# Properties

 Label 4400.2.a.bs.1.2 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## 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: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ 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: $$1$$ Twist minimal: no (minimal twist has level 275) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +O(q^{10})$$ $$q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} -6.90833 q^{17} +1.00000 q^{19} +1.60555 q^{21} -7.30278 q^{23} -1.60555 q^{27} +0.908327 q^{29} -10.2111 q^{31} +2.30278 q^{33} -2.39445 q^{37} -11.5139 q^{39} -5.60555 q^{41} +7.21110 q^{43} -3.00000 q^{47} -6.51388 q^{49} -15.9083 q^{51} +1.30278 q^{53} +2.30278 q^{57} +14.2111 q^{59} -7.90833 q^{61} +1.60555 q^{63} -4.00000 q^{67} -16.8167 q^{69} +2.60555 q^{71} +7.90833 q^{73} +0.697224 q^{77} +10.9083 q^{79} -10.6056 q^{81} -3.51388 q^{83} +2.09167 q^{87} +1.69722 q^{89} -3.48612 q^{91} -23.5139 q^{93} -15.3028 q^{97} +2.30278 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 5 q^{7} + q^{9}+O(q^{10})$$ 2 * q + q^3 + 5 * q^7 + q^9 $$2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51} - q^{53} + q^{57} + 14 q^{59} - 5 q^{61} - 4 q^{63} - 8 q^{67} - 12 q^{69} - 2 q^{71} + 5 q^{73} + 5 q^{77} + 11 q^{79} - 14 q^{81} + 11 q^{83} + 15 q^{87} + 7 q^{89} - 25 q^{91} - 29 q^{93} - 27 q^{97} + q^{99}+O(q^{100})$$ 2 * q + q^3 + 5 * q^7 + q^9 + 2 * q^11 - 10 * q^13 - 3 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 + 4 * q^27 - 9 * q^29 - 6 * q^31 + q^33 - 12 * q^37 - 5 * q^39 - 4 * q^41 - 6 * q^47 + 5 * q^49 - 21 * q^51 - q^53 + q^57 + 14 * q^59 - 5 * q^61 - 4 * q^63 - 8 * q^67 - 12 * q^69 - 2 * q^71 + 5 * q^73 + 5 * q^77 + 11 * q^79 - 14 * q^81 + 11 * q^83 + 15 * q^87 + 7 * q^89 - 25 * q^91 - 29 * q^93 - 27 * q^97 + q^99

## 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
$$3$$ 2.30278 1.32951 0.664754 0.747062i $$-0.268536\pi$$
0.664754 + 0.747062i $$0.268536\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.697224 0.263526 0.131763 0.991281i $$-0.457936\pi$$
0.131763 + 0.991281i $$0.457936\pi$$
$$8$$ 0 0
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −5.00000 −1.38675 −0.693375 0.720577i $$-0.743877\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.90833 −1.67552 −0.837758 0.546042i $$-0.816134\pi$$
−0.837758 + 0.546042i $$0.816134\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 1.60555 0.350360
$$22$$ 0 0
$$23$$ −7.30278 −1.52273 −0.761367 0.648321i $$-0.775471\pi$$
−0.761367 + 0.648321i $$0.775471\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.60555 −0.308988
$$28$$ 0 0
$$29$$ 0.908327 0.168672 0.0843360 0.996437i $$-0.473123\pi$$
0.0843360 + 0.996437i $$0.473123\pi$$
$$30$$ 0 0
$$31$$ −10.2111 −1.83397 −0.916984 0.398924i $$-0.869384\pi$$
−0.916984 + 0.398924i $$0.869384\pi$$
$$32$$ 0 0
$$33$$ 2.30278 0.400862
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.39445 −0.393645 −0.196822 0.980439i $$-0.563062\pi$$
−0.196822 + 0.980439i $$0.563062\pi$$
$$38$$ 0 0
$$39$$ −11.5139 −1.84370
$$40$$ 0 0
$$41$$ −5.60555 −0.875440 −0.437720 0.899111i $$-0.644214\pi$$
−0.437720 + 0.899111i $$0.644214\pi$$
$$42$$ 0 0
$$43$$ 7.21110 1.09968 0.549841 0.835269i $$-0.314688\pi$$
0.549841 + 0.835269i $$0.314688\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ −6.51388 −0.930554
$$50$$ 0 0
$$51$$ −15.9083 −2.22761
$$52$$ 0 0
$$53$$ 1.30278 0.178950 0.0894750 0.995989i $$-0.471481\pi$$
0.0894750 + 0.995989i $$0.471481\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.30278 0.305010
$$58$$ 0 0
$$59$$ 14.2111 1.85013 0.925064 0.379811i $$-0.124011\pi$$
0.925064 + 0.379811i $$0.124011\pi$$
$$60$$ 0 0
$$61$$ −7.90833 −1.01256 −0.506279 0.862370i $$-0.668979\pi$$
−0.506279 + 0.862370i $$0.668979\pi$$
$$62$$ 0 0
$$63$$ 1.60555 0.202280
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −16.8167 −2.02449
$$70$$ 0 0
$$71$$ 2.60555 0.309222 0.154611 0.987975i $$-0.450588\pi$$
0.154611 + 0.987975i $$0.450588\pi$$
$$72$$ 0 0
$$73$$ 7.90833 0.925600 0.462800 0.886463i $$-0.346845\pi$$
