# Properties

 Label 338.4.b.a.337.1 Level $338$ Weight $4$ Character 338.337 Analytic conductor $19.943$ 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] = [338,4,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, 4, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$19.9426455819$$ 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: $$1$$ 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 338.4.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)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +2.00000i q^{5} +6.00000i q^{6} +5.00000i q^{7} +8.00000i q^{8} -18.0000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +2.00000i q^{5} +6.00000i q^{6} +5.00000i q^{7} +8.00000i q^{8} -18.0000 q^{9} +4.00000 q^{10} -13.0000i q^{11} +12.0000 q^{12} +10.0000 q^{14} -6.00000i q^{15} +16.0000 q^{16} -27.0000 q^{17} +36.0000i q^{18} +75.0000i q^{19} -8.00000i q^{20} -15.0000i q^{21} -26.0000 q^{22} +187.000 q^{23} -24.0000i q^{24} +121.000 q^{25} +135.000 q^{27} -20.0000i q^{28} -13.0000 q^{29} -12.0000 q^{30} -104.000i q^{31} -32.0000i q^{32} +39.0000i q^{33} +54.0000i q^{34} -10.0000 q^{35} +72.0000 q^{36} -423.000i q^{37} +150.000 q^{38} -16.0000 q^{40} +195.000i q^{41} -30.0000 q^{42} -199.000 q^{43} +52.0000i q^{44} -36.0000i q^{45} -374.000i q^{46} -388.000i q^{47} -48.0000 q^{48} +318.000 q^{49} -242.000i q^{50} +81.0000 q^{51} +618.000 q^{53} -270.000i q^{54} +26.0000 q^{55} -40.0000 q^{56} -225.000i q^{57} +26.0000i q^{58} -491.000i q^{59} +24.0000i q^{60} +175.000 q^{61} -208.000 q^{62} -90.0000i q^{63} -64.0000 q^{64} +78.0000 q^{66} +817.000i q^{67} +108.000 q^{68} -561.000 q^{69} +20.0000i q^{70} +79.0000i q^{71} -144.000i q^{72} -230.000i q^{73} -846.000 q^{74} -363.000 q^{75} -300.000i q^{76} +65.0000 q^{77} +764.000 q^{79} +32.0000i q^{80} +81.0000 q^{81} +390.000 q^{82} -732.000i q^{83} +60.0000i q^{84} -54.0000i q^{85} +398.000i q^{86} +39.0000 q^{87} +104.000 q^{88} +1041.00i q^{89} -72.0000 q^{90} -748.000 q^{92} +312.000i q^{93} -776.000 q^{94} -150.000 q^{95} +96.0000i q^{96} -97.0000i q^{97} -636.000i q^{98} +234.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{4} - 36 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^4 - 36 * q^9 $$2 q - 6 q^{3} - 8 q^{4} - 36 q^{9} + 8 q^{10} + 24 q^{12} + 20 q^{14} + 32 q^{16} - 54 q^{17} - 52 q^{22} + 374 q^{23} + 242 q^{25} + 270 q^{27} - 26 q^{29} - 24 q^{30} - 20 q^{35} + 144 q^{36} + 300 q^{38} - 32 q^{40} - 60 q^{42} - 398 q^{43} - 96 q^{48} + 636 q^{49} + 162 q^{51} + 1236 q^{53} + 52 q^{55} - 80 q^{56} + 350 q^{61} - 416 q^{62} - 128 q^{64} + 156 q^{66} + 216 q^{68} - 1122 q^{69} - 1692 q^{74} - 726 q^{75} + 130 q^{77} + 1528 q^{79} + 162 q^{81} + 780 q^{82} + 78 q^{87} + 208 q^{88} - 144 q^{90} - 1496 q^{92} - 1552 q^{94} - 300 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 - 36 * q^9 + 8 * q^10 + 24 * q^12 + 20 * q^14 + 32 * q^16 - 54 * q^17 - 52 * q^22 + 374 * q^23 + 242 * q^25 + 270 * q^27 - 26 * q^29 - 24 * q^30 - 20 * q^35 + 144 * q^36 + 300 * q^38 - 32 * q^40 - 60 * q^42 - 398 * q^43 - 96 * q^48 + 636 * q^49 + 162 * q^51 + 1236 * q^53 + 52 * q^55 - 80 * q^56 + 350 * q^61 - 416 * q^62 - 128 * q^64 + 156 * q^66 + 216 * q^68 - 1122 * q^69 - 1692 * q^74 - 726 * q^75 + 130 * q^77 + 1528 * q^79 + 162 * q^81 + 780 * q^82 + 78 * q^87 + 208 * q^88 - 144 * q^90 - 1496 * q^92 - 1552 * q^94 - 300 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i
$$3$$ −3.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ −4.00000 −0.500000
$$5$$ 2.00000i 0.178885i 0.995992 + 0.0894427i $$0.0285086\pi$$
−0.995992 + 0.0894427i $$0.971491\pi$$
$$6$$ 6.00000i 0.408248i
$$7$$ 5.00000i 0.269975i 0.990847 + 0.134987i $$0.0430994\pi$$
−0.990847 + 0.134987i $$0.956901\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ −18.0000 −0.666667
$$10$$ 4.00000 0.126491
$$11$$ − 13.0000i − 0.356332i −0.984000 0.178166i $$-0.942984\pi$$
0.984000 0.178166i $$-0.0570163\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 0 0
$$14$$ 10.0000 0.190901
$$15$$ − 6.00000i − 0.103280i
$$16$$ 16.0000 0.250000
$$17$$ −27.0000 −0.385204 −0.192602 0.981277i $$-0.561693\pi$$
−0.192602 + 0.981277i $$0.561693\pi$$
$$18$$ 36.0000i 0.471405i
$$19$$ 75.0000i 0.905588i 0.891615 + 0.452794i $$0.149573\pi$$
−0.891615 + 0.452794i $$0.850427\pi$$
$$20$$ − 8.00000i − 0.0894427i
$$21$$ − 15.0000i − 0.155870i
$$22$$ −26.0000 −0.251964
$$23$$ 187.000 1.69531 0.847656 0.530546i $$-0.178013\pi$$
0.847656 + 0.530546i $$0.178013\pi$$
$$24$$ − 24.0000i − 0.204124i
$$25$$ 121.000 0.968000
$$26$$ 0 0
$$27$$ 135.000 0.962250
$$28$$ − 20.0000i − 0.134987i
$$29$$ −13.0000 −0.0832427 −0.0416214 0.999133i $$-0.513252\pi$$
−0.0416214 + 0.999133i $$0.513252\pi$$
$$30$$ −12.0000 −0.0730297
$$31$$ − 104.000i − 0.602547i −0.953538 0.301273i $$-0.902588\pi$$
0.953538 0.301273i $$-0.0974117\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ 39.0000i 0.205728i
$$34$$ 54.0000i 0.272380i
$$35$$ −10.0000 −0.0482945
$$36$$ 72.0000 0.333333
$$37$$ − 423.000i − 1.87948i −0.341890 0.939740i $$-0.611067\pi$$
0.341890 0.939740i $$-0.388933\pi$$
$$38$$ 150.000 0.640348
$$39$$ 0 0
$$40$$ −16.0000 −0.0632456
$$41$$ 195.000i 0.742778i 0.928477 + 0.371389i $$0.121118\pi$$
−0.928477 + 0.371389i $$0.878882\pi$$
$$42$$ −30.0000 −0.110217
$$43$$ −199.000 −0.705749 −0.352875 0.935671i $$-0.614796\pi$$
−0.352875 + 0.935671i $$0.614796\pi$$
$$44$$ 52.0000i 0.178166i
$$45$$ − 36.0000i − 0.119257i
