# Properties

 Label 507.4.b.d.337.1 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ 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] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.d.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +3.00000 q^{3} +7.00000 q^{4} +7.00000i q^{5} -3.00000i q^{6} +10.0000i q^{7} -15.0000i q^{8} +9.00000 q^{9} +7.00000 q^{10} +22.0000i q^{11} +21.0000 q^{12} +10.0000 q^{14} +21.0000i q^{15} +41.0000 q^{16} -37.0000 q^{17} -9.00000i q^{18} +30.0000i q^{19} +49.0000i q^{20} +30.0000i q^{21} +22.0000 q^{22} +162.000 q^{23} -45.0000i q^{24} +76.0000 q^{25} +27.0000 q^{27} +70.0000i q^{28} -113.000 q^{29} +21.0000 q^{30} +196.000i q^{31} -161.000i q^{32} +66.0000i q^{33} +37.0000i q^{34} -70.0000 q^{35} +63.0000 q^{36} -13.0000i q^{37} +30.0000 q^{38} +105.000 q^{40} +285.000i q^{41} +30.0000 q^{42} +246.000 q^{43} +154.000i q^{44} +63.0000i q^{45} -162.000i q^{46} +462.000i q^{47} +123.000 q^{48} +243.000 q^{49} -76.0000i q^{50} -111.000 q^{51} -537.000 q^{53} -27.0000i q^{54} -154.000 q^{55} +150.000 q^{56} +90.0000i q^{57} +113.000i q^{58} -576.000i q^{59} +147.000i q^{60} -635.000 q^{61} +196.000 q^{62} +90.0000i q^{63} +167.000 q^{64} +66.0000 q^{66} +202.000i q^{67} -259.000 q^{68} +486.000 q^{69} +70.0000i q^{70} -1086.00i q^{71} -135.000i q^{72} +805.000i q^{73} -13.0000 q^{74} +228.000 q^{75} +210.000i q^{76} -220.000 q^{77} +884.000 q^{79} +287.000i q^{80} +81.0000 q^{81} +285.000 q^{82} +518.000i q^{83} +210.000i q^{84} -259.000i q^{85} -246.000i q^{86} -339.000 q^{87} +330.000 q^{88} -194.000i q^{89} +63.0000 q^{90} +1134.00 q^{92} +588.000i q^{93} +462.000 q^{94} -210.000 q^{95} -483.000i q^{96} -1202.00i q^{97} -243.000i q^{98} +198.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 $$2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 + 14 * q^10 + 42 * q^12 + 20 * q^14 + 82 * q^16 - 74 * q^17 + 44 * q^22 + 324 * q^23 + 152 * q^25 + 54 * q^27 - 226 * q^29 + 42 * q^30 - 140 * q^35 + 126 * q^36 + 60 * q^38 + 210 * q^40 + 60 * q^42 + 492 * q^43 + 246 * q^48 + 486 * q^49 - 222 * q^51 - 1074 * q^53 - 308 * q^55 + 300 * q^56 - 1270 * q^61 + 392 * q^62 + 334 * q^64 + 132 * q^66 - 518 * q^68 + 972 * q^69 - 26 * q^74 + 456 * q^75 - 440 * q^77 + 1768 * q^79 + 162 * q^81 + 570 * q^82 - 678 * q^87 + 660 * q^88 + 126 * q^90 + 2268 * q^92 + 924 * q^94 - 420 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.353553i −0.984251 0.176777i $$-0.943433\pi$$
0.984251 0.176777i $$-0.0565670\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 7.00000 0.875000
$$5$$ 7.00000i 0.626099i 0.949737 + 0.313050i $$0.101351\pi$$
−0.949737 + 0.313050i $$0.898649\pi$$
$$6$$ − 3.00000i − 0.204124i
$$7$$ 10.0000i 0.539949i 0.962867 + 0.269975i $$0.0870153\pi$$
−0.962867 + 0.269975i $$0.912985\pi$$
$$8$$ − 15.0000i − 0.662913i
$$9$$ 9.00000 0.333333
$$10$$ 7.00000 0.221359
$$11$$ 22.0000i 0.603023i 0.953463 + 0.301511i $$0.0974911\pi$$
−0.953463 + 0.301511i $$0.902509\pi$$
$$12$$ 21.0000 0.505181
$$13$$ 0 0
$$14$$ 10.0000 0.190901
$$15$$ 21.0000i 0.361478i
$$16$$ 41.0000 0.640625
$$17$$ −37.0000 −0.527872 −0.263936 0.964540i $$-0.585021\pi$$
−0.263936 + 0.964540i $$0.585021\pi$$
$$18$$ − 9.00000i − 0.117851i
$$19$$ 30.0000i 0.362235i 0.983461 + 0.181118i $$0.0579715\pi$$
−0.983461 + 0.181118i $$0.942029\pi$$
$$20$$ 49.0000i 0.547837i
$$21$$ 30.0000i 0.311740i
$$22$$ 22.0000 0.213201
$$23$$ 162.000 1.46867 0.734333 0.678789i $$-0.237495\pi$$
0.734333 + 0.678789i $$0.237495\pi$$
$$24$$ − 45.0000i − 0.382733i
$$25$$ 76.0000 0.608000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 70.0000i 0.472456i
$$29$$ −113.000 −0.723571 −0.361786 0.932261i $$-0.617833\pi$$
−0.361786 + 0.932261i $$0.617833\pi$$
$$30$$ 21.0000 0.127802
$$31$$ 196.000i 1.13557i 0.823177 + 0.567785i $$0.192199\pi$$
−0.823177 + 0.567785i $$0.807801\pi$$
$$32$$ − 161.000i − 0.889408i
$$33$$ 66.0000i 0.348155i
$$34$$ 37.0000i 0.186631i
$$35$$ −70.0000 −0.338062
$$36$$ 63.0000 0.291667
$$37$$ − 13.0000i − 0.0577618i −0.999583 0.0288809i $$-0.990806\pi$$
0.999583 0.0288809i $$-0.00919436\pi$$
$$38$$ 30.0000 0.128070
$$39$$ 0 0
$$40$$ 105.000 0.415049
$$41$$ 285.000i 1.08560i 0.839863 + 0.542799i $$0.182635\pi$$
−0.839863 + 0.542799i $$0.817365\pi$$
$$42$$ 30.0000 0.110217
$$43$$ 246.000 0.872434 0.436217 0.899842i $$-0.356318\pi$$
0.436217 + 0.899842i $$0.356318\pi$$
$$44$$ 154.000i 0.527645i
$$45$$ 63.0000i 0.208700i
$$46$$ − 162.000i − 0.519252i
$$47$$ 462.000i 1.43382i 0.697165 + 0.716911i $$0.254445\pi$$
−0.697165 + 0.716911i $$0.745555\pi$$
$$48$$ 123.000 0.369865
$$49$$ 243.000 0.708455
$$50$$ − 76.0000i − 0.214960i
$$51$$ −111.000 −0.304767
$$52$$ 0 0
$$53$$ −537.000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$54$$ − 27.0000i − 0.0680414i
$$55$$ −154.000 −0.377552
$$56$$ 150.000 0.357939
$$57$$ 90.0000i 0.209137i
$$58$$ 113.000i 0.255821i
$$59$$ − 576.000i − 1.27100i −0.772102 0.635498i $$-0.780795\pi$$
0.772102 0.635498i $$-0.219205\pi$$
$$60$$ 147.000i 0.316294i
$$61$$ −635.000 −1.33284 −0.666421 0.745575i $$-0.732175\pi$$
−0.666421 + 0.745575i $$0.732175\pi$$
$$62$$ 196.000 0.401484
$$63$$ 90.0000i 0.179983i
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ 66.0000 0.123091