0.462800 + 0.886463i $$0.346845\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0.697224 0.0794561
$$78$$ 0 0
$$79$$ 10.9083 1.22728 0.613641 0.789585i $$-0.289704\pi$$
0.613641 + 0.789585i $$0.289704\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 0 0
$$83$$ −3.51388 −0.385698 −0.192849 0.981228i $$-0.561773\pi$$
−0.192849 + 0.981228i $$0.561773\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.09167 0.224251
$$88$$ 0 0
$$89$$ 1.69722 0.179905 0.0899527 0.995946i $$-0.471328\pi$$
0.0899527 + 0.995946i $$0.471328\pi$$
$$90$$ 0 0
$$91$$ −3.48612 −0.365445
$$92$$ 0 0
$$93$$ −23.5139 −2.43828
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −15.3028 −1.55376 −0.776881 0.629648i $$-0.783199\pi$$
−0.776881 + 0.629648i $$0.783199\pi$$
$$98$$ 0 0
$$99$$ 2.30278 0.231438
$$100$$ 0 0
$$101$$ −0.513878 −0.0511328 −0.0255664 0.999673i $$-0.508139\pi$$
−0.0255664 + 0.999673i $$0.508139\pi$$
$$102$$ 0 0
$$103$$ 2.90833 0.286566 0.143283 0.989682i $$-0.454234\pi$$
0.143283 + 0.989682i $$0.454234\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ 11.5139 1.10283 0.551415 0.834231i $$-0.314088\pi$$
0.551415 + 0.834231i $$0.314088\pi$$
$$110$$ 0 0
$$111$$ −5.51388 −0.523354
$$112$$ 0 0
$$113$$ 10.8167 1.01755 0.508773 0.860901i $$-0.330099\pi$$
0.508773 + 0.860901i $$0.330099\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −11.5139 −1.06446
$$118$$ 0 0
$$119$$ −4.81665 −0.441542
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −12.9083 −1.16390
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.11943 0.720483 0.360241 0.932859i $$-0.382694\pi$$
0.360241 + 0.932859i $$0.382694\pi$$
$$128$$ 0 0
$$129$$ 16.6056 1.46204
$$130$$ 0 0
$$131$$ 9.90833 0.865695 0.432847 0.901467i $$-0.357509\pi$$
0.432847 + 0.901467i $$0.357509\pi$$
$$132$$ 0 0
$$133$$ 0.697224 0.0604570
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.9083 1.10283 0.551416 0.834230i $$-0.314088\pi$$
0.551416 + 0.834230i $$0.314088\pi$$
$$138$$ 0 0
$$139$$ 6.21110 0.526819 0.263409 0.964684i $$-0.415153\pi$$
0.263409 + 0.964684i $$0.415153\pi$$
$$140$$ 0 0
$$141$$ −6.90833 −0.581786
$$142$$ 0 0
$$143$$ −5.00000 −0.418121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −15.0000 −1.23718
$$148$$ 0 0
$$149$$ 17.2111 1.40999 0.704994 0.709213i $$-0.250950\pi$$
0.704994 + 0.709213i $$0.250950\pi$$
$$150$$ 0 0
$$151$$ −0.816654 −0.0664583 −0.0332292 0.999448i $$-0.510579\pi$$
−0.0332292 + 0.999448i $$0.510579\pi$$
$$152$$ 0 0
$$153$$ −15.9083 −1.28611
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −19.2111 −1.53321 −0.766606 0.642117i $$-0.778056\pi$$
−0.766606 + 0.642117i $$0.778056\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −5.09167 −0.401280
$$162$$ 0 0
$$163$$ 9.30278 0.728650 0.364325 0.931272i $$-0.381300\pi$$
0.364325 + 0.931272i $$0.381300\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 13.4222 1.03864 0.519321 0.854579i $$-0.326185\pi$$
0.519321 + 0.854579i $$0.326185\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 2.30278 0.176098
$$172$$ 0 0
$$173$$ 4.81665 0.366203 0.183102 0.983094i $$-0.441386\pi$$
0.183102 + 0.983094i $$0.441386\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 32.7250 2.45976
$$178$$ 0 0
$$179$$ −12.5139 −0.935331 −0.467666 0.883905i $$-0.654905\pi$$
−0.467666 + 0.883905i $$0.654905\pi$$
$$180$$ 0 0
$$181$$ −19.9083 −1.47977 −0.739887 0.672731i $$-0.765121\pi$$
−0.739887 + 0.672731i $$0.765121\pi$$
$$182$$ 0 0
$$183$$ −18.2111 −1.34620
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.90833 −0.505187
$$188$$ 0 0
$$189$$ −1.11943 −0.0814265
$$190$$ 0 0
$$191$$ −10.3028 −0.745483 −0.372741 0.927935i $$-0.621582\pi$$
−0.372741 + 0.927935i $$0.621582\pi$$
$$192$$ 0 0
$$193$$ −13.2111 −0.950956 −0.475478 0.879728i $$-0.657725\pi$$
−0.475478 + 0.879728i $$0.657725\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.3028 −0.947784 −0.473892 0.880583i $$-0.657151\pi$$
−0.473892 + 0.880583i $$0.657151\pi$$
$$198$$ 0 0
$$199$$ 6.48612 0.459789 0.229894 0.973216i $$-0.426162\pi$$
