# Properties

 Label 510.2.a.h.1.2 Level $510$ Weight $2$ Character 510.1 Self dual yes Analytic conductor $4.072$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [510,2,Mod(1,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$510 = 2 \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 510.a (trivial)

## 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: yes Analytic conductor: $$4.07237050309$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 510.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.89898 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -2.89898 q^{13} -4.89898 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -4.89898 q^{21} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.89898 q^{26} -1.00000 q^{27} +4.89898 q^{28} +6.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +4.89898 q^{35} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.89898 q^{39} -1.00000 q^{40} +6.89898 q^{41} +4.89898 q^{42} -0.898979 q^{43} +1.00000 q^{45} +4.00000 q^{46} -9.79796 q^{47} -1.00000 q^{48} +17.0000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -2.89898 q^{52} +11.7980 q^{53} +1.00000 q^{54} -4.89898 q^{56} -4.00000 q^{57} -6.00000 q^{58} -4.89898 q^{59} -1.00000 q^{60} -7.79796 q^{61} -4.00000 q^{62} +4.89898 q^{63} +1.00000 q^{64} -2.89898 q^{65} -8.89898 q^{67} -1.00000 q^{68} +4.00000 q^{69} -4.89898 q^{70} +0.898979 q^{71} -1.00000 q^{72} -1.10102 q^{73} -6.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.89898 q^{78} +13.7980 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.89898 q^{82} -5.79796 q^{83} -4.89898 q^{84} -1.00000 q^{85} +0.898979 q^{86} -6.00000 q^{87} +11.7980 q^{89} -1.00000 q^{90} -14.2020 q^{91} -4.00000 q^{92} -4.00000 q^{93} +9.79796 q^{94} +4.00000 q^{95} +1.00000 q^{96} +16.6969 q^{97} -17.0000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 8 q^{19} + 2 q^{20} - 8 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} - 2 q^{27} + 12 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} + 2 q^{34} + 2 q^{36} + 12 q^{37} - 8 q^{38} - 4 q^{39} - 2 q^{40} + 4 q^{41} + 8 q^{43} + 2 q^{45} + 8 q^{46} - 2 q^{48} + 34 q^{49} - 2 q^{50} + 2 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 8 q^{57} - 12 q^{58} - 2 q^{60} + 4 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 2 q^{68} + 8 q^{69} - 8 q^{71} - 2 q^{72} - 12 q^{73} - 12 q^{74} - 2 q^{75} + 8 q^{76} + 4 q^{78} + 8 q^{79} + 2 q^{80} + 2 q^{81} - 4 q^{82} + 8 q^{83} - 2 q^{85} - 8 q^{86} - 12 q^{87} + 4 q^{89} - 2 q^{90} - 48 q^{91} - 8 q^{92} - 8 q^{93} + 8 q^{95} + 2 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 4 * q^13 - 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 8 * q^19 + 2 * q^20 - 8 * q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 - 2 * q^27 + 12 * q^29 + 2 * q^30 + 8 * q^31 - 2 * q^32 + 2 * q^34 + 2 * q^36 + 12 * q^37 - 8 * q^38 - 4 * q^39 - 2 * q^40 + 4 * q^41 + 8 * q^43 + 2 * q^45 + 8 * q^46 - 2 * q^48 + 34 * q^49 - 2 * q^50 + 2 * q^51 + 4 * q^52 + 4 * q^53 + 2 * q^54 - 8 * q^57 - 12 * q^58 - 2 * q^60 + 4 * q^61 - 8 * q^62 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 2 * q^68 + 8 * q^69 - 8 * q^71 - 2 * q^72 - 12 * q^73 - 12 * q^74 - 2 * q^75 + 8 * q^76 + 4 * q^78 + 8 * q^79 + 2 * q^80 + 2 * q^81 - 4 * q^82 + 8 * q^83 - 2 * q^85 - 8 * q^86 - 12 * q^87 + 4 * q^89 - 2 * q^90 - 48 * q^91 - 8 * q^92 - 8 * q^93 + 8 * q^95 + 2 * q^96 + 4 * q^97 - 34 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 4.89898 1.85164 0.925820 0.377964i $$-0.123376\pi$$
0.925820 + 0.377964i $$0.123376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −2.89898 −0.804032 −0.402016 0.915633i $$-0.631690\pi$$
−0.402016 + 0.915633i $$0.631690\pi$$
$$14$$ −4.89898 −1.30931
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −4.89898 −1.06904
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 2.89898 0.568537
$$27$$ −1.00000 −0.192450
$$28$$ 4.89898 0.925820
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 4.89898 0.828079
$$36$$ 1.00000 0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 2.89898 0.464208
$$40$$ −1.00000 −0.158114
$$41$$ 6.89898 1.07744 0.538720 0.842485i $$-0.318908\pi$$
0.538720 + 0.842485i $$0.318908\pi$$
$$42$$ 4.89898 0.755929
$$43$$ −0.898979 −0.137093 −0.0685465 0.997648i $$-0.521836\pi$$
−0.0685465 + 0.997648i $$0.521836\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 4.00000 0.589768
$$47$$ −9.79796 −1.42918 −0.714590 0.699544i $$-0.753387\pi$$
−0.714590 + 0.699544i $$0.753387\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 17.0000 2.42857
$$50$$ −1.00000 −0.141421
$$51$$ 1.00000 0.140028
$$52$$ −2.89898 −0.402016
$$53$$ 11.7980 1.62057 0.810287 0.586033i $$-0.199311\pi$$
0.810287 + 0.586033i $$0.199311\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.89898 −0.654654
