# Properties

 Label 2028.4.b.a.337.1 Level $2028$ Weight $4$ Character 2028.337 Analytic conductor $119.656$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (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: $$119.655873492$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 156) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2028.337 Dual form 2028.4.b.a.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -6.00000i q^{5} +4.00000i q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -6.00000i q^{5} +4.00000i q^{7} +9.00000 q^{9} -36.0000i q^{11} +18.0000i q^{15} -66.0000 q^{17} +56.0000i q^{19} -12.0000i q^{21} -96.0000 q^{23} +89.0000 q^{25} -27.0000 q^{27} +222.000 q^{29} +260.000i q^{31} +108.000i q^{33} +24.0000 q^{35} +106.000i q^{37} -90.0000i q^{41} -44.0000 q^{43} -54.0000i q^{45} -168.000i q^{47} +327.000 q^{49} +198.000 q^{51} +30.0000 q^{53} -216.000 q^{55} -168.000i q^{57} -348.000i q^{59} -346.000 q^{61} +36.0000i q^{63} -256.000i q^{67} +288.000 q^{69} -168.000i q^{71} +814.000i q^{73} -267.000 q^{75} +144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} +1236.00i q^{83} +396.000i q^{85} -666.000 q^{87} -318.000i q^{89} -780.000i q^{93} +336.000 q^{95} -502.000i q^{97} -324.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 18 * q^9 $$2 q - 6 q^{3} + 18 q^{9} - 132 q^{17} - 192 q^{23} + 178 q^{25} - 54 q^{27} + 444 q^{29} + 48 q^{35} - 88 q^{43} + 654 q^{49} + 396 q^{51} + 60 q^{53} - 432 q^{55} - 692 q^{61} + 576 q^{69} - 534 q^{75} + 288 q^{77} + 400 q^{79} + 162 q^{81} - 1332 q^{87} + 672 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 + 18 * q^9 - 132 * q^17 - 192 * q^23 + 178 * q^25 - 54 * q^27 + 444 * q^29 + 48 * q^35 - 88 * q^43 + 654 * q^49 + 396 * q^51 + 60 * q^53 - 432 * q^55 - 692 * q^61 + 576 * q^69 - 534 * q^75 + 288 * q^77 + 400 * q^79 + 162 * q^81 - 1332 * q^87 + 672 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ − 6.00000i − 0.536656i −0.963328 0.268328i $$-0.913529\pi$$
0.963328 0.268328i $$-0.0864711\pi$$
$$6$$ 0 0
$$7$$ 4.00000i 0.215980i 0.994152 + 0.107990i $$0.0344414\pi$$
−0.994152 + 0.107990i $$0.965559\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ − 36.0000i − 0.986764i −0.869813 0.493382i $$-0.835760\pi$$
0.869813 0.493382i $$-0.164240\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 18.0000i 0.309839i
$$16$$ 0 0
$$17$$ −66.0000 −0.941609 −0.470804 0.882238i $$-0.656036\pi$$
−0.470804 + 0.882238i $$0.656036\pi$$
$$18$$ 0 0
$$19$$ 56.0000i 0.676173i 0.941115 + 0.338086i $$0.109780\pi$$
−0.941115 + 0.338086i $$0.890220\pi$$
$$20$$ 0 0
$$21$$ − 12.0000i − 0.124696i
$$22$$ 0 0
$$23$$ −96.0000 −0.870321 −0.435161 0.900353i $$-0.643308\pi$$
−0.435161 + 0.900353i $$0.643308\pi$$
$$24$$ 0 0
$$25$$ 89.0000 0.712000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 222.000 1.42153 0.710765 0.703430i $$-0.248349\pi$$
0.710765 + 0.703430i $$0.248349\pi$$
$$30$$ 0 0
$$31$$ 260.000i 1.50637i 0.657810 + 0.753184i $$0.271483\pi$$
−0.657810 + 0.753184i $$0.728517\pi$$
$$32$$ 0 0
$$33$$ 108.000i 0.569709i
$$34$$ 0 0
$$35$$ 24.0000 0.115907
$$36$$ 0 0
$$37$$ 106.000i 0.470981i 0.971877 + 0.235490i $$0.0756696\pi$$
−0.971877 + 0.235490i $$0.924330\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 90.0000i − 0.342820i −0.985200 0.171410i $$-0.945168\pi$$
0.985200 0.171410i $$-0.0548323\pi$$
$$42$$ 0 0
$$43$$ −44.0000 −0.156045 −0.0780225 0.996952i $$-0.524861\pi$$
−0.0780225 + 0.996952i $$0.524861\pi$$
$$44$$ 0 0
$$45$$ − 54.0000i − 0.178885i
$$46$$ 0 0
$$47$$ − 168.000i − 0.521390i −0.965421 0.260695i $$-0.916048\pi$$
0.965421 0.260695i $$-0.0839517\pi$$
$$48$$ 0 0
$$49$$ 327.000 0.953353
$$50$$ 0 0
$$51$$ 198.000 0.543638
$$52$$ 0 0
$$53$$ 30.0000 0.0777513 0.0388756 0.999244i $$-0.487622\pi$$
0.0388756 + 0.999244i $$0.487622\pi$$
$$54$$ 0 0
$$55$$ −216.000 −0.529553
$$56$$ 0 0
$$57$$ − 168.000i − 0.390388i
$$58$$ 0 0
$$59$$ − 348.000i − 0.767894i −0.923355 0.383947i $$-0.874565\pi$$
0.923355 0.383947i $$-0.125435\pi$$
$$60$$ 0 0
$$61$$ −346.000 −0.726242 −0.363121 0.931742i $$-0.618289\pi$$
−0.363121 + 0.931742i $$0.618289\pi$$
$$62$$ 0 0
$$63$$ 36.0000i 0.0719932i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 256.000i − 0.466797i −0.972381 0.233398i $$-0.925015\pi$$
