# Properties

 Label 441.2.c.a.440.2 Level $441$ Weight $2$ Character 441.440 Analytic conductor $3.521$ 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] = [441,2,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.440");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (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: $$3.52140272914$$ 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 440.2 Root $$1.22474 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 441.440 Dual form 441.2.c.a.440.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} +O(q^{10})$$ $$q-1.41421i q^{2} +2.44949 q^{5} -2.82843i q^{8} -3.46410i q^{10} -1.41421i q^{11} +5.19615i q^{13} -4.00000 q^{16} +4.89898 q^{17} -1.73205i q^{19} -2.00000 q^{22} -5.65685i q^{23} +1.00000 q^{25} +7.34847 q^{26} +2.82843i q^{29} +1.73205i q^{31} -6.92820i q^{34} -1.00000 q^{37} -2.44949 q^{38} -6.92820i q^{40} -7.34847 q^{41} -1.00000 q^{43} -8.00000 q^{46} -12.2474 q^{47} -1.41421i q^{50} +2.82843i q^{53} -3.46410i q^{55} +4.00000 q^{58} +4.89898 q^{59} +3.46410i q^{61} +2.44949 q^{62} -8.00000 q^{64} +12.7279i q^{65} +11.0000 q^{67} +7.07107i q^{71} +1.73205i q^{73} +1.41421i q^{74} +5.00000 q^{79} -9.79796 q^{80} +10.3923i q^{82} +7.34847 q^{83} +12.0000 q^{85} +1.41421i q^{86} -4.00000 q^{88} -4.89898 q^{89} +17.3205i q^{94} -4.24264i q^{95} +10.3923i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{16} - 8 q^{22} + 4 q^{25} - 4 q^{37} - 4 q^{43} - 32 q^{46} + 16 q^{58} - 32 q^{64} + 44 q^{67} + 20 q^{79} + 48 q^{85} - 16 q^{88}+O(q^{100})$$ 4 * q - 16 * q^16 - 8 * q^22 + 4 * q^25 - 4 * q^37 - 4 * q^43 - 32 * q^46 + 16 * q^58 - 32 * q^64 + 44 * q^67 + 20 * q^79 + 48 * q^85 - 16 * q^88

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.41421i − 1.00000i −0.866025 0.500000i $$-0.833333\pi$$
0.866025 0.500000i $$-0.166667\pi$$
$$3$$ 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
$$8$$ − 2.82843i − 1.00000i
$$9$$ 0 0
$$10$$ − 3.46410i − 1.09545i
$$11$$ − 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 $$0.256123\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 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
$$22$$ −2.00000 −0.426401
$$23$$ − 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
$$26$$ 7.34847 1.44115
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.82843i 0.525226i 0.964901 + 0.262613i $$0.0845842\pi$$
−0.964901 + 0.262613i $$0.915416\pi$$
$$30$$ 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
$$34$$ − 6.92820i − 1.18818i
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ −2.44949 −0.397360
$$39$$ 0 0
$$40$$ − 6.92820i − 1.09545i
$$41$$ −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 $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ −12.2474 −1.78647 −0.893237 0.449586i $$-0.851571\pi$$
−0.893237 + 0.449586i $$0.851571\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ − 1.41421i − 0.200000i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.82843i 0.388514i 0.980951 + 0.194257i $$0.0622296\pi$$
−0.980951 + 0.194257i $$0.937770\pi$$
$$54$$ 0 0
$$55$$ − 3.46410i − 0.467099i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 4.00000 0.525226
$$59$$ 4.89898 0.637793 0.318896 0.947790i $$-0.396688\pi$$
0.318896 + 0.947790i $$0.396688\pi$$
$$60$$ 0 0
$$61$$ 3.46410i 0.443533i 0.975100 + 0.221766i $$0.0711822\pi$$
−0.975100 + 0.221766i $$0.928818\pi$$
$$62$$ 2.44949 0.311086
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 12.7279i 1.57870i
$$66$$ 0 0
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ 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 $$0.0323196\pi$$
−0.994850 + 0.101361i $$0.967680\pi$$
$$74$$ 1.41421i 0.164399i
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ −9.79796 −1.09545
