# Properties

 Label 338.6.b.a.337.1 Level $338$ Weight $6$ Character 338.337 Analytic conductor $54.210$ 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] = [338,6,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$54.2097310968$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 338.337 Dual form 338.6.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)

 $$f(q)$$ $$=$$ $$q-4.00000i q^{2} -16.0000 q^{4} -14.0000i q^{5} +170.000i q^{7} +64.0000i q^{8} -243.000 q^{9} +O(q^{10})$$ $$q-4.00000i q^{2} -16.0000 q^{4} -14.0000i q^{5} +170.000i q^{7} +64.0000i q^{8} -243.000 q^{9} -56.0000 q^{10} +250.000i q^{11} +680.000 q^{14} +256.000 q^{16} -1062.00 q^{17} +972.000i q^{18} -78.0000i q^{19} +224.000i q^{20} +1000.00 q^{22} -1576.00 q^{23} +2929.00 q^{25} -2720.00i q^{28} +2578.00 q^{29} -8654.00i q^{31} -1024.00i q^{32} +4248.00i q^{34} +2380.00 q^{35} +3888.00 q^{36} -10986.0i q^{37} -312.000 q^{38} +896.000 q^{40} +1050.00i q^{41} +5900.00 q^{43} -4000.00i q^{44} +3402.00i q^{45} +6304.00i q^{46} +5962.00i q^{47} -12093.0 q^{49} -11716.0i q^{50} +29046.0 q^{53} +3500.00 q^{55} -10880.0 q^{56} -10312.0i q^{58} +13922.0i q^{59} -32882.0 q^{61} -34616.0 q^{62} -41310.0i q^{63} -4096.00 q^{64} -69566.0i q^{67} +16992.0 q^{68} -9520.00i q^{70} -50542.0i q^{71} -15552.0i q^{72} +46750.0i q^{73} -43944.0 q^{74} +1248.00i q^{76} -42500.0 q^{77} -19348.0 q^{79} -3584.00i q^{80} +59049.0 q^{81} +4200.00 q^{82} -87438.0i q^{83} +14868.0i q^{85} -23600.0i q^{86} -16000.0 q^{88} -94170.0i q^{89} +13608.0 q^{90} +25216.0 q^{92} +23848.0 q^{94} -1092.00 q^{95} +182786. i q^{97} +48372.0i q^{98} -60750.0i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 486 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 486 * q^9 $$2 q - 32 q^{4} - 486 q^{9} - 112 q^{10} + 1360 q^{14} + 512 q^{16} - 2124 q^{17} + 2000 q^{22} - 3152 q^{23} + 5858 q^{25} + 5156 q^{29} + 4760 q^{35} + 7776 q^{36} - 624 q^{38} + 1792 q^{40} + 11800 q^{43} - 24186 q^{49} + 58092 q^{53} + 7000 q^{55} - 21760 q^{56} - 65764 q^{61} - 69232 q^{62} - 8192 q^{64} + 33984 q^{68} - 87888 q^{74} - 85000 q^{77} - 38696 q^{79} + 118098 q^{81} + 8400 q^{82} - 32000 q^{88} + 27216 q^{90} + 50432 q^{92} + 47696 q^{94} - 2184 q^{95}+O(q^{100})$$ 2 * q - 32 * q^4 - 486 * q^9 - 112 * q^10 + 1360 * q^14 + 512 * q^16 - 2124 * q^17 + 2000 * q^22 - 3152 * q^23 + 5858 * q^25 + 5156 * q^29 + 4760 * q^35 + 7776 * q^36 - 624 * q^38 + 1792 * q^40 + 11800 * q^43 - 24186 * q^49 + 58092 * q^53 + 7000 * q^55 - 21760 * q^56 - 65764 * q^61 - 69232 * q^62 - 8192 * q^64 + 33984 * q^68 - 87888 * q^74 - 85000 * q^77 - 38696 * q^79 + 118098 * q^81 + 8400 * q^82 - 32000 * q^88 + 27216 * q^90 + 50432 * q^92 + 47696 * q^94 - 2184 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 4.00000i − 0.707107i
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −16.0000 −0.500000
$$5$$ − 14.0000i − 0.250440i −0.992129 0.125220i $$-0.960036\pi$$
0.992129 0.125220i $$-0.0399636\pi$$
$$6$$ 0 0
$$7$$ 170.000i 1.31131i 0.755063 + 0.655653i $$0.227606\pi$$
−0.755063 + 0.655653i $$0.772394\pi$$
$$8$$ 64.0000i 0.353553i
$$9$$ −243.000 −1.00000
$$10$$ −56.0000 −0.177088
$$11$$ 250.000i 0.622957i 0.950253 + 0.311479i $$0.100824\pi$$
−0.950253 + 0.311479i $$0.899176\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 680.000 0.927233
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −1062.00 −0.891255 −0.445628 0.895218i $$-0.647019\pi$$
−0.445628 + 0.895218i $$0.647019\pi$$
$$18$$ 972.000i 0.707107i
$$19$$ − 78.0000i − 0.0495691i −0.999693 0.0247845i $$-0.992110\pi$$
0.999693 0.0247845i $$-0.00788997\pi$$
$$20$$ 224.000i 0.125220i
$$21$$ 0 0
$$22$$ 1000.00 0.440497
$$23$$ −1576.00 −0.621207 −0.310604 0.950539i $$-0.600531\pi$$
−0.310604 + 0.950539i $$0.600531\pi$$
$$24$$ 0 0
$$25$$ 2929.00 0.937280
$$26$$ 0 0
$$27$$ 0 0
$$28$$ − 2720.00i − 0.655653i
$$29$$ 2578.00 0.569230 0.284615 0.958642i $$-0.408134\pi$$
0.284615 + 0.958642i $$0.408134\pi$$
$$30$$ 0 0
$$31$$ − 8654.00i − 1.61738i −0.588234 0.808691i $$-0.700176\pi$$
0.588234 0.808691i $$-0.299824\pi$$
$$32$$ − 1024.00i − 0.176777i
$$33$$ 0 0
$$34$$ 4248.00i 0.630213i
$$35$$ 2380.00 0.328403
$$36$$ 3888.00 0.500000
$$37$$ − 10986.0i − 1.31927i −0.751584 0.659637i $$-0.770710\pi$$
0.751584 0.659637i $$-0.229290\pi$$
$$38$$ −312.000 −0.0350506
$$39$$ 0 0
$$40$$ 896.000 0.0885438
$$41$$ 1050.00i 0.0975505i 0.998810 + 0.0487753i $$0.0155318\pi$$
−0.998810 + 0.0487753i $$0.984468\pi$$
$$42$$ 0 0
$$43$$ 5900.00 0.486610 0.243305 0.969950i $$-0.421768\pi$$
0.243305 + 0.969950i $$0.421768\pi$$
$$44$$ − 4000.00i − 0.311479i
$$45$$ 3402.00i 0.250440i
$$46$$ 6304.00i 0.439260i
$$47$$ 5962.00i 0.393684i 0.980435 + 0.196842i $$0.0630685\pi$$
−0.980435 + 0.196842i $$0.936931\pi$$
$$48$$ 0 0
$$49$$ −12093.0 −0.719522
$$50$$ − 11716.0i − 0.662757i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 29046.0 1.42035 0.710177 0.704023i $$-0.248615\pi$$
0.710177 + 0.704023i $$0.248615\pi$$
$$54$$ 0 0
$$55$$ 3500.00 0.156013