$$46$$ − 374.000i − 1.19877i
$$47$$ − 388.000i − 1.20416i −0.798435 0.602081i $$-0.794338\pi$$
0.798435 0.602081i $$-0.205662\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ 318.000 0.927114
$$50$$ − 242.000i − 0.684479i
$$51$$ 81.0000 0.222397
$$52$$ 0 0
$$53$$ 618.000 1.60168 0.800838 0.598881i $$-0.204388\pi$$
0.800838 + 0.598881i $$0.204388\pi$$
$$54$$ − 270.000i − 0.680414i
$$55$$ 26.0000 0.0637425
$$56$$ −40.0000 −0.0954504
$$57$$ − 225.000i − 0.522842i
$$58$$ 26.0000i 0.0588615i
$$59$$ − 491.000i − 1.08344i −0.840560 0.541718i $$-0.817774\pi$$
0.840560 0.541718i $$-0.182226\pi$$
$$60$$ 24.0000i 0.0516398i
$$61$$ 175.000 0.367319 0.183659 0.982990i $$-0.441206\pi$$
0.183659 + 0.982990i $$0.441206\pi$$
$$62$$ −208.000 −0.426065
$$63$$ − 90.0000i − 0.179983i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 78.0000 0.145472
$$67$$ 817.000i 1.48974i 0.667211 + 0.744869i $$0.267488\pi$$
−0.667211 + 0.744869i $$0.732512\pi$$
$$68$$ 108.000 0.192602
$$69$$ −561.000 −0.978789
$$70$$ 20.0000i 0.0341494i
$$71$$ 79.0000i 0.132050i 0.997818 + 0.0660252i $$0.0210318\pi$$
−0.997818 + 0.0660252i $$0.978968\pi$$
$$72$$ − 144.000i − 0.235702i
$$73$$ − 230.000i − 0.368760i −0.982855 0.184380i $$-0.940972\pi$$
0.982855 0.184380i $$-0.0590277\pi$$
$$74$$ −846.000 −1.32899
$$75$$ −363.000 −0.558875
$$76$$ − 300.000i − 0.452794i
$$77$$ 65.0000 0.0962005
$$78$$ 0 0
$$79$$ 764.000 1.08806 0.544030 0.839066i $$-0.316898\pi$$
0.544030 + 0.839066i $$0.316898\pi$$
$$80$$ 32.0000i 0.0447214i
$$81$$ 81.0000 0.111111
$$82$$ 390.000 0.525223
$$83$$ − 732.000i − 0.968041i −0.875057 0.484021i $$-0.839176\pi$$
0.875057 0.484021i $$-0.160824\pi$$
$$84$$ 60.0000i 0.0779350i
$$85$$ − 54.0000i − 0.0689073i
$$86$$ 398.000i 0.499040i
$$87$$ 39.0000 0.0480602
$$88$$ 104.000 0.125982
$$89$$ 1041.00i 1.23984i 0.784665 + 0.619920i $$0.212835\pi$$
−0.784665 + 0.619920i $$0.787165\pi$$
$$90$$ −72.0000 −0.0843274
$$91$$ 0 0
$$92$$ −748.000 −0.847656
$$93$$ 312.000i 0.347881i
$$94$$ −776.000 −0.851471
$$95$$ −150.000 −0.161997
$$96$$ 96.0000i 0.102062i
$$97$$ − 97.0000i − 0.101535i −0.998711 0.0507673i $$-0.983833\pi$$
0.998711 0.0507673i $$-0.0161667\pi$$
$$98$$ − 636.000i − 0.655568i
$$99$$ 234.000i 0.237554i
$$100$$ −484.000 −0.484000
$$101$$ 809.000 0.797015 0.398507 0.917165i $$-0.369528\pi$$
0.398507 + 0.917165i $$0.369528\pi$$
$$102$$ − 162.000i − 0.157259i
$$103$$ −1288.00 −1.23214 −0.616070 0.787691i $$-0.711276\pi$$
−0.616070 + 0.787691i $$0.711276\pi$$
$$104$$ 0 0
$$105$$ 30.0000 0.0278829
$$106$$ − 1236.00i − 1.13256i
$$107$$ 1277.00 1.15376 0.576880 0.816829i $$-0.304270\pi$$
0.576880 + 0.816829i $$0.304270\pi$$
$$108$$ −540.000 −0.481125
$$109$$ 826.000i 0.725839i 0.931820 + 0.362920i $$0.118220\pi$$
−0.931820 + 0.362920i $$0.881780\pi$$
$$110$$ − 52.0000i − 0.0450728i
$$111$$ 1269.00i 1.08512i
$$112$$ 80.0000i 0.0674937i
$$113$$ 947.000 0.788374 0.394187 0.919030i $$-0.371026\pi$$
0.394187 + 0.919030i $$0.371026\pi$$
$$114$$ −450.000 −0.369705
$$115$$ 374.000i 0.303267i
$$116$$ 52.0000 0.0416214
$$117$$ 0 0
$$118$$ −982.000 −0.766105
$$119$$ − 135.000i − 0.103995i
$$120$$ 48.0000 0.0365148
$$121$$ 1162.00 0.873028
$$122$$ − 350.000i − 0.259734i
$$123$$ − 585.000i − 0.428843i
$$124$$ 416.000i 0.301273i
$$125$$ 492.000i 0.352047i
$$126$$ −180.000 −0.127267
$$127$$ −1177.00 −0.822377 −0.411188 0.911550i $$-0.634886\pi$$
−0.411188 + 0.911550i $$0.634886\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ 597.000 0.407464
$$130$$ 0 0
$$131$$ −1420.00 −0.947069 −0.473534 0.880775i $$-0.657022\pi$$
−0.473534 + 0.880775i $$0.657022\pi$$
$$132$$ − 156.000i − 0.102864i
$$133$$ −375.000 −0.244486
$$134$$ 1634.00 1.05340
$$135$$ 270.000i 0.172133i
$$136$$ − 216.000i − 0.136190i
$$137$$ 2409.00i 1.50230i 0.660133 + 0.751149i $$0.270500\pi$$
−0.660133 + 0.751149i $$0.729500\pi$$
$$138$$ 1122.00i 0.692109i
$$139$$ 2827.00 1.72506 0.862529 0.506008i $$-0.168879\pi$$
0.862529 + 0.506008i $$0.168879\pi$$
$$140$$ 40.0000 0.0241473
$$141$$ 1164.00i 0.695223i
$$142$$ 158.000 0.0933737
$$143$$ 0 0
$$144$$ −288.000 −0.166667
$$145$$ − 26.0000i − 0.0148909i
$$146$$ −460.000 −0.260753
$$147$$ −954.000 −0.535269
$$148$$ 1692.00i 0.939740i
$$149$$ 855.000i 0.470096i 0.971984 + 0.235048i $$0.0755248\pi$$
−0.971984 + 0.235048i $$0.924475\pi$$
$$150$$ 726.000i 0.395184i
$$151$$ − 2064.00i − 1.11236i −0.831063 0.556179i $$-0.812267\pi$$
0.831063 0.556179i $$-0.187733\pi$$
$$152$$ −600.000 −0.320174
$$153$$ 486.000 0.256802
$$154$$ − 130.000i − 0.0680240i
$$155$$ 208.000 0.107787
$$156$$ 0 0
$$157$$ −1894.00 −0.962788 −0.481394 0.876504i $$-0.659869\pi$$
−0.481394 + 0.876504i $$0.659869\pi$$
$$158$$ − 1528.00i − 0.769374i
$$159$$ −1854.00 −0.924728
$$160$$ 64.0000 0.0316228
$$161$$ 935.000i 0.457691i
$$162$$ − 162.000i − 0.0785674i
$$163$$ 985.000i 0.473320i 0.971593 + 0.236660i $$0.0760527\pi$$
−0.971593 + 0.236660i $$0.923947\pi$$
$$164$$ − 780.000i − 0.371389i
$$165$$ −78.0000 −0.0368018
$$166$$ −1464.00 −0.684509
$$167$$ 2355.00i 1.09123i 0.838036 + 0.545615i $$0.183704\pi$$
−0.838036 + 0.545615i $$0.816296\pi$$
$$168$$ 120.000 0.0551083
$$169$$ 0 0
$$170$$ −108.000 −0.0487248
$$171$$ − 1350.00i − 0.603726i
$$172$$ 796.000 0.352875
$$173$$ 3889.00 1.70911 0.854553 0.519365i $$-0.173831\pi$$
0.854553 + 0.519365i $$0.173831\pi$$
$$174$$ − 78.0000i − 0.0339837i
$$175$$ 605.000i 0.261335i
$$176$$ − 208.000i − 0.0890829i
