# Properties

 Label 169.4.b.b.168.2 Level $169$ Weight $4$ Character 169.168 Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, 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("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 168.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 169.168 Dual form 169.4.b.b.168.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} +13.8564i q^{5} -24.2487i q^{6} +22.5167i q^{7} +13.8564i q^{8} +22.0000 q^{9} +O(q^{10})$$ $$q+3.46410i q^{2} -7.00000 q^{3} -4.00000 q^{4} +13.8564i q^{5} -24.2487i q^{6} +22.5167i q^{7} +13.8564i q^{8} +22.0000 q^{9} -48.0000 q^{10} +22.5167i q^{11} +28.0000 q^{12} -78.0000 q^{14} -96.9948i q^{15} -80.0000 q^{16} +27.0000 q^{17} +76.2102i q^{18} -88.3346i q^{19} -55.4256i q^{20} -157.617i q^{21} -78.0000 q^{22} +57.0000 q^{23} -96.9948i q^{24} -67.0000 q^{25} +35.0000 q^{27} -90.0666i q^{28} -69.0000 q^{29} +336.000 q^{30} -72.7461i q^{31} -166.277i q^{32} -157.617i q^{33} +93.5307i q^{34} -312.000 q^{35} -88.0000 q^{36} +39.8372i q^{37} +306.000 q^{38} -192.000 q^{40} +393.176i q^{41} +546.000 q^{42} -85.0000 q^{43} -90.0666i q^{44} +304.841i q^{45} +197.454i q^{46} -342.946i q^{47} +560.000 q^{48} -164.000 q^{49} -232.095i q^{50} -189.000 q^{51} +426.000 q^{53} +121.244i q^{54} -312.000 q^{55} -312.000 q^{56} +618.342i q^{57} -239.023i q^{58} -19.0526i q^{59} +387.979i q^{60} -17.0000 q^{61} +252.000 q^{62} +495.367i q^{63} -64.0000 q^{64} +546.000 q^{66} -164.545i q^{67} -108.000 q^{68} -399.000 q^{69} -1080.80i q^{70} +583.701i q^{71} +304.841i q^{72} +1004.59i q^{73} -138.000 q^{74} +469.000 q^{75} +353.338i q^{76} -507.000 q^{77} -1244.00 q^{79} -1108.51i q^{80} -839.000 q^{81} -1362.00 q^{82} -426.084i q^{83} +630.466i q^{84} +374.123i q^{85} -294.449i q^{86} +483.000 q^{87} -312.000 q^{88} -306.573i q^{89} -1056.00 q^{90} -228.000 q^{92} +509.223i q^{93} +1188.00 q^{94} +1224.00 q^{95} +1163.94i q^{96} +1234.95i q^{97} -568.113i q^{98} +495.367i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 8 q^{4} + 44 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 $$2 q - 14 q^{3} - 8 q^{4} + 44 q^{9} - 96 q^{10} + 56 q^{12} - 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} - 134 q^{25} + 70 q^{27} - 138 q^{29} + 672 q^{30} - 624 q^{35} - 176 q^{36} + 612 q^{38} - 384 q^{40} + 1092 q^{42} - 170 q^{43} + 1120 q^{48} - 328 q^{49} - 378 q^{51} + 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} - 128 q^{64} + 1092 q^{66} - 216 q^{68} - 798 q^{69} - 276 q^{74} + 938 q^{75} - 1014 q^{77} - 2488 q^{79} - 1678 q^{81} - 2724 q^{82} + 966 q^{87} - 624 q^{88} - 2112 q^{90} - 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100})$$ 2 * q - 14 * q^3 - 8 * q^4 + 44 * q^9 - 96 * q^10 + 56 * q^12 - 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 - 134 * q^25 + 70 * q^27 - 138 * q^29 + 672 * q^30 - 624 * q^35 - 176 * q^36 + 612 * q^38 - 384 * q^40 + 1092 * q^42 - 170 * q^43 + 1120 * q^48 - 328 * q^49 - 378 * q^51 + 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 - 128 * q^64 + 1092 * q^66 - 216 * q^68 - 798 * q^69 - 276 * q^74 + 938 * q^75 - 1014 * q^77 - 2488 * q^79 - 1678 * q^81 - 2724 * q^82 + 966 * q^87 - 624 * q^88 - 2112 * q^90 - 456 * q^92 + 2376 * q^94 + 2448 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.46410i 1.22474i 0.790569 + 0.612372i $$0.209785\pi$$
−0.790569 + 0.612372i $$0.790215\pi$$
$$3$$ −7.00000 −1.34715 −0.673575 0.739119i $$-0.735242\pi$$
−0.673575 + 0.739119i $$0.735242\pi$$
$$4$$ −4.00000 −0.500000
$$5$$ 13.8564i 1.23935i 0.784857 + 0.619677i $$0.212737\pi$$
−0.784857 + 0.619677i $$0.787263\pi$$
$$6$$ − 24.2487i − 1.64992i
$$7$$ 22.5167i 1.21579i 0.794019 + 0.607893i $$0.207985\pi$$
−0.794019 + 0.607893i $$0.792015\pi$$
$$8$$ 13.8564i 0.612372i
$$9$$ 22.0000 0.814815
$$10$$ −48.0000 −1.51789
$$11$$ 22.5167i 0.617184i 0.951194 + 0.308592i $$0.0998578\pi$$
−0.951194 + 0.308592i $$0.900142\pi$$
$$12$$ 28.0000 0.673575
$$13$$ 0 0
$$14$$ −78.0000 −1.48903
$$15$$ − 96.9948i − 1.66960i
$$16$$ −80.0000 −1.25000
$$17$$ 27.0000 0.385204 0.192602 0.981277i $$-0.438307\pi$$
0.192602 + 0.981277i $$0.438307\pi$$
$$18$$ 76.2102i 0.997940i
$$19$$ − 88.3346i − 1.06660i −0.845927 0.533299i $$-0.820952\pi$$
0.845927 0.533299i $$-0.179048\pi$$
$$20$$ − 55.4256i − 0.619677i
$$21$$ − 157.617i − 1.63785i
$$22$$ −78.0000 −0.755893
$$23$$ 57.0000 0.516753 0.258377 0.966044i $$-0.416812\pi$$
0.258377 + 0.966044i $$0.416812\pi$$
$$24$$ − 96.9948i − 0.824958i
$$25$$ −67.0000 −0.536000
$$26$$ 0 0
$$27$$ 35.0000 0.249472
$$28$$ − 90.0666i − 0.607893i
$$29$$ −69.0000 −0.441827 −0.220913 0.975293i $$-0.570904\pi$$
−0.220913 + 0.975293i $$0.570904\pi$$
$$30$$ 336.000 2.04483
$$31$$ − 72.7461i − 0.421471i −0.977543 0.210735i $$-0.932414\pi$$
0.977543 0.210735i $$-0.0675858\pi$$
$$32$$ − 166.277i − 0.918559i
$$33$$ − 157.617i − 0.831440i
$$34$$ 93.5307i 0.471776i
$$35$$ −312.000 −1.50679
$$36$$ −88.0000 −0.407407
$$37$$ 39.8372i 0.177005i 0.996076 + 0.0885026i $$0.0282081\pi$$
−0.996076 + 0.0885026i $$0.971792\pi$$
$$38$$ 306.000 1.30631
$$39$$ 0 0
$$40$$ −192.000 −0.758947
$$41$$ 393.176i 1.49765i 0.662767 + 0.748826i $$0.269382\pi$$
−0.662767 + 0.748826i $$0.730618\pi$$
$$42$$ 546.000 2.00594
$$43$$ −85.0000 −0.301451 −0.150725 0.988576i $$-0.548161\pi$$
−0.150725 + 0.988576i $$0.548161\pi$$
$$44$$ − 90.0666i − 0.308592i
$$45$$ 304.841i 1.00984i
