# Properties

 Label 2550.2.d.s.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ 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] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (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: $$20.3618525154$$ 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 510) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.s.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{39} -10.0000 q^{41} -8.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +1.00000i q^{48} +7.00000 q^{49} +1.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +1.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} +2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} -8.00000 q^{86} -2.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -4.00000i q^{92} -4.00000i q^{93} +1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} + 4 q^{26} - 4 q^{29} - 8 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} - 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} + 2 q^{51} - 2 q^{54} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} + 16 q^{71} + 12 q^{74} - 8 q^{76} - 8 q^{79} + 2 q^{81} - 16 q^{86} + 28 q^{89} + 2 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 + 4 * q^26 - 4 * q^29 - 8 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 - 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 + 2 * q^51 - 2 * q^54 + 16 * q^59 + 20 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 + 16 * q^71 + 12 * q^74 - 8 * q^76 - 8 * q^79 + 2 * q^81 - 16 * q^86 + 28 * q^89 + 2 * q^96 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 − 0.707107i
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 2.00000i 0.226455i
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ − 2.00000i − 0.214423i
$$88$$ 4.00000i 0.426401i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 2.00000i − 0.184900i
$$118$$ − 8.00000i − 0.736460i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 10.0000i − 0.901670i
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ 8.00000i 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 7.00000i 0.577350i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ 1.00000i 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 8.00000i 0.609994i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 8.00000i 0.601317i
$$178$$ − 14.0000i − 1.04934i
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ − 4.00000i − 0.292509i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ − 4.00000i − 0.278019i
$$208$$ 2.00000i 0.138675i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 8.00000i 0.548151i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 14.0000i − 0.948200i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 6.00000i 0.402694i
$$223$$ 24.0000i 1.60716i 0.595198 + 0.803579i $$0.297074\pi$$
−0.595198 + 0.803579i $$0.702926\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 2.00000i − 0.131306i
$$233$$ − 30.0000i − 1.96537i −0.185296 0.982683i $$-0.559325\pi$$
0.185296 0.982683i $$-0.440675\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 8.00000i 0.509028i
$$248$$ − 4.00000i − 0.254000i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 4.00000i − 0.247121i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.0000i 0.856786i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 2.00000i 0.117041i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ − 4.00000i − 0.232104i
$$298$$ 2.00000i 0.115857i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 2.00000i − 0.113047i −0.998401 0.0565233i $$-0.981998\pi$$
0.998401 0.0565233i $$-0.0180015\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ − 26.0000i − 1.46031i −0.683284 0.730153i $$-0.739449\pi$$
0.683284 0.730153i $$-0.260551\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 14.0000i 0.774202i
$$328$$ − 10.0000i − 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 6.00000i − 0.326841i −0.986557 0.163420i $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 4.00000i 0.216295i
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000i 0.735824i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 4.00000i 0.207390i
$$373$$ 18.0000i 0.932005i 0.884783 + 0.466002i $$0.154306\pi$$
−0.884783 + 0.466002i $$0.845694\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 8.00000i 0.406663i
$$388$$ − 10.0000i − 0.507673i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 7.00000i 0.353553i
$$393$$ 4.00000i 0.201773i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ − 8.00000i − 0.398508i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000i 1.18964i
$$408$$ 1.00000i 0.0495074i
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 4.00000i 0.195881i
$$418$$ − 16.0000i − 0.782586i
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 6.00000i − 0.288342i −0.989553 0.144171i $$-0.953949\pi$$
0.989553 0.144171i $$-0.0460515\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 16.0000i 0.765384i
$$438$$ − 2.00000i − 0.0955637i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 2.00000i − 0.0951303i