$$67$$ 202.000i 0.368332i 0.982895 + 0.184166i $$0.0589584\pi$$
−0.982895 + 0.184166i $$0.941042\pi$$
$$68$$ −259.000 −0.461888
$$69$$ 486.000 0.847935
$$70$$ 70.0000i 0.119523i
$$71$$ − 1086.00i − 1.81527i −0.419755 0.907637i $$-0.637884\pi$$
0.419755 0.907637i $$-0.362116\pi$$
$$72$$ − 135.000i − 0.220971i
$$73$$ 805.000i 1.29066i 0.763904 + 0.645330i $$0.223280\pi$$
−0.763904 + 0.645330i $$0.776720\pi$$
$$74$$ −13.0000 −0.0204219
$$75$$ 228.000 0.351029
$$76$$ 210.000i 0.316956i
$$77$$ −220.000 −0.325602
$$78$$ 0 0
$$79$$ 884.000 1.25896 0.629480 0.777017i $$-0.283268\pi$$
0.629480 + 0.777017i $$0.283268\pi$$
$$80$$ 287.000i 0.401095i
$$81$$ 81.0000 0.111111
$$82$$ 285.000 0.383817
$$83$$ 518.000i 0.685035i 0.939511 + 0.342517i $$0.111280\pi$$
−0.939511 + 0.342517i $$0.888720\pi$$
$$84$$ 210.000i 0.272772i
$$85$$ − 259.000i − 0.330500i
$$86$$ − 246.000i − 0.308452i
$$87$$ −339.000 −0.417754
$$88$$ 330.000 0.399751
$$89$$ − 194.000i − 0.231056i −0.993304 0.115528i $$-0.963144\pi$$
0.993304 0.115528i $$-0.0368560\pi$$
$$90$$ 63.0000 0.0737865
$$91$$ 0 0
$$92$$ 1134.00 1.28508
$$93$$ 588.000i 0.655621i
$$94$$ 462.000 0.506933
$$95$$ −210.000 −0.226795
$$96$$ − 483.000i − 0.513500i
$$97$$ − 1202.00i − 1.25819i −0.777328 0.629096i $$-0.783425\pi$$
0.777328 0.629096i $$-0.216575\pi$$
$$98$$ − 243.000i − 0.250477i
$$99$$ 198.000i 0.201008i
$$100$$ 532.000 0.532000
$$101$$ 429.000 0.422645 0.211322 0.977416i $$-0.432223\pi$$
0.211322 + 0.977416i $$0.432223\pi$$
$$102$$ 111.000i 0.107751i
$$103$$ 1302.00 1.24553 0.622766 0.782408i $$-0.286009\pi$$
0.622766 + 0.782408i $$0.286009\pi$$
$$104$$ 0 0
$$105$$ −210.000 −0.195180
$$106$$ 537.000i 0.492057i
$$107$$ −1338.00 −1.20887 −0.604436 0.796654i $$-0.706602\pi$$
−0.604436 + 0.796654i $$0.706602\pi$$
$$108$$ 189.000 0.168394
$$109$$ − 1034.00i − 0.908617i −0.890844 0.454308i $$-0.849886\pi$$
0.890844 0.454308i $$-0.150114\pi$$
$$110$$ 154.000i 0.133485i
$$111$$ − 39.0000i − 0.0333488i
$$112$$ 410.000i 0.345905i
$$113$$ 1077.00 0.896599 0.448299 0.893884i $$-0.352030\pi$$
0.448299 + 0.893884i $$0.352030\pi$$
$$114$$ 90.0000 0.0739410
$$115$$ 1134.00i 0.919531i
$$116$$ −791.000 −0.633125
$$117$$ 0 0
$$118$$ −576.000 −0.449365
$$119$$ − 370.000i − 0.285024i
$$120$$ 315.000 0.239629
$$121$$ 847.000 0.636364
$$122$$ 635.000i 0.471231i
$$123$$ 855.000i 0.626770i
$$124$$ 1372.00i 0.993623i
$$125$$ 1407.00i 1.00677i
$$126$$ 90.0000 0.0636336
$$127$$ 988.000 0.690321 0.345161 0.938544i $$-0.387824\pi$$
0.345161 + 0.938544i $$0.387824\pi$$
$$128$$ − 1455.00i − 1.00473i
$$129$$ 738.000 0.503700
$$130$$ 0 0
$$131$$ 560.000 0.373492 0.186746 0.982408i $$-0.440206\pi$$
0.186746 + 0.982408i $$0.440206\pi$$
$$132$$ 462.000i 0.304636i
$$133$$ −300.000 −0.195589
$$134$$ 202.000 0.130225
$$135$$ 189.000i 0.120493i
$$136$$ 555.000i 0.349933i
$$137$$ 519.000i 0.323658i 0.986819 + 0.161829i $$0.0517393\pi$$
−0.986819 + 0.161829i $$0.948261\pi$$
$$138$$ − 486.000i − 0.299790i
$$139$$ −348.000 −0.212352 −0.106176 0.994347i $$-0.533861\pi$$
−0.106176 + 0.994347i $$0.533861\pi$$
$$140$$ −490.000 −0.295804
$$141$$ 1386.00i 0.827817i
$$142$$ −1086.00 −0.641796
$$143$$ 0 0
$$144$$ 369.000 0.213542
$$145$$ − 791.000i − 0.453027i
$$146$$ 805.000 0.456317
$$147$$ 729.000 0.409027
$$148$$ − 91.0000i − 0.0505416i
$$149$$ − 645.000i − 0.354634i −0.984154 0.177317i $$-0.943258\pi$$
0.984154 0.177317i $$-0.0567418\pi$$
$$150$$ − 228.000i − 0.124107i
$$151$$ − 2914.00i − 1.57045i −0.619211 0.785225i $$-0.712547\pi$$
0.619211 0.785225i $$-0.287453\pi$$
$$152$$ 450.000 0.240130
$$153$$ −333.000 −0.175957
$$154$$ 220.000i 0.115118i
$$155$$ −1372.00 −0.710979
$$156$$ 0 0
$$157$$ −2079.00 −1.05683 −0.528415 0.848986i $$-0.677213\pi$$
−0.528415 + 0.848986i $$0.677213\pi$$
$$158$$ − 884.000i − 0.445109i
$$159$$ −1611.00 −0.803526
$$160$$ 1127.00 0.556857
$$161$$ 1620.00i 0.793006i
$$162$$ − 81.0000i − 0.0392837i
$$163$$ − 1700.00i − 0.816897i −0.912781 0.408449i $$-0.866070\pi$$
0.912781 0.408449i $$-0.133930\pi$$
$$164$$ 1995.00i 0.949898i
$$165$$ −462.000 −0.217980
$$166$$ 518.000 0.242196
$$167$$ − 3680.00i − 1.70519i −0.522571 0.852596i $$-0.675027\pi$$
0.522571 0.852596i $$-0.324973\pi$$
$$168$$ 450.000 0.206656
$$169$$ 0 0
$$170$$ −259.000 −0.116849
$$171$$ 270.000i 0.120745i
$$172$$ 1722.00 0.763379
$$173$$ −4146.00 −1.82205 −0.911025 0.412352i $$-0.864707\pi$$
−0.911025 + 0.412352i $$0.864707\pi$$
$$174$$ 339.000i 0.147698i
$$175$$ 760.000i 0.328289i
$$176$$ 902.000i 0.386311i
$$177$$ − 1728.00i − 0.733810i
$$178$$ −194.000 −0.0816905
$$179$$ −3674.00 −1.53412 −0.767060 0.641575i $$-0.778281\pi$$
−0.767060 + 0.641575i $$0.778281\pi$$
$$180$$ 441.000i 0.182612i
$$181$$ 3283.00 1.34820 0.674098 0.738642i $$-0.264533\pi$$
0.674098 + 0.738642i $$0.264533\pi$$
$$182$$ 0 0
$$183$$ −1905.00 −0.769517
$$184$$ − 2430.00i − 0.973598i
$$185$$ 91.0000 0.0361646
$$186$$ 588.000 0.231797
$$187$$ − 814.000i − 0.318319i
$$188$$ 3234.00i 1.25459i
$$189$$ 270.000i 0.103913i
$$190$$ 210.000i 0.0801842i
$$191$$ −596.000 −0.225786 −0.112893 0.993607i $$-0.536012\pi$$
−0.112893 + 0.993607i $$0.536012\pi$$
$$192$$ 501.000 0.188315
$$193$$ 393.000i 0.146574i 0.997311 + 0.0732869i $$0.0233489\pi$$
−0.997311 + 0.0732869i $$0.976651\pi$$
