Properties

 Label 275.2.b.b.199.2 Level $275$ Weight $2$ Character 275.199 Analytic conductor $2.196$ 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] = [275,2,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("275.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.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: $$2.19588605559$$ 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 55) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 275.199 Dual form 275.2.b.b.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} -1.00000 q^{16} +6.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} -4.00000i q^{23} +2.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} -6.00000 q^{34} +3.00000 q^{36} -2.00000i q^{37} +4.00000i q^{38} +2.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} +7.00000 q^{49} -2.00000i q^{52} +2.00000i q^{53} -6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} -8.00000i q^{62} -7.00000 q^{64} -16.0000i q^{67} +6.00000i q^{68} +8.00000 q^{71} +9.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} +9.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -3.00000i q^{88} -10.0000 q^{89} -4.00000i q^{92} +12.0000 q^{94} +10.0000i q^{97} +7.00000i q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} - 16 q^{31} - 12 q^{34} + 6 q^{36} + 4 q^{41} - 2 q^{44} + 8 q^{46} + 14 q^{49} - 8 q^{59} - 20 q^{61} - 14 q^{64} + 16 q^{71} + 4 q^{74} + 8 q^{76} - 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} + 24 q^{94} - 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 - 2 * q^11 - 2 * q^16 + 8 * q^19 + 4 * q^26 - 12 * q^29 - 16 * q^31 - 12 * q^34 + 6 * q^36 + 4 * q^41 - 2 * q^44 + 8 * q^46 + 14 * q^49 - 8 * q^59 - 20 * q^61 - 14 * q^64 + 16 * q^71 + 4 * q^74 + 8 * q^76 - 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 + 24 * q^94 - 6 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

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 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 3.00000i 0.707107i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 1.00000i − 0.213201i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 16.0000i − 1.95471i −0.211604 0.977356i $$-0.567869\pi$$
0.211604 0.977356i $$-0.432131\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 9.00000i 1.06066i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 2.00000i 0.220863i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ − 3.00000i − 0.319801i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ − 6.00000i − 0.554700i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 10.0000i − 0.905357i
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ 2.00000i 0.167248i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 12.0000i 0.973329i
$$153$$ 18.0000i 1.45521i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000i 0.707107i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ − 4.00000i − 0.304997i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.00000i − 0.438763i
$$188$$ − 12.0000i − 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i 0.352636 + 0.935760i $$0.385285\pi$$
−0.352636 + 0.935760i $$0.614715\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ − 3.00000i − 0.213201i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 10.0000i − 0.703598i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 12.0000i − 0.834058i
$$208$$ 2.00000i 0.138675i
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 18.0000i 1.21911i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 18.0000i − 1.18176i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 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 $$0.395619\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 8.00000i − 0.509028i
$$248$$ − 24.0000i − 1.52400i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 4.00000i 0.251478i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 16.0000i − 0.977356i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ 15.0000i 0.883883i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 14.0000i − 0.819288i
$$293$$ − 10.0000i − 0.584206i −0.956387 0.292103i $$-0.905645\pi$$
