# Properties

 Label 2925.2.c.m.2224.1 Level $2925$ Weight $2$ Character 2925.2224 Analytic conductor $23.356$ 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] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.c (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: $$23.3562425912$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2224.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2925.2224 Dual form 2925.2.c.m.2224.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{4} -1.00000i q^{7} +O(q^{10})$$ $$q+2.00000 q^{4} -1.00000i q^{7} +3.00000 q^{11} -1.00000i q^{13} +4.00000 q^{16} +3.00000i q^{17} +4.00000 q^{19} -9.00000i q^{23} -2.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{37} +3.00000 q^{41} -2.00000i q^{43} +6.00000 q^{44} +6.00000i q^{47} +6.00000 q^{49} -2.00000i q^{52} +9.00000i q^{53} -12.0000 q^{59} +5.00000 q^{61} +8.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} -9.00000 q^{71} -14.0000i q^{73} +8.00000 q^{76} -3.00000i q^{77} +7.00000 q^{79} +15.0000 q^{89} -1.00000 q^{91} -18.0000i q^{92} +5.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4}+O(q^{10})$$ 2 * q + 4 * q^4 $$2 q + 4 q^{4} + 6 q^{11} + 8 q^{16} + 8 q^{19} - 12 q^{29} + 4 q^{31} + 6 q^{41} + 12 q^{44} + 12 q^{49} - 24 q^{59} + 10 q^{61} + 16 q^{64} - 18 q^{71} + 16 q^{76} + 14 q^{79} + 30 q^{89} - 2 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 6 * q^11 + 8 * q^16 + 8 * q^19 - 12 * q^29 + 4 * q^31 + 6 * q^41 + 12 * q^44 + 12 * q^49 - 24 * q^59 + 10 * q^61 + 16 * q^64 - 18 * q^71 + 16 * q^76 + 14 * q^79 + 30 * q^89 - 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 9.00000i − 1.87663i −0.345782 0.938315i $$-0.612386\pi$$
0.345782 0.938315i $$-0.387614\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ − 2.00000i − 0.377964i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −9.00000 −1.06810 −0.534052 0.845452i $$-0.679331\pi$$
−0.534052 + 0.845452i $$0.679331\pi$$
$$72$$ 0 0
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ − 3.00000i − 0.341882i
$$78$$ 0 0
$$79$$ 7.00000 0.787562 0.393781 0.919204i $$-0.371167\pi$$
0.393781 + 0.919204i $$0.371167\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ − 18.0000i − 1.87663i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000i 0.507673i 0.967247 + 0.253837i $$0.0816925\pi$$
−0.967247 + 0.253837i $$0.918307\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 3.00000i − 0.290021i −0.989430 0.145010i $$-0.953678\pi$$
0.989430 0.145010i $$-0.0463216\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −12.0000 −1.11417
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.0000i 1.77471i 0.461084 + 0.887357i $$0.347461\pi$$
−0.461084 + 0.887357i $$0.652539\pi$$
$$128$$ 0 0
$$129$$ 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
$$133$$ − 4.00000i − 0.346844i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 3.00000i − 0.250873i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.00000i 0.638470i 0.947676 + 0.319235i $$0.103426\pi$$
−0.947676 + 0.319235i $$0.896574\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −9.00000 −0.709299
$$162$$ 0 0
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000i 0.464294i 0.972681 + 0.232147i $$0.0745750\pi$$
−0.972681 + 0.232147i $$0.925425\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 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
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ 12.0000i 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 13.0000i 0.935760i 0.883792 + 0.467880i $$0.154982\pi$$
−0.883792 + 0.467880i $$0.845018\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 12.0000 0.857143
$$197$$ − 24.0000i − 1.70993i −0.518686 0.854965i $$-0.673579\pi$$
0.518686 0.854965i $$-0.326421\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ − 4.00000i − 0.277350i
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 18.0000i 1.23625i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 2.00000i − 0.135769i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3.00000i 0.196537i 0.995160 + 0.0982683i $$0.0313303\pi$$
−0.995160 + 0.0982683i $$0.968670\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −24.0000 −1.56227
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ 0 0
$$249$$ 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