$$194$$ −1202.00 −0.444838
$$195$$ 0 0
$$196$$ 1701.00 0.619898
$$197$$ − 3522.00i − 1.27377i −0.770960 0.636884i $$-0.780223\pi$$
0.770960 0.636884i $$-0.219777\pi$$
$$198$$ 198.000 0.0710669
$$199$$ −2018.00 −0.718855 −0.359428 0.933173i $$-0.617028\pi$$
−0.359428 + 0.933173i $$0.617028\pi$$
$$200$$ − 1140.00i − 0.403051i
$$201$$ 606.000i 0.212656i
$$202$$ − 429.000i − 0.149427i
$$203$$ − 1130.00i − 0.390692i
$$204$$ −777.000 −0.266671
$$205$$ −1995.00 −0.679692
$$206$$ − 1302.00i − 0.440362i
$$207$$ 1458.00 0.489556
$$208$$ 0 0
$$209$$ −660.000 −0.218436
$$210$$ 210.000i 0.0690066i
$$211$$ 160.000 0.0522031 0.0261016 0.999659i $$-0.491691\pi$$
0.0261016 + 0.999659i $$0.491691\pi$$
$$212$$ −3759.00 −1.21778
$$213$$ − 3258.00i − 1.04805i
$$214$$ 1338.00i 0.427401i
$$215$$ 1722.00i 0.546230i
$$216$$ − 405.000i − 0.127578i
$$217$$ −1960.00 −0.613150
$$218$$ −1034.00 −0.321245
$$219$$ 2415.00i 0.745162i
$$220$$ −1078.00 −0.330358
$$221$$ 0 0
$$222$$ −39.0000 −0.0117906
$$223$$ 4072.00i 1.22279i 0.791327 + 0.611393i $$0.209391\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$224$$ 1610.00 0.480235
$$225$$ 684.000 0.202667
$$226$$ − 1077.00i − 0.316995i
$$227$$ − 5794.00i − 1.69410i −0.531511 0.847051i $$-0.678376\pi$$
0.531511 0.847051i $$-0.321624\pi$$
$$228$$ 630.000i 0.182995i
$$229$$ − 6482.00i − 1.87049i −0.353999 0.935246i $$-0.615178\pi$$
0.353999 0.935246i $$-0.384822\pi$$
$$230$$ 1134.00 0.325103
$$231$$ −660.000 −0.187986
$$232$$ 1695.00i 0.479665i
$$233$$ −6890.00 −1.93725 −0.968624 0.248530i $$-0.920053\pi$$
−0.968624 + 0.248530i $$0.920053\pi$$
$$234$$ 0 0
$$235$$ −3234.00 −0.897714
$$236$$ − 4032.00i − 1.11212i
$$237$$ 2652.00 0.726860
$$238$$ −370.000 −0.100771
$$239$$ 2466.00i 0.667415i 0.942677 + 0.333708i $$0.108300\pi$$
−0.942677 + 0.333708i $$0.891700\pi$$
$$240$$ 861.000i 0.231572i
$$241$$ 3617.00i 0.966770i 0.875408 + 0.483385i $$0.160593\pi$$
−0.875408 + 0.483385i $$0.839407\pi$$
$$242$$ − 847.000i − 0.224989i
$$243$$ 243.000 0.0641500
$$244$$ −4445.00 −1.16624
$$245$$ 1701.00i 0.443563i
$$246$$ 855.000 0.221597
$$247$$ 0 0
$$248$$ 2940.00 0.752783
$$249$$ 1554.00i 0.395505i
$$250$$ 1407.00 0.355946
$$251$$ −4860.00 −1.22215 −0.611077 0.791571i $$-0.709263\pi$$
−0.611077 + 0.791571i $$0.709263\pi$$
$$252$$ 630.000i 0.157485i
$$253$$ 3564.00i 0.885639i
$$254$$ − 988.000i − 0.244065i
$$255$$ − 777.000i − 0.190814i
$$256$$ −119.000 −0.0290527
$$257$$ −565.000 −0.137135 −0.0685676 0.997646i $$-0.521843\pi$$
−0.0685676 + 0.997646i $$0.521843\pi$$
$$258$$ − 738.000i − 0.178085i
$$259$$ 130.000 0.0311884
$$260$$ 0 0
$$261$$ −1017.00 −0.241190
$$262$$ − 560.000i − 0.132049i
$$263$$ −498.000 −0.116760 −0.0583802 0.998294i $$-0.518594\pi$$
−0.0583802 + 0.998294i $$0.518594\pi$$
$$264$$ 990.000 0.230797
$$265$$ − 3759.00i − 0.871372i
$$266$$ 300.000i 0.0691511i
$$267$$ − 582.000i − 0.133400i
$$268$$ 1414.00i 0.322290i
$$269$$ 5546.00 1.25705 0.628523 0.777791i $$-0.283660\pi$$
0.628523 + 0.777791i $$0.283660\pi$$
$$270$$ 189.000 0.0426006
$$271$$ 2256.00i 0.505691i 0.967507 + 0.252845i $$0.0813664\pi$$
−0.967507 + 0.252845i $$0.918634\pi$$
$$272$$ −1517.00 −0.338168
$$273$$ 0 0
$$274$$ 519.000 0.114430
$$275$$ 1672.00i 0.366638i
$$276$$ 3402.00 0.741943
$$277$$ −2309.00 −0.500846 −0.250423 0.968137i $$-0.580570\pi$$
−0.250423 + 0.968137i $$0.580570\pi$$
$$278$$ 348.000i 0.0750779i
$$279$$ 1764.00i 0.378523i
$$280$$ 1050.00i 0.224105i
$$281$$ − 5833.00i − 1.23832i −0.785265 0.619159i $$-0.787473\pi$$
0.785265 0.619159i $$-0.212527\pi$$
$$282$$ 1386.00 0.292678
$$283$$ −1650.00 −0.346581 −0.173290 0.984871i $$-0.555440\pi$$
−0.173290 + 0.984871i $$0.555440\pi$$
$$284$$ − 7602.00i − 1.58837i
$$285$$ −630.000 −0.130940
$$286$$ 0 0
$$287$$ −2850.00 −0.586168
$$288$$ − 1449.00i − 0.296469i
$$289$$ −3544.00 −0.721352
$$290$$ −791.000 −0.160169
$$291$$ − 3606.00i − 0.726417i
$$292$$ 5635.00i 1.12933i
$$293$$ − 2991.00i − 0.596369i −0.954508 0.298184i $$-0.903619\pi$$
0.954508 0.298184i $$-0.0963811\pi$$
$$294$$ − 729.000i − 0.144613i
$$295$$ 4032.00 0.795770
$$296$$ −195.000 −0.0382910
$$297$$ 594.000i 0.116052i
$$298$$ −645.000 −0.125382
$$299$$ 0 0
$$300$$ 1596.00 0.307150
$$301$$ 2460.00i 0.471070i
$$302$$ −2914.00 −0.555238
$$303$$ 1287.00 0.244014
$$304$$ 1230.00i 0.232057i
$$305$$ − 4445.00i − 0.834492i
$$306$$ 333.000i 0.0622103i
$$307$$ 2422.00i 0.450263i 0.974328 + 0.225132i $$0.0722812\pi$$
−0.974328 + 0.225132i $$0.927719\pi$$
$$308$$ −1540.00 −0.284901
$$309$$ 3906.00 0.719109
$$310$$ 1372.00i 0.251369i
$$311$$ 3402.00 0.620288 0.310144 0.950690i $$-0.399623\pi$$
0.310144 + 0.950690i $$0.399623\pi$$
$$312$$ 0 0
$$313$$ 2310.00 0.417153 0.208577 0.978006i $$-0.433117\pi$$
0.208577 + 0.978006i $$0.433117\pi$$
$$314$$ 2079.00i 0.373646i
$$315$$ −630.000 −0.112687
$$316$$ 6188.00 1.10159
$$317$$ − 257.000i − 0.0455349i −0.999741 0.0227674i $$-0.992752\pi$$
0.999741 0.0227674i $$-0.00724773\pi$$
$$318$$ 1611.00i 0.284089i
$$319$$ − 2486.00i − 0.436330i
$$320$$ 1169.00i 0.204216i
$$321$$ −4014.00 −0.697943
$$322$$ 1620.00 0.280370
$$323$$ − 1110.00i − 0.191214i
$$324$$ 567.000 0.0972222
$$325$$ 0 0
$$326$$ −1700.00 −0.288817
$$327$$ − 3102.00i − 0.524590i
$$328$$ 4275.00 0.719657
$$329$$ −4620.00 −0.774191
