# Properties

 Label 2175.2.c.e.349.2 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ 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] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.e.349.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.00000 q^{4} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.00000 q^{4} -2.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +4.00000 q^{16} -2.00000 q^{19} +2.00000 q^{21} +3.00000i q^{23} -1.00000i q^{27} -4.00000i q^{28} +1.00000 q^{29} +8.00000 q^{31} +3.00000i q^{33} -2.00000 q^{36} +1.00000i q^{37} -2.00000 q^{39} -3.00000 q^{41} -1.00000i q^{43} +6.00000 q^{44} +6.00000i q^{47} +4.00000i q^{48} +3.00000 q^{49} +4.00000i q^{52} -3.00000i q^{53} -2.00000i q^{57} +12.0000 q^{59} +8.00000 q^{61} +2.00000i q^{63} +8.00000 q^{64} -14.0000i q^{67} -3.00000 q^{69} -6.00000 q^{71} -7.00000i q^{73} -4.00000 q^{76} -6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +9.00000i q^{83} +4.00000 q^{84} +1.00000i q^{87} +6.00000 q^{89} +4.00000 q^{91} +6.00000i q^{92} +8.00000i q^{93} -11.0000i q^{97} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^9 $$2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{29} + 16 q^{31} - 4 q^{36} - 4 q^{39} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 24 q^{59} + 16 q^{61} + 16 q^{64} - 6 q^{69} - 12 q^{71} - 8 q^{76} + 8 q^{79} + 2 q^{81} + 8 q^{84} + 12 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^9 + 6 * q^11 + 8 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^29 + 16 * q^31 - 4 * q^36 - 4 * q^39 - 6 * q^41 + 12 * q^44 + 6 * q^49 + 24 * q^59 + 16 * q^61 + 16 * q^64 - 6 * q^69 - 12 * q^71 - 8 * q^76 + 8 * q^79 + 2 * q^81 + 8 * q^84 + 12 * q^89 + 8 * q^91 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 2.00000i 0.577350i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 3.00000i 0.625543i 0.949828 + 0.312772i $$0.101257\pi$$
−0.949828 + 0.312772i $$0.898743\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 4.00000i − 0.755929i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 3.00000i 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 1.00000i 0.164399i 0.996616 + 0.0821995i $$0.0261945\pi$$
−0.996616 + 0.0821995i $$0.973806\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 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 $$0.855830\pi$$
$$48$$ 4.00000i 0.577350i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.00000i 0.554700i
$$53$$ − 3.00000i − 0.412082i −0.978543 0.206041i $$-0.933942\pi$$
0.978543 0.206041i $$-0.0660580\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 14.0000i − 1.71037i −0.518321 0.855186i $$-0.673443\pi$$
0.518321 0.855186i $$-0.326557\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ − 7.00000i − 0.819288i −0.912245 0.409644i $$-0.865653\pi$$
0.912245 0.409644i $$-0.134347\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ − 6.00000i − 0.683763i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.00000i 0.107211i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 6.00000i 0.625543i
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 11.0000i − 1.11688i −0.829545 0.558440i $$-0.811400\pi$$
0.829545 0.558440i $$-0.188600\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 2.00000i − 0.192450i
$$109$$ 13.0000 1.24517 0.622587 0.782551i $$-0.286082\pi$$
0.622587 + 0.782551i $$0.286082\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ − 8.00000i − 0.755929i
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ − 3.00000i − 0.270501i
$$124$$ 16.0000 1.43684
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 5.00000i − 0.443678i −0.975083 0.221839i $$-0.928794\pi$$
0.975083 0.221839i $$-0.0712060\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ 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
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 6.00000i 0.501745i
$$144$$ −4.00000 −0.333333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.00000i 0.247436i
$$148$$ 2.00000i 0.164399i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ − 19.0000i − 1.48819i −0.668071 0.744097i $$-0.732880\pi$$
0.668071 0.744097i $$-0.267120\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ − 2.00000i − 0.152499i
$$173$$ 21.0000i 1.59660i 0.602260 + 0.798300i $$0.294267\pi$$
−0.602260 + 0.798300i $$0.705733\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 12.0000i 0.901975i
