# Properties

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{16} -6.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} +1.00000i q^{22} +4.00000i q^{23} +2.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -6.00000 q^{34} +3.00000 q^{36} +2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{41} +4.00000i q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} +7.00000 q^{49} +2.00000i q^{52} -2.00000i q^{53} +6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} +8.00000i q^{62} -7.00000 q^{64} +16.0000i q^{67} -6.00000i q^{68} +8.00000 q^{71} -9.00000i q^{72} +14.0000i q^{73} +2.00000 q^{74} +4.00000 q^{76} -8.00000 q^{79} +9.00000 q^{81} -2.00000i q^{82} -4.00000i q^{83} +4.00000 q^{86} +3.00000i q^{88} -10.0000 q^{89} +4.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} -7.00000i q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 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 $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 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 $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ − 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 $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 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 $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ − 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 $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 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 $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 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 $$0.432131\pi$$
−0.211604 + 0.977356i $$0.567869\pi$$
$$68$$ − 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 $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 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 $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 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 $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 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 $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 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 $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 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 $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 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 $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 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 $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 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 $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 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 $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 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 $$-0.614715\pi$$
0.352636 0.935760i $$-0.385285\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 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 $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 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 $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 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 $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 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 $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 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 $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 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 $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 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 $$0.0943550\pi$$
−0.956387 + 0.292103i $$0.905645\pi$$
$$294$$ 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 $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 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 $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 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 $$-0.947747\pi$$
0.986557 0.163420i $$-0.0522527\pi$$
$$338$$ − 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 $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 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 $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 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 $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ − 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 $$0.154306\pi$$
−0.884783 + 0.466002i $$0.845694\pi$$
$$374$$ 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 $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 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 $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ 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 $$-0.822707\pi$$
0.848855 0.528626i $$-0.177293\pi$$
$$434$$ 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 $$0.0608636\pi$$
−0.981775 + 0.190046i $$0.939136\pi$$
$$444$$ 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 $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ − 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 $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ 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 $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ 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 $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 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 $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −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 $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ − 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 $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ 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 $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 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 $$0.151411\pi$$