$$46$$ 197.454i 0.632891i
$$47$$ − 342.946i − 1.06434i −0.846639 0.532168i $$-0.821377\pi$$
0.846639 0.532168i $$-0.178623\pi$$
$$48$$ 560.000 1.68394
$$49$$ −164.000 −0.478134
$$50$$ − 232.095i − 0.656463i
$$51$$ −189.000 −0.518927
$$52$$ 0 0
$$53$$ 426.000 1.10407 0.552034 0.833822i $$-0.313852\pi$$
0.552034 + 0.833822i $$0.313852\pi$$
$$54$$ 121.244i 0.305540i
$$55$$ −312.000 −0.764910
$$56$$ −312.000 −0.744513
$$57$$ 618.342i 1.43687i
$$58$$ − 239.023i − 0.541125i
$$59$$ − 19.0526i − 0.0420412i −0.999779 0.0210206i $$-0.993308\pi$$
0.999779 0.0210206i $$-0.00669156\pi$$
$$60$$ 387.979i 0.834799i
$$61$$ −17.0000 −0.0356824 −0.0178412 0.999841i $$-0.505679\pi$$
−0.0178412 + 0.999841i $$0.505679\pi$$
$$62$$ 252.000 0.516194
$$63$$ 495.367i 0.990640i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 546.000 1.01830
$$67$$ − 164.545i − 0.300035i −0.988683 0.150018i $$-0.952067\pi$$
0.988683 0.150018i $$-0.0479330\pi$$
$$68$$ −108.000 −0.192602
$$69$$ −399.000 −0.696144
$$70$$ − 1080.80i − 1.84543i
$$71$$ 583.701i 0.975670i 0.872936 + 0.487835i $$0.162213\pi$$
−0.872936 + 0.487835i $$0.837787\pi$$
$$72$$ 304.841i 0.498970i
$$73$$ 1004.59i 1.61066i 0.592826 + 0.805331i $$0.298012\pi$$
−0.592826 + 0.805331i $$0.701988\pi$$
$$74$$ −138.000 −0.216786
$$75$$ 469.000 0.722073
$$76$$ 353.338i 0.533299i
$$77$$ −507.000 −0.750364
$$78$$ 0 0
$$79$$ −1244.00 −1.77166 −0.885829 0.464012i $$-0.846409\pi$$
−0.885829 + 0.464012i $$0.846409\pi$$
$$80$$ − 1108.51i − 1.54919i
$$81$$ −839.000 −1.15089
$$82$$ −1362.00 −1.83424
$$83$$ − 426.084i − 0.563480i −0.959491 0.281740i $$-0.909088\pi$$
0.959491 0.281740i $$-0.0909116\pi$$
$$84$$ 630.466i 0.818923i
$$85$$ 374.123i 0.477404i
$$86$$ − 294.449i − 0.369200i
$$87$$ 483.000 0.595207
$$88$$ −312.000 −0.377947
$$89$$ − 306.573i − 0.365131i −0.983194 0.182566i $$-0.941560\pi$$
0.983194 0.182566i $$-0.0584402\pi$$
$$90$$ −1056.00 −1.23680
$$91$$ 0 0
$$92$$ −228.000 −0.258377
$$93$$ 509.223i 0.567785i
$$94$$ 1188.00 1.30354
$$95$$ 1224.00 1.32189
$$96$$ 1163.94i 1.23744i
$$97$$ 1234.95i 1.29268i 0.763048 + 0.646342i $$0.223702\pi$$
−0.763048 + 0.646342i $$0.776298\pi$$
$$98$$ − 568.113i − 0.585592i
$$99$$ 495.367i 0.502891i
$$100$$ 268.000 0.268000
$$101$$ 1959.00 1.92998 0.964989 0.262290i $$-0.0844778\pi$$
0.964989 + 0.262290i $$0.0844778\pi$$
$$102$$ − 654.715i − 0.635554i
$$103$$ 1856.00 1.77551 0.887753 0.460320i $$-0.152265\pi$$
0.887753 + 0.460320i $$0.152265\pi$$
$$104$$ 0 0
$$105$$ 2184.00 2.02987
$$106$$ 1475.71i 1.35220i
$$107$$ −255.000 −0.230390 −0.115195 0.993343i $$-0.536749\pi$$
−0.115195 + 0.993343i $$0.536749\pi$$
$$108$$ −140.000 −0.124736
$$109$$ − 609.682i − 0.535752i −0.963453 0.267876i $$-0.913678\pi$$
0.963453 0.267876i $$-0.0863217\pi$$
$$110$$ − 1080.80i − 0.936820i
$$111$$ − 278.860i − 0.238453i
$$112$$ − 1801.33i − 1.51973i
$$113$$ 411.000 0.342156 0.171078 0.985257i $$-0.445275\pi$$
0.171078 + 0.985257i $$0.445275\pi$$
$$114$$ −2142.00 −1.75980
$$115$$ 789.815i 0.640440i
$$116$$ 276.000 0.220913
$$117$$ 0 0
$$118$$ 66.0000 0.0514898
$$119$$ 607.950i 0.468325i
$$120$$ 1344.00 1.02242
$$121$$ 824.000 0.619083
$$122$$ − 58.8897i − 0.0437018i
$$123$$ − 2752.23i − 2.01756i
$$124$$ 290.985i 0.210735i
$$125$$ 803.672i 0.575061i
$$126$$ −1716.00 −1.21328
$$127$$ −2243.00 −1.56720 −0.783599 0.621267i $$-0.786618\pi$$
−0.783599 + 0.621267i $$0.786618\pi$$
$$128$$ − 1551.92i − 1.07165i
$$129$$ 595.000 0.406099
$$130$$ 0 0
$$131$$ −372.000 −0.248105 −0.124053 0.992276i $$-0.539589\pi$$
−0.124053 + 0.992276i $$0.539589\pi$$
$$132$$ 630.466i 0.415720i
$$133$$ 1989.00 1.29675
$$134$$ 570.000 0.367466
$$135$$ 484.974i 0.309185i
$$136$$ 374.123i 0.235888i
$$137$$ − 1189.92i − 0.742056i −0.928622 0.371028i $$-0.879005\pi$$
0.928622 0.371028i $$-0.120995\pi$$
$$138$$ − 1382.18i − 0.852599i
$$139$$ −2545.00 −1.55298 −0.776490 0.630130i $$-0.783002\pi$$
−0.776490 + 0.630130i $$0.783002\pi$$
$$140$$ 1248.00 0.753395
$$141$$ 2400.62i 1.43382i
$$142$$ −2022.00 −1.19495
$$143$$ 0 0
$$144$$ −1760.00 −1.01852
$$145$$ − 956.092i − 0.547580i
$$146$$ −3480.00 −1.97265
$$147$$ 1148.00 0.644119
$$148$$ − 159.349i − 0.0885026i
$$149$$ 1304.23i 0.717094i 0.933512 + 0.358547i $$0.116728\pi$$
−0.933512 + 0.358547i $$0.883272\pi$$
$$150$$ 1624.66i 0.884355i
$$151$$ − 86.6025i − 0.0466729i −0.999728 0.0233365i $$-0.992571\pi$$
0.999728 0.0233365i $$-0.00742890\pi$$
$$152$$ 1224.00 0.653155
$$153$$ 594.000 0.313870
$$154$$ − 1756.30i − 0.919004i
$$155$$ 1008.00 0.522352
$$156$$ 0 0
$$157$$ −1534.00 −0.779787 −0.389893 0.920860i $$-0.627488\pi$$
−0.389893 + 0.920860i $$0.627488\pi$$
$$158$$ − 4309.34i − 2.16983i
$$159$$ −2982.00 −1.48735
$$160$$ 2304.00 1.13842
$$161$$ 1283.45i 0.628261i
$$162$$ − 2906.38i − 1.40955i
$$163$$ − 1633.32i − 0.784858i −0.919782 0.392429i $$-0.871635\pi$$
0.919782 0.392429i $$-0.128365\pi$$
$$164$$ − 1572.70i − 0.748826i
$$165$$ 2184.00 1.03045
$$166$$ 1476.00 0.690119
$$167$$ − 1626.40i − 0.753618i −0.926291 0.376809i $$-0.877021\pi$$
0.926291 0.376809i $$-0.122979\pi$$
$$168$$ 2184.00 1.00297
$$169$$ 0 0
$$170$$ −1296.00 −0.584698
$$171$$ − 1943.36i − 0.869079i
$$172$$ 340.000 0.150725
$$173$$ −873.000 −0.383659 −0.191829 0.981428i $$-0.561442\pi$$
−0.191829 + 0.981428i $$0.561442\pi$$
$$174$$ 1673.16i 0.728977i
$$175$$ − 1508.62i − 0.651661i
$$176$$ − 1801.33i − 0.771481i
$$177$$ 133.368i 0.0566359i
$$178$$ 1062.00 0.447193