0.956387 0.292103i $$-0.0943550\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 10.0000i 0.579284i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −18.0000 −1.02899
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 6.00000i − 0.328798i
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.00000i 0.326841i 0.986557 + 0.163420i $$0.0522527\pi$$
−0.986557 + 0.163420i $$0.947747\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 12.0000i 0.648886i
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 5.00000i − 0.266501i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 10.0000i − 0.525588i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.00000i 0.208798i 0.994535 + 0.104399i $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 36.0000 1.85656
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 8.00000i 0.409316i
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.0000 −1.32337
$$387$$ − 12.0000i − 0.609994i
$$388$$ 10.0000i 0.507673i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 21.0000i 1.06066i
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ 30.0000i 1.50566i 0.658217 + 0.752828i $$0.271311\pi$$
−0.658217 + 0.752828i $$0.728689\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.00000i 0.0991363i
$$408$$ 0 0
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ 12.0000 0.589768
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ − 4.00000i − 0.195646i
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ − 36.0000i − 1.75038i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 22.0000i 1.05725i 0.848855 + 0.528626i $$0.177293\pi$$
−0.848855 + 0.528626i $$0.822707\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ − 16.0000i − 0.765384i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 21.0000 1.00000
$$442$$ 12.0000i 0.570782i
$$443$$ − 8.00000i − 0.380091i −0.981775 0.190046i $$-0.939136\pi$$
0.981775 0.190046i $$-0.0608636\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 6.00000i 0.282216i
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 12.0000i − 0.552345i
$$473$$ 4.00000i 0.183920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ − 8.00000i − 0.365911i
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ − 30.0000i − 1.35804i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ − 36.0000i − 1.62136i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 12.0000i 0.527759i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ − 18.0000i − 0.787839i
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ − 48.0000i − 2.09091i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ − 4.00000i − 0.173259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 48.0000 2.07328
$$537$$ 0 0
$$538$$ 18.0000i 0.776035i
$$539$$ −7.00000 −0.301511
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −30.0000 −1.28624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000i 0.513083i 0.966533 + 0.256541i $$0.0825830\pi$$
−0.966533 + 0.256541i $$0.917417\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ −30.0000 −1.28037
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 10.0000i 0.423714i 0.977301 + 0.211857i $$0.0679510\pi$$
−0.977301 + 0.211857i $$0.932049\pi$$
$$558$$ − 24.0000i − 1.01600i
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000i 0.759284i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 24.0000i 1.00702i
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ − 22.0000i − 0.915872i −0.888985 0.457936i $$-0.848589\pi$$
0.888985 0.457936i $$-0.151411\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 2.00000i − 0.0828315i
$$584$$ 42.0000 1.73797
$$585$$ 0 0
$$586$$ 10.0000 0.413096
$$587$$ − 24.0000i − 0.990586i −0.868726 0.495293i $$-0.835061\pi$$
0.868726 0.495293i $$-0.164939\pi$$
$$588$$ 0 0
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ − 22.0000i − 0.903432i −0.892162 0.451716i $$-0.850812\pi$$
0.892162 0.451716i $$-0.149188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ − 8.00000i − 0.327144i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ − 48.0000i − 1.95471i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 20.0000i 0.811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 18.0000i 0.727607i