$$330$$ 462.000i 0.0770675i
$$331$$ 1028.00i 0.170707i 0.996351 + 0.0853535i $$0.0272019\pi$$
−0.996351 + 0.0853535i $$0.972798\pi$$
$$332$$ 3626.00i 0.599405i
$$333$$ − 117.000i − 0.0192539i
$$334$$ −3680.00 −0.602876
$$335$$ −1414.00 −0.230612
$$336$$ 1230.00i 0.199708i
$$337$$ −2487.00 −0.402005 −0.201002 0.979591i $$-0.564420\pi$$
−0.201002 + 0.979591i $$0.564420\pi$$
$$338$$ 0 0
$$339$$ 3231.00 0.517651
$$340$$ − 1813.00i − 0.289187i
$$341$$ −4312.00 −0.684774
$$342$$ 270.000 0.0426898
$$343$$ 5860.00i 0.922479i
$$344$$ − 3690.00i − 0.578347i
$$345$$ 3402.00i 0.530891i
$$346$$ 4146.00i 0.644192i
$$347$$ −2850.00 −0.440911 −0.220455 0.975397i $$-0.570754\pi$$
−0.220455 + 0.975397i $$0.570754\pi$$
$$348$$ −2373.00 −0.365535
$$349$$ 2018.00i 0.309516i 0.987952 + 0.154758i $$0.0494598\pi$$
−0.987952 + 0.154758i $$0.950540\pi$$
$$350$$ 760.000 0.116068
$$351$$ 0 0
$$352$$ 3542.00 0.536333
$$353$$ − 5287.00i − 0.797163i −0.917133 0.398582i $$-0.869503\pi$$
0.917133 0.398582i $$-0.130497\pi$$
$$354$$ −1728.00 −0.259441
$$355$$ 7602.00 1.13654
$$356$$ − 1358.00i − 0.202174i
$$357$$ − 1110.00i − 0.164559i
$$358$$ 3674.00i 0.542394i
$$359$$ 7278.00i 1.06997i 0.844863 + 0.534983i $$0.179682\pi$$
−0.844863 + 0.534983i $$0.820318\pi$$
$$360$$ 945.000 0.138350
$$361$$ 5959.00 0.868786
$$362$$ − 3283.00i − 0.476659i
$$363$$ 2541.00 0.367405
$$364$$ 0 0
$$365$$ −5635.00 −0.808080
$$366$$ 1905.00i 0.272065i
$$367$$ −4202.00 −0.597664 −0.298832 0.954306i $$-0.596597\pi$$
−0.298832 + 0.954306i $$0.596597\pi$$
$$368$$ 6642.00 0.940865
$$369$$ 2565.00i 0.361866i
$$370$$ − 91.0000i − 0.0127861i
$$371$$ − 5370.00i − 0.751473i
$$372$$ 4116.00i 0.573668i
$$373$$ −1583.00 −0.219744 −0.109872 0.993946i $$-0.535044\pi$$
−0.109872 + 0.993946i $$0.535044\pi$$
$$374$$ −814.000 −0.112543
$$375$$ 4221.00i 0.581257i
$$376$$ 6930.00 0.950499
$$377$$ 0 0
$$378$$ 270.000 0.0367389
$$379$$ − 2052.00i − 0.278111i −0.990285 0.139056i $$-0.955593\pi$$
0.990285 0.139056i $$-0.0444067\pi$$
$$380$$ −1470.00 −0.198446
$$381$$ 2964.00 0.398557
$$382$$ 596.000i 0.0798273i
$$383$$ − 6872.00i − 0.916822i −0.888740 0.458411i $$-0.848419\pi$$
0.888740 0.458411i $$-0.151581\pi$$
$$384$$ − 4365.00i − 0.580079i
$$385$$ − 1540.00i − 0.203859i
$$386$$ 393.000 0.0518217
$$387$$ 2214.00 0.290811
$$388$$ − 8414.00i − 1.10092i
$$389$$ 11653.0 1.51884 0.759422 0.650598i $$-0.225482\pi$$
0.759422 + 0.650598i $$0.225482\pi$$
$$390$$ 0 0
$$391$$ −5994.00 −0.775268
$$392$$ − 3645.00i − 0.469644i
$$393$$ 1680.00 0.215636
$$394$$ −3522.00 −0.450345
$$395$$ 6188.00i 0.788233i
$$396$$ 1386.00i 0.175882i
$$397$$ − 6134.00i − 0.775458i −0.921774 0.387729i $$-0.873260\pi$$
0.921774 0.387729i $$-0.126740\pi$$
$$398$$ 2018.00i 0.254154i
$$399$$ −900.000 −0.112923
$$400$$ 3116.00 0.389500
$$401$$ 10795.0i 1.34433i 0.740401 + 0.672165i $$0.234636\pi$$
−0.740401 + 0.672165i $$0.765364\pi$$
$$402$$ 606.000 0.0751854
$$403$$ 0 0
$$404$$ 3003.00 0.369814
$$405$$ 567.000i 0.0695666i
$$406$$ −1130.00 −0.138130
$$407$$ 286.000 0.0348317
$$408$$ 1665.00i 0.202034i
$$409$$ − 8489.00i − 1.02629i −0.858301 0.513147i $$-0.828480\pi$$
0.858301 0.513147i $$-0.171520\pi$$
$$410$$ 1995.00i 0.240307i
$$411$$ 1557.00i 0.186864i
$$412$$ 9114.00 1.08984
$$413$$ 5760.00 0.686274
$$414$$ − 1458.00i − 0.173084i
$$415$$ −3626.00 −0.428900
$$416$$ 0 0
$$417$$ −1044.00 −0.122602
$$418$$ 660.000i 0.0772288i
$$419$$ 1496.00 0.174426 0.0872129 0.996190i $$-0.472204\pi$$
0.0872129 + 0.996190i $$0.472204\pi$$
$$420$$ −1470.00 −0.170783
$$421$$ − 11695.0i − 1.35387i −0.736043 0.676935i $$-0.763308\pi$$
0.736043 0.676935i $$-0.236692\pi$$
$$422$$ − 160.000i − 0.0184566i
$$423$$ 4158.00i 0.477941i
$$424$$ 8055.00i 0.922607i
$$425$$ −2812.00 −0.320946
$$426$$ −3258.00 −0.370541
$$427$$ − 6350.00i − 0.719668i
$$428$$ −9366.00 −1.05776
$$429$$ 0 0
$$430$$ 1722.00 0.193121
$$431$$ 10590.0i 1.18353i 0.806110 + 0.591766i $$0.201569\pi$$
−0.806110 + 0.591766i $$0.798431\pi$$
$$432$$ 1107.00 0.123288
$$433$$ 13949.0 1.54814 0.774072 0.633098i $$-0.218217\pi$$
0.774072 + 0.633098i $$0.218217\pi$$
$$434$$ 1960.00i 0.216781i
$$435$$ − 2373.00i − 0.261555i
$$436$$ − 7238.00i − 0.795040i
$$437$$ 4860.00i 0.532003i
$$438$$ 2415.00 0.263455
$$439$$ 10726.0 1.16611 0.583057 0.812431i $$-0.301856\pi$$
0.583057 + 0.812431i $$0.301856\pi$$
$$440$$ 2310.00i 0.250284i
$$441$$ 2187.00 0.236152
$$442$$ 0 0
$$443$$ 16228.0 1.74044 0.870221 0.492662i $$-0.163976\pi$$
0.870221 + 0.492662i $$0.163976\pi$$
$$444$$ − 273.000i − 0.0291802i
$$445$$ 1358.00 0.144664
$$446$$ 4072.00 0.432320
$$447$$ − 1935.00i − 0.204748i
$$448$$ 1670.00i 0.176116i
$$449$$ − 7538.00i − 0.792294i −0.918187 0.396147i $$-0.870347\pi$$
0.918187 0.396147i $$-0.129653\pi$$
$$450$$ − 684.000i − 0.0716535i
$$451$$ −6270.00 −0.654640
$$452$$ 7539.00 0.784524
$$453$$ − 8742.00i − 0.906700i
$$454$$ −5794.00 −0.598956
$$455$$ 0 0
$$456$$ 1350.00 0.138639
$$457$$ 15539.0i 1.59056i 0.606245 + 0.795278i $$0.292675\pi$$
−0.606245 + 0.795278i $$0.707325\pi$$
$$458$$ −6482.00 −0.661319
$$459$$ −999.000 −0.101589
$$460$$ 7938.00i 0.804589i
$$461$$ 4811.00i 0.486053i 0.970020 + 0.243027i $$0.0781403\pi$$
−0.970020 + 0.243027i $$0.921860\pi$$
$$462$$ 660.000i 0.0664632i