$$179$$ −1287.00 −0.537402 −0.268701 0.963224i $$-0.586594\pi$$
−0.268701 + 0.963224i $$0.586594\pi$$
$$180$$ − 1219.36i − 0.504922i
$$181$$ 2.00000 0.000821319 0 0.000410660 1.00000i $$-0.499869\pi$$
0.000410660 1.00000i $$0.499869\pi$$
$$182$$ 0 0
$$183$$ 119.000 0.0480696
$$184$$ 789.815i 0.316445i
$$185$$ −552.000 −0.219372
$$186$$ −1764.00 −0.695391
$$187$$ 607.950i 0.237742i
$$188$$ 1371.78i 0.532168i
$$189$$ 788.083i 0.303305i
$$190$$ 4240.06i 1.61898i
$$191$$ −2841.00 −1.07627 −0.538135 0.842859i $$-0.680871\pi$$
−0.538135 + 0.842859i $$0.680871\pi$$
$$192$$ 448.000 0.168394
$$193$$ 4245.26i 1.58332i 0.610964 + 0.791659i $$0.290782\pi$$
−0.610964 + 0.791659i $$0.709218\pi$$
$$194$$ −4278.00 −1.58321
$$195$$ 0 0
$$196$$ 656.000 0.239067
$$197$$ − 2752.23i − 0.995371i −0.867357 0.497686i $$-0.834183\pi$$
0.867357 0.497686i $$-0.165817\pi$$
$$198$$ −1716.00 −0.615913
$$199$$ −1685.00 −0.600234 −0.300117 0.953902i $$-0.597026\pi$$
−0.300117 + 0.953902i $$0.597026\pi$$
$$200$$ − 928.379i − 0.328232i
$$201$$ 1151.81i 0.404192i
$$202$$ 6786.18i 2.36373i
$$203$$ − 1553.65i − 0.537167i
$$204$$ 756.000 0.259464
$$205$$ −5448.00 −1.85612
$$206$$ 6429.37i 2.17454i
$$207$$ 1254.00 0.421058
$$208$$ 0 0
$$209$$ 1989.00 0.658287
$$210$$ 7565.60i 2.48608i
$$211$$ 1681.00 0.548459 0.274229 0.961664i $$-0.411577\pi$$
0.274229 + 0.961664i $$0.411577\pi$$
$$212$$ −1704.00 −0.552034
$$213$$ − 4085.91i − 1.31437i
$$214$$ − 883.346i − 0.282170i
$$215$$ − 1177.79i − 0.373604i
$$216$$ 484.974i 0.152770i
$$217$$ 1638.00 0.512418
$$218$$ 2112.00 0.656159
$$219$$ − 7032.13i − 2.16980i
$$220$$ 1248.00 0.382455
$$221$$ 0 0
$$222$$ 966.000 0.292044
$$223$$ 4096.30i 1.23008i 0.788495 + 0.615042i $$0.210861\pi$$
−0.788495 + 0.615042i $$0.789139\pi$$
$$224$$ 3744.00 1.11677
$$225$$ −1474.00 −0.436741
$$226$$ 1423.75i 0.419054i
$$227$$ 438.209i 0.128128i 0.997946 + 0.0640638i $$0.0204061\pi$$
−0.997946 + 0.0640638i $$0.979594\pi$$
$$228$$ − 2473.37i − 0.718433i
$$229$$ 180.133i 0.0519805i 0.999662 + 0.0259903i $$0.00827389\pi$$
−0.999662 + 0.0259903i $$0.991726\pi$$
$$230$$ −2736.00 −0.784376
$$231$$ 3549.00 1.01085
$$232$$ − 956.092i − 0.270563i
$$233$$ −5778.00 −1.62459 −0.812295 0.583247i $$-0.801782\pi$$
−0.812295 + 0.583247i $$0.801782\pi$$
$$234$$ 0 0
$$235$$ 4752.00 1.31909
$$236$$ 76.2102i 0.0210206i
$$237$$ 8708.00 2.38669
$$238$$ −2106.00 −0.573579
$$239$$ − 1860.22i − 0.503464i −0.967797 0.251732i $$-0.919000\pi$$
0.967797 0.251732i $$-0.0810001\pi$$
$$240$$ 7759.59i 2.08700i
$$241$$ 2059.41i 0.550449i 0.961380 + 0.275224i $$0.0887521\pi$$
−0.961380 + 0.275224i $$0.911248\pi$$
$$242$$ 2854.42i 0.758219i
$$243$$ 4928.00 1.30095
$$244$$ 68.0000 0.0178412
$$245$$ − 2272.45i − 0.592578i
$$246$$ 9534.00 2.47100
$$247$$ 0 0
$$248$$ 1008.00 0.258097
$$249$$ 2982.59i 0.759093i
$$250$$ −2784.00 −0.704302
$$251$$ 4491.00 1.12936 0.564680 0.825310i $$-0.309000\pi$$
0.564680 + 0.825310i $$0.309000\pi$$
$$252$$ − 1981.47i − 0.495320i
$$253$$ 1283.45i 0.318932i
$$254$$ − 7769.98i − 1.91942i
$$255$$ − 2618.86i − 0.643135i
$$256$$ 4864.00 1.18750
$$257$$ 5451.00 1.32305 0.661525 0.749923i $$-0.269909\pi$$
0.661525 + 0.749923i $$0.269909\pi$$
$$258$$ 2061.14i 0.497368i
$$259$$ −897.000 −0.215200
$$260$$ 0 0
$$261$$ −1518.00 −0.360007
$$262$$ − 1288.65i − 0.303866i
$$263$$ −783.000 −0.183581 −0.0917906 0.995778i $$-0.529259\pi$$
−0.0917906 + 0.995778i $$0.529259\pi$$
$$264$$ 2184.00 0.509151
$$265$$ 5902.83i 1.36833i
$$266$$ 6890.10i 1.58819i
$$267$$ 2146.01i 0.491887i
$$268$$ 658.179i 0.150018i
$$269$$ −5085.00 −1.15256 −0.576279 0.817253i $$-0.695496\pi$$
−0.576279 + 0.817253i $$0.695496\pi$$
$$270$$ −1680.00 −0.378672
$$271$$ 1325.02i 0.297008i 0.988912 + 0.148504i $$0.0474458\pi$$
−0.988912 + 0.148504i $$0.952554\pi$$
$$272$$ −2160.00 −0.481505
$$273$$ 0 0
$$274$$ 4122.00 0.908829
$$275$$ − 1508.62i − 0.330811i
$$276$$ 1596.00 0.348072
$$277$$ −3421.00 −0.742050 −0.371025 0.928623i $$-0.620994\pi$$
−0.371025 + 0.928623i $$0.620994\pi$$
$$278$$ − 8816.14i − 1.90200i
$$279$$ − 1600.41i − 0.343421i
$$280$$ − 4323.20i − 0.922716i
$$281$$ 810.600i 0.172087i 0.996291 + 0.0860433i $$0.0274223\pi$$
−0.996291 + 0.0860433i $$0.972578\pi$$
$$282$$ −8316.00 −1.75607
$$283$$ 7177.00 1.50752 0.753760 0.657149i $$-0.228238\pi$$
0.753760 + 0.657149i $$0.228238\pi$$
$$284$$ − 2334.80i − 0.487835i
$$285$$ −8568.00 −1.78079
$$286$$ 0 0
$$287$$ −8853.00 −1.82082
$$288$$ − 3658.09i − 0.748455i
$$289$$ −4184.00 −0.851618
$$290$$ 3312.00 0.670646
$$291$$ − 8644.67i − 1.74144i
$$292$$ − 4018.36i − 0.805331i
$$293$$ 9313.24i 1.85695i 0.371400 + 0.928473i $$0.378878\pi$$
−0.371400 + 0.928473i $$0.621122\pi$$
$$294$$ 3976.79i 0.788881i
$$295$$ 264.000 0.0521040
$$296$$ −552.000 −0.108393
$$297$$ 788.083i 0.153970i
$$298$$ −4518.00 −0.878257
$$299$$ 0 0
$$300$$ −1876.00 −0.361036
$$301$$ − 1913.92i − 0.366499i
$$302$$ 300.000 0.0571625
$$303$$ −13713.0 −2.59997
$$304$$ 7066.77i 1.33325i
$$305$$ − 235.559i − 0.0442232i
$$306$$ 2057.68i 0.384410i
$$307$$ − 4777.00i − 0.888070i −0.896009 0.444035i $$-0.853547\pi$$
0.896009 0.444035i $$-0.146453\pi$$
$$308$$ 2028.00 0.375182
$$309$$ −12992.0 −2.39187
$$310$$ 3491.81i 0.639748i
$$311$$ 6192.00 1.12899 0.564495 0.825436i $$-0.309071\pi$$
0.564495 + 0.825436i $$0.309071\pi$$
$$312$$ 0 0
$$313$$ −770.000 −0.139051 −0.0695255 0.997580i $$-0.522149\pi$$
