −0.975100 0.221766i $$-0.928818\pi$$
0.975100 0.221766i $$-0.0711822\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 12.7279i − 1.57870i
$$66$$ 0 0
$$67$$ −11.0000 −1.34386 −0.671932 0.740613i $$-0.734535\pi$$
−0.671932 + 0.740613i $$0.734535\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.07107i 0.839181i 0.907713 + 0.419591i $$0.137826\pi$$
−0.907713 + 0.419591i $$0.862174\pi$$
$$72$$ 0 0
$$73$$ − 1.73205i − 0.202721i −0.994850 0.101361i $$-0.967680\pi$$
0.994850 0.101361i $$-0.0323196\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.00000 −0.562544 −0.281272 0.959628i $$-0.590756\pi$$
−0.281272 + 0.959628i $$0.590756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −7.34847 −0.806599 −0.403300 0.915068i $$-0.632137\pi$$
−0.403300 + 0.915068i $$0.632137\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −4.89898 −0.519291 −0.259645 0.965704i $$-0.583606\pi$$
−0.259645 + 0.965704i $$0.583606\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 4.24264i − 0.435286i
$$96$$ 0 0
$$97$$ − 10.3923i − 1.05518i −0.849500 0.527589i $$-0.823096\pi$$
0.849500 0.527589i $$-0.176904\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −17.1464 −1.70613 −0.853067 0.521802i $$-0.825260\pi$$
−0.853067 + 0.521802i $$0.825260\pi$$
$$102$$ 0 0
$$103$$ 8.66025i 0.853320i 0.904412 + 0.426660i $$0.140310\pi$$
−0.904412 + 0.426660i $$0.859690\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.82843i 0.273434i 0.990610 + 0.136717i $$0.0436552\pi$$
−0.990610 + 0.136717i $$0.956345\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.41421i 0.133038i 0.997785 + 0.0665190i $$0.0211893\pi$$
−0.997785 + 0.0665190i $$0.978811\pi$$
$$114$$ 0 0
$$115$$ − 13.8564i − 1.29212i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 9.00000 0.818182
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −9.79796 −0.876356
$$126$$ 0 0
$$127$$ −11.0000 −0.976092 −0.488046 0.872818i $$-0.662290\pi$$
−0.488046 + 0.872818i $$0.662290\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2.44949 −0.214013 −0.107006 0.994258i $$-0.534127\pi$$
−0.107006 + 0.994258i $$0.534127\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 11.3137i − 0.966595i −0.875456 0.483298i $$-0.839439\pi$$
0.875456 0.483298i $$-0.160561\pi$$
$$138$$ 0 0
$$139$$ 5.19615i 0.440732i 0.975417 + 0.220366i $$0.0707252\pi$$
−0.975417 + 0.220366i $$0.929275\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −7.34847 −0.614510
$$144$$ 0 0
$$145$$ − 6.92820i − 0.575356i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 5.65685i 0.463428i 0.972784 + 0.231714i $$0.0744333\pi$$
−0.972784 + 0.231714i $$0.925567\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.24264i 0.340777i
$$156$$ 0 0
$$157$$ − 17.3205i − 1.38233i −0.722698 0.691164i $$-0.757098\pi$$
0.722698 0.691164i $$-0.242902\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 10.0000 0.783260 0.391630 0.920123i $$-0.371911\pi$$
0.391630 + 0.920123i $$0.371911\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.34847 0.568642 0.284321 0.958729i $$-0.408232\pi$$
0.284321 + 0.958729i $$0.408232\pi$$
$$168$$ 0 0
$$169$$ −14.0000 −1.07692
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 9.79796 0.744925 0.372463 0.928047i $$-0.378514\pi$$
0.372463 + 0.928047i $$0.378514\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 9.89949i − 0.739923i −0.929047 0.369961i $$-0.879371\pi$$