$$81$$ 0 0
$$82$$ 10.3923i 1.14764i
$$83$$ 7.34847 0.806599 0.403300 0.915068i $$-0.367863\pi$$
0.403300 + 0.915068i $$0.367863\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ 1.41421i 0.152499i
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ −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
$$94$$ 17.3205i 1.78647i
$$95$$ − 4.24264i − 0.435286i
$$96$$ 0 0
$$97$$ 10.3923i 1.05518i 0.849500 + 0.527589i $$0.176904\pi$$
−0.849500 + 0.527589i $$0.823096\pi$$
$$98$$ 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 $$0.859690\pi$$
$$104$$ 14.6969 1.44115
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 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 $$0.515250\pi$$
$$110$$ −4.89898 −0.467099
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 1.41421i − 0.133038i −0.997785 0.0665190i $$-0.978811\pi$$
0.997785 0.0665190i $$-0.0211893\pi$$
$$114$$ 0 0
$$115$$ − 13.8564i − 1.29212i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ − 6.92820i − 0.637793i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 9.00000 0.818182
$$122$$ 4.89898 0.443533
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −9.79796 −0.876356
$$126$$ 0 0
$$127$$ 11.0000 0.976092 0.488046 0.872818i $$-0.337710\pi$$
0.488046 + 0.872818i $$0.337710\pi$$
$$128$$ 11.3137i 1.00000i
$$129$$ 0 0
$$130$$ 18.0000 1.57870
$$131$$ 2.44949 0.214013 0.107006 0.994258i $$-0.465873\pi$$
0.107006 + 0.994258i $$0.465873\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ − 15.5563i − 1.34386i
$$135$$ 0 0
$$136$$ − 13.8564i − 1.18818i
$$137$$ 11.3137i 0.966595i 0.875456 + 0.483298i $$0.160561\pi$$
−0.875456 + 0.483298i $$0.839439\pi$$
$$138$$ 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
$$142$$ 10.0000 0.839181
$$143$$ 7.34847 0.614510
$$144$$ 0 0
$$145$$ 6.92820i 0.575356i
$$146$$ 2.44949 0.202721
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 5.65685i − 0.463428i −0.972784 0.231714i $$-0.925567\pi$$
0.972784 0.231714i $$-0.0744333\pi$$
$$150$$ 0 0
$$151$$ −22.0000 −1.79033 −0.895167 0.445730i $$-0.852944\pi$$
−0.895167 + 0.445730i $$0.852944\pi$$
$$152$$ −4.89898 −0.397360
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.24264i 0.340777i
$$156$$ 0 0
$$157$$ 17.3205i 1.38233i 0.722698 + 0.691164i $$0.242902\pi$$
−0.722698 + 0.691164i $$0.757098\pi$$
$$158$$ − 7.07107i − 0.562544i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ − 10.3923i − 0.806599i
$$167$$ −7.34847 −0.568642 −0.284321 0.958729i $$-0.591768\pi$$
−0.284321 + 0.958729i $$0.591768\pi$$
$$168$$ 0 0
$$169$$ −14.0000 −1.07692
$$170$$ − 16.9706i − 1.30158i
$$171$$ 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
$$176$$ 5.65685i 0.426401i
$$177$$ 0 0
$$178$$ 6.92820i 0.519291i
$$179$$ − 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 $$-0.803310\pi$$
0.815086 0.579340i $$-0.196690\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −16.0000 −1.17954
$$185$$ −2.44949 −0.180090
$$186$$ 0 0
$$187$$ − 6.92820i − 0.506640i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ − 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 $$0.370433\pi$$
$$194$$ 14.6969 1.05518
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 19.7990i 1.41062i 0.708899 + 0.705310i $$0.249192\pi$$
−0.708899 + 0.705310i $$0.750808\pi$$
$$198$$ 0 0
$$199$$ − 13.8564i − 0.982255i −0.871088 0.491127i $$-0.836585\pi$$
0.871088 0.491127i $$-0.163415\pi$$
$$200$$ − 2.82843i − 0.200000i
$$201$$ 0 0
$$202$$ 24.2487i 1.70613i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −18.0000 −1.25717
$$206$$ 12.2474 0.853320
$$207$$ 0 0
$$208$$ − 20.7846i − 1.44115i
$$209$$ −2.44949 −0.169435
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ −2.44949 −0.167054