−0.0695255 + 0.997580i $$0.522149\pi$$
$$314$$ − 5313.93i − 0.955040i
$$315$$ −6864.00 −1.22775
$$316$$ 4976.00 0.885829
$$317$$ 8057.50i 1.42762i 0.700341 + 0.713808i $$0.253031\pi$$
−0.700341 + 0.713808i $$0.746969\pi$$
$$318$$ − 10330.0i − 1.82162i
$$319$$ − 1553.65i − 0.272689i
$$320$$ − 886.810i − 0.154919i
$$321$$ 1785.00 0.310371
$$322$$ −4446.00 −0.769459
$$323$$ − 2385.03i − 0.410857i
$$324$$ 3356.00 0.575446
$$325$$ 0 0
$$326$$ 5658.00 0.961250
$$327$$ 4267.77i 0.721738i
$$328$$ −5448.00 −0.917120
$$329$$ 7722.00 1.29400
$$330$$ 7565.60i 1.26204i
$$331$$ − 5277.56i − 0.876377i −0.898883 0.438189i $$-0.855620\pi$$
0.898883 0.438189i $$-0.144380\pi$$
$$332$$ 1704.34i 0.281740i
$$333$$ 876.418i 0.144226i
$$334$$ 5634.00 0.922990
$$335$$ 2280.00 0.371850
$$336$$ 12609.3i 2.04731i
$$337$$ 8278.00 1.33808 0.669038 0.743228i $$-0.266706\pi$$
0.669038 + 0.743228i $$0.266706\pi$$
$$338$$ 0 0
$$339$$ −2877.00 −0.460936
$$340$$ − 1496.49i − 0.238702i
$$341$$ 1638.00 0.260125
$$342$$ 6732.00 1.06440
$$343$$ 4030.48i 0.634477i
$$344$$ − 1177.79i − 0.184600i
$$345$$ − 5528.71i − 0.862770i
$$346$$ − 3024.16i − 0.469884i
$$347$$ 6867.00 1.06236 0.531181 0.847258i $$-0.321748\pi$$
0.531181 + 0.847258i $$0.321748\pi$$
$$348$$ −1932.00 −0.297604
$$349$$ − 12153.8i − 1.86412i −0.362303 0.932060i $$-0.618010\pi$$
0.362303 0.932060i $$-0.381990\pi$$
$$350$$ 5226.00 0.798118
$$351$$ 0 0
$$352$$ 3744.00 0.566920
$$353$$ − 5807.57i − 0.875653i −0.899059 0.437827i $$-0.855748\pi$$
0.899059 0.437827i $$-0.144252\pi$$
$$354$$ −462.000 −0.0693645
$$355$$ −8088.00 −1.20920
$$356$$ 1226.29i 0.182566i
$$357$$ − 4255.65i − 0.630904i
$$358$$ − 4458.30i − 0.658180i
$$359$$ 1340.61i 0.197088i 0.995133 + 0.0985439i $$0.0314185\pi$$
−0.995133 + 0.0985439i $$0.968581\pi$$
$$360$$ −4224.00 −0.618401
$$361$$ −944.000 −0.137629
$$362$$ 6.92820i 0.00100591i
$$363$$ −5768.00 −0.833999
$$364$$ 0 0
$$365$$ −13920.0 −1.99618
$$366$$ 412.228i 0.0588730i
$$367$$ 3665.00 0.521285 0.260642 0.965435i $$-0.416066\pi$$
0.260642 + 0.965435i $$0.416066\pi$$
$$368$$ −4560.00 −0.645941
$$369$$ 8649.86i 1.22031i
$$370$$ − 1912.18i − 0.268675i
$$371$$ 9592.10i 1.34231i
$$372$$ − 2036.89i − 0.283892i
$$373$$ 5371.00 0.745576 0.372788 0.927917i $$-0.378402\pi$$
0.372788 + 0.927917i $$0.378402\pi$$
$$374$$ −2106.00 −0.291173
$$375$$ − 5625.70i − 0.774693i
$$376$$ 4752.00 0.651770
$$377$$ 0 0
$$378$$ −2730.00 −0.371471
$$379$$ 11509.5i 1.55990i 0.625842 + 0.779950i $$0.284756\pi$$
−0.625842 + 0.779950i $$0.715244\pi$$
$$380$$ −4896.00 −0.660946
$$381$$ 15701.0 2.11125
$$382$$ − 9841.51i − 1.31816i
$$383$$ 2419.67i 0.322819i 0.986888 + 0.161409i $$0.0516040\pi$$
−0.986888 + 0.161409i $$0.948396\pi$$
$$384$$ 10863.4i 1.44368i
$$385$$ − 7025.20i − 0.929967i
$$386$$ −14706.0 −1.93916
$$387$$ −1870.00 −0.245626
$$388$$ − 4939.81i − 0.646342i
$$389$$ −9858.00 −1.28489 −0.642443 0.766334i $$-0.722079\pi$$
−0.642443 + 0.766334i $$0.722079\pi$$
$$390$$ 0 0
$$391$$ 1539.00 0.199055
$$392$$ − 2272.45i − 0.292796i
$$393$$ 2604.00 0.334235
$$394$$ 9534.00 1.21908
$$395$$ − 17237.4i − 2.19571i
$$396$$ − 1981.47i − 0.251446i
$$397$$ − 8720.88i − 1.10249i −0.834344 0.551245i $$-0.814153\pi$$
0.834344 0.551245i $$-0.185847\pi$$
$$398$$ − 5837.01i − 0.735133i
$$399$$ −13923.0 −1.74692
$$400$$ 5360.00 0.670000
$$401$$ 7584.65i 0.944537i 0.881455 + 0.472269i $$0.156565\pi$$
−0.881455 + 0.472269i $$0.843435\pi$$
$$402$$ −3990.00 −0.495033
$$403$$ 0 0
$$404$$ −7836.00 −0.964989
$$405$$ − 11625.5i − 1.42636i
$$406$$ 5382.00 0.657892
$$407$$ −897.000 −0.109245
$$408$$ − 2618.86i − 0.317777i
$$409$$ 4304.15i 0.520358i 0.965560 + 0.260179i $$0.0837815\pi$$
−0.965560 + 0.260179i $$0.916218\pi$$
$$410$$ − 18872.4i − 2.27327i
$$411$$ 8329.43i 0.999661i
$$412$$ −7424.00 −0.887753
$$413$$ 429.000 0.0511131
$$414$$ 4343.98i 0.515689i
$$415$$ 5904.00 0.698352
$$416$$ 0 0
$$417$$ 17815.0 2.09210
$$418$$ 6890.10i 0.806234i
$$419$$ 5397.00 0.629262 0.314631 0.949214i $$-0.398119\pi$$
0.314631 + 0.949214i $$0.398119\pi$$
$$420$$ −8736.00 −1.01494
$$421$$ 7260.76i 0.840541i 0.907399 + 0.420270i $$0.138065\pi$$
−0.907399 + 0.420270i $$0.861935\pi$$
$$422$$ 5823.15i 0.671722i
$$423$$ − 7544.81i − 0.867237i
$$424$$ 5902.83i 0.676101i
$$425$$ −1809.00 −0.206469
$$426$$ 14154.0 1.60977
$$427$$ − 382.783i − 0.0433822i
$$428$$ 1020.00 0.115195
$$429$$ 0 0
$$430$$ 4080.00 0.457570
$$431$$ 486.706i 0.0543940i 0.999630 + 0.0271970i $$0.00865814\pi$$
−0.999630 + 0.0271970i $$0.991342\pi$$
$$432$$ −2800.00 −0.311840
$$433$$ 12139.0 1.34726 0.673629 0.739069i $$-0.264734\pi$$
0.673629 + 0.739069i $$0.264734\pi$$
$$434$$ 5674.20i 0.627581i
$$435$$ 6692.64i 0.737673i
$$436$$ 2438.73i 0.267876i
$$437$$ − 5035.07i − 0.551167i
$$438$$ 24360.0 2.65746
$$439$$ 461.000 0.0501192 0.0250596 0.999686i $$-0.492022\pi$$
0.0250596 + 0.999686i $$0.492022\pi$$
$$440$$ − 4323.20i − 0.468410i
$$441$$ −3608.00 −0.389591
$$442$$ 0 0
$$443$$ 12156.0 1.30372 0.651861 0.758338i $$-0.273988\pi$$
0.651861 + 0.758338i $$0.273988\pi$$
$$444$$ 1115.44i 0.119226i
$$445$$ 4248.00 0.452527
$$446$$ −14190.0 −1.50654
$$447$$ − 9129.64i − 0.966034i
$$448$$ − 1441.07i − 0.151973i
$$449$$ − 296.181i − 0.0311306i −0.999879 0.0155653i $$-0.995045\pi$$
0.999879 0.0155653i $$-0.00495479\pi$$
$$450$$ − 5106.09i − 0.534896i
$$451$$ −8853.00 −0.924327
$$452$$ −1644.00 −0.171078
$$453$$ 606.218i 0.0628755i
$$454$$ −1518.00 −0.156924