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −7.79796 −0.998426 −0.499213 0.866479i $$-0.666378\pi$$
−0.499213 + 0.866479i $$0.666378\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 4.89898 0.617213
$$64$$ 1.00000 0.125000
$$65$$ −2.89898 −0.359574
$$66$$ 0 0
$$67$$ −8.89898 −1.08718 −0.543592 0.839350i $$-0.682936\pi$$
−0.543592 + 0.839350i $$0.682936\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 4.00000 0.481543
$$70$$ −4.89898 −0.585540
$$71$$ 0.898979 0.106689 0.0533446 0.998576i $$-0.483012\pi$$
0.0533446 + 0.998576i $$0.483012\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −1.10102 −0.128865 −0.0644324 0.997922i $$-0.520524\pi$$
−0.0644324 + 0.997922i $$0.520524\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −2.89898 −0.328245
$$79$$ 13.7980 1.55239 0.776196 0.630492i $$-0.217147\pi$$
0.776196 + 0.630492i $$0.217147\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.89898 −0.761865
$$83$$ −5.79796 −0.636409 −0.318204 0.948022i $$-0.603080\pi$$
−0.318204 + 0.948022i $$0.603080\pi$$
$$84$$ −4.89898 −0.534522
$$85$$ −1.00000 −0.108465
$$86$$ 0.898979 0.0969395
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 11.7980 1.25058 0.625291 0.780392i $$-0.284980\pi$$
0.625291 + 0.780392i $$0.284980\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −14.2020 −1.48878
$$92$$ −4.00000 −0.417029
$$93$$ −4.00000 −0.414781
$$94$$ 9.79796 1.01058
$$95$$ 4.00000 0.410391
$$96$$ 1.00000 0.102062
$$97$$ 16.6969 1.69532 0.847659 0.530542i $$-0.178012\pi$$
0.847659 + 0.530542i $$0.178012\pi$$
$$98$$ −17.0000 −1.71726
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −9.10102 −0.905585 −0.452793 0.891616i $$-0.649572\pi$$
−0.452793 + 0.891616i $$0.649572\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 2.89898 0.284268
$$105$$ −4.89898 −0.478091
$$106$$ −11.7980 −1.14592
$$107$$ −13.7980 −1.33390 −0.666950 0.745103i $$-0.732400\pi$$
−0.666950 + 0.745103i $$0.732400\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −7.79796 −0.746909 −0.373455 0.927648i $$-0.621827\pi$$
−0.373455 + 0.927648i $$0.621827\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 4.89898 0.462910
$$113$$ −11.7980 −1.10986 −0.554929 0.831898i $$-0.687255\pi$$
−0.554929 + 0.831898i $$0.687255\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −4.00000 −0.373002
$$116$$ 6.00000 0.557086
$$117$$ −2.89898 −0.268011
$$118$$ 4.89898 0.450988
$$119$$ −4.89898 −0.449089
$$120$$ 1.00000 0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ 7.79796 0.705994
$$123$$ −6.89898 −0.622060
$$124$$ 4.00000 0.359211
$$125$$ 1.00000 0.0894427
$$126$$ −4.89898 −0.436436
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0.898979 0.0791507
$$130$$ 2.89898 0.254257
$$131$$ −9.79796 −0.856052 −0.428026 0.903767i $$-0.640791\pi$$
−0.428026 + 0.903767i $$0.640791\pi$$
$$132$$ 0 0
$$133$$ 19.5959 1.69918
$$134$$ 8.89898 0.768755
$$135$$ −1.00000 −0.0860663
$$136$$ 1.00000 0.0857493
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −5.79796 −0.491776 −0.245888 0.969298i $$-0.579080\pi$$
−0.245888 + 0.969298i $$0.579080\pi$$
$$140$$ 4.89898 0.414039
$$141$$ 9.79796 0.825137
$$142$$ −0.898979 −0.0754407
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ 1.10102 0.0911211
$$147$$ −17.0000 −1.40214
$$148$$ 6.00000 0.493197
$$149$$ −9.10102 −0.745585 −0.372792 0.927915i $$-0.621600\pi$$
−0.372792 + 0.927915i $$0.621600\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 9.79796 0.797347 0.398673 0.917093i $$-0.369471\pi$$
0.398673 + 0.917093i $$0.369471\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 2.89898 0.232104
$$157$$ −10.8990 −0.869833 −0.434917 0.900471i $$-0.643222\pi$$
−0.434917 + 0.900471i $$0.643222\pi$$
$$158$$ −13.7980 −1.09771
$$159$$ −11.7980 −0.935639
$$160$$ −1.00000 −0.0790569
$$161$$ −19.5959 −1.54437
$$162$$ −1.00000 −0.0785674
$$163$$ −21.7980 −1.70735 −0.853674 0.520808i $$-0.825631\pi$$
−0.853674 + 0.520808i $$0.825631\pi$$
$$164$$ 6.89898 0.538720
$$165$$ 0 0
$$166$$ 5.79796 0.450009
$$167$$ 21.7980 1.68678 0.843388 0.537304i $$-0.180557\pi$$
0.843388 + 0.537304i $$0.180557\pi$$
$$168$$ 4.89898 0.377964
$$169$$ −4.59592 −0.353532
$$170$$ 1.00000 0.0766965
$$171$$ 4.00000 0.305888
$$172$$ −0.898979 −0.0685465
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 4.89898 0.370328
$$176$$ 0 0
$$177$$ 4.89898 0.368230
$$178$$ −11.7980 −0.884294
$$179$$ 4.89898 0.366167 0.183083 0.983097i $$-0.441392\pi$$
0.183083 + 0.983097i $$0.441392\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −23.7980 −1.76889 −0.884444 0.466646i $$-0.845462\pi$$
−0.884444 + 0.466646i $$0.845462\pi$$
$$182$$ 14.2020 1.05273
$$183$$ 7.79796 0.576442
$$184$$ 4.00000 0.294884
$$185$$ 6.00000 0.441129
$$186$$ 4.00000 0.293294
$$187$$ 0 0
$$188$$ −9.79796 −0.714590
$$189$$ −4.89898 −0.356348
$$190$$ −4.00000 −0.290191
$$191$$ 9.79796 0.708955 0.354478 0.935064i $$-0.384659\pi$$