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 1.41421i 0.0957826i
$$219$$ 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
$$226$$ −2.00000 −0.133038
$$227$$ 26.9444 1.78836 0.894181 0.447706i $$-0.147759\pi$$
0.894181 + 0.447706i $$0.147759\pi$$
$$228$$ 0 0
$$229$$ − 22.5167i − 1.48794i −0.668211 0.743971i $$-0.732940\pi$$
0.668211 0.743971i $$-0.267060\pi$$
$$230$$ −19.5959 −1.29212
$$231$$ 0 0
$$232$$ 8.00000 0.525226
$$233$$ − 9.89949i − 0.648537i −0.945965 0.324269i $$-0.894882\pi$$
0.945965 0.324269i $$-0.105118\pi$$
$$234$$ 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 $$-0.852747\pi$$
0.894891 0.446285i $$-0.147253\pi$$
$$242$$ − 12.7279i − 0.818182i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9.00000 0.572656
$$248$$ 4.89898 0.311086
$$249$$ 0 0
$$250$$ 13.8564i 0.876356i
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ − 15.5563i − 0.976092i
$$255$$ 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
$$262$$ − 3.46410i − 0.214013i
$$263$$ − 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 $$0.861729\pi$$
$$272$$ −19.5959 −1.18818
$$273$$ 0 0
$$274$$ 16.0000 0.966595
$$275$$ − 1.41421i − 0.0852803i
$$276$$ 0 0
$$277$$ 23.0000 1.38194 0.690968 0.722885i $$-0.257185\pi$$
0.690968 + 0.722885i $$0.257185\pi$$
$$278$$ 7.34847 0.440732
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 22.6274i − 1.34984i −0.737892 0.674919i $$-0.764178\pi$$
0.737892 0.674919i $$-0.235822\pi$$
$$282$$ 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
$$286$$ − 10.3923i − 0.614510i
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 7.00000 0.411765
$$290$$ 9.79796 0.575356
$$291$$ 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
$$296$$ 2.82843i 0.164399i
$$297$$ 0 0
$$298$$ −8.00000 −0.463428
$$299$$ 29.3939 1.69989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 31.1127i 1.79033i
$$303$$ 0 0
$$304$$ 6.92820i 0.397360i
$$305$$ 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
$$310$$ 6.00000 0.340777
$$311$$ −17.1464 −0.972285 −0.486142 0.873880i $$-0.661596\pi$$
−0.486142 + 0.873880i $$0.661596\pi$$
$$312$$ 0 0
$$313$$ − 12.1244i − 0.685309i −0.939461 0.342655i $$-0.888674\pi$$
0.939461 0.342655i $$-0.111326\pi$$
$$314$$ 24.4949 1.38233
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.1421i − 0.794301i −0.917753 0.397151i $$-0.869999\pi$$
0.917753 0.397151i $$-0.130001\pi$$
$$318$$ 0 0
$$319$$ 4.00000 0.223957
$$320$$ −19.5959 −1.09545
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 8.48528i − 0.472134i
$$324$$ 0 0
$$325$$ 5.19615i 0.288231i
$$326$$ 14.1421i 0.783260i
$$327$$ 0 0
$$328$$ 20.7846i 1.14764i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −31.0000 −1.70391 −0.851957 0.523612i $$-0.824584\pi$$
−0.851957 + 0.523612i $$0.824584\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 10.3923i 0.568642i
$$335$$ 26.9444 1.47213
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 19.7990i 1.07692i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.44949 0.132647
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.82843i 0.152499i
$$345$$ 0 0
$$346$$ − 13.8564i − 0.744925i
$$347$$ − 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 $$0.0897191\pi$$
−0.960539 + 0.278144i $$0.910281\pi$$
$$350$$ 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
$$358$$ −14.0000 −0.739923
$$359$$ 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
$$362$$ −22.0454 −1.15868
$$363$$ 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 $$0.985606\pi$$
$$368$$ 22.6274i 1.17954i
$$369$$ 0 0
$$370$$ 3.46410i 0.180090i
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ −9.79796 −0.506640
$$375$$ 0 0
$$376$$ 34.6410i 1.78647i
$$377$$ −14.6969 −0.756931
$$378$$ 0 0
$$379$$ −7.00000 −0.359566 −0.179783 0.983706i $$-0.557540\pi$$