$$177$$ 1473.00i 0.625522i
$$178$$ 2082.00 0.876699
$$179$$ −2229.00 −0.930745 −0.465372 0.885115i $$-0.654080\pi$$
−0.465372 + 0.885115i $$0.654080\pi$$
$$180$$ 144.000i 0.0596285i
$$181$$ 1038.00 0.426265 0.213132 0.977023i $$-0.431633\pi$$
0.213132 + 0.977023i $$0.431633\pi$$
$$182$$ 0 0
$$183$$ −525.000 −0.212072
$$184$$ 1496.00i 0.599384i
$$185$$ 846.000 0.336212
$$186$$ 624.000 0.245989
$$187$$ 351.000i 0.137260i
$$188$$ 1552.00i 0.602081i
$$189$$ 675.000i 0.259783i
$$190$$ 300.000i 0.114549i
$$191$$ −2141.00 −0.811085 −0.405543 0.914076i $$-0.632917\pi$$
−0.405543 + 0.914076i $$0.632917\pi$$
$$192$$ 192.000 0.0721688
$$193$$ − 2627.00i − 0.979770i −0.871787 0.489885i $$-0.837039\pi$$
0.871787 0.489885i $$-0.162961\pi$$
$$194$$ −194.000 −0.0717958
$$195$$ 0 0
$$196$$ −1272.00 −0.463557
$$197$$ 1203.00i 0.435077i 0.976052 + 0.217539i $$0.0698028\pi$$
−0.976052 + 0.217539i $$0.930197\pi$$
$$198$$ 468.000 0.167976
$$199$$ −743.000 −0.264673 −0.132336 0.991205i $$-0.542248\pi$$
−0.132336 + 0.991205i $$0.542248\pi$$
$$200$$ 968.000i 0.342240i
$$201$$ − 2451.00i − 0.860101i
$$202$$ − 1618.00i − 0.563575i
$$203$$ − 65.0000i − 0.0224734i
$$204$$ −324.000 −0.111199
$$205$$ −390.000 −0.132872
$$206$$ 2576.00i 0.871254i
$$207$$ −3366.00 −1.13021
$$208$$ 0 0
$$209$$ 975.000 0.322690
$$210$$ − 60.0000i − 0.0197162i
$$211$$ −355.000 −0.115826 −0.0579128 0.998322i $$-0.518445\pi$$
−0.0579128 + 0.998322i $$0.518445\pi$$
$$212$$ −2472.00 −0.800838
$$213$$ − 237.000i − 0.0762393i
$$214$$ − 2554.00i − 0.815831i
$$215$$ − 398.000i − 0.126248i
$$216$$ 1080.00i 0.340207i
$$217$$ 520.000 0.162672
$$218$$ 1652.00 0.513246
$$219$$ 690.000i 0.212904i
$$220$$ −104.000 −0.0318713
$$221$$ 0 0
$$222$$ 2538.00 0.767295
$$223$$ − 2283.00i − 0.685565i −0.939415 0.342782i $$-0.888631\pi$$
0.939415 0.342782i $$-0.111369\pi$$
$$224$$ 160.000 0.0477252
$$225$$ −2178.00 −0.645333
$$226$$ − 1894.00i − 0.557465i
$$227$$ 2451.00i 0.716646i 0.933598 + 0.358323i $$0.116651\pi$$
−0.933598 + 0.358323i $$0.883349\pi$$
$$228$$ 900.000i 0.261421i
$$229$$ 1878.00i 0.541929i 0.962589 + 0.270964i $$0.0873426\pi$$
−0.962589 + 0.270964i $$0.912657\pi$$
$$230$$ 748.000 0.214442
$$231$$ −195.000 −0.0555414
$$232$$ − 104.000i − 0.0294308i
$$233$$ −1630.00 −0.458304 −0.229152 0.973391i $$-0.573595\pi$$
−0.229152 + 0.973391i $$0.573595\pi$$
$$234$$ 0 0
$$235$$ 776.000 0.215407
$$236$$ 1964.00i 0.541718i
$$237$$ −2292.00 −0.628192
$$238$$ −270.000 −0.0735357
$$239$$ − 5544.00i − 1.50047i −0.661173 0.750233i $$-0.729941\pi$$
0.661173 0.750233i $$-0.270059\pi$$
$$240$$ − 96.0000i − 0.0258199i
$$241$$ − 5523.00i − 1.47621i −0.674683 0.738107i $$-0.735720\pi$$
0.674683 0.738107i $$-0.264280\pi$$
$$242$$ − 2324.00i − 0.617324i
$$243$$ −3888.00 −1.02640
$$244$$ −700.000 −0.183659
$$245$$ 636.000i 0.165847i
$$246$$ −1170.00 −0.303238
$$247$$ 0 0
$$248$$ 832.000 0.213032
$$249$$ 2196.00i 0.558899i
$$250$$ 984.000 0.248934
$$251$$ −2175.00 −0.546951 −0.273476 0.961879i $$-0.588173\pi$$
−0.273476 + 0.961879i $$0.588173\pi$$
$$252$$ 360.000i 0.0899915i
$$253$$ − 2431.00i − 0.604094i
$$254$$ 2354.00i 0.581508i
$$255$$ 162.000i 0.0397837i
$$256$$ 256.000 0.0625000
$$257$$ 5685.00 1.37985 0.689923 0.723883i $$-0.257644\pi$$
0.689923 + 0.723883i $$0.257644\pi$$
$$258$$ − 1194.00i − 0.288121i
$$259$$ 2115.00 0.507412
$$260$$ 0 0
$$261$$ 234.000 0.0554952
$$262$$ 2840.00i 0.669679i
$$263$$ 6117.00 1.43418 0.717092 0.696979i $$-0.245473\pi$$
0.717092 + 0.696979i $$0.245473\pi$$
$$264$$ −312.000 −0.0727359
$$265$$ 1236.00i 0.286517i
$$266$$ 750.000i 0.172878i
$$267$$ − 3123.00i − 0.715822i
$$268$$ − 3268.00i − 0.744869i
$$269$$ −5109.00 −1.15800 −0.578999 0.815329i $$-0.696556\pi$$
−0.578999 + 0.815329i $$0.696556\pi$$
$$270$$ 540.000 0.121716
$$271$$ − 7549.00i − 1.69214i −0.533074 0.846068i $$-0.678963\pi$$
0.533074 0.846068i $$-0.321037\pi$$
$$272$$ −432.000 −0.0963009
$$273$$ 0 0
$$274$$ 4818.00 1.06228
$$275$$ − 1573.00i − 0.344929i
$$276$$ 2244.00 0.489395
$$277$$ 981.000 0.212789 0.106395 0.994324i $$-0.466069\pi$$
0.106395 + 0.994324i $$0.466069\pi$$
$$278$$ − 5654.00i − 1.21980i
$$279$$ 1872.00i 0.401698i
$$280$$ − 80.0000i − 0.0170747i
$$281$$ 2762.00i 0.586360i 0.956057 + 0.293180i $$0.0947135\pi$$
−0.956057 + 0.293180i $$0.905287\pi$$
$$282$$ 2328.00 0.491597
$$283$$ −3925.00 −0.824442 −0.412221 0.911084i $$-0.635247\pi$$
−0.412221 + 0.911084i $$0.635247\pi$$
$$284$$ − 316.000i − 0.0660252i
$$285$$ 450.000 0.0935288
$$286$$ 0 0
$$287$$ −975.000 −0.200531
$$288$$ 576.000i 0.117851i
$$289$$ −4184.00 −0.851618
$$290$$ −52.0000 −0.0105295
$$291$$ 291.000i 0.0586210i
$$292$$ 920.000i 0.184380i
$$293$$ − 7711.00i − 1.53748i −0.639562 0.768740i $$-0.720884\pi$$
0.639562 0.768740i $$-0.279116\pi$$
$$294$$ 1908.00i 0.378493i
$$295$$ 982.000 0.193811
$$296$$ 3384.00 0.664497
$$297$$ − 1755.00i − 0.342880i
$$298$$ 1710.00 0.332408
$$299$$ 0 0
$$300$$ 1452.00 0.279438
$$301$$ − 995.000i − 0.190534i
$$302$$ −4128.00 −0.786555
$$303$$ −2427.00 −0.460157
$$304$$ 1200.00i 0.226397i
$$305$$ 350.000i 0.0657080i
$$306$$ − 972.000i − 0.181587i
$$307$$ − 10388.0i − 1.93119i −0.260056 0.965594i $$-0.583741\pi$$
0.260056 0.965594i $$-0.416259\pi$$
$$308$$ −260.000 −0.0481002
$$309$$ 3864.00 0.711376
$$310$$ − 416.000i − 0.0762168i
$$311$$ 7272.00 1.32591 0.662954 0.748660i $$-0.269303\pi$$
0.662954 + 0.748660i $$0.269303\pi$$
$$312$$ 0 0
$$313$$ 7910.00 1.42843 0.714217 0.699925i $$-0.246783\pi$$