0.929047 0.369961i $$-0.120629\pi$$
$$180$$ 0 0
$$181$$ 15.5885i 1.15868i 0.815086 + 0.579340i $$0.196690\pi$$
−0.815086 + 0.579340i $$0.803310\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.44949 −0.180090
$$186$$ 0 0
$$187$$ − 6.92820i − 0.506640i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 1.41421i − 0.102329i −0.998690 0.0511645i $$-0.983707\pi$$
0.998690 0.0511645i $$-0.0162933\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 19.7990i − 1.41062i −0.708899 0.705310i $$-0.750808\pi$$
0.708899 0.705310i $$-0.249192\pi$$
$$198$$ 0 0
$$199$$ − 13.8564i − 0.982255i −0.871088 0.491127i $$-0.836585\pi$$
0.871088 0.491127i $$-0.163415\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −18.0000 −1.25717
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.44949 −0.169435
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 2.44949 0.167054
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 25.4558i − 1.71235i
$$222$$ 0 0
$$223$$ − 20.7846i − 1.39184i −0.718119 0.695920i $$-0.754997\pi$$
0.718119 0.695920i $$-0.245003\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −26.9444 −1.78836 −0.894181 0.447706i $$-0.852241\pi$$
−0.894181 + 0.447706i $$0.852241\pi$$
$$228$$ 0 0
$$229$$ 22.5167i 1.48794i 0.668211 + 0.743971i $$0.267060\pi$$
−0.668211 + 0.743971i $$0.732940\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 9.89949i 0.648537i 0.945965 + 0.324269i $$0.105118\pi$$
−0.945965 + 0.324269i $$0.894882\pi$$
$$234$$ 0 0
$$235$$ 30.0000 1.95698
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 26.8701i − 1.73808i −0.494742 0.869040i $$-0.664738\pi$$
0.494742 0.869040i $$-0.335262\pi$$
$$240$$ 0 0
$$241$$ 13.8564i 0.892570i 0.894891 + 0.446285i $$0.147253\pi$$
−0.894891 + 0.446285i $$0.852747\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.00000 −0.572656
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.44949 0.152795 0.0763975 0.997077i $$-0.475658\pi$$
0.0763975 + 0.997077i $$0.475658\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 14.1421i − 0.872041i −0.899937 0.436021i $$-0.856387\pi$$
0.899937 0.436021i $$-0.143613\pi$$
$$264$$ 0 0
$$265$$ − 6.92820i − 0.425596i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −17.1464 −1.04544 −0.522718 0.852506i $$-0.675082\pi$$
−0.522718 + 0.852506i $$0.675082\pi$$
$$270$$ 0 0
$$271$$ 13.8564i 0.841717i 0.907126 + 0.420858i $$0.138271\pi$$
−0.907126 + 0.420858i $$0.861729\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 1.41421i − 0.0852803i
$$276$$ 0 0
$$277$$ 23.0000 1.38194 0.690968 0.722885i $$-0.257185\pi$$
0.690968 + 0.722885i $$0.257185\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.6274i 1.34984i 0.737892 + 0.674919i $$0.235822\pi$$
−0.737892 + 0.674919i $$0.764178\pi$$
$$282$$ 0 0
$$283$$ 1.73205i 0.102960i 0.998674 + 0.0514799i $$0.0163938\pi$$
−0.998674 + 0.0514799i $$0.983606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 7.00000 0.411765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 14.6969 0.858604 0.429302 0.903161i $$-0.358760\pi$$
0.429302 + 0.903161i $$0.358760\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −29.3939 −1.69989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 8.48528i − 0.485866i
$$306$$ 0 0
$$307$$ 15.5885i 0.889680i 0.895610 + 0.444840i $$0.146740\pi$$
−0.895610 + 0.444840i $$0.853260\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 17.1464 0.972285 0.486142 0.873880i $$-0.338404\pi$$