−0.179783 + 0.983706i $$0.557540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −2.00000 −0.102329
$$383$$ −19.5959 −1.00130 −0.500652 0.865648i $$-0.666906\pi$$
−0.500652 + 0.865648i $$0.666906\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ − 15.5563i − 0.791797i
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 26.8701i − 1.36237i −0.732113 0.681183i $$-0.761466\pi$$
0.732113 0.681183i $$-0.238534\pi$$
$$390$$ 0 0
$$391$$ − 27.7128i − 1.40150i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 28.0000 1.41062
$$395$$ 12.2474 0.616236
$$396$$ 0 0
$$397$$ − 1.73205i − 0.0869291i −0.999055 0.0434646i $$-0.986160\pi$$
0.999055 0.0434646i $$-0.0138396\pi$$
$$398$$ −19.5959 −0.982255
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ 19.7990i 0.988714i 0.869259 + 0.494357i $$0.164597\pi$$
−0.869259 + 0.494357i $$0.835403\pi$$
$$402$$ 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 $$0.302507\pi$$
−0.581394 + 0.813622i $$0.697493\pi$$
$$410$$ 25.4558i 1.25717i
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 18.0000 0.883585
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 3.46410i 0.169435i
$$419$$ 36.7423 1.79498 0.897491 0.441034i $$-0.145388\pi$$
0.897491 + 0.441034i $$0.145388\pi$$
$$420$$ 0 0
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ 31.1127i 1.51454i
$$423$$ 0 0
$$424$$ 8.00000 0.388514
$$425$$ 4.89898 0.237635
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 3.46410i 0.167054i
$$431$$ 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 $$-0.877791\pi$$
0.927200 0.374567i $$-0.122209\pi$$
$$434$$ 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 $$-0.230008\pi$$
$$440$$ −9.79796 −0.467099
$$441$$ 0 0
$$442$$ 36.0000 1.71235
$$443$$ − 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
$$446$$ −29.3939 −1.39184
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 7.07107i 0.333704i 0.985982 + 0.166852i $$0.0533603\pi$$
−0.985982 + 0.166852i $$0.946640\pi$$
$$450$$ 0 0
$$451$$ 10.3923i 0.489355i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ − 38.1051i − 1.78836i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.00000 0.233890 0.116945 0.993138i $$-0.462690\pi$$
0.116945 + 0.993138i $$0.462690\pi$$
$$458$$ −31.8434 −1.48794
$$459$$ 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 $$-0.597681\pi$$
−0.302081 + 0.953282i $$0.597681\pi$$
$$464$$ − 11.3137i − 0.525226i
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ −26.9444 −1.24684 −0.623419 0.781888i $$-0.714257\pi$$
−0.623419 + 0.781888i $$0.714257\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 42.4264i 1.95698i
$$471$$ 0 0
$$472$$ − 13.8564i − 0.637793i
$$473$$ 1.41421i 0.0650256i
$$474$$ 0 0
$$475$$ − 1.73205i − 0.0794719i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −38.0000 −1.73808
$$479$$ 4.89898 0.223840 0.111920 0.993717i $$-0.464300\pi$$
0.111920 + 0.993717i $$0.464300\pi$$
$$480$$ 0 0
$$481$$ − 5.19615i − 0.236924i
$$482$$ −19.5959 −0.892570
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 25.4558i 1.15589i
$$486$$ 0 0
$$487$$ 17.0000 0.770344 0.385172 0.922845i $$-0.374142\pi$$
0.385172 + 0.922845i $$0.374142\pi$$
$$488$$ 9.79796 0.443533
$$489$$ 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
$$494$$ − 12.7279i − 0.572656i
$$495$$ 0 0
$$496$$ − 6.92820i − 0.311086i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 22.0454 0.982956 0.491478 0.870890i $$-0.336457\pi$$
0.491478 + 0.870890i $$0.336457\pi$$
$$504$$ 0 0
$$505$$ −42.0000 −1.86898
$$506$$ 11.3137i 0.502956i
$$507$$ 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
$$512$$ 22.6274i 1.00000i
$$513$$ 0 0
$$514$$ − 3.46410i − 0.152795i
$$515$$ 21.2132i 0.934765i
$$516$$ 0 0