0.229894 + 0.973216i $$0.426162\pi$$
$$200$$ 0 0
$$201$$ −9.21110 −0.649701
$$202$$ 0 0
$$203$$ 0.633308 0.0444495
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −16.8167 −1.16884
$$208$$ 0 0
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 25.2389 1.73751 0.868757 0.495238i $$-0.164919\pi$$
0.868757 + 0.495238i $$0.164919\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −7.11943 −0.483298
$$218$$ 0 0
$$219$$ 18.2111 1.23059
$$220$$ 0 0
$$221$$ 34.5416 2.32352
$$222$$ 0 0
$$223$$ −22.6333 −1.51564 −0.757819 0.652465i $$-0.773735\pi$$
−0.757819 + 0.652465i $$0.773735\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1.69722 0.112649 0.0563244 0.998413i $$-0.482062\pi$$
0.0563244 + 0.998413i $$0.482062\pi$$
$$228$$ 0 0
$$229$$ −18.7250 −1.23738 −0.618691 0.785635i $$-0.712337\pi$$
−0.618691 + 0.785635i $$0.712337\pi$$
$$230$$ 0 0
$$231$$ 1.60555 0.105638
$$232$$ 0 0
$$233$$ −15.9083 −1.04219 −0.521095 0.853499i $$-0.674476\pi$$
−0.521095 + 0.853499i $$0.674476\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 25.1194 1.63168
$$238$$ 0 0
$$239$$ −21.1194 −1.36610 −0.683051 0.730371i $$-0.739347\pi$$
−0.683051 + 0.730371i $$0.739347\pi$$
$$240$$ 0 0
$$241$$ 21.9361 1.41303 0.706514 0.707699i $$-0.250267\pi$$
0.706514 + 0.707699i $$0.250267\pi$$
$$242$$ 0 0
$$243$$ −19.6056 −1.25770
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 0 0
$$249$$ −8.09167 −0.512789
$$250$$ 0 0
$$251$$ −6.90833 −0.436050 −0.218025 0.975943i $$-0.569961\pi$$
−0.218025 + 0.975943i $$0.569961\pi$$
$$252$$ 0 0
$$253$$ −7.30278 −0.459122
$$254$$ 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
$$259$$ −1.66947 −0.103736
$$260$$ 0 0
$$261$$ 2.09167 0.129471
$$262$$ 0 0
$$263$$ −22.8167 −1.40694 −0.703468 0.710727i $$-0.748366\pi$$
−0.703468 + 0.710727i $$0.748366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 3.90833 0.239186
$$268$$ 0 0
$$269$$ 8.72498 0.531971 0.265986 0.963977i $$-0.414303\pi$$
0.265986 + 0.963977i $$0.414303\pi$$
$$270$$ 0 0
$$271$$ 0.211103 0.0128236 0.00641178 0.999979i $$-0.497959\pi$$
0.00641178 + 0.999979i $$0.497959\pi$$
$$272$$ 0 0
$$273$$ −8.02776 −0.485862
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.3944 −0.864879 −0.432439 0.901663i $$-0.642347\pi$$
−0.432439 + 0.901663i $$0.642347\pi$$
$$278$$ 0 0
$$279$$ −23.5139 −1.40774
$$280$$ 0 0
$$281$$ −1.18335 −0.0705925 −0.0352963 0.999377i $$-0.511237\pi$$
−0.0352963 + 0.999377i $$0.511237\pi$$
$$282$$ 0 0
$$283$$ 6.30278 0.374661 0.187331 0.982297i $$-0.440016\pi$$
0.187331 + 0.982297i $$0.440016\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3.90833 −0.230701
$$288$$ 0 0
$$289$$ 30.7250 1.80735
$$290$$ 0 0
$$291$$ −35.2389 −2.06574
$$292$$ 0 0
$$293$$ −0.788897 −0.0460879 −0.0230439 0.999734i $$-0.507336\pi$$
−0.0230439 + 0.999734i $$0.507336\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1.60555 −0.0931635
$$298$$ 0 0
$$299$$ 36.5139 2.11165
$$300$$ 0 0
$$301$$ 5.02776 0.289795
$$302$$ 0 0
$$303$$ −1.18335 −0.0679815
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.9083 −0.965009 −0.482505 0.875893i $$-0.660273\pi$$
−0.482505 + 0.875893i $$0.660273\pi$$
$$308$$ 0 0
$$309$$ 6.69722 0.380992
$$310$$ 0 0
$$311$$ −4.81665 −0.273127 −0.136564 0.990631i $$-0.543606\pi$$
−0.136564 + 0.990631i $$0.543606\pi$$
$$312$$ 0 0
$$313$$ −0.183346 −0.0103633 −0.00518167 0.999987i $$-0.501649\pi$$
−0.00518167 + 0.999987i $$0.501649\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0.908327 0.0510167 0.0255084 0.999675i $$-0.491880\pi$$
0.0255084 + 0.999675i $$0.491880\pi$$
$$318$$ 0 0
$$319$$ 0.908327 0.0508565
$$320$$ 0 0
$$321$$ −6.90833 −0.385585
$$322$$ 0 0
$$323$$ −6.90833 −0.384390
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 26.5139 1.46622
$$328$$ 0 0
$$329$$ −2.09167 −0.115318
$$330$$ 0 0
$$331$$ 21.6056 1.18755 0.593774 0.804632i $$-0.297637\pi$$
0.593774 + 0.804632i $$0.297637\pi$$
$$332$$ 0 0
$$333$$ −5.51388 −0.302159
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 30.8444 1.68020 0.840101 0.542430i $$-0.182496\pi$$