$$56$$ −10880.0 −0.463616
$$57$$ 0 0
$$58$$ − 10312.0i − 0.402507i
$$59$$ 13922.0i 0.520681i 0.965517 + 0.260340i $$0.0838348\pi$$
−0.965517 + 0.260340i $$0.916165\pi$$
$$60$$ 0 0
$$61$$ −32882.0 −1.13145 −0.565723 0.824596i $$-0.691403\pi$$
−0.565723 + 0.824596i $$0.691403\pi$$
$$62$$ −34616.0 −1.14366
$$63$$ − 41310.0i − 1.31131i
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 69566.0i − 1.89326i −0.322324 0.946629i $$-0.604464\pi$$
0.322324 0.946629i $$-0.395536\pi$$
$$68$$ 16992.0 0.445628
$$69$$ 0 0
$$70$$ − 9520.00i − 0.232216i
$$71$$ − 50542.0i − 1.18989i −0.803767 0.594945i $$-0.797174\pi$$
0.803767 0.594945i $$-0.202826\pi$$
$$72$$ − 15552.0i − 0.353553i
$$73$$ 46750.0i 1.02677i 0.858157 + 0.513387i $$0.171609\pi$$
−0.858157 + 0.513387i $$0.828391\pi$$
$$74$$ −43944.0 −0.932868
$$75$$ 0 0
$$76$$ 1248.00i 0.0247845i
$$77$$ −42500.0 −0.816887
$$78$$ 0 0
$$79$$ −19348.0 −0.348793 −0.174397 0.984675i $$-0.555798\pi$$
−0.174397 + 0.984675i $$0.555798\pi$$
$$80$$ − 3584.00i − 0.0626099i
$$81$$ 59049.0 1.00000
$$82$$ 4200.00 0.0689786
$$83$$ − 87438.0i − 1.39317i −0.717473 0.696586i $$-0.754701\pi$$
0.717473 0.696586i $$-0.245299\pi$$
$$84$$ 0 0
$$85$$ 14868.0i 0.223206i
$$86$$ − 23600.0i − 0.344085i
$$87$$ 0 0
$$88$$ −16000.0 −0.220249
$$89$$ − 94170.0i − 1.26019i −0.776516 0.630097i $$-0.783015\pi$$
0.776516 0.630097i $$-0.216985\pi$$
$$90$$ 13608.0 0.177088
$$91$$ 0 0
$$92$$ 25216.0 0.310604
$$93$$ 0 0
$$94$$ 23848.0 0.278376
$$95$$ −1092.00 −0.0124141
$$96$$ 0 0
$$97$$ 182786.i 1.97248i 0.165307 + 0.986242i $$0.447139\pi$$
−0.165307 + 0.986242i $$0.552861\pi$$
$$98$$ 48372.0i 0.508779i
$$99$$ − 60750.0i − 0.622957i
$$100$$ −46864.0 −0.468640
$$101$$ 18514.0 0.180591 0.0902957 0.995915i $$-0.471219\pi$$
0.0902957 + 0.995915i $$0.471219\pi$$
$$102$$ 0 0
$$103$$ −116056. −1.07789 −0.538945 0.842341i $$-0.681177\pi$$
−0.538945 + 0.842341i $$0.681177\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ − 116184.i − 1.00434i
$$107$$ 153520. 1.29630 0.648150 0.761513i $$-0.275543\pi$$
0.648150 + 0.761513i $$0.275543\pi$$
$$108$$ 0 0
$$109$$ − 178622.i − 1.44002i −0.693963 0.720010i $$-0.744137\pi$$
0.693963 0.720010i $$-0.255863\pi$$
$$110$$ − 14000.0i − 0.110318i
$$111$$ 0 0
$$112$$ 43520.0i 0.327826i
$$113$$ −244754. −1.80316 −0.901579 0.432615i $$-0.857591\pi$$
−0.901579 + 0.432615i $$0.857591\pi$$
$$114$$ 0 0
$$115$$ 22064.0i 0.155575i
$$116$$ −41248.0 −0.284615
$$117$$ 0 0
$$118$$ 55688.0 0.368177
$$119$$ − 180540.i − 1.16871i
$$120$$ 0 0
$$121$$ 98551.0 0.611924
$$122$$ 131528.i 0.800053i
$$123$$ 0 0
$$124$$ 138464.i 0.808691i
$$125$$ − 84756.0i − 0.485172i
$$126$$ −165240. −0.927233
$$127$$ −256600. −1.41172 −0.705858 0.708353i $$-0.749438\pi$$
−0.705858 + 0.708353i $$0.749438\pi$$
$$128$$ 16384.0i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −262736. −1.33765 −0.668823 0.743421i $$-0.733202\pi$$
−0.668823 + 0.743421i $$0.733202\pi$$
$$132$$ 0 0
$$133$$ 13260.0 0.0650002
$$134$$ −278264. −1.33874
$$135$$ 0 0
$$136$$ − 67968.0i − 0.315106i
$$137$$ 38286.0i 0.174276i 0.996196 + 0.0871382i $$0.0277722\pi$$
−0.996196 + 0.0871382i $$0.972228\pi$$
$$138$$ 0 0
$$139$$ −57776.0 −0.253636 −0.126818 0.991926i $$-0.540476\pi$$
−0.126818 + 0.991926i $$0.540476\pi$$
$$140$$ −38080.0 −0.164201
$$141$$ 0 0
$$142$$ −202168. −0.841379
$$143$$ 0 0
$$144$$ −62208.0 −0.250000
$$145$$ − 36092.0i − 0.142558i
$$146$$ 187000. 0.726038
$$147$$ 0 0
$$148$$ 175776.i 0.659637i
$$149$$ 28866.0i 0.106517i 0.998581 + 0.0532587i $$0.0169608\pi$$
−0.998581 + 0.0532587i $$0.983039\pi$$
$$150$$ 0 0
$$151$$ − 39870.0i − 0.142300i −0.997466 0.0711498i $$-0.977333\pi$$
0.997466 0.0711498i $$-0.0226668\pi$$
$$152$$ 4992.00 0.0175253
$$153$$ 258066. 0.891255
$$154$$ 170000.i 0.577627i
$$155$$ −121156. −0.405057
$$156$$ 0 0
$$157$$ 161042. 0.521423 0.260711 0.965417i $$-0.416043\pi$$
0.260711 + 0.965417i $$0.416043\pi$$
$$158$$ 77392.0i 0.246634i
$$159$$ 0 0
$$160$$ −14336.0 −0.0442719
$$161$$ − 267920.i − 0.814593i
$$162$$ − 236196.i − 0.707107i
$$163$$ − 312830.i − 0.922230i −0.887340 0.461115i $$-0.847450\pi$$
0.887340 0.461115i $$-0.152550\pi$$
$$164$$ − 16800.0i − 0.0487753i
$$165$$ 0 0
$$166$$ −349752. −0.985122
$$167$$ − 532926.i − 1.47869i −0.673329 0.739343i $$-0.735136\pi$$
0.673329 0.739343i $$-0.264864\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 59472.0 0.157830
$$171$$ 18954.0i 0.0495691i
$$172$$ −94400.0 −0.243305
$$173$$ 630458. 1.60155 0.800776 0.598964i $$-0.204421\pi$$
0.800776 + 0.598964i $$0.204421\pi$$
$$174$$ 0 0
$$175$$ 497930.i 1.22906i
$$176$$ 64000.0i 0.155739i
$$177$$ 0 0
$$178$$ −376680. −0.891092
$$179$$ 674916. 1.57441 0.787204 0.616693i $$-0.211528\pi$$
0.787204 + 0.616693i $$0.211528\pi$$
$$180$$ − 54432.0i − 0.125220i
$$181$$ −186282. −0.422644 −0.211322 0.977417i $$-0.567777\pi$$
−0.211322 + 0.977417i $$0.567777\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ − 100864.i − 0.219630i
$$185$$ −153804. −0.330399
$$186$$ 0 0
$$187$$ − 265500.i − 0.555214i