$$455$$ 0 0
$$456$$ −8568.00 −0.879898
$$457$$ − 611.414i − 0.0625837i −0.999510 0.0312918i $$-0.990038\pi$$
0.999510 0.0312918i $$-0.00996213\pi$$
$$458$$ −624.000 −0.0636629
$$459$$ 945.000 0.0960977
$$460$$ − 3159.26i − 0.320220i
$$461$$ − 13127.2i − 1.32624i −0.748514 0.663119i $$-0.769233\pi$$
0.748514 0.663119i $$-0.230767\pi$$
$$462$$ 12294.1i 1.23804i
$$463$$ − 834.848i − 0.0837985i −0.999122 0.0418992i $$-0.986659\pi$$
0.999122 0.0418992i $$-0.0133408\pi$$
$$464$$ 5520.00 0.552284
$$465$$ −7056.00 −0.703686
$$466$$ − 20015.6i − 1.98971i
$$467$$ 14496.0 1.43639 0.718196 0.695841i $$-0.244968\pi$$
0.718196 + 0.695841i $$0.244968\pi$$
$$468$$ 0 0
$$469$$ 3705.00 0.364778
$$470$$ 16461.4i 1.61555i
$$471$$ 10738.0 1.05049
$$472$$ 264.000 0.0257449
$$473$$ − 1913.92i − 0.186051i
$$474$$ 30165.4i 2.92309i
$$475$$ 5918.42i 0.571696i
$$476$$ − 2431.80i − 0.234162i
$$477$$ 9372.00 0.899611
$$478$$ 6444.00 0.616614
$$479$$ 8897.54i 0.848725i 0.905492 + 0.424362i $$0.139502\pi$$
−0.905492 + 0.424362i $$0.860498\pi$$
$$480$$ −16128.0 −1.53362
$$481$$ 0 0
$$482$$ −7134.00 −0.674159
$$483$$ − 8984.15i − 0.846362i
$$484$$ −3296.00 −0.309542
$$485$$ −17112.0 −1.60209
$$486$$ 17071.1i 1.59333i
$$487$$ 4754.48i 0.442394i 0.975229 + 0.221197i $$0.0709964\pi$$
−0.975229 + 0.221197i $$0.929004\pi$$
$$488$$ − 235.559i − 0.0218509i
$$489$$ 11433.3i 1.05732i
$$490$$ 7872.00 0.725757
$$491$$ −1635.00 −0.150278 −0.0751390 0.997173i $$-0.523940\pi$$
−0.0751390 + 0.997173i $$0.523940\pi$$
$$492$$ 11008.9i 1.00878i
$$493$$ −1863.00 −0.170193
$$494$$ 0 0
$$495$$ −6864.00 −0.623260
$$496$$ 5819.69i 0.526838i
$$497$$ −13143.0 −1.18621
$$498$$ −10332.0 −0.929695
$$499$$ − 14434.9i − 1.29498i −0.762074 0.647490i $$-0.775819\pi$$
0.762074 0.647490i $$-0.224181\pi$$
$$500$$ − 3214.69i − 0.287530i
$$501$$ 11384.8i 1.01524i
$$502$$ 15557.3i 1.38318i
$$503$$ 12687.0 1.12462 0.562312 0.826925i $$-0.309912\pi$$
0.562312 + 0.826925i $$0.309912\pi$$
$$504$$ −6864.00 −0.606641
$$505$$ 27144.7i 2.39193i
$$506$$ −4446.00 −0.390610
$$507$$ 0 0
$$508$$ 8972.00 0.783599
$$509$$ 5748.68i 0.500600i 0.968168 + 0.250300i $$0.0805293\pi$$
−0.968168 + 0.250300i $$0.919471\pi$$
$$510$$ 9072.00 0.787676
$$511$$ −22620.0 −1.95822
$$512$$ 4434.05i 0.382733i
$$513$$ − 3091.71i − 0.266086i
$$514$$ 18882.8i 1.62040i
$$515$$ 25717.5i 2.20048i
$$516$$ −2380.00 −0.203050
$$517$$ 7722.00 0.656892
$$518$$ − 3107.30i − 0.263565i
$$519$$ 6111.00 0.516846
$$520$$ 0 0
$$521$$ 6054.00 0.509080 0.254540 0.967062i $$-0.418076\pi$$
0.254540 + 0.967062i $$0.418076\pi$$
$$522$$ − 5258.51i − 0.440917i
$$523$$ −14803.0 −1.23765 −0.618824 0.785530i $$-0.712391\pi$$
−0.618824 + 0.785530i $$0.712391\pi$$
$$524$$ 1488.00 0.124053
$$525$$ 10560.3i 0.877885i
$$526$$ − 2712.39i − 0.224840i
$$527$$ − 1964.15i − 0.162352i
$$528$$ 12609.3i 1.03930i
$$529$$ −8918.00 −0.732966
$$530$$ −20448.0 −1.67586
$$531$$ − 419.156i − 0.0342558i
$$532$$ −7956.00 −0.648377
$$533$$ 0 0
$$534$$ −7434.00 −0.602436
$$535$$ − 3533.38i − 0.285536i
$$536$$ 2280.00 0.183733
$$537$$ 9009.00 0.723961
$$538$$ − 17615.0i − 1.41159i
$$539$$ − 3692.73i − 0.295097i
$$540$$ − 1939.90i − 0.154592i
$$541$$ − 21470.5i − 1.70626i −0.521695 0.853132i $$-0.674700\pi$$
0.521695 0.853132i $$-0.325300\pi$$
$$542$$ −4590.00 −0.363759
$$543$$ −14.0000 −0.00110644
$$544$$ − 4489.48i − 0.353832i
$$545$$ 8448.00 0.663986
$$546$$ 0 0
$$547$$ −13516.0 −1.05649 −0.528247 0.849091i $$-0.677151\pi$$
−0.528247 + 0.849091i $$0.677151\pi$$
$$548$$ 4759.68i 0.371028i
$$549$$ −374.000 −0.0290746
$$550$$ 5226.00 0.405159
$$551$$ 6095.09i 0.471251i
$$552$$ − 5528.71i − 0.426300i
$$553$$ − 28010.7i − 2.15396i
$$554$$ − 11850.7i − 0.908822i
$$555$$ 3864.00 0.295527
$$556$$ 10180.0 0.776490
$$557$$ − 2890.79i − 0.219905i −0.993937 0.109952i $$-0.964930\pi$$
0.993937 0.109952i $$-0.0350698\pi$$
$$558$$ 5544.00 0.420603
$$559$$ 0 0
$$560$$ 24960.0 1.88349
$$561$$ − 4255.65i − 0.320274i
$$562$$ −2808.00 −0.210762
$$563$$ 11583.0 0.867079 0.433539 0.901135i $$-0.357265\pi$$
0.433539 + 0.901135i $$0.357265\pi$$
$$564$$ − 9602.49i − 0.716911i
$$565$$ 5694.98i 0.424053i
$$566$$ 24861.9i 1.84633i
$$567$$ − 18891.5i − 1.39924i
$$568$$ −8088.00 −0.597473
$$569$$ 12879.0 0.948885 0.474443 0.880286i $$-0.342650\pi$$
0.474443 + 0.880286i $$0.342650\pi$$
$$570$$ − 29680.4i − 2.18101i
$$571$$ −11636.0 −0.852805 −0.426402 0.904534i $$-0.640219\pi$$
−0.426402 + 0.904534i $$0.640219\pi$$
$$572$$ 0 0
$$573$$ 19887.0 1.44990
$$574$$ − 30667.7i − 2.23004i
$$575$$ −3819.00 −0.276980
$$576$$ −1408.00 −0.101852
$$577$$ 12311.4i 0.888269i 0.895960 + 0.444134i $$0.146489\pi$$
−0.895960 + 0.444134i $$0.853511\pi$$
$$578$$ − 14493.8i − 1.04301i
$$579$$ − 29716.8i − 2.13297i
$$580$$ 3824.37i 0.273790i
$$581$$ 9594.00 0.685071
$$582$$ 29946.0 2.13282
$$583$$ 9592.10i 0.681414i
$$584$$ −13920.0 −0.986325
$$585$$ 0 0
$$586$$ −32262.0 −2.27428
$$587$$ 15645.6i 1.10011i 0.835129 + 0.550054i $$0.185393\pi$$
−0.835129 + 0.550054i $$0.814607\pi$$
$$588$$ −4592.00 −0.322059
$$589$$ −6426.00 −0.449539
$$590$$ 914.523i 0.0638141i
$$591$$ 19265.6i 1.34092i
$$592$$ − 3186.97i − 0.221256i
$$593$$ 25821.4i 1.78813i 0.447942 + 0.894063i $$0.352157\pi$$
−0.447942 + 0.894063i $$0.647843\pi$$
$$594$$ −2730.00 −0.188575
$$595$$ −8424.00 −0.580421
$$596$$ − 5216.94i − 0.358547i
$$597$$ 11795.0 0.808605
$$598$$ 0 0