0.840101 + 0.542430i $$0.182496\pi$$
$$338$$ 0 0
$$339$$ 24.9083 1.35283
$$340$$ 0 0
$$341$$ −10.2111 −0.552962
$$342$$ 0 0
$$343$$ −9.42221 −0.508751
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.5139 −0.671780 −0.335890 0.941901i $$-0.609037\pi$$
−0.335890 + 0.941901i $$0.609037\pi$$
$$348$$ 0 0
$$349$$ −5.18335 −0.277458 −0.138729 0.990330i $$-0.544302\pi$$
−0.138729 + 0.990330i $$0.544302\pi$$
$$350$$ 0 0
$$351$$ 8.02776 0.428490
$$352$$ 0 0
$$353$$ −18.6333 −0.991751 −0.495875 0.868394i $$-0.665153\pi$$
−0.495875 + 0.868394i $$0.665153\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −11.0917 −0.587034
$$358$$ 0 0
$$359$$ −0.788897 −0.0416364 −0.0208182 0.999783i $$-0.506627\pi$$
−0.0208182 + 0.999783i $$0.506627\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 2.30278 0.120864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −20.6972 −1.08039 −0.540193 0.841541i $$-0.681649\pi$$
−0.540193 + 0.841541i $$0.681649\pi$$
$$368$$ 0 0
$$369$$ −12.9083 −0.671981
$$370$$ 0 0
$$371$$ 0.908327 0.0471580
$$372$$ 0 0
$$373$$ −27.4222 −1.41987 −0.709934 0.704268i $$-0.751275\pi$$
−0.709934 + 0.704268i $$0.751275\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.54163 −0.233906
$$378$$ 0 0
$$379$$ −3.18335 −0.163518 −0.0817588 0.996652i $$-0.526054\pi$$
−0.0817588 + 0.996652i $$0.526054\pi$$
$$380$$ 0 0
$$381$$ 18.6972 0.957888
$$382$$ 0 0
$$383$$ 21.6333 1.10541 0.552705 0.833377i $$-0.313596\pi$$
0.552705 + 0.833377i $$0.313596\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 16.6056 0.844108
$$388$$ 0 0
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 50.4500 2.55136
$$392$$ 0 0
$$393$$ 22.8167 1.15095
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −21.6972 −1.08895 −0.544476 0.838776i $$-0.683272\pi$$
−0.544476 + 0.838776i $$0.683272\pi$$
$$398$$ 0 0
$$399$$ 1.60555 0.0803781
$$400$$ 0 0
$$401$$ −12.7889 −0.638647 −0.319324 0.947646i $$-0.603456\pi$$
−0.319324 + 0.947646i $$0.603456\pi$$
$$402$$ 0 0
$$403$$ 51.0555 2.54326
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.39445 −0.118688
$$408$$ 0 0
$$409$$ −6.21110 −0.307119 −0.153560 0.988139i $$-0.549074\pi$$
−0.153560 + 0.988139i $$0.549074\pi$$
$$410$$ 0 0
$$411$$ 29.7250 1.46623
$$412$$ 0 0
$$413$$ 9.90833 0.487557
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.3028 0.700410
$$418$$ 0 0
$$419$$ −6.39445 −0.312389 −0.156195 0.987726i $$-0.549923\pi$$
−0.156195 + 0.987726i $$0.549923\pi$$
$$420$$ 0 0
$$421$$ 0.697224 0.0339806 0.0169903 0.999856i $$-0.494592\pi$$
0.0169903 + 0.999856i $$0.494592\pi$$
$$422$$ 0 0
$$423$$ −6.90833 −0.335894
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.51388 −0.266835
$$428$$ 0 0
$$429$$ −11.5139 −0.555895
$$430$$ 0 0
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ 0 0
$$433$$ −5.00000 −0.240285 −0.120142 0.992757i $$-0.538335\pi$$
−0.120142 + 0.992757i $$0.538335\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.30278 −0.349339
$$438$$ 0 0
$$439$$ −24.3028 −1.15991 −0.579954 0.814649i $$-0.696930\pi$$
−0.579954 + 0.814649i $$0.696930\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ −8.60555 −0.408862 −0.204431 0.978881i $$-0.565534\pi$$
−0.204431 + 0.978881i $$0.565534\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 39.6333 1.87459
$$448$$ 0 0
$$449$$ 23.4861 1.10838 0.554189 0.832391i $$-0.313028\pi$$
0.554189 + 0.832391i $$0.313028\pi$$
$$450$$ 0 0
$$451$$ −5.60555 −0.263955
$$452$$ 0 0
$$453$$ −1.88057 −0.0883569
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.6972 0.968175 0.484088 0.875020i $$-0.339152\pi$$
0.484088 + 0.875020i $$0.339152\pi$$
$$458$$ 0 0
$$459$$ 11.0917 0.517715
$$460$$ 0 0
$$461$$ 32.2111 1.50022 0.750110 0.661313i $$-0.230000\pi$$
0.750110 + 0.661313i $$0.230000\pi$$
$$462$$ 0 0
$$463$$ 11.7889 0.547877 0.273938 0.961747i $$-0.411674\pi$$
0.273938 + 0.961747i $$0.411674\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18.6333 −0.862247 −0.431123 0.902293i $$-0.641883\pi$$
−0.431123 + 0.902293i $$0.641883\pi$$
$$468$$ 0 0