$$188$$ − 95392.0i − 0.196842i
$$189$$ 0 0
$$190$$ 4368.00i 0.00877806i
$$191$$ 812180. 1.61090 0.805451 0.592663i $$-0.201923\pi$$
0.805451 + 0.592663i $$0.201923\pi$$
$$192$$ 0 0
$$193$$ 150142.i 0.290141i 0.989421 + 0.145070i $$0.0463409\pi$$
−0.989421 + 0.145070i $$0.953659\pi$$
$$194$$ 731144. 1.39476
$$195$$ 0 0
$$196$$ 193488. 0.359761
$$197$$ 236394.i 0.433981i 0.976174 + 0.216991i $$0.0696241\pi$$
−0.976174 + 0.216991i $$0.930376\pi$$
$$198$$ −243000. −0.440497
$$199$$ 39376.0 0.0704854 0.0352427 0.999379i $$-0.488780\pi$$
0.0352427 + 0.999379i $$0.488780\pi$$
$$200$$ 187456.i 0.331379i
$$201$$ 0 0
$$202$$ − 74056.0i − 0.127697i
$$203$$ 438260.i 0.746435i
$$204$$ 0 0
$$205$$ 14700.0 0.0244305
$$206$$ 464224.i 0.762183i
$$207$$ 382968. 0.621207
$$208$$ 0 0
$$209$$ 19500.0 0.0308794
$$210$$ 0 0
$$211$$ −410776. −0.635183 −0.317592 0.948228i $$-0.602874\pi$$
−0.317592 + 0.948228i $$0.602874\pi$$
$$212$$ −464736. −0.710177
$$213$$ 0 0
$$214$$ − 614080.i − 0.916623i
$$215$$ − 82600.0i − 0.121866i
$$216$$ 0 0
$$217$$ 1.47118e6 2.12088
$$218$$ −714488. −1.01825
$$219$$ 0 0
$$220$$ −56000.0 −0.0780066
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 1.08688e6i 1.46359i 0.681523 + 0.731796i $$0.261318\pi$$
−0.681523 + 0.731796i $$0.738682\pi$$
$$224$$ 174080. 0.231808
$$225$$ −711747. −0.937280
$$226$$ 979016.i 1.27502i
$$227$$ − 256470.i − 0.330348i −0.986264 0.165174i $$-0.947181\pi$$
0.986264 0.165174i $$-0.0528186\pi$$
$$228$$ 0 0
$$229$$ 298110.i 0.375654i 0.982202 + 0.187827i $$0.0601444\pi$$
−0.982202 + 0.187827i $$0.939856\pi$$
$$230$$ 88256.0 0.110008
$$231$$ 0 0
$$232$$ 164992.i 0.201253i
$$233$$ 611926. 0.738430 0.369215 0.929344i $$-0.379627\pi$$
0.369215 + 0.929344i $$0.379627\pi$$
$$234$$ 0 0
$$235$$ 83468.0 0.0985940
$$236$$ − 222752.i − 0.260340i
$$237$$ 0 0
$$238$$ −722160. −0.826401
$$239$$ 36570.0i 0.0414124i 0.999786 + 0.0207062i $$0.00659146\pi$$
−0.999786 + 0.0207062i $$0.993409\pi$$
$$240$$ 0 0
$$241$$ − 380922.i − 0.422468i −0.977436 0.211234i $$-0.932252\pi$$
0.977436 0.211234i $$-0.0677482\pi$$
$$242$$ − 394204.i − 0.432696i
$$243$$ 0 0
$$244$$ 526112. 0.565723
$$245$$ 169302.i 0.180197i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 553856. 0.571831
$$249$$ 0 0
$$250$$ −339024. −0.343068
$$251$$ 1.22807e6 1.23038 0.615188 0.788380i $$-0.289080\pi$$
0.615188 + 0.788380i $$0.289080\pi$$
$$252$$ 660960.i 0.655653i
$$253$$ − 394000.i − 0.386986i
$$254$$ 1.02640e6i 0.998234i
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 439278. 0.414865 0.207432 0.978249i $$-0.433489\pi$$
0.207432 + 0.978249i $$0.433489\pi$$
$$258$$ 0 0
$$259$$ 1.86762e6 1.72997
$$260$$ 0 0
$$261$$ −626454. −0.569230
$$262$$ 1.05094e6i 0.945859i
$$263$$ −1.67987e6 −1.49757 −0.748783 0.662816i $$-0.769361\pi$$
−0.748783 + 0.662816i $$0.769361\pi$$
$$264$$ 0 0
$$265$$ − 406644.i − 0.355713i
$$266$$ − 53040.0i − 0.0459621i
$$267$$ 0 0
$$268$$ 1.11306e6i 0.946629i
$$269$$ 1.93840e6 1.63329 0.816645 0.577141i $$-0.195832\pi$$
0.816645 + 0.577141i $$0.195832\pi$$
$$270$$ 0 0
$$271$$ 695498.i 0.575271i 0.957740 + 0.287636i $$0.0928692\pi$$
−0.957740 + 0.287636i $$0.907131\pi$$
$$272$$ −271872. −0.222814
$$273$$ 0 0
$$274$$ 153144. 0.123232
$$275$$ 732250.i 0.583885i
$$276$$ 0 0
$$277$$ 1.13138e6 0.885948 0.442974 0.896534i $$-0.353923\pi$$
0.442974 + 0.896534i $$0.353923\pi$$
$$278$$ 231104.i 0.179348i
$$279$$ 2.10292e6i 1.61738i
$$280$$ 152320.i 0.116108i
$$281$$ − 1.73122e6i − 1.30793i −0.756523 0.653967i $$-0.773103\pi$$
0.756523 0.653967i $$-0.226897\pi$$
$$282$$ 0 0
$$283$$ 1.47124e6 1.09199 0.545995 0.837788i $$-0.316152\pi$$
0.545995 + 0.837788i $$0.316152\pi$$
$$284$$ 808672.i 0.594945i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −178500. −0.127919
$$288$$ 248832.i 0.176777i
$$289$$ −292013. −0.205664
$$290$$ −144368. −0.100804
$$291$$ 0 0
$$292$$ − 748000.i − 0.513387i
$$293$$ − 2.88855e6i − 1.96567i −0.184491 0.982834i $$-0.559064\pi$$
0.184491 0.982834i $$-0.440936\pi$$
$$294$$ 0 0
$$295$$ 194908. 0.130399
$$296$$ 703104. 0.466434
$$297$$ 0 0
$$298$$ 115464. 0.0753192
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00300e6i 0.638094i
$$302$$ −159480. −0.100621
$$303$$ 0 0
$$304$$ − 19968.0i − 0.0123923i
$$305$$ 460348.i 0.283359i
$$306$$ − 1.03226e6i − 0.630213i
$$307$$ − 874118.i − 0.529327i −0.964341 0.264664i $$-0.914739\pi$$
0.964341 0.264664i $$-0.0852609\pi$$
$$308$$ 680000. 0.408444
$$309$$ 0 0
$$310$$ 484624.i 0.286418i
$$311$$ −2.68224e6 −1.57252 −0.786261 0.617895i $$-0.787986\pi$$
−0.786261 + 0.617895i $$0.787986\pi$$
$$312$$ 0 0
$$313$$ −1.34459e6 −0.775761 −0.387880 0.921710i $$-0.626793\pi$$
−0.387880 + 0.921710i $$0.626793\pi$$
$$314$$ − 644168.i − 0.368702i
$$315$$ −578340. −0.328403
$$316$$ 309568. 0.174397
$$317$$ 1.32074e6i 0.738191i 0.929392 + 0.369095i $$0.120332\pi$$
−0.929392 + 0.369095i $$0.879668\pi$$
$$318$$ 0 0
$$319$$ 644500.i 0.354606i
$$320$$ 57344.0i 0.0313050i
$$321$$ 0 0
$$322$$ −1.07168e6 −0.576004
$$323$$ 82836.0i 0.0441787i
$$324$$ −944784. −0.500000