$$613$$ − 34.0000i − 1.37325i −0.727013 0.686624i $$-0.759092\pi$$
0.727013 0.686624i $$-0.240908\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ − 24.0000i − 0.954669i
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 14.0000i − 0.554700i
$$638$$ 6.00000i 0.237542i
$$639$$ 24.0000 0.949425
$$640$$ 0 0
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 0 0
$$643$$ 16.0000i 0.630978i 0.948929 + 0.315489i $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 20.0000i 0.786281i 0.919478 + 0.393141i $$0.128611\pi$$
−0.919478 + 0.393141i $$0.871389\pi$$
$$648$$ 27.0000i 1.06066i
$$649$$ 4.00000 0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 16.0000i − 0.626608i
$$653$$ 10.0000i 0.391330i 0.980671 + 0.195665i $$0.0626866\pi$$
−0.980671 + 0.195665i $$0.937313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ − 42.0000i − 1.63858i
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 24.0000i 0.929284i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 8.00000i 0.306336i
$$683$$ 16.0000i 0.612223i 0.951996 + 0.306111i $$0.0990280\pi$$
−0.951996 + 0.306111i $$0.900972\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000i 0.454532i
$$698$$ 10.0000i 0.378506i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 7.00000 0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −24.0000 −0.900070
$$712$$ − 30.0000i − 1.12430i
$$713$$ 32.0000i 1.19841i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ 32.0000i 1.19423i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.0000i 1.92857i 0.264861 + 0.964287i $$0.414674\pi$$
−0.264861 + 0.964287i $$0.585326\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ − 42.0000i − 1.55131i −0.631160 0.775653i $$-0.717421\pi$$
0.631160 0.775653i $$-0.282579\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 16.0000i 0.589368i
$$738$$ 6.00000i 0.220863i
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 40.0000i − 1.46746i −0.679442 0.733729i $$-0.737778\pi$$
0.679442 0.733729i $$-0.262222\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 18.0000 0.659027
$$747$$ 12.0000i 0.439057i
$$748$$ − 6.00000i − 0.219382i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.00000i 0.218074i 0.994038 + 0.109037i $$0.0347767\pi$$
−0.994038 + 0.109037i $$0.965223\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000i 0.935760i
$$773$$ − 14.0000i − 0.503545i −0.967786 0.251773i $$-0.918987\pi$$
0.967786 0.251773i $$-0.0810135\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 24.0000i 0.858238i
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 52.0000i 1.85360i 0.375555 + 0.926800i $$0.377452\pi$$
−0.375555 + 0.926800i $$0.622548\pi$$
$$788$$ 2.00000i 0.0712470i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 9.00000i − 0.319801i
$$793$$ 20.0000i 0.710221i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 2.00000i − 0.0708436i −0.999372 0.0354218i $$-0.988723\pi$$
0.999372 0.0354218i $$-0.0112775\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 2.00000i 0.0706225i
$$803$$ 14.0000i 0.494049i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 0 0
$$808$$ − 30.0000i − 1.05540i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −2.00000 −0.0701000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ 6.00000i 0.209785i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ 36.0000i 1.25488i 0.778664 + 0.627441i $$0.215897\pi$$
−0.778664 + 0.627441i $$0.784103\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 12.0000i − 0.417029i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 14.0000i 0.485363i
$$833$$ 42.0000i 1.45521i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 28.0000i 0.967244i
$$839$$ 32.0000 1.10476 0.552381 0.833592i $$-0.313719\pi$$
0.552381 + 0.833592i $$0.313719\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 6.00000i 0.206774i
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 36.0000 1.23771
$$847$$ 0 0
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 2.00000i − 0.0684787i −0.999414 0.0342393i $$-0.989099\pi$$
0.999414 0.0342393i $$-0.0109009\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ − 4.00000i − 0.136162i −0.997680 0.0680808i $$-0.978312\pi$$