$$469$$ −2.78890 −0.128779
$$470$$ 0 0
$$471$$ −44.2389 −2.03842
$$472$$ 0 0
$$473$$ 7.21110 0.331567
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3.00000 0.137361
$$478$$ 0 0
$$479$$ 34.8167 1.59081 0.795407 0.606076i $$-0.207257\pi$$
0.795407 + 0.606076i $$0.207257\pi$$
$$480$$ 0 0
$$481$$ 11.9722 0.545887
$$482$$ 0 0
$$483$$ −11.7250 −0.533505
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.21110 0.190823 0.0954116 0.995438i $$-0.469583\pi$$
0.0954116 + 0.995438i $$0.469583\pi$$
$$488$$ 0 0
$$489$$ 21.4222 0.968746
$$490$$ 0 0
$$491$$ 9.78890 0.441767 0.220883 0.975300i $$-0.429106\pi$$
0.220883 + 0.975300i $$0.429106\pi$$
$$492$$ 0 0
$$493$$ −6.27502 −0.282613
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.81665 0.0814881
$$498$$ 0 0
$$499$$ 3.48612 0.156060 0.0780301 0.996951i $$-0.475137\pi$$
0.0780301 + 0.996951i $$0.475137\pi$$
$$500$$ 0 0
$$501$$ 30.9083 1.38088
$$502$$ 0 0
$$503$$ 9.39445 0.418878 0.209439 0.977822i $$-0.432836\pi$$
0.209439 + 0.977822i $$0.432836\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 27.6333 1.22724
$$508$$ 0 0
$$509$$ −22.6972 −1.00604 −0.503018 0.864276i $$-0.667777\pi$$
−0.503018 + 0.864276i $$0.667777\pi$$
$$510$$ 0 0
$$511$$ 5.51388 0.243920
$$512$$ 0 0
$$513$$ −1.60555 −0.0708868
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.00000 −0.131940
$$518$$ 0 0
$$519$$ 11.0917 0.486870
$$520$$ 0 0
$$521$$ −41.4500 −1.81596 −0.907978 0.419018i $$-0.862374\pi$$
−0.907978 + 0.419018i $$0.862374\pi$$
$$522$$ 0 0
$$523$$ −32.4222 −1.41772 −0.708862 0.705347i $$-0.750791\pi$$
−0.708862 + 0.705347i $$0.750791\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 70.5416 3.07284
$$528$$ 0 0
$$529$$ 30.3305 1.31872
$$530$$ 0 0
$$531$$ 32.7250 1.42014
$$532$$ 0 0
$$533$$ 28.0278 1.21402
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −28.8167 −1.24353
$$538$$ 0 0
$$539$$ −6.51388 −0.280573
$$540$$ 0 0
$$541$$ 25.7250 1.10600 0.553002 0.833180i $$-0.313482\pi$$
0.553002 + 0.833180i $$0.313482\pi$$
$$542$$ 0 0
$$543$$ −45.8444 −1.96737
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7.11943 −0.304405 −0.152202 0.988349i $$-0.548637\pi$$
−0.152202 + 0.988349i $$0.548637\pi$$
$$548$$ 0 0
$$549$$ −18.2111 −0.777231
$$550$$ 0 0
$$551$$ 0.908327 0.0386960
$$552$$ 0 0
$$553$$ 7.60555 0.323421
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −19.4222 −0.822945 −0.411473 0.911422i $$-0.634985\pi$$
−0.411473 + 0.911422i $$0.634985\pi$$
$$558$$ 0 0
$$559$$ −36.0555 −1.52499
$$560$$ 0 0
$$561$$ −15.9083 −0.671650
$$562$$ 0 0
$$563$$ 8.09167 0.341023 0.170512 0.985356i $$-0.445458\pi$$
0.170512 + 0.985356i $$0.445458\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −7.39445 −0.310538
$$568$$ 0 0
$$569$$ −46.1472 −1.93459 −0.967295 0.253653i $$-0.918368\pi$$
−0.967295 + 0.253653i $$0.918368\pi$$
$$570$$ 0 0
$$571$$ −22.3305 −0.934504 −0.467252 0.884124i $$-0.654756\pi$$
−0.467252 + 0.884124i $$0.654756\pi$$
$$572$$ 0 0
$$573$$ −23.7250 −0.991125
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −44.3583 −1.84666 −0.923330 0.384008i $$-0.874544\pi$$
−0.923330 + 0.384008i $$0.874544\pi$$
$$578$$ 0 0
$$579$$ −30.4222 −1.26430
$$580$$ 0 0
$$581$$ −2.44996 −0.101642
$$582$$ 0 0
$$583$$ 1.30278 0.0539555
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 16.5416 0.682746 0.341373 0.939928i $$-0.389108\pi$$
0.341373 + 0.939928i $$0.389108\pi$$
$$588$$ 0 0
$$589$$ −10.2111 −0.420741
$$590$$ 0 0
$$591$$ −30.6333 −1.26009
$$592$$ 0 0
$$593$$ 6.39445 0.262589 0.131294 0.991343i $$-0.458087\pi$$
0.131294 + 0.991343i $$0.458087\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 14.9361 0.611293
$$598$$ 0 0
$$599$$ 24.9083 1.01773 0.508863 0.860847i $$-0.330066\pi$$
0.508863 + 0.860847i $$0.330066\pi$$
$$600$$ 0 0
$$601$$ −1.90833 −0.0778423 −0.0389211 0.999242i $$-0.512392\pi$$
−0.0389211 + 0.999242i $$0.512392\pi$$
$$602$$ 0 0
$$603$$ −9.21110 −0.375105
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.21110 0.292690 0.146345 0.989234i $$-0.453249\pi$$