$$517$$ 17.3205i 0.761755i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 36.0000 1.57870
$$521$$ 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
$$526$$ −20.0000 −0.872041
$$527$$ 8.48528i 0.369625i
$$528$$ 0 0
$$529$$ −9.00000 −0.391304
$$530$$ 9.79796 0.425596
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 38.1838i − 1.65392i
$$534$$ 0 0
$$535$$ 6.92820i 0.299532i
$$536$$ − 31.1127i − 1.34386i
$$537$$ 0 0
$$538$$ 24.2487i 1.04544i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 19.5959 0.841717
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.44949 −0.104925
$$546$$ 0 0
$$547$$ −10.0000 −0.427569 −0.213785 0.976881i $$-0.568579\pi$$
−0.213785 + 0.976881i $$0.568579\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ 4.89898 0.208704
$$552$$ 0 0
$$553$$ 0 0
$$554$$ − 32.5269i − 1.38194i
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15.5563i 0.659144i 0.944131 + 0.329572i $$0.106904\pi$$
−0.944131 + 0.329572i $$0.893096\pi$$
$$558$$ 0 0
$$559$$ − 5.19615i − 0.219774i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −32.0000 −1.34984
$$563$$ 26.9444 1.13557 0.567785 0.823177i $$-0.307800\pi$$
0.567785 + 0.823177i $$0.307800\pi$$
$$564$$ 0 0
$$565$$ − 3.46410i − 0.145736i
$$566$$ 2.44949 0.102960
$$567$$ 0 0
$$568$$ 20.0000 0.839181
$$569$$ − 1.41421i − 0.0592869i −0.999561 0.0296435i $$-0.990563\pi$$
0.999561 0.0296435i $$-0.00943719\pi$$
$$570$$ 0 0
$$571$$ 11.0000 0.460336 0.230168 0.973151i $$-0.426072\pi$$
0.230168 + 0.973151i $$0.426072\pi$$
$$572$$ 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 $$0.0114785\pi$$
−0.999350 + 0.0360531i $$0.988521\pi$$
$$578$$ − 9.89949i − 0.411765i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ 4.89898 0.202721
$$585$$ 0 0
$$586$$ − 20.7846i − 0.858604i
$$587$$ −14.6969 −0.606608 −0.303304 0.952894i $$-0.598090\pi$$
−0.303304 + 0.952894i $$0.598090\pi$$
$$588$$ 0 0
$$589$$ 3.00000 0.123613
$$590$$ − 16.9706i − 0.698667i
$$591$$ 0 0
$$592$$ 4.00000 0.164399
$$593$$ 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
$$598$$ − 41.5692i − 1.69989i
$$599$$ − 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 $$0.177766\pi$$
−0.848067 + 0.529889i $$0.822234\pi$$
$$602$$ 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
$$610$$ 12.0000 0.485866
$$611$$ − 63.6396i − 2.57458i
$$612$$ 0 0
$$613$$ 8.00000 0.323117 0.161558 0.986863i $$-0.448348\pi$$
0.161558 + 0.986863i $$0.448348\pi$$
$$614$$ 22.0454 0.889680
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 24.0416i 0.967880i 0.875101 + 0.483940i $$0.160795\pi$$
−0.875101 + 0.483940i $$0.839205\pi$$
$$618$$ 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
$$622$$ 24.2487i 0.972285i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ −17.1464 −0.685309
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4.89898 −0.195335
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ − 14.1421i − 0.562544i
$$633$$ 0 0
$$634$$ −20.0000 −0.794301
$$635$$ 26.9444 1.06926
$$636$$ 0 0
$$637$$ 0 0
$$638$$ − 5.65685i − 0.223957i
$$639$$ 0 0
$$640$$ 27.7128i 1.09545i
$$641$$ − 14.1421i − 0.558581i −0.960207 0.279290i $$-0.909901\pi$$
0.960207 0.279290i $$-0.0900992\pi$$
$$642$$ 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
$$646$$ −12.0000 −0.472134
$$647$$ −17.1464 −0.674096 −0.337048 0.941488i $$-0.609428\pi$$
−0.337048 + 0.941488i $$0.609428\pi$$
$$648$$ 0 0
$$649$$ − 6.92820i − 0.271956i
$$650$$ 7.34847 0.288231
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 7.07107i 0.276712i 0.990383 + 0.138356i $$0.0441819\pi$$
−0.990383 + 0.138356i $$0.955818\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 29.3939 1.14764
$$657$$ 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 $$-0.805921\pi$$