0.354478 + 0.935064i $$0.384659\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −10.8990 −0.784526 −0.392263 0.919853i $$-0.628308\pi$$
−0.392263 + 0.919853i $$0.628308\pi$$
$$194$$ −16.6969 −1.19877
$$195$$ 2.89898 0.207600
$$196$$ 17.0000 1.21429
$$197$$ 25.5959 1.82363 0.911817 0.410597i $$-0.134680\pi$$
0.911817 + 0.410597i $$0.134680\pi$$
$$198$$ 0 0
$$199$$ −23.5959 −1.67267 −0.836335 0.548219i $$-0.815306\pi$$
−0.836335 + 0.548219i $$0.815306\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 8.89898 0.627686
$$202$$ 9.10102 0.640346
$$203$$ 29.3939 2.06305
$$204$$ 1.00000 0.0700140
$$205$$ 6.89898 0.481846
$$206$$ −4.00000 −0.278693
$$207$$ −4.00000 −0.278019
$$208$$ −2.89898 −0.201008
$$209$$ 0 0
$$210$$ 4.89898 0.338062
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 11.7980 0.810287
$$213$$ −0.898979 −0.0615971
$$214$$ 13.7980 0.943209
$$215$$ −0.898979 −0.0613099
$$216$$ 1.00000 0.0680414
$$217$$ 19.5959 1.33026
$$218$$ 7.79796 0.528144
$$219$$ 1.10102 0.0744001
$$220$$ 0 0
$$221$$ 2.89898 0.195006
$$222$$ 6.00000 0.402694
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ −4.89898 −0.327327
$$225$$ 1.00000 0.0666667
$$226$$ 11.7980 0.784789
$$227$$ −2.20204 −0.146155 −0.0730773 0.997326i $$-0.523282\pi$$
−0.0730773 + 0.997326i $$0.523282\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ 25.5959 1.69143 0.845713 0.533638i $$-0.179176\pi$$
0.845713 + 0.533638i $$0.179176\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 2.89898 0.189512
$$235$$ −9.79796 −0.639148
$$236$$ −4.89898 −0.318896
$$237$$ −13.7980 −0.896274
$$238$$ 4.89898 0.317554
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −25.5959 −1.64878 −0.824389 0.566024i $$-0.808481\pi$$
−0.824389 + 0.566024i $$0.808481\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ −7.79796 −0.499213
$$245$$ 17.0000 1.08609
$$246$$ 6.89898 0.439863
$$247$$ −11.5959 −0.737831
$$248$$ −4.00000 −0.254000
$$249$$ 5.79796 0.367431
$$250$$ −1.00000 −0.0632456
$$251$$ −4.89898 −0.309221 −0.154610 0.987976i $$-0.549412\pi$$
−0.154610 + 0.987976i $$0.549412\pi$$
$$252$$ 4.89898 0.308607
$$253$$ 0 0
$$254$$ 12.0000 0.752947
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −0.898979 −0.0559680
$$259$$ 29.3939 1.82645
$$260$$ −2.89898 −0.179787
$$261$$ 6.00000 0.371391
$$262$$ 9.79796 0.605320
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 11.7980 0.724743
$$266$$ −19.5959 −1.20150
$$267$$ −11.7980 −0.722023
$$268$$ −8.89898 −0.543592
$$269$$ −29.5959 −1.80449 −0.902247 0.431219i $$-0.858084\pi$$
−0.902247 + 0.431219i $$0.858084\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 17.7980 1.08115 0.540575 0.841296i $$-0.318207\pi$$
0.540575 + 0.841296i $$0.318207\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 14.2020 0.859547
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ −11.7980 −0.708871 −0.354435 0.935081i $$-0.615327\pi$$
−0.354435 + 0.935081i $$0.615327\pi$$
$$278$$ 5.79796 0.347738
$$279$$ 4.00000 0.239474
$$280$$ −4.89898 −0.292770
$$281$$ 8.20204 0.489293 0.244646 0.969612i $$-0.421328\pi$$
0.244646 + 0.969612i $$0.421328\pi$$
$$282$$ −9.79796 −0.583460
$$283$$ −15.5959 −0.927081 −0.463541 0.886076i $$-0.653421\pi$$
−0.463541 + 0.886076i $$0.653421\pi$$
$$284$$ 0.898979 0.0533446
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ 33.7980 1.99503
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ −6.00000 −0.352332
$$291$$ −16.6969 −0.978792
$$292$$ −1.10102 −0.0644324
$$293$$ −25.5959 −1.49533 −0.747665 0.664076i $$-0.768825\pi$$
−0.747665 + 0.664076i $$0.768825\pi$$
$$294$$ 17.0000 0.991460
$$295$$ −4.89898 −0.285230
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ 9.10102 0.527208
$$299$$ 11.5959 0.670609
$$300$$ −1.00000 −0.0577350
$$301$$ −4.40408 −0.253847
$$302$$ −9.79796 −0.563809
$$303$$ 9.10102 0.522840
$$304$$ 4.00000 0.229416
$$305$$ −7.79796 −0.446510
$$306$$ 1.00000 0.0571662
$$307$$ −8.89898 −0.507892 −0.253946 0.967218i $$-0.581728\pi$$
−0.253946 + 0.967218i $$0.581728\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ −4.00000 −0.227185
$$311$$ −16.8990 −0.958253 −0.479127 0.877746i $$-0.659047\pi$$
−0.479127 + 0.877746i $$0.659047\pi$$
$$312$$ −2.89898 −0.164122
$$313$$ −2.89898 −0.163860 −0.0819300 0.996638i $$-0.526108\pi$$
−0.0819300 + 0.996638i $$0.526108\pi$$
$$314$$ 10.8990 0.615065
$$315$$ 4.89898 0.276026
$$316$$ 13.7980 0.776196
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 11.7980 0.661597
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 13.7980 0.770127
$$322$$ 19.5959 1.09204
$$323$$ −4.00000 −0.222566
$$324$$ 1.00000 0.0555556
$$325$$ −2.89898 −0.160806
$$326$$ 21.7980 1.20728
$$327$$ 7.79796 0.431228
$$328$$ −6.89898 −0.380932
$$329$$ −48.0000 −2.64633
$$330$$ 0 0
$$331$$ 13.7980 0.758404 0.379202 0.925314i $$-0.376198\pi$$