0.146345 + 0.989234i $$0.453249\pi$$
$$608$$ 0 0
$$609$$ 1.45837 0.0590959
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 0 0
$$613$$ 15.8806 0.641410 0.320705 0.947179i $$-0.396080\pi$$
0.320705 + 0.947179i $$0.396080\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −3.39445 −0.136655 −0.0683277 0.997663i $$-0.521766\pi$$
−0.0683277 + 0.997663i $$0.521766\pi$$
$$618$$ 0 0
$$619$$ 11.4222 0.459097 0.229549 0.973297i $$-0.426275\pi$$
0.229549 + 0.973297i $$0.426275\pi$$
$$620$$ 0 0
$$621$$ 11.7250 0.470507
$$622$$ 0 0
$$623$$ 1.18335 0.0474098
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 2.30278 0.0919640
$$628$$ 0 0
$$629$$ 16.5416 0.659558
$$630$$ 0 0
$$631$$ −6.93608 −0.276121 −0.138061 0.990424i $$-0.544087\pi$$
−0.138061 + 0.990424i $$0.544087\pi$$
$$632$$ 0 0
$$633$$ 58.1194 2.31004
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 32.5694 1.29045
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 27.7889 1.09760 0.548798 0.835955i $$-0.315086\pi$$
0.548798 + 0.835955i $$0.315086\pi$$
$$642$$ 0 0
$$643$$ −22.0000 −0.867595 −0.433798 0.901010i $$-0.642827\pi$$
−0.433798 + 0.901010i $$0.642827\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −33.2389 −1.30675 −0.653377 0.757033i $$-0.726648\pi$$
−0.653377 + 0.757033i $$0.726648\pi$$
$$648$$ 0 0
$$649$$ 14.2111 0.557835
$$650$$ 0 0
$$651$$ −16.3944 −0.642549
$$652$$ 0 0
$$653$$ −6.11943 −0.239472 −0.119736 0.992806i $$-0.538205\pi$$
−0.119736 + 0.992806i $$0.538205\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.2111 0.710483
$$658$$ 0 0
$$659$$ 30.9083 1.20402 0.602009 0.798489i $$-0.294367\pi$$
0.602009 + 0.798489i $$0.294367\pi$$
$$660$$ 0 0
$$661$$ −8.81665 −0.342928 −0.171464 0.985190i $$-0.554850\pi$$
−0.171464 + 0.985190i $$0.554850\pi$$
$$662$$ 0 0
$$663$$ 79.5416 3.08914
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.63331 −0.256843
$$668$$ 0 0
$$669$$ −52.1194 −2.01505
$$670$$ 0 0
$$671$$ −7.90833 −0.305298
$$672$$ 0 0
$$673$$ −30.0278 −1.15748 −0.578742 0.815510i $$-0.696456\pi$$
−0.578742 + 0.815510i $$0.696456\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −24.2389 −0.931575 −0.465788 0.884897i $$-0.654229\pi$$
−0.465788 + 0.884897i $$0.654229\pi$$
$$678$$ 0 0
$$679$$ −10.6695 −0.409457
$$680$$ 0 0
$$681$$ 3.90833 0.149767
$$682$$ 0 0
$$683$$ 47.8444 1.83072 0.915358 0.402642i $$-0.131908\pi$$
0.915358 + 0.402642i $$0.131908\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −43.1194 −1.64511
$$688$$ 0 0
$$689$$ −6.51388 −0.248159
$$690$$ 0 0
$$691$$ −27.5416 −1.04773 −0.523867 0.851800i $$-0.675511\pi$$
−0.523867 + 0.851800i $$0.675511\pi$$
$$692$$ 0 0
$$693$$ 1.60555 0.0609898
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 38.7250 1.46681
$$698$$ 0 0
$$699$$ −36.6333 −1.38560
$$700$$ 0 0
$$701$$ −41.2111 −1.55652 −0.778261 0.627941i $$-0.783898\pi$$
−0.778261 + 0.627941i $$0.783898\pi$$
$$702$$ 0 0
$$703$$ −2.39445 −0.0903083
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −0.358288 −0.0134748
$$708$$ 0 0
$$709$$ −31.6333 −1.18801 −0.594007 0.804460i $$-0.702455\pi$$
−0.594007 + 0.804460i $$0.702455\pi$$
$$710$$ 0 0
$$711$$ 25.1194 0.942052
$$712$$ 0 0
$$713$$ 74.5694 2.79265
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −48.6333 −1.81624
$$718$$ 0 0
$$719$$ −7.18335 −0.267894 −0.133947 0.990989i $$-0.542765\pi$$
−0.133947 + 0.990989i $$0.542765\pi$$
$$720$$ 0 0
$$721$$ 2.02776 0.0755176
$$722$$ 0 0
$$723$$ 50.5139 1.87863
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −39.3305 −1.45869 −0.729344 0.684147i $$-0.760175\pi$$
−0.729344 + 0.684147i $$0.760175\pi$$
$$728$$ 0 0
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ −49.8167 −1.84254
$$732$$ 0 0
$$733$$ −19.6056 −0.724148 −0.362074 0.932149i $$-0.617931\pi$$
−0.362074 + 0.932149i $$0.617931\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ −35.1194 −1.29189 −0.645945 0.763384i $$-0.723536\pi$$
−0.645945 + 0.763384i $$0.723536\pi$$
$$740$$ 0 0
$$741$$ −11.5139 −0.422973
$$742$$ 0 0
$$743$$ 40.6972 1.49304 0.746518 0.665365i $$-0.231724\pi$$