0.819810 0.572636i $$-0.194079\pi$$
$$662$$ 43.8406i 1.70391i
$$663$$ 0 0
$$664$$ − 20.7846i − 0.806599i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ 0 0
$$670$$ − 38.1051i − 1.47213i
$$671$$ 4.89898 0.189123
$$672$$ 0 0
$$673$$ 35.0000 1.34915 0.674575 0.738206i $$-0.264327\pi$$
0.674575 + 0.738206i $$0.264327\pi$$
$$674$$ − 32.5269i − 1.25289i
$$675$$ 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
$$680$$ − 33.9411i − 1.30158i
$$681$$ 0 0
$$682$$ − 3.46410i − 0.132647i
$$683$$ − 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
$$688$$ 4.00000 0.152499
$$689$$ −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
$$694$$ −44.0000 −1.67022
$$695$$ 12.7279i 0.482798i
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 14.6969 0.556287
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 5.65685i − 0.213656i −0.994277 0.106828i $$-0.965931\pi$$
0.994277 0.106828i $$-0.0340695\pi$$
$$702$$ 0 0
$$703$$ 1.73205i 0.0653255i
$$704$$ 11.3137i 0.426401i
$$705$$ 0 0
$$706$$ 24.2487i 0.912612i
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −40.0000 −1.50223 −0.751116 0.660171i $$-0.770484\pi$$
−0.751116 + 0.660171i $$0.770484\pi$$
$$710$$ 24.4949 0.919277
$$711$$ 0 0
$$712$$ 13.8564i 0.519291i
$$713$$ 9.79796 0.366936
$$714$$ 0 0
$$715$$ 18.0000 0.673162
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 40.0000 1.49279
$$719$$ −26.9444 −1.00486 −0.502428 0.864619i $$-0.667560\pi$$
−0.502428 + 0.864619i $$0.667560\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 22.6274i − 0.842105i
$$723$$ 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
$$730$$ 6.00000 0.222070
$$731$$ −4.89898 −0.181195
$$732$$ 0 0
$$733$$ 39.8372i 1.47142i 0.677297 + 0.735710i $$0.263151\pi$$
−0.677297 + 0.735710i $$0.736849\pi$$
$$734$$ 2.44949 0.0904123
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 15.5563i − 0.573025i
$$738$$ 0 0
$$739$$ −1.00000 −0.0367856 −0.0183928 0.999831i $$-0.505855\pi$$
−0.0183928 + 0.999831i $$0.505855\pi$$
$$740$$ 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
$$746$$ − 41.0122i − 1.50156i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29.0000 1.05823 0.529113 0.848552i $$-0.322525\pi$$
0.529113 + 0.848552i $$0.322525\pi$$
$$752$$ 48.9898 1.78647
$$753$$ 0 0
$$754$$ 20.7846i 0.756931i
$$755$$ −53.8888 −1.96121
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 9.89949i 0.359566i
$$759$$ 0 0
$$760$$ −12.0000 −0.435286
$$761$$ 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
$$766$$ 27.7128i 1.00130i
$$767$$ 25.4558i 0.919157i
$$768$$ 0 0
$$769$$ − 25.9808i − 0.936890i −0.883493 0.468445i $$-0.844814\pi$$
0.883493 0.468445i $$-0.155186\pi$$
$$770$$ 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
$$776$$ 29.3939 1.05518
$$777$$ 0 0
$$778$$ −38.0000 −1.36237
$$779$$ 12.7279i 0.456025i
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ −39.1918 −1.40150
$$783$$ 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
$$790$$ − 17.3205i − 0.616236i
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −18.0000 −0.639199
$$794$$ −2.44949 −0.0869291
$$795$$ 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
$$802$$ 28.0000 0.988714
$$803$$ 2.44949 0.0864406
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 12.7279i 0.448322i
$$807$$ 0 0
$$808$$ 48.4974i 1.70613i
$$809$$ 41.0122i 1.44191i 0.692981 + 0.720956i $$0.256297\pi$$
−0.692981 + 0.720956i $$0.743703\pi$$
$$810$$ 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
$$814$$ 2.00000 0.0701000
$$815$$ −24.4949 −0.858019
$$816$$ 0 0
$$817$$ 1.73205i 0.0605968i
$$818$$ 46.5403 1.62724
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.0416i 0.839059i 0.907742 + 0.419529i $$0.137805\pi$$
−0.907742 + 0.419529i $$0.862195\pi$$
$$822$$ 0 0