$$599$$ 1668.00 0.113777 0.0568887 0.998381i $$-0.481882\pi$$
0.0568887 + 0.998381i $$0.481882\pi$$
$$600$$ 6498.65i 0.442177i
$$601$$ 13699.0 0.929773 0.464887 0.885370i $$-0.346095\pi$$
0.464887 + 0.885370i $$0.346095\pi$$
$$602$$ 6630.00 0.448868
$$603$$ − 3619.99i − 0.244473i
$$604$$ 346.410i 0.0233365i
$$605$$ 11417.7i 0.767264i
$$606$$ − 47503.2i − 3.18430i
$$607$$ −23173.0 −1.54953 −0.774764 0.632251i $$-0.782131\pi$$
−0.774764 + 0.632251i $$0.782131\pi$$
$$608$$ −14688.0 −0.979732
$$609$$ 10875.5i 0.723644i
$$610$$ 816.000 0.0541621
$$611$$ 0 0
$$612$$ −2376.00 −0.156935
$$613$$ − 16615.6i − 1.09477i −0.836880 0.547387i $$-0.815623\pi$$
0.836880 0.547387i $$-0.184377\pi$$
$$614$$ 16548.0 1.08766
$$615$$ 38136.0 2.50047
$$616$$ − 7025.20i − 0.459502i
$$617$$ 28393.5i 1.85264i 0.376736 + 0.926321i $$0.377046\pi$$
−0.376736 + 0.926321i $$0.622954\pi$$
$$618$$ − 45005.6i − 2.92944i
$$619$$ − 6245.78i − 0.405556i −0.979225 0.202778i $$-0.935003\pi$$
0.979225 0.202778i $$-0.0649969\pi$$
$$620$$ −4032.00 −0.261176
$$621$$ 1995.00 0.128916
$$622$$ 21449.7i 1.38273i
$$623$$ 6903.00 0.443921
$$624$$ 0 0
$$625$$ −19511.0 −1.24870
$$626$$ − 2667.36i − 0.170302i
$$627$$ −13923.0 −0.886812
$$628$$ 6136.00 0.389893
$$629$$ 1075.60i 0.0681830i
$$630$$ − 23777.6i − 1.50369i
$$631$$ 22379.8i 1.41193i 0.708247 + 0.705964i $$0.249486\pi$$
−0.708247 + 0.705964i $$0.750514\pi$$
$$632$$ − 17237.4i − 1.08491i
$$633$$ −11767.0 −0.738857
$$634$$ −27912.0 −1.74847
$$635$$ − 31079.9i − 1.94231i
$$636$$ 11928.0 0.743673
$$637$$ 0 0
$$638$$ 5382.00 0.333974
$$639$$ 12841.4i 0.794990i
$$640$$ 21504.0 1.32816
$$641$$ 19827.0 1.22172 0.610858 0.791740i $$-0.290825\pi$$
0.610858 + 0.791740i $$0.290825\pi$$
$$642$$ 6183.42i 0.380125i
$$643$$ − 8450.68i − 0.518293i −0.965838 0.259146i $$-0.916559\pi$$
0.965838 0.259146i $$-0.0834412\pi$$
$$644$$ − 5133.80i − 0.314130i
$$645$$ 8244.56i 0.503301i
$$646$$ 8262.00 0.503195
$$647$$ 2949.00 0.179192 0.0895959 0.995978i $$-0.471442\pi$$
0.0895959 + 0.995978i $$0.471442\pi$$
$$648$$ − 11625.5i − 0.704774i
$$649$$ 429.000 0.0259472
$$650$$ 0 0
$$651$$ −11466.0 −0.690304
$$652$$ 6533.30i 0.392429i
$$653$$ 12039.0 0.721474 0.360737 0.932668i $$-0.382525\pi$$
0.360737 + 0.932668i $$0.382525\pi$$
$$654$$ −14784.0 −0.883945
$$655$$ − 5154.58i − 0.307490i
$$656$$ − 31454.0i − 1.87206i
$$657$$ 22101.0i 1.31239i
$$658$$ 26749.8i 1.58483i
$$659$$ 3363.00 0.198792 0.0993960 0.995048i $$-0.468309\pi$$
0.0993960 + 0.995048i $$0.468309\pi$$
$$660$$ −8736.00 −0.515225
$$661$$ − 10158.5i − 0.597759i −0.954291 0.298880i $$-0.903387\pi$$
0.954291 0.298880i $$-0.0966129\pi$$
$$662$$ 18282.0 1.07334
$$663$$ 0 0
$$664$$ 5904.00 0.345060
$$665$$ 27560.4i 1.60714i
$$666$$ −3036.00 −0.176641
$$667$$ −3933.00 −0.228315
$$668$$ 6505.58i 0.376809i
$$669$$ − 28674.1i − 1.65711i
$$670$$ 7898.15i 0.455421i
$$671$$ − 382.783i − 0.0220226i
$$672$$ −26208.0 −1.50446
$$673$$ −18169.0 −1.04066 −0.520329 0.853966i $$-0.674191\pi$$
−0.520329 + 0.853966i $$0.674191\pi$$
$$674$$ 28675.8i 1.63880i
$$675$$ −2345.00 −0.133717
$$676$$ 0 0
$$677$$ 9042.00 0.513312 0.256656 0.966503i $$-0.417379\pi$$
0.256656 + 0.966503i $$0.417379\pi$$
$$678$$ − 9966.22i − 0.564529i
$$679$$ −27807.0 −1.57163
$$680$$ −5184.00 −0.292349
$$681$$ − 3067.46i − 0.172607i
$$682$$ 5674.20i 0.318587i
$$683$$ − 12462.1i − 0.698169i −0.937091 0.349084i $$-0.886493\pi$$
0.937091 0.349084i $$-0.113507\pi$$
$$684$$ 7773.44i 0.434540i
$$685$$ 16488.0 0.919670
$$686$$ −13962.0 −0.777072
$$687$$ − 1260.93i − 0.0700256i
$$688$$ 6800.00 0.376813
$$689$$ 0 0
$$690$$ 19152.0 1.05667
$$691$$ 4318.00i 0.237720i 0.992911 + 0.118860i $$0.0379240\pi$$
−0.992911 + 0.118860i $$0.962076\pi$$
$$692$$ 3492.00 0.191829
$$693$$ −11154.0 −0.611408
$$694$$ 23788.0i 1.30112i
$$695$$ − 35264.6i − 1.92469i
$$696$$ 6692.64i 0.364489i
$$697$$ 10615.7i 0.576901i
$$698$$ 42102.0 2.28307
$$699$$ 40446.0 2.18857
$$700$$ 6034.47i 0.325830i
$$701$$ −18270.0 −0.984377 −0.492189 0.870489i $$-0.663803\pi$$
−0.492189 + 0.870489i $$0.663803\pi$$
$$702$$ 0 0
$$703$$ 3519.00 0.188793
$$704$$ − 1441.07i − 0.0771481i
$$705$$ −33264.0 −1.77701
$$706$$ 20118.0 1.07245
$$707$$ 44110.1i 2.34644i
$$708$$ − 533.472i − 0.0283179i
$$709$$ 1629.86i 0.0863338i 0.999068 + 0.0431669i $$0.0137447\pi$$
−0.999068 + 0.0431669i $$0.986255\pi$$
$$710$$ − 28017.7i − 1.48096i
$$711$$ −27368.0 −1.44357
$$712$$ 4248.00 0.223596
$$713$$ − 4146.53i − 0.217796i
$$714$$ 14742.0 0.772697
$$715$$ 0 0
$$716$$ 5148.00 0.268701
$$717$$ 13021.6i 0.678241i
$$718$$ −4644.00 −0.241382
$$719$$ 9831.00 0.509923 0.254961 0.966951i $$-0.417937\pi$$
0.254961 + 0.966951i $$0.417937\pi$$
$$720$$ − 24387.3i − 1.26231i
$$721$$ 41790.9i 2.15863i
$$722$$ − 3270.11i − 0.168561i
$$723$$ − 14415.9i − 0.741537i
$$724$$ −8.00000 −0.000410660 0
$$725$$ 4623.00 0.236819
$$726$$ − 19980.9i − 1.02144i
$$727$$ −15464.0 −0.788897 −0.394448 0.918918i $$-0.629064\pi$$
−0.394448 + 0.918918i $$0.629064\pi$$
$$728$$ 0 0
$$729$$ −11843.0 −0.601687
$$730$$ − 48220.3i − 2.44481i
$$731$$ −2295.00 −0.116120
$$732$$ −476.000 −0.0240348
$$733$$ − 12616.3i − 0.635733i −0.948136 0.317866i $$-0.897034\pi$$
0.948136 0.317866i $$-0.102966\pi$$
$$734$$ 12695.9i 0.638441i
$$735$$ 15907.2i 0.798291i
$$736$$ − 9477.78i − 0.474668i
$$737$$ 3705.00 0.185177
$$738$$ −29964.0 −1.49457
$$739$$ 16283.0i 0.810528i 0.914200 + 0.405264i $$0.132820\pi$$