0.714217 + 0.699925i $$0.246783\pi$$
$$314$$ 3788.00i 0.680794i
$$315$$ 180.000 0.0321964
$$316$$ −3056.00 −0.544030
$$317$$ 7398.00i 1.31077i 0.755296 + 0.655383i $$0.227493\pi$$
−0.755296 + 0.655383i $$0.772507\pi$$
$$318$$ 3708.00i 0.653881i
$$319$$ 169.000i 0.0296620i
$$320$$ − 128.000i − 0.0223607i
$$321$$ −3831.00 −0.666123
$$322$$ 1870.00 0.323637
$$323$$ − 2025.00i − 0.348836i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ 1970.00 0.334688
$$327$$ − 2478.00i − 0.419063i
$$328$$ −1560.00 −0.262612
$$329$$ 1940.00 0.325093
$$330$$ 156.000i 0.0260228i
$$331$$ − 2377.00i − 0.394718i −0.980331 0.197359i $$-0.936764\pi$$
0.980331 0.197359i $$-0.0632365\pi$$
$$332$$ 2928.00i 0.484021i
$$333$$ 7614.00i 1.25299i
$$334$$ 4710.00 0.771616
$$335$$ −1634.00 −0.266492
$$336$$ − 240.000i − 0.0389675i
$$337$$ 7618.00 1.23139 0.615696 0.787984i $$-0.288875\pi$$
0.615696 + 0.787984i $$0.288875\pi$$
$$338$$ 0 0
$$339$$ −2841.00 −0.455168
$$340$$ 216.000i 0.0344537i
$$341$$ −1352.00 −0.214706
$$342$$ −2700.00 −0.426898
$$343$$ 3305.00i 0.520272i
$$344$$ − 1592.00i − 0.249520i
$$345$$ − 1122.00i − 0.175091i
$$346$$ − 7778.00i − 1.20852i
$$347$$ 375.000 0.0580146 0.0290073 0.999579i $$-0.490765\pi$$
0.0290073 + 0.999579i $$0.490765\pi$$
$$348$$ −156.000 −0.0240301
$$349$$ − 9727.00i − 1.49190i −0.666000 0.745952i $$-0.731995\pi$$
0.666000 0.745952i $$-0.268005\pi$$
$$350$$ 1210.00 0.184792
$$351$$ 0 0
$$352$$ −416.000 −0.0629911
$$353$$ 2263.00i 0.341211i 0.985339 + 0.170605i $$0.0545723\pi$$
−0.985339 + 0.170605i $$0.945428\pi$$
$$354$$ 2946.00 0.442311
$$355$$ −158.000 −0.0236219
$$356$$ − 4164.00i − 0.619920i
$$357$$ 405.000i 0.0600417i
$$358$$ 4458.00i 0.658136i
$$359$$ 4488.00i 0.659798i 0.944016 + 0.329899i $$0.107015\pi$$
−0.944016 + 0.329899i $$0.892985\pi$$
$$360$$ 288.000 0.0421637
$$361$$ 1234.00 0.179910
$$362$$ − 2076.00i − 0.301415i
$$363$$ −3486.00 −0.504043
$$364$$ 0 0
$$365$$ 460.000 0.0659658
$$366$$ 1050.00i 0.149957i
$$367$$ −1627.00 −0.231413 −0.115707 0.993283i $$-0.536913\pi$$
−0.115707 + 0.993283i $$0.536913\pi$$
$$368$$ 2992.00 0.423828
$$369$$ − 3510.00i − 0.495185i
$$370$$ − 1692.00i − 0.237738i
$$371$$ 3090.00i 0.432412i
$$372$$ − 1248.00i − 0.173940i
$$373$$ 2987.00 0.414641 0.207320 0.978273i $$-0.433526\pi$$
0.207320 + 0.978273i $$0.433526\pi$$
$$374$$ 702.000 0.0970576
$$375$$ − 1476.00i − 0.203254i
$$376$$ 3104.00 0.425736
$$377$$ 0 0
$$378$$ 1350.00 0.183694
$$379$$ − 8867.00i − 1.20176i −0.799339 0.600880i $$-0.794817\pi$$
0.799339 0.600880i $$-0.205183\pi$$
$$380$$ 600.000 0.0809983
$$381$$ 3531.00 0.474800
$$382$$ 4282.00i 0.573524i
$$383$$ 11403.0i 1.52132i 0.649150 + 0.760661i $$0.275125\pi$$
−0.649150 + 0.760661i $$0.724875\pi$$
$$384$$ − 384.000i − 0.0510310i
$$385$$ 130.000i 0.0172089i
$$386$$ −5254.00 −0.692802
$$387$$ 3582.00 0.470499
$$388$$ 388.000i 0.0507673i
$$389$$ −2622.00 −0.341750 −0.170875 0.985293i $$-0.554659\pi$$
−0.170875 + 0.985293i $$0.554659\pi$$
$$390$$ 0 0
$$391$$ −5049.00 −0.653041
$$392$$ 2544.00i 0.327784i
$$393$$ 4260.00 0.546790
$$394$$ 2406.00 0.307646
$$395$$ 1528.00i 0.194638i
$$396$$ − 936.000i − 0.118777i
$$397$$ − 659.000i − 0.0833105i −0.999132 0.0416552i $$-0.986737\pi$$
0.999132 0.0416552i $$-0.0132631\pi$$
$$398$$ 1486.00i 0.187152i
$$399$$ 1125.00 0.141154
$$400$$ 1936.00 0.242000
$$401$$ 14685.0i 1.82876i 0.404854 + 0.914381i $$0.367322\pi$$
−0.404854 + 0.914381i $$0.632678\pi$$
$$402$$ −4902.00 −0.608183
$$403$$ 0 0
$$404$$ −3236.00 −0.398507
$$405$$ 162.000i 0.0198762i
$$406$$ −130.000 −0.0158911
$$407$$ −5499.00 −0.669718
$$408$$ 648.000i 0.0786294i
$$409$$ − 7829.00i − 0.946502i −0.880928 0.473251i $$-0.843080\pi$$
0.880928 0.473251i $$-0.156920\pi$$
$$410$$ 780.000i 0.0939548i
$$411$$ − 7227.00i − 0.867352i
$$412$$ 5152.00 0.616070
$$413$$ 2455.00 0.292500
$$414$$ 6732.00i 0.799178i
$$415$$ 1464.00 0.173169
$$416$$ 0 0
$$417$$ −8481.00 −0.995962
$$418$$ − 1950.00i − 0.228176i
$$419$$ −2919.00 −0.340340 −0.170170 0.985415i $$-0.554432\pi$$
−0.170170 + 0.985415i $$0.554432\pi$$
$$420$$ −120.000 −0.0139414
$$421$$ − 3110.00i − 0.360029i −0.983664 0.180014i $$-0.942386\pi$$
0.983664 0.180014i $$-0.0576144\pi$$
$$422$$ 710.000i 0.0819011i
$$423$$ 6984.00i 0.802775i
$$424$$ 4944.00i 0.566278i
$$425$$ −3267.00 −0.372877
$$426$$ −474.000 −0.0539093
$$427$$ 875.000i 0.0991668i
$$428$$ −5108.00 −0.576880
$$429$$ 0 0
$$430$$ −796.000 −0.0892710
$$431$$ − 9135.00i − 1.02092i −0.859901 0.510461i $$-0.829475\pi$$
0.859901 0.510461i $$-0.170525\pi$$
$$432$$ 2160.00 0.240563
$$433$$ 11669.0 1.29510 0.647548 0.762025i $$-0.275795\pi$$
0.647548 + 0.762025i $$0.275795\pi$$
$$434$$ − 1040.00i − 0.115027i
$$435$$ 78.0000i 0.00859727i
$$436$$ − 3304.00i − 0.362920i
$$437$$ 14025.0i 1.53526i
$$438$$ 1380.00 0.150546
$$439$$ −13529.0 −1.47085 −0.735426 0.677605i $$-0.763018\pi$$
−0.735426 + 0.677605i $$0.763018\pi$$
$$440$$ 208.000i 0.0225364i
$$441$$ −5724.00 −0.618076
$$442$$ 0 0
$$443$$ −1932.00 −0.207206 −0.103603 0.994619i $$-0.533037\pi$$
−0.103603 + 0.994619i $$0.533037\pi$$
$$444$$ − 5076.00i − 0.542559i
$$445$$ −2082.00 −0.221789
$$446$$ −4566.00 −0.484768
$$447$$ − 2565.00i − 0.271410i
$$448$$ − 320.000i − 0.0337468i
$$449$$ 5357.00i 0.563057i 0.959553 + 0.281528i $$0.0908413\pi$$
−0.959553 + 0.281528i $$0.909159\pi$$
$$450$$ 4356.00i 0.456320i
$$451$$ 2535.00 0.264675
$$452$$ −3788.00 −0.394187
$$453$$ 6192.00i 0.642220i