$$823$$ −34.0000 −1.18517 −0.592583 0.805510i $$-0.701892\pi$$
−0.592583 + 0.805510i $$0.701892\pi$$
$$824$$ 24.4949 0.853320
$$825$$ 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 $$0.00957567\pi$$
−0.999548 + 0.0300783i $$0.990424\pi$$
$$830$$ − 25.4558i − 0.883585i
$$831$$ 0 0
$$832$$ − 41.5692i − 1.44115i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −18.0000 −0.622916
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 51.9615i − 1.79498i
$$839$$ −14.6969 −0.507395 −0.253697 0.967284i $$-0.581647\pi$$
−0.253697 + 0.967284i $$0.581647\pi$$
$$840$$ 0 0
$$841$$ 21.0000 0.724138
$$842$$ 1.41421i 0.0487370i
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −34.2929 −1.17971
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 11.3137i − 0.388514i
$$849$$ 0 0
$$850$$ − 6.92820i − 0.237635i
$$851$$ 5.65685i 0.193914i
$$852$$ 0 0
$$853$$ − 36.3731i − 1.24539i −0.782465 0.622695i $$-0.786038\pi$$
0.782465 0.622695i $$-0.213962\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ −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
$$862$$ 22.0000 0.749323
$$863$$ 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
$$866$$ −22.0454 −0.749133
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 7.07107i − 0.239870i
$$870$$ 0 0
$$871$$ 57.1577i 1.93671i
$$872$$ 2.82843i 0.0957826i
$$873$$ 0 0
$$874$$ 13.8564i 0.468700i
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000 0.675352 0.337676 0.941262i $$-0.390359\pi$$
0.337676 + 0.941262i $$0.390359\pi$$
$$878$$ −39.1918 −1.32266
$$879$$ 0 0
$$880$$ 13.8564i 0.467099i
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 11.0000 0.370179 0.185090 0.982722i $$-0.440742\pi$$
0.185090 + 0.982722i $$0.440742\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −56.0000 −1.88136
$$887$$ −41.6413 −1.39818 −0.699089 0.715034i $$-0.746411\pi$$
−0.699089 + 0.715034i $$0.746411\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 16.9706i 0.568855i
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 21.2132i 0.709873i
$$894$$ 0 0
$$895$$ − 24.2487i − 0.810545i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 10.0000 0.333704
$$899$$ −4.89898 −0.163390
$$900$$ 0 0
$$901$$ 13.8564i 0.461624i
$$902$$ 14.6969 0.489355
$$903$$ 0 0
$$904$$ −4.00000 −0.133038
$$905$$ − 38.1838i − 1.26927i
$$906$$ 0 0
$$907$$ 5.00000 0.166022 0.0830111 0.996549i $$-0.473546\pi$$
0.0830111 + 0.996549i $$0.473546\pi$$
$$908$$ 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
$$914$$ − 7.07107i − 0.233890i
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 17.0000 0.560778 0.280389 0.959886i $$-0.409536\pi$$
0.280389 + 0.959886i $$0.409536\pi$$
$$920$$ −39.1918 −1.29212
$$921$$ 0 0
$$922$$ 20.7846i 0.684505i
$$923$$ −36.7423 −1.20939
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 18.3848i 0.604161i
$$927$$ 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
$$934$$ 38.1051i 1.24684i
$$935$$ − 16.9706i − 0.554997i
$$936$$ 0 0
$$937$$ − 46.7654i − 1.52776i −0.645359 0.763879i $$-0.723292\pi$$
0.645359 0.763879i $$-0.276708\pi$$
$$938$$ 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
$$944$$ −19.5959 −0.637793
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ 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
$$950$$ −2.44949 −0.0794719
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 5.65685i − 0.183243i −0.995794 0.0916217i $$-0.970795\pi$$
0.995794 0.0916217i $$-0.0292051\pi$$
$$954$$ 0 0
$$955$$ − 3.46410i − 0.112096i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ − 6.92820i − 0.223840i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 28.0000 0.903226
$$962$$ −7.34847 −0.236924
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 26.9444 0.867371
$$966$$ 0 0