−0.888985 + 0.457936i $$0.848589\pi$$
$$578$$ 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 $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 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 $$0.149188\pi$$
−0.892162 + 0.451716i $$0.850812\pi$$
$$594$$ 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 $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ − 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 $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 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 $$-0.897831\pi$$
0.948929 0.315489i $$-0.102169\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ − 20.0000i − 0.786281i −0.919478 0.393141i $$-0.871389\pi$$
0.919478 0.393141i $$-0.128611\pi$$
$$648$$ − 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 $$-0.937313\pi$$
0.980671 0.195665i $$-0.0626866\pi$$
$$654$$ 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 $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 38.0000i 1.46046i 0.683202 + 0.730229i $$0.260587\pi$$
−0.683202 + 0.730229i $$0.739413\pi$$
$$678$$ 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 $$-0.900972\pi$$
0.951996 0.306111i $$-0.0990280\pi$$
$$684$$ 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 $$-0.585326\pi$$
0.264861 0.964287i $$-0.414674\pi$$
$$728$$ 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 $$0.282579\pi$$
−0.631160 + 0.775653i $$0.717421\pi$$
$$734$$ −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 $$0.262222\pi$$
−0.679442 + 0.733729i $$0.737778\pi$$
$$744$$ 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 $$-0.965223\pi$$
0.994038 0.109037i $$-0.0347767\pi$$
$$758$$ 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 $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ 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 $$-0.622548\pi$$
0.375555 0.926800i $$-0.377452\pi$$
$$788$$ − 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 $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 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 $$-0.784103\pi$$
0.778664 0.627441i $$-0.215897\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 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 $$0.0109009\pi$$
−0.999414 + 0.0342393i $$0.989099\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 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 $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ 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 $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 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 $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 8.00000 0.268765
$$887$$ − 56.0000i − 1.88030i −0.340766 0.940148i $$-0.610687\pi$$
0.340766 0.940148i $$-0.389313\pi$$
$$888$$ 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 $$0.231180\pi$$
−0.747653 + 0.664089i $$0.768820\pi$$
$$908$$ 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 $$-0.968754\pi$$
0.995186 0.0980057i $$-0.0312463\pi$$
$$938$$ 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 $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ −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 $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ − 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 $$-0.948862\pi$$
0.987123 0.159964i $$-0.0511379\pi$$
$$978$$ 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 $$-0.979681\pi$$
0.997963 0.0637901i $$-0.0203188\pi$$
$$984$$ 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 $$0.259744\pi$$
−0.685134 + 0.728417i $$0.740256\pi$$
$$998$$ − 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.199.1 2
3.2 odd 2 2475.2.c.f.199.2 2
4.3 odd 2 4400.2.b.n.4049.2 2
5.2 odd 4 55.2.a.a.1.1 1
5.3 odd 4 275.2.a.a.1.1 1
5.4 even 2 inner 275.2.b.b.199.2 2
15.2 even 4 495.2.a.a.1.1 1
15.8 even 4 2475.2.a.i.1.1 1
15.14 odd 2 2475.2.c.f.199.1 2
20.3 even 4 4400.2.a.p.1.1 1
20.7 even 4 880.2.a.h.1.1 1
20.19 odd 2 4400.2.b.n.4049.1 2
35.27 even 4 2695.2.a.c.1.1 1
40.27 even 4 3520.2.a.n.1.1 1
40.37 odd 4 3520.2.a.p.1.1 1
55.2 even 20 605.2.g.c.81.1 4
55.7 even 20 605.2.g.c.511.1 4
55.17 even 20 605.2.g.c.366.1 4
55.27 odd 20 605.2.g.a.366.1 4
55.32 even 4 605.2.a.b.1.1 1
55.37 odd 20 605.2.g.a.511.1 4
55.42 odd 20 605.2.g.a.81.1 4
55.43 even 4 3025.2.a.f.1.1 1
55.47 odd 20 605.2.g.a.251.1 4
55.52 even 20 605.2.g.c.251.1 4
60.47 odd 4 7920.2.a.i.1.1 1
65.12 odd 4 9295.2.a.b.1.1 1
165.32 odd 4 5445.2.a.i.1.1 1
220.87 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 5.2 odd 4
275.2.a.a.1.1 1 5.3 odd 4
275.2.b.b.199.1 2 1.1 even 1 trivial
275.2.b.b.199.2 2 5.4 even 2 inner
495.2.a.a.1.1 1 15.2 even 4
605.2.a.b.1.1 1 55.32 even 4
605.2.g.a.81.1 4 55.42 odd 20
605.2.g.a.251.1 4 55.47 odd 20
605.2.g.a.366.1 4 55.27 odd 20
605.2.g.a.511.1 4 55.37 odd 20
605.2.g.c.81.1 4 55.2 even 20
605.2.g.c.251.1 4 55.52 even 20
605.2.g.c.366.1 4 55.17 even 20
605.2.g.c.511.1 4 55.7 even 20
880.2.a.h.1.1 1 20.7 even 4
2475.2.a.i.1.1 1 15.8 even 4
2475.2.c.f.199.1 2 15.14 odd 2
2475.2.c.f.199.2 2 3.2 odd 2
2695.2.a.c.1.1 1 35.27 even 4
3025.2.a.f.1.1 1 55.43 even 4
3520.2.a.n.1.1 1 40.27 even 4
3520.2.a.p.1.1 1 40.37 odd 4
4400.2.a.p.1.1 1 20.3 even 4
4400.2.b.n.4049.1 2 20.19 odd 2
4400.2.b.n.4049.2 2 4.3 odd 2
5445.2.a.i.1.1 1 165.32 odd 4
7920.2.a.i.1.1 1 60.47 odd 4
9295.2.a.b.1.1 1 65.12 odd 4
9680.2.a.r.1.1 1 220.87 odd 4