0.486142 + 0.873880i $$0.338404\pi$$
$$312$$ 0 0
$$313$$ 12.1244i 0.685309i 0.939461 + 0.342655i $$0.111326\pi$$
−0.939461 + 0.342655i $$0.888674\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 14.1421i 0.794301i 0.917753 + 0.397151i $$0.130001\pi$$
−0.917753 + 0.397151i $$0.869999\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 8.48528i − 0.472134i
$$324$$ 0 0
$$325$$ − 5.19615i − 0.288231i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 31.0000 1.70391 0.851957 0.523612i $$-0.175416\pi$$
0.851957 + 0.523612i $$0.175416\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −26.9444 −1.47213
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.44949 0.132647
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 31.1127i − 1.67022i −0.550085 0.835109i $$-0.685405\pi$$
0.550085 0.835109i $$-0.314595\pi$$
$$348$$ 0 0
$$349$$ − 10.3923i − 0.556287i −0.960539 0.278144i $$-0.910281\pi$$
0.960539 0.278144i $$-0.0897191\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −17.1464 −0.912612 −0.456306 0.889823i $$-0.650828\pi$$
−0.456306 + 0.889823i $$0.650828\pi$$
$$354$$ 0 0
$$355$$ 17.3205i 0.919277i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 28.2843i 1.49279i 0.665505 + 0.746393i $$0.268216\pi$$
−0.665505 + 0.746393i $$0.731784\pi$$
$$360$$ 0 0
$$361$$ 16.0000 0.842105
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 4.24264i − 0.222070i
$$366$$ 0 0
$$367$$ 1.73205i 0.0904123i 0.998978 + 0.0452062i $$0.0143945\pi$$
−0.998978 + 0.0452062i $$0.985606\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −14.6969 −0.756931
$$378$$ 0 0
$$379$$ 7.00000 0.359566 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 19.5959 1.00130 0.500652 0.865648i $$-0.333094\pi$$
0.500652 + 0.865648i $$0.333094\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 26.8701i 1.36237i 0.732113 + 0.681183i $$0.238534\pi$$
−0.732113 + 0.681183i $$0.761466\pi$$
$$390$$ 0 0
$$391$$ − 27.7128i − 1.40150i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −12.2474 −0.616236
$$396$$ 0 0
$$397$$ 1.73205i 0.0869291i 0.999055 + 0.0434646i $$0.0138396\pi$$
−0.999055 + 0.0434646i $$0.986160\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 19.7990i − 0.988714i −0.869259 0.494357i $$-0.835403\pi$$
0.869259 0.494357i $$-0.164597\pi$$
$$402$$ 0 0
$$403$$ 9.00000 0.448322
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.41421i 0.0701000i
$$408$$ 0 0
$$409$$ − 32.9090i − 1.62724i −0.581394 0.813622i $$-0.697493\pi$$
0.581394 0.813622i $$-0.302507\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −18.0000 −0.883585
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −36.7423 −1.79498 −0.897491 0.441034i $$-0.854612\pi$$
−0.897491 + 0.441034i $$0.854612\pi$$
$$420$$ 0 0
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4.89898 0.237635
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.5563i 0.749323i 0.927162 + 0.374661i $$0.122241\pi$$
−0.927162 + 0.374661i $$0.877759\pi$$
$$432$$ 0 0
$$433$$ 15.5885i 0.749133i 0.927200 + 0.374567i $$0.122209\pi$$
−0.927200 + 0.374567i $$0.877791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −9.79796 −0.468700
$$438$$ 0 0
$$439$$ − 27.7128i − 1.32266i −0.750095 0.661330i $$-0.769992\pi$$
0.750095 0.661330i $$-0.230008\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 39.5980i − 1.88136i −0.339300 0.940678i $$-0.610190\pi$$