0.379202 + 0.925314i $$0.376198\pi$$
$$332$$ −5.79796 −0.318204
$$333$$ 6.00000 0.328798
$$334$$ −21.7980 −1.19273
$$335$$ −8.89898 −0.486203
$$336$$ −4.89898 −0.267261
$$337$$ 3.30306 0.179929 0.0899646 0.995945i $$-0.471325\pi$$
0.0899646 + 0.995945i $$0.471325\pi$$
$$338$$ 4.59592 0.249985
$$339$$ 11.7980 0.640777
$$340$$ −1.00000 −0.0542326
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 48.9898 2.64520
$$344$$ 0.898979 0.0484697
$$345$$ 4.00000 0.215353
$$346$$ −6.00000 −0.322562
$$347$$ 2.20204 0.118212 0.0591059 0.998252i $$-0.481175\pi$$
0.0591059 + 0.998252i $$0.481175\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 23.7980 1.27388 0.636938 0.770915i $$-0.280201\pi$$
0.636938 + 0.770915i $$0.280201\pi$$
$$350$$ −4.89898 −0.261861
$$351$$ 2.89898 0.154736
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ −4.89898 −0.260378
$$355$$ 0.898979 0.0477129
$$356$$ 11.7980 0.625291
$$357$$ 4.89898 0.259281
$$358$$ −4.89898 −0.258919
$$359$$ −21.3939 −1.12913 −0.564563 0.825390i $$-0.690955\pi$$
−0.564563 + 0.825390i $$0.690955\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 23.7980 1.25079
$$363$$ 11.0000 0.577350
$$364$$ −14.2020 −0.744389
$$365$$ −1.10102 −0.0576300
$$366$$ −7.79796 −0.407606
$$367$$ −36.8990 −1.92611 −0.963056 0.269303i $$-0.913207\pi$$
−0.963056 + 0.269303i $$0.913207\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 6.89898 0.359147
$$370$$ −6.00000 −0.311925
$$371$$ 57.7980 3.00072
$$372$$ −4.00000 −0.207390
$$373$$ −4.69694 −0.243198 −0.121599 0.992579i $$-0.538802\pi$$
−0.121599 + 0.992579i $$0.538802\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 9.79796 0.505291
$$377$$ −17.3939 −0.895830
$$378$$ 4.89898 0.251976
$$379$$ −10.2020 −0.524044 −0.262022 0.965062i $$-0.584389\pi$$
−0.262022 + 0.965062i $$0.584389\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 12.0000 0.614779
$$382$$ −9.79796 −0.501307
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 10.8990 0.554743
$$387$$ −0.898979 −0.0456977
$$388$$ 16.6969 0.847659
$$389$$ 10.4949 0.532112 0.266056 0.963958i $$-0.414279\pi$$
0.266056 + 0.963958i $$0.414279\pi$$
$$390$$ −2.89898 −0.146796
$$391$$ 4.00000 0.202289
$$392$$ −17.0000 −0.858630
$$393$$ 9.79796 0.494242
$$394$$ −25.5959 −1.28950
$$395$$ 13.7980 0.694251
$$396$$ 0 0
$$397$$ −37.5959 −1.88689 −0.943443 0.331536i $$-0.892433\pi$$
−0.943443 + 0.331536i $$0.892433\pi$$
$$398$$ 23.5959 1.18276
$$399$$ −19.5959 −0.981023
$$400$$ 1.00000 0.0500000
$$401$$ 13.1010 0.654234 0.327117 0.944984i $$-0.393923\pi$$
0.327117 + 0.944984i $$0.393923\pi$$
$$402$$ −8.89898 −0.443841
$$403$$ −11.5959 −0.577634
$$404$$ −9.10102 −0.452793
$$405$$ 1.00000 0.0496904
$$406$$ −29.3939 −1.45879
$$407$$ 0 0
$$408$$ −1.00000 −0.0495074
$$409$$ 21.5959 1.06785 0.533925 0.845532i $$-0.320717\pi$$
0.533925 + 0.845532i $$0.320717\pi$$
$$410$$ −6.89898 −0.340716
$$411$$ −14.0000 −0.690569
$$412$$ 4.00000 0.197066
$$413$$ −24.0000 −1.18096
$$414$$ 4.00000 0.196589
$$415$$ −5.79796 −0.284611
$$416$$ 2.89898 0.142134
$$417$$ 5.79796 0.283927
$$418$$ 0 0
$$419$$ 1.79796 0.0878360 0.0439180 0.999035i $$-0.486016\pi$$
0.0439180 + 0.999035i $$0.486016\pi$$
$$420$$ −4.89898 −0.239046
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 12.0000 0.584151
$$423$$ −9.79796 −0.476393
$$424$$ −11.7980 −0.572960
$$425$$ −1.00000 −0.0485071
$$426$$ 0.898979 0.0435557
$$427$$ −38.2020 −1.84873
$$428$$ −13.7980 −0.666950
$$429$$ 0 0
$$430$$ 0.898979 0.0433526
$$431$$ −32.8990 −1.58469 −0.792344 0.610075i $$-0.791139\pi$$
−0.792344 + 0.610075i $$0.791139\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 23.3939 1.12424 0.562119 0.827056i $$-0.309986\pi$$
0.562119 + 0.827056i $$0.309986\pi$$
$$434$$ −19.5959 −0.940634
$$435$$ −6.00000 −0.287678
$$436$$ −7.79796 −0.373455
$$437$$ −16.0000 −0.765384
$$438$$ −1.10102 −0.0526088
$$439$$ −13.7980 −0.658541 −0.329270 0.944236i $$-0.606803\pi$$
−0.329270 + 0.944236i $$0.606803\pi$$
$$440$$ 0 0
$$441$$ 17.0000 0.809524
$$442$$ −2.89898 −0.137890
$$443$$ 18.2020 0.864805 0.432403 0.901681i $$-0.357666\pi$$
0.432403 + 0.901681i $$0.357666\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 11.7980 0.559277
$$446$$ 4.00000 0.189405
$$447$$ 9.10102 0.430463
$$448$$ 4.89898 0.231455
$$449$$ −2.89898 −0.136811 −0.0684057 0.997658i $$-0.521791\pi$$
−0.0684057 + 0.997658i $$0.521791\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −11.7980 −0.554929
$$453$$ −9.79796 −0.460348
$$454$$ 2.20204 0.103347
$$455$$ −14.2020 −0.665802
$$456$$ 4.00000 0.187317
$$457$$ 35.7980 1.67456 0.837279 0.546776i $$-0.184145\pi$$
0.837279 + 0.546776i $$0.184145\pi$$
$$458$$ −25.5959 −1.19602
$$459$$ 1.00000 0.0466760
$$460$$ −4.00000 −0.186501
$$461$$ 0.696938 0.0324597 0.0162298 0.999868i $$-0.494834\pi$$
0.0162298 + 0.999868i $$0.494834\pi$$
$$462$$ 0 0