$$325$$ 0 0
$$326$$ −1.25132e6 −0.652115
$$327$$ 0 0
$$328$$ −67200.0 −0.0344893
$$329$$ −1.01354e6 −0.516239
$$330$$ 0 0
$$331$$ − 2.05728e6i − 1.03210i −0.856558 0.516051i $$-0.827401\pi$$
0.856558 0.516051i $$-0.172599\pi$$
$$332$$ 1.39901e6i 0.696586i
$$333$$ 2.66960e6i 1.31927i
$$334$$ −2.13170e6 −1.04559
$$335$$ −973924. −0.474147
$$336$$ 0 0
$$337$$ −453398. −0.217473 −0.108736 0.994071i $$-0.534680\pi$$
−0.108736 + 0.994071i $$0.534680\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ − 237888.i − 0.111603i
$$341$$ 2.16350e6 1.00756
$$342$$ 75816.0 0.0350506
$$343$$ 801380.i 0.367793i
$$344$$ 377600.i 0.172043i
$$345$$ 0 0
$$346$$ − 2.52183e6i − 1.13247i
$$347$$ −1.23065e6 −0.548669 −0.274334 0.961634i $$-0.588457\pi$$
−0.274334 + 0.961634i $$0.588457\pi$$
$$348$$ 0 0
$$349$$ 2.43825e6i 1.07155i 0.844360 + 0.535777i $$0.179981\pi$$
−0.844360 + 0.535777i $$0.820019\pi$$
$$350$$ 1.99172e6 0.869077
$$351$$ 0 0
$$352$$ 256000. 0.110124
$$353$$ − 2.68315e6i − 1.14606i −0.819534 0.573031i $$-0.805767\pi$$
0.819534 0.573031i $$-0.194233\pi$$
$$354$$ 0 0
$$355$$ −707588. −0.297995
$$356$$ 1.50672e6i 0.630097i
$$357$$ 0 0
$$358$$ − 2.69966e6i − 1.11327i
$$359$$ − 1.58693e6i − 0.649864i −0.945737 0.324932i $$-0.894659\pi$$
0.945737 0.324932i $$-0.105341\pi$$
$$360$$ −217728. −0.0885438
$$361$$ 2.47002e6 0.997543
$$362$$ 745128.i 0.298854i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 654500. 0.257145
$$366$$ 0 0
$$367$$ −60052.0 −0.0232735 −0.0116368 0.999932i $$-0.503704\pi$$
−0.0116368 + 0.999932i $$0.503704\pi$$
$$368$$ −403456. −0.155302
$$369$$ − 255150.i − 0.0975505i
$$370$$ 615216.i 0.233627i
$$371$$ 4.93782e6i 1.86252i
$$372$$ 0 0
$$373$$ −4.01853e6 −1.49553 −0.747766 0.663963i $$-0.768873\pi$$
−0.747766 + 0.663963i $$0.768873\pi$$
$$374$$ −1.06200e6 −0.392596
$$375$$ 0 0
$$376$$ −381568. −0.139188
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1.67581e6i 0.599276i 0.954053 + 0.299638i $$0.0968659\pi$$
−0.954053 + 0.299638i $$0.903134\pi$$
$$380$$ 17472.0 0.00620703
$$381$$ 0 0
$$382$$ − 3.24872e6i − 1.13908i
$$383$$ 687258.i 0.239399i 0.992810 + 0.119700i $$0.0381932\pi$$
−0.992810 + 0.119700i $$0.961807\pi$$
$$384$$ 0 0
$$385$$ 595000.i 0.204581i
$$386$$ 600568. 0.205161
$$387$$ −1.43370e6 −0.486610
$$388$$ − 2.92458e6i − 0.986242i
$$389$$ −1.37611e6 −0.461082 −0.230541 0.973063i $$-0.574050\pi$$
−0.230541 + 0.973063i $$0.574050\pi$$
$$390$$ 0 0
$$391$$ 1.67371e6 0.553655
$$392$$ − 773952.i − 0.254389i
$$393$$ 0 0
$$394$$ 945576. 0.306871
$$395$$ 270872.i 0.0873517i
$$396$$ 972000.i 0.311479i
$$397$$ 721198.i 0.229656i 0.993385 + 0.114828i $$0.0366317\pi$$
−0.993385 + 0.114828i $$0.963368\pi$$
$$398$$ − 157504.i − 0.0498407i
$$399$$ 0 0
$$400$$ 749824. 0.234320
$$401$$ − 2.22681e6i − 0.691548i −0.938318 0.345774i $$-0.887616\pi$$
0.938318 0.345774i $$-0.112384\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −296224. −0.0902957
$$405$$ − 826686.i − 0.250440i
$$406$$ 1.75304e6 0.527809
$$407$$ 2.74650e6 0.821852
$$408$$ 0 0
$$409$$ 2.00783e6i 0.593496i 0.954956 + 0.296748i $$0.0959021\pi$$
−0.954956 + 0.296748i $$0.904098\pi$$
$$410$$ − 58800.0i − 0.0172750i
$$411$$ 0 0
$$412$$ 1.85690e6 0.538945
$$413$$ −2.36674e6 −0.682772
$$414$$ − 1.53187e6i − 0.439260i
$$415$$ −1.22413e6 −0.348906
$$416$$ 0 0
$$417$$ 0 0
$$418$$ − 78000.0i − 0.0218350i
$$419$$ 5.99378e6 1.66788 0.833942 0.551852i $$-0.186079\pi$$
0.833942 + 0.551852i $$0.186079\pi$$
$$420$$ 0 0
$$421$$ − 5.32737e6i − 1.46490i −0.680822 0.732449i $$-0.738377\pi$$
0.680822 0.732449i $$-0.261623\pi$$
$$422$$ 1.64310e6i 0.449142i
$$423$$ − 1.44877e6i − 0.393684i
$$424$$ 1.85894e6i 0.502171i
$$425$$ −3.11060e6 −0.835356
$$426$$ 0 0
$$427$$ − 5.58994e6i − 1.48367i
$$428$$ −2.45632e6 −0.648150
$$429$$ 0 0
$$430$$ −330400. −0.0861725
$$431$$ − 5.42972e6i − 1.40794i −0.710230 0.703970i $$-0.751409\pi$$
0.710230 0.703970i $$-0.248591\pi$$
$$432$$ 0 0
$$433$$ −7.43979e6 −1.90696 −0.953479 0.301459i $$-0.902526\pi$$
−0.953479 + 0.301459i $$0.902526\pi$$
$$434$$ − 5.88472e6i − 1.49969i
$$435$$ 0 0
$$436$$ 2.85795e6i 0.720010i
$$437$$ 122928.i 0.0307927i
$$438$$ 0 0
$$439$$ −6.86418e6 −1.69991 −0.849957 0.526852i $$-0.823372\pi$$
−0.849957 + 0.526852i $$0.823372\pi$$
$$440$$ 224000.i 0.0551590i
$$441$$ 2.93860e6 0.719522
$$442$$ 0 0
$$443$$ −3.46630e6 −0.839182 −0.419591 0.907713i $$-0.637827\pi$$
−0.419591 + 0.907713i $$0.637827\pi$$
$$444$$ 0 0
$$445$$ −1.31838e6 −0.315603
$$446$$ 4.34753e6 1.03492
$$447$$ 0 0
$$448$$ − 696320.i − 0.163913i
$$449$$ 1.40426e6i 0.328725i 0.986400 + 0.164362i $$0.0525566\pi$$
−0.986400 + 0.164362i $$0.947443\pi$$
$$450$$ 2.84699e6i 0.662757i
$$451$$ −262500. −0.0607698
$$452$$ 3.91606e6 0.901579
$$453$$ 0 0
$$454$$ −1.02588e6 −0.233591
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 5.95072e6i − 1.33284i −0.745575 0.666421i $$-0.767825\pi$$
0.745575 0.666421i $$-0.232175\pi$$
$$458$$ 1.19244e6 0.265627
$$459$$ 0 0
$$460$$ − 353024.i − 0.0777875i