$$253$$ − 27.0000i − 1.69748i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 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
$$268$$ − 8.00000i − 0.488678i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 12.0000i 0.727607i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ −18.0000 −1.06810
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 3.00000i − 0.177084i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 28.0000i − 1.63858i
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −9.00000 −0.520483
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 16.0000 0.917663
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1.00000i − 0.0570730i −0.999593 0.0285365i $$-0.990915\pi$$
0.999593 0.0285365i $$-0.00908469\pi$$
$$308$$ − 6.00000i − 0.341882i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 28.0000i 1.58265i 0.611393 + 0.791327i $$0.290609\pi$$
−0.611393 + 0.791327i $$0.709391\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ − 13.0000i − 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 33.0000i 1.77153i 0.464131 + 0.885766i $$0.346367\pi$$
−0.464131 + 0.885766i $$0.653633\pi$$
$$348$$ 0 0
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 30.0000 1.59000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ − 36.0000i − 1.87663i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9.00000 0.467257
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000i 0.309016i
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 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
$$391$$ 27.0000 1.36545
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 13.0000i − 0.652451i −0.945292 0.326226i $$-0.894223\pi$$
0.945292 0.326226i $$-0.105777\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 2.00000i − 0.0996271i
$$404$$ 24.0000 1.19404
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3.00000i − 0.148704i
$$408$$ 0 0
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 28.0000i − 1.37946i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 5.00000i − 0.241967i
$$428$$ − 6.00000i − 0.290021i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 28.0000i 1.34559i 0.739827 + 0.672797i $$0.234907\pi$$
−0.739827 + 0.672797i $$0.765093\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 20.0000 0.957826
$$437$$ − 36.0000i − 1.72211i
$$438$$ 0 0
$$439$$ −11.0000 −0.525001 −0.262501 0.964932i $$-0.584547\pi$$
−0.262501 + 0.964932i $$0.584547\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 15.0000i 0.712672i 0.934358 + 0.356336i $$0.115974\pi$$
−0.934358 + 0.356336i $$0.884026\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ − 8.00000i − 0.377964i
$$449$$ 21.0000 0.991051 0.495526 0.868593i $$-0.334975\pi$$
0.495526 + 0.868593i $$0.334975\pi$$
$$450$$ 0 0
$$451$$ 9.00000 0.423793
$$452$$ − 12.0000i − 0.564433i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 0 0
$$463$$ 1.00000i 0.0464739i 0.999730 + 0.0232370i $$0.00739722\pi$$
−0.999730 + 0.0232370i $$0.992603\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 15.0000i − 0.694117i −0.937843 0.347059i $$-0.887180\pi$$
0.937843 0.347059i $$-0.112820\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 6.00000i − 0.275880i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 6.00000 0.275010
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −9.00000 −0.411220 −0.205610 0.978634i $$-0.565918\pi$$
−0.205610 + 0.978634i $$0.565918\pi$$
$$480$$ 0 0
$$481$$ −1.00000 −0.0455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 43.0000i − 1.94852i −0.225436 0.974258i $$-0.572381\pi$$
0.225436 0.974258i $$-0.427619\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 0 0
$$493$$ − 18.0000i − 0.810679i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 9.00000i 0.403705i
$$498$$ 0 0
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 40.0000i 1.77471i
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000i 0.261364i
$$528$$ 0 0
$$529$$ −58.0000 −2.52174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 8.00000i − 0.346844i
$$533$$ − 3.00000i − 0.129944i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 38.0000i 1.62476i 0.583127 + 0.812381i $$0.301829\pi$$
−0.583127 + 0.812381i $$0.698171\pi$$
$$548$$ 24.0000i 1.02523i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ − 7.00000i − 0.297670i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 2.00000 0.0848189