$$463$$ − 562.000i − 0.0564111i −0.999602 0.0282056i $$-0.991021\pi$$
0.999602 0.0282056i $$-0.00897930\pi$$
$$464$$ −4633.00 −0.463538
$$465$$ −4116.00 −0.410484
$$466$$ 6890.00i 0.684921i
$$467$$ −4914.00 −0.486922 −0.243461 0.969911i $$-0.578283\pi$$
−0.243461 + 0.969911i $$0.578283\pi$$
$$468$$ 0 0
$$469$$ −2020.00 −0.198880
$$470$$ 3234.00i 0.317390i
$$471$$ −6237.00 −0.610161
$$472$$ −8640.00 −0.842560
$$473$$ 5412.00i 0.526097i
$$474$$ − 2652.00i − 0.256984i
$$475$$ 2280.00i 0.220239i
$$476$$ − 2590.00i − 0.249396i
$$477$$ −4833.00 −0.463916
$$478$$ 2466.00 0.235967
$$479$$ 3600.00i 0.343399i 0.985149 + 0.171700i $$0.0549258\pi$$
−0.985149 + 0.171700i $$0.945074\pi$$
$$480$$ 3381.00 0.321502
$$481$$ 0 0
$$482$$ 3617.00 0.341805
$$483$$ 4860.00i 0.457842i
$$484$$ 5929.00 0.556818
$$485$$ 8414.00 0.787753
$$486$$ − 243.000i − 0.0226805i
$$487$$ − 17130.0i − 1.59391i −0.604038 0.796955i $$-0.706443\pi$$
0.604038 0.796955i $$-0.293557\pi$$
$$488$$ 9525.00i 0.883558i
$$489$$ − 5100.00i − 0.471636i
$$490$$ 1701.00 0.156823
$$491$$ 11838.0 1.08807 0.544034 0.839063i $$-0.316896\pi$$
0.544034 + 0.839063i $$0.316896\pi$$
$$492$$ 5985.00i 0.548424i
$$493$$ 4181.00 0.381953
$$494$$ 0 0
$$495$$ −1386.00 −0.125851
$$496$$ 8036.00i 0.727474i
$$497$$ 10860.0 0.980156
$$498$$ 1554.00 0.139832
$$499$$ 8976.00i 0.805252i 0.915364 + 0.402626i $$0.131903\pi$$
−0.915364 + 0.402626i $$0.868097\pi$$
$$500$$ 9849.00i 0.880921i
$$501$$ − 11040.0i − 0.984493i
$$502$$ 4860.00i 0.432096i
$$503$$ 1682.00 0.149099 0.0745494 0.997217i $$-0.476248\pi$$
0.0745494 + 0.997217i $$0.476248\pi$$
$$504$$ 1350.00 0.119313
$$505$$ 3003.00i 0.264617i
$$506$$ 3564.00 0.313121
$$507$$ 0 0
$$508$$ 6916.00 0.604031
$$509$$ 15167.0i 1.32076i 0.750933 + 0.660379i $$0.229604\pi$$
−0.750933 + 0.660379i $$0.770396\pi$$
$$510$$ −777.000 −0.0674630
$$511$$ −8050.00 −0.696890
$$512$$ − 11521.0i − 0.994455i
$$513$$ 810.000i 0.0697122i
$$514$$ 565.000i 0.0484846i
$$515$$ 9114.00i 0.779827i
$$516$$ 5166.00 0.440737
$$517$$ −10164.0 −0.864627
$$518$$ − 130.000i − 0.0110268i
$$519$$ −12438.0 −1.05196
$$520$$ 0 0
$$521$$ −6783.00 −0.570381 −0.285191 0.958471i $$-0.592057\pi$$
−0.285191 + 0.958471i $$0.592057\pi$$
$$522$$ 1017.00i 0.0852737i
$$523$$ −13918.0 −1.16366 −0.581828 0.813312i $$-0.697662\pi$$
−0.581828 + 0.813312i $$0.697662\pi$$
$$524$$ 3920.00 0.326805
$$525$$ 2280.00i 0.189538i
$$526$$ 498.000i 0.0412810i
$$527$$ − 7252.00i − 0.599435i
$$528$$ 2706.00i 0.223037i
$$529$$ 14077.0 1.15698
$$530$$ −3759.00 −0.308076
$$531$$ − 5184.00i − 0.423666i
$$532$$ −2100.00 −0.171140
$$533$$ 0 0
$$534$$ −582.000 −0.0471641
$$535$$ − 9366.00i − 0.756874i
$$536$$ 3030.00 0.244172
$$537$$ −11022.0 −0.885725
$$538$$ − 5546.00i − 0.444433i
$$539$$ 5346.00i 0.427214i
$$540$$ 1323.00i 0.105431i
$$541$$ 1335.00i 0.106093i 0.998592 + 0.0530463i $$0.0168931\pi$$
−0.998592 + 0.0530463i $$0.983107\pi$$
$$542$$ 2256.00 0.178789
$$543$$ 9849.00 0.778381
$$544$$ 5957.00i 0.469493i
$$545$$ 7238.00 0.568884
$$546$$ 0 0
$$547$$ −3806.00 −0.297501 −0.148750 0.988875i $$-0.547525\pi$$
−0.148750 + 0.988875i $$0.547525\pi$$
$$548$$ 3633.00i 0.283201i
$$549$$ −5715.00 −0.444281
$$550$$ 1672.00 0.129626
$$551$$ − 3390.00i − 0.262103i
$$552$$ − 7290.00i − 0.562107i
$$553$$ 8840.00i 0.679774i
$$554$$ 2309.00i 0.177076i
$$555$$ 273.000 0.0208796
$$556$$ −2436.00 −0.185808
$$557$$ 1905.00i 0.144915i 0.997372 + 0.0724573i $$0.0230841\pi$$
−0.997372 + 0.0724573i $$0.976916\pi$$
$$558$$ 1764.00 0.133828
$$559$$ 0 0
$$560$$ −2870.00 −0.216571
$$561$$ − 2442.00i − 0.183781i
$$562$$ −5833.00 −0.437812
$$563$$ 4800.00 0.359318 0.179659 0.983729i $$-0.442501\pi$$
0.179659 + 0.983729i $$0.442501\pi$$
$$564$$ 9702.00i 0.724340i
$$565$$ 7539.00i 0.561359i
$$566$$ 1650.00i 0.122535i
$$567$$ 810.000i 0.0599944i
$$568$$ −16290.0 −1.20337
$$569$$ −14678.0 −1.08143 −0.540715 0.841206i $$-0.681846\pi$$
−0.540715 + 0.841206i $$0.681846\pi$$
$$570$$ 630.000i 0.0462944i
$$571$$ 586.000 0.0429481 0.0214740 0.999769i $$-0.493164\pi$$
0.0214740 + 0.999769i $$0.493164\pi$$
$$572$$ 0 0
$$573$$ −1788.00 −0.130357
$$574$$ 2850.00i 0.207242i
$$575$$ 12312.0 0.892949
$$576$$ 1503.00 0.108724
$$577$$ 8939.00i 0.644949i 0.946578 + 0.322474i $$0.104515\pi$$
−0.946578 + 0.322474i $$0.895485\pi$$
$$578$$ 3544.00i 0.255036i
$$579$$ 1179.00i 0.0846245i
$$580$$ − 5537.00i − 0.396399i
$$581$$ −5180.00 −0.369884
$$582$$ −3606.00 −0.256827
$$583$$ − 11814.0i − 0.839255i
$$584$$ 12075.0 0.855594
$$585$$ 0 0
$$586$$ −2991.00 −0.210848
$$587$$ − 13792.0i − 0.969773i −0.874577 0.484887i $$-0.838861\pi$$
0.874577 0.484887i $$-0.161139\pi$$
$$588$$ 5103.00 0.357898
$$589$$ −5880.00 −0.411343
$$590$$ − 4032.00i − 0.281347i
$$591$$ − 10566.0i − 0.735410i
$$592$$ − 533.000i − 0.0370037i
$$593$$ − 9569.00i − 0.662650i −0.943517 0.331325i $$-0.892504\pi$$
0.943517 0.331325i $$-0.107496\pi$$
$$594$$ 594.000 0.0410305
$$595$$ 2590.00 0.178453
$$596$$ − 4515.00i − 0.310305i
$$597$$ −6054.00 −0.415031
$$598$$ 0 0
$$599$$ −5192.00 −0.354156 −0.177078 0.984197i $$-0.556664\pi$$
−0.177078 + 0.984197i $$0.556664\pi$$
$$600$$ − 3420.00i − 0.232702i
$$601$$ −3677.00 −0.249564 −0.124782 0.992184i $$-0.539823\pi$$
−0.124782 + 0.992184i $$0.539823\pi$$
$$602$$ 2460.00 0.166548