0.339300 0.940678i $$-0.389810\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 7.07107i − 0.333704i −0.985982 0.166852i $$-0.946640\pi$$
0.985982 0.166852i $$-0.0533603\pi$$
$$450$$ 0 0
$$451$$ 10.3923i 0.489355i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.00000 0.233890 0.116945 0.993138i $$-0.462690\pi$$
0.116945 + 0.993138i $$0.462690\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.6969 −0.684505 −0.342252 0.939608i $$-0.611190\pi$$
−0.342252 + 0.939608i $$0.611190\pi$$
$$462$$ 0 0
$$463$$ 13.0000 0.604161 0.302081 0.953282i $$-0.402319\pi$$
0.302081 + 0.953282i $$0.402319\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 26.9444 1.24684 0.623419 0.781888i $$-0.285743\pi$$
0.623419 + 0.781888i $$0.285743\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 1.41421i − 0.0650256i
$$474$$ 0 0
$$475$$ − 1.73205i − 0.0794719i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −4.89898 −0.223840 −0.111920 0.993717i $$-0.535700\pi$$
−0.111920 + 0.993717i $$0.535700\pi$$
$$480$$ 0 0
$$481$$ 5.19615i 0.236924i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 25.4558i − 1.15589i
$$486$$ 0 0
$$487$$ −17.0000 −0.770344 −0.385172 0.922845i $$-0.625858\pi$$
−0.385172 + 0.922845i $$0.625858\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 11.3137i 0.510581i 0.966864 + 0.255290i $$0.0821710\pi$$
−0.966864 + 0.255290i $$0.917829\pi$$
$$492$$ 0 0
$$493$$ − 13.8564i − 0.624061i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 25.0000 1.11915 0.559577 0.828778i $$-0.310964\pi$$
0.559577 + 0.828778i $$0.310964\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −22.0454 −0.982956 −0.491478 0.870890i $$-0.663543\pi$$
−0.491478 + 0.870890i $$0.663543\pi$$
$$504$$ 0 0
$$505$$ −42.0000 −1.86898
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 2.44949 0.108572 0.0542859 0.998525i $$-0.482712\pi$$
0.0542859 + 0.998525i $$0.482712\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 21.2132i 0.934765i
$$516$$ 0 0
$$517$$ − 17.3205i − 0.761755i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4.89898 0.214628 0.107314 0.994225i $$-0.465775\pi$$
0.107314 + 0.994225i $$0.465775\pi$$
$$522$$ 0 0
$$523$$ − 1.73205i − 0.0757373i −0.999283 0.0378686i $$-0.987943\pi$$
0.999283 0.0378686i $$-0.0120568\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.48528i 0.369625i
$$528$$ 0 0
$$529$$ −9.00000 −0.391304
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 38.1838i 1.65392i
$$534$$ 0 0
$$535$$ 6.92820i 0.299532i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.44949 −0.104925
$$546$$ 0 0
$$547$$ 10.0000 0.427569 0.213785 0.976881i $$-0.431421\pi$$
0.213785 + 0.976881i $$0.431421\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.89898 −0.208704
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 15.5563i − 0.659144i −0.944131 0.329572i $$-0.893096\pi$$
0.944131 0.329572i $$-0.106904\pi$$
$$558$$ 0 0
$$559$$ − 5.19615i − 0.219774i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −26.9444 −1.13557 −0.567785 0.823177i $$-0.692200\pi$$
−0.567785 + 0.823177i $$0.692200\pi$$
$$564$$ 0 0
$$565$$ 3.46410i 0.145736i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.41421i 0.0592869i 0.999561 + 0.0296435i $$0.00943719\pi$$
−0.999561 + 0.0296435i $$0.990563\pi$$
$$570$$ 0 0
$$571$$ −11.0000 −0.460336 −0.230168 0.973151i $$-0.573928\pi$$