0.972381 0.233398i $$-0.0749846\pi$$
$$68$$ 0 0
$$69$$ 288.000 0.502480
$$70$$ 0 0
$$71$$ − 168.000i − 0.280816i −0.990094 0.140408i $$-0.955159\pi$$
0.990094 0.140408i $$-0.0448414\pi$$
$$72$$ 0 0
$$73$$ 814.000i 1.30509i 0.757750 + 0.652544i $$0.226298\pi$$
−0.757750 + 0.652544i $$0.773702\pi$$
$$74$$ 0 0
$$75$$ −267.000 −0.411073
$$76$$ 0 0
$$77$$ 144.000 0.213121
$$78$$ 0 0
$$79$$ 200.000 0.284832 0.142416 0.989807i $$-0.454513\pi$$
0.142416 + 0.989807i $$0.454513\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1236.00i 1.63456i 0.576240 + 0.817281i $$0.304520\pi$$
−0.576240 + 0.817281i $$0.695480\pi$$
$$84$$ 0 0
$$85$$ 396.000i 0.505320i
$$86$$ 0 0
$$87$$ −666.000 −0.820721
$$88$$ 0 0
$$89$$ − 318.000i − 0.378741i −0.981906 0.189370i $$-0.939355\pi$$
0.981906 0.189370i $$-0.0606447\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 780.000i − 0.869701i
$$94$$ 0 0
$$95$$ 336.000 0.362872
$$96$$ 0 0
$$97$$ − 502.000i − 0.525468i −0.964868 0.262734i $$-0.915376\pi$$
0.964868 0.262734i $$-0.0846241\pi$$
$$98$$ 0 0
$$99$$ − 324.000i − 0.328921i
$$100$$ 0 0
$$101$$ −1062.00 −1.04627 −0.523133 0.852251i $$-0.675237\pi$$
−0.523133 + 0.852251i $$0.675237\pi$$
$$102$$ 0 0
$$103$$ 64.0000 0.0612243 0.0306122 0.999531i $$-0.490254\pi$$
0.0306122 + 0.999531i $$0.490254\pi$$
$$104$$ 0 0
$$105$$ −72.0000 −0.0669189
$$106$$ 0 0
$$107$$ −444.000 −0.401150 −0.200575 0.979678i $$-0.564281\pi$$
−0.200575 + 0.979678i $$0.564281\pi$$
$$108$$ 0 0
$$109$$ 1382.00i 1.21442i 0.794542 + 0.607209i $$0.207711\pi$$
−0.794542 + 0.607209i $$0.792289\pi$$
$$110$$ 0 0
$$111$$ − 318.000i − 0.271921i
$$112$$ 0 0
$$113$$ −870.000 −0.724272 −0.362136 0.932125i $$-0.617952\pi$$
−0.362136 + 0.932125i $$0.617952\pi$$
$$114$$ 0 0
$$115$$ 576.000i 0.467063i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 264.000i − 0.203368i
$$120$$ 0 0
$$121$$ 35.0000 0.0262960
$$122$$ 0 0
$$123$$ 270.000i 0.197927i
$$124$$ 0 0
$$125$$ − 1284.00i − 0.918756i
$$126$$ 0 0
$$127$$ −464.000 −0.324200 −0.162100 0.986774i $$-0.551827\pi$$
−0.162100 + 0.986774i $$0.551827\pi$$
$$128$$ 0 0
$$129$$ 132.000 0.0900927
$$130$$ 0 0
$$131$$ 1548.00 1.03244 0.516219 0.856457i $$-0.327339\pi$$
0.516219 + 0.856457i $$0.327339\pi$$
$$132$$ 0 0
$$133$$ −224.000 −0.146040
$$134$$ 0 0
$$135$$ 162.000i 0.103280i
$$136$$ 0 0
$$137$$ − 294.000i − 0.183344i −0.995789 0.0916720i $$-0.970779\pi$$
0.995789 0.0916720i $$-0.0292211\pi$$
$$138$$ 0 0
$$139$$ 2564.00 1.56457 0.782286 0.622919i $$-0.214053\pi$$
0.782286 + 0.622919i $$0.214053\pi$$
$$140$$ 0 0
$$141$$ 504.000i 0.301025i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 1332.00i − 0.762873i
$$146$$ 0 0
$$147$$ −981.000 −0.550418
$$148$$ 0 0
$$149$$ 114.000i 0.0626795i 0.999509 + 0.0313397i $$0.00997738\pi$$
−0.999509 + 0.0313397i $$0.990023\pi$$
$$150$$ 0 0
$$151$$ − 2036.00i − 1.09727i −0.836063 0.548634i $$-0.815148\pi$$
0.836063 0.548634i $$-0.184852\pi$$
$$152$$ 0 0
$$153$$ −594.000 −0.313870
$$154$$ 0 0
$$155$$ 1560.00 0.808401
$$156$$ 0 0
$$157$$ 2870.00 1.45892 0.729462 0.684022i $$-0.239771\pi$$
0.729462 + 0.684022i $$0.239771\pi$$
$$158$$ 0 0
$$159$$ −90.0000 −0.0448897
$$160$$ 0 0
$$161$$ − 384.000i − 0.187972i
$$162$$ 0 0
$$163$$ − 1472.00i − 0.707337i −0.935371 0.353669i $$-0.884934\pi$$
0.935371 0.353669i $$-0.115066\pi$$
$$164$$ 0 0
$$165$$ 648.000 0.305738
$$166$$ 0 0
$$167$$ 240.000i 0.111208i 0.998453 + 0.0556041i $$0.0177085\pi$$
−0.998453 + 0.0556041i $$0.982292\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 504.000i 0.225391i
$$172$$ 0 0
$$173$$ 306.000 0.134478 0.0672392 0.997737i $$-0.478581\pi$$
0.0672392 + 0.997737i $$0.478581\pi$$
$$174$$ 0 0
$$175$$ 356.000i 0.153778i
$$176$$ 0 0
$$177$$ 1044.00i 0.443344i
$$178$$ 0 0
$$179$$ 2052.00 0.856836 0.428418 0.903581i $$-0.359071\pi$$
0.428418 + 0.903581i $$0.359071\pi$$
$$180$$ 0 0
$$181$$ 4498.00 1.84715 0.923574 0.383421i $$-0.125254\pi$$
0.923574 + 0.383421i $$0.125254\pi$$
$$182$$ 0 0
$$183$$ 1038.00 0.419296
$$184$$ 0 0
$$185$$ 636.000 0.252755
$$186$$ 0 0
$$187$$ 2376.00i 0.929146i
$$188$$ 0 0
$$189$$ − 108.000i − 0.0415653i
$$190$$ 0 0
$$191$$ −4056.00 −1.53655 −0.768277 0.640117i $$-0.778886\pi$$
−0.768277 + 0.640117i $$0.778886\pi$$
$$192$$ 0 0
$$193$$ 2062.00i 0.769047i 0.923115 + 0.384523i $$0.125634\pi$$