0.997680 0.0680808i $$-0.0216876\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −22.0000 −0.747590
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ 54.0000i 1.82867i
$$873$$ 30.0000i 1.01535i
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 21.0000i 0.707107i
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 8.00000 0.268765
$$887$$ 56.0000i 1.88030i 0.340766 + 0.940148i $$0.389313\pi$$
−0.340766 + 0.940148i $$0.610687\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 4.00000i 0.133930i
$$893$$ − 48.0000i − 1.60626i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ − 2.00000i − 0.0665927i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 40.0000i − 1.32818i −0.747653 0.664089i $$-0.768820\pi$$
0.747653 0.664089i $$-0.231180\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ − 4.00000i − 0.132381i
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 34.0000i − 1.11973i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 12.0000i 0.394132i
$$928$$ − 30.0000i − 0.984798i
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ − 6.00000i − 0.196537i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ 6.00000i 0.196011i 0.995186 + 0.0980057i $$0.0312463\pi$$
−0.995186 + 0.0980057i $$0.968754\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 0 0
$$943$$ − 8.00000i − 0.260516i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 42.0000i 1.36051i 0.732974 + 0.680257i $$0.238132\pi$$
−0.732974 + 0.680257i $$0.761868\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 4.00000i − 0.128965i
$$963$$ 36.0000i 1.16008i
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ 3.00000i 0.0964237i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 10.0000i 0.319928i 0.987123 + 0.159964i $$0.0511379\pi$$
−0.987123 + 0.159964i $$0.948862\pi$$
$$978$$ 0 0
$$979$$ 10.0000 0.319601
$$980$$ 0 0
$$981$$ 54.0000 1.72409
$$982$$ − 28.0000i − 0.893516i
$$983$$ 4.00000i 0.127580i 0.997963 + 0.0637901i $$0.0203188\pi$$
−0.997963 + 0.0637901i $$0.979681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ − 40.0000i − 1.27000i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 46.0000i − 1.45683i −0.685134 0.728417i $$-0.740256\pi$$
0.685134 0.728417i $$-0.259744\pi$$
$$998$$ 36.0000i 1.13956i
$$999$$ 0 0
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 275.2.b.b.199.2 2
3.2 odd 2 2475.2.c.f.199.1 2
4.3 odd 2 4400.2.b.n.4049.1 2
5.2 odd 4 275.2.a.a.1.1 1
5.3 odd 4 55.2.a.a.1.1 1
5.4 even 2 inner 275.2.b.b.199.1 2
15.2 even 4 2475.2.a.i.1.1 1
15.8 even 4 495.2.a.a.1.1 1
15.14 odd 2 2475.2.c.f.199.2 2
20.3 even 4 880.2.a.h.1.1 1
20.7 even 4 4400.2.a.p.1.1 1
20.19 odd 2 4400.2.b.n.4049.2 2
35.13 even 4 2695.2.a.c.1.1 1
40.3 even 4 3520.2.a.n.1.1 1
40.13 odd 4 3520.2.a.p.1.1 1
55.3 odd 20 605.2.g.a.251.1 4
55.8 even 20 605.2.g.c.251.1 4
55.13 even 20 605.2.g.c.81.1 4
55.18 even 20 605.2.g.c.511.1 4
55.28 even 20 605.2.g.c.366.1 4
55.32 even 4 3025.2.a.f.1.1 1
55.38 odd 20 605.2.g.a.366.1 4
55.43 even 4 605.2.a.b.1.1 1
55.48 odd 20 605.2.g.a.511.1 4
55.53 odd 20 605.2.g.a.81.1 4
60.23 odd 4 7920.2.a.i.1.1 1
65.38 odd 4 9295.2.a.b.1.1 1
165.98 odd 4 5445.2.a.i.1.1 1
220.43 odd 4 9680.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.a.1.1 1 5.3 odd 4
275.2.a.a.1.1 1 5.2 odd 4
275.2.b.b.199.1 2 5.4 even 2 inner
275.2.b.b.199.2 2 1.1 even 1 trivial
495.2.a.a.1.1 1 15.8 even 4
605.2.a.b.1.1 1 55.43 even 4
605.2.g.a.81.1 4 55.53 odd 20
605.2.g.a.251.1 4 55.3 odd 20
605.2.g.a.366.1 4 55.38 odd 20
605.2.g.a.511.1 4 55.48 odd 20
605.2.g.c.81.1 4 55.13 even 20
605.2.g.c.251.1 4 55.8 even 20
605.2.g.c.366.1 4 55.28 even 20
605.2.g.c.511.1 4 55.18 even 20
880.2.a.h.1.1 1 20.3 even 4
2475.2.a.i.1.1 1 15.2 even 4
2475.2.c.f.199.1 2 3.2 odd 2
2475.2.c.f.199.2 2 15.14 odd 2
2695.2.a.c.1.1 1 35.13 even 4
3025.2.a.f.1.1 1 55.32 even 4
3520.2.a.n.1.1 1 40.3 even 4
3520.2.a.p.1.1 1 40.13 odd 4
4400.2.a.p.1.1 1 20.7 even 4
4400.2.b.n.4049.1 2 4.3 odd 2
4400.2.b.n.4049.2 2 20.19 odd 2
5445.2.a.i.1.1 1 165.98 odd 4
7920.2.a.i.1.1 1 60.23 odd 4
9295.2.a.b.1.1 1 65.38 odd 4
9680.2.a.r.1.1 1 220.43 odd 4