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 8.00000i 0.591377i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 6.00000 0.428571
$$197$$ − 21.0000i − 1.49619i −0.663593 0.748094i $$-0.730969\pi$$
0.663593 0.748094i $$-0.269031\pi$$
$$198$$ 0 0
$$199$$ 1.00000 0.0708881 0.0354441 0.999372i $$-0.488715\pi$$
0.0354441 + 0.999372i $$0.488715\pi$$
$$200$$ 0 0
$$201$$ 14.0000 0.987484
$$202$$ 0 0
$$203$$ − 2.00000i − 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 3.00000i − 0.208514i
$$208$$ 8.00000i 0.554700i
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 6.00000i − 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 16.0000i − 1.08615i
$$218$$ 0 0
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 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
$$226$$ 0 0
$$227$$ 21.0000i 1.39382i 0.717159 + 0.696909i $$0.245442\pi$$
−0.717159 + 0.696909i $$0.754558\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 15.0000i 0.982683i 0.870967 + 0.491341i $$0.163493\pi$$
−0.870967 + 0.491341i $$0.836507\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 24.0000 1.56227
$$237$$ 4.00000i 0.259828i
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 16.0000 1.02430
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ 0 0
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 9.00000i 0.565825i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 3.00000i − 0.187135i −0.995613 0.0935674i $$-0.970173\pi$$
0.995613 0.0935674i $$-0.0298271\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 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
$$267$$ 6.00000i 0.367194i
$$268$$ − 28.0000i − 1.71037i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 0 0
$$273$$ 4.00000i 0.242091i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ − 8.00000i − 0.480673i −0.970690 0.240337i $$-0.922742\pi$$
0.970690 0.240337i $$-0.0772579\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ − 10.0000i − 0.594438i −0.954809 0.297219i $$-0.903941\pi$$
0.954809 0.297219i $$-0.0960592\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000i 0.354169i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 11.0000 0.644831
$$292$$ − 14.0000i − 0.819288i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 3.00000i − 0.174078i
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 0 0
$$303$$ − 9.00000i − 0.517036i
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.00000i 0.399511i 0.979846 + 0.199756i $$0.0640148\pi$$
−0.979846 + 0.199756i $$0.935985\pi$$
$$308$$ − 12.0000i − 0.683763i
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ − 4.00000i − 0.226093i −0.993590 0.113047i $$-0.963939\pi$$
0.993590 0.113047i $$-0.0360610\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ − 24.0000i − 1.34797i −0.738743 0.673987i $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ 0 0
$$319$$ 3.00000 0.167968
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 13.0000i 0.718902i
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 18.0000i 0.987878i
$$333$$ − 1.00000i − 0.0547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 8.00000 0.436436
$$337$$ − 2.00000i − 0.108947i −0.998515 0.0544735i $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 3.00000i − 0.161048i −0.996753 0.0805242i $$-0.974341\pi$$
0.996753 0.0805242i $$-0.0256594\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 25.0000 1.33822 0.669110 0.743164i $$-0.266676\pi$$
0.669110 + 0.743164i $$0.266676\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 2.00000i − 0.104973i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 19.0000i 0.991792i 0.868382 + 0.495896i $$0.165160\pi$$
−0.868382 + 0.495896i $$0.834840\pi$$
$$368$$ 12.0000i 0.625543i
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 16.0000i 0.829561i
$$373$$ 32.0000i 1.65690i 0.560065 + 0.828449i $$0.310776\pi$$
−0.560065 + 0.828449i $$0.689224\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.00000i 0.103005i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 5.00000 0.256158
$$382$$ 0 0
$$383$$ − 21.0000i − 1.07305i −0.843884 0.536525i $$-0.819737\pi$$
0.843884 0.536525i $$-0.180263\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.00000i 0.0508329i
$$388$$ − 22.0000i − 1.11688i
$$389$$ −33.0000 −1.67317 −0.836583 0.547840i $$-0.815450\pi$$
−0.836583 + 0.547840i $$0.815450\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 12.0000i − 0.605320i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.00000i 0.148704i