0.746518 + 0.665365i $$0.231724\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −8.09167 −0.296059
$$748$$ 0 0
$$749$$ −2.09167 −0.0764281
$$750$$ 0 0
$$751$$ 45.3305 1.65413 0.827067 0.562103i $$-0.190008\pi$$
0.827067 + 0.562103i $$0.190008\pi$$
$$752$$ 0 0
$$753$$ −15.9083 −0.579732
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −49.0555 −1.78295 −0.891476 0.453067i $$-0.850330\pi$$
−0.891476 + 0.453067i $$0.850330\pi$$
$$758$$ 0 0
$$759$$ −16.8167 −0.610406
$$760$$ 0 0
$$761$$ −13.5778 −0.492195 −0.246097 0.969245i $$-0.579148\pi$$
−0.246097 + 0.969245i $$0.579148\pi$$
$$762$$ 0 0
$$763$$ 8.02776 0.290624
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −71.0555 −2.56567
$$768$$ 0 0
$$769$$ −5.18335 −0.186916 −0.0934581 0.995623i $$-0.529792\pi$$
−0.0934581 + 0.995623i $$0.529792\pi$$
$$770$$ 0 0
$$771$$ 41.4500 1.49278
$$772$$ 0 0
$$773$$ −3.11943 −0.112198 −0.0560990 0.998425i $$-0.517866\pi$$
−0.0560990 + 0.998425i $$0.517866\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −3.84441 −0.137917
$$778$$ 0 0
$$779$$ −5.60555 −0.200840
$$780$$ 0 0
$$781$$ 2.60555 0.0932340
$$782$$ 0 0
$$783$$ −1.45837 −0.0521177
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 10.2111 0.363986 0.181993 0.983300i $$-0.441745\pi$$
0.181993 + 0.983300i $$0.441745\pi$$
$$788$$ 0 0
$$789$$ −52.5416 −1.87053
$$790$$ 0 0
$$791$$ 7.54163 0.268150
$$792$$ 0 0
$$793$$ 39.5416 1.40416
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3.51388 −0.124468 −0.0622340 0.998062i $$-0.519822\pi$$
−0.0622340 + 0.998062i $$0.519822\pi$$
$$798$$ 0 0
$$799$$ 20.7250 0.733197
$$800$$ 0 0
$$801$$ 3.90833 0.138094
$$802$$ 0 0
$$803$$ 7.90833 0.279079
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.0917 0.707260
$$808$$ 0 0
$$809$$ 39.6333 1.39343 0.696716 0.717347i $$-0.254644\pi$$
0.696716 + 0.717347i $$0.254644\pi$$
$$810$$ 0 0
$$811$$ −38.8722 −1.36499 −0.682493 0.730892i $$-0.739104\pi$$
−0.682493 + 0.730892i $$0.739104\pi$$
$$812$$ 0 0
$$813$$ 0.486122 0.0170490
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.21110 0.252285
$$818$$ 0 0
$$819$$ −8.02776 −0.280513
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 0 0
$$823$$ 18.4222 0.642158 0.321079 0.947052i $$-0.395955\pi$$
0.321079 + 0.947052i $$0.395955\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 13.8167 0.480452 0.240226 0.970717i $$-0.422778\pi$$
0.240226 + 0.970717i $$0.422778\pi$$
$$828$$ 0 0
$$829$$ 29.7527 1.03336 0.516678 0.856180i $$-0.327169\pi$$
0.516678 + 0.856180i $$0.327169\pi$$
$$830$$ 0 0
$$831$$ −33.1472 −1.14986
$$832$$ 0 0
$$833$$ 45.0000 1.55916
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 16.3944 0.566675
$$838$$ 0 0
$$839$$ 9.11943 0.314838 0.157419 0.987532i $$-0.449683\pi$$
0.157419 + 0.987532i $$0.449683\pi$$
$$840$$ 0 0
$$841$$ −28.1749 −0.971550
$$842$$ 0 0
$$843$$ −2.72498 −0.0938533
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.697224 0.0239569
$$848$$ 0 0
$$849$$ 14.5139 0.498115
$$850$$ 0 0
$$851$$ 17.4861 0.599417
$$852$$ 0 0
$$853$$ 12.7250 0.435695 0.217848 0.975983i $$-0.430096\pi$$
0.217848 + 0.975983i $$0.430096\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −41.3944 −1.41236 −0.706180 0.708032i $$-0.749583\pi$$
−0.706180 + 0.708032i $$0.749583\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 0 0
$$863$$ −12.3944 −0.421912 −0.210956 0.977496i $$-0.567658\pi$$
−0.210956 + 0.977496i $$0.567658\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 70.7527 2.40289
$$868$$ 0 0
$$869$$ 10.9083 0.370040
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ −35.2389 −1.19265
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −20.0000 −0.675352 −0.337676 0.941262i $$-0.609641\pi$$
−0.337676 + 0.941262i $$0.609641\pi$$
$$878$$ 0 0
$$879$$ −1.81665 −0.0612742
$$880$$ 0 0
$$881$$ 19.5416 0.658374 0.329187 0.944265i $$-0.393225\pi$$
0.329187 + 0.944265i $$0.393225\pi$$
$$882$$ 0 0
$$883$$ 52.4500 1.76508 0.882541 0.470236i $$-0.155831\pi$$