$$463$$ −31.5959 −1.46839 −0.734193 0.678940i $$-0.762439\pi$$
−0.734193 + 0.678940i $$0.762439\pi$$
$$464$$ 6.00000 0.278543
$$465$$ −4.00000 −0.185496
$$466$$ −14.0000 −0.648537
$$467$$ 31.5959 1.46208 0.731042 0.682332i $$-0.239034\pi$$
0.731042 + 0.682332i $$0.239034\pi$$
$$468$$ −2.89898 −0.134005
$$469$$ −43.5959 −2.01307
$$470$$ 9.79796 0.451946
$$471$$ 10.8990 0.502198
$$472$$ 4.89898 0.225494
$$473$$ 0 0
$$474$$ 13.7980 0.633761
$$475$$ 4.00000 0.183533
$$476$$ −4.89898 −0.224544
$$477$$ 11.7980 0.540191
$$478$$ 0 0
$$479$$ −23.1010 −1.05551 −0.527756 0.849396i $$-0.676967\pi$$
−0.527756 + 0.849396i $$0.676967\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −17.3939 −0.793093
$$482$$ 25.5959 1.16586
$$483$$ 19.5959 0.891645
$$484$$ −11.0000 −0.500000
$$485$$ 16.6969 0.758169
$$486$$ 1.00000 0.0453609
$$487$$ −11.1010 −0.503035 −0.251518 0.967853i $$-0.580930\pi$$
−0.251518 + 0.967853i $$0.580930\pi$$
$$488$$ 7.79796 0.352997
$$489$$ 21.7980 0.985738
$$490$$ −17.0000 −0.767982
$$491$$ −30.6969 −1.38533 −0.692667 0.721258i $$-0.743564\pi$$
−0.692667 + 0.721258i $$0.743564\pi$$
$$492$$ −6.89898 −0.311030
$$493$$ −6.00000 −0.270226
$$494$$ 11.5959 0.521725
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 4.40408 0.197550
$$498$$ −5.79796 −0.259813
$$499$$ −0.404082 −0.0180892 −0.00904460 0.999959i $$-0.502879\pi$$
−0.00904460 + 0.999959i $$0.502879\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −21.7980 −0.973861
$$502$$ 4.89898 0.218652
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ −4.89898 −0.218218
$$505$$ −9.10102 −0.404990
$$506$$ 0 0
$$507$$ 4.59592 0.204112
$$508$$ −12.0000 −0.532414
$$509$$ −44.6969 −1.98116 −0.990578 0.136946i $$-0.956271\pi$$
−0.990578 + 0.136946i $$0.956271\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ −5.39388 −0.238611
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 2.00000 0.0882162
$$515$$ 4.00000 0.176261
$$516$$ 0.898979 0.0395754
$$517$$ 0 0
$$518$$ −29.3939 −1.29149
$$519$$ −6.00000 −0.263371
$$520$$ 2.89898 0.127129
$$521$$ 28.2929 1.23953 0.619766 0.784786i $$-0.287227\pi$$
0.619766 + 0.784786i $$0.287227\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −10.6969 −0.467744 −0.233872 0.972267i $$-0.575140\pi$$
−0.233872 + 0.972267i $$0.575140\pi$$
$$524$$ −9.79796 −0.428026
$$525$$ −4.89898 −0.213809
$$526$$ −8.00000 −0.348817
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −11.7980 −0.512471
$$531$$ −4.89898 −0.212598
$$532$$ 19.5959 0.849591
$$533$$ −20.0000 −0.866296
$$534$$ 11.7980 0.510548
$$535$$ −13.7980 −0.596538
$$536$$ 8.89898 0.384377
$$537$$ −4.89898 −0.211407
$$538$$ 29.5959 1.27597
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ 11.7980 0.507234 0.253617 0.967305i $$-0.418380\pi$$
0.253617 + 0.967305i $$0.418380\pi$$
$$542$$ −17.7980 −0.764488
$$543$$ 23.7980 1.02127
$$544$$ 1.00000 0.0428746
$$545$$ −7.79796 −0.334028
$$546$$ −14.2020 −0.607791
$$547$$ 39.5959 1.69300 0.846500 0.532389i $$-0.178706\pi$$
0.846500 + 0.532389i $$0.178706\pi$$
$$548$$ 14.0000 0.598050
$$549$$ −7.79796 −0.332809
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ −4.00000 −0.170251
$$553$$ 67.5959 2.87447
$$554$$ 11.7980 0.501247
$$555$$ −6.00000 −0.254686
$$556$$ −5.79796 −0.245888
$$557$$ 0.202041 0.00856075 0.00428038 0.999991i $$-0.498638\pi$$
0.00428038 + 0.999991i $$0.498638\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 2.60612 0.110227
$$560$$ 4.89898 0.207020
$$561$$ 0 0
$$562$$ −8.20204 −0.345982
$$563$$ 2.20204 0.0928050 0.0464025 0.998923i $$-0.485224\pi$$
0.0464025 + 0.998923i $$0.485224\pi$$
$$564$$ 9.79796 0.412568
$$565$$ −11.7980 −0.496344
$$566$$ 15.5959 0.655545
$$567$$ 4.89898 0.205738
$$568$$ −0.898979 −0.0377203
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 10.2020 0.426942 0.213471 0.976949i $$-0.431523\pi$$
0.213471 + 0.976949i $$0.431523\pi$$
$$572$$ 0 0
$$573$$ −9.79796 −0.409316
$$574$$ −33.7980 −1.41070
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 31.3939 1.30694 0.653472 0.756951i $$-0.273312\pi$$
0.653472 + 0.756951i $$0.273312\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 10.8990 0.452946
$$580$$ 6.00000 0.249136
$$581$$ −28.4041 −1.17840
$$582$$ 16.6969 0.692110
$$583$$ 0 0
$$584$$ 1.10102 0.0455606
$$585$$ −2.89898 −0.119858
$$586$$ 25.5959 1.05736
$$587$$ 17.3939 0.717922 0.358961 0.933353i $$-0.383131\pi$$
0.358961 + 0.933353i $$0.383131\pi$$
$$588$$ −17.0000 −0.701068
$$589$$ 16.0000 0.659269
$$590$$ 4.89898 0.201688
$$591$$ −25.5959 −1.05288
$$592$$ 6.00000 0.246598
$$593$$ −37.5959 −1.54388 −0.771940 0.635696i $$-0.780713\pi$$
−0.771940 + 0.635696i $$0.780713\pi$$
$$594$$ 0 0
$$595$$ −4.89898 −0.200839
$$596$$ −9.10102 −0.372792
$$597$$ 23.5959 0.965717
$$598$$ −11.5959 −0.474192
$$599$$ −37.3939 −1.52787 −0.763936 0.645292i $$-0.776736\pi$$