−0.923115 + 0.384523i $$0.874366\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 4374.00i − 1.58190i −0.611880 0.790951i $$-0.709586\pi$$
0.611880 0.790951i $$-0.290414\pi$$
$$198$$ 0 0
$$199$$ 2536.00 0.903378 0.451689 0.892175i $$-0.350822\pi$$
0.451689 + 0.892175i $$0.350822\pi$$
$$200$$ 0 0
$$201$$ 768.000i 0.269505i
$$202$$ 0 0
$$203$$ 888.000i 0.307022i
$$204$$ 0 0
$$205$$ −540.000 −0.183977
$$206$$ 0 0
$$207$$ −864.000 −0.290107
$$208$$ 0 0
$$209$$ 2016.00 0.667223
$$210$$ 0 0
$$211$$ −4444.00 −1.44994 −0.724971 0.688780i $$-0.758147\pi$$
−0.724971 + 0.688780i $$0.758147\pi$$
$$212$$ 0 0
$$213$$ 504.000i 0.162129i
$$214$$ 0 0
$$215$$ 264.000i 0.0837426i
$$216$$ 0 0
$$217$$ −1040.00 −0.325345
$$218$$ 0 0
$$219$$ − 2442.00i − 0.753493i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 2716.00i − 0.815591i −0.913073 0.407796i $$-0.866298\pi$$
0.913073 0.407796i $$-0.133702\pi$$
$$224$$ 0 0
$$225$$ 801.000 0.237333
$$226$$ 0 0
$$227$$ − 4692.00i − 1.37189i −0.727653 0.685945i $$-0.759389\pi$$
0.727653 0.685945i $$-0.240611\pi$$
$$228$$ 0 0
$$229$$ − 6446.00i − 1.86010i −0.367429 0.930052i $$-0.619762\pi$$
0.367429 0.930052i $$-0.380238\pi$$
$$230$$ 0 0
$$231$$ −432.000 −0.123046
$$232$$ 0 0
$$233$$ 3102.00 0.872184 0.436092 0.899902i $$-0.356362\pi$$
0.436092 + 0.899902i $$0.356362\pi$$
$$234$$ 0 0
$$235$$ −1008.00 −0.279807
$$236$$ 0 0
$$237$$ −600.000 −0.164448
$$238$$ 0 0
$$239$$ 816.000i 0.220848i 0.993885 + 0.110424i $$0.0352209\pi$$
−0.993885 + 0.110424i $$0.964779\pi$$
$$240$$ 0 0
$$241$$ − 3818.00i − 1.02049i −0.860028 0.510247i $$-0.829554\pi$$
0.860028 0.510247i $$-0.170446\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ − 1962.00i − 0.511623i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 3708.00i − 0.943715i
$$250$$ 0 0
$$251$$ 6612.00 1.66273 0.831366 0.555725i $$-0.187559\pi$$
0.831366 + 0.555725i $$0.187559\pi$$
$$252$$ 0 0
$$253$$ 3456.00i 0.858802i
$$254$$ 0 0
$$255$$ − 1188.00i − 0.291747i
$$256$$ 0 0
$$257$$ 4806.00 1.16650 0.583249 0.812293i $$-0.301781\pi$$
0.583249 + 0.812293i $$0.301781\pi$$
$$258$$ 0 0
$$259$$ −424.000 −0.101722
$$260$$ 0 0
$$261$$ 1998.00 0.473843
$$262$$ 0 0
$$263$$ −4584.00 −1.07476 −0.537379 0.843341i $$-0.680586\pi$$
−0.537379 + 0.843341i $$0.680586\pi$$
$$264$$ 0 0
$$265$$ − 180.000i − 0.0417257i
$$266$$ 0 0
$$267$$ 954.000i 0.218666i
$$268$$ 0 0
$$269$$ 7134.00 1.61698 0.808490 0.588510i $$-0.200285\pi$$
0.808490 + 0.588510i $$0.200285\pi$$
$$270$$ 0 0
$$271$$ − 3140.00i − 0.703843i −0.936030 0.351921i $$-0.885528\pi$$
0.936030 0.351921i $$-0.114472\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3204.00i − 0.702576i
$$276$$ 0 0
$$277$$ 4786.00 1.03813 0.519067 0.854734i $$-0.326280\pi$$
0.519067 + 0.854734i $$0.326280\pi$$
$$278$$ 0 0
$$279$$ 2340.00i 0.502122i
$$280$$ 0 0
$$281$$ − 3798.00i − 0.806298i −0.915134 0.403149i $$-0.867916\pi$$
0.915134 0.403149i $$-0.132084\pi$$
$$282$$ 0 0
$$283$$ −3572.00 −0.750295 −0.375147 0.926965i $$-0.622408\pi$$
−0.375147 + 0.926965i $$0.622408\pi$$
$$284$$ 0 0
$$285$$ −1008.00 −0.209504
$$286$$ 0 0
$$287$$ 360.000 0.0740423
$$288$$ 0 0
$$289$$ −557.000 −0.113373
$$290$$ 0 0
$$291$$ 1506.00i 0.303379i
$$292$$ 0 0
$$293$$ − 7122.00i − 1.42004i −0.704182 0.710020i $$-0.748686\pi$$
0.704182 0.710020i $$-0.251314\pi$$
$$294$$ 0 0
$$295$$ −2088.00 −0.412095
$$296$$ 0 0
$$297$$ 972.000i 0.189903i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 176.000i − 0.0337026i
$$302$$ 0 0
$$303$$ 3186.00 0.604062
$$304$$ 0 0
$$305$$ 2076.00i 0.389742i
$$306$$ 0 0
$$307$$ 6856.00i 1.27457i 0.770629 + 0.637284i $$0.219942\pi$$
−0.770629 + 0.637284i $$0.780058\pi$$
$$308$$ 0 0
$$309$$ −192.000 −0.0353479
$$310$$ 0 0
$$311$$ 8832.00 1.61034 0.805172 0.593042i $$-0.202073\pi$$
0.805172 + 0.593042i $$0.202073\pi$$
$$312$$ 0 0
$$313$$ 3626.00 0.654804 0.327402 0.944885i $$-0.393827\pi$$
0.327402 + 0.944885i $$0.393827\pi$$
$$314$$ 0 0
$$315$$ 216.000 0.0386356
$$316$$ 0 0
$$317$$ 10146.0i 1.79765i 0.438304 + 0.898827i $$0.355579\pi$$
−0.438304 + 0.898827i $$0.644421\pi$$
$$318$$ 0 0
$$319$$ − 7992.00i − 1.40272i
$$320$$ 0 0
$$321$$ 1332.00 0.231604
$$322$$ 0 0
$$323$$ − 3696.00i − 0.636690i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4146.00i − 0.701145i
$$328$$ 0 0