$$967$$ −13.0000 −0.418052 −0.209026 0.977910i $$-0.567029\pi$$
−0.209026 + 0.977910i $$0.567029\pi$$
$$968$$ − 25.4558i − 0.818182i
$$969$$ 0 0
$$970$$ 36.0000 1.15589
$$971$$ 39.1918 1.25773 0.628863 0.777516i $$-0.283521\pi$$
0.628863 + 0.777516i $$0.283521\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ − 24.0416i − 0.770344i
$$975$$ 0 0
$$976$$ − 13.8564i − 0.443533i
$$977$$ 7.07107i 0.226224i 0.993582 + 0.113112i $$0.0360818\pi$$
−0.993582 + 0.113112i $$0.963918\pi$$
$$978$$ 0 0
$$979$$ 6.92820i 0.221426i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 16.0000 0.510581
$$983$$ 4.89898 0.156253 0.0781266 0.996943i $$-0.475106\pi$$
0.0781266 + 0.996943i $$0.475106\pi$$
$$984$$ 0 0
$$985$$ 48.4974i 1.54526i
$$986$$ 19.5959 0.624061
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 5.65685i 0.179878i
$$990$$ 0 0
$$991$$ 35.0000 1.11181 0.555906 0.831245i $$-0.312372\pi$$
0.555906 + 0.831245i $$0.312372\pi$$
$$992$$ 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 $$-0.845600\pi$$
0.884646 0.466264i $$-0.154400\pi$$
$$998$$ 35.3553i 1.11915i
$$999$$ 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 441.2.c.a.440.2 4
3.2 odd 2 inner 441.2.c.a.440.3 4
4.3 odd 2 7056.2.k.b.881.4 4
7.2 even 3 441.2.p.a.80.1 4
7.3 odd 6 441.2.p.a.215.2 4
7.4 even 3 63.2.p.a.26.2 yes 4
7.5 odd 6 63.2.p.a.17.1 4
7.6 odd 2 inner 441.2.c.a.440.1 4
12.11 even 2 7056.2.k.b.881.1 4
21.2 odd 6 441.2.p.a.80.2 4
21.5 even 6 63.2.p.a.17.2 yes 4
21.11 odd 6 63.2.p.a.26.1 yes 4
21.17 even 6 441.2.p.a.215.1 4
21.20 even 2 inner 441.2.c.a.440.4 4
28.11 odd 6 1008.2.bt.b.593.1 4
28.19 even 6 1008.2.bt.b.17.2 4
28.27 even 2 7056.2.k.b.881.2 4
35.4 even 6 1575.2.bk.c.26.1 4
35.12 even 12 1575.2.bc.a.899.1 8
35.18 odd 12 1575.2.bc.a.1349.3 8
35.19 odd 6 1575.2.bk.c.1151.2 4
35.32 odd 12 1575.2.bc.a.1349.2 8
35.33 even 12 1575.2.bc.a.899.4 8
63.4 even 3 567.2.s.d.26.1 4
63.5 even 6 567.2.i.d.269.2 4
63.11 odd 6 567.2.i.d.215.2 4
63.25 even 3 567.2.i.d.215.1 4
63.32 odd 6 567.2.s.d.26.2 4
63.40 odd 6 567.2.i.d.269.1 4
63.47 even 6 567.2.s.d.458.1 4
63.61 odd 6 567.2.s.d.458.2 4
84.11 even 6 1008.2.bt.b.593.2 4
84.47 odd 6 1008.2.bt.b.17.1 4
84.83 odd 2 7056.2.k.b.881.3 4
105.32 even 12 1575.2.bc.a.1349.4 8
105.47 odd 12 1575.2.bc.a.899.3 8
105.53 even 12 1575.2.bc.a.1349.1 8
105.68 odd 12 1575.2.bc.a.899.2 8
105.74 odd 6 1575.2.bk.c.26.2 4
105.89 even 6 1575.2.bk.c.1151.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.p.a.17.1 4 7.5 odd 6
63.2.p.a.17.2 yes 4 21.5 even 6
63.2.p.a.26.1 yes 4 21.11 odd 6
63.2.p.a.26.2 yes 4 7.4 even 3
441.2.c.a.440.1 4 7.6 odd 2 inner
441.2.c.a.440.2 4 1.1 even 1 trivial
441.2.c.a.440.3 4 3.2 odd 2 inner
441.2.c.a.440.4 4 21.20 even 2 inner
441.2.p.a.80.1 4 7.2 even 3
441.2.p.a.80.2 4 21.2 odd 6
441.2.p.a.215.1 4 21.17 even 6
441.2.p.a.215.2 4 7.3 odd 6
567.2.i.d.215.1 4 63.25 even 3
567.2.i.d.215.2 4 63.11 odd 6
567.2.i.d.269.1 4 63.40 odd 6
567.2.i.d.269.2 4 63.5 even 6
567.2.s.d.26.1 4 63.4 even 3
567.2.s.d.26.2 4 63.32 odd 6
567.2.s.d.458.1 4 63.47 even 6
567.2.s.d.458.2 4 63.61 odd 6
1008.2.bt.b.17.1 4 84.47 odd 6
1008.2.bt.b.17.2 4 28.19 even 6
1008.2.bt.b.593.1 4 28.11 odd 6
1008.2.bt.b.593.2 4 84.11 even 6
1575.2.bc.a.899.1 8 35.12 even 12
1575.2.bc.a.899.2 8 105.68 odd 12
1575.2.bc.a.899.3 8 105.47 odd 12
1575.2.bc.a.899.4 8 35.33 even 12
1575.2.bc.a.1349.1 8 105.53 even 12
1575.2.bc.a.1349.2 8 35.32 odd 12
1575.2.bc.a.1349.3 8 35.18 odd 12
1575.2.bc.a.1349.4 8 105.32 even 12
1575.2.bk.c.26.1 4 35.4 even 6
1575.2.bk.c.26.2 4 105.74 odd 6
1575.2.bk.c.1151.1 4 105.89 even 6
1575.2.bk.c.1151.2 4 35.19 odd 6
7056.2.k.b.881.1 4 12.11 even 2
7056.2.k.b.881.2 4 28.27 even 2
7056.2.k.b.881.3 4 84.83 odd 2
7056.2.k.b.881.4 4 4.3 odd 2