$$461$$ − 6.25465e6i − 1.37073i −0.728202 0.685363i $$-0.759644\pi$$
0.728202 0.685363i $$-0.240356\pi$$
$$462$$ 0 0
$$463$$ 1.55055e6i 0.336149i 0.985774 + 0.168075i $$0.0537550\pi$$
−0.985774 + 0.168075i $$0.946245\pi$$
$$464$$ 659968. 0.142308
$$465$$ 0 0
$$466$$ − 2.44770e6i − 0.522149i
$$467$$ 1.80480e6 0.382945 0.191472 0.981498i $$-0.438674\pi$$
0.191472 + 0.981498i $$0.438674\pi$$
$$468$$ 0 0
$$469$$ 1.18262e7 2.48264
$$470$$ − 333872.i − 0.0697165i
$$471$$ 0 0
$$472$$ −891008. −0.184088
$$473$$ 1.47500e6i 0.303137i
$$474$$ 0 0
$$475$$ − 228462.i − 0.0464601i
$$476$$ 2.88864e6i 0.584354i
$$477$$ −7.05818e6 −1.42035
$$478$$ 146280. 0.0292830
$$479$$ − 2.21809e6i − 0.441712i −0.975306 0.220856i $$-0.929115\pi$$
0.975306 0.220856i $$-0.0708851\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −1.52369e6 −0.298730
$$483$$ 0 0
$$484$$ −1.57682e6 −0.305962
$$485$$ 2.55900e6 0.493988
$$486$$ 0 0
$$487$$ − 6.14268e6i − 1.17364i −0.809717 0.586821i $$-0.800379\pi$$
0.809717 0.586821i $$-0.199621\pi$$
$$488$$ − 2.10445e6i − 0.400026i
$$489$$ 0 0
$$490$$ 677208. 0.127418
$$491$$ −6.44486e6 −1.20645 −0.603226 0.797571i $$-0.706118\pi$$
−0.603226 + 0.797571i $$0.706118\pi$$
$$492$$ 0 0
$$493$$ −2.73784e6 −0.507330
$$494$$ 0 0
$$495$$ −850500. −0.156013
$$496$$ − 2.21542e6i − 0.404346i
$$497$$ 8.59214e6 1.56031
$$498$$ 0 0
$$499$$ 4.25838e6i 0.765584i 0.923835 + 0.382792i $$0.125037\pi$$
−0.923835 + 0.382792i $$0.874963\pi$$
$$500$$ 1.35610e6i 0.242586i
$$501$$ 0 0
$$502$$ − 4.91227e6i − 0.870008i
$$503$$ −3.56242e6 −0.627806 −0.313903 0.949455i $$-0.601637\pi$$
−0.313903 + 0.949455i $$0.601637\pi$$
$$504$$ 2.64384e6 0.463616
$$505$$ − 259196.i − 0.0452272i
$$506$$ −1.57600e6 −0.273640
$$507$$ 0 0
$$508$$ 4.10560e6 0.705858
$$509$$ 4.23936e6i 0.725281i 0.931929 + 0.362640i $$0.118125\pi$$
−0.931929 + 0.362640i $$0.881875\pi$$
$$510$$ 0 0
$$511$$ −7.94750e6 −1.34641
$$512$$ − 262144.i − 0.0441942i
$$513$$ 0 0
$$514$$ − 1.75711e6i − 0.293354i
$$515$$ 1.62478e6i 0.269946i
$$516$$ 0 0
$$517$$ −1.49050e6 −0.245248
$$518$$ − 7.47048e6i − 1.22328i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2.38657e6 0.385194 0.192597 0.981278i $$-0.438309\pi$$
0.192597 + 0.981278i $$0.438309\pi$$
$$522$$ 2.50582e6i 0.402507i
$$523$$ −8.84129e6 −1.41339 −0.706694 0.707519i $$-0.749814\pi$$
−0.706694 + 0.707519i $$0.749814\pi$$
$$524$$ 4.20378e6 0.668823
$$525$$ 0 0
$$526$$ 6.71947e6i 1.05894i
$$527$$ 9.19055e6i 1.44150i
$$528$$ 0 0
$$529$$ −3.95257e6 −0.614101
$$530$$ −1.62658e6 −0.251527
$$531$$ − 3.38305e6i − 0.520681i
$$532$$ −212160. −0.0325001
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 2.14928e6i − 0.324645i
$$536$$ 4.45222e6 0.669368
$$537$$ 0 0
$$538$$ − 7.75361e6i − 1.15491i
$$539$$ − 3.02325e6i − 0.448231i
$$540$$ 0 0
$$541$$ − 70058.0i − 0.0102912i −0.999987 0.00514558i $$-0.998362\pi$$
0.999987 0.00514558i $$-0.00163790\pi$$
$$542$$ 2.78199e6 0.406778
$$543$$ 0 0
$$544$$ 1.08749e6i 0.157553i
$$545$$ −2.50071e6 −0.360638
$$546$$ 0 0
$$547$$ −6.60752e6 −0.944213 −0.472107 0.881541i $$-0.656506\pi$$
−0.472107 + 0.881541i $$0.656506\pi$$
$$548$$ − 612576.i − 0.0871382i
$$549$$ 7.99033e6 1.13145
$$550$$ 2.92900e6 0.412869
$$551$$ − 201084.i − 0.0282162i
$$552$$ 0 0
$$553$$ − 3.28916e6i − 0.457375i
$$554$$ − 4.52551e6i − 0.626460i
$$555$$ 0 0
$$556$$ 924416. 0.126818
$$557$$ 1.10726e7i 1.51221i 0.654448 + 0.756107i $$0.272901\pi$$
−0.654448 + 0.756107i $$0.727099\pi$$
$$558$$ 8.41169e6 1.14366
$$559$$ 0 0
$$560$$ 609280. 0.0821007
$$561$$ 0 0
$$562$$ −6.92487e6 −0.924849
$$563$$ 1.43532e6 0.190843 0.0954216 0.995437i $$-0.469580\pi$$
0.0954216 + 0.995437i $$0.469580\pi$$
$$564$$ 0 0
$$565$$ 3.42656e6i 0.451582i
$$566$$ − 5.88498e6i − 0.772153i
$$567$$ 1.00383e7i 1.31131i
$$568$$ 3.23469e6 0.420689
$$569$$ 1.17051e7 1.51564 0.757818 0.652466i $$-0.226266\pi$$
0.757818 + 0.652466i $$0.226266\pi$$
$$570$$ 0 0
$$571$$ −4.81885e6 −0.618519 −0.309260 0.950978i $$-0.600081\pi$$
−0.309260 + 0.950978i $$0.600081\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 714000.i 0.0904521i
$$575$$ −4.61610e6 −0.582245
$$576$$ 995328. 0.125000
$$577$$ − 1.35572e6i − 0.169523i −0.996401 0.0847617i $$-0.972987\pi$$
0.996401 0.0847617i $$-0.0270129\pi$$
$$578$$ 1.16805e6i 0.145426i
$$579$$ 0 0
$$580$$ 577472.i 0.0712789i
$$581$$ 1.48645e7 1.82687
$$582$$ 0 0
$$583$$ 7.26150e6i 0.884820i
$$584$$ −2.99200e6 −0.363019
$$585$$ 0 0
$$586$$ −1.15542e7 −1.38994
$$587$$ 5.03941e6i 0.603649i 0.953364 + 0.301824i $$0.0975957\pi$$
−0.953364 + 0.301824i $$0.902404\pi$$
$$588$$ 0 0
$$589$$ −675012. −0.0801721
$$590$$ − 779632.i − 0.0922061i
$$591$$ 0 0
$$592$$ − 2.81242e6i − 0.329819i
$$593$$ − 9.16124e6i − 1.06984i −0.844904 0.534919i $$-0.820342\pi$$
0.844904 0.534919i $$-0.179658\pi$$
$$594$$ 0 0
$$595$$ −2.52756e6 −0.292691
$$596$$ − 461856.i − 0.0532587i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.46635e6 −0.736363 −0.368182 0.929754i $$-0.620020\pi$$
−0.368182 + 0.929754i $$0.620020\pi$$
$$600$$ 0 0