−0.763936 + 0.645292i $$0.776736\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 35.7980 1.46023 0.730115 0.683325i $$-0.239467\pi$$
0.730115 + 0.683325i $$0.239467\pi$$
$$602$$ 4.40408 0.179497
$$603$$ −8.89898 −0.362394
$$604$$ 9.79796 0.398673
$$605$$ −11.0000 −0.447214
$$606$$ −9.10102 −0.369704
$$607$$ 16.4949 0.669507 0.334754 0.942306i $$-0.391347\pi$$
0.334754 + 0.942306i $$0.391347\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ −29.3939 −1.19110
$$610$$ 7.79796 0.315730
$$611$$ 28.4041 1.14911
$$612$$ −1.00000 −0.0404226
$$613$$ 34.4949 1.39324 0.696618 0.717442i $$-0.254687\pi$$
0.696618 + 0.717442i $$0.254687\pi$$
$$614$$ 8.89898 0.359134
$$615$$ −6.89898 −0.278194
$$616$$ 0 0
$$617$$ 1.59592 0.0642492 0.0321246 0.999484i $$-0.489773\pi$$
0.0321246 + 0.999484i $$0.489773\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 4.00000 0.160514
$$622$$ 16.8990 0.677587
$$623$$ 57.7980 2.31563
$$624$$ 2.89898 0.116052
$$625$$ 1.00000 0.0400000
$$626$$ 2.89898 0.115867
$$627$$ 0 0
$$628$$ −10.8990 −0.434917
$$629$$ −6.00000 −0.239236
$$630$$ −4.89898 −0.195180
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −13.7980 −0.548853
$$633$$ 12.0000 0.476957
$$634$$ −14.0000 −0.556011
$$635$$ −12.0000 −0.476205
$$636$$ −11.7980 −0.467820
$$637$$ −49.2827 −1.95265
$$638$$ 0 0
$$639$$ 0.898979 0.0355631
$$640$$ −1.00000 −0.0395285
$$641$$ 8.69694 0.343508 0.171754 0.985140i $$-0.445057\pi$$
0.171754 + 0.985140i $$0.445057\pi$$
$$642$$ −13.7980 −0.544562
$$643$$ 10.2020 0.402329 0.201165 0.979557i $$-0.435527\pi$$
0.201165 + 0.979557i $$0.435527\pi$$
$$644$$ −19.5959 −0.772187
$$645$$ 0.898979 0.0353973
$$646$$ 4.00000 0.157378
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 2.89898 0.113707
$$651$$ −19.5959 −0.768025
$$652$$ −21.7980 −0.853674
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ −7.79796 −0.304924
$$655$$ −9.79796 −0.382838
$$656$$ 6.89898 0.269360
$$657$$ −1.10102 −0.0429549
$$658$$ 48.0000 1.87123
$$659$$ 1.30306 0.0507601 0.0253800 0.999678i $$-0.491920\pi$$
0.0253800 + 0.999678i $$0.491920\pi$$
$$660$$ 0 0
$$661$$ −0.202041 −0.00785849 −0.00392924 0.999992i $$-0.501251\pi$$
−0.00392924 + 0.999992i $$0.501251\pi$$
$$662$$ −13.7980 −0.536273
$$663$$ −2.89898 −0.112587
$$664$$ 5.79796 0.225004
$$665$$ 19.5959 0.759897
$$666$$ −6.00000 −0.232495
$$667$$ −24.0000 −0.929284
$$668$$ 21.7980 0.843388
$$669$$ 4.00000 0.154649
$$670$$ 8.89898 0.343798
$$671$$ 0 0
$$672$$ 4.89898 0.188982
$$673$$ −42.8990 −1.65363 −0.826817 0.562471i $$-0.809851\pi$$
−0.826817 + 0.562471i $$0.809851\pi$$
$$674$$ −3.30306 −0.127229
$$675$$ −1.00000 −0.0384900
$$676$$ −4.59592 −0.176766
$$677$$ 25.5959 0.983731 0.491866 0.870671i $$-0.336315\pi$$
0.491866 + 0.870671i $$0.336315\pi$$
$$678$$ −11.7980 −0.453098
$$679$$ 81.7980 3.13912
$$680$$ 1.00000 0.0383482
$$681$$ 2.20204 0.0843824
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 14.0000 0.534913
$$686$$ −48.9898 −1.87044
$$687$$ −25.5959 −0.976545
$$688$$ −0.898979 −0.0342733
$$689$$ −34.2020 −1.30299
$$690$$ −4.00000 −0.152277
$$691$$ 51.1918 1.94743 0.973715 0.227772i $$-0.0731439\pi$$
0.973715 + 0.227772i $$0.0731439\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −2.20204 −0.0835883
$$695$$ −5.79796 −0.219929
$$696$$ 6.00000 0.227429
$$697$$ −6.89898 −0.261317
$$698$$ −23.7980 −0.900766
$$699$$ −14.0000 −0.529529
$$700$$ 4.89898 0.185164
$$701$$ 5.10102 0.192663 0.0963314 0.995349i $$-0.469289\pi$$
0.0963314 + 0.995349i $$0.469289\pi$$
$$702$$ −2.89898 −0.109415
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 9.79796 0.369012
$$706$$ 26.0000 0.978523
$$707$$ −44.5857 −1.67682
$$708$$ 4.89898 0.184115
$$709$$ −12.2020 −0.458257 −0.229129 0.973396i $$-0.573588\pi$$
−0.229129 + 0.973396i $$0.573588\pi$$
$$710$$ −0.898979 −0.0337381
$$711$$ 13.7980 0.517464
$$712$$ −11.7980 −0.442147
$$713$$ −16.0000 −0.599205
$$714$$ −4.89898 −0.183340
$$715$$ 0 0
$$716$$ 4.89898 0.183083
$$717$$ 0 0
$$718$$ 21.3939 0.798412
$$719$$ 36.4949 1.36103 0.680515 0.732734i $$-0.261756\pi$$
0.680515 + 0.732734i $$0.261756\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 19.5959 0.729790
$$722$$ 3.00000 0.111648
$$723$$ 25.5959 0.951922
$$724$$ −23.7980 −0.884444
$$725$$ 6.00000 0.222834
$$726$$ −11.0000 −0.408248
$$727$$ 39.5959 1.46853 0.734266 0.678862i $$-0.237527\pi$$
0.734266 + 0.678862i $$0.237527\pi$$
$$728$$ 14.2020 0.526363
$$729$$ 1.00000 0.0370370
$$730$$ 1.10102 0.0407506
$$731$$ 0.898979 0.0332500
$$732$$ 7.79796 0.288221
$$733$$ 2.49490 0.0921511 0.0460756 0.998938i $$-0.485328\pi$$
0.0460756 + 0.998938i $$0.485328\pi$$
$$734$$ 36.8990 1.36197
$$735$$ −17.0000 −0.627054
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ −6.89898 −0.253955
$$739$$ 49.3939 1.81698 0.908492 0.417903i $$-0.137235\pi$$
0.908492 + 0.417903i $$0.137235\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 11.5959 0.425987