−0.230168 + 0.973151i $$0.573928\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 5.65685i − 0.235907i
$$576$$ 0 0
$$577$$ − 1.73205i − 0.0721062i −0.999350 0.0360531i $$-0.988521\pi$$
0.999350 0.0360531i $$-0.0114785\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.6969 0.606608 0.303304 0.952894i $$-0.401910\pi$$
0.303304 + 0.952894i $$0.401910\pi$$
$$588$$ 0 0
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 17.1464 0.704119 0.352060 0.935978i $$-0.385481\pi$$
0.352060 + 0.935978i $$0.385481\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 5.65685i − 0.231133i −0.993300 0.115566i $$-0.963132\pi$$
0.993300 0.115566i $$-0.0368683\pi$$
$$600$$ 0 0
$$601$$ − 25.9808i − 1.05978i −0.848067 0.529889i $$-0.822234\pi$$
0.848067 0.529889i $$-0.177766\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 22.0454 0.896273
$$606$$ 0 0
$$607$$ 39.8372i 1.61694i 0.588537 + 0.808470i $$0.299704\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 63.6396i − 2.57458i
$$612$$ 0 0
$$613$$ 8.00000 0.323117 0.161558 0.986863i $$-0.448348\pi$$
0.161558 + 0.986863i $$0.448348\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 24.0416i − 0.967880i −0.875101 0.483940i $$-0.839205\pi$$
0.875101 0.483940i $$-0.160795\pi$$
$$618$$ 0 0
$$619$$ − 29.4449i − 1.18349i −0.806126 0.591744i $$-0.798439\pi$$
0.806126 0.591744i $$-0.201561\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.89898 −0.195335
$$630$$ 0 0
$$631$$ −38.0000 −1.51276 −0.756378 0.654135i $$-0.773033\pi$$
−0.756378 + 0.654135i $$0.773033\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −26.9444 −1.06926
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14.1421i 0.558581i 0.960207 + 0.279290i $$0.0900992\pi$$
−0.960207 + 0.279290i $$0.909901\pi$$
$$642$$ 0 0
$$643$$ − 25.9808i − 1.02458i −0.858812 0.512291i $$-0.828797\pi$$
0.858812 0.512291i $$-0.171203\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.1464 0.674096 0.337048 0.941488i $$-0.390572\pi$$
0.337048 + 0.941488i $$0.390572\pi$$
$$648$$ 0 0
$$649$$ 6.92820i 0.271956i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 7.07107i − 0.276712i −0.990383 0.138356i $$-0.955818\pi$$
0.990383 0.138356i $$-0.0441819\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 22.6274i − 0.881439i −0.897645 0.440720i $$-0.854723\pi$$
0.897645 0.440720i $$-0.145277\pi$$
$$660$$ 0 0
$$661$$ 29.4449i 1.14527i 0.819810 + 0.572636i $$0.194079\pi$$
−0.819810 + 0.572636i $$0.805921\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.89898 −0.189123
$$672$$ 0 0
$$673$$ 35.0000 1.34915 0.674575 0.738206i $$-0.264327\pi$$
0.674575 + 0.738206i $$0.264327\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 9.79796 0.376566 0.188283 0.982115i $$-0.439708\pi$$
0.188283 + 0.982115i $$0.439708\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 48.0833i − 1.83985i −0.392089 0.919927i $$-0.628247\pi$$
0.392089 0.919927i $$-0.371753\pi$$
$$684$$ 0 0
$$685$$ − 27.7128i − 1.05885i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −14.6969 −0.559909
$$690$$ 0 0
$$691$$ − 43.3013i − 1.64726i −0.567129 0.823629i $$-0.691946\pi$$
0.567129 0.823629i $$-0.308054\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.7279i 0.482798i
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5.65685i 0.213656i 0.994277 + 0.106828i $$0.0340695\pi$$
−0.994277 + 0.106828i $$0.965931\pi$$
$$702$$ 0 0