$$601$$ −1.18021e7 −1.33282 −0.666411 0.745585i $$-0.732170\pi$$
−0.666411 + 0.745585i $$0.732170\pi$$
$$602$$ 4.01200e6 0.451201
$$603$$ 1.69045e7i 1.89326i
$$604$$ 637920.i 0.0711498i
$$605$$ − 1.37971e6i − 0.153250i
$$606$$ 0 0
$$607$$ 2.25748e6 0.248686 0.124343 0.992239i $$-0.460318\pi$$
0.124343 + 0.992239i $$0.460318\pi$$
$$608$$ −79872.0 −0.00876265
$$609$$ 0 0
$$610$$ 1.84139e6 0.200365
$$611$$ 0 0
$$612$$ −4.12906e6 −0.445628
$$613$$ − 2.75378e6i − 0.295991i −0.988988 0.147995i $$-0.952718\pi$$
0.988988 0.147995i $$-0.0472821\pi$$
$$614$$ −3.49647e6 −0.374291
$$615$$ 0 0
$$616$$ − 2.72000e6i − 0.288813i
$$617$$ 3.41607e6i 0.361255i 0.983552 + 0.180627i $$0.0578128\pi$$
−0.983552 + 0.180627i $$0.942187\pi$$
$$618$$ 0 0
$$619$$ − 9.43169e6i − 0.989379i −0.869070 0.494690i $$-0.835282\pi$$
0.869070 0.494690i $$-0.164718\pi$$
$$620$$ 1.93850e6 0.202528
$$621$$ 0 0
$$622$$ 1.07290e7i 1.11194i
$$623$$ 1.60089e7 1.65250
$$624$$ 0 0
$$625$$ 7.96654e6 0.815774
$$626$$ 5.37834e6i 0.548546i
$$627$$ 0 0
$$628$$ −2.57667e6 −0.260711
$$629$$ 1.16671e7i 1.17581i
$$630$$ 2.31336e6i 0.232216i
$$631$$ 4.87474e6i 0.487391i 0.969852 + 0.243696i $$0.0783598\pi$$
−0.969852 + 0.243696i $$0.921640\pi$$
$$632$$ − 1.23827e6i − 0.123317i
$$633$$ 0 0
$$634$$ 5.28295e6 0.521980
$$635$$ 3.59240e6i 0.353550i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 2.57800e6 0.250744
$$639$$ 1.22817e7i 1.18989i
$$640$$ 229376. 0.0221359
$$641$$ −9.74279e6 −0.936566 −0.468283 0.883579i $$-0.655127\pi$$
−0.468283 + 0.883579i $$0.655127\pi$$
$$642$$ 0 0
$$643$$ 1.63894e6i 0.156327i 0.996941 + 0.0781637i $$0.0249057\pi$$
−0.996941 + 0.0781637i $$0.975094\pi$$
$$644$$ 4.28672e6i 0.407296i
$$645$$ 0 0
$$646$$ 331344. 0.0312390
$$647$$ 1.59069e6 0.149391 0.0746955 0.997206i $$-0.476202\pi$$
0.0746955 + 0.997206i $$0.476202\pi$$
$$648$$ 3.77914e6i 0.353553i
$$649$$ −3.48050e6 −0.324362
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 5.00528e6i 0.461115i
$$653$$ 1.59778e7 1.46634 0.733170 0.680045i $$-0.238040\pi$$
0.733170 + 0.680045i $$0.238040\pi$$
$$654$$ 0 0
$$655$$ 3.67830e6i 0.335000i
$$656$$ 268800.i 0.0243876i
$$657$$ − 1.13602e7i − 1.02677i
$$658$$ 4.05416e6i 0.365036i
$$659$$ −6.02458e6 −0.540397 −0.270199 0.962805i $$-0.587089\pi$$
−0.270199 + 0.962805i $$0.587089\pi$$
$$660$$ 0 0
$$661$$ 2.00705e7i 1.78671i 0.449352 + 0.893355i $$0.351655\pi$$
−0.449352 + 0.893355i $$0.648345\pi$$
$$662$$ −8.22911e6 −0.729807
$$663$$ 0 0
$$664$$ 5.59603e6 0.492561
$$665$$ − 185640.i − 0.0162786i
$$666$$ 1.06784e7 0.932868
$$667$$ −4.06293e6 −0.353610
$$668$$ 8.52682e6i 0.739343i
$$669$$ 0 0
$$670$$ 3.89570e6i 0.335273i
$$671$$ − 8.22050e6i − 0.704842i
$$672$$ 0 0
$$673$$ 5.48575e6 0.466873 0.233436 0.972372i $$-0.425003\pi$$
0.233436 + 0.972372i $$0.425003\pi$$
$$674$$ 1.81359e6i 0.153776i
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −4.74926e6 −0.398248 −0.199124 0.979974i $$-0.563810\pi$$
−0.199124 + 0.979974i $$0.563810\pi$$
$$678$$ 0 0
$$679$$ −3.10736e7 −2.58653
$$680$$ −951552. −0.0789151
$$681$$ 0 0
$$682$$ − 8.65400e6i − 0.712453i
$$683$$ 6.13964e6i 0.503606i 0.967778 + 0.251803i $$0.0810236\pi$$
−0.967778 + 0.251803i $$0.918976\pi$$
$$684$$ − 303264.i − 0.0247845i
$$685$$ 536004. 0.0436457
$$686$$ 3.20552e6 0.260069
$$687$$ 0 0
$$688$$ 1.51040e6 0.121652
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.57617e7i 1.25577i 0.778308 + 0.627883i $$0.216078\pi$$
−0.778308 + 0.627883i $$0.783922\pi$$
$$692$$ −1.00873e7 −0.800776
$$693$$ 1.03275e7 0.816887
$$694$$ 4.92259e6i 0.387967i
$$695$$ 808864.i 0.0635204i
$$696$$ 0 0
$$697$$ − 1.11510e6i − 0.0869424i
$$698$$ 9.75298e6 0.757703
$$699$$ 0 0
$$700$$ − 7.96688e6i − 0.614530i
$$701$$ 1.42036e7 1.09170 0.545851 0.837882i $$-0.316207\pi$$
0.545851 + 0.837882i $$0.316207\pi$$
$$702$$ 0 0
$$703$$ −856908. −0.0653952
$$704$$ − 1.02400e6i − 0.0778697i
$$705$$ 0 0
$$706$$ −1.07326e7 −0.810388
$$707$$ 3.14738e6i 0.236810i
$$708$$ 0 0
$$709$$ − 1.60718e7i − 1.20074i −0.799723 0.600369i $$-0.795020\pi$$
0.799723 0.600369i $$-0.204980\pi$$
$$710$$ 2.83035e6i 0.210715i
$$711$$ 4.70156e6 0.348793
$$712$$ 6.02688e6 0.445546
$$713$$ 1.36387e7i 1.00473i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −1.07987e7 −0.787204
$$717$$ 0 0
$$718$$ −6.34774e6 −0.459524
$$719$$ 2.07078e7 1.49387 0.746933 0.664900i $$-0.231526\pi$$
0.746933 + 0.664900i $$0.231526\pi$$
$$720$$ 870912.i 0.0626099i
$$721$$ − 1.97295e7i − 1.41344i
$$722$$ − 9.88006e6i − 0.705369i
$$723$$ 0 0
$$724$$ 2.98051e6 0.211322
$$725$$ 7.55096e6 0.533528
$$726$$ 0 0
$$727$$ −5.04803e6 −0.354231 −0.177115 0.984190i $$-0.556677\pi$$
−0.177115 + 0.984190i $$0.556677\pi$$
$$728$$ 0 0
$$729$$ −1.43489e7 −1.00000
$$730$$ − 2.61800e6i − 0.181829i
$$731$$ −6.26580e6 −0.433694
$$732$$ 0 0
$$733$$ − 2.10377e7i − 1.44623i −0.690728 0.723115i $$-0.742710\pi$$
0.690728 0.723115i $$-0.257290\pi$$
$$734$$ 240208.i 0.0164569i
$$735$$ 0 0
$$736$$ 1.61382e6i 0.109815i
$$737$$ 1.73915e7 1.17942
$$738$$ −1.02060e6 −0.0689786