$$408$$ 0 0
$$409$$ −20.0000 −0.988936 −0.494468 0.869196i $$-0.664637\pi$$
−0.494468 + 0.869196i $$0.664637\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 28.0000i 1.37946i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 5.00000i − 0.244851i
$$418$$ 0 0
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ − 6.00000i − 0.291730i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 16.0000i − 0.774294i
$$428$$ 24.0000i 1.16008i
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ 5.00000i 0.240285i 0.992757 + 0.120142i $$0.0383351\pi$$
−0.992757 + 0.120142i $$0.961665\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 26.0000 1.24517
$$437$$ − 6.00000i − 0.287019i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 18.0000i − 0.851371i
$$448$$ − 16.0000i − 0.755929i
$$449$$ −27.0000 −1.27421 −0.637104 0.770778i $$-0.719868\pi$$
−0.637104 + 0.770778i $$0.719868\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 0 0
$$453$$ − 19.0000i − 0.892698i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.0000i 0.748448i 0.927338 + 0.374224i $$0.122091\pi$$
−0.927338 + 0.374224i $$0.877909\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 33.0000 1.53696 0.768482 0.639872i $$-0.221013\pi$$
0.768482 + 0.639872i $$0.221013\pi$$
$$462$$ 0 0
$$463$$ − 34.0000i − 1.58011i −0.613033 0.790057i $$-0.710051\pi$$
0.613033 0.790057i $$-0.289949\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6.00000i − 0.277647i −0.990317 0.138823i $$-0.955668\pi$$
0.990317 0.138823i $$-0.0443321\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ −28.0000 −1.29292
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ − 3.00000i − 0.137940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3.00000i 0.137361i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ 0 0
$$489$$ 19.0000 0.859210
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 32.0000 1.43684
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 18.0000i − 0.802580i −0.915951 0.401290i $$-0.868562\pi$$
0.915951 0.401290i $$-0.131438\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ − 10.0000i − 0.443678i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ 0 0
$$513$$ 2.00000i 0.0883022i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ −21.0000 −0.921798
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ −24.0000 −1.04844
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 12.0000i 0.522233i
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 8.00000i 0.346844i
$$533$$ − 6.00000i − 0.259889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 6.00000i − 0.258919i
$$538$$ 0 0
$$539$$ 9.00000 0.387657
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ − 7.00000i − 0.300399i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 44.0000i − 1.88130i −0.339372 0.940652i $$-0.610215\pi$$
0.339372 0.940652i $$-0.389785\pi$$
$$548$$ 24.0000i 1.02523i
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ − 8.00000i − 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ 27.0000i 1.14403i 0.820244 + 0.572013i $$0.193837\pi$$
−0.820244 + 0.572013i $$0.806163\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 30.0000i − 1.26435i −0.774826 0.632175i $$-0.782163\pi$$
0.774826 0.632175i $$-0.217837\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −43.0000 −1.79949 −0.899747 0.436412i $$-0.856249\pi$$
−0.899747 + 0.436412i $$0.856249\pi$$
$$572$$ 12.0000i 0.501745i
$$573$$ 15.0000i 0.626634i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ − 9.00000i − 0.372742i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 6.00000i 0.247436i
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 21.0000 0.863825
$$592$$ 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
$$596$$ −36.0000 −1.47462
$$597$$ 1.00000i 0.0409273i
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 20.0000 0.815817 0.407909 0.913023i $$-0.366258\pi$$
0.407909 + 0.913023i $$0.366258\pi$$
$$602$$ 0 0
$$603$$ 14.0000i 0.570124i
$$604$$ −38.0000 −1.54620
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ − 12.0000i − 0.480770i
$$624$$ −8.00000 −0.320256
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 6.00000i − 0.239617i
$$628$$ − 4.00000i − 0.159617i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ − 10.0000i − 0.397464i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ 32.0000i 1.26196i 0.775800 + 0.630978i $$0.217346\pi$$