$$443$$ − 28.0000i − 1.33032i −0.746701 0.665160i $$-0.768363\pi$$
0.746701 0.665160i $$-0.231637\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ − 2.00000i − 0.0945968i
$$448$$ 0 0
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ − 18.0000i − 0.846649i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ − 2.00000i − 0.0935561i −0.998905 0.0467780i $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −38.0000 −1.76984 −0.884918 0.465746i $$-0.845786\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −30.0000 −1.38972
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 8.00000i 0.368230i
$$473$$ − 32.0000i − 1.47136i
$$474$$ −4.00000 −0.183726
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 8.00000i 0.365911i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ − 10.0000i − 0.455488i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 40.0000i 1.81257i 0.422664 + 0.906287i $$0.361095\pi$$
−0.422664 + 0.906287i $$0.638905\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 2.00000i 0.0900755i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 4.00000i 0.179244i
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ − 28.0000i − 1.24846i −0.781241 0.624229i $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ 9.00000i 0.399704i
$$508$$ − 16.0000i − 0.709885i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ − 40.0000i − 1.74908i −0.484955 0.874539i $$-0.661164\pi$$
0.484955 0.874539i $$-0.338836\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 4.00000i 0.174243i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ − 20.0000i − 0.866296i
$$534$$ 14.0000 0.605839
$$535$$ 0 0
$$536$$ −8.00000 −0.345547
$$537$$ 0 0
$$538$$ − 14.0000i − 0.603583i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ − 14.0000i − 0.600798i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ − 4.00000i − 0.170251i
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 30.0000i 1.26547i
$$563$$ 20.0000i 0.842900i 0.906852 + 0.421450i $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ 8.00000i 0.335673i
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 6.00000i 0.249783i 0.992170 + 0.124892i $$0.0398583\pi$$
−0.992170 + 0.124892i $$0.960142\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 10.0000i 0.414513i
$$583$$ 24.0000i 0.993978i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 6.00000i 0.246598i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 4.00000i 0.163709i
$$598$$ 8.00000i 0.327144i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 34.0000 1.38689 0.693444 0.720510i $$-0.256092\pi$$
0.693444 + 0.720510i $$0.256092\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 1.00000i − 0.0404226i
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 8.00000i 0.320771i
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −2.00000 −0.0799361
$$627$$ 16.0000i 0.638978i
$$628$$ − 6.00000i − 0.239426i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 12.0000i 0.476957i
$$634$$ −26.0000 −1.03259
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 14.0000i 0.554700i
$$638$$ 8.00000i 0.316723i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 32.0000 1.25611
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 12.0000i − 0.469956i
$$653$$ 34.0000i 1.33052i 0.746611 + 0.665261i $$0.231680\pi$$
−0.746611 + 0.665261i $$0.768320\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 2.00000i 0.0776736i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 8.00000i − 0.309761i
$$668$$ 12.0000i 0.464294i
$$669$$ −24.0000 −0.927894
$$670$$ 0 0
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 16.0000i 0.612672i
$$683$$ 44.0000i 1.68361i 0.539779 + 0.841807i $$0.318508\pi$$
−0.539779 + 0.841807i $$0.681492\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 6.00000i − 0.228914i
$$688$$ − 8.00000i − 0.304997i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 10.0000i 0.378777i
$$698$$ − 34.0000i − 1.28692i
$$699$$ 30.0000 1.13470
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 24.0000i 0.905177i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ − 8.00000i − 0.300658i
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 14.0000i 0.524672i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 8.00000i − 0.298765i
$$718$$ 32.0000i 1.19423i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ 10.0000i 0.371904i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 40.0000i − 1.48352i −0.670667 0.741759i $$-0.733992\pi$$
0.670667 0.741759i $$-0.266008\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ − 10.0000i − 0.369611i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 32.0000i 1.17874i
$$738$$ − 10.0000i − 0.368105i
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 28.0000i 1.02722i 0.858024 + 0.513610i $$0.171692\pi$$
−0.858024 + 0.513610i $$0.828308\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 18.0000 0.659027
$$747$$ − 4.00000i − 0.146352i