0.882541 + 0.470236i $$0.155831\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −3.23886 −0.108750 −0.0543751 0.998521i $$-0.517317\pi$$
−0.0543751 + 0.998521i $$0.517317\pi$$
$$888$$ 0 0
$$889$$ 5.66106 0.189866
$$890$$ 0 0
$$891$$ −10.6056 −0.355299
$$892$$ 0 0
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 84.0833 2.80746
$$898$$ 0 0
$$899$$ −9.27502 −0.309339
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 11.5778 0.385285
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ 0 0
$$909$$ −1.18335 −0.0392491
$$910$$ 0 0
$$911$$ 24.7889 0.821293 0.410646 0.911795i $$-0.365303\pi$$
0.410646 + 0.911795i $$0.365303\pi$$
$$912$$ 0 0
$$913$$ −3.51388 −0.116292
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 6.90833 0.228133
$$918$$ 0 0
$$919$$ −26.7889 −0.883684 −0.441842 0.897093i $$-0.645675\pi$$
−0.441842 + 0.897093i $$0.645675\pi$$
$$920$$ 0 0
$$921$$ −38.9361 −1.28299
$$922$$ 0 0
$$923$$ −13.0278 −0.428814
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 6.69722 0.219966
$$928$$ 0 0
$$929$$ 53.6056 1.75874 0.879371 0.476138i $$-0.157964\pi$$
0.879371 + 0.476138i $$0.157964\pi$$
$$930$$ 0 0
$$931$$ −6.51388 −0.213484
$$932$$ 0 0
$$933$$ −11.0917 −0.363125
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 9.21110 0.300914 0.150457 0.988617i $$-0.451926\pi$$
0.150457 + 0.988617i $$0.451926\pi$$
$$938$$ 0 0
$$939$$ −0.422205 −0.0137781
$$940$$ 0 0
$$941$$ 59.6056 1.94309 0.971543 0.236864i $$-0.0761197\pi$$
0.971543 + 0.236864i $$0.0761197\pi$$
$$942$$ 0 0
$$943$$ 40.9361 1.33306
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.63331 0.215554 0.107777 0.994175i $$-0.465627\pi$$
0.107777 + 0.994175i $$0.465627\pi$$
$$948$$ 0 0
$$949$$ −39.5416 −1.28358
$$950$$ 0 0
$$951$$ 2.09167 0.0678271
$$952$$ 0 0
$$953$$ −37.2666 −1.20718 −0.603592 0.797293i $$-0.706264\pi$$
−0.603592 + 0.797293i $$0.706264\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 2.09167 0.0676142
$$958$$ 0 0
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ 73.2666 2.36344
$$962$$ 0 0
$$963$$ −6.90833 −0.222618
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.9083 0.479419 0.239710 0.970845i $$-0.422948\pi$$
0.239710 + 0.970845i $$0.422948\pi$$
$$968$$ 0 0
$$969$$ −15.9083 −0.511049
$$970$$ 0 0
$$971$$ −45.3583 −1.45562 −0.727808 0.685781i $$-0.759461\pi$$
−0.727808 + 0.685781i $$0.759461\pi$$
$$972$$ 0 0
$$973$$ 4.33053 0.138830
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.0278 1.66452 0.832258 0.554389i $$-0.187048\pi$$
0.832258 + 0.554389i $$0.187048\pi$$
$$978$$ 0 0
$$979$$ 1.69722 0.0542435
$$980$$ 0 0
$$981$$ 26.5139 0.846523
$$982$$ 0 0
$$983$$ 8.84441 0.282093 0.141046 0.990003i $$-0.454953\pi$$
0.141046 + 0.990003i $$0.454953\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −4.81665 −0.153316
$$988$$ 0 0
$$989$$ −52.6611 −1.67452
$$990$$ 0 0
$$991$$ 16.9083 0.537111 0.268555 0.963264i $$-0.413454\pi$$
0.268555 + 0.963264i $$0.413454\pi$$
$$992$$ 0 0
$$993$$ 49.7527 1.57886
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −46.7250 −1.47979 −0.739897 0.672720i $$-0.765126\pi$$
−0.739897 + 0.672720i $$0.765126\pi$$
$$998$$ 0 0
$$999$$ 3.84441 0.121632
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 4400.2.a.bs.1.2 2
4.3 odd 2 275.2.a.e.1.2 2
5.2 odd 4 4400.2.b.y.4049.1 4
5.3 odd 4 4400.2.b.y.4049.4 4
5.4 even 2 4400.2.a.bh.1.1 2
12.11 even 2 2475.2.a.t.1.1 2
20.3 even 4 275.2.b.c.199.2 4
20.7 even 4 275.2.b.c.199.3 4
20.19 odd 2 275.2.a.f.1.1 yes 2
44.43 even 2 3025.2.a.n.1.1 2
60.23 odd 4 2475.2.c.k.199.3 4
60.47 odd 4 2475.2.c.k.199.2 4
60.59 even 2 2475.2.a.o.1.2 2
220.219 even 2 3025.2.a.h.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 4.3 odd 2
275.2.a.f.1.1 yes 2 20.19 odd 2
275.2.b.c.199.2 4 20.3 even 4
275.2.b.c.199.3 4 20.7 even 4
2475.2.a.o.1.2 2 60.59 even 2
2475.2.a.t.1.1 2 12.11 even 2
2475.2.c.k.199.2 4 60.47 odd 4
2475.2.c.k.199.3 4 60.23 odd 4
3025.2.a.h.1.2 2 220.219 even 2
3025.2.a.n.1.1 2 44.43 even 2
4400.2.a.bh.1.1 2 5.4 even 2
4400.2.a.bs.1.2 2 1.1 even 1 trivial
4400.2.b.y.4049.1 4 5.2 odd 4
4400.2.b.y.4049.4 4 5.3 odd 4