$$742$$ −57.7980 −2.12183
$$743$$ 49.3939 1.81209 0.906043 0.423186i $$-0.139088\pi$$
0.906043 + 0.423186i $$0.139088\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −9.10102 −0.333436
$$746$$ 4.69694 0.171967
$$747$$ −5.79796 −0.212136
$$748$$ 0 0
$$749$$ −67.5959 −2.46990
$$750$$ 1.00000 0.0365148
$$751$$ 2.20204 0.0803536 0.0401768 0.999193i $$-0.487208\pi$$
0.0401768 + 0.999193i $$0.487208\pi$$
$$752$$ −9.79796 −0.357295
$$753$$ 4.89898 0.178529
$$754$$ 17.3939 0.633448
$$755$$ 9.79796 0.356584
$$756$$ −4.89898 −0.178174
$$757$$ 24.6969 0.897625 0.448813 0.893626i $$-0.351847\pi$$
0.448813 + 0.893626i $$0.351847\pi$$
$$758$$ 10.2020 0.370555
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ 37.5959 1.36285 0.681425 0.731888i $$-0.261360\pi$$
0.681425 + 0.731888i $$0.261360\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ −38.2020 −1.38301
$$764$$ 9.79796 0.354478
$$765$$ −1.00000 −0.0361551
$$766$$ −8.00000 −0.289052
$$767$$ 14.2020 0.512806
$$768$$ −1.00000 −0.0360844
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ −10.8990 −0.392263
$$773$$ 23.3939 0.841419 0.420710 0.907195i $$-0.361781\pi$$
0.420710 + 0.907195i $$0.361781\pi$$
$$774$$ 0.898979 0.0323132
$$775$$ 4.00000 0.143684
$$776$$ −16.6969 −0.599385
$$777$$ −29.3939 −1.05450
$$778$$ −10.4949 −0.376260
$$779$$ 27.5959 0.988726
$$780$$ 2.89898 0.103800
$$781$$ 0 0
$$782$$ −4.00000 −0.143040
$$783$$ −6.00000 −0.214423
$$784$$ 17.0000 0.607143
$$785$$ −10.8990 −0.389001
$$786$$ −9.79796 −0.349482
$$787$$ −26.2020 −0.934002 −0.467001 0.884257i $$-0.654666\pi$$
−0.467001 + 0.884257i $$0.654666\pi$$
$$788$$ 25.5959 0.911817
$$789$$ −8.00000 −0.284808
$$790$$ −13.7980 −0.490909
$$791$$ −57.7980 −2.05506
$$792$$ 0 0
$$793$$ 22.6061 0.802767
$$794$$ 37.5959 1.33423
$$795$$ −11.7980 −0.418430
$$796$$ −23.5959 −0.836335
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 19.5959 0.693688
$$799$$ 9.79796 0.346627
$$800$$ −1.00000 −0.0353553
$$801$$ 11.7980 0.416860
$$802$$ −13.1010 −0.462613
$$803$$ 0 0
$$804$$ 8.89898 0.313843
$$805$$ −19.5959 −0.690665
$$806$$ 11.5959 0.408449
$$807$$ 29.5959 1.04183
$$808$$ 9.10102 0.320173
$$809$$ 3.30306 0.116129 0.0580647 0.998313i $$-0.481507\pi$$
0.0580647 + 0.998313i $$0.481507\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −33.3939 −1.17262 −0.586309 0.810088i $$-0.699419\pi$$
−0.586309 + 0.810088i $$0.699419\pi$$
$$812$$ 29.3939 1.03152
$$813$$ −17.7980 −0.624202
$$814$$ 0 0
$$815$$ −21.7980 −0.763549
$$816$$ 1.00000 0.0350070
$$817$$ −3.59592 −0.125805
$$818$$ −21.5959 −0.755084
$$819$$ −14.2020 −0.496259
$$820$$ 6.89898 0.240923
$$821$$ −3.79796 −0.132550 −0.0662748 0.997801i $$-0.521111\pi$$
−0.0662748 + 0.997801i $$0.521111\pi$$
$$822$$ 14.0000 0.488306
$$823$$ 11.1010 0.386957 0.193479 0.981104i $$-0.438023\pi$$
0.193479 + 0.981104i $$0.438023\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ −4.00000 −0.139094 −0.0695468 0.997579i $$-0.522155\pi$$
−0.0695468 + 0.997579i $$0.522155\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −55.3939 −1.92391 −0.961954 0.273210i $$-0.911915\pi$$
−0.961954 + 0.273210i $$0.911915\pi$$
$$830$$ 5.79796 0.201250
$$831$$ 11.7980 0.409267
$$832$$ −2.89898 −0.100504
$$833$$ −17.0000 −0.589015
$$834$$ −5.79796 −0.200767
$$835$$ 21.7980 0.754349
$$836$$ 0 0
$$837$$ −4.00000 −0.138260
$$838$$ −1.79796 −0.0621095
$$839$$ −16.8990 −0.583418 −0.291709 0.956507i $$-0.594224\pi$$
−0.291709 + 0.956507i $$0.594224\pi$$
$$840$$ 4.89898 0.169031
$$841$$ 7.00000 0.241379
$$842$$ −14.0000 −0.482472
$$843$$ −8.20204 −0.282493
$$844$$ −12.0000 −0.413057
$$845$$ −4.59592 −0.158104
$$846$$ 9.79796 0.336861
$$847$$ −53.8888 −1.85164
$$848$$ 11.7980 0.405144
$$849$$ 15.5959 0.535251
$$850$$ 1.00000 0.0342997
$$851$$ −24.0000 −0.822709
$$852$$ −0.898979 −0.0307985
$$853$$ −47.3939 −1.62274 −0.811368 0.584536i $$-0.801277\pi$$
−0.811368 + 0.584536i $$0.801277\pi$$
$$854$$ 38.2020 1.30725
$$855$$ 4.00000 0.136797
$$856$$ 13.7980 0.471605
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ −13.7980 −0.470780 −0.235390 0.971901i $$-0.575637\pi$$
−0.235390 + 0.971901i $$0.575637\pi$$
$$860$$ −0.898979 −0.0306549
$$861$$ −33.7980 −1.15183
$$862$$ 32.8990 1.12054
$$863$$ 13.3939 0.455933 0.227966 0.973669i $$-0.426792\pi$$
0.227966 + 0.973669i $$0.426792\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 6.00000 0.204006
$$866$$ −23.3939 −0.794956
$$867$$ −1.00000 −0.0339618
$$868$$ 19.5959 0.665129
$$869$$ 0 0
$$870$$ 6.00000 0.203419
$$871$$ 25.7980 0.874130
$$872$$ 7.79796 0.264072
$$873$$ 16.6969 0.565106
$$874$$ 16.0000 0.541208
$$875$$ 4.89898 0.165616
$$876$$ 1.10102 0.0372000
$$877$$ 27.3939 0.925025 0.462513 0.886613i $$-0.346948\pi$$
0.462513 + 0.886613i $$0.346948\pi$$
$$878$$ 13.7980 0.465659
$$879$$ 25.5959 0.863329