$$454$$ 4902.00 0.506745
$$455$$ 0 0
$$456$$ 1800.00 0.184852
$$457$$ 19399.0i 1.98566i 0.119532 + 0.992830i $$0.461861\pi$$
−0.119532 + 0.992830i $$0.538139\pi$$
$$458$$ 3756.00 0.383202
$$459$$ −3645.00 −0.370662
$$460$$ − 1496.00i − 0.151633i
$$461$$ − 15549.0i − 1.57091i −0.618919 0.785455i $$-0.712429\pi$$
0.618919 0.785455i $$-0.287571\pi$$
$$462$$ 390.000i 0.0392737i
$$463$$ − 4072.00i − 0.408730i −0.978895 0.204365i $$-0.934487\pi$$
0.978895 0.204365i $$-0.0655129\pi$$
$$464$$ −208.000 −0.0208107
$$465$$ −624.000 −0.0622308
$$466$$ 3260.00i 0.324070i
$$467$$ −15224.0 −1.50853 −0.754264 0.656571i $$-0.772006\pi$$
−0.754264 + 0.656571i $$0.772006\pi$$
$$468$$ 0 0
$$469$$ −4085.00 −0.402191
$$470$$ − 1552.00i − 0.152316i
$$471$$ 5682.00 0.555866
$$472$$ 3928.00 0.383053
$$473$$ 2587.00i 0.251481i
$$474$$ 4584.00i 0.444199i
$$475$$ 9075.00i 0.876610i
$$476$$ 540.000i 0.0519976i
$$477$$ −11124.0 −1.06778
$$478$$ −11088.0 −1.06099
$$479$$ 10335.0i 0.985842i 0.870074 + 0.492921i $$0.164071\pi$$
−0.870074 + 0.492921i $$0.835929\pi$$
$$480$$ −192.000 −0.0182574
$$481$$ 0 0
$$482$$ −11046.0 −1.04384
$$483$$ − 2805.00i − 0.264248i
$$484$$ −4648.00 −0.436514
$$485$$ 194.000 0.0181631
$$486$$ 7776.00i 0.725775i
$$487$$ 6455.00i 0.600624i 0.953841 + 0.300312i $$0.0970908\pi$$
−0.953841 + 0.300312i $$0.902909\pi$$
$$488$$ 1400.00i 0.129867i
$$489$$ − 2955.00i − 0.273271i
$$490$$ 1272.00 0.117272
$$491$$ −7777.00 −0.714809 −0.357404 0.933950i $$-0.616338\pi$$
−0.357404 + 0.933950i $$0.616338\pi$$
$$492$$ 2340.00i 0.214421i
$$493$$ 351.000 0.0320654
$$494$$ 0 0
$$495$$ −468.000 −0.0424950
$$496$$ − 1664.00i − 0.150637i
$$497$$ −395.000 −0.0356502
$$498$$ 4392.00 0.395201
$$499$$ − 3044.00i − 0.273082i −0.990634 0.136541i $$-0.956401\pi$$
0.990634 0.136541i $$-0.0435986\pi$$
$$500$$ − 1968.00i − 0.176023i
$$501$$ − 7065.00i − 0.630022i
$$502$$ 4350.00i 0.386753i
$$503$$ 11347.0 1.00584 0.502920 0.864333i $$-0.332259\pi$$
0.502920 + 0.864333i $$0.332259\pi$$
$$504$$ 720.000 0.0636336
$$505$$ 1618.00i 0.142574i
$$506$$ −4862.00 −0.427159
$$507$$ 0 0
$$508$$ 4708.00 0.411188
$$509$$ 727.000i 0.0633079i 0.999499 + 0.0316539i $$0.0100774\pi$$
−0.999499 + 0.0316539i $$0.989923\pi$$
$$510$$ 324.000 0.0281313
$$511$$ 1150.00 0.0995558
$$512$$ − 512.000i − 0.0441942i
$$513$$ 10125.0i 0.871403i
$$514$$ − 11370.0i − 0.975699i
$$515$$ − 2576.00i − 0.220412i
$$516$$ −2388.00 −0.203732
$$517$$ −5044.00 −0.429081
$$518$$ − 4230.00i − 0.358794i
$$519$$ −11667.0 −0.986752
$$520$$ 0 0
$$521$$ 9582.00 0.805749 0.402874 0.915255i $$-0.368011\pi$$
0.402874 + 0.915255i $$0.368011\pi$$
$$522$$ − 468.000i − 0.0392410i
$$523$$ −10383.0 −0.868101 −0.434051 0.900889i $$-0.642916\pi$$
−0.434051 + 0.900889i $$0.642916\pi$$
$$524$$ 5680.00 0.473534
$$525$$ − 1815.00i − 0.150882i
$$526$$ − 12234.0i − 1.01412i
$$527$$ 2808.00i 0.232103i
$$528$$ 624.000i 0.0514320i
$$529$$ 22802.0 1.87409
$$530$$ 2472.00 0.202598
$$531$$ 8838.00i 0.722291i
$$532$$ 1500.00 0.122243
$$533$$ 0 0
$$534$$ −6246.00 −0.506163
$$535$$ 2554.00i 0.206391i
$$536$$ −6536.00 −0.526702
$$537$$ 6687.00 0.537366
$$538$$ 10218.0i 0.818828i
$$539$$ − 4134.00i − 0.330360i
$$540$$ − 1080.00i − 0.0860663i
$$541$$ − 12230.0i − 0.971920i −0.873981 0.485960i $$-0.838470\pi$$
0.873981 0.485960i $$-0.161530\pi$$
$$542$$ −15098.0 −1.19652
$$543$$ −3114.00 −0.246104
$$544$$ 864.000i 0.0680950i
$$545$$ −1652.00 −0.129842
$$546$$ 0 0
$$547$$ −14636.0 −1.14404 −0.572020 0.820239i $$-0.693840\pi$$
−0.572020 + 0.820239i $$0.693840\pi$$
$$548$$ − 9636.00i − 0.751149i
$$549$$ −3150.00 −0.244879
$$550$$ −3146.00 −0.243902
$$551$$ − 975.000i − 0.0753837i
$$552$$ − 4488.00i − 0.346054i
$$553$$ 3820.00i 0.293749i
$$554$$ − 1962.00i − 0.150465i
$$555$$ −2538.00 −0.194112
$$556$$ −11308.0 −0.862529
$$557$$ 765.000i 0.0581941i 0.999577 + 0.0290970i $$0.00926318\pi$$
−0.999577 + 0.0290970i $$0.990737\pi$$
$$558$$ 3744.00 0.284043
$$559$$ 0 0
$$560$$ −160.000 −0.0120736
$$561$$ − 1053.00i − 0.0792472i
$$562$$ 5524.00 0.414619
$$563$$ −5915.00 −0.442784 −0.221392 0.975185i $$-0.571060\pi$$
−0.221392 + 0.975185i $$0.571060\pi$$
$$564$$ − 4656.00i − 0.347612i
$$565$$ 1894.00i 0.141029i
$$566$$ 7850.00i 0.582968i
$$567$$ 405.000i 0.0299972i
$$568$$ −632.000 −0.0466869
$$569$$ 1217.00 0.0896648 0.0448324 0.998995i $$-0.485725\pi$$
0.0448324 + 0.998995i $$0.485725\pi$$
$$570$$ − 900.000i − 0.0661348i
$$571$$ 23436.0 1.71763 0.858814 0.512287i $$-0.171202\pi$$
0.858814 + 0.512287i $$0.171202\pi$$
$$572$$ 0 0
$$573$$ 6423.00 0.468280
$$574$$ 1950.00i 0.141797i
$$575$$ 22627.0 1.64106
$$576$$ 1152.00 0.0833333
$$577$$ 7854.00i 0.566666i 0.959022 + 0.283333i $$0.0914402\pi$$
−0.959022 + 0.283333i $$0.908560\pi$$
$$578$$ 8368.00i 0.602185i
$$579$$ 7881.00i 0.565670i
$$580$$ 104.000i 0.00744546i
$$581$$ 3660.00 0.261347
$$582$$ 582.000 0.0414513
$$583$$ − 8034.00i − 0.570728i
$$584$$ 1840.00 0.130376
$$585$$ 0 0
$$586$$ −15422.0 −1.08716
$$587$$ 17033.0i 1.19766i 0.800876 + 0.598831i $$0.204368\pi$$
−0.800876 + 0.598831i $$0.795632\pi$$
$$588$$ 3816.00 0.267635
$$589$$ 7800.00 0.545659
$$590$$ − 1964.00i − 0.137045i
$$591$$ − 3609.00i − 0.251192i
$$592$$ − 6768.00i − 0.469870i
$$593$$ 14506.0i 1.00454i 0.864712 + 0.502268i $$0.167501\pi$$
−0.864712 + 0.502268i $$0.832499\pi$$
$$594$$ −3510.00 −0.242453
$$595$$ 270.000 0.0186032
$$596$$ − 3420.00i − 0.235048i
$$597$$ 2229.00 0.152809
$$598$$ 0 0