−0.775800 + 0.630978i $$0.782654\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.0000i 1.06148i 0.847535 + 0.530740i $$0.178086\pi$$
−0.847535 + 0.530740i $$0.821914\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ − 38.0000i − 1.48819i
$$653$$ 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
$$657$$ 7.00000i 0.273096i
$$658$$ 0 0
$$659$$ −27.0000 −1.05177 −0.525885 0.850555i $$-0.676266\pi$$
−0.525885 + 0.850555i $$0.676266\pi$$
$$660$$ 0 0
$$661$$ 23.0000 0.894596 0.447298 0.894385i $$-0.352386\pi$$
0.447298 + 0.894385i $$0.352386\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.00000i 0.116160i
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ − 22.0000i − 0.848038i −0.905653 0.424019i $$-0.860619\pi$$
0.905653 0.424019i $$-0.139381\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 18.0000 0.692308
$$677$$ − 36.0000i − 1.38359i −0.722093 0.691796i $$-0.756820\pi$$
0.722093 0.691796i $$-0.243180\pi$$
$$678$$ 0 0
$$679$$ −22.0000 −0.844283
$$680$$ 0 0
$$681$$ −21.0000 −0.804722
$$682$$ 0 0
$$683$$ 15.0000i 0.573959i 0.957937 + 0.286980i $$0.0926512\pi$$
−0.957937 + 0.286980i $$0.907349\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 14.0000i − 0.534133i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 42.0000i 1.59660i
$$693$$ 6.00000i 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −15.0000 −0.567352
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 24.0000 0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.0000i 0.676960i
$$708$$ 24.0000i 0.901975i
$$709$$ 1.00000 0.0375558 0.0187779 0.999824i $$-0.494022\pi$$
0.0187779 + 0.999824i $$0.494022\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 24.0000i 0.898807i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 6.00000i − 0.224074i
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 0 0
$$723$$ − 7.00000i − 0.260333i
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 16.0000i 0.591377i
$$733$$ 50.0000i 1.84679i 0.383849 + 0.923396i $$0.374598\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 42.0000i − 1.54709i
$$738$$ 0 0
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ − 42.0000i − 1.54083i −0.637542 0.770415i $$-0.720049\pi$$
0.637542 0.770415i $$-0.279951\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 9.00000i − 0.329293i
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ 24.0000i 0.875190i
$$753$$ 12.0000i 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 43.0000i 1.56286i 0.623992 + 0.781431i $$0.285510\pi$$
−0.623992 + 0.781431i $$0.714490\pi$$
$$758$$ 0 0
$$759$$ −9.00000 −0.326679
$$760$$ 0 0
$$761$$ −24.0000 −0.869999 −0.435000 0.900431i $$-0.643252\pi$$
−0.435000 + 0.900431i $$0.643252\pi$$
$$762$$ 0 0
$$763$$ − 26.0000i − 0.941263i
$$764$$ 30.0000 1.08536
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000i 0.866590i
$$768$$ 16.0000i 0.577350i
$$769$$ 16.0000 0.576975 0.288487 0.957484i $$-0.406848\pi$$
0.288487 + 0.957484i $$0.406848\pi$$
$$770$$ 0 0
$$771$$ 3.00000 0.108042
$$772$$ 28.0000i 1.00774i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2.00000i 0.0717496i
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ − 1.00000i − 0.0357371i
$$784$$ 12.0000 0.428571
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 42.0000i − 1.49619i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 16.0000i 0.568177i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ − 21.0000i − 0.741074i
$$804$$ 28.0000 0.987484
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18.0000i 0.633630i
$$808$$ 0 0
$$809$$ 27.0000 0.949269 0.474635 0.880183i $$-0.342580\pi$$
0.474635 + 0.880183i $$0.342580\pi$$
$$810$$ 0 0
$$811$$ −49.0000 −1.72062 −0.860311 0.509769i $$-0.829731\pi$$
−0.860311 + 0.509769i $$0.829731\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 2.00000i 0.0701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.00000i 0.0699711i
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ 0 0
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 18.0000i 0.625921i 0.949766 + 0.312961i $$0.101321\pi$$
−0.949766 + 0.312961i $$0.898679\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 16.0000i 0.554700i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ − 8.00000i − 0.276520i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.00000i 0.137442i
$$848$$ − 12.0000i − 0.412082i