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 15.0000i 0.632175i 0.948730 + 0.316087i $$0.102369\pi$$
−0.948730 + 0.316087i $$0.897631\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ − 6.00000i − 0.250873i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 41.0000i 1.70685i 0.521214 + 0.853426i $$0.325479\pi$$
−0.521214 + 0.853426i $$0.674521\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 27.0000i 1.11823i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 4.00000i − 0.164399i
$$593$$ 42.0000i 1.72473i 0.506284 + 0.862367i $$0.331019\pi$$
−0.506284 + 0.862367i $$0.668981\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 30.0000 1.22885
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −32.0000 −1.30206
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 0 0
$$613$$ 31.0000i 1.25208i 0.779792 + 0.626039i $$0.215325\pi$$
−0.779792 + 0.626039i $$0.784675\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 15.0000i − 0.600962i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 16.0000i 0.638470i
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −34.0000 −1.35352 −0.676759 0.736204i $$-0.736616\pi$$
−0.676759 + 0.736204i $$0.736616\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ − 23.0000i − 0.907031i −0.891248 0.453516i $$-0.850170\pi$$
0.891248 0.453516i $$-0.149830\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 39.0000i 1.53325i 0.642096 + 0.766624i $$0.278065\pi$$
−0.642096 + 0.766624i $$0.721935\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 22.0000i − 0.861586i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 12.0000 0.468521
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 54.0000i 2.09089i
$$668$$ 12.0000i 0.464294i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 15.0000 0.579069
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −2.00000 −0.0769231
$$677$$ 3.00000i 0.115299i 0.998337 + 0.0576497i $$0.0183606\pi$$
−0.998337 + 0.0576497i $$0.981639\pi$$
$$678$$ 0 0
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ − 8.00000i − 0.304997i
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 9.00000i 0.340899i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 12.0000i − 0.451306i
$$708$$ 0 0
$$709$$ −44.0000 −1.65245 −0.826227 0.563337i $$-0.809517\pi$$
−0.826227 + 0.563337i $$0.809517\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 18.0000i − 0.674105i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −36.0000 −1.34538
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.00000 0.221918
$$732$$ 0 0
$$733$$ 49.0000i 1.80986i 0.425564 + 0.904928i $$0.360076\pi$$
−0.425564 + 0.904928i $$0.639924\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 12.0000i − 0.442026i
$$738$$ 0 0
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 18.0000i 0.658145i
$$749$$ −3.00000 −0.109618
$$750$$ 0 0
$$751$$ 11.0000 0.401396 0.200698 0.979653i $$-0.435679\pi$$
0.200698 + 0.979653i $$0.435679\pi$$
$$752$$ 24.0000i 0.875190i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.0000i 0.508839i 0.967094 + 0.254419i $$0.0818843\pi$$
−0.967094 + 0.254419i $$0.918116\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ − 10.0000i − 0.362024i
$$764$$ 36.0000 1.30243
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000i 0.433295i
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000i 0.935760i
$$773$$ − 48.0000i − 1.72644i −0.504828 0.863220i $$-0.668444\pi$$
0.504828 0.863220i $$-0.331556\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ −27.0000 −0.966136
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 24.0000 0.857143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 52.0000i − 1.85360i −0.375555 0.926800i $$-0.622548\pi$$
0.375555 0.926800i $$-0.377452\pi$$
$$788$$ − 48.0000i − 1.70993i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ − 5.00000i − 0.177555i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 32.0000 1.13421
$$797$$ − 39.0000i − 1.38145i −0.723117 0.690725i $$-0.757291\pi$$
0.723117 0.690725i $$-0.242709\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 42.0000i − 1.48215i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −48.0000 −1.68759 −0.843795 0.536666i $$-0.819684\pi$$
−0.843795 + 0.536666i $$0.819684\pi$$
$$810$$ 0 0
$$811$$ −22.0000 −0.772524 −0.386262 0.922389i $$-0.626234\pi$$