−0.914200 + 0.405264i $$0.867180\pi$$
$$740$$ 2208.00 0.109686
$$741$$ 0 0
$$742$$ −33228.0 −1.64399
$$743$$ − 10806.3i − 0.533571i −0.963756 0.266786i $$-0.914038\pi$$
0.963756 0.266786i $$-0.0859616\pi$$
$$744$$ −7056.00 −0.347696
$$745$$ −18072.0 −0.888734
$$746$$ 18605.7i 0.913140i
$$747$$ − 9373.86i − 0.459132i
$$748$$ − 2431.80i − 0.118871i
$$749$$ − 5741.75i − 0.280105i
$$750$$ 19488.0 0.948802
$$751$$ −13615.0 −0.661542 −0.330771 0.943711i $$-0.607309\pi$$
−0.330771 + 0.943711i $$0.607309\pi$$
$$752$$ 27435.7i 1.33042i
$$753$$ −31437.0 −1.52142
$$754$$ 0 0
$$755$$ 1200.00 0.0578443
$$756$$ − 3152.33i − 0.151652i
$$757$$ 5551.00 0.266519 0.133259 0.991081i $$-0.457456\pi$$
0.133259 + 0.991081i $$0.457456\pi$$
$$758$$ −39870.0 −1.91048
$$759$$ − 8984.15i − 0.429649i
$$760$$ 16960.2i 0.809490i
$$761$$ 10082.3i 0.480265i 0.970740 + 0.240133i $$0.0771909\pi$$
−0.970740 + 0.240133i $$0.922809\pi$$
$$762$$ 54389.9i 2.58574i
$$763$$ 13728.0 0.651359
$$764$$ 11364.0 0.538135
$$765$$ 8230.71i 0.388996i
$$766$$ −8382.00 −0.395371
$$767$$ 0 0
$$768$$ −34048.0 −1.59974
$$769$$ − 29758.4i − 1.39547i −0.716357 0.697733i $$-0.754192\pi$$
0.716357 0.697733i $$-0.245808\pi$$
$$770$$ 24336.0 1.13897
$$771$$ −38157.0 −1.78235
$$772$$ − 16981.0i − 0.791659i
$$773$$ 27735.3i 1.29052i 0.763964 + 0.645259i $$0.223251\pi$$
−0.763964 + 0.645259i $$0.776749\pi$$
$$774$$ − 6477.87i − 0.300830i
$$775$$ 4873.99i 0.225908i
$$776$$ −17112.0 −0.791604
$$777$$ 6279.00 0.289907
$$778$$ − 34149.1i − 1.57366i
$$779$$ 34731.0 1.59739
$$780$$ 0 0
$$781$$ −13143.0 −0.602168
$$782$$ 5331.25i 0.243792i
$$783$$ −2415.00 −0.110224
$$784$$ 13120.0 0.597668
$$785$$ − 21255.7i − 0.966432i
$$786$$ 9020.52i 0.409353i
$$787$$ − 31549.3i − 1.42899i −0.699643 0.714493i $$-0.746658\pi$$
0.699643 0.714493i $$-0.253342\pi$$
$$788$$ 11008.9i 0.497686i
$$789$$ 5481.00 0.247311
$$790$$ 59712.0 2.68919
$$791$$ 9254.35i 0.415988i
$$792$$ −6864.00 −0.307957
$$793$$ 0 0
$$794$$ 30210.0 1.35027
$$795$$ − 41319.8i − 1.84335i
$$796$$ 6740.00 0.300117
$$797$$ 1455.00 0.0646659 0.0323330 0.999477i $$-0.489706\pi$$
0.0323330 + 0.999477i $$0.489706\pi$$
$$798$$ − 48230.7i − 2.13953i
$$799$$ − 9259.54i − 0.409986i
$$800$$ 11140.6i 0.492347i
$$801$$ − 6744.61i − 0.297514i
$$802$$ −26274.0 −1.15682
$$803$$ −22620.0 −0.994075
$$804$$ − 4607.26i − 0.202096i
$$805$$ −17784.0 −0.778638
$$806$$ 0 0
$$807$$ 35595.0 1.55267
$$808$$ 27144.7i 1.18187i
$$809$$ 1659.00 0.0720981 0.0360490 0.999350i $$-0.488523\pi$$
0.0360490 + 0.999350i $$0.488523\pi$$
$$810$$ 40272.0 1.74693
$$811$$ 4402.87i 0.190636i 0.995447 + 0.0953180i $$0.0303868\pi$$
−0.995447 + 0.0953180i $$0.969613\pi$$
$$812$$ 6214.60i 0.268583i
$$813$$ − 9275.13i − 0.400114i
$$814$$ − 3107.30i − 0.133797i
$$815$$ 22632.0 0.972717
$$816$$ 15120.0 0.648659
$$817$$ 7508.44i 0.321526i
$$818$$ −14910.0 −0.637306
$$819$$ 0 0
$$820$$ 21792.0 0.928061
$$821$$ 28701.8i 1.22010i 0.792364 + 0.610049i $$0.208850\pi$$
−0.792364 + 0.610049i $$0.791150\pi$$
$$822$$ −28854.0 −1.22433
$$823$$ 15779.0 0.668313 0.334156 0.942518i $$-0.391549\pi$$
0.334156 + 0.942518i $$0.391549\pi$$
$$824$$ 25717.5i 1.08727i
$$825$$ 10560.3i 0.445652i
$$826$$ 1486.10i 0.0626005i
$$827$$ − 7354.29i − 0.309231i −0.987975 0.154615i $$-0.950586\pi$$
0.987975 0.154615i $$-0.0494138\pi$$
$$828$$ −5016.00 −0.210529
$$829$$ 17371.0 0.727768 0.363884 0.931444i $$-0.381450\pi$$
0.363884 + 0.931444i $$0.381450\pi$$
$$830$$ 20452.1i 0.855303i
$$831$$ 23947.0 0.999654
$$832$$ 0 0
$$833$$ −4428.00 −0.184179
$$834$$ 61713.0i 2.56228i
$$835$$ 22536.0 0.934001
$$836$$ −7956.00 −0.329144
$$837$$ − 2546.11i − 0.105145i
$$838$$ 18695.8i 0.770685i
$$839$$ 29474.3i 1.21283i 0.795148 + 0.606416i $$0.207393\pi$$
−0.795148 + 0.606416i $$0.792607\pi$$
$$840$$ 30262.4i 1.24304i
$$841$$ −19628.0 −0.804789
$$842$$ −25152.0 −1.02945
$$843$$ − 5674.20i − 0.231827i
$$844$$ −6724.00 −0.274229
$$845$$ 0 0
$$846$$ 26136.0 1.06214
$$847$$ 18553.7i 0.752673i
$$848$$ −34080.0 −1.38008
$$849$$ −50239.0 −2.03086
$$850$$ − 6266.56i − 0.252872i
$$851$$ 2270.72i 0.0914680i
$$852$$ 16343.6i 0.657187i
$$853$$ 2909.85i 0.116801i 0.998293 + 0.0584005i $$0.0186000\pi$$
−0.998293 + 0.0584005i $$0.981400\pi$$
$$854$$ 1326.00 0.0531321
$$855$$ 26928.0 1.07710
$$856$$ − 3533.38i − 0.141085i
$$857$$ −5346.00 −0.213087 −0.106544 0.994308i $$-0.533978\pi$$
−0.106544 + 0.994308i $$0.533978\pi$$
$$858$$ 0 0
$$859$$ 24244.0 0.962974 0.481487 0.876453i $$-0.340097\pi$$
0.481487 + 0.876453i $$0.340097\pi$$
$$860$$ 4711.18i 0.186802i
$$861$$ 61971.0 2.45292
$$862$$ −1686.00 −0.0666188
$$863$$ 32780.8i 1.29301i 0.762908 + 0.646507i $$0.223771\pi$$
−0.762908 + 0.646507i $$0.776229\pi$$
$$864$$ − 5819.69i − 0.229155i
$$865$$ − 12096.6i − 0.475489i
$$866$$ 42050.7i 1.65005i
$$867$$ 29288.0 1.14726
$$868$$ −6552.00 −0.256209
$$869$$ − 28010.7i − 1.09344i
$$870$$ −23184.0 −0.903461
$$871$$ 0 0
$$872$$ 8448.00 0.328080
$$873$$ 27168.9i 1.05330i
$$874$$ 17442.0 0.675039
$$875$$ −18096.0 −0.699150
$$876$$ 28128.5i 1.08490i
$$877$$ − 4543.17i − 0.174928i −0.996168 0.0874640i $$-0.972124\pi$$
0.996168 0.0874640i $$-0.0278763\pi$$
$$878$$ 1596.95i 0.0613832i
$$879$$ − 65192.7i − 2.50159i
$$880$$ 24960.0 0.956138
$$881$$ −20517.0 −0.784603 −0.392302 0.919837i $$-0.628321\pi$$
−0.392302 + 0.919837i $$0.628321\pi$$
$$882$$ − 12498.5i − 0.477149i
$$883$$ −23852.0 −0.909042 −0.454521 0.890736i $$-0.650189\pi$$