$$599$$ 15388.0 1.04964 0.524822 0.851212i $$-0.324132\pi$$
0.524822 + 0.851212i $$0.324132\pi$$
$$600$$ − 2904.00i − 0.197592i
$$601$$ −6077.00 −0.412456 −0.206228 0.978504i $$-0.566119\pi$$
−0.206228 + 0.978504i $$0.566119\pi$$
$$602$$ −1990.00 −0.134728
$$603$$ − 14706.0i − 0.993159i
$$604$$ 8256.00i 0.556179i
$$605$$ 2324.00i 0.156172i
$$606$$ 4854.00i 0.325380i
$$607$$ 10215.0 0.683054 0.341527 0.939872i $$-0.389056\pi$$
0.341527 + 0.939872i $$0.389056\pi$$
$$608$$ 2400.00 0.160087
$$609$$ 195.000i 0.0129750i
$$610$$ 700.000 0.0464626
$$611$$ 0 0
$$612$$ −1944.00 −0.128401
$$613$$ − 3457.00i − 0.227776i −0.993494 0.113888i $$-0.963669\pi$$
0.993494 0.113888i $$-0.0363306\pi$$
$$614$$ −20776.0 −1.36556
$$615$$ 1170.00 0.0767137
$$616$$ 520.000i 0.0340120i
$$617$$ − 7169.00i − 0.467768i −0.972264 0.233884i $$-0.924856\pi$$
0.972264 0.233884i $$-0.0751437\pi$$
$$618$$ − 7728.00i − 0.503019i
$$619$$ − 20212.0i − 1.31242i −0.754578 0.656211i $$-0.772158\pi$$
0.754578 0.656211i $$-0.227842\pi$$
$$620$$ −832.000 −0.0538934
$$621$$ 25245.0 1.63132
$$622$$ − 14544.0i − 0.937558i
$$623$$ −5205.00 −0.334725
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ − 15820.0i − 1.01005i
$$627$$ −2925.00 −0.186305
$$628$$ 7576.00 0.481394
$$629$$ 11421.0i 0.723983i
$$630$$ − 360.000i − 0.0227663i
$$631$$ 8945.00i 0.564334i 0.959365 + 0.282167i $$0.0910532\pi$$
−0.959365 + 0.282167i $$0.908947\pi$$
$$632$$ 6112.00i 0.384687i
$$633$$ 1065.00 0.0668720
$$634$$ 14796.0 0.926852
$$635$$ − 2354.00i − 0.147111i
$$636$$ 7416.00 0.462364
$$637$$ 0 0
$$638$$ 338.000 0.0209742
$$639$$ − 1422.00i − 0.0880336i
$$640$$ −256.000 −0.0158114
$$641$$ −28243.0 −1.74030 −0.870149 0.492788i $$-0.835978\pi$$
−0.870149 + 0.492788i $$0.835978\pi$$
$$642$$ 7662.00i 0.471020i
$$643$$ 5231.00i 0.320825i 0.987050 + 0.160413i $$0.0512825\pi$$
−0.987050 + 0.160413i $$0.948718\pi$$
$$644$$ − 3740.00i − 0.228846i
$$645$$ 1194.00i 0.0728895i
$$646$$ −4050.00 −0.246664
$$647$$ 4871.00 0.295980 0.147990 0.988989i $$-0.452720\pi$$
0.147990 + 0.988989i $$0.452720\pi$$
$$648$$ 648.000i 0.0392837i
$$649$$ −6383.00 −0.386063
$$650$$ 0 0
$$651$$ −1560.00 −0.0939189
$$652$$ − 3940.00i − 0.236660i
$$653$$ 12255.0 0.734418 0.367209 0.930138i $$-0.380313\pi$$
0.367209 + 0.930138i $$0.380313\pi$$
$$654$$ −4956.00 −0.296323
$$655$$ − 2840.00i − 0.169417i
$$656$$ 3120.00i 0.185694i
$$657$$ 4140.00i 0.245840i
$$658$$ − 3880.00i − 0.229876i
$$659$$ −2145.00 −0.126794 −0.0633971 0.997988i $$-0.520193\pi$$
−0.0633971 + 0.997988i $$0.520193\pi$$
$$660$$ 312.000 0.0184009
$$661$$ − 2111.00i − 0.124218i −0.998069 0.0621092i $$-0.980217\pi$$
0.998069 0.0621092i $$-0.0197827\pi$$
$$662$$ −4754.00 −0.279108
$$663$$ 0 0
$$664$$ 5856.00 0.342254
$$665$$ − 750.000i − 0.0437350i
$$666$$ 15228.0 0.885996
$$667$$ −2431.00 −0.141122
$$668$$ − 9420.00i − 0.545615i
$$669$$ 6849.00i 0.395811i
$$670$$ 3268.00i 0.188439i
$$671$$ − 2275.00i − 0.130887i
$$672$$ −480.000 −0.0275542
$$673$$ 23273.0 1.33300 0.666499 0.745506i $$-0.267792\pi$$
0.666499 + 0.745506i $$0.267792\pi$$
$$674$$ − 15236.0i − 0.870725i
$$675$$ 16335.0 0.931458
$$676$$ 0 0
$$677$$ −5910.00 −0.335509 −0.167755 0.985829i $$-0.553652\pi$$
−0.167755 + 0.985829i $$0.553652\pi$$
$$678$$ 5682.00i 0.321852i
$$679$$ 485.000 0.0274118
$$680$$ 432.000 0.0243624
$$681$$ − 7353.00i − 0.413756i
$$682$$ 2704.00i 0.151820i
$$683$$ − 16747.0i − 0.938223i −0.883139 0.469111i $$-0.844574\pi$$
0.883139 0.469111i $$-0.155426\pi$$
$$684$$ 5400.00i 0.301863i
$$685$$ −4818.00 −0.268739
$$686$$ 6610.00 0.367888
$$687$$ − 5634.00i − 0.312883i
$$688$$ −3184.00 −0.176437
$$689$$ 0 0
$$690$$ −2244.00 −0.123808
$$691$$ 10309.0i 0.567544i 0.958892 + 0.283772i $$0.0915859\pi$$
−0.958892 + 0.283772i $$0.908414\pi$$
$$692$$ −15556.0 −0.854553
$$693$$ −1170.00 −0.0641337
$$694$$ − 750.000i − 0.0410225i
$$695$$ 5654.00i 0.308588i
$$696$$ 312.000i 0.0169919i
$$697$$ − 5265.00i − 0.286121i
$$698$$ −19454.0 −1.05494
$$699$$ 4890.00 0.264602
$$700$$ − 2420.00i − 0.130668i
$$701$$ 24294.0 1.30895 0.654473 0.756085i $$-0.272890\pi$$
0.654473 + 0.756085i $$0.272890\pi$$
$$702$$ 0 0
$$703$$ 31725.0 1.70204
$$704$$ 832.000i 0.0445414i
$$705$$ −2328.00 −0.124365
$$706$$ 4526.00 0.241272
$$707$$ 4045.00i 0.215174i
$$708$$ − 5892.00i − 0.312761i
$$709$$ − 12659.0i − 0.670548i −0.942121 0.335274i $$-0.891171\pi$$
0.942121 0.335274i $$-0.108829\pi$$
$$710$$ 316.000i 0.0167032i
$$711$$ −13752.0 −0.725373
$$712$$ −8328.00 −0.438350
$$713$$ − 19448.0i − 1.02151i
$$714$$ 810.000 0.0424559
$$715$$ 0 0
$$716$$ 8916.00 0.465372
$$717$$ 16632.0i 0.866295i
$$718$$ 8976.00 0.466548
$$719$$ −13091.0 −0.679015 −0.339508 0.940603i $$-0.610260\pi$$
−0.339508 + 0.940603i $$0.610260\pi$$
$$720$$ − 576.000i − 0.0298142i
$$721$$ − 6440.00i − 0.332647i
$$722$$ − 2468.00i − 0.127215i
$$723$$ 16569.0i 0.852293i
$$724$$ −4152.00 −0.213132
$$725$$ −1573.00 −0.0805790
$$726$$ 6972.00i 0.356412i
$$727$$ −10792.0 −0.550555 −0.275277 0.961365i $$-0.588770\pi$$
−0.275277 + 0.961365i $$0.588770\pi$$
$$728$$ 0 0
$$729$$ 9477.00 0.481481
$$730$$ − 920.000i − 0.0466448i
$$731$$ 5373.00 0.271857
$$732$$ 2100.00 0.106036
$$733$$ − 2698.00i − 0.135952i −0.997687 0.0679761i $$-0.978346\pi$$
0.997687 0.0679761i $$-0.0216542\pi$$
$$734$$ 3254.00i 0.163634i
$$735$$ − 1908.00i − 0.0957519i
$$736$$ − 5984.00i − 0.299692i
$$737$$ 10621.0 0.530841
$$738$$ −7020.00 −0.350149