−0.386262 + 0.922389i $$0.626234\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 8.00000i − 0.279885i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 15.0000 0.523504 0.261752 0.965135i $$-0.415700\pi$$
0.261752 + 0.965135i $$0.415700\pi$$
$$822$$ 0 0
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 18.0000i 0.625921i 0.949766 + 0.312961i $$0.101321\pi$$
−0.949766 + 0.312961i $$0.898679\pi$$
$$828$$ 0 0
$$829$$ 22.0000 0.764092 0.382046 0.924143i $$-0.375220\pi$$
0.382046 + 0.924143i $$0.375220\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 8.00000i − 0.277350i
$$833$$ 18.0000i 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 24.0000 0.830057
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −51.0000 −1.76072 −0.880358 0.474310i $$-0.842698\pi$$
−0.880358 + 0.474310i $$0.842698\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000i 0.0687208i
$$848$$ 36.0000i 1.23625i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −9.00000 −0.308516
$$852$$ 0 0
$$853$$ − 17.0000i − 0.582069i −0.956713 0.291034i $$-0.906001\pi$$
0.956713 0.291034i $$-0.0939994\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33.0000i 1.12726i 0.826028 + 0.563629i $$0.190595\pi$$
−0.826028 + 0.563629i $$0.809405\pi$$
$$858$$ 0 0
$$859$$ 13.0000 0.443554 0.221777 0.975097i $$-0.428814\pi$$
0.221777 + 0.975097i $$0.428814\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 18.0000i − 0.612727i −0.951915 0.306364i $$-0.900888\pi$$
0.951915 0.306364i $$-0.0991123\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ − 4.00000i − 0.135769i
$$869$$ 21.0000 0.712376
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 0 0
$$883$$ 10.0000i 0.336527i 0.985742 + 0.168263i $$0.0538159\pi$$
−0.985742 + 0.168263i $$0.946184\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 9.00000i − 0.302190i −0.988519 0.151095i $$-0.951720\pi$$
0.988519 0.151095i $$-0.0482800\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 24.0000i 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −27.0000 −0.899500
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 50.0000i 1.66022i 0.557598 + 0.830111i $$0.311723\pi$$
−0.557598 + 0.830111i $$0.688277\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 8.00000 0.264327
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 9.00000i 0.296239i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 24.0000 0.786568
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 4.00000i − 0.130674i −0.997863 0.0653372i $$-0.979188\pi$$
0.997863 0.0653372i $$-0.0208123\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 33.0000 1.07577 0.537885 0.843018i $$-0.319224\pi$$
0.537885 + 0.843018i $$0.319224\pi$$
$$942$$ 0 0
$$943$$ − 27.0000i − 0.879241i
$$944$$ −48.0000 −1.56227
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 42.0000i − 1.36482i −0.730971 0.682408i $$-0.760933\pi$$
0.730971 0.682408i $$-0.239067\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.00000i 0.291539i 0.989319 + 0.145769i $$0.0465657\pi$$
−0.989319 + 0.145769i $$0.953434\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ − 1.00000i − 0.0320585i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 20.0000 0.640184
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ −18.0000 −0.572367
$$990$$ 0 0
$$991$$ 23.0000 0.730619 0.365310 0.930886i $$-0.380963\pi$$
0.365310 + 0.930886i $$0.380963\pi$$
$$992$$ 0 0
$$993$$ 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 $$-0.259744\pi$$
$$998$$ 0 0
$$999$$ 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 2925.2.c.m.2224.1 2
3.2 odd 2 2925.2.c.l.2224.1 2
5.2 odd 4 2925.2.a.k.1.1 1
5.3 odd 4 585.2.a.e.1.1 1
5.4 even 2 inner 2925.2.c.m.2224.2 2
15.2 even 4 2925.2.a.i.1.1 1
15.8 even 4 585.2.a.f.1.1 yes 1
15.14 odd 2 2925.2.c.l.2224.2 2
20.3 even 4 9360.2.a.r.1.1 1
60.23 odd 4 9360.2.a.bu.1.1 1
65.38 odd 4 7605.2.a.m.1.1 1
195.38 even 4 7605.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.e.1.1 1 5.3 odd 4
585.2.a.f.1.1 yes 1 15.8 even 4
2925.2.a.i.1.1 1 15.2 even 4
2925.2.a.k.1.1 1 5.2 odd 4
2925.2.c.l.2224.1 2 3.2 odd 2
2925.2.c.l.2224.2 2 15.14 odd 2
2925.2.c.m.2224.1 2 1.1 even 1 trivial
2925.2.c.m.2224.2 2 5.4 even 2 inner
7605.2.a.j.1.1 1 195.38 even 4
7605.2.a.m.1.1 1 65.38 odd 4
9360.2.a.r.1.1 1 20.3 even 4
9360.2.a.bu.1.1 1 60.23 odd 4