−0.454521 + 0.890736i $$0.650189\pi$$
$$884$$ 0 0
$$885$$ −1848.00 −0.0701919
$$886$$ 42109.6i 1.59673i
$$887$$ 38757.0 1.46712 0.733558 0.679626i $$-0.237858\pi$$
0.733558 + 0.679626i $$0.237858\pi$$
$$888$$ 3864.00 0.146022
$$889$$ − 50504.9i − 1.90538i
$$890$$ 14715.5i 0.554230i
$$891$$ − 18891.5i − 0.710312i
$$892$$ − 16385.2i − 0.615042i
$$893$$ −30294.0 −1.13522
$$894$$ 31626.0 1.18315
$$895$$ − 17833.2i − 0.666031i
$$896$$ 34944.0 1.30290
$$897$$ 0 0
$$898$$ 1026.00 0.0381270
$$899$$ 5019.48i 0.186217i
$$900$$ 5896.00 0.218370
$$901$$ 11502.0 0.425291
$$902$$ − 30667.7i − 1.13206i
$$903$$ 13397.4i 0.493730i
$$904$$ 5694.98i 0.209527i
$$905$$ 27.7128i 0.00101791i
$$906$$ −2100.00 −0.0770064
$$907$$ −39071.0 −1.43035 −0.715177 0.698943i $$-0.753654\pi$$
−0.715177 + 0.698943i $$0.753654\pi$$
$$908$$ − 1752.84i − 0.0640638i
$$909$$ 43098.0 1.57257
$$910$$ 0 0
$$911$$ −53040.0 −1.92897 −0.964486 0.264134i $$-0.914914\pi$$
−0.964486 + 0.264134i $$0.914914\pi$$
$$912$$ − 49467.4i − 1.79608i
$$913$$ 9594.00 0.347771
$$914$$ 2118.00 0.0766490
$$915$$ 1648.91i 0.0595753i
$$916$$ − 720.533i − 0.0259903i
$$917$$ − 8376.20i − 0.301643i
$$918$$ 3273.58i 0.117695i
$$919$$ 367.000 0.0131732 0.00658662 0.999978i $$-0.497903\pi$$
0.00658662 + 0.999978i $$0.497903\pi$$
$$920$$ −10944.0 −0.392188
$$921$$ 33439.0i 1.19636i
$$922$$ 45474.0 1.62430
$$923$$ 0 0
$$924$$ −14196.0 −0.505427
$$925$$ − 2669.09i − 0.0948748i
$$926$$ 2892.00 0.102632
$$927$$ 40832.0 1.44671
$$928$$ 11473.1i 0.405844i
$$929$$ − 29935.0i − 1.05720i −0.848872 0.528599i $$-0.822718\pi$$
0.848872 0.528599i $$-0.177282\pi$$
$$930$$ − 24442.7i − 0.861836i
$$931$$ 14486.9i 0.509976i
$$932$$ 23112.0 0.812295
$$933$$ −43344.0 −1.52092
$$934$$ 50215.6i 1.75921i
$$935$$ −8424.00 −0.294646
$$936$$ 0 0
$$937$$ 42166.0 1.47012 0.735060 0.678002i $$-0.237154\pi$$
0.735060 + 0.678002i $$0.237154\pi$$
$$938$$ 12834.5i 0.446760i
$$939$$ 5390.00 0.187323
$$940$$ −19008.0 −0.659545
$$941$$ − 35022.1i − 1.21327i −0.794981 0.606635i $$-0.792519\pi$$
0.794981 0.606635i $$-0.207481\pi$$
$$942$$ 37197.5i 1.28658i
$$943$$ 22411.0i 0.773916i
$$944$$ 1524.20i 0.0525515i
$$945$$ −10920.0 −0.375902
$$946$$ 6630.00 0.227865
$$947$$ 2599.81i 0.0892106i 0.999005 + 0.0446053i $$0.0142030\pi$$
−0.999005 + 0.0446053i $$0.985797\pi$$
$$948$$ −34832.0 −1.19334
$$949$$ 0 0
$$950$$ −20502.0 −0.700182
$$951$$ − 56402.5i − 1.92321i
$$952$$ −8424.00 −0.286789
$$953$$ 10623.0 0.361084 0.180542 0.983567i $$-0.442215\pi$$
0.180542 + 0.983567i $$0.442215\pi$$
$$954$$ 32465.6i 1.10179i
$$955$$ − 39366.1i − 1.33388i
$$956$$ 7440.89i 0.251732i
$$957$$ 10875.5i 0.367353i
$$958$$ −30822.0 −1.03947
$$959$$ 26793.0 0.902180
$$960$$ 6207.67i 0.208700i
$$961$$ 24499.0 0.822362
$$962$$ 0 0
$$963$$ −5610.00 −0.187726
$$964$$ − 8237.63i − 0.275224i
$$965$$ −58824.0 −1.96229
$$966$$ 31122.0 1.03658
$$967$$ 20199.2i 0.671729i 0.941910 + 0.335864i $$0.109028\pi$$
−0.941910 + 0.335864i $$0.890972\pi$$
$$968$$ 11417.7i 0.379110i
$$969$$ 16695.2i 0.553486i
$$970$$ − 59277.7i − 1.96216i
$$971$$ −2325.00 −0.0768412 −0.0384206 0.999262i $$-0.512233\pi$$
−0.0384206 + 0.999262i $$0.512233\pi$$
$$972$$ −19712.0 −0.650476
$$973$$ − 57304.9i − 1.88809i
$$974$$ −16470.0 −0.541820
$$975$$ 0 0
$$976$$ 1360.00 0.0446030
$$977$$ − 32938.4i − 1.07860i −0.842113 0.539300i $$-0.818689\pi$$
0.842113 0.539300i $$-0.181311\pi$$
$$978$$ −39606.0 −1.29495
$$979$$ 6903.00 0.225353
$$980$$ 9089.80i 0.296289i
$$981$$ − 13413.0i − 0.436538i
$$982$$ − 5663.81i − 0.184052i
$$983$$ 42702.0i 1.38554i 0.721161 + 0.692768i $$0.243609\pi$$
−0.721161 + 0.692768i $$0.756391\pi$$
$$984$$ 38136.0 1.23550
$$985$$ 38136.0 1.23362
$$986$$ − 6453.62i − 0.208443i
$$987$$ −54054.0 −1.74322
$$988$$ 0 0
$$989$$ −4845.00 −0.155776
$$990$$ − 23777.6i − 0.763335i
$$991$$ −4843.00 −0.155240 −0.0776201 0.996983i $$-0.524732\pi$$
−0.0776201 + 0.996983i $$0.524732\pi$$
$$992$$ −12096.0 −0.387146
$$993$$ 36942.9i 1.18061i
$$994$$ − 45528.7i − 1.45280i
$$995$$ − 23348.0i − 0.743902i
$$996$$ − 11930.4i − 0.379546i
$$997$$ 10943.0 0.347611 0.173806 0.984780i $$-0.444394\pi$$
0.173806 + 0.984780i $$0.444394\pi$$
$$998$$ 50004.0 1.58602
$$999$$ 1394.30i 0.0441579i
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 169.4.b.b.168.2 2
13.2 odd 12 169.4.c.i.22.1 4
13.3 even 3 13.4.e.a.4.1 2
13.4 even 6 13.4.e.a.10.1 yes 2
13.5 odd 4 169.4.a.h.1.2 2
13.6 odd 12 169.4.c.i.146.1 4
13.7 odd 12 169.4.c.i.146.2 4
13.8 odd 4 169.4.a.h.1.1 2
13.9 even 3 169.4.e.b.23.1 2
13.10 even 6 169.4.e.b.147.1 2
13.11 odd 12 169.4.c.i.22.2 4
13.12 even 2 inner 169.4.b.b.168.1 2
39.5 even 4 1521.4.a.q.1.1 2
39.8 even 4 1521.4.a.q.1.2 2
39.17 odd 6 117.4.q.c.10.1 2
39.29 odd 6 117.4.q.c.82.1 2
52.3 odd 6 208.4.w.a.17.1 2
52.43 odd 6 208.4.w.a.49.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.a.4.1 2 13.3 even 3
13.4.e.a.10.1 yes 2 13.4 even 6
117.4.q.c.10.1 2 39.17 odd 6
117.4.q.c.82.1 2 39.29 odd 6
169.4.a.h.1.1 2 13.8 odd 4
169.4.a.h.1.2 2 13.5 odd 4
169.4.b.b.168.1 2 13.12 even 2 inner
169.4.b.b.168.2 2 1.1 even 1 trivial
169.4.c.i.22.1 4 13.2 odd 12
169.4.c.i.22.2 4 13.11 odd 12
169.4.c.i.146.1 4 13.6 odd 12
169.4.c.i.146.2 4 13.7 odd 12
169.4.e.b.23.1 2 13.9 even 3
169.4.e.b.147.1 2 13.10 even 6
208.4.w.a.17.1 2 52.3 odd 6
208.4.w.a.49.1 2 52.43 odd 6
1521.4.a.q.1.1 2 39.5 even 4
1521.4.a.q.1.2 2 39.8 even 4