$$329$$ 672.000 0.112610
$$330$$ 0 0
$$331$$ 6536.00i 1.08535i 0.839943 + 0.542675i $$0.182589\pi$$
−0.839943 + 0.542675i $$0.817411\pi$$
$$332$$ 0 0
$$333$$ 954.000i 0.156994i
$$334$$ 0 0
$$335$$ −1536.00 −0.250509
$$336$$ 0 0
$$337$$ 6094.00 0.985048 0.492524 0.870299i $$-0.336074\pi$$
0.492524 + 0.870299i $$0.336074\pi$$
$$338$$ 0 0
$$339$$ 2610.00 0.418159
$$340$$ 0 0
$$341$$ 9360.00 1.48643
$$342$$ 0 0
$$343$$ 2680.00i 0.421885i
$$344$$ 0 0
$$345$$ − 1728.00i − 0.269659i
$$346$$ 0 0
$$347$$ −2724.00 −0.421418 −0.210709 0.977549i $$-0.567577\pi$$
−0.210709 + 0.977549i $$0.567577\pi$$
$$348$$ 0 0
$$349$$ 1522.00i 0.233441i 0.993165 + 0.116720i $$0.0372381\pi$$
−0.993165 + 0.116720i $$0.962762\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 1362.00i − 0.205360i −0.994714 0.102680i $$-0.967258\pi$$
0.994714 0.102680i $$-0.0327417\pi$$
$$354$$ 0 0
$$355$$ −1008.00 −0.150702
$$356$$ 0 0
$$357$$ 792.000i 0.117415i
$$358$$ 0 0
$$359$$ − 8880.00i − 1.30548i −0.757581 0.652742i $$-0.773619\pi$$
0.757581 0.652742i $$-0.226381\pi$$
$$360$$ 0 0
$$361$$ 3723.00 0.542790
$$362$$ 0 0
$$363$$ −105.000 −0.0151820
$$364$$ 0 0
$$365$$ 4884.00 0.700384
$$366$$ 0 0
$$367$$ −3712.00 −0.527970 −0.263985 0.964527i $$-0.585037\pi$$
−0.263985 + 0.964527i $$0.585037\pi$$
$$368$$ 0 0
$$369$$ − 810.000i − 0.114273i
$$370$$ 0 0
$$371$$ 120.000i 0.0167927i
$$372$$ 0 0
$$373$$ 5726.00 0.794855 0.397428 0.917634i $$-0.369903\pi$$
0.397428 + 0.917634i $$0.369903\pi$$
$$374$$ 0 0
$$375$$ 3852.00i 0.530444i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 13168.0i − 1.78468i −0.451361 0.892341i $$-0.649061\pi$$
0.451361 0.892341i $$-0.350939\pi$$
$$380$$ 0 0
$$381$$ 1392.00 0.187177
$$382$$ 0 0
$$383$$ 4872.00i 0.649994i 0.945715 + 0.324997i $$0.105363\pi$$
−0.945715 + 0.324997i $$0.894637\pi$$
$$384$$ 0 0
$$385$$ − 864.000i − 0.114373i
$$386$$ 0 0
$$387$$ −396.000 −0.0520150
$$388$$ 0 0
$$389$$ 1266.00 0.165010 0.0825048 0.996591i $$-0.473708\pi$$
0.0825048 + 0.996591i $$0.473708\pi$$
$$390$$ 0 0
$$391$$ 6336.00 0.819502
$$392$$ 0 0
$$393$$ −4644.00 −0.596078
$$394$$ 0 0
$$395$$ − 1200.00i − 0.152857i
$$396$$ 0 0
$$397$$ 4882.00i 0.617180i 0.951195 + 0.308590i $$0.0998571\pi$$
−0.951195 + 0.308590i $$0.900143\pi$$
$$398$$ 0 0
$$399$$ 672.000 0.0843160
$$400$$ 0 0
$$401$$ 90.0000i 0.0112079i 0.999984 + 0.00560397i $$0.00178381\pi$$
−0.999984 + 0.00560397i $$0.998216\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 486.000i − 0.0596285i
$$406$$ 0 0
$$407$$ 3816.00 0.464747
$$408$$ 0 0
$$409$$ 2354.00i 0.284591i 0.989824 + 0.142296i $$0.0454484\pi$$
−0.989824 + 0.142296i $$0.954552\pi$$
$$410$$ 0 0
$$411$$ 882.000i 0.105854i
$$412$$ 0 0
$$413$$ 1392.00 0.165849
$$414$$ 0 0
$$415$$ 7416.00 0.877198
$$416$$ 0 0
$$417$$ −7692.00 −0.903307
$$418$$ 0 0
$$419$$ 7020.00 0.818495 0.409248 0.912423i $$-0.365791\pi$$
0.409248 + 0.912423i $$0.365791\pi$$
$$420$$ 0 0
$$421$$ 302.000i 0.0349610i 0.999847 + 0.0174805i $$0.00556450\pi$$
−0.999847 + 0.0174805i $$0.994436\pi$$
$$422$$ 0 0
$$423$$ − 1512.00i − 0.173797i
$$424$$ 0 0
$$425$$ −5874.00 −0.670426
$$426$$ 0 0
$$427$$ − 1384.00i − 0.156854i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9816.00i 1.09703i 0.836141 + 0.548515i $$0.184807\pi$$
−0.836141 + 0.548515i $$0.815193\pi$$
$$432$$ 0 0
$$433$$ 14782.0 1.64059 0.820297 0.571937i $$-0.193808\pi$$
0.820297 + 0.571937i $$0.193808\pi$$
$$434$$ 0 0
$$435$$ 3996.00i 0.440445i
$$436$$ 0 0
$$437$$ − 5376.00i − 0.588487i
$$438$$ 0 0
$$439$$ −3584.00 −0.389647 −0.194823 0.980838i $$-0.562413\pi$$
−0.194823 + 0.980838i $$0.562413\pi$$
$$440$$ 0 0
$$441$$ 2943.00 0.317784
$$442$$ 0 0
$$443$$ 180.000 0.0193049 0.00965244 0.999953i $$-0.496927\pi$$
0.00965244 + 0.999953i $$0.496927\pi$$
$$444$$ 0 0
$$445$$ −1908.00 −0.203254
$$446$$ 0 0
$$447$$ − 342.000i − 0.0361880i
$$448$$ 0 0
$$449$$ 3450.00i 0.362618i 0.983426 + 0.181309i $$0.0580334\pi$$
−0.983426 + 0.181309i $$0.941967\pi$$
$$450$$ 0 0
$$451$$ −3240.00 −0.338283
$$452$$ 0 0
$$453$$ 6108.00i 0.633507i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 16654.0i − 1.70469i −0.522984 0.852343i $$-0.675181\pi$$
0.522984 0.852343i $$-0.324819\pi$$
$$458$$ 0 0
$$459$$ 1782.00 0.181213
$$460$$ 0 0
$$461$$ − 14046.0i − 1.41906i −0.704674 0.709531i $$-0.748907\pi$$