$$739$$ − 1.38992e7i − 0.936218i −0.883671 0.468109i $$-0.844935\pi$$
0.883671 0.468109i $$-0.155065\pi$$
$$740$$ 2.46086e6 0.165199
$$741$$ 0 0
$$742$$ 1.97513e7 1.31700
$$743$$ 1.23267e6i 0.0819169i 0.999161 + 0.0409584i $$0.0130411\pi$$
−0.999161 + 0.0409584i $$0.986959\pi$$
$$744$$ 0 0
$$745$$ 404124. 0.0266762
$$746$$ 1.60741e7i 1.05750i
$$747$$ 2.12474e7i 1.39317i
$$748$$ 4.24800e6i 0.277607i
$$749$$ 2.60984e7i 1.69985i
$$750$$ 0 0
$$751$$ 1.62624e6 0.105217 0.0526084 0.998615i $$-0.483247\pi$$
0.0526084 + 0.998615i $$0.483247\pi$$
$$752$$ 1.52627e6i 0.0984209i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −558180. −0.0356375
$$756$$ 0 0
$$757$$ 3.49882e6 0.221913 0.110956 0.993825i $$-0.464609\pi$$
0.110956 + 0.993825i $$0.464609\pi$$
$$758$$ 6.70324e6 0.423752
$$759$$ 0 0
$$760$$ − 69888.0i − 0.00438903i
$$761$$ − 2.21713e7i − 1.38781i −0.720067 0.693905i $$-0.755889\pi$$
0.720067 0.693905i $$-0.244111\pi$$
$$762$$ 0 0
$$763$$ 3.03657e7 1.88831
$$764$$ −1.29949e7 −0.805451
$$765$$ − 3.61292e6i − 0.223206i
$$766$$ 2.74903e6 0.169281
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1.08955e6i 0.0664400i 0.999448 + 0.0332200i $$0.0105762\pi$$
−0.999448 + 0.0332200i $$0.989424\pi$$
$$770$$ 2.38000e6 0.144661
$$771$$ 0 0
$$772$$ − 2.40227e6i − 0.145070i
$$773$$ 1.95219e6i 0.117510i 0.998272 + 0.0587549i $$0.0187130\pi$$
−0.998272 + 0.0587549i $$0.981287\pi$$
$$774$$ 5.73480e6i 0.344085i
$$775$$ − 2.53476e7i − 1.51594i
$$776$$ −1.16983e7 −0.697379
$$777$$ 0 0
$$778$$ 5.50442e6i 0.326034i
$$779$$ 81900.0 0.00483549
$$780$$ 0 0
$$781$$ 1.26355e7 0.741250
$$782$$ − 6.69485e6i − 0.391493i
$$783$$ 0 0
$$784$$ −3.09581e6 −0.179880
$$785$$ − 2.25459e6i − 0.130585i
$$786$$ 0 0
$$787$$ 1.44531e7i 0.831809i 0.909408 + 0.415904i $$0.136535\pi$$
−0.909408 + 0.415904i $$0.863465\pi$$
$$788$$ − 3.78230e6i − 0.216991i
$$789$$ 0 0
$$790$$ 1.08349e6 0.0617670
$$791$$ − 4.16082e7i − 2.36449i
$$792$$ 3.88800e6 0.220249
$$793$$ 0 0
$$794$$ 2.88479e6 0.162391
$$795$$ 0 0
$$796$$ −630016. −0.0352427
$$797$$ −1.23500e7 −0.688685 −0.344343 0.938844i $$-0.611898\pi$$
−0.344343 + 0.938844i $$0.611898\pi$$
$$798$$ 0 0
$$799$$ − 6.33164e6i − 0.350873i
$$800$$ − 2.99930e6i − 0.165689i
$$801$$ 2.28833e7i 1.26019i
$$802$$ −8.90724e6 −0.488998
$$803$$ −1.16875e7 −0.639636
$$804$$ 0 0
$$805$$ −3.75088e6 −0.204006
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 1.18490e6i 0.0638487i
$$809$$ 1.15968e7 0.622970 0.311485 0.950251i $$-0.399174\pi$$
0.311485 + 0.950251i $$0.399174\pi$$
$$810$$ −3.30674e6 −0.177088
$$811$$ − 2.47534e7i − 1.32155i −0.750585 0.660774i $$-0.770228\pi$$
0.750585 0.660774i $$-0.229772\pi$$
$$812$$ − 7.01216e6i − 0.373217i
$$813$$ 0 0
$$814$$ − 1.09860e7i − 0.581137i
$$815$$ −4.37962e6 −0.230963
$$816$$ 0 0
$$817$$ − 460200.i − 0.0241208i
$$818$$ 8.03130e6 0.419665
$$819$$ 0 0
$$820$$ −235200. −0.0122153
$$821$$ − 2.47470e6i − 0.128134i −0.997946 0.0640671i $$-0.979593\pi$$
0.997946 0.0640671i $$-0.0204072\pi$$
$$822$$ 0 0
$$823$$ 7.84754e6 0.403863 0.201932 0.979400i $$-0.435278\pi$$
0.201932 + 0.979400i $$0.435278\pi$$
$$824$$ − 7.42758e6i − 0.381092i
$$825$$ 0 0
$$826$$ 9.46696e6i 0.482792i
$$827$$ 2.26192e7i 1.15004i 0.818140 + 0.575020i $$0.195006\pi$$
−0.818140 + 0.575020i $$0.804994\pi$$
$$828$$ −6.12749e6 −0.310604
$$829$$ 1.73912e7 0.878907 0.439454 0.898265i $$-0.355172\pi$$
0.439454 + 0.898265i $$0.355172\pi$$
$$830$$ 4.89653e6i 0.246714i
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.28428e7 0.641278
$$834$$ 0 0
$$835$$ −7.46096e6 −0.370321
$$836$$ −312000. −0.0154397
$$837$$ 0 0
$$838$$ − 2.39751e7i − 1.17937i
$$839$$ 3.43825e7i 1.68629i 0.537684 + 0.843147i $$0.319299\pi$$
−0.537684 + 0.843147i $$0.680701\pi$$
$$840$$ 0 0
$$841$$ −1.38651e7 −0.675977
$$842$$ −2.13095e7 −1.03584
$$843$$ 0 0
$$844$$ 6.57242e6 0.317592
$$845$$ 0 0
$$846$$ −5.79506e6 −0.278376
$$847$$ 1.67537e7i 0.802419i
$$848$$ 7.43578e6 0.355089
$$849$$ 0 0
$$850$$ 1.24424e7i 0.590686i
$$851$$ 1.73139e7i 0.819543i
$$852$$ 0 0
$$853$$ − 2.31007e7i − 1.08706i −0.839391 0.543528i $$-0.817088\pi$$
0.839391 0.543528i $$-0.182912\pi$$
$$854$$ −2.23598e7 −1.04911
$$855$$ 265356. 0.0124141
$$856$$ 9.82528e6i 0.458311i
$$857$$ 7.02305e6 0.326643 0.163322 0.986573i $$-0.447779\pi$$
0.163322 + 0.986573i $$0.447779\pi$$
$$858$$ 0 0
$$859$$ 8.82135e6 0.407899 0.203949 0.978981i $$-0.434622\pi$$
0.203949 + 0.978981i $$0.434622\pi$$
$$860$$ 1.32160e6i 0.0609332i
$$861$$ 0 0
$$862$$ −2.17189e7 −0.995564
$$863$$ 2.39560e7i 1.09493i 0.836828 + 0.547466i $$0.184408\pi$$
−0.836828 + 0.547466i $$0.815592\pi$$
$$864$$ 0 0
$$865$$ − 8.82641e6i − 0.401092i
$$866$$ 2.97592e7i 1.34842i
$$867$$ 0 0
$$868$$ −2.35389e7 −1.06044
$$869$$ − 4.83700e6i − 0.217283i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 1.14318e7 0.509124
$$873$$ − 4.44170e7i − 1.97248i
$$874$$ 491712. 0.0217737
$$875$$ 1.44085e7 0.636208
$$876$$ 0 0
$$877$$ − 5.79805e6i − 0.254556i −0.991867 0.127278i $$-0.959376\pi$$
0.991867 0.127278i $$-0.0406240\pi$$
$$878$$ 2.74567e7i 1.20202i