$$603$$ 1818.00i 0.122777i
$$604$$ − 20398.0i − 1.37414i
$$605$$ 5929.00i 0.398427i
$$606$$ − 1287.00i − 0.0862719i
$$607$$ −10960.0 −0.732871 −0.366435 0.930443i $$-0.619422\pi$$
−0.366435 + 0.930443i $$0.619422\pi$$
$$608$$ 4830.00 0.322175
$$609$$ − 3390.00i − 0.225566i
$$610$$ −4445.00 −0.295037
$$611$$ 0 0
$$612$$ −2331.00 −0.153963
$$613$$ − 26027.0i − 1.71488i −0.514585 0.857439i $$-0.672054\pi$$
0.514585 0.857439i $$-0.327946\pi$$
$$614$$ 2422.00 0.159192
$$615$$ −5985.00 −0.392420
$$616$$ 3300.00i 0.215845i
$$617$$ 17681.0i 1.15366i 0.816863 + 0.576832i $$0.195711\pi$$
−0.816863 + 0.576832i $$0.804289\pi$$
$$618$$ − 3906.00i − 0.254243i
$$619$$ − 3192.00i − 0.207265i −0.994616 0.103633i $$-0.966953\pi$$
0.994616 0.103633i $$-0.0330467\pi$$
$$620$$ −9604.00 −0.622106
$$621$$ 4374.00 0.282645
$$622$$ − 3402.00i − 0.219305i
$$623$$ 1940.00 0.124758
$$624$$ 0 0
$$625$$ −349.000 −0.0223360
$$626$$ − 2310.00i − 0.147486i
$$627$$ −1980.00 −0.126114
$$628$$ −14553.0 −0.924726
$$629$$ 481.000i 0.0304908i
$$630$$ 630.000i 0.0398410i
$$631$$ − 7580.00i − 0.478217i −0.970993 0.239109i $$-0.923145\pi$$
0.970993 0.239109i $$-0.0768552\pi$$
$$632$$ − 13260.0i − 0.834580i
$$633$$ 480.000 0.0301395
$$634$$ −257.000 −0.0160990
$$635$$ 6916.00i 0.432210i
$$636$$ −11277.0 −0.703085
$$637$$ 0 0
$$638$$ −2486.00 −0.154266
$$639$$ − 9774.00i − 0.605091i
$$640$$ 10185.0 0.629059
$$641$$ 27707.0 1.70727 0.853635 0.520871i $$-0.174393\pi$$
0.853635 + 0.520871i $$0.174393\pi$$
$$642$$ 4014.00i 0.246760i
$$643$$ 11216.0i 0.687894i 0.938989 + 0.343947i $$0.111764\pi$$
−0.938989 + 0.343947i $$0.888236\pi$$
$$644$$ 11340.0i 0.693880i
$$645$$ 5166.00i 0.315366i
$$646$$ −1110.00 −0.0676043
$$647$$ 2536.00 0.154097 0.0770483 0.997027i $$-0.475450\pi$$
0.0770483 + 0.997027i $$0.475450\pi$$
$$648$$ − 1215.00i − 0.0736570i
$$649$$ 12672.0 0.766440
$$650$$ 0 0
$$651$$ −5880.00 −0.354002
$$652$$ − 11900.0i − 0.714785i
$$653$$ 17730.0 1.06252 0.531262 0.847207i $$-0.321718\pi$$
0.531262 + 0.847207i $$0.321718\pi$$
$$654$$ −3102.00 −0.185471
$$655$$ 3920.00i 0.233843i
$$656$$ 11685.0i 0.695461i
$$657$$ 7245.00i 0.430220i
$$658$$ 4620.00i 0.273718i
$$659$$ 18920.0 1.11839 0.559195 0.829036i $$-0.311110\pi$$
0.559195 + 0.829036i $$0.311110\pi$$
$$660$$ −3234.00 −0.190732
$$661$$ − 5241.00i − 0.308398i −0.988040 0.154199i $$-0.950720\pi$$
0.988040 0.154199i $$-0.0492797\pi$$
$$662$$ 1028.00 0.0603540
$$663$$ 0 0
$$664$$ 7770.00 0.454118
$$665$$ − 2100.00i − 0.122458i
$$666$$ −117.000 −0.00680729
$$667$$ −18306.0 −1.06269
$$668$$ − 25760.0i − 1.49204i
$$669$$ 12216.0i 0.705976i
$$670$$ 1414.00i 0.0815337i
$$671$$ − 13970.0i − 0.803735i
$$672$$ 4830.00 0.277264
$$673$$ −20467.0 −1.17228 −0.586140 0.810210i $$-0.699353\pi$$
−0.586140 + 0.810210i $$0.699353\pi$$
$$674$$ 2487.00i 0.142130i
$$675$$ 2052.00 0.117010
$$676$$ 0 0
$$677$$ −70.0000 −0.00397388 −0.00198694 0.999998i $$-0.500632\pi$$
−0.00198694 + 0.999998i $$0.500632\pi$$
$$678$$ − 3231.00i − 0.183017i
$$679$$ 12020.0 0.679360
$$680$$ −3885.00 −0.219093
$$681$$ − 17382.0i − 0.978091i
$$682$$ 4312.00i 0.242104i
$$683$$ − 6432.00i − 0.360342i −0.983635 0.180171i $$-0.942335\pi$$
0.983635 0.180171i $$-0.0576651\pi$$
$$684$$ 1890.00i 0.105652i
$$685$$ −3633.00 −0.202642
$$686$$ 5860.00 0.326146
$$687$$ − 19446.0i − 1.07993i
$$688$$ 10086.0 0.558903
$$689$$ 0 0
$$690$$ 3402.00 0.187698
$$691$$ − 6666.00i − 0.366985i −0.983021 0.183492i $$-0.941260\pi$$
0.983021 0.183492i $$-0.0587403\pi$$
$$692$$ −29022.0 −1.59429
$$693$$ −1980.00 −0.108534
$$694$$ 2850.00i 0.155885i
$$695$$ − 2436.00i − 0.132954i
$$696$$ 5085.00i 0.276935i
$$697$$ − 10545.0i − 0.573056i
$$698$$ 2018.00 0.109430
$$699$$ −20670.0 −1.11847
$$700$$ 5320.00i 0.287253i
$$701$$ 14054.0 0.757221 0.378611 0.925556i $$-0.376402\pi$$
0.378611 + 0.925556i $$0.376402\pi$$
$$702$$ 0 0
$$703$$ 390.000 0.0209234
$$704$$ 3674.00i 0.196689i
$$705$$ −9702.00 −0.518296
$$706$$ −5287.00 −0.281840
$$707$$ 4290.00i 0.228207i
$$708$$ − 12096.0i − 0.642084i
$$709$$ 71.0000i 0.00376088i 0.999998 + 0.00188044i $$0.000598562\pi$$
−0.999998 + 0.00188044i $$0.999401\pi$$
$$710$$ − 7602.00i − 0.401828i
$$711$$ 7956.00 0.419653
$$712$$ −2910.00 −0.153170
$$713$$ 31752.0i 1.66777i
$$714$$ −1110.00 −0.0581803
$$715$$ 0 0
$$716$$ −25718.0 −1.34236
$$717$$ 7398.00i 0.385332i
$$718$$ 7278.00 0.378290
$$719$$ −3936.00 −0.204156 −0.102078 0.994776i $$-0.532549\pi$$
−0.102078 + 0.994776i $$0.532549\pi$$
$$720$$ 2583.00i 0.133698i
$$721$$ 13020.0i 0.672524i
$$722$$ − 5959.00i − 0.307162i
$$723$$ 10851.0i 0.558165i
$$724$$ 22981.0 1.17967
$$725$$ −8588.00 −0.439931
$$726$$ − 2541.00i − 0.129897i
$$727$$ −34202.0 −1.74482 −0.872409 0.488777i $$-0.837443\pi$$
−0.872409 + 0.488777i $$0.837443\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 5635.00i 0.285700i
$$731$$ −9102.00 −0.460533
$$732$$ −13335.0 −0.673328
$$733$$ − 27363.0i − 1.37882i −0.724371 0.689410i $$-0.757870\pi$$
0.724371 0.689410i $$-0.242130\pi$$
$$734$$ 4202.00i 0.211306i
$$735$$ 5103.00i 0.256091i
$$736$$ − 26082.0i − 1.30624i
$$737$$ −4444.00 −0.222112
$$738$$ 2565.00 0.127939
$$739$$ − 21776.0i − 1.08396i −0.840393 0.541978i $$-0.817676\pi$$
0.840393 0.541978i $$-0.182324\pi$$
$$740$$ 637.000 0.0316440