0.704674 0.709531i $$-0.251093\pi$$
$$462$$ 0 0
$$463$$ 4588.00i 0.460524i 0.973129 + 0.230262i $$0.0739583\pi$$
−0.973129 + 0.230262i $$0.926042\pi$$
$$464$$ 0 0
$$465$$ −4680.00 −0.466731
$$466$$ 0 0
$$467$$ 15372.0 1.52319 0.761597 0.648051i $$-0.224416\pi$$
0.761597 + 0.648051i $$0.224416\pi$$
$$468$$ 0 0
$$469$$ 1024.00 0.100819
$$470$$ 0 0
$$471$$ −8610.00 −0.842310
$$472$$ 0 0
$$473$$ 1584.00i 0.153980i
$$474$$ 0 0
$$475$$ 4984.00i 0.481435i
$$476$$ 0 0
$$477$$ 270.000 0.0259171
$$478$$ 0 0
$$479$$ − 12864.0i − 1.22708i −0.789664 0.613540i $$-0.789745\pi$$
0.789664 0.613540i $$-0.210255\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 1152.00i 0.108525i
$$484$$ 0 0
$$485$$ −3012.00 −0.281996
$$486$$ 0 0
$$487$$ − 10276.0i − 0.956160i −0.878316 0.478080i $$-0.841333\pi$$
0.878316 0.478080i $$-0.158667\pi$$
$$488$$ 0 0
$$489$$ 4416.00i 0.408381i
$$490$$ 0 0
$$491$$ 11220.0 1.03127 0.515633 0.856810i $$-0.327557\pi$$
0.515633 + 0.856810i $$0.327557\pi$$
$$492$$ 0 0
$$493$$ −14652.0 −1.33853
$$494$$ 0 0
$$495$$ −1944.00 −0.176518
$$496$$ 0 0
$$497$$ 672.000 0.0606505
$$498$$ 0 0
$$499$$ 17264.0i 1.54878i 0.632707 + 0.774392i $$0.281944\pi$$
−0.632707 + 0.774392i $$0.718056\pi$$
$$500$$ 0 0
$$501$$ − 720.000i − 0.0642060i
$$502$$ 0 0
$$503$$ −1896.00 −0.168069 −0.0840343 0.996463i $$-0.526781\pi$$
−0.0840343 + 0.996463i $$0.526781\pi$$
$$504$$ 0 0
$$505$$ 6372.00i 0.561486i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 5010.00i 0.436276i 0.975918 + 0.218138i $$0.0699982\pi$$
−0.975918 + 0.218138i $$0.930002\pi$$
$$510$$ 0 0
$$511$$ −3256.00 −0.281873
$$512$$ 0 0
$$513$$ − 1512.00i − 0.130129i
$$514$$ 0 0
$$515$$ − 384.000i − 0.0328564i
$$516$$ 0 0
$$517$$ −6048.00 −0.514489
$$518$$ 0 0
$$519$$ −918.000 −0.0776411
$$520$$ 0 0
$$521$$ 8610.00 0.724013 0.362007 0.932176i $$-0.382092\pi$$
0.362007 + 0.932176i $$0.382092\pi$$
$$522$$ 0 0
$$523$$ −5308.00 −0.443791 −0.221895 0.975070i $$-0.571224\pi$$
−0.221895 + 0.975070i $$0.571224\pi$$
$$524$$ 0 0
$$525$$ − 1068.00i − 0.0887835i
$$526$$ 0 0
$$527$$ − 17160.0i − 1.41841i
$$528$$ 0 0
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ − 3132.00i − 0.255965i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 2664.00i 0.215280i
$$536$$ 0 0
$$537$$ −6156.00 −0.494695
$$538$$ 0 0
$$539$$ − 11772.0i − 0.940735i
$$540$$ 0 0
$$541$$ − 6182.00i − 0.491285i −0.969361 0.245642i $$-0.921001\pi$$
0.969361 0.245642i $$-0.0789989\pi$$
$$542$$ 0 0
$$543$$ −13494.0 −1.06645
$$544$$ 0 0
$$545$$ 8292.00 0.651725
$$546$$ 0 0
$$547$$ 1292.00 0.100991 0.0504954 0.998724i $$-0.483920\pi$$
0.0504954 + 0.998724i $$0.483920\pi$$
$$548$$ 0 0
$$549$$ −3114.00 −0.242081
$$550$$ 0 0
$$551$$ 12432.0i 0.961200i
$$552$$ 0 0
$$553$$ 800.000i 0.0615180i
$$554$$ 0 0
$$555$$ −1908.00 −0.145928
$$556$$ 0 0
$$557$$ 12774.0i 0.971727i 0.874035 + 0.485863i $$0.161495\pi$$
−0.874035 + 0.485863i $$0.838505\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 7128.00i − 0.536443i
$$562$$ 0 0
$$563$$ −16908.0 −1.26570 −0.632848 0.774276i $$-0.718114\pi$$
−0.632848 + 0.774276i $$0.718114\pi$$
$$564$$ 0 0
$$565$$ 5220.00i 0.388685i
$$566$$ 0 0
$$567$$ 324.000i 0.0239977i
$$568$$ 0 0
$$569$$ 11214.0 0.826213 0.413107 0.910683i $$-0.364444\pi$$
0.413107 + 0.910683i $$0.364444\pi$$
$$570$$ 0 0
$$571$$ −25220.0 −1.84838 −0.924189 0.381935i $$-0.875258\pi$$
−0.924189 + 0.381935i $$0.875258\pi$$
$$572$$ 0 0
$$573$$ 12168.0 0.887130
$$574$$ 0 0
$$575$$ −8544.00 −0.619669
$$576$$ 0 0
$$577$$ − 17710.0i − 1.27778i −0.769300 0.638888i $$-0.779395\pi$$
0.769300 0.638888i $$-0.220605\pi$$
$$578$$ 0 0
$$579$$ − 6186.00i − 0.444009i
$$580$$ 0 0
$$581$$ −4944.00 −0.353032
$$582$$ 0 0
$$583$$ − 1080.00i − 0.0767222i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20028.0i 1.40825i 0.710075 + 0.704126i $$0.248661\pi$$
−0.710075 + 0.704126i $$0.751339\pi$$
$$588$$ 0 0
$$589$$ −14560.0 −1.01856
$$590$$ 0 0
$$591$$ 13122.0i 0.913311i
$$592$$ 0 0
$$593$$ − 19926.0i − 1.37987i −0.723871 0.689935i $$-0.757639\pi$$
0.723871 0.689935i $$-0.242361\pi$$
$$594$$ 0 0
$$595$$ −1584.00 −0.109139
$$596$$ 0 0
$$597$$ −7608.00 −0.521566
$$598$$ 0 0
$$599$$ −1704.00 −0.116233 −0.0581165 0.998310i $$-0.518509\pi$$
−0.0581165 + 0.998310i $$0.518509\pi$$
$$600$$ 0 0