$$703$$ 1.73205i 0.0653255i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −40.0000 −1.50223 −0.751116 0.660171i $$-0.770484\pi$$
−0.751116 + 0.660171i $$0.770484\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 9.79796 0.366936
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 26.9444 1.00486 0.502428 0.864619i $$-0.332440\pi$$
0.502428 + 0.864619i $$0.332440\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 2.82843i − 0.105045i
$$726$$ 0 0
$$727$$ 25.9808i 0.963573i 0.876289 + 0.481787i $$0.160012\pi$$
−0.876289 + 0.481787i $$0.839988\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 4.89898 0.181195
$$732$$ 0 0
$$733$$ − 39.8372i − 1.47142i −0.677297 0.735710i $$-0.736849\pi$$
0.677297 0.735710i $$-0.263151\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15.5563i 0.573025i
$$738$$ 0 0
$$739$$ 1.00000 0.0367856 0.0183928 0.999831i $$-0.494145\pi$$
0.0183928 + 0.999831i $$0.494145\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0416i 0.882002i 0.897507 + 0.441001i $$0.145376\pi$$
−0.897507 + 0.441001i $$0.854624\pi$$
$$744$$ 0 0
$$745$$ 13.8564i 0.507659i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −29.0000 −1.05823 −0.529113 0.848552i $$-0.677475\pi$$
−0.529113 + 0.848552i $$0.677475\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 53.8888 1.96121
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 24.4949 0.887939 0.443970 0.896042i $$-0.353570\pi$$
0.443970 + 0.896042i $$0.353570\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 25.4558i 0.919157i
$$768$$ 0 0
$$769$$ 25.9808i 0.936890i 0.883493 + 0.468445i $$0.155186\pi$$
−0.883493 + 0.468445i $$0.844814\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 26.9444 0.969122 0.484561 0.874757i $$-0.338979\pi$$
0.484561 + 0.874757i $$0.338979\pi$$
$$774$$ 0 0
$$775$$ 1.73205i 0.0622171i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.7279i 0.456025i
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 42.4264i − 1.51426i
$$786$$ 0 0
$$787$$ − 45.0333i − 1.60526i −0.596474 0.802632i $$-0.703432\pi$$
0.596474 0.802632i $$-0.296568\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −18.0000 −0.639199
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −29.3939 −1.04118 −0.520592 0.853805i $$-0.674289\pi$$
−0.520592 + 0.853805i $$0.674289\pi$$
$$798$$ 0 0
$$799$$ 60.0000 2.12265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −2.44949 −0.0864406
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 41.0122i − 1.44191i −0.692981 0.720956i $$-0.743703\pi$$
0.692981 0.720956i $$-0.256297\pi$$
$$810$$ 0 0
$$811$$ 31.1769i 1.09477i 0.836881 + 0.547385i $$0.184377\pi$$
−0.836881 + 0.547385i $$0.815623\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 24.4949 0.858019
$$816$$ 0 0
$$817$$ − 1.73205i − 0.0605968i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 24.0416i − 0.839059i −0.907742 0.419529i $$-0.862195\pi$$
0.907742 0.419529i $$-0.137805\pi$$
$$822$$ 0 0
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.07107i 0.245885i 0.992414 + 0.122943i $$0.0392331\pi$$
−0.992414 + 0.122943i $$0.960767\pi$$
$$828$$ 0 0
$$829$$ − 1.73205i − 0.0601566i −0.999548 0.0300783i $$-0.990424\pi$$
0.999548 0.0300783i $$-0.00957567\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 18.0000 0.622916
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 14.6969 0.507395 0.253697 0.967284i $$-0.418353\pi$$
0.253697 + 0.967284i $$0.418353\pi$$