$$879$$ 0 0
$$880$$ 896000. 0.0390033
$$881$$ 1.30527e7 0.566580 0.283290 0.959034i $$-0.408574\pi$$
0.283290 + 0.959034i $$0.408574\pi$$
$$882$$ − 1.17544e7i − 0.508779i
$$883$$ −4.73009e6 −0.204159 −0.102079 0.994776i $$-0.532550\pi$$
−0.102079 + 0.994776i $$0.532550\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 1.38652e7i 0.593392i
$$887$$ 2.80737e7 1.19809 0.599046 0.800714i $$-0.295547\pi$$
0.599046 + 0.800714i $$0.295547\pi$$
$$888$$ 0 0
$$889$$ − 4.36220e7i − 1.85119i
$$890$$ 5.27352e6i 0.223165i
$$891$$ 1.47622e7i 0.622957i
$$892$$ − 1.73901e7i − 0.731796i
$$893$$ 465036. 0.0195145
$$894$$ 0 0
$$895$$ − 9.44882e6i − 0.394294i
$$896$$ −2.78528e6 −0.115904
$$897$$ 0 0
$$898$$ 5.61705e6 0.232443
$$899$$ − 2.23100e7i − 0.920663i
$$900$$ 1.13880e7 0.468640
$$901$$ −3.08469e7 −1.26590
$$902$$ 1.05000e6i 0.0429708i
$$903$$ 0 0
$$904$$ − 1.56643e7i − 0.637512i
$$905$$ 2.60795e6i 0.105847i
$$906$$ 0 0
$$907$$ −2.28552e7 −0.922500 −0.461250 0.887270i $$-0.652599\pi$$
−0.461250 + 0.887270i $$0.652599\pi$$
$$908$$ 4.10352e6i 0.165174i
$$909$$ −4.49890e6 −0.180591
$$910$$ 0 0
$$911$$ −3.27335e7 −1.30676 −0.653381 0.757029i $$-0.726650\pi$$
−0.653381 + 0.757029i $$0.726650\pi$$
$$912$$ 0 0
$$913$$ 2.18595e7 0.867887
$$914$$ −2.38029e7 −0.942462
$$915$$ 0 0
$$916$$ − 4.76976e6i − 0.187827i
$$917$$ − 4.46651e7i − 1.75406i
$$918$$ 0 0
$$919$$ −1.27717e7 −0.498839 −0.249419 0.968396i $$-0.580240\pi$$
−0.249419 + 0.968396i $$0.580240\pi$$
$$920$$ −1.41210e6 −0.0550040
$$921$$ 0 0
$$922$$ −2.50186e7 −0.969249
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 3.21780e7i − 1.23653i
$$926$$ 6.20218e6 0.237693
$$927$$ 2.82016e7 1.07789
$$928$$ − 2.63987e6i − 0.100627i
$$929$$ 3.48297e7i 1.32407i 0.749473 + 0.662034i $$0.230307\pi$$
−0.749473 + 0.662034i $$0.769693\pi$$
$$930$$ 0 0
$$931$$ 943254.i 0.0356660i
$$932$$ −9.79082e6 −0.369215
$$933$$ 0 0
$$934$$ − 7.21918e6i − 0.270783i
$$935$$ −3.71700e6 −0.139048
$$936$$ 0 0
$$937$$ 3.00172e7 1.11692 0.558459 0.829532i $$-0.311393\pi$$
0.558459 + 0.829532i $$0.311393\pi$$
$$938$$ − 4.73049e7i − 1.75549i
$$939$$ 0 0
$$940$$ −1.33549e6 −0.0492970
$$941$$ − 4.50649e7i − 1.65907i −0.558457 0.829534i $$-0.688606\pi$$
0.558457 0.829534i $$-0.311394\pi$$
$$942$$ 0 0
$$943$$ − 1.65480e6i − 0.0605991i
$$944$$ 3.56403e6i 0.130170i
$$945$$ 0 0
$$946$$ 5.90000e6 0.214350
$$947$$ − 2.99276e7i − 1.08442i −0.840243 0.542210i $$-0.817588\pi$$
0.840243 0.542210i $$-0.182412\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −913848. −0.0328522
$$951$$ 0 0
$$952$$ 1.15546e7 0.413201
$$953$$ 4.25147e7 1.51638 0.758188 0.652036i $$-0.226085\pi$$
0.758188 + 0.652036i $$0.226085\pi$$
$$954$$ 2.82327e7i 1.00434i
$$955$$ − 1.13705e7i − 0.403433i
$$956$$ − 585120.i − 0.0207062i
$$957$$ 0 0
$$958$$ −8.87234e6 −0.312338
$$959$$ −6.50862e6 −0.228530
$$960$$ 0 0
$$961$$ −4.62626e7 −1.61593
$$962$$ 0 0
$$963$$ −3.73054e7 −1.29630
$$964$$ 6.09475e6i 0.211234i
$$965$$ 2.10199e6 0.0726628
$$966$$ 0 0
$$967$$ − 3.00251e7i − 1.03257i −0.856417 0.516284i $$-0.827315\pi$$
0.856417 0.516284i $$-0.172685\pi$$
$$968$$ 6.30726e6i 0.216348i
$$969$$ 0 0
$$970$$ − 1.02360e7i − 0.349302i
$$971$$ −4.00864e7 −1.36442 −0.682211 0.731155i $$-0.738982\pi$$
−0.682211 + 0.731155i $$0.738982\pi$$
$$972$$ 0 0
$$973$$ − 9.82192e6i − 0.332594i
$$974$$ −2.45707e7 −0.829890
$$975$$ 0 0
$$976$$ −8.41779e6 −0.282861
$$977$$ 5.12151e7i 1.71657i 0.513174 + 0.858284i $$0.328469\pi$$
−0.513174 + 0.858284i $$0.671531\pi$$
$$978$$ 0 0
$$979$$ 2.35425e7 0.785047
$$980$$ − 2.70883e6i − 0.0900984i
$$981$$ 4.34051e7i 1.44002i
$$982$$ 2.57794e7i 0.853090i
$$983$$ − 1.82382e7i − 0.602004i −0.953624 0.301002i $$-0.902679\pi$$
0.953624 0.301002i $$-0.0973211\pi$$
$$984$$ 0 0
$$985$$ 3.30952e6 0.108686
$$986$$ 1.09513e7i 0.358736i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.29840e6 −0.302286
$$990$$ 3.40200e6i 0.110318i
$$991$$ −3.24103e7 −1.04833 −0.524166 0.851616i $$-0.675623\pi$$
−0.524166 + 0.851616i $$0.675623\pi$$
$$992$$ −8.86170e6 −0.285915
$$993$$ 0 0
$$994$$ − 3.43686e7i − 1.10330i
$$995$$ − 551264.i − 0.0176523i
$$996$$ 0 0
$$997$$ −2.07867e7 −0.662289 −0.331145 0.943580i $$-0.607435\pi$$
−0.331145 + 0.943580i $$0.607435\pi$$
$$998$$ 1.70335e7 0.541350
$$999$$ 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 338.6.b.a.337.1 2
13.5 odd 4 26.6.a.a.1.1 1
13.8 odd 4 338.6.a.d.1.1 1
13.12 even 2 inner 338.6.b.a.337.2 2
39.5 even 4 234.6.a.g.1.1 1
52.31 even 4 208.6.a.b.1.1 1
65.18 even 4 650.6.b.a.599.2 2
65.44 odd 4 650.6.a.a.1.1 1
65.57 even 4 650.6.b.a.599.1 2
104.5 odd 4 832.6.a.d.1.1 1
104.83 even 4 832.6.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.a.a.1.1 1 13.5 odd 4
208.6.a.b.1.1 1 52.31 even 4
234.6.a.g.1.1 1 39.5 even 4
338.6.a.d.1.1 1 13.8 odd 4
338.6.b.a.337.1 2 1.1 even 1 trivial
338.6.b.a.337.2 2 13.12 even 2 inner
650.6.a.a.1.1 1 65.44 odd 4
650.6.b.a.599.1 2 65.57 even 4
650.6.b.a.599.2 2 65.18 even 4
832.6.a.d.1.1 1 104.5 odd 4
832.6.a.e.1.1 1 104.83 even 4