$$601$$ 11018.0 0.747810 0.373905 0.927467i $$-0.378019\pi$$
0.373905 + 0.927467i $$0.378019\pi$$
$$602$$ 0 0
$$603$$ − 2304.00i − 0.155599i
$$604$$ 0 0
$$605$$ − 210.000i − 0.0141119i
$$606$$ 0 0
$$607$$ −448.000 −0.0299568 −0.0149784 0.999888i $$-0.504768\pi$$
−0.0149784 + 0.999888i $$0.504768\pi$$
$$608$$ 0 0
$$609$$ − 2664.00i − 0.177259i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 12586.0i − 0.829272i −0.909987 0.414636i $$-0.863909\pi$$
0.909987 0.414636i $$-0.136091\pi$$
$$614$$ 0 0
$$615$$ 1620.00 0.106219
$$616$$ 0 0
$$617$$ − 29610.0i − 1.93202i −0.258513 0.966008i $$-0.583232\pi$$
0.258513 0.966008i $$-0.416768\pi$$
$$618$$ 0 0
$$619$$ 7120.00i 0.462321i 0.972916 + 0.231161i $$0.0742523\pi$$
−0.972916 + 0.231161i $$0.925748\pi$$
$$620$$ 0 0
$$621$$ 2592.00 0.167493
$$622$$ 0 0
$$623$$ 1272.00 0.0818003
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ −6048.00 −0.385221
$$628$$ 0 0
$$629$$ − 6996.00i − 0.443480i
$$630$$ 0 0
$$631$$ 15580.0i 0.982932i 0.870897 + 0.491466i $$0.163539\pi$$
−0.870897 + 0.491466i $$0.836461\pi$$
$$632$$ 0 0
$$633$$ 13332.0 0.837124
$$634$$ 0 0
$$635$$ 2784.00i 0.173984i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 1512.00i − 0.0936053i
$$640$$ 0 0
$$641$$ 19806.0 1.22042 0.610211 0.792239i $$-0.291085\pi$$
0.610211 + 0.792239i $$0.291085\pi$$
$$642$$ 0 0
$$643$$ 24032.0i 1.47392i 0.675937 + 0.736959i $$0.263739\pi$$
−0.675937 + 0.736959i $$0.736261\pi$$
$$644$$ 0 0
$$645$$ − 792.000i − 0.0483488i
$$646$$ 0 0
$$647$$ 2808.00 0.170624 0.0853121 0.996354i $$-0.472811\pi$$
0.0853121 + 0.996354i $$0.472811\pi$$
$$648$$ 0 0
$$649$$ −12528.0 −0.757730
$$650$$ 0 0
$$651$$ 3120.00 0.187838
$$652$$ 0 0
$$653$$ 23886.0 1.43144 0.715721 0.698386i $$-0.246098\pi$$
0.715721 + 0.698386i $$0.246098\pi$$
$$654$$ 0 0
$$655$$ − 9288.00i − 0.554064i
$$656$$ 0 0
$$657$$ 7326.00i 0.435030i
$$658$$ 0 0
$$659$$ −3948.00 −0.233372 −0.116686 0.993169i $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$660$$ 0 0
$$661$$ − 5750.00i − 0.338350i −0.985586 0.169175i $$-0.945890\pi$$
0.985586 0.169175i $$-0.0541102\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1344.00i 0.0783731i
$$666$$ 0 0
$$667$$ −21312.0 −1.23719
$$668$$ 0 0
$$669$$ 8148.00i 0.470882i
$$670$$ 0 0
$$671$$ 12456.0i 0.716630i
$$672$$ 0 0
$$673$$ −28082.0 −1.60844 −0.804221 0.594330i $$-0.797417\pi$$
−0.804221 + 0.594330i $$0.797417\pi$$
$$674$$ 0 0
$$675$$ −2403.00 −0.137024
$$676$$ 0 0
$$677$$ −27954.0 −1.58694 −0.793471 0.608608i $$-0.791728\pi$$
−0.793471 + 0.608608i $$0.791728\pi$$
$$678$$ 0 0
$$679$$ 2008.00 0.113490
$$680$$ 0 0
$$681$$ 14076.0i 0.792061i
$$682$$ 0 0
$$683$$ 28428.0i 1.59263i 0.604881 + 0.796316i $$0.293221\pi$$
−0.604881 + 0.796316i $$0.706779\pi$$
$$684$$ 0 0
$$685$$ −1764.00 −0.0983927
$$686$$ 0 0
$$687$$ 19338.0i 1.07393i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 21680.0i 1.19355i 0.802407 + 0.596777i $$0.203552\pi$$
−0.802407 + 0.596777i $$0.796448\pi$$
$$692$$ 0 0
$$693$$ 1296.00 0.0710404
$$694$$ 0 0
$$695$$ − 15384.0i − 0.839638i
$$696$$ 0 0
$$697$$ 5940.00i 0.322803i
$$698$$ 0 0
$$699$$ −9306.00 −0.503555
$$700$$ 0 0
$$701$$ 7482.00 0.403126 0.201563 0.979476i $$-0.435398\pi$$
0.201563 + 0.979476i $$0.435398\pi$$
$$702$$ 0 0
$$703$$ −5936.00 −0.318464
$$704$$ 0 0
$$705$$ 3024.00 0.161547
$$706$$ 0 0
$$707$$ − 4248.00i − 0.225972i
$$708$$ 0 0
$$709$$ − 2270.00i − 0.120242i −0.998191 0.0601210i $$-0.980851\pi$$
0.998191 0.0601210i $$-0.0191487\pi$$
$$710$$ 0 0
$$711$$ 1800.00 0.0949441
$$712$$ 0 0
$$713$$ − 24960.0i − 1.31102i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 2448.00i − 0.127507i
$$718$$ 0 0
$$719$$ −36024.0 −1.86852 −0.934262 0.356588i $$-0.883940\pi$$
−0.934262 + 0.356588i $$0.883940\pi$$
$$720$$ 0 0
$$721$$ 256.000i 0.0132232i
$$722$$ 0 0
$$723$$ 11454.0i 0.589182i
$$724$$ 0 0
$$725$$ 19758.0 1.01213
$$726$$ 0 0
$$727$$ 21544.0 1.09907 0.549534 0.835471i $$-0.314805\pi$$
0.549534 + 0.835471i $$0.314805\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 2904.00 0.146933
$$732$$ 0 0
$$733$$ − 1018.00i − 0.0512970i −0.999671 0.0256485i $$-0.991835\pi$$
0.999671 0.0256485i $$-0.00816506\pi$$
$$734$$ 0 0
$$735$$ 5886.00i 0.295386i
$$736$$ 0 0
$$737$$ −9216.00 −0.460618
$$738$$ 0 0
$$739$$ 24568.0i 1.22293i 0.791270 + 0.611467i $$0.209420\pi$$