$$840$$ 0 0
$$841$$ 21.0000 0.724138
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −34.2929 −1.17971
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 5.65685i 0.193914i
$$852$$ 0 0
$$853$$ 36.3731i 1.24539i 0.782465 + 0.622695i $$0.213962\pi$$
−0.782465 + 0.622695i $$0.786038\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −39.1918 −1.33877 −0.669384 0.742917i $$-0.733442\pi$$
−0.669384 + 0.742917i $$0.733442\pi$$
$$858$$ 0 0
$$859$$ − 38.1051i − 1.30013i −0.759879 0.650065i $$-0.774742\pi$$
0.759879 0.650065i $$-0.225258\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 41.0122i 1.39607i 0.716063 + 0.698036i $$0.245942\pi$$
−0.716063 + 0.698036i $$0.754058\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 7.07107i 0.239870i
$$870$$ 0 0
$$871$$ 57.1577i 1.93671i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000 0.675352 0.337676 0.941262i $$-0.390359\pi$$
0.337676 + 0.941262i $$0.390359\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −11.0000 −0.370179 −0.185090 0.982722i $$-0.559258\pi$$
−0.185090 + 0.982722i $$0.559258\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 41.6413 1.39818 0.699089 0.715034i $$-0.253589\pi$$
0.699089 + 0.715034i $$0.253589\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 21.2132i − 0.709873i
$$894$$ 0 0
$$895$$ − 24.2487i − 0.810545i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 4.89898 0.163390
$$900$$ 0 0
$$901$$ − 13.8564i − 0.461624i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 38.1838i 1.26927i
$$906$$ 0 0
$$907$$ −5.00000 −0.166022 −0.0830111 0.996549i $$-0.526454\pi$$
−0.0830111 + 0.996549i $$0.526454\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 53.7401i 1.78049i 0.455483 + 0.890245i $$0.349467\pi$$
−0.455483 + 0.890245i $$0.650533\pi$$
$$912$$ 0 0
$$913$$ 10.3923i 0.343935i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −17.0000 −0.560778 −0.280389 0.959886i $$-0.590464\pi$$
−0.280389 + 0.959886i $$0.590464\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 36.7423 1.20939
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −56.3383 −1.84840 −0.924199 0.381911i $$-0.875266\pi$$
−0.924199 + 0.381911i $$0.875266\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 16.9706i − 0.554997i
$$936$$ 0 0
$$937$$ 46.7654i 1.52776i 0.645359 + 0.763879i $$0.276708\pi$$
−0.645359 + 0.763879i $$0.723292\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 48.9898 1.59702 0.798511 0.601980i $$-0.205621\pi$$
0.798511 + 0.601980i $$0.205621\pi$$
$$942$$ 0 0
$$943$$ 41.5692i 1.35368i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 49.4975i 1.60845i 0.594324 + 0.804226i $$0.297420\pi$$
−0.594324 + 0.804226i $$0.702580\pi$$
$$948$$ 0 0
$$949$$ −9.00000 −0.292152
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 5.65685i 0.183243i 0.995794 + 0.0916217i $$0.0292051\pi$$
−0.995794 + 0.0916217i $$0.970795\pi$$
$$954$$ 0 0
$$955$$ − 3.46410i − 0.112096i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 28.0000 0.903226
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 26.9444 0.867371
$$966$$ 0 0
$$967$$ 13.0000 0.418052 0.209026 0.977910i $$-0.432971\pi$$
0.209026 + 0.977910i $$0.432971\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −39.1918 −1.25773 −0.628863 0.777516i $$-0.716479\pi$$
−0.628863 + 0.777516i $$0.716479\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 7.07107i − 0.226224i −0.993582 0.113112i $$-0.963918\pi$$