$$880$$ 0 0
$$881$$ −14.4949 −0.488346 −0.244173 0.969732i $$-0.578516\pi$$
−0.244173 + 0.969732i $$0.578516\pi$$
$$882$$ −17.0000 −0.572420
$$883$$ −2.69694 −0.0907592 −0.0453796 0.998970i $$-0.514450\pi$$
−0.0453796 + 0.998970i $$0.514450\pi$$
$$884$$ 2.89898 0.0975032
$$885$$ 4.89898 0.164677
$$886$$ −18.2020 −0.611510
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −58.7878 −1.97168
$$890$$ −11.7980 −0.395468
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ −39.1918 −1.31150
$$894$$ −9.10102 −0.304384
$$895$$ 4.89898 0.163755
$$896$$ −4.89898 −0.163663
$$897$$ −11.5959 −0.387176
$$898$$ 2.89898 0.0967402
$$899$$ 24.0000 0.800445
$$900$$ 1.00000 0.0333333
$$901$$ −11.7980 −0.393047
$$902$$ 0 0
$$903$$ 4.40408 0.146559
$$904$$ 11.7980 0.392394
$$905$$ −23.7980 −0.791071
$$906$$ 9.79796 0.325515
$$907$$ 25.3939 0.843190 0.421595 0.906784i $$-0.361470\pi$$
0.421595 + 0.906784i $$0.361470\pi$$
$$908$$ −2.20204 −0.0730773
$$909$$ −9.10102 −0.301862
$$910$$ 14.2020 0.470793
$$911$$ 32.8990 1.08999 0.544996 0.838439i $$-0.316531\pi$$
0.544996 + 0.838439i $$0.316531\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ −35.7980 −1.18409
$$915$$ 7.79796 0.257793
$$916$$ 25.5959 0.845713
$$917$$ −48.0000 −1.58510
$$918$$ −1.00000 −0.0330049
$$919$$ −17.7980 −0.587100 −0.293550 0.955944i $$-0.594837\pi$$
−0.293550 + 0.955944i $$0.594837\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 8.89898 0.293231
$$922$$ −0.696938 −0.0229524
$$923$$ −2.60612 −0.0857816
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 31.5959 1.03831
$$927$$ 4.00000 0.131377
$$928$$ −6.00000 −0.196960
$$929$$ −32.2929 −1.05949 −0.529747 0.848156i $$-0.677713\pi$$
−0.529747 + 0.848156i $$0.677713\pi$$
$$930$$ 4.00000 0.131165
$$931$$ 68.0000 2.22861
$$932$$ 14.0000 0.458585
$$933$$ 16.8990 0.553248
$$934$$ −31.5959 −1.03385
$$935$$ 0 0
$$936$$ 2.89898 0.0947561
$$937$$ −19.3939 −0.633570 −0.316785 0.948497i $$-0.602603\pi$$
−0.316785 + 0.948497i $$0.602603\pi$$
$$938$$ 43.5959 1.42346
$$939$$ 2.89898 0.0946046
$$940$$ −9.79796 −0.319574
$$941$$ −35.7980 −1.16698 −0.583490 0.812120i $$-0.698313\pi$$
−0.583490 + 0.812120i $$0.698313\pi$$
$$942$$ −10.8990 −0.355108
$$943$$ −27.5959 −0.898647
$$944$$ −4.89898 −0.159448
$$945$$ −4.89898 −0.159364
$$946$$ 0 0
$$947$$ 2.20204 0.0715567 0.0357784 0.999360i $$-0.488609\pi$$
0.0357784 + 0.999360i $$0.488609\pi$$
$$948$$ −13.7980 −0.448137
$$949$$ 3.19184 0.103611
$$950$$ −4.00000 −0.129777
$$951$$ −14.0000 −0.453981
$$952$$ 4.89898 0.158777
$$953$$ 37.1918 1.20476 0.602381 0.798209i $$-0.294219\pi$$
0.602381 + 0.798209i $$0.294219\pi$$
$$954$$ −11.7980 −0.381973
$$955$$ 9.79796 0.317055
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 23.1010 0.746360
$$959$$ 68.5857 2.21475
$$960$$ −1.00000 −0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 17.3939 0.560801
$$963$$ −13.7980 −0.444633
$$964$$ −25.5959 −0.824389
$$965$$ −10.8990 −0.350851
$$966$$ −19.5959 −0.630488
$$967$$ 17.3939 0.559349 0.279675 0.960095i $$-0.409773\pi$$
0.279675 + 0.960095i $$0.409773\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 4.00000 0.128499
$$970$$ −16.6969 −0.536106
$$971$$ 9.30306 0.298549 0.149275 0.988796i $$-0.452306\pi$$
0.149275 + 0.988796i $$0.452306\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −28.4041 −0.910593
$$974$$ 11.1010 0.355700
$$975$$ 2.89898 0.0928416
$$976$$ −7.79796 −0.249607
$$977$$ 9.59592 0.307001 0.153500 0.988149i $$-0.450945\pi$$
0.153500 + 0.988149i $$0.450945\pi$$
$$978$$ −21.7980 −0.697022
$$979$$ 0 0
$$980$$ 17.0000 0.543045
$$981$$ −7.79796 −0.248970
$$982$$ 30.6969 0.979579
$$983$$ −49.3939 −1.57542 −0.787710 0.616046i $$-0.788733\pi$$
−0.787710 + 0.616046i $$0.788733\pi$$
$$984$$ 6.89898 0.219931
$$985$$ 25.5959 0.815554
$$986$$ 6.00000 0.191079
$$987$$ 48.0000 1.52786
$$988$$ −11.5959 −0.368915
$$989$$ 3.59592 0.114344
$$990$$ 0 0
$$991$$ 23.5959 0.749549 0.374775 0.927116i $$-0.377720\pi$$
0.374775 + 0.927116i $$0.377720\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −13.7980 −0.437865
$$994$$ −4.40408 −0.139689
$$995$$ −23.5959 −0.748041
$$996$$ 5.79796 0.183715
$$997$$ 17.5959 0.557268 0.278634 0.960397i $$-0.410118\pi$$
0.278634 + 0.960397i $$0.410118\pi$$
$$998$$ 0.404082 0.0127910
$$999$$ −6.00000 −0.189832
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 510.2.a.h.1.2 2
3.2 odd 2 1530.2.a.s.1.2 2
4.3 odd 2 4080.2.a.bq.1.1 2
5.2 odd 4 2550.2.d.u.2449.2 4
5.3 odd 4 2550.2.d.u.2449.3 4
5.4 even 2 2550.2.a.bl.1.1 2
15.14 odd 2 7650.2.a.cu.1.1 2
17.16 even 2 8670.2.a.be.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.2 2 1.1 even 1 trivial
1530.2.a.s.1.2 2 3.2 odd 2
2550.2.a.bl.1.1 2 5.4 even 2
2550.2.d.u.2449.2 4 5.2 odd 4
2550.2.d.u.2449.3 4 5.3 odd 4
4080.2.a.bq.1.1 2 4.3 odd 2
7650.2.a.cu.1.1 2 15.14 odd 2
8670.2.a.be.1.1 2 17.16 even 2