$$849$$ 10.0000 0.343199
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ − 12.0000i − 0.411113i
$$853$$ 17.0000i 0.582069i 0.956713 + 0.291034i $$0.0939994\pi$$
−0.956713 + 0.291034i $$0.906001\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 39.0000i 1.33221i 0.745856 + 0.666107i $$0.232041\pi$$
−0.745856 + 0.666107i $$0.767959\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 17.0000i 0.577350i
$$868$$ − 32.0000i − 1.08615i
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ 28.0000 0.948744
$$872$$ 0 0
$$873$$ 11.0000i 0.372294i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 28.0000i 0.945493i 0.881199 + 0.472746i $$0.156737\pi$$
−0.881199 + 0.472746i $$0.843263\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 3.00000 0.101073 0.0505363 0.998722i $$-0.483907\pi$$
0.0505363 + 0.998722i $$0.483907\pi$$
$$882$$ 0 0
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 42.0000i − 1.41022i −0.709097 0.705111i $$-0.750897\pi$$
0.709097 0.705111i $$-0.249103\pi$$
$$888$$ 0 0
$$889$$ −10.0000 −0.335389
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ − 8.00000i − 0.267860i
$$893$$ − 12.0000i − 0.401565i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 2.00000i − 0.0665558i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 19.0000i 0.630885i 0.948945 + 0.315442i $$0.102153\pi$$
−0.948945 + 0.315442i $$0.897847\pi$$
$$908$$ 42.0000i 1.39382i
$$909$$ 9.00000 0.298511
$$910$$ 0 0
$$911$$ 33.0000 1.09334 0.546669 0.837349i $$-0.315895\pi$$
0.546669 + 0.837349i $$0.315895\pi$$
$$912$$ − 8.00000i − 0.264906i
$$913$$ 27.0000i 0.893570i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −28.0000 −0.925146
$$917$$ 24.0000i 0.792550i
$$918$$ 0 0
$$919$$ 52.0000 1.71532 0.857661 0.514216i $$-0.171917\pi$$
0.857661 + 0.514216i $$0.171917\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 0 0
$$923$$ − 12.0000i − 0.394985i
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 14.0000i − 0.459820i
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 30.0000i 0.982683i
$$933$$ − 21.0000i − 0.687509i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 52.0000i 1.69877i 0.527777 + 0.849383i $$0.323026\pi$$
−0.527777 + 0.849383i $$0.676974\pi$$
$$938$$ 0 0
$$939$$ 4.00000 0.130535
$$940$$ 0 0
$$941$$ 54.0000 1.76035 0.880175 0.474650i $$-0.157425\pi$$
0.880175 + 0.474650i $$0.157425\pi$$
$$942$$ 0 0
$$943$$ − 9.00000i − 0.293080i
$$944$$ 48.0000 1.56227
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 30.0000i − 0.974869i −0.873160 0.487435i $$-0.837933\pi$$
0.873160 0.487435i $$-0.162067\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ 24.0000 0.778253
$$952$$ 0 0
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 3.00000i 0.0969762i
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 41.0000i − 1.31847i −0.751936 0.659236i $$-0.770880\pi$$
0.751936 0.659236i $$-0.229120\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −57.0000 −1.82922 −0.914609 0.404341i $$-0.867501\pi$$
−0.914609 + 0.404341i $$0.867501\pi$$
$$972$$ 2.00000i 0.0641500i
$$973$$ 10.0000i 0.320585i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 32.0000 1.02430
$$977$$ − 57.0000i − 1.82359i −0.410644 0.911796i $$-0.634696\pi$$
0.410644 0.911796i $$-0.365304\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ −13.0000 −0.415058
$$982$$ 0 0
$$983$$ 30.0000i 0.956851i 0.878128 + 0.478426i $$0.158792\pi$$
−0.878128 + 0.478426i $$0.841208\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 12.0000i 0.381964i
$$988$$ − 8.00000i − 0.254514i
$$989$$ 3.00000 0.0953945
$$990$$ 0 0
$$991$$ −25.0000 −0.794151 −0.397076 0.917786i $$-0.629975\pi$$
−0.397076 + 0.917786i $$0.629975\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −18.0000 −0.570352
$$997$$ 37.0000i 1.17180i 0.810383 + 0.585901i $$0.199259\pi$$
−0.810383 + 0.585901i $$0.800741\pi$$
$$998$$ 0 0
$$999$$ 1.00000 0.0316386
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 2175.2.c.e.349.2 2
5.2 odd 4 435.2.a.c.1.1 1
5.3 odd 4 2175.2.a.d.1.1 1
5.4 even 2 inner 2175.2.c.e.349.1 2
15.2 even 4 1305.2.a.d.1.1 1
15.8 even 4 6525.2.a.f.1.1 1
20.7 even 4 6960.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.c.1.1 1 5.2 odd 4
1305.2.a.d.1.1 1 15.2 even 4
2175.2.a.d.1.1 1 5.3 odd 4
2175.2.c.e.349.1 2 5.4 even 2 inner
2175.2.c.e.349.2 2 1.1 even 1 trivial
6525.2.a.f.1.1 1 15.8 even 4
6960.2.a.b.1.1 1 20.7 even 4