0.993582 0.113112i $$-0.0360818\pi$$
$$978$$ 0 0
$$979$$ 6.92820i 0.221426i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −4.89898 −0.156253 −0.0781266 0.996943i $$-0.524894\pi$$
−0.0781266 + 0.996943i $$0.524894\pi$$
$$984$$ 0 0
$$985$$ − 48.4974i − 1.54526i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 5.65685i − 0.179878i
$$990$$ 0 0
$$991$$ −35.0000 −1.11181 −0.555906 0.831245i $$-0.687628\pi$$
−0.555906 + 0.831245i $$0.687628\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 33.9411i − 1.07601i
$$996$$ 0 0
$$997$$ 29.4449i 0.932528i 0.884646 + 0.466264i $$0.154400\pi$$
−0.884646 + 0.466264i $$0.845600\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 7056.2.k.b.881.3 4
3.2 odd 2 inner 7056.2.k.b.881.2 4
4.3 odd 2 441.2.c.a.440.4 4
7.2 even 3 1008.2.bt.b.17.1 4
7.3 odd 6 1008.2.bt.b.593.2 4
7.6 odd 2 inner 7056.2.k.b.881.1 4
12.11 even 2 441.2.c.a.440.1 4
21.2 odd 6 1008.2.bt.b.17.2 4
21.17 even 6 1008.2.bt.b.593.1 4
21.20 even 2 inner 7056.2.k.b.881.4 4
28.3 even 6 63.2.p.a.26.1 yes 4
28.11 odd 6 441.2.p.a.215.1 4
28.19 even 6 441.2.p.a.80.2 4
28.23 odd 6 63.2.p.a.17.2 yes 4
28.27 even 2 441.2.c.a.440.3 4
84.11 even 6 441.2.p.a.215.2 4
84.23 even 6 63.2.p.a.17.1 4
84.47 odd 6 441.2.p.a.80.1 4
84.59 odd 6 63.2.p.a.26.2 yes 4
84.83 odd 2 441.2.c.a.440.2 4
140.3 odd 12 1575.2.bc.a.1349.1 8
140.23 even 12 1575.2.bc.a.899.2 8
140.59 even 6 1575.2.bk.c.26.2 4
140.79 odd 6 1575.2.bk.c.1151.1 4
140.87 odd 12 1575.2.bc.a.1349.4 8
140.107 even 12 1575.2.bc.a.899.3 8
252.23 even 6 567.2.i.d.269.1 4
252.31 even 6 567.2.s.d.26.2 4
252.59 odd 6 567.2.s.d.26.1 4
252.79 odd 6 567.2.s.d.458.1 4
252.115 even 6 567.2.i.d.215.2 4
252.191 even 6 567.2.s.d.458.2 4
252.227 odd 6 567.2.i.d.215.1 4
252.247 odd 6 567.2.i.d.269.2 4
420.23 odd 12 1575.2.bc.a.899.4 8
420.59 odd 6 1575.2.bk.c.26.1 4
420.107 odd 12 1575.2.bc.a.899.1 8
420.143 even 12 1575.2.bc.a.1349.3 8
420.227 even 12 1575.2.bc.a.1349.2 8
420.359 even 6 1575.2.bk.c.1151.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.p.a.17.1 4 84.23 even 6
63.2.p.a.17.2 yes 4 28.23 odd 6
63.2.p.a.26.1 yes 4 28.3 even 6
63.2.p.a.26.2 yes 4 84.59 odd 6
441.2.c.a.440.1 4 12.11 even 2
441.2.c.a.440.2 4 84.83 odd 2
441.2.c.a.440.3 4 28.27 even 2
441.2.c.a.440.4 4 4.3 odd 2
441.2.p.a.80.1 4 84.47 odd 6
441.2.p.a.80.2 4 28.19 even 6
441.2.p.a.215.1 4 28.11 odd 6
441.2.p.a.215.2 4 84.11 even 6
567.2.i.d.215.1 4 252.227 odd 6
567.2.i.d.215.2 4 252.115 even 6
567.2.i.d.269.1 4 252.23 even 6
567.2.i.d.269.2 4 252.247 odd 6
567.2.s.d.26.1 4 252.59 odd 6
567.2.s.d.26.2 4 252.31 even 6
567.2.s.d.458.1 4 252.79 odd 6
567.2.s.d.458.2 4 252.191 even 6
1008.2.bt.b.17.1 4 7.2 even 3
1008.2.bt.b.17.2 4 21.2 odd 6
1008.2.bt.b.593.1 4 21.17 even 6
1008.2.bt.b.593.2 4 7.3 odd 6
1575.2.bc.a.899.1 8 420.107 odd 12
1575.2.bc.a.899.2 8 140.23 even 12
1575.2.bc.a.899.3 8 140.107 even 12
1575.2.bc.a.899.4 8 420.23 odd 12
1575.2.bc.a.1349.1 8 140.3 odd 12
1575.2.bc.a.1349.2 8 420.227 even 12
1575.2.bc.a.1349.3 8 420.143 even 12
1575.2.bc.a.1349.4 8 140.87 odd 12
1575.2.bk.c.26.1 4 420.59 odd 6
1575.2.bk.c.26.2 4 140.59 even 6
1575.2.bk.c.1151.1 4 140.79 odd 6
1575.2.bk.c.1151.2 4 420.359 even 6
7056.2.k.b.881.1 4 7.6 odd 2 inner
7056.2.k.b.881.2 4 3.2 odd 2 inner
7056.2.k.b.881.3 4 1.1 even 1 trivial
7056.2.k.b.881.4 4 21.20 even 2 inner