$$741$$ 0 0
$$742$$ −5370.00 −0.265686
$$743$$ 2484.00i 0.122650i 0.998118 + 0.0613251i $$0.0195326\pi$$
−0.998118 + 0.0613251i $$0.980467\pi$$
$$744$$ 8820.00 0.434619
$$745$$ 4515.00 0.222036
$$746$$ 1583.00i 0.0776914i
$$747$$ 4662.00i 0.228345i
$$748$$ − 5698.00i − 0.278529i
$$749$$ − 13380.0i − 0.652730i
$$750$$ 4221.00 0.205506
$$751$$ −32906.0 −1.59888 −0.799439 0.600748i $$-0.794870\pi$$
−0.799439 + 0.600748i $$0.794870\pi$$
$$752$$ 18942.0i 0.918542i
$$753$$ −14580.0 −0.705611
$$754$$ 0 0
$$755$$ 20398.0 0.983257
$$756$$ 1890.00i 0.0909241i
$$757$$ −3914.00 −0.187922 −0.0939609 0.995576i $$-0.529953\pi$$
−0.0939609 + 0.995576i $$0.529953\pi$$
$$758$$ −2052.00 −0.0983272
$$759$$ 10692.0i 0.511324i
$$760$$ 3150.00i 0.150345i
$$761$$ 33038.0i 1.57375i 0.617110 + 0.786877i $$0.288303\pi$$
−0.617110 + 0.786877i $$0.711697\pi$$
$$762$$ − 2964.00i − 0.140911i
$$763$$ 10340.0 0.490607
$$764$$ −4172.00 −0.197562
$$765$$ − 2331.00i − 0.110167i
$$766$$ −6872.00 −0.324145
$$767$$ 0 0
$$768$$ −357.000 −0.0167736
$$769$$ − 17586.0i − 0.824665i −0.911033 0.412332i $$-0.864714\pi$$
0.911033 0.412332i $$-0.135286\pi$$
$$770$$ −1540.00 −0.0720750
$$771$$ −1695.00 −0.0791750
$$772$$ 2751.00i 0.128252i
$$773$$ 18314.0i 0.852146i 0.904689 + 0.426073i $$0.140103\pi$$
−0.904689 + 0.426073i $$0.859897\pi$$
$$774$$ − 2214.00i − 0.102817i
$$775$$ 14896.0i 0.690426i
$$776$$ −18030.0 −0.834071
$$777$$ 390.000 0.0180067
$$778$$ − 11653.0i − 0.536993i
$$779$$ −8550.00 −0.393242
$$780$$ 0 0
$$781$$ 23892.0 1.09465
$$782$$ 5994.00i 0.274098i
$$783$$ −3051.00 −0.139251
$$784$$ 9963.00 0.453854
$$785$$ − 14553.0i − 0.661680i
$$786$$ − 1680.00i − 0.0762387i
$$787$$ 42068.0i 1.90542i 0.303888 + 0.952708i $$0.401715\pi$$
−0.303888 + 0.952708i $$0.598285\pi$$
$$788$$ − 24654.0i − 1.11455i
$$789$$ −1494.00 −0.0674117
$$790$$ 6188.00 0.278682
$$791$$ 10770.0i 0.484118i
$$792$$ 2970.00 0.133250
$$793$$ 0 0
$$794$$ −6134.00 −0.274166
$$795$$ − 11277.0i − 0.503087i
$$796$$ −14126.0 −0.628998
$$797$$ 4282.00 0.190309 0.0951545 0.995463i $$-0.469665\pi$$
0.0951545 + 0.995463i $$0.469665\pi$$
$$798$$ 900.000i 0.0399244i
$$799$$ − 17094.0i − 0.756874i
$$800$$ − 12236.0i − 0.540760i
$$801$$ − 1746.00i − 0.0770186i
$$802$$ 10795.0 0.475293
$$803$$ −17710.0 −0.778297
$$804$$ 4242.00i 0.186074i
$$805$$ −11340.0 −0.496500
$$806$$ 0 0
$$807$$ 16638.0 0.725756
$$808$$ − 6435.00i − 0.280176i
$$809$$ 40221.0 1.74795 0.873977 0.485967i $$-0.161532\pi$$
0.873977 + 0.485967i $$0.161532\pi$$
$$810$$ 567.000 0.0245955
$$811$$ − 7084.00i − 0.306724i −0.988170 0.153362i $$-0.950990\pi$$
0.988170 0.153362i $$-0.0490100\pi$$
$$812$$ − 7910.00i − 0.341855i
$$813$$ 6768.00i 0.291961i
$$814$$ − 286.000i − 0.0123149i
$$815$$ 11900.0 0.511459
$$816$$ −4551.00 −0.195241
$$817$$ 7380.00i 0.316026i
$$818$$ −8489.00 −0.362850
$$819$$ 0 0
$$820$$ −13965.0 −0.594730
$$821$$ 17338.0i 0.737028i 0.929622 + 0.368514i $$0.120133\pi$$
−0.929622 + 0.368514i $$0.879867\pi$$
$$822$$ 1557.00 0.0660664
$$823$$ −35496.0 −1.50342 −0.751709 0.659495i $$-0.770770\pi$$
−0.751709 + 0.659495i $$0.770770\pi$$
$$824$$ − 19530.0i − 0.825679i
$$825$$ 5016.00i 0.211678i
$$826$$ − 5760.00i − 0.242634i
$$827$$ 14992.0i 0.630378i 0.949029 + 0.315189i $$0.102068\pi$$
−0.949029 + 0.315189i $$0.897932\pi$$
$$828$$ 10206.0 0.428361
$$829$$ 20659.0 0.865521 0.432760 0.901509i $$-0.357540\pi$$
0.432760 + 0.901509i $$0.357540\pi$$
$$830$$ 3626.00i 0.151639i
$$831$$ −6927.00 −0.289164
$$832$$ 0 0
$$833$$ −8991.00 −0.373973
$$834$$ 1044.00i 0.0433462i
$$835$$ 25760.0 1.06762
$$836$$ −4620.00 −0.191132
$$837$$ 5292.00i 0.218540i
$$838$$ − 1496.00i − 0.0616688i
$$839$$ − 28716.0i − 1.18163i −0.806808 0.590814i $$-0.798807\pi$$
0.806808 0.590814i $$-0.201193\pi$$
$$840$$ 3150.00i 0.129387i
$$841$$ −11620.0 −0.476444
$$842$$ −11695.0 −0.478665
$$843$$ − 17499.0i − 0.714944i
$$844$$ 1120.00 0.0456777
$$845$$ 0 0
$$846$$ 4158.00 0.168978
$$847$$ 8470.00i 0.343604i
$$848$$ −22017.0 −0.891588
$$849$$ −4950.00 −0.200098
$$850$$ 2812.00i 0.113472i
$$851$$ − 2106.00i − 0.0848328i
$$852$$ − 22806.0i − 0.917043i
$$853$$ − 13377.0i − 0.536952i −0.963286 0.268476i $$-0.913480\pi$$
0.963286 0.268476i $$-0.0865199\pi$$
$$854$$ −6350.00 −0.254441
$$855$$ −1890.00 −0.0755984
$$856$$ 20070.0i 0.801377i
$$857$$ 27419.0 1.09290 0.546450 0.837492i $$-0.315979\pi$$
0.546450 + 0.837492i $$0.315979\pi$$
$$858$$ 0 0
$$859$$ 2422.00 0.0962021 0.0481010 0.998842i $$-0.484683\pi$$
0.0481010 + 0.998842i $$0.484683\pi$$
$$860$$ 12054.0i 0.477951i
$$861$$ −8550.00 −0.338424
$$862$$ 10590.0 0.418442
$$863$$ − 34522.0i − 1.36169i −0.732425 0.680847i $$-0.761612\pi$$
0.732425 0.680847i $$-0.238388\pi$$
$$864$$ − 4347.00i − 0.171167i
$$865$$ − 29022.0i − 1.14078i
$$866$$ − 13949.0i − 0.547351i
$$867$$ −10632.0 −0.416472
$$868$$ −13720.0 −0.536506
$$869$$ 19448.0i 0.759181i
$$870$$ −2373.00 −0.0924738
$$871$$ 0 0
$$872$$ −15510.0 −0.602334
$$873$$ − 10818.0i − 0.419397i
$$874$$ 4860.00 0.188091
$$875$$ −14070.0 −0.543603
$$876$$ 16905.0i 0.652017i
$$877$$ 13733.0i 0.528769i 0.964417 + 0.264385i $$0.0851688\pi$$
−0.964417 + 0.264385i $$0.914831\pi$$
$$878$$ − 10726.0i − 0.412284i
$$879$$ − 8973.00i − 0.344314i
$$880$$ −6314.00 −0.241869