$$739$$ − 2841.00i − 0.141418i −0.997497 0.0707090i $$-0.977474\pi$$
0.997497 0.0707090i $$-0.0225262\pi$$
$$740$$ −3384.00 −0.168106
$$741$$ 0 0
$$742$$ 6180.00 0.305761
$$743$$ − 9191.00i − 0.453816i −0.973916 0.226908i $$-0.927138\pi$$
0.973916 0.226908i $$-0.0728616\pi$$
$$744$$ −2496.00 −0.122994
$$745$$ −1710.00 −0.0840934
$$746$$ − 5974.00i − 0.293195i
$$747$$ 13176.0i 0.645361i
$$748$$ − 1404.00i − 0.0686301i
$$749$$ 6385.00i 0.311486i
$$750$$ −2952.00 −0.143722
$$751$$ 1659.00 0.0806095 0.0403048 0.999187i $$-0.487167\pi$$
0.0403048 + 0.999187i $$0.487167\pi$$
$$752$$ − 6208.00i − 0.301041i
$$753$$ 6525.00 0.315782
$$754$$ 0 0
$$755$$ 4128.00 0.198985
$$756$$ − 2700.00i − 0.129892i
$$757$$ −13929.0 −0.668769 −0.334384 0.942437i $$-0.608528\pi$$
−0.334384 + 0.942437i $$0.608528\pi$$
$$758$$ −17734.0 −0.849773
$$759$$ 7293.00i 0.348774i
$$760$$ − 1200.00i − 0.0572744i
$$761$$ − 4587.00i − 0.218500i −0.994014 0.109250i $$-0.965155\pi$$
0.994014 0.109250i $$-0.0348449\pi$$
$$762$$ − 7062.00i − 0.335734i
$$763$$ −4130.00 −0.195958
$$764$$ 8564.00 0.405543
$$765$$ 972.000i 0.0459382i
$$766$$ 22806.0 1.07574
$$767$$ 0 0
$$768$$ −768.000 −0.0360844
$$769$$ 14499.0i 0.679905i 0.940443 + 0.339953i $$0.110411\pi$$
−0.940443 + 0.339953i $$0.889589\pi$$
$$770$$ 260.000 0.0121685
$$771$$ −17055.0 −0.796655
$$772$$ 10508.0i 0.489885i
$$773$$ 3059.00i 0.142335i 0.997464 + 0.0711673i $$0.0226724\pi$$
−0.997464 + 0.0711673i $$0.977328\pi$$
$$774$$ − 7164.00i − 0.332693i
$$775$$ − 12584.0i − 0.583265i
$$776$$ 776.000 0.0358979
$$777$$ −6345.00 −0.292954
$$778$$ 5244.00i 0.241654i
$$779$$ −14625.0 −0.672651
$$780$$ 0 0
$$781$$ 1027.00 0.0470537
$$782$$ 10098.0i 0.461769i
$$783$$ −1755.00 −0.0801004
$$784$$ 5088.00 0.231778
$$785$$ − 3788.00i − 0.172229i
$$786$$ − 8520.00i − 0.386639i
$$787$$ − 36407.0i − 1.64901i −0.565856 0.824504i $$-0.691454\pi$$
0.565856 0.824504i $$-0.308546\pi$$
$$788$$ − 4812.00i − 0.217539i
$$789$$ −18351.0 −0.828026
$$790$$ 3056.00 0.137630
$$791$$ 4735.00i 0.212841i
$$792$$ −1872.00 −0.0839882
$$793$$ 0 0
$$794$$ −1318.00 −0.0589094
$$795$$ − 3708.00i − 0.165420i
$$796$$ 2972.00 0.132336
$$797$$ 13137.0 0.583860 0.291930 0.956440i $$-0.405703\pi$$
0.291930 + 0.956440i $$0.405703\pi$$
$$798$$ − 2250.00i − 0.0998109i
$$799$$ 10476.0i 0.463848i
$$800$$ − 3872.00i − 0.171120i
$$801$$ − 18738.0i − 0.826560i
$$802$$ 29370.0 1.29313
$$803$$ −2990.00 −0.131401
$$804$$ 9804.00i 0.430050i
$$805$$ −1870.00 −0.0818743
$$806$$ 0 0
$$807$$ 15327.0 0.668570
$$808$$ 6472.00i 0.281787i
$$809$$ 15411.0 0.669743 0.334871 0.942264i $$-0.391307\pi$$
0.334871 + 0.942264i $$0.391307\pi$$
$$810$$ 324.000 0.0140546
$$811$$ − 27664.0i − 1.19780i −0.800824 0.598899i $$-0.795605\pi$$
0.800824 0.598899i $$-0.204395\pi$$
$$812$$ 260.000i 0.0112367i
$$813$$ 22647.0i 0.976956i
$$814$$ 10998.0i 0.473562i
$$815$$ −1970.00 −0.0846700
$$816$$ 1296.00 0.0555994
$$817$$ − 14925.0i − 0.639118i
$$818$$ −15658.0 −0.669278
$$819$$ 0 0
$$820$$ 1560.00 0.0664361
$$821$$ − 21397.0i − 0.909574i −0.890600 0.454787i $$-0.849715\pi$$
0.890600 0.454787i $$-0.150285\pi$$
$$822$$ −14454.0 −0.613310
$$823$$ 24249.0 1.02706 0.513528 0.858073i $$-0.328338\pi$$
0.513528 + 0.858073i $$0.328338\pi$$
$$824$$ − 10304.0i − 0.435627i
$$825$$ 4719.00i 0.199145i
$$826$$ − 4910.00i − 0.206829i
$$827$$ − 14028.0i − 0.589844i −0.955521 0.294922i $$-0.904706\pi$$
0.955521 0.294922i $$-0.0952937\pi$$
$$828$$ 13464.0 0.565104
$$829$$ −30451.0 −1.27576 −0.637881 0.770135i $$-0.720189\pi$$
−0.637881 + 0.770135i $$0.720189\pi$$
$$830$$ − 2928.00i − 0.122449i
$$831$$ −2943.00 −0.122854
$$832$$ 0 0
$$833$$ −8586.00 −0.357128
$$834$$ 16962.0i 0.704252i
$$835$$ −4710.00 −0.195205
$$836$$ −3900.00 −0.161345
$$837$$ − 14040.0i − 0.579801i
$$838$$ 5838.00i 0.240657i
$$839$$ − 20591.0i − 0.847295i −0.905827 0.423647i $$-0.860750\pi$$
0.905827 0.423647i $$-0.139250\pi$$
$$840$$ 240.000i 0.00985808i
$$841$$ −24220.0 −0.993071
$$842$$ −6220.00 −0.254579
$$843$$ − 8286.00i − 0.338535i
$$844$$ 1420.00 0.0579128
$$845$$ 0 0
$$846$$ 13968.0 0.567647
$$847$$ 5810.00i 0.235695i
$$848$$ 9888.00 0.400419
$$849$$ 11775.0 0.475992
$$850$$ 6534.00i 0.263664i
$$851$$ − 79101.0i − 3.18631i
$$852$$ 948.000i 0.0381197i
$$853$$ 5798.00i 0.232731i 0.993206 + 0.116366i $$0.0371244\pi$$
−0.993206 + 0.116366i $$0.962876\pi$$
$$854$$ 1750.00 0.0701215
$$855$$ 2700.00 0.107998
$$856$$ 10216.0i 0.407916i
$$857$$ −5686.00 −0.226640 −0.113320 0.993559i $$-0.536148\pi$$
−0.113320 + 0.993559i $$0.536148\pi$$
$$858$$ 0 0
$$859$$ −46708.0 −1.85525 −0.927623 0.373518i $$-0.878152\pi$$
−0.927623 + 0.373518i $$0.878152\pi$$
$$860$$ 1592.00i 0.0631241i
$$861$$ 2925.00 0.115777
$$862$$ −18270.0 −0.721901
$$863$$ 25168.0i 0.992733i 0.868113 + 0.496367i $$0.165333\pi$$
−0.868113 + 0.496367i $$0.834667\pi$$
$$864$$ − 4320.00i − 0.170103i
$$865$$ 7778.00i 0.305734i
$$866$$ − 23338.0i − 0.915771i
$$867$$ 12552.0 0.491682
$$868$$ −2080.00 −0.0813362
$$869$$ − 9932.00i − 0.387710i
$$870$$ 156.000 0.00607919
$$871$$ 0 0
$$872$$ −6608.00 −0.256623
$$873$$ 1746.00i 0.0676897i
$$874$$ 28050.0 1.08559
$$875$$ −2460.00 −0.0950436
$$876$$ − 2760.00i − 0.106452i
$$877$$ 18663.0i 0.718591i 0.933224 + 0.359296i $$0.116983\pi$$
−0.933224 + 0.359296i $$0.883017\pi$$
$$878$$ 27058.0i 1.04005i
$$879$$ 23133.0i 0.887664i
$$880$$ 416.000 0.0159356
$$881$$ −4971.00 −0.190099 −0.0950495 0.995473i $$-0.530301\pi$$
−0.0950495 + 0.995473i $$0.530301\pi$$