−0.791270 + 0.611467i $$0.790580\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16968.0i 0.837814i 0.908029 + 0.418907i $$0.137587\pi$$
−0.908029 + 0.418907i $$0.862413\pi$$
$$744$$ 0 0
$$745$$ 684.000 0.0336373
$$746$$ 0 0
$$747$$ 11124.0i 0.544854i
$$748$$ 0 0
$$749$$ − 1776.00i − 0.0866404i
$$750$$ 0 0
$$751$$ −3224.00 −0.156652 −0.0783259 0.996928i $$-0.524957\pi$$
−0.0783259 + 0.996928i $$0.524957\pi$$
$$752$$ 0 0
$$753$$ −19836.0 −0.959979
$$754$$ 0 0
$$755$$ −12216.0 −0.588855
$$756$$ 0 0
$$757$$ −31570.0 −1.51576 −0.757881 0.652393i $$-0.773765\pi$$
−0.757881 + 0.652393i $$0.773765\pi$$
$$758$$ 0 0
$$759$$ − 10368.0i − 0.495829i
$$760$$ 0 0
$$761$$ 34890.0i 1.66197i 0.556293 + 0.830987i $$0.312223\pi$$
−0.556293 + 0.830987i $$0.687777\pi$$
$$762$$ 0 0
$$763$$ −5528.00 −0.262290
$$764$$ 0 0
$$765$$ 3564.00i 0.168440i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 11522.0i 0.540304i 0.962818 + 0.270152i $$0.0870740\pi$$
−0.962818 + 0.270152i $$0.912926\pi$$
$$770$$ 0 0
$$771$$ −14418.0 −0.673478
$$772$$ 0 0
$$773$$ − 28158.0i − 1.31018i −0.755549 0.655092i $$-0.772630\pi$$
0.755549 0.655092i $$-0.227370\pi$$
$$774$$ 0 0
$$775$$ 23140.0i 1.07253i
$$776$$ 0 0
$$777$$ 1272.00 0.0587294
$$778$$ 0 0
$$779$$ 5040.00 0.231806
$$780$$ 0 0
$$781$$ −6048.00 −0.277099
$$782$$ 0 0
$$783$$ −5994.00 −0.273574
$$784$$ 0 0
$$785$$ − 17220.0i − 0.782940i
$$786$$ 0 0
$$787$$ − 14504.0i − 0.656940i −0.944514 0.328470i $$-0.893467\pi$$
0.944514 0.328470i $$-0.106533\pi$$
$$788$$ 0 0
$$789$$ 13752.0 0.620512
$$790$$ 0 0
$$791$$ − 3480.00i − 0.156428i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 540.000i 0.0240903i
$$796$$ 0 0
$$797$$ 18090.0 0.803991 0.401995 0.915642i $$-0.368317\pi$$
0.401995 + 0.915642i $$0.368317\pi$$
$$798$$ 0 0
$$799$$ 11088.0i 0.490945i
$$800$$ 0 0
$$801$$ − 2862.00i − 0.126247i
$$802$$ 0 0
$$803$$ 29304.0 1.28782
$$804$$ 0 0
$$805$$ −2304.00 −0.100876
$$806$$ 0 0
$$807$$ −21402.0 −0.933564
$$808$$ 0 0
$$809$$ 36402.0 1.58199 0.790993 0.611826i $$-0.209565\pi$$
0.790993 + 0.611826i $$0.209565\pi$$
$$810$$ 0 0
$$811$$ − 32368.0i − 1.40147i −0.713420 0.700736i $$-0.752855\pi$$
0.713420 0.700736i $$-0.247145\pi$$
$$812$$ 0 0
$$813$$ 9420.00i 0.406364i
$$814$$ 0 0
$$815$$ −8832.00 −0.379597
$$816$$ 0 0
$$817$$ − 2464.00i − 0.105513i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 35778.0i 1.52090i 0.649395 + 0.760451i $$0.275022\pi$$
−0.649395 + 0.760451i $$0.724978\pi$$
$$822$$ 0 0
$$823$$ 10240.0 0.433711 0.216855 0.976204i $$-0.430420\pi$$
0.216855 + 0.976204i $$0.430420\pi$$
$$824$$ 0 0
$$825$$ 9612.00i 0.405633i
$$826$$ 0 0
$$827$$ 16284.0i 0.684704i 0.939572 + 0.342352i $$0.111224\pi$$
−0.939572 + 0.342352i $$0.888776\pi$$
$$828$$ 0 0
$$829$$ −14150.0 −0.592822 −0.296411 0.955060i $$-0.595790\pi$$
−0.296411 + 0.955060i $$0.595790\pi$$
$$830$$ 0 0
$$831$$ −14358.0 −0.599366
$$832$$ 0 0
$$833$$ −21582.0 −0.897685
$$834$$ 0 0
$$835$$ 1440.00 0.0596805
$$836$$ 0 0
$$837$$ − 7020.00i − 0.289900i
$$838$$ 0 0
$$839$$ − 39576.0i − 1.62850i −0.580511 0.814252i $$-0.697147\pi$$
0.580511 0.814252i $$-0.302853\pi$$
$$840$$ 0 0
$$841$$ 24895.0 1.02075
$$842$$ 0 0
$$843$$ 11394.0i 0.465516i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 140.000i 0.00567941i
$$848$$ 0 0
$$849$$ 10716.0 0.433183
$$850$$ 0 0
$$851$$ − 10176.0i − 0.409905i
$$852$$ 0 0
$$853$$ 6922.00i 0.277848i 0.990303 + 0.138924i $$0.0443645\pi$$
−0.990303 + 0.138924i $$0.955636\pi$$
$$854$$ 0 0
$$855$$ 3024.00 0.120957
$$856$$ 0 0
$$857$$ −48162.0 −1.91970 −0.959850 0.280514i $$-0.909495\pi$$
−0.959850 + 0.280514i $$0.909495\pi$$
$$858$$ 0 0
$$859$$ −27652.0 −1.09834 −0.549170 0.835711i $$-0.685056\pi$$
−0.549170 + 0.835711i $$0.685056\pi$$
$$860$$ 0 0
$$861$$ −1080.00 −0.0427483
$$862$$ 0 0
$$863$$ 648.000i 0.0255599i 0.999918 + 0.0127799i $$0.00406809\pi$$
−0.999918 + 0.0127799i $$0.995932\pi$$
$$864$$ 0 0
$$865$$ − 1836.00i − 0.0721686i
$$866$$ 0 0
$$867$$ 1671.00 0.0654558
$$868$$ 0 0
$$869$$ − 7200.00i − 0.281062i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 4518.00i − 0.175156i
$$874$$ 0 0
$$875$$ 5136.00 0.198433
$$876$$ 0 0
$$877$$ 7166.00i 0.275916i 0.990438 + 0.137958i $$0.0440540\pi$$
−0.990438 + 0.137958i $$0.955946\pi$$
$$878$$ 0 0