$$881$$ 22759.0 0.870341 0.435170 0.900348i $$-0.356688\pi$$
0.435170 + 0.900348i $$0.356688\pi$$
$$882$$ − 2187.00i − 0.0834922i
$$883$$ 2168.00 0.0826263 0.0413131 0.999146i $$-0.486846\pi$$
0.0413131 + 0.999146i $$0.486846\pi$$
$$884$$ 0 0
$$885$$ 12096.0 0.459438
$$886$$ − 16228.0i − 0.615339i
$$887$$ 15888.0 0.601428 0.300714 0.953714i $$-0.402775\pi$$
0.300714 + 0.953714i $$0.402775\pi$$
$$888$$ −585.000 −0.0221073
$$889$$ 9880.00i 0.372739i
$$890$$ − 1358.00i − 0.0511464i
$$891$$ 1782.00i 0.0670025i
$$892$$ 28504.0i 1.06994i
$$893$$ −13860.0 −0.519381
$$894$$ −1935.00 −0.0723894
$$895$$ − 25718.0i − 0.960512i
$$896$$ 14550.0 0.542502
$$897$$ 0 0
$$898$$ −7538.00 −0.280118
$$899$$ − 22148.0i − 0.821665i
$$900$$ 4788.00 0.177333
$$901$$ 19869.0 0.734664
$$902$$ 6270.00i 0.231450i
$$903$$ 7380.00i 0.271972i
$$904$$ − 16155.0i − 0.594366i
$$905$$ 22981.0i 0.844104i
$$906$$ −8742.00 −0.320567
$$907$$ 11628.0 0.425691 0.212845 0.977086i $$-0.431727\pi$$
0.212845 + 0.977086i $$0.431727\pi$$
$$908$$ − 40558.0i − 1.48234i
$$909$$ 3861.00 0.140882
$$910$$ 0 0
$$911$$ −12584.0 −0.457658 −0.228829 0.973467i $$-0.573490\pi$$
−0.228829 + 0.973467i $$0.573490\pi$$
$$912$$ 3690.00i 0.133978i
$$913$$ −11396.0 −0.413092
$$914$$ 15539.0 0.562346
$$915$$ − 13335.0i − 0.481794i
$$916$$ − 45374.0i − 1.63668i
$$917$$ 5600.00i 0.201667i
$$918$$ 999.000i 0.0359171i
$$919$$ 17184.0 0.616809 0.308405 0.951255i $$-0.400205\pi$$
0.308405 + 0.951255i $$0.400205\pi$$
$$920$$ 17010.0 0.609569
$$921$$ 7266.00i 0.259960i
$$922$$ 4811.00 0.171846
$$923$$ 0 0
$$924$$ −4620.00 −0.164488
$$925$$ − 988.000i − 0.0351192i
$$926$$ −562.000 −0.0199443
$$927$$ 11718.0 0.415178
$$928$$ 18193.0i 0.643550i
$$929$$ 12777.0i 0.451238i 0.974216 + 0.225619i $$0.0724404\pi$$
−0.974216 + 0.225619i $$0.927560\pi$$
$$930$$ 4116.00i 0.145128i
$$931$$ 7290.00i 0.256627i
$$932$$ −48230.0 −1.69509
$$933$$ 10206.0 0.358124
$$934$$ 4914.00i 0.172153i
$$935$$ 5698.00 0.199299
$$936$$ 0 0
$$937$$ 9191.00 0.320445 0.160222 0.987081i $$-0.448779\pi$$
0.160222 + 0.987081i $$0.448779\pi$$
$$938$$ 2020.00i 0.0703149i
$$939$$ 6930.00 0.240843
$$940$$ −22638.0 −0.785500
$$941$$ 50498.0i 1.74940i 0.484662 + 0.874701i $$0.338942\pi$$
−0.484662 + 0.874701i $$0.661058\pi$$
$$942$$ 6237.00i 0.215724i
$$943$$ 46170.0i 1.59438i
$$944$$ − 23616.0i − 0.814232i
$$945$$ −1890.00 −0.0650600
$$946$$ 5412.00 0.186003
$$947$$ − 1560.00i − 0.0535303i −0.999642 0.0267651i $$-0.991479\pi$$
0.999642 0.0267651i $$-0.00852063\pi$$
$$948$$ 18564.0 0.636003
$$949$$ 0 0
$$950$$ 2280.00 0.0778663
$$951$$ − 771.000i − 0.0262896i
$$952$$ −5550.00 −0.188946
$$953$$ 21498.0 0.730733 0.365366 0.930864i $$-0.380944\pi$$
0.365366 + 0.930864i $$0.380944\pi$$
$$954$$ 4833.00i 0.164019i
$$955$$ − 4172.00i − 0.141364i
$$956$$ 17262.0i 0.583988i
$$957$$ − 7458.00i − 0.251915i
$$958$$ 3600.00 0.121410
$$959$$ −5190.00 −0.174759
$$960$$ 3507.00i 0.117904i
$$961$$ −8625.00 −0.289517
$$962$$ 0 0
$$963$$ −12042.0 −0.402957
$$964$$ 25319.0i 0.845923i
$$965$$ −2751.00 −0.0917698
$$966$$ 4860.00 0.161872
$$967$$ − 418.000i − 0.0139007i −0.999976 0.00695035i $$-0.997788\pi$$
0.999976 0.00695035i $$-0.00221238\pi$$
$$968$$ − 12705.0i − 0.421853i
$$969$$ − 3330.00i − 0.110397i
$$970$$ − 8414.00i − 0.278513i
$$971$$ 18132.0 0.599262 0.299631 0.954055i $$-0.403136\pi$$
0.299631 + 0.954055i $$0.403136\pi$$
$$972$$ 1701.00 0.0561313
$$973$$ − 3480.00i − 0.114659i
$$974$$ −17130.0 −0.563532
$$975$$ 0 0
$$976$$ −26035.0 −0.853853
$$977$$ 12501.0i 0.409358i 0.978829 + 0.204679i $$0.0656150\pi$$
−0.978829 + 0.204679i $$0.934385\pi$$
$$978$$ −5100.00 −0.166748
$$979$$ 4268.00 0.139332
$$980$$ 11907.0i 0.388118i
$$981$$ − 9306.00i − 0.302872i
$$982$$ − 11838.0i − 0.384690i
$$983$$ 43708.0i 1.41818i 0.705119 + 0.709089i $$0.250894\pi$$
−0.705119 + 0.709089i $$0.749106\pi$$
$$984$$ 12825.0 0.415494
$$985$$ 24654.0 0.797504
$$986$$ − 4181.00i − 0.135041i
$$987$$ −13860.0 −0.446979
$$988$$ 0 0
$$989$$ 39852.0 1.28131
$$990$$ 1386.00i 0.0444949i
$$991$$ −39614.0 −1.26981 −0.634904 0.772591i $$-0.718960\pi$$
−0.634904 + 0.772591i $$0.718960\pi$$
$$992$$ 31556.0 1.00998
$$993$$ 3084.00i 0.0985577i
$$994$$ − 10860.0i − 0.346538i
$$995$$ − 14126.0i − 0.450075i
$$996$$ 10878.0i 0.346067i
$$997$$ −36503.0 −1.15954 −0.579770 0.814780i $$-0.696858\pi$$
−0.579770 + 0.814780i $$0.696858\pi$$
$$998$$ 8976.00 0.284700
$$999$$ − 351.000i − 0.0111163i
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 507.4.b.d.337.1 2
13.2 odd 12 39.4.e.b.22.1 yes 2
13.5 odd 4 507.4.a.b.1.1 1
13.6 odd 12 39.4.e.b.16.1 2
13.8 odd 4 507.4.a.d.1.1 1
13.12 even 2 inner 507.4.b.d.337.2 2
39.2 even 12 117.4.g.a.100.1 2
39.5 even 4 1521.4.a.h.1.1 1
39.8 even 4 1521.4.a.e.1.1 1
39.32 even 12 117.4.g.a.55.1 2
52.15 even 12 624.4.q.c.529.1 2
52.19 even 12 624.4.q.c.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.b.16.1 2 13.6 odd 12
39.4.e.b.22.1 yes 2 13.2 odd 12
117.4.g.a.55.1 2 39.32 even 12
117.4.g.a.100.1 2 39.2 even 12
507.4.a.b.1.1 1 13.5 odd 4
507.4.a.d.1.1 1 13.8 odd 4
507.4.b.d.337.1 2 1.1 even 1 trivial
507.4.b.d.337.2 2 13.12 even 2 inner
624.4.q.c.289.1 2 52.19 even 12
624.4.q.c.529.1 2 52.15 even 12
1521.4.a.e.1.1 1 39.8 even 4
1521.4.a.h.1.1 1 39.5 even 4