$$882$$ 11448.0i 0.437046i
$$883$$ −6892.00 −0.262666 −0.131333 0.991338i $$-0.541926\pi$$
−0.131333 + 0.991338i $$0.541926\pi$$
$$884$$ 0 0
$$885$$ −2946.00 −0.111897
$$886$$ 3864.00i 0.146516i
$$887$$ −24047.0 −0.910281 −0.455140 0.890420i $$-0.650411\pi$$
−0.455140 + 0.890420i $$0.650411\pi$$
$$888$$ −10152.0 −0.383647
$$889$$ − 5885.00i − 0.222021i
$$890$$ 4164.00i 0.156829i
$$891$$ − 1053.00i − 0.0395924i
$$892$$ 9132.00i 0.342782i
$$893$$ 29100.0 1.09048
$$894$$ −5130.00 −0.191916
$$895$$ − 4458.00i − 0.166497i
$$896$$ −640.000 −0.0238626
$$897$$ 0 0
$$898$$ 10714.0 0.398141
$$899$$ 1352.00i 0.0501576i
$$900$$ 8712.00 0.322667
$$901$$ −16686.0 −0.616971
$$902$$ − 5070.00i − 0.187154i
$$903$$ 2985.00i 0.110005i
$$904$$ 7576.00i 0.278732i
$$905$$ 2076.00i 0.0762526i
$$906$$ 12384.0 0.454118
$$907$$ 12843.0 0.470171 0.235085 0.971975i $$-0.424463\pi$$
0.235085 + 0.971975i $$0.424463\pi$$
$$908$$ − 9804.00i − 0.358323i
$$909$$ −14562.0 −0.531343
$$910$$ 0 0
$$911$$ −144.000 −0.00523703 −0.00261851 0.999997i $$-0.500833\pi$$
−0.00261851 + 0.999997i $$0.500833\pi$$
$$912$$ − 3600.00i − 0.130710i
$$913$$ −9516.00 −0.344944
$$914$$ 38798.0 1.40407
$$915$$ − 1050.00i − 0.0379365i
$$916$$ − 7512.00i − 0.270964i
$$917$$ − 7100.00i − 0.255684i
$$918$$ 7290.00i 0.262098i
$$919$$ −11061.0 −0.397028 −0.198514 0.980098i $$-0.563612\pi$$
−0.198514 + 0.980098i $$0.563612\pi$$
$$920$$ −2992.00 −0.107221
$$921$$ 31164.0i 1.11497i
$$922$$ −31098.0 −1.11080
$$923$$ 0 0
$$924$$ 780.000 0.0277707
$$925$$ − 51183.0i − 1.81934i
$$926$$ −8144.00 −0.289016
$$927$$ 23184.0 0.821427
$$928$$ 416.000i 0.0147154i
$$929$$ 26307.0i 0.929069i 0.885555 + 0.464534i $$0.153778\pi$$
−0.885555 + 0.464534i $$0.846222\pi$$
$$930$$ 1248.00i 0.0440038i
$$931$$ 23850.0i 0.839583i
$$932$$ 6520.00 0.229152
$$933$$ −21816.0 −0.765513
$$934$$ 30448.0i 1.06669i
$$935$$ −702.000 −0.0245539
$$936$$ 0 0
$$937$$ −46074.0 −1.60637 −0.803187 0.595727i $$-0.796864\pi$$
−0.803187 + 0.595727i $$0.796864\pi$$
$$938$$ 8170.00i 0.284392i
$$939$$ −23730.0 −0.824706
$$940$$ −3104.00 −0.107704
$$941$$ 36118.0i 1.25124i 0.780130 + 0.625618i $$0.215153\pi$$
−0.780130 + 0.625618i $$0.784847\pi$$
$$942$$ − 11364.0i − 0.393056i
$$943$$ 36465.0i 1.25924i
$$944$$ − 7856.00i − 0.270859i
$$945$$ −1350.00 −0.0464714
$$946$$ 5174.00 0.177824
$$947$$ 55515.0i 1.90496i 0.304604 + 0.952479i $$0.401476\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$948$$ 9168.00 0.314096
$$949$$ 0 0
$$950$$ 18150.0 0.619857
$$951$$ − 22194.0i − 0.756772i
$$952$$ 1080.00 0.0367679
$$953$$ 5353.00 0.181952 0.0909762 0.995853i $$-0.471001\pi$$
0.0909762 + 0.995853i $$0.471001\pi$$
$$954$$ 22248.0i 0.755037i
$$955$$ − 4282.00i − 0.145091i
$$956$$ 22176.0i 0.750233i
$$957$$ − 507.000i − 0.0171254i
$$958$$ 20670.0 0.697095
$$959$$ −12045.0 −0.405582
$$960$$ 384.000i 0.0129099i
$$961$$ 18975.0 0.636937
$$962$$ 0 0
$$963$$ −22986.0 −0.769173
$$964$$ 22092.0i 0.738107i
$$965$$ 5254.00 0.175267
$$966$$ −5610.00 −0.186852
$$967$$ − 5488.00i − 0.182505i −0.995828 0.0912524i $$-0.970913\pi$$
0.995828 0.0912524i $$-0.0290870\pi$$
$$968$$ 9296.00i 0.308662i
$$969$$ 6075.00i 0.201401i
$$970$$ − 388.000i − 0.0128432i
$$971$$ −37353.0 −1.23452 −0.617258 0.786761i $$-0.711756\pi$$
−0.617258 + 0.786761i $$0.711756\pi$$
$$972$$ 15552.0 0.513200
$$973$$ 14135.0i 0.465722i
$$974$$ 12910.0 0.424705
$$975$$ 0 0
$$976$$ 2800.00 0.0918297
$$977$$ − 12729.0i − 0.416824i −0.978041 0.208412i $$-0.933171\pi$$
0.978041 0.208412i $$-0.0668294\pi$$
$$978$$ −5910.00 −0.193232
$$979$$ 13533.0 0.441794
$$980$$ − 2544.00i − 0.0829236i
$$981$$ − 14868.0i − 0.483893i
$$982$$ 15554.0i 0.505446i
$$983$$ 56128.0i 1.82116i 0.413327 + 0.910582i $$0.364367\pi$$
−0.413327 + 0.910582i $$0.635633\pi$$
$$984$$ 4680.00 0.151619
$$985$$ −2406.00 −0.0778290
$$986$$ − 702.000i − 0.0226737i
$$987$$ −5820.00 −0.187693
$$988$$ 0 0
$$989$$ −37213.0 −1.19647
$$990$$ 936.000i 0.0300485i
$$991$$ 47001.0 1.50660 0.753298 0.657680i $$-0.228462\pi$$
0.753298 + 0.657680i $$0.228462\pi$$
$$992$$ −3328.00 −0.106516
$$993$$ 7131.00i 0.227891i
$$994$$ 790.000i 0.0252085i
$$995$$ − 1486.00i − 0.0473461i
$$996$$ − 8784.00i − 0.279449i
$$997$$ −24433.0 −0.776129 −0.388065 0.921632i $$-0.626856\pi$$
−0.388065 + 0.921632i $$0.626856\pi$$
$$998$$ −6088.00 −0.193098
$$999$$ − 57105.0i − 1.80853i
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 338.4.b.a.337.1 2
13.2 odd 12 26.4.c.a.9.1 yes 2
13.3 even 3 338.4.e.d.147.2 4
13.4 even 6 338.4.e.d.23.2 4
13.5 odd 4 338.4.a.a.1.1 1
13.6 odd 12 26.4.c.a.3.1 2
13.7 odd 12 338.4.c.d.315.1 2
13.8 odd 4 338.4.a.d.1.1 1
13.9 even 3 338.4.e.d.23.1 4
13.10 even 6 338.4.e.d.147.1 4
13.11 odd 12 338.4.c.d.191.1 2
13.12 even 2 inner 338.4.b.a.337.2 2
39.2 even 12 234.4.h.b.217.1 2
39.32 even 12 234.4.h.b.55.1 2
52.15 even 12 208.4.i.a.113.1 2
52.19 even 12 208.4.i.a.81.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.c.a.3.1 2 13.6 odd 12
26.4.c.a.9.1 yes 2 13.2 odd 12
208.4.i.a.81.1 2 52.19 even 12
208.4.i.a.113.1 2 52.15 even 12
234.4.h.b.55.1 2 39.32 even 12
234.4.h.b.217.1 2 39.2 even 12
338.4.a.a.1.1 1 13.5 odd 4
338.4.a.d.1.1 1 13.8 odd 4
338.4.b.a.337.1 2 1.1 even 1 trivial
338.4.b.a.337.2 2 13.12 even 2 inner
338.4.c.d.191.1 2 13.11 odd 12
338.4.c.d.315.1 2 13.7 odd 12
338.4.e.d.23.1 4 13.9 even 3
338.4.e.d.23.2 4 13.4 even 6
338.4.e.d.147.1 4 13.10 even 6
338.4.e.d.147.2 4 13.3 even 3