$$748$$ 4.00000i 0.146254i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.0000i 0.508839i 0.967094 + 0.254419i $$0.0818843\pi$$
−0.967094 + 0.254419i $$0.918116\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 16.0000i 0.577727i
$$768$$ 1.00000i 0.0360844i
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 6.00000i − 0.215945i
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ − 30.0000i − 1.07555i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ − 4.00000i − 0.143040i
$$783$$ 2.00000i 0.0714742i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 4.00000 0.142675
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ 20.0000i 0.710221i
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ − 6.00000i − 0.211867i
$$803$$ − 8.00000i − 0.282314i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 14.0000i 0.492823i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ − 32.0000i − 1.11954i
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 16.0000i 0.557725i 0.960331 + 0.278862i $$0.0899574\pi$$
−0.960331 + 0.278862i $$0.910043\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ −54.0000 −1.87550 −0.937749 0.347314i $$-0.887094\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 7.00000i − 0.242536i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 4.00000i 0.138260i
$$838$$ 28.0000i 0.967244i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 30.0000i − 1.03387i
$$843$$ − 30.0000i − 1.03325i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000i 0.206041i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 54.0000i − 1.84892i −0.381273 0.924462i $$-0.624514\pi$$
0.381273 0.924462i $$-0.375486\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 26.0000i − 0.888143i −0.895991 0.444072i $$-0.853534\pi$$
0.895991 0.444072i $$-0.146466\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 16.0000i 0.544962i
$$863$$ − 8.00000i − 0.272323i −0.990687 0.136162i $$-0.956523\pi$$
0.990687 0.136162i $$-0.0434766\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −6.00000 −0.203888
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 14.0000i 0.474100i
$$873$$ − 10.0000i − 0.338449i
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ − 50.0000i − 1.68838i −0.536044 0.844190i $$-0.680082\pi$$
0.536044 0.844190i $$-0.319918\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ −28.0000 −0.940678
$$887$$ − 12.0000i − 0.402921i −0.979497 0.201460i $$-0.935431\pi$$
0.979497 0.201460i $$-0.0645687\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ − 24.0000i − 0.803579i
$$893$$ 0 0
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 8.00000i − 0.267112i
$$898$$ − 26.0000i − 0.867631i
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 6.00000 0.199889
$$902$$ 40.0000i 1.33185i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 36.0000i 1.19536i 0.801735 + 0.597680i $$0.203911\pi$$
−0.801735 + 0.597680i $$0.796089\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 16.0000i 0.529523i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 1.00000i 0.0330049i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 16.0000 0.527218
$$922$$ 38.0000i 1.25146i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −24.0000 −0.788689
$$927$$ − 16.0000i − 0.525509i
$$928$$ 2.00000i 0.0656532i
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ 30.0000i 0.982683i
$$933$$ − 8.00000i − 0.261908i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 0 0
$$939$$ 2.00000 0.0652675
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ − 40.0000i − 1.30258i
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −32.0000 −1.04041
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 4.00000i 0.129914i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 26.0000 0.843108
$$952$$ 0 0
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ − 8.00000i − 0.258603i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 12.0000i 0.386896i
$$963$$ 12.0000i 0.386695i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 24.0000i − 0.771788i −0.922543 0.385894i $$-0.873893\pi$$
0.922543 0.385894i $$-0.126107\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 12.0000i 0.383718i
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ − 24.0000i − 0.765871i
$$983$$ − 44.0000i − 1.40338i −0.712481 0.701691i $$-0.752429\pi$$
0.712481 0.701691i $$-0.247571\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 2.00000 0.0636930
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 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 2550.2.d.s.2449.1 2
5.2 odd 4 510.2.a.f.1.1 1
5.3 odd 4 2550.2.a.d.1.1 1
5.4 even 2 inner 2550.2.d.s.2449.2 2
15.2 even 4 1530.2.a.f.1.1 1
15.8 even 4 7650.2.a.bw.1.1 1
20.7 even 4 4080.2.a.c.1.1 1
85.67 odd 4 8670.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.f.1.1 1 5.2 odd 4
1530.2.a.f.1.1 1 15.2 even 4
2550.2.a.d.1.1 1 5.3 odd 4
2550.2.d.s.2449.1 2 1.1 even 1 trivial
2550.2.d.s.2449.2 2 5.4 even 2 inner
4080.2.a.c.1.1 1 20.7 even 4
7650.2.a.bw.1.1 1 15.8 even 4
8670.2.a.r.1.1 1 85.67 odd 4