$$879$$ 21366.0i 0.819860i
$$880$$ 0 0
$$881$$ 37062.0 1.41731 0.708655 0.705555i $$-0.249302\pi$$
0.708655 + 0.705555i $$0.249302\pi$$
$$882$$ 0 0
$$883$$ −24716.0 −0.941970 −0.470985 0.882141i $$-0.656101\pi$$
−0.470985 + 0.882141i $$0.656101\pi$$
$$884$$ 0 0
$$885$$ 6264.00 0.237923
$$886$$ 0 0
$$887$$ 48672.0 1.84244 0.921221 0.389040i $$-0.127193\pi$$
0.921221 + 0.389040i $$0.127193\pi$$
$$888$$ 0 0
$$889$$ − 1856.00i − 0.0700205i
$$890$$ 0 0
$$891$$ − 2916.00i − 0.109640i
$$892$$ 0 0
$$893$$ 9408.00 0.352550
$$894$$ 0 0
$$895$$ − 12312.0i − 0.459827i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 57720.0i 2.14135i
$$900$$ 0 0
$$901$$ −1980.00 −0.0732113
$$902$$ 0 0
$$903$$ 528.000i 0.0194582i
$$904$$ 0 0
$$905$$ − 26988.0i − 0.991283i
$$906$$ 0 0
$$907$$ 9484.00 0.347201 0.173600 0.984816i $$-0.444460\pi$$
0.173600 + 0.984816i $$0.444460\pi$$
$$908$$ 0 0
$$909$$ −9558.00 −0.348756
$$910$$ 0 0
$$911$$ 12792.0 0.465223 0.232611 0.972570i $$-0.425273\pi$$
0.232611 + 0.972570i $$0.425273\pi$$
$$912$$ 0 0
$$913$$ 44496.0 1.61293
$$914$$ 0 0
$$915$$ − 6228.00i − 0.225018i
$$916$$ 0 0
$$917$$ 6192.00i 0.222986i
$$918$$ 0 0
$$919$$ −18592.0 −0.667349 −0.333674 0.942688i $$-0.608289\pi$$
−0.333674 + 0.942688i $$0.608289\pi$$
$$920$$ 0 0
$$921$$ − 20568.0i − 0.735873i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 9434.00i 0.335338i
$$926$$ 0 0
$$927$$ 576.000 0.0204081
$$928$$ 0 0
$$929$$ − 15378.0i − 0.543096i −0.962425 0.271548i $$-0.912464\pi$$
0.962425 0.271548i $$-0.0875355\pi$$
$$930$$ 0 0
$$931$$ 18312.0i 0.644631i
$$932$$ 0 0
$$933$$ −26496.0 −0.929732
$$934$$ 0 0
$$935$$ 14256.0 0.498632
$$936$$ 0 0
$$937$$ −37078.0 −1.29273 −0.646364 0.763030i $$-0.723711\pi$$
−0.646364 + 0.763030i $$0.723711\pi$$
$$938$$ 0 0
$$939$$ −10878.0 −0.378051
$$940$$ 0 0
$$941$$ 10842.0i 0.375599i 0.982207 + 0.187800i $$0.0601356\pi$$
−0.982207 + 0.187800i $$0.939864\pi$$
$$942$$ 0 0
$$943$$ 8640.00i 0.298364i
$$944$$ 0 0
$$945$$ −648.000 −0.0223063
$$946$$ 0 0
$$947$$ − 41508.0i − 1.42432i −0.702018 0.712159i $$-0.747718\pi$$
0.702018 0.712159i $$-0.252282\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 30438.0i − 1.03788i
$$952$$ 0 0
$$953$$ −38706.0 −1.31565 −0.657823 0.753173i $$-0.728522\pi$$
−0.657823 + 0.753173i $$0.728522\pi$$
$$954$$ 0 0
$$955$$ 24336.0i 0.824602i
$$956$$ 0 0
$$957$$ 23976.0i 0.809858i
$$958$$ 0 0
$$959$$ 1176.00 0.0395986
$$960$$ 0 0
$$961$$ −37809.0 −1.26914
$$962$$ 0 0
$$963$$ −3996.00 −0.133717
$$964$$ 0 0
$$965$$ 12372.0 0.412714
$$966$$ 0 0
$$967$$ − 24388.0i − 0.811029i −0.914089 0.405515i $$-0.867092\pi$$
0.914089 0.405515i $$-0.132908\pi$$
$$968$$ 0 0
$$969$$ 11088.0i 0.367593i
$$970$$ 0 0
$$971$$ −14100.0 −0.466005 −0.233002 0.972476i $$-0.574855\pi$$
−0.233002 + 0.972476i $$0.574855\pi$$
$$972$$ 0 0
$$973$$ 10256.0i 0.337916i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 44838.0i 1.46826i 0.679006 + 0.734132i $$0.262411\pi$$
−0.679006 + 0.734132i $$0.737589\pi$$
$$978$$ 0 0
$$979$$ −11448.0 −0.373728
$$980$$ 0 0
$$981$$ 12438.0i 0.404806i
$$982$$ 0 0
$$983$$ − 13176.0i − 0.427517i −0.976887 0.213758i $$-0.931429\pi$$
0.976887 0.213758i $$-0.0685706\pi$$
$$984$$ 0 0
$$985$$ −26244.0 −0.848937
$$986$$ 0 0
$$987$$ −2016.00 −0.0650152
$$988$$ 0 0
$$989$$ 4224.00 0.135809
$$990$$ 0 0
$$991$$ −43648.0 −1.39912 −0.699558 0.714576i $$-0.746620\pi$$
−0.699558 + 0.714576i $$0.746620\pi$$
$$992$$ 0 0
$$993$$ − 19608.0i − 0.626627i
$$994$$ 0 0
$$995$$ − 15216.0i − 0.484804i
$$996$$ 0 0
$$997$$ 62750.0 1.99329 0.996646 0.0818317i $$-0.0260770\pi$$
0.996646 + 0.0818317i $$0.0260770\pi$$
$$998$$ 0 0
$$999$$ − 2862.00i − 0.0906403i
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 2028.4.b.a.337.1 2
13.5 odd 4 156.4.a.a.1.1 1
13.8 odd 4 2028.4.a.a.1.1 1
13.12 even 2 inner 2028.4.b.a.337.2 2
39.5 even 4 468.4.a.b.1.1 1
52.31 even 4 624.4.a.h.1.1 1
104.5 odd 4 2496.4.a.n.1.1 1
104.83 even 4 2496.4.a.e.1.1 1
156.83 odd 4 1872.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.a.1.1 1 13.5 odd 4
468.4.a.b.1.1 1 39.5 even 4
624.4.a.h.1.1 1 52.31 even 4
1872.4.a.j.1.1 1 156.83 odd 4
2028.4.a.a.1.1 1 13.8 odd 4
2028.4.b.a.337.1 2 1.1 even 1 trivial
2028.4.b.a.337.2 2 13.12 even 2 inner
2496.4.a.e.1.1 1 104.83 even 4
2496.4.a.n.1